from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2169, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,11]))
pari: [g,chi] = znchar(Mod(97,2169))
Basic properties
Modulus: | \(2169\) | |
Conductor: | \(2169\) | 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 2169.cw
\(\chi_{2169}(97,\cdot)\) \(\chi_{2169}(151,\cdot)\) \(\chi_{2169}(331,\cdot)\) \(\chi_{2169}(337,\cdot)\) \(\chi_{2169}(385,\cdot)\) \(\chi_{2169}(400,\cdot)\) \(\chi_{2169}(805,\cdot)\) \(\chi_{2169}(868,\cdot)\) \(\chi_{2169}(1087,\cdot)\) \(\chi_{2169}(1312,\cdot)\) \(\chi_{2169}(1339,\cdot)\) \(\chi_{2169}(1363,\cdot)\) \(\chi_{2169}(1564,\cdot)\) \(\chi_{2169}(1678,\cdot)\) \(\chi_{2169}(1696,\cdot)\) \(\chi_{2169}(2011,\cdot)\)
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
\((965,730)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{11}{60}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 2169 }(97, a) \) | \(1\) | \(1\) | \(-1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(-1\) | \(e\left(\frac{2}{15}\right)\) | \(i\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)