from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9295, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([15,36,35]))
pari: [g,chi] = znchar(Mod(427,9295))
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 \(\chi_{715}(427,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9295.dl
\(\chi_{9295}(427,\cdot)\) \(\chi_{9295}(1263,\cdot)\) \(\chi_{9295}(2117,\cdot)\) \(\chi_{9295}(2347,\cdot)\) \(\chi_{9295}(2953,\cdot)\) \(\chi_{9295}(2962,\cdot)\) \(\chi_{9295}(3568,\cdot)\) \(\chi_{9295}(3798,\cdot)\) \(\chi_{9295}(4882,\cdot)\) \(\chi_{9295}(6103,\cdot)\) \(\chi_{9295}(6572,\cdot)\) \(\chi_{9295}(7187,\cdot)\) \(\chi_{9295}(7417,\cdot)\) \(\chi_{9295}(7793,\cdot)\) \(\chi_{9295}(8023,\cdot)\) \(\chi_{9295}(8638,\cdot)\)
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
\((7437,4226,6931)\) → \((i,e\left(\frac{3}{5}\right),e\left(\frac{7}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(14\) | \(16\) |
\( \chi_{ 9295 }(427, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(-i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) |
sage: chi.jacobi_sum(n)