Properties

Label 407.213
Modulus $407$
Conductor $407$
Order $90$
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(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,85]))
 
pari: [g,chi] = znchar(Mod(213,407))
 

Basic properties

Modulus: \(407\)
Conductor: \(407\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
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.bg

\(\chi_{407}(3,\cdot)\) \(\chi_{407}(4,\cdot)\) \(\chi_{407}(25,\cdot)\) \(\chi_{407}(58,\cdot)\) \(\chi_{407}(102,\cdot)\) \(\chi_{407}(104,\cdot)\) \(\chi_{407}(114,\cdot)\) \(\chi_{407}(115,\cdot)\) \(\chi_{407}(136,\cdot)\) \(\chi_{407}(141,\cdot)\) \(\chi_{407}(152,\cdot)\) \(\chi_{407}(169,\cdot)\) \(\chi_{407}(213,\cdot)\) \(\chi_{407}(225,\cdot)\) \(\chi_{407}(247,\cdot)\) \(\chi_{407}(262,\cdot)\) \(\chi_{407}(280,\cdot)\) \(\chi_{407}(284,\cdot)\) \(\chi_{407}(289,\cdot)\) \(\chi_{407}(300,\cdot)\) \(\chi_{407}(317,\cdot)\) \(\chi_{407}(324,\cdot)\) \(\chi_{407}(361,\cdot)\) \(\chi_{407}(400,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((112,298)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{17}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 407 }(213, a) \) \(1\)\(1\)\(e\left(\frac{13}{90}\right)\)\(e\left(\frac{7}{45}\right)\)\(e\left(\frac{13}{45}\right)\)\(e\left(\frac{47}{90}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 407 }(213,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 407 }(213,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 407 }(213,·),\chi_{ 407 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 407 }(213,·)) \;\) at \(\; a,b = \) e.g. 1,2