from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4332, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,19,32]))
pari: [g,chi] = znchar(Mod(191,4332))
Basic properties
| Modulus: | \(4332\) | |
| Conductor: | \(4332\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | 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 4332.be
\(\chi_{4332}(191,\cdot)\) \(\chi_{4332}(419,\cdot)\) \(\chi_{4332}(647,\cdot)\) \(\chi_{4332}(875,\cdot)\) \(\chi_{4332}(1103,\cdot)\) \(\chi_{4332}(1331,\cdot)\) \(\chi_{4332}(1559,\cdot)\) \(\chi_{4332}(1787,\cdot)\) \(\chi_{4332}(2015,\cdot)\) \(\chi_{4332}(2243,\cdot)\) \(\chi_{4332}(2471,\cdot)\) \(\chi_{4332}(2699,\cdot)\) \(\chi_{4332}(2927,\cdot)\) \(\chi_{4332}(3155,\cdot)\) \(\chi_{4332}(3383,\cdot)\) \(\chi_{4332}(3839,\cdot)\) \(\chi_{4332}(4067,\cdot)\) \(\chi_{4332}(4295,\cdot)\)
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
\((2167,1445,3973)\) → \((-1,-1,e\left(\frac{16}{19}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4332 }(191, a) \) | \(1\) | \(1\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) |
sage: chi.jacobi_sum(n)