Properties

Label 7728.2.a.cd
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.24197.1
Defining polynomial: \(x^{4} - 6 x^{2} - x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( 1 - \beta_{3} ) q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( 1 - \beta_{3} ) q^{5} - q^{7} + q^{9} + ( 1 + \beta_{2} - \beta_{3} ) q^{11} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( 1 - \beta_{3} ) q^{15} + ( 3 + \beta_{1} - \beta_{2} ) q^{17} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{19} - q^{21} + q^{23} + ( 1 + \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{25} + q^{27} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{29} + ( -1 - \beta_{1} + \beta_{2} ) q^{31} + ( 1 + \beta_{2} - \beta_{3} ) q^{33} + ( -1 + \beta_{3} ) q^{35} + ( 5 + 3 \beta_{1} - \beta_{2} ) q^{37} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{39} + ( 1 + \beta_{2} + \beta_{3} ) q^{41} + ( -2 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + ( 1 - \beta_{3} ) q^{45} + ( -2 + \beta_{1} - \beta_{3} ) q^{47} + q^{49} + ( 3 + \beta_{1} - \beta_{2} ) q^{51} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{53} + ( 5 - 2 \beta_{1} - \beta_{3} ) q^{55} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{57} + ( 1 - 7 \beta_{1} + 2 \beta_{3} ) q^{59} + ( 6 - 2 \beta_{1} - \beta_{2} ) q^{61} - q^{63} + ( -3 - 5 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{65} + ( -1 - 2 \beta_{1} - 3 \beta_{3} ) q^{67} + q^{69} + ( 4 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{71} + ( 5 - \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{73} + ( 1 + \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{75} + ( -1 - \beta_{2} + \beta_{3} ) q^{77} + ( 3 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{79} + q^{81} + ( -1 - \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{83} + ( 3 + 4 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{85} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{87} + ( -\beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{89} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{91} + ( -1 - \beta_{1} + \beta_{2} ) q^{93} + ( 1 - 2 \beta_{2} + \beta_{3} ) q^{95} + ( -1 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{97} + ( 1 + \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 5q^{5} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 5q^{5} - 4q^{7} + 4q^{9} + 5q^{11} + 7q^{13} + 5q^{15} + 12q^{17} - 3q^{19} - 4q^{21} + 4q^{23} + 7q^{25} + 4q^{27} + 6q^{29} - 4q^{31} + 5q^{33} - 5q^{35} + 20q^{37} + 7q^{39} + 3q^{41} - 9q^{43} + 5q^{45} - 7q^{47} + 4q^{49} + 12q^{51} - 6q^{53} + 21q^{55} - 3q^{57} + 2q^{59} + 24q^{61} - 4q^{63} - 14q^{65} - q^{67} + 4q^{69} + 17q^{71} + 16q^{73} + 7q^{75} - 5q^{77} + 10q^{79} + 4q^{81} - 8q^{83} + 17q^{85} + 6q^{87} - 3q^{89} - 7q^{91} - 4q^{93} + 3q^{95} - 2q^{97} + 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 6 x^{2} - x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1} + 1\)

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
−0.700017
2.46506
−2.27460
0.509552
0 1.00000 0 −1.15706 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −0.653724 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 2.39532 0 −1.00000 0 1.00000 0
1.4 0 1.00000 0 4.41546 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.cd 4
4.b odd 2 1 483.2.a.i 4
12.b even 2 1 1449.2.a.p 4
28.d even 2 1 3381.2.a.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.i 4 4.b odd 2 1
1449.2.a.p 4 12.b even 2 1
3381.2.a.w 4 28.d even 2 1
7728.2.a.cd 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{4} - 5 T_{5}^{3} - T_{5}^{2} + 14 T_{5} + 8 \)
\( T_{11}^{4} - 5 T_{11}^{3} - 9 T_{11}^{2} + 65 T_{11} - 68 \)
\( T_{13}^{4} - 7 T_{13}^{3} - 11 T_{13}^{2} + 152 T_{13} - 236 \)
\( T_{17}^{4} - 12 T_{17}^{3} + 37 T_{17}^{2} + 10 T_{17} - 104 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( 8 + 14 T - T^{2} - 5 T^{3} + T^{4} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( -68 + 65 T - 9 T^{2} - 5 T^{3} + T^{4} \)
$13$ \( -236 + 152 T - 11 T^{2} - 7 T^{3} + T^{4} \)
$17$ \( -104 + 10 T + 37 T^{2} - 12 T^{3} + T^{4} \)
$19$ \( 52 - 51 T - 37 T^{2} + 3 T^{3} + T^{4} \)
$23$ \( ( -1 + T )^{4} \)
$29$ \( -436 + 264 T - 31 T^{2} - 6 T^{3} + T^{4} \)
$31$ \( -16 - 46 T - 11 T^{2} + 4 T^{3} + T^{4} \)
$37$ \( -1868 + 280 T + 91 T^{2} - 20 T^{3} + T^{4} \)
$41$ \( -34 + 71 T - 27 T^{2} - 3 T^{3} + T^{4} \)
$43$ \( 272 - 184 T - 39 T^{2} + 9 T^{3} + T^{4} \)
$47$ \( -32 - 28 T + 6 T^{2} + 7 T^{3} + T^{4} \)
$53$ \( -202 - 249 T - 68 T^{2} + 6 T^{3} + T^{4} \)
$59$ \( 17564 + 263 T - 278 T^{2} - 2 T^{3} + T^{4} \)
$61$ \( -1286 - 191 T + 172 T^{2} - 24 T^{3} + T^{4} \)
$67$ \( 4208 + 24 T - 141 T^{2} + T^{3} + T^{4} \)
$71$ \( 64 - 128 T + 79 T^{2} - 17 T^{3} + T^{4} \)
$73$ \( -6656 + 2270 T - 95 T^{2} - 16 T^{3} + T^{4} \)
$79$ \( 128 + 214 T - 17 T^{2} - 10 T^{3} + T^{4} \)
$83$ \( -7256 - 2450 T - 195 T^{2} + 8 T^{3} + T^{4} \)
$89$ \( -92 + 380 T - 113 T^{2} + 3 T^{3} + T^{4} \)
$97$ \( 4 - 356 T - 87 T^{2} + 2 T^{3} + T^{4} \)
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