from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(775, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([57,22]))
pari: [g,chi] = znchar(Mod(13,775))
Basic properties
| Modulus: | \(775\) | |
| Conductor: | \(775\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 775.cu
\(\chi_{775}(13,\cdot)\) \(\chi_{775}(17,\cdot)\) \(\chi_{775}(52,\cdot)\) \(\chi_{775}(137,\cdot)\) \(\chi_{775}(197,\cdot)\) \(\chi_{775}(198,\cdot)\) \(\chi_{775}(208,\cdot)\) \(\chi_{775}(228,\cdot)\) \(\chi_{775}(272,\cdot)\) \(\chi_{775}(303,\cdot)\) \(\chi_{775}(313,\cdot)\) \(\chi_{775}(358,\cdot)\) \(\chi_{775}(437,\cdot)\) \(\chi_{775}(477,\cdot)\) \(\chi_{775}(548,\cdot)\) \(\chi_{775}(642,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((652,251)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{11}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 775 }(13, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{1}{60}\right)\) | \(-1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(i\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)