Properties

Label 384.125
Modulus $384$
Conductor $384$
Order $32$
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(384, base_ring=CyclotomicField(32))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,16]))
 
pari: [g,chi] = znchar(Mod(125,384))
 

Basic properties

Modulus: \(384\)
Conductor: \(384\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
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 384.x

\(\chi_{384}(5,\cdot)\) \(\chi_{384}(29,\cdot)\) \(\chi_{384}(53,\cdot)\) \(\chi_{384}(77,\cdot)\) \(\chi_{384}(101,\cdot)\) \(\chi_{384}(125,\cdot)\) \(\chi_{384}(149,\cdot)\) \(\chi_{384}(173,\cdot)\) \(\chi_{384}(197,\cdot)\) \(\chi_{384}(221,\cdot)\) \(\chi_{384}(245,\cdot)\) \(\chi_{384}(269,\cdot)\) \(\chi_{384}(293,\cdot)\) \(\chi_{384}(317,\cdot)\) \(\chi_{384}(341,\cdot)\) \(\chi_{384}(365,\cdot)\)

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: \(\Q(\zeta_{32})\)
Fixed field: 32.0.135104323545903136978453058557785670637514001130337144105502507008.1

Values on generators

\((127,133,257)\) → \((1,e\left(\frac{3}{32}\right),-1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{19}{32}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{1}{32}\right)\)\(-i\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 384 }(125,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{384}(125,\cdot)) = \sum_{r\in \Z/384\Z} \chi_{384}(125,r) e\left(\frac{r}{192}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 384 }(125,·),\chi_{ 384 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{384}(125,\cdot),\chi_{384}(1,\cdot)) = \sum_{r\in \Z/384\Z} \chi_{384}(125,r) \chi_{384}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 384 }(125,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{384}(125,·)) = \sum_{r \in \Z/384\Z} \chi_{384}(125,r) e\left(\frac{1 r + 2 r^{-1}}{384}\right) = 0.0 \)