from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9898, base_ring=CyclotomicField(700))
M = H._module
chi = DirichletCharacter(H, M([50,49]))
chi.galois_orbit()
[g,chi] = znchar(Mod(27,9898))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(9898\) | |
| Conductor: | \(4949\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(700\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 4949.co | 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: | $\Q(\zeta_{700})$ |
| Fixed field: | Number field defined by a degree 700 polynomial (not computed) |
First 31 of 240 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{9898}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{631}{700}\right)\) | \(e\left(\frac{263}{350}\right)\) | \(e\left(\frac{281}{350}\right)\) | \(e\left(\frac{537}{700}\right)\) | \(e\left(\frac{171}{175}\right)\) | \(e\left(\frac{457}{700}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{257}{350}\right)\) | \(e\left(\frac{88}{175}\right)\) |
| \(\chi_{9898}(55,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{700}\right)\) | \(e\left(\frac{183}{350}\right)\) | \(e\left(\frac{121}{350}\right)\) | \(e\left(\frac{367}{700}\right)\) | \(e\left(\frac{111}{175}\right)\) | \(e\left(\frac{487}{700}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{87}{350}\right)\) | \(e\left(\frac{8}{175}\right)\) |
| \(\chi_{9898}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{437}{700}\right)\) | \(e\left(\frac{201}{350}\right)\) | \(e\left(\frac{87}{350}\right)\) | \(e\left(\frac{99}{700}\right)\) | \(e\left(\frac{142}{175}\right)\) | \(e\left(\frac{139}{700}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{239}{350}\right)\) | \(e\left(\frac{26}{175}\right)\) |
| \(\chi_{9898}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{201}{700}\right)\) | \(e\left(\frac{223}{350}\right)\) | \(e\left(\frac{201}{350}\right)\) | \(e\left(\frac{627}{700}\right)\) | \(e\left(\frac{141}{175}\right)\) | \(e\left(\frac{647}{700}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{347}{350}\right)\) | \(e\left(\frac{48}{175}\right)\) |
| \(\chi_{9898}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{700}\right)\) | \(e\left(\frac{297}{350}\right)\) | \(e\left(\frac{139}{350}\right)\) | \(e\left(\frac{653}{700}\right)\) | \(e\left(\frac{74}{175}\right)\) | \(e\left(\frac{33}{700}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{233}{350}\right)\) | \(e\left(\frac{122}{175}\right)\) |
| \(\chi_{9898}(209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{697}{700}\right)\) | \(e\left(\frac{331}{350}\right)\) | \(e\left(\frac{347}{350}\right)\) | \(e\left(\frac{419}{700}\right)\) | \(e\left(\frac{152}{175}\right)\) | \(e\left(\frac{659}{700}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{209}{350}\right)\) | \(e\left(\frac{156}{175}\right)\) |
| \(\chi_{9898}(237,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{700}\right)\) | \(e\left(\frac{97}{350}\right)\) | \(e\left(\frac{89}{350}\right)\) | \(e\left(\frac{403}{700}\right)\) | \(e\left(\frac{99}{175}\right)\) | \(e\left(\frac{283}{700}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{333}{350}\right)\) | \(e\left(\frac{97}{175}\right)\) |
| \(\chi_{9898}(265,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{700}\right)\) | \(e\left(\frac{73}{350}\right)\) | \(e\left(\frac{251}{350}\right)\) | \(e\left(\frac{177}{700}\right)\) | \(e\left(\frac{116}{175}\right)\) | \(e\left(\frac{397}{700}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{247}{350}\right)\) | \(e\left(\frac{73}{175}\right)\) |
| \(\chi_{9898}(321,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{687}{700}\right)\) | \(e\left(\frac{151}{350}\right)\) | \(e\left(\frac{337}{350}\right)\) | \(e\left(\frac{649}{700}\right)\) | \(e\left(\frac{17}{175}\right)\) | \(e\left(\frac{289}{700}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{89}{350}\right)\) | \(e\left(\frac{151}{175}\right)\) |
| \(\chi_{9898}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{471}{700}\right)\) | \(e\left(\frac{183}{350}\right)\) | \(e\left(\frac{121}{350}\right)\) | \(e\left(\frac{17}{700}\right)\) | \(e\left(\frac{111}{175}\right)\) | \(e\left(\frac{137}{700}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{87}{350}\right)\) | \(e\left(\frac{8}{175}\right)\) |
| \(\chi_{9898}(377,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{381}{700}\right)\) | \(e\left(\frac{313}{350}\right)\) | \(e\left(\frac{31}{350}\right)\) | \(e\left(\frac{687}{700}\right)\) | \(e\left(\frac{121}{175}\right)\) | \(e\left(\frac{307}{700}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{57}{350}\right)\) | \(e\left(\frac{138}{175}\right)\) |
| \(\chi_{9898}(419,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{700}\right)\) | \(e\left(\frac{137}{350}\right)\) | \(e\left(\frac{169}{350}\right)\) | \(e\left(\frac{663}{700}\right)\) | \(e\left(\frac{129}{175}\right)\) | \(e\left(\frac{443}{700}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{243}{350}\right)\) | \(e\left(\frac{137}{175}\right)\) |
| \(\chi_{9898}(433,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{700}\right)\) | \(e\left(\frac{69}{350}\right)\) | \(e\left(\frac{103}{350}\right)\) | \(e\left(\frac{81}{700}\right)\) | \(e\left(\frac{148}{175}\right)\) | \(e\left(\frac{241}{700}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{291}{350}\right)\) | \(e\left(\frac{69}{175}\right)\) |
| \(\chi_{9898}(503,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{683}{700}\right)\) | \(e\left(\frac{9}{350}\right)\) | \(e\left(\frac{333}{350}\right)\) | \(e\left(\frac{41}{700}\right)\) | \(e\left(\frac{103}{175}\right)\) | \(e\left(\frac{1}{700}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{251}{350}\right)\) | \(e\left(\frac{9}{175}\right)\) |
| \(\chi_{9898}(517,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{700}\right)\) | \(e\left(\frac{39}{350}\right)\) | \(e\left(\frac{43}{350}\right)\) | \(e\left(\frac{61}{700}\right)\) | \(e\left(\frac{38}{175}\right)\) | \(e\left(\frac{121}{700}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{271}{350}\right)\) | \(e\left(\frac{39}{175}\right)\) |
| \(\chi_{9898}(531,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{411}{700}\right)\) | \(e\left(\frac{153}{350}\right)\) | \(e\left(\frac{61}{350}\right)\) | \(e\left(\frac{697}{700}\right)\) | \(e\left(\frac{1}{175}\right)\) | \(e\left(\frac{17}{700}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{67}{350}\right)\) | \(e\left(\frac{153}{175}\right)\) |
| \(\chi_{9898}(545,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{700}\right)\) | \(e\left(\frac{43}{350}\right)\) | \(e\left(\frac{191}{350}\right)\) | \(e\left(\frac{157}{700}\right)\) | \(e\left(\frac{6}{175}\right)\) | \(e\left(\frac{277}{700}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{227}{350}\right)\) | \(e\left(\frac{43}{175}\right)\) |
| \(\chi_{9898}(657,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{700}\right)\) | \(e\left(\frac{241}{350}\right)\) | \(e\left(\frac{167}{350}\right)\) | \(e\left(\frac{9}{700}\right)\) | \(e\left(\frac{172}{175}\right)\) | \(e\left(\frac{649}{700}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{149}{350}\right)\) | \(e\left(\frac{66}{175}\right)\) |
| \(\chi_{9898}(699,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{249}{700}\right)\) | \(e\left(\frac{177}{350}\right)\) | \(e\left(\frac{249}{350}\right)\) | \(e\left(\frac{223}{700}\right)\) | \(e\left(\frac{159}{175}\right)\) | \(e\left(\frac{603}{700}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{153}{350}\right)\) | \(e\left(\frac{2}{175}\right)\) |
| \(\chi_{9898}(741,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{700}\right)\) | \(e\left(\frac{29}{350}\right)\) | \(e\left(\frac{23}{350}\right)\) | \(e\left(\frac{521}{700}\right)\) | \(e\left(\frac{118}{175}\right)\) | \(e\left(\frac{81}{700}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{31}{350}\right)\) | \(e\left(\frac{29}{175}\right)\) |
| \(\chi_{9898}(755,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{700}\right)\) | \(e\left(\frac{157}{350}\right)\) | \(e\left(\frac{209}{350}\right)\) | \(e\left(\frac{443}{700}\right)\) | \(e\left(\frac{144}{175}\right)\) | \(e\left(\frac{523}{700}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{23}{350}\right)\) | \(e\left(\frac{157}{175}\right)\) |
| \(\chi_{9898}(797,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{700}\right)\) | \(e\left(\frac{317}{350}\right)\) | \(e\left(\frac{179}{350}\right)\) | \(e\left(\frac{433}{700}\right)\) | \(e\left(\frac{89}{175}\right)\) | \(e\left(\frac{113}{700}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{13}{350}\right)\) | \(e\left(\frac{142}{175}\right)\) |
| \(\chi_{9898}(811,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{477}{700}\right)\) | \(e\left(\frac{221}{350}\right)\) | \(e\left(\frac{127}{350}\right)\) | \(e\left(\frac{579}{700}\right)\) | \(e\left(\frac{157}{175}\right)\) | \(e\left(\frac{219}{700}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{19}{350}\right)\) | \(e\left(\frac{46}{175}\right)\) |
| \(\chi_{9898}(867,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{700}\right)\) | \(e\left(\frac{61}{350}\right)\) | \(e\left(\frac{157}{350}\right)\) | \(e\left(\frac{239}{700}\right)\) | \(e\left(\frac{37}{175}\right)\) | \(e\left(\frac{279}{700}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{29}{350}\right)\) | \(e\left(\frac{61}{175}\right)\) |
| \(\chi_{9898}(937,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{700}\right)\) | \(e\left(\frac{99}{350}\right)\) | \(e\left(\frac{163}{350}\right)\) | \(e\left(\frac{101}{700}\right)\) | \(e\left(\frac{83}{175}\right)\) | \(e\left(\frac{361}{700}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{311}{350}\right)\) | \(e\left(\frac{99}{175}\right)\) |
| \(\chi_{9898}(951,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{307}{700}\right)\) | \(e\left(\frac{311}{350}\right)\) | \(e\left(\frac{307}{350}\right)\) | \(e\left(\frac{289}{700}\right)\) | \(e\left(\frac{137}{175}\right)\) | \(e\left(\frac{229}{700}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{79}{350}\right)\) | \(e\left(\frac{136}{175}\right)\) |
| \(\chi_{9898}(1007,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{700}\right)\) | \(e\left(\frac{221}{350}\right)\) | \(e\left(\frac{127}{350}\right)\) | \(e\left(\frac{229}{700}\right)\) | \(e\left(\frac{157}{175}\right)\) | \(e\left(\frac{569}{700}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{19}{350}\right)\) | \(e\left(\frac{46}{175}\right)\) |
| \(\chi_{9898}(1021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{700}\right)\) | \(e\left(\frac{167}{350}\right)\) | \(e\left(\frac{229}{350}\right)\) | \(e\left(\frac{683}{700}\right)\) | \(e\left(\frac{64}{175}\right)\) | \(e\left(\frac{563}{700}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{263}{350}\right)\) | \(e\left(\frac{167}{175}\right)\) |
| \(\chi_{9898}(1063,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{700}\right)\) | \(e\left(\frac{257}{350}\right)\) | \(e\left(\frac{59}{350}\right)\) | \(e\left(\frac{393}{700}\right)\) | \(e\left(\frac{44}{175}\right)\) | \(e\left(\frac{573}{700}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{323}{350}\right)\) | \(e\left(\frac{82}{175}\right)\) |
| \(\chi_{9898}(1119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{299}{700}\right)\) | \(e\left(\frac{27}{350}\right)\) | \(e\left(\frac{299}{350}\right)\) | \(e\left(\frac{473}{700}\right)\) | \(e\left(\frac{134}{175}\right)\) | \(e\left(\frac{353}{700}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{53}{350}\right)\) | \(e\left(\frac{27}{175}\right)\) |
| \(\chi_{9898}(1161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{700}\right)\) | \(e\left(\frac{341}{350}\right)\) | \(e\left(\frac{17}{350}\right)\) | \(e\left(\frac{659}{700}\right)\) | \(e\left(\frac{72}{175}\right)\) | \(e\left(\frac{699}{700}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{99}{350}\right)\) | \(e\left(\frac{166}{175}\right)\) |