from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8008, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,30,40,54,55]))
pari: [g,chi] = znchar(Mod(501,8008))
Basic properties
| Modulus: | \(8008\) | |
| Conductor: | \(8008\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 8008.qi
\(\chi_{8008}(501,\cdot)\) \(\chi_{8008}(765,\cdot)\) \(\chi_{8008}(821,\cdot)\) \(\chi_{8008}(1229,\cdot)\) \(\chi_{8008}(1493,\cdot)\) \(\chi_{8008}(2125,\cdot)\) \(\chi_{8008}(3005,\cdot)\) \(\chi_{8008}(3581,\cdot)\) \(\chi_{8008}(4309,\cdot)\) \(\chi_{8008}(4461,\cdot)\) \(\chi_{8008}(4869,\cdot)\) \(\chi_{8008}(5133,\cdot)\) \(\chi_{8008}(5189,\cdot)\) \(\chi_{8008}(7053,\cdot)\) \(\chi_{8008}(7317,\cdot)\) \(\chi_{8008}(7949,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((6007,4005,3433,4369,4929)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{9}{10}\right),e\left(\frac{11}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
| \( \chi_{ 8008 }(501, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(-1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{15}\right)\) |
sage: chi.jacobi_sum(n)