Properties

Conductor 311
Order 31
Real No
Primitive Yes
Parity Even
Orbit Label 311.e

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(311)
sage: chi = H[7]
pari: [g,chi] = znchar(Mod(7,311))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 311
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 31
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 = 311.e
Orbit index = 5

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 ]

\(\chi_{311}(7,\cdot)\) \(\chi_{311}(13,\cdot)\) \(\chi_{311}(15,\cdot)\) \(\chi_{311}(18,\cdot)\) \(\chi_{311}(20,\cdot)\) \(\chi_{311}(24,\cdot)\) \(\chi_{311}(32,\cdot)\) \(\chi_{311}(47,\cdot)\) \(\chi_{311}(49,\cdot)\) \(\chi_{311}(83,\cdot)\) \(\chi_{311}(89,\cdot)\) \(\chi_{311}(91,\cdot)\) \(\chi_{311}(105,\cdot)\) \(\chi_{311}(113,\cdot)\) \(\chi_{311}(121,\cdot)\) \(\chi_{311}(126,\cdot)\) \(\chi_{311}(140,\cdot)\) \(\chi_{311}(146,\cdot)\) \(\chi_{311}(168,\cdot)\) \(\chi_{311}(169,\cdot)\) \(\chi_{311}(195,\cdot)\) \(\chi_{311}(224,\cdot)\) \(\chi_{311}(225,\cdot)\) \(\chi_{311}(234,\cdot)\) \(\chi_{311}(243,\cdot)\) \(\chi_{311}(250,\cdot)\) \(\chi_{311}(260,\cdot)\) \(\chi_{311}(265,\cdot)\) \(\chi_{311}(270,\cdot)\) \(\chi_{311}(300,\cdot)\)

Values on generators

\(17\) → \(e\left(\frac{14}{31}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{29}{31}\right)\)\(e\left(\frac{2}{31}\right)\)\(e\left(\frac{27}{31}\right)\)\(e\left(\frac{16}{31}\right)\)\(1\)\(e\left(\frac{7}{31}\right)\)\(e\left(\frac{25}{31}\right)\)\(e\left(\frac{4}{31}\right)\)\(e\left(\frac{14}{31}\right)\)\(e\left(\frac{30}{31}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{31})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 311 }(7,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{311}(7,\cdot)) = \sum_{r\in \Z/311\Z} \chi_{311}(7,r) e\left(\frac{2r}{311}\right) = 16.6937830618+5.6848577013i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 311 }(7,·),\chi_{ 311 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{311}(7,\cdot),\chi_{311}(1,\cdot)) = \sum_{r\in \Z/311\Z} \chi_{311}(7,r) \chi_{311}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 311 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{311}(7,·)) = \sum_{r \in \Z/311\Z} \chi_{311}(7,r) e\left(\frac{1 r + 2 r^{-1}}{311}\right) = 1.7817898884+-0.3661671303i \)