Properties

Label 5520.2.a.bq
Level $5520$
Weight $2$
Character orbit 5520.a
Self dual yes
Analytic conductor $44.077$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} -\beta q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + q^{5} -\beta q^{7} + q^{9} -4 q^{11} + ( -2 + 2 \beta ) q^{13} + q^{15} + ( 2 + \beta ) q^{17} -4 q^{19} -\beta q^{21} - q^{23} + q^{25} + q^{27} + ( -2 - \beta ) q^{29} -\beta q^{31} -4 q^{33} -\beta q^{35} + ( -2 - \beta ) q^{37} + ( -2 + 2 \beta ) q^{39} + ( 2 - 3 \beta ) q^{41} + 4 q^{43} + q^{45} + 2 \beta q^{47} + ( 1 + \beta ) q^{49} + ( 2 + \beta ) q^{51} + ( -2 + \beta ) q^{53} -4 q^{55} -4 q^{57} + ( -12 + \beta ) q^{59} -2 q^{61} -\beta q^{63} + ( -2 + 2 \beta ) q^{65} + ( -12 - \beta ) q^{67} - q^{69} -\beta q^{71} + ( -6 - 2 \beta ) q^{73} + q^{75} + 4 \beta q^{77} + 4 \beta q^{79} + q^{81} + ( -4 + 5 \beta ) q^{83} + ( 2 + \beta ) q^{85} + ( -2 - \beta ) q^{87} + ( 2 - 2 \beta ) q^{89} -16 q^{91} -\beta q^{93} -4 q^{95} + ( 2 + 4 \beta ) q^{97} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{5} - q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{5} - q^{7} + 2q^{9} - 8q^{11} - 2q^{13} + 2q^{15} + 5q^{17} - 8q^{19} - q^{21} - 2q^{23} + 2q^{25} + 2q^{27} - 5q^{29} - q^{31} - 8q^{33} - q^{35} - 5q^{37} - 2q^{39} + q^{41} + 8q^{43} + 2q^{45} + 2q^{47} + 3q^{49} + 5q^{51} - 3q^{53} - 8q^{55} - 8q^{57} - 23q^{59} - 4q^{61} - q^{63} - 2q^{65} - 25q^{67} - 2q^{69} - q^{71} - 14q^{73} + 2q^{75} + 4q^{77} + 4q^{79} + 2q^{81} - 3q^{83} + 5q^{85} - 5q^{87} + 2q^{89} - 32q^{91} - q^{93} - 8q^{95} + 8q^{97} - 8q^{99} + O(q^{100}) \)

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
3.37228
−2.37228
0 1.00000 0 1.00000 0 −3.37228 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 2.37228 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.bq 2
4.b odd 2 1 2760.2.a.o 2
12.b even 2 1 8280.2.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.o 2 4.b odd 2 1
5520.2.a.bq 2 1.a even 1 1 trivial
8280.2.a.z 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\):

\( T_{7}^{2} + T_{7} - 8 \)
\( T_{11} + 4 \)
\( T_{13}^{2} + 2 T_{13} - 32 \)
\( T_{17}^{2} - 5 T_{17} - 2 \)
\( T_{19} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -8 + T + T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( -32 + 2 T + T^{2} \)
$17$ \( -2 - 5 T + T^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -2 + 5 T + T^{2} \)
$31$ \( -8 + T + T^{2} \)
$37$ \( -2 + 5 T + T^{2} \)
$41$ \( -74 - T + T^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( -32 - 2 T + T^{2} \)
$53$ \( -6 + 3 T + T^{2} \)
$59$ \( 124 + 23 T + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 148 + 25 T + T^{2} \)
$71$ \( -8 + T + T^{2} \)
$73$ \( 16 + 14 T + T^{2} \)
$79$ \( -128 - 4 T + T^{2} \)
$83$ \( -204 + 3 T + T^{2} \)
$89$ \( -32 - 2 T + T^{2} \)
$97$ \( -116 - 8 T + T^{2} \)
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