Properties

Label 4140.2.a.s
Level $4140$
Weight $2$
Character orbit 4140.a
Self dual yes
Analytic conductor $33.058$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.0580664368\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
Defining polynomial: \( x^{3} - x^{2} - 16x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1380)
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.

\(f(q)\) \(=\) \( q + q^{5} + ( - \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + ( - \beta_1 + 1) q^{7} + ( - \beta_{2} - 1) q^{11} + ( - \beta_{2} + 1) q^{13} + (\beta_{2} - \beta_1) q^{17} + (\beta_{2} + 3) q^{19} - q^{23} + q^{25} + ( - \beta_{2} + \beta_1) q^{29} + (\beta_1 + 3) q^{31} + ( - \beta_1 + 1) q^{35} + ( - 2 \beta_{2} + \beta_1 + 1) q^{37} + (\beta_{2} - 3 \beta_1 - 2) q^{41} + ( - 2 \beta_1 + 6) q^{43} + ( - \beta_{2} + 2 \beta_1 + 1) q^{47} + (2 \beta_{2} - \beta_1 + 4) q^{49} + (\beta_{2} - 3 \beta_1 - 2) q^{53} + ( - \beta_{2} - 1) q^{55} + (\beta_{2} - \beta_1) q^{59} + ( - \beta_{2} + 2 \beta_1 + 3) q^{61} + ( - \beta_{2} + 1) q^{65} + (\beta_1 + 3) q^{67} + ( - \beta_{2} + 3 \beta_1 + 2) q^{71} + (3 \beta_{2} + 5) q^{73} + ( - \beta_{2} + 4 \beta_1 + 3) q^{77} + (2 \beta_1 + 4) q^{79} + (\beta_{2} + \beta_1 - 4) q^{83} + (\beta_{2} - \beta_1) q^{85} + (2 \beta_{2} + 2 \beta_1 - 2) q^{89} + ( - \beta_{2} + 2 \beta_1 + 5) q^{91} + (\beta_{2} + 3) q^{95} + (2 \beta_{2} - 4 \beta_1 - 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{5} + 2 q^{7} - 4 q^{11} + 2 q^{13} + 10 q^{19} - 3 q^{23} + 3 q^{25} + 10 q^{31} + 2 q^{35} + 2 q^{37} - 8 q^{41} + 16 q^{43} + 4 q^{47} + 13 q^{49} - 8 q^{53} - 4 q^{55} + 10 q^{61} + 2 q^{65} + 10 q^{67} + 8 q^{71} + 18 q^{73} + 12 q^{77} + 14 q^{79} - 10 q^{83} - 2 q^{89} + 16 q^{91} + 10 q^{95} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 16x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - \nu - 10 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + \beta _1 + 10 \) 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
4.73549
−0.526440
−3.20905
0 0 0 1.00000 0 −3.73549 0 0 0
1.2 0 0 0 1.00000 0 1.52644 0 0 0
1.3 0 0 0 1.00000 0 4.20905 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(23\) \(1\)

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 4140.2.a.s 3
3.b odd 2 1 1380.2.a.j 3
12.b even 2 1 5520.2.a.bv 3
15.d odd 2 1 6900.2.a.x 3
15.e even 4 2 6900.2.f.r 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.j 3 3.b odd 2 1
4140.2.a.s 3 1.a even 1 1 trivial
5520.2.a.bv 3 12.b even 2 1
6900.2.a.x 3 15.d odd 2 1
6900.2.f.r 6 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4140))\):

\( T_{7}^{3} - 2T_{7}^{2} - 15T_{7} + 24 \) Copy content Toggle raw display
\( T_{11}^{3} + 4T_{11}^{2} - 14T_{11} - 48 \) Copy content Toggle raw display
\( T_{13}^{3} - 2T_{13}^{2} - 18T_{13} - 12 \) Copy content Toggle raw display
\( T_{17}^{3} - 21T_{17} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} - 15 T + 24 \) Copy content Toggle raw display
$11$ \( T^{3} + 4 T^{2} - 14 T - 48 \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} - 18 T - 12 \) Copy content Toggle raw display
$17$ \( T^{3} - 21T - 18 \) Copy content Toggle raw display
$19$ \( T^{3} - 10 T^{2} + 14 T + 52 \) Copy content Toggle raw display
$23$ \( (T + 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 21T + 18 \) Copy content Toggle raw display
$31$ \( T^{3} - 10 T^{2} + 17 T + 4 \) Copy content Toggle raw display
$37$ \( T^{3} - 2 T^{2} - 63 T - 108 \) Copy content Toggle raw display
$41$ \( T^{3} + 8 T^{2} - 101 T - 582 \) Copy content Toggle raw display
$43$ \( T^{3} - 16 T^{2} + 20 T + 304 \) Copy content Toggle raw display
$47$ \( T^{3} - 4 T^{2} - 50 T + 216 \) Copy content Toggle raw display
$53$ \( T^{3} + 8 T^{2} - 101 T - 582 \) Copy content Toggle raw display
$59$ \( T^{3} - 21T - 18 \) Copy content Toggle raw display
$61$ \( T^{3} - 10 T^{2} - 22 T + 292 \) Copy content Toggle raw display
$67$ \( T^{3} - 10 T^{2} + 17 T + 4 \) Copy content Toggle raw display
$71$ \( T^{3} - 8 T^{2} - 101 T + 582 \) Copy content Toggle raw display
$73$ \( T^{3} - 18 T^{2} - 66 T + 1492 \) Copy content Toggle raw display
$79$ \( T^{3} - 14T^{2} + 96 \) Copy content Toggle raw display
$83$ \( T^{3} + 10 T^{2} - 17 T - 228 \) Copy content Toggle raw display
$89$ \( T^{3} + 2 T^{2} - 200 T - 912 \) Copy content Toggle raw display
$97$ \( T^{3} + 20 T^{2} - 88 T - 2336 \) Copy content Toggle raw display
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