Properties

Label 3640.2.a.p
Level $3640$
Weight $2$
Character orbit 3640.a
Self dual yes
Analytic conductor $29.066$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.1101.1
Defining polynomial: \( x^{3} - x^{2} - 9x + 12 \) 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - q^{5} + q^{7} + (\beta_{2} - \beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - q^{5} + q^{7} + (\beta_{2} - \beta_1 + 4) q^{9} - q^{13} + \beta_1 q^{15} + (\beta_{2} - \beta_1 + 1) q^{17} + (\beta_{2} - \beta_1 - 1) q^{19} - \beta_1 q^{21} + (2 \beta_{2} + 2) q^{23} + q^{25} + ( - \beta_{2} - 2 \beta_1 + 5) q^{27} + ( - 2 \beta_{2} + \beta_1 - 4) q^{29} + ( - \beta_1 + 4) q^{31} - q^{35} + ( - 2 \beta_{2} - \beta_1) q^{37} + \beta_1 q^{39} + (\beta_{2} - \beta_1 + 1) q^{41} + (2 \beta_1 - 4) q^{43} + ( - \beta_{2} + \beta_1 - 4) q^{45} + ( - 2 \beta_{2} + 2) q^{47} + q^{49} + ( - \beta_{2} - 2 \beta_1 + 5) q^{51} + 2 q^{53} + ( - \beta_{2} + 5) q^{57} + ( - 3 \beta_1 + 4) q^{59} - 10 q^{61} + (\beta_{2} - \beta_1 + 4) q^{63} + q^{65} + (3 \beta_{2} + \beta_1 + 1) q^{67} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{69} + 2 \beta_1 q^{71} + (2 \beta_{2} - 4 \beta_1 + 4) q^{73} - \beta_1 q^{75} + ( - \beta_{2} + \beta_1 + 1) q^{79} + (\beta_{2} - 4 \beta_1 + 4) q^{81} + ( - 2 \beta_{2} - 2) q^{83} + ( - \beta_{2} + \beta_1 - 1) q^{85} + (3 \beta_{2} + 5 \beta_1 - 3) q^{87} + ( - 2 \beta_{2} - 3 \beta_1) q^{89} - q^{91} + (\beta_{2} - 5 \beta_1 + 7) q^{93} + ( - \beta_{2} + \beta_1 + 1) q^{95} + (4 \beta_{2} + 2 \beta_1 + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - 3 q^{5} + 3 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} - 3 q^{5} + 3 q^{7} + 10 q^{9} - 3 q^{13} + q^{15} + q^{17} - 5 q^{19} - q^{21} + 4 q^{23} + 3 q^{25} + 14 q^{27} - 9 q^{29} + 11 q^{31} - 3 q^{35} + q^{37} + q^{39} + q^{41} - 10 q^{43} - 10 q^{45} + 8 q^{47} + 3 q^{49} + 14 q^{51} + 6 q^{53} + 16 q^{57} + 9 q^{59} - 30 q^{61} + 10 q^{63} + 3 q^{65} + q^{67} - 10 q^{69} + 2 q^{71} + 6 q^{73} - q^{75} + 5 q^{79} + 7 q^{81} - 4 q^{83} - q^{85} - 7 q^{87} - q^{89} - 3 q^{91} + 15 q^{93} + 5 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 9x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 7 \) 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
2.68740
1.43163
−3.11903
0 −2.68740 0 −1.00000 0 1.00000 0 4.22212 0
1.2 0 −1.43163 0 −1.00000 0 1.00000 0 −0.950444 0
1.3 0 3.11903 0 −1.00000 0 1.00000 0 6.72833 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.p 3
4.b odd 2 1 7280.2.a.bn 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.p 3 1.a even 1 1 trivial
7280.2.a.bn 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3640))\):

\( T_{3}^{3} + T_{3}^{2} - 9T_{3} - 12 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 9T - 12 \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( (T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - T^{2} - 15 T + 18 \) Copy content Toggle raw display
$19$ \( T^{3} + 5 T^{2} - 7 T - 8 \) Copy content Toggle raw display
$23$ \( T^{3} - 4 T^{2} - 36 T + 48 \) Copy content Toggle raw display
$29$ \( T^{3} + 9 T^{2} - 15 T - 202 \) Copy content Toggle raw display
$31$ \( T^{3} - 11 T^{2} + 31 T - 24 \) Copy content Toggle raw display
$37$ \( T^{3} - T^{2} - 59 T + 186 \) Copy content Toggle raw display
$41$ \( T^{3} - T^{2} - 15 T + 18 \) Copy content Toggle raw display
$43$ \( T^{3} + 10 T^{2} - 4 T - 16 \) Copy content Toggle raw display
$47$ \( T^{3} - 8 T^{2} - 20 T + 96 \) Copy content Toggle raw display
$53$ \( (T - 2)^{3} \) Copy content Toggle raw display
$59$ \( T^{3} - 9 T^{2} - 57 T - 16 \) Copy content Toggle raw display
$61$ \( (T + 10)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} - T^{2} - 115 T - 332 \) Copy content Toggle raw display
$71$ \( T^{3} - 2 T^{2} - 36 T + 96 \) Copy content Toggle raw display
$73$ \( T^{3} - 6 T^{2} - 144 T - 128 \) Copy content Toggle raw display
$79$ \( T^{3} - 5 T^{2} - 7 T + 8 \) Copy content Toggle raw display
$83$ \( T^{3} + 4 T^{2} - 36 T - 48 \) Copy content Toggle raw display
$89$ \( T^{3} + T^{2} - 151 T + 386 \) Copy content Toggle raw display
$97$ \( T^{3} - 4 T^{2} - 232 T - 1016 \) Copy content Toggle raw display
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