from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5796, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,44,36]))
pari: [g,chi] = znchar(Mod(179,5796))
Basic properties
| Modulus: | \(5796\) | |
| Conductor: | \(1932\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1932}(179,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5796.ep
\(\chi_{5796}(179,\cdot)\) \(\chi_{5796}(611,\cdot)\) \(\chi_{5796}(683,\cdot)\) \(\chi_{5796}(863,\cdot)\) \(\chi_{5796}(1439,\cdot)\) \(\chi_{5796}(1619,\cdot)\) \(\chi_{5796}(1691,\cdot)\) \(\chi_{5796}(1871,\cdot)\) \(\chi_{5796}(2375,\cdot)\) \(\chi_{5796}(2447,\cdot)\) \(\chi_{5796}(2699,\cdot)\) \(\chi_{5796}(2879,\cdot)\) \(\chi_{5796}(3131,\cdot)\) \(\chi_{5796}(3203,\cdot)\) \(\chi_{5796}(3383,\cdot)\) \(\chi_{5796}(3707,\cdot)\) \(\chi_{5796}(3959,\cdot)\) \(\chi_{5796}(4211,\cdot)\) \(\chi_{5796}(5147,\cdot)\) \(\chi_{5796}(5651,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((2899,1289,829,4789)\) → \((-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{6}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5796 }(179, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) |
sage: chi.jacobi_sum(n)