from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2548, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,6,21]))
pari: [g,chi] = znchar(Mod(83,2548))
Basic properties
| Modulus: | \(2548\) | |
| Conductor: | \(2548\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2548.cu
\(\chi_{2548}(83,\cdot)\) \(\chi_{2548}(307,\cdot)\) \(\chi_{2548}(447,\cdot)\) \(\chi_{2548}(671,\cdot)\) \(\chi_{2548}(811,\cdot)\) \(\chi_{2548}(1035,\cdot)\) \(\chi_{2548}(1399,\cdot)\) \(\chi_{2548}(1539,\cdot)\) \(\chi_{2548}(1903,\cdot)\) \(\chi_{2548}(2127,\cdot)\) \(\chi_{2548}(2267,\cdot)\) \(\chi_{2548}(2491,\cdot)\)
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: | 28.0.119275783420208128178399298306069209896021994054064945982597375019211292672.1 |
Values on generators
\((1275,885,197)\) → \((-1,e\left(\frac{3}{14}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 2548 }(83, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(-i\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)