from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3528, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,7,39]))
pari: [g,chi] = znchar(Mod(461,3528))
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.ev
\(\chi_{3528}(461,\cdot)\) \(\chi_{3528}(797,\cdot)\) \(\chi_{3528}(965,\cdot)\) \(\chi_{3528}(1301,\cdot)\) \(\chi_{3528}(1805,\cdot)\) \(\chi_{3528}(1973,\cdot)\) \(\chi_{3528}(2309,\cdot)\) \(\chi_{3528}(2477,\cdot)\) \(\chi_{3528}(2813,\cdot)\) \(\chi_{3528}(2981,\cdot)\) \(\chi_{3528}(3317,\cdot)\) \(\chi_{3528}(3485,\cdot)\)
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: | 42.42.25461370422593899025010993639968319493979540256968222705473361226986223705326760477191735540542813852487110361088.1 |
Values on generators
\((2647,1765,785,1081)\) → \((1,-1,e\left(\frac{1}{6}\right),e\left(\frac{13}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3528 }(461, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{14}\right)\) |
sage: chi.jacobi_sum(n)