from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2304, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,3,40]))
pari: [g,chi] = znchar(Mod(113,2304))
Basic properties
Modulus: | \(2304\) | |
Conductor: | \(576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{576}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2304.br
\(\chi_{2304}(113,\cdot)\) \(\chi_{2304}(209,\cdot)\) \(\chi_{2304}(401,\cdot)\) \(\chi_{2304}(497,\cdot)\) \(\chi_{2304}(689,\cdot)\) \(\chi_{2304}(785,\cdot)\) \(\chi_{2304}(977,\cdot)\) \(\chi_{2304}(1073,\cdot)\) \(\chi_{2304}(1265,\cdot)\) \(\chi_{2304}(1361,\cdot)\) \(\chi_{2304}(1553,\cdot)\) \(\chi_{2304}(1649,\cdot)\) \(\chi_{2304}(1841,\cdot)\) \(\chi_{2304}(1937,\cdot)\) \(\chi_{2304}(2129,\cdot)\) \(\chi_{2304}(2225,\cdot)\)
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
\((1279,2053,1793)\) → \((1,e\left(\frac{1}{16}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 2304 }(113, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)