Properties

Label 4028.2.a.c
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \( -\beta_{10} q^{5} \) \( + ( -1 + \beta_{8} ) q^{7} \) \( + ( 1 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \( -\beta_{10} q^{5} \) \( + ( -1 + \beta_{8} ) q^{7} \) \( + ( 1 + \beta_{2} ) q^{9} \) \( + \beta_{12} q^{11} \) \( + ( -1 + \beta_{13} ) q^{13} \) \( + ( -2 \beta_{2} - \beta_{3} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{14} + \beta_{17} ) q^{15} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{13} - \beta_{17} ) q^{17} \) \(+ q^{19}\) \( + ( -1 + 2 \beta_{1} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} + \beta_{14} ) q^{21} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{23} \) \( + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} - 2 \beta_{17} + \beta_{18} ) q^{25} \) \( + ( -1 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{7} - 3 \beta_{9} - 2 \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{27} \) \( + ( -\beta_{3} - \beta_{8} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} ) q^{29} \) \( + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{15} + 2 \beta_{17} - \beta_{18} ) q^{31} \) \( + ( 2 \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{11} - 2 \beta_{12} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{33} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{35} \) \( + ( -1 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{17} + \beta_{18} ) q^{37} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{17} + \beta_{18} ) q^{39} \) \( + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{17} ) q^{41} \) \( + ( -3 - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{15} + \beta_{17} - 2 \beta_{18} ) q^{43} \) \( + ( 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{45} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{3} - \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{13} + \beta_{16} - \beta_{17} ) q^{47} \) \( + ( 1 - \beta_{5} - \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{49} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} + 2 \beta_{17} - \beta_{18} ) q^{51} \) \(- q^{53}\) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{55} \) \( -\beta_{1} q^{57} \) \( + ( -3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{12} + 3 \beta_{13} + \beta_{15} - \beta_{16} + \beta_{18} ) q^{59} \) \( + ( -2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{14} - \beta_{15} + \beta_{17} - \beta_{18} ) q^{61} \) \( + ( -3 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} + 3 \beta_{12} + \beta_{15} - 2 \beta_{17} + \beta_{18} ) q^{63} \) \( + ( -3 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} - \beta_{12} + 2 \beta_{13} - 2 \beta_{15} - \beta_{16} + 3 \beta_{17} - 2 \beta_{18} ) q^{65} \) \( + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{14} - \beta_{16} ) q^{67} \) \( + ( 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{8} - \beta_{10} + 2 \beta_{12} + \beta_{13} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{69} \) \( + ( -1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - \beta_{10} + 2 \beta_{13} - \beta_{14} + \beta_{17} + \beta_{18} ) q^{71} \) \( + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{15} - \beta_{16} - 3 \beta_{17} + \beta_{18} ) q^{73} \) \( + ( -2 + \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{12} + \beta_{14} + \beta_{16} - \beta_{17} ) q^{75} \) \( + ( -2 + \beta_{2} - \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{11} + 2 \beta_{13} - \beta_{15} + \beta_{17} - \beta_{18} ) q^{77} \) \( + ( 1 - 2 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} - \beta_{12} - 3 \beta_{14} + \beta_{15} + 2 \beta_{16} + 2 \beta_{17} + \beta_{18} ) q^{79} \) \( + ( 4 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} - 3 \beta_{13} + \beta_{16} - \beta_{17} ) q^{81} \) \( + ( 1 + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} + 2 \beta_{11} + \beta_{14} + 2 \beta_{15} + \beta_{18} ) q^{83} \) \( + ( -2 + 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{17} - \beta_{18} ) q^{85} \) \( + ( -3 + 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - \beta_{5} + \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 3 \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} - 4 \beta_{17} + 2 \beta_{18} ) q^{87} \) \( + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{10} + \beta_{11} - \beta_{14} + \beta_{15} ) q^{89} \) \( + ( 1 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{7} - 5 \beta_{8} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} + 3 \beta_{17} - \beta_{18} ) q^{91} \) \( + ( -4 + \beta_{2} + \beta_{3} + \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} - \beta_{18} ) q^{93} \) \( -\beta_{10} q^{95} \) \( + ( 1 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{11} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{97} \) \( + ( -1 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + 3 \beta_{12} + 2 \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(19q \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 22q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(19q \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 22q^{9} \) \(\mathstrut -\mathstrut 3q^{11} \) \(\mathstrut -\mathstrut 23q^{13} \) \(\mathstrut -\mathstrut 18q^{15} \) \(\mathstrut -\mathstrut 7q^{17} \) \(\mathstrut +\mathstrut 19q^{19} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut 6q^{23} \) \(\mathstrut +\mathstrut 18q^{25} \) \(\mathstrut -\mathstrut 17q^{27} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut -\mathstrut 30q^{31} \) \(\mathstrut -\mathstrut 10q^{33} \) \(\mathstrut -\mathstrut q^{35} \) \(\mathstrut -\mathstrut 31q^{37} \) \(\mathstrut +\mathstrut 5q^{39} \) \(\mathstrut -\mathstrut 15q^{41} \) \(\mathstrut -\mathstrut 29q^{43} \) \(\mathstrut +\mathstrut 6q^{45} \) \(\mathstrut -\mathstrut 18q^{47} \) \(\mathstrut +\mathstrut 23q^{49} \) \(\mathstrut -\mathstrut 5q^{51} \) \(\mathstrut -\mathstrut 19q^{53} \) \(\mathstrut -\mathstrut 19q^{55} \) \(\mathstrut -\mathstrut 5q^{57} \) \(\mathstrut +\mathstrut 8q^{59} \) \(\mathstrut -\mathstrut 4q^{61} \) \(\mathstrut -\mathstrut 64q^{63} \) \(\mathstrut -\mathstrut 26q^{65} \) \(\mathstrut -\mathstrut 62q^{67} \) \(\mathstrut +\mathstrut 3q^{69} \) \(\mathstrut -\mathstrut 17q^{71} \) \(\mathstrut +\mathstrut q^{73} \) \(\mathstrut -\mathstrut 40q^{75} \) \(\mathstrut -\mathstrut 14q^{77} \) \(\mathstrut -\mathstrut 28q^{79} \) \(\mathstrut +\mathstrut 11q^{81} \) \(\mathstrut +\mathstrut 4q^{83} \) \(\mathstrut -\mathstrut 31q^{85} \) \(\mathstrut -\mathstrut 20q^{87} \) \(\mathstrut +\mathstrut 33q^{89} \) \(\mathstrut -\mathstrut 29q^{91} \) \(\mathstrut -\mathstrut 59q^{93} \) \(\mathstrut -\mathstrut 5q^{95} \) \(\mathstrut +\mathstrut 5q^{97} \) \(\mathstrut -\mathstrut 48q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19}\mathstrut -\mathstrut \) \(5\) \(x^{18}\mathstrut -\mathstrut \) \(27\) \(x^{17}\mathstrut +\mathstrut \) \(161\) \(x^{16}\mathstrut +\mathstrut \) \(253\) \(x^{15}\mathstrut -\mathstrut \) \(2103\) \(x^{14}\mathstrut -\mathstrut \) \(683\) \(x^{13}\mathstrut +\mathstrut \) \(14442\) \(x^{12}\mathstrut -\mathstrut \) \(4144\) \(x^{11}\mathstrut -\mathstrut \) \(56325\) \(x^{10}\mathstrut +\mathstrut \) \(37245\) \(x^{9}\mathstrut +\mathstrut \) \(124233\) \(x^{8}\mathstrut -\mathstrut \) \(117486\) \(x^{7}\mathstrut -\mathstrut \) \(140498\) \(x^{6}\mathstrut +\mathstrut \) \(176987\) \(x^{5}\mathstrut +\mathstrut \) \(54044\) \(x^{4}\mathstrut -\mathstrut \) \(115318\) \(x^{3}\mathstrut +\mathstrut \) \(16295\) \(x^{2}\mathstrut +\mathstrut \) \(17097\) \(x\mathstrut -\mathstrut \) \(4088\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(230932252029763600\) \(\nu^{18}\mathstrut +\mathstrut \) \(771352569808624807\) \(\nu^{17}\mathstrut +\mathstrut \) \(7517636986446276940\) \(\nu^{16}\mathstrut -\mathstrut \) \(24699782190210839422\) \(\nu^{15}\mathstrut -\mathstrut \) \(99544755445357807412\) \(\nu^{14}\mathstrut +\mathstrut \) \(320463374700700034614\) \(\nu^{13}\mathstrut +\mathstrut \) \(691930720592944211094\) \(\nu^{12}\mathstrut -\mathstrut \) \(2188724235972491204649\) \(\nu^{11}\mathstrut -\mathstrut \) \(2695926299077568146661\) \(\nu^{10}\mathstrut +\mathstrut \) \(8556776683472723020036\) \(\nu^{9}\mathstrut +\mathstrut \) \(5690315242078193294026\) \(\nu^{8}\mathstrut -\mathstrut \) \(19368333579818318722637\) \(\nu^{7}\mathstrut -\mathstrut \) \(5213664175573591564835\) \(\nu^{6}\mathstrut +\mathstrut \) \(24094476382138391977005\) \(\nu^{5}\mathstrut -\mathstrut \) \(685712570418515344286\) \(\nu^{4}\mathstrut -\mathstrut \) \(13946465697384201972598\) \(\nu^{3}\mathstrut +\mathstrut \) \(3418786095418652393640\) \(\nu^{2}\mathstrut +\mathstrut \) \(2029605603818657467720\) \(\nu\mathstrut -\mathstrut \) \(559000528160505516701\)\()/\)\(1020617365548765537\)
\(\beta_{4}\)\(=\)\((\)\(875528915528587834\) \(\nu^{18}\mathstrut -\mathstrut \) \(2945866034282633359\) \(\nu^{17}\mathstrut -\mathstrut \) \(28441627765216453390\) \(\nu^{16}\mathstrut +\mathstrut \) \(94381307734636366351\) \(\nu^{15}\mathstrut +\mathstrut \) \(375454115240758987523\) \(\nu^{14}\mathstrut -\mathstrut \) \(1225261283186139166069\) \(\nu^{13}\mathstrut -\mathstrut \) \(2597845709815004915076\) \(\nu^{12}\mathstrut +\mathstrut \) \(8373044592396267831255\) \(\nu^{11}\mathstrut +\mathstrut \) \(10049173965293108768405\) \(\nu^{10}\mathstrut -\mathstrut \) \(32745652666429505413411\) \(\nu^{9}\mathstrut -\mathstrut \) \(20932286542992443572798\) \(\nu^{8}\mathstrut +\mathstrut \) \(74113858854044883613301\) \(\nu^{7}\mathstrut +\mathstrut \) \(18460104059299133986604\) \(\nu^{6}\mathstrut -\mathstrut \) \(92132517355351897342965\) \(\nu^{5}\mathstrut +\mathstrut \) \(3924111297750415041182\) \(\nu^{4}\mathstrut +\mathstrut \) \(53271141345181872007837\) \(\nu^{3}\mathstrut -\mathstrut \) \(13497952352124495714738\) \(\nu^{2}\mathstrut -\mathstrut \) \(7785439224171119199046\) \(\nu\mathstrut +\mathstrut \) \(2180611944454599942083\)\()/\)\(1020617365548765537\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(354590216384385574\) \(\nu^{18}\mathstrut +\mathstrut \) \(1193212023299130290\) \(\nu^{17}\mathstrut +\mathstrut \) \(11515854784202914891\) \(\nu^{16}\mathstrut -\mathstrut \) \(38217006504532794497\) \(\nu^{15}\mathstrut -\mathstrut \) \(151979838020528594576\) \(\nu^{14}\mathstrut +\mathstrut \) \(495946530917679222852\) \(\nu^{13}\mathstrut +\mathstrut \) \(1051378356346381623577\) \(\nu^{12}\mathstrut -\mathstrut \) \(3387616336327301966957\) \(\nu^{11}\mathstrut -\mathstrut \) \(4067037719755339521505\) \(\nu^{10}\mathstrut +\mathstrut \) \(13241702215893487137163\) \(\nu^{9}\mathstrut +\mathstrut \) \(8476128656919013795683\) \(\nu^{8}\mathstrut -\mathstrut \) \(29953839378652224181147\) \(\nu^{7}\mathstrut -\mathstrut \) \(7496065746793099537451\) \(\nu^{6}\mathstrut +\mathstrut \) \(37215617224851103731443\) \(\nu^{5}\mathstrut -\mathstrut \) \(1538590024286327600901\) \(\nu^{4}\mathstrut -\mathstrut \) \(21507554118729812700555\) \(\nu^{3}\mathstrut +\mathstrut \) \(5425974211475896529426\) \(\nu^{2}\mathstrut +\mathstrut \) \(3143855434471526581062\) \(\nu\mathstrut -\mathstrut \) \(876861559834659737548\)\()/\)\(340205788516255179\)
\(\beta_{6}\)\(=\)\((\)\(424763042929344671\) \(\nu^{18}\mathstrut -\mathstrut \) \(1422271446904556952\) \(\nu^{17}\mathstrut -\mathstrut \) \(13825062070214422379\) \(\nu^{16}\mathstrut +\mathstrut \) \(45588979619200733250\) \(\nu^{15}\mathstrut +\mathstrut \) \(182942762604429687361\) \(\nu^{14}\mathstrut -\mathstrut \) \(592165637902043239231\) \(\nu^{13}\mathstrut -\mathstrut \) \(1269627646876099575979\) \(\nu^{12}\mathstrut +\mathstrut \) \(4049280450544040165240\) \(\nu^{11}\mathstrut +\mathstrut \) \(4930560933092364936771\) \(\nu^{10}\mathstrut -\mathstrut \) \(15847871506981178564528\) \(\nu^{9}\mathstrut -\mathstrut \) \(10333187508119749146331\) \(\nu^{8}\mathstrut +\mathstrut \) \(35899535980524742129167\) \(\nu^{7}\mathstrut +\mathstrut \) \(9260123456218507240234\) \(\nu^{6}\mathstrut -\mathstrut \) \(44670836748007913730407\) \(\nu^{5}\mathstrut +\mathstrut \) \(1659333251477190613799\) \(\nu^{4}\mathstrut +\mathstrut \) \(25853779804305224834854\) \(\nu^{3}\mathstrut -\mathstrut \) \(6483453041824550668805\) \(\nu^{2}\mathstrut -\mathstrut \) \(3777271293160294271680\) \(\nu\mathstrut +\mathstrut \) \(1054539004068349328928\)\()/\)\(340205788516255179\)
\(\beta_{7}\)\(=\)\((\)\(448953286040227951\) \(\nu^{18}\mathstrut -\mathstrut \) \(1506284609611833844\) \(\nu^{17}\mathstrut -\mathstrut \) \(14606018616498591766\) \(\nu^{16}\mathstrut +\mathstrut \) \(48294761928196158835\) \(\nu^{15}\mathstrut +\mathstrut \) \(193157170064257025927\) \(\nu^{14}\mathstrut -\mathstrut \) \(627526426707466350334\) \(\nu^{13}\mathstrut -\mathstrut \) \(1339318450802426804415\) \(\nu^{12}\mathstrut +\mathstrut \) \(4292894423206683218979\) \(\nu^{11}\mathstrut +\mathstrut \) \(5194073468977857664784\) \(\nu^{10}\mathstrut -\mathstrut \) \(16809593509175084354404\) \(\nu^{9}\mathstrut -\mathstrut \) \(10857950095125161190265\) \(\nu^{8}\mathstrut +\mathstrut \) \(38098792288020567015341\) \(\nu^{7}\mathstrut +\mathstrut \) \(9655200960637739189294\) \(\nu^{6}\mathstrut -\mathstrut \) \(47434147415442238691622\) \(\nu^{5}\mathstrut +\mathstrut \) \(1902372273823297919282\) \(\nu^{4}\mathstrut +\mathstrut \) \(27467030702365949468806\) \(\nu^{3}\mathstrut -\mathstrut \) \(6930820499387568470826\) \(\nu^{2}\mathstrut -\mathstrut \) \(4012879229066312890531\) \(\nu\mathstrut +\mathstrut \) \(1122963597203859608204\)\()/\)\(340205788516255179\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(1352757971844910124\) \(\nu^{18}\mathstrut +\mathstrut \) \(4546829191032612965\) \(\nu^{17}\mathstrut +\mathstrut \) \(43958121862800815375\) \(\nu^{16}\mathstrut -\mathstrut \) \(145666372513542861773\) \(\nu^{15}\mathstrut -\mathstrut \) \(580531058423908743019\) \(\nu^{14}\mathstrut +\mathstrut \) \(1890904205523604106279\) \(\nu^{13}\mathstrut +\mathstrut \) \(4019202283259812456725\) \(\nu^{12}\mathstrut -\mathstrut \) \(12920575027075918284777\) \(\nu^{11}\mathstrut -\mathstrut \) \(15561347823271401992551\) \(\nu^{10}\mathstrut +\mathstrut \) \(50525013113390133253205\) \(\nu^{9}\mathstrut +\mathstrut \) \(32466710694107373188738\) \(\nu^{8}\mathstrut -\mathstrut \) \(114345660884479411286191\) \(\nu^{7}\mathstrut -\mathstrut \) \(28765709569083740736907\) \(\nu^{6}\mathstrut +\mathstrut \) \(142147026963392467635678\) \(\nu^{5}\mathstrut -\mathstrut \) \(5839122245555926977304\) \(\nu^{4}\mathstrut -\mathstrut \) \(82198564402970606687183\) \(\nu^{3}\mathstrut +\mathstrut \) \(20790967687992497367723\) \(\nu^{2}\mathstrut +\mathstrut \) \(12008964709865310368579\) \(\nu\mathstrut -\mathstrut \) \(3367653529973470405570\)\()/\)\(1020617365548765537\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(468681528281495967\) \(\nu^{18}\mathstrut +\mathstrut \) \(1571097931032411689\) \(\nu^{17}\mathstrut +\mathstrut \) \(15245015657238045153\) \(\nu^{16}\mathstrut -\mathstrut \) \(50347397246559756989\) \(\nu^{15}\mathstrut -\mathstrut \) \(201565543479763955640\) \(\nu^{14}\mathstrut +\mathstrut \) \(653770006309333434286\) \(\nu^{13}\mathstrut +\mathstrut \) \(1397328944188422045536\) \(\nu^{12}\mathstrut -\mathstrut \) \(4468742364147898555525\) \(\nu^{11}\mathstrut -\mathstrut \) \(5418317931607116439915\) \(\nu^{10}\mathstrut +\mathstrut \) \(17480772177635977906707\) \(\nu^{9}\mathstrut +\mathstrut \) \(11329183922870921688688\) \(\nu^{8}\mathstrut -\mathstrut \) \(39574683632789554874854\) \(\nu^{7}\mathstrut -\mathstrut \) \(10096955513201770033524\) \(\nu^{6}\mathstrut +\mathstrut \) \(49210397307649992951946\) \(\nu^{5}\mathstrut -\mathstrut \) \(1914034028660342209205\) \(\nu^{4}\mathstrut -\mathstrut \) \(28461272687761586169598\) \(\nu^{3}\mathstrut +\mathstrut \) \(7162213491645045692482\) \(\nu^{2}\mathstrut +\mathstrut \) \(4156065193005307389205\) \(\nu\mathstrut -\mathstrut \) \(1162504230026762236243\)\()/\)\(340205788516255179\)
\(\beta_{10}\)\(=\)\((\)\(562129532956561075\) \(\nu^{18}\mathstrut -\mathstrut \) \(1884412109942047461\) \(\nu^{17}\mathstrut -\mathstrut \) \(18288938794118670352\) \(\nu^{16}\mathstrut +\mathstrut \) \(60404777319665689905\) \(\nu^{15}\mathstrut +\mathstrut \) \(241880665108478187509\) \(\nu^{14}\mathstrut -\mathstrut \) \(784661186164824524036\) \(\nu^{13}\mathstrut -\mathstrut \) \(1677357692806188201260\) \(\nu^{12}\mathstrut +\mathstrut \) \(5366078884174232700148\) \(\nu^{11}\mathstrut +\mathstrut \) \(6506283809211215397069\) \(\nu^{10}\mathstrut -\mathstrut \) \(21004181895593013804073\) \(\nu^{9}\mathstrut -\mathstrut \) \(13606308175872567119777\) \(\nu^{8}\mathstrut +\mathstrut \) \(47587935912139108988382\) \(\nu^{7}\mathstrut +\mathstrut \) \(12115464462762931679411\) \(\nu^{6}\mathstrut -\mathstrut \) \(59227527613831439391730\) \(\nu^{5}\mathstrut +\mathstrut \) \(2346179152438763354344\) \(\nu^{4}\mathstrut +\mathstrut \) \(34287271839024809236385\) \(\nu^{3}\mathstrut -\mathstrut \) \(8655422383362684278899\) \(\nu^{2}\mathstrut -\mathstrut \) \(5011081694545955018021\) \(\nu\mathstrut +\mathstrut \) \(1404718427564086952373\)\()/\)\(340205788516255179\)
\(\beta_{11}\)\(=\)\((\)\(85119069142506145\) \(\nu^{18}\mathstrut -\mathstrut \) \(285165608490116031\) \(\nu^{17}\mathstrut -\mathstrut \) \(2770267905550830082\) \(\nu^{16}\mathstrut +\mathstrut \) \(9142366611591555024\) \(\nu^{15}\mathstrut +\mathstrut \) \(36652693546506143903\) \(\nu^{14}\mathstrut -\mathstrut \) \(118779762545868164951\) \(\nu^{13}\mathstrut -\mathstrut \) \(254294963984078886713\) \(\nu^{12}\mathstrut +\mathstrut \) \(812441324135528233288\) \(\nu^{11}\mathstrut +\mathstrut \) \(986998703102590405530\) \(\nu^{10}\mathstrut -\mathstrut \) \(3180598889868353209927\) \(\nu^{9}\mathstrut -\mathstrut \) \(2066223278519077476896\) \(\nu^{8}\mathstrut +\mathstrut \) \(7206983780436683612706\) \(\nu^{7}\mathstrut +\mathstrut \) \(1845523179179075533697\) \(\nu^{6}\mathstrut -\mathstrut \) \(8970411594462544668655\) \(\nu^{5}\mathstrut +\mathstrut \) \(344404977921557791603\) \(\nu^{4}\mathstrut +\mathstrut \) \(5193137183696587044533\) \(\nu^{3}\mathstrut -\mathstrut \) \(1305703211617004113555\) \(\nu^{2}\mathstrut -\mathstrut \) \(758989349069120752892\) \(\nu\mathstrut +\mathstrut \) \(212059742701487858145\)\()/\)\(48600826930893597\)
\(\beta_{12}\)\(=\)\((\)\(1873974907296708614\) \(\nu^{18}\mathstrut -\mathstrut \) \(6283922313079958432\) \(\nu^{17}\mathstrut -\mathstrut \) \(60943644288332340656\) \(\nu^{16}\mathstrut +\mathstrut \) \(201345019980433073510\) \(\nu^{15}\mathstrut +\mathstrut \) \(805621674808816695688\) \(\nu^{14}\mathstrut -\mathstrut \) \(2614075030937030707880\) \(\nu^{13}\mathstrut -\mathstrut \) \(5583917894989463673705\) \(\nu^{12}\mathstrut +\mathstrut \) \(17865125891581362586482\) \(\nu^{11}\mathstrut +\mathstrut \) \(21649748643248381838097\) \(\nu^{10}\mathstrut -\mathstrut \) \(69874443705711848619527\) \(\nu^{9}\mathstrut -\mathstrut \) \(45263944234660747660067\) \(\nu^{8}\mathstrut +\mathstrut \) \(158174564295299740135210\) \(\nu^{7}\mathstrut +\mathstrut \) \(40329818361882135806509\) \(\nu^{6}\mathstrut -\mathstrut \) \(196690298786461654073838\) \(\nu^{5}\mathstrut +\mathstrut \) \(7690190752562843363047\) \(\nu^{4}\mathstrut +\mathstrut \) \(113779077026578856297756\) \(\nu^{3}\mathstrut -\mathstrut \) \(28666956234162005008317\) \(\nu^{2}\mathstrut -\mathstrut \) \(16626640105199122180016\) \(\nu\mathstrut +\mathstrut \) \(4654685085746666817805\)\()/\)\(1020617365548765537\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(2144892591088627034\) \(\nu^{18}\mathstrut +\mathstrut \) \(7194469518027730034\) \(\nu^{17}\mathstrut +\mathstrut \) \(69730404974741217923\) \(\nu^{16}\mathstrut -\mathstrut \) \(230437366136199165995\) \(\nu^{15}\mathstrut -\mathstrut \) \(921469411676204542558\) \(\nu^{14}\mathstrut +\mathstrut \) \(2990487515085089939468\) \(\nu^{13}\mathstrut +\mathstrut \) \(6385303391287405847445\) \(\nu^{12}\mathstrut -\mathstrut \) \(20427301609477794239322\) \(\nu^{11}\mathstrut -\mathstrut \) \(24756390176926830053503\) \(\nu^{10}\mathstrut +\mathstrut \) \(79850878293857396396654\) \(\nu^{9}\mathstrut +\mathstrut \) \(51788411949832743202817\) \(\nu^{8}\mathstrut -\mathstrut \) \(180649471065960940359055\) \(\nu^{7}\mathstrut -\mathstrut \) \(46283070502099205057266\) \(\nu^{6}\mathstrut +\mathstrut \) \(224496760642673981731902\) \(\nu^{5}\mathstrut -\mathstrut \) \(8452079588993243282797\) \(\nu^{4}\mathstrut -\mathstrut \) \(129780856901449777235285\) \(\nu^{3}\mathstrut +\mathstrut \) \(32518140527160546461946\) \(\nu^{2}\mathstrut +\mathstrut \) \(18954556397781083030699\) \(\nu\mathstrut -\mathstrut \) \(5286569058612135617575\)\()/\)\(1020617365548765537\)
\(\beta_{14}\)\(=\)\((\)\(2457833589618540821\) \(\nu^{18}\mathstrut -\mathstrut \) \(8230807208306680178\) \(\nu^{17}\mathstrut -\mathstrut \) \(79981413968377317875\) \(\nu^{16}\mathstrut +\mathstrut \) \(263789798794718894636\) \(\nu^{15}\mathstrut +\mathstrut \) \(1058091060765007141555\) \(\nu^{14}\mathstrut -\mathstrut \) \(3425765874336692174384\) \(\nu^{13}\mathstrut -\mathstrut \) \(7340584968184626475839\) \(\nu^{12}\mathstrut +\mathstrut \) \(23419718163023151414243\) \(\nu^{11}\mathstrut +\mathstrut \) \(28493582837939112455179\) \(\nu^{10}\mathstrut -\mathstrut \) \(91629382793921658431342\) \(\nu^{9}\mathstrut -\mathstrut \) \(59677381013277746326763\) \(\nu^{8}\mathstrut +\mathstrut \) \(207484979222026518073606\) \(\nu^{7}\mathstrut +\mathstrut \) \(53421344508988625098015\) \(\nu^{6}\mathstrut -\mathstrut \) \(258073333821884028257892\) \(\nu^{5}\mathstrut +\mathstrut \) \(9644745708255936896053\) \(\nu^{4}\mathstrut +\mathstrut \) \(149308891279037428721921\) \(\nu^{3}\mathstrut -\mathstrut \) \(37448517732094712720502\) \(\nu^{2}\mathstrut -\mathstrut \) \(21814096022736375674984\) \(\nu\mathstrut +\mathstrut \) \(6091484169428763616294\)\()/\)\(1020617365548765537\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(2480478071381031934\) \(\nu^{18}\mathstrut +\mathstrut \) \(8317049413702779763\) \(\nu^{17}\mathstrut +\mathstrut \) \(80694059490133678921\) \(\nu^{16}\mathstrut -\mathstrut \) \(266586666707022673075\) \(\nu^{15}\mathstrut -\mathstrut \) \(1067096206066221493565\) \(\nu^{14}\mathstrut +\mathstrut \) \(3462721764936389780893\) \(\nu^{13}\mathstrut +\mathstrut \) \(7399068896730559425402\) \(\nu^{12}\mathstrut -\mathstrut \) \(23678606437711504111974\) \(\nu^{11}\mathstrut -\mathstrut \) \(28697173832378925715760\) \(\nu^{10}\mathstrut +\mathstrut \) \(92676052053730455930559\) \(\nu^{9}\mathstrut +\mathstrut \) \(60008490368648521259134\) \(\nu^{8}\mathstrut -\mathstrut \) \(209956117447002986908421\) \(\nu^{7}\mathstrut -\mathstrut \) \(53431366446972392674742\) \(\nu^{6}\mathstrut +\mathstrut \) \(261304500629775154221276\) \(\nu^{5}\mathstrut -\mathstrut \) \(10344792388029446712416\) \(\nu^{4}\mathstrut -\mathstrut \) \(151282696826724566269381\) \(\nu^{3}\mathstrut +\mathstrut \) \(38166899838158590967094\) \(\nu^{2}\mathstrut +\mathstrut \) \(22115082331848633584194\) \(\nu\mathstrut -\mathstrut \) \(6191111997121590472409\)\()/\)\(1020617365548765537\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(2782406347958195911\) \(\nu^{18}\mathstrut +\mathstrut \) \(9321270578210196910\) \(\nu^{17}\mathstrut +\mathstrut \) \(90540346893836227708\) \(\nu^{16}\mathstrut -\mathstrut \) \(298751519361850754959\) \(\nu^{15}\mathstrut -\mathstrut \) \(1197777629536310623775\) \(\nu^{14}\mathstrut +\mathstrut \) \(3880099917466543232167\) \(\nu^{13}\mathstrut +\mathstrut \) \(8310256235737560923667\) \(\nu^{12}\mathstrut -\mathstrut \) \(26528937275279513428785\) \(\nu^{11}\mathstrut -\mathstrut \) \(32263716832136550391982\) \(\nu^{10}\mathstrut +\mathstrut \) \(103813020415943587502146\) \(\nu^{9}\mathstrut +\mathstrut \) \(67600246629408794257744\) \(\nu^{8}\mathstrut -\mathstrut \) \(235132067871666249092903\) \(\nu^{7}\mathstrut -\mathstrut \) \(60565986399289712681852\) \(\nu^{6}\mathstrut +\mathstrut \) \(292550270288526975464949\) \(\nu^{5}\mathstrut -\mathstrut \) \(10879870475710539887606\) \(\nu^{4}\mathstrut -\mathstrut \) \(169303999527494574248281\) \(\nu^{3}\mathstrut +\mathstrut \) \(42455085917591594483679\) \(\nu^{2}\mathstrut +\mathstrut \) \(24732045323581350761809\) \(\nu\mathstrut -\mathstrut \) \(6910046618358011013662\)\()/\)\(1020617365548765537\)
\(\beta_{17}\)\(=\)\((\)\(3593665303542336791\) \(\nu^{18}\mathstrut -\mathstrut \) \(12052509696078272852\) \(\nu^{17}\mathstrut -\mathstrut \) \(116882896455933794909\) \(\nu^{16}\mathstrut +\mathstrut \) \(386251417947279105581\) \(\nu^{15}\mathstrut +\mathstrut \) \(1545281791577690052052\) \(\nu^{14}\mathstrut -\mathstrut \) \(5015937220697357742740\) \(\nu^{13}\mathstrut -\mathstrut \) \(10712007976527151119663\) \(\nu^{12}\mathstrut +\mathstrut \) \(34290120858500709390786\) \(\nu^{11}\mathstrut +\mathstrut \) \(41536867472258232226480\) \(\nu^{10}\mathstrut -\mathstrut \) \(134162524884195174575648\) \(\nu^{9}\mathstrut -\mathstrut \) \(86847960660814927475009\) \(\nu^{8}\mathstrut +\mathstrut \) \(303816755038514222896822\) \(\nu^{7}\mathstrut +\mathstrut \) \(77368349629137635834443\) \(\nu^{6}\mathstrut -\mathstrut \) \(377933402290562240744280\) \(\nu^{5}\mathstrut +\mathstrut \) \(14807555707531629201949\) \(\nu^{4}\mathstrut +\mathstrut \) \(218685828516861787263617\) \(\nu^{3}\mathstrut -\mathstrut \) \(55055491039399386968535\) \(\nu^{2}\mathstrut -\mathstrut \) \(31963998409418137583369\) \(\nu\mathstrut +\mathstrut \) \(8931999866080954709410\)\()/\)\(1020617365548765537\)
\(\beta_{18}\)\(=\)\((\)\(5075731687814602463\) \(\nu^{18}\mathstrut -\mathstrut \) \(17031966165042810164\) \(\nu^{17}\mathstrut -\mathstrut \) \(165024155254798231928\) \(\nu^{16}\mathstrut +\mathstrut \) \(545675914726858560707\) \(\nu^{15}\mathstrut +\mathstrut \) \(2180829431863158362884\) \(\nu^{14}\mathstrut -\mathstrut \) \(7083872605568085309362\) \(\nu^{13}\mathstrut -\mathstrut \) \(15111057917705477193000\) \(\nu^{12}\mathstrut +\mathstrut \) \(48408404673936521132841\) \(\nu^{11}\mathstrut +\mathstrut \) \(58569446326550598616000\) \(\nu^{10}\mathstrut -\mathstrut \) \(189323011977443783912135\) \(\nu^{9}\mathstrut -\mathstrut \) \(122407348473336739035173\) \(\nu^{8}\mathstrut +\mathstrut \) \(428551132417492891548211\) \(\nu^{7}\mathstrut +\mathstrut \) \(108971836267219933096339\) \(\nu^{6}\mathstrut -\mathstrut \) \(532887729738232799998188\) \(\nu^{5}\mathstrut +\mathstrut \) \(20976768851486207992069\) \(\nu^{4}\mathstrut +\mathstrut \) \(308237991426416883886049\) \(\nu^{3}\mathstrut -\mathstrut \) \(77657300251984486959579\) \(\nu^{2}\mathstrut -\mathstrut \) \(45029812483252704236399\) \(\nu\mathstrut +\mathstrut \) \(12602055865532474888050\)\()/\)\(1020617365548765537\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{18}\mathstrut -\mathstrut \) \(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{16}\mathstrut +\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(3\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(27\)
\(\nu^{5}\)\(=\)\(13\) \(\beta_{18}\mathstrut -\mathstrut \) \(15\) \(\beta_{17}\mathstrut +\mathstrut \) \(15\) \(\beta_{16}\mathstrut +\mathstrut \) \(12\) \(\beta_{15}\mathstrut +\mathstrut \) \(13\) \(\beta_{14}\mathstrut -\mathstrut \) \(15\) \(\beta_{13}\mathstrut -\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(27\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(38\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(13\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(26\) \(\beta_{3}\mathstrut +\mathstrut \) \(39\) \(\beta_{2}\mathstrut +\mathstrut \) \(73\) \(\beta_{1}\mathstrut +\mathstrut \) \(17\)
\(\nu^{6}\)\(=\)\(-\)\(12\) \(\beta_{17}\mathstrut +\mathstrut \) \(20\) \(\beta_{16}\mathstrut +\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(44\) \(\beta_{13}\mathstrut -\mathstrut \) \(20\) \(\beta_{12}\mathstrut +\mathstrut \) \(16\) \(\beta_{11}\mathstrut +\mathstrut \) \(17\) \(\beta_{10}\mathstrut +\mathstrut \) \(29\) \(\beta_{9}\mathstrut -\mathstrut \) \(21\) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(19\) \(\beta_{6}\mathstrut +\mathstrut \) \(27\) \(\beta_{5}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(41\) \(\beta_{3}\mathstrut +\mathstrut \) \(107\) \(\beta_{2}\mathstrut +\mathstrut \) \(63\) \(\beta_{1}\mathstrut +\mathstrut \) \(223\)
\(\nu^{7}\)\(=\)\(135\) \(\beta_{18}\mathstrut -\mathstrut \) \(175\) \(\beta_{17}\mathstrut +\mathstrut \) \(180\) \(\beta_{16}\mathstrut +\mathstrut \) \(114\) \(\beta_{15}\mathstrut +\mathstrut \) \(140\) \(\beta_{14}\mathstrut -\mathstrut \) \(183\) \(\beta_{13}\mathstrut -\mathstrut \) \(46\) \(\beta_{12}\mathstrut +\mathstrut \) \(296\) \(\beta_{11}\mathstrut +\mathstrut \) \(26\) \(\beta_{10}\mathstrut +\mathstrut \) \(402\) \(\beta_{9}\mathstrut -\mathstrut \) \(45\) \(\beta_{8}\mathstrut +\mathstrut \) \(152\) \(\beta_{7}\mathstrut +\mathstrut \) \(42\) \(\beta_{6}\mathstrut +\mathstrut \) \(133\) \(\beta_{5}\mathstrut -\mathstrut \) \(49\) \(\beta_{4}\mathstrut +\mathstrut \) \(284\) \(\beta_{3}\mathstrut +\mathstrut \) \(415\) \(\beta_{2}\mathstrut +\mathstrut \) \(702\) \(\beta_{1}\mathstrut +\mathstrut \) \(208\)
\(\nu^{8}\)\(=\)\(9\) \(\beta_{18}\mathstrut -\mathstrut \) \(142\) \(\beta_{17}\mathstrut +\mathstrut \) \(296\) \(\beta_{16}\mathstrut +\mathstrut \) \(20\) \(\beta_{15}\mathstrut +\mathstrut \) \(38\) \(\beta_{14}\mathstrut -\mathstrut \) \(535\) \(\beta_{13}\mathstrut -\mathstrut \) \(296\) \(\beta_{12}\mathstrut +\mathstrut \) \(216\) \(\beta_{11}\mathstrut +\mathstrut \) \(235\) \(\beta_{10}\mathstrut +\mathstrut \) \(365\) \(\beta_{9}\mathstrut -\mathstrut \) \(311\) \(\beta_{8}\mathstrut +\mathstrut \) \(171\) \(\beta_{7}\mathstrut +\mathstrut \) \(262\) \(\beta_{6}\mathstrut +\mathstrut \) \(293\) \(\beta_{5}\mathstrut -\mathstrut \) \(205\) \(\beta_{4}\mathstrut +\mathstrut \) \(482\) \(\beta_{3}\mathstrut +\mathstrut \) \(1049\) \(\beta_{2}\mathstrut +\mathstrut \) \(808\) \(\beta_{1}\mathstrut +\mathstrut \) \(2008\)
\(\nu^{9}\)\(=\)\(1343\) \(\beta_{18}\mathstrut -\mathstrut \) \(1912\) \(\beta_{17}\mathstrut +\mathstrut \) \(2038\) \(\beta_{16}\mathstrut +\mathstrut \) \(1026\) \(\beta_{15}\mathstrut +\mathstrut \) \(1444\) \(\beta_{14}\mathstrut -\mathstrut \) \(2100\) \(\beta_{13}\mathstrut -\mathstrut \) \(746\) \(\beta_{12}\mathstrut +\mathstrut \) \(3087\) \(\beta_{11}\mathstrut +\mathstrut \) \(432\) \(\beta_{10}\mathstrut +\mathstrut \) \(4085\) \(\beta_{9}\mathstrut -\mathstrut \) \(710\) \(\beta_{8}\mathstrut +\mathstrut \) \(1566\) \(\beta_{7}\mathstrut +\mathstrut \) \(649\) \(\beta_{6}\mathstrut +\mathstrut \) \(1284\) \(\beta_{5}\mathstrut -\mathstrut \) \(828\) \(\beta_{4}\mathstrut +\mathstrut \) \(3006\) \(\beta_{3}\mathstrut +\mathstrut \) \(4229\) \(\beta_{2}\mathstrut +\mathstrut \) \(6932\) \(\beta_{1}\mathstrut +\mathstrut \) \(2338\)
\(\nu^{10}\)\(=\)\(251\) \(\beta_{18}\mathstrut -\mathstrut \) \(1775\) \(\beta_{17}\mathstrut +\mathstrut \) \(3923\) \(\beta_{16}\mathstrut +\mathstrut \) \(262\) \(\beta_{15}\mathstrut +\mathstrut \) \(773\) \(\beta_{14}\mathstrut -\mathstrut \) \(6220\) \(\beta_{13}\mathstrut -\mathstrut \) \(3926\) \(\beta_{12}\mathstrut +\mathstrut \) \(2762\) \(\beta_{11}\mathstrut +\mathstrut \) \(2981\) \(\beta_{10}\mathstrut +\mathstrut \) \(4439\) \(\beta_{9}\mathstrut -\mathstrut \) \(4053\) \(\beta_{8}\mathstrut +\mathstrut \) \(2048\) \(\beta_{7}\mathstrut +\mathstrut \) \(3272\) \(\beta_{6}\mathstrut +\mathstrut \) \(2996\) \(\beta_{5}\mathstrut -\mathstrut \) \(3288\) \(\beta_{4}\mathstrut +\mathstrut \) \(5549\) \(\beta_{3}\mathstrut +\mathstrut \) \(10511\) \(\beta_{2}\mathstrut +\mathstrut \) \(9732\) \(\beta_{1}\mathstrut +\mathstrut \) \(18825\)
\(\nu^{11}\)\(=\)\(13304\) \(\beta_{18}\mathstrut -\mathstrut \) \(20503\) \(\beta_{17}\mathstrut +\mathstrut \) \(22707\) \(\beta_{16}\mathstrut +\mathstrut \) \(9079\) \(\beta_{15}\mathstrut +\mathstrut \) \(14824\) \(\beta_{14}\mathstrut -\mathstrut \) \(23638\) \(\beta_{13}\mathstrut -\mathstrut \) \(10524\) \(\beta_{12}\mathstrut +\mathstrut \) \(31826\) \(\beta_{11}\mathstrut +\mathstrut \) \(6110\) \(\beta_{10}\mathstrut +\mathstrut \) \(41310\) \(\beta_{9}\mathstrut -\mathstrut \) \(9805\) \(\beta_{8}\mathstrut +\mathstrut \) \(16062\) \(\beta_{7}\mathstrut +\mathstrut \) \(8876\) \(\beta_{6}\mathstrut +\mathstrut \) \(12235\) \(\beta_{5}\mathstrut -\mathstrut \) \(12015\) \(\beta_{4}\mathstrut +\mathstrut \) \(31795\) \(\beta_{3}\mathstrut +\mathstrut \) \(42823\) \(\beta_{2}\mathstrut +\mathstrut \) \(69711\) \(\beta_{1}\mathstrut +\mathstrut \) \(25525\)
\(\nu^{12}\)\(=\)\(4617\) \(\beta_{18}\mathstrut -\mathstrut \) \(22638\) \(\beta_{17}\mathstrut +\mathstrut \) \(49266\) \(\beta_{16}\mathstrut +\mathstrut \) \(2821\) \(\beta_{15}\mathstrut +\mathstrut \) \(12294\) \(\beta_{14}\mathstrut -\mathstrut \) \(71324\) \(\beta_{13}\mathstrut -\mathstrut \) \(49250\) \(\beta_{12}\mathstrut +\mathstrut \) \(34294\) \(\beta_{11}\mathstrut +\mathstrut \) \(36104\) \(\beta_{10}\mathstrut +\mathstrut \) \(53020\) \(\beta_{9}\mathstrut -\mathstrut \) \(49772\) \(\beta_{8}\mathstrut +\mathstrut \) \(24392\) \(\beta_{7}\mathstrut +\mathstrut \) \(39391\) \(\beta_{6}\mathstrut +\mathstrut \) \(30014\) \(\beta_{5}\mathstrut -\mathstrut \) \(45978\) \(\beta_{4}\mathstrut +\mathstrut \) \(63843\) \(\beta_{3}\mathstrut +\mathstrut \) \(107398\) \(\beta_{2}\mathstrut +\mathstrut \) \(113961\) \(\beta_{1}\mathstrut +\mathstrut \) \(180504\)
\(\nu^{13}\)\(=\)\(132613\) \(\beta_{18}\mathstrut -\mathstrut \) \(219348\) \(\beta_{17}\mathstrut +\mathstrut \) \(252299\) \(\beta_{16}\mathstrut +\mathstrut \) \(79711\) \(\beta_{15}\mathstrut +\mathstrut \) \(153304\) \(\beta_{14}\mathstrut -\mathstrut \) \(264805\) \(\beta_{13}\mathstrut -\mathstrut \) \(138010\) \(\beta_{12}\mathstrut +\mathstrut \) \(328475\) \(\beta_{11}\mathstrut +\mathstrut \) \(80182\) \(\beta_{10}\mathstrut +\mathstrut \) \(420500\) \(\beta_{9}\mathstrut -\mathstrut \) \(126753\) \(\beta_{8}\mathstrut +\mathstrut \) \(166387\) \(\beta_{7}\mathstrut +\mathstrut \) \(114066\) \(\beta_{6}\mathstrut +\mathstrut \) \(116663\) \(\beta_{5}\mathstrut -\mathstrut \) \(161023\) \(\beta_{4}\mathstrut +\mathstrut \) \(338772\) \(\beta_{3}\mathstrut +\mathstrut \) \(435818\) \(\beta_{2}\mathstrut +\mathstrut \) \(711597\) \(\beta_{1}\mathstrut +\mathstrut \) \(275496\)
\(\nu^{14}\)\(=\)\(71072\) \(\beta_{18}\mathstrut -\mathstrut \) \(287660\) \(\beta_{17}\mathstrut +\mathstrut \) \(599952\) \(\beta_{16}\mathstrut +\mathstrut \) \(26545\) \(\beta_{15}\mathstrut +\mathstrut \) \(172772\) \(\beta_{14}\mathstrut -\mathstrut \) \(813934\) \(\beta_{13}\mathstrut -\mathstrut \) \(597693\) \(\beta_{12}\mathstrut +\mathstrut \) \(417514\) \(\beta_{11}\mathstrut +\mathstrut \) \(425638\) \(\beta_{10}\mathstrut +\mathstrut \) \(625325\) \(\beta_{9}\mathstrut -\mathstrut \) \(592146\) \(\beta_{8}\mathstrut +\mathstrut \) \(289263\) \(\beta_{7}\mathstrut +\mathstrut \) \(466712\) \(\beta_{6}\mathstrut +\mathstrut \) \(298456\) \(\beta_{5}\mathstrut -\mathstrut \) \(599693\) \(\beta_{4}\mathstrut +\mathstrut \) \(735699\) \(\beta_{3}\mathstrut +\mathstrut \) \(1115229\) \(\beta_{2}\mathstrut +\mathstrut \) \(1314386\) \(\beta_{1}\mathstrut +\mathstrut \) \(1757947\)
\(\nu^{15}\)\(=\)\(1334363\) \(\beta_{18}\mathstrut -\mathstrut \) \(2355999\) \(\beta_{17}\mathstrut +\mathstrut \) \(2807394\) \(\beta_{16}\mathstrut +\mathstrut \) \(695010\) \(\beta_{15}\mathstrut +\mathstrut \) \(1602340\) \(\beta_{14}\mathstrut -\mathstrut \) \(2967240\) \(\beta_{13}\mathstrut -\mathstrut \) \(1732233\) \(\beta_{12}\mathstrut +\mathstrut \) \(3409972\) \(\beta_{11}\mathstrut +\mathstrut \) \(1009407\) \(\beta_{10}\mathstrut +\mathstrut \) \(4324648\) \(\beta_{9}\mathstrut -\mathstrut \) \(1576502\) \(\beta_{8}\mathstrut +\mathstrut \) \(1747725\) \(\beta_{7}\mathstrut +\mathstrut \) \(1413874\) \(\beta_{6}\mathstrut +\mathstrut \) \(1118165\) \(\beta_{5}\mathstrut -\mathstrut \) \(2058194\) \(\beta_{4}\mathstrut +\mathstrut \) \(3642479\) \(\beta_{3}\mathstrut +\mathstrut \) \(4474920\) \(\beta_{2}\mathstrut +\mathstrut \) \(7359778\) \(\beta_{1}\mathstrut +\mathstrut \) \(2960971\)
\(\nu^{16}\)\(=\)\(994020\) \(\beta_{18}\mathstrut -\mathstrut \) \(3609537\) \(\beta_{17}\mathstrut +\mathstrut \) \(7165673\) \(\beta_{16}\mathstrut +\mathstrut \) \(218190\) \(\beta_{15}\mathstrut +\mathstrut \) \(2263170\) \(\beta_{14}\mathstrut -\mathstrut \) \(9271397\) \(\beta_{13}\mathstrut -\mathstrut \) \(7100292\) \(\beta_{12}\mathstrut +\mathstrut \) \(5010451\) \(\beta_{11}\mathstrut +\mathstrut \) \(4936534\) \(\beta_{10}\mathstrut +\mathstrut \) \(7307651\) \(\beta_{9}\mathstrut -\mathstrut \) \(6915112\) \(\beta_{8}\mathstrut +\mathstrut \) \(3415992\) \(\beta_{7}\mathstrut +\mathstrut \) \(5482136\) \(\beta_{6}\mathstrut +\mathstrut \) \(2961968\) \(\beta_{5}\mathstrut -\mathstrut \) \(7509610\) \(\beta_{4}\mathstrut +\mathstrut \) \(8484738\) \(\beta_{3}\mathstrut +\mathstrut \) \(11739393\) \(\beta_{2}\mathstrut +\mathstrut \) \(15026138\) \(\beta_{1}\mathstrut +\mathstrut \) \(17341065\)
\(\nu^{17}\)\(=\)\(13567889\) \(\beta_{18}\mathstrut -\mathstrut \) \(25465968\) \(\beta_{17}\mathstrut +\mathstrut \) \(31318019\) \(\beta_{16}\mathstrut +\mathstrut \) \(6003260\) \(\beta_{15}\mathstrut +\mathstrut \) \(16929851\) \(\beta_{14}\mathstrut -\mathstrut \) \(33308178\) \(\beta_{13}\mathstrut -\mathstrut \) \(21130030\) \(\beta_{12}\mathstrut +\mathstrut \) \(35671399\) \(\beta_{11}\mathstrut +\mathstrut \) \(12376619\) \(\beta_{10}\mathstrut +\mathstrut \) \(44979060\) \(\beta_{9}\mathstrut -\mathstrut \) \(19121302\) \(\beta_{8}\mathstrut +\mathstrut \) \(18617976\) \(\beta_{7}\mathstrut +\mathstrut \) \(17128527\) \(\beta_{6}\mathstrut +\mathstrut \) \(10789112\) \(\beta_{5}\mathstrut -\mathstrut \) \(25510859\) \(\beta_{4}\mathstrut +\mathstrut \) \(39512408\) \(\beta_{3}\mathstrut +\mathstrut \) \(46413527\) \(\beta_{2}\mathstrut +\mathstrut \) \(77019927\) \(\beta_{1}\mathstrut +\mathstrut \) \(31805551\)
\(\nu^{18}\)\(=\)\(13117686\) \(\beta_{18}\mathstrut -\mathstrut \) \(44646481\) \(\beta_{17}\mathstrut +\mathstrut \) \(84471055\) \(\beta_{16}\mathstrut +\mathstrut \) \(1453099\) \(\beta_{15}\mathstrut +\mathstrut \) \(28391930\) \(\beta_{14}\mathstrut -\mathstrut \) \(105525778\) \(\beta_{13}\mathstrut -\mathstrut \) \(83139156\) \(\beta_{12}\mathstrut +\mathstrut \) \(59468870\) \(\beta_{11}\mathstrut +\mathstrut \) \(56673742\) \(\beta_{10}\mathstrut +\mathstrut \) \(84827496\) \(\beta_{9}\mathstrut -\mathstrut \) \(79820651\) \(\beta_{8}\mathstrut +\mathstrut \) \(40179778\) \(\beta_{7}\mathstrut +\mathstrut \) \(64021137\) \(\beta_{6}\mathstrut +\mathstrut \) \(29420558\) \(\beta_{5}\mathstrut -\mathstrut \) \(91614456\) \(\beta_{4}\mathstrut +\mathstrut \) \(97842864\) \(\beta_{3}\mathstrut +\mathstrut \) \(125031276\) \(\beta_{2}\mathstrut +\mathstrut \) \(170869755\) \(\beta_{1}\mathstrut +\mathstrut \) \(173050415\)

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
3.36626
3.10589
3.02356
2.14522
1.86683
1.64644
1.64534
1.24916
0.889948
0.700061
0.279555
−0.440747
−1.34880
−1.49075
−1.88079
−1.93680
−2.03161
−2.74056
−3.04822
0 −3.36626 0 3.30411 0 −2.65381 0 8.33173 0
1.2 0 −3.10589 0 −2.95976 0 −2.66591 0 6.64656 0
1.3 0 −3.02356 0 1.31787 0 3.08066 0 6.14194 0
1.4 0 −2.14522 0 3.02869 0 −3.82352 0 1.60196 0
1.5 0 −1.86683 0 −2.50480 0 4.61610 0 0.485062 0
1.6 0 −1.64644 0 1.56330 0 −1.03398 0 −0.289225 0
1.7 0 −1.64534 0 −3.46538 0 −5.08170 0 −0.292845 0
1.8 0 −1.24916 0 −0.269755 0 1.00361 0 −1.43961 0
1.9 0 −0.889948 0 −0.485054 0 1.10304 0 −2.20799 0
1.10 0 −0.700061 0 3.71173 0 3.38907 0 −2.50991 0
1.11 0 −0.279555 0 −3.34909 0 −2.71416 0 −2.92185 0
1.12 0 0.440747 0 0.359379 0 0.921983 0 −2.80574 0
1.13 0 1.34880 0 0.901606 0 0.205950 0 −1.18075 0
1.14 0 1.49075 0 −0.450569 0 1.77312 0 −0.777652 0
1.15 0 1.88079 0 2.88434 0 −4.36689 0 0.537360 0
1.16 0 1.93680 0 −2.93047 0 −0.385104 0 0.751200 0
1.17 0 2.03161 0 −4.10213 0 2.65539 0 1.12742 0
1.18 0 2.74056 0 −0.391247 0 −2.92798 0 4.51068 0
1.19 0 3.04822 0 −1.16277 0 −3.09586 0 6.29166 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4028))\):

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