from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6045, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,25,58]))
pari: [g,chi] = znchar(Mod(331,6045))
Basic properties
Modulus: | \(6045\) | |
Conductor: | \(403\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{403}(331,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6045.ng
\(\chi_{6045}(331,\cdot)\) \(\chi_{6045}(631,\cdot)\) \(\chi_{6045}(1501,\cdot)\) \(\chi_{6045}(1636,\cdot)\) \(\chi_{6045}(1696,\cdot)\) \(\chi_{6045}(2191,\cdot)\) \(\chi_{6045}(3031,\cdot)\) \(\chi_{6045}(3196,\cdot)\) \(\chi_{6045}(3361,\cdot)\) \(\chi_{6045}(3391,\cdot)\) \(\chi_{6045}(3556,\cdot)\) \(\chi_{6045}(4171,\cdot)\) \(\chi_{6045}(4591,\cdot)\) \(\chi_{6045}(4786,\cdot)\) \(\chi_{6045}(4816,\cdot)\) \(\chi_{6045}(5566,\cdot)\)
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
\((4031,4837,1861,2731)\) → \((1,1,e\left(\frac{5}{12}\right),e\left(\frac{29}{30}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
\( \chi_{ 6045 }(331, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) |
sage: chi.jacobi_sum(n)