from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1035, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,42]))
pari: [g,chi] = znchar(Mod(37,1035))
Basic properties
Modulus: | \(1035\) | |
Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{115}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1035.bj
\(\chi_{1035}(28,\cdot)\) \(\chi_{1035}(37,\cdot)\) \(\chi_{1035}(172,\cdot)\) \(\chi_{1035}(217,\cdot)\) \(\chi_{1035}(343,\cdot)\) \(\chi_{1035}(352,\cdot)\) \(\chi_{1035}(388,\cdot)\) \(\chi_{1035}(433,\cdot)\) \(\chi_{1035}(442,\cdot)\) \(\chi_{1035}(523,\cdot)\) \(\chi_{1035}(613,\cdot)\) \(\chi_{1035}(658,\cdot)\) \(\chi_{1035}(757,\cdot)\) \(\chi_{1035}(793,\cdot)\) \(\chi_{1035}(802,\cdot)\) \(\chi_{1035}(838,\cdot)\) \(\chi_{1035}(847,\cdot)\) \(\chi_{1035}(937,\cdot)\) \(\chi_{1035}(973,\cdot)\) \(\chi_{1035}(1027,\cdot)\)
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: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((461,622,856)\) → \((1,i,e\left(\frac{21}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 1035 }(37, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) |
sage: chi.jacobi_sum(n)