Properties

Label 2520.2.a.x
Level $2520$
Weight $2$
Character orbit 2520.a
Self dual yes
Analytic conductor $20.122$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.1223013094\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} - q^{7} +O(q^{10})\) \( q + q^{5} - q^{7} + ( -3 - \beta ) q^{11} + ( 1 + \beta ) q^{13} + ( -3 + \beta ) q^{17} + ( 2 - 2 \beta ) q^{19} + ( -2 + 2 \beta ) q^{23} + q^{25} + ( 1 + \beta ) q^{29} -8 q^{31} - q^{35} -2 q^{37} -2 \beta q^{41} + ( -2 - 2 \beta ) q^{43} + ( -3 + 3 \beta ) q^{47} + q^{49} + ( -4 - 2 \beta ) q^{53} + ( -3 - \beta ) q^{55} -8 q^{59} + ( 4 - 2 \beta ) q^{61} + ( 1 + \beta ) q^{65} -4 q^{67} -8 q^{71} -6 q^{73} + ( 3 + \beta ) q^{77} + ( 5 + 3 \beta ) q^{79} + ( -4 + 4 \beta ) q^{83} + ( -3 + \beta ) q^{85} + ( -8 - 2 \beta ) q^{89} + ( -1 - \beta ) q^{91} + ( 2 - 2 \beta ) q^{95} + ( 7 - 5 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - 2q^{7} + O(q^{10}) \) \( 2q + 2q^{5} - 2q^{7} - 7q^{11} + 3q^{13} - 5q^{17} + 2q^{19} - 2q^{23} + 2q^{25} + 3q^{29} - 16q^{31} - 2q^{35} - 4q^{37} - 2q^{41} - 6q^{43} - 3q^{47} + 2q^{49} - 10q^{53} - 7q^{55} - 16q^{59} + 6q^{61} + 3q^{65} - 8q^{67} - 16q^{71} - 12q^{73} + 7q^{77} + 13q^{79} - 4q^{83} - 5q^{85} - 18q^{89} - 3q^{91} + 2q^{95} + 9q^{97} + O(q^{100}) \)

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
3.37228
−2.37228
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 2520.2.a.x 2
3.b odd 2 1 280.2.a.c 2
4.b odd 2 1 5040.2.a.by 2
12.b even 2 1 560.2.a.h 2
15.d odd 2 1 1400.2.a.r 2
15.e even 4 2 1400.2.g.i 4
21.c even 2 1 1960.2.a.s 2
21.g even 6 2 1960.2.q.r 4
21.h odd 6 2 1960.2.q.t 4
24.f even 2 1 2240.2.a.bg 2
24.h odd 2 1 2240.2.a.bk 2
60.h even 2 1 2800.2.a.bk 2
60.l odd 4 2 2800.2.g.r 4
84.h odd 2 1 3920.2.a.bt 2
105.g even 2 1 9800.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.c 2 3.b odd 2 1
560.2.a.h 2 12.b even 2 1
1400.2.a.r 2 15.d odd 2 1
1400.2.g.i 4 15.e even 4 2
1960.2.a.s 2 21.c even 2 1
1960.2.q.r 4 21.g even 6 2
1960.2.q.t 4 21.h odd 6 2
2240.2.a.bg 2 24.f even 2 1
2240.2.a.bk 2 24.h odd 2 1
2520.2.a.x 2 1.a even 1 1 trivial
2800.2.a.bk 2 60.h even 2 1
2800.2.g.r 4 60.l odd 4 2
3920.2.a.bt 2 84.h odd 2 1
5040.2.a.by 2 4.b odd 2 1
9800.2.a.bu 2 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2520))\):

\( T_{11}^{2} + 7 T_{11} + 4 \)
\( T_{13}^{2} - 3 T_{13} - 6 \)
\( T_{17}^{2} + 5 T_{17} - 2 \)
\( T_{19}^{2} - 2 T_{19} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 4 + 7 T + T^{2} \)
$13$ \( -6 - 3 T + T^{2} \)
$17$ \( -2 + 5 T + T^{2} \)
$19$ \( -32 - 2 T + T^{2} \)
$23$ \( -32 + 2 T + T^{2} \)
$29$ \( -6 - 3 T + T^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( ( 2 + T )^{2} \)
$41$ \( -32 + 2 T + T^{2} \)
$43$ \( -24 + 6 T + T^{2} \)
$47$ \( -72 + 3 T + T^{2} \)
$53$ \( -8 + 10 T + T^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( -24 - 6 T + T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( -32 - 13 T + T^{2} \)
$83$ \( -128 + 4 T + T^{2} \)
$89$ \( 48 + 18 T + T^{2} \)
$97$ \( -186 - 9 T + T^{2} \)
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