from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,0,49]))
pari: [g,chi] = znchar(Mod(191,4020))
Basic properties
Modulus: | \(4020\) | |
Conductor: | \(804\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{804}(191,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4020.de
\(\chi_{4020}(11,\cdot)\) \(\chi_{4020}(191,\cdot)\) \(\chi_{4020}(251,\cdot)\) \(\chi_{4020}(731,\cdot)\) \(\chi_{4020}(1151,\cdot)\) \(\chi_{4020}(1391,\cdot)\) \(\chi_{4020}(1451,\cdot)\) \(\chi_{4020}(1811,\cdot)\) \(\chi_{4020}(1991,\cdot)\) \(\chi_{4020}(2051,\cdot)\) \(\chi_{4020}(2111,\cdot)\) \(\chi_{4020}(2231,\cdot)\) \(\chi_{4020}(2291,\cdot)\) \(\chi_{4020}(2711,\cdot)\) \(\chi_{4020}(3011,\cdot)\) \(\chi_{4020}(3311,\cdot)\) \(\chi_{4020}(3491,\cdot)\) \(\chi_{4020}(3731,\cdot)\) \(\chi_{4020}(3851,\cdot)\) \(\chi_{4020}(3971,\cdot)\)
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
\((2011,2681,3217,1141)\) → \((-1,-1,1,e\left(\frac{49}{66}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 4020 }(191, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{28}{33}\right)\) |
sage: chi.jacobi_sum(n)