from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3136, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,8]))
pari: [g,chi] = znchar(Mod(289,3136))
Basic properties
| Modulus: | \(3136\) | |
| 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 \(\chi_{392}(93,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3136.bx
\(\chi_{3136}(289,\cdot)\) \(\chi_{3136}(417,\cdot)\) \(\chi_{3136}(737,\cdot)\) \(\chi_{3136}(865,\cdot)\) \(\chi_{3136}(1185,\cdot)\) \(\chi_{3136}(1313,\cdot)\) \(\chi_{3136}(1633,\cdot)\) \(\chi_{3136}(1761,\cdot)\) \(\chi_{3136}(2081,\cdot)\) \(\chi_{3136}(2209,\cdot)\) \(\chi_{3136}(2657,\cdot)\) \(\chi_{3136}(2977,\cdot)\)
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: | 42.42.155718699466313184257207094263668545441599708733396657696588937331033553383727300608.1 |
Values on generators
\((1471,197,1473)\) → \((1,-1,e\left(\frac{4}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 3136 }(289, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) |
sage: chi.jacobi_sum(n)