Properties

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

Newform invariants

Self dual: yes
Analytic conductor: \(84.6826568613\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q - 9q^{3} - 92q^{5} + 26q^{7} + 81q^{9} + O(q^{10}) \) \( q - 9q^{3} - 92q^{5} + 26q^{7} + 81q^{9} - 121q^{11} - 692q^{13} + 828q^{15} - 1442q^{17} - 2160q^{19} - 234q^{21} + 1582q^{23} + 5339q^{25} - 729q^{27} - 5526q^{29} - 4792q^{31} + 1089q^{33} - 2392q^{35} - 10194q^{37} + 6228q^{39} - 10622q^{41} - 8580q^{43} - 7452q^{45} + 2362q^{47} - 16131q^{49} + 12978q^{51} - 30804q^{53} + 11132q^{55} + 19440q^{57} - 6416q^{59} + 42096q^{61} + 2106q^{63} + 63664q^{65} + 28444q^{67} - 14238q^{69} - 45690q^{71} - 18374q^{73} - 48051q^{75} - 3146q^{77} + 105214q^{79} + 6561q^{81} - 62292q^{83} + 132664q^{85} + 49734q^{87} - 72246q^{89} - 17992q^{91} + 43128q^{93} + 198720q^{95} + 79262q^{97} - 9801q^{99} + O(q^{100}) \)

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 −92.0000 0 26.0000 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.a 1
4.b odd 2 1 33.6.a.b 1
12.b even 2 1 99.6.a.a 1
20.d odd 2 1 825.6.a.a 1
44.c even 2 1 363.6.a.b 1
132.d odd 2 1 1089.6.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.b 1 4.b odd 2 1
99.6.a.a 1 12.b even 2 1
363.6.a.b 1 44.c even 2 1
528.6.a.a 1 1.a even 1 1 trivial
825.6.a.a 1 20.d odd 2 1
1089.6.a.h 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(528))\):

\( T_{5} + 92 \)
\( T_{7} - 26 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 9 + T \)
$5$ \( 92 + T \)
$7$ \( -26 + T \)
$11$ \( 121 + T \)
$13$ \( 692 + T \)
$17$ \( 1442 + T \)
$19$ \( 2160 + T \)
$23$ \( -1582 + T \)
$29$ \( 5526 + T \)
$31$ \( 4792 + T \)
$37$ \( 10194 + T \)
$41$ \( 10622 + T \)
$43$ \( 8580 + T \)
$47$ \( -2362 + T \)
$53$ \( 30804 + T \)
$59$ \( 6416 + T \)
$61$ \( -42096 + T \)
$67$ \( -28444 + T \)
$71$ \( 45690 + T \)
$73$ \( 18374 + T \)
$79$ \( -105214 + T \)
$83$ \( 62292 + T \)
$89$ \( 72246 + T \)
$97$ \( -79262 + T \)
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