from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9240, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,0,15,0,8]))
pari: [g,chi] = znchar(Mod(1303,9240))
Basic properties
Modulus: | \(9240\) | |
Conductor: | \(220\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{220}(203,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9240.kj
\(\chi_{9240}(1303,\cdot)\) \(\chi_{9240}(1807,\cdot)\) \(\chi_{9240}(2143,\cdot)\) \(\chi_{9240}(4327,\cdot)\) \(\chi_{9240}(5503,\cdot)\) \(\chi_{9240}(6847,\cdot)\) \(\chi_{9240}(7687,\cdot)\) \(\chi_{9240}(8023,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((2311,4621,6161,3697,5281,2521)\) → \((-1,1,1,-i,1,e\left(\frac{2}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 9240 }(1303, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-i\) | \(e\left(\frac{9}{20}\right)\) |
sage: chi.jacobi_sum(n)