Properties

Label 429.109
Modulus $429$
Conductor $143$
Order $4$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(429, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,2,3]))
 
pari: [g,chi] = znchar(Mod(109,429))
 

Basic properties

Modulus: \(429\)
Conductor: \(143\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{143}(109,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 429.m

\(\chi_{429}(109,\cdot)\) \(\chi_{429}(307,\cdot)\)

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

Values on generators

\((287,79,67)\) → \((1,-1,-i)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(14\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(i\)\(-1\)\(-i\)\(-i\)\(-i\)\(1\)\(1\)\(1\)\(1\)\(i\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\sqrt{-1}) \)
Fixed field: 4.4.265837.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 429 }(109,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{429}(109,\cdot)) = \sum_{r\in \Z/429\Z} \chi_{429}(109,r) e\left(\frac{2r}{429}\right) = -3.4653144092+11.4451560079i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 429 }(109,·),\chi_{ 429 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{429}(109,\cdot),\chi_{429}(1,\cdot)) = \sum_{r\in \Z/429\Z} \chi_{429}(109,r) \chi_{429}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 429 }(109,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{429}(109,·)) = \sum_{r \in \Z/429\Z} \chi_{429}(109,r) e\left(\frac{1 r + 2 r^{-1}}{429}\right) = 0.0 \)