Properties

Label 147.121
Modulus $147$
Conductor $49$
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(147, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,38]))
 
pari: [g,chi] = znchar(Mod(121,147))
 

Basic properties

Modulus: \(147\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(23,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 147.m

\(\chi_{147}(4,\cdot)\) \(\chi_{147}(16,\cdot)\) \(\chi_{147}(25,\cdot)\) \(\chi_{147}(37,\cdot)\) \(\chi_{147}(46,\cdot)\) \(\chi_{147}(58,\cdot)\) \(\chi_{147}(88,\cdot)\) \(\chi_{147}(100,\cdot)\) \(\chi_{147}(109,\cdot)\) \(\chi_{147}(121,\cdot)\) \(\chi_{147}(130,\cdot)\) \(\chi_{147}(142,\cdot)\)

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: \(\Q(\zeta_{49})^+\)

Values on generators

\((50,52)\) → \((1,e\left(\frac{19}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{2}{3}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 147 }(121,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{147}(121,\cdot)) = \sum_{r\in \Z/147\Z} \chi_{147}(121,r) e\left(\frac{2r}{147}\right) = 4.0048166209+-5.7412057822i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 147 }(121,·),\chi_{ 147 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{147}(121,\cdot),\chi_{147}(1,\cdot)) = \sum_{r\in \Z/147\Z} \chi_{147}(121,r) \chi_{147}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 147 }(121,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{147}(121,·)) = \sum_{r \in \Z/147\Z} \chi_{147}(121,r) e\left(\frac{1 r + 2 r^{-1}}{147}\right) = -0.0 \)