Properties

Label 270.23
Modulus $270$
Conductor $135$
Order $36$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,27]))
 
pari: [g,chi] = znchar(Mod(23,270))
 

Basic properties

Modulus: \(270\)
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 \(\chi_{135}(23,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 270.r

\(\chi_{270}(23,\cdot)\) \(\chi_{270}(47,\cdot)\) \(\chi_{270}(77,\cdot)\) \(\chi_{270}(83,\cdot)\) \(\chi_{270}(113,\cdot)\) \(\chi_{270}(137,\cdot)\) \(\chi_{270}(167,\cdot)\) \(\chi_{270}(173,\cdot)\) \(\chi_{270}(203,\cdot)\) \(\chi_{270}(227,\cdot)\) \(\chi_{270}(257,\cdot)\) \(\chi_{270}(263,\cdot)\)

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

\((191,217)\) → \((e\left(\frac{11}{18}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 270 }(23, a) \) \(1\)\(1\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 270 }(23,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 270 }(23,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 270 }(23,·),\chi_{ 270 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 270 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2