Properties

Label 525.2
Modulus $525$
Conductor $525$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(525)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([30,3,20]))
 
pari: [g,chi] = znchar(Mod(2,525))
 

Basic properties

Modulus: \(525\)
Conductor: \(525\)
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 525.bs

\(\chi_{525}(2,\cdot)\) \(\chi_{525}(23,\cdot)\) \(\chi_{525}(53,\cdot)\) \(\chi_{525}(128,\cdot)\) \(\chi_{525}(137,\cdot)\) \(\chi_{525}(158,\cdot)\) \(\chi_{525}(212,\cdot)\) \(\chi_{525}(233,\cdot)\) \(\chi_{525}(242,\cdot)\) \(\chi_{525}(263,\cdot)\) \(\chi_{525}(317,\cdot)\) \(\chi_{525}(338,\cdot)\) \(\chi_{525}(347,\cdot)\) \(\chi_{525}(422,\cdot)\) \(\chi_{525}(452,\cdot)\) \(\chi_{525}(473,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((176,127,451)\) → \((-1,e\left(\frac{1}{20}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(1\)\(1\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{43}{60}\right)\)
value at e.g. 2

Related number fields

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 525 }(2,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{525}(2,\cdot)) = \sum_{r\in \Z/525\Z} \chi_{525}(2,r) e\left(\frac{2r}{525}\right) = -7.5235232274+-21.6424720919i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 525 }(2,·),\chi_{ 525 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{525}(2,\cdot),\chi_{525}(1,\cdot)) = \sum_{r\in \Z/525\Z} \chi_{525}(2,r) \chi_{525}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 525 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{525}(2,·)) = \sum_{r \in \Z/525\Z} \chi_{525}(2,r) e\left(\frac{1 r + 2 r^{-1}}{525}\right) = 0.0 \)