from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4851, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,20,27]))
pari: [g,chi] = znchar(Mod(116,4851))
Basic properties
Modulus: | \(4851\) | |
Conductor: | \(231\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{231}(116,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4851.df
\(\chi_{4851}(116,\cdot)\) \(\chi_{4851}(557,\cdot)\) \(\chi_{4851}(998,\cdot)\) \(\chi_{4851}(1304,\cdot)\) \(\chi_{4851}(1745,\cdot)\) \(\chi_{4851}(2186,\cdot)\) \(\chi_{4851}(3203,\cdot)\) \(\chi_{4851}(4391,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((4313,199,442)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{9}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 4851 }(116, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) |
sage: chi.jacobi_sum(n)