Properties

Label 920.517
Modulus $920$
Conductor $920$
Order $44$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(920, base_ring=CyclotomicField(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,22,11,18]))
 
pari: [g,chi] = znchar(Mod(517,920))
 

Basic properties

Modulus: \(920\)
Conductor: \(920\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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 920.bo

\(\chi_{920}(37,\cdot)\) \(\chi_{920}(53,\cdot)\) \(\chi_{920}(157,\cdot)\) \(\chi_{920}(237,\cdot)\) \(\chi_{920}(293,\cdot)\) \(\chi_{920}(333,\cdot)\) \(\chi_{920}(373,\cdot)\) \(\chi_{920}(477,\cdot)\) \(\chi_{920}(493,\cdot)\) \(\chi_{920}(517,\cdot)\) \(\chi_{920}(557,\cdot)\) \(\chi_{920}(573,\cdot)\) \(\chi_{920}(613,\cdot)\) \(\chi_{920}(677,\cdot)\) \(\chi_{920}(733,\cdot)\) \(\chi_{920}(757,\cdot)\) \(\chi_{920}(773,\cdot)\) \(\chi_{920}(797,\cdot)\) \(\chi_{920}(893,\cdot)\) \(\chi_{920}(917,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{44})\)
Fixed field: 44.44.13383169230192059253459701104387771124501004765020501667165784506368000000000000000000000000000000000.1

Values on generators

\((231,461,737,281)\) → \((1,-1,i,e\left(\frac{9}{22}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{4}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 920 }(517,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 920 }(517,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 920 }(517,·),\chi_{ 920 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 920 }(517,·)) \;\) at \(\; a,b = \) e.g. 1,2