Properties

Conductor 31
Order 30
Real No
Primitive Yes
Parity Odd
Orbit Label 31.h

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(31)
sage: chi = H[12]
pari: [g,chi] = znchar(Mod(12,31))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 31
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 30
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = Yes
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Odd
Orbit label = 31.h
Orbit index = 8

Galois orbit

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

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

Values on generators

\(3\) → \(e\left(\frac{19}{30}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{17}{30}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{15})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 31 }(12,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{31}(12,\cdot)) = \sum_{r\in \Z/31\Z} \chi_{31}(12,r) e\left(\frac{2r}{31}\right) = 0.9796244011+-5.4809064973i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 31 }(12,·),\chi_{ 31 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{31}(12,\cdot),\chi_{31}(1,\cdot)) = \sum_{r\in \Z/31\Z} \chi_{31}(12,r) \chi_{31}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 31 }(12,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{31}(12,·)) = \sum_{r \in \Z/31\Z} \chi_{31}(12,r) e\left(\frac{1 r + 2 r^{-1}}{31}\right) = -3.6940758395+5.0844591983i \)