Properties

Conductor 127
Order 63
Real No
Primitive Yes
Parity Even
Orbit Label 127.k

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(127)
sage: chi = H[15]
pari: [g,chi] = znchar(Mod(15,127))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 127
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 63
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 = 127.k
Orbit index = 11

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 ]

\(\chi_{127}(9,\cdot)\) \(\chi_{127}(11,\cdot)\) \(\chi_{127}(13,\cdot)\) \(\chi_{127}(15,\cdot)\) \(\chi_{127}(17,\cdot)\) \(\chi_{127}(18,\cdot)\) \(\chi_{127}(21,\cdot)\) \(\chi_{127}(26,\cdot)\) \(\chi_{127}(30,\cdot)\) \(\chi_{127}(31,\cdot)\) \(\chi_{127}(34,\cdot)\) \(\chi_{127}(35,\cdot)\) \(\chi_{127}(36,\cdot)\) \(\chi_{127}(41,\cdot)\) \(\chi_{127}(42,\cdot)\) \(\chi_{127}(44,\cdot)\) \(\chi_{127}(49,\cdot)\) \(\chi_{127}(60,\cdot)\) \(\chi_{127}(62,\cdot)\) \(\chi_{127}(69,\cdot)\) \(\chi_{127}(70,\cdot)\) \(\chi_{127}(71,\cdot)\) \(\chi_{127}(72,\cdot)\) \(\chi_{127}(74,\cdot)\) \(\chi_{127}(79,\cdot)\) \(\chi_{127}(81,\cdot)\) \(\chi_{127}(82,\cdot)\) \(\chi_{127}(84,\cdot)\) \(\chi_{127}(88,\cdot)\) \(\chi_{127}(98,\cdot)\) ...

Values on generators

\(3\) → \(e\left(\frac{44}{63}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{44}{63}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{62}{63}\right)\)\(e\left(\frac{20}{63}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{25}{63}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{31}{63}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{63})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 127 }(15,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{127}(15,\cdot)) = \sum_{r\in \Z/127\Z} \chi_{127}(15,r) e\left(\frac{2r}{127}\right) = -7.5672097513+8.3508883707i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 127 }(15,·),\chi_{ 127 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{127}(15,\cdot),\chi_{127}(1,\cdot)) = \sum_{r\in \Z/127\Z} \chi_{127}(15,r) \chi_{127}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 127 }(15,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{127}(15,·)) = \sum_{r \in \Z/127\Z} \chi_{127}(15,r) e\left(\frac{1 r + 2 r^{-1}}{127}\right) = 8.9767435397+11.2564803602i \)