from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4025, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,44,54]))
pari: [g,chi] = znchar(Mod(949,4025))
Basic properties
| Modulus: | \(4025\) | |
| Conductor: | \(805\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{805}(144,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4025.cl
\(\chi_{4025}(324,\cdot)\) \(\chi_{4025}(499,\cdot)\) \(\chi_{4025}(949,\cdot)\) \(\chi_{4025}(1024,\cdot)\) \(\chi_{4025}(1199,\cdot)\) \(\chi_{4025}(1474,\cdot)\) \(\chi_{4025}(1549,\cdot)\) \(\chi_{4025}(1649,\cdot)\) \(\chi_{4025}(1899,\cdot)\) \(\chi_{4025}(2074,\cdot)\) \(\chi_{4025}(2174,\cdot)\) \(\chi_{4025}(2249,\cdot)\) \(\chi_{4025}(2349,\cdot)\) \(\chi_{4025}(2424,\cdot)\) \(\chi_{4025}(2699,\cdot)\) \(\chi_{4025}(3049,\cdot)\) \(\chi_{4025}(3224,\cdot)\) \(\chi_{4025}(3399,\cdot)\) \(\chi_{4025}(3574,\cdot)\) \(\chi_{4025}(3824,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((2577,1151,3501)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{9}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 4025 }(949, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{29}{33}\right)\) |
sage: chi.jacobi_sum(n)