from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([32,15,0]))
pari: [g,chi] = znchar(Mod(5591,8041))
Basic properties
| Modulus: | \(8041\) | |
| Conductor: | \(187\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{187}(168,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8041.cu
\(\chi_{8041}(603,\cdot)\) \(\chi_{8041}(818,\cdot)\) \(\chi_{8041}(1334,\cdot)\) \(\chi_{8041}(1549,\cdot)\) \(\chi_{8041}(2280,\cdot)\) \(\chi_{8041}(2667,\cdot)\) \(\chi_{8041}(3613,\cdot)\) \(\chi_{8041}(4129,\cdot)\) \(\chi_{8041}(4860,\cdot)\) \(\chi_{8041}(5075,\cdot)\) \(\chi_{8041}(5591,\cdot)\) \(\chi_{8041}(5806,\cdot)\) \(\chi_{8041}(6451,\cdot)\) \(\chi_{8041}(6537,\cdot)\) \(\chi_{8041}(7397,\cdot)\) \(\chi_{8041}(7913,\cdot)\)
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: | 40.40.24562817038400928776197921239227357886542077974183334844678041435576602047153.1 |
Values on generators
\((6580,2366,562)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{3}{8}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 8041 }(5591, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)