Properties

Label 225.103
Modulus $225$
Conductor $225$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

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([20,21]))
 
pari: [g,chi] = znchar(Mod(103,225))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 225.x

\(\chi_{225}(13,\cdot)\) \(\chi_{225}(22,\cdot)\) \(\chi_{225}(52,\cdot)\) \(\chi_{225}(58,\cdot)\) \(\chi_{225}(67,\cdot)\) \(\chi_{225}(88,\cdot)\) \(\chi_{225}(97,\cdot)\) \(\chi_{225}(103,\cdot)\) \(\chi_{225}(112,\cdot)\) \(\chi_{225}(133,\cdot)\) \(\chi_{225}(142,\cdot)\) \(\chi_{225}(148,\cdot)\) \(\chi_{225}(178,\cdot)\) \(\chi_{225}(187,\cdot)\) \(\chi_{225}(202,\cdot)\) \(\chi_{225}(223,\cdot)\)

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

\((101,127)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{20}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\(-1\)\(1\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{3}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 225 }(103,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 225 }(103,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 225 }(103,·),\chi_{ 225 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 225 }(103,·)) \;\) at \(\; a,b = \) e.g. 1,2