Properties

Label 2541.376
Modulus $2541$
Conductor $847$
Order $330$
Real no
Primitive no
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([0,275,303]))
 
pari: [g,chi] = znchar(Mod(376,2541))
 

Basic properties

Modulus: \(2541\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{847}(376,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2541.ck

\(\chi_{2541}(19,\cdot)\) \(\chi_{2541}(52,\cdot)\) \(\chi_{2541}(61,\cdot)\) \(\chi_{2541}(73,\cdot)\) \(\chi_{2541}(145,\cdot)\) \(\chi_{2541}(178,\cdot)\) \(\chi_{2541}(250,\cdot)\) \(\chi_{2541}(271,\cdot)\) \(\chi_{2541}(283,\cdot)\) \(\chi_{2541}(292,\cdot)\) \(\chi_{2541}(304,\cdot)\) \(\chi_{2541}(325,\cdot)\) \(\chi_{2541}(376,\cdot)\) \(\chi_{2541}(409,\cdot)\) \(\chi_{2541}(502,\cdot)\) \(\chi_{2541}(514,\cdot)\) \(\chi_{2541}(523,\cdot)\) \(\chi_{2541}(535,\cdot)\) \(\chi_{2541}(556,\cdot)\) \(\chi_{2541}(607,\cdot)\) \(\chi_{2541}(640,\cdot)\) \(\chi_{2541}(712,\cdot)\) \(\chi_{2541}(733,\cdot)\) \(\chi_{2541}(745,\cdot)\) \(\chi_{2541}(754,\cdot)\) \(\chi_{2541}(787,\cdot)\) \(\chi_{2541}(871,\cdot)\) \(\chi_{2541}(943,\cdot)\) \(\chi_{2541}(964,\cdot)\) \(\chi_{2541}(976,\cdot)\) ...

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

\((848,1816,2059)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{101}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 2541 }(376, a) \) \(1\)\(1\)\(e\left(\frac{193}{330}\right)\)\(e\left(\frac{28}{165}\right)\)\(e\left(\frac{37}{330}\right)\)\(e\left(\frac{83}{110}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{13}{55}\right)\)\(e\left(\frac{56}{165}\right)\)\(e\left(\frac{136}{165}\right)\)\(e\left(\frac{62}{165}\right)\)\(e\left(\frac{31}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2541 }(376,a) \;\) at \(\;a = \) e.g. 2