from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1035, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,33,42]))
pari: [g,chi] = znchar(Mod(59,1035))
Basic properties
| Modulus: | \(1035\) | |
| Conductor: | \(1035\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1035.bp
\(\chi_{1035}(29,\cdot)\) \(\chi_{1035}(59,\cdot)\) \(\chi_{1035}(104,\cdot)\) \(\chi_{1035}(119,\cdot)\) \(\chi_{1035}(164,\cdot)\) \(\chi_{1035}(209,\cdot)\) \(\chi_{1035}(239,\cdot)\) \(\chi_{1035}(284,\cdot)\) \(\chi_{1035}(374,\cdot)\) \(\chi_{1035}(464,\cdot)\) \(\chi_{1035}(509,\cdot)\) \(\chi_{1035}(524,\cdot)\) \(\chi_{1035}(554,\cdot)\) \(\chi_{1035}(614,\cdot)\) \(\chi_{1035}(749,\cdot)\) \(\chi_{1035}(794,\cdot)\) \(\chi_{1035}(869,\cdot)\) \(\chi_{1035}(929,\cdot)\) \(\chi_{1035}(959,\cdot)\) \(\chi_{1035}(974,\cdot)\)
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
\((461,622,856)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1035 }(59, a) \) | \(-1\) | \(1\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)