Properties

Label 1470.2.m.d
Level $1470$
Weight $2$
Character orbit 1470.m
Analytic conductor $11.738$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.7380090971\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(1434138866353 \nu^{15} - 74224360591111 \nu^{14} + 217326371958939 \nu^{13} - 648847305484636 \nu^{12} + 2680829374033629 \nu^{11} - 1607350163227359 \nu^{10} - 900494454550979 \nu^{9} - 8792384561334166 \nu^{8} + 3536477625795241 \nu^{7} - 17495309067416083 \nu^{6} - 6622116382602757 \nu^{5} - 26263502656698172 \nu^{4} - 5327429661771194 \nu^{3} - 19428846797347660 \nu^{2} + 4742345058394342 \nu - 493738919956596\)\()/ 5173992472766390 \)
\(\beta_{2}\)\(=\)\((\)\(-55858235064187 \nu^{15} + 212060352259669 \nu^{14} - 656112655892906 \nu^{13} + 2669557685090689 \nu^{12} - 3545813042004261 \nu^{11} + 1994047041344521 \nu^{10} - 8175048259429059 \nu^{9} + 8255439822249079 \nu^{8} - 20520974134633139 \nu^{7} + 12424277744931352 \nu^{6} - 26721889665279852 \nu^{5} + 17459665701401863 \nu^{4} - 24068206993748769 \nu^{3} + 27360555914506800 \nu^{2} - 11408642698798268 \nu + 5144228213119549\)\()/ 2586996236383195 \)
\(\beta_{3}\)\(=\)\((\)\(5636832106551 \nu^{15} - 21709778642242 \nu^{14} + 64379091580243 \nu^{13} - 260750579275672 \nu^{12} + 338028777532663 \nu^{11} - 82000957355918 \nu^{10} + 645517707426777 \nu^{9} - 887847768153872 \nu^{8} + 1842581756644927 \nu^{7} - 728335947587826 \nu^{6} + 1876259187302601 \nu^{5} - 1583918378255464 \nu^{4} + 1938755903365772 \nu^{3} - 1633661181069180 \nu^{2} + 652747806115874 \nu - 76799728323202\)\()/ 28585593772190 \)
\(\beta_{4}\)\(=\)\((\)\(1388226594503844 \nu^{15} - 4482742599545073 \nu^{14} + 12897697280548092 \nu^{13} - 55652842856452568 \nu^{12} + 47104585380200152 \nu^{11} + 15706565729160203 \nu^{10} + 163461317318294988 \nu^{9} - 120145869848106398 \nu^{8} + 359627912073759638 \nu^{7} + 60829584239638731 \nu^{6} + 460236117588043124 \nu^{5} - 115390999404586036 \nu^{4} + 354828260557698958 \nu^{3} - 163384514126303100 \nu^{2} + 17755959327575476 \nu + 1775647992717362\)\()/ 5173992472766390 \)
\(\beta_{5}\)\(=\)\((\)\(1396750425716719 \nu^{15} - 5709959401678228 \nu^{14} + 17042093946215907 \nu^{13} - 67821946469399298 \nu^{12} + 97458011489969627 \nu^{11} - 33077669933831672 \nu^{10} + 159690429016687003 \nu^{9} - 261287156101646078 \nu^{8} + 485503526029668103 \nu^{7} - 273169157095358754 \nu^{6} + 465868812665474149 \nu^{5} - 515739665353898456 \nu^{4} + 510037937936129188 \nu^{3} - 497908252550261020 \nu^{2} + 211262518593440286 \nu - 37754197084838088\)\()/ 5173992472766390 \)
\(\beta_{6}\)\(=\)\((\)\(12275800546287 \nu^{15} - 43031097335949 \nu^{14} + 125528996510051 \nu^{13} - 525506789430144 \nu^{12} + 557872546858571 \nu^{11} + 1339880959479 \nu^{10} + 1431391471112369 \nu^{9} - 1459228421868034 \nu^{8} + 3531210809259339 \nu^{7} - 411948255026437 \nu^{6} + 4061950178518467 \nu^{5} - 2133791584817908 \nu^{4} + 3580211964980104 \nu^{3} - 2397988262146480 \nu^{2} + 706764140969698 \nu - 79799128103174\)\()/ 16530327389030 \)
\(\beta_{7}\)\(=\)\((\)\(-14132761546 \nu^{15} + 55431822137 \nu^{14} - 164129630598 \nu^{13} + 661673701232 \nu^{12} - 884127813478 \nu^{11} + 222597696723 \nu^{10} - 1603632172602 \nu^{9} + 2372154768512 \nu^{8} - 4647202615862 \nu^{7} + 2054556195351 \nu^{6} - 4588566438706 \nu^{5} + 4405332898494 \nu^{4} - 4814314934552 \nu^{3} + 4319514489740 \nu^{2} - 1674276466044 \nu + 253006704102\)\()/ 18195794890 \)
\(\beta_{8}\)\(=\)\((\)\(19482487322 \nu^{15} - 71829768307 \nu^{14} + 211530176114 \nu^{13} - 869648666130 \nu^{12} + 1035106924390 \nu^{11} - 152629602893 \nu^{10} + 2258419134670 \nu^{9} - 2719627979686 \nu^{8} + 6018321521990 \nu^{7} - 1643327598977 \nu^{6} + 6516717498206 \nu^{5} - 4492931472132 \nu^{4} + 6274129258728 \nu^{3} - 4768566345236 \nu^{2} + 1753345508964 \nu - 258618081490\)\()/ 18265555126 \)
\(\beta_{9}\)\(=\)\((\)\(1366533886371797 \nu^{15} - 5035011274628262 \nu^{14} + 14813991158747755 \nu^{13} - 60928317813212046 \nu^{12} + 72359889876527225 \nu^{11} - 10090918515881632 \nu^{10} + 158025827977532887 \nu^{9} - 190516441697877034 \nu^{8} + 420105007354616681 \nu^{7} - 113257420421837586 \nu^{6} + 454688571510523463 \nu^{5} - 314563215075476670 \nu^{4} + 435667370982289430 \nu^{3} - 330372575997770256 \nu^{2} + 121436904472934918 \nu - 16648434234825310\)\()/ 1034798494553278 \)
\(\beta_{10}\)\(=\)\((\)\(22618335117691 \nu^{15} - 83577207461092 \nu^{14} + 245635592946633 \nu^{13} - 1009765720131312 \nu^{12} + 1204607056198763 \nu^{11} - 162996168916358 \nu^{10} + 2607172638462137 \nu^{9} - 3187962371740152 \nu^{8} + 6931602070615967 \nu^{7} - 1928596324772176 \nu^{6} + 7433966473131041 \nu^{5} - 5332295356853984 \nu^{4} + 7099325683672802 \nu^{3} - 5582878062443860 \nu^{2} + 1945090799381534 \nu - 267195717113322\)\()/ 16530327389030 \)
\(\beta_{11}\)\(=\)\((\)\(-7882832156422018 \nu^{15} + 28959198858731031 \nu^{14} - 85188792443935384 \nu^{13} + 350751002477513236 \nu^{12} - 414220467622600394 \nu^{11} + 56073657866986719 \nu^{10} - 914518104851475506 \nu^{9} + 1089592584043389166 \nu^{8} - 2414819720487606666 \nu^{7} + 635877106539153013 \nu^{6} - 2631702420839099108 \nu^{5} + 1795791817509202122 \nu^{4} - 2501280039312637946 \nu^{3} + 1900561425363104360 \nu^{2} - 694421692945999592 \nu + 101703749105246726\)\()/ 5173992472766390 \)
\(\beta_{12}\)\(=\)\((\)\(-816410724118240 \nu^{15} + 2903996383855008 \nu^{14} - 8474568273332767 \nu^{13} + 35311035098566922 \nu^{12} - 38703849749202449 \nu^{11} + 948339094597864 \nu^{10} - 94441025844222703 \nu^{9} + 102078709515329683 \nu^{8} - 237140850924336357 \nu^{7} + 37997222485986755 \nu^{6} - 265784362843393858 \nu^{5} + 156062122425352126 \nu^{4} - 237837937538603482 \nu^{3} + 169685821655658166 \nu^{2} - 49463522029391302 \nu + 4466387218106494\)\()/ 517399247276639 \)
\(\beta_{13}\)\(=\)\((\)\(-8322113848090681 \nu^{15} + 30844250285030922 \nu^{14} - 90790589547879283 \nu^{13} + 372754874346223932 \nu^{12} - 448036214233635153 \nu^{11} + 67782479284162768 \nu^{10} - 962451816547164107 \nu^{9} + 1184529056271977382 \nu^{8} - 2573319600199631357 \nu^{7} + 739956875788202846 \nu^{6} - 2759629630188815891 \nu^{5} + 1992887058679050064 \nu^{4} - 2663465958893009432 \nu^{3} + 2078126208376925060 \nu^{2} - 752306823225839714 \nu + 110913864163531712\)\()/ 5173992472766390 \)
\(\beta_{14}\)\(=\)\((\)\(-11852855978577777 \nu^{15} + 44468339679473764 \nu^{14} - 131080267767082101 \nu^{13} + 535971469924936804 \nu^{12} - 659857989050043931 \nu^{11} + 115643332677662656 \nu^{10} - 1363956092616422119 \nu^{9} + 1747295344817322224 \nu^{8} - 3713021397053398749 \nu^{7} + 1192592904282470072 \nu^{6} - 3915931487261010437 \nu^{5} + 3000809553656731158 \nu^{4} - 3841451134085191824 \nu^{3} + 3088672852403722840 \nu^{2} - 1143052848607575738 \nu + 163229867812904994\)\()/ 5173992472766390 \)
\(\beta_{15}\)\(=\)\((\)\(-8339297950504309 \nu^{15} + 30850369744641918 \nu^{14} - 90694752433004392 \nu^{13} + 372655125611272543 \nu^{12} - 445643597511712467 \nu^{11} + 61742048132006817 \nu^{10} - 960630466246120233 \nu^{9} + 1179126300462522803 \nu^{8} - 2561330206078081448 \nu^{7} + 718144087600701779 \nu^{6} - 2739531278541562474 \nu^{5} + 1974449770501803281 \nu^{4} - 2627419461123962203 \nu^{3} + 2059854583464111220 \nu^{2} - 717610521890578526 \nu + 98820879076274058\)\()/ 2586996236383195 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + 2 \beta_{6} - 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_{1} + 2\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - 3 \beta_{10} + 3 \beta_{9} - 3 \beta_{8} - \beta_{7} + 4 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{15} - \beta_{14} - 2 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} - 6 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{2} - 2 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-9 \beta_{15} + 13 \beta_{14} + 5 \beta_{13} + 7 \beta_{12} - 3 \beta_{11} + 13 \beta_{10} - 7 \beta_{9} + 13 \beta_{8} + 11 \beta_{7} + \beta_{6} + 19 \beta_{5} - 6 \beta_{4} - 12 \beta_{3} + 22 \beta_{2} - 2 \beta_{1} + 23\)
\(\nu^{5}\)\(=\)\(-14 \beta_{15} + 16 \beta_{14} + 48 \beta_{13} - 21 \beta_{11} - 28 \beta_{10} - 14 \beta_{9} + 47 \beta_{8} - 25 \beta_{7} + 50 \beta_{6} + 19 \beta_{5} - 42 \beta_{4} - 34 \beta_{3} + 14 \beta_{2} + 7 \beta_{1} + 25\)
\(\nu^{6}\)\(=\)\(22 \beta_{15} - 111 \beta_{14} - 22 \beta_{13} + 43 \beta_{12} + 99 \beta_{11} - 46 \beta_{10} + 127 \beta_{9} - 199 \beta_{8} - 22 \beta_{7} + 167 \beta_{6} - 133 \beta_{5} - 39 \beta_{4} + 46 \beta_{3} - 167 \beta_{2} + 4 \beta_{1} - 91\)
\(\nu^{7}\)\(=\)\(-66 \beta_{15} + 66 \beta_{14} - 291 \beta_{13} + 250 \beta_{12} + 285 \beta_{11} + 569 \beta_{10} + 129 \beta_{9} - 307 \beta_{8} + 474 \beta_{7} - 85 \beta_{6} + 244 \beta_{5} + 129 \beta_{4} + 72 \beta_{3} + 266 \beta_{2} - 150 \beta_{1} + 228\)
\(\nu^{8}\)\(=\)\(-623 \beta_{15} + 1537 \beta_{14} + 987 \beta_{13} + 49 \beta_{12} - 865 \beta_{11} + 758 \beta_{10} - 1305 \beta_{9} + 2564 \beta_{8} + 383 \beta_{7} - 86 \beta_{6} + 2289 \beta_{5} - 672 \beta_{4} - 1076 \beta_{3} + 2451 \beta_{2} - 202 \beta_{1} + 1699\)
\(\nu^{9}\)\(=\)\(318 \beta_{15} - 795 \beta_{14} + 3492 \beta_{13} - 981 \beta_{12} - 981 \beta_{11} - 4267 \beta_{10} + 2025 \beta_{8} - 3790 \beta_{7} + 5164 \beta_{6} - 1093 \beta_{5} - 3325 \beta_{4} - 914 \beta_{3} - 2094 \beta_{2} + 795 \beta_{1} - 1895\)
\(\nu^{10}\)\(=\)\(4732 \beta_{15} - 12213 \beta_{14} - 10318 \beta_{13} + 4199 \beta_{12} + 13706 \beta_{11} + 2939 \beta_{10} + 13885 \beta_{9} - 25133 \beta_{8} + 3404 \beta_{7} + 7305 \beta_{6} - 12798 \beta_{5} + 3465 \beta_{4} + 10361 \beta_{3} - 15678 \beta_{2} - 1843 \beta_{1} - 12701\)
\(\nu^{11}\)\(=\)\(-9261 \beta_{15} + 27968 \beta_{14} - 20977 \beta_{13} + 15303 \beta_{12} + 9446 \beta_{11} + 64544 \beta_{10} - 13074 \beta_{9} + 12715 \beta_{8} + 48769 \beta_{7} - 33565 \beta_{6} + 53470 \beta_{5} + 15992 \beta_{4} - 1057 \beta_{3} + 60384 \beta_{2} - 17113 \beta_{1} + 36280\)
\(\nu^{12}\)\(=\)\(-40556 \beta_{15} + 130306 \beta_{14} + 143366 \beta_{13} - 40556 \beta_{12} - 123066 \beta_{11} - 22185 \beta_{10} - 143366 \beta_{9} + 297114 \beta_{8} - 46815 \beta_{7} + 18371 \beta_{6} + 180042 \beta_{5} - 89750 \beta_{4} - 105726 \beta_{3} + 186301 \beta_{2} + 2916 \beta_{1} + 105726\)
\(\nu^{13}\)\(=\)\(155166 \beta_{15} - 341023 \beta_{14} + 134815 \beta_{13} - 99171 \beta_{12} + 87130 \beta_{11} - 482685 \beta_{10} + 228792 \beta_{9} - 255939 \beta_{8} - 396295 \beta_{7} + 500473 \beta_{6} - 463017 \beta_{5} - 186301 \beta_{4} + 131899 \beta_{3} - 594916 \beta_{2} + 86686 \beta_{1} - 502797\)
\(\nu^{14}\)\(=\)\(407950 \beta_{15} - 918652 \beta_{14} - 1566704 \beta_{13} + 541254 \beta_{12} + 1452297 \beta_{11} + 1152997 \beta_{10} + 1210240 \beta_{9} - 2530660 \beta_{8} + 1037204 \beta_{7} - 249726 \beta_{6} - 798589 \beta_{5} + 897778 \beta_{4} + 1151145 \beta_{3} - 905387 \beta_{2} - 369789 \beta_{1} - 864228\)
\(\nu^{15}\)\(=\)\(-1560582 \beta_{15} + 4825341 \beta_{14} + 529500 \beta_{12} - 1560582 \beta_{11} + 5527039 \beta_{10} - 3827007 \beta_{9} + 6028621 \beta_{8} + 3827007 \beta_{7} - 4281011 \beta_{6} + 7536105 \beta_{5} + 765039 \beta_{4} - 1824039 \beta_{3} + 8595105 \beta_{2} - 1355382 \beta_{1} + 5340011\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−0.709944 0.925217i
0.339278 0.0446668i
0.792206 + 1.03242i
2.69978 0.355433i
−0.424637 3.22544i
0.277956 0.213283i
0.117630 + 0.893490i
−1.09227 + 0.838128i
−0.709944 + 0.925217i
0.339278 + 0.0446668i
0.792206 1.03242i
2.69978 + 0.355433i
−0.424637 + 3.22544i
0.277956 + 0.213283i
0.117630 0.893490i
−1.09227 0.838128i
−0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −2.17041 0.537883i 1.00000i 0 0.707107 0.707107i 1.00000i 1.15437 + 1.91505i
97.2 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −1.03078 + 1.98431i 1.00000i 0 0.707107 0.707107i 1.00000i 2.13199 0.674247i
97.3 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 1.91159 1.16009i 1.00000i 0 0.707107 0.707107i 1.00000i −2.17201 0.531389i
97.4 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 1.99671 + 1.00656i 1.00000i 0 0.707107 0.707107i 1.00000i −0.700141 2.12363i
97.5 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −2.15899 + 0.582041i 1.00000i 0 −0.707107 + 0.707107i 1.00000i −1.93820 1.11507i
97.6 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −1.36519 + 1.77095i 1.00000i 0 −0.707107 + 0.707107i 1.00000i −2.21758 + 0.286912i
97.7 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 1.19306 + 1.89119i 1.00000i 0 −0.707107 + 0.707107i 1.00000i −0.493652 + 2.18090i
97.8 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 1.62401 1.53707i 1.00000i 0 −0.707107 + 0.707107i 1.00000i 2.23522 + 0.0614757i
1273.1 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −2.17041 + 0.537883i 1.00000i 0 0.707107 + 0.707107i 1.00000i 1.15437 1.91505i
1273.2 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −1.03078 1.98431i 1.00000i 0 0.707107 + 0.707107i 1.00000i 2.13199 + 0.674247i
1273.3 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 1.91159 + 1.16009i 1.00000i 0 0.707107 + 0.707107i 1.00000i −2.17201 + 0.531389i
1273.4 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 1.99671 1.00656i 1.00000i 0 0.707107 + 0.707107i 1.00000i −0.700141 + 2.12363i
1273.5 0.707107 0.707107i −0.707107 + 0.707107i 1.00000i −2.15899 0.582041i 1.00000i 0 −0.707107 0.707107i 1.00000i −1.93820 + 1.11507i
1273.6 0.707107 0.707107i −0.707107 + 0.707107i 1.00000i −1.36519 1.77095i 1.00000i 0 −0.707107 0.707107i 1.00000i −2.21758 0.286912i
1273.7 0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 1.19306 1.89119i 1.00000i 0 −0.707107 0.707107i 1.00000i −0.493652 2.18090i
1273.8 0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 1.62401 + 1.53707i 1.00000i 0 −0.707107 0.707107i 1.00000i 2.23522 0.0614757i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1273.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.m.d 16
5.c odd 4 1 1470.2.m.e 16
7.b odd 2 1 1470.2.m.e 16
7.c even 3 1 210.2.u.a 16
7.d odd 6 1 210.2.u.b yes 16
21.g even 6 1 630.2.bv.b 16
21.h odd 6 1 630.2.bv.a 16
35.f even 4 1 inner 1470.2.m.d 16
35.i odd 6 1 1050.2.bc.g 16
35.j even 6 1 1050.2.bc.h 16
35.k even 12 1 210.2.u.a 16
35.k even 12 1 1050.2.bc.h 16
35.l odd 12 1 210.2.u.b yes 16
35.l odd 12 1 1050.2.bc.g 16
105.w odd 12 1 630.2.bv.a 16
105.x even 12 1 630.2.bv.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.u.a 16 7.c even 3 1
210.2.u.a 16 35.k even 12 1
210.2.u.b yes 16 7.d odd 6 1
210.2.u.b yes 16 35.l odd 12 1
630.2.bv.a 16 21.h odd 6 1
630.2.bv.a 16 105.w odd 12 1
630.2.bv.b 16 21.g even 6 1
630.2.bv.b 16 105.x even 12 1
1050.2.bc.g 16 35.i odd 6 1
1050.2.bc.g 16 35.l odd 12 1
1050.2.bc.h 16 35.j even 6 1
1050.2.bc.h 16 35.k even 12 1
1470.2.m.d 16 1.a even 1 1 trivial
1470.2.m.d 16 35.f even 4 1 inner
1470.2.m.e 16 5.c odd 4 1
1470.2.m.e 16 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1470, [\chi])\):

\(T_{11}^{8} + \cdots\)
\(T_{13}^{16} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{4} \)
$3$ \( ( 1 + T^{4} )^{4} \)
$5$ \( 390625 - 125000 T^{2} + 25000 T^{3} + 40625 T^{4} + 1000 T^{5} - 8600 T^{6} + 240 T^{7} + 2544 T^{8} + 48 T^{9} - 344 T^{10} + 8 T^{11} + 65 T^{12} + 8 T^{13} - 8 T^{14} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( ( -383 + 100 T + 1202 T^{2} + 868 T^{3} - 6 T^{4} - 156 T^{5} - 30 T^{6} + 4 T^{7} + T^{8} )^{2} \)
$13$ \( 171295744 + 51933184 T + 7872512 T^{2} + 9624320 T^{3} + 18339968 T^{4} + 7707968 T^{5} + 1764384 T^{6} + 370872 T^{7} + 304225 T^{8} + 126936 T^{9} + 29856 T^{10} + 4424 T^{11} + 1442 T^{12} + 536 T^{13} + 128 T^{14} + 16 T^{15} + T^{16} \)
$17$ \( 16384 - 1114112 T + 37879808 T^{2} - 99602432 T^{3} + 134277120 T^{4} - 114138112 T^{5} + 66732544 T^{6} - 27702400 T^{7} + 8298512 T^{8} - 1816960 T^{9} + 314624 T^{10} - 53632 T^{11} + 10888 T^{12} - 2080 T^{13} + 288 T^{14} - 24 T^{15} + T^{16} \)
$19$ \( ( 76768 - 128672 T + 76568 T^{2} - 16784 T^{3} - 735 T^{4} + 792 T^{5} - 66 T^{6} - 8 T^{7} + T^{8} )^{2} \)
$23$ \( 82925569024 + 45448261632 T + 12454207488 T^{2} - 4487743232 T^{3} + 1419753088 T^{4} + 267723456 T^{5} + 54935840 T^{6} - 16832280 T^{7} + 3871713 T^{8} + 476528 T^{9} + 82240 T^{10} - 21024 T^{11} + 3170 T^{12} + 176 T^{13} + 32 T^{14} - 8 T^{15} + T^{16} \)
$29$ \( 939790336 + 2601994240 T^{2} + 1378025856 T^{4} + 231567136 T^{6} + 18367617 T^{8} + 781240 T^{10} + 18194 T^{12} + 216 T^{14} + T^{16} \)
$31$ \( 85229824 + 81020309760 T^{2} + 38694454880 T^{4} + 4521080144 T^{6} + 218447649 T^{8} + 5201252 T^{10} + 64710 T^{12} + 404 T^{14} + T^{16} \)
$37$ \( 1586310022144 + 1018633590784 T + 327052838912 T^{2} + 12906720512 T^{3} + 2843166848 T^{4} + 3716892416 T^{5} + 1853091360 T^{6} - 1491080 T^{7} - 12515151 T^{8} - 1994200 T^{9} + 2320800 T^{10} - 273752 T^{11} + 16178 T^{12} - 152 T^{13} + 128 T^{14} - 16 T^{15} + T^{16} \)
$41$ \( 729780649984 + 406165422080 T^{2} + 84574220800 T^{4} + 8100985984 T^{6} + 363545169 T^{8} + 8080772 T^{10} + 90070 T^{12} + 484 T^{14} + T^{16} \)
$43$ \( 396169216 - 132481024 T + 22151168 T^{2} + 916025344 T^{3} + 2977536512 T^{4} + 1507826688 T^{5} + 388311040 T^{6} + 22278400 T^{7} + 8241168 T^{8} + 3516480 T^{9} + 806528 T^{10} + 56352 T^{11} + 5416 T^{12} + 1424 T^{13} + 288 T^{14} + 24 T^{15} + T^{16} \)
$47$ \( 405330862336 - 103759648256 T + 13280588288 T^{2} - 3026181248 T^{3} + 4081700064 T^{4} - 1131058336 T^{5} + 167097376 T^{6} - 22218424 T^{7} + 11510129 T^{8} - 3221152 T^{9} + 508736 T^{10} - 47440 T^{11} + 7762 T^{12} - 1792 T^{13} + 288 T^{14} - 24 T^{15} + T^{16} \)
$53$ \( 7800599041 - 27523649872 T + 48557251712 T^{2} - 3862366064 T^{3} - 62096732 T^{4} + 136395280 T^{5} + 861483392 T^{6} - 24470480 T^{7} - 663226 T^{8} + 6229264 T^{9} + 2315648 T^{10} + 271280 T^{11} + 16548 T^{12} + 432 T^{13} + 128 T^{14} + 16 T^{15} + T^{16} \)
$59$ \( ( -41408 - 92416 T + 21744 T^{2} + 46944 T^{3} + 6729 T^{4} - 1288 T^{5} - 172 T^{6} + 8 T^{7} + T^{8} )^{2} \)
$61$ \( 4676942565376 + 4224911327232 T^{2} + 647886133760 T^{4} + 39449228800 T^{6} + 1165358352 T^{8} + 18079264 T^{10} + 149112 T^{12} + 616 T^{14} + T^{16} \)
$67$ \( 13679819493376 + 4136304369664 T + 625337704448 T^{2} - 388620863488 T^{3} + 73711322624 T^{4} + 2720497152 T^{5} + 2973096448 T^{6} - 1504722304 T^{7} + 303918096 T^{8} - 31355328 T^{9} + 3623552 T^{10} - 754464 T^{11} + 139624 T^{12} - 16016 T^{13} + 1152 T^{14} - 48 T^{15} + T^{16} \)
$71$ \( ( -17030912 - 5972992 T + 328960 T^{2} + 253696 T^{3} + 6240 T^{4} - 3520 T^{5} - 176 T^{6} + 16 T^{7} + T^{8} )^{2} \)
$73$ \( 34279432458496 - 29104201071616 T + 12355142125568 T^{2} - 3008313656832 T^{3} + 562196047104 T^{4} - 130085705984 T^{5} + 39819912704 T^{6} - 9159952000 T^{7} + 1404051296 T^{8} - 145710400 T^{9} + 15288704 T^{10} - 2120224 T^{11} + 282064 T^{12} - 26032 T^{13} + 1568 T^{14} - 56 T^{15} + T^{16} \)
$79$ \( 31950847504 + 29670868576 T^{2} + 9774143928 T^{4} + 1414463176 T^{6} + 94628721 T^{8} + 3105820 T^{10} + 49934 T^{12} + 372 T^{14} + T^{16} \)
$83$ \( 8632854701584 - 15282725367680 T + 13527489036800 T^{2} - 6608651071808 T^{3} + 1960957443688 T^{4} - 320767035488 T^{5} + 24615111808 T^{6} + 317475888 T^{7} + 128886705 T^{8} - 45923808 T^{9} + 4262656 T^{10} + 437632 T^{11} + 27802 T^{12} - 1280 T^{13} + 128 T^{14} + 16 T^{15} + T^{16} \)
$89$ \( ( -8192 + 45056 T + 35328 T^{2} - 14592 T^{3} - 2544 T^{4} + 1088 T^{5} - 16 T^{6} - 16 T^{7} + T^{8} )^{2} \)
$97$ \( 67108864 + 301989888 T + 679477248 T^{2} + 313262080 T^{3} - 156729344 T^{4} - 386531328 T^{5} + 578642432 T^{6} - 309264096 T^{7} + 93564321 T^{8} - 15942868 T^{9} + 1838152 T^{10} - 261300 T^{11} + 69470 T^{12} - 10732 T^{13} + 968 T^{14} - 44 T^{15} + T^{16} \)
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