from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3513, base_ring=CyclotomicField(1170))
M = H._module
chi = DirichletCharacter(H, M([0,599]))
chi.galois_orbit()
[g,chi] = znchar(Mod(10,3513))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3513\) | |
| Conductor: | \(1171\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1170\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1171.x | 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: | $\Q(\zeta_{585})$ |
| Fixed field: | Number field defined by a degree 1170 polynomial (not computed) |
First 31 of 288 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3513}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{599}{1170}\right)\) | \(e\left(\frac{14}{585}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{31}{234}\right)\) | \(e\left(\frac{209}{390}\right)\) | \(e\left(\frac{781}{1170}\right)\) | \(e\left(\frac{353}{390}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{28}{585}\right)\) |
| \(\chi_{3513}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{841}{1170}\right)\) | \(e\left(\frac{256}{585}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{149}{234}\right)\) | \(e\left(\frac{61}{390}\right)\) | \(e\left(\frac{659}{1170}\right)\) | \(e\left(\frac{187}{390}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{512}{585}\right)\) |
| \(\chi_{3513}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{601}{1170}\right)\) | \(e\left(\frac{16}{585}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{119}{234}\right)\) | \(e\left(\frac{211}{390}\right)\) | \(e\left(\frac{809}{1170}\right)\) | \(e\left(\frac{97}{390}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{32}{585}\right)\) |
| \(\chi_{3513}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{719}{1170}\right)\) | \(e\left(\frac{134}{585}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{163}{234}\right)\) | \(e\left(\frac{329}{390}\right)\) | \(e\left(\frac{121}{1170}\right)\) | \(e\left(\frac{203}{390}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{268}{585}\right)\) |
| \(\chi_{3513}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{349}{1170}\right)\) | \(e\left(\frac{349}{585}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{29}{234}\right)\) | \(e\left(\frac{349}{390}\right)\) | \(e\left(\frac{791}{1170}\right)\) | \(e\left(\frac{373}{390}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{113}{585}\right)\) |
| \(\chi_{3513}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{799}{1170}\right)\) | \(e\left(\frac{214}{585}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{173}{234}\right)\) | \(e\left(\frac{19}{390}\right)\) | \(e\left(\frac{71}{1170}\right)\) | \(e\left(\frac{103}{390}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{428}{585}\right)\) |
| \(\chi_{3513}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{1170}\right)\) | \(e\left(\frac{53}{585}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{109}{234}\right)\) | \(e\left(\frac{53}{390}\right)\) | \(e\left(\frac{157}{1170}\right)\) | \(e\left(\frac{41}{390}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{106}{585}\right)\) |
| \(\chi_{3513}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{1170}\right)\) | \(e\left(\frac{127}{585}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{89}{234}\right)\) | \(e\left(\frac{127}{390}\right)\) | \(e\left(\frac{23}{1170}\right)\) | \(e\left(\frac{319}{390}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{254}{585}\right)\) |
| \(\chi_{3513}(94,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1147}{1170}\right)\) | \(e\left(\frac{562}{585}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{41}{234}\right)\) | \(e\left(\frac{367}{390}\right)\) | \(e\left(\frac{263}{1170}\right)\) | \(e\left(\frac{19}{390}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{539}{585}\right)\) |
| \(\chi_{3513}(112,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{809}{1170}\right)\) | \(e\left(\frac{224}{585}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{145}{234}\right)\) | \(e\left(\frac{29}{390}\right)\) | \(e\left(\frac{211}{1170}\right)\) | \(e\left(\frac{383}{390}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{448}{585}\right)\) |
| \(\chi_{3513}(118,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{881}{1170}\right)\) | \(e\left(\frac{296}{585}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{37}{234}\right)\) | \(e\left(\frac{101}{390}\right)\) | \(e\left(\frac{49}{1170}\right)\) | \(e\left(\frac{137}{390}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{7}{585}\right)\) |
| \(\chi_{3513}(136,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{671}{1170}\right)\) | \(e\left(\frac{86}{585}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{157}{234}\right)\) | \(e\left(\frac{281}{390}\right)\) | \(e\left(\frac{619}{1170}\right)\) | \(e\left(\frac{107}{390}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{172}{585}\right)\) |
| \(\chi_{3513}(154,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1007}{1170}\right)\) | \(e\left(\frac{422}{585}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{199}{234}\right)\) | \(e\left(\frac{227}{390}\right)\) | \(e\left(\frac{643}{1170}\right)\) | \(e\left(\frac{389}{390}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{259}{585}\right)\) |
| \(\chi_{3513}(172,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{721}{1170}\right)\) | \(e\left(\frac{136}{585}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{17}{234}\right)\) | \(e\left(\frac{331}{390}\right)\) | \(e\left(\frac{149}{1170}\right)\) | \(e\left(\frac{337}{390}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{272}{585}\right)\) |
| \(\chi_{3513}(187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{869}{1170}\right)\) | \(e\left(\frac{284}{585}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{211}{234}\right)\) | \(e\left(\frac{89}{390}\right)\) | \(e\left(\frac{1051}{1170}\right)\) | \(e\left(\frac{113}{390}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{568}{585}\right)\) |
| \(\chi_{3513}(190,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{331}{1170}\right)\) | \(e\left(\frac{331}{585}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{173}{234}\right)\) | \(e\left(\frac{331}{390}\right)\) | \(e\left(\frac{539}{1170}\right)\) | \(e\left(\frac{337}{390}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{77}{585}\right)\) |
| \(\chi_{3513}(217,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{707}{1170}\right)\) | \(e\left(\frac{122}{585}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{103}{234}\right)\) | \(e\left(\frac{317}{390}\right)\) | \(e\left(\frac{1123}{1170}\right)\) | \(e\left(\frac{179}{390}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{244}{585}\right)\) |
| \(\chi_{3513}(226,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{1170}\right)\) | \(e\left(\frac{113}{585}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{175}{234}\right)\) | \(e\left(\frac{113}{390}\right)\) | \(e\left(\frac{997}{1170}\right)\) | \(e\left(\frac{161}{390}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{226}{585}\right)\) |
| \(\chi_{3513}(241,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{1170}\right)\) | \(e\left(\frac{71}{585}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{199}{234}\right)\) | \(e\left(\frac{71}{390}\right)\) | \(e\left(\frac{409}{1170}\right)\) | \(e\left(\frac{77}{390}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{142}{585}\right)\) |
| \(\chi_{3513}(265,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{409}{1170}\right)\) | \(e\left(\frac{409}{585}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{95}{234}\right)\) | \(e\left(\frac{19}{390}\right)\) | \(e\left(\frac{461}{1170}\right)\) | \(e\left(\frac{103}{390}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{233}{585}\right)\) |
| \(\chi_{3513}(337,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1033}{1170}\right)\) | \(e\left(\frac{448}{585}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{173}{234}\right)\) | \(e\left(\frac{253}{390}\right)\) | \(e\left(\frac{1007}{1170}\right)\) | \(e\left(\frac{181}{390}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{311}{585}\right)\) |
| \(\chi_{3513}(346,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{967}{1170}\right)\) | \(e\left(\frac{382}{585}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{77}{234}\right)\) | \(e\left(\frac{187}{390}\right)\) | \(e\left(\frac{83}{1170}\right)\) | \(e\left(\frac{49}{390}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{179}{585}\right)\) |
| \(\chi_{3513}(358,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{493}{1170}\right)\) | \(e\left(\frac{493}{585}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{47}{234}\right)\) | \(e\left(\frac{103}{390}\right)\) | \(e\left(\frac{467}{1170}\right)\) | \(e\left(\frac{271}{390}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{401}{585}\right)\) |
| \(\chi_{3513}(379,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1099}{1170}\right)\) | \(e\left(\frac{514}{585}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{35}{234}\right)\) | \(e\left(\frac{319}{390}\right)\) | \(e\left(\frac{761}{1170}\right)\) | \(e\left(\frac{313}{390}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{443}{585}\right)\) |
| \(\chi_{3513}(406,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{991}{1170}\right)\) | \(e\left(\frac{406}{585}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{197}{234}\right)\) | \(e\left(\frac{211}{390}\right)\) | \(e\left(\frac{419}{1170}\right)\) | \(e\left(\frac{97}{390}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{227}{585}\right)\) |
| \(\chi_{3513}(409,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{653}{1170}\right)\) | \(e\left(\frac{68}{585}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{67}{234}\right)\) | \(e\left(\frac{263}{390}\right)\) | \(e\left(\frac{367}{1170}\right)\) | \(e\left(\frac{71}{390}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{136}{585}\right)\) |
| \(\chi_{3513}(412,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{1170}\right)\) | \(e\left(\frac{119}{585}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{205}{234}\right)\) | \(e\left(\frac{119}{390}\right)\) | \(e\left(\frac{1081}{1170}\right)\) | \(e\left(\frac{173}{390}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{238}{585}\right)\) |
| \(\chi_{3513}(448,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{811}{1170}\right)\) | \(e\left(\frac{226}{585}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{233}{234}\right)\) | \(e\left(\frac{31}{390}\right)\) | \(e\left(\frac{239}{1170}\right)\) | \(e\left(\frac{127}{390}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{452}{585}\right)\) |
| \(\chi_{3513}(469,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{727}{1170}\right)\) | \(e\left(\frac{142}{585}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{47}{234}\right)\) | \(e\left(\frac{337}{390}\right)\) | \(e\left(\frac{233}{1170}\right)\) | \(e\left(\frac{349}{390}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{284}{585}\right)\) |
| \(\chi_{3513}(472,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{883}{1170}\right)\) | \(e\left(\frac{298}{585}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{125}{234}\right)\) | \(e\left(\frac{103}{390}\right)\) | \(e\left(\frac{77}{1170}\right)\) | \(e\left(\frac{271}{390}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{11}{585}\right)\) |
| \(\chi_{3513}(478,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{731}{1170}\right)\) | \(e\left(\frac{146}{585}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{223}{234}\right)\) | \(e\left(\frac{341}{390}\right)\) | \(e\left(\frac{289}{1170}\right)\) | \(e\left(\frac{227}{390}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{292}{585}\right)\) |