Properties

Label 407.34
Modulus $407$
Conductor $37$
Order $9$
Real no
Primitive no
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(407, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,4]))
 
pari: [g,chi] = znchar(Mod(34,407))
 

Basic properties

Modulus: \(407\)
Conductor: \(37\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{37}(34,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 407.l

\(\chi_{407}(12,\cdot)\) \(\chi_{407}(34,\cdot)\) \(\chi_{407}(144,\cdot)\) \(\chi_{407}(155,\cdot)\) \(\chi_{407}(342,\cdot)\) \(\chi_{407}(386,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: 9.9.3512479453921.1

Values on generators

\((112,298)\) → \((1,e\left(\frac{2}{9}\right))\)

Values

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 407 }(34,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{407}(34,\cdot)) = \sum_{r\in \Z/407\Z} \chi_{407}(34,r) e\left(\frac{2r}{407}\right) = 5.5662079377+2.4530244994i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 407 }(34,·),\chi_{ 407 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{407}(34,\cdot),\chi_{407}(1,\cdot)) = \sum_{r\in \Z/407\Z} \chi_{407}(34,r) \chi_{407}(1,1-r) = -9 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 407 }(34,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{407}(34,·)) = \sum_{r \in \Z/407\Z} \chi_{407}(34,r) e\left(\frac{1 r + 2 r^{-1}}{407}\right) = 0.0256808926+0.0215488275i \)