Properties

Conductor 177
Order 2
Real Yes
Primitive No
Parity Even
Orbit Label 354.c

Related objects

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Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(354)
 
sage: chi = H[353]
 
pari: [g,chi] = znchar(Mod(353,354))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 177
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 2
Real = Yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = No
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 354.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 ]
 

\(\chi_{354}(353,\cdot)\)

Inducing primitive character

\(\chi_{177}(176,\cdot)\) = \(\displaystyle\left(\frac{177}{\bullet}\right)\)

Values on generators

\((119,61)\) → \((-1,-1)\)

Values

-11571113171923252931
\(1\)\(1\)\(-1\)\(1\)\(1\)\(-1\)\(-1\)\(1\)\(1\)\(1\)\(-1\)\(-1\)
value at  e.g. 2

Related number fields

Field of values \(\Q\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 354 }(353,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{354}(353,\cdot)) = \sum_{r\in \Z/354\Z} \chi_{354}(353,r) e\left(\frac{r}{177}\right) = 13.3041346957 \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 354 }(353,·),\chi_{ 354 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{354}(353,\cdot),\chi_{354}(1,\cdot)) = \sum_{r\in \Z/354\Z} \chi_{354}(353,r) \chi_{354}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 354 }(353,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{354}(353,·)) = \sum_{r \in \Z/354\Z} \chi_{354}(353,r) e\left(\frac{1 r + 2 r^{-1}}{354}\right) = 0.0 \)