Properties

Label 169.3
Modulus $169$
Conductor $169$
Order $39$
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(78))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([62]))
 
pari: [g,chi] = znchar(Mod(3,169))
 

Basic properties

Modulus: \(169\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(39\)
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.i

\(\chi_{169}(3,\cdot)\) \(\chi_{169}(9,\cdot)\) \(\chi_{169}(16,\cdot)\) \(\chi_{169}(29,\cdot)\) \(\chi_{169}(35,\cdot)\) \(\chi_{169}(42,\cdot)\) \(\chi_{169}(48,\cdot)\) \(\chi_{169}(55,\cdot)\) \(\chi_{169}(61,\cdot)\) \(\chi_{169}(68,\cdot)\) \(\chi_{169}(74,\cdot)\) \(\chi_{169}(81,\cdot)\) \(\chi_{169}(87,\cdot)\) \(\chi_{169}(94,\cdot)\) \(\chi_{169}(100,\cdot)\) \(\chi_{169}(107,\cdot)\) \(\chi_{169}(113,\cdot)\) \(\chi_{169}(120,\cdot)\) \(\chi_{169}(126,\cdot)\) \(\chi_{169}(133,\cdot)\) \(\chi_{169}(139,\cdot)\) \(\chi_{169}(152,\cdot)\) \(\chi_{169}(159,\cdot)\) \(\chi_{169}(165,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: 39.39.27027636582498189040621249864144468324898507852136260989871841246090732111847218889.1

Values on generators

\(2\) → \(e\left(\frac{31}{39}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{31}{39}\right)\)\(e\left(\frac{22}{39}\right)\)\(e\left(\frac{23}{39}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{37}{39}\right)\)\(e\left(\frac{34}{39}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 169 }(3,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{169}(3,\cdot)) = \sum_{r\in \Z/169\Z} \chi_{169}(3,r) e\left(\frac{2r}{169}\right) = 12.964078078+0.9657533778i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 169 }(3,·),\chi_{ 169 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{169}(3,\cdot),\chi_{169}(1,\cdot)) = \sum_{r\in \Z/169\Z} \chi_{169}(3,r) \chi_{169}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 169 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{169}(3,·)) = \sum_{r \in \Z/169\Z} \chi_{169}(3,r) e\left(\frac{1 r + 2 r^{-1}}{169}\right) = -7.8500365606+5.8989197897i \)