Properties

Label 5472.2.a.bt
Level $5472$
Weight $2$
Character orbit 5472.a
Self dual yes
Analytic conductor $43.694$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5472.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.6941399860\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 608)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{5} - \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{5} - \beta_{3} q^{7} + (\beta_{2} - \beta_1 + 2) q^{11} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{13} + (2 \beta_{3} - \beta_1 - 1) q^{17} - q^{19} + ( - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{23} + (2 \beta_{3} + \beta_{2} - \beta_1 + 5) q^{25} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{29} + ( - 2 \beta_{3} + 2 \beta_{2} + 2) q^{31} + ( - 3 \beta_{2} + \beta_1) q^{35} + ( - 2 \beta_{3} + 2 \beta_{2} + 4) q^{37} - 2 \beta_1 q^{41} + (\beta_{2} + \beta_1) q^{43} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{47} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{49} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{53} + (2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 10) q^{55} + ( - \beta_{2} + 2 \beta_1 + 5) q^{59} + ( - 3 \beta_{2} + \beta_1 - 2) q^{61} + ( - 2 \beta_{3} + 8 \beta_{2} - 2 \beta_1 + 6) q^{65} + (\beta_{2} + 2 \beta_1 - 1) q^{67} + ( - 2 \beta_{3} - 8) q^{71} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{73} + ( - 2 \beta_{3} - 3 \beta_{2} + \beta_1) q^{77} + (2 \beta_{3} - 4 \beta_{2} - 4) q^{79} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{83} + (7 \beta_{2} - \beta_1 + 8) q^{85} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 6) q^{89} + ( - 4 \beta_{3} + \beta_{2} - 3) q^{91} + ( - \beta_{2} + \beta_1) q^{95} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{5} + q^{7} + 7 q^{11} + 10 q^{13} - 5 q^{17} - 4 q^{19} - 8 q^{23} + 17 q^{25} - 10 q^{29} + 6 q^{31} + 5 q^{35} + 14 q^{37} + 2 q^{41} - 3 q^{43} - 3 q^{47} + 7 q^{49} - 4 q^{53} + 35 q^{55} + 20 q^{59} - 3 q^{61} + 12 q^{65} - 8 q^{67} - 30 q^{71} + 9 q^{73} + 7 q^{77} - 10 q^{79} + 4 q^{83} + 19 q^{85} + 16 q^{89} - 10 q^{91} + q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + \beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + \beta_{2} + 3\beta _1 + 5 \) 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
−0.329727
2.69353
1.32973
−1.69353
0 0 0 −3.40617 0 −2.50407 0 0 0
1.2 0 0 0 −2.64453 0 0.180969 0 0 0
1.3 0 0 0 0.844614 0 5.06562 0 0 0
1.4 0 0 0 4.20608 0 −1.74252 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(19\) \(1\)

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 5472.2.a.bt 4
3.b odd 2 1 608.2.a.i 4
4.b odd 2 1 5472.2.a.bs 4
12.b even 2 1 608.2.a.j yes 4
24.f even 2 1 1216.2.a.w 4
24.h odd 2 1 1216.2.a.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.a.i 4 3.b odd 2 1
608.2.a.j yes 4 12.b even 2 1
1216.2.a.w 4 24.f even 2 1
1216.2.a.x 4 24.h odd 2 1
5472.2.a.bs 4 4.b odd 2 1
5472.2.a.bt 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5472))\):

\( T_{5}^{4} + T_{5}^{3} - 18T_{5}^{2} - 24T_{5} + 32 \) Copy content Toggle raw display
\( T_{7}^{4} - T_{7}^{3} - 17T_{7}^{2} - 19T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 7T_{11}^{3} + 28T_{11} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} - 10T_{13}^{3} + 7T_{13}^{2} + 158T_{13} - 344 \) Copy content Toggle raw display
\( T_{23}^{4} + 8T_{23}^{3} - 17T_{23}^{2} - 132T_{23} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} - 18 T^{2} - 24 T + 32 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} - 17 T^{2} - 19 T + 4 \) Copy content Toggle raw display
$11$ \( T^{4} - 7 T^{3} + 28 T + 16 \) Copy content Toggle raw display
$13$ \( T^{4} - 10 T^{3} + 7 T^{2} + 158 T - 344 \) Copy content Toggle raw display
$17$ \( T^{4} + 5 T^{3} - 51 T^{2} - 141 T + 698 \) Copy content Toggle raw display
$19$ \( (T + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 8 T^{3} - 17 T^{2} - 132 T + 64 \) Copy content Toggle raw display
$29$ \( T^{4} + 10 T^{3} + 7 T^{2} - 158 T - 344 \) Copy content Toggle raw display
$31$ \( T^{4} - 6 T^{3} - 56 T^{2} + 416 T - 512 \) Copy content Toggle raw display
$37$ \( T^{4} - 14 T^{3} + 4 T^{2} + \cdots - 1504 \) Copy content Toggle raw display
$41$ \( T^{4} - 2 T^{3} - 72 T^{2} + 192 T + 512 \) Copy content Toggle raw display
$43$ \( T^{4} + 3 T^{3} - 32 T^{2} - 96 T - 64 \) Copy content Toggle raw display
$47$ \( T^{4} + 3 T^{3} - 108 T^{2} - 176 T - 64 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} - 69 T^{2} - 384 T - 508 \) Copy content Toggle raw display
$59$ \( T^{4} - 20 T^{3} + 85 T^{2} + \cdots - 152 \) Copy content Toggle raw display
$61$ \( T^{4} + 3 T^{3} - 66 T^{2} - 28 T + 344 \) Copy content Toggle raw display
$67$ \( T^{4} + 8 T^{3} - 75 T^{2} - 602 T - 824 \) Copy content Toggle raw display
$71$ \( T^{4} + 30 T^{3} + 268 T^{2} + \cdots - 2432 \) Copy content Toggle raw display
$73$ \( T^{4} - 9 T^{3} - 47 T^{2} + 309 T + 2 \) Copy content Toggle raw display
$79$ \( T^{4} + 10 T^{3} - 100 T^{2} + \cdots - 3136 \) Copy content Toggle raw display
$83$ \( T^{4} - 4 T^{3} - 116 T^{2} + \cdots - 1024 \) Copy content Toggle raw display
$89$ \( T^{4} - 16 T^{3} - 204 T^{2} + \cdots - 15296 \) Copy content Toggle raw display
$97$ \( T^{4} + 6 T^{3} - 60 T^{2} - 88 T + 608 \) Copy content Toggle raw display
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