from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4004, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,40,24,25]))
pari: [g,chi] = znchar(Mod(1215,4004))
Basic properties
Modulus: | \(4004\) | |
Conductor: | \(4004\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | 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 4004.ip
\(\chi_{4004}(487,\cdot)\) \(\chi_{4004}(851,\cdot)\) \(\chi_{4004}(1215,\cdot)\) \(\chi_{4004}(1367,\cdot)\) \(\chi_{4004}(1411,\cdot)\) \(\chi_{4004}(1675,\cdot)\) \(\chi_{4004}(1731,\cdot)\) \(\chi_{4004}(1775,\cdot)\) \(\chi_{4004}(2039,\cdot)\) \(\chi_{4004}(2095,\cdot)\) \(\chi_{4004}(2139,\cdot)\) \(\chi_{4004}(2403,\cdot)\) \(\chi_{4004}(2671,\cdot)\) \(\chi_{4004}(3551,\cdot)\) \(\chi_{4004}(3595,\cdot)\) \(\chi_{4004}(3859,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((2003,3433,365,925)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{2}{5}\right),e\left(\frac{5}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
\( \chi_{ 4004 }(1215, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{15}\right)\) |
sage: chi.jacobi_sum(n)