from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3751, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([9,4]))
pari: [g,chi] = znchar(Mod(360,3751))
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 \(\chi_{341}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3751.br
\(\chi_{3751}(360,\cdot)\) \(\chi_{3751}(524,\cdot)\) \(\chi_{3751}(596,\cdot)\) \(\chi_{3751}(723,\cdot)\) \(\chi_{3751}(820,\cdot)\) \(\chi_{3751}(1322,\cdot)\) \(\chi_{3751}(2272,\cdot)\) \(\chi_{3751}(3428,\cdot)\)
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: | 30.0.7511556299133379092165966152351244631135155455755771848525381828328211.2 |
Values on generators
\((2543,2421)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{2}{15}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 3751 }(360, a) \) | \(-1\) | \(1\) | \(-1\) | \(e\left(\frac{8}{15}\right)\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) |
sage: chi.jacobi_sum(n)