LMFDB - Dirichlet character orbit 578.i

Show commands: Magma / Pari/GP / SageMath

comment:Define the Dirichlet character orbit

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(578, base_ring=CyclotomicField(136)) M = H._module chi = DirichletCharacter(H, M([1])) chi.galois_orbit()

gp:[g,chi] = znchar(Mod(9, 578)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("578.9"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };

Basic properties

Modulus:	$578$	comment:Modulus sage:chi.modulus() gp:g[1][1] magma:Modulus(chi);
Conductor:	$289$	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	$136$	comment:Order sage:chi.multiplicative_order() gp:charorder(g,chi) magma:Order(chi);
Real:	no	comment:Whether the character is real sage:chi.multiplicative_order() <= 2 gp:charorder(g,chi) <= 2 magma:Order(chi) le 2;
Primitive:	no, induced from 289.i	comment:If the character is primitive sage:chi.is_primitive() gp:#znconreyconductor(g,chi)==1 magma:IsPrimitive(chi);
Minimal:	yes
Parity:	even	comment:Parity sage:chi.is_odd() gp:zncharisodd(g,chi) magma:IsOdd(chi);

Related number fields

Field of values:	$\Q(\zeta_{136})$	comment:Field of values of chi sage:CyclotomicField(chi.multiplicative_order()) gp:nfinit(polcyclo(charorder(g,chi))) magma:CyclotomicField(Order(chi));
Fixed field:	Number field defined by a degree 136 polynomial (not computed)	comment:Fixed field sage:chi.fixed_field()

First 31 of 64 characters in Galois orbit

Character	$-1$	$1$	$3$	$5$	$7$	$9$	$11$	$13$	$15$	$19$	$21$	$23$
$\chi_{578}(9,\cdot)$	$1$	$1$	$e\left(\frac{1}{136}\right)$	$e\left(\frac{93}{136}\right)$	$e\left(\frac{19}{136}\right)$	$e\left(\frac{1}{68}\right)$	$e\left(\frac{23}{136}\right)$	$e\left(\frac{15}{34}\right)$	$e\left(\frac{47}{68}\right)$	$e\left(\frac{7}{68}\right)$	$e\left(\frac{5}{34}\right)$	$e\left(\frac{87}{136}\right)$
$\chi_{578}(15,\cdot)$	$1$	$1$	$e\left(\frac{115}{136}\right)$	$e\left(\frac{87}{136}\right)$	$e\left(\frac{9}{136}\right)$	$e\left(\frac{47}{68}\right)$	$e\left(\frac{61}{136}\right)$	$e\left(\frac{25}{34}\right)$	$e\left(\frac{33}{68}\right)$	$e\left(\frac{57}{68}\right)$	$e\left(\frac{31}{34}\right)$	$e\left(\frac{77}{136}\right)$
$\chi_{578}(19,\cdot)$	$1$	$1$	$e\left(\frac{7}{136}\right)$	$e\left(\frac{107}{136}\right)$	$e\left(\frac{133}{136}\right)$	$e\left(\frac{7}{68}\right)$	$e\left(\frac{25}{136}\right)$	$e\left(\frac{3}{34}\right)$	$e\left(\frac{57}{68}\right)$	$e\left(\frac{49}{68}\right)$	$e\left(\frac{1}{34}\right)$	$e\left(\frac{65}{136}\right)$
$\chi_{578}(25,\cdot)$	$1$	$1$	$e\left(\frac{93}{136}\right)$	$e\left(\frac{81}{136}\right)$	$e\left(\frac{135}{136}\right)$	$e\left(\frac{25}{68}\right)$	$e\left(\frac{99}{136}\right)$	$e\left(\frac{1}{34}\right)$	$e\left(\frac{19}{68}\right)$	$e\left(\frac{39}{68}\right)$	$e\left(\frac{23}{34}\right)$	$e\left(\frac{67}{136}\right)$
$\chi_{578}(43,\cdot)$	$1$	$1$	$e\left(\frac{113}{136}\right)$	$e\left(\frac{37}{136}\right)$	$e\left(\frac{107}{136}\right)$	$e\left(\frac{45}{68}\right)$	$e\left(\frac{15}{136}\right)$	$e\left(\frac{29}{34}\right)$	$e\left(\frac{7}{68}\right)$	$e\left(\frac{43}{68}\right)$	$e\left(\frac{21}{34}\right)$	$e\left(\frac{39}{136}\right)$
$\chi_{578}(49,\cdot)$	$1$	$1$	$e\left(\frac{19}{136}\right)$	$e\left(\frac{135}{136}\right)$	$e\left(\frac{89}{136}\right)$	$e\left(\frac{19}{68}\right)$	$e\left(\frac{29}{136}\right)$	$e\left(\frac{13}{34}\right)$	$e\left(\frac{9}{68}\right)$	$e\left(\frac{65}{68}\right)$	$e\left(\frac{27}{34}\right)$	$e\left(\frac{21}{136}\right)$
$\chi_{578}(53,\cdot)$	$1$	$1$	$e\left(\frac{103}{136}\right)$	$e\left(\frac{59}{136}\right)$	$e\left(\frac{53}{136}\right)$	$e\left(\frac{35}{68}\right)$	$e\left(\frac{57}{136}\right)$	$e\left(\frac{15}{34}\right)$	$e\left(\frac{13}{68}\right)$	$e\left(\frac{41}{68}\right)$	$e\left(\frac{5}{34}\right)$	$e\left(\frac{121}{136}\right)$
$\chi_{578}(59,\cdot)$	$1$	$1$	$e\left(\frac{117}{136}\right)$	$e\left(\frac{1}{136}\right)$	$e\left(\frac{47}{136}\right)$	$e\left(\frac{49}{68}\right)$	$e\left(\frac{107}{136}\right)$	$e\left(\frac{21}{34}\right)$	$e\left(\frac{59}{68}\right)$	$e\left(\frac{3}{68}\right)$	$e\left(\frac{7}{34}\right)$	$e\left(\frac{115}{136}\right)$
$\chi_{578}(77,\cdot)$	$1$	$1$	$e\left(\frac{89}{136}\right)$	$e\left(\frac{117}{136}\right)$	$e\left(\frac{59}{136}\right)$	$e\left(\frac{21}{68}\right)$	$e\left(\frac{7}{136}\right)$	$e\left(\frac{9}{34}\right)$	$e\left(\frac{35}{68}\right)$	$e\left(\frac{11}{68}\right)$	$e\left(\frac{3}{34}\right)$	$e\left(\frac{127}{136}\right)$
$\chi_{578}(83,\cdot)$	$1$	$1$	$e\left(\frac{59}{136}\right)$	$e\left(\frac{47}{136}\right)$	$e\left(\frac{33}{136}\right)$	$e\left(\frac{59}{68}\right)$	$e\left(\frac{133}{136}\right)$	$e\left(\frac{1}{34}\right)$	$e\left(\frac{53}{68}\right)$	$e\left(\frac{5}{68}\right)$	$e\left(\frac{23}{34}\right)$	$e\left(\frac{101}{136}\right)$
$\chi_{578}(87,\cdot)$	$1$	$1$	$e\left(\frac{63}{136}\right)$	$e\left(\frac{11}{136}\right)$	$e\left(\frac{109}{136}\right)$	$e\left(\frac{63}{68}\right)$	$e\left(\frac{89}{136}\right)$	$e\left(\frac{27}{34}\right)$	$e\left(\frac{37}{68}\right)$	$e\left(\frac{33}{68}\right)$	$e\left(\frac{9}{34}\right)$	$e\left(\frac{41}{136}\right)$
$\chi_{578}(93,\cdot)$	$1$	$1$	$e\left(\frac{5}{136}\right)$	$e\left(\frac{57}{136}\right)$	$e\left(\frac{95}{136}\right)$	$e\left(\frac{5}{68}\right)$	$e\left(\frac{115}{136}\right)$	$e\left(\frac{7}{34}\right)$	$e\left(\frac{31}{68}\right)$	$e\left(\frac{35}{68}\right)$	$e\left(\frac{25}{34}\right)$	$e\left(\frac{27}{136}\right)$
$\chi_{578}(111,\cdot)$	$1$	$1$	$e\left(\frac{65}{136}\right)$	$e\left(\frac{61}{136}\right)$	$e\left(\frac{11}{136}\right)$	$e\left(\frac{65}{68}\right)$	$e\left(\frac{135}{136}\right)$	$e\left(\frac{23}{34}\right)$	$e\left(\frac{63}{68}\right)$	$e\left(\frac{47}{68}\right)$	$e\left(\frac{19}{34}\right)$	$e\left(\frac{79}{136}\right)$
$\chi_{578}(117,\cdot)$	$1$	$1$	$e\left(\frac{99}{136}\right)$	$e\left(\frac{95}{136}\right)$	$e\left(\frac{113}{136}\right)$	$e\left(\frac{31}{68}\right)$	$e\left(\frac{101}{136}\right)$	$e\left(\frac{23}{34}\right)$	$e\left(\frac{29}{68}\right)$	$e\left(\frac{13}{68}\right)$	$e\left(\frac{19}{34}\right)$	$e\left(\frac{45}{136}\right)$
$\chi_{578}(121,\cdot)$	$1$	$1$	$e\left(\frac{23}{136}\right)$	$e\left(\frac{99}{136}\right)$	$e\left(\frac{29}{136}\right)$	$e\left(\frac{23}{68}\right)$	$e\left(\frac{121}{136}\right)$	$e\left(\frac{5}{34}\right)$	$e\left(\frac{61}{68}\right)$	$e\left(\frac{25}{68}\right)$	$e\left(\frac{13}{34}\right)$	$e\left(\frac{97}{136}\right)$
$\chi_{578}(127,\cdot)$	$1$	$1$	$e\left(\frac{29}{136}\right)$	$e\left(\frac{113}{136}\right)$	$e\left(\frac{7}{136}\right)$	$e\left(\frac{29}{68}\right)$	$e\left(\frac{123}{136}\right)$	$e\left(\frac{27}{34}\right)$	$e\left(\frac{3}{68}\right)$	$e\left(\frac{67}{68}\right)$	$e\left(\frac{9}{34}\right)$	$e\left(\frac{75}{136}\right)$
$\chi_{578}(145,\cdot)$	$1$	$1$	$e\left(\frac{41}{136}\right)$	$e\left(\frac{5}{136}\right)$	$e\left(\frac{99}{136}\right)$	$e\left(\frac{41}{68}\right)$	$e\left(\frac{127}{136}\right)$	$e\left(\frac{3}{34}\right)$	$e\left(\frac{23}{68}\right)$	$e\left(\frac{15}{68}\right)$	$e\left(\frac{1}{34}\right)$	$e\left(\frac{31}{136}\right)$
$\chi_{578}(151,\cdot)$	$1$	$1$	$e\left(\frac{3}{136}\right)$	$e\left(\frac{7}{136}\right)$	$e\left(\frac{57}{136}\right)$	$e\left(\frac{3}{68}\right)$	$e\left(\frac{69}{136}\right)$	$e\left(\frac{11}{34}\right)$	$e\left(\frac{5}{68}\right)$	$e\left(\frac{21}{68}\right)$	$e\left(\frac{15}{34}\right)$	$e\left(\frac{125}{136}\right)$
$\chi_{578}(161,\cdot)$	$1$	$1$	$e\left(\frac{53}{136}\right)$	$e\left(\frac{33}{136}\right)$	$e\left(\frac{55}{136}\right)$	$e\left(\frac{53}{68}\right)$	$e\left(\frac{131}{136}\right)$	$e\left(\frac{13}{34}\right)$	$e\left(\frac{43}{68}\right)$	$e\left(\frac{31}{68}\right)$	$e\left(\frac{27}{34}\right)$	$e\left(\frac{123}{136}\right)$
$\chi_{578}(185,\cdot)$	$1$	$1$	$e\left(\frac{43}{136}\right)$	$e\left(\frac{55}{136}\right)$	$e\left(\frac{1}{136}\right)$	$e\left(\frac{43}{68}\right)$	$e\left(\frac{37}{136}\right)$	$e\left(\frac{33}{34}\right)$	$e\left(\frac{49}{68}\right)$	$e\left(\frac{29}{68}\right)$	$e\left(\frac{11}{34}\right)$	$e\left(\frac{69}{136}\right)$
$\chi_{578}(189,\cdot)$	$1$	$1$	$e\left(\frac{79}{136}\right)$	$e\left(\frac{3}{136}\right)$	$e\left(\frac{5}{136}\right)$	$e\left(\frac{11}{68}\right)$	$e\left(\frac{49}{136}\right)$	$e\left(\frac{29}{34}\right)$	$e\left(\frac{41}{68}\right)$	$e\left(\frac{9}{68}\right)$	$e\left(\frac{21}{34}\right)$	$e\left(\frac{73}{136}\right)$
$\chi_{578}(195,\cdot)$	$1$	$1$	$e\left(\frac{77}{136}\right)$	$e\left(\frac{89}{136}\right)$	$e\left(\frac{103}{136}\right)$	$e\left(\frac{9}{68}\right)$	$e\left(\frac{3}{136}\right)$	$e\left(\frac{33}{34}\right)$	$e\left(\frac{15}{68}\right)$	$e\left(\frac{63}{68}\right)$	$e\left(\frac{11}{34}\right)$	$e\left(\frac{35}{136}\right)$
$\chi_{578}(213,\cdot)$	$1$	$1$	$e\left(\frac{129}{136}\right)$	$e\left(\frac{29}{136}\right)$	$e\left(\frac{3}{136}\right)$	$e\left(\frac{61}{68}\right)$	$e\left(\frac{111}{136}\right)$	$e\left(\frac{31}{34}\right)$	$e\left(\frac{11}{68}\right)$	$e\left(\frac{19}{68}\right)$	$e\left(\frac{33}{34}\right)$	$e\left(\frac{71}{136}\right)$
$\chi_{578}(219,\cdot)$	$1$	$1$	$e\left(\frac{83}{136}\right)$	$e\left(\frac{103}{136}\right)$	$e\left(\frac{81}{136}\right)$	$e\left(\frac{15}{68}\right)$	$e\left(\frac{5}{136}\right)$	$e\left(\frac{21}{34}\right)$	$e\left(\frac{25}{68}\right)$	$e\left(\frac{37}{68}\right)$	$e\left(\frac{7}{34}\right)$	$e\left(\frac{13}{136}\right)$
$\chi_{578}(223,\cdot)$	$1$	$1$	$e\left(\frac{39}{136}\right)$	$e\left(\frac{91}{136}\right)$	$e\left(\frac{61}{136}\right)$	$e\left(\frac{39}{68}\right)$	$e\left(\frac{81}{136}\right)$	$e\left(\frac{7}{34}\right)$	$e\left(\frac{65}{68}\right)$	$e\left(\frac{1}{68}\right)$	$e\left(\frac{25}{34}\right)$	$e\left(\frac{129}{136}\right)$
$\chi_{578}(229,\cdot)$	$1$	$1$	$e\left(\frac{101}{136}\right)$	$e\left(\frac{9}{136}\right)$	$e\left(\frac{15}{136}\right)$	$e\left(\frac{33}{68}\right)$	$e\left(\frac{11}{136}\right)$	$e\left(\frac{19}{34}\right)$	$e\left(\frac{55}{68}\right)$	$e\left(\frac{27}{68}\right)$	$e\left(\frac{29}{34}\right)$	$e\left(\frac{83}{136}\right)$
$\chi_{578}(247,\cdot)$	$1$	$1$	$e\left(\frac{105}{136}\right)$	$e\left(\frac{109}{136}\right)$	$e\left(\frac{91}{136}\right)$	$e\left(\frac{37}{68}\right)$	$e\left(\frac{103}{136}\right)$	$e\left(\frac{11}{34}\right)$	$e\left(\frac{39}{68}\right)$	$e\left(\frac{55}{68}\right)$	$e\left(\frac{15}{34}\right)$	$e\left(\frac{23}{136}\right)$
$\chi_{578}(253,\cdot)$	$1$	$1$	$e\left(\frac{123}{136}\right)$	$e\left(\frac{15}{136}\right)$	$e\left(\frac{25}{136}\right)$	$e\left(\frac{55}{68}\right)$	$e\left(\frac{109}{136}\right)$	$e\left(\frac{9}{34}\right)$	$e\left(\frac{1}{68}\right)$	$e\left(\frac{45}{68}\right)$	$e\left(\frac{3}{34}\right)$	$e\left(\frac{93}{136}\right)$
$\chi_{578}(257,\cdot)$	$1$	$1$	$e\left(\frac{135}{136}\right)$	$e\left(\frac{43}{136}\right)$	$e\left(\frac{117}{136}\right)$	$e\left(\frac{67}{68}\right)$	$e\left(\frac{113}{136}\right)$	$e\left(\frac{19}{34}\right)$	$e\left(\frac{21}{68}\right)$	$e\left(\frac{61}{68}\right)$	$e\left(\frac{29}{34}\right)$	$e\left(\frac{49}{136}\right)$
$\chi_{578}(263,\cdot)$	$1$	$1$	$e\left(\frac{125}{136}\right)$	$e\left(\frac{65}{136}\right)$	$e\left(\frac{63}{136}\right)$	$e\left(\frac{57}{68}\right)$	$e\left(\frac{19}{136}\right)$	$e\left(\frac{5}{34}\right)$	$e\left(\frac{27}{68}\right)$	$e\left(\frac{59}{68}\right)$	$e\left(\frac{13}{34}\right)$	$e\left(\frac{131}{136}\right)$
$\chi_{578}(281,\cdot)$	$1$	$1$	$e\left(\frac{81}{136}\right)$	$e\left(\frac{53}{136}\right)$	$e\left(\frac{43}{136}\right)$	$e\left(\frac{13}{68}\right)$	$e\left(\frac{95}{136}\right)$	$e\left(\frac{25}{34}\right)$	$e\left(\frac{67}{68}\right)$	$e\left(\frac{23}{68}\right)$	$e\left(\frac{31}{34}\right)$	$e\left(\frac{111}{136}\right)$

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	\(-1\)	\(1\)	\(3\)	\(5\)	\(7\)	\(9\)	\(11\)	\(13\)	\(15\)	\(19\)	\(21\)	\(23\)
\(\chi_{578}(9,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{1}{136}\right)\)	\(e\left(\frac{93}{136}\right)\)	\(e\left(\frac{19}{136}\right)\)	\(e\left(\frac{1}{68}\right)\)	\(e\left(\frac{23}{136}\right)\)	\(e\left(\frac{15}{34}\right)\)	\(e\left(\frac{47}{68}\right)\)	\(e\left(\frac{7}{68}\right)\)	\(e\left(\frac{5}{34}\right)\)	\(e\left(\frac{87}{136}\right)\)
\(\chi_{578}(15,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{115}{136}\right)\)	\(e\left(\frac{87}{136}\right)\)	\(e\left(\frac{9}{136}\right)\)	\(e\left(\frac{47}{68}\right)\)	\(e\left(\frac{61}{136}\right)\)	\(e\left(\frac{25}{34}\right)\)	\(e\left(\frac{33}{68}\right)\)	\(e\left(\frac{57}{68}\right)\)	\(e\left(\frac{31}{34}\right)\)	\(e\left(\frac{77}{136}\right)\)
\(\chi_{578}(19,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{7}{136}\right)\)	\(e\left(\frac{107}{136}\right)\)	\(e\left(\frac{133}{136}\right)\)	\(e\left(\frac{7}{68}\right)\)	\(e\left(\frac{25}{136}\right)\)	\(e\left(\frac{3}{34}\right)\)	\(e\left(\frac{57}{68}\right)\)	\(e\left(\frac{49}{68}\right)\)	\(e\left(\frac{1}{34}\right)\)	\(e\left(\frac{65}{136}\right)\)
\(\chi_{578}(25,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{93}{136}\right)\)	\(e\left(\frac{81}{136}\right)\)	\(e\left(\frac{135}{136}\right)\)	\(e\left(\frac{25}{68}\right)\)	\(e\left(\frac{99}{136}\right)\)	\(e\left(\frac{1}{34}\right)\)	\(e\left(\frac{19}{68}\right)\)	\(e\left(\frac{39}{68}\right)\)	\(e\left(\frac{23}{34}\right)\)	\(e\left(\frac{67}{136}\right)\)
\(\chi_{578}(43,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{113}{136}\right)\)	\(e\left(\frac{37}{136}\right)\)	\(e\left(\frac{107}{136}\right)\)	\(e\left(\frac{45}{68}\right)\)	\(e\left(\frac{15}{136}\right)\)	\(e\left(\frac{29}{34}\right)\)	\(e\left(\frac{7}{68}\right)\)	\(e\left(\frac{43}{68}\right)\)	\(e\left(\frac{21}{34}\right)\)	\(e\left(\frac{39}{136}\right)\)
\(\chi_{578}(49,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{19}{136}\right)\)	\(e\left(\frac{135}{136}\right)\)	\(e\left(\frac{89}{136}\right)\)	\(e\left(\frac{19}{68}\right)\)	\(e\left(\frac{29}{136}\right)\)	\(e\left(\frac{13}{34}\right)\)	\(e\left(\frac{9}{68}\right)\)	\(e\left(\frac{65}{68}\right)\)	\(e\left(\frac{27}{34}\right)\)	\(e\left(\frac{21}{136}\right)\)
\(\chi_{578}(53,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{103}{136}\right)\)	\(e\left(\frac{59}{136}\right)\)	\(e\left(\frac{53}{136}\right)\)	\(e\left(\frac{35}{68}\right)\)	\(e\left(\frac{57}{136}\right)\)	\(e\left(\frac{15}{34}\right)\)	\(e\left(\frac{13}{68}\right)\)	\(e\left(\frac{41}{68}\right)\)	\(e\left(\frac{5}{34}\right)\)	\(e\left(\frac{121}{136}\right)\)
\(\chi_{578}(59,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{117}{136}\right)\)	\(e\left(\frac{1}{136}\right)\)	\(e\left(\frac{47}{136}\right)\)	\(e\left(\frac{49}{68}\right)\)	\(e\left(\frac{107}{136}\right)\)	\(e\left(\frac{21}{34}\right)\)	\(e\left(\frac{59}{68}\right)\)	\(e\left(\frac{3}{68}\right)\)	\(e\left(\frac{7}{34}\right)\)	\(e\left(\frac{115}{136}\right)\)
\(\chi_{578}(77,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{89}{136}\right)\)	\(e\left(\frac{117}{136}\right)\)	\(e\left(\frac{59}{136}\right)\)	\(e\left(\frac{21}{68}\right)\)	\(e\left(\frac{7}{136}\right)\)	\(e\left(\frac{9}{34}\right)\)	\(e\left(\frac{35}{68}\right)\)	\(e\left(\frac{11}{68}\right)\)	\(e\left(\frac{3}{34}\right)\)	\(e\left(\frac{127}{136}\right)\)
\(\chi_{578}(83,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{59}{136}\right)\)	\(e\left(\frac{47}{136}\right)\)	\(e\left(\frac{33}{136}\right)\)	\(e\left(\frac{59}{68}\right)\)	\(e\left(\frac{133}{136}\right)\)	\(e\left(\frac{1}{34}\right)\)	\(e\left(\frac{53}{68}\right)\)	\(e\left(\frac{5}{68}\right)\)	\(e\left(\frac{23}{34}\right)\)	\(e\left(\frac{101}{136}\right)\)
\(\chi_{578}(87,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{63}{136}\right)\)	\(e\left(\frac{11}{136}\right)\)	\(e\left(\frac{109}{136}\right)\)	\(e\left(\frac{63}{68}\right)\)	\(e\left(\frac{89}{136}\right)\)	\(e\left(\frac{27}{34}\right)\)	\(e\left(\frac{37}{68}\right)\)	\(e\left(\frac{33}{68}\right)\)	\(e\left(\frac{9}{34}\right)\)	\(e\left(\frac{41}{136}\right)\)
\(\chi_{578}(93,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{5}{136}\right)\)	\(e\left(\frac{57}{136}\right)\)	\(e\left(\frac{95}{136}\right)\)	\(e\left(\frac{5}{68}\right)\)	\(e\left(\frac{115}{136}\right)\)	\(e\left(\frac{7}{34}\right)\)	\(e\left(\frac{31}{68}\right)\)	\(e\left(\frac{35}{68}\right)\)	\(e\left(\frac{25}{34}\right)\)	\(e\left(\frac{27}{136}\right)\)
\(\chi_{578}(111,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{65}{136}\right)\)	\(e\left(\frac{61}{136}\right)\)	\(e\left(\frac{11}{136}\right)\)	\(e\left(\frac{65}{68}\right)\)	\(e\left(\frac{135}{136}\right)\)	\(e\left(\frac{23}{34}\right)\)	\(e\left(\frac{63}{68}\right)\)	\(e\left(\frac{47}{68}\right)\)	\(e\left(\frac{19}{34}\right)\)	\(e\left(\frac{79}{136}\right)\)
\(\chi_{578}(117,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{99}{136}\right)\)	\(e\left(\frac{95}{136}\right)\)	\(e\left(\frac{113}{136}\right)\)	\(e\left(\frac{31}{68}\right)\)	\(e\left(\frac{101}{136}\right)\)	\(e\left(\frac{23}{34}\right)\)	\(e\left(\frac{29}{68}\right)\)	\(e\left(\frac{13}{68}\right)\)	\(e\left(\frac{19}{34}\right)\)	\(e\left(\frac{45}{136}\right)\)
\(\chi_{578}(121,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{23}{136}\right)\)	\(e\left(\frac{99}{136}\right)\)	\(e\left(\frac{29}{136}\right)\)	\(e\left(\frac{23}{68}\right)\)	\(e\left(\frac{121}{136}\right)\)	\(e\left(\frac{5}{34}\right)\)	\(e\left(\frac{61}{68}\right)\)	\(e\left(\frac{25}{68}\right)\)	\(e\left(\frac{13}{34}\right)\)	\(e\left(\frac{97}{136}\right)\)
\(\chi_{578}(127,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{29}{136}\right)\)	\(e\left(\frac{113}{136}\right)\)	\(e\left(\frac{7}{136}\right)\)	\(e\left(\frac{29}{68}\right)\)	\(e\left(\frac{123}{136}\right)\)	\(e\left(\frac{27}{34}\right)\)	\(e\left(\frac{3}{68}\right)\)	\(e\left(\frac{67}{68}\right)\)	\(e\left(\frac{9}{34}\right)\)	\(e\left(\frac{75}{136}\right)\)
\(\chi_{578}(145,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{41}{136}\right)\)	\(e\left(\frac{5}{136}\right)\)	\(e\left(\frac{99}{136}\right)\)	\(e\left(\frac{41}{68}\right)\)	\(e\left(\frac{127}{136}\right)\)	\(e\left(\frac{3}{34}\right)\)	\(e\left(\frac{23}{68}\right)\)	\(e\left(\frac{15}{68}\right)\)	\(e\left(\frac{1}{34}\right)\)	\(e\left(\frac{31}{136}\right)\)
\(\chi_{578}(151,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{3}{136}\right)\)	\(e\left(\frac{7}{136}\right)\)	\(e\left(\frac{57}{136}\right)\)	\(e\left(\frac{3}{68}\right)\)	\(e\left(\frac{69}{136}\right)\)	\(e\left(\frac{11}{34}\right)\)	\(e\left(\frac{5}{68}\right)\)	\(e\left(\frac{21}{68}\right)\)	\(e\left(\frac{15}{34}\right)\)	\(e\left(\frac{125}{136}\right)\)
\(\chi_{578}(161,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{53}{136}\right)\)	\(e\left(\frac{33}{136}\right)\)	\(e\left(\frac{55}{136}\right)\)	\(e\left(\frac{53}{68}\right)\)	\(e\left(\frac{131}{136}\right)\)	\(e\left(\frac{13}{34}\right)\)	\(e\left(\frac{43}{68}\right)\)	\(e\left(\frac{31}{68}\right)\)	\(e\left(\frac{27}{34}\right)\)	\(e\left(\frac{123}{136}\right)\)
\(\chi_{578}(185,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{43}{136}\right)\)	\(e\left(\frac{55}{136}\right)\)	\(e\left(\frac{1}{136}\right)\)	\(e\left(\frac{43}{68}\right)\)	\(e\left(\frac{37}{136}\right)\)	\(e\left(\frac{33}{34}\right)\)	\(e\left(\frac{49}{68}\right)\)	\(e\left(\frac{29}{68}\right)\)	\(e\left(\frac{11}{34}\right)\)	\(e\left(\frac{69}{136}\right)\)
\(\chi_{578}(189,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{79}{136}\right)\)	\(e\left(\frac{3}{136}\right)\)	\(e\left(\frac{5}{136}\right)\)	\(e\left(\frac{11}{68}\right)\)	\(e\left(\frac{49}{136}\right)\)	\(e\left(\frac{29}{34}\right)\)	\(e\left(\frac{41}{68}\right)\)	\(e\left(\frac{9}{68}\right)\)	\(e\left(\frac{21}{34}\right)\)	\(e\left(\frac{73}{136}\right)\)
\(\chi_{578}(195,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{77}{136}\right)\)	\(e\left(\frac{89}{136}\right)\)	\(e\left(\frac{103}{136}\right)\)	\(e\left(\frac{9}{68}\right)\)	\(e\left(\frac{3}{136}\right)\)	\(e\left(\frac{33}{34}\right)\)	\(e\left(\frac{15}{68}\right)\)	\(e\left(\frac{63}{68}\right)\)	\(e\left(\frac{11}{34}\right)\)	\(e\left(\frac{35}{136}\right)\)
\(\chi_{578}(213,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{129}{136}\right)\)	\(e\left(\frac{29}{136}\right)\)	\(e\left(\frac{3}{136}\right)\)	\(e\left(\frac{61}{68}\right)\)	\(e\left(\frac{111}{136}\right)\)	\(e\left(\frac{31}{34}\right)\)	\(e\left(\frac{11}{68}\right)\)	\(e\left(\frac{19}{68}\right)\)	\(e\left(\frac{33}{34}\right)\)	\(e\left(\frac{71}{136}\right)\)
\(\chi_{578}(219,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{83}{136}\right)\)	\(e\left(\frac{103}{136}\right)\)	\(e\left(\frac{81}{136}\right)\)	\(e\left(\frac{15}{68}\right)\)	\(e\left(\frac{5}{136}\right)\)	\(e\left(\frac{21}{34}\right)\)	\(e\left(\frac{25}{68}\right)\)	\(e\left(\frac{37}{68}\right)\)	\(e\left(\frac{7}{34}\right)\)	\(e\left(\frac{13}{136}\right)\)
\(\chi_{578}(223,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{39}{136}\right)\)	\(e\left(\frac{91}{136}\right)\)	\(e\left(\frac{61}{136}\right)\)	\(e\left(\frac{39}{68}\right)\)	\(e\left(\frac{81}{136}\right)\)	\(e\left(\frac{7}{34}\right)\)	\(e\left(\frac{65}{68}\right)\)	\(e\left(\frac{1}{68}\right)\)	\(e\left(\frac{25}{34}\right)\)	\(e\left(\frac{129}{136}\right)\)
\(\chi_{578}(229,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{101}{136}\right)\)	\(e\left(\frac{9}{136}\right)\)	\(e\left(\frac{15}{136}\right)\)	\(e\left(\frac{33}{68}\right)\)	\(e\left(\frac{11}{136}\right)\)	\(e\left(\frac{19}{34}\right)\)	\(e\left(\frac{55}{68}\right)\)	\(e\left(\frac{27}{68}\right)\)	\(e\left(\frac{29}{34}\right)\)	\(e\left(\frac{83}{136}\right)\)
\(\chi_{578}(247,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{105}{136}\right)\)	\(e\left(\frac{109}{136}\right)\)	\(e\left(\frac{91}{136}\right)\)	\(e\left(\frac{37}{68}\right)\)	\(e\left(\frac{103}{136}\right)\)	\(e\left(\frac{11}{34}\right)\)	\(e\left(\frac{39}{68}\right)\)	\(e\left(\frac{55}{68}\right)\)	\(e\left(\frac{15}{34}\right)\)	\(e\left(\frac{23}{136}\right)\)
\(\chi_{578}(253,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{123}{136}\right)\)	\(e\left(\frac{15}{136}\right)\)	\(e\left(\frac{25}{136}\right)\)	\(e\left(\frac{55}{68}\right)\)	\(e\left(\frac{109}{136}\right)\)	\(e\left(\frac{9}{34}\right)\)	\(e\left(\frac{1}{68}\right)\)	\(e\left(\frac{45}{68}\right)\)	\(e\left(\frac{3}{34}\right)\)	\(e\left(\frac{93}{136}\right)\)
\(\chi_{578}(257,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{135}{136}\right)\)	\(e\left(\frac{43}{136}\right)\)	\(e\left(\frac{117}{136}\right)\)	\(e\left(\frac{67}{68}\right)\)	\(e\left(\frac{113}{136}\right)\)	\(e\left(\frac{19}{34}\right)\)	\(e\left(\frac{21}{68}\right)\)	\(e\left(\frac{61}{68}\right)\)	\(e\left(\frac{29}{34}\right)\)	\(e\left(\frac{49}{136}\right)\)
\(\chi_{578}(263,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{125}{136}\right)\)	\(e\left(\frac{65}{136}\right)\)	\(e\left(\frac{63}{136}\right)\)	\(e\left(\frac{57}{68}\right)\)	\(e\left(\frac{19}{136}\right)\)	\(e\left(\frac{5}{34}\right)\)	\(e\left(\frac{27}{68}\right)\)	\(e\left(\frac{59}{68}\right)\)	\(e\left(\frac{13}{34}\right)\)	\(e\left(\frac{131}{136}\right)\)
\(\chi_{578}(281,\cdot)\)	\(1\)	\(1\)	\(e\left(\frac{81}{136}\right)\)	\(e\left(\frac{53}{136}\right)\)	\(e\left(\frac{43}{136}\right)\)	\(e\left(\frac{13}{68}\right)\)	\(e\left(\frac{95}{136}\right)\)	\(e\left(\frac{25}{34}\right)\)	\(e\left(\frac{67}{68}\right)\)	\(e\left(\frac{23}{68}\right)\)	\(e\left(\frac{31}{34}\right)\)	\(e\left(\frac{111}{136}\right)\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Basic properties

Related number fields

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	578.i
Modulus	$578$
Conductor	$289$
Order	$136$
Real	no
Primitive	no
Minimal	yes
Parity	even

Modulus:	\(578\)	comment:Modulus sage:chi.modulus() gp:g[1][1] magma:Modulus(chi);
Conductor:	\(289\)	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	\(136\)	comment:Order sage:chi.multiplicative_order() gp:charorder(g,chi) magma:Order(chi);
Real:	no	comment:Whether the character is real sage:chi.multiplicative_order() <= 2 gp:charorder(g,chi) <= 2 magma:Order(chi) le 2;
Primitive:	no, induced from 289.i	comment:If the character is primitive sage:chi.is_primitive() gp:#znconreyconductor(g,chi)==1 magma:IsPrimitive(chi);
Minimal:	yes
Parity:	even	comment:Parity sage:chi.is_odd() gp:zncharisodd(g,chi) magma:IsOdd(chi);