Properties

Label 2640.2.a.be
Level $2640$
Weight $2$
Character orbit 2640.a
Self dual yes
Analytic conductor $21.081$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2640.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.0805061336\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
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.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} -\beta_{2} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{5} -\beta_{2} q^{7} + q^{9} - q^{11} + ( -1 - \beta_{1} ) q^{13} - q^{15} + ( 2 \beta_{1} + \beta_{2} ) q^{17} + ( -3 - \beta_{1} + \beta_{2} ) q^{19} + \beta_{2} q^{21} + 2 \beta_{2} q^{23} + q^{25} - q^{27} + ( -3 + \beta_{1} + \beta_{2} ) q^{29} + ( -2 + 2 \beta_{1} ) q^{31} + q^{33} -\beta_{2} q^{35} -2 q^{37} + ( 1 + \beta_{1} ) q^{39} + ( -5 - \beta_{1} - \beta_{2} ) q^{41} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{43} + q^{45} + ( 2 - 2 \beta_{1} ) q^{47} + ( 3 - 2 \beta_{1} - 2 \beta_{2} ) q^{49} + ( -2 \beta_{1} - \beta_{2} ) q^{51} + ( -2 + 2 \beta_{2} ) q^{53} - q^{55} + ( 3 + \beta_{1} - \beta_{2} ) q^{57} + ( -4 - 2 \beta_{2} ) q^{59} + ( -2 - 2 \beta_{2} ) q^{61} -\beta_{2} q^{63} + ( -1 - \beta_{1} ) q^{65} + ( 2 + 2 \beta_{1} ) q^{67} -2 \beta_{2} q^{69} + ( -2 + 2 \beta_{1} ) q^{71} + ( -5 - \beta_{1} - 2 \beta_{2} ) q^{73} - q^{75} + \beta_{2} q^{77} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( 1 + 3 \beta_{1} ) q^{83} + ( 2 \beta_{1} + \beta_{2} ) q^{85} + ( 3 - \beta_{1} - \beta_{2} ) q^{87} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -2 + 2 \beta_{1} ) q^{91} + ( 2 - 2 \beta_{1} ) q^{93} + ( -3 - \beta_{1} + \beta_{2} ) q^{95} + ( 8 + 2 \beta_{1} ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} - 3 q^{11} - 2 q^{13} - 3 q^{15} - 2 q^{17} - 8 q^{19} + 3 q^{25} - 3 q^{27} - 10 q^{29} - 8 q^{31} + 3 q^{33} - 6 q^{37} + 2 q^{39} - 14 q^{41} - 4 q^{43} + 3 q^{45} + 8 q^{47} + 11 q^{49} + 2 q^{51} - 6 q^{53} - 3 q^{55} + 8 q^{57} - 12 q^{59} - 6 q^{61} - 2 q^{65} + 4 q^{67} - 8 q^{71} - 14 q^{73} - 3 q^{75} - 12 q^{79} + 3 q^{81} - 2 q^{85} + 10 q^{87} - 10 q^{89} - 8 q^{91} + 8 q^{93} - 8 q^{95} + 22 q^{97} - 3 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 5\)\()/2\)

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
−1.48119
2.17009
0.311108
0 −1.00000 0 1.00000 0 −3.35026 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −1.07838 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 4.42864 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(11\) \(1\)

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 2640.2.a.be 3
3.b odd 2 1 7920.2.a.cj 3
4.b odd 2 1 165.2.a.c 3
12.b even 2 1 495.2.a.e 3
20.d odd 2 1 825.2.a.k 3
20.e even 4 2 825.2.c.g 6
28.d even 2 1 8085.2.a.bk 3
44.c even 2 1 1815.2.a.m 3
60.h even 2 1 2475.2.a.bb 3
60.l odd 4 2 2475.2.c.r 6
132.d odd 2 1 5445.2.a.z 3
220.g even 2 1 9075.2.a.cf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.c 3 4.b odd 2 1
495.2.a.e 3 12.b even 2 1
825.2.a.k 3 20.d odd 2 1
825.2.c.g 6 20.e even 4 2
1815.2.a.m 3 44.c even 2 1
2475.2.a.bb 3 60.h even 2 1
2475.2.c.r 6 60.l odd 4 2
2640.2.a.be 3 1.a even 1 1 trivial
5445.2.a.z 3 132.d odd 2 1
7920.2.a.cj 3 3.b odd 2 1
8085.2.a.bk 3 28.d even 2 1
9075.2.a.cf 3 220.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2640))\):

\( T_{7}^{3} - 16 T_{7} - 16 \)
\( T_{13}^{3} + 2 T_{13}^{2} - 12 T_{13} - 8 \)
\( T_{17}^{3} + 2 T_{17}^{2} - 52 T_{17} - 184 \)
\( T_{19}^{3} + 8 T_{19}^{2} - 16 T_{19} - 160 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( -16 - 16 T + T^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( -8 - 12 T + 2 T^{2} + T^{3} \)
$17$ \( -184 - 52 T + 2 T^{2} + T^{3} \)
$19$ \( -160 - 16 T + 8 T^{2} + T^{3} \)
$23$ \( 128 - 64 T + T^{3} \)
$29$ \( -40 + 12 T + 10 T^{2} + T^{3} \)
$31$ \( -128 - 32 T + 8 T^{2} + T^{3} \)
$37$ \( ( 2 + T )^{3} \)
$41$ \( 8 + 44 T + 14 T^{2} + T^{3} \)
$43$ \( -400 - 80 T + 4 T^{2} + T^{3} \)
$47$ \( 128 - 32 T - 8 T^{2} + T^{3} \)
$53$ \( 8 - 52 T + 6 T^{2} + T^{3} \)
$59$ \( -320 - 16 T + 12 T^{2} + T^{3} \)
$61$ \( -248 - 52 T + 6 T^{2} + T^{3} \)
$67$ \( 64 - 48 T - 4 T^{2} + T^{3} \)
$71$ \( -128 - 32 T + 8 T^{2} + T^{3} \)
$73$ \( -344 + 4 T + 14 T^{2} + T^{3} \)
$79$ \( -800 - 64 T + 12 T^{2} + T^{3} \)
$83$ \( -16 - 120 T + T^{3} \)
$89$ \( -200 - 52 T + 10 T^{2} + T^{3} \)
$97$ \( -8 + 108 T - 22 T^{2} + T^{3} \)
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