Properties

Conductor 403
Order 60
Real No
Primitive Yes
Parity Even
Orbit Label 403.cd

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(403)
 
sage: chi = H[21]
 
pari: [g,chi] = znchar(Mod(21,403))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 403
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 60
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 = 403.cd
Orbit index = 56

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 ]
 

\(\chi_{403}(21,\cdot)\) \(\chi_{403}(34,\cdot)\) \(\chi_{403}(44,\cdot)\) \(\chi_{403}(73,\cdot)\) \(\chi_{403}(83,\cdot)\) \(\chi_{403}(86,\cdot)\) \(\chi_{403}(96,\cdot)\) \(\chi_{403}(135,\cdot)\) \(\chi_{403}(148,\cdot)\) \(\chi_{403}(177,\cdot)\) \(\chi_{403}(203,\cdot)\) \(\chi_{403}(229,\cdot)\) \(\chi_{403}(239,\cdot)\) \(\chi_{403}(265,\cdot)\) \(\chi_{403}(291,\cdot)\) \(\chi_{403}(385,\cdot)\)

Values on generators

\((249,313)\) → \((i,e\left(\frac{29}{30}\right))\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{59}{60}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{60})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 403 }(21,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{403}(21,\cdot)) = \sum_{r\in \Z/403\Z} \chi_{403}(21,r) e\left(\frac{2r}{403}\right) = -16.5090421919+11.4215378083i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 403 }(21,·),\chi_{ 403 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{403}(21,\cdot),\chi_{403}(1,\cdot)) = \sum_{r\in \Z/403\Z} \chi_{403}(21,r) \chi_{403}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 403 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{403}(21,·)) = \sum_{r \in \Z/403\Z} \chi_{403}(21,r) e\left(\frac{1 r + 2 r^{-1}}{403}\right) = 0.0323162854+0.2040369959i \)