from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,33,10]))
pari: [g,chi] = znchar(Mod(73,5600))
Basic properties
Modulus: | \(5600\) | |
Conductor: | \(2800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2800}(2173,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5600.hw
\(\chi_{5600}(73,\cdot)\) \(\chi_{5600}(537,\cdot)\) \(\chi_{5600}(873,\cdot)\) \(\chi_{5600}(1977,\cdot)\) \(\chi_{5600}(2313,\cdot)\) \(\chi_{5600}(2777,\cdot)\) \(\chi_{5600}(3097,\cdot)\) \(\chi_{5600}(3113,\cdot)\) \(\chi_{5600}(3433,\cdot)\) \(\chi_{5600}(3897,\cdot)\) \(\chi_{5600}(4217,\cdot)\) \(\chi_{5600}(4233,\cdot)\) \(\chi_{5600}(4553,\cdot)\) \(\chi_{5600}(5017,\cdot)\) \(\chi_{5600}(5337,\cdot)\) \(\chi_{5600}(5353,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((351,4901,5377,801)\) → \((1,-i,e\left(\frac{11}{20}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 5600 }(73, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) |
sage: chi.jacobi_sum(n)