Properties

Label 4028.2.a.e
Level 4028
Weight 2
Character orbit 4028.a
Self dual Yes
Analytic conductor 32.164
Analytic rank 0
Dimension 19
CM No
Inner twists 1

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Newspace parameters

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

Newform invariants

Self dual: Yes
Analytic conductor: \(32.1637419342\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \( + \beta_{3} q^{5} \) \( -\beta_{18} q^{7} \) \( + ( 1 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( + \beta_{3} q^{5} \) \( -\beta_{18} q^{7} \) \( + ( 1 + \beta_{2} ) q^{9} \) \( + \beta_{12} q^{11} \) \( + ( 1 - \beta_{6} ) q^{13} \) \( + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{12} - \beta_{15} + \beta_{18} ) q^{15} \) \( -\beta_{15} q^{17} \) \(+ q^{19}\) \( + ( \beta_{1} - \beta_{9} ) q^{21} \) \( + ( 2 + \beta_{6} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{18} ) q^{23} \) \( + ( 1 + \beta_{17} - \beta_{18} ) q^{25} \) \( + ( \beta_{1} - \beta_{10} + \beta_{11} ) q^{27} \) \( + ( \beta_{4} + \beta_{6} + \beta_{10} - \beta_{14} + \beta_{15} - \beta_{18} ) q^{29} \) \( + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{17} ) q^{31} \) \( + ( -2 + 2 \beta_{1} + \beta_{9} - \beta_{10} + \beta_{12} + \beta_{14} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{33} \) \( + ( \beta_{3} + \beta_{6} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{17} ) q^{35} \) \( + ( 1 - \beta_{4} + \beta_{12} + \beta_{14} ) q^{37} \) \( + ( -1 + 2 \beta_{1} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{17} + \beta_{18} ) q^{39} \) \( + ( -\beta_{2} + \beta_{6} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{18} ) q^{41} \) \( + ( 2 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{14} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{43} \) \( + ( 2 \beta_{2} + 2 \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - 2 \beta_{12} - 2 \beta_{15} - \beta_{17} + 2 \beta_{18} ) q^{45} \) \( + ( \beta_{3} - \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{47} \) \( + ( 1 - \beta_{2} + \beta_{10} - \beta_{12} + \beta_{15} + \beta_{16} - \beta_{18} ) q^{49} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{16} ) q^{51} \) \(+ q^{53}\) \( + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{5} + \beta_{8} - \beta_{10} + \beta_{12} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{55} \) \( + \beta_{1} q^{57} \) \( + ( \beta_{3} + \beta_{8} + \beta_{9} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{59} \) \( + ( \beta_{1} - \beta_{3} - \beta_{7} + \beta_{9} + \beta_{14} ) q^{61} \) \( + ( 3 - \beta_{5} + \beta_{11} - \beta_{12} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{63} \) \( + ( -2 + \beta_{1} + \beta_{3} + \beta_{6} + \beta_{8} + \beta_{12} - \beta_{13} + \beta_{15} - \beta_{17} ) q^{65} \) \( + ( 3 - \beta_{3} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{18} ) q^{67} \) \( + ( 1 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{13} - \beta_{14} + 2 \beta_{17} - 3 \beta_{18} ) q^{69} \) \( + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{10} + 2 \beta_{12} - \beta_{13} - \beta_{18} ) q^{71} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{8} - \beta_{10} + \beta_{12} - \beta_{16} ) q^{73} \) \( + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{6} + \beta_{8} - \beta_{11} + \beta_{13} - \beta_{17} ) q^{75} \) \( + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{12} - \beta_{18} ) q^{77} \) \( + ( 2 - 2 \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{15} + \beta_{17} - \beta_{18} ) q^{79} \) \( + ( 1 + \beta_{1} + \beta_{5} + \beta_{7} - \beta_{9} + \beta_{13} ) q^{81} \) \( + ( 2 - \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{12} - \beta_{14} - \beta_{16} + 2 \beta_{17} + \beta_{18} ) q^{83} \) \( + ( \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{9} - \beta_{14} + \beta_{15} ) q^{85} \) \( + ( 2 + \beta_{1} - \beta_{2} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{87} \) \( + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} + \beta_{18} ) q^{89} \) \( + ( 3 + \beta_{5} + \beta_{8} + \beta_{10} - \beta_{12} + \beta_{14} - \beta_{15} - 2 \beta_{18} ) q^{91} \) \( + ( -1 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} - \beta_{17} ) q^{93} \) \( + \beta_{3} q^{95} \) \( + ( 2 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{12} + \beta_{13} - 2 \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{97} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{6} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{14} - 2 \beta_{15} - \beta_{16} + \beta_{17} + 3 \beta_{18} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(19q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 22q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(19q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 22q^{9} \) \(\mathstrut +\mathstrut 5q^{11} \) \(\mathstrut +\mathstrut 25q^{13} \) \(\mathstrut +\mathstrut 20q^{15} \) \(\mathstrut -\mathstrut 7q^{17} \) \(\mathstrut +\mathstrut 19q^{19} \) \(\mathstrut +\mathstrut 2q^{21} \) \(\mathstrut +\mathstrut 18q^{23} \) \(\mathstrut +\mathstrut 22q^{25} \) \(\mathstrut +\mathstrut 15q^{27} \) \(\mathstrut +\mathstrut 4q^{29} \) \(\mathstrut +\mathstrut 12q^{31} \) \(\mathstrut -\mathstrut 8q^{33} \) \(\mathstrut +\mathstrut 11q^{35} \) \(\mathstrut +\mathstrut 19q^{37} \) \(\mathstrut +\mathstrut 9q^{39} \) \(\mathstrut -\mathstrut 9q^{41} \) \(\mathstrut +\mathstrut 31q^{43} \) \(\mathstrut -\mathstrut 2q^{45} \) \(\mathstrut -\mathstrut 2q^{47} \) \(\mathstrut +\mathstrut 7q^{49} \) \(\mathstrut +\mathstrut 5q^{51} \) \(\mathstrut +\mathstrut 19q^{53} \) \(\mathstrut +\mathstrut 11q^{55} \) \(\mathstrut +\mathstrut 3q^{57} \) \(\mathstrut +\mathstrut 2q^{59} \) \(\mathstrut +\mathstrut 6q^{61} \) \(\mathstrut +\mathstrut 52q^{63} \) \(\mathstrut -\mathstrut 6q^{65} \) \(\mathstrut +\mathstrut 50q^{67} \) \(\mathstrut -\mathstrut 7q^{69} \) \(\mathstrut +\mathstrut 25q^{71} \) \(\mathstrut -\mathstrut 5q^{73} \) \(\mathstrut +\mathstrut 22q^{75} \) \(\mathstrut -\mathstrut 14q^{77} \) \(\mathstrut +\mathstrut 36q^{79} \) \(\mathstrut +\mathstrut 11q^{81} \) \(\mathstrut +\mathstrut 20q^{83} \) \(\mathstrut +\mathstrut 5q^{85} \) \(\mathstrut +\mathstrut 18q^{87} \) \(\mathstrut +\mathstrut 9q^{89} \) \(\mathstrut +\mathstrut 61q^{91} \) \(\mathstrut +\mathstrut q^{93} \) \(\mathstrut +\mathstrut 3q^{95} \) \(\mathstrut +\mathstrut 7q^{97} \) \(\mathstrut +\mathstrut 48q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19}\mathstrut -\mathstrut \) \(3\) \(x^{18}\mathstrut -\mathstrut \) \(35\) \(x^{17}\mathstrut +\mathstrut \) \(103\) \(x^{16}\mathstrut +\mathstrut \) \(501\) \(x^{15}\mathstrut -\mathstrut \) \(1437\) \(x^{14}\mathstrut -\mathstrut \) \(3775\) \(x^{13}\mathstrut +\mathstrut \) \(10450\) \(x^{12}\mathstrut +\mathstrut \) \(16076\) \(x^{11}\mathstrut -\mathstrut \) \(42255\) \(x^{10}\mathstrut -\mathstrut \) \(38701\) \(x^{9}\mathstrut +\mathstrut \) \(93907\) \(x^{8}\mathstrut +\mathstrut \) \(49522\) \(x^{7}\mathstrut -\mathstrut \) \(106284\) \(x^{6}\mathstrut -\mathstrut \) \(26703\) \(x^{5}\mathstrut +\mathstrut \) \(50522\) \(x^{4}\mathstrut -\mathstrut \) \(446\) \(x^{3}\mathstrut -\mathstrut \) \(6265\) \(x^{2}\mathstrut +\mathstrut \) \(719\) \(x\mathstrut -\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(21100945968499127\) \(\nu^{18}\mathstrut +\mathstrut \) \(43592264861399349\) \(\nu^{17}\mathstrut +\mathstrut \) \(731231114527023890\) \(\nu^{16}\mathstrut -\mathstrut \) \(1332358868879285565\) \(\nu^{15}\mathstrut -\mathstrut \) \(10306695488928063533\) \(\nu^{14}\mathstrut +\mathstrut \) \(15701162516973550226\) \(\nu^{13}\mathstrut +\mathstrut \) \(75764391261288070328\) \(\nu^{12}\mathstrut -\mathstrut \) \(87335047704835019838\) \(\nu^{11}\mathstrut -\mathstrut \) \(310498381159206139915\) \(\nu^{10}\mathstrut +\mathstrut \) \(213095283104129536061\) \(\nu^{9}\mathstrut +\mathstrut \) \(712331831191207224631\) \(\nu^{8}\mathstrut -\mathstrut \) \(78592192324395588250\) \(\nu^{7}\mathstrut -\mathstrut \) \(920228840015359809035\) \(\nu^{6}\mathstrut -\mathstrut \) \(421785253951289381280\) \(\nu^{5}\mathstrut +\mathstrut \) \(774922243897831498622\) \(\nu^{4}\mathstrut +\mathstrut \) \(389266259778245706769\) \(\nu^{3}\mathstrut -\mathstrut \) \(500153876929998375905\) \(\nu^{2}\mathstrut -\mathstrut \) \(37805424441014071291\) \(\nu\mathstrut +\mathstrut \) \(50053918559643374453\)\()/\)\(17909267846343676161\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(136908262179951951\) \(\nu^{18}\mathstrut -\mathstrut \) \(53492348310275221\) \(\nu^{17}\mathstrut +\mathstrut \) \(5818529815243362336\) \(\nu^{16}\mathstrut +\mathstrut \) \(1934335817930349716\) \(\nu^{15}\mathstrut -\mathstrut \) \(101099181631066455927\) \(\nu^{14}\mathstrut -\mathstrut \) \(27683630729732305251\) \(\nu^{13}\mathstrut +\mathstrut \) \(923598210947245605091\) \(\nu^{12}\mathstrut +\mathstrut \) \(191396659797679950034\) \(\nu^{11}\mathstrut -\mathstrut \) \(4739508222026592018835\) \(\nu^{10}\mathstrut -\mathstrut \) \(578802595597291051935\) \(\nu^{9}\mathstrut +\mathstrut \) \(13477529062375417591031\) \(\nu^{8}\mathstrut -\mathstrut \) \(6667990765502063008\) \(\nu^{7}\mathstrut -\mathstrut \) \(19507513440017244676054\) \(\nu^{6}\mathstrut +\mathstrut \) \(3972430629289733498703\) \(\nu^{5}\mathstrut +\mathstrut \) \(12056748511931200977565\) \(\nu^{4}\mathstrut -\mathstrut \) \(6900433131526733913542\) \(\nu^{3}\mathstrut -\mathstrut \) \(2674568218517795494802\) \(\nu^{2}\mathstrut +\mathstrut \) \(1847407378417465238077\) \(\nu\mathstrut +\mathstrut \) \(59868927860611210344\)\()/\)\(89546339231718380805\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(246661825946525039\) \(\nu^{18}\mathstrut +\mathstrut \) \(747482118132434871\) \(\nu^{17}\mathstrut +\mathstrut \) \(8863297965208146954\) \(\nu^{16}\mathstrut -\mathstrut \) \(25687506993920713671\) \(\nu^{15}\mathstrut -\mathstrut \) \(132581292907709915218\) \(\nu^{14}\mathstrut +\mathstrut \) \(356910054429108976046\) \(\nu^{13}\mathstrut +\mathstrut \) \(1077343350655289335334\) \(\nu^{12}\mathstrut -\mathstrut \) \(2559852021800000764684\) \(\nu^{11}\mathstrut -\mathstrut \) \(5226317071881411034190\) \(\nu^{10}\mathstrut +\mathstrut \) \(10006872130670858050620\) \(\nu^{9}\mathstrut +\mathstrut \) \(15667393649386334523559\) \(\nu^{8}\mathstrut -\mathstrut \) \(20546184059828063675412\) \(\nu^{7}\mathstrut -\mathstrut \) \(28414629129491753599741\) \(\nu^{6}\mathstrut +\mathstrut \) \(19036438182072509750812\) \(\nu^{5}\mathstrut +\mathstrut \) \(26965112106894850487975\) \(\nu^{4}\mathstrut -\mathstrut \) \(4720975944200840804073\) \(\nu^{3}\mathstrut -\mathstrut \) \(8299838499205240708533\) \(\nu^{2}\mathstrut +\mathstrut \) \(442042278782650978138\) \(\nu\mathstrut +\mathstrut \) \(470528835784796148676\)\()/\)\(89546339231718380805\)
\(\beta_{6}\)\(=\)\((\)\(8588788932901922\) \(\nu^{18}\mathstrut -\mathstrut \) \(24598790822708248\) \(\nu^{17}\mathstrut -\mathstrut \) \(303188994633903047\) \(\nu^{16}\mathstrut +\mathstrut \) \(847046746016347583\) \(\nu^{15}\mathstrut +\mathstrut \) \(4381895191306129934\) \(\nu^{14}\mathstrut -\mathstrut \) \(11893572304258799408\) \(\nu^{13}\mathstrut -\mathstrut \) \(33324966021946750867\) \(\nu^{12}\mathstrut +\mathstrut \) \(87597216503076317482\) \(\nu^{11}\mathstrut +\mathstrut \) \(142580551464474467085\) \(\nu^{10}\mathstrut -\mathstrut \) \(363063184919444913230\) \(\nu^{9}\mathstrut -\mathstrut \) \(339085461074023098317\) \(\nu^{8}\mathstrut +\mathstrut \) \(847209467500464756561\) \(\nu^{7}\mathstrut +\mathstrut \) \(406328890927976691268\) \(\nu^{6}\mathstrut -\mathstrut \) \(1058563054017739692936\) \(\nu^{5}\mathstrut -\mathstrut \) \(165604750305068776765\) \(\nu^{4}\mathstrut +\mathstrut \) \(618012436023074859429\) \(\nu^{3}\mathstrut -\mathstrut \) \(41925793310811650286\) \(\nu^{2}\mathstrut -\mathstrut \) \(111693658459827030349\) \(\nu\mathstrut +\mathstrut \) \(8522304274770198037\)\()/\)\(2420171330586983265\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(338797414629417969\) \(\nu^{18}\mathstrut +\mathstrut \) \(1341260311365471236\) \(\nu^{17}\mathstrut +\mathstrut \) \(10985639744011807319\) \(\nu^{16}\mathstrut -\mathstrut \) \(46433893433427580681\) \(\nu^{15}\mathstrut -\mathstrut \) \(139955971988660452678\) \(\nu^{14}\mathstrut +\mathstrut \) \(654916323810790627881\) \(\nu^{13}\mathstrut +\mathstrut \) \(867352781157848356579\) \(\nu^{12}\mathstrut -\mathstrut \) \(4835318122752433520779\) \(\nu^{11}\mathstrut -\mathstrut \) \(2499861396570081957465\) \(\nu^{10}\mathstrut +\mathstrut \) \(20001223717907642511560\) \(\nu^{9}\mathstrut +\mathstrut \) \(1513526672381725270814\) \(\nu^{8}\mathstrut -\mathstrut \) \(46122936216411564172562\) \(\nu^{7}\mathstrut +\mathstrut \) \(7743698781981419813329\) \(\nu^{6}\mathstrut +\mathstrut \) \(55585183354213545284387\) \(\nu^{5}\mathstrut -\mathstrut \) \(15851763276516705836150\) \(\nu^{4}\mathstrut -\mathstrut \) \(29115456858802182462598\) \(\nu^{3}\mathstrut +\mathstrut \) \(8540003351396893639907\) \(\nu^{2}\mathstrut +\mathstrut \) \(3302934883729144679873\) \(\nu\mathstrut -\mathstrut \) \(115646767234654745624\)\()/\)\(89546339231718380805\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(398134173783196662\) \(\nu^{18}\mathstrut +\mathstrut \) \(984801587358537278\) \(\nu^{17}\mathstrut +\mathstrut \) \(14408372617516244937\) \(\nu^{16}\mathstrut -\mathstrut \) \(33669928343851728583\) \(\nu^{15}\mathstrut -\mathstrut \) \(215090490793182655419\) \(\nu^{14}\mathstrut +\mathstrut \) \(468266667173622203328\) \(\nu^{13}\mathstrut +\mathstrut \) \(1708344212020960053037\) \(\nu^{12}\mathstrut -\mathstrut \) \(3399549999677453720987\) \(\nu^{11}\mathstrut -\mathstrut \) \(7766477653414113674140\) \(\nu^{10}\mathstrut +\mathstrut \) \(13739388104667722323290\) \(\nu^{9}\mathstrut +\mathstrut \) \(20223446896964222545877\) \(\nu^{8}\mathstrut -\mathstrut \) \(30441794153420757999886\) \(\nu^{7}\mathstrut -\mathstrut \) \(28293696025386100897963\) \(\nu^{6}\mathstrut +\mathstrut \) \(33644953760222426936856\) \(\nu^{5}\mathstrut +\mathstrut \) \(17258018194941992096965\) \(\nu^{4}\mathstrut -\mathstrut \) \(13974343239846319564469\) \(\nu^{3}\mathstrut -\mathstrut \) \(1200103315978137133649\) \(\nu^{2}\mathstrut +\mathstrut \) \(403350709421123440879\) \(\nu\mathstrut -\mathstrut \) \(203848413030310005582\)\()/\)\(89546339231718380805\)
\(\beta_{9}\)\(=\)\((\)\(501664824208012753\) \(\nu^{18}\mathstrut -\mathstrut \) \(1169630015521112147\) \(\nu^{17}\mathstrut -\mathstrut \) \(18151377362431438813\) \(\nu^{16}\mathstrut +\mathstrut \) \(39454549507419281917\) \(\nu^{15}\mathstrut +\mathstrut \) \(270609158879168522341\) \(\nu^{14}\mathstrut -\mathstrut \) \(537949826052023811577\) \(\nu^{13}\mathstrut -\mathstrut \) \(2144352993792124886003\) \(\nu^{12}\mathstrut +\mathstrut \) \(3791849464230420946433\) \(\nu^{11}\mathstrut +\mathstrut \) \(9724455581595308114535\) \(\nu^{10}\mathstrut -\mathstrut \) \(14655366041370904607680\) \(\nu^{9}\mathstrut -\mathstrut \) \(25321280438043727089343\) \(\nu^{8}\mathstrut +\mathstrut \) \(30331096702516302045339\) \(\nu^{7}\mathstrut +\mathstrut \) \(35801188177838838802712\) \(\nu^{6}\mathstrut -\mathstrut \) \(30323047383843330196194\) \(\nu^{5}\mathstrut -\mathstrut \) \(23062526339424253887365\) \(\nu^{4}\mathstrut +\mathstrut \) \(11226089102017029373341\) \(\nu^{3}\mathstrut +\mathstrut \) \(3331822192664065395081\) \(\nu^{2}\mathstrut -\mathstrut \) \(806473089461851587716\) \(\nu\mathstrut +\mathstrut \) \(7279743352830115568\)\()/\)\(89546339231718380805\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(581536435015724399\) \(\nu^{18}\mathstrut +\mathstrut \) \(1733264810074488531\) \(\nu^{17}\mathstrut +\mathstrut \) \(20711067403505933469\) \(\nu^{16}\mathstrut -\mathstrut \) \(59714168387891579916\) \(\nu^{15}\mathstrut -\mathstrut \) \(304598937001726506163\) \(\nu^{14}\mathstrut +\mathstrut \) \(835325981323216141781\) \(\nu^{13}\mathstrut +\mathstrut \) \(2395706337003663537134\) \(\nu^{12}\mathstrut -\mathstrut \) \(6075324531451742474734\) \(\nu^{11}\mathstrut -\mathstrut \) \(10931541022809763218215\) \(\nu^{10}\mathstrut +\mathstrut \) \(24400329377254798141410\) \(\nu^{9}\mathstrut +\mathstrut \) \(29426389312706891914039\) \(\nu^{8}\mathstrut -\mathstrut \) \(52885655380179115484637\) \(\nu^{7}\mathstrut -\mathstrut \) \(45072768894167527458946\) \(\nu^{6}\mathstrut +\mathstrut \) \(55465765959505412571052\) \(\nu^{5}\mathstrut +\mathstrut \) \(33633014694997936394870\) \(\nu^{4}\mathstrut -\mathstrut \) \(20869754937754579492938\) \(\nu^{3}\mathstrut -\mathstrut \) \(6400722858329954893278\) \(\nu^{2}\mathstrut +\mathstrut \) \(1965624848092621841353\) \(\nu\mathstrut +\mathstrut \) \(240670229350970720491\)\()/\)\(89546339231718380805\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(581536435015724399\) \(\nu^{18}\mathstrut +\mathstrut \) \(1733264810074488531\) \(\nu^{17}\mathstrut +\mathstrut \) \(20711067403505933469\) \(\nu^{16}\mathstrut -\mathstrut \) \(59714168387891579916\) \(\nu^{15}\mathstrut -\mathstrut \) \(304598937001726506163\) \(\nu^{14}\mathstrut +\mathstrut \) \(835325981323216141781\) \(\nu^{13}\mathstrut +\mathstrut \) \(2395706337003663537134\) \(\nu^{12}\mathstrut -\mathstrut \) \(6075324531451742474734\) \(\nu^{11}\mathstrut -\mathstrut \) \(10931541022809763218215\) \(\nu^{10}\mathstrut +\mathstrut \) \(24400329377254798141410\) \(\nu^{9}\mathstrut +\mathstrut \) \(29426389312706891914039\) \(\nu^{8}\mathstrut -\mathstrut \) \(52885655380179115484637\) \(\nu^{7}\mathstrut -\mathstrut \) \(45072768894167527458946\) \(\nu^{6}\mathstrut +\mathstrut \) \(55465765959505412571052\) \(\nu^{5}\mathstrut +\mathstrut \) \(33633014694997936394870\) \(\nu^{4}\mathstrut -\mathstrut \) \(20780208598522861112133\) \(\nu^{3}\mathstrut -\mathstrut \) \(6400722858329954893278\) \(\nu^{2}\mathstrut +\mathstrut \) \(1338800473470593175718\) \(\nu\mathstrut +\mathstrut \) \(240670229350970720491\)\()/\)\(89546339231718380805\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(138546269126020126\) \(\nu^{18}\mathstrut +\mathstrut \) \(416756918291755202\) \(\nu^{17}\mathstrut +\mathstrut \) \(4838948341285928874\) \(\nu^{16}\mathstrut -\mathstrut \) \(14248850530452564640\) \(\nu^{15}\mathstrut -\mathstrut \) \(69102876481004751170\) \(\nu^{14}\mathstrut +\mathstrut \) \(197620369717702837408\) \(\nu^{13}\mathstrut +\mathstrut \) \(519372680911788769703\) \(\nu^{12}\mathstrut -\mathstrut \) \(1425031869992511913312\) \(\nu^{11}\mathstrut -\mathstrut \) \(2207557821760331266553\) \(\nu^{10}\mathstrut +\mathstrut \) \(5693919640062491936901\) \(\nu^{9}\mathstrut +\mathstrut \) \(5327239318269943774621\) \(\nu^{8}\mathstrut -\mathstrut \) \(12458245417387377814606\) \(\nu^{7}\mathstrut -\mathstrut \) \(6973347773023659077559\) \(\nu^{6}\mathstrut +\mathstrut \) \(13896768224972629243244\) \(\nu^{5}\mathstrut +\mathstrut \) \(4261376958461069513489\) \(\nu^{4}\mathstrut -\mathstrut \) \(6692547192241831737536\) \(\nu^{3}\mathstrut -\mathstrut \) \(615672453502425739109\) \(\nu^{2}\mathstrut +\mathstrut \) \(993960595031413313406\) \(\nu\mathstrut -\mathstrut \) \(19332067752171448891\)\()/\)\(17909267846343676161\)
\(\beta_{13}\)\(=\)\((\)\(29381731480647453\) \(\nu^{18}\mathstrut -\mathstrut \) \(88064120135649142\) \(\nu^{17}\mathstrut -\mathstrut \) \(1027035542477064678\) \(\nu^{16}\mathstrut +\mathstrut \) \(3015566214453177737\) \(\nu^{15}\mathstrut +\mathstrut \) \(14679633075014564601\) \(\nu^{14}\mathstrut -\mathstrut \) \(41885843359241173392\) \(\nu^{13}\mathstrut -\mathstrut \) \(110514841232574664268\) \(\nu^{12}\mathstrut +\mathstrut \) \(302351881318455546808\) \(\nu^{11}\mathstrut +\mathstrut \) \(471638758109373002870\) \(\nu^{10}\mathstrut -\mathstrut \) \(1207120591620254193780\) \(\nu^{9}\mathstrut -\mathstrut \) \(1148708128643561807668\) \(\nu^{8}\mathstrut +\mathstrut \) \(2621627485912322429549\) \(\nu^{7}\mathstrut +\mathstrut \) \(1526273473658085745652\) \(\nu^{6}\mathstrut -\mathstrut \) \(2836342403246740141389\) \(\nu^{5}\mathstrut -\mathstrut \) \(921252130555964328605\) \(\nu^{4}\mathstrut +\mathstrut \) \(1217905997432974395676\) \(\nu^{3}\mathstrut +\mathstrut \) \(61776764523971541526\) \(\nu^{2}\mathstrut -\mathstrut \) \(125432340302847719636\) \(\nu\mathstrut +\mathstrut \) \(9966713206930696188\)\()/\)\(2420171330586983265\)
\(\beta_{14}\)\(=\)\((\)\(1096208497783710026\) \(\nu^{18}\mathstrut -\mathstrut \) \(3614869478562196699\) \(\nu^{17}\mathstrut -\mathstrut \) \(37670387054244046116\) \(\nu^{16}\mathstrut +\mathstrut \) \(124698625386912973469\) \(\nu^{15}\mathstrut +\mathstrut \) \(526437510692263603327\) \(\nu^{14}\mathstrut -\mathstrut \) \(1751475188544804105494\) \(\nu^{13}\mathstrut -\mathstrut \) \(3839507559212732113831\) \(\nu^{12}\mathstrut +\mathstrut \) \(12861131680448799365666\) \(\nu^{11}\mathstrut +\mathstrut \) \(15617363378635828841845\) \(\nu^{10}\mathstrut -\mathstrut \) \(52761532933220689374195\) \(\nu^{9}\mathstrut -\mathstrut \) \(35175593801667285923546\) \(\nu^{8}\mathstrut +\mathstrut \) \(119935047108322924176623\) \(\nu^{7}\mathstrut +\mathstrut \) \(40728455079016355766174\) \(\nu^{6}\mathstrut -\mathstrut \) \(140904736026859429717258\) \(\nu^{5}\mathstrut -\mathstrut \) \(17827255666631800839070\) \(\nu^{4}\mathstrut +\mathstrut \) \(71551209854246334401422\) \(\nu^{3}\mathstrut -\mathstrut \) \(3557883376045746009563\) \(\nu^{2}\mathstrut -\mathstrut \) \(9769543979097126961947\) \(\nu\mathstrut +\mathstrut \) \(961486293397867783946\)\()/\)\(89546339231718380805\)
\(\beta_{15}\)\(=\)\((\)\(1465562625942357974\) \(\nu^{18}\mathstrut -\mathstrut \) \(3842050644086289351\) \(\nu^{17}\mathstrut -\mathstrut \) \(52725702990042041519\) \(\nu^{16}\mathstrut +\mathstrut \) \(131032285862337631761\) \(\nu^{15}\mathstrut +\mathstrut \) \(782561225002809539978\) \(\nu^{14}\mathstrut -\mathstrut \) \(1810571355011973384116\) \(\nu^{13}\mathstrut -\mathstrut \) \(6190780166295440083229\) \(\nu^{12}\mathstrut +\mathstrut \) \(12973988000394429291819\) \(\nu^{11}\mathstrut +\mathstrut \) \(28185894051570646131400\) \(\nu^{10}\mathstrut -\mathstrut \) \(51196944557976578474390\) \(\nu^{9}\mathstrut -\mathstrut \) \(74473300611049090638484\) \(\nu^{8}\mathstrut +\mathstrut \) \(108835500850120004008447\) \(\nu^{7}\mathstrut +\mathstrut \) \(108833332171744581114941\) \(\nu^{6}\mathstrut -\mathstrut \) \(112518073099102594456407\) \(\nu^{5}\mathstrut -\mathstrut \) \(74332235800712066909225\) \(\nu^{4}\mathstrut +\mathstrut \) \(43237837765373125969433\) \(\nu^{3}\mathstrut +\mathstrut \) \(11137948520344239072023\) \(\nu^{2}\mathstrut -\mathstrut \) \(3919529859912068632418\) \(\nu\mathstrut +\mathstrut \) \(232467753078314024479\)\()/\)\(89546339231718380805\)
\(\beta_{16}\)\(=\)\((\)\(1539998023311586228\) \(\nu^{18}\mathstrut -\mathstrut \) \(4281635936569828972\) \(\nu^{17}\mathstrut -\mathstrut \) \(54867623361036009418\) \(\nu^{16}\mathstrut +\mathstrut \) \(146566736057406318962\) \(\nu^{15}\mathstrut +\mathstrut \) \(804513534220761118126\) \(\nu^{14}\mathstrut -\mathstrut \) \(2036143956540307370617\) \(\nu^{13}\mathstrut -\mathstrut \) \(6267527100976675086393\) \(\nu^{12}\mathstrut +\mathstrut \) \(14713543388037137529733\) \(\nu^{11}\mathstrut +\mathstrut \) \(27982179057637497024310\) \(\nu^{10}\mathstrut -\mathstrut \) \(58898071313913429014200\) \(\nu^{9}\mathstrut -\mathstrut \) \(72089072996606911551798\) \(\nu^{8}\mathstrut +\mathstrut \) \(128595800088562798658179\) \(\nu^{7}\mathstrut +\mathstrut \) \(101807075726982577692792\) \(\nu^{6}\mathstrut -\mathstrut \) \(140435038155015301988979\) \(\nu^{5}\mathstrut -\mathstrut \) \(65495608990244865764295\) \(\nu^{4}\mathstrut +\mathstrut \) \(61085453612229519133616\) \(\nu^{3}\mathstrut +\mathstrut \) \(7129462529688796405166\) \(\nu^{2}\mathstrut -\mathstrut \) \(5846174546367191438361\) \(\nu\mathstrut +\mathstrut \) \(288001577397998411783\)\()/\)\(89546339231718380805\)
\(\beta_{17}\)\(=\)\((\)\(1761025671704815092\) \(\nu^{18}\mathstrut -\mathstrut \) \(5059631075919888638\) \(\nu^{17}\mathstrut -\mathstrut \) \(62444927053137929667\) \(\nu^{16}\mathstrut +\mathstrut \) \(173814346789842070423\) \(\nu^{15}\mathstrut +\mathstrut \) \(910213189478705432934\) \(\nu^{14}\mathstrut -\mathstrut \) \(2426946406221586966923\) \(\nu^{13}\mathstrut -\mathstrut \) \(7037267211630338997877\) \(\nu^{12}\mathstrut +\mathstrut \) \(17665019450059976330822\) \(\nu^{11}\mathstrut +\mathstrut \) \(31104294625143844667890\) \(\nu^{10}\mathstrut -\mathstrut \) \(71453292883603523032980\) \(\nu^{9}\mathstrut -\mathstrut \) \(79047792267450725547377\) \(\nu^{8}\mathstrut +\mathstrut \) \(158409503502853794536026\) \(\nu^{7}\mathstrut +\mathstrut \) \(109528974055237865924908\) \(\nu^{6}\mathstrut -\mathstrut \) \(177212422210484656678476\) \(\nu^{5}\mathstrut -\mathstrut \) \(68176534072717537330015\) \(\nu^{4}\mathstrut +\mathstrut \) \(81370456846738939817189\) \(\nu^{3}\mathstrut +\mathstrut \) \(5751393501139334545514\) \(\nu^{2}\mathstrut -\mathstrut \) \(10018061049117193845829\) \(\nu\mathstrut +\mathstrut \) \(579626577137598468522\)\()/\)\(89546339231718380805\)
\(\beta_{18}\)\(=\)\((\)\(1819935838207528892\) \(\nu^{18}\mathstrut -\mathstrut \) \(4958142690414573923\) \(\nu^{17}\mathstrut -\mathstrut \) \(64867384352784623367\) \(\nu^{16}\mathstrut +\mathstrut \) \(169302013972944037063\) \(\nu^{15}\mathstrut +\mathstrut \) \(951242404449391256809\) \(\nu^{14}\mathstrut -\mathstrut \) \(2344638640625050495463\) \(\nu^{13}\mathstrut -\mathstrut \) \(7408207615285445378877\) \(\nu^{12}\mathstrut +\mathstrut \) \(16873976515476552035397\) \(\nu^{11}\mathstrut +\mathstrut \) \(33049137999254655414225\) \(\nu^{10}\mathstrut -\mathstrut \) \(67176933261863825216925\) \(\nu^{9}\mathstrut -\mathstrut \) \(85088702915840480256972\) \(\nu^{8}\mathstrut +\mathstrut \) \(145583434320510688571701\) \(\nu^{7}\mathstrut +\mathstrut \) \(120457959282229547834963\) \(\nu^{6}\mathstrut -\mathstrut \) \(157628872450210161954616\) \(\nu^{5}\mathstrut -\mathstrut \) \(78920794071498974199270\) \(\nu^{4}\mathstrut +\mathstrut \) \(68884272078496520794259\) \(\nu^{3}\mathstrut +\mathstrut \) \(10414397718176471487509\) \(\nu^{2}\mathstrut -\mathstrut \) \(8070075833706103113299\) \(\nu\mathstrut +\mathstrut \) \(412514438977643304827\)\()/\)\(89546339231718380805\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(28\)
\(\nu^{5}\)\(=\)\(-\)\(3\) \(\beta_{18}\mathstrut +\mathstrut \) \(2\) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(2\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(10\) \(\beta_{11}\mathstrut -\mathstrut \) \(11\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(53\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{6}\)\(=\)\(-\)\(3\) \(\beta_{15}\mathstrut -\mathstrut \) \(2\) \(\beta_{14}\mathstrut +\mathstrut \) \(15\) \(\beta_{13}\mathstrut -\mathstrut \) \(2\) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{10}\mathstrut -\mathstrut \) \(12\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(15\) \(\beta_{5}\mathstrut +\mathstrut \) \(79\) \(\beta_{2}\mathstrut +\mathstrut \) \(18\) \(\beta_{1}\mathstrut +\mathstrut \) \(223\)
\(\nu^{7}\)\(=\)\(-\)\(48\) \(\beta_{18}\mathstrut +\mathstrut \) \(4\) \(\beta_{17}\mathstrut +\mathstrut \) \(29\) \(\beta_{16}\mathstrut -\mathstrut \) \(18\) \(\beta_{15}\mathstrut -\mathstrut \) \(5\) \(\beta_{14}\mathstrut +\mathstrut \) \(36\) \(\beta_{13}\mathstrut -\mathstrut \) \(22\) \(\beta_{12}\mathstrut +\mathstrut \) \(86\) \(\beta_{11}\mathstrut -\mathstrut \) \(100\) \(\beta_{10}\mathstrut +\mathstrut \) \(30\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(35\) \(\beta_{7}\mathstrut +\mathstrut \) \(19\) \(\beta_{6}\mathstrut +\mathstrut \) \(18\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(41\) \(\beta_{2}\mathstrut +\mathstrut \) \(419\) \(\beta_{1}\mathstrut +\mathstrut \) \(99\)
\(\nu^{8}\)\(=\)\(-\)\(5\) \(\beta_{18}\mathstrut +\mathstrut \) \(\beta_{17}\mathstrut +\mathstrut \) \(2\) \(\beta_{16}\mathstrut -\mathstrut \) \(57\) \(\beta_{15}\mathstrut -\mathstrut \) \(43\) \(\beta_{14}\mathstrut +\mathstrut \) \(179\) \(\beta_{13}\mathstrut -\mathstrut \) \(48\) \(\beta_{12}\mathstrut -\mathstrut \) \(18\) \(\beta_{11}\mathstrut -\mathstrut \) \(49\) \(\beta_{10}\mathstrut -\mathstrut \) \(115\) \(\beta_{9}\mathstrut -\mathstrut \) \(25\) \(\beta_{8}\mathstrut +\mathstrut \) \(160\) \(\beta_{7}\mathstrut +\mathstrut \) \(46\) \(\beta_{6}\mathstrut +\mathstrut \) \(173\) \(\beta_{5}\mathstrut +\mathstrut \) \(6\) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(706\) \(\beta_{2}\mathstrut +\mathstrut \) \(233\) \(\beta_{1}\mathstrut +\mathstrut \) \(1891\)
\(\nu^{9}\)\(=\)\(-\)\(559\) \(\beta_{18}\mathstrut +\mathstrut \) \(84\) \(\beta_{17}\mathstrut +\mathstrut \) \(315\) \(\beta_{16}\mathstrut -\mathstrut \) \(245\) \(\beta_{15}\mathstrut -\mathstrut \) \(119\) \(\beta_{14}\mathstrut +\mathstrut \) \(473\) \(\beta_{13}\mathstrut -\mathstrut \) \(334\) \(\beta_{12}\mathstrut +\mathstrut \) \(714\) \(\beta_{11}\mathstrut -\mathstrut \) \(863\) \(\beta_{10}\mathstrut +\mathstrut \) \(324\) \(\beta_{9}\mathstrut -\mathstrut \) \(63\) \(\beta_{8}\mathstrut +\mathstrut \) \(448\) \(\beta_{7}\mathstrut +\mathstrut \) \(250\) \(\beta_{6}\mathstrut +\mathstrut \) \(241\) \(\beta_{5}\mathstrut +\mathstrut \) \(47\) \(\beta_{4}\mathstrut +\mathstrut \) \(74\) \(\beta_{3}\mathstrut +\mathstrut \) \(591\) \(\beta_{2}\mathstrut +\mathstrut \) \(3410\) \(\beta_{1}\mathstrut +\mathstrut \) \(1412\)
\(\nu^{10}\)\(=\)\(-\)\(122\) \(\beta_{18}\mathstrut +\mathstrut \) \(37\) \(\beta_{17}\mathstrut +\mathstrut \) \(45\) \(\beta_{16}\mathstrut -\mathstrut \) \(788\) \(\beta_{15}\mathstrut -\mathstrut \) \(647\) \(\beta_{14}\mathstrut +\mathstrut \) \(1974\) \(\beta_{13}\mathstrut -\mathstrut \) \(797\) \(\beta_{12}\mathstrut -\mathstrut \) \(228\) \(\beta_{11}\mathstrut -\mathstrut \) \(573\) \(\beta_{10}\mathstrut -\mathstrut \) \(1032\) \(\beta_{9}\mathstrut -\mathstrut \) \(426\) \(\beta_{8}\mathstrut +\mathstrut \) \(1723\) \(\beta_{7}\mathstrut +\mathstrut \) \(683\) \(\beta_{6}\mathstrut +\mathstrut \) \(1826\) \(\beta_{5}\mathstrut +\mathstrut \) \(146\) \(\beta_{4}\mathstrut +\mathstrut \) \(256\) \(\beta_{3}\mathstrut +\mathstrut \) \(6451\) \(\beta_{2}\mathstrut +\mathstrut \) \(2638\) \(\beta_{1}\mathstrut +\mathstrut \) \(16664\)
\(\nu^{11}\)\(=\)\(-\)\(5763\) \(\beta_{18}\mathstrut +\mathstrut \) \(1200\) \(\beta_{17}\mathstrut +\mathstrut \) \(3096\) \(\beta_{16}\mathstrut -\mathstrut \) \(3008\) \(\beta_{15}\mathstrut -\mathstrut \) \(1906\) \(\beta_{14}\mathstrut +\mathstrut \) \(5519\) \(\beta_{13}\mathstrut -\mathstrut \) \(4380\) \(\beta_{12}\mathstrut +\mathstrut \) \(5865\) \(\beta_{11}\mathstrut -\mathstrut \) \(7308\) \(\beta_{10}\mathstrut +\mathstrut \) \(3073\) \(\beta_{9}\mathstrut -\mathstrut \) \(1171\) \(\beta_{8}\mathstrut +\mathstrut \) \(5130\) \(\beta_{7}\mathstrut +\mathstrut \) \(2874\) \(\beta_{6}\mathstrut +\mathstrut \) \(2900\) \(\beta_{5}\mathstrut +\mathstrut \) \(757\) \(\beta_{4}\mathstrut +\mathstrut \) \(1254\) \(\beta_{3}\mathstrut +\mathstrut \) \(7427\) \(\beta_{2}\mathstrut +\mathstrut \) \(28337\) \(\beta_{1}\mathstrut +\mathstrut \) \(17568\)
\(\nu^{12}\)\(=\)\(-\)\(1956\) \(\beta_{18}\mathstrut +\mathstrut \) \(764\) \(\beta_{17}\mathstrut +\mathstrut \) \(714\) \(\beta_{16}\mathstrut -\mathstrut \) \(9675\) \(\beta_{15}\mathstrut -\mathstrut \) \(8425\) \(\beta_{14}\mathstrut +\mathstrut \) \(20972\) \(\beta_{13}\mathstrut -\mathstrut \) \(11240\) \(\beta_{12}\mathstrut -\mathstrut \) \(2517\) \(\beta_{11}\mathstrut -\mathstrut \) \(5886\) \(\beta_{10}\mathstrut -\mathstrut \) \(9030\) \(\beta_{9}\mathstrut -\mathstrut \) \(6061\) \(\beta_{8}\mathstrut +\mathstrut \) \(18130\) \(\beta_{7}\mathstrut +\mathstrut \) \(8495\) \(\beta_{6}\mathstrut +\mathstrut \) \(18614\) \(\beta_{5}\mathstrut +\mathstrut \) \(2403\) \(\beta_{4}\mathstrut +\mathstrut \) \(4382\) \(\beta_{3}\mathstrut +\mathstrut \) \(60155\) \(\beta_{2}\mathstrut +\mathstrut \) \(27828\) \(\beta_{1}\mathstrut +\mathstrut \) \(150827\)
\(\nu^{13}\)\(=\)\(-\)\(55928\) \(\beta_{18}\mathstrut +\mathstrut \) \(14610\) \(\beta_{17}\mathstrut +\mathstrut \) \(29148\) \(\beta_{16}\mathstrut -\mathstrut \) \(35040\) \(\beta_{15}\mathstrut -\mathstrut \) \(25796\) \(\beta_{14}\mathstrut +\mathstrut \) \(60792\) \(\beta_{13}\mathstrut -\mathstrut \) \(53261\) \(\beta_{12}\mathstrut +\mathstrut \) \(47918\) \(\beta_{11}\mathstrut -\mathstrut \) \(61383\) \(\beta_{10}\mathstrut +\mathstrut \) \(27274\) \(\beta_{9}\mathstrut -\mathstrut \) \(17352\) \(\beta_{8}\mathstrut +\mathstrut \) \(55876\) \(\beta_{7}\mathstrut +\mathstrut \) \(30985\) \(\beta_{6}\mathstrut +\mathstrut \) \(33198\) \(\beta_{5}\mathstrut +\mathstrut \) \(10404\) \(\beta_{4}\mathstrut +\mathstrut \) \(18085\) \(\beta_{3}\mathstrut +\mathstrut \) \(87052\) \(\beta_{2}\mathstrut +\mathstrut \) \(239191\) \(\beta_{1}\mathstrut +\mathstrut \) \(203218\)
\(\nu^{14}\)\(=\)\(-\)\(25949\) \(\beta_{18}\mathstrut +\mathstrut \) \(12153\) \(\beta_{17}\mathstrut +\mathstrut \) \(9854\) \(\beta_{16}\mathstrut -\mathstrut \) \(112233\) \(\beta_{15}\mathstrut -\mathstrut \) \(101598\) \(\beta_{14}\mathstrut +\mathstrut \) \(218320\) \(\beta_{13}\mathstrut -\mathstrut \) \(144395\) \(\beta_{12}\mathstrut -\mathstrut \) \(26072\) \(\beta_{11}\mathstrut -\mathstrut \) \(56678\) \(\beta_{10}\mathstrut -\mathstrut \) \(78061\) \(\beta_{9}\mathstrut -\mathstrut \) \(77707\) \(\beta_{8}\mathstrut +\mathstrut \) \(188850\) \(\beta_{7}\mathstrut +\mathstrut \) \(96473\) \(\beta_{6}\mathstrut +\mathstrut \) \(187339\) \(\beta_{5}\mathstrut +\mathstrut \) \(33477\) \(\beta_{4}\mathstrut +\mathstrut \) \(62981\) \(\beta_{3}\mathstrut +\mathstrut \) \(570721\) \(\beta_{2}\mathstrut +\mathstrut \) \(281404\) \(\beta_{1}\mathstrut +\mathstrut \) \(1392607\)
\(\nu^{15}\)\(=\)\(-\)\(524427\) \(\beta_{18}\mathstrut +\mathstrut \) \(163536\) \(\beta_{17}\mathstrut +\mathstrut \) \(269128\) \(\beta_{16}\mathstrut -\mathstrut \) \(395485\) \(\beta_{15}\mathstrut -\mathstrut \) \(318575\) \(\beta_{14}\mathstrut +\mathstrut \) \(648670\) \(\beta_{13}\mathstrut -\mathstrut \) \(618844\) \(\beta_{12}\mathstrut +\mathstrut \) \(389321\) \(\beta_{11}\mathstrut -\mathstrut \) \(513606\) \(\beta_{10}\mathstrut +\mathstrut \) \(232973\) \(\beta_{9}\mathstrut -\mathstrut \) \(227616\) \(\beta_{8}\mathstrut +\mathstrut \) \(593439\) \(\beta_{7}\mathstrut +\mathstrut \) \(322451\) \(\beta_{6}\mathstrut +\mathstrut \) \(369681\) \(\beta_{5}\mathstrut +\mathstrut \) \(131102\) \(\beta_{4}\mathstrut +\mathstrut \) \(238100\) \(\beta_{3}\mathstrut +\mathstrut \) \(979532\) \(\beta_{2}\mathstrut +\mathstrut \) \(2043548\) \(\beta_{1}\mathstrut +\mathstrut \) \(2250122\)
\(\nu^{16}\)\(=\)\(-\)\(308306\) \(\beta_{18}\mathstrut +\mathstrut \) \(167148\) \(\beta_{17}\mathstrut +\mathstrut \) \(125634\) \(\beta_{16}\mathstrut -\mathstrut \) \(1261579\) \(\beta_{15}\mathstrut -\mathstrut \) \(1168911\) \(\beta_{14}\mathstrut +\mathstrut \) \(2245560\) \(\beta_{13}\mathstrut -\mathstrut \) \(1747135\) \(\beta_{12}\mathstrut -\mathstrut \) \(262139\) \(\beta_{11}\mathstrut -\mathstrut \) \(526143\) \(\beta_{10}\mathstrut -\mathstrut \) \(669837\) \(\beta_{9}\mathstrut -\mathstrut \) \(932844\) \(\beta_{8}\mathstrut +\mathstrut \) \(1957530\) \(\beta_{7}\mathstrut +\mathstrut \) \(1038871\) \(\beta_{6}\mathstrut +\mathstrut \) \(1879885\) \(\beta_{5}\mathstrut +\mathstrut \) \(425573\) \(\beta_{4}\mathstrut +\mathstrut \) \(821619\) \(\beta_{3}\mathstrut +\mathstrut \) \(5492530\) \(\beta_{2}\mathstrut +\mathstrut \) \(2769379\) \(\beta_{1}\mathstrut +\mathstrut \) \(13057426\)
\(\nu^{17}\)\(=\)\(-\)\(4812041\) \(\beta_{18}\mathstrut +\mathstrut \) \(1741784\) \(\beta_{17}\mathstrut +\mathstrut \) \(2465143\) \(\beta_{16}\mathstrut -\mathstrut \) \(4370834\) \(\beta_{15}\mathstrut -\mathstrut \) \(3718850\) \(\beta_{14}\mathstrut +\mathstrut \) \(6790451\) \(\beta_{13}\mathstrut -\mathstrut \) \(6975182\) \(\beta_{12}\mathstrut +\mathstrut \) \(3139156\) \(\beta_{11}\mathstrut -\mathstrut \) \(4289618\) \(\beta_{10}\mathstrut +\mathstrut \) \(1944110\) \(\beta_{9}\mathstrut -\mathstrut \) \(2773579\) \(\beta_{8}\mathstrut +\mathstrut \) \(6216545\) \(\beta_{7}\mathstrut +\mathstrut \) \(3284097\) \(\beta_{6}\mathstrut +\mathstrut \) \(4044823\) \(\beta_{5}\mathstrut +\mathstrut \) \(1564957\) \(\beta_{4}\mathstrut +\mathstrut \) \(2955058\) \(\beta_{3}\mathstrut +\mathstrut \) \(10737016\) \(\beta_{2}\mathstrut +\mathstrut \) \(17627416\) \(\beta_{1}\mathstrut +\mathstrut \) \(24221515\)
\(\nu^{18}\)\(=\)\(-\)\(3403569\) \(\beta_{18}\mathstrut +\mathstrut \) \(2099037\) \(\beta_{17}\mathstrut +\mathstrut \) \(1516987\) \(\beta_{16}\mathstrut -\mathstrut \) \(13904015\) \(\beta_{15}\mathstrut -\mathstrut \) \(13032858\) \(\beta_{14}\mathstrut +\mathstrut \) \(22924352\) \(\beta_{13}\mathstrut -\mathstrut \) \(20288048\) \(\beta_{12}\mathstrut -\mathstrut \) \(2603519\) \(\beta_{11}\mathstrut -\mathstrut \) \(4775363\) \(\beta_{10}\mathstrut -\mathstrut \) \(5714635\) \(\beta_{9}\mathstrut -\mathstrut \) \(10716645\) \(\beta_{8}\mathstrut +\mathstrut \) \(20233843\) \(\beta_{7}\mathstrut +\mathstrut \) \(10815204\) \(\beta_{6}\mathstrut +\mathstrut \) \(18888628\) \(\beta_{5}\mathstrut +\mathstrut \) \(5108951\) \(\beta_{4}\mathstrut +\mathstrut \) \(10086678\) \(\beta_{3}\mathstrut +\mathstrut \) \(53473521\) \(\beta_{2}\mathstrut +\mathstrut \) \(26760509\) \(\beta_{1}\mathstrut +\mathstrut \) \(123923583\)

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
−2.88114
−2.82248
−2.59429
−2.37182
−1.65183
−1.30887
−1.13995
−0.430685
0.00586283
0.121614
0.439460
0.589982
1.59810
1.68826
2.24158
2.42761
2.89772
3.01880
3.17208
0 −2.88114 0 −1.52019 0 −1.29303 0 5.30096 0
1.2 0 −2.82248 0 0.0936572 0 −0.650141 0 4.96640 0
1.3 0 −2.59429 0 −4.28452 0 2.47228 0 3.73036 0
1.4 0 −2.37182 0 1.99718 0 3.57468 0 2.62553 0
1.5 0 −1.65183 0 0.480995 0 −0.613677 0 −0.271464 0
1.6 0 −1.30887 0 3.64710 0 1.67706 0 −1.28687 0
1.7 0 −1.13995 0 −2.77941 0 1.98761 0 −1.70052 0
1.8 0 −0.430685 0 −1.65287 0 −2.73487 0 −2.81451 0
1.9 0 0.00586283 0 2.78153 0 −4.08250 0 −2.99997 0
1.10 0 0.121614 0 2.17287 0 3.48429 0 −2.98521 0
1.11 0 0.439460 0 −0.780123 0 −4.19801 0 −2.80687 0
1.12 0 0.589982 0 −1.83325 0 3.69467 0 −2.65192 0
1.13 0 1.59810 0 −2.51381 0 −3.65683 0 −0.446065 0
1.14 0 1.68826 0 3.67694 0 −0.618763 0 −0.149789 0
1.15 0 2.24158 0 −1.11384 0 −1.62199 0 2.02467 0
1.16 0 2.42761 0 2.44173 0 4.41683 0 2.89329 0
1.17 0 2.89772 0 2.58527 0 0.456152 0 5.39677 0
1.18 0 3.01880 0 −3.20894 0 2.67289 0 6.11314 0
1.19 0 3.17208 0 2.80966 0 1.03334 0 7.06208 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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

Hecke kernels

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

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