from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5376, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,9,24,32]))
pari: [g,chi] = znchar(Mod(431,5376))
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 \(\chi_{1344}(515,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5376.df
\(\chi_{5376}(431,\cdot)\) \(\chi_{5376}(527,\cdot)\) \(\chi_{5376}(1103,\cdot)\) \(\chi_{5376}(1199,\cdot)\) \(\chi_{5376}(1775,\cdot)\) \(\chi_{5376}(1871,\cdot)\) \(\chi_{5376}(2447,\cdot)\) \(\chi_{5376}(2543,\cdot)\) \(\chi_{5376}(3119,\cdot)\) \(\chi_{5376}(3215,\cdot)\) \(\chi_{5376}(3791,\cdot)\) \(\chi_{5376}(3887,\cdot)\) \(\chi_{5376}(4463,\cdot)\) \(\chi_{5376}(4559,\cdot)\) \(\chi_{5376}(5135,\cdot)\) \(\chi_{5376}(5231,\cdot)\)
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
\((2815,5125,1793,4609)\) → \((-1,e\left(\frac{3}{16}\right),-1,e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 5376 }(431, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{48}\right)\) |
sage: chi.jacobi_sum(n)