Properties

Label 528.6.a.o
Level $528$
Weight $6$
Character orbit 528.a
Self dual yes
Analytic conductor $84.683$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{33}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -9 q^{3} + ( 29 - 5 \beta ) q^{5} + ( -73 + 31 \beta ) q^{7} + 81 q^{9} +O(q^{10})\) \( q -9 q^{3} + ( 29 - 5 \beta ) q^{5} + ( -73 + 31 \beta ) q^{7} + 81 q^{9} + 121 q^{11} + ( -65 + 37 \beta ) q^{13} + ( -261 + 45 \beta ) q^{15} + ( -364 - 186 \beta ) q^{17} + ( 414 - 426 \beta ) q^{19} + ( 657 - 279 \beta ) q^{21} + ( 119 - 165 \beta ) q^{23} + ( -1459 - 290 \beta ) q^{25} -729 q^{27} + ( 348 + 746 \beta ) q^{29} + ( 5240 + 800 \beta ) q^{31} -1089 q^{33} + ( -7232 + 1264 \beta ) q^{35} + ( -954 + 1408 \beta ) q^{37} + ( 585 - 333 \beta ) q^{39} + ( 18242 + 4 \beta ) q^{41} + ( -4884 - 1556 \beta ) q^{43} + ( 2349 - 405 \beta ) q^{45} + ( -21871 - 195 \beta ) q^{47} + ( 20235 - 4526 \beta ) q^{49} + ( 3276 + 1674 \beta ) q^{51} + ( -6087 + 3551 \beta ) q^{53} + ( 3509 - 605 \beta ) q^{55} + ( -3726 + 3834 \beta ) q^{57} + ( 1394 + 990 \beta ) q^{59} + ( -12651 - 1013 \beta ) q^{61} + ( -5913 + 2511 \beta ) q^{63} + ( -7990 + 1398 \beta ) q^{65} + ( 20260 - 6352 \beta ) q^{67} + ( -1071 + 1485 \beta ) q^{69} + ( -15693 - 2177 \beta ) q^{71} + ( -23390 - 2784 \beta ) q^{73} + ( 13131 + 2610 \beta ) q^{75} + ( -8833 + 3751 \beta ) q^{77} + ( 8425 + 5713 \beta ) q^{79} + 6561 q^{81} + ( -39720 + 10980 \beta ) q^{83} + ( 20134 - 3574 \beta ) q^{85} + ( -3132 - 6714 \beta ) q^{87} + ( -27102 - 13352 \beta ) q^{89} + ( 42596 - 4716 \beta ) q^{91} + ( -47160 - 7200 \beta ) q^{93} + ( 82296 - 14424 \beta ) q^{95} + ( -120784 - 4962 \beta ) q^{97} + 9801 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 18q^{3} + 58q^{5} - 146q^{7} + 162q^{9} + O(q^{10}) \) \( 2q - 18q^{3} + 58q^{5} - 146q^{7} + 162q^{9} + 242q^{11} - 130q^{13} - 522q^{15} - 728q^{17} + 828q^{19} + 1314q^{21} + 238q^{23} - 2918q^{25} - 1458q^{27} + 696q^{29} + 10480q^{31} - 2178q^{33} - 14464q^{35} - 1908q^{37} + 1170q^{39} + 36484q^{41} - 9768q^{43} + 4698q^{45} - 43742q^{47} + 40470q^{49} + 6552q^{51} - 12174q^{53} + 7018q^{55} - 7452q^{57} + 2788q^{59} - 25302q^{61} - 11826q^{63} - 15980q^{65} + 40520q^{67} - 2142q^{69} - 31386q^{71} - 46780q^{73} + 26262q^{75} - 17666q^{77} + 16850q^{79} + 13122q^{81} - 79440q^{83} + 40268q^{85} - 6264q^{87} - 54204q^{89} + 85192q^{91} - 94320q^{93} + 164592q^{95} - 241568q^{97} + 19602q^{99} + O(q^{100}) \)

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
3.37228
−2.37228
0 −9.00000 0 0.277187 0 105.081 0 81.0000 0
1.2 0 −9.00000 0 57.7228 0 −251.081 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.o 2
4.b odd 2 1 33.6.a.e 2
12.b even 2 1 99.6.a.d 2
20.d odd 2 1 825.6.a.c 2
44.c even 2 1 363.6.a.f 2
132.d odd 2 1 1089.6.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.e 2 4.b odd 2 1
99.6.a.d 2 12.b even 2 1
363.6.a.f 2 44.c even 2 1
528.6.a.o 2 1.a even 1 1 trivial
825.6.a.c 2 20.d odd 2 1
1089.6.a.p 2 132.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(528))\):

\( T_{5}^{2} - 58 T_{5} + 16 \)
\( T_{7}^{2} + 146 T_{7} - 26384 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 9 + T )^{2} \)
$5$ \( 16 - 58 T + T^{2} \)
$7$ \( -26384 + 146 T + T^{2} \)
$11$ \( ( -121 + T )^{2} \)
$13$ \( -40952 + 130 T + T^{2} \)
$17$ \( -1009172 + 728 T + T^{2} \)
$19$ \( -5817312 - 828 T + T^{2} \)
$23$ \( -884264 - 238 T + T^{2} \)
$29$ \( -18243924 - 696 T + T^{2} \)
$31$ \( 6337600 - 10480 T + T^{2} \)
$37$ \( -64511196 + 1908 T + T^{2} \)
$41$ \( 332770036 - 36484 T + T^{2} \)
$43$ \( -56044032 + 9768 T + T^{2} \)
$47$ \( 477085816 + 43742 T + T^{2} \)
$53$ \( -379065264 + 12174 T + T^{2} \)
$59$ \( -30400064 - 2788 T + T^{2} \)
$61$ \( 126184224 + 25302 T + T^{2} \)
$67$ \( -921013232 - 40520 T + T^{2} \)
$71$ \( 89872392 + 31386 T + T^{2} \)
$73$ \( 291320452 + 46780 T + T^{2} \)
$79$ \( -1006085552 - 16850 T + T^{2} \)
$83$ \( -2400814800 + 79440 T + T^{2} \)
$89$ \( -5148586428 + 54204 T + T^{2} \)
$97$ \( 13776267004 + 241568 T + T^{2} \)
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