Properties

Conductor 45
Order 12
Real No
Primitive No
Parity Even
Orbit Label 90.l

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(90)
sage: chi = H[77]
pari: [g,chi] = znchar(Mod(77,90))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 45
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 12
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = No
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Even
Orbit label = 90.l
Orbit index = 12

Galois orbit

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

\(\chi_{90}(23,\cdot)\) \(\chi_{90}(47,\cdot)\) \(\chi_{90}(77,\cdot)\) \(\chi_{90}(83,\cdot)\)

Inducing primitive character

\(\chi_{45}(32,\cdot)\)

Values on generators

\((11,37)\) → \((e\left(\frac{5}{6}\right),i)\)

Values

-117111317192329313741
\(1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{12}\right)\)\(-i\)\(-1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(i\)\(e\left(\frac{1}{6}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{12})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 90 }(77,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{90}(77,\cdot)) = \sum_{r\in \Z/90\Z} \chi_{90}(77,r) e\left(\frac{r}{45}\right) = 5.0072399573+4.4640282269i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 90 }(77,·),\chi_{ 90 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{90}(77,\cdot),\chi_{90}(1,\cdot)) = \sum_{r\in \Z/90\Z} \chi_{90}(77,r) \chi_{90}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 90 }(77,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{90}(77,·)) = \sum_{r \in \Z/90\Z} \chi_{90}(77,r) e\left(\frac{1 r + 2 r^{-1}}{90}\right) = 7.7950041702+2.0886650724i \)