Newform invariants
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 \(x^{8} - 4 x^{7} + 200 x^{6} - 586 x^{5} + 10949 x^{4} - 20926 x^{3} + 78946 x^{2} - 68580 x + 222900\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( -432 \nu^{6} + 1296 \nu^{5} - 76896 \nu^{4} + 151632 \nu^{3} - 3639600 \nu^{2} + 3564000 \nu - 12572496 \)\()/403\) |
| \(\beta_{2}\) | \(=\) | \((\)\( 422 \nu^{7} - 466693 \nu^{6} + 1475675 \nu^{5} - 68583127 \nu^{4} + 138435443 \nu^{3} - 536132632 \nu^{2} + 473860092 \nu + 89182264770 \)\()/5257910\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -3798 \nu^{7} + 13293 \nu^{6} - 720243 \nu^{5} + 1767375 \nu^{4} - 35892171 \nu^{3} + 52077528 \nu^{2} - 94544604 \nu + 38651310 \)\()/5257910\) |
| \(\beta_{4}\) | \(=\) | \((\)\( -1288 \nu^{7} + 4508 \nu^{6} - 269884 \nu^{5} + 663440 \nu^{4} - 16358356 \nu^{3} + 23876348 \nu^{2} - 177074040 \nu + 84579636 \)\()/976469\) |
| \(\beta_{5}\) | \(=\) | \((\)\( 10064 \nu^{7} - 49762 \nu^{6} + 2067878 \nu^{5} - 8127346 \nu^{4} + 120250730 \nu^{3} - 342411844 \nu^{2} + 1396161240 \nu - 1195854900 \)\()/3307395\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -10064 \nu^{7} - 81080 \nu^{6} - 1675352 \nu^{5} - 20265368 \nu^{4} - 64119512 \nu^{3} - 1244701616 \nu^{2} + 162952032 \nu - 4311284880 \)\()/3307395\) |
| \(\beta_{7}\) | \(=\) | \((\)\( 3072 \nu^{7} - 10752 \nu^{6} + 609024 \nu^{5} - 1495680 \nu^{4} + 31491840 \nu^{3} - 45747456 \nu^{2} + 82874880 \nu - 33862464 \)\()/2423\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(-\beta_{6} + 8 \beta_{5} + 12 \beta_{4} + 16 \beta_{3} + 432\)\()/864\) |
| \(\nu^{2}\) | \(=\) | \((\)\(36 \beta_{6} + 144 \beta_{5} + 144 \beta_{4} + 195 \beta_{3} + 27 \beta_{2} - 4 \beta_{1} - 497664\)\()/10368\) |
| \(\nu^{3}\) | \(=\) | \((\)\(-9 \beta_{7} + 652 \beta_{6} - 4784 \beta_{5} - 10992 \beta_{4} - 18621 \beta_{3} + 27 \beta_{2} - 4 \beta_{1} - 499392\)\()/6912\) |
| \(\nu^{4}\) | \(=\) | \((\)\(-27 \beta_{7} - 9312 \beta_{6} - 25728 \beta_{5} - 33120 \beta_{4} - 56340 \beta_{3} - 2484 \beta_{2} + 616 \beta_{1} + 42814656\)\()/10368\) |
| \(\nu^{5}\) | \(=\) | \((\)\(4275 \beta_{7} - 212796 \beta_{6} + 1199088 \beta_{5} + 2870928 \beta_{4} + 8207181 \beta_{3} - 12555 \beta_{2} + 3100 \beta_{1} + 216571968\)\()/20736\) |
| \(\nu^{6}\) | \(=\) | \((\)\(540 \beta_{7} + 106610 \beta_{6} + 286520 \beta_{5} + 365772 \beta_{4} + 1039705 \beta_{3} + 17505 \beta_{2} - 6923 \beta_{1} - 302102784\)\()/864\) |
| \(\nu^{7}\) | \(=\) | \((\)\(-535311 \beta_{7} + 23742708 \beta_{6} - 92469072 \beta_{5} - 236122704 \beta_{4} - 1026485487 \beta_{3} + 1514457 \beta_{2} - 592396 \beta_{1} - 26136385344\)\()/20736\) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).
| \(n\) |
\(127\) |
\(133\) |
\(257\) |
| \(\chi(n)\) |
\(-1\) |
\(-1\) |
\(1\) |
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 \( T_{5}^{4} + 682272 T_{5}^{2} + 84192825600 \) acting on \(S_{9}^{\mathrm{new}}(384, [\chi])\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{8} \) |
| $3$ |
\( ( -2187 + T^{2} )^{4} \) |
| $5$ |
\( ( 84192825600 + 682272 T^{2} + T^{4} )^{2} \) |
| $7$ |
\( ( 3746736 + T^{2} )^{4} \) |
| $11$ |
\( ( 84020850951158016 - 612159840 T^{2} + T^{4} )^{2} \) |
| $13$ |
\( ( 9971438023839744 + 1922702976 T^{2} + T^{4} )^{2} \) |
| $17$ |
\( ( -7909979324 - 23260 T + T^{2} )^{4} \) |
| $19$ |
\( ( 456063477443078400 - 2542536288 T^{2} + T^{4} )^{2} \) |
| $23$ |
\( ( \)\(93\!\cdots\!84\)\( + 237198222720 T^{2} + T^{4} )^{2} \) |
| $29$ |
\( ( \)\(27\!\cdots\!00\)\( + 1490396210208 T^{2} + T^{4} )^{2} \) |
| $31$ |
\( ( \)\(15\!\cdots\!00\)\( + 2886948519264 T^{2} + T^{4} )^{2} \) |
| $37$ |
\( ( \)\(57\!\cdots\!56\)\( + 7501371886080 T^{2} + T^{4} )^{2} \) |
| $41$ |
\( ( -17633303147132 - 744644 T + T^{2} )^{4} \) |
| $43$ |
\( ( \)\(30\!\cdots\!04\)\( - 12049696420704 T^{2} + T^{4} )^{2} \) |
| $47$ |
\( ( \)\(19\!\cdots\!84\)\( + 91745074569600 T^{2} + T^{4} )^{2} \) |
| $53$ |
\( ( \)\(14\!\cdots\!44\)\( + 88167062209824 T^{2} + T^{4} )^{2} \) |
| $59$ |
\( ( \)\(22\!\cdots\!24\)\( - 389367404122464 T^{2} + T^{4} )^{2} \) |
| $61$ |
\( ( \)\(66\!\cdots\!56\)\( + 463016972620800 T^{2} + T^{4} )^{2} \) |
| $67$ |
\( ( \)\(62\!\cdots\!36\)\( - 50799407453280 T^{2} + T^{4} )^{2} \) |
| $71$ |
\( ( \)\(13\!\cdots\!24\)\( + 908938263991680 T^{2} + T^{4} )^{2} \) |
| $73$ |
\( ( 323128538436100 - 47847164 T + T^{2} )^{4} \) |
| $79$ |
\( ( \)\(12\!\cdots\!00\)\( + 1858332724601184 T^{2} + T^{4} )^{2} \) |
| $83$ |
\( ( \)\(27\!\cdots\!00\)\( - 1499067578583648 T^{2} + T^{4} )^{2} \) |
| $89$ |
\( ( 2172311553439300 - 105718108 T + T^{2} )^{4} \) |
| $97$ |
\( ( -10698595013335484 - 7842140 T + T^{2} )^{4} \) |
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