Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) 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 \(x^{9} - 4 x^{8} - 46 x^{7} + 145 x^{6} + 680 x^{5} - 1501 x^{4} - 3203 x^{3} + 4784 x^{2} + 3584 x - 4096\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \((\)\( -6901 \nu^{8} - 12396 \nu^{7} + 501446 \nu^{6} + 728955 \nu^{5} - 11530440 \nu^{4} - 12738943 \nu^{3} + 91159455 \nu^{2} + 52058000 \nu - 149293824 \)\()/8071424\) |
| \(\beta_{3}\) | \(=\) | \((\)\( -3843 \nu^{8} + 11372 \nu^{7} + 195178 \nu^{6} - 384275 \nu^{5} - 3297016 \nu^{4} + 4265751 \nu^{3} + 16793257 \nu^{2} - 22791952 \nu - 6050816 \)\()/4035712\) |
| \(\beta_{4}\) | \(=\) | \((\)\( -595 \nu^{8} + 2737 \nu^{7} + 19422 \nu^{6} - 72705 \nu^{5} - 171803 \nu^{4} + 485407 \nu^{3} + 252488 \nu^{2} - 1604391 \nu - 646464 \)\()/504464\) |
| \(\beta_{5}\) | \(=\) | \((\)\( -23843 \nu^{8} + 103372 \nu^{7} + 1059978 \nu^{6} - 3803155 \nu^{5} - 14845688 \nu^{4} + 38793527 \nu^{3} + 67400873 \nu^{2} - 97179408 \nu - 92826880 \)\()/8071424\) |
| \(\beta_{6}\) | \(=\) | \((\)\( -32107 \nu^{8} + 90940 \nu^{7} + 1548474 \nu^{6} - 2570427 \nu^{5} - 23878504 \nu^{4} + 12527327 \nu^{3} + 111900273 \nu^{2} + 12391680 \nu - 122506496 \)\()/8071424\) |
| \(\beta_{7}\) | \(=\) | \((\)\( -53515 \nu^{8} + 88524 \nu^{7} + 2837370 \nu^{6} - 2210427 \nu^{5} - 46218904 \nu^{4} + 3078399 \nu^{3} + 217315329 \nu^{2} + 51740944 \nu - 202361088 \)\()/8071424\) |
| \(\beta_{8}\) | \(=\) | \((\)\( 27975 \nu^{8} - 97156 \nu^{7} - 1304226 \nu^{6} + 3186791 \nu^{5} + 19362096 \nu^{4} - 25660427 \nu^{3} - 85614861 \nu^{2} + 34322440 \nu + 59238144 \)\()/4035712\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{8} + \beta_{6} + \beta_{5} + 2 \beta_{1} + 12\) |
| \(\nu^{3}\) | \(=\) | \(3 \beta_{8} + 3 \beta_{6} + 2 \beta_{5} + 4 \beta_{3} - \beta_{2} + 23 \beta_{1} + 12\) |
| \(\nu^{4}\) | \(=\) | \(32 \beta_{8} + 37 \beta_{6} + 27 \beta_{5} - 5 \beta_{4} + 8 \beta_{3} - 8 \beta_{2} + 77 \beta_{1} + 260\) |
| \(\nu^{5}\) | \(=\) | \(128 \beta_{8} - 10 \beta_{7} + 166 \beta_{6} + 75 \beta_{5} - 18 \beta_{4} + 138 \beta_{3} - 45 \beta_{2} + 636 \beta_{1} + 604\) |
| \(\nu^{6}\) | \(=\) | \(1004 \beta_{8} - 46 \beta_{7} + 1315 \beta_{6} + 748 \beta_{5} - 299 \beta_{4} + 490 \beta_{3} - 339 \beta_{2} + 2746 \beta_{1} + 6752\) |
| \(\nu^{7}\) | \(=\) | \(4677 \beta_{8} - 644 \beta_{7} + 6896 \beta_{6} + 2684 \beta_{5} - 1299 \beta_{4} + 4588 \beta_{3} - 1762 \beta_{2} + 19224 \beta_{1} + 23360\) |
| \(\nu^{8}\) | \(=\) | \(32278 \beta_{8} - 3242 \beta_{7} + 46550 \beta_{6} + 21858 \beta_{5} - 12940 \beta_{4} + 21190 \beta_{3} - 12178 \beta_{2} + 95033 \beta_{1} + 192772\) |
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
|
| \(13\) |
\(1\) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{9} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(169))\).
| $p$ |
$F_p(T)$ |
| $2$ |
\( -344 + 1464 T + 5898 T^{2} - 1257 T^{3} - 2310 T^{4} + 486 T^{5} + 205 T^{6} - 42 T^{7} - 5 T^{8} + T^{9} \) |
| $3$ |
\( -143717 - 375830 T + 472696 T^{2} - 97525 T^{3} - 24453 T^{4} + 7245 T^{5} + 359 T^{6} - 154 T^{7} - T^{8} + T^{9} \) |
| $5$ |
\( -3059376152 - 79067576 T + 128609278 T^{2} + 1389429 T^{3} - 1945112 T^{4} + 11193 T^{5} + 12686 T^{6} - 266 T^{7} - 30 T^{8} + T^{9} \) |
| $7$ |
\( 27715644424 - 12058090668 T + 1223096892 T^{2} + 80750883 T^{3} - 15288900 T^{4} + 47784 T^{5} + 49156 T^{6} - 1023 T^{7} - 38 T^{8} + T^{9} \) |
| $11$ |
\( -276199564381 - 179311762994 T - 26882546428 T^{2} + 738228615 T^{3} + 167003035 T^{4} - 5844107 T^{5} - 165265 T^{6} + 10834 T^{7} - 181 T^{8} + T^{9} \) |
| $13$ |
\( T^{9} \) |
| $17$ |
\( 5572934105557 - 13634726454440 T - 8401319709328 T^{2} - 224267154861 T^{3} + 6767325927 T^{4} + 147895979 T^{5} - 1146115 T^{6} - 23436 T^{7} + 55 T^{8} + T^{9} \) |
| $19$ |
\( -865058822963419 + 17958379163746 T + 7470263773658 T^{2} - 8601436031 T^{3} - 7019204857 T^{4} + 20615523 T^{5} + 2116695 T^{6} - 11292 T^{7} - 161 T^{8} + T^{9} \) |
| $23$ |
\( 68005016808744392 + 1677405262433836 T - 96771175504120 T^{2} - 1742693252577 T^{3} + 41479717140 T^{4} + 357001151 T^{5} - 5407664 T^{6} - 29131 T^{7} + 204 T^{8} + T^{9} \) |
| $29$ |
\( 56071014835079896 + 5762250822516012 T + 56291593140298 T^{2} - 6977746485675 T^{3} - 132290324688 T^{4} + 976035630 T^{5} + 12568120 T^{6} - 48535 T^{7} - 280 T^{8} + T^{9} \) |
| $31$ |
\( 3096277304854445656 - 65749435529097852 T - 1985161101259952 T^{2} + 25565731686363 T^{3} + 228326104668 T^{4} - 3609315492 T^{5} + 1130104 T^{6} + 148229 T^{7} - 706 T^{8} + T^{9} \) |
| $37$ |
\( \)\(31\!\cdots\!48\)\( - 8896933236559731448 T + 45875333850181822 T^{2} + 594222222755749 T^{3} - 6293535923688 T^{4} + 7976918585 T^{5} + 87939738 T^{6} - 232098 T^{7} - 298 T^{8} + T^{9} \) |
| $41$ |
\( -61873581397572169253 + 1619792110119036588 T + 1616425926085138 T^{2} - 421646173565359 T^{3} + 4652224924458 T^{4} - 14140274357 T^{5} - 56821158 T^{6} + 494135 T^{7} - 1201 T^{8} + T^{9} \) |
| $43$ |
\( -\)\(35\!\cdots\!77\)\( - 4678442324800455822 T + 70886710557000845 T^{2} + 1406432909501023 T^{3} + 5706993788145 T^{4} - 14526437912 T^{5} - 136823024 T^{6} - 153079 T^{7} + 533 T^{8} + T^{9} \) |
| $47$ |
\( -11464353104027851384 + 1329362167362062820 T - 32303753105700258 T^{2} - 49762716955757 T^{3} + 6545051933164 T^{4} - 53107853183 T^{5} + 127738074 T^{6} + 136612 T^{7} - 956 T^{8} + T^{9} \) |
| $53$ |
\( -\)\(12\!\cdots\!76\)\( + \)\(24\!\cdots\!24\)\( T - 412869644476868336 T^{2} - 9184309039838179 T^{3} + 15501811028404 T^{4} + 112216880490 T^{5} - 123482118 T^{6} - 566375 T^{7} + 278 T^{8} + T^{9} \) |
| $59$ |
\( -\)\(27\!\cdots\!23\)\( + \)\(10\!\cdots\!24\)\( T - 1114668162573142982 T^{2} + 2587489249146439 T^{3} + 30104186322184 T^{4} - 204479067637 T^{5} + 372100382 T^{6} + 252933 T^{7} - 1377 T^{8} + T^{9} \) |
| $61$ |
\( -\)\(27\!\cdots\!32\)\( - \)\(11\!\cdots\!08\)\( T - 13633889619625467612 T^{2} - 42822229764922651 T^{3} + 89968446968294 T^{4} + 403231012325 T^{5} - 187548706 T^{6} - 1131006 T^{7} + 136 T^{8} + T^{9} \) |
| $67$ |
\( -\)\(32\!\cdots\!99\)\( + \)\(52\!\cdots\!26\)\( T - 10821001517180519169 T^{2} - 92063039319250813 T^{3} + 233825752125911 T^{4} + 495065391476 T^{5} - 890936272 T^{6} - 1141335 T^{7} + 931 T^{8} + T^{9} \) |
| $71$ |
\( -\)\(44\!\cdots\!72\)\( + \)\(19\!\cdots\!12\)\( T + 49455590099883069580 T^{2} + 78182065417258095 T^{3} - 405356603728390 T^{4} - 434754292148 T^{5} + 1448605362 T^{6} + 308287 T^{7} - 2046 T^{8} + T^{9} \) |
| $73$ |
\( \)\(31\!\cdots\!21\)\( - \)\(15\!\cdots\!50\)\( T - 17310298486778544639 T^{2} - 32424645086193109 T^{3} + 181643944920255 T^{4} + 609692221608 T^{5} - 173876306 T^{6} - 1609687 T^{7} + 45 T^{8} + T^{9} \) |
| $79$ |
\( \)\(63\!\cdots\!88\)\( + \)\(79\!\cdots\!44\)\( T + 556477632229543614 T^{2} - 21981799979683177 T^{3} - 36065298040498 T^{4} + 234613432336 T^{5} + 282452944 T^{6} - 1032469 T^{7} - 412 T^{8} + T^{9} \) |
| $83$ |
\( -\)\(14\!\cdots\!61\)\( - \)\(26\!\cdots\!46\)\( T + \)\(47\!\cdots\!03\)\( T^{2} - 2443290848701223317 T^{3} + 5791016389917361 T^{4} - 6314534308340 T^{5} + 1123208654 T^{6} + 4107377 T^{7} - 3709 T^{8} + T^{9} \) |
| $89$ |
\( -\)\(43\!\cdots\!23\)\( + \)\(55\!\cdots\!94\)\( T - \)\(27\!\cdots\!79\)\( T^{2} + 599122717986710443 T^{3} - 259318680361049 T^{4} - 1289685028236 T^{5} + 2134074534 T^{6} - 271263 T^{7} - 1663 T^{8} + T^{9} \) |
| $97$ |
\( -\)\(20\!\cdots\!89\)\( + \)\(11\!\cdots\!26\)\( T + \)\(14\!\cdots\!87\)\( T^{2} - 1296433223643319245 T^{3} + 484271211396037 T^{4} + 3490144221316 T^{5} - 1955124078 T^{6} - 3079171 T^{7} + 1087 T^{8} + T^{9} \) |
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