Properties

Label 45120.5329
Modulus $45120$
Conductor $3760$
Order $92$
Real no
Primitive no
Minimal no
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45120, base_ring=CyclotomicField(92)) M = H._module chi = DirichletCharacter(H, M([0,69,0,46,24]))
 
Copy content pari:[g,chi] = znchar(Mod(5329,45120))
 

Basic properties

Modulus: \(45120\)
Conductor: \(3760\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(92\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3760}(2509,\cdot)\)
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 45120.gy

\(\chi_{45120}(49,\cdot)\) \(\chi_{45120}(529,\cdot)\) \(\chi_{45120}(1489,\cdot)\) \(\chi_{45120}(1969,\cdot)\) \(\chi_{45120}(3409,\cdot)\) \(\chi_{45120}(4849,\cdot)\) \(\chi_{45120}(5329,\cdot)\) \(\chi_{45120}(5809,\cdot)\) \(\chi_{45120}(7729,\cdot)\) \(\chi_{45120}(9169,\cdot)\) \(\chi_{45120}(9649,\cdot)\) \(\chi_{45120}(10129,\cdot)\) \(\chi_{45120}(10609,\cdot)\) \(\chi_{45120}(11569,\cdot)\) \(\chi_{45120}(12049,\cdot)\) \(\chi_{45120}(12529,\cdot)\) \(\chi_{45120}(13009,\cdot)\) \(\chi_{45120}(15889,\cdot)\) \(\chi_{45120}(18289,\cdot)\) \(\chi_{45120}(18769,\cdot)\) \(\chi_{45120}(19729,\cdot)\) \(\chi_{45120}(20689,\cdot)\) \(\chi_{45120}(22609,\cdot)\) \(\chi_{45120}(23089,\cdot)\) \(\chi_{45120}(24049,\cdot)\) \(\chi_{45120}(24529,\cdot)\) \(\chi_{45120}(25969,\cdot)\) \(\chi_{45120}(27409,\cdot)\) \(\chi_{45120}(27889,\cdot)\) \(\chi_{45120}(28369,\cdot)\) ...

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: $\Q(\zeta_{92})$
Fixed field: Number field defined by a degree 92 polynomial

Values on generators

\((43711,2821,15041,36097,18241)\) → \((1,-i,1,-1,e\left(\frac{6}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 45120 }(5329, a) \) \(1\)\(1\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{53}{92}\right)\)\(e\left(\frac{57}{92}\right)\)\(e\left(\frac{31}{46}\right)\)\(e\left(\frac{91}{92}\right)\)\(e\left(\frac{7}{23}\right)\)\(e\left(\frac{35}{92}\right)\)\(e\left(\frac{18}{23}\right)\)\(e\left(\frac{19}{92}\right)\)\(e\left(\frac{19}{46}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 45120 }(5329,a) \;\) at \(\;a = \) e.g. 2