from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(975, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,21,5]))
pari: [g,chi] = znchar(Mod(28,975))
Basic properties
Modulus: | \(975\) | |
Conductor: | \(325\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{325}(28,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 975.cq
\(\chi_{975}(28,\cdot)\) \(\chi_{975}(37,\cdot)\) \(\chi_{975}(58,\cdot)\) \(\chi_{975}(202,\cdot)\) \(\chi_{975}(223,\cdot)\) \(\chi_{975}(253,\cdot)\) \(\chi_{975}(397,\cdot)\) \(\chi_{975}(427,\cdot)\) \(\chi_{975}(448,\cdot)\) \(\chi_{975}(592,\cdot)\) \(\chi_{975}(613,\cdot)\) \(\chi_{975}(622,\cdot)\) \(\chi_{975}(787,\cdot)\) \(\chi_{975}(808,\cdot)\) \(\chi_{975}(817,\cdot)\) \(\chi_{975}(838,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((326,352,301)\) → \((1,e\left(\frac{7}{20}\right),e\left(\frac{1}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
\( \chi_{ 975 }(28, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)