from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9196, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,9,20]))
pari: [g,chi] = znchar(Mod(239,9196))
Basic properties
Modulus: | \(9196\) | |
Conductor: | \(836\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{836}(239,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9196.bp
\(\chi_{9196}(239,\cdot)\) \(\chi_{9196}(2823,\cdot)\) \(\chi_{9196}(2895,\cdot)\) \(\chi_{9196}(4111,\cdot)\) \(\chi_{9196}(4571,\cdot)\) \(\chi_{9196}(6695,\cdot)\) \(\chi_{9196}(8219,\cdot)\) \(\chi_{9196}(8443,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((4599,3269,8229)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
\( \chi_{ 9196 }(239, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{15}\right)\) |
sage: chi.jacobi_sum(n)