from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38025, base_ring=CyclotomicField(390))
M = H._module
chi = DirichletCharacter(H, M([130,39,5]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,38025))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(38025\) | |
| Conductor: | \(38025\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(390\) | 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: | $\Q(\zeta_{195})$ |
| Fixed field: | Number field defined by a degree 390 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{38025}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{33}{130}\right)\) | \(e\left(\frac{127}{195}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{67}{390}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{38025}(439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{89}{130}\right)\) | \(e\left(\frac{116}{195}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{161}{390}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) |
| \(\chi_{38025}(589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{119}{130}\right)\) | \(e\left(\frac{1}{195}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{151}{390}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) |
| \(\chi_{38025}(1609,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{111}{130}\right)\) | \(e\left(\frac{179}{195}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{119}{390}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) |
| \(\chi_{38025}(1759,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{139}{195}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{319}{390}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) |
| \(\chi_{38025}(2194,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{113}{195}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{293}{390}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) |
| \(\chi_{38025}(2779,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{3}{130}\right)\) | \(e\left(\frac{47}{195}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{77}{390}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{38025}(2929,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{82}{195}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{97}{390}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{38025}(3364,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{79}{130}\right)\) | \(e\left(\frac{176}{195}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{251}{390}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) |
| \(\chi_{38025}(3514,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{29}{130}\right)\) | \(e\left(\frac{151}{195}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{181}{390}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) |
| \(\chi_{38025}(4534,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{44}{195}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{209}{390}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) |
| \(\chi_{38025}(4684,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{71}{130}\right)\) | \(e\left(\frac{94}{195}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{349}{390}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) |
| \(\chi_{38025}(5119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{47}{130}\right)\) | \(e\left(\frac{173}{195}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{383}{390}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) |
| \(\chi_{38025}(5269,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{27}{130}\right)\) | \(e\left(\frac{163}{195}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{43}{390}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) |
| \(\chi_{38025}(5704,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{123}{130}\right)\) | \(e\left(\frac{107}{195}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{167}{390}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{38025}(5854,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{37}{195}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{127}{390}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{38025}(6289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{41}{195}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{341}{390}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) |
| \(\chi_{38025}(6439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{106}{195}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{211}{390}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) |
| \(\chi_{38025}(7609,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{111}{130}\right)\) | \(e\left(\frac{49}{195}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{379}{390}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) |
| \(\chi_{38025}(8044,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{37}{130}\right)\) | \(e\left(\frac{38}{195}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{83}{390}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) |
| \(\chi_{38025}(8194,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{67}{130}\right)\) | \(e\left(\frac{118}{195}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{73}{390}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) |
| \(\chi_{38025}(8629,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{167}{195}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{257}{390}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{38025}(8779,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{23}{130}\right)\) | \(e\left(\frac{187}{195}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{157}{390}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{38025}(9214,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{59}{130}\right)\) | \(e\left(\frac{101}{195}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{41}{390}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) |
| \(\chi_{38025}(9364,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{61}{195}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{241}{390}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) |
| \(\chi_{38025}(10384,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{164}{195}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{389}{390}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) |
| \(\chi_{38025}(10534,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{21}{130}\right)\) | \(e\left(\frac{4}{195}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{19}{390}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) |
| \(\chi_{38025}(10969,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{27}{130}\right)\) | \(e\left(\frac{98}{195}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{173}{390}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) |
| \(\chi_{38025}(11119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{73}{195}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{103}{390}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) |
| \(\chi_{38025}(11554,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{103}{130}\right)\) | \(e\left(\frac{32}{195}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{347}{390}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{38025}(11704,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{142}{195}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{187}{390}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) |