Properties

Label 4012.2.a.i
Level 4012
Weight 2
Character orbit 4012.a
Self dual Yes
Analytic conductor 32.036
Analytic rank 0
Dimension 18
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4012.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)  \(=\)  \( q\) \( + \beta_{1} q^{3} \) \( - \beta_{9} q^{5} \) \( - \beta_{8} q^{7} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( - \beta_{9} q^{5} \) \( - \beta_{8} q^{7} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{9} \) \( + ( 1 + \beta_{4} ) q^{11} \) \( + \beta_{7} q^{13} \) \( + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{9} + \beta_{12} + \beta_{13} - \beta_{17} ) q^{15} \) \(- q^{17}\) \( + ( - \beta_{6} + \beta_{9} + \beta_{17} ) q^{19} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} + \beta_{11} + \beta_{12} - 2 \beta_{15} - \beta_{17} ) q^{21} \) \( + ( 2 + \beta_{11} - \beta_{13} - \beta_{14} ) q^{23} \) \( + ( \beta_{1} + \beta_{2} + \beta_{9} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{16} + \beta_{17} ) q^{25} \) \( + ( 1 + 2 \beta_{1} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{15} - \beta_{17} ) q^{27} \) \( + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} + \beta_{17} ) q^{29} \) \( + ( - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{31} \) \( + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{13} - \beta_{17} ) q^{33} \) \( + ( 3 - 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{35} \) \( + ( 1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{9} + \beta_{12} - \beta_{14} + 2 \beta_{15} + \beta_{16} ) q^{37} \) \( + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - \beta_{16} ) q^{39} \) \( + ( 3 - \beta_{1} - \beta_{2} + \beta_{11} + \beta_{15} + \beta_{16} ) q^{41} \) \( + ( -2 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{8} + 2 \beta_{9} - \beta_{11} - \beta_{12} + 3 \beta_{14} - 3 \beta_{15} - \beta_{16} ) q^{43} \) \( + ( 5 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} - 3 \beta_{9} - \beta_{11} + 3 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} - 2 \beta_{17} ) q^{45} \) \( + ( 3 + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{14} + 2 \beta_{15} + \beta_{17} ) q^{47} \) \( + ( 1 + 2 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + \beta_{17} ) q^{49} \) \( - \beta_{1} q^{51} \) \( + ( -3 - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{14} ) q^{53} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{16} - \beta_{17} ) q^{55} \) \( + ( \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{9} - \beta_{12} - \beta_{13} - \beta_{15} + \beta_{17} ) q^{57} \) \(- q^{59}\) \( + ( 2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{14} + \beta_{15} - \beta_{17} ) q^{61} \) \( + ( 7 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{9} + \beta_{11} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{17} ) q^{63} \) \( + ( -3 + 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} + \beta_{10} - \beta_{12} - 3 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{65} \) \( + ( 5 - \beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{13} - 2 \beta_{14} - \beta_{17} ) q^{67} \) \( + ( -1 + 3 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} + 2 \beta_{9} + 2 \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{69} \) \( + ( -3 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - 3 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} ) q^{71} \) \( + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{73} \) \( + ( 1 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{9} + \beta_{12} - 2 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{75} \) \( + ( 1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{12} + 2 \beta_{13} + \beta_{15} + 2 \beta_{16} ) q^{77} \) \( + ( 3 + \beta_{1} - \beta_{2} + 2 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - 3 \beta_{14} + \beta_{15} ) q^{79} \) \( + ( 3 - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{7} + \beta_{8} - 3 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + 3 \beta_{15} + \beta_{16} - \beta_{17} ) q^{81} \) \( + ( 1 + 3 \beta_{3} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{13} + \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{83} \) \( + \beta_{9} q^{85} \) \( + ( -2 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{7} + 3 \beta_{9} - \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{87} \) \( + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} + 2 \beta_{14} - 3 \beta_{15} - \beta_{16} ) q^{89} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{11} + \beta_{12} - 3 \beta_{15} - \beta_{17} ) q^{91} \) \( + ( -3 + \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{10} - \beta_{11} - 2 \beta_{13} + \beta_{14} + 2 \beta_{17} ) q^{93} \) \( + ( -2 \beta_{1} - \beta_{3} + \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{15} ) q^{95} \) \( + ( 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + 4 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{97} \) \( + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - 4 \beta_{9} + \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + \beta_{15} + \beta_{16} - 3 \beta_{17} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(18q \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 18q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(18q \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 18q^{9} \) \(\mathstrut +\mathstrut 12q^{11} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut 18q^{17} \) \(\mathstrut +\mathstrut 5q^{19} \) \(\mathstrut -\mathstrut 3q^{21} \) \(\mathstrut +\mathstrut 21q^{23} \) \(\mathstrut +\mathstrut 16q^{25} \) \(\mathstrut +\mathstrut 26q^{27} \) \(\mathstrut +\mathstrut 14q^{29} \) \(\mathstrut +\mathstrut 15q^{31} \) \(\mathstrut +\mathstrut 19q^{33} \) \(\mathstrut +\mathstrut 20q^{35} \) \(\mathstrut +\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut 14q^{39} \) \(\mathstrut +\mathstrut 34q^{41} \) \(\mathstrut +\mathstrut 21q^{43} \) \(\mathstrut +\mathstrut 49q^{45} \) \(\mathstrut +\mathstrut 69q^{47} \) \(\mathstrut +\mathstrut 28q^{49} \) \(\mathstrut -\mathstrut 8q^{51} \) \(\mathstrut -\mathstrut 4q^{53} \) \(\mathstrut +\mathstrut 18q^{55} \) \(\mathstrut +\mathstrut 5q^{57} \) \(\mathstrut -\mathstrut 18q^{59} \) \(\mathstrut +\mathstrut 11q^{61} \) \(\mathstrut +\mathstrut 35q^{63} \) \(\mathstrut +\mathstrut 27q^{65} \) \(\mathstrut +\mathstrut 34q^{67} \) \(\mathstrut -\mathstrut 4q^{69} \) \(\mathstrut +\mathstrut 37q^{71} \) \(\mathstrut +\mathstrut 18q^{73} \) \(\mathstrut +\mathstrut 72q^{75} \) \(\mathstrut +\mathstrut 11q^{77} \) \(\mathstrut +\mathstrut 11q^{79} \) \(\mathstrut +\mathstrut 30q^{81} \) \(\mathstrut +\mathstrut 28q^{83} \) \(\mathstrut -\mathstrut 4q^{85} \) \(\mathstrut +\mathstrut 7q^{87} \) \(\mathstrut +\mathstrut 44q^{89} \) \(\mathstrut -\mathstrut 23q^{91} \) \(\mathstrut -\mathstrut 3q^{93} \) \(\mathstrut -\mathstrut 11q^{95} \) \(\mathstrut +\mathstrut 11q^{97} \) \(\mathstrut +\mathstrut 56q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18}\mathstrut -\mathstrut \) \(8\) \(x^{17}\mathstrut -\mathstrut \) \(4\) \(x^{16}\mathstrut +\mathstrut \) \(178\) \(x^{15}\mathstrut -\mathstrut \) \(265\) \(x^{14}\mathstrut -\mathstrut \) \(1405\) \(x^{13}\mathstrut +\mathstrut \) \(3503\) \(x^{12}\mathstrut +\mathstrut \) \(4295\) \(x^{11}\mathstrut -\mathstrut \) \(17374\) \(x^{10}\mathstrut -\mathstrut \) \(893\) \(x^{9}\mathstrut +\mathstrut \) \(38112\) \(x^{8}\mathstrut -\mathstrut \) \(18700\) \(x^{7}\mathstrut -\mathstrut \) \(32137\) \(x^{6}\mathstrut +\mathstrut \) \(26381\) \(x^{5}\mathstrut +\mathstrut \) \(3964\) \(x^{4}\mathstrut -\mathstrut \) \(5788\) \(x^{3}\mathstrut -\mathstrut \) \(108\) \(x^{2}\mathstrut +\mathstrut \) \(232\) \(x\mathstrut -\mathstrut \) \(16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\(721980238514\) \(\nu^{17}\mathstrut -\mathstrut \) \(8792294540969\) \(\nu^{16}\mathstrut +\mathstrut \) \(19922000071949\) \(\nu^{15}\mathstrut +\mathstrut \) \(155380660037965\) \(\nu^{14}\mathstrut -\mathstrut \) \(735707335127267\) \(\nu^{13}\mathstrut -\mathstrut \) \(568622846364540\) \(\nu^{12}\mathstrut +\mathstrut \) \(7445947298023033\) \(\nu^{11}\mathstrut -\mathstrut \) \(4273813195078958\) \(\nu^{10}\mathstrut -\mathstrut \) \(33371400511887937\) \(\nu^{9}\mathstrut +\mathstrut \) \(38339043861291599\) \(\nu^{8}\mathstrut +\mathstrut \) \(68469249736557868\) \(\nu^{7}\mathstrut -\mathstrut \) \(103056900210181944\) \(\nu^{6}\mathstrut -\mathstrut \) \(52510563287426852\) \(\nu^{5}\mathstrut +\mathstrut \) \(102057820349258193\) \(\nu^{4}\mathstrut +\mathstrut \) \(3198625341800514\) \(\nu^{3}\mathstrut -\mathstrut \) \(23238733137161126\) \(\nu^{2}\mathstrut -\mathstrut \) \(2215838727764722\) \(\nu\mathstrut +\mathstrut \) \(926602245958234\)\()/\)\(221648864308394\)
\(\beta_{4}\)\(=\)\((\)\(3830861910649\) \(\nu^{17}\mathstrut +\mathstrut \) \(47591230451132\) \(\nu^{16}\mathstrut -\mathstrut \) \(573472883791908\) \(\nu^{15}\mathstrut -\mathstrut \) \(137289585232790\) \(\nu^{14}\mathstrut +\mathstrut \) \(12325036003989759\) \(\nu^{13}\mathstrut -\mathstrut \) \(14148964215534769\) \(\nu^{12}\mathstrut -\mathstrut \) \(107106555866144333\) \(\nu^{11}\mathstrut +\mathstrut \) \(184409875945857643\) \(\nu^{10}\mathstrut +\mathstrut \) \(442914820232424934\) \(\nu^{9}\mathstrut -\mathstrut \) \(930141316065082005\) \(\nu^{8}\mathstrut -\mathstrut \) \(849682139411626348\) \(\nu^{7}\mathstrut +\mathstrut \) \(2136821502467077292\) \(\nu^{6}\mathstrut +\mathstrut \) \(583790399310322567\) \(\nu^{5}\mathstrut -\mathstrut \) \(2048651270396240855\) \(\nu^{4}\mathstrut +\mathstrut \) \(4680051383184224\) \(\nu^{3}\mathstrut +\mathstrut \) \(525804686675066276\) \(\nu^{2}\mathstrut +\mathstrut \) \(49267910585755140\) \(\nu\mathstrut -\mathstrut \) \(14341710559057800\)\()/\)\(886595457233576\)
\(\beta_{5}\)\(=\)\((\)\(3229865338098\) \(\nu^{17}\mathstrut -\mathstrut \) \(28341808006461\) \(\nu^{16}\mathstrut +\mathstrut \) \(7224068968562\) \(\nu^{15}\mathstrut +\mathstrut \) \(591300911310204\) \(\nu^{14}\mathstrut -\mathstrut \) \(1336425942773296\) \(\nu^{13}\mathstrut -\mathstrut \) \(4004054182384929\) \(\nu^{12}\mathstrut +\mathstrut \) \(15632584348897933\) \(\nu^{11}\mathstrut +\mathstrut \) \(5978431644745797\) \(\nu^{10}\mathstrut -\mathstrut \) \(74099469131188689\) \(\nu^{9}\mathstrut +\mathstrut \) \(38791752370189050\) \(\nu^{8}\mathstrut +\mathstrut \) \(156577499294520397\) \(\nu^{7}\mathstrut -\mathstrut \) \(158561937111261718\) \(\nu^{6}\mathstrut -\mathstrut \) \(122359738200449602\) \(\nu^{5}\mathstrut +\mathstrut \) \(178581023652807271\) \(\nu^{4}\mathstrut +\mathstrut \) \(6230795980002505\) \(\nu^{3}\mathstrut -\mathstrut \) \(37354205935957182\) \(\nu^{2}\mathstrut -\mathstrut \) \(2171567585544804\) \(\nu\mathstrut -\mathstrut \) \(773148687785764\)\()/\)\(443297728616788\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(3998990215870\) \(\nu^{17}\mathstrut +\mathstrut \) \(48822273723019\) \(\nu^{16}\mathstrut -\mathstrut \) \(95914145833494\) \(\nu^{15}\mathstrut -\mathstrut \) \(926486305202038\) \(\nu^{14}\mathstrut +\mathstrut \) \(3758734853696590\) \(\nu^{13}\mathstrut +\mathstrut \) \(4674117759559001\) \(\nu^{12}\mathstrut -\mathstrut \) \(38722068240010909\) \(\nu^{11}\mathstrut +\mathstrut \) \(10159045369925169\) \(\nu^{10}\mathstrut +\mathstrut \) \(176233209704042289\) \(\nu^{9}\mathstrut -\mathstrut \) \(156608643808120130\) \(\nu^{8}\mathstrut -\mathstrut \) \(369918339657289877\) \(\nu^{7}\mathstrut +\mathstrut \) \(465163701156950644\) \(\nu^{6}\mathstrut +\mathstrut \) \(302735513077342338\) \(\nu^{5}\mathstrut -\mathstrut \) \(488011641581764889\) \(\nu^{4}\mathstrut -\mathstrut \) \(42529357188719131\) \(\nu^{3}\mathstrut +\mathstrut \) \(117684572825314984\) \(\nu^{2}\mathstrut +\mathstrut \) \(15555876709359224\) \(\nu\mathstrut -\mathstrut \) \(2577127790916876\)\()/\)\(443297728616788\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(13022680202683\) \(\nu^{17}\mathstrut +\mathstrut \) \(118335650003618\) \(\nu^{16}\mathstrut -\mathstrut \) \(44771164178744\) \(\nu^{15}\mathstrut -\mathstrut \) \(2504859808443810\) \(\nu^{14}\mathstrut +\mathstrut \) \(5858001428716411\) \(\nu^{13}\mathstrut +\mathstrut \) \(17617430837766009\) \(\nu^{12}\mathstrut -\mathstrut \) \(68767243551113023\) \(\nu^{11}\mathstrut -\mathstrut \) \(33470024699481755\) \(\nu^{10}\mathstrut +\mathstrut \) \(335301431568089064\) \(\nu^{9}\mathstrut -\mathstrut \) \(124869146258752505\) \(\nu^{8}\mathstrut -\mathstrut \) \(759407407344539158\) \(\nu^{7}\mathstrut +\mathstrut \) \(593603282669244732\) \(\nu^{6}\mathstrut +\mathstrut \) \(729456897339086163\) \(\nu^{5}\mathstrut -\mathstrut \) \(727242628798399153\) \(\nu^{4}\mathstrut -\mathstrut \) \(213089357105068366\) \(\nu^{3}\mathstrut +\mathstrut \) \(210304169983562996\) \(\nu^{2}\mathstrut +\mathstrut \) \(37001723229781356\) \(\nu\mathstrut -\mathstrut \) \(5195292962197536\)\()/\)\(886595457233576\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(10740292040922\) \(\nu^{17}\mathstrut +\mathstrut \) \(79117903472021\) \(\nu^{16}\mathstrut +\mathstrut \) \(92870769217390\) \(\nu^{15}\mathstrut -\mathstrut \) \(1850035038846512\) \(\nu^{14}\mathstrut +\mathstrut \) \(1667867302396844\) \(\nu^{13}\mathstrut +\mathstrut \) \(16086137984186553\) \(\nu^{12}\mathstrut -\mathstrut \) \(27196147917604641\) \(\nu^{11}\mathstrut -\mathstrut \) \(62934754460213401\) \(\nu^{10}\mathstrut +\mathstrut \) \(144258069159704345\) \(\nu^{9}\mathstrut +\mathstrut \) \(100056788353315650\) \(\nu^{8}\mathstrut -\mathstrut \) \(334405675482316269\) \(\nu^{7}\mathstrut -\mathstrut \) \(12326711344075138\) \(\nu^{6}\mathstrut +\mathstrut \) \(312468213116224454\) \(\nu^{5}\mathstrut -\mathstrut \) \(79163093196098791\) \(\nu^{4}\mathstrut -\mathstrut \) \(71491791608438813\) \(\nu^{3}\mathstrut +\mathstrut \) \(13826534439622850\) \(\nu^{2}\mathstrut +\mathstrut \) \(5554471852155572\) \(\nu\mathstrut -\mathstrut \) \(813298060986948\)\()/\)\(443297728616788\)
\(\beta_{9}\)\(=\)\((\)\(33959676594129\) \(\nu^{17}\mathstrut -\mathstrut \) \(244226734403362\) \(\nu^{16}\mathstrut -\mathstrut \) \(339891834715816\) \(\nu^{15}\mathstrut +\mathstrut \) \(5819918185918742\) \(\nu^{14}\mathstrut -\mathstrut \) \(4245767604975569\) \(\nu^{13}\mathstrut -\mathstrut \) \(52276748204292455\) \(\nu^{12}\mathstrut +\mathstrut \) \(77888712930711393\) \(\nu^{11}\mathstrut +\mathstrut \) \(218104344572336437\) \(\nu^{10}\mathstrut -\mathstrut \) \(431291327425516436\) \(\nu^{9}\mathstrut -\mathstrut \) \(410563817675315645\) \(\nu^{8}\mathstrut +\mathstrut \) \(1050176863097033118\) \(\nu^{7}\mathstrut +\mathstrut \) \(240664778693316116\) \(\nu^{6}\mathstrut -\mathstrut \) \(1081581921979083169\) \(\nu^{5}\mathstrut +\mathstrut \) \(90985224423031575\) \(\nu^{4}\mathstrut +\mathstrut \) \(340304923142133006\) \(\nu^{3}\mathstrut -\mathstrut \) \(34519210394347964\) \(\nu^{2}\mathstrut -\mathstrut \) \(19663175760623644\) \(\nu\mathstrut +\mathstrut \) \(3721085582082608\)\()/\)\(886595457233576\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(25214595273041\) \(\nu^{17}\mathstrut +\mathstrut \) \(189190565282336\) \(\nu^{16}\mathstrut +\mathstrut \) \(190733503488708\) \(\nu^{15}\mathstrut -\mathstrut \) \(4362707258208572\) \(\nu^{14}\mathstrut +\mathstrut \) \(4551293117275079\) \(\nu^{13}\mathstrut +\mathstrut \) \(36959384933538211\) \(\nu^{12}\mathstrut -\mathstrut \) \(69373780767620221\) \(\nu^{11}\mathstrut -\mathstrut \) \(136233524368167603\) \(\nu^{10}\mathstrut +\mathstrut \) \(360617565778697220\) \(\nu^{9}\mathstrut +\mathstrut \) \(174131370845942773\) \(\nu^{8}\mathstrut -\mathstrut \) \(823035531340428794\) \(\nu^{7}\mathstrut +\mathstrut \) \(117468256302145770\) \(\nu^{6}\mathstrut +\mathstrut \) \(750442873636541309\) \(\nu^{5}\mathstrut -\mathstrut \) \(335379413156274269\) \(\nu^{4}\mathstrut -\mathstrut \) \(153579131812724200\) \(\nu^{3}\mathstrut +\mathstrut \) \(74616740535521018\) \(\nu^{2}\mathstrut +\mathstrut \) \(7869748119362996\) \(\nu\mathstrut -\mathstrut \) \(2555424121174580\)\()/\)\(443297728616788\)
\(\beta_{11}\)\(=\)\((\)\(60700263486941\) \(\nu^{17}\mathstrut -\mathstrut \) \(478320509610002\) \(\nu^{16}\mathstrut -\mathstrut \) \(288622338255540\) \(\nu^{15}\mathstrut +\mathstrut \) \(10692032400216810\) \(\nu^{14}\mathstrut -\mathstrut \) \(14938440419646921\) \(\nu^{13}\mathstrut -\mathstrut \) \(85238794749149239\) \(\nu^{12}\mathstrut +\mathstrut \) \(201539076508820437\) \(\nu^{11}\mathstrut +\mathstrut \) \(268704783294678477\) \(\nu^{10}\mathstrut -\mathstrut \) \(1002553424780760044\) \(\nu^{9}\mathstrut -\mathstrut \) \(109607014231966189\) \(\nu^{8}\mathstrut +\mathstrut \) \(2192698808647176082\) \(\nu^{7}\mathstrut -\mathstrut \) \(981512517810791100\) \(\nu^{6}\mathstrut -\mathstrut \) \(1831423361688449061\) \(\nu^{5}\mathstrut +\mathstrut \) \(1411410964749415411\) \(\nu^{4}\mathstrut +\mathstrut \) \(220519163542968282\) \(\nu^{3}\mathstrut -\mathstrut \) \(260745438390303108\) \(\nu^{2}\mathstrut -\mathstrut \) \(23357849237344556\) \(\nu\mathstrut +\mathstrut \) \(3826534657429544\)\()/\)\(886595457233576\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(64223237008785\) \(\nu^{17}\mathstrut +\mathstrut \) \(539084204709562\) \(\nu^{16}\mathstrut +\mathstrut \) \(81357560610632\) \(\nu^{15}\mathstrut -\mathstrut \) \(11710397156422866\) \(\nu^{14}\mathstrut +\mathstrut \) \(21166954016967573\) \(\nu^{13}\mathstrut +\mathstrut \) \(87837555810305059\) \(\nu^{12}\mathstrut -\mathstrut \) \(261703414914380485\) \(\nu^{11}\mathstrut -\mathstrut \) \(227073674080421397\) \(\nu^{10}\mathstrut +\mathstrut \) \(1264964784404379456\) \(\nu^{9}\mathstrut -\mathstrut \) \(206997387858497275\) \(\nu^{8}\mathstrut -\mathstrut \) \(2711299217705068346\) \(\nu^{7}\mathstrut +\mathstrut \) \(1796551460295109904\) \(\nu^{6}\mathstrut +\mathstrut \) \(2182131551810517545\) \(\nu^{5}\mathstrut -\mathstrut \) \(2191000025769834679\) \(\nu^{4}\mathstrut -\mathstrut \) \(177198874843194846\) \(\nu^{3}\mathstrut +\mathstrut \) \(415845133191300240\) \(\nu^{2}\mathstrut +\mathstrut \) \(23424340673115172\) \(\nu\mathstrut -\mathstrut \) \(9963283153582664\)\()/\)\(886595457233576\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(37688647737868\) \(\nu^{17}\mathstrut +\mathstrut \) \(267971533358753\) \(\nu^{16}\mathstrut +\mathstrut \) \(388679230699790\) \(\nu^{15}\mathstrut -\mathstrut \) \(6358068531340576\) \(\nu^{14}\mathstrut +\mathstrut \) \(4330758126391390\) \(\nu^{13}\mathstrut +\mathstrut \) \(56693825410091331\) \(\nu^{12}\mathstrut -\mathstrut \) \(81374284776496999\) \(\nu^{11}\mathstrut -\mathstrut \) \(233289544590126007\) \(\nu^{10}\mathstrut +\mathstrut \) \(444417916482658529\) \(\nu^{9}\mathstrut +\mathstrut \) \(425485922166118924\) \(\nu^{8}\mathstrut -\mathstrut \) \(1042263821748804701\) \(\nu^{7}\mathstrut -\mathstrut \) \(220021774917612470\) \(\nu^{6}\mathstrut +\mathstrut \) \(975909258044316960\) \(\nu^{5}\mathstrut -\mathstrut \) \(111684278643633409\) \(\nu^{4}\mathstrut -\mathstrut \) \(207875405300109133\) \(\nu^{3}\mathstrut +\mathstrut \) \(6733463685372634\) \(\nu^{2}\mathstrut +\mathstrut \) \(1177219773180376\) \(\nu\mathstrut -\mathstrut \) \(1739958772516472\)\()/\)\(443297728616788\)
\(\beta_{14}\)\(=\)\((\)\(20738042771486\) \(\nu^{17}\mathstrut -\mathstrut \) \(150148962527149\) \(\nu^{16}\mathstrut -\mathstrut \) \(193241956632582\) \(\nu^{15}\mathstrut +\mathstrut \) \(3518231938451895\) \(\nu^{14}\mathstrut -\mathstrut \) \(2860459078067425\) \(\nu^{13}\mathstrut -\mathstrut \) \(30701744481925376\) \(\nu^{12}\mathstrut +\mathstrut \) \(48866970211736716\) \(\nu^{11}\mathstrut +\mathstrut \) \(121031595408829947\) \(\nu^{10}\mathstrut -\mathstrut \) \(260172497518722414\) \(\nu^{9}\mathstrut -\mathstrut \) \(197112789215297770\) \(\nu^{8}\mathstrut +\mathstrut \) \(597183645669358292\) \(\nu^{7}\mathstrut +\mathstrut \) \(41320595955814197\) \(\nu^{6}\mathstrut -\mathstrut \) \(538072198616116060\) \(\nu^{5}\mathstrut +\mathstrut \) \(123850088209191205\) \(\nu^{4}\mathstrut +\mathstrut \) \(96432842921397039\) \(\nu^{3}\mathstrut -\mathstrut \) \(6433339283442403\) \(\nu^{2}\mathstrut -\mathstrut \) \(613174527785290\) \(\nu\mathstrut +\mathstrut \) \(150607842162690\)\()/\)\(221648864308394\)
\(\beta_{15}\)\(=\)\((\)\(2295821306687\) \(\nu^{17}\mathstrut -\mathstrut \) \(17156001995758\) \(\nu^{16}\mathstrut -\mathstrut \) \(17887868681760\) \(\nu^{15}\mathstrut +\mathstrut \) \(397008940133822\) \(\nu^{14}\mathstrut -\mathstrut \) \(403834534801947\) \(\nu^{13}\mathstrut -\mathstrut \) \(3384225865140037\) \(\nu^{12}\mathstrut +\mathstrut \) \(6248449737443027\) \(\nu^{11}\mathstrut +\mathstrut \) \(12644424093024211\) \(\nu^{10}\mathstrut -\mathstrut \) \(32751398149840752\) \(\nu^{9}\mathstrut -\mathstrut \) \(17011627017926707\) \(\nu^{8}\mathstrut +\mathstrut \) \(75624975123556534\) \(\nu^{7}\mathstrut -\mathstrut \) \(8189006740723728\) \(\nu^{6}\mathstrut -\mathstrut \) \(70785359758478719\) \(\nu^{5}\mathstrut +\mathstrut \) \(28358086895742889\) \(\nu^{4}\mathstrut +\mathstrut \) \(16444614084075122\) \(\nu^{3}\mathstrut -\mathstrut \) \(6470040163960736\) \(\nu^{2}\mathstrut -\mathstrut \) \(1228312191468324\) \(\nu\mathstrut +\mathstrut \) \(240505445116088\)\()/\)\(20618499005432\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(83524578661994\) \(\nu^{17}\mathstrut +\mathstrut \) \(599721831065959\) \(\nu^{16}\mathstrut +\mathstrut \) \(816991773268750\) \(\nu^{15}\mathstrut -\mathstrut \) \(14153817184728346\) \(\nu^{14}\mathstrut +\mathstrut \) \(10718678082361394\) \(\nu^{13}\mathstrut +\mathstrut \) \(125010733622999949\) \(\nu^{12}\mathstrut -\mathstrut \) \(191483409046659589\) \(\nu^{11}\mathstrut -\mathstrut \) \(504321692370034015\) \(\nu^{10}\mathstrut +\mathstrut \) \(1040978183727296037\) \(\nu^{9}\mathstrut +\mathstrut \) \(870661205166077602\) \(\nu^{8}\mathstrut -\mathstrut \) \(2463306567788116773\) \(\nu^{7}\mathstrut -\mathstrut \) \(306122927946215072\) \(\nu^{6}\mathstrut +\mathstrut \) \(2390048740897147574\) \(\nu^{5}\mathstrut -\mathstrut \) \(446980437277724669\) \(\nu^{4}\mathstrut -\mathstrut \) \(625847651795991035\) \(\nu^{3}\mathstrut +\mathstrut \) \(90322232452229724\) \(\nu^{2}\mathstrut +\mathstrut \) \(41997132578678184\) \(\nu\mathstrut -\mathstrut \) \(4915948560145472\)\()/\)\(443297728616788\)
\(\beta_{17}\)\(=\)\((\)\(-\)\(180069207323109\) \(\nu^{17}\mathstrut +\mathstrut \) \(1359427611894492\) \(\nu^{16}\mathstrut +\mathstrut \) \(1313595443605624\) \(\nu^{15}\mathstrut -\mathstrut \) \(31312823898472886\) \(\nu^{14}\mathstrut +\mathstrut \) \(33756602890160281\) \(\nu^{13}\mathstrut +\mathstrut \) \(264706668860550137\) \(\nu^{12}\mathstrut -\mathstrut \) \(507981876728244887\) \(\nu^{11}\mathstrut -\mathstrut \) \(971137358684465475\) \(\nu^{10}\mathstrut +\mathstrut \) \(2636145445743970938\) \(\nu^{9}\mathstrut +\mathstrut \) \(1221007760565960533\) \(\nu^{8}\mathstrut -\mathstrut \) \(6020469856393215568\) \(\nu^{7}\mathstrut +\mathstrut \) \(884958557799693236\) \(\nu^{6}\mathstrut +\mathstrut \) \(5492208908050632229\) \(\nu^{5}\mathstrut -\mathstrut \) \(2390650266780158997\) \(\nu^{4}\mathstrut -\mathstrut \) \(1111520133995208436\) \(\nu^{3}\mathstrut +\mathstrut \) \(481468355205050148\) \(\nu^{2}\mathstrut +\mathstrut \) \(62720763267435044\) \(\nu\mathstrut -\mathstrut \) \(17682230665574032\)\()/\)\(886595457233576\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{16}\mathstrut +\mathstrut \) \(3\) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(30\)
\(\nu^{5}\)\(=\)\(-\)\(12\) \(\beta_{17}\mathstrut +\mathstrut \) \(2\) \(\beta_{16}\mathstrut -\mathstrut \) \(7\) \(\beta_{15}\mathstrut +\mathstrut \) \(3\) \(\beta_{14}\mathstrut +\mathstrut \) \(3\) \(\beta_{13}\mathstrut +\mathstrut \) \(13\) \(\beta_{12}\mathstrut +\mathstrut \) \(10\) \(\beta_{11}\mathstrut +\mathstrut \) \(10\) \(\beta_{10}\mathstrut -\mathstrut \) \(15\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(13\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(63\) \(\beta_{1}\mathstrut +\mathstrut \) \(18\)
\(\nu^{6}\)\(=\)\(-\)\(16\) \(\beta_{17}\mathstrut +\mathstrut \) \(15\) \(\beta_{16}\mathstrut +\mathstrut \) \(41\) \(\beta_{15}\mathstrut -\mathstrut \) \(7\) \(\beta_{14}\mathstrut +\mathstrut \) \(17\) \(\beta_{13}\mathstrut +\mathstrut \) \(17\) \(\beta_{12}\mathstrut +\mathstrut \) \(12\) \(\beta_{11}\mathstrut +\mathstrut \) \(13\) \(\beta_{10}\mathstrut -\mathstrut \) \(50\) \(\beta_{9}\mathstrut +\mathstrut \) \(16\) \(\beta_{8}\mathstrut -\mathstrut \) \(22\) \(\beta_{7}\mathstrut -\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(30\) \(\beta_{4}\mathstrut +\mathstrut \) \(28\) \(\beta_{3}\mathstrut +\mathstrut \) \(33\) \(\beta_{2}\mathstrut +\mathstrut \) \(80\) \(\beta_{1}\mathstrut +\mathstrut \) \(246\)
\(\nu^{7}\)\(=\)\(-\)\(127\) \(\beta_{17}\mathstrut +\mathstrut \) \(37\) \(\beta_{16}\mathstrut -\mathstrut \) \(29\) \(\beta_{15}\mathstrut +\mathstrut \) \(52\) \(\beta_{14}\mathstrut +\mathstrut \) \(57\) \(\beta_{13}\mathstrut +\mathstrut \) \(146\) \(\beta_{12}\mathstrut +\mathstrut \) \(90\) \(\beta_{11}\mathstrut +\mathstrut \) \(90\) \(\beta_{10}\mathstrut -\mathstrut \) \(193\) \(\beta_{9}\mathstrut +\mathstrut \) \(31\) \(\beta_{8}\mathstrut -\mathstrut \) \(148\) \(\beta_{7}\mathstrut -\mathstrut \) \(22\) \(\beta_{6}\mathstrut -\mathstrut \) \(87\) \(\beta_{5}\mathstrut +\mathstrut \) \(43\) \(\beta_{4}\mathstrut +\mathstrut \) \(43\) \(\beta_{3}\mathstrut -\mathstrut \) \(46\) \(\beta_{2}\mathstrut +\mathstrut \) \(508\) \(\beta_{1}\mathstrut +\mathstrut \) \(227\)
\(\nu^{8}\)\(=\)\(-\)\(222\) \(\beta_{17}\mathstrut +\mathstrut \) \(185\) \(\beta_{16}\mathstrut +\mathstrut \) \(456\) \(\beta_{15}\mathstrut -\mathstrut \) \(20\) \(\beta_{14}\mathstrut +\mathstrut \) \(238\) \(\beta_{13}\mathstrut +\mathstrut \) \(240\) \(\beta_{12}\mathstrut +\mathstrut \) \(129\) \(\beta_{11}\mathstrut +\mathstrut \) \(144\) \(\beta_{10}\mathstrut -\mathstrut \) \(663\) \(\beta_{9}\mathstrut +\mathstrut \) \(186\) \(\beta_{8}\mathstrut -\mathstrut \) \(326\) \(\beta_{7}\mathstrut -\mathstrut \) \(53\) \(\beta_{6}\mathstrut -\mathstrut \) \(18\) \(\beta_{5}\mathstrut +\mathstrut \) \(362\) \(\beta_{4}\mathstrut +\mathstrut \) \(321\) \(\beta_{3}\mathstrut +\mathstrut \) \(126\) \(\beta_{2}\mathstrut +\mathstrut \) \(719\) \(\beta_{1}\mathstrut +\mathstrut \) \(2115\)
\(\nu^{9}\)\(=\)\(-\)\(1349\) \(\beta_{17}\mathstrut +\mathstrut \) \(511\) \(\beta_{16}\mathstrut +\mathstrut \) \(86\) \(\beta_{15}\mathstrut +\mathstrut \) \(637\) \(\beta_{14}\mathstrut +\mathstrut \) \(824\) \(\beta_{13}\mathstrut +\mathstrut \) \(1582\) \(\beta_{12}\mathstrut +\mathstrut \) \(826\) \(\beta_{11}\mathstrut +\mathstrut \) \(814\) \(\beta_{10}\mathstrut -\mathstrut \) \(2344\) \(\beta_{9}\mathstrut +\mathstrut \) \(377\) \(\beta_{8}\mathstrut -\mathstrut \) \(1635\) \(\beta_{7}\mathstrut -\mathstrut \) \(297\) \(\beta_{6}\mathstrut -\mathstrut \) \(716\) \(\beta_{5}\mathstrut +\mathstrut \) \(668\) \(\beta_{4}\mathstrut +\mathstrut \) \(637\) \(\beta_{3}\mathstrut -\mathstrut \) \(741\) \(\beta_{2}\mathstrut +\mathstrut \) \(4180\) \(\beta_{1}\mathstrut +\mathstrut \) \(2618\)
\(\nu^{10}\)\(=\)\(-\)\(2892\) \(\beta_{17}\mathstrut +\mathstrut \) \(2146\) \(\beta_{16}\mathstrut +\mathstrut \) \(4803\) \(\beta_{15}\mathstrut +\mathstrut \) \(250\) \(\beta_{14}\mathstrut +\mathstrut \) \(3098\) \(\beta_{13}\mathstrut +\mathstrut \) \(3110\) \(\beta_{12}\mathstrut +\mathstrut \) \(1413\) \(\beta_{11}\mathstrut +\mathstrut \) \(1546\) \(\beta_{10}\mathstrut -\mathstrut \) \(8077\) \(\beta_{9}\mathstrut +\mathstrut \) \(1940\) \(\beta_{8}\mathstrut -\mathstrut \) \(4175\) \(\beta_{7}\mathstrut -\mathstrut \) \(661\) \(\beta_{6}\mathstrut -\mathstrut \) \(185\) \(\beta_{5}\mathstrut +\mathstrut \) \(4110\) \(\beta_{4}\mathstrut +\mathstrut \) \(3477\) \(\beta_{3}\mathstrut -\mathstrut \) \(436\) \(\beta_{2}\mathstrut +\mathstrut \) \(6511\) \(\beta_{1}\mathstrut +\mathstrut \) \(18881\)
\(\nu^{11}\)\(=\)\(-\)\(14580\) \(\beta_{17}\mathstrut +\mathstrut \) \(6321\) \(\beta_{16}\mathstrut +\mathstrut \) \(4094\) \(\beta_{15}\mathstrut +\mathstrut \) \(6863\) \(\beta_{14}\mathstrut +\mathstrut \) \(10740\) \(\beta_{13}\mathstrut +\mathstrut \) \(16972\) \(\beta_{12}\mathstrut +\mathstrut \) \(7912\) \(\beta_{11}\mathstrut +\mathstrut \) \(7552\) \(\beta_{10}\mathstrut -\mathstrut \) \(27508\) \(\beta_{9}\mathstrut +\mathstrut \) \(4182\) \(\beta_{8}\mathstrut -\mathstrut \) \(17889\) \(\beta_{7}\mathstrut -\mathstrut \) \(3321\) \(\beta_{6}\mathstrut -\mathstrut \) \(5633\) \(\beta_{5}\mathstrut +\mathstrut \) \(9031\) \(\beta_{4}\mathstrut +\mathstrut \) \(8116\) \(\beta_{3}\mathstrut -\mathstrut \) \(10320\) \(\beta_{2}\mathstrut +\mathstrut \) \(34945\) \(\beta_{1}\mathstrut +\mathstrut \) \(29208\)
\(\nu^{12}\)\(=\)\(-\)\(35985\) \(\beta_{17}\mathstrut +\mathstrut \) \(24210\) \(\beta_{16}\mathstrut +\mathstrut \) \(49797\) \(\beta_{15}\mathstrut +\mathstrut \) \(6043\) \(\beta_{14}\mathstrut +\mathstrut \) \(38481\) \(\beta_{13}\mathstrut +\mathstrut \) \(38185\) \(\beta_{12}\mathstrut +\mathstrut \) \(15877\) \(\beta_{11}\mathstrut +\mathstrut \) \(16448\) \(\beta_{10}\mathstrut -\mathstrut \) \(94249\) \(\beta_{9}\mathstrut +\mathstrut \) \(19291\) \(\beta_{8}\mathstrut -\mathstrut \) \(49876\) \(\beta_{7}\mathstrut -\mathstrut \) \(7169\) \(\beta_{6}\mathstrut -\mathstrut \) \(1155\) \(\beta_{5}\mathstrut +\mathstrut \) \(45651\) \(\beta_{4}\mathstrut +\mathstrut \) \(36934\) \(\beta_{3}\mathstrut -\mathstrut \) \(19306\) \(\beta_{2}\mathstrut +\mathstrut \) \(59197\) \(\beta_{1}\mathstrut +\mathstrut \) \(174104\)
\(\nu^{13}\)\(=\)\(-\)\(159475\) \(\beta_{17}\mathstrut +\mathstrut \) \(74086\) \(\beta_{16}\mathstrut +\mathstrut \) \(68613\) \(\beta_{15}\mathstrut +\mathstrut \) \(69730\) \(\beta_{14}\mathstrut +\mathstrut \) \(132398\) \(\beta_{13}\mathstrut +\mathstrut \) \(181738\) \(\beta_{12}\mathstrut +\mathstrut \) \(78951\) \(\beta_{11}\mathstrut +\mathstrut \) \(71986\) \(\beta_{10}\mathstrut -\mathstrut \) \(315341\) \(\beta_{9}\mathstrut +\mathstrut \) \(44109\) \(\beta_{8}\mathstrut -\mathstrut \) \(194856\) \(\beta_{7}\mathstrut -\mathstrut \) \(33746\) \(\beta_{6}\mathstrut -\mathstrut \) \(41832\) \(\beta_{5}\mathstrut +\mathstrut \) \(113140\) \(\beta_{4}\mathstrut +\mathstrut \) \(95878\) \(\beta_{3}\mathstrut -\mathstrut \) \(132962\) \(\beta_{2}\mathstrut +\mathstrut \) \(295770\) \(\beta_{1}\mathstrut +\mathstrut \) \(320459\)
\(\nu^{14}\)\(=\)\(-\)\(432410\) \(\beta_{17}\mathstrut +\mathstrut \) \(268942\) \(\beta_{16}\mathstrut +\mathstrut \) \(514908\) \(\beta_{15}\mathstrut +\mathstrut \) \(87030\) \(\beta_{14}\mathstrut +\mathstrut \) \(461917\) \(\beta_{13}\mathstrut +\mathstrut \) \(451976\) \(\beta_{12}\mathstrut +\mathstrut \) \(180676\) \(\beta_{11}\mathstrut +\mathstrut \) \(174265\) \(\beta_{10}\mathstrut -\mathstrut \) \(1072029\) \(\beta_{9}\mathstrut +\mathstrut \) \(187274\) \(\beta_{8}\mathstrut -\mathstrut \) \(573694\) \(\beta_{7}\mathstrut -\mathstrut \) \(72160\) \(\beta_{6}\mathstrut +\mathstrut \) \(1302\) \(\beta_{5}\mathstrut +\mathstrut \) \(502291\) \(\beta_{4}\mathstrut +\mathstrut \) \(389798\) \(\beta_{3}\mathstrut -\mathstrut \) \(329743\) \(\beta_{2}\mathstrut +\mathstrut \) \(538929\) \(\beta_{1}\mathstrut +\mathstrut \) \(1650712\)
\(\nu^{15}\)\(=\)\(-\)\(1753792\) \(\beta_{17}\mathstrut +\mathstrut \) \(842636\) \(\beta_{16}\mathstrut +\mathstrut \) \(926227\) \(\beta_{15}\mathstrut +\mathstrut \) \(688583\) \(\beta_{14}\mathstrut +\mathstrut \) \(1575225\) \(\beta_{13}\mathstrut +\mathstrut \) \(1947310\) \(\beta_{12}\mathstrut +\mathstrut \) \(813383\) \(\beta_{11}\mathstrut +\mathstrut \) \(702648\) \(\beta_{10}\mathstrut -\mathstrut \) \(3554459\) \(\beta_{9}\mathstrut +\mathstrut \) \(450659\) \(\beta_{8}\mathstrut -\mathstrut \) \(2116396\) \(\beta_{7}\mathstrut -\mathstrut \) \(323851\) \(\beta_{6}\mathstrut -\mathstrut \) \(281984\) \(\beta_{5}\mathstrut +\mathstrut \) \(1352851\) \(\beta_{4}\mathstrut +\mathstrut \) \(1086363\) \(\beta_{3}\mathstrut -\mathstrut \) \(1633101\) \(\beta_{2}\mathstrut +\mathstrut \) \(2526944\) \(\beta_{1}\mathstrut +\mathstrut \) \(3479739\)
\(\nu^{16}\)\(=\)\(-\)\(5060507\) \(\beta_{17}\mathstrut +\mathstrut \) \(2959032\) \(\beta_{16}\mathstrut +\mathstrut \) \(5336145\) \(\beta_{15}\mathstrut +\mathstrut \) \(1066828\) \(\beta_{14}\mathstrut +\mathstrut \) \(5404657\) \(\beta_{13}\mathstrut +\mathstrut \) \(5213807\) \(\beta_{12}\mathstrut +\mathstrut \) \(2059918\) \(\beta_{11}\mathstrut +\mathstrut \) \(1841870\) \(\beta_{10}\mathstrut -\mathstrut \) \(11992809\) \(\beta_{9}\mathstrut +\mathstrut \) \(1794652\) \(\beta_{8}\mathstrut -\mathstrut \) \(6451975\) \(\beta_{7}\mathstrut -\mathstrut \) \(692272\) \(\beta_{6}\mathstrut +\mathstrut \) \(185229\) \(\beta_{5}\mathstrut +\mathstrut \) \(5498809\) \(\beta_{4}\mathstrut +\mathstrut \) \(4108399\) \(\beta_{3}\mathstrut -\mathstrut \) \(4580523\) \(\beta_{2}\mathstrut +\mathstrut \) \(4902760\) \(\beta_{1}\mathstrut +\mathstrut \) \(16021039\)
\(\nu^{17}\)\(=\)\(-\)\(19305219\) \(\beta_{17}\mathstrut +\mathstrut \) \(9411007\) \(\beta_{16}\mathstrut +\mathstrut \) \(11390705\) \(\beta_{15}\mathstrut +\mathstrut \) \(6711080\) \(\beta_{14}\mathstrut +\mathstrut \) \(18284331\) \(\beta_{13}\mathstrut +\mathstrut \) \(20889069\) \(\beta_{12}\mathstrut +\mathstrut \) \(8568643\) \(\beta_{11}\mathstrut +\mathstrut \) \(6992451\) \(\beta_{10}\mathstrut -\mathstrut \) \(39568172\) \(\beta_{9}\mathstrut +\mathstrut \) \(4507699\) \(\beta_{8}\mathstrut -\mathstrut \) \(22935540\) \(\beta_{7}\mathstrut -\mathstrut \) \(2988949\) \(\beta_{6}\mathstrut -\mathstrut \) \(1535353\) \(\beta_{5}\mathstrut +\mathstrut \) \(15690584\) \(\beta_{4}\mathstrut +\mathstrut \) \(12014672\) \(\beta_{3}\mathstrut -\mathstrut \) \(19426758\) \(\beta_{2}\mathstrut +\mathstrut \) \(21736118\) \(\beta_{1}\mathstrut +\mathstrut \) \(37517387\)

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
−2.80641
−2.57211
−2.15409
−1.74374
−1.35164
−0.397638
−0.267006
0.0880346
0.154866
0.839537
0.877348
1.33821
1.56216
2.54274
2.60297
2.97166
3.05303
3.26206
0 −2.80641 0 3.19831 0 4.58875 0 4.87591 0
1.2 0 −2.57211 0 −0.393688 0 −1.17366 0 3.61576 0
1.3 0 −2.15409 0 −0.228433 0 2.56317 0 1.64008 0
1.4 0 −1.74374 0 0.394783 0 −1.78199 0 0.0406123 0
1.5 0 −1.35164 0 1.29976 0 −4.65863 0 −1.17306 0
1.6 0 −0.397638 0 3.51710 0 2.09256 0 −2.84188 0
1.7 0 −0.267006 0 −2.18930 0 1.68463 0 −2.92871 0
1.8 0 0.0880346 0 −2.20461 0 0.606938 0 −2.99225 0
1.9 0 0.154866 0 −1.21070 0 −0.213015 0 −2.97602 0
1.10 0 0.839537 0 −4.04363 0 −2.98927 0 −2.29518 0
1.11 0 0.877348 0 −1.13091 0 −4.50700 0 −2.23026 0
1.12 0 1.33821 0 3.45044 0 −1.09754 0 −1.20921 0
1.13 0 1.56216 0 −0.592812 0 4.71818 0 −0.559654 0
1.14 0 2.54274 0 3.21926 0 1.46547 0 3.46555 0
1.15 0 2.60297 0 0.183919 0 3.24534 0 3.77545 0
1.16 0 2.97166 0 −4.20066 0 0.274921 0 5.83079 0
1.17 0 3.05303 0 2.19982 0 −4.80385 0 6.32102 0
1.18 0 3.26206 0 2.73134 0 1.98500 0 7.64104 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)
\(59\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\):

\(T_{3}^{18} - \cdots\)
\(T_{5}^{18} - \cdots\)