from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3360, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,9,0,0,4]))
pari: [g,chi] = znchar(Mod(451,3360))
Basic properties
Modulus: | \(3360\) | |
Conductor: | \(224\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{224}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3360.hm
\(\chi_{3360}(451,\cdot)\) \(\chi_{3360}(691,\cdot)\) \(\chi_{3360}(1291,\cdot)\) \(\chi_{3360}(1531,\cdot)\) \(\chi_{3360}(2131,\cdot)\) \(\chi_{3360}(2371,\cdot)\) \(\chi_{3360}(2971,\cdot)\) \(\chi_{3360}(3211,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
Fixed field: | 24.24.790224330201082600125157415256880139617697792.1 |
Values on generators
\((1471,421,1121,2017,1921)\) → \((-1,e\left(\frac{3}{8}\right),1,1,e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 3360 }(451, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) |
sage: chi.jacobi_sum(n)