from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5776, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,19,1]))
pari: [g,chi] = znchar(Mod(151,5776))
Basic properties
| Modulus: | \(5776\) | |
| Conductor: | \(2888\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2888}(1595,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5776.br
\(\chi_{5776}(151,\cdot)\) \(\chi_{5776}(455,\cdot)\) \(\chi_{5776}(759,\cdot)\) \(\chi_{5776}(1063,\cdot)\) \(\chi_{5776}(1367,\cdot)\) \(\chi_{5776}(1671,\cdot)\) \(\chi_{5776}(1975,\cdot)\) \(\chi_{5776}(2279,\cdot)\) \(\chi_{5776}(2583,\cdot)\) \(\chi_{5776}(3191,\cdot)\) \(\chi_{5776}(3495,\cdot)\) \(\chi_{5776}(3799,\cdot)\) \(\chi_{5776}(4103,\cdot)\) \(\chi_{5776}(4407,\cdot)\) \(\chi_{5776}(4711,\cdot)\) \(\chi_{5776}(5015,\cdot)\) \(\chi_{5776}(5319,\cdot)\) \(\chi_{5776}(5623,\cdot)\)
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: | 38.38.321901219811890081790219546628722051791865953039568238015939027374467326085267423464178688376545784307644366848.1 |
Values on generators
\((5055,1445,2529)\) → \((-1,-1,e\left(\frac{1}{38}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 5776 }(151, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) |
sage: chi.jacobi_sum(n)