Properties

Label 429.7
Modulus $429$
Conductor $143$
Order $60$
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(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,42,55]))
 
pari: [g,chi] = znchar(Mod(7,429))
 

Basic properties

Modulus: \(429\)
Conductor: \(143\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{143}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 429.bs

\(\chi_{429}(7,\cdot)\) \(\chi_{429}(19,\cdot)\) \(\chi_{429}(28,\cdot)\) \(\chi_{429}(46,\cdot)\) \(\chi_{429}(85,\cdot)\) \(\chi_{429}(106,\cdot)\) \(\chi_{429}(145,\cdot)\) \(\chi_{429}(184,\cdot)\) \(\chi_{429}(193,\cdot)\) \(\chi_{429}(271,\cdot)\) \(\chi_{429}(292,\cdot)\) \(\chi_{429}(310,\cdot)\) \(\chi_{429}(349,\cdot)\) \(\chi_{429}(358,\cdot)\) \(\chi_{429}(370,\cdot)\) \(\chi_{429}(409,\cdot)\)

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

\((287,79,67)\) → \((1,e\left(\frac{7}{10}\right),e\left(\frac{11}{12}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(14\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{41}{60}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 429 }(7,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{429}(7,\cdot)) = \sum_{r\in \Z/429\Z} \chi_{429}(7,r) e\left(\frac{2r}{429}\right) = 0.5229289726+11.9468215559i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 429 }(7,·),\chi_{ 429 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{429}(7,\cdot),\chi_{429}(1,\cdot)) = \sum_{r\in \Z/429\Z} \chi_{429}(7,r) \chi_{429}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 429 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{429}(7,·)) = \sum_{r \in \Z/429\Z} \chi_{429}(7,r) e\left(\frac{1 r + 2 r^{-1}}{429}\right) = 5.5104327425+-14.3551680791i \)