Properties

Label 700.79
Modulus $700$
Conductor $700$
Order $30$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(700, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([15,3,10]))
 
pari: [g,chi] = znchar(Mod(79,700))
 

Basic properties

Modulus: \(700\)
Conductor: \(700\)
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 700.bp

\(\chi_{700}(39,\cdot)\) \(\chi_{700}(79,\cdot)\) \(\chi_{700}(179,\cdot)\) \(\chi_{700}(219,\cdot)\) \(\chi_{700}(319,\cdot)\) \(\chi_{700}(359,\cdot)\) \(\chi_{700}(459,\cdot)\) \(\chi_{700}(639,\cdot)\)

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: 30.0.38047917507940292835235595703125000000000000000000000000000000.1

Values on generators

\((351,477,101)\) → \((-1,e\left(\frac{1}{10}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{19}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 700 }(79,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 700 }(79,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 700 }(79,·),\chi_{ 700 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 700 }(79,·)) \;\) at \(\; a,b = \) e.g. 1,2