Properties

Label 225.113
Modulus $225$
Conductor $225$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(225, base_ring=CyclotomicField(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([50,57]))
 
pari: [g,chi] = znchar(Mod(113,225))
 

Basic properties

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

Galois orbit 225.w

\(\chi_{225}(2,\cdot)\) \(\chi_{225}(23,\cdot)\) \(\chi_{225}(38,\cdot)\) \(\chi_{225}(47,\cdot)\) \(\chi_{225}(77,\cdot)\) \(\chi_{225}(83,\cdot)\) \(\chi_{225}(92,\cdot)\) \(\chi_{225}(113,\cdot)\) \(\chi_{225}(122,\cdot)\) \(\chi_{225}(128,\cdot)\) \(\chi_{225}(137,\cdot)\) \(\chi_{225}(158,\cdot)\) \(\chi_{225}(167,\cdot)\) \(\chi_{225}(173,\cdot)\) \(\chi_{225}(203,\cdot)\) \(\chi_{225}(212,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ 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

\((101,127)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{19}{20}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 225 }(113, a) \) \(1\)\(1\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{1}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 225 }(113,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 225 }(113,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 225 }(113,·),\chi_{ 225 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 225 }(113,·)) \;\) at \(\; a,b = \) e.g. 1,2