Properties

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

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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} -\beta q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + q^{5} -\beta q^{7} + q^{9} -2 \beta q^{11} + 2 q^{13} + q^{15} + ( 2 - \beta ) q^{17} + 2 q^{19} -\beta q^{21} - q^{23} + q^{25} + q^{27} + ( 2 - 3 \beta ) q^{29} + 3 \beta q^{31} -2 \beta q^{33} -\beta q^{35} + ( 4 + \beta ) q^{37} + 2 q^{39} + ( 2 + 3 \beta ) q^{41} + q^{45} -4 q^{47} + ( -3 + \beta ) q^{49} + ( 2 - \beta ) q^{51} + ( 6 - \beta ) q^{53} -2 \beta q^{55} + 2 q^{57} + ( -2 + \beta ) q^{59} + ( 6 - 2 \beta ) q^{61} -\beta q^{63} + 2 q^{65} + ( -4 + 3 \beta ) q^{67} - q^{69} + ( -2 - \beta ) q^{71} + ( 2 + 4 \beta ) q^{73} + q^{75} + ( 8 + 2 \beta ) q^{77} + ( 6 + 2 \beta ) q^{79} + q^{81} + ( -8 + 3 \beta ) q^{83} + ( 2 - \beta ) q^{85} + ( 2 - 3 \beta ) q^{87} + ( -2 + 2 \beta ) q^{89} -2 \beta q^{91} + 3 \beta q^{93} + 2 q^{95} + ( 8 - 2 \beta ) q^{97} -2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{5} - q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{5} - q^{7} + 2q^{9} - 2q^{11} + 4q^{13} + 2q^{15} + 3q^{17} + 4q^{19} - q^{21} - 2q^{23} + 2q^{25} + 2q^{27} + q^{29} + 3q^{31} - 2q^{33} - q^{35} + 9q^{37} + 4q^{39} + 7q^{41} + 2q^{45} - 8q^{47} - 5q^{49} + 3q^{51} + 11q^{53} - 2q^{55} + 4q^{57} - 3q^{59} + 10q^{61} - q^{63} + 4q^{65} - 5q^{67} - 2q^{69} - 5q^{71} + 8q^{73} + 2q^{75} + 18q^{77} + 14q^{79} + 2q^{81} - 13q^{83} + 3q^{85} + q^{87} - 2q^{89} - 2q^{91} + 3q^{93} + 4q^{95} + 14q^{97} - 2q^{99} + O(q^{100}) \)

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
2.56155
−1.56155
0 1.00000 0 1.00000 0 −2.56155 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 1.56155 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.br 2
4.b odd 2 1 1380.2.a.g 2
12.b even 2 1 4140.2.a.n 2
20.d odd 2 1 6900.2.a.u 2
20.e even 4 2 6900.2.f.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.g 2 4.b odd 2 1
4140.2.a.n 2 12.b even 2 1
5520.2.a.br 2 1.a even 1 1 trivial
6900.2.a.u 2 20.d odd 2 1
6900.2.f.p 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -4 + T + T^{2} \)
$11$ \( -16 + 2 T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -2 - 3 T + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -38 - T + T^{2} \)
$31$ \( -36 - 3 T + T^{2} \)
$37$ \( 16 - 9 T + T^{2} \)
$41$ \( -26 - 7 T + T^{2} \)
$43$ \( T^{2} \)
$47$ \( ( 4 + T )^{2} \)
$53$ \( 26 - 11 T + T^{2} \)
$59$ \( -2 + 3 T + T^{2} \)
$61$ \( 8 - 10 T + T^{2} \)
$67$ \( -32 + 5 T + T^{2} \)
$71$ \( 2 + 5 T + T^{2} \)
$73$ \( -52 - 8 T + T^{2} \)
$79$ \( 32 - 14 T + T^{2} \)
$83$ \( 4 + 13 T + T^{2} \)
$89$ \( -16 + 2 T + T^{2} \)
$97$ \( 32 - 14 T + T^{2} \)
show more
show less