Properties

Label 7728.2.a.cc
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.75645.1
Defining polynomial: \(x^{4} - x^{3} - 13 x^{2} + 22 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{1} q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{1} q^{5} + q^{7} + q^{9} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} + \beta_{3} q^{13} + \beta_{1} q^{15} + ( 2 - \beta_{1} - \beta_{3} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{19} + q^{21} + q^{23} + ( 1 - 2 \beta_{2} + \beta_{3} ) q^{25} + q^{27} + ( \beta_{1} - \beta_{3} ) q^{29} + ( \beta_{1} - \beta_{3} ) q^{31} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{33} + \beta_{1} q^{35} + ( \beta_{1} + \beta_{3} ) q^{37} + \beta_{3} q^{39} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{41} + ( 2 + 2 \beta_{2} - \beta_{3} ) q^{43} + \beta_{1} q^{45} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{47} + q^{49} + ( 2 - \beta_{1} - \beta_{3} ) q^{51} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{53} + ( 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{55} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{57} + ( 3 + \beta_{1} + \beta_{2} ) q^{59} + ( 1 - \beta_{2} - \beta_{3} ) q^{61} + q^{63} + ( -2 + \beta_{1} + \beta_{3} ) q^{65} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{67} + q^{69} + ( 6 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{71} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{73} + ( 1 - 2 \beta_{2} + \beta_{3} ) q^{75} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{77} + ( 4 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{79} + q^{81} + ( 4 + \beta_{1} + 3 \beta_{3} ) q^{83} + ( -4 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{85} + ( \beta_{1} - \beta_{3} ) q^{87} + ( 4 - 2 \beta_{1} + \beta_{3} ) q^{89} + \beta_{3} q^{91} + ( \beta_{1} - \beta_{3} ) q^{93} + ( -3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{95} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{97} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + q^{5} + 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + q^{5} + 4q^{7} + 4q^{9} + 7q^{11} + q^{13} + q^{15} + 6q^{17} + q^{19} + 4q^{21} + 4q^{23} + 7q^{25} + 4q^{27} + 7q^{33} + q^{35} + 2q^{37} + q^{39} - q^{41} + 5q^{43} + q^{45} + 7q^{47} + 4q^{49} + 6q^{51} + 4q^{53} + 9q^{55} + q^{57} + 12q^{59} + 4q^{61} + 4q^{63} - 6q^{65} + 5q^{67} + 4q^{69} + 19q^{71} + 4q^{73} + 7q^{75} + 7q^{77} + 18q^{79} + 4q^{81} + 20q^{83} - 19q^{85} + 15q^{89} + q^{91} - 7q^{95} + 2q^{97} + 7q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 13 x^{2} + 22 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 13 \nu + 16 \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} - 13 \nu + 10 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 2 \beta_{2} + 6\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 13 \beta_{1} - 10\)

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
−3.90542
0.537835
1.20386
3.16373
0 1.00000 0 −3.90542 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 0.537835 0 1.00000 0 1.00000 0
1.3 0 1.00000 0 1.20386 0 1.00000 0 1.00000 0
1.4 0 1.00000 0 3.16373 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.cc 4
4.b odd 2 1 3864.2.a.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.s 4 4.b odd 2 1
7728.2.a.cc 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{4} - T_{5}^{3} - 13 T_{5}^{2} + 22 T_{5} - 8 \)
\( T_{11}^{4} - 7 T_{11}^{3} - T_{11}^{2} + 31 T_{11} - 8 \)
\( T_{13}^{4} - T_{13}^{3} - 13 T_{13}^{2} + 22 T_{13} - 8 \)
\( T_{17}^{2} - 3 T_{17} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( -8 + 22 T - 13 T^{2} - T^{3} + T^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( -8 + 31 T - T^{2} - 7 T^{3} + T^{4} \)
$13$ \( -8 + 22 T - 13 T^{2} - T^{3} + T^{4} \)
$17$ \( ( -8 - 3 T + T^{2} )^{2} \)
$19$ \( 10 + 25 T - 19 T^{2} - T^{3} + T^{4} \)
$23$ \( ( -1 + T )^{4} \)
$29$ \( 180 - 33 T^{2} + T^{4} \)
$31$ \( 180 - 33 T^{2} + T^{4} \)
$37$ \( ( -10 - T + T^{2} )^{2} \)
$41$ \( -86 + 187 T - 87 T^{2} + T^{3} + T^{4} \)
$43$ \( 736 + 260 T - 67 T^{2} - 5 T^{3} + T^{4} \)
$47$ \( -584 + 352 T - 40 T^{2} - 7 T^{3} + T^{4} \)
$53$ \( 802 + 73 T - 94 T^{2} - 4 T^{3} + T^{4} \)
$59$ \( -788 + 291 T + 8 T^{2} - 12 T^{3} + T^{4} \)
$61$ \( -254 + 199 T - 34 T^{2} - 4 T^{3} + T^{4} \)
$67$ \( 1024 - 4 T - 109 T^{2} - 5 T^{3} + T^{4} \)
$71$ \( -1712 + 352 T + 71 T^{2} - 19 T^{3} + T^{4} \)
$73$ \( 664 + 110 T - 51 T^{2} - 4 T^{3} + T^{4} \)
$79$ \( -4448 + 1014 T + 23 T^{2} - 18 T^{3} + T^{4} \)
$83$ \( -2972 + 896 T + 35 T^{2} - 20 T^{3} + T^{4} \)
$89$ \( 328 + 246 T + 5 T^{2} - 15 T^{3} + T^{4} \)
$97$ \( -920 + 590 T - 97 T^{2} - 2 T^{3} + T^{4} \)
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