from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9196, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,36,11]))
pari: [g,chi] = znchar(Mod(331,9196))
Basic properties
| Modulus: | \(9196\) | |
| Conductor: | \(9196\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | 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)
|
Galois orbit 9196.cb
\(\chi_{9196}(331,\cdot)\) \(\chi_{9196}(639,\cdot)\) \(\chi_{9196}(1167,\cdot)\) \(\chi_{9196}(1475,\cdot)\) \(\chi_{9196}(2003,\cdot)\) \(\chi_{9196}(2311,\cdot)\) \(\chi_{9196}(2839,\cdot)\) \(\chi_{9196}(3675,\cdot)\) \(\chi_{9196}(3983,\cdot)\) \(\chi_{9196}(4511,\cdot)\) \(\chi_{9196}(4819,\cdot)\) \(\chi_{9196}(5347,\cdot)\) \(\chi_{9196}(5655,\cdot)\) \(\chi_{9196}(6183,\cdot)\) \(\chi_{9196}(6491,\cdot)\) \(\chi_{9196}(7327,\cdot)\) \(\chi_{9196}(7855,\cdot)\) \(\chi_{9196}(8163,\cdot)\) \(\chi_{9196}(8691,\cdot)\) \(\chi_{9196}(8999,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((4599,3269,8229)\) → \((-1,e\left(\frac{6}{11}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 9196 }(331, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) |
sage: chi.jacobi_sum(n)