from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3630, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,3]))
pari: [g,chi] = znchar(Mod(461,3630))
Basic properties
Modulus: | \(3630\) | |
Conductor: | \(363\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{363}(98,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3630.bd
\(\chi_{3630}(131,\cdot)\) \(\chi_{3630}(461,\cdot)\) \(\chi_{3630}(791,\cdot)\) \(\chi_{3630}(1121,\cdot)\) \(\chi_{3630}(1781,\cdot)\) \(\chi_{3630}(2111,\cdot)\) \(\chi_{3630}(2441,\cdot)\) \(\chi_{3630}(2771,\cdot)\) \(\chi_{3630}(3101,\cdot)\) \(\chi_{3630}(3431,\cdot)\)
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: | Number field defined by a degree 22 polynomial |
Values on generators
\((1211,727,3511)\) → \((-1,1,e\left(\frac{3}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 3630 }(461, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) |
sage: chi.jacobi_sum(n)