from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9405, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([30,9,18,10]))
pari: [g,chi] = znchar(Mod(32,9405))
Basic properties
| Modulus: | \(9405\) | |
| Conductor: | \(9405\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | 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 9405.ko
\(\chi_{9405}(32,\cdot)\) \(\chi_{9405}(428,\cdot)\) \(\chi_{9405}(857,\cdot)\) \(\chi_{9405}(1022,\cdot)\) \(\chi_{9405}(1352,\cdot)\) \(\chi_{9405}(1913,\cdot)\) \(\chi_{9405}(2738,\cdot)\) \(\chi_{9405}(2903,\cdot)\) \(\chi_{9405}(3233,\cdot)\) \(\chi_{9405}(6797,\cdot)\) \(\chi_{9405}(7952,\cdot)\) \(\chi_{9405}(8678,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((1046,1882,5986,496)\) → \((e\left(\frac{5}{6}\right),i,-1,e\left(\frac{5}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(23\) | \(26\) |
| \( \chi_{ 9405 }(32, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)