Properties

Label 7728.2.a.bn
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: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). 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 \( q + q^{3} + \beta q^{5} + q^{7} + q^{9} + (2 \beta - 1) q^{11} + ( - \beta - 3) q^{13} + \beta q^{15} + ( - 6 \beta + 3) q^{17} + ( - 4 \beta + 1) q^{19} + q^{21} + q^{23} + (\beta - 4) q^{25} + q^{27} + (2 \beta - 7) q^{29} + (6 \beta - 3) q^{31} + (2 \beta - 1) q^{33} + \beta q^{35} - 11 q^{37} + ( - \beta - 3) q^{39} + (4 \beta + 1) q^{41} + ( - \beta + 1) q^{43} + \beta q^{45} + ( - 2 \beta + 6) q^{47} + q^{49} + ( - 6 \beta + 3) q^{51} + (5 \beta - 10) q^{53} + (\beta + 2) q^{55} + ( - 4 \beta + 1) q^{57} + ( - \beta - 10) q^{59} + (3 \beta - 3) q^{61} + q^{63} + ( - 4 \beta - 1) q^{65} + ( - 5 \beta + 2) q^{67} + q^{69} + ( - \beta - 5) q^{71} + (6 \beta - 9) q^{73} + (\beta - 4) q^{75} + (2 \beta - 1) q^{77} + ( - 4 \beta + 7) q^{79} + q^{81} + ( - 10 \beta + 3) q^{83} + ( - 3 \beta - 6) q^{85} + (2 \beta - 7) q^{87} + ( - \beta + 11) q^{89} + ( - \beta - 3) q^{91} + (6 \beta - 3) q^{93} + ( - 3 \beta - 4) q^{95} + ( - 12 \beta + 3) q^{97} + (2 \beta - 1) q^{99}+O(q^{100}) \) 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 \( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} - 7 q^{13} + q^{15} - 2 q^{19} + 2 q^{21} + 2 q^{23} - 7 q^{25} + 2 q^{27} - 12 q^{29} + q^{35} - 22 q^{37} - 7 q^{39} + 6 q^{41} + q^{43} + q^{45} + 10 q^{47} + 2 q^{49} - 15 q^{53} + 5 q^{55} - 2 q^{57} - 21 q^{59} - 3 q^{61} + 2 q^{63} - 6 q^{65} - q^{67} + 2 q^{69} - 11 q^{71} - 12 q^{73} - 7 q^{75} + 10 q^{79} + 2 q^{81} - 4 q^{83} - 15 q^{85} - 12 q^{87} + 21 q^{89} - 7 q^{91} - 11 q^{95} - 6 q^{97}+O(q^{100}) \) 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
−0.618034
1.61803
0 1.00000 0 −0.618034 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 1.61803 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.bn 2
4.b odd 2 1 483.2.a.d 2
12.b even 2 1 1449.2.a.h 2
28.d even 2 1 3381.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.d 2 4.b odd 2 1
1449.2.a.h 2 12.b even 2 1
3381.2.a.r 2 28.d even 2 1
7728.2.a.bn 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{2} - T_{5} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 5 \) Copy content Toggle raw display
\( T_{13}^{2} + 7T_{13} + 11 \) Copy content Toggle raw display
\( T_{17}^{2} - 45 \) 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
$5$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 5 \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 11 \) Copy content Toggle raw display
$17$ \( T^{2} - 45 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 12T + 31 \) Copy content Toggle raw display
$31$ \( T^{2} - 45 \) Copy content Toggle raw display
$37$ \( (T + 11)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$43$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$47$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$53$ \( T^{2} + 15T + 25 \) Copy content Toggle raw display
$59$ \( T^{2} + 21T + 109 \) Copy content Toggle raw display
$61$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$67$ \( T^{2} + T - 31 \) Copy content Toggle raw display
$71$ \( T^{2} + 11T + 29 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T - 9 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T + 5 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 121 \) Copy content Toggle raw display
$89$ \( T^{2} - 21T + 109 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T - 171 \) Copy content Toggle raw display
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