Properties

Label 5550.107
Modulus $5550$
Conductor $555$
Order $36$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
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)
 
\( \chi_{ 5550 }(107,a) \;\) at \(\;a = \) e.g. 2