from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,22,33,8]))
pari: [g,chi] = znchar(Mod(143,4020))
Basic properties
| Modulus: | \(4020\) | |
| Conductor: | \(4020\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4020.co
\(\chi_{4020}(107,\cdot)\) \(\chi_{4020}(143,\cdot)\) \(\chi_{4020}(263,\cdot)\) \(\chi_{4020}(863,\cdot)\) \(\chi_{4020}(947,\cdot)\) \(\chi_{4020}(1067,\cdot)\) \(\chi_{4020}(1163,\cdot)\) \(\chi_{4020}(1667,\cdot)\) \(\chi_{4020}(1823,\cdot)\) \(\chi_{4020}(1967,\cdot)\) \(\chi_{4020}(2303,\cdot)\) \(\chi_{4020}(2543,\cdot)\) \(\chi_{4020}(2627,\cdot)\) \(\chi_{4020}(2903,\cdot)\) \(\chi_{4020}(2963,\cdot)\) \(\chi_{4020}(3107,\cdot)\) \(\chi_{4020}(3323,\cdot)\) \(\chi_{4020}(3347,\cdot)\) \(\chi_{4020}(3707,\cdot)\) \(\chi_{4020}(3767,\cdot)\)
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
\((2011,2681,3217,1141)\) → \((-1,-1,-i,e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4020 }(143, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(-i\) | \(e\left(\frac{3}{22}\right)\) |
sage: chi.jacobi_sum(n)