Properties

Label 2541.1303
Modulus $2541$
Conductor $121$
Order $55$
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([0,0,64]))
 
pari: [g,chi] = znchar(Mod(1303,2541))
 

Basic properties

Modulus: \(2541\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(55\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{121}(93,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2541.bp

\(\chi_{2541}(64,\cdot)\) \(\chi_{2541}(169,\cdot)\) \(\chi_{2541}(190,\cdot)\) \(\chi_{2541}(295,\cdot)\) \(\chi_{2541}(379,\cdot)\) \(\chi_{2541}(400,\cdot)\) \(\chi_{2541}(421,\cdot)\) \(\chi_{2541}(526,\cdot)\) \(\chi_{2541}(610,\cdot)\) \(\chi_{2541}(631,\cdot)\) \(\chi_{2541}(652,\cdot)\) \(\chi_{2541}(757,\cdot)\) \(\chi_{2541}(841,\cdot)\) \(\chi_{2541}(862,\cdot)\) \(\chi_{2541}(883,\cdot)\) \(\chi_{2541}(988,\cdot)\) \(\chi_{2541}(1072,\cdot)\) \(\chi_{2541}(1093,\cdot)\) \(\chi_{2541}(1114,\cdot)\) \(\chi_{2541}(1303,\cdot)\) \(\chi_{2541}(1324,\cdot)\) \(\chi_{2541}(1345,\cdot)\) \(\chi_{2541}(1450,\cdot)\) \(\chi_{2541}(1534,\cdot)\) \(\chi_{2541}(1555,\cdot)\) \(\chi_{2541}(1681,\cdot)\) \(\chi_{2541}(1765,\cdot)\) \(\chi_{2541}(1786,\cdot)\) \(\chi_{2541}(1807,\cdot)\) \(\chi_{2541}(1912,\cdot)\) ...

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 55 polynomial

Values on generators

\((848,1816,2059)\) → \((1,1,e\left(\frac{32}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 2541 }(1303, a) \) \(1\)\(1\)\(e\left(\frac{32}{55}\right)\)\(e\left(\frac{9}{55}\right)\)\(e\left(\frac{3}{55}\right)\)\(e\left(\frac{41}{55}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{42}{55}\right)\)\(e\left(\frac{18}{55}\right)\)\(e\left(\frac{28}{55}\right)\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{12}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2541 }(1303,a) \;\) at \(\;a = \) e.g. 2