Properties

Label 3040.297
Modulus $3040$
Conductor $1520$
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(3040, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,3,10]))
 
pari: [g,chi] = znchar(Mod(297,3040))
 

Basic properties

Modulus: \(3040\)
Conductor: \(1520\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1520}(1437,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3040.dr

\(\chi_{3040}(297,\cdot)\) \(\chi_{3040}(1433,\cdot)\) \(\chi_{3040}(1737,\cdot)\) \(\chi_{3040}(3033,\cdot)\)

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: 12.12.102862222917439062016000000000.2

Values on generators

\((191,2661,1217,1921)\) → \((1,-i,i,e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 3040 }(297, a) \) \(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(-i\)\(e\left(\frac{2}{3}\right)\)\(-i\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(-1\)\(e\left(\frac{11}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3040 }(297,a) \;\) at \(\;a = \) e.g. 2