from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5472, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,12,22]))
pari: [g,chi] = znchar(Mod(4423,5472))
Basic properties
Modulus: | \(5472\) | |
Conductor: | \(2736\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2736}(1003,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5472.iv
\(\chi_{5472}(439,\cdot)\) \(\chi_{5472}(1447,\cdot)\) \(\chi_{5472}(1591,\cdot)\) \(\chi_{5472}(1687,\cdot)\) \(\chi_{5472}(2119,\cdot)\) \(\chi_{5472}(2407,\cdot)\) \(\chi_{5472}(3175,\cdot)\) \(\chi_{5472}(4183,\cdot)\) \(\chi_{5472}(4327,\cdot)\) \(\chi_{5472}(4423,\cdot)\) \(\chi_{5472}(4855,\cdot)\) \(\chi_{5472}(5143,\cdot)\)
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
\((4447,2053,1217,3745)\) → \((-1,i,e\left(\frac{1}{3}\right),e\left(\frac{11}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 5472 }(4423, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{25}{36}\right)\) |
sage: chi.jacobi_sum(n)