Properties

Label 4016.2.a.l
Level 4016
Weight 2
Character orbit 4016.a
Self dual Yes
Analytic conductor 32.068
Analytic rank 0
Dimension 19
CM No
Inner twists 1

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Newspace parameters

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

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)  \(=\)  \( q\) \( + \beta_{1} q^{3} \) \( + \beta_{4} q^{5} \) \( + ( 1 + \beta_{13} ) q^{7} \) \( + ( 1 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( + \beta_{4} q^{5} \) \( + ( 1 + \beta_{13} ) q^{7} \) \( + ( 1 + \beta_{2} ) q^{9} \) \( + ( 1 - \beta_{12} ) q^{11} \) \( + ( \beta_{1} - \beta_{2} - \beta_{4} + \beta_{7} - \beta_{12} + \beta_{13} ) q^{13} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{7} - \beta_{13} - \beta_{15} ) q^{15} \) \( + ( -1 + \beta_{1} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{11} + \beta_{13} ) q^{17} \) \( + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{11} + \beta_{14} + \beta_{15} ) q^{19} \) \( + ( \beta_{1} + \beta_{7} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{21} \) \( + ( 1 + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} ) q^{23} \) \( + ( - \beta_{3} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{25} \) \( + ( 2 + \beta_{1} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} + \beta_{14} + \beta_{15} + \beta_{18} ) q^{27} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} - \beta_{9} - 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} + \beta_{18} ) q^{29} \) \( + ( 3 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{7} - \beta_{9} - \beta_{14} - 2 \beta_{15} + \beta_{16} - \beta_{18} ) q^{31} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{10} + 2 \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} ) q^{33} \) \( + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{12} + \beta_{15} - \beta_{16} ) q^{35} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{10} + \beta_{11} + 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{18} ) q^{37} \) \( + ( 2 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{39} \) \( + ( -1 + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{10} + 2 \beta_{12} + \beta_{15} - \beta_{16} - \beta_{18} ) q^{41} \) \( + ( 2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{43} \) \( + ( 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} - 2 \beta_{13} + \beta_{14} - \beta_{16} + \beta_{18} ) q^{45} \) \( + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{5} + \beta_{6} - 3 \beta_{7} + 2 \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{47} \) \( + ( 2 + \beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{10} + \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{49} \) \( + ( \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + 2 \beta_{12} - \beta_{14} + 2 \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{51} \) \( + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{17} - \beta_{18} ) q^{53} \) \( + ( 3 + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} + \beta_{16} ) q^{55} \) \( + ( -1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{8} + \beta_{11} + \beta_{12} - 3 \beta_{13} + \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{57} \) \( + ( 3 + 2 \beta_{3} - \beta_{5} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{59} \) \( + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{61} \) \( + ( 4 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{10} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{63} \) \( + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{8} - \beta_{11} - 2 \beta_{14} - \beta_{15} + \beta_{17} ) q^{65} \) \( + ( 2 + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{16} ) q^{67} \) \( + ( -2 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} + \beta_{18} ) q^{69} \) \( + ( 2 + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{8} - \beta_{9} + \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{71} \) \( + ( 1 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{15} + \beta_{16} ) q^{73} \) \( + ( -1 + 4 \beta_{1} - 4 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + 3 \beta_{13} + 3 \beta_{15} + \beta_{17} + \beta_{18} ) q^{75} \) \( + ( -2 - 2 \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{17} + \beta_{18} ) q^{77} \) \( + ( 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} + 3 \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{79} \) \( + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - 4 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} - \beta_{16} + 2 \beta_{18} ) q^{81} \) \( + ( 4 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - \beta_{13} - \beta_{14} - 3 \beta_{15} + 2 \beta_{16} ) q^{83} \) \( + ( -2 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} - \beta_{15} + 2 \beta_{16} - 3 \beta_{17} - \beta_{18} ) q^{85} \) \( + ( 3 + \beta_{1} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{87} \) \( + ( -1 + \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{17} ) q^{89} \) \( + ( 2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + 3 \beta_{17} ) q^{91} \) \( + ( -1 + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} - 3 \beta_{15} + \beta_{16} - \beta_{17} - 3 \beta_{18} ) q^{93} \) \( + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} - 3 \beta_{13} - \beta_{15} + 2 \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{95} \) \( + ( 3 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} - 4 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} - \beta_{15} - 2 \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{97} \) \( + ( 2 - 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{10} + \beta_{12} - 2 \beta_{13} - \beta_{16} - \beta_{18} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(19q \) \(\mathstrut +\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 8q^{5} \) \(\mathstrut +\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 21q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(19q \) \(\mathstrut +\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 8q^{5} \) \(\mathstrut +\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 21q^{9} \) \(\mathstrut +\mathstrut 15q^{11} \) \(\mathstrut -\mathstrut 8q^{13} \) \(\mathstrut +\mathstrut 17q^{15} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 14q^{19} \) \(\mathstrut -\mathstrut 9q^{21} \) \(\mathstrut +\mathstrut 28q^{23} \) \(\mathstrut +\mathstrut 25q^{25} \) \(\mathstrut +\mathstrut 21q^{27} \) \(\mathstrut -\mathstrut 13q^{29} \) \(\mathstrut +\mathstrut 20q^{31} \) \(\mathstrut -\mathstrut 6q^{33} \) \(\mathstrut +\mathstrut 32q^{35} \) \(\mathstrut -\mathstrut 16q^{37} \) \(\mathstrut +\mathstrut 27q^{39} \) \(\mathstrut +\mathstrut 2q^{41} \) \(\mathstrut +\mathstrut 28q^{43} \) \(\mathstrut -\mathstrut 29q^{45} \) \(\mathstrut +\mathstrut 37q^{47} \) \(\mathstrut +\mathstrut 36q^{49} \) \(\mathstrut +\mathstrut 35q^{51} \) \(\mathstrut -\mathstrut 37q^{53} \) \(\mathstrut +\mathstrut 24q^{55} \) \(\mathstrut -\mathstrut 11q^{57} \) \(\mathstrut +\mathstrut 32q^{59} \) \(\mathstrut -\mathstrut 7q^{61} \) \(\mathstrut +\mathstrut 45q^{63} \) \(\mathstrut +\mathstrut q^{65} \) \(\mathstrut +\mathstrut 45q^{67} \) \(\mathstrut -\mathstrut 12q^{69} \) \(\mathstrut +\mathstrut 49q^{71} \) \(\mathstrut +\mathstrut 16q^{73} \) \(\mathstrut +\mathstrut 35q^{75} \) \(\mathstrut -\mathstrut 40q^{77} \) \(\mathstrut +\mathstrut 33q^{79} \) \(\mathstrut +\mathstrut 15q^{81} \) \(\mathstrut +\mathstrut 43q^{83} \) \(\mathstrut -\mathstrut 28q^{85} \) \(\mathstrut +\mathstrut 48q^{87} \) \(\mathstrut +\mathstrut 3q^{89} \) \(\mathstrut +\mathstrut 56q^{91} \) \(\mathstrut -\mathstrut 48q^{93} \) \(\mathstrut +\mathstrut 43q^{95} \) \(\mathstrut +\mathstrut 8q^{97} \) \(\mathstrut +\mathstrut 74q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19}\mathstrut -\mathstrut \) \(6\) \(x^{18}\mathstrut -\mathstrut \) \(21\) \(x^{17}\mathstrut +\mathstrut \) \(179\) \(x^{16}\mathstrut +\mathstrut \) \(90\) \(x^{15}\mathstrut -\mathstrut \) \(2109\) \(x^{14}\mathstrut +\mathstrut \) \(926\) \(x^{13}\mathstrut +\mathstrut \) \(12681\) \(x^{12}\mathstrut -\mathstrut \) \(10845\) \(x^{11}\mathstrut -\mathstrut \) \(41921\) \(x^{10}\mathstrut +\mathstrut \) \(43551\) \(x^{9}\mathstrut +\mathstrut \) \(76260\) \(x^{8}\mathstrut -\mathstrut \) \(80907\) \(x^{7}\mathstrut -\mathstrut \) \(72526\) \(x^{6}\mathstrut +\mathstrut \) \(64793\) \(x^{5}\mathstrut +\mathstrut \) \(33209\) \(x^{4}\mathstrut -\mathstrut \) \(13777\) \(x^{3}\mathstrut -\mathstrut \) \(7607\) \(x^{2}\mathstrut -\mathstrut \) \(747\) \(x\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\(196521312495034\) \(\nu^{18}\mathstrut -\mathstrut \) \(1441579400369817\) \(\nu^{17}\mathstrut -\mathstrut \) \(2924913574354401\) \(\nu^{16}\mathstrut +\mathstrut \) \(41912041347671166\) \(\nu^{15}\mathstrut -\mathstrut \) \(17818217143386024\) \(\nu^{14}\mathstrut -\mathstrut \) \(475669164938282307\) \(\nu^{13}\mathstrut +\mathstrut \) \(595092550551867088\) \(\nu^{12}\mathstrut +\mathstrut \) \(2706294034633503240\) \(\nu^{11}\mathstrut -\mathstrut \) \(4580166278975943512\) \(\nu^{10}\mathstrut -\mathstrut \) \(8221073571569711829\) \(\nu^{9}\mathstrut +\mathstrut \) \(16553976937434110332\) \(\nu^{8}\mathstrut +\mathstrut \) \(13024341630912266379\) \(\nu^{7}\mathstrut -\mathstrut \) \(30398145885453740584\) \(\nu^{6}\mathstrut -\mathstrut \) \(9681015167402184673\) \(\nu^{5}\mathstrut +\mathstrut \) \(26640229615794321610\) \(\nu^{4}\mathstrut +\mathstrut \) \(2970308817404871287\) \(\nu^{3}\mathstrut -\mathstrut \) \(8517014376848128207\) \(\nu^{2}\mathstrut -\mathstrut \) \(1027269288097940330\) \(\nu\mathstrut +\mathstrut \) \(166591551288078888\)\()/\)\(77904810267289528\)
\(\beta_{4}\)\(=\)\((\)\(209544026239557\) \(\nu^{18}\mathstrut -\mathstrut \) \(1435803919056848\) \(\nu^{17}\mathstrut -\mathstrut \) \(3871628649572152\) \(\nu^{16}\mathstrut +\mathstrut \) \(43016014683542135\) \(\nu^{15}\mathstrut +\mathstrut \) \(3243544109696637\) \(\nu^{14}\mathstrut -\mathstrut \) \(509405238497308593\) \(\nu^{13}\mathstrut +\mathstrut \) \(372341170757747395\) \(\nu^{12}\mathstrut +\mathstrut \) \(3079098801819725010\) \(\nu^{11}\mathstrut -\mathstrut \) \(3266269544593699975\) \(\nu^{10}\mathstrut -\mathstrut \) \(10210653079904899581\) \(\nu^{9}\mathstrut +\mathstrut \) \(11898633774999893396\) \(\nu^{8}\mathstrut +\mathstrut \) \(18475628716806910011\) \(\nu^{7}\mathstrut -\mathstrut \) \(20363135569951304978\) \(\nu^{6}\mathstrut -\mathstrut \) \(17067609862621199939\) \(\nu^{5}\mathstrut +\mathstrut \) \(14446277403237981944\) \(\nu^{4}\mathstrut +\mathstrut \) \(7316444036690675014\) \(\nu^{3}\mathstrut -\mathstrut \) \(2070932208707001660\) \(\nu^{2}\mathstrut -\mathstrut \) \(1800728050905473725\) \(\nu\mathstrut -\mathstrut \) \(242077884568730940\)\()/\)\(77904810267289528\)
\(\beta_{5}\)\(=\)\((\)\(325315462336008\) \(\nu^{18}\mathstrut -\mathstrut \) \(1976289568186404\) \(\nu^{17}\mathstrut -\mathstrut \) \(6464820675496617\) \(\nu^{16}\mathstrut +\mathstrut \) \(57670999367330371\) \(\nu^{15}\mathstrut +\mathstrut \) \(19676691355848852\) \(\nu^{14}\mathstrut -\mathstrut \) \(658008988812589996\) \(\nu^{13}\mathstrut +\mathstrut \) \(394842138098307975\) \(\nu^{12}\mathstrut +\mathstrut \) \(3769404406780378010\) \(\nu^{11}\mathstrut -\mathstrut \) \(3952782696839577594\) \(\nu^{10}\mathstrut -\mathstrut \) \(11513787143158348190\) \(\nu^{9}\mathstrut +\mathstrut \) \(15060880540333249297\) \(\nu^{8}\mathstrut +\mathstrut \) \(18022917578774451206\) \(\nu^{7}\mathstrut -\mathstrut \) \(26919821315765788637\) \(\nu^{6}\mathstrut -\mathstrut \) \(11837322142659575168\) \(\nu^{5}\mathstrut +\mathstrut \) \(20551698355449594859\) \(\nu^{4}\mathstrut +\mathstrut \) \(912217879033491144\) \(\nu^{3}\mathstrut -\mathstrut \) \(3803526816531730763\) \(\nu^{2}\mathstrut -\mathstrut \) \(12176117032348777\) \(\nu\mathstrut -\mathstrut \) \(47302041294770420\)\()/\)\(77904810267289528\)
\(\beta_{6}\)\(=\)\((\)\(346551847676251\) \(\nu^{18}\mathstrut -\mathstrut \) \(1668795238793348\) \(\nu^{17}\mathstrut -\mathstrut \) \(9554492206151678\) \(\nu^{16}\mathstrut +\mathstrut \) \(52690311990224015\) \(\nu^{15}\mathstrut +\mathstrut \) \(99268216390112767\) \(\nu^{14}\mathstrut -\mathstrut \) \(672831196084720603\) \(\nu^{13}\mathstrut -\mathstrut \) \(480330669882914705\) \(\nu^{12}\mathstrut +\mathstrut \) \(4531249610863146126\) \(\nu^{11}\mathstrut +\mathstrut \) \(1011790432893158971\) \(\nu^{10}\mathstrut -\mathstrut \) \(17538233581202119839\) \(\nu^{9}\mathstrut -\mathstrut \) \(194355797236659890\) \(\nu^{8}\mathstrut +\mathstrut \) \(39545581195445075145\) \(\nu^{7}\mathstrut -\mathstrut \) \(2411607128241522824\) \(\nu^{6}\mathstrut -\mathstrut \) \(49510080137058458545\) \(\nu^{5}\mathstrut +\mathstrut \) \(2855882369841072346\) \(\nu^{4}\mathstrut +\mathstrut \) \(29568840637183398650\) \(\nu^{3}\mathstrut -\mathstrut \) \(308192843555754662\) \(\nu^{2}\mathstrut -\mathstrut \) \(5219928908967227585\) \(\nu\mathstrut -\mathstrut \) \(498446106796647004\)\()/\)\(77904810267289528\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(498188113457669\) \(\nu^{18}\mathstrut +\mathstrut \) \(2523550044875629\) \(\nu^{17}\mathstrut +\mathstrut \) \(12303917053161873\) \(\nu^{16}\mathstrut -\mathstrut \) \(76188368510367967\) \(\nu^{15}\mathstrut -\mathstrut \) \(98542977570888683\) \(\nu^{14}\mathstrut +\mathstrut \) \(909421304377374380\) \(\nu^{13}\mathstrut +\mathstrut \) \(150593332231996867\) \(\nu^{12}\mathstrut -\mathstrut \) \(5531017263463239106\) \(\nu^{11}\mathstrut +\mathstrut \) \(1903876238659999871\) \(\nu^{10}\mathstrut +\mathstrut \) \(18353616170262585332\) \(\nu^{9}\mathstrut -\mathstrut \) \(11013728035806195900\) \(\nu^{8}\mathstrut -\mathstrut \) \(32825601884278571158\) \(\nu^{7}\mathstrut +\mathstrut \) \(23418204013206537144\) \(\nu^{6}\mathstrut +\mathstrut \) \(29170695205661226350\) \(\nu^{5}\mathstrut -\mathstrut \) \(20413894654710609824\) \(\nu^{4}\mathstrut -\mathstrut \) \(11312765533213992939\) \(\nu^{3}\mathstrut +\mathstrut \) \(4920039105746828385\) \(\nu^{2}\mathstrut +\mathstrut \) \(2667131240989337419\) \(\nu\mathstrut -\mathstrut \) \(4071618367057092\)\()/\)\(77904810267289528\)
\(\beta_{8}\)\(=\)\((\)\(1007706538158390\) \(\nu^{18}\mathstrut -\mathstrut \) \(6067612651914337\) \(\nu^{17}\mathstrut -\mathstrut \) \(19728860249241516\) \(\nu^{16}\mathstrut +\mathstrut \) \(175301510143739749\) \(\nu^{15}\mathstrut +\mathstrut \) \(51557910709593326\) \(\nu^{14}\mathstrut -\mathstrut \) \(1963532947901539733\) \(\nu^{13}\mathstrut +\mathstrut \) \(1333989630839258891\) \(\nu^{12}\mathstrut +\mathstrut \) \(10860911091044278840\) \(\nu^{11}\mathstrut -\mathstrut \) \(12827190398504637216\) \(\nu^{10}\mathstrut -\mathstrut \) \(30953673001625573677\) \(\nu^{9}\mathstrut +\mathstrut \) \(47778169234604056599\) \(\nu^{8}\mathstrut +\mathstrut \) \(41554386482378362657\) \(\nu^{7}\mathstrut -\mathstrut \) \(82552732261583313803\) \(\nu^{6}\mathstrut -\mathstrut \) \(16139376886382327193\) \(\nu^{5}\mathstrut +\mathstrut \) \(58885576401540912769\) \(\nu^{4}\mathstrut -\mathstrut \) \(7789723814283011947\) \(\nu^{3}\mathstrut -\mathstrut \) \(8788558770665667400\) \(\nu^{2}\mathstrut +\mathstrut \) \(701698033358479469\) \(\nu\mathstrut +\mathstrut \) \(10329328802699356\)\()/\)\(77904810267289528\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(1027594418075442\) \(\nu^{18}\mathstrut +\mathstrut \) \(6762123352898153\) \(\nu^{17}\mathstrut +\mathstrut \) \(17708158431050788\) \(\nu^{16}\mathstrut -\mathstrut \) \(195121779192913155\) \(\nu^{15}\mathstrut +\mathstrut \) \(22140646155625798\) \(\nu^{14}\mathstrut +\mathstrut \) \(2178765600486528667\) \(\nu^{13}\mathstrut -\mathstrut \) \(2280762022625764797\) \(\nu^{12}\mathstrut -\mathstrut \) \(11959308405924989292\) \(\nu^{11}\mathstrut +\mathstrut \) \(18873883305931959538\) \(\nu^{10}\mathstrut +\mathstrut \) \(33399622787542986827\) \(\nu^{9}\mathstrut -\mathstrut \) \(68537331180212418241\) \(\nu^{8}\mathstrut -\mathstrut \) \(41967676115055920471\) \(\nu^{7}\mathstrut +\mathstrut \) \(119852409134664945067\) \(\nu^{6}\mathstrut +\mathstrut \) \(9494499845645210107\) \(\nu^{5}\mathstrut -\mathstrut \) \(88948367973291215153\) \(\nu^{4}\mathstrut +\mathstrut \) \(15105377852532462783\) \(\nu^{3}\mathstrut +\mathstrut \) \(14887804797132335270\) \(\nu^{2}\mathstrut -\mathstrut \) \(817380982643208987\) \(\nu\mathstrut -\mathstrut \) \(90972821278178156\)\()/\)\(77904810267289528\)
\(\beta_{10}\)\(=\)\((\)\(1083690820042657\) \(\nu^{18}\mathstrut -\mathstrut \) \(6973741260199556\) \(\nu^{17}\mathstrut -\mathstrut \) \(20471590021611702\) \(\nu^{16}\mathstrut +\mathstrut \) \(205820671089989371\) \(\nu^{15}\mathstrut +\mathstrut \) \(29617048668770011\) \(\nu^{14}\mathstrut -\mathstrut \) \(2387368572651253167\) \(\nu^{13}\mathstrut +\mathstrut \) \(1798484996667138455\) \(\nu^{12}\mathstrut +\mathstrut \) \(14027991543956850112\) \(\nu^{11}\mathstrut -\mathstrut \) \(16481112740348660517\) \(\nu^{10}\mathstrut -\mathstrut \) \(44788609308024169219\) \(\nu^{9}\mathstrut +\mathstrut \) \(62515975677943484674\) \(\nu^{8}\mathstrut +\mathstrut \) \(77126164322063243227\) \(\nu^{7}\mathstrut -\mathstrut \) \(114320497483603387644\) \(\nu^{6}\mathstrut -\mathstrut \) \(67030073201629306155\) \(\nu^{5}\mathstrut +\mathstrut \) \(92460718390550759524\) \(\nu^{4}\mathstrut +\mathstrut \) \(26968630910803904880\) \(\nu^{3}\mathstrut -\mathstrut \) \(21158349312845396762\) \(\nu^{2}\mathstrut -\mathstrut \) \(6540496659385454301\) \(\nu\mathstrut -\mathstrut \) \(355943311226414196\)\()/\)\(77904810267289528\)
\(\beta_{11}\)\(=\)\((\)\(1092816357613181\) \(\nu^{18}\mathstrut -\mathstrut \) \(7459601496024570\) \(\nu^{17}\mathstrut -\mathstrut \) \(18319998171235509\) \(\nu^{16}\mathstrut +\mathstrut \) \(216809473764291880\) \(\nu^{15}\mathstrut -\mathstrut \) \(38084178623578787\) \(\nu^{14}\mathstrut -\mathstrut \) \(2452948852753917143\) \(\nu^{13}\mathstrut +\mathstrut \) \(2587612671357407948\) \(\nu^{12}\mathstrut +\mathstrut \) \(13809479980337519780\) \(\nu^{11}\mathstrut -\mathstrut \) \(21008885863502095005\) \(\nu^{10}\mathstrut -\mathstrut \) \(40699585786793027837\) \(\nu^{9}\mathstrut +\mathstrut \) \(76099498049635194623\) \(\nu^{8}\mathstrut +\mathstrut \) \(58998155312108487979\) \(\nu^{7}\mathstrut -\mathstrut \) \(134698131321362958305\) \(\nu^{6}\mathstrut -\mathstrut \) \(31609719806534699185\) \(\nu^{5}\mathstrut +\mathstrut \) \(105275876872953735805\) \(\nu^{4}\mathstrut -\mathstrut \) \(1609513481442969668\) \(\nu^{3}\mathstrut -\mathstrut \) \(23331520877142669999\) \(\nu^{2}\mathstrut -\mathstrut \) \(1314890913361810684\) \(\nu\mathstrut +\mathstrut \) \(316411219035010592\)\()/\)\(77904810267289528\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(1134567769505457\) \(\nu^{18}\mathstrut +\mathstrut \) \(7266804211212438\) \(\nu^{17}\mathstrut +\mathstrut \) \(21427225666173599\) \(\nu^{16}\mathstrut -\mathstrut \) \(213680552404619654\) \(\nu^{15}\mathstrut -\mathstrut \) \(31574624618047029\) \(\nu^{14}\mathstrut +\mathstrut \) \(2462130918680185093\) \(\nu^{13}\mathstrut -\mathstrut \) \(1860936011562635530\) \(\nu^{12}\mathstrut -\mathstrut \) \(14289017657424015228\) \(\nu^{11}\mathstrut +\mathstrut \) \(16964033962802824291\) \(\nu^{10}\mathstrut +\mathstrut \) \(44531006101484114293\) \(\nu^{9}\mathstrut -\mathstrut \) \(63643830928625667821\) \(\nu^{8}\mathstrut -\mathstrut \) \(72903731379051850569\) \(\nu^{7}\mathstrut +\mathstrut \) \(114267928088836612117\) \(\nu^{6}\mathstrut +\mathstrut \) \(56413798233430700797\) \(\nu^{5}\mathstrut -\mathstrut \) \(90005221911718988253\) \(\nu^{4}\mathstrut -\mathstrut \) \(17474895021582544010\) \(\nu^{3}\mathstrut +\mathstrut \) \(20358214123365373471\) \(\nu^{2}\mathstrut +\mathstrut \) \(4862472104716683620\) \(\nu\mathstrut -\mathstrut \) \(12617454281865112\)\()/\)\(77904810267289528\)
\(\beta_{13}\)\(=\)\((\)\(1372031837712026\) \(\nu^{18}\mathstrut -\mathstrut \) \(8632704946027114\) \(\nu^{17}\mathstrut -\mathstrut \) \(26488322998174035\) \(\nu^{16}\mathstrut +\mathstrut \) \(254137595755781765\) \(\nu^{15}\mathstrut +\mathstrut \) \(54680083798284962\) \(\nu^{14}\mathstrut -\mathstrut \) \(2933166057645453758\) \(\nu^{13}\mathstrut +\mathstrut \) \(2068849535716815029\) \(\nu^{12}\mathstrut +\mathstrut \) \(17062702713254432978\) \(\nu^{11}\mathstrut -\mathstrut \) \(19537063246727999042\) \(\nu^{10}\mathstrut -\mathstrut \) \(53342262959495678406\) \(\nu^{9}\mathstrut +\mathstrut \) \(74236047532850061909\) \(\nu^{8}\mathstrut +\mathstrut \) \(87637830612841723000\) \(\nu^{7}\mathstrut -\mathstrut \) \(134126430474056569771\) \(\nu^{6}\mathstrut -\mathstrut \) \(67877444561900174468\) \(\nu^{5}\mathstrut +\mathstrut \) \(104965677147146286421\) \(\nu^{4}\mathstrut +\mathstrut \) \(20562863803007876548\) \(\nu^{3}\mathstrut -\mathstrut \) \(21757429321116481975\) \(\nu^{2}\mathstrut -\mathstrut \) \(5320072813767842773\) \(\nu\mathstrut -\mathstrut \) \(364829756993871316\)\()/\)\(77904810267289528\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(1426213381500197\) \(\nu^{18}\mathstrut +\mathstrut \) \(8799653434478100\) \(\nu^{17}\mathstrut +\mathstrut \) \(28760184662207652\) \(\nu^{16}\mathstrut -\mathstrut \) \(261180023056501289\) \(\nu^{15}\mathstrut -\mathstrut \) \(93169367682922185\) \(\nu^{14}\mathstrut +\mathstrut \) \(3052987536609974455\) \(\nu^{13}\mathstrut -\mathstrut \) \(1727735731011966637\) \(\nu^{12}\mathstrut -\mathstrut \) \(18126754975236576178\) \(\nu^{11}\mathstrut +\mathstrut \) \(17826983466581622477\) \(\nu^{10}\mathstrut +\mathstrut \) \(58672937274694536375\) \(\nu^{9}\mathstrut -\mathstrut \) \(69370865243696314662\) \(\nu^{8}\mathstrut -\mathstrut \) \(102777671504280390989\) \(\nu^{7}\mathstrut +\mathstrut \) \(126833847987152230770\) \(\nu^{6}\mathstrut +\mathstrut \) \(90812240101347596701\) \(\nu^{5}\mathstrut -\mathstrut \) \(100607382747545578412\) \(\nu^{4}\mathstrut -\mathstrut \) \(36001730803095990114\) \(\nu^{3}\mathstrut +\mathstrut \) \(22114795187908175950\) \(\nu^{2}\mathstrut +\mathstrut \) \(7656742856427456895\) \(\nu\mathstrut +\mathstrut \) \(266086065397816924\)\()/\)\(77904810267289528\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(741068081655316\) \(\nu^{18}\mathstrut +\mathstrut \) \(4595827585193675\) \(\nu^{17}\mathstrut +\mathstrut \) \(14706488707551162\) \(\nu^{16}\mathstrut -\mathstrut \) \(135847265665372052\) \(\nu^{15}\mathstrut -\mathstrut \) \(41251315050697064\) \(\nu^{14}\mathstrut +\mathstrut \) \(1577439360532800966\) \(\nu^{13}\mathstrut -\mathstrut \) \(983893018901486730\) \(\nu^{12}\mathstrut -\mathstrut \) \(9260428297436121382\) \(\nu^{11}\mathstrut +\mathstrut \) \(9800513948355364761\) \(\nu^{10}\mathstrut +\mathstrut \) \(29356222080806208834\) \(\nu^{9}\mathstrut -\mathstrut \) \(37923471534717328802\) \(\nu^{8}\mathstrut -\mathstrut \) \(49289123370662958684\) \(\nu^{7}\mathstrut +\mathstrut \) \(69525859366441445447\) \(\nu^{6}\mathstrut +\mathstrut \) \(39555619296920917818\) \(\nu^{5}\mathstrut -\mathstrut \) \(55645495433960070451\) \(\nu^{4}\mathstrut -\mathstrut \) \(12687570570163284801\) \(\nu^{3}\mathstrut +\mathstrut \) \(12445583835862380927\) \(\nu^{2}\mathstrut +\mathstrut \) \(3058107440142454636\) \(\nu\mathstrut -\mathstrut \) \(18145595185768772\)\()/\)\(38952405133644764\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(1788992130708019\) \(\nu^{18}\mathstrut +\mathstrut \) \(11616156032253848\) \(\nu^{17}\mathstrut +\mathstrut \) \(32536276431610078\) \(\nu^{16}\mathstrut -\mathstrut \) \(339399271749954761\) \(\nu^{15}\mathstrut -\mathstrut \) \(11582361937006627\) \(\nu^{14}\mathstrut +\mathstrut \) \(3869531900522449111\) \(\nu^{13}\mathstrut -\mathstrut \) \(3397394588597605237\) \(\nu^{12}\mathstrut -\mathstrut \) \(22042316492135969886\) \(\nu^{11}\mathstrut +\mathstrut \) \(29610043641973866985\) \(\nu^{10}\mathstrut +\mathstrut \) \(66273975827091356685\) \(\nu^{9}\mathstrut -\mathstrut \) \(109891497445168112760\) \(\nu^{8}\mathstrut -\mathstrut \) \(100149415988085444129\) \(\nu^{7}\mathstrut +\mathstrut \) \(196461996303947941470\) \(\nu^{6}\mathstrut +\mathstrut \) \(61210305129251803963\) \(\nu^{5}\mathstrut -\mathstrut \) \(153155685352928992056\) \(\nu^{4}\mathstrut -\mathstrut \) \(4144258394197102016\) \(\nu^{3}\mathstrut +\mathstrut \) \(32758860467229390110\) \(\nu^{2}\mathstrut +\mathstrut \) \(1617997536471123617\) \(\nu\mathstrut -\mathstrut \) \(577666401596282284\)\()/\)\(77904810267289528\)
\(\beta_{17}\)\(=\)\((\)\(-\)\(2405904627279449\) \(\nu^{18}\mathstrut +\mathstrut \) \(15658637998627299\) \(\nu^{17}\mathstrut +\mathstrut \) \(44007385279792924\) \(\nu^{16}\mathstrut -\mathstrut \) \(458604356413412660\) \(\nu^{15}\mathstrut -\mathstrut \) \(24696620256853403\) \(\nu^{14}\mathstrut +\mathstrut \) \(5253377279220361316\) \(\nu^{13}\mathstrut -\mathstrut \) \(4439755889786174314\) \(\nu^{12}\mathstrut -\mathstrut \) \(30218043560971088396\) \(\nu^{11}\mathstrut +\mathstrut \) \(38921551280019068667\) \(\nu^{10}\mathstrut +\mathstrut \) \(92817108697767086806\) \(\nu^{9}\mathstrut -\mathstrut \) \(144639030879757924651\) \(\nu^{8}\mathstrut -\mathstrut \) \(147891536233421275612\) \(\nu^{7}\mathstrut +\mathstrut \) \(259387225697679758869\) \(\nu^{6}\mathstrut +\mathstrut \) \(107332204391096892346\) \(\nu^{5}\mathstrut -\mathstrut \) \(204811156944413783707\) \(\nu^{4}\mathstrut -\mathstrut \) \(26845110033970579257\) \(\nu^{3}\mathstrut +\mathstrut \) \(46303300320837429364\) \(\nu^{2}\mathstrut +\mathstrut \) \(6903621981002853558\) \(\nu\mathstrut -\mathstrut \) \(440523358723489168\)\()/\)\(77904810267289528\)
\(\beta_{18}\)\(=\)\((\)\(2792699668717361\) \(\nu^{18}\mathstrut -\mathstrut \) \(16416910358581216\) \(\nu^{17}\mathstrut -\mathstrut \) \(59062739523594601\) \(\nu^{16}\mathstrut +\mathstrut \) \(486780419195914490\) \(\nu^{15}\mathstrut +\mathstrut \) \(264505957024420359\) \(\nu^{14}\mathstrut -\mathstrut \) \(5678792202416026997\) \(\nu^{13}\mathstrut +\mathstrut \) \(2420943149231987866\) \(\nu^{12}\mathstrut +\mathstrut \) \(33581661759727533114\) \(\nu^{11}\mathstrut -\mathstrut \) \(29233897625988081233\) \(\nu^{10}\mathstrut -\mathstrut \) \(107791343874420349483\) \(\nu^{9}\mathstrut +\mathstrut \) \(117984898012449172935\) \(\nu^{8}\mathstrut +\mathstrut \) \(185420740613861372835\) \(\nu^{7}\mathstrut -\mathstrut \) \(219246910241231539663\) \(\nu^{6}\mathstrut -\mathstrut \) \(157164230605325737913\) \(\nu^{5}\mathstrut +\mathstrut \) \(175208861191622404069\) \(\nu^{4}\mathstrut +\mathstrut \) \(57037782759786383952\) \(\nu^{3}\mathstrut -\mathstrut \) \(37801683907989238741\) \(\nu^{2}\mathstrut -\mathstrut \) \(12531931956461062566\) \(\nu\mathstrut -\mathstrut \) \(498882579668991336\)\()/\)\(77904810267289528\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{18}\mathstrut +\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{18}\mathstrut -\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(2\) \(\beta_{15}\mathstrut +\mathstrut \) \(3\) \(\beta_{14}\mathstrut -\mathstrut \) \(4\) \(\beta_{13}\mathstrut -\mathstrut \) \(3\) \(\beta_{12}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(30\)
\(\nu^{5}\)\(=\)\(14\) \(\beta_{18}\mathstrut -\mathstrut \) \(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{16}\mathstrut +\mathstrut \) \(12\) \(\beta_{15}\mathstrut +\mathstrut \) \(14\) \(\beta_{14}\mathstrut -\mathstrut \) \(2\) \(\beta_{13}\mathstrut -\mathstrut \) \(10\) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(12\) \(\beta_{10}\mathstrut -\mathstrut \) \(12\) \(\beta_{9}\mathstrut -\mathstrut \) \(12\) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(54\) \(\beta_{1}\mathstrut +\mathstrut \) \(26\)
\(\nu^{6}\)\(=\)\(32\) \(\beta_{18}\mathstrut -\mathstrut \) \(\beta_{17}\mathstrut -\mathstrut \) \(16\) \(\beta_{16}\mathstrut -\mathstrut \) \(30\) \(\beta_{15}\mathstrut +\mathstrut \) \(47\) \(\beta_{14}\mathstrut -\mathstrut \) \(64\) \(\beta_{13}\mathstrut -\mathstrut \) \(40\) \(\beta_{12}\mathstrut -\mathstrut \) \(33\) \(\beta_{11}\mathstrut -\mathstrut \) \(9\) \(\beta_{10}\mathstrut -\mathstrut \) \(28\) \(\beta_{9}\mathstrut -\mathstrut \) \(13\) \(\beta_{8}\mathstrut +\mathstrut \) \(59\) \(\beta_{7}\mathstrut +\mathstrut \) \(24\) \(\beta_{6}\mathstrut -\mathstrut \) \(29\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(15\) \(\beta_{3}\mathstrut +\mathstrut \) \(114\) \(\beta_{2}\mathstrut -\mathstrut \) \(23\) \(\beta_{1}\mathstrut +\mathstrut \) \(259\)
\(\nu^{7}\)\(=\)\(166\) \(\beta_{18}\mathstrut -\mathstrut \) \(11\) \(\beta_{17}\mathstrut -\mathstrut \) \(20\) \(\beta_{16}\mathstrut +\mathstrut \) \(120\) \(\beta_{15}\mathstrut +\mathstrut \) \(164\) \(\beta_{14}\mathstrut -\mathstrut \) \(42\) \(\beta_{13}\mathstrut -\mathstrut \) \(96\) \(\beta_{12}\mathstrut -\mathstrut \) \(21\) \(\beta_{11}\mathstrut -\mathstrut \) \(123\) \(\beta_{10}\mathstrut -\mathstrut \) \(127\) \(\beta_{9}\mathstrut -\mathstrut \) \(124\) \(\beta_{8}\mathstrut +\mathstrut \) \(171\) \(\beta_{7}\mathstrut +\mathstrut \) \(123\) \(\beta_{6}\mathstrut -\mathstrut \) \(41\) \(\beta_{5}\mathstrut +\mathstrut \) \(57\) \(\beta_{4}\mathstrut +\mathstrut \) \(149\) \(\beta_{3}\mathstrut +\mathstrut \) \(34\) \(\beta_{2}\mathstrut +\mathstrut \) \(444\) \(\beta_{1}\mathstrut +\mathstrut \) \(293\)
\(\nu^{8}\)\(=\)\(406\) \(\beta_{18}\mathstrut -\mathstrut \) \(20\) \(\beta_{17}\mathstrut -\mathstrut \) \(197\) \(\beta_{16}\mathstrut -\mathstrut \) \(358\) \(\beta_{15}\mathstrut +\mathstrut \) \(568\) \(\beta_{14}\mathstrut -\mathstrut \) \(810\) \(\beta_{13}\mathstrut -\mathstrut \) \(436\) \(\beta_{12}\mathstrut -\mathstrut \) \(423\) \(\beta_{11}\mathstrut -\mathstrut \) \(60\) \(\beta_{10}\mathstrut -\mathstrut \) \(319\) \(\beta_{9}\mathstrut -\mathstrut \) \(139\) \(\beta_{8}\mathstrut +\mathstrut \) \(703\) \(\beta_{7}\mathstrut +\mathstrut \) \(223\) \(\beta_{6}\mathstrut -\mathstrut \) \(339\) \(\beta_{5}\mathstrut +\mathstrut \) \(230\) \(\beta_{4}\mathstrut +\mathstrut \) \(185\) \(\beta_{3}\mathstrut +\mathstrut \) \(1167\) \(\beta_{2}\mathstrut -\mathstrut \) \(341\) \(\beta_{1}\mathstrut +\mathstrut \) \(2390\)
\(\nu^{9}\)\(=\)\(1866\) \(\beta_{18}\mathstrut -\mathstrut \) \(79\) \(\beta_{17}\mathstrut -\mathstrut \) \(301\) \(\beta_{16}\mathstrut +\mathstrut \) \(1162\) \(\beta_{15}\mathstrut +\mathstrut \) \(1811\) \(\beta_{14}\mathstrut -\mathstrut \) \(646\) \(\beta_{13}\mathstrut -\mathstrut \) \(941\) \(\beta_{12}\mathstrut -\mathstrut \) \(315\) \(\beta_{11}\mathstrut -\mathstrut \) \(1206\) \(\beta_{10}\mathstrut -\mathstrut \) \(1311\) \(\beta_{9}\mathstrut -\mathstrut \) \(1243\) \(\beta_{8}\mathstrut +\mathstrut \) \(1973\) \(\beta_{7}\mathstrut +\mathstrut \) \(1215\) \(\beta_{6}\mathstrut -\mathstrut \) \(583\) \(\beta_{5}\mathstrut +\mathstrut \) \(807\) \(\beta_{4}\mathstrut +\mathstrut \) \(1644\) \(\beta_{3}\mathstrut +\mathstrut \) \(449\) \(\beta_{2}\mathstrut +\mathstrut \) \(3798\) \(\beta_{1}\mathstrut +\mathstrut \) \(3163\)
\(\nu^{10}\)\(=\)\(4762\) \(\beta_{18}\mathstrut -\mathstrut \) \(277\) \(\beta_{17}\mathstrut -\mathstrut \) \(2224\) \(\beta_{16}\mathstrut -\mathstrut \) \(3940\) \(\beta_{15}\mathstrut +\mathstrut \) \(6298\) \(\beta_{14}\mathstrut -\mathstrut \) \(9446\) \(\beta_{13}\mathstrut -\mathstrut \) \(4498\) \(\beta_{12}\mathstrut -\mathstrut \) \(4928\) \(\beta_{11}\mathstrut -\mathstrut \) \(300\) \(\beta_{10}\mathstrut -\mathstrut \) \(3441\) \(\beta_{9}\mathstrut -\mathstrut \) \(1427\) \(\beta_{8}\mathstrut +\mathstrut \) \(7849\) \(\beta_{7}\mathstrut +\mathstrut \) \(1901\) \(\beta_{6}\mathstrut -\mathstrut \) \(3719\) \(\beta_{5}\mathstrut +\mathstrut \) \(3387\) \(\beta_{4}\mathstrut +\mathstrut \) \(2136\) \(\beta_{3}\mathstrut +\mathstrut \) \(11916\) \(\beta_{2}\mathstrut -\mathstrut \) \(4299\) \(\beta_{1}\mathstrut +\mathstrut \) \(22878\)
\(\nu^{11}\)\(=\)\(20481\) \(\beta_{18}\mathstrut -\mathstrut \) \(392\) \(\beta_{17}\mathstrut -\mathstrut \) \(3957\) \(\beta_{16}\mathstrut +\mathstrut \) \(11201\) \(\beta_{15}\mathstrut +\mathstrut \) \(19469\) \(\beta_{14}\mathstrut -\mathstrut \) \(8838\) \(\beta_{13}\mathstrut -\mathstrut \) \(9396\) \(\beta_{12}\mathstrut -\mathstrut \) \(4154\) \(\beta_{11}\mathstrut -\mathstrut \) \(11610\) \(\beta_{10}\mathstrut -\mathstrut \) \(13476\) \(\beta_{9}\mathstrut -\mathstrut \) \(12397\) \(\beta_{8}\mathstrut +\mathstrut \) \(22055\) \(\beta_{7}\mathstrut +\mathstrut \) \(11820\) \(\beta_{6}\mathstrut -\mathstrut \) \(7246\) \(\beta_{5}\mathstrut +\mathstrut \) \(10246\) \(\beta_{4}\mathstrut +\mathstrut \) \(17718\) \(\beta_{3}\mathstrut +\mathstrut \) \(5541\) \(\beta_{2}\mathstrut +\mathstrut \) \(33286\) \(\beta_{1}\mathstrut +\mathstrut \) \(33634\)
\(\nu^{12}\)\(=\)\(53886\) \(\beta_{18}\mathstrut -\mathstrut \) \(3302\) \(\beta_{17}\mathstrut -\mathstrut \) \(24225\) \(\beta_{16}\mathstrut -\mathstrut \) \(41620\) \(\beta_{15}\mathstrut +\mathstrut \) \(67437\) \(\beta_{14}\mathstrut -\mathstrut \) \(105894\) \(\beta_{13}\mathstrut -\mathstrut \) \(45613\) \(\beta_{12}\mathstrut -\mathstrut \) \(54679\) \(\beta_{11}\mathstrut -\mathstrut \) \(388\) \(\beta_{10}\mathstrut -\mathstrut \) \(36426\) \(\beta_{9}\mathstrut -\mathstrut \) \(14628\) \(\beta_{8}\mathstrut +\mathstrut \) \(85296\) \(\beta_{7}\mathstrut +\mathstrut \) \(15588\) \(\beta_{6}\mathstrut -\mathstrut \) \(39822\) \(\beta_{5}\mathstrut +\mathstrut \) \(43424\) \(\beta_{4}\mathstrut +\mathstrut \) \(23880\) \(\beta_{3}\mathstrut +\mathstrut \) \(121645\) \(\beta_{2}\mathstrut -\mathstrut \) \(50134\) \(\beta_{1}\mathstrut +\mathstrut \) \(223931\)
\(\nu^{13}\)\(=\)\(222039\) \(\beta_{18}\mathstrut -\mathstrut \) \(343\) \(\beta_{17}\mathstrut -\mathstrut \) \(48163\) \(\beta_{16}\mathstrut +\mathstrut \) \(108145\) \(\beta_{15}\mathstrut +\mathstrut \) \(206563\) \(\beta_{14}\mathstrut -\mathstrut \) \(113512\) \(\beta_{13}\mathstrut -\mathstrut \) \(95038\) \(\beta_{12}\mathstrut -\mathstrut \) \(51238\) \(\beta_{11}\mathstrut -\mathstrut \) \(110758\) \(\beta_{10}\mathstrut -\mathstrut \) \(138617\) \(\beta_{9}\mathstrut -\mathstrut \) \(123924\) \(\beta_{8}\mathstrut +\mathstrut \) \(241896\) \(\beta_{7}\mathstrut +\mathstrut \) \(113917\) \(\beta_{6}\mathstrut -\mathstrut \) \(84495\) \(\beta_{5}\mathstrut +\mathstrut \) \(123217\) \(\beta_{4}\mathstrut +\mathstrut \) \(187688\) \(\beta_{3}\mathstrut +\mathstrut \) \(66692\) \(\beta_{2}\mathstrut +\mathstrut \) \(296007\) \(\beta_{1}\mathstrut +\mathstrut \) \(355806\)
\(\nu^{14}\)\(=\)\(598270\) \(\beta_{18}\mathstrut -\mathstrut \) \(36416\) \(\beta_{17}\mathstrut -\mathstrut \) \(259618\) \(\beta_{16}\mathstrut -\mathstrut \) \(428934\) \(\beta_{15}\mathstrut +\mathstrut \) \(711545\) \(\beta_{14}\mathstrut -\mathstrut \) \(1160924\) \(\beta_{13}\mathstrut -\mathstrut \) \(460874\) \(\beta_{12}\mathstrut -\mathstrut \) \(589931\) \(\beta_{11}\mathstrut +\mathstrut \) \(18141\) \(\beta_{10}\mathstrut -\mathstrut \) \(382725\) \(\beta_{9}\mathstrut -\mathstrut \) \(151458\) \(\beta_{8}\mathstrut +\mathstrut \) \(914878\) \(\beta_{7}\mathstrut +\mathstrut \) \(124921\) \(\beta_{6}\mathstrut -\mathstrut \) \(421960\) \(\beta_{5}\mathstrut +\mathstrut \) \(519212\) \(\beta_{4}\mathstrut +\mathstrut \) \(261957\) \(\beta_{3}\mathstrut +\mathstrut \) \(1242366\) \(\beta_{2}\mathstrut -\mathstrut \) \(560071\) \(\beta_{1}\mathstrut +\mathstrut \) \(2224383\)
\(\nu^{15}\)\(=\)\(2390667\) \(\beta_{18}\mathstrut +\mathstrut \) \(24909\) \(\beta_{17}\mathstrut -\mathstrut \) \(559761\) \(\beta_{16}\mathstrut +\mathstrut \) \(1047245\) \(\beta_{15}\mathstrut +\mathstrut \) \(2178127\) \(\beta_{14}\mathstrut -\mathstrut \) \(1399172\) \(\beta_{13}\mathstrut -\mathstrut \) \(970862\) \(\beta_{12}\mathstrut -\mathstrut \) \(607311\) \(\beta_{11}\mathstrut -\mathstrut \) \(1051467\) \(\beta_{10}\mathstrut -\mathstrut \) \(1428413\) \(\beta_{9}\mathstrut -\mathstrut \) \(1244605\) \(\beta_{8}\mathstrut +\mathstrut \) \(2622420\) \(\beta_{7}\mathstrut +\mathstrut \) \(1090639\) \(\beta_{6}\mathstrut -\mathstrut \) \(951943\) \(\beta_{5}\mathstrut +\mathstrut \) \(1435219\) \(\beta_{4}\mathstrut +\mathstrut \) \(1963587\) \(\beta_{3}\mathstrut +\mathstrut \) \(790976\) \(\beta_{2}\mathstrut +\mathstrut \) \(2654266\) \(\beta_{1}\mathstrut +\mathstrut \) \(3759565\)
\(\nu^{16}\)\(=\)\(6568957\) \(\beta_{18}\mathstrut -\mathstrut \) \(383804\) \(\beta_{17}\mathstrut -\mathstrut \) \(2760886\) \(\beta_{16}\mathstrut -\mathstrut \) \(4349501\) \(\beta_{15}\mathstrut +\mathstrut \) \(7464710\) \(\beta_{14}\mathstrut -\mathstrut \) \(12553596\) \(\beta_{13}\mathstrut -\mathstrut \) \(4665496\) \(\beta_{12}\mathstrut -\mathstrut \) \(6258198\) \(\beta_{11}\mathstrut +\mathstrut \) \(366287\) \(\beta_{10}\mathstrut -\mathstrut \) \(4008652\) \(\beta_{9}\mathstrut -\mathstrut \) \(1587514\) \(\beta_{8}\mathstrut +\mathstrut \) \(9746604\) \(\beta_{7}\mathstrut +\mathstrut \) \(981564\) \(\beta_{6}\mathstrut -\mathstrut \) \(4449797\) \(\beta_{5}\mathstrut +\mathstrut \) \(5966974\) \(\beta_{4}\mathstrut +\mathstrut \) \(2839020\) \(\beta_{3}\mathstrut +\mathstrut \) \(12697934\) \(\beta_{2}\mathstrut -\mathstrut \) \(6097556\) \(\beta_{1}\mathstrut +\mathstrut \) \(22327897\)
\(\nu^{17}\)\(=\)\(25636785\) \(\beta_{18}\mathstrut +\mathstrut \) \(460550\) \(\beta_{17}\mathstrut -\mathstrut \) \(6322641\) \(\beta_{16}\mathstrut +\mathstrut \) \(10172705\) \(\beta_{15}\mathstrut +\mathstrut \) \(22913109\) \(\beta_{14}\mathstrut -\mathstrut \) \(16748054\) \(\beta_{13}\mathstrut -\mathstrut \) \(10001968\) \(\beta_{12}\mathstrut -\mathstrut \) \(7016597\) \(\beta_{11}\mathstrut -\mathstrut \) \(9954693\) \(\beta_{10}\mathstrut -\mathstrut \) \(14748904\) \(\beta_{9}\mathstrut -\mathstrut \) \(12566728\) \(\beta_{8}\mathstrut +\mathstrut \) \(28228483\) \(\beta_{7}\mathstrut +\mathstrut \) \(10391540\) \(\beta_{6}\mathstrut -\mathstrut \) \(10511760\) \(\beta_{5}\mathstrut +\mathstrut \) \(16371454\) \(\beta_{4}\mathstrut +\mathstrut \) \(20367469\) \(\beta_{3}\mathstrut +\mathstrut \) \(9267444\) \(\beta_{2}\mathstrut +\mathstrut \) \(23894792\) \(\beta_{1}\mathstrut +\mathstrut \) \(39746289\)
\(\nu^{18}\)\(=\)\(71626570\) \(\beta_{18}\mathstrut -\mathstrut \) \(3932010\) \(\beta_{17}\mathstrut -\mathstrut \) \(29251285\) \(\beta_{16}\mathstrut -\mathstrut \) \(43615626\) \(\beta_{15}\mathstrut +\mathstrut \) \(78189603\) \(\beta_{14}\mathstrut -\mathstrut \) \(134550690\) \(\beta_{13}\mathstrut -\mathstrut \) \(47424330\) \(\beta_{12}\mathstrut -\mathstrut \) \(65709230\) \(\beta_{11}\mathstrut +\mathstrut \) \(5223217\) \(\beta_{10}\mathstrut -\mathstrut \) \(41938766\) \(\beta_{9}\mathstrut -\mathstrut \) \(16824361\) \(\beta_{8}\mathstrut +\mathstrut \) \(103464713\) \(\beta_{7}\mathstrut +\mathstrut \) \(7529642\) \(\beta_{6}\mathstrut -\mathstrut \) \(46824177\) \(\beta_{5}\mathstrut +\mathstrut \) \(66932680\) \(\beta_{4}\mathstrut +\mathstrut \) \(30526998\) \(\beta_{3}\mathstrut +\mathstrut \) \(129912594\) \(\beta_{2}\mathstrut -\mathstrut \) \(65303830\) \(\beta_{1}\mathstrut +\mathstrut \) \(225892133\)

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
−3.12892
−2.63379
−2.08912
−1.99202
−1.53958
−1.31693
−0.455245
−0.312548
−0.157639
0.00508866
0.749639
1.27063
1.56031
2.01204
2.43794
2.50185
2.79120
3.03361
3.26349
0 −3.12892 0 −3.70734 0 −1.25809 0 6.79011 0
1.2 0 −2.63379 0 −2.05129 0 4.98850 0 3.93685 0
1.3 0 −2.08912 0 −2.23992 0 1.92906 0 1.36442 0
1.4 0 −1.99202 0 2.61608 0 4.58459 0 0.968146 0
1.5 0 −1.53958 0 −0.551255 0 −1.16112 0 −0.629704 0
1.6 0 −1.31693 0 −1.28443 0 1.02036 0 −1.26569 0
1.7 0 −0.455245 0 2.37571 0 1.84768 0 −2.79275 0
1.8 0 −0.312548 0 0.780222 0 −4.05339 0 −2.90231 0
1.9 0 −0.157639 0 −0.360777 0 −0.00399840 0 −2.97515 0
1.10 0 0.00508866 0 −3.22565 0 −4.03772 0 −2.99997 0
1.11 0 0.749639 0 0.796495 0 1.70851 0 −2.43804 0
1.12 0 1.27063 0 −4.07798 0 2.09947 0 −1.38551 0
1.13 0 1.56031 0 3.44545 0 1.77285 0 −0.565441 0
1.14 0 2.01204 0 3.16436 0 2.95084 0 1.04829 0
1.15 0 2.43794 0 −3.80787 0 −4.24792 0 2.94357 0
1.16 0 2.50185 0 0.708265 0 −3.63078 0 3.25924 0
1.17 0 2.79120 0 −3.17175 0 1.46800 0 4.79080 0
1.18 0 3.03361 0 −0.0964405 0 5.16490 0 6.20277 0
1.19 0 3.26349 0 2.68811 0 −0.141739 0 7.65038 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{19} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).