from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4004, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,40,18,25]))
pari: [g,chi] = znchar(Mod(305,4004))
Basic properties
| Modulus: | \(4004\) | |
| Conductor: | \(1001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1001}(305,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4004.iq
\(\chi_{4004}(305,\cdot)\) \(\chi_{4004}(457,\cdot)\) \(\chi_{4004}(501,\cdot)\) \(\chi_{4004}(765,\cdot)\) \(\chi_{4004}(821,\cdot)\) \(\chi_{4004}(865,\cdot)\) \(\chi_{4004}(1129,\cdot)\) \(\chi_{4004}(1185,\cdot)\) \(\chi_{4004}(1229,\cdot)\) \(\chi_{4004}(1493,\cdot)\) \(\chi_{4004}(2125,\cdot)\) \(\chi_{4004}(3005,\cdot)\) \(\chi_{4004}(3049,\cdot)\) \(\chi_{4004}(3313,\cdot)\) \(\chi_{4004}(3581,\cdot)\) \(\chi_{4004}(3945,\cdot)\)
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
\((2003,3433,365,925)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{3}{10}\right),e\left(\frac{5}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
| \( \chi_{ 4004 }(305, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(-1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{23}{30}\right)\) |
sage: chi.jacobi_sum(n)