Properties

Conductor 403
Order 12
Real No
Primitive Yes
Parity Even
Orbit Label 403.be

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(403)
 
sage: chi = H[57]
 
pari: [g,chi] = znchar(Mod(57,403))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 403
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 = Even
Orbit label = 403.be
Orbit index = 31

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 ]
 

\(\chi_{403}(57,\cdot)\) \(\chi_{403}(99,\cdot)\) \(\chi_{403}(161,\cdot)\) \(\chi_{403}(398,\cdot)\)

Values on generators

\((249,313)\) → \((-i,e\left(\frac{1}{6}\right))\)

Values

-11234567891011
\(1\)\(1\)\(-i\)\(e\left(\frac{1}{6}\right)\)\(-1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(i\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{12}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 403 }(57,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{403}(57,\cdot)) = \sum_{r\in \Z/403\Z} \chi_{403}(57,r) e\left(\frac{2r}{403}\right) = -0.984344566+20.0507123508i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 403 }(57,·),\chi_{ 403 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{403}(57,\cdot),\chi_{403}(1,\cdot)) = \sum_{r\in \Z/403\Z} \chi_{403}(57,r) \chi_{403}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 403 }(57,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{403}(57,·)) = \sum_{r \in \Z/403\Z} \chi_{403}(57,r) e\left(\frac{1 r + 2 r^{-1}}{403}\right) = 2.2694425321+-2.2694425321i \)