Properties

Label 4100.107
Modulus $4100$
Conductor $820$
Order $20$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4100, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,5,2]))
 
pari: [g,chi] = znchar(Mod(107,4100))
 

Basic properties

Modulus: \(4100\)
Conductor: \(820\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{820}(107,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4100.dk

\(\chi_{4100}(107,\cdot)\) \(\chi_{4100}(843,\cdot)\) \(\chi_{4100}(1007,\cdot)\) \(\chi_{4100}(1343,\cdot)\) \(\chi_{4100}(1507,\cdot)\) \(\chi_{4100}(3243,\cdot)\) \(\chi_{4100}(3407,\cdot)\) \(\chi_{4100}(4043,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.20.3429725790961017099403304624672000000000000000.1

Values on generators

\((2051,1477,3901)\) → \((-1,i,e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4100 }(107, a) \) \(1\)\(1\)\(-i\)\(e\left(\frac{13}{20}\right)\)\(-1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{17}{20}\right)\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4100 }(107,a) \;\) at \(\;a = \) e.g. 2