Properties

Label 450.23
Modulus $450$
Conductor $225$
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(450, base_ring=CyclotomicField(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([50,33]))
 
pari: [g,chi] = znchar(Mod(23,450))
 

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 \(\chi_{225}(23,\cdot)\)
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()
 
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

\((101,127)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{11}{20}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{11}{30}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 450 }(23,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{450}(23,\cdot)) = \sum_{r\in \Z/450\Z} \chi_{450}(23,r) e\left(\frac{r}{225}\right) = 11.2862236334+9.8803419018i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 450 }(23,·),\chi_{ 450 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{450}(23,\cdot),\chi_{450}(1,\cdot)) = \sum_{r\in \Z/450\Z} \chi_{450}(23,r) \chi_{450}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 450 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{450}(23,·)) = \sum_{r \in \Z/450\Z} \chi_{450}(23,r) e\left(\frac{1 r + 2 r^{-1}}{450}\right) = 4.269752257+11.1230849136i \)