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Properties

Label 2205.1942
Modulus $2205$
Conductor $315$
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(2205, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([8,3,2]))
 
Copy content pari:[g,chi] = znchar(Mod(1942,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(315\)
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 \(\chi_{315}(52,\cdot)\)
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 2205.bt

\(\chi_{2205}(178,\cdot)\) \(\chi_{2205}(607,\cdot)\) \(\chi_{2205}(1048,\cdot)\) \(\chi_{2205}(1942,\cdot)\)

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: 12.12.23749283658415095703125.1

Values on generators

\((1226,442,1081)\) → \((e\left(\frac{2}{3}\right),i,e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 2205 }(1942, a) \) \(1\)\(1\)\(i\)\(-1\)\(-i\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{12}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2205 }(1942,a) \;\) at \(\;a = \) e.g. 2