LMFDB - Group of Dirichlet characters of modulus 70

Show commands: PariGP / SageMath

sage: H = DirichletGroup(70)

pari: g = idealstar(,70,2)

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	=	$\chi_{70}(57,\cdot)$, $\chi_{70}(31,\cdot)$

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	$-1$	$1$	$3$	$9$	$11$	$13$	$17$	$19$	$23$	$27$	$29$	$31$
$\chi_{70}(1,\cdot)$	70.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{70}(3,\cdot)$	70.k	12	no	$1$	$1$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$-i$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{7}{12}\right)$	$i$	$-1$	$e\left(\frac{1}{6}\right)$
$\chi_{70}(9,\cdot)$	70.i	6	no	$1$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$
$\chi_{70}(11,\cdot)$	70.e	3	no	$1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$e\left(\frac{2}{3}\right)$
$\chi_{70}(13,\cdot)$	70.g	4	no	$1$	$1$	$-i$	$-1$	$1$	$-i$	$i$	$1$	$i$	$i$	$-1$	$-1$
$\chi_{70}(17,\cdot)$	70.k	12	no	$1$	$1$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$i$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{12}\right)$	$-i$	$-1$	$e\left(\frac{1}{6}\right)$
$\chi_{70}(19,\cdot)$	70.h	6	no	$-1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{6}\right)$	$1$	$1$	$e\left(\frac{5}{6}\right)$
$\chi_{70}(23,\cdot)$	70.l	12	no	$-1$	$1$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$i$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{11}{12}\right)$	$-i$	$-1$	$e\left(\frac{1}{3}\right)$
$\chi_{70}(27,\cdot)$	70.g	4	no	$1$	$1$	$i$	$-1$	$1$	$i$	$-i$	$1$	$-i$	$-i$	$-1$	$-1$
$\chi_{70}(29,\cdot)$	70.c	2	no	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$1$
$\chi_{70}(31,\cdot)$	70.j	6	no	$-1$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{3}\right)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$
$\chi_{70}(33,\cdot)$	70.k	12	no	$1$	$1$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$-i$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{11}{12}\right)$	$i$	$-1$	$e\left(\frac{5}{6}\right)$
$\chi_{70}(37,\cdot)$	70.l	12	no	$-1$	$1$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$-i$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{12}\right)$	$i$	$-1$	$e\left(\frac{1}{3}\right)$
$\chi_{70}(39,\cdot)$	70.i	6	no	$1$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$
$\chi_{70}(41,\cdot)$	70.b	2	no	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$
$\chi_{70}(43,\cdot)$	70.f	4	no	$-1$	$1$	$i$	$-1$	$1$	$i$	$-i$	$-1$	$i$	$-i$	$-1$	$1$
$\chi_{70}(47,\cdot)$	70.k	12	no	$1$	$1$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$i$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{12}\right)$	$-i$	$-1$	$e\left(\frac{5}{6}\right)$
$\chi_{70}(51,\cdot)$	70.e	3	no	$1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$e\left(\frac{1}{3}\right)$
$\chi_{70}(53,\cdot)$	70.l	12	no	$-1$	$1$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$i$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{7}{12}\right)$	$-i$	$-1$	$e\left(\frac{2}{3}\right)$
$\chi_{70}(57,\cdot)$	70.f	4	no	$-1$	$1$	$-i$	$-1$	$1$	$-i$	$i$	$-1$	$-i$	$i$	$-1$	$1$
$\chi_{70}(59,\cdot)$	70.h	6	no	$-1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$	$1$	$1$	$e\left(\frac{1}{6}\right)$
$\chi_{70}(61,\cdot)$	70.j	6	no	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$1$	$e\left(\frac{5}{6}\right)$
$\chi_{70}(67,\cdot)$	70.l	12	no	$-1$	$1$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$-i$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{12}\right)$	$i$	$-1$	$e\left(\frac{2}{3}\right)$
$\chi_{70}(69,\cdot)$	70.d	2	no	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$

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	$70$
Structure	\(C_{2}\times C_{12}\)
Order	$24$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(9\)	\(11\)	\(13\)	\(17\)	\(19\)	\(23\)	\(27\)	\(29\)	\(31\)
\(\chi_{70}(1,\cdot)\)	70.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{70}(3,\cdot)\)	70.k	12	no	\(1\)	\(1\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-i\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(i\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{70}(9,\cdot)\)	70.i	6	no	\(1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{70}(11,\cdot)\)	70.e	3	no	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{70}(13,\cdot)\)	70.g	4	no	\(1\)	\(1\)	\(-i\)	\(-1\)	\(1\)	\(-i\)	\(i\)	\(1\)	\(i\)	\(i\)	\(-1\)	\(-1\)
\(\chi_{70}(17,\cdot)\)	70.k	12	no	\(1\)	\(1\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(i\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(-i\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{70}(19,\cdot)\)	70.h	6	no	\(-1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{70}(23,\cdot)\)	70.l	12	no	\(-1\)	\(1\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(i\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(-i\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{70}(27,\cdot)\)	70.g	4	no	\(1\)	\(1\)	\(i\)	\(-1\)	\(1\)	\(i\)	\(-i\)	\(1\)	\(-i\)	\(-i\)	\(-1\)	\(-1\)
\(\chi_{70}(29,\cdot)\)	70.c	2	no	\(1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(1\)
\(\chi_{70}(31,\cdot)\)	70.j	6	no	\(-1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{70}(33,\cdot)\)	70.k	12	no	\(1\)	\(1\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-i\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(i\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{70}(37,\cdot)\)	70.l	12	no	\(-1\)	\(1\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-i\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(i\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{70}(39,\cdot)\)	70.i	6	no	\(1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{70}(41,\cdot)\)	70.b	2	no	\(-1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)
\(\chi_{70}(43,\cdot)\)	70.f	4	no	\(-1\)	\(1\)	\(i\)	\(-1\)	\(1\)	\(i\)	\(-i\)	\(-1\)	\(i\)	\(-i\)	\(-1\)	\(1\)
\(\chi_{70}(47,\cdot)\)	70.k	12	no	\(1\)	\(1\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(i\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(-i\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{70}(51,\cdot)\)	70.e	3	no	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{70}(53,\cdot)\)	70.l	12	no	\(-1\)	\(1\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(i\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(-i\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{70}(57,\cdot)\)	70.f	4	no	\(-1\)	\(1\)	\(-i\)	\(-1\)	\(1\)	\(-i\)	\(i\)	\(-1\)	\(-i\)	\(i\)	\(-1\)	\(1\)
\(\chi_{70}(59,\cdot)\)	70.h	6	no	\(-1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{70}(61,\cdot)\)	70.j	6	no	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{70}(67,\cdot)\)	70.l	12	no	\(-1\)	\(1\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-i\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(i\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{70}(69,\cdot)\)	70.d	2	no	\(-1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(-1\)

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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.