from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1035, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,11,14]))
pari: [g,chi] = znchar(Mod(17,1035))
Basic properties
| Modulus: | \(1035\) | |
| Conductor: | \(345\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{345}(17,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1035.bh
\(\chi_{1035}(17,\cdot)\) \(\chi_{1035}(53,\cdot)\) \(\chi_{1035}(107,\cdot)\) \(\chi_{1035}(143,\cdot)\) \(\chi_{1035}(152,\cdot)\) \(\chi_{1035}(287,\cdot)\) \(\chi_{1035}(332,\cdot)\) \(\chi_{1035}(458,\cdot)\) \(\chi_{1035}(467,\cdot)\) \(\chi_{1035}(503,\cdot)\) \(\chi_{1035}(548,\cdot)\) \(\chi_{1035}(557,\cdot)\) \(\chi_{1035}(638,\cdot)\) \(\chi_{1035}(728,\cdot)\) \(\chi_{1035}(773,\cdot)\) \(\chi_{1035}(872,\cdot)\) \(\chi_{1035}(908,\cdot)\) \(\chi_{1035}(917,\cdot)\) \(\chi_{1035}(953,\cdot)\) \(\chi_{1035}(962,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((461,622,856)\) → \((-1,i,e\left(\frac{7}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1035 }(17, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) |
sage: chi.jacobi_sum(n)