Properties

Label 3640.2.a.o
Level $3640$
Weight $2$
Character orbit 3640.a
Self dual yes
Analytic conductor $29.066$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{3} - q^{5} - q^{7} + (\beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{3} - q^{5} - q^{7} + (\beta_1 + 1) q^{9} + (\beta_{2} + \beta_1) q^{11} + q^{13} + (\beta_{2} + 1) q^{15} + ( - \beta_{2} - 2 \beta_1 - 2) q^{17} - \beta_{2} q^{19} + (\beta_{2} + 1) q^{21} + (\beta_{2} - 3 \beta_1 + 2) q^{23} + q^{25} + (2 \beta_{2} - 2 \beta_1 + 1) q^{27} + (2 \beta_{2} + 3 \beta_1 - 1) q^{29} + (\beta_{2} + 5) q^{31} + (\beta_{2} - 3 \beta_1 - 4) q^{33} + q^{35} + (\beta_{2} - 2 \beta_1 - 5) q^{37} + ( - \beta_{2} - 1) q^{39} + ( - 3 \beta_1 + 2) q^{41} + (2 \beta_{2} - 2) q^{43} + ( - \beta_1 - 1) q^{45} + ( - 3 \beta_{2} + 5 \beta_1 + 2) q^{47} + q^{49} + (\beta_{2} + 5 \beta_1 + 7) q^{51} + (2 \beta_{2} + 2 \beta_1 - 2) q^{53} + ( - \beta_{2} - \beta_1) q^{55} + ( - \beta_{2} + \beta_1 + 3) q^{57} + (4 \beta_{2} - 3 \beta_1 - 1) q^{59} + ( - \beta_{2} - \beta_1 - 6) q^{61} + ( - \beta_1 - 1) q^{63} - q^{65} + (\beta_1 + 4) q^{67} + ( - \beta_{2} + 5 \beta_1 - 2) q^{69} + (4 \beta_{2} + 2 \beta_1 + 2) q^{71} + ( - \beta_{2} + 7 \beta_1) q^{73} + ( - \beta_{2} - 1) q^{75} + ( - \beta_{2} - \beta_1) q^{77} + (2 \beta_{2} + \beta_1 + 4) q^{79} + (\beta_{2} - \beta_1 - 8) q^{81} + ( - 4 \beta_1 - 6) q^{83} + (\beta_{2} + 2 \beta_1 + 2) q^{85} + (3 \beta_{2} - 8 \beta_1 - 8) q^{87} + ( - 2 \beta_{2} + 3 \beta_1 - 5) q^{89} - q^{91} + ( - 4 \beta_{2} - \beta_1 - 8) q^{93} + \beta_{2} q^{95} + ( - 5 \beta_{2} + \beta_1 - 4) q^{97} + (2 \beta_{2} + 2 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} - q^{11} + 3 q^{13} + 2 q^{15} - 5 q^{17} + q^{19} + 2 q^{21} + 5 q^{23} + 3 q^{25} + q^{27} - 5 q^{29} + 14 q^{31} - 13 q^{33} + 3 q^{35} - 16 q^{37} - 2 q^{39} + 6 q^{41} - 8 q^{43} - 3 q^{45} + 9 q^{47} + 3 q^{49} + 20 q^{51} - 8 q^{53} + q^{55} + 10 q^{57} - 7 q^{59} - 17 q^{61} - 3 q^{63} - 3 q^{65} + 12 q^{67} - 5 q^{69} + 2 q^{71} + q^{73} - 2 q^{75} + q^{77} + 10 q^{79} - 25 q^{81} - 18 q^{83} + 5 q^{85} - 27 q^{87} - 13 q^{89} - 3 q^{91} - 20 q^{93} - q^{95} - 7 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
2.11491
−1.86081
−0.254102
0 −2.47283 0 −1.00000 0 −1.00000 0 3.11491 0
1.2 0 −1.46260 0 −1.00000 0 −1.00000 0 −0.860806 0
1.3 0 1.93543 0 −1.00000 0 −1.00000 0 0.745898 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.o 3
4.b odd 2 1 7280.2.a.bp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.o 3 1.a even 1 1 trivial
7280.2.a.bp 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3640))\):

\( T_{3}^{3} + 2T_{3}^{2} - 4T_{3} - 7 \) Copy content Toggle raw display
\( T_{11}^{3} + T_{11}^{2} - 12T_{11} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} - 4 T - 7 \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + T^{2} - 12 T - 16 \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 5 T^{2} - 19 T + 14 \) Copy content Toggle raw display
$19$ \( T^{3} - T^{2} - 5T - 2 \) Copy content Toggle raw display
$23$ \( T^{3} - 5 T^{2} - 24 T - 4 \) Copy content Toggle raw display
$29$ \( T^{3} + 5 T^{2} - 67 T - 358 \) Copy content Toggle raw display
$31$ \( T^{3} - 14 T^{2} + 60 T - 73 \) Copy content Toggle raw display
$37$ \( T^{3} + 16 T^{2} + 70 T + 47 \) Copy content Toggle raw display
$41$ \( T^{3} - 6 T^{2} - 24 T + 91 \) Copy content Toggle raw display
$43$ \( T^{3} + 8T^{2} - 8 \) Copy content Toggle raw display
$47$ \( T^{3} - 9 T^{2} - 76 T + 676 \) Copy content Toggle raw display
$53$ \( T^{3} + 8 T^{2} - 28 T - 208 \) Copy content Toggle raw display
$59$ \( T^{3} + 7 T^{2} - 69 T - 112 \) Copy content Toggle raw display
$61$ \( T^{3} + 17 T^{2} + 84 T + 124 \) Copy content Toggle raw display
$67$ \( T^{3} - 12 T^{2} + 44 T - 49 \) Copy content Toggle raw display
$71$ \( T^{3} - 2 T^{2} - 124 T + 16 \) Copy content Toggle raw display
$73$ \( T^{3} - T^{2} - 180 T + 208 \) Copy content Toggle raw display
$79$ \( T^{3} - 10 T^{2} + 2 T + 59 \) Copy content Toggle raw display
$83$ \( T^{3} + 18 T^{2} + 44 T - 104 \) Copy content Toggle raw display
$89$ \( T^{3} + 13 T^{2} + 17 T - 2 \) Copy content Toggle raw display
$97$ \( T^{3} + 7 T^{2} - 106 T - 788 \) Copy content Toggle raw display
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