from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4225, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,19]))
pari: [g,chi] = znchar(Mod(51,4225))
Basic properties
| Modulus: | \(4225\) | |
| Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{169}(51,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4225.bh
\(\chi_{4225}(51,\cdot)\) \(\chi_{4225}(376,\cdot)\) \(\chi_{4225}(701,\cdot)\) \(\chi_{4225}(1026,\cdot)\) \(\chi_{4225}(1676,\cdot)\) \(\chi_{4225}(2001,\cdot)\) \(\chi_{4225}(2326,\cdot)\) \(\chi_{4225}(2651,\cdot)\) \(\chi_{4225}(2976,\cdot)\) \(\chi_{4225}(3301,\cdot)\) \(\chi_{4225}(3626,\cdot)\) \(\chi_{4225}(3951,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
| Fixed field: | 26.26.3830224792147131369362629348887201408953937846517364173.1 |
Values on generators
\((677,3551)\) → \((1,e\left(\frac{19}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 4225 }(51, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) |
sage: chi.jacobi_sum(n)