from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9450, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,3,30]))
pari: [g,chi] = znchar(Mod(2827,9450))
Basic properties
Modulus: | \(9450\) | |
Conductor: | \(1575\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1575}(727,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9450.fv
\(\chi_{9450}(937,\cdot)\) \(\chi_{9450}(1063,\cdot)\) \(\chi_{9450}(2197,\cdot)\) \(\chi_{9450}(2827,\cdot)\) \(\chi_{9450}(2953,\cdot)\) \(\chi_{9450}(3583,\cdot)\) \(\chi_{9450}(4087,\cdot)\) \(\chi_{9450}(4717,\cdot)\) \(\chi_{9450}(5473,\cdot)\) \(\chi_{9450}(5977,\cdot)\) \(\chi_{9450}(6733,\cdot)\) \(\chi_{9450}(7363,\cdot)\) \(\chi_{9450}(7867,\cdot)\) \(\chi_{9450}(8497,\cdot)\) \(\chi_{9450}(8623,\cdot)\) \(\chi_{9450}(9253,\cdot)\)
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
\((9101,6427,6751)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{20}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 9450 }(2827, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)