from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1004, base_ring=CyclotomicField(250))
M = H._module
chi = DirichletCharacter(H, M([125,16]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,1004))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(1004\) | |
Conductor: | \(1004\) | 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{1004}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{250}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{93}{250}\right)\) | \(e\left(\frac{6}{125}\right)\) | \(e\left(\frac{1}{250}\right)\) | \(e\left(\frac{81}{125}\right)\) | \(e\left(\frac{211}{250}\right)\) | \(e\left(\frac{92}{125}\right)\) | \(e\left(\frac{233}{250}\right)\) | \(e\left(\frac{112}{125}\right)\) |
\(\chi_{1004}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{93}{250}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{129}{250}\right)\) | \(e\left(\frac{93}{125}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{68}{125}\right)\) | \(e\left(\frac{83}{250}\right)\) | \(e\left(\frac{51}{125}\right)\) | \(e\left(\frac{49}{250}\right)\) | \(e\left(\frac{111}{125}\right)\) |
\(\chi_{1004}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{250}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{83}{250}\right)\) | \(e\left(\frac{86}{125}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{36}{125}\right)\) | \(e\left(\frac{191}{250}\right)\) | \(e\left(\frac{27}{125}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{22}{125}\right)\) |
\(\chi_{1004}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{250}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{77}{250}\right)\) | \(e\left(\frac{9}{125}\right)\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{59}{125}\right)\) | \(e\left(\frac{129}{250}\right)\) | \(e\left(\frac{13}{125}\right)\) | \(e\left(\frac{37}{250}\right)\) | \(e\left(\frac{43}{125}\right)\) |
\(\chi_{1004}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{250}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{29}{250}\right)\) | \(e\left(\frac{18}{125}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{118}{125}\right)\) | \(e\left(\frac{133}{250}\right)\) | \(e\left(\frac{26}{125}\right)\) | \(e\left(\frac{199}{250}\right)\) | \(e\left(\frac{86}{125}\right)\) |
\(\chi_{1004}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{250}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{121}{250}\right)\) | \(e\left(\frac{51}{125}\right)\) | \(e\left(\frac{31}{250}\right)\) | \(e\left(\frac{7}{125}\right)\) | \(e\left(\frac{193}{250}\right)\) | \(e\left(\frac{52}{125}\right)\) |
\(\chi_{1004}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{119}{250}\right)\) | \(e\left(\frac{48}{125}\right)\) | \(e\left(\frac{133}{250}\right)\) | \(e\left(\frac{23}{125}\right)\) | \(e\left(\frac{63}{250}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{239}{250}\right)\) | \(e\left(\frac{21}{125}\right)\) |
\(\chi_{1004}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{250}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{229}{250}\right)\) | \(e\left(\frac{43}{125}\right)\) | \(e\left(\frac{153}{250}\right)\) | \(e\left(\frac{18}{125}\right)\) | \(e\left(\frac{33}{250}\right)\) | \(e\left(\frac{76}{125}\right)\) | \(e\left(\frac{149}{250}\right)\) | \(e\left(\frac{11}{125}\right)\) |
\(\chi_{1004}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{250}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{39}{125}\right)\) | \(e\left(\frac{69}{250}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{59}{250}\right)\) | \(e\left(\frac{98}{125}\right)\) | \(e\left(\frac{77}{250}\right)\) | \(e\left(\frac{103}{125}\right)\) |
\(\chi_{1004}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{250}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{73}{250}\right)\) | \(e\left(\frac{41}{125}\right)\) | \(e\left(\frac{111}{250}\right)\) | \(e\left(\frac{116}{125}\right)\) | \(e\left(\frac{171}{250}\right)\) | \(e\left(\frac{87}{125}\right)\) | \(e\left(\frac{113}{250}\right)\) | \(e\left(\frac{57}{125}\right)\) |
\(\chi_{1004}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{250}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{191}{250}\right)\) | \(e\left(\frac{97}{125}\right)\) | \(e\left(\frac{37}{250}\right)\) | \(e\left(\frac{122}{125}\right)\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{29}{125}\right)\) | \(e\left(\frac{121}{250}\right)\) | \(e\left(\frac{19}{125}\right)\) |
\(\chi_{1004}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{129}{250}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{187}{250}\right)\) | \(e\left(\frac{4}{125}\right)\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{54}{125}\right)\) | \(e\left(\frac{99}{250}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{197}{250}\right)\) | \(e\left(\frac{33}{125}\right)\) |
\(\chi_{1004}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{207}{250}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{21}{250}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{97}{250}\right)\) | \(e\left(\frac{107}{125}\right)\) | \(e\left(\frac{217}{250}\right)\) | \(e\left(\frac{49}{125}\right)\) | \(e\left(\frac{101}{250}\right)\) | \(e\left(\frac{114}{125}\right)\) |
\(\chi_{1004}(115,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{250}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{67}{250}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{119}{250}\right)\) | \(e\left(\frac{14}{125}\right)\) | \(e\left(\frac{109}{250}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{78}{125}\right)\) |
\(\chi_{1004}(119,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{250}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{231}{250}\right)\) | \(e\left(\frac{27}{125}\right)\) | \(e\left(\frac{67}{250}\right)\) | \(e\left(\frac{52}{125}\right)\) | \(e\left(\frac{137}{250}\right)\) | \(e\left(\frac{39}{125}\right)\) | \(e\left(\frac{111}{250}\right)\) | \(e\left(\frac{4}{125}\right)\) |
\(\chi_{1004}(131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{241}{250}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{116}{125}\right)\) | \(e\left(\frac{61}{250}\right)\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{121}{250}\right)\) | \(e\left(\frac{112}{125}\right)\) | \(e\left(\frac{213}{250}\right)\) | \(e\left(\frac{82}{125}\right)\) |
\(\chi_{1004}(135,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{223}{250}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{19}{250}\right)\) | \(e\left(\frac{98}{125}\right)\) | \(e\left(\frac{183}{250}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{113}{250}\right)\) | \(e\left(\frac{86}{125}\right)\) | \(e\left(\frac{139}{250}\right)\) | \(e\left(\frac{121}{125}\right)\) |
\(\chi_{1004}(147,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{250}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{101}{250}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{157}{250}\right)\) | \(e\left(\frac{92}{125}\right)\) | \(e\left(\frac{127}{250}\right)\) | \(e\left(\frac{69}{125}\right)\) | \(e\left(\frac{81}{250}\right)\) | \(e\left(\frac{84}{125}\right)\) |
\(\chi_{1004}(155,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{243}{250}\right)\) | \(e\left(\frac{56}{125}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{6}{125}\right)\) | \(e\left(\frac{11}{250}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{133}{250}\right)\) | \(e\left(\frac{87}{125}\right)\) |
\(\chi_{1004}(175,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{109}{250}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{63}{250}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{43}{250}\right)\) | \(e\left(\frac{46}{125}\right)\) | \(e\left(\frac{179}{250}\right)\) | \(e\left(\frac{56}{125}\right)\) |
\(\chi_{1004}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{151}{250}\right)\) | \(e\left(\frac{42}{125}\right)\) | \(e\left(\frac{7}{250}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{19}{125}\right)\) | \(e\left(\frac{131}{250}\right)\) | \(e\left(\frac{34}{125}\right)\) |
\(\chi_{1004}(195,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{123}{250}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{219}{250}\right)\) | \(e\left(\frac{123}{125}\right)\) | \(e\left(\frac{83}{250}\right)\) | \(e\left(\frac{98}{125}\right)\) | \(e\left(\frac{13}{250}\right)\) | \(e\left(\frac{11}{125}\right)\) | \(e\left(\frac{89}{250}\right)\) | \(e\left(\frac{46}{125}\right)\) |
\(\chi_{1004}(207,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{250}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{13}{250}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{191}{250}\right)\) | \(e\left(\frac{96}{125}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{72}{125}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{17}{125}\right)\) |
\(\chi_{1004}(227,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{250}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{183}{250}\right)\) | \(e\left(\frac{36}{125}\right)\) | \(e\left(\frac{131}{250}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{141}{250}\right)\) | \(e\left(\frac{52}{125}\right)\) | \(e\left(\frac{23}{250}\right)\) | \(e\left(\frac{47}{125}\right)\) |
\(\chi_{1004}(263,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{250}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{153}{250}\right)\) | \(e\left(\frac{26}{125}\right)\) | \(e\left(\frac{171}{250}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{81}{250}\right)\) | \(e\left(\frac{107}{125}\right)\) | \(e\left(\frac{93}{250}\right)\) | \(e\left(\frac{27}{125}\right)\) |
\(\chi_{1004}(279,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{250}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{113}{125}\right)\) | \(e\left(\frac{123}{250}\right)\) | \(e\left(\frac{88}{125}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{159}{250}\right)\) | \(e\left(\frac{26}{125}\right)\) |
\(\chi_{1004}(287,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{250}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{121}{250}\right)\) | \(e\left(\frac{32}{125}\right)\) | \(e\left(\frac{47}{250}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{201}{250}\right)\) | \(e\left(\frac{14}{125}\right)\) |
\(\chi_{1004}(299,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{171}{250}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{213}{250}\right)\) | \(e\left(\frac{46}{125}\right)\) | \(e\left(\frac{91}{250}\right)\) | \(e\left(\frac{121}{125}\right)\) | \(e\left(\frac{201}{250}\right)\) | \(e\left(\frac{122}{125}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{67}{125}\right)\) |
\(\chi_{1004}(303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{71}{250}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{197}{250}\right)\) | \(e\left(\frac{82}{125}\right)\) | \(e\left(\frac{67}{250}\right)\) | \(e\left(\frac{124}{125}\right)\) | \(e\left(\frac{151}{250}\right)\) | \(e\left(\frac{64}{125}\right)\) |
\(\chi_{1004}(311,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{231}{250}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{143}{250}\right)\) | \(e\left(\frac{106}{125}\right)\) | \(e\left(\frac{101}{250}\right)\) | \(e\left(\frac{56}{125}\right)\) | \(e\left(\frac{61}{250}\right)\) | \(e\left(\frac{42}{125}\right)\) | \(e\left(\frac{33}{250}\right)\) | \(e\left(\frac{62}{125}\right)\) |
\(\chi_{1004}(319,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{250}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{37}{250}\right)\) | \(e\left(\frac{79}{125}\right)\) | \(e\left(\frac{159}{250}\right)\) | \(e\left(\frac{4}{125}\right)\) | \(e\left(\frac{49}{250}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{47}{250}\right)\) | \(e\left(\frac{58}{125}\right)\) |