Properties

Label 3640.2.a.z
Level $3640$
Weight $2$
Character orbit 3640.a
Self dual yes
Analytic conductor $29.066$
Analytic rank $0$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.2112217.1
Defining polynomial: \(x^{5} - 10 x^{3} - x^{2} + 10 x + 4\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + q^{5} + q^{7} + ( 1 + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + q^{5} + q^{7} + ( 1 + \beta_{2} - \beta_{3} ) q^{9} + ( 1 - \beta_{1} + \beta_{4} ) q^{11} - q^{13} -\beta_{1} q^{15} + ( 1 - \beta_{3} ) q^{17} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{19} -\beta_{1} q^{21} + ( 1 - \beta_{1} - \beta_{4} ) q^{23} + q^{25} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{27} + ( 1 + 2 \beta_{1} + \beta_{4} ) q^{29} + ( 2 - \beta_{1} ) q^{31} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{33} + q^{35} + ( 2 + \beta_{2} - \beta_{4} ) q^{37} + \beta_{1} q^{39} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{41} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{43} + ( 1 + \beta_{2} - \beta_{3} ) q^{45} + ( -1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{47} + q^{49} + ( -2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{51} + ( 6 - 2 \beta_{4} ) q^{53} + ( 1 - \beta_{1} + \beta_{4} ) q^{55} + ( -4 - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{57} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{59} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{61} + ( 1 + \beta_{2} - \beta_{3} ) q^{63} - q^{65} + ( 4 + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{67} + ( 5 + \beta_{2} - 2 \beta_{3} ) q^{69} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{71} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{73} -\beta_{1} q^{75} + ( 1 - \beta_{1} + \beta_{4} ) q^{77} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{79} + ( 5 - \beta_{3} + \beta_{4} ) q^{81} + ( -2 \beta_{2} - 2 \beta_{4} ) q^{83} + ( 1 - \beta_{3} ) q^{85} + ( -9 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{87} + ( 1 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{89} - q^{91} + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{95} + ( -1 - 2 \beta_{1} - \beta_{2} - 4 \beta_{4} ) q^{97} + ( 4 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9} + O(q^{10}) \) \( 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9} + 7 q^{11} - 5 q^{13} + 3 q^{17} + 3 q^{19} + 3 q^{23} + 5 q^{25} - 3 q^{27} + 7 q^{29} + 10 q^{31} + 17 q^{33} + 5 q^{35} + 10 q^{37} + 4 q^{43} + 5 q^{45} - 3 q^{47} + 5 q^{49} + 26 q^{53} + 7 q^{55} - 20 q^{57} + 3 q^{59} + 13 q^{61} + 5 q^{63} - 5 q^{65} + 12 q^{67} + 23 q^{69} + 8 q^{71} + q^{73} + 7 q^{77} - 6 q^{79} + 25 q^{81} - 8 q^{83} + 3 q^{85} - 43 q^{87} + 3 q^{89} - 5 q^{91} + 20 q^{93} + 3 q^{95} - 15 q^{97} + 16 q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - \nu^{3} - 9 \nu^{2} + 8 \nu + 5 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 10 \nu^{2} + 8 \nu + 9 \)
\(\beta_{4}\)\(=\)\( 2 \nu^{4} - \nu^{3} - 19 \nu^{2} + 8 \nu + 13 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{4} - \beta_{3} - \beta_{2} + 8 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} - 10 \beta_{3} + 9 \beta_{2} + 32\)

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
3.01413
1.19075
−0.492015
−0.769609
−2.94326
0 −3.01413 0 1.00000 0 1.00000 0 6.08498 0
1.2 0 −1.19075 0 1.00000 0 1.00000 0 −1.58211 0
1.3 0 0.492015 0 1.00000 0 1.00000 0 −2.75792 0
1.4 0 0.769609 0 1.00000 0 1.00000 0 −2.40770 0
1.5 0 2.94326 0 1.00000 0 1.00000 0 5.66276 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.z 5
4.b odd 2 1 7280.2.a.cd 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.z 5 1.a even 1 1 trivial
7280.2.a.cd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3640))\):

\( T_{3}^{5} - 10 T_{3}^{3} + T_{3}^{2} + 10 T_{3} - 4 \)
\( T_{11}^{5} - 7 T_{11}^{4} - 8 T_{11}^{3} + 68 T_{11}^{2} + 80 T_{11} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( -4 + 10 T + T^{2} - 10 T^{3} + T^{5} \)
$5$ \( ( -1 + T )^{5} \)
$7$ \( ( -1 + T )^{5} \)
$11$ \( -16 + 80 T + 68 T^{2} - 8 T^{3} - 7 T^{4} + T^{5} \)
$13$ \( ( 1 + T )^{5} \)
$17$ \( -128 + 48 T + 50 T^{2} - 17 T^{3} - 3 T^{4} + T^{5} \)
$19$ \( -3352 + 1032 T + 212 T^{2} - 67 T^{3} - 3 T^{4} + T^{5} \)
$23$ \( -448 + 208 T + 76 T^{2} - 30 T^{3} - 3 T^{4} + T^{5} \)
$29$ \( -952 + 572 T + 194 T^{2} - 47 T^{3} - 7 T^{4} + T^{5} \)
$31$ \( 28 - 34 T - 19 T^{2} + 30 T^{3} - 10 T^{4} + T^{5} \)
$37$ \( -124 - 228 T + 149 T^{2} + 6 T^{3} - 10 T^{4} + T^{5} \)
$41$ \( 256 - 528 T + 341 T^{2} - 74 T^{3} + T^{5} \)
$43$ \( 3808 + 2400 T + 120 T^{2} - 108 T^{3} - 4 T^{4} + T^{5} \)
$47$ \( 256 + 800 T - 364 T^{2} - 110 T^{3} + 3 T^{4} + T^{5} \)
$53$ \( 3584 - 2496 T - 64 T^{2} + 188 T^{3} - 26 T^{4} + T^{5} \)
$59$ \( 224 - 508 T + 370 T^{2} - 83 T^{3} - 3 T^{4} + T^{5} \)
$61$ \( -1984 - 704 T + 460 T^{2} - 8 T^{3} - 13 T^{4} + T^{5} \)
$67$ \( -57808 - 2076 T + 3005 T^{2} - 202 T^{3} - 12 T^{4} + T^{5} \)
$71$ \( -22016 + 464 T + 1536 T^{2} - 156 T^{3} - 8 T^{4} + T^{5} \)
$73$ \( -4336 + 2376 T + 184 T^{2} - 178 T^{3} - T^{4} + T^{5} \)
$79$ \( 256 - 2128 T - 1069 T^{2} - 118 T^{3} + 6 T^{4} + T^{5} \)
$83$ \( -128 - 11328 T - 3336 T^{2} - 240 T^{3} + 8 T^{4} + T^{5} \)
$89$ \( -22400 + 6640 T + 1082 T^{2} - 223 T^{3} - 3 T^{4} + T^{5} \)
$97$ \( 16576 + 13136 T - 5620 T^{2} - 404 T^{3} + 15 T^{4} + T^{5} \)
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