from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8020, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,15,2]))
pari: [g,chi] = znchar(Mod(83,8020))
Basic properties
| Modulus: | \(8020\) | |
| Conductor: | \(8020\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8020.bz
\(\chi_{8020}(83,\cdot)\) \(\chi_{8020}(763,\cdot)\) \(\chi_{8020}(1687,\cdot)\) \(\chi_{8020}(2367,\cdot)\) \(\chi_{8020}(5643,\cdot)\) \(\chi_{8020}(5943,\cdot)\) \(\chi_{8020}(7247,\cdot)\) \(\chi_{8020}(7547,\cdot)\)
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: | Number field defined by a degree 20 polynomial |
Values on generators
\((4011,6417,7221)\) → \((-1,-i,e\left(\frac{1}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 8020 }(83, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) |
sage: chi.jacobi_sum(n)