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Properties

Label 1210.403
Modulus $1210$
Conductor $55$
Order $20$
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(1210, base_ring=CyclotomicField(20)) M = H._module chi = DirichletCharacter(H, M([15,14]))
 
Copy content pari:[g,chi] = znchar(Mod(403,1210))
 

Basic properties

Modulus: \(1210\)
Conductor: \(55\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(20\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{55}(18,\cdot)\)
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 1210.l

\(\chi_{1210}(233,\cdot)\) \(\chi_{1210}(403,\cdot)\) \(\chi_{1210}(457,\cdot)\) \(\chi_{1210}(717,\cdot)\) \(\chi_{1210}(723,\cdot)\) \(\chi_{1210}(887,\cdot)\) \(\chi_{1210}(1183,\cdot)\) \(\chi_{1210}(1207,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{55})^+\)

Values on generators

\((727,1091)\) → \((-i,e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 1210 }(403, a) \) \(1\)\(1\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(-1\)\(i\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{2}{5}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1210 }(403,a) \;\) at \(\;a = \) e.g. 2