Properties

Label 6003.2.a.s
Level 6003
Weight 2
Character orbit 6003.a
Self dual Yes
Analytic conductor 47.934
Analytic rank 0
Dimension 20
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6003.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{7} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 2 + \beta_{2} ) q^{4} \) \( -\beta_{6} q^{5} \) \( -\beta_{14} q^{7} \) \( + ( -2 \beta_{1} - \beta_{3} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 2 + \beta_{2} ) q^{4} \) \( -\beta_{6} q^{5} \) \( -\beta_{14} q^{7} \) \( + ( -2 \beta_{1} - \beta_{3} ) q^{8} \) \( -\beta_{12} q^{10} \) \( + \beta_{8} q^{11} \) \( + ( 1 + \beta_{18} ) q^{13} \) \( + ( 1 - \beta_{1} + \beta_{14} + \beta_{18} + \beta_{19} ) q^{14} \) \( + ( 3 + \beta_{2} + \beta_{4} ) q^{16} \) \( + \beta_{16} q^{17} \) \( + \beta_{7} q^{19} \) \( + ( \beta_{1} + \beta_{3} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} - \beta_{16} - \beta_{19} ) q^{20} \) \( + ( 3 - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{14} - \beta_{15} + \beta_{19} ) q^{22} \) \(- q^{23}\) \( + ( 2 - \beta_{10} ) q^{25} \) \( + ( -2 \beta_{1} - \beta_{3} - \beta_{5} + \beta_{19} ) q^{26} \) \( + ( -1 - \beta_{5} - \beta_{14} + \beta_{15} - \beta_{19} ) q^{28} \) \(- q^{29}\) \( + ( 1 + \beta_{3} + \beta_{15} ) q^{31} \) \( + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{32} \) \( + ( 1 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{13} + \beta_{15} - \beta_{18} ) q^{34} \) \( + ( -1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{35} \) \( + ( 3 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{37} \) \( + ( \beta_{3} + \beta_{4} - \beta_{8} + \beta_{9} - \beta_{17} + \beta_{18} ) q^{38} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{40} \) \( + ( 1 - \beta_{1} + \beta_{5} - \beta_{7} - \beta_{13} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{41} \) \( + ( 2 - 2 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} + \beta_{18} + \beta_{19} ) q^{43} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{19} ) q^{44} \) \( + \beta_{1} q^{46} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} - \beta_{19} ) q^{47} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{13} - 2 \beta_{14} + \beta_{15} - \beta_{18} - \beta_{19} ) q^{49} \) \( + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{16} - 2 \beta_{18} - \beta_{19} ) q^{50} \) \( + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{13} - 2 \beta_{14} + \beta_{15} - 2 \beta_{16} + 2 \beta_{18} ) q^{52} \) \( + ( 1 + 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{53} \) \( + ( 2 - \beta_{1} - 2 \beta_{3} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{18} - \beta_{19} ) q^{55} \) \( + ( 4 - \beta_{1} + \beta_{5} - \beta_{7} - 2 \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} + \beta_{18} + 2 \beta_{19} ) q^{56} \) \( + \beta_{1} q^{58} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{6} - \beta_{15} ) q^{59} \) \( + ( 1 - \beta_{1} - \beta_{3} + \beta_{7} + \beta_{12} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{61} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{7} + \beta_{16} + \beta_{19} ) q^{62} \) \( + ( 7 - 3 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} + 2 \beta_{14} - \beta_{15} - \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{64} \) \( + ( 3 \beta_{1} + \beta_{3} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{19} ) q^{65} \) \( + ( -2 \beta_{1} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} - \beta_{14} + \beta_{15} - \beta_{17} ) q^{67} \) \( + ( 1 - 3 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{13} + \beta_{15} + 3 \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{68} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{14} - \beta_{15} ) q^{70} \) \( + ( -1 - \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{10} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{71} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{16} - \beta_{18} + \beta_{19} ) q^{73} \) \( + ( -3 + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} - \beta_{18} - \beta_{19} ) q^{74} \) \( + ( -3 - \beta_{2} - \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{12} - \beta_{13} - \beta_{15} - 2 \beta_{16} + \beta_{17} - \beta_{19} ) q^{76} \) \( + ( -2 + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{11} + 2 \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{77} \) \( + ( \beta_{3} + \beta_{4} - 2 \beta_{14} - \beta_{16} - \beta_{18} - \beta_{19} ) q^{79} \) \( + ( 7 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} - \beta_{18} - \beta_{19} ) q^{80} \) \( + ( -3 + \beta_{1} - \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{8} - 2 \beta_{14} + \beta_{15} + \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{82} \) \( + ( 1 + 2 \beta_{2} + \beta_{3} - \beta_{6} - \beta_{8} - \beta_{11} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{18} ) q^{83} \) \( + ( -2 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{10} - 2 \beta_{12} + \beta_{13} + \beta_{15} - \beta_{19} ) q^{85} \) \( + ( \beta_{1} - \beta_{5} + \beta_{6} + \beta_{11} + \beta_{12} - \beta_{14} - \beta_{16} - \beta_{18} - \beta_{19} ) q^{86} \) \( + ( 6 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{88} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{89} \) \( + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{6} - \beta_{7} - \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} - \beta_{16} - 2 \beta_{19} ) q^{91} \) \( + ( -2 - \beta_{2} ) q^{92} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} - 3 \beta_{16} ) q^{94} \) \( + ( -1 - 4 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} + 2 \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{95} \) \( + ( 2 - 3 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{16} - \beta_{17} + \beta_{19} ) q^{97} \) \( + ( 2 + \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + 3 \beta_{14} - \beta_{15} - 2 \beta_{16} + \beta_{17} + 2 \beta_{19} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(20q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 30q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(20q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 30q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 7q^{10} \) \(\mathstrut +\mathstrut 21q^{13} \) \(\mathstrut +\mathstrut q^{14} \) \(\mathstrut +\mathstrut 58q^{16} \) \(\mathstrut +\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 7q^{19} \) \(\mathstrut +\mathstrut 20q^{20} \) \(\mathstrut +\mathstrut 7q^{22} \) \(\mathstrut -\mathstrut 20q^{23} \) \(\mathstrut +\mathstrut 47q^{25} \) \(\mathstrut -\mathstrut 8q^{26} \) \(\mathstrut +\mathstrut 11q^{28} \) \(\mathstrut -\mathstrut 20q^{29} \) \(\mathstrut +\mathstrut 28q^{31} \) \(\mathstrut -\mathstrut 14q^{32} \) \(\mathstrut +\mathstrut 16q^{34} \) \(\mathstrut -\mathstrut 9q^{35} \) \(\mathstrut +\mathstrut 14q^{37} \) \(\mathstrut +\mathstrut 20q^{38} \) \(\mathstrut +\mathstrut 34q^{40} \) \(\mathstrut -\mathstrut 7q^{41} \) \(\mathstrut +\mathstrut 3q^{43} \) \(\mathstrut +\mathstrut q^{44} \) \(\mathstrut +\mathstrut 2q^{46} \) \(\mathstrut -\mathstrut 3q^{47} \) \(\mathstrut +\mathstrut 35q^{49} \) \(\mathstrut +\mathstrut 24q^{50} \) \(\mathstrut +\mathstrut 73q^{52} \) \(\mathstrut +\mathstrut 19q^{53} \) \(\mathstrut +\mathstrut 29q^{55} \) \(\mathstrut +\mathstrut 30q^{56} \) \(\mathstrut +\mathstrut 2q^{58} \) \(\mathstrut -\mathstrut 20q^{59} \) \(\mathstrut +\mathstrut 15q^{61} \) \(\mathstrut -\mathstrut 12q^{62} \) \(\mathstrut +\mathstrut 82q^{64} \) \(\mathstrut +\mathstrut 28q^{65} \) \(\mathstrut +\mathstrut 20q^{67} \) \(\mathstrut +\mathstrut 23q^{68} \) \(\mathstrut -\mathstrut 24q^{70} \) \(\mathstrut -\mathstrut 63q^{71} \) \(\mathstrut +\mathstrut 19q^{73} \) \(\mathstrut -\mathstrut 16q^{74} \) \(\mathstrut -\mathstrut 44q^{76} \) \(\mathstrut +\mathstrut 7q^{77} \) \(\mathstrut +\mathstrut 32q^{79} \) \(\mathstrut +\mathstrut 56q^{80} \) \(\mathstrut -\mathstrut 20q^{82} \) \(\mathstrut +\mathstrut 21q^{83} \) \(\mathstrut +\mathstrut 4q^{85} \) \(\mathstrut +\mathstrut 6q^{86} \) \(\mathstrut +\mathstrut 55q^{88} \) \(\mathstrut +\mathstrut 13q^{89} \) \(\mathstrut +\mathstrut 70q^{91} \) \(\mathstrut -\mathstrut 30q^{92} \) \(\mathstrut -\mathstrut 12q^{94} \) \(\mathstrut -\mathstrut 9q^{95} \) \(\mathstrut -\mathstrut 9q^{97} \) \(\mathstrut -\mathstrut 31q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20}\mathstrut -\mathstrut \) \(2\) \(x^{19}\mathstrut -\mathstrut \) \(33\) \(x^{18}\mathstrut +\mathstrut \) \(64\) \(x^{17}\mathstrut +\mathstrut \) \(453\) \(x^{16}\mathstrut -\mathstrut \) \(846\) \(x^{15}\mathstrut -\mathstrut \) \(3353\) \(x^{14}\mathstrut +\mathstrut \) \(5985\) \(x^{13}\mathstrut +\mathstrut \) \(14484\) \(x^{12}\mathstrut -\mathstrut \) \(24566\) \(x^{11}\mathstrut -\mathstrut \) \(36791\) \(x^{10}\mathstrut +\mathstrut \) \(59410\) \(x^{9}\mathstrut +\mathstrut \) \(52109\) \(x^{8}\mathstrut -\mathstrut \) \(82362\) \(x^{7}\mathstrut -\mathstrut \) \(34967\) \(x^{6}\mathstrut +\mathstrut \) \(60661\) \(x^{5}\mathstrut +\mathstrut \) \(5201\) \(x^{4}\mathstrut -\mathstrut \) \(19624\) \(x^{3}\mathstrut +\mathstrut \) \(2768\) \(x^{2}\mathstrut +\mathstrut \) \(1408\) \(x\mathstrut -\mathstrut \) \(256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 7 \nu^{2} + 5 \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(9977\) \(\nu^{19}\mathstrut -\mathstrut \) \(284189\) \(\nu^{18}\mathstrut -\mathstrut \) \(706\) \(\nu^{17}\mathstrut +\mathstrut \) \(9004494\) \(\nu^{16}\mathstrut +\mathstrut \) \(5676465\) \(\nu^{15}\mathstrut -\mathstrut \) \(116705443\) \(\nu^{14}\mathstrut -\mathstrut \) \(93997884\) \(\nu^{13}\mathstrut +\mathstrut \) \(796777255\) \(\nu^{12}\mathstrut +\mathstrut \) \(679406561\) \(\nu^{11}\mathstrut -\mathstrut \) \(3077890179\) \(\nu^{10}\mathstrut -\mathstrut \) \(2551075658\) \(\nu^{9}\mathstrut +\mathstrut \) \(6745265900\) \(\nu^{8}\mathstrut +\mathstrut \) \(4995760963\) \(\nu^{7}\mathstrut -\mathstrut \) \(8060495321\) \(\nu^{6}\mathstrut -\mathstrut \) \(4647412912\) \(\nu^{5}\mathstrut +\mathstrut \) \(4886184991\) \(\nu^{4}\mathstrut +\mathstrut \) \(1615059980\) \(\nu^{3}\mathstrut -\mathstrut \) \(1279285860\) \(\nu^{2}\mathstrut -\mathstrut \) \(73689968\) \(\nu\mathstrut +\mathstrut \) \(67052416\)\()/3404224\)
\(\beta_{6}\)\(=\)\((\)\(59655\) \(\nu^{19}\mathstrut +\mathstrut \) \(83835\) \(\nu^{18}\mathstrut -\mathstrut \) \(1977318\) \(\nu^{17}\mathstrut -\mathstrut \) \(2493558\) \(\nu^{16}\mathstrut +\mathstrut \) \(26985321\) \(\nu^{15}\mathstrut +\mathstrut \) \(29771565\) \(\nu^{14}\mathstrut -\mathstrut \) \(195494072\) \(\nu^{13}\mathstrut -\mathstrut \) \(181802773\) \(\nu^{12}\mathstrut +\mathstrut \) \(808522329\) \(\nu^{11}\mathstrut +\mathstrut \) \(596421017\) \(\nu^{10}\mathstrut -\mathstrut \) \(1915046350\) \(\nu^{9}\mathstrut -\mathstrut \) \(993555752\) \(\nu^{8}\mathstrut +\mathstrut \) \(2487007763\) \(\nu^{7}\mathstrut +\mathstrut \) \(636510631\) \(\nu^{6}\mathstrut -\mathstrut \) \(1618911100\) \(\nu^{5}\mathstrut +\mathstrut \) \(152363091\) \(\nu^{4}\mathstrut +\mathstrut \) \(447758060\) \(\nu^{3}\mathstrut -\mathstrut \) \(274311304\) \(\nu^{2}\mathstrut -\mathstrut \) \(39969248\) \(\nu\mathstrut +\mathstrut \) \(42998144\)\()/6808448\)
\(\beta_{7}\)\(=\)\((\)\(32190\) \(\nu^{19}\mathstrut -\mathstrut \) \(101041\) \(\nu^{18}\mathstrut -\mathstrut \) \(1039401\) \(\nu^{17}\mathstrut +\mathstrut \) \(3106330\) \(\nu^{16}\mathstrut +\mathstrut \) \(13951816\) \(\nu^{15}\mathstrut -\mathstrut \) \(38936723\) \(\nu^{14}\mathstrut -\mathstrut \) \(101021583\) \(\nu^{13}\mathstrut +\mathstrut \) \(256304866\) \(\nu^{12}\mathstrut +\mathstrut \) \(427557079\) \(\nu^{11}\mathstrut -\mathstrut \) \(951947129\) \(\nu^{10}\mathstrut -\mathstrut \) \(1066095561\) \(\nu^{9}\mathstrut +\mathstrut \) \(1998865072\) \(\nu^{8}\mathstrut +\mathstrut \) \(1483887288\) \(\nu^{7}\mathstrut -\mathstrut \) \(2270424875\) \(\nu^{6}\mathstrut -\mathstrut \) \(990251463\) \(\nu^{5}\mathstrut +\mathstrut \) \(1290653668\) \(\nu^{4}\mathstrut +\mathstrut \) \(195615347\) \(\nu^{3}\mathstrut -\mathstrut \) \(326105000\) \(\nu^{2}\mathstrut +\mathstrut \) \(28151104\) \(\nu\mathstrut +\mathstrut \) \(21780992\)\()/3404224\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(113117\) \(\nu^{19}\mathstrut -\mathstrut \) \(286133\) \(\nu^{18}\mathstrut +\mathstrut \) \(3436418\) \(\nu^{17}\mathstrut +\mathstrut \) \(9067866\) \(\nu^{16}\mathstrut -\mathstrut \) \(41870219\) \(\nu^{15}\mathstrut -\mathstrut \) \(117469803\) \(\nu^{14}\mathstrut +\mathstrut \) \(259419552\) \(\nu^{13}\mathstrut +\mathstrut \) \(800884431\) \(\nu^{12}\mathstrut -\mathstrut \) \(846624863\) \(\nu^{11}\mathstrut -\mathstrut \) \(3086463419\) \(\nu^{10}\mathstrut +\mathstrut \) \(1308880446\) \(\nu^{9}\mathstrut +\mathstrut \) \(6744972372\) \(\nu^{8}\mathstrut -\mathstrut \) \(477260749\) \(\nu^{7}\mathstrut -\mathstrut \) \(8047603685\) \(\nu^{6}\mathstrut -\mathstrut \) \(765017784\) \(\nu^{5}\mathstrut +\mathstrut \) \(4881910035\) \(\nu^{4}\mathstrut +\mathstrut \) \(543546028\) \(\nu^{3}\mathstrut -\mathstrut \) \(1275029840\) \(\nu^{2}\mathstrut -\mathstrut \) \(6205632\) \(\nu\mathstrut +\mathstrut \) \(65443584\)\()/6808448\)
\(\beta_{9}\)\(=\)\((\)\(132421\) \(\nu^{19}\mathstrut +\mathstrut \) \(61925\) \(\nu^{18}\mathstrut -\mathstrut \) \(4039074\) \(\nu^{17}\mathstrut -\mathstrut \) \(1758962\) \(\nu^{16}\mathstrut +\mathstrut \) \(49733347\) \(\nu^{15}\mathstrut +\mathstrut \) \(18817259\) \(\nu^{14}\mathstrut -\mathstrut \) \(315679888\) \(\nu^{13}\mathstrut -\mathstrut \) \(86908655\) \(\nu^{12}\mathstrut +\mathstrut \) \(1090655743\) \(\nu^{11}\mathstrut +\mathstrut \) \(87630643\) \(\nu^{10}\mathstrut -\mathstrut \) \(1976507934\) \(\nu^{9}\mathstrut +\mathstrut \) \(645077396\) \(\nu^{8}\mathstrut +\mathstrut \) \(1618257989\) \(\nu^{7}\mathstrut -\mathstrut \) \(2223325131\) \(\nu^{6}\mathstrut -\mathstrut \) \(337309496\) \(\nu^{5}\mathstrut +\mathstrut \) \(2337172669\) \(\nu^{4}\mathstrut -\mathstrut \) \(167806212\) \(\nu^{3}\mathstrut -\mathstrut \) \(665008808\) \(\nu^{2}\mathstrut +\mathstrut \) \(110604448\) \(\nu\mathstrut -\mathstrut \) \(8160896\)\()/6808448\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(161261\) \(\nu^{19}\mathstrut +\mathstrut \) \(371971\) \(\nu^{18}\mathstrut +\mathstrut \) \(5495374\) \(\nu^{17}\mathstrut -\mathstrut \) \(11857542\) \(\nu^{16}\mathstrut -\mathstrut \) \(78622707\) \(\nu^{15}\mathstrut +\mathstrut \) \(155862669\) \(\nu^{14}\mathstrut +\mathstrut \) \(614092532\) \(\nu^{13}\mathstrut -\mathstrut \) \(1093593473\) \(\nu^{12}\mathstrut -\mathstrut \) \(2847831847\) \(\nu^{11}\mathstrut +\mathstrut \) \(4435330241\) \(\nu^{10}\mathstrut +\mathstrut \) \(7971368478\) \(\nu^{9}\mathstrut -\mathstrut \) \(10545189400\) \(\nu^{8}\mathstrut -\mathstrut \) \(13052790841\) \(\nu^{7}\mathstrut +\mathstrut \) \(14281125551\) \(\nu^{6}\mathstrut +\mathstrut \) \(11462171224\) \(\nu^{5}\mathstrut -\mathstrut \) \(10197457825\) \(\nu^{4}\mathstrut -\mathstrut \) \(4446760952\) \(\nu^{3}\mathstrut +\mathstrut \) \(3175929024\) \(\nu^{2}\mathstrut +\mathstrut \) \(397653248\) \(\nu\mathstrut -\mathstrut \) \(221220224\)\()/6808448\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(88710\) \(\nu^{19}\mathstrut +\mathstrut \) \(348943\) \(\nu^{18}\mathstrut +\mathstrut \) \(3305005\) \(\nu^{17}\mathstrut -\mathstrut \) \(10913794\) \(\nu^{16}\mathstrut -\mathstrut \) \(52040672\) \(\nu^{15}\mathstrut +\mathstrut \) \(139461381\) \(\nu^{14}\mathstrut +\mathstrut \) \(448941315\) \(\nu^{13}\mathstrut -\mathstrut \) \(938038438\) \(\nu^{12}\mathstrut -\mathstrut \) \(2296222593\) \(\nu^{11}\mathstrut +\mathstrut \) \(3571430473\) \(\nu^{10}\mathstrut +\mathstrut \) \(7028716107\) \(\nu^{9}\mathstrut -\mathstrut \) \(7739376550\) \(\nu^{8}\mathstrut -\mathstrut \) \(12364008590\) \(\nu^{7}\mathstrut +\mathstrut \) \(9238883527\) \(\nu^{6}\mathstrut +\mathstrut \) \(11341153205\) \(\nu^{5}\mathstrut -\mathstrut \) \(5750325066\) \(\nu^{4}\mathstrut -\mathstrut \) \(4484529451\) \(\nu^{3}\mathstrut +\mathstrut \) \(1641458868\) \(\nu^{2}\mathstrut +\mathstrut \) \(476989360\) \(\nu\mathstrut -\mathstrut \) \(107471168\)\()/3404224\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(203145\) \(\nu^{19}\mathstrut +\mathstrut \) \(8703\) \(\nu^{18}\mathstrut +\mathstrut \) \(6311478\) \(\nu^{17}\mathstrut +\mathstrut \) \(38394\) \(\nu^{16}\mathstrut -\mathstrut \) \(80239695\) \(\nu^{15}\mathstrut -\mathstrut \) \(4529143\) \(\nu^{14}\mathstrut +\mathstrut \) \(538837948\) \(\nu^{13}\mathstrut +\mathstrut \) \(55520691\) \(\nu^{12}\mathstrut -\mathstrut \) \(2061905747\) \(\nu^{11}\mathstrut -\mathstrut \) \(279720755\) \(\nu^{10}\mathstrut +\mathstrut \) \(4537659302\) \(\nu^{9}\mathstrut +\mathstrut \) \(621554632\) \(\nu^{8}\mathstrut -\mathstrut \) \(5549815741\) \(\nu^{7}\mathstrut -\mathstrut \) \(467045285\) \(\nu^{6}\mathstrut +\mathstrut \) \(3466368864\) \(\nu^{5}\mathstrut -\mathstrut \) \(137492405\) \(\nu^{4}\mathstrut -\mathstrut \) \(896358416\) \(\nu^{3}\mathstrut +\mathstrut \) \(205094288\) \(\nu^{2}\mathstrut +\mathstrut \) \(40996096\) \(\nu\mathstrut -\mathstrut \) \(15271680\)\()/6808448\)
\(\beta_{13}\)\(=\)\((\)\(294337\) \(\nu^{19}\mathstrut -\mathstrut \) \(270767\) \(\nu^{18}\mathstrut -\mathstrut \) \(9616398\) \(\nu^{17}\mathstrut +\mathstrut \) \(8494062\) \(\nu^{16}\mathstrut +\mathstrut \) \(130255487\) \(\nu^{15}\mathstrut -\mathstrut \) \(109626129\) \(\nu^{14}\mathstrut -\mathstrut \) \(947801932\) \(\nu^{13}\mathstrut +\mathstrut \) \(753143013\) \(\nu^{12}\mathstrut +\mathstrut \) \(4013917123\) \(\nu^{11}\mathstrut -\mathstrut \) \(2978708653\) \(\nu^{10}\mathstrut -\mathstrut \) \(10021968214\) \(\nu^{9}\mathstrut +\mathstrut \) \(6853628544\) \(\nu^{8}\mathstrut +\mathstrut \) \(14266108021\) \(\nu^{7}\mathstrut -\mathstrut \) \(8841784179\) \(\nu^{6}\mathstrut -\mathstrut \) \(10620817160\) \(\nu^{5}\mathstrut +\mathstrut \) \(5854037837\) \(\nu^{4}\mathstrut +\mathstrut \) \(3454356288\) \(\nu^{3}\mathstrut -\mathstrut \) \(1635642432\) \(\nu^{2}\mathstrut -\mathstrut \) \(316927872\) \(\nu\mathstrut +\mathstrut \) \(110542464\)\()/6808448\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(148537\) \(\nu^{19}\mathstrut +\mathstrut \) \(338681\) \(\nu^{18}\mathstrut +\mathstrut \) \(5041136\) \(\nu^{17}\mathstrut -\mathstrut \) \(10572042\) \(\nu^{16}\mathstrut -\mathstrut \) \(71767335\) \(\nu^{15}\mathstrut +\mathstrut \) \(135044723\) \(\nu^{14}\mathstrut +\mathstrut \) \(556857158\) \(\nu^{13}\mathstrut -\mathstrut \) \(910556129\) \(\nu^{12}\mathstrut -\mathstrut \) \(2556880073\) \(\nu^{11}\mathstrut +\mathstrut \) \(3490925219\) \(\nu^{10}\mathstrut +\mathstrut \) \(7037682580\) \(\nu^{9}\mathstrut -\mathstrut \) \(7663102560\) \(\nu^{8}\mathstrut -\mathstrut \) \(11171608865\) \(\nu^{7}\mathstrut +\mathstrut \) \(9303904093\) \(\nu^{6}\mathstrut +\mathstrut \) \(9240832926\) \(\nu^{5}\mathstrut -\mathstrut \) \(5836685909\) \(\nu^{4}\mathstrut -\mathstrut \) \(3213222030\) \(\nu^{3}\mathstrut +\mathstrut \) \(1626363316\) \(\nu^{2}\mathstrut +\mathstrut \) \(260140592\) \(\nu\mathstrut -\mathstrut \) \(97520832\)\()/3404224\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(380439\) \(\nu^{19}\mathstrut +\mathstrut \) \(254997\) \(\nu^{18}\mathstrut +\mathstrut \) \(12633678\) \(\nu^{17}\mathstrut -\mathstrut \) \(8079138\) \(\nu^{16}\mathstrut -\mathstrut \) \(174716497\) \(\nu^{15}\mathstrut +\mathstrut \) \(105738795\) \(\nu^{14}\mathstrut +\mathstrut \) \(1306063448\) \(\nu^{13}\mathstrut -\mathstrut \) \(741027003\) \(\nu^{12}\mathstrut -\mathstrut \) \(5732647449\) \(\nu^{11}\mathstrut +\mathstrut \) \(3015967983\) \(\nu^{10}\mathstrut +\mathstrut \) \(15028081270\) \(\nu^{9}\mathstrut -\mathstrut \) \(7238359288\) \(\nu^{8}\mathstrut -\mathstrut \) \(22897375291\) \(\nu^{7}\mathstrut +\mathstrut \) \(9964491865\) \(\nu^{6}\mathstrut +\mathstrut \) \(18755286572\) \(\nu^{5}\mathstrut -\mathstrut \) \(7338969931\) \(\nu^{4}\mathstrut -\mathstrut \) \(6950567148\) \(\nu^{3}\mathstrut +\mathstrut \) \(2443697432\) \(\nu^{2}\mathstrut +\mathstrut \) \(760925792\) \(\nu\mathstrut -\mathstrut \) \(178908672\)\()/6808448\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(202867\) \(\nu^{19}\mathstrut +\mathstrut \) \(156362\) \(\nu^{18}\mathstrut +\mathstrut \) \(6774905\) \(\nu^{17}\mathstrut -\mathstrut \) \(4911368\) \(\nu^{16}\mathstrut -\mathstrut \) \(94271295\) \(\nu^{15}\mathstrut +\mathstrut \) \(63450778\) \(\nu^{14}\mathstrut +\mathstrut \) \(709058137\) \(\nu^{13}\mathstrut -\mathstrut \) \(436217767\) \(\nu^{12}\mathstrut -\mathstrut \) \(3127379984\) \(\nu^{11}\mathstrut +\mathstrut \) \(1727035600\) \(\nu^{10}\mathstrut +\mathstrut \) \(8203441237\) \(\nu^{9}\mathstrut -\mathstrut \) \(3991457880\) \(\nu^{8}\mathstrut -\mathstrut \) \(12375653553\) \(\nu^{7}\mathstrut +\mathstrut \) \(5243719104\) \(\nu^{6}\mathstrut +\mathstrut \) \(9802374313\) \(\nu^{5}\mathstrut -\mathstrut \) \(3672857369\) \(\nu^{4}\mathstrut -\mathstrut \) \(3330475213\) \(\nu^{3}\mathstrut +\mathstrut \) \(1148408812\) \(\nu^{2}\mathstrut +\mathstrut \) \(272653424\) \(\nu\mathstrut -\mathstrut \) \(69421504\)\()/3404224\)
\(\beta_{17}\)\(=\)\((\)\(-\)\(423491\) \(\nu^{19}\mathstrut +\mathstrut \) \(589261\) \(\nu^{18}\mathstrut +\mathstrut \) \(14300798\) \(\nu^{17}\mathstrut -\mathstrut \) \(18316426\) \(\nu^{16}\mathstrut -\mathstrut \) \(202063709\) \(\nu^{15}\mathstrut +\mathstrut \) \(232872267\) \(\nu^{14}\mathstrut +\mathstrut \) \(1551417160\) \(\nu^{13}\mathstrut -\mathstrut \) \(1562401159\) \(\nu^{12}\mathstrut -\mathstrut \) \(7027134489\) \(\nu^{11}\mathstrut +\mathstrut \) \(5963688187\) \(\nu^{10}\mathstrut +\mathstrut \) \(19037717322\) \(\nu^{9}\mathstrut -\mathstrut \) \(13077406708\) \(\nu^{8}\mathstrut -\mathstrut \) \(29747755059\) \(\nu^{7}\mathstrut +\mathstrut \) \(16061786213\) \(\nu^{6}\mathstrut +\mathstrut \) \(24293520488\) \(\nu^{5}\mathstrut -\mathstrut \) \(10588454955\) \(\nu^{4}\mathstrut -\mathstrut \) \(8318188780\) \(\nu^{3}\mathstrut +\mathstrut \) \(3293324216\) \(\nu^{2}\mathstrut +\mathstrut \) \(584394208\) \(\nu\mathstrut -\mathstrut \) \(197504512\)\()/6808448\)
\(\beta_{18}\)\(=\)\((\)\(-\)\(595707\) \(\nu^{19}\mathstrut +\mathstrut \) \(195465\) \(\nu^{18}\mathstrut +\mathstrut \) \(19683950\) \(\nu^{17}\mathstrut -\mathstrut \) \(6229090\) \(\nu^{16}\mathstrut -\mathstrut \) \(270259309\) \(\nu^{15}\mathstrut +\mathstrut \) \(82762231\) \(\nu^{14}\mathstrut +\mathstrut \) \(1999221168\) \(\nu^{13}\mathstrut -\mathstrut \) \(597242311\) \(\nu^{12}\mathstrut -\mathstrut \) \(8642079917\) \(\nu^{11}\mathstrut +\mathstrut \) \(2553180667\) \(\nu^{10}\mathstrut +\mathstrut \) \(22150191358\) \(\nu^{9}\mathstrut -\mathstrut \) \(6590508888\) \(\nu^{8}\mathstrut -\mathstrut \) \(32604889607\) \(\nu^{7}\mathstrut +\mathstrut \) \(9966835125\) \(\nu^{6}\mathstrut +\mathstrut \) \(25189860044\) \(\nu^{5}\mathstrut -\mathstrut \) \(8106916351\) \(\nu^{4}\mathstrut -\mathstrut \) \(8224828108\) \(\nu^{3}\mathstrut +\mathstrut \) \(2852863952\) \(\nu^{2}\mathstrut +\mathstrut \) \(555519872\) \(\nu\mathstrut -\mathstrut \) \(174423552\)\()/6808448\)
\(\beta_{19}\)\(=\)\((\)\(975995\) \(\nu^{19}\mathstrut -\mathstrut \) \(593997\) \(\nu^{18}\mathstrut -\mathstrut \) \(31897570\) \(\nu^{17}\mathstrut +\mathstrut \) \(18413026\) \(\nu^{16}\mathstrut +\mathstrut \) \(432558821\) \(\nu^{15}\mathstrut -\mathstrut \) \(235226483\) \(\nu^{14}\mathstrut -\mathstrut \) \(3156059852\) \(\nu^{13}\mathstrut +\mathstrut \) \(1607414239\) \(\nu^{12}\mathstrut +\mathstrut \) \(13439770617\) \(\nu^{11}\mathstrut -\mathstrut \) \(6389315479\) \(\nu^{10}\mathstrut -\mathstrut \) \(33902595298\) \(\nu^{9}\mathstrut +\mathstrut \) \(15053725344\) \(\nu^{8}\mathstrut +\mathstrut \) \(49088306735\) \(\nu^{7}\mathstrut -\mathstrut \) \(20480764017\) \(\nu^{6}\mathstrut -\mathstrut \) \(37324091800\) \(\nu^{5}\mathstrut +\mathstrut \) \(14898925983\) \(\nu^{4}\mathstrut +\mathstrut \) \(12074218624\) \(\nu^{3}\mathstrut -\mathstrut \) \(4763008568\) \(\nu^{2}\mathstrut -\mathstrut \) \(845754080\) \(\nu\mathstrut +\mathstrut \) \(286605824\)\()/6808448\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(23\)
\(\nu^{5}\)\(=\)\(\beta_{19}\mathstrut +\mathstrut \) \(\beta_{18}\mathstrut -\mathstrut \) \(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{16}\mathstrut +\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(37\) \(\beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{19}\mathstrut +\mathstrut \) \(2\) \(\beta_{18}\mathstrut -\mathstrut \) \(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(2\) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(47\) \(\beta_{2}\mathstrut -\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(149\)
\(\nu^{7}\)\(=\)\(15\) \(\beta_{19}\mathstrut +\mathstrut \) \(17\) \(\beta_{18}\mathstrut -\mathstrut \) \(13\) \(\beta_{17}\mathstrut +\mathstrut \) \(12\) \(\beta_{16}\mathstrut +\mathstrut \) \(12\) \(\beta_{15}\mathstrut +\mathstrut \) \(13\) \(\beta_{14}\mathstrut -\mathstrut \) \(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(12\) \(\beta_{12}\mathstrut -\mathstrut \) \(13\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(14\) \(\beta_{9}\mathstrut -\mathstrut \) \(14\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(16\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(13\) \(\beta_{4}\mathstrut +\mathstrut \) \(70\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(236\) \(\beta_{1}\mathstrut -\mathstrut \) \(2\)
\(\nu^{8}\)\(=\)\(15\) \(\beta_{19}\mathstrut +\mathstrut \) \(29\) \(\beta_{18}\mathstrut -\mathstrut \) \(16\) \(\beta_{17}\mathstrut +\mathstrut \) \(5\) \(\beta_{16}\mathstrut -\mathstrut \) \(16\) \(\beta_{15}\mathstrut +\mathstrut \) \(31\) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(15\) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(15\) \(\beta_{10}\mathstrut +\mathstrut \) \(15\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(28\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(31\) \(\beta_{5}\mathstrut +\mathstrut \) \(97\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(322\) \(\beta_{2}\mathstrut -\mathstrut \) \(47\) \(\beta_{1}\mathstrut +\mathstrut \) \(1019\)
\(\nu^{9}\)\(=\)\(163\) \(\beta_{19}\mathstrut +\mathstrut \) \(195\) \(\beta_{18}\mathstrut -\mathstrut \) \(125\) \(\beta_{17}\mathstrut +\mathstrut \) \(109\) \(\beta_{16}\mathstrut +\mathstrut \) \(107\) \(\beta_{15}\mathstrut +\mathstrut \) \(128\) \(\beta_{14}\mathstrut -\mathstrut \) \(38\) \(\beta_{13}\mathstrut +\mathstrut \) \(110\) \(\beta_{12}\mathstrut -\mathstrut \) \(126\) \(\beta_{11}\mathstrut +\mathstrut \) \(20\) \(\beta_{10}\mathstrut +\mathstrut \) \(144\) \(\beta_{9}\mathstrut -\mathstrut \) \(139\) \(\beta_{8}\mathstrut -\mathstrut \) \(49\) \(\beta_{7}\mathstrut +\mathstrut \) \(177\) \(\beta_{6}\mathstrut +\mathstrut \) \(35\) \(\beta_{5}\mathstrut +\mathstrut \) \(128\) \(\beta_{4}\mathstrut +\mathstrut \) \(527\) \(\beta_{3}\mathstrut -\mathstrut \) \(38\) \(\beta_{2}\mathstrut +\mathstrut \) \(1550\) \(\beta_{1}\mathstrut -\mathstrut \) \(32\)
\(\nu^{10}\)\(=\)\(161\) \(\beta_{19}\mathstrut +\mathstrut \) \(298\) \(\beta_{18}\mathstrut -\mathstrut \) \(177\) \(\beta_{17}\mathstrut +\mathstrut \) \(101\) \(\beta_{16}\mathstrut -\mathstrut \) \(176\) \(\beta_{15}\mathstrut +\mathstrut \) \(341\) \(\beta_{14}\mathstrut +\mathstrut \) \(23\) \(\beta_{13}\mathstrut -\mathstrut \) \(159\) \(\beta_{12}\mathstrut -\mathstrut \) \(25\) \(\beta_{11}\mathstrut +\mathstrut \) \(163\) \(\beta_{10}\mathstrut +\mathstrut \) \(163\) \(\beta_{9}\mathstrut -\mathstrut \) \(39\) \(\beta_{8}\mathstrut -\mathstrut \) \(280\) \(\beta_{7}\mathstrut +\mathstrut \) \(39\) \(\beta_{6}\mathstrut +\mathstrut \) \(339\) \(\beta_{5}\mathstrut +\mathstrut \) \(800\) \(\beta_{4}\mathstrut -\mathstrut \) \(22\) \(\beta_{3}\mathstrut +\mathstrut \) \(2262\) \(\beta_{2}\mathstrut -\mathstrut \) \(522\) \(\beta_{1}\mathstrut +\mathstrut \) \(7189\)
\(\nu^{11}\)\(=\)\(1554\) \(\beta_{19}\mathstrut +\mathstrut \) \(1914\) \(\beta_{18}\mathstrut -\mathstrut \) \(1080\) \(\beta_{17}\mathstrut +\mathstrut \) \(900\) \(\beta_{16}\mathstrut +\mathstrut \) \(862\) \(\beta_{15}\mathstrut +\mathstrut \) \(1144\) \(\beta_{14}\mathstrut -\mathstrut \) \(476\) \(\beta_{13}\mathstrut +\mathstrut \) \(920\) \(\beta_{12}\mathstrut -\mathstrut \) \(1108\) \(\beta_{11}\mathstrut +\mathstrut \) \(262\) \(\beta_{10}\mathstrut +\mathstrut \) \(1326\) \(\beta_{9}\mathstrut -\mathstrut \) \(1218\) \(\beta_{8}\mathstrut -\mathstrut \) \(544\) \(\beta_{7}\mathstrut +\mathstrut \) \(1704\) \(\beta_{6}\mathstrut +\mathstrut \) \(414\) \(\beta_{5}\mathstrut +\mathstrut \) \(1145\) \(\beta_{4}\mathstrut +\mathstrut \) \(3940\) \(\beta_{3}\mathstrut -\mathstrut \) \(472\) \(\beta_{2}\mathstrut +\mathstrut \) \(10431\) \(\beta_{1}\mathstrut -\mathstrut \) \(361\)
\(\nu^{12}\)\(=\)\(1517\) \(\beta_{19}\mathstrut +\mathstrut \) \(2679\) \(\beta_{18}\mathstrut -\mathstrut \) \(1689\) \(\beta_{17}\mathstrut +\mathstrut \) \(1337\) \(\beta_{16}\mathstrut -\mathstrut \) \(1661\) \(\beta_{15}\mathstrut +\mathstrut \) \(3273\) \(\beta_{14}\mathstrut +\mathstrut \) \(336\) \(\beta_{13}\mathstrut -\mathstrut \) \(1479\) \(\beta_{12}\mathstrut -\mathstrut \) \(377\) \(\beta_{11}\mathstrut +\mathstrut \) \(1564\) \(\beta_{10}\mathstrut +\mathstrut \) \(1559\) \(\beta_{9}\mathstrut -\mathstrut \) \(487\) \(\beta_{8}\mathstrut -\mathstrut \) \(2472\) \(\beta_{7}\mathstrut +\mathstrut \) \(517\) \(\beta_{6}\mathstrut +\mathstrut \) \(3224\) \(\beta_{5}\mathstrut +\mathstrut \) \(6419\) \(\beta_{4}\mathstrut -\mathstrut \) \(306\) \(\beta_{3}\mathstrut +\mathstrut \) \(16243\) \(\beta_{2}\mathstrut -\mathstrut \) \(5047\) \(\beta_{1}\mathstrut +\mathstrut \) \(51776\)
\(\nu^{13}\)\(=\)\(13817\) \(\beta_{19}\mathstrut +\mathstrut \) \(17346\) \(\beta_{18}\mathstrut -\mathstrut \) \(8891\) \(\beta_{17}\mathstrut +\mathstrut \) \(7126\) \(\beta_{16}\mathstrut +\mathstrut \) \(6667\) \(\beta_{15}\mathstrut +\mathstrut \) \(9750\) \(\beta_{14}\mathstrut -\mathstrut \) \(4998\) \(\beta_{13}\mathstrut +\mathstrut \) \(7381\) \(\beta_{12}\mathstrut -\mathstrut \) \(9352\) \(\beta_{11}\mathstrut +\mathstrut \) \(2857\) \(\beta_{10}\mathstrut +\mathstrut \) \(11597\) \(\beta_{9}\mathstrut -\mathstrut \) \(10098\) \(\beta_{8}\mathstrut -\mathstrut \) \(5144\) \(\beta_{7}\mathstrut +\mathstrut \) \(15336\) \(\beta_{6}\mathstrut +\mathstrut \) \(4158\) \(\beta_{5}\mathstrut +\mathstrut \) \(9812\) \(\beta_{4}\mathstrut +\mathstrut \) \(29497\) \(\beta_{3}\mathstrut -\mathstrut \) \(4892\) \(\beta_{2}\mathstrut +\mathstrut \) \(71651\) \(\beta_{1}\mathstrut -\mathstrut \) \(3564\)
\(\nu^{14}\)\(=\)\(13384\) \(\beta_{19}\mathstrut +\mathstrut \) \(22530\) \(\beta_{18}\mathstrut -\mathstrut \) \(14956\) \(\beta_{17}\mathstrut +\mathstrut \) \(14755\) \(\beta_{16}\mathstrut -\mathstrut \) \(14479\) \(\beta_{15}\mathstrut +\mathstrut \) \(29274\) \(\beta_{14}\mathstrut +\mathstrut \) \(4010\) \(\beta_{13}\mathstrut -\mathstrut \) \(12905\) \(\beta_{12}\mathstrut -\mathstrut \) \(4542\) \(\beta_{11}\mathstrut +\mathstrut \) \(14095\) \(\beta_{10}\mathstrut +\mathstrut \) \(13955\) \(\beta_{9}\mathstrut -\mathstrut \) \(5009\) \(\beta_{8}\mathstrut -\mathstrut \) \(20601\) \(\beta_{7}\mathstrut +\mathstrut \) \(5797\) \(\beta_{6}\mathstrut +\mathstrut \) \(28526\) \(\beta_{5}\mathstrut +\mathstrut \) \(50833\) \(\beta_{4}\mathstrut -\mathstrut \) \(3497\) \(\beta_{3}\mathstrut +\mathstrut \) \(118747\) \(\beta_{2}\mathstrut -\mathstrut \) \(45393\) \(\beta_{1}\mathstrut +\mathstrut \) \(378519\)
\(\nu^{15}\)\(=\)\(117785\) \(\beta_{19}\mathstrut +\mathstrut \) \(150054\) \(\beta_{18}\mathstrut -\mathstrut \) \(71434\) \(\beta_{17}\mathstrut +\mathstrut \) \(55290\) \(\beta_{16}\mathstrut +\mathstrut \) \(50746\) \(\beta_{15}\mathstrut +\mathstrut \) \(80782\) \(\beta_{14}\mathstrut -\mathstrut \) \(47767\) \(\beta_{13}\mathstrut +\mathstrut \) \(57941\) \(\beta_{12}\mathstrut -\mathstrut \) \(77366\) \(\beta_{11}\mathstrut +\mathstrut \) \(28192\) \(\beta_{10}\mathstrut +\mathstrut \) \(98627\) \(\beta_{9}\mathstrut -\mathstrut \) \(81521\) \(\beta_{8}\mathstrut -\mathstrut \) \(44718\) \(\beta_{7}\mathstrut +\mathstrut \) \(132887\) \(\beta_{6}\mathstrut +\mathstrut \) \(38293\) \(\beta_{5}\mathstrut +\mathstrut \) \(82225\) \(\beta_{4}\mathstrut +\mathstrut \) \(221776\) \(\beta_{3}\mathstrut -\mathstrut \) \(46055\) \(\beta_{2}\mathstrut +\mathstrut \) \(500887\) \(\beta_{1}\mathstrut -\mathstrut \) \(32922\)
\(\nu^{16}\)\(=\)\(113673\) \(\beta_{19}\mathstrut +\mathstrut \) \(182464\) \(\beta_{18}\mathstrut -\mathstrut \) \(126943\) \(\beta_{17}\mathstrut +\mathstrut \) \(147497\) \(\beta_{16}\mathstrut -\mathstrut \) \(120512\) \(\beta_{15}\mathstrut +\mathstrut \) \(251097\) \(\beta_{14}\mathstrut +\mathstrut \) \(42710\) \(\beta_{13}\mathstrut -\mathstrut \) \(108603\) \(\beta_{12}\mathstrut -\mathstrut \) \(48321\) \(\beta_{11}\mathstrut +\mathstrut \) \(122482\) \(\beta_{10}\mathstrut +\mathstrut \) \(120137\) \(\beta_{9}\mathstrut -\mathstrut \) \(46421\) \(\beta_{8}\mathstrut -\mathstrut \) \(166712\) \(\beta_{7}\mathstrut +\mathstrut \) \(59163\) \(\beta_{6}\mathstrut +\mathstrut \) \(241951\) \(\beta_{5}\mathstrut +\mathstrut \) \(399854\) \(\beta_{4}\mathstrut -\mathstrut \) \(35925\) \(\beta_{3}\mathstrut +\mathstrut \) \(880615\) \(\beta_{2}\mathstrut -\mathstrut \) \(391163\) \(\beta_{1}\mathstrut +\mathstrut \) \(2799099\)
\(\nu^{17}\)\(=\)\(977165\) \(\beta_{19}\mathstrut +\mathstrut \) \(1260560\) \(\beta_{18}\mathstrut -\mathstrut \) \(566566\) \(\beta_{17}\mathstrut +\mathstrut \) \(424748\) \(\beta_{16}\mathstrut +\mathstrut \) \(384398\) \(\beta_{15}\mathstrut +\mathstrut \) \(656953\) \(\beta_{14}\mathstrut -\mathstrut \) \(431490\) \(\beta_{13}\mathstrut +\mathstrut \) \(449294\) \(\beta_{12}\mathstrut -\mathstrut \) \(632839\) \(\beta_{11}\mathstrut +\mathstrut \) \(261711\) \(\beta_{10}\mathstrut +\mathstrut \) \(824566\) \(\beta_{9}\mathstrut -\mathstrut \) \(649433\) \(\beta_{8}\mathstrut -\mathstrut \) \(370153\) \(\beta_{7}\mathstrut +\mathstrut \) \(1124485\) \(\beta_{6}\mathstrut +\mathstrut \) \(334904\) \(\beta_{5}\mathstrut +\mathstrut \) \(680038\) \(\beta_{4}\mathstrut +\mathstrut \) \(1675901\) \(\beta_{3}\mathstrut -\mathstrut \) \(409578\) \(\beta_{2}\mathstrut +\mathstrut \) \(3555127\) \(\beta_{1}\mathstrut -\mathstrut \) \(292518\)
\(\nu^{18}\)\(=\)\(942530\) \(\beta_{19}\mathstrut +\mathstrut \) \(1444290\) \(\beta_{18}\mathstrut -\mathstrut \) \(1050191\) \(\beta_{17}\mathstrut +\mathstrut \) \(1387506\) \(\beta_{16}\mathstrut -\mathstrut \) \(974521\) \(\beta_{15}\mathstrut +\mathstrut \) \(2096617\) \(\beta_{14}\mathstrut +\mathstrut \) \(423211\) \(\beta_{13}\mathstrut -\mathstrut \) \(893741\) \(\beta_{12}\mathstrut -\mathstrut \) \(475861\) \(\beta_{11}\mathstrut +\mathstrut \) \(1039783\) \(\beta_{10}\mathstrut +\mathstrut \) \(1009051\) \(\beta_{9}\mathstrut -\mathstrut \) \(404469\) \(\beta_{8}\mathstrut -\mathstrut \) \(1328183\) \(\beta_{7}\mathstrut +\mathstrut \) \(568587\) \(\beta_{6}\mathstrut +\mathstrut \) \(1998264\) \(\beta_{5}\mathstrut +\mathstrut \) \(3134246\) \(\beta_{4}\mathstrut -\mathstrut \) \(345522\) \(\beta_{3}\mathstrut +\mathstrut \) \(6604802\) \(\beta_{2}\mathstrut -\mathstrut \) \(3279990\) \(\beta_{1}\mathstrut +\mathstrut \) \(20887382\)
\(\nu^{19}\)\(=\)\(7960315\) \(\beta_{19}\mathstrut +\mathstrut \) \(10386058\) \(\beta_{18}\mathstrut -\mathstrut \) \(4462429\) \(\beta_{17}\mathstrut +\mathstrut \) \(3248175\) \(\beta_{16}\mathstrut +\mathstrut \) \(2912926\) \(\beta_{15}\mathstrut +\mathstrut \) \(5273968\) \(\beta_{14}\mathstrut -\mathstrut \) \(3758411\) \(\beta_{13}\mathstrut +\mathstrut \) \(3459400\) \(\beta_{12}\mathstrut -\mathstrut \) \(5139162\) \(\beta_{11}\mathstrut +\mathstrut \) \(2333897\) \(\beta_{10}\mathstrut +\mathstrut \) \(6815068\) \(\beta_{9}\mathstrut -\mathstrut \) \(5138883\) \(\beta_{8}\mathstrut -\mathstrut \) \(2972779\) \(\beta_{7}\mathstrut +\mathstrut \) \(9364341\) \(\beta_{6}\mathstrut +\mathstrut \) \(2833876\) \(\beta_{5}\mathstrut +\mathstrut \) \(5575805\) \(\beta_{4}\mathstrut +\mathstrut \) \(12727477\) \(\beta_{3}\mathstrut -\mathstrut \) \(3513325\) \(\beta_{2}\mathstrut +\mathstrut \) \(25568807\) \(\beta_{1}\mathstrut -\mathstrut \) \(2533741\)

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
2.80377
2.62822
2.45061
2.44527
2.23669
1.74461
1.20801
0.762638
0.603983
0.458217
0.194668
−0.302778
−0.852674
−1.19744
−1.41510
−1.94926
−1.97506
−2.40422
−2.63931
−2.80084
−2.80377 0 5.86110 −2.52828 0 −2.67086 −10.8256 0 7.08870
1.2 −2.62822 0 4.90753 4.03060 0 −1.30880 −7.64164 0 −10.5933
1.3 −2.45061 0 4.00547 −1.21123 0 0.971463 −4.91461 0 2.96824
1.4 −2.44527 0 3.97935 1.74520 0 5.10283 −4.84005 0 −4.26748
1.5 −2.23669 0 3.00280 −1.57347 0 −1.14044 −2.24296 0 3.51938
1.6 −1.74461 0 1.04367 −0.892571 0 3.89655 1.66842 0 1.55719
1.7 −1.20801 0 −0.540713 −2.72981 0 −1.05258 3.06921 0 3.29764
1.8 −0.762638 0 −1.41838 4.10409 0 −2.37592 2.60699 0 −3.12993
1.9 −0.603983 0 −1.63520 2.98438 0 4.14035 2.19560 0 −1.80252
1.10 −0.458217 0 −1.79004 0.388274 0 −4.22376 1.73666 0 −0.177914
1.11 −0.194668 0 −1.96210 −4.10517 0 4.22277 0.771295 0 0.799144
1.12 0.302778 0 −1.90833 −3.34986 0 0.428154 −1.18336 0 −1.01426
1.13 0.852674 0 −1.27295 0.278169 0 2.70488 −2.79076 0 0.237188
1.14 1.19744 0 −0.566132 2.06209 0 −5.04192 −3.07279 0 2.46923
1.15 1.41510 0 0.00250280 −0.0661489 0 1.68271 −2.82665 0 −0.0936071
1.16 1.94926 0 1.79960 −3.69296 0 −2.45473 −0.390627 0 −7.19852
1.17 1.97506 0 1.90086 3.88400 0 2.64382 −0.195801 0 7.67114
1.18 2.40422 0 3.78028 −2.94625 0 2.89217 4.28018 0 −7.08343
1.19 2.63931 0 4.96597 1.13791 0 1.99715 7.82811 0 3.00330
1.20 2.80084 0 5.84471 3.48103 0 −1.41384 10.7684 0 9.74980
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\(T_{2}^{20} + \cdots\)
\(T_{5}^{20} - \cdots\)