Properties

Label 7728.2.a.by
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.1229.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 6\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{1} q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{1} q^{5} - q^{7} + q^{9} + ( 1 - \beta_{1} - \beta_{2} ) q^{11} + ( -1 + \beta_{2} ) q^{13} + \beta_{1} q^{15} + ( 1 - \beta_{1} + \beta_{2} ) q^{17} + ( 3 - \beta_{1} + \beta_{2} ) q^{19} - q^{21} - q^{23} + \beta_{2} q^{25} + q^{27} + ( -1 + \beta_{1} - \beta_{2} ) q^{29} + ( -3 - \beta_{1} - \beta_{2} ) q^{31} + ( 1 - \beta_{1} - \beta_{2} ) q^{33} -\beta_{1} q^{35} + ( 3 + 3 \beta_{1} - \beta_{2} ) q^{37} + ( -1 + \beta_{2} ) q^{39} + ( 5 + \beta_{1} + \beta_{2} ) q^{41} + ( 3 - \beta_{2} ) q^{43} + \beta_{1} q^{45} + ( -6 + 4 \beta_{1} + 2 \beta_{2} ) q^{47} + q^{49} + ( 1 - \beta_{1} + \beta_{2} ) q^{51} + ( 6 - 3 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -4 - \beta_{1} - 2 \beta_{2} ) q^{55} + ( 3 - \beta_{1} + \beta_{2} ) q^{57} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{59} + ( 5 + 4 \beta_{1} - \beta_{2} ) q^{61} - q^{63} + ( -1 + \beta_{1} + \beta_{2} ) q^{65} + ( 4 + \beta_{1} - 2 \beta_{2} ) q^{67} - q^{69} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{71} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{73} + \beta_{2} q^{75} + ( -1 + \beta_{1} + \beta_{2} ) q^{77} + ( 7 - \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( -1 + \beta_{1} + \beta_{2} ) q^{83} + ( -6 + 3 \beta_{1} ) q^{85} + ( -1 + \beta_{1} - \beta_{2} ) q^{87} + ( 5 + 2 \beta_{1} - \beta_{2} ) q^{89} + ( 1 - \beta_{2} ) q^{91} + ( -3 - \beta_{1} - \beta_{2} ) q^{93} + ( -6 + 5 \beta_{1} ) q^{95} + ( 7 - \beta_{1} - \beta_{2} ) q^{97} + ( 1 - \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + q^{5} - 3q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + q^{5} - 3q^{7} + 3q^{9} + 2q^{11} - 3q^{13} + q^{15} + 2q^{17} + 8q^{19} - 3q^{21} - 3q^{23} + 3q^{27} - 2q^{29} - 10q^{31} + 2q^{33} - q^{35} + 12q^{37} - 3q^{39} + 16q^{41} + 9q^{43} + q^{45} - 14q^{47} + 3q^{49} + 2q^{51} + 15q^{53} - 13q^{55} + 8q^{57} + 11q^{59} + 19q^{61} - 3q^{63} - 2q^{65} + 13q^{67} - 3q^{69} + 13q^{71} - 10q^{73} - 2q^{77} + 20q^{79} + 3q^{81} - 2q^{83} - 15q^{85} - 2q^{87} + 17q^{89} + 3q^{91} - 10q^{93} - 13q^{95} + 20q^{97} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 7 x + 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)

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
−2.59261
0.841083
2.75153
0 1.00000 0 −2.59261 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 0.841083 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 2.75153 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.by 3
4.b odd 2 1 3864.2.a.n 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.n 3 4.b odd 2 1
7728.2.a.by 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{3} - T_{5}^{2} - 7 T_{5} + 6 \)
\( T_{11}^{3} - 2 T_{11}^{2} - 19 T_{11} + 36 \)
\( T_{13}^{3} + 3 T_{13}^{2} - 11 T_{13} + 6 \)
\( T_{17}^{3} - 2 T_{17}^{2} - 21 T_{17} + 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( 6 - 7 T - T^{2} + T^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( 36 - 19 T - 2 T^{2} + T^{3} \)
$13$ \( 6 - 11 T + 3 T^{2} + T^{3} \)
$17$ \( 18 - 21 T - 2 T^{2} + T^{3} \)
$19$ \( 44 - T - 8 T^{2} + T^{3} \)
$23$ \( ( 1 + T )^{3} \)
$29$ \( -18 - 21 T + 2 T^{2} + T^{3} \)
$31$ \( -8 + 13 T + 10 T^{2} + T^{3} \)
$37$ \( 554 - 35 T - 12 T^{2} + T^{3} \)
$41$ \( -66 + 65 T - 16 T^{2} + T^{3} \)
$43$ \( -4 + 13 T - 9 T^{2} + T^{3} \)
$47$ \( -1472 - 100 T + 14 T^{2} + T^{3} \)
$53$ \( 922 - 41 T - 15 T^{2} + T^{3} \)
$59$ \( 348 - 25 T - 11 T^{2} + T^{3} \)
$61$ \( 1206 - 15 T - 19 T^{2} + T^{3} \)
$67$ \( 44 - 9 T - 13 T^{2} + T^{3} \)
$71$ \( 24 + 11 T - 13 T^{2} + T^{3} \)
$73$ \( -678 - 97 T + 10 T^{2} + T^{3} \)
$79$ \( -144 + 111 T - 20 T^{2} + T^{3} \)
$83$ \( -36 - 19 T + 2 T^{2} + T^{3} \)
$89$ \( 166 + 51 T - 17 T^{2} + T^{3} \)
$97$ \( -138 + 113 T - 20 T^{2} + T^{3} \)
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