Properties

Label 712.49
Modulus $712$
Conductor $89$
Order $44$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(712, base_ring=CyclotomicField(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,37]))
 
pari: [g,chi] = znchar(Mod(49,712))
 

Basic properties

Modulus: \(712\)
Conductor: \(89\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{89}(49,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 712.ba

\(\chi_{712}(9,\cdot)\) \(\chi_{712}(17,\cdot)\) \(\chi_{712}(49,\cdot)\) \(\chi_{712}(129,\cdot)\) \(\chi_{712}(161,\cdot)\) \(\chi_{712}(169,\cdot)\) \(\chi_{712}(225,\cdot)\) \(\chi_{712}(249,\cdot)\) \(\chi_{712}(257,\cdot)\) \(\chi_{712}(361,\cdot)\) \(\chi_{712}(377,\cdot)\) \(\chi_{712}(409,\cdot)\) \(\chi_{712}(425,\cdot)\) \(\chi_{712}(465,\cdot)\) \(\chi_{712}(481,\cdot)\) \(\chi_{712}(513,\cdot)\) \(\chi_{712}(529,\cdot)\) \(\chi_{712}(633,\cdot)\) \(\chi_{712}(641,\cdot)\) \(\chi_{712}(665,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: \(\Q(\zeta_{89})^+\)

Values on generators

\((535,357,537)\) → \((1,1,e\left(\frac{37}{44}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{21}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 712 }(49,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 712 }(49,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 712 }(49,·),\chi_{ 712 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 712 }(49,·)) \;\) at \(\; a,b = \) e.g. 1,2