from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5733, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,22,7]))
pari: [g,chi] = znchar(Mod(307,5733))
Basic properties
Modulus: | \(5733\) | |
Conductor: | \(637\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{637}(307,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5733.hh
\(\chi_{5733}(307,\cdot)\) \(\chi_{5733}(811,\cdot)\) \(\chi_{5733}(1630,\cdot)\) \(\chi_{5733}(1945,\cdot)\) \(\chi_{5733}(2764,\cdot)\) \(\chi_{5733}(3268,\cdot)\) \(\chi_{5733}(3583,\cdot)\) \(\chi_{5733}(4087,\cdot)\) \(\chi_{5733}(4402,\cdot)\) \(\chi_{5733}(4906,\cdot)\) \(\chi_{5733}(5221,\cdot)\) \(\chi_{5733}(5725,\cdot)\)
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: | 28.28.444336918816745758721229800232012606769882120393458552593728061837.1 |
Values on generators
\((2549,1522,5293)\) → \((1,e\left(\frac{11}{14}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 5733 }(307, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(-i\) | \(e\left(\frac{11}{28}\right)\) |
sage: chi.jacobi_sum(n)