Properties

Label 546.137
Modulus $546$
Conductor $273$
Order $12$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(546, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,8,11]))
 
pari: [g,chi] = znchar(Mod(137,546))
 

Basic properties

Modulus: \(546\)
Conductor: \(273\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{273}(137,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 546.ch

\(\chi_{546}(137,\cdot)\) \(\chi_{546}(275,\cdot)\) \(\chi_{546}(305,\cdot)\) \(\chi_{546}(401,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.12.7531625615112866003373.1

Values on generators

\((365,157,379)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{11}{12}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(1\)\(e\left(\frac{11}{12}\right)\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(-i\)\(e\left(\frac{5}{12}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 546 }(137,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{546}(137,\cdot)) = \sum_{r\in \Z/546\Z} \chi_{546}(137,r) e\left(\frac{r}{273}\right) = 7.7178390907+-14.6094133958i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 546 }(137,·),\chi_{ 546 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{546}(137,\cdot),\chi_{546}(1,\cdot)) = \sum_{r\in \Z/546\Z} \chi_{546}(137,r) \chi_{546}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 546 }(137,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{546}(137,·)) = \sum_{r \in \Z/546\Z} \chi_{546}(137,r) e\left(\frac{1 r + 2 r^{-1}}{546}\right) = -0.0 \)