Properties

Label 328.7
Modulus $328$
Conductor $164$
Order $40$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(328, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,0,39]))
 
pari: [g,chi] = znchar(Mod(7,328))
 

Basic properties

Modulus: \(328\)
Conductor: \(164\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{164}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 328.bc

\(\chi_{328}(7,\cdot)\) \(\chi_{328}(15,\cdot)\) \(\chi_{328}(47,\cdot)\) \(\chi_{328}(63,\cdot)\) \(\chi_{328}(71,\cdot)\) \(\chi_{328}(95,\cdot)\) \(\chi_{328}(111,\cdot)\) \(\chi_{328}(135,\cdot)\) \(\chi_{328}(151,\cdot)\) \(\chi_{328}(175,\cdot)\) \(\chi_{328}(183,\cdot)\) \(\chi_{328}(199,\cdot)\) \(\chi_{328}(231,\cdot)\) \(\chi_{328}(239,\cdot)\) \(\chi_{328}(263,\cdot)\) \(\chi_{328}(311,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: \(\Q(\zeta_{164})^+\)

Values on generators

\((247,165,129)\) → \((-1,1,e\left(\frac{39}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 328 }(7, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{21}{40}\right)\)\(i\)\(e\left(\frac{17}{40}\right)\)\(e\left(\frac{9}{40}\right)\)\(e\left(\frac{23}{40}\right)\)\(e\left(\frac{7}{40}\right)\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{13}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 328 }(7,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 328 }(7,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 328 }(7,·),\chi_{ 328 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 328 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2