from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7744, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,4]))
pari: [g,chi] = znchar(Mod(177,7744))
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 \(\chi_{1936}(661,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7744.bu
\(\chi_{7744}(177,\cdot)\) \(\chi_{7744}(529,\cdot)\) \(\chi_{7744}(881,\cdot)\) \(\chi_{7744}(1233,\cdot)\) \(\chi_{7744}(1585,\cdot)\) \(\chi_{7744}(2289,\cdot)\) \(\chi_{7744}(2641,\cdot)\) \(\chi_{7744}(2993,\cdot)\) \(\chi_{7744}(3345,\cdot)\) \(\chi_{7744}(3697,\cdot)\) \(\chi_{7744}(4049,\cdot)\) \(\chi_{7744}(4401,\cdot)\) \(\chi_{7744}(4753,\cdot)\) \(\chi_{7744}(5105,\cdot)\) \(\chi_{7744}(5457,\cdot)\) \(\chi_{7744}(6161,\cdot)\) \(\chi_{7744}(6513,\cdot)\) \(\chi_{7744}(6865,\cdot)\) \(\chi_{7744}(7217,\cdot)\) \(\chi_{7744}(7569,\cdot)\)
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: | Number field defined by a degree 44 polynomial |
Values on generators
\((5567,4357,6657)\) → \((1,i,e\left(\frac{1}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 7744 }(177, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(-1\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) |
sage: chi.jacobi_sum(n)