from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7942, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,17]))
pari: [g,chi] = znchar(Mod(417,7942))
Basic properties
| Modulus: | \(7942\) | |
| Conductor: | \(3971\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3971}(417,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7942.x
\(\chi_{7942}(417,\cdot)\) \(\chi_{7942}(835,\cdot)\) \(\chi_{7942}(1253,\cdot)\) \(\chi_{7942}(1671,\cdot)\) \(\chi_{7942}(2089,\cdot)\) \(\chi_{7942}(2507,\cdot)\) \(\chi_{7942}(2925,\cdot)\) \(\chi_{7942}(3343,\cdot)\) \(\chi_{7942}(3761,\cdot)\) \(\chi_{7942}(4179,\cdot)\) \(\chi_{7942}(4597,\cdot)\) \(\chi_{7942}(5015,\cdot)\) \(\chi_{7942}(5433,\cdot)\) \(\chi_{7942}(5851,\cdot)\) \(\chi_{7942}(6269,\cdot)\) \(\chi_{7942}(6687,\cdot)\) \(\chi_{7942}(7105,\cdot)\) \(\chi_{7942}(7523,\cdot)\)
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
\((5777,6139)\) → \((-1,e\left(\frac{17}{38}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 7942 }(417, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) |
sage: chi.jacobi_sum(n)