Properties

Label 48.24.a.l
Level $48$
Weight $24$
Character orbit 48.a
Self dual yes
Analytic conductor $160.898$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(160.897937926\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \( x^{3} - x^{2} - 313478447x - 3858843765 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 177147 q^{3} + (\beta_1 - 25741722) q^{5} + (\beta_{2} + 43 \beta_1 - 1113542624) q^{7} + 31381059609 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 177147 q^{3} + (\beta_1 - 25741722) q^{5} + (\beta_{2} + 43 \beta_1 - 1113542624) q^{7} + 31381059609 q^{9} + (176 \beta_{2} - 2882 \beta_1 + 22293915380) q^{11} + ( - 53 \beta_{2} - 44054 \beta_1 + 1712906606870) q^{13} + (177147 \beta_1 - 4560068827134) q^{15} + ( - 36774 \beta_{2} + 1346618 \beta_1 - 77654745279966) q^{17} + ( - 82802 \beta_{2} - 2764386 \beta_1 + 13017214452204) q^{19} + (177147 \beta_{2} + 7617321 \beta_1 - 197260735213728) q^{21} + ( - 169702 \beta_{2} - 49099786 \beta_1 + 14\!\cdots\!88) q^{23}+ \cdots + (5523066491184 \beta_{2} + \cdots + 69\!\cdots\!20) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 531441 q^{3} - 77225166 q^{5} - 3340627872 q^{7} + 94143178827 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 531441 q^{3} - 77225166 q^{5} - 3340627872 q^{7} + 94143178827 q^{9} + 66881746140 q^{11} + 5138719820610 q^{13} - 13680206481402 q^{15} - 232964235839898 q^{17} + 39051643356612 q^{19} - 591782205641184 q^{21} + 43\!\cdots\!64 q^{23}+ \cdots + 20\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 313478447x - 3858843765 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -192\nu^{2} + 1771392\nu + 40124650816 ) / 391 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8448\nu^{2} + 95024640\nu - 1765542291200 ) / 391 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 44\beta _1 + 147456 ) / 442368 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4613\beta_{2} - 247460\beta _1 + 46224277954560 ) / 221184 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−17698.7
17712.0
−12.3098
0 177147. 0 −1.57121e8 0 −8.81163e9 0 3.13811e10 0
1.2 0 177147. 0 3.07262e6 0 6.69271e9 0 3.13811e10 0
1.3 0 177147. 0 7.68230e7 0 −1.22170e9 0 3.13811e10 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 48.24.a.l 3
4.b odd 2 1 24.24.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.24.a.b 3 4.b odd 2 1
48.24.a.l 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 77225166T_{5}^{2} - 12317220461555220T_{5} + 37088028673642780985000 \) acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(48))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 177147)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 77225166 T^{2} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} + 3340627872 T^{2} + \cdots - 72\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{3} - 66881746140 T^{2} + \cdots + 47\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{3} - 5138719820610 T^{2} + \cdots + 18\!\cdots\!92 \) Copy content Toggle raw display
$17$ \( T^{3} + 232964235839898 T^{2} + \cdots - 11\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{3} - 39051643356612 T^{2} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 25\!\cdots\!20 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots - 21\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 70\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 35\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 35\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 12\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 38\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 58\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 47\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 29\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 93\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 16\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
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