from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1035, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([11,0,9]))
pari: [g,chi] = znchar(Mod(56,1035))
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 \(\chi_{207}(56,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1035.bm
\(\chi_{1035}(11,\cdot)\) \(\chi_{1035}(56,\cdot)\) \(\chi_{1035}(86,\cdot)\) \(\chi_{1035}(176,\cdot)\) \(\chi_{1035}(191,\cdot)\) \(\chi_{1035}(221,\cdot)\) \(\chi_{1035}(281,\cdot)\) \(\chi_{1035}(356,\cdot)\) \(\chi_{1035}(401,\cdot)\) \(\chi_{1035}(536,\cdot)\) \(\chi_{1035}(596,\cdot)\) \(\chi_{1035}(626,\cdot)\) \(\chi_{1035}(641,\cdot)\) \(\chi_{1035}(686,\cdot)\) \(\chi_{1035}(776,\cdot)\) \(\chi_{1035}(866,\cdot)\) \(\chi_{1035}(911,\cdot)\) \(\chi_{1035}(941,\cdot)\) \(\chi_{1035}(986,\cdot)\) \(\chi_{1035}(1031,\cdot)\)
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
\((461,622,856)\) → \((e\left(\frac{1}{6}\right),1,e\left(\frac{3}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1035 }(56, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) |
sage: chi.jacobi_sum(n)