Properties

Conductor 67
Order 33
Real No
Primitive Yes
Parity Even
Orbit Label 67.g

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(67)
sage: chi = H[10]
pari: [g,chi] = znchar(Mod(10,67))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 67
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 33
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = Yes
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Even
Orbit label = 67.g
Orbit index = 7

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 ]

\(\chi_{67}(4,\cdot)\) \(\chi_{67}(6,\cdot)\) \(\chi_{67}(10,\cdot)\) \(\chi_{67}(16,\cdot)\) \(\chi_{67}(17,\cdot)\) \(\chi_{67}(19,\cdot)\) \(\chi_{67}(21,\cdot)\) \(\chi_{67}(23,\cdot)\) \(\chi_{67}(26,\cdot)\) \(\chi_{67}(33,\cdot)\) \(\chi_{67}(35,\cdot)\) \(\chi_{67}(36,\cdot)\) \(\chi_{67}(39,\cdot)\) \(\chi_{67}(47,\cdot)\) \(\chi_{67}(49,\cdot)\) \(\chi_{67}(54,\cdot)\) \(\chi_{67}(55,\cdot)\) \(\chi_{67}(56,\cdot)\) \(\chi_{67}(60,\cdot)\) \(\chi_{67}(65,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{8}{33}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{10}{33}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{33})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 67 }(10,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{67}(10,\cdot)) = \sum_{r\in \Z/67\Z} \chi_{67}(10,r) e\left(\frac{2r}{67}\right) = -8.0917024982+1.234645974i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 67 }(10,·),\chi_{ 67 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{67}(10,\cdot),\chi_{67}(1,\cdot)) = \sum_{r\in \Z/67\Z} \chi_{67}(10,r) \chi_{67}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 67 }(10,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{67}(10,·)) = \sum_{r \in \Z/67\Z} \chi_{67}(10,r) e\left(\frac{1 r + 2 r^{-1}}{67}\right) = 7.0178423385+6.691499154i \)