LMFDB - Dirichlet character orbit 1309.db

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1309, base_ring=CyclotomicField(240)) M = H._module chi = DirichletCharacter(H, M([80,48,15])) chi.galois_orbit()

pari:[g,chi] = znchar(Mod(37,1309)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

Basic properties

Modulus:	$1309$
Conductor:	$1309$	sage:chi.conductor() pari:znconreyconductor(g,chi)
Order:	$240$	sage:chi.multiplicative_order() pari:charorder(g,chi)
Real:	no
Primitive:	yes	sage:chi.is_primitive() pari:#znconreyconductor(g,chi)==1
Minimal:	yes
Parity:	odd	sage:chi.is_odd() pari:zncharisodd(g,chi)

Related number fields

Field of values:	$\Q(\zeta_{240})$
Fixed field:	Number field defined by a degree 240 polynomial (not computed)

First 31 of 64 characters in Galois orbit

Character	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$8$	$9$	$10$	$12$	$13$
$\chi_{1309}(37,\cdot)$	$-1$	$1$	$e\left(\frac{89}{120}\right)$	$e\left(\frac{239}{240}\right)$	$e\left(\frac{29}{60}\right)$	$e\left(\frac{187}{240}\right)$	$e\left(\frac{59}{80}\right)$	$e\left(\frac{9}{40}\right)$	$e\left(\frac{119}{120}\right)$	$e\left(\frac{25}{48}\right)$	$e\left(\frac{23}{48}\right)$	$e\left(\frac{9}{20}\right)$
$\chi_{1309}(58,\cdot)$	$-1$	$1$	$e\left(\frac{11}{120}\right)$	$e\left(\frac{101}{240}\right)$	$e\left(\frac{11}{60}\right)$	$e\left(\frac{73}{240}\right)$	$e\left(\frac{41}{80}\right)$	$e\left(\frac{11}{40}\right)$	$e\left(\frac{101}{120}\right)$	$e\left(\frac{19}{48}\right)$	$e\left(\frac{29}{48}\right)$	$e\left(\frac{11}{20}\right)$
$\chi_{1309}(114,\cdot)$	$-1$	$1$	$e\left(\frac{29}{120}\right)$	$e\left(\frac{179}{240}\right)$	$e\left(\frac{29}{60}\right)$	$e\left(\frac{127}{240}\right)$	$e\left(\frac{79}{80}\right)$	$e\left(\frac{29}{40}\right)$	$e\left(\frac{59}{120}\right)$	$e\left(\frac{37}{48}\right)$	$e\left(\frac{11}{48}\right)$	$e\left(\frac{9}{20}\right)$
$\chi_{1309}(130,\cdot)$	$-1$	$1$	$e\left(\frac{7}{120}\right)$	$e\left(\frac{217}{240}\right)$	$e\left(\frac{7}{60}\right)$	$e\left(\frac{221}{240}\right)$	$e\left(\frac{77}{80}\right)$	$e\left(\frac{7}{40}\right)$	$e\left(\frac{97}{120}\right)$	$e\left(\frac{47}{48}\right)$	$e\left(\frac{1}{48}\right)$	$e\left(\frac{7}{20}\right)$
$\chi_{1309}(158,\cdot)$	$-1$	$1$	$e\left(\frac{109}{120}\right)$	$e\left(\frac{139}{240}\right)$	$e\left(\frac{49}{60}\right)$	$e\left(\frac{167}{240}\right)$	$e\left(\frac{39}{80}\right)$	$e\left(\frac{29}{40}\right)$	$e\left(\frac{19}{120}\right)$	$e\left(\frac{29}{48}\right)$	$e\left(\frac{19}{48}\right)$	$e\left(\frac{9}{20}\right)$
$\chi_{1309}(163,\cdot)$	$-1$	$1$	$e\left(\frac{107}{120}\right)$	$e\left(\frac{77}{240}\right)$	$e\left(\frac{47}{60}\right)$	$e\left(\frac{1}{240}\right)$	$e\left(\frac{17}{80}\right)$	$e\left(\frac{27}{40}\right)$	$e\left(\frac{77}{120}\right)$	$e\left(\frac{43}{48}\right)$	$e\left(\frac{5}{48}\right)$	$e\left(\frac{7}{20}\right)$
$\chi_{1309}(207,\cdot)$	$-1$	$1$	$e\left(\frac{97}{120}\right)$	$e\left(\frac{127}{240}\right)$	$e\left(\frac{37}{60}\right)$	$e\left(\frac{11}{240}\right)$	$e\left(\frac{27}{80}\right)$	$e\left(\frac{17}{40}\right)$	$e\left(\frac{7}{120}\right)$	$e\left(\frac{41}{48}\right)$	$e\left(\frac{7}{48}\right)$	$e\left(\frac{17}{20}\right)$
$\chi_{1309}(214,\cdot)$	$-1$	$1$	$e\left(\frac{43}{120}\right)$	$e\left(\frac{13}{240}\right)$	$e\left(\frac{43}{60}\right)$	$e\left(\frac{209}{240}\right)$	$e\left(\frac{33}{80}\right)$	$e\left(\frac{3}{40}\right)$	$e\left(\frac{13}{120}\right)$	$e\left(\frac{11}{48}\right)$	$e\left(\frac{37}{48}\right)$	$e\left(\frac{3}{20}\right)$
$\chi_{1309}(235,\cdot)$	$-1$	$1$	$e\left(\frac{49}{120}\right)$	$e\left(\frac{199}{240}\right)$	$e\left(\frac{49}{60}\right)$	$e\left(\frac{227}{240}\right)$	$e\left(\frac{19}{80}\right)$	$e\left(\frac{9}{40}\right)$	$e\left(\frac{79}{120}\right)$	$e\left(\frac{17}{48}\right)$	$e\left(\frac{31}{48}\right)$	$e\left(\frac{9}{20}\right)$
$\chi_{1309}(284,\cdot)$	$-1$	$1$	$e\left(\frac{37}{120}\right)$	$e\left(\frac{67}{240}\right)$	$e\left(\frac{37}{60}\right)$	$e\left(\frac{191}{240}\right)$	$e\left(\frac{47}{80}\right)$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{67}{120}\right)$	$e\left(\frac{5}{48}\right)$	$e\left(\frac{43}{48}\right)$	$e\left(\frac{17}{20}\right)$
$\chi_{1309}(312,\cdot)$	$-1$	$1$	$e\left(\frac{79}{120}\right)$	$e\left(\frac{49}{240}\right)$	$e\left(\frac{19}{60}\right)$	$e\left(\frac{197}{240}\right)$	$e\left(\frac{69}{80}\right)$	$e\left(\frac{39}{40}\right)$	$e\left(\frac{49}{120}\right)$	$e\left(\frac{23}{48}\right)$	$e\left(\frac{25}{48}\right)$	$e\left(\frac{19}{20}\right)$
$\chi_{1309}(317,\cdot)$	$-1$	$1$	$e\left(\frac{47}{120}\right)$	$e\left(\frac{137}{240}\right)$	$e\left(\frac{47}{60}\right)$	$e\left(\frac{61}{240}\right)$	$e\left(\frac{77}{80}\right)$	$e\left(\frac{7}{40}\right)$	$e\left(\frac{17}{120}\right)$	$e\left(\frac{31}{48}\right)$	$e\left(\frac{17}{48}\right)$	$e\left(\frac{7}{20}\right)$
$\chi_{1309}(333,\cdot)$	$-1$	$1$	$e\left(\frac{91}{120}\right)$	$e\left(\frac{61}{240}\right)$	$e\left(\frac{31}{60}\right)$	$e\left(\frac{113}{240}\right)$	$e\left(\frac{1}{80}\right)$	$e\left(\frac{11}{40}\right)$	$e\left(\frac{61}{120}\right)$	$e\left(\frac{11}{48}\right)$	$e\left(\frac{37}{48}\right)$	$e\left(\frac{11}{20}\right)$
$\chi_{1309}(345,\cdot)$	$-1$	$1$	$e\left(\frac{29}{120}\right)$	$e\left(\frac{59}{240}\right)$	$e\left(\frac{29}{60}\right)$	$e\left(\frac{7}{240}\right)$	$e\left(\frac{39}{80}\right)$	$e\left(\frac{29}{40}\right)$	$e\left(\frac{59}{120}\right)$	$e\left(\frac{13}{48}\right)$	$e\left(\frac{35}{48}\right)$	$e\left(\frac{9}{20}\right)$
$\chi_{1309}(368,\cdot)$	$-1$	$1$	$e\left(\frac{103}{120}\right)$	$e\left(\frac{73}{240}\right)$	$e\left(\frac{43}{60}\right)$	$e\left(\frac{29}{240}\right)$	$e\left(\frac{13}{80}\right)$	$e\left(\frac{23}{40}\right)$	$e\left(\frac{73}{120}\right)$	$e\left(\frac{47}{48}\right)$	$e\left(\frac{1}{48}\right)$	$e\left(\frac{3}{20}\right)$
$\chi_{1309}(394,\cdot)$	$-1$	$1$	$e\left(\frac{17}{120}\right)$	$e\left(\frac{47}{240}\right)$	$e\left(\frac{17}{60}\right)$	$e\left(\frac{91}{240}\right)$	$e\left(\frac{27}{80}\right)$	$e\left(\frac{17}{40}\right)$	$e\left(\frac{47}{120}\right)$	$e\left(\frac{25}{48}\right)$	$e\left(\frac{23}{48}\right)$	$e\left(\frac{17}{20}\right)$
$\chi_{1309}(401,\cdot)$	$-1$	$1$	$e\left(\frac{83}{120}\right)$	$e\left(\frac{173}{240}\right)$	$e\left(\frac{23}{60}\right)$	$e\left(\frac{49}{240}\right)$	$e\left(\frac{33}{80}\right)$	$e\left(\frac{3}{40}\right)$	$e\left(\frac{53}{120}\right)$	$e\left(\frac{43}{48}\right)$	$e\left(\frac{5}{48}\right)$	$e\left(\frac{3}{20}\right)$
$\chi_{1309}(422,\cdot)$	$-1$	$1$	$e\left(\frac{89}{120}\right)$	$e\left(\frac{119}{240}\right)$	$e\left(\frac{29}{60}\right)$	$e\left(\frac{67}{240}\right)$	$e\left(\frac{19}{80}\right)$	$e\left(\frac{9}{40}\right)$	$e\left(\frac{119}{120}\right)$	$e\left(\frac{1}{48}\right)$	$e\left(\frac{47}{48}\right)$	$e\left(\frac{9}{20}\right)$
$\chi_{1309}(445,\cdot)$	$-1$	$1$	$e\left(\frac{73}{120}\right)$	$e\left(\frac{223}{240}\right)$	$e\left(\frac{13}{60}\right)$	$e\left(\frac{59}{240}\right)$	$e\left(\frac{43}{80}\right)$	$e\left(\frac{33}{40}\right)$	$e\left(\frac{103}{120}\right)$	$e\left(\frac{41}{48}\right)$	$e\left(\frac{7}{48}\right)$	$e\left(\frac{13}{20}\right)$
$\chi_{1309}(466,\cdot)$	$-1$	$1$	$e\left(\frac{19}{120}\right)$	$e\left(\frac{229}{240}\right)$	$e\left(\frac{19}{60}\right)$	$e\left(\frac{137}{240}\right)$	$e\left(\frac{9}{80}\right)$	$e\left(\frac{19}{40}\right)$	$e\left(\frac{109}{120}\right)$	$e\left(\frac{35}{48}\right)$	$e\left(\frac{13}{48}\right)$	$e\left(\frac{19}{20}\right)$
$\chi_{1309}(471,\cdot)$	$-1$	$1$	$e\left(\frac{77}{120}\right)$	$e\left(\frac{227}{240}\right)$	$e\left(\frac{17}{60}\right)$	$e\left(\frac{31}{240}\right)$	$e\left(\frac{47}{80}\right)$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{107}{120}\right)$	$e\left(\frac{37}{48}\right)$	$e\left(\frac{11}{48}\right)$	$e\left(\frac{17}{20}\right)$
$\chi_{1309}(487,\cdot)$	$-1$	$1$	$e\left(\frac{31}{120}\right)$	$e\left(\frac{121}{240}\right)$	$e\left(\frac{31}{60}\right)$	$e\left(\frac{173}{240}\right)$	$e\left(\frac{61}{80}\right)$	$e\left(\frac{31}{40}\right)$	$e\left(\frac{1}{120}\right)$	$e\left(\frac{47}{48}\right)$	$e\left(\frac{1}{48}\right)$	$e\left(\frac{11}{20}\right)$
$\chi_{1309}(499,\cdot)$	$-1$	$1$	$e\left(\frac{119}{120}\right)$	$e\left(\frac{209}{240}\right)$	$e\left(\frac{59}{60}\right)$	$e\left(\frac{37}{240}\right)$	$e\left(\frac{69}{80}\right)$	$e\left(\frac{39}{40}\right)$	$e\left(\frac{89}{120}\right)$	$e\left(\frac{7}{48}\right)$	$e\left(\frac{41}{48}\right)$	$e\left(\frac{19}{20}\right)$
$\chi_{1309}(515,\cdot)$	$-1$	$1$	$e\left(\frac{37}{120}\right)$	$e\left(\frac{187}{240}\right)$	$e\left(\frac{37}{60}\right)$	$e\left(\frac{71}{240}\right)$	$e\left(\frac{7}{80}\right)$	$e\left(\frac{37}{40}\right)$	$e\left(\frac{67}{120}\right)$	$e\left(\frac{29}{48}\right)$	$e\left(\frac{19}{48}\right)$	$e\left(\frac{17}{20}\right)$
$\chi_{1309}(520,\cdot)$	$-1$	$1$	$e\left(\frac{11}{120}\right)$	$e\left(\frac{221}{240}\right)$	$e\left(\frac{11}{60}\right)$	$e\left(\frac{193}{240}\right)$	$e\left(\frac{1}{80}\right)$	$e\left(\frac{11}{40}\right)$	$e\left(\frac{101}{120}\right)$	$e\left(\frac{43}{48}\right)$	$e\left(\frac{5}{48}\right)$	$e\left(\frac{11}{20}\right)$
$\chi_{1309}(522,\cdot)$	$-1$	$1$	$e\left(\frac{13}{120}\right)$	$e\left(\frac{163}{240}\right)$	$e\left(\frac{13}{60}\right)$	$e\left(\frac{239}{240}\right)$	$e\left(\frac{63}{80}\right)$	$e\left(\frac{13}{40}\right)$	$e\left(\frac{43}{120}\right)$	$e\left(\frac{5}{48}\right)$	$e\left(\frac{43}{48}\right)$	$e\left(\frac{13}{20}\right)$
$\chi_{1309}(555,\cdot)$	$-1$	$1$	$e\left(\frac{23}{120}\right)$	$e\left(\frac{233}{240}\right)$	$e\left(\frac{23}{60}\right)$	$e\left(\frac{109}{240}\right)$	$e\left(\frac{13}{80}\right)$	$e\left(\frac{23}{40}\right)$	$e\left(\frac{113}{120}\right)$	$e\left(\frac{31}{48}\right)$	$e\left(\frac{17}{48}\right)$	$e\left(\frac{3}{20}\right)$
$\chi_{1309}(564,\cdot)$	$-1$	$1$	$e\left(\frac{1}{120}\right)$	$e\left(\frac{31}{240}\right)$	$e\left(\frac{1}{60}\right)$	$e\left(\frac{203}{240}\right)$	$e\left(\frac{11}{80}\right)$	$e\left(\frac{1}{40}\right)$	$e\left(\frac{31}{120}\right)$	$e\left(\frac{41}{48}\right)$	$e\left(\frac{7}{48}\right)$	$e\left(\frac{1}{20}\right)$
$\chi_{1309}(592,\cdot)$	$-1$	$1$	$e\left(\frac{97}{120}\right)$	$e\left(\frac{7}{240}\right)$	$e\left(\frac{37}{60}\right)$	$e\left(\frac{131}{240}\right)$	$e\left(\frac{67}{80}\right)$	$e\left(\frac{17}{40}\right)$	$e\left(\frac{7}{120}\right)$	$e\left(\frac{17}{48}\right)$	$e\left(\frac{31}{48}\right)$	$e\left(\frac{17}{20}\right)$
$\chi_{1309}(632,\cdot)$	$-1$	$1$	$e\left(\frac{113}{120}\right)$	$e\left(\frac{143}{240}\right)$	$e\left(\frac{53}{60}\right)$	$e\left(\frac{139}{240}\right)$	$e\left(\frac{43}{80}\right)$	$e\left(\frac{33}{40}\right)$	$e\left(\frac{23}{120}\right)$	$e\left(\frac{25}{48}\right)$	$e\left(\frac{23}{48}\right)$	$e\left(\frac{13}{20}\right)$
$\chi_{1309}(641,\cdot)$	$-1$	$1$	$e\left(\frac{61}{120}\right)$	$e\left(\frac{211}{240}\right)$	$e\left(\frac{1}{60}\right)$	$e\left(\frac{143}{240}\right)$	$e\left(\frac{31}{80}\right)$	$e\left(\frac{21}{40}\right)$	$e\left(\frac{91}{120}\right)$	$e\left(\frac{5}{48}\right)$	$e\left(\frac{43}{48}\right)$	$e\left(\frac{1}{20}\right)$

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

Character	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(8\)	\(9\)	\(10\)	\(12\)	\(13\)
\(\chi_{1309}(37,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{89}{120}\right)\)	\(e\left(\frac{239}{240}\right)\)	\(e\left(\frac{29}{60}\right)\)	\(e\left(\frac{187}{240}\right)\)	\(e\left(\frac{59}{80}\right)\)	\(e\left(\frac{9}{40}\right)\)	\(e\left(\frac{119}{120}\right)\)	\(e\left(\frac{25}{48}\right)\)	\(e\left(\frac{23}{48}\right)\)	\(e\left(\frac{9}{20}\right)\)
\(\chi_{1309}(58,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{11}{120}\right)\)	\(e\left(\frac{101}{240}\right)\)	\(e\left(\frac{11}{60}\right)\)	\(e\left(\frac{73}{240}\right)\)	\(e\left(\frac{41}{80}\right)\)	\(e\left(\frac{11}{40}\right)\)	\(e\left(\frac{101}{120}\right)\)	\(e\left(\frac{19}{48}\right)\)	\(e\left(\frac{29}{48}\right)\)	\(e\left(\frac{11}{20}\right)\)
\(\chi_{1309}(114,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{29}{120}\right)\)	\(e\left(\frac{179}{240}\right)\)	\(e\left(\frac{29}{60}\right)\)	\(e\left(\frac{127}{240}\right)\)	\(e\left(\frac{79}{80}\right)\)	\(e\left(\frac{29}{40}\right)\)	\(e\left(\frac{59}{120}\right)\)	\(e\left(\frac{37}{48}\right)\)	\(e\left(\frac{11}{48}\right)\)	\(e\left(\frac{9}{20}\right)\)
\(\chi_{1309}(130,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{7}{120}\right)\)	\(e\left(\frac{217}{240}\right)\)	\(e\left(\frac{7}{60}\right)\)	\(e\left(\frac{221}{240}\right)\)	\(e\left(\frac{77}{80}\right)\)	\(e\left(\frac{7}{40}\right)\)	\(e\left(\frac{97}{120}\right)\)	\(e\left(\frac{47}{48}\right)\)	\(e\left(\frac{1}{48}\right)\)	\(e\left(\frac{7}{20}\right)\)
\(\chi_{1309}(158,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{109}{120}\right)\)	\(e\left(\frac{139}{240}\right)\)	\(e\left(\frac{49}{60}\right)\)	\(e\left(\frac{167}{240}\right)\)	\(e\left(\frac{39}{80}\right)\)	\(e\left(\frac{29}{40}\right)\)	\(e\left(\frac{19}{120}\right)\)	\(e\left(\frac{29}{48}\right)\)	\(e\left(\frac{19}{48}\right)\)	\(e\left(\frac{9}{20}\right)\)
\(\chi_{1309}(163,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{107}{120}\right)\)	\(e\left(\frac{77}{240}\right)\)	\(e\left(\frac{47}{60}\right)\)	\(e\left(\frac{1}{240}\right)\)	\(e\left(\frac{17}{80}\right)\)	\(e\left(\frac{27}{40}\right)\)	\(e\left(\frac{77}{120}\right)\)	\(e\left(\frac{43}{48}\right)\)	\(e\left(\frac{5}{48}\right)\)	\(e\left(\frac{7}{20}\right)\)
\(\chi_{1309}(207,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{97}{120}\right)\)	\(e\left(\frac{127}{240}\right)\)	\(e\left(\frac{37}{60}\right)\)	\(e\left(\frac{11}{240}\right)\)	\(e\left(\frac{27}{80}\right)\)	\(e\left(\frac{17}{40}\right)\)	\(e\left(\frac{7}{120}\right)\)	\(e\left(\frac{41}{48}\right)\)	\(e\left(\frac{7}{48}\right)\)	\(e\left(\frac{17}{20}\right)\)
\(\chi_{1309}(214,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{43}{120}\right)\)	\(e\left(\frac{13}{240}\right)\)	\(e\left(\frac{43}{60}\right)\)	\(e\left(\frac{209}{240}\right)\)	\(e\left(\frac{33}{80}\right)\)	\(e\left(\frac{3}{40}\right)\)	\(e\left(\frac{13}{120}\right)\)	\(e\left(\frac{11}{48}\right)\)	\(e\left(\frac{37}{48}\right)\)	\(e\left(\frac{3}{20}\right)\)
\(\chi_{1309}(235,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{49}{120}\right)\)	\(e\left(\frac{199}{240}\right)\)	\(e\left(\frac{49}{60}\right)\)	\(e\left(\frac{227}{240}\right)\)	\(e\left(\frac{19}{80}\right)\)	\(e\left(\frac{9}{40}\right)\)	\(e\left(\frac{79}{120}\right)\)	\(e\left(\frac{17}{48}\right)\)	\(e\left(\frac{31}{48}\right)\)	\(e\left(\frac{9}{20}\right)\)
\(\chi_{1309}(284,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{37}{120}\right)\)	\(e\left(\frac{67}{240}\right)\)	\(e\left(\frac{37}{60}\right)\)	\(e\left(\frac{191}{240}\right)\)	\(e\left(\frac{47}{80}\right)\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{67}{120}\right)\)	\(e\left(\frac{5}{48}\right)\)	\(e\left(\frac{43}{48}\right)\)	\(e\left(\frac{17}{20}\right)\)
\(\chi_{1309}(312,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{79}{120}\right)\)	\(e\left(\frac{49}{240}\right)\)	\(e\left(\frac{19}{60}\right)\)	\(e\left(\frac{197}{240}\right)\)	\(e\left(\frac{69}{80}\right)\)	\(e\left(\frac{39}{40}\right)\)	\(e\left(\frac{49}{120}\right)\)	\(e\left(\frac{23}{48}\right)\)	\(e\left(\frac{25}{48}\right)\)	\(e\left(\frac{19}{20}\right)\)
\(\chi_{1309}(317,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{47}{120}\right)\)	\(e\left(\frac{137}{240}\right)\)	\(e\left(\frac{47}{60}\right)\)	\(e\left(\frac{61}{240}\right)\)	\(e\left(\frac{77}{80}\right)\)	\(e\left(\frac{7}{40}\right)\)	\(e\left(\frac{17}{120}\right)\)	\(e\left(\frac{31}{48}\right)\)	\(e\left(\frac{17}{48}\right)\)	\(e\left(\frac{7}{20}\right)\)
\(\chi_{1309}(333,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{91}{120}\right)\)	\(e\left(\frac{61}{240}\right)\)	\(e\left(\frac{31}{60}\right)\)	\(e\left(\frac{113}{240}\right)\)	\(e\left(\frac{1}{80}\right)\)	\(e\left(\frac{11}{40}\right)\)	\(e\left(\frac{61}{120}\right)\)	\(e\left(\frac{11}{48}\right)\)	\(e\left(\frac{37}{48}\right)\)	\(e\left(\frac{11}{20}\right)\)
\(\chi_{1309}(345,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{29}{120}\right)\)	\(e\left(\frac{59}{240}\right)\)	\(e\left(\frac{29}{60}\right)\)	\(e\left(\frac{7}{240}\right)\)	\(e\left(\frac{39}{80}\right)\)	\(e\left(\frac{29}{40}\right)\)	\(e\left(\frac{59}{120}\right)\)	\(e\left(\frac{13}{48}\right)\)	\(e\left(\frac{35}{48}\right)\)	\(e\left(\frac{9}{20}\right)\)
\(\chi_{1309}(368,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{103}{120}\right)\)	\(e\left(\frac{73}{240}\right)\)	\(e\left(\frac{43}{60}\right)\)	\(e\left(\frac{29}{240}\right)\)	\(e\left(\frac{13}{80}\right)\)	\(e\left(\frac{23}{40}\right)\)	\(e\left(\frac{73}{120}\right)\)	\(e\left(\frac{47}{48}\right)\)	\(e\left(\frac{1}{48}\right)\)	\(e\left(\frac{3}{20}\right)\)
\(\chi_{1309}(394,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{17}{120}\right)\)	\(e\left(\frac{47}{240}\right)\)	\(e\left(\frac{17}{60}\right)\)	\(e\left(\frac{91}{240}\right)\)	\(e\left(\frac{27}{80}\right)\)	\(e\left(\frac{17}{40}\right)\)	\(e\left(\frac{47}{120}\right)\)	\(e\left(\frac{25}{48}\right)\)	\(e\left(\frac{23}{48}\right)\)	\(e\left(\frac{17}{20}\right)\)
\(\chi_{1309}(401,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{83}{120}\right)\)	\(e\left(\frac{173}{240}\right)\)	\(e\left(\frac{23}{60}\right)\)	\(e\left(\frac{49}{240}\right)\)	\(e\left(\frac{33}{80}\right)\)	\(e\left(\frac{3}{40}\right)\)	\(e\left(\frac{53}{120}\right)\)	\(e\left(\frac{43}{48}\right)\)	\(e\left(\frac{5}{48}\right)\)	\(e\left(\frac{3}{20}\right)\)
\(\chi_{1309}(422,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{89}{120}\right)\)	\(e\left(\frac{119}{240}\right)\)	\(e\left(\frac{29}{60}\right)\)	\(e\left(\frac{67}{240}\right)\)	\(e\left(\frac{19}{80}\right)\)	\(e\left(\frac{9}{40}\right)\)	\(e\left(\frac{119}{120}\right)\)	\(e\left(\frac{1}{48}\right)\)	\(e\left(\frac{47}{48}\right)\)	\(e\left(\frac{9}{20}\right)\)
\(\chi_{1309}(445,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{73}{120}\right)\)	\(e\left(\frac{223}{240}\right)\)	\(e\left(\frac{13}{60}\right)\)	\(e\left(\frac{59}{240}\right)\)	\(e\left(\frac{43}{80}\right)\)	\(e\left(\frac{33}{40}\right)\)	\(e\left(\frac{103}{120}\right)\)	\(e\left(\frac{41}{48}\right)\)	\(e\left(\frac{7}{48}\right)\)	\(e\left(\frac{13}{20}\right)\)
\(\chi_{1309}(466,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{19}{120}\right)\)	\(e\left(\frac{229}{240}\right)\)	\(e\left(\frac{19}{60}\right)\)	\(e\left(\frac{137}{240}\right)\)	\(e\left(\frac{9}{80}\right)\)	\(e\left(\frac{19}{40}\right)\)	\(e\left(\frac{109}{120}\right)\)	\(e\left(\frac{35}{48}\right)\)	\(e\left(\frac{13}{48}\right)\)	\(e\left(\frac{19}{20}\right)\)
\(\chi_{1309}(471,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{77}{120}\right)\)	\(e\left(\frac{227}{240}\right)\)	\(e\left(\frac{17}{60}\right)\)	\(e\left(\frac{31}{240}\right)\)	\(e\left(\frac{47}{80}\right)\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{107}{120}\right)\)	\(e\left(\frac{37}{48}\right)\)	\(e\left(\frac{11}{48}\right)\)	\(e\left(\frac{17}{20}\right)\)
\(\chi_{1309}(487,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{31}{120}\right)\)	\(e\left(\frac{121}{240}\right)\)	\(e\left(\frac{31}{60}\right)\)	\(e\left(\frac{173}{240}\right)\)	\(e\left(\frac{61}{80}\right)\)	\(e\left(\frac{31}{40}\right)\)	\(e\left(\frac{1}{120}\right)\)	\(e\left(\frac{47}{48}\right)\)	\(e\left(\frac{1}{48}\right)\)	\(e\left(\frac{11}{20}\right)\)
\(\chi_{1309}(499,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{119}{120}\right)\)	\(e\left(\frac{209}{240}\right)\)	\(e\left(\frac{59}{60}\right)\)	\(e\left(\frac{37}{240}\right)\)	\(e\left(\frac{69}{80}\right)\)	\(e\left(\frac{39}{40}\right)\)	\(e\left(\frac{89}{120}\right)\)	\(e\left(\frac{7}{48}\right)\)	\(e\left(\frac{41}{48}\right)\)	\(e\left(\frac{19}{20}\right)\)
\(\chi_{1309}(515,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{37}{120}\right)\)	\(e\left(\frac{187}{240}\right)\)	\(e\left(\frac{37}{60}\right)\)	\(e\left(\frac{71}{240}\right)\)	\(e\left(\frac{7}{80}\right)\)	\(e\left(\frac{37}{40}\right)\)	\(e\left(\frac{67}{120}\right)\)	\(e\left(\frac{29}{48}\right)\)	\(e\left(\frac{19}{48}\right)\)	\(e\left(\frac{17}{20}\right)\)
\(\chi_{1309}(520,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{11}{120}\right)\)	\(e\left(\frac{221}{240}\right)\)	\(e\left(\frac{11}{60}\right)\)	\(e\left(\frac{193}{240}\right)\)	\(e\left(\frac{1}{80}\right)\)	\(e\left(\frac{11}{40}\right)\)	\(e\left(\frac{101}{120}\right)\)	\(e\left(\frac{43}{48}\right)\)	\(e\left(\frac{5}{48}\right)\)	\(e\left(\frac{11}{20}\right)\)
\(\chi_{1309}(522,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{13}{120}\right)\)	\(e\left(\frac{163}{240}\right)\)	\(e\left(\frac{13}{60}\right)\)	\(e\left(\frac{239}{240}\right)\)	\(e\left(\frac{63}{80}\right)\)	\(e\left(\frac{13}{40}\right)\)	\(e\left(\frac{43}{120}\right)\)	\(e\left(\frac{5}{48}\right)\)	\(e\left(\frac{43}{48}\right)\)	\(e\left(\frac{13}{20}\right)\)
\(\chi_{1309}(555,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{23}{120}\right)\)	\(e\left(\frac{233}{240}\right)\)	\(e\left(\frac{23}{60}\right)\)	\(e\left(\frac{109}{240}\right)\)	\(e\left(\frac{13}{80}\right)\)	\(e\left(\frac{23}{40}\right)\)	\(e\left(\frac{113}{120}\right)\)	\(e\left(\frac{31}{48}\right)\)	\(e\left(\frac{17}{48}\right)\)	\(e\left(\frac{3}{20}\right)\)
\(\chi_{1309}(564,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{1}{120}\right)\)	\(e\left(\frac{31}{240}\right)\)	\(e\left(\frac{1}{60}\right)\)	\(e\left(\frac{203}{240}\right)\)	\(e\left(\frac{11}{80}\right)\)	\(e\left(\frac{1}{40}\right)\)	\(e\left(\frac{31}{120}\right)\)	\(e\left(\frac{41}{48}\right)\)	\(e\left(\frac{7}{48}\right)\)	\(e\left(\frac{1}{20}\right)\)
\(\chi_{1309}(592,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{97}{120}\right)\)	\(e\left(\frac{7}{240}\right)\)	\(e\left(\frac{37}{60}\right)\)	\(e\left(\frac{131}{240}\right)\)	\(e\left(\frac{67}{80}\right)\)	\(e\left(\frac{17}{40}\right)\)	\(e\left(\frac{7}{120}\right)\)	\(e\left(\frac{17}{48}\right)\)	\(e\left(\frac{31}{48}\right)\)	\(e\left(\frac{17}{20}\right)\)
\(\chi_{1309}(632,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{113}{120}\right)\)	\(e\left(\frac{143}{240}\right)\)	\(e\left(\frac{53}{60}\right)\)	\(e\left(\frac{139}{240}\right)\)	\(e\left(\frac{43}{80}\right)\)	\(e\left(\frac{33}{40}\right)\)	\(e\left(\frac{23}{120}\right)\)	\(e\left(\frac{25}{48}\right)\)	\(e\left(\frac{23}{48}\right)\)	\(e\left(\frac{13}{20}\right)\)
\(\chi_{1309}(641,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{61}{120}\right)\)	\(e\left(\frac{211}{240}\right)\)	\(e\left(\frac{1}{60}\right)\)	\(e\left(\frac{143}{240}\right)\)	\(e\left(\frac{31}{80}\right)\)	\(e\left(\frac{21}{40}\right)\)	\(e\left(\frac{91}{120}\right)\)	\(e\left(\frac{5}{48}\right)\)	\(e\left(\frac{43}{48}\right)\)	\(e\left(\frac{1}{20}\right)\)

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First 31 of 64 characters in Galois orbit

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

Label	1309.db
Modulus	$1309$
Conductor	$1309$
Order	$240$
Real	no
Primitive	yes
Minimal	yes
Parity	odd

Modulus:	\(1309\)
Conductor:	\(1309\)	sage:chi.conductor() pari:znconreyconductor(g,chi)
Order:	\(240\)	sage:chi.multiplicative_order() pari:charorder(g,chi)
Real:	no
Primitive:	yes	sage:chi.is_primitive() pari:#znconreyconductor(g,chi)==1
Minimal:	yes
Parity:	odd	sage:chi.is_odd() pari:zncharisodd(g,chi)