from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7605, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,0,21]))
pari: [g,chi] = znchar(Mod(181,7605))
Basic properties
Modulus: | \(7605\) | |
Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{169}(12,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7605.ed
\(\chi_{7605}(181,\cdot)\) \(\chi_{7605}(766,\cdot)\) \(\chi_{7605}(1936,\cdot)\) \(\chi_{7605}(2521,\cdot)\) \(\chi_{7605}(3106,\cdot)\) \(\chi_{7605}(3691,\cdot)\) \(\chi_{7605}(4276,\cdot)\) \(\chi_{7605}(4861,\cdot)\) \(\chi_{7605}(5446,\cdot)\) \(\chi_{7605}(6031,\cdot)\) \(\chi_{7605}(6616,\cdot)\) \(\chi_{7605}(7201,\cdot)\)
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: | 26.26.3830224792147131369362629348887201408953937846517364173.1 |
Values on generators
\((6761,1522,6931)\) → \((1,1,e\left(\frac{21}{26}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
\( \chi_{ 7605 }(181, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(-1\) | \(1\) |
sage: chi.jacobi_sum(n)