Properties

Label 2624.65
Modulus $2624$
Conductor $41$
Order $40$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2624, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,13]))
 
pari: [g,chi] = znchar(Mod(65,2624))
 

Basic properties

Modulus: \(2624\)
Conductor: \(41\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{41}(24,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2624.dq

\(\chi_{2624}(65,\cdot)\) \(\chi_{2624}(129,\cdot)\) \(\chi_{2624}(193,\cdot)\) \(\chi_{2624}(257,\cdot)\) \(\chi_{2624}(321,\cdot)\) \(\chi_{2624}(641,\cdot)\) \(\chi_{2624}(833,\cdot)\) \(\chi_{2624}(1217,\cdot)\) \(\chi_{2624}(1409,\cdot)\) \(\chi_{2624}(1729,\cdot)\) \(\chi_{2624}(1793,\cdot)\) \(\chi_{2624}(1857,\cdot)\) \(\chi_{2624}(1921,\cdot)\) \(\chi_{2624}(1985,\cdot)\) \(\chi_{2624}(2113,\cdot)\) \(\chi_{2624}(2561,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: Number field defined by a degree 40 polynomial

Values on generators

\((575,1477,129)\) → \((1,1,e\left(\frac{13}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 2624 }(65, a) \) \(-1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{27}{40}\right)\)\(-i\)\(e\left(\frac{39}{40}\right)\)\(e\left(\frac{3}{40}\right)\)\(e\left(\frac{1}{40}\right)\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{37}{40}\right)\)\(e\left(\frac{11}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2624 }(65,a) \;\) at \(\;a = \) e.g. 2