Properties

Label 7728.2.a.bj
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta q^{5} - q^{7} + q^{9} + (\beta - 1) q^{11} + (\beta + 2) q^{13} + \beta q^{15} + 4 q^{17} + (\beta - 5) q^{19} - q^{21} - q^{23} + (\beta + 3) q^{25} + q^{27} + (\beta + 2) q^{29} + 6 q^{31} + (\beta - 1) q^{33} - \beta q^{35} + (\beta - 6) q^{37} + (\beta + 2) q^{39} + ( - 2 \beta + 1) q^{41} - \beta q^{43} + \beta q^{45} + ( - 2 \beta + 3) q^{47} + q^{49} + 4 q^{51} + (3 \beta - 3) q^{53} + 8 q^{55} + (\beta - 5) q^{57} + (\beta - 7) q^{59} + ( - \beta - 5) q^{61} - q^{63} + (3 \beta + 8) q^{65} + 4 q^{67} - q^{69} + (2 \beta - 8) q^{71} + 12 q^{73} + (\beta + 3) q^{75} + ( - \beta + 1) q^{77} + ( - 2 \beta - 2) q^{79} + q^{81} + (2 \beta + 2) q^{83} + 4 \beta q^{85} + (\beta + 2) q^{87} + ( - 4 \beta + 6) q^{89} + ( - \beta - 2) q^{91} + 6 q^{93} + ( - 4 \beta + 8) q^{95} + ( - 3 \beta - 2) q^{97} + (\beta - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} - q^{11} + 5 q^{13} + q^{15} + 8 q^{17} - 9 q^{19} - 2 q^{21} - 2 q^{23} + 7 q^{25} + 2 q^{27} + 5 q^{29} + 12 q^{31} - q^{33} - q^{35} - 11 q^{37} + 5 q^{39} - q^{43} + q^{45} + 4 q^{47} + 2 q^{49} + 8 q^{51} - 3 q^{53} + 16 q^{55} - 9 q^{57} - 13 q^{59} - 11 q^{61} - 2 q^{63} + 19 q^{65} + 8 q^{67} - 2 q^{69} - 14 q^{71} + 24 q^{73} + 7 q^{75} + q^{77} - 6 q^{79} + 2 q^{81} + 6 q^{83} + 4 q^{85} + 5 q^{87} + 8 q^{89} - 5 q^{91} + 12 q^{93} + 12 q^{95} - 7 q^{97} - q^{99}+O(q^{100}) \) 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
−2.37228
3.37228
0 1.00000 0 −2.37228 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 3.37228 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.bj 2
4.b odd 2 1 3864.2.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.i 2 4.b odd 2 1
7728.2.a.bj 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{2} - T_{5} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} + T_{11} - 8 \) Copy content Toggle raw display
\( T_{13}^{2} - 5T_{13} - 2 \) Copy content Toggle raw display
\( T_{17} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T - 2 \) Copy content Toggle raw display
$17$ \( (T - 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 9T + 12 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 5T - 2 \) Copy content Toggle raw display
$31$ \( (T - 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 22 \) Copy content Toggle raw display
$41$ \( T^{2} - 33 \) Copy content Toggle raw display
$43$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 29 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T - 72 \) Copy content Toggle raw display
$59$ \( T^{2} + 13T + 34 \) Copy content Toggle raw display
$61$ \( T^{2} + 11T + 22 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 14T + 16 \) Copy content Toggle raw display
$73$ \( (T - 12)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 6T - 24 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$89$ \( T^{2} - 8T - 116 \) Copy content Toggle raw display
$97$ \( T^{2} + 7T - 62 \) Copy content Toggle raw display
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