from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1502, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([22]))
pari: [g,chi] = znchar(Mod(51,1502))
Basic properties
| Modulus: | \(1502\) | |
| 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 \(\chi_{751}(51,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1502.h
\(\chi_{1502}(51,\cdot)\) \(\chi_{1502}(53,\cdot)\) \(\chi_{1502}(117,\cdot)\) \(\chi_{1502}(171,\cdot)\) \(\chi_{1502}(179,\cdot)\) \(\chi_{1502}(193,\cdot)\) \(\chi_{1502}(325,\cdot)\) \(\chi_{1502}(475,\cdot)\) \(\chi_{1502}(481,\cdot)\) \(\chi_{1502}(485,\cdot)\) \(\chi_{1502}(499,\cdot)\) \(\chi_{1502}(703,\cdot)\) \(\chi_{1502}(913,\cdot)\) \(\chi_{1502}(1099,\cdot)\) \(\chi_{1502}(1171,\cdot)\) \(\chi_{1502}(1201,\cdot)\) \(\chi_{1502}(1217,\cdot)\) \(\chi_{1502}(1307,\cdot)\) \(\chi_{1502}(1417,\cdot)\) \(\chi_{1502}(1461,\cdot)\)
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
\(3\) → \(e\left(\frac{11}{25}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1502 }(51, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) |
sage: chi.jacobi_sum(n)