from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8027, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([2,11]))
pari: [g,chi] = znchar(Mod(6146,8027))
Basic properties
Modulus: | \(8027\) | |
Conductor: | \(8027\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8027.s
\(\chi_{8027}(136,\cdot)\) \(\chi_{8027}(562,\cdot)\) \(\chi_{8027}(911,\cdot)\) \(\chi_{8027}(1183,\cdot)\) \(\chi_{8027}(1532,\cdot)\) \(\chi_{8027}(2307,\cdot)\) \(\chi_{8027}(2656,\cdot)\) \(\chi_{8027}(2928,\cdot)\) \(\chi_{8027}(3005,\cdot)\) \(\chi_{8027}(3277,\cdot)\) \(\chi_{8027}(3354,\cdot)\) \(\chi_{8027}(3626,\cdot)\) \(\chi_{8027}(3975,\cdot)\) \(\chi_{8027}(5448,\cdot)\) \(\chi_{8027}(6069,\cdot)\) \(\chi_{8027}(6146,\cdot)\) \(\chi_{8027}(6767,\cdot)\) \(\chi_{8027}(7193,\cdot)\) \(\chi_{8027}(7542,\cdot)\) \(\chi_{8027}(7814,\cdot)\)
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
\((350,5935)\) → \((e\left(\frac{1}{22}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 8027 }(6146, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) |
sage: chi.jacobi_sum(n)