Properties

Label 3640.2.a.y
Level $3640$
Weight $2$
Character orbit 3640.a
Self dual yes
Analytic conductor $29.066$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.1194649.1
Defining polynomial: \(x^{5} - 8 x^{3} - 3 x^{2} + 10 x + 4\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} - q^{5} + q^{7} + ( \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} - q^{5} + q^{7} + ( \beta_{1} + \beta_{2} ) q^{9} + ( -1 + \beta_{3} ) q^{11} - q^{13} + \beta_{1} q^{15} + ( -\beta_{2} + \beta_{4} ) q^{17} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{19} -\beta_{1} q^{21} + ( -1 - 2 \beta_{2} - \beta_{3} ) q^{23} + q^{25} + ( -2 - \beta_{2} - \beta_{3} ) q^{27} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{29} + ( 2 + \beta_{1} - 2 \beta_{4} ) q^{31} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{33} - q^{35} + ( -2 - \beta_{3} - \beta_{4} ) q^{37} + \beta_{1} q^{39} + ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{41} + ( 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{43} + ( -\beta_{1} - \beta_{2} ) q^{45} + ( -3 + 3 \beta_{1} + \beta_{4} ) q^{47} + q^{49} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{51} + ( -4 - 2 \beta_{1} + 2 \beta_{4} ) q^{53} + ( 1 - \beta_{3} ) q^{55} + ( -3 - 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{57} + ( -1 + 2 \beta_{2} + \beta_{4} ) q^{59} + ( -3 - \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{61} + ( \beta_{1} + \beta_{2} ) q^{63} + q^{65} + ( -5 - 3 \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{67} + ( -3 + 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{69} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{71} + ( 5 + 3 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{73} -\beta_{1} q^{75} + ( -1 + \beta_{3} ) q^{77} + ( -1 + 5 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} ) q^{79} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{81} + ( 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( \beta_{2} - \beta_{4} ) q^{85} + ( 4 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{87} + ( -5 + 5 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{89} - q^{91} + ( -3 - 5 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} ) q^{93} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{95} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{97} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{5} + 5 q^{7} + q^{9} + O(q^{10}) \) \( 5 q - 5 q^{5} + 5 q^{7} + q^{9} - 7 q^{11} - 5 q^{13} + q^{17} + 3 q^{19} - 5 q^{23} + 5 q^{25} - 9 q^{27} - 9 q^{29} + 6 q^{31} + q^{33} - 5 q^{35} - 10 q^{37} - 8 q^{41} + 6 q^{43} - q^{45} - 13 q^{47} + 5 q^{49} - 4 q^{51} - 16 q^{53} + 7 q^{55} - 10 q^{57} - q^{59} - 11 q^{61} + q^{63} + 5 q^{65} - 30 q^{67} - 15 q^{69} - 12 q^{71} + 23 q^{73} - 7 q^{77} + 2 q^{79} - 11 q^{81} - 2 q^{83} - q^{85} - 5 q^{87} - 25 q^{89} - 5 q^{91} - 22 q^{93} - 3 q^{95} - 5 q^{97} - 12 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 8 x^{3} - 3 x^{2} + 10 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 7 \nu^{2} + 3 \nu + 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 6 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 8 \beta_{2} + 10 \beta_{1} + 16\)

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
2.75291
1.21592
−0.402509
−1.32400
−2.24233
0 −2.75291 0 −1.00000 0 1.00000 0 4.57851 0
1.2 0 −1.21592 0 −1.00000 0 1.00000 0 −1.52153 0
1.3 0 0.402509 0 −1.00000 0 1.00000 0 −2.83799 0
1.4 0 1.32400 0 −1.00000 0 1.00000 0 −1.24704 0
1.5 0 2.24233 0 −1.00000 0 1.00000 0 2.02804 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.y 5
4.b odd 2 1 7280.2.a.cc 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.y 5 1.a even 1 1 trivial
7280.2.a.cc 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3640))\):

\( T_{3}^{5} - 8 T_{3}^{3} + 3 T_{3}^{2} + 10 T_{3} - 4 \)
\( T_{11}^{5} + 7 T_{11}^{4} - 10 T_{11}^{3} - 84 T_{11}^{2} + 96 T_{11} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( -4 + 10 T + 3 T^{2} - 8 T^{3} + T^{5} \)
$5$ \( ( 1 + T )^{5} \)
$7$ \( ( -1 + T )^{5} \)
$11$ \( 64 + 96 T - 84 T^{2} - 10 T^{3} + 7 T^{4} + T^{5} \)
$13$ \( ( 1 + T )^{5} \)
$17$ \( 776 + 412 T - 54 T^{2} - 47 T^{3} - T^{4} + T^{5} \)
$19$ \( -2168 + 720 T + 160 T^{2} - 55 T^{3} - 3 T^{4} + T^{5} \)
$23$ \( 1328 - 584 T - 520 T^{2} - 70 T^{3} + 5 T^{4} + T^{5} \)
$29$ \( 8168 + 652 T - 582 T^{2} - 59 T^{3} + 9 T^{4} + T^{5} \)
$31$ \( -3076 + 1198 T + 421 T^{2} - 82 T^{3} - 6 T^{4} + T^{5} \)
$37$ \( 20 - 24 T - 45 T^{2} + 10 T^{4} + T^{5} \)
$41$ \( -3340 + 2596 T - 259 T^{2} - 90 T^{3} + 8 T^{4} + T^{5} \)
$43$ \( -13472 + 3792 T + 968 T^{2} - 164 T^{3} - 6 T^{4} + T^{5} \)
$47$ \( -256 + 672 T - 364 T^{2} - 16 T^{3} + 13 T^{4} + T^{5} \)
$53$ \( 12928 - 432 T - 992 T^{2} - 24 T^{3} + 16 T^{4} + T^{5} \)
$59$ \( 400 + 188 T - 230 T^{2} - 71 T^{3} + T^{4} + T^{5} \)
$61$ \( -4112 - 3536 T - 936 T^{2} - 48 T^{3} + 11 T^{4} + T^{5} \)
$67$ \( -38288 - 12596 T - 421 T^{2} + 246 T^{3} + 30 T^{4} + T^{5} \)
$71$ \( -640 - 1456 T - 464 T^{2} - 4 T^{3} + 12 T^{4} + T^{5} \)
$73$ \( -1216 - 6208 T + 1708 T^{2} + 36 T^{3} - 23 T^{4} + T^{5} \)
$79$ \( 1328 + 2036 T + 307 T^{2} - 246 T^{3} - 2 T^{4} + T^{5} \)
$83$ \( 2432 + 1792 T - 1160 T^{2} - 212 T^{3} + 2 T^{4} + T^{5} \)
$89$ \( -100376 - 35444 T - 3590 T^{2} + 45 T^{3} + 25 T^{4} + T^{5} \)
$97$ \( -3376 + 3480 T - 288 T^{2} - 118 T^{3} + 5 T^{4} + T^{5} \)
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