Properties

Label 7728.2.a.ce
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: 4.4.15317.1
Defining polynomial: \(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2\)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( 1 + \beta_{3} ) q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( 1 + \beta_{3} ) q^{5} + q^{7} + q^{9} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{11} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( 1 + \beta_{3} ) q^{15} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{17} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{19} + q^{21} - q^{23} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{25} + q^{27} + ( -1 - 3 \beta_{2} ) q^{29} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{31} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{33} + ( 1 + \beta_{3} ) q^{35} + ( 5 + \beta_{2} - 2 \beta_{3} ) q^{37} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{39} + ( 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{41} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + ( 1 + \beta_{3} ) q^{45} + ( 7 - 3 \beta_{1} - \beta_{3} ) q^{47} + q^{49} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{51} + ( 4 + \beta_{1} + 4 \beta_{2} ) q^{53} + ( -3 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{55} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{57} + ( 6 + \beta_{1} ) q^{59} + ( 3 + \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{61} + q^{63} + ( 3 + 4 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{65} + ( 3 - 2 \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{67} - q^{69} + ( 4 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{71} + ( 5 - 2 \beta_{1} + 3 \beta_{2} ) q^{73} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{75} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{77} + ( 5 - 2 \beta_{1} + 5 \beta_{2} ) q^{79} + q^{81} + ( -3 + 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{83} + ( -1 - 6 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{85} + ( -1 - 3 \beta_{2} ) q^{87} + ( -6 + 6 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{89} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{91} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{93} + ( 3 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{95} + ( 9 + \beta_{2} - 2 \beta_{3} ) q^{97} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 5q^{5} + 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 5q^{5} + 4q^{7} + 4q^{9} - q^{11} + 7q^{13} + 5q^{15} + 2q^{17} + q^{19} + 4q^{21} - 4q^{23} + 11q^{25} + 4q^{27} + 2q^{29} - 6q^{31} - q^{33} + 5q^{35} + 16q^{37} + 7q^{39} + 5q^{41} - 9q^{43} + 5q^{45} + 21q^{47} + 4q^{49} + 2q^{51} + 10q^{53} - 17q^{55} + q^{57} + 26q^{59} + 2q^{61} + 4q^{63} + 26q^{65} - 5q^{67} - 4q^{69} + 19q^{71} + 10q^{73} + 11q^{75} - q^{77} + 6q^{79} + 4q^{81} + 2q^{83} - 7q^{85} + 2q^{87} - 17q^{89} + 7q^{91} - 6q^{93} + 17q^{95} + 32q^{97} - q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 2\)

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
1.32973
−1.69353
−0.329727
2.69353
0 1.00000 0 −2.73589 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 1.04900 0 1.00000 0 1.00000 0
1.3 0 1.00000 0 3.17434 0 1.00000 0 1.00000 0
1.4 0 1.00000 0 3.51256 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.ce 4
4.b odd 2 1 483.2.a.j 4
12.b even 2 1 1449.2.a.o 4
28.d even 2 1 3381.2.a.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.j 4 4.b odd 2 1
1449.2.a.o 4 12.b even 2 1
3381.2.a.x 4 28.d even 2 1
7728.2.a.ce 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{4} - 5 T_{5}^{3} - 3 T_{5}^{2} + 38 T_{5} - 32 \)
\( T_{11}^{4} + T_{11}^{3} - 17 T_{11}^{2} + 19 T_{11} + 4 \)
\( T_{13}^{4} - 7 T_{13}^{3} - 17 T_{13}^{2} + 164 T_{13} - 188 \)
\( T_{17}^{4} - 2 T_{17}^{3} - 29 T_{17}^{2} + 98 T_{17} - 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( -32 + 38 T - 3 T^{2} - 5 T^{3} + T^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( 4 + 19 T - 17 T^{2} + T^{3} + T^{4} \)
$13$ \( -188 + 164 T - 17 T^{2} - 7 T^{3} + T^{4} \)
$17$ \( -64 + 98 T - 29 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( 4 - 19 T - 17 T^{2} - T^{3} + T^{4} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( ( -38 - T + T^{2} )^{2} \)
$31$ \( 16 - 10 T - 17 T^{2} + 6 T^{3} + T^{4} \)
$37$ \( -188 + 72 T + 55 T^{2} - 16 T^{3} + T^{4} \)
$41$ \( -134 - 183 T - 71 T^{2} - 5 T^{3} + T^{4} \)
$43$ \( -16 - 128 T - 3 T^{2} + 9 T^{3} + T^{4} \)
$47$ \( -1696 + 212 T + 102 T^{2} - 21 T^{3} + T^{4} \)
$53$ \( 3578 + 577 T - 104 T^{2} - 10 T^{3} + T^{4} \)
$59$ \( 1556 - 1027 T + 248 T^{2} - 26 T^{3} + T^{4} \)
$61$ \( 1114 + 929 T - 180 T^{2} - 2 T^{3} + T^{4} \)
$67$ \( 11008 - 1396 T - 281 T^{2} + 5 T^{3} + T^{4} \)
$71$ \( 32 - 116 T + 83 T^{2} - 19 T^{3} + T^{4} \)
$73$ \( 608 + 226 T - 61 T^{2} - 10 T^{3} + T^{4} \)
$79$ \( 8992 + 350 T - 221 T^{2} - 6 T^{3} + T^{4} \)
$83$ \( 8152 + 130 T - 197 T^{2} - 2 T^{3} + T^{4} \)
$89$ \( 11188 - 2312 T - 175 T^{2} + 17 T^{3} + T^{4} \)
$97$ \( 1684 - 1392 T + 343 T^{2} - 32 T^{3} + T^{4} \)
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