from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2205, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([28,21,33]))
pari: [g,chi] = znchar(Mod(1924,2205))
Basic properties
| Modulus: | \(2205\) | |
| Conductor: | \(2205\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2205.dj
\(\chi_{2205}(34,\cdot)\) \(\chi_{2205}(139,\cdot)\) \(\chi_{2205}(349,\cdot)\) \(\chi_{2205}(454,\cdot)\) \(\chi_{2205}(664,\cdot)\) \(\chi_{2205}(769,\cdot)\) \(\chi_{2205}(1084,\cdot)\) \(\chi_{2205}(1294,\cdot)\) \(\chi_{2205}(1399,\cdot)\) \(\chi_{2205}(1609,\cdot)\) \(\chi_{2205}(1924,\cdot)\) \(\chi_{2205}(2029,\cdot)\)
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 42 polynomial |
Values on generators
\((1226,442,1081)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{11}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 2205 }(1924, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(-1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) |
sage: chi.jacobi_sum(n)