Properties

Label 5520.2.a.bp
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: \(\Q(\sqrt{3}) \)
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 1380)
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.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + (2 \beta - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{5} + (2 \beta - 1) q^{7} + q^{9} + ( - 3 \beta + 1) q^{11} + (\beta - 5) q^{13} + q^{15} + ( - \beta - 6) q^{17} + ( - \beta + 1) q^{19} + (2 \beta - 1) q^{21} + q^{23} + q^{25} + q^{27} + (3 \beta + 2) q^{29} + q^{31} + ( - 3 \beta + 1) q^{33} + (2 \beta - 1) q^{35} + ( - 6 \beta - 1) q^{37} + (\beta - 5) q^{39} + ( - \beta - 6) q^{41} - 2 \beta q^{43} + q^{45} + ( - \beta + 1) q^{47} + ( - 4 \beta + 6) q^{49} + ( - \beta - 6) q^{51} + (5 \beta - 2) q^{53} + ( - 3 \beta + 1) q^{55} + ( - \beta + 1) q^{57} + (3 \beta + 2) q^{59} + ( - \beta - 9) q^{61} + (2 \beta - 1) q^{63} + (\beta - 5) q^{65} - 5 q^{67} + q^{69} + (7 \beta + 2) q^{71} + (\beta - 13) q^{73} + q^{75} + (5 \beta - 19) q^{77} - 4 q^{79} + q^{81} + (5 \beta - 2) q^{83} + ( - \beta - 6) q^{85} + (3 \beta + 2) q^{87} + ( - 6 \beta - 6) q^{89} + ( - 11 \beta + 11) q^{91} + q^{93} + ( - \beta + 1) q^{95} + 4 q^{97} + ( - 3 \beta + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - 10 q^{13} + 2 q^{15} - 12 q^{17} + 2 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 2 q^{31} + 2 q^{33} - 2 q^{35} - 2 q^{37} - 10 q^{39} - 12 q^{41} + 2 q^{45} + 2 q^{47} + 12 q^{49} - 12 q^{51} - 4 q^{53} + 2 q^{55} + 2 q^{57} + 4 q^{59} - 18 q^{61} - 2 q^{63} - 10 q^{65} - 10 q^{67} + 2 q^{69} + 4 q^{71} - 26 q^{73} + 2 q^{75} - 38 q^{77} - 8 q^{79} + 2 q^{81} - 4 q^{83} - 12 q^{85} + 4 q^{87} - 12 q^{89} + 22 q^{91} + 2 q^{93} + 2 q^{95} + 8 q^{97} + 2 q^{99}+O(q^{100}) \) 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 −4.46410 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 2.46410 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.bp 2
4.b odd 2 1 1380.2.a.h 2
12.b even 2 1 4140.2.a.o 2
20.d odd 2 1 6900.2.a.s 2
20.e even 4 2 6900.2.f.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.h 2 4.b odd 2 1
4140.2.a.o 2 12.b even 2 1
5520.2.a.bp 2 1.a even 1 1 trivial
6900.2.a.s 2 20.d odd 2 1
6900.2.f.i 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\):

\( T_{7}^{2} + 2T_{7} - 11 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 26 \) Copy content Toggle raw display
\( T_{13}^{2} + 10T_{13} + 22 \) Copy content Toggle raw display
\( T_{17}^{2} + 12T_{17} + 33 \) Copy content Toggle raw display
\( T_{19}^{2} - 2T_{19} - 2 \) 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
$7$ \( T^{2} + 2T - 11 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 26 \) Copy content Toggle raw display
$13$ \( T^{2} + 10T + 22 \) Copy content Toggle raw display
$17$ \( T^{2} + 12T + 33 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 4T - 23 \) Copy content Toggle raw display
$31$ \( (T - 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 107 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 33 \) Copy content Toggle raw display
$43$ \( T^{2} - 12 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 71 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 23 \) Copy content Toggle raw display
$61$ \( T^{2} + 18T + 78 \) Copy content Toggle raw display
$67$ \( (T + 5)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 143 \) Copy content Toggle raw display
$73$ \( T^{2} + 26T + 166 \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 71 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T - 72 \) Copy content Toggle raw display
$97$ \( (T - 4)^{2} \) Copy content Toggle raw display
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