from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3150, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,3,20]))
pari: [g,chi] = znchar(Mod(527,3150))
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 \(\chi_{1575}(527,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3150.ee
\(\chi_{3150}(23,\cdot)\) \(\chi_{3150}(137,\cdot)\) \(\chi_{3150}(263,\cdot)\) \(\chi_{3150}(527,\cdot)\) \(\chi_{3150}(653,\cdot)\) \(\chi_{3150}(767,\cdot)\) \(\chi_{3150}(1283,\cdot)\) \(\chi_{3150}(1397,\cdot)\) \(\chi_{3150}(1523,\cdot)\) \(\chi_{3150}(1787,\cdot)\) \(\chi_{3150}(1913,\cdot)\) \(\chi_{3150}(2027,\cdot)\) \(\chi_{3150}(2153,\cdot)\) \(\chi_{3150}(2417,\cdot)\) \(\chi_{3150}(2783,\cdot)\) \(\chi_{3150}(3047,\cdot)\)
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
\((2801,127,451)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{20}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3150 }(527, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)