Properties

Label 6480.2.a.br
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: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 405)
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.

\(f(q)\) \(=\) \( q + q^{5} + (\beta + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + (\beta + 3) q^{7} + (\beta + 4) q^{11} + (2 \beta - 2) q^{13} + ( - \beta - 1) q^{17} + ( - 2 \beta - 1) q^{19} + 2 \beta q^{23} + q^{25} + (3 \beta - 2) q^{29} + 3 q^{31} + (\beta + 3) q^{35} + ( - \beta - 1) q^{37} + ( - 3 \beta - 2) q^{41} + ( - 3 \beta + 5) q^{43} + (\beta + 7) q^{47} + (6 \beta + 5) q^{49} + (\beta + 5) q^{53} + (\beta + 4) q^{55} + ( - \beta + 10) q^{59} + 4 q^{61} + (2 \beta - 2) q^{65} - 2 \beta q^{67} + (\beta + 2) q^{71} + ( - 5 \beta + 1) q^{73} + (7 \beta + 15) q^{77} + ( - 2 \beta - 12) q^{79} + ( - 3 \beta + 3) q^{83} + ( - \beta - 1) q^{85} + 3 \beta q^{89} + 4 \beta q^{91} + ( - 2 \beta - 1) q^{95} + ( - 5 \beta - 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 6 q^{7} + 8 q^{11} - 4 q^{13} - 2 q^{17} - 2 q^{19} + 2 q^{25} - 4 q^{29} + 6 q^{31} + 6 q^{35} - 2 q^{37} - 4 q^{41} + 10 q^{43} + 14 q^{47} + 10 q^{49} + 10 q^{53} + 8 q^{55} + 20 q^{59} + 8 q^{61} - 4 q^{65} + 4 q^{71} + 2 q^{73} + 30 q^{77} - 24 q^{79} + 6 q^{83} - 2 q^{85} - 2 q^{95} - 2 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
0 0 0 1.00000 0 1.26795 0 0 0
1.2 0 0 0 1.00000 0 4.73205 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.br 2
3.b odd 2 1 6480.2.a.bi 2
4.b odd 2 1 405.2.a.g 2
12.b even 2 1 405.2.a.h yes 2
20.d odd 2 1 2025.2.a.m 2
20.e even 4 2 2025.2.b.g 4
36.f odd 6 2 405.2.e.l 4
36.h even 6 2 405.2.e.i 4
60.h even 2 1 2025.2.a.g 2
60.l odd 4 2 2025.2.b.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.g 2 4.b odd 2 1
405.2.a.h yes 2 12.b even 2 1
405.2.e.i 4 36.h even 6 2
405.2.e.l 4 36.f odd 6 2
2025.2.a.g 2 60.h even 2 1
2025.2.a.m 2 20.d odd 2 1
2025.2.b.g 4 20.e even 4 2
2025.2.b.h 4 60.l odd 4 2
6480.2.a.bi 2 3.b odd 2 1
6480.2.a.br 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{7}^{2} - 6T_{7} + 6 \) Copy content Toggle raw display
\( T_{11}^{2} - 8T_{11} + 13 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 8 \) Copy content Toggle raw display
\( T_{17}^{2} + 2T_{17} - 2 \) Copy content Toggle raw display
\( T_{19}^{2} + 2T_{19} - 11 \) Copy content Toggle raw display
\( T_{23}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$11$ \( T^{2} - 8T + 13 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 11 \) Copy content Toggle raw display
$23$ \( T^{2} - 12 \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 23 \) Copy content Toggle raw display
$31$ \( (T - 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 23 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T - 2 \) Copy content Toggle raw display
$47$ \( T^{2} - 14T + 46 \) Copy content Toggle raw display
$53$ \( T^{2} - 10T + 22 \) Copy content Toggle raw display
$59$ \( T^{2} - 20T + 97 \) Copy content Toggle raw display
$61$ \( (T - 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 12 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T + 1 \) Copy content Toggle raw display
$73$ \( T^{2} - 2T - 74 \) Copy content Toggle raw display
$79$ \( T^{2} + 24T + 132 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T - 18 \) Copy content Toggle raw display
$89$ \( T^{2} - 27 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T - 74 \) Copy content Toggle raw display
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