from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8008, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,20,12,35]))
pari: [g,chi] = znchar(Mod(2039,8008))
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 \(\chi_{4004}(2039,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8008.rf
\(\chi_{8008}(487,\cdot)\) \(\chi_{8008}(1215,\cdot)\) \(\chi_{8008}(1367,\cdot)\) \(\chi_{8008}(1775,\cdot)\) \(\chi_{8008}(2039,\cdot)\) \(\chi_{8008}(2095,\cdot)\) \(\chi_{8008}(2671,\cdot)\) \(\chi_{8008}(3551,\cdot)\) \(\chi_{8008}(4855,\cdot)\) \(\chi_{8008}(5415,\cdot)\) \(\chi_{8008}(5679,\cdot)\) \(\chi_{8008}(5735,\cdot)\) \(\chi_{8008}(6143,\cdot)\) \(\chi_{8008}(6407,\cdot)\) \(\chi_{8008}(7599,\cdot)\) \(\chi_{8008}(7863,\cdot)\)
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
\((6007,4005,3433,4369,4929)\) → \((-1,1,e\left(\frac{1}{3}\right),e\left(\frac{1}{5}\right),e\left(\frac{7}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
| \( \chi_{ 8008 }(2039, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) |
sage: chi.jacobi_sum(n)