Properties

Label 2601.8
Modulus $2601$
Conductor $867$
Order $136$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2601, base_ring=CyclotomicField(136))
 
M = H._module
 
chi = DirichletCharacter(H, M([68,13]))
 
pari: [g,chi] = znchar(Mod(8,2601))
 

Basic properties

Modulus: \(2601\)
Conductor: \(867\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(136\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{867}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2601.bf

\(\chi_{2601}(8,\cdot)\) \(\chi_{2601}(26,\cdot)\) \(\chi_{2601}(53,\cdot)\) \(\chi_{2601}(161,\cdot)\) \(\chi_{2601}(206,\cdot)\) \(\chi_{2601}(287,\cdot)\) \(\chi_{2601}(314,\cdot)\) \(\chi_{2601}(332,\cdot)\) \(\chi_{2601}(359,\cdot)\) \(\chi_{2601}(440,\cdot)\) \(\chi_{2601}(467,\cdot)\) \(\chi_{2601}(485,\cdot)\) \(\chi_{2601}(512,\cdot)\) \(\chi_{2601}(593,\cdot)\) \(\chi_{2601}(620,\cdot)\) \(\chi_{2601}(638,\cdot)\) \(\chi_{2601}(665,\cdot)\) \(\chi_{2601}(746,\cdot)\) \(\chi_{2601}(773,\cdot)\) \(\chi_{2601}(791,\cdot)\) \(\chi_{2601}(818,\cdot)\) \(\chi_{2601}(899,\cdot)\) \(\chi_{2601}(926,\cdot)\) \(\chi_{2601}(944,\cdot)\) \(\chi_{2601}(971,\cdot)\) \(\chi_{2601}(1052,\cdot)\) \(\chi_{2601}(1079,\cdot)\) \(\chi_{2601}(1097,\cdot)\) \(\chi_{2601}(1124,\cdot)\) \(\chi_{2601}(1205,\cdot)\) ...

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: $\Q(\zeta_{136})$
Fixed field: Number field defined by a degree 136 polynomial (not computed)

Values on generators

\((290,2026)\) → \((-1,e\left(\frac{13}{136}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 2601 }(8, a) \) \(-1\)\(1\)\(e\left(\frac{45}{68}\right)\)\(e\left(\frac{11}{34}\right)\)\(e\left(\frac{53}{136}\right)\)\(e\left(\frac{111}{136}\right)\)\(e\left(\frac{67}{68}\right)\)\(e\left(\frac{7}{136}\right)\)\(e\left(\frac{95}{136}\right)\)\(e\left(\frac{25}{34}\right)\)\(e\left(\frac{65}{136}\right)\)\(e\left(\frac{11}{17}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2601 }(8,a) \;\) at \(\;a = \) e.g. 2