from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(637, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([11,21]))
pari: [g,chi] = znchar(Mod(12,637))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 637.br
\(\chi_{637}(12,\cdot)\) \(\chi_{637}(38,\cdot)\) \(\chi_{637}(103,\cdot)\) \(\chi_{637}(194,\cdot)\) \(\chi_{637}(220,\cdot)\) \(\chi_{637}(285,\cdot)\) \(\chi_{637}(311,\cdot)\) \(\chi_{637}(376,\cdot)\) \(\chi_{637}(402,\cdot)\) \(\chi_{637}(467,\cdot)\) \(\chi_{637}(493,\cdot)\) \(\chi_{637}(584,\cdot)\)
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: | 42.0.29198428620782310880522337720254845955751250559410488348634029682058779274295867292920491.1 |
Values on generators
\((248,197)\) → \((e\left(\frac{11}{42}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 637 }(12, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{37}{42}\right)\) |
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)