Properties

Label 5520.2.a.bt
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + ( 1 + 2 \beta ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + q^{5} + ( 1 + 2 \beta ) q^{7} + q^{9} -\beta q^{11} + ( 2 + \beta ) q^{13} + q^{15} + ( -1 + \beta ) q^{17} + ( 6 - \beta ) q^{19} + ( 1 + 2 \beta ) q^{21} + q^{23} + q^{25} + q^{27} + ( -5 - \beta ) q^{29} + q^{31} -\beta q^{33} + ( 1 + 2 \beta ) q^{35} + ( 1 + 6 \beta ) q^{37} + ( 2 + \beta ) q^{39} + ( 3 - 3 \beta ) q^{41} + ( 2 - 4 \beta ) q^{43} + q^{45} + ( 2 - 3 \beta ) q^{47} + ( 2 + 4 \beta ) q^{49} + ( -1 + \beta ) q^{51} + ( 1 - 3 \beta ) q^{53} -\beta q^{55} + ( 6 - \beta ) q^{57} + ( -1 + 9 \beta ) q^{59} -\beta q^{61} + ( 1 + 2 \beta ) q^{63} + ( 2 + \beta ) q^{65} + ( 7 - 6 \beta ) q^{67} + q^{69} + ( 7 - 5 \beta ) q^{71} + ( -2 - 3 \beta ) q^{73} + q^{75} + ( -4 - \beta ) q^{77} + 12 \beta q^{79} + q^{81} + ( -7 - 3 \beta ) q^{83} + ( -1 + \beta ) q^{85} + ( -5 - \beta ) q^{87} + ( -12 - 2 \beta ) q^{89} + ( 6 + 5 \beta ) q^{91} + q^{93} + ( 6 - \beta ) q^{95} + ( 4 - 10 \beta ) q^{97} -\beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{5} + 2q^{7} + 2q^{9} + 4q^{13} + 2q^{15} - 2q^{17} + 12q^{19} + 2q^{21} + 2q^{23} + 2q^{25} + 2q^{27} - 10q^{29} + 2q^{31} + 2q^{35} + 2q^{37} + 4q^{39} + 6q^{41} + 4q^{43} + 2q^{45} + 4q^{47} + 4q^{49} - 2q^{51} + 2q^{53} + 12q^{57} - 2q^{59} + 2q^{63} + 4q^{65} + 14q^{67} + 2q^{69} + 14q^{71} - 4q^{73} + 2q^{75} - 8q^{77} + 2q^{81} - 14q^{83} - 2q^{85} - 10q^{87} - 24q^{89} + 12q^{91} + 2q^{93} + 12q^{95} + 8q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
0 1.00000 0 1.00000 0 −1.82843 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 3.82843 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.bt 2
4.b odd 2 1 2760.2.a.n 2
12.b even 2 1 8280.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.n 2 4.b odd 2 1
5520.2.a.bt 2 1.a even 1 1 trivial
8280.2.a.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\):

\( T_{7}^{2} - 2 T_{7} - 7 \)
\( T_{11}^{2} - 2 \)
\( T_{13}^{2} - 4 T_{13} + 2 \)
\( T_{17}^{2} + 2 T_{17} - 1 \)
\( T_{19}^{2} - 12 T_{19} + 34 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -7 - 2 T + T^{2} \)
$11$ \( -2 + T^{2} \)
$13$ \( 2 - 4 T + T^{2} \)
$17$ \( -1 + 2 T + T^{2} \)
$19$ \( 34 - 12 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 23 + 10 T + T^{2} \)
$31$ \( ( -1 + T )^{2} \)
$37$ \( -71 - 2 T + T^{2} \)
$41$ \( -9 - 6 T + T^{2} \)
$43$ \( -28 - 4 T + T^{2} \)
$47$ \( -14 - 4 T + T^{2} \)
$53$ \( -17 - 2 T + T^{2} \)
$59$ \( -161 + 2 T + T^{2} \)
$61$ \( -2 + T^{2} \)
$67$ \( -23 - 14 T + T^{2} \)
$71$ \( -1 - 14 T + T^{2} \)
$73$ \( -14 + 4 T + T^{2} \)
$79$ \( -288 + T^{2} \)
$83$ \( 31 + 14 T + T^{2} \)
$89$ \( 136 + 24 T + T^{2} \)
$97$ \( -184 - 8 T + T^{2} \)
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