Properties

Label 48.24.a.d
Level $48$
Weight $24$
Character orbit 48.a
Self dual yes
Analytic conductor $160.898$
Analytic rank $0$
Dimension $1$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 177147 q^{3} + 196251270 q^{5} + 8131131904 q^{7} + 31381059609 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 177147 q^{3} + 196251270 q^{5} + 8131131904 q^{7} + 31381059609 q^{9} - 413280901452 q^{11} - 5925134073898 q^{13} + 34765323726690 q^{15} + 44831041228386 q^{17} - 869707616627540 q^{19} + 14\!\cdots\!88 q^{21}+ \cdots - 12\!\cdots\!68 q^{99}+O(q^{100}) \) 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.

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
0 177147. 0 1.96251e8 0 8.13113e9 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.d 1
4.b odd 2 1 6.24.a.a 1
12.b even 2 1 18.24.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.24.a.a 1 4.b odd 2 1
18.24.a.b 1 12.b even 2 1
48.24.a.d 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 196251270 \) acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(48))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 177147 \) Copy content Toggle raw display
$5$ \( T - 196251270 \) Copy content Toggle raw display
$7$ \( T - 8131131904 \) Copy content Toggle raw display
$11$ \( T + 413280901452 \) Copy content Toggle raw display
$13$ \( T + 5925134073898 \) Copy content Toggle raw display
$17$ \( T - 44831041228386 \) Copy content Toggle raw display
$19$ \( T + 869707616627540 \) Copy content Toggle raw display
$23$ \( T - 3916239181103928 \) Copy content Toggle raw display
$29$ \( T - 99\!\cdots\!10 \) Copy content Toggle raw display
$31$ \( T + 56\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T - 85\!\cdots\!66 \) Copy content Toggle raw display
$41$ \( T - 19\!\cdots\!02 \) Copy content Toggle raw display
$43$ \( T - 18\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T + 16\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T - 63\!\cdots\!22 \) Copy content Toggle raw display
$59$ \( T - 70\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T - 19\!\cdots\!02 \) Copy content Toggle raw display
$67$ \( T + 42\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T + 49\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T - 44\!\cdots\!22 \) Copy content Toggle raw display
$79$ \( T - 16\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T - 19\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T + 77\!\cdots\!90 \) Copy content Toggle raw display
$97$ \( T + 93\!\cdots\!34 \) Copy content Toggle raw display
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