Properties

Label 2601.31
Modulus $2601$
Conductor $2601$
Order $816$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2601, base_ring=CyclotomicField(816))
 
M = H._module
 
chi = DirichletCharacter(H, M([272,27]))
 
pari: [g,chi] = znchar(Mod(31,2601))
 

Basic properties

Modulus: \(2601\)
Conductor: \(2601\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(816\)
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 2601.bm

\(\chi_{2601}(7,\cdot)\) \(\chi_{2601}(22,\cdot)\) \(\chi_{2601}(31,\cdot)\) \(\chi_{2601}(58,\cdot)\) \(\chi_{2601}(61,\cdot)\) \(\chi_{2601}(79,\cdot)\) \(\chi_{2601}(88,\cdot)\) \(\chi_{2601}(97,\cdot)\) \(\chi_{2601}(112,\cdot)\) \(\chi_{2601}(124,\cdot)\) \(\chi_{2601}(130,\cdot)\) \(\chi_{2601}(133,\cdot)\) \(\chi_{2601}(139,\cdot)\) \(\chi_{2601}(142,\cdot)\) \(\chi_{2601}(148,\cdot)\) \(\chi_{2601}(160,\cdot)\) \(\chi_{2601}(175,\cdot)\) \(\chi_{2601}(184,\cdot)\) \(\chi_{2601}(193,\cdot)\) \(\chi_{2601}(211,\cdot)\) \(\chi_{2601}(232,\cdot)\) \(\chi_{2601}(241,\cdot)\) \(\chi_{2601}(250,\cdot)\) \(\chi_{2601}(265,\cdot)\) \(\chi_{2601}(277,\cdot)\) \(\chi_{2601}(283,\cdot)\) \(\chi_{2601}(286,\cdot)\) \(\chi_{2601}(292,\cdot)\) \(\chi_{2601}(295,\cdot)\) \(\chi_{2601}(301,\cdot)\) ...

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: $\Q(\zeta_{816})$
Fixed field: Number field defined by a degree 816 polynomial (not computed)

Values on generators

\((290,2026)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{9}{272}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 2601 }(31, a) \) \(-1\)\(1\)\(e\left(\frac{253}{408}\right)\)\(e\left(\frac{49}{204}\right)\)\(e\left(\frac{199}{816}\right)\)\(e\left(\frac{377}{816}\right)\)\(e\left(\frac{117}{136}\right)\)\(e\left(\frac{235}{272}\right)\)\(e\left(\frac{77}{816}\right)\)\(e\left(\frac{31}{204}\right)\)\(e\left(\frac{67}{816}\right)\)\(e\left(\frac{49}{102}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2601 }(31,a) \;\) at \(\;a = \) e.g. 2