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

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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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.

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

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

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

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:

Inner twists

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

Atkin-Lehner signs

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

Hecke kernels

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

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