Properties

Label 93.53
Modulus $93$
Conductor $93$
Order $30$
Real no
Primitive yes
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(93, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([15,17]))
 
pari: [g,chi] = znchar(Mod(53,93))
 

Basic properties

Modulus: \(93\)
Conductor: \(93\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 93.p

\(\chi_{93}(11,\cdot)\) \(\chi_{93}(17,\cdot)\) \(\chi_{93}(44,\cdot)\) \(\chi_{93}(53,\cdot)\) \(\chi_{93}(65,\cdot)\) \(\chi_{93}(74,\cdot)\) \(\chi_{93}(83,\cdot)\) \(\chi_{93}(86,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: \(\Q(\zeta_{93})^+\)

Values on generators

\((32,34)\) → \((-1,e\left(\frac{17}{30}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{2}{5}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 93 }(53,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{93}(53,\cdot)) = \sum_{r\in \Z/93\Z} \chi_{93}(53,r) e\left(\frac{2r}{93}\right) = 9.6319668945+-0.4745669013i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 93 }(53,·),\chi_{ 93 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{93}(53,\cdot),\chi_{93}(1,\cdot)) = \sum_{r\in \Z/93\Z} \chi_{93}(53,r) \chi_{93}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 93 }(53,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{93}(53,·)) = \sum_{r \in \Z/93\Z} \chi_{93}(53,r) e\left(\frac{1 r + 2 r^{-1}}{93}\right) = -0.0 \)