Properties

Label 612.77
Modulus $612$
Conductor $153$
Order $24$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(612, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,20,3]))
 
pari: [g,chi] = znchar(Mod(77,612))
 

Basic properties

Modulus: \(612\)
Conductor: \(153\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{153}(77,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 612.bg

\(\chi_{612}(77,\cdot)\) \(\chi_{612}(185,\cdot)\) \(\chi_{612}(257,\cdot)\) \(\chi_{612}(281,\cdot)\) \(\chi_{612}(365,\cdot)\) \(\chi_{612}(389,\cdot)\) \(\chi_{612}(461,\cdot)\) \(\chi_{612}(569,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: 24.0.10370328622637411153913943764610276201876257.1

Values on generators

\((307,137,37)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{1}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 612 }(77, a) \) \(-1\)\(1\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{1}{6}\right)\)\(-i\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{19}{24}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 612 }(77,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 612 }(77,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 612 }(77,·),\chi_{ 612 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 612 }(77,·)) \;\) at \(\; a,b = \) e.g. 1,2