from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1225, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([56,20]))
chi.galois_orbit()
[g,chi] = znchar(Mod(36,1225))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1225\) | |
| Conductor: | \(1225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(35\) | 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: | $\Q(\zeta_{35})$ |
| Fixed field: | Number field defined by a degree 35 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1225}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) |
| \(\chi_{1225}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) |
| \(\chi_{1225}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) |
| \(\chi_{1225}(141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) |
| \(\chi_{1225}(211,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) |
| \(\chi_{1225}(281,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) |
| \(\chi_{1225}(316,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) |
| \(\chi_{1225}(386,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) |
| \(\chi_{1225}(421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) |
| \(\chi_{1225}(456,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) |
| \(\chi_{1225}(561,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) |
| \(\chi_{1225}(596,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) |
| \(\chi_{1225}(631,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) |
| \(\chi_{1225}(666,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) |
| \(\chi_{1225}(771,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) |
| \(\chi_{1225}(806,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) |
| \(\chi_{1225}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) |
| \(\chi_{1225}(911,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) |
| \(\chi_{1225}(946,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) |
| \(\chi_{1225}(1016,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) |
| \(\chi_{1225}(1086,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) |
| \(\chi_{1225}(1121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) |
| \(\chi_{1225}(1156,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) |
| \(\chi_{1225}(1191,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) |