from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9295, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([15,6,10]))
pari: [g,chi] = znchar(Mod(992,9295))
Basic properties
| Modulus: | \(9295\) | |
| Conductor: | \(715\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{715}(277,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9295.ds
\(\chi_{9295}(992,\cdot)\) \(\chi_{9295}(1668,\cdot)\) \(\chi_{9295}(1713,\cdot)\) \(\chi_{9295}(2558,\cdot)\) \(\chi_{9295}(3527,\cdot)\) \(\chi_{9295}(3572,\cdot)\) \(\chi_{9295}(4248,\cdot)\) \(\chi_{9295}(4417,\cdot)\) \(\chi_{9295}(5893,\cdot)\) \(\chi_{9295}(6107,\cdot)\) \(\chi_{9295}(6738,\cdot)\) \(\chi_{9295}(6783,\cdot)\) \(\chi_{9295}(7752,\cdot)\) \(\chi_{9295}(8428,\cdot)\) \(\chi_{9295}(8597,\cdot)\) \(\chi_{9295}(8642,\cdot)\)
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
\((7437,4226,6931)\) → \((i,e\left(\frac{1}{10}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(14\) | \(16\) |
| \( \chi_{ 9295 }(992, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{15}\right)\) |
sage: chi.jacobi_sum(n)