from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2541, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,55,6]))
pari: [g,chi] = znchar(Mod(782,2541))
Basic properties
Modulus: | \(2541\) | |
Conductor: | \(2541\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | 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 2541.bt
\(\chi_{2541}(89,\cdot)\) \(\chi_{2541}(320,\cdot)\) \(\chi_{2541}(353,\cdot)\) \(\chi_{2541}(551,\cdot)\) \(\chi_{2541}(584,\cdot)\) \(\chi_{2541}(782,\cdot)\) \(\chi_{2541}(815,\cdot)\) \(\chi_{2541}(1013,\cdot)\) \(\chi_{2541}(1046,\cdot)\) \(\chi_{2541}(1244,\cdot)\) \(\chi_{2541}(1277,\cdot)\) \(\chi_{2541}(1475,\cdot)\) \(\chi_{2541}(1508,\cdot)\) \(\chi_{2541}(1706,\cdot)\) \(\chi_{2541}(1739,\cdot)\) \(\chi_{2541}(1970,\cdot)\) \(\chi_{2541}(2168,\cdot)\) \(\chi_{2541}(2201,\cdot)\) \(\chi_{2541}(2399,\cdot)\) \(\chi_{2541}(2432,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((848,1816,2059)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{1}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 2541 }(782, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) |
sage: chi.jacobi_sum(n)