Properties

Label 6480.2.a.bs
Level $6480$
Weight $2$
Character orbit 6480.a
Self dual yes
Analytic conductor $51.743$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
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.

\(f(q)\) \(=\) \( q - q^{5} + (\beta_1 - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + (\beta_1 - 2) q^{7} + ( - \beta_{2} - 1) q^{11} + ( - \beta_{2} + 1) q^{13} + (\beta_{2} + 1) q^{17} + (\beta_{2} - 1) q^{19} + (2 \beta_{2} - \beta_1 + 2) q^{23} + q^{25} + ( - 2 \beta_{2} + 2 \beta_1 + 1) q^{29} + (\beta_{2} - 4 \beta_1 - 1) q^{31} + ( - \beta_1 + 2) q^{35} + (\beta_{2} - 2 \beta_1 + 3) q^{37} + (\beta_{2} + 2 \beta_1 + 4) q^{41} + ( - \beta_{2} - 2 \beta_1 - 3) q^{43} + ( - \beta_{2} - 3 \beta_1 + 5) q^{47} + (\beta_{2} - 4 \beta_1 + 1) q^{49} + 2 \beta_1 q^{53} + (\beta_{2} + 1) q^{55} - 2 \beta_1 q^{59} + (\beta_{2} + 2 \beta_1) q^{61} + (\beta_{2} - 1) q^{65} + ( - 3 \beta_{2} + \beta_1 - 5) q^{67} + (3 \beta_{2} + 2 \beta_1 - 3) q^{71} + (4 \beta_1 - 4) q^{73} + (\beta_{2} - 2 \beta_1 + 1) q^{77} + (4 \beta_1 - 2) q^{79} + (3 \beta_1 - 6) q^{83} + ( - \beta_{2} - 1) q^{85} + 3 q^{89} + (\beta_{2} - 3) q^{91} + ( - \beta_{2} + 1) q^{95} + ( - 4 \beta_{2} + 2 \beta_1 - 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} - 5 q^{7} - 2 q^{11} + 4 q^{13} + 2 q^{17} - 4 q^{19} + 3 q^{23} + 3 q^{25} + 7 q^{29} - 8 q^{31} + 5 q^{35} + 6 q^{37} + 13 q^{41} - 10 q^{43} + 13 q^{47} - 2 q^{49} + 2 q^{53} + 2 q^{55} - 2 q^{59} + q^{61} - 4 q^{65} - 11 q^{67} - 10 q^{71} - 8 q^{73} - 2 q^{79} - 15 q^{83} - 2 q^{85} + 9 q^{89} - 10 q^{91} + 4 q^{95} - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
−2.08613
0.571993
2.51414
0 0 0 −1.00000 0 −4.08613 0 0 0
1.2 0 0 0 −1.00000 0 −1.42801 0 0 0
1.3 0 0 0 −1.00000 0 0.514137 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.bs 3
3.b odd 2 1 6480.2.a.bv 3
4.b odd 2 1 405.2.a.i 3
9.c even 3 2 2160.2.q.k 6
9.d odd 6 2 720.2.q.i 6
12.b even 2 1 405.2.a.j 3
20.d odd 2 1 2025.2.a.o 3
20.e even 4 2 2025.2.b.m 6
36.f odd 6 2 135.2.e.b 6
36.h even 6 2 45.2.e.b 6
60.h even 2 1 2025.2.a.n 3
60.l odd 4 2 2025.2.b.l 6
180.n even 6 2 225.2.e.b 6
180.p odd 6 2 675.2.e.b 6
180.v odd 12 4 225.2.k.b 12
180.x even 12 4 675.2.k.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.b 6 36.h even 6 2
135.2.e.b 6 36.f odd 6 2
225.2.e.b 6 180.n even 6 2
225.2.k.b 12 180.v odd 12 4
405.2.a.i 3 4.b odd 2 1
405.2.a.j 3 12.b even 2 1
675.2.e.b 6 180.p odd 6 2
675.2.k.b 12 180.x even 12 4
720.2.q.i 6 9.d odd 6 2
2025.2.a.n 3 60.h even 2 1
2025.2.a.o 3 20.d odd 2 1
2025.2.b.l 6 60.l odd 4 2
2025.2.b.m 6 20.e even 4 2
2160.2.q.k 6 9.c even 3 2
6480.2.a.bs 3 1.a even 1 1 trivial
6480.2.a.bv 3 3.b odd 2 1

Hecke kernels

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

\( T_{7}^{3} + 5T_{7}^{2} + 3T_{7} - 3 \) Copy content Toggle raw display
\( T_{11}^{3} + 2T_{11}^{2} - 8T_{11} - 12 \) Copy content Toggle raw display
\( T_{13}^{3} - 4T_{13}^{2} - 4T_{13} + 4 \) Copy content Toggle raw display
\( T_{17}^{3} - 2T_{17}^{2} - 8T_{17} + 12 \) Copy content Toggle raw display
\( T_{19}^{3} + 4T_{19}^{2} - 4T_{19} - 4 \) Copy content Toggle raw display
\( T_{23}^{3} - 3T_{23}^{2} - 33T_{23} + 117 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 5 T^{2} + 3 T - 3 \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} - 8 T - 12 \) Copy content Toggle raw display
$13$ \( T^{3} - 4 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} - 8 T + 12 \) Copy content Toggle raw display
$19$ \( T^{3} + 4 T^{2} - 4 T - 4 \) Copy content Toggle raw display
$23$ \( T^{3} - 3 T^{2} - 33 T + 117 \) Copy content Toggle raw display
$29$ \( T^{3} - 7 T^{2} - 29 T + 51 \) Copy content Toggle raw display
$31$ \( T^{3} + 8 T^{2} - 60 T - 468 \) Copy content Toggle raw display
$37$ \( T^{3} - 6 T^{2} - 12 T + 4 \) Copy content Toggle raw display
$41$ \( T^{3} - 13 T^{2} + 19 T - 3 \) Copy content Toggle raw display
$43$ \( T^{3} + 10 T^{2} - 4 T - 4 \) Copy content Toggle raw display
$47$ \( T^{3} - 13 T^{2} - 11 T + 369 \) Copy content Toggle raw display
$53$ \( T^{3} - 2 T^{2} - 20 T + 24 \) Copy content Toggle raw display
$59$ \( T^{3} + 2 T^{2} - 20 T - 24 \) Copy content Toggle raw display
$61$ \( T^{3} - T^{2} - 37 T - 71 \) Copy content Toggle raw display
$67$ \( T^{3} + 11 T^{2} - 39 T - 507 \) Copy content Toggle raw display
$71$ \( T^{3} + 10 T^{2} - 92 T - 708 \) Copy content Toggle raw display
$73$ \( T^{3} + 8 T^{2} - 64 T - 128 \) Copy content Toggle raw display
$79$ \( T^{3} + 2 T^{2} - 84 T + 24 \) Copy content Toggle raw display
$83$ \( T^{3} + 15 T^{2} + 27 T - 81 \) Copy content Toggle raw display
$89$ \( (T - 3)^{3} \) Copy content Toggle raw display
$97$ \( T^{3} + 18 T^{2} - 36 T - 1304 \) Copy content Toggle raw display
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