from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3004, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,4]))
pari: [g,chi] = znchar(Mod(53,3004))
Basic properties
| Modulus: | \(3004\) | |
| Conductor: | \(751\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(25\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{751}(53,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3004.n
\(\chi_{3004}(53,\cdot)\) \(\chi_{3004}(117,\cdot)\) \(\chi_{3004}(193,\cdot)\) \(\chi_{3004}(325,\cdot)\) \(\chi_{3004}(481,\cdot)\) \(\chi_{3004}(485,\cdot)\) \(\chi_{3004}(913,\cdot)\) \(\chi_{3004}(1201,\cdot)\) \(\chi_{3004}(1217,\cdot)\) \(\chi_{3004}(1417,\cdot)\) \(\chi_{3004}(1461,\cdot)\) \(\chi_{3004}(1553,\cdot)\) \(\chi_{3004}(1673,\cdot)\) \(\chi_{3004}(1681,\cdot)\) \(\chi_{3004}(1977,\cdot)\) \(\chi_{3004}(2001,\cdot)\) \(\chi_{3004}(2205,\cdot)\) \(\chi_{3004}(2601,\cdot)\) \(\chi_{3004}(2673,\cdot)\) \(\chi_{3004}(2809,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | Number field defined by a degree 25 polynomial |
Values on generators
\((1503,1505)\) → \((1,e\left(\frac{2}{25}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 3004 }(53, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) |
sage: chi.jacobi_sum(n)