Properties

Label 528.2.a.a
Level 528
Weight 2
Character orbit 528.a
Self dual yes
Analytic conductor 4.216
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 528.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.21610122672\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0
0 −1.00000 0 −4.00000 0 2.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 528.2.a.a 1
3.b odd 2 1 1584.2.a.s 1
4.b odd 2 1 66.2.a.c 1
8.b even 2 1 2112.2.a.bd 1
8.d odd 2 1 2112.2.a.n 1
11.b odd 2 1 5808.2.a.b 1
12.b even 2 1 198.2.a.c 1
20.d odd 2 1 1650.2.a.c 1
20.e even 4 2 1650.2.c.m 2
24.f even 2 1 6336.2.a.c 1
24.h odd 2 1 6336.2.a.d 1
28.d even 2 1 3234.2.a.s 1
36.f odd 6 2 1782.2.e.l 2
36.h even 6 2 1782.2.e.n 2
44.c even 2 1 726.2.a.d 1
44.g even 10 4 726.2.e.m 4
44.h odd 10 4 726.2.e.e 4
60.h even 2 1 4950.2.a.bo 1
60.l odd 4 2 4950.2.c.d 2
84.h odd 2 1 9702.2.a.a 1
132.d odd 2 1 2178.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.c 1 4.b odd 2 1
198.2.a.c 1 12.b even 2 1
528.2.a.a 1 1.a even 1 1 trivial
726.2.a.d 1 44.c even 2 1
726.2.e.e 4 44.h odd 10 4
726.2.e.m 4 44.g even 10 4
1584.2.a.s 1 3.b odd 2 1
1650.2.a.c 1 20.d odd 2 1
1650.2.c.m 2 20.e even 4 2
1782.2.e.l 2 36.f odd 6 2
1782.2.e.n 2 36.h even 6 2
2112.2.a.n 1 8.d odd 2 1
2112.2.a.bd 1 8.b even 2 1
2178.2.a.m 1 132.d odd 2 1
3234.2.a.s 1 28.d even 2 1
4950.2.a.bo 1 60.h even 2 1
4950.2.c.d 2 60.l odd 4 2
5808.2.a.b 1 11.b odd 2 1
6336.2.a.c 1 24.f even 2 1
6336.2.a.d 1 24.h odd 2 1
9702.2.a.a 1 84.h odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(11\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(528))\):

\( T_{5} + 4 \)
\( T_{7} - 2 \)
\( T_{13} - 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + T \)
$5$ \( 1 + 4 T + 5 T^{2} \)
$7$ \( 1 - 2 T + 7 T^{2} \)
$11$ \( 1 + T \)
$13$ \( 1 - 4 T + 13 T^{2} \)
$17$ \( 1 + 2 T + 17 T^{2} \)
$19$ \( 1 + 19 T^{2} \)
$23$ \( 1 - 6 T + 23 T^{2} \)
$29$ \( 1 - 10 T + 29 T^{2} \)
$31$ \( 1 - 8 T + 31 T^{2} \)
$37$ \( 1 + 2 T + 37 T^{2} \)
$41$ \( 1 - 2 T + 41 T^{2} \)
$43$ \( 1 + 4 T + 43 T^{2} \)
$47$ \( 1 - 2 T + 47 T^{2} \)
$53$ \( 1 - 4 T + 53 T^{2} \)
$59$ \( 1 + 59 T^{2} \)
$61$ \( 1 + 8 T + 61 T^{2} \)
$67$ \( 1 - 12 T + 67 T^{2} \)
$71$ \( 1 + 2 T + 71 T^{2} \)
$73$ \( 1 + 6 T + 73 T^{2} \)
$79$ \( 1 + 10 T + 79 T^{2} \)
$83$ \( 1 + 4 T + 83 T^{2} \)
$89$ \( 1 - 10 T + 89 T^{2} \)
$97$ \( 1 + 2 T + 97 T^{2} \)
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