Properties

Label 6480.2.a.bp
Level $6480$
Weight $2$
Character orbit 6480.a
Self dual yes
Analytic conductor $51.743$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6480.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.7430605098\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1620)
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.

\(f(q)\) \(=\) \( q + q^{5} + ( 1 + \beta ) q^{7} +O(q^{10})\) \( q + q^{5} + ( 1 + \beta ) q^{7} + \beta q^{11} + ( 2 + 2 \beta ) q^{13} + ( 3 + \beta ) q^{17} + ( 1 + 2 \beta ) q^{19} + 2 \beta q^{23} + q^{25} + ( 6 + \beta ) q^{29} + ( 1 - 4 \beta ) q^{31} + ( 1 + \beta ) q^{35} + ( -1 - 3 \beta ) q^{37} + ( 6 + 3 \beta ) q^{41} + ( -5 + \beta ) q^{43} + ( -3 + \beta ) q^{47} + ( -3 + 2 \beta ) q^{49} + ( 9 - \beta ) q^{53} + \beta q^{55} + ( -6 - \beta ) q^{59} -4 q^{61} + ( 2 + 2 \beta ) q^{65} + ( 4 - 6 \beta ) q^{67} + ( -6 - 3 \beta ) q^{71} + ( 5 - 3 \beta ) q^{73} + ( 3 + \beta ) q^{77} + ( 4 + 6 \beta ) q^{79} + ( -3 - 7 \beta ) q^{83} + ( 3 + \beta ) q^{85} -3 \beta q^{89} + ( 8 + 4 \beta ) q^{91} + ( 1 + 2 \beta ) q^{95} + ( -1 + \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 2 q^{7} + O(q^{10}) \) \( 2 q + 2 q^{5} + 2 q^{7} + 4 q^{13} + 6 q^{17} + 2 q^{19} + 2 q^{25} + 12 q^{29} + 2 q^{31} + 2 q^{35} - 2 q^{37} + 12 q^{41} - 10 q^{43} - 6 q^{47} - 6 q^{49} + 18 q^{53} - 12 q^{59} - 8 q^{61} + 4 q^{65} + 8 q^{67} - 12 q^{71} + 10 q^{73} + 6 q^{77} + 8 q^{79} - 6 q^{83} + 6 q^{85} + 16 q^{91} + 2 q^{95} - 2 q^{97} + O(q^{100}) \)

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
−1.73205
1.73205
0 0 0 1.00000 0 −0.732051 0 0 0
1.2 0 0 0 1.00000 0 2.73205 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 6480.2.a.bp 2
3.b odd 2 1 6480.2.a.bh 2
4.b odd 2 1 1620.2.a.h yes 2
12.b even 2 1 1620.2.a.g 2
20.d odd 2 1 8100.2.a.s 2
20.e even 4 2 8100.2.d.l 4
36.f odd 6 2 1620.2.i.m 4
36.h even 6 2 1620.2.i.n 4
60.h even 2 1 8100.2.a.t 2
60.l odd 4 2 8100.2.d.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.g 2 12.b even 2 1
1620.2.a.h yes 2 4.b odd 2 1
1620.2.i.m 4 36.f odd 6 2
1620.2.i.n 4 36.h even 6 2
6480.2.a.bh 2 3.b odd 2 1
6480.2.a.bp 2 1.a even 1 1 trivial
8100.2.a.s 2 20.d odd 2 1
8100.2.a.t 2 60.h even 2 1
8100.2.d.l 4 20.e even 4 2
8100.2.d.m 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6480))\):

\( T_{7}^{2} - 2 T_{7} - 2 \)
\( T_{11}^{2} - 3 \)
\( T_{13}^{2} - 4 T_{13} - 8 \)
\( T_{17}^{2} - 6 T_{17} + 6 \)
\( T_{19}^{2} - 2 T_{19} - 11 \)
\( T_{23}^{2} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -2 - 2 T + T^{2} \)
$11$ \( -3 + T^{2} \)
$13$ \( -8 - 4 T + T^{2} \)
$17$ \( 6 - 6 T + T^{2} \)
$19$ \( -11 - 2 T + T^{2} \)
$23$ \( -12 + T^{2} \)
$29$ \( 33 - 12 T + T^{2} \)
$31$ \( -47 - 2 T + T^{2} \)
$37$ \( -26 + 2 T + T^{2} \)
$41$ \( 9 - 12 T + T^{2} \)
$43$ \( 22 + 10 T + T^{2} \)
$47$ \( 6 + 6 T + T^{2} \)
$53$ \( 78 - 18 T + T^{2} \)
$59$ \( 33 + 12 T + T^{2} \)
$61$ \( ( 4 + T )^{2} \)
$67$ \( -92 - 8 T + T^{2} \)
$71$ \( 9 + 12 T + T^{2} \)
$73$ \( -2 - 10 T + T^{2} \)
$79$ \( -92 - 8 T + T^{2} \)
$83$ \( -138 + 6 T + T^{2} \)
$89$ \( -27 + T^{2} \)
$97$ \( -2 + 2 T + T^{2} \)
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