Properties

Label 405.152
Modulus $405$
Conductor $135$
Order $36$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([34,9]))
 
pari: [g,chi] = znchar(Mod(152,405))
 

Basic properties

Modulus: \(405\)
Conductor: \(135\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{135}(122,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 405.r

\(\chi_{405}(8,\cdot)\) \(\chi_{405}(17,\cdot)\) \(\chi_{405}(62,\cdot)\) \(\chi_{405}(98,\cdot)\) \(\chi_{405}(143,\cdot)\) \(\chi_{405}(152,\cdot)\) \(\chi_{405}(197,\cdot)\) \(\chi_{405}(233,\cdot)\) \(\chi_{405}(278,\cdot)\) \(\chi_{405}(287,\cdot)\) \(\chi_{405}(332,\cdot)\) \(\chi_{405}(368,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: \(\Q(\zeta_{135})^+\)

Values on generators

\((326,82)\) → \((e\left(\frac{17}{18}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 405 }(152, a) \) \(1\)\(1\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 405 }(152,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 405 }(152,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 405 }(152,·),\chi_{ 405 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 405 }(152,·)) \;\) at \(\; a,b = \) e.g. 1,2