from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2240, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,5,4,8]))
pari: [g,chi] = znchar(Mod(1077,2240))
Basic properties
Modulus: | \(2240\) | |
Conductor: | \(2240\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(16\) | 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.du
\(\chi_{2240}(13,\cdot)\) \(\chi_{2240}(517,\cdot)\) \(\chi_{2240}(573,\cdot)\) \(\chi_{2240}(1077,\cdot)\) \(\chi_{2240}(1133,\cdot)\) \(\chi_{2240}(1637,\cdot)\) \(\chi_{2240}(1693,\cdot)\) \(\chi_{2240}(2197,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
Fixed field: | 16.16.850734469462919174923747328000000000000.2 |
Values on generators
\((1471,1541,897,1921)\) → \((1,e\left(\frac{5}{16}\right),i,-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 2240 }(1077, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(-1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)