from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(198, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([20,24]))
pari: [g,chi] = znchar(Mod(25,198))
Basic properties
Modulus: | \(198\) | |
Conductor: | \(99\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{99}(25,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 198.m
\(\chi_{198}(25,\cdot)\) \(\chi_{198}(31,\cdot)\) \(\chi_{198}(49,\cdot)\) \(\chi_{198}(97,\cdot)\) \(\chi_{198}(103,\cdot)\) \(\chi_{198}(115,\cdot)\) \(\chi_{198}(157,\cdot)\) \(\chi_{198}(169,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | 15.15.10943023107606534329121.1 |
Values on generators
\((155,145)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{4}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 198 }(25, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)