Properties

Label 420.73
Modulus $420$
Conductor $35$
Order $12$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(420, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,9,2]))
 
pari: [g,chi] = znchar(Mod(73,420))
 

Basic properties

Modulus: \(420\)
Conductor: \(35\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{35}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 420.bo

\(\chi_{420}(73,\cdot)\) \(\chi_{420}(157,\cdot)\) \(\chi_{420}(313,\cdot)\) \(\chi_{420}(397,\cdot)\)

sage: chi.galois_orbit()
 
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: \(\Q(\zeta_{35})^+\)

Values on generators

\((211,281,337,241)\) → \((1,1,-i,e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 420 }(73, a) \) \(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(-i\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(-1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{12}\right)\)\(-1\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 420 }(73,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 420 }(73,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 420 }(73,·),\chi_{ 420 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 420 }(73,·)) \;\) at \(\; a,b = \) e.g. 1,2