Properties

Label 1521.46
Modulus $1521$
Conductor $169$
Order $156$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, base_ring=CyclotomicField(156))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,131]))
 
pari: [g,chi] = znchar(Mod(46,1521))
 

Basic properties

Modulus: \(1521\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{169}(46,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1521.ca

\(\chi_{1521}(28,\cdot)\) \(\chi_{1521}(37,\cdot)\) \(\chi_{1521}(46,\cdot)\) \(\chi_{1521}(136,\cdot)\) \(\chi_{1521}(145,\cdot)\) \(\chi_{1521}(154,\cdot)\) \(\chi_{1521}(163,\cdot)\) \(\chi_{1521}(253,\cdot)\) \(\chi_{1521}(262,\cdot)\) \(\chi_{1521}(271,\cdot)\) \(\chi_{1521}(280,\cdot)\) \(\chi_{1521}(370,\cdot)\) \(\chi_{1521}(379,\cdot)\) \(\chi_{1521}(388,\cdot)\) \(\chi_{1521}(397,\cdot)\) \(\chi_{1521}(487,\cdot)\) \(\chi_{1521}(496,\cdot)\) \(\chi_{1521}(505,\cdot)\) \(\chi_{1521}(514,\cdot)\) \(\chi_{1521}(604,\cdot)\) \(\chi_{1521}(613,\cdot)\) \(\chi_{1521}(622,\cdot)\) \(\chi_{1521}(631,\cdot)\) \(\chi_{1521}(721,\cdot)\) \(\chi_{1521}(730,\cdot)\) \(\chi_{1521}(739,\cdot)\) \(\chi_{1521}(748,\cdot)\) \(\chi_{1521}(838,\cdot)\) \(\chi_{1521}(847,\cdot)\) \(\chi_{1521}(856,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((677,847)\) → \((1,e\left(\frac{131}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\( \chi_{ 1521 }(46, a) \) \(-1\)\(1\)\(e\left(\frac{131}{156}\right)\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{29}{52}\right)\)\(e\left(\frac{133}{156}\right)\)\(e\left(\frac{27}{52}\right)\)\(e\left(\frac{31}{78}\right)\)\(e\left(\frac{77}{156}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{47}{78}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1521 }(46,a) \;\) at \(\;a = \) e.g. 2