Properties

Modulus 74
Conductor 37
Order 36
Real no
Primitive no
Minimal yes
Parity odd
Orbit label 74.i

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(74)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(39,74))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 74
Conductor = 37
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 36
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 74.i
Orbit index = 9

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 ]
 

\(\chi_{74}(5,\cdot)\) \(\chi_{74}(13,\cdot)\) \(\chi_{74}(15,\cdot)\) \(\chi_{74}(17,\cdot)\) \(\chi_{74}(19,\cdot)\) \(\chi_{74}(35,\cdot)\) \(\chi_{74}(39,\cdot)\) \(\chi_{74}(55,\cdot)\) \(\chi_{74}(57,\cdot)\) \(\chi_{74}(59,\cdot)\) \(\chi_{74}(61,\cdot)\) \(\chi_{74}(69,\cdot)\)

Values on generators

\(39\) → \(e\left(\frac{1}{36}\right)\)

Values

-113579111315171921
\(-1\)\(1\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{11}{18}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{36})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 74 }(39,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{74}(39,\cdot)) = \sum_{r\in \Z/74\Z} \chi_{74}(39,r) e\left(\frac{r}{37}\right) = 5.2235050658+3.1168886454i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 74 }(39,·),\chi_{ 74 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{74}(39,\cdot),\chi_{74}(1,\cdot)) = \sum_{r\in \Z/74\Z} \chi_{74}(39,r) \chi_{74}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 74 }(39,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{74}(39,·)) = \sum_{r \in \Z/74\Z} \chi_{74}(39,r) e\left(\frac{1 r + 2 r^{-1}}{74}\right) = 0.0272322491+-0.3112660317i \)