from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8004, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,17,0]))
pari: [g,chi] = znchar(Mod(1625,8004))
Basic properties
Modulus: | \(8004\) | |
Conductor: | \(69\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{69}(38,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8004.bs
\(\chi_{8004}(1625,\cdot)\) \(\chi_{8004}(2321,\cdot)\) \(\chi_{8004}(3365,\cdot)\) \(\chi_{8004}(3713,\cdot)\) \(\chi_{8004}(4757,\cdot)\) \(\chi_{8004}(5801,\cdot)\) \(\chi_{8004}(6497,\cdot)\) \(\chi_{8004}(6845,\cdot)\) \(\chi_{8004}(7193,\cdot)\) \(\chi_{8004}(7541,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
Fixed field: | \(\Q(\zeta_{69})^+\) |
Values on generators
\((4003,2669,3133,553)\) → \((1,-1,e\left(\frac{17}{22}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
\( \chi_{ 8004 }(1625, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) |
sage: chi.jacobi_sum(n)