Properties

Label 407.338
Modulus $407$
Conductor $407$
Order $180$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(407, base_ring=CyclotomicField(180))
 
M = H._module
 
chi = DirichletCharacter(H, M([54,115]))
 
pari: [g,chi] = znchar(Mod(338,407))
 

Basic properties

Modulus: \(407\)
Conductor: \(407\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
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.bj

\(\chi_{407}(2,\cdot)\) \(\chi_{407}(13,\cdot)\) \(\chi_{407}(17,\cdot)\) \(\chi_{407}(18,\cdot)\) \(\chi_{407}(19,\cdot)\) \(\chi_{407}(24,\cdot)\) \(\chi_{407}(35,\cdot)\) \(\chi_{407}(39,\cdot)\) \(\chi_{407}(50,\cdot)\) \(\chi_{407}(52,\cdot)\) \(\chi_{407}(57,\cdot)\) \(\chi_{407}(61,\cdot)\) \(\chi_{407}(72,\cdot)\) \(\chi_{407}(79,\cdot)\) \(\chi_{407}(94,\cdot)\) \(\chi_{407}(96,\cdot)\) \(\chi_{407}(106,\cdot)\) \(\chi_{407}(116,\cdot)\) \(\chi_{407}(128,\cdot)\) \(\chi_{407}(129,\cdot)\) \(\chi_{407}(150,\cdot)\) \(\chi_{407}(161,\cdot)\) \(\chi_{407}(167,\cdot)\) \(\chi_{407}(172,\cdot)\) \(\chi_{407}(183,\cdot)\) \(\chi_{407}(200,\cdot)\) \(\chi_{407}(204,\cdot)\) \(\chi_{407}(205,\cdot)\) \(\chi_{407}(217,\cdot)\) \(\chi_{407}(227,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((112,298)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{23}{36}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 407 }(338, a) \) \(1\)\(1\)\(e\left(\frac{169}{180}\right)\)\(e\left(\frac{1}{90}\right)\)\(e\left(\frac{79}{90}\right)\)\(e\left(\frac{161}{180}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{49}{90}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{8}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 407 }(338,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 407 }(338,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 407 }(338,·),\chi_{ 407 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 407 }(338,·)) \;\) at \(\; a,b = \) e.g. 1,2