Properties

Label 5520.2.a.bz
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: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2760)
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.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{5} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{7} + q^{9} + ( -2 - \beta_{1} + \beta_{2} ) q^{11} + ( \beta_{1} - \beta_{2} ) q^{13} - q^{15} + ( -1 - \beta_{1} ) q^{17} + ( -\beta_{1} - \beta_{2} ) q^{19} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{21} + q^{23} + q^{25} - q^{27} + ( 7 - \beta_{1} ) q^{29} + ( -5 - 2 \beta_{1} - 3 \beta_{2} ) q^{31} + ( 2 + \beta_{1} - \beta_{2} ) q^{33} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{35} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{37} + ( -\beta_{1} + \beta_{2} ) q^{39} + ( -1 - 3 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 2 - 2 \beta_{2} ) q^{43} + q^{45} + ( -4 + \beta_{1} + \beta_{2} ) q^{47} + ( 8 + 2 \beta_{1} - 3 \beta_{2} ) q^{49} + ( 1 + \beta_{1} ) q^{51} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{53} + ( -2 - \beta_{1} + \beta_{2} ) q^{55} + ( \beta_{1} + \beta_{2} ) q^{57} + ( -5 - 3 \beta_{1} - 4 \beta_{2} ) q^{59} + ( -6 - \beta_{1} + 3 \beta_{2} ) q^{61} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{63} + ( \beta_{1} - \beta_{2} ) q^{65} + ( -3 - 2 \beta_{1} - 5 \beta_{2} ) q^{67} - q^{69} + ( 3 - 5 \beta_{1} - 2 \beta_{2} ) q^{71} + ( -4 + 5 \beta_{1} + 3 \beta_{2} ) q^{73} - q^{75} + ( -2 - 9 \beta_{1} - 7 \beta_{2} ) q^{77} -4 \beta_{1} q^{79} + q^{81} + ( -7 - \beta_{1} ) q^{83} + ( -1 - \beta_{1} ) q^{85} + ( -7 + \beta_{1} ) q^{87} + ( -4 - 2 \beta_{1} - 4 \beta_{2} ) q^{89} + ( 4 + 5 \beta_{1} + 5 \beta_{2} ) q^{91} + ( 5 + 2 \beta_{1} + 3 \beta_{2} ) q^{93} + ( -\beta_{1} - \beta_{2} ) q^{95} + ( 6 \beta_{1} + 4 \beta_{2} ) q^{97} + ( -2 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 3q^{5} - 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 3q^{5} - 2q^{7} + 3q^{9} - 8q^{11} + 2q^{13} - 3q^{15} - 4q^{17} + 2q^{21} + 3q^{23} + 3q^{25} - 3q^{27} + 20q^{29} - 14q^{31} + 8q^{33} - 2q^{35} - 4q^{37} - 2q^{39} - 8q^{41} + 8q^{43} + 3q^{45} - 12q^{47} + 29q^{49} + 4q^{51} - 8q^{55} - 14q^{59} - 22q^{61} - 2q^{63} + 2q^{65} - 6q^{67} - 3q^{69} + 6q^{71} - 10q^{73} - 3q^{75} - 8q^{77} - 4q^{79} + 3q^{81} - 22q^{83} - 4q^{85} - 20q^{87} - 10q^{89} + 12q^{91} + 14q^{93} + 2q^{97} - 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 6 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 4\)

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
−0.363328
−1.76156
3.12489
0 −1.00000 0 1.00000 0 −4.86799 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −1.89692 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 4.76491 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.bz 3
4.b odd 2 1 2760.2.a.u 3
12.b even 2 1 8280.2.a.bi 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.u 3 4.b odd 2 1
5520.2.a.bz 3 1.a even 1 1 trivial
8280.2.a.bi 3 12.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\):

\( T_{7}^{3} + 2 T_{7}^{2} - 23 T_{7} - 44 \)
\( T_{11}^{3} + 8 T_{11}^{2} + 2 T_{11} - 64 \)
\( T_{13}^{3} - 2 T_{13}^{2} - 18 T_{13} + 44 \)
\( T_{17}^{3} + 4 T_{17}^{2} - T_{17} - 2 \)
\( T_{19}^{3} - 10 T_{19} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( -44 - 23 T + 2 T^{2} + T^{3} \)
$11$ \( -64 + 2 T + 8 T^{2} + T^{3} \)
$13$ \( 44 - 18 T - 2 T^{2} + T^{3} \)
$17$ \( -2 - T + 4 T^{2} + T^{3} \)
$19$ \( -8 - 10 T + T^{3} \)
$23$ \( ( -1 + T )^{3} \)
$29$ \( -250 + 127 T - 20 T^{2} + T^{3} \)
$31$ \( -472 - 7 T + 14 T^{2} + T^{3} \)
$37$ \( -2 - 19 T + 4 T^{2} + T^{3} \)
$41$ \( -670 - 97 T + 8 T^{2} + T^{3} \)
$43$ \( 80 - 12 T - 8 T^{2} + T^{3} \)
$47$ \( 32 + 38 T + 12 T^{2} + T^{3} \)
$53$ \( -122 - 49 T + T^{3} \)
$59$ \( -1100 - 69 T + 14 T^{2} + T^{3} \)
$61$ \( -580 + 66 T + 22 T^{2} + T^{3} \)
$67$ \( -1156 - 175 T + 6 T^{2} + T^{3} \)
$71$ \( 848 - 133 T - 6 T^{2} + T^{3} \)
$73$ \( -764 - 130 T + 10 T^{2} + T^{3} \)
$79$ \( 128 - 96 T + 4 T^{2} + T^{3} \)
$83$ \( 352 + 155 T + 22 T^{2} + T^{3} \)
$89$ \( -848 - 88 T + 10 T^{2} + T^{3} \)
$97$ \( -16 - 248 T - 2 T^{2} + T^{3} \)
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