Properties

Label 407.234
Modulus $407$
Conductor $407$
Order $45$
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(407, base_ring=CyclotomicField(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([72,70]))
 
pari: [g,chi] = znchar(Mod(234,407))
 

Basic properties

Modulus: \(407\)
Conductor: \(407\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(45\)
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.bc

\(\chi_{407}(9,\cdot)\) \(\chi_{407}(16,\cdot)\) \(\chi_{407}(49,\cdot)\) \(\chi_{407}(53,\cdot)\) \(\chi_{407}(70,\cdot)\) \(\chi_{407}(71,\cdot)\) \(\chi_{407}(81,\cdot)\) \(\chi_{407}(86,\cdot)\) \(\chi_{407}(108,\cdot)\) \(\chi_{407}(157,\cdot)\) \(\chi_{407}(181,\cdot)\) \(\chi_{407}(192,\cdot)\) \(\chi_{407}(201,\cdot)\) \(\chi_{407}(218,\cdot)\) \(\chi_{407}(229,\cdot)\) \(\chi_{407}(234,\cdot)\) \(\chi_{407}(256,\cdot)\) \(\chi_{407}(268,\cdot)\) \(\chi_{407}(312,\cdot)\) \(\chi_{407}(345,\cdot)\) \(\chi_{407}(366,\cdot)\) \(\chi_{407}(367,\cdot)\) \(\chi_{407}(377,\cdot)\) \(\chi_{407}(379,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: 45.45.16527441490674381067348948889146522048718287580785548225593089356497792068001085725808918264262962961.1

Values on generators

\((112,298)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{7}{9}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(1\)\(1\)\(e\left(\frac{26}{45}\right)\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{7}{45}\right)\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{11}{45}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{9}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 407 }(234,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{407}(234,\cdot)) = \sum_{r\in \Z/407\Z} \chi_{407}(234,r) e\left(\frac{2r}{407}\right) = 13.796147258+14.7195896966i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 407 }(234,·),\chi_{ 407 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{407}(234,\cdot),\chi_{407}(1,\cdot)) = \sum_{r\in \Z/407\Z} \chi_{407}(234,r) \chi_{407}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 407 }(234,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{407}(234,·)) = \sum_{r \in \Z/407\Z} \chi_{407}(234,r) e\left(\frac{1 r + 2 r^{-1}}{407}\right) = -0.0012582758+0.0050466688i \)