Properties

Label 232.219
Modulus $232$
Conductor $232$
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(232, base_ring=CyclotomicField(14))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([7,7,2]))
 
pari: [g,chi] = znchar(Mod(219,232))
 

Basic properties

Modulus: \(232\)
Conductor: \(232\)
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 232.p

\(\chi_{232}(83,\cdot)\) \(\chi_{232}(107,\cdot)\) \(\chi_{232}(123,\cdot)\) \(\chi_{232}(139,\cdot)\) \(\chi_{232}(219,\cdot)\) \(\chi_{232}(227,\cdot)\)

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: 14.0.742003380228915810271232.1

Values on generators

\((175,117,89)\) → \((-1,-1,e\left(\frac{1}{7}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(-1\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{5}{14}\right)\)\(1\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{13}{14}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 232 }(219,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{232}(219,\cdot)) = \sum_{r\in \Z/232\Z} \chi_{232}(219,r) e\left(\frac{r}{116}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 232 }(219,·),\chi_{ 232 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{232}(219,\cdot),\chi_{232}(1,\cdot)) = \sum_{r\in \Z/232\Z} \chi_{232}(219,r) \chi_{232}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 232 }(219,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{232}(219,·)) = \sum_{r \in \Z/232\Z} \chi_{232}(219,r) e\left(\frac{1 r + 2 r^{-1}}{232}\right) = 2.4069750119+-4.9981351132i \)