from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8280, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,55,0,63]))
pari: [g,chi] = znchar(Mod(221,8280))
Basic properties
Modulus: | \(8280\) | |
Conductor: | \(1656\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1656}(221,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8280.fq
\(\chi_{8280}(221,\cdot)\) \(\chi_{8280}(941,\cdot)\) \(\chi_{8280}(1661,\cdot)\) \(\chi_{8280}(1901,\cdot)\) \(\chi_{8280}(2021,\cdot)\) \(\chi_{8280}(2261,\cdot)\) \(\chi_{8280}(2981,\cdot)\) \(\chi_{8280}(3101,\cdot)\) \(\chi_{8280}(3461,\cdot)\) \(\chi_{8280}(3701,\cdot)\) \(\chi_{8280}(4421,\cdot)\) \(\chi_{8280}(4541,\cdot)\) \(\chi_{8280}(4781,\cdot)\) \(\chi_{8280}(5261,\cdot)\) \(\chi_{8280}(5861,\cdot)\) \(\chi_{8280}(6221,\cdot)\) \(\chi_{8280}(7301,\cdot)\) \(\chi_{8280}(7421,\cdot)\) \(\chi_{8280}(7781,\cdot)\) \(\chi_{8280}(8021,\cdot)\)
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
\((2071,4141,4601,1657,3961)\) → \((1,-1,e\left(\frac{5}{6}\right),1,e\left(\frac{21}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 8280 }(221, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) |
sage: chi.jacobi_sum(n)