from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2057, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([24,5]))
pari: [g,chi] = znchar(Mod(9,2057))
Basic properties
Modulus: | \(2057\) | |
Conductor: | \(187\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{187}(9,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2057.v
\(\chi_{2057}(9,\cdot)\) \(\chi_{2057}(202,\cdot)\) \(\chi_{2057}(366,\cdot)\) \(\chi_{2057}(372,\cdot)\) \(\chi_{2057}(444,\cdot)\) \(\chi_{2057}(614,\cdot)\) \(\chi_{2057}(729,\cdot)\) \(\chi_{2057}(807,\cdot)\) \(\chi_{2057}(971,\cdot)\) \(\chi_{2057}(977,\cdot)\) \(\chi_{2057}(995,\cdot)\) \(\chi_{2057}(1334,\cdot)\) \(\chi_{2057}(1358,\cdot)\) \(\chi_{2057}(1600,\cdot)\) \(\chi_{2057}(1896,\cdot)\) \(\chi_{2057}(1963,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | 40.40.24562817038400928776197921239227357886542077974183334844678041435576602047153.1 |
Values on generators
\((970,122)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{1}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 2057 }(9, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) |
sage: chi.jacobi_sum(n)