from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6040, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([0,0,225,242]))
chi.galois_orbit()
[g,chi] = znchar(Mod(233,6040))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6040\) | |
| Conductor: | \(755\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(300\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 755.bi | 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: | $\Q(\zeta_{300})$ |
| Fixed field: | Number field defined by a degree 300 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6040}(233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{100}\right)\) | \(e\left(\frac{239}{300}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{97}{300}\right)\) | \(e\left(\frac{161}{300}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{77}{100}\right)\) |
| \(\chi_{6040}(257,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{100}\right)\) | \(e\left(\frac{41}{300}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{43}{300}\right)\) | \(e\left(\frac{59}{300}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{63}{100}\right)\) |
| \(\chi_{6040}(297,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{100}\right)\) | \(e\left(\frac{233}{300}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{259}{300}\right)\) | \(e\left(\frac{167}{300}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{19}{100}\right)\) |
| \(\chi_{6040}(337,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{100}\right)\) | \(e\left(\frac{61}{300}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{203}{300}\right)\) | \(e\left(\frac{139}{300}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{23}{100}\right)\) |
| \(\chi_{6040}(353,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{100}\right)\) | \(e\left(\frac{251}{300}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{73}{300}\right)\) | \(e\left(\frac{149}{300}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{93}{100}\right)\) |
| \(\chi_{6040}(417,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{193}{300}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{239}{300}\right)\) | \(e\left(\frac{7}{300}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{99}{100}\right)\) |
| \(\chi_{6040}(593,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{71}{300}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{133}{300}\right)\) | \(e\left(\frac{29}{300}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{53}{100}\right)\) |
| \(\chi_{6040}(617,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{100}\right)\) | \(e\left(\frac{169}{300}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{287}{300}\right)\) | \(e\left(\frac{31}{300}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{67}{100}\right)\) |
| \(\chi_{6040}(697,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{100}\right)\) | \(e\left(\frac{281}{300}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{163}{300}\right)\) | \(e\left(\frac{119}{300}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{83}{100}\right)\) |
| \(\chi_{6040}(713,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{187}{300}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{101}{300}\right)\) | \(e\left(\frac{13}{300}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{41}{100}\right)\) |
| \(\chi_{6040}(737,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{100}\right)\) | \(e\left(\frac{113}{300}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{199}{300}\right)\) | \(e\left(\frac{287}{300}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{59}{100}\right)\) |
| \(\chi_{6040}(857,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{100}\right)\) | \(e\left(\frac{181}{300}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{263}{300}\right)\) | \(e\left(\frac{19}{300}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{83}{100}\right)\) |
| \(\chi_{6040}(913,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{100}\right)\) | \(e\left(\frac{203}{300}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{169}{300}\right)\) | \(e\left(\frac{197}{300}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{29}{100}\right)\) |
| \(\chi_{6040}(977,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{100}\right)\) | \(e\left(\frac{253}{300}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{119}{300}\right)\) | \(e\left(\frac{247}{300}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{79}{100}\right)\) |
| \(\chi_{6040}(1017,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{173}{300}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{79}{300}\right)\) | \(e\left(\frac{227}{300}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{39}{100}\right)\) |
| \(\chi_{6040}(1113,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{100}\right)\) | \(e\left(\frac{143}{300}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{289}{300}\right)\) | \(e\left(\frac{257}{300}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{49}{100}\right)\) |
| \(\chi_{6040}(1153,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{79}{300}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{17}{300}\right)\) | \(e\left(\frac{121}{300}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{97}{100}\right)\) |
| \(\chi_{6040}(1177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{77}{300}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{271}{300}\right)\) | \(e\left(\frac{23}{300}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{11}{100}\right)\) |
| \(\chi_{6040}(1297,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{100}\right)\) | \(e\left(\frac{157}{300}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{11}{300}\right)\) | \(e\left(\frac{43}{300}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{51}{100}\right)\) |
| \(\chi_{6040}(1337,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{100}\right)\) | \(e\left(\frac{1}{300}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{23}{300}\right)\) | \(e\left(\frac{199}{300}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{43}{100}\right)\) |
| \(\chi_{6040}(1473,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{119}{300}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{37}{300}\right)\) | \(e\left(\frac{281}{300}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{17}{100}\right)\) |
| \(\chi_{6040}(1673,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{100}\right)\) | \(e\left(\frac{139}{300}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{197}{300}\right)\) | \(e\left(\frac{61}{300}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{77}{100}\right)\) |
| \(\chi_{6040}(1713,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{179}{300}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{217}{300}\right)\) | \(e\left(\frac{221}{300}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{97}{100}\right)\) |
| \(\chi_{6040}(1873,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{271}{300}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{233}{300}\right)\) | \(e\left(\frac{229}{300}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{53}{100}\right)\) |
| \(\chi_{6040}(1953,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{100}\right)\) | \(e\left(\frac{163}{300}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{149}{300}\right)\) | \(e\left(\frac{37}{300}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{9}{100}\right)\) |
| \(\chi_{6040}(1977,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{100}\right)\) | \(e\left(\frac{133}{300}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{59}{300}\right)\) | \(e\left(\frac{67}{300}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{19}{100}\right)\) |
| \(\chi_{6040}(1993,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{100}\right)\) | \(e\left(\frac{67}{300}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{41}{300}\right)\) | \(e\left(\frac{133}{300}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{81}{100}\right)\) |
| \(\chi_{6040}(2017,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{100}\right)\) | \(e\left(\frac{17}{300}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{91}{300}\right)\) | \(e\left(\frac{83}{300}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{31}{100}\right)\) |
| \(\chi_{6040}(2097,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{100}\right)\) | \(e\left(\frac{197}{300}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{31}{300}\right)\) | \(e\left(\frac{203}{300}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{71}{100}\right)\) |
| \(\chi_{6040}(2177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{100}\right)\) | \(e\left(\frac{161}{300}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{103}{300}\right)\) | \(e\left(\frac{239}{300}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{23}{100}\right)\) |
| \(\chi_{6040}(2313,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{299}{300}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{277}{300}\right)\) | \(e\left(\frac{101}{300}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{57}{100}\right)\) |