Properties

Conductor 99
Order 30
Real No
Primitive Yes
Parity Odd
Orbit Label 99.n

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(99)
sage: chi = H[20]
pari: [g,chi] = znchar(Mod(20,99))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 99
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 30
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 = Odd
Orbit label = 99.n
Orbit index = 14

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 ]

\(\chi_{99}(5,\cdot)\) \(\chi_{99}(14,\cdot)\) \(\chi_{99}(20,\cdot)\) \(\chi_{99}(38,\cdot)\) \(\chi_{99}(47,\cdot)\) \(\chi_{99}(59,\cdot)\) \(\chi_{99}(86,\cdot)\) \(\chi_{99}(92,\cdot)\)

Values on generators

\((56,46)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{3}{5}\right))\)

Values

-11245781013141617
\(-1\)\(1\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{3}{10}\right)\)\(1\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{9}{10}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{15})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 99 }(20,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{99}(20,\cdot)) = \sum_{r\in \Z/99\Z} \chi_{99}(20,r) e\left(\frac{2r}{99}\right) = 0.2393067105+9.9469961445i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 99 }(20,·),\chi_{ 99 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{99}(20,\cdot),\chi_{99}(1,\cdot)) = \sum_{r\in \Z/99\Z} \chi_{99}(20,r) \chi_{99}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 99 }(20,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{99}(20,·)) = \sum_{r \in \Z/99\Z} \chi_{99}(20,r) e\left(\frac{1 r + 2 r^{-1}}{99}\right) = -8.6228160033+-9.5766073664i \)