from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1287, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,54,35]))
pari: [g,chi] = znchar(Mod(1051,1287))
Basic properties
Modulus: | \(1287\) | |
Conductor: | \(1287\) | 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 1287.en
\(\chi_{1287}(85,\cdot)\) \(\chi_{1287}(106,\cdot)\) \(\chi_{1287}(292,\cdot)\) \(\chi_{1287}(349,\cdot)\) \(\chi_{1287}(409,\cdot)\) \(\chi_{1287}(436,\cdot)\) \(\chi_{1287}(457,\cdot)\) \(\chi_{1287}(574,\cdot)\) \(\chi_{1287}(700,\cdot)\) \(\chi_{1287}(787,\cdot)\) \(\chi_{1287}(877,\cdot)\) \(\chi_{1287}(904,\cdot)\) \(\chi_{1287}(1042,\cdot)\) \(\chi_{1287}(1051,\cdot)\) \(\chi_{1287}(1168,\cdot)\) \(\chi_{1287}(1228,\cdot)\)
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
\((1145,937,496)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{9}{10}\right),e\left(\frac{7}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 1287 }(1051, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) |
sage: chi.jacobi_sum(n)