from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([54,45,50]))
pari: [g,chi] = znchar(Mod(1942,8041))
Basic properties
Modulus: | \(8041\) | |
Conductor: | \(8041\) | 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 8041.dn
\(\chi_{8041}(123,\cdot)\) \(\chi_{8041}(480,\cdot)\) \(\chi_{8041}(523,\cdot)\) \(\chi_{8041}(854,\cdot)\) \(\chi_{8041}(897,\cdot)\) \(\chi_{8041}(1942,\cdot)\) \(\chi_{8041}(2316,\cdot)\) \(\chi_{8041}(5640,\cdot)\) \(\chi_{8041}(6014,\cdot)\) \(\chi_{8041}(6371,\cdot)\) \(\chi_{8041}(6745,\cdot)\) \(\chi_{8041}(7059,\cdot)\) \(\chi_{8041}(7102,\cdot)\) \(\chi_{8041}(7433,\cdot)\) \(\chi_{8041}(7476,\cdot)\) \(\chi_{8041}(7790,\cdot)\)
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
\((6580,2366,562)\) → \((e\left(\frac{9}{10}\right),-i,e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 8041 }(1942, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)