Properties

Label 7728.2.a.cb
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
Defining polynomial: \(x^{4} - x^{3} - 5 x^{2} + 2 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 - \beta_{3} ) q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 - \beta_{3} ) q^{5} - q^{7} + q^{9} + ( -1 - \beta_{1} + \beta_{2} ) q^{11} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( -1 - \beta_{3} ) q^{15} + ( -2 + \beta_{1} - \beta_{3} ) q^{17} + ( 1 + \beta_{1} - \beta_{2} ) q^{19} - q^{21} + q^{23} + ( 1 + \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{25} + q^{27} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{29} + ( 4 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{31} + ( -1 - \beta_{1} + \beta_{2} ) q^{33} + ( 1 + \beta_{3} ) q^{35} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{37} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{39} + ( -2 - \beta_{1} + \beta_{3} ) q^{41} + ( -3 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -1 - \beta_{3} ) q^{45} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{47} + q^{49} + ( -2 + \beta_{1} - \beta_{3} ) q^{51} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{53} + ( -\beta_{2} + 2 \beta_{3} ) q^{55} + ( 1 + \beta_{1} - \beta_{2} ) q^{57} + ( -2 \beta_{1} - \beta_{2} ) q^{59} + ( 3 + 3 \beta_{1} + 4 \beta_{3} ) q^{61} - q^{63} + ( -6 - \beta_{1} - \beta_{3} ) q^{65} + ( -1 + 2 \beta_{2} + \beta_{3} ) q^{67} + q^{69} + ( \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{71} + ( -6 - 3 \beta_{1} - \beta_{3} ) q^{73} + ( 1 + \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{75} + ( 1 + \beta_{1} - \beta_{2} ) q^{77} + ( -4 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{79} + q^{81} + ( -3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{83} + ( 8 + 2 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{85} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{87} + ( -6 + \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{89} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{91} + ( 4 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{93} + ( \beta_{2} - 2 \beta_{3} ) q^{95} + ( -6 - 3 \beta_{1} - \beta_{3} ) q^{97} + ( -1 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 3q^{5} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 3q^{5} - 4q^{7} + 4q^{9} - 2q^{11} + q^{13} - 3q^{15} - 8q^{17} + 2q^{19} - 4q^{21} + 4q^{23} - q^{25} + 4q^{27} - 10q^{29} + 10q^{31} - 2q^{33} + 3q^{35} + q^{39} - 8q^{41} - 13q^{43} - 3q^{45} + 6q^{47} + 4q^{49} - 8q^{51} + 5q^{53} - 3q^{55} + 2q^{57} + q^{59} + 5q^{61} - 4q^{63} - 22q^{65} - 3q^{67} + 4q^{69} - 5q^{71} - 20q^{73} - q^{75} + 2q^{77} - 10q^{79} + 4q^{81} - 18q^{83} + 25q^{85} - 10q^{87} - 31q^{89} - q^{91} + 10q^{93} + 3q^{95} - 20q^{97} - 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 5 x^{2} + 2 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - \nu^{2} + 7 \nu + 4 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 3 \nu^{2} - 3 \nu + 6 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + 2 \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{3} - \beta_{2} + 2 \beta_{1} + 11\)\()/4\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + 3 \beta_{1} + 3\)

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
−0.820249
1.13856
−1.75660
2.43828
0 1.00000 0 −3.94523 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −1.08564 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 0.703671 0 −1.00000 0 1.00000 0
1.4 0 1.00000 0 1.32719 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.cb 4
4.b odd 2 1 3864.2.a.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.r 4 4.b odd 2 1
7728.2.a.cb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{4} + 3 T_{5}^{3} - 5 T_{5}^{2} - 4 T_{5} + 4 \)
\( T_{11}^{4} + 2 T_{11}^{3} - 15 T_{11}^{2} - 36 T_{11} - 16 \)
\( T_{13}^{4} - T_{13}^{3} - 29 T_{13}^{2} + 24 T_{13} + 76 \)
\( T_{17}^{4} + 8 T_{17}^{3} + 3 T_{17}^{2} - 92 T_{17} - 164 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( 4 - 4 T - 5 T^{2} + 3 T^{3} + T^{4} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( -16 - 36 T - 15 T^{2} + 2 T^{3} + T^{4} \)
$13$ \( 76 + 24 T - 29 T^{2} - T^{3} + T^{4} \)
$17$ \( -164 - 92 T + 3 T^{2} + 8 T^{3} + T^{4} \)
$19$ \( -16 + 36 T - 15 T^{2} - 2 T^{3} + T^{4} \)
$23$ \( ( -1 + T )^{4} \)
$29$ \( 1444 - 420 T - 59 T^{2} + 10 T^{3} + T^{4} \)
$31$ \( 1024 + 480 T - 59 T^{2} - 10 T^{3} + T^{4} \)
$37$ \( -76 - 120 T - 41 T^{2} + T^{4} \)
$41$ \( -4 - 12 T + 3 T^{2} + 8 T^{3} + T^{4} \)
$43$ \( 1136 - 412 T - 37 T^{2} + 13 T^{3} + T^{4} \)
$47$ \( 256 + 256 T - 68 T^{2} - 6 T^{3} + T^{4} \)
$53$ \( 3884 + 500 T - 139 T^{2} - 5 T^{3} + T^{4} \)
$59$ \( 304 - 128 T - 75 T^{2} - T^{3} + T^{4} \)
$61$ \( -1076 + 940 T - 191 T^{2} - 5 T^{3} + T^{4} \)
$67$ \( 16 - 188 T - 71 T^{2} + 3 T^{3} + T^{4} \)
$71$ \( 64 + 280 T - 111 T^{2} + 5 T^{3} + T^{4} \)
$73$ \( -2900 - 700 T + 55 T^{2} + 20 T^{3} + T^{4} \)
$79$ \( 1024 - 480 T - 59 T^{2} + 10 T^{3} + T^{4} \)
$83$ \( 464 - 444 T + 25 T^{2} + 18 T^{3} + T^{4} \)
$89$ \( -22364 - 2736 T + 157 T^{2} + 31 T^{3} + T^{4} \)
$97$ \( -2900 - 700 T + 55 T^{2} + 20 T^{3} + T^{4} \)
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