from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1035, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,0,3]))
pari: [g,chi] = znchar(Mod(511,1035))
Basic properties
Modulus: | \(1035\) | |
Conductor: | \(207\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{207}(97,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1035.bq
\(\chi_{1035}(61,\cdot)\) \(\chi_{1035}(76,\cdot)\) \(\chi_{1035}(106,\cdot)\) \(\chi_{1035}(166,\cdot)\) \(\chi_{1035}(241,\cdot)\) \(\chi_{1035}(286,\cdot)\) \(\chi_{1035}(421,\cdot)\) \(\chi_{1035}(481,\cdot)\) \(\chi_{1035}(511,\cdot)\) \(\chi_{1035}(526,\cdot)\) \(\chi_{1035}(571,\cdot)\) \(\chi_{1035}(661,\cdot)\) \(\chi_{1035}(751,\cdot)\) \(\chi_{1035}(796,\cdot)\) \(\chi_{1035}(826,\cdot)\) \(\chi_{1035}(871,\cdot)\) \(\chi_{1035}(916,\cdot)\) \(\chi_{1035}(931,\cdot)\) \(\chi_{1035}(976,\cdot)\) \(\chi_{1035}(1006,\cdot)\)
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
\((461,622,856)\) → \((e\left(\frac{2}{3}\right),1,e\left(\frac{1}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 1035 }(511, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) |
sage: chi.jacobi_sum(n)