from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2205, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,40]))
pari: [g,chi] = znchar(Mod(2069,2205))
Basic properties
Modulus: | \(2205\) | |
Conductor: | \(735\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{735}(599,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2205.dr
\(\chi_{2205}(44,\cdot)\) \(\chi_{2205}(179,\cdot)\) \(\chi_{2205}(359,\cdot)\) \(\chi_{2205}(494,\cdot)\) \(\chi_{2205}(674,\cdot)\) \(\chi_{2205}(809,\cdot)\) \(\chi_{2205}(989,\cdot)\) \(\chi_{2205}(1124,\cdot)\) \(\chi_{2205}(1619,\cdot)\) \(\chi_{2205}(1754,\cdot)\) \(\chi_{2205}(1934,\cdot)\) \(\chi_{2205}(2069,\cdot)\)
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
\((1226,442,1081)\) → \((-1,-1,e\left(\frac{20}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
\( \chi_{ 2205 }(2069, a) \) | \(-1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{4}{21}\right)\) |
sage: chi.jacobi_sum(n)