Properties

Label 5520.2.a.ca
Level $5520$
Weight $2$
Character orbit 5520.a
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} + \beta_{1} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{5} + \beta_{1} q^{7} + q^{9} + ( 1 - \beta_{2} ) q^{13} - q^{15} + ( 2 - \beta_{1} ) q^{17} -\beta_{1} q^{21} - q^{23} + q^{25} - q^{27} + ( 2 + \beta_{1} ) q^{29} + ( -1 - \beta_{1} - \beta_{2} ) q^{31} + \beta_{1} q^{35} + ( 4 + \beta_{1} + 2 \beta_{2} ) q^{37} + ( -1 + \beta_{2} ) q^{39} + ( 1 + \beta_{1} - \beta_{2} ) q^{41} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{43} + q^{45} + 2 \beta_{1} q^{47} + ( 2 + \beta_{1} + \beta_{2} ) q^{49} + ( -2 + \beta_{1} ) q^{51} + ( 5 + \beta_{1} - \beta_{2} ) q^{53} + ( -1 + \beta_{1} - \beta_{2} ) q^{59} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{61} + \beta_{1} q^{63} + ( 1 - \beta_{2} ) q^{65} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{67} + q^{69} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{71} + ( 2 - 2 \beta_{1} ) q^{73} - q^{75} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{83} + ( 2 - \beta_{1} ) q^{85} + ( -2 - \beta_{1} ) q^{87} + ( 8 + 2 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 2 + 2 \beta_{2} ) q^{91} + ( 1 + \beta_{1} + \beta_{2} ) q^{93} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 3q^{5} - q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 3q^{5} - q^{7} + 3q^{9} + 4q^{13} - 3q^{15} + 7q^{17} + q^{21} - 3q^{23} + 3q^{25} - 3q^{27} + 5q^{29} - q^{31} - q^{35} + 9q^{37} - 4q^{39} + 3q^{41} + 3q^{45} - 2q^{47} + 4q^{49} - 7q^{51} + 15q^{53} - 3q^{59} - 6q^{61} - q^{63} + 4q^{65} + 3q^{67} + 3q^{69} - 5q^{71} + 8q^{73} - 3q^{75} - 8q^{79} + 3q^{81} - 7q^{83} + 7q^{85} - 5q^{87} + 20q^{89} + 4q^{91} + q^{93} + 6q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 4 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 3 \)
\(\beta_{2}\)\(=\)\( -2 \nu^{2} + 2 \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{2} + 2 \beta_{1} + 11\)\()/4\)

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
−0.254102
−1.86081
2.11491
0 −1.00000 0 1.00000 0 −3.18953 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −1.39821 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 3.58774 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.ca 3
4.b odd 2 1 2760.2.a.t 3
12.b even 2 1 8280.2.a.bh 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.t 3 4.b odd 2 1
5520.2.a.ca 3 1.a even 1 1 trivial
8280.2.a.bh 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\):

\( T_{7}^{3} + T_{7}^{2} - 12 T_{7} - 16 \)
\( T_{11} \)
\( T_{13}^{3} - 4 T_{13}^{2} - 20 T_{13} + 16 \)
\( T_{17}^{3} - 7 T_{17}^{2} + 4 T_{17} + 28 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( -16 - 12 T + T^{2} + T^{3} \)
$11$ \( T^{3} \)
$13$ \( 16 - 20 T - 4 T^{2} + T^{3} \)
$17$ \( 28 + 4 T - 7 T^{2} + T^{3} \)
$19$ \( T^{3} \)
$23$ \( ( 1 + T )^{3} \)
$29$ \( 4 - 4 T - 5 T^{2} + T^{3} \)
$31$ \( -64 - 32 T + T^{2} + T^{3} \)
$37$ \( 676 - 76 T - 9 T^{2} + T^{3} \)
$41$ \( 148 - 40 T - 3 T^{2} + T^{3} \)
$43$ \( 64 - 64 T + T^{3} \)
$47$ \( -128 - 48 T + 2 T^{2} + T^{3} \)
$53$ \( 196 + 32 T - 15 T^{2} + T^{3} \)
$59$ \( 64 - 40 T + 3 T^{2} + T^{3} \)
$61$ \( -184 - 52 T + 6 T^{2} + T^{3} \)
$67$ \( 496 - 100 T - 3 T^{2} + T^{3} \)
$71$ \( 112 - 116 T + 5 T^{2} + T^{3} \)
$73$ \( 208 - 28 T - 8 T^{2} + T^{3} \)
$79$ \( -448 - 64 T + 8 T^{2} + T^{3} \)
$83$ \( -592 - 108 T + 7 T^{2} + T^{3} \)
$89$ \( 992 + 4 T - 20 T^{2} + T^{3} \)
$97$ \( 184 - 52 T - 6 T^{2} + T^{3} \)
show more
show less