from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4004, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,0,12,5]))
pari: [g,chi] = znchar(Mod(225,4004))
Basic properties
| Modulus: | \(4004\) | |
| Conductor: | \(143\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{143}(82,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4004.hq
\(\chi_{4004}(225,\cdot)\) \(\chi_{4004}(1037,\cdot)\) \(\chi_{4004}(1401,\cdot)\) \(\chi_{4004}(1681,\cdot)\) \(\chi_{4004}(1765,\cdot)\) \(\chi_{4004}(3221,\cdot)\) \(\chi_{4004}(3501,\cdot)\) \(\chi_{4004}(3865,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.30.69503752297329754905479727341904896738456941915804813.1 |
Values on generators
\((2003,3433,365,925)\) → \((1,1,e\left(\frac{2}{5}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
| \( \chi_{ 4004 }(225, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) |
sage: chi.jacobi_sum(n)