from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3751, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,26]))
pari: [g,chi] = znchar(Mod(483,3751))
Basic properties
Modulus: | \(3751\) | |
Conductor: | \(341\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{341}(142,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3751.cc
\(\chi_{3751}(483,\cdot)\) \(\chi_{3751}(846,\cdot)\) \(\chi_{3751}(1330,\cdot)\) \(\chi_{3751}(1693,\cdot)\) \(\chi_{3751}(2056,\cdot)\) \(\chi_{3751}(2177,\cdot)\) \(\chi_{3751}(2903,\cdot)\) \(\chi_{3751}(3145,\cdot)\)
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
\((2543,2421)\) → \((-1,e\left(\frac{13}{15}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 3751 }(483, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) |
sage: chi.jacobi_sum(n)