Properties

Label 129.68
Modulus $129$
Conductor $129$
Order $42$
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(129, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,8]))
 
pari: [g,chi] = znchar(Mod(68,129))
 

Basic properties

Modulus: \(129\)
Conductor: \(129\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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 129.o

\(\chi_{129}(14,\cdot)\) \(\chi_{129}(17,\cdot)\) \(\chi_{129}(23,\cdot)\) \(\chi_{129}(38,\cdot)\) \(\chi_{129}(53,\cdot)\) \(\chi_{129}(56,\cdot)\) \(\chi_{129}(68,\cdot)\) \(\chi_{129}(74,\cdot)\) \(\chi_{129}(83,\cdot)\) \(\chi_{129}(95,\cdot)\) \(\chi_{129}(101,\cdot)\) \(\chi_{129}(110,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.0.2281836760183646137444154412268560109828024514076489472840222217265158917203.1

Values on generators

\((44,46)\) → \((-1,e\left(\frac{4}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{4}{7}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 129 }(68,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{129}(68,\cdot)) = \sum_{r\in \Z/129\Z} \chi_{129}(68,r) e\left(\frac{2r}{129}\right) = 11.2448833236+1.5976855258i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 129 }(68,·),\chi_{ 129 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{129}(68,\cdot),\chi_{129}(1,\cdot)) = \sum_{r\in \Z/129\Z} \chi_{129}(68,r) \chi_{129}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 129 }(68,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{129}(68,·)) = \sum_{r \in \Z/129\Z} \chi_{129}(68,r) e\left(\frac{1 r + 2 r^{-1}}{129}\right) = 0.0 \)