Properties

Label 3640.2.a.l
Level $3640$
Weight $2$
Character orbit 3640.a
Self dual yes
Analytic conductor $29.066$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + q^{5} + q^{7} + \beta q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + q^{5} + q^{7} + \beta q^{9} + (2 \beta - 2) q^{11} - q^{13} - \beta q^{15} + ( - \beta + 5) q^{17} + (\beta - 5) q^{19} - \beta q^{21} - 8 q^{23} + q^{25} + (2 \beta - 3) q^{27} + ( - 3 \beta - 2) q^{29} + ( - \beta - 2) q^{31} - 6 q^{33} + q^{35} + 3 \beta q^{37} + \beta q^{39} + (\beta - 3) q^{41} - 2 q^{43} + \beta q^{45} + (2 \beta - 2) q^{47} + q^{49} + ( - 4 \beta + 3) q^{51} + 2 \beta q^{53} + (2 \beta - 2) q^{55} + (4 \beta - 3) q^{57} + ( - 7 \beta + 2) q^{59} + (4 \beta + 4) q^{61} + \beta q^{63} - q^{65} + ( - 3 \beta + 5) q^{67} + 8 \beta q^{69} + 2 \beta q^{71} + (2 \beta - 6) q^{73} - \beta q^{75} + (2 \beta - 2) q^{77} + ( - 5 \beta - 3) q^{79} + ( - 2 \beta - 6) q^{81} + (6 \beta - 4) q^{83} + ( - \beta + 5) q^{85} + (5 \beta + 9) q^{87} + (\beta - 6) q^{89} - q^{91} + (3 \beta + 3) q^{93} + (\beta - 5) q^{95} + (4 \beta - 8) q^{97} + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{5} + 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 2 q^{5} + 2 q^{7} + q^{9} - 2 q^{11} - 2 q^{13} - q^{15} + 9 q^{17} - 9 q^{19} - q^{21} - 16 q^{23} + 2 q^{25} - 4 q^{27} - 7 q^{29} - 5 q^{31} - 12 q^{33} + 2 q^{35} + 3 q^{37} + q^{39} - 5 q^{41} - 4 q^{43} + q^{45} - 2 q^{47} + 2 q^{49} + 2 q^{51} + 2 q^{53} - 2 q^{55} - 2 q^{57} - 3 q^{59} + 12 q^{61} + q^{63} - 2 q^{65} + 7 q^{67} + 8 q^{69} + 2 q^{71} - 10 q^{73} - q^{75} - 2 q^{77} - 11 q^{79} - 14 q^{81} - 2 q^{83} + 9 q^{85} + 23 q^{87} - 11 q^{89} - 2 q^{91} + 9 q^{93} - 9 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 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
2.30278
−1.30278
0 −2.30278 0 1.00000 0 1.00000 0 2.30278 0
1.2 0 1.30278 0 1.00000 0 1.00000 0 −1.30278 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.l 2
4.b odd 2 1 7280.2.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.l 2 1.a even 1 1 trivial
7280.2.a.bh 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3640))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 9T + 17 \) Copy content Toggle raw display
$19$ \( T^{2} + 9T + 17 \) Copy content Toggle raw display
$23$ \( (T + 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 7T - 17 \) Copy content Toggle raw display
$31$ \( T^{2} + 5T + 3 \) Copy content Toggle raw display
$37$ \( T^{2} - 3T - 27 \) Copy content Toggle raw display
$41$ \( T^{2} + 5T + 3 \) Copy content Toggle raw display
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$53$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$59$ \( T^{2} + 3T - 157 \) Copy content Toggle raw display
$61$ \( T^{2} - 12T - 16 \) Copy content Toggle raw display
$67$ \( T^{2} - 7T - 17 \) Copy content Toggle raw display
$71$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$73$ \( T^{2} + 10T + 12 \) Copy content Toggle raw display
$79$ \( T^{2} + 11T - 51 \) Copy content Toggle raw display
$83$ \( T^{2} + 2T - 116 \) Copy content Toggle raw display
$89$ \( T^{2} + 11T + 27 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T - 16 \) Copy content Toggle raw display
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