Properties

Label 2541.2410
Modulus $2541$
Conductor $847$
Order $33$
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(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,22,12]))
 
pari: [g,chi] = znchar(Mod(2410,2541))
 

Basic properties

Modulus: \(2541\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{847}(716,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2541.bo

\(\chi_{2541}(67,\cdot)\) \(\chi_{2541}(100,\cdot)\) \(\chi_{2541}(298,\cdot)\) \(\chi_{2541}(331,\cdot)\) \(\chi_{2541}(529,\cdot)\) \(\chi_{2541}(562,\cdot)\) \(\chi_{2541}(760,\cdot)\) \(\chi_{2541}(793,\cdot)\) \(\chi_{2541}(991,\cdot)\) \(\chi_{2541}(1024,\cdot)\) \(\chi_{2541}(1222,\cdot)\) \(\chi_{2541}(1255,\cdot)\) \(\chi_{2541}(1486,\cdot)\) \(\chi_{2541}(1684,\cdot)\) \(\chi_{2541}(1717,\cdot)\) \(\chi_{2541}(1915,\cdot)\) \(\chi_{2541}(1948,\cdot)\) \(\chi_{2541}(2146,\cdot)\) \(\chi_{2541}(2377,\cdot)\) \(\chi_{2541}(2410,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 33 polynomial

Values on generators

\((848,1816,2059)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{2}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 2541 }(2410, a) \) \(1\)\(1\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{9}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2541 }(2410,a) \;\) at \(\;a = \) e.g. 2