Properties

Label 473.47
Modulus $473$
Conductor $473$
Order $35$
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(473, base_ring=CyclotomicField(70))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([56,20]))
 
pari: [g,chi] = znchar(Mod(47,473))
 

Basic properties

Modulus: \(473\)
Conductor: \(473\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(35\)
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 473.v

\(\chi_{473}(4,\cdot)\) \(\chi_{473}(16,\cdot)\) \(\chi_{473}(47,\cdot)\) \(\chi_{473}(59,\cdot)\) \(\chi_{473}(64,\cdot)\) \(\chi_{473}(97,\cdot)\) \(\chi_{473}(102,\cdot)\) \(\chi_{473}(170,\cdot)\) \(\chi_{473}(207,\cdot)\) \(\chi_{473}(213,\cdot)\) \(\chi_{473}(236,\cdot)\) \(\chi_{473}(256,\cdot)\) \(\chi_{473}(262,\cdot)\) \(\chi_{473}(269,\cdot)\) \(\chi_{473}(279,\cdot)\) \(\chi_{473}(312,\cdot)\) \(\chi_{473}(317,\cdot)\) \(\chi_{473}(322,\cdot)\) \(\chi_{473}(355,\cdot)\) \(\chi_{473}(379,\cdot)\) \(\chi_{473}(422,\cdot)\) \(\chi_{473}(434,\cdot)\) \(\chi_{473}(465,\cdot)\) \(\chi_{473}(471,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: 35.35.1455622807785591094953547155658149343464416905925495778881639129313848614446169.1

Values on generators

\((431,89)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{2}{7}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(1\)\(1\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{1}{35}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{19}{35}\right)\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{5}{7}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 473 }(47,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{473}(47,\cdot)) = \sum_{r\in \Z/473\Z} \chi_{473}(47,r) e\left(\frac{2r}{473}\right) = 21.5310436192+-3.0682504246i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 473 }(47,·),\chi_{ 473 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{473}(47,\cdot),\chi_{473}(1,\cdot)) = \sum_{r\in \Z/473\Z} \chi_{473}(47,r) \chi_{473}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 473 }(47,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{473}(47,·)) = \sum_{r \in \Z/473\Z} \chi_{473}(47,r) e\left(\frac{1 r + 2 r^{-1}}{473}\right) = 0.3355587734+-7.4717966266i \)