from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7744, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,18]))
pari: [g,chi] = znchar(Mod(175,7744))
Basic properties
| Modulus: | \(7744\) | |
| Conductor: | \(1936\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1936}(659,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7744.bv
\(\chi_{7744}(175,\cdot)\) \(\chi_{7744}(527,\cdot)\) \(\chi_{7744}(879,\cdot)\) \(\chi_{7744}(1231,\cdot)\) \(\chi_{7744}(1583,\cdot)\) \(\chi_{7744}(2287,\cdot)\) \(\chi_{7744}(2639,\cdot)\) \(\chi_{7744}(2991,\cdot)\) \(\chi_{7744}(3343,\cdot)\) \(\chi_{7744}(3695,\cdot)\) \(\chi_{7744}(4047,\cdot)\) \(\chi_{7744}(4399,\cdot)\) \(\chi_{7744}(4751,\cdot)\) \(\chi_{7744}(5103,\cdot)\) \(\chi_{7744}(5455,\cdot)\) \(\chi_{7744}(6159,\cdot)\) \(\chi_{7744}(6511,\cdot)\) \(\chi_{7744}(6863,\cdot)\) \(\chi_{7744}(7215,\cdot)\) \(\chi_{7744}(7567,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | 44.44.65891540065531696512311406077777585030540198005846477706480206115552831500546427599154745891192168020941904820029716692992.1 |
Values on generators
\((5567,4357,6657)\) → \((-1,-i,e\left(\frac{9}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 7744 }(175, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(-1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) |
sage: chi.jacobi_sum(n)