from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8008, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,40,12,25]))
pari: [g,chi] = znchar(Mod(851,8008))
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.sc
\(\chi_{8008}(851,\cdot)\) \(\chi_{8008}(1411,\cdot)\) \(\chi_{8008}(1675,\cdot)\) \(\chi_{8008}(1731,\cdot)\) \(\chi_{8008}(2139,\cdot)\) \(\chi_{8008}(2403,\cdot)\) \(\chi_{8008}(3595,\cdot)\) \(\chi_{8008}(3859,\cdot)\) \(\chi_{8008}(4491,\cdot)\) \(\chi_{8008}(5219,\cdot)\) \(\chi_{8008}(5371,\cdot)\) \(\chi_{8008}(5779,\cdot)\) \(\chi_{8008}(6043,\cdot)\) \(\chi_{8008}(6099,\cdot)\) \(\chi_{8008}(6675,\cdot)\) \(\chi_{8008}(7555,\cdot)\)
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
\((6007,4005,3433,4369,4929)\) → \((-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{1}{5}\right),e\left(\frac{5}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
\( \chi_{ 8008 }(851, a) \) | \(1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{30}\right)\) |
sage: chi.jacobi_sum(n)