Properties

Label 129.25
Modulus $129$
Conductor $43$
Order $21$
Real no
Primitive no
Minimal yes
Parity even

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([0,8]))
 
pari: [g,chi] = znchar(Mod(25,129))
 

Basic properties

Modulus: \(129\)
Conductor: \(43\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{43}(25,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 129.m

\(\chi_{129}(10,\cdot)\) \(\chi_{129}(13,\cdot)\) \(\chi_{129}(25,\cdot)\) \(\chi_{129}(31,\cdot)\) \(\chi_{129}(40,\cdot)\) \(\chi_{129}(52,\cdot)\) \(\chi_{129}(58,\cdot)\) \(\chi_{129}(67,\cdot)\) \(\chi_{129}(100,\cdot)\) \(\chi_{129}(103,\cdot)\) \(\chi_{129}(109,\cdot)\) \(\chi_{129}(124,\cdot)\)

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: \(\Q(\zeta_{43})^+\)

Values on generators

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

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{4}{7}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 129 }(25,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{129}(25,\cdot)) = \sum_{r\in \Z/129\Z} \chi_{129}(25,r) e\left(\frac{2r}{129}\right) = 0.9224241684+-6.4922364139i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 129 }(25,·),\chi_{ 129 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{129}(25,\cdot),\chi_{129}(1,\cdot)) = \sum_{r\in \Z/129\Z} \chi_{129}(25,r) \chi_{129}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 129 }(25,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{129}(25,·)) = \sum_{r \in \Z/129\Z} \chi_{129}(25,r) e\left(\frac{1 r + 2 r^{-1}}{129}\right) = 12.1312487213+5.8421014786i \)