Properties

Label 1296.35
Modulus $1296$
Conductor $432$
Order $36$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,27,26]))
 
pari: [g,chi] = znchar(Mod(35,1296))
 

Basic properties

Modulus: \(1296\)
Conductor: \(432\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{432}(227,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1296.bk

\(\chi_{1296}(35,\cdot)\) \(\chi_{1296}(179,\cdot)\) \(\chi_{1296}(251,\cdot)\) \(\chi_{1296}(395,\cdot)\) \(\chi_{1296}(467,\cdot)\) \(\chi_{1296}(611,\cdot)\) \(\chi_{1296}(683,\cdot)\) \(\chi_{1296}(827,\cdot)\) \(\chi_{1296}(899,\cdot)\) \(\chi_{1296}(1043,\cdot)\) \(\chi_{1296}(1115,\cdot)\) \(\chi_{1296}(1259,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.36.5532004127928253705369187176396364210546696053048780432717505515499814912.1

Values on generators

\((1135,325,1217)\) → \((-1,-i,e\left(\frac{13}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 1296 }(35, a) \) \(1\)\(1\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{17}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1296 }(35,a) \;\) at \(\;a = \) e.g. 2