from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3920, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,7,7,6]))
pari: [g,chi] = znchar(Mod(3317,3920))
Basic properties
Modulus: | \(3920\) | |
Conductor: | \(3920\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3920.dt
\(\chi_{3920}(13,\cdot)\) \(\chi_{3920}(517,\cdot)\) \(\chi_{3920}(573,\cdot)\) \(\chi_{3920}(1133,\cdot)\) \(\chi_{3920}(1637,\cdot)\) \(\chi_{3920}(1693,\cdot)\) \(\chi_{3920}(2197,\cdot)\) \(\chi_{3920}(2757,\cdot)\) \(\chi_{3920}(2813,\cdot)\) \(\chi_{3920}(3317,\cdot)\) \(\chi_{3920}(3373,\cdot)\) \(\chi_{3920}(3877,\cdot)\)
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: | 28.28.129593063934491976647600951409586130589894568495749019992064000000000000000000000.1 |
Values on generators
\((1471,981,3137,3041)\) → \((1,i,i,e\left(\frac{3}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 3920 }(3317, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(-i\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)