Properties

Label 2541.821
Modulus $2541$
Conductor $2541$
Order $330$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2541, base_ring=CyclotomicField(330))
 
M = H._module
 
chi = DirichletCharacter(H, M([165,110,141]))
 
pari: [g,chi] = znchar(Mod(821,2541))
 

Basic properties

Modulus: \(2541\)
Conductor: \(2541\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2541.cg

\(\chi_{2541}(2,\cdot)\) \(\chi_{2541}(74,\cdot)\) \(\chi_{2541}(95,\cdot)\) \(\chi_{2541}(107,\cdot)\) \(\chi_{2541}(116,\cdot)\) \(\chi_{2541}(128,\cdot)\) \(\chi_{2541}(149,\cdot)\) \(\chi_{2541}(200,\cdot)\) \(\chi_{2541}(305,\cdot)\) \(\chi_{2541}(326,\cdot)\) \(\chi_{2541}(338,\cdot)\) \(\chi_{2541}(347,\cdot)\) \(\chi_{2541}(359,\cdot)\) \(\chi_{2541}(380,\cdot)\) \(\chi_{2541}(431,\cdot)\) \(\chi_{2541}(464,\cdot)\) \(\chi_{2541}(536,\cdot)\) \(\chi_{2541}(557,\cdot)\) \(\chi_{2541}(569,\cdot)\) \(\chi_{2541}(590,\cdot)\) \(\chi_{2541}(611,\cdot)\) \(\chi_{2541}(662,\cdot)\) \(\chi_{2541}(695,\cdot)\) \(\chi_{2541}(767,\cdot)\) \(\chi_{2541}(788,\cdot)\) \(\chi_{2541}(800,\cdot)\) \(\chi_{2541}(809,\cdot)\) \(\chi_{2541}(821,\cdot)\) \(\chi_{2541}(842,\cdot)\) \(\chi_{2541}(893,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((848,1816,2059)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{47}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 2541 }(821, a) \) \(1\)\(1\)\(e\left(\frac{98}{165}\right)\)\(e\left(\frac{31}{165}\right)\)\(e\left(\frac{259}{330}\right)\)\(e\left(\frac{43}{55}\right)\)\(e\left(\frac{25}{66}\right)\)\(e\left(\frac{17}{110}\right)\)\(e\left(\frac{62}{165}\right)\)\(e\left(\frac{127}{165}\right)\)\(e\left(\frac{43}{330}\right)\)\(e\left(\frac{107}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2541 }(821,a) \;\) at \(\;a = \) e.g. 2