from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2205, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([7,0,22]))
pari: [g,chi] = znchar(Mod(1271,2205))
Basic properties
Modulus: | \(2205\) | |
Conductor: | \(441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{441}(389,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2205.dx
\(\chi_{2205}(11,\cdot)\) \(\chi_{2205}(86,\cdot)\) \(\chi_{2205}(326,\cdot)\) \(\chi_{2205}(401,\cdot)\) \(\chi_{2205}(641,\cdot)\) \(\chi_{2205}(956,\cdot)\) \(\chi_{2205}(1031,\cdot)\) \(\chi_{2205}(1271,\cdot)\) \(\chi_{2205}(1346,\cdot)\) \(\chi_{2205}(1661,\cdot)\) \(\chi_{2205}(1901,\cdot)\) \(\chi_{2205}(1976,\cdot)\)
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
\((1226,442,1081)\) → \((e\left(\frac{1}{6}\right),1,e\left(\frac{11}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
\( \chi_{ 2205 }(1271, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) |
sage: chi.jacobi_sum(n)