Properties

Label 97.68
Modulus $97$
Conductor $97$
Order $96$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(97, base_ring=CyclotomicField(96))
 
M = H._module
 
chi = DirichletCharacter(H, M([61]))
 
pari: [g,chi] = znchar(Mod(68,97))
 

Basic properties

Modulus: \(97\)
Conductor: \(97\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(96\)
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 97.l

\(\chi_{97}(5,\cdot)\) \(\chi_{97}(7,\cdot)\) \(\chi_{97}(10,\cdot)\) \(\chi_{97}(13,\cdot)\) \(\chi_{97}(14,\cdot)\) \(\chi_{97}(15,\cdot)\) \(\chi_{97}(17,\cdot)\) \(\chi_{97}(21,\cdot)\) \(\chi_{97}(23,\cdot)\) \(\chi_{97}(26,\cdot)\) \(\chi_{97}(29,\cdot)\) \(\chi_{97}(37,\cdot)\) \(\chi_{97}(38,\cdot)\) \(\chi_{97}(39,\cdot)\) \(\chi_{97}(40,\cdot)\) \(\chi_{97}(41,\cdot)\) \(\chi_{97}(56,\cdot)\) \(\chi_{97}(57,\cdot)\) \(\chi_{97}(58,\cdot)\) \(\chi_{97}(59,\cdot)\) \(\chi_{97}(60,\cdot)\) \(\chi_{97}(68,\cdot)\) \(\chi_{97}(71,\cdot)\) \(\chi_{97}(74,\cdot)\) \(\chi_{97}(76,\cdot)\) \(\chi_{97}(80,\cdot)\) \(\chi_{97}(82,\cdot)\) \(\chi_{97}(83,\cdot)\) \(\chi_{97}(84,\cdot)\) \(\chi_{97}(87,\cdot)\) ...

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: $\Q(\zeta_{96})$
Fixed field: Number field defined by a degree 96 polynomial

Values on generators

\(5\) → \(e\left(\frac{61}{96}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 97 }(68, a) \) \(-1\)\(1\)\(e\left(\frac{29}{48}\right)\)\(e\left(\frac{23}{48}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{61}{96}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{67}{96}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{23}{96}\right)\)\(e\left(\frac{31}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 97 }(68,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 97 }(68,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 97 }(68,·),\chi_{ 97 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 97 }(68,·)) \;\) at \(\; a,b = \) e.g. 1,2