from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1210, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,10]))
pari: [g,chi] = znchar(Mod(43,1210))
Basic properties
| Modulus: | \(1210\) | |
| Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{605}(43,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1210.q
\(\chi_{1210}(43,\cdot)\) \(\chi_{1210}(87,\cdot)\) \(\chi_{1210}(153,\cdot)\) \(\chi_{1210}(197,\cdot)\) \(\chi_{1210}(263,\cdot)\) \(\chi_{1210}(307,\cdot)\) \(\chi_{1210}(373,\cdot)\) \(\chi_{1210}(417,\cdot)\) \(\chi_{1210}(527,\cdot)\) \(\chi_{1210}(593,\cdot)\) \(\chi_{1210}(637,\cdot)\) \(\chi_{1210}(703,\cdot)\) \(\chi_{1210}(747,\cdot)\) \(\chi_{1210}(813,\cdot)\) \(\chi_{1210}(857,\cdot)\) \(\chi_{1210}(923,\cdot)\) \(\chi_{1210}(1033,\cdot)\) \(\chi_{1210}(1077,\cdot)\) \(\chi_{1210}(1143,\cdot)\) \(\chi_{1210}(1187,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{44})\) |
| Fixed field: | 44.44.2885428559557085084648615903962269104974580506944665166312236845353556846511909399754484184086322784423828125.1 |
Values on generators
\((727,1091)\) → \((-i,e\left(\frac{5}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1210 }(43, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{15}{44}\right)\) | \(-1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(-i\) | \(e\left(\frac{4}{11}\right)\) |
sage: chi.jacobi_sum(n)