from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2240, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,3,36,16]))
pari: [g,chi] = znchar(Mod(443,2240))
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.fk
\(\chi_{2240}(67,\cdot)\) \(\chi_{2240}(123,\cdot)\) \(\chi_{2240}(387,\cdot)\) \(\chi_{2240}(443,\cdot)\) \(\chi_{2240}(627,\cdot)\) \(\chi_{2240}(683,\cdot)\) \(\chi_{2240}(947,\cdot)\) \(\chi_{2240}(1003,\cdot)\) \(\chi_{2240}(1187,\cdot)\) \(\chi_{2240}(1243,\cdot)\) \(\chi_{2240}(1507,\cdot)\) \(\chi_{2240}(1563,\cdot)\) \(\chi_{2240}(1747,\cdot)\) \(\chi_{2240}(1803,\cdot)\) \(\chi_{2240}(2067,\cdot)\) \(\chi_{2240}(2123,\cdot)\)
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
\((1471,1541,897,1921)\) → \((-1,e\left(\frac{1}{16}\right),-i,e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 2240 }(443, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)