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Properties

Label 8649.2.a.bh
Level $8649$
Weight $2$
Character orbit 8649.a
Self dual yes
Analytic conductor $69.063$
Analytic rank $0$
Dimension $8$
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] = [8649,2,Mod(1,8649)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8649, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8649.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 8649 = 3^{2} \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8649.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-1,0,13,-3,0,-1,-9,0,6,5,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(69.0626127082\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 14x^{6} + 10x^{5} + 62x^{4} - 31x^{3} - 82x^{2} + 42x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 93)
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} - \beta_{3} q^{5} + ( - \beta_{5} + \beta_{3} - 1) q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{8} + (\beta_{4} - \beta_{3} + \beta_{2} + 2) q^{10} + ( - \beta_{7} - \beta_{5}) q^{11}+ \cdots + (\beta_{7} + \beta_{6} - 5 \beta_{5} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + 13 q^{4} - 3 q^{5} - q^{7} - 9 q^{8} + 6 q^{10} + 5 q^{11} - 4 q^{13} - 3 q^{14} + 19 q^{16} - 2 q^{17} - 7 q^{19} - q^{20} + 9 q^{23} + 15 q^{25} + 8 q^{26} - 3 q^{28} + q^{29} - 54 q^{32}+ \cdots + 41 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 14x^{6} + 10x^{5} + 62x^{4} - 31x^{3} - 82x^{2} + 42x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 7\nu^{2} - 2\nu + 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - \nu^{6} - 11\nu^{5} + 7\nu^{4} + 35\nu^{3} - 10\nu^{2} - 25\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - \nu^{5} - 9\nu^{4} + 7\nu^{3} + 19\nu^{2} - 12\nu - 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + \nu^{6} + 17\nu^{5} - 7\nu^{4} - 83\nu^{3} - 2\nu^{2} + 91\nu ) / 6 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 7\beta_{2} + 2\beta _1 + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + \beta_{5} + 8\beta_{3} + 10\beta_{2} + 29\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} + 2\beta_{6} + \beta_{5} + 9\beta_{4} + \beta_{3} + 47\beta_{2} + 24\beta _1 + 143 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12\beta_{7} + 2\beta_{6} + 18\beta_{5} + 2\beta_{4} + 54\beta_{3} + 83\beta_{2} + 179\beta _1 + 163 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
2.71677
2.60827
1.19735
0.773712
−0.165325
−1.67120
−2.13076
−2.32882
−2.71677 0 5.38085 −2.08737 0 0.469339 −9.18499 0 5.67092
1.2 −2.60827 0 4.80309 −0.899861 0 1.51790 −7.31122 0 2.34708
1.3 −1.19735 0 −0.566358 2.70381 0 −4.32185 3.07282 0 −3.23741
1.4 −0.773712 0 −1.40137 1.00402 0 −0.385990 2.63168 0 −0.776825
1.5 0.165325 0 −1.97267 −3.79477 0 2.17674 −0.656780 0 −0.627369
1.6 1.67120 0 0.792899 −3.89560 0 4.51364 −2.01730 0 −6.51032
1.7 2.13076 0 2.54014 0.560297 0 −2.17833 1.15092 0 1.19386
1.8 2.32882 0 3.42341 3.40947 0 −2.79144 3.31488 0 7.94006
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(31\) \( +1 \)

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 8649.2.a.bh 8
3.b odd 2 1 2883.2.a.p 8
31.b odd 2 1 8649.2.a.bg 8
31.d even 5 2 279.2.i.c 16
93.c even 2 1 2883.2.a.o 8
93.l odd 10 2 93.2.f.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.2.f.b ✓ 16 93.l odd 10 2
279.2.i.c 16 31.d even 5 2
2883.2.a.o 8 93.c even 2 1
2883.2.a.p 8 3.b odd 2 1
8649.2.a.bg 8 31.b odd 2 1
8649.2.a.bh 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8649))\):

\( T_{2}^{8} + T_{2}^{7} - 14T_{2}^{6} - 10T_{2}^{5} + 62T_{2}^{4} + 31T_{2}^{3} - 82T_{2}^{2} - 42T_{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{8} + 3T_{5}^{7} - 23T_{5}^{6} - 57T_{5}^{5} + 159T_{5}^{4} + 268T_{5}^{3} - 304T_{5}^{2} - 192T_{5} + 144 \) Copy content Toggle raw display
\( T_{7}^{8} + T_{7}^{7} - 29T_{7}^{6} - 27T_{7}^{5} + 204T_{7}^{4} + 103T_{7}^{3} - 437T_{7}^{2} + 12T_{7} + 71 \) Copy content Toggle raw display
\( T_{11}^{8} - 5T_{11}^{7} - 28T_{11}^{6} + 108T_{11}^{5} + 217T_{11}^{4} - 512T_{11}^{3} - 328T_{11}^{2} + 384T_{11} + 144 \) Copy content Toggle raw display
\( T_{13}^{8} + 4T_{13}^{7} - 41T_{13}^{6} - 145T_{13}^{5} + 272T_{13}^{4} + 1129T_{13}^{3} + 365T_{13}^{2} - 795T_{13} - 99 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} - 14 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 3 T^{7} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} + \cdots + 71 \) Copy content Toggle raw display
$11$ \( T^{8} - 5 T^{7} + \cdots + 144 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots - 99 \) Copy content Toggle raw display
$17$ \( T^{8} + 2 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( T^{8} + 7 T^{7} + \cdots + 8725 \) Copy content Toggle raw display
$23$ \( T^{8} - 9 T^{7} + \cdots + 10224 \) Copy content Toggle raw display
$29$ \( T^{8} - T^{7} + \cdots + 25920 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 14 T^{7} + \cdots + 23216 \) Copy content Toggle raw display
$41$ \( T^{8} - 6 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$43$ \( T^{8} - 36 T^{7} + \cdots + 21089 \) Copy content Toggle raw display
$47$ \( T^{8} + 35 T^{7} + \cdots + 763344 \) Copy content Toggle raw display
$53$ \( T^{8} + 8 T^{7} + \cdots - 306576 \) Copy content Toggle raw display
$59$ \( T^{8} - 28 T^{7} + \cdots + 2980080 \) Copy content Toggle raw display
$61$ \( T^{8} - 154 T^{6} + \cdots - 508939 \) Copy content Toggle raw display
$67$ \( T^{8} - 30 T^{7} + \cdots + 417024 \) Copy content Toggle raw display
$71$ \( T^{8} - 6 T^{7} + \cdots + 146736 \) Copy content Toggle raw display
$73$ \( T^{8} + T^{7} + \cdots - 631 \) Copy content Toggle raw display
$79$ \( T^{8} - 36 T^{7} + \cdots + 16475 \) Copy content Toggle raw display
$83$ \( T^{8} - 22 T^{7} + \cdots - 1123056 \) Copy content Toggle raw display
$89$ \( T^{8} - T^{7} + \cdots + 3102480 \) Copy content Toggle raw display
$97$ \( T^{8} - 29 T^{7} + \cdots - 67051 \) Copy content Toggle raw display
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