from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2205, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,21,20]))
pari: [g,chi] = znchar(Mod(2074,2205))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2205.dy
\(\chi_{2205}(184,\cdot)\) \(\chi_{2205}(499,\cdot)\) \(\chi_{2205}(529,\cdot)\) \(\chi_{2205}(844,\cdot)\) \(\chi_{2205}(1129,\cdot)\) \(\chi_{2205}(1159,\cdot)\) \(\chi_{2205}(1444,\cdot)\) \(\chi_{2205}(1474,\cdot)\) \(\chi_{2205}(1759,\cdot)\) \(\chi_{2205}(1789,\cdot)\) \(\chi_{2205}(2074,\cdot)\) \(\chi_{2205}(2104,\cdot)\)
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
\((1226,442,1081)\) → \((e\left(\frac{1}{3}\right),-1,e\left(\frac{10}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
\( \chi_{ 2205 }(2074, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) |
sage: chi.jacobi_sum(n)