from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2925, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,11,10]))
pari: [g,chi] = znchar(Mod(298,2925))
Basic properties
| Modulus: | \(2925\) | |
| Conductor: | \(325\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{325}(298,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2925.er
\(\chi_{2925}(298,\cdot)\) \(\chi_{2925}(883,\cdot)\) \(\chi_{2925}(1117,\cdot)\) \(\chi_{2925}(1702,\cdot)\) \(\chi_{2925}(2053,\cdot)\) \(\chi_{2925}(2287,\cdot)\) \(\chi_{2925}(2638,\cdot)\) \(\chi_{2925}(2872,\cdot)\)
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: | 20.0.401221017379430122673511505126953125.1 |
Values on generators
\((326,352,2251)\) → \((1,e\left(\frac{11}{20}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 2925 }(298, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(i\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) |
sage: chi.jacobi_sum(n)