Properties

Conductor 37
Order 12
Real No
Primitive Yes
Parity Odd
Orbit Label 37.g

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(37)
sage: chi = H[8]
pari: [g,chi] = znchar(Mod(8,37))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 37
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 12
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 = 37.g
Orbit index = 7

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 ]

\(\chi_{37}(8,\cdot)\) \(\chi_{37}(14,\cdot)\) \(\chi_{37}(23,\cdot)\) \(\chi_{37}(29,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{1}{12}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(i\)\(e\left(\frac{2}{3}\right)\)\(i\)\(e\left(\frac{1}{3}\right)\)\(1\)\(-1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{12})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 37 }(8,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{37}(8,\cdot)) = \sum_{r\in \Z/37\Z} \chi_{37}(8,r) e\left(\frac{2r}{37}\right) = -4.0148590516+4.5695630859i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 37 }(8,·),\chi_{ 37 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{37}(8,\cdot),\chi_{37}(1,\cdot)) = \sum_{r\in \Z/37\Z} \chi_{37}(8,r) \chi_{37}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 37 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{37}(8,·)) = \sum_{r \in \Z/37\Z} \chi_{37}(8,r) e\left(\frac{1 r + 2 r^{-1}}{37}\right) = -0.1201679671+0.4484729587i \)