from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3150, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([5,12,0]))
pari: [g,chi] = znchar(Mod(281,3150))
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 \(\chi_{225}(56,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3150.dj
\(\chi_{3150}(281,\cdot)\) \(\chi_{3150}(491,\cdot)\) \(\chi_{3150}(911,\cdot)\) \(\chi_{3150}(1121,\cdot)\) \(\chi_{3150}(1541,\cdot)\) \(\chi_{3150}(2171,\cdot)\) \(\chi_{3150}(2381,\cdot)\) \(\chi_{3150}(3011,\cdot)\)
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: | 30.0.10495827164017277150673379537693108431994915008544921875.1 |
Values on generators
\((2801,127,451)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{2}{5}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 3150 }(281, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)