Properties

Label 5760.2.a.cd
Level $5760$
Weight $2$
Character orbit 5760.a
Self dual yes
Analytic conductor $45.994$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5760 = 2^{7} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5760.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.9938315643\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 640)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + ( 1 + \beta ) q^{7} +O(q^{10})\) \( q - q^{5} + ( 1 + \beta ) q^{7} + 2 q^{11} + 2 \beta q^{13} -2 \beta q^{17} -2 \beta q^{19} + ( -7 + \beta ) q^{23} + q^{25} -2 q^{29} + ( -2 - 2 \beta ) q^{31} + ( -1 - \beta ) q^{35} + ( 2 - 4 \beta ) q^{37} + ( -8 - 2 \beta ) q^{41} + ( -1 - 3 \beta ) q^{43} + ( -5 - \beta ) q^{47} + ( -1 + 2 \beta ) q^{49} + ( -4 + 2 \beta ) q^{53} -2 q^{55} + ( 4 + 2 \beta ) q^{59} + 6 q^{61} -2 \beta q^{65} + ( 1 + 3 \beta ) q^{67} + ( 2 - 2 \beta ) q^{71} + 2 \beta q^{73} + ( 2 + 2 \beta ) q^{77} + ( -4 - 4 \beta ) q^{79} + ( -3 + 3 \beta ) q^{83} + 2 \beta q^{85} + ( 6 + 4 \beta ) q^{89} + ( 10 + 2 \beta ) q^{91} + 2 \beta q^{95} + ( -12 - 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 2 q^{7} + O(q^{10}) \) \( 2 q - 2 q^{5} + 2 q^{7} + 4 q^{11} - 14 q^{23} + 2 q^{25} - 4 q^{29} - 4 q^{31} - 2 q^{35} + 4 q^{37} - 16 q^{41} - 2 q^{43} - 10 q^{47} - 2 q^{49} - 8 q^{53} - 4 q^{55} + 8 q^{59} + 12 q^{61} + 2 q^{67} + 4 q^{71} + 4 q^{77} - 8 q^{79} - 6 q^{83} + 12 q^{89} + 20 q^{91} - 24 q^{97} + O(q^{100}) \)

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
−0.618034
1.61803
0 0 0 −1.00000 0 −1.23607 0 0 0
1.2 0 0 0 −1.00000 0 3.23607 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

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 5760.2.a.cd 2
3.b odd 2 1 640.2.a.j yes 2
4.b odd 2 1 5760.2.a.bw 2
8.b even 2 1 5760.2.a.ci 2
8.d odd 2 1 5760.2.a.ch 2
12.b even 2 1 640.2.a.l yes 2
15.d odd 2 1 3200.2.a.bk 2
15.e even 4 2 3200.2.c.v 4
24.f even 2 1 640.2.a.i 2
24.h odd 2 1 640.2.a.k yes 2
48.i odd 4 2 1280.2.d.k 4
48.k even 4 2 1280.2.d.m 4
60.h even 2 1 3200.2.a.bf 2
60.l odd 4 2 3200.2.c.x 4
120.i odd 2 1 3200.2.a.be 2
120.m even 2 1 3200.2.a.bl 2
120.q odd 4 2 3200.2.c.u 4
120.w even 4 2 3200.2.c.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.a.i 2 24.f even 2 1
640.2.a.j yes 2 3.b odd 2 1
640.2.a.k yes 2 24.h odd 2 1
640.2.a.l yes 2 12.b even 2 1
1280.2.d.k 4 48.i odd 4 2
1280.2.d.m 4 48.k even 4 2
3200.2.a.be 2 120.i odd 2 1
3200.2.a.bf 2 60.h even 2 1
3200.2.a.bk 2 15.d odd 2 1
3200.2.a.bl 2 120.m even 2 1
3200.2.c.u 4 120.q odd 4 2
3200.2.c.v 4 15.e even 4 2
3200.2.c.w 4 120.w even 4 2
3200.2.c.x 4 60.l odd 4 2
5760.2.a.bw 2 4.b odd 2 1
5760.2.a.cd 2 1.a even 1 1 trivial
5760.2.a.ch 2 8.d odd 2 1
5760.2.a.ci 2 8.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5760))\):

\( T_{7}^{2} - 2 T_{7} - 4 \)
\( T_{11} - 2 \)
\( T_{13}^{2} - 20 \)
\( T_{17}^{2} - 20 \)
\( T_{29} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -4 - 2 T + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( -20 + T^{2} \)
$17$ \( -20 + T^{2} \)
$19$ \( -20 + T^{2} \)
$23$ \( 44 + 14 T + T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( -16 + 4 T + T^{2} \)
$37$ \( -76 - 4 T + T^{2} \)
$41$ \( 44 + 16 T + T^{2} \)
$43$ \( -44 + 2 T + T^{2} \)
$47$ \( 20 + 10 T + T^{2} \)
$53$ \( -4 + 8 T + T^{2} \)
$59$ \( -4 - 8 T + T^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( -44 - 2 T + T^{2} \)
$71$ \( -16 - 4 T + T^{2} \)
$73$ \( -20 + T^{2} \)
$79$ \( -64 + 8 T + T^{2} \)
$83$ \( -36 + 6 T + T^{2} \)
$89$ \( -44 - 12 T + T^{2} \)
$97$ \( 124 + 24 T + T^{2} \)
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