LMFDB - Dirichlet character orbit 2301.bu

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comment:Define the Dirichlet character orbit

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2301, base_ring=CyclotomicField(348)) M = H._module chi = DirichletCharacter(H, M([174,29,6])) chi.galois_orbit()

gp:[g,chi] = znchar(Mod(2, 2301)) 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("2301.2"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };

Basic properties

Modulus:	$2301$	comment:Modulus sage:chi.modulus() gp:g[1][1] magma:Modulus(chi);
Conductor:	$2301$	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	$348$	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:	yes	comment:If the character is primitive sage:chi.is_primitive() gp:#znconreyconductor(g,chi)==1 magma:IsPrimitive(chi);
Minimal:	yes
Parity:	odd	comment:Parity sage:chi.is_odd() gp:zncharisodd(g,chi) magma:IsOdd(chi);

Related number fields

Field of values:	$\Q(\zeta_{348})$	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 348 polynomial (not computed)	comment:Fixed field sage:chi.fixed_field()

First 31 of 112 characters in Galois orbit

Character	$-1$	$1$	$2$	$4$	$5$	$7$	$8$	$10$	$11$	$14$	$16$	$17$
$\chi_{2301}(2,\cdot)$	$-1$	$1$	$e\left(\frac{209}{348}\right)$	$e\left(\frac{35}{174}\right)$	$e\left(\frac{41}{116}\right)$	$e\left(\frac{79}{348}\right)$	$e\left(\frac{93}{116}\right)$	$e\left(\frac{83}{87}\right)$	$e\left(\frac{179}{348}\right)$	$e\left(\frac{24}{29}\right)$	$e\left(\frac{35}{87}\right)$	$e\left(\frac{31}{87}\right)$
$\chi_{2301}(11,\cdot)$	$-1$	$1$	$e\left(\frac{179}{348}\right)$	$e\left(\frac{5}{174}\right)$	$e\left(\frac{39}{116}\right)$	$e\left(\frac{61}{348}\right)$	$e\left(\frac{63}{116}\right)$	$e\left(\frac{74}{87}\right)$	$e\left(\frac{125}{348}\right)$	$e\left(\frac{20}{29}\right)$	$e\left(\frac{5}{87}\right)$	$e\left(\frac{79}{87}\right)$
$\chi_{2301}(32,\cdot)$	$-1$	$1$	$e\left(\frac{1}{348}\right)$	$e\left(\frac{1}{174}\right)$	$e\left(\frac{89}{116}\right)$	$e\left(\frac{47}{348}\right)$	$e\left(\frac{1}{116}\right)$	$e\left(\frac{67}{87}\right)$	$e\left(\frac{199}{348}\right)$	$e\left(\frac{4}{29}\right)$	$e\left(\frac{1}{87}\right)$	$e\left(\frac{68}{87}\right)$
$\chi_{2301}(50,\cdot)$	$-1$	$1$	$e\left(\frac{107}{348}\right)$	$e\left(\frac{107}{174}\right)$	$e\left(\frac{11}{116}\right)$	$e\left(\frac{157}{348}\right)$	$e\left(\frac{107}{116}\right)$	$e\left(\frac{35}{87}\right)$	$e\left(\frac{65}{348}\right)$	$e\left(\frac{22}{29}\right)$	$e\left(\frac{20}{87}\right)$	$e\left(\frac{55}{87}\right)$
$\chi_{2301}(89,\cdot)$	$-1$	$1$	$e\left(\frac{23}{348}\right)$	$e\left(\frac{23}{174}\right)$	$e\left(\frac{75}{116}\right)$	$e\left(\frac{37}{348}\right)$	$e\left(\frac{23}{116}\right)$	$e\left(\frac{62}{87}\right)$	$e\left(\frac{53}{348}\right)$	$e\left(\frac{5}{29}\right)$	$e\left(\frac{23}{87}\right)$	$e\left(\frac{85}{87}\right)$
$\chi_{2301}(98,\cdot)$	$-1$	$1$	$e\left(\frac{19}{348}\right)$	$e\left(\frac{19}{174}\right)$	$e\left(\frac{67}{116}\right)$	$e\left(\frac{197}{348}\right)$	$e\left(\frac{19}{116}\right)$	$e\left(\frac{55}{87}\right)$	$e\left(\frac{301}{348}\right)$	$e\left(\frac{18}{29}\right)$	$e\left(\frac{19}{87}\right)$	$e\left(\frac{74}{87}\right)$
$\chi_{2301}(128,\cdot)$	$-1$	$1$	$e\left(\frac{71}{348}\right)$	$e\left(\frac{71}{174}\right)$	$e\left(\frac{55}{116}\right)$	$e\left(\frac{205}{348}\right)$	$e\left(\frac{71}{116}\right)$	$e\left(\frac{59}{87}\right)$	$e\left(\frac{209}{348}\right)$	$e\left(\frac{23}{29}\right)$	$e\left(\frac{71}{87}\right)$	$e\left(\frac{43}{87}\right)$
$\chi_{2301}(149,\cdot)$	$-1$	$1$	$e\left(\frac{265}{348}\right)$	$e\left(\frac{91}{174}\right)$	$e\left(\frac{37}{116}\right)$	$e\left(\frac{275}{348}\right)$	$e\left(\frac{33}{116}\right)$	$e\left(\frac{7}{87}\right)$	$e\left(\frac{187}{348}\right)$	$e\left(\frac{16}{29}\right)$	$e\left(\frac{4}{87}\right)$	$e\left(\frac{11}{87}\right)$
$\chi_{2301}(158,\cdot)$	$-1$	$1$	$e\left(\frac{257}{348}\right)$	$e\left(\frac{83}{174}\right)$	$e\left(\frac{21}{116}\right)$	$e\left(\frac{247}{348}\right)$	$e\left(\frac{25}{116}\right)$	$e\left(\frac{80}{87}\right)$	$e\left(\frac{335}{348}\right)$	$e\left(\frac{13}{29}\right)$	$e\left(\frac{83}{87}\right)$	$e\left(\frac{76}{87}\right)$
$\chi_{2301}(188,\cdot)$	$-1$	$1$	$e\left(\frac{121}{348}\right)$	$e\left(\frac{121}{174}\right)$	$e\left(\frac{97}{116}\right)$	$e\left(\frac{119}{348}\right)$	$e\left(\frac{5}{116}\right)$	$e\left(\frac{16}{87}\right)$	$e\left(\frac{67}{348}\right)$	$e\left(\frac{20}{29}\right)$	$e\left(\frac{34}{87}\right)$	$e\left(\frac{50}{87}\right)$
$\chi_{2301}(215,\cdot)$	$-1$	$1$	$e\left(\frac{31}{348}\right)$	$e\left(\frac{31}{174}\right)$	$e\left(\frac{91}{116}\right)$	$e\left(\frac{65}{348}\right)$	$e\left(\frac{31}{116}\right)$	$e\left(\frac{76}{87}\right)$	$e\left(\frac{253}{348}\right)$	$e\left(\frac{8}{29}\right)$	$e\left(\frac{31}{87}\right)$	$e\left(\frac{20}{87}\right)$
$\chi_{2301}(227,\cdot)$	$-1$	$1$	$e\left(\frac{49}{348}\right)$	$e\left(\frac{49}{174}\right)$	$e\left(\frac{69}{116}\right)$	$e\left(\frac{215}{348}\right)$	$e\left(\frac{49}{116}\right)$	$e\left(\frac{64}{87}\right)$	$e\left(\frac{7}{348}\right)$	$e\left(\frac{22}{29}\right)$	$e\left(\frac{49}{87}\right)$	$e\left(\frac{26}{87}\right)$
$\chi_{2301}(254,\cdot)$	$-1$	$1$	$e\left(\frac{55}{348}\right)$	$e\left(\frac{55}{174}\right)$	$e\left(\frac{23}{116}\right)$	$e\left(\frac{149}{348}\right)$	$e\left(\frac{55}{116}\right)$	$e\left(\frac{31}{87}\right)$	$e\left(\frac{157}{348}\right)$	$e\left(\frac{17}{29}\right)$	$e\left(\frac{55}{87}\right)$	$e\left(\frac{86}{87}\right)$
$\chi_{2301}(266,\cdot)$	$-1$	$1$	$e\left(\frac{313}{348}\right)$	$e\left(\frac{139}{174}\right)$	$e\left(\frac{17}{116}\right)$	$e\left(\frac{95}{348}\right)$	$e\left(\frac{81}{116}\right)$	$e\left(\frac{4}{87}\right)$	$e\left(\frac{343}{348}\right)$	$e\left(\frac{5}{29}\right)$	$e\left(\frac{52}{87}\right)$	$e\left(\frac{56}{87}\right)$
$\chi_{2301}(275,\cdot)$	$-1$	$1$	$e\left(\frac{77}{348}\right)$	$e\left(\frac{77}{174}\right)$	$e\left(\frac{9}{116}\right)$	$e\left(\frac{139}{348}\right)$	$e\left(\frac{77}{116}\right)$	$e\left(\frac{26}{87}\right)$	$e\left(\frac{11}{348}\right)$	$e\left(\frac{18}{29}\right)$	$e\left(\frac{77}{87}\right)$	$e\left(\frac{16}{87}\right)$
$\chi_{2301}(305,\cdot)$	$-1$	$1$	$e\left(\frac{13}{348}\right)$	$e\left(\frac{13}{174}\right)$	$e\left(\frac{113}{116}\right)$	$e\left(\frac{263}{348}\right)$	$e\left(\frac{13}{116}\right)$	$e\left(\frac{1}{87}\right)$	$e\left(\frac{151}{348}\right)$	$e\left(\frac{23}{29}\right)$	$e\left(\frac{13}{87}\right)$	$e\left(\frac{14}{87}\right)$
$\chi_{2301}(332,\cdot)$	$-1$	$1$	$e\left(\frac{127}{348}\right)$	$e\left(\frac{127}{174}\right)$	$e\left(\frac{51}{116}\right)$	$e\left(\frac{53}{348}\right)$	$e\left(\frac{11}{116}\right)$	$e\left(\frac{70}{87}\right)$	$e\left(\frac{217}{348}\right)$	$e\left(\frac{15}{29}\right)$	$e\left(\frac{40}{87}\right)$	$e\left(\frac{23}{87}\right)$
$\chi_{2301}(362,\cdot)$	$-1$	$1$	$e\left(\frac{47}{348}\right)$	$e\left(\frac{47}{174}\right)$	$e\left(\frac{7}{116}\right)$	$e\left(\frac{121}{348}\right)$	$e\left(\frac{47}{116}\right)$	$e\left(\frac{17}{87}\right)$	$e\left(\frac{305}{348}\right)$	$e\left(\frac{14}{29}\right)$	$e\left(\frac{47}{87}\right)$	$e\left(\frac{64}{87}\right)$
$\chi_{2301}(392,\cdot)$	$-1$	$1$	$e\left(\frac{89}{348}\right)$	$e\left(\frac{89}{174}\right)$	$e\left(\frac{33}{116}\right)$	$e\left(\frac{7}{348}\right)$	$e\left(\frac{89}{116}\right)$	$e\left(\frac{47}{87}\right)$	$e\left(\frac{311}{348}\right)$	$e\left(\frac{8}{29}\right)$	$e\left(\frac{2}{87}\right)$	$e\left(\frac{49}{87}\right)$
$\chi_{2301}(401,\cdot)$	$-1$	$1$	$e\left(\frac{167}{348}\right)$	$e\left(\frac{167}{174}\right)$	$e\left(\frac{15}{116}\right)$	$e\left(\frac{193}{348}\right)$	$e\left(\frac{51}{116}\right)$	$e\left(\frac{53}{87}\right)$	$e\left(\frac{173}{348}\right)$	$e\left(\frac{1}{29}\right)$	$e\left(\frac{80}{87}\right)$	$e\left(\frac{46}{87}\right)$
$\chi_{2301}(410,\cdot)$	$-1$	$1$	$e\left(\frac{271}{348}\right)$	$e\left(\frac{97}{174}\right)$	$e\left(\frac{107}{116}\right)$	$e\left(\frac{209}{348}\right)$	$e\left(\frac{39}{116}\right)$	$e\left(\frac{61}{87}\right)$	$e\left(\frac{337}{348}\right)$	$e\left(\frac{11}{29}\right)$	$e\left(\frac{10}{87}\right)$	$e\left(\frac{71}{87}\right)$
$\chi_{2301}(431,\cdot)$	$-1$	$1$	$e\left(\frac{113}{348}\right)$	$e\left(\frac{113}{174}\right)$	$e\left(\frac{81}{116}\right)$	$e\left(\frac{91}{348}\right)$	$e\left(\frac{113}{116}\right)$	$e\left(\frac{2}{87}\right)$	$e\left(\frac{215}{348}\right)$	$e\left(\frac{17}{29}\right)$	$e\left(\frac{26}{87}\right)$	$e\left(\frac{28}{87}\right)$
$\chi_{2301}(509,\cdot)$	$-1$	$1$	$e\left(\frac{185}{348}\right)$	$e\left(\frac{11}{174}\right)$	$e\left(\frac{109}{116}\right)$	$e\left(\frac{343}{348}\right)$	$e\left(\frac{69}{116}\right)$	$e\left(\frac{41}{87}\right)$	$e\left(\frac{275}{348}\right)$	$e\left(\frac{15}{29}\right)$	$e\left(\frac{11}{87}\right)$	$e\left(\frac{52}{87}\right)$
$\chi_{2301}(527,\cdot)$	$-1$	$1$	$e\left(\frac{331}{348}\right)$	$e\left(\frac{157}{174}\right)$	$e\left(\frac{111}{116}\right)$	$e\left(\frac{245}{348}\right)$	$e\left(\frac{99}{116}\right)$	$e\left(\frac{79}{87}\right)$	$e\left(\frac{97}{348}\right)$	$e\left(\frac{19}{29}\right)$	$e\left(\frac{70}{87}\right)$	$e\left(\frac{62}{87}\right)$
$\chi_{2301}(539,\cdot)$	$-1$	$1$	$e\left(\frac{337}{348}\right)$	$e\left(\frac{163}{174}\right)$	$e\left(\frac{65}{116}\right)$	$e\left(\frac{179}{348}\right)$	$e\left(\frac{105}{116}\right)$	$e\left(\frac{46}{87}\right)$	$e\left(\frac{247}{348}\right)$	$e\left(\frac{14}{29}\right)$	$e\left(\frac{76}{87}\right)$	$e\left(\frac{35}{87}\right)$
$\chi_{2301}(578,\cdot)$	$-1$	$1$	$e\left(\frac{109}{348}\right)$	$e\left(\frac{109}{174}\right)$	$e\left(\frac{73}{116}\right)$	$e\left(\frac{251}{348}\right)$	$e\left(\frac{109}{116}\right)$	$e\left(\frac{82}{87}\right)$	$e\left(\frac{115}{348}\right)$	$e\left(\frac{1}{29}\right)$	$e\left(\frac{22}{87}\right)$	$e\left(\frac{17}{87}\right)$
$\chi_{2301}(587,\cdot)$	$-1$	$1$	$e\left(\frac{329}{348}\right)$	$e\left(\frac{155}{174}\right)$	$e\left(\frac{49}{116}\right)$	$e\left(\frac{151}{348}\right)$	$e\left(\frac{97}{116}\right)$	$e\left(\frac{32}{87}\right)$	$e\left(\frac{47}{348}\right)$	$e\left(\frac{11}{29}\right)$	$e\left(\frac{68}{87}\right)$	$e\left(\frac{13}{87}\right)$
$\chi_{2301}(596,\cdot)$	$-1$	$1$	$e\left(\frac{335}{348}\right)$	$e\left(\frac{161}{174}\right)$	$e\left(\frac{3}{116}\right)$	$e\left(\frac{85}{348}\right)$	$e\left(\frac{103}{116}\right)$	$e\left(\frac{86}{87}\right)$	$e\left(\frac{197}{348}\right)$	$e\left(\frac{6}{29}\right)$	$e\left(\frac{74}{87}\right)$	$e\left(\frac{73}{87}\right)$
$\chi_{2301}(644,\cdot)$	$-1$	$1$	$e\left(\frac{7}{348}\right)$	$e\left(\frac{7}{174}\right)$	$e\left(\frac{43}{116}\right)$	$e\left(\frac{329}{348}\right)$	$e\left(\frac{7}{116}\right)$	$e\left(\frac{34}{87}\right)$	$e\left(\frac{1}{348}\right)$	$e\left(\frac{28}{29}\right)$	$e\left(\frac{7}{87}\right)$	$e\left(\frac{41}{87}\right)$
$\chi_{2301}(683,\cdot)$	$-1$	$1$	$e\left(\frac{43}{348}\right)$	$e\left(\frac{43}{174}\right)$	$e\left(\frac{115}{116}\right)$	$e\left(\frac{281}{348}\right)$	$e\left(\frac{43}{116}\right)$	$e\left(\frac{10}{87}\right)$	$e\left(\frac{205}{348}\right)$	$e\left(\frac{27}{29}\right)$	$e\left(\frac{43}{87}\right)$	$e\left(\frac{53}{87}\right)$
$\chi_{2301}(704,\cdot)$	$-1$	$1$	$e\left(\frac{41}{348}\right)$	$e\left(\frac{41}{174}\right)$	$e\left(\frac{53}{116}\right)$	$e\left(\frac{187}{348}\right)$	$e\left(\frac{41}{116}\right)$	$e\left(\frac{50}{87}\right)$	$e\left(\frac{155}{348}\right)$	$e\left(\frac{19}{29}\right)$	$e\left(\frac{41}{87}\right)$	$e\left(\frac{4}{87}\right)$

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	\(-1\)	\(1\)	\(2\)	\(4\)	\(5\)	\(7\)	\(8\)	\(10\)	\(11\)	\(14\)	\(16\)	\(17\)
\(\chi_{2301}(2,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{209}{348}\right)\)	\(e\left(\frac{35}{174}\right)\)	\(e\left(\frac{41}{116}\right)\)	\(e\left(\frac{79}{348}\right)\)	\(e\left(\frac{93}{116}\right)\)	\(e\left(\frac{83}{87}\right)\)	\(e\left(\frac{179}{348}\right)\)	\(e\left(\frac{24}{29}\right)\)	\(e\left(\frac{35}{87}\right)\)	\(e\left(\frac{31}{87}\right)\)
\(\chi_{2301}(11,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{179}{348}\right)\)	\(e\left(\frac{5}{174}\right)\)	\(e\left(\frac{39}{116}\right)\)	\(e\left(\frac{61}{348}\right)\)	\(e\left(\frac{63}{116}\right)\)	\(e\left(\frac{74}{87}\right)\)	\(e\left(\frac{125}{348}\right)\)	\(e\left(\frac{20}{29}\right)\)	\(e\left(\frac{5}{87}\right)\)	\(e\left(\frac{79}{87}\right)\)
\(\chi_{2301}(32,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{1}{348}\right)\)	\(e\left(\frac{1}{174}\right)\)	\(e\left(\frac{89}{116}\right)\)	\(e\left(\frac{47}{348}\right)\)	\(e\left(\frac{1}{116}\right)\)	\(e\left(\frac{67}{87}\right)\)	\(e\left(\frac{199}{348}\right)\)	\(e\left(\frac{4}{29}\right)\)	\(e\left(\frac{1}{87}\right)\)	\(e\left(\frac{68}{87}\right)\)
\(\chi_{2301}(50,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{107}{348}\right)\)	\(e\left(\frac{107}{174}\right)\)	\(e\left(\frac{11}{116}\right)\)	\(e\left(\frac{157}{348}\right)\)	\(e\left(\frac{107}{116}\right)\)	\(e\left(\frac{35}{87}\right)\)	\(e\left(\frac{65}{348}\right)\)	\(e\left(\frac{22}{29}\right)\)	\(e\left(\frac{20}{87}\right)\)	\(e\left(\frac{55}{87}\right)\)
\(\chi_{2301}(89,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{23}{348}\right)\)	\(e\left(\frac{23}{174}\right)\)	\(e\left(\frac{75}{116}\right)\)	\(e\left(\frac{37}{348}\right)\)	\(e\left(\frac{23}{116}\right)\)	\(e\left(\frac{62}{87}\right)\)	\(e\left(\frac{53}{348}\right)\)	\(e\left(\frac{5}{29}\right)\)	\(e\left(\frac{23}{87}\right)\)	\(e\left(\frac{85}{87}\right)\)
\(\chi_{2301}(98,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{19}{348}\right)\)	\(e\left(\frac{19}{174}\right)\)	\(e\left(\frac{67}{116}\right)\)	\(e\left(\frac{197}{348}\right)\)	\(e\left(\frac{19}{116}\right)\)	\(e\left(\frac{55}{87}\right)\)	\(e\left(\frac{301}{348}\right)\)	\(e\left(\frac{18}{29}\right)\)	\(e\left(\frac{19}{87}\right)\)	\(e\left(\frac{74}{87}\right)\)
\(\chi_{2301}(128,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{71}{348}\right)\)	\(e\left(\frac{71}{174}\right)\)	\(e\left(\frac{55}{116}\right)\)	\(e\left(\frac{205}{348}\right)\)	\(e\left(\frac{71}{116}\right)\)	\(e\left(\frac{59}{87}\right)\)	\(e\left(\frac{209}{348}\right)\)	\(e\left(\frac{23}{29}\right)\)	\(e\left(\frac{71}{87}\right)\)	\(e\left(\frac{43}{87}\right)\)
\(\chi_{2301}(149,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{265}{348}\right)\)	\(e\left(\frac{91}{174}\right)\)	\(e\left(\frac{37}{116}\right)\)	\(e\left(\frac{275}{348}\right)\)	\(e\left(\frac{33}{116}\right)\)	\(e\left(\frac{7}{87}\right)\)	\(e\left(\frac{187}{348}\right)\)	\(e\left(\frac{16}{29}\right)\)	\(e\left(\frac{4}{87}\right)\)	\(e\left(\frac{11}{87}\right)\)
\(\chi_{2301}(158,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{257}{348}\right)\)	\(e\left(\frac{83}{174}\right)\)	\(e\left(\frac{21}{116}\right)\)	\(e\left(\frac{247}{348}\right)\)	\(e\left(\frac{25}{116}\right)\)	\(e\left(\frac{80}{87}\right)\)	\(e\left(\frac{335}{348}\right)\)	\(e\left(\frac{13}{29}\right)\)	\(e\left(\frac{83}{87}\right)\)	\(e\left(\frac{76}{87}\right)\)
\(\chi_{2301}(188,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{121}{348}\right)\)	\(e\left(\frac{121}{174}\right)\)	\(e\left(\frac{97}{116}\right)\)	\(e\left(\frac{119}{348}\right)\)	\(e\left(\frac{5}{116}\right)\)	\(e\left(\frac{16}{87}\right)\)	\(e\left(\frac{67}{348}\right)\)	\(e\left(\frac{20}{29}\right)\)	\(e\left(\frac{34}{87}\right)\)	\(e\left(\frac{50}{87}\right)\)
\(\chi_{2301}(215,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{31}{348}\right)\)	\(e\left(\frac{31}{174}\right)\)	\(e\left(\frac{91}{116}\right)\)	\(e\left(\frac{65}{348}\right)\)	\(e\left(\frac{31}{116}\right)\)	\(e\left(\frac{76}{87}\right)\)	\(e\left(\frac{253}{348}\right)\)	\(e\left(\frac{8}{29}\right)\)	\(e\left(\frac{31}{87}\right)\)	\(e\left(\frac{20}{87}\right)\)
\(\chi_{2301}(227,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{49}{348}\right)\)	\(e\left(\frac{49}{174}\right)\)	\(e\left(\frac{69}{116}\right)\)	\(e\left(\frac{215}{348}\right)\)	\(e\left(\frac{49}{116}\right)\)	\(e\left(\frac{64}{87}\right)\)	\(e\left(\frac{7}{348}\right)\)	\(e\left(\frac{22}{29}\right)\)	\(e\left(\frac{49}{87}\right)\)	\(e\left(\frac{26}{87}\right)\)
\(\chi_{2301}(254,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{55}{348}\right)\)	\(e\left(\frac{55}{174}\right)\)	\(e\left(\frac{23}{116}\right)\)	\(e\left(\frac{149}{348}\right)\)	\(e\left(\frac{55}{116}\right)\)	\(e\left(\frac{31}{87}\right)\)	\(e\left(\frac{157}{348}\right)\)	\(e\left(\frac{17}{29}\right)\)	\(e\left(\frac{55}{87}\right)\)	\(e\left(\frac{86}{87}\right)\)
\(\chi_{2301}(266,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{313}{348}\right)\)	\(e\left(\frac{139}{174}\right)\)	\(e\left(\frac{17}{116}\right)\)	\(e\left(\frac{95}{348}\right)\)	\(e\left(\frac{81}{116}\right)\)	\(e\left(\frac{4}{87}\right)\)	\(e\left(\frac{343}{348}\right)\)	\(e\left(\frac{5}{29}\right)\)	\(e\left(\frac{52}{87}\right)\)	\(e\left(\frac{56}{87}\right)\)
\(\chi_{2301}(275,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{77}{348}\right)\)	\(e\left(\frac{77}{174}\right)\)	\(e\left(\frac{9}{116}\right)\)	\(e\left(\frac{139}{348}\right)\)	\(e\left(\frac{77}{116}\right)\)	\(e\left(\frac{26}{87}\right)\)	\(e\left(\frac{11}{348}\right)\)	\(e\left(\frac{18}{29}\right)\)	\(e\left(\frac{77}{87}\right)\)	\(e\left(\frac{16}{87}\right)\)
\(\chi_{2301}(305,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{13}{348}\right)\)	\(e\left(\frac{13}{174}\right)\)	\(e\left(\frac{113}{116}\right)\)	\(e\left(\frac{263}{348}\right)\)	\(e\left(\frac{13}{116}\right)\)	\(e\left(\frac{1}{87}\right)\)	\(e\left(\frac{151}{348}\right)\)	\(e\left(\frac{23}{29}\right)\)	\(e\left(\frac{13}{87}\right)\)	\(e\left(\frac{14}{87}\right)\)
\(\chi_{2301}(332,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{127}{348}\right)\)	\(e\left(\frac{127}{174}\right)\)	\(e\left(\frac{51}{116}\right)\)	\(e\left(\frac{53}{348}\right)\)	\(e\left(\frac{11}{116}\right)\)	\(e\left(\frac{70}{87}\right)\)	\(e\left(\frac{217}{348}\right)\)	\(e\left(\frac{15}{29}\right)\)	\(e\left(\frac{40}{87}\right)\)	\(e\left(\frac{23}{87}\right)\)
\(\chi_{2301}(362,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{47}{348}\right)\)	\(e\left(\frac{47}{174}\right)\)	\(e\left(\frac{7}{116}\right)\)	\(e\left(\frac{121}{348}\right)\)	\(e\left(\frac{47}{116}\right)\)	\(e\left(\frac{17}{87}\right)\)	\(e\left(\frac{305}{348}\right)\)	\(e\left(\frac{14}{29}\right)\)	\(e\left(\frac{47}{87}\right)\)	\(e\left(\frac{64}{87}\right)\)
\(\chi_{2301}(392,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{89}{348}\right)\)	\(e\left(\frac{89}{174}\right)\)	\(e\left(\frac{33}{116}\right)\)	\(e\left(\frac{7}{348}\right)\)	\(e\left(\frac{89}{116}\right)\)	\(e\left(\frac{47}{87}\right)\)	\(e\left(\frac{311}{348}\right)\)	\(e\left(\frac{8}{29}\right)\)	\(e\left(\frac{2}{87}\right)\)	\(e\left(\frac{49}{87}\right)\)
\(\chi_{2301}(401,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{167}{348}\right)\)	\(e\left(\frac{167}{174}\right)\)	\(e\left(\frac{15}{116}\right)\)	\(e\left(\frac{193}{348}\right)\)	\(e\left(\frac{51}{116}\right)\)	\(e\left(\frac{53}{87}\right)\)	\(e\left(\frac{173}{348}\right)\)	\(e\left(\frac{1}{29}\right)\)	\(e\left(\frac{80}{87}\right)\)	\(e\left(\frac{46}{87}\right)\)
\(\chi_{2301}(410,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{271}{348}\right)\)	\(e\left(\frac{97}{174}\right)\)	\(e\left(\frac{107}{116}\right)\)	\(e\left(\frac{209}{348}\right)\)	\(e\left(\frac{39}{116}\right)\)	\(e\left(\frac{61}{87}\right)\)	\(e\left(\frac{337}{348}\right)\)	\(e\left(\frac{11}{29}\right)\)	\(e\left(\frac{10}{87}\right)\)	\(e\left(\frac{71}{87}\right)\)
\(\chi_{2301}(431,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{113}{348}\right)\)	\(e\left(\frac{113}{174}\right)\)	\(e\left(\frac{81}{116}\right)\)	\(e\left(\frac{91}{348}\right)\)	\(e\left(\frac{113}{116}\right)\)	\(e\left(\frac{2}{87}\right)\)	\(e\left(\frac{215}{348}\right)\)	\(e\left(\frac{17}{29}\right)\)	\(e\left(\frac{26}{87}\right)\)	\(e\left(\frac{28}{87}\right)\)
\(\chi_{2301}(509,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{185}{348}\right)\)	\(e\left(\frac{11}{174}\right)\)	\(e\left(\frac{109}{116}\right)\)	\(e\left(\frac{343}{348}\right)\)	\(e\left(\frac{69}{116}\right)\)	\(e\left(\frac{41}{87}\right)\)	\(e\left(\frac{275}{348}\right)\)	\(e\left(\frac{15}{29}\right)\)	\(e\left(\frac{11}{87}\right)\)	\(e\left(\frac{52}{87}\right)\)
\(\chi_{2301}(527,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{331}{348}\right)\)	\(e\left(\frac{157}{174}\right)\)	\(e\left(\frac{111}{116}\right)\)	\(e\left(\frac{245}{348}\right)\)	\(e\left(\frac{99}{116}\right)\)	\(e\left(\frac{79}{87}\right)\)	\(e\left(\frac{97}{348}\right)\)	\(e\left(\frac{19}{29}\right)\)	\(e\left(\frac{70}{87}\right)\)	\(e\left(\frac{62}{87}\right)\)
\(\chi_{2301}(539,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{337}{348}\right)\)	\(e\left(\frac{163}{174}\right)\)	\(e\left(\frac{65}{116}\right)\)	\(e\left(\frac{179}{348}\right)\)	\(e\left(\frac{105}{116}\right)\)	\(e\left(\frac{46}{87}\right)\)	\(e\left(\frac{247}{348}\right)\)	\(e\left(\frac{14}{29}\right)\)	\(e\left(\frac{76}{87}\right)\)	\(e\left(\frac{35}{87}\right)\)
\(\chi_{2301}(578,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{109}{348}\right)\)	\(e\left(\frac{109}{174}\right)\)	\(e\left(\frac{73}{116}\right)\)	\(e\left(\frac{251}{348}\right)\)	\(e\left(\frac{109}{116}\right)\)	\(e\left(\frac{82}{87}\right)\)	\(e\left(\frac{115}{348}\right)\)	\(e\left(\frac{1}{29}\right)\)	\(e\left(\frac{22}{87}\right)\)	\(e\left(\frac{17}{87}\right)\)
\(\chi_{2301}(587,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{329}{348}\right)\)	\(e\left(\frac{155}{174}\right)\)	\(e\left(\frac{49}{116}\right)\)	\(e\left(\frac{151}{348}\right)\)	\(e\left(\frac{97}{116}\right)\)	\(e\left(\frac{32}{87}\right)\)	\(e\left(\frac{47}{348}\right)\)	\(e\left(\frac{11}{29}\right)\)	\(e\left(\frac{68}{87}\right)\)	\(e\left(\frac{13}{87}\right)\)
\(\chi_{2301}(596,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{335}{348}\right)\)	\(e\left(\frac{161}{174}\right)\)	\(e\left(\frac{3}{116}\right)\)	\(e\left(\frac{85}{348}\right)\)	\(e\left(\frac{103}{116}\right)\)	\(e\left(\frac{86}{87}\right)\)	\(e\left(\frac{197}{348}\right)\)	\(e\left(\frac{6}{29}\right)\)	\(e\left(\frac{74}{87}\right)\)	\(e\left(\frac{73}{87}\right)\)
\(\chi_{2301}(644,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{7}{348}\right)\)	\(e\left(\frac{7}{174}\right)\)	\(e\left(\frac{43}{116}\right)\)	\(e\left(\frac{329}{348}\right)\)	\(e\left(\frac{7}{116}\right)\)	\(e\left(\frac{34}{87}\right)\)	\(e\left(\frac{1}{348}\right)\)	\(e\left(\frac{28}{29}\right)\)	\(e\left(\frac{7}{87}\right)\)	\(e\left(\frac{41}{87}\right)\)
\(\chi_{2301}(683,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{43}{348}\right)\)	\(e\left(\frac{43}{174}\right)\)	\(e\left(\frac{115}{116}\right)\)	\(e\left(\frac{281}{348}\right)\)	\(e\left(\frac{43}{116}\right)\)	\(e\left(\frac{10}{87}\right)\)	\(e\left(\frac{205}{348}\right)\)	\(e\left(\frac{27}{29}\right)\)	\(e\left(\frac{43}{87}\right)\)	\(e\left(\frac{53}{87}\right)\)
\(\chi_{2301}(704,\cdot)\)	\(-1\)	\(1\)	\(e\left(\frac{41}{348}\right)\)	\(e\left(\frac{41}{174}\right)\)	\(e\left(\frac{53}{116}\right)\)	\(e\left(\frac{187}{348}\right)\)	\(e\left(\frac{41}{116}\right)\)	\(e\left(\frac{50}{87}\right)\)	\(e\left(\frac{155}{348}\right)\)	\(e\left(\frac{19}{29}\right)\)	\(e\left(\frac{41}{87}\right)\)	\(e\left(\frac{4}{87}\right)\)

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First 31 of 112 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	2301.bu
Modulus	$2301$
Conductor	$2301$
Order	$348$
Real	no
Primitive	yes
Minimal	yes
Parity	odd

Modulus:	\(2301\)	comment:Modulus sage:chi.modulus() gp:g[1][1] magma:Modulus(chi);
Conductor:	\(2301\)	comment:Conductor sage:chi.conductor() gp:znconreyconductor(g,chi) magma:Conductor(chi);
Order:	\(348\)	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:	yes	comment:If the character is primitive sage:chi.is_primitive() gp:#znconreyconductor(g,chi)==1 magma:IsPrimitive(chi);
Minimal:	yes
Parity:	odd	comment:Parity sage:chi.is_odd() gp:zncharisodd(g,chi) magma:IsOdd(chi);