Newspace parameters
Level: | \( N \) | \(=\) | \( 216 = 2^{3} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 216.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.88557371018\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - 2x^{14} - 2x^{12} - 56x^{10} + 400x^{8} - 896x^{6} - 512x^{4} - 8192x^{2} + 65536 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{12}\cdot 3^{8} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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 \( x^{16} - 2x^{14} - 2x^{12} - 56x^{10} + 400x^{8} - 896x^{6} - 512x^{4} - 8192x^{2} + 65536 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{15} + 2\nu^{13} + 2\nu^{11} + 56\nu^{9} - 400\nu^{7} + 896\nu^{5} + 512\nu^{3} + 8192\nu ) / 4096 \) |
\(\beta_{5}\) | \(=\) | \( ( \nu^{14} - 2\nu^{12} - 2\nu^{10} - 56\nu^{8} + 400\nu^{6} - 896\nu^{4} - 512\nu^{2} - 6144 ) / 2048 \) |
\(\beta_{6}\) | \(=\) | \( ( \nu^{15} - 10\nu^{13} + 78\nu^{11} - 168\nu^{9} + 720\nu^{7} - 3584\nu^{5} + 15872\nu^{3} - 36864\nu ) / 8192 \) |
\(\beta_{7}\) | \(=\) | \( ( 3\nu^{14} - 22\nu^{12} + 90\nu^{10} - 264\nu^{8} + 944\nu^{6} - 6528\nu^{4} + 19968\nu^{2} - 28672 ) / 4096 \) |
\(\beta_{8}\) | \(=\) | \( ( \nu^{14} - 10\nu^{12} + 14\nu^{10} - 40\nu^{8} + 336\nu^{6} - 2048\nu^{4} + 3584\nu^{2} + 7168 ) / 1024 \) |
\(\beta_{9}\) | \(=\) | \( ( -3\nu^{15} - 10\nu^{13} + 166\nu^{11} - 568\nu^{9} + 464\nu^{7} - 1664\nu^{5} + 46592\nu^{3} - 139264\nu ) / 8192 \) |
\(\beta_{10}\) | \(=\) | \( ( -3\nu^{15} - 6\nu^{13} + 62\nu^{11} - 128\nu^{9} - 80\nu^{7} - 1344\nu^{5} + 18944\nu^{3} - 38912\nu ) / 4096 \) |
\(\beta_{11}\) | \(=\) | \( ( -3\nu^{15} + 22\nu^{13} - 90\nu^{11} + 264\nu^{9} - 944\nu^{7} + 6528\nu^{5} - 19968\nu^{3} + 28672\nu ) / 4096 \) |
\(\beta_{12}\) | \(=\) | \( ( -\nu^{14} - 6\nu^{12} + 82\nu^{10} - 312\nu^{8} + 432\nu^{6} - 1280\nu^{4} + 23040\nu^{2} - 73728 ) / 2048 \) |
\(\beta_{13}\) | \(=\) | \( ( -\nu^{15} + 5\nu^{13} + 4\nu^{11} - 30\nu^{9} - 200\nu^{7} + 752\nu^{5} + 2048\nu^{3} - 15872\nu ) / 1024 \) |
\(\beta_{14}\) | \(=\) | \( ( -7\nu^{14} + 14\nu^{12} + 78\nu^{10} - 248\nu^{8} - 880\nu^{6} + 1664\nu^{4} + 27136\nu^{2} - 102400 ) / 4096 \) |
\(\beta_{15}\) | \(=\) | \( ( 5\nu^{14} - 58\nu^{12} + 278\nu^{10} - 568\nu^{8} + 3280\nu^{6} - 16000\nu^{4} + 61952\nu^{2} - 106496 ) / 4096 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} \) |
\(\nu^{4}\) | \(=\) | \( -\beta_{14} + \beta_{12} - \beta_{7} - \beta_{5} + 1 \) |
\(\nu^{5}\) | \(=\) | \( -\beta_{13} + \beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{4} \) |
\(\nu^{6}\) | \(=\) | \( 2\beta_{15} - 6\beta_{7} + 4\beta_{5} + 22 \) |
\(\nu^{7}\) | \(=\) | \( -2\beta_{13} + 6\beta_{11} + 2\beta_{10} + 12\beta_{6} - 10\beta_{4} + 2\beta_{3} + 20\beta_1 \) |
\(\nu^{8}\) | \(=\) | \( 8\beta_{15} - 2\beta_{14} - 6\beta_{12} - 4\beta_{8} - 10\beta_{7} - 10\beta_{5} + 20\beta_{2} - 130 \) |
\(\nu^{9}\) | \(=\) | \( -6\beta_{13} + 14\beta_{11} + 2\beta_{10} - 12\beta_{9} + 56\beta_{6} + 22\beta_{4} + 28\beta_{3} - 168\beta_1 \) |
\(\nu^{10}\) | \(=\) | \( 20\beta_{15} - 28\beta_{14} + 12\beta_{12} - 40\beta_{8} + 8\beta_{7} - 68\beta_{5} - 168\beta_{2} + 384 \) |
\(\nu^{11}\) | \(=\) | \( -8\beta_{13} + 32\beta_{11} - 48\beta_{10} + 24\beta_{9} + 200\beta_{6} + 144\beta_{4} - 148\beta_{3} + 216\beta_1 \) |
\(\nu^{12}\) | \(=\) | \( 40\beta_{15} + 84\beta_{14} - 132\beta_{12} - 216\beta_{8} + 188\beta_{7} + 212\beta_{5} + 216\beta_{2} + 1852 \) |
\(\nu^{13}\) | \(=\) | \( 260 \beta_{13} + 28 \beta_{11} - 92 \beta_{10} - 264 \beta_{9} + 672 \beta_{6} - 116 \beta_{4} + 256 \beta_{3} + 1728 \beta_1 \) |
\(\nu^{14}\) | \(=\) | \( - 232 \beta_{15} - 896 \beta_{14} + 320 \beta_{12} - 736 \beta_{8} + 1336 \beta_{7} - 720 \beta_{5} + 1728 \beta_{2} - 4568 \) |
\(\nu^{15}\) | \(=\) | \( 72 \beta_{13} - 600 \beta_{11} - 1864 \beta_{10} + 640 \beta_{9} + 80 \beta_{6} + 2088 \beta_{4} + 1496 \beta_{3} - 5328 \beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).
\(n\) | \(55\) | \(109\) | \(137\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
163.1 |
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−1.98451 | − | 0.248465i | 0 | 3.87653 | + | 0.986159i | 8.47512i | 0 | 7.86423i | −7.44797 | − | 2.92022i | 0 | 2.10577 | − | 16.8189i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.2 | −1.98451 | + | 0.248465i | 0 | 3.87653 | − | 0.986159i | − | 8.47512i | 0 | − | 7.86423i | −7.44797 | + | 2.92022i | 0 | 2.10577 | + | 16.8189i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.3 | −1.77866 | − | 0.914534i | 0 | 2.32725 | + | 3.25329i | − | 3.96500i | 0 | 3.99887i | −1.16415 | − | 7.91484i | 0 | −3.62613 | + | 7.05238i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.4 | −1.77866 | + | 0.914534i | 0 | 2.32725 | − | 3.25329i | 3.96500i | 0 | − | 3.99887i | −1.16415 | + | 7.91484i | 0 | −3.62613 | − | 7.05238i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.5 | −0.862987 | − | 1.80423i | 0 | −2.51051 | + | 3.11406i | − | 1.42436i | 0 | 4.94379i | 7.78502 | + | 1.84214i | 0 | −2.56987 | + | 1.22920i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.6 | −0.862987 | + | 1.80423i | 0 | −2.51051 | − | 3.11406i | 1.42436i | 0 | − | 4.94379i | 7.78502 | − | 1.84214i | 0 | −2.56987 | − | 1.22920i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.7 | −0.808307 | − | 1.82938i | 0 | −2.69328 | + | 2.95740i | 5.51565i | 0 | − | 11.2126i | 7.58722 | + | 2.53655i | 0 | 10.0902 | − | 4.45834i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.8 | −0.808307 | + | 1.82938i | 0 | −2.69328 | − | 2.95740i | − | 5.51565i | 0 | 11.2126i | 7.58722 | − | 2.53655i | 0 | 10.0902 | + | 4.45834i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.9 | 0.808307 | − | 1.82938i | 0 | −2.69328 | − | 2.95740i | 5.51565i | 0 | 11.2126i | −7.58722 | + | 2.53655i | 0 | 10.0902 | + | 4.45834i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.10 | 0.808307 | + | 1.82938i | 0 | −2.69328 | + | 2.95740i | − | 5.51565i | 0 | − | 11.2126i | −7.58722 | − | 2.53655i | 0 | 10.0902 | − | 4.45834i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.11 | 0.862987 | − | 1.80423i | 0 | −2.51051 | − | 3.11406i | − | 1.42436i | 0 | − | 4.94379i | −7.78502 | + | 1.84214i | 0 | −2.56987 | − | 1.22920i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.12 | 0.862987 | + | 1.80423i | 0 | −2.51051 | + | 3.11406i | 1.42436i | 0 | 4.94379i | −7.78502 | − | 1.84214i | 0 | −2.56987 | + | 1.22920i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.13 | 1.77866 | − | 0.914534i | 0 | 2.32725 | − | 3.25329i | − | 3.96500i | 0 | − | 3.99887i | 1.16415 | − | 7.91484i | 0 | −3.62613 | − | 7.05238i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.14 | 1.77866 | + | 0.914534i | 0 | 2.32725 | + | 3.25329i | 3.96500i | 0 | 3.99887i | 1.16415 | + | 7.91484i | 0 | −3.62613 | + | 7.05238i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.15 | 1.98451 | − | 0.248465i | 0 | 3.87653 | − | 0.986159i | 8.47512i | 0 | − | 7.86423i | 7.44797 | − | 2.92022i | 0 | 2.10577 | + | 16.8189i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
163.16 | 1.98451 | + | 0.248465i | 0 | 3.87653 | + | 0.986159i | − | 8.47512i | 0 | 7.86423i | 7.44797 | + | 2.92022i | 0 | 2.10577 | − | 16.8189i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
24.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 216.3.b.b | ✓ | 16 |
3.b | odd | 2 | 1 | inner | 216.3.b.b | ✓ | 16 |
4.b | odd | 2 | 1 | 864.3.b.b | 16 | ||
8.b | even | 2 | 1 | 864.3.b.b | 16 | ||
8.d | odd | 2 | 1 | inner | 216.3.b.b | ✓ | 16 |
12.b | even | 2 | 1 | 864.3.b.b | 16 | ||
24.f | even | 2 | 1 | inner | 216.3.b.b | ✓ | 16 |
24.h | odd | 2 | 1 | 864.3.b.b | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
216.3.b.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
216.3.b.b | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
216.3.b.b | ✓ | 16 | 8.d | odd | 2 | 1 | inner |
216.3.b.b | ✓ | 16 | 24.f | even | 2 | 1 | inner |
864.3.b.b | 16 | 4.b | odd | 2 | 1 | ||
864.3.b.b | 16 | 8.b | even | 2 | 1 | ||
864.3.b.b | 16 | 12.b | even | 2 | 1 | ||
864.3.b.b | 16 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 120T_{5}^{6} + 4032T_{5}^{4} + 42048T_{5}^{2} + 69696 \)
acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} - 2 T^{14} - 2 T^{12} + \cdots + 65536 \)
$3$
\( T^{16} \)
$5$
\( (T^{8} + 120 T^{6} + 4032 T^{4} + \cdots + 69696)^{2} \)
$7$
\( (T^{8} + 228 T^{6} + 15750 T^{4} + \cdots + 3038913)^{2} \)
$11$
\( (T^{8} - 568 T^{6} + 94272 T^{4} + \cdots + 31294528)^{2} \)
$13$
\( (T^{8} + 732 T^{6} + 172134 T^{4} + \cdots + 300808737)^{2} \)
$17$
\( (T^{8} - 1000 T^{6} + 141504 T^{4} + \cdots + 33082432)^{2} \)
$19$
\( (T^{4} + 16 T^{3} - 750 T^{2} + \cdots + 59233)^{4} \)
$23$
\( (T^{8} + 1704 T^{6} + 643392 T^{4} + \cdots + 655769664)^{2} \)
$29$
\( (T^{8} + 3360 T^{6} + \cdots + 89942409216)^{2} \)
$31$
\( (T^{8} + 4080 T^{6} + \cdots + 64927957248)^{2} \)
$37$
\( (T^{8} + 7596 T^{6} + \cdots + 225812702097)^{2} \)
$41$
\( (T^{8} - 8896 T^{6} + \cdots + 11517064118272)^{2} \)
$43$
\( (T^{4} + 16 T^{3} - 3936 T^{2} + \cdots + 417856)^{4} \)
$47$
\( (T^{8} + 7464 T^{6} + \cdots + 54006041664)^{2} \)
$53$
\( (T^{8} + 17568 T^{6} + \cdots + 13649064247296)^{2} \)
$59$
\( (T^{8} - 21496 T^{6} + \cdots + 12544953873472)^{2} \)
$61$
\( (T^{8} + 11388 T^{6} + \cdots + 199870871553)^{2} \)
$67$
\( (T^{2} - 16 T + 37)^{8} \)
$71$
\( (T^{8} + 19584 T^{6} + \cdots + 516742447104)^{2} \)
$73$
\( (T^{4} - 20 T^{3} - 4458 T^{2} + \cdots + 3184753)^{4} \)
$79$
\( (T^{8} + 33780 T^{6} + \cdots + 593133122547153)^{2} \)
$83$
\( (T^{8} - 38464 T^{6} + \cdots + 4297326592)^{2} \)
$89$
\( (T^{8} - 35368 T^{6} + \cdots + 44\!\cdots\!12)^{2} \)
$97$
\( (T^{4} - 20 T^{3} - 23394 T^{2} + \cdots + 34042441)^{4} \)
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