Properties

Label 8030.2.a.bl
Level 8030
Weight 2
Character orbit 8030.a
Self dual Yes
Analytic conductor 64.120
Analytic rank 0
Dimension 19
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8030.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1198728231\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \( + ( 1 - \beta_{1} ) q^{3} \) \(+ q^{4}\) \(+ q^{5}\) \( + ( 1 - \beta_{1} ) q^{6} \) \( -\beta_{10} q^{7} \) \(+ q^{8}\) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \( + ( 1 - \beta_{1} ) q^{3} \) \(+ q^{4}\) \(+ q^{5}\) \( + ( 1 - \beta_{1} ) q^{6} \) \( -\beta_{10} q^{7} \) \(+ q^{8}\) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{9} \) \(+ q^{10}\) \(+ q^{11}\) \( + ( 1 - \beta_{1} ) q^{12} \) \( + ( 1 + \beta_{5} ) q^{13} \) \( -\beta_{10} q^{14} \) \( + ( 1 - \beta_{1} ) q^{15} \) \(+ q^{16}\) \( + ( 1 - \beta_{15} ) q^{17} \) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{18} \) \( + ( 1 - \beta_{8} ) q^{19} \) \(+ q^{20}\) \( + ( -\beta_{1} - \beta_{17} - \beta_{18} ) q^{21} \) \(+ q^{22}\) \( + ( 1 - \beta_{4} ) q^{23} \) \( + ( 1 - \beta_{1} ) q^{24} \) \(+ q^{25}\) \( + ( 1 + \beta_{5} ) q^{26} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{27} \) \( -\beta_{10} q^{28} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} - \beta_{10} + \beta_{11} + \beta_{15} - \beta_{16} + \beta_{17} + \beta_{18} ) q^{29} \) \( + ( 1 - \beta_{1} ) q^{30} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} - \beta_{9} + \beta_{13} - \beta_{14} ) q^{31} \) \(+ q^{32}\) \( + ( 1 - \beta_{1} ) q^{33} \) \( + ( 1 - \beta_{15} ) q^{34} \) \( -\beta_{10} q^{35} \) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{36} \) \( + ( 2 - \beta_{1} + \beta_{3} - \beta_{6} - \beta_{13} ) q^{37} \) \( + ( 1 - \beta_{8} ) q^{38} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{5} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} ) q^{39} \) \(+ q^{40}\) \( + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{8} + \beta_{14} ) q^{41} \) \( + ( -\beta_{1} - \beta_{17} - \beta_{18} ) q^{42} \) \( + ( 1 + \beta_{4} + \beta_{9} + \beta_{17} ) q^{43} \) \(+ q^{44}\) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{45} \) \( + ( 1 - \beta_{4} ) q^{46} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{47} \) \( + ( 1 - \beta_{1} ) q^{48} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{7} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} + \beta_{18} ) q^{49} \) \(+ q^{50}\) \( + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{51} \) \( + ( 1 + \beta_{5} ) q^{52} \) \( + ( 2 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{10} + \beta_{13} + \beta_{18} ) q^{53} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{54} \) \(+ q^{55}\) \( -\beta_{10} q^{56} \) \( + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{16} - \beta_{17} - \beta_{18} ) q^{57} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} - \beta_{10} + \beta_{11} + \beta_{15} - \beta_{16} + \beta_{17} + \beta_{18} ) q^{58} \) \( + ( 1 - 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{17} ) q^{59} \) \( + ( 1 - \beta_{1} ) q^{60} \) \( + ( 2 - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{16} - \beta_{17} - \beta_{18} ) q^{61} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} - \beta_{9} + \beta_{13} - \beta_{14} ) q^{62} \) \( + ( -1 + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{13} - \beta_{17} - \beta_{18} ) q^{63} \) \(+ q^{64}\) \( + ( 1 + \beta_{5} ) q^{65} \) \( + ( 1 - \beta_{1} ) q^{66} \) \( + ( 1 - \beta_{7} + \beta_{10} + \beta_{15} + \beta_{16} ) q^{67} \) \( + ( 1 - \beta_{15} ) q^{68} \) \( + ( 1 - 2 \beta_{1} - \beta_{4} + \beta_{10} + \beta_{14} ) q^{69} \) \( -\beta_{10} q^{70} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{71} \) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{72} \) \(- q^{73}\) \( + ( 2 - \beta_{1} + \beta_{3} - \beta_{6} - \beta_{13} ) q^{74} \) \( + ( 1 - \beta_{1} ) q^{75} \) \( + ( 1 - \beta_{8} ) q^{76} \) \( -\beta_{10} q^{77} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{5} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} ) q^{78} \) \( + ( 1 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} + \beta_{12} + \beta_{17} ) q^{79} \) \(+ q^{80}\) \( + ( 2 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14} - \beta_{15} ) q^{81} \) \( + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{8} + \beta_{14} ) q^{82} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} + \beta_{11} - \beta_{13} + \beta_{15} - \beta_{16} ) q^{83} \) \( + ( -\beta_{1} - \beta_{17} - \beta_{18} ) q^{84} \) \( + ( 1 - \beta_{15} ) q^{85} \) \( + ( 1 + \beta_{4} + \beta_{9} + \beta_{17} ) q^{86} \) \( + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{18} ) q^{87} \) \(+ q^{88}\) \( + ( 1 - \beta_{2} - \beta_{3} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{17} + \beta_{18} ) q^{89} \) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{90} \) \( + ( 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{91} \) \( + ( 1 - \beta_{4} ) q^{92} \) \( + ( -\beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{16} + \beta_{18} ) q^{93} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{94} \) \( + ( 1 - \beta_{8} ) q^{95} \) \( + ( 1 - \beta_{1} ) q^{96} \) \( + ( -2 - \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{15} - \beta_{18} ) q^{97} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{7} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} + \beta_{18} ) q^{98} \) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(19q \) \(\mathstrut +\mathstrut 19q^{2} \) \(\mathstrut +\mathstrut 10q^{3} \) \(\mathstrut +\mathstrut 19q^{4} \) \(\mathstrut +\mathstrut 19q^{5} \) \(\mathstrut +\mathstrut 10q^{6} \) \(\mathstrut +\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 19q^{8} \) \(\mathstrut +\mathstrut 27q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(19q \) \(\mathstrut +\mathstrut 19q^{2} \) \(\mathstrut +\mathstrut 10q^{3} \) \(\mathstrut +\mathstrut 19q^{4} \) \(\mathstrut +\mathstrut 19q^{5} \) \(\mathstrut +\mathstrut 10q^{6} \) \(\mathstrut +\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 19q^{8} \) \(\mathstrut +\mathstrut 27q^{9} \) \(\mathstrut +\mathstrut 19q^{10} \) \(\mathstrut +\mathstrut 19q^{11} \) \(\mathstrut +\mathstrut 10q^{12} \) \(\mathstrut +\mathstrut 16q^{13} \) \(\mathstrut +\mathstrut 8q^{14} \) \(\mathstrut +\mathstrut 10q^{15} \) \(\mathstrut +\mathstrut 19q^{16} \) \(\mathstrut +\mathstrut 12q^{17} \) \(\mathstrut +\mathstrut 27q^{18} \) \(\mathstrut +\mathstrut 12q^{19} \) \(\mathstrut +\mathstrut 19q^{20} \) \(\mathstrut +\mathstrut 3q^{21} \) \(\mathstrut +\mathstrut 19q^{22} \) \(\mathstrut +\mathstrut 26q^{23} \) \(\mathstrut +\mathstrut 10q^{24} \) \(\mathstrut +\mathstrut 19q^{25} \) \(\mathstrut +\mathstrut 16q^{26} \) \(\mathstrut +\mathstrut 25q^{27} \) \(\mathstrut +\mathstrut 8q^{28} \) \(\mathstrut +\mathstrut q^{29} \) \(\mathstrut +\mathstrut 10q^{30} \) \(\mathstrut +\mathstrut 24q^{31} \) \(\mathstrut +\mathstrut 19q^{32} \) \(\mathstrut +\mathstrut 10q^{33} \) \(\mathstrut +\mathstrut 12q^{34} \) \(\mathstrut +\mathstrut 8q^{35} \) \(\mathstrut +\mathstrut 27q^{36} \) \(\mathstrut +\mathstrut 23q^{37} \) \(\mathstrut +\mathstrut 12q^{38} \) \(\mathstrut -\mathstrut 5q^{39} \) \(\mathstrut +\mathstrut 19q^{40} \) \(\mathstrut +\mathstrut 3q^{42} \) \(\mathstrut +\mathstrut 8q^{43} \) \(\mathstrut +\mathstrut 19q^{44} \) \(\mathstrut +\mathstrut 27q^{45} \) \(\mathstrut +\mathstrut 26q^{46} \) \(\mathstrut +\mathstrut 34q^{47} \) \(\mathstrut +\mathstrut 10q^{48} \) \(\mathstrut +\mathstrut 27q^{49} \) \(\mathstrut +\mathstrut 19q^{50} \) \(\mathstrut +\mathstrut 15q^{51} \) \(\mathstrut +\mathstrut 16q^{52} \) \(\mathstrut +\mathstrut 25q^{53} \) \(\mathstrut +\mathstrut 25q^{54} \) \(\mathstrut +\mathstrut 19q^{55} \) \(\mathstrut +\mathstrut 8q^{56} \) \(\mathstrut +\mathstrut q^{57} \) \(\mathstrut +\mathstrut q^{58} \) \(\mathstrut +\mathstrut 24q^{59} \) \(\mathstrut +\mathstrut 10q^{60} \) \(\mathstrut +\mathstrut 31q^{61} \) \(\mathstrut +\mathstrut 24q^{62} \) \(\mathstrut +\mathstrut 15q^{63} \) \(\mathstrut +\mathstrut 19q^{64} \) \(\mathstrut +\mathstrut 16q^{65} \) \(\mathstrut +\mathstrut 10q^{66} \) \(\mathstrut +\mathstrut 24q^{67} \) \(\mathstrut +\mathstrut 12q^{68} \) \(\mathstrut +\mathstrut q^{69} \) \(\mathstrut +\mathstrut 8q^{70} \) \(\mathstrut +\mathstrut 5q^{71} \) \(\mathstrut +\mathstrut 27q^{72} \) \(\mathstrut -\mathstrut 19q^{73} \) \(\mathstrut +\mathstrut 23q^{74} \) \(\mathstrut +\mathstrut 10q^{75} \) \(\mathstrut +\mathstrut 12q^{76} \) \(\mathstrut +\mathstrut 8q^{77} \) \(\mathstrut -\mathstrut 5q^{78} \) \(\mathstrut +\mathstrut 18q^{79} \) \(\mathstrut +\mathstrut 19q^{80} \) \(\mathstrut +\mathstrut 11q^{81} \) \(\mathstrut +\mathstrut 12q^{83} \) \(\mathstrut +\mathstrut 3q^{84} \) \(\mathstrut +\mathstrut 12q^{85} \) \(\mathstrut +\mathstrut 8q^{86} \) \(\mathstrut +\mathstrut 12q^{87} \) \(\mathstrut +\mathstrut 19q^{88} \) \(\mathstrut +\mathstrut 27q^{90} \) \(\mathstrut +\mathstrut 23q^{91} \) \(\mathstrut +\mathstrut 26q^{92} \) \(\mathstrut +\mathstrut 18q^{93} \) \(\mathstrut +\mathstrut 34q^{94} \) \(\mathstrut +\mathstrut 12q^{95} \) \(\mathstrut +\mathstrut 10q^{96} \) \(\mathstrut +\mathstrut 15q^{97} \) \(\mathstrut +\mathstrut 27q^{98} \) \(\mathstrut +\mathstrut 27q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19}\mathstrut -\mathstrut \) \(9\) \(x^{18}\mathstrut -\mathstrut \) \(x^{17}\mathstrut +\mathstrut \) \(200\) \(x^{16}\mathstrut -\mathstrut \) \(263\) \(x^{15}\mathstrut -\mathstrut \) \(1900\) \(x^{14}\mathstrut +\mathstrut \) \(3165\) \(x^{13}\mathstrut +\mathstrut \) \(10217\) \(x^{12}\mathstrut -\mathstrut \) \(16393\) \(x^{11}\mathstrut -\mathstrut \) \(33875\) \(x^{10}\mathstrut +\mathstrut \) \(44043\) \(x^{9}\mathstrut +\mathstrut \) \(68537\) \(x^{8}\mathstrut -\mathstrut \) \(63620\) \(x^{7}\mathstrut -\mathstrut \) \(79890\) \(x^{6}\mathstrut +\mathstrut \) \(48388\) \(x^{5}\mathstrut +\mathstrut \) \(49353\) \(x^{4}\mathstrut -\mathstrut \) \(17806\) \(x^{3}\mathstrut -\mathstrut \) \(14287\) \(x^{2}\mathstrut +\mathstrut \) \(2436\) \(x\mathstrut +\mathstrut \) \(1388\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 5 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\(67767703269323\) \(\nu^{18}\mathstrut -\mathstrut \) \(461540381624216\) \(\nu^{17}\mathstrut -\mathstrut \) \(1829047994384574\) \(\nu^{16}\mathstrut +\mathstrut \) \(17992642377892825\) \(\nu^{15}\mathstrut +\mathstrut \) \(4196398387883617\) \(\nu^{14}\mathstrut -\mathstrut \) \(240093375209630295\) \(\nu^{13}\mathstrut +\mathstrut \) \(175689678614282352\) \(\nu^{12}\mathstrut +\mathstrut \) \(1545621783813084880\) \(\nu^{11}\mathstrut -\mathstrut \) \(1685623549305064110\) \(\nu^{10}\mathstrut -\mathstrut \) \(5321431444591112515\) \(\nu^{9}\mathstrut +\mathstrut \) \(6451066205133237889\) \(\nu^{8}\mathstrut +\mathstrut \) \(9802479332703309879\) \(\nu^{7}\mathstrut -\mathstrut \) \(12066538545176504503\) \(\nu^{6}\mathstrut -\mathstrut \) \(8887864938918716648\) \(\nu^{5}\mathstrut +\mathstrut \) \(10817375807991639982\) \(\nu^{4}\mathstrut +\mathstrut \) \(3127244755625591552\) \(\nu^{3}\mathstrut -\mathstrut \) \(3853629647903613676\) \(\nu^{2}\mathstrut -\mathstrut \) \(218044026178024810\) \(\nu\mathstrut +\mathstrut \) \(344274665371602070\)\()/\)\(6452776078088694\)
\(\beta_{5}\)\(=\)\((\)\(50630052638212\) \(\nu^{18}\mathstrut -\mathstrut \) \(424185699767877\) \(\nu^{17}\mathstrut -\mathstrut \) \(370381254330593\) \(\nu^{16}\mathstrut +\mathstrut \) \(10547153500587325\) \(\nu^{15}\mathstrut -\mathstrut \) \(8281369604976276\) \(\nu^{14}\mathstrut -\mathstrut \) \(110003180498292848\) \(\nu^{13}\mathstrut +\mathstrut \) \(131861973015827473\) \(\nu^{12}\mathstrut +\mathstrut \) \(623020664935530527\) \(\nu^{11}\mathstrut -\mathstrut \) \(759293886794764662\) \(\nu^{10}\mathstrut -\mathstrut \) \(2040915187917304795\) \(\nu^{9}\mathstrut +\mathstrut \) \(2137905924757903123\) \(\nu^{8}\mathstrut +\mathstrut \) \(3741636168265147766\) \(\nu^{7}\mathstrut -\mathstrut \) \(3089662924077936893\) \(\nu^{6}\mathstrut -\mathstrut \) \(3457327076988766834\) \(\nu^{5}\mathstrut +\mathstrut \) \(2208238538690460199\) \(\nu^{4}\mathstrut +\mathstrut \) \(1284067887242840180\) \(\nu^{3}\mathstrut -\mathstrut \) \(671551593523850225\) \(\nu^{2}\mathstrut -\mathstrut \) \(109683999941140110\) \(\nu\mathstrut +\mathstrut \) \(57522920151092322\)\()/\)\(2150925359362898\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(177313727903810\) \(\nu^{18}\mathstrut +\mathstrut \) \(1665474227812742\) \(\nu^{17}\mathstrut -\mathstrut \) \(817566705574623\) \(\nu^{16}\mathstrut -\mathstrut \) \(31776613488142648\) \(\nu^{15}\mathstrut +\mathstrut \) \(56123555563441991\) \(\nu^{14}\mathstrut +\mathstrut \) \(253115387929776351\) \(\nu^{13}\mathstrut -\mathstrut \) \(520137757020581049\) \(\nu^{12}\mathstrut -\mathstrut \) \(1153838640643625854\) \(\nu^{11}\mathstrut +\mathstrut \) \(2041932300231603969\) \(\nu^{10}\mathstrut +\mathstrut \) \(3414330698132732743\) \(\nu^{9}\mathstrut -\mathstrut \) \(3498938180013604810\) \(\nu^{8}\mathstrut -\mathstrut \) \(6406569444902143959\) \(\nu^{7}\mathstrut +\mathstrut \) \(1481111167068662677\) \(\nu^{6}\mathstrut +\mathstrut \) \(6760547362112289872\) \(\nu^{5}\mathstrut +\mathstrut \) \(2095831201089977810\) \(\nu^{4}\mathstrut -\mathstrut \) \(3386200950438267341\) \(\nu^{3}\mathstrut -\mathstrut \) \(1849969278868735331\) \(\nu^{2}\mathstrut +\mathstrut \) \(572749823366703520\) \(\nu\mathstrut +\mathstrut \) \(304977081039327422\)\()/\)\(6452776078088694\)
\(\beta_{7}\)\(=\)\((\)\(180707121536345\) \(\nu^{18}\mathstrut -\mathstrut \) \(1757289921571220\) \(\nu^{17}\mathstrut +\mathstrut \) \(1067457298155204\) \(\nu^{16}\mathstrut +\mathstrut \) \(35556578692171072\) \(\nu^{15}\mathstrut -\mathstrut \) \(73017026724538451\) \(\nu^{14}\mathstrut -\mathstrut \) \(294942816649205871\) \(\nu^{13}\mathstrut +\mathstrut \) \(786804273096354387\) \(\nu^{12}\mathstrut +\mathstrut \) \(1321201035967726273\) \(\nu^{11}\mathstrut -\mathstrut \) \(3947967467767076349\) \(\nu^{10}\mathstrut -\mathstrut \) \(3502847447286564085\) \(\nu^{9}\mathstrut +\mathstrut \) \(10643436601064984017\) \(\nu^{8}\mathstrut +\mathstrut \) \(5411386613391953151\) \(\nu^{7}\mathstrut -\mathstrut \) \(15720637772997710776\) \(\nu^{6}\mathstrut -\mathstrut \) \(4267988730186030875\) \(\nu^{5}\mathstrut +\mathstrut \) \(12057502327232884855\) \(\nu^{4}\mathstrut +\mathstrut \) \(1155556121353730039\) \(\nu^{3}\mathstrut -\mathstrut \) \(4062319619472367228\) \(\nu^{2}\mathstrut +\mathstrut \) \(56895231626467904\) \(\nu\mathstrut +\mathstrut \) \(385112694063467830\)\()/\)\(6452776078088694\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(187046570499115\) \(\nu^{18}\mathstrut +\mathstrut \) \(2126250099756004\) \(\nu^{17}\mathstrut -\mathstrut \) \(4677278383371342\) \(\nu^{16}\mathstrut -\mathstrut \) \(28518355699180013\) \(\nu^{15}\mathstrut +\mathstrut \) \(123529702428607693\) \(\nu^{14}\mathstrut +\mathstrut \) \(85741378995503835\) \(\nu^{13}\mathstrut -\mathstrut \) \(951237276959235936\) \(\nu^{12}\mathstrut +\mathstrut \) \(381420734027553856\) \(\nu^{11}\mathstrut +\mathstrut \) \(3311818712545587714\) \(\nu^{10}\mathstrut -\mathstrut \) \(2913706569760999807\) \(\nu^{9}\mathstrut -\mathstrut \) \(5225985194999967611\) \(\nu^{8}\mathstrut +\mathstrut \) \(6637289161386080289\) \(\nu^{7}\mathstrut +\mathstrut \) \(2340312350837746631\) \(\nu^{6}\mathstrut -\mathstrut \) \(6142632963029006402\) \(\nu^{5}\mathstrut +\mathstrut \) \(1999002300765653128\) \(\nu^{4}\mathstrut +\mathstrut \) \(1737283757101114346\) \(\nu^{3}\mathstrut -\mathstrut \) \(1476421741857246556\) \(\nu^{2}\mathstrut +\mathstrut \) \(41948543187362648\) \(\nu\mathstrut +\mathstrut \) \(141481739842397824\)\()/\)\(6452776078088694\)
\(\beta_{9}\)\(=\)\((\)\(199422333353506\) \(\nu^{18}\mathstrut -\mathstrut \) \(2014465387386478\) \(\nu^{17}\mathstrut +\mathstrut \) \(1906181846145771\) \(\nu^{16}\mathstrut +\mathstrut \) \(38816044176556868\) \(\nu^{15}\mathstrut -\mathstrut \) \(95268980193765529\) \(\nu^{14}\mathstrut -\mathstrut \) \(295818729450298563\) \(\nu^{13}\mathstrut +\mathstrut \) \(988191079157380581\) \(\nu^{12}\mathstrut +\mathstrut \) \(1145424685650821264\) \(\nu^{11}\mathstrut -\mathstrut \) \(4873494582387198855\) \(\nu^{10}\mathstrut -\mathstrut \) \(2379354798586710119\) \(\nu^{9}\mathstrut +\mathstrut \) \(13019442257136659414\) \(\nu^{8}\mathstrut +\mathstrut \) \(2371301826252860343\) \(\nu^{7}\mathstrut -\mathstrut \) \(19097171245233324467\) \(\nu^{6}\mathstrut -\mathstrut \) \(354529966492831108\) \(\nu^{5}\mathstrut +\mathstrut \) \(14503700024022309248\) \(\nu^{4}\mathstrut -\mathstrut \) \(997716900791892905\) \(\nu^{3}\mathstrut -\mathstrut \) \(4826059871666993303\) \(\nu^{2}\mathstrut +\mathstrut \) \(381719549166996556\) \(\nu\mathstrut +\mathstrut \) \(450056685971050898\)\()/\)\(6452776078088694\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(204668395194571\) \(\nu^{18}\mathstrut +\mathstrut \) \(1889070473055328\) \(\nu^{17}\mathstrut -\mathstrut \) \(231033388022982\) \(\nu^{16}\mathstrut -\mathstrut \) \(41001173088474143\) \(\nu^{15}\mathstrut +\mathstrut \) \(64680973088016400\) \(\nu^{14}\mathstrut +\mathstrut \) \(371446109160217413\) \(\nu^{13}\mathstrut -\mathstrut \) \(755575182193119126\) \(\nu^{12}\mathstrut -\mathstrut \) \(1841774213553957773\) \(\nu^{11}\mathstrut +\mathstrut \) \(3902115469933026549\) \(\nu^{10}\mathstrut +\mathstrut \) \(5399031921639471008\) \(\nu^{9}\mathstrut -\mathstrut \) \(10511273964711378593\) \(\nu^{8}\mathstrut -\mathstrut \) \(9075453858322075833\) \(\nu^{7}\mathstrut +\mathstrut \) \(15099732646720254755\) \(\nu^{6}\mathstrut +\mathstrut \) \(7724333910488621041\) \(\nu^{5}\mathstrut -\mathstrut \) \(10967228393769351437\) \(\nu^{4}\mathstrut -\mathstrut \) \(2474695556966677747\) \(\nu^{3}\mathstrut +\mathstrut \) \(3375054409069307021\) \(\nu^{2}\mathstrut +\mathstrut \) \(97252565481551060\) \(\nu\mathstrut -\mathstrut \) \(271173877669587878\)\()/\)\(6452776078088694\)
\(\beta_{11}\)\(=\)\((\)\(88141905048141\) \(\nu^{18}\mathstrut -\mathstrut \) \(814908283378919\) \(\nu^{17}\mathstrut +\mathstrut \) \(200828455194863\) \(\nu^{16}\mathstrut +\mathstrut \) \(16733442992079219\) \(\nu^{15}\mathstrut -\mathstrut \) \(26811469029011869\) \(\nu^{14}\mathstrut -\mathstrut \) \(144626362341102703\) \(\nu^{13}\mathstrut +\mathstrut \) \(283523942468175634\) \(\nu^{12}\mathstrut +\mathstrut \) \(701700799538067486\) \(\nu^{11}\mathstrut -\mathstrut \) \(1315528579217468117\) \(\nu^{10}\mathstrut -\mathstrut \) \(2089937946939639246\) \(\nu^{9}\mathstrut +\mathstrut \) \(3079051604823160300\) \(\nu^{8}\mathstrut +\mathstrut \) \(3695360919986239977\) \(\nu^{7}\mathstrut -\mathstrut \) \(3625264049187407313\) \(\nu^{6}\mathstrut -\mathstrut \) \(3436752265005777145\) \(\nu^{5}\mathstrut +\mathstrut \) \(1973479196948448564\) \(\nu^{4}\mathstrut +\mathstrut \) \(1347661156488340979\) \(\nu^{3}\mathstrut -\mathstrut \) \(385387455575350897\) \(\nu^{2}\mathstrut -\mathstrut \) \(142460742911687256\) \(\nu\mathstrut +\mathstrut \) \(1915931615769778\)\()/\)\(2150925359362898\)
\(\beta_{12}\)\(=\)\((\)\(269915381623373\) \(\nu^{18}\mathstrut -\mathstrut \) \(2841938440979513\) \(\nu^{17}\mathstrut +\mathstrut \) \(4006034675112768\) \(\nu^{16}\mathstrut +\mathstrut \) \(48474452496874237\) \(\nu^{15}\mathstrut -\mathstrut \) \(145023275287357724\) \(\nu^{14}\mathstrut -\mathstrut \) \(304461867691627542\) \(\nu^{13}\mathstrut +\mathstrut \) \(1336620421453371411\) \(\nu^{12}\mathstrut +\mathstrut \) \(836905206374923726\) \(\nu^{11}\mathstrut -\mathstrut \) \(5896813044614868540\) \(\nu^{10}\mathstrut -\mathstrut \) \(759150774994407175\) \(\nu^{9}\mathstrut +\mathstrut \) \(13966417645290300604\) \(\nu^{8}\mathstrut -\mathstrut \) \(898502344381930698\) \(\nu^{7}\mathstrut -\mathstrut \) \(17955552886676810950\) \(\nu^{6}\mathstrut +\mathstrut \) \(2376869868707732887\) \(\nu^{5}\mathstrut +\mathstrut \) \(11949942655653332698\) \(\nu^{4}\mathstrut -\mathstrut \) \(1626954965148382672\) \(\nu^{3}\mathstrut -\mathstrut \) \(3678775171845158002\) \(\nu^{2}\mathstrut +\mathstrut \) \(344884458126327884\) \(\nu\mathstrut +\mathstrut \) \(378983067491770858\)\()/\)\(6452776078088694\)
\(\beta_{13}\)\(=\)\((\)\(93524537835642\) \(\nu^{18}\mathstrut -\mathstrut \) \(945600665621253\) \(\nu^{17}\mathstrut +\mathstrut \) \(1032129142109091\) \(\nu^{16}\mathstrut +\mathstrut \) \(16817935178870461\) \(\nu^{15}\mathstrut -\mathstrut \) \(42675098815703659\) \(\nu^{14}\mathstrut -\mathstrut \) \(116412043724817898\) \(\nu^{13}\mathstrut +\mathstrut \) \(394913671407158125\) \(\nu^{12}\mathstrut +\mathstrut \) \(411311671591537732\) \(\nu^{11}\mathstrut -\mathstrut \) \(1698653466570678497\) \(\nu^{10}\mathstrut -\mathstrut \) \(841761953649657192\) \(\nu^{9}\mathstrut +\mathstrut \) \(3778937125039004947\) \(\nu^{8}\mathstrut +\mathstrut \) \(1060259039041748670\) \(\nu^{7}\mathstrut -\mathstrut \) \(4289355055114182239\) \(\nu^{6}\mathstrut -\mathstrut \) \(761297357669862713\) \(\nu^{5}\mathstrut +\mathstrut \) \(2250678952906410976\) \(\nu^{4}\mathstrut +\mathstrut \) \(211833884566188009\) \(\nu^{3}\mathstrut -\mathstrut \) \(437051403902960670\) \(\nu^{2}\mathstrut +\mathstrut \) \(7580132412691114\) \(\nu\mathstrut +\mathstrut \) \(15578359166866210\)\()/\)\(2150925359362898\)
\(\beta_{14}\)\(=\)\((\)\(353037342994262\) \(\nu^{18}\mathstrut -\mathstrut \) \(3650350764170579\) \(\nu^{17}\mathstrut +\mathstrut \) \(4670135112051207\) \(\nu^{16}\mathstrut +\mathstrut \) \(63020477436189709\) \(\nu^{15}\mathstrut -\mathstrut \) \(176015712085932995\) \(\nu^{14}\mathstrut -\mathstrut \) \(410241211393342356\) \(\nu^{13}\mathstrut +\mathstrut \) \(1608814341703530915\) \(\nu^{12}\mathstrut +\mathstrut \) \(1267066623942905602\) \(\nu^{11}\mathstrut -\mathstrut \) \(6927915966275822439\) \(\nu^{10}\mathstrut -\mathstrut \) \(1932658671597026008\) \(\nu^{9}\mathstrut +\mathstrut \) \(15669158218445098021\) \(\nu^{8}\mathstrut +\mathstrut \) \(1320296595139900590\) \(\nu^{7}\mathstrut -\mathstrut \) \(18573635771452756933\) \(\nu^{6}\mathstrut -\mathstrut \) \(186101728292982383\) \(\nu^{5}\mathstrut +\mathstrut \) \(10749933689944044970\) \(\nu^{4}\mathstrut -\mathstrut \) \(172262366523370591\) \(\nu^{3}\mathstrut -\mathstrut \) \(2624901258638514130\) \(\nu^{2}\mathstrut +\mathstrut \) \(88392750804068876\) \(\nu\mathstrut +\mathstrut \) \(177112305531767554\)\()/\)\(6452776078088694\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(144967756131270\) \(\nu^{18}\mathstrut +\mathstrut \) \(1518332100275248\) \(\nu^{17}\mathstrut -\mathstrut \) \(2043400743394654\) \(\nu^{16}\mathstrut -\mathstrut \) \(26419688004122481\) \(\nu^{15}\mathstrut +\mathstrut \) \(76971523571758580\) \(\nu^{14}\mathstrut +\mathstrut \) \(171792510258985088\) \(\nu^{13}\mathstrut -\mathstrut \) \(723954769683912741\) \(\nu^{12}\mathstrut -\mathstrut \) \(507326951336781899\) \(\nu^{11}\mathstrut +\mathstrut \) \(3267982838716931709\) \(\nu^{10}\mathstrut +\mathstrut \) \(589488230410150038\) \(\nu^{9}\mathstrut -\mathstrut \) \(7971847715108751644\) \(\nu^{8}\mathstrut +\mathstrut \) \(234780707290912976\) \(\nu^{7}\mathstrut +\mathstrut \) \(10655251783486461969\) \(\nu^{6}\mathstrut -\mathstrut \) \(1211330927599255329\) \(\nu^{5}\mathstrut -\mathstrut \) \(7436375629403471659\) \(\nu^{4}\mathstrut +\mathstrut \) \(1000265952470454345\) \(\nu^{3}\mathstrut +\mathstrut \) \(2377886858485232662\) \(\nu^{2}\mathstrut -\mathstrut \) \(250483038850930136\) \(\nu\mathstrut -\mathstrut \) \(241598324807555192\)\()/\)\(2150925359362898\)
\(\beta_{16}\)\(=\)\((\)\(174828361439158\) \(\nu^{18}\mathstrut -\mathstrut \) \(1670891983134690\) \(\nu^{17}\mathstrut +\mathstrut \) \(937145544465880\) \(\nu^{16}\mathstrut +\mathstrut \) \(32696588872638296\) \(\nu^{15}\mathstrut -\mathstrut \) \(63028458125406149\) \(\nu^{14}\mathstrut -\mathstrut \) \(263609998575206724\) \(\nu^{13}\mathstrut +\mathstrut \) \(632146897252809814\) \(\nu^{12}\mathstrut +\mathstrut \) \(1172201476817415999\) \(\nu^{11}\mathstrut -\mathstrut \) \(2854157122966359203\) \(\nu^{10}\mathstrut -\mathstrut \) \(3207399300370227501\) \(\nu^{9}\mathstrut +\mathstrut \) \(6553291837500947322\) \(\nu^{8}\mathstrut +\mathstrut \) \(5344968114777729854\) \(\nu^{7}\mathstrut -\mathstrut \) \(7557497765747525586\) \(\nu^{6}\mathstrut -\mathstrut \) \(4862970742421685071\) \(\nu^{5}\mathstrut +\mathstrut \) \(3967442057212468607\) \(\nu^{4}\mathstrut +\mathstrut \) \(1968215908875671143\) \(\nu^{3}\mathstrut -\mathstrut \) \(763009136103245645\) \(\nu^{2}\mathstrut -\mathstrut \) \(229861114910833352\) \(\nu\mathstrut +\mathstrut \) \(21725368230805266\)\()/\)\(2150925359362898\)
\(\beta_{17}\)\(=\)\((\)\(655027066204972\) \(\nu^{18}\mathstrut -\mathstrut \) \(7110694252781680\) \(\nu^{17}\mathstrut +\mathstrut \) \(12095069929418247\) \(\nu^{16}\mathstrut +\mathstrut \) \(113053018902722006\) \(\nu^{15}\mathstrut -\mathstrut \) \(387019060792266505\) \(\nu^{14}\mathstrut -\mathstrut \) \(605415309481864821\) \(\nu^{13}\mathstrut +\mathstrut \) \(3397385355004093485\) \(\nu^{12}\mathstrut +\mathstrut \) \(927136883649285980\) \(\nu^{11}\mathstrut -\mathstrut \) \(14280966115401152403\) \(\nu^{10}\mathstrut +\mathstrut \) \(2436901544615426671\) \(\nu^{9}\mathstrut +\mathstrut \) \(31935385298806960094\) \(\nu^{8}\mathstrut -\mathstrut \) \(10540251563429677311\) \(\nu^{7}\mathstrut -\mathstrut \) \(38152982159858021813\) \(\nu^{6}\mathstrut +\mathstrut \) \(13433577279653527748\) \(\nu^{5}\mathstrut +\mathstrut \) \(23122456789069359998\) \(\nu^{4}\mathstrut -\mathstrut \) \(6443205707102733899\) \(\nu^{3}\mathstrut -\mathstrut \) \(6476202794540901323\) \(\nu^{2}\mathstrut +\mathstrut \) \(967405796552558206\) \(\nu\mathstrut +\mathstrut \) \(607129024872538562\)\()/\)\(6452776078088694\)
\(\beta_{18}\)\(=\)\((\)\(-\)\(302250125901244\) \(\nu^{18}\mathstrut +\mathstrut \) \(3145155503018187\) \(\nu^{17}\mathstrut -\mathstrut \) \(4086203089293762\) \(\nu^{16}\mathstrut -\mathstrut \) \(54969125714346792\) \(\nu^{15}\mathstrut +\mathstrut \) \(156374625196583464\) \(\nu^{14}\mathstrut +\mathstrut \) \(361553710014794715\) \(\nu^{13}\mathstrut -\mathstrut \) \(1467427772448728915\) \(\nu^{12}\mathstrut -\mathstrut \) \(1105299188237222633\) \(\nu^{11}\mathstrut +\mathstrut \) \(6572397183636933523\) \(\nu^{10}\mathstrut +\mathstrut \) \(1486398070726977459\) \(\nu^{9}\mathstrut -\mathstrut \) \(15799521068882191827\) \(\nu^{8}\mathstrut -\mathstrut \) \(204643879778015419\) \(\nu^{7}\mathstrut +\mathstrut \) \(20626446329394644239\) \(\nu^{6}\mathstrut -\mathstrut \) \(1548503094023485606\) \(\nu^{5}\mathstrut -\mathstrut \) \(13905329644636565417\) \(\nu^{4}\mathstrut +\mathstrut \) \(1412593728633760119\) \(\nu^{3}\mathstrut +\mathstrut \) \(4226034000091164387\) \(\nu^{2}\mathstrut -\mathstrut \) \(368001446724494306\) \(\nu\mathstrut -\mathstrut \) \(387460878357396996\)\()/\)\(2150925359362898\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(16\) \(\beta_{1}\mathstrut +\mathstrut \) \(33\)
\(\nu^{5}\)\(=\)\(\beta_{18}\mathstrut +\mathstrut \) \(3\) \(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut -\mathstrut \) \(5\) \(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{13}\mathstrut -\mathstrut \) \(3\) \(\beta_{11}\mathstrut -\mathstrut \) \(7\) \(\beta_{10}\mathstrut -\mathstrut \) \(4\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{3}\mathstrut +\mathstrut \) \(34\) \(\beta_{2}\mathstrut +\mathstrut \) \(73\) \(\beta_{1}\mathstrut +\mathstrut \) \(77\)
\(\nu^{6}\)\(=\)\(5\) \(\beta_{18}\mathstrut +\mathstrut \) \(13\) \(\beta_{17}\mathstrut -\mathstrut \) \(5\) \(\beta_{16}\mathstrut -\mathstrut \) \(10\) \(\beta_{15}\mathstrut -\mathstrut \) \(27\) \(\beta_{14}\mathstrut +\mathstrut \) \(6\) \(\beta_{13}\mathstrut +\mathstrut \) \(4\) \(\beta_{12}\mathstrut -\mathstrut \) \(19\) \(\beta_{11}\mathstrut -\mathstrut \) \(41\) \(\beta_{10}\mathstrut -\mathstrut \) \(24\) \(\beta_{9}\mathstrut -\mathstrut \) \(6\) \(\beta_{7}\mathstrut -\mathstrut \) \(16\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(38\) \(\beta_{3}\mathstrut +\mathstrut \) \(149\) \(\beta_{2}\mathstrut +\mathstrut \) \(222\) \(\beta_{1}\mathstrut +\mathstrut \) \(338\)
\(\nu^{7}\)\(=\)\(33\) \(\beta_{18}\mathstrut +\mathstrut \) \(87\) \(\beta_{17}\mathstrut -\mathstrut \) \(33\) \(\beta_{16}\mathstrut -\mathstrut \) \(6\) \(\beta_{15}\mathstrut -\mathstrut \) \(118\) \(\beta_{14}\mathstrut +\mathstrut \) \(45\) \(\beta_{13}\mathstrut +\mathstrut \) \(22\) \(\beta_{12}\mathstrut -\mathstrut \) \(67\) \(\beta_{11}\mathstrut -\mathstrut \) \(199\) \(\beta_{10}\mathstrut -\mathstrut \) \(97\) \(\beta_{9}\mathstrut -\mathstrut \) \(38\) \(\beta_{7}\mathstrut -\mathstrut \) \(43\) \(\beta_{6}\mathstrut +\mathstrut \) \(29\) \(\beta_{5}\mathstrut +\mathstrut \) \(28\) \(\beta_{4}\mathstrut +\mathstrut \) \(171\) \(\beta_{3}\mathstrut +\mathstrut \) \(492\) \(\beta_{2}\mathstrut +\mathstrut \) \(883\) \(\beta_{1}\mathstrut +\mathstrut \) \(987\)
\(\nu^{8}\)\(=\)\(148\) \(\beta_{18}\mathstrut +\mathstrut \) \(373\) \(\beta_{17}\mathstrut -\mathstrut \) \(150\) \(\beta_{16}\mathstrut -\mathstrut \) \(37\) \(\beta_{15}\mathstrut -\mathstrut \) \(515\) \(\beta_{14}\mathstrut +\mathstrut \) \(168\) \(\beta_{13}\mathstrut +\mathstrut \) \(150\) \(\beta_{12}\mathstrut -\mathstrut \) \(308\) \(\beta_{11}\mathstrut -\mathstrut \) \(927\) \(\beta_{10}\mathstrut -\mathstrut \) \(451\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(191\) \(\beta_{7}\mathstrut -\mathstrut \) \(212\) \(\beta_{6}\mathstrut +\mathstrut \) \(115\) \(\beta_{5}\mathstrut +\mathstrut \) \(120\) \(\beta_{4}\mathstrut +\mathstrut \) \(578\) \(\beta_{3}\mathstrut +\mathstrut \) \(1950\) \(\beta_{2}\mathstrut +\mathstrut \) \(3006\) \(\beta_{1}\mathstrut +\mathstrut \) \(3770\)
\(\nu^{9}\)\(=\)\(718\) \(\beta_{18}\mathstrut +\mathstrut \) \(1816\) \(\beta_{17}\mathstrut -\mathstrut \) \(736\) \(\beta_{16}\mathstrut +\mathstrut \) \(189\) \(\beta_{15}\mathstrut -\mathstrut \) \(2120\) \(\beta_{14}\mathstrut +\mathstrut \) \(852\) \(\beta_{13}\mathstrut +\mathstrut \) \(732\) \(\beta_{12}\mathstrut -\mathstrut \) \(1154\) \(\beta_{11}\mathstrut -\mathstrut \) \(4036\) \(\beta_{10}\mathstrut -\mathstrut \) \(1823\) \(\beta_{9}\mathstrut +\mathstrut \) \(16\) \(\beta_{8}\mathstrut -\mathstrut \) \(935\) \(\beta_{7}\mathstrut -\mathstrut \) \(653\) \(\beta_{6}\mathstrut +\mathstrut \) \(656\) \(\beta_{5}\mathstrut +\mathstrut \) \(601\) \(\beta_{4}\mathstrut +\mathstrut \) \(2313\) \(\beta_{3}\mathstrut +\mathstrut \) \(6899\) \(\beta_{2}\mathstrut +\mathstrut \) \(11419\) \(\beta_{1}\mathstrut +\mathstrut \) \(12091\)
\(\nu^{10}\)\(=\)\(3054\) \(\beta_{18}\mathstrut +\mathstrut \) \(7613\) \(\beta_{17}\mathstrut -\mathstrut \) \(3175\) \(\beta_{16}\mathstrut +\mathstrut \) \(1017\) \(\beta_{15}\mathstrut -\mathstrut \) \(8664\) \(\beta_{14}\mathstrut +\mathstrut \) \(3397\) \(\beta_{13}\mathstrut +\mathstrut \) \(3682\) \(\beta_{12}\mathstrut -\mathstrut \) \(4771\) \(\beta_{11}\mathstrut -\mathstrut \) \(17260\) \(\beta_{10}\mathstrut -\mathstrut \) \(7746\) \(\beta_{9}\mathstrut +\mathstrut \) \(111\) \(\beta_{8}\mathstrut -\mathstrut \) \(4254\) \(\beta_{7}\mathstrut -\mathstrut \) \(2663\) \(\beta_{6}\mathstrut +\mathstrut \) \(2760\) \(\beta_{5}\mathstrut +\mathstrut \) \(2562\) \(\beta_{4}\mathstrut +\mathstrut \) \(8280\) \(\beta_{3}\mathstrut +\mathstrut \) \(26384\) \(\beta_{2}\mathstrut +\mathstrut \) \(40826\) \(\beta_{1}\mathstrut +\mathstrut \) \(43769\)
\(\nu^{11}\)\(=\)\(13256\) \(\beta_{18}\mathstrut +\mathstrut \) \(33126\) \(\beta_{17}\mathstrut -\mathstrut \) \(13971\) \(\beta_{16}\mathstrut +\mathstrut \) \(7420\) \(\beta_{15}\mathstrut -\mathstrut \) \(34623\) \(\beta_{14}\mathstrut +\mathstrut \) \(15021\) \(\beta_{13}\mathstrut +\mathstrut \) \(16630\) \(\beta_{12}\mathstrut -\mathstrut \) \(18223\) \(\beta_{11}\mathstrut -\mathstrut \) \(71541\) \(\beta_{10}\mathstrut -\mathstrut \) \(31059\) \(\beta_{9}\mathstrut +\mathstrut \) \(616\) \(\beta_{8}\mathstrut -\mathstrut \) \(18877\) \(\beta_{7}\mathstrut -\mathstrut \) \(8713\) \(\beta_{6}\mathstrut +\mathstrut \) \(13015\) \(\beta_{5}\mathstrut +\mathstrut \) \(11377\) \(\beta_{4}\mathstrut +\mathstrut \) \(31750\) \(\beta_{3}\mathstrut +\mathstrut \) \(96373\) \(\beta_{2}\mathstrut +\mathstrut \) \(152906\) \(\beta_{1}\mathstrut +\mathstrut \) \(146598\)
\(\nu^{12}\)\(=\)\(54525\) \(\beta_{18}\mathstrut +\mathstrut \) \(135787\) \(\beta_{17}\mathstrut -\mathstrut \) \(58230\) \(\beta_{16}\mathstrut +\mathstrut \) \(35171\) \(\beta_{15}\mathstrut -\mathstrut \) \(137565\) \(\beta_{14}\mathstrut +\mathstrut \) \(60562\) \(\beta_{13}\mathstrut +\mathstrut \) \(74771\) \(\beta_{12}\mathstrut -\mathstrut \) \(72209\) \(\beta_{11}\mathstrut -\mathstrut \) \(293260\) \(\beta_{10}\mathstrut -\mathstrut \) \(126557\) \(\beta_{9}\mathstrut +\mathstrut \) \(3202\) \(\beta_{8}\mathstrut -\mathstrut \) \(81036\) \(\beta_{7}\mathstrut -\mathstrut \) \(32570\) \(\beta_{6}\mathstrut +\mathstrut \) \(54570\) \(\beta_{5}\mathstrut +\mathstrut \) \(47440\) \(\beta_{4}\mathstrut +\mathstrut \) \(116419\) \(\beta_{3}\mathstrut +\mathstrut \) \(364117\) \(\beta_{2}\mathstrut +\mathstrut \) \(559476\) \(\beta_{1}\mathstrut +\mathstrut \) \(519838\)
\(\nu^{13}\)\(=\)\(224702\) \(\beta_{18}\mathstrut +\mathstrut \) \(561263\) \(\beta_{17}\mathstrut -\mathstrut \) \(243129\) \(\beta_{16}\mathstrut +\mathstrut \) \(174115\) \(\beta_{15}\mathstrut -\mathstrut \) \(540922\) \(\beta_{14}\mathstrut +\mathstrut \) \(252439\) \(\beta_{13}\mathstrut +\mathstrut \) \(321241\) \(\beta_{12}\mathstrut -\mathstrut \) \(277027\) \(\beta_{11}\mathstrut -\mathstrut \) \(1182524\) \(\beta_{10}\mathstrut -\mathstrut \) \(502340\) \(\beta_{9}\mathstrut +\mathstrut \) \(15336\) \(\beta_{8}\mathstrut -\mathstrut \) \(341719\) \(\beta_{7}\mathstrut -\mathstrut \) \(109336\) \(\beta_{6}\mathstrut +\mathstrut \) \(236461\) \(\beta_{5}\mathstrut +\mathstrut \) \(199199\) \(\beta_{4}\mathstrut +\mathstrut \) \(439412\) \(\beta_{3}\mathstrut +\mathstrut \) \(1350786\) \(\beta_{2}\mathstrut +\mathstrut \) \(2089400\) \(\beta_{1}\mathstrut +\mathstrut \) \(1780558\)
\(\nu^{14}\)\(=\)\(903293\) \(\beta_{18}\mathstrut +\mathstrut \) \(2258486\) \(\beta_{17}\mathstrut -\mathstrut \) \(989217\) \(\beta_{16}\mathstrut +\mathstrut \) \(769054\) \(\beta_{15}\mathstrut -\mathstrut \) \(2118285\) \(\beta_{14}\mathstrut +\mathstrut \) \(1012380\) \(\beta_{13}\mathstrut +\mathstrut \) \(1366034\) \(\beta_{12}\mathstrut -\mathstrut \) \(1077174\) \(\beta_{11}\mathstrut -\mathstrut \) \(4731453\) \(\beta_{10}\mathstrut -\mathstrut \) \(2001167\) \(\beta_{9}\mathstrut +\mathstrut \) \(70841\) \(\beta_{8}\mathstrut -\mathstrut \) \(1414889\) \(\beta_{7}\mathstrut -\mathstrut \) \(391376\) \(\beta_{6}\mathstrut +\mathstrut \) \(974508\) \(\beta_{5}\mathstrut +\mathstrut \) \(813161\) \(\beta_{4}\mathstrut +\mathstrut \) \(1628907\) \(\beta_{3}\mathstrut +\mathstrut \) \(5086973\) \(\beta_{2}\mathstrut +\mathstrut \) \(7736668\) \(\beta_{1}\mathstrut +\mathstrut \) \(6272763\)
\(\nu^{15}\)\(=\)\(3619501\) \(\beta_{18}\mathstrut +\mathstrut \) \(9081824\) \(\beta_{17}\mathstrut -\mathstrut \) \(4010750\) \(\beta_{16}\mathstrut +\mathstrut \) \(3398286\) \(\beta_{15}\mathstrut -\mathstrut \) \(8249447\) \(\beta_{14}\mathstrut +\mathstrut \) \(4096382\) \(\beta_{13}\mathstrut +\mathstrut \) \(5676865\) \(\beta_{12}\mathstrut -\mathstrut \) \(4130858\) \(\beta_{11}\mathstrut -\mathstrut \) \(18751178\) \(\beta_{10}\mathstrut -\mathstrut \) \(7869401\) \(\beta_{9}\mathstrut +\mathstrut \) \(314221\) \(\beta_{8}\mathstrut -\mathstrut \) \(5786516\) \(\beta_{7}\mathstrut -\mathstrut \) \(1327147\) \(\beta_{6}\mathstrut +\mathstrut \) \(4042291\) \(\beta_{5}\mathstrut +\mathstrut \) \(3311304\) \(\beta_{4}\mathstrut +\mathstrut \) \(6115199\) \(\beta_{3}\mathstrut +\mathstrut \) \(19023991\) \(\beta_{2}\mathstrut +\mathstrut \) \(28919588\) \(\beta_{1}\mathstrut +\mathstrut \) \(21772458\)
\(\nu^{16}\)\(=\)\(14318368\) \(\beta_{18}\mathstrut +\mathstrut \) \(36028449\) \(\beta_{17}\mathstrut -\mathstrut \) \(16039115\) \(\beta_{16}\mathstrut +\mathstrut \) \(14364222\) \(\beta_{15}\mathstrut -\mathstrut \) \(32032891\) \(\beta_{14}\mathstrut +\mathstrut \) \(16281919\) \(\beta_{13}\mathstrut +\mathstrut \) \(23366682\) \(\beta_{12}\mathstrut -\mathstrut \) \(15913967\) \(\beta_{11}\mathstrut -\mathstrut \) \(73895209\) \(\beta_{10}\mathstrut -\mathstrut \) \(30925352\) \(\beta_{9}\mathstrut +\mathstrut \) \(1362582\) \(\beta_{8}\mathstrut -\mathstrut \) \(23397889\) \(\beta_{7}\mathstrut -\mathstrut \) \(4642555\) \(\beta_{6}\mathstrut +\mathstrut \) \(16372699\) \(\beta_{5}\mathstrut +\mathstrut \) \(13290225\) \(\beta_{4}\mathstrut +\mathstrut \) \(22796495\) \(\beta_{3}\mathstrut +\mathstrut \) \(71639883\) \(\beta_{2}\mathstrut +\mathstrut \) \(107807386\) \(\beta_{1}\mathstrut +\mathstrut \) \(76694101\)
\(\nu^{17}\)\(=\)\(56413703\) \(\beta_{18}\mathstrut +\mathstrut \) \(142495824\) \(\beta_{17}\mathstrut -\mathstrut \) \(63856265\) \(\beta_{16}\mathstrut +\mathstrut \) \(60259176\) \(\beta_{15}\mathstrut -\mathstrut \) \(123971633\) \(\beta_{14}\mathstrut +\mathstrut \) \(64760198\) \(\beta_{13}\mathstrut +\mathstrut \) \(94906578\) \(\beta_{12}\mathstrut -\mathstrut \) \(60924245\) \(\beta_{11}\mathstrut -\mathstrut \) \(289457735\) \(\beta_{10}\mathstrut -\mathstrut \) \(120674380\) \(\beta_{9}\mathstrut +\mathstrut \) \(5772761\) \(\beta_{8}\mathstrut -\mathstrut \) \(93797958\) \(\beta_{7}\mathstrut -\mathstrut \) \(15791541\) \(\beta_{6}\mathstrut +\mathstrut \) \(66205883\) \(\beta_{5}\mathstrut +\mathstrut \) \(53106013\) \(\beta_{4}\mathstrut +\mathstrut \) \(85477884\) \(\beta_{3}\mathstrut +\mathstrut \) \(269128095\) \(\beta_{2}\mathstrut +\mathstrut \) \(403768568\) \(\beta_{1}\mathstrut +\mathstrut \) \(268664383\)
\(\nu^{18}\)\(=\)\(220637586\) \(\beta_{18}\mathstrut +\mathstrut \) \(559286244\) \(\beta_{17}\mathstrut -\mathstrut \) \(252174614\) \(\beta_{16}\mathstrut +\mathstrut \) \(247440195\) \(\beta_{15}\mathstrut -\mathstrut \) \(478755658\) \(\beta_{14}\mathstrut +\mathstrut \) \(255195414\) \(\beta_{13}\mathstrut +\mathstrut \) \(382514580\) \(\beta_{12}\mathstrut -\mathstrut \) \(233528207\) \(\beta_{11}\mathstrut -\mathstrut \) \(1129159776\) \(\beta_{10}\mathstrut -\mathstrut \) \(470005455\) \(\beta_{9}\mathstrut +\mathstrut \) \(24088210\) \(\beta_{8}\mathstrut -\mathstrut \) \(373166748\) \(\beta_{7}\mathstrut -\mathstrut \) \(54522627\) \(\beta_{6}\mathstrut +\mathstrut \) \(264335503\) \(\beta_{5}\mathstrut +\mathstrut \) \(210424521\) \(\beta_{4}\mathstrut +\mathstrut \) \(319707626\) \(\beta_{3}\mathstrut +\mathstrut \) \(1014435142\) \(\beta_{2}\mathstrut +\mathstrut \) \(1511463626\) \(\beta_{1}\mathstrut +\mathstrut \) \(949180335\)

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
3.78636
3.44816
3.39598
3.23865
2.75123
1.60398
1.26056
0.855860
0.767725
0.489749
−0.328384
−0.730825
−0.769853
−1.34302
−1.49380
−1.51882
−1.87423
−2.08616
−2.45315
1.00000 −2.78636 1.00000 1.00000 −2.78636 4.72864 1.00000 4.76382 1.00000
1.2 1.00000 −2.44816 1.00000 1.00000 −2.44816 2.12370 1.00000 2.99346 1.00000
1.3 1.00000 −2.39598 1.00000 1.00000 −2.39598 −2.58609 1.00000 2.74074 1.00000
1.4 1.00000 −2.23865 1.00000 1.00000 −2.23865 −3.60651 1.00000 2.01156 1.00000
1.5 1.00000 −1.75123 1.00000 1.00000 −1.75123 −0.581330 1.00000 0.0668066 1.00000
1.6 1.00000 −0.603975 1.00000 1.00000 −0.603975 4.28177 1.00000 −2.63521 1.00000
1.7 1.00000 −0.260557 1.00000 1.00000 −0.260557 −0.332615 1.00000 −2.93211 1.00000
1.8 1.00000 0.144140 1.00000 1.00000 0.144140 −1.20931 1.00000 −2.97922 1.00000
1.9 1.00000 0.232275 1.00000 1.00000 0.232275 3.40525 1.00000 −2.94605 1.00000
1.10 1.00000 0.510251 1.00000 1.00000 0.510251 −0.810786 1.00000 −2.73964 1.00000
1.11 1.00000 1.32838 1.00000 1.00000 1.32838 −1.94037 1.00000 −1.23540 1.00000
1.12 1.00000 1.73082 1.00000 1.00000 1.73082 −4.26460 1.00000 −0.00424589 1.00000
1.13 1.00000 1.76985 1.00000 1.00000 1.76985 3.59218 1.00000 0.132380 1.00000
1.14 1.00000 2.34302 1.00000 1.00000 2.34302 2.12832 1.00000 2.48975 1.00000
1.15 1.00000 2.49380 1.00000 1.00000 2.49380 2.61865 1.00000 3.21904 1.00000
1.16 1.00000 2.51882 1.00000 1.00000 2.51882 3.87973 1.00000 3.34446 1.00000
1.17 1.00000 2.87423 1.00000 1.00000 2.87423 −4.32557 1.00000 5.26121 1.00000
1.18 1.00000 3.08616 1.00000 1.00000 3.08616 −0.0642466 1.00000 6.52440 1.00000
1.19 1.00000 3.45315 1.00000 1.00000 3.45315 0.963209 1.00000 8.92423 1.00000
\(n\): 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
\(2\) \(-1\)
\(5\) \(-1\)
\(11\) \(-1\)
\(73\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\(T_{3}^{19} - \cdots\)
\(T_{7}^{19} - \cdots\)