from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,33,64]))
pari: [g,chi] = znchar(Mod(419,4020))
Basic properties
Modulus: | \(4020\) | |
Conductor: | \(4020\) | 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 4020.di
\(\chi_{4020}(419,\cdot)\) \(\chi_{4020}(479,\cdot)\) \(\chi_{4020}(659,\cdot)\) \(\chi_{4020}(719,\cdot)\) \(\chi_{4020}(839,\cdot)\) \(\chi_{4020}(959,\cdot)\) \(\chi_{4020}(1199,\cdot)\) \(\chi_{4020}(1379,\cdot)\) \(\chi_{4020}(1679,\cdot)\) \(\chi_{4020}(1979,\cdot)\) \(\chi_{4020}(2399,\cdot)\) \(\chi_{4020}(2459,\cdot)\) \(\chi_{4020}(2579,\cdot)\) \(\chi_{4020}(2639,\cdot)\) \(\chi_{4020}(2699,\cdot)\) \(\chi_{4020}(2879,\cdot)\) \(\chi_{4020}(3239,\cdot)\) \(\chi_{4020}(3299,\cdot)\) \(\chi_{4020}(3539,\cdot)\) \(\chi_{4020}(3959,\cdot)\)
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
\((2011,2681,3217,1141)\) → \((-1,-1,-1,e\left(\frac{32}{33}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 4020 }(419, a) \) | \(1\) | \(1\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{59}{66}\right)\) |
sage: chi.jacobi_sum(n)