from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2205, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([7,21,40]))
pari: [g,chi] = znchar(Mod(1334,2205))
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.cy
\(\chi_{2205}(74,\cdot)\) \(\chi_{2205}(149,\cdot)\) \(\chi_{2205}(389,\cdot)\) \(\chi_{2205}(464,\cdot)\) \(\chi_{2205}(779,\cdot)\) \(\chi_{2205}(1019,\cdot)\) \(\chi_{2205}(1094,\cdot)\) \(\chi_{2205}(1334,\cdot)\) \(\chi_{2205}(1409,\cdot)\) \(\chi_{2205}(1649,\cdot)\) \(\chi_{2205}(1724,\cdot)\) \(\chi_{2205}(1964,\cdot)\)
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
\((1226,442,1081)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{20}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
\( \chi_{ 2205 }(1334, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) |
sage: chi.jacobi_sum(n)