Properties

Label 4560.2.a.br
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: 3.3.564.1
Defining polynomial: \(x^{3} - x^{2} - 5 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1140)
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.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} + ( -1 + \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{5} + ( -1 + \beta_{2} ) q^{7} + q^{9} + ( -2 - \beta_{1} + \beta_{2} ) q^{11} + ( 1 + \beta_{2} ) q^{13} - q^{15} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{17} + q^{19} + ( 1 - \beta_{2} ) q^{21} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{23} + q^{25} - q^{27} + ( -\beta_{1} + \beta_{2} ) q^{29} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{31} + ( 2 + \beta_{1} - \beta_{2} ) q^{33} + ( -1 + \beta_{2} ) q^{35} + ( -1 + 3 \beta_{2} ) q^{37} + ( -1 - \beta_{2} ) q^{39} + ( 4 - \beta_{1} + \beta_{2} ) q^{41} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{43} + q^{45} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{47} + ( \beta_{1} - 4 \beta_{2} ) q^{49} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{51} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -2 - \beta_{1} + \beta_{2} ) q^{55} - q^{57} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( 5 + \beta_{1} ) q^{61} + ( -1 + \beta_{2} ) q^{63} + ( 1 + \beta_{2} ) q^{65} + ( 4 + 4 \beta_{2} ) q^{67} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{69} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -4 - 2 \beta_{1} ) q^{73} - q^{75} + ( 5 + \beta_{1} - 6 \beta_{2} ) q^{77} + ( 6 + 2 \beta_{1} ) q^{79} + q^{81} + ( -1 - \beta_{1} ) q^{83} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{85} + ( \beta_{1} - \beta_{2} ) q^{87} + ( -\beta_{1} + \beta_{2} ) q^{89} + ( 5 + \beta_{1} - 2 \beta_{2} ) q^{91} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{93} + q^{95} + ( -3 + \beta_{2} ) q^{97} + ( -2 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 6 q^{11} + 2 q^{13} - 3 q^{15} + 2 q^{17} + 3 q^{19} + 4 q^{21} - 6 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{33} - 4 q^{35} - 6 q^{37} - 2 q^{39} + 12 q^{41} + 8 q^{43} + 3 q^{45} - 6 q^{47} + 3 q^{49} - 2 q^{51} + 8 q^{53} - 6 q^{55} - 3 q^{57} + 4 q^{59} + 14 q^{61} - 4 q^{63} + 2 q^{65} + 8 q^{67} + 6 q^{69} - 16 q^{71} - 10 q^{73} - 3 q^{75} + 20 q^{77} + 16 q^{79} + 3 q^{81} - 2 q^{83} + 2 q^{85} + 16 q^{91} + 3 q^{95} - 10 q^{97} - 6 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 5 x + 3\):

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

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
0.571993
−2.08613
2.51414
0 −1.00000 0 1.00000 0 −4.67282 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −0.648061 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 1.32088 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.br 3
4.b odd 2 1 1140.2.a.g 3
12.b even 2 1 3420.2.a.m 3
20.d odd 2 1 5700.2.a.w 3
20.e even 4 2 5700.2.f.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.a.g 3 4.b odd 2 1
3420.2.a.m 3 12.b even 2 1
4560.2.a.br 3 1.a even 1 1 trivial
5700.2.a.w 3 20.d odd 2 1
5700.2.f.q 6 20.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4560))\):

\( T_{7}^{3} + 4 T_{7}^{2} - 4 T_{7} - 4 \)
\( T_{11}^{3} + 6 T_{11}^{2} - 12 T_{11} - 76 \)
\( T_{13}^{3} - 2 T_{13}^{2} - 8 T_{13} + 12 \)
\( T_{17}^{3} - 2 T_{17}^{2} - 44 T_{17} + 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( -4 - 4 T + 4 T^{2} + T^{3} \)
$11$ \( -76 - 12 T + 6 T^{2} + T^{3} \)
$13$ \( 12 - 8 T - 2 T^{2} + T^{3} \)
$17$ \( 72 - 44 T - 2 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( -344 - 60 T + 6 T^{2} + T^{3} \)
$29$ \( -36 - 24 T + T^{3} \)
$31$ \( 208 - 72 T + T^{3} \)
$37$ \( 4 - 72 T + 6 T^{2} + T^{3} \)
$41$ \( -4 + 24 T - 12 T^{2} + T^{3} \)
$43$ \( 468 - 60 T - 8 T^{2} + T^{3} \)
$47$ \( 72 - 60 T + 6 T^{2} + T^{3} \)
$53$ \( 1488 - 168 T - 8 T^{2} + T^{3} \)
$59$ \( 864 - 144 T - 4 T^{2} + T^{3} \)
$61$ \( 8 + 44 T - 14 T^{2} + T^{3} \)
$67$ \( 768 - 128 T - 8 T^{2} + T^{3} \)
$71$ \( -1312 - 64 T + 16 T^{2} + T^{3} \)
$73$ \( -328 - 52 T + 10 T^{2} + T^{3} \)
$79$ \( 384 - 16 T^{2} + T^{3} \)
$83$ \( -24 - 20 T + 2 T^{2} + T^{3} \)
$89$ \( -36 - 24 T + T^{3} \)
$97$ \( 12 + 24 T + 10 T^{2} + T^{3} \)
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