Properties

Label 731.148
Modulus $731$
Conductor $731$
Order $336$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(336))
 
M = H._module
 
chi = DirichletCharacter(H, M([273,152]))
 
pari: [g,chi] = znchar(Mod(148,731))
 

Basic properties

Modulus: \(731\)
Conductor: \(731\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(336\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 731.bm

\(\chi_{731}(3,\cdot)\) \(\chi_{731}(5,\cdot)\) \(\chi_{731}(12,\cdot)\) \(\chi_{731}(20,\cdot)\) \(\chi_{731}(28,\cdot)\) \(\chi_{731}(29,\cdot)\) \(\chi_{731}(46,\cdot)\) \(\chi_{731}(48,\cdot)\) \(\chi_{731}(61,\cdot)\) \(\chi_{731}(62,\cdot)\) \(\chi_{731}(63,\cdot)\) \(\chi_{731}(71,\cdot)\) \(\chi_{731}(73,\cdot)\) \(\chi_{731}(91,\cdot)\) \(\chi_{731}(105,\cdot)\) \(\chi_{731}(112,\cdot)\) \(\chi_{731}(114,\cdot)\) \(\chi_{731}(116,\cdot)\) \(\chi_{731}(141,\cdot)\) \(\chi_{731}(147,\cdot)\) \(\chi_{731}(148,\cdot)\) \(\chi_{731}(158,\cdot)\) \(\chi_{731}(159,\cdot)\) \(\chi_{731}(163,\cdot)\) \(\chi_{731}(175,\cdot)\) \(\chi_{731}(177,\cdot)\) \(\chi_{731}(184,\cdot)\) \(\chi_{731}(190,\cdot)\) \(\chi_{731}(192,\cdot)\) \(\chi_{731}(198,\cdot)\) ...

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: $\Q(\zeta_{336})$
Fixed field: Number field defined by a degree 336 polynomial (not computed)

Values on generators

\((173,562)\) → \((e\left(\frac{13}{16}\right),e\left(\frac{19}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 731 }(148, a) \) \(1\)\(1\)\(e\left(\frac{33}{56}\right)\)\(e\left(\frac{89}{336}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{125}{336}\right)\)\(e\left(\frac{41}{48}\right)\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{43}{56}\right)\)\(e\left(\frac{89}{168}\right)\)\(e\left(\frac{323}{336}\right)\)\(e\left(\frac{29}{112}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 731 }(148,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 731 }(148,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 731 }(148,·),\chi_{ 731 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 731 }(148,·)) \;\) at \(\; a,b = \) e.g. 1,2