from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8020, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,10,39]))
pari: [g,chi] = znchar(Mod(287,8020))
Basic properties
| Modulus: | \(8020\) | |
| Conductor: | \(8020\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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.cg
\(\chi_{8020}(287,\cdot)\) \(\chi_{8020}(503,\cdot)\) \(\chi_{8020}(767,\cdot)\) \(\chi_{8020}(1903,\cdot)\) \(\chi_{8020}(2167,\cdot)\) \(\chi_{8020}(3447,\cdot)\) \(\chi_{8020}(3483,\cdot)\) \(\chi_{8020}(4443,\cdot)\) \(\chi_{8020}(4847,\cdot)\) \(\chi_{8020}(4963,\cdot)\) \(\chi_{8020}(5327,\cdot)\) \(\chi_{8020}(5463,\cdot)\) \(\chi_{8020}(5827,\cdot)\) \(\chi_{8020}(5983,\cdot)\) \(\chi_{8020}(6943,\cdot)\) \(\chi_{8020}(7807,\cdot)\)
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
\((4011,6417,7221)\) → \((-1,i,e\left(\frac{39}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 8020 }(287, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) |
sage: chi.jacobi_sum(n)