from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1575, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,57,50]))
pari: [g,chi] = znchar(Mod(1363,1575))
Basic properties
Modulus: | \(1575\) | |
Conductor: | \(1575\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | 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 1575.ep
\(\chi_{1575}(52,\cdot)\) \(\chi_{1575}(103,\cdot)\) \(\chi_{1575}(178,\cdot)\) \(\chi_{1575}(292,\cdot)\) \(\chi_{1575}(367,\cdot)\) \(\chi_{1575}(733,\cdot)\) \(\chi_{1575}(808,\cdot)\) \(\chi_{1575}(922,\cdot)\) \(\chi_{1575}(997,\cdot)\) \(\chi_{1575}(1048,\cdot)\) \(\chi_{1575}(1123,\cdot)\) \(\chi_{1575}(1237,\cdot)\) \(\chi_{1575}(1312,\cdot)\) \(\chi_{1575}(1363,\cdot)\) \(\chi_{1575}(1438,\cdot)\) \(\chi_{1575}(1552,\cdot)\)
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
\((1226,127,451)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{19}{20}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
\( \chi_{ 1575 }(1363, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) |
sage: chi.jacobi_sum(n)