Properties

Label 5241.26
Modulus $5241$
Conductor $5241$
Order $1746$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5241, base_ring=CyclotomicField(1746))
 
M = H._module
 
chi = DirichletCharacter(H, M([873,1256]))
 
pari: [g,chi] = znchar(Mod(26,5241))
 

Basic properties

Modulus: \(5241\)
Conductor: \(5241\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1746\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5241.w

\(\chi_{5241}(14,\cdot)\) \(\chi_{5241}(17,\cdot)\) \(\chi_{5241}(23,\cdot)\) \(\chi_{5241}(26,\cdot)\) \(\chi_{5241}(29,\cdot)\) \(\chi_{5241}(56,\cdot)\) \(\chi_{5241}(68,\cdot)\) \(\chi_{5241}(74,\cdot)\) \(\chi_{5241}(77,\cdot)\) \(\chi_{5241}(104,\cdot)\) \(\chi_{5241}(113,\cdot)\) \(\chi_{5241}(137,\cdot)\) \(\chi_{5241}(140,\cdot)\) \(\chi_{5241}(143,\cdot)\) \(\chi_{5241}(146,\cdot)\) \(\chi_{5241}(158,\cdot)\) \(\chi_{5241}(164,\cdot)\) \(\chi_{5241}(170,\cdot)\) \(\chi_{5241}(179,\cdot)\) \(\chi_{5241}(191,\cdot)\) \(\chi_{5241}(197,\cdot)\) \(\chi_{5241}(230,\cdot)\) \(\chi_{5241}(233,\cdot)\) \(\chi_{5241}(239,\cdot)\) \(\chi_{5241}(257,\cdot)\) \(\chi_{5241}(260,\cdot)\) \(\chi_{5241}(266,\cdot)\) \(\chi_{5241}(269,\cdot)\) \(\chi_{5241}(290,\cdot)\) \(\chi_{5241}(296,\cdot)\) ...

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: $\Q(\zeta_{873})$
Fixed field: Number field defined by a degree 1746 polynomial (not computed)

Values on generators

\((1748,3496)\) → \((-1,e\left(\frac{628}{873}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 5241 }(26, a) \) \(-1\)\(1\)\(e\left(\frac{383}{1746}\right)\)\(e\left(\frac{383}{873}\right)\)\(e\left(\frac{991}{1746}\right)\)\(e\left(\frac{736}{873}\right)\)\(e\left(\frac{383}{582}\right)\)\(e\left(\frac{229}{291}\right)\)\(e\left(\frac{367}{582}\right)\)\(e\left(\frac{694}{873}\right)\)\(e\left(\frac{109}{1746}\right)\)\(e\left(\frac{766}{873}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5241 }(26,a) \;\) at \(\;a = \) e.g. 2