Properties

Label 4028.2.a.f
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \( + \beta_{7} q^{5} \) \( + ( 1 + \beta_{8} ) q^{7} \) \( + ( 1 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( + \beta_{7} q^{5} \) \( + ( 1 + \beta_{8} ) q^{7} \) \( + ( 1 + \beta_{2} ) q^{9} \) \( -\beta_{14} q^{11} \) \( + \beta_{16} q^{13} \) \( + ( \beta_{1} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{11} - \beta_{14} - \beta_{17} ) q^{15} \) \( + ( -\beta_{10} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{17} \) \(- q^{19}\) \( + ( 1 - \beta_{6} + \beta_{11} - \beta_{17} + \beta_{18} ) q^{21} \) \( + ( 1 - \beta_{6} + \beta_{7} + \beta_{13} - \beta_{15} ) q^{23} \) \( + ( 1 + 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} - \beta_{14} - \beta_{16} ) q^{25} \) \( + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{6} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{27} \) \( + ( -\beta_{5} + \beta_{13} - \beta_{15} ) q^{29} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} + \beta_{14} + \beta_{16} + \beta_{17} ) q^{31} \) \( + ( -\beta_{3} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{14} - \beta_{17} ) q^{33} \) \( + ( 1 - \beta_{1} - \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{35} \) \( + ( 1 + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{11} - \beta_{12} + 2 \beta_{14} + \beta_{15} + \beta_{17} ) q^{37} \) \( + ( 2 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{11} - \beta_{12} + 3 \beta_{14} + \beta_{15} + \beta_{17} ) q^{39} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{8} - \beta_{11} - \beta_{13} + 2 \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{41} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{18} ) q^{43} \) \( + ( 1 + 2 \beta_{1} + \beta_{6} + \beta_{10} - \beta_{11} - \beta_{14} + \beta_{15} - 2 \beta_{16} - \beta_{18} ) q^{45} \) \( + ( 1 + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{12} ) q^{47} \) \( + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{49} \) \( + ( 2 + \beta_{1} + \beta_{4} - \beta_{8} - \beta_{10} - \beta_{11} + \beta_{14} - \beta_{15} + \beta_{18} ) q^{51} \) \(- q^{53}\) \( + ( 1 + \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{55} \) \( -\beta_{1} q^{57} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{15} - \beta_{17} + \beta_{18} ) q^{59} \) \( + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{18} ) q^{61} \) \( + ( -\beta_{1} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{18} ) q^{63} \) \( + ( -\beta_{2} - \beta_{4} + \beta_{9} + \beta_{10} + \beta_{13} - \beta_{15} + \beta_{16} ) q^{65} \) \( + ( 5 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{12} + 2 \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{18} ) q^{67} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{13} + \beta_{15} - \beta_{17} ) q^{69} \) \( + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{13} - \beta_{15} ) q^{71} \) \( + ( -1 - \beta_{1} - 2 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{17} + \beta_{18} ) q^{73} \) \( + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{16} + 2 \beta_{17} ) q^{75} \) \( + ( 3 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - 3 \beta_{14} - \beta_{15} ) q^{77} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{14} - 2 \beta_{15} + 2 \beta_{16} + \beta_{18} ) q^{79} \) \( + ( 1 + 4 \beta_{1} + 2 \beta_{3} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - 3 \beta_{15} + \beta_{18} ) q^{81} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} + 2 \beta_{17} - \beta_{18} ) q^{83} \) \( + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{10} - \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{17} - \beta_{18} ) q^{85} \) \( + ( 1 + 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{8} + \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} - 5 \beta_{14} - \beta_{16} - 3 \beta_{17} ) q^{87} \) \( + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{9} + \beta_{11} + \beta_{14} + \beta_{18} ) q^{89} \) \( + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{91} \) \( + ( 7 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{14} - \beta_{15} - 2 \beta_{16} - \beta_{17} ) q^{93} \) \( -\beta_{7} q^{95} \) \( + ( -2 \beta_{1} - \beta_{3} - \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{97} \) \( + ( 1 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} - \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(19q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 13q^{7} \) \(\mathstrut +\mathstrut 19q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(19q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 13q^{7} \) \(\mathstrut +\mathstrut 19q^{9} \) \(\mathstrut +\mathstrut q^{11} \) \(\mathstrut -\mathstrut q^{13} \) \(\mathstrut +\mathstrut 8q^{15} \) \(\mathstrut +\mathstrut 3q^{17} \) \(\mathstrut -\mathstrut 19q^{19} \) \(\mathstrut +\mathstrut 8q^{21} \) \(\mathstrut +\mathstrut 10q^{23} \) \(\mathstrut +\mathstrut 21q^{25} \) \(\mathstrut +\mathstrut 28q^{27} \) \(\mathstrut +\mathstrut 2q^{29} \) \(\mathstrut +\mathstrut 25q^{31} \) \(\mathstrut +\mathstrut q^{33} \) \(\mathstrut +\mathstrut 20q^{35} \) \(\mathstrut +\mathstrut 19q^{37} \) \(\mathstrut +\mathstrut 37q^{39} \) \(\mathstrut -\mathstrut 9q^{41} \) \(\mathstrut +\mathstrut 35q^{43} \) \(\mathstrut +\mathstrut 37q^{45} \) \(\mathstrut +\mathstrut 23q^{47} \) \(\mathstrut +\mathstrut 30q^{49} \) \(\mathstrut +\mathstrut 34q^{51} \) \(\mathstrut -\mathstrut 19q^{53} \) \(\mathstrut +\mathstrut 40q^{55} \) \(\mathstrut -\mathstrut 4q^{57} \) \(\mathstrut +\mathstrut 16q^{59} \) \(\mathstrut +\mathstrut 21q^{61} \) \(\mathstrut +\mathstrut 3q^{63} \) \(\mathstrut -\mathstrut 10q^{65} \) \(\mathstrut +\mathstrut 67q^{67} \) \(\mathstrut +\mathstrut 23q^{69} \) \(\mathstrut +\mathstrut 18q^{71} \) \(\mathstrut -\mathstrut 20q^{73} \) \(\mathstrut +\mathstrut 33q^{75} \) \(\mathstrut +\mathstrut 37q^{77} \) \(\mathstrut +\mathstrut 2q^{79} \) \(\mathstrut +\mathstrut 23q^{81} \) \(\mathstrut +\mathstrut 38q^{83} \) \(\mathstrut +\mathstrut 8q^{85} \) \(\mathstrut +\mathstrut 18q^{87} \) \(\mathstrut -\mathstrut q^{89} \) \(\mathstrut -\mathstrut 9q^{91} \) \(\mathstrut +\mathstrut 14q^{93} \) \(\mathstrut -\mathstrut 4q^{95} \) \(\mathstrut -\mathstrut 21q^{97} \) \(\mathstrut +\mathstrut 33q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19}\mathstrut -\mathstrut \) \(4\) \(x^{18}\mathstrut -\mathstrut \) \(30\) \(x^{17}\mathstrut +\mathstrut \) \(124\) \(x^{16}\mathstrut +\mathstrut \) \(364\) \(x^{15}\mathstrut -\mathstrut \) \(1554\) \(x^{14}\mathstrut -\mathstrut \) \(2310\) \(x^{13}\mathstrut +\mathstrut \) \(10113\) \(x^{12}\mathstrut +\mathstrut \) \(8368\) \(x^{11}\mathstrut -\mathstrut \) \(36567\) \(x^{10}\mathstrut -\mathstrut \) \(18074\) \(x^{9}\mathstrut +\mathstrut \) \(72868\) \(x^{8}\mathstrut +\mathstrut \) \(23819\) \(x^{7}\mathstrut -\mathstrut \) \(73816\) \(x^{6}\mathstrut -\mathstrut \) \(17877\) \(x^{5}\mathstrut +\mathstrut \) \(30973\) \(x^{4}\mathstrut +\mathstrut \) \(5074\) \(x^{3}\mathstrut -\mathstrut \) \(4299\) \(x^{2}\mathstrut -\mathstrut \) \(454\) \(x\mathstrut +\mathstrut \) \(139\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(1013664970932510359\) \(\nu^{18}\mathstrut +\mathstrut \) \(6383747831994437549\) \(\nu^{17}\mathstrut +\mathstrut \) \(32803688712224653227\) \(\nu^{16}\mathstrut -\mathstrut \) \(228419041336493582073\) \(\nu^{15}\mathstrut -\mathstrut \) \(464929689331149538134\) \(\nu^{14}\mathstrut +\mathstrut \) \(3427742929074332447811\) \(\nu^{13}\mathstrut +\mathstrut \) \(3904702873986854643991\) \(\nu^{12}\mathstrut -\mathstrut \) \(27975363851656900935375\) \(\nu^{11}\mathstrut -\mathstrut \) \(21632651398845249336587\) \(\nu^{10}\mathstrut +\mathstrut \) \(134639173642389041239377\) \(\nu^{9}\mathstrut +\mathstrut \) \(78774350849770433685435\) \(\nu^{8}\mathstrut -\mathstrut \) \(386387527001845234846065\) \(\nu^{7}\mathstrut -\mathstrut \) \(173273403885715628219601\) \(\nu^{6}\mathstrut +\mathstrut \) \(628050699125868748022486\) \(\nu^{5}\mathstrut +\mathstrut \) \(199437779162941782881844\) \(\nu^{4}\mathstrut -\mathstrut \) \(486508306277487080639637\) \(\nu^{3}\mathstrut -\mathstrut \) \(95863405384260650578292\) \(\nu^{2}\mathstrut +\mathstrut \) \(90955650844496841415443\) \(\nu\mathstrut +\mathstrut \) \(9549212368394962571709\)\()/\)\(34\!\cdots\!67\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(539666137422440460\) \(\nu^{18}\mathstrut +\mathstrut \) \(3504924600062958270\) \(\nu^{17}\mathstrut +\mathstrut \) \(13579116821905178591\) \(\nu^{16}\mathstrut -\mathstrut \) \(113206383041965395179\) \(\nu^{15}\mathstrut -\mathstrut \) \(122100450800907992989\) \(\nu^{14}\mathstrut +\mathstrut \) \(1491567589135895220569\) \(\nu^{13}\mathstrut +\mathstrut \) \(425689341513369750520\) \(\nu^{12}\mathstrut -\mathstrut \) \(10269789484355555618923\) \(\nu^{11}\mathstrut -\mathstrut \) \(101997297017966586210\) \(\nu^{10}\mathstrut +\mathstrut \) \(39188082322529543612125\) \(\nu^{9}\mathstrut -\mathstrut \) \(2126285279123905811034\) \(\nu^{8}\mathstrut -\mathstrut \) \(80261934709718916524839\) \(\nu^{7}\mathstrut +\mathstrut \) \(1974045361748312551583\) \(\nu^{6}\mathstrut +\mathstrut \) \(75554189902004790177990\) \(\nu^{5}\mathstrut +\mathstrut \) \(1526271337661236731546\) \(\nu^{4}\mathstrut -\mathstrut \) \(17339171104576617579725\) \(\nu^{3}\mathstrut +\mathstrut \) \(2463678410242470840173\) \(\nu^{2}\mathstrut -\mathstrut \) \(2596169087667389090749\) \(\nu\mathstrut -\mathstrut \) \(705048640521550753042\)\()/\)\(26\!\cdots\!59\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(8082393497238969918\) \(\nu^{18}\mathstrut +\mathstrut \) \(50880960665791710281\) \(\nu^{17}\mathstrut +\mathstrut \) \(149810450509960698612\) \(\nu^{16}\mathstrut -\mathstrut \) \(1492920352180888609775\) \(\nu^{15}\mathstrut -\mathstrut \) \(106264229225035104222\) \(\nu^{14}\mathstrut +\mathstrut \) \(17414925175315486603474\) \(\nu^{13}\mathstrut -\mathstrut \) \(16346936089987529154161\) \(\nu^{12}\mathstrut -\mathstrut \) \(103019545288071940947301\) \(\nu^{11}\mathstrut +\mathstrut \) \(156111636410667783119248\) \(\nu^{10}\mathstrut +\mathstrut \) \(328269697028020825287738\) \(\nu^{9}\mathstrut -\mathstrut \) \(642414053222074731899579\) \(\nu^{8}\mathstrut -\mathstrut \) \(560547808941881549889402\) \(\nu^{7}\mathstrut +\mathstrut \) \(1313330090608220013976793\) \(\nu^{6}\mathstrut +\mathstrut \) \(498719127530284526140679\) \(\nu^{5}\mathstrut -\mathstrut \) \(1251133748221990524239808\) \(\nu^{4}\mathstrut -\mathstrut \) \(227160207345741293466219\) \(\nu^{3}\mathstrut +\mathstrut \) \(409776759302680392011964\) \(\nu^{2}\mathstrut +\mathstrut \) \(30696628439091594640205\) \(\nu\mathstrut -\mathstrut \) \(27761816022030137605193\)\()/\)\(34\!\cdots\!67\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(9012255479615042506\) \(\nu^{18}\mathstrut +\mathstrut \) \(57926039052261232522\) \(\nu^{17}\mathstrut +\mathstrut \) \(192444453247398587102\) \(\nu^{16}\mathstrut -\mathstrut \) \(1774315165620840355805\) \(\nu^{15}\mathstrut -\mathstrut \) \(955354606015150023233\) \(\nu^{14}\mathstrut +\mathstrut \) \(21925206662121998889480\) \(\nu^{13}\mathstrut -\mathstrut \) \(6799678529958853409030\) \(\nu^{12}\mathstrut -\mathstrut \) \(140062845135953860708398\) \(\nu^{11}\mathstrut +\mathstrut \) \(91128082993182265376372\) \(\nu^{10}\mathstrut +\mathstrut \) \(492197278713807875634399\) \(\nu^{9}\mathstrut -\mathstrut \) \(377008214805426624184168\) \(\nu^{8}\mathstrut -\mathstrut \) \(929821432914397749569504\) \(\nu^{7}\mathstrut +\mathstrut \) \(706422386390206413846371\) \(\nu^{6}\mathstrut +\mathstrut \) \(823567850258607734691640\) \(\nu^{5}\mathstrut -\mathstrut \) \(596066650789253892258349\) \(\nu^{4}\mathstrut -\mathstrut \) \(192745004320722586135416\) \(\nu^{3}\mathstrut +\mathstrut \) \(208367847416842091060431\) \(\nu^{2}\mathstrut -\mathstrut \) \(31727649377226224441820\) \(\nu\mathstrut -\mathstrut \) \(10377632966827837230239\)\()/\)\(34\!\cdots\!67\)
\(\beta_{7}\)\(=\)\((\)\(12279512726675507013\) \(\nu^{18}\mathstrut -\mathstrut \) \(75947048383905243843\) \(\nu^{17}\mathstrut -\mathstrut \) \(279042796602546905350\) \(\nu^{16}\mathstrut +\mathstrut \) \(2340433431752486897347\) \(\nu^{15}\mathstrut +\mathstrut \) \(1830443128106006764405\) \(\nu^{14}\mathstrut -\mathstrut \) \(29172991237797969942832\) \(\nu^{13}\mathstrut +\mathstrut \) \(2617817530125671121032\) \(\nu^{12}\mathstrut +\mathstrut \) \(188792773780458340703538\) \(\nu^{11}\mathstrut -\mathstrut \) \(81682667251389768988723\) \(\nu^{10}\mathstrut -\mathstrut \) \(677647246369639428662873\) \(\nu^{9}\mathstrut +\mathstrut \) \(369120574768379920994051\) \(\nu^{8}\mathstrut +\mathstrut \) \(1334621888720261551464630\) \(\nu^{7}\mathstrut -\mathstrut \) \(713224542559588122446075\) \(\nu^{6}\mathstrut -\mathstrut \) \(1321405476326058704896156\) \(\nu^{5}\mathstrut +\mathstrut \) \(625724025506759966537372\) \(\nu^{4}\mathstrut +\mathstrut \) \(519748650162003555440563\) \(\nu^{3}\mathstrut -\mathstrut \) \(234975891591382458211335\) \(\nu^{2}\mathstrut -\mathstrut \) \(57209576941038616681447\) \(\nu\mathstrut +\mathstrut \) \(15374816921928772387641\)\()/\)\(34\!\cdots\!67\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(12314233074030226871\) \(\nu^{18}\mathstrut +\mathstrut \) \(51660710244299223716\) \(\nu^{17}\mathstrut +\mathstrut \) \(335701280791985868818\) \(\nu^{16}\mathstrut -\mathstrut \) \(1550580703664022927300\) \(\nu^{15}\mathstrut -\mathstrut \) \(3348151150958097388752\) \(\nu^{14}\mathstrut +\mathstrut \) \(18593376947389501907842\) \(\nu^{13}\mathstrut +\mathstrut \) \(12819124770709822621272\) \(\nu^{12}\mathstrut -\mathstrut \) \(113935358339789835760057\) \(\nu^{11}\mathstrut +\mathstrut \) \(9828945992025776641311\) \(\nu^{10}\mathstrut +\mathstrut \) \(380259478726129276646478\) \(\nu^{9}\mathstrut -\mathstrut \) \(230459265695755928256657\) \(\nu^{8}\mathstrut -\mathstrut \) \(686917593354695955136902\) \(\nu^{7}\mathstrut +\mathstrut \) \(688097937337630306295553\) \(\nu^{6}\mathstrut +\mathstrut \) \(636223365416096458777228\) \(\nu^{5}\mathstrut -\mathstrut \) \(785844487179013523942315\) \(\nu^{4}\mathstrut -\mathstrut \) \(268415003301992744582801\) \(\nu^{3}\mathstrut +\mathstrut \) \(267273604049339077042987\) \(\nu^{2}\mathstrut +\mathstrut \) \(23840706565177952489302\) \(\nu\mathstrut -\mathstrut \) \(15853588812280295893235\)\()/\)\(34\!\cdots\!67\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(14065420372976239962\) \(\nu^{18}\mathstrut +\mathstrut \) \(56121672911821323774\) \(\nu^{17}\mathstrut +\mathstrut \) \(392201561392315144161\) \(\nu^{16}\mathstrut -\mathstrut \) \(1691009143344334751529\) \(\nu^{15}\mathstrut -\mathstrut \) \(4076081956140805156372\) \(\nu^{14}\mathstrut +\mathstrut \) \(20488657664749485257454\) \(\nu^{13}\mathstrut +\mathstrut \) \(17408020904773011271906\) \(\nu^{12}\mathstrut -\mathstrut \) \(128638473769920492132139\) \(\nu^{11}\mathstrut -\mathstrut \) \(3016754497444860404492\) \(\nu^{10}\mathstrut +\mathstrut \) \(453114600578601704464099\) \(\nu^{9}\mathstrut -\mathstrut \) \(230905172735736471947896\) \(\nu^{8}\mathstrut -\mathstrut \) \(917172825624740629980752\) \(\nu^{7}\mathstrut +\mathstrut \) \(772620603092657335620073\) \(\nu^{6}\mathstrut +\mathstrut \) \(1056358072940628201207339\) \(\nu^{5}\mathstrut -\mathstrut \) \(946705048039643406597082\) \(\nu^{4}\mathstrut -\mathstrut \) \(622937693458533872437694\) \(\nu^{3}\mathstrut +\mathstrut \) \(350995415744919634018826\) \(\nu^{2}\mathstrut +\mathstrut \) \(93150844803515409831409\) \(\nu\mathstrut -\mathstrut \) \(28540922552046344408613\)\()/\)\(34\!\cdots\!67\)
\(\beta_{10}\)\(=\)\((\)\(16072902599680908164\) \(\nu^{18}\mathstrut -\mathstrut \) \(91966583363222535462\) \(\nu^{17}\mathstrut -\mathstrut \) \(394095446186709103493\) \(\nu^{16}\mathstrut +\mathstrut \) \(2857419119741421916394\) \(\nu^{15}\mathstrut +\mathstrut \) \(3252798257142975761126\) \(\nu^{14}\mathstrut -\mathstrut \) \(36042418775253329863245\) \(\nu^{13}\mathstrut -\mathstrut \) \(6868373979917272840906\) \(\nu^{12}\mathstrut +\mathstrut \) \(237678327165159094674093\) \(\nu^{11}\mathstrut -\mathstrut \) \(42092416564172393413605\) \(\nu^{10}\mathstrut -\mathstrut \) \(880863656053346109871933\) \(\nu^{9}\mathstrut +\mathstrut \) \(250297060329871709304802\) \(\nu^{8}\mathstrut +\mathstrut \) \(1836166095699660343346776\) \(\nu^{7}\mathstrut -\mathstrut \) \(448004696282821758586931\) \(\nu^{6}\mathstrut -\mathstrut \) \(2016364357746995762333272\) \(\nu^{5}\mathstrut +\mathstrut \) \(265372142123995710209070\) \(\nu^{4}\mathstrut +\mathstrut \) \(968395259204709981854911\) \(\nu^{3}\mathstrut -\mathstrut \) \(39479411448802007040965\) \(\nu^{2}\mathstrut -\mathstrut \) \(139956340876172755996760\) \(\nu\mathstrut +\mathstrut \) \(262481099435760130156\)\()/\)\(34\!\cdots\!67\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(30920828473910836670\) \(\nu^{18}\mathstrut +\mathstrut \) \(119883384573197873174\) \(\nu^{17}\mathstrut +\mathstrut \) \(950561701022030194182\) \(\nu^{16}\mathstrut -\mathstrut \) \(3740690915509558295648\) \(\nu^{15}\mathstrut -\mathstrut \) \(11963509787421702685666\) \(\nu^{14}\mathstrut +\mathstrut \) \(47245043857622210665744\) \(\nu^{13}\mathstrut +\mathstrut \) \(80170999974298067452935\) \(\nu^{12}\mathstrut -\mathstrut \) \(310036125983548622083787\) \(\nu^{11}\mathstrut -\mathstrut \) \(313689368711917707970084\) \(\nu^{10}\mathstrut +\mathstrut \) \(1128173850045330519126837\) \(\nu^{9}\mathstrut +\mathstrut \) \(744046916765897486514004\) \(\nu^{8}\mathstrut -\mathstrut \) \(2239724891565664824511173\) \(\nu^{7}\mathstrut -\mathstrut \) \(1058318120853383259078135\) \(\nu^{6}\mathstrut +\mathstrut \) \(2180901680454689586443078\) \(\nu^{5}\mathstrut +\mathstrut \) \(791993991768000869119075\) \(\nu^{4}\mathstrut -\mathstrut \) \(760900293840186507497893\) \(\nu^{3}\mathstrut -\mathstrut \) \(179606709602492481236308\) \(\nu^{2}\mathstrut +\mathstrut \) \(49020417227303367772001\) \(\nu\mathstrut +\mathstrut \) \(11724294891590491667200\)\()/\)\(34\!\cdots\!67\)
\(\beta_{12}\)\(=\)\((\)\(2810991684136139728\) \(\nu^{18}\mathstrut -\mathstrut \) \(11193644850226234750\) \(\nu^{17}\mathstrut -\mathstrut \) \(83094520879379915030\) \(\nu^{16}\mathstrut +\mathstrut \) \(343539561760627625000\) \(\nu^{15}\mathstrut +\mathstrut \) \(981957540622806796858\) \(\nu^{14}\mathstrut -\mathstrut \) \(4248080377837881910399\) \(\nu^{13}\mathstrut -\mathstrut \) \(5923377110657869476954\) \(\nu^{12}\mathstrut +\mathstrut \) \(27142175615340616021689\) \(\nu^{11}\mathstrut +\mathstrut \) \(19341402264776327604609\) \(\nu^{10}\mathstrut -\mathstrut \) \(95597437847383147171360\) \(\nu^{9}\mathstrut -\mathstrut \) \(33475877988139306145394\) \(\nu^{8}\mathstrut +\mathstrut \) \(182933524201420133860161\) \(\nu^{7}\mathstrut +\mathstrut \) \(26888038421775232406730\) \(\nu^{6}\mathstrut -\mathstrut \) \(172099709947074829946152\) \(\nu^{5}\mathstrut -\mathstrut \) \(2496555985410708691581\) \(\nu^{4}\mathstrut +\mathstrut \) \(59232482816746354496069\) \(\nu^{3}\mathstrut -\mathstrut \) \(9804675742813541314583\) \(\nu^{2}\mathstrut -\mathstrut \) \(3661243503266635980568\) \(\nu\mathstrut +\mathstrut \) \(1702381593629708054002\)\()/\)\(26\!\cdots\!59\)
\(\beta_{13}\)\(=\)\((\)\(2843677740234168613\) \(\nu^{18}\mathstrut -\mathstrut \) \(11349244696320364505\) \(\nu^{17}\mathstrut -\mathstrut \) \(86108385019162773642\) \(\nu^{16}\mathstrut +\mathstrut \) \(353490448330280482628\) \(\nu^{15}\mathstrut +\mathstrut \) \(1061232596093721452725\) \(\nu^{14}\mathstrut -\mathstrut \) \(4456501003206093014746\) \(\nu^{13}\mathstrut -\mathstrut \) \(6918026893091333773760\) \(\nu^{12}\mathstrut +\mathstrut \) \(29205864935468996547239\) \(\nu^{11}\mathstrut +\mathstrut \) \(26237875672467871584328\) \(\nu^{10}\mathstrut -\mathstrut \) \(106323157980190747086538\) \(\nu^{9}\mathstrut -\mathstrut \) \(60921824645513997515995\) \(\nu^{8}\mathstrut +\mathstrut \) \(212283232873006321803260\) \(\nu^{7}\mathstrut +\mathstrut \) \(87958646698080945085998\) \(\nu^{6}\mathstrut -\mathstrut \) \(211203652985939143543248\) \(\nu^{5}\mathstrut -\mathstrut \) \(71019286251562866791270\) \(\nu^{4}\mathstrut +\mathstrut \) \(80504940697230352291768\) \(\nu^{3}\mathstrut +\mathstrut \) \(19433210783559378052114\) \(\nu^{2}\mathstrut -\mathstrut \) \(7840357659589812655154\) \(\nu\mathstrut -\mathstrut \) \(677863154453489493139\)\()/\)\(26\!\cdots\!59\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(37978614188213183709\) \(\nu^{18}\mathstrut +\mathstrut \) \(159449436967766400861\) \(\nu^{17}\mathstrut +\mathstrut \) \(1120729210703677838193\) \(\nu^{16}\mathstrut -\mathstrut \) \(4935523460323896609933\) \(\nu^{15}\mathstrut -\mathstrut \) \(13340827181414440749721\) \(\nu^{14}\mathstrut +\mathstrut \) \(61679278046621658447683\) \(\nu^{13}\mathstrut +\mathstrut \) \(83153170959761190732914\) \(\nu^{12}\mathstrut -\mathstrut \) \(398916321013084576590828\) \(\nu^{11}\mathstrut -\mathstrut \) \(299779358367535526932361\) \(\nu^{10}\mathstrut +\mathstrut \) \(1421761132329473018744018\) \(\nu^{9}\mathstrut +\mathstrut \) \(670278700785601694237276\) \(\nu^{8}\mathstrut -\mathstrut \) \(2735691555041965902330833\) \(\nu^{7}\mathstrut -\mathstrut \) \(972178791369964142248127\) \(\nu^{6}\mathstrut +\mathstrut \) \(2523077391398104754291646\) \(\nu^{5}\mathstrut +\mathstrut \) \(820001318022588237116666\) \(\nu^{4}\mathstrut -\mathstrut \) \(762538587432516170978728\) \(\nu^{3}\mathstrut -\mathstrut \) \(236425795844853268020665\) \(\nu^{2}\mathstrut +\mathstrut \) \(23092845362882775090641\) \(\nu\mathstrut +\mathstrut \) \(18658611605802252907286\)\()/\)\(34\!\cdots\!67\)
\(\beta_{15}\)\(=\)\((\)\(40260786094638775650\) \(\nu^{18}\mathstrut -\mathstrut \) \(175567919116157259598\) \(\nu^{17}\mathstrut -\mathstrut \) \(1146351490718162460997\) \(\nu^{16}\mathstrut +\mathstrut \) \(5416654855583962870974\) \(\nu^{15}\mathstrut +\mathstrut \) \(12745454425647953208988\) \(\nu^{14}\mathstrut -\mathstrut \) \(67477650887082219075842\) \(\nu^{13}\mathstrut -\mathstrut \) \(69043082073825151131539\) \(\nu^{12}\mathstrut +\mathstrut \) \(435638704099058343964752\) \(\nu^{11}\mathstrut +\mathstrut \) \(181268757821956946544273\) \(\nu^{10}\mathstrut -\mathstrut \) \(1557247265878266347682706\) \(\nu^{9}\mathstrut -\mathstrut \) \(170599429571660391077600\) \(\nu^{8}\mathstrut +\mathstrut \) \(3047221868344953263091880\) \(\nu^{7}\mathstrut -\mathstrut \) \(118537219926767515035213\) \(\nu^{6}\mathstrut -\mathstrut \) \(2983358528152073364113123\) \(\nu^{5}\mathstrut +\mathstrut \) \(291400103820138825132342\) \(\nu^{4}\mathstrut +\mathstrut \) \(1146406281163259773760714\) \(\nu^{3}\mathstrut -\mathstrut \) \(134077632441870097460973\) \(\nu^{2}\mathstrut -\mathstrut \) \(113067419744114310145875\) \(\nu\mathstrut +\mathstrut \) \(556041491076014525149\)\()/\)\(34\!\cdots\!67\)
\(\beta_{16}\)\(=\)\((\)\(48161166165251651829\) \(\nu^{18}\mathstrut -\mathstrut \) \(214183933471934477069\) \(\nu^{17}\mathstrut -\mathstrut \) \(1354654963745146272946\) \(\nu^{16}\mathstrut +\mathstrut \) \(6594657794653130538136\) \(\nu^{15}\mathstrut +\mathstrut \) \(14770922372314263261637\) \(\nu^{14}\mathstrut -\mathstrut \) \(81978738033841550109093\) \(\nu^{13}\mathstrut -\mathstrut \) \(77276600504063208667028\) \(\nu^{12}\mathstrut +\mathstrut \) \(528414033628651687262510\) \(\nu^{11}\mathstrut +\mathstrut \) \(187312962802747769677365\) \(\nu^{10}\mathstrut -\mathstrut \) \(1890216667950960941706585\) \(\nu^{9}\mathstrut -\mathstrut \) \(116988033924023668491712\) \(\nu^{8}\mathstrut +\mathstrut \) \(3727403505749017453402842\) \(\nu^{7}\mathstrut -\mathstrut \) \(283478766103769053494878\) \(\nu^{6}\mathstrut -\mathstrut \) \(3747246844666897741476244\) \(\nu^{5}\mathstrut +\mathstrut \) \(487122365658243491064730\) \(\nu^{4}\mathstrut +\mathstrut \) \(1557597897636070494502675\) \(\nu^{3}\mathstrut -\mathstrut \) \(243669449289305650873372\) \(\nu^{2}\mathstrut -\mathstrut \) \(183519035076096841260150\) \(\nu\mathstrut +\mathstrut \) \(15780177101141905520027\)\()/\)\(34\!\cdots\!67\)
\(\beta_{17}\)\(=\)\((\)\(50398135494972326796\) \(\nu^{18}\mathstrut -\mathstrut \) \(205635435554699590005\) \(\nu^{17}\mathstrut -\mathstrut \) \(1452874990210943747302\) \(\nu^{16}\mathstrut +\mathstrut \) \(6232225832453837317484\) \(\nu^{15}\mathstrut +\mathstrut \) \(16540275904376039157620\) \(\nu^{14}\mathstrut -\mathstrut \) \(75769169127617525350201\) \(\nu^{13}\mathstrut -\mathstrut \) \(94140475891623742215449\) \(\nu^{12}\mathstrut +\mathstrut \) \(473026411609922601696842\) \(\nu^{11}\mathstrut +\mathstrut \) \(279312317316166220169993\) \(\nu^{10}\mathstrut -\mathstrut \) \(1614284798142203414385807\) \(\nu^{9}\mathstrut -\mathstrut \) \(408904352130513796813489\) \(\nu^{8}\mathstrut +\mathstrut \) \(2962668592805649058520512\) \(\nu^{7}\mathstrut +\mathstrut \) \(240849246121361947629705\) \(\nu^{6}\mathstrut -\mathstrut \) \(2646330299676823810790046\) \(\nu^{5}\mathstrut -\mathstrut \) \(6987864269487478484626\) \(\nu^{4}\mathstrut +\mathstrut \) \(859923878877118650833277\) \(\nu^{3}\mathstrut -\mathstrut \) \(31233698366619744425441\) \(\nu^{2}\mathstrut -\mathstrut \) \(41897785519851591635960\) \(\nu\mathstrut -\mathstrut \) \(1760874733024935089596\)\()/\)\(34\!\cdots\!67\)
\(\beta_{18}\)\(=\)\((\)\(74710486437446437192\) \(\nu^{18}\mathstrut -\mathstrut \) \(301318492504557167969\) \(\nu^{17}\mathstrut -\mathstrut \) \(2234608040469850149678\) \(\nu^{16}\mathstrut +\mathstrut \) \(9332831270331460449619\) \(\nu^{15}\mathstrut +\mathstrut \) \(27005489836129121170361\) \(\nu^{14}\mathstrut -\mathstrut \) \(116715759953417738577203\) \(\nu^{13}\mathstrut -\mathstrut \) \(170512673658002814491048\) \(\nu^{12}\mathstrut +\mathstrut \) \(755874140813028078170070\) \(\nu^{11}\mathstrut +\mathstrut \) \(614094686929332164171070\) \(\nu^{10}\mathstrut -\mathstrut \) \(2703288083749504306601356\) \(\nu^{9}\mathstrut -\mathstrut \) \(1319563541418099291012535\) \(\nu^{8}\mathstrut +\mathstrut \) \(5253982706384872413598083\) \(\nu^{7}\mathstrut +\mathstrut \) \(1732825690188432852621703\) \(\nu^{6}\mathstrut -\mathstrut \) \(5009650161716357552256666\) \(\nu^{5}\mathstrut -\mathstrut \) \(1282054769126796767569368\) \(\nu^{4}\mathstrut +\mathstrut \) \(1757835191063551020382195\) \(\nu^{3}\mathstrut +\mathstrut \) \(327642677232636835042171\) \(\nu^{2}\mathstrut -\mathstrut \) \(140612089369578959957683\) \(\nu\mathstrut -\mathstrut \) \(25629137576845305236733\)\()/\)\(34\!\cdots\!67\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{18}\mathstrut -\mathstrut \) \(3\) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(28\)
\(\nu^{5}\)\(=\)\(2\) \(\beta_{18}\mathstrut -\mathstrut \) \(2\) \(\beta_{16}\mathstrut -\mathstrut \) \(4\) \(\beta_{15}\mathstrut -\mathstrut \) \(12\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(13\) \(\beta_{11}\mathstrut +\mathstrut \) \(13\) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(10\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(73\) \(\beta_{1}\mathstrut +\mathstrut \) \(17\)
\(\nu^{6}\)\(=\)\(16\) \(\beta_{18}\mathstrut +\mathstrut \) \(5\) \(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(41\) \(\beta_{15}\mathstrut +\mathstrut \) \(20\) \(\beta_{14}\mathstrut +\mathstrut \) \(7\) \(\beta_{13}\mathstrut -\mathstrut \) \(20\) \(\beta_{12}\mathstrut -\mathstrut \) \(21\) \(\beta_{11}\mathstrut +\mathstrut \) \(17\) \(\beta_{10}\mathstrut -\mathstrut \) \(17\) \(\beta_{9}\mathstrut -\mathstrut \) \(9\) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(4\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(33\) \(\beta_{3}\mathstrut +\mathstrut \) \(78\) \(\beta_{2}\mathstrut +\mathstrut \) \(70\) \(\beta_{1}\mathstrut +\mathstrut \) \(226\)
\(\nu^{7}\)\(=\)\(41\) \(\beta_{18}\mathstrut +\mathstrut \) \(5\) \(\beta_{17}\mathstrut -\mathstrut \) \(28\) \(\beta_{16}\mathstrut -\mathstrut \) \(74\) \(\beta_{15}\mathstrut +\mathstrut \) \(19\) \(\beta_{14}\mathstrut -\mathstrut \) \(119\) \(\beta_{13}\mathstrut -\mathstrut \) \(31\) \(\beta_{12}\mathstrut -\mathstrut \) \(149\) \(\beta_{11}\mathstrut +\mathstrut \) \(139\) \(\beta_{10}\mathstrut -\mathstrut \) \(53\) \(\beta_{9}\mathstrut -\mathstrut \) \(27\) \(\beta_{8}\mathstrut +\mathstrut \) \(54\) \(\beta_{7}\mathstrut +\mathstrut \) \(87\) \(\beta_{6}\mathstrut +\mathstrut \) \(33\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut +\mathstrut \) \(151\) \(\beta_{3}\mathstrut +\mathstrut \) \(37\) \(\beta_{2}\mathstrut +\mathstrut \) \(692\) \(\beta_{1}\mathstrut +\mathstrut \) \(231\)
\(\nu^{8}\)\(=\)\(207\) \(\beta_{18}\mathstrut +\mathstrut \) \(103\) \(\beta_{17}\mathstrut +\mathstrut \) \(24\) \(\beta_{16}\mathstrut -\mathstrut \) \(463\) \(\beta_{15}\mathstrut +\mathstrut \) \(302\) \(\beta_{14}\mathstrut +\mathstrut \) \(16\) \(\beta_{13}\mathstrut -\mathstrut \) \(283\) \(\beta_{12}\mathstrut -\mathstrut \) \(314\) \(\beta_{11}\mathstrut +\mathstrut \) \(219\) \(\beta_{10}\mathstrut -\mathstrut \) \(214\) \(\beta_{9}\mathstrut -\mathstrut \) \(55\) \(\beta_{8}\mathstrut +\mathstrut \) \(160\) \(\beta_{7}\mathstrut +\mathstrut \) \(70\) \(\beta_{6}\mathstrut +\mathstrut \) \(39\) \(\beta_{5}\mathstrut -\mathstrut \) \(77\) \(\beta_{4}\mathstrut +\mathstrut \) \(421\) \(\beta_{3}\mathstrut +\mathstrut \) \(681\) \(\beta_{2}\mathstrut +\mathstrut \) \(931\) \(\beta_{1}\mathstrut +\mathstrut \) \(1977\)
\(\nu^{9}\)\(=\)\(599\) \(\beta_{18}\mathstrut +\mathstrut \) \(140\) \(\beta_{17}\mathstrut -\mathstrut \) \(277\) \(\beta_{16}\mathstrut -\mathstrut \) \(1010\) \(\beta_{15}\mathstrut +\mathstrut \) \(472\) \(\beta_{14}\mathstrut -\mathstrut \) \(1147\) \(\beta_{13}\mathstrut -\mathstrut \) \(560\) \(\beta_{12}\mathstrut -\mathstrut \) \(1672\) \(\beta_{11}\mathstrut +\mathstrut \) \(1413\) \(\beta_{10}\mathstrut -\mathstrut \) \(696\) \(\beta_{9}\mathstrut -\mathstrut \) \(249\) \(\beta_{8}\mathstrut +\mathstrut \) \(687\) \(\beta_{7}\mathstrut +\mathstrut \) \(735\) \(\beta_{6}\mathstrut +\mathstrut \) \(402\) \(\beta_{5}\mathstrut +\mathstrut \) \(161\) \(\beta_{4}\mathstrut +\mathstrut \) \(1688\) \(\beta_{3}\mathstrut +\mathstrut \) \(494\) \(\beta_{2}\mathstrut +\mathstrut \) \(6691\) \(\beta_{1}\mathstrut +\mathstrut \) \(2842\)
\(\nu^{10}\)\(=\)\(2507\) \(\beta_{18}\mathstrut +\mathstrut \) \(1506\) \(\beta_{17}\mathstrut +\mathstrut \) \(386\) \(\beta_{16}\mathstrut -\mathstrut \) \(4994\) \(\beta_{15}\mathstrut +\mathstrut \) \(4028\) \(\beta_{14}\mathstrut -\mathstrut \) \(394\) \(\beta_{13}\mathstrut -\mathstrut \) \(3568\) \(\beta_{12}\mathstrut -\mathstrut \) \(4139\) \(\beta_{11}\mathstrut +\mathstrut \) \(2577\) \(\beta_{10}\mathstrut -\mathstrut \) \(2449\) \(\beta_{9}\mathstrut -\mathstrut \) \(162\) \(\beta_{8}\mathstrut +\mathstrut \) \(1716\) \(\beta_{7}\mathstrut +\mathstrut \) \(865\) \(\beta_{6}\mathstrut +\mathstrut \) \(531\) \(\beta_{5}\mathstrut -\mathstrut \) \(1052\) \(\beta_{4}\mathstrut +\mathstrut \) \(4938\) \(\beta_{3}\mathstrut +\mathstrut \) \(6003\) \(\beta_{2}\mathstrut +\mathstrut \) \(11218\) \(\beta_{1}\mathstrut +\mathstrut \) \(18218\)
\(\nu^{11}\)\(=\)\(7733\) \(\beta_{18}\mathstrut +\mathstrut \) \(2547\) \(\beta_{17}\mathstrut -\mathstrut \) \(2271\) \(\beta_{16}\mathstrut -\mathstrut \) \(12358\) \(\beta_{15}\mathstrut +\mathstrut \) \(8016\) \(\beta_{14}\mathstrut -\mathstrut \) \(11158\) \(\beta_{13}\mathstrut -\mathstrut \) \(8238\) \(\beta_{12}\mathstrut -\mathstrut \) \(18683\) \(\beta_{11}\mathstrut +\mathstrut \) \(14181\) \(\beta_{10}\mathstrut -\mathstrut \) \(8207\) \(\beta_{9}\mathstrut -\mathstrut \) \(1753\) \(\beta_{8}\mathstrut +\mathstrut \) \(7669\) \(\beta_{7}\mathstrut +\mathstrut \) \(6174\) \(\beta_{6}\mathstrut +\mathstrut \) \(4359\) \(\beta_{5}\mathstrut +\mathstrut \) \(1195\) \(\beta_{4}\mathstrut +\mathstrut \) \(18527\) \(\beta_{3}\mathstrut +\mathstrut \) \(5800\) \(\beta_{2}\mathstrut +\mathstrut \) \(65667\) \(\beta_{1}\mathstrut +\mathstrut \) \(33065\)
\(\nu^{12}\)\(=\)\(29496\) \(\beta_{18}\mathstrut +\mathstrut \) \(19295\) \(\beta_{17}\mathstrut +\mathstrut \) \(5343\) \(\beta_{16}\mathstrut -\mathstrut \) \(53290\) \(\beta_{15}\mathstrut +\mathstrut \) \(50228\) \(\beta_{14}\mathstrut -\mathstrut \) \(9201\) \(\beta_{13}\mathstrut -\mathstrut \) \(42712\) \(\beta_{12}\mathstrut -\mathstrut \) \(51146\) \(\beta_{11}\mathstrut +\mathstrut \) \(29102\) \(\beta_{10}\mathstrut -\mathstrut \) \(27072\) \(\beta_{9}\mathstrut +\mathstrut \) \(2241\) \(\beta_{8}\mathstrut +\mathstrut \) \(17963\) \(\beta_{7}\mathstrut +\mathstrut \) \(9253\) \(\beta_{6}\mathstrut +\mathstrut \) \(6308\) \(\beta_{5}\mathstrut -\mathstrut \) \(12636\) \(\beta_{4}\mathstrut +\mathstrut \) \(55872\) \(\beta_{3}\mathstrut +\mathstrut \) \(53392\) \(\beta_{2}\mathstrut +\mathstrut \) \(128885\) \(\beta_{1}\mathstrut +\mathstrut \) \(174103\)
\(\nu^{13}\)\(=\)\(94228\) \(\beta_{18}\mathstrut +\mathstrut \) \(38386\) \(\beta_{17}\mathstrut -\mathstrut \) \(15205\) \(\beta_{16}\mathstrut -\mathstrut \) \(143737\) \(\beta_{15}\mathstrut +\mathstrut \) \(115928\) \(\beta_{14}\mathstrut -\mathstrut \) \(110631\) \(\beta_{13}\mathstrut -\mathstrut \) \(109371\) \(\beta_{12}\mathstrut -\mathstrut \) \(208329\) \(\beta_{11}\mathstrut +\mathstrut \) \(142341\) \(\beta_{10}\mathstrut -\mathstrut \) \(92230\) \(\beta_{9}\mathstrut -\mathstrut \) \(7003\) \(\beta_{8}\mathstrut +\mathstrut \) \(80490\) \(\beta_{7}\mathstrut +\mathstrut \) \(51731\) \(\beta_{6}\mathstrut +\mathstrut \) \(44603\) \(\beta_{5}\mathstrut +\mathstrut \) \(5169\) \(\beta_{4}\mathstrut +\mathstrut \) \(201521\) \(\beta_{3}\mathstrut +\mathstrut \) \(63756\) \(\beta_{2}\mathstrut +\mathstrut \) \(652905\) \(\beta_{1}\mathstrut +\mathstrut \) \(372147\)
\(\nu^{14}\)\(=\)\(341593\) \(\beta_{18}\mathstrut +\mathstrut \) \(231871\) \(\beta_{17}\mathstrut +\mathstrut \) \(68875\) \(\beta_{16}\mathstrut -\mathstrut \) \(568501\) \(\beta_{15}\mathstrut +\mathstrut \) \(602069\) \(\beta_{14}\mathstrut -\mathstrut \) \(141725\) \(\beta_{13}\mathstrut -\mathstrut \) \(497375\) \(\beta_{12}\mathstrut -\mathstrut \) \(608076\) \(\beta_{11}\mathstrut +\mathstrut \) \(321422\) \(\beta_{10}\mathstrut -\mathstrut \) \(295529\) \(\beta_{9}\mathstrut +\mathstrut \) \(61074\) \(\beta_{8}\mathstrut +\mathstrut \) \(185971\) \(\beta_{7}\mathstrut +\mathstrut \) \(91071\) \(\beta_{6}\mathstrut +\mathstrut \) \(70012\) \(\beta_{5}\mathstrut -\mathstrut \) \(143207\) \(\beta_{4}\mathstrut +\mathstrut \) \(620996\) \(\beta_{3}\mathstrut +\mathstrut \) \(478922\) \(\beta_{2}\mathstrut +\mathstrut \) \(1441982\) \(\beta_{1}\mathstrut +\mathstrut \) \(1709345\)
\(\nu^{15}\)\(=\)\(1112893\) \(\beta_{18}\mathstrut +\mathstrut \) \(521542\) \(\beta_{17}\mathstrut -\mathstrut \) \(63831\) \(\beta_{16}\mathstrut -\mathstrut \) \(1627497\) \(\beta_{15}\mathstrut +\mathstrut \) \(1536252\) \(\beta_{14}\mathstrut -\mathstrut \) \(1119294\) \(\beta_{13}\mathstrut -\mathstrut \) \(1368510\) \(\beta_{12}\mathstrut -\mathstrut \) \(2317456\) \(\beta_{11}\mathstrut +\mathstrut \) \(1435963\) \(\beta_{10}\mathstrut -\mathstrut \) \(1013225\) \(\beta_{9}\mathstrut +\mathstrut \) \(56185\) \(\beta_{8}\mathstrut +\mathstrut \) \(817935\) \(\beta_{7}\mathstrut +\mathstrut \) \(430423\) \(\beta_{6}\mathstrut +\mathstrut \) \(441520\) \(\beta_{5}\mathstrut -\mathstrut \) \(31328\) \(\beta_{4}\mathstrut +\mathstrut \) \(2182651\) \(\beta_{3}\mathstrut +\mathstrut \) \(674506\) \(\beta_{2}\mathstrut +\mathstrut \) \(6568063\) \(\beta_{1}\mathstrut +\mathstrut \) \(4105878\)
\(\nu^{16}\)\(=\)\(3915869\) \(\beta_{18}\mathstrut +\mathstrut \) \(2691446\) \(\beta_{17}\mathstrut +\mathstrut \) \(854084\) \(\beta_{16}\mathstrut -\mathstrut \) \(6080900\) \(\beta_{15}\mathstrut +\mathstrut \) \(7042028\) \(\beta_{14}\mathstrut -\mathstrut \) \(1893310\) \(\beta_{13}\mathstrut -\mathstrut \) \(5696665\) \(\beta_{12}\mathstrut -\mathstrut \) \(7050409\) \(\beta_{11}\mathstrut +\mathstrut \) \(3503369\) \(\beta_{10}\mathstrut -\mathstrut \) \(3212783\) \(\beta_{9}\mathstrut +\mathstrut \) \(1008594\) \(\beta_{8}\mathstrut +\mathstrut \) \(1912114\) \(\beta_{7}\mathstrut +\mathstrut \) \(842610\) \(\beta_{6}\mathstrut +\mathstrut \) \(745765\) \(\beta_{5}\mathstrut -\mathstrut \) \(1580109\) \(\beta_{4}\mathstrut +\mathstrut \) \(6835032\) \(\beta_{3}\mathstrut +\mathstrut \) \(4331414\) \(\beta_{2}\mathstrut +\mathstrut \) \(15876273\) \(\beta_{1}\mathstrut +\mathstrut \) \(17133185\)
\(\nu^{17}\)\(=\)\(12898673\) \(\beta_{18}\mathstrut +\mathstrut \) \(6646398\) \(\beta_{17}\mathstrut +\mathstrut \) \(313034\) \(\beta_{16}\mathstrut -\mathstrut \) \(18144121\) \(\beta_{15}\mathstrut +\mathstrut \) \(19290684\) \(\beta_{14}\mathstrut -\mathstrut \) \(11533401\) \(\beta_{13}\mathstrut -\mathstrut \) \(16492848\) \(\beta_{12}\mathstrut -\mathstrut \) \(25708684\) \(\beta_{11}\mathstrut +\mathstrut \) \(14585142\) \(\beta_{10}\mathstrut -\mathstrut \) \(11014630\) \(\beta_{9}\mathstrut +\mathstrut \) \(1981433\) \(\beta_{8}\mathstrut +\mathstrut \) \(8163606\) \(\beta_{7}\mathstrut +\mathstrut \) \(3523507\) \(\beta_{6}\mathstrut +\mathstrut \) \(4278885\) \(\beta_{5}\mathstrut -\mathstrut \) \(1227289\) \(\beta_{4}\mathstrut +\mathstrut \) \(23596786\) \(\beta_{3}\mathstrut +\mathstrut \) \(6967379\) \(\beta_{2}\mathstrut +\mathstrut \) \(66768800\) \(\beta_{1}\mathstrut +\mathstrut \) \(44764082\)
\(\nu^{18}\)\(=\)\(44557628\) \(\beta_{18}\mathstrut +\mathstrut \) \(30617557\) \(\beta_{17}\mathstrut +\mathstrut \) \(10343134\) \(\beta_{16}\mathstrut -\mathstrut \) \(65241437\) \(\beta_{15}\mathstrut +\mathstrut \) \(81056867\) \(\beta_{14}\mathstrut -\mathstrut \) \(23575127\) \(\beta_{13}\mathstrut -\mathstrut \) \(64539806\) \(\beta_{12}\mathstrut -\mathstrut \) \(80361253\) \(\beta_{11}\mathstrut +\mathstrut \) \(37871708\) \(\beta_{10}\mathstrut -\mathstrut \) \(34891524\) \(\beta_{9}\mathstrut +\mathstrut \) \(14280489\) \(\beta_{8}\mathstrut +\mathstrut \) \(19544704\) \(\beta_{7}\mathstrut +\mathstrut \) \(7352832\) \(\beta_{6}\mathstrut +\mathstrut \) \(7714365\) \(\beta_{5}\mathstrut -\mathstrut \) \(17240032\) \(\beta_{4}\mathstrut +\mathstrut \) \(74799612\) \(\beta_{3}\mathstrut +\mathstrut \) \(39491384\) \(\beta_{2}\mathstrut +\mathstrut \) \(173039514\) \(\beta_{1}\mathstrut +\mathstrut \) \(174525936\)

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
−2.84270
−2.63586
−2.55329
−1.93943
−1.53086
−1.37886
−0.828283
−0.414343
−0.284621
0.154590
0.416989
0.667966
1.62054
1.87845
1.95331
2.30353
2.97835
3.13171
3.30281
0 −2.84270 0 0.245225 0 0.879635 0 5.08095 0
1.2 0 −2.63586 0 −2.63773 0 −2.07266 0 3.94775 0
1.3 0 −2.55329 0 3.69573 0 1.87818 0 3.51931 0
1.4 0 −1.93943 0 0.842053 0 −2.29879 0 0.761401 0
1.5 0 −1.53086 0 2.27832 0 5.10455 0 −0.656480 0
1.6 0 −1.37886 0 −2.19516 0 1.41910 0 −1.09875 0
1.7 0 −0.828283 0 0.378111 0 −3.91567 0 −2.31395 0
1.8 0 −0.414343 0 −2.73706 0 4.71163 0 −2.82832 0
1.9 0 −0.284621 0 1.91371 0 0.801952 0 −2.91899 0
1.10 0 0.154590 0 0.882499 0 −1.05782 0 −2.97610 0
1.11 0 0.416989 0 −2.91551 0 3.12812 0 −2.82612 0
1.12 0 0.667966 0 −3.08795 0 −2.50903 0 −2.55382 0
1.13 0 1.62054 0 4.01966 0 2.03129 0 −0.373838 0
1.14 0 1.87845 0 3.30275 0 4.61837 0 0.528558 0
1.15 0 1.95331 0 −3.65783 0 −1.44461 0 0.815417 0
1.16 0 2.30353 0 0.631218 0 −0.267329 0 2.30625 0
1.17 0 2.97835 0 1.17616 0 3.75385 0 5.87055 0
1.18 0 3.13171 0 3.22705 0 −4.22839 0 6.80761 0
1.19 0 3.30281 0 −1.36124 0 2.46762 0 7.90858 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4028))\):

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