Properties

Label 31.11
Modulus $31$
Conductor $31$
Order $30$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(31)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([23]))
 
pari: [g,chi] = znchar(Mod(11,31))
 

Basic properties

Modulus: \(31\)
Conductor: \(31\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 31.h

\(\chi_{31}(3,\cdot)\) \(\chi_{31}(11,\cdot)\) \(\chi_{31}(12,\cdot)\) \(\chi_{31}(13,\cdot)\) \(\chi_{31}(17,\cdot)\) \(\chi_{31}(21,\cdot)\) \(\chi_{31}(22,\cdot)\) \(\chi_{31}(24,\cdot)\)

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

Values on generators

\(3\) → \(e\left(\frac{23}{30}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{19}{30}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: \(\Q(\zeta_{31})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 31 }(11,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{31}(11,\cdot)) = \sum_{r\in \Z/31\Z} \chi_{31}(11,r) e\left(\frac{2r}{31}\right) = -3.6421482497+4.2112653832i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 31 }(11,·),\chi_{ 31 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{31}(11,\cdot),\chi_{31}(1,\cdot)) = \sum_{r\in \Z/31\Z} \chi_{31}(11,r) \chi_{31}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 31 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{31}(11,·)) = \sum_{r \in \Z/31\Z} \chi_{31}(11,r) e\left(\frac{1 r + 2 r^{-1}}{31}\right) = 1.0681204561+-0.3470533741i \)