Understanding Output in SageMath Regarding Dirichlet Characters












1














p=7
G = DirichletGroup(p); G

m=3; n=ZZ((p-1)/m); print m,n

c=G[1]

c1=c^n;c1


The output is:



Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1


Can anyone explain what zeta6 is? Is this the Riemann-Zeta function? Is this the whole group of units? Is there a relation to the Eisenstein primes? I'm still a bit weak in this material and am having trouble grasping some of these sage outputs. Thank you in advance!



EDIT: If I take the log and then exponentiate I get, for p=7;



$$ -frac{3i}{2}sqrt{3} +frac{1}{2}$$



Not sure what this is exactly.










share|cite|improve this question




















  • 1




    the best place to ask this question is here
    – Masacroso
    Dec 5 '18 at 17:12






  • 2




    $3$ is a generator of $(mathbb{Z}/7mathbb{Z})^times$ and the Dirichlet character is defined by "$chi(3) = zeta_6,chi(n+7) = chi(n)$" which implies $chi(7n) = 0, chi(3^l+7n) = zeta_6^l$ where $zeta_6 = e^{2i pi /6} $ or any primitive $6$-th root of unity (and it is an element of the ring of Eisenstein integers). Because $(mathbb{Z}/7mathbb{Z})^times$ is cyclic so is the group of Dirichlet characters modulo $7$ and $chi$ is a generator of it.
    – reuns
    Dec 5 '18 at 18:12






  • 1




    Indeed, there is an accepted answer to this question at ask.sagemath.org/question/44593/…
    – kcrisman
    Dec 7 '18 at 18:14










  • I know. I posted the question :)
    – Nicklovn
    Dec 7 '18 at 23:20










  • I suppose you could post that answer here ... apparently you aren't supposed to close your own questions for this reason though.
    – kcrisman
    Dec 20 '18 at 12:42
















1














p=7
G = DirichletGroup(p); G

m=3; n=ZZ((p-1)/m); print m,n

c=G[1]

c1=c^n;c1


The output is:



Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1


Can anyone explain what zeta6 is? Is this the Riemann-Zeta function? Is this the whole group of units? Is there a relation to the Eisenstein primes? I'm still a bit weak in this material and am having trouble grasping some of these sage outputs. Thank you in advance!



EDIT: If I take the log and then exponentiate I get, for p=7;



$$ -frac{3i}{2}sqrt{3} +frac{1}{2}$$



Not sure what this is exactly.










share|cite|improve this question




















  • 1




    the best place to ask this question is here
    – Masacroso
    Dec 5 '18 at 17:12






  • 2




    $3$ is a generator of $(mathbb{Z}/7mathbb{Z})^times$ and the Dirichlet character is defined by "$chi(3) = zeta_6,chi(n+7) = chi(n)$" which implies $chi(7n) = 0, chi(3^l+7n) = zeta_6^l$ where $zeta_6 = e^{2i pi /6} $ or any primitive $6$-th root of unity (and it is an element of the ring of Eisenstein integers). Because $(mathbb{Z}/7mathbb{Z})^times$ is cyclic so is the group of Dirichlet characters modulo $7$ and $chi$ is a generator of it.
    – reuns
    Dec 5 '18 at 18:12






  • 1




    Indeed, there is an accepted answer to this question at ask.sagemath.org/question/44593/…
    – kcrisman
    Dec 7 '18 at 18:14










  • I know. I posted the question :)
    – Nicklovn
    Dec 7 '18 at 23:20










  • I suppose you could post that answer here ... apparently you aren't supposed to close your own questions for this reason though.
    – kcrisman
    Dec 20 '18 at 12:42














1












1








1







p=7
G = DirichletGroup(p); G

m=3; n=ZZ((p-1)/m); print m,n

c=G[1]

c1=c^n;c1


The output is:



Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1


Can anyone explain what zeta6 is? Is this the Riemann-Zeta function? Is this the whole group of units? Is there a relation to the Eisenstein primes? I'm still a bit weak in this material and am having trouble grasping some of these sage outputs. Thank you in advance!



EDIT: If I take the log and then exponentiate I get, for p=7;



$$ -frac{3i}{2}sqrt{3} +frac{1}{2}$$



Not sure what this is exactly.










share|cite|improve this question















p=7
G = DirichletGroup(p); G

m=3; n=ZZ((p-1)/m); print m,n

c=G[1]

c1=c^n;c1


The output is:



Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1


Can anyone explain what zeta6 is? Is this the Riemann-Zeta function? Is this the whole group of units? Is there a relation to the Eisenstein primes? I'm still a bit weak in this material and am having trouble grasping some of these sage outputs. Thank you in advance!



EDIT: If I take the log and then exponentiate I get, for p=7;



$$ -frac{3i}{2}sqrt{3} +frac{1}{2}$$



Not sure what this is exactly.







abstract-algebra number-theory sagemath






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 5 '18 at 17:29







Nicklovn

















asked Dec 5 '18 at 17:09









NicklovnNicklovn

343110




343110








  • 1




    the best place to ask this question is here
    – Masacroso
    Dec 5 '18 at 17:12






  • 2




    $3$ is a generator of $(mathbb{Z}/7mathbb{Z})^times$ and the Dirichlet character is defined by "$chi(3) = zeta_6,chi(n+7) = chi(n)$" which implies $chi(7n) = 0, chi(3^l+7n) = zeta_6^l$ where $zeta_6 = e^{2i pi /6} $ or any primitive $6$-th root of unity (and it is an element of the ring of Eisenstein integers). Because $(mathbb{Z}/7mathbb{Z})^times$ is cyclic so is the group of Dirichlet characters modulo $7$ and $chi$ is a generator of it.
    – reuns
    Dec 5 '18 at 18:12






  • 1




    Indeed, there is an accepted answer to this question at ask.sagemath.org/question/44593/…
    – kcrisman
    Dec 7 '18 at 18:14










  • I know. I posted the question :)
    – Nicklovn
    Dec 7 '18 at 23:20










  • I suppose you could post that answer here ... apparently you aren't supposed to close your own questions for this reason though.
    – kcrisman
    Dec 20 '18 at 12:42














  • 1




    the best place to ask this question is here
    – Masacroso
    Dec 5 '18 at 17:12






  • 2




    $3$ is a generator of $(mathbb{Z}/7mathbb{Z})^times$ and the Dirichlet character is defined by "$chi(3) = zeta_6,chi(n+7) = chi(n)$" which implies $chi(7n) = 0, chi(3^l+7n) = zeta_6^l$ where $zeta_6 = e^{2i pi /6} $ or any primitive $6$-th root of unity (and it is an element of the ring of Eisenstein integers). Because $(mathbb{Z}/7mathbb{Z})^times$ is cyclic so is the group of Dirichlet characters modulo $7$ and $chi$ is a generator of it.
    – reuns
    Dec 5 '18 at 18:12






  • 1




    Indeed, there is an accepted answer to this question at ask.sagemath.org/question/44593/…
    – kcrisman
    Dec 7 '18 at 18:14










  • I know. I posted the question :)
    – Nicklovn
    Dec 7 '18 at 23:20










  • I suppose you could post that answer here ... apparently you aren't supposed to close your own questions for this reason though.
    – kcrisman
    Dec 20 '18 at 12:42








1




1




the best place to ask this question is here
– Masacroso
Dec 5 '18 at 17:12




the best place to ask this question is here
– Masacroso
Dec 5 '18 at 17:12




2




2




$3$ is a generator of $(mathbb{Z}/7mathbb{Z})^times$ and the Dirichlet character is defined by "$chi(3) = zeta_6,chi(n+7) = chi(n)$" which implies $chi(7n) = 0, chi(3^l+7n) = zeta_6^l$ where $zeta_6 = e^{2i pi /6} $ or any primitive $6$-th root of unity (and it is an element of the ring of Eisenstein integers). Because $(mathbb{Z}/7mathbb{Z})^times$ is cyclic so is the group of Dirichlet characters modulo $7$ and $chi$ is a generator of it.
– reuns
Dec 5 '18 at 18:12




$3$ is a generator of $(mathbb{Z}/7mathbb{Z})^times$ and the Dirichlet character is defined by "$chi(3) = zeta_6,chi(n+7) = chi(n)$" which implies $chi(7n) = 0, chi(3^l+7n) = zeta_6^l$ where $zeta_6 = e^{2i pi /6} $ or any primitive $6$-th root of unity (and it is an element of the ring of Eisenstein integers). Because $(mathbb{Z}/7mathbb{Z})^times$ is cyclic so is the group of Dirichlet characters modulo $7$ and $chi$ is a generator of it.
– reuns
Dec 5 '18 at 18:12




1




1




Indeed, there is an accepted answer to this question at ask.sagemath.org/question/44593/…
– kcrisman
Dec 7 '18 at 18:14




Indeed, there is an accepted answer to this question at ask.sagemath.org/question/44593/…
– kcrisman
Dec 7 '18 at 18:14












I know. I posted the question :)
– Nicklovn
Dec 7 '18 at 23:20




I know. I posted the question :)
– Nicklovn
Dec 7 '18 at 23:20












I suppose you could post that answer here ... apparently you aren't supposed to close your own questions for this reason though.
– kcrisman
Dec 20 '18 at 12:42




I suppose you could post that answer here ... apparently you aren't supposed to close your own questions for this reason though.
– kcrisman
Dec 20 '18 at 12:42










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