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Properties

Label 8464.2.a.cf
Level $8464$
Weight $2$
Character orbit 8464.a
Self dual yes
Analytic conductor $67.585$
Analytic rank $1$
Dimension $12$
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 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")
 
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);
 
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 = [12,0,-8,0,0,0,0,0,8,0,0,0,16] 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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 20x^{10} + 157x^{8} - 616x^{6} + 1264x^{4} - 1272x^{2} + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 4232)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + \beta_{9} q^{5} + ( - \beta_{9} + \beta_{5} + \cdots - \beta_{2}) q^{7} + (\beta_{6} - \beta_{4} - \beta_1 + 1) q^{9} + ( - \beta_{9} - \beta_{8} + \beta_{3}) q^{11} + ( - \beta_{10} + \beta_{4} - 2 \beta_1 + 2) q^{13}+ \cdots + (3 \beta_{9} + 2 \beta_{7} + \cdots - 7 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{3} + 8 q^{9} + 16 q^{13} + 4 q^{25} - 8 q^{27} - 8 q^{31} - 56 q^{35} - 64 q^{39} - 40 q^{41} - 32 q^{47} + 28 q^{49} - 64 q^{55} - 60 q^{59} + 32 q^{71} + 28 q^{73} - 16 q^{75} + 24 q^{77}+ \cdots - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 20x^{10} + 157x^{8} - 616x^{6} + 1264x^{4} - 1272x^{2} + 484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{11} + 29\nu^{9} - 138\nu^{7} + 231\nu^{5} - 64\nu^{3} - 30\nu ) / 44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{11} + 69\nu^{9} - 452\nu^{7} + 1419\nu^{5} - 2152\nu^{3} + 1282\nu ) / 44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 14\nu^{6} + 65\nu^{4} - 114\nu^{2} + 60 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{11} + 89\nu^{9} - 587\nu^{7} + 1771\nu^{5} - 2426\nu^{3} + 1234\nu ) / 44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{8} - 14\nu^{6} + 67\nu^{4} - 128\nu^{2} + 78 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{11} + 129\nu^{9} - 901\nu^{7} + 2959\nu^{5} - 4470\nu^{3} + 2370\nu ) / 44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9\nu^{11} - 158\nu^{9} + 1039\nu^{7} - 3190\nu^{5} + 4534\nu^{3} - 2252\nu ) / 44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\nu^{11} - 305\nu^{9} + 1902\nu^{7} - 5379\nu^{5} + 6736\nu^{3} - 2898\nu ) / 44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{10} + 17\nu^{8} - 105\nu^{6} + 287\nu^{4} - 336\nu^{2} + 134 ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{10} - 17\nu^{8} + 107\nu^{6} - 309\nu^{4} + 404\nu^{2} - 188 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{8} + 3\beta_{7} - \beta_{5} - \beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - \beta_{4} + 7\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} + 9\beta_{8} + 18\beta_{7} - 7\beta_{5} - 8\beta_{3} + 20\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} + \beta_{10} + 11\beta_{6} - 11\beta_{4} + 43\beta _1 + 57 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11\beta_{9} + 45\beta_{8} + 109\beta_{7} - 40\beta_{5} - 56\beta_{3} + 132\beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14\beta_{11} + 14\beta_{10} + 89\beta_{6} - 87\beta_{4} + 261\beta _1 + 300 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 89\beta_{9} + 244\beta_{8} + 669\beta_{7} - 215\beta_{5} - 380\beta_{3} + 855\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 133\beta_{11} + 131\beta_{10} + 645\beta_{6} - 611\beta_{4} + 1595\beta _1 + 1685 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 647\beta_{9} + 1401\beta_{8} + 4155\beta_{7} - 1134\beta_{5} - 2538\beta_{3} + 5474\beta_{2} \) 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
−0.984153
0.984153
−1.31154
1.31154
1.66287
−1.66287
−1.90564
1.90564
2.12938
−2.12938
−2.52595
2.52595
0 −3.03144 0 −1.23569 0 3.74592 0 6.18964 0
1.2 0 −3.03144 0 1.23569 0 −3.74592 0 6.18964 0
1.3 0 −2.27985 0 −1.62262 0 3.69933 0 2.19774 0
1.4 0 −2.27985 0 1.62262 0 −3.69933 0 2.19774 0
1.5 0 −1.23487 0 −4.08929 0 5.04976 0 −1.47508 0
1.6 0 −1.23487 0 4.08929 0 −5.04976 0 −1.47508 0
1.7 0 −0.368525 0 −1.72190 0 0.634565 0 −2.86419 0
1.8 0 −0.368525 0 1.72190 0 −0.634565 0 −2.86419 0
1.9 0 0.534266 0 −2.85360 0 −1.52458 0 −2.71456 0
1.10 0 0.534266 0 2.85360 0 1.52458 0 −2.71456 0
1.11 0 2.38043 0 −0.0992850 0 −0.236341 0 2.66645 0
1.12 0 2.38043 0 0.0992850 0 0.236341 0 2.66645 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.cf 12
4.b odd 2 1 4232.2.a.x ✓ 12
23.b odd 2 1 inner 8464.2.a.cf 12
92.b even 2 1 4232.2.a.x ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4232.2.a.x ✓ 12 4.b odd 2 1
4232.2.a.x ✓ 12 92.b even 2 1
8464.2.a.cf 12 1.a even 1 1 trivial
8464.2.a.cf 12 23.b odd 2 1 inner

Hecke kernels

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

\( T_{3}^{6} + 4T_{3}^{5} - 3T_{3}^{4} - 24T_{3}^{3} - 16T_{3}^{2} + 8T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{12} - 32T_{5}^{10} + 330T_{5}^{8} - 1392T_{5}^{6} + 2537T_{5}^{4} - 1648T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{12} - 56T_{7}^{10} + 1048T_{7}^{8} - 7456T_{7}^{6} + 14608T_{7}^{4} - 5376T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{13}^{6} - 8T_{13}^{5} - 30T_{13}^{4} + 240T_{13}^{3} + 197T_{13}^{2} - 448T_{13} - 368 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 4 T^{5} - 3 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} - 32 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{12} - 56 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{12} - 100 T^{10} + \cdots + 2096704 \) Copy content Toggle raw display
$13$ \( (T^{6} - 8 T^{5} + \cdots - 368)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 140 T^{10} + \cdots + 39589264 \) Copy content Toggle raw display
$19$ \( T^{12} - 124 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( (T^{6} - 50 T^{4} + \cdots - 104)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 4 T^{5} + \cdots - 5312)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 360696064 \) Copy content Toggle raw display
$41$ \( (T^{6} + 20 T^{5} + \cdots - 30848)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 149426176 \) Copy content Toggle raw display
$47$ \( (T^{6} + 16 T^{5} + \cdots + 49936)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 190219264 \) Copy content Toggle raw display
$59$ \( (T^{6} + 30 T^{5} + \cdots + 274900)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 576384064 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 72244838656 \) Copy content Toggle raw display
$71$ \( (T^{6} - 16 T^{5} + \cdots + 16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 14 T^{5} + \cdots + 4297)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} - 272 T^{10} + \cdots + 262144 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 5587861504 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 2062431396 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 3836811364 \) Copy content Toggle raw display
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