Properties

Label 7728.2.a.bq
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.733.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 8\)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( -1 - \beta_{2} ) q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -1 - \beta_{2} ) q^{5} + q^{7} + q^{9} + ( 1 - \beta_{1} - \beta_{2} ) q^{11} + ( 2 - \beta_{1} ) q^{13} + ( 1 + \beta_{2} ) q^{15} + ( -1 - \beta_{1} + \beta_{2} ) q^{17} + ( 5 - \beta_{1} - \beta_{2} ) q^{19} - q^{21} - q^{23} + ( 3 - \beta_{1} ) q^{25} - q^{27} + ( -1 - \beta_{1} + \beta_{2} ) q^{29} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{31} + ( -1 + \beta_{1} + \beta_{2} ) q^{33} + ( -1 - \beta_{2} ) q^{35} + ( -1 - 3 \beta_{1} - \beta_{2} ) q^{37} + ( -2 + \beta_{1} ) q^{39} + ( 3 + \beta_{1} + \beta_{2} ) q^{41} + ( 8 - 3 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -1 - \beta_{2} ) q^{45} + 2 \beta_{1} q^{47} + q^{49} + ( 1 + \beta_{1} - \beta_{2} ) q^{51} + ( -5 + 2 \beta_{1} - \beta_{2} ) q^{53} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{55} + ( -5 + \beta_{1} + \beta_{2} ) q^{57} + ( -3 + 4 \beta_{1} + \beta_{2} ) q^{59} + ( -6 + \beta_{1} - 2 \beta_{2} ) q^{61} + q^{63} + ( -5 + 3 \beta_{1} - \beta_{2} ) q^{65} + ( 3 + 4 \beta_{1} + \beta_{2} ) q^{67} + q^{69} -\beta_{1} q^{71} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{73} + ( -3 + \beta_{1} ) q^{75} + ( 1 - \beta_{1} - \beta_{2} ) q^{77} + ( 5 + \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( -1 - 3 \beta_{1} - 3 \beta_{2} ) q^{83} + ( -9 + 4 \beta_{1} + 3 \beta_{2} ) q^{85} + ( 1 + \beta_{1} - \beta_{2} ) q^{87} + ( -2 - 5 \beta_{1} ) q^{89} + ( 2 - \beta_{1} ) q^{91} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{93} + ( -1 + 2 \beta_{1} - 5 \beta_{2} ) q^{95} + ( 3 - \beta_{1} + \beta_{2} ) q^{97} + ( 1 - \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} - 3q^{5} + 3q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} - 3q^{5} + 3q^{7} + 3q^{9} + 2q^{11} + 5q^{13} + 3q^{15} - 4q^{17} + 14q^{19} - 3q^{21} - 3q^{23} + 8q^{25} - 3q^{27} - 4q^{29} + 6q^{31} - 2q^{33} - 3q^{35} - 6q^{37} - 5q^{39} + 10q^{41} + 21q^{43} - 3q^{45} + 2q^{47} + 3q^{49} + 4q^{51} - 13q^{53} + 11q^{55} - 14q^{57} - 5q^{59} - 17q^{61} + 3q^{63} - 12q^{65} + 13q^{67} + 3q^{69} - q^{71} + 6q^{73} - 8q^{75} + 2q^{77} + 16q^{79} + 3q^{81} - 6q^{83} - 23q^{85} + 4q^{87} - 11q^{89} + 5q^{91} - 6q^{93} - q^{95} + 8q^{97} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 7 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)

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
−2.69639
2.51820
1.17819
0 −1.00000 0 −3.27053 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 −2.34132 0 1.00000 0 1.00000 0
1.3 0 −1.00000 0 2.61186 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.bq 3
4.b odd 2 1 3864.2.a.o 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.o 3 4.b odd 2 1
7728.2.a.bq 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{3} + 3 T_{5}^{2} - 7 T_{5} - 20 \)
\( T_{11}^{3} - 2 T_{11}^{2} - 9 T_{11} + 14 \)
\( T_{13}^{3} - 5 T_{13}^{2} + T_{13} + 2 \)
\( T_{17}^{3} + 4 T_{17}^{2} - 19 T_{17} - 50 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( -20 - 7 T + 3 T^{2} + T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( 14 - 9 T - 2 T^{2} + T^{3} \)
$13$ \( 2 + T - 5 T^{2} + T^{3} \)
$17$ \( -50 - 19 T + 4 T^{2} + T^{3} \)
$19$ \( -46 + 55 T - 14 T^{2} + T^{3} \)
$23$ \( ( 1 + T )^{3} \)
$29$ \( -50 - 19 T + 4 T^{2} + T^{3} \)
$31$ \( 160 - 43 T - 6 T^{2} + T^{3} \)
$37$ \( -44 - 43 T + 6 T^{2} + T^{3} \)
$41$ \( -10 + 23 T - 10 T^{2} + T^{3} \)
$43$ \( 302 + 83 T - 21 T^{2} + T^{3} \)
$47$ \( 64 - 28 T - 2 T^{2} + T^{3} \)
$53$ \( -16 + 3 T + 13 T^{2} + T^{3} \)
$59$ \( -184 - 91 T + 5 T^{2} + T^{3} \)
$61$ \( -196 + 35 T + 17 T^{2} + T^{3} \)
$67$ \( 326 - 43 T - 13 T^{2} + T^{3} \)
$71$ \( -8 - 7 T + T^{2} + T^{3} \)
$73$ \( 550 - 85 T - 6 T^{2} + T^{3} \)
$79$ \( -104 + 75 T - 16 T^{2} + T^{3} \)
$83$ \( 22 - 81 T + 6 T^{2} + T^{3} \)
$89$ \( -1322 - 143 T + 11 T^{2} + T^{3} \)
$97$ \( 26 - 3 T - 8 T^{2} + T^{3} \)
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