Properties

Label 8004.2.a.i
Level $8004$
Weight $2$
Character orbit 8004.a
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} - 17844 x^{3} + 14352 x^{2} + 72 x - 208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{1} q^{5} -\beta_{12} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta_{1} q^{5} -\beta_{12} q^{7} + q^{9} -\beta_{4} q^{11} + \beta_{5} q^{13} -\beta_{1} q^{15} -\beta_{13} q^{17} + ( -\beta_{5} + \beta_{8} - \beta_{9} ) q^{19} + \beta_{12} q^{21} + q^{23} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{10} - \beta_{13} + \beta_{14} ) q^{25} - q^{27} + q^{29} + ( -\beta_{4} - \beta_{7} ) q^{31} + \beta_{4} q^{33} + ( 2 - \beta_{1} + \beta_{3} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{35} + ( 1 - \beta_{14} ) q^{37} -\beta_{5} q^{39} + ( -1 - \beta_{3} + \beta_{8} ) q^{41} + ( -1 + \beta_{1} - \beta_{4} - \beta_{8} ) q^{43} + \beta_{1} q^{45} + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{10} - \beta_{15} ) q^{47} + ( 1 + \beta_{1} + \beta_{4} + \beta_{7} + \beta_{9} - \beta_{13} - \beta_{15} ) q^{49} + \beta_{13} q^{51} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{53} + ( -2 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{55} + ( \beta_{5} - \beta_{8} + \beta_{9} ) q^{57} + ( \beta_{1} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{15} ) q^{59} + ( 1 + \beta_{2} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{61} -\beta_{12} q^{63} + ( 2 + 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{12} + \beta_{13} ) q^{65} + ( -1 + \beta_{1} + \beta_{2} + \beta_{7} + \beta_{9} + \beta_{11} + \beta_{13} + \beta_{15} ) q^{67} - q^{69} + ( \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{14} + \beta_{15} ) q^{71} + ( -2 + \beta_{1} + \beta_{2} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{13} ) q^{73} + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{10} + \beta_{13} - \beta_{14} ) q^{75} + ( -\beta_{1} + \beta_{4} + \beta_{6} - \beta_{11} - \beta_{13} ) q^{77} + ( -1 + \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} + \beta_{15} ) q^{79} + q^{81} + ( -\beta_{3} + \beta_{4} + 2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{83} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} - 2 \beta_{8} - \beta_{11} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{85} - q^{87} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} - \beta_{14} ) q^{89} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{91} + ( \beta_{4} + \beta_{7} ) q^{93} + ( 2 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} - 2 \beta_{12} - \beta_{15} ) q^{95} + ( 2 \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{97} -\beta_{4} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + 5q^{11} + 8q^{13} - 5q^{15} + 7q^{17} + q^{19} + 4q^{21} + 16q^{23} + 31q^{25} - 16q^{27} + 16q^{29} - 2q^{31} - 5q^{33} + 5q^{35} + 14q^{37} - 8q^{39} - q^{41} - 13q^{43} + 5q^{45} - 4q^{47} + 30q^{49} - 7q^{51} + 19q^{53} - 37q^{55} - q^{57} + 12q^{59} + 21q^{61} - 4q^{63} + 26q^{65} - 11q^{67} - 16q^{69} + 7q^{71} - 13q^{73} - 31q^{75} + 4q^{77} - 18q^{79} + 16q^{81} + 25q^{83} + 48q^{85} - 16q^{87} + 12q^{89} - 11q^{91} + 2q^{93} - q^{95} + 5q^{97} + 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} - 17844 x^{3} + 14352 x^{2} + 72 x - 208\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(3040780320993915014 \nu^{15} + 11451848272286613279 \nu^{14} - 232887392825327605241 \nu^{13} - 552866610872686710043 \nu^{12} + 6701952172673579845542 \nu^{11} + 10014744839360784183052 \nu^{10} - 92816762444366568951910 \nu^{9} - 85086576675455429657055 \nu^{8} + 652807380563049726096164 \nu^{7} + 327739783995970876724887 \nu^{6} - 2206166321766519707780335 \nu^{5} - 345737863611574611652196 \nu^{4} + 2820087633149276715197960 \nu^{3} - 505057097539218663987944 \nu^{2} - 265212571902372090704692 \nu + 52042966114613282035144\)\()/ \)\(82\!\cdots\!44\)\( \)
\(\beta_{3}\)\(=\)\((\)\(1133155627719690365 \nu^{15} - 6183981961343391225 \nu^{14} - 45962507843656667153 \nu^{13} + 285577742288708681086 \nu^{12} + 594553806911671718002 \nu^{11} - 4841993240933967346590 \nu^{10} - 1931243557888597965827 \nu^{9} + 37694840050022447013852 \nu^{8} - 15054873801417991803487 \nu^{7} - 133459508425382606930219 \nu^{6} + 118176839691438596424822 \nu^{5} + 162630129553180087835490 \nu^{4} - 217538448934933635766102 \nu^{3} + 28814439094940890866468 \nu^{2} + 20345563209993771194972 \nu - 181736188397342579200\)\()/ \)\(13\!\cdots\!24\)\( \)
\(\beta_{4}\)\(=\)\((\)\(10327108820298630749 \nu^{15} - 46623029354138300703 \nu^{14} - 468648207638402451527 \nu^{13} + 2199331736699944090170 \nu^{12} + 7701026401458349054308 \nu^{11} - 38562172136739934951652 \nu^{10} - 55997241963193925731391 \nu^{9} + 317505626992755431744994 \nu^{8} + 160070546734796190670629 \nu^{7} - 1253648947739701211892215 \nu^{6} + 15498431467411784950720 \nu^{5} + 2076442970619250704115952 \nu^{4} - 572372392754759088010752 \nu^{3} - 871645844830868408937972 \nu^{2} + 75547605968724778430888 \nu + 39550714308552523443072\)\()/ \)\(54\!\cdots\!96\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-15783173244444101129 \nu^{15} + 71891299794190027998 \nu^{14} + 710848605201316221158 \nu^{13} - 3385320715110873005891 \nu^{12} - 11475349920655983031686 \nu^{11} + 59064258354862485012260 \nu^{10} + 79549194381055089354751 \nu^{9} - 480110131920749519232459 \nu^{8} - 187404565471993343666351 \nu^{7} + 1833305564855172683452274 \nu^{6} - 268495880029625550510797 \nu^{5} - 2745160541549121158528824 \nu^{4} + 1224989810538650711651380 \nu^{3} + 637548574623595049191988 \nu^{2} - 28435539046608856852172 \nu - 48618750485435144983000\)\()/ \)\(82\!\cdots\!44\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-43409510040342958591 \nu^{15} + 204436784173408593213 \nu^{14} + 1933384284718598061661 \nu^{13} - 9609952842332358293734 \nu^{12} - 30738946990350785496204 \nu^{11} + 167641296806600010566452 \nu^{10} + 209050863408559362327029 \nu^{9} - 1369734242025953674106622 \nu^{8} - 481998571182840860681887 \nu^{7} + 5334598757017222508511037 \nu^{6} - 669736051113647402565376 \nu^{5} - 8528134675336330855060592 \nu^{4} + 2968897987381385534458088 \nu^{3} + 2965875364568320786841260 \nu^{2} + 9531422201946669156536 \nu - 115448866564292248468160\)\()/ \)\(16\!\cdots\!88\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-14876422501216426365 \nu^{15} + 63444477320010917489 \nu^{14} + 680444530030849926369 \nu^{13} - 2960246772843543741264 \nu^{12} - 11324818386317160387376 \nu^{11} + 50934833529316434627484 \nu^{10} + 84152959587324240424423 \nu^{9} - 405344514391113240217288 \nu^{8} - 252413267047417095561809 \nu^{7} + 1494626491148611763355897 \nu^{6} + 27305100885580799650810 \nu^{5} - 2074495686819045701186224 \nu^{4} + 799934920651196090914288 \nu^{3} + 246595002574247516330436 \nu^{2} - 32576648691634350903024 \nu + 430803475028090582512\)\()/ \)\(54\!\cdots\!96\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-7603870522084452615 \nu^{15} + 32046172073721707913 \nu^{14} + 353456303235461540053 \nu^{13} - 1511395690869302031178 \nu^{12} - 6050691028509434703612 \nu^{11} + 26465002897883635403568 \nu^{10} + 47517289986971177956637 \nu^{9} - 217058424842370164675842 \nu^{8} - 165337223101262763914851 \nu^{7} + 849163607976476235455277 \nu^{6} + 154277444344167389409560 \nu^{5} - 1376527448283916327600720 \nu^{4} + 189737204442907705092132 \nu^{3} + 538400774762995113598740 \nu^{2} + 25492596062475603023440 \nu - 27182643118546951377232\)\()/ \)\(27\!\cdots\!48\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-15813712956832157237 \nu^{15} + 70916958369897065433 \nu^{14} + 719012486847903488913 \nu^{13} - 3344745531914834911664 \nu^{12} - 11838796361473972966656 \nu^{11} + 58582378625148598351460 \nu^{10} + 86125788536977874427823 \nu^{9} - 480661070940613526614664 \nu^{8} - 244010727226114098109473 \nu^{7} + 1878957650573029373491257 \nu^{6} - 43320182299082538137646 \nu^{5} - 3016049778386644474401072 \nu^{4} + 910802804643754812688216 \nu^{3} + 1071305771301748204569172 \nu^{2} - 83620908096978852457952 \nu - 31358839502922458309168\)\()/ \)\(54\!\cdots\!96\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-48750376286856419701 \nu^{15} + 229053636521692224375 \nu^{14} + 2151043186106097123631 \nu^{13} - 10697144276185392156634 \nu^{12} - 33512675596998272085780 \nu^{11} + 184493534161900674094564 \nu^{10} + 215733079036281931399607 \nu^{9} - 1476019645175755861347138 \nu^{8} - 369662096234080721052685 \nu^{7} + 5502983380583967143683615 \nu^{6} - 1501117875210636465270160 \nu^{5} - 7825684947137813046148304 \nu^{4} + 4671490927199216453556224 \nu^{3} + 1130638678095749386304068 \nu^{2} - 387205328131515237238792 \nu - 65047851739085236716416\)\()/ \)\(16\!\cdots\!88\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-19819065957461023811 \nu^{15} + 105296243094269658907 \nu^{14} + 825924130302220654971 \nu^{13} - 4912947016947780053268 \nu^{12} - 11381973623510737504984 \nu^{11} + 84635925481199523199764 \nu^{10} + 49699417845194961077353 \nu^{9} - 675789922002448975704604 \nu^{8} + 145894747281510593499017 \nu^{7} + 2503760282664111920367379 \nu^{6} - 1670543280670879404138622 \nu^{5} - 3441103630357287161444048 \nu^{4} + 3297458934296158318389232 \nu^{3} + 124510648830211096930172 \nu^{2} - 256144554369799347906592 \nu + 83494722494782046960\)\()/ \)\(54\!\cdots\!96\)\( \)
\(\beta_{12}\)\(=\)\((\)\(15344073927299171897 \nu^{15} - 76517980267269701535 \nu^{14} - 662035163879610543401 \nu^{13} + 3585945259827232385072 \nu^{12} + 9839491871465361076638 \nu^{11} - 62181408508523380600868 \nu^{10} - 55554094321534472370211 \nu^{9} + 501622126176053457192078 \nu^{8} + 17345193658302384622799 \nu^{7} - 1894731945704220803403863 \nu^{6} + 833413740406888323847562 \nu^{5} + 2755519788258439982119666 \nu^{4} - 1934597359389313500952252 \nu^{3} - 426428181856744945912580 \nu^{2} + 121576357609022372496944 \nu + 4443088667932115690440\)\()/ \)\(41\!\cdots\!72\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-70933075913422809427 \nu^{15} + 319190212777976757561 \nu^{14} + 3190335724091810475001 \nu^{13} - 14936160860061056201230 \nu^{12} - 51494181033152023858860 \nu^{11} + 258321554986471115357812 \nu^{10} + 358664574332552237611265 \nu^{9} - 2075255844526952664807366 \nu^{8} - 869371686203195894627251 \nu^{7} + 7797419486270366874993913 \nu^{6} - 1086937561234508913856240 \nu^{5} - 11349378748818973102168016 \nu^{4} + 5541104129287872654006968 \nu^{3} + 2151780320120874407422252 \nu^{2} - 662086086998971531518376 \nu - 31359800394432217976480\)\()/ \)\(16\!\cdots\!88\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-32225813913671340095 \nu^{15} + 159924296878592499289 \nu^{14} + 1389566703547393943857 \nu^{13} - 7479591954295351894242 \nu^{12} - 20617773136724243749644 \nu^{11} + 129273187698150131786804 \nu^{10} + 115592584583412047647733 \nu^{9} - 1037020682429358346607922 \nu^{8} - 23473862923459495877375 \nu^{7} + 3874435369925228060759257 \nu^{6} - 1830412119817231334118652 \nu^{5} - 5468684705975812969273176 \nu^{4} + 4274698916700503360392296 \nu^{3} + 543054530270901314400956 \nu^{2} - 386578636465775801362264 \nu - 12820290579032488312704\)\()/ \)\(54\!\cdots\!96\)\( \)
\(\beta_{15}\)\(=\)\((\)\(45395358841873163252 \nu^{15} - 216603173312831897727 \nu^{14} - 1997551184811081394793 \nu^{13} + 10150483883027433779483 \nu^{12} + 30918959212470415787136 \nu^{11} - 175989519979563028243244 \nu^{10} - 195236339850544654368244 \nu^{9} + 1419614721106134505598337 \nu^{8} + 293196484616290703750216 \nu^{7} - 5367934888868512540382867 \nu^{6} + 1602889278627538618860671 \nu^{5} + 7880544085015915696563274 \nu^{4} - 4618652122090728228850804 \nu^{3} - 1480438220894389050010748 \nu^{2} + 319835708049430771492988 \nu + 39675257796285985779040\)\()/ \)\(41\!\cdots\!72\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{14} - \beta_{13} - \beta_{10} - \beta_{8} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_{1} + 6\)
\(\nu^{3}\)\(=\)\(-\beta_{12} + \beta_{10} + \beta_{9} - 2 \beta_{7} - \beta_{6} + \beta_{4} - \beta_{2} + 11 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-3 \beta_{15} + 16 \beta_{14} - 20 \beta_{13} - 22 \beta_{10} + 3 \beta_{9} - 20 \beta_{8} + 3 \beta_{7} - 2 \beta_{6} + 14 \beta_{5} - 18 \beta_{4} - \beta_{3} - 18 \beta_{2} + 20 \beta_{1} + 72\)
\(\nu^{5}\)\(=\)\(-4 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - 13 \beta_{12} + 27 \beta_{10} + 20 \beta_{9} + 12 \beta_{8} - 43 \beta_{7} - 24 \beta_{6} - 9 \beta_{5} + 28 \beta_{4} - \beta_{3} - 16 \beta_{2} + 143 \beta_{1} + 65\)
\(\nu^{6}\)\(=\)\(-96 \beta_{15} + 268 \beta_{14} - 381 \beta_{13} + 12 \beta_{12} - 22 \beta_{11} - 413 \beta_{10} + 68 \beta_{9} - 381 \beta_{8} + 70 \beta_{7} - 50 \beta_{6} + 214 \beta_{5} - 320 \beta_{4} - 34 \beta_{3} - 318 \beta_{2} + 345 \beta_{1} + 1014\)
\(\nu^{7}\)\(=\)\(-94 \beta_{15} - 82 \beta_{14} - 40 \beta_{13} - 131 \beta_{12} + 33 \beta_{11} + 593 \beta_{10} + 384 \beta_{9} + 385 \beta_{8} - 763 \beta_{7} - 484 \beta_{6} - 290 \beta_{5} + 591 \beta_{4} - 23 \beta_{3} - 200 \beta_{2} + 2058 \beta_{1} + 1084\)
\(\nu^{8}\)\(=\)\(-2185 \beta_{15} + 4636 \beta_{14} - 7081 \beta_{13} + 416 \beta_{12} - 736 \beta_{11} - 7451 \beta_{10} + 1312 \beta_{9} - 7090 \beta_{8} + 1421 \beta_{7} - 963 \beta_{6} + 3459 \beta_{5} - 5678 \beta_{4} - 761 \beta_{3} - 5594 \beta_{2} + 5735 \beta_{1} + 15516\)
\(\nu^{9}\)\(=\)\(-1625 \beta_{15} - 2295 \beta_{14} - 470 \beta_{13} - 1017 \beta_{12} + 1194 \beta_{11} + 12215 \beta_{10} + 7305 \beta_{9} + 9254 \beta_{8} - 12962 \beta_{7} - 9158 \beta_{6} - 6943 \beta_{5} + 11651 \beta_{4} - 485 \beta_{3} - 1982 \beta_{2} + 31367 \beta_{1} + 16430\)
\(\nu^{10}\)\(=\)\(-43998 \beta_{15} + 81591 \beta_{14} - 129666 \beta_{13} + 10392 \beta_{12} - 17679 \beta_{11} - 133030 \beta_{10} + 24157 \beta_{9} - 129934 \beta_{8} + 28119 \beta_{7} - 16928 \beta_{6} + 57781 \beta_{5} - 101022 \beta_{4} - 15125 \beta_{3} - 98180 \beta_{2} + 94302 \beta_{1} + 248780\)
\(\nu^{11}\)\(=\)\(-24501 \beta_{15} - 54903 \beta_{14} - 95 \beta_{13} - 1533 \beta_{12} + 29931 \beta_{11} + 244048 \beta_{10} + 136881 \beta_{9} + 200061 \beta_{8} - 219178 \beta_{7} - 167791 \beta_{6} - 148686 \beta_{5} + 224655 \beta_{4} - 10169 \beta_{3} - 8900 \beta_{2} + 494758 \beta_{1} + 235142\)
\(\nu^{12}\)\(=\)\(-837746 \beta_{15} + 1448480 \beta_{14} - 2352986 \beta_{13} + 229352 \beta_{12} - 373284 \beta_{11} - 2372878 \beta_{10} + 436378 \beta_{9} - 2360164 \beta_{8} + 551556 \beta_{7} - 284806 \beta_{6} + 986829 \beta_{5} - 1803570 \beta_{4} - 288483 \beta_{3} - 1722157 \beta_{2} + 1546720 \beta_{1} + 4102327\)
\(\nu^{13}\)\(=\)\(-331408 \beta_{15} - 1212971 \beta_{14} + 196253 \beta_{13} + 189349 \beta_{12} + 652250 \beta_{11} + 4785042 \beta_{10} + 2528884 \beta_{9} + 4105260 \beta_{8} - 3733385 \beta_{7} - 3017877 \beta_{6} - 3011680 \beta_{5} + 4293382 \beta_{4} - 208889 \beta_{3} + 274111 \beta_{2} + 7978295 \beta_{1} + 3180682\)
\(\nu^{14}\)\(=\)\(-15503403 \beta_{15} + 25829971 \beta_{14} - 42461733 \beta_{13} + 4762320 \beta_{12} - 7391012 \beta_{11} - 42417933 \beta_{10} + 7808976 \beta_{9} - 42674857 \beta_{8} + 10738719 \beta_{7} - 4664994 \beta_{6} + 17134871 \beta_{5} - 32299338 \beta_{4} - 5412584 \beta_{3} - 30230530 \beta_{2} + 25381774 \beta_{1} + 68876016\)
\(\nu^{15}\)\(=\)\(-3861844 \beta_{15} - 25604851 \beta_{14} + 7668573 \beta_{13} + 6343023 \beta_{12} + 13314661 \beta_{11} + 92598498 \beta_{10} + 46210791 \beta_{9} + 81763455 \beta_{8} - 64261165 \beta_{7} - 53681443 \beta_{6} - 59078108 \beta_{5} + 81652998 \beta_{4} - 4173985 \beta_{3} + 11987477 \beta_{2} + 130625667 \beta_{1} + 39683032\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.28632
−3.33956
−2.66915
−2.58465
−1.83485
−0.309856
−0.123882
0.144883
0.282383
1.27509
1.38601
2.30095
3.01816
3.56414
4.02229
4.15437
0 −1.00000 0 −4.28632 0 1.70119 0 1.00000 0
1.2 0 −1.00000 0 −3.33956 0 −2.20985 0 1.00000 0
1.3 0 −1.00000 0 −2.66915 0 −2.40707 0 1.00000 0
1.4 0 −1.00000 0 −2.58465 0 −3.26725 0 1.00000 0
1.5 0 −1.00000 0 −1.83485 0 1.71276 0 1.00000 0
1.6 0 −1.00000 0 −0.309856 0 −1.15608 0 1.00000 0
1.7 0 −1.00000 0 −0.123882 0 3.12124 0 1.00000 0
1.8 0 −1.00000 0 0.144883 0 −2.05535 0 1.00000 0
1.9 0 −1.00000 0 0.282383 0 5.02462 0 1.00000 0
1.10 0 −1.00000 0 1.27509 0 1.05879 0 1.00000 0
1.11 0 −1.00000 0 1.38601 0 −3.43605 0 1.00000 0
1.12 0 −1.00000 0 2.30095 0 −4.14667 0 1.00000 0
1.13 0 −1.00000 0 3.01816 0 3.11628 0 1.00000 0
1.14 0 −1.00000 0 3.56414 0 −0.955621 0 1.00000 0
1.15 0 −1.00000 0 4.02229 0 4.04331 0 1.00000 0
1.16 0 −1.00000 0 4.15437 0 −4.14427 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(-1\)
\(29\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8004.2.a.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.a.i 16 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\(T_{5}^{16} - \cdots\)
\(T_{7}^{16} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 1 + T )^{16} \)
$5$ \( -208 + 72 T + 14352 T^{2} - 17844 T^{3} - 138024 T^{4} + 172218 T^{5} + 60168 T^{6} - 121665 T^{7} - 137 T^{8} + 32512 T^{9} - 3483 T^{10} - 4048 T^{11} + 634 T^{12} + 234 T^{13} - 43 T^{14} - 5 T^{15} + T^{16} \)
$7$ \( -1420576 - 1621888 T + 2367696 T^{2} + 3019440 T^{3} - 1161776 T^{4} - 1972024 T^{5} + 124216 T^{6} + 599660 T^{7} + 48966 T^{8} - 91842 T^{9} - 14299 T^{10} + 7244 T^{11} + 1440 T^{12} - 277 T^{13} - 63 T^{14} + 4 T^{15} + T^{16} \)
$11$ \( -211968 + 198912 T + 3673576 T^{2} - 397328 T^{3} - 9395570 T^{4} + 6335456 T^{5} + 1851269 T^{6} - 2212563 T^{7} + 28441 T^{8} + 292167 T^{9} - 32133 T^{10} - 17826 T^{11} + 2784 T^{12} + 493 T^{13} - 90 T^{14} - 5 T^{15} + T^{16} \)
$13$ \( -52214944 + 2843184 T + 250861204 T^{2} - 122931304 T^{3} - 129928985 T^{4} + 84743050 T^{5} + 13390555 T^{6} - 15366197 T^{7} + 121954 T^{8} + 1251051 T^{9} - 87131 T^{10} - 51535 T^{11} + 5114 T^{12} + 1039 T^{13} - 120 T^{14} - 8 T^{15} + T^{16} \)
$17$ \( 1623072768 + 3339657216 T - 1517934080 T^{2} - 2456117760 T^{3} + 503901152 T^{4} + 558021728 T^{5} - 86191850 T^{6} - 57843218 T^{7} + 8111505 T^{8} + 3116876 T^{9} - 427315 T^{10} - 89258 T^{11} + 12319 T^{12} + 1269 T^{13} - 178 T^{14} - 7 T^{15} + T^{16} \)
$19$ \( -1855731712 + 593961984 T + 5592510976 T^{2} + 4127068224 T^{3} + 149879600 T^{4} - 748126760 T^{5} - 167328940 T^{6} + 47895038 T^{7} + 16569134 T^{8} - 1359033 T^{9} - 734851 T^{10} + 15191 T^{11} + 17117 T^{12} + 13 T^{13} - 205 T^{14} - T^{15} + T^{16} \)
$23$ \( ( -1 + T )^{16} \)
$29$ \( ( -1 + T )^{16} \)
$31$ \( -5833216 + 15487104 T + 139669360 T^{2} + 66657512 T^{3} - 121577708 T^{4} - 78171354 T^{5} + 25274110 T^{6} + 24028821 T^{7} + 814346 T^{8} - 2148925 T^{9} - 302300 T^{10} + 63161 T^{11} + 12631 T^{12} - 685 T^{13} - 192 T^{14} + 2 T^{15} + T^{16} \)
$37$ \( -258933952 - 764298912 T + 611468596 T^{2} + 1170475478 T^{3} - 1128972376 T^{4} + 21761409 T^{5} + 236799364 T^{6} - 51245460 T^{7} - 14615936 T^{8} + 5036474 T^{9} + 252639 T^{10} - 186222 T^{11} + 4275 T^{12} + 2843 T^{13} - 159 T^{14} - 14 T^{15} + T^{16} \)
$41$ \( 254378586688 - 373275066080 T - 194027287168 T^{2} + 97088060912 T^{3} + 39180381628 T^{4} - 9202636234 T^{5} - 3418940164 T^{6} + 423677627 T^{7} + 154416507 T^{8} - 10169934 T^{9} - 3848753 T^{10} + 123476 T^{11} + 52642 T^{12} - 656 T^{13} - 365 T^{14} + T^{15} + T^{16} \)
$43$ \( 19590801152 - 17560930432 T - 11642184256 T^{2} + 8746066080 T^{3} + 2536509072 T^{4} - 1648049688 T^{5} - 278496580 T^{6} + 154181354 T^{7} + 17627454 T^{8} - 7766579 T^{9} - 688419 T^{10} + 210389 T^{11} + 16454 T^{12} - 2787 T^{13} - 212 T^{14} + 13 T^{15} + T^{16} \)
$47$ \( 241789371392 + 338556414464 T - 275408458368 T^{2} - 61532882304 T^{3} + 57116486176 T^{4} + 2878288704 T^{5} - 4852263144 T^{6} + 33969032 T^{7} + 207120762 T^{8} - 5515022 T^{9} - 4819923 T^{10} + 138961 T^{11} + 61689 T^{12} - 1328 T^{13} - 402 T^{14} + 4 T^{15} + T^{16} \)
$53$ \( -4489400316672 - 48507319485120 T + 20111087614912 T^{2} + 3231228498960 T^{3} - 2336877751440 T^{4} + 86565915568 T^{5} + 90201530184 T^{6} - 10034375088 T^{7} - 1349463064 T^{8} + 249021880 T^{9} + 4684434 T^{10} - 2545465 T^{11} + 55067 T^{12} + 11496 T^{13} - 473 T^{14} - 19 T^{15} + T^{16} \)
$59$ \( 23868801024 + 7198029312 T - 42486993152 T^{2} - 13360938752 T^{3} + 17729754304 T^{4} + 5622657472 T^{5} - 2550838656 T^{6} - 750436304 T^{7} + 149957996 T^{8} + 36434286 T^{9} - 4294337 T^{10} - 665780 T^{11} + 62799 T^{12} + 4850 T^{13} - 419 T^{14} - 12 T^{15} + T^{16} \)
$61$ \( -674407936 - 4148738816 T + 14053671008 T^{2} - 2802119536 T^{3} - 14390449644 T^{4} + 5870736154 T^{5} + 2806986210 T^{6} - 771558957 T^{7} - 164089809 T^{8} + 41327189 T^{9} + 3043316 T^{10} - 979520 T^{11} + 3676 T^{12} + 8319 T^{13} - 288 T^{14} - 21 T^{15} + T^{16} \)
$67$ \( 3408931808 - 20323155872 T - 67303291956 T^{2} - 57238406888 T^{3} - 8441057944 T^{4} + 11444751736 T^{5} + 6017802531 T^{6} + 779798186 T^{7} - 149012504 T^{8} - 45640208 T^{9} - 694500 T^{10} + 740443 T^{11} + 45481 T^{12} - 4802 T^{13} - 395 T^{14} + 11 T^{15} + T^{16} \)
$71$ \( -7539615417450496 + 4465101971112960 T - 206140027672064 T^{2} - 274570835485952 T^{3} + 29958839157888 T^{4} + 6870609027941 T^{5} - 877886501433 T^{6} - 91068170682 T^{7} + 12332564897 T^{8} + 693369667 T^{9} - 96431784 T^{10} - 3044147 T^{11} + 429418 T^{12} + 7164 T^{13} - 1019 T^{14} - 7 T^{15} + T^{16} \)
$73$ \( -75334284986368 - 60238310709248 T + 2522576110592 T^{2} + 7991502840832 T^{3} + 397807756480 T^{4} - 430447894112 T^{5} - 31964287456 T^{6} + 12087113392 T^{7} + 985119740 T^{8} - 189361874 T^{9} - 15554334 T^{10} + 1645257 T^{11} + 132586 T^{12} - 7337 T^{13} - 577 T^{14} + 13 T^{15} + T^{16} \)
$79$ \( -39769115858944 + 60446999295104 T + 20799197360512 T^{2} - 16116729553312 T^{3} - 7977550854656 T^{4} - 640173331032 T^{5} + 208992939488 T^{6} + 35959103410 T^{7} - 1353382418 T^{8} - 565552841 T^{9} - 9401122 T^{10} + 4053920 T^{11} + 156730 T^{12} - 13777 T^{13} - 685 T^{14} + 18 T^{15} + T^{16} \)
$83$ \( 918869999616 + 7563443576832 T + 11405679911936 T^{2} - 2308653355008 T^{3} - 2794299169088 T^{4} + 458752244320 T^{5} + 127566460384 T^{6} - 22720418912 T^{7} - 2001338868 T^{8} + 454200198 T^{9} + 7919380 T^{10} - 4091141 T^{11} + 57759 T^{12} + 16678 T^{13} - 509 T^{14} - 25 T^{15} + T^{16} \)
$89$ \( 848564373554432 - 640155366472320 T + 113503360595872 T^{2} + 28700863829728 T^{3} - 11881904091856 T^{4} + 477148539376 T^{5} + 307055020794 T^{6} - 39080928814 T^{7} - 2268078741 T^{8} + 645819106 T^{9} - 11152431 T^{10} - 4227859 T^{11} + 196169 T^{12} + 11929 T^{13} - 778 T^{14} - 12 T^{15} + T^{16} \)
$97$ \( -1191522349056 - 1179043501056 T + 1705223894656 T^{2} + 504771995072 T^{3} - 814331653904 T^{4} + 136027717272 T^{5} + 49055969512 T^{6} - 14182771292 T^{7} - 237905300 T^{8} + 346487586 T^{9} - 19018236 T^{10} - 2545545 T^{11} + 217301 T^{12} + 6697 T^{13} - 806 T^{14} - 5 T^{15} + T^{16} \)
show more
show less