LMFDB - Group of Dirichlet characters of modulus 449

sage:H = DirichletGroup(449)

pari:g = idealstar(,449,2)

Character group

sage:G.order() pari:g.no
Order	=	448
sage:H.invariants() pari:g.cyc
Structure	=	$C_{448}$
sage:H.gens() pari:g.gen
Generators	=	$\chi_{449}(3,\cdot)$

First 32 of 448 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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{449}(1,\cdot)$	449.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{449}(2,\cdot)$	449.m	224	yes	$1$	$1$	$e\left(\frac{1}{112}\right)$	$e\left(\frac{127}{224}\right)$	$e\left(\frac{1}{56}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{129}{224}\right)$	$e\left(\frac{53}{56}\right)$	$e\left(\frac{3}{112}\right)$	$e\left(\frac{15}{112}\right)$	$e\left(\frac{7}{16}\right)$	$e\left(\frac{19}{28}\right)$
$\chi_{449}(3,\cdot)$	449.n	448	yes	$-1$	$1$	$e\left(\frac{127}{224}\right)$	$e\left(\frac{1}{448}\right)$	$e\left(\frac{15}{112}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{255}{448}\right)$	$e\left(\frac{11}{112}\right)$	$e\left(\frac{157}{224}\right)$	$e\left(\frac{1}{224}\right)$	$e\left(\frac{25}{32}\right)$	$e\left(\frac{33}{56}\right)$
$\chi_{449}(4,\cdot)$	449.l	112	yes	$1$	$1$	$e\left(\frac{1}{56}\right)$	$e\left(\frac{15}{112}\right)$	$e\left(\frac{1}{28}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{17}{112}\right)$	$e\left(\frac{25}{28}\right)$	$e\left(\frac{3}{56}\right)$	$e\left(\frac{15}{56}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{5}{14}\right)$
$\chi_{449}(5,\cdot)$	449.f	14	yes	$1$	$1$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{3}{7}\right)$	$1$	$e\left(\frac{4}{7}\right)$
$\chi_{449}(6,\cdot)$	449.n	448	yes	$-1$	$1$	$e\left(\frac{129}{224}\right)$	$e\left(\frac{255}{448}\right)$	$e\left(\frac{17}{112}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{65}{448}\right)$	$e\left(\frac{5}{112}\right)$	$e\left(\frac{163}{224}\right)$	$e\left(\frac{31}{224}\right)$	$e\left(\frac{7}{32}\right)$	$e\left(\frac{15}{56}\right)$
$\chi_{449}(7,\cdot)$	449.l	112	yes	$1$	$1$	$e\left(\frac{53}{56}\right)$	$e\left(\frac{11}{112}\right)$	$e\left(\frac{25}{28}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{5}{112}\right)$	$e\left(\frac{9}{28}\right)$	$e\left(\frac{47}{56}\right)$	$e\left(\frac{11}{56}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{13}{14}\right)$
$\chi_{449}(8,\cdot)$	449.m	224	yes	$1$	$1$	$e\left(\frac{3}{112}\right)$	$e\left(\frac{157}{224}\right)$	$e\left(\frac{3}{56}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{163}{224}\right)$	$e\left(\frac{47}{56}\right)$	$e\left(\frac{9}{112}\right)$	$e\left(\frac{45}{112}\right)$	$e\left(\frac{5}{16}\right)$	$e\left(\frac{1}{28}\right)$
$\chi_{449}(9,\cdot)$	449.m	224	yes	$1$	$1$	$e\left(\frac{15}{112}\right)$	$e\left(\frac{1}{224}\right)$	$e\left(\frac{15}{56}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{31}{224}\right)$	$e\left(\frac{11}{56}\right)$	$e\left(\frac{45}{112}\right)$	$e\left(\frac{1}{112}\right)$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{5}{28}\right)$
$\chi_{449}(10,\cdot)$	449.i	32	yes	$1$	$1$	$e\left(\frac{7}{16}\right)$	$e\left(\frac{25}{32}\right)$	$e\left(\frac{7}{8}\right)$	$1$	$e\left(\frac{7}{32}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{5}{16}\right)$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{7}{16}\right)$	$i$
$\chi_{449}(11,\cdot)$	449.j	56	yes	$1$	$1$	$e\left(\frac{19}{28}\right)$	$e\left(\frac{33}{56}\right)$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{15}{56}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{1}{28}\right)$	$e\left(\frac{5}{28}\right)$	$i$	$e\left(\frac{4}{7}\right)$
$\chi_{449}(12,\cdot)$	449.n	448	yes	$-1$	$1$	$e\left(\frac{131}{224}\right)$	$e\left(\frac{61}{448}\right)$	$e\left(\frac{19}{112}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{323}{448}\right)$	$e\left(\frac{111}{112}\right)$	$e\left(\frac{169}{224}\right)$	$e\left(\frac{61}{224}\right)$	$e\left(\frac{21}{32}\right)$	$e\left(\frac{53}{56}\right)$
$\chi_{449}(13,\cdot)$	449.n	448	yes	$-1$	$1$	$e\left(\frac{45}{224}\right)$	$e\left(\frac{115}{448}\right)$	$e\left(\frac{45}{112}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{205}{448}\right)$	$e\left(\frac{33}{112}\right)$	$e\left(\frac{135}{224}\right)$	$e\left(\frac{115}{224}\right)$	$e\left(\frac{27}{32}\right)$	$e\left(\frac{43}{56}\right)$
$\chi_{449}(14,\cdot)$	449.m	224	yes	$1$	$1$	$e\left(\frac{107}{112}\right)$	$e\left(\frac{149}{224}\right)$	$e\left(\frac{51}{56}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{139}{224}\right)$	$e\left(\frac{15}{56}\right)$	$e\left(\frac{97}{112}\right)$	$e\left(\frac{37}{112}\right)$	$e\left(\frac{13}{16}\right)$	$e\left(\frac{17}{28}\right)$
$\chi_{449}(15,\cdot)$	449.n	448	yes	$-1$	$1$	$e\left(\frac{223}{224}\right)$	$e\left(\frac{97}{448}\right)$	$e\left(\frac{111}{112}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{95}{448}\right)$	$e\left(\frac{59}{112}\right)$	$e\left(\frac{221}{224}\right)$	$e\left(\frac{97}{224}\right)$	$e\left(\frac{25}{32}\right)$	$e\left(\frac{9}{56}\right)$
$\chi_{449}(16,\cdot)$	449.j	56	yes	$1$	$1$	$e\left(\frac{1}{28}\right)$	$e\left(\frac{15}{56}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{17}{56}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{3}{28}\right)$	$e\left(\frac{15}{28}\right)$	$-i$	$e\left(\frac{5}{7}\right)$
$\chi_{449}(17,\cdot)$	449.n	448	yes	$-1$	$1$	$e\left(\frac{165}{224}\right)$	$e\left(\frac{347}{448}\right)$	$e\left(\frac{53}{112}\right)$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{229}{448}\right)$	$e\left(\frac{9}{112}\right)$	$e\left(\frac{47}{224}\right)$	$e\left(\frac{123}{224}\right)$	$e\left(\frac{3}{32}\right)$	$e\left(\frac{27}{56}\right)$
$\chi_{449}(18,\cdot)$	449.d	7	yes	$1$	$1$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{1}{7}\right)$	$1$	$e\left(\frac{6}{7}\right)$
$\chi_{449}(19,\cdot)$	449.n	448	yes	$-1$	$1$	$e\left(\frac{79}{224}\right)$	$e\left(\frac{177}{448}\right)$	$e\left(\frac{79}{112}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{335}{448}\right)$	$e\left(\frac{43}{112}\right)$	$e\left(\frac{13}{224}\right)$	$e\left(\frac{177}{224}\right)$	$e\left(\frac{9}{32}\right)$	$e\left(\frac{17}{56}\right)$
$\chi_{449}(20,\cdot)$	449.l	112	yes	$1$	$1$	$e\left(\frac{25}{56}\right)$	$e\left(\frac{39}{112}\right)$	$e\left(\frac{25}{28}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{89}{112}\right)$	$e\left(\frac{9}{28}\right)$	$e\left(\frac{19}{56}\right)$	$e\left(\frac{39}{56}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{13}{14}\right)$
$\chi_{449}(21,\cdot)$	449.n	448	yes	$-1$	$1$	$e\left(\frac{115}{224}\right)$	$e\left(\frac{45}{448}\right)$	$e\left(\frac{3}{112}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{275}{448}\right)$	$e\left(\frac{47}{112}\right)$	$e\left(\frac{121}{224}\right)$	$e\left(\frac{45}{224}\right)$	$e\left(\frac{5}{32}\right)$	$e\left(\frac{29}{56}\right)$
$\chi_{449}(22,\cdot)$	449.i	32	yes	$1$	$1$	$e\left(\frac{11}{16}\right)$	$e\left(\frac{5}{32}\right)$	$e\left(\frac{3}{8}\right)$	$1$	$e\left(\frac{27}{32}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{16}\right)$	$e\left(\frac{5}{16}\right)$	$e\left(\frac{11}{16}\right)$	$i$
$\chi_{449}(23,\cdot)$	449.l	112	yes	$1$	$1$	$e\left(\frac{13}{56}\right)$	$e\left(\frac{27}{112}\right)$	$e\left(\frac{13}{28}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{53}{112}\right)$	$e\left(\frac{17}{28}\right)$	$e\left(\frac{39}{56}\right)$	$e\left(\frac{27}{56}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{9}{14}\right)$
$\chi_{449}(24,\cdot)$	449.k	64	yes	$-1$	$1$	$e\left(\frac{19}{32}\right)$	$e\left(\frac{45}{64}\right)$	$e\left(\frac{3}{16}\right)$	$-1$	$e\left(\frac{19}{64}\right)$	$e\left(\frac{15}{16}\right)$	$e\left(\frac{25}{32}\right)$	$e\left(\frac{13}{32}\right)$	$e\left(\frac{3}{32}\right)$	$e\left(\frac{5}{8}\right)$
$\chi_{449}(25,\cdot)$	449.d	7	yes	$1$	$1$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{6}{7}\right)$	$1$	$e\left(\frac{1}{7}\right)$
$\chi_{449}(26,\cdot)$	449.n	448	yes	$-1$	$1$	$e\left(\frac{47}{224}\right)$	$e\left(\frac{369}{448}\right)$	$e\left(\frac{47}{112}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{15}{448}\right)$	$e\left(\frac{27}{112}\right)$	$e\left(\frac{141}{224}\right)$	$e\left(\frac{145}{224}\right)$	$e\left(\frac{9}{32}\right)$	$e\left(\frac{25}{56}\right)$
$\chi_{449}(27,\cdot)$	449.n	448	yes	$-1$	$1$	$e\left(\frac{157}{224}\right)$	$e\left(\frac{3}{448}\right)$	$e\left(\frac{45}{112}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{317}{448}\right)$	$e\left(\frac{33}{112}\right)$	$e\left(\frac{23}{224}\right)$	$e\left(\frac{3}{224}\right)$	$e\left(\frac{11}{32}\right)$	$e\left(\frac{43}{56}\right)$
$\chi_{449}(28,\cdot)$	449.j	56	yes	$1$	$1$	$e\left(\frac{27}{28}\right)$	$e\left(\frac{13}{56}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{11}{56}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{25}{28}\right)$	$e\left(\frac{13}{28}\right)$	$i$	$e\left(\frac{2}{7}\right)$
$\chi_{449}(29,\cdot)$	449.n	448	yes	$-1$	$1$	$e\left(\frac{215}{224}\right)$	$e\left(\frac{425}{448}\right)$	$e\left(\frac{103}{112}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{407}{448}\right)$	$e\left(\frac{83}{112}\right)$	$e\left(\frac{197}{224}\right)$	$e\left(\frac{201}{224}\right)$	$e\left(\frac{1}{32}\right)$	$e\left(\frac{25}{56}\right)$
$\chi_{449}(30,\cdot)$	449.n	448	yes	$-1$	$1$	$e\left(\frac{1}{224}\right)$	$e\left(\frac{351}{448}\right)$	$e\left(\frac{1}{112}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{353}{448}\right)$	$e\left(\frac{53}{112}\right)$	$e\left(\frac{3}{224}\right)$	$e\left(\frac{127}{224}\right)$	$e\left(\frac{7}{32}\right)$	$e\left(\frac{47}{56}\right)$
$\chi_{449}(31,\cdot)$	449.n	448	yes	$-1$	$1$	$e\left(\frac{9}{224}\right)$	$e\left(\frac{23}{448}\right)$	$e\left(\frac{9}{112}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{41}{448}\right)$	$e\left(\frac{29}{112}\right)$	$e\left(\frac{27}{224}\right)$	$e\left(\frac{23}{224}\right)$	$e\left(\frac{31}{32}\right)$	$e\left(\frac{31}{56}\right)$
$\chi_{449}(32,\cdot)$	449.m	224	yes	$1$	$1$	$e\left(\frac{5}{112}\right)$	$e\left(\frac{187}{224}\right)$	$e\left(\frac{5}{56}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{197}{224}\right)$	$e\left(\frac{41}{56}\right)$	$e\left(\frac{15}{112}\right)$	$e\left(\frac{75}{112}\right)$	$e\left(\frac{3}{16}\right)$	$e\left(\frac{11}{28}\right)$

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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 $\F_{q}$
Belyi maps

Modulus	$449$
Structure	\(C_{448}\)
Order	$448$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\chi_{449}(1,\cdot)\)	449.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{449}(2,\cdot)\)	449.m	224	yes	\(1\)	\(1\)	\(e\left(\frac{1}{112}\right)\)	\(e\left(\frac{127}{224}\right)\)	\(e\left(\frac{1}{56}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{129}{224}\right)\)	\(e\left(\frac{53}{56}\right)\)	\(e\left(\frac{3}{112}\right)\)	\(e\left(\frac{15}{112}\right)\)	\(e\left(\frac{7}{16}\right)\)	\(e\left(\frac{19}{28}\right)\)
\(\chi_{449}(3,\cdot)\)	449.n	448	yes	\(-1\)	\(1\)	\(e\left(\frac{127}{224}\right)\)	\(e\left(\frac{1}{448}\right)\)	\(e\left(\frac{15}{112}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{255}{448}\right)\)	\(e\left(\frac{11}{112}\right)\)	\(e\left(\frac{157}{224}\right)\)	\(e\left(\frac{1}{224}\right)\)	\(e\left(\frac{25}{32}\right)\)	\(e\left(\frac{33}{56}\right)\)
\(\chi_{449}(4,\cdot)\)	449.l	112	yes	\(1\)	\(1\)	\(e\left(\frac{1}{56}\right)\)	\(e\left(\frac{15}{112}\right)\)	\(e\left(\frac{1}{28}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{17}{112}\right)\)	\(e\left(\frac{25}{28}\right)\)	\(e\left(\frac{3}{56}\right)\)	\(e\left(\frac{15}{56}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{5}{14}\right)\)
\(\chi_{449}(5,\cdot)\)	449.f	14	yes	\(1\)	\(1\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(1\)	\(e\left(\frac{4}{7}\right)\)
\(\chi_{449}(6,\cdot)\)	449.n	448	yes	\(-1\)	\(1\)	\(e\left(\frac{129}{224}\right)\)	\(e\left(\frac{255}{448}\right)\)	\(e\left(\frac{17}{112}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{65}{448}\right)\)	\(e\left(\frac{5}{112}\right)\)	\(e\left(\frac{163}{224}\right)\)	\(e\left(\frac{31}{224}\right)\)	\(e\left(\frac{7}{32}\right)\)	\(e\left(\frac{15}{56}\right)\)
\(\chi_{449}(7,\cdot)\)	449.l	112	yes	\(1\)	\(1\)	\(e\left(\frac{53}{56}\right)\)	\(e\left(\frac{11}{112}\right)\)	\(e\left(\frac{25}{28}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{5}{112}\right)\)	\(e\left(\frac{9}{28}\right)\)	\(e\left(\frac{47}{56}\right)\)	\(e\left(\frac{11}{56}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{13}{14}\right)\)
\(\chi_{449}(8,\cdot)\)	449.m	224	yes	\(1\)	\(1\)	\(e\left(\frac{3}{112}\right)\)	\(e\left(\frac{157}{224}\right)\)	\(e\left(\frac{3}{56}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{163}{224}\right)\)	\(e\left(\frac{47}{56}\right)\)	\(e\left(\frac{9}{112}\right)\)	\(e\left(\frac{45}{112}\right)\)	\(e\left(\frac{5}{16}\right)\)	\(e\left(\frac{1}{28}\right)\)
\(\chi_{449}(9,\cdot)\)	449.m	224	yes	\(1\)	\(1\)	\(e\left(\frac{15}{112}\right)\)	\(e\left(\frac{1}{224}\right)\)	\(e\left(\frac{15}{56}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{31}{224}\right)\)	\(e\left(\frac{11}{56}\right)\)	\(e\left(\frac{45}{112}\right)\)	\(e\left(\frac{1}{112}\right)\)	\(e\left(\frac{9}{16}\right)\)	\(e\left(\frac{5}{28}\right)\)
\(\chi_{449}(10,\cdot)\)	449.i	32	yes	\(1\)	\(1\)	\(e\left(\frac{7}{16}\right)\)	\(e\left(\frac{25}{32}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(1\)	\(e\left(\frac{7}{32}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{5}{16}\right)\)	\(e\left(\frac{9}{16}\right)\)	\(e\left(\frac{7}{16}\right)\)	\(i\)
\(\chi_{449}(11,\cdot)\)	449.j	56	yes	\(1\)	\(1\)	\(e\left(\frac{19}{28}\right)\)	\(e\left(\frac{33}{56}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{15}{56}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{1}{28}\right)\)	\(e\left(\frac{5}{28}\right)\)	\(i\)	\(e\left(\frac{4}{7}\right)\)
\(\chi_{449}(12,\cdot)\)	449.n	448	yes	\(-1\)	\(1\)	\(e\left(\frac{131}{224}\right)\)	\(e\left(\frac{61}{448}\right)\)	\(e\left(\frac{19}{112}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{323}{448}\right)\)	\(e\left(\frac{111}{112}\right)\)	\(e\left(\frac{169}{224}\right)\)	\(e\left(\frac{61}{224}\right)\)	\(e\left(\frac{21}{32}\right)\)	\(e\left(\frac{53}{56}\right)\)
\(\chi_{449}(13,\cdot)\)	449.n	448	yes	\(-1\)	\(1\)	\(e\left(\frac{45}{224}\right)\)	\(e\left(\frac{115}{448}\right)\)	\(e\left(\frac{45}{112}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{205}{448}\right)\)	\(e\left(\frac{33}{112}\right)\)	\(e\left(\frac{135}{224}\right)\)	\(e\left(\frac{115}{224}\right)\)	\(e\left(\frac{27}{32}\right)\)	\(e\left(\frac{43}{56}\right)\)
\(\chi_{449}(14,\cdot)\)	449.m	224	yes	\(1\)	\(1\)	\(e\left(\frac{107}{112}\right)\)	\(e\left(\frac{149}{224}\right)\)	\(e\left(\frac{51}{56}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{139}{224}\right)\)	\(e\left(\frac{15}{56}\right)\)	\(e\left(\frac{97}{112}\right)\)	\(e\left(\frac{37}{112}\right)\)	\(e\left(\frac{13}{16}\right)\)	\(e\left(\frac{17}{28}\right)\)
\(\chi_{449}(15,\cdot)\)	449.n	448	yes	\(-1\)	\(1\)	\(e\left(\frac{223}{224}\right)\)	\(e\left(\frac{97}{448}\right)\)	\(e\left(\frac{111}{112}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{95}{448}\right)\)	\(e\left(\frac{59}{112}\right)\)	\(e\left(\frac{221}{224}\right)\)	\(e\left(\frac{97}{224}\right)\)	\(e\left(\frac{25}{32}\right)\)	\(e\left(\frac{9}{56}\right)\)
\(\chi_{449}(16,\cdot)\)	449.j	56	yes	\(1\)	\(1\)	\(e\left(\frac{1}{28}\right)\)	\(e\left(\frac{15}{56}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{17}{56}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{3}{28}\right)\)	\(e\left(\frac{15}{28}\right)\)	\(-i\)	\(e\left(\frac{5}{7}\right)\)
\(\chi_{449}(17,\cdot)\)	449.n	448	yes	\(-1\)	\(1\)	\(e\left(\frac{165}{224}\right)\)	\(e\left(\frac{347}{448}\right)\)	\(e\left(\frac{53}{112}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{229}{448}\right)\)	\(e\left(\frac{9}{112}\right)\)	\(e\left(\frac{47}{224}\right)\)	\(e\left(\frac{123}{224}\right)\)	\(e\left(\frac{3}{32}\right)\)	\(e\left(\frac{27}{56}\right)\)
\(\chi_{449}(18,\cdot)\)	449.d	7	yes	\(1\)	\(1\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(1\)	\(e\left(\frac{6}{7}\right)\)
\(\chi_{449}(19,\cdot)\)	449.n	448	yes	\(-1\)	\(1\)	\(e\left(\frac{79}{224}\right)\)	\(e\left(\frac{177}{448}\right)\)	\(e\left(\frac{79}{112}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{335}{448}\right)\)	\(e\left(\frac{43}{112}\right)\)	\(e\left(\frac{13}{224}\right)\)	\(e\left(\frac{177}{224}\right)\)	\(e\left(\frac{9}{32}\right)\)	\(e\left(\frac{17}{56}\right)\)
\(\chi_{449}(20,\cdot)\)	449.l	112	yes	\(1\)	\(1\)	\(e\left(\frac{25}{56}\right)\)	\(e\left(\frac{39}{112}\right)\)	\(e\left(\frac{25}{28}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{89}{112}\right)\)	\(e\left(\frac{9}{28}\right)\)	\(e\left(\frac{19}{56}\right)\)	\(e\left(\frac{39}{56}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{13}{14}\right)\)
\(\chi_{449}(21,\cdot)\)	449.n	448	yes	\(-1\)	\(1\)	\(e\left(\frac{115}{224}\right)\)	\(e\left(\frac{45}{448}\right)\)	\(e\left(\frac{3}{112}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{275}{448}\right)\)	\(e\left(\frac{47}{112}\right)\)	\(e\left(\frac{121}{224}\right)\)	\(e\left(\frac{45}{224}\right)\)	\(e\left(\frac{5}{32}\right)\)	\(e\left(\frac{29}{56}\right)\)
\(\chi_{449}(22,\cdot)\)	449.i	32	yes	\(1\)	\(1\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{5}{32}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(1\)	\(e\left(\frac{27}{32}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{16}\right)\)	\(e\left(\frac{5}{16}\right)\)	\(e\left(\frac{11}{16}\right)\)	\(i\)
\(\chi_{449}(23,\cdot)\)	449.l	112	yes	\(1\)	\(1\)	\(e\left(\frac{13}{56}\right)\)	\(e\left(\frac{27}{112}\right)\)	\(e\left(\frac{13}{28}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{53}{112}\right)\)	\(e\left(\frac{17}{28}\right)\)	\(e\left(\frac{39}{56}\right)\)	\(e\left(\frac{27}{56}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{9}{14}\right)\)
\(\chi_{449}(24,\cdot)\)	449.k	64	yes	\(-1\)	\(1\)	\(e\left(\frac{19}{32}\right)\)	\(e\left(\frac{45}{64}\right)\)	\(e\left(\frac{3}{16}\right)\)	\(-1\)	\(e\left(\frac{19}{64}\right)\)	\(e\left(\frac{15}{16}\right)\)	\(e\left(\frac{25}{32}\right)\)	\(e\left(\frac{13}{32}\right)\)	\(e\left(\frac{3}{32}\right)\)	\(e\left(\frac{5}{8}\right)\)
\(\chi_{449}(25,\cdot)\)	449.d	7	yes	\(1\)	\(1\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(1\)	\(e\left(\frac{1}{7}\right)\)
\(\chi_{449}(26,\cdot)\)	449.n	448	yes	\(-1\)	\(1\)	\(e\left(\frac{47}{224}\right)\)	\(e\left(\frac{369}{448}\right)\)	\(e\left(\frac{47}{112}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{15}{448}\right)\)	\(e\left(\frac{27}{112}\right)\)	\(e\left(\frac{141}{224}\right)\)	\(e\left(\frac{145}{224}\right)\)	\(e\left(\frac{9}{32}\right)\)	\(e\left(\frac{25}{56}\right)\)
\(\chi_{449}(27,\cdot)\)	449.n	448	yes	\(-1\)	\(1\)	\(e\left(\frac{157}{224}\right)\)	\(e\left(\frac{3}{448}\right)\)	\(e\left(\frac{45}{112}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{317}{448}\right)\)	\(e\left(\frac{33}{112}\right)\)	\(e\left(\frac{23}{224}\right)\)	\(e\left(\frac{3}{224}\right)\)	\(e\left(\frac{11}{32}\right)\)	\(e\left(\frac{43}{56}\right)\)
\(\chi_{449}(28,\cdot)\)	449.j	56	yes	\(1\)	\(1\)	\(e\left(\frac{27}{28}\right)\)	\(e\left(\frac{13}{56}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{11}{56}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{25}{28}\right)\)	\(e\left(\frac{13}{28}\right)\)	\(i\)	\(e\left(\frac{2}{7}\right)\)
\(\chi_{449}(29,\cdot)\)	449.n	448	yes	\(-1\)	\(1\)	\(e\left(\frac{215}{224}\right)\)	\(e\left(\frac{425}{448}\right)\)	\(e\left(\frac{103}{112}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{407}{448}\right)\)	\(e\left(\frac{83}{112}\right)\)	\(e\left(\frac{197}{224}\right)\)	\(e\left(\frac{201}{224}\right)\)	\(e\left(\frac{1}{32}\right)\)	\(e\left(\frac{25}{56}\right)\)
\(\chi_{449}(30,\cdot)\)	449.n	448	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{224}\right)\)	\(e\left(\frac{351}{448}\right)\)	\(e\left(\frac{1}{112}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{353}{448}\right)\)	\(e\left(\frac{53}{112}\right)\)	\(e\left(\frac{3}{224}\right)\)	\(e\left(\frac{127}{224}\right)\)	\(e\left(\frac{7}{32}\right)\)	\(e\left(\frac{47}{56}\right)\)
\(\chi_{449}(31,\cdot)\)	449.n	448	yes	\(-1\)	\(1\)	\(e\left(\frac{9}{224}\right)\)	\(e\left(\frac{23}{448}\right)\)	\(e\left(\frac{9}{112}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{41}{448}\right)\)	\(e\left(\frac{29}{112}\right)\)	\(e\left(\frac{27}{224}\right)\)	\(e\left(\frac{23}{224}\right)\)	\(e\left(\frac{31}{32}\right)\)	\(e\left(\frac{31}{56}\right)\)
\(\chi_{449}(32,\cdot)\)	449.m	224	yes	\(1\)	\(1\)	\(e\left(\frac{5}{112}\right)\)	\(e\left(\frac{187}{224}\right)\)	\(e\left(\frac{5}{56}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{197}{224}\right)\)	\(e\left(\frac{41}{56}\right)\)	\(e\left(\frac{15}{112}\right)\)	\(e\left(\frac{75}{112}\right)\)	\(e\left(\frac{3}{16}\right)\)	\(e\left(\frac{11}{28}\right)\)

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First 32 of 448 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.