from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3660, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,0,0,23]))
pari: [g,chi] = znchar(Mod(3211,3660))
Basic properties
| Modulus: | \(3660\) | |
| Conductor: | \(244\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{244}(39,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3660.fd
\(\chi_{3660}(751,\cdot)\) \(\chi_{3660}(1591,\cdot)\) \(\chi_{3660}(1651,\cdot)\) \(\chi_{3660}(2611,\cdot)\) \(\chi_{3660}(2791,\cdot)\) \(\chi_{3660}(2851,\cdot)\) \(\chi_{3660}(3211,\cdot)\) \(\chi_{3660}(3391,\cdot)\)
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: | 30.0.6389473676049168697910808817736968627185974114850268443049984.1 |
Values on generators
\((1831,2441,2197,3601)\) → \((-1,1,1,e\left(\frac{23}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3660 }(3211, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)