from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9450, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,27,20]))
pari: [g,chi] = znchar(Mod(37,9450))
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 \(\chi_{1575}(562,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9450.fw
\(\chi_{9450}(37,\cdot)\) \(\chi_{9450}(613,\cdot)\) \(\chi_{9450}(1747,\cdot)\) \(\chi_{9450}(1927,\cdot)\) \(\chi_{9450}(2503,\cdot)\) \(\chi_{9450}(2683,\cdot)\) \(\chi_{9450}(3637,\cdot)\) \(\chi_{9450}(3817,\cdot)\) \(\chi_{9450}(4573,\cdot)\) \(\chi_{9450}(5527,\cdot)\) \(\chi_{9450}(6283,\cdot)\) \(\chi_{9450}(6463,\cdot)\) \(\chi_{9450}(7417,\cdot)\) \(\chi_{9450}(7597,\cdot)\) \(\chi_{9450}(8173,\cdot)\) \(\chi_{9450}(8353,\cdot)\)
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
\((9101,6427,6751)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{9}{20}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 9450 }(37, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)