Properties

Label 97.28
Modulus $97$
Conductor $97$
Order $32$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(97)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(28,97))
 

Basic properties

Modulus: \(97\)
Conductor: \(97\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
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.j

\(\chi_{97}(19,\cdot)\) \(\chi_{97}(20,\cdot)\) \(\chi_{97}(28,\cdot)\) \(\chi_{97}(30,\cdot)\) \(\chi_{97}(34,\cdot)\) \(\chi_{97}(42,\cdot)\) \(\chi_{97}(45,\cdot)\) \(\chi_{97}(46,\cdot)\) \(\chi_{97}(51,\cdot)\) \(\chi_{97}(52,\cdot)\) \(\chi_{97}(55,\cdot)\) \(\chi_{97}(63,\cdot)\) \(\chi_{97}(67,\cdot)\) \(\chi_{97}(69,\cdot)\) \(\chi_{97}(77,\cdot)\) \(\chi_{97}(78,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\(5\) → \(e\left(\frac{1}{32}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{1}{32}\right)\)\(i\)\(e\left(\frac{31}{32}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{11}{16}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{32})\)
Fixed field: 32.0.38897685648686306961584993323026037365043690280067733586177953.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 97 }(28,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{97}(28,\cdot)) = \sum_{r\in \Z/97\Z} \chi_{97}(28,r) e\left(\frac{2r}{97}\right) = -8.6162383807+4.7707898893i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 97 }(28,·),\chi_{ 97 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{97}(28,\cdot),\chi_{97}(1,\cdot)) = \sum_{r\in \Z/97\Z} \chi_{97}(28,r) \chi_{97}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 97 }(28,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{97}(28,·)) = \sum_{r \in \Z/97\Z} \chi_{97}(28,r) e\left(\frac{1 r + 2 r^{-1}}{97}\right) = -1.266698677+6.3681242836i \)