Properties

Label 421.377
Modulus $421$
Conductor $421$
Order $5$
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(421, base_ring=CyclotomicField(10))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4]))
 
pari: [g,chi] = znchar(Mod(377,421))
 

Basic properties

Modulus: \(421\)
Conductor: \(421\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(5\)
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 421.e

\(\chi_{421}(252,\cdot)\) \(\chi_{421}(279,\cdot)\) \(\chi_{421}(354,\cdot)\) \(\chi_{421}(377,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 5.5.31414372081.1

Values on generators

\(2\) → \(e\left(\frac{2}{5}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 421 }(377,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{421}(377,\cdot)) = \sum_{r\in \Z/421\Z} \chi_{421}(377,r) e\left(\frac{2r}{421}\right) = 7.3545014418+19.1549290926i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 421 }(377,·),\chi_{ 421 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{421}(377,\cdot),\chi_{421}(1,\cdot)) = \sum_{r\in \Z/421\Z} \chi_{421}(377,r) \chi_{421}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 421 }(377,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{421}(377,·)) = \sum_{r \in \Z/421\Z} \chi_{421}(377,r) e\left(\frac{1 r + 2 r^{-1}}{421}\right) = 5.591422866+17.2086301041i \)