Properties

Label 90.43
Modulus $90$
Conductor $45$
Order $12$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([8,9]))
 
pari: [g,chi] = znchar(Mod(43,90))
 

Basic properties

Modulus: \(90\)
Conductor: \(45\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{45}(43,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 90.k

\(\chi_{90}(7,\cdot)\) \(\chi_{90}(13,\cdot)\) \(\chi_{90}(43,\cdot)\) \(\chi_{90}(67,\cdot)\)

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: 12.0.84075626953125.1

Values on generators

\((11,37)\) → \((e\left(\frac{2}{3}\right),-i)\)

Values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 90 }(43, a) \) \(-1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(-i\)\(-1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(-i\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 90 }(43,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 90 }(43,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 90 }(43,·),\chi_{ 90 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 90 }(43,·)) \;\) at \(\; a,b = \) e.g. 1,2