Properties

Label 384.19
Modulus $384$
Conductor $128$
Order $32$
Real no
Primitive no
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([16,23,0]))
 
pari: [g,chi] = znchar(Mod(19,384))
 

Basic properties

Modulus: \(384\)
Conductor: \(128\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{128}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 384.u

\(\chi_{384}(19,\cdot)\) \(\chi_{384}(43,\cdot)\) \(\chi_{384}(67,\cdot)\) \(\chi_{384}(91,\cdot)\) \(\chi_{384}(115,\cdot)\) \(\chi_{384}(139,\cdot)\) \(\chi_{384}(163,\cdot)\) \(\chi_{384}(187,\cdot)\) \(\chi_{384}(211,\cdot)\) \(\chi_{384}(235,\cdot)\) \(\chi_{384}(259,\cdot)\) \(\chi_{384}(283,\cdot)\) \(\chi_{384}(307,\cdot)\) \(\chi_{384}(331,\cdot)\) \(\chi_{384}(355,\cdot)\) \(\chi_{384}(379,\cdot)\)

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: 32.0.3138550867693340381917894711603833208051177722232017256448.1

Values on generators

\((127,133,257)\) → \((-1,e\left(\frac{23}{32}\right),1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{23}{32}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{19}{32}\right)\)\(e\left(\frac{25}{32}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{1}{32}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{13}{32}\right)\)\(i\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 384 }(19,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{384}(19,\cdot)) = \sum_{r\in \Z/384\Z} \chi_{384}(19,r) e\left(\frac{r}{192}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 384 }(19,·),\chi_{ 384 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{384}(19,\cdot),\chi_{384}(1,\cdot)) = \sum_{r\in \Z/384\Z} \chi_{384}(19,r) \chi_{384}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 384 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{384}(19,·)) = \sum_{r \in \Z/384\Z} \chi_{384}(19,r) e\left(\frac{1 r + 2 r^{-1}}{384}\right) = -9.6744673887+-20.4549426972i \)