Properties

Label 648.4.a.k
Level $648$
Weight $4$
Character orbit 648.a
Self dual yes
Analytic conductor $38.233$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 648.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.2332376837\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - x^{4} - 19 x^{3} + 4 x^{2} + 81 x + 12\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 72)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{5} + ( 1 + \beta_{1} ) q^{7} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{5} + ( 1 + \beta_{1} ) q^{7} + ( 5 - 2 \beta_{2} + \beta_{4} ) q^{11} + ( 7 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{13} + ( -6 - 3 \beta_{2} + \beta_{3} - \beta_{4} ) q^{17} + ( 12 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{19} + ( -17 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{23} + ( 62 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{25} + ( -23 + 6 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{29} + ( 49 - \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{31} + ( 41 - 11 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} - 6 \beta_{4} ) q^{35} + ( 72 - 6 \beta_{1} + 12 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} ) q^{37} + ( -33 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} + \beta_{4} ) q^{41} + ( 161 - 8 \beta_{2} - \beta_{3} + 6 \beta_{4} ) q^{43} + ( 99 + 7 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} + 10 \beta_{4} ) q^{47} + ( 218 + 4 \beta_{1} + 11 \beta_{2} - 5 \beta_{3} + 9 \beta_{4} ) q^{49} + ( -68 + 10 \beta_{1} + 16 \beta_{2} - 7 \beta_{3} + 7 \beta_{4} ) q^{53} + ( 289 - 5 \beta_{1} - 18 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} ) q^{55} + ( 121 - 6 \beta_{1} + 12 \beta_{2} - 5 \beta_{3} ) q^{59} + ( 267 - 2 \beta_{1} + 23 \beta_{2} + 5 \beta_{3} - 13 \beta_{4} ) q^{61} + ( -121 - 32 \beta_{1} + 9 \beta_{2} + 7 \beta_{3} - 11 \beta_{4} ) q^{65} + ( 305 - 6 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - 15 \beta_{4} ) q^{67} + ( 170 + 18 \beta_{1} + 18 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{71} + ( 390 + 8 \beta_{1} - 23 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} ) q^{73} + ( -187 + 4 \beta_{1} + 3 \beta_{2} + 20 \beta_{3} - 4 \beta_{4} ) q^{77} + ( 359 + 17 \beta_{1} - 18 \beta_{2} - 17 \beta_{3} + 3 \beta_{4} ) q^{79} + ( 215 + 23 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} ) q^{83} + ( 576 + 6 \beta_{1} - 12 \beta_{2} + 17 \beta_{3} - \beta_{4} ) q^{85} + ( -464 - 22 \beta_{1} - 14 \beta_{2} - 17 \beta_{3} + 9 \beta_{4} ) q^{89} + ( 613 + 37 \beta_{1} + 76 \beta_{2} - 3 \beta_{3} + 13 \beta_{4} ) q^{91} + ( 408 + 20 \beta_{1} - 16 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} ) q^{95} + ( 577 - 2 \beta_{1} - 48 \beta_{2} - 7 \beta_{3} + 11 \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{5} + 3q^{7} + O(q^{10}) \) \( 5q - 5q^{5} + 3q^{7} + 25q^{11} + 29q^{13} - 28q^{17} + 64q^{19} - 89q^{23} + 322q^{25} - 129q^{29} + 241q^{31} + 243q^{35} + 366q^{37} - 171q^{41} + 803q^{43} + 477q^{47} + 1072q^{49} - 374q^{53} + 1469q^{55} + 607q^{59} + 1349q^{61} - 527q^{65} + 1549q^{67} + 812q^{71} + 1928q^{73} - 903q^{77} + 1727q^{79} + 1025q^{83} + 2902q^{85} - 2310q^{89} + 2985q^{91} + 2012q^{95} + 2875q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 19 x^{3} + 4 x^{2} + 81 x + 12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{4} - 7 \nu^{3} - 14 \nu^{2} + 43 \nu - 19 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{4} - 7 \nu^{3} - 20 \nu^{2} + 49 \nu + 27 \)
\(\beta_{3}\)\(=\)\( 6 \nu^{4} - 18 \nu^{3} - 60 \nu^{2} + 126 \nu + 58 \)
\(\beta_{4}\)\(=\)\( 10 \nu^{4} - 38 \nu^{3} - 88 \nu^{2} + 290 \nu + 60 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3} - 6 \beta_{2} - 2 \beta_{1} + 6\)\()/36\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} + \beta_{3} - 12 \beta_{2} + 4 \beta_{1} + 282\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(7 \beta_{4} + 19 \beta_{3} - 78 \beta_{2} - 14 \beta_{1} + 318\)\()/36\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{4} + 26 \beta_{3} - 114 \beta_{2} + 20 \beta_{1} + 1650\)\()/18\)

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
−0.150045
−2.80998
−2.71954
2.50994
4.16963
0 0 0 −20.2222 0 −24.7425 0 0 0
1.2 0 0 0 −12.3969 0 30.6325 0 0 0
1.3 0 0 0 2.98196 0 11.7109 0 0 0
1.4 0 0 0 6.31871 0 −29.5796 0 0 0
1.5 0 0 0 18.3184 0 14.9787 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

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 648.4.a.k 5
3.b odd 2 1 648.4.a.l 5
4.b odd 2 1 1296.4.a.bc 5
9.c even 3 2 216.4.i.b 10
9.d odd 6 2 72.4.i.b 10
12.b even 2 1 1296.4.a.bd 5
36.f odd 6 2 432.4.i.f 10
36.h even 6 2 144.4.i.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.i.b 10 9.d odd 6 2
144.4.i.f 10 36.h even 6 2
216.4.i.b 10 9.c even 3 2
432.4.i.f 10 36.f odd 6 2
648.4.a.k 5 1.a even 1 1 trivial
648.4.a.l 5 3.b odd 2 1
1296.4.a.bc 5 4.b odd 2 1
1296.4.a.bd 5 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} + 5 T_{5}^{4} - 461 T_{5}^{3} - 1097 T_{5}^{2} + 36176 T_{5} - 86528 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(648))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( T^{5} \)
$5$ \( -86528 + 36176 T - 1097 T^{2} - 461 T^{3} + 5 T^{4} + T^{5} \)
$7$ \( -3932604 + 434844 T + 6615 T^{2} - 1389 T^{3} - 3 T^{4} + T^{5} \)
$11$ \( 2450383 + 2671697 T + 77482 T^{2} - 3890 T^{3} - 25 T^{4} + T^{5} \)
$13$ \( 11723252 + 2020232 T - 43435 T^{2} - 5405 T^{3} - 29 T^{4} + T^{5} \)
$17$ \( 465026264 + 9589868 T - 387766 T^{2} - 8927 T^{3} + 28 T^{4} + T^{5} \)
$19$ \( -219796928 + 25870544 T + 365212 T^{2} - 10133 T^{3} - 64 T^{4} + T^{5} \)
$23$ \( 54657912052 + 580906268 T - 4432661 T^{2} - 48365 T^{3} + 89 T^{4} + T^{5} \)
$29$ \( -2405376324 + 204856704 T - 2562561 T^{2} - 36093 T^{3} + 129 T^{4} + T^{5} \)
$31$ \( -2225150096 + 158063312 T + 3945385 T^{2} - 33665 T^{3} - 241 T^{4} + T^{5} \)
$37$ \( -525481299072 - 3133248768 T + 69985944 T^{2} - 163524 T^{3} - 366 T^{4} + T^{5} \)
$41$ \( -816147765 - 56379231 T - 1096182 T^{2} + 222 T^{3} + 171 T^{4} + T^{5} \)
$43$ \( 236167016117 - 5502242383 T + 18306062 T^{2} + 143086 T^{3} - 803 T^{4} + T^{5} \)
$47$ \( 134565454332 - 9738525204 T + 94739769 T^{2} - 184845 T^{3} - 477 T^{4} + T^{5} \)
$53$ \( 758189779072 - 3898947712 T - 95073272 T^{2} - 242948 T^{3} + 374 T^{4} + T^{5} \)
$59$ \( 804795801121 - 19481707399 T + 112363102 T^{2} - 93914 T^{3} - 607 T^{4} + T^{5} \)
$61$ \( -4452174344512 - 64161140320 T + 348549065 T^{2} + 110179 T^{3} - 1349 T^{4} + T^{5} \)
$67$ \( -6065728349525 - 39525998239 T + 147555010 T^{2} + 428158 T^{3} - 1549 T^{4} + T^{5} \)
$71$ \( -24420797440 + 6494698496 T + 243239552 T^{2} - 329744 T^{3} - 812 T^{4} + T^{5} \)
$73$ \( 28702127905256 - 130341479668 T - 56743786 T^{2} + 1119481 T^{3} - 1928 T^{4} + T^{5} \)
$79$ \( 85346901126224 - 861490732960 T + 1726168679 T^{2} - 226097 T^{3} - 1727 T^{4} + T^{5} \)
$83$ \( -70154840960560 + 156684449096 T + 655571969 T^{2} - 735857 T^{3} - 1025 T^{4} + T^{5} \)
$89$ \( -887800297007520 - 3596560713072 T - 4312041264 T^{2} - 277992 T^{3} + 2310 T^{4} + T^{5} \)
$97$ \( 27675431349397 - 286750735375 T + 373099366 T^{2} + 1887694 T^{3} - 2875 T^{4} + T^{5} \)
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