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Properties

Label 8464.2.a.cb
Level $8464$
Weight $2$
Character orbit 8464.a
Self dual yes
Analytic conductor $67.585$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8464 = 2^{4} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8464.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-4,0,0,0,0,0,20,0,0,0,12] 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: \(67.5853802708\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.819879542784.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 151x^{4} - 440x^{3} - 298x^{2} + 532x - 146 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1058)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + ( - \beta_{4} - \beta_{2}) q^{5} + ( - \beta_{6} - \beta_{3} + \beta_{2}) q^{7} + (\beta_{5} - \beta_1 + 2) q^{9} + (\beta_{6} + \beta_{2}) q^{11} + ( - \beta_{5} + \beta_1 + 2) q^{13}+ \cdots + ( - \beta_{6} - 2 \beta_{4} + \cdots + 5 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 20 q^{9} + 12 q^{13} + 32 q^{25} - 40 q^{27} + 12 q^{35} + 36 q^{39} + 12 q^{41} + 12 q^{47} + 32 q^{49} - 12 q^{55} - 24 q^{59} + 12 q^{71} + 12 q^{73} + 8 q^{75} - 16 q^{81} - 36 q^{85}+ \cdots + 84 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 151x^{4} - 440x^{3} - 298x^{2} + 532x - 146 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 13\nu^{2} + 14\nu + 19 ) / 13 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6\nu^{7} - 21\nu^{6} - 131\nu^{5} + 380\nu^{4} + 889\nu^{3} - 1724\nu^{2} - 1845\nu + 1223 ) / 299 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} + 29\nu^{6} + 90\nu^{5} - 562\nu^{4} - 553\nu^{3} + 2855\nu^{2} + 1572\nu - 2196 ) / 299 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{7} + 28\nu^{6} + 190\nu^{5} - 545\nu^{4} - 1469\nu^{3} + 2912\nu^{2} + 3541\nu - 3371 ) / 299 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\nu^{7} - 56\nu^{6} - 380\nu^{5} + 1090\nu^{4} + 2938\nu^{3} - 5525\nu^{2} - 7082\nu + 4649 ) / 299 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\nu^{7} - 56\nu^{6} - 380\nu^{5} + 1090\nu^{4} + 2938\nu^{3} - 5525\nu^{2} - 7381\nu + 4649 ) / 299 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 42\nu^{7} - 147\nu^{6} - 986\nu^{5} + 2821\nu^{4} + 7672\nu^{3} - 14230\nu^{2} - 19884\nu + 12287 ) / 299 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{6} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{4} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - 13\beta_{6} + 10\beta_{5} + 3\beta_{4} - 2\beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{7} - 12\beta_{6} + 19\beta_{5} + 32\beta_{4} - 4\beta_{2} + 15\beta _1 + 82 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 47\beta_{7} - 200\beta_{6} + 122\beta_{5} + 75\beta_{4} - 21\beta_{2} + 56\beta _1 + 131 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 131\beta_{7} - 374\beta_{6} + 319\beta_{5} + 512\beta_{4} + 26\beta_{3} - 66\beta_{2} + 397\beta _1 + 1125 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 935\beta_{7} - 3297\beta_{6} + 1690\beta_{5} + 1533\beta_{4} + 91\beta_{3} - 90\beta_{2} + 1514\beta _1 + 2671 \) 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
2.62472
3.66000
4.30747
0.443768
−3.30747
0.556232
−2.66000
−1.62472
0 −3.14236 0 −0.305248 0 −3.02975 0 6.87441 0
1.2 0 −3.14236 0 0.305248 0 3.02975 0 6.87441 0
1.3 0 −2.37562 0 −4.07171 0 −1.94542 0 2.64357 0
1.4 0 −2.37562 0 4.07171 0 1.94542 0 2.64357 0
1.5 0 1.37562 0 −3.17513 0 −3.35963 0 −1.10767 0
1.6 0 1.37562 0 3.17513 0 3.35963 0 −1.10767 0
1.7 0 2.14236 0 −3.04082 0 4.44396 0 1.58969 0
1.8 0 2.14236 0 3.04082 0 −4.44396 0 1.58969 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(23\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8464.2.a.cb 8
4.b odd 2 1 1058.2.a.n ✓ 8
12.b even 2 1 9522.2.a.ce 8
23.b odd 2 1 inner 8464.2.a.cb 8
92.b even 2 1 1058.2.a.n ✓ 8
276.h odd 2 1 9522.2.a.ce 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1058.2.a.n ✓ 8 4.b odd 2 1
1058.2.a.n ✓ 8 92.b even 2 1
8464.2.a.cb 8 1.a even 1 1 trivial
8464.2.a.cb 8 23.b odd 2 1 inner
9522.2.a.ce 8 12.b even 2 1
9522.2.a.ce 8 276.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8464))\):

\( T_{3}^{4} + 2T_{3}^{3} - 9T_{3}^{2} - 10T_{3} + 22 \) Copy content Toggle raw display
\( T_{5}^{8} - 36T_{5}^{6} + 417T_{5}^{4} - 1584T_{5}^{2} + 144 \) Copy content Toggle raw display
\( T_{7}^{8} - 44T_{7}^{6} + 660T_{7}^{4} - 3968T_{7}^{2} + 7744 \) Copy content Toggle raw display
\( T_{13}^{4} - 6T_{13}^{3} - 3T_{13}^{2} + 48T_{13} - 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 2 T^{3} - 9 T^{2} + \cdots + 22)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 36 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{8} - 44 T^{6} + \cdots + 7744 \) Copy content Toggle raw display
$11$ \( (T^{2} - 6)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 6 T^{3} - 3 T^{2} + \cdots - 48)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 96 T^{6} + \cdots + 125316 \) Copy content Toggle raw display
$19$ \( T^{8} - 56 T^{6} + \cdots + 21316 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} - 63 T^{2} + \cdots - 264)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} - 44 T^{6} + \cdots + 7744 \) Copy content Toggle raw display
$41$ \( (T^{4} - 6 T^{3} + \cdots + 384)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 164 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$47$ \( (T^{4} - 6 T^{3} + \cdots - 552)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 132 T^{6} + \cdots + 82944 \) Copy content Toggle raw display
$59$ \( (T^{4} + 12 T^{3} - 33 T^{2} + \cdots + 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 572 T^{6} + \cdots + 183439936 \) Copy content Toggle raw display
$67$ \( T^{8} - 116 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$71$ \( (T^{4} - 6 T^{3} + \cdots + 15864)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 6 T^{3} + \cdots + 1293)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 464 T^{6} + \cdots + 22429696 \) Copy content Toggle raw display
$83$ \( T^{8} - 84 T^{6} + \cdots + 69696 \) Copy content Toggle raw display
$89$ \( T^{8} - 360 T^{6} + \cdots + 527076 \) Copy content Toggle raw display
$97$ \( T^{8} - 224 T^{6} + \cdots + 662596 \) Copy content Toggle raw display
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