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: \(x^{2} - x - 3\)
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.

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

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

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

Hecke characteristic polynomials

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