from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,1]))
pari: [g,chi] = znchar(Mod(1141,4020))
Basic properties
Modulus: | \(4020\) | |
Conductor: | \(67\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{67}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4020.cv
\(\chi_{4020}(61,\cdot)\) \(\chi_{4020}(481,\cdot)\) \(\chi_{4020}(721,\cdot)\) \(\chi_{4020}(781,\cdot)\) \(\chi_{4020}(1141,\cdot)\) \(\chi_{4020}(1321,\cdot)\) \(\chi_{4020}(1381,\cdot)\) \(\chi_{4020}(1441,\cdot)\) \(\chi_{4020}(1561,\cdot)\) \(\chi_{4020}(1621,\cdot)\) \(\chi_{4020}(2041,\cdot)\) \(\chi_{4020}(2341,\cdot)\) \(\chi_{4020}(2641,\cdot)\) \(\chi_{4020}(2821,\cdot)\) \(\chi_{4020}(3061,\cdot)\) \(\chi_{4020}(3181,\cdot)\) \(\chi_{4020}(3301,\cdot)\) \(\chi_{4020}(3361,\cdot)\) \(\chi_{4020}(3541,\cdot)\) \(\chi_{4020}(3601,\cdot)\)
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
\((2011,2681,3217,1141)\) → \((1,1,1,e\left(\frac{1}{66}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 4020 }(1141, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{53}{66}\right)\) |
sage: chi.jacobi_sum(n)