from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7942, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,2]))
pari: [g,chi] = znchar(Mod(419,7942))
Basic properties
| Modulus: | \(7942\) | |
| Conductor: | \(361\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(19\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{361}(58,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7942.r
\(\chi_{7942}(419,\cdot)\) \(\chi_{7942}(837,\cdot)\) \(\chi_{7942}(1255,\cdot)\) \(\chi_{7942}(1673,\cdot)\) \(\chi_{7942}(2091,\cdot)\) \(\chi_{7942}(2509,\cdot)\) \(\chi_{7942}(2927,\cdot)\) \(\chi_{7942}(3345,\cdot)\) \(\chi_{7942}(3763,\cdot)\) \(\chi_{7942}(4181,\cdot)\) \(\chi_{7942}(4599,\cdot)\) \(\chi_{7942}(5017,\cdot)\) \(\chi_{7942}(5435,\cdot)\) \(\chi_{7942}(5853,\cdot)\) \(\chi_{7942}(6271,\cdot)\) \(\chi_{7942}(6689,\cdot)\) \(\chi_{7942}(7107,\cdot)\) \(\chi_{7942}(7525,\cdot)\)
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: | 19.19.10842505080063916320800450434338728415281531281.1 |
Values on generators
\((5777,6139)\) → \((1,e\left(\frac{1}{19}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 7942 }(419, a) \) | \(1\) | \(1\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) |
sage: chi.jacobi_sum(n)