from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4334, base_ring=CyclotomicField(980))
M = H._module
chi = DirichletCharacter(H, M([98,125]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,4334))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4334\) | |
| Conductor: | \(2167\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(980\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2167.bj | 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: | $\Q(\zeta_{980})$ |
| Fixed field: | Number field defined by a degree 980 polynomial (not computed) |
First 31 of 336 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4334}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{869}{980}\right)\) | \(e\left(\frac{737}{980}\right)\) | \(e\left(\frac{79}{245}\right)\) | \(e\left(\frac{379}{490}\right)\) | \(e\left(\frac{283}{980}\right)\) | \(e\left(\frac{313}{490}\right)\) | \(e\left(\frac{177}{980}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{41}{196}\right)\) | \(e\left(\frac{15}{49}\right)\) |
| \(\chi_{4334}(17,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{980}\right)\) | \(e\left(\frac{783}{980}\right)\) | \(e\left(\frac{181}{245}\right)\) | \(e\left(\frac{31}{490}\right)\) | \(e\left(\frac{177}{980}\right)\) | \(e\left(\frac{407}{490}\right)\) | \(e\left(\frac{83}{980}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{151}{196}\right)\) | \(e\left(\frac{17}{49}\right)\) |
| \(\chi_{4334}(35,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{799}{980}\right)\) | \(e\left(\frac{107}{980}\right)\) | \(e\left(\frac{184}{245}\right)\) | \(e\left(\frac{309}{490}\right)\) | \(e\left(\frac{73}{980}\right)\) | \(e\left(\frac{453}{490}\right)\) | \(e\left(\frac{527}{980}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{111}{196}\right)\) | \(e\left(\frac{43}{49}\right)\) |
| \(\chi_{4334}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{159}{980}\right)\) | \(e\left(\frac{507}{980}\right)\) | \(e\left(\frac{59}{245}\right)\) | \(e\left(\frac{159}{490}\right)\) | \(e\left(\frac{813}{980}\right)\) | \(e\left(\frac{333}{490}\right)\) | \(e\left(\frac{647}{980}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{79}{196}\right)\) | \(e\left(\frac{5}{49}\right)\) |
| \(\chi_{4334}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{153}{980}\right)\) | \(e\left(\frac{229}{980}\right)\) | \(e\left(\frac{103}{245}\right)\) | \(e\left(\frac{153}{490}\right)\) | \(e\left(\frac{431}{980}\right)\) | \(e\left(\frac{191}{490}\right)\) | \(e\left(\frac{789}{980}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{113}{196}\right)\) | \(e\left(\frac{27}{49}\right)\) |
| \(\chi_{4334}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{599}{980}\right)\) | \(e\left(\frac{967}{980}\right)\) | \(e\left(\frac{99}{245}\right)\) | \(e\left(\frac{109}{490}\right)\) | \(e\left(\frac{733}{980}\right)\) | \(e\left(\frac{293}{490}\right)\) | \(e\left(\frac{687}{980}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{3}{196}\right)\) | \(e\left(\frac{25}{49}\right)\) |
| \(\chi_{4334}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{980}\right)\) | \(e\left(\frac{139}{980}\right)\) | \(e\left(\frac{223}{245}\right)\) | \(e\left(\frac{3}{490}\right)\) | \(e\left(\frac{681}{980}\right)\) | \(e\left(\frac{71}{490}\right)\) | \(e\left(\frac{419}{980}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{179}{196}\right)\) | \(e\left(\frac{38}{49}\right)\) |
| \(\chi_{4334}(117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{963}{980}\right)\) | \(e\left(\frac{519}{980}\right)\) | \(e\left(\frac{43}{245}\right)\) | \(e\left(\frac{473}{490}\right)\) | \(e\left(\frac{61}{980}\right)\) | \(e\left(\frac{251}{490}\right)\) | \(e\left(\frac{239}{980}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{31}{196}\right)\) | \(e\left(\frac{46}{49}\right)\) |
| \(\chi_{4334}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{519}{980}\right)\) | \(e\left(\frac{527}{980}\right)\) | \(e\left(\frac{114}{245}\right)\) | \(e\left(\frac{29}{490}\right)\) | \(e\left(\frac{213}{980}\right)\) | \(e\left(\frac{33}{490}\right)\) | \(e\left(\frac{947}{980}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{195}{196}\right)\) | \(e\left(\frac{8}{49}\right)\) |
| \(\chi_{4334}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{263}{980}\right)\) | \(e\left(\frac{99}{980}\right)\) | \(e\left(\frac{113}{245}\right)\) | \(e\left(\frac{263}{490}\right)\) | \(e\left(\frac{901}{980}\right)\) | \(e\left(\frac{181}{490}\right)\) | \(e\left(\frac{799}{980}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{143}{196}\right)\) | \(e\left(\frac{32}{49}\right)\) |
| \(\chi_{4334}(145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{980}\right)\) | \(e\left(\frac{157}{980}\right)\) | \(e\left(\frac{199}{245}\right)\) | \(e\left(\frac{229}{490}\right)\) | \(e\left(\frac{43}{980}\right)\) | \(e\left(\frac{193}{490}\right)\) | \(e\left(\frac{297}{980}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{9}{196}\right)\) | \(e\left(\frac{26}{49}\right)\) |
| \(\chi_{4334}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{531}{980}\right)\) | \(e\left(\frac{103}{980}\right)\) | \(e\left(\frac{26}{245}\right)\) | \(e\left(\frac{41}{490}\right)\) | \(e\left(\frac{977}{980}\right)\) | \(e\left(\frac{317}{490}\right)\) | \(e\left(\frac{663}{980}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{127}{196}\right)\) | \(e\left(\frac{13}{49}\right)\) |
| \(\chi_{4334}(151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{627}{980}\right)\) | \(e\left(\frac{631}{980}\right)\) | \(e\left(\frac{57}{245}\right)\) | \(e\left(\frac{137}{490}\right)\) | \(e\left(\frac{229}{980}\right)\) | \(e\left(\frac{139}{490}\right)\) | \(e\left(\frac{351}{980}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{171}{196}\right)\) | \(e\left(\frac{4}{49}\right)\) |
| \(\chi_{4334}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{549}{980}\right)\) | \(e\left(\frac{937}{980}\right)\) | \(e\left(\frac{139}{245}\right)\) | \(e\left(\frac{59}{490}\right)\) | \(e\left(\frac{163}{980}\right)\) | \(e\left(\frac{253}{490}\right)\) | \(e\left(\frac{237}{980}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{25}{196}\right)\) | \(e\left(\frac{45}{49}\right)\) |
| \(\chi_{4334}(189,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{980}\right)\) | \(e\left(\frac{257}{980}\right)\) | \(e\left(\frac{229}{245}\right)\) | \(e\left(\frac{69}{490}\right)\) | \(e\left(\frac{963}{980}\right)\) | \(e\left(\frac{163}{490}\right)\) | \(e\left(\frac{817}{980}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{1}{196}\right)\) | \(e\left(\frac{41}{49}\right)\) |
| \(\chi_{4334}(195,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{807}{980}\right)\) | \(e\left(\frac{151}{980}\right)\) | \(e\left(\frac{207}{245}\right)\) | \(e\left(\frac{317}{490}\right)\) | \(e\left(\frac{909}{980}\right)\) | \(e\left(\frac{479}{490}\right)\) | \(e\left(\frac{11}{980}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{131}{196}\right)\) | \(e\left(\frac{30}{49}\right)\) |
| \(\chi_{4334}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{363}{980}\right)\) | \(e\left(\frac{159}{980}\right)\) | \(e\left(\frac{33}{245}\right)\) | \(e\left(\frac{363}{490}\right)\) | \(e\left(\frac{81}{980}\right)\) | \(e\left(\frac{261}{490}\right)\) | \(e\left(\frac{719}{980}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{99}{196}\right)\) | \(e\left(\frac{41}{49}\right)\) |
| \(\chi_{4334}(215,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{411}{980}\right)\) | \(e\left(\frac{423}{980}\right)\) | \(e\left(\frac{171}{245}\right)\) | \(e\left(\frac{411}{490}\right)\) | \(e\left(\frac{197}{980}\right)\) | \(e\left(\frac{417}{490}\right)\) | \(e\left(\frac{563}{980}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{23}{196}\right)\) | \(e\left(\frac{12}{49}\right)\) |
| \(\chi_{4334}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{843}{980}\right)\) | \(e\left(\frac{839}{980}\right)\) | \(e\left(\frac{188}{245}\right)\) | \(e\left(\frac{353}{490}\right)\) | \(e\left(\frac{261}{980}\right)\) | \(e\left(\frac{351}{490}\right)\) | \(e\left(\frac{139}{980}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{123}{196}\right)\) | \(e\left(\frac{45}{49}\right)\) |
| \(\chi_{4334}(249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{523}{980}\right)\) | \(e\left(\frac{59}{980}\right)\) | \(e\left(\frac{3}{245}\right)\) | \(e\left(\frac{33}{490}\right)\) | \(e\left(\frac{141}{980}\right)\) | \(e\left(\frac{291}{490}\right)\) | \(e\left(\frac{199}{980}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{107}{196}\right)\) | \(e\left(\frac{26}{49}\right)\) |
| \(\chi_{4334}(255,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{949}{980}\right)\) | \(e\left(\frac{197}{980}\right)\) | \(e\left(\frac{64}{245}\right)\) | \(e\left(\frac{459}{490}\right)\) | \(e\left(\frac{803}{980}\right)\) | \(e\left(\frac{83}{490}\right)\) | \(e\left(\frac{897}{980}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{45}{196}\right)\) | \(e\left(\frac{32}{49}\right)\) |
| \(\chi_{4334}(271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{813}{980}\right)\) | \(e\left(\frac{429}{980}\right)\) | \(e\left(\frac{163}{245}\right)\) | \(e\left(\frac{323}{490}\right)\) | \(e\left(\frac{311}{980}\right)\) | \(e\left(\frac{131}{490}\right)\) | \(e\left(\frac{849}{980}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{97}{196}\right)\) | \(e\left(\frac{8}{49}\right)\) |
| \(\chi_{4334}(277,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{669}{980}\right)\) | \(e\left(\frac{617}{980}\right)\) | \(e\left(\frac{239}{245}\right)\) | \(e\left(\frac{179}{490}\right)\) | \(e\left(\frac{943}{980}\right)\) | \(e\left(\frac{153}{490}\right)\) | \(e\left(\frac{337}{980}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{129}{196}\right)\) | \(e\left(\frac{46}{49}\right)\) |
| \(\chi_{4334}(283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{347}{980}\right)\) | \(e\left(\frac{71}{980}\right)\) | \(e\left(\frac{232}{245}\right)\) | \(e\left(\frac{347}{490}\right)\) | \(e\left(\frac{369}{980}\right)\) | \(e\left(\frac{209}{490}\right)\) | \(e\left(\frac{771}{980}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{59}{196}\right)\) | \(e\left(\frac{18}{49}\right)\) |
| \(\chi_{4334}(299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{689}{980}\right)\) | \(e\left(\frac{237}{980}\right)\) | \(e\left(\frac{174}{245}\right)\) | \(e\left(\frac{199}{490}\right)\) | \(e\left(\frac{583}{980}\right)\) | \(e\left(\frac{463}{490}\right)\) | \(e\left(\frac{517}{980}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{81}{196}\right)\) | \(e\left(\frac{38}{49}\right)\) |
| \(\chi_{4334}(303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{601}{980}\right)\) | \(e\left(\frac{733}{980}\right)\) | \(e\left(\frac{166}{245}\right)\) | \(e\left(\frac{111}{490}\right)\) | \(e\left(\frac{207}{980}\right)\) | \(e\left(\frac{177}{490}\right)\) | \(e\left(\frac{313}{980}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{57}{196}\right)\) | \(e\left(\frac{34}{49}\right)\) |
| \(\chi_{4334}(305,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{677}{980}\right)\) | \(e\left(\frac{661}{980}\right)\) | \(e\left(\frac{17}{245}\right)\) | \(e\left(\frac{187}{490}\right)\) | \(e\left(\frac{799}{980}\right)\) | \(e\left(\frac{179}{490}\right)\) | \(e\left(\frac{801}{980}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{149}{196}\right)\) | \(e\left(\frac{33}{49}\right)\) |
| \(\chi_{4334}(315,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{893}{980}\right)\) | \(e\left(\frac{869}{980}\right)\) | \(e\left(\frac{148}{245}\right)\) | \(e\left(\frac{403}{490}\right)\) | \(e\left(\frac{831}{980}\right)\) | \(e\left(\frac{391}{490}\right)\) | \(e\left(\frac{589}{980}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{101}{196}\right)\) | \(e\left(\frac{25}{49}\right)\) |
| \(\chi_{4334}(321,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{839}{980}\right)\) | \(e\left(\frac{327}{980}\right)\) | \(e\left(\frac{54}{245}\right)\) | \(e\left(\frac{349}{490}\right)\) | \(e\left(\frac{333}{980}\right)\) | \(e\left(\frac{93}{490}\right)\) | \(e\left(\frac{887}{980}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{15}{196}\right)\) | \(e\left(\frac{27}{49}\right)\) |
| \(\chi_{4334}(327,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{587}{980}\right)\) | \(e\left(\frac{411}{980}\right)\) | \(e\left(\frac{187}{245}\right)\) | \(e\left(\frac{97}{490}\right)\) | \(e\left(\frac{949}{980}\right)\) | \(e\left(\frac{9}{490}\right)\) | \(e\left(\frac{971}{980}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{71}{196}\right)\) | \(e\left(\frac{20}{49}\right)\) |
| \(\chi_{4334}(337,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{453}{980}\right)\) | \(e\left(\frac{409}{980}\right)\) | \(e\left(\frac{108}{245}\right)\) | \(e\left(\frac{453}{490}\right)\) | \(e\left(\frac{911}{980}\right)\) | \(e\left(\frac{431}{490}\right)\) | \(e\left(\frac{549}{980}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{177}{196}\right)\) | \(e\left(\frac{5}{49}\right)\) |