Properties

Label 403.24
Modulus $403$
Conductor $403$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(403, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([35,26]))
 
pari: [g,chi] = znchar(Mod(24,403))
 

Basic properties

Modulus: \(403\)
Conductor: \(403\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 403.cc

\(\chi_{403}(24,\cdot)\) \(\chi_{403}(84,\cdot)\) \(\chi_{403}(137,\cdot)\) \(\chi_{403}(141,\cdot)\) \(\chi_{403}(158,\cdot)\) \(\chi_{403}(167,\cdot)\) \(\chi_{403}(176,\cdot)\) \(\chi_{403}(210,\cdot)\) \(\chi_{403}(228,\cdot)\) \(\chi_{403}(292,\cdot)\) \(\chi_{403}(327,\cdot)\) \(\chi_{403}(331,\cdot)\) \(\chi_{403}(332,\cdot)\) \(\chi_{403}(353,\cdot)\) \(\chi_{403}(375,\cdot)\) \(\chi_{403}(383,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((249,313)\) → \((e\left(\frac{7}{12}\right),e\left(\frac{13}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 403 }(24, a) \) \(1\)\(1\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{11}{12}\right)\)\(-i\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 403 }(24,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 403 }(24,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 403 }(24,·),\chi_{ 403 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 403 }(24,·)) \;\) at \(\; a,b = \) e.g. 1,2