Properties

Label 7728.2.a.u
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 3 q^{5} - q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} + 3 q^{5} - q^{7} + q^{9} - 4 q^{11} + 3 q^{13} + 3 q^{15} - q^{21} + q^{23} + 4 q^{25} + q^{27} + q^{29} + 2 q^{31} - 4 q^{33} - 3 q^{35} - 5 q^{37} + 3 q^{39} + 5 q^{41} + 7 q^{43} + 3 q^{45} + 3 q^{47} + q^{49} + 12 q^{53} - 12 q^{55} + 2 q^{59} - 6 q^{61} - q^{63} + 9 q^{65} + 12 q^{67} + q^{69} - 10 q^{71} + 4 q^{75} + 4 q^{77} - 4 q^{79} + q^{81} - 4 q^{83} + q^{87} + 10 q^{89} - 3 q^{91} + 2 q^{93} + 19 q^{97} - 4 q^{99} + O(q^{100}) \)

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 1.00000 0 3.00000 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)
\(23\) \(-1\)

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 7728.2.a.u 1
4.b odd 2 1 966.2.a.d 1
12.b even 2 1 2898.2.a.k 1
28.d even 2 1 6762.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.d 1 4.b odd 2 1
2898.2.a.k 1 12.b even 2 1
6762.2.a.l 1 28.d even 2 1
7728.2.a.u 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5} - 3 \)
\( T_{11} + 4 \)
\( T_{13} - 3 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( -3 + T \)
$7$ \( 1 + T \)
$11$ \( 4 + T \)
$13$ \( -3 + T \)
$17$ \( T \)
$19$ \( T \)
$23$ \( -1 + T \)
$29$ \( -1 + T \)
$31$ \( -2 + T \)
$37$ \( 5 + T \)
$41$ \( -5 + T \)
$43$ \( -7 + T \)
$47$ \( -3 + T \)
$53$ \( -12 + T \)
$59$ \( -2 + T \)
$61$ \( 6 + T \)
$67$ \( -12 + T \)
$71$ \( 10 + T \)
$73$ \( T \)
$79$ \( 4 + T \)
$83$ \( 4 + T \)
$89$ \( -10 + T \)
$97$ \( -19 + T \)
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