Properties

Label 31.27
Modulus $31$
Conductor $31$
Order $10$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(31, base_ring=CyclotomicField(10))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(27,31))
 

Basic properties

Modulus: \(31\)
Conductor: \(31\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
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.f

\(\chi_{31}(15,\cdot)\) \(\chi_{31}(23,\cdot)\) \(\chi_{31}(27,\cdot)\) \(\chi_{31}(29,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 10.0.26439622160671.1

Values on generators

\(3\) → \(e\left(\frac{1}{10}\right)\)

Values

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 31 }(27,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{31}(27,\cdot)) = \sum_{r\in \Z/31\Z} \chi_{31}(27,r) e\left(\frac{2r}{31}\right) = 5.560077191+0.292475008i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 31 }(27,·),\chi_{ 31 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{31}(27,\cdot),\chi_{31}(1,\cdot)) = \sum_{r\in \Z/31\Z} \chi_{31}(27,r) \chi_{31}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 31 }(27,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{31}(27,·)) = \sum_{r \in \Z/31\Z} \chi_{31}(27,r) e\left(\frac{1 r + 2 r^{-1}}{31}\right) = -2.6110266491+0.8483739857i \)