from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5915, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,0,1]))
pari: [g,chi] = znchar(Mod(64,5915))
Basic properties
Modulus: | \(5915\) | |
Conductor: | \(845\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{845}(64,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5915.dz
\(\chi_{5915}(64,\cdot)\) \(\chi_{5915}(519,\cdot)\) \(\chi_{5915}(974,\cdot)\) \(\chi_{5915}(1429,\cdot)\) \(\chi_{5915}(1884,\cdot)\) \(\chi_{5915}(2339,\cdot)\) \(\chi_{5915}(2794,\cdot)\) \(\chi_{5915}(3249,\cdot)\) \(\chi_{5915}(3704,\cdot)\) \(\chi_{5915}(4159,\cdot)\) \(\chi_{5915}(4614,\cdot)\) \(\chi_{5915}(5524,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
Fixed field: | Number field defined by a degree 26 polynomial |
Values on generators
\((2367,5071,1016)\) → \((-1,1,e\left(\frac{1}{26}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(16\) | \(17\) |
\( \chi_{ 5915 }(64, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) |
sage: chi.jacobi_sum(n)