from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(1932))
M = H._module
chi = DirichletCharacter(H, M([1449,1242,896]))
pari: [g,chi] = znchar(Mod(13,40310))
Basic properties
| Modulus: | \(40310\) | |
| Conductor: | \(20155\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1932\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{20155}(13,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 40310.fy
\(\chi_{40310}(13,\cdot)\) \(\chi_{40310}(67,\cdot)\) \(\chi_{40310}(167,\cdot)\) \(\chi_{40310}(283,\cdot)\) \(\chi_{40310}(303,\cdot)\) \(\chi_{40310}(593,\cdot)\) \(\chi_{40310}(673,\cdot)\) \(\chi_{40310}(817,\cdot)\) \(\chi_{40310}(847,\cdot)\) \(\chi_{40310}(863,\cdot)\) \(\chi_{40310}(883,\cdot)\) \(\chi_{40310}(903,\cdot)\) \(\chi_{40310}(933,\cdot)\) \(\chi_{40310}(1137,\cdot)\) \(\chi_{40310}(1153,\cdot)\) \(\chi_{40310}(1193,\cdot)\) \(\chi_{40310}(1397,\cdot)\) \(\chi_{40310}(1427,\cdot)\) \(\chi_{40310}(1517,\cdot)\)
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: | $\Q(\zeta_{1932})$ |
| Fixed field: | Number field defined by a degree 1932 polynomial (not computed) |
Values on generators
\((24187,19461,16821)\) → \((-i,e\left(\frac{9}{14}\right),e\left(\frac{32}{69}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(13, a) \) | \(-1\) | \(1\) | \(e\left(\frac{925}{1932}\right)\) | \(e\left(\frac{1261}{1932}\right)\) | \(e\left(\frac{925}{966}\right)\) | \(e\left(\frac{307}{966}\right)\) | \(e\left(\frac{971}{1932}\right)\) | \(e\left(\frac{241}{276}\right)\) | \(e\left(\frac{278}{483}\right)\) | \(e\left(\frac{127}{966}\right)\) | \(e\left(\frac{405}{644}\right)\) | \(e\left(\frac{281}{644}\right)\) |
sage: chi.jacobi_sum(n)