Properties

Label 975.341
Modulus $975$
Conductor $975$
Order $30$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,6,10]))
 
pari: [g,chi] = znchar(Mod(341,975))
 

Basic properties

Modulus: \(975\)
Conductor: \(975\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 975.cj

\(\chi_{975}(146,\cdot)\) \(\chi_{975}(191,\cdot)\) \(\chi_{975}(341,\cdot)\) \(\chi_{975}(386,\cdot)\) \(\chi_{975}(536,\cdot)\) \(\chi_{975}(581,\cdot)\) \(\chi_{975}(731,\cdot)\) \(\chi_{975}(971,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((326,352,301)\) → \((-1,e\left(\frac{1}{5}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(14\)\(16\)\(17\)\(19\)\(22\)
\( \chi_{ 975 }(341, a) \) \(-1\)\(1\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{1}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 975 }(341,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 975 }(341,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 975 }(341,·),\chi_{ 975 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 975 }(341,·)) \;\) at \(\; a,b = \) e.g. 1,2