Properties

Label 5520.2.a.bu
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + 3 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{5} + 3 q^{7} + q^{9} + (\beta + 1) q^{11} + (3 \beta + 1) q^{13} + q^{15} + (\beta - 4) q^{17} + ( - \beta + 5) q^{19} + 3 q^{21} - q^{23} + q^{25} + q^{27} + ( - \beta - 2) q^{29} + (4 \beta + 3) q^{31} + (\beta + 1) q^{33} + 3 q^{35} + ( - 4 \beta + 5) q^{37} + (3 \beta + 1) q^{39} + ( - \beta - 2) q^{41} + (2 \beta + 8) q^{43} + q^{45} + ( - 5 \beta + 1) q^{47} + 2 q^{49} + (\beta - 4) q^{51} + ( - \beta - 4) q^{53} + (\beta + 1) q^{55} + ( - \beta + 5) q^{57} + ( - 3 \beta - 2) q^{59} + ( - 7 \beta + 1) q^{61} + 3 q^{63} + (3 \beta + 1) q^{65} + (2 \beta + 3) q^{67} - q^{69} + (\beta + 2) q^{71} + ( - 5 \beta + 1) q^{73} + q^{75} + (3 \beta + 3) q^{77} - 4 \beta q^{79} + q^{81} + (\beta + 12) q^{83} + (\beta - 4) q^{85} + ( - \beta - 2) q^{87} + ( - 2 \beta + 6) q^{89} + (9 \beta + 3) q^{91} + (4 \beta + 3) q^{93} + ( - \beta + 5) q^{95} - 4 q^{97} + (\beta + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{15} - 8 q^{17} + 10 q^{19} + 6 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 6 q^{31} + 2 q^{33} + 6 q^{35} + 10 q^{37} + 2 q^{39} - 4 q^{41} + 16 q^{43} + 2 q^{45} + 2 q^{47} + 4 q^{49} - 8 q^{51} - 8 q^{53} + 2 q^{55} + 10 q^{57} - 4 q^{59} + 2 q^{61} + 6 q^{63} + 2 q^{65} + 6 q^{67} - 2 q^{69} + 4 q^{71} + 2 q^{73} + 2 q^{75} + 6 q^{77} + 2 q^{81} + 24 q^{83} - 8 q^{85} - 4 q^{87} + 12 q^{89} + 6 q^{91} + 6 q^{93} + 10 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100}) \) 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
−1.73205
1.73205
0 1.00000 0 1.00000 0 3.00000 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 3.00000 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.bu 2
4.b odd 2 1 345.2.a.g 2
12.b even 2 1 1035.2.a.l 2
20.d odd 2 1 1725.2.a.bd 2
20.e even 4 2 1725.2.b.p 4
60.h even 2 1 5175.2.a.bd 2
92.b even 2 1 7935.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.a.g 2 4.b odd 2 1
1035.2.a.l 2 12.b even 2 1
1725.2.a.bd 2 20.d odd 2 1
1725.2.b.p 4 20.e even 4 2
5175.2.a.bd 2 60.h even 2 1
5520.2.a.bu 2 1.a even 1 1 trivial
7935.2.a.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\):

\( T_{7} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 26 \) Copy content Toggle raw display
\( T_{17}^{2} + 8T_{17} + 13 \) Copy content Toggle raw display
\( T_{19}^{2} - 10T_{19} + 22 \) 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
$7$ \( (T - 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 26 \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 13 \) Copy content Toggle raw display
$19$ \( T^{2} - 10T + 22 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} - 6T - 39 \) Copy content Toggle raw display
$37$ \( T^{2} - 10T - 23 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T + 1 \) Copy content Toggle raw display
$43$ \( T^{2} - 16T + 52 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 74 \) Copy content Toggle raw display
$53$ \( T^{2} + 8T + 13 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 23 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T - 146 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T - 3 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T + 1 \) Copy content Toggle raw display
$73$ \( T^{2} - 2T - 74 \) Copy content Toggle raw display
$79$ \( T^{2} - 48 \) Copy content Toggle raw display
$83$ \( T^{2} - 24T + 141 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$97$ \( (T + 4)^{2} \) Copy content Toggle raw display
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