from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7605, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([52,39,111]))
chi.galois_orbit()
[g,chi] = znchar(Mod(112,7605))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(7605\) | |
| Conductor: | \(7605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(156\) | 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: | $\Q(\zeta_{156})$ |
| Fixed field: | Number field defined by a degree 156 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{7605}(112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{97}{156}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(-i\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{7605}(148,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{55}{156}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(i\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{7605}(502,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{17}{156}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{7605}(538,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{7605}(697,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(-i\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{7605}(733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{19}{156}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(i\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{7605}(1087,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{53}{156}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{7605}(1123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{47}{156}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{7605}(1318,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(i\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{7605}(1672,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{7605}(1708,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{11}{156}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{7605}(1867,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{49}{156}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(-i\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{7605}(1903,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(i\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{7605}(2257,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{125}{156}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{7605}(2293,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{131}{156}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{7605}(2452,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{85}{156}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(-i\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{7605}(2488,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(i\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{7605}(2842,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{5}{156}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{7605}(2878,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{95}{156}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{7605}(3037,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(-i\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{7605}(3073,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{31}{156}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(i\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{7605}(3427,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{41}{156}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{7605}(3463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{7605}(3622,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(-i\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{7605}(3658,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(i\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{7605}(4012,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{77}{156}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{7605}(4048,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{23}{156}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{7605}(4207,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(-i\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{7605}(4243,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(i\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{7605}(4597,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{113}{156}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{7605}(4792,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(-i\) | \(e\left(\frac{11}{12}\right)\) |