from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7920, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,20,0,12]))
pari: [g,chi] = znchar(Mod(301,7920))
Basic properties
Modulus: | \(7920\) | |
Conductor: | \(1584\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1584}(301,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7920.kp
\(\chi_{7920}(301,\cdot)\) \(\chi_{7920}(421,\cdot)\) \(\chi_{7920}(1021,\cdot)\) \(\chi_{7920}(1501,\cdot)\) \(\chi_{7920}(1741,\cdot)\) \(\chi_{7920}(2821,\cdot)\) \(\chi_{7920}(2941,\cdot)\) \(\chi_{7920}(3661,\cdot)\) \(\chi_{7920}(4261,\cdot)\) \(\chi_{7920}(4381,\cdot)\) \(\chi_{7920}(4981,\cdot)\) \(\chi_{7920}(5461,\cdot)\) \(\chi_{7920}(5701,\cdot)\) \(\chi_{7920}(6781,\cdot)\) \(\chi_{7920}(6901,\cdot)\) \(\chi_{7920}(7621,\cdot)\)
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
\((991,5941,3521,6337,6481)\) → \((1,-i,e\left(\frac{1}{3}\right),1,e\left(\frac{1}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 7920 }(301, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)