from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1003, base_ring=CyclotomicField(232))
M = H._module
chi = DirichletCharacter(H, M([203,4]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1003))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1003\) | |
| Conductor: | \(1003\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(232\) | 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: | $\Q(\zeta_{232})$ |
| Fixed field: | Number field defined by a degree 232 polynomial (not computed) |
First 31 of 112 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1003}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{116}\right)\) | \(e\left(\frac{171}{232}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{111}{232}\right)\) | \(e\left(\frac{1}{232}\right)\) | \(e\left(\frac{217}{232}\right)\) | \(e\left(\frac{93}{116}\right)\) | \(e\left(\frac{55}{116}\right)\) | \(e\left(\frac{173}{232}\right)\) | \(e\left(\frac{129}{232}\right)\) |
| \(\chi_{1003}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{93}{116}\right)\) | \(e\left(\frac{49}{232}\right)\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{101}{232}\right)\) | \(e\left(\frac{3}{232}\right)\) | \(e\left(\frac{187}{232}\right)\) | \(e\left(\frac{47}{116}\right)\) | \(e\left(\frac{49}{116}\right)\) | \(e\left(\frac{55}{232}\right)\) | \(e\left(\frac{155}{232}\right)\) |
| \(\chi_{1003}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{116}\right)\) | \(e\left(\frac{159}{232}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{91}{232}\right)\) | \(e\left(\frac{5}{232}\right)\) | \(e\left(\frac{157}{232}\right)\) | \(e\left(\frac{1}{116}\right)\) | \(e\left(\frac{43}{116}\right)\) | \(e\left(\frac{169}{232}\right)\) | \(e\left(\frac{181}{232}\right)\) |
| \(\chi_{1003}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{116}\right)\) | \(e\left(\frac{25}{232}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{61}{232}\right)\) | \(e\left(\frac{11}{232}\right)\) | \(e\left(\frac{67}{232}\right)\) | \(e\left(\frac{95}{116}\right)\) | \(e\left(\frac{25}{116}\right)\) | \(e\left(\frac{47}{232}\right)\) | \(e\left(\frac{27}{232}\right)\) |
| \(\chi_{1003}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{116}\right)\) | \(e\left(\frac{133}{232}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{9}{232}\right)\) | \(e\left(\frac{207}{232}\right)\) | \(e\left(\frac{143}{232}\right)\) | \(e\left(\frac{111}{116}\right)\) | \(e\left(\frac{17}{116}\right)\) | \(e\left(\frac{83}{232}\right)\) | \(e\left(\frac{23}{232}\right)\) |
| \(\chi_{1003}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{116}\right)\) | \(e\left(\frac{99}{232}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{223}{232}\right)\) | \(e\left(\frac{25}{232}\right)\) | \(e\left(\frac{89}{232}\right)\) | \(e\left(\frac{5}{116}\right)\) | \(e\left(\frac{99}{116}\right)\) | \(e\left(\frac{149}{232}\right)\) | \(e\left(\frac{209}{232}\right)\) |
| \(\chi_{1003}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{116}\right)\) | \(e\left(\frac{45}{232}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{17}{232}\right)\) | \(e\left(\frac{159}{232}\right)\) | \(e\left(\frac{167}{232}\right)\) | \(e\left(\frac{55}{116}\right)\) | \(e\left(\frac{45}{116}\right)\) | \(e\left(\frac{131}{232}\right)\) | \(e\left(\frac{95}{232}\right)\) |
| \(\chi_{1003}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{116}\right)\) | \(e\left(\frac{15}{232}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{83}{232}\right)\) | \(e\left(\frac{53}{232}\right)\) | \(e\left(\frac{133}{232}\right)\) | \(e\left(\frac{57}{116}\right)\) | \(e\left(\frac{15}{116}\right)\) | \(e\left(\frac{121}{232}\right)\) | \(e\left(\frac{109}{232}\right)\) |
| \(\chi_{1003}(93,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{116}\right)\) | \(e\left(\frac{225}{232}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{85}{232}\right)\) | \(e\left(\frac{99}{232}\right)\) | \(e\left(\frac{139}{232}\right)\) | \(e\left(\frac{43}{116}\right)\) | \(e\left(\frac{109}{116}\right)\) | \(e\left(\frac{191}{232}\right)\) | \(e\left(\frac{11}{232}\right)\) |
| \(\chi_{1003}(111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{116}\right)\) | \(e\left(\frac{149}{232}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{113}{232}\right)\) | \(e\left(\frac{47}{232}\right)\) | \(e\left(\frac{223}{232}\right)\) | \(e\left(\frac{79}{116}\right)\) | \(e\left(\frac{33}{116}\right)\) | \(e\left(\frac{11}{232}\right)\) | \(e\left(\frac{31}{232}\right)\) |
| \(\chi_{1003}(128,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{116}\right)\) | \(e\left(\frac{37}{232}\right)\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{81}{232}\right)\) | \(e\left(\frac{7}{232}\right)\) | \(e\left(\frac{127}{232}\right)\) | \(e\left(\frac{71}{116}\right)\) | \(e\left(\frac{37}{116}\right)\) | \(e\left(\frac{51}{232}\right)\) | \(e\left(\frac{207}{232}\right)\) |
| \(\chi_{1003}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{116}\right)\) | \(e\left(\frac{7}{232}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{147}{232}\right)\) | \(e\left(\frac{133}{232}\right)\) | \(e\left(\frac{93}{232}\right)\) | \(e\left(\frac{73}{116}\right)\) | \(e\left(\frac{7}{116}\right)\) | \(e\left(\frac{41}{232}\right)\) | \(e\left(\frac{221}{232}\right)\) |
| \(\chi_{1003}(155,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{116}\right)\) | \(e\left(\frac{67}{232}\right)\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{15}{232}\right)\) | \(e\left(\frac{113}{232}\right)\) | \(e\left(\frac{161}{232}\right)\) | \(e\left(\frac{69}{116}\right)\) | \(e\left(\frac{67}{116}\right)\) | \(e\left(\frac{61}{232}\right)\) | \(e\left(\frac{193}{232}\right)\) |
| \(\chi_{1003}(161,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{116}\right)\) | \(e\left(\frac{17}{232}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{125}{232}\right)\) | \(e\left(\frac{91}{232}\right)\) | \(e\left(\frac{27}{232}\right)\) | \(e\left(\frac{111}{116}\right)\) | \(e\left(\frac{17}{116}\right)\) | \(e\left(\frac{199}{232}\right)\) | \(e\left(\frac{139}{232}\right)\) |
| \(\chi_{1003}(162,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{116}\right)\) | \(e\left(\frac{93}{232}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{97}{232}\right)\) | \(e\left(\frac{143}{232}\right)\) | \(e\left(\frac{175}{232}\right)\) | \(e\left(\frac{75}{116}\right)\) | \(e\left(\frac{93}{116}\right)\) | \(e\left(\frac{147}{232}\right)\) | \(e\left(\frac{119}{232}\right)\) |
| \(\chi_{1003}(168,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{116}\right)\) | \(e\left(\frac{135}{232}\right)\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{51}{232}\right)\) | \(e\left(\frac{13}{232}\right)\) | \(e\left(\frac{37}{232}\right)\) | \(e\left(\frac{49}{116}\right)\) | \(e\left(\frac{19}{116}\right)\) | \(e\left(\frac{161}{232}\right)\) | \(e\left(\frac{53}{232}\right)\) |
| \(\chi_{1003}(172,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{99}{116}\right)\) | \(e\left(\frac{11}{232}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{231}{232}\right)\) | \(e\left(\frac{209}{232}\right)\) | \(e\left(\frac{113}{232}\right)\) | \(e\left(\frac{65}{116}\right)\) | \(e\left(\frac{11}{116}\right)\) | \(e\left(\frac{197}{232}\right)\) | \(e\left(\frac{49}{232}\right)\) |
| \(\chi_{1003}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{116}\right)\) | \(e\left(\frac{229}{232}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{169}{232}\right)\) | \(e\left(\frac{175}{232}\right)\) | \(e\left(\frac{159}{232}\right)\) | \(e\left(\frac{35}{116}\right)\) | \(e\left(\frac{113}{116}\right)\) | \(e\left(\frac{115}{232}\right)\) | \(e\left(\frac{71}{232}\right)\) |
| \(\chi_{1003}(185,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{116}\right)\) | \(e\left(\frac{223}{232}\right)\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{43}{232}\right)\) | \(e\left(\frac{61}{232}\right)\) | \(e\left(\frac{13}{232}\right)\) | \(e\left(\frac{105}{116}\right)\) | \(e\left(\frac{107}{116}\right)\) | \(e\left(\frac{113}{232}\right)\) | \(e\left(\frac{213}{232}\right)\) |
| \(\chi_{1003}(195,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{116}\right)\) | \(e\left(\frac{161}{232}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{133}{232}\right)\) | \(e\left(\frac{43}{232}\right)\) | \(e\left(\frac{51}{232}\right)\) | \(e\left(\frac{55}{116}\right)\) | \(e\left(\frac{45}{116}\right)\) | \(e\left(\frac{15}{232}\right)\) | \(e\left(\frac{211}{232}\right)\) |
| \(\chi_{1003}(219,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{116}\right)\) | \(e\left(\frac{199}{232}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{3}{232}\right)\) | \(e\left(\frac{69}{232}\right)\) | \(e\left(\frac{125}{232}\right)\) | \(e\left(\frac{37}{116}\right)\) | \(e\left(\frac{83}{116}\right)\) | \(e\left(\frac{105}{232}\right)\) | \(e\left(\frac{85}{232}\right)\) |
| \(\chi_{1003}(229,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{116}\right)\) | \(e\left(\frac{33}{232}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{229}{232}\right)\) | \(e\left(\frac{163}{232}\right)\) | \(e\left(\frac{107}{232}\right)\) | \(e\left(\frac{79}{116}\right)\) | \(e\left(\frac{33}{116}\right)\) | \(e\left(\frac{127}{232}\right)\) | \(e\left(\frac{147}{232}\right)\) |
| \(\chi_{1003}(246,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{116}\right)\) | \(e\left(\frac{153}{232}\right)\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{197}{232}\right)\) | \(e\left(\frac{123}{232}\right)\) | \(e\left(\frac{11}{232}\right)\) | \(e\left(\frac{71}{116}\right)\) | \(e\left(\frac{37}{116}\right)\) | \(e\left(\frac{167}{232}\right)\) | \(e\left(\frac{91}{232}\right)\) |
| \(\chi_{1003}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{116}\right)\) | \(e\left(\frac{157}{232}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{49}{232}\right)\) | \(e\left(\frac{199}{232}\right)\) | \(e\left(\frac{31}{232}\right)\) | \(e\left(\frac{63}{116}\right)\) | \(e\left(\frac{41}{116}\right)\) | \(e\left(\frac{91}{232}\right)\) | \(e\left(\frac{151}{232}\right)\) |
| \(\chi_{1003}(270,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{111}{116}\right)\) | \(e\left(\frac{167}{232}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{27}{232}\right)\) | \(e\left(\frac{157}{232}\right)\) | \(e\left(\frac{197}{232}\right)\) | \(e\left(\frac{101}{116}\right)\) | \(e\left(\frac{51}{116}\right)\) | \(e\left(\frac{17}{232}\right)\) | \(e\left(\frac{69}{232}\right)\) |
| \(\chi_{1003}(274,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{116}\right)\) | \(e\left(\frac{115}{232}\right)\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{95}{232}\right)\) | \(e\left(\frac{97}{232}\right)\) | \(e\left(\frac{169}{232}\right)\) | \(e\left(\frac{89}{116}\right)\) | \(e\left(\frac{115}{116}\right)\) | \(e\left(\frac{77}{232}\right)\) | \(e\left(\frac{217}{232}\right)\) |
| \(\chi_{1003}(280,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{116}\right)\) | \(e\left(\frac{209}{232}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{213}{232}\right)\) | \(e\left(\frac{27}{232}\right)\) | \(e\left(\frac{59}{232}\right)\) | \(e\left(\frac{75}{116}\right)\) | \(e\left(\frac{93}{116}\right)\) | \(e\left(\frac{31}{232}\right)\) | \(e\left(\frac{3}{232}\right)\) |
| \(\chi_{1003}(291,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{116}\right)\) | \(e\left(\frac{139}{232}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{135}{232}\right)\) | \(e\left(\frac{89}{232}\right)\) | \(e\left(\frac{57}{232}\right)\) | \(e\left(\frac{41}{116}\right)\) | \(e\left(\frac{23}{116}\right)\) | \(e\left(\frac{85}{232}\right)\) | \(e\left(\frac{113}{232}\right)\) |
| \(\chi_{1003}(297,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{116}\right)\) | \(e\left(\frac{113}{232}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{53}{232}\right)\) | \(e\left(\frac{59}{232}\right)\) | \(e\left(\frac{43}{232}\right)\) | \(e\left(\frac{35}{116}\right)\) | \(e\left(\frac{113}{116}\right)\) | \(e\left(\frac{231}{232}\right)\) | \(e\left(\frac{187}{232}\right)\) |
| \(\chi_{1003}(308,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{116}\right)\) | \(e\left(\frac{155}{232}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{7}{232}\right)\) | \(e\left(\frac{161}{232}\right)\) | \(e\left(\frac{137}{232}\right)\) | \(e\left(\frac{9}{116}\right)\) | \(e\left(\frac{39}{116}\right)\) | \(e\left(\frac{13}{232}\right)\) | \(e\left(\frac{121}{232}\right)\) |
| \(\chi_{1003}(325,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{116}\right)\) | \(e\left(\frac{3}{232}\right)\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{63}{232}\right)\) | \(e\left(\frac{57}{232}\right)\) | \(e\left(\frac{73}{232}\right)\) | \(e\left(\frac{81}{116}\right)\) | \(e\left(\frac{3}{116}\right)\) | \(e\left(\frac{117}{232}\right)\) | \(e\left(\frac{161}{232}\right)\) |