from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8624, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,40,18]))
pari: [g,chi] = znchar(Mod(459,8624))
Basic properties
Modulus: | \(8624\) | |
Conductor: | \(1232\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1232}(459,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8624.fd
\(\chi_{8624}(459,\cdot)\) \(\chi_{8624}(667,\cdot)\) \(\chi_{8624}(1635,\cdot)\) \(\chi_{8624}(1843,\cdot)\) \(\chi_{8624}(2235,\cdot)\) \(\chi_{8624}(2811,\cdot)\) \(\chi_{8624}(3203,\cdot)\) \(\chi_{8624}(3803,\cdot)\) \(\chi_{8624}(4771,\cdot)\) \(\chi_{8624}(4979,\cdot)\) \(\chi_{8624}(5947,\cdot)\) \(\chi_{8624}(6155,\cdot)\) \(\chi_{8624}(6547,\cdot)\) \(\chi_{8624}(7123,\cdot)\) \(\chi_{8624}(7515,\cdot)\) \(\chi_{8624}(8115,\cdot)\)
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
\((5391,6469,7745,3137)\) → \((-1,i,e\left(\frac{2}{3}\right),e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 8624 }(459, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) |
sage: chi.jacobi_sum(n)