Properties

Label 392.213
Modulus $392$
Conductor $392$
Order $42$
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(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,21,25]))
 
pari: [g,chi] = znchar(Mod(213,392))
 

Basic properties

Modulus: \(392\)
Conductor: \(392\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 392.bf

\(\chi_{392}(5,\cdot)\) \(\chi_{392}(45,\cdot)\) \(\chi_{392}(61,\cdot)\) \(\chi_{392}(101,\cdot)\) \(\chi_{392}(157,\cdot)\) \(\chi_{392}(173,\cdot)\) \(\chi_{392}(213,\cdot)\) \(\chi_{392}(229,\cdot)\) \(\chi_{392}(269,\cdot)\) \(\chi_{392}(285,\cdot)\) \(\chi_{392}(341,\cdot)\) \(\chi_{392}(381,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.0.1090030896264192289800449659845679818091197961133776603876122561317234873686091104256.1

Values on generators

\((295,197,297)\) → \((1,-1,e\left(\frac{25}{42}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(-1\)\(1\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{11}{21}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 392 }(213,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{392}(213,\cdot)) = \sum_{r\in \Z/392\Z} \chi_{392}(213,r) e\left(\frac{r}{196}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 392 }(213,·),\chi_{ 392 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{392}(213,\cdot),\chi_{392}(1,\cdot)) = \sum_{r\in \Z/392\Z} \chi_{392}(213,r) \chi_{392}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 392 }(213,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{392}(213,·)) = \sum_{r \in \Z/392\Z} \chi_{392}(213,r) e\left(\frac{1 r + 2 r^{-1}}{392}\right) = 0.0 \)