Properties

Label 3640.2.a.s
Level $3640$
Weight $2$
Character orbit 3640.a
Self dual yes
Analytic conductor $29.066$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
Defining polynomial: \(x^{4} - x^{3} - 5 x^{2} + 2 x + 4\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 5 x^{2} + 2 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 4 \beta_{1} + 2\)

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
2.43828
1.13856
−0.820249
−1.75660
0 −2.43828 0 1.00000 0 −1.00000 0 2.94523 0
1.2 0 −1.13856 0 1.00000 0 −1.00000 0 −1.70367 0
1.3 0 0.820249 0 1.00000 0 −1.00000 0 −2.32719 0
1.4 0 1.75660 0 1.00000 0 −1.00000 0 0.0856374 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.s 4
4.b odd 2 1 7280.2.a.ca 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.s 4 1.a even 1 1 trivial
7280.2.a.ca 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3640))\):

\( T_{3}^{4} + T_{3}^{3} - 5 T_{3}^{2} - 2 T_{3} + 4 \)
\( T_{11}^{2} + 2 T_{11} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 4 - 2 T - 5 T^{2} + T^{3} + T^{4} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( ( -4 + 2 T + T^{2} )^{2} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( 4 + 8 T - 15 T^{2} + T^{3} + T^{4} \)
$19$ \( 4 - 26 T - 5 T^{2} + 7 T^{3} + T^{4} \)
$23$ \( 64 - 32 T - 20 T^{2} + 6 T^{3} + T^{4} \)
$29$ \( -284 + 236 T - 53 T^{2} - T^{3} + T^{4} \)
$31$ \( -1604 - 586 T - 23 T^{2} + 11 T^{3} + T^{4} \)
$37$ \( -4 - 12 T - 7 T^{2} + 3 T^{3} + T^{4} \)
$41$ \( 124 - 40 T - 29 T^{2} + 5 T^{3} + T^{4} \)
$43$ \( 16 - 56 T - 8 T^{2} + 6 T^{3} + T^{4} \)
$47$ \( 1024 - 832 T - 140 T^{2} + 6 T^{3} + T^{4} \)
$53$ \( 16 - 88 T - 32 T^{2} + 2 T^{3} + T^{4} \)
$59$ \( 10636 - 194 T - 219 T^{2} + T^{3} + T^{4} \)
$61$ \( 64 - 48 T - 20 T^{2} + 4 T^{3} + T^{4} \)
$67$ \( 16 - 44 T - 9 T^{2} + 11 T^{3} + T^{4} \)
$71$ \( -16 + 48 T + 60 T^{2} + 16 T^{3} + T^{4} \)
$73$ \( 2096 + 72 T - 96 T^{2} - 2 T^{3} + T^{4} \)
$79$ \( -5776 - 1900 T - 99 T^{2} + 15 T^{3} + T^{4} \)
$83$ \( 1216 - 192 T - 116 T^{2} + 2 T^{3} + T^{4} \)
$89$ \( 6156 - 648 T - 261 T^{2} + 3 T^{3} + T^{4} \)
$97$ \( 64 - 16 T - 60 T^{2} - 8 T^{3} + T^{4} \)
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