Properties

Label 915.7
Modulus $915$
Conductor $305$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(915, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,15,49]))
 
pari: [g,chi] = znchar(Mod(7,915))
 

Basic properties

Modulus: \(915\)
Conductor: \(305\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{305}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 915.db

\(\chi_{915}(7,\cdot)\) \(\chi_{915}(43,\cdot)\) \(\chi_{915}(112,\cdot)\) \(\chi_{915}(193,\cdot)\) \(\chi_{915}(262,\cdot)\) \(\chi_{915}(298,\cdot)\) \(\chi_{915}(307,\cdot)\) \(\chi_{915}(433,\cdot)\) \(\chi_{915}(457,\cdot)\) \(\chi_{915}(523,\cdot)\) \(\chi_{915}(532,\cdot)\) \(\chi_{915}(688,\cdot)\) \(\chi_{915}(697,\cdot)\) \(\chi_{915}(763,\cdot)\) \(\chi_{915}(787,\cdot)\) \(\chi_{915}(913,\cdot)\)

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

\((611,367,856)\) → \((1,i,e\left(\frac{49}{60}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 915 }(7, a) \) \(1\)\(1\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(i\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{11}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 915 }(7,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 915 }(7,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 915 }(7,·),\chi_{ 915 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 915 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2