Properties

Label 7728.2.a.bp
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: 3.3.229.1
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.

\(f(q)\) \(=\) \( q - q^{3} + ( -1 + \beta_{1} ) q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -1 + \beta_{1} ) q^{5} - q^{7} + q^{9} + ( 1 + 2 \beta_{2} ) q^{11} + ( -4 - \beta_{2} ) q^{13} + ( 1 - \beta_{1} ) q^{15} + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{17} + ( 1 + 2 \beta_{1} - 2 \beta_{2} ) q^{19} + q^{21} + q^{23} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{25} - q^{27} + ( -1 + \beta_{1} + \beta_{2} ) q^{29} + ( -1 - 3 \beta_{1} - \beta_{2} ) q^{31} + ( -1 - 2 \beta_{2} ) q^{33} + ( 1 - \beta_{1} ) q^{35} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{37} + ( 4 + \beta_{2} ) q^{39} + ( -3 - 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{43} + ( -1 + \beta_{1} ) q^{45} + ( 3 \beta_{1} + \beta_{2} ) q^{47} + q^{49} + ( -1 + 3 \beta_{1} - 3 \beta_{2} ) q^{51} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{53} + ( 1 + 3 \beta_{1} - 2 \beta_{2} ) q^{55} + ( -1 - 2 \beta_{1} + 2 \beta_{2} ) q^{57} + ( -3 + 4 \beta_{1} + \beta_{2} ) q^{59} + ( -8 - \beta_{1} - 2 \beta_{2} ) q^{61} - q^{63} + ( 3 - 5 \beta_{1} + \beta_{2} ) q^{65} + ( 5 - \beta_{1} + 4 \beta_{2} ) q^{67} - q^{69} + ( 4 + 2 \beta_{1} + 3 \beta_{2} ) q^{71} + ( -3 - 3 \beta_{1} + \beta_{2} ) q^{73} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{75} + ( -1 - 2 \beta_{2} ) q^{77} + ( -1 - 3 \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( 5 - 5 \beta_{1} - \beta_{2} ) q^{83} + ( -7 + 7 \beta_{1} - 6 \beta_{2} ) q^{85} + ( 1 - \beta_{1} - \beta_{2} ) q^{87} + ( -4 + 4 \beta_{1} - 3 \beta_{2} ) q^{89} + ( 4 + \beta_{2} ) q^{91} + ( 1 + 3 \beta_{1} + \beta_{2} ) q^{93} + ( 3 - 3 \beta_{1} + 4 \beta_{2} ) q^{95} + ( -1 + 9 \beta_{1} - \beta_{2} ) q^{97} + ( 1 + 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} - 3q^{5} - 3q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} - 3q^{5} - 3q^{7} + 3q^{9} + q^{11} - 11q^{13} + 3q^{15} + 5q^{19} + 3q^{21} + 3q^{23} - 4q^{25} - 3q^{27} - 4q^{29} - 2q^{31} - q^{33} + 3q^{35} - 8q^{37} + 11q^{39} - 11q^{41} + 11q^{43} - 3q^{45} - q^{47} + 3q^{49} - 4q^{53} + 5q^{55} - 5q^{57} - 10q^{59} - 22q^{61} - 3q^{63} + 8q^{65} + 11q^{67} - 3q^{69} + 9q^{71} - 10q^{73} + 4q^{75} - q^{77} - 2q^{79} + 3q^{81} + 16q^{83} - 15q^{85} + 4q^{87} - 9q^{89} + 11q^{91} + 2q^{93} + 5q^{95} - 2q^{97} + q^{99} + O(q^{100}) \)

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
−1.86081
−0.254102
2.11491
0 −1.00000 0 −2.86081 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 −1.25410 0 −1.00000 0 1.00000 0
1.3 0 −1.00000 0 1.11491 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.bp 3
4.b odd 2 1 3864.2.a.p 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.p 3 4.b odd 2 1
7728.2.a.bp 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{3} + 3 T_{5}^{2} - T_{5} - 4 \)
\( T_{11}^{3} - T_{11}^{2} - 21 T_{11} + 37 \)
\( T_{13}^{3} + 11 T_{13}^{2} + 35 T_{13} + 26 \)
\( T_{17}^{3} - 57 T_{17} - 52 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( -4 - T + 3 T^{2} + T^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( 37 - 21 T - T^{2} + T^{3} \)
$13$ \( 26 + 35 T + 11 T^{2} + T^{3} \)
$17$ \( -52 - 57 T + T^{3} \)
$19$ \( 53 - 17 T - 5 T^{2} + T^{3} \)
$23$ \( ( -1 + T )^{3} \)
$29$ \( -26 - 7 T + 4 T^{2} + T^{3} \)
$31$ \( 98 - 49 T + 2 T^{2} + T^{3} \)
$37$ \( -14 - 11 T + 8 T^{2} + T^{3} \)
$41$ \( -59 + 15 T + 11 T^{2} + T^{3} \)
$43$ \( -4 + 13 T - 11 T^{2} + T^{3} \)
$47$ \( -148 - 50 T + T^{2} + T^{3} \)
$53$ \( -89 - 22 T + 4 T^{2} + T^{3} \)
$59$ \( -481 - 48 T + 10 T^{2} + T^{3} \)
$61$ \( 173 + 130 T + 22 T^{2} + T^{3} \)
$67$ \( 496 - 37 T - 11 T^{2} + T^{3} \)
$71$ \( 112 - 55 T - 9 T^{2} + T^{3} \)
$73$ \( -124 + T + 10 T^{2} + T^{3} \)
$79$ \( 98 - 49 T + 2 T^{2} + T^{3} \)
$83$ \( 898 - 35 T - 16 T^{2} + T^{3} \)
$89$ \( 2 - 49 T + 9 T^{2} + T^{3} \)
$97$ \( -106 - 301 T + 2 T^{2} + T^{3} \)
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