from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2500, base_ring=CyclotomicField(250))
M = H._module
chi = DirichletCharacter(H, M([125,209]))
chi.galois_orbit()
[g,chi] = znchar(Mod(19,2500))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2500\) | |
| Conductor: | \(2500\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(250\) | 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: | $\Q(\zeta_{125})$ |
| Fixed field: | Number field defined by a degree 250 polynomial (not computed) |
First 31 of 100 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2500}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{109}{250}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{157}{250}\right)\) | \(e\left(\frac{237}{250}\right)\) | \(e\left(\frac{114}{125}\right)\) | \(e\left(\frac{102}{125}\right)\) | \(e\left(\frac{107}{125}\right)\) |
| \(\chi_{2500}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{18}{125}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{36}{125}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{97}{250}\right)\) | \(e\left(\frac{29}{250}\right)\) | \(e\left(\frac{39}{250}\right)\) | \(e\left(\frac{33}{125}\right)\) | \(e\left(\frac{69}{125}\right)\) | \(e\left(\frac{54}{125}\right)\) |
| \(\chi_{2500}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{125}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{44}{125}\right)\) | \(e\left(\frac{17}{250}\right)\) | \(e\left(\frac{63}{250}\right)\) | \(e\left(\frac{91}{250}\right)\) | \(e\left(\frac{131}{250}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{1}{125}\right)\) | \(e\left(\frac{66}{125}\right)\) |
| \(\chi_{2500}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{81}{125}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{37}{125}\right)\) | \(e\left(\frac{91}{250}\right)\) | \(e\left(\frac{249}{250}\right)\) | \(e\left(\frac{193}{250}\right)\) | \(e\left(\frac{113}{250}\right)\) | \(e\left(\frac{86}{125}\right)\) | \(e\left(\frac{123}{125}\right)\) | \(e\left(\frac{118}{125}\right)\) |
| \(\chi_{2500}(119,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{125}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{78}{125}\right)\) | \(e\left(\frac{229}{250}\right)\) | \(e\left(\frac{231}{250}\right)\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{147}{250}\right)\) | \(e\left(\frac{9}{125}\right)\) | \(e\left(\frac{87}{125}\right)\) | \(e\left(\frac{117}{125}\right)\) |
| \(\chi_{2500}(139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{88}{125}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{51}{125}\right)\) | \(e\left(\frac{193}{250}\right)\) | \(e\left(\frac{127}{250}\right)\) | \(e\left(\frac{239}{250}\right)\) | \(e\left(\frac{149}{250}\right)\) | \(e\left(\frac{78}{125}\right)\) | \(e\left(\frac{4}{125}\right)\) | \(e\left(\frac{14}{125}\right)\) |
| \(\chi_{2500}(159,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{117}{125}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{109}{125}\right)\) | \(e\left(\frac{187}{250}\right)\) | \(e\left(\frac{193}{250}\right)\) | \(e\left(\frac{1}{250}\right)\) | \(e\left(\frac{191}{250}\right)\) | \(e\left(\frac{27}{125}\right)\) | \(e\left(\frac{11}{125}\right)\) | \(e\left(\frac{101}{125}\right)\) |
| \(\chi_{2500}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{76}{125}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{27}{125}\right)\) | \(e\left(\frac{161}{250}\right)\) | \(e\left(\frac{229}{250}\right)\) | \(e\left(\frac{53}{250}\right)\) | \(e\left(\frac{123}{250}\right)\) | \(e\left(\frac{56}{125}\right)\) | \(e\left(\frac{83}{125}\right)\) | \(e\left(\frac{103}{125}\right)\) |
| \(\chi_{2500}(219,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{84}{125}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{43}{125}\right)\) | \(e\left(\frac{99}{250}\right)\) | \(e\left(\frac{161}{250}\right)\) | \(e\left(\frac{177}{250}\right)\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{29}{125}\right)\) | \(e\left(\frac{72}{125}\right)\) | \(e\left(\frac{2}{125}\right)\) |
| \(\chi_{2500}(239,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{125}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{213}{250}\right)\) | \(e\left(\frac{157}{250}\right)\) | \(e\left(\frac{199}{250}\right)\) | \(e\left(\frac{9}{250}\right)\) | \(e\left(\frac{123}{125}\right)\) | \(e\left(\frac{64}{125}\right)\) | \(e\left(\frac{99}{125}\right)\) |
| \(\chi_{2500}(259,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{87}{125}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{49}{125}\right)\) | \(e\left(\frac{107}{250}\right)\) | \(e\left(\frac{73}{250}\right)\) | \(e\left(\frac{161}{250}\right)\) | \(e\left(\frac{1}{250}\right)\) | \(e\left(\frac{97}{125}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{11}{125}\right)\) |
| \(\chi_{2500}(279,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{125}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{17}{125}\right)\) | \(e\left(\frac{231}{250}\right)\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{133}{250}\right)\) | \(e\left(\frac{26}{125}\right)\) | \(e\left(\frac{43}{125}\right)\) | \(e\left(\frac{88}{125}\right)\) |
| \(\chi_{2500}(319,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{125}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{8}{125}\right)\) | \(e\left(\frac{219}{250}\right)\) | \(e\left(\frac{91}{250}\right)\) | \(e\left(\frac{187}{250}\right)\) | \(e\left(\frac{217}{250}\right)\) | \(e\left(\frac{49}{125}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{12}{125}\right)\) |
| \(\chi_{2500}(339,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{81}{125}\right)\) | \(e\left(\frac{233}{250}\right)\) | \(e\left(\frac{187}{250}\right)\) | \(e\left(\frac{159}{250}\right)\) | \(e\left(\frac{119}{250}\right)\) | \(e\left(\frac{43}{125}\right)\) | \(e\left(\frac{124}{125}\right)\) | \(e\left(\frac{59}{125}\right)\) |
| \(\chi_{2500}(359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{114}{125}\right)\) | \(e\left(\frac{27}{250}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{71}{250}\right)\) | \(e\left(\frac{61}{250}\right)\) | \(e\left(\frac{42}{125}\right)\) | \(e\left(\frac{31}{125}\right)\) | \(e\left(\frac{46}{125}\right)\) |
| \(\chi_{2500}(379,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{7}{125}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{23}{250}\right)\) | \(e\left(\frac{143}{250}\right)\) | \(e\left(\frac{121}{125}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{73}{125}\right)\) |
| \(\chi_{2500}(419,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{125}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{98}{125}\right)\) | \(e\left(\frac{89}{250}\right)\) | \(e\left(\frac{21}{250}\right)\) | \(e\left(\frac{197}{250}\right)\) | \(e\left(\frac{127}{250}\right)\) | \(e\left(\frac{69}{125}\right)\) | \(e\left(\frac{42}{125}\right)\) | \(e\left(\frac{22}{125}\right)\) |
| \(\chi_{2500}(439,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{48}{125}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{96}{125}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{217}{250}\right)\) | \(e\left(\frac{119}{250}\right)\) | \(e\left(\frac{229}{250}\right)\) | \(e\left(\frac{88}{125}\right)\) | \(e\left(\frac{59}{125}\right)\) | \(e\left(\frac{19}{125}\right)\) |
| \(\chi_{2500}(459,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{125}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{54}{125}\right)\) | \(e\left(\frac{197}{250}\right)\) | \(e\left(\frac{83}{250}\right)\) | \(e\left(\frac{231}{250}\right)\) | \(e\left(\frac{121}{250}\right)\) | \(e\left(\frac{112}{125}\right)\) | \(e\left(\frac{41}{125}\right)\) | \(e\left(\frac{81}{125}\right)\) |
| \(\chi_{2500}(479,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{125}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{122}{125}\right)\) | \(e\left(\frac{121}{250}\right)\) | \(e\left(\frac{169}{250}\right)\) | \(e\left(\frac{133}{250}\right)\) | \(e\left(\frac{153}{250}\right)\) | \(e\left(\frac{91}{125}\right)\) | \(e\left(\frac{88}{125}\right)\) | \(e\left(\frac{58}{125}\right)\) |
| \(\chi_{2500}(519,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{94}{125}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{63}{125}\right)\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{201}{250}\right)\) | \(e\left(\frac{207}{250}\right)\) | \(e\left(\frac{37}{250}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{27}{125}\right)\) | \(e\left(\frac{32}{125}\right)\) |
| \(\chi_{2500}(539,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{118}{125}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{23}{250}\right)\) | \(e\left(\frac{247}{250}\right)\) | \(e\left(\frac{79}{250}\right)\) | \(e\left(\frac{89}{250}\right)\) | \(e\left(\frac{8}{125}\right)\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{104}{125}\right)\) |
| \(\chi_{2500}(559,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{122}{125}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{117}{250}\right)\) | \(e\left(\frac{213}{250}\right)\) | \(e\left(\frac{141}{250}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{51}{125}\right)\) | \(e\left(\frac{116}{125}\right)\) |
| \(\chi_{2500}(579,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{56}{125}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{112}{125}\right)\) | \(e\left(\frac{191}{250}\right)\) | \(e\left(\frac{149}{250}\right)\) | \(e\left(\frac{243}{250}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{61}{125}\right)\) | \(e\left(\frac{48}{125}\right)\) | \(e\left(\frac{43}{125}\right)\) |
| \(\chi_{2500}(619,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{125}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{28}{125}\right)\) | \(e\left(\frac{79}{250}\right)\) | \(e\left(\frac{131}{250}\right)\) | \(e\left(\frac{217}{250}\right)\) | \(e\left(\frac{197}{250}\right)\) | \(e\left(\frac{109}{125}\right)\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{42}{125}\right)\) |
| \(\chi_{2500}(639,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{125}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{1}{125}\right)\) | \(e\left(\frac{43}{250}\right)\) | \(e\left(\frac{27}{250}\right)\) | \(e\left(\frac{39}{250}\right)\) | \(e\left(\frac{199}{250}\right)\) | \(e\left(\frac{53}{125}\right)\) | \(e\left(\frac{54}{125}\right)\) | \(e\left(\frac{64}{125}\right)\) |
| \(\chi_{2500}(659,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{92}{125}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{59}{125}\right)\) | \(e\left(\frac{37}{250}\right)\) | \(e\left(\frac{93}{250}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{241}{250}\right)\) | \(e\left(\frac{2}{125}\right)\) | \(e\left(\frac{61}{125}\right)\) | \(e\left(\frac{26}{125}\right)\) |
| \(\chi_{2500}(679,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{125}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{102}{125}\right)\) | \(e\left(\frac{11}{250}\right)\) | \(e\left(\frac{129}{250}\right)\) | \(e\left(\frac{103}{250}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{31}{125}\right)\) | \(e\left(\frac{8}{125}\right)\) | \(e\left(\frac{28}{125}\right)\) |
| \(\chi_{2500}(719,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{125}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{118}{125}\right)\) | \(e\left(\frac{199}{250}\right)\) | \(e\left(\frac{61}{250}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{107}{250}\right)\) | \(e\left(\frac{4}{125}\right)\) | \(e\left(\frac{122}{125}\right)\) | \(e\left(\frac{52}{125}\right)\) |
| \(\chi_{2500}(739,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{125}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{16}{125}\right)\) | \(e\left(\frac{63}{250}\right)\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{249}{250}\right)\) | \(e\left(\frac{59}{250}\right)\) | \(e\left(\frac{98}{125}\right)\) | \(e\left(\frac{114}{125}\right)\) | \(e\left(\frac{24}{125}\right)\) |
| \(\chi_{2500}(759,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{62}{125}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{124}{125}\right)\) | \(e\left(\frac{207}{250}\right)\) | \(e\left(\frac{223}{250}\right)\) | \(e\left(\frac{211}{250}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{72}{125}\right)\) | \(e\left(\frac{71}{125}\right)\) | \(e\left(\frac{61}{125}\right)\) |