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: \(x^{3} - 4 x - 1\)
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.

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

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

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3640))\):

\( T_{3}^{3} + 2 T_{3}^{2} - 4 T_{3} - 7 \)
\( T_{11}^{3} + T_{11}^{2} - 12 T_{11} - 16 \)

Hecke characteristic polynomials

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