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Properties

Label 9800.3657
Modulus $9800$
Conductor $35$
Order $12$
Real no
Primitive no
Minimal no
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9800, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([0,0,3,2]))
 
Copy content pari:[g,chi] = znchar(Mod(3657,9800))
 

Basic properties

Modulus: \(9800\)
Conductor: \(35\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(12\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{35}(17,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 9800.cf

\(\chi_{9800}(3057,\cdot)\) \(\chi_{9800}(3657,\cdot)\) \(\chi_{9800}(6193,\cdot)\) \(\chi_{9800}(6793,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: \(\Q(\zeta_{35})^+\)

Values on generators

\((7351,4901,1177,5001)\) → \((1,1,i,e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 9800 }(3657, a) \) \(1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(i\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(-i\)\(-1\)\(e\left(\frac{1}{6}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 9800 }(3657,a) \;\) at \(\;a = \) e.g. 2