Properties

Conductor 675
Order 45
Real No
Primitive Yes
Parity Even
Orbit Label 675.bc

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(675)
sage: chi = H[211]
pari: [g,chi] = znchar(Mod(211,675))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 675
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 45
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 = 675.bc
Orbit index = 29

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 ]

\(\chi_{675}(16,\cdot)\) \(\chi_{675}(31,\cdot)\) \(\chi_{675}(61,\cdot)\) \(\chi_{675}(106,\cdot)\) \(\chi_{675}(121,\cdot)\) \(\chi_{675}(166,\cdot)\) \(\chi_{675}(196,\cdot)\) \(\chi_{675}(211,\cdot)\) \(\chi_{675}(241,\cdot)\) \(\chi_{675}(256,\cdot)\) \(\chi_{675}(286,\cdot)\) \(\chi_{675}(331,\cdot)\) \(\chi_{675}(346,\cdot)\) \(\chi_{675}(391,\cdot)\) \(\chi_{675}(421,\cdot)\) \(\chi_{675}(436,\cdot)\) \(\chi_{675}(466,\cdot)\) \(\chi_{675}(481,\cdot)\) \(\chi_{675}(511,\cdot)\) \(\chi_{675}(556,\cdot)\) \(\chi_{675}(571,\cdot)\) \(\chi_{675}(616,\cdot)\) \(\chi_{675}(646,\cdot)\) \(\chi_{675}(661,\cdot)\)

Values on generators

\((326,352)\) → \((e\left(\frac{7}{9}\right),e\left(\frac{4}{5}\right))\)

Values

-112478111314161719
\(1\)\(1\)\(e\left(\frac{26}{45}\right)\)\(e\left(\frac{7}{45}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{19}{45}\right)\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{11}{15}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{45})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 675 }(211,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{675}(211,\cdot)) = \sum_{r\in \Z/675\Z} \chi_{675}(211,r) e\left(\frac{2r}{675}\right) = -25.9553704855+-1.1483653426i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 675 }(211,·),\chi_{ 675 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{675}(211,\cdot),\chi_{675}(1,\cdot)) = \sum_{r\in \Z/675\Z} \chi_{675}(211,r) \chi_{675}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 675 }(211,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{675}(211,·)) = \sum_{r \in \Z/675\Z} \chi_{675}(211,r) e\left(\frac{1 r + 2 r^{-1}}{675}\right) = -0.0 \)