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Properties

Label 840.97
Modulus $840$
Conductor $35$
Order $4$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([0,0,0,1,2]))
 
Copy content pari:[g,chi] = znchar(Mod(97,840))
 

Basic properties

Modulus: \(840\)
Conductor: \(35\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(4\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{35}(27,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 840.bt

\(\chi_{840}(97,\cdot)\) \(\chi_{840}(433,\cdot)\)

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

Related number fields

Field of values: \(\mathbb{Q}(i)\)
Fixed field: 4.4.6125.1

Values on generators

\((631,421,281,337,241)\) → \((1,1,1,i,-1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 840 }(97, a) \) \(1\)\(1\)\(1\)\(i\)\(-i\)\(1\)\(-i\)\(-1\)\(-1\)\(i\)\(-1\)\(-i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 840 }(97,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 840 }(97,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 840 }(97,·),\chi_{ 840 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 840 }(97,·)) \;\) at \(\; a,b = \) e.g. 1,2