Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{16}\) 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 \(x^{17} - 5 x^{16} - 14 x^{15} + 102 x^{14} + 26 x^{13} - 792 x^{12} + 474 x^{11} + 2887 x^{10} - 3021 x^{9} - 4835 x^{8} + 6673 x^{7} + 2880 x^{6} - 5373 x^{5} - 164 x^{4} + 1075 x^{3} + 75 x^{2} - 41 x - 4\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \( \nu^{2} - 3 \) |
| \(\beta_{3}\) | \(=\) | \( \nu^{3} - 5 \nu \) |
| \(\beta_{4}\) | \(=\) | \((\)\(6881 \nu^{16} + 3046496 \nu^{15} - 12143208 \nu^{14} - 53358172 \nu^{13} + 248098852 \nu^{12} + 302832280 \nu^{11} - 1946294566 \nu^{10} - 358261227 \nu^{9} + 7225993314 \nu^{8} - 2198561585 \nu^{7} - 12507143628 \nu^{6} + 7020646464 \nu^{5} + 8001174573 \nu^{4} - 5454831045 \nu^{3} - 528604566 \nu^{2} + 464407341 \nu - 349096\)\()/13854332\) |
| \(\beta_{5}\) | \(=\) | \((\)\(129907 \nu^{16} - 673231 \nu^{15} - 842951 \nu^{14} + 11316825 \nu^{13} - 15879081 \nu^{12} - 58049485 \nu^{11} + 202090863 \nu^{10} + 16498676 \nu^{9} - 839916930 \nu^{8} + 693906061 \nu^{7} + 1360000699 \nu^{6} - 1874089153 \nu^{5} - 446520802 \nu^{4} + 1367630449 \nu^{3} - 385718169 \nu^{2} - 18197482 \nu + 33470372\)\()/13854332\) |
| \(\beta_{6}\) | \(=\) | \((\)\(342344 \nu^{16} - 1539407 \nu^{15} - 4675055 \nu^{14} + 29695925 \nu^{13} + 6728395 \nu^{12} - 208650675 \nu^{11} + 175130105 \nu^{10} + 610889171 \nu^{9} - 1046385666 \nu^{8} - 440334094 \nu^{7} + 2146286917 \nu^{6} - 1020314197 \nu^{5} - 1394056621 \nu^{4} + 1432645940 \nu^{3} + 26091993 \nu^{2} - 345190511 \nu - 16651472\)\()/13854332\) |
| \(\beta_{7}\) | \(=\) | \((\)\(954477 \nu^{16} - 6048190 \nu^{15} - 9436362 \nu^{14} + 122225562 \nu^{13} - 56644828 \nu^{12} - 933243792 \nu^{11} + 1102309566 \nu^{10} + 3290717187 \nu^{9} - 5364723646 \nu^{8} - 5084510469 \nu^{7} + 10918796292 \nu^{6} + 2175158294 \nu^{5} - 8563127797 \nu^{4} + 550576315 \nu^{3} + 1738656858 \nu^{2} + 84175177 \nu - 60860676\)\()/13854332\) |
| \(\beta_{8}\) | \(=\) | \((\)\(1423805 \nu^{16} - 5123424 \nu^{15} - 25701018 \nu^{14} + 104173792 \nu^{13} + 157565816 \nu^{12} - 803930404 \nu^{11} - 296651690 \nu^{10} + 2889057543 \nu^{9} - 534122266 \nu^{8} - 4627216617 \nu^{7} + 2410932626 \nu^{6} + 2127467628 \nu^{5} - 2034349445 \nu^{4} + 787373203 \nu^{3} + 301820608 \nu^{2} - 187476637 \nu - 38280072\)\()/13854332\) |
| \(\beta_{9}\) | \(=\) | \((\)\(-1624725 \nu^{16} + 8295022 \nu^{15} + 22421616 \nu^{14} - 168218710 \nu^{13} - 38246072 \nu^{12} + 1292185188 \nu^{11} - 770532978 \nu^{10} - 4606979421 \nu^{9} + 4723757470 \nu^{8} + 7287976453 \nu^{7} - 9907626526 \nu^{6} - 3380595882 \nu^{5} + 7039828559 \nu^{4} - 765969227 \nu^{3} - 752233888 \nu^{2} + 32793065 \nu - 16113024\)\()/13854332\) |
| \(\beta_{10}\) | \(=\) | \((\)\(-2320607 \nu^{16} + 13871476 \nu^{15} + 23761238 \nu^{14} - 276884452 \nu^{13} + 119779992 \nu^{12} + 2072568454 \nu^{11} - 2524139110 \nu^{10} - 7035651475 \nu^{9} + 12362382076 \nu^{8} + 9816264405 \nu^{7} - 24980012280 \nu^{6} - 1752525352 \nu^{5} + 18977545907 \nu^{4} - 3709594629 \nu^{3} - 3249870492 \nu^{2} + 334787557 \nu + 100990532\)\()/13854332\) |
| \(\beta_{11}\) | \(=\) | \((\)\(1177751 \nu^{16} - 4876786 \nu^{15} - 19898736 \nu^{14} + 100486682 \nu^{13} + 101420316 \nu^{12} - 791693437 \nu^{11} - 6972284 \nu^{10} + 2952339580 \nu^{9} - 1397185719 \nu^{8} - 5145466896 \nu^{7} + 3888315285 \nu^{6} + 3337708828 \nu^{5} - 3336736615 \nu^{4} - 213355823 \nu^{3} + 708875774 \nu^{2} - 40686922 \nu - 38449256\)\()/6927166\) |
| \(\beta_{12}\) | \(=\) | \((\)\(1430016 \nu^{16} - 7943686 \nu^{15} - 16588569 \nu^{14} + 158865051 \nu^{13} - 33046398 \nu^{12} - 1193366371 \nu^{11} + 1225665635 \nu^{10} + 4081521360 \nu^{9} - 6331983519 \nu^{8} - 5820649228 \nu^{7} + 12956859425 \nu^{6} + 1355364934 \nu^{5} - 9771397341 \nu^{4} + 1891397865 \nu^{3} + 1628902717 \nu^{2} - 173708857 \nu - 53459238\)\()/6927166\) |
| \(\beta_{13}\) | \(=\) | \((\)\(2212232 \nu^{16} - 10666320 \nu^{15} - 31922034 \nu^{14} + 216378870 \nu^{13} + 78707123 \nu^{12} - 1664310761 \nu^{11} + 862799245 \nu^{10} + 5955878758 \nu^{9} - 5873224414 \nu^{8} - 9524615238 \nu^{7} + 12943391397 \nu^{6} + 4629682174 \nu^{5} - 9935315824 \nu^{4} + 917431766 \nu^{3} + 1598499815 \nu^{2} - 167916467 \nu - 54985350\)\()/6927166\) |
| \(\beta_{14}\) | \(=\) | \((\)\(-4464588 \nu^{16} + 22316577 \nu^{15} + 62453009 \nu^{14} - 453309389 \nu^{13} - 117535949 \nu^{12} + 3492081761 \nu^{11} - 2076095689 \nu^{10} - 12518312453 \nu^{9} + 13167816710 \nu^{8} + 20045296036 \nu^{7} - 28679333929 \nu^{6} - 9681635945 \nu^{5} + 22306957823 \nu^{4} - 2117117842 \nu^{3} - 3979676103 \nu^{2} + 424686951 \nu + 161461920\)\()/13854332\) |
| \(\beta_{15}\) | \(=\) | \((\)\(-2258564 \nu^{16} + 11405411 \nu^{15} + 30768077 \nu^{14} - 230738128 \nu^{13} - 42311710 \nu^{12} + 1766179249 \nu^{11} - 1184928143 \nu^{10} - 6260382996 \nu^{9} + 7147568361 \nu^{8} + 9783571190 \nu^{7} - 15251807704 \nu^{6} - 4310704213 \nu^{5} + 11481073899 \nu^{4} - 1222783277 \nu^{3} - 1709938046 \nu^{2} + 45368196 \nu + 41232606\)\()/6927166\) |
| \(\beta_{16}\) | \(=\) | \((\)\(-5588767 \nu^{16} + 28694781 \nu^{15} + 74918371 \nu^{14} - 580575835 \nu^{13} - 79913865 \nu^{12} + 4445515351 \nu^{11} - 3127113413 \nu^{10} - 15773479316 \nu^{9} + 18433074534 \nu^{8} + 24731166679 \nu^{7} - 39180778403 \nu^{6} - 11084633469 \nu^{5} + 29732956670 \nu^{4} - 2992752301 \nu^{3} - 4755103951 \nu^{2} + 276723354 \nu + 155763180\)\()/13854332\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{2} + 3\) |
| \(\nu^{3}\) | \(=\) | \(\beta_{3} + 5 \beta_{1}\) |
| \(\nu^{4}\) | \(=\) | \(-\beta_{13} - \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 6 \beta_{2} + 15\) |
| \(\nu^{5}\) | \(=\) | \(-\beta_{16} - \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} + \beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + 9 \beta_{3} + 28 \beta_{1} + 2\) |
| \(\nu^{6}\) | \(=\) | \(-\beta_{16} - 2 \beta_{14} - 11 \beta_{13} - 2 \beta_{11} - 8 \beta_{10} - 8 \beta_{9} + \beta_{8} + \beta_{7} + 10 \beta_{6} + 11 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} + 36 \beta_{2} + \beta_{1} + 86\) |
| \(\nu^{7}\) | \(=\) | \(-15 \beta_{16} + \beta_{15} - 14 \beta_{14} - 16 \beta_{13} - 13 \beta_{12} - 14 \beta_{11} - \beta_{10} + 12 \beta_{9} + 13 \beta_{8} - \beta_{7} - 9 \beta_{6} + 14 \beta_{5} + 2 \beta_{4} + 67 \beta_{3} + \beta_{2} + 166 \beta_{1} + 32\) |
| \(\nu^{8}\) | \(=\) | \(-17 \beta_{16} - 3 \beta_{15} - 32 \beta_{14} - 96 \beta_{13} - 7 \beta_{12} - 36 \beta_{11} - 53 \beta_{10} - 47 \beta_{9} + 20 \beta_{8} + 12 \beta_{7} + 77 \beta_{6} + 92 \beta_{5} + 79 \beta_{4} + 82 \beta_{3} + 220 \beta_{2} + 17 \beta_{1} + 538\) |
| \(\nu^{9}\) | \(=\) | \(-160 \beta_{16} + 13 \beta_{15} - 139 \beta_{14} - 179 \beta_{13} - 126 \beta_{12} - 148 \beta_{11} - 18 \beta_{10} + 106 \beta_{9} + 134 \beta_{8} - 14 \beta_{7} - 58 \beta_{6} + 135 \beta_{5} + 35 \beta_{4} + 476 \beta_{3} + 19 \beta_{2} + 1025 \beta_{1} + 352\) |
| \(\nu^{10}\) | \(=\) | \(-202 \beta_{16} - 53 \beta_{15} - 354 \beta_{14} - 785 \beta_{13} - 129 \beta_{12} - 433 \beta_{11} - 341 \beta_{10} - 229 \beta_{9} + 263 \beta_{8} + 101 \beta_{7} + 540 \beta_{6} + 694 \beta_{5} + 582 \beta_{4} + 639 \beta_{3} + 1371 \beta_{2} + 206 \beta_{1} + 3572\) |
| \(\nu^{11}\) | \(=\) | \(-1481 \beta_{16} + 105 \beta_{15} - 1229 \beta_{14} - 1730 \beta_{13} - 1102 \beta_{12} - 1406 \beta_{11} - 216 \beta_{10} + 847 \beta_{9} + 1264 \beta_{8} - 144 \beta_{7} - 316 \beta_{6} + 1126 \beta_{5} + 410 \beta_{4} + 3347 \beta_{3} + 242 \beta_{2} + 6530 \beta_{1} + 3364\) |
| \(\nu^{12}\) | \(=\) | \(-2070 \beta_{16} - 622 \beta_{15} - 3377 \beta_{14} - 6286 \beta_{13} - 1573 \beta_{12} - 4385 \beta_{11} - 2219 \beta_{10} - 844 \beta_{9} + 2848 \beta_{8} + 720 \beta_{7} + 3632 \beta_{6} + 4999 \beta_{5} + 4194 \beta_{4} + 4908 \beta_{3} + 8716 \beta_{2} + 2160 \beta_{1} + 24725\) |
| \(\nu^{13}\) | \(=\) | \(-12736 \beta_{16} + 616 \beta_{15} - 10358 \beta_{14} - 15482 \beta_{13} - 9192 \beta_{12} - 12629 \beta_{11} - 2166 \beta_{10} + 6541 \beta_{9} + 11315 \beta_{8} - 1339 \beta_{7} - 1469 \beta_{6} + 8775 \beta_{5} + 4064 \beta_{4} + 23595 \beta_{3} + 2598 \beta_{2} + 42675 \beta_{1} + 30022\) |
| \(\nu^{14}\) | \(=\) | \(-19563 \beta_{16} - 6149 \beta_{15} - 29869 \beta_{14} - 50019 \beta_{13} - 16060 \beta_{12} - 40482 \beta_{11} - 14795 \beta_{10} - 677 \beta_{9} + 27643 \beta_{8} + 4547 \beta_{7} + 23977 \beta_{6} + 35301 \beta_{5} + 30136 \beta_{4} + 37611 \beta_{3} + 56538 \beta_{2} + 20845 \beta_{1} + 176300\) |
| \(\nu^{15}\) | \(=\) | \(-105024 \beta_{16} + 2128 \beta_{15} - 85333 \beta_{14} - 132384 \beta_{13} - 74733 \beta_{12} - 109542 \beta_{11} - 19696 \beta_{10} + 50118 \beta_{9} + 97739 \beta_{8} - 11906 \beta_{7} - 5116 \beta_{6} + 66169 \beta_{5} + 36934 \beta_{4} + 167666 \beta_{3} + 25381 \beta_{2} + 285081 \beta_{1} + 257719\) |
| \(\nu^{16}\) | \(=\) | \(-175725 \beta_{16} - 55584 \beta_{15} - 252927 \beta_{14} - 397350 \beta_{13} - 149019 \beta_{12} - 353544 \beta_{11} - 101394 \beta_{10} + 30387 \beta_{9} + 250809 \beta_{8} + 25227 \beta_{7} + 157342 \beta_{6} + 247838 \beta_{5} + 217784 \beta_{4} + 288865 \beta_{3} + 374172 \beta_{2} + 190567 \beta_{1} + 1284423\) |
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 does not admit any (nontrivial) inner twists.
| \( p \) |
Sign
|
| \(349\) |
\(-1\) |
This newform can be constructed as the kernel of the linear operator \(T_{2}^{17} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(349))\).
