Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) 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 \(x^{10} - x^{9} + 8 x^{8} + 7 x^{7} + 41 x^{6} + 18 x^{5} + 58 x^{4} + 28 x^{3} + 64 x^{2} + 16 x + 4\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \((\)\( -364 \nu^{9} + 176 \nu^{8} - 220 \nu^{7} - 5913 \nu^{6} + 880 \nu^{5} + 6908 \nu^{4} + 84549 \nu^{3} + 9416 \nu^{2} + 2376 \nu + 30518 \)\()/118350\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -983 \nu^{9} - 7328 \nu^{8} + 9160 \nu^{7} - 87336 \nu^{6} - 36640 \nu^{5} - 287624 \nu^{4} - 39747 \nu^{3} - 392048 \nu^{2} - 98928 \nu - 22604 \)\()/118350\) |
| \(\beta_{4}\) | \(=\) | \((\)\( -1159 \nu^{9} - 9844 \nu^{8} + 12305 \nu^{7} - 109053 \nu^{6} - 49220 \nu^{5} - 386377 \nu^{4} + 25194 \nu^{3} - 526654 \nu^{2} - 132894 \nu - 348592 \)\()/118350\) |
| \(\beta_{5}\) | \(=\) | \((\)\( 916 \nu^{9} - 3044 \nu^{8} + 3805 \nu^{7} - 3978 \nu^{6} - 15220 \nu^{5} - 119477 \nu^{4} - 129531 \nu^{3} - 162854 \nu^{2} - 41094 \nu - 180842 \)\()/59175\) |
| \(\beta_{6}\) | \(=\) | \((\)\( 2249 \nu^{9} - 9541 \nu^{8} + 26720 \nu^{7} - 34617 \nu^{6} + 31195 \nu^{5} - 152578 \nu^{4} + 39066 \nu^{3} - 46906 \nu^{2} - 20316 \nu - 3988 \)\()/78900\) |
| \(\beta_{7}\) | \(=\) | \((\)\( -15259 \nu^{9} + 14531 \nu^{8} - 121720 \nu^{7} - 107253 \nu^{6} - 637445 \nu^{5} - 272902 \nu^{4} - 871206 \nu^{3} - 258154 \nu^{2} - 957744 \nu - 2692 \)\()/236700\) |
| \(\beta_{8}\) | \(=\) | \((\)\( 18139 \nu^{9} - 21776 \nu^{8} + 145570 \nu^{7} + 96813 \nu^{6} + 719570 \nu^{5} + 92092 \nu^{4} + 981276 \nu^{3} + 255184 \nu^{2} + 1362924 \nu + 532 \)\()/118350\) |
| \(\beta_{9}\) | \(=\) | \((\)\( 1058 \nu^{9} - 1552 \nu^{8} + 9041 \nu^{7} + 3726 \nu^{6} + 39580 \nu^{5} + 2993 \nu^{4} + 53832 \nu^{3} + 11648 \nu^{2} + 50058 \nu - 136 \)\()/4734\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{9} + 3 \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 3\) |
| \(\nu^{3}\) | \(=\) | \(\beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} - 4\) |
| \(\nu^{4}\) | \(=\) | \(-8 \beta_{9} + 2 \beta_{8} - 19 \beta_{7} + 9 \beta_{6} - 13 \beta_{1}\) |
| \(\nu^{5}\) | \(=\) | \(-22 \beta_{9} + 9 \beta_{8} - 45 \beta_{7} + 23 \beta_{6} - 9 \beta_{5} + 22 \beta_{4} - 23 \beta_{3} - 47 \beta_{2} - 47 \beta_{1} + 45\) |
| \(\nu^{6}\) | \(=\) | \(-23 \beta_{5} + 70 \beta_{4} - 78 \beta_{3} - 128 \beta_{2} + 154\) |
| \(\nu^{7}\) | \(=\) | \(206 \beta_{9} - 78 \beta_{8} + 431 \beta_{7} - 221 \beta_{6} + 407 \beta_{1}\) |
| \(\nu^{8}\) | \(=\) | \(628 \beta_{9} - 221 \beta_{8} + 1349 \beta_{7} - 691 \beta_{6} + 221 \beta_{5} - 628 \beta_{4} + 691 \beta_{3} + 1187 \beta_{2} + 1187 \beta_{1} - 1349\) |
| \(\nu^{9}\) | \(=\) | \(691 \beta_{5} - 1878 \beta_{4} + 2036 \beta_{3} + 3634 \beta_{2} - 3968\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).
| \(n\) |
\(15\) |
\(66\) |
| \(\chi(n)\) |
\(1\) |
\(-\beta_{7}\) |
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.
This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( 9 - 36 T + 195 T^{2} + 210 T^{3} + 265 T^{4} + 116 T^{5} + 81 T^{6} + 30 T^{7} + 17 T^{8} + 4 T^{9} + T^{10} \) |
| $3$ |
\( 16 - 64 T + 256 T^{2} - 72 T^{3} + 144 T^{4} - 4 T^{5} + 65 T^{6} + 9 T^{8} + T^{10} \) |
| $5$ |
\( 2304 + 2304 T + 3264 T^{2} + 480 T^{3} + 1024 T^{4} + 156 T^{5} + 217 T^{6} + 10 T^{7} + 19 T^{8} + 2 T^{9} + T^{10} \) |
| $7$ |
\( 16807 - 2401 T + 2058 T^{2} - 833 T^{3} + 119 T^{4} - 204 T^{5} + 17 T^{6} - 17 T^{7} + 6 T^{8} - T^{9} + T^{10} \) |
| $11$ |
\( 1089 + 1485 T + 2751 T^{2} + 1386 T^{3} + 2467 T^{4} + 1749 T^{5} + 1099 T^{6} + 352 T^{7} + 85 T^{8} + 11 T^{9} + T^{10} \) |
| $13$ |
\( ( -1 + T )^{10} \) |
| $17$ |
\( 184041 - 39897 T + 54123 T^{2} - 9018 T^{3} + 11137 T^{4} - 1831 T^{5} + 921 T^{6} - 102 T^{7} + 47 T^{8} - 5 T^{9} + T^{10} \) |
| $19$ |
\( 49729 + 38579 T + 69177 T^{2} - 24204 T^{3} + 31391 T^{4} - 427 T^{5} + 1607 T^{6} + 226 T^{7} + 95 T^{8} + 9 T^{9} + T^{10} \) |
| $23$ |
\( 144 + 144 T + 456 T^{2} + 432 T^{3} + 1168 T^{4} + 1034 T^{5} + 713 T^{6} + 258 T^{7} + 69 T^{8} + 10 T^{9} + T^{10} \) |
| $29$ |
\( ( -108 + 144 T - 19 T^{2} - 25 T^{3} + 3 T^{4} + T^{5} )^{2} \) |
| $31$ |
\( 126736 + 180848 T + 221752 T^{2} + 95248 T^{3} + 43528 T^{4} + 230 T^{5} + 3825 T^{6} + 162 T^{7} + 97 T^{8} - 6 T^{9} + T^{10} \) |
| $37$ |
\( 49505296 + 4643760 T + 5206008 T^{2} + 1114512 T^{3} + 504800 T^{4} + 77014 T^{5} + 14373 T^{6} + 912 T^{7} + 127 T^{8} + 4 T^{9} + T^{10} \) |
| $41$ |
\( ( -1584 - 2544 T + 940 T^{2} - 28 T^{3} - 14 T^{4} + T^{5} )^{2} \) |
| $43$ |
\( ( 64 - 288 T + 308 T^{2} - 72 T^{3} - 2 T^{4} + T^{5} )^{2} \) |
| $47$ |
\( 26718561 + 14530059 T + 8036115 T^{2} + 1208826 T^{3} + 344071 T^{4} + 2771 T^{5} + 12591 T^{6} - 72 T^{7} + 125 T^{8} + T^{9} + T^{10} \) |
| $53$ |
\( 398361681 + 254656881 T + 114371547 T^{2} + 27999402 T^{3} + 5280613 T^{4} + 593371 T^{5} + 59477 T^{6} + 3594 T^{7} + 363 T^{8} + 17 T^{9} + T^{10} \) |
| $59$ |
\( 1089 + 1485 T + 2751 T^{2} + 1386 T^{3} + 2467 T^{4} + 1749 T^{5} + 1099 T^{6} + 352 T^{7} + 85 T^{8} + 11 T^{9} + T^{10} \) |
| $61$ |
\( 71588521 + 49759141 T + 28105035 T^{2} + 6569330 T^{3} + 1397309 T^{4} + 44391 T^{5} + 17429 T^{6} - 190 T^{7} + 243 T^{8} - 11 T^{9} + T^{10} \) |
| $67$ |
\( 515244601 - 13415109 T + 49379121 T^{2} + 8631036 T^{3} + 4274771 T^{4} + 387985 T^{5} + 54915 T^{6} + 2214 T^{7} + 331 T^{8} + 13 T^{9} + T^{10} \) |
| $71$ |
\( ( -6336 - 456 T + 853 T^{2} - 25 T^{3} - 15 T^{4} + T^{5} )^{2} \) |
| $73$ |
\( 506944 + 498400 T + 519904 T^{2} + 77400 T^{3} + 54264 T^{4} + 3862 T^{5} + 4925 T^{6} + 84 T^{7} + 75 T^{8} + T^{10} \) |
| $79$ |
\( 1000000 + 1500000 T + 2060000 T^{2} + 559000 T^{3} + 239600 T^{4} - 31030 T^{5} + 16889 T^{6} - 654 T^{7} + 141 T^{8} + 2 T^{9} + T^{10} \) |
| $83$ |
\( ( -7488 + 2688 T + 308 T^{2} - 124 T^{3} - 6 T^{4} + T^{5} )^{2} \) |
| $89$ |
\( 59166864 - 16522416 T + 9952152 T^{2} - 893808 T^{3} + 783808 T^{4} - 98078 T^{5} + 24653 T^{6} - 768 T^{7} + 171 T^{8} - 4 T^{9} + T^{10} \) |
| $97$ |
\( ( -2384 - 2240 T - 612 T^{2} - 16 T^{3} + 12 T^{4} + T^{5} )^{2} \) |
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