from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9295, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([15,18,20]))
pari: [g,chi] = znchar(Mod(822,9295))
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 \(\chi_{715}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9295.dq
\(\chi_{9295}(822,\cdot)\) \(\chi_{9295}(1498,\cdot)\) \(\chi_{9295}(1667,\cdot)\) \(\chi_{9295}(1712,\cdot)\) \(\chi_{9295}(3357,\cdot)\) \(\chi_{9295}(4033,\cdot)\) \(\chi_{9295}(4078,\cdot)\) \(\chi_{9295}(4923,\cdot)\) \(\chi_{9295}(5892,\cdot)\) \(\chi_{9295}(5937,\cdot)\) \(\chi_{9295}(6613,\cdot)\) \(\chi_{9295}(6782,\cdot)\) \(\chi_{9295}(8258,\cdot)\) \(\chi_{9295}(8472,\cdot)\) \(\chi_{9295}(9103,\cdot)\) \(\chi_{9295}(9148,\cdot)\)
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
\((7437,4226,6931)\) → \((i,e\left(\frac{3}{10}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(14\) | \(16\) |
\( \chi_{ 9295 }(822, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(i\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) |
sage: chi.jacobi_sum(n)