from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3528, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,0,41]))
pari: [g,chi] = znchar(Mod(1405,3528))
Basic properties
| Modulus: | \(3528\) | |
| Conductor: | \(392\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{392}(229,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3528.fb
\(\chi_{3528}(397,\cdot)\) \(\chi_{3528}(829,\cdot)\) \(\chi_{3528}(1333,\cdot)\) \(\chi_{3528}(1405,\cdot)\) \(\chi_{3528}(1837,\cdot)\) \(\chi_{3528}(1909,\cdot)\) \(\chi_{3528}(2341,\cdot)\) \(\chi_{3528}(2413,\cdot)\) \(\chi_{3528}(2845,\cdot)\) \(\chi_{3528}(2917,\cdot)\) \(\chi_{3528}(3349,\cdot)\) \(\chi_{3528}(3421,\cdot)\)
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: | 42.0.1090030896264192289800449659845679818091197961133776603876122561317234873686091104256.1 |
Values on generators
\((2647,1765,785,1081)\) → \((1,-1,1,e\left(\frac{41}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3528 }(1405, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{31}{42}\right)\) |
sage: chi.jacobi_sum(n)