from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8512, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,24,4]))
pari: [g,chi] = znchar(Mod(1201,8512))
Basic properties
| Modulus: | \(8512\) | |
| Conductor: | \(2128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2128}(1733,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8512.iq
\(\chi_{8512}(625,\cdot)\) \(\chi_{8512}(1201,\cdot)\) \(\chi_{8512}(1297,\cdot)\) \(\chi_{8512}(1745,\cdot)\) \(\chi_{8512}(3217,\cdot)\) \(\chi_{8512}(4113,\cdot)\) \(\chi_{8512}(4881,\cdot)\) \(\chi_{8512}(5457,\cdot)\) \(\chi_{8512}(5553,\cdot)\) \(\chi_{8512}(6001,\cdot)\) \(\chi_{8512}(7473,\cdot)\) \(\chi_{8512}(8369,\cdot)\)
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
\((5055,6917,7297,3137)\) → \((1,i,e\left(\frac{2}{3}\right),e\left(\frac{1}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 8512 }(1201, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(i\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)