Properties

Label 7728.2.a.bf
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: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta - 2) q^{5} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta - 2) q^{5} + q^{7} + q^{9} + (2 \beta - 3) q^{11} + ( - \beta + 1) q^{13} + ( - \beta - 2) q^{15} + ( - 2 \beta - 3) q^{17} + (2 \beta + 3) q^{19} + q^{21} + q^{23} + (5 \beta + 2) q^{25} + q^{27} + (2 \beta - 3) q^{29} + ( - 2 \beta + 1) q^{31} + (2 \beta - 3) q^{33} + ( - \beta - 2) q^{35} + ( - 2 \beta + 3) q^{37} + ( - \beta + 1) q^{39} + ( - 2 \beta + 1) q^{41} + (\beta - 3) q^{43} + ( - \beta - 2) q^{45} + ( - 2 \beta + 2) q^{47} + q^{49} + ( - 2 \beta - 3) q^{51} + ( - 3 \beta + 2) q^{53} - 3 \beta q^{55} + (2 \beta + 3) q^{57} + 3 \beta q^{59} + ( - 3 \beta + 1) q^{61} + q^{63} + (2 \beta + 1) q^{65} + (\beta - 12) q^{67} + q^{69} + (3 \beta - 9) q^{71} + 11 q^{73} + (5 \beta + 2) q^{75} + (2 \beta - 3) q^{77} + (8 \beta - 3) q^{79} + q^{81} + (4 \beta - 11) q^{83} + (9 \beta + 12) q^{85} + (2 \beta - 3) q^{87} + (5 \beta - 9) q^{89} + ( - \beta + 1) q^{91} + ( - 2 \beta + 1) q^{93} + ( - 9 \beta - 12) q^{95} + (4 \beta + 9) q^{97} + (2 \beta - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{11} + q^{13} - 5 q^{15} - 8 q^{17} + 8 q^{19} + 2 q^{21} + 2 q^{23} + 9 q^{25} + 2 q^{27} - 4 q^{29} - 4 q^{33} - 5 q^{35} + 4 q^{37} + q^{39} - 5 q^{43} - 5 q^{45} + 2 q^{47} + 2 q^{49} - 8 q^{51} + q^{53} - 3 q^{55} + 8 q^{57} + 3 q^{59} - q^{61} + 2 q^{63} + 4 q^{65} - 23 q^{67} + 2 q^{69} - 15 q^{71} + 22 q^{73} + 9 q^{75} - 4 q^{77} + 2 q^{79} + 2 q^{81} - 18 q^{83} + 33 q^{85} - 4 q^{87} - 13 q^{89} + q^{91} - 33 q^{95} + 22 q^{97} - 4 q^{99}+O(q^{100}) \) 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
2.30278
−1.30278
0 1.00000 0 −4.30278 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 −0.697224 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.bf 2
4.b odd 2 1 1932.2.a.c 2
12.b even 2 1 5796.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.2.a.c 2 4.b odd 2 1
5796.2.a.m 2 12.b even 2 1
7728.2.a.bf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{2} + 5T_{5} + 3 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 9 \) Copy content Toggle raw display
\( T_{13}^{2} - T_{13} - 3 \) Copy content Toggle raw display
\( T_{17}^{2} + 8T_{17} + 3 \) 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
$5$ \( T^{2} + 5T + 3 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$13$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 3 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 3 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$31$ \( T^{2} - 13 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 9 \) Copy content Toggle raw display
$41$ \( T^{2} - 13 \) Copy content Toggle raw display
$43$ \( T^{2} + 5T + 3 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$53$ \( T^{2} - T - 29 \) Copy content Toggle raw display
$59$ \( T^{2} - 3T - 27 \) Copy content Toggle raw display
$61$ \( T^{2} + T - 29 \) Copy content Toggle raw display
$67$ \( T^{2} + 23T + 129 \) Copy content Toggle raw display
$71$ \( T^{2} + 15T + 27 \) Copy content Toggle raw display
$73$ \( (T - 11)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 2T - 207 \) Copy content Toggle raw display
$83$ \( T^{2} + 18T + 29 \) Copy content Toggle raw display
$89$ \( T^{2} + 13T - 39 \) Copy content Toggle raw display
$97$ \( T^{2} - 22T + 69 \) Copy content Toggle raw display
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