from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2057, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([4,5]))
pari: [g,chi] = znchar(Mod(81,2057))
Basic properties
Modulus: | \(2057\) | |
Conductor: | \(187\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{187}(81,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2057.q
\(\chi_{2057}(81,\cdot)\) \(\chi_{2057}(251,\cdot)\) \(\chi_{2057}(565,\cdot)\) \(\chi_{2057}(608,\cdot)\) \(\chi_{2057}(735,\cdot)\) \(\chi_{2057}(1092,\cdot)\) \(\chi_{2057}(1237,\cdot)\) \(\chi_{2057}(1721,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((970,122)\) → \((e\left(\frac{1}{5}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 2057 }(81, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-i\) | \(i\) |
sage: chi.jacobi_sum(n)