Properties

Label 29.21
Modulus $29$
Conductor $29$
Order $28$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(29, base_ring=CyclotomicField(28))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([17]))
 
pari: [g,chi] = znchar(Mod(21,29))
 

Basic properties

Modulus: \(29\)
Conductor: \(29\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 29.f

\(\chi_{29}(2,\cdot)\) \(\chi_{29}(3,\cdot)\) \(\chi_{29}(8,\cdot)\) \(\chi_{29}(10,\cdot)\) \(\chi_{29}(11,\cdot)\) \(\chi_{29}(14,\cdot)\) \(\chi_{29}(15,\cdot)\) \(\chi_{29}(18,\cdot)\) \(\chi_{29}(19,\cdot)\) \(\chi_{29}(21,\cdot)\) \(\chi_{29}(26,\cdot)\) \(\chi_{29}(27,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\(2\) → \(e\left(\frac{17}{28}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{5}{28}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{28})\)
Fixed field: \(\Q(\zeta_{29})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 29 }(21,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{29}(21,\cdot)) = \sum_{r\in \Z/29\Z} \chi_{29}(21,r) e\left(\frac{2r}{29}\right) = 0.2795934337+-5.377901776i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 29 }(21,·),\chi_{ 29 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{29}(21,\cdot),\chi_{29}(1,\cdot)) = \sum_{r\in \Z/29\Z} \chi_{29}(21,r) \chi_{29}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 29 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{29}(21,·)) = \sum_{r \in \Z/29\Z} \chi_{29}(21,r) e\left(\frac{1 r + 2 r^{-1}}{29}\right) = 6.1166906546+2.1403226297i \)