Properties

Label 7728.2.a.cg
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: \( x^{6} - x^{5} - 13x^{4} + 7x^{3} + 31x^{2} - 17x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{5} q^{5} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + \beta_{5} q^{5} + q^{7} + q^{9} + ( - \beta_{4} - 1) q^{11} + (\beta_{5} + \beta_{2}) q^{13} - \beta_{5} q^{15} + ( - \beta_{2} + 2) q^{17} + ( - \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{19} - q^{21} - q^{23} + (\beta_{4} + \beta_{3} - \beta_1 + 3) q^{25} - q^{27} + ( - \beta_{4} + \beta_1) q^{29} + \beta_{2} q^{31} + (\beta_{4} + 1) q^{33} + \beta_{5} q^{35} + ( - \beta_{4} + \beta_1) q^{37} + ( - \beta_{5} - \beta_{2}) q^{39} + (\beta_{5} - \beta_{4} - \beta_{3} + 1) q^{41} + (2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{43} + \beta_{5} q^{45} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{47} + q^{49} + (\beta_{2} - 2) q^{51} + (\beta_{5} + \beta_1 - 1) q^{53} + ( - 2 \beta_{5} + \beta_{3} + 2) q^{55} + (\beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{57} + (\beta_{5} + \beta_1 - 3) q^{59} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} + 3) q^{61} + q^{63} + (\beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 6) q^{65} + (2 \beta_{5} + \beta_{3} - 2) q^{67} + q^{69} + (\beta_{5} + \beta_{4} + \beta_1 - 2) q^{71} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 + 4) q^{73} + ( - \beta_{4} - \beta_{3} + \beta_1 - 3) q^{75} + ( - \beta_{4} - 1) q^{77} + (3 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 2) q^{79} + q^{81} + ( - 2 \beta_{5} - \beta_{2}) q^{83} + (2 \beta_{5} - \beta_{3} + 2 \beta_{2} + 2) q^{85} + (\beta_{4} - \beta_1) q^{87} + (2 \beta_{5} - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{89} + (\beta_{5} + \beta_{2}) q^{91} - \beta_{2} q^{93} + ( - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{95} + ( - \beta_{5} + \beta_{3} - \beta_{2} + 6) q^{97} + ( - \beta_{4} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9} - 5 q^{11} + 2 q^{13} + 10 q^{17} - 3 q^{19} - 6 q^{21} - 6 q^{23} + 18 q^{25} - 6 q^{27} + 3 q^{29} + 2 q^{31} + 5 q^{33} + 3 q^{37} - 2 q^{39} + 4 q^{41} - 6 q^{43} + 2 q^{47} + 6 q^{49} - 10 q^{51} - 4 q^{53} + 15 q^{55} + 3 q^{57} - 16 q^{59} + 22 q^{61} + 6 q^{63} + 35 q^{65} - 9 q^{67} + 6 q^{69} - 11 q^{71} + 24 q^{73} - 18 q^{75} - 5 q^{77} - 18 q^{79} + 6 q^{81} - 2 q^{83} + 13 q^{85} - 3 q^{87} + 7 q^{89} + 2 q^{91} - 2 q^{93} - 5 q^{95} + 37 q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 13x^{4} + 7x^{3} + 31x^{2} - 17x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} - 12\nu^{3} + 5\nu^{2} + 21\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 14\nu^{3} - 5\nu^{2} + 32\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + \nu^{4} + 14\nu^{3} - 7\nu^{2} - 39\nu + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{5} - 2\nu^{4} - 40\nu^{3} + 9\nu^{2} + 98\nu - 26 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 3\beta _1 + 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{5} + 7\beta_{4} - 5\beta_{3} - 3\beta_{2} - 3\beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 19\beta_{5} + 27\beta_{4} - 11\beta_{3} - 19\beta_{2} + 29\beta _1 + 166 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 113\beta_{5} + 169\beta_{4} - 105\beta_{3} - 57\beta_{2} - 37\beta _1 + 346 ) / 4 \) 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
0.174924
−1.90697
1.67441
3.52361
−2.82561
0.359645
0 −1.00000 0 −4.39875 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 −2.43390 0 1.00000 0 1.00000 0
1.3 0 −1.00000 0 −0.344522 0 1.00000 0 1.00000 0
1.4 0 −1.00000 0 1.16465 0 1.00000 0 1.00000 0
1.5 0 −1.00000 0 1.74592 0 1.00000 0 1.00000 0
1.6 0 −1.00000 0 4.26660 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.cg 6
4.b odd 2 1 3864.2.a.x 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.x 6 4.b odd 2 1
7728.2.a.cg 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{6} - 24T_{5}^{4} + 5T_{5}^{3} + 100T_{5}^{2} - 60T_{5} - 32 \) Copy content Toggle raw display
\( T_{11}^{6} + 5T_{11}^{5} - 33T_{11}^{4} - 145T_{11}^{3} + 100T_{11}^{2} + 464T_{11} - 64 \) Copy content Toggle raw display
\( T_{13}^{6} - 2T_{13}^{5} - 58T_{13}^{4} + 81T_{13}^{3} + 934T_{13}^{2} - 660T_{13} - 3160 \) Copy content Toggle raw display
\( T_{17}^{6} - 10T_{17}^{5} - 7T_{17}^{4} + 372T_{17}^{3} - 1412T_{17}^{2} + 1936T_{17} - 896 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 24 T^{4} + 5 T^{3} + 100 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$7$ \( (T - 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 5 T^{5} - 33 T^{4} - 145 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$13$ \( T^{6} - 2 T^{5} - 58 T^{4} + \cdots - 3160 \) Copy content Toggle raw display
$17$ \( T^{6} - 10 T^{5} - 7 T^{4} + 372 T^{3} + \cdots - 896 \) Copy content Toggle raw display
$19$ \( T^{6} + 3 T^{5} - 81 T^{4} + \cdots + 6272 \) Copy content Toggle raw display
$23$ \( (T + 1)^{6} \) Copy content Toggle raw display
$29$ \( T^{6} - 3 T^{5} - 75 T^{4} + 31 T^{3} + \cdots - 184 \) Copy content Toggle raw display
$31$ \( T^{6} - 2 T^{5} - 47 T^{4} - 76 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$37$ \( T^{6} - 3 T^{5} - 75 T^{4} + 31 T^{3} + \cdots - 184 \) Copy content Toggle raw display
$41$ \( T^{6} - 4 T^{5} - 122 T^{4} + \cdots - 268 \) Copy content Toggle raw display
$43$ \( T^{6} + 6 T^{5} - 134 T^{4} + \cdots - 5312 \) Copy content Toggle raw display
$47$ \( T^{6} - 2 T^{5} - 249 T^{4} + \cdots - 184000 \) Copy content Toggle raw display
$53$ \( T^{6} + 4 T^{5} - 90 T^{4} + \cdots + 2800 \) Copy content Toggle raw display
$59$ \( T^{6} + 16 T^{5} + 10 T^{4} + \cdots + 14048 \) Copy content Toggle raw display
$61$ \( T^{6} - 22 T^{5} + 14 T^{4} + \cdots - 89480 \) Copy content Toggle raw display
$67$ \( T^{6} + 9 T^{5} - 177 T^{4} + \cdots + 220928 \) Copy content Toggle raw display
$71$ \( T^{6} + 11 T^{5} - 97 T^{4} + \cdots + 17920 \) Copy content Toggle raw display
$73$ \( T^{6} - 24 T^{5} + 47 T^{4} + \cdots + 219520 \) Copy content Toggle raw display
$79$ \( T^{6} + 18 T^{5} - 89 T^{4} + \cdots + 145408 \) Copy content Toggle raw display
$83$ \( T^{6} + 2 T^{5} - 117 T^{4} + \cdots - 3968 \) Copy content Toggle raw display
$89$ \( T^{6} - 7 T^{5} - 177 T^{4} + \cdots - 22832 \) Copy content Toggle raw display
$97$ \( T^{6} - 37 T^{5} + 469 T^{4} + \cdots - 4840 \) Copy content Toggle raw display
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