from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(915, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,59]))
pari: [g,chi] = znchar(Mod(31,915))
Basic properties
| Modulus: | \(915\) | |
| Conductor: | \(61\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{61}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 915.cv
\(\chi_{915}(31,\cdot)\) \(\chi_{915}(91,\cdot)\) \(\chi_{915}(166,\cdot)\) \(\chi_{915}(181,\cdot)\) \(\chi_{915}(226,\cdot)\) \(\chi_{915}(331,\cdot)\) \(\chi_{915}(376,\cdot)\) \(\chi_{915}(421,\cdot)\) \(\chi_{915}(481,\cdot)\) \(\chi_{915}(556,\cdot)\) \(\chi_{915}(616,\cdot)\) \(\chi_{915}(661,\cdot)\) \(\chi_{915}(706,\cdot)\) \(\chi_{915}(811,\cdot)\) \(\chi_{915}(856,\cdot)\) \(\chi_{915}(871,\cdot)\)
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
\((611,367,856)\) → \((1,1,e\left(\frac{59}{60}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 915 }(31, a) \) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) |
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)