Properties

Label 5520.2.a.bi
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: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 345)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} - q^{5} + q^{7} + q^{9} -\beta q^{11} + ( 2 + \beta ) q^{13} + q^{15} + ( -3 - \beta ) q^{17} + ( -2 - \beta ) q^{19} - q^{21} + q^{23} + q^{25} - q^{27} + ( 3 + 3 \beta ) q^{29} + ( -5 + 2 \beta ) q^{31} + \beta q^{33} - q^{35} + ( -1 + 2 \beta ) q^{37} + ( -2 - \beta ) q^{39} + ( 3 + \beta ) q^{41} -2 q^{43} - q^{45} + ( -6 - \beta ) q^{47} -6 q^{49} + ( 3 + \beta ) q^{51} + ( 3 - \beta ) q^{53} + \beta q^{55} + ( 2 + \beta ) q^{57} + ( -3 - 3 \beta ) q^{59} + ( 8 - 3 \beta ) q^{61} + q^{63} + ( -2 - \beta ) q^{65} + 7 q^{67} - q^{69} + ( -3 + 3 \beta ) q^{71} + ( 2 - 3 \beta ) q^{73} - q^{75} -\beta q^{77} + 4 q^{79} + q^{81} + ( -3 - 5 \beta ) q^{83} + ( 3 + \beta ) q^{85} + ( -3 - 3 \beta ) q^{87} + ( -12 - 2 \beta ) q^{89} + ( 2 + \beta ) q^{91} + ( 5 - 2 \beta ) q^{93} + ( 2 + \beta ) q^{95} + ( 8 - 2 \beta ) q^{97} -\beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + 4q^{13} + 2q^{15} - 6q^{17} - 4q^{19} - 2q^{21} + 2q^{23} + 2q^{25} - 2q^{27} + 6q^{29} - 10q^{31} - 2q^{35} - 2q^{37} - 4q^{39} + 6q^{41} - 4q^{43} - 2q^{45} - 12q^{47} - 12q^{49} + 6q^{51} + 6q^{53} + 4q^{57} - 6q^{59} + 16q^{61} + 2q^{63} - 4q^{65} + 14q^{67} - 2q^{69} - 6q^{71} + 4q^{73} - 2q^{75} + 8q^{79} + 2q^{81} - 6q^{83} + 6q^{85} - 6q^{87} - 24q^{89} + 4q^{91} + 10q^{93} + 4q^{95} + 16q^{97} + O(q^{100}) \)

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
2.44949
−2.44949
0 −1.00000 0 −1.00000 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 1.00000 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.bi 2
4.b odd 2 1 345.2.a.i 2
12.b even 2 1 1035.2.a.k 2
20.d odd 2 1 1725.2.a.y 2
20.e even 4 2 1725.2.b.m 4
60.h even 2 1 5175.2.a.bl 2
92.b even 2 1 7935.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.a.i 2 4.b odd 2 1
1035.2.a.k 2 12.b even 2 1
1725.2.a.y 2 20.d odd 2 1
1725.2.b.m 4 20.e even 4 2
5175.2.a.bl 2 60.h even 2 1
5520.2.a.bi 2 1.a even 1 1 trivial
7935.2.a.t 2 92.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\):

\( T_{7} - 1 \)
\( T_{11}^{2} - 6 \)
\( T_{13}^{2} - 4 T_{13} - 2 \)
\( T_{17}^{2} + 6 T_{17} + 3 \)
\( T_{19}^{2} + 4 T_{19} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -6 + T^{2} \)
$13$ \( -2 - 4 T + T^{2} \)
$17$ \( 3 + 6 T + T^{2} \)
$19$ \( -2 + 4 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( -45 - 6 T + T^{2} \)
$31$ \( 1 + 10 T + T^{2} \)
$37$ \( -23 + 2 T + T^{2} \)
$41$ \( 3 - 6 T + T^{2} \)
$43$ \( ( 2 + T )^{2} \)
$47$ \( 30 + 12 T + T^{2} \)
$53$ \( 3 - 6 T + T^{2} \)
$59$ \( -45 + 6 T + T^{2} \)
$61$ \( 10 - 16 T + T^{2} \)
$67$ \( ( -7 + T )^{2} \)
$71$ \( -45 + 6 T + T^{2} \)
$73$ \( -50 - 4 T + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( -141 + 6 T + T^{2} \)
$89$ \( 120 + 24 T + T^{2} \)
$97$ \( 40 - 16 T + T^{2} \)
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