Properties

Label 528.6.a.s
Level $528$
Weight $6$
Character orbit 528.a
Self dual yes
Analytic conductor $84.683$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
Defining polynomial: \(x^{2} - x - 44\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\beta = \sqrt{177}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 q^{3} + ( 29 - 5 \beta ) q^{5} + ( 143 - 5 \beta ) q^{7} + 81 q^{9} +O(q^{10})\) \( q + 9 q^{3} + ( 29 - 5 \beta ) q^{5} + ( 143 - 5 \beta ) q^{7} + 81 q^{9} + 121 q^{11} + ( -83 + 19 \beta ) q^{13} + ( 261 - 45 \beta ) q^{15} + ( -400 + 30 \beta ) q^{17} + ( 738 + 6 \beta ) q^{19} + ( 1287 - 45 \beta ) q^{21} + ( 1685 - 183 \beta ) q^{23} + ( 2141 - 290 \beta ) q^{25} + 729 q^{27} + ( 3300 + 206 \beta ) q^{29} + ( 3764 - 172 \beta ) q^{31} + 1089 q^{33} + ( 8572 - 860 \beta ) q^{35} + ( -14958 - 68 \beta ) q^{37} + ( -747 + 171 \beta ) q^{39} + ( -2890 - 356 \beta ) q^{41} + ( 8328 + 748 \beta ) q^{43} + ( 2349 - 405 \beta ) q^{45} + ( -3925 + 1263 \beta ) q^{47} + ( 8067 - 1430 \beta ) q^{49} + ( -3600 + 270 \beta ) q^{51} + ( 7089 + 1103 \beta ) q^{53} + ( 3509 - 605 \beta ) q^{55} + ( 6642 + 54 \beta ) q^{57} + ( -8650 + 738 \beta ) q^{59} + ( -1473 + 1165 \beta ) q^{61} + ( 11583 - 405 \beta ) q^{63} + ( -19222 + 966 \beta ) q^{65} + ( -15668 - 1600 \beta ) q^{67} + ( 15165 - 1647 \beta ) q^{69} + ( 16905 + 1549 \beta ) q^{71} + ( 30322 + 3768 \beta ) q^{73} + ( 19269 - 2610 \beta ) q^{75} + ( 17303 - 605 \beta ) q^{77} + ( -935 + 4741 \beta ) q^{79} + 6561 q^{81} + ( 29148 - 1296 \beta ) q^{83} + ( -38150 + 2870 \beta ) q^{85} + ( 29700 + 1854 \beta ) q^{87} + ( 46194 + 7528 \beta ) q^{89} + ( -28684 + 3132 \beta ) q^{91} + ( 33876 - 1548 \beta ) q^{93} + ( 16092 - 3516 \beta ) q^{95} + ( 3560 - 10650 \beta ) q^{97} + 9801 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 18q^{3} + 58q^{5} + 286q^{7} + 162q^{9} + O(q^{10}) \) \( 2q + 18q^{3} + 58q^{5} + 286q^{7} + 162q^{9} + 242q^{11} - 166q^{13} + 522q^{15} - 800q^{17} + 1476q^{19} + 2574q^{21} + 3370q^{23} + 4282q^{25} + 1458q^{27} + 6600q^{29} + 7528q^{31} + 2178q^{33} + 17144q^{35} - 29916q^{37} - 1494q^{39} - 5780q^{41} + 16656q^{43} + 4698q^{45} - 7850q^{47} + 16134q^{49} - 7200q^{51} + 14178q^{53} + 7018q^{55} + 13284q^{57} - 17300q^{59} - 2946q^{61} + 23166q^{63} - 38444q^{65} - 31336q^{67} + 30330q^{69} + 33810q^{71} + 60644q^{73} + 38538q^{75} + 34606q^{77} - 1870q^{79} + 13122q^{81} + 58296q^{83} - 76300q^{85} + 59400q^{87} + 92388q^{89} - 57368q^{91} + 67752q^{93} + 32184q^{95} + 7120q^{97} + 19602q^{99} + O(q^{100}) \)

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
7.15207
−6.15207
0 9.00000 0 −37.5207 0 76.4793 0 81.0000 0
1.2 0 9.00000 0 95.5207 0 209.521 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.s 2
4.b odd 2 1 33.6.a.c 2
12.b even 2 1 99.6.a.f 2
20.d odd 2 1 825.6.a.e 2
44.c even 2 1 363.6.a.j 2
132.d odd 2 1 1089.6.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.c 2 4.b odd 2 1
99.6.a.f 2 12.b even 2 1
363.6.a.j 2 44.c even 2 1
528.6.a.s 2 1.a even 1 1 trivial
825.6.a.e 2 20.d odd 2 1
1089.6.a.j 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(528))\):

\( T_{5}^{2} - 58 T_{5} - 3584 \)
\( T_{7}^{2} - 286 T_{7} + 16024 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -9 + T )^{2} \)
$5$ \( -3584 - 58 T + T^{2} \)
$7$ \( 16024 - 286 T + T^{2} \)
$11$ \( ( -121 + T )^{2} \)
$13$ \( -57008 + 166 T + T^{2} \)
$17$ \( 700 + 800 T + T^{2} \)
$19$ \( 538272 - 1476 T + T^{2} \)
$23$ \( -3088328 - 3370 T + T^{2} \)
$29$ \( 3378828 - 6600 T + T^{2} \)
$31$ \( 8931328 - 7528 T + T^{2} \)
$37$ \( 222923316 + 29916 T + T^{2} \)
$41$ \( -14080172 + 5780 T + T^{2} \)
$43$ \( -29676624 - 16656 T + T^{2} \)
$47$ \( -266939288 + 7850 T + T^{2} \)
$53$ \( -165085872 - 14178 T + T^{2} \)
$59$ \( -21579488 + 17300 T + T^{2} \)
$61$ \( -238059096 + 2946 T + T^{2} \)
$67$ \( -207633776 + 31336 T + T^{2} \)
$71$ \( -138914952 - 33810 T + T^{2} \)
$73$ \( -1593591164 - 60644 T + T^{2} \)
$79$ \( -3977569112 + 1870 T + T^{2} \)
$83$ \( 552313872 - 58296 T + T^{2} \)
$89$ \( -7896843132 - 92388 T + T^{2} \)
$97$ \( -20063108900 - 7120 T + T^{2} \)
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