from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3528, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,14,12]))
pari: [g,chi] = znchar(Mod(85,3528))
Basic properties
Modulus: | \(3528\) | |
Conductor: | \(3528\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | 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 3528.ef
\(\chi_{3528}(85,\cdot)\) \(\chi_{3528}(421,\cdot)\) \(\chi_{3528}(925,\cdot)\) \(\chi_{3528}(1093,\cdot)\) \(\chi_{3528}(1429,\cdot)\) \(\chi_{3528}(1597,\cdot)\) \(\chi_{3528}(1933,\cdot)\) \(\chi_{3528}(2101,\cdot)\) \(\chi_{3528}(2437,\cdot)\) \(\chi_{3528}(2605,\cdot)\) \(\chi_{3528}(3109,\cdot)\) \(\chi_{3528}(3445,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((2647,1765,785,1081)\) → \((1,-1,e\left(\frac{1}{3}\right),e\left(\frac{2}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 3528 }(85, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(-1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{9}{14}\right)\) |
sage: chi.jacobi_sum(n)