Properties

Label 5520.2.a.bh
Level $5520$
Weight $2$
Character orbit 5520.a
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.56155
2.56155
0 −1.00000 0 −1.00000 0 −1.56155 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 2.56155 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.bh 2
4.b odd 2 1 2760.2.a.p 2
12.b even 2 1 8280.2.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.p 2 4.b odd 2 1
5520.2.a.bh 2 1.a even 1 1 trivial
8280.2.a.be 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\):

\( T_{7}^{2} - T_{7} - 4 \)
\( T_{11}^{2} - 2 T_{11} - 16 \)
\( T_{13} - 2 \)
\( T_{17}^{2} - 5 T_{17} + 2 \)
\( T_{19}^{2} - 6 T_{19} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -4 - T + T^{2} \)
$11$ \( -16 - 2 T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( 2 - 5 T + T^{2} \)
$19$ \( -8 - 6 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 38 - 13 T + T^{2} \)
$31$ \( -32 + 5 T + T^{2} \)
$37$ \( -2 + 3 T + T^{2} \)
$41$ \( 26 - 11 T + T^{2} \)
$43$ \( -64 + 4 T + T^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( -2 + 3 T + T^{2} \)
$59$ \( 16 + 9 T + T^{2} \)
$61$ \( -152 - 2 T + T^{2} \)
$67$ \( -4 - T + T^{2} \)
$71$ \( -8 + 11 T + T^{2} \)
$73$ \( ( -2 + T )^{2} \)
$79$ \( -64 + 4 T + T^{2} \)
$83$ \( 4 - 13 T + T^{2} \)
$89$ \( -16 - 2 T + T^{2} \)
$97$ \( ( 6 + T )^{2} \)
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