Properties

Label 2100.1993
Modulus $2100$
Conductor $35$
Order $12$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,9,10]))
 
pari: [g,chi] = znchar(Mod(1993,2100))
 

Basic properties

Modulus: \(2100\)
Conductor: \(35\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{35}(33,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2100.ce

\(\chi_{2100}(157,\cdot)\) \(\chi_{2100}(493,\cdot)\) \(\chi_{2100}(1657,\cdot)\) \(\chi_{2100}(1993,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: \(\Q(\zeta_{35})^+\)

Values on generators

\((1051,701,1177,1501)\) → \((1,1,-i,e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 2100 }(1993, a) \) \(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(-i\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{12}\right)\)\(-1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{12}\right)\)\(-1\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2100 }(1993,a) \;\) at \(\;a = \) e.g. 2