Properties

Label 407.306
Modulus $407$
Conductor $407$
Order $15$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(407, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([18,20]))
 
pari: [g,chi] = znchar(Mod(306,407))
 

Basic properties

Modulus: \(407\)
Conductor: \(407\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
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 407.r

\(\chi_{407}(26,\cdot)\) \(\chi_{407}(47,\cdot)\) \(\chi_{407}(137,\cdot)\) \(\chi_{407}(158,\cdot)\) \(\chi_{407}(174,\cdot)\) \(\chi_{407}(269,\cdot)\) \(\chi_{407}(306,\cdot)\) \(\chi_{407}(322,\cdot)\)

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: 15.15.15091397646253318362996493129.1

Values on generators

\((112,298)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(1\)\(1\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{4}{15}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 407 }(306,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{407}(306,\cdot)) = \sum_{r\in \Z/407\Z} \chi_{407}(306,r) e\left(\frac{2r}{407}\right) = 20.0260102339+-2.4410887145i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 407 }(306,·),\chi_{ 407 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{407}(306,\cdot),\chi_{407}(1,\cdot)) = \sum_{r\in \Z/407\Z} \chi_{407}(306,r) \chi_{407}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 407 }(306,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{407}(306,·)) = \sum_{r \in \Z/407\Z} \chi_{407}(306,r) e\left(\frac{1 r + 2 r^{-1}}{407}\right) = -7.3536635949+-8.1670708183i \)