from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8008, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,50,54,55]))
pari: [g,chi] = znchar(Mod(215,8008))
Basic properties
| Modulus: | \(8008\) | |
| Conductor: | \(4004\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4004}(215,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8008.qb
\(\chi_{8008}(215,\cdot)\) \(\chi_{8008}(479,\cdot)\) \(\chi_{8008}(535,\cdot)\) \(\chi_{8008}(943,\cdot)\) \(\chi_{8008}(1207,\cdot)\) \(\chi_{8008}(1839,\cdot)\) \(\chi_{8008}(2719,\cdot)\) \(\chi_{8008}(3295,\cdot)\) \(\chi_{8008}(4023,\cdot)\) \(\chi_{8008}(4175,\cdot)\) \(\chi_{8008}(4583,\cdot)\) \(\chi_{8008}(4847,\cdot)\) \(\chi_{8008}(4903,\cdot)\) \(\chi_{8008}(6767,\cdot)\) \(\chi_{8008}(7031,\cdot)\) \(\chi_{8008}(7663,\cdot)\)
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
\((6007,4005,3433,4369,4929)\) → \((-1,1,e\left(\frac{5}{6}\right),e\left(\frac{9}{10}\right),e\left(\frac{11}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
| \( \chi_{ 8008 }(215, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) |
sage: chi.jacobi_sum(n)