from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5550, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,20]))
pari: [g,chi] = znchar(Mod(107,5550))
Basic properties
Modulus: | \(5550\) | |
Conductor: | \(555\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{555}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5550.df
\(\chi_{5550}(107,\cdot)\) \(\chi_{5550}(293,\cdot)\) \(\chi_{5550}(1043,\cdot)\) \(\chi_{5550}(1193,\cdot)\) \(\chi_{5550}(1307,\cdot)\) \(\chi_{5550}(1607,\cdot)\) \(\chi_{5550}(2957,\cdot)\) \(\chi_{5550}(2993,\cdot)\) \(\chi_{5550}(3707,\cdot)\) \(\chi_{5550}(3857,\cdot)\) \(\chi_{5550}(4193,\cdot)\) \(\chi_{5550}(4493,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((3701,1777,2851)\) → \((-1,i,e\left(\frac{5}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
\( \chi_{ 5550 }(107, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)