from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3136, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,7,20]))
pari: [g,chi] = znchar(Mod(113,3136))
Basic properties
| Modulus: | \(3136\) | |
| Conductor: | \(784\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{784}(309,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3136.bt
\(\chi_{3136}(113,\cdot)\) \(\chi_{3136}(337,\cdot)\) \(\chi_{3136}(561,\cdot)\) \(\chi_{3136}(1009,\cdot)\) \(\chi_{3136}(1233,\cdot)\) \(\chi_{3136}(1457,\cdot)\) \(\chi_{3136}(1681,\cdot)\) \(\chi_{3136}(1905,\cdot)\) \(\chi_{3136}(2129,\cdot)\) \(\chi_{3136}(2577,\cdot)\) \(\chi_{3136}(2801,\cdot)\) \(\chi_{3136}(3025,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((1471,197,1473)\) → \((1,i,e\left(\frac{5}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 3136 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(-i\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) |
sage: chi.jacobi_sum(n)