Properties

Label 5550.47
Modulus $5550$
Conductor $2775$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5550, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,51,40]))
 
pari: [g,chi] = znchar(Mod(47,5550))
 

Basic properties

Modulus: \(5550\)
Conductor: \(2775\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2775}(47,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5550.ds

\(\chi_{5550}(47,\cdot)\) \(\chi_{5550}(137,\cdot)\) \(\chi_{5550}(713,\cdot)\) \(\chi_{5550}(803,\cdot)\) \(\chi_{5550}(1247,\cdot)\) \(\chi_{5550}(1823,\cdot)\) \(\chi_{5550}(1913,\cdot)\) \(\chi_{5550}(2267,\cdot)\) \(\chi_{5550}(2933,\cdot)\) \(\chi_{5550}(3023,\cdot)\) \(\chi_{5550}(3377,\cdot)\) \(\chi_{5550}(3467,\cdot)\) \(\chi_{5550}(4133,\cdot)\) \(\chi_{5550}(4487,\cdot)\) \(\chi_{5550}(4577,\cdot)\) \(\chi_{5550}(5153,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((3701,1777,2851)\) → \((-1,e\left(\frac{17}{20}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(41\)\(43\)
\( \chi_{ 5550 }(47, a) \) \(1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{30}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5550 }(47,a) \;\) at \(\;a = \) e.g. 2