LMFDB - Group of Dirichlet characters of modulus 52

Show commands: PariGP / SageMath

sage: H = DirichletGroup(52)

pari: g = idealstar(,52,2)

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	=	$\chi_{52}(27,\cdot)$, $\chi_{52}(41,\cdot)$

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	$-1$	$1$	$3$	$5$	$7$	$9$	$11$	$15$	$17$	$19$	$21$	$23$
$\chi_{52}(1,\cdot)$	52.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{52}(3,\cdot)$	52.j	6	yes	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$	$1$	$e\left(\frac{5}{6}\right)$
$\chi_{52}(5,\cdot)$	52.g	4	no	$-1$	$1$	$1$	$-i$	$i$	$1$	$i$	$-i$	$-1$	$-i$	$i$	$-1$
$\chi_{52}(7,\cdot)$	52.l	12	yes	$1$	$1$	$e\left(\frac{1}{6}\right)$	$i$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{12}\right)$	$-i$	$e\left(\frac{2}{3}\right)$
$\chi_{52}(9,\cdot)$	52.e	3	no	$1$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$
$\chi_{52}(11,\cdot)$	52.l	12	yes	$1$	$1$	$e\left(\frac{5}{6}\right)$	$i$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{12}\right)$	$-i$	$e\left(\frac{1}{3}\right)$
$\chi_{52}(15,\cdot)$	52.l	12	yes	$1$	$1$	$e\left(\frac{5}{6}\right)$	$-i$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{11}{12}\right)$	$i$	$e\left(\frac{1}{3}\right)$
$\chi_{52}(17,\cdot)$	52.h	6	no	$1$	$1$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$	$-1$	$e\left(\frac{2}{3}\right)$
$\chi_{52}(19,\cdot)$	52.l	12	yes	$1$	$1$	$e\left(\frac{1}{6}\right)$	$-i$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{7}{12}\right)$	$i$	$e\left(\frac{2}{3}\right)$
$\chi_{52}(21,\cdot)$	52.g	4	no	$-1$	$1$	$1$	$i$	$-i$	$1$	$-i$	$i$	$-1$	$i$	$-i$	$-1$
$\chi_{52}(23,\cdot)$	52.i	6	yes	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{5}{6}\right)$
$\chi_{52}(25,\cdot)$	52.d	2	no	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$
$\chi_{52}(27,\cdot)$	52.c	2	no	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$
$\chi_{52}(29,\cdot)$	52.e	3	no	$1$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$
$\chi_{52}(31,\cdot)$	52.f	4	yes	$1$	$1$	$-1$	$-i$	$-i$	$1$	$-i$	$i$	$-1$	$i$	$i$	$1$
$\chi_{52}(33,\cdot)$	52.k	12	no	$-1$	$1$	$e\left(\frac{2}{3}\right)$	$i$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{7}{12}\right)$	$-i$	$e\left(\frac{1}{6}\right)$
$\chi_{52}(35,\cdot)$	52.j	6	yes	$-1$	$1$	$e\left(\frac{1}{6}\right)$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{1}{6}\right)$
$\chi_{52}(37,\cdot)$	52.k	12	no	$-1$	$1$	$e\left(\frac{1}{3}\right)$	$i$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{11}{12}\right)$	$-i$	$e\left(\frac{5}{6}\right)$
$\chi_{52}(41,\cdot)$	52.k	12	no	$-1$	$1$	$e\left(\frac{1}{3}\right)$	$-i$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{12}\right)$	$i$	$e\left(\frac{5}{6}\right)$
$\chi_{52}(43,\cdot)$	52.i	6	yes	$-1$	$1$	$e\left(\frac{1}{6}\right)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{1}{6}\right)$
$\chi_{52}(45,\cdot)$	52.k	12	no	$-1$	$1$	$e\left(\frac{2}{3}\right)$	$-i$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{12}\right)$	$i$	$e\left(\frac{1}{6}\right)$
$\chi_{52}(47,\cdot)$	52.f	4	yes	$1$	$1$	$-1$	$i$	$i$	$1$	$i$	$-i$	$-1$	$-i$	$-i$	$1$
$\chi_{52}(49,\cdot)$	52.h	6	no	$1$	$1$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$	$-1$	$e\left(\frac{1}{3}\right)$
$\chi_{52}(51,\cdot)$	52.b	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	$52$
Structure	\(C_{2}\times C_{12}\)
Order	$24$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(5\)	\(7\)	\(9\)	\(11\)	\(15\)	\(17\)	\(19\)	\(21\)	\(23\)
\(\chi_{52}(1,\cdot)\)	52.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{52}(3,\cdot)\)	52.j	6	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{52}(5,\cdot)\)	52.g	4	no	\(-1\)	\(1\)	\(1\)	\(-i\)	\(i\)	\(1\)	\(i\)	\(-i\)	\(-1\)	\(-i\)	\(i\)	\(-1\)
\(\chi_{52}(7,\cdot)\)	52.l	12	yes	\(1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(i\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(-i\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{52}(9,\cdot)\)	52.e	3	no	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{52}(11,\cdot)\)	52.l	12	yes	\(1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(i\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(-i\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{52}(15,\cdot)\)	52.l	12	yes	\(1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(-i\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(i\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{52}(17,\cdot)\)	52.h	6	no	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{52}(19,\cdot)\)	52.l	12	yes	\(1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(-i\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(i\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{52}(21,\cdot)\)	52.g	4	no	\(-1\)	\(1\)	\(1\)	\(i\)	\(-i\)	\(1\)	\(-i\)	\(i\)	\(-1\)	\(i\)	\(-i\)	\(-1\)
\(\chi_{52}(23,\cdot)\)	52.i	6	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{52}(25,\cdot)\)	52.d	2	no	\(1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)
\(\chi_{52}(27,\cdot)\)	52.c	2	no	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)
\(\chi_{52}(29,\cdot)\)	52.e	3	no	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{52}(31,\cdot)\)	52.f	4	yes	\(1\)	\(1\)	\(-1\)	\(-i\)	\(-i\)	\(1\)	\(-i\)	\(i\)	\(-1\)	\(i\)	\(i\)	\(1\)
\(\chi_{52}(33,\cdot)\)	52.k	12	no	\(-1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(i\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(-i\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{52}(35,\cdot)\)	52.j	6	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{52}(37,\cdot)\)	52.k	12	no	\(-1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(i\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(-i\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{52}(41,\cdot)\)	52.k	12	no	\(-1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(-i\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(i\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{52}(43,\cdot)\)	52.i	6	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{52}(45,\cdot)\)	52.k	12	no	\(-1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(-i\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(i\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{52}(47,\cdot)\)	52.f	4	yes	\(1\)	\(1\)	\(-1\)	\(i\)	\(i\)	\(1\)	\(i\)	\(-i\)	\(-1\)	\(-i\)	\(-i\)	\(1\)
\(\chi_{52}(49,\cdot)\)	52.h	6	no	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{52}(51,\cdot)\)	52.b	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.