from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9025, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,24]))
pari: [g,chi] = znchar(Mod(324,9025))
Basic properties
| Modulus: | \(9025\) | |
| Conductor: | \(1805\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1805}(324,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9025.bf
\(\chi_{9025}(324,\cdot)\) \(\chi_{9025}(799,\cdot)\) \(\chi_{9025}(1274,\cdot)\) \(\chi_{9025}(1749,\cdot)\) \(\chi_{9025}(2224,\cdot)\) \(\chi_{9025}(2699,\cdot)\) \(\chi_{9025}(3174,\cdot)\) \(\chi_{9025}(3649,\cdot)\) \(\chi_{9025}(4124,\cdot)\) \(\chi_{9025}(4599,\cdot)\) \(\chi_{9025}(5074,\cdot)\) \(\chi_{9025}(5549,\cdot)\) \(\chi_{9025}(6024,\cdot)\) \(\chi_{9025}(6974,\cdot)\) \(\chi_{9025}(7449,\cdot)\) \(\chi_{9025}(7924,\cdot)\) \(\chi_{9025}(8399,\cdot)\) \(\chi_{9025}(8874,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
| Fixed field: | Number field defined by a degree 38 polynomial |
Values on generators
\((5777,3251)\) → \((-1,e\left(\frac{12}{19}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 9025 }(324, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{11}{38}\right)\) |
sage: chi.jacobi_sum(n)