Properties

Label 528.6.a.q
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: \(\Q(\sqrt{313}) \)
Defining polynomial: \(x^{2} - x - 78\)
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 \(\beta = \sqrt{313}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 q^{3} + ( -19 - 5 \beta ) q^{5} + ( 9 + \beta ) q^{7} + 81 q^{9} +O(q^{10})\) \( q + 9 q^{3} + ( -19 - 5 \beta ) q^{5} + ( 9 + \beta ) q^{7} + 81 q^{9} -121 q^{11} + ( -33 + 53 \beta ) q^{13} + ( -171 - 45 \beta ) q^{15} + ( -460 - 2 \beta ) q^{17} + ( 1466 + 2 \beta ) q^{19} + ( 81 + 9 \beta ) q^{21} + ( -2623 + 13 \beta ) q^{23} + ( 5061 + 190 \beta ) q^{25} + 729 q^{27} + ( -6300 + 66 \beta ) q^{29} + ( -4968 + 304 \beta ) q^{31} -1089 q^{33} + ( -1736 - 64 \beta ) q^{35} + ( 2998 + 160 \beta ) q^{37} + ( -297 + 477 \beta ) q^{39} + ( 12122 + 364 \beta ) q^{41} + ( -10180 + 620 \beta ) q^{43} + ( -1539 - 405 \beta ) q^{45} + ( 2903 - 389 \beta ) q^{47} + ( -16413 + 18 \beta ) q^{49} + ( -4140 - 18 \beta ) q^{51} + ( 20385 - 297 \beta ) q^{53} + ( 2299 + 605 \beta ) q^{55} + ( 13194 + 18 \beta ) q^{57} + ( -9106 - 1838 \beta ) q^{59} + ( -5699 - 1373 \beta ) q^{61} + ( 729 + 81 \beta ) q^{63} + ( -82318 - 842 \beta ) q^{65} + ( -32684 + 384 \beta ) q^{67} + ( -23607 + 117 \beta ) q^{69} + ( -30723 - 1551 \beta ) q^{71} + ( 26706 - 160 \beta ) q^{73} + ( 45549 + 1710 \beta ) q^{75} + ( -1089 - 121 \beta ) q^{77} + ( -8561 - 1065 \beta ) q^{79} + 6561 q^{81} + ( 7152 - 1764 \beta ) q^{83} + ( 11870 + 2338 \beta ) q^{85} + ( -56700 + 594 \beta ) q^{87} + ( -29070 - 1512 \beta ) q^{89} + ( 16292 + 444 \beta ) q^{91} + ( -44712 + 2736 \beta ) q^{93} + ( -30984 - 7368 \beta ) q^{95} + ( -91528 - 546 \beta ) q^{97} -9801 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 18q^{3} - 38q^{5} + 18q^{7} + 162q^{9} + O(q^{10}) \) \( 2q + 18q^{3} - 38q^{5} + 18q^{7} + 162q^{9} - 242q^{11} - 66q^{13} - 342q^{15} - 920q^{17} + 2932q^{19} + 162q^{21} - 5246q^{23} + 10122q^{25} + 1458q^{27} - 12600q^{29} - 9936q^{31} - 2178q^{33} - 3472q^{35} + 5996q^{37} - 594q^{39} + 24244q^{41} - 20360q^{43} - 3078q^{45} + 5806q^{47} - 32826q^{49} - 8280q^{51} + 40770q^{53} + 4598q^{55} + 26388q^{57} - 18212q^{59} - 11398q^{61} + 1458q^{63} - 164636q^{65} - 65368q^{67} - 47214q^{69} - 61446q^{71} + 53412q^{73} + 91098q^{75} - 2178q^{77} - 17122q^{79} + 13122q^{81} + 14304q^{83} + 23740q^{85} - 113400q^{87} - 58140q^{89} + 32584q^{91} - 89424q^{93} - 61968q^{95} - 183056q^{97} - 19602q^{99} + O(q^{100}) \)

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
9.34590
−8.34590
0 9.00000 0 −107.459 0 26.6918 0 81.0000 0
1.2 0 9.00000 0 69.4590 0 −8.69181 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.q 2
4.b odd 2 1 33.6.a.d 2
12.b even 2 1 99.6.a.e 2
20.d odd 2 1 825.6.a.d 2
44.c even 2 1 363.6.a.g 2
132.d odd 2 1 1089.6.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.d 2 4.b odd 2 1
99.6.a.e 2 12.b even 2 1
363.6.a.g 2 44.c even 2 1
528.6.a.q 2 1.a even 1 1 trivial
825.6.a.d 2 20.d odd 2 1
1089.6.a.o 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(528))\):

\( T_{5}^{2} + 38 T_{5} - 7464 \)
\( T_{7}^{2} - 18 T_{7} - 232 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -9 + T )^{2} \)
$5$ \( -7464 + 38 T + T^{2} \)
$7$ \( -232 - 18 T + T^{2} \)
$11$ \( ( 121 + T )^{2} \)
$13$ \( -878128 + 66 T + T^{2} \)
$17$ \( 210348 + 920 T + T^{2} \)
$19$ \( 2147904 - 2932 T + T^{2} \)
$23$ \( 6827232 + 5246 T + T^{2} \)
$29$ \( 38326572 + 12600 T + T^{2} \)
$31$ \( -4245184 + 9936 T + T^{2} \)
$37$ \( 975204 - 5996 T + T^{2} \)
$41$ \( 105471636 - 24244 T + T^{2} \)
$43$ \( -16684800 + 20360 T + T^{2} \)
$47$ \( -38936064 - 5806 T + T^{2} \)
$53$ \( 387938808 - 40770 T + T^{2} \)
$59$ \( -974471136 + 18212 T + T^{2} \)
$61$ \( -557566776 + 11398 T + T^{2} \)
$67$ \( 1022090128 + 65368 T + T^{2} \)
$71$ \( 190949616 + 61446 T + T^{2} \)
$73$ \( 705197636 - 53412 T + T^{2} \)
$79$ \( -281721704 + 17122 T + T^{2} \)
$83$ \( -922809744 - 14304 T + T^{2} \)
$89$ \( 129501828 + 58140 T + T^{2} \)
$97$ \( 8284064476 + 183056 T + T^{2} \)
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