Properties

Label 6040.2.a.p
Level 6040
Weight 2
Character orbit 6040.a
Self dual Yes
Analytic conductor 48.230
Analytic rank 1
Dimension 19
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6040.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2296428209\)
Analytic rank: \(1\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \(+ q^{5}\) \( + \beta_{3} q^{7} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \(+ q^{5}\) \( + \beta_{3} q^{7} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{9} \) \( + ( -1 - \beta_{18} ) q^{11} \) \( + ( \beta_{1} - \beta_{14} ) q^{13} \) \( -\beta_{1} q^{15} \) \( -\beta_{13} q^{17} \) \( + ( \beta_{3} + \beta_{8} + \beta_{12} - \beta_{17} ) q^{19} \) \( + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{9} ) q^{21} \) \( + ( -3 + \beta_{1} - 2 \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{17} + \beta_{18} ) q^{23} \) \(+ q^{25}\) \( + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{14} + \beta_{15} + \beta_{16} ) q^{27} \) \( + ( -2 - \beta_{2} - \beta_{3} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{16} + \beta_{17} + \beta_{18} ) q^{29} \) \( + ( -1 - \beta_{1} - \beta_{8} - \beta_{16} + \beta_{18} ) q^{31} \) \( + ( \beta_{1} - \beta_{3} + \beta_{9} + \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{18} ) q^{33} \) \( + \beta_{3} q^{35} \) \( + ( -\beta_{2} + \beta_{3} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} - \beta_{18} ) q^{37} \) \( + ( -5 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{17} + \beta_{18} ) q^{39} \) \( + ( -1 - \beta_{3} - \beta_{5} + \beta_{7} - \beta_{11} + \beta_{14} + \beta_{18} ) q^{41} \) \( + ( -\beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{12} - \beta_{17} ) q^{43} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{45} \) \( + ( \beta_{1} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} + \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{14} - \beta_{15} - \beta_{18} ) q^{47} \) \( + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} + 2 \beta_{9} - \beta_{10} + \beta_{16} ) q^{49} \) \( + ( 1 - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} - 3 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + \beta_{12} + 2 \beta_{16} - \beta_{17} ) q^{51} \) \( + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} + 2 \beta_{7} - 2 \beta_{9} + \beta_{11} + \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{53} \) \( + ( -1 - \beta_{18} ) q^{55} \) \( + ( -1 + 2 \beta_{1} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{18} ) q^{57} \) \( + ( -3 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} + \beta_{17} ) q^{59} \) \( + ( -5 + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} + \beta_{16} ) q^{61} \) \( + ( -3 + \beta_{1} + \beta_{4} - \beta_{5} + 3 \beta_{7} - 2 \beta_{9} + \beta_{11} - 2 \beta_{16} + \beta_{18} ) q^{63} \) \( + ( \beta_{1} - \beta_{14} ) q^{65} \) \( + ( -\beta_{1} + 2 \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - 2 \beta_{17} - 2 \beta_{18} ) q^{67} \) \( + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{12} - 2 \beta_{14} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{69} \) \( + ( -5 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + \beta_{11} - 2 \beta_{12} + 2 \beta_{17} + \beta_{18} ) q^{71} \) \( + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{73} \) \( -\beta_{1} q^{75} \) \( + ( -3 + \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{14} + 2 \beta_{15} + \beta_{17} + \beta_{18} ) q^{77} \) \( + ( -5 + 2 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - 3 \beta_{9} - \beta_{10} + 4 \beta_{12} + 2 \beta_{13} + \beta_{15} + \beta_{16} ) q^{79} \) \( + ( 6 + 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{81} \) \( + ( -6 + 3 \beta_{1} - 4 \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{7} - \beta_{8} - 4 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{83} \) \( -\beta_{13} q^{85} \) \( + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{7} - \beta_{8} - 4 \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + \beta_{15} - 4 \beta_{16} + 2 \beta_{17} + \beta_{18} ) q^{87} \) \( + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{15} + 2 \beta_{16} - \beta_{17} - \beta_{18} ) q^{89} \) \( + ( 2 \beta_{2} + 3 \beta_{3} - \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} - \beta_{17} ) q^{91} \) \( + ( 3 + \beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{15} - 2 \beta_{18} ) q^{93} \) \( + ( \beta_{3} + \beta_{8} + \beta_{12} - \beta_{17} ) q^{95} \) \( + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} + 2 \beta_{14} + \beta_{15} + \beta_{18} ) q^{97} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{13} - 3 \beta_{14} - \beta_{15} - \beta_{17} - 4 \beta_{18} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(19q \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut +\mathstrut 19q^{5} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 26q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(19q \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut +\mathstrut 19q^{5} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 26q^{9} \) \(\mathstrut -\mathstrut 18q^{11} \) \(\mathstrut +\mathstrut 5q^{13} \) \(\mathstrut -\mathstrut 5q^{15} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut -\mathstrut 27q^{19} \) \(\mathstrut -\mathstrut 18q^{21} \) \(\mathstrut -\mathstrut 25q^{23} \) \(\mathstrut +\mathstrut 19q^{25} \) \(\mathstrut -\mathstrut 35q^{27} \) \(\mathstrut -\mathstrut 35q^{29} \) \(\mathstrut -\mathstrut 26q^{31} \) \(\mathstrut -\mathstrut 8q^{35} \) \(\mathstrut -\mathstrut 10q^{37} \) \(\mathstrut -\mathstrut 48q^{39} \) \(\mathstrut -\mathstrut 14q^{41} \) \(\mathstrut -\mathstrut 21q^{43} \) \(\mathstrut +\mathstrut 26q^{45} \) \(\mathstrut -\mathstrut 40q^{47} \) \(\mathstrut +\mathstrut 23q^{49} \) \(\mathstrut -\mathstrut 32q^{51} \) \(\mathstrut -\mathstrut 3q^{53} \) \(\mathstrut -\mathstrut 18q^{55} \) \(\mathstrut -\mathstrut 13q^{57} \) \(\mathstrut -\mathstrut 28q^{59} \) \(\mathstrut -\mathstrut 46q^{61} \) \(\mathstrut -\mathstrut 53q^{63} \) \(\mathstrut +\mathstrut 5q^{65} \) \(\mathstrut -\mathstrut 42q^{67} \) \(\mathstrut -\mathstrut 31q^{69} \) \(\mathstrut -\mathstrut 46q^{71} \) \(\mathstrut +\mathstrut 31q^{73} \) \(\mathstrut -\mathstrut 5q^{75} \) \(\mathstrut +\mathstrut 15q^{77} \) \(\mathstrut -\mathstrut 56q^{79} \) \(\mathstrut +\mathstrut 31q^{81} \) \(\mathstrut -\mathstrut 25q^{83} \) \(\mathstrut -\mathstrut 4q^{85} \) \(\mathstrut -\mathstrut 20q^{87} \) \(\mathstrut -\mathstrut 7q^{89} \) \(\mathstrut -\mathstrut 61q^{91} \) \(\mathstrut +\mathstrut 29q^{93} \) \(\mathstrut -\mathstrut 27q^{95} \) \(\mathstrut +\mathstrut 39q^{97} \) \(\mathstrut -\mathstrut 52q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19}\mathstrut -\mathstrut \) \(5\) \(x^{18}\mathstrut -\mathstrut \) \(29\) \(x^{17}\mathstrut +\mathstrut \) \(165\) \(x^{16}\mathstrut +\mathstrut \) \(325\) \(x^{15}\mathstrut -\mathstrut \) \(2208\) \(x^{14}\mathstrut -\mathstrut \) \(1891\) \(x^{13}\mathstrut +\mathstrut \) \(15895\) \(x^{12}\mathstrut +\mathstrut \) \(6652\) \(x^{11}\mathstrut -\mathstrut \) \(67665\) \(x^{10}\mathstrut -\mathstrut \) \(17345\) \(x^{9}\mathstrut +\mathstrut \) \(174105\) \(x^{8}\mathstrut +\mathstrut \) \(41499\) \(x^{7}\mathstrut -\mathstrut \) \(262172\) \(x^{6}\mathstrut -\mathstrut \) \(80919\) \(x^{5}\mathstrut +\mathstrut \) \(206783\) \(x^{4}\mathstrut +\mathstrut \) \(91643\) \(x^{3}\mathstrut -\mathstrut \) \(59750\) \(x^{2}\mathstrut -\mathstrut \) \(40224\) \(x\mathstrut -\mathstrut \) \(5628\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\(55395572792623\) \(\nu^{18}\mathstrut -\mathstrut \) \(1282570164104816\) \(\nu^{17}\mathstrut +\mathstrut \) \(3843828599512745\) \(\nu^{16}\mathstrut +\mathstrut \) \(33567815192362180\) \(\nu^{15}\mathstrut -\mathstrut \) \(144066698105804193\) \(\nu^{14}\mathstrut -\mathstrut \) \(325977195535059477\) \(\nu^{13}\mathstrut +\mathstrut \) \(1792759675120172710\) \(\nu^{12}\mathstrut +\mathstrut \) \(1588052770850214607\) \(\nu^{11}\mathstrut -\mathstrut \) \(11294005761518533393\) \(\nu^{10}\mathstrut -\mathstrut \) \(4500209660384611576\) \(\nu^{9}\mathstrut +\mathstrut \) \(40067118650304619217\) \(\nu^{8}\mathstrut +\mathstrut \) \(8934620047045302660\) \(\nu^{7}\mathstrut -\mathstrut \) \(80173003627501509915\) \(\nu^{6}\mathstrut -\mathstrut \) \(14636171244761239999\) \(\nu^{5}\mathstrut +\mathstrut \) \(82631139058765852268\) \(\nu^{4}\mathstrut +\mathstrut \) \(16339925852923382945\) \(\nu^{3}\mathstrut -\mathstrut \) \(33212601804754250690\) \(\nu^{2}\mathstrut -\mathstrut \) \(7576366659179760236\) \(\nu\mathstrut +\mathstrut \) \(1011634103784965924\)\()/\)\(228771461625160096\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(825471425164009\) \(\nu^{18}\mathstrut +\mathstrut \) \(5148768030504856\) \(\nu^{17}\mathstrut +\mathstrut \) \(16656356771269549\) \(\nu^{16}\mathstrut -\mathstrut \) \(150533195307394516\) \(\nu^{15}\mathstrut -\mathstrut \) \(67958043505442033\) \(\nu^{14}\mathstrut +\mathstrut \) \(1739613051424907459\) \(\nu^{13}\mathstrut -\mathstrut \) \(567866092541121870\) \(\nu^{12}\mathstrut -\mathstrut \) \(10775530281108362853\) \(\nu^{11}\mathstrut +\mathstrut \) \(6541382809435982627\) \(\nu^{10}\mathstrut +\mathstrut \) \(39992548162303710512\) \(\nu^{9}\mathstrut -\mathstrut \) \(27241571945505753767\) \(\nu^{8}\mathstrut -\mathstrut \) \(91839653932711203196\) \(\nu^{7}\mathstrut +\mathstrut \) \(58655996588257041457\) \(\nu^{6}\mathstrut +\mathstrut \) \(126453980894546361929\) \(\nu^{5}\mathstrut -\mathstrut \) \(62145344231637892636\) \(\nu^{4}\mathstrut -\mathstrut \) \(93399395100110890643\) \(\nu^{3}\mathstrut +\mathstrut \) \(20011205006996581446\) \(\nu^{2}\mathstrut +\mathstrut \) \(28137440905699534180\) \(\nu\mathstrut +\mathstrut \) \(3451667624814453428\)\()/\)\(457542923250320192\)
\(\beta_{5}\)\(=\)\((\)\(831149248013573\) \(\nu^{18}\mathstrut -\mathstrut \) \(6529221237850736\) \(\nu^{17}\mathstrut -\mathstrut \) \(8468181688802937\) \(\nu^{16}\mathstrut +\mathstrut \) \(172205964148202772\) \(\nu^{15}\mathstrut -\mathstrut \) \(143534913928110307\) \(\nu^{14}\mathstrut -\mathstrut \) \(1694597100662116159\) \(\nu^{13}\mathstrut +\mathstrut \) \(2487208350510041006\) \(\nu^{12}\mathstrut +\mathstrut \) \(8381720526370062953\) \(\nu^{11}\mathstrut -\mathstrut \) \(14128580590427639255\) \(\nu^{10}\mathstrut -\mathstrut \) \(23375799369623611568\) \(\nu^{9}\mathstrut +\mathstrut \) \(37623785569757123579\) \(\nu^{8}\mathstrut +\mathstrut \) \(39546479291651782084\) \(\nu^{7}\mathstrut -\mathstrut \) \(47987975251419114109\) \(\nu^{6}\mathstrut -\mathstrut \) \(43115468783407000837\) \(\nu^{5}\mathstrut +\mathstrut \) \(27556736624880712228\) \(\nu^{4}\mathstrut +\mathstrut \) \(28276765452349644391\) \(\nu^{3}\mathstrut -\mathstrut \) \(9151291982405240270\) \(\nu^{2}\mathstrut -\mathstrut \) \(6077448197860154164\) \(\nu\mathstrut +\mathstrut \) \(1614654699627908572\)\()/\)\(457542923250320192\)
\(\beta_{6}\)\(=\)\((\)\(893262017742841\) \(\nu^{18}\mathstrut -\mathstrut \) \(6709911317038048\) \(\nu^{17}\mathstrut -\mathstrut \) \(4374937933861837\) \(\nu^{16}\mathstrut +\mathstrut \) \(143581904507289700\) \(\nu^{15}\mathstrut -\mathstrut \) \(236985751269460111\) \(\nu^{14}\mathstrut -\mathstrut \) \(857864446451730955\) \(\nu^{13}\mathstrut +\mathstrut \) \(2836792745171684454\) \(\nu^{12}\mathstrut -\mathstrut \) \(100863235120512307\) \(\nu^{11}\mathstrut -\mathstrut \) \(11438341612028607971\) \(\nu^{10}\mathstrut +\mathstrut \) \(18713949331749328624\) \(\nu^{9}\mathstrut +\mathstrut \) \(13366198949241451367\) \(\nu^{8}\mathstrut -\mathstrut \) \(72912510979539031644\) \(\nu^{7}\mathstrut +\mathstrut \) \(23448483493090406239\) \(\nu^{6}\mathstrut +\mathstrut \) \(119709595019567528583\) \(\nu^{5}\mathstrut -\mathstrut \) \(64647877824025594908\) \(\nu^{4}\mathstrut -\mathstrut \) \(91149929208499445773\) \(\nu^{3}\mathstrut +\mathstrut \) \(34787223627735288026\) \(\nu^{2}\mathstrut +\mathstrut \) \(29631649698934677180\) \(\nu\mathstrut +\mathstrut \) \(1722308039210366348\)\()/\)\(457542923250320192\)
\(\beta_{7}\)\(=\)\((\)\(56636909905056\) \(\nu^{18}\mathstrut -\mathstrut \) \(466859136090342\) \(\nu^{17}\mathstrut -\mathstrut \) \(472631481848911\) \(\nu^{16}\mathstrut +\mathstrut \) \(12662979091017916\) \(\nu^{15}\mathstrut -\mathstrut \) \(14351726530935302\) \(\nu^{14}\mathstrut -\mathstrut \) \(129569857849845676\) \(\nu^{13}\mathstrut +\mathstrut \) \(242072346115647165\) \(\nu^{12}\mathstrut +\mathstrut \) \(676539570406064409\) \(\nu^{11}\mathstrut -\mathstrut \) \(1504470703145431741\) \(\nu^{10}\mathstrut -\mathstrut \) \(2096524180851425584\) \(\nu^{9}\mathstrut +\mathstrut \) \(4638648378150119924\) \(\nu^{8}\mathstrut +\mathstrut \) \(4537703098655126976\) \(\nu^{7}\mathstrut -\mathstrut \) \(7232763185988054269\) \(\nu^{6}\mathstrut -\mathstrut \) \(7699778770407449598\) \(\nu^{5}\mathstrut +\mathstrut \) \(4793176495819915334\) \(\nu^{4}\mathstrut +\mathstrut \) \(8624873138914767615\) \(\nu^{3}\mathstrut -\mathstrut \) \(30204773087751264\) \(\nu^{2}\mathstrut -\mathstrut \) \(3889761727382701440\) \(\nu\mathstrut -\mathstrut \) \(982608926580490792\)\()/\)\(28596432703145012\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(1842573741871917\) \(\nu^{18}\mathstrut +\mathstrut \) \(7487676163763712\) \(\nu^{17}\mathstrut +\mathstrut \) \(56247718798349417\) \(\nu^{16}\mathstrut -\mathstrut \) \(234583820154108356\) \(\nu^{15}\mathstrut -\mathstrut \) \(689632753169362133\) \(\nu^{14}\mathstrut +\mathstrut \) \(2894698946932575287\) \(\nu^{13}\mathstrut +\mathstrut \) \(4616023233902091338\) \(\nu^{12}\mathstrut -\mathstrut \) \(18426961314159695961\) \(\nu^{11}\mathstrut -\mathstrut \) \(19088000936660001737\) \(\nu^{10}\mathstrut +\mathstrut \) \(65042439574070369888\) \(\nu^{9}\mathstrut +\mathstrut \) \(51035187199415557773\) \(\nu^{8}\mathstrut -\mathstrut \) \(124009163196399080772\) \(\nu^{7}\mathstrut -\mathstrut \) \(85870096282012514947\) \(\nu^{6}\mathstrut +\mathstrut \) \(107127077210264354573\) \(\nu^{5}\mathstrut +\mathstrut \) \(80227629513941781164\) \(\nu^{4}\mathstrut -\mathstrut \) \(9135263353863909511\) \(\nu^{3}\mathstrut -\mathstrut \) \(27312540334604478130\) \(\nu^{2}\mathstrut -\mathstrut \) \(25973098539530318860\) \(\nu\mathstrut -\mathstrut \) \(6696186093159290140\)\()/\)\(457542923250320192\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(265246968233581\) \(\nu^{18}\mathstrut +\mathstrut \) \(1683217593650907\) \(\nu^{17}\mathstrut +\mathstrut \) \(5145929270516660\) \(\nu^{16}\mathstrut -\mathstrut \) \(48909518613942212\) \(\nu^{15}\mathstrut -\mathstrut \) \(14899268175785925\) \(\nu^{14}\mathstrut +\mathstrut \) \(555872474701520570\) \(\nu^{13}\mathstrut -\mathstrut \) \(271304883452175172\) \(\nu^{12}\mathstrut -\mathstrut \) \(3312637470646319049\) \(\nu^{11}\mathstrut +\mathstrut \) \(2635534383536689278\) \(\nu^{10}\mathstrut +\mathstrut \) \(11382041130194319182\) \(\nu^{9}\mathstrut -\mathstrut \) \(10194286201079735993\) \(\nu^{8}\mathstrut -\mathstrut \) \(22851621137466968613\) \(\nu^{7}\mathstrut +\mathstrut \) \(20015000123231956768\) \(\nu^{6}\mathstrut +\mathstrut \) \(25437466164583338201\) \(\nu^{5}\mathstrut -\mathstrut \) \(19436518983654857783\) \(\nu^{4}\mathstrut -\mathstrut \) \(13657286033777995806\) \(\nu^{3}\mathstrut +\mathstrut \) \(7179337289577138652\) \(\nu^{2}\mathstrut +\mathstrut \) \(2818443801556378476\) \(\nu\mathstrut -\mathstrut \) \(60581224214440872\)\()/\)\(57192865406290024\)
\(\beta_{10}\)\(=\)\((\)\(2144199181458703\) \(\nu^{18}\mathstrut -\mathstrut \) \(14007918034232216\) \(\nu^{17}\mathstrut -\mathstrut \) \(38711864397166811\) \(\nu^{16}\mathstrut +\mathstrut \) \(401669051946076876\) \(\nu^{15}\mathstrut +\mathstrut \) \(46336575550144007\) \(\nu^{14}\mathstrut -\mathstrut \) \(4507542403554445333\) \(\nu^{13}\mathstrut +\mathstrut \) \(2875138811829736290\) \(\nu^{12}\mathstrut +\mathstrut \) \(26864902225484161459\) \(\nu^{11}\mathstrut -\mathstrut \) \(24123074281514233269\) \(\nu^{10}\mathstrut -\mathstrut \) \(95642613213777855920\) \(\nu^{9}\mathstrut +\mathstrut \) \(86611245129153271489\) \(\nu^{8}\mathstrut +\mathstrut \) \(213623456478839512564\) \(\nu^{7}\mathstrut -\mathstrut \) \(155286643777086364727\) \(\nu^{6}\mathstrut -\mathstrut \) \(297154724962151344431\) \(\nu^{5}\mathstrut +\mathstrut \) \(124888418158671406260\) \(\nu^{4}\mathstrut +\mathstrut \) \(233698178817215866373\) \(\nu^{3}\mathstrut -\mathstrut \) \(18400403424873851690\) \(\nu^{2}\mathstrut -\mathstrut \) \(77031313802425172604\) \(\nu\mathstrut -\mathstrut \) \(16633657644450319404\)\()/\)\(457542923250320192\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(2543053966709621\) \(\nu^{18}\mathstrut +\mathstrut \) \(19553203998112072\) \(\nu^{17}\mathstrut +\mathstrut \) \(30277054135873193\) \(\nu^{16}\mathstrut -\mathstrut \) \(548476037235782756\) \(\nu^{15}\mathstrut +\mathstrut \) \(419594042994086787\) \(\nu^{14}\mathstrut +\mathstrut \) \(5903216243609510583\) \(\nu^{13}\mathstrut -\mathstrut \) \(9019156893641800262\) \(\nu^{12}\mathstrut -\mathstrut \) \(32741676166555832273\) \(\nu^{11}\mathstrut +\mathstrut \) \(62547489597013713767\) \(\nu^{10}\mathstrut +\mathstrut \) \(104760723853281959600\) \(\nu^{9}\mathstrut -\mathstrut \) \(217003369085973033403\) \(\nu^{8}\mathstrut -\mathstrut \) \(206900403791732026524\) \(\nu^{7}\mathstrut +\mathstrut \) \(399813773255867433917\) \(\nu^{6}\mathstrut +\mathstrut \) \(267992302397578881109\) \(\nu^{5}\mathstrut -\mathstrut \) \(363351139346577945276\) \(\nu^{4}\mathstrut -\mathstrut \) \(225766875158450189399\) \(\nu^{3}\mathstrut +\mathstrut \) \(116871191622166077710\) \(\nu^{2}\mathstrut +\mathstrut \) \(89834398748893076436\) \(\nu\mathstrut +\mathstrut \) \(11925598053551983396\)\()/\)\(457542923250320192\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(648499956243255\) \(\nu^{18}\mathstrut +\mathstrut \) \(5012064203537208\) \(\nu^{17}\mathstrut +\mathstrut \) \(8667195040728045\) \(\nu^{16}\mathstrut -\mathstrut \) \(145923623282507952\) \(\nu^{15}\mathstrut +\mathstrut \) \(84746839523124309\) \(\nu^{14}\mathstrut +\mathstrut \) \(1666477337278914385\) \(\nu^{13}\mathstrut -\mathstrut \) \(2131813093126046692\) \(\nu^{12}\mathstrut -\mathstrut \) \(10077000035553477193\) \(\nu^{11}\mathstrut +\mathstrut \) \(15555986639844284415\) \(\nu^{10}\mathstrut +\mathstrut \) \(36144995093235280244\) \(\nu^{9}\mathstrut -\mathstrut \) \(56032071984249745465\) \(\nu^{8}\mathstrut -\mathstrut \) \(80756436732202726424\) \(\nu^{7}\mathstrut +\mathstrut \) \(106048900728093493181\) \(\nu^{6}\mathstrut +\mathstrut \) \(112635352391755816023\) \(\nu^{5}\mathstrut -\mathstrut \) \(96314212157456324132\) \(\nu^{4}\mathstrut -\mathstrut \) \(90911545845393234887\) \(\nu^{3}\mathstrut +\mathstrut \) \(26840551255631058438\) \(\nu^{2}\mathstrut +\mathstrut \) \(32005685014482075124\) \(\nu\mathstrut +\mathstrut \) \(5986425026293859236\)\()/\)\(114385730812580048\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(1374745276971833\) \(\nu^{18}\mathstrut +\mathstrut \) \(6233607352857088\) \(\nu^{17}\mathstrut +\mathstrut \) \(40860651477333453\) \(\nu^{16}\mathstrut -\mathstrut \) \(201481782286309668\) \(\nu^{15}\mathstrut -\mathstrut \) \(474806648528782897\) \(\nu^{14}\mathstrut +\mathstrut \) \(2607313444268021163\) \(\nu^{13}\mathstrut +\mathstrut \) \(2870798724368680730\) \(\nu^{12}\mathstrut -\mathstrut \) \(17811058676487844173\) \(\nu^{11}\mathstrut -\mathstrut \) \(9974466268086971869\) \(\nu^{10}\mathstrut +\mathstrut \) \(69983881375983107312\) \(\nu^{9}\mathstrut +\mathstrut \) \(20776420739350952313\) \(\nu^{8}\mathstrut -\mathstrut \) \(159794259178220365380\) \(\nu^{7}\mathstrut -\mathstrut \) \(27366939526710095807\) \(\nu^{6}\mathstrut +\mathstrut \) \(202499225907393933433\) \(\nu^{5}\mathstrut +\mathstrut \) \(27096772152993291772\) \(\nu^{4}\mathstrut -\mathstrut \) \(126908161268216007379\) \(\nu^{3}\mathstrut -\mathstrut \) \(22574056397359268538\) \(\nu^{2}\mathstrut +\mathstrut \) \(30638708633486468964\) \(\nu\mathstrut +\mathstrut \) \(8167086807839149108\)\()/\)\(228771461625160096\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(3039888107633601\) \(\nu^{18}\mathstrut +\mathstrut \) \(21385926549656952\) \(\nu^{17}\mathstrut +\mathstrut \) \(43569315816672797\) \(\nu^{16}\mathstrut -\mathstrut \) \(584514148862704228\) \(\nu^{15}\mathstrut +\mathstrut \) \(237007316844629479\) \(\nu^{14}\mathstrut +\mathstrut \) \(6020226093986778971\) \(\nu^{13}\mathstrut -\mathstrut \) \(6978674237471914710\) \(\nu^{12}\mathstrut -\mathstrut \) \(31024857777623091109\) \(\nu^{11}\mathstrut +\mathstrut \) \(46387313396762744579\) \(\nu^{10}\mathstrut +\mathstrut \) \(87392079449956939424\) \(\nu^{9}\mathstrut -\mathstrut \) \(141094929426519309839\) \(\nu^{8}\mathstrut -\mathstrut \) \(138163079917435060844\) \(\nu^{7}\mathstrut +\mathstrut \) \(205522033580513668913\) \(\nu^{6}\mathstrut +\mathstrut \) \(127604654796804573089\) \(\nu^{5}\mathstrut -\mathstrut \) \(121490766031181554524\) \(\nu^{4}\mathstrut -\mathstrut \) \(77599544991083627219\) \(\nu^{3}\mathstrut +\mathstrut \) \(11835540468138859814\) \(\nu^{2}\mathstrut +\mathstrut \) \(24582797901195324452\) \(\nu\mathstrut +\mathstrut \) \(6670908614249021236\)\()/\)\(457542923250320192\)
\(\beta_{15}\)\(=\)\((\)\(1652069990496175\) \(\nu^{18}\mathstrut -\mathstrut \) \(12540283023594488\) \(\nu^{17}\mathstrut -\mathstrut \) \(21564673027731835\) \(\nu^{16}\mathstrut +\mathstrut \) \(357014723638780620\) \(\nu^{15}\mathstrut -\mathstrut \) \(219304735752279001\) \(\nu^{14}\mathstrut -\mathstrut \) \(3937975463808284437\) \(\nu^{13}\mathstrut +\mathstrut \) \(5278789286287108706\) \(\nu^{12}\mathstrut +\mathstrut \) \(22691217218032930995\) \(\nu^{11}\mathstrut -\mathstrut \) \(37297691424213731637\) \(\nu^{10}\mathstrut -\mathstrut \) \(76764217475065407952\) \(\nu^{9}\mathstrut +\mathstrut \) \(129528865730130573569\) \(\nu^{8}\mathstrut +\mathstrut \) \(162421707726185401652\) \(\nu^{7}\mathstrut -\mathstrut \) \(235389831702239684215\) \(\nu^{6}\mathstrut -\mathstrut \) \(221308435501997123887\) \(\nu^{5}\mathstrut +\mathstrut \) \(205953568009256349140\) \(\nu^{4}\mathstrut +\mathstrut \) \(182585464307163511781\) \(\nu^{3}\mathstrut -\mathstrut \) \(58440175368525168138\) \(\nu^{2}\mathstrut -\mathstrut \) \(65490892598179360316\) \(\nu\mathstrut -\mathstrut \) \(10388116111708522412\)\()/\)\(228771461625160096\)
\(\beta_{16}\)\(=\)\((\)\(4125642740291903\) \(\nu^{18}\mathstrut -\mathstrut \) \(27103994474524504\) \(\nu^{17}\mathstrut -\mathstrut \) \(75818776899204467\) \(\nu^{16}\mathstrut +\mathstrut \) \(789591293776478876\) \(\nu^{15}\mathstrut +\mathstrut \) \(86357606179246823\) \(\nu^{14}\mathstrut -\mathstrut \) \(8966005037752738805\) \(\nu^{13}\mathstrut +\mathstrut \) \(6088557687263850570\) \(\nu^{12}\mathstrut +\mathstrut \) \(53110027235529915035\) \(\nu^{11}\mathstrut -\mathstrut \) \(52220481059478395469\) \(\nu^{10}\mathstrut -\mathstrut \) \(181821342301299894016\) \(\nu^{9}\mathstrut +\mathstrut \) \(191189145105819027985\) \(\nu^{8}\mathstrut +\mathstrut \) \(373141090695751187588\) \(\nu^{7}\mathstrut -\mathstrut \) \(349667716662878613727\) \(\nu^{6}\mathstrut -\mathstrut \) \(457820635907237577503\) \(\nu^{5}\mathstrut +\mathstrut \) \(296565179915255069556\) \(\nu^{4}\mathstrut +\mathstrut \) \(316755291677928609789\) \(\nu^{3}\mathstrut -\mathstrut \) \(72028452155976973882\) \(\nu^{2}\mathstrut -\mathstrut \) \(97455190035628303772\) \(\nu\mathstrut -\mathstrut \) \(17922009821745376204\)\()/\)\(457542923250320192\)
\(\beta_{17}\)\(=\)\((\)\(-\)\(5734380684972101\) \(\nu^{18}\mathstrut +\mathstrut \) \(32134720242086384\) \(\nu^{17}\mathstrut +\mathstrut \) \(141604639245533105\) \(\nu^{16}\mathstrut -\mathstrut \) \(1005499581532864900\) \(\nu^{15}\mathstrut -\mathstrut \) \(1114941497406725869\) \(\nu^{14}\mathstrut +\mathstrut \) \(12534591592790226543\) \(\nu^{13}\mathstrut +\mathstrut \) \(1861805423561134778\) \(\nu^{12}\mathstrut -\mathstrut \) \(82659828787906528913\) \(\nu^{11}\mathstrut +\mathstrut \) \(18907168699394828847\) \(\nu^{10}\mathstrut +\mathstrut \) \(316724397005442684960\) \(\nu^{9}\mathstrut -\mathstrut \) \(114197716146189228923\) \(\nu^{8}\mathstrut -\mathstrut \) \(719721317464493735924\) \(\nu^{7}\mathstrut +\mathstrut \) \(254723974714131028581\) \(\nu^{6}\mathstrut +\mathstrut \) \(939373349287456002501\) \(\nu^{5}\mathstrut -\mathstrut \) \(234702087701587701668\) \(\nu^{4}\mathstrut -\mathstrut \) \(638524446302924062879\) \(\nu^{3}\mathstrut +\mathstrut \) \(39594315574452729118\) \(\nu^{2}\mathstrut +\mathstrut \) \(175451898025721634132\) \(\nu\mathstrut +\mathstrut \) \(33803425560306121028\)\()/\)\(457542923250320192\)
\(\beta_{18}\)\(=\)\((\)\(6000740936115901\) \(\nu^{18}\mathstrut -\mathstrut \) \(37872303245196736\) \(\nu^{17}\mathstrut -\mathstrut \) \(117375998014468801\) \(\nu^{16}\mathstrut +\mathstrut \) \(1104923052372439220\) \(\nu^{15}\mathstrut +\mathstrut \) \(346768879365177429\) \(\nu^{14}\mathstrut -\mathstrut \) \(12588330322487702519\) \(\nu^{13}\mathstrut +\mathstrut \) \(6177222909748312958\) \(\nu^{12}\mathstrut +\mathstrut \) \(74989229416502241633\) \(\nu^{11}\mathstrut -\mathstrut \) \(59542132916546240239\) \(\nu^{10}\mathstrut -\mathstrut \) \(258328728461468122512\) \(\nu^{9}\mathstrut +\mathstrut \) \(223294169369261280803\) \(\nu^{8}\mathstrut +\mathstrut \) \(530761454648421512276\) \(\nu^{7}\mathstrut -\mathstrut \) \(410391423955509687781\) \(\nu^{6}\mathstrut -\mathstrut \) \(641770077230588731133\) \(\nu^{5}\mathstrut +\mathstrut \) \(348640318779203430292\) \(\nu^{4}\mathstrut +\mathstrut \) \(426181604695001605439\) \(\nu^{3}\mathstrut -\mathstrut \) \(88184304972892466494\) \(\nu^{2}\mathstrut -\mathstrut \) \(122429703947055203924\) \(\nu\mathstrut -\mathstrut \) \(19426824517150598020\)\()/\)\(457542923250320192\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(-\)\(2\) \(\beta_{14}\mathstrut -\mathstrut \) \(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(33\)
\(\nu^{5}\)\(=\)\(\beta_{17}\mathstrut -\mathstrut \) \(17\) \(\beta_{16}\mathstrut -\mathstrut \) \(12\) \(\beta_{15}\mathstrut -\mathstrut \) \(14\) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(3\) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(11\) \(\beta_{10}\mathstrut -\mathstrut \) \(16\) \(\beta_{9}\mathstrut -\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(17\) \(\beta_{7}\mathstrut -\mathstrut \) \(14\) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(15\) \(\beta_{2}\mathstrut +\mathstrut \) \(61\) \(\beta_{1}\mathstrut +\mathstrut \) \(34\)
\(\nu^{6}\)\(=\)\(-\)\(3\) \(\beta_{18}\mathstrut -\mathstrut \) \(\beta_{17}\mathstrut +\mathstrut \) \(4\) \(\beta_{15}\mathstrut -\mathstrut \) \(32\) \(\beta_{14}\mathstrut -\mathstrut \) \(34\) \(\beta_{13}\mathstrut +\mathstrut \) \(32\) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(10\) \(\beta_{10}\mathstrut +\mathstrut \) \(28\) \(\beta_{9}\mathstrut +\mathstrut \) \(32\) \(\beta_{8}\mathstrut -\mathstrut \) \(29\) \(\beta_{7}\mathstrut -\mathstrut \) \(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(13\) \(\beta_{5}\mathstrut -\mathstrut \) \(16\) \(\beta_{4}\mathstrut +\mathstrut \) \(65\) \(\beta_{3}\mathstrut +\mathstrut \) \(135\) \(\beta_{2}\mathstrut +\mathstrut \) \(114\) \(\beta_{1}\mathstrut +\mathstrut \) \(323\)
\(\nu^{7}\)\(=\)\(-\)\(5\) \(\beta_{18}\mathstrut +\mathstrut \) \(14\) \(\beta_{17}\mathstrut -\mathstrut \) \(222\) \(\beta_{16}\mathstrut -\mathstrut \) \(128\) \(\beta_{15}\mathstrut -\mathstrut \) \(174\) \(\beta_{14}\mathstrut -\mathstrut \) \(22\) \(\beta_{13}\mathstrut -\mathstrut \) \(46\) \(\beta_{12}\mathstrut +\mathstrut \) \(33\) \(\beta_{11}\mathstrut +\mathstrut \) \(112\) \(\beta_{10}\mathstrut -\mathstrut \) \(204\) \(\beta_{9}\mathstrut -\mathstrut \) \(46\) \(\beta_{8}\mathstrut +\mathstrut \) \(216\) \(\beta_{7}\mathstrut -\mathstrut \) \(166\) \(\beta_{6}\mathstrut -\mathstrut \) \(5\) \(\beta_{5}\mathstrut -\mathstrut \) \(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(172\) \(\beta_{3}\mathstrut +\mathstrut \) \(178\) \(\beta_{2}\mathstrut +\mathstrut \) \(597\) \(\beta_{1}\mathstrut +\mathstrut \) \(351\)
\(\nu^{8}\)\(=\)\(-\)\(64\) \(\beta_{18}\mathstrut -\mathstrut \) \(25\) \(\beta_{17}\mathstrut -\mathstrut \) \(6\) \(\beta_{16}\mathstrut +\mathstrut \) \(92\) \(\beta_{15}\mathstrut -\mathstrut \) \(411\) \(\beta_{14}\mathstrut -\mathstrut \) \(444\) \(\beta_{13}\mathstrut +\mathstrut \) \(417\) \(\beta_{12}\mathstrut +\mathstrut \) \(41\) \(\beta_{11}\mathstrut +\mathstrut \) \(81\) \(\beta_{10}\mathstrut +\mathstrut \) \(317\) \(\beta_{9}\mathstrut +\mathstrut \) \(411\) \(\beta_{8}\mathstrut -\mathstrut \) \(348\) \(\beta_{7}\mathstrut -\mathstrut \) \(142\) \(\beta_{6}\mathstrut +\mathstrut \) \(133\) \(\beta_{5}\mathstrut -\mathstrut \) \(210\) \(\beta_{4}\mathstrut +\mathstrut \) \(829\) \(\beta_{3}\mathstrut +\mathstrut \) \(1509\) \(\beta_{2}\mathstrut +\mathstrut \) \(1189\) \(\beta_{1}\mathstrut +\mathstrut \) \(3380\)
\(\nu^{9}\)\(=\)\(-\)\(121\) \(\beta_{18}\mathstrut +\mathstrut \) \(146\) \(\beta_{17}\mathstrut -\mathstrut \) \(2662\) \(\beta_{16}\mathstrut -\mathstrut \) \(1350\) \(\beta_{15}\mathstrut -\mathstrut \) \(2074\) \(\beta_{14}\mathstrut -\mathstrut \) \(321\) \(\beta_{13}\mathstrut -\mathstrut \) \(541\) \(\beta_{12}\mathstrut +\mathstrut \) \(427\) \(\beta_{11}\mathstrut +\mathstrut \) \(1161\) \(\beta_{10}\mathstrut -\mathstrut \) \(2439\) \(\beta_{9}\mathstrut -\mathstrut \) \(548\) \(\beta_{8}\mathstrut +\mathstrut \) \(2534\) \(\beta_{7}\mathstrut -\mathstrut \) \(1887\) \(\beta_{6}\mathstrut -\mathstrut \) \(131\) \(\beta_{5}\mathstrut -\mathstrut \) \(123\) \(\beta_{4}\mathstrut +\mathstrut \) \(2018\) \(\beta_{3}\mathstrut +\mathstrut \) \(1988\) \(\beta_{2}\mathstrut +\mathstrut \) \(6212\) \(\beta_{1}\mathstrut +\mathstrut \) \(3583\)
\(\nu^{10}\)\(=\)\(-\)\(959\) \(\beta_{18}\mathstrut -\mathstrut \) \(409\) \(\beta_{17}\mathstrut -\mathstrut \) \(118\) \(\beta_{16}\mathstrut +\mathstrut \) \(1463\) \(\beta_{15}\mathstrut -\mathstrut \) \(4922\) \(\beta_{14}\mathstrut -\mathstrut \) \(5324\) \(\beta_{13}\mathstrut +\mathstrut \) \(5077\) \(\beta_{12}\mathstrut +\mathstrut \) \(603\) \(\beta_{11}\mathstrut +\mathstrut \) \(621\) \(\beta_{10}\mathstrut +\mathstrut \) \(3470\) \(\beta_{9}\mathstrut +\mathstrut \) \(4940\) \(\beta_{8}\mathstrut -\mathstrut \) \(4038\) \(\beta_{7}\mathstrut -\mathstrut \) \(1498\) \(\beta_{6}\mathstrut +\mathstrut \) \(1272\) \(\beta_{5}\mathstrut -\mathstrut \) \(2610\) \(\beta_{4}\mathstrut +\mathstrut \) \(9788\) \(\beta_{3}\mathstrut +\mathstrut \) \(16847\) \(\beta_{2}\mathstrut +\mathstrut \) \(12560\) \(\beta_{1}\mathstrut +\mathstrut \) \(36456\)
\(\nu^{11}\)\(=\)\(-\)\(1920\) \(\beta_{18}\mathstrut +\mathstrut \) \(1412\) \(\beta_{17}\mathstrut -\mathstrut \) \(30868\) \(\beta_{16}\mathstrut -\mathstrut \) \(14337\) \(\beta_{15}\mathstrut -\mathstrut \) \(24113\) \(\beta_{14}\mathstrut -\mathstrut \) \(3992\) \(\beta_{13}\mathstrut -\mathstrut \) \(5950\) \(\beta_{12}\mathstrut +\mathstrut \) \(5170\) \(\beta_{11}\mathstrut +\mathstrut \) \(12315\) \(\beta_{10}\mathstrut -\mathstrut \) \(28447\) \(\beta_{9}\mathstrut -\mathstrut \) \(6145\) \(\beta_{8}\mathstrut +\mathstrut \) \(28984\) \(\beta_{7}\mathstrut -\mathstrut \) \(21179\) \(\beta_{6}\mathstrut -\mathstrut \) \(2293\) \(\beta_{5}\mathstrut -\mathstrut \) \(2047\) \(\beta_{4}\mathstrut +\mathstrut \) \(23051\) \(\beta_{3}\mathstrut +\mathstrut \) \(21755\) \(\beta_{2}\mathstrut +\mathstrut \) \(66682\) \(\beta_{1}\mathstrut +\mathstrut \) \(36668\)
\(\nu^{12}\)\(=\)\(-\)\(12470\) \(\beta_{18}\mathstrut -\mathstrut \) \(5672\) \(\beta_{17}\mathstrut -\mathstrut \) \(1427\) \(\beta_{16}\mathstrut +\mathstrut \) \(20174\) \(\beta_{15}\mathstrut -\mathstrut \) \(57021\) \(\beta_{14}\mathstrut -\mathstrut \) \(61588\) \(\beta_{13}\mathstrut +\mathstrut \) \(59950\) \(\beta_{12}\mathstrut +\mathstrut \) \(7862\) \(\beta_{11}\mathstrut +\mathstrut \) \(4546\) \(\beta_{10}\mathstrut +\mathstrut \) \(38202\) \(\beta_{9}\mathstrut +\mathstrut \) \(57796\) \(\beta_{8}\mathstrut -\mathstrut \) \(46614\) \(\beta_{7}\mathstrut -\mathstrut \) \(15628\) \(\beta_{6}\mathstrut +\mathstrut \) \(11930\) \(\beta_{5}\mathstrut -\mathstrut \) \(31564\) \(\beta_{4}\mathstrut +\mathstrut \) \(111958\) \(\beta_{3}\mathstrut +\mathstrut \) \(188003\) \(\beta_{2}\mathstrut +\mathstrut \) \(134116\) \(\beta_{1}\mathstrut +\mathstrut \) \(399005\)
\(\nu^{13}\)\(=\)\(-\)\(25505\) \(\beta_{18}\mathstrut +\mathstrut \) \(13564\) \(\beta_{17}\mathstrut -\mathstrut \) \(352686\) \(\beta_{16}\mathstrut -\mathstrut \) \(153886\) \(\beta_{15}\mathstrut -\mathstrut \) \(275852\) \(\beta_{14}\mathstrut -\mathstrut \) \(45889\) \(\beta_{13}\mathstrut -\mathstrut \) \(64801\) \(\beta_{12}\mathstrut +\mathstrut \) \(61073\) \(\beta_{11}\mathstrut +\mathstrut \) \(132740\) \(\beta_{10}\mathstrut -\mathstrut \) \(327797\) \(\beta_{9}\mathstrut -\mathstrut \) \(68239\) \(\beta_{8}\mathstrut +\mathstrut \) \(328790\) \(\beta_{7}\mathstrut -\mathstrut \) \(236702\) \(\beta_{6}\mathstrut -\mathstrut \) \(33859\) \(\beta_{5}\mathstrut -\mathstrut \) \(28878\) \(\beta_{4}\mathstrut +\mathstrut \) \(259007\) \(\beta_{3}\mathstrut +\mathstrut \) \(236089\) \(\beta_{2}\mathstrut +\mathstrut \) \(727192\) \(\beta_{1}\mathstrut +\mathstrut \) \(376952\)
\(\nu^{14}\)\(=\)\(-\)\(150682\) \(\beta_{18}\mathstrut -\mathstrut \) \(72638\) \(\beta_{17}\mathstrut -\mathstrut \) \(12258\) \(\beta_{16}\mathstrut +\mathstrut \) \(259294\) \(\beta_{15}\mathstrut -\mathstrut \) \(648026\) \(\beta_{14}\mathstrut -\mathstrut \) \(700250\) \(\beta_{13}\mathstrut +\mathstrut \) \(696622\) \(\beta_{12}\mathstrut +\mathstrut \) \(96978\) \(\beta_{11}\mathstrut +\mathstrut \) \(29927\) \(\beta_{10}\mathstrut +\mathstrut \) \(426198\) \(\beta_{9}\mathstrut +\mathstrut \) \(667481\) \(\beta_{8}\mathstrut -\mathstrut \) \(537800\) \(\beta_{7}\mathstrut -\mathstrut \) \(161710\) \(\beta_{6}\mathstrut +\mathstrut \) \(111779\) \(\beta_{5}\mathstrut -\mathstrut \) \(374403\) \(\beta_{4}\mathstrut +\mathstrut \) \(1262091\) \(\beta_{3}\mathstrut +\mathstrut \) \(2097637\) \(\beta_{2}\mathstrut +\mathstrut \) \(1443169\) \(\beta_{1}\mathstrut +\mathstrut \) \(4400719\)
\(\nu^{15}\)\(=\)\(-\)\(308282\) \(\beta_{18}\mathstrut +\mathstrut \) \(132598\) \(\beta_{17}\mathstrut -\mathstrut \) \(4000805\) \(\beta_{16}\mathstrut -\mathstrut \) \(1669352\) \(\beta_{15}\mathstrut -\mathstrut \) \(3122385\) \(\beta_{14}\mathstrut -\mathstrut \) \(504470\) \(\beta_{13}\mathstrut -\mathstrut \) \(711853\) \(\beta_{12}\mathstrut +\mathstrut \) \(712758\) \(\beta_{11}\mathstrut +\mathstrut \) \(1445104\) \(\beta_{10}\mathstrut -\mathstrut \) \(3750518\) \(\beta_{9}\mathstrut -\mathstrut \) \(761410\) \(\beta_{8}\mathstrut +\mathstrut \) \(3720259\) \(\beta_{7}\mathstrut -\mathstrut \) \(2641192\) \(\beta_{6}\mathstrut -\mathstrut \) \(456732\) \(\beta_{5}\mathstrut -\mathstrut \) \(372564\) \(\beta_{4}\mathstrut +\mathstrut \) \(2881845\) \(\beta_{3}\mathstrut +\mathstrut \) \(2551366\) \(\beta_{2}\mathstrut +\mathstrut \) \(7995998\) \(\beta_{1}\mathstrut +\mathstrut \) \(3889559\)
\(\nu^{16}\)\(=\)\(-\)\(1743576\) \(\beta_{18}\mathstrut -\mathstrut \) \(891156\) \(\beta_{17}\mathstrut -\mathstrut \) \(53088\) \(\beta_{16}\mathstrut +\mathstrut \) \(3203127\) \(\beta_{15}\mathstrut -\mathstrut \) \(7274769\) \(\beta_{14}\mathstrut -\mathstrut \) \(7890282\) \(\beta_{13}\mathstrut +\mathstrut \) \(8019421\) \(\beta_{12}\mathstrut +\mathstrut \) \(1161585\) \(\beta_{11}\mathstrut +\mathstrut \) \(140965\) \(\beta_{10}\mathstrut +\mathstrut \) \(4814026\) \(\beta_{9}\mathstrut +\mathstrut \) \(7654625\) \(\beta_{8}\mathstrut -\mathstrut \) \(6202090\) \(\beta_{7}\mathstrut -\mathstrut \) \(1657891\) \(\beta_{6}\mathstrut +\mathstrut \) \(1055320\) \(\beta_{5}\mathstrut -\mathstrut \) \(4375233\) \(\beta_{4}\mathstrut +\mathstrut \) \(14127675\) \(\beta_{3}\mathstrut +\mathstrut \) \(23403975\) \(\beta_{2}\mathstrut +\mathstrut \) \(15608893\) \(\beta_{1}\mathstrut +\mathstrut \) \(48752267\)
\(\nu^{17}\)\(=\)\(-\)\(3518378\) \(\beta_{18}\mathstrut +\mathstrut \) \(1327790\) \(\beta_{17}\mathstrut -\mathstrut \) \(45212269\) \(\beta_{16}\mathstrut -\mathstrut \) \(18279525\) \(\beta_{15}\mathstrut -\mathstrut \) \(35092958\) \(\beta_{14}\mathstrut -\mathstrut \) \(5388501\) \(\beta_{13}\mathstrut -\mathstrut \) \(7921385\) \(\beta_{12}\mathstrut +\mathstrut \) \(8256408\) \(\beta_{11}\mathstrut +\mathstrut \) \(15829354\) \(\beta_{10}\mathstrut -\mathstrut \) \(42713866\) \(\beta_{9}\mathstrut -\mathstrut \) \(8561836\) \(\beta_{8}\mathstrut +\mathstrut \) \(42071022\) \(\beta_{7}\mathstrut -\mathstrut \) \(29449614\) \(\beta_{6}\mathstrut -\mathstrut \) \(5838612\) \(\beta_{5}\mathstrut -\mathstrut \) \(4550413\) \(\beta_{4}\mathstrut +\mathstrut \) \(31882688\) \(\beta_{3}\mathstrut +\mathstrut \) \(27500351\) \(\beta_{2}\mathstrut +\mathstrut \) \(88319490\) \(\beta_{1}\mathstrut +\mathstrut \) \(40226199\)
\(\nu^{18}\)\(=\)\(-\)\(19618273\) \(\beta_{18}\mathstrut -\mathstrut \) \(10663514\) \(\beta_{17}\mathstrut +\mathstrut \) \(691708\) \(\beta_{16}\mathstrut +\mathstrut \) \(38617512\) \(\beta_{15}\mathstrut -\mathstrut \) \(80990164\) \(\beta_{14}\mathstrut -\mathstrut \) \(88464006\) \(\beta_{13}\mathstrut +\mathstrut \) \(91778379\) \(\beta_{12}\mathstrut +\mathstrut \) \(13673570\) \(\beta_{11}\mathstrut -\mathstrut \) \(318919\) \(\beta_{10}\mathstrut +\mathstrut \) \(54898170\) \(\beta_{9}\mathstrut +\mathstrut \) \(87414312\) \(\beta_{8}\mathstrut -\mathstrut \) \(71469117\) \(\beta_{7}\mathstrut -\mathstrut \) \(16809780\) \(\beta_{6}\mathstrut +\mathstrut \) \(10083123\) \(\beta_{5}\mathstrut -\mathstrut \) \(50542513\) \(\beta_{4}\mathstrut +\mathstrut \) \(157595999\) \(\beta_{3}\mathstrut +\mathstrut \) \(261149835\) \(\beta_{2}\mathstrut +\mathstrut \) \(169370911\) \(\beta_{1}\mathstrut +\mathstrut \) \(541608246\)

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
3.33655
3.31016
2.89238
2.44352
2.34269
1.90812
1.68584
1.40330
1.02734
−0.237601
−0.435585
−0.815820
−0.942024
−1.62033
−1.76710
−1.97307
−2.06929
−2.11583
−3.37325
0 −3.33655 0 1.00000 0 2.35293 0 8.13258 0
1.2 0 −3.31016 0 1.00000 0 −0.477457 0 7.95713 0
1.3 0 −2.89238 0 1.00000 0 0.451178 0 5.36583 0
1.4 0 −2.44352 0 1.00000 0 −4.90700 0 2.97081 0
1.5 0 −2.34269 0 1.00000 0 −1.53409 0 2.48818 0
1.6 0 −1.90812 0 1.00000 0 1.15158 0 0.640908 0
1.7 0 −1.68584 0 1.00000 0 −4.78926 0 −0.157936 0
1.8 0 −1.40330 0 1.00000 0 3.27194 0 −1.03074 0
1.9 0 −1.02734 0 1.00000 0 3.96391 0 −1.94457 0
1.10 0 0.237601 0 1.00000 0 4.27350 0 −2.94355 0
1.11 0 0.435585 0 1.00000 0 −2.87898 0 −2.81027 0
1.12 0 0.815820 0 1.00000 0 −2.49382 0 −2.33444 0
1.13 0 0.942024 0 1.00000 0 1.85968 0 −2.11259 0
1.14 0 1.62033 0 1.00000 0 1.05807 0 −0.374546 0
1.15 0 1.76710 0 1.00000 0 0.0387940 0 0.122653 0
1.16 0 1.97307 0 1.00000 0 −0.188446 0 0.892989 0
1.17 0 2.06929 0 1.00000 0 −0.835530 0 1.28196 0
1.18 0 2.11583 0 1.00000 0 −4.51128 0 1.47674 0
1.19 0 3.37325 0 1.00000 0 −3.80572 0 8.37885 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(151\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

\(T_{3}^{19} + \cdots\)
\(T_{7}^{19} + \cdots\)
\(T_{11}^{19} + \cdots\)