from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([3,0,25]))
pari: [g,chi] = znchar(Mod(222,8041))
Basic properties
Modulus: | \(8041\) | |
Conductor: | \(473\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{473}(222,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8041.cl
\(\chi_{8041}(222,\cdot)\) \(\chi_{8041}(596,\cdot)\) \(\chi_{8041}(953,\cdot)\) \(\chi_{8041}(1327,\cdot)\) \(\chi_{8041}(2415,\cdot)\) \(\chi_{8041}(2789,\cdot)\) \(\chi_{8041}(7532,\cdot)\) \(\chi_{8041}(7906,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | 30.30.900147933423976629543618770038416552630751039074980685343506012294353.1 |
Values on generators
\((6580,2366,562)\) → \((e\left(\frac{1}{10}\right),1,e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 8041 }(222, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)