Properties

Label 517.m
Modulus $517$
Conductor $517$
Order $115$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(517, base_ring=CyclotomicField(230))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([184,100]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(3,517))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(517\)
Conductor: \(517\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(115\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: $\Q(\zeta_{115})$
Fixed field: Number field defined by a degree 115 polynomial (not computed)

First 31 of 88 characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(12\)
\(\chi_{517}(3,\cdot)\) \(1\) \(1\) \(e\left(\frac{72}{115}\right)\) \(e\left(\frac{11}{115}\right)\) \(e\left(\frac{29}{115}\right)\) \(e\left(\frac{73}{115}\right)\) \(e\left(\frac{83}{115}\right)\) \(e\left(\frac{59}{115}\right)\) \(e\left(\frac{101}{115}\right)\) \(e\left(\frac{22}{115}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{8}{23}\right)\)
\(\chi_{517}(4,\cdot)\) \(1\) \(1\) \(e\left(\frac{33}{115}\right)\) \(e\left(\frac{29}{115}\right)\) \(e\left(\frac{66}{115}\right)\) \(e\left(\frac{67}{115}\right)\) \(e\left(\frac{62}{115}\right)\) \(e\left(\frac{51}{115}\right)\) \(e\left(\frac{99}{115}\right)\) \(e\left(\frac{58}{115}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{19}{23}\right)\)
\(\chi_{517}(9,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{115}\right)\) \(e\left(\frac{22}{115}\right)\) \(e\left(\frac{58}{115}\right)\) \(e\left(\frac{31}{115}\right)\) \(e\left(\frac{51}{115}\right)\) \(e\left(\frac{3}{115}\right)\) \(e\left(\frac{87}{115}\right)\) \(e\left(\frac{44}{115}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{16}{23}\right)\)
\(\chi_{517}(14,\cdot)\) \(1\) \(1\) \(e\left(\frac{42}{115}\right)\) \(e\left(\frac{16}{115}\right)\) \(e\left(\frac{84}{115}\right)\) \(e\left(\frac{33}{115}\right)\) \(e\left(\frac{58}{115}\right)\) \(e\left(\frac{44}{115}\right)\) \(e\left(\frac{11}{115}\right)\) \(e\left(\frac{32}{115}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{20}{23}\right)\)
\(\chi_{517}(16,\cdot)\) \(1\) \(1\) \(e\left(\frac{66}{115}\right)\) \(e\left(\frac{58}{115}\right)\) \(e\left(\frac{17}{115}\right)\) \(e\left(\frac{19}{115}\right)\) \(e\left(\frac{9}{115}\right)\) \(e\left(\frac{102}{115}\right)\) \(e\left(\frac{83}{115}\right)\) \(e\left(\frac{1}{115}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{15}{23}\right)\)
\(\chi_{517}(25,\cdot)\) \(1\) \(1\) \(e\left(\frac{67}{115}\right)\) \(e\left(\frac{31}{115}\right)\) \(e\left(\frac{19}{115}\right)\) \(e\left(\frac{28}{115}\right)\) \(e\left(\frac{98}{115}\right)\) \(e\left(\frac{114}{115}\right)\) \(e\left(\frac{86}{115}\right)\) \(e\left(\frac{62}{115}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{10}{23}\right)\)
\(\chi_{517}(27,\cdot)\) \(1\) \(1\) \(e\left(\frac{101}{115}\right)\) \(e\left(\frac{33}{115}\right)\) \(e\left(\frac{87}{115}\right)\) \(e\left(\frac{104}{115}\right)\) \(e\left(\frac{19}{115}\right)\) \(e\left(\frac{62}{115}\right)\) \(e\left(\frac{73}{115}\right)\) \(e\left(\frac{66}{115}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{1}{23}\right)\)
\(\chi_{517}(36,\cdot)\) \(1\) \(1\) \(e\left(\frac{62}{115}\right)\) \(e\left(\frac{51}{115}\right)\) \(e\left(\frac{9}{115}\right)\) \(e\left(\frac{98}{115}\right)\) \(e\left(\frac{113}{115}\right)\) \(e\left(\frac{54}{115}\right)\) \(e\left(\frac{71}{115}\right)\) \(e\left(\frac{102}{115}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{12}{23}\right)\)
\(\chi_{517}(37,\cdot)\) \(1\) \(1\) \(e\left(\frac{73}{115}\right)\) \(e\left(\frac{99}{115}\right)\) \(e\left(\frac{31}{115}\right)\) \(e\left(\frac{82}{115}\right)\) \(e\left(\frac{57}{115}\right)\) \(e\left(\frac{71}{115}\right)\) \(e\left(\frac{104}{115}\right)\) \(e\left(\frac{83}{115}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{3}{23}\right)\)
\(\chi_{517}(42,\cdot)\) \(1\) \(1\) \(e\left(\frac{114}{115}\right)\) \(e\left(\frac{27}{115}\right)\) \(e\left(\frac{113}{115}\right)\) \(e\left(\frac{106}{115}\right)\) \(e\left(\frac{26}{115}\right)\) \(e\left(\frac{103}{115}\right)\) \(e\left(\frac{112}{115}\right)\) \(e\left(\frac{54}{115}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{5}{23}\right)\)
\(\chi_{517}(49,\cdot)\) \(1\) \(1\) \(e\left(\frac{51}{115}\right)\) \(e\left(\frac{3}{115}\right)\) \(e\left(\frac{102}{115}\right)\) \(e\left(\frac{114}{115}\right)\) \(e\left(\frac{54}{115}\right)\) \(e\left(\frac{37}{115}\right)\) \(e\left(\frac{38}{115}\right)\) \(e\left(\frac{6}{115}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{21}{23}\right)\)
\(\chi_{517}(53,\cdot)\) \(1\) \(1\) \(e\left(\frac{54}{115}\right)\) \(e\left(\frac{37}{115}\right)\) \(e\left(\frac{108}{115}\right)\) \(e\left(\frac{26}{115}\right)\) \(e\left(\frac{91}{115}\right)\) \(e\left(\frac{73}{115}\right)\) \(e\left(\frac{47}{115}\right)\) \(e\left(\frac{74}{115}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{6}{23}\right)\)
\(\chi_{517}(59,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{115}\right)\) \(e\left(\frac{109}{115}\right)\) \(e\left(\frac{26}{115}\right)\) \(e\left(\frac{2}{115}\right)\) \(e\left(\frac{7}{115}\right)\) \(e\left(\frac{41}{115}\right)\) \(e\left(\frac{39}{115}\right)\) \(e\left(\frac{103}{115}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{4}{23}\right)\)
\(\chi_{517}(64,\cdot)\) \(1\) \(1\) \(e\left(\frac{99}{115}\right)\) \(e\left(\frac{87}{115}\right)\) \(e\left(\frac{83}{115}\right)\) \(e\left(\frac{86}{115}\right)\) \(e\left(\frac{71}{115}\right)\) \(e\left(\frac{38}{115}\right)\) \(e\left(\frac{67}{115}\right)\) \(e\left(\frac{59}{115}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{11}{23}\right)\)
\(\chi_{517}(71,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{115}\right)\) \(e\left(\frac{43}{115}\right)\) \(e\left(\frac{82}{115}\right)\) \(e\left(\frac{24}{115}\right)\) \(e\left(\frac{84}{115}\right)\) \(e\left(\frac{32}{115}\right)\) \(e\left(\frac{8}{115}\right)\) \(e\left(\frac{86}{115}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{2}{23}\right)\)
\(\chi_{517}(75,\cdot)\) \(1\) \(1\) \(e\left(\frac{24}{115}\right)\) \(e\left(\frac{42}{115}\right)\) \(e\left(\frac{48}{115}\right)\) \(e\left(\frac{101}{115}\right)\) \(e\left(\frac{66}{115}\right)\) \(e\left(\frac{58}{115}\right)\) \(e\left(\frac{72}{115}\right)\) \(e\left(\frac{84}{115}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{18}{23}\right)\)
\(\chi_{517}(81,\cdot)\) \(1\) \(1\) \(e\left(\frac{58}{115}\right)\) \(e\left(\frac{44}{115}\right)\) \(e\left(\frac{1}{115}\right)\) \(e\left(\frac{62}{115}\right)\) \(e\left(\frac{102}{115}\right)\) \(e\left(\frac{6}{115}\right)\) \(e\left(\frac{59}{115}\right)\) \(e\left(\frac{88}{115}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{9}{23}\right)\)
\(\chi_{517}(97,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{115}\right)\) \(e\left(\frac{57}{115}\right)\) \(e\left(\frac{98}{115}\right)\) \(e\left(\frac{96}{115}\right)\) \(e\left(\frac{106}{115}\right)\) \(e\left(\frac{13}{115}\right)\) \(e\left(\frac{32}{115}\right)\) \(e\left(\frac{114}{115}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{8}{23}\right)\)
\(\chi_{517}(102,\cdot)\) \(1\) \(1\) \(e\left(\frac{107}{115}\right)\) \(e\left(\frac{101}{115}\right)\) \(e\left(\frac{99}{115}\right)\) \(e\left(\frac{43}{115}\right)\) \(e\left(\frac{93}{115}\right)\) \(e\left(\frac{19}{115}\right)\) \(e\left(\frac{91}{115}\right)\) \(e\left(\frac{87}{115}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{17}{23}\right)\)
\(\chi_{517}(103,\cdot)\) \(1\) \(1\) \(e\left(\frac{98}{115}\right)\) \(e\left(\frac{114}{115}\right)\) \(e\left(\frac{81}{115}\right)\) \(e\left(\frac{77}{115}\right)\) \(e\left(\frac{97}{115}\right)\) \(e\left(\frac{26}{115}\right)\) \(e\left(\frac{64}{115}\right)\) \(e\left(\frac{113}{115}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{16}{23}\right)\)
\(\chi_{517}(108,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{115}\right)\) \(e\left(\frac{62}{115}\right)\) \(e\left(\frac{38}{115}\right)\) \(e\left(\frac{56}{115}\right)\) \(e\left(\frac{81}{115}\right)\) \(e\left(\frac{113}{115}\right)\) \(e\left(\frac{57}{115}\right)\) \(e\left(\frac{9}{115}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{20}{23}\right)\)
\(\chi_{517}(115,\cdot)\) \(1\) \(1\) \(e\left(\frac{86}{115}\right)\) \(e\left(\frac{93}{115}\right)\) \(e\left(\frac{57}{115}\right)\) \(e\left(\frac{84}{115}\right)\) \(e\left(\frac{64}{115}\right)\) \(e\left(\frac{112}{115}\right)\) \(e\left(\frac{28}{115}\right)\) \(e\left(\frac{71}{115}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{7}{23}\right)\)
\(\chi_{517}(119,\cdot)\) \(1\) \(1\) \(e\left(\frac{44}{115}\right)\) \(e\left(\frac{77}{115}\right)\) \(e\left(\frac{88}{115}\right)\) \(e\left(\frac{51}{115}\right)\) \(e\left(\frac{6}{115}\right)\) \(e\left(\frac{68}{115}\right)\) \(e\left(\frac{17}{115}\right)\) \(e\left(\frac{39}{115}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{10}{23}\right)\)
\(\chi_{517}(126,\cdot)\) \(1\) \(1\) \(e\left(\frac{71}{115}\right)\) \(e\left(\frac{38}{115}\right)\) \(e\left(\frac{27}{115}\right)\) \(e\left(\frac{64}{115}\right)\) \(e\left(\frac{109}{115}\right)\) \(e\left(\frac{47}{115}\right)\) \(e\left(\frac{98}{115}\right)\) \(e\left(\frac{76}{115}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{13}{23}\right)\)
\(\chi_{517}(130,\cdot)\) \(1\) \(1\) \(e\left(\frac{39}{115}\right)\) \(e\left(\frac{97}{115}\right)\) \(e\left(\frac{78}{115}\right)\) \(e\left(\frac{6}{115}\right)\) \(e\left(\frac{21}{115}\right)\) \(e\left(\frac{8}{115}\right)\) \(e\left(\frac{2}{115}\right)\) \(e\left(\frac{79}{115}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{12}{23}\right)\)
\(\chi_{517}(136,\cdot)\) \(1\) \(1\) \(e\left(\frac{68}{115}\right)\) \(e\left(\frac{4}{115}\right)\) \(e\left(\frac{21}{115}\right)\) \(e\left(\frac{37}{115}\right)\) \(e\left(\frac{72}{115}\right)\) \(e\left(\frac{11}{115}\right)\) \(e\left(\frac{89}{115}\right)\) \(e\left(\frac{8}{115}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{5}{23}\right)\)
\(\chi_{517}(147,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{115}\right)\) \(e\left(\frac{14}{115}\right)\) \(e\left(\frac{16}{115}\right)\) \(e\left(\frac{72}{115}\right)\) \(e\left(\frac{22}{115}\right)\) \(e\left(\frac{96}{115}\right)\) \(e\left(\frac{24}{115}\right)\) \(e\left(\frac{28}{115}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{6}{23}\right)\)
\(\chi_{517}(148,\cdot)\) \(1\) \(1\) \(e\left(\frac{106}{115}\right)\) \(e\left(\frac{13}{115}\right)\) \(e\left(\frac{97}{115}\right)\) \(e\left(\frac{34}{115}\right)\) \(e\left(\frac{4}{115}\right)\) \(e\left(\frac{7}{115}\right)\) \(e\left(\frac{88}{115}\right)\) \(e\left(\frac{26}{115}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{22}{23}\right)\)
\(\chi_{517}(157,\cdot)\) \(1\) \(1\) \(e\left(\frac{112}{115}\right)\) \(e\left(\frac{81}{115}\right)\) \(e\left(\frac{109}{115}\right)\) \(e\left(\frac{88}{115}\right)\) \(e\left(\frac{78}{115}\right)\) \(e\left(\frac{79}{115}\right)\) \(e\left(\frac{106}{115}\right)\) \(e\left(\frac{47}{115}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{15}{23}\right)\)
\(\chi_{517}(158,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{115}\right)\) \(e\left(\frac{64}{115}\right)\) \(e\left(\frac{106}{115}\right)\) \(e\left(\frac{17}{115}\right)\) \(e\left(\frac{2}{115}\right)\) \(e\left(\frac{61}{115}\right)\) \(e\left(\frac{44}{115}\right)\) \(e\left(\frac{13}{115}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{11}{23}\right)\)
\(\chi_{517}(159,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{115}\right)\) \(e\left(\frac{48}{115}\right)\) \(e\left(\frac{22}{115}\right)\) \(e\left(\frac{99}{115}\right)\) \(e\left(\frac{59}{115}\right)\) \(e\left(\frac{17}{115}\right)\) \(e\left(\frac{33}{115}\right)\) \(e\left(\frac{96}{115}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{14}{23}\right)\)