from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3696, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,0,20,12]))
pari: [g,chi] = znchar(Mod(37,3696))
Basic properties
Modulus: | \(3696\) | |
Conductor: | \(1232\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1232}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3696.hj
\(\chi_{3696}(37,\cdot)\) \(\chi_{3696}(445,\cdot)\) \(\chi_{3696}(709,\cdot)\) \(\chi_{3696}(949,\cdot)\) \(\chi_{3696}(1213,\cdot)\) \(\chi_{3696}(1285,\cdot)\) \(\chi_{3696}(1549,\cdot)\) \(\chi_{3696}(1621,\cdot)\) \(\chi_{3696}(1885,\cdot)\) \(\chi_{3696}(2293,\cdot)\) \(\chi_{3696}(2557,\cdot)\) \(\chi_{3696}(2797,\cdot)\) \(\chi_{3696}(3061,\cdot)\) \(\chi_{3696}(3133,\cdot)\) \(\chi_{3696}(3397,\cdot)\) \(\chi_{3696}(3469,\cdot)\)
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
\((463,2773,2465,1585,673)\) → \((1,i,1,e\left(\frac{1}{3}\right),e\left(\frac{1}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3696 }(37, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) |
sage: chi.jacobi_sum(n)