Properties

Label 169.43
Modulus $169$
Conductor $169$
Order $78$
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([61]))
 
pari: [g,chi] = znchar(Mod(43,169))
 

Basic properties

Modulus: \(169\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
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.k

\(\chi_{169}(4,\cdot)\) \(\chi_{169}(10,\cdot)\) \(\chi_{169}(17,\cdot)\) \(\chi_{169}(30,\cdot)\) \(\chi_{169}(36,\cdot)\) \(\chi_{169}(43,\cdot)\) \(\chi_{169}(49,\cdot)\) \(\chi_{169}(56,\cdot)\) \(\chi_{169}(62,\cdot)\) \(\chi_{169}(69,\cdot)\) \(\chi_{169}(75,\cdot)\) \(\chi_{169}(82,\cdot)\) \(\chi_{169}(88,\cdot)\) \(\chi_{169}(95,\cdot)\) \(\chi_{169}(101,\cdot)\) \(\chi_{169}(108,\cdot)\) \(\chi_{169}(114,\cdot)\) \(\chi_{169}(121,\cdot)\) \(\chi_{169}(127,\cdot)\) \(\chi_{169}(134,\cdot)\) \(\chi_{169}(140,\cdot)\) \(\chi_{169}(153,\cdot)\) \(\chi_{169}(160,\cdot)\) \(\chi_{169}(166,\cdot)\)

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: Number field defined by a degree 78 polynomial

Values on generators

\(2\) → \(e\left(\frac{61}{78}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{61}{78}\right)\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{22}{39}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{59}{78}\right)\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{37}{39}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{43}{78}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 169 }(43,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{169}(43,\cdot)) = \sum_{r\in \Z/169\Z} \chi_{169}(43,r) e\left(\frac{2r}{169}\right) = 12.0523839879+-4.8723752122i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 169 }(43,·),\chi_{ 169 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{169}(43,\cdot),\chi_{169}(1,\cdot)) = \sum_{r\in \Z/169\Z} \chi_{169}(43,r) \chi_{169}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 169 }(43,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{169}(43,·)) = \sum_{r \in \Z/169\Z} \chi_{169}(43,r) e\left(\frac{1 r + 2 r^{-1}}{169}\right) = 0.0 \)