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Properties

Label 3675.1832
Modulus $3675$
Conductor $105$
Order $12$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3675, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([6,3,10]))
 
Copy content pari:[g,chi] = znchar(Mod(1832,3675))
 

Basic properties

Modulus: \(3675\)
Conductor: \(105\)
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 \(\chi_{105}(47,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 3675.bf

\(\chi_{3675}(68,\cdot)\) \(\chi_{3675}(668,\cdot)\) \(\chi_{3675}(1832,\cdot)\) \(\chi_{3675}(2432,\cdot)\)

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: 12.0.402196204142578125.1

Values on generators

\((1226,1177,2551)\) → \((-1,i,e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 3675 }(1832, a) \) \(-1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(i\)\(e\left(\frac{5}{6}\right)\)\(i\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(i\)\(e\left(\frac{11}{12}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 3675 }(1832,a) \;\) at \(\;a = \) e.g. 2