Properties

Label 648.4.a.e
Level $648$
Weight $4$
Character orbit 648.a
Self dual yes
Analytic conductor $38.233$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{129}) \)
Defining polynomial: \(x^{2} - x - 32\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{129}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 + \beta ) q^{5} + ( -3 - \beta ) q^{7} +O(q^{10})\) \( q + ( 2 + \beta ) q^{5} + ( -3 - \beta ) q^{7} + ( 5 + 3 \beta ) q^{11} + ( -20 - 5 \beta ) q^{13} + ( -17 - 4 \beta ) q^{17} + ( 17 - 5 \beta ) q^{19} + ( -49 - 7 \beta ) q^{23} + ( 8 + 4 \beta ) q^{25} + ( -60 + 11 \beta ) q^{29} + ( -100 + 8 \beta ) q^{31} + ( -135 - 5 \beta ) q^{35} + ( 240 + 7 \beta ) q^{37} + ( -96 + 10 \beta ) q^{41} + ( -167 - 25 \beta ) q^{43} + ( -150 - 38 \beta ) q^{47} + ( -205 + 6 \beta ) q^{49} + ( -34 + 32 \beta ) q^{53} + ( 397 + 11 \beta ) q^{55} + ( -310 + 6 \beta ) q^{59} + ( 100 + 35 \beta ) q^{61} + ( -685 - 30 \beta ) q^{65} + ( -703 + 11 \beta ) q^{67} + ( -695 + 3 \beta ) q^{71} + ( 901 - 18 \beta ) q^{73} + ( -402 - 14 \beta ) q^{77} + ( -167 + 79 \beta ) q^{79} + ( 250 + 74 \beta ) q^{83} + ( -550 - 25 \beta ) q^{85} + ( -45 - 24 \beta ) q^{89} + ( 705 + 35 \beta ) q^{91} + ( -611 + 7 \beta ) q^{95} + ( -100 - 122 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 6q^{7} + O(q^{10}) \) \( 2q + 4q^{5} - 6q^{7} + 10q^{11} - 40q^{13} - 34q^{17} + 34q^{19} - 98q^{23} + 16q^{25} - 120q^{29} - 200q^{31} - 270q^{35} + 480q^{37} - 192q^{41} - 334q^{43} - 300q^{47} - 410q^{49} - 68q^{53} + 794q^{55} - 620q^{59} + 200q^{61} - 1370q^{65} - 1406q^{67} - 1390q^{71} + 1802q^{73} - 804q^{77} - 334q^{79} + 500q^{83} - 1100q^{85} - 90q^{89} + 1410q^{91} - 1222q^{95} - 200q^{97} + O(q^{100}) \)

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
−5.17891
6.17891
0 0 0 −9.35782 0 8.35782 0 0 0
1.2 0 0 0 13.3578 0 −14.3578 0 0 0
\(n\): e.g. 2-40 or 990-1000
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.e yes 2
3.b odd 2 1 648.4.a.d 2
4.b odd 2 1 1296.4.a.p 2
9.c even 3 2 648.4.i.p 4
9.d odd 6 2 648.4.i.q 4
12.b even 2 1 1296.4.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.d 2 3.b odd 2 1
648.4.a.e yes 2 1.a even 1 1 trivial
648.4.i.p 4 9.c even 3 2
648.4.i.q 4 9.d odd 6 2
1296.4.a.n 2 12.b even 2 1
1296.4.a.p 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 4 T_{5} - 125 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(648))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -125 - 4 T + T^{2} \)
$7$ \( -120 + 6 T + T^{2} \)
$11$ \( -1136 - 10 T + T^{2} \)
$13$ \( -2825 + 40 T + T^{2} \)
$17$ \( -1775 + 34 T + T^{2} \)
$19$ \( -2936 - 34 T + T^{2} \)
$23$ \( -3920 + 98 T + T^{2} \)
$29$ \( -12009 + 120 T + T^{2} \)
$31$ \( 1744 + 200 T + T^{2} \)
$37$ \( 51279 - 480 T + T^{2} \)
$41$ \( -3684 + 192 T + T^{2} \)
$43$ \( -52736 + 334 T + T^{2} \)
$47$ \( -163776 + 300 T + T^{2} \)
$53$ \( -130940 + 68 T + T^{2} \)
$59$ \( 91456 + 620 T + T^{2} \)
$61$ \( -148025 - 200 T + T^{2} \)
$67$ \( 478600 + 1406 T + T^{2} \)
$71$ \( 481864 + 1390 T + T^{2} \)
$73$ \( 770005 - 1802 T + T^{2} \)
$79$ \( -777200 + 334 T + T^{2} \)
$83$ \( -643904 - 500 T + T^{2} \)
$89$ \( -72279 + 90 T + T^{2} \)
$97$ \( -1910036 + 200 T + T^{2} \)
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