Properties

Label 169.14
Modulus $169$
Conductor $169$
Order $13$
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(169, base_ring=CyclotomicField(26))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([18]))
 
pari: [g,chi] = znchar(Mod(14,169))
 

Basic properties

Modulus: \(169\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(13\)
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 169.g

\(\chi_{169}(14,\cdot)\) \(\chi_{169}(27,\cdot)\) \(\chi_{169}(40,\cdot)\) \(\chi_{169}(53,\cdot)\) \(\chi_{169}(66,\cdot)\) \(\chi_{169}(79,\cdot)\) \(\chi_{169}(92,\cdot)\) \(\chi_{169}(105,\cdot)\) \(\chi_{169}(118,\cdot)\) \(\chi_{169}(131,\cdot)\) \(\chi_{169}(144,\cdot)\) \(\chi_{169}(157,\cdot)\)

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: \(\Q(\zeta_{13})\)
Fixed field: 13.13.542800770374370512771595361.1

Values on generators

\(2\) → \(e\left(\frac{9}{13}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{4}{13}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 169 }(14,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{169}(14,\cdot)) = \sum_{r\in \Z/169\Z} \chi_{169}(14,r) e\left(\frac{2r}{169}\right) = 6.1481288204+11.4542792007i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 169 }(14,·),\chi_{ 169 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{169}(14,\cdot),\chi_{169}(1,\cdot)) = \sum_{r\in \Z/169\Z} \chi_{169}(14,r) \chi_{169}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 169 }(14,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{169}(14,·)) = \sum_{r \in \Z/169\Z} \chi_{169}(14,r) e\left(\frac{1 r + 2 r^{-1}}{169}\right) = 14.72887146+-21.3384541888i \)