from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4018, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,27]))
pari: [g,chi] = znchar(Mod(361,4018))
Basic properties
Modulus: | \(4018\) | |
Conductor: | \(287\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{287}(74,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4018.bp
\(\chi_{4018}(361,\cdot)\) \(\chi_{4018}(459,\cdot)\) \(\chi_{4018}(471,\cdot)\) \(\chi_{4018}(569,\cdot)\) \(\chi_{4018}(863,\cdot)\) \(\chi_{4018}(1537,\cdot)\) \(\chi_{4018}(1635,\cdot)\) \(\chi_{4018}(1843,\cdot)\) \(\chi_{4018}(1929,\cdot)\) \(\chi_{4018}(2137,\cdot)\) \(\chi_{4018}(2235,\cdot)\) \(\chi_{4018}(2909,\cdot)\) \(\chi_{4018}(3203,\cdot)\) \(\chi_{4018}(3301,\cdot)\) \(\chi_{4018}(3313,\cdot)\) \(\chi_{4018}(3411,\cdot)\)
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
\((493,785)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{9}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 4018 }(361, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) |
sage: chi.jacobi_sum(n)