Properties

Label 252.139
Modulus $252$
Conductor $252$
Order $6$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3,2,3]))
 
pari: [g,chi] = znchar(Mod(139,252))
 

Basic properties

Modulus: \(252\)
Conductor: \(252\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 252.bi

\(\chi_{252}(139,\cdot)\) \(\chi_{252}(223,\cdot)\)

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: \(\Q(\sqrt{-3}) \)
Fixed field: 6.6.144027072.1

Values on generators

\((127,29,73)\) → \((-1,e\left(\frac{1}{3}\right),-1)\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(-1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(1\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 252 }(139,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{252}(139,\cdot)) = \sum_{r\in \Z/252\Z} \chi_{252}(139,r) e\left(\frac{r}{126}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 252 }(139,·),\chi_{ 252 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{252}(139,\cdot),\chi_{252}(1,\cdot)) = \sum_{r\in \Z/252\Z} \chi_{252}(139,r) \chi_{252}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 252 }(139,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{252}(139,·)) = \sum_{r \in \Z/252\Z} \chi_{252}(139,r) e\left(\frac{1 r + 2 r^{-1}}{252}\right) = 9.8975937642+17.1431352723i \)