Properties

Label 507.40
Modulus $507$
Conductor $169$
Order $13$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(507, base_ring=CyclotomicField(26))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,2]))
 
pari: [g,chi] = znchar(Mod(40,507))
 

Basic properties

Modulus: \(507\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(13\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{169}(40,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 507.m

\(\chi_{507}(40,\cdot)\) \(\chi_{507}(79,\cdot)\) \(\chi_{507}(118,\cdot)\) \(\chi_{507}(157,\cdot)\) \(\chi_{507}(196,\cdot)\) \(\chi_{507}(235,\cdot)\) \(\chi_{507}(274,\cdot)\) \(\chi_{507}(313,\cdot)\) \(\chi_{507}(352,\cdot)\) \(\chi_{507}(391,\cdot)\) \(\chi_{507}(430,\cdot)\) \(\chi_{507}(469,\cdot)\)

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

\((170,340)\) → \((1,e\left(\frac{1}{13}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{3}{13}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 507 }(40,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{507}(40,\cdot)) = \sum_{r\in \Z/507\Z} \chi_{507}(40,r) e\left(\frac{2r}{507}\right) = 10.9005232267+7.0836850144i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 507 }(40,·),\chi_{ 507 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{507}(40,\cdot),\chi_{507}(1,\cdot)) = \sum_{r\in \Z/507\Z} \chi_{507}(40,r) \chi_{507}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 507 }(40,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{507}(40,·)) = \sum_{r \in \Z/507\Z} \chi_{507}(40,r) e\left(\frac{1 r + 2 r^{-1}}{507}\right) = 48.5388620257+11.9637549861i \)