Properties

Conductor 41
Order 4
Real No
Primitive Yes
Parity Even
Orbit Label 41.c

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(41)
sage: chi = H[9]
pari: [g,chi] = znchar(Mod(9,41))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 41
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 4
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 = Even
Orbit label = 41.c
Orbit index = 3

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 ]

\(\chi_{41}(9,\cdot)\) \(\chi_{41}(32,\cdot)\)

Values on generators

\(6\) → \(-i\)

Values

-11234567891011
\(1\)\(1\)\(-1\)\(i\)\(1\)\(-1\)\(-i\)\(i\)\(-1\)\(-1\)\(1\)\(i\)
value at  e.g. 2

Related number fields

Field of values \(\Q(i)\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 41 }(9,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{41}(9,\cdot)) = \sum_{r\in \Z/41\Z} \chi_{41}(9,r) e\left(\frac{2r}{41}\right) = 2.1194785695+-6.0421693615i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 41 }(9,·),\chi_{ 41 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{41}(9,\cdot),\chi_{41}(1,\cdot)) = \sum_{r\in \Z/41\Z} \chi_{41}(9,r) \chi_{41}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 41 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{41}(9,·)) = \sum_{r \in \Z/41\Z} \chi_{41}(9,r) e\left(\frac{1 r + 2 r^{-1}}{41}\right) = -1.9804646361i \)