LMFDB - Group of Dirichlet characters of modulus 353

Show commands: PariGP / SageMath

sage: H = DirichletGroup(353)

pari: g = idealstar(,353,2)

Character group

sage: G.order() pari: g.no
Order	=	352
sage: H.invariants() pari: g.cyc
Structure	=	$C_{352}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{353}(3,\cdot)$

First 32 of 352 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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{353}(1,\cdot)$	353.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{353}(2,\cdot)$	353.j	88	yes	$1$	$1$	$e\left(\frac{9}{22}\right)$	$e\left(\frac{41}{88}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{85}{88}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{5}{22}\right)$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{21}{22}\right)$
$\chi_{353}(3,\cdot)$	353.l	352	yes	$-1$	$1$	$e\left(\frac{41}{88}\right)$	$e\left(\frac{1}{352}\right)$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{309}{352}\right)$	$e\left(\frac{15}{32}\right)$	$e\left(\frac{9}{32}\right)$	$e\left(\frac{35}{88}\right)$	$e\left(\frac{1}{176}\right)$	$e\left(\frac{11}{32}\right)$	$e\left(\frac{15}{88}\right)$
$\chi_{353}(4,\cdot)$	353.i	44	yes	$1$	$1$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{41}{44}\right)$	$-i$	$i$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{19}{22}\right)$	$-i$	$e\left(\frac{10}{11}\right)$
$\chi_{353}(5,\cdot)$	353.l	352	yes	$-1$	$1$	$e\left(\frac{85}{88}\right)$	$e\left(\frac{309}{352}\right)$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{89}{352}\right)$	$e\left(\frac{27}{32}\right)$	$e\left(\frac{29}{32}\right)$	$e\left(\frac{79}{88}\right)$	$e\left(\frac{133}{176}\right)$	$e\left(\frac{7}{32}\right)$	$e\left(\frac{59}{88}\right)$
$\chi_{353}(6,\cdot)$	353.h	32	yes	$-1$	$1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{15}{32}\right)$	$-i$	$e\left(\frac{27}{32}\right)$	$e\left(\frac{11}{32}\right)$	$e\left(\frac{13}{32}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{15}{16}\right)$	$e\left(\frac{23}{32}\right)$	$e\left(\frac{1}{8}\right)$
$\chi_{353}(7,\cdot)$	353.h	32	yes	$-1$	$1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{9}{32}\right)$	$i$	$e\left(\frac{29}{32}\right)$	$e\left(\frac{13}{32}\right)$	$e\left(\frac{27}{32}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{1}{32}\right)$	$e\left(\frac{7}{8}\right)$
$\chi_{353}(8,\cdot)$	353.j	88	yes	$1$	$1$	$e\left(\frac{5}{22}\right)$	$e\left(\frac{35}{88}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{79}{88}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{15}{22}\right)$	$e\left(\frac{35}{44}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{19}{22}\right)$
$\chi_{353}(9,\cdot)$	353.k	176	yes	$1$	$1$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{1}{176}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{133}{176}\right)$	$e\left(\frac{15}{16}\right)$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{35}{44}\right)$	$e\left(\frac{1}{88}\right)$	$e\left(\frac{11}{16}\right)$	$e\left(\frac{15}{44}\right)$
$\chi_{353}(10,\cdot)$	353.h	32	yes	$-1$	$1$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{11}{32}\right)$	$-i$	$e\left(\frac{7}{32}\right)$	$e\left(\frac{23}{32}\right)$	$e\left(\frac{1}{32}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{11}{16}\right)$	$e\left(\frac{19}{32}\right)$	$e\left(\frac{5}{8}\right)$
$\chi_{353}(11,\cdot)$	353.j	88	yes	$1$	$1$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{15}{88}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{59}{88}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{15}{44}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{5}{22}\right)$
$\chi_{353}(12,\cdot)$	353.l	352	yes	$-1$	$1$	$e\left(\frac{25}{88}\right)$	$e\left(\frac{329}{352}\right)$	$e\left(\frac{25}{44}\right)$	$e\left(\frac{285}{352}\right)$	$e\left(\frac{7}{32}\right)$	$e\left(\frac{17}{32}\right)$	$e\left(\frac{75}{88}\right)$	$e\left(\frac{153}{176}\right)$	$e\left(\frac{3}{32}\right)$	$e\left(\frac{7}{88}\right)$
$\chi_{353}(13,\cdot)$	353.l	352	yes	$-1$	$1$	$e\left(\frac{1}{88}\right)$	$e\left(\frac{337}{352}\right)$	$e\left(\frac{1}{44}\right)$	$e\left(\frac{293}{352}\right)$	$e\left(\frac{31}{32}\right)$	$e\left(\frac{25}{32}\right)$	$e\left(\frac{3}{88}\right)$	$e\left(\frac{161}{176}\right)$	$e\left(\frac{27}{32}\right)$	$e\left(\frac{39}{88}\right)$
$\chi_{353}(14,\cdot)$	353.l	352	yes	$-1$	$1$	$e\left(\frac{47}{88}\right)$	$e\left(\frac{263}{352}\right)$	$e\left(\frac{3}{44}\right)$	$e\left(\frac{307}{352}\right)$	$e\left(\frac{9}{32}\right)$	$e\left(\frac{31}{32}\right)$	$e\left(\frac{53}{88}\right)$	$e\left(\frac{87}{176}\right)$	$e\left(\frac{13}{32}\right)$	$e\left(\frac{73}{88}\right)$
$\chi_{353}(15,\cdot)$	353.k	176	yes	$1$	$1$	$e\left(\frac{19}{44}\right)$	$e\left(\frac{155}{176}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{23}{176}\right)$	$e\left(\frac{5}{16}\right)$	$e\left(\frac{3}{16}\right)$	$e\left(\frac{13}{44}\right)$	$e\left(\frac{67}{88}\right)$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{37}{44}\right)$
$\chi_{353}(16,\cdot)$	353.g	22	yes	$1$	$1$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{19}{22}\right)$	$-1$	$-1$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{8}{11}\right)$	$-1$	$e\left(\frac{9}{11}\right)$
$\chi_{353}(17,\cdot)$	353.j	88	yes	$1$	$1$	$e\left(\frac{5}{22}\right)$	$e\left(\frac{57}{88}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{13}{88}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{15}{22}\right)$	$e\left(\frac{13}{44}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{19}{22}\right)$
$\chi_{353}(18,\cdot)$	353.k	176	yes	$1$	$1$	$e\left(\frac{15}{44}\right)$	$e\left(\frac{83}{176}\right)$	$e\left(\frac{15}{22}\right)$	$e\left(\frac{127}{176}\right)$	$e\left(\frac{13}{16}\right)$	$e\left(\frac{11}{16}\right)$	$e\left(\frac{1}{44}\right)$	$e\left(\frac{83}{88}\right)$	$e\left(\frac{1}{16}\right)$	$e\left(\frac{13}{44}\right)$
$\chi_{353}(19,\cdot)$	353.k	176	yes	$1$	$1$	$e\left(\frac{27}{44}\right)$	$e\left(\frac{35}{176}\right)$	$e\left(\frac{5}{22}\right)$	$e\left(\frac{79}{176}\right)$	$e\left(\frac{13}{16}\right)$	$e\left(\frac{11}{16}\right)$	$e\left(\frac{37}{44}\right)$	$e\left(\frac{35}{88}\right)$	$e\left(\frac{1}{16}\right)$	$e\left(\frac{41}{44}\right)$
$\chi_{353}(20,\cdot)$	353.l	352	yes	$-1$	$1$	$e\left(\frac{69}{88}\right)$	$e\left(\frac{285}{352}\right)$	$e\left(\frac{25}{44}\right)$	$e\left(\frac{65}{352}\right)$	$e\left(\frac{19}{32}\right)$	$e\left(\frac{5}{32}\right)$	$e\left(\frac{31}{88}\right)$	$e\left(\frac{109}{176}\right)$	$e\left(\frac{31}{32}\right)$	$e\left(\frac{51}{88}\right)$
$\chi_{353}(21,\cdot)$	353.j	88	yes	$1$	$1$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{25}{88}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{69}{88}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{17}{22}\right)$	$e\left(\frac{25}{44}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{1}{22}\right)$
$\chi_{353}(22,\cdot)$	353.e	11	yes	$1$	$1$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{7}{11}\right)$	$1$	$1$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{3}{11}\right)$	$1$	$e\left(\frac{2}{11}\right)$
$\chi_{353}(23,\cdot)$	353.k	176	yes	$1$	$1$	$e\left(\frac{35}{44}\right)$	$e\left(\frac{3}{176}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{47}{176}\right)$	$e\left(\frac{13}{16}\right)$	$e\left(\frac{11}{16}\right)$	$e\left(\frac{17}{44}\right)$	$e\left(\frac{3}{88}\right)$	$e\left(\frac{1}{16}\right)$	$e\left(\frac{1}{44}\right)$
$\chi_{353}(24,\cdot)$	353.l	352	yes	$-1$	$1$	$e\left(\frac{61}{88}\right)$	$e\left(\frac{141}{352}\right)$	$e\left(\frac{17}{44}\right)$	$e\left(\frac{273}{352}\right)$	$e\left(\frac{3}{32}\right)$	$e\left(\frac{21}{32}\right)$	$e\left(\frac{7}{88}\right)$	$e\left(\frac{141}{176}\right)$	$e\left(\frac{15}{32}\right)$	$e\left(\frac{3}{88}\right)$
$\chi_{353}(25,\cdot)$	353.k	176	yes	$1$	$1$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{133}{176}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{89}{176}\right)$	$e\left(\frac{11}{16}\right)$	$e\left(\frac{13}{16}\right)$	$e\left(\frac{35}{44}\right)$	$e\left(\frac{45}{88}\right)$	$e\left(\frac{7}{16}\right)$	$e\left(\frac{15}{44}\right)$
$\chi_{353}(26,\cdot)$	353.l	352	yes	$-1$	$1$	$e\left(\frac{37}{88}\right)$	$e\left(\frac{149}{352}\right)$	$e\left(\frac{37}{44}\right)$	$e\left(\frac{281}{352}\right)$	$e\left(\frac{27}{32}\right)$	$e\left(\frac{29}{32}\right)$	$e\left(\frac{23}{88}\right)$	$e\left(\frac{149}{176}\right)$	$e\left(\frac{7}{32}\right)$	$e\left(\frac{35}{88}\right)$
$\chi_{353}(27,\cdot)$	353.l	352	yes	$-1$	$1$	$e\left(\frac{35}{88}\right)$	$e\left(\frac{3}{352}\right)$	$e\left(\frac{35}{44}\right)$	$e\left(\frac{223}{352}\right)$	$e\left(\frac{13}{32}\right)$	$e\left(\frac{27}{32}\right)$	$e\left(\frac{17}{88}\right)$	$e\left(\frac{3}{176}\right)$	$e\left(\frac{1}{32}\right)$	$e\left(\frac{45}{88}\right)$
$\chi_{353}(28,\cdot)$	353.l	352	yes	$-1$	$1$	$e\left(\frac{83}{88}\right)$	$e\left(\frac{75}{352}\right)$	$e\left(\frac{39}{44}\right)$	$e\left(\frac{295}{352}\right)$	$e\left(\frac{5}{32}\right)$	$e\left(\frac{3}{32}\right)$	$e\left(\frac{73}{88}\right)$	$e\left(\frac{75}{176}\right)$	$e\left(\frac{25}{32}\right)$	$e\left(\frac{69}{88}\right)$
$\chi_{353}(29,\cdot)$	353.j	88	yes	$1$	$1$	$e\left(\frac{15}{22}\right)$	$e\left(\frac{39}{88}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{83}{88}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{39}{44}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{13}{22}\right)$
$\chi_{353}(30,\cdot)$	353.k	176	yes	$1$	$1$	$e\left(\frac{37}{44}\right)$	$e\left(\frac{61}{176}\right)$	$e\left(\frac{15}{22}\right)$	$e\left(\frac{17}{176}\right)$	$e\left(\frac{3}{16}\right)$	$e\left(\frac{5}{16}\right)$	$e\left(\frac{23}{44}\right)$	$e\left(\frac{61}{88}\right)$	$e\left(\frac{15}{16}\right)$	$e\left(\frac{35}{44}\right)$
$\chi_{353}(31,\cdot)$	353.l	352	yes	$-1$	$1$	$e\left(\frac{29}{88}\right)$	$e\left(\frac{93}{352}\right)$	$e\left(\frac{29}{44}\right)$	$e\left(\frac{225}{352}\right)$	$e\left(\frac{19}{32}\right)$	$e\left(\frac{5}{32}\right)$	$e\left(\frac{87}{88}\right)$	$e\left(\frac{93}{176}\right)$	$e\left(\frac{31}{32}\right)$	$e\left(\frac{75}{88}\right)$
$\chi_{353}(32,\cdot)$	353.j	88	yes	$1$	$1$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{29}{88}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{73}{88}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{22}\right)$	$e\left(\frac{29}{44}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{17}{22}\right)$

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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	$353$
Structure	\(C_{352}\)
Order	$352$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\chi_{353}(1,\cdot)\)	353.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{353}(2,\cdot)\)	353.j	88	yes	\(1\)	\(1\)	\(e\left(\frac{9}{22}\right)\)	\(e\left(\frac{41}{88}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{85}{88}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{5}{22}\right)\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{21}{22}\right)\)
\(\chi_{353}(3,\cdot)\)	353.l	352	yes	\(-1\)	\(1\)	\(e\left(\frac{41}{88}\right)\)	\(e\left(\frac{1}{352}\right)\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{309}{352}\right)\)	\(e\left(\frac{15}{32}\right)\)	\(e\left(\frac{9}{32}\right)\)	\(e\left(\frac{35}{88}\right)\)	\(e\left(\frac{1}{176}\right)\)	\(e\left(\frac{11}{32}\right)\)	\(e\left(\frac{15}{88}\right)\)
\(\chi_{353}(4,\cdot)\)	353.i	44	yes	\(1\)	\(1\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{41}{44}\right)\)	\(-i\)	\(i\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(-i\)	\(e\left(\frac{10}{11}\right)\)
\(\chi_{353}(5,\cdot)\)	353.l	352	yes	\(-1\)	\(1\)	\(e\left(\frac{85}{88}\right)\)	\(e\left(\frac{309}{352}\right)\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{89}{352}\right)\)	\(e\left(\frac{27}{32}\right)\)	\(e\left(\frac{29}{32}\right)\)	\(e\left(\frac{79}{88}\right)\)	\(e\left(\frac{133}{176}\right)\)	\(e\left(\frac{7}{32}\right)\)	\(e\left(\frac{59}{88}\right)\)
\(\chi_{353}(6,\cdot)\)	353.h	32	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{15}{32}\right)\)	\(-i\)	\(e\left(\frac{27}{32}\right)\)	\(e\left(\frac{11}{32}\right)\)	\(e\left(\frac{13}{32}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{15}{16}\right)\)	\(e\left(\frac{23}{32}\right)\)	\(e\left(\frac{1}{8}\right)\)
\(\chi_{353}(7,\cdot)\)	353.h	32	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{9}{32}\right)\)	\(i\)	\(e\left(\frac{29}{32}\right)\)	\(e\left(\frac{13}{32}\right)\)	\(e\left(\frac{27}{32}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{9}{16}\right)\)	\(e\left(\frac{1}{32}\right)\)	\(e\left(\frac{7}{8}\right)\)
\(\chi_{353}(8,\cdot)\)	353.j	88	yes	\(1\)	\(1\)	\(e\left(\frac{5}{22}\right)\)	\(e\left(\frac{35}{88}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{79}{88}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{15}{22}\right)\)	\(e\left(\frac{35}{44}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{19}{22}\right)\)
\(\chi_{353}(9,\cdot)\)	353.k	176	yes	\(1\)	\(1\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{1}{176}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{133}{176}\right)\)	\(e\left(\frac{15}{16}\right)\)	\(e\left(\frac{9}{16}\right)\)	\(e\left(\frac{35}{44}\right)\)	\(e\left(\frac{1}{88}\right)\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{15}{44}\right)\)
\(\chi_{353}(10,\cdot)\)	353.h	32	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{11}{32}\right)\)	\(-i\)	\(e\left(\frac{7}{32}\right)\)	\(e\left(\frac{23}{32}\right)\)	\(e\left(\frac{1}{32}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{19}{32}\right)\)	\(e\left(\frac{5}{8}\right)\)
\(\chi_{353}(11,\cdot)\)	353.j	88	yes	\(1\)	\(1\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{15}{88}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{59}{88}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{15}{44}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{5}{22}\right)\)
\(\chi_{353}(12,\cdot)\)	353.l	352	yes	\(-1\)	\(1\)	\(e\left(\frac{25}{88}\right)\)	\(e\left(\frac{329}{352}\right)\)	\(e\left(\frac{25}{44}\right)\)	\(e\left(\frac{285}{352}\right)\)	\(e\left(\frac{7}{32}\right)\)	\(e\left(\frac{17}{32}\right)\)	\(e\left(\frac{75}{88}\right)\)	\(e\left(\frac{153}{176}\right)\)	\(e\left(\frac{3}{32}\right)\)	\(e\left(\frac{7}{88}\right)\)
\(\chi_{353}(13,\cdot)\)	353.l	352	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{88}\right)\)	\(e\left(\frac{337}{352}\right)\)	\(e\left(\frac{1}{44}\right)\)	\(e\left(\frac{293}{352}\right)\)	\(e\left(\frac{31}{32}\right)\)	\(e\left(\frac{25}{32}\right)\)	\(e\left(\frac{3}{88}\right)\)	\(e\left(\frac{161}{176}\right)\)	\(e\left(\frac{27}{32}\right)\)	\(e\left(\frac{39}{88}\right)\)
\(\chi_{353}(14,\cdot)\)	353.l	352	yes	\(-1\)	\(1\)	\(e\left(\frac{47}{88}\right)\)	\(e\left(\frac{263}{352}\right)\)	\(e\left(\frac{3}{44}\right)\)	\(e\left(\frac{307}{352}\right)\)	\(e\left(\frac{9}{32}\right)\)	\(e\left(\frac{31}{32}\right)\)	\(e\left(\frac{53}{88}\right)\)	\(e\left(\frac{87}{176}\right)\)	\(e\left(\frac{13}{32}\right)\)	\(e\left(\frac{73}{88}\right)\)
\(\chi_{353}(15,\cdot)\)	353.k	176	yes	\(1\)	\(1\)	\(e\left(\frac{19}{44}\right)\)	\(e\left(\frac{155}{176}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{23}{176}\right)\)	\(e\left(\frac{5}{16}\right)\)	\(e\left(\frac{3}{16}\right)\)	\(e\left(\frac{13}{44}\right)\)	\(e\left(\frac{67}{88}\right)\)	\(e\left(\frac{9}{16}\right)\)	\(e\left(\frac{37}{44}\right)\)
\(\chi_{353}(16,\cdot)\)	353.g	22	yes	\(1\)	\(1\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(-1\)	\(-1\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(-1\)	\(e\left(\frac{9}{11}\right)\)
\(\chi_{353}(17,\cdot)\)	353.j	88	yes	\(1\)	\(1\)	\(e\left(\frac{5}{22}\right)\)	\(e\left(\frac{57}{88}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{13}{88}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{15}{22}\right)\)	\(e\left(\frac{13}{44}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{19}{22}\right)\)
\(\chi_{353}(18,\cdot)\)	353.k	176	yes	\(1\)	\(1\)	\(e\left(\frac{15}{44}\right)\)	\(e\left(\frac{83}{176}\right)\)	\(e\left(\frac{15}{22}\right)\)	\(e\left(\frac{127}{176}\right)\)	\(e\left(\frac{13}{16}\right)\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{1}{44}\right)\)	\(e\left(\frac{83}{88}\right)\)	\(e\left(\frac{1}{16}\right)\)	\(e\left(\frac{13}{44}\right)\)
\(\chi_{353}(19,\cdot)\)	353.k	176	yes	\(1\)	\(1\)	\(e\left(\frac{27}{44}\right)\)	\(e\left(\frac{35}{176}\right)\)	\(e\left(\frac{5}{22}\right)\)	\(e\left(\frac{79}{176}\right)\)	\(e\left(\frac{13}{16}\right)\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{37}{44}\right)\)	\(e\left(\frac{35}{88}\right)\)	\(e\left(\frac{1}{16}\right)\)	\(e\left(\frac{41}{44}\right)\)
\(\chi_{353}(20,\cdot)\)	353.l	352	yes	\(-1\)	\(1\)	\(e\left(\frac{69}{88}\right)\)	\(e\left(\frac{285}{352}\right)\)	\(e\left(\frac{25}{44}\right)\)	\(e\left(\frac{65}{352}\right)\)	\(e\left(\frac{19}{32}\right)\)	\(e\left(\frac{5}{32}\right)\)	\(e\left(\frac{31}{88}\right)\)	\(e\left(\frac{109}{176}\right)\)	\(e\left(\frac{31}{32}\right)\)	\(e\left(\frac{51}{88}\right)\)
\(\chi_{353}(21,\cdot)\)	353.j	88	yes	\(1\)	\(1\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{25}{88}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{69}{88}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{25}{44}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{1}{22}\right)\)
\(\chi_{353}(22,\cdot)\)	353.e	11	yes	\(1\)	\(1\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(1\)	\(1\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(1\)	\(e\left(\frac{2}{11}\right)\)
\(\chi_{353}(23,\cdot)\)	353.k	176	yes	\(1\)	\(1\)	\(e\left(\frac{35}{44}\right)\)	\(e\left(\frac{3}{176}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{47}{176}\right)\)	\(e\left(\frac{13}{16}\right)\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{17}{44}\right)\)	\(e\left(\frac{3}{88}\right)\)	\(e\left(\frac{1}{16}\right)\)	\(e\left(\frac{1}{44}\right)\)
\(\chi_{353}(24,\cdot)\)	353.l	352	yes	\(-1\)	\(1\)	\(e\left(\frac{61}{88}\right)\)	\(e\left(\frac{141}{352}\right)\)	\(e\left(\frac{17}{44}\right)\)	\(e\left(\frac{273}{352}\right)\)	\(e\left(\frac{3}{32}\right)\)	\(e\left(\frac{21}{32}\right)\)	\(e\left(\frac{7}{88}\right)\)	\(e\left(\frac{141}{176}\right)\)	\(e\left(\frac{15}{32}\right)\)	\(e\left(\frac{3}{88}\right)\)
\(\chi_{353}(25,\cdot)\)	353.k	176	yes	\(1\)	\(1\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{133}{176}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{89}{176}\right)\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{13}{16}\right)\)	\(e\left(\frac{35}{44}\right)\)	\(e\left(\frac{45}{88}\right)\)	\(e\left(\frac{7}{16}\right)\)	\(e\left(\frac{15}{44}\right)\)
\(\chi_{353}(26,\cdot)\)	353.l	352	yes	\(-1\)	\(1\)	\(e\left(\frac{37}{88}\right)\)	\(e\left(\frac{149}{352}\right)\)	\(e\left(\frac{37}{44}\right)\)	\(e\left(\frac{281}{352}\right)\)	\(e\left(\frac{27}{32}\right)\)	\(e\left(\frac{29}{32}\right)\)	\(e\left(\frac{23}{88}\right)\)	\(e\left(\frac{149}{176}\right)\)	\(e\left(\frac{7}{32}\right)\)	\(e\left(\frac{35}{88}\right)\)
\(\chi_{353}(27,\cdot)\)	353.l	352	yes	\(-1\)	\(1\)	\(e\left(\frac{35}{88}\right)\)	\(e\left(\frac{3}{352}\right)\)	\(e\left(\frac{35}{44}\right)\)	\(e\left(\frac{223}{352}\right)\)	\(e\left(\frac{13}{32}\right)\)	\(e\left(\frac{27}{32}\right)\)	\(e\left(\frac{17}{88}\right)\)	\(e\left(\frac{3}{176}\right)\)	\(e\left(\frac{1}{32}\right)\)	\(e\left(\frac{45}{88}\right)\)
\(\chi_{353}(28,\cdot)\)	353.l	352	yes	\(-1\)	\(1\)	\(e\left(\frac{83}{88}\right)\)	\(e\left(\frac{75}{352}\right)\)	\(e\left(\frac{39}{44}\right)\)	\(e\left(\frac{295}{352}\right)\)	\(e\left(\frac{5}{32}\right)\)	\(e\left(\frac{3}{32}\right)\)	\(e\left(\frac{73}{88}\right)\)	\(e\left(\frac{75}{176}\right)\)	\(e\left(\frac{25}{32}\right)\)	\(e\left(\frac{69}{88}\right)\)
\(\chi_{353}(29,\cdot)\)	353.j	88	yes	\(1\)	\(1\)	\(e\left(\frac{15}{22}\right)\)	\(e\left(\frac{39}{88}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{83}{88}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{39}{44}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{13}{22}\right)\)
\(\chi_{353}(30,\cdot)\)	353.k	176	yes	\(1\)	\(1\)	\(e\left(\frac{37}{44}\right)\)	\(e\left(\frac{61}{176}\right)\)	\(e\left(\frac{15}{22}\right)\)	\(e\left(\frac{17}{176}\right)\)	\(e\left(\frac{3}{16}\right)\)	\(e\left(\frac{5}{16}\right)\)	\(e\left(\frac{23}{44}\right)\)	\(e\left(\frac{61}{88}\right)\)	\(e\left(\frac{15}{16}\right)\)	\(e\left(\frac{35}{44}\right)\)
\(\chi_{353}(31,\cdot)\)	353.l	352	yes	\(-1\)	\(1\)	\(e\left(\frac{29}{88}\right)\)	\(e\left(\frac{93}{352}\right)\)	\(e\left(\frac{29}{44}\right)\)	\(e\left(\frac{225}{352}\right)\)	\(e\left(\frac{19}{32}\right)\)	\(e\left(\frac{5}{32}\right)\)	\(e\left(\frac{87}{88}\right)\)	\(e\left(\frac{93}{176}\right)\)	\(e\left(\frac{31}{32}\right)\)	\(e\left(\frac{75}{88}\right)\)
\(\chi_{353}(32,\cdot)\)	353.j	88	yes	\(1\)	\(1\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{29}{88}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{73}{88}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{3}{22}\right)\)	\(e\left(\frac{29}{44}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{17}{22}\right)\)

Introduction

L-functions

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Character group

First 32 of 352 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.