from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,0,47]))
pari: [g,chi] = znchar(Mod(31,4020))
Basic properties
Modulus: | \(4020\) | |
Conductor: | \(268\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{268}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4020.dh
\(\chi_{4020}(31,\cdot)\) \(\chi_{4020}(331,\cdot)\) \(\chi_{4020}(631,\cdot)\) \(\chi_{4020}(811,\cdot)\) \(\chi_{4020}(1051,\cdot)\) \(\chi_{4020}(1171,\cdot)\) \(\chi_{4020}(1291,\cdot)\) \(\chi_{4020}(1351,\cdot)\) \(\chi_{4020}(1531,\cdot)\) \(\chi_{4020}(1591,\cdot)\) \(\chi_{4020}(2071,\cdot)\) \(\chi_{4020}(2491,\cdot)\) \(\chi_{4020}(2731,\cdot)\) \(\chi_{4020}(2791,\cdot)\) \(\chi_{4020}(3151,\cdot)\) \(\chi_{4020}(3331,\cdot)\) \(\chi_{4020}(3391,\cdot)\) \(\chi_{4020}(3451,\cdot)\) \(\chi_{4020}(3571,\cdot)\) \(\chi_{4020}(3631,\cdot)\)
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
\((2011,2681,3217,1141)\) → \((-1,1,1,e\left(\frac{47}{66}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 4020 }(31, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{49}{66}\right)\) |
sage: chi.jacobi_sum(n)