from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3100, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,1,16]))
pari: [g,chi] = znchar(Mod(777,3100))
Basic properties
| Modulus: | \(3100\) | |
| Conductor: | \(775\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{775}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3100.dk
\(\chi_{3100}(97,\cdot)\) \(\chi_{3100}(777,\cdot)\) \(\chi_{3100}(1217,\cdot)\) \(\chi_{3100}(1273,\cdot)\) \(\chi_{3100}(2333,\cdot)\) \(\chi_{3100}(2453,\cdot)\) \(\chi_{3100}(2713,\cdot)\) \(\chi_{3100}(2837,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | 20.0.2117079920563803938649597694166004657745361328125.1 |
Values on generators
\((1551,2977,1801)\) → \((1,e\left(\frac{1}{20}\right),e\left(\frac{4}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 3100 }(777, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-i\) | \(i\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) |
sage: chi.jacobi_sum(n)