Properties

Label 99.85
Modulus $99$
Conductor $99$
Order $30$
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(99, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,9]))
 
pari: [g,chi] = znchar(Mod(85,99))
 

Basic properties

Modulus: \(99\)
Conductor: \(99\)
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 99.o

\(\chi_{99}(7,\cdot)\) \(\chi_{99}(13,\cdot)\) \(\chi_{99}(40,\cdot)\) \(\chi_{99}(52,\cdot)\) \(\chi_{99}(61,\cdot)\) \(\chi_{99}(79,\cdot)\) \(\chi_{99}(85,\cdot)\) \(\chi_{99}(94,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: 30.0.159386923550435671074967363509984324121230045171.1

Values on generators

\((56,46)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{3}{10}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\(-1\)\(1\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{9}{10}\right)\)\(-1\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{7}{10}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 99 }(85,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{99}(85,\cdot)) = \sum_{r\in \Z/99\Z} \chi_{99}(85,r) e\left(\frac{2r}{99}\right) = 1.7305174735+9.7982299051i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 99 }(85,·),\chi_{ 99 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{99}(85,\cdot),\chi_{99}(1,\cdot)) = \sum_{r\in \Z/99\Z} \chi_{99}(85,r) \chi_{99}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 99 }(85,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{99}(85,·)) = \sum_{r \in \Z/99\Z} \chi_{99}(85,r) e\left(\frac{1 r + 2 r^{-1}}{99}\right) = 7.807108742+3.4759487613i \)