Properties

Label 216.3.m.b
Level $216$
Weight $3$
Character orbit 216.m
Analytic conductor $5.886$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.m (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.19269881856.9
Defining polynomial: \(x^{8} - 2 x^{7} + 15 x^{6} - 2 x^{5} + 133 x^{4} - 84 x^{3} + 276 x^{2} + 144 x + 144\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{5} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{6} ) q^{7} +O(q^{10})\) \( q + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{5} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{6} ) q^{7} + ( -2 - \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{11} + ( 1 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{13} + ( -3 \beta_{1} - 4 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{17} + ( -2 - 5 \beta_{1} - \beta_{3} + 4 \beta_{7} ) q^{19} + ( 14 - 2 \beta_{1} - 8 \beta_{2} + 4 \beta_{4} + 3 \beta_{6} + 3 \beta_{7} ) q^{23} + ( 7 - 6 \beta_{1} - 8 \beta_{2} + 5 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} ) q^{25} + ( 10 + 2 \beta_{1} + 9 \beta_{2} - \beta_{3} - 3 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{29} + ( -12 \beta_{2} - 6 \beta_{4} - 12 \beta_{5} - \beta_{6} + \beta_{7} ) q^{31} + ( 7 - \beta_{1} - 12 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{35} + ( 16 + 2 \beta_{1} - 6 \beta_{4} + 6 \beta_{5} - 2 \beta_{7} ) q^{37} + ( -42 + 21 \beta_{2} - 2 \beta_{6} - 2 \beta_{7} ) q^{41} + ( -14 + 7 \beta_{1} + 14 \beta_{2} - 7 \beta_{3} + 12 \beta_{4} + 6 \beta_{5} + 7 \beta_{6} ) q^{43} + ( -15 - 3 \beta_{1} - 10 \beta_{2} - 2 \beta_{3} + 6 \beta_{5} - 3 \beta_{6} + 6 \beta_{7} ) q^{47} + ( -1 + \beta_{1} + 22 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{49} + ( 10 + 2 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 12 \beta_{6} + 6 \beta_{7} ) q^{53} + ( -27 + 7 \beta_{1} + 6 \beta_{3} - 12 \beta_{4} + 12 \beta_{5} - \beta_{7} ) q^{55} + ( 21 + 9 \beta_{1} - 6 \beta_{2} - 14 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{59} + ( 2 + 12 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} + 18 \beta_{4} + 9 \beta_{5} - 2 \beta_{6} ) q^{61} + ( 12 + 6 \beta_{1} + 21 \beta_{2} - 15 \beta_{3} + 7 \beta_{5} + 6 \beta_{6} - 12 \beta_{7} ) q^{65} + ( -5 + 5 \beta_{1} - 10 \beta_{3} + 5 \beta_{6} - 5 \beta_{7} ) q^{67} + ( -42 + 12 \beta_{1} + 84 \beta_{2} - 12 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{71} + ( -4 + 11 \beta_{1} - 7 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} - 18 \beta_{7} ) q^{73} + ( -7 + 7 \beta_{1} + 7 \beta_{2} - 9 \beta_{4} - 4 \beta_{6} - 4 \beta_{7} ) q^{77} + ( 13 - 5 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} + 24 \beta_{4} + 12 \beta_{5} - 7 \beta_{6} ) q^{79} + ( -19 + \beta_{1} - 22 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{83} + ( 6 - 6 \beta_{1} - 60 \beta_{2} + 12 \beta_{3} - 10 \beta_{6} + 10 \beta_{7} ) q^{85} + ( -2 + 14 \beta_{1} - 16 \beta_{2} - 24 \beta_{3} + 20 \beta_{6} - 10 \beta_{7} ) q^{89} + ( 15 + 3 \beta_{1} - 4 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} - 7 \beta_{7} ) q^{91} + ( 82 + 2 \beta_{1} - 40 \beta_{2} + 2 \beta_{4} - 6 \beta_{6} - 6 \beta_{7} ) q^{95} + ( -63 + 10 \beta_{1} + 67 \beta_{2} - 6 \beta_{3} + 2 \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 6q^{5} + 6q^{7} + O(q^{10}) \) \( 8q + 6q^{5} + 6q^{7} - 36q^{11} + 14q^{13} + 4q^{19} + 102q^{23} + 10q^{25} + 114q^{29} - 50q^{31} + 120q^{37} - 264q^{41} - 28q^{43} - 150q^{47} + 94q^{49} - 244q^{55} + 108q^{59} + 14q^{61} + 198q^{65} - 20q^{67} - 76q^{73} - 66q^{77} + 26q^{79} - 246q^{83} - 224q^{85} + 108q^{91} + 456q^{95} - 236q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 15 x^{6} - 2 x^{5} + 133 x^{4} - 84 x^{3} + 276 x^{2} + 144 x + 144\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -473 \nu^{7} + 198 \nu^{6} - 5547 \nu^{5} - 7826 \nu^{4} - 75285 \nu^{3} - 29928 \nu^{2} + 140724 \nu - 256788 \)\()/159300\)
\(\beta_{2}\)\(=\)\((\)\( 677 \nu^{7} - 1827 \nu^{6} + 10353 \nu^{5} - 6901 \nu^{4} + 82215 \nu^{3} - 132153 \nu^{2} + 156924 \nu + 78912 \)\()/159300\)
\(\beta_{3}\)\(=\)\((\)\( -473 \nu^{7} + 198 \nu^{6} - 5547 \nu^{5} - 7826 \nu^{4} - 75285 \nu^{3} - 29928 \nu^{2} - 98226 \nu - 177138 \)\()/79650\)
\(\beta_{4}\)\(=\)\((\)\( 1013 \nu^{7} - 2688 \nu^{6} + 16707 \nu^{5} - 19444 \nu^{4} + 143085 \nu^{3} - 232782 \nu^{2} + 247356 \nu - 401472 \)\()/159300\)
\(\beta_{5}\)\(=\)\((\)\( 644 \nu^{7} - 2019 \nu^{6} + 9966 \nu^{5} - 7447 \nu^{4} + 68730 \nu^{3} - 107691 \nu^{2} + 129078 \nu + 194364 \)\()/79650\)
\(\beta_{6}\)\(=\)\((\)\( -2407 \nu^{7} + 7182 \nu^{6} - 45123 \nu^{5} + 47066 \nu^{4} - 389565 \nu^{3} + 475248 \nu^{2} - 1245834 \nu + 209808 \)\()/159300\)
\(\beta_{7}\)\(=\)\((\)\( 5753 \nu^{7} - 4878 \nu^{6} + 67467 \nu^{5} + 95186 \nu^{4} + 657585 \nu^{3} + 364008 \nu^{2} + 544536 \nu + 2015568 \)\()/318600\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(6 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 18 \beta_{2} + \beta_{1} - 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-6 \beta_{7} + 6 \beta_{5} - 6 \beta_{4} - 13 \beta_{3} - 7 \beta_{1} - 32\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-12 \beta_{6} - 45 \beta_{5} - 90 \beta_{4} + 25 \beta_{3} + 186 \beta_{2} - 38 \beta_{1} - 199\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(90 \beta_{7} - 90 \beta_{6} - 240 \beta_{5} - 120 \beta_{4} + 338 \beta_{3} + 288 \beta_{2} - 169 \beta_{1} + 169\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(240 \beta_{7} - 627 \beta_{5} + 627 \beta_{4} + 445 \beta_{3} + 205 \beta_{1} + 2912\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(1254 \beta_{6} + 1962 \beta_{5} + 3924 \beta_{4} - 2293 \beta_{3} - 5316 \beta_{2} + 3332 \beta_{1} + 6355\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.831167 + 1.43962i
1.91950 + 3.32468i
−0.331167 0.573598i
−1.41950 2.45865i
0.831167 1.43962i
1.91950 3.32468i
−0.331167 + 0.573598i
−1.41950 + 2.45865i
0 0 0 −3.44299 1.98781i 0 −1.80469 3.12582i 0 0 0
17.2 0 0 0 −1.80902 1.04444i 0 −0.781452 1.35351i 0 0 0
17.3 0 0 0 0.0440114 + 0.0254100i 0 4.52944 + 7.84521i 0 0 0
17.4 0 0 0 8.20800 + 4.73889i 0 1.05671 + 1.83027i 0 0 0
89.1 0 0 0 −3.44299 + 1.98781i 0 −1.80469 + 3.12582i 0 0 0
89.2 0 0 0 −1.80902 + 1.04444i 0 −0.781452 + 1.35351i 0 0 0
89.3 0 0 0 0.0440114 0.0254100i 0 4.52944 7.84521i 0 0 0
89.4 0 0 0 8.20800 4.73889i 0 1.05671 1.83027i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.m.b 8
3.b odd 2 1 72.3.m.b 8
4.b odd 2 1 432.3.q.e 8
8.b even 2 1 1728.3.q.j 8
8.d odd 2 1 1728.3.q.i 8
9.c even 3 1 72.3.m.b 8
9.c even 3 1 648.3.e.c 8
9.d odd 6 1 inner 216.3.m.b 8
9.d odd 6 1 648.3.e.c 8
12.b even 2 1 144.3.q.e 8
24.f even 2 1 576.3.q.j 8
24.h odd 2 1 576.3.q.i 8
36.f odd 6 1 144.3.q.e 8
36.f odd 6 1 1296.3.e.i 8
36.h even 6 1 432.3.q.e 8
36.h even 6 1 1296.3.e.i 8
72.j odd 6 1 1728.3.q.j 8
72.l even 6 1 1728.3.q.i 8
72.n even 6 1 576.3.q.i 8
72.p odd 6 1 576.3.q.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.m.b 8 3.b odd 2 1
72.3.m.b 8 9.c even 3 1
144.3.q.e 8 12.b even 2 1
144.3.q.e 8 36.f odd 6 1
216.3.m.b 8 1.a even 1 1 trivial
216.3.m.b 8 9.d odd 6 1 inner
432.3.q.e 8 4.b odd 2 1
432.3.q.e 8 36.h even 6 1
576.3.q.i 8 24.h odd 2 1
576.3.q.i 8 72.n even 6 1
576.3.q.j 8 24.f even 2 1
576.3.q.j 8 72.p odd 6 1
648.3.e.c 8 9.c even 3 1
648.3.e.c 8 9.d odd 6 1
1296.3.e.i 8 36.f odd 6 1
1296.3.e.i 8 36.h even 6 1
1728.3.q.i 8 8.d odd 2 1
1728.3.q.i 8 72.l even 6 1
1728.3.q.j 8 8.b even 2 1
1728.3.q.j 8 72.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{8} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 16 - 528 T + 5612 T^{2} + 6468 T^{3} + 2661 T^{4} + 294 T^{5} - 37 T^{6} - 6 T^{7} + T^{8} \)
$7$ \( 11664 + 3888 T + 4860 T^{2} + 108 T^{3} + 1197 T^{4} + 126 T^{5} + 69 T^{6} - 6 T^{7} + T^{8} \)
$11$ \( 105616729 + 58455576 T + 11688824 T^{2} + 500544 T^{3} - 50235 T^{4} - 3168 T^{5} + 344 T^{6} + 36 T^{7} + T^{8} \)
$13$ \( 2611456 - 6270080 T + 14487184 T^{2} - 1407128 T^{3} + 179137 T^{4} - 2846 T^{5} + 547 T^{6} - 14 T^{7} + T^{8} \)
$17$ \( 7020428944 + 120824648 T^{2} + 687753 T^{4} + 1454 T^{6} + T^{8} \)
$19$ \( ( 226348 - 4004 T - 1179 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$23$ \( 11198718976 - 3042651648 T + 228044192 T^{2} + 12909648 T^{3} - 670143 T^{4} - 45798 T^{5} + 3917 T^{6} - 102 T^{7} + T^{8} \)
$29$ \( 106450807824 - 9055894608 T - 37821492 T^{2} + 25063668 T^{3} + 86949 T^{4} - 102942 T^{5} + 5235 T^{6} - 114 T^{7} + T^{8} \)
$31$ \( 152712134656 - 12067409920 T + 1224387712 T^{2} - 17678560 T^{3} + 1633465 T^{4} + 27110 T^{5} + 3193 T^{6} + 50 T^{7} + T^{8} \)
$37$ \( ( 206496 + 27360 T - 276 T^{2} - 60 T^{3} + T^{4} )^{2} \)
$41$ \( 1919025613521 + 435967071768 T + 44376688026 T^{2} + 2581267824 T^{3} + 93582171 T^{4} + 2165328 T^{5} + 31434 T^{6} + 264 T^{7} + T^{8} \)
$43$ \( 1352729498761 + 211873953592 T + 28244463112 T^{2} + 838981528 T^{3} + 24309277 T^{4} + 245392 T^{5} + 5032 T^{6} + 28 T^{7} + T^{8} \)
$47$ \( 4615347568896 + 475641590400 T + 16526225232 T^{2} + 19261800 T^{3} - 8914095 T^{4} - 13050 T^{5} + 7413 T^{6} + 150 T^{7} + T^{8} \)
$53$ \( 78435844096 + 2590797824 T^{2} + 12396816 T^{4} + 7016 T^{6} + T^{8} \)
$59$ \( 127589696809 - 43823785536 T + 6086181872 T^{2} - 367082496 T^{3} + 4892493 T^{4} + 323136 T^{5} + 896 T^{6} - 108 T^{7} + T^{8} \)
$61$ \( 133593174016 + 46766967808 T + 13427579584 T^{2} + 1020419248 T^{3} + 63457201 T^{4} + 368674 T^{5} + 8251 T^{6} - 14 T^{7} + T^{8} \)
$67$ \( 17391015625 - 4879375000 T + 1606375000 T^{2} + 61325000 T^{3} + 3848125 T^{4} + 38000 T^{5} + 2200 T^{6} + 20 T^{7} + T^{8} \)
$71$ \( 114698616545536 + 406952754944 T^{2} + 206774880 T^{4} + 26864 T^{6} + T^{8} \)
$73$ \( ( 2961976 - 487192 T - 9831 T^{2} + 38 T^{3} + T^{4} )^{2} \)
$79$ \( 103529078405776 - 749891898800 T + 123470983324 T^{2} + 1384089748 T^{3} + 122492077 T^{4} + 449026 T^{5} + 12277 T^{6} - 26 T^{7} + T^{8} \)
$83$ \( 1085363908864 + 263110694016 T + 28654548944 T^{2} + 1792361544 T^{3} + 70034865 T^{4} + 1745862 T^{5} + 27269 T^{6} + 246 T^{7} + T^{8} \)
$89$ \( 309931236458496 + 1618052474880 T^{2} + 407712528 T^{4} + 34920 T^{6} + T^{8} \)
$97$ \( 3435006304129 + 837036947756 T + 171307640890 T^{2} + 7083794672 T^{3} + 202097299 T^{4} + 3255536 T^{5} + 38074 T^{6} + 236 T^{7} + T^{8} \)
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