from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,22,11,34]))
pari: [g,chi] = znchar(Mod(3377,4020))
Basic properties
Modulus: | \(4020\) | |
Conductor: | \(1005\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1005}(362,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4020.cq
\(\chi_{4020}(53,\cdot)\) \(\chi_{4020}(137,\cdot)\) \(\chi_{4020}(377,\cdot)\) \(\chi_{4020}(713,\cdot)\) \(\chi_{4020}(857,\cdot)\) \(\chi_{4020}(1013,\cdot)\) \(\chi_{4020}(1517,\cdot)\) \(\chi_{4020}(1613,\cdot)\) \(\chi_{4020}(1733,\cdot)\) \(\chi_{4020}(1817,\cdot)\) \(\chi_{4020}(2417,\cdot)\) \(\chi_{4020}(2537,\cdot)\) \(\chi_{4020}(2573,\cdot)\) \(\chi_{4020}(2933,\cdot)\) \(\chi_{4020}(2993,\cdot)\) \(\chi_{4020}(3353,\cdot)\) \(\chi_{4020}(3377,\cdot)\) \(\chi_{4020}(3593,\cdot)\) \(\chi_{4020}(3737,\cdot)\) \(\chi_{4020}(3797,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((2011,2681,3217,1141)\) → \((1,-1,i,e\left(\frac{17}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 4020 }(3377, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(i\) | \(e\left(\frac{5}{11}\right)\) |
sage: chi.jacobi_sum(n)