Properties

Label 58.19
Modulus $58$
Conductor $29$
Order $28$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(58, base_ring=CyclotomicField(28))
 
M = H._module
 
chi = DirichletCharacter(H, M([9]))
 
pari: [g,chi] = znchar(Mod(19,58))
 

Basic properties

Modulus: \(58\)
Conductor: \(29\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{29}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 58.f

\(\chi_{58}(3,\cdot)\) \(\chi_{58}(11,\cdot)\) \(\chi_{58}(15,\cdot)\) \(\chi_{58}(19,\cdot)\) \(\chi_{58}(21,\cdot)\) \(\chi_{58}(27,\cdot)\) \(\chi_{58}(31,\cdot)\) \(\chi_{58}(37,\cdot)\) \(\chi_{58}(39,\cdot)\) \(\chi_{58}(43,\cdot)\) \(\chi_{58}(47,\cdot)\) \(\chi_{58}(55,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: Number field defined by a degree 28 polynomial

Values on generators

\(31\) → \(e\left(\frac{9}{28}\right)\)

Values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 58 }(19, a) \) \(-1\)\(1\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{19}{28}\right)\)\(-i\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{13}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 58 }(19,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 58 }(19,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 58 }(19,·),\chi_{ 58 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 58 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2