Properties

Label 7728.2.a.bd
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
0 −1.00000 0 2.00000 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 2.00000 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.bd 2
4.b odd 2 1 966.2.a.p 2
12.b even 2 1 2898.2.a.v 2
28.d even 2 1 6762.2.a.bz 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.p 2 4.b odd 2 1
2898.2.a.v 2 12.b even 2 1
6762.2.a.bz 2 28.d even 2 1
7728.2.a.bd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5} - 2 \)
\( T_{11} \)
\( T_{13}^{2} - 20 \)
\( T_{17}^{2} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -2 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( -20 + T^{2} \)
$17$ \( -20 + T^{2} \)
$19$ \( -16 - 4 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( -16 + 4 T + T^{2} \)
$37$ \( -76 + 4 T + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( -64 + 8 T + T^{2} \)
$47$ \( -16 + 4 T + T^{2} \)
$53$ \( -76 + 4 T + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( ( 6 + T )^{2} \)
$67$ \( -64 - 8 T + T^{2} \)
$71$ \( -64 - 8 T + T^{2} \)
$73$ \( -44 - 12 T + T^{2} \)
$79$ \( -64 + 8 T + T^{2} \)
$83$ \( -16 - 4 T + T^{2} \)
$89$ \( -164 + 8 T + T^{2} \)
$97$ \( -4 - 8 T + T^{2} \)
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