Properties

Label 450.353
Modulus $450$
Conductor $225$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,21]))
 
pari: [g,chi] = znchar(Mod(353,450))
 

Basic properties

Modulus: \(450\)
Conductor: \(225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{225}(128,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 450.w

\(\chi_{450}(23,\cdot)\) \(\chi_{450}(47,\cdot)\) \(\chi_{450}(77,\cdot)\) \(\chi_{450}(83,\cdot)\) \(\chi_{450}(113,\cdot)\) \(\chi_{450}(137,\cdot)\) \(\chi_{450}(167,\cdot)\) \(\chi_{450}(173,\cdot)\) \(\chi_{450}(203,\cdot)\) \(\chi_{450}(227,\cdot)\) \(\chi_{450}(263,\cdot)\) \(\chi_{450}(317,\cdot)\) \(\chi_{450}(347,\cdot)\) \(\chi_{450}(353,\cdot)\) \(\chi_{450}(383,\cdot)\) \(\chi_{450}(437,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((101,127)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{7}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 450 }(353, a) \) \(1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{7}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 450 }(353,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 450 }(353,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 450 }(353,·),\chi_{ 450 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 450 }(353,·)) \;\) at \(\; a,b = \) e.g. 1,2