from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3960, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,0,5,0,6]))
pari: [g,chi] = znchar(Mod(191,3960))
Basic properties
| Modulus: | \(3960\) | |
| Conductor: | \(396\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{396}(191,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3960.gd
\(\chi_{3960}(191,\cdot)\) \(\chi_{3960}(311,\cdot)\) \(\chi_{3960}(911,\cdot)\) \(\chi_{3960}(1391,\cdot)\) \(\chi_{3960}(1631,\cdot)\) \(\chi_{3960}(2711,\cdot)\) \(\chi_{3960}(2831,\cdot)\) \(\chi_{3960}(3551,\cdot)\)
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: | 30.30.31245017777306374823059337284350587021473200026746355712.1 |
Values on generators
\((991,1981,3521,2377,2521)\) → \((-1,1,e\left(\frac{1}{6}\right),1,e\left(\frac{1}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3960 }(191, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)