from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3150, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,3,40]))
pari: [g,chi] = znchar(Mod(3077,3150))
Basic properties
| Modulus: | \(3150\) | |
| Conductor: | \(525\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{525}(452,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3150.ed
\(\chi_{3150}(53,\cdot)\) \(\chi_{3150}(233,\cdot)\) \(\chi_{3150}(683,\cdot)\) \(\chi_{3150}(737,\cdot)\) \(\chi_{3150}(863,\cdot)\) \(\chi_{3150}(1187,\cdot)\) \(\chi_{3150}(1313,\cdot)\) \(\chi_{3150}(1367,\cdot)\) \(\chi_{3150}(1817,\cdot)\) \(\chi_{3150}(1997,\cdot)\) \(\chi_{3150}(2123,\cdot)\) \(\chi_{3150}(2447,\cdot)\) \(\chi_{3150}(2573,\cdot)\) \(\chi_{3150}(2627,\cdot)\) \(\chi_{3150}(2753,\cdot)\) \(\chi_{3150}(3077,\cdot)\)
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
\((2801,127,451)\) → \((-1,e\left(\frac{1}{20}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3150 }(3077, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)