Properties

Label 154.93
Modulus $154$
Conductor $77$
Order $15$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(154, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,12]))
 
pari: [g,chi] = znchar(Mod(93,154))
 

Basic properties

Modulus: \(154\)
Conductor: \(77\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{77}(16,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 154.m

\(\chi_{154}(9,\cdot)\) \(\chi_{154}(25,\cdot)\) \(\chi_{154}(37,\cdot)\) \(\chi_{154}(53,\cdot)\) \(\chi_{154}(81,\cdot)\) \(\chi_{154}(93,\cdot)\) \(\chi_{154}(135,\cdot)\) \(\chi_{154}(137,\cdot)\)

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: 15.15.886528337182930278529.1

Values on generators

\((45,57)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 154 }(93, a) \) \(1\)\(1\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{3}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 154 }(93,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 154 }(93,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 154 }(93,·),\chi_{ 154 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 154 }(93,·)) \;\) at \(\; a,b = \) e.g. 1,2