Properties

Label 572.2.a.e
Level $572$
Weight $2$
Character orbit 572.a
Self dual yes
Analytic conductor $4.567$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.56744299562\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{7} + ( 1 + \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{7} + ( 1 + \beta_{1} ) q^{9} + q^{11} - q^{13} + ( -4 - 3 \beta_{1} + \beta_{2} ) q^{15} + ( -2 - 2 \beta_{1} ) q^{17} + ( -1 + 3 \beta_{1} ) q^{19} + ( 1 + 3 \beta_{1} + 3 \beta_{2} ) q^{21} + ( -4 + \beta_{1} - 2 \beta_{2} ) q^{23} + ( 3 + 3 \beta_{1} - \beta_{2} ) q^{25} + ( 1 - 2 \beta_{1} + 2 \beta_{2} ) q^{27} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -8 + \beta_{1} - \beta_{2} ) q^{31} + ( -1 - \beta_{2} ) q^{33} + ( -4 - 2 \beta_{1} - 6 \beta_{2} ) q^{35} + ( -4 - \beta_{1} - \beta_{2} ) q^{37} + ( 1 + \beta_{2} ) q^{39} + ( -4 \beta_{1} + \beta_{2} ) q^{41} + ( -6 + 2 \beta_{1} - 4 \beta_{2} ) q^{43} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{45} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 8 + 5 \beta_{1} - 2 \beta_{2} ) q^{49} + ( 4 + 4 \beta_{1} + 2 \beta_{2} ) q^{51} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{53} + ( \beta_{1} + \beta_{2} ) q^{55} + ( -2 - 6 \beta_{1} + \beta_{2} ) q^{57} + ( -2 + 5 \beta_{1} - 5 \beta_{2} ) q^{59} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{61} + ( -7 - 3 \beta_{1} - \beta_{2} ) q^{63} + ( -\beta_{1} - \beta_{2} ) q^{65} + ( -2 + 5 \beta_{1} + 3 \beta_{2} ) q^{67} + ( 9 + 2 \beta_{2} ) q^{69} + ( 4 - \beta_{1} + 5 \beta_{2} ) q^{71} + ( 3 - 3 \beta_{1} + 6 \beta_{2} ) q^{73} + ( -3 - 5 \beta_{1} - 4 \beta_{2} ) q^{75} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{77} + ( 2 - 6 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -8 - \beta_{1} + \beta_{2} ) q^{81} + ( 2 + \beta_{2} ) q^{83} + ( -8 - 4 \beta_{1} - 4 \beta_{2} ) q^{85} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{87} + ( 8 - 5 \beta_{1} - \beta_{2} ) q^{89} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{91} + ( 10 - \beta_{1} + 7 \beta_{2} ) q^{93} + ( 12 + 2 \beta_{1} + 2 \beta_{2} ) q^{95} + ( \beta_{1} + \beta_{2} ) q^{97} + ( 1 + \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{3} - q^{5} - 7q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 2q^{3} - q^{5} - 7q^{7} + 3q^{9} + 3q^{11} - 3q^{13} - 13q^{15} - 6q^{17} - 3q^{19} - 10q^{23} + 10q^{25} + q^{27} - 14q^{29} - 23q^{31} - 2q^{33} - 6q^{35} - 11q^{37} + 2q^{39} - q^{41} - 14q^{43} + 10q^{45} + 8q^{47} + 26q^{49} + 10q^{51} + 5q^{53} - q^{55} - 7q^{57} - q^{59} + 4q^{61} - 20q^{63} + q^{65} - 9q^{67} + 25q^{69} + 7q^{71} + 3q^{73} - 5q^{75} - 7q^{77} + 4q^{79} - 25q^{81} + 5q^{83} - 20q^{85} - 6q^{87} + 25q^{89} + 7q^{91} + 23q^{93} + 34q^{95} - q^{97} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 4 x - 1\):

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

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
2.11491
−1.86081
−0.254102
0 −2.47283 0 3.58774 0 −4.75698 0 3.11491 0
1.2 0 −1.46260 0 −1.39821 0 2.18421 0 −0.860806 0
1.3 0 1.93543 0 −3.18953 0 −4.42723 0 0.745898 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(-1\)
\(13\) \(1\)

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 572.2.a.e 3
3.b odd 2 1 5148.2.a.o 3
4.b odd 2 1 2288.2.a.w 3
8.b even 2 1 9152.2.a.cc 3
8.d odd 2 1 9152.2.a.bv 3
11.b odd 2 1 6292.2.a.p 3
13.b even 2 1 7436.2.a.l 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.a.e 3 1.a even 1 1 trivial
2288.2.a.w 3 4.b odd 2 1
5148.2.a.o 3 3.b odd 2 1
6292.2.a.p 3 11.b odd 2 1
7436.2.a.l 3 13.b even 2 1
9152.2.a.bv 3 8.d odd 2 1
9152.2.a.cc 3 8.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(572))\):

\( T_{3}^{3} + 2 T_{3}^{2} - 4 T_{3} - 7 \)
\( T_{5}^{3} + T_{5}^{2} - 12 T_{5} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -7 - 4 T + 2 T^{2} + T^{3} \)
$5$ \( -16 - 12 T + T^{2} + T^{3} \)
$7$ \( -46 + T + 7 T^{2} + T^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( ( 1 + T )^{3} \)
$17$ \( -16 - 4 T + 6 T^{2} + T^{3} \)
$19$ \( -62 - 33 T + 3 T^{2} + T^{3} \)
$23$ \( -53 + 14 T + 10 T^{2} + T^{3} \)
$29$ \( -32 + 40 T + 14 T^{2} + T^{3} \)
$31$ \( 404 + 170 T + 23 T^{2} + T^{3} \)
$37$ \( 16 + 28 T + 11 T^{2} + T^{3} \)
$41$ \( -106 - 57 T + T^{2} + T^{3} \)
$43$ \( -464 - 12 T + 14 T^{2} + T^{3} \)
$47$ \( 208 - 28 T - 8 T^{2} + T^{3} \)
$53$ \( 2 - 11 T - 5 T^{2} + T^{3} \)
$59$ \( 188 - 158 T + T^{2} + T^{3} \)
$61$ \( 232 - 120 T - 4 T^{2} + T^{3} \)
$67$ \( -1556 - 166 T + 9 T^{2} + T^{3} \)
$71$ \( 788 - 106 T - 7 T^{2} + T^{3} \)
$73$ \( 864 - 171 T - 3 T^{2} + T^{3} \)
$79$ \( -256 - 124 T - 4 T^{2} + T^{3} \)
$83$ \( 8 + 3 T - 5 T^{2} + T^{3} \)
$89$ \( 832 + 88 T - 25 T^{2} + T^{3} \)
$97$ \( -16 - 12 T + T^{2} + T^{3} \)
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