Properties

Label 6027.2.a.bm
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 1
Dimension 16
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6027.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1258372982\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \(+ q^{3}\) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( -1 + \beta_{5} ) q^{5} \) \( -\beta_{1} q^{6} \) \( + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} ) q^{8} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \(+ q^{3}\) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( -1 + \beta_{5} ) q^{5} \) \( -\beta_{1} q^{6} \) \( + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} ) q^{8} \) \(+ q^{9}\) \( + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{10} \) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{11} \) \( + ( 1 + \beta_{2} ) q^{12} \) \( + ( -\beta_{5} - \beta_{15} ) q^{13} \) \( + ( -1 + \beta_{5} ) q^{15} \) \( + ( \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} ) q^{16} \) \( + ( -1 + \beta_{4} - \beta_{8} - \beta_{11} - \beta_{14} + \beta_{15} ) q^{17} \) \( -\beta_{1} q^{18} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{8} + 2 \beta_{11} + \beta_{13} + \beta_{14} ) q^{19} \) \( + ( -2 - \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{13} + \beta_{14} ) q^{20} \) \( + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{22} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{23} \) \( + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} ) q^{24} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{25} \) \( + ( -1 - \beta_{3} + \beta_{4} - \beta_{8} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{15} ) q^{26} \) \(+ q^{27}\) \( + ( -1 + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{29} \) \( + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{30} \) \( + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{13} - \beta_{15} ) q^{31} \) \( + ( -4 + \beta_{1} - 3 \beta_{2} - \beta_{5} - 2 \beta_{10} + \beta_{13} ) q^{32} \) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{33} \) \( + ( \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{13} - \beta_{14} ) q^{34} \) \( + ( 1 + \beta_{2} ) q^{36} \) \( + ( -4 - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} + 2 \beta_{13} - \beta_{15} ) q^{37} \) \( + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{10} + \beta_{11} + 3 \beta_{12} - \beta_{13} - \beta_{14} ) q^{38} \) \( + ( -\beta_{5} - \beta_{15} ) q^{39} \) \( + ( 5 + 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - 3 \beta_{13} - \beta_{15} ) q^{40} \) \(+ q^{41}\) \( + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{8} - \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{43} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{44} \) \( + ( -1 + \beta_{5} ) q^{45} \) \( + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{46} \) \( + ( -4 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{47} \) \( + ( \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} ) q^{48} \) \( + ( -3 + \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{50} \) \( + ( -1 + \beta_{4} - \beta_{8} - \beta_{11} - \beta_{14} + \beta_{15} ) q^{51} \) \( + ( 2 \beta_{2} + \beta_{6} + 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{52} \) \( + ( -5 - \beta_{2} + \beta_{4} + 3 \beta_{5} - \beta_{6} - 4 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{53} \) \( -\beta_{1} q^{54} \) \( + ( -1 - \beta_{2} - \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{12} + 3 \beta_{13} + \beta_{14} + \beta_{15} ) q^{55} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{8} + 2 \beta_{11} + \beta_{13} + \beta_{14} ) q^{57} \) \( + ( -3 + \beta_{1} - \beta_{3} - \beta_{8} - \beta_{9} - \beta_{11} - \beta_{13} ) q^{58} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{12} + \beta_{14} ) q^{59} \) \( + ( -2 - \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{13} + \beta_{14} ) q^{60} \) \( + ( 2 - 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} + 2 \beta_{14} - \beta_{15} ) q^{61} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{7} + \beta_{8} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{62} \) \( + ( 3 + 3 \beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{13} ) q^{64} \) \( + ( -2 + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{65} \) \( + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{66} \) \( + ( -2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{10} - 4 \beta_{12} + \beta_{13} ) q^{67} \) \( + ( \beta_{1} - 3 \beta_{2} - 3 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{68} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{69} \) \( + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{71} \) \( + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} ) q^{72} \) \( + ( -2 + \beta_{1} + 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{73} \) \( + ( 1 + 3 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{74} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{75} \) \( + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{11} + \beta_{12} + 4 \beta_{13} + \beta_{15} ) q^{76} \) \( + ( -1 - \beta_{3} + \beta_{4} - \beta_{8} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{15} ) q^{78} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{79} \) \( + ( -6 - 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{12} + 3 \beta_{13} - \beta_{14} + \beta_{15} ) q^{80} \) \(+ q^{81}\) \( -\beta_{1} q^{82} \) \( + ( -3 + \beta_{1} - \beta_{5} - 2 \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{83} \) \( + ( -3 + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{85} \) \( + ( -1 + 2 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{14} ) q^{86} \) \( + ( -1 + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{87} \) \( + ( -5 - 2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{88} \) \( + ( \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{89} \) \( + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{90} \) \( + ( -4 + \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} + 3 \beta_{13} + \beta_{15} ) q^{92} \) \( + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{13} - \beta_{15} ) q^{93} \) \( + ( 2 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 4 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - \beta_{15} ) q^{94} \) \( + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{95} \) \( + ( -4 + \beta_{1} - 3 \beta_{2} - \beta_{5} - 2 \beta_{10} + \beta_{13} ) q^{96} \) \( + ( 4 + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{97} \) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 16q^{9} \) \(\mathstrut -\mathstrut 4q^{10} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 12q^{12} \) \(\mathstrut -\mathstrut 12q^{15} \) \(\mathstrut -\mathstrut 8q^{17} \) \(\mathstrut -\mathstrut 4q^{18} \) \(\mathstrut +\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 20q^{20} \) \(\mathstrut -\mathstrut 16q^{22} \) \(\mathstrut -\mathstrut 12q^{23} \) \(\mathstrut -\mathstrut 12q^{24} \) \(\mathstrut -\mathstrut 8q^{25} \) \(\mathstrut -\mathstrut 8q^{26} \) \(\mathstrut +\mathstrut 16q^{27} \) \(\mathstrut -\mathstrut 16q^{29} \) \(\mathstrut -\mathstrut 4q^{30} \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut -\mathstrut 48q^{32} \) \(\mathstrut -\mathstrut 4q^{33} \) \(\mathstrut +\mathstrut 16q^{34} \) \(\mathstrut +\mathstrut 12q^{36} \) \(\mathstrut -\mathstrut 48q^{37} \) \(\mathstrut -\mathstrut 4q^{38} \) \(\mathstrut +\mathstrut 56q^{40} \) \(\mathstrut +\mathstrut 16q^{41} \) \(\mathstrut -\mathstrut 16q^{43} \) \(\mathstrut -\mathstrut 12q^{45} \) \(\mathstrut -\mathstrut 4q^{46} \) \(\mathstrut -\mathstrut 36q^{47} \) \(\mathstrut -\mathstrut 8q^{50} \) \(\mathstrut -\mathstrut 8q^{51} \) \(\mathstrut -\mathstrut 60q^{53} \) \(\mathstrut -\mathstrut 4q^{54} \) \(\mathstrut +\mathstrut 8q^{55} \) \(\mathstrut +\mathstrut 4q^{57} \) \(\mathstrut -\mathstrut 36q^{58} \) \(\mathstrut -\mathstrut 36q^{59} \) \(\mathstrut -\mathstrut 20q^{60} \) \(\mathstrut -\mathstrut 4q^{61} \) \(\mathstrut -\mathstrut 12q^{62} \) \(\mathstrut +\mathstrut 52q^{64} \) \(\mathstrut -\mathstrut 36q^{65} \) \(\mathstrut -\mathstrut 16q^{66} \) \(\mathstrut -\mathstrut 52q^{67} \) \(\mathstrut -\mathstrut 8q^{68} \) \(\mathstrut -\mathstrut 12q^{69} \) \(\mathstrut -\mathstrut 12q^{71} \) \(\mathstrut -\mathstrut 12q^{72} \) \(\mathstrut -\mathstrut 16q^{73} \) \(\mathstrut +\mathstrut 4q^{74} \) \(\mathstrut -\mathstrut 8q^{75} \) \(\mathstrut +\mathstrut 16q^{76} \) \(\mathstrut -\mathstrut 8q^{78} \) \(\mathstrut -\mathstrut 36q^{79} \) \(\mathstrut -\mathstrut 68q^{80} \) \(\mathstrut +\mathstrut 16q^{81} \) \(\mathstrut -\mathstrut 4q^{82} \) \(\mathstrut -\mathstrut 32q^{83} \) \(\mathstrut -\mathstrut 28q^{85} \) \(\mathstrut -\mathstrut 8q^{86} \) \(\mathstrut -\mathstrut 16q^{87} \) \(\mathstrut -\mathstrut 36q^{88} \) \(\mathstrut -\mathstrut 12q^{89} \) \(\mathstrut -\mathstrut 4q^{90} \) \(\mathstrut -\mathstrut 36q^{92} \) \(\mathstrut -\mathstrut 4q^{93} \) \(\mathstrut +\mathstrut 24q^{94} \) \(\mathstrut -\mathstrut 20q^{95} \) \(\mathstrut -\mathstrut 48q^{96} \) \(\mathstrut +\mathstrut 48q^{97} \) \(\mathstrut -\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(4\) \(x^{15}\mathstrut -\mathstrut \) \(14\) \(x^{14}\mathstrut +\mathstrut \) \(68\) \(x^{13}\mathstrut +\mathstrut \) \(64\) \(x^{12}\mathstrut -\mathstrut \) \(456\) \(x^{11}\mathstrut -\mathstrut \) \(54\) \(x^{10}\mathstrut +\mathstrut \) \(1532\) \(x^{9}\mathstrut -\mathstrut \) \(400\) \(x^{8}\mathstrut -\mathstrut \) \(2708\) \(x^{7}\mathstrut +\mathstrut \) \(1218\) \(x^{6}\mathstrut +\mathstrut \) \(2424\) \(x^{5}\mathstrut -\mathstrut \) \(1276\) \(x^{4}\mathstrut -\mathstrut \) \(960\) \(x^{3}\mathstrut +\mathstrut \) \(500\) \(x^{2}\mathstrut +\mathstrut \) \(112\) \(x\mathstrut -\mathstrut \) \(49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(2\) \(\nu^{15}\mathstrut +\mathstrut \) \(316\) \(\nu^{14}\mathstrut -\mathstrut \) \(525\) \(\nu^{13}\mathstrut -\mathstrut \) \(5554\) \(\nu^{12}\mathstrut +\mathstrut \) \(7439\) \(\nu^{11}\mathstrut +\mathstrut \) \(39832\) \(\nu^{10}\mathstrut -\mathstrut \) \(38854\) \(\nu^{9}\mathstrut -\mathstrut \) \(146235\) \(\nu^{8}\mathstrut +\mathstrut \) \(97946\) \(\nu^{7}\mathstrut +\mathstrut \) \(285731\) \(\nu^{6}\mathstrut -\mathstrut \) \(129437\) \(\nu^{5}\mathstrut -\mathstrut \) \(281488\) \(\nu^{4}\mathstrut +\mathstrut \) \(89961\) \(\nu^{3}\mathstrut +\mathstrut \) \(120024\) \(\nu^{2}\mathstrut -\mathstrut \) \(26662\) \(\nu\mathstrut -\mathstrut \) \(16891\)\()/1659\)
\(\beta_{4}\)\(=\)\((\)\(344\) \(\nu^{15}\mathstrut -\mathstrut \) \(711\) \(\nu^{14}\mathstrut -\mathstrut \) \(6475\) \(\nu^{13}\mathstrut +\mathstrut \) \(11870\) \(\nu^{12}\mathstrut +\mathstrut \) \(48798\) \(\nu^{11}\mathstrut -\mathstrut \) \(77407\) \(\nu^{10}\mathstrut -\mathstrut \) \(186478\) \(\nu^{9}\mathstrut +\mathstrut \) \(251383\) \(\nu^{8}\mathstrut +\mathstrut \) \(380344\) \(\nu^{7}\mathstrut -\mathstrut \) \(426432\) \(\nu^{6}\mathstrut -\mathstrut \) \(399882\) \(\nu^{5}\mathstrut +\mathstrut \) \(360236\) \(\nu^{4}\mathstrut +\mathstrut \) \(189327\) \(\nu^{3}\mathstrut -\mathstrut \) \(129487\) \(\nu^{2}\mathstrut -\mathstrut \) \(27262\) \(\nu\mathstrut +\mathstrut \) \(13062\)\()/553\)
\(\beta_{5}\)\(=\)\((\)\(615\) \(\nu^{15}\mathstrut -\mathstrut \) \(1501\) \(\nu^{14}\mathstrut -\mathstrut \) \(10822\) \(\nu^{13}\mathstrut +\mathstrut \) \(24523\) \(\nu^{12}\mathstrut +\mathstrut \) \(75753\) \(\nu^{11}\mathstrut -\mathstrut \) \(155889\) \(\nu^{10}\mathstrut -\mathstrut \) \(266506\) \(\nu^{9}\mathstrut +\mathstrut \) \(489196\) \(\nu^{8}\mathstrut +\mathstrut \) \(494579\) \(\nu^{7}\mathstrut -\mathstrut \) \(790497\) \(\nu^{6}\mathstrut -\mathstrut \) \(467859\) \(\nu^{5}\mathstrut +\mathstrut \) \(623019\) \(\nu^{4}\mathstrut +\mathstrut \) \(198515\) \(\nu^{3}\mathstrut -\mathstrut \) \(202558\) \(\nu^{2}\mathstrut -\mathstrut \) \(24545\) \(\nu\mathstrut +\mathstrut \) \(17073\)\()/553\)
\(\beta_{6}\)\(=\)\((\)\(615\) \(\nu^{15}\mathstrut -\mathstrut \) \(1501\) \(\nu^{14}\mathstrut -\mathstrut \) \(10822\) \(\nu^{13}\mathstrut +\mathstrut \) \(24523\) \(\nu^{12}\mathstrut +\mathstrut \) \(75753\) \(\nu^{11}\mathstrut -\mathstrut \) \(155889\) \(\nu^{10}\mathstrut -\mathstrut \) \(266506\) \(\nu^{9}\mathstrut +\mathstrut \) \(489196\) \(\nu^{8}\mathstrut +\mathstrut \) \(494579\) \(\nu^{7}\mathstrut -\mathstrut \) \(790497\) \(\nu^{6}\mathstrut -\mathstrut \) \(467859\) \(\nu^{5}\mathstrut +\mathstrut \) \(623019\) \(\nu^{4}\mathstrut +\mathstrut \) \(199068\) \(\nu^{3}\mathstrut -\mathstrut \) \(203111\) \(\nu^{2}\mathstrut -\mathstrut \) \(26757\) \(\nu\mathstrut +\mathstrut \) \(18179\)\()/553\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(137\) \(\nu^{15}\mathstrut +\mathstrut \) \(316\) \(\nu^{14}\mathstrut +\mathstrut \) \(2471\) \(\nu^{13}\mathstrut -\mathstrut \) \(5199\) \(\nu^{12}\mathstrut -\mathstrut \) \(17793\) \(\nu^{11}\mathstrut +\mathstrut \) \(33328\) \(\nu^{10}\mathstrut +\mathstrut \) \(64712\) \(\nu^{9}\mathstrut -\mathstrut \) \(105797\) \(\nu^{8}\mathstrut -\mathstrut \) \(124989\) \(\nu^{7}\mathstrut +\mathstrut \) \(173926\) \(\nu^{6}\mathstrut +\mathstrut \) \(123884\) \(\nu^{5}\mathstrut -\mathstrut \) \(141097\) \(\nu^{4}\mathstrut -\mathstrut \) \(55090\) \(\nu^{3}\mathstrut +\mathstrut \) \(48620\) \(\nu^{2}\mathstrut +\mathstrut \) \(7243\) \(\nu\mathstrut -\mathstrut \) \(4621\)\()/79\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(230\) \(\nu^{15}\mathstrut +\mathstrut \) \(553\) \(\nu^{14}\mathstrut +\mathstrut \) \(4089\) \(\nu^{13}\mathstrut -\mathstrut \) \(9080\) \(\nu^{12}\mathstrut -\mathstrut \) \(28999\) \(\nu^{11}\mathstrut +\mathstrut \) \(58088\) \(\nu^{10}\mathstrut +\mathstrut \) \(103836\) \(\nu^{9}\mathstrut -\mathstrut \) \(183812\) \(\nu^{8}\mathstrut -\mathstrut \) \(197609\) \(\nu^{7}\mathstrut +\mathstrut \) \(300321\) \(\nu^{6}\mathstrut +\mathstrut \) \(193949\) \(\nu^{5}\mathstrut -\mathstrut \) \(240319\) \(\nu^{4}\mathstrut -\mathstrut \) \(86909\) \(\nu^{3}\mathstrut +\mathstrut \) \(80223\) \(\nu^{2}\mathstrut +\mathstrut \) \(12184\) \(\nu\mathstrut -\mathstrut \) \(7202\)\()/79\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(5042\) \(\nu^{15}\mathstrut +\mathstrut \) \(11929\) \(\nu^{14}\mathstrut +\mathstrut \) \(89943\) \(\nu^{13}\mathstrut -\mathstrut \) \(195436\) \(\nu^{12}\mathstrut -\mathstrut \) \(639991\) \(\nu^{11}\mathstrut +\mathstrut \) \(1246429\) \(\nu^{10}\mathstrut +\mathstrut \) \(2297459\) \(\nu^{9}\mathstrut -\mathstrut \) \(3928944\) \(\nu^{8}\mathstrut -\mathstrut \) \(4373500\) \(\nu^{7}\mathstrut +\mathstrut \) \(6390473\) \(\nu^{6}\mathstrut +\mathstrut \) \(4268299\) \(\nu^{5}\mathstrut -\mathstrut \) \(5090446\) \(\nu^{4}\mathstrut -\mathstrut \) \(1874925\) \(\nu^{3}\mathstrut +\mathstrut \) \(1695969\) \(\nu^{2}\mathstrut +\mathstrut \) \(248333\) \(\nu\mathstrut -\mathstrut \) \(153286\)\()/1659\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(5042\) \(\nu^{15}\mathstrut +\mathstrut \) \(11929\) \(\nu^{14}\mathstrut +\mathstrut \) \(89943\) \(\nu^{13}\mathstrut -\mathstrut \) \(195436\) \(\nu^{12}\mathstrut -\mathstrut \) \(639991\) \(\nu^{11}\mathstrut +\mathstrut \) \(1246429\) \(\nu^{10}\mathstrut +\mathstrut \) \(2297459\) \(\nu^{9}\mathstrut -\mathstrut \) \(3928944\) \(\nu^{8}\mathstrut -\mathstrut \) \(4373500\) \(\nu^{7}\mathstrut +\mathstrut \) \(6390473\) \(\nu^{6}\mathstrut +\mathstrut \) \(4268299\) \(\nu^{5}\mathstrut -\mathstrut \) \(5088787\) \(\nu^{4}\mathstrut -\mathstrut \) \(1876584\) \(\nu^{3}\mathstrut +\mathstrut \) \(1686015\) \(\nu^{2}\mathstrut +\mathstrut \) \(253310\) \(\nu\mathstrut -\mathstrut \) \(144991\)\()/1659\)
\(\beta_{11}\)\(=\)\((\)\(2533\) \(\nu^{15}\mathstrut -\mathstrut \) \(5925\) \(\nu^{14}\mathstrut -\mathstrut \) \(45416\) \(\nu^{13}\mathstrut +\mathstrut \) \(97309\) \(\nu^{12}\mathstrut +\mathstrut \) \(324855\) \(\nu^{11}\mathstrut -\mathstrut \) \(622481\) \(\nu^{10}\mathstrut -\mathstrut \) \(1172474\) \(\nu^{9}\mathstrut +\mathstrut \) \(1970354\) \(\nu^{8}\mathstrut +\mathstrut \) \(2243872\) \(\nu^{7}\mathstrut -\mathstrut \) \(3224355\) \(\nu^{6}\mathstrut -\mathstrut \) \(2197811\) \(\nu^{5}\mathstrut +\mathstrut \) \(2591188\) \(\nu^{4}\mathstrut +\mathstrut \) \(960374\) \(\nu^{3}\mathstrut -\mathstrut \) \(872178\) \(\nu^{2}\mathstrut -\mathstrut \) \(122629\) \(\nu\mathstrut +\mathstrut \) \(78449\)\()/553\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(2547\) \(\nu^{15}\mathstrut +\mathstrut \) \(5925\) \(\nu^{14}\mathstrut +\mathstrut \) \(45612\) \(\nu^{13}\mathstrut -\mathstrut \) \(96924\) \(\nu^{12}\mathstrut -\mathstrut \) \(325870\) \(\nu^{11}\mathstrut +\mathstrut \) \(617063\) \(\nu^{10}\mathstrut +\mathstrut \) \(1174231\) \(\nu^{9}\mathstrut -\mathstrut \) \(1941640\) \(\nu^{8}\mathstrut -\mathstrut \) \(2241205\) \(\nu^{7}\mathstrut +\mathstrut \) \(3152381\) \(\nu^{6}\mathstrut +\mathstrut \) \(2185400\) \(\nu^{5}\mathstrut -\mathstrut \) \(2504444\) \(\nu^{4}\mathstrut -\mathstrut \) \(949454\) \(\nu^{3}\mathstrut +\mathstrut \) \(828099\) \(\nu^{2}\mathstrut +\mathstrut \) \(120697\) \(\nu\mathstrut -\mathstrut \) \(72814\)\()/553\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(1177\) \(\nu^{15}\mathstrut +\mathstrut \) \(2765\) \(\nu^{14}\mathstrut +\mathstrut \) \(21060\) \(\nu^{13}\mathstrut -\mathstrut \) \(45329\) \(\nu^{12}\mathstrut -\mathstrut \) \(150389\) \(\nu^{11}\mathstrut +\mathstrut \) \(289313\) \(\nu^{10}\mathstrut +\mathstrut \) \(542200\) \(\nu^{9}\mathstrut -\mathstrut \) \(912900\) \(\nu^{8}\mathstrut -\mathstrut \) \(1037609\) \(\nu^{7}\mathstrut +\mathstrut \) \(1487065\) \(\nu^{6}\mathstrut +\mathstrut \) \(1018766\) \(\nu^{5}\mathstrut -\mathstrut \) \(1186931\) \(\nu^{4}\mathstrut -\mathstrut \) \(449193\) \(\nu^{3}\mathstrut +\mathstrut \) \(395619\) \(\nu^{2}\mathstrut +\mathstrut \) \(58774\) \(\nu\mathstrut -\mathstrut \) \(35294\)\()/237\)
\(\beta_{14}\)\(=\)\((\)\(9529\) \(\nu^{15}\mathstrut -\mathstrut \) \(22436\) \(\nu^{14}\mathstrut -\mathstrut \) \(170457\) \(\nu^{13}\mathstrut +\mathstrut \) \(367622\) \(\nu^{12}\mathstrut +\mathstrut \) \(1218296\) \(\nu^{11}\mathstrut -\mathstrut \) \(2345744\) \(\nu^{10}\mathstrut -\mathstrut \) \(4404277\) \(\nu^{9}\mathstrut +\mathstrut \) \(7401420\) \(\nu^{8}\mathstrut +\mathstrut \) \(8477201\) \(\nu^{7}\mathstrut -\mathstrut \) \(12056686\) \(\nu^{6}\mathstrut -\mathstrut \) \(8416079\) \(\nu^{5}\mathstrut +\mathstrut \) \(9619586\) \(\nu^{4}\mathstrut +\mathstrut \) \(3790041\) \(\nu^{3}\mathstrut -\mathstrut \) \(3202236\) \(\nu^{2}\mathstrut -\mathstrut \) \(517798\) \(\nu\mathstrut +\mathstrut \) \(286916\)\()/1659\)
\(\beta_{15}\)\(=\)\((\)\(5834\) \(\nu^{15}\mathstrut -\mathstrut \) \(13746\) \(\nu^{14}\mathstrut -\mathstrut \) \(104349\) \(\nu^{13}\mathstrut +\mathstrut \) \(225401\) \(\nu^{12}\mathstrut +\mathstrut \) \(745048\) \(\nu^{11}\mathstrut -\mathstrut \) \(1438573\) \(\nu^{10}\mathstrut -\mathstrut \) \(2687259\) \(\nu^{9}\mathstrut +\mathstrut \) \(4536144\) \(\nu^{8}\mathstrut +\mathstrut \) \(5150163\) \(\nu^{7}\mathstrut -\mathstrut \) \(7373853\) \(\nu^{6}\mathstrut -\mathstrut \) \(5072879\) \(\nu^{5}\mathstrut +\mathstrut \) \(5857389\) \(\nu^{4}\mathstrut +\mathstrut \) \(2249010\) \(\nu^{3}\mathstrut -\mathstrut \) \(1934572\) \(\nu^{2}\mathstrut -\mathstrut \) \(295615\) \(\nu\mathstrut +\mathstrut \) \(170730\)\()/553\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(8\) \(\beta_{6}\mathstrut -\mathstrut \) \(7\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(19\) \(\beta_{1}\mathstrut +\mathstrut \) \(12\)
\(\nu^{6}\)\(=\)\(-\)\(\beta_{13}\mathstrut +\mathstrut \) \(10\) \(\beta_{10}\mathstrut -\mathstrut \) \(8\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut -\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(48\) \(\beta_{2}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(79\)
\(\nu^{7}\)\(=\)\(-\)\(10\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(23\) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(58\) \(\beta_{6}\mathstrut -\mathstrut \) \(45\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(96\) \(\beta_{2}\mathstrut +\mathstrut \) \(103\) \(\beta_{1}\mathstrut +\mathstrut \) \(109\)
\(\nu^{8}\)\(=\)\(\beta_{14}\mathstrut -\mathstrut \) \(12\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(80\) \(\beta_{10}\mathstrut -\mathstrut \) \(54\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(9\) \(\beta_{7}\mathstrut +\mathstrut \) \(110\) \(\beta_{6}\mathstrut -\mathstrut \) \(81\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(337\) \(\beta_{2}\mathstrut +\mathstrut \) \(123\) \(\beta_{1}\mathstrut +\mathstrut \) \(495\)
\(\nu^{9}\)\(=\)\(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(73\) \(\beta_{13}\mathstrut -\mathstrut \) \(12\) \(\beta_{11}\mathstrut +\mathstrut \) \(199\) \(\beta_{10}\mathstrut -\mathstrut \) \(44\) \(\beta_{9}\mathstrut +\mathstrut \) \(3\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(419\) \(\beta_{6}\mathstrut -\mathstrut \) \(296\) \(\beta_{5}\mathstrut +\mathstrut \) \(28\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(770\) \(\beta_{2}\mathstrut +\mathstrut \) \(619\) \(\beta_{1}\mathstrut +\mathstrut \) \(894\)
\(\nu^{10}\)\(=\)\(3\) \(\beta_{15}\mathstrut +\mathstrut \) \(15\) \(\beta_{14}\mathstrut -\mathstrut \) \(99\) \(\beta_{13}\mathstrut +\mathstrut \) \(12\) \(\beta_{12}\mathstrut -\mathstrut \) \(3\) \(\beta_{11}\mathstrut +\mathstrut \) \(605\) \(\beta_{10}\mathstrut -\mathstrut \) \(355\) \(\beta_{9}\mathstrut +\mathstrut \) \(30\) \(\beta_{8}\mathstrut +\mathstrut \) \(53\) \(\beta_{7}\mathstrut +\mathstrut \) \(916\) \(\beta_{6}\mathstrut -\mathstrut \) \(618\) \(\beta_{5}\mathstrut +\mathstrut \) \(108\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2413\) \(\beta_{2}\mathstrut +\mathstrut \) \(1028\) \(\beta_{1}\mathstrut +\mathstrut \) \(3297\)
\(\nu^{11}\)\(=\)\(19\) \(\beta_{15}\mathstrut +\mathstrut \) \(24\) \(\beta_{14}\mathstrut -\mathstrut \) \(469\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(98\) \(\beta_{11}\mathstrut +\mathstrut \) \(1571\) \(\beta_{10}\mathstrut -\mathstrut \) \(441\) \(\beta_{9}\mathstrut +\mathstrut \) \(55\) \(\beta_{8}\mathstrut -\mathstrut \) \(25\) \(\beta_{7}\mathstrut +\mathstrut \) \(3055\) \(\beta_{6}\mathstrut -\mathstrut \) \(2006\) \(\beta_{5}\mathstrut +\mathstrut \) \(274\) \(\beta_{4}\mathstrut -\mathstrut \) \(15\) \(\beta_{3}\mathstrut +\mathstrut \) \(5940\) \(\beta_{2}\mathstrut +\mathstrut \) \(4001\) \(\beta_{1}\mathstrut +\mathstrut \) \(6978\)
\(\nu^{12}\)\(=\)\(58\) \(\beta_{15}\mathstrut +\mathstrut \) \(161\) \(\beta_{14}\mathstrut -\mathstrut \) \(689\) \(\beta_{13}\mathstrut +\mathstrut \) \(94\) \(\beta_{12}\mathstrut -\mathstrut \) \(50\) \(\beta_{11}\mathstrut +\mathstrut \) \(4503\) \(\beta_{10}\mathstrut -\mathstrut \) \(2348\) \(\beta_{9}\mathstrut +\mathstrut \) \(310\) \(\beta_{8}\mathstrut +\mathstrut \) \(224\) \(\beta_{7}\mathstrut +\mathstrut \) \(7288\) \(\beta_{6}\mathstrut -\mathstrut \) \(4598\) \(\beta_{5}\mathstrut +\mathstrut \) \(888\) \(\beta_{4}\mathstrut -\mathstrut \) \(20\) \(\beta_{3}\mathstrut +\mathstrut \) \(17500\) \(\beta_{2}\mathstrut +\mathstrut \) \(8086\) \(\beta_{1}\mathstrut +\mathstrut \) \(22773\)
\(\nu^{13}\)\(=\)\(239\) \(\beta_{15}\mathstrut +\mathstrut \) \(339\) \(\beta_{14}\mathstrut -\mathstrut \) \(2779\) \(\beta_{13}\mathstrut +\mathstrut \) \(12\) \(\beta_{12}\mathstrut -\mathstrut \) \(686\) \(\beta_{11}\mathstrut +\mathstrut \) \(11965\) \(\beta_{10}\mathstrut -\mathstrut \) \(3795\) \(\beta_{9}\mathstrut +\mathstrut \) \(661\) \(\beta_{8}\mathstrut -\mathstrut \) \(385\) \(\beta_{7}\mathstrut +\mathstrut \) \(22482\) \(\beta_{6}\mathstrut -\mathstrut \) \(13893\) \(\beta_{5}\mathstrut +\mathstrut \) \(2351\) \(\beta_{4}\mathstrut -\mathstrut \) \(152\) \(\beta_{3}\mathstrut +\mathstrut \) \(44933\) \(\beta_{2}\mathstrut +\mathstrut \) \(27110\) \(\beta_{1}\mathstrut +\mathstrut \) \(53012\)
\(\nu^{14}\)\(=\)\(730\) \(\beta_{15}\mathstrut +\mathstrut \) \(1547\) \(\beta_{14}\mathstrut -\mathstrut \) \(4238\) \(\beta_{13}\mathstrut +\mathstrut \) \(599\) \(\beta_{12}\mathstrut -\mathstrut \) \(527\) \(\beta_{11}\mathstrut +\mathstrut \) \(33376\) \(\beta_{10}\mathstrut -\mathstrut \) \(15762\) \(\beta_{9}\mathstrut +\mathstrut \) \(2807\) \(\beta_{8}\mathstrut +\mathstrut \) \(313\) \(\beta_{7}\mathstrut +\mathstrut \) \(56597\) \(\beta_{6}\mathstrut -\mathstrut \) \(33753\) \(\beta_{5}\mathstrut +\mathstrut \) \(7042\) \(\beta_{4}\mathstrut -\mathstrut \) \(251\) \(\beta_{3}\mathstrut +\mathstrut \) \(127913\) \(\beta_{2}\mathstrut +\mathstrut \) \(61557\) \(\beta_{1}\mathstrut +\mathstrut \) \(160772\)
\(\nu^{15}\)\(=\)\(2528\) \(\beta_{15}\mathstrut +\mathstrut \) \(3805\) \(\beta_{14}\mathstrut -\mathstrut \) \(15189\) \(\beta_{13}\mathstrut +\mathstrut \) \(48\) \(\beta_{12}\mathstrut -\mathstrut \) \(4449\) \(\beta_{11}\mathstrut +\mathstrut \) \(89759\) \(\beta_{10}\mathstrut -\mathstrut \) \(30258\) \(\beta_{9}\mathstrut +\mathstrut \) \(6660\) \(\beta_{8}\mathstrut -\mathstrut \) \(4736\) \(\beta_{7}\mathstrut +\mathstrut \) \(166624\) \(\beta_{6}\mathstrut -\mathstrut \) \(97558\) \(\beta_{5}\mathstrut +\mathstrut \) \(19039\) \(\beta_{4}\mathstrut -\mathstrut \) \(1329\) \(\beta_{3}\mathstrut +\mathstrut \) \(336425\) \(\beta_{2}\mathstrut +\mathstrut \) \(189077\) \(\beta_{1}\mathstrut +\mathstrut \) \(396492\)

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
2.72726
2.69180
1.83908
1.64639
1.63508
1.44011
1.04648
0.631061
0.304034
−0.404293
−0.819942
−1.10456
−1.49558
−1.87592
−2.11474
−2.14627
−2.72726 1.00000 5.43794 −0.416230 −2.72726 0 −9.37616 1.00000 1.13517
1.2 −2.69180 1.00000 5.24578 −3.62847 −2.69180 0 −8.73699 1.00000 9.76712
1.3 −1.83908 1.00000 1.38223 −2.07594 −1.83908 0 1.13613 1.00000 3.81783
1.4 −1.64639 1.00000 0.710595 0.0457395 −1.64639 0 2.12286 1.00000 −0.0753050
1.5 −1.63508 1.00000 0.673497 1.18651 −1.63508 0 2.16894 1.00000 −1.94005
1.6 −1.44011 1.00000 0.0739309 3.20961 −1.44011 0 2.77376 1.00000 −4.62221
1.7 −1.04648 1.00000 −0.904885 1.81328 −1.04648 0 3.03990 1.00000 −1.89755
1.8 −0.631061 1.00000 −1.60176 −1.65938 −0.631061 0 2.27293 1.00000 1.04717
1.9 −0.304034 1.00000 −1.90756 −0.824488 −0.304034 0 1.18803 1.00000 0.250672
1.10 0.404293 1.00000 −1.83655 −3.62653 0.404293 0 −1.55109 1.00000 −1.46618
1.11 0.819942 1.00000 −1.32769 0.332064 0.819942 0 −2.72852 1.00000 0.272273
1.12 1.10456 1.00000 −0.779950 −3.67747 1.10456 0 −3.07062 1.00000 −4.06198
1.13 1.49558 1.00000 0.236756 1.30108 1.49558 0 −2.63707 1.00000 1.94586
1.14 1.87592 1.00000 1.51907 −1.33107 1.87592 0 −0.902182 1.00000 −2.49697
1.15 2.11474 1.00000 2.47212 −0.285010 2.11474 0 0.998417 1.00000 −0.602721
1.16 2.14627 1.00000 2.60647 −2.36369 2.14627 0 1.30166 1.00000 −5.07312
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(41\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2}^{16} + \cdots\)
\(T_{5}^{16} + \cdots\)
\(T_{13}^{16} - \cdots\)