from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1925, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([51,40,6]))
pari: [g,chi] = znchar(Mod(1047,1925))
Basic properties
Modulus: | \(1925\) | |
Conductor: | \(1925\) | 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 1925.gf
\(\chi_{1925}(513,\cdot)\) \(\chi_{1925}(723,\cdot)\) \(\chi_{1925}(772,\cdot)\) \(\chi_{1925}(788,\cdot)\) \(\chi_{1925}(842,\cdot)\) \(\chi_{1925}(877,\cdot)\) \(\chi_{1925}(998,\cdot)\) \(\chi_{1925}(1003,\cdot)\) \(\chi_{1925}(1047,\cdot)\) \(\chi_{1925}(1117,\cdot)\) \(\chi_{1925}(1152,\cdot)\) \(\chi_{1925}(1278,\cdot)\) \(\chi_{1925}(1437,\cdot)\) \(\chi_{1925}(1458,\cdot)\) \(\chi_{1925}(1712,\cdot)\) \(\chi_{1925}(1733,\cdot)\)
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
\((1002,276,1751)\) → \((e\left(\frac{17}{20}\right),e\left(\frac{2}{3}\right),e\left(\frac{1}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
\( \chi_{ 1925 }(1047, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(i\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) |
sage: chi.jacobi_sum(n)