from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9898, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([50,33]))
chi.galois_orbit()
[g,chi] = znchar(Mod(129,9898))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(9898\) | |
| Conductor: | \(707\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(300\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 707.bi | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{9898}(129,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{300}\right)\) | \(e\left(\frac{71}{150}\right)\) | \(e\left(\frac{77}{150}\right)\) | \(e\left(\frac{29}{300}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{23}{100}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{59}{150}\right)\) | \(e\left(\frac{119}{150}\right)\) | \(e\left(\frac{71}{75}\right)\) |
| \(\chi_{9898}(411,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{300}\right)\) | \(e\left(\frac{49}{150}\right)\) | \(e\left(\frac{13}{150}\right)\) | \(e\left(\frac{151}{300}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{121}{150}\right)\) | \(e\left(\frac{61}{150}\right)\) | \(e\left(\frac{49}{75}\right)\) |
| \(\chi_{9898}(705,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{300}\right)\) | \(e\left(\frac{61}{150}\right)\) | \(e\left(\frac{7}{150}\right)\) | \(e\left(\frac{289}{300}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{43}{100}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{19}{150}\right)\) | \(e\left(\frac{79}{150}\right)\) | \(e\left(\frac{61}{75}\right)\) |
| \(\chi_{9898}(815,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{300}\right)\) | \(e\left(\frac{149}{150}\right)\) | \(e\left(\frac{113}{150}\right)\) | \(e\left(\frac{251}{300}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{71}{150}\right)\) | \(e\left(\frac{11}{150}\right)\) | \(e\left(\frac{74}{75}\right)\) |
| \(\chi_{9898}(901,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{300}\right)\) | \(e\left(\frac{133}{150}\right)\) | \(e\left(\frac{121}{150}\right)\) | \(e\left(\frac{67}{300}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{29}{100}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{150}\right)\) | \(e\left(\frac{37}{150}\right)\) | \(e\left(\frac{58}{75}\right)\) |
| \(\chi_{9898}(999,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{300}\right)\) | \(e\left(\frac{43}{150}\right)\) | \(e\left(\frac{91}{150}\right)\) | \(e\left(\frac{157}{300}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{59}{100}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{97}{150}\right)\) | \(e\left(\frac{127}{150}\right)\) | \(e\left(\frac{43}{75}\right)\) |
| \(\chi_{9898}(1109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{300}\right)\) | \(e\left(\frac{11}{150}\right)\) | \(e\left(\frac{107}{150}\right)\) | \(e\left(\frac{89}{300}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{43}{100}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{119}{150}\right)\) | \(e\left(\frac{29}{150}\right)\) | \(e\left(\frac{11}{75}\right)\) |
| \(\chi_{9898}(1305,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{221}{300}\right)\) | \(e\left(\frac{83}{150}\right)\) | \(e\left(\frac{71}{150}\right)\) | \(e\left(\frac{167}{300}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{29}{100}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{107}{150}\right)\) | \(e\left(\frac{137}{150}\right)\) | \(e\left(\frac{8}{75}\right)\) |
| \(\chi_{9898}(1403,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{300}\right)\) | \(e\left(\frac{143}{150}\right)\) | \(e\left(\frac{41}{150}\right)\) | \(e\left(\frac{257}{300}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{59}{100}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{47}{150}\right)\) | \(e\left(\frac{77}{150}\right)\) | \(e\left(\frac{68}{75}\right)\) |
| \(\chi_{9898}(1489,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{300}\right)\) | \(e\left(\frac{37}{150}\right)\) | \(e\left(\frac{19}{150}\right)\) | \(e\left(\frac{163}{300}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{81}{100}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{73}{150}\right)\) | \(e\left(\frac{43}{150}\right)\) | \(e\left(\frac{37}{75}\right)\) |
| \(\chi_{9898}(1587,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{300}\right)\) | \(e\left(\frac{1}{150}\right)\) | \(e\left(\frac{37}{150}\right)\) | \(e\left(\frac{199}{300}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{79}{150}\right)\) | \(e\left(\frac{139}{150}\right)\) | \(e\left(\frac{1}{75}\right)\) |
| \(\chi_{9898}(1783,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{300}\right)\) | \(e\left(\frac{13}{150}\right)\) | \(e\left(\frac{31}{150}\right)\) | \(e\left(\frac{37}{300}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{127}{150}\right)\) | \(e\left(\frac{7}{150}\right)\) | \(e\left(\frac{13}{75}\right)\) |
| \(\chi_{9898}(1881,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{300}\right)\) | \(e\left(\frac{67}{150}\right)\) | \(e\left(\frac{79}{150}\right)\) | \(e\left(\frac{133}{300}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{71}{100}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{43}{150}\right)\) | \(e\left(\frac{13}{150}\right)\) | \(e\left(\frac{67}{75}\right)\) |
| \(\chi_{9898}(1893,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{269}{300}\right)\) | \(e\left(\frac{137}{150}\right)\) | \(e\left(\frac{119}{150}\right)\) | \(e\left(\frac{263}{300}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{81}{100}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{23}{150}\right)\) | \(e\left(\frac{143}{150}\right)\) | \(e\left(\frac{62}{75}\right)\) |
| \(\chi_{9898}(1991,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{300}\right)\) | \(e\left(\frac{101}{150}\right)\) | \(e\left(\frac{137}{150}\right)\) | \(e\left(\frac{299}{300}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{29}{150}\right)\) | \(e\left(\frac{89}{150}\right)\) | \(e\left(\frac{26}{75}\right)\) |
| \(\chi_{9898}(2187,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{300}\right)\) | \(e\left(\frac{113}{150}\right)\) | \(e\left(\frac{131}{150}\right)\) | \(e\left(\frac{137}{300}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{77}{150}\right)\) | \(e\left(\frac{107}{150}\right)\) | \(e\left(\frac{38}{75}\right)\) |
| \(\chi_{9898}(2273,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{300}\right)\) | \(e\left(\frac{139}{150}\right)\) | \(e\left(\frac{43}{150}\right)\) | \(e\left(\frac{61}{300}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{7}{100}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{31}{150}\right)\) | \(e\left(\frac{121}{150}\right)\) | \(e\left(\frac{64}{75}\right)\) |
| \(\chi_{9898}(2285,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{300}\right)\) | \(e\left(\frac{17}{150}\right)\) | \(e\left(\frac{29}{150}\right)\) | \(e\left(\frac{233}{300}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{71}{100}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{143}{150}\right)\) | \(e\left(\frac{113}{150}\right)\) | \(e\left(\frac{17}{75}\right)\) |
| \(\chi_{9898}(2371,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{300}\right)\) | \(e\left(\frac{103}{150}\right)\) | \(e\left(\frac{61}{150}\right)\) | \(e\left(\frac{247}{300}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{89}{100}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{37}{150}\right)\) | \(e\left(\frac{67}{150}\right)\) | \(e\left(\frac{28}{75}\right)\) |
| \(\chi_{9898}(2567,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{300}\right)\) | \(e\left(\frac{19}{150}\right)\) | \(e\left(\frac{103}{150}\right)\) | \(e\left(\frac{181}{300}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{150}\right)\) | \(e\left(\frac{91}{150}\right)\) | \(e\left(\frac{19}{75}\right)\) |
| \(\chi_{9898}(2677,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{300}\right)\) | \(e\left(\frac{89}{150}\right)\) | \(e\left(\frac{143}{150}\right)\) | \(e\left(\frac{161}{300}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{7}{100}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{131}{150}\right)\) | \(e\left(\frac{71}{150}\right)\) | \(e\left(\frac{14}{75}\right)\) |
| \(\chi_{9898}(2775,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{300}\right)\) | \(e\left(\frac{53}{150}\right)\) | \(e\left(\frac{11}{150}\right)\) | \(e\left(\frac{47}{300}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{89}{100}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{137}{150}\right)\) | \(e\left(\frac{17}{150}\right)\) | \(e\left(\frac{53}{75}\right)\) |
| \(\chi_{9898}(2971,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{203}{300}\right)\) | \(e\left(\frac{119}{150}\right)\) | \(e\left(\frac{53}{150}\right)\) | \(e\left(\frac{281}{300}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{101}{150}\right)\) | \(e\left(\frac{41}{150}\right)\) | \(e\left(\frac{44}{75}\right)\) |
| \(\chi_{9898}(3057,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{300}\right)\) | \(e\left(\frac{127}{150}\right)\) | \(e\left(\frac{49}{150}\right)\) | \(e\left(\frac{73}{300}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{133}{150}\right)\) | \(e\left(\frac{103}{150}\right)\) | \(e\left(\frac{52}{75}\right)\) |
| \(\chi_{9898}(3351,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{300}\right)\) | \(e\left(\frac{79}{150}\right)\) | \(e\left(\frac{73}{150}\right)\) | \(e\left(\frac{121}{300}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{27}{100}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{91}{150}\right)\) | \(e\left(\frac{31}{150}\right)\) | \(e\left(\frac{4}{75}\right)\) |
| \(\chi_{9898}(3449,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{300}\right)\) | \(e\left(\frac{73}{150}\right)\) | \(e\left(\frac{1}{150}\right)\) | \(e\left(\frac{127}{300}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{67}{150}\right)\) | \(e\left(\frac{97}{150}\right)\) | \(e\left(\frac{73}{75}\right)\) |
| \(\chi_{9898}(3461,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{299}{300}\right)\) | \(e\left(\frac{77}{150}\right)\) | \(e\left(\frac{149}{150}\right)\) | \(e\left(\frac{173}{300}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{83}{150}\right)\) | \(e\left(\frac{53}{150}\right)\) | \(e\left(\frac{2}{75}\right)\) |
| \(\chi_{9898}(3547,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{247}{300}\right)\) | \(e\left(\frac{31}{150}\right)\) | \(e\left(\frac{97}{150}\right)\) | \(e\left(\frac{169}{300}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{3}{100}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{49}{150}\right)\) | \(e\left(\frac{109}{150}\right)\) | \(e\left(\frac{31}{75}\right)\) |
| \(\chi_{9898}(3755,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{300}\right)\) | \(e\left(\frac{29}{150}\right)\) | \(e\left(\frac{23}{150}\right)\) | \(e\left(\frac{221}{300}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{27}{100}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{41}{150}\right)\) | \(e\left(\frac{131}{150}\right)\) | \(e\left(\frac{29}{75}\right)\) |
| \(\chi_{9898}(3841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{300}\right)\) | \(e\left(\frac{109}{150}\right)\) | \(e\left(\frac{133}{150}\right)\) | \(e\left(\frac{91}{300}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{17}{100}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{61}{150}\right)\) | \(e\left(\frac{1}{150}\right)\) | \(e\left(\frac{34}{75}\right)\) |
| \(\chi_{9898}(3853,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{300}\right)\) | \(e\left(\frac{23}{150}\right)\) | \(e\left(\frac{101}{150}\right)\) | \(e\left(\frac{227}{300}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{17}{150}\right)\) | \(e\left(\frac{47}{150}\right)\) | \(e\left(\frac{23}{75}\right)\) |