Properties

Label 4028.2.a.d
Level 4028
Weight 2
Character orbit 4028.a
Self dual Yes
Analytic conductor 32.164
Analytic rank 1
Dimension 19
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4028.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.1637419342\)
Analytic rank: \(1\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -\beta_{6} q^{5} + ( -1 + \beta_{11} ) q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -\beta_{6} q^{5} + ( -1 + \beta_{11} ) q^{7} + ( 1 + \beta_{2} ) q^{9} + ( -1 - \beta_{4} ) q^{11} + \beta_{16} q^{13} + ( -1 + \beta_{1} - \beta_{2} - \beta_{7} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{15} + ( \beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{13} - \beta_{15} ) q^{17} - q^{19} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{21} + ( -2 + \beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} - \beta_{9} - \beta_{11} + \beta_{13} ) q^{23} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{25} + ( -\beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{16} + \beta_{17} ) q^{27} + ( -1 + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} - \beta_{11} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{29} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{31} + ( 1 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{16} ) q^{33} + ( -\beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{35} + ( -1 + \beta_{1} - \beta_{8} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{18} ) q^{37} + ( -2 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} - \beta_{16} + \beta_{17} ) q^{39} + ( 2 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{41} + ( \beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{9} + \beta_{10} - \beta_{13} + \beta_{15} - \beta_{18} ) q^{43} + ( -2 + 3 \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{16} - \beta_{17} - \beta_{18} ) q^{45} + ( -1 - \beta_{2} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{18} ) q^{47} + ( \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{11} + \beta_{13} - \beta_{16} ) q^{49} + ( -4 - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} + 3 \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{51} + q^{53} + ( -3 + 2 \beta_{1} + 2 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14} - 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{55} + \beta_{1} q^{57} + ( -3 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{17} + \beta_{18} ) q^{59} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{8} - \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{16} + 2 \beta_{18} ) q^{61} + ( -3 + \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{63} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{65} + ( -2 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{67} + ( -4 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{10} + \beta_{12} + 4 \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{69} + ( -\beta_{3} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{16} ) q^{71} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{10} + \beta_{11} - 3 \beta_{13} + \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{73} + ( 1 - 2 \beta_{1} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{14} + \beta_{15} + \beta_{17} ) q^{75} + ( 2 - \beta_{2} + \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{13} - \beta_{17} - \beta_{18} ) q^{77} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{16} ) q^{79} + ( 1 - 2 \beta_{1} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} + 3 \beta_{16} - \beta_{17} ) q^{81} + ( -1 + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{13} - \beta_{14} - \beta_{17} + \beta_{18} ) q^{83} + ( -5 + \beta_{3} - 3 \beta_{4} + \beta_{6} - 3 \beta_{9} + 2 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{85} + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{14} + 2 \beta_{16} + \beta_{17} - \beta_{18} ) q^{87} + ( -3 + \beta_{1} - \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{14} + \beta_{17} - 2 \beta_{18} ) q^{89} + ( -4 + \beta_{1} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} - \beta_{13} + \beta_{17} - \beta_{18} ) q^{91} + ( -3 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{8} + 2 \beta_{10} - 2 \beta_{13} + 3 \beta_{15} - 2 \beta_{16} + \beta_{17} + \beta_{18} ) q^{93} + \beta_{6} q^{95} + ( -3 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{16} + 2 \beta_{17} - \beta_{18} ) q^{97} + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} + \beta_{16} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19q - 4q^{3} - 2q^{5} - 11q^{7} + 19q^{9} + O(q^{10}) \) \( 19q - 4q^{3} - 2q^{5} - 11q^{7} + 19q^{9} - 11q^{11} + q^{13} - 20q^{15} - q^{17} - 19q^{19} - 10q^{21} - 16q^{23} + 21q^{25} - 4q^{27} - 9q^{31} + 7q^{33} - 25q^{37} - 25q^{39} + q^{41} - 41q^{43} - 27q^{45} - 29q^{47} + 14q^{49} - 24q^{51} + 19q^{53} - 28q^{55} + 4q^{57} - 42q^{59} + q^{61} - 41q^{63} + 2q^{65} - 41q^{67} - 25q^{69} - 20q^{73} + 11q^{75} - 19q^{77} - 38q^{79} + 23q^{81} - 36q^{83} - 58q^{85} - 30q^{87} - 25q^{89} - 55q^{91} - 38q^{93} + 2q^{95} - 13q^{97} + q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19} - 4 x^{18} - 30 x^{17} + 132 x^{16} + 332 x^{15} - 1714 x^{14} - 1598 x^{13} + 11179 x^{12} + 2544 x^{11} - 38897 x^{10} + 3416 x^{9} + 71354 x^{8} - 10941 x^{7} - 64854 x^{6} + 219 x^{5} + 26515 x^{4} + 5796 x^{3} - 2375 x^{2} - 826 x - 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\(1746095294249905525 \nu^{18} - 9882149061894242460 \nu^{17} - 43375197057086977459 \nu^{16} + 320664495983429416571 \nu^{15} + 289170464821686701843 \nu^{14} - 4057421456854802221336 \nu^{13} + 855756579969074611934 \nu^{12} + 25399920257422537055929 \nu^{11} - 18076037847766854541528 \nu^{10} - 82627527926263228597648 \nu^{9} + 77255857497718219255495 \nu^{8} + 135462042252905651618449 \nu^{7} - 128499761875924242856430 \nu^{6} - 103779970138095754748039 \nu^{5} + 69513017742215683190015 \nu^{4} + 39595263533206333032434 \nu^{3} - 6989008533635252643771 \nu^{2} - 4164221605907178213362 \nu - 36307996555354849920\)\()/ 57367504414897716263 \)
\(\beta_{4}\)\(=\)\((\)\(-2943780406006641765 \nu^{18} + 12694984241530833769 \nu^{17} + 84076917574960256843 \nu^{16} - 413624666729155238549 \nu^{15} - 839866884795173919115 \nu^{14} + 5268285624886140485261 \nu^{13} + 2963377924552730128550 \nu^{12} - 33332936791207512527838 \nu^{11} + 3431892679435488472801 \nu^{10} + 110332407390778069031044 \nu^{9} - 45746760297992265654926 \nu^{8} - 186042123550325361577403 \nu^{7} + 91287776562331314493241 \nu^{6} + 148143090411297334733782 \nu^{5} - 46789161173125655425506 \nu^{4} - 55943767001779800289086 \nu^{3} + 874953622100007023511 \nu^{2} + 5689248964462977037139 \nu + 483921811060713145533\)\()/ 57367504414897716263 \)
\(\beta_{5}\)\(=\)\((\)\(5367366175361990662 \nu^{18} - 21770149302925042489 \nu^{17} - 157775075461770482860 \nu^{16} + 713001287691853722889 \nu^{15} + 1674137724859097326569 \nu^{14} - 9154097347335590647765 \nu^{13} - 7164617461856096041454 \nu^{12} + 58657910015823290199629 \nu^{11} + 4317230792299849092388 \nu^{10} - 198272510073702647686792 \nu^{9} + 51617732662797384306580 \nu^{8} + 346388332779359323049071 \nu^{7} - 122717780039396478438628 \nu^{6} - 291230576907575657777941 \nu^{5} + 64803520076310100110799 \nu^{4} + 111813032790893018762455 \nu^{3} + 1557379993306038135353 \nu^{2} - 10716774429709574464823 \nu - 1163592902846952508077\)\()/ 57367504414897716263 \)
\(\beta_{6}\)\(=\)\((\)\(5986540821363048083 \nu^{18} - 26680747861818192613 \nu^{17} - 168178430493991923146 \nu^{16} + 868798848549651489109 \nu^{15} + 1616710294741622158139 \nu^{14} - 11055794589174623184396 \nu^{13} - 4865977009331272204910 \nu^{12} + 69849775976451597633276 \nu^{11} - 14283739980853695730737 \nu^{10} - 230632984523399313959543 \nu^{9} + 116850638684083221365791 \nu^{8} + 387071868444202644915812 \nu^{7} - 224004286752909750898195 \nu^{6} - 304946572937529267890946 \nu^{5} + 121318731949279369464324 \nu^{4} + 112359898898568469548753 \nu^{3} - 7902964447253402551451 \nu^{2} - 10555286174427386953037 \nu - 767982636210474333887\)\()/ 57367504414897716263 \)
\(\beta_{7}\)\(=\)\((\)\(7634075689117315443 \nu^{18} - 32234704197001569255 \nu^{17} - 218246461204391413265 \nu^{16} + 1048160645118184606647 \nu^{15} + 2182009360343441981983 \nu^{14} - 13310513813794211859868 \nu^{13} - 7685249755764291736826 \nu^{12} + 83812009843926202606782 \nu^{11} - 9376119487851360027860 \nu^{10} - 275051090647484797769281 \nu^{9} + 122805265553987816991443 \nu^{8} + 455968562412609108317925 \nu^{7} - 251510394749040653365510 \nu^{6} - 349870200047557082743712 \nu^{5} + 143986447003899364514933 \nu^{4} + 124479998890880575687817 \nu^{3} - 13988656566371121836817 \nu^{2} - 11787538564722251354047 \nu - 425409066286894832191\)\()/ 57367504414897716263 \)
\(\beta_{8}\)\(=\)\((\)\(10583488570732583080 \nu^{18} - 46317271424766431009 \nu^{17} - 300153156956505182157 \nu^{16} + 1509736642100354109756 \nu^{15} + 2948870386732726476653 \nu^{14} - 19241747310288014109107 \nu^{13} - 9727120821862602810878 \nu^{12} + 121870727971300884271619 \nu^{11} - 18448630739731778650887 \nu^{10} - 404093377305337244125151 \nu^{9} + 185842642247020067427678 \nu^{8} + 683279953541843260445629 \nu^{7} - 366628352549431740785521 \nu^{6} - 545424751792344385783695 \nu^{5} + 199384122108507089701670 \nu^{4} + 203416584074287209867702 \nu^{3} - 11348298894438304802815 \nu^{2} - 20001792233774411403989 \nu - 1571164249019956057369\)\()/ 57367504414897716263 \)
\(\beta_{9}\)\(=\)\((\)\(-11670781974338432578 \nu^{18} + 50707552311051149897 \nu^{17} + 333006147432104940562 \nu^{16} - 1656236244441767709207 \nu^{15} - 3314682900513242189374 \nu^{14} + 21174240238043555130948 \nu^{13} + 11471880999951571487763 \nu^{12} - 134764644450956180041717 \nu^{11} + 16183010811149986450309 \nu^{10} + 450451640075255791543425 \nu^{9} - 194246883387390224925121 \nu^{8} - 772172666021715621775968 \nu^{7} + 395419269034118877063649 \nu^{6} + 629862347828509404003707 \nu^{5} - 225126564890101466278021 \nu^{4} - 236977996666831952475416 \nu^{3} + 17288575867973303529433 \nu^{2} + 22881633776118358856299 \nu + 1423735123658409096612\)\()/ 57367504414897716263 \)
\(\beta_{10}\)\(=\)\((\)\(12887337244309102310 \nu^{18} - 56320378776851402420 \nu^{17} - 366733316124510233831 \nu^{16} + 1838393784492939905825 \nu^{15} + 3630523757720748227307 \nu^{14} - 23480695927427830562029 \nu^{13} - 12335770759364466632990 \nu^{12} + 149226602100555301111447 \nu^{11} - 19508279410526756576822 \nu^{10} - 497659654230141998572598 \nu^{9} + 217340664863567212748095 \nu^{8} + 850195825097354486442429 \nu^{7} - 433707429116295140456111 \nu^{6} - 691072682416700603098456 \nu^{5} + 235680140090945263367131 \nu^{4} + 261986926110365744568328 \nu^{3} - 11570439594905860465419 \nu^{2} - 26260223104617142553418 \nu - 2227351681199340420998\)\()/ 57367504414897716263 \)
\(\beta_{11}\)\(=\)\((\)\(-2774197342220715149 \nu^{18} + 12176056338712085544 \nu^{17} + 78708560151114998678 \nu^{16} - 397380567530873041216 \nu^{15} - 773719932918447123382 \nu^{14} + 5074222031303082478672 \nu^{13} + 2554374730253101640076 \nu^{12} - 32235027301369914602315 \nu^{11} + 4852906604258270437399 \nu^{10} + 107424995269575245141776 \nu^{9} - 49008248157728226951319 \nu^{8} - 183275383715474459093238 \nu^{7} + 97207718232382746830615 \nu^{6} + 148560170897994524266440 \nu^{5} - 53721696104914722775357 \nu^{4} - 56048267335871887838477 \nu^{3} + 3325967357404050000805 \nu^{2} + 5506401536397342947630 \nu + 422929458039533903359\)\()/ 8195357773556816609 \)
\(\beta_{12}\)\(=\)\((\)\(19791322111485126204 \nu^{18} - 87950799774851853575 \nu^{17} - 558589882809057011560 \nu^{16} + 2868546645954305222135 \nu^{15} + 5426248967727490103921 \nu^{14} - 36591913083644724216675 \nu^{13} - 17060810333536869116116 \nu^{12} + 232074251927807865451673 \nu^{11} - 41705165044881695029030 \nu^{10} - 771230824813105073147062 \nu^{9} + 371603559829635867753852 \nu^{8} + 1309212673190232652360595 \nu^{7} - 725952232729224716384505 \nu^{6} - 1051964580499689009862854 \nu^{5} + 401836188374829621113285 \nu^{4} + 393788704637746081556702 \nu^{3} - 27127912924828904331696 \nu^{2} - 38505505587758768950801 \nu - 2899363062180072381215\)\()/ 57367504414897716263 \)
\(\beta_{13}\)\(=\)\((\)\(-22289867140185408278 \nu^{18} + 97050618546137680209 \nu^{17} + 633802788787607721493 \nu^{16} - 3165720527951712937811 \nu^{15} - 6261465310879068713631 \nu^{14} + 40391910531833465482195 \nu^{13} + 21079302661832812973500 \nu^{12} - 256270775630394694065260 \nu^{11} + 35640582461401394679617 \nu^{10} + 852117377936432109055784 \nu^{9} - 383718486771389100286389 \nu^{8} - 1447524651396993508147989 \nu^{7} + 767925470706985433199518 \nu^{6} + 1163341240616084616448015 \nu^{5} - 428362716991615958333398 \nu^{4} - 434094089528405914136182 \nu^{3} + 29895819272271390186238 \nu^{2} + 42072549651667420730118 \nu + 2872552231573007321058\)\()/ 57367504414897716263 \)
\(\beta_{14}\)\(=\)\((\)\(-24998369798017516548 \nu^{18} + 109803036871893909119 \nu^{17} + 707647289178142272786 \nu^{16} - 3579202409104159368513 \nu^{15} - 6921202134071797919604 \nu^{14} + 45618730798133385926743 \nu^{13} + 22393926429258917147637 \nu^{12} - 288938484730705145961233 \nu^{11} + 47451143554738106380962 \nu^{10} + 958002424674232127312982 \nu^{9} - 452810635090310303890993 \nu^{8} - 1619413654701392804562061 \nu^{7} + 892123991041871910598122 \nu^{6} + 1291257565269313627230584 \nu^{5} - 495044373763106739194155 \nu^{4} - 480279123762301493691939 \nu^{3} + 35926639666028662100641 \nu^{2} + 47174575583883340369582 \nu + 3061469864996299447894\)\()/ 57367504414897716263 \)
\(\beta_{15}\)\(=\)\((\)\(-29681376895996197944 \nu^{18} + 130670739174724825698 \nu^{17} + 840355296619823602257 \nu^{16} - 4261845844975398059130 \nu^{15} - 8222505495437665605209 \nu^{14} + 54366704678379851026705 \nu^{13} + 26650501384434821329586 \nu^{12} - 344834623061650905277647 \nu^{11} + 55934569543284961625347 \nu^{10} + 1146171567520701030543707 \nu^{9} - 536095155374584599520342 \nu^{8} - 1946576197255065457127915 \nu^{7} + 1055431957751715212449711 \nu^{6} + 1566178638821494910920206 \nu^{5} - 581346941904714610628111 \nu^{4} - 588597017763378213232673 \nu^{3} + 37102300925453892343376 \nu^{2} + 58078743700198732636821 \nu + 4320986960026640338519\)\()/ 57367504414897716263 \)
\(\beta_{16}\)\(=\)\((\)\(32126750325000417763 \nu^{18} - 140598400564189312527 \nu^{17} - 912383701329069111656 \nu^{16} + 4587401677629310734476 \nu^{15} + 8989055740957247679851 \nu^{14} - 58553492280121244504793 \nu^{13} - 29944337197098411648676 \nu^{12} + 371721318395513424412158 \nu^{11} - 53976877019970239794073 \nu^{10} - 1237272726567812173162473 \nu^{9} + 560778203360560864323400 \nu^{8} + 2105826776065409602738968 \nu^{7} - 1116865682365930558280660 \nu^{6} - 1698478842811708781813908 \nu^{5} + 620498805560384532184000 \nu^{4} + 636235023035425463017489 \nu^{3} - 41733388729028138072672 \nu^{2} - 62045186303231683408650 \nu - 4269370263534605129033\)\()/ 57367504414897716263 \)
\(\beta_{17}\)\(=\)\((\)\(34139576512779733606 \nu^{18} - 148902808973472428057 \nu^{17} - 968844809569255216252 \nu^{16} + 4853538593332586374159 \nu^{15} + 9529404779309219804362 \nu^{14} - 61857820411663542311015 \nu^{13} - 31532631822481732262996 \nu^{12} + 391759239964210643251937 \nu^{11} - 59144568971774230229251 \nu^{10} - 1298635512620329536916196 \nu^{9} + 601711962492870212285946 \nu^{8} + 2193802864638411868169019 \nu^{7} - 1196825848752823800956711 \nu^{6} - 1745435831325722933254093 \nu^{5} + 668892019683462644248792 \nu^{4} + 645175837406879853982374 \nu^{3} - 49686465088446824450421 \nu^{2} - 62369066267108048404051 \nu - 4043661676142879980322\)\()/ 57367504414897716263 \)
\(\beta_{18}\)\(=\)\((\)\(36723603444103622524 \nu^{18} - 159238822068501653493 \nu^{17} - 1047361719567607781222 \nu^{16} + 5197633440043365121890 \nu^{15} + 10418007751645399158371 \nu^{14} - 66383634490649735076473 \nu^{13} - 36017717369322207106991 \nu^{12} + 421858313389172862318606 \nu^{11} - 50686847457296408089964 \nu^{10} - 1406601328489586758783901 \nu^{9} + 606407207630109272285654 \nu^{8} + 2401546521589485076123624 \nu^{7} - 1224609347834873327463388 \nu^{6} - 1947900714257381645846663 \nu^{5} + 678401614172280322061840 \nu^{4} + 733853893038264699002650 \nu^{3} - 41271629821614059947905 \nu^{2} - 72413223583601264960432 \nu - 5256536700393855125760\)\()/ 57367504414897716263 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{17} + \beta_{16} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} + \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{17} + 3 \beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{11} - 2 \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 9 \beta_{2} - 2 \beta_{1} + 28\)
\(\nu^{5}\)\(=\)\(-\beta_{18} - 14 \beta_{17} + 14 \beta_{16} - \beta_{15} - 12 \beta_{14} + 12 \beta_{13} + 14 \beta_{12} + \beta_{11} - 14 \beta_{10} + 13 \beta_{9} - \beta_{8} + 15 \beta_{7} - 13 \beta_{6} + 16 \beta_{5} + 13 \beta_{4} + 2 \beta_{3} + \beta_{2} + 56 \beta_{1} - 2\)
\(\nu^{6}\)\(=\)\(-3 \beta_{18} - 13 \beta_{17} + 51 \beta_{16} - 21 \beta_{15} + 15 \beta_{14} + 29 \beta_{13} + 15 \beta_{11} - 33 \beta_{10} + 14 \beta_{9} + 13 \beta_{8} + 18 \beta_{7} - 18 \beta_{6} + 6 \beta_{5} + 2 \beta_{4} + 84 \beta_{2} - 31 \beta_{1} + 220\)
\(\nu^{7}\)\(=\)\(-15 \beta_{18} - 160 \beta_{17} + 163 \beta_{16} - 20 \beta_{15} - 123 \beta_{14} + 118 \beta_{13} + 158 \beta_{12} + 27 \beta_{11} - 157 \beta_{10} + 150 \beta_{9} - 19 \beta_{8} + 175 \beta_{7} - 148 \beta_{6} + 192 \beta_{5} + 147 \beta_{4} + 36 \beta_{3} + 25 \beta_{2} + 479 \beta_{1} - 13\)
\(\nu^{8}\)\(=\)\(-64 \beta_{18} - 143 \beta_{17} + 674 \beta_{16} - 318 \beta_{15} + 175 \beta_{14} + 479 \beta_{13} - 4 \beta_{12} + 195 \beta_{11} - 425 \beta_{10} + 157 \beta_{9} + 135 \beta_{8} + 242 \beta_{7} - 238 \beta_{6} + 126 \beta_{5} + 51 \beta_{4} - 6 \beta_{3} + 809 \beta_{2} - 374 \beta_{1} + 1845\)
\(\nu^{9}\)\(=\)\(-186 \beta_{18} - 1716 \beta_{17} + 1810 \beta_{16} - 311 \beta_{15} - 1213 \beta_{14} + 1137 \beta_{13} + 1665 \beta_{12} + 460 \beta_{11} - 1668 \beta_{10} + 1645 \beta_{9} - 243 \beta_{8} + 1893 \beta_{7} - 1617 \beta_{6} + 2130 \beta_{5} + 1611 \beta_{4} + 481 \beta_{3} + 403 \beta_{2} + 4269 \beta_{1} + 18\)
\(\nu^{10}\)\(=\)\(-963 \beta_{18} - 1529 \beta_{17} + 8115 \beta_{16} - 4215 \beta_{15} + 1882 \beta_{14} + 6487 \beta_{13} - 82 \beta_{12} + 2422 \beta_{11} - 5033 \beta_{10} + 1673 \beta_{9} + 1317 \beta_{8} + 2916 \beta_{7} - 2835 \beta_{6} + 1879 \beta_{5} + 876 \beta_{4} - 172 \beta_{3} + 7939 \beta_{2} - 4168 \beta_{1} + 16183\)
\(\nu^{11}\)\(=\)\(-2222 \beta_{18} - 17931 \beta_{17} + 19781 \beta_{16} - 4373 \beta_{15} - 11876 \beta_{14} + 11262 \beta_{13} + 17055 \beta_{12} + 6505 \beta_{11} - 17514 \beta_{10} + 17530 \beta_{9} - 2659 \beta_{8} + 19893 \beta_{7} - 17406 \beta_{6} + 22998 \beta_{5} + 17433 \beta_{4} + 5754 \beta_{3} + 5465 \beta_{2} + 39070 \beta_{1} + 1667\)
\(\nu^{12}\)\(=\)\(-12567 \beta_{18} - 16352 \beta_{17} + 93240 \beta_{16} - 51976 \beta_{15} + 19521 \beta_{14} + 79899 \beta_{13} - 1081 \beta_{12} + 29186 \beta_{11} - 57334 \beta_{10} + 17665 \beta_{9} + 12565 \beta_{8} + 33361 \beta_{7} - 32252 \beta_{6} + 24558 \beta_{5} + 12689 \beta_{4} - 3068 \beta_{3} + 78916 \beta_{2} - 44959 \beta_{1} + 146604\)
\(\nu^{13}\)\(=\)\(-26256 \beta_{18} - 185243 \beta_{17} + 214944 \beta_{16} - 57740 \beta_{15} - 116593 \beta_{14} + 115702 \beta_{13} + 172554 \beta_{12} + 83565 \beta_{11} - 184020 \beta_{10} + 183851 \beta_{9} - 26951 \beta_{8} + 206620 \beta_{7} - 186246 \beta_{6} + 245676 \beta_{5} + 187232 \beta_{4} + 65137 \beta_{3} + 67968 \beta_{2} + 364091 \beta_{1} + 29492\)
\(\nu^{14}\)\(=\)\(-152326 \beta_{18} - 175877 \beta_{17} + 1042772 \beta_{16} - 612969 \beta_{15} + 198689 \beta_{14} + 933588 \beta_{13} - 11355 \beta_{12} + 343172 \beta_{11} - 639139 \beta_{10} + 187139 \beta_{9} + 118913 \beta_{8} + 371384 \beta_{7} - 358565 \beta_{6} + 300573 \beta_{5} + 167529 \beta_{4} - 44284 \beta_{3} + 792232 \beta_{2} - 477330 \beta_{1} + 1360642\)
\(\nu^{15}\)\(=\)\(-307369 \beta_{18} - 1905027 \beta_{17} + 2330726 \beta_{16} - 728959 \beta_{15} - 1150881 \beta_{14} + 1226306 \beta_{13} + 1737498 \beta_{12} + 1014949 \beta_{11} - 1940848 \beta_{10} + 1912096 \beta_{9} - 262265 \beta_{8} + 2137376 \beta_{7} - 1986879 \beta_{6} + 2611948 \beta_{5} + 2000591 \beta_{4} + 714199 \beta_{3} + 805094 \beta_{2} + 3437395 \beta_{1} + 401589\)
\(\nu^{16}\)\(=\)\(-1768769 \beta_{18} - 1901705 \beta_{17} + 11466798 \beta_{16} - 7018125 \beta_{15} + 2000621 \beta_{14} + 10563867 \beta_{13} - 98886 \beta_{12} + 3954773 \beta_{11} - 7028045 \beta_{10} + 1995140 \beta_{9} + 1123630 \beta_{8} + 4071289 \beta_{7} - 3937297 \beta_{6} + 3542969 \beta_{5} + 2089967 \beta_{4} - 568040 \beta_{3} + 8017265 \beta_{2} - 5022238 \beta_{1} + 12868814\)
\(\nu^{17}\)\(=\)\(-3559419 \beta_{18} - 19569956 \beta_{17} + 25251168 \beta_{16} - 8902203 \beta_{15} - 11423098 \beta_{14} + 13286046 \beta_{13} + 17479989 \beta_{12} + 11897330 \beta_{11} - 20550023 \beta_{10} + 19808271 \beta_{9} - 2496977 \beta_{8} + 22101561 \beta_{7} - 21154091 \beta_{6} + 27701411 \beta_{5} + 21299332 \beta_{4} + 7674698 \beta_{3} + 9258227 \beta_{2} + 32774287 \beta_{1} + 4915976\)
\(\nu^{18}\)\(=\)\(-19994652 \beta_{18} - 20638398 \beta_{17} + 124696919 \beta_{16} - 78717542 \beta_{15} + 20014792 \beta_{14} + 117066253 \beta_{13} - 660934 \beta_{12} + 44843477 \beta_{11} - 76560272 \beta_{10} + 21401468 \beta_{9} + 10640001 \beta_{8} + 44228607 \beta_{7} - 42930433 \beta_{6} + 40781914 \beta_{5} + 25117599 \beta_{4} - 6767736 \beta_{3} + 81676523 \beta_{2} - 52525018 \beta_{1} + 123576982\)

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
3.25955
2.97522
2.44269
2.40459
2.05170
1.96979
1.76613
1.01911
0.383819
−0.0800487
−0.346400
−0.365086
−0.537191
−0.755832
−1.67288
−1.94972
−2.53438
−2.85652
−3.17454
0 −3.25955 0 3.01882 0 1.05463 0 7.62467 0
1.2 0 −2.97522 0 0.840694 0 −4.22061 0 5.85193 0
1.3 0 −2.44269 0 −3.59133 0 0.459217 0 2.96674 0
1.4 0 −2.40459 0 −0.489610 0 4.26722 0 2.78204 0
1.5 0 −2.05170 0 2.67326 0 0.688434 0 1.20949 0
1.6 0 −1.96979 0 −1.29651 0 −5.21755 0 0.880060 0
1.7 0 −1.76613 0 −1.36988 0 0.174249 0 0.119213 0
1.8 0 −1.01911 0 3.39604 0 −2.78127 0 −1.96142 0
1.9 0 −0.383819 0 −4.10226 0 0.287206 0 −2.85268 0
1.10 0 0.0800487 0 0.442813 0 2.60518 0 −2.99359 0
1.11 0 0.346400 0 0.720634 0 −2.22237 0 −2.88001 0
1.12 0 0.365086 0 1.83829 0 2.27559 0 −2.86671 0
1.13 0 0.537191 0 −2.35184 0 −1.00979 0 −2.71143 0
1.14 0 0.755832 0 4.26919 0 −2.09147 0 −2.42872 0
1.15 0 1.67288 0 −0.778334 0 3.24553 0 −0.201474 0
1.16 0 1.94972 0 1.42435 0 −4.01869 0 0.801405 0
1.17 0 2.53438 0 −2.27936 0 1.27718 0 3.42306 0
1.18 0 2.85652 0 −0.0885731 0 −1.03776 0 5.15970 0
1.19 0 3.17454 0 −4.27638 0 −4.73492 0 7.07773 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(1\)
\(53\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4028))\):

\(T_{3}^{19} + \cdots\)
\(T_{5}^{19} + \cdots\)