Properties

Label 5520.2.a.bv
Level $5520$
Weight $2$
Character orbit 5520.a
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
Defining polynomial: \( x^{3} - x^{2} - 16x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1380)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + (\beta_1 - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{5} + (\beta_1 - 1) q^{7} + q^{9} + ( - \beta_{2} - 1) q^{11} + ( - \beta_{2} + 1) q^{13} + q^{15} + ( - \beta_{2} + \beta_1) q^{17} + ( - \beta_{2} - 3) q^{19} + ( - \beta_1 + 1) q^{21} - q^{23} + q^{25} - q^{27} + (\beta_{2} - \beta_1) q^{29} + ( - \beta_1 - 3) q^{31} + (\beta_{2} + 1) q^{33} + ( - \beta_1 + 1) q^{35} + ( - 2 \beta_{2} + \beta_1 + 1) q^{37} + (\beta_{2} - 1) q^{39} + ( - \beta_{2} + 3 \beta_1 + 2) q^{41} + (2 \beta_1 - 6) q^{43} - q^{45} + ( - \beta_{2} + 2 \beta_1 + 1) q^{47} + (2 \beta_{2} - \beta_1 + 4) q^{49} + (\beta_{2} - \beta_1) q^{51} + ( - \beta_{2} + 3 \beta_1 + 2) q^{53} + (\beta_{2} + 1) q^{55} + (\beta_{2} + 3) q^{57} + (\beta_{2} - \beta_1) q^{59} + ( - \beta_{2} + 2 \beta_1 + 3) q^{61} + (\beta_1 - 1) q^{63} + (\beta_{2} - 1) q^{65} + ( - \beta_1 - 3) q^{67} + q^{69} + ( - \beta_{2} + 3 \beta_1 + 2) q^{71} + (3 \beta_{2} + 5) q^{73} - q^{75} + (\beta_{2} - 4 \beta_1 - 3) q^{77} + ( - 2 \beta_1 - 4) q^{79} + q^{81} + (\beta_{2} + \beta_1 - 4) q^{83} + (\beta_{2} - \beta_1) q^{85} + ( - \beta_{2} + \beta_1) q^{87} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{89} + (\beta_{2} - 2 \beta_1 - 5) q^{91} + (\beta_1 + 3) q^{93} + (\beta_{2} + 3) q^{95} + (2 \beta_{2} - 4 \beta_1 - 6) q^{97} + ( - \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 4 q^{11} + 2 q^{13} + 3 q^{15} - 10 q^{19} + 2 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 10 q^{31} + 4 q^{33} + 2 q^{35} + 2 q^{37} - 2 q^{39} + 8 q^{41} - 16 q^{43} - 3 q^{45} + 4 q^{47} + 13 q^{49} + 8 q^{53} + 4 q^{55} + 10 q^{57} + 10 q^{61} - 2 q^{63} - 2 q^{65} - 10 q^{67} + 3 q^{69} + 8 q^{71} + 18 q^{73} - 3 q^{75} - 12 q^{77} - 14 q^{79} + 3 q^{81} - 10 q^{83} + 2 q^{89} - 16 q^{91} + 10 q^{93} + 10 q^{95} - 20 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 16x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - \nu - 10 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + \beta _1 + 10 \) 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
−3.20905
−0.526440
4.73549
0 −1.00000 0 −1.00000 0 −4.20905 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 −1.52644 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 3.73549 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(23\) \(1\)

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 5520.2.a.bv 3
4.b odd 2 1 1380.2.a.j 3
12.b even 2 1 4140.2.a.s 3
20.d odd 2 1 6900.2.a.x 3
20.e even 4 2 6900.2.f.r 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.j 3 4.b odd 2 1
4140.2.a.s 3 12.b even 2 1
5520.2.a.bv 3 1.a even 1 1 trivial
6900.2.a.x 3 20.d odd 2 1
6900.2.f.r 6 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\):

\( T_{7}^{3} + 2T_{7}^{2} - 15T_{7} - 24 \) Copy content Toggle raw display
\( T_{11}^{3} + 4T_{11}^{2} - 14T_{11} - 48 \) Copy content Toggle raw display
\( T_{13}^{3} - 2T_{13}^{2} - 18T_{13} - 12 \) Copy content Toggle raw display
\( T_{17}^{3} - 21T_{17} + 18 \) Copy content Toggle raw display
\( T_{19}^{3} + 10T_{19}^{2} + 14T_{19} - 52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} - 15 T - 24 \) Copy content Toggle raw display
$11$ \( T^{3} + 4 T^{2} - 14 T - 48 \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} - 18 T - 12 \) Copy content Toggle raw display
$17$ \( T^{3} - 21T + 18 \) Copy content Toggle raw display
$19$ \( T^{3} + 10 T^{2} + 14 T - 52 \) Copy content Toggle raw display
$23$ \( (T + 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 21T - 18 \) Copy content Toggle raw display
$31$ \( T^{3} + 10 T^{2} + 17 T - 4 \) Copy content Toggle raw display
$37$ \( T^{3} - 2 T^{2} - 63 T - 108 \) Copy content Toggle raw display
$41$ \( T^{3} - 8 T^{2} - 101 T + 582 \) Copy content Toggle raw display
$43$ \( T^{3} + 16 T^{2} + 20 T - 304 \) Copy content Toggle raw display
$47$ \( T^{3} - 4 T^{2} - 50 T + 216 \) Copy content Toggle raw display
$53$ \( T^{3} - 8 T^{2} - 101 T + 582 \) Copy content Toggle raw display
$59$ \( T^{3} - 21T - 18 \) Copy content Toggle raw display
$61$ \( T^{3} - 10 T^{2} - 22 T + 292 \) Copy content Toggle raw display
$67$ \( T^{3} + 10 T^{2} + 17 T - 4 \) Copy content Toggle raw display
$71$ \( T^{3} - 8 T^{2} - 101 T + 582 \) Copy content Toggle raw display
$73$ \( T^{3} - 18 T^{2} - 66 T + 1492 \) Copy content Toggle raw display
$79$ \( T^{3} + 14T^{2} - 96 \) Copy content Toggle raw display
$83$ \( T^{3} + 10 T^{2} - 17 T - 228 \) Copy content Toggle raw display
$89$ \( T^{3} - 2 T^{2} - 200 T + 912 \) Copy content Toggle raw display
$97$ \( T^{3} + 20 T^{2} - 88 T - 2336 \) Copy content Toggle raw display
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