Properties

Label 7728.2.a.bb
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: \( 1 \)
Twist minimal: no (minimal twist has level 1932)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( -1 + 3 \beta ) q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -1 + 3 \beta ) q^{5} - q^{7} + q^{9} -3 q^{11} + ( -2 + \beta ) q^{13} + ( 1 - 3 \beta ) q^{15} + ( 5 - 4 \beta ) q^{17} + ( 3 - 2 \beta ) q^{19} + q^{21} + q^{23} + ( 5 + 3 \beta ) q^{25} - q^{27} -5 q^{29} + ( -9 + 2 \beta ) q^{31} + 3 q^{33} + ( 1 - 3 \beta ) q^{35} + ( -1 - 2 \beta ) q^{37} + ( 2 - \beta ) q^{39} + 3 q^{41} + \beta q^{43} + ( -1 + 3 \beta ) q^{45} + 2 \beta q^{47} + q^{49} + ( -5 + 4 \beta ) q^{51} + ( 9 - 7 \beta ) q^{53} + ( 3 - 9 \beta ) q^{55} + ( -3 + 2 \beta ) q^{57} + ( -1 + 9 \beta ) q^{59} -3 \beta q^{61} - q^{63} + ( 5 - 4 \beta ) q^{65} + ( 9 - 7 \beta ) q^{67} - q^{69} + ( -6 - \beta ) q^{71} + ( 7 - 8 \beta ) q^{73} + ( -5 - 3 \beta ) q^{75} + 3 q^{77} + ( 3 + 4 \beta ) q^{79} + q^{81} + ( -9 - 2 \beta ) q^{83} + ( -17 + 7 \beta ) q^{85} + 5 q^{87} + ( 8 - 9 \beta ) q^{89} + ( 2 - \beta ) q^{91} + ( 9 - 2 \beta ) q^{93} + ( -9 + 5 \beta ) q^{95} + ( 11 - 8 \beta ) q^{97} -3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + q^{5} - 2q^{7} + 2q^{9} - 6q^{11} - 3q^{13} - q^{15} + 6q^{17} + 4q^{19} + 2q^{21} + 2q^{23} + 13q^{25} - 2q^{27} - 10q^{29} - 16q^{31} + 6q^{33} - q^{35} - 4q^{37} + 3q^{39} + 6q^{41} + q^{43} + q^{45} + 2q^{47} + 2q^{49} - 6q^{51} + 11q^{53} - 3q^{55} - 4q^{57} + 7q^{59} - 3q^{61} - 2q^{63} + 6q^{65} + 11q^{67} - 2q^{69} - 13q^{71} + 6q^{73} - 13q^{75} + 6q^{77} + 10q^{79} + 2q^{81} - 20q^{83} - 27q^{85} + 10q^{87} + 7q^{89} + 3q^{91} + 16q^{93} - 13q^{95} + 14q^{97} - 6q^{99} + O(q^{100}) \)

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
−0.618034
1.61803
0 −1.00000 0 −2.85410 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 3.85410 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.bb 2
4.b odd 2 1 1932.2.a.h 2
12.b even 2 1 5796.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.2.a.h 2 4.b odd 2 1
5796.2.a.j 2 12.b even 2 1
7728.2.a.bb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{2} - T_{5} - 11 \)
\( T_{11} + 3 \)
\( T_{13}^{2} + 3 T_{13} + 1 \)
\( T_{17}^{2} - 6 T_{17} - 11 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -11 - T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( 1 + 3 T + T^{2} \)
$17$ \( -11 - 6 T + T^{2} \)
$19$ \( -1 - 4 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( 59 + 16 T + T^{2} \)
$37$ \( -1 + 4 T + T^{2} \)
$41$ \( ( -3 + T )^{2} \)
$43$ \( -1 - T + T^{2} \)
$47$ \( -4 - 2 T + T^{2} \)
$53$ \( -31 - 11 T + T^{2} \)
$59$ \( -89 - 7 T + T^{2} \)
$61$ \( -9 + 3 T + T^{2} \)
$67$ \( -31 - 11 T + T^{2} \)
$71$ \( 41 + 13 T + T^{2} \)
$73$ \( -71 - 6 T + T^{2} \)
$79$ \( 5 - 10 T + T^{2} \)
$83$ \( 95 + 20 T + T^{2} \)
$89$ \( -89 - 7 T + T^{2} \)
$97$ \( -31 - 14 T + T^{2} \)
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