Properties

Label 525.73
Modulus $525$
Conductor $175$
Order $60$
Real no
Primitive no
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([0,33,10]))
 
pari: [g,chi] = znchar(Mod(73,525))
 

Basic properties

Modulus: \(525\)
Conductor: \(175\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{175}(73,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 525.bv

\(\chi_{525}(52,\cdot)\) \(\chi_{525}(73,\cdot)\) \(\chi_{525}(103,\cdot)\) \(\chi_{525}(178,\cdot)\) \(\chi_{525}(187,\cdot)\) \(\chi_{525}(208,\cdot)\) \(\chi_{525}(262,\cdot)\) \(\chi_{525}(283,\cdot)\) \(\chi_{525}(292,\cdot)\) \(\chi_{525}(313,\cdot)\) \(\chi_{525}(367,\cdot)\) \(\chi_{525}(388,\cdot)\) \(\chi_{525}(397,\cdot)\) \(\chi_{525}(472,\cdot)\) \(\chi_{525}(502,\cdot)\) \(\chi_{525}(523,\cdot)\)

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

\((176,127,451)\) → \((1,e\left(\frac{11}{20}\right),e\left(\frac{1}{6}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(1\)\(1\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{23}{60}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 525 }(73,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{525}(73,\cdot)) = \sum_{r\in \Z/525\Z} \chi_{525}(73,r) e\left(\frac{2r}{525}\right) = -12.7453014894+3.5436266656i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 525 }(73,·),\chi_{ 525 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{525}(73,\cdot),\chi_{525}(1,\cdot)) = \sum_{r\in \Z/525\Z} \chi_{525}(73,r) \chi_{525}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 525 }(73,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{525}(73,·)) = \sum_{r \in \Z/525\Z} \chi_{525}(73,r) e\left(\frac{1 r + 2 r^{-1}}{525}\right) = 0.0 \)