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: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_{3} - 1) q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_{3} - 1) q^{5} - q^{7} + q^{9} + (\beta_{2} - \beta_1 - 1) q^{11} + (\beta_{3} + \beta_{2} - \beta_1) q^{13} + ( - \beta_{3} - 1) q^{15} + ( - \beta_{3} + \beta_1 - 2) q^{17} + ( - \beta_{2} + \beta_1 + 1) q^{19} - q^{21} + q^{23} + (3 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{25} + q^{27} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{29} + (\beta_{3} - 2 \beta_{2} + 3 \beta_1 + 4) q^{31} + (\beta_{2} - \beta_1 - 1) q^{33} + (\beta_{3} + 1) q^{35} + (2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{37} + (\beta_{3} + \beta_{2} - \beta_1) q^{39} + (\beta_{3} - \beta_1 - 2) q^{41} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{43} + ( - \beta_{3} - 1) q^{45} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{47} + q^{49} + ( - \beta_{3} + \beta_1 - 2) q^{51} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{53} + (2 \beta_{3} - \beta_{2}) q^{55} + ( - \beta_{2} + \beta_1 + 1) q^{57} + ( - \beta_{2} - 2 \beta_1) q^{59} + (4 \beta_{3} + 3 \beta_1 + 3) q^{61} - q^{63} + ( - \beta_{3} - \beta_1 - 6) q^{65} + (\beta_{3} + 2 \beta_{2} - 1) q^{67} + q^{69} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{71} + ( - \beta_{3} - 3 \beta_1 - 6) q^{73} + (3 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{75} + ( - \beta_{2} + \beta_1 + 1) q^{77} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 4) q^{79} + q^{81} + (2 \beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{83} + (4 \beta_{3} - \beta_{2} + 2 \beta_1 + 8) q^{85} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{87} + (5 \beta_{3} - \beta_{2} + \beta_1 - 6) q^{89} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{91} + (\beta_{3} - 2 \beta_{2} + 3 \beta_1 + 4) q^{93} + ( - 2 \beta_{3} + \beta_{2}) q^{95} + ( - \beta_{3} - 3 \beta_1 - 6) q^{97} + (\beta_{2} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} - 3 q^{15} - 8 q^{17} + 2 q^{19} - 4 q^{21} + 4 q^{23} - q^{25} + 4 q^{27} - 10 q^{29} + 10 q^{31} - 2 q^{33} + 3 q^{35} + q^{39} - 8 q^{41} - 13 q^{43} - 3 q^{45} + 6 q^{47} + 4 q^{49} - 8 q^{51} + 5 q^{53} - 3 q^{55} + 2 q^{57} + q^{59} + 5 q^{61} - 4 q^{63} - 22 q^{65} - 3 q^{67} + 4 q^{69} - 5 q^{71} - 20 q^{73} - q^{75} + 2 q^{77} - 10 q^{79} + 4 q^{81} - 18 q^{83} + 25 q^{85} - 10 q^{87} - 31 q^{89} - q^{91} + 10 q^{93} + 3 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu^{2} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} - \nu^{2} + 7\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 3\nu^{2} - 3\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + 2\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{3} - \beta_{2} + 2\beta _1 + 11 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + 3\beta _1 + 3 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

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

Hecke characteristic polynomials

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