from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(425, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([32,35]))
pari: [g,chi] = znchar(Mod(36,425))
Basic properties
| Modulus: | \(425\) | |
| Conductor: | \(425\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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 425.be
\(\chi_{425}(36,\cdot)\) \(\chi_{425}(66,\cdot)\) \(\chi_{425}(111,\cdot)\) \(\chi_{425}(121,\cdot)\) \(\chi_{425}(161,\cdot)\) \(\chi_{425}(196,\cdot)\) \(\chi_{425}(206,\cdot)\) \(\chi_{425}(236,\cdot)\) \(\chi_{425}(246,\cdot)\) \(\chi_{425}(281,\cdot)\) \(\chi_{425}(291,\cdot)\) \(\chi_{425}(321,\cdot)\) \(\chi_{425}(331,\cdot)\) \(\chi_{425}(366,\cdot)\) \(\chi_{425}(406,\cdot)\) \(\chi_{425}(416,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((52,326)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{7}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 425 }(36, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) |
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)