from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7623, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([110,165,303]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,7623))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(7623\) | |
| Conductor: | \(7623\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(330\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{165})$ |
| Fixed field: | Number field defined by a degree 330 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{7623}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{330}\right)\) | \(e\left(\frac{83}{165}\right)\) | \(e\left(\frac{37}{330}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{1}{165}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{203}{330}\right)\) |
| \(\chi_{7623}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{311}{330}\right)\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{79}{330}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{41}{330}\right)\) |
| \(\chi_{7623}(160,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{330}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{323}{330}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{247}{330}\right)\) |
| \(\chi_{7623}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{330}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{71}{330}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{229}{330}\right)\) |
| \(\chi_{7623}(391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{73}{330}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{71}{165}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{17}{330}\right)\) |
| \(\chi_{7623}(580,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{330}\right)\) | \(e\left(\frac{59}{165}\right)\) | \(e\left(\frac{241}{330}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{92}{165}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{29}{330}\right)\) |
| \(\chi_{7623}(601,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{330}\right)\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{59}{330}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{181}{330}\right)\) |
| \(\chi_{7623}(706,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{330}\right)\) | \(e\left(\frac{8}{165}\right)\) | \(e\left(\frac{97}{330}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{74}{165}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{113}{330}\right)\) |
| \(\chi_{7623}(832,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{330}\right)\) | \(e\left(\frac{101}{165}\right)\) | \(e\left(\frac{49}{330}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{68}{165}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{251}{330}\right)\) |
| \(\chi_{7623}(853,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{330}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{233}{330}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{149}{165}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{217}{330}\right)\) |
| \(\chi_{7623}(1042,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{19}{165}\right)\) | \(e\left(\frac{251}{330}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{289}{330}\right)\) |
| \(\chi_{7623}(1084,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{330}\right)\) | \(e\left(\frac{2}{165}\right)\) | \(e\left(\frac{313}{330}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{101}{165}\right)\) | \(e\left(\frac{4}{165}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{317}{330}\right)\) |
| \(\chi_{7623}(1168,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{330}\right)\) | \(e\left(\frac{13}{165}\right)\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{26}{165}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{163}{330}\right)\) |
| \(\chi_{7623}(1273,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{91}{330}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{163}{165}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{89}{330}\right)\) |
| \(\chi_{7623}(1294,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{330}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{61}{330}\right)\) |
| \(\chi_{7623}(1399,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{263}{330}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{157}{330}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{31}{165}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{23}{330}\right)\) |
| \(\chi_{7623}(1525,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{221}{330}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{131}{330}\right)\) |
| \(\chi_{7623}(1735,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{289}{330}\right)\) | \(e\left(\frac{124}{165}\right)\) | \(e\left(\frac{101}{330}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{83}{165}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{19}{330}\right)\) |
| \(\chi_{7623}(1777,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{330}\right)\) | \(e\left(\frac{32}{165}\right)\) | \(e\left(\frac{223}{330}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{287}{330}\right)\) |
| \(\chi_{7623}(1861,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{330}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{197}{330}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{4}{165}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{73}{330}\right)\) |
| \(\chi_{7623}(1966,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{269}{330}\right)\) | \(e\left(\frac{104}{165}\right)\) | \(e\left(\frac{271}{330}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{137}{165}\right)\) | \(e\left(\frac{43}{165}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{149}{330}\right)\) |
| \(\chi_{7623}(1987,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{301}{330}\right)\) | \(e\left(\frac{136}{165}\right)\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{107}{165}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{271}{330}\right)\) |
| \(\chi_{7623}(2092,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{330}\right)\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{217}{330}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{46}{165}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{263}{330}\right)\) |
| \(\chi_{7623}(2239,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{217}{330}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{53}{330}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{104}{165}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{157}{330}\right)\) |
| \(\chi_{7623}(2428,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{330}\right)\) | \(e\left(\frac{64}{165}\right)\) | \(e\left(\frac{281}{330}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{128}{165}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{79}{330}\right)\) |
| \(\chi_{7623}(2470,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{330}\right)\) | \(e\left(\frac{62}{165}\right)\) | \(e\left(\frac{133}{330}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{124}{165}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{257}{330}\right)\) |
| \(\chi_{7623}(2554,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{330}\right)\) | \(e\left(\frac{28}{165}\right)\) | \(e\left(\frac{257}{330}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{313}{330}\right)\) |
| \(\chi_{7623}(2680,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{330}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{299}{330}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{151}{330}\right)\) |
| \(\chi_{7623}(2785,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{330}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{14}{165}\right)\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{173}{330}\right)\) |
| \(\chi_{7623}(2911,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{330}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{289}{330}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{97}{165}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{221}{330}\right)\) |
| \(\chi_{7623}(2932,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{247}{330}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{293}{330}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{16}{165}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{127}{330}\right)\) |