from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5915, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,0,14]))
pari: [g,chi] = znchar(Mod(274,5915))
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 \(\chi_{845}(274,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5915.ec
\(\chi_{5915}(274,\cdot)\) \(\chi_{5915}(729,\cdot)\) \(\chi_{5915}(1639,\cdot)\) \(\chi_{5915}(2094,\cdot)\) \(\chi_{5915}(2549,\cdot)\) \(\chi_{5915}(3004,\cdot)\) \(\chi_{5915}(3459,\cdot)\) \(\chi_{5915}(3914,\cdot)\) \(\chi_{5915}(4369,\cdot)\) \(\chi_{5915}(4824,\cdot)\) \(\chi_{5915}(5279,\cdot)\) \(\chi_{5915}(5734,\cdot)\)
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
\((2367,5071,1016)\) → \((-1,1,e\left(\frac{7}{13}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(16\) | \(17\) |
\( \chi_{ 5915 }(274, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) |
sage: chi.jacobi_sum(n)