Properties

Label 5040.2.a.bt
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: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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.

\(f(q)\) \(=\) \( q - q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + q^{7} + \beta q^{11} + (\beta + 2) q^{13} + (\beta + 2) q^{17} + ( - 2 \beta + 4) q^{19} - 2 \beta q^{23} + q^{25} + (3 \beta - 2) q^{29} - q^{35} + 6 q^{37} + (2 \beta - 2) q^{41} + ( - 2 \beta - 4) q^{43} + (3 \beta - 4) q^{47} + q^{49} + ( - 2 \beta + 2) q^{53} - \beta q^{55} - 4 q^{59} + ( - 6 \beta + 6) q^{61} + ( - \beta - 2) q^{65} + (4 \beta - 4) q^{67} + 8 q^{71} + (4 \beta - 6) q^{73} + \beta q^{77} + (\beta + 4) q^{79} + 4 q^{83} + ( - \beta - 2) q^{85} + ( - 2 \beta - 2) q^{89} + (\beta + 2) q^{91} + (2 \beta - 4) q^{95} + ( - 5 \beta - 2) q^{97} +O(q^{100}) \) 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 \( 2 q - 2 q^{5} + 2 q^{7} + q^{11} + 5 q^{13} + 5 q^{17} + 6 q^{19} - 2 q^{23} + 2 q^{25} - q^{29} - 2 q^{35} + 12 q^{37} - 2 q^{41} - 10 q^{43} - 5 q^{47} + 2 q^{49} + 2 q^{53} - q^{55} - 8 q^{59} + 6 q^{61} - 5 q^{65} - 4 q^{67} + 16 q^{71} - 8 q^{73} + q^{77} + 9 q^{79} + 8 q^{83} - 5 q^{85} - 6 q^{89} + 5 q^{91} - 6 q^{95} - 9 q^{97}+O(q^{100}) \) 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
−1.56155
2.56155
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.bt 2
3.b odd 2 1 560.2.a.i 2
4.b odd 2 1 315.2.a.e 2
12.b even 2 1 35.2.a.b 2
15.d odd 2 1 2800.2.a.bi 2
15.e even 4 2 2800.2.g.t 4
20.d odd 2 1 1575.2.a.p 2
20.e even 4 2 1575.2.d.e 4
21.c even 2 1 3920.2.a.bs 2
24.f even 2 1 2240.2.a.bh 2
24.h odd 2 1 2240.2.a.bd 2
28.d even 2 1 2205.2.a.x 2
60.h even 2 1 175.2.a.f 2
60.l odd 4 2 175.2.b.b 4
84.h odd 2 1 245.2.a.d 2
84.j odd 6 2 245.2.e.h 4
84.n even 6 2 245.2.e.i 4
132.d odd 2 1 4235.2.a.m 2
156.h even 2 1 5915.2.a.l 2
420.o odd 2 1 1225.2.a.s 2
420.w even 4 2 1225.2.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 12.b even 2 1
175.2.a.f 2 60.h even 2 1
175.2.b.b 4 60.l odd 4 2
245.2.a.d 2 84.h odd 2 1
245.2.e.h 4 84.j odd 6 2
245.2.e.i 4 84.n even 6 2
315.2.a.e 2 4.b odd 2 1
560.2.a.i 2 3.b odd 2 1
1225.2.a.s 2 420.o odd 2 1
1225.2.b.f 4 420.w even 4 2
1575.2.a.p 2 20.d odd 2 1
1575.2.d.e 4 20.e even 4 2
2205.2.a.x 2 28.d even 2 1
2240.2.a.bd 2 24.h odd 2 1
2240.2.a.bh 2 24.f even 2 1
2800.2.a.bi 2 15.d odd 2 1
2800.2.g.t 4 15.e even 4 2
3920.2.a.bs 2 21.c even 2 1
4235.2.a.m 2 132.d odd 2 1
5040.2.a.bt 2 1.a even 1 1 trivial
5915.2.a.l 2 156.h even 2 1

Hecke kernels

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

\( T_{11}^{2} - T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 5T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 5T_{17} + 2 \) Copy content Toggle raw display
\( T_{19}^{2} - 6T_{19} - 8 \) Copy content Toggle raw display
\( T_{29}^{2} + T_{29} - 38 \) 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
$11$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$47$ \( T^{2} + 5T - 32 \) Copy content Toggle raw display
$53$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 6T - 144 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 52 \) Copy content Toggle raw display
$79$ \( T^{2} - 9T + 16 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$97$ \( T^{2} + 9T - 86 \) Copy content Toggle raw display
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