from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8004, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,14,0,25]))
pari: [g,chi] = znchar(Mod(185,8004))
Basic properties
| Modulus: | \(8004\) | |
| Conductor: | \(87\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{87}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8004.cf
\(\chi_{8004}(185,\cdot)\) \(\chi_{8004}(461,\cdot)\) \(\chi_{8004}(1013,\cdot)\) \(\chi_{8004}(1841,\cdot)\) \(\chi_{8004}(2393,\cdot)\) \(\chi_{8004}(3221,\cdot)\) \(\chi_{8004}(3773,\cdot)\) \(\chi_{8004}(4049,\cdot)\) \(\chi_{8004}(4601,\cdot)\) \(\chi_{8004}(5705,\cdot)\) \(\chi_{8004}(6533,\cdot)\) \(\chi_{8004}(7637,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | \(\Q(\zeta_{87})^+\) |
Values on generators
\((4003,2669,3133,553)\) → \((1,-1,1,e\left(\frac{25}{28}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 8004 }(185, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(i\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{19}{28}\right)\) |
sage: chi.jacobi_sum(n)