Properties

Label 1728.49
Modulus $1728$
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(1728, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,28]))
 
pari: [g,chi] = znchar(Mod(49,1728))
 

Basic properties

Modulus: \(1728\)
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 \(\chi_{432}(373,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1728.bs

\(\chi_{1728}(49,\cdot)\) \(\chi_{1728}(241,\cdot)\) \(\chi_{1728}(337,\cdot)\) \(\chi_{1728}(529,\cdot)\) \(\chi_{1728}(625,\cdot)\) \(\chi_{1728}(817,\cdot)\) \(\chi_{1728}(913,\cdot)\) \(\chi_{1728}(1105,\cdot)\) \(\chi_{1728}(1201,\cdot)\) \(\chi_{1728}(1393,\cdot)\) \(\chi_{1728}(1489,\cdot)\) \(\chi_{1728}(1681,\cdot)\)

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: 36.36.614667125325361522818798575155151578949632894783197825857500612833312768.1

Values on generators

\((703,325,1217)\) → \((1,i,e\left(\frac{7}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 1728 }(49, a) \) \(1\)\(1\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{5}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1728 }(49,a) \;\) at \(\;a = \) e.g. 2