from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9025, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([9,10]))
pari: [g,chi] = znchar(Mod(2887,9025))
Basic properties
Modulus: | \(9025\) | |
Conductor: | \(475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{475}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9025.w
\(\chi_{9025}(2887,\cdot)\) \(\chi_{9025}(3248,\cdot)\) \(\chi_{9025}(4692,\cdot)\) \(\chi_{9025}(5053,\cdot)\) \(\chi_{9025}(6497,\cdot)\) \(\chi_{9025}(6858,\cdot)\) \(\chi_{9025}(8302,\cdot)\) \(\chi_{9025}(8663,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | 20.20.17843751288604107685387134552001953125.1 |
Values on generators
\((5777,3251)\) → \((e\left(\frac{9}{20}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 9025 }(2887, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(i\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) |
sage: chi.jacobi_sum(n)