Properties

Label 572.2.a.d
Level $572$
Weight $2$
Character orbit 572.a
Self dual yes
Analytic conductor $4.567$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \(x^{2} - x - 5\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.79129
2.79129
0 −1.79129 0 2.00000 0 4.79129 0 0.208712 0
1.2 0 2.79129 0 2.00000 0 0.208712 0 4.79129 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.d 2
3.b odd 2 1 5148.2.a.g 2
4.b odd 2 1 2288.2.a.n 2
8.b even 2 1 9152.2.a.bl 2
8.d odd 2 1 9152.2.a.bn 2
11.b odd 2 1 6292.2.a.o 2
13.b even 2 1 7436.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.a.d 2 1.a even 1 1 trivial
2288.2.a.n 2 4.b odd 2 1
5148.2.a.g 2 3.b odd 2 1
6292.2.a.o 2 11.b odd 2 1
7436.2.a.h 2 13.b even 2 1
9152.2.a.bl 2 8.b even 2 1
9152.2.a.bn 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(572))\):

\( T_{3}^{2} - T_{3} - 5 \)
\( T_{5} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -5 - T + T^{2} \)
$5$ \( ( -2 + T )^{2} \)
$7$ \( 1 - 5 T + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( ( 4 + T )^{2} \)
$19$ \( 7 - 7 T + T^{2} \)
$23$ \( 1 - 5 T + T^{2} \)
$29$ \( -20 - 2 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -4 + T )^{2} \)
$41$ \( -5 + T + T^{2} \)
$43$ \( -80 - 4 T + T^{2} \)
$47$ \( -20 + 2 T + T^{2} \)
$53$ \( -41 + 5 T + T^{2} \)
$59$ \( -20 + 2 T + T^{2} \)
$61$ \( 28 - 14 T + T^{2} \)
$67$ \( -80 - 4 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 67 + 17 T + T^{2} \)
$79$ \( -188 + 2 T + T^{2} \)
$83$ \( 9 - 15 T + T^{2} \)
$89$ \( 4 - 10 T + T^{2} \)
$97$ \( -164 - 10 T + T^{2} \)
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