from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7225, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([0,22]))
pari: [g,chi] = znchar(Mod(426,7225))
Basic properties
| Modulus: | \(7225\) | |
| Conductor: | \(289\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(17\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{289}(137,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7225.w
\(\chi_{7225}(426,\cdot)\) \(\chi_{7225}(851,\cdot)\) \(\chi_{7225}(1276,\cdot)\) \(\chi_{7225}(1701,\cdot)\) \(\chi_{7225}(2126,\cdot)\) \(\chi_{7225}(2551,\cdot)\) \(\chi_{7225}(2976,\cdot)\) \(\chi_{7225}(3401,\cdot)\) \(\chi_{7225}(3826,\cdot)\) \(\chi_{7225}(4251,\cdot)\) \(\chi_{7225}(4676,\cdot)\) \(\chi_{7225}(5101,\cdot)\) \(\chi_{7225}(5526,\cdot)\) \(\chi_{7225}(5951,\cdot)\) \(\chi_{7225}(6376,\cdot)\) \(\chi_{7225}(6801,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
| Fixed field: | Number field defined by a degree 17 polynomial |
Values on generators
\((2602,2026)\) → \((1,e\left(\frac{11}{17}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 7225 }(426, a) \) | \(1\) | \(1\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) |
sage: chi.jacobi_sum(n)