Properties

Label 2541.1646
Modulus $2541$
Conductor $363$
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([55,0,37]))
 
pari: [g,chi] = znchar(Mod(1646,2541))
 

Basic properties

Modulus: \(2541\)
Conductor: \(363\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{363}(194,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2541.cc

\(\chi_{2541}(8,\cdot)\) \(\chi_{2541}(29,\cdot)\) \(\chi_{2541}(50,\cdot)\) \(\chi_{2541}(134,\cdot)\) \(\chi_{2541}(260,\cdot)\) \(\chi_{2541}(281,\cdot)\) \(\chi_{2541}(365,\cdot)\) \(\chi_{2541}(470,\cdot)\) \(\chi_{2541}(491,\cdot)\) \(\chi_{2541}(512,\cdot)\) \(\chi_{2541}(701,\cdot)\) \(\chi_{2541}(722,\cdot)\) \(\chi_{2541}(743,\cdot)\) \(\chi_{2541}(827,\cdot)\) \(\chi_{2541}(932,\cdot)\) \(\chi_{2541}(953,\cdot)\) \(\chi_{2541}(974,\cdot)\) \(\chi_{2541}(1058,\cdot)\) \(\chi_{2541}(1163,\cdot)\) \(\chi_{2541}(1184,\cdot)\) \(\chi_{2541}(1205,\cdot)\) \(\chi_{2541}(1289,\cdot)\) \(\chi_{2541}(1394,\cdot)\) \(\chi_{2541}(1415,\cdot)\) \(\chi_{2541}(1436,\cdot)\) \(\chi_{2541}(1520,\cdot)\) \(\chi_{2541}(1625,\cdot)\) \(\chi_{2541}(1646,\cdot)\) \(\chi_{2541}(1751,\cdot)\) \(\chi_{2541}(1856,\cdot)\) ...

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

\((848,1816,2059)\) → \((-1,1,e\left(\frac{37}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 2541 }(1646, a) \) \(1\)\(1\)\(e\left(\frac{46}{55}\right)\)\(e\left(\frac{37}{55}\right)\)\(e\left(\frac{43}{110}\right)\)\(e\left(\frac{28}{55}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{107}{110}\right)\)\(e\left(\frac{19}{55}\right)\)\(e\left(\frac{54}{55}\right)\)\(e\left(\frac{101}{110}\right)\)\(e\left(\frac{7}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2541 }(1646,a) \;\) at \(\;a = \) e.g. 2