Properties

Label 5520.2.a.br
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: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1380)
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.

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

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
2.56155
−1.56155
0 1.00000 0 1.00000 0 −2.56155 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 1.56155 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.br 2
4.b odd 2 1 1380.2.a.g 2
12.b even 2 1 4140.2.a.n 2
20.d odd 2 1 6900.2.a.u 2
20.e even 4 2 6900.2.f.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.g 2 4.b odd 2 1
4140.2.a.n 2 12.b even 2 1
5520.2.a.br 2 1.a even 1 1 trivial
6900.2.a.u 2 20.d odd 2 1
6900.2.f.p 4 20.e even 4 2

Hecke kernels

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

\( T_{7}^{2} + T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 16 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 3T_{17} - 2 \) Copy content Toggle raw display
\( T_{19} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$31$ \( T^{2} - 3T - 36 \) Copy content Toggle raw display
$37$ \( T^{2} - 9T + 16 \) Copy content Toggle raw display
$41$ \( T^{2} - 7T - 26 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 11T + 26 \) Copy content Toggle raw display
$59$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$67$ \( T^{2} + 5T - 32 \) Copy content Toggle raw display
$71$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 52 \) Copy content Toggle raw display
$79$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$83$ \( T^{2} + 13T + 4 \) Copy content Toggle raw display
$89$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$97$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
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