from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4176, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,14,36]))
pari: [g,chi] = znchar(Mod(49,4176))
Basic properties
| Modulus: | \(4176\) | |
| Conductor: | \(261\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{261}(49,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4176.ds
\(\chi_{4176}(49,\cdot)\) \(\chi_{4176}(529,\cdot)\) \(\chi_{4176}(625,\cdot)\) \(\chi_{4176}(1921,\cdot)\) \(\chi_{4176}(2113,\cdot)\) \(\chi_{4176}(2257,\cdot)\) \(\chi_{4176}(2401,\cdot)\) \(\chi_{4176}(2833,\cdot)\) \(\chi_{4176}(3409,\cdot)\) \(\chi_{4176}(3505,\cdot)\) \(\chi_{4176}(3649,\cdot)\) \(\chi_{4176}(3793,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 21 polynomial |
Values on generators
\((1567,1045,929,4033)\) → \((1,1,e\left(\frac{1}{3}\right),e\left(\frac{6}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
| \( \chi_{ 4176 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)