from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4025, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,55,57]))
pari: [g,chi] = znchar(Mod(2399,4025))
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 \(\chi_{805}(789,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4025.cm
\(\chi_{4025}(199,\cdot)\) \(\chi_{4025}(474,\cdot)\) \(\chi_{4025}(549,\cdot)\) \(\chi_{4025}(649,\cdot)\) \(\chi_{4025}(724,\cdot)\) \(\chi_{4025}(824,\cdot)\) \(\chi_{4025}(999,\cdot)\) \(\chi_{4025}(1249,\cdot)\) \(\chi_{4025}(1349,\cdot)\) \(\chi_{4025}(1424,\cdot)\) \(\chi_{4025}(1699,\cdot)\) \(\chi_{4025}(1874,\cdot)\) \(\chi_{4025}(1949,\cdot)\) \(\chi_{4025}(2399,\cdot)\) \(\chi_{4025}(2574,\cdot)\) \(\chi_{4025}(3099,\cdot)\) \(\chi_{4025}(3349,\cdot)\) \(\chi_{4025}(3524,\cdot)\) \(\chi_{4025}(3699,\cdot)\) \(\chi_{4025}(3874,\cdot)\)
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
\((2577,1151,3501)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{19}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
\( \chi_{ 4025 }(2399, a) \) | \(1\) | \(1\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{19}{33}\right)\) |
sage: chi.jacobi_sum(n)