Properties

Conductor 49
Order 42
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 49.h

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(49)
 
sage: chi = H[3]
 
pari: [g,chi] = znchar(Mod(3,49))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 49
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 42
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 49.h
Orbit index = 8

Galois orbit

sage: chi.sage_character().galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{49}(3,\cdot)\) \(\chi_{49}(5,\cdot)\) \(\chi_{49}(10,\cdot)\) \(\chi_{49}(12,\cdot)\) \(\chi_{49}(17,\cdot)\) \(\chi_{49}(24,\cdot)\) \(\chi_{49}(26,\cdot)\) \(\chi_{49}(33,\cdot)\) \(\chi_{49}(38,\cdot)\) \(\chi_{49}(40,\cdot)\) \(\chi_{49}(45,\cdot)\) \(\chi_{49}(47,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{1}{42}\right)\)

Values

-112345689101112
\(-1\)\(1\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{11}{42}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{21})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 49 }(3,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{49}(3,\cdot)) = \sum_{r\in \Z/49\Z} \chi_{49}(3,r) e\left(\frac{2r}{49}\right) = -5.6980845968+4.0659355538i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 49 }(3,·),\chi_{ 49 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{49}(3,\cdot),\chi_{49}(1,\cdot)) = \sum_{r\in \Z/49\Z} \chi_{49}(3,r) \chi_{49}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 49 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{49}(3,·)) = \sum_{r \in \Z/49\Z} \chi_{49}(3,r) e\left(\frac{1 r + 2 r^{-1}}{49}\right) = -9.4580390467+-3.7120067881i \)