from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7920, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,10,45,6]))
pari: [g,chi] = znchar(Mod(5843,7920))
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.lu
\(\chi_{7920}(83,\cdot)\) \(\chi_{7920}(347,\cdot)\) \(\chi_{7920}(563,\cdot)\) \(\chi_{7920}(1283,\cdot)\) \(\chi_{7920}(2723,\cdot)\) \(\chi_{7920}(2747,\cdot)\) \(\chi_{7920}(3467,\cdot)\) \(\chi_{7920}(4187,\cdot)\) \(\chi_{7920}(5123,\cdot)\) \(\chi_{7920}(5387,\cdot)\) \(\chi_{7920}(5627,\cdot)\) \(\chi_{7920}(5843,\cdot)\) \(\chi_{7920}(6107,\cdot)\) \(\chi_{7920}(6563,\cdot)\) \(\chi_{7920}(6827,\cdot)\) \(\chi_{7920}(7763,\cdot)\)
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
\((991,5941,3521,6337,6481)\) → \((-1,-i,e\left(\frac{1}{6}\right),-i,e\left(\frac{1}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7920 }(5843, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)