Properties

Label 8001.2.a.v
Level 8001
Weight 2
Character orbit 8001.a
Self dual Yes
Analytic conductor 63.888
Analytic rank 0
Dimension 19
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8001.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + \beta_{10} q^{5} \) \(+ q^{7}\) \( + ( -\beta_{1} - \beta_{3} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + \beta_{10} q^{5} \) \(+ q^{7}\) \( + ( -\beta_{1} - \beta_{3} ) q^{8} \) \( + ( \beta_{1} - \beta_{6} - \beta_{12} + \beta_{17} ) q^{10} \) \( + \beta_{8} q^{11} \) \( + ( 1 + \beta_{12} ) q^{13} \) \( -\beta_{1} q^{14} \) \( + ( \beta_{2} + \beta_{8} + \beta_{9} ) q^{16} \) \( + ( -1 + \beta_{7} - \beta_{16} ) q^{17} \) \( + ( 2 - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{11} - \beta_{16} ) q^{19} \) \( + ( -\beta_{5} + \beta_{10} + \beta_{12} ) q^{20} \) \( + ( -\beta_{1} - \beta_{3} - \beta_{7} + \beta_{15} ) q^{22} \) \( + ( 1 + \beta_{2} - \beta_{6} + \beta_{10} + \beta_{18} ) q^{23} \) \( + ( 3 + \beta_{2} - \beta_{9} - \beta_{15} - \beta_{16} + \beta_{18} ) q^{25} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} - 2 \beta_{10} - \beta_{12} - \beta_{13} - \beta_{16} - \beta_{18} ) q^{26} \) \( + ( 1 + \beta_{2} ) q^{28} \) \( + ( 1 - \beta_{1} - \beta_{7} + \beta_{10} + \beta_{11} + \beta_{18} ) q^{29} \) \( + ( 1 + \beta_{10} - \beta_{13} - \beta_{17} ) q^{31} \) \( + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{5} - \beta_{7} + \beta_{14} + \beta_{15} ) q^{32} \) \( + ( 1 + \beta_{1} - \beta_{5} - \beta_{10} - \beta_{12} - \beta_{13} + \beta_{15} - \beta_{18} ) q^{34} \) \( + \beta_{10} q^{35} \) \( + ( 3 - \beta_{6} + \beta_{10} + \beta_{14} + \beta_{18} ) q^{37} \) \( + ( -1 - \beta_{1} + \beta_{5} - 2 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{15} - \beta_{18} ) q^{38} \) \( + ( -2 + \beta_{2} + \beta_{4} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - \beta_{13} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{40} \) \( + ( 1 + \beta_{1} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{41} \) \( + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{10} + \beta_{12} + \beta_{16} - \beta_{17} ) q^{43} \) \( + ( 1 + 3 \beta_{2} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{13} - 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{44} \) \( + ( 1 - 2 \beta_{1} - \beta_{6} + 2 \beta_{10} + \beta_{13} + \beta_{16} ) q^{46} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{10} + \beta_{13} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{47} \) \(+ q^{49}\) \( + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{50} \) \( + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} + \beta_{13} - \beta_{15} + \beta_{17} ) q^{52} \) \( + ( -2 + \beta_{1} + \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{15} - \beta_{16} - \beta_{18} ) q^{53} \) \( + ( 2 + \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{17} + \beta_{18} ) q^{55} \) \( + ( -\beta_{1} - \beta_{3} ) q^{56} \) \( + ( \beta_{1} + \beta_{5} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{13} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{58} \) \( + ( -1 - \beta_{1} + \beta_{3} - \beta_{6} + 2 \beta_{10} + 2 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} + \beta_{18} ) q^{59} \) \( + ( 1 - \beta_{1} + \beta_{5} - \beta_{7} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{16} ) q^{61} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{13} + \beta_{16} + 2 \beta_{17} ) q^{62} \) \( + ( 4 - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} + \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{64} \) \( + ( -1 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} + 4 \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{65} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{67} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{68} \) \( + ( \beta_{1} - \beta_{6} - \beta_{12} + \beta_{17} ) q^{70} \) \( + ( 2 + \beta_{2} + \beta_{3} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} - \beta_{13} - \beta_{16} - \beta_{18} ) q^{71} \) \( + ( 3 + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{9} - \beta_{11} - \beta_{15} + \beta_{17} ) q^{73} \) \( + ( -2 \beta_{1} + \beta_{3} - \beta_{6} + \beta_{9} + 2 \beta_{10} + \beta_{13} - \beta_{14} + \beta_{16} ) q^{74} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{15} + \beta_{16} ) q^{76} \) \( + \beta_{8} q^{77} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{15} - \beta_{17} ) q^{79} \) \( + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{10} + \beta_{13} - \beta_{14} - \beta_{15} + \beta_{17} ) q^{80} \) \( + ( -2 - 2 \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{12} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{82} \) \( + ( \beta_{2} - \beta_{3} + \beta_{6} - \beta_{8} - 2 \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{15} + \beta_{17} - \beta_{18} ) q^{83} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{16} - \beta_{17} ) q^{85} \) \( + ( 2 \beta_{2} + \beta_{9} - 3 \beta_{10} - 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{86} \) \( + ( -1 - 6 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{11} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{17} - \beta_{18} ) q^{88} \) \( + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{14} - \beta_{15} - 2 \beta_{16} + \beta_{17} - \beta_{18} ) q^{89} \) \( + ( 1 + \beta_{12} ) q^{91} \) \( + ( 4 - \beta_{1} + 3 \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{11} - \beta_{12} ) q^{92} \) \( + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{17} + \beta_{18} ) q^{94} \) \( + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{95} \) \( + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{97} \) \( -\beta_{1} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(19q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 22q^{4} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut 19q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(19q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 22q^{4} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut 19q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 9q^{11} \) \(\mathstrut +\mathstrut 24q^{13} \) \(\mathstrut -\mathstrut 4q^{14} \) \(\mathstrut +\mathstrut 20q^{16} \) \(\mathstrut -\mathstrut 17q^{17} \) \(\mathstrut +\mathstrut 23q^{19} \) \(\mathstrut -\mathstrut 5q^{20} \) \(\mathstrut -\mathstrut 3q^{22} \) \(\mathstrut +\mathstrut 17q^{23} \) \(\mathstrut +\mathstrut 38q^{25} \) \(\mathstrut -\mathstrut 28q^{26} \) \(\mathstrut +\mathstrut 22q^{28} \) \(\mathstrut -\mathstrut 2q^{29} \) \(\mathstrut +\mathstrut 16q^{31} \) \(\mathstrut -\mathstrut 17q^{32} \) \(\mathstrut +\mathstrut 29q^{34} \) \(\mathstrut -\mathstrut 5q^{35} \) \(\mathstrut +\mathstrut 56q^{37} \) \(\mathstrut -\mathstrut 2q^{38} \) \(\mathstrut -\mathstrut 13q^{40} \) \(\mathstrut +\mathstrut 7q^{41} \) \(\mathstrut +\mathstrut 19q^{43} \) \(\mathstrut +\mathstrut 29q^{44} \) \(\mathstrut +\mathstrut 10q^{46} \) \(\mathstrut -\mathstrut 25q^{47} \) \(\mathstrut +\mathstrut 19q^{49} \) \(\mathstrut +\mathstrut 9q^{50} \) \(\mathstrut +\mathstrut 16q^{52} \) \(\mathstrut -\mathstrut 18q^{53} \) \(\mathstrut +\mathstrut 10q^{55} \) \(\mathstrut -\mathstrut 9q^{56} \) \(\mathstrut +\mathstrut 31q^{58} \) \(\mathstrut -\mathstrut 11q^{59} \) \(\mathstrut +\mathstrut 26q^{61} \) \(\mathstrut -\mathstrut 26q^{62} \) \(\mathstrut +\mathstrut 45q^{64} \) \(\mathstrut -\mathstrut 27q^{65} \) \(\mathstrut +\mathstrut 24q^{67} \) \(\mathstrut -\mathstrut 14q^{68} \) \(\mathstrut +\mathstrut 32q^{71} \) \(\mathstrut +\mathstrut 51q^{73} \) \(\mathstrut +\mathstrut 12q^{76} \) \(\mathstrut +\mathstrut 9q^{77} \) \(\mathstrut +\mathstrut 30q^{79} \) \(\mathstrut +\mathstrut 30q^{80} \) \(\mathstrut -\mathstrut 52q^{82} \) \(\mathstrut -\mathstrut q^{83} \) \(\mathstrut +\mathstrut 44q^{85} \) \(\mathstrut +\mathstrut 24q^{86} \) \(\mathstrut -\mathstrut 30q^{88} \) \(\mathstrut -\mathstrut 5q^{89} \) \(\mathstrut +\mathstrut 24q^{91} \) \(\mathstrut +\mathstrut 88q^{92} \) \(\mathstrut +\mathstrut 7q^{94} \) \(\mathstrut +\mathstrut 24q^{95} \) \(\mathstrut +\mathstrut 5q^{97} \) \(\mathstrut -\mathstrut 4q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19}\mathstrut -\mathstrut \) \(4\) \(x^{18}\mathstrut -\mathstrut \) \(22\) \(x^{17}\mathstrut +\mathstrut \) \(101\) \(x^{16}\mathstrut +\mathstrut \) \(178\) \(x^{15}\mathstrut -\mathstrut \) \(1035\) \(x^{14}\mathstrut -\mathstrut \) \(583\) \(x^{13}\mathstrut +\mathstrut \) \(5572\) \(x^{12}\mathstrut +\mathstrut \) \(21\) \(x^{11}\mathstrut -\mathstrut \) \(17032\) \(x^{10}\mathstrut +\mathstrut \) \(4985\) \(x^{9}\mathstrut +\mathstrut \) \(29792\) \(x^{8}\mathstrut -\mathstrut \) \(13249\) \(x^{7}\mathstrut -\mathstrut \) \(28600\) \(x^{6}\mathstrut +\mathstrut \) \(14000\) \(x^{5}\mathstrut +\mathstrut \) \(13725\) \(x^{4}\mathstrut -\mathstrut \) \(5723\) \(x^{3}\mathstrut -\mathstrut \) \(2913\) \(x^{2}\mathstrut +\mathstrut \) \(608\) \(x\mathstrut +\mathstrut \) \(210\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\((\)\(11967908\) \(\nu^{18}\mathstrut +\mathstrut \) \(10330919\) \(\nu^{17}\mathstrut -\mathstrut \) \(422890858\) \(\nu^{16}\mathstrut -\mathstrut \) \(185012986\) \(\nu^{15}\mathstrut +\mathstrut \) \(5929350010\) \(\nu^{14}\mathstrut +\mathstrut \) \(1055361561\) \(\nu^{13}\mathstrut -\mathstrut \) \(43092462154\) \(\nu^{12}\mathstrut -\mathstrut \) \(1159028944\) \(\nu^{11}\mathstrut +\mathstrut \) \(177057585547\) \(\nu^{10}\mathstrut -\mathstrut \) \(8859733933\) \(\nu^{9}\mathstrut -\mathstrut \) \(419706704682\) \(\nu^{8}\mathstrut +\mathstrut \) \(32463907935\) \(\nu^{7}\mathstrut +\mathstrut \) \(560761757568\) \(\nu^{6}\mathstrut -\mathstrut \) \(42523921020\) \(\nu^{5}\mathstrut -\mathstrut \) \(394138444469\) \(\nu^{4}\mathstrut +\mathstrut \) \(26352010178\) \(\nu^{3}\mathstrut +\mathstrut \) \(124055335687\) \(\nu^{2}\mathstrut -\mathstrut \) \(9280440934\) \(\nu\mathstrut -\mathstrut \) \(9272820218\)\()/\)\(579103703\)
\(\beta_{5}\)\(=\)\((\)\(65114179\) \(\nu^{18}\mathstrut -\mathstrut \) \(132926940\) \(\nu^{17}\mathstrut -\mathstrut \) \(1671831523\) \(\nu^{16}\mathstrut +\mathstrut \) \(3285269092\) \(\nu^{15}\mathstrut +\mathstrut \) \(17368062950\) \(\nu^{14}\mathstrut -\mathstrut \) \(32561720490\) \(\nu^{13}\mathstrut -\mathstrut \) \(94107501027\) \(\nu^{12}\mathstrut +\mathstrut \) \(166375659285\) \(\nu^{11}\mathstrut +\mathstrut \) \(285664374904\) \(\nu^{10}\mathstrut -\mathstrut \) \(469100157707\) \(\nu^{9}\mathstrut -\mathstrut \) \(486402019943\) \(\nu^{8}\mathstrut +\mathstrut \) \(726626199106\) \(\nu^{7}\mathstrut +\mathstrut \) \(448072525313\) \(\nu^{6}\mathstrut -\mathstrut \) \(582478309828\) \(\nu^{5}\mathstrut -\mathstrut \) \(222447224597\) \(\nu^{4}\mathstrut +\mathstrut \) \(206901778527\) \(\nu^{3}\mathstrut +\mathstrut \) \(71510619076\) \(\nu^{2}\mathstrut -\mathstrut \) \(23752461453\) \(\nu\mathstrut -\mathstrut \) \(6128233325\)\()/\)\(2895518515\)
\(\beta_{6}\)\(=\)\((\)\(138645363\) \(\nu^{18}\mathstrut -\mathstrut \) \(343698145\) \(\nu^{17}\mathstrut -\mathstrut \) \(3866824621\) \(\nu^{16}\mathstrut +\mathstrut \) \(9112645479\) \(\nu^{15}\mathstrut +\mathstrut \) \(44954282930\) \(\nu^{14}\mathstrut -\mathstrut \) \(98469596705\) \(\nu^{13}\mathstrut -\mathstrut \) \(284325273789\) \(\nu^{12}\mathstrut +\mathstrut \) \(559933954855\) \(\nu^{11}\mathstrut +\mathstrut \) \(1071360077368\) \(\nu^{10}\mathstrut -\mathstrut \) \(1803665054044\) \(\nu^{9}\mathstrut -\mathstrut \) \(2478737837371\) \(\nu^{8}\mathstrut +\mathstrut \) \(3295314382092\) \(\nu^{7}\mathstrut +\mathstrut \) \(3500770767616\) \(\nu^{6}\mathstrut -\mathstrut \) \(3225526895581\) \(\nu^{5}\mathstrut -\mathstrut \) \(2872131807524\) \(\nu^{4}\mathstrut +\mathstrut \) \(1473343414389\) \(\nu^{3}\mathstrut +\mathstrut \) \(1168583011302\) \(\nu^{2}\mathstrut -\mathstrut \) \(242700042466\) \(\nu\mathstrut -\mathstrut \) \(133199717500\)\()/\)\(5791037030\)
\(\beta_{7}\)\(=\)\((\)\(318716503\) \(\nu^{18}\mathstrut -\mathstrut \) \(1140866135\) \(\nu^{17}\mathstrut -\mathstrut \) \(7501449651\) \(\nu^{16}\mathstrut +\mathstrut \) \(29542139284\) \(\nu^{15}\mathstrut +\mathstrut \) \(67595392300\) \(\nu^{14}\mathstrut -\mathstrut \) \(309929331465\) \(\nu^{13}\mathstrut -\mathstrut \) \(281749529274\) \(\nu^{12}\mathstrut +\mathstrut \) \(1696786234950\) \(\nu^{11}\mathstrut +\mathstrut \) \(451103298313\) \(\nu^{10}\mathstrut -\mathstrut \) \(5196857369249\) \(\nu^{9}\mathstrut +\mathstrut \) \(375217402964\) \(\nu^{8}\mathstrut +\mathstrut \) \(8851478759162\) \(\nu^{7}\mathstrut -\mathstrut \) \(2162801145399\) \(\nu^{6}\mathstrut -\mathstrut \) \(7808662348931\) \(\nu^{5}\mathstrut +\mathstrut \) \(2393312892281\) \(\nu^{4}\mathstrut +\mathstrut \) \(2961548762254\) \(\nu^{3}\mathstrut -\mathstrut \) \(839166774003\) \(\nu^{2}\mathstrut -\mathstrut \) \(274887004146\) \(\nu\mathstrut +\mathstrut \) \(54702171430\)\()/\)\(11582074060\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(93609108\) \(\nu^{18}\mathstrut +\mathstrut \) \(195011795\) \(\nu^{17}\mathstrut +\mathstrut \) \(2566480776\) \(\nu^{16}\mathstrut -\mathstrut \) \(4911478344\) \(\nu^{15}\mathstrut -\mathstrut \) \(29294218685\) \(\nu^{14}\mathstrut +\mathstrut \) \(49897035585\) \(\nu^{13}\mathstrut +\mathstrut \) \(181459391204\) \(\nu^{12}\mathstrut -\mathstrut \) \(263787283760\) \(\nu^{11}\mathstrut -\mathstrut \) \(664790856488\) \(\nu^{10}\mathstrut +\mathstrut \) \(781891378959\) \(\nu^{9}\mathstrut +\mathstrut \) \(1466329373136\) \(\nu^{8}\mathstrut -\mathstrut \) \(1310268250057\) \(\nu^{7}\mathstrut -\mathstrut \) \(1887181446716\) \(\nu^{6}\mathstrut +\mathstrut \) \(1200884819526\) \(\nu^{5}\mathstrut +\mathstrut \) \(1292110192984\) \(\nu^{4}\mathstrut -\mathstrut \) \(547670340074\) \(\nu^{3}\mathstrut -\mathstrut \) \(388056127817\) \(\nu^{2}\mathstrut +\mathstrut \) \(88500571266\) \(\nu\mathstrut +\mathstrut \) \(40044597930\)\()/\)\(2895518515\)
\(\beta_{9}\)\(=\)\((\)\(93609108\) \(\nu^{18}\mathstrut -\mathstrut \) \(195011795\) \(\nu^{17}\mathstrut -\mathstrut \) \(2566480776\) \(\nu^{16}\mathstrut +\mathstrut \) \(4911478344\) \(\nu^{15}\mathstrut +\mathstrut \) \(29294218685\) \(\nu^{14}\mathstrut -\mathstrut \) \(49897035585\) \(\nu^{13}\mathstrut -\mathstrut \) \(181459391204\) \(\nu^{12}\mathstrut +\mathstrut \) \(263787283760\) \(\nu^{11}\mathstrut +\mathstrut \) \(664790856488\) \(\nu^{10}\mathstrut -\mathstrut \) \(781891378959\) \(\nu^{9}\mathstrut -\mathstrut \) \(1466329373136\) \(\nu^{8}\mathstrut +\mathstrut \) \(1310268250057\) \(\nu^{7}\mathstrut +\mathstrut \) \(1887181446716\) \(\nu^{6}\mathstrut -\mathstrut \) \(1200884819526\) \(\nu^{5}\mathstrut -\mathstrut \) \(1289214674469\) \(\nu^{4}\mathstrut +\mathstrut \) \(547670340074\) \(\nu^{3}\mathstrut +\mathstrut \) \(367787498212\) \(\nu^{2}\mathstrut -\mathstrut \) \(88500571266\) \(\nu\mathstrut -\mathstrut \) \(19775968325\)\()/\)\(2895518515\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(37953488\) \(\nu^{18}\mathstrut +\mathstrut \) \(92196111\) \(\nu^{17}\mathstrut +\mathstrut \) \(991696752\) \(\nu^{16}\mathstrut -\mathstrut \) \(2324375148\) \(\nu^{15}\mathstrut -\mathstrut \) \(10672704577\) \(\nu^{14}\mathstrut +\mathstrut \) \(23745817195\) \(\nu^{13}\mathstrut +\mathstrut \) \(61534319327\) \(\nu^{12}\mathstrut -\mathstrut \) \(126913768361\) \(\nu^{11}\mathstrut -\mathstrut \) \(206178478291\) \(\nu^{10}\mathstrut +\mathstrut \) \(381112300142\) \(\nu^{9}\mathstrut +\mathstrut \) \(404304603883\) \(\nu^{8}\mathstrut -\mathstrut \) \(638066586796\) \(\nu^{7}\mathstrut -\mathstrut \) \(439470286205\) \(\nu^{6}\mathstrut +\mathstrut \) \(548407227881\) \(\nu^{5}\mathstrut +\mathstrut \) \(230702575789\) \(\nu^{4}\mathstrut -\mathstrut \) \(194336671842\) \(\nu^{3}\mathstrut -\mathstrut \) \(44204269770\) \(\nu^{2}\mathstrut +\mathstrut \) \(15157612018\) \(\nu\mathstrut +\mathstrut \) \(2068831656\)\()/\)\(1158207406\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(240105167\) \(\nu^{18}\mathstrut +\mathstrut \) \(881338715\) \(\nu^{17}\mathstrut +\mathstrut \) \(5495659804\) \(\nu^{16}\mathstrut -\mathstrut \) \(21651992771\) \(\nu^{15}\mathstrut -\mathstrut \) \(49786456290\) \(\nu^{14}\mathstrut +\mathstrut \) \(215066182570\) \(\nu^{13}\mathstrut +\mathstrut \) \(231300589556\) \(\nu^{12}\mathstrut -\mathstrut \) \(1119608505200\) \(\nu^{11}\mathstrut -\mathstrut \) \(605499617407\) \(\nu^{10}\mathstrut +\mathstrut \) \(3307955295931\) \(\nu^{9}\mathstrut +\mathstrut \) \(961255585349\) \(\nu^{8}\mathstrut -\mathstrut \) \(5591713835683\) \(\nu^{7}\mathstrut -\mathstrut \) \(1060121848224\) \(\nu^{6}\mathstrut +\mathstrut \) \(5135218235504\) \(\nu^{5}\mathstrut +\mathstrut \) \(906752365551\) \(\nu^{4}\mathstrut -\mathstrut \) \(2201652718176\) \(\nu^{3}\mathstrut -\mathstrut \) \(455951257588\) \(\nu^{2}\mathstrut +\mathstrut \) \(287009377064\) \(\nu\mathstrut +\mathstrut \) \(53652831630\)\()/\)\(5791037030\)
\(\beta_{12}\)\(=\)\((\)\(290773938\) \(\nu^{18}\mathstrut -\mathstrut \) \(662122355\) \(\nu^{17}\mathstrut -\mathstrut \) \(7697023186\) \(\nu^{16}\mathstrut +\mathstrut \) \(16815829469\) \(\nu^{15}\mathstrut +\mathstrut \) \(83341546495\) \(\nu^{14}\mathstrut -\mathstrut \) \(172281371545\) \(\nu^{13}\mathstrut -\mathstrut \) \(477184016914\) \(\nu^{12}\mathstrut +\mathstrut \) \(916160179920\) \(\nu^{11}\mathstrut +\mathstrut \) \(1554464292428\) \(\nu^{10}\mathstrut -\mathstrut \) \(2705691492119\) \(\nu^{9}\mathstrut -\mathstrut \) \(2868279746356\) \(\nu^{8}\mathstrut +\mathstrut \) \(4389554678512\) \(\nu^{7}\mathstrut +\mathstrut \) \(2773105119646\) \(\nu^{6}\mathstrut -\mathstrut \) \(3584941413081\) \(\nu^{5}\mathstrut -\mathstrut \) \(1121219155374\) \(\nu^{4}\mathstrut +\mathstrut \) \(1145884621839\) \(\nu^{3}\mathstrut +\mathstrut \) \(63474192337\) \(\nu^{2}\mathstrut -\mathstrut \) \(53779704136\) \(\nu\mathstrut +\mathstrut \) \(19309791560\)\()/\)\(5791037030\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(485961931\) \(\nu^{18}\mathstrut +\mathstrut \) \(1113420900\) \(\nu^{17}\mathstrut +\mathstrut \) \(13007283232\) \(\nu^{16}\mathstrut -\mathstrut \) \(28691667118\) \(\nu^{15}\mathstrut -\mathstrut \) \(142770006155\) \(\nu^{14}\mathstrut +\mathstrut \) \(298637498085\) \(\nu^{13}\mathstrut +\mathstrut \) \(833859292088\) \(\nu^{12}\mathstrut -\mathstrut \) \(1614947153800\) \(\nu^{11}\mathstrut -\mathstrut \) \(2816533502951\) \(\nu^{10}\mathstrut +\mathstrut \) \(4855571864208\) \(\nu^{9}\mathstrut +\mathstrut \) \(5615791269077\) \(\nu^{8}\mathstrut -\mathstrut \) \(8046688000049\) \(\nu^{7}\mathstrut -\mathstrut \) \(6493467022012\) \(\nu^{6}\mathstrut +\mathstrut \) \(6795149697847\) \(\nu^{5}\mathstrut +\mathstrut \) \(4085986343623\) \(\nu^{4}\mathstrut -\mathstrut \) \(2351933693773\) \(\nu^{3}\mathstrut -\mathstrut \) \(1176825672339\) \(\nu^{2}\mathstrut +\mathstrut \) \(161926194492\) \(\nu\mathstrut +\mathstrut \) \(74147034150\)\()/\)\(5791037030\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(244538816\) \(\nu^{18}\mathstrut +\mathstrut \) \(640007340\) \(\nu^{17}\mathstrut +\mathstrut \) \(6214873087\) \(\nu^{16}\mathstrut -\mathstrut \) \(15917066553\) \(\nu^{15}\mathstrut -\mathstrut \) \(64356454145\) \(\nu^{14}\mathstrut +\mathstrut \) \(159447001730\) \(\nu^{13}\mathstrut +\mathstrut \) \(351910167043\) \(\nu^{12}\mathstrut -\mathstrut \) \(829200724505\) \(\nu^{11}\mathstrut -\mathstrut \) \(1098123323401\) \(\nu^{10}\mathstrut +\mathstrut \) \(2402070934223\) \(\nu^{9}\mathstrut +\mathstrut \) \(1964936315422\) \(\nu^{8}\mathstrut -\mathstrut \) \(3854034717714\) \(\nu^{7}\mathstrut -\mathstrut \) \(1924408194587\) \(\nu^{6}\mathstrut +\mathstrut \) \(3182220496297\) \(\nu^{5}\mathstrut +\mathstrut \) \(959561891823\) \(\nu^{4}\mathstrut -\mathstrut \) \(1107518683308\) \(\nu^{3}\mathstrut -\mathstrut \) \(255693379414\) \(\nu^{2}\mathstrut +\mathstrut \) \(88860693382\) \(\nu\mathstrut +\mathstrut \) \(28681664520\)\()/\)\(2895518515\)
\(\beta_{15}\)\(=\)\((\)\(1036415051\) \(\nu^{18}\mathstrut -\mathstrut \) \(3169187735\) \(\nu^{17}\mathstrut -\mathstrut \) \(25673615907\) \(\nu^{16}\mathstrut +\mathstrut \) \(80069329128\) \(\nu^{15}\mathstrut +\mathstrut \) \(255548957080\) \(\nu^{14}\mathstrut -\mathstrut \) \(817470456425\) \(\nu^{13}\mathstrut -\mathstrut \) \(1312960193338\) \(\nu^{12}\mathstrut +\mathstrut \) \(4348086495830\) \(\nu^{11}\mathstrut +\mathstrut \) \(3700939092301\) \(\nu^{10}\mathstrut -\mathstrut \) \(12928740475313\) \(\nu^{9}\mathstrut -\mathstrut \) \(5538919778952\) \(\nu^{8}\mathstrut +\mathstrut \) \(21361112833594\) \(\nu^{7}\mathstrut +\mathstrut \) \(3742541531697\) \(\nu^{6}\mathstrut -\mathstrut \) \(18219213168867\) \(\nu^{5}\mathstrut -\mathstrut \) \(555145776623\) \(\nu^{4}\mathstrut +\mathstrut \) \(6668255047918\) \(\nu^{3}\mathstrut -\mathstrut \) \(102435732651\) \(\nu^{2}\mathstrut -\mathstrut \) \(709051042762\) \(\nu\mathstrut -\mathstrut \) \(23929479290\)\()/\)\(11582074060\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(1177420721\) \(\nu^{18}\mathstrut +\mathstrut \) \(3022806135\) \(\nu^{17}\mathstrut +\mathstrut \) \(30214864317\) \(\nu^{16}\mathstrut -\mathstrut \) \(76041669698\) \(\nu^{15}\mathstrut -\mathstrut \) \(316167323730\) \(\nu^{14}\mathstrut +\mathstrut \) \(773094156925\) \(\nu^{13}\mathstrut +\mathstrut \) \(1747199273478\) \(\nu^{12}\mathstrut -\mathstrut \) \(4098812389930\) \(\nu^{11}\mathstrut -\mathstrut \) \(5505446366291\) \(\nu^{10}\mathstrut +\mathstrut \) \(12181293389813\) \(\nu^{9}\mathstrut +\mathstrut \) \(9922760972072\) \(\nu^{8}\mathstrut -\mathstrut \) \(20239239560654\) \(\nu^{7}\mathstrut -\mathstrut \) \(9698265109967\) \(\nu^{6}\mathstrut +\mathstrut \) \(17589452904587\) \(\nu^{5}\mathstrut +\mathstrut \) \(4616300955633\) \(\nu^{4}\mathstrut -\mathstrut \) \(6763494721768\) \(\nu^{3}\mathstrut -\mathstrut \) \(1017780451669\) \(\nu^{2}\mathstrut +\mathstrut \) \(804092581722\) \(\nu\mathstrut +\mathstrut \) \(101431035930\)\()/\)\(11582074060\)
\(\beta_{17}\)\(=\)\((\)\(727508506\) \(\nu^{18}\mathstrut -\mathstrut \) \(1789420580\) \(\nu^{17}\mathstrut -\mathstrut \) \(19108483507\) \(\nu^{16}\mathstrut +\mathstrut \) \(45513393513\) \(\nu^{15}\mathstrut +\mathstrut \) \(205976043850\) \(\nu^{14}\mathstrut -\mathstrut \) \(467788147365\) \(\nu^{13}\mathstrut -\mathstrut \) \(1184324624578\) \(\nu^{12}\mathstrut +\mathstrut \) \(2503001409990\) \(\nu^{11}\mathstrut +\mathstrut \) \(3952381907166\) \(\nu^{10}\mathstrut -\mathstrut \) \(7476870253978\) \(\nu^{9}\mathstrut -\mathstrut \) \(7810236222227\) \(\nu^{8}\mathstrut +\mathstrut \) \(12396449304189\) \(\nu^{7}\mathstrut +\mathstrut \) \(8959188531857\) \(\nu^{6}\mathstrut -\mathstrut \) \(10620725347607\) \(\nu^{5}\mathstrut -\mathstrut \) \(5626225717688\) \(\nu^{4}\mathstrut +\mathstrut \) \(3926288444198\) \(\nu^{3}\mathstrut +\mathstrut \) \(1709061696269\) \(\nu^{2}\mathstrut -\mathstrut \) \(427993545432\) \(\nu\mathstrut -\mathstrut \) \(153741088340\)\()/\)\(5791037030\)
\(\beta_{18}\)\(=\)\((\)\(3366745827\) \(\nu^{18}\mathstrut -\mathstrut \) \(9263333215\) \(\nu^{17}\mathstrut -\mathstrut \) \(85049828069\) \(\nu^{16}\mathstrut +\mathstrut \) \(231373969696\) \(\nu^{15}\mathstrut +\mathstrut \) \(877139208820\) \(\nu^{14}\mathstrut -\mathstrut \) \(2334094800655\) \(\nu^{13}\mathstrut -\mathstrut \) \(4806101920096\) \(\nu^{12}\mathstrut +\mathstrut \) \(12274426272960\) \(\nu^{11}\mathstrut +\mathstrut \) \(15248263236747\) \(\nu^{10}\mathstrut -\mathstrut \) \(36169011220831\) \(\nu^{9}\mathstrut -\mathstrut \) \(28613582641044\) \(\nu^{8}\mathstrut +\mathstrut \) \(59507326593548\) \(\nu^{7}\mathstrut +\mathstrut \) \(31177087450109\) \(\nu^{6}\mathstrut -\mathstrut \) \(50962697977249\) \(\nu^{5}\mathstrut -\mathstrut \) \(18711356928001\) \(\nu^{4}\mathstrut +\mathstrut \) \(18937119806716\) \(\nu^{3}\mathstrut +\mathstrut \) \(5559535596393\) \(\nu^{2}\mathstrut -\mathstrut \) \(1938290594534\) \(\nu\mathstrut -\mathstrut \) \(509625256950\)\()/\)\(11582074060\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(30\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{6}\)\(=\)\(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(2\) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(11\) \(\beta_{9}\mathstrut +\mathstrut \) \(11\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(46\) \(\beta_{2}\mathstrut +\mathstrut \) \(80\)
\(\nu^{7}\)\(=\)\(\beta_{18}\mathstrut +\mathstrut \) \(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(14\) \(\beta_{15}\mathstrut -\mathstrut \) \(12\) \(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{7}\mathstrut -\mathstrut \) \(14\) \(\beta_{5}\mathstrut +\mathstrut \) \(70\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(195\) \(\beta_{1}\mathstrut +\mathstrut \) \(14\)
\(\nu^{8}\)\(=\)\(2\) \(\beta_{18}\mathstrut +\mathstrut \) \(13\) \(\beta_{17}\mathstrut -\mathstrut \) \(14\) \(\beta_{16}\mathstrut -\mathstrut \) \(31\) \(\beta_{15}\mathstrut -\mathstrut \) \(13\) \(\beta_{14}\mathstrut +\mathstrut \) \(16\) \(\beta_{13}\mathstrut +\mathstrut \) \(4\) \(\beta_{12}\mathstrut +\mathstrut \) \(31\) \(\beta_{11}\mathstrut +\mathstrut \) \(4\) \(\beta_{10}\mathstrut +\mathstrut \) \(96\) \(\beta_{9}\mathstrut +\mathstrut \) \(95\) \(\beta_{8}\mathstrut +\mathstrut \) \(17\) \(\beta_{7}\mathstrut +\mathstrut \) \(30\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(29\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(305\) \(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(508\)
\(\nu^{9}\)\(=\)\(17\) \(\beta_{18}\mathstrut +\mathstrut \) \(17\) \(\beta_{17}\mathstrut -\mathstrut \) \(16\) \(\beta_{16}\mathstrut -\mathstrut \) \(145\) \(\beta_{15}\mathstrut -\mathstrut \) \(109\) \(\beta_{14}\mathstrut +\mathstrut \) \(34\) \(\beta_{13}\mathstrut +\mathstrut \) \(8\) \(\beta_{12}\mathstrut +\mathstrut \) \(32\) \(\beta_{11}\mathstrut +\mathstrut \) \(54\) \(\beta_{10}\mathstrut +\mathstrut \) \(16\) \(\beta_{9}\mathstrut +\mathstrut \) \(20\) \(\beta_{8}\mathstrut +\mathstrut \) \(142\) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(143\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(525\) \(\beta_{3}\mathstrut -\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(1321\) \(\beta_{1}\mathstrut +\mathstrut \) \(141\)
\(\nu^{10}\)\(=\)\(38\) \(\beta_{18}\mathstrut +\mathstrut \) \(123\) \(\beta_{17}\mathstrut -\mathstrut \) \(137\) \(\beta_{16}\mathstrut -\mathstrut \) \(338\) \(\beta_{15}\mathstrut -\mathstrut \) \(125\) \(\beta_{14}\mathstrut +\mathstrut \) \(181\) \(\beta_{13}\mathstrut +\mathstrut \) \(78\) \(\beta_{12}\mathstrut +\mathstrut \) \(334\) \(\beta_{11}\mathstrut +\mathstrut \) \(80\) \(\beta_{10}\mathstrut +\mathstrut \) \(777\) \(\beta_{9}\mathstrut +\mathstrut \) \(762\) \(\beta_{8}\mathstrut +\mathstrut \) \(199\) \(\beta_{7}\mathstrut +\mathstrut \) \(318\) \(\beta_{6}\mathstrut -\mathstrut \) \(21\) \(\beta_{5}\mathstrut -\mathstrut \) \(295\) \(\beta_{4}\mathstrut +\mathstrut \) \(39\) \(\beta_{3}\mathstrut +\mathstrut \) \(2061\) \(\beta_{2}\mathstrut -\mathstrut \) \(28\) \(\beta_{1}\mathstrut +\mathstrut \) \(3420\)
\(\nu^{11}\)\(=\)\(196\) \(\beta_{18}\mathstrut +\mathstrut \) \(201\) \(\beta_{17}\mathstrut -\mathstrut \) \(175\) \(\beta_{16}\mathstrut -\mathstrut \) \(1338\) \(\beta_{15}\mathstrut -\mathstrut \) \(902\) \(\beta_{14}\mathstrut +\mathstrut \) \(401\) \(\beta_{13}\mathstrut +\mathstrut \) \(165\) \(\beta_{12}\mathstrut +\mathstrut \) \(354\) \(\beta_{11}\mathstrut +\mathstrut \) \(651\) \(\beta_{10}\mathstrut +\mathstrut \) \(185\) \(\beta_{9}\mathstrut +\mathstrut \) \(258\) \(\beta_{8}\mathstrut +\mathstrut \) \(1281\) \(\beta_{7}\mathstrut +\mathstrut \) \(63\) \(\beta_{6}\mathstrut -\mathstrut \) \(1291\) \(\beta_{5}\mathstrut -\mathstrut \) \(22\) \(\beta_{4}\mathstrut +\mathstrut \) \(3896\) \(\beta_{3}\mathstrut -\mathstrut \) \(92\) \(\beta_{2}\mathstrut +\mathstrut \) \(9180\) \(\beta_{1}\mathstrut +\mathstrut \) \(1270\)
\(\nu^{12}\)\(=\)\(472\) \(\beta_{18}\mathstrut +\mathstrut \) \(1052\) \(\beta_{17}\mathstrut -\mathstrut \) \(1168\) \(\beta_{16}\mathstrut -\mathstrut \) \(3209\) \(\beta_{15}\mathstrut -\mathstrut \) \(1087\) \(\beta_{14}\mathstrut +\mathstrut \) \(1783\) \(\beta_{13}\mathstrut +\mathstrut \) \(1001\) \(\beta_{12}\mathstrut +\mathstrut \) \(3109\) \(\beta_{11}\mathstrut +\mathstrut \) \(1036\) \(\beta_{10}\mathstrut +\mathstrut \) \(6089\) \(\beta_{9}\mathstrut +\mathstrut \) \(5939\) \(\beta_{8}\mathstrut +\mathstrut \) \(1997\) \(\beta_{7}\mathstrut +\mathstrut \) \(2936\) \(\beta_{6}\mathstrut -\mathstrut \) \(291\) \(\beta_{5}\mathstrut -\mathstrut \) \(2617\) \(\beta_{4}\mathstrut +\mathstrut \) \(509\) \(\beta_{3}\mathstrut +\mathstrut \) \(14191\) \(\beta_{2}\mathstrut -\mathstrut \) \(222\) \(\beta_{1}\mathstrut +\mathstrut \) \(23845\)
\(\nu^{13}\)\(=\)\(1930\) \(\beta_{18}\mathstrut +\mathstrut \) \(2048\) \(\beta_{17}\mathstrut -\mathstrut \) \(1628\) \(\beta_{16}\mathstrut -\mathstrut \) \(11637\) \(\beta_{15}\mathstrut -\mathstrut \) \(7176\) \(\beta_{14}\mathstrut +\mathstrut \) \(4070\) \(\beta_{13}\mathstrut +\mathstrut \) \(2196\) \(\beta_{12}\mathstrut +\mathstrut \) \(3394\) \(\beta_{11}\mathstrut +\mathstrut \) \(6618\) \(\beta_{10}\mathstrut +\mathstrut \) \(1887\) \(\beta_{9}\mathstrut +\mathstrut \) \(2764\) \(\beta_{8}\mathstrut +\mathstrut \) \(10931\) \(\beta_{7}\mathstrut +\mathstrut \) \(856\) \(\beta_{6}\mathstrut -\mathstrut \) \(10946\) \(\beta_{5}\mathstrut -\mathstrut \) \(312\) \(\beta_{4}\mathstrut +\mathstrut \) \(28856\) \(\beta_{3}\mathstrut -\mathstrut \) \(508\) \(\beta_{2}\mathstrut +\mathstrut \) \(64948\) \(\beta_{1}\mathstrut +\mathstrut \) \(10927\)
\(\nu^{14}\)\(=\)\(4884\) \(\beta_{18}\mathstrut +\mathstrut \) \(8685\) \(\beta_{17}\mathstrut -\mathstrut \) \(9323\) \(\beta_{16}\mathstrut -\mathstrut \) \(28414\) \(\beta_{15}\mathstrut -\mathstrut \) \(9063\) \(\beta_{14}\mathstrut +\mathstrut \) \(16327\) \(\beta_{13}\mathstrut +\mathstrut \) \(10700\) \(\beta_{12}\mathstrut +\mathstrut \) \(26856\) \(\beta_{11}\mathstrut +\mathstrut \) \(11110\) \(\beta_{10}\mathstrut +\mathstrut \) \(46978\) \(\beta_{9}\mathstrut +\mathstrut \) \(45726\) \(\beta_{8}\mathstrut +\mathstrut \) \(18471\) \(\beta_{7}\mathstrut +\mathstrut \) \(25254\) \(\beta_{6}\mathstrut -\mathstrut \) \(3357\) \(\beta_{5}\mathstrut -\mathstrut \) \(21718\) \(\beta_{4}\mathstrut +\mathstrut \) \(5607\) \(\beta_{3}\mathstrut +\mathstrut \) \(99346\) \(\beta_{2}\mathstrut -\mathstrut \) \(954\) \(\beta_{1}\mathstrut +\mathstrut \) \(170076\)
\(\nu^{15}\)\(=\)\(17519\) \(\beta_{18}\mathstrut +\mathstrut \) \(19249\) \(\beta_{17}\mathstrut -\mathstrut \) \(13913\) \(\beta_{16}\mathstrut -\mathstrut \) \(97749\) \(\beta_{15}\mathstrut -\mathstrut \) \(56041\) \(\beta_{14}\mathstrut +\mathstrut \) \(38110\) \(\beta_{13}\mathstrut +\mathstrut \) \(24088\) \(\beta_{12}\mathstrut +\mathstrut \) \(30340\) \(\beta_{11}\mathstrut +\mathstrut \) \(61369\) \(\beta_{10}\mathstrut +\mathstrut \) \(18027\) \(\beta_{9}\mathstrut +\mathstrut \) \(26842\) \(\beta_{8}\mathstrut +\mathstrut \) \(90467\) \(\beta_{7}\mathstrut +\mathstrut \) \(9628\) \(\beta_{6}\mathstrut -\mathstrut \) \(89511\) \(\beta_{5}\mathstrut -\mathstrut \) \(3656\) \(\beta_{4}\mathstrut +\mathstrut \) \(214022\) \(\beta_{3}\mathstrut -\mathstrut \) \(1303\) \(\beta_{2}\mathstrut +\mathstrut \) \(465784\) \(\beta_{1}\mathstrut +\mathstrut \) \(92083\)
\(\nu^{16}\)\(=\)\(45824\) \(\beta_{18}\mathstrut +\mathstrut \) \(70814\) \(\beta_{17}\mathstrut -\mathstrut \) \(71931\) \(\beta_{16}\mathstrut -\mathstrut \) \(241742\) \(\beta_{15}\mathstrut -\mathstrut \) \(74068\) \(\beta_{14}\mathstrut +\mathstrut \) \(142969\) \(\beta_{13}\mathstrut +\mathstrut \) \(103396\) \(\beta_{12}\mathstrut +\mathstrut \) \(222271\) \(\beta_{11}\mathstrut +\mathstrut \) \(107566\) \(\beta_{10}\mathstrut +\mathstrut \) \(359574\) \(\beta_{9}\mathstrut +\mathstrut \) \(350215\) \(\beta_{8}\mathstrut +\mathstrut \) \(162701\) \(\beta_{7}\mathstrut +\mathstrut \) \(208480\) \(\beta_{6}\mathstrut -\mathstrut \) \(34917\) \(\beta_{5}\mathstrut -\mathstrut \) \(173795\) \(\beta_{4}\mathstrut +\mathstrut \) \(56387\) \(\beta_{3}\mathstrut +\mathstrut \) \(705414\) \(\beta_{2}\mathstrut +\mathstrut \) \(4037\) \(\beta_{1}\mathstrut +\mathstrut \) \(1232416\)
\(\nu^{17}\)\(=\)\(151648\) \(\beta_{18}\mathstrut +\mathstrut \) \(171997\) \(\beta_{17}\mathstrut -\mathstrut \) \(113356\) \(\beta_{16}\mathstrut -\mathstrut \) \(803134\) \(\beta_{15}\mathstrut -\mathstrut \) \(433642\) \(\beta_{14}\mathstrut +\mathstrut \) \(339563\) \(\beta_{13}\mathstrut +\mathstrut \) \(237420\) \(\beta_{12}\mathstrut +\mathstrut \) \(260928\) \(\beta_{11}\mathstrut +\mathstrut \) \(538331\) \(\beta_{10}\mathstrut +\mathstrut \) \(165372\) \(\beta_{9}\mathstrut +\mathstrut \) \(245930\) \(\beta_{8}\mathstrut +\mathstrut \) \(734926\) \(\beta_{7}\mathstrut +\mathstrut \) \(97683\) \(\beta_{6}\mathstrut -\mathstrut \) \(715742\) \(\beta_{5}\mathstrut -\mathstrut \) \(38593\) \(\beta_{4}\mathstrut +\mathstrut \) \(1591650\) \(\beta_{3}\mathstrut +\mathstrut \) \(15758\) \(\beta_{2}\mathstrut +\mathstrut \) \(3376342\) \(\beta_{1}\mathstrut +\mathstrut \) \(767873\)
\(\nu^{18}\)\(=\)\(405545\) \(\beta_{18}\mathstrut +\mathstrut \) \(574450\) \(\beta_{17}\mathstrut -\mathstrut \) \(545365\) \(\beta_{16}\mathstrut -\mathstrut \) \(2006325\) \(\beta_{15}\mathstrut -\mathstrut \) \(599014\) \(\beta_{14}\mathstrut +\mathstrut \) \(1215030\) \(\beta_{13}\mathstrut +\mathstrut \) \(938630\) \(\beta_{12}\mathstrut +\mathstrut \) \(1792211\) \(\beta_{11}\mathstrut +\mathstrut \) \(978901\) \(\beta_{10}\mathstrut +\mathstrut \) \(2741034\) \(\beta_{9}\mathstrut +\mathstrut \) \(2676791\) \(\beta_{8}\mathstrut +\mathstrut \) \(1388627\) \(\beta_{7}\mathstrut +\mathstrut \) \(1677626\) \(\beta_{6}\mathstrut -\mathstrut \) \(339940\) \(\beta_{5}\mathstrut -\mathstrut \) \(1361630\) \(\beta_{4}\mathstrut +\mathstrut \) \(536340\) \(\beta_{3}\mathstrut +\mathstrut \) \(5069187\) \(\beta_{2}\mathstrut +\mathstrut \) \(153210\) \(\beta_{1}\mathstrut +\mathstrut \) \(9034683\)

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
2.79341
2.49323
2.42782
2.26573
1.88777
1.49024
1.35771
1.20823
0.823573
0.395540
−0.249163
−0.407951
−0.782325
−1.30029
−1.48888
−1.91759
−1.94177
−2.35941
−2.69590
−2.79341 0 5.80316 1.63704 0 1.00000 −10.6238 0 −4.57293
1.2 −2.49323 0 4.21621 0.262973 0 1.00000 −5.52554 0 −0.655653
1.3 −2.42782 0 3.89430 −4.13136 0 1.00000 −4.59902 0 10.0302
1.4 −2.26573 0 3.13354 0.263451 0 1.00000 −2.56830 0 −0.596909
1.5 −1.88777 0 1.56369 3.95777 0 1.00000 0.823654 0 −7.47138
1.6 −1.49024 0 0.220813 −2.04744 0 1.00000 2.65141 0 3.05118
1.7 −1.35771 0 −0.156613 −4.06757 0 1.00000 2.92806 0 5.52260
1.8 −1.20823 0 −0.540168 −0.486834 0 1.00000 3.06912 0 0.588209
1.9 −0.823573 0 −1.32173 2.87784 0 1.00000 2.73569 0 −2.37011
1.10 −0.395540 0 −1.84355 −1.95228 0 1.00000 1.52028 0 0.772204
1.11 0.249163 0 −1.93792 −0.989651 0 1.00000 −0.981182 0 −0.246584
1.12 0.407951 0 −1.83358 1.08992 0 1.00000 −1.56391 0 0.444635
1.13 0.782325 0 −1.38797 2.85652 0 1.00000 −2.65049 0 2.23472
1.14 1.30029 0 −0.309249 −3.89453 0 1.00000 −3.00269 0 −5.06401
1.15 1.48888 0 0.216778 2.91159 0 1.00000 −2.65501 0 4.33502
1.16 1.91759 0 1.67713 −1.63006 0 1.00000 −0.619124 0 −3.12578
1.17 1.94177 0 1.77048 −3.66642 0 1.00000 −0.445683 0 −7.11935
1.18 2.35941 0 3.56679 3.48367 0 1.00000 3.69670 0 8.21939
1.19 2.69590 0 5.26787 −1.47463 0 1.00000 8.80983 0 −3.97544
\(n\): 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
\(3\) \(-1\)
\(7\) \(-1\)
\(127\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{19} + \cdots\)
\(T_{5}^{19} + \cdots\)