Properties

Label 4140.2.a.r
Level $4140$
Weight $2$
Character orbit 4140.a
Self dual yes
Analytic conductor $33.058$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) 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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + q^{7} + ( - \beta + 2) q^{11} - \beta q^{13} + ( - \beta + 1) q^{17} - 3 \beta q^{19} - q^{23} + q^{25} + ( - \beta + 7) q^{29} + (2 \beta + 3) q^{31} + q^{35} + (2 \beta - 5) q^{37} + (\beta + 3) q^{41} + (4 \beta + 2) q^{43} + 3 \beta q^{47} - 6 q^{49} + (3 \beta + 3) q^{53} + ( - \beta + 2) q^{55} + (\beta + 9) q^{59} + ( - \beta - 2) q^{61} - \beta q^{65} + ( - 4 \beta - 1) q^{67} + (3 \beta + 5) q^{71} + 3 \beta q^{73} + ( - \beta + 2) q^{77} + (4 \beta - 4) q^{79} + ( - \beta + 13) q^{83} + ( - \beta + 1) q^{85} + (2 \beta + 8) q^{89} - \beta q^{91} - 3 \beta q^{95} - 2 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 2 q^{7} + 4 q^{11} + 2 q^{17} - 2 q^{23} + 2 q^{25} + 14 q^{29} + 6 q^{31} + 2 q^{35} - 10 q^{37} + 6 q^{41} + 4 q^{43} - 12 q^{49} + 6 q^{53} + 4 q^{55} + 18 q^{59} - 4 q^{61} - 2 q^{67} + 10 q^{71} + 4 q^{77} - 8 q^{79} + 26 q^{83} + 2 q^{85} + 16 q^{89}+O(q^{100}) \) 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
2.44949
−2.44949
0 0 0 1.00000 0 1.00000 0 0 0
1.2 0 0 0 1.00000 0 1.00000 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.r 2
3.b odd 2 1 1380.2.a.f 2
12.b even 2 1 5520.2.a.bl 2
15.d odd 2 1 6900.2.a.r 2
15.e even 4 2 6900.2.f.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.f 2 3.b odd 2 1
4140.2.a.r 2 1.a even 1 1 trivial
5520.2.a.bl 2 12.b even 2 1
6900.2.a.r 2 15.d odd 2 1
6900.2.f.h 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4140))\):

\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 6 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 2 \) Copy content Toggle raw display
$13$ \( T^{2} - 6 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 5 \) Copy content Toggle raw display
$19$ \( T^{2} - 54 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 14T + 43 \) Copy content Toggle raw display
$31$ \( T^{2} - 6T - 15 \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 3 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 92 \) Copy content Toggle raw display
$47$ \( T^{2} - 54 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T - 45 \) Copy content Toggle raw display
$59$ \( T^{2} - 18T + 75 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T - 2 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T - 95 \) Copy content Toggle raw display
$71$ \( T^{2} - 10T - 29 \) Copy content Toggle raw display
$73$ \( T^{2} - 54 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T - 80 \) Copy content Toggle raw display
$83$ \( T^{2} - 26T + 163 \) Copy content Toggle raw display
$89$ \( T^{2} - 16T + 40 \) Copy content Toggle raw display
$97$ \( T^{2} - 24 \) Copy content Toggle raw display
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