from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1035, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([11,33,57]))
pari: [g,chi] = znchar(Mod(1019,1035))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1035.br
\(\chi_{1035}(14,\cdot)\) \(\chi_{1035}(74,\cdot)\) \(\chi_{1035}(149,\cdot)\) \(\chi_{1035}(194,\cdot)\) \(\chi_{1035}(329,\cdot)\) \(\chi_{1035}(389,\cdot)\) \(\chi_{1035}(419,\cdot)\) \(\chi_{1035}(434,\cdot)\) \(\chi_{1035}(479,\cdot)\) \(\chi_{1035}(569,\cdot)\) \(\chi_{1035}(659,\cdot)\) \(\chi_{1035}(704,\cdot)\) \(\chi_{1035}(734,\cdot)\) \(\chi_{1035}(779,\cdot)\) \(\chi_{1035}(824,\cdot)\) \(\chi_{1035}(839,\cdot)\) \(\chi_{1035}(884,\cdot)\) \(\chi_{1035}(914,\cdot)\) \(\chi_{1035}(1004,\cdot)\) \(\chi_{1035}(1019,\cdot)\)
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
\((461,622,856)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{19}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1035 }(1019, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) |
sage: chi.jacobi_sum(n)