from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3648, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,18,20]))
pari: [g,chi] = znchar(Mod(17,3648))
Basic properties
Modulus: | \(3648\) | |
Conductor: | \(912\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{912}(701,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3648.dp
\(\chi_{3648}(17,\cdot)\) \(\chi_{3648}(593,\cdot)\) \(\chi_{3648}(689,\cdot)\) \(\chi_{3648}(785,\cdot)\) \(\chi_{3648}(1073,\cdot)\) \(\chi_{3648}(1745,\cdot)\) \(\chi_{3648}(1841,\cdot)\) \(\chi_{3648}(2417,\cdot)\) \(\chi_{3648}(2513,\cdot)\) \(\chi_{3648}(2609,\cdot)\) \(\chi_{3648}(2897,\cdot)\) \(\chi_{3648}(3569,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((2623,2053,1217,1921)\) → \((1,-i,-1,e\left(\frac{5}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 3648 }(17, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{35}{36}\right)\) |
sage: chi.jacobi_sum(n)