Properties

Label 5520.2.a.cb
Level $5520$
Weight $2$
Character orbit 5520.a
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.54764.1
Defining polynomial: \(x^{4} - x^{3} - 9 x^{2} + 3 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + \beta_{1} q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} - q^{5} + \beta_{1} q^{7} + q^{9} + \beta_{3} q^{11} + ( -1 + \beta_{1} - \beta_{2} ) q^{13} - q^{15} + ( 1 + \beta_{2} + 2 \beta_{3} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} ) q^{19} + \beta_{1} q^{21} - q^{23} + q^{25} + q^{27} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{29} + ( 1 - \beta_{2} - \beta_{3} ) q^{31} + \beta_{3} q^{33} -\beta_{1} q^{35} + ( \beta_{1} + 2 \beta_{2} ) q^{37} + ( -1 + \beta_{1} - \beta_{2} ) q^{39} + ( 2 - \beta_{1} + \beta_{3} ) q^{41} + ( 6 + 2 \beta_{2} ) q^{43} - q^{45} + ( -1 + \beta_{1} + \beta_{2} ) q^{47} + ( 2 - \beta_{2} - 3 \beta_{3} ) q^{49} + ( 1 + \beta_{2} + 2 \beta_{3} ) q^{51} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{53} -\beta_{3} q^{55} + ( 1 - \beta_{1} - \beta_{2} ) q^{57} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{59} + ( 2 \beta_{2} + \beta_{3} ) q^{61} + \beta_{1} q^{63} + ( 1 - \beta_{1} + \beta_{2} ) q^{65} + ( 7 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{67} - q^{69} + ( \beta_{1} - \beta_{3} ) q^{71} + ( -5 + \beta_{1} - \beta_{2} ) q^{73} + q^{75} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{77} + ( 3 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{79} + q^{81} + ( 7 + \beta_{2} - 2 \beta_{3} ) q^{83} + ( -1 - \beta_{2} - 2 \beta_{3} ) q^{85} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{87} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{89} + ( 9 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{91} + ( 1 - \beta_{2} - \beta_{3} ) q^{93} + ( -1 + \beta_{1} + \beta_{2} ) q^{95} + ( -4 - 2 \beta_{2} - 2 \beta_{3} ) q^{97} + \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 4q^{5} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 4q^{5} + 4q^{9} - 2q^{13} - 4q^{15} + 2q^{17} + 6q^{19} - 4q^{23} + 4q^{25} + 4q^{27} + 4q^{29} + 6q^{31} - 4q^{37} - 2q^{39} + 8q^{41} + 20q^{43} - 4q^{45} - 6q^{47} + 10q^{49} + 2q^{51} - 4q^{53} + 6q^{57} + 4q^{59} - 4q^{61} + 2q^{65} + 26q^{67} - 4q^{69} - 18q^{73} + 4q^{75} + 6q^{77} + 18q^{79} + 4q^{81} + 26q^{83} - 2q^{85} + 4q^{87} + 14q^{89} + 38q^{91} + 6q^{93} - 6q^{95} - 12q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 9 x^{2} + 3 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 11 \nu - 2 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 2 \nu^{2} + 9 \nu - 8 \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 8 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 3 \beta_{2} + \beta_{1} + 11\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(11 \beta_{3} + 11 \beta_{2} - 7 \beta_{1} + 15\)\()/2\)

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
0.655762
3.36007
−0.339102
−2.67673
0 1.00000 0 −1.00000 0 −4.46569 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −0.512641 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 0.845563 0 1.00000 0
1.4 0 1.00000 0 −1.00000 0 4.13277 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.cb 4
4.b odd 2 1 2760.2.a.v 4
12.b even 2 1 8280.2.a.bq 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.v 4 4.b odd 2 1
5520.2.a.cb 4 1.a even 1 1 trivial
8280.2.a.bq 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\):

\( T_{7}^{4} - 19 T_{7}^{2} + 6 T_{7} + 8 \)
\( T_{11}^{4} - 22 T_{11}^{2} - 12 T_{11} + 80 \)
\( T_{13}^{4} + 2 T_{13}^{3} - 38 T_{13}^{2} - 56 T_{13} + 296 \)
\( T_{17}^{4} - 2 T_{17}^{3} - 71 T_{17}^{2} + 58 T_{17} + 1024 \)
\( T_{19}^{4} - 6 T_{19}^{3} - 26 T_{19}^{2} + 112 T_{19} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( 8 + 6 T - 19 T^{2} + T^{4} \)
$11$ \( 80 - 12 T - 22 T^{2} + T^{4} \)
$13$ \( 296 - 56 T - 38 T^{2} + 2 T^{3} + T^{4} \)
$17$ \( 1024 + 58 T - 71 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( 256 + 112 T - 26 T^{2} - 6 T^{3} + T^{4} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( -164 + 400 T - 75 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( -128 + 104 T - 11 T^{2} - 6 T^{3} + T^{4} \)
$37$ \( 2104 - 146 T - 95 T^{2} + 4 T^{3} + T^{4} \)
$41$ \( -380 + 204 T - 11 T^{2} - 8 T^{3} + T^{4} \)
$43$ \( -2048 + 464 T + 68 T^{2} - 20 T^{3} + T^{4} \)
$47$ \( 256 - 112 T - 26 T^{2} + 6 T^{3} + T^{4} \)
$53$ \( 1468 + 176 T - 147 T^{2} + 4 T^{3} + T^{4} \)
$59$ \( 320 - 84 T - 75 T^{2} - 4 T^{3} + T^{4} \)
$61$ \( -400 - 364 T - 62 T^{2} + 4 T^{3} + T^{4} \)
$67$ \( -6976 + 900 T + 141 T^{2} - 26 T^{3} + T^{4} \)
$71$ \( -64 - 96 T - 35 T^{2} + T^{4} \)
$73$ \( -152 - 8 T + 82 T^{2} + 18 T^{3} + T^{4} \)
$79$ \( 512 + 1536 T - 56 T^{2} - 18 T^{3} + T^{4} \)
$83$ \( -9176 + 1486 T + 109 T^{2} - 26 T^{3} + T^{4} \)
$89$ \( -64 + 48 T + 24 T^{2} - 14 T^{3} + T^{4} \)
$97$ \( 64 - 128 T - 44 T^{2} + 12 T^{3} + T^{4} \)
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