Properties

Label 2541.818
Modulus $2541$
Conductor $2541$
Order $110$
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(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([55,55,72]))
 
pari: [g,chi] = znchar(Mod(818,2541))
 

Basic properties

Modulus: \(2541\)
Conductor: \(2541\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
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.ca

\(\chi_{2541}(20,\cdot)\) \(\chi_{2541}(104,\cdot)\) \(\chi_{2541}(125,\cdot)\) \(\chi_{2541}(146,\cdot)\) \(\chi_{2541}(335,\cdot)\) \(\chi_{2541}(356,\cdot)\) \(\chi_{2541}(377,\cdot)\) \(\chi_{2541}(482,\cdot)\) \(\chi_{2541}(566,\cdot)\) \(\chi_{2541}(587,\cdot)\) \(\chi_{2541}(713,\cdot)\) \(\chi_{2541}(797,\cdot)\) \(\chi_{2541}(818,\cdot)\) \(\chi_{2541}(839,\cdot)\) \(\chi_{2541}(944,\cdot)\) \(\chi_{2541}(1028,\cdot)\) \(\chi_{2541}(1070,\cdot)\) \(\chi_{2541}(1175,\cdot)\) \(\chi_{2541}(1259,\cdot)\) \(\chi_{2541}(1280,\cdot)\) \(\chi_{2541}(1301,\cdot)\) \(\chi_{2541}(1406,\cdot)\) \(\chi_{2541}(1490,\cdot)\) \(\chi_{2541}(1511,\cdot)\) \(\chi_{2541}(1532,\cdot)\) \(\chi_{2541}(1637,\cdot)\) \(\chi_{2541}(1742,\cdot)\) \(\chi_{2541}(1763,\cdot)\) \(\chi_{2541}(1868,\cdot)\) \(\chi_{2541}(1952,\cdot)\) ...

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

\((848,1816,2059)\) → \((-1,-1,e\left(\frac{36}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 2541 }(818, a) \) \(1\)\(1\)\(e\left(\frac{17}{110}\right)\)\(e\left(\frac{17}{55}\right)\)\(e\left(\frac{24}{55}\right)\)\(e\left(\frac{51}{110}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{67}{110}\right)\)\(e\left(\frac{34}{55}\right)\)\(e\left(\frac{4}{55}\right)\)\(e\left(\frac{91}{110}\right)\)\(e\left(\frac{41}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2541 }(818,a) \;\) at \(\;a = \) e.g. 2