from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3960, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,0,5,15,18]))
pari: [g,chi] = znchar(Mod(119,3960))
Basic properties
Modulus: | \(3960\) | |
Conductor: | \(1980\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1980}(119,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3960.gi
\(\chi_{3960}(119,\cdot)\) \(\chi_{3960}(599,\cdot)\) \(\chi_{3960}(839,\cdot)\) \(\chi_{3960}(1919,\cdot)\) \(\chi_{3960}(2039,\cdot)\) \(\chi_{3960}(2759,\cdot)\) \(\chi_{3960}(3359,\cdot)\) \(\chi_{3960}(3479,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((991,1981,3521,2377,2521)\) → \((-1,1,e\left(\frac{1}{6}\right),-1,e\left(\frac{3}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 3960 }(119, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)