Properties

Label 392.349
Modulus $392$
Conductor $392$
Order $14$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(392, base_ring=CyclotomicField(14))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,7,9]))
 
pari: [g,chi] = znchar(Mod(349,392))
 

Basic properties

Modulus: \(392\)
Conductor: \(392\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 392.r

\(\chi_{392}(13,\cdot)\) \(\chi_{392}(69,\cdot)\) \(\chi_{392}(125,\cdot)\) \(\chi_{392}(181,\cdot)\) \(\chi_{392}(237,\cdot)\) \(\chi_{392}(349,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{7})\)
Fixed field: 14.0.2812424737865523319657201664.1

Values on generators

\((295,197,297)\) → \((1,-1,e\left(\frac{9}{14}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(-1\)\(1\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{1}{14}\right)\)\(1\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{2}{7}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 392 }(349,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{392}(349,\cdot)) = \sum_{r\in \Z/392\Z} \chi_{392}(349,r) e\left(\frac{r}{196}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 392 }(349,·),\chi_{ 392 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{392}(349,\cdot),\chi_{392}(1,\cdot)) = \sum_{r\in \Z/392\Z} \chi_{392}(349,r) \chi_{392}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 392 }(349,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{392}(349,·)) = \sum_{r \in \Z/392\Z} \chi_{392}(349,r) e\left(\frac{1 r + 2 r^{-1}}{392}\right) = 30.5621087173+24.3724684102i \)