LMFDB - Group of Dirichlet characters of modulus 39

Show commands: PariGP / SageMath

sage: H = DirichletGroup(39)

pari: g = idealstar(,39,2)

Character group

sage: G.order() pari: g.no
Order	=	24
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{12}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{39}(14,\cdot)$, $\chi_{39}(28,\cdot)$

Characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

Character	Orbit	Order	Primitive	$-1$	$1$	$2$	$4$	$5$	$7$	$8$	$10$	$11$	$14$	$16$	$17$
$\chi_{39}(1,\cdot)$	39.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{39}(2,\cdot)$	39.k	12	yes	$1$	$1$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{6}\right)$	$i$	$e\left(\frac{11}{12}\right)$	$-i$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{12}\right)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{39}(4,\cdot)$	39.j	6	no	$1$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{5}{6}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{39}(5,\cdot)$	39.f	4	yes	$1$	$1$	$i$	$-1$	$i$	$i$	$-i$	$-1$	$-i$	$-1$	$1$	$1$
$\chi_{39}(7,\cdot)$	39.l	12	no	$-1$	$1$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{6}\right)$	$i$	$e\left(\frac{1}{12}\right)$	$-i$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{12}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{39}(8,\cdot)$	39.f	4	yes	$1$	$1$	$-i$	$-1$	$-i$	$-i$	$i$	$-1$	$i$	$-1$	$1$	$1$
$\chi_{39}(10,\cdot)$	39.j	6	no	$1$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{1}{6}\right)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{39}(11,\cdot)$	39.k	12	yes	$1$	$1$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{6}\right)$	$-i$	$e\left(\frac{5}{12}\right)$	$i$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{7}{12}\right)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{39}(14,\cdot)$	39.c	2	no	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$
$\chi_{39}(16,\cdot)$	39.e	3	no	$1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{39}(17,\cdot)$	39.h	6	yes	$-1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{39}(19,\cdot)$	39.l	12	no	$-1$	$1$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{5}{6}\right)$	$-i$	$e\left(\frac{7}{12}\right)$	$i$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{11}{12}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{39}(20,\cdot)$	39.k	12	yes	$1$	$1$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{5}{6}\right)$	$-i$	$e\left(\frac{1}{12}\right)$	$i$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{11}{12}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{39}(22,\cdot)$	39.e	3	no	$1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{39}(23,\cdot)$	39.h	6	yes	$-1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{6}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{39}(25,\cdot)$	39.b	2	no	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$
$\chi_{39}(28,\cdot)$	39.l	12	no	$-1$	$1$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{6}\right)$	$-i$	$e\left(\frac{11}{12}\right)$	$i$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{7}{12}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{39}(29,\cdot)$	39.i	6	yes	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{39}(31,\cdot)$	39.g	4	no	$-1$	$1$	$-i$	$-1$	$-i$	$i$	$i$	$-1$	$i$	$1$	$1$	$-1$
$\chi_{39}(32,\cdot)$	39.k	12	yes	$1$	$1$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{6}\right)$	$i$	$e\left(\frac{7}{12}\right)$	$-i$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{12}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{39}(34,\cdot)$	39.g	4	no	$-1$	$1$	$i$	$-1$	$i$	$-i$	$-i$	$-1$	$-i$	$1$	$1$	$-1$
$\chi_{39}(35,\cdot)$	39.i	6	yes	$-1$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{39}(37,\cdot)$	39.l	12	no	$-1$	$1$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{6}\right)$	$i$	$e\left(\frac{5}{12}\right)$	$-i$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{12}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{39}(38,\cdot)$	39.d	2	yes	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Modulus	$39$
Structure	\(C_{2}\times C_{12}\)
Order	$24$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(4\)	\(5\)	\(7\)	\(8\)	\(10\)	\(11\)	\(14\)	\(16\)	\(17\)
\(\chi_{39}(1,\cdot)\)	39.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{39}(2,\cdot)\)	39.k	12	yes	\(1\)	\(1\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(i\)	\(e\left(\frac{11}{12}\right)\)	\(-i\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{39}(4,\cdot)\)	39.j	6	no	\(1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{39}(5,\cdot)\)	39.f	4	yes	\(1\)	\(1\)	\(i\)	\(-1\)	\(i\)	\(i\)	\(-i\)	\(-1\)	\(-i\)	\(-1\)	\(1\)	\(1\)
\(\chi_{39}(7,\cdot)\)	39.l	12	no	\(-1\)	\(1\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(i\)	\(e\left(\frac{1}{12}\right)\)	\(-i\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{39}(8,\cdot)\)	39.f	4	yes	\(1\)	\(1\)	\(-i\)	\(-1\)	\(-i\)	\(-i\)	\(i\)	\(-1\)	\(i\)	\(-1\)	\(1\)	\(1\)
\(\chi_{39}(10,\cdot)\)	39.j	6	no	\(1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{39}(11,\cdot)\)	39.k	12	yes	\(1\)	\(1\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-i\)	\(e\left(\frac{5}{12}\right)\)	\(i\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{39}(14,\cdot)\)	39.c	2	no	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)
\(\chi_{39}(16,\cdot)\)	39.e	3	no	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{39}(17,\cdot)\)	39.h	6	yes	\(-1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{39}(19,\cdot)\)	39.l	12	no	\(-1\)	\(1\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-i\)	\(e\left(\frac{7}{12}\right)\)	\(i\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{39}(20,\cdot)\)	39.k	12	yes	\(1\)	\(1\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-i\)	\(e\left(\frac{1}{12}\right)\)	\(i\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{39}(22,\cdot)\)	39.e	3	no	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{39}(23,\cdot)\)	39.h	6	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{39}(25,\cdot)\)	39.b	2	no	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(1\)
\(\chi_{39}(28,\cdot)\)	39.l	12	no	\(-1\)	\(1\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-i\)	\(e\left(\frac{11}{12}\right)\)	\(i\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{39}(29,\cdot)\)	39.i	6	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{39}(31,\cdot)\)	39.g	4	no	\(-1\)	\(1\)	\(-i\)	\(-1\)	\(-i\)	\(i\)	\(i\)	\(-1\)	\(i\)	\(1\)	\(1\)	\(-1\)
\(\chi_{39}(32,\cdot)\)	39.k	12	yes	\(1\)	\(1\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(i\)	\(e\left(\frac{7}{12}\right)\)	\(-i\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{39}(34,\cdot)\)	39.g	4	no	\(-1\)	\(1\)	\(i\)	\(-1\)	\(i\)	\(-i\)	\(-i\)	\(-1\)	\(-i\)	\(1\)	\(1\)	\(-1\)
\(\chi_{39}(35,\cdot)\)	39.i	6	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{39}(37,\cdot)\)	39.l	12	no	\(-1\)	\(1\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(i\)	\(e\left(\frac{5}{12}\right)\)	\(-i\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{39}(38,\cdot)\)	39.d	2	yes	\(-1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.