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Properties

Label 30345.24142
Modulus $30345$
Conductor $595$
Order $8$
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(30345, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([0,2,4,7]))
 
Copy content pari:[g,chi] = znchar(Mod(24142,30345))
 

Basic properties

Modulus: \(30345\)
Conductor: \(595\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(8\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{595}(342,\cdot)\)
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 30345.ce

\(\chi_{30345}(9358,\cdot)\) \(\chi_{30345}(18052,\cdot)\) \(\chi_{30345}(20698,\cdot)\) \(\chi_{30345}(24142,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.8.15394111779265625.2

Values on generators

\((20231,24277,4336,28036)\) → \((1,i,-1,e\left(\frac{7}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(19\)\(22\)\(23\)\(26\)
\( \chi_{ 30345 }(24142, a) \) \(1\)\(1\)\(-1\)\(1\)\(-1\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(1\)\(i\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 30345 }(24142,a) \;\) at \(\;a = \) e.g. 2