from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7920, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,40,45,48]))
pari: [g,chi] = znchar(Mod(223,7920))
Basic properties
Modulus: | \(7920\) | |
Conductor: | \(1980\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1980}(223,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7920.ld
\(\chi_{7920}(223,\cdot)\) \(\chi_{7920}(367,\cdot)\) \(\chi_{7920}(1087,\cdot)\) \(\chi_{7920}(1807,\cdot)\) \(\chi_{7920}(2623,\cdot)\) \(\chi_{7920}(4063,\cdot)\) \(\chi_{7920}(4207,\cdot)\) \(\chi_{7920}(4783,\cdot)\) \(\chi_{7920}(5263,\cdot)\) \(\chi_{7920}(5503,\cdot)\) \(\chi_{7920}(5647,\cdot)\) \(\chi_{7920}(6367,\cdot)\) \(\chi_{7920}(6703,\cdot)\) \(\chi_{7920}(6847,\cdot)\) \(\chi_{7920}(7087,\cdot)\) \(\chi_{7920}(7423,\cdot)\)
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
\((991,5941,3521,6337,6481)\) → \((-1,1,e\left(\frac{2}{3}\right),-i,e\left(\frac{4}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 7920 }(223, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)