Properties

Label 344.331
Modulus $344$
Conductor $344$
Order $42$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(344)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,21,11]))
 
pari: [g,chi] = znchar(Mod(331,344))
 

Basic properties

Modulus: \(344\)
Conductor: \(344\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 344.bb

\(\chi_{344}(3,\cdot)\) \(\chi_{344}(19,\cdot)\) \(\chi_{344}(91,\cdot)\) \(\chi_{344}(115,\cdot)\) \(\chi_{344}(147,\cdot)\) \(\chi_{344}(155,\cdot)\) \(\chi_{344}(163,\cdot)\) \(\chi_{344}(227,\cdot)\) \(\chi_{344}(235,\cdot)\) \(\chi_{344}(243,\cdot)\) \(\chi_{344}(291,\cdot)\) \(\chi_{344}(331,\cdot)\)

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

Values on generators

\((87,173,89)\) → \((-1,-1,e\left(\frac{11}{42}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{13}{14}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.42.86515994746897550947675385197225985831622982825258543026271873735800731799783414431744.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 344 }(331,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{344}(331,\cdot)) = \sum_{r\in \Z/344\Z} \chi_{344}(331,r) e\left(\frac{r}{172}\right) = -0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 344 }(331,·),\chi_{ 344 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{344}(331,\cdot),\chi_{344}(1,\cdot)) = \sum_{r\in \Z/344\Z} \chi_{344}(331,r) \chi_{344}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 344 }(331,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{344}(331,·)) = \sum_{r \in \Z/344\Z} \chi_{344}(331,r) e\left(\frac{1 r + 2 r^{-1}}{344}\right) = -9.9227129817+-2.2647944863i \)