from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8008, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,6,5]))
pari: [g,chi] = znchar(Mod(1289,8008))
Basic properties
| Modulus: | \(8008\) | |
| Conductor: | \(143\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{143}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8008.ry
\(\chi_{8008}(1289,\cdot)\) \(\chi_{8008}(1905,\cdot)\) \(\chi_{8008}(2745,\cdot)\) \(\chi_{8008}(2801,\cdot)\) \(\chi_{8008}(3361,\cdot)\) \(\chi_{8008}(3417,\cdot)\) \(\chi_{8008}(3473,\cdot)\) \(\chi_{8008}(4089,\cdot)\) \(\chi_{8008}(4985,\cdot)\) \(\chi_{8008}(5601,\cdot)\) \(\chi_{8008}(6441,\cdot)\) \(\chi_{8008}(7057,\cdot)\) \(\chi_{8008}(7113,\cdot)\) \(\chi_{8008}(7169,\cdot)\) \(\chi_{8008}(7729,\cdot)\) \(\chi_{8008}(7785,\cdot)\)
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
\((6007,4005,3433,4369,4929)\) → \((1,1,1,e\left(\frac{1}{10}\right),e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
| \( \chi_{ 8008 }(1289, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{30}\right)\) |
sage: chi.jacobi_sum(n)