from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1960, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,0,26]))
pari: [g,chi] = znchar(Mod(51,1960))
Basic properties
| Modulus: | \(1960\) | |
| Conductor: | \(392\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{392}(51,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1960.di
\(\chi_{1960}(11,\cdot)\) \(\chi_{1960}(51,\cdot)\) \(\chi_{1960}(291,\cdot)\) \(\chi_{1960}(331,\cdot)\) \(\chi_{1960}(571,\cdot)\) \(\chi_{1960}(611,\cdot)\) \(\chi_{1960}(891,\cdot)\) \(\chi_{1960}(1131,\cdot)\) \(\chi_{1960}(1171,\cdot)\) \(\chi_{1960}(1411,\cdot)\) \(\chi_{1960}(1691,\cdot)\) \(\chi_{1960}(1731,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.0.155718699466313184257207094263668545441599708733396657696588937331033553383727300608.1 |
Values on generators
\((1471,981,1177,1081)\) → \((-1,-1,1,e\left(\frac{13}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 1960 }(51, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)