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Properties

Label 168.6.a.j
Level $168$
Weight $6$
Character orbit 168.a
Self dual yes
Analytic conductor $26.944$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [168,6,Mod(1,168)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("168.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 168.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,18,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.9444817286\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1129}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 282 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{1129}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 q^{3} - \beta q^{5} - 49 q^{7} + 81 q^{9} + (\beta + 50) q^{11} + ( - 8 \beta + 270) q^{13} - 9 \beta q^{15} + ( - 3 \beta + 576) q^{17} + (18 \beta + 1472) q^{19} - 441 q^{21} + (49 \beta + 1342) q^{23} + \cdots + (81 \beta + 4050) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 18 q^{3} - 98 q^{7} + 162 q^{9} + 100 q^{11} + 540 q^{13} + 1152 q^{17} + 2944 q^{19} - 882 q^{21} + 2684 q^{23} + 2782 q^{25} + 1458 q^{27} + 996 q^{29} + 2616 q^{31} + 900 q^{33} - 3492 q^{37} + 4860 q^{39}+ \cdots + 8100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
17.3003
−16.3003
0 9.00000 0 −67.2012 0 −49.0000 0 81.0000 0
1.2 0 9.00000 0 67.2012 0 −49.0000 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.6.a.j ✓ 2
3.b odd 2 1 504.6.a.o 2
4.b odd 2 1 336.6.a.t 2
12.b even 2 1 1008.6.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.6.a.j ✓ 2 1.a even 1 1 trivial
336.6.a.t 2 4.b odd 2 1
504.6.a.o 2 3.b odd 2 1
1008.6.a.bn 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 4516 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(168))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 4516 \) Copy content Toggle raw display
$7$ \( (T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 100T - 2016 \) Copy content Toggle raw display
$13$ \( T^{2} - 540T - 216124 \) Copy content Toggle raw display
$17$ \( T^{2} - 1152 T + 291132 \) Copy content Toggle raw display
$19$ \( T^{2} - 2944 T + 703600 \) Copy content Toggle raw display
$23$ \( T^{2} - 2684 T - 9041952 \) Copy content Toggle raw display
$29$ \( T^{2} - 996 T - 3816396 \) Copy content Toggle raw display
$31$ \( T^{2} - 2616 T - 15648640 \) Copy content Toggle raw display
$37$ \( T^{2} + 3492 T - 66389500 \) Copy content Toggle raw display
$41$ \( T^{2} - 17016 T + 41275340 \) Copy content Toggle raw display
$43$ \( T^{2} - 13640 T - 14254896 \) Copy content Toggle raw display
$47$ \( T^{2} - 11832 T - 256355200 \) Copy content Toggle raw display
$53$ \( T^{2} - 37548 T + 92341476 \) Copy content Toggle raw display
$59$ \( T^{2} + 6320 T - 649007184 \) Copy content Toggle raw display
$61$ \( T^{2} - 39764 T + 288192468 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 2746394832 \) Copy content Toggle raw display
$71$ \( T^{2} - 35460 T - 463415200 \) Copy content Toggle raw display
$73$ \( T^{2} - 64188 T + 240971252 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 2285442560 \) Copy content Toggle raw display
$83$ \( T^{2} + 5896 T - 101210672 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 5624489484 \) Copy content Toggle raw display
$97$ \( T^{2} + 11876 T - 480666060 \) Copy content Toggle raw display
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