from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(637, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([10,35]))
pari: [g,chi] = znchar(Mod(543,637))
Basic properties
| Modulus: | \(637\) | |
| Conductor: | \(637\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | 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 637.bp
\(\chi_{637}(88,\cdot)\) \(\chi_{637}(121,\cdot)\) \(\chi_{637}(179,\cdot)\) \(\chi_{637}(212,\cdot)\) \(\chi_{637}(270,\cdot)\) \(\chi_{637}(303,\cdot)\) \(\chi_{637}(394,\cdot)\) \(\chi_{637}(452,\cdot)\) \(\chi_{637}(485,\cdot)\) \(\chi_{637}(543,\cdot)\) \(\chi_{637}(576,\cdot)\) \(\chi_{637}(634,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.42.16423600478713504434070778628293678810006717122913176085381268066336462525553883868883157384200587461557.2 |
Values on generators
\((248,197)\) → \((e\left(\frac{5}{21}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 637 }(543, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{13}{21}\right)\) |
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)