from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4864, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,23,16]))
pari: [g,chi] = znchar(Mod(151,4864))
Basic properties
| Modulus: | \(4864\) | |
| Conductor: | \(2432\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2432}(531,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4864.bv
\(\chi_{4864}(151,\cdot)\) \(\chi_{4864}(455,\cdot)\) \(\chi_{4864}(759,\cdot)\) \(\chi_{4864}(1063,\cdot)\) \(\chi_{4864}(1367,\cdot)\) \(\chi_{4864}(1671,\cdot)\) \(\chi_{4864}(1975,\cdot)\) \(\chi_{4864}(2279,\cdot)\) \(\chi_{4864}(2583,\cdot)\) \(\chi_{4864}(2887,\cdot)\) \(\chi_{4864}(3191,\cdot)\) \(\chi_{4864}(3495,\cdot)\) \(\chi_{4864}(3799,\cdot)\) \(\chi_{4864}(4103,\cdot)\) \(\chi_{4864}(4407,\cdot)\) \(\chi_{4864}(4711,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | 32.32.905288048831351058796666807211863041216387224344298280390835989733155786457088.1 |
Values on generators
\((3839,2053,4353)\) → \((-1,e\left(\frac{23}{32}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 4864 }(151, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) |
sage: chi.jacobi_sum(n)