Properties

Label 8001.2.a.w
Level 8001
Weight 2
Character orbit 8001.a
Self dual Yes
Analytic conductor 63.888
Analytic rank 1
Dimension 20
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: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + \beta_{16} q^{5} \) \(+ q^{7}\) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + \beta_{16} q^{5} \) \(+ q^{7}\) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{8} \) \( + ( -\beta_{2} + \beta_{4} + \beta_{7} - \beta_{10} - \beta_{15} + \beta_{19} ) q^{10} \) \( + ( -1 - \beta_{5} ) q^{11} \) \( -\beta_{14} q^{13} \) \( -\beta_{1} q^{14} \) \( + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{10} + \beta_{14} + \beta_{15} + \beta_{18} ) q^{16} \) \( + ( -1 + \beta_{5} + \beta_{12} + \beta_{17} + \beta_{19} ) q^{17} \) \( + ( -1 + \beta_{1} - \beta_{4} + \beta_{10} - \beta_{16} - \beta_{19} ) q^{19} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} - \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{20} \) \( + ( -1 + 2 \beta_{1} + \beta_{5} + \beta_{15} ) q^{22} \) \( + ( -2 + \beta_{4} + \beta_{7} + \beta_{8} - \beta_{11} ) q^{23} \) \( + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{11} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{25} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{8} + 2 \beta_{9} - \beta_{11} + \beta_{14} ) q^{26} \) \( + ( 1 + \beta_{2} ) q^{28} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{10} + \beta_{13} - \beta_{14} ) q^{29} \) \( + ( \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{15} + \beta_{18} ) q^{31} \) \( + ( -1 + \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{15} - 2 \beta_{16} - \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{32} \) \( + ( \beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{15} - \beta_{17} - \beta_{19} ) q^{34} \) \( + \beta_{16} q^{35} \) \( + ( 1 - \beta_{1} + \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} + \beta_{19} ) q^{37} \) \( + ( \beta_{1} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{13} + 2 \beta_{15} - 2 \beta_{16} - \beta_{17} - \beta_{19} ) q^{38} \) \( + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{14} - 2 \beta_{15} - \beta_{16} ) q^{40} \) \( + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{10} + 2 \beta_{11} + \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{41} \) \( + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} ) q^{43} \) \( + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{8} + 2 \beta_{9} + \beta_{14} - 2 \beta_{15} - \beta_{16} - \beta_{17} - \beta_{19} ) q^{44} \) \( + ( 2 + 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{14} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{46} \) \( + ( -1 + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{14} - \beta_{15} + \beta_{18} ) q^{47} \) \(+ q^{49}\) \( + ( -1 - \beta_{1} + \beta_{2} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{14} - \beta_{16} + \beta_{18} ) q^{50} \) \( + ( 2 + \beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{15} - \beta_{18} ) q^{52} \) \( + ( -3 + 3 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{16} - \beta_{19} ) q^{53} \) \( + ( -\beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{10} + \beta_{12} + \beta_{14} - 3 \beta_{16} - \beta_{18} ) q^{55} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{56} \) \( + ( 1 - \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{17} ) q^{58} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{14} + \beta_{18} ) q^{59} \) \( + ( -2 + \beta_{1} - \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{61} \) \( + ( \beta_{1} + \beta_{2} - 3 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{16} - \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{62} \) \( + ( -2 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{64} \) \( + ( -5 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} + 3 \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{13} + 2 \beta_{15} - \beta_{16} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{65} \) \( + ( 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{12} - \beta_{13} + \beta_{15} - \beta_{16} - \beta_{17} - \beta_{19} ) q^{67} \) \( + ( -\beta_{1} + \beta_{2} + 2 \beta_{5} + \beta_{7} + 2 \beta_{9} + \beta_{12} + \beta_{13} + \beta_{15} + \beta_{16} + \beta_{17} + 2 \beta_{19} ) q^{68} \) \( + ( -\beta_{2} + \beta_{4} + \beta_{7} - \beta_{10} - \beta_{15} + \beta_{19} ) q^{70} \) \( + ( -4 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{12} + \beta_{15} + 2 \beta_{19} ) q^{71} \) \( + ( -1 + \beta_{2} - \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} - \beta_{17} - \beta_{19} ) q^{73} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{12} - 3 \beta_{14} + 2 \beta_{16} + \beta_{17} + \beta_{19} ) q^{74} \) \( + ( -2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} - \beta_{18} ) q^{76} \) \( + ( -1 - \beta_{5} ) q^{77} \) \( + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} - \beta_{18} + \beta_{19} ) q^{79} \) \( + ( 2 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + 3 \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{12} - 3 \beta_{14} + \beta_{15} + \beta_{16} ) q^{80} \) \( + ( -4 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} - 2 \beta_{14} + 3 \beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{82} \) \( + ( -2 + 3 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} ) q^{83} \) \( + ( -2 - 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{16} + \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{85} \) \( + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{10} + \beta_{12} + \beta_{13} - 3 \beta_{14} + 2 \beta_{16} + 2 \beta_{17} ) q^{86} \) \( + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{8} - 4 \beta_{9} + \beta_{10} + \beta_{12} - 3 \beta_{14} + 2 \beta_{15} + \beta_{16} + 2 \beta_{17} + \beta_{19} ) q^{88} \) \( + ( 1 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{89} \) \( -\beta_{14} q^{91} \) \( + ( -2 - \beta_{1} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{11} + \beta_{13} + \beta_{16} + 2 \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{92} \) \( + ( -\beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} + 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{94} \) \( + ( -6 + \beta_{1} - 2 \beta_{3} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} + \beta_{17} + \beta_{19} ) q^{95} \) \( + ( -2 + 2 \beta_{1} + 2 \beta_{3} - \beta_{6} + \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{13} - \beta_{16} - \beta_{17} - \beta_{19} ) q^{97} \) \( -\beta_{1} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(20q \) \(\mathstrut -\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 24q^{4} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 20q^{7} \) \(\mathstrut -\mathstrut 24q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(20q \) \(\mathstrut -\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 24q^{4} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 20q^{7} \) \(\mathstrut -\mathstrut 24q^{8} \) \(\mathstrut -\mathstrut 8q^{10} \) \(\mathstrut -\mathstrut 26q^{11} \) \(\mathstrut -\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut 8q^{14} \) \(\mathstrut +\mathstrut 24q^{16} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut q^{19} \) \(\mathstrut +\mathstrut 2q^{20} \) \(\mathstrut +\mathstrut q^{22} \) \(\mathstrut -\mathstrut 31q^{23} \) \(\mathstrut +\mathstrut 27q^{25} \) \(\mathstrut -\mathstrut 4q^{26} \) \(\mathstrut +\mathstrut 24q^{28} \) \(\mathstrut -\mathstrut 16q^{29} \) \(\mathstrut +\mathstrut 6q^{31} \) \(\mathstrut -\mathstrut 41q^{32} \) \(\mathstrut -\mathstrut 10q^{34} \) \(\mathstrut -\mathstrut 3q^{35} \) \(\mathstrut +\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut 3q^{38} \) \(\mathstrut -\mathstrut 38q^{40} \) \(\mathstrut -\mathstrut 25q^{41} \) \(\mathstrut +\mathstrut 13q^{43} \) \(\mathstrut -\mathstrut 66q^{44} \) \(\mathstrut +\mathstrut 20q^{46} \) \(\mathstrut -\mathstrut 19q^{47} \) \(\mathstrut +\mathstrut 20q^{49} \) \(\mathstrut +\mathstrut 4q^{50} \) \(\mathstrut +\mathstrut 20q^{52} \) \(\mathstrut -\mathstrut 24q^{53} \) \(\mathstrut -\mathstrut 3q^{55} \) \(\mathstrut -\mathstrut 24q^{56} \) \(\mathstrut +\mathstrut 12q^{58} \) \(\mathstrut -\mathstrut 23q^{59} \) \(\mathstrut -\mathstrut 27q^{61} \) \(\mathstrut -\mathstrut 7q^{62} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 26q^{65} \) \(\mathstrut +\mathstrut 9q^{67} \) \(\mathstrut +\mathstrut 25q^{68} \) \(\mathstrut -\mathstrut 8q^{70} \) \(\mathstrut -\mathstrut 63q^{71} \) \(\mathstrut -\mathstrut 21q^{73} \) \(\mathstrut -\mathstrut 21q^{74} \) \(\mathstrut -\mathstrut 10q^{76} \) \(\mathstrut -\mathstrut 26q^{77} \) \(\mathstrut +\mathstrut 18q^{79} \) \(\mathstrut +\mathstrut 23q^{80} \) \(\mathstrut -\mathstrut 42q^{82} \) \(\mathstrut +\mathstrut q^{83} \) \(\mathstrut -\mathstrut 41q^{85} \) \(\mathstrut +\mathstrut 12q^{86} \) \(\mathstrut +\mathstrut 57q^{88} \) \(\mathstrut +\mathstrut 16q^{89} \) \(\mathstrut -\mathstrut 4q^{91} \) \(\mathstrut -\mathstrut 17q^{92} \) \(\mathstrut +\mathstrut 7q^{94} \) \(\mathstrut -\mathstrut 75q^{95} \) \(\mathstrut -\mathstrut 32q^{97} \) \(\mathstrut -\mathstrut 8q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20}\mathstrut -\mathstrut \) \(8\) \(x^{19}\mathstrut +\mathstrut \) \(152\) \(x^{17}\mathstrut -\mathstrut \) \(274\) \(x^{16}\mathstrut -\mathstrut \) \(1061\) \(x^{15}\mathstrut +\mathstrut \) \(3125\) \(x^{14}\mathstrut +\mathstrut \) \(2977\) \(x^{13}\mathstrut -\mathstrut \) \(15474\) \(x^{12}\mathstrut -\mathstrut \) \(56\) \(x^{11}\mathstrut +\mathstrut \) \(39579\) \(x^{10}\mathstrut -\mathstrut \) \(17664\) \(x^{9}\mathstrut -\mathstrut \) \(52271\) \(x^{8}\mathstrut +\mathstrut \) \(35701\) \(x^{7}\mathstrut +\mathstrut \) \(32493\) \(x^{6}\mathstrut -\mathstrut \) \(25504\) \(x^{5}\mathstrut -\mathstrut \) \(8607\) \(x^{4}\mathstrut +\mathstrut \) \(6812\) \(x^{3}\mathstrut +\mathstrut \) \(609\) \(x^{2}\mathstrut -\mathstrut \) \(425\) \(x\mathstrut +\mathstrut \) \(31\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 2 \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(80574\) \(\nu^{19}\mathstrut +\mathstrut \) \(88265\) \(\nu^{18}\mathstrut +\mathstrut \) \(4448116\) \(\nu^{17}\mathstrut -\mathstrut \) \(11956345\) \(\nu^{16}\mathstrut -\mathstrut \) \(63248616\) \(\nu^{15}\mathstrut +\mathstrut \) \(232480171\) \(\nu^{14}\mathstrut +\mathstrut \) \(345773311\) \(\nu^{13}\mathstrut -\mathstrut \) \(1910206802\) \(\nu^{12}\mathstrut -\mathstrut \) \(405883602\) \(\nu^{11}\mathstrut +\mathstrut \) \(7970416630\) \(\nu^{10}\mathstrut -\mathstrut \) \(3293131310\) \(\nu^{9}\mathstrut -\mathstrut \) \(17070805117\) \(\nu^{8}\mathstrut +\mathstrut \) \(13762066276\) \(\nu^{7}\mathstrut +\mathstrut \) \(16264479010\) \(\nu^{6}\mathstrut -\mathstrut \) \(19092142347\) \(\nu^{5}\mathstrut -\mathstrut \) \(3272911391\) \(\nu^{4}\mathstrut +\mathstrut \) \(8444727149\) \(\nu^{3}\mathstrut -\mathstrut \) \(1113016448\) \(\nu^{2}\mathstrut -\mathstrut \) \(888775901\) \(\nu\mathstrut +\mathstrut \) \(123446515\)\()/4334246\)
\(\beta_{5}\)\(=\)\((\)\(112041\) \(\nu^{19}\mathstrut -\mathstrut \) \(1197675\) \(\nu^{18}\mathstrut +\mathstrut \) \(2264887\) \(\nu^{17}\mathstrut +\mathstrut \) \(17783216\) \(\nu^{16}\mathstrut -\mathstrut \) \(73262421\) \(\nu^{15}\mathstrut -\mathstrut \) \(57671117\) \(\nu^{14}\mathstrut +\mathstrut \) \(647447518\) \(\nu^{13}\mathstrut -\mathstrut \) \(364390775\) \(\nu^{12}\mathstrut -\mathstrut \) \(2623374358\) \(\nu^{11}\mathstrut +\mathstrut \) \(3210092650\) \(\nu^{10}\mathstrut +\mathstrut \) \(5054834351\) \(\nu^{9}\mathstrut -\mathstrut \) \(9103752501\) \(\nu^{8}\mathstrut -\mathstrut \) \(3527560932\) \(\nu^{7}\mathstrut +\mathstrut \) \(10907649962\) \(\nu^{6}\mathstrut -\mathstrut \) \(969053095\) \(\nu^{5}\mathstrut -\mathstrut \) \(4410403111\) \(\nu^{4}\mathstrut +\mathstrut \) \(1105027913\) \(\nu^{3}\mathstrut +\mathstrut \) \(377557653\) \(\nu^{2}\mathstrut -\mathstrut \) \(99590538\) \(\nu\mathstrut +\mathstrut \) \(2196193\)\()/2167123\)
\(\beta_{6}\)\(=\)\((\)\(16028\) \(\nu^{19}\mathstrut -\mathstrut \) \(143864\) \(\nu^{18}\mathstrut +\mathstrut \) \(122280\) \(\nu^{17}\mathstrut +\mathstrut \) \(2470224\) \(\nu^{16}\mathstrut -\mathstrut \) \(6859382\) \(\nu^{15}\mathstrut -\mathstrut \) \(12972651\) \(\nu^{14}\mathstrut +\mathstrut \) \(67984425\) \(\nu^{13}\mathstrut -\mathstrut \) \(1910609\) \(\nu^{12}\mathstrut -\mathstrut \) \(295889912\) \(\nu^{11}\mathstrut +\mathstrut \) \(245045373\) \(\nu^{10}\mathstrut +\mathstrut \) \(599709512\) \(\nu^{9}\mathstrut -\mathstrut \) \(851828022\) \(\nu^{8}\mathstrut -\mathstrut \) \(418277501\) \(\nu^{7}\mathstrut +\mathstrut \) \(1092553461\) \(\nu^{6}\mathstrut -\mathstrut \) \(169615722\) \(\nu^{5}\mathstrut -\mathstrut \) \(415699503\) \(\nu^{4}\mathstrut +\mathstrut \) \(165471717\) \(\nu^{3}\mathstrut +\mathstrut \) \(18340672\) \(\nu^{2}\mathstrut -\mathstrut \) \(13862243\) \(\nu\mathstrut +\mathstrut \) \(1383690\)\()/309589\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(248879\) \(\nu^{19}\mathstrut +\mathstrut \) \(3978132\) \(\nu^{18}\mathstrut -\mathstrut \) \(13988931\) \(\nu^{17}\mathstrut -\mathstrut \) \(48139158\) \(\nu^{16}\mathstrut +\mathstrut \) \(338768723\) \(\nu^{15}\mathstrut -\mathstrut \) \(38956001\) \(\nu^{14}\mathstrut -\mathstrut \) \(2759046818\) \(\nu^{13}\mathstrut +\mathstrut \) \(3143549108\) \(\nu^{12}\mathstrut +\mathstrut \) \(10402831226\) \(\nu^{11}\mathstrut -\mathstrut \) \(18846899174\) \(\nu^{10}\mathstrut -\mathstrut \) \(17448853443\) \(\nu^{9}\mathstrut +\mathstrut \) \(48415589078\) \(\nu^{8}\mathstrut +\mathstrut \) \(5425063462\) \(\nu^{7}\mathstrut -\mathstrut \) \(55791961163\) \(\nu^{6}\mathstrut +\mathstrut \) \(14568997447\) \(\nu^{5}\mathstrut +\mathstrut \) \(22437701529\) \(\nu^{4}\mathstrut -\mathstrut \) \(8633213124\) \(\nu^{3}\mathstrut -\mathstrut \) \(1967839611\) \(\nu^{2}\mathstrut +\mathstrut \) \(676572227\) \(\nu\mathstrut -\mathstrut \) \(44644594\)\()/4334246\)
\(\beta_{8}\)\(=\)\((\)\(130444\) \(\nu^{19}\mathstrut -\mathstrut \) \(1062848\) \(\nu^{18}\mathstrut +\mathstrut \) \(203009\) \(\nu^{17}\mathstrut +\mathstrut \) \(19282840\) \(\nu^{16}\mathstrut -\mathstrut \) \(38105845\) \(\nu^{15}\mathstrut -\mathstrut \) \(122444455\) \(\nu^{14}\mathstrut +\mathstrut \) \(401582731\) \(\nu^{13}\mathstrut +\mathstrut \) \(251551762\) \(\nu^{12}\mathstrut -\mathstrut \) \(1805118745\) \(\nu^{11}\mathstrut +\mathstrut \) \(509262882\) \(\nu^{10}\mathstrut +\mathstrut \) \(3897308205\) \(\nu^{9}\mathstrut -\mathstrut \) \(3070652687\) \(\nu^{8}\mathstrut -\mathstrut \) \(3512303592\) \(\nu^{7}\mathstrut +\mathstrut \) \(4434806592\) \(\nu^{6}\mathstrut +\mathstrut \) \(364613036\) \(\nu^{5}\mathstrut -\mathstrut \) \(1821783462\) \(\nu^{4}\mathstrut +\mathstrut \) \(433924998\) \(\nu^{3}\mathstrut +\mathstrut \) \(117630811\) \(\nu^{2}\mathstrut -\mathstrut \) \(52984391\) \(\nu\mathstrut +\mathstrut \) \(4552939\)\()/2167123\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(146869\) \(\nu^{19}\mathstrut +\mathstrut \) \(1305396\) \(\nu^{18}\mathstrut -\mathstrut \) \(1062848\) \(\nu^{17}\mathstrut -\mathstrut \) \(22121079\) \(\nu^{16}\mathstrut +\mathstrut \) \(59524946\) \(\nu^{15}\mathstrut +\mathstrut \) \(117722164\) \(\nu^{14}\mathstrut -\mathstrut \) \(581410080\) \(\nu^{13}\mathstrut -\mathstrut \) \(35646282\) \(\nu^{12}\mathstrut +\mathstrut \) \(2524202668\) \(\nu^{11}\mathstrut -\mathstrut \) \(1796894081\) \(\nu^{10}\mathstrut -\mathstrut \) \(5303665269\) \(\nu^{9}\mathstrut +\mathstrut \) \(6491602221\) \(\nu^{8}\mathstrut +\mathstrut \) \(4606336812\) \(\nu^{7}\mathstrut -\mathstrut \) \(8755673761\) \(\nu^{6}\mathstrut -\mathstrut \) \(337407825\) \(\nu^{5}\mathstrut +\mathstrut \) \(4110360012\) \(\nu^{4}\mathstrut -\mathstrut \) \(557681979\) \(\nu^{3}\mathstrut -\mathstrut \) \(566546630\) \(\nu^{2}\mathstrut +\mathstrut \) \(28187590\) \(\nu\mathstrut +\mathstrut \) \(9434934\)\()/2167123\)
\(\beta_{10}\)\(=\)\((\)\(452378\) \(\nu^{19}\mathstrut -\mathstrut \) \(3636757\) \(\nu^{18}\mathstrut +\mathstrut \) \(479452\) \(\nu^{17}\mathstrut +\mathstrut \) \(64942855\) \(\nu^{16}\mathstrut -\mathstrut \) \(120306036\) \(\nu^{15}\mathstrut -\mathstrut \) \(413198975\) \(\nu^{14}\mathstrut +\mathstrut \) \(1229305933\) \(\nu^{13}\mathstrut +\mathstrut \) \(962389260\) \(\nu^{12}\mathstrut -\mathstrut \) \(5289287134\) \(\nu^{11}\mathstrut +\mathstrut \) \(625710296\) \(\nu^{10}\mathstrut +\mathstrut \) \(10741270610\) \(\nu^{9}\mathstrut -\mathstrut \) \(6399002549\) \(\nu^{8}\mathstrut -\mathstrut \) \(8503960458\) \(\nu^{7}\mathstrut +\mathstrut \) \(8810093974\) \(\nu^{6}\mathstrut -\mathstrut \) \(662651141\) \(\nu^{5}\mathstrut -\mathstrut \) \(2565501807\) \(\nu^{4}\mathstrut +\mathstrut \) \(1987510173\) \(\nu^{3}\mathstrut -\mathstrut \) \(333016752\) \(\nu^{2}\mathstrut -\mathstrut \) \(247405745\) \(\nu\mathstrut +\mathstrut \) \(51386023\)\()/4334246\)
\(\beta_{11}\)\(=\)\((\)\(239299\) \(\nu^{19}\mathstrut -\mathstrut \) \(839918\) \(\nu^{18}\mathstrut -\mathstrut \) \(6727183\) \(\nu^{17}\mathstrut +\mathstrut \) \(25057160\) \(\nu^{16}\mathstrut +\mathstrut \) \(75577807\) \(\nu^{15}\mathstrut -\mathstrut \) \(306448870\) \(\nu^{14}\mathstrut -\mathstrut \) \(430391229\) \(\nu^{13}\mathstrut +\mathstrut \) \(1985694746\) \(\nu^{12}\mathstrut +\mathstrut \) \(1274010673\) \(\nu^{11}\mathstrut -\mathstrut \) \(7326712947\) \(\nu^{10}\mathstrut -\mathstrut \) \(1640919567\) \(\nu^{9}\mathstrut +\mathstrut \) \(15286073938\) \(\nu^{8}\mathstrut -\mathstrut \) \(166042331\) \(\nu^{7}\mathstrut -\mathstrut \) \(16629421700\) \(\nu^{6}\mathstrut +\mathstrut \) \(1992882190\) \(\nu^{5}\mathstrut +\mathstrut \) \(7760740247\) \(\nu^{4}\mathstrut -\mathstrut \) \(741296302\) \(\nu^{3}\mathstrut -\mathstrut \) \(1279308585\) \(\nu^{2}\mathstrut -\mathstrut \) \(50188832\) \(\nu\mathstrut +\mathstrut \) \(33626793\)\()/2167123\)
\(\beta_{12}\)\(=\)\((\)\(513169\) \(\nu^{19}\mathstrut -\mathstrut \) \(3359961\) \(\nu^{18}\mathstrut -\mathstrut \) \(4253023\) \(\nu^{17}\mathstrut +\mathstrut \) \(67902593\) \(\nu^{16}\mathstrut -\mathstrut \) \(49716311\) \(\nu^{15}\mathstrut -\mathstrut \) \(524918352\) \(\nu^{14}\mathstrut +\mathstrut \) \(801626921\) \(\nu^{13}\mathstrut +\mathstrut \) \(1873963420\) \(\nu^{12}\mathstrut -\mathstrut \) \(4115132892\) \(\nu^{11}\mathstrut -\mathstrut \) \(2626187800\) \(\nu^{10}\mathstrut +\mathstrut \) \(9547027119\) \(\nu^{9}\mathstrut -\mathstrut \) \(992559901\) \(\nu^{8}\mathstrut -\mathstrut \) \(8895240438\) \(\nu^{7}\mathstrut +\mathstrut \) \(5258752701\) \(\nu^{6}\mathstrut +\mathstrut \) \(511422678\) \(\nu^{5}\mathstrut -\mathstrut \) \(2384056922\) \(\nu^{4}\mathstrut +\mathstrut \) \(1615698465\) \(\nu^{3}\mathstrut +\mathstrut \) \(53574147\) \(\nu^{2}\mathstrut -\mathstrut \) \(261345936\) \(\nu\mathstrut +\mathstrut \) \(14104789\)\()/4334246\)
\(\beta_{13}\)\(=\)\((\)\(600491\) \(\nu^{19}\mathstrut -\mathstrut \) \(5014091\) \(\nu^{18}\mathstrut +\mathstrut \) \(1433657\) \(\nu^{17}\mathstrut +\mathstrut \) \(92851141\) \(\nu^{16}\mathstrut -\mathstrut \) \(196388497\) \(\nu^{15}\mathstrut -\mathstrut \) \(595632982\) \(\nu^{14}\mathstrut +\mathstrut \) \(2117597099\) \(\nu^{13}\mathstrut +\mathstrut \) \(1139674102\) \(\nu^{12}\mathstrut -\mathstrut \) \(9839915902\) \(\nu^{11}\mathstrut +\mathstrut \) \(3476162040\) \(\nu^{10}\mathstrut +\mathstrut \) \(22105769119\) \(\nu^{9}\mathstrut -\mathstrut \) \(18817836025\) \(\nu^{8}\mathstrut -\mathstrut \) \(21121831192\) \(\nu^{7}\mathstrut +\mathstrut \) \(27608913433\) \(\nu^{6}\mathstrut +\mathstrut \) \(3331895996\) \(\nu^{5}\mathstrut -\mathstrut \) \(12057424380\) \(\nu^{4}\mathstrut +\mathstrut \) \(2030109247\) \(\nu^{3}\mathstrut +\mathstrut \) \(1034875291\) \(\nu^{2}\mathstrut -\mathstrut \) \(246014754\) \(\nu\mathstrut +\mathstrut \) \(34267467\)\()/4334246\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(410993\) \(\nu^{19}\mathstrut +\mathstrut \) \(2213625\) \(\nu^{18}\mathstrut +\mathstrut \) \(6917810\) \(\nu^{17}\mathstrut -\mathstrut \) \(52563575\) \(\nu^{16}\mathstrut -\mathstrut \) \(29025299\) \(\nu^{15}\mathstrut +\mathstrut \) \(513862951\) \(\nu^{14}\mathstrut -\mathstrut \) \(139289961\) \(\nu^{13}\mathstrut -\mathstrut \) \(2668384327\) \(\nu^{12}\mathstrut +\mathstrut \) \(1733798795\) \(\nu^{11}\mathstrut +\mathstrut \) \(7888472785\) \(\nu^{10}\mathstrut -\mathstrut \) \(6649332298\) \(\nu^{9}\mathstrut -\mathstrut \) \(13110138889\) \(\nu^{8}\mathstrut +\mathstrut \) \(12281615752\) \(\nu^{7}\mathstrut +\mathstrut \) \(11099392631\) \(\nu^{6}\mathstrut -\mathstrut \) \(10831511067\) \(\nu^{5}\mathstrut -\mathstrut \) \(3600113430\) \(\nu^{4}\mathstrut +\mathstrut \) \(3719573150\) \(\nu^{3}\mathstrut +\mathstrut \) \(165170286\) \(\nu^{2}\mathstrut -\mathstrut \) \(303545763\) \(\nu\mathstrut +\mathstrut \) \(31062379\)\()/2167123\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(413388\) \(\nu^{19}\mathstrut +\mathstrut \) \(3462562\) \(\nu^{18}\mathstrut -\mathstrut \) \(1511903\) \(\nu^{17}\mathstrut -\mathstrut \) \(60346403\) \(\nu^{16}\mathstrut +\mathstrut \) \(134466805\) \(\nu^{15}\mathstrut +\mathstrut \) \(354990510\) \(\nu^{14}\mathstrut -\mathstrut \) \(1345384350\) \(\nu^{13}\mathstrut -\mathstrut \) \(525261149\) \(\nu^{12}\mathstrut +\mathstrut \) \(5839741304\) \(\nu^{11}\mathstrut -\mathstrut \) \(2589729038\) \(\nu^{10}\mathstrut -\mathstrut \) \(12179494628\) \(\nu^{9}\mathstrut +\mathstrut \) \(11432686680\) \(\nu^{8}\mathstrut +\mathstrut \) \(10435235153\) \(\nu^{7}\mathstrut -\mathstrut \) \(15517251270\) \(\nu^{6}\mathstrut -\mathstrut \) \(583856352\) \(\nu^{5}\mathstrut +\mathstrut \) \(6479767911\) \(\nu^{4}\mathstrut -\mathstrut \) \(1490693552\) \(\nu^{3}\mathstrut -\mathstrut \) \(545381160\) \(\nu^{2}\mathstrut +\mathstrut \) \(147237033\) \(\nu\mathstrut -\mathstrut \) \(3502341\)\()/2167123\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(1008515\) \(\nu^{19}\mathstrut +\mathstrut \) \(6530918\) \(\nu^{18}\mathstrut +\mathstrut \) \(8850087\) \(\nu^{17}\mathstrut -\mathstrut \) \(132965184\) \(\nu^{16}\mathstrut +\mathstrut \) \(86306181\) \(\nu^{15}\mathstrut +\mathstrut \) \(1050649075\) \(\nu^{14}\mathstrut -\mathstrut \) \(1488281912\) \(\nu^{13}\mathstrut -\mathstrut \) \(3982849598\) \(\nu^{12}\mathstrut +\mathstrut \) \(7914662926\) \(\nu^{11}\mathstrut +\mathstrut \) \(6906341856\) \(\nu^{10}\mathstrut -\mathstrut \) \(19645694201\) \(\nu^{9}\mathstrut -\mathstrut \) \(2787853840\) \(\nu^{8}\mathstrut +\mathstrut \) \(22331737484\) \(\nu^{7}\mathstrut -\mathstrut \) \(4642008039\) \(\nu^{6}\mathstrut -\mathstrut \) \(8848072497\) \(\nu^{5}\mathstrut +\mathstrut \) \(2556273551\) \(\nu^{4}\mathstrut +\mathstrut \) \(817071694\) \(\nu^{3}\mathstrut -\mathstrut \) \(21377165\) \(\nu^{2}\mathstrut -\mathstrut \) \(16996363\) \(\nu\mathstrut -\mathstrut \) \(10476190\)\()/4334246\)
\(\beta_{17}\)\(=\)\((\)\(-\)\(579954\) \(\nu^{19}\mathstrut +\mathstrut \) \(3614942\) \(\nu^{18}\mathstrut +\mathstrut \) \(6341910\) \(\nu^{17}\mathstrut -\mathstrut \) \(77052369\) \(\nu^{16}\mathstrut +\mathstrut \) \(25536710\) \(\nu^{15}\mathstrut +\mathstrut \) \(659355662\) \(\nu^{14}\mathstrut -\mathstrut \) \(695262384\) \(\nu^{13}\mathstrut -\mathstrut \) \(2896884557\) \(\nu^{12}\mathstrut +\mathstrut \) \(4231996085\) \(\nu^{11}\mathstrut +\mathstrut \) \(6882461020\) \(\nu^{10}\mathstrut -\mathstrut \) \(12292561414\) \(\nu^{9}\mathstrut -\mathstrut \) \(8381823921\) \(\nu^{8}\mathstrut +\mathstrut \) \(18384933518\) \(\nu^{7}\mathstrut +\mathstrut \) \(4109230177\) \(\nu^{6}\mathstrut -\mathstrut \) \(13340039557\) \(\nu^{5}\mathstrut +\mathstrut \) \(136389714\) \(\nu^{4}\mathstrut +\mathstrut \) \(4022677293\) \(\nu^{3}\mathstrut -\mathstrut \) \(483654127\) \(\nu^{2}\mathstrut -\mathstrut \) \(315155203\) \(\nu\mathstrut +\mathstrut \) \(42358963\)\()/2167123\)
\(\beta_{18}\)\(=\)\((\)\(170912\) \(\nu^{19}\mathstrut -\mathstrut \) \(1102231\) \(\nu^{18}\mathstrut -\mathstrut \) \(1613038\) \(\nu^{17}\mathstrut +\mathstrut \) \(22982443\) \(\nu^{16}\mathstrut -\mathstrut \) \(12939568\) \(\nu^{15}\mathstrut -\mathstrut \) \(189215421\) \(\nu^{14}\mathstrut +\mathstrut \) \(248577527\) \(\nu^{13}\mathstrut +\mathstrut \) \(774985956\) \(\nu^{12}\mathstrut -\mathstrut \) \(1408256152\) \(\nu^{11}\mathstrut -\mathstrut \) \(1603313970\) \(\nu^{10}\mathstrut +\mathstrut \) \(3845197606\) \(\nu^{9}\mathstrut +\mathstrut \) \(1393415281\) \(\nu^{8}\mathstrut -\mathstrut \) \(5275677336\) \(\nu^{7}\mathstrut +\mathstrut \) \(3660472\) \(\nu^{6}\mathstrut +\mathstrut \) \(3356197997\) \(\nu^{5}\mathstrut -\mathstrut \) \(455638987\) \(\nu^{4}\mathstrut -\mathstrut \) \(921371945\) \(\nu^{3}\mathstrut +\mathstrut \) \(151871254\) \(\nu^{2}\mathstrut +\mathstrut \) \(82480027\) \(\nu\mathstrut -\mathstrut \) \(10261733\)\()/619178\)
\(\beta_{19}\)\(=\)\((\)\(1492257\) \(\nu^{19}\mathstrut -\mathstrut \) \(9628117\) \(\nu^{18}\mathstrut -\mathstrut \) \(13332635\) \(\nu^{17}\mathstrut +\mathstrut \) \(194372481\) \(\nu^{16}\mathstrut -\mathstrut \) \(107507193\) \(\nu^{15}\mathstrut -\mathstrut \) \(1560069588\) \(\nu^{14}\mathstrut +\mathstrut \) \(1932311183\) \(\nu^{13}\mathstrut +\mathstrut \) \(6369622840\) \(\nu^{12}\mathstrut -\mathstrut \) \(10456617166\) \(\nu^{11}\mathstrut -\mathstrut \) \(13947586220\) \(\nu^{10}\mathstrut +\mathstrut \) \(27726528907\) \(\nu^{9}\mathstrut +\mathstrut \) \(15505936931\) \(\nu^{8}\mathstrut -\mathstrut \) \(38183605866\) \(\nu^{7}\mathstrut -\mathstrut \) \(6618531811\) \(\nu^{6}\mathstrut +\mathstrut \) \(25857674064\) \(\nu^{5}\mathstrut -\mathstrut \) \(907539212\) \(\nu^{4}\mathstrut -\mathstrut \) \(7654017971\) \(\nu^{3}\mathstrut +\mathstrut \) \(1064221961\) \(\nu^{2}\mathstrut +\mathstrut \) \(698367060\) \(\nu\mathstrut -\mathstrut \) \(78687283\)\()/4334246\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{18}\mathstrut +\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(\beta_{19}\mathstrut +\mathstrut \) \(2\) \(\beta_{18}\mathstrut +\mathstrut \) \(\beta_{17}\mathstrut +\mathstrut \) \(2\) \(\beta_{16}\mathstrut +\mathstrut \) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(27\) \(\beta_{1}\mathstrut +\mathstrut \) \(9\)
\(\nu^{6}\)\(=\)\(12\) \(\beta_{18}\mathstrut +\mathstrut \) \(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{16}\mathstrut +\mathstrut \) \(11\) \(\beta_{15}\mathstrut +\mathstrut \) \(9\) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(12\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(58\) \(\beta_{2}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(74\)
\(\nu^{7}\)\(=\)\(11\) \(\beta_{19}\mathstrut +\mathstrut \) \(25\) \(\beta_{18}\mathstrut +\mathstrut \) \(12\) \(\beta_{17}\mathstrut +\mathstrut \) \(23\) \(\beta_{16}\mathstrut +\mathstrut \) \(14\) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(14\) \(\beta_{11}\mathstrut +\mathstrut \) \(15\) \(\beta_{10}\mathstrut -\mathstrut \) \(15\) \(\beta_{9}\mathstrut +\mathstrut \) \(14\) \(\beta_{8}\mathstrut +\mathstrut \) \(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(8\) \(\beta_{6}\mathstrut -\mathstrut \) \(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(26\) \(\beta_{4}\mathstrut +\mathstrut \) \(70\) \(\beta_{3}\mathstrut +\mathstrut \) \(95\) \(\beta_{2}\mathstrut +\mathstrut \) \(159\) \(\beta_{1}\mathstrut +\mathstrut \) \(69\)
\(\nu^{8}\)\(=\)\(3\) \(\beta_{19}\mathstrut +\mathstrut \) \(107\) \(\beta_{18}\mathstrut +\mathstrut \) \(16\) \(\beta_{17}\mathstrut +\mathstrut \) \(17\) \(\beta_{16}\mathstrut +\mathstrut \) \(95\) \(\beta_{15}\mathstrut +\mathstrut \) \(61\) \(\beta_{14}\mathstrut -\mathstrut \) \(12\) \(\beta_{13}\mathstrut +\mathstrut \) \(12\) \(\beta_{12}\mathstrut -\mathstrut \) \(31\) \(\beta_{11}\mathstrut +\mathstrut \) \(108\) \(\beta_{10}\mathstrut -\mathstrut \) \(18\) \(\beta_{9}\mathstrut +\mathstrut \) \(16\) \(\beta_{8}\mathstrut +\mathstrut \) \(16\) \(\beta_{7}\mathstrut -\mathstrut \) \(15\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(34\) \(\beta_{4}\mathstrut +\mathstrut \) \(110\) \(\beta_{3}\mathstrut +\mathstrut \) \(415\) \(\beta_{2}\mathstrut +\mathstrut \) \(122\) \(\beta_{1}\mathstrut +\mathstrut \) \(427\)
\(\nu^{9}\)\(=\)\(94\) \(\beta_{19}\mathstrut +\mathstrut \) \(233\) \(\beta_{18}\mathstrut +\mathstrut \) \(112\) \(\beta_{17}\mathstrut +\mathstrut \) \(200\) \(\beta_{16}\mathstrut +\mathstrut \) \(143\) \(\beta_{15}\mathstrut -\mathstrut \) \(18\) \(\beta_{14}\mathstrut -\mathstrut \) \(14\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(142\) \(\beta_{11}\mathstrut +\mathstrut \) \(159\) \(\beta_{10}\mathstrut -\mathstrut \) \(157\) \(\beta_{9}\mathstrut +\mathstrut \) \(137\) \(\beta_{8}\mathstrut +\mathstrut \) \(125\) \(\beta_{7}\mathstrut +\mathstrut \) \(42\) \(\beta_{6}\mathstrut -\mathstrut \) \(90\) \(\beta_{5}\mathstrut +\mathstrut \) \(249\) \(\beta_{4}\mathstrut +\mathstrut \) \(522\) \(\beta_{3}\mathstrut +\mathstrut \) \(760\) \(\beta_{2}\mathstrut +\mathstrut \) \(998\) \(\beta_{1}\mathstrut +\mathstrut \) \(508\)
\(\nu^{10}\)\(=\)\(55\) \(\beta_{19}\mathstrut +\mathstrut \) \(861\) \(\beta_{18}\mathstrut +\mathstrut \) \(181\) \(\beta_{17}\mathstrut +\mathstrut \) \(195\) \(\beta_{16}\mathstrut +\mathstrut \) \(758\) \(\beta_{15}\mathstrut +\mathstrut \) \(368\) \(\beta_{14}\mathstrut -\mathstrut \) \(104\) \(\beta_{13}\mathstrut +\mathstrut \) \(107\) \(\beta_{12}\mathstrut -\mathstrut \) \(335\) \(\beta_{11}\mathstrut +\mathstrut \) \(881\) \(\beta_{10}\mathstrut -\mathstrut \) \(223\) \(\beta_{9}\mathstrut +\mathstrut \) \(178\) \(\beta_{8}\mathstrut +\mathstrut \) \(183\) \(\beta_{7}\mathstrut -\mathstrut \) \(159\) \(\beta_{6}\mathstrut -\mathstrut \) \(29\) \(\beta_{5}\mathstrut +\mathstrut \) \(388\) \(\beta_{4}\mathstrut +\mathstrut \) \(915\) \(\beta_{3}\mathstrut +\mathstrut \) \(2969\) \(\beta_{2}\mathstrut +\mathstrut \) \(1029\) \(\beta_{1}\mathstrut +\mathstrut \) \(2627\)
\(\nu^{11}\)\(=\)\(749\) \(\beta_{19}\mathstrut +\mathstrut \) \(1951\) \(\beta_{18}\mathstrut +\mathstrut \) \(964\) \(\beta_{17}\mathstrut +\mathstrut \) \(1587\) \(\beta_{16}\mathstrut +\mathstrut \) \(1289\) \(\beta_{15}\mathstrut -\mathstrut \) \(215\) \(\beta_{14}\mathstrut -\mathstrut \) \(130\) \(\beta_{13}\mathstrut +\mathstrut \) \(28\) \(\beta_{12}\mathstrut -\mathstrut \) \(1272\) \(\beta_{11}\mathstrut +\mathstrut \) \(1465\) \(\beta_{10}\mathstrut -\mathstrut \) \(1424\) \(\beta_{9}\mathstrut +\mathstrut \) \(1164\) \(\beta_{8}\mathstrut +\mathstrut \) \(1082\) \(\beta_{7}\mathstrut +\mathstrut \) \(139\) \(\beta_{6}\mathstrut -\mathstrut \) \(655\) \(\beta_{5}\mathstrut +\mathstrut \) \(2129\) \(\beta_{4}\mathstrut +\mathstrut \) \(3829\) \(\beta_{3}\mathstrut +\mathstrut \) \(5886\) \(\beta_{2}\mathstrut +\mathstrut \) \(6543\) \(\beta_{1}\mathstrut +\mathstrut \) \(3700\)
\(\nu^{12}\)\(=\)\(682\) \(\beta_{19}\mathstrut +\mathstrut \) \(6626\) \(\beta_{18}\mathstrut +\mathstrut \) \(1783\) \(\beta_{17}\mathstrut +\mathstrut \) \(1903\) \(\beta_{16}\mathstrut +\mathstrut \) \(5847\) \(\beta_{15}\mathstrut +\mathstrut \) \(2054\) \(\beta_{14}\mathstrut -\mathstrut \) \(784\) \(\beta_{13}\mathstrut +\mathstrut \) \(866\) \(\beta_{12}\mathstrut -\mathstrut \) \(3132\) \(\beta_{11}\mathstrut +\mathstrut \) \(6881\) \(\beta_{10}\mathstrut -\mathstrut \) \(2321\) \(\beta_{9}\mathstrut +\mathstrut \) \(1695\) \(\beta_{8}\mathstrut +\mathstrut \) \(1823\) \(\beta_{7}\mathstrut -\mathstrut \) \(1465\) \(\beta_{6}\mathstrut -\mathstrut \) \(274\) \(\beta_{5}\mathstrut +\mathstrut \) \(3757\) \(\beta_{4}\mathstrut +\mathstrut \) \(7272\) \(\beta_{3}\mathstrut +\mathstrut \) \(21318\) \(\beta_{2}\mathstrut +\mathstrut \) \(8274\) \(\beta_{1}\mathstrut +\mathstrut \) \(16903\)
\(\nu^{13}\)\(=\)\(5849\) \(\beta_{19}\mathstrut +\mathstrut \) \(15528\) \(\beta_{18}\mathstrut +\mathstrut \) \(8007\) \(\beta_{17}\mathstrut +\mathstrut \) \(12128\) \(\beta_{16}\mathstrut +\mathstrut \) \(10902\) \(\beta_{15}\mathstrut -\mathstrut \) \(2161\) \(\beta_{14}\mathstrut -\mathstrut \) \(997\) \(\beta_{13}\mathstrut +\mathstrut \) \(443\) \(\beta_{12}\mathstrut -\mathstrut \) \(10725\) \(\beta_{11}\mathstrut +\mathstrut \) \(12567\) \(\beta_{10}\mathstrut -\mathstrut \) \(12026\) \(\beta_{9}\mathstrut +\mathstrut \) \(9231\) \(\beta_{8}\mathstrut +\mathstrut \) \(8945\) \(\beta_{7}\mathstrut -\mathstrut \) \(237\) \(\beta_{6}\mathstrut -\mathstrut \) \(4470\) \(\beta_{5}\mathstrut +\mathstrut \) \(17254\) \(\beta_{4}\mathstrut +\mathstrut \) \(27907\) \(\beta_{3}\mathstrut +\mathstrut \) \(44880\) \(\beta_{2}\mathstrut +\mathstrut \) \(44173\) \(\beta_{1}\mathstrut +\mathstrut \) \(26897\)
\(\nu^{14}\)\(=\)\(7165\) \(\beta_{19}\mathstrut +\mathstrut \) \(49927\) \(\beta_{18}\mathstrut +\mathstrut \) \(16332\) \(\beta_{17}\mathstrut +\mathstrut \) \(17066\) \(\beta_{16}\mathstrut +\mathstrut \) \(44367\) \(\beta_{15}\mathstrut +\mathstrut \) \(10564\) \(\beta_{14}\mathstrut -\mathstrut \) \(5421\) \(\beta_{13}\mathstrut +\mathstrut \) \(6771\) \(\beta_{12}\mathstrut -\mathstrut \) \(27216\) \(\beta_{11}\mathstrut +\mathstrut \) \(52601\) \(\beta_{10}\mathstrut -\mathstrut \) \(21850\) \(\beta_{9}\mathstrut +\mathstrut \) \(14829\) \(\beta_{8}\mathstrut +\mathstrut \) \(16853\) \(\beta_{7}\mathstrut -\mathstrut \) \(12569\) \(\beta_{6}\mathstrut -\mathstrut \) \(2112\) \(\beta_{5}\mathstrut +\mathstrut \) \(33413\) \(\beta_{4}\mathstrut +\mathstrut \) \(56412\) \(\beta_{3}\mathstrut +\mathstrut \) \(153805\) \(\beta_{2}\mathstrut +\mathstrut \) \(64837\) \(\beta_{1}\mathstrut +\mathstrut \) \(112151\)
\(\nu^{15}\)\(=\)\(45504\) \(\beta_{19}\mathstrut +\mathstrut \) \(120296\) \(\beta_{18}\mathstrut +\mathstrut \) \(65244\) \(\beta_{17}\mathstrut +\mathstrut \) \(91153\) \(\beta_{16}\mathstrut +\mathstrut \) \(88859\) \(\beta_{15}\mathstrut -\mathstrut \) \(19819\) \(\beta_{14}\mathstrut -\mathstrut \) \(6721\) \(\beta_{13}\mathstrut +\mathstrut \) \(5434\) \(\beta_{12}\mathstrut -\mathstrut \) \(87508\) \(\beta_{11}\mathstrut +\mathstrut \) \(103472\) \(\beta_{10}\mathstrut -\mathstrut \) \(97615\) \(\beta_{9}\mathstrut +\mathstrut \) \(70564\) \(\beta_{8}\mathstrut +\mathstrut \) \(72322\) \(\beta_{7}\mathstrut -\mathstrut \) \(10263\) \(\beta_{6}\mathstrut -\mathstrut \) \(29212\) \(\beta_{5}\mathstrut +\mathstrut \) \(135979\) \(\beta_{4}\mathstrut +\mathstrut \) \(203042\) \(\beta_{3}\mathstrut +\mathstrut \) \(339458\) \(\beta_{2}\mathstrut +\mathstrut \) \(304274\) \(\beta_{1}\mathstrut +\mathstrut \) \(195625\)
\(\nu^{16}\)\(=\)\(68741\) \(\beta_{19}\mathstrut +\mathstrut \) \(372420\) \(\beta_{18}\mathstrut +\mathstrut \) \(143070\) \(\beta_{17}\mathstrut +\mathstrut \) \(145540\) \(\beta_{16}\mathstrut +\mathstrut \) \(333767\) \(\beta_{15}\mathstrut +\mathstrut \) \(47771\) \(\beta_{14}\mathstrut -\mathstrut \) \(34977\) \(\beta_{13}\mathstrut +\mathstrut \) \(52411\) \(\beta_{12}\mathstrut -\mathstrut \) \(226950\) \(\beta_{11}\mathstrut +\mathstrut \) \(397689\) \(\beta_{10}\mathstrut -\mathstrut \) \(193095\) \(\beta_{9}\mathstrut +\mathstrut \) \(123205\) \(\beta_{8}\mathstrut +\mathstrut \) \(148769\) \(\beta_{7}\mathstrut -\mathstrut \) \(103573\) \(\beta_{6}\mathstrut -\mathstrut \) \(14172\) \(\beta_{5}\mathstrut +\mathstrut \) \(282868\) \(\beta_{4}\mathstrut +\mathstrut \) \(431591\) \(\beta_{3}\mathstrut +\mathstrut \) \(1115180\) \(\beta_{2}\mathstrut +\mathstrut \) \(500189\) \(\beta_{1}\mathstrut +\mathstrut \) \(760028\)
\(\nu^{17}\)\(=\)\(354430\) \(\beta_{19}\mathstrut +\mathstrut \) \(917916\) \(\beta_{18}\mathstrut +\mathstrut \) \(525239\) \(\beta_{17}\mathstrut +\mathstrut \) \(680149\) \(\beta_{16}\mathstrut +\mathstrut \) \(707465\) \(\beta_{15}\mathstrut -\mathstrut \) \(172044\) \(\beta_{14}\mathstrut -\mathstrut \) \(40089\) \(\beta_{13}\mathstrut +\mathstrut \) \(57847\) \(\beta_{12}\mathstrut -\mathstrut \) \(700514\) \(\beta_{11}\mathstrut +\mathstrut \) \(830610\) \(\beta_{10}\mathstrut -\mathstrut \) \(773876\) \(\beta_{9}\mathstrut +\mathstrut \) \(528454\) \(\beta_{8}\mathstrut +\mathstrut \) \(578015\) \(\beta_{7}\mathstrut -\mathstrut \) \(131379\) \(\beta_{6}\mathstrut -\mathstrut \) \(184387\) \(\beta_{5}\mathstrut +\mathstrut \) \(1055444\) \(\beta_{4}\mathstrut +\mathstrut \) \(1478128\) \(\beta_{3}\mathstrut +\mathstrut \) \(2556417\) \(\beta_{2}\mathstrut +\mathstrut \) \(2126070\) \(\beta_{1}\mathstrut +\mathstrut \) \(1424146\)
\(\nu^{18}\)\(=\)\(623089\) \(\beta_{19}\mathstrut +\mathstrut \) \(2765411\) \(\beta_{18}\mathstrut +\mathstrut \) \(1215932\) \(\beta_{17}\mathstrut +\mathstrut \) \(1201403\) \(\beta_{16}\mathstrut +\mathstrut \) \(2498983\) \(\beta_{15}\mathstrut +\mathstrut \) \(158548\) \(\beta_{14}\mathstrut -\mathstrut \) \(210305\) \(\beta_{13}\mathstrut +\mathstrut \) \(405455\) \(\beta_{12}\mathstrut -\mathstrut \) \(1846013\) \(\beta_{11}\mathstrut +\mathstrut \) \(2990011\) \(\beta_{10}\mathstrut -\mathstrut \) \(1636228\) \(\beta_{9}\mathstrut +\mathstrut \) \(989690\) \(\beta_{8}\mathstrut +\mathstrut \) \(1273224\) \(\beta_{7}\mathstrut -\mathstrut \) \(832969\) \(\beta_{6}\mathstrut -\mathstrut \) \(83371\) \(\beta_{5}\mathstrut +\mathstrut \) \(2322373\) \(\beta_{4}\mathstrut +\mathstrut \) \(3274565\) \(\beta_{3}\mathstrut +\mathstrut \) \(8123538\) \(\beta_{2}\mathstrut +\mathstrut \) \(3819421\) \(\beta_{1}\mathstrut +\mathstrut \) \(5228188\)
\(\nu^{19}\)\(=\)\(2765954\) \(\beta_{19}\mathstrut +\mathstrut \) \(6943503\) \(\beta_{18}\mathstrut +\mathstrut \) \(4191957\) \(\beta_{17}\mathstrut +\mathstrut \) \(5061055\) \(\beta_{16}\mathstrut +\mathstrut \) \(5543561\) \(\beta_{15}\mathstrut -\mathstrut \) \(1440901\) \(\beta_{14}\mathstrut -\mathstrut \) \(201460\) \(\beta_{13}\mathstrut +\mathstrut \) \(563630\) \(\beta_{12}\mathstrut -\mathstrut \) \(5542441\) \(\beta_{11}\mathstrut +\mathstrut \) \(6557790\) \(\beta_{10}\mathstrut -\mathstrut \) \(6045266\) \(\beta_{9}\mathstrut +\mathstrut \) \(3911178\) \(\beta_{8}\mathstrut +\mathstrut \) \(4589665\) \(\beta_{7}\mathstrut -\mathstrut \) \(1330431\) \(\beta_{6}\mathstrut -\mathstrut \) \(1126086\) \(\beta_{5}\mathstrut +\mathstrut \) \(8123077\) \(\beta_{4}\mathstrut +\mathstrut \) \(10779432\) \(\beta_{3}\mathstrut +\mathstrut \) \(19206088\) \(\beta_{2}\mathstrut +\mathstrut \) \(15014219\) \(\beta_{1}\mathstrut +\mathstrut \) \(10376694\)

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
2.75954
2.70334
2.58654
2.24329
2.11976
2.02020
1.78791
1.72132
0.831296
0.697877
0.170762
0.102309
−0.343534
−0.723268
−0.731838
−1.60447
−1.86440
−1.86718
−2.12365
−2.48579
−2.75954 0 5.61505 −0.308409 0 1.00000 −9.97588 0 0.851066
1.2 −2.70334 0 5.30805 3.37987 0 1.00000 −8.94278 0 −9.13693
1.3 −2.58654 0 4.69021 −1.18795 0 1.00000 −6.95835 0 3.07267
1.4 −2.24329 0 3.03233 1.67114 0 1.00000 −2.31581 0 −3.74884
1.5 −2.11976 0 2.49337 −3.20666 0 1.00000 −1.04583 0 6.79735
1.6 −2.02020 0 2.08119 3.61502 0 1.00000 −0.164016 0 −7.30304
1.7 −1.78791 0 1.19664 −2.61034 0 1.00000 1.43634 0 4.66707
1.8 −1.72132 0 0.962958 −2.01516 0 1.00000 1.78509 0 3.46875
1.9 −0.831296 0 −1.30895 1.71517 0 1.00000 2.75071 0 −1.42582
1.10 −0.697877 0 −1.51297 0.682552 0 1.00000 2.45162 0 −0.476337
1.11 −0.170762 0 −1.97084 −2.09248 0 1.00000 0.678071 0 0.357317
1.12 −0.102309 0 −1.98953 −2.62692 0 1.00000 0.408164 0 0.268756
1.13 0.343534 0 −1.88198 4.21457 0 1.00000 −1.33360 0 1.44785
1.14 0.723268 0 −1.47688 1.71672 0 1.00000 −2.51472 0 1.24165
1.15 0.731838 0 −1.46441 −3.93264 0 1.00000 −2.53539 0 −2.87806
1.16 1.60447 0 0.574339 −0.248535 0 1.00000 −2.28744 0 −0.398768
1.17 1.86440 0 1.47599 1.88221 0 1.00000 −0.976959 0 3.50920
1.18 1.86718 0 1.48637 −2.52504 0 1.00000 −0.959047 0 −4.71472
1.19 2.12365 0 2.50989 2.22935 0 1.00000 1.08284 0 4.73436
1.20 2.48579 0 4.17918 −3.35246 0 1.00000 5.41698 0 −8.33354
\(n\): 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
\(3\) \(-1\)
\(7\) \(-1\)
\(127\) \(1\)

Hecke kernels

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

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