from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(469, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,14]))
pari: [g,chi] = znchar(Mod(36,469))
Basic properties
| Modulus: | \(469\) | |
| Conductor: | \(67\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{67}(36,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 469.z
\(\chi_{469}(36,\cdot)\) \(\chi_{469}(71,\cdot)\) \(\chi_{469}(106,\cdot)\) \(\chi_{469}(127,\cdot)\) \(\chi_{469}(155,\cdot)\) \(\chi_{469}(169,\cdot)\) \(\chi_{469}(183,\cdot)\) \(\chi_{469}(190,\cdot)\) \(\chi_{469}(211,\cdot)\) \(\chi_{469}(218,\cdot)\) \(\chi_{469}(274,\cdot)\) \(\chi_{469}(323,\cdot)\) \(\chi_{469}(351,\cdot)\) \(\chi_{469}(358,\cdot)\) \(\chi_{469}(400,\cdot)\) \(\chi_{469}(421,\cdot)\) \(\chi_{469}(428,\cdot)\) \(\chi_{469}(435,\cdot)\) \(\chi_{469}(449,\cdot)\) \(\chi_{469}(456,\cdot)\)
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 33 polynomial |
Values on generators
\((269,337)\) → \((1,e\left(\frac{7}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 469 }(36, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) |
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)