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: \( x^{4} - x^{3} - 6x^{2} + x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} - q^{5} - q^{7} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} - q^{5} - q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{11} + q^{13} + (\beta_1 - 1) q^{15} + (\beta_{2} + \beta_1 + 2) q^{17} + (\beta_{3} + 1) q^{19} + (\beta_1 - 1) q^{21} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{23} + q^{25} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{27} + ( - \beta_{3} - \beta_{2} - 2 \beta_1) q^{29} + ( - 2 \beta_{3} - \beta_1 + 1) q^{31} - 4 \beta_1 q^{33} + q^{35} + (3 \beta_{3} - \beta_{2} + 2 \beta_1 - 4) q^{37} + ( - \beta_1 + 1) q^{39} + (4 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{41} + (\beta_{3} + \beta_{2} - \beta_1 + 3) q^{43} + ( - \beta_{2} + \beta_1 - 1) q^{45} + (2 \beta_{3} + 2 \beta_1) q^{47} + q^{49} + ( - \beta_{3} - 5 \beta_1) q^{51} + ( - 2 \beta_{3} - 2 \beta_1 + 4) q^{53} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{55} + (\beta_{3} - \beta_1 + 2) q^{57} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{59} + ( - 4 \beta_{3} - 4 \beta_1 + 2) q^{61} + ( - \beta_{2} + \beta_1 - 1) q^{63} - q^{65} + (\beta_{2} + 3 \beta_1 + 4) q^{67} + (2 \beta_1 + 2) q^{69} + ( - 5 \beta_{3} + \beta_{2} - 7 \beta_1 + 1) q^{71} + (2 \beta_{3} - 2 \beta_1 + 2) q^{73} + ( - \beta_1 + 1) q^{75} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{77} + (2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 2) q^{79} + ( - 3 \beta_{3} - 5 \beta_1 + 3) q^{81} + (6 \beta_{3} + 2 \beta_1) q^{83} + ( - \beta_{2} - \beta_1 - 2) q^{85} + (\beta_{2} + 3 \beta_1 + 4) q^{87} + ( - 3 \beta_{3} - \beta_{2} + 4 \beta_1 + 6) q^{89} - q^{91} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{93} + ( - \beta_{3} - 1) q^{95} + ( - 6 \beta_{3} + 2 \beta_{2} + 6) q^{97} + ( - 3 \beta_{3} + \beta_{2} - 3 \beta_1 + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9} + 6 q^{11} + 4 q^{13} - 3 q^{15} + 9 q^{17} + 5 q^{19} - 3 q^{21} + 2 q^{23} + 4 q^{25} + 6 q^{27} - 3 q^{29} + q^{31} - 4 q^{33} + 4 q^{35} - 11 q^{37} + 3 q^{39} - 5 q^{41} + 12 q^{43} - 3 q^{45} + 4 q^{47} + 4 q^{49} - 6 q^{51} + 12 q^{53} - 6 q^{55} + 8 q^{57} + 11 q^{59} - 3 q^{63} - 4 q^{65} + 19 q^{67} + 10 q^{69} - 8 q^{71} + 8 q^{73} + 3 q^{75} - 6 q^{77} + 13 q^{79} + 4 q^{81} + 8 q^{83} - 9 q^{85} + 19 q^{87} + 25 q^{89} - 4 q^{91} + 5 q^{93} - 5 q^{95} + 18 q^{97} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 6\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 7\beta _1 + 2 \) 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
2.90570
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3640))\):

\( T_{3}^{4} - 3T_{3}^{3} - 3T_{3}^{2} + 10T_{3} - 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 6T_{11}^{3} - 12T_{11}^{2} + 80T_{11} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

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