Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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 \(x^{4} - 2 x^{3} - 2411 x^{2} + 2412 x + 1444802\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \((\)\( -2 \nu^{3} + 3 \nu^{2} + 2419 \nu - 1210 \)\()/687\) |
| \(\beta_{2}\) | \(=\) | \((\)\( -4 \nu^{3} + 6 \nu^{2} + 14456 \nu - 7229 \)\()/229\) |
| \(\beta_{3}\) | \(=\) | \((\)\( \nu^{3} + 1029 \nu^{2} - 2240 \nu - 1242178 \)\()/687\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(\beta_{2} - 6 \beta_{1} + 21\)\()/42\) |
| \(\nu^{2}\) | \(=\) | \((\)\(28 \beta_{3} + \beta_{2} + 8 \beta_{1} + 50673\)\()/42\) |
| \(\nu^{3}\) | \(=\) | \((\)\(6 \beta_{3} + 173 \beta_{2} - 3096 \beta_{1} + 10857\)\()/6\) |
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.
| \( p \) |
Sign
|
| \(2\) |
\(1\) |
| \(7\) |
\(1\) |
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 47322 T_{3}^{2} + 387774864 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( ( 16 + T )^{4} \) |
| $3$ |
\( 387774864 - 47322 T^{2} + T^{4} \) |
| $5$ |
\( 4951092496 - 4389322 T^{2} + T^{4} \) |
| $7$ |
\( T^{4} \) |
| $11$ |
\( ( -2902464824 - 4774 T + T^{2} )^{2} \) |
| $13$ |
\( \)\(20\!\cdots\!96\)\( - 28441135314 T^{2} + T^{4} \) |
| $17$ |
\( \)\(32\!\cdots\!76\)\( - 364177565476 T^{2} + T^{4} \) |
| $19$ |
\( \)\(42\!\cdots\!44\)\( - 760834631722 T^{2} + T^{4} \) |
| $23$ |
\( ( -514307412752 + 1418368 T + T^{2} )^{2} \) |
| $29$ |
\( ( 14857994423056 - 8334986 T + T^{2} )^{2} \) |
| $31$ |
\( \)\(12\!\cdots\!84\)\( - 74580188894856 T^{2} + T^{4} \) |
| $37$ |
\( ( -172221385813328 - 1613186 T + T^{2} )^{2} \) |
| $41$ |
\( \)\(15\!\cdots\!36\)\( - 524653229945812 T^{2} + T^{4} \) |
| $43$ |
\( ( 187841499446728 + 38240102 T + T^{2} )^{2} \) |
| $47$ |
\( \)\(26\!\cdots\!56\)\( - 2630861410847464 T^{2} + T^{4} \) |
| $53$ |
\( ( -7190189657832788 - 6854056 T + T^{2} )^{2} \) |
| $59$ |
\( \)\(12\!\cdots\!24\)\( - 6371386772799514 T^{2} + T^{4} \) |
| $61$ |
\( \)\(24\!\cdots\!96\)\( - 13170727678904314 T^{2} + T^{4} \) |
| $67$ |
\( ( 31073956702009344 + 407210388 T + T^{2} )^{2} \) |
| $71$ |
\( ( 107691169551243264 + 662334372 T + T^{2} )^{2} \) |
| $73$ |
\( \)\(45\!\cdots\!24\)\( - 201160054502509972 T^{2} + T^{4} \) |
| $79$ |
\( ( 24380563916903968 + 334205588 T + T^{2} )^{2} \) |
| $83$ |
\( \)\(89\!\cdots\!96\)\( - 280763948299304794 T^{2} + T^{4} \) |
| $89$ |
\( \)\(19\!\cdots\!84\)\( - 989995707342536644 T^{2} + T^{4} \) |
| $97$ |
\( \)\(62\!\cdots\!56\)\( - 1576090761528077764 T^{2} + T^{4} \) |
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