Properties

Label 528.6.a.i
Level $528$
Weight $6$
Character orbit 528.a
Self dual yes
Analytic conductor $84.683$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 528.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(84.6826568613\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 9q^{3} + 46q^{5} - 148q^{7} + 81q^{9} + O(q^{10}) \) \( q + 9q^{3} + 46q^{5} - 148q^{7} + 81q^{9} - 121q^{11} + 574q^{13} + 414q^{15} - 722q^{17} - 2160q^{19} - 1332q^{21} + 2536q^{23} - 1009q^{25} + 729q^{27} + 4650q^{29} - 5032q^{31} - 1089q^{33} - 6808q^{35} + 8118q^{37} + 5166q^{39} - 5138q^{41} - 8304q^{43} + 3726q^{45} - 24728q^{47} + 5097q^{49} - 6498q^{51} - 28746q^{53} - 5566q^{55} - 19440q^{57} + 5860q^{59} - 53658q^{61} - 11988q^{63} + 26404q^{65} - 30908q^{67} + 22824q^{69} + 69648q^{71} - 18446q^{73} - 9081q^{75} + 17908q^{77} + 25300q^{79} + 6561q^{81} + 17556q^{83} - 33212q^{85} + 41850q^{87} + 132570q^{89} - 84952q^{91} - 45288q^{93} - 99360q^{95} + 70658q^{97} - 9801q^{99} + O(q^{100}) \)

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 9.00000 0 46.0000 0 −148.000 0 81.0000 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 528.6.a.i 1
4.b odd 2 1 33.6.a.a 1
12.b even 2 1 99.6.a.b 1
20.d odd 2 1 825.6.a.b 1
44.c even 2 1 363.6.a.c 1
132.d odd 2 1 1089.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.a 1 4.b odd 2 1
99.6.a.b 1 12.b even 2 1
363.6.a.c 1 44.c even 2 1
528.6.a.i 1 1.a even 1 1 trivial
825.6.a.b 1 20.d odd 2 1
1089.6.a.d 1 132.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(528))\):

\( T_{5} - 46 \)
\( T_{7} + 148 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -9 + T \)
$5$ \( -46 + T \)
$7$ \( 148 + T \)
$11$ \( 121 + T \)
$13$ \( -574 + T \)
$17$ \( 722 + T \)
$19$ \( 2160 + T \)
$23$ \( -2536 + T \)
$29$ \( -4650 + T \)
$31$ \( 5032 + T \)
$37$ \( -8118 + T \)
$41$ \( 5138 + T \)
$43$ \( 8304 + T \)
$47$ \( 24728 + T \)
$53$ \( 28746 + T \)
$59$ \( -5860 + T \)
$61$ \( 53658 + T \)
$67$ \( 30908 + T \)
$71$ \( -69648 + T \)
$73$ \( 18446 + T \)
$79$ \( -25300 + T \)
$83$ \( -17556 + T \)
$89$ \( -132570 + T \)
$97$ \( -70658 + T \)
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