from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3513, base_ring=CyclotomicField(130))
M = H._module
chi = DirichletCharacter(H, M([65,14]))
chi.galois_orbit()
[g,chi] = znchar(Mod(41,3513))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3513\) | |
| Conductor: | \(3513\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(130\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{65})$ |
| Fixed field: | Number field defined by a degree 130 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3513}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{130}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{19}{130}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{28}{65}\right)\) |
| \(\chi_{3513}(158,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{130}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{12}{65}\right)\) |
| \(\chi_{3513}(236,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{130}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{99}{130}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{3}{130}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{65}\right)\) |
| \(\chi_{3513}(308,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{130}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{11}{130}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{87}{130}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{29}{65}\right)\) |
| \(\chi_{3513}(389,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{79}{130}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{93}{130}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{31}{65}\right)\) |
| \(\chi_{3513}(461,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{130}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{97}{130}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{54}{65}\right)\) |
| \(\chi_{3513}(467,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{130}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{46}{65}\right)\) |
| \(\chi_{3513}(482,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{89}{130}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{16}{65}\right)\) |
| \(\chi_{3513}(491,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{130}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{33}{130}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{1}{130}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{22}{65}\right)\) |
| \(\chi_{3513}(521,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{130}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{127}{130}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{64}{65}\right)\) |
| \(\chi_{3513}(533,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{130}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{129}{130}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{21}{65}\right)\) |
| \(\chi_{3513}(653,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{130}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{29}{130}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{123}{130}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{41}{65}\right)\) |
| \(\chi_{3513}(659,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{130}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{3}{130}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{71}{130}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{65}\right)\) |
| \(\chi_{3513}(707,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{130}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{61}{130}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{42}{65}\right)\) |
| \(\chi_{3513}(1073,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{130}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{17}{130}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{99}{130}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{33}{65}\right)\) |
| \(\chi_{3513}(1118,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{67}{130}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{23}{65}\right)\) |
| \(\chi_{3513}(1121,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{130}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{9}{130}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{6}{65}\right)\) |
| \(\chi_{3513}(1184,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{130}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{87}{130}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{58}{65}\right)\) |
| \(\chi_{3513}(1340,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{123}{130}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{23}{130}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{51}{65}\right)\) |
| \(\chi_{3513}(1433,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{93}{130}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{121}{130}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{62}{65}\right)\) |
| \(\chi_{3513}(1454,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{111}{130}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{27}{65}\right)\) |
| \(\chi_{3513}(1463,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{130}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{51}{130}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{37}{130}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{34}{65}\right)\) |
| \(\chi_{3513}(1544,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{130}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{7}{130}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{24}{65}\right)\) |
| \(\chi_{3513}(1628,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{130}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{23}{130}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{111}{130}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{37}{65}\right)\) |
| \(\chi_{3513}(1637,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{32}{65}\right)\) |
| \(\chi_{3513}(1946,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{41}{130}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{17}{130}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{49}{65}\right)\) |
| \(\chi_{3513}(2111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{129}{130}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{127}{130}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{59}{130}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{63}{65}\right)\) |
| \(\chi_{3513}(2123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{99}{130}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{37}{130}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{9}{130}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{65}\right)\) |
| \(\chi_{3513}(2174,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{61}{130}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{19}{65}\right)\) |
| \(\chi_{3513}(2180,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{77}{130}\right)\) | \(e\left(\frac{28}{65}\right)\) | \(e\left(\frac{89}{130}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{8}{65}\right)\) |
| \(\chi_{3513}(2252,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{43}{130}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{21}{130}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{65}\right)\) |