Properties

Label 4028.2.a.d
Level 4028
Weight 2
Character orbit 4028.a
Self dual Yes
Analytic conductor 32.164
Analytic rank 1
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: \(1\)
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \( -\beta_{6} q^{5} \) \( + ( -1 + \beta_{11} ) q^{7} \) \( + ( 1 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \( -\beta_{6} q^{5} \) \( + ( -1 + \beta_{11} ) q^{7} \) \( + ( 1 + \beta_{2} ) q^{9} \) \( + ( -1 - \beta_{4} ) q^{11} \) \( + \beta_{16} q^{13} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{7} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{15} \) \( + ( \beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{13} - \beta_{15} ) q^{17} \) \(- q^{19}\) \( + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{21} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} - \beta_{9} - \beta_{11} + \beta_{13} ) q^{23} \) \( + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{25} \) \( + ( -\beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{16} + \beta_{17} ) q^{27} \) \( + ( -1 + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} - \beta_{11} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{29} \) \( + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{31} \) \( + ( 1 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{16} ) q^{33} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{35} \) \( + ( -1 + \beta_{1} - \beta_{8} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{18} ) q^{37} \) \( + ( -2 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} - \beta_{16} + \beta_{17} ) q^{39} \) \( + ( 2 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{41} \) \( + ( \beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{9} + \beta_{10} - \beta_{13} + \beta_{15} - \beta_{18} ) q^{43} \) \( + ( -2 + 3 \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{16} - \beta_{17} - \beta_{18} ) q^{45} \) \( + ( -1 - \beta_{2} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{18} ) q^{47} \) \( + ( \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{11} + \beta_{13} - \beta_{16} ) q^{49} \) \( + ( -4 - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} + 3 \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{51} \) \(+ q^{53}\) \( + ( -3 + 2 \beta_{1} + 2 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14} - 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{55} \) \( + \beta_{1} q^{57} \) \( + ( -3 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{17} + \beta_{18} ) q^{59} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{8} - \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{16} + 2 \beta_{18} ) q^{61} \) \( + ( -3 + \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{63} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{65} \) \( + ( -2 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{67} \) \( + ( -4 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{10} + \beta_{12} + 4 \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{69} \) \( + ( -\beta_{3} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{16} ) q^{71} \) \( + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{10} + \beta_{11} - 3 \beta_{13} + \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{73} \) \( + ( 1 - 2 \beta_{1} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{14} + \beta_{15} + \beta_{17} ) q^{75} \) \( + ( 2 - \beta_{2} + \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{13} - \beta_{17} - \beta_{18} ) q^{77} \) \( + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{16} ) q^{79} \) \( + ( 1 - 2 \beta_{1} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} + 3 \beta_{16} - \beta_{17} ) q^{81} \) \( + ( -1 + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{13} - \beta_{14} - \beta_{17} + \beta_{18} ) q^{83} \) \( + ( -5 + \beta_{3} - 3 \beta_{4} + \beta_{6} - 3 \beta_{9} + 2 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{85} \) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{14} + 2 \beta_{16} + \beta_{17} - \beta_{18} ) q^{87} \) \( + ( -3 + \beta_{1} - \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{14} + \beta_{17} - 2 \beta_{18} ) q^{89} \) \( + ( -4 + \beta_{1} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} - \beta_{13} + \beta_{17} - \beta_{18} ) q^{91} \) \( + ( -3 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{8} + 2 \beta_{10} - 2 \beta_{13} + 3 \beta_{15} - 2 \beta_{16} + \beta_{17} + \beta_{18} ) q^{93} \) \( + \beta_{6} q^{95} \) \( + ( -3 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{16} + 2 \beta_{17} - \beta_{18} ) q^{97} \) \( + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} + \beta_{16} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(19q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 19q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(19q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 19q^{9} \) \(\mathstrut -\mathstrut 11q^{11} \) \(\mathstrut +\mathstrut q^{13} \) \(\mathstrut -\mathstrut 20q^{15} \) \(\mathstrut -\mathstrut q^{17} \) \(\mathstrut -\mathstrut 19q^{19} \) \(\mathstrut -\mathstrut 10q^{21} \) \(\mathstrut -\mathstrut 16q^{23} \) \(\mathstrut +\mathstrut 21q^{25} \) \(\mathstrut -\mathstrut 4q^{27} \) \(\mathstrut -\mathstrut 9q^{31} \) \(\mathstrut +\mathstrut 7q^{33} \) \(\mathstrut -\mathstrut 25q^{37} \) \(\mathstrut -\mathstrut 25q^{39} \) \(\mathstrut +\mathstrut q^{41} \) \(\mathstrut -\mathstrut 41q^{43} \) \(\mathstrut -\mathstrut 27q^{45} \) \(\mathstrut -\mathstrut 29q^{47} \) \(\mathstrut +\mathstrut 14q^{49} \) \(\mathstrut -\mathstrut 24q^{51} \) \(\mathstrut +\mathstrut 19q^{53} \) \(\mathstrut -\mathstrut 28q^{55} \) \(\mathstrut +\mathstrut 4q^{57} \) \(\mathstrut -\mathstrut 42q^{59} \) \(\mathstrut +\mathstrut q^{61} \) \(\mathstrut -\mathstrut 41q^{63} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut -\mathstrut 41q^{67} \) \(\mathstrut -\mathstrut 25q^{69} \) \(\mathstrut -\mathstrut 20q^{73} \) \(\mathstrut +\mathstrut 11q^{75} \) \(\mathstrut -\mathstrut 19q^{77} \) \(\mathstrut -\mathstrut 38q^{79} \) \(\mathstrut +\mathstrut 23q^{81} \) \(\mathstrut -\mathstrut 36q^{83} \) \(\mathstrut -\mathstrut 58q^{85} \) \(\mathstrut -\mathstrut 30q^{87} \) \(\mathstrut -\mathstrut 25q^{89} \) \(\mathstrut -\mathstrut 55q^{91} \) \(\mathstrut -\mathstrut 38q^{93} \) \(\mathstrut +\mathstrut 2q^{95} \) \(\mathstrut -\mathstrut 13q^{97} \) \(\mathstrut +\mathstrut q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19}\mathstrut -\mathstrut \) \(4\) \(x^{18}\mathstrut -\mathstrut \) \(30\) \(x^{17}\mathstrut +\mathstrut \) \(132\) \(x^{16}\mathstrut +\mathstrut \) \(332\) \(x^{15}\mathstrut -\mathstrut \) \(1714\) \(x^{14}\mathstrut -\mathstrut \) \(1598\) \(x^{13}\mathstrut +\mathstrut \) \(11179\) \(x^{12}\mathstrut +\mathstrut \) \(2544\) \(x^{11}\mathstrut -\mathstrut \) \(38897\) \(x^{10}\mathstrut +\mathstrut \) \(3416\) \(x^{9}\mathstrut +\mathstrut \) \(71354\) \(x^{8}\mathstrut -\mathstrut \) \(10941\) \(x^{7}\mathstrut -\mathstrut \) \(64854\) \(x^{6}\mathstrut +\mathstrut \) \(219\) \(x^{5}\mathstrut +\mathstrut \) \(26515\) \(x^{4}\mathstrut +\mathstrut \) \(5796\) \(x^{3}\mathstrut -\mathstrut \) \(2375\) \(x^{2}\mathstrut -\mathstrut \) \(826\) \(x\mathstrut -\mathstrut \) \(49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\(1746095294249905525\) \(\nu^{18}\mathstrut -\mathstrut \) \(9882149061894242460\) \(\nu^{17}\mathstrut -\mathstrut \) \(43375197057086977459\) \(\nu^{16}\mathstrut +\mathstrut \) \(320664495983429416571\) \(\nu^{15}\mathstrut +\mathstrut \) \(289170464821686701843\) \(\nu^{14}\mathstrut -\mathstrut \) \(4057421456854802221336\) \(\nu^{13}\mathstrut +\mathstrut \) \(855756579969074611934\) \(\nu^{12}\mathstrut +\mathstrut \) \(25399920257422537055929\) \(\nu^{11}\mathstrut -\mathstrut \) \(18076037847766854541528\) \(\nu^{10}\mathstrut -\mathstrut \) \(82627527926263228597648\) \(\nu^{9}\mathstrut +\mathstrut \) \(77255857497718219255495\) \(\nu^{8}\mathstrut +\mathstrut \) \(135462042252905651618449\) \(\nu^{7}\mathstrut -\mathstrut \) \(128499761875924242856430\) \(\nu^{6}\mathstrut -\mathstrut \) \(103779970138095754748039\) \(\nu^{5}\mathstrut +\mathstrut \) \(69513017742215683190015\) \(\nu^{4}\mathstrut +\mathstrut \) \(39595263533206333032434\) \(\nu^{3}\mathstrut -\mathstrut \) \(6989008533635252643771\) \(\nu^{2}\mathstrut -\mathstrut \) \(4164221605907178213362\) \(\nu\mathstrut -\mathstrut \) \(36307996555354849920\)\()/\)\(57367504414897716263\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(2943780406006641765\) \(\nu^{18}\mathstrut +\mathstrut \) \(12694984241530833769\) \(\nu^{17}\mathstrut +\mathstrut \) \(84076917574960256843\) \(\nu^{16}\mathstrut -\mathstrut \) \(413624666729155238549\) \(\nu^{15}\mathstrut -\mathstrut \) \(839866884795173919115\) \(\nu^{14}\mathstrut +\mathstrut \) \(5268285624886140485261\) \(\nu^{13}\mathstrut +\mathstrut \) \(2963377924552730128550\) \(\nu^{12}\mathstrut -\mathstrut \) \(33332936791207512527838\) \(\nu^{11}\mathstrut +\mathstrut \) \(3431892679435488472801\) \(\nu^{10}\mathstrut +\mathstrut \) \(110332407390778069031044\) \(\nu^{9}\mathstrut -\mathstrut \) \(45746760297992265654926\) \(\nu^{8}\mathstrut -\mathstrut \) \(186042123550325361577403\) \(\nu^{7}\mathstrut +\mathstrut \) \(91287776562331314493241\) \(\nu^{6}\mathstrut +\mathstrut \) \(148143090411297334733782\) \(\nu^{5}\mathstrut -\mathstrut \) \(46789161173125655425506\) \(\nu^{4}\mathstrut -\mathstrut \) \(55943767001779800289086\) \(\nu^{3}\mathstrut +\mathstrut \) \(874953622100007023511\) \(\nu^{2}\mathstrut +\mathstrut \) \(5689248964462977037139\) \(\nu\mathstrut +\mathstrut \) \(483921811060713145533\)\()/\)\(57367504414897716263\)
\(\beta_{5}\)\(=\)\((\)\(5367366175361990662\) \(\nu^{18}\mathstrut -\mathstrut \) \(21770149302925042489\) \(\nu^{17}\mathstrut -\mathstrut \) \(157775075461770482860\) \(\nu^{16}\mathstrut +\mathstrut \) \(713001287691853722889\) \(\nu^{15}\mathstrut +\mathstrut \) \(1674137724859097326569\) \(\nu^{14}\mathstrut -\mathstrut \) \(9154097347335590647765\) \(\nu^{13}\mathstrut -\mathstrut \) \(7164617461856096041454\) \(\nu^{12}\mathstrut +\mathstrut \) \(58657910015823290199629\) \(\nu^{11}\mathstrut +\mathstrut \) \(4317230792299849092388\) \(\nu^{10}\mathstrut -\mathstrut \) \(198272510073702647686792\) \(\nu^{9}\mathstrut +\mathstrut \) \(51617732662797384306580\) \(\nu^{8}\mathstrut +\mathstrut \) \(346388332779359323049071\) \(\nu^{7}\mathstrut -\mathstrut \) \(122717780039396478438628\) \(\nu^{6}\mathstrut -\mathstrut \) \(291230576907575657777941\) \(\nu^{5}\mathstrut +\mathstrut \) \(64803520076310100110799\) \(\nu^{4}\mathstrut +\mathstrut \) \(111813032790893018762455\) \(\nu^{3}\mathstrut +\mathstrut \) \(1557379993306038135353\) \(\nu^{2}\mathstrut -\mathstrut \) \(10716774429709574464823\) \(\nu\mathstrut -\mathstrut \) \(1163592902846952508077\)\()/\)\(57367504414897716263\)
\(\beta_{6}\)\(=\)\((\)\(5986540821363048083\) \(\nu^{18}\mathstrut -\mathstrut \) \(26680747861818192613\) \(\nu^{17}\mathstrut -\mathstrut \) \(168178430493991923146\) \(\nu^{16}\mathstrut +\mathstrut \) \(868798848549651489109\) \(\nu^{15}\mathstrut +\mathstrut \) \(1616710294741622158139\) \(\nu^{14}\mathstrut -\mathstrut \) \(11055794589174623184396\) \(\nu^{13}\mathstrut -\mathstrut \) \(4865977009331272204910\) \(\nu^{12}\mathstrut +\mathstrut \) \(69849775976451597633276\) \(\nu^{11}\mathstrut -\mathstrut \) \(14283739980853695730737\) \(\nu^{10}\mathstrut -\mathstrut \) \(230632984523399313959543\) \(\nu^{9}\mathstrut +\mathstrut \) \(116850638684083221365791\) \(\nu^{8}\mathstrut +\mathstrut \) \(387071868444202644915812\) \(\nu^{7}\mathstrut -\mathstrut \) \(224004286752909750898195\) \(\nu^{6}\mathstrut -\mathstrut \) \(304946572937529267890946\) \(\nu^{5}\mathstrut +\mathstrut \) \(121318731949279369464324\) \(\nu^{4}\mathstrut +\mathstrut \) \(112359898898568469548753\) \(\nu^{3}\mathstrut -\mathstrut \) \(7902964447253402551451\) \(\nu^{2}\mathstrut -\mathstrut \) \(10555286174427386953037\) \(\nu\mathstrut -\mathstrut \) \(767982636210474333887\)\()/\)\(57367504414897716263\)
\(\beta_{7}\)\(=\)\((\)\(7634075689117315443\) \(\nu^{18}\mathstrut -\mathstrut \) \(32234704197001569255\) \(\nu^{17}\mathstrut -\mathstrut \) \(218246461204391413265\) \(\nu^{16}\mathstrut +\mathstrut \) \(1048160645118184606647\) \(\nu^{15}\mathstrut +\mathstrut \) \(2182009360343441981983\) \(\nu^{14}\mathstrut -\mathstrut \) \(13310513813794211859868\) \(\nu^{13}\mathstrut -\mathstrut \) \(7685249755764291736826\) \(\nu^{12}\mathstrut +\mathstrut \) \(83812009843926202606782\) \(\nu^{11}\mathstrut -\mathstrut \) \(9376119487851360027860\) \(\nu^{10}\mathstrut -\mathstrut \) \(275051090647484797769281\) \(\nu^{9}\mathstrut +\mathstrut \) \(122805265553987816991443\) \(\nu^{8}\mathstrut +\mathstrut \) \(455968562412609108317925\) \(\nu^{7}\mathstrut -\mathstrut \) \(251510394749040653365510\) \(\nu^{6}\mathstrut -\mathstrut \) \(349870200047557082743712\) \(\nu^{5}\mathstrut +\mathstrut \) \(143986447003899364514933\) \(\nu^{4}\mathstrut +\mathstrut \) \(124479998890880575687817\) \(\nu^{3}\mathstrut -\mathstrut \) \(13988656566371121836817\) \(\nu^{2}\mathstrut -\mathstrut \) \(11787538564722251354047\) \(\nu\mathstrut -\mathstrut \) \(425409066286894832191\)\()/\)\(57367504414897716263\)
\(\beta_{8}\)\(=\)\((\)\(10583488570732583080\) \(\nu^{18}\mathstrut -\mathstrut \) \(46317271424766431009\) \(\nu^{17}\mathstrut -\mathstrut \) \(300153156956505182157\) \(\nu^{16}\mathstrut +\mathstrut \) \(1509736642100354109756\) \(\nu^{15}\mathstrut +\mathstrut \) \(2948870386732726476653\) \(\nu^{14}\mathstrut -\mathstrut \) \(19241747310288014109107\) \(\nu^{13}\mathstrut -\mathstrut \) \(9727120821862602810878\) \(\nu^{12}\mathstrut +\mathstrut \) \(121870727971300884271619\) \(\nu^{11}\mathstrut -\mathstrut \) \(18448630739731778650887\) \(\nu^{10}\mathstrut -\mathstrut \) \(404093377305337244125151\) \(\nu^{9}\mathstrut +\mathstrut \) \(185842642247020067427678\) \(\nu^{8}\mathstrut +\mathstrut \) \(683279953541843260445629\) \(\nu^{7}\mathstrut -\mathstrut \) \(366628352549431740785521\) \(\nu^{6}\mathstrut -\mathstrut \) \(545424751792344385783695\) \(\nu^{5}\mathstrut +\mathstrut \) \(199384122108507089701670\) \(\nu^{4}\mathstrut +\mathstrut \) \(203416584074287209867702\) \(\nu^{3}\mathstrut -\mathstrut \) \(11348298894438304802815\) \(\nu^{2}\mathstrut -\mathstrut \) \(20001792233774411403989\) \(\nu\mathstrut -\mathstrut \) \(1571164249019956057369\)\()/\)\(57367504414897716263\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(11670781974338432578\) \(\nu^{18}\mathstrut +\mathstrut \) \(50707552311051149897\) \(\nu^{17}\mathstrut +\mathstrut \) \(333006147432104940562\) \(\nu^{16}\mathstrut -\mathstrut \) \(1656236244441767709207\) \(\nu^{15}\mathstrut -\mathstrut \) \(3314682900513242189374\) \(\nu^{14}\mathstrut +\mathstrut \) \(21174240238043555130948\) \(\nu^{13}\mathstrut +\mathstrut \) \(11471880999951571487763\) \(\nu^{12}\mathstrut -\mathstrut \) \(134764644450956180041717\) \(\nu^{11}\mathstrut +\mathstrut \) \(16183010811149986450309\) \(\nu^{10}\mathstrut +\mathstrut \) \(450451640075255791543425\) \(\nu^{9}\mathstrut -\mathstrut \) \(194246883387390224925121\) \(\nu^{8}\mathstrut -\mathstrut \) \(772172666021715621775968\) \(\nu^{7}\mathstrut +\mathstrut \) \(395419269034118877063649\) \(\nu^{6}\mathstrut +\mathstrut \) \(629862347828509404003707\) \(\nu^{5}\mathstrut -\mathstrut \) \(225126564890101466278021\) \(\nu^{4}\mathstrut -\mathstrut \) \(236977996666831952475416\) \(\nu^{3}\mathstrut +\mathstrut \) \(17288575867973303529433\) \(\nu^{2}\mathstrut +\mathstrut \) \(22881633776118358856299\) \(\nu\mathstrut +\mathstrut \) \(1423735123658409096612\)\()/\)\(57367504414897716263\)
\(\beta_{10}\)\(=\)\((\)\(12887337244309102310\) \(\nu^{18}\mathstrut -\mathstrut \) \(56320378776851402420\) \(\nu^{17}\mathstrut -\mathstrut \) \(366733316124510233831\) \(\nu^{16}\mathstrut +\mathstrut \) \(1838393784492939905825\) \(\nu^{15}\mathstrut +\mathstrut \) \(3630523757720748227307\) \(\nu^{14}\mathstrut -\mathstrut \) \(23480695927427830562029\) \(\nu^{13}\mathstrut -\mathstrut \) \(12335770759364466632990\) \(\nu^{12}\mathstrut +\mathstrut \) \(149226602100555301111447\) \(\nu^{11}\mathstrut -\mathstrut \) \(19508279410526756576822\) \(\nu^{10}\mathstrut -\mathstrut \) \(497659654230141998572598\) \(\nu^{9}\mathstrut +\mathstrut \) \(217340664863567212748095\) \(\nu^{8}\mathstrut +\mathstrut \) \(850195825097354486442429\) \(\nu^{7}\mathstrut -\mathstrut \) \(433707429116295140456111\) \(\nu^{6}\mathstrut -\mathstrut \) \(691072682416700603098456\) \(\nu^{5}\mathstrut +\mathstrut \) \(235680140090945263367131\) \(\nu^{4}\mathstrut +\mathstrut \) \(261986926110365744568328\) \(\nu^{3}\mathstrut -\mathstrut \) \(11570439594905860465419\) \(\nu^{2}\mathstrut -\mathstrut \) \(26260223104617142553418\) \(\nu\mathstrut -\mathstrut \) \(2227351681199340420998\)\()/\)\(57367504414897716263\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(2774197342220715149\) \(\nu^{18}\mathstrut +\mathstrut \) \(12176056338712085544\) \(\nu^{17}\mathstrut +\mathstrut \) \(78708560151114998678\) \(\nu^{16}\mathstrut -\mathstrut \) \(397380567530873041216\) \(\nu^{15}\mathstrut -\mathstrut \) \(773719932918447123382\) \(\nu^{14}\mathstrut +\mathstrut \) \(5074222031303082478672\) \(\nu^{13}\mathstrut +\mathstrut \) \(2554374730253101640076\) \(\nu^{12}\mathstrut -\mathstrut \) \(32235027301369914602315\) \(\nu^{11}\mathstrut +\mathstrut \) \(4852906604258270437399\) \(\nu^{10}\mathstrut +\mathstrut \) \(107424995269575245141776\) \(\nu^{9}\mathstrut -\mathstrut \) \(49008248157728226951319\) \(\nu^{8}\mathstrut -\mathstrut \) \(183275383715474459093238\) \(\nu^{7}\mathstrut +\mathstrut \) \(97207718232382746830615\) \(\nu^{6}\mathstrut +\mathstrut \) \(148560170897994524266440\) \(\nu^{5}\mathstrut -\mathstrut \) \(53721696104914722775357\) \(\nu^{4}\mathstrut -\mathstrut \) \(56048267335871887838477\) \(\nu^{3}\mathstrut +\mathstrut \) \(3325967357404050000805\) \(\nu^{2}\mathstrut +\mathstrut \) \(5506401536397342947630\) \(\nu\mathstrut +\mathstrut \) \(422929458039533903359\)\()/\)\(8195357773556816609\)
\(\beta_{12}\)\(=\)\((\)\(19791322111485126204\) \(\nu^{18}\mathstrut -\mathstrut \) \(87950799774851853575\) \(\nu^{17}\mathstrut -\mathstrut \) \(558589882809057011560\) \(\nu^{16}\mathstrut +\mathstrut \) \(2868546645954305222135\) \(\nu^{15}\mathstrut +\mathstrut \) \(5426248967727490103921\) \(\nu^{14}\mathstrut -\mathstrut \) \(36591913083644724216675\) \(\nu^{13}\mathstrut -\mathstrut \) \(17060810333536869116116\) \(\nu^{12}\mathstrut +\mathstrut \) \(232074251927807865451673\) \(\nu^{11}\mathstrut -\mathstrut \) \(41705165044881695029030\) \(\nu^{10}\mathstrut -\mathstrut \) \(771230824813105073147062\) \(\nu^{9}\mathstrut +\mathstrut \) \(371603559829635867753852\) \(\nu^{8}\mathstrut +\mathstrut \) \(1309212673190232652360595\) \(\nu^{7}\mathstrut -\mathstrut \) \(725952232729224716384505\) \(\nu^{6}\mathstrut -\mathstrut \) \(1051964580499689009862854\) \(\nu^{5}\mathstrut +\mathstrut \) \(401836188374829621113285\) \(\nu^{4}\mathstrut +\mathstrut \) \(393788704637746081556702\) \(\nu^{3}\mathstrut -\mathstrut \) \(27127912924828904331696\) \(\nu^{2}\mathstrut -\mathstrut \) \(38505505587758768950801\) \(\nu\mathstrut -\mathstrut \) \(2899363062180072381215\)\()/\)\(57367504414897716263\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(22289867140185408278\) \(\nu^{18}\mathstrut +\mathstrut \) \(97050618546137680209\) \(\nu^{17}\mathstrut +\mathstrut \) \(633802788787607721493\) \(\nu^{16}\mathstrut -\mathstrut \) \(3165720527951712937811\) \(\nu^{15}\mathstrut -\mathstrut \) \(6261465310879068713631\) \(\nu^{14}\mathstrut +\mathstrut \) \(40391910531833465482195\) \(\nu^{13}\mathstrut +\mathstrut \) \(21079302661832812973500\) \(\nu^{12}\mathstrut -\mathstrut \) \(256270775630394694065260\) \(\nu^{11}\mathstrut +\mathstrut \) \(35640582461401394679617\) \(\nu^{10}\mathstrut +\mathstrut \) \(852117377936432109055784\) \(\nu^{9}\mathstrut -\mathstrut \) \(383718486771389100286389\) \(\nu^{8}\mathstrut -\mathstrut \) \(1447524651396993508147989\) \(\nu^{7}\mathstrut +\mathstrut \) \(767925470706985433199518\) \(\nu^{6}\mathstrut +\mathstrut \) \(1163341240616084616448015\) \(\nu^{5}\mathstrut -\mathstrut \) \(428362716991615958333398\) \(\nu^{4}\mathstrut -\mathstrut \) \(434094089528405914136182\) \(\nu^{3}\mathstrut +\mathstrut \) \(29895819272271390186238\) \(\nu^{2}\mathstrut +\mathstrut \) \(42072549651667420730118\) \(\nu\mathstrut +\mathstrut \) \(2872552231573007321058\)\()/\)\(57367504414897716263\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(24998369798017516548\) \(\nu^{18}\mathstrut +\mathstrut \) \(109803036871893909119\) \(\nu^{17}\mathstrut +\mathstrut \) \(707647289178142272786\) \(\nu^{16}\mathstrut -\mathstrut \) \(3579202409104159368513\) \(\nu^{15}\mathstrut -\mathstrut \) \(6921202134071797919604\) \(\nu^{14}\mathstrut +\mathstrut \) \(45618730798133385926743\) \(\nu^{13}\mathstrut +\mathstrut \) \(22393926429258917147637\) \(\nu^{12}\mathstrut -\mathstrut \) \(288938484730705145961233\) \(\nu^{11}\mathstrut +\mathstrut \) \(47451143554738106380962\) \(\nu^{10}\mathstrut +\mathstrut \) \(958002424674232127312982\) \(\nu^{9}\mathstrut -\mathstrut \) \(452810635090310303890993\) \(\nu^{8}\mathstrut -\mathstrut \) \(1619413654701392804562061\) \(\nu^{7}\mathstrut +\mathstrut \) \(892123991041871910598122\) \(\nu^{6}\mathstrut +\mathstrut \) \(1291257565269313627230584\) \(\nu^{5}\mathstrut -\mathstrut \) \(495044373763106739194155\) \(\nu^{4}\mathstrut -\mathstrut \) \(480279123762301493691939\) \(\nu^{3}\mathstrut +\mathstrut \) \(35926639666028662100641\) \(\nu^{2}\mathstrut +\mathstrut \) \(47174575583883340369582\) \(\nu\mathstrut +\mathstrut \) \(3061469864996299447894\)\()/\)\(57367504414897716263\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(29681376895996197944\) \(\nu^{18}\mathstrut +\mathstrut \) \(130670739174724825698\) \(\nu^{17}\mathstrut +\mathstrut \) \(840355296619823602257\) \(\nu^{16}\mathstrut -\mathstrut \) \(4261845844975398059130\) \(\nu^{15}\mathstrut -\mathstrut \) \(8222505495437665605209\) \(\nu^{14}\mathstrut +\mathstrut \) \(54366704678379851026705\) \(\nu^{13}\mathstrut +\mathstrut \) \(26650501384434821329586\) \(\nu^{12}\mathstrut -\mathstrut \) \(344834623061650905277647\) \(\nu^{11}\mathstrut +\mathstrut \) \(55934569543284961625347\) \(\nu^{10}\mathstrut +\mathstrut \) \(1146171567520701030543707\) \(\nu^{9}\mathstrut -\mathstrut \) \(536095155374584599520342\) \(\nu^{8}\mathstrut -\mathstrut \) \(1946576197255065457127915\) \(\nu^{7}\mathstrut +\mathstrut \) \(1055431957751715212449711\) \(\nu^{6}\mathstrut +\mathstrut \) \(1566178638821494910920206\) \(\nu^{5}\mathstrut -\mathstrut \) \(581346941904714610628111\) \(\nu^{4}\mathstrut -\mathstrut \) \(588597017763378213232673\) \(\nu^{3}\mathstrut +\mathstrut \) \(37102300925453892343376\) \(\nu^{2}\mathstrut +\mathstrut \) \(58078743700198732636821\) \(\nu\mathstrut +\mathstrut \) \(4320986960026640338519\)\()/\)\(57367504414897716263\)
\(\beta_{16}\)\(=\)\((\)\(32126750325000417763\) \(\nu^{18}\mathstrut -\mathstrut \) \(140598400564189312527\) \(\nu^{17}\mathstrut -\mathstrut \) \(912383701329069111656\) \(\nu^{16}\mathstrut +\mathstrut \) \(4587401677629310734476\) \(\nu^{15}\mathstrut +\mathstrut \) \(8989055740957247679851\) \(\nu^{14}\mathstrut -\mathstrut \) \(58553492280121244504793\) \(\nu^{13}\mathstrut -\mathstrut \) \(29944337197098411648676\) \(\nu^{12}\mathstrut +\mathstrut \) \(371721318395513424412158\) \(\nu^{11}\mathstrut -\mathstrut \) \(53976877019970239794073\) \(\nu^{10}\mathstrut -\mathstrut \) \(1237272726567812173162473\) \(\nu^{9}\mathstrut +\mathstrut \) \(560778203360560864323400\) \(\nu^{8}\mathstrut +\mathstrut \) \(2105826776065409602738968\) \(\nu^{7}\mathstrut -\mathstrut \) \(1116865682365930558280660\) \(\nu^{6}\mathstrut -\mathstrut \) \(1698478842811708781813908\) \(\nu^{5}\mathstrut +\mathstrut \) \(620498805560384532184000\) \(\nu^{4}\mathstrut +\mathstrut \) \(636235023035425463017489\) \(\nu^{3}\mathstrut -\mathstrut \) \(41733388729028138072672\) \(\nu^{2}\mathstrut -\mathstrut \) \(62045186303231683408650\) \(\nu\mathstrut -\mathstrut \) \(4269370263534605129033\)\()/\)\(57367504414897716263\)
\(\beta_{17}\)\(=\)\((\)\(34139576512779733606\) \(\nu^{18}\mathstrut -\mathstrut \) \(148902808973472428057\) \(\nu^{17}\mathstrut -\mathstrut \) \(968844809569255216252\) \(\nu^{16}\mathstrut +\mathstrut \) \(4853538593332586374159\) \(\nu^{15}\mathstrut +\mathstrut \) \(9529404779309219804362\) \(\nu^{14}\mathstrut -\mathstrut \) \(61857820411663542311015\) \(\nu^{13}\mathstrut -\mathstrut \) \(31532631822481732262996\) \(\nu^{12}\mathstrut +\mathstrut \) \(391759239964210643251937\) \(\nu^{11}\mathstrut -\mathstrut \) \(59144568971774230229251\) \(\nu^{10}\mathstrut -\mathstrut \) \(1298635512620329536916196\) \(\nu^{9}\mathstrut +\mathstrut \) \(601711962492870212285946\) \(\nu^{8}\mathstrut +\mathstrut \) \(2193802864638411868169019\) \(\nu^{7}\mathstrut -\mathstrut \) \(1196825848752823800956711\) \(\nu^{6}\mathstrut -\mathstrut \) \(1745435831325722933254093\) \(\nu^{5}\mathstrut +\mathstrut \) \(668892019683462644248792\) \(\nu^{4}\mathstrut +\mathstrut \) \(645175837406879853982374\) \(\nu^{3}\mathstrut -\mathstrut \) \(49686465088446824450421\) \(\nu^{2}\mathstrut -\mathstrut \) \(62369066267108048404051\) \(\nu\mathstrut -\mathstrut \) \(4043661676142879980322\)\()/\)\(57367504414897716263\)
\(\beta_{18}\)\(=\)\((\)\(36723603444103622524\) \(\nu^{18}\mathstrut -\mathstrut \) \(159238822068501653493\) \(\nu^{17}\mathstrut -\mathstrut \) \(1047361719567607781222\) \(\nu^{16}\mathstrut +\mathstrut \) \(5197633440043365121890\) \(\nu^{15}\mathstrut +\mathstrut \) \(10418007751645399158371\) \(\nu^{14}\mathstrut -\mathstrut \) \(66383634490649735076473\) \(\nu^{13}\mathstrut -\mathstrut \) \(36017717369322207106991\) \(\nu^{12}\mathstrut +\mathstrut \) \(421858313389172862318606\) \(\nu^{11}\mathstrut -\mathstrut \) \(50686847457296408089964\) \(\nu^{10}\mathstrut -\mathstrut \) \(1406601328489586758783901\) \(\nu^{9}\mathstrut +\mathstrut \) \(606407207630109272285654\) \(\nu^{8}\mathstrut +\mathstrut \) \(2401546521589485076123624\) \(\nu^{7}\mathstrut -\mathstrut \) \(1224609347834873327463388\) \(\nu^{6}\mathstrut -\mathstrut \) \(1947900714257381645846663\) \(\nu^{5}\mathstrut +\mathstrut \) \(678401614172280322061840\) \(\nu^{4}\mathstrut +\mathstrut \) \(733853893038264699002650\) \(\nu^{3}\mathstrut -\mathstrut \) \(41271629821614059947905\) \(\nu^{2}\mathstrut -\mathstrut \) \(72413223583601264960432\) \(\nu\mathstrut -\mathstrut \) \(5256536700393855125760\)\()/\)\(57367504414897716263\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{17}\mathstrut +\mathstrut \) \(3\) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(28\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{18}\mathstrut -\mathstrut \) \(14\) \(\beta_{17}\mathstrut +\mathstrut \) \(14\) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut -\mathstrut \) \(12\) \(\beta_{14}\mathstrut +\mathstrut \) \(12\) \(\beta_{13}\mathstrut +\mathstrut \) \(14\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(14\) \(\beta_{10}\mathstrut +\mathstrut \) \(13\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(15\) \(\beta_{7}\mathstrut -\mathstrut \) \(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(13\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(56\) \(\beta_{1}\mathstrut -\mathstrut \) \(2\)
\(\nu^{6}\)\(=\)\(-\)\(3\) \(\beta_{18}\mathstrut -\mathstrut \) \(13\) \(\beta_{17}\mathstrut +\mathstrut \) \(51\) \(\beta_{16}\mathstrut -\mathstrut \) \(21\) \(\beta_{15}\mathstrut +\mathstrut \) \(15\) \(\beta_{14}\mathstrut +\mathstrut \) \(29\) \(\beta_{13}\mathstrut +\mathstrut \) \(15\) \(\beta_{11}\mathstrut -\mathstrut \) \(33\) \(\beta_{10}\mathstrut +\mathstrut \) \(14\) \(\beta_{9}\mathstrut +\mathstrut \) \(13\) \(\beta_{8}\mathstrut +\mathstrut \) \(18\) \(\beta_{7}\mathstrut -\mathstrut \) \(18\) \(\beta_{6}\mathstrut +\mathstrut \) \(6\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(84\) \(\beta_{2}\mathstrut -\mathstrut \) \(31\) \(\beta_{1}\mathstrut +\mathstrut \) \(220\)
\(\nu^{7}\)\(=\)\(-\)\(15\) \(\beta_{18}\mathstrut -\mathstrut \) \(160\) \(\beta_{17}\mathstrut +\mathstrut \) \(163\) \(\beta_{16}\mathstrut -\mathstrut \) \(20\) \(\beta_{15}\mathstrut -\mathstrut \) \(123\) \(\beta_{14}\mathstrut +\mathstrut \) \(118\) \(\beta_{13}\mathstrut +\mathstrut \) \(158\) \(\beta_{12}\mathstrut +\mathstrut \) \(27\) \(\beta_{11}\mathstrut -\mathstrut \) \(157\) \(\beta_{10}\mathstrut +\mathstrut \) \(150\) \(\beta_{9}\mathstrut -\mathstrut \) \(19\) \(\beta_{8}\mathstrut +\mathstrut \) \(175\) \(\beta_{7}\mathstrut -\mathstrut \) \(148\) \(\beta_{6}\mathstrut +\mathstrut \) \(192\) \(\beta_{5}\mathstrut +\mathstrut \) \(147\) \(\beta_{4}\mathstrut +\mathstrut \) \(36\) \(\beta_{3}\mathstrut +\mathstrut \) \(25\) \(\beta_{2}\mathstrut +\mathstrut \) \(479\) \(\beta_{1}\mathstrut -\mathstrut \) \(13\)
\(\nu^{8}\)\(=\)\(-\)\(64\) \(\beta_{18}\mathstrut -\mathstrut \) \(143\) \(\beta_{17}\mathstrut +\mathstrut \) \(674\) \(\beta_{16}\mathstrut -\mathstrut \) \(318\) \(\beta_{15}\mathstrut +\mathstrut \) \(175\) \(\beta_{14}\mathstrut +\mathstrut \) \(479\) \(\beta_{13}\mathstrut -\mathstrut \) \(4\) \(\beta_{12}\mathstrut +\mathstrut \) \(195\) \(\beta_{11}\mathstrut -\mathstrut \) \(425\) \(\beta_{10}\mathstrut +\mathstrut \) \(157\) \(\beta_{9}\mathstrut +\mathstrut \) \(135\) \(\beta_{8}\mathstrut +\mathstrut \) \(242\) \(\beta_{7}\mathstrut -\mathstrut \) \(238\) \(\beta_{6}\mathstrut +\mathstrut \) \(126\) \(\beta_{5}\mathstrut +\mathstrut \) \(51\) \(\beta_{4}\mathstrut -\mathstrut \) \(6\) \(\beta_{3}\mathstrut +\mathstrut \) \(809\) \(\beta_{2}\mathstrut -\mathstrut \) \(374\) \(\beta_{1}\mathstrut +\mathstrut \) \(1845\)
\(\nu^{9}\)\(=\)\(-\)\(186\) \(\beta_{18}\mathstrut -\mathstrut \) \(1716\) \(\beta_{17}\mathstrut +\mathstrut \) \(1810\) \(\beta_{16}\mathstrut -\mathstrut \) \(311\) \(\beta_{15}\mathstrut -\mathstrut \) \(1213\) \(\beta_{14}\mathstrut +\mathstrut \) \(1137\) \(\beta_{13}\mathstrut +\mathstrut \) \(1665\) \(\beta_{12}\mathstrut +\mathstrut \) \(460\) \(\beta_{11}\mathstrut -\mathstrut \) \(1668\) \(\beta_{10}\mathstrut +\mathstrut \) \(1645\) \(\beta_{9}\mathstrut -\mathstrut \) \(243\) \(\beta_{8}\mathstrut +\mathstrut \) \(1893\) \(\beta_{7}\mathstrut -\mathstrut \) \(1617\) \(\beta_{6}\mathstrut +\mathstrut \) \(2130\) \(\beta_{5}\mathstrut +\mathstrut \) \(1611\) \(\beta_{4}\mathstrut +\mathstrut \) \(481\) \(\beta_{3}\mathstrut +\mathstrut \) \(403\) \(\beta_{2}\mathstrut +\mathstrut \) \(4269\) \(\beta_{1}\mathstrut +\mathstrut \) \(18\)
\(\nu^{10}\)\(=\)\(-\)\(963\) \(\beta_{18}\mathstrut -\mathstrut \) \(1529\) \(\beta_{17}\mathstrut +\mathstrut \) \(8115\) \(\beta_{16}\mathstrut -\mathstrut \) \(4215\) \(\beta_{15}\mathstrut +\mathstrut \) \(1882\) \(\beta_{14}\mathstrut +\mathstrut \) \(6487\) \(\beta_{13}\mathstrut -\mathstrut \) \(82\) \(\beta_{12}\mathstrut +\mathstrut \) \(2422\) \(\beta_{11}\mathstrut -\mathstrut \) \(5033\) \(\beta_{10}\mathstrut +\mathstrut \) \(1673\) \(\beta_{9}\mathstrut +\mathstrut \) \(1317\) \(\beta_{8}\mathstrut +\mathstrut \) \(2916\) \(\beta_{7}\mathstrut -\mathstrut \) \(2835\) \(\beta_{6}\mathstrut +\mathstrut \) \(1879\) \(\beta_{5}\mathstrut +\mathstrut \) \(876\) \(\beta_{4}\mathstrut -\mathstrut \) \(172\) \(\beta_{3}\mathstrut +\mathstrut \) \(7939\) \(\beta_{2}\mathstrut -\mathstrut \) \(4168\) \(\beta_{1}\mathstrut +\mathstrut \) \(16183\)
\(\nu^{11}\)\(=\)\(-\)\(2222\) \(\beta_{18}\mathstrut -\mathstrut \) \(17931\) \(\beta_{17}\mathstrut +\mathstrut \) \(19781\) \(\beta_{16}\mathstrut -\mathstrut \) \(4373\) \(\beta_{15}\mathstrut -\mathstrut \) \(11876\) \(\beta_{14}\mathstrut +\mathstrut \) \(11262\) \(\beta_{13}\mathstrut +\mathstrut \) \(17055\) \(\beta_{12}\mathstrut +\mathstrut \) \(6505\) \(\beta_{11}\mathstrut -\mathstrut \) \(17514\) \(\beta_{10}\mathstrut +\mathstrut \) \(17530\) \(\beta_{9}\mathstrut -\mathstrut \) \(2659\) \(\beta_{8}\mathstrut +\mathstrut \) \(19893\) \(\beta_{7}\mathstrut -\mathstrut \) \(17406\) \(\beta_{6}\mathstrut +\mathstrut \) \(22998\) \(\beta_{5}\mathstrut +\mathstrut \) \(17433\) \(\beta_{4}\mathstrut +\mathstrut \) \(5754\) \(\beta_{3}\mathstrut +\mathstrut \) \(5465\) \(\beta_{2}\mathstrut +\mathstrut \) \(39070\) \(\beta_{1}\mathstrut +\mathstrut \) \(1667\)
\(\nu^{12}\)\(=\)\(-\)\(12567\) \(\beta_{18}\mathstrut -\mathstrut \) \(16352\) \(\beta_{17}\mathstrut +\mathstrut \) \(93240\) \(\beta_{16}\mathstrut -\mathstrut \) \(51976\) \(\beta_{15}\mathstrut +\mathstrut \) \(19521\) \(\beta_{14}\mathstrut +\mathstrut \) \(79899\) \(\beta_{13}\mathstrut -\mathstrut \) \(1081\) \(\beta_{12}\mathstrut +\mathstrut \) \(29186\) \(\beta_{11}\mathstrut -\mathstrut \) \(57334\) \(\beta_{10}\mathstrut +\mathstrut \) \(17665\) \(\beta_{9}\mathstrut +\mathstrut \) \(12565\) \(\beta_{8}\mathstrut +\mathstrut \) \(33361\) \(\beta_{7}\mathstrut -\mathstrut \) \(32252\) \(\beta_{6}\mathstrut +\mathstrut \) \(24558\) \(\beta_{5}\mathstrut +\mathstrut \) \(12689\) \(\beta_{4}\mathstrut -\mathstrut \) \(3068\) \(\beta_{3}\mathstrut +\mathstrut \) \(78916\) \(\beta_{2}\mathstrut -\mathstrut \) \(44959\) \(\beta_{1}\mathstrut +\mathstrut \) \(146604\)
\(\nu^{13}\)\(=\)\(-\)\(26256\) \(\beta_{18}\mathstrut -\mathstrut \) \(185243\) \(\beta_{17}\mathstrut +\mathstrut \) \(214944\) \(\beta_{16}\mathstrut -\mathstrut \) \(57740\) \(\beta_{15}\mathstrut -\mathstrut \) \(116593\) \(\beta_{14}\mathstrut +\mathstrut \) \(115702\) \(\beta_{13}\mathstrut +\mathstrut \) \(172554\) \(\beta_{12}\mathstrut +\mathstrut \) \(83565\) \(\beta_{11}\mathstrut -\mathstrut \) \(184020\) \(\beta_{10}\mathstrut +\mathstrut \) \(183851\) \(\beta_{9}\mathstrut -\mathstrut \) \(26951\) \(\beta_{8}\mathstrut +\mathstrut \) \(206620\) \(\beta_{7}\mathstrut -\mathstrut \) \(186246\) \(\beta_{6}\mathstrut +\mathstrut \) \(245676\) \(\beta_{5}\mathstrut +\mathstrut \) \(187232\) \(\beta_{4}\mathstrut +\mathstrut \) \(65137\) \(\beta_{3}\mathstrut +\mathstrut \) \(67968\) \(\beta_{2}\mathstrut +\mathstrut \) \(364091\) \(\beta_{1}\mathstrut +\mathstrut \) \(29492\)
\(\nu^{14}\)\(=\)\(-\)\(152326\) \(\beta_{18}\mathstrut -\mathstrut \) \(175877\) \(\beta_{17}\mathstrut +\mathstrut \) \(1042772\) \(\beta_{16}\mathstrut -\mathstrut \) \(612969\) \(\beta_{15}\mathstrut +\mathstrut \) \(198689\) \(\beta_{14}\mathstrut +\mathstrut \) \(933588\) \(\beta_{13}\mathstrut -\mathstrut \) \(11355\) \(\beta_{12}\mathstrut +\mathstrut \) \(343172\) \(\beta_{11}\mathstrut -\mathstrut \) \(639139\) \(\beta_{10}\mathstrut +\mathstrut \) \(187139\) \(\beta_{9}\mathstrut +\mathstrut \) \(118913\) \(\beta_{8}\mathstrut +\mathstrut \) \(371384\) \(\beta_{7}\mathstrut -\mathstrut \) \(358565\) \(\beta_{6}\mathstrut +\mathstrut \) \(300573\) \(\beta_{5}\mathstrut +\mathstrut \) \(167529\) \(\beta_{4}\mathstrut -\mathstrut \) \(44284\) \(\beta_{3}\mathstrut +\mathstrut \) \(792232\) \(\beta_{2}\mathstrut -\mathstrut \) \(477330\) \(\beta_{1}\mathstrut +\mathstrut \) \(1360642\)
\(\nu^{15}\)\(=\)\(-\)\(307369\) \(\beta_{18}\mathstrut -\mathstrut \) \(1905027\) \(\beta_{17}\mathstrut +\mathstrut \) \(2330726\) \(\beta_{16}\mathstrut -\mathstrut \) \(728959\) \(\beta_{15}\mathstrut -\mathstrut \) \(1150881\) \(\beta_{14}\mathstrut +\mathstrut \) \(1226306\) \(\beta_{13}\mathstrut +\mathstrut \) \(1737498\) \(\beta_{12}\mathstrut +\mathstrut \) \(1014949\) \(\beta_{11}\mathstrut -\mathstrut \) \(1940848\) \(\beta_{10}\mathstrut +\mathstrut \) \(1912096\) \(\beta_{9}\mathstrut -\mathstrut \) \(262265\) \(\beta_{8}\mathstrut +\mathstrut \) \(2137376\) \(\beta_{7}\mathstrut -\mathstrut \) \(1986879\) \(\beta_{6}\mathstrut +\mathstrut \) \(2611948\) \(\beta_{5}\mathstrut +\mathstrut \) \(2000591\) \(\beta_{4}\mathstrut +\mathstrut \) \(714199\) \(\beta_{3}\mathstrut +\mathstrut \) \(805094\) \(\beta_{2}\mathstrut +\mathstrut \) \(3437395\) \(\beta_{1}\mathstrut +\mathstrut \) \(401589\)
\(\nu^{16}\)\(=\)\(-\)\(1768769\) \(\beta_{18}\mathstrut -\mathstrut \) \(1901705\) \(\beta_{17}\mathstrut +\mathstrut \) \(11466798\) \(\beta_{16}\mathstrut -\mathstrut \) \(7018125\) \(\beta_{15}\mathstrut +\mathstrut \) \(2000621\) \(\beta_{14}\mathstrut +\mathstrut \) \(10563867\) \(\beta_{13}\mathstrut -\mathstrut \) \(98886\) \(\beta_{12}\mathstrut +\mathstrut \) \(3954773\) \(\beta_{11}\mathstrut -\mathstrut \) \(7028045\) \(\beta_{10}\mathstrut +\mathstrut \) \(1995140\) \(\beta_{9}\mathstrut +\mathstrut \) \(1123630\) \(\beta_{8}\mathstrut +\mathstrut \) \(4071289\) \(\beta_{7}\mathstrut -\mathstrut \) \(3937297\) \(\beta_{6}\mathstrut +\mathstrut \) \(3542969\) \(\beta_{5}\mathstrut +\mathstrut \) \(2089967\) \(\beta_{4}\mathstrut -\mathstrut \) \(568040\) \(\beta_{3}\mathstrut +\mathstrut \) \(8017265\) \(\beta_{2}\mathstrut -\mathstrut \) \(5022238\) \(\beta_{1}\mathstrut +\mathstrut \) \(12868814\)
\(\nu^{17}\)\(=\)\(-\)\(3559419\) \(\beta_{18}\mathstrut -\mathstrut \) \(19569956\) \(\beta_{17}\mathstrut +\mathstrut \) \(25251168\) \(\beta_{16}\mathstrut -\mathstrut \) \(8902203\) \(\beta_{15}\mathstrut -\mathstrut \) \(11423098\) \(\beta_{14}\mathstrut +\mathstrut \) \(13286046\) \(\beta_{13}\mathstrut +\mathstrut \) \(17479989\) \(\beta_{12}\mathstrut +\mathstrut \) \(11897330\) \(\beta_{11}\mathstrut -\mathstrut \) \(20550023\) \(\beta_{10}\mathstrut +\mathstrut \) \(19808271\) \(\beta_{9}\mathstrut -\mathstrut \) \(2496977\) \(\beta_{8}\mathstrut +\mathstrut \) \(22101561\) \(\beta_{7}\mathstrut -\mathstrut \) \(21154091\) \(\beta_{6}\mathstrut +\mathstrut \) \(27701411\) \(\beta_{5}\mathstrut +\mathstrut \) \(21299332\) \(\beta_{4}\mathstrut +\mathstrut \) \(7674698\) \(\beta_{3}\mathstrut +\mathstrut \) \(9258227\) \(\beta_{2}\mathstrut +\mathstrut \) \(32774287\) \(\beta_{1}\mathstrut +\mathstrut \) \(4915976\)
\(\nu^{18}\)\(=\)\(-\)\(19994652\) \(\beta_{18}\mathstrut -\mathstrut \) \(20638398\) \(\beta_{17}\mathstrut +\mathstrut \) \(124696919\) \(\beta_{16}\mathstrut -\mathstrut \) \(78717542\) \(\beta_{15}\mathstrut +\mathstrut \) \(20014792\) \(\beta_{14}\mathstrut +\mathstrut \) \(117066253\) \(\beta_{13}\mathstrut -\mathstrut \) \(660934\) \(\beta_{12}\mathstrut +\mathstrut \) \(44843477\) \(\beta_{11}\mathstrut -\mathstrut \) \(76560272\) \(\beta_{10}\mathstrut +\mathstrut \) \(21401468\) \(\beta_{9}\mathstrut +\mathstrut \) \(10640001\) \(\beta_{8}\mathstrut +\mathstrut \) \(44228607\) \(\beta_{7}\mathstrut -\mathstrut \) \(42930433\) \(\beta_{6}\mathstrut +\mathstrut \) \(40781914\) \(\beta_{5}\mathstrut +\mathstrut \) \(25117599\) \(\beta_{4}\mathstrut -\mathstrut \) \(6767736\) \(\beta_{3}\mathstrut +\mathstrut \) \(81676523\) \(\beta_{2}\mathstrut -\mathstrut \) \(52525018\) \(\beta_{1}\mathstrut +\mathstrut \) \(123576982\)

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
3.25955
2.97522
2.44269
2.40459
2.05170
1.96979
1.76613
1.01911
0.383819
−0.0800487
−0.346400
−0.365086
−0.537191
−0.755832
−1.67288
−1.94972
−2.53438
−2.85652
−3.17454
0 −3.25955 0 3.01882 0 1.05463 0 7.62467 0
1.2 0 −2.97522 0 0.840694 0 −4.22061 0 5.85193 0
1.3 0 −2.44269 0 −3.59133 0 0.459217 0 2.96674 0
1.4 0 −2.40459 0 −0.489610 0 4.26722 0 2.78204 0
1.5 0 −2.05170 0 2.67326 0 0.688434 0 1.20949 0
1.6 0 −1.96979 0 −1.29651 0 −5.21755 0 0.880060 0
1.7 0 −1.76613 0 −1.36988 0 0.174249 0 0.119213 0
1.8 0 −1.01911 0 3.39604 0 −2.78127 0 −1.96142 0
1.9 0 −0.383819 0 −4.10226 0 0.287206 0 −2.85268 0
1.10 0 0.0800487 0 0.442813 0 2.60518 0 −2.99359 0
1.11 0 0.346400 0 0.720634 0 −2.22237 0 −2.88001 0
1.12 0 0.365086 0 1.83829 0 2.27559 0 −2.86671 0
1.13 0 0.537191 0 −2.35184 0 −1.00979 0 −2.71143 0
1.14 0 0.755832 0 4.26919 0 −2.09147 0 −2.42872 0
1.15 0 1.67288 0 −0.778334 0 3.24553 0 −0.201474 0
1.16 0 1.94972 0 1.42435 0 −4.01869 0 0.801405 0
1.17 0 2.53438 0 −2.27936 0 1.27718 0 3.42306 0
1.18 0 2.85652 0 −0.0885731 0 −1.03776 0 5.15970 0
1.19 0 3.17454 0 −4.27638 0 −4.73492 0 7.07773 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4028))\):

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