LMFDB - Group of Dirichlet characters of modulus 1600

Show commands: PariGP / SageMath

sage: H = DirichletGroup(1600)

pari: g = idealstar(,1600,2)

Character group

sage: G.order() pari: g.no
Order	=	640
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{4}\times C_{80}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{1600}(1151,\cdot)$, $\chi_{1600}(901,\cdot)$, $\chi_{1600}(577,\cdot)$

First 32 of 640 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	$-1$	$1$	$3$	$7$	$9$	$11$	$13$	$17$	$19$	$21$	$23$	$27$
$\chi_{1600}(1,\cdot)$	1600.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{1600}(3,\cdot)$	1600.cm	80	yes	$1$	$1$	$e\left(\frac{41}{80}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{1}{40}\right)$	$e\left(\frac{3}{80}\right)$	$e\left(\frac{37}{80}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{9}{80}\right)$	$e\left(\frac{51}{80}\right)$	$e\left(\frac{39}{40}\right)$	$e\left(\frac{43}{80}\right)$
$\chi_{1600}(7,\cdot)$	1600.bb	8	no	$1$	$1$	$e\left(\frac{1}{8}\right)$	$1$	$i$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{1}{8}\right)$	$-i$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{1}{8}\right)$	$1$	$e\left(\frac{3}{8}\right)$
$\chi_{1600}(9,\cdot)$	1600.cg	40	no	$1$	$1$	$e\left(\frac{1}{40}\right)$	$i$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{3}{40}\right)$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{9}{40}\right)$	$e\left(\frac{11}{40}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{3}{40}\right)$
$\chi_{1600}(11,\cdot)$	1600.cp	80	yes	$-1$	$1$	$e\left(\frac{3}{80}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{40}\right)$	$e\left(\frac{69}{80}\right)$	$e\left(\frac{71}{80}\right)$	$e\left(\frac{3}{20}\right)$	$e\left(\frac{7}{80}\right)$	$e\left(\frac{53}{80}\right)$	$e\left(\frac{27}{40}\right)$	$e\left(\frac{9}{80}\right)$
$\chi_{1600}(13,\cdot)$	1600.cn	80	yes	$-1$	$1$	$e\left(\frac{37}{80}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{71}{80}\right)$	$e\left(\frac{9}{80}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{53}{80}\right)$	$e\left(\frac{47}{80}\right)$	$e\left(\frac{23}{40}\right)$	$e\left(\frac{31}{80}\right)$
$\chi_{1600}(17,\cdot)$	1600.bs	20	no	$-1$	$1$	$e\left(\frac{4}{5}\right)$	$-i$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{3}{20}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{9}{20}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{2}{5}\right)$
$\chi_{1600}(19,\cdot)$	1600.co	80	yes	$-1$	$1$	$e\left(\frac{9}{80}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{9}{40}\right)$	$e\left(\frac{7}{80}\right)$	$e\left(\frac{53}{80}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{61}{80}\right)$	$e\left(\frac{39}{80}\right)$	$e\left(\frac{21}{40}\right)$	$e\left(\frac{27}{80}\right)$
$\chi_{1600}(21,\cdot)$	1600.cr	80	yes	$1$	$1$	$e\left(\frac{51}{80}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{11}{40}\right)$	$e\left(\frac{53}{80}\right)$	$e\left(\frac{47}{80}\right)$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{39}{80}\right)$	$e\left(\frac{61}{80}\right)$	$e\left(\frac{39}{40}\right)$	$e\left(\frac{73}{80}\right)$
$\chi_{1600}(23,\cdot)$	1600.cl	40	no	$1$	$1$	$e\left(\frac{39}{40}\right)$	$1$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{27}{40}\right)$	$e\left(\frac{23}{40}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{21}{40}\right)$	$e\left(\frac{39}{40}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{37}{40}\right)$
$\chi_{1600}(27,\cdot)$	1600.cm	80	yes	$1$	$1$	$e\left(\frac{43}{80}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{3}{40}\right)$	$e\left(\frac{9}{80}\right)$	$e\left(\frac{31}{80}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{27}{80}\right)$	$e\left(\frac{73}{80}\right)$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{49}{80}\right)$
$\chi_{1600}(29,\cdot)$	1600.cq	80	yes	$1$	$1$	$e\left(\frac{61}{80}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{21}{40}\right)$	$e\left(\frac{3}{80}\right)$	$e\left(\frac{17}{80}\right)$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{49}{80}\right)$	$e\left(\frac{11}{80}\right)$	$e\left(\frac{29}{40}\right)$	$e\left(\frac{23}{80}\right)$
$\chi_{1600}(31,\cdot)$	1600.bd	10	no	$-1$	$1$	$e\left(\frac{4}{5}\right)$	$-1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{2}{5}\right)$
$\chi_{1600}(33,\cdot)$	1600.bz	20	no	$-1$	$1$	$e\left(\frac{11}{20}\right)$	$-i$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{13}{20}\right)$
$\chi_{1600}(37,\cdot)$	1600.cn	80	yes	$-1$	$1$	$e\left(\frac{67}{80}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{27}{40}\right)$	$e\left(\frac{1}{80}\right)$	$e\left(\frac{79}{80}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{3}{80}\right)$	$e\left(\frac{57}{80}\right)$	$e\left(\frac{33}{40}\right)$	$e\left(\frac{41}{80}\right)$
$\chi_{1600}(39,\cdot)$	1600.ch	40	no	$-1$	$1$	$e\left(\frac{39}{40}\right)$	$i$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{23}{40}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{31}{40}\right)$	$e\left(\frac{9}{40}\right)$	$e\left(\frac{11}{20}\right)$	$e\left(\frac{37}{40}\right)$
$\chi_{1600}(41,\cdot)$	1600.ci	40	no	$1$	$1$	$e\left(\frac{1}{40}\right)$	$-i$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{23}{40}\right)$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{29}{40}\right)$	$e\left(\frac{31}{40}\right)$	$e\left(\frac{9}{20}\right)$	$e\left(\frac{3}{40}\right)$
$\chi_{1600}(43,\cdot)$	1600.bl	16	no	$1$	$1$	$e\left(\frac{3}{16}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{7}{16}\right)$	$-1$	$e\left(\frac{11}{16}\right)$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{9}{16}\right)$
$\chi_{1600}(47,\cdot)$	1600.cc	20	no	$1$	$1$	$e\left(\frac{7}{10}\right)$	$i$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{17}{20}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{1}{10}\right)$
$\chi_{1600}(49,\cdot)$	1600.q	4	no	$1$	$1$	$i$	$1$	$-1$	$i$	$i$	$-1$	$-i$	$i$	$1$	$-i$
$\chi_{1600}(51,\cdot)$	1600.bo	16	no	$-1$	$1$	$e\left(\frac{5}{16}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{16}\right)$	$e\left(\frac{1}{16}\right)$	$i$	$e\left(\frac{1}{16}\right)$	$e\left(\frac{3}{16}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{15}{16}\right)$
$\chi_{1600}(53,\cdot)$	1600.ct	80	yes	$-1$	$1$	$e\left(\frac{31}{80}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{31}{40}\right)$	$e\left(\frac{13}{80}\right)$	$e\left(\frac{27}{80}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{39}{80}\right)$	$e\left(\frac{21}{80}\right)$	$e\left(\frac{9}{40}\right)$	$e\left(\frac{13}{80}\right)$
$\chi_{1600}(57,\cdot)$	1600.w	8	no	$-1$	$1$	$e\left(\frac{5}{8}\right)$	$-1$	$i$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{1}{8}\right)$	$-i$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{8}\right)$	$-1$	$e\left(\frac{7}{8}\right)$
$\chi_{1600}(59,\cdot)$	1600.co	80	yes	$-1$	$1$	$e\left(\frac{47}{80}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{7}{40}\right)$	$e\left(\frac{1}{80}\right)$	$e\left(\frac{19}{80}\right)$	$e\left(\frac{17}{20}\right)$	$e\left(\frac{43}{80}\right)$	$e\left(\frac{17}{80}\right)$	$e\left(\frac{3}{40}\right)$	$e\left(\frac{61}{80}\right)$
$\chi_{1600}(61,\cdot)$	1600.cr	80	yes	$1$	$1$	$e\left(\frac{13}{80}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{13}{40}\right)$	$e\left(\frac{59}{80}\right)$	$e\left(\frac{1}{80}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{57}{80}\right)$	$e\left(\frac{3}{80}\right)$	$e\left(\frac{17}{40}\right)$	$e\left(\frac{39}{80}\right)$
$\chi_{1600}(63,\cdot)$	1600.by	20	no	$1$	$1$	$e\left(\frac{3}{20}\right)$	$i$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{9}{20}\right)$
$\chi_{1600}(67,\cdot)$	1600.cs	80	yes	$1$	$1$	$e\left(\frac{49}{80}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{9}{40}\right)$	$e\left(\frac{67}{80}\right)$	$e\left(\frac{13}{80}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{41}{80}\right)$	$e\left(\frac{19}{80}\right)$	$e\left(\frac{11}{40}\right)$	$e\left(\frac{67}{80}\right)$
$\chi_{1600}(69,\cdot)$	1600.cq	80	yes	$1$	$1$	$e\left(\frac{39}{80}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{39}{40}\right)$	$e\left(\frac{57}{80}\right)$	$e\left(\frac{3}{80}\right)$	$e\left(\frac{9}{20}\right)$	$e\left(\frac{51}{80}\right)$	$e\left(\frac{49}{80}\right)$	$e\left(\frac{31}{40}\right)$	$e\left(\frac{37}{80}\right)$
$\chi_{1600}(71,\cdot)$	1600.cj	40	no	$-1$	$1$	$e\left(\frac{23}{40}\right)$	$-i$	$e\left(\frac{3}{20}\right)$	$e\left(\frac{9}{40}\right)$	$e\left(\frac{31}{40}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{27}{40}\right)$	$e\left(\frac{13}{40}\right)$	$e\left(\frac{17}{20}\right)$	$e\left(\frac{29}{40}\right)$
$\chi_{1600}(73,\cdot)$	1600.ce	40	no	$-1$	$1$	$e\left(\frac{39}{40}\right)$	$-1$	$e\left(\frac{19}{20}\right)$	$e\left(\frac{27}{40}\right)$	$e\left(\frac{3}{40}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{21}{40}\right)$	$e\left(\frac{19}{40}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{37}{40}\right)$
$\chi_{1600}(77,\cdot)$	1600.ct	80	yes	$-1$	$1$	$e\left(\frac{13}{80}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{13}{40}\right)$	$e\left(\frac{39}{80}\right)$	$e\left(\frac{1}{80}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{37}{80}\right)$	$e\left(\frac{63}{80}\right)$	$e\left(\frac{27}{40}\right)$	$e\left(\frac{39}{80}\right)$
$\chi_{1600}(79,\cdot)$	1600.bv	20	no	$-1$	$1$	$e\left(\frac{19}{20}\right)$	$-1$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{7}{20}\right)$	$e\left(\frac{13}{20}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{1}{20}\right)$	$e\left(\frac{9}{20}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{17}{20}\right)$

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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	$1600$
Structure	\(C_{2}\times C_{4}\times C_{80}\)
Order	$640$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(7\)	\(9\)	\(11\)	\(13\)	\(17\)	\(19\)	\(21\)	\(23\)	\(27\)
\(\chi_{1600}(1,\cdot)\)	1600.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{1600}(3,\cdot)\)	1600.cm	80	yes	\(1\)	\(1\)	\(e\left(\frac{41}{80}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{1}{40}\right)\)	\(e\left(\frac{3}{80}\right)\)	\(e\left(\frac{37}{80}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{9}{80}\right)\)	\(e\left(\frac{51}{80}\right)\)	\(e\left(\frac{39}{40}\right)\)	\(e\left(\frac{43}{80}\right)\)
\(\chi_{1600}(7,\cdot)\)	1600.bb	8	no	\(1\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(1\)	\(i\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(-i\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(1\)	\(e\left(\frac{3}{8}\right)\)
\(\chi_{1600}(9,\cdot)\)	1600.cg	40	no	\(1\)	\(1\)	\(e\left(\frac{1}{40}\right)\)	\(i\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{3}{40}\right)\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{9}{40}\right)\)	\(e\left(\frac{11}{40}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{3}{40}\right)\)
\(\chi_{1600}(11,\cdot)\)	1600.cp	80	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{80}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{3}{40}\right)\)	\(e\left(\frac{69}{80}\right)\)	\(e\left(\frac{71}{80}\right)\)	\(e\left(\frac{3}{20}\right)\)	\(e\left(\frac{7}{80}\right)\)	\(e\left(\frac{53}{80}\right)\)	\(e\left(\frac{27}{40}\right)\)	\(e\left(\frac{9}{80}\right)\)
\(\chi_{1600}(13,\cdot)\)	1600.cn	80	yes	\(-1\)	\(1\)	\(e\left(\frac{37}{80}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{71}{80}\right)\)	\(e\left(\frac{9}{80}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{53}{80}\right)\)	\(e\left(\frac{47}{80}\right)\)	\(e\left(\frac{23}{40}\right)\)	\(e\left(\frac{31}{80}\right)\)
\(\chi_{1600}(17,\cdot)\)	1600.bs	20	no	\(-1\)	\(1\)	\(e\left(\frac{4}{5}\right)\)	\(-i\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{3}{20}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{9}{20}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{2}{5}\right)\)
\(\chi_{1600}(19,\cdot)\)	1600.co	80	yes	\(-1\)	\(1\)	\(e\left(\frac{9}{80}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{9}{40}\right)\)	\(e\left(\frac{7}{80}\right)\)	\(e\left(\frac{53}{80}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{61}{80}\right)\)	\(e\left(\frac{39}{80}\right)\)	\(e\left(\frac{21}{40}\right)\)	\(e\left(\frac{27}{80}\right)\)
\(\chi_{1600}(21,\cdot)\)	1600.cr	80	yes	\(1\)	\(1\)	\(e\left(\frac{51}{80}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{11}{40}\right)\)	\(e\left(\frac{53}{80}\right)\)	\(e\left(\frac{47}{80}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{39}{80}\right)\)	\(e\left(\frac{61}{80}\right)\)	\(e\left(\frac{39}{40}\right)\)	\(e\left(\frac{73}{80}\right)\)
\(\chi_{1600}(23,\cdot)\)	1600.cl	40	no	\(1\)	\(1\)	\(e\left(\frac{39}{40}\right)\)	\(1\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{27}{40}\right)\)	\(e\left(\frac{23}{40}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{21}{40}\right)\)	\(e\left(\frac{39}{40}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{37}{40}\right)\)
\(\chi_{1600}(27,\cdot)\)	1600.cm	80	yes	\(1\)	\(1\)	\(e\left(\frac{43}{80}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{3}{40}\right)\)	\(e\left(\frac{9}{80}\right)\)	\(e\left(\frac{31}{80}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{27}{80}\right)\)	\(e\left(\frac{73}{80}\right)\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{49}{80}\right)\)
\(\chi_{1600}(29,\cdot)\)	1600.cq	80	yes	\(1\)	\(1\)	\(e\left(\frac{61}{80}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{21}{40}\right)\)	\(e\left(\frac{3}{80}\right)\)	\(e\left(\frac{17}{80}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{49}{80}\right)\)	\(e\left(\frac{11}{80}\right)\)	\(e\left(\frac{29}{40}\right)\)	\(e\left(\frac{23}{80}\right)\)
\(\chi_{1600}(31,\cdot)\)	1600.bd	10	no	\(-1\)	\(1\)	\(e\left(\frac{4}{5}\right)\)	\(-1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)
\(\chi_{1600}(33,\cdot)\)	1600.bz	20	no	\(-1\)	\(1\)	\(e\left(\frac{11}{20}\right)\)	\(-i\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{13}{20}\right)\)
\(\chi_{1600}(37,\cdot)\)	1600.cn	80	yes	\(-1\)	\(1\)	\(e\left(\frac{67}{80}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{27}{40}\right)\)	\(e\left(\frac{1}{80}\right)\)	\(e\left(\frac{79}{80}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{3}{80}\right)\)	\(e\left(\frac{57}{80}\right)\)	\(e\left(\frac{33}{40}\right)\)	\(e\left(\frac{41}{80}\right)\)
\(\chi_{1600}(39,\cdot)\)	1600.ch	40	no	\(-1\)	\(1\)	\(e\left(\frac{39}{40}\right)\)	\(i\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{23}{40}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{31}{40}\right)\)	\(e\left(\frac{9}{40}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(e\left(\frac{37}{40}\right)\)
\(\chi_{1600}(41,\cdot)\)	1600.ci	40	no	\(1\)	\(1\)	\(e\left(\frac{1}{40}\right)\)	\(-i\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{23}{40}\right)\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{29}{40}\right)\)	\(e\left(\frac{31}{40}\right)\)	\(e\left(\frac{9}{20}\right)\)	\(e\left(\frac{3}{40}\right)\)
\(\chi_{1600}(43,\cdot)\)	1600.bl	16	no	\(1\)	\(1\)	\(e\left(\frac{3}{16}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{9}{16}\right)\)	\(e\left(\frac{7}{16}\right)\)	\(-1\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{9}{16}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{9}{16}\right)\)
\(\chi_{1600}(47,\cdot)\)	1600.cc	20	no	\(1\)	\(1\)	\(e\left(\frac{7}{10}\right)\)	\(i\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{1}{10}\right)\)
\(\chi_{1600}(49,\cdot)\)	1600.q	4	no	\(1\)	\(1\)	\(i\)	\(1\)	\(-1\)	\(i\)	\(i\)	\(-1\)	\(-i\)	\(i\)	\(1\)	\(-i\)
\(\chi_{1600}(51,\cdot)\)	1600.bo	16	no	\(-1\)	\(1\)	\(e\left(\frac{5}{16}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{3}{16}\right)\)	\(e\left(\frac{1}{16}\right)\)	\(i\)	\(e\left(\frac{1}{16}\right)\)	\(e\left(\frac{3}{16}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{15}{16}\right)\)
\(\chi_{1600}(53,\cdot)\)	1600.ct	80	yes	\(-1\)	\(1\)	\(e\left(\frac{31}{80}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{31}{40}\right)\)	\(e\left(\frac{13}{80}\right)\)	\(e\left(\frac{27}{80}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{39}{80}\right)\)	\(e\left(\frac{21}{80}\right)\)	\(e\left(\frac{9}{40}\right)\)	\(e\left(\frac{13}{80}\right)\)
\(\chi_{1600}(57,\cdot)\)	1600.w	8	no	\(-1\)	\(1\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)	\(i\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(-i\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(-1\)	\(e\left(\frac{7}{8}\right)\)
\(\chi_{1600}(59,\cdot)\)	1600.co	80	yes	\(-1\)	\(1\)	\(e\left(\frac{47}{80}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{7}{40}\right)\)	\(e\left(\frac{1}{80}\right)\)	\(e\left(\frac{19}{80}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(e\left(\frac{43}{80}\right)\)	\(e\left(\frac{17}{80}\right)\)	\(e\left(\frac{3}{40}\right)\)	\(e\left(\frac{61}{80}\right)\)
\(\chi_{1600}(61,\cdot)\)	1600.cr	80	yes	\(1\)	\(1\)	\(e\left(\frac{13}{80}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{13}{40}\right)\)	\(e\left(\frac{59}{80}\right)\)	\(e\left(\frac{1}{80}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{57}{80}\right)\)	\(e\left(\frac{3}{80}\right)\)	\(e\left(\frac{17}{40}\right)\)	\(e\left(\frac{39}{80}\right)\)
\(\chi_{1600}(63,\cdot)\)	1600.by	20	no	\(1\)	\(1\)	\(e\left(\frac{3}{20}\right)\)	\(i\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{9}{20}\right)\)
\(\chi_{1600}(67,\cdot)\)	1600.cs	80	yes	\(1\)	\(1\)	\(e\left(\frac{49}{80}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{9}{40}\right)\)	\(e\left(\frac{67}{80}\right)\)	\(e\left(\frac{13}{80}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{41}{80}\right)\)	\(e\left(\frac{19}{80}\right)\)	\(e\left(\frac{11}{40}\right)\)	\(e\left(\frac{67}{80}\right)\)
\(\chi_{1600}(69,\cdot)\)	1600.cq	80	yes	\(1\)	\(1\)	\(e\left(\frac{39}{80}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{39}{40}\right)\)	\(e\left(\frac{57}{80}\right)\)	\(e\left(\frac{3}{80}\right)\)	\(e\left(\frac{9}{20}\right)\)	\(e\left(\frac{51}{80}\right)\)	\(e\left(\frac{49}{80}\right)\)	\(e\left(\frac{31}{40}\right)\)	\(e\left(\frac{37}{80}\right)\)
\(\chi_{1600}(71,\cdot)\)	1600.cj	40	no	\(-1\)	\(1\)	\(e\left(\frac{23}{40}\right)\)	\(-i\)	\(e\left(\frac{3}{20}\right)\)	\(e\left(\frac{9}{40}\right)\)	\(e\left(\frac{31}{40}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{27}{40}\right)\)	\(e\left(\frac{13}{40}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(e\left(\frac{29}{40}\right)\)
\(\chi_{1600}(73,\cdot)\)	1600.ce	40	no	\(-1\)	\(1\)	\(e\left(\frac{39}{40}\right)\)	\(-1\)	\(e\left(\frac{19}{20}\right)\)	\(e\left(\frac{27}{40}\right)\)	\(e\left(\frac{3}{40}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{21}{40}\right)\)	\(e\left(\frac{19}{40}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{37}{40}\right)\)
\(\chi_{1600}(77,\cdot)\)	1600.ct	80	yes	\(-1\)	\(1\)	\(e\left(\frac{13}{80}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{13}{40}\right)\)	\(e\left(\frac{39}{80}\right)\)	\(e\left(\frac{1}{80}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{37}{80}\right)\)	\(e\left(\frac{63}{80}\right)\)	\(e\left(\frac{27}{40}\right)\)	\(e\left(\frac{39}{80}\right)\)
\(\chi_{1600}(79,\cdot)\)	1600.bv	20	no	\(-1\)	\(1\)	\(e\left(\frac{19}{20}\right)\)	\(-1\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{9}{20}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{17}{20}\right)\)

Introduction

L-functions

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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.