from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5635, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([7,6,0]))
pari: [g,chi] = znchar(Mod(622,5635))
Basic properties
Modulus: | \(5635\) | |
Conductor: | \(245\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{245}(132,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5635.br
\(\chi_{5635}(622,\cdot)\) \(\chi_{5635}(1427,\cdot)\) \(\chi_{5635}(1588,\cdot)\) \(\chi_{5635}(2232,\cdot)\) \(\chi_{5635}(2393,\cdot)\) \(\chi_{5635}(3198,\cdot)\) \(\chi_{5635}(3842,\cdot)\) \(\chi_{5635}(4003,\cdot)\) \(\chi_{5635}(4647,\cdot)\) \(\chi_{5635}(4808,\cdot)\) \(\chi_{5635}(5452,\cdot)\) \(\chi_{5635}(5613,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((3382,346,2696)\) → \((i,e\left(\frac{3}{14}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
\( \chi_{ 5635 }(622, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) |
sage: chi.jacobi_sum(n)