from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2205, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,0,10]))
pari: [g,chi] = znchar(Mod(886,2205))
Basic properties
| Modulus: | \(2205\) | |
| Conductor: | \(441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{441}(4,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2205.cr
\(\chi_{2205}(16,\cdot)\) \(\chi_{2205}(256,\cdot)\) \(\chi_{2205}(331,\cdot)\) \(\chi_{2205}(571,\cdot)\) \(\chi_{2205}(646,\cdot)\) \(\chi_{2205}(886,\cdot)\) \(\chi_{2205}(1201,\cdot)\) \(\chi_{2205}(1276,\cdot)\) \(\chi_{2205}(1516,\cdot)\) \(\chi_{2205}(1591,\cdot)\) \(\chi_{2205}(1906,\cdot)\) \(\chi_{2205}(2146,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 21 polynomial |
Values on generators
\((1226,442,1081)\) → \((e\left(\frac{1}{3}\right),1,e\left(\frac{5}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 2205 }(886, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) |
sage: chi.jacobi_sum(n)