Properties

Label 4016.2.a.k
Level 4016
Weight 2
Character orbit 4016.a
Self dual Yes
Analytic conductor 32.068
Analytic rank 0
Dimension 17
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4016.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)  \(=\)  \( q\) \( + \beta_{4} q^{3} \) \( + \beta_{6} q^{5} \) \( + \beta_{13} q^{7} \) \( + ( 1 + \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{4} q^{3} \) \( + \beta_{6} q^{5} \) \( + \beta_{13} q^{7} \) \( + ( 1 + \beta_{3} ) q^{9} \) \( + ( 1 + \beta_{1} + \beta_{6} - \beta_{11} - \beta_{16} ) q^{11} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{6} - \beta_{10} ) q^{13} \) \( + ( \beta_{3} - \beta_{4} - \beta_{10} + \beta_{11} - \beta_{15} ) q^{15} \) \( + ( -1 - \beta_{1} + \beta_{5} + \beta_{6} - \beta_{15} ) q^{17} \) \( + ( -2 - \beta_{1} + \beta_{2} - \beta_{6} + \beta_{11} + \beta_{12} + \beta_{16} ) q^{19} \) \( + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{16} ) q^{21} \) \( + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{14} ) q^{23} \) \( + ( 2 - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{15} ) q^{25} \) \( + ( - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{9} + \beta_{14} ) q^{27} \) \( + ( 2 + \beta_{7} + \beta_{8} - \beta_{14} ) q^{29} \) \( + ( \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{31} \) \( + ( \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} + 2 \beta_{15} ) q^{33} \) \( + ( 2 - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{11} + \beta_{13} - 2 \beta_{14} ) q^{35} \) \( + ( - \beta_{1} + 2 \beta_{4} + 3 \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{14} + \beta_{16} ) q^{37} \) \( + ( - \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + \beta_{10} - \beta_{11} - \beta_{16} ) q^{39} \) \( + ( 1 + \beta_{1} - \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{41} \) \( + ( \beta_{1} + \beta_{3} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{12} + \beta_{14} ) q^{43} \) \( + ( -2 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - 3 \beta_{14} + \beta_{16} ) q^{45} \) \( + ( 2 - \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{13} - \beta_{14} ) q^{47} \) \( + ( 5 + 2 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{8} + \beta_{9} - \beta_{11} + \beta_{15} - \beta_{16} ) q^{49} \) \( + ( -1 - \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{14} ) q^{51} \) \( + ( - \beta_{2} + \beta_{6} - \beta_{8} - \beta_{14} - \beta_{15} - \beta_{16} ) q^{53} \) \( + ( -1 + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{11} + \beta_{15} + \beta_{16} ) q^{55} \) \( + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{14} - \beta_{16} ) q^{57} \) \( + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{9} - \beta_{10} + \beta_{14} - 2 \beta_{15} + \beta_{16} ) q^{59} \) \( + ( 7 + 2 \beta_{1} - \beta_{3} - 3 \beta_{5} - 2 \beta_{6} - \beta_{8} + 3 \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{14} + \beta_{15} ) q^{61} \) \( + ( 5 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{13} + 3 \beta_{15} ) q^{63} \) \( + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} + \beta_{13} + \beta_{16} ) q^{65} \) \( + ( - \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{67} \) \( + ( -1 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{8} - 2 \beta_{9} + \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{69} \) \( + ( 4 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{9} - 3 \beta_{11} + \beta_{15} - \beta_{16} ) q^{71} \) \( + ( - \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{7} - 2 \beta_{11} - \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{73} \) \( + ( 2 + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{75} \) \( + ( -3 - \beta_{1} + 2 \beta_{4} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} + 2 \beta_{16} ) q^{77} \) \( + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{13} - \beta_{15} ) q^{79} \) \( + ( \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{11} - 2 \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{81} \) \( + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{15} + \beta_{16} ) q^{83} \) \( + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{85} \) \( + ( -2 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} - \beta_{12} + 2 \beta_{14} - 2 \beta_{15} ) q^{87} \) \( + ( 4 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{11} - \beta_{12} + 2 \beta_{14} - 2 \beta_{16} ) q^{89} \) \( + ( -1 - \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} + 3 \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{15} + \beta_{16} ) q^{91} \) \( + ( 4 + 2 \beta_{1} + \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} ) q^{93} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - 3 \beta_{8} + 2 \beta_{11} + \beta_{12} - 3 \beta_{14} ) q^{95} \) \( + ( -3 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - 3 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{14} + 2 \beta_{16} ) q^{97} \) \( + ( 6 + 4 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{16} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(17q \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 25q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(17q \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 25q^{9} \) \(\mathstrut +\mathstrut q^{11} \) \(\mathstrut +\mathstrut 22q^{13} \) \(\mathstrut +\mathstrut 8q^{15} \) \(\mathstrut -\mathstrut q^{17} \) \(\mathstrut -\mathstrut 13q^{19} \) \(\mathstrut +\mathstrut 25q^{21} \) \(\mathstrut +\mathstrut 2q^{23} \) \(\mathstrut +\mathstrut 32q^{25} \) \(\mathstrut +\mathstrut 15q^{27} \) \(\mathstrut +\mathstrut 28q^{29} \) \(\mathstrut -\mathstrut 12q^{31} \) \(\mathstrut -\mathstrut 16q^{33} \) \(\mathstrut +\mathstrut 15q^{35} \) \(\mathstrut +\mathstrut 27q^{37} \) \(\mathstrut -\mathstrut 13q^{39} \) \(\mathstrut -\mathstrut q^{41} \) \(\mathstrut -\mathstrut 9q^{43} \) \(\mathstrut -\mathstrut 7q^{45} \) \(\mathstrut +\mathstrut 20q^{47} \) \(\mathstrut +\mathstrut 32q^{49} \) \(\mathstrut +\mathstrut 2q^{51} \) \(\mathstrut +\mathstrut q^{53} \) \(\mathstrut +\mathstrut 11q^{55} \) \(\mathstrut -\mathstrut 24q^{57} \) \(\mathstrut +\mathstrut 20q^{59} \) \(\mathstrut +\mathstrut 59q^{61} \) \(\mathstrut +\mathstrut 41q^{63} \) \(\mathstrut -\mathstrut 14q^{65} \) \(\mathstrut -\mathstrut 15q^{67} \) \(\mathstrut +\mathstrut 38q^{69} \) \(\mathstrut +\mathstrut 26q^{71} \) \(\mathstrut +\mathstrut 8q^{73} \) \(\mathstrut +\mathstrut 20q^{75} \) \(\mathstrut -\mathstrut 33q^{79} \) \(\mathstrut +\mathstrut 29q^{81} \) \(\mathstrut +\mathstrut 67q^{85} \) \(\mathstrut +\mathstrut 11q^{87} \) \(\mathstrut +\mathstrut 11q^{89} \) \(\mathstrut +\mathstrut 2q^{91} \) \(\mathstrut +\mathstrut 28q^{93} \) \(\mathstrut +\mathstrut 8q^{95} \) \(\mathstrut -\mathstrut 10q^{97} \) \(\mathstrut +\mathstrut 36q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{17}\mathstrut -\mathstrut \) \(2\) \(x^{16}\mathstrut -\mathstrut \) \(28\) \(x^{15}\mathstrut +\mathstrut \) \(54\) \(x^{14}\mathstrut +\mathstrut \) \(317\) \(x^{13}\mathstrut -\mathstrut \) \(582\) \(x^{12}\mathstrut -\mathstrut \) \(1867\) \(x^{11}\mathstrut +\mathstrut \) \(3178\) \(x^{10}\mathstrut +\mathstrut \) \(6186\) \(x^{9}\mathstrut -\mathstrut \) \(9216\) \(x^{8}\mathstrut -\mathstrut \) \(11921\) \(x^{7}\mathstrut +\mathstrut \) \(13680\) \(x^{6}\mathstrut +\mathstrut \) \(13752\) \(x^{5}\mathstrut -\mathstrut \) \(9400\) \(x^{4}\mathstrut -\mathstrut \) \(8800\) \(x^{3}\mathstrut +\mathstrut \) \(1920\) \(x^{2}\mathstrut +\mathstrut \) \(2240\) \(x\mathstrut +\mathstrut \) \(256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(22\) \(\nu^{16}\mathstrut +\mathstrut \) \(15\) \(\nu^{15}\mathstrut +\mathstrut \) \(647\) \(\nu^{14}\mathstrut -\mathstrut \) \(374\) \(\nu^{13}\mathstrut -\mathstrut \) \(7676\) \(\nu^{12}\mathstrut +\mathstrut \) \(3513\) \(\nu^{11}\mathstrut +\mathstrut \) \(46913\) \(\nu^{10}\mathstrut -\mathstrut \) \(14575\) \(\nu^{9}\mathstrut -\mathstrut \) \(156321\) \(\nu^{8}\mathstrut +\mathstrut \) \(19540\) \(\nu^{7}\mathstrut +\mathstrut \) \(276564\) \(\nu^{6}\mathstrut +\mathstrut \) \(28861\) \(\nu^{5}\mathstrut -\mathstrut \) \(231693\) \(\nu^{4}\mathstrut -\mathstrut \) \(78844\) \(\nu^{3}\mathstrut +\mathstrut \) \(59404\) \(\nu^{2}\mathstrut +\mathstrut \) \(34224\) \(\nu\mathstrut +\mathstrut \) \(4304\)\()/304\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(85\) \(\nu^{16}\mathstrut +\mathstrut \) \(178\) \(\nu^{15}\mathstrut +\mathstrut \) \(2264\) \(\nu^{14}\mathstrut -\mathstrut \) \(4694\) \(\nu^{13}\mathstrut -\mathstrut \) \(23617\) \(\nu^{12}\mathstrut +\mathstrut \) \(48558\) \(\nu^{11}\mathstrut +\mathstrut \) \(120979\) \(\nu^{10}\mathstrut -\mathstrut \) \(246626\) \(\nu^{9}\mathstrut -\mathstrut \) \(312422\) \(\nu^{8}\mathstrut +\mathstrut \) \(626264\) \(\nu^{7}\mathstrut +\mathstrut \) \(384541\) \(\nu^{6}\mathstrut -\mathstrut \) \(723064\) \(\nu^{5}\mathstrut -\mathstrut \) \(234500\) \(\nu^{4}\mathstrut +\mathstrut \) \(335368\) \(\nu^{3}\mathstrut +\mathstrut \) \(66416\) \(\nu^{2}\mathstrut -\mathstrut \) \(42336\) \(\nu\mathstrut -\mathstrut \) \(3200\)\()/1216\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(69\) \(\nu^{16}\mathstrut +\mathstrut \) \(212\) \(\nu^{15}\mathstrut +\mathstrut \) \(1752\) \(\nu^{14}\mathstrut -\mathstrut \) \(5638\) \(\nu^{13}\mathstrut -\mathstrut \) \(17157\) \(\nu^{12}\mathstrut +\mathstrut \) \(59424\) \(\nu^{11}\mathstrut +\mathstrut \) \(79979\) \(\nu^{10}\mathstrut -\mathstrut \) \(313888\) \(\nu^{9}\mathstrut -\mathstrut \) \(174414\) \(\nu^{8}\mathstrut +\mathstrut \) \(866052\) \(\nu^{7}\mathstrut +\mathstrut \) \(146661\) \(\nu^{6}\mathstrut -\mathstrut \) \(1198330\) \(\nu^{5}\mathstrut -\mathstrut \) \(54216\) \(\nu^{4}\mathstrut +\mathstrut \) \(779784\) \(\nu^{3}\mathstrut +\mathstrut \) \(55216\) \(\nu^{2}\mathstrut -\mathstrut \) \(183520\) \(\nu\mathstrut -\mathstrut \) \(27776\)\()/1216\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{16} - \nu^{15} + 31 \nu^{14} + 30 \nu^{13} - 391 \nu^{12} - 363 \nu^{11} + 2570 \nu^{10} + 2275 \nu^{9} - 9331 \nu^{8} - 7882 \nu^{7} + 18119 \nu^{6} + 14855 \nu^{5} - 16325 \nu^{4} - 13706 \nu^{3} + 4320 \nu^{2} + 4336 \nu + 448 \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(42\) \(\nu^{16}\mathstrut -\mathstrut \) \(37\) \(\nu^{15}\mathstrut +\mathstrut \) \(1306\) \(\nu^{14}\mathstrut +\mathstrut \) \(1148\) \(\nu^{13}\mathstrut -\mathstrut \) \(16568\) \(\nu^{12}\mathstrut -\mathstrut \) \(14297\) \(\nu^{11}\mathstrut +\mathstrut \) \(109772\) \(\nu^{10}\mathstrut +\mathstrut \) \(91723\) \(\nu^{9}\mathstrut -\mathstrut \) \(401886\) \(\nu^{8}\mathstrut -\mathstrut \) \(323534\) \(\nu^{7}\mathstrut +\mathstrut \) \(783066\) \(\nu^{6}\mathstrut +\mathstrut \) \(619761\) \(\nu^{5}\mathstrut -\mathstrut \) \(694624\) \(\nu^{4}\mathstrut -\mathstrut \) \(587700\) \(\nu^{3}\mathstrut +\mathstrut \) \(170912\) \(\nu^{2}\mathstrut +\mathstrut \) \(190336\) \(\nu\mathstrut +\mathstrut \) \(23776\)\()/608\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(235\) \(\nu^{16}\mathstrut +\mathstrut \) \(244\) \(\nu^{15}\mathstrut +\mathstrut \) \(6608\) \(\nu^{14}\mathstrut -\mathstrut \) \(6066\) \(\nu^{13}\mathstrut -\mathstrut \) \(74651\) \(\nu^{12}\mathstrut +\mathstrut \) \(58072\) \(\nu^{11}\mathstrut +\mathstrut \) \(432413\) \(\nu^{10}\mathstrut -\mathstrut \) \(261584\) \(\nu^{9}\mathstrut -\mathstrut \) \(1360330\) \(\nu^{8}\mathstrut +\mathstrut \) \(517908\) \(\nu^{7}\mathstrut +\mathstrut \) \(2278651\) \(\nu^{6}\mathstrut -\mathstrut \) \(214814\) \(\nu^{5}\mathstrut -\mathstrut \) \(1860528\) \(\nu^{4}\mathstrut -\mathstrut \) \(368928\) \(\nu^{3}\mathstrut +\mathstrut \) \(508704\) \(\nu^{2}\mathstrut +\mathstrut \) \(220000\) \(\nu\mathstrut +\mathstrut \) \(19264\)\()/1216\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(55\) \(\nu^{16}\mathstrut +\mathstrut \) \(66\) \(\nu^{15}\mathstrut +\mathstrut \) \(1551\) \(\nu^{14}\mathstrut -\mathstrut \) \(1676\) \(\nu^{13}\mathstrut -\mathstrut \) \(17575\) \(\nu^{12}\mathstrut +\mathstrut \) \(16544\) \(\nu^{11}\mathstrut +\mathstrut \) \(102092\) \(\nu^{10}\mathstrut -\mathstrut \) \(78608\) \(\nu^{9}\mathstrut -\mathstrut \) \(321823\) \(\nu^{8}\mathstrut +\mathstrut \) \(176682\) \(\nu^{7}\mathstrut +\mathstrut \) \(539429\) \(\nu^{6}\mathstrut -\mathstrut \) \(143716\) \(\nu^{5}\mathstrut -\mathstrut \) \(441435\) \(\nu^{4}\mathstrut -\mathstrut \) \(12012\) \(\nu^{3}\mathstrut +\mathstrut \) \(125672\) \(\nu^{2}\mathstrut +\mathstrut \) \(30688\) \(\nu\mathstrut +\mathstrut \) \(1184\)\()/304\)
\(\beta_{9}\)\(=\)\((\)\(217\) \(\nu^{16}\mathstrut -\mathstrut \) \(420\) \(\nu^{15}\mathstrut -\mathstrut \) \(5804\) \(\nu^{14}\mathstrut +\mathstrut \) \(10814\) \(\nu^{13}\mathstrut +\mathstrut \) \(61465\) \(\nu^{12}\mathstrut -\mathstrut \) \(109080\) \(\nu^{11}\mathstrut -\mathstrut \) \(326267\) \(\nu^{10}\mathstrut +\mathstrut \) \(539504\) \(\nu^{9}\mathstrut +\mathstrut \) \(908994\) \(\nu^{8}\mathstrut -\mathstrut \) \(1330756\) \(\nu^{7}\mathstrut -\mathstrut \) \(1298289\) \(\nu^{6}\mathstrut +\mathstrut \) \(1475730\) \(\nu^{5}\mathstrut +\mathstrut \) \(933932\) \(\nu^{4}\mathstrut -\mathstrut \) \(616832\) \(\nu^{3}\mathstrut -\mathstrut \) \(264000\) \(\nu^{2}\mathstrut +\mathstrut \) \(60736\) \(\nu\mathstrut +\mathstrut \) \(8384\)\()/1216\)
\(\beta_{10}\)\(=\)\((\)\(273\) \(\nu^{16}\mathstrut -\mathstrut \) \(54\) \(\nu^{15}\mathstrut -\mathstrut \) \(8052\) \(\nu^{14}\mathstrut +\mathstrut \) \(670\) \(\nu^{13}\mathstrut +\mathstrut \) \(96501\) \(\nu^{12}\mathstrut +\mathstrut \) \(3222\) \(\nu^{11}\mathstrut -\mathstrut \) \(601779\) \(\nu^{10}\mathstrut -\mathstrut \) \(92994\) \(\nu^{9}\mathstrut +\mathstrut \) \(2071842\) \(\nu^{8}\mathstrut +\mathstrut \) \(586448\) \(\nu^{7}\mathstrut -\mathstrut \) \(3830305\) \(\nu^{6}\mathstrut -\mathstrut \) \(1609452\) \(\nu^{5}\mathstrut +\mathstrut \) \(3345112\) \(\nu^{4}\mathstrut +\mathstrut \) \(1879504\) \(\nu^{3}\mathstrut -\mathstrut \) \(907248\) \(\nu^{2}\mathstrut -\mathstrut \) \(668096\) \(\nu\mathstrut -\mathstrut \) \(63040\)\()/1216\)
\(\beta_{11}\)\(=\)\((\)\(81\) \(\nu^{16}\mathstrut -\mathstrut \) \(120\) \(\nu^{15}\mathstrut -\mathstrut \) \(2231\) \(\nu^{14}\mathstrut +\mathstrut \) \(3068\) \(\nu^{13}\mathstrut +\mathstrut \) \(24529\) \(\nu^{12}\mathstrut -\mathstrut \) \(30574\) \(\nu^{11}\mathstrut -\mathstrut \) \(136930\) \(\nu^{10}\mathstrut +\mathstrut \) \(147646\) \(\nu^{9}\mathstrut +\mathstrut \) \(409343\) \(\nu^{8}\mathstrut -\mathstrut \) \(344458\) \(\nu^{7}\mathstrut -\mathstrut \) \(642219\) \(\nu^{6}\mathstrut +\mathstrut \) \(322886\) \(\nu^{5}\mathstrut +\mathstrut \) \(496127\) \(\nu^{4}\mathstrut -\mathstrut \) \(58112\) \(\nu^{3}\mathstrut -\mathstrut \) \(132396\) \(\nu^{2}\mathstrut -\mathstrut \) \(24512\) \(\nu\mathstrut -\mathstrut \) \(3424\)\()/304\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(81\) \(\nu^{16}\mathstrut +\mathstrut \) \(120\) \(\nu^{15}\mathstrut +\mathstrut \) \(2231\) \(\nu^{14}\mathstrut -\mathstrut \) \(3068\) \(\nu^{13}\mathstrut -\mathstrut \) \(24529\) \(\nu^{12}\mathstrut +\mathstrut \) \(30574\) \(\nu^{11}\mathstrut +\mathstrut \) \(136930\) \(\nu^{10}\mathstrut -\mathstrut \) \(147646\) \(\nu^{9}\mathstrut -\mathstrut \) \(409343\) \(\nu^{8}\mathstrut +\mathstrut \) \(344458\) \(\nu^{7}\mathstrut +\mathstrut \) \(642219\) \(\nu^{6}\mathstrut -\mathstrut \) \(322886\) \(\nu^{5}\mathstrut -\mathstrut \) \(496127\) \(\nu^{4}\mathstrut +\mathstrut \) \(58112\) \(\nu^{3}\mathstrut +\mathstrut \) \(132396\) \(\nu^{2}\mathstrut +\mathstrut \) \(25120\) \(\nu\mathstrut +\mathstrut \) \(3424\)\()/304\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(112\) \(\nu^{16}\mathstrut +\mathstrut \) \(85\) \(\nu^{15}\mathstrut +\mathstrut \) \(3204\) \(\nu^{14}\mathstrut -\mathstrut \) \(2018\) \(\nu^{13}\mathstrut -\mathstrut \) \(36974\) \(\nu^{12}\mathstrut +\mathstrut \) \(17893\) \(\nu^{11}\mathstrut +\mathstrut \) \(219968\) \(\nu^{10}\mathstrut -\mathstrut \) \(68557\) \(\nu^{9}\mathstrut -\mathstrut \) \(715256\) \(\nu^{8}\mathstrut +\mathstrut \) \(72676\) \(\nu^{7}\mathstrut +\mathstrut \) \(1242448\) \(\nu^{6}\mathstrut +\mathstrut \) \(185079\) \(\nu^{5}\mathstrut -\mathstrut \) \(1036154\) \(\nu^{4}\mathstrut -\mathstrut \) \(411582\) \(\nu^{3}\mathstrut +\mathstrut \) \(273948\) \(\nu^{2}\mathstrut +\mathstrut \) \(173264\) \(\nu\mathstrut +\mathstrut \) \(17904\)\()/304\)
\(\beta_{14}\)\(=\)\((\)\(455\) \(\nu^{16}\mathstrut -\mathstrut \) \(508\) \(\nu^{15}\mathstrut -\mathstrut \) \(12812\) \(\nu^{14}\mathstrut +\mathstrut \) \(12770\) \(\nu^{13}\mathstrut +\mathstrut \) \(144951\) \(\nu^{12}\mathstrut -\mathstrut \) \(124248\) \(\nu^{11}\mathstrut -\mathstrut \) \(840781\) \(\nu^{10}\mathstrut +\mathstrut \) \(576016\) \(\nu^{9}\mathstrut +\mathstrut \) \(2647622\) \(\nu^{8}\mathstrut -\mathstrut \) \(1224636\) \(\nu^{7}\mathstrut -\mathstrut \) \(4436367\) \(\nu^{6}\mathstrut +\mathstrut \) \(789678\) \(\nu^{5}\mathstrut +\mathstrut \) \(3625052\) \(\nu^{4}\mathstrut +\mathstrut \) \(418192\) \(\nu^{3}\mathstrut -\mathstrut \) \(1002880\) \(\nu^{2}\mathstrut -\mathstrut \) \(348832\) \(\nu\mathstrut -\mathstrut \) \(31296\)\()/1216\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(483\) \(\nu^{16}\mathstrut +\mathstrut \) \(268\) \(\nu^{15}\mathstrut +\mathstrut \) \(13936\) \(\nu^{14}\mathstrut -\mathstrut \) \(5874\) \(\nu^{13}\mathstrut -\mathstrut \) \(162659\) \(\nu^{12}\mathstrut +\mathstrut \) \(44784\) \(\nu^{11}\mathstrut +\mathstrut \) \(982565\) \(\nu^{10}\mathstrut -\mathstrut \) \(107976\) \(\nu^{9}\mathstrut -\mathstrut \) \(3259674\) \(\nu^{8}\mathstrut -\mathstrut \) \(264940\) \(\nu^{7}\mathstrut +\mathstrut \) \(5799123\) \(\nu^{6}\mathstrut +\mathstrut \) \(1718170\) \(\nu^{5}\mathstrut -\mathstrut \) \(4930848\) \(\nu^{4}\mathstrut -\mathstrut \) \(2477584\) \(\nu^{3}\mathstrut +\mathstrut \) \(1307936\) \(\nu^{2}\mathstrut +\mathstrut \) \(940640\) \(\nu\mathstrut +\mathstrut \) \(98624\)\()/1216\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(332\) \(\nu^{16}\mathstrut +\mathstrut \) \(159\) \(\nu^{15}\mathstrut +\mathstrut \) \(9636\) \(\nu^{14}\mathstrut -\mathstrut \) \(3364\) \(\nu^{13}\mathstrut -\mathstrut \) \(113202\) \(\nu^{12}\mathstrut +\mathstrut \) \(23535\) \(\nu^{11}\mathstrut +\mathstrut \) \(688604\) \(\nu^{10}\mathstrut -\mathstrut \) \(33845\) \(\nu^{9}\mathstrut -\mathstrut \) \(2300544\) \(\nu^{8}\mathstrut -\mathstrut \) \(307434\) \(\nu^{7}\mathstrut +\mathstrut \) \(4115096\) \(\nu^{6}\mathstrut +\mathstrut \) \(1400433\) \(\nu^{5}\mathstrut -\mathstrut \) \(3494786\) \(\nu^{4}\mathstrut -\mathstrut \) \(1908228\) \(\nu^{3}\mathstrut +\mathstrut \) \(903176\) \(\nu^{2}\mathstrut +\mathstrut \) \(714624\) \(\nu\mathstrut +\mathstrut \) \(80704\)\()/608\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\((\)\(5\) \(\beta_{12}\mathstrut +\mathstrut \) \(3\) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\)\()/2\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(23\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(2\) \(\beta_{14}\mathstrut -\mathstrut \) \(4\) \(\beta_{13}\mathstrut +\mathstrut \) \(29\) \(\beta_{12}\mathstrut +\mathstrut \) \(11\) \(\beta_{11}\mathstrut -\mathstrut \) \(22\) \(\beta_{10}\mathstrut +\mathstrut \) \(18\) \(\beta_{9}\mathstrut -\mathstrut \) \(20\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(18\) \(\beta_{6}\mathstrut -\mathstrut \) \(22\) \(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(18\)\()/2\)
\(\nu^{6}\)\(=\)\(-\)\(\beta_{16}\mathstrut +\mathstrut \) \(\beta_{15}\mathstrut -\mathstrut \) \(12\) \(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{13}\mathstrut -\mathstrut \) \(10\) \(\beta_{11}\mathstrut -\mathstrut \) \(12\) \(\beta_{10}\mathstrut +\mathstrut \) \(12\) \(\beta_{9}\mathstrut -\mathstrut \) \(24\) \(\beta_{8}\mathstrut -\mathstrut \) \(12\) \(\beta_{7}\mathstrut -\mathstrut \) \(12\) \(\beta_{6}\mathstrut -\mathstrut \) \(14\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(48\) \(\beta_{1}\mathstrut +\mathstrut \) \(147\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(2\) \(\beta_{16}\mathstrut +\mathstrut \) \(6\) \(\beta_{15}\mathstrut -\mathstrut \) \(30\) \(\beta_{14}\mathstrut -\mathstrut \) \(52\) \(\beta_{13}\mathstrut +\mathstrut \) \(177\) \(\beta_{12}\mathstrut +\mathstrut \) \(37\) \(\beta_{11}\mathstrut -\mathstrut \) \(192\) \(\beta_{10}\mathstrut +\mathstrut \) \(144\) \(\beta_{9}\mathstrut -\mathstrut \) \(174\) \(\beta_{8}\mathstrut +\mathstrut \) \(26\) \(\beta_{7}\mathstrut -\mathstrut \) \(144\) \(\beta_{6}\mathstrut -\mathstrut \) \(200\) \(\beta_{5}\mathstrut -\mathstrut \) \(52\) \(\beta_{4}\mathstrut +\mathstrut \) \(50\) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(58\) \(\beta_{1}\mathstrut +\mathstrut \) \(162\)\()/2\)
\(\nu^{8}\)\(=\)\(-\)\(16\) \(\beta_{16}\mathstrut +\mathstrut \) \(18\) \(\beta_{15}\mathstrut -\mathstrut \) \(112\) \(\beta_{14}\mathstrut +\mathstrut \) \(28\) \(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{12}\mathstrut -\mathstrut \) \(82\) \(\beta_{11}\mathstrut -\mathstrut \) \(110\) \(\beta_{10}\mathstrut +\mathstrut \) \(114\) \(\beta_{9}\mathstrut -\mathstrut \) \(226\) \(\beta_{8}\mathstrut -\mathstrut \) \(110\) \(\beta_{7}\mathstrut -\mathstrut \) \(108\) \(\beta_{6}\mathstrut -\mathstrut \) \(144\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(339\) \(\beta_{1}\mathstrut +\mathstrut \) \(986\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(30\) \(\beta_{16}\mathstrut +\mathstrut \) \(102\) \(\beta_{15}\mathstrut -\mathstrut \) \(322\) \(\beta_{14}\mathstrut -\mathstrut \) \(496\) \(\beta_{13}\mathstrut +\mathstrut \) \(1113\) \(\beta_{12}\mathstrut +\mathstrut \) \(67\) \(\beta_{11}\mathstrut -\mathstrut \) \(1546\) \(\beta_{10}\mathstrut +\mathstrut \) \(1126\) \(\beta_{9}\mathstrut -\mathstrut \) \(1448\) \(\beta_{8}\mathstrut +\mathstrut \) \(250\) \(\beta_{7}\mathstrut -\mathstrut \) \(1118\) \(\beta_{6}\mathstrut -\mathstrut \) \(1694\) \(\beta_{5}\mathstrut -\mathstrut \) \(484\) \(\beta_{4}\mathstrut +\mathstrut \) \(466\) \(\beta_{3}\mathstrut +\mathstrut \) \(68\) \(\beta_{2}\mathstrut +\mathstrut \) \(614\) \(\beta_{1}\mathstrut +\mathstrut \) \(1436\)\()/2\)
\(\nu^{10}\)\(=\)\(-\)\(176\) \(\beta_{16}\mathstrut +\mathstrut \) \(222\) \(\beta_{15}\mathstrut -\mathstrut \) \(953\) \(\beta_{14}\mathstrut +\mathstrut \) \(272\) \(\beta_{13}\mathstrut +\mathstrut \) \(34\) \(\beta_{12}\mathstrut -\mathstrut \) \(647\) \(\beta_{11}\mathstrut -\mathstrut \) \(923\) \(\beta_{10}\mathstrut +\mathstrut \) \(1007\) \(\beta_{9}\mathstrut -\mathstrut \) \(1956\) \(\beta_{8}\mathstrut -\mathstrut \) \(911\) \(\beta_{7}\mathstrut -\mathstrut \) \(885\) \(\beta_{6}\mathstrut -\mathstrut \) \(1321\) \(\beta_{5}\mathstrut +\mathstrut \) \(50\) \(\beta_{4}\mathstrut +\mathstrut \) \(146\) \(\beta_{3}\mathstrut +\mathstrut \) \(38\) \(\beta_{2}\mathstrut +\mathstrut \) \(2461\) \(\beta_{1}\mathstrut +\mathstrut \) \(6825\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(314\) \(\beta_{16}\mathstrut +\mathstrut \) \(1214\) \(\beta_{15}\mathstrut -\mathstrut \) \(3060\) \(\beta_{14}\mathstrut -\mathstrut \) \(4220\) \(\beta_{13}\mathstrut +\mathstrut \) \(7161\) \(\beta_{12}\mathstrut -\mathstrut \) \(557\) \(\beta_{11}\mathstrut -\mathstrut \) \(11994\) \(\beta_{10}\mathstrut +\mathstrut \) \(8786\) \(\beta_{9}\mathstrut -\mathstrut \) \(11814\) \(\beta_{8}\mathstrut +\mathstrut \) \(2120\) \(\beta_{7}\mathstrut -\mathstrut \) \(8586\) \(\beta_{6}\mathstrut -\mathstrut \) \(13870\) \(\beta_{5}\mathstrut -\mathstrut \) \(3944\) \(\beta_{4}\mathstrut +\mathstrut \) \(3898\) \(\beta_{3}\mathstrut +\mathstrut \) \(784\) \(\beta_{2}\mathstrut +\mathstrut \) \(5778\) \(\beta_{1}\mathstrut +\mathstrut \) \(12424\)\()/2\)
\(\nu^{12}\)\(=\)\(-\)\(1665\) \(\beta_{16}\mathstrut +\mathstrut \) \(2343\) \(\beta_{15}\mathstrut -\mathstrut \) \(7772\) \(\beta_{14}\mathstrut +\mathstrut \) \(2270\) \(\beta_{13}\mathstrut +\mathstrut \) \(390\) \(\beta_{12}\mathstrut -\mathstrut \) \(5088\) \(\beta_{11}\mathstrut -\mathstrut \) \(7458\) \(\beta_{10}\mathstrut +\mathstrut \) \(8622\) \(\beta_{9}\mathstrut -\mathstrut \) \(16278\) \(\beta_{8}\mathstrut -\mathstrut \) \(7178\) \(\beta_{7}\mathstrut -\mathstrut \) \(6984\) \(\beta_{6}\mathstrut -\mathstrut \) \(11498\) \(\beta_{5}\mathstrut +\mathstrut \) \(734\) \(\beta_{4}\mathstrut +\mathstrut \) \(1375\) \(\beta_{3}\mathstrut +\mathstrut \) \(474\) \(\beta_{2}\mathstrut +\mathstrut \) \(18270\) \(\beta_{1}\mathstrut +\mathstrut \) \(48397\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(2872\) \(\beta_{16}\mathstrut +\mathstrut \) \(12548\) \(\beta_{15}\mathstrut -\mathstrut \) \(27424\) \(\beta_{14}\mathstrut -\mathstrut \) \(33972\) \(\beta_{13}\mathstrut +\mathstrut \) \(46965\) \(\beta_{12}\mathstrut -\mathstrut \) \(9867\) \(\beta_{11}\mathstrut -\mathstrut \) \(91272\) \(\beta_{10}\mathstrut +\mathstrut \) \(68804\) \(\beta_{9}\mathstrut -\mathstrut \) \(95412\) \(\beta_{8}\mathstrut +\mathstrut \) \(16732\) \(\beta_{7}\mathstrut -\mathstrut \) \(65684\) \(\beta_{6}\mathstrut -\mathstrut \) \(111572\) \(\beta_{5}\mathstrut -\mathstrut \) \(29944\) \(\beta_{4}\mathstrut +\mathstrut \) \(30964\) \(\beta_{3}\mathstrut +\mathstrut \) \(7696\) \(\beta_{2}\mathstrut +\mathstrut \) \(51344\) \(\beta_{1}\mathstrut +\mathstrut \) \(105308\)\()/2\)
\(\nu^{14}\)\(=\)\(-\)\(14572\) \(\beta_{16}\mathstrut +\mathstrut \) \(22760\) \(\beta_{15}\mathstrut -\mathstrut \) \(62088\) \(\beta_{14}\mathstrut +\mathstrut \) \(17404\) \(\beta_{13}\mathstrut +\mathstrut \) \(3808\) \(\beta_{12}\mathstrut -\mathstrut \) \(40160\) \(\beta_{11}\mathstrut -\mathstrut \) \(59188\) \(\beta_{10}\mathstrut +\mathstrut \) \(72596\) \(\beta_{9}\mathstrut -\mathstrut \) \(132704\) \(\beta_{8}\mathstrut -\mathstrut \) \(55032\) \(\beta_{7}\mathstrut -\mathstrut \) \(54300\) \(\beta_{6}\mathstrut -\mathstrut \) \(97396\) \(\beta_{5}\mathstrut +\mathstrut \) \(8556\) \(\beta_{4}\mathstrut +\mathstrut \) \(12332\) \(\beta_{3}\mathstrut +\mathstrut \) \(4968\) \(\beta_{2}\mathstrut +\mathstrut \) \(138009\) \(\beta_{1}\mathstrut +\mathstrut \) \(349916\)
\(\nu^{15}\)\(=\)\((\)\(-\)\(24712\) \(\beta_{16}\mathstrut +\mathstrut \) \(120508\) \(\beta_{15}\mathstrut -\mathstrut \) \(237700\) \(\beta_{14}\mathstrut -\mathstrut \) \(265304\) \(\beta_{13}\mathstrut +\mathstrut \) \(313117\) \(\beta_{12}\mathstrut -\mathstrut \) \(106437\) \(\beta_{11}\mathstrut -\mathstrut \) \(687462\) \(\beta_{10}\mathstrut +\mathstrut \) \(541310\) \(\beta_{9}\mathstrut -\mathstrut \) \(766130\) \(\beta_{8}\mathstrut +\mathstrut \) \(125856\) \(\beta_{7}\mathstrut -\mathstrut \) \(502070\) \(\beta_{6}\mathstrut -\mathstrut \) \(888926\) \(\beta_{5}\mathstrut -\mathstrut \) \(217320\) \(\beta_{4}\mathstrut +\mathstrut \) \(239312\) \(\beta_{3}\mathstrut +\mathstrut \) \(69500\) \(\beta_{2}\mathstrut +\mathstrut \) \(441796\) \(\beta_{1}\mathstrut +\mathstrut \) \(879042\)\()/2\)
\(\nu^{16}\)\(=\)\(-\)\(121836\) \(\beta_{16}\mathstrut +\mathstrut \) \(210152\) \(\beta_{15}\mathstrut -\mathstrut \) \(491137\) \(\beta_{14}\mathstrut +\mathstrut \) \(125988\) \(\beta_{13}\mathstrut +\mathstrut \) \(34252\) \(\beta_{12}\mathstrut -\mathstrut \) \(318049\) \(\beta_{11}\mathstrut -\mathstrut \) \(465593\) \(\beta_{10}\mathstrut +\mathstrut \) \(604689\) \(\beta_{9}\mathstrut -\mathstrut \) \(1069530\) \(\beta_{8}\mathstrut -\mathstrut \) \(415321\) \(\beta_{7}\mathstrut -\mathstrut \) \(420437\) \(\beta_{6}\mathstrut -\mathstrut \) \(812357\) \(\beta_{5}\mathstrut +\mathstrut \) \(88088\) \(\beta_{4}\mathstrut +\mathstrut \) \(107424\) \(\beta_{3}\mathstrut +\mathstrut \) \(47652\) \(\beta_{2}\mathstrut +\mathstrut \) \(1056139\) \(\beta_{1}\mathstrut +\mathstrut \) \(2569939\)

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
−2.32547
−0.139956
1.37907
1.37191
−0.787554
2.64128
−2.27410
2.18124
−2.51582
1.84638
2.33247
−1.09599
−0.622810
−2.65791
2.82015
−0.932399
0.779516
0 −3.27059 0 −2.03057 0 −1.64874 0 7.69673 0
1.2 0 −2.55238 0 2.96478 0 −0.820250 0 3.51462 0
1.3 0 −2.46273 0 1.31548 0 3.48956 0 3.06502 0
1.4 0 −2.36731 0 −1.32016 0 −4.19189 0 2.60414 0
1.5 0 −1.89108 0 −0.652808 0 −1.12991 0 0.576182 0
1.6 0 −1.66988 0 −3.99830 0 2.26241 0 −0.211508 0
1.7 0 −0.935470 0 3.41593 0 −3.69332 0 −2.12490 0
1.8 0 −0.923498 0 1.20531 0 1.96022 0 −2.14715 0
1.9 0 0.505139 0 −3.97764 0 −1.36760 0 −2.74483 0
1.10 0 0.508487 0 3.51373 0 −0.924114 0 −2.74144 0
1.11 0 0.826533 0 −1.37815 0 −4.67534 0 −2.31684 0
1.12 0 1.16074 0 4.05107 0 4.08218 0 −1.65268 0
1.13 0 1.30185 0 −1.69081 0 −3.93205 0 −1.30520 0
1.14 0 2.62368 0 1.15029 0 2.51426 0 3.88372 0
1.15 0 2.95607 0 2.29008 0 2.82675 0 5.73835 0
1.16 0 3.04458 0 −3.52221 0 4.32039 0 6.26949 0
1.17 0 3.14584 0 1.66398 0 −2.07256 0 6.89631 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(251\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{17} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).