from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4025, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,0,18]))
pari: [g,chi] = znchar(Mod(218,4025))
Basic properties
Modulus: | \(4025\) | |
Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{115}(103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4025.cc
\(\chi_{4025}(43,\cdot)\) \(\chi_{4025}(57,\cdot)\) \(\chi_{4025}(218,\cdot)\) \(\chi_{4025}(582,\cdot)\) \(\chi_{4025}(743,\cdot)\) \(\chi_{4025}(757,\cdot)\) \(\chi_{4025}(918,\cdot)\) \(\chi_{4025}(1282,\cdot)\) \(\chi_{4025}(1443,\cdot)\) \(\chi_{4025}(2682,\cdot)\) \(\chi_{4025}(2843,\cdot)\) \(\chi_{4025}(2857,\cdot)\) \(\chi_{4025}(3018,\cdot)\) \(\chi_{4025}(3032,\cdot)\) \(\chi_{4025}(3193,\cdot)\) \(\chi_{4025}(3207,\cdot)\) \(\chi_{4025}(3368,\cdot)\) \(\chi_{4025}(3557,\cdot)\) \(\chi_{4025}(3718,\cdot)\) \(\chi_{4025}(3907,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
Fixed field: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((2577,1151,3501)\) → \((-i,1,e\left(\frac{9}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
\( \chi_{ 4025 }(218, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) |
sage: chi.jacobi_sum(n)