Properties

Label 517.o
Modulus $517$
Conductor $517$
Order $230$
Real no
Primitive yes
Minimal yes
Parity odd

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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([23,90]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(2,517))
 
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: \(230\)
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: $\Q(\zeta_{115})$
Fixed field: Number field defined by a degree 230 polynomial (not computed)

First 31 of 88 characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(12\)
\(\chi_{517}(2,\cdot)\) \(-1\) \(1\) \(e\left(\frac{33}{230}\right)\) \(e\left(\frac{72}{115}\right)\) \(e\left(\frac{33}{115}\right)\) \(e\left(\frac{91}{115}\right)\) \(e\left(\frac{177}{230}\right)\) \(e\left(\frac{51}{230}\right)\) \(e\left(\frac{99}{230}\right)\) \(e\left(\frac{29}{115}\right)\) \(e\left(\frac{43}{46}\right)\) \(e\left(\frac{21}{23}\right)\)
\(\chi_{517}(6,\cdot)\) \(-1\) \(1\) \(e\left(\frac{177}{230}\right)\) \(e\left(\frac{83}{115}\right)\) \(e\left(\frac{62}{115}\right)\) \(e\left(\frac{49}{115}\right)\) \(e\left(\frac{113}{230}\right)\) \(e\left(\frac{169}{230}\right)\) \(e\left(\frac{71}{230}\right)\) \(e\left(\frac{51}{115}\right)\) \(e\left(\frac{9}{46}\right)\) \(e\left(\frac{6}{23}\right)\)
\(\chi_{517}(7,\cdot)\) \(-1\) \(1\) \(e\left(\frac{51}{230}\right)\) \(e\left(\frac{59}{115}\right)\) \(e\left(\frac{51}{115}\right)\) \(e\left(\frac{57}{115}\right)\) \(e\left(\frac{169}{230}\right)\) \(e\left(\frac{37}{230}\right)\) \(e\left(\frac{153}{230}\right)\) \(e\left(\frac{3}{115}\right)\) \(e\left(\frac{33}{46}\right)\) \(e\left(\frac{22}{23}\right)\)
\(\chi_{517}(8,\cdot)\) \(-1\) \(1\) \(e\left(\frac{99}{230}\right)\) \(e\left(\frac{101}{115}\right)\) \(e\left(\frac{99}{115}\right)\) \(e\left(\frac{43}{115}\right)\) \(e\left(\frac{71}{230}\right)\) \(e\left(\frac{153}{230}\right)\) \(e\left(\frac{67}{230}\right)\) \(e\left(\frac{87}{115}\right)\) \(e\left(\frac{37}{46}\right)\) \(e\left(\frac{17}{23}\right)\)
\(\chi_{517}(17,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{230}\right)\) \(e\left(\frac{18}{115}\right)\) \(e\left(\frac{37}{115}\right)\) \(e\left(\frac{109}{115}\right)\) \(e\left(\frac{73}{230}\right)\) \(e\left(\frac{99}{230}\right)\) \(e\left(\frac{111}{230}\right)\) \(e\left(\frac{36}{115}\right)\) \(e\left(\frac{5}{46}\right)\) \(e\left(\frac{11}{23}\right)\)
\(\chi_{517}(18,\cdot)\) \(-1\) \(1\) \(e\left(\frac{91}{230}\right)\) \(e\left(\frac{94}{115}\right)\) \(e\left(\frac{91}{115}\right)\) \(e\left(\frac{7}{115}\right)\) \(e\left(\frac{49}{230}\right)\) \(e\left(\frac{57}{230}\right)\) \(e\left(\frac{43}{230}\right)\) \(e\left(\frac{73}{115}\right)\) \(e\left(\frac{21}{46}\right)\) \(e\left(\frac{14}{23}\right)\)
\(\chi_{517}(24,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{230}\right)\) \(e\left(\frac{112}{115}\right)\) \(e\left(\frac{13}{115}\right)\) \(e\left(\frac{1}{115}\right)\) \(e\left(\frac{7}{230}\right)\) \(e\left(\frac{41}{230}\right)\) \(e\left(\frac{39}{230}\right)\) \(e\left(\frac{109}{115}\right)\) \(e\left(\frac{3}{46}\right)\) \(e\left(\frac{2}{23}\right)\)
\(\chi_{517}(28,\cdot)\) \(-1\) \(1\) \(e\left(\frac{117}{230}\right)\) \(e\left(\frac{88}{115}\right)\) \(e\left(\frac{2}{115}\right)\) \(e\left(\frac{9}{115}\right)\) \(e\left(\frac{63}{230}\right)\) \(e\left(\frac{139}{230}\right)\) \(e\left(\frac{121}{230}\right)\) \(e\left(\frac{61}{115}\right)\) \(e\left(\frac{27}{46}\right)\) \(e\left(\frac{18}{23}\right)\)
\(\chi_{517}(50,\cdot)\) \(-1\) \(1\) \(e\left(\frac{167}{230}\right)\) \(e\left(\frac{103}{115}\right)\) \(e\left(\frac{52}{115}\right)\) \(e\left(\frac{4}{115}\right)\) \(e\left(\frac{143}{230}\right)\) \(e\left(\frac{49}{230}\right)\) \(e\left(\frac{41}{230}\right)\) \(e\left(\frac{91}{115}\right)\) \(e\left(\frac{35}{46}\right)\) \(e\left(\frac{8}{23}\right)\)
\(\chi_{517}(51,\cdot)\) \(-1\) \(1\) \(e\left(\frac{181}{230}\right)\) \(e\left(\frac{29}{115}\right)\) \(e\left(\frac{66}{115}\right)\) \(e\left(\frac{67}{115}\right)\) \(e\left(\frac{9}{230}\right)\) \(e\left(\frac{217}{230}\right)\) \(e\left(\frac{83}{230}\right)\) \(e\left(\frac{58}{115}\right)\) \(e\left(\frac{17}{46}\right)\) \(e\left(\frac{19}{23}\right)\)
\(\chi_{517}(61,\cdot)\) \(-1\) \(1\) \(e\left(\frac{107}{230}\right)\) \(e\left(\frac{108}{115}\right)\) \(e\left(\frac{107}{115}\right)\) \(e\left(\frac{79}{115}\right)\) \(e\left(\frac{93}{230}\right)\) \(e\left(\frac{19}{230}\right)\) \(e\left(\frac{91}{230}\right)\) \(e\left(\frac{101}{115}\right)\) \(e\left(\frac{7}{46}\right)\) \(e\left(\frac{20}{23}\right)\)
\(\chi_{517}(63,\cdot)\) \(-1\) \(1\) \(e\left(\frac{109}{230}\right)\) \(e\left(\frac{81}{115}\right)\) \(e\left(\frac{109}{115}\right)\) \(e\left(\frac{88}{115}\right)\) \(e\left(\frac{41}{230}\right)\) \(e\left(\frac{43}{230}\right)\) \(e\left(\frac{97}{230}\right)\) \(e\left(\frac{47}{115}\right)\) \(e\left(\frac{11}{46}\right)\) \(e\left(\frac{15}{23}\right)\)
\(\chi_{517}(68,\cdot)\) \(-1\) \(1\) \(e\left(\frac{103}{230}\right)\) \(e\left(\frac{47}{115}\right)\) \(e\left(\frac{103}{115}\right)\) \(e\left(\frac{61}{115}\right)\) \(e\left(\frac{197}{230}\right)\) \(e\left(\frac{201}{230}\right)\) \(e\left(\frac{79}{230}\right)\) \(e\left(\frac{94}{115}\right)\) \(e\left(\frac{45}{46}\right)\) \(e\left(\frac{7}{23}\right)\)
\(\chi_{517}(72,\cdot)\) \(-1\) \(1\) \(e\left(\frac{157}{230}\right)\) \(e\left(\frac{8}{115}\right)\) \(e\left(\frac{42}{115}\right)\) \(e\left(\frac{74}{115}\right)\) \(e\left(\frac{173}{230}\right)\) \(e\left(\frac{159}{230}\right)\) \(e\left(\frac{11}{230}\right)\) \(e\left(\frac{16}{115}\right)\) \(e\left(\frac{15}{46}\right)\) \(e\left(\frac{10}{23}\right)\)
\(\chi_{517}(74,\cdot)\) \(-1\) \(1\) \(e\left(\frac{179}{230}\right)\) \(e\left(\frac{56}{115}\right)\) \(e\left(\frac{64}{115}\right)\) \(e\left(\frac{58}{115}\right)\) \(e\left(\frac{61}{230}\right)\) \(e\left(\frac{193}{230}\right)\) \(e\left(\frac{77}{230}\right)\) \(e\left(\frac{112}{115}\right)\) \(e\left(\frac{13}{46}\right)\) \(e\left(\frac{1}{23}\right)\)
\(\chi_{517}(79,\cdot)\) \(-1\) \(1\) \(e\left(\frac{73}{230}\right)\) \(e\left(\frac{107}{115}\right)\) \(e\left(\frac{73}{115}\right)\) \(e\left(\frac{41}{115}\right)\) \(e\left(\frac{57}{230}\right)\) \(e\left(\frac{71}{230}\right)\) \(e\left(\frac{219}{230}\right)\) \(e\left(\frac{99}{115}\right)\) \(e\left(\frac{31}{46}\right)\) \(e\left(\frac{13}{23}\right)\)
\(\chi_{517}(83,\cdot)\) \(-1\) \(1\) \(e\left(\frac{147}{230}\right)\) \(e\left(\frac{28}{115}\right)\) \(e\left(\frac{32}{115}\right)\) \(e\left(\frac{29}{115}\right)\) \(e\left(\frac{203}{230}\right)\) \(e\left(\frac{39}{230}\right)\) \(e\left(\frac{211}{230}\right)\) \(e\left(\frac{56}{115}\right)\) \(e\left(\frac{41}{46}\right)\) \(e\left(\frac{12}{23}\right)\)
\(\chi_{517}(84,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{230}\right)\) \(e\left(\frac{99}{115}\right)\) \(e\left(\frac{31}{115}\right)\) \(e\left(\frac{82}{115}\right)\) \(e\left(\frac{229}{230}\right)\) \(e\left(\frac{27}{230}\right)\) \(e\left(\frac{93}{230}\right)\) \(e\left(\frac{83}{115}\right)\) \(e\left(\frac{39}{46}\right)\) \(e\left(\frac{3}{23}\right)\)
\(\chi_{517}(96,\cdot)\) \(-1\) \(1\) \(e\left(\frac{79}{230}\right)\) \(e\left(\frac{26}{115}\right)\) \(e\left(\frac{79}{115}\right)\) \(e\left(\frac{68}{115}\right)\) \(e\left(\frac{131}{230}\right)\) \(e\left(\frac{143}{230}\right)\) \(e\left(\frac{7}{230}\right)\) \(e\left(\frac{52}{115}\right)\) \(e\left(\frac{43}{46}\right)\) \(e\left(\frac{21}{23}\right)\)
\(\chi_{517}(101,\cdot)\) \(-1\) \(1\) \(e\left(\frac{143}{230}\right)\) \(e\left(\frac{82}{115}\right)\) \(e\left(\frac{28}{115}\right)\) \(e\left(\frac{11}{115}\right)\) \(e\left(\frac{77}{230}\right)\) \(e\left(\frac{221}{230}\right)\) \(e\left(\frac{199}{230}\right)\) \(e\left(\frac{49}{115}\right)\) \(e\left(\frac{33}{46}\right)\) \(e\left(\frac{22}{23}\right)\)
\(\chi_{517}(106,\cdot)\) \(-1\) \(1\) \(e\left(\frac{141}{230}\right)\) \(e\left(\frac{109}{115}\right)\) \(e\left(\frac{26}{115}\right)\) \(e\left(\frac{2}{115}\right)\) \(e\left(\frac{129}{230}\right)\) \(e\left(\frac{197}{230}\right)\) \(e\left(\frac{193}{230}\right)\) \(e\left(\frac{103}{115}\right)\) \(e\left(\frac{29}{46}\right)\) \(e\left(\frac{4}{23}\right)\)
\(\chi_{517}(112,\cdot)\) \(-1\) \(1\) \(e\left(\frac{183}{230}\right)\) \(e\left(\frac{2}{115}\right)\) \(e\left(\frac{68}{115}\right)\) \(e\left(\frac{76}{115}\right)\) \(e\left(\frac{187}{230}\right)\) \(e\left(\frac{11}{230}\right)\) \(e\left(\frac{89}{230}\right)\) \(e\left(\frac{4}{115}\right)\) \(e\left(\frac{21}{46}\right)\) \(e\left(\frac{14}{23}\right)\)
\(\chi_{517}(118,\cdot)\) \(-1\) \(1\) \(e\left(\frac{59}{230}\right)\) \(e\left(\frac{66}{115}\right)\) \(e\left(\frac{59}{115}\right)\) \(e\left(\frac{93}{115}\right)\) \(e\left(\frac{191}{230}\right)\) \(e\left(\frac{133}{230}\right)\) \(e\left(\frac{177}{230}\right)\) \(e\left(\frac{17}{115}\right)\) \(e\left(\frac{3}{46}\right)\) \(e\left(\frac{2}{23}\right)\)
\(\chi_{517}(128,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{230}\right)\) \(e\left(\frac{44}{115}\right)\) \(e\left(\frac{1}{115}\right)\) \(e\left(\frac{62}{115}\right)\) \(e\left(\frac{89}{230}\right)\) \(e\left(\frac{127}{230}\right)\) \(e\left(\frac{3}{230}\right)\) \(e\left(\frac{88}{115}\right)\) \(e\left(\frac{25}{46}\right)\) \(e\left(\frac{9}{23}\right)\)
\(\chi_{517}(145,\cdot)\) \(-1\) \(1\) \(e\left(\frac{43}{230}\right)\) \(e\left(\frac{52}{115}\right)\) \(e\left(\frac{43}{115}\right)\) \(e\left(\frac{21}{115}\right)\) \(e\left(\frac{147}{230}\right)\) \(e\left(\frac{171}{230}\right)\) \(e\left(\frac{129}{230}\right)\) \(e\left(\frac{104}{115}\right)\) \(e\left(\frac{17}{46}\right)\) \(e\left(\frac{19}{23}\right)\)
\(\chi_{517}(149,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{230}\right)\) \(e\left(\frac{78}{115}\right)\) \(e\left(\frac{7}{115}\right)\) \(e\left(\frac{89}{115}\right)\) \(e\left(\frac{163}{230}\right)\) \(e\left(\frac{199}{230}\right)\) \(e\left(\frac{21}{230}\right)\) \(e\left(\frac{41}{115}\right)\) \(e\left(\frac{37}{46}\right)\) \(e\left(\frac{17}{23}\right)\)
\(\chi_{517}(150,\cdot)\) \(-1\) \(1\) \(e\left(\frac{81}{230}\right)\) \(e\left(\frac{114}{115}\right)\) \(e\left(\frac{81}{115}\right)\) \(e\left(\frac{77}{115}\right)\) \(e\left(\frac{79}{230}\right)\) \(e\left(\frac{167}{230}\right)\) \(e\left(\frac{13}{230}\right)\) \(e\left(\frac{113}{115}\right)\) \(e\left(\frac{1}{46}\right)\) \(e\left(\frac{16}{23}\right)\)
\(\chi_{517}(162,\cdot)\) \(-1\) \(1\) \(e\left(\frac{149}{230}\right)\) \(e\left(\frac{1}{115}\right)\) \(e\left(\frac{34}{115}\right)\) \(e\left(\frac{38}{115}\right)\) \(e\left(\frac{151}{230}\right)\) \(e\left(\frac{63}{230}\right)\) \(e\left(\frac{217}{230}\right)\) \(e\left(\frac{2}{115}\right)\) \(e\left(\frac{45}{46}\right)\) \(e\left(\frac{7}{23}\right)\)
\(\chi_{517}(173,\cdot)\) \(-1\) \(1\) \(e\left(\frac{119}{230}\right)\) \(e\left(\frac{61}{115}\right)\) \(e\left(\frac{4}{115}\right)\) \(e\left(\frac{18}{115}\right)\) \(e\left(\frac{11}{230}\right)\) \(e\left(\frac{163}{230}\right)\) \(e\left(\frac{127}{230}\right)\) \(e\left(\frac{7}{115}\right)\) \(e\left(\frac{31}{46}\right)\) \(e\left(\frac{13}{23}\right)\)
\(\chi_{517}(178,\cdot)\) \(-1\) \(1\) \(e\left(\frac{123}{230}\right)\) \(e\left(\frac{7}{115}\right)\) \(e\left(\frac{8}{115}\right)\) \(e\left(\frac{36}{115}\right)\) \(e\left(\frac{137}{230}\right)\) \(e\left(\frac{211}{230}\right)\) \(e\left(\frac{139}{230}\right)\) \(e\left(\frac{14}{115}\right)\) \(e\left(\frac{39}{46}\right)\) \(e\left(\frac{3}{23}\right)\)
\(\chi_{517}(183,\cdot)\) \(-1\) \(1\) \(e\left(\frac{21}{230}\right)\) \(e\left(\frac{4}{115}\right)\) \(e\left(\frac{21}{115}\right)\) \(e\left(\frac{37}{115}\right)\) \(e\left(\frac{29}{230}\right)\) \(e\left(\frac{137}{230}\right)\) \(e\left(\frac{63}{230}\right)\) \(e\left(\frac{8}{115}\right)\) \(e\left(\frac{19}{46}\right)\) \(e\left(\frac{5}{23}\right)\)