Properties

Label 7728.2.a.bw
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{2} q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{2} q^{5} + q^{7} + q^{9} + ( -1 - \beta_{1} - \beta_{2} ) q^{11} + ( 1 + \beta_{1} ) q^{13} + \beta_{2} q^{15} + ( -3 - \beta_{1} - \beta_{2} ) q^{17} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{19} + q^{21} - q^{23} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{25} + q^{27} + ( -5 + \beta_{1} - 3 \beta_{2} ) q^{29} + ( -1 - \beta_{1} + \beta_{2} ) q^{31} + ( -1 - \beta_{1} - \beta_{2} ) q^{33} + \beta_{2} q^{35} + ( 1 - 5 \beta_{1} + \beta_{2} ) q^{37} + ( 1 + \beta_{1} ) q^{39} + ( -5 - 3 \beta_{1} + \beta_{2} ) q^{41} + ( 1 - \beta_{1} ) q^{43} + \beta_{2} q^{45} + ( -6 - 2 \beta_{1} ) q^{47} + q^{49} + ( -3 - \beta_{1} - \beta_{2} ) q^{51} + ( -4 - \beta_{2} ) q^{53} + ( -4 - 2 \beta_{1} + \beta_{2} ) q^{55} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{57} + ( -6 - 4 \beta_{1} + \beta_{2} ) q^{59} + ( -5 + 3 \beta_{1} - 6 \beta_{2} ) q^{61} + q^{63} + ( 1 + \beta_{1} + \beta_{2} ) q^{65} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{67} - q^{69} + ( 1 + 7 \beta_{1} - 6 \beta_{2} ) q^{71} + ( 3 + 5 \beta_{1} + \beta_{2} ) q^{73} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{75} + ( -1 - \beta_{1} - \beta_{2} ) q^{77} + ( 5 + \beta_{1} - 3 \beta_{2} ) q^{79} + q^{81} + ( -1 - 5 \beta_{1} + \beta_{2} ) q^{83} + ( -4 - 2 \beta_{1} - \beta_{2} ) q^{85} + ( -5 + \beta_{1} - 3 \beta_{2} ) q^{87} + ( 1 + \beta_{1} + 6 \beta_{2} ) q^{89} + ( 1 + \beta_{1} ) q^{91} + ( -1 - \beta_{1} + \beta_{2} ) q^{93} + ( 6 + 4 \beta_{1} - 3 \beta_{2} ) q^{95} + ( -5 + 5 \beta_{1} + \beta_{2} ) q^{97} + ( -1 - \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} - q^{5} + 3q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} - q^{5} + 3q^{7} + 3q^{9} - 2q^{11} + 3q^{13} - q^{15} - 8q^{17} - 4q^{19} + 3q^{21} - 3q^{23} - 4q^{25} + 3q^{27} - 12q^{29} - 4q^{31} - 2q^{33} - q^{35} + 2q^{37} + 3q^{39} - 16q^{41} + 3q^{43} - q^{45} - 18q^{47} + 3q^{49} - 8q^{51} - 11q^{53} - 13q^{55} - 4q^{57} - 19q^{59} - 9q^{61} + 3q^{63} + 2q^{65} - 3q^{67} - 3q^{69} + 9q^{71} + 8q^{73} - 4q^{75} - 2q^{77} + 18q^{79} + 3q^{81} - 4q^{83} - 11q^{85} - 12q^{87} - 3q^{89} + 3q^{91} - 4q^{93} + 21q^{95} - 16q^{97} - 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 4 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
−0.254102
−1.86081
2.11491
0 1.00000 0 −2.93543 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 0.462598 0 1.00000 0 1.00000 0
1.3 0 1.00000 0 1.47283 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.bw 3
4.b odd 2 1 3864.2.a.l 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.l 3 4.b odd 2 1
7728.2.a.bw 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{3} + T_{5}^{2} - 5 T_{5} + 2 \)
\( T_{11}^{3} + 2 T_{11}^{2} - 11 T_{11} + 4 \)
\( T_{13}^{3} - 3 T_{13}^{2} - T_{13} + 2 \)
\( T_{17}^{3} + 8 T_{17}^{2} + 9 T_{17} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( 2 - 5 T + T^{2} + T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( 4 - 11 T + 2 T^{2} + T^{3} \)
$13$ \( 2 - T - 3 T^{2} + T^{3} \)
$17$ \( -2 + 9 T + 8 T^{2} + T^{3} \)
$19$ \( -196 - 45 T + 4 T^{2} + T^{3} \)
$23$ \( ( 1 + T )^{3} \)
$29$ \( -214 + 5 T + 12 T^{2} + T^{3} \)
$31$ \( -8 - T + 4 T^{2} + T^{3} \)
$37$ \( -58 - 89 T - 2 T^{2} + T^{3} \)
$41$ \( -74 + 53 T + 16 T^{2} + T^{3} \)
$43$ \( 4 - T - 3 T^{2} + T^{3} \)
$47$ \( 128 + 92 T + 18 T^{2} + T^{3} \)
$53$ \( 26 + 35 T + 11 T^{2} + T^{3} \)
$59$ \( -196 + 63 T + 19 T^{2} + T^{3} \)
$61$ \( -1186 - 147 T + 9 T^{2} + T^{3} \)
$67$ \( -188 - 79 T + 3 T^{2} + T^{3} \)
$71$ \( 1736 - 235 T - 9 T^{2} + T^{3} \)
$73$ \( -106 - 99 T - 8 T^{2} + T^{3} \)
$79$ \( -64 + 65 T - 18 T^{2} + T^{3} \)
$83$ \( -236 - 85 T + 4 T^{2} + T^{3} \)
$89$ \( 386 - 211 T + 3 T^{2} + T^{3} \)
$97$ \( -898 - 35 T + 16 T^{2} + T^{3} \)
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