Properties

Label 7728.2.a.ch
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 3 x^{5} - 11 x^{4} + 23 x^{3} + 9 x^{2} - 23 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 11 x^{4} + 23 x^{3} + 9 x^{2} - 23 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 12 \nu^{3} + 9 \nu^{2} + 6 \nu - 4 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 12 \nu^{2} + 5 \nu + 10 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{5} + 5 \nu^{4} + 40 \nu^{3} - 17 \nu^{2} - 53 \nu + 6 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -4 \nu^{5} + 7 \nu^{4} + 52 \nu^{3} - 24 \nu^{2} - 61 \nu \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{5} + 9 \nu^{4} + 66 \nu^{3} - 37 \nu^{2} - 87 \nu + 16 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - 2 \beta_{3} - \beta_{2} - \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + 4 \beta_{4} - 6 \beta_{3} - \beta_{2} + 3 \beta_{1} + 21\)\()/4\)
\(\nu^{3}\)\(=\)\(4 \beta_{5} + \beta_{4} - 8 \beta_{3} - 3 \beta_{2} + 7\)
\(\nu^{4}\)\(=\)\((\)\(39 \beta_{5} + 56 \beta_{4} - 126 \beta_{3} - 23 \beta_{2} + 41 \beta_{1} + 263\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(255 \beta_{5} + 124 \beta_{4} - 570 \beta_{3} - 175 \beta_{2} + 69 \beta_{1} + 683\)\()/4\)

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
1.35141
−1.07044
0.0909082
−2.87740
1.23871
4.26681
0 1.00000 0 −2.60318 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 −2.48633 0 1.00000 0 1.00000 0
1.3 0 1.00000 0 −0.535873 0 1.00000 0 1.00000 0
1.4 0 1.00000 0 0.351005 0 1.00000 0 1.00000 0
1.5 0 1.00000 0 3.34309 0 1.00000 0 1.00000 0
1.6 0 1.00000 0 3.93128 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.ch 6
4.b odd 2 1 3864.2.a.w 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.w 6 4.b odd 2 1
7728.2.a.ch 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{6} - 2 T_{5}^{5} - 18 T_{5}^{4} + 17 T_{5}^{3} + 92 T_{5}^{2} + 12 T_{5} - 16 \)
\( T_{11}^{6} - 3 T_{11}^{5} - 41 T_{11}^{4} + 75 T_{11}^{3} + 560 T_{11}^{2} - 416 T_{11} - 2368 \)
\( T_{13}^{6} - 66 T_{13}^{4} - 11 T_{13}^{3} + 1094 T_{13}^{2} + 92 T_{13} - 2264 \)
\( T_{17}^{6} - 6 T_{17}^{5} - 63 T_{17}^{4} + 284 T_{17}^{3} + 1252 T_{17}^{2} - 3200 T_{17} - 7168 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( ( -1 + T )^{6} \)
$5$ \( -16 + 12 T + 92 T^{2} + 17 T^{3} - 18 T^{4} - 2 T^{5} + T^{6} \)
$7$ \( ( -1 + T )^{6} \)
$11$ \( -2368 - 416 T + 560 T^{2} + 75 T^{3} - 41 T^{4} - 3 T^{5} + T^{6} \)
$13$ \( -2264 + 92 T + 1094 T^{2} - 11 T^{3} - 66 T^{4} + T^{6} \)
$17$ \( -7168 - 3200 T + 1252 T^{2} + 284 T^{3} - 63 T^{4} - 6 T^{5} + T^{6} \)
$19$ \( -2368 - 416 T + 560 T^{2} + 75 T^{3} - 41 T^{4} - 3 T^{5} + T^{6} \)
$23$ \( ( -1 + T )^{6} \)
$29$ \( -5800 - 1620 T + 1374 T^{2} + 181 T^{3} - 75 T^{4} - 5 T^{5} + T^{6} \)
$31$ \( -12032 - 7584 T + 2988 T^{2} + 612 T^{3} - 137 T^{4} - 4 T^{5} + T^{6} \)
$37$ \( -296 - 788 T + 334 T^{2} + 157 T^{3} - 63 T^{4} - T^{5} + T^{6} \)
$41$ \( 67604 - 29140 T - 2547 T^{2} + 1808 T^{3} - 114 T^{4} - 12 T^{5} + T^{6} \)
$43$ \( -6784 - 8304 T + 2836 T^{2} + 719 T^{3} - 130 T^{4} - 6 T^{5} + T^{6} \)
$47$ \( 4864 - 2272 T - 876 T^{2} + 610 T^{3} - 73 T^{4} - 6 T^{5} + T^{6} \)
$53$ \( -800 + 2140 T - 1944 T^{2} + 689 T^{3} - 62 T^{4} - 10 T^{5} + T^{6} \)
$59$ \( 4256 - 1840 T - 1378 T^{2} + 779 T^{3} - 56 T^{4} - 14 T^{5} + T^{6} \)
$61$ \( -9208 - 4892 T + 2222 T^{2} + 267 T^{3} - 94 T^{4} - 4 T^{5} + T^{6} \)
$67$ \( -140288 + 5824 T + 11520 T^{2} - 704 T^{3} - 189 T^{4} + 7 T^{5} + T^{6} \)
$71$ \( -68096 + 28352 T + 7720 T^{2} - 886 T^{3} - 171 T^{4} + 7 T^{5} + T^{6} \)
$73$ \( 82304 - 56528 T - 1900 T^{2} + 2704 T^{3} - 203 T^{4} - 10 T^{5} + T^{6} \)
$79$ \( 2048 + 2176 T - 904 T^{2} - 870 T^{3} - 37 T^{4} + 14 T^{5} + T^{6} \)
$83$ \( 295808 + 155584 T + 1856 T^{2} - 3686 T^{3} - 221 T^{4} + 14 T^{5} + T^{6} \)
$89$ \( 24016 + 62912 T - 30816 T^{2} + 3924 T^{3} + 7 T^{4} - 25 T^{5} + T^{6} \)
$97$ \( 23896 + 756 T - 14754 T^{2} + 3795 T^{3} - 227 T^{4} - 11 T^{5} + T^{6} \)
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