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: |
\( x^{8} - 2x^{7} + 15x^{6} - 2x^{5} + 133x^{4} - 84x^{3} + 276x^{2} + 144x + 144 \)
|
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.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{8} - 2x^{7} + 15x^{6} - 2x^{5} + 133x^{4} - 84x^{3} + 276x^{2} + 144x + 144 \)
:
\(\beta_{1}\) | \(=\) |
\( ( -473\nu^{7} + 198\nu^{6} - 5547\nu^{5} - 7826\nu^{4} - 75285\nu^{3} - 29928\nu^{2} + 140724\nu - 256788 ) / 159300 \)
|
\(\beta_{2}\) | \(=\) |
\( ( 677\nu^{7} - 1827\nu^{6} + 10353\nu^{5} - 6901\nu^{4} + 82215\nu^{3} - 132153\nu^{2} + 156924\nu + 78912 ) / 159300 \)
|
\(\beta_{3}\) | \(=\) |
\( ( -473\nu^{7} + 198\nu^{6} - 5547\nu^{5} - 7826\nu^{4} - 75285\nu^{3} - 29928\nu^{2} - 98226\nu - 177138 ) / 79650 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 1013 \nu^{7} - 2688 \nu^{6} + 16707 \nu^{5} - 19444 \nu^{4} + 143085 \nu^{3} - 232782 \nu^{2} + 247356 \nu - 401472 ) / 159300 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 644\nu^{7} - 2019\nu^{6} + 9966\nu^{5} - 7447\nu^{4} + 68730\nu^{3} - 107691\nu^{2} + 129078\nu + 194364 ) / 79650 \)
|
\(\beta_{6}\) | \(=\) |
\( ( - 2407 \nu^{7} + 7182 \nu^{6} - 45123 \nu^{5} + 47066 \nu^{4} - 389565 \nu^{3} + 475248 \nu^{2} - 1245834 \nu + 209808 ) / 159300 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 5753 \nu^{7} - 4878 \nu^{6} + 67467 \nu^{5} + 95186 \nu^{4} + 657585 \nu^{3} + 364008 \nu^{2} + 544536 \nu + 2015568 ) / 318600 \)
|
\(\nu\) | \(=\) |
\( ( -\beta_{3} + 2\beta _1 + 1 ) / 3 \)
|
\(\nu^{2}\) | \(=\) |
\( ( 6\beta_{5} + 3\beta_{4} - 2\beta_{3} - 18\beta_{2} + \beta _1 - 1 ) / 3 \)
|
\(\nu^{3}\) | \(=\) |
\( ( -6\beta_{7} + 6\beta_{5} - 6\beta_{4} - 13\beta_{3} - 7\beta _1 - 32 ) / 3 \)
|
\(\nu^{4}\) | \(=\) |
\( ( -12\beta_{6} - 45\beta_{5} - 90\beta_{4} + 25\beta_{3} + 186\beta_{2} - 38\beta _1 - 199 ) / 3 \)
|
\(\nu^{5}\) | \(=\) |
\( ( 90\beta_{7} - 90\beta_{6} - 240\beta_{5} - 120\beta_{4} + 338\beta_{3} + 288\beta_{2} - 169\beta _1 + 169 ) / 3 \)
|
\(\nu^{6}\) | \(=\) |
\( ( 240\beta_{7} - 627\beta_{5} + 627\beta_{4} + 445\beta_{3} + 205\beta _1 + 2912 ) / 3 \)
|
\(\nu^{7}\) | \(=\) |
\( ( 1254\beta_{6} + 1962\beta_{5} + 3924\beta_{4} - 2293\beta_{3} - 5316\beta_{2} + 3332\beta _1 + 6355 ) / 3 \)
|
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 | 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 | |||||||||||||||||||||||||||||||||||||||
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
\( T_{5}^{8} - 6T_{5}^{7} - 37T_{5}^{6} + 294T_{5}^{5} + 2661T_{5}^{4} + 6468T_{5}^{3} + 5612T_{5}^{2} - 528T_{5} + 16 \)
acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} \)
$5$
\( T^{8} - 6 T^{7} - 37 T^{6} + 294 T^{5} + \cdots + 16 \)
$7$
\( T^{8} - 6 T^{7} + 69 T^{6} + \cdots + 11664 \)
$11$
\( T^{8} + 36 T^{7} + \cdots + 105616729 \)
$13$
\( T^{8} - 14 T^{7} + 547 T^{6} + \cdots + 2611456 \)
$17$
\( T^{8} + 1454 T^{6} + \cdots + 7020428944 \)
$19$
\( (T^{4} - 2 T^{3} - 1179 T^{2} + \cdots + 226348)^{2} \)
$23$
\( T^{8} - 102 T^{7} + \cdots + 11198718976 \)
$29$
\( T^{8} - 114 T^{7} + \cdots + 106450807824 \)
$31$
\( T^{8} + 50 T^{7} + \cdots + 152712134656 \)
$37$
\( (T^{4} - 60 T^{3} - 276 T^{2} + \cdots + 206496)^{2} \)
$41$
\( T^{8} + 264 T^{7} + \cdots + 1919025613521 \)
$43$
\( T^{8} + 28 T^{7} + \cdots + 1352729498761 \)
$47$
\( T^{8} + 150 T^{7} + \cdots + 4615347568896 \)
$53$
\( T^{8} + 7016 T^{6} + \cdots + 78435844096 \)
$59$
\( T^{8} - 108 T^{7} + \cdots + 127589696809 \)
$61$
\( T^{8} - 14 T^{7} + \cdots + 133593174016 \)
$67$
\( T^{8} + 20 T^{7} + \cdots + 17391015625 \)
$71$
\( T^{8} + \cdots + 114698616545536 \)
$73$
\( (T^{4} + 38 T^{3} - 9831 T^{2} + \cdots + 2961976)^{2} \)
$79$
\( T^{8} + \cdots + 103529078405776 \)
$83$
\( T^{8} + 246 T^{7} + \cdots + 1085363908864 \)
$89$
\( T^{8} + \cdots + 309931236458496 \)
$97$
\( T^{8} + 236 T^{7} + \cdots + 3435006304129 \)
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