from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3150, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([5,3,0]))
pari: [g,chi] = znchar(Mod(29,3150))
Basic properties
| Modulus: | \(3150\) | |
| Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{225}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3150.dd
\(\chi_{3150}(29,\cdot)\) \(\chi_{3150}(239,\cdot)\) \(\chi_{3150}(659,\cdot)\) \(\chi_{3150}(869,\cdot)\) \(\chi_{3150}(1289,\cdot)\) \(\chi_{3150}(1919,\cdot)\) \(\chi_{3150}(2129,\cdot)\) \(\chi_{3150}(2759,\cdot)\)
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: | 30.0.1311978395502159643834172442211638553999364376068115234375.1 |
Values on generators
\((2801,127,451)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{1}{10}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3150 }(29, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)