Properties

Label 900.287
Modulus $900$
Conductor $300$
Order $20$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(900, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,10,9]))
 
pari: [g,chi] = znchar(Mod(287,900))
 

Basic properties

Modulus: \(900\)
Conductor: \(300\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{300}(287,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 900.bh

\(\chi_{900}(287,\cdot)\) \(\chi_{900}(323,\cdot)\) \(\chi_{900}(467,\cdot)\) \(\chi_{900}(503,\cdot)\) \(\chi_{900}(647,\cdot)\) \(\chi_{900}(683,\cdot)\) \(\chi_{900}(827,\cdot)\) \(\chi_{900}(863,\cdot)\)

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: 20.0.180203247070312500000000000000000000.1

Values on generators

\((451,101,577)\) → \((-1,-1,e\left(\frac{9}{20}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\(-1\)\(1\)\(-i\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{3}{10}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 900 }(287,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{900}(287,\cdot)) = \sum_{r\in \Z/900\Z} \chi_{900}(287,r) e\left(\frac{r}{450}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 900 }(287,·),\chi_{ 900 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{900}(287,\cdot),\chi_{900}(1,\cdot)) = \sum_{r\in \Z/900\Z} \chi_{900}(287,r) \chi_{900}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 900 }(287,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{900}(287,·)) = \sum_{r \in \Z/900\Z} \chi_{900}(287,r) e\left(\frac{1 r + 2 r^{-1}}{900}\right) = 0.0 \)