from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2736, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,30,32]))
pari: [g,chi] = znchar(Mod(347,2736))
Basic properties
Modulus: | \(2736\) | |
Conductor: | \(2736\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | 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 2736.gq
\(\chi_{2736}(131,\cdot)\) \(\chi_{2736}(347,\cdot)\) \(\chi_{2736}(443,\cdot)\) \(\chi_{2736}(491,\cdot)\) \(\chi_{2736}(731,\cdot)\) \(\chi_{2736}(1163,\cdot)\) \(\chi_{2736}(1499,\cdot)\) \(\chi_{2736}(1715,\cdot)\) \(\chi_{2736}(1811,\cdot)\) \(\chi_{2736}(1859,\cdot)\) \(\chi_{2736}(2099,\cdot)\) \(\chi_{2736}(2531,\cdot)\)
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
\((1711,2053,1217,1009)\) → \((-1,i,e\left(\frac{5}{6}\right),e\left(\frac{8}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 2736 }(347, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(i\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(-1\) | \(e\left(\frac{11}{36}\right)\) |
sage: chi.jacobi_sum(n)