Properties

Label 5040.2.a.bq
Level $5040$
Weight $2$
Character orbit 5040.a
Self dual yes
Analytic conductor $40.245$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} - q^{7} +O(q^{10})\) \( q - q^{5} - q^{7} -\beta q^{11} + ( 2 - 3 \beta ) q^{13} + ( -6 + \beta ) q^{17} + ( 4 - 2 \beta ) q^{19} -2 \beta q^{23} + q^{25} + ( -2 - \beta ) q^{29} + 4 \beta q^{31} + q^{35} + ( -2 + 4 \beta ) q^{37} + ( -2 - 2 \beta ) q^{41} + ( 4 - 2 \beta ) q^{43} + ( 4 + \beta ) q^{47} + q^{49} + ( 10 - 2 \beta ) q^{53} + \beta q^{55} -4 q^{59} + ( -10 - 2 \beta ) q^{61} + ( -2 + 3 \beta ) q^{65} + ( 4 + 4 \beta ) q^{67} + ( 2 + 4 \beta ) q^{73} + \beta q^{77} + ( 4 + 3 \beta ) q^{79} + 12 q^{83} + ( 6 - \beta ) q^{85} + ( -2 + 2 \beta ) q^{89} + ( -2 + 3 \beta ) q^{91} + ( -4 + 2 \beta ) q^{95} + ( 6 + 3 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} - 2q^{7} + O(q^{10}) \) \( 2q - 2q^{5} - 2q^{7} - q^{11} + q^{13} - 11q^{17} + 6q^{19} - 2q^{23} + 2q^{25} - 5q^{29} + 4q^{31} + 2q^{35} - 6q^{41} + 6q^{43} + 9q^{47} + 2q^{49} + 18q^{53} + q^{55} - 8q^{59} - 22q^{61} - q^{65} + 12q^{67} + 8q^{73} + q^{77} + 11q^{79} + 24q^{83} + 11q^{85} - 2q^{89} - q^{91} - 6q^{95} + 15q^{97} + O(q^{100}) \)

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
2.56155
−1.56155
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
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

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 5040.2.a.bq 2
3.b odd 2 1 560.2.a.g 2
4.b odd 2 1 2520.2.a.w 2
12.b even 2 1 280.2.a.d 2
15.d odd 2 1 2800.2.a.bn 2
15.e even 4 2 2800.2.g.u 4
21.c even 2 1 3920.2.a.bu 2
24.f even 2 1 2240.2.a.be 2
24.h odd 2 1 2240.2.a.bi 2
60.h even 2 1 1400.2.a.p 2
60.l odd 4 2 1400.2.g.k 4
84.h odd 2 1 1960.2.a.r 2
84.j odd 6 2 1960.2.q.u 4
84.n even 6 2 1960.2.q.s 4
420.o odd 2 1 9800.2.a.by 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.d 2 12.b even 2 1
560.2.a.g 2 3.b odd 2 1
1400.2.a.p 2 60.h even 2 1
1400.2.g.k 4 60.l odd 4 2
1960.2.a.r 2 84.h odd 2 1
1960.2.q.s 4 84.n even 6 2
1960.2.q.u 4 84.j odd 6 2
2240.2.a.be 2 24.f even 2 1
2240.2.a.bi 2 24.h odd 2 1
2520.2.a.w 2 4.b odd 2 1
2800.2.a.bn 2 15.d odd 2 1
2800.2.g.u 4 15.e even 4 2
3920.2.a.bu 2 21.c even 2 1
5040.2.a.bq 2 1.a even 1 1 trivial
9800.2.a.by 2 420.o odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5040))\):

\( T_{11}^{2} + T_{11} - 4 \)
\( T_{13}^{2} - T_{13} - 38 \)
\( T_{17}^{2} + 11 T_{17} + 26 \)
\( T_{19}^{2} - 6 T_{19} - 8 \)
\( T_{29}^{2} + 5 T_{29} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -4 + T + T^{2} \)
$13$ \( -38 - T + T^{2} \)
$17$ \( 26 + 11 T + T^{2} \)
$19$ \( -8 - 6 T + T^{2} \)
$23$ \( -16 + 2 T + T^{2} \)
$29$ \( 2 + 5 T + T^{2} \)
$31$ \( -64 - 4 T + T^{2} \)
$37$ \( -68 + T^{2} \)
$41$ \( -8 + 6 T + T^{2} \)
$43$ \( -8 - 6 T + T^{2} \)
$47$ \( 16 - 9 T + T^{2} \)
$53$ \( 64 - 18 T + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( 104 + 22 T + T^{2} \)
$67$ \( -32 - 12 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( -52 - 8 T + T^{2} \)
$79$ \( -8 - 11 T + T^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( -16 + 2 T + T^{2} \)
$97$ \( 18 - 15 T + T^{2} \)
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