from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7920, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,20,15,36]))
pari: [g,chi] = znchar(Mod(427,7920))
Basic properties
Modulus: | \(7920\) | |
Conductor: | \(7920\) | 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 7920.kw
\(\chi_{7920}(427,\cdot)\) \(\chi_{7920}(643,\cdot)\) \(\chi_{7920}(907,\cdot)\) \(\chi_{7920}(1147,\cdot)\) \(\chi_{7920}(2083,\cdot)\) \(\chi_{7920}(2347,\cdot)\) \(\chi_{7920}(2803,\cdot)\) \(\chi_{7920}(3067,\cdot)\) \(\chi_{7920}(3283,\cdot)\) \(\chi_{7920}(3523,\cdot)\) \(\chi_{7920}(3787,\cdot)\) \(\chi_{7920}(4723,\cdot)\) \(\chi_{7920}(5443,\cdot)\) \(\chi_{7920}(6163,\cdot)\) \(\chi_{7920}(6187,\cdot)\) \(\chi_{7920}(7627,\cdot)\)
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
\((991,5941,3521,6337,6481)\) → \((-1,i,e\left(\frac{1}{3}\right),i,e\left(\frac{3}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 7920 }(427, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)