Properties

Label 528.2.a.d
Level $528$
Weight $2$
Character orbit 528.a
Self dual yes
Analytic conductor $4.216$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - q^{3} - 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - 2 q^{7} + q^{9} + q^{11} - 4 q^{13} - 6 q^{17} + 4 q^{19} + 2 q^{21} - 6 q^{23} - 5 q^{25} - q^{27} + 6 q^{29} - 8 q^{31} - q^{33} - 10 q^{37} + 4 q^{39} + 6 q^{41} - 8 q^{43} + 6 q^{47} - 3 q^{49} + 6 q^{51} - 4 q^{57} + 8 q^{61} - 2 q^{63} + 4 q^{67} + 6 q^{69} - 6 q^{71} + 2 q^{73} + 5 q^{75} - 2 q^{77} - 14 q^{79} + q^{81} + 12 q^{83} - 6 q^{87} - 6 q^{89} + 8 q^{91} + 8 q^{93} + 14 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

Atkin-Lehner signs

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

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

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T - 6 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 8 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T + 6 \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T + 14 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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