Properties

Label 4560.2.a.bt
Level $4560$
Weight $2$
Character orbit 4560.a
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2280)
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.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + ( 1 + \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} - q^{5} + ( 1 + \beta_{2} ) q^{7} + q^{9} + ( -1 + \beta_{2} ) q^{11} + ( -2 + \beta_{1} ) q^{13} - q^{15} + 2 \beta_{1} q^{17} - q^{19} + ( 1 + \beta_{2} ) q^{21} + ( 3 + \beta_{1} + \beta_{2} ) q^{23} + q^{25} + q^{27} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{29} + ( 2 + 2 \beta_{1} ) q^{31} + ( -1 + \beta_{2} ) q^{33} + ( -1 - \beta_{2} ) q^{35} + ( -2 - 3 \beta_{1} ) q^{37} + ( -2 + \beta_{1} ) q^{39} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{41} -3 \beta_{1} q^{43} - q^{45} + ( 5 - \beta_{1} - \beta_{2} ) q^{47} + ( 6 + \beta_{1} - \beta_{2} ) q^{49} + 2 \beta_{1} q^{51} + ( 7 - \beta_{1} - \beta_{2} ) q^{53} + ( 1 - \beta_{2} ) q^{55} - q^{57} + ( -7 - 3 \beta_{1} + \beta_{2} ) q^{59} + ( 1 + 3 \beta_{1} + \beta_{2} ) q^{61} + ( 1 + \beta_{2} ) q^{63} + ( 2 - \beta_{1} ) q^{65} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( 3 + \beta_{1} + \beta_{2} ) q^{69} + ( 2 - 4 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{73} + q^{75} + ( 11 + \beta_{1} - 3 \beta_{2} ) q^{77} + ( 3 + \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{83} -2 \beta_{1} q^{85} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{87} + ( 8 + 3 \beta_{1} ) q^{89} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{91} + ( 2 + 2 \beta_{1} ) q^{93} + q^{95} + ( -2 - 3 \beta_{1} ) q^{97} + ( -1 + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} - 3 q^{15} - 3 q^{19} + 2 q^{21} + 8 q^{23} + 3 q^{25} + 3 q^{27} + 10 q^{29} + 6 q^{31} - 4 q^{33} - 2 q^{35} - 6 q^{37} - 6 q^{39} + 4 q^{41} - 3 q^{45} + 16 q^{47} + 19 q^{49} + 22 q^{53} + 4 q^{55} - 3 q^{57} - 22 q^{59} + 2 q^{61} + 2 q^{63} + 6 q^{65} - 4 q^{67} + 8 q^{69} + 8 q^{71} + 10 q^{73} + 3 q^{75} + 36 q^{77} + 10 q^{79} + 3 q^{81} - 2 q^{83} + 10 q^{87} + 24 q^{89} - 8 q^{91} + 6 q^{93} + 3 q^{95} - 6 q^{97} - 4 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 6 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + 3 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 3 \beta_{1} + 9\)\()/2\)

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
−1.76156
−0.363328
3.12489
0 1.00000 0 −1.00000 0 −4.38776 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 2.77801 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 3.60975 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(19\) \(1\)

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 4560.2.a.bt 3
4.b odd 2 1 2280.2.a.r 3
12.b even 2 1 6840.2.a.bk 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.r 3 4.b odd 2 1
4560.2.a.bt 3 1.a even 1 1 trivial
6840.2.a.bk 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4560))\):

\( T_{7}^{3} - 2 T_{7}^{2} - 18 T_{7} + 44 \)
\( T_{11}^{3} + 4 T_{11}^{2} - 14 T_{11} + 8 \)
\( T_{13}^{3} + 6 T_{13}^{2} + 2 T_{13} - 4 \)
\( T_{17}^{3} - 40 T_{17} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( 44 - 18 T - 2 T^{2} + T^{3} \)
$11$ \( 8 - 14 T + 4 T^{2} + T^{3} \)
$13$ \( -4 + 2 T + 6 T^{2} + T^{3} \)
$17$ \( 64 - 40 T + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( 16 - 4 T - 8 T^{2} + T^{3} \)
$29$ \( 268 - 18 T - 10 T^{2} + T^{3} \)
$31$ \( 136 - 28 T - 6 T^{2} + T^{3} \)
$37$ \( -388 - 78 T + 6 T^{2} + T^{3} \)
$41$ \( 400 - 90 T - 4 T^{2} + T^{3} \)
$43$ \( -216 - 90 T + T^{3} \)
$47$ \( 16 + 60 T - 16 T^{2} + T^{3} \)
$53$ \( -176 + 136 T - 22 T^{2} + T^{3} \)
$59$ \( -976 + 40 T + 22 T^{2} + T^{3} \)
$61$ \( -160 - 96 T - 2 T^{2} + T^{3} \)
$67$ \( 128 - 96 T + 4 T^{2} + T^{3} \)
$71$ \( 1600 - 184 T - 8 T^{2} + T^{3} \)
$73$ \( 488 - 100 T - 10 T^{2} + T^{3} \)
$79$ \( 64 - 10 T^{2} + T^{3} \)
$83$ \( -328 - 100 T + 2 T^{2} + T^{3} \)
$89$ \( 424 + 102 T - 24 T^{2} + T^{3} \)
$97$ \( -388 - 78 T + 6 T^{2} + T^{3} \)
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