from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,0,11,42]))
pari: [g,chi] = znchar(Mod(3727,4020))
Basic properties
Modulus: | \(4020\) | |
Conductor: | \(1340\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1340}(1047,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4020.cn
\(\chi_{4020}(43,\cdot)\) \(\chi_{4020}(187,\cdot)\) \(\chi_{4020}(343,\cdot)\) \(\chi_{4020}(847,\cdot)\) \(\chi_{4020}(943,\cdot)\) \(\chi_{4020}(1063,\cdot)\) \(\chi_{4020}(1147,\cdot)\) \(\chi_{4020}(1747,\cdot)\) \(\chi_{4020}(1867,\cdot)\) \(\chi_{4020}(1903,\cdot)\) \(\chi_{4020}(2263,\cdot)\) \(\chi_{4020}(2323,\cdot)\) \(\chi_{4020}(2683,\cdot)\) \(\chi_{4020}(2707,\cdot)\) \(\chi_{4020}(2923,\cdot)\) \(\chi_{4020}(3067,\cdot)\) \(\chi_{4020}(3127,\cdot)\) \(\chi_{4020}(3403,\cdot)\) \(\chi_{4020}(3487,\cdot)\) \(\chi_{4020}(3727,\cdot)\)
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
\((2011,2681,3217,1141)\) → \((-1,1,i,e\left(\frac{21}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 4020 }(3727, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(-1\) | \(e\left(\frac{4}{11}\right)\) | \(i\) | \(e\left(\frac{13}{22}\right)\) |
sage: chi.jacobi_sum(n)