from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3630, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,15]))
pari: [g,chi] = znchar(Mod(2309,3630))
Basic properties
| Modulus: | \(3630\) | |
| Conductor: | \(1815\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1815}(494,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3630.bb
\(\chi_{3630}(329,\cdot)\) \(\chi_{3630}(659,\cdot)\) \(\chi_{3630}(989,\cdot)\) \(\chi_{3630}(1319,\cdot)\) \(\chi_{3630}(1649,\cdot)\) \(\chi_{3630}(1979,\cdot)\) \(\chi_{3630}(2309,\cdot)\) \(\chi_{3630}(2639,\cdot)\) \(\chi_{3630}(2969,\cdot)\) \(\chi_{3630}(3299,\cdot)\)
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: | 22.22.43062966214595858730535497699537467025089813545751953125.1 |
Values on generators
\((1211,727,3511)\) → \((-1,-1,e\left(\frac{15}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3630 }(2309, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)