Properties

Label 5520.2.a.by
Level $5520$
Weight $2$
Character orbit 5520.a
Self dual yes
Analytic conductor $44.077$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 345)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
2.34292
−1.81361
0.470683
0 −1.00000 0 1.00000 0 −4.48929 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −2.28917 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 0.778457 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.by 3
4.b odd 2 1 345.2.a.j 3
12.b even 2 1 1035.2.a.n 3
20.d odd 2 1 1725.2.a.bi 3
20.e even 4 2 1725.2.b.u 6
60.h even 2 1 5175.2.a.br 3
92.b even 2 1 7935.2.a.u 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.a.j 3 4.b odd 2 1
1035.2.a.n 3 12.b even 2 1
1725.2.a.bi 3 20.d odd 2 1
1725.2.b.u 6 20.e even 4 2
5175.2.a.br 3 60.h even 2 1
5520.2.a.by 3 1.a even 1 1 trivial
7935.2.a.u 3 92.b even 2 1

Hecke kernels

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

\( T_{7}^{3} + 6 T_{7}^{2} + 5 T_{7} - 8 \)
\( T_{11}^{3} - 2 T_{11}^{2} - 6 T_{11} + 8 \)
\( T_{13}^{3} - 4 T_{13}^{2} - 2 T_{13} + 4 \)
\( T_{17}^{3} + 2 T_{17}^{2} - 3 T_{17} - 2 \)
\( T_{19}^{3} + 2 T_{19}^{2} - 14 T_{19} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( -8 + 5 T + 6 T^{2} + T^{3} \)
$11$ \( 8 - 6 T - 2 T^{2} + T^{3} \)
$13$ \( 4 - 2 T - 4 T^{2} + T^{3} \)
$17$ \( -2 - 3 T + 2 T^{2} + T^{3} \)
$19$ \( -32 - 14 T + 2 T^{2} + T^{3} \)
$23$ \( ( 1 + T )^{3} \)
$29$ \( -86 - 27 T + 6 T^{2} + T^{3} \)
$31$ \( -32 - 15 T + 2 T^{2} + T^{3} \)
$37$ \( -2 - 7 T + T^{3} \)
$41$ \( -218 - 131 T + 2 T^{2} + T^{3} \)
$43$ \( 176 + 116 T + 20 T^{2} + T^{3} \)
$47$ \( -16 - 22 T - 6 T^{2} + T^{3} \)
$53$ \( -54 - 27 T + 6 T^{2} + T^{3} \)
$59$ \( -32 - 43 T - 12 T^{2} + T^{3} \)
$61$ \( 4 - 10 T + 4 T^{2} + T^{3} \)
$67$ \( 724 - 127 T - 6 T^{2} + T^{3} \)
$71$ \( 148 - 19 T - 8 T^{2} + T^{3} \)
$73$ \( -932 - 138 T + 8 T^{2} + T^{3} \)
$79$ \( -64 - 112 T - 4 T^{2} + T^{3} \)
$83$ \( -92 - 3 T + 8 T^{2} + T^{3} \)
$89$ \( -16 - 160 T - 6 T^{2} + T^{3} \)
$97$ \( 2176 - 160 T - 14 T^{2} + T^{3} \)
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