Properties

Modulus 99
Conductor 99
Order 15
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 99.m

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(99)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,3]))
 
pari: [g,chi] = znchar(Mod(70,99))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 99
Conductor = 99
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 15
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 99.m
Orbit index = 13

Galois orbit

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

\(\chi_{99}(4,\cdot)\) \(\chi_{99}(16,\cdot)\) \(\chi_{99}(25,\cdot)\) \(\chi_{99}(31,\cdot)\) \(\chi_{99}(49,\cdot)\) \(\chi_{99}(58,\cdot)\) \(\chi_{99}(70,\cdot)\) \(\chi_{99}(97,\cdot)\)

Values on generators

\((56,46)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{5}\right))\)

Values

-11245781013141617
\(1\)\(1\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(1\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{4}{5}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{15})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 99 }(70,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{99}(70,\cdot)) = \sum_{r\in \Z/99\Z} \chi_{99}(70,r) e\left(\frac{2r}{99}\right) = 9.9393312949+-0.457922931i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 99 }(70,·),\chi_{ 99 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{99}(70,\cdot),\chi_{99}(1,\cdot)) = \sum_{r\in \Z/99\Z} \chi_{99}(70,r) \chi_{99}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 99 }(70,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{99}(70,·)) = \sum_{r \in \Z/99\Z} \chi_{99}(70,r) e\left(\frac{1 r + 2 r^{-1}}{99}\right) = 8.0387669469+-3.5790896393i \)