Properties

Label 6480.2.a.bo
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: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3240)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} -\beta q^{7} +O(q^{10})\) \( q + q^{5} -\beta q^{7} + ( 1 - \beta ) q^{11} -\beta q^{13} -2 \beta q^{17} - q^{19} + ( -2 - \beta ) q^{23} + q^{25} + ( -1 + \beta ) q^{29} + ( -1 - 3 \beta ) q^{31} -\beta q^{35} + 6 q^{37} + ( 7 - 2 \beta ) q^{41} + ( -2 + 2 \beta ) q^{43} + ( 6 + \beta ) q^{47} + ( 1 + \beta ) q^{49} + 3 \beta q^{53} + ( 1 - \beta ) q^{55} -5 q^{59} + ( 6 + 2 \beta ) q^{61} -\beta q^{65} + ( -6 + 2 \beta ) q^{67} + ( 1 + \beta ) q^{71} + ( 8 + 2 \beta ) q^{73} + 8 q^{77} + ( -4 + 2 \beta ) q^{79} + 10 q^{83} -2 \beta q^{85} + ( -1 - \beta ) q^{89} + ( 8 + \beta ) q^{91} - q^{95} + ( 2 - 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - q^{7} + O(q^{10}) \) \( 2 q + 2 q^{5} - q^{7} + q^{11} - q^{13} - 2 q^{17} - 2 q^{19} - 5 q^{23} + 2 q^{25} - q^{29} - 5 q^{31} - q^{35} + 12 q^{37} + 12 q^{41} - 2 q^{43} + 13 q^{47} + 3 q^{49} + 3 q^{53} + q^{55} - 10 q^{59} + 14 q^{61} - q^{65} - 10 q^{67} + 3 q^{71} + 18 q^{73} + 16 q^{77} - 6 q^{79} + 20 q^{83} - 2 q^{85} - 3 q^{89} + 17 q^{91} - 2 q^{95} + 2 q^{97} + O(q^{100}) \)

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
3.37228
−2.37228
0 0 0 1.00000 0 −3.37228 0 0 0
1.2 0 0 0 1.00000 0 2.37228 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.bo 2
3.b odd 2 1 6480.2.a.bd 2
4.b odd 2 1 3240.2.a.m yes 2
12.b even 2 1 3240.2.a.i 2
36.f odd 6 2 3240.2.q.ba 4
36.h even 6 2 3240.2.q.bd 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.2.a.i 2 12.b even 2 1
3240.2.a.m yes 2 4.b odd 2 1
3240.2.q.ba 4 36.f odd 6 2
3240.2.q.bd 4 36.h even 6 2
6480.2.a.bd 2 3.b odd 2 1
6480.2.a.bo 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6480))\):

\( T_{7}^{2} + T_{7} - 8 \)
\( T_{11}^{2} - T_{11} - 8 \)
\( T_{13}^{2} + T_{13} - 8 \)
\( T_{17}^{2} + 2 T_{17} - 32 \)
\( T_{19} + 1 \)
\( T_{23}^{2} + 5 T_{23} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -8 + T + T^{2} \)
$11$ \( -8 - T + T^{2} \)
$13$ \( -8 + T + T^{2} \)
$17$ \( -32 + 2 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -2 + 5 T + T^{2} \)
$29$ \( -8 + T + T^{2} \)
$31$ \( -68 + 5 T + T^{2} \)
$37$ \( ( -6 + T )^{2} \)
$41$ \( 3 - 12 T + T^{2} \)
$43$ \( -32 + 2 T + T^{2} \)
$47$ \( 34 - 13 T + T^{2} \)
$53$ \( -72 - 3 T + T^{2} \)
$59$ \( ( 5 + T )^{2} \)
$61$ \( 16 - 14 T + T^{2} \)
$67$ \( -8 + 10 T + T^{2} \)
$71$ \( -6 - 3 T + T^{2} \)
$73$ \( 48 - 18 T + T^{2} \)
$79$ \( -24 + 6 T + T^{2} \)
$83$ \( ( -10 + T )^{2} \)
$89$ \( -6 + 3 T + T^{2} \)
$97$ \( -32 - 2 T + T^{2} \)
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