from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4032, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,45,0,32]))
pari: [g,chi] = znchar(Mod(2125,4032))
Basic properties
Modulus: | \(4032\) | |
Conductor: | \(448\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{448}(333,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4032.hf
\(\chi_{4032}(37,\cdot)\) \(\chi_{4032}(109,\cdot)\) \(\chi_{4032}(541,\cdot)\) \(\chi_{4032}(613,\cdot)\) \(\chi_{4032}(1045,\cdot)\) \(\chi_{4032}(1117,\cdot)\) \(\chi_{4032}(1549,\cdot)\) \(\chi_{4032}(1621,\cdot)\) \(\chi_{4032}(2053,\cdot)\) \(\chi_{4032}(2125,\cdot)\) \(\chi_{4032}(2557,\cdot)\) \(\chi_{4032}(2629,\cdot)\) \(\chi_{4032}(3061,\cdot)\) \(\chi_{4032}(3133,\cdot)\) \(\chi_{4032}(3565,\cdot)\) \(\chi_{4032}(3637,\cdot)\)
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
\((127,3781,1793,577)\) → \((1,e\left(\frac{15}{16}\right),1,e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 4032 }(2125, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{37}{48}\right)\) |
sage: chi.jacobi_sum(n)