LMFDB - Group of Dirichlet characters of modulus 100010

Show commands: PariGP / SageMath

sage: H = DirichletGroup(100010)

pari: g = idealstar(,100010,2)

Character group

sage: G.order() pari: g.no
Order	=	39168
sage: H.invariants() pari: g.cyc
Structure	=	$C_{4}\times C_{8}\times C_{1224}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{100010}(60007,\cdot)$, $\chi_{100010}(64391,\cdot)$, $\chi_{100010}(97821,\cdot)$

First 32 of 39168 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	$-1$	$1$	$3$	$7$	$9$	$11$	$13$	$17$	$19$	$21$	$23$	$27$
$\chi_{100010}(1,\cdot)$	100010.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{100010}(3,\cdot)$	100010.oz	408	no	$1$	$1$	$e\left(\frac{103}{136}\right)$	$e\left(\frac{55}{68}\right)$	$e\left(\frac{35}{68}\right)$	$e\left(\frac{49}{102}\right)$	$e\left(\frac{143}{408}\right)$	$e\left(\frac{53}{68}\right)$	$e\left(\frac{1}{204}\right)$	$e\left(\frac{77}{136}\right)$	$e\left(\frac{1}{408}\right)$	$e\left(\frac{37}{136}\right)$
$\chi_{100010}(7,\cdot)$	100010.py	408	no	$1$	$1$	$e\left(\frac{55}{68}\right)$	$e\left(\frac{47}{136}\right)$	$e\left(\frac{21}{34}\right)$	$e\left(\frac{361}{408}\right)$	$e\left(\frac{209}{408}\right)$	$e\left(\frac{83}{136}\right)$	$e\left(\frac{25}{204}\right)$	$e\left(\frac{21}{136}\right)$	$e\left(\frac{89}{204}\right)$	$e\left(\frac{29}{68}\right)$
$\chi_{100010}(9,\cdot)$	100010.oc	204	no	$1$	$1$	$e\left(\frac{35}{68}\right)$	$e\left(\frac{21}{34}\right)$	$e\left(\frac{1}{34}\right)$	$e\left(\frac{49}{51}\right)$	$e\left(\frac{143}{204}\right)$	$e\left(\frac{19}{34}\right)$	$e\left(\frac{1}{102}\right)$	$e\left(\frac{9}{68}\right)$	$e\left(\frac{1}{204}\right)$	$e\left(\frac{37}{68}\right)$
$\chi_{100010}(11,\cdot)$	100010.ro	1224	no	$-1$	$1$	$e\left(\frac{49}{102}\right)$	$e\left(\frac{361}{408}\right)$	$e\left(\frac{49}{51}\right)$	$e\left(\frac{557}{1224}\right)$	$e\left(\frac{607}{1224}\right)$	$e\left(\frac{53}{408}\right)$	$e\left(\frac{383}{612}\right)$	$e\left(\frac{149}{408}\right)$	$e\left(\frac{83}{306}\right)$	$e\left(\frac{15}{34}\right)$
$\chi_{100010}(13,\cdot)$	100010.sy	1224	no	$-1$	$1$	$e\left(\frac{143}{408}\right)$	$e\left(\frac{209}{408}\right)$	$e\left(\frac{143}{204}\right)$	$e\left(\frac{607}{1224}\right)$	$e\left(\frac{59}{306}\right)$	$e\left(\frac{385}{408}\right)$	$e\left(\frac{233}{306}\right)$	$e\left(\frac{44}{51}\right)$	$e\left(\frac{1129}{1224}\right)$	$e\left(\frac{7}{136}\right)$
$\chi_{100010}(17,\cdot)$	100010.qb	408	no	$1$	$1$	$e\left(\frac{53}{68}\right)$	$e\left(\frac{83}{136}\right)$	$e\left(\frac{19}{34}\right)$	$e\left(\frac{53}{408}\right)$	$e\left(\frac{385}{408}\right)$	$e\left(\frac{135}{136}\right)$	$e\left(\frac{89}{204}\right)$	$e\left(\frac{53}{136}\right)$	$e\left(\frac{19}{204}\right)$	$e\left(\frac{23}{68}\right)$
$\chi_{100010}(19,\cdot)$	100010.qw	612	no	$1$	$1$	$e\left(\frac{1}{204}\right)$	$e\left(\frac{25}{204}\right)$	$e\left(\frac{1}{102}\right)$	$e\left(\frac{383}{612}\right)$	$e\left(\frac{233}{306}\right)$	$e\left(\frac{89}{204}\right)$	$e\left(\frac{145}{153}\right)$	$e\left(\frac{13}{102}\right)$	$e\left(\frac{239}{612}\right)$	$e\left(\frac{1}{68}\right)$
$\chi_{100010}(21,\cdot)$	100010.ps	408	no	$1$	$1$	$e\left(\frac{77}{136}\right)$	$e\left(\frac{21}{136}\right)$	$e\left(\frac{9}{68}\right)$	$e\left(\frac{149}{408}\right)$	$e\left(\frac{44}{51}\right)$	$e\left(\frac{53}{136}\right)$	$e\left(\frac{13}{102}\right)$	$e\left(\frac{49}{68}\right)$	$e\left(\frac{179}{408}\right)$	$e\left(\frac{95}{136}\right)$
$\chi_{100010}(23,\cdot)$	100010.sq	1224	no	$1$	$1$	$e\left(\frac{1}{408}\right)$	$e\left(\frac{89}{204}\right)$	$e\left(\frac{1}{204}\right)$	$e\left(\frac{83}{306}\right)$	$e\left(\frac{1129}{1224}\right)$	$e\left(\frac{19}{204}\right)$	$e\left(\frac{239}{612}\right)$	$e\left(\frac{179}{408}\right)$	$e\left(\frac{647}{1224}\right)$	$e\left(\frac{1}{136}\right)$
$\chi_{100010}(27,\cdot)$	100010.mo	136	no	$1$	$1$	$e\left(\frac{37}{136}\right)$	$e\left(\frac{29}{68}\right)$	$e\left(\frac{37}{68}\right)$	$e\left(\frac{15}{34}\right)$	$e\left(\frac{7}{136}\right)$	$e\left(\frac{23}{68}\right)$	$e\left(\frac{1}{68}\right)$	$e\left(\frac{95}{136}\right)$	$e\left(\frac{1}{136}\right)$	$e\left(\frac{111}{136}\right)$
$\chi_{100010}(29,\cdot)$	100010.sx	1224	no	$1$	$1$	$e\left(\frac{35}{408}\right)$	$e\left(\frac{263}{408}\right)$	$e\left(\frac{35}{204}\right)$	$e\left(\frac{451}{1224}\right)$	$e\left(\frac{139}{153}\right)$	$e\left(\frac{55}{408}\right)$	$e\left(\frac{281}{306}\right)$	$e\left(\frac{149}{204}\right)$	$e\left(\frac{613}{1224}\right)$	$e\left(\frac{35}{136}\right)$
$\chi_{100010}(31,\cdot)$	100010.ta	1224	no	$1$	$1$	$e\left(\frac{185}{408}\right)$	$e\left(\frac{239}{408}\right)$	$e\left(\frac{185}{204}\right)$	$e\left(\frac{1087}{1224}\right)$	$e\left(\frac{265}{612}\right)$	$e\left(\frac{247}{408}\right)$	$e\left(\frac{25}{153}\right)$	$e\left(\frac{2}{51}\right)$	$e\left(\frac{151}{1224}\right)$	$e\left(\frac{49}{136}\right)$
$\chi_{100010}(33,\cdot)$	100010.se	1224	no	$-1$	$1$	$e\left(\frac{97}{408}\right)$	$e\left(\frac{283}{408}\right)$	$e\left(\frac{97}{204}\right)$	$e\left(\frac{1145}{1224}\right)$	$e\left(\frac{259}{306}\right)$	$e\left(\frac{371}{408}\right)$	$e\left(\frac{193}{306}\right)$	$e\left(\frac{95}{102}\right)$	$e\left(\frac{335}{1224}\right)$	$e\left(\frac{97}{136}\right)$
$\chi_{100010}(37,\cdot)$	100010.ic	36	no	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{7}{18}\right)$	$-1$
$\chi_{100010}(39,\cdot)$	100010.rp	1224	no	$-1$	$1$	$e\left(\frac{11}{102}\right)$	$e\left(\frac{131}{408}\right)$	$e\left(\frac{11}{51}\right)$	$e\left(\frac{1195}{1224}\right)$	$e\left(\frac{665}{1224}\right)$	$e\left(\frac{295}{408}\right)$	$e\left(\frac{469}{612}\right)$	$e\left(\frac{175}{408}\right)$	$e\left(\frac{283}{306}\right)$	$e\left(\frac{11}{34}\right)$
$\chi_{100010}(41,\cdot)$	100010.kd	72	no	$-1$	$1$	$e\left(\frac{5}{24}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{29}{36}\right)$	$e\left(\frac{11}{72}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{25}{36}\right)$	$e\left(\frac{19}{24}\right)$	$e\left(\frac{67}{72}\right)$	$e\left(\frac{5}{8}\right)$
$\chi_{100010}(43,\cdot)$	100010.pa	408	no	$-1$	$1$	$e\left(\frac{29}{136}\right)$	$e\left(\frac{11}{136}\right)$	$e\left(\frac{29}{68}\right)$	$e\left(\frac{397}{408}\right)$	$e\left(\frac{89}{102}\right)$	$e\left(\frac{99}{136}\right)$	$e\left(\frac{23}{102}\right)$	$e\left(\frac{5}{17}\right)$	$e\left(\frac{403}{408}\right)$	$e\left(\frac{87}{136}\right)$
$\chi_{100010}(47,\cdot)$	100010.st	1224	no	$-1$	$1$	$e\left(\frac{193}{408}\right)$	$e\left(\frac{133}{408}\right)$	$e\left(\frac{193}{204}\right)$	$e\left(\frac{887}{1224}\right)$	$e\left(\frac{395}{612}\right)$	$e\left(\frac{245}{408}\right)$	$e\left(\frac{95}{153}\right)$	$e\left(\frac{163}{204}\right)$	$e\left(\frac{23}{1224}\right)$	$e\left(\frac{57}{136}\right)$
$\chi_{100010}(49,\cdot)$	100010.og	204	no	$1$	$1$	$e\left(\frac{21}{34}\right)$	$e\left(\frac{47}{68}\right)$	$e\left(\frac{4}{17}\right)$	$e\left(\frac{157}{204}\right)$	$e\left(\frac{5}{204}\right)$	$e\left(\frac{15}{68}\right)$	$e\left(\frac{25}{102}\right)$	$e\left(\frac{21}{68}\right)$	$e\left(\frac{89}{102}\right)$	$e\left(\frac{29}{34}\right)$
$\chi_{100010}(51,\cdot)$	100010.lv	136	no	$1$	$1$	$e\left(\frac{73}{136}\right)$	$e\left(\frac{57}{136}\right)$	$e\left(\frac{5}{68}\right)$	$e\left(\frac{83}{136}\right)$	$e\left(\frac{5}{17}\right)$	$e\left(\frac{105}{136}\right)$	$e\left(\frac{15}{34}\right)$	$e\left(\frac{65}{68}\right)$	$e\left(\frac{13}{136}\right)$	$e\left(\frac{83}{136}\right)$
$\chi_{100010}(53,\cdot)$	100010.se	1224	no	$-1$	$1$	$e\left(\frac{257}{408}\right)$	$e\left(\frac{203}{408}\right)$	$e\left(\frac{53}{204}\right)$	$e\left(\frac{1}{1224}\right)$	$e\left(\frac{233}{306}\right)$	$e\left(\frac{331}{408}\right)$	$e\left(\frac{137}{306}\right)$	$e\left(\frac{13}{102}\right)$	$e\left(\frac{631}{1224}\right)$	$e\left(\frac{121}{136}\right)$
$\chi_{100010}(57,\cdot)$	100010.rt	1224	no	$1$	$1$	$e\left(\frac{311}{408}\right)$	$e\left(\frac{95}{102}\right)$	$e\left(\frac{107}{204}\right)$	$e\left(\frac{65}{612}\right)$	$e\left(\frac{137}{1224}\right)$	$e\left(\frac{11}{51}\right)$	$e\left(\frac{583}{612}\right)$	$e\left(\frac{283}{408}\right)$	$e\left(\frac{481}{1224}\right)$	$e\left(\frac{39}{136}\right)$
$\chi_{100010}(59,\cdot)$	100010.rx	1224	no	$-1$	$1$	$e\left(\frac{139}{204}\right)$	$e\left(\frac{371}{408}\right)$	$e\left(\frac{37}{102}\right)$	$e\left(\frac{139}{1224}\right)$	$e\left(\frac{875}{1224}\right)$	$e\left(\frac{7}{408}\right)$	$e\left(\frac{295}{612}\right)$	$e\left(\frac{241}{408}\right)$	$e\left(\frac{173}{612}\right)$	$e\left(\frac{3}{68}\right)$
$\chi_{100010}(61,\cdot)$	100010.ra	612	no	$1$	$1$	$e\left(\frac{167}{204}\right)$	$e\left(\frac{197}{204}\right)$	$e\left(\frac{65}{102}\right)$	$e\left(\frac{313}{612}\right)$	$e\left(\frac{49}{306}\right)$	$e\left(\frac{73}{204}\right)$	$e\left(\frac{41}{153}\right)$	$e\left(\frac{40}{51}\right)$	$e\left(\frac{133}{612}\right)$	$e\left(\frac{31}{68}\right)$
$\chi_{100010}(63,\cdot)$	100010.ne	136	no	$1$	$1$	$e\left(\frac{11}{34}\right)$	$e\left(\frac{131}{136}\right)$	$e\left(\frac{11}{17}\right)$	$e\left(\frac{115}{136}\right)$	$e\left(\frac{29}{136}\right)$	$e\left(\frac{23}{136}\right)$	$e\left(\frac{9}{68}\right)$	$e\left(\frac{39}{136}\right)$	$e\left(\frac{15}{34}\right)$	$e\left(\frac{33}{34}\right)$
$\chi_{100010}(67,\cdot)$	100010.sq	1224	no	$1$	$1$	$e\left(\frac{143}{408}\right)$	$e\left(\frac{79}{204}\right)$	$e\left(\frac{143}{204}\right)$	$e\left(\frac{37}{306}\right)$	$e\left(\frac{695}{1224}\right)$	$e\left(\frac{65}{204}\right)$	$e\left(\frac{313}{612}\right)$	$e\left(\frac{301}{408}\right)$	$e\left(\frac{1129}{1224}\right)$	$e\left(\frac{7}{136}\right)$
$\chi_{100010}(69,\cdot)$	100010.qu	612	no	$1$	$1$	$e\left(\frac{155}{204}\right)$	$e\left(\frac{25}{102}\right)$	$e\left(\frac{53}{102}\right)$	$e\left(\frac{115}{153}\right)$	$e\left(\frac{167}{612}\right)$	$e\left(\frac{89}{102}\right)$	$e\left(\frac{121}{306}\right)$	$e\left(\frac{1}{204}\right)$	$e\left(\frac{325}{612}\right)$	$e\left(\frac{19}{68}\right)$
$\chi_{100010}(71,\cdot)$	100010.sn	1224	no	$-1$	$1$	$e\left(\frac{59}{408}\right)$	$e\left(\frac{49}{204}\right)$	$e\left(\frac{59}{204}\right)$	$e\left(\frac{563}{612}\right)$	$e\left(\frac{5}{1224}\right)$	$e\left(\frac{203}{204}\right)$	$e\left(\frac{535}{612}\right)$	$e\left(\frac{157}{408}\right)$	$e\left(\frac{1045}{1224}\right)$	$e\left(\frac{59}{136}\right)$
$\chi_{100010}(77,\cdot)$	100010.qx	612	no	$-1$	$1$	$e\left(\frac{59}{204}\right)$	$e\left(\frac{47}{204}\right)$	$e\left(\frac{59}{102}\right)$	$e\left(\frac{52}{153}\right)$	$e\left(\frac{5}{612}\right)$	$e\left(\frac{151}{204}\right)$	$e\left(\frac{229}{306}\right)$	$e\left(\frac{53}{102}\right)$	$e\left(\frac{433}{612}\right)$	$e\left(\frac{59}{68}\right)$
$\chi_{100010}(79,\cdot)$	100010.rv	1224	no	$-1$	$1$	$e\left(\frac{371}{408}\right)$	$e\left(\frac{11}{102}\right)$	$e\left(\frac{167}{204}\right)$	$e\left(\frac{91}{306}\right)$	$e\left(\frac{47}{1224}\right)$	$e\left(\frac{41}{51}\right)$	$e\left(\frac{133}{612}\right)$	$e\left(\frac{7}{408}\right)$	$e\left(\frac{949}{1224}\right)$	$e\left(\frac{99}{136}\right)$
$\chi_{100010}(81,\cdot)$	100010.lh	102	no	$1$	$1$	$e\left(\frac{1}{34}\right)$	$e\left(\frac{4}{17}\right)$	$e\left(\frac{1}{17}\right)$	$e\left(\frac{47}{51}\right)$	$e\left(\frac{41}{102}\right)$	$e\left(\frac{2}{17}\right)$	$e\left(\frac{1}{51}\right)$	$e\left(\frac{9}{34}\right)$	$e\left(\frac{1}{102}\right)$	$e\left(\frac{3}{34}\right)$

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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	$100010$
Structure	\(C_{4}\times C_{8}\times C_{1224}\)
Order	$39168$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(7\)	\(9\)	\(11\)	\(13\)	\(17\)	\(19\)	\(21\)	\(23\)	\(27\)
\(\chi_{100010}(1,\cdot)\)	100010.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{100010}(3,\cdot)\)	100010.oz	408	no	\(1\)	\(1\)	\(e\left(\frac{103}{136}\right)\)	\(e\left(\frac{55}{68}\right)\)	\(e\left(\frac{35}{68}\right)\)	\(e\left(\frac{49}{102}\right)\)	\(e\left(\frac{143}{408}\right)\)	\(e\left(\frac{53}{68}\right)\)	\(e\left(\frac{1}{204}\right)\)	\(e\left(\frac{77}{136}\right)\)	\(e\left(\frac{1}{408}\right)\)	\(e\left(\frac{37}{136}\right)\)
\(\chi_{100010}(7,\cdot)\)	100010.py	408	no	\(1\)	\(1\)	\(e\left(\frac{55}{68}\right)\)	\(e\left(\frac{47}{136}\right)\)	\(e\left(\frac{21}{34}\right)\)	\(e\left(\frac{361}{408}\right)\)	\(e\left(\frac{209}{408}\right)\)	\(e\left(\frac{83}{136}\right)\)	\(e\left(\frac{25}{204}\right)\)	\(e\left(\frac{21}{136}\right)\)	\(e\left(\frac{89}{204}\right)\)	\(e\left(\frac{29}{68}\right)\)
\(\chi_{100010}(9,\cdot)\)	100010.oc	204	no	\(1\)	\(1\)	\(e\left(\frac{35}{68}\right)\)	\(e\left(\frac{21}{34}\right)\)	\(e\left(\frac{1}{34}\right)\)	\(e\left(\frac{49}{51}\right)\)	\(e\left(\frac{143}{204}\right)\)	\(e\left(\frac{19}{34}\right)\)	\(e\left(\frac{1}{102}\right)\)	\(e\left(\frac{9}{68}\right)\)	\(e\left(\frac{1}{204}\right)\)	\(e\left(\frac{37}{68}\right)\)
\(\chi_{100010}(11,\cdot)\)	100010.ro	1224	no	\(-1\)	\(1\)	\(e\left(\frac{49}{102}\right)\)	\(e\left(\frac{361}{408}\right)\)	\(e\left(\frac{49}{51}\right)\)	\(e\left(\frac{557}{1224}\right)\)	\(e\left(\frac{607}{1224}\right)\)	\(e\left(\frac{53}{408}\right)\)	\(e\left(\frac{383}{612}\right)\)	\(e\left(\frac{149}{408}\right)\)	\(e\left(\frac{83}{306}\right)\)	\(e\left(\frac{15}{34}\right)\)
\(\chi_{100010}(13,\cdot)\)	100010.sy	1224	no	\(-1\)	\(1\)	\(e\left(\frac{143}{408}\right)\)	\(e\left(\frac{209}{408}\right)\)	\(e\left(\frac{143}{204}\right)\)	\(e\left(\frac{607}{1224}\right)\)	\(e\left(\frac{59}{306}\right)\)	\(e\left(\frac{385}{408}\right)\)	\(e\left(\frac{233}{306}\right)\)	\(e\left(\frac{44}{51}\right)\)	\(e\left(\frac{1129}{1224}\right)\)	\(e\left(\frac{7}{136}\right)\)
\(\chi_{100010}(17,\cdot)\)	100010.qb	408	no	\(1\)	\(1\)	\(e\left(\frac{53}{68}\right)\)	\(e\left(\frac{83}{136}\right)\)	\(e\left(\frac{19}{34}\right)\)	\(e\left(\frac{53}{408}\right)\)	\(e\left(\frac{385}{408}\right)\)	\(e\left(\frac{135}{136}\right)\)	\(e\left(\frac{89}{204}\right)\)	\(e\left(\frac{53}{136}\right)\)	\(e\left(\frac{19}{204}\right)\)	\(e\left(\frac{23}{68}\right)\)
\(\chi_{100010}(19,\cdot)\)	100010.qw	612	no	\(1\)	\(1\)	\(e\left(\frac{1}{204}\right)\)	\(e\left(\frac{25}{204}\right)\)	\(e\left(\frac{1}{102}\right)\)	\(e\left(\frac{383}{612}\right)\)	\(e\left(\frac{233}{306}\right)\)	\(e\left(\frac{89}{204}\right)\)	\(e\left(\frac{145}{153}\right)\)	\(e\left(\frac{13}{102}\right)\)	\(e\left(\frac{239}{612}\right)\)	\(e\left(\frac{1}{68}\right)\)
\(\chi_{100010}(21,\cdot)\)	100010.ps	408	no	\(1\)	\(1\)	\(e\left(\frac{77}{136}\right)\)	\(e\left(\frac{21}{136}\right)\)	\(e\left(\frac{9}{68}\right)\)	\(e\left(\frac{149}{408}\right)\)	\(e\left(\frac{44}{51}\right)\)	\(e\left(\frac{53}{136}\right)\)	\(e\left(\frac{13}{102}\right)\)	\(e\left(\frac{49}{68}\right)\)	\(e\left(\frac{179}{408}\right)\)	\(e\left(\frac{95}{136}\right)\)
\(\chi_{100010}(23,\cdot)\)	100010.sq	1224	no	\(1\)	\(1\)	\(e\left(\frac{1}{408}\right)\)	\(e\left(\frac{89}{204}\right)\)	\(e\left(\frac{1}{204}\right)\)	\(e\left(\frac{83}{306}\right)\)	\(e\left(\frac{1129}{1224}\right)\)	\(e\left(\frac{19}{204}\right)\)	\(e\left(\frac{239}{612}\right)\)	\(e\left(\frac{179}{408}\right)\)	\(e\left(\frac{647}{1224}\right)\)	\(e\left(\frac{1}{136}\right)\)
\(\chi_{100010}(27,\cdot)\)	100010.mo	136	no	\(1\)	\(1\)	\(e\left(\frac{37}{136}\right)\)	\(e\left(\frac{29}{68}\right)\)	\(e\left(\frac{37}{68}\right)\)	\(e\left(\frac{15}{34}\right)\)	\(e\left(\frac{7}{136}\right)\)	\(e\left(\frac{23}{68}\right)\)	\(e\left(\frac{1}{68}\right)\)	\(e\left(\frac{95}{136}\right)\)	\(e\left(\frac{1}{136}\right)\)	\(e\left(\frac{111}{136}\right)\)
\(\chi_{100010}(29,\cdot)\)	100010.sx	1224	no	\(1\)	\(1\)	\(e\left(\frac{35}{408}\right)\)	\(e\left(\frac{263}{408}\right)\)	\(e\left(\frac{35}{204}\right)\)	\(e\left(\frac{451}{1224}\right)\)	\(e\left(\frac{139}{153}\right)\)	\(e\left(\frac{55}{408}\right)\)	\(e\left(\frac{281}{306}\right)\)	\(e\left(\frac{149}{204}\right)\)	\(e\left(\frac{613}{1224}\right)\)	\(e\left(\frac{35}{136}\right)\)
\(\chi_{100010}(31,\cdot)\)	100010.ta	1224	no	\(1\)	\(1\)	\(e\left(\frac{185}{408}\right)\)	\(e\left(\frac{239}{408}\right)\)	\(e\left(\frac{185}{204}\right)\)	\(e\left(\frac{1087}{1224}\right)\)	\(e\left(\frac{265}{612}\right)\)	\(e\left(\frac{247}{408}\right)\)	\(e\left(\frac{25}{153}\right)\)	\(e\left(\frac{2}{51}\right)\)	\(e\left(\frac{151}{1224}\right)\)	\(e\left(\frac{49}{136}\right)\)
\(\chi_{100010}(33,\cdot)\)	100010.se	1224	no	\(-1\)	\(1\)	\(e\left(\frac{97}{408}\right)\)	\(e\left(\frac{283}{408}\right)\)	\(e\left(\frac{97}{204}\right)\)	\(e\left(\frac{1145}{1224}\right)\)	\(e\left(\frac{259}{306}\right)\)	\(e\left(\frac{371}{408}\right)\)	\(e\left(\frac{193}{306}\right)\)	\(e\left(\frac{95}{102}\right)\)	\(e\left(\frac{335}{1224}\right)\)	\(e\left(\frac{97}{136}\right)\)
\(\chi_{100010}(37,\cdot)\)	100010.ic	36	no	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(-1\)
\(\chi_{100010}(39,\cdot)\)	100010.rp	1224	no	\(-1\)	\(1\)	\(e\left(\frac{11}{102}\right)\)	\(e\left(\frac{131}{408}\right)\)	\(e\left(\frac{11}{51}\right)\)	\(e\left(\frac{1195}{1224}\right)\)	\(e\left(\frac{665}{1224}\right)\)	\(e\left(\frac{295}{408}\right)\)	\(e\left(\frac{469}{612}\right)\)	\(e\left(\frac{175}{408}\right)\)	\(e\left(\frac{283}{306}\right)\)	\(e\left(\frac{11}{34}\right)\)
\(\chi_{100010}(41,\cdot)\)	100010.kd	72	no	\(-1\)	\(1\)	\(e\left(\frac{5}{24}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{29}{36}\right)\)	\(e\left(\frac{11}{72}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{25}{36}\right)\)	\(e\left(\frac{19}{24}\right)\)	\(e\left(\frac{67}{72}\right)\)	\(e\left(\frac{5}{8}\right)\)
\(\chi_{100010}(43,\cdot)\)	100010.pa	408	no	\(-1\)	\(1\)	\(e\left(\frac{29}{136}\right)\)	\(e\left(\frac{11}{136}\right)\)	\(e\left(\frac{29}{68}\right)\)	\(e\left(\frac{397}{408}\right)\)	\(e\left(\frac{89}{102}\right)\)	\(e\left(\frac{99}{136}\right)\)	\(e\left(\frac{23}{102}\right)\)	\(e\left(\frac{5}{17}\right)\)	\(e\left(\frac{403}{408}\right)\)	\(e\left(\frac{87}{136}\right)\)
\(\chi_{100010}(47,\cdot)\)	100010.st	1224	no	\(-1\)	\(1\)	\(e\left(\frac{193}{408}\right)\)	\(e\left(\frac{133}{408}\right)\)	\(e\left(\frac{193}{204}\right)\)	\(e\left(\frac{887}{1224}\right)\)	\(e\left(\frac{395}{612}\right)\)	\(e\left(\frac{245}{408}\right)\)	\(e\left(\frac{95}{153}\right)\)	\(e\left(\frac{163}{204}\right)\)	\(e\left(\frac{23}{1224}\right)\)	\(e\left(\frac{57}{136}\right)\)
\(\chi_{100010}(49,\cdot)\)	100010.og	204	no	\(1\)	\(1\)	\(e\left(\frac{21}{34}\right)\)	\(e\left(\frac{47}{68}\right)\)	\(e\left(\frac{4}{17}\right)\)	\(e\left(\frac{157}{204}\right)\)	\(e\left(\frac{5}{204}\right)\)	\(e\left(\frac{15}{68}\right)\)	\(e\left(\frac{25}{102}\right)\)	\(e\left(\frac{21}{68}\right)\)	\(e\left(\frac{89}{102}\right)\)	\(e\left(\frac{29}{34}\right)\)
\(\chi_{100010}(51,\cdot)\)	100010.lv	136	no	\(1\)	\(1\)	\(e\left(\frac{73}{136}\right)\)	\(e\left(\frac{57}{136}\right)\)	\(e\left(\frac{5}{68}\right)\)	\(e\left(\frac{83}{136}\right)\)	\(e\left(\frac{5}{17}\right)\)	\(e\left(\frac{105}{136}\right)\)	\(e\left(\frac{15}{34}\right)\)	\(e\left(\frac{65}{68}\right)\)	\(e\left(\frac{13}{136}\right)\)	\(e\left(\frac{83}{136}\right)\)
\(\chi_{100010}(53,\cdot)\)	100010.se	1224	no	\(-1\)	\(1\)	\(e\left(\frac{257}{408}\right)\)	\(e\left(\frac{203}{408}\right)\)	\(e\left(\frac{53}{204}\right)\)	\(e\left(\frac{1}{1224}\right)\)	\(e\left(\frac{233}{306}\right)\)	\(e\left(\frac{331}{408}\right)\)	\(e\left(\frac{137}{306}\right)\)	\(e\left(\frac{13}{102}\right)\)	\(e\left(\frac{631}{1224}\right)\)	\(e\left(\frac{121}{136}\right)\)
\(\chi_{100010}(57,\cdot)\)	100010.rt	1224	no	\(1\)	\(1\)	\(e\left(\frac{311}{408}\right)\)	\(e\left(\frac{95}{102}\right)\)	\(e\left(\frac{107}{204}\right)\)	\(e\left(\frac{65}{612}\right)\)	\(e\left(\frac{137}{1224}\right)\)	\(e\left(\frac{11}{51}\right)\)	\(e\left(\frac{583}{612}\right)\)	\(e\left(\frac{283}{408}\right)\)	\(e\left(\frac{481}{1224}\right)\)	\(e\left(\frac{39}{136}\right)\)
\(\chi_{100010}(59,\cdot)\)	100010.rx	1224	no	\(-1\)	\(1\)	\(e\left(\frac{139}{204}\right)\)	\(e\left(\frac{371}{408}\right)\)	\(e\left(\frac{37}{102}\right)\)	\(e\left(\frac{139}{1224}\right)\)	\(e\left(\frac{875}{1224}\right)\)	\(e\left(\frac{7}{408}\right)\)	\(e\left(\frac{295}{612}\right)\)	\(e\left(\frac{241}{408}\right)\)	\(e\left(\frac{173}{612}\right)\)	\(e\left(\frac{3}{68}\right)\)
\(\chi_{100010}(61,\cdot)\)	100010.ra	612	no	\(1\)	\(1\)	\(e\left(\frac{167}{204}\right)\)	\(e\left(\frac{197}{204}\right)\)	\(e\left(\frac{65}{102}\right)\)	\(e\left(\frac{313}{612}\right)\)	\(e\left(\frac{49}{306}\right)\)	\(e\left(\frac{73}{204}\right)\)	\(e\left(\frac{41}{153}\right)\)	\(e\left(\frac{40}{51}\right)\)	\(e\left(\frac{133}{612}\right)\)	\(e\left(\frac{31}{68}\right)\)
\(\chi_{100010}(63,\cdot)\)	100010.ne	136	no	\(1\)	\(1\)	\(e\left(\frac{11}{34}\right)\)	\(e\left(\frac{131}{136}\right)\)	\(e\left(\frac{11}{17}\right)\)	\(e\left(\frac{115}{136}\right)\)	\(e\left(\frac{29}{136}\right)\)	\(e\left(\frac{23}{136}\right)\)	\(e\left(\frac{9}{68}\right)\)	\(e\left(\frac{39}{136}\right)\)	\(e\left(\frac{15}{34}\right)\)	\(e\left(\frac{33}{34}\right)\)
\(\chi_{100010}(67,\cdot)\)	100010.sq	1224	no	\(1\)	\(1\)	\(e\left(\frac{143}{408}\right)\)	\(e\left(\frac{79}{204}\right)\)	\(e\left(\frac{143}{204}\right)\)	\(e\left(\frac{37}{306}\right)\)	\(e\left(\frac{695}{1224}\right)\)	\(e\left(\frac{65}{204}\right)\)	\(e\left(\frac{313}{612}\right)\)	\(e\left(\frac{301}{408}\right)\)	\(e\left(\frac{1129}{1224}\right)\)	\(e\left(\frac{7}{136}\right)\)
\(\chi_{100010}(69,\cdot)\)	100010.qu	612	no	\(1\)	\(1\)	\(e\left(\frac{155}{204}\right)\)	\(e\left(\frac{25}{102}\right)\)	\(e\left(\frac{53}{102}\right)\)	\(e\left(\frac{115}{153}\right)\)	\(e\left(\frac{167}{612}\right)\)	\(e\left(\frac{89}{102}\right)\)	\(e\left(\frac{121}{306}\right)\)	\(e\left(\frac{1}{204}\right)\)	\(e\left(\frac{325}{612}\right)\)	\(e\left(\frac{19}{68}\right)\)
\(\chi_{100010}(71,\cdot)\)	100010.sn	1224	no	\(-1\)	\(1\)	\(e\left(\frac{59}{408}\right)\)	\(e\left(\frac{49}{204}\right)\)	\(e\left(\frac{59}{204}\right)\)	\(e\left(\frac{563}{612}\right)\)	\(e\left(\frac{5}{1224}\right)\)	\(e\left(\frac{203}{204}\right)\)	\(e\left(\frac{535}{612}\right)\)	\(e\left(\frac{157}{408}\right)\)	\(e\left(\frac{1045}{1224}\right)\)	\(e\left(\frac{59}{136}\right)\)
\(\chi_{100010}(77,\cdot)\)	100010.qx	612	no	\(-1\)	\(1\)	\(e\left(\frac{59}{204}\right)\)	\(e\left(\frac{47}{204}\right)\)	\(e\left(\frac{59}{102}\right)\)	\(e\left(\frac{52}{153}\right)\)	\(e\left(\frac{5}{612}\right)\)	\(e\left(\frac{151}{204}\right)\)	\(e\left(\frac{229}{306}\right)\)	\(e\left(\frac{53}{102}\right)\)	\(e\left(\frac{433}{612}\right)\)	\(e\left(\frac{59}{68}\right)\)
\(\chi_{100010}(79,\cdot)\)	100010.rv	1224	no	\(-1\)	\(1\)	\(e\left(\frac{371}{408}\right)\)	\(e\left(\frac{11}{102}\right)\)	\(e\left(\frac{167}{204}\right)\)	\(e\left(\frac{91}{306}\right)\)	\(e\left(\frac{47}{1224}\right)\)	\(e\left(\frac{41}{51}\right)\)	\(e\left(\frac{133}{612}\right)\)	\(e\left(\frac{7}{408}\right)\)	\(e\left(\frac{949}{1224}\right)\)	\(e\left(\frac{99}{136}\right)\)
\(\chi_{100010}(81,\cdot)\)	100010.lh	102	no	\(1\)	\(1\)	\(e\left(\frac{1}{34}\right)\)	\(e\left(\frac{4}{17}\right)\)	\(e\left(\frac{1}{17}\right)\)	\(e\left(\frac{47}{51}\right)\)	\(e\left(\frac{41}{102}\right)\)	\(e\left(\frac{2}{17}\right)\)	\(e\left(\frac{1}{51}\right)\)	\(e\left(\frac{9}{34}\right)\)	\(e\left(\frac{1}{102}\right)\)	\(e\left(\frac{3}{34}\right)\)

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.