from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2240, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,45,36,32]))
pari: [g,chi] = znchar(Mod(1523,2240))
Basic properties
Modulus: | \(2240\) | |
Conductor: | \(2240\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2240.ez
\(\chi_{2240}(107,\cdot)\) \(\chi_{2240}(163,\cdot)\) \(\chi_{2240}(347,\cdot)\) \(\chi_{2240}(403,\cdot)\) \(\chi_{2240}(667,\cdot)\) \(\chi_{2240}(723,\cdot)\) \(\chi_{2240}(907,\cdot)\) \(\chi_{2240}(963,\cdot)\) \(\chi_{2240}(1227,\cdot)\) \(\chi_{2240}(1283,\cdot)\) \(\chi_{2240}(1467,\cdot)\) \(\chi_{2240}(1523,\cdot)\) \(\chi_{2240}(1787,\cdot)\) \(\chi_{2240}(1843,\cdot)\) \(\chi_{2240}(2027,\cdot)\) \(\chi_{2240}(2083,\cdot)\)
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
\((1471,1541,897,1921)\) → \((-1,e\left(\frac{15}{16}\right),-i,e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 2240 }(1523, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)