from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3528, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([0,7,7,1]))
pari: [g,chi] = znchar(Mod(125,3528))
Basic properties
Modulus: | \(3528\) | |
Conductor: | \(1176\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(14\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1176}(125,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3528.dk
\(\chi_{3528}(125,\cdot)\) \(\chi_{3528}(629,\cdot)\) \(\chi_{3528}(1133,\cdot)\) \(\chi_{3528}(1637,\cdot)\) \(\chi_{3528}(2141,\cdot)\) \(\chi_{3528}(3149,\cdot)\)
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: | \(\Q(\zeta_{7})\) |
Fixed field: | 14.14.6150772901711899500090300039168.1 |
Values on generators
\((2647,1765,785,1081)\) → \((1,-1,-1,e\left(\frac{1}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 3528 }(125, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(-1\) | \(e\left(\frac{11}{14}\right)\) |
sage: chi.jacobi_sum(n)