from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4100, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,20,33]))
pari: [g,chi] = znchar(Mod(99,4100))
Basic properties
| Modulus: | \(4100\) | |
| Conductor: | \(820\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{820}(99,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4100.hf
\(\chi_{4100}(99,\cdot)\) \(\chi_{4100}(199,\cdot)\) \(\chi_{4100}(299,\cdot)\) \(\chi_{4100}(399,\cdot)\) \(\chi_{4100}(499,\cdot)\) \(\chi_{4100}(999,\cdot)\) \(\chi_{4100}(1299,\cdot)\) \(\chi_{4100}(1899,\cdot)\) \(\chi_{4100}(2199,\cdot)\) \(\chi_{4100}(2699,\cdot)\) \(\chi_{4100}(2799,\cdot)\) \(\chi_{4100}(2899,\cdot)\) \(\chi_{4100}(2999,\cdot)\) \(\chi_{4100}(3099,\cdot)\) \(\chi_{4100}(3299,\cdot)\) \(\chi_{4100}(3999,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.40.83017628936247865363646717578368932999206418219130479061598801598873600000000000000000000.1 |
Values on generators
\((2051,1477,3901)\) → \((-1,-1,e\left(\frac{33}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 4100 }(99, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(-i\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) |
sage: chi.jacobi_sum(n)