from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5776, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,14]))
pari: [g,chi] = znchar(Mod(299,5776))
Basic properties
| Modulus: | \(5776\) | |
| Conductor: | \(304\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{304}(299,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5776.bh
\(\chi_{5776}(299,\cdot)\) \(\chi_{5776}(307,\cdot)\) \(\chi_{5776}(1571,\cdot)\) \(\chi_{5776}(2067,\cdot)\) \(\chi_{5776}(2499,\cdot)\) \(\chi_{5776}(2643,\cdot)\) \(\chi_{5776}(3187,\cdot)\) \(\chi_{5776}(3195,\cdot)\) \(\chi_{5776}(4459,\cdot)\) \(\chi_{5776}(4955,\cdot)\) \(\chi_{5776}(5387,\cdot)\) \(\chi_{5776}(5531,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.36.19036714782161565107424425435655777110146017378670996611401194085493506048.1 |
Values on generators
\((5055,1445,2529)\) → \((-1,i,e\left(\frac{7}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 5776 }(299, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) |
sage: chi.jacobi_sum(n)