from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3150, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,51,50]))
pari: [g,chi] = znchar(Mod(47,3150))
Basic properties
| Modulus: | \(3150\) | |
| Conductor: | \(1575\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1575}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3150.ep
\(\chi_{3150}(47,\cdot)\) \(\chi_{3150}(173,\cdot)\) \(\chi_{3150}(437,\cdot)\) \(\chi_{3150}(563,\cdot)\) \(\chi_{3150}(677,\cdot)\) \(\chi_{3150}(803,\cdot)\) \(\chi_{3150}(1067,\cdot)\) \(\chi_{3150}(1433,\cdot)\) \(\chi_{3150}(1697,\cdot)\) \(\chi_{3150}(1823,\cdot)\) \(\chi_{3150}(1937,\cdot)\) \(\chi_{3150}(2063,\cdot)\) \(\chi_{3150}(2327,\cdot)\) \(\chi_{3150}(2453,\cdot)\) \(\chi_{3150}(2567,\cdot)\) \(\chi_{3150}(3083,\cdot)\)
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
\((2801,127,451)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{17}{20}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3150 }(47, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)