from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5376, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,21,24,8]))
pari: [g,chi] = znchar(Mod(17,5376))
Basic properties
Modulus: | \(5376\) | |
Conductor: | \(1344\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1344}(941,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5376.de
\(\chi_{5376}(17,\cdot)\) \(\chi_{5376}(593,\cdot)\) \(\chi_{5376}(689,\cdot)\) \(\chi_{5376}(1265,\cdot)\) \(\chi_{5376}(1361,\cdot)\) \(\chi_{5376}(1937,\cdot)\) \(\chi_{5376}(2033,\cdot)\) \(\chi_{5376}(2609,\cdot)\) \(\chi_{5376}(2705,\cdot)\) \(\chi_{5376}(3281,\cdot)\) \(\chi_{5376}(3377,\cdot)\) \(\chi_{5376}(3953,\cdot)\) \(\chi_{5376}(4049,\cdot)\) \(\chi_{5376}(4625,\cdot)\) \(\chi_{5376}(4721,\cdot)\) \(\chi_{5376}(5297,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((2815,5125,1793,4609)\) → \((1,e\left(\frac{7}{16}\right),-1,e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 5376 }(17, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{48}\right)\) |
sage: chi.jacobi_sum(n)