Properties

Label 3640.2.a.x
Level $3640$
Weight $2$
Character orbit 3640.a
Self dual yes
Analytic conductor $29.066$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3640.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.0655463357\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4913.1
Defining polynomial: \(x^{4} - x^{3} - 6 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{3} - q^{5} - q^{7} + ( 1 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{3} - q^{5} - q^{7} + ( 1 - \beta_{1} + \beta_{2} ) q^{9} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + q^{13} + ( -1 + \beta_{1} ) q^{15} + ( 2 + \beta_{1} + \beta_{2} ) q^{17} + ( 1 + \beta_{3} ) q^{19} + ( -1 + \beta_{1} ) q^{21} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{23} + q^{25} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{27} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{29} + ( 1 - \beta_{1} - 2 \beta_{3} ) q^{31} -4 \beta_{1} q^{33} + q^{35} + ( -4 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{37} + ( 1 - \beta_{1} ) q^{39} + ( -2 - \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{41} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{43} + ( -1 + \beta_{1} - \beta_{2} ) q^{45} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{47} + q^{49} + ( -5 \beta_{1} - \beta_{3} ) q^{51} + ( 4 - 2 \beta_{1} - 2 \beta_{3} ) q^{53} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{55} + ( 2 - \beta_{1} + \beta_{3} ) q^{57} + ( 3 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 2 - 4 \beta_{1} - 4 \beta_{3} ) q^{61} + ( -1 + \beta_{1} - \beta_{2} ) q^{63} - q^{65} + ( 4 + 3 \beta_{1} + \beta_{2} ) q^{67} + ( 2 + 2 \beta_{1} ) q^{69} + ( 1 - 7 \beta_{1} + \beta_{2} - 5 \beta_{3} ) q^{71} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{73} + ( 1 - \beta_{1} ) q^{75} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{77} + ( 2 + 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{79} + ( 3 - 5 \beta_{1} - 3 \beta_{3} ) q^{81} + ( 2 \beta_{1} + 6 \beta_{3} ) q^{83} + ( -2 - \beta_{1} - \beta_{2} ) q^{85} + ( 4 + 3 \beta_{1} + \beta_{2} ) q^{87} + ( 6 + 4 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{89} - q^{91} + ( 2 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{93} + ( -1 - \beta_{3} ) q^{95} + ( 6 + 2 \beta_{2} - 6 \beta_{3} ) q^{97} + ( 9 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9} + O(q^{10}) \) \( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9} + 6 q^{11} + 4 q^{13} - 3 q^{15} + 9 q^{17} + 5 q^{19} - 3 q^{21} + 2 q^{23} + 4 q^{25} + 6 q^{27} - 3 q^{29} + q^{31} - 4 q^{33} + 4 q^{35} - 11 q^{37} + 3 q^{39} - 5 q^{41} + 12 q^{43} - 3 q^{45} + 4 q^{47} + 4 q^{49} - 6 q^{51} + 12 q^{53} - 6 q^{55} + 8 q^{57} + 11 q^{59} - 3 q^{63} - 4 q^{65} + 19 q^{67} + 10 q^{69} - 8 q^{71} + 8 q^{73} + 3 q^{75} - 6 q^{77} + 13 q^{79} + 4 q^{81} + 8 q^{83} - 9 q^{85} + 19 q^{87} + 25 q^{89} - 4 q^{91} + 5 q^{93} - 5 q^{95} + 18 q^{97} + 30 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 6 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 6 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 7 \beta_{1} + 2\)

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
2.90570
0.487928
−0.344151
−2.04948
0 −1.90570 0 −1.00000 0 −1.00000 0 0.631706 0
1.2 0 0.512072 0 −1.00000 0 −1.00000 0 −2.73778 0
1.3 0 1.34415 0 −1.00000 0 −1.00000 0 −1.19326 0
1.4 0 3.04948 0 −1.00000 0 −1.00000 0 6.29934 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(7\) \(1\)
\(13\) \(-1\)

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 3640.2.a.x 4
4.b odd 2 1 7280.2.a.br 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.x 4 1.a even 1 1 trivial
7280.2.a.br 4 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3640))\):

\( T_{3}^{4} - 3 T_{3}^{3} - 3 T_{3}^{2} + 10 T_{3} - 4 \)
\( T_{11}^{4} - 6 T_{11}^{3} - 12 T_{11}^{2} + 80 T_{11} - 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( -4 + 10 T - 3 T^{2} - 3 T^{3} + T^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( -64 + 80 T - 12 T^{2} - 6 T^{3} + T^{4} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( 16 + 32 T + 7 T^{2} - 9 T^{3} + T^{4} \)
$19$ \( -4 + 6 T + 3 T^{2} - 5 T^{3} + T^{4} \)
$23$ \( 16 + 8 T - 24 T^{2} - 2 T^{3} + T^{4} \)
$29$ \( -4 + 24 T - 37 T^{2} + 3 T^{3} + T^{4} \)
$31$ \( 52 + 18 T - 23 T^{2} - T^{3} + T^{4} \)
$37$ \( -676 - 464 T - 29 T^{2} + 11 T^{3} + T^{4} \)
$41$ \( -1024 - 992 T - 133 T^{2} + 5 T^{3} + T^{4} \)
$43$ \( -208 + 96 T + 20 T^{2} - 12 T^{3} + T^{4} \)
$47$ \( ( -16 - 2 T + T^{2} )^{2} \)
$53$ \( ( -8 - 6 T + T^{2} )^{2} \)
$59$ \( -268 + 294 T - 63 T^{2} - 11 T^{3} + T^{4} \)
$61$ \( ( -68 + T^{2} )^{2} \)
$67$ \( 16 - 60 T + 61 T^{2} - 19 T^{3} + T^{4} \)
$71$ \( 15248 - 2144 T - 316 T^{2} + 8 T^{3} + T^{4} \)
$73$ \( 832 + 240 T - 44 T^{2} - 8 T^{3} + T^{4} \)
$79$ \( -4096 + 2112 T - 147 T^{2} - 13 T^{3} + T^{4} \)
$83$ \( 832 + 512 T - 180 T^{2} - 8 T^{3} + T^{4} \)
$89$ \( 3008 + 2008 T + 7 T^{2} - 25 T^{3} + T^{4} \)
$97$ \( -24496 + 5016 T - 176 T^{2} - 18 T^{3} + T^{4} \)
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