from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3520, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,15,20,24]))
pari: [g,chi] = znchar(Mod(9,3520))
Basic properties
| Modulus: | \(3520\) | |
| Conductor: | \(1760\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1760}(669,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3520.em
\(\chi_{3520}(9,\cdot)\) \(\chi_{3520}(169,\cdot)\) \(\chi_{3520}(489,\cdot)\) \(\chi_{3520}(729,\cdot)\) \(\chi_{3520}(889,\cdot)\) \(\chi_{3520}(1049,\cdot)\) \(\chi_{3520}(1369,\cdot)\) \(\chi_{3520}(1609,\cdot)\) \(\chi_{3520}(1769,\cdot)\) \(\chi_{3520}(1929,\cdot)\) \(\chi_{3520}(2249,\cdot)\) \(\chi_{3520}(2489,\cdot)\) \(\chi_{3520}(2649,\cdot)\) \(\chi_{3520}(2809,\cdot)\) \(\chi_{3520}(3129,\cdot)\) \(\chi_{3520}(3369,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((2751,1541,2817,321)\) → \((1,e\left(\frac{3}{8}\right),-1,e\left(\frac{3}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 3520 }(9, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) |
sage: chi.jacobi_sum(n)