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Properties

Label 1110.103
Modulus $1110$
Conductor $185$
Order $12$
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(1110, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([0,9,7]))
 
Copy content pari:[g,chi] = znchar(Mod(103,1110))
 

Basic properties

Modulus: \(1110\)
Conductor: \(185\)
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 \(\chi_{185}(103,\cdot)\)
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 1110.bf

\(\chi_{1110}(97,\cdot)\) \(\chi_{1110}(103,\cdot)\) \(\chi_{1110}(637,\cdot)\) \(\chi_{1110}(643,\cdot)\)

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: 12.12.347495355038008619140625.1

Values on generators

\((371,667,631)\) → \((1,-i,e\left(\frac{7}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(41\)\(43\)
\( \chi_{ 1110 }(103, a) \) \(1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(1\)\(-i\)\(i\)\(e\left(\frac{1}{6}\right)\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1110 }(103,a) \;\) at \(\;a = \) e.g. 2