Properties

Label 4.25.b.b
Level $4$
Weight $25$
Character orbit 4.b
Analytic conductor $14.599$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 25 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.5986860903\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 3 x^{9} + 1157886 x^{8} - 1182314620 x^{7} + 1715110302918 x^{6} - 2255628699630474 x^{5} + 1518357032488149568 x^{4} - 1731829168106243136300 x^{3} + 1460524813513291203716721 x^{2} - 1269695210566625624700394443 x + 1028204700608421275722350402186\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{90}\cdot 3^{8}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( -621 + \beta_{1} ) q^{2} + ( -3 - 15 \beta_{1} + \beta_{2} ) q^{3} + ( -3319485 - 761 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{4} + ( 5675512 - 1492 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{5} + ( 242821904 + 9136 \beta_{1} + 359 \beta_{2} + 10 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} ) q^{6} + ( 4644 + 23219 \beta_{1} + 782 \beta_{2} + 17 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} ) q^{7} + ( -15982548246 - 4002033 \beta_{1} - 5298 \beta_{2} + 1136 \beta_{3} + 123 \beta_{4} - 17 \beta_{5} - 27 \beta_{6} + 6 \beta_{7} - 6 \beta_{8} - 2 \beta_{9} ) q^{8} + ( -93909906177 - 22247043 \beta_{1} - 13877 \beta_{2} + 4351 \beta_{3} + 586 \beta_{4} - 160 \beta_{5} - 92 \beta_{6} + 91 \beta_{8} ) q^{9} +O(q^{10})\) \( q + ( -621 + \beta_{1} ) q^{2} + ( -3 - 15 \beta_{1} + \beta_{2} ) q^{3} + ( -3319485 - 761 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{4} + ( 5675512 - 1492 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{5} + ( 242821904 + 9136 \beta_{1} + 359 \beta_{2} + 10 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} ) q^{6} + ( 4644 + 23219 \beta_{1} + 782 \beta_{2} + 17 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} ) q^{7} + ( -15982548246 - 4002033 \beta_{1} - 5298 \beta_{2} + 1136 \beta_{3} + 123 \beta_{4} - 17 \beta_{5} - 27 \beta_{6} + 6 \beta_{7} - 6 \beta_{8} - 2 \beta_{9} ) q^{8} + ( -93909906177 - 22247043 \beta_{1} - 13877 \beta_{2} + 4351 \beta_{3} + 586 \beta_{4} - 160 \beta_{5} - 92 \beta_{6} + 91 \beta_{8} ) q^{9} + ( -27983823783 + 4757022 \beta_{1} - 41406 \beta_{2} + 2643 \beta_{3} + 545 \beta_{4} - 1036 \beta_{5} - 140 \beta_{6} + 256 \beta_{7} - 252 \beta_{8} + 56 \beta_{9} ) q^{10} + ( -17008825 - 85043963 \beta_{1} + 787367 \beta_{2} + 46494 \beta_{3} - 34 \beta_{4} - 480 \beta_{5} + 384 \beta_{6} - 2022 \beta_{7} + 48 \beta_{8} - 26 \beta_{9} ) q^{11} + ( 1243426111308 + 295845420 \beta_{1} - 1036176 \beta_{2} - 28060 \beta_{3} - 4744 \beta_{4} + 21388 \beta_{5} + 1316 \beta_{6} + 3000 \beta_{7} - 3016 \beta_{8} - 168 \beta_{9} ) q^{12} + ( -2479916133820 - 548034282 \beta_{1} - 439715 \beta_{2} - 495401 \beta_{3} - 4044 \beta_{4} + 617 \beta_{5} + 9352 \beta_{6} + 12582 \beta_{8} ) q^{13} + ( -382951363136 - 16306848 \beta_{1} + 9770146 \beta_{2} - 24212 \beta_{3} + 3730 \beta_{4} - 169024 \beta_{5} + 14400 \beta_{6} + 8370 \beta_{7} - 8142 \beta_{8} - 1664 \beta_{9} ) q^{14} + ( -206407244 - 1031929999 \beta_{1} + 26288802 \beta_{2} + 5435779 \beta_{3} + 106515 \beta_{4} - 31808 \beta_{5} + 32000 \beta_{6} + 4093 \beta_{7} - 96 \beta_{8} + 3 \beta_{9} ) q^{15} + ( -58890153435768 - 18578498556 \beta_{1} - 56505752 \beta_{2} + 5612272 \beta_{3} + 399268 \beta_{4} + 460356 \beta_{5} + 69612 \beta_{6} - 45592 \beta_{7} + 45144 \beta_{8} + 7432 \beta_{9} ) q^{16} + ( -162514173681696 - 51801449771 \beta_{1} - 35727317 \beta_{2} - 12173809 \beta_{3} + 941498 \beta_{4} - 272568 \beta_{5} + 58884 \beta_{6} - 279261 \beta_{8} ) q^{17} + ( -306397703923839 - 107906150151 \beta_{1} + 978441716 \beta_{2} + 24279374 \beta_{3} + 1807610 \beta_{4} - 472760 \beta_{5} + 327752 \beta_{6} - 274944 \beta_{7} + 274664 \beta_{8} + 19632 \beta_{9} ) q^{18} + ( 4395413693 + 21979620023 \beta_{1} + 353333405 \beta_{2} + 105441338 \beta_{3} + 2617466 \beta_{4} - 566816 \beta_{5} + 597632 \beta_{6} + 647566 \beta_{7} - 15408 \beta_{8} + 9842 \beta_{9} ) q^{19} + ( -312139542851002 - 39764708338 \beta_{1} - 3781785224 \beta_{2} + 23526014 \beta_{3} + 687360 \beta_{4} - 8773216 \beta_{5} + 1286880 \beta_{6} - 39360 \beta_{7} + 25920 \beta_{8} - 127680 \beta_{9} ) q^{20} + ( -285052873921908 - 232216022886 \beta_{1} - 178663330 \beta_{2} - 166023754 \beta_{3} + 406292 \beta_{4} + 375544 \beta_{5} + 2640200 \beta_{6} + 1858742 \beta_{8} ) q^{21} + ( 1391673258418864 + 49730686032 \beta_{1} + 16150371025 \beta_{2} - 25420922 \beta_{3} - 12716359 \beta_{4} + 26644096 \beta_{5} + 1526144 \beta_{6} + 2731897 \beta_{7} - 2753415 \beta_{8} - 87808 \beta_{9} ) q^{22} + ( 129347713076 + 646744135589 \beta_{1} + 1324453114 \beta_{2} - 223115249 \beta_{3} + 4457055 \beta_{4} - 13248 \beta_{5} - 441600 \beta_{6} - 9487983 \beta_{7} + 227424 \beta_{8} - 215441 \beta_{9} ) q^{23} + ( 1066932777992096 + 1322677024336 \beta_{1} - 51729638880 \beta_{2} - 62890048 \beta_{3} - 33566256 \beta_{4} + 3183440 \beta_{5} - 9504656 \beta_{6} + 5730848 \beta_{7} - 4811040 \beta_{8} + 1237152 \beta_{9} ) q^{24} + ( 7116674122539975 - 1559066024850 \beta_{1} - 794380550 \beta_{2} + 1439810050 \beta_{3} + 70357500 \beta_{4} - 4898200 \beta_{5} - 25241000 \beta_{6} + 2693250 \beta_{8} ) q^{25} + ( -7443582638367987 - 2839303886354 \beta_{1} + 102443040698 \beta_{2} + 149371079 \beta_{3} - 101409059 \beta_{4} - 159819484 \beta_{5} - 37239132 \beta_{6} - 9591552 \beta_{7} + 9732084 \beta_{8} - 426664 \beta_{9} ) q^{26} + ( 3916350786822 + 19581846086904 \beta_{1} - 53999510814 \beta_{2} - 7493469978 \beta_{3} + 101504166 \beta_{4} + 44514336 \beta_{5} - 41784960 \beta_{6} + 55598034 \beta_{7} - 1364688 \beta_{8} + 2628654 \beta_{9} ) q^{27} + ( 5792089686526760 - 149199178904 \beta_{1} - 125195007456 \beta_{2} + 401178744 \beta_{3} - 546980208 \beta_{4} + 442967720 \beta_{5} - 18815240 \beta_{6} - 56760816 \beta_{7} + 38616336 \beta_{8} - 7302960 \beta_{9} ) q^{28} + ( 156559626135582664 - 6144915734188 \beta_{1} - 4553085337 \beta_{2} - 3292517399 \beta_{3} + 73856592 \beta_{4} + 195499249 \beta_{5} + 27596064 \beta_{6} - 107909928 \beta_{8} ) q^{29} + ( 16845980473408704 + 633783881376 \beta_{1} + 25210932718 \beta_{2} + 471457652 \beta_{3} - 1564391170 \beta_{4} + 26297152 \beta_{5} + 109488320 \beta_{6} - 31039650 \beta_{7} + 31071710 \beta_{8} + 8645760 \beta_{9} ) q^{30} + ( 26473093918840 + 132367433549600 \beta_{1} + 19215359384 \beta_{2} + 3700288376 \beta_{3} + 1912683576 \beta_{4} - 86245312 \beta_{5} + 81179392 \beta_{6} - 86236488 \beta_{7} + 2532960 \beta_{8} - 21836472 \beta_{9} ) q^{31} + ( -201145207132966304 - 68047530833232 \beta_{1} + 832781942752 \beta_{2} + 19164148800 \beta_{3} - 1925509328 \beta_{4} - 1812817488 \beta_{5} + 109401744 \beta_{6} + 272998368 \beta_{7} - 69855456 \beta_{8} + 23910752 \beta_{9} ) q^{32} + ( -311463573663779262 - 378353241918333 \beta_{1} - 253695013979 \beta_{2} - 38294238191 \beta_{3} + 7332493366 \beta_{4} - 1376683792 \beta_{5} + 101414236 \beta_{6} + 622173541 \beta_{8} ) q^{33} + ( -748270056594979844 - 194275534846406 \beta_{1} - 2155629433948 \beta_{2} + 79672845398 \beta_{3} - 2281183278 \beta_{4} + 2959579944 \beta_{5} - 95394264 \beta_{6} + 419464704 \beta_{7} - 422553528 \beta_{8} - 58317840 \beta_{9} ) q^{34} + ( 136118031829124 + 680601085848824 \beta_{1} + 546212977948 \beta_{2} + 74750663836 \beta_{3} + 11147756700 \beta_{4} - 756259392 \beta_{5} + 728044800 \beta_{6} - 733598988 \beta_{7} + 14107296 \beta_{8} + 131687692 \beta_{9} ) q^{35} + ( -481553276773001565 - 332836463198361 \beta_{1} + 5436173159548 \beta_{2} + 94564436575 \beta_{3} - 11405602304 \beta_{4} - 3023991232 \beta_{5} + 686143168 \beta_{6} - 728135040 \beta_{7} - 795357056 \beta_{8} - 9838464 \beta_{9} ) q^{36} + ( 1810736672462033108 - 791341177541490 \beta_{1} - 542733222619 \beta_{2} - 151732058497 \beta_{3} + 13789877412 \beta_{4} + 4863802345 \beta_{5} + 1077853928 \beta_{6} - 1108790802 \beta_{8} ) q^{37} + ( -361131080839574896 - 12948187543440 \beta_{1} - 3962590394229 \beta_{2} + 171277346 \beta_{3} - 30884553725 \beta_{4} - 7236945024 \beta_{5} + 2023066752 \beta_{6} - 1418043837 \beta_{7} + 1425335619 \beta_{8} + 210246400 \beta_{9} ) q^{38} + ( 474189993460956 + 2370974621989269 \beta_{1} - 3596416713310 \beta_{2} - 388833452505 \beta_{3} + 23759802039 \beta_{4} + 1824955392 \beta_{5} - 1640607744 \beta_{6} + 4518451785 \beta_{7} - 92173824 \beta_{8} - 585701961 \beta_{9} ) q^{39} + ( -329796889529944812 - 331917465584642 \beta_{1} - 301448334244 \beta_{2} + 38294950112 \beta_{3} - 37771664810 \beta_{4} + 30265057726 \beta_{5} - 1679107990 \beta_{6} + 1592944844 \beta_{7} + 6625038132 \beta_{8} - 291330116 \beta_{9} ) q^{40} + ( 7508043627864639798 - 496196931678026 \beta_{1} - 292080672158 \beta_{2} + 181092041258 \beta_{3} + 17869476236 \beta_{4} - 19620408648 \beta_{5} - 4758286344 \beta_{6} - 3910928454 \beta_{8} ) q^{41} + ( -3629603687700402492 - 432002808528048 \beta_{1} + 14837928642136 \beta_{2} + 220163133124 \beta_{3} - 35567100116 \beta_{4} - 27947513872 \beta_{5} - 10067183120 \beta_{6} - 55323648 \beta_{7} + 82256560 \beta_{8} - 298830432 \beta_{9} ) q^{42} + ( 519515893509387 + 2597584317682163 \beta_{1} - 4780583988673 \beta_{2} - 1274559020092 \beta_{3} + 8230435396 \beta_{4} + 6324321984 \beta_{5} - 6563823360 \beta_{6} - 6979138804 \beta_{7} + 119750688 \beta_{8} + 1869776116 \beta_{9} ) q^{43} + ( 12082319061532587284 + 1890178780781876 \beta_{1} - 42392246658288 \beta_{2} - 17661759812 \beta_{3} - 45473111480 \beta_{4} - 17698356524 \beta_{5} - 10676825604 \beta_{6} - 8901593592 \beta_{7} - 24035323512 \beta_{8} + 1304055656 \beta_{9} ) q^{44} + ( -8100968049410224596 - 1346286970525614 \beta_{1} - 763004940767 \beta_{2} + 909204021067 \beta_{3} + 39814654060 \beta_{4} + 106204576457 \beta_{5} - 10696314440 \beta_{6} + 23030064010 \beta_{8} ) q^{45} + ( -10605643738132137792 - 414797329426400 \beta_{1} + 68457154665942 \beta_{2} - 937412598684 \beta_{3} + 49089753254 \beta_{4} + 136912779328 \beta_{5} - 1911561280 \beta_{6} + 13906734214 \beta_{7} - 14032276218 \beta_{8} - 841339264 \beta_{9} ) q^{46} + ( -2086785676613712 - 10434006718541594 \beta_{1} + 23087229675236 \beta_{2} + 2479560242026 \beta_{3} - 87739974518 \beta_{4} - 14424373824 \beta_{5} + 13176280320 \beta_{6} - 22859814138 \beta_{7} + 624046752 \beta_{8} - 3766180614 \beta_{9} ) q^{47} + ( 16848274266241396608 + 1749477189690816 \beta_{1} - 103344376947328 \beta_{2} - 976971057920 \beta_{3} + 289890836416 \beta_{4} - 222963641920 \beta_{5} + 25101251392 \beta_{6} + 52349476224 \beta_{7} + 46227533440 \beta_{8} - 1652122752 \beta_{9} ) q^{48} + ( -27497779195701141863 + 14776198499381028 \beta_{1} + 9205401573596 \beta_{2} - 3527926682452 \beta_{3} - 384764900472 \beta_{4} - 432671972960 \beta_{5} + 68030446160 \beta_{6} - 23358907332 \beta_{8} ) q^{49} + ( -29980243200311989275 + 6138758009707475 \beta_{1} + 52654845144200 \beta_{2} + 1838636213900 \beta_{3} + 424486226500 \beta_{4} + 58505109200 \beta_{5} + 90698274000 \beta_{6} - 31237555200 \beta_{7} + 31138544400 \beta_{8} + 5075544800 \beta_{9} ) q^{50} + ( -9287675255095946 - 46438791335413792 \beta_{1} + 8899433354658 \beta_{2} + 11153254031278 \beta_{3} - 404452263954 \beta_{4} - 37333295840 \beta_{5} + 42351506816 \beta_{6} + 105514990666 \beta_{7} - 2509105488 \beta_{8} + 1540176822 \beta_{9} ) q^{51} + ( 142117931460195142406 - 2269956644767538 \beta_{1} + 202258699013752 \beta_{2} - 100476570818 \beta_{3} + 830526834944 \beta_{4} + 316790678304 \beta_{5} + 75107676000 \beta_{6} - 170336281280 \beta_{7} - 42175406016 \beta_{8} - 6388225984 \beta_{9} ) q^{52} + ( -110745529109228498500 + 35327254689118218 \beta_{1} + 23750351279145 \beta_{2} + 5238748113459 \beta_{3} - 695095847588 \beta_{4} + 1300297825481 \beta_{5} - 24283021032 \beta_{6} - 112440038382 \beta_{8} ) q^{53} + ( -320825167507638096288 - 12058136462052384 \beta_{1} - 547511501310834 \beta_{2} - 12758058240300 \beta_{3} + 1866071370366 \beta_{4} - 1144047936384 \beta_{5} - 122859676800 \beta_{6} - 38106776514 \beta_{7} + 39235151166 \beta_{8} - 8549640960 \beta_{9} ) q^{54} + ( -21026466608794620 - 105133797001489805 \beta_{1} + 133502123545390 \beta_{2} - 40077723215 \beta_{3} - 1426505868895 \beta_{4} + 38527818240 \beta_{5} - 39615744000 \beta_{6} - 42750851105 \beta_{7} + 543962880 \beta_{8} + 19541768225 \beta_{9} ) q^{55} + ( 452413976067770454720 + 22957098478707808 \beta_{1} + 621106268333504 \beta_{2} - 10047865959296 \beta_{3} + 1447732823392 \beta_{4} + 1060673714784 \beta_{5} - 189299082720 \beta_{6} + 237746533824 \beta_{7} + 38016124992 \beta_{8} + 30693040320 \beta_{9} ) q^{56} + ( -119593205866227697290 + 85477875118594065 \beta_{1} + 60310086377895 \beta_{2} + 23979057121035 \beta_{3} - 1179808881102 \beta_{4} - 3489269222448 \beta_{5} - 260371083852 \beta_{6} + 303936915303 \beta_{8} ) q^{57} + ( -197954781567915448743 + 152916722233004382 \beta_{1} - 923532492819006 \beta_{2} + 13085690962323 \beta_{3} - 797329568287 \beta_{4} + 816411721460 \beta_{5} - 96390561164 \beta_{6} + 244055826688 \beta_{7} - 244026724860 \beta_{8} - 6657166792 \beta_{9} ) q^{58} + ( -13541489834397441 - 67709253708656509 \beta_{1} - 478795190594021 \beta_{2} - 44854011382024 \beta_{3} - 2001715836808 \beta_{4} + 217907579136 \beta_{5} - 247385478144 \beta_{6} - 552380686008 \beta_{7} + 14738949504 \beta_{8} - 76481159496 \beta_{9} ) q^{59} + ( 1402119440720424171544 + 70315997576077784 \beta_{1} + 326943709269728 \beta_{2} - 23314258296504 \beta_{3} - 1534203078160 \beta_{4} - 1920321152872 \beta_{5} - 274681254200 \beta_{6} + 212918177392 \beta_{7} - 144276213904 \beta_{8} - 33791415888 \beta_{9} ) q^{60} + ( 751763261552655329204 + 114045607796582958 \beta_{1} + 73087713721909 \beta_{2} - 559457679569 \beta_{3} - 3086334513564 \beta_{4} + 8115398545145 \beta_{5} + 380334421096 \beta_{6} + 228807334926 \beta_{8} ) q^{61} + ( -2169744606191334450048 - 82300090282098048 \beta_{1} + 305090803907432 \beta_{2} - 113891505597840 \beta_{3} - 3703374015320 \beta_{4} + 4997545070080 \beta_{5} - 3628789248 \beta_{6} - 74764849752 \beta_{7} + 67250801064 \beta_{8} + 45013642240 \beta_{9} ) q^{62} + ( 69850640846769132 + 349257276081347655 \beta_{1} + 870173567826318 \beta_{2} - 28687400845467 \beta_{3} + 4433405401173 \beta_{4} + 174802496064 \beta_{5} - 122800231680 \beta_{6} + 967604388891 \beta_{7} - 26001132192 \beta_{8} + 141777251301 \beta_{9} ) q^{63} + ( 2763029956102979444864 - 100768402022843840 \beta_{1} - 2034471740325248 \beta_{2} + 44994166283008 \beta_{3} - 6525367111616 \beta_{4} - 4692623241664 \beta_{5} + 564674735296 \beta_{6} - 1377043963264 \beta_{7} - 129988275840 \beta_{8} - 106381597568 \beta_{9} ) q^{64} + ( -225950956986952522800 - 678551870735624950 \beta_{1} - 451239575238850 \beta_{2} - 59934906514250 \beta_{3} + 14727058853620 \beta_{4} - 12344611476120 \beta_{5} - 613882757880 \beta_{6} - 1538799438330 \beta_{8} ) q^{65} + ( -6009093143379568367070 - 547740663582962520 \beta_{1} + 7255621167484844 \beta_{2} + 459575271219746 \beta_{3} + 167988415958 \beta_{4} - 5285487342344 \beta_{5} - 346158453256 \beta_{6} - 1204241432064 \beta_{7} + 1204117742168 \beta_{8} - 27693702960 \beta_{9} ) q^{66} + ( 240207117794030241 + 1201054746447667003 \beta_{1} - 3524294722153775 \beta_{2} + 150578300717482 \beta_{3} + 19089996492650 \beta_{4} - 1198929707424 \beta_{5} + 1258378704000 \beta_{6} + 1346572519294 \beta_{7} - 29724498288 \beta_{8} - 78327259006 \beta_{9} ) q^{67} + ( 4422806637375930969798 - 596997275763253298 \beta_{1} - 6474551153055240 \beta_{2} + 185049363082430 \beta_{3} - 4880899192320 \beta_{4} + 8612853927232 \beta_{5} + 732244427712 \beta_{6} + 2221989400704 \beta_{7} + 2479385345664 \beta_{8} + 395432808064 \beta_{9} ) q^{68} + ( -352577322789316719516 - 1579945730919589170 \beta_{1} - 1067602327263238 \beta_{2} - 203060432230078 \beta_{3} + 29763560456252 \beta_{4} + 6037619242120 \beta_{5} + 779068905944 \beta_{6} - 522311434078 \beta_{8} ) q^{69} + ( -11157549155289365275584 - 423815515162490816 \beta_{1} + 4828242499843332 \beta_{2} - 594157052084712 \beta_{3} - 33050255616220 \beta_{4} - 14380570456832 \beta_{5} + 4580960756480 \beta_{6} + 1507965956260 \beta_{7} - 1459854677340 \beta_{8} - 100162915840 \beta_{9} ) q^{70} + ( 280980253313395100 + 1404926742020542503 \beta_{1} + 11852969289741822 \beta_{2} + 122348178095477 \beta_{3} + 24467409993445 \beta_{4} - 1961130196032 \beta_{5} + 1706284055808 \beta_{6} - 5123830476501 \beta_{7} + 127423070112 \beta_{8} - 312887181611 \beta_{9} ) q^{71} + ( 15083690638118060412330 + 78630842463524655 \beta_{1} + 1066191060913550 \beta_{2} + 122595554441840 \beta_{3} - 31792372556005 \beta_{4} + 20422381689871 \beta_{5} + 1116756951365 \beta_{6} - 2880143809722 \beta_{7} - 6171391292230 \beta_{8} - 249110973378 \beta_{9} ) q^{72} + ( 5150915713327505938904 + 11261021344061649 \beta_{1} - 49552346202697 \beta_{2} - 329844860893189 \beta_{3} - 11942081348814 \beta_{4} + 18385202373184 \beta_{5} + 7064585130932 \beta_{6} + 7870953559527 \beta_{8} ) q^{73} + ( -14097140537845838157027 + 1320610492558713966 \beta_{1} - 6537078571694438 \beta_{2} + 1079031203720215 \beta_{3} - 19725969919411 \beta_{4} + 2879662745572 \beta_{5} - 4246433628828 \beta_{6} + 4051958494464 \beta_{7} - 4037062195788 \beta_{8} - 67382424104 \beta_{9} ) q^{74} + ( 221081732208896175 + 1105400872453268175 \beta_{1} - 28455313867794525 \beta_{2} - 713894539731300 \beta_{3} - 5584250854500 \beta_{4} + 4310335185600 \beta_{5} - 4235695968000 \beta_{6} + 546112542900 \beta_{7} - 37319608800 \beta_{8} + 1046190765900 \beta_{9} ) q^{75} + ( 27419122818687561024284 + 665997821779218044 \beta_{1} + 18522163291464368 \beta_{2} - 487471860945964 \beta_{3} - 20451126230632 \beta_{4} - 36496897649060 \beta_{5} - 3597819526252 \beta_{6} + 7106859546328 \beta_{7} + 3480086109528 \beta_{8} - 1157788838920 \beta_{9} ) q^{76} + ( -1974176768341021363724 + 1381880858975801734 \beta_{1} + 1093263204163042 \beta_{2} + 1117654575201098 \beta_{3} + 1695936889900 \beta_{4} - 68728247023832 \beta_{5} - 17189891522760 \beta_{6} - 766354538070 \beta_{8} ) q^{77} + ( -38853497639246278009280 - 1464142186640581216 \beta_{1} - 45930481200665042 \beta_{2} - 1733146655334796 \beta_{3} + 65756307589054 \beta_{4} + 54306893645376 \beta_{5} - 15205218552384 \beta_{6} - 9981313377122 \beta_{7} + 9660237349150 \beta_{8} + 995080460928 \beta_{9} ) q^{78} + ( -1056322906438979768 - 5281698429797692458 \beta_{1} + 48560119204670460 \beta_{2} - 1107453256385358 \beta_{3} - 86841716923246 \beta_{4} + 9061586329856 \beta_{5} - 8663688983552 \beta_{6} + 10258956795694 \beta_{7} - 198948673152 \beta_{8} - 1770480074542 \beta_{9} ) q^{79} + ( 33774364647077580143632 + 934996944533041288 \beta_{1} + 40034181165404624 \beta_{2} - 341216669473824 \beta_{3} + 113351292271560 \beta_{4} - 65046184342264 \beta_{5} - 14885843385000 \beta_{6} - 6797817837360 \beta_{7} + 6168654986160 \beta_{8} + 2642320214800 \beta_{9} ) q^{80} + ( -2116333335542020879221 + 6859673548807464783 \beta_{1} + 4706705368923897 \beta_{2} + 1500650212877589 \beta_{3} - 113042206460274 \beta_{4} + 167323321332528 \beta_{5} - 16963747218996 \beta_{6} - 38122404189639 \beta_{8} ) q^{81} + ( -12797196690669146868974 + 7204770538005655090 \beta_{1} - 25878471018779320 \beta_{2} + 797410403021228 \beta_{3} + 51441393724068 \beta_{4} + 75752128748112 \beta_{5} + 20952714870864 \beta_{6} - 3888123724800 \beta_{7} + 3759321517200 \beta_{8} - 585109602336 \beta_{9} ) q^{82} + ( -2464738878045721639 - 12323857449254038595 \beta_{1} - 109834451130557779 \beta_{2} + 1593499662373632 \beta_{3} - 159449184181888 \beta_{4} - 4127622390144 \beta_{5} + 4110801784320 \beta_{6} - 2176375669536 \beta_{7} + 8410302912 \beta_{8} + 1817536078624 \beta_{9} ) q^{83} + ( 46622832220239343847040 - 1949634994234553472 \beta_{1} + 8368324980383744 \beta_{2} - 237247031452288 \beta_{3} + 212089073957888 \beta_{4} + 104109040416640 \beta_{5} + 19025489383040 \beta_{6} - 31398361238784 \beta_{7} - 88956254464 \beta_{8} - 333839587584 \beta_{9} ) q^{84} + ( -15692961305889544179260 + 12575491757624141310 \beta_{1} + 7833941034049680 \beta_{2} - 2824840581980580 \beta_{3} - 344559925788180 \beta_{4} - 279875984955750 \beta_{5} + 66617869147320 \beta_{6} + 35953328576970 \beta_{8} ) q^{85} + ( -42566278206381928760720 - 1620374360217665712 \beta_{1} + 37512233921134105 \beta_{2} - 2234839353177482 \beta_{3} + 341195185170433 \beta_{4} - 276896421596416 \beta_{5} + 5670824090880 \beta_{6} + 29590124220033 \beta_{7} - 27592987585407 \beta_{8} - 5363894858240 \beta_{9} ) q^{86} + ( -4672934818086068300 - 23364806935107802255 \beta_{1} + 269750998423008738 \beta_{2} + 6405136013574787 \beta_{3} - 131638946724333 \beta_{4} - 30341822388032 \beta_{5} + 30628586912000 \beta_{6} + 5729851202749 \beta_{7} - 143382261984 \beta_{8} + 387791975235 \beta_{9} ) q^{87} + ( 27390137950117094714720 + 13433338219272180784 \beta_{1} - 156806831081068832 \beta_{2} - 1607828507804096 \beta_{3} + 217258286224560 \beta_{4} + 148796674088752 \beta_{5} + 51015639421712 \beta_{6} + 93358827933408 \beta_{7} - 4021526515680 \beta_{8} - 5864245865376 \beta_{9} ) q^{88} + ( -81791095550061312736248 - 7850013975042879431 \beta_{1} - 5187782524923281 \beta_{2} + 224456555011475 \beta_{3} + 130870060757378 \beta_{4} + 330295669447392 \beta_{5} + 8082606543060 \beta_{6} + 101299422487743 \beta_{8} ) q^{89} + ( -17041107990992150013291 - 8975657272811861106 \beta_{1} + 207634139877263978 \beta_{2} + 738075845915471 \beta_{3} + 329997186235445 \beta_{4} - 312051956226812 \beta_{5} + 17146888099460 \beta_{6} - 19138832732928 \beta_{7} + 19705106293076 \beta_{8} + 10666086338712 \beta_{9} ) q^{90} + ( -562672809061253644 - 2813358757259014608 \beta_{1} - 384698347486687716 \beta_{2} + 4838991091173780 \beta_{3} - 17560094324588 \beta_{4} - 37620958940096 \beta_{5} + 32960584590080 \beta_{6} - 91492529440420 \beta_{7} + 2330187175008 \beta_{8} - 7928790026588 \beta_{9} ) q^{91} + ( -46116916845975749812168 - 12563636691453274376 \beta_{1} - 209778740028740256 \beta_{2} + 2386788971955624 \beta_{3} + 46191273801520 \beta_{4} - 4680523475272 \beta_{5} - 49696739130072 \beta_{6} - 45464963223120 \beta_{7} - 109045942075728 \beta_{8} + 7447913454576 \beta_{9} ) q^{92} + ( 25136059183138712336784 - 7651274602977819624 \beta_{1} - 4513929528697112 \beta_{2} + 2454090922445512 \beta_{3} + 310381773055408 \beta_{4} - 377583382632256 \beta_{5} - 92925454137632 \beta_{6} - 192836238116312 \beta_{8} ) q^{93} + ( 170962990473256623889664 + 6438630770575993408 \beta_{1} + 221644371410646140 \beta_{2} + 7144926810975272 \beta_{3} - 579936753209316 \beta_{4} + 943647899203968 \beta_{5} - 1240497525120 \beta_{6} - 4268596575396 \beta_{7} - 6406815037860 \beta_{8} + 6657940891392 \beta_{9} ) q^{94} + ( 7190033078749658860 + 35950429041852187905 \beta_{1} + 648437843202395610 \beta_{2} - 15502411618211765 \beta_{3} + 306940528738715 \beta_{4} + 83463309116160 \beta_{5} - 81090007833600 \beta_{6} + 30764190900645 \beta_{7} - 1186650641280 \beta_{8} + 19866236460635 \beta_{9} ) q^{95} + ( -325409340714386049988096 + 4913863775895211264 \beta_{1} - 7139173931044352 \beta_{2} + 2739868389350400 \beta_{3} - 577308605414144 \beta_{4} - 561367144178432 \beta_{5} - 72934457327872 \beta_{6} - 121712335441408 \beta_{7} + 292252750700032 \beta_{8} - 1157474723328 \beta_{9} ) q^{96} + ( 223305769546217777239976 - 7505642221876718463 \beta_{1} - 4973160287699233 \beta_{2} - 331608863867533 \beta_{3} + 165517193373138 \beta_{4} + 81529193003848 \beta_{5} - 12329117768716 \beta_{6} - 40780939437801 \beta_{8} ) q^{97} + ( 259315545587404625026059 - 18264764495169913183 \beta_{1} - 312288542870527280 \beta_{2} - 18199632624181832 \beta_{3} - 1454099719849624 \beta_{4} + 469490195776288 \beta_{5} - 177522914124000 \beta_{6} - 5221566547968 \beta_{7} + 3576129182112 \beta_{8} - 40451917338944 \beta_{9} ) q^{98} + ( 29524024158297246267 + 147621227965096740927 \beta_{1} - 1459732533346964313 \beta_{2} - 30126892547043480 \beta_{3} + 1099643896885992 \beta_{4} + 184196567953920 \beta_{5} - 161504856385536 \beta_{6} + 504873487280280 \beta_{7} - 11345855784192 \beta_{8} - 20783640488088 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 6212q^{2} - 33193328q^{4} + 56758100q^{5} + 2428200768q^{6} - 159817477952q^{8} - 939054565686q^{9} + O(q^{10}) \) \( 10q - 6212q^{2} - 33193328q^{4} + 56758100q^{5} + 2428200768q^{6} - 159817477952q^{8} - 939054565686q^{9} - 279847745800q^{10} + 12433669328640q^{12} - 24798065342764q^{13} - 3829480368768q^{14} - 588864378801920q^{16} - 1625038130621164q^{17} - 3063761223370692q^{18} - 3121315867314400q^{20} - 2850064322141184q^{21} + 13916632990560960q^{22} + 10666682388698112q^{24} + 71169859613334750q^{25} - 74430147208248328q^{26} + 57921192425034240q^{28} + 1565608550658446804q^{29} + 168458533397673600q^{30} - 2011315973261284352q^{32} - 3113879011632238080q^{33} - 7482312029820836488q^{34} - 4814867106633363312q^{36} + 18108949413006125396q^{37} - 3611284956625913280q^{38} - 3297305262340163200q^{40} + 75081428813585343380q^{41} - 36295172789431572480q^{42} + \)\(12\!\cdots\!40\)\(q^{44} - 81006988268625062700q^{45} - \)\(10\!\cdots\!12\)\(q^{46} + \)\(16\!\cdots\!20\)\(q^{48} - \)\(27\!\cdots\!66\)\(q^{49} - \)\(29\!\cdots\!00\)\(q^{50} + \)\(14\!\cdots\!76\)\(q^{52} - \)\(11\!\cdots\!44\)\(q^{53} - \)\(32\!\cdots\!36\)\(q^{54} + \)\(45\!\cdots\!48\)\(q^{56} - \)\(11\!\cdots\!40\)\(q^{57} - \)\(19\!\cdots\!64\)\(q^{58} + \)\(14\!\cdots\!00\)\(q^{60} + \)\(75\!\cdots\!80\)\(q^{61} - \)\(21\!\cdots\!40\)\(q^{62} + \)\(27\!\cdots\!12\)\(q^{64} - \)\(22\!\cdots\!00\)\(q^{65} - \)\(60\!\cdots\!40\)\(q^{66} + \)\(44\!\cdots\!76\)\(q^{68} - \)\(35\!\cdots\!16\)\(q^{69} - \)\(11\!\cdots\!00\)\(q^{70} + \)\(15\!\cdots\!08\)\(q^{72} + \)\(51\!\cdots\!96\)\(q^{73} - \)\(14\!\cdots\!08\)\(q^{74} + \)\(27\!\cdots\!60\)\(q^{76} - \)\(19\!\cdots\!00\)\(q^{77} - \)\(38\!\cdots\!40\)\(q^{78} + \)\(33\!\cdots\!00\)\(q^{80} - \)\(21\!\cdots\!74\)\(q^{81} - \)\(12\!\cdots\!44\)\(q^{82} + \)\(46\!\cdots\!64\)\(q^{84} - \)\(15\!\cdots\!00\)\(q^{85} - \)\(42\!\cdots\!92\)\(q^{86} + \)\(27\!\cdots\!60\)\(q^{88} - \)\(81\!\cdots\!56\)\(q^{89} - \)\(17\!\cdots\!00\)\(q^{90} - \)\(46\!\cdots\!60\)\(q^{92} + \)\(25\!\cdots\!80\)\(q^{93} + \)\(17\!\cdots\!52\)\(q^{94} - \)\(32\!\cdots\!12\)\(q^{96} + \)\(22\!\cdots\!36\)\(q^{97} + \)\(25\!\cdots\!08\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 3 x^{9} + 1157886 x^{8} - 1182314620 x^{7} + 1715110302918 x^{6} - 2255628699630474 x^{5} + 1518357032488149568 x^{4} - 1731829168106243136300 x^{3} + 1460524813513291203716721 x^{2} - 1269695210566625624700394443 x + 1028204700608421275722350402186\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\nu^{9} - 158 \nu^{8} + 1182376 \nu^{7} - 1365582900 \nu^{6} + 1926775652418 \nu^{5} - 2554278925755264 \nu^{4} + 1914270265980215488 \nu^{3} - 2028541059333176536940 \nu^{2} + 1774948677709933566942421 \nu - 1357126522116494148203335874\)\()/ \)\(30\!\cdots\!44\)\( \)
\(\beta_{2}\)\(=\)\((\)\(116131909 \nu^{9} - 164190034582 \nu^{8} + 258894897225544 \nu^{7} - 250916861271892484 \nu^{6} + 353358878725539520970 \nu^{5} - 352561762478215110545792 \nu^{4} + 367837377315613706694267840 \nu^{3} - 270784545525938919964994152476 \nu^{2} + 129050528219804263437766633205097 \nu - 38218486633532921582008437600277578\)\()/ \)\(85\!\cdots\!56\)\( \)
\(\beta_{3}\)\(=\)\((\)\(1071558713 \nu^{9} + 1596009168082 \nu^{8} + 410701483466600 \nu^{7} + 360866639681368172 \nu^{6} - 293295494272873222478 \nu^{5} + 49138671695413342328960 \nu^{4} - 1947081895186060780438746432 \nu^{3} + 759210586974008932396840890612 \nu^{2} - 620136577296581628696639305966739 \nu - 6440563485986328707124230967089970\)\()/ \)\(85\!\cdots\!56\)\( \)
\(\beta_{4}\)\(=\)\((\)\(732859613 \nu^{9} + 436776275866 \nu^{8} + 845431847719688 \nu^{7} - 146258461642240228 \nu^{6} + 357286577934501581050 \nu^{5} - 153438663608347974722944 \nu^{4} - 746204463862983780736825920 \nu^{3} + 437299380731049034354823036868 \nu^{2} + 2314400397447898003846199506051041 \nu + 618447585734024908264859527779856518\)\()/ \)\(12\!\cdots\!08\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-1915129217 \nu^{9} - 485066975842 \nu^{8} + 5621882820952 \nu^{7} - 5699593475059490252 \nu^{6} + 4084570868396157715646 \nu^{5} - 2403333032252027647024256 \nu^{4} + 7544118004687672298186735424 \nu^{3} - 7724089079964083907904806461076 \nu^{2} + 8509490821567084578727342781065131 \nu - 4612943170438339101318163550424429630\)\()/ \)\(10\!\cdots\!32\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-9465306485 \nu^{9} - 3666444407242 \nu^{8} - 9659565766731464 \nu^{7} - 3192891318134804540 \nu^{6} - 952471492873220773930 \nu^{5} + 176237967757027646470528 \nu^{4} + 13393138831101815481452103744 \nu^{3} + 10478251858305635474112520284252 \nu^{2} + 15342583405573330878643483876651431 \nu - 6347391857521786862730808963492027734\)\()/ \)\(10\!\cdots\!32\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-153798607993 \nu^{9} + 377481835692718 \nu^{8} - 346252805906213224 \nu^{7} + 633063219563902594196 \nu^{6} - 409909619109451931281202 \nu^{5} + 762745109237977625433966464 \nu^{4} - 610093383465463784034775349952 \nu^{3} + 498027772286055417926205210597900 \nu^{2} - 257766092318222197734642544457036973 \nu - 18749649399902573203713191089328606286\)\()/ \)\(85\!\cdots\!56\)\( \)
\(\beta_{8}\)\(=\)\((\)\(399504925291 \nu^{9} + 14488026038774 \nu^{8} - 86609785759808456 \nu^{7} + 193024584906543068740 \nu^{6} - 297838874778756190101610 \nu^{5} + 241123302857201730150984064 \nu^{4} - 579254715233383238543675723712 \nu^{3} + 777137885632381085379771297977820 \nu^{2} - 740300232490585769705322953238325497 \nu + 438072388435341752615237471404871121642\)\()/ \)\(85\!\cdots\!56\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-5798090099 \nu^{9} - 3295412110086 \nu^{8} - 15767261924856184 \nu^{7} + 5483315099921712988 \nu^{6} - 5527833378217181497254 \nu^{5} + 9832146012768417367306880 \nu^{4} - 11801203530132836283888547392 \nu^{3} + 8595715620330726771115699644804 \nu^{2} - 3342597091810490516725003820297391 \nu - 1224020898076655578386758123908386522\)\()/ \)\(59\!\cdots\!24\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} - 5 \beta_{3} - 34 \beta_{2} + 14820 \beta_{1} + 81607\)\()/262144\)
\(\nu^{2}\)\(=\)\((\)\(128 \beta_{6} - 128 \beta_{5} + 95 \beta_{4} + 6821 \beta_{3} + 54370 \beta_{2} - 22637284 \beta_{1} - 60710865063\)\()/262144\)
\(\nu^{3}\)\(=\)\((\)\(-8192 \beta_{9} + 23808 \beta_{8} + 24576 \beta_{7} + 49152 \beta_{6} + 132608 \beta_{5} - 622897 \beta_{4} - 7779595 \beta_{3} - 24084606 \beta_{2} + 26653728348 \beta_{1} + 92713157194985\)\()/262144\)
\(\nu^{4}\)\(=\)\((\)\(-2531328 \beta_{9} + 29634816 \beta_{8} + 31449088 \beta_{7} - 72194176 \beta_{6} + 505853568 \beta_{5} + 509668355 \beta_{4} - 5960046479 \beta_{3} + 42417056154 \beta_{2} - 28340980068564 \beta_{1} - 109185895331455787\)\()/262144\)
\(\nu^{5}\)\(=\)\((\)\(10051059712 \beta_{9} - 14366767104 \beta_{8} + 39610155008 \beta_{7} + 42742323456 \beta_{6} + 20704057088 \beta_{5} - 263573653265 \beta_{4} + 11731844349013 \beta_{3} + 168501166078530 \beta_{2} - 24877171096256804 \beta_{1} + 116063282556833309385\)\()/262144\)
\(\nu^{6}\)\(=\)\((\)\(-1016716279808 \beta_{9} - 11861240065536 \beta_{8} + 38129545363456 \beta_{7} - 21442424033152 \beta_{6} - 462667894707328 \beta_{5} + 76070557542363 \beta_{4} - 11987760185858247 \beta_{3} - 26771353648665622 \beta_{2} + 58929036916745462156 \beta_{1} + 101866608651817520529597\)\()/262144\)
\(\nu^{7}\)\(=\)\((\)\(-7041866209452032 \beta_{9} - 19841610429360384 \beta_{8} - 11442054278602752 \beta_{7} - 15867130863702528 \beta_{6} - 172602169993026560 \beta_{5} + 791130566254871255 \beta_{4} + 1383957575638941389 \beta_{3} + 40699179559404929138 \beta_{2} - 49064408726070117460612 \beta_{1} - 241445914300524941676566239\)\()/262144\)
\(\nu^{8}\)\(=\)\((\)\(-3499016235740782592 \beta_{9} - 9406459662492532992 \beta_{8} + 13983910992267583488 \beta_{7} + 84162205315081148800 \beta_{6} + 136645515530868877440 \beta_{5} - 991291787022217239101 \beta_{4} + 14686797480478360283057 \beta_{3} + 12070497101820789754906 \beta_{2} - 2130793952250605610934740 \beta_{1} + 200954594364919006206813306645\)\()/262144\)
\(\nu^{9}\)\(=\)\((\)\(-3765274209582778761216 \beta_{9} + 63578693284985752035840 \beta_{8} + 24772077837580962349056 \beta_{7} - 98418751433012444140800 \beta_{6} + 332557517541550092833024 \beta_{5} + 431681120643352076823967 \beta_{4} - 15910708362984825651945243 \beta_{3} - 82347571346795996944503582 \beta_{2} + 69670681850064810592434995100 \beta_{1} + 8804044893017324866469678437785\)\()/262144\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

Embeddings

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
848.477 203.983i
848.477 + 203.983i
472.934 808.873i
472.934 + 808.873i
112.122 988.545i
112.122 + 988.545i
−697.391 868.555i
−697.391 + 868.555i
−734.642 844.151i
−734.642 + 844.151i
−4013.91 815.932i 666259.i 1.54457e7 + 6.55016e6i 9.84947e7 −5.43622e8 + 2.67430e9i 1.39617e10i −5.66533e10 3.88944e10i −1.61471e11 −3.95349e11 8.03650e10i
3.2 −4013.91 + 815.932i 666259.i 1.54457e7 6.55016e6i 9.84947e7 −5.43622e8 2.67430e9i 1.39617e10i −5.66533e10 + 3.88944e10i −1.61471e11 −3.95349e11 + 8.03650e10i
3.3 −2511.74 3235.49i 386500.i −4.15959e6 + 1.62534e7i −3.41994e8 1.25052e9 9.70785e8i 1.51829e10i 6.30355e10 2.73659e10i 1.33047e11 8.58998e11 + 1.10652e12i
3.4 −2511.74 + 3235.49i 386500.i −4.15959e6 1.62534e7i −3.41994e8 1.25052e9 + 9.70785e8i 1.51829e10i 6.30355e10 + 2.73659e10i 1.33047e11 8.58998e11 1.10652e12i
3.5 −1068.49 3954.18i 193889.i −1.44939e7 + 8.44998e6i 3.74285e8 −7.66672e8 + 2.07168e8i 2.47654e10i 4.88993e10 + 4.82828e10i 2.44837e11 −3.99918e11 1.47999e12i
3.6 −1068.49 + 3954.18i 193889.i −1.44939e7 8.44998e6i 3.74285e8 −7.66672e8 2.07168e8i 2.47654e10i 4.88993e10 4.82828e10i 2.44837e11 −3.99918e11 + 1.47999e12i
3.7 2169.56 3474.22i 569070.i −7.36320e6 1.50751e7i −2.26723e8 −1.97707e9 1.23463e9i 6.45252e9i −6.83491e10 7.12502e9i −4.14111e10 −4.91891e11 + 7.87687e11i
3.8 2169.56 + 3474.22i 569070.i −7.36320e6 + 1.50751e7i −2.26723e8 −1.97707e9 + 1.23463e9i 6.45252e9i −6.83491e10 + 7.12502e9i −4.14111e10 −4.91891e11 7.87687e11i
3.9 2318.57 3376.61i 962787.i −6.02571e6 1.56578e7i 1.24317e8 3.25095e9 + 2.23229e9i 3.87327e9i −6.68411e10 1.59571e10i −6.44529e11 2.88236e11 4.19768e11i
3.10 2318.57 + 3376.61i 962787.i −6.02571e6 + 1.56578e7i 1.24317e8 3.25095e9 2.23229e9i 3.87327e9i −6.68411e10 + 1.59571e10i −6.44529e11 2.88236e11 + 4.19768e11i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.25.b.b 10
3.b odd 2 1 36.25.d.b 10
4.b odd 2 1 inner 4.25.b.b 10
8.b even 2 1 64.25.c.d 10
8.d odd 2 1 64.25.c.d 10
12.b even 2 1 36.25.d.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.25.b.b 10 1.a even 1 1 trivial
4.25.b.b 10 4.b odd 2 1 inner
36.25.d.b 10 3.b odd 2 1
36.25.d.b 10 12.b even 2 1
64.25.c.d 10 8.b even 2 1
64.25.c.d 10 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + \)\(18\!\cdots\!48\)\( T_{3}^{8} + \)\(11\!\cdots\!16\)\( T_{3}^{6} + \)\(30\!\cdots\!40\)\( T_{3}^{4} + \)\(29\!\cdots\!20\)\( T_{3}^{2} + \)\(74\!\cdots\!00\)\( \) acting on \(S_{25}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( \)\(13\!\cdots\!76\)\( + \)\(49\!\cdots\!32\)\( T + \)\(16\!\cdots\!56\)\( T^{2} + \)\(55\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!28\)\( T^{4} + 4824488218655195136 T^{5} + 998139424145408 T^{6} + 196323389440 T^{7} + 35891136 T^{8} + 6212 T^{9} + T^{10} \)
$3$ \( \)\(74\!\cdots\!00\)\( + \)\(29\!\cdots\!20\)\( T^{2} + \)\(30\!\cdots\!40\)\( T^{4} + \)\(11\!\cdots\!16\)\( T^{6} + 1881674965248 T^{8} + T^{10} \)
$5$ \( ( -\)\(35\!\cdots\!00\)\( + \)\(48\!\cdots\!00\)\( T + \)\(35\!\cdots\!00\)\( T^{2} - 166401391602359000 T^{3} - 28379050 T^{4} + T^{5} )^{2} \)
$7$ \( \)\(17\!\cdots\!00\)\( + \)\(17\!\cdots\!00\)\( T^{2} + \)\(45\!\cdots\!00\)\( T^{4} + \)\(36\!\cdots\!36\)\( T^{6} + \)\(10\!\cdots\!88\)\( T^{8} + T^{10} \)
$11$ \( \)\(12\!\cdots\!00\)\( + \)\(30\!\cdots\!00\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{4} + \)\(13\!\cdots\!20\)\( T^{6} + \)\(63\!\cdots\!40\)\( T^{8} + T^{10} \)
$13$ \( ( \)\(81\!\cdots\!00\)\( + \)\(70\!\cdots\!60\)\( T - \)\(20\!\cdots\!08\)\( T^{2} - \)\(16\!\cdots\!24\)\( T^{3} + 12399032671382 T^{4} + T^{5} )^{2} \)
$17$ \( ( -\)\(30\!\cdots\!00\)\( + \)\(21\!\cdots\!80\)\( T - \)\(23\!\cdots\!08\)\( T^{2} - \)\(75\!\cdots\!04\)\( T^{3} + 812519065310582 T^{4} + T^{5} )^{2} \)
$19$ \( \)\(18\!\cdots\!00\)\( + \)\(28\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{4} + \)\(29\!\cdots\!20\)\( T^{6} + \)\(28\!\cdots\!40\)\( T^{8} + T^{10} \)
$23$ \( \)\(83\!\cdots\!00\)\( + \)\(60\!\cdots\!20\)\( T^{2} + \)\(10\!\cdots\!60\)\( T^{4} + \)\(65\!\cdots\!36\)\( T^{6} + \)\(14\!\cdots\!08\)\( T^{8} + T^{10} \)
$29$ \( ( \)\(91\!\cdots\!48\)\( - \)\(50\!\cdots\!64\)\( T + \)\(22\!\cdots\!96\)\( T^{2} + \)\(12\!\cdots\!12\)\( T^{3} - 782804275329223402 T^{4} + T^{5} )^{2} \)
$31$ \( \)\(37\!\cdots\!00\)\( + \)\(94\!\cdots\!00\)\( T^{2} + \)\(80\!\cdots\!00\)\( T^{4} + \)\(27\!\cdots\!20\)\( T^{6} + \)\(32\!\cdots\!40\)\( T^{8} + T^{10} \)
$37$ \( ( \)\(21\!\cdots\!00\)\( - \)\(25\!\cdots\!60\)\( T + \)\(34\!\cdots\!12\)\( T^{2} - \)\(63\!\cdots\!04\)\( T^{3} - 9054474706503062698 T^{4} + T^{5} )^{2} \)
$41$ \( ( -\)\(17\!\cdots\!52\)\( - \)\(48\!\cdots\!00\)\( T + \)\(12\!\cdots\!20\)\( T^{2} + \)\(20\!\cdots\!60\)\( T^{3} - 37540714406792671690 T^{4} + T^{5} )^{2} \)
$43$ \( \)\(55\!\cdots\!00\)\( + \)\(22\!\cdots\!00\)\( T^{2} + \)\(33\!\cdots\!00\)\( T^{4} + \)\(23\!\cdots\!96\)\( T^{6} + \)\(78\!\cdots\!28\)\( T^{8} + T^{10} \)
$47$ \( \)\(17\!\cdots\!00\)\( + \)\(20\!\cdots\!20\)\( T^{2} + \)\(25\!\cdots\!20\)\( T^{4} + \)\(60\!\cdots\!56\)\( T^{6} + \)\(43\!\cdots\!28\)\( T^{8} + T^{10} \)
$53$ \( ( \)\(14\!\cdots\!00\)\( + \)\(13\!\cdots\!80\)\( T - \)\(24\!\cdots\!28\)\( T^{2} - \)\(39\!\cdots\!44\)\( T^{3} + \)\(55\!\cdots\!22\)\( T^{4} + T^{5} )^{2} \)
$59$ \( \)\(22\!\cdots\!00\)\( + \)\(94\!\cdots\!00\)\( T^{2} + \)\(89\!\cdots\!00\)\( T^{4} + \)\(69\!\cdots\!20\)\( T^{6} + \)\(16\!\cdots\!80\)\( T^{8} + T^{10} \)
$61$ \( ( \)\(19\!\cdots\!48\)\( - \)\(60\!\cdots\!00\)\( T + \)\(48\!\cdots\!20\)\( T^{2} - \)\(78\!\cdots\!40\)\( T^{3} - \)\(37\!\cdots\!90\)\( T^{4} + T^{5} )^{2} \)
$67$ \( \)\(92\!\cdots\!00\)\( + \)\(29\!\cdots\!20\)\( T^{2} + \)\(16\!\cdots\!40\)\( T^{4} + \)\(37\!\cdots\!16\)\( T^{6} + \)\(33\!\cdots\!48\)\( T^{8} + T^{10} \)
$71$ \( \)\(15\!\cdots\!00\)\( + \)\(21\!\cdots\!00\)\( T^{2} + \)\(59\!\cdots\!00\)\( T^{4} + \)\(42\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!80\)\( T^{8} + T^{10} \)
$73$ \( ( -\)\(16\!\cdots\!00\)\( + \)\(39\!\cdots\!40\)\( T + \)\(13\!\cdots\!72\)\( T^{2} - \)\(56\!\cdots\!64\)\( T^{3} - \)\(25\!\cdots\!98\)\( T^{4} + T^{5} )^{2} \)
$79$ \( \)\(88\!\cdots\!00\)\( + \)\(26\!\cdots\!00\)\( T^{2} + \)\(25\!\cdots\!00\)\( T^{4} + \)\(98\!\cdots\!20\)\( T^{6} + \)\(16\!\cdots\!40\)\( T^{8} + T^{10} \)
$83$ \( \)\(49\!\cdots\!00\)\( + \)\(27\!\cdots\!20\)\( T^{2} + \)\(25\!\cdots\!80\)\( T^{4} + \)\(66\!\cdots\!56\)\( T^{6} + \)\(49\!\cdots\!68\)\( T^{8} + T^{10} \)
$89$ \( ( -\)\(26\!\cdots\!52\)\( + \)\(13\!\cdots\!16\)\( T - \)\(15\!\cdots\!24\)\( T^{2} - \)\(36\!\cdots\!08\)\( T^{3} + \)\(40\!\cdots\!78\)\( T^{4} + T^{5} )^{2} \)
$97$ \( ( -\)\(23\!\cdots\!00\)\( + \)\(86\!\cdots\!20\)\( T - \)\(96\!\cdots\!88\)\( T^{2} + \)\(47\!\cdots\!96\)\( T^{3} - \)\(11\!\cdots\!18\)\( T^{4} + T^{5} )^{2} \)
show more
show less