Properties

Label 450.17
Modulus $450$
Conductor $75$
Order $20$
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(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,13]))
 
pari: [g,chi] = znchar(Mod(17,450))
 

Basic properties

Modulus: \(450\)
Conductor: \(75\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{75}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 450.s

\(\chi_{450}(17,\cdot)\) \(\chi_{450}(53,\cdot)\) \(\chi_{450}(197,\cdot)\) \(\chi_{450}(233,\cdot)\) \(\chi_{450}(287,\cdot)\) \(\chi_{450}(323,\cdot)\) \(\chi_{450}(377,\cdot)\) \(\chi_{450}(413,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{75})^+\)

Values on generators

\((101,127)\) → \((-1,e\left(\frac{13}{20}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(i\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{1}{10}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 450 }(17,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{450}(17,\cdot)) = \sum_{r\in \Z/450\Z} \chi_{450}(17,r) e\left(\frac{r}{225}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 450 }(17,·),\chi_{ 450 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{450}(17,\cdot),\chi_{450}(1,\cdot)) = \sum_{r\in \Z/450\Z} \chi_{450}(17,r) \chi_{450}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 450 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{450}(17,·)) = \sum_{r \in \Z/450\Z} \chi_{450}(17,r) e\left(\frac{1 r + 2 r^{-1}}{450}\right) = 0.0 \)