from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,0,11,12]))
pari: [g,chi] = znchar(Mod(73,7728))
Basic properties
| Modulus: | \(7728\) | |
| Conductor: | \(1288\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1288}(717,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7728.gg
\(\chi_{7728}(73,\cdot)\) \(\chi_{7728}(409,\cdot)\) \(\chi_{7728}(745,\cdot)\) \(\chi_{7728}(2329,\cdot)\) \(\chi_{7728}(3337,\cdot)\) \(\chi_{7728}(3433,\cdot)\) \(\chi_{7728}(3673,\cdot)\) \(\chi_{7728}(4441,\cdot)\) \(\chi_{7728}(4681,\cdot)\) \(\chi_{7728}(4777,\cdot)\) \(\chi_{7728}(5017,\cdot)\) \(\chi_{7728}(5689,\cdot)\) \(\chi_{7728}(5785,\cdot)\) \(\chi_{7728}(6121,\cdot)\) \(\chi_{7728}(6361,\cdot)\) \(\chi_{7728}(6697,\cdot)\) \(\chi_{7728}(6793,\cdot)\) \(\chi_{7728}(7033,\cdot)\) \(\chi_{7728}(7369,\cdot)\) \(\chi_{7728}(7465,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((4831,5797,5153,6625,6721)\) → \((1,-1,1,e\left(\frac{1}{6}\right),e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7728 }(73, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) |
sage: chi.jacobi_sum(n)