LMFDB - Group of Dirichlet characters of modulus 249

Show commands: PariGP / SageMath

sage: H = DirichletGroup(249)

pari: g = idealstar(,249,2)

Character group

sage: G.order() pari: g.no
Order	=	164
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{82}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{249}(167,\cdot)$, $\chi_{249}(85,\cdot)$

First 32 of 164 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	$-1$	$1$	$2$	$4$	$5$	$7$	$8$	$10$	$11$	$13$	$14$	$16$
$\chi_{249}(1,\cdot)$	249.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{249}(2,\cdot)$	249.f	82	yes	$1$	$1$	$e\left(\frac{21}{41}\right)$	$e\left(\frac{1}{41}\right)$	$e\left(\frac{34}{41}\right)$	$e\left(\frac{4}{41}\right)$	$e\left(\frac{22}{41}\right)$	$e\left(\frac{14}{41}\right)$	$e\left(\frac{65}{82}\right)$	$e\left(\frac{77}{82}\right)$	$e\left(\frac{25}{41}\right)$	$e\left(\frac{2}{41}\right)$
$\chi_{249}(4,\cdot)$	249.e	41	no	$1$	$1$	$e\left(\frac{1}{41}\right)$	$e\left(\frac{2}{41}\right)$	$e\left(\frac{27}{41}\right)$	$e\left(\frac{8}{41}\right)$	$e\left(\frac{3}{41}\right)$	$e\left(\frac{28}{41}\right)$	$e\left(\frac{24}{41}\right)$	$e\left(\frac{36}{41}\right)$	$e\left(\frac{9}{41}\right)$	$e\left(\frac{4}{41}\right)$
$\chi_{249}(5,\cdot)$	249.f	82	yes	$1$	$1$	$e\left(\frac{34}{41}\right)$	$e\left(\frac{27}{41}\right)$	$e\left(\frac{16}{41}\right)$	$e\left(\frac{26}{41}\right)$	$e\left(\frac{20}{41}\right)$	$e\left(\frac{9}{41}\right)$	$e\left(\frac{33}{82}\right)$	$e\left(\frac{29}{82}\right)$	$e\left(\frac{19}{41}\right)$	$e\left(\frac{13}{41}\right)$
$\chi_{249}(7,\cdot)$	249.e	41	no	$1$	$1$	$e\left(\frac{4}{41}\right)$	$e\left(\frac{8}{41}\right)$	$e\left(\frac{26}{41}\right)$	$e\left(\frac{32}{41}\right)$	$e\left(\frac{12}{41}\right)$	$e\left(\frac{30}{41}\right)$	$e\left(\frac{14}{41}\right)$	$e\left(\frac{21}{41}\right)$	$e\left(\frac{36}{41}\right)$	$e\left(\frac{16}{41}\right)$
$\chi_{249}(8,\cdot)$	249.f	82	yes	$1$	$1$	$e\left(\frac{22}{41}\right)$	$e\left(\frac{3}{41}\right)$	$e\left(\frac{20}{41}\right)$	$e\left(\frac{12}{41}\right)$	$e\left(\frac{25}{41}\right)$	$e\left(\frac{1}{41}\right)$	$e\left(\frac{31}{82}\right)$	$e\left(\frac{67}{82}\right)$	$e\left(\frac{34}{41}\right)$	$e\left(\frac{6}{41}\right)$
$\chi_{249}(10,\cdot)$	249.e	41	no	$1$	$1$	$e\left(\frac{14}{41}\right)$	$e\left(\frac{28}{41}\right)$	$e\left(\frac{9}{41}\right)$	$e\left(\frac{30}{41}\right)$	$e\left(\frac{1}{41}\right)$	$e\left(\frac{23}{41}\right)$	$e\left(\frac{8}{41}\right)$	$e\left(\frac{12}{41}\right)$	$e\left(\frac{3}{41}\right)$	$e\left(\frac{15}{41}\right)$
$\chi_{249}(11,\cdot)$	249.h	82	yes	$-1$	$1$	$e\left(\frac{65}{82}\right)$	$e\left(\frac{24}{41}\right)$	$e\left(\frac{33}{82}\right)$	$e\left(\frac{14}{41}\right)$	$e\left(\frac{31}{82}\right)$	$e\left(\frac{8}{41}\right)$	$e\left(\frac{43}{82}\right)$	$e\left(\frac{22}{41}\right)$	$e\left(\frac{11}{82}\right)$	$e\left(\frac{7}{41}\right)$
$\chi_{249}(13,\cdot)$	249.g	82	no	$-1$	$1$	$e\left(\frac{77}{82}\right)$	$e\left(\frac{36}{41}\right)$	$e\left(\frac{29}{82}\right)$	$e\left(\frac{21}{41}\right)$	$e\left(\frac{67}{82}\right)$	$e\left(\frac{12}{41}\right)$	$e\left(\frac{22}{41}\right)$	$e\left(\frac{25}{82}\right)$	$e\left(\frac{37}{82}\right)$	$e\left(\frac{31}{41}\right)$
$\chi_{249}(14,\cdot)$	249.f	82	yes	$1$	$1$	$e\left(\frac{25}{41}\right)$	$e\left(\frac{9}{41}\right)$	$e\left(\frac{19}{41}\right)$	$e\left(\frac{36}{41}\right)$	$e\left(\frac{34}{41}\right)$	$e\left(\frac{3}{41}\right)$	$e\left(\frac{11}{82}\right)$	$e\left(\frac{37}{82}\right)$	$e\left(\frac{20}{41}\right)$	$e\left(\frac{18}{41}\right)$
$\chi_{249}(16,\cdot)$	249.e	41	no	$1$	$1$	$e\left(\frac{2}{41}\right)$	$e\left(\frac{4}{41}\right)$	$e\left(\frac{13}{41}\right)$	$e\left(\frac{16}{41}\right)$	$e\left(\frac{6}{41}\right)$	$e\left(\frac{15}{41}\right)$	$e\left(\frac{7}{41}\right)$	$e\left(\frac{31}{41}\right)$	$e\left(\frac{18}{41}\right)$	$e\left(\frac{8}{41}\right)$
$\chi_{249}(17,\cdot)$	249.h	82	yes	$-1$	$1$	$e\left(\frac{15}{82}\right)$	$e\left(\frac{15}{41}\right)$	$e\left(\frac{77}{82}\right)$	$e\left(\frac{19}{41}\right)$	$e\left(\frac{45}{82}\right)$	$e\left(\frac{5}{41}\right)$	$e\left(\frac{73}{82}\right)$	$e\left(\frac{24}{41}\right)$	$e\left(\frac{53}{82}\right)$	$e\left(\frac{30}{41}\right)$
$\chi_{249}(19,\cdot)$	249.g	82	no	$-1$	$1$	$e\left(\frac{47}{82}\right)$	$e\left(\frac{6}{41}\right)$	$e\left(\frac{39}{82}\right)$	$e\left(\frac{24}{41}\right)$	$e\left(\frac{59}{82}\right)$	$e\left(\frac{2}{41}\right)$	$e\left(\frac{31}{41}\right)$	$e\left(\frac{11}{82}\right)$	$e\left(\frac{13}{82}\right)$	$e\left(\frac{12}{41}\right)$
$\chi_{249}(20,\cdot)$	249.f	82	yes	$1$	$1$	$e\left(\frac{35}{41}\right)$	$e\left(\frac{29}{41}\right)$	$e\left(\frac{2}{41}\right)$	$e\left(\frac{34}{41}\right)$	$e\left(\frac{23}{41}\right)$	$e\left(\frac{37}{41}\right)$	$e\left(\frac{81}{82}\right)$	$e\left(\frac{19}{82}\right)$	$e\left(\frac{28}{41}\right)$	$e\left(\frac{17}{41}\right)$
$\chi_{249}(22,\cdot)$	249.g	82	no	$-1$	$1$	$e\left(\frac{25}{82}\right)$	$e\left(\frac{25}{41}\right)$	$e\left(\frac{19}{82}\right)$	$e\left(\frac{18}{41}\right)$	$e\left(\frac{75}{82}\right)$	$e\left(\frac{22}{41}\right)$	$e\left(\frac{13}{41}\right)$	$e\left(\frac{39}{82}\right)$	$e\left(\frac{61}{82}\right)$	$e\left(\frac{9}{41}\right)$
$\chi_{249}(23,\cdot)$	249.h	82	yes	$-1$	$1$	$e\left(\frac{19}{82}\right)$	$e\left(\frac{19}{41}\right)$	$e\left(\frac{21}{82}\right)$	$e\left(\frac{35}{41}\right)$	$e\left(\frac{57}{82}\right)$	$e\left(\frac{20}{41}\right)$	$e\left(\frac{5}{82}\right)$	$e\left(\frac{14}{41}\right)$	$e\left(\frac{7}{82}\right)$	$e\left(\frac{38}{41}\right)$
$\chi_{249}(25,\cdot)$	249.e	41	no	$1$	$1$	$e\left(\frac{27}{41}\right)$	$e\left(\frac{13}{41}\right)$	$e\left(\frac{32}{41}\right)$	$e\left(\frac{11}{41}\right)$	$e\left(\frac{40}{41}\right)$	$e\left(\frac{18}{41}\right)$	$e\left(\frac{33}{41}\right)$	$e\left(\frac{29}{41}\right)$	$e\left(\frac{38}{41}\right)$	$e\left(\frac{26}{41}\right)$
$\chi_{249}(26,\cdot)$	249.h	82	yes	$-1$	$1$	$e\left(\frac{37}{82}\right)$	$e\left(\frac{37}{41}\right)$	$e\left(\frac{15}{82}\right)$	$e\left(\frac{25}{41}\right)$	$e\left(\frac{29}{82}\right)$	$e\left(\frac{26}{41}\right)$	$e\left(\frac{27}{82}\right)$	$e\left(\frac{10}{41}\right)$	$e\left(\frac{5}{82}\right)$	$e\left(\frac{33}{41}\right)$
$\chi_{249}(28,\cdot)$	249.e	41	no	$1$	$1$	$e\left(\frac{5}{41}\right)$	$e\left(\frac{10}{41}\right)$	$e\left(\frac{12}{41}\right)$	$e\left(\frac{40}{41}\right)$	$e\left(\frac{15}{41}\right)$	$e\left(\frac{17}{41}\right)$	$e\left(\frac{38}{41}\right)$	$e\left(\frac{16}{41}\right)$	$e\left(\frac{4}{41}\right)$	$e\left(\frac{20}{41}\right)$
$\chi_{249}(29,\cdot)$	249.h	82	yes	$-1$	$1$	$e\left(\frac{53}{82}\right)$	$e\left(\frac{12}{41}\right)$	$e\left(\frac{37}{82}\right)$	$e\left(\frac{7}{41}\right)$	$e\left(\frac{77}{82}\right)$	$e\left(\frac{4}{41}\right)$	$e\left(\frac{1}{82}\right)$	$e\left(\frac{11}{41}\right)$	$e\left(\frac{67}{82}\right)$	$e\left(\frac{24}{41}\right)$
$\chi_{249}(31,\cdot)$	249.e	41	no	$1$	$1$	$e\left(\frac{19}{41}\right)$	$e\left(\frac{38}{41}\right)$	$e\left(\frac{21}{41}\right)$	$e\left(\frac{29}{41}\right)$	$e\left(\frac{16}{41}\right)$	$e\left(\frac{40}{41}\right)$	$e\left(\frac{5}{41}\right)$	$e\left(\frac{28}{41}\right)$	$e\left(\frac{7}{41}\right)$	$e\left(\frac{35}{41}\right)$
$\chi_{249}(32,\cdot)$	249.f	82	yes	$1$	$1$	$e\left(\frac{23}{41}\right)$	$e\left(\frac{5}{41}\right)$	$e\left(\frac{6}{41}\right)$	$e\left(\frac{20}{41}\right)$	$e\left(\frac{28}{41}\right)$	$e\left(\frac{29}{41}\right)$	$e\left(\frac{79}{82}\right)$	$e\left(\frac{57}{82}\right)$	$e\left(\frac{2}{41}\right)$	$e\left(\frac{10}{41}\right)$
$\chi_{249}(34,\cdot)$	249.g	82	no	$-1$	$1$	$e\left(\frac{57}{82}\right)$	$e\left(\frac{16}{41}\right)$	$e\left(\frac{63}{82}\right)$	$e\left(\frac{23}{41}\right)$	$e\left(\frac{7}{82}\right)$	$e\left(\frac{19}{41}\right)$	$e\left(\frac{28}{41}\right)$	$e\left(\frac{43}{82}\right)$	$e\left(\frac{21}{82}\right)$	$e\left(\frac{32}{41}\right)$
$\chi_{249}(35,\cdot)$	249.f	82	yes	$1$	$1$	$e\left(\frac{38}{41}\right)$	$e\left(\frac{35}{41}\right)$	$e\left(\frac{1}{41}\right)$	$e\left(\frac{17}{41}\right)$	$e\left(\frac{32}{41}\right)$	$e\left(\frac{39}{41}\right)$	$e\left(\frac{61}{82}\right)$	$e\left(\frac{71}{82}\right)$	$e\left(\frac{14}{41}\right)$	$e\left(\frac{29}{41}\right)$
$\chi_{249}(37,\cdot)$	249.e	41	no	$1$	$1$	$e\left(\frac{10}{41}\right)$	$e\left(\frac{20}{41}\right)$	$e\left(\frac{24}{41}\right)$	$e\left(\frac{39}{41}\right)$	$e\left(\frac{30}{41}\right)$	$e\left(\frac{34}{41}\right)$	$e\left(\frac{35}{41}\right)$	$e\left(\frac{32}{41}\right)$	$e\left(\frac{8}{41}\right)$	$e\left(\frac{40}{41}\right)$
$\chi_{249}(38,\cdot)$	249.h	82	yes	$-1$	$1$	$e\left(\frac{7}{82}\right)$	$e\left(\frac{7}{41}\right)$	$e\left(\frac{25}{82}\right)$	$e\left(\frac{28}{41}\right)$	$e\left(\frac{21}{82}\right)$	$e\left(\frac{16}{41}\right)$	$e\left(\frac{45}{82}\right)$	$e\left(\frac{3}{41}\right)$	$e\left(\frac{63}{82}\right)$	$e\left(\frac{14}{41}\right)$
$\chi_{249}(40,\cdot)$	249.e	41	no	$1$	$1$	$e\left(\frac{15}{41}\right)$	$e\left(\frac{30}{41}\right)$	$e\left(\frac{36}{41}\right)$	$e\left(\frac{38}{41}\right)$	$e\left(\frac{4}{41}\right)$	$e\left(\frac{10}{41}\right)$	$e\left(\frac{32}{41}\right)$	$e\left(\frac{7}{41}\right)$	$e\left(\frac{12}{41}\right)$	$e\left(\frac{19}{41}\right)$
$\chi_{249}(41,\cdot)$	249.h	82	yes	$-1$	$1$	$e\left(\frac{81}{82}\right)$	$e\left(\frac{40}{41}\right)$	$e\left(\frac{55}{82}\right)$	$e\left(\frac{37}{41}\right)$	$e\left(\frac{79}{82}\right)$	$e\left(\frac{27}{41}\right)$	$e\left(\frac{17}{82}\right)$	$e\left(\frac{23}{41}\right)$	$e\left(\frac{73}{82}\right)$	$e\left(\frac{39}{41}\right)$
$\chi_{249}(43,\cdot)$	249.g	82	no	$-1$	$1$	$e\left(\frac{71}{82}\right)$	$e\left(\frac{30}{41}\right)$	$e\left(\frac{31}{82}\right)$	$e\left(\frac{38}{41}\right)$	$e\left(\frac{49}{82}\right)$	$e\left(\frac{10}{41}\right)$	$e\left(\frac{32}{41}\right)$	$e\left(\frac{55}{82}\right)$	$e\left(\frac{65}{82}\right)$	$e\left(\frac{19}{41}\right)$
$\chi_{249}(44,\cdot)$	249.h	82	yes	$-1$	$1$	$e\left(\frac{67}{82}\right)$	$e\left(\frac{26}{41}\right)$	$e\left(\frac{5}{82}\right)$	$e\left(\frac{22}{41}\right)$	$e\left(\frac{37}{82}\right)$	$e\left(\frac{36}{41}\right)$	$e\left(\frac{9}{82}\right)$	$e\left(\frac{17}{41}\right)$	$e\left(\frac{29}{82}\right)$	$e\left(\frac{11}{41}\right)$
$\chi_{249}(46,\cdot)$	249.g	82	no	$-1$	$1$	$e\left(\frac{61}{82}\right)$	$e\left(\frac{20}{41}\right)$	$e\left(\frac{7}{82}\right)$	$e\left(\frac{39}{41}\right)$	$e\left(\frac{19}{82}\right)$	$e\left(\frac{34}{41}\right)$	$e\left(\frac{35}{41}\right)$	$e\left(\frac{23}{82}\right)$	$e\left(\frac{57}{82}\right)$	$e\left(\frac{40}{41}\right)$
$\chi_{249}(47,\cdot)$	249.f	82	yes	$1$	$1$	$e\left(\frac{32}{41}\right)$	$e\left(\frac{23}{41}\right)$	$e\left(\frac{3}{41}\right)$	$e\left(\frac{10}{41}\right)$	$e\left(\frac{14}{41}\right)$	$e\left(\frac{35}{41}\right)$	$e\left(\frac{19}{82}\right)$	$e\left(\frac{49}{82}\right)$	$e\left(\frac{1}{41}\right)$	$e\left(\frac{5}{41}\right)$

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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	$249$
Structure	\(C_{2}\times C_{82}\)
Order	$164$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(4\)	\(5\)	\(7\)	\(8\)	\(10\)	\(11\)	\(13\)	\(14\)	\(16\)
\(\chi_{249}(1,\cdot)\)	249.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{249}(2,\cdot)\)	249.f	82	yes	\(1\)	\(1\)	\(e\left(\frac{21}{41}\right)\)	\(e\left(\frac{1}{41}\right)\)	\(e\left(\frac{34}{41}\right)\)	\(e\left(\frac{4}{41}\right)\)	\(e\left(\frac{22}{41}\right)\)	\(e\left(\frac{14}{41}\right)\)	\(e\left(\frac{65}{82}\right)\)	\(e\left(\frac{77}{82}\right)\)	\(e\left(\frac{25}{41}\right)\)	\(e\left(\frac{2}{41}\right)\)
\(\chi_{249}(4,\cdot)\)	249.e	41	no	\(1\)	\(1\)	\(e\left(\frac{1}{41}\right)\)	\(e\left(\frac{2}{41}\right)\)	\(e\left(\frac{27}{41}\right)\)	\(e\left(\frac{8}{41}\right)\)	\(e\left(\frac{3}{41}\right)\)	\(e\left(\frac{28}{41}\right)\)	\(e\left(\frac{24}{41}\right)\)	\(e\left(\frac{36}{41}\right)\)	\(e\left(\frac{9}{41}\right)\)	\(e\left(\frac{4}{41}\right)\)
\(\chi_{249}(5,\cdot)\)	249.f	82	yes	\(1\)	\(1\)	\(e\left(\frac{34}{41}\right)\)	\(e\left(\frac{27}{41}\right)\)	\(e\left(\frac{16}{41}\right)\)	\(e\left(\frac{26}{41}\right)\)	\(e\left(\frac{20}{41}\right)\)	\(e\left(\frac{9}{41}\right)\)	\(e\left(\frac{33}{82}\right)\)	\(e\left(\frac{29}{82}\right)\)	\(e\left(\frac{19}{41}\right)\)	\(e\left(\frac{13}{41}\right)\)
\(\chi_{249}(7,\cdot)\)	249.e	41	no	\(1\)	\(1\)	\(e\left(\frac{4}{41}\right)\)	\(e\left(\frac{8}{41}\right)\)	\(e\left(\frac{26}{41}\right)\)	\(e\left(\frac{32}{41}\right)\)	\(e\left(\frac{12}{41}\right)\)	\(e\left(\frac{30}{41}\right)\)	\(e\left(\frac{14}{41}\right)\)	\(e\left(\frac{21}{41}\right)\)	\(e\left(\frac{36}{41}\right)\)	\(e\left(\frac{16}{41}\right)\)
\(\chi_{249}(8,\cdot)\)	249.f	82	yes	\(1\)	\(1\)	\(e\left(\frac{22}{41}\right)\)	\(e\left(\frac{3}{41}\right)\)	\(e\left(\frac{20}{41}\right)\)	\(e\left(\frac{12}{41}\right)\)	\(e\left(\frac{25}{41}\right)\)	\(e\left(\frac{1}{41}\right)\)	\(e\left(\frac{31}{82}\right)\)	\(e\left(\frac{67}{82}\right)\)	\(e\left(\frac{34}{41}\right)\)	\(e\left(\frac{6}{41}\right)\)
\(\chi_{249}(10,\cdot)\)	249.e	41	no	\(1\)	\(1\)	\(e\left(\frac{14}{41}\right)\)	\(e\left(\frac{28}{41}\right)\)	\(e\left(\frac{9}{41}\right)\)	\(e\left(\frac{30}{41}\right)\)	\(e\left(\frac{1}{41}\right)\)	\(e\left(\frac{23}{41}\right)\)	\(e\left(\frac{8}{41}\right)\)	\(e\left(\frac{12}{41}\right)\)	\(e\left(\frac{3}{41}\right)\)	\(e\left(\frac{15}{41}\right)\)
\(\chi_{249}(11,\cdot)\)	249.h	82	yes	\(-1\)	\(1\)	\(e\left(\frac{65}{82}\right)\)	\(e\left(\frac{24}{41}\right)\)	\(e\left(\frac{33}{82}\right)\)	\(e\left(\frac{14}{41}\right)\)	\(e\left(\frac{31}{82}\right)\)	\(e\left(\frac{8}{41}\right)\)	\(e\left(\frac{43}{82}\right)\)	\(e\left(\frac{22}{41}\right)\)	\(e\left(\frac{11}{82}\right)\)	\(e\left(\frac{7}{41}\right)\)
\(\chi_{249}(13,\cdot)\)	249.g	82	no	\(-1\)	\(1\)	\(e\left(\frac{77}{82}\right)\)	\(e\left(\frac{36}{41}\right)\)	\(e\left(\frac{29}{82}\right)\)	\(e\left(\frac{21}{41}\right)\)	\(e\left(\frac{67}{82}\right)\)	\(e\left(\frac{12}{41}\right)\)	\(e\left(\frac{22}{41}\right)\)	\(e\left(\frac{25}{82}\right)\)	\(e\left(\frac{37}{82}\right)\)	\(e\left(\frac{31}{41}\right)\)
\(\chi_{249}(14,\cdot)\)	249.f	82	yes	\(1\)	\(1\)	\(e\left(\frac{25}{41}\right)\)	\(e\left(\frac{9}{41}\right)\)	\(e\left(\frac{19}{41}\right)\)	\(e\left(\frac{36}{41}\right)\)	\(e\left(\frac{34}{41}\right)\)	\(e\left(\frac{3}{41}\right)\)	\(e\left(\frac{11}{82}\right)\)	\(e\left(\frac{37}{82}\right)\)	\(e\left(\frac{20}{41}\right)\)	\(e\left(\frac{18}{41}\right)\)
\(\chi_{249}(16,\cdot)\)	249.e	41	no	\(1\)	\(1\)	\(e\left(\frac{2}{41}\right)\)	\(e\left(\frac{4}{41}\right)\)	\(e\left(\frac{13}{41}\right)\)	\(e\left(\frac{16}{41}\right)\)	\(e\left(\frac{6}{41}\right)\)	\(e\left(\frac{15}{41}\right)\)	\(e\left(\frac{7}{41}\right)\)	\(e\left(\frac{31}{41}\right)\)	\(e\left(\frac{18}{41}\right)\)	\(e\left(\frac{8}{41}\right)\)
\(\chi_{249}(17,\cdot)\)	249.h	82	yes	\(-1\)	\(1\)	\(e\left(\frac{15}{82}\right)\)	\(e\left(\frac{15}{41}\right)\)	\(e\left(\frac{77}{82}\right)\)	\(e\left(\frac{19}{41}\right)\)	\(e\left(\frac{45}{82}\right)\)	\(e\left(\frac{5}{41}\right)\)	\(e\left(\frac{73}{82}\right)\)	\(e\left(\frac{24}{41}\right)\)	\(e\left(\frac{53}{82}\right)\)	\(e\left(\frac{30}{41}\right)\)
\(\chi_{249}(19,\cdot)\)	249.g	82	no	\(-1\)	\(1\)	\(e\left(\frac{47}{82}\right)\)	\(e\left(\frac{6}{41}\right)\)	\(e\left(\frac{39}{82}\right)\)	\(e\left(\frac{24}{41}\right)\)	\(e\left(\frac{59}{82}\right)\)	\(e\left(\frac{2}{41}\right)\)	\(e\left(\frac{31}{41}\right)\)	\(e\left(\frac{11}{82}\right)\)	\(e\left(\frac{13}{82}\right)\)	\(e\left(\frac{12}{41}\right)\)
\(\chi_{249}(20,\cdot)\)	249.f	82	yes	\(1\)	\(1\)	\(e\left(\frac{35}{41}\right)\)	\(e\left(\frac{29}{41}\right)\)	\(e\left(\frac{2}{41}\right)\)	\(e\left(\frac{34}{41}\right)\)	\(e\left(\frac{23}{41}\right)\)	\(e\left(\frac{37}{41}\right)\)	\(e\left(\frac{81}{82}\right)\)	\(e\left(\frac{19}{82}\right)\)	\(e\left(\frac{28}{41}\right)\)	\(e\left(\frac{17}{41}\right)\)
\(\chi_{249}(22,\cdot)\)	249.g	82	no	\(-1\)	\(1\)	\(e\left(\frac{25}{82}\right)\)	\(e\left(\frac{25}{41}\right)\)	\(e\left(\frac{19}{82}\right)\)	\(e\left(\frac{18}{41}\right)\)	\(e\left(\frac{75}{82}\right)\)	\(e\left(\frac{22}{41}\right)\)	\(e\left(\frac{13}{41}\right)\)	\(e\left(\frac{39}{82}\right)\)	\(e\left(\frac{61}{82}\right)\)	\(e\left(\frac{9}{41}\right)\)
\(\chi_{249}(23,\cdot)\)	249.h	82	yes	\(-1\)	\(1\)	\(e\left(\frac{19}{82}\right)\)	\(e\left(\frac{19}{41}\right)\)	\(e\left(\frac{21}{82}\right)\)	\(e\left(\frac{35}{41}\right)\)	\(e\left(\frac{57}{82}\right)\)	\(e\left(\frac{20}{41}\right)\)	\(e\left(\frac{5}{82}\right)\)	\(e\left(\frac{14}{41}\right)\)	\(e\left(\frac{7}{82}\right)\)	\(e\left(\frac{38}{41}\right)\)
\(\chi_{249}(25,\cdot)\)	249.e	41	no	\(1\)	\(1\)	\(e\left(\frac{27}{41}\right)\)	\(e\left(\frac{13}{41}\right)\)	\(e\left(\frac{32}{41}\right)\)	\(e\left(\frac{11}{41}\right)\)	\(e\left(\frac{40}{41}\right)\)	\(e\left(\frac{18}{41}\right)\)	\(e\left(\frac{33}{41}\right)\)	\(e\left(\frac{29}{41}\right)\)	\(e\left(\frac{38}{41}\right)\)	\(e\left(\frac{26}{41}\right)\)
\(\chi_{249}(26,\cdot)\)	249.h	82	yes	\(-1\)	\(1\)	\(e\left(\frac{37}{82}\right)\)	\(e\left(\frac{37}{41}\right)\)	\(e\left(\frac{15}{82}\right)\)	\(e\left(\frac{25}{41}\right)\)	\(e\left(\frac{29}{82}\right)\)	\(e\left(\frac{26}{41}\right)\)	\(e\left(\frac{27}{82}\right)\)	\(e\left(\frac{10}{41}\right)\)	\(e\left(\frac{5}{82}\right)\)	\(e\left(\frac{33}{41}\right)\)
\(\chi_{249}(28,\cdot)\)	249.e	41	no	\(1\)	\(1\)	\(e\left(\frac{5}{41}\right)\)	\(e\left(\frac{10}{41}\right)\)	\(e\left(\frac{12}{41}\right)\)	\(e\left(\frac{40}{41}\right)\)	\(e\left(\frac{15}{41}\right)\)	\(e\left(\frac{17}{41}\right)\)	\(e\left(\frac{38}{41}\right)\)	\(e\left(\frac{16}{41}\right)\)	\(e\left(\frac{4}{41}\right)\)	\(e\left(\frac{20}{41}\right)\)
\(\chi_{249}(29,\cdot)\)	249.h	82	yes	\(-1\)	\(1\)	\(e\left(\frac{53}{82}\right)\)	\(e\left(\frac{12}{41}\right)\)	\(e\left(\frac{37}{82}\right)\)	\(e\left(\frac{7}{41}\right)\)	\(e\left(\frac{77}{82}\right)\)	\(e\left(\frac{4}{41}\right)\)	\(e\left(\frac{1}{82}\right)\)	\(e\left(\frac{11}{41}\right)\)	\(e\left(\frac{67}{82}\right)\)	\(e\left(\frac{24}{41}\right)\)
\(\chi_{249}(31,\cdot)\)	249.e	41	no	\(1\)	\(1\)	\(e\left(\frac{19}{41}\right)\)	\(e\left(\frac{38}{41}\right)\)	\(e\left(\frac{21}{41}\right)\)	\(e\left(\frac{29}{41}\right)\)	\(e\left(\frac{16}{41}\right)\)	\(e\left(\frac{40}{41}\right)\)	\(e\left(\frac{5}{41}\right)\)	\(e\left(\frac{28}{41}\right)\)	\(e\left(\frac{7}{41}\right)\)	\(e\left(\frac{35}{41}\right)\)
\(\chi_{249}(32,\cdot)\)	249.f	82	yes	\(1\)	\(1\)	\(e\left(\frac{23}{41}\right)\)	\(e\left(\frac{5}{41}\right)\)	\(e\left(\frac{6}{41}\right)\)	\(e\left(\frac{20}{41}\right)\)	\(e\left(\frac{28}{41}\right)\)	\(e\left(\frac{29}{41}\right)\)	\(e\left(\frac{79}{82}\right)\)	\(e\left(\frac{57}{82}\right)\)	\(e\left(\frac{2}{41}\right)\)	\(e\left(\frac{10}{41}\right)\)
\(\chi_{249}(34,\cdot)\)	249.g	82	no	\(-1\)	\(1\)	\(e\left(\frac{57}{82}\right)\)	\(e\left(\frac{16}{41}\right)\)	\(e\left(\frac{63}{82}\right)\)	\(e\left(\frac{23}{41}\right)\)	\(e\left(\frac{7}{82}\right)\)	\(e\left(\frac{19}{41}\right)\)	\(e\left(\frac{28}{41}\right)\)	\(e\left(\frac{43}{82}\right)\)	\(e\left(\frac{21}{82}\right)\)	\(e\left(\frac{32}{41}\right)\)
\(\chi_{249}(35,\cdot)\)	249.f	82	yes	\(1\)	\(1\)	\(e\left(\frac{38}{41}\right)\)	\(e\left(\frac{35}{41}\right)\)	\(e\left(\frac{1}{41}\right)\)	\(e\left(\frac{17}{41}\right)\)	\(e\left(\frac{32}{41}\right)\)	\(e\left(\frac{39}{41}\right)\)	\(e\left(\frac{61}{82}\right)\)	\(e\left(\frac{71}{82}\right)\)	\(e\left(\frac{14}{41}\right)\)	\(e\left(\frac{29}{41}\right)\)
\(\chi_{249}(37,\cdot)\)	249.e	41	no	\(1\)	\(1\)	\(e\left(\frac{10}{41}\right)\)	\(e\left(\frac{20}{41}\right)\)	\(e\left(\frac{24}{41}\right)\)	\(e\left(\frac{39}{41}\right)\)	\(e\left(\frac{30}{41}\right)\)	\(e\left(\frac{34}{41}\right)\)	\(e\left(\frac{35}{41}\right)\)	\(e\left(\frac{32}{41}\right)\)	\(e\left(\frac{8}{41}\right)\)	\(e\left(\frac{40}{41}\right)\)
\(\chi_{249}(38,\cdot)\)	249.h	82	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{82}\right)\)	\(e\left(\frac{7}{41}\right)\)	\(e\left(\frac{25}{82}\right)\)	\(e\left(\frac{28}{41}\right)\)	\(e\left(\frac{21}{82}\right)\)	\(e\left(\frac{16}{41}\right)\)	\(e\left(\frac{45}{82}\right)\)	\(e\left(\frac{3}{41}\right)\)	\(e\left(\frac{63}{82}\right)\)	\(e\left(\frac{14}{41}\right)\)
\(\chi_{249}(40,\cdot)\)	249.e	41	no	\(1\)	\(1\)	\(e\left(\frac{15}{41}\right)\)	\(e\left(\frac{30}{41}\right)\)	\(e\left(\frac{36}{41}\right)\)	\(e\left(\frac{38}{41}\right)\)	\(e\left(\frac{4}{41}\right)\)	\(e\left(\frac{10}{41}\right)\)	\(e\left(\frac{32}{41}\right)\)	\(e\left(\frac{7}{41}\right)\)	\(e\left(\frac{12}{41}\right)\)	\(e\left(\frac{19}{41}\right)\)
\(\chi_{249}(41,\cdot)\)	249.h	82	yes	\(-1\)	\(1\)	\(e\left(\frac{81}{82}\right)\)	\(e\left(\frac{40}{41}\right)\)	\(e\left(\frac{55}{82}\right)\)	\(e\left(\frac{37}{41}\right)\)	\(e\left(\frac{79}{82}\right)\)	\(e\left(\frac{27}{41}\right)\)	\(e\left(\frac{17}{82}\right)\)	\(e\left(\frac{23}{41}\right)\)	\(e\left(\frac{73}{82}\right)\)	\(e\left(\frac{39}{41}\right)\)
\(\chi_{249}(43,\cdot)\)	249.g	82	no	\(-1\)	\(1\)	\(e\left(\frac{71}{82}\right)\)	\(e\left(\frac{30}{41}\right)\)	\(e\left(\frac{31}{82}\right)\)	\(e\left(\frac{38}{41}\right)\)	\(e\left(\frac{49}{82}\right)\)	\(e\left(\frac{10}{41}\right)\)	\(e\left(\frac{32}{41}\right)\)	\(e\left(\frac{55}{82}\right)\)	\(e\left(\frac{65}{82}\right)\)	\(e\left(\frac{19}{41}\right)\)
\(\chi_{249}(44,\cdot)\)	249.h	82	yes	\(-1\)	\(1\)	\(e\left(\frac{67}{82}\right)\)	\(e\left(\frac{26}{41}\right)\)	\(e\left(\frac{5}{82}\right)\)	\(e\left(\frac{22}{41}\right)\)	\(e\left(\frac{37}{82}\right)\)	\(e\left(\frac{36}{41}\right)\)	\(e\left(\frac{9}{82}\right)\)	\(e\left(\frac{17}{41}\right)\)	\(e\left(\frac{29}{82}\right)\)	\(e\left(\frac{11}{41}\right)\)
\(\chi_{249}(46,\cdot)\)	249.g	82	no	\(-1\)	\(1\)	\(e\left(\frac{61}{82}\right)\)	\(e\left(\frac{20}{41}\right)\)	\(e\left(\frac{7}{82}\right)\)	\(e\left(\frac{39}{41}\right)\)	\(e\left(\frac{19}{82}\right)\)	\(e\left(\frac{34}{41}\right)\)	\(e\left(\frac{35}{41}\right)\)	\(e\left(\frac{23}{82}\right)\)	\(e\left(\frac{57}{82}\right)\)	\(e\left(\frac{40}{41}\right)\)
\(\chi_{249}(47,\cdot)\)	249.f	82	yes	\(1\)	\(1\)	\(e\left(\frac{32}{41}\right)\)	\(e\left(\frac{23}{41}\right)\)	\(e\left(\frac{3}{41}\right)\)	\(e\left(\frac{10}{41}\right)\)	\(e\left(\frac{14}{41}\right)\)	\(e\left(\frac{35}{41}\right)\)	\(e\left(\frac{19}{82}\right)\)	\(e\left(\frac{49}{82}\right)\)	\(e\left(\frac{1}{41}\right)\)	\(e\left(\frac{5}{41}\right)\)

Introduction

L-functions

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Character group

First 32 of 164 characters

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