from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5148, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([0,4,6,3]))
pari: [g,chi] = znchar(Mod(1165,5148))
Basic properties
| Modulus: | \(5148\) | |
| Conductor: | \(1287\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1287}(1165,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5148.fe
\(\chi_{5148}(1165,\cdot)\) \(\chi_{5148}(1825,\cdot)\) \(\chi_{5148}(3541,\cdot)\) \(\chi_{5148}(4597,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
| Fixed field: | 12.12.808697977975202916871413.1 |
Values on generators
\((2575,1145,937,4357)\) → \((1,e\left(\frac{1}{3}\right),-1,i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 5148 }(1165, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(-1\) | \(-i\) |
sage: chi.jacobi_sum(n)