Properties

Label 2541.1084
Modulus $2541$
Conductor $847$
Order $110$
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(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,55,19]))
 
pari: [g,chi] = znchar(Mod(1084,2541))
 

Basic properties

Modulus: \(2541\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{847}(237,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2541.by

\(\chi_{2541}(13,\cdot)\) \(\chi_{2541}(139,\cdot)\) \(\chi_{2541}(160,\cdot)\) \(\chi_{2541}(244,\cdot)\) \(\chi_{2541}(349,\cdot)\) \(\chi_{2541}(370,\cdot)\) \(\chi_{2541}(391,\cdot)\) \(\chi_{2541}(580,\cdot)\) \(\chi_{2541}(601,\cdot)\) \(\chi_{2541}(622,\cdot)\) \(\chi_{2541}(706,\cdot)\) \(\chi_{2541}(811,\cdot)\) \(\chi_{2541}(832,\cdot)\) \(\chi_{2541}(853,\cdot)\) \(\chi_{2541}(937,\cdot)\) \(\chi_{2541}(1042,\cdot)\) \(\chi_{2541}(1063,\cdot)\) \(\chi_{2541}(1084,\cdot)\) \(\chi_{2541}(1168,\cdot)\) \(\chi_{2541}(1273,\cdot)\) \(\chi_{2541}(1294,\cdot)\) \(\chi_{2541}(1315,\cdot)\) \(\chi_{2541}(1399,\cdot)\) \(\chi_{2541}(1504,\cdot)\) \(\chi_{2541}(1525,\cdot)\) \(\chi_{2541}(1630,\cdot)\) \(\chi_{2541}(1735,\cdot)\) \(\chi_{2541}(1756,\cdot)\) \(\chi_{2541}(1777,\cdot)\) \(\chi_{2541}(1861,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((848,1816,2059)\) → \((1,-1,e\left(\frac{19}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 2541 }(1084, a) \) \(1\)\(1\)\(e\left(\frac{19}{110}\right)\)\(e\left(\frac{19}{55}\right)\)\(e\left(\frac{31}{110}\right)\)\(e\left(\frac{57}{110}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{52}{55}\right)\)\(e\left(\frac{38}{55}\right)\)\(e\left(\frac{53}{55}\right)\)\(e\left(\frac{46}{55}\right)\)\(e\left(\frac{69}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2541 }(1084,a) \;\) at \(\;a = \) e.g. 2