from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7220, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,19,31]))
pari: [g,chi] = znchar(Mod(379,7220))
Basic properties
| Modulus: | \(7220\) | |
| Conductor: | \(7220\) | 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 7220.bp
\(\chi_{7220}(379,\cdot)\) \(\chi_{7220}(759,\cdot)\) \(\chi_{7220}(1139,\cdot)\) \(\chi_{7220}(1519,\cdot)\) \(\chi_{7220}(1899,\cdot)\) \(\chi_{7220}(2279,\cdot)\) \(\chi_{7220}(2659,\cdot)\) \(\chi_{7220}(3039,\cdot)\) \(\chi_{7220}(3419,\cdot)\) \(\chi_{7220}(3799,\cdot)\) \(\chi_{7220}(4179,\cdot)\) \(\chi_{7220}(4559,\cdot)\) \(\chi_{7220}(4939,\cdot)\) \(\chi_{7220}(5319,\cdot)\) \(\chi_{7220}(5699,\cdot)\) \(\chi_{7220}(6079,\cdot)\) \(\chi_{7220}(6459,\cdot)\) \(\chi_{7220}(6839,\cdot)\)
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
\((3611,5777,6861)\) → \((-1,-1,e\left(\frac{31}{38}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 7220 }(379, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{33}{38}\right)\) |
sage: chi.jacobi_sum(n)