Properties

Label 8004.2.a.k
Level $8004$
Weight $2$
Character orbit 8004.a
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $1$

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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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + 76571 x^{10} - 655793 x^{9} - 114554 x^{8} + 2789438 x^{7} - 855636 x^{6} - 6184176 x^{5} + 4260960 x^{4} + 5001480 x^{3} - 5659296 x^{2} + 1509552 x - 115488\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{1} q^{5} + \beta_{7} q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{1} q^{5} + \beta_{7} q^{7} + q^{9} -\beta_{9} q^{11} + \beta_{8} q^{13} + \beta_{1} q^{15} -\beta_{13} q^{17} + ( 1 + \beta_{4} ) q^{19} + \beta_{7} q^{21} - q^{23} + ( 3 + \beta_{2} ) q^{25} + q^{27} + q^{29} + ( 1 + \beta_{1} - \beta_{6} - \beta_{8} + \beta_{14} ) q^{31} -\beta_{9} q^{33} + ( \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{35} + ( 1 + \beta_{16} ) q^{37} + \beta_{8} q^{39} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{41} + ( 1 + \beta_{15} ) q^{43} + \beta_{1} q^{45} + ( -1 - \beta_{6} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{47} + ( 3 + \beta_{6} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{17} ) q^{49} -\beta_{13} q^{51} + ( 2 - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{53} + ( 1 - \beta_{1} - 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{55} + ( 1 + \beta_{4} ) q^{57} + ( \beta_{1} - \beta_{2} + \beta_{6} + \beta_{13} ) q^{59} + ( -\beta_{2} - \beta_{5} - \beta_{12} - \beta_{13} ) q^{61} + \beta_{7} q^{63} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} - \beta_{14} + \beta_{17} ) q^{65} + ( 1 - \beta_{1} - \beta_{2} + \beta_{6} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{17} ) q^{67} - q^{69} + ( -\beta_{3} + \beta_{4} - \beta_{8} - \beta_{13} ) q^{71} + ( -\beta_{4} + \beta_{6} + \beta_{8} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{73} + ( 3 + \beta_{2} ) q^{75} + ( -1 + \beta_{1} - \beta_{4} - 2 \beta_{9} + \beta_{10} - \beta_{12} + \beta_{15} + \beta_{16} ) q^{77} + ( \beta_{8} + \beta_{13} - \beta_{17} ) q^{79} + q^{81} + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{16} ) q^{83} + ( 1 - \beta_{3} + \beta_{6} + \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{85} + q^{87} + ( 1 + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{89} + ( 3 + \beta_{1} + 2 \beta_{3} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{91} + ( 1 + \beta_{1} - \beta_{6} - \beta_{8} + \beta_{14} ) q^{93} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{95} + ( 4 + 2 \beta_{1} + \beta_{2} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{97} -\beta_{9} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 18q^{3} + 5q^{5} + 6q^{7} + 18q^{9} + O(q^{10}) \) \( 18q + 18q^{3} + 5q^{5} + 6q^{7} + 18q^{9} + 5q^{11} + 6q^{13} + 5q^{15} + 7q^{17} + 15q^{19} + 6q^{21} - 18q^{23} + 45q^{25} + 18q^{27} + 18q^{29} + 10q^{31} + 5q^{33} - 7q^{35} + 22q^{37} + 6q^{39} + 17q^{41} + 25q^{43} + 5q^{45} - 6q^{47} + 64q^{49} + 7q^{51} + 21q^{53} + 3q^{55} + 15q^{57} + 6q^{59} + 7q^{61} + 6q^{63} + 44q^{65} + 35q^{67} - 18q^{69} + q^{71} + 27q^{73} + 45q^{75} + 4q^{77} + 18q^{81} + 13q^{83} + 24q^{85} + 18q^{87} + 30q^{89} + 53q^{91} + 10q^{93} + 17q^{95} + 27q^{97} + 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + 76571 x^{10} - 655793 x^{9} - 114554 x^{8} + 2789438 x^{7} - 855636 x^{6} - 6184176 x^{5} + 4260960 x^{4} + 5001480 x^{3} - 5659296 x^{2} + 1509552 x - 115488\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 8 \)
\(\beta_{3}\)\(=\)\((\)\(2927284725800794255337 \nu^{17} + 11207570645722098877963 \nu^{16} - 258457330225268973228733 \nu^{15} - 605877232513046416552126 \nu^{14} + 8810425247375992739248360 \nu^{13} + 13256749203470338601883184 \nu^{12} - 154879314725306781562364529 \nu^{11} - 148490565710478946852143558 \nu^{10} + 1539624796713181155670596745 \nu^{9} + 877757215823034564311919639 \nu^{8} - 8751517417440225088359606592 \nu^{7} - 2437178316183588229476738912 \nu^{6} + 26795700891125903440393467120 \nu^{5} + 1490503315719243326926821168 \nu^{4} - 36435941553545754661166169408 \nu^{3} + 4381380361918556505105405528 \nu^{2} + 9713974267365832529985440160 \nu - 2924380632926426855889005136\)\()/ \)\(34\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-15856898046747700680671 \nu^{17} + 20481619523436301822646 \nu^{16} + 1158425410995214634885614 \nu^{15} - 1558114069513149064931567 \nu^{14} - 34245685489149462402114880 \nu^{13} + 47118126192082374503085128 \nu^{12} + 529806407574936006061495707 \nu^{11} - 744048668514131700022046961 \nu^{10} - 4600025462646453193930691635 \nu^{9} + 6674535572415086879154761238 \nu^{8} + 22070720640860118940025133811 \nu^{7} - 34093655015930802739541252754 \nu^{6} - 51815390154058981684358411460 \nu^{5} + 91328367074137931967426591756 \nu^{4} + 33691959097630028455317049464 \nu^{3} - 96757902891444698299605604224 \nu^{2} + 38674162220774257482841017720 \nu - 5257001432083204378379490912\)\()/ \)\(34\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-19834097645988095110793 \nu^{17} + 50260217613457649535518 \nu^{16} + 975993069880071227811562 \nu^{15} - 1738114125253927213119461 \nu^{14} - 18690205504873808659084540 \nu^{13} + 10935750146712076138961624 \nu^{12} + 171388383393455234023881081 \nu^{11} + 283654137960019281643287837 \nu^{10} - 733187440012258688665178305 \nu^{9} - 5389983465140058970299406446 \nu^{8} + 1410447997834672133763623113 \nu^{7} + 37182686942633677528041243018 \nu^{6} - 6839508136673965432165924380 \nu^{5} - 115592059513900897373079693552 \nu^{4} + 46698395583386654989372541112 \nu^{3} + 131671159993904411990283618408 \nu^{2} - 90209877165370623418178176440 \nu + 10695232811231753481967711104\)\()/ \)\(34\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(41412917201663495765029 \nu^{17} - 242158390012619827991104 \nu^{16} - 1962686869754752843704686 \nu^{15} + 12677064418113551600564233 \nu^{14} + 35986054589389871064974270 \nu^{13} - 269664845965959593069360572 \nu^{12} - 309995813495353279322649693 \nu^{11} + 2995662900716414698105237689 \nu^{10} + 1059451342617790598718407015 \nu^{9} - 18559515190192405343680079712 \nu^{8} + 1733250830043557095125535111 \nu^{7} + 63119926944263630390491559796 \nu^{6} - 24356837027641604097167219460 \nu^{5} - 106229023753839406893910752744 \nu^{4} + 67246741158110098928910237264 \nu^{3} + 62697508676133717281912410776 \nu^{2} - 61660011299197642888375646280 \nu + 8636480002243850177143274688\)\()/ \)\(34\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-43333738854790069367731 \nu^{17} + 159157061769265235549581 \nu^{16} + 2431529415250997934497279 \nu^{15} - 8452255223194401946523362 \nu^{14} - 56323942595387279210912330 \nu^{13} + 183420481011514812288520708 \nu^{12} + 692826517348627412394938427 \nu^{11} - 2102329179432804940444823796 \nu^{10} - 4822124902841509064269946885 \nu^{9} + 13740225505480615539555664893 \nu^{8} + 18455128601695451156405460296 \nu^{7} - 51386911406851359650899423494 \nu^{6} - 32780948216448336579167996760 \nu^{5} + 102362362568865716687445020016 \nu^{4} + 6400046343306190805161804704 \nu^{3} - 83071322267623402723018709064 \nu^{2} + 36895192385313984461759761920 \nu - 3999378399434756613970855632\)\()/ \)\(34\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-44718954707623369104061 \nu^{17} + 139307793084177255893761 \nu^{16} + 2600708965267858817240399 \nu^{15} - 7171121978121236543096572 \nu^{14} - 63334249081299279545880230 \nu^{13} + 148845418283002898662051348 \nu^{12} + 835895921149036982929059537 \nu^{11} - 1599623366103212950263091626 \nu^{10} - 6449064309911489713591369835 \nu^{9} + 9523526946947109432701706333 \nu^{8} + 29161315269817013716219877426 \nu^{7} - 31224180682863786864906001914 \nu^{6} - 72956988236190067499062374360 \nu^{5} + 52502575807393264684246688796 \nu^{4} + 85756363701189933343131301224 \nu^{3} - 37524938006783953794801845184 \nu^{2} - 27283903064457130077361534080 \nu + 6040669928408817177236365008\)\()/ \)\(34\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-51853525011373513017361 \nu^{17} + 92927380737163194056686 \nu^{16} + 3297035103635045498715224 \nu^{15} - 4873239849396061715171647 \nu^{14} - 87281697845317926776793230 \nu^{13} + 104492819933826308740565248 \nu^{12} + 1238799475775927751864882537 \nu^{11} - 1196491429946642089001785251 \nu^{10} - 10105200954165235091298156935 \nu^{9} + 8073008091972462186683944458 \nu^{8} + 47040810137737578862956680351 \nu^{7} - 33261156855348442216504401864 \nu^{6} - 115261731745435518607127291160 \nu^{5} + 79631386952190925576037748096 \nu^{4} + 115083049679871086523463233024 \nu^{3} - 85374333119104489519707116184 \nu^{2} - 1947118300677229386765953880 \nu + 3008097760086639408616881408\)\()/ \)\(34\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(14033228130829226246429 \nu^{17} - 31995528324989864271104 \nu^{16} - 889402633050528900276736 \nu^{15} + 1801536096414810102638408 \nu^{14} + 23367301898416757955738445 \nu^{13} - 41944197979170167366928197 \nu^{12} - 327118924112844966934060443 \nu^{11} + 524448178573778345675274639 \nu^{10} + 2604920670161152338084814165 \nu^{9} - 3822807420872789710614397737 \nu^{8} - 11594313592548550448496538039 \nu^{7} + 16353771186501568785007642521 \nu^{6} + 25717986162282620789330129565 \nu^{5} - 38051816390374258619272801644 \nu^{4} - 17846465682043634395873270686 \nu^{3} + 36578590925784190753196736876 \nu^{2} - 11820928042756235246108674680 \nu + 1128076375034555595788640588\)\()/ \)\(85\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(2043657375292274692832 \nu^{17} - 4132887294948038749297 \nu^{16} - 134035659386486971516363 \nu^{15} + 241274852048796945543259 \nu^{14} + 3649111112421406449063920 \nu^{13} - 5839633611313119245563386 \nu^{12} - 53144243972831898803072104 \nu^{11} + 76002317937248245684942997 \nu^{10} + 443986295934300244350812190 \nu^{9} - 575770363835084394423021311 \nu^{8} - 2108208898351544702370197467 \nu^{7} + 2546930838129243530860438458 \nu^{6} + 5181563055234374441047816730 \nu^{5} - 6097135374116470568948991732 \nu^{4} - 4728364354650308547608786748 \nu^{3} + 6088158990970631638713408288 \nu^{2} - 1120246991235604368346854960 \nu - 30698113361234511387302736\)\()/ \)\(11\!\cdots\!40\)\( \)
\(\beta_{12}\)\(=\)\((\)\(4995941761396610786931 \nu^{17} - 13287830637189807515156 \nu^{16} - 310484410604219777652934 \nu^{15} + 739666407122397339495607 \nu^{14} + 8049631490531154088016450 \nu^{13} - 16989568684753651016724188 \nu^{12} - 112169083731814423593511687 \nu^{11} + 208604337202095783110453411 \nu^{10} + 899747708956589888775125945 \nu^{9} - 1480746492115686220737314128 \nu^{8} - 4102827281786056450482443111 \nu^{7} + 6096668891211704391560605624 \nu^{6} + 9594786857378271634723498540 \nu^{5} - 13516888294155117008705914116 \nu^{4} - 7801609743285289466141240424 \nu^{3} + 12421021416576272215336647024 \nu^{2} - 3318033069803232178388716680 \nu + 256355102982801246997433712\)\()/ \)\(22\!\cdots\!80\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-75583454060712120333914 \nu^{17} + 247359119346637422602489 \nu^{16} + 4499212025046647057467651 \nu^{15} - 13621216980384200787892703 \nu^{14} - 111548002620698527873025470 \nu^{13} + 308918686637788241218752752 \nu^{12} + 1483458306376376908666109838 \nu^{11} - 3734779360628703213576040299 \nu^{10} - 11305719829974373633144705240 \nu^{9} + 25997863081338479397499348917 \nu^{8} + 48375436972321660440373605049 \nu^{7} - 104351645621829317437821724536 \nu^{6} - 101794241554796089134885020040 \nu^{5} + 223571380953424962748150067604 \nu^{4} + 55058426415419523236831591376 \nu^{3} - 195042416927436587160124849416 \nu^{2} + 74031494629762249738102621080 \nu - 7052986200113986695186856608\)\()/ \)\(34\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-25375359856559521468813 \nu^{17} + 50706308357888305174438 \nu^{16} + 1621391099742668687407242 \nu^{15} - 2787059132695762560041201 \nu^{14} - 43062987187257089314087440 \nu^{13} + 63124567150850560089419284 \nu^{12} + 612247323123168837291118721 \nu^{11} - 766444611082574756244411983 \nu^{10} - 4996130029700500791392041605 \nu^{9} + 5438953100444389328700375114 \nu^{8} + 23238852539553548864635576433 \nu^{7} - 22875642795226859604150349662 \nu^{6} - 56795024945993943716391326080 \nu^{5} + 53267848295130048169108835868 \nu^{4} + 56203037642536860065732144592 \nu^{3} - 53002926748150652601934426272 \nu^{2} - 320043913744284501580902840 \nu + 1694910894136937542040729664\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-129493678954510444776307 \nu^{17} + 431773041270738106994782 \nu^{16} + 7430842493517809979493238 \nu^{15} - 22876400833337976918678139 \nu^{14} - 176391017767978015129757960 \nu^{13} + 495073566837989110647323776 \nu^{12} + 2224551856745135776014595119 \nu^{11} - 5662871533265845603953326637 \nu^{10} - 15845377455265245439711524695 \nu^{9} + 37064898126902626404185670246 \nu^{8} + 61819842828166455730611430487 \nu^{7} - 139996025850649010318350558818 \nu^{6} - 111926710864861937694075735420 \nu^{5} + 285458046825356976426144078252 \nu^{4} + 27593374043982656861939793288 \nu^{3} - 241090682282959142754718657008 \nu^{2} + 114258078842512541583720498840 \nu - 11905134249553848674588421504\)\()/ \)\(34\!\cdots\!00\)\( \)
\(\beta_{16}\)\(=\)\((\)\(51630065955629125660147 \nu^{17} - 120497952335555890666272 \nu^{16} - 3301094205914111580435448 \nu^{15} + 6974997754173978279065069 \nu^{14} + 87406529355025077003262660 \nu^{13} - 167826455047710272981218396 \nu^{12} - 1231346651820545843191010399 \nu^{11} + 2179566689407526344392923527 \nu^{10} + 9839305921704552663533784895 \nu^{9} - 16553666775885714098903130716 \nu^{8} - 43601062938602665583605596777 \nu^{7} + 73708886232166849341296927778 \nu^{6} + 93384499139860437324993568320 \nu^{5} - 177373672886268339449404300692 \nu^{4} - 47142890042521339455822020448 \nu^{3} + 173857065163518210547271807568 \nu^{2} - 79273804513783476304358061240 \nu + 8960410328410972391900913984\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-29816274986940605729983 \nu^{17} + 96084227747052021743608 \nu^{16} + 1714342125406004953414072 \nu^{15} - 5012332624848445488632241 \nu^{14} - 40912295647699343410768990 \nu^{13} + 106083697589212727903688994 \nu^{12} + 522015234781145833499003461 \nu^{11} - 1174501743296248588497210103 \nu^{10} - 3809427495290295870543010255 \nu^{9} + 7327142908386201768821339124 \nu^{8} + 15667914927487977019971647903 \nu^{7} - 25833094437610044877549984242 \nu^{6} - 32697004031765348651644703630 \nu^{5} + 48029100724302248127293500938 \nu^{4} + 23299424567320790024088093072 \nu^{3} - 36322685239692333567154626252 \nu^{2} + 9444633811713278028244127160 \nu - 1409401304765454177605790576\)\()/ \)\(56\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 8\)
\(\nu^{3}\)\(=\)\(\beta_{14} + \beta_{12} - \beta_{11} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + 2 \beta_{2} + 11 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(\beta_{16} + 2 \beta_{15} + \beta_{14} + \beta_{13} - \beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{8} - 4 \beta_{7} - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 16 \beta_{2} + 4 \beta_{1} + 92\)
\(\nu^{5}\)\(=\)\(-\beta_{17} + 2 \beta_{16} + 2 \beta_{15} + 24 \beta_{14} - 6 \beta_{13} + 17 \beta_{12} - 21 \beta_{11} - 5 \beta_{10} - 3 \beta_{9} - 26 \beta_{8} - 21 \beta_{7} - 25 \beta_{6} + \beta_{5} - 18 \beta_{4} - 7 \beta_{3} + 45 \beta_{2} + 139 \beta_{1} + 88\)
\(\nu^{6}\)\(=\)\(29 \beta_{16} + 42 \beta_{15} + 49 \beta_{14} - 9 \beta_{12} - 19 \beta_{11} - 42 \beta_{10} - 31 \beta_{9} - 74 \beta_{8} - 88 \beta_{7} - 24 \beta_{6} - 21 \beta_{5} - 61 \beta_{4} - 84 \beta_{3} + 252 \beta_{2} + 120 \beta_{1} + 1229\)
\(\nu^{7}\)\(=\)\(-39 \beta_{17} + 72 \beta_{16} + 59 \beta_{15} + 481 \beta_{14} - 202 \beta_{13} + 238 \beta_{12} - 360 \beta_{11} - 200 \beta_{10} - 96 \beta_{9} - 538 \beta_{8} - 355 \beta_{7} - 513 \beta_{6} + 30 \beta_{5} - 279 \beta_{4} - 233 \beta_{3} + 859 \beta_{2} + 1932 \beta_{1} + 1892\)
\(\nu^{8}\)\(=\)\(-34 \beta_{17} + 637 \beta_{16} + 732 \beta_{15} + 1345 \beta_{14} - 460 \beta_{13} - 272 \beta_{12} - 375 \beta_{11} - 1177 \beta_{10} - 718 \beta_{9} - 1821 \beta_{8} - 1556 \beta_{7} - 907 \beta_{6} - 352 \beta_{5} - 1067 \beta_{4} - 1819 \beta_{3} + 4137 \beta_{2} + 2691 \beta_{1} + 17949\)
\(\nu^{9}\)\(=\)\(-967 \beta_{17} + 1816 \beta_{16} + 1311 \beta_{15} + 9196 \beta_{14} - 4954 \beta_{13} + 3130 \beta_{12} - 5887 \beta_{11} - 5416 \beta_{10} - 2364 \beta_{9} - 10412 \beta_{8} - 5793 \beta_{7} - 9885 \beta_{6} + 582 \beta_{5} - 4424 \beta_{4} - 5799 \beta_{3} + 15771 \beta_{2} + 28717 \beta_{1} + 37154\)
\(\nu^{10}\)\(=\)\(-1479 \beta_{17} + 12606 \beta_{16} + 12264 \beta_{15} + 30580 \beta_{14} - 16021 \beta_{13} - 5919 \beta_{12} - 7643 \beta_{11} - 27770 \beta_{10} - 15233 \beta_{9} - 38674 \beta_{8} - 26072 \beta_{7} - 23797 \beta_{6} - 5487 \beta_{5} - 18525 \beta_{4} - 36643 \beta_{3} + 70769 \beta_{2} + 54584 \beta_{1} + 278596\)
\(\nu^{11}\)\(=\)\(-20278 \beta_{17} + 39789 \beta_{16} + 26568 \beta_{15} + 173546 \beta_{14} - 108389 \beta_{13} + 39319 \beta_{12} - 95331 \beta_{11} - 125262 \beta_{10} - 53547 \beta_{9} - 196193 \beta_{8} - 95117 \beta_{7} - 185651 \beta_{6} + 9143 \beta_{5} - 74128 \beta_{4} - 129617 \beta_{3} + 287384 \beta_{2} + 448718 \beta_{1} + 706338\)
\(\nu^{12}\)\(=\)\(-42563 \beta_{17} + 238053 \beta_{16} + 205200 \beta_{15} + 639849 \beta_{14} - 406287 \beta_{13} - 114274 \beta_{12} - 153251 \beta_{11} - 600305 \beta_{10} - 312170 \beta_{9} - 768222 \beta_{8} - 433100 \beta_{7} - 539007 \beta_{6} - 83802 \beta_{5} - 328162 \beta_{4} - 720973 \beta_{3} + 1249065 \beta_{2} + 1060825 \beta_{1} + 4526448\)
\(\nu^{13}\)\(=\)\(-395183 \beta_{17} + 812873 \beta_{16} + 518825 \beta_{15} + 3266881 \beta_{14} - 2248369 \beta_{13} + 464538 \beta_{12} - 1549484 \beta_{11} - 2677765 \beta_{10} - 1162405 \beta_{9} - 3653181 \beta_{8} - 1589595 \beta_{7} - 3452058 \beta_{6} + 121871 \beta_{5} - 1306554 \beta_{4} - 2743697 \beta_{3} + 5250205 \beta_{2} + 7292502 \beta_{1} + 13272318\)
\(\nu^{14}\)\(=\)\(-1025003 \beta_{17} + 4403594 \beta_{16} + 3474899 \beta_{15} + 12834092 \beta_{14} - 9080592 \beta_{13} - 2093208 \beta_{12} - 2996334 \beta_{11} - 12365414 \beta_{10} - 6287539 \beta_{9} - 14765614 \beta_{8} - 7252720 \beta_{7} - 11334893 \beta_{6} - 1291171 \beta_{5} - 5942542 \beta_{4} - 14066176 \beta_{3} + 22534058 \beta_{2} + 20213449 \beta_{1} + 76218350\)
\(\nu^{15}\)\(=\)\(-7452832 \beta_{17} + 15986248 \beta_{16} + 9963642 \beta_{15} + 61539929 \beta_{14} - 45313575 \beta_{13} + 4904857 \beta_{12} - 25434568 \beta_{11} - 54809437 \beta_{10} - 24510099 \beta_{9} - 67694373 \beta_{8} - 27107101 \beta_{7} - 64033603 \beta_{6} + 1304803 \beta_{5} - 23875828 \beta_{4} - 56234711 \beta_{3} + 96458883 \beta_{2} + 122356044 \beta_{1} + 248546557\)
\(\nu^{16}\)\(=\)\(-22438351 \beta_{17} + 80865737 \beta_{16} + 59837840 \beta_{15} + 251519875 \beta_{14} - 190432856 \beta_{13} - 37477972 \beta_{12} - 57432396 \beta_{11} - 247669390 \beta_{10} - 125290239 \beta_{9} - 279207716 \beta_{8} - 123254528 \beta_{7} - 228688349 \beta_{6} - 20452043 \beta_{5} - 109509687 \beta_{4} - 273376116 \beta_{3} + 412666916 \beta_{2} + 381618956 \beta_{1} + 1320066906\)
\(\nu^{17}\)\(=\)\(-138668160 \beta_{17} + 307761935 \beta_{16} + 189811401 \beta_{15} + 1161068731 \beta_{14} - 897610896 \beta_{13} + 38935284 \beta_{12} - 422962621 \beta_{11} - 1093591518 \beta_{10} - 505467695 \beta_{9} - 1253119679 \beta_{8} - 471319633 \beta_{7} - 1189246853 \beta_{6} + 7389645 \beta_{5} - 446193700 \beta_{4} - 1128869710 \beta_{3} + 1783059963 \beta_{2} + 2107186314 \beta_{1} + 4654580796\)

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
−3.70317
−3.44612
−3.07005
−2.76979
−2.66067
−2.25234
−1.83157
0.140190
0.235801
0.767523
1.69179
1.90345
2.40801
2.45476
3.38212
3.50248
3.88351
4.36408
0 1.00000 0 −3.70317 0 −1.62014 0 1.00000 0
1.2 0 1.00000 0 −3.44612 0 4.67774 0 1.00000 0
1.3 0 1.00000 0 −3.07005 0 −0.993636 0 1.00000 0
1.4 0 1.00000 0 −2.76979 0 −3.16982 0 1.00000 0
1.5 0 1.00000 0 −2.66067 0 4.50596 0 1.00000 0
1.6 0 1.00000 0 −2.25234 0 3.02254 0 1.00000 0
1.7 0 1.00000 0 −1.83157 0 −3.10909 0 1.00000 0
1.8 0 1.00000 0 0.140190 0 −1.18735 0 1.00000 0
1.9 0 1.00000 0 0.235801 0 1.32081 0 1.00000 0
1.10 0 1.00000 0 0.767523 0 −4.22279 0 1.00000 0
1.11 0 1.00000 0 1.69179 0 3.04338 0 1.00000 0
1.12 0 1.00000 0 1.90345 0 2.67198 0 1.00000 0
1.13 0 1.00000 0 2.40801 0 3.93112 0 1.00000 0
1.14 0 1.00000 0 2.45476 0 0.345256 0 1.00000 0
1.15 0 1.00000 0 3.38212 0 5.02591 0 1.00000 0
1.16 0 1.00000 0 3.50248 0 −4.71048 0 1.00000 0
1.17 0 1.00000 0 3.88351 0 −3.85235 0 1.00000 0
1.18 0 1.00000 0 4.36408 0 0.320970 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
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.k 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.a.k 18 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\(T_{5}^{18} - \cdots\)
\(T_{7}^{18} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \)
$3$ \( ( -1 + T )^{18} \)
$5$ \( -115488 + 1509552 T - 5659296 T^{2} + 5001480 T^{3} + 4260960 T^{4} - 6184176 T^{5} - 855636 T^{6} + 2789438 T^{7} - 114554 T^{8} - 655793 T^{9} + 76571 T^{10} + 88434 T^{11} - 13745 T^{12} - 6888 T^{13} + 1222 T^{14} + 288 T^{15} - 55 T^{16} - 5 T^{17} + T^{18} \)
$7$ \( 2162624 - 10553984 T + 2287648 T^{2} + 34280000 T^{3} - 4020656 T^{4} - 31927888 T^{5} + 2685216 T^{6} + 12561120 T^{7} - 1269308 T^{8} - 2471372 T^{9} + 299782 T^{10} + 264892 T^{11} - 36453 T^{12} - 15706 T^{13} + 2356 T^{14} + 483 T^{15} - 77 T^{16} - 6 T^{17} + T^{18} \)
$11$ \( -467213184 + 2172769920 T - 3012531264 T^{2} + 903532320 T^{3} + 1115491824 T^{4} - 759079296 T^{5} - 90837594 T^{6} + 165709044 T^{7} - 10210257 T^{8} - 17731491 T^{9} + 2351263 T^{10} + 1063959 T^{11} - 178259 T^{12} - 36454 T^{13} + 6778 T^{14} + 665 T^{15} - 130 T^{16} - 5 T^{17} + T^{18} \)
$13$ \( 4082087840 - 6649020208 T - 1324821612 T^{2} + 5019939384 T^{3} - 275881737 T^{4} - 1538619000 T^{5} + 192192730 T^{6} + 250033711 T^{7} - 37729073 T^{8} - 23679170 T^{9} + 3822809 T^{10} + 1344826 T^{11} - 222839 T^{12} - 44888 T^{13} + 7504 T^{14} + 807 T^{15} - 135 T^{16} - 6 T^{17} + T^{18} \)
$17$ \( 33177600 - 274599936 T + 794151936 T^{2} - 935973120 T^{3} + 190444896 T^{4} + 520363712 T^{5} - 395438602 T^{6} + 13041782 T^{7} + 77297437 T^{8} - 21090818 T^{9} - 3752706 T^{10} + 2079742 T^{11} - 38220 T^{12} - 77765 T^{13} + 6561 T^{14} + 1238 T^{15} - 147 T^{16} - 7 T^{17} + T^{18} \)
$19$ \( 138259290112 + 52498134016 T - 220248086272 T^{2} - 36646288960 T^{3} + 94929656464 T^{4} - 1789288712 T^{5} - 15625760564 T^{6} + 1785249006 T^{7} + 1178992866 T^{8} - 212645369 T^{9} - 41708601 T^{10} + 10737146 T^{11} + 562266 T^{12} - 266920 T^{13} + 3124 T^{14} + 3216 T^{15} - 148 T^{16} - 15 T^{17} + T^{18} \)
$23$ \( ( 1 + T )^{18} \)
$29$ \( ( -1 + T )^{18} \)
$31$ \( 47539851264 + 282378161664 T + 374838720768 T^{2} - 6768128016 T^{3} - 220555694016 T^{4} - 46948384188 T^{5} + 40012646496 T^{6} + 9645447994 T^{7} - 3126768030 T^{8} - 698243135 T^{9} + 131614968 T^{10} + 23553759 T^{11} - 3241374 T^{12} - 396035 T^{13} + 45805 T^{14} + 3211 T^{15} - 338 T^{16} - 10 T^{17} + T^{18} \)
$37$ \( 88011603968 + 612183150304 T + 1348975141648 T^{2} + 1163017711470 T^{3} + 173398189978 T^{4} - 270267208675 T^{5} - 86670166950 T^{6} + 28439098787 T^{7} + 8054184430 T^{8} - 1901030400 T^{9} - 252621007 T^{10} + 63273110 T^{11} + 2777898 T^{12} - 1029375 T^{13} + 3436 T^{14} + 7833 T^{15} - 234 T^{16} - 22 T^{17} + T^{18} \)
$41$ \( 1065212050560 - 1872974505024 T - 6967419072896 T^{2} + 2664621143712 T^{3} + 2228590326648 T^{4} - 802389201724 T^{5} - 206630496032 T^{6} + 81815407322 T^{7} + 7301572438 T^{8} - 3767666801 T^{9} - 74961835 T^{10} + 89208762 T^{11} - 1403639 T^{12} - 1121678 T^{13} + 41212 T^{14} + 7040 T^{15} - 353 T^{16} - 17 T^{17} + T^{18} \)
$43$ \( -2872980099072 - 12073137588864 T - 7888411077120 T^{2} + 4767861579360 T^{3} + 3452248470880 T^{4} - 625444684904 T^{5} - 447962312472 T^{6} + 52642748026 T^{7} + 25048521320 T^{8} - 3046527669 T^{9} - 642065789 T^{10} + 93661036 T^{11} + 7000111 T^{12} - 1414338 T^{13} - 14988 T^{14} + 9874 T^{15} - 219 T^{16} - 25 T^{17} + T^{18} \)
$47$ \( -200448000 + 16580275200 T - 265056759040 T^{2} - 110209927936 T^{3} + 181795982656 T^{4} + 80997676672 T^{5} - 28680786768 T^{6} - 16255995216 T^{7} + 852335572 T^{8} + 1179438972 T^{9} + 54585698 T^{10} - 35993728 T^{11} - 3183241 T^{12} + 495951 T^{13} + 56355 T^{14} - 2948 T^{15} - 402 T^{16} + 6 T^{17} + T^{18} \)
$53$ \( -2582846976 - 28785499392 T + 200690848768 T^{2} - 224646159488 T^{3} - 240318026976 T^{4} + 292952926384 T^{5} - 71217811504 T^{6} - 14486014160 T^{7} + 7873408168 T^{8} - 294604856 T^{9} - 274259640 T^{10} + 31991708 T^{11} + 3677352 T^{12} - 733069 T^{13} - 7505 T^{14} + 6634 T^{15} - 189 T^{16} - 21 T^{17} + T^{18} \)
$59$ \( 1081728691200 - 155990292480 T - 2363129116416 T^{2} + 1088573927040 T^{3} + 830039084352 T^{4} - 525379225920 T^{5} - 28463024448 T^{6} + 55030404264 T^{7} - 3101388696 T^{8} - 2409027696 T^{9} + 231438881 T^{10} + 52694054 T^{11} - 6139740 T^{12} - 593858 T^{13} + 77582 T^{14} + 3178 T^{15} - 460 T^{16} - 6 T^{17} + T^{18} \)
$61$ \( 7259060789248 - 49479681497088 T - 47451713021440 T^{2} + 80236880867968 T^{3} - 11201096713856 T^{4} - 10196171340976 T^{5} + 1868484354344 T^{6} + 478097285826 T^{7} - 91762126392 T^{8} - 11229667611 T^{9} + 2144122247 T^{10} + 147567147 T^{11} - 27059168 T^{12} - 1102868 T^{13} + 188544 T^{14} + 4353 T^{15} - 682 T^{16} - 7 T^{17} + T^{18} \)
$67$ \( -134264805632 - 2427830729216 T - 4794645607648 T^{2} - 1530387461872 T^{3} + 2242842486876 T^{4} + 1075923243816 T^{5} - 393625170908 T^{6} - 137820817136 T^{7} + 47686790521 T^{8} + 1899684434 T^{9} - 1628674406 T^{10} + 63338336 T^{11} + 21786278 T^{12} - 1738179 T^{13} - 104721 T^{14} + 13966 T^{15} - 47 T^{16} - 35 T^{17} + T^{18} \)
$71$ \( -188938579968 + 68861592576 T + 368600009472 T^{2} - 140138911296 T^{3} - 228537620964 T^{4} + 74655411517 T^{5} + 59431031609 T^{6} - 12044453251 T^{7} - 7463463536 T^{8} + 518813809 T^{9} + 374823633 T^{10} - 8316434 T^{11} - 8587374 T^{12} + 49579 T^{13} + 96277 T^{14} + 29 T^{15} - 510 T^{16} - T^{17} + T^{18} \)
$73$ \( -149749399552 + 1689202497536 T - 463128142848 T^{2} - 2550088344064 T^{3} + 288292329728 T^{4} + 833607214528 T^{5} - 157415388160 T^{6} - 91163085728 T^{7} + 27735442912 T^{8} + 1086308020 T^{9} - 1075784968 T^{10} + 67763698 T^{11} + 13210754 T^{12} - 1567735 T^{13} - 40614 T^{14} + 11345 T^{15} - 201 T^{16} - 27 T^{17} + T^{18} \)
$79$ \( -1201848130386688 + 821326165196160 T + 640025930244160 T^{2} - 390779087585696 T^{3} - 115638427411632 T^{4} + 44421727753416 T^{5} + 11753331774916 T^{6} - 1418268425746 T^{7} - 442805709856 T^{8} + 17919852623 T^{9} + 7824588350 T^{10} - 99663575 T^{11} - 73533464 T^{12} + 238865 T^{13} + 374837 T^{14} - 191 T^{15} - 972 T^{16} + T^{18} \)
$83$ \( 17352937279193088 - 77090214783123456 T + 412861949057024 T^{2} + 10612939990423552 T^{3} - 574711788739712 T^{4} - 493528671739968 T^{5} + 33100278606912 T^{6} + 11327498328384 T^{7} - 816228248904 T^{8} - 147122944692 T^{9} + 10946570992 T^{10} + 1134562630 T^{11} - 85823374 T^{12} - 5145237 T^{13} + 392773 T^{14} + 12656 T^{15} - 971 T^{16} - 13 T^{17} + T^{18} \)
$89$ \( -928864270080 - 13528622001024 T + 30362465794080 T^{2} - 5548245267552 T^{3} - 17900833033872 T^{4} + 7792880065440 T^{5} + 271839120138 T^{6} - 647416113658 T^{7} + 92593276603 T^{8} + 8583046350 T^{9} - 2898551800 T^{10} + 92207355 T^{11} + 29172098 T^{12} - 2457510 T^{13} - 82729 T^{14} + 15393 T^{15} - 219 T^{16} - 30 T^{17} + T^{18} \)
$97$ \( -2155993322127360 - 5834426594746368 T - 2476972490185728 T^{2} + 2387892467128320 T^{3} + 229687538461824 T^{4} - 224894267762624 T^{5} - 7077309210864 T^{6} + 8922074246200 T^{7} + 59750062536 T^{8} - 171756259660 T^{9} + 1497156388 T^{10} + 1723724938 T^{11} - 33174900 T^{12} - 9236177 T^{13} + 247957 T^{14} + 25047 T^{15} - 816 T^{16} - 27 T^{17} + T^{18} \)
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