Properties

Label 300.23
Modulus $300$
Conductor $300$
Order $20$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(300, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,10,11]))
 
pari: [g,chi] = znchar(Mod(23,300))
 

Basic properties

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

Galois orbit 300.u

\(\chi_{300}(23,\cdot)\) \(\chi_{300}(47,\cdot)\) \(\chi_{300}(83,\cdot)\) \(\chi_{300}(167,\cdot)\) \(\chi_{300}(203,\cdot)\) \(\chi_{300}(227,\cdot)\) \(\chi_{300}(263,\cdot)\) \(\chi_{300}(287,\cdot)\)

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

\((151,101,277)\) → \((-1,-1,e\left(\frac{11}{20}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\(-1\)\(1\)\(i\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{7}{10}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: 20.0.180203247070312500000000000000000000.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 300 }(23,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{300}(23,\cdot)) = \sum_{r\in \Z/300\Z} \chi_{300}(23,r) e\left(\frac{r}{150}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 300 }(23,·),\chi_{ 300 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{300}(23,\cdot),\chi_{300}(1,\cdot)) = \sum_{r\in \Z/300\Z} \chi_{300}(23,r) \chi_{300}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 300 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{300}(23,·)) = \sum_{r \in \Z/300\Z} \chi_{300}(23,r) e\left(\frac{1 r + 2 r^{-1}}{300}\right) = -0.0 \)