from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4830, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,20]))
pari: [g,chi] = znchar(Mod(3991,4830))
Basic properties
Modulus: | \(4830\) | |
Conductor: | \(23\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(11\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{23}(12,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4830.bo
\(\chi_{4830}(211,\cdot)\) \(\chi_{4830}(841,\cdot)\) \(\chi_{4830}(1051,\cdot)\) \(\chi_{4830}(1681,\cdot)\) \(\chi_{4830}(2101,\cdot)\) \(\chi_{4830}(3361,\cdot)\) \(\chi_{4830}(3571,\cdot)\) \(\chi_{4830}(3781,\cdot)\) \(\chi_{4830}(3991,\cdot)\) \(\chi_{4830}(4411,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
Fixed field: | \(\Q(\zeta_{23})^+\) |
Values on generators
\((3221,967,2761,1891)\) → \((1,1,1,e\left(\frac{10}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 4830 }(3991, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)