from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(221760, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,15,20,0,0,8]))
pari: [g,chi] = znchar(Mod(25271,221760))
Basic properties
| Modulus: | \(221760\) | |
| Conductor: | \(1056\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1056}(323,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 221760.ckb
\(\chi_{221760}(71,\cdot)\) \(\chi_{221760}(10151,\cdot)\) \(\chi_{221760}(25271,\cdot)\) \(\chi_{221760}(50471,\cdot)\) \(\chi_{221760}(55511,\cdot)\) \(\chi_{221760}(65591,\cdot)\) \(\chi_{221760}(80711,\cdot)\) \(\chi_{221760}(105911,\cdot)\) \(\chi_{221760}(110951,\cdot)\) \(\chi_{221760}(121031,\cdot)\) \(\chi_{221760}(136151,\cdot)\) \(\chi_{221760}(161351,\cdot)\) \(\chi_{221760}(166391,\cdot)\) \(\chi_{221760}(176471,\cdot)\) \(\chi_{221760}(191591,\cdot)\) \(\chi_{221760}(216791,\cdot)\)
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: | Number field defined by a degree 40 polynomial |
Values on generators
\((48511,124741,98561,133057,190081,141121)\) → \((-1,e\left(\frac{3}{8}\right),-1,1,1,e\left(\frac{1}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 221760 }(25271, a) \) | \(1\) | \(1\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(i\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) |
sage: chi.jacobi_sum(n)