Properties

Label 2736.3.o.r
Level $2736$
Weight $3$
Character orbit 2736.o
Analytic conductor $74.551$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} + 80 x^{18} - 152 x^{17} + 4326 x^{16} - 10096 x^{15} + 70116 x^{14} - 93436 x^{13} + 597327 x^{12} - 837564 x^{11} + 2838686 x^{10} - 2898276 x^{9} + 7652658 x^{8} - 6988600 x^{7} + 12843126 x^{6} - 6054124 x^{5} + 7434909 x^{4} - 621764 x^{3} + 3099194 x^{2} - 329080 x + 36100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{37} \)
Twist minimal: no (minimal twist has level 456)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{4} q^{5} + ( 1 + \beta_{3} ) q^{7} +O(q^{10})\) \( q -\beta_{4} q^{5} + ( 1 + \beta_{3} ) q^{7} + ( 1 + \beta_{1} - \beta_{4} ) q^{11} -\beta_{13} q^{13} + ( -2 + \beta_{2} ) q^{17} + ( -2 + \beta_{3} - \beta_{15} ) q^{19} + ( 3 + \beta_{3} + \beta_{4} - \beta_{5} ) q^{23} + ( 4 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{10} ) q^{25} + ( -\beta_{9} - \beta_{13} - \beta_{15} - \beta_{17} ) q^{29} + ( \beta_{8} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{18} - \beta_{19} ) q^{31} + ( -10 - \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{7} - \beta_{10} + \beta_{15} - \beta_{17} ) q^{35} + ( \beta_{8} + \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{15} + \beta_{17} + \beta_{18} ) q^{37} + ( -\beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{18} ) q^{41} + ( -2 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{15} - \beta_{17} ) q^{43} + ( 3 - \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{10} + \beta_{15} - \beta_{17} ) q^{47} + ( 1 - \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{10} + \beta_{15} - \beta_{17} ) q^{49} + ( -\beta_{9} - \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{53} + ( 16 - 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{10} ) q^{55} + ( \beta_{9} + \beta_{11} - 2 \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{59} + ( 9 + 3 \beta_{1} - 3 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{10} - \beta_{15} + \beta_{17} ) q^{61} + ( \beta_{8} - 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{14} + \beta_{15} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{65} + ( -\beta_{8} + 2 \beta_{9} - \beta_{11} + 2 \beta_{12} + 4 \beta_{13} - 3 \beta_{14} - \beta_{18} ) q^{67} + ( -2 \beta_{8} + \beta_{9} + 5 \beta_{12} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{71} + ( 8 + \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + \beta_{10} + 2 \beta_{15} - 2 \beta_{17} ) q^{73} + ( -2 - 3 \beta_{2} - 2 \beta_{6} - 2 \beta_{10} + 2 \beta_{15} - 2 \beta_{17} ) q^{77} + ( 2 \beta_{8} + 2 \beta_{9} - 4 \beta_{11} - 2 \beta_{12} - \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - 2 \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{79} + ( 9 + 3 \beta_{2} - 9 \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{10} - \beta_{15} + \beta_{17} ) q^{83} + ( -4 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{10} + \beta_{15} - \beta_{17} ) q^{85} + ( -2 \beta_{8} + \beta_{9} - 3 \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{89} + ( -2 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} + 4 \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{16} - \beta_{18} - 2 \beta_{19} ) q^{91} + ( -18 + \beta_{1} - 5 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{16} + 2 \beta_{17} - \beta_{19} ) q^{95} + ( -3 \beta_{9} - \beta_{12} - 2 \beta_{13} - 3 \beta_{14} - 3 \beta_{15} - \beta_{16} - 3 \beta_{17} + 2 \beta_{18} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 16q^{7} + O(q^{10}) \) \( 20q + 16q^{7} + 16q^{11} - 32q^{17} - 40q^{19} + 64q^{23} + 68q^{25} - 208q^{35} - 64q^{43} + 48q^{47} + 20q^{49} + 336q^{55} + 184q^{61} + 104q^{73} - 88q^{77} + 224q^{83} - 136q^{85} - 320q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 4 x^{19} + 80 x^{18} - 152 x^{17} + 4326 x^{16} - 10096 x^{15} + 70116 x^{14} - 93436 x^{13} + 597327 x^{12} - 837564 x^{11} + 2838686 x^{10} - 2898276 x^{9} + 7652658 x^{8} - 6988600 x^{7} + 12843126 x^{6} - 6054124 x^{5} + 7434909 x^{4} - 621764 x^{3} + 3099194 x^{2} - 329080 x + 36100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(16\!\cdots\!99\)\( \nu^{19} + \)\(14\!\cdots\!19\)\( \nu^{18} - \)\(16\!\cdots\!24\)\( \nu^{17} + \)\(87\!\cdots\!11\)\( \nu^{16} - \)\(85\!\cdots\!67\)\( \nu^{15} + \)\(50\!\cdots\!46\)\( \nu^{14} - \)\(20\!\cdots\!37\)\( \nu^{13} + \)\(69\!\cdots\!03\)\( \nu^{12} - \)\(17\!\cdots\!59\)\( \nu^{11} + \)\(58\!\cdots\!86\)\( \nu^{10} - \)\(11\!\cdots\!51\)\( \nu^{9} + \)\(25\!\cdots\!65\)\( \nu^{8} - \)\(34\!\cdots\!80\)\( \nu^{7} + \)\(65\!\cdots\!01\)\( \nu^{6} - \)\(71\!\cdots\!85\)\( \nu^{5} + \)\(10\!\cdots\!26\)\( \nu^{4} - \)\(29\!\cdots\!66\)\( \nu^{3} + \)\(50\!\cdots\!32\)\( \nu^{2} - \)\(53\!\cdots\!40\)\( \nu + \)\(42\!\cdots\!16\)\(\)\()/ \)\(27\!\cdots\!64\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(38\!\cdots\!16\)\( \nu^{19} - \)\(12\!\cdots\!85\)\( \nu^{18} - \)\(16\!\cdots\!40\)\( \nu^{17} - \)\(17\!\cdots\!65\)\( \nu^{16} - \)\(98\!\cdots\!28\)\( \nu^{15} - \)\(85\!\cdots\!70\)\( \nu^{14} + \)\(16\!\cdots\!46\)\( \nu^{13} - \)\(18\!\cdots\!89\)\( \nu^{12} + \)\(23\!\cdots\!08\)\( \nu^{11} - \)\(14\!\cdots\!50\)\( \nu^{10} + \)\(29\!\cdots\!14\)\( \nu^{9} - \)\(78\!\cdots\!47\)\( \nu^{8} + \)\(11\!\cdots\!64\)\( \nu^{7} - \)\(21\!\cdots\!07\)\( \nu^{6} + \)\(28\!\cdots\!22\)\( \nu^{5} - \)\(40\!\cdots\!34\)\( \nu^{4} + \)\(26\!\cdots\!64\)\( \nu^{3} - \)\(20\!\cdots\!60\)\( \nu^{2} + \)\(21\!\cdots\!00\)\( \nu - \)\(81\!\cdots\!16\)\(\)\()/ \)\(27\!\cdots\!64\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(55\!\cdots\!12\)\( \nu^{19} + \)\(35\!\cdots\!53\)\( \nu^{18} - \)\(49\!\cdots\!36\)\( \nu^{17} + \)\(18\!\cdots\!29\)\( \nu^{16} - \)\(25\!\cdots\!76\)\( \nu^{15} + \)\(11\!\cdots\!14\)\( \nu^{14} - \)\(49\!\cdots\!02\)\( \nu^{13} + \)\(14\!\cdots\!53\)\( \nu^{12} - \)\(41\!\cdots\!28\)\( \nu^{11} + \)\(12\!\cdots\!94\)\( \nu^{10} - \)\(23\!\cdots\!82\)\( \nu^{9} + \)\(53\!\cdots\!63\)\( \nu^{8} - \)\(71\!\cdots\!56\)\( \nu^{7} + \)\(13\!\cdots\!79\)\( \nu^{6} - \)\(14\!\cdots\!18\)\( \nu^{5} + \)\(19\!\cdots\!70\)\( \nu^{4} - \)\(10\!\cdots\!72\)\( \nu^{3} + \)\(97\!\cdots\!68\)\( \nu^{2} - \)\(10\!\cdots\!60\)\( \nu + \)\(19\!\cdots\!08\)\(\)\()/ \)\(27\!\cdots\!64\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(58\!\cdots\!16\)\( \nu^{19} + \)\(36\!\cdots\!09\)\( \nu^{18} - \)\(51\!\cdots\!32\)\( \nu^{17} + \)\(19\!\cdots\!61\)\( \nu^{16} - \)\(26\!\cdots\!72\)\( \nu^{15} + \)\(11\!\cdots\!14\)\( \nu^{14} - \)\(50\!\cdots\!54\)\( \nu^{13} + \)\(14\!\cdots\!37\)\( \nu^{12} - \)\(42\!\cdots\!20\)\( \nu^{11} + \)\(12\!\cdots\!46\)\( \nu^{10} - \)\(23\!\cdots\!58\)\( \nu^{9} + \)\(52\!\cdots\!47\)\( \nu^{8} - \)\(70\!\cdots\!76\)\( \nu^{7} + \)\(13\!\cdots\!79\)\( \nu^{6} - \)\(13\!\cdots\!38\)\( \nu^{5} + \)\(18\!\cdots\!46\)\( \nu^{4} - \)\(96\!\cdots\!68\)\( \nu^{3} + \)\(93\!\cdots\!68\)\( \nu^{2} - \)\(99\!\cdots\!60\)\( \nu + \)\(21\!\cdots\!12\)\(\)\()/ \)\(27\!\cdots\!64\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(90\!\cdots\!71\)\( \nu^{19} - \)\(42\!\cdots\!79\)\( \nu^{18} + \)\(72\!\cdots\!28\)\( \nu^{17} - \)\(17\!\cdots\!51\)\( \nu^{16} + \)\(38\!\cdots\!47\)\( \nu^{15} - \)\(11\!\cdots\!30\)\( \nu^{14} + \)\(60\!\cdots\!77\)\( \nu^{13} - \)\(11\!\cdots\!43\)\( \nu^{12} + \)\(46\!\cdots\!79\)\( \nu^{11} - \)\(10\!\cdots\!46\)\( \nu^{10} + \)\(19\!\cdots\!11\)\( \nu^{9} - \)\(32\!\cdots\!09\)\( \nu^{8} + \)\(42\!\cdots\!24\)\( \nu^{7} - \)\(86\!\cdots\!37\)\( \nu^{6} + \)\(50\!\cdots\!57\)\( \nu^{5} - \)\(58\!\cdots\!62\)\( \nu^{4} - \)\(29\!\cdots\!10\)\( \nu^{3} - \)\(28\!\cdots\!80\)\( \nu^{2} + \)\(29\!\cdots\!00\)\( \nu + \)\(59\!\cdots\!52\)\(\)\()/ \)\(27\!\cdots\!64\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(10\!\cdots\!43\)\( \nu^{19} + \)\(32\!\cdots\!62\)\( \nu^{18} - \)\(77\!\cdots\!72\)\( \nu^{17} + \)\(64\!\cdots\!30\)\( \nu^{16} - \)\(41\!\cdots\!15\)\( \nu^{15} + \)\(58\!\cdots\!72\)\( \nu^{14} - \)\(47\!\cdots\!83\)\( \nu^{13} - \)\(45\!\cdots\!82\)\( \nu^{12} - \)\(29\!\cdots\!79\)\( \nu^{11} + \)\(59\!\cdots\!20\)\( \nu^{10} - \)\(10\!\cdots\!29\)\( \nu^{9} - \)\(19\!\cdots\!42\)\( \nu^{8} + \)\(30\!\cdots\!76\)\( \nu^{7} - \)\(62\!\cdots\!78\)\( \nu^{6} + \)\(13\!\cdots\!93\)\( \nu^{5} - \)\(20\!\cdots\!40\)\( \nu^{4} + \)\(21\!\cdots\!02\)\( \nu^{3} - \)\(10\!\cdots\!56\)\( \nu^{2} + \)\(11\!\cdots\!20\)\( \nu - \)\(32\!\cdots\!64\)\(\)\()/ \)\(27\!\cdots\!64\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(40\!\cdots\!32\)\( \nu^{19} + \)\(24\!\cdots\!27\)\( \nu^{18} - \)\(35\!\cdots\!56\)\( \nu^{17} + \)\(12\!\cdots\!43\)\( \nu^{16} - \)\(18\!\cdots\!56\)\( \nu^{15} + \)\(76\!\cdots\!02\)\( \nu^{14} - \)\(34\!\cdots\!38\)\( \nu^{13} + \)\(90\!\cdots\!59\)\( \nu^{12} - \)\(28\!\cdots\!68\)\( \nu^{11} + \)\(78\!\cdots\!58\)\( \nu^{10} - \)\(15\!\cdots\!10\)\( \nu^{9} + \)\(30\!\cdots\!13\)\( \nu^{8} - \)\(41\!\cdots\!72\)\( \nu^{7} + \)\(78\!\cdots\!05\)\( \nu^{6} - \)\(74\!\cdots\!26\)\( \nu^{5} + \)\(10\!\cdots\!82\)\( \nu^{4} - \)\(24\!\cdots\!56\)\( \nu^{3} + \)\(49\!\cdots\!44\)\( \nu^{2} - \)\(53\!\cdots\!80\)\( \nu + \)\(25\!\cdots\!40\)\(\)\()/ \)\(90\!\cdots\!88\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(12\!\cdots\!97\)\( \nu^{19} + \)\(17\!\cdots\!59\)\( \nu^{18} + \)\(73\!\cdots\!70\)\( \nu^{17} + \)\(33\!\cdots\!97\)\( \nu^{16} + \)\(44\!\cdots\!95\)\( \nu^{15} + \)\(15\!\cdots\!00\)\( \nu^{14} + \)\(25\!\cdots\!75\)\( \nu^{13} + \)\(30\!\cdots\!59\)\( \nu^{12} + \)\(24\!\cdots\!47\)\( \nu^{11} + \)\(23\!\cdots\!20\)\( \nu^{10} - \)\(12\!\cdots\!95\)\( \nu^{9} + \)\(98\!\cdots\!97\)\( \nu^{8} - \)\(52\!\cdots\!18\)\( \nu^{7} + \)\(22\!\cdots\!03\)\( \nu^{6} - \)\(20\!\cdots\!67\)\( \nu^{5} + \)\(30\!\cdots\!52\)\( \nu^{4} - \)\(13\!\cdots\!90\)\( \nu^{3} + \)\(29\!\cdots\!80\)\( \nu^{2} - \)\(94\!\cdots\!76\)\( \nu + \)\(50\!\cdots\!60\)\(\)\()/ \)\(27\!\cdots\!64\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(18\!\cdots\!41\)\( \nu^{19} - \)\(77\!\cdots\!45\)\( \nu^{18} + \)\(15\!\cdots\!14\)\( \nu^{17} - \)\(30\!\cdots\!19\)\( \nu^{16} + \)\(80\!\cdots\!83\)\( \nu^{15} - \)\(19\!\cdots\!20\)\( \nu^{14} + \)\(12\!\cdots\!39\)\( \nu^{13} - \)\(17\!\cdots\!33\)\( \nu^{12} + \)\(10\!\cdots\!79\)\( \nu^{11} - \)\(15\!\cdots\!68\)\( \nu^{10} + \)\(49\!\cdots\!93\)\( \nu^{9} - \)\(47\!\cdots\!91\)\( \nu^{8} + \)\(12\!\cdots\!82\)\( \nu^{7} - \)\(11\!\cdots\!17\)\( \nu^{6} + \)\(19\!\cdots\!69\)\( \nu^{5} - \)\(56\!\cdots\!28\)\( \nu^{4} + \)\(56\!\cdots\!38\)\( \nu^{3} + \)\(21\!\cdots\!36\)\( \nu^{2} - \)\(14\!\cdots\!48\)\( \nu + \)\(98\!\cdots\!80\)\(\)\()/ \)\(27\!\cdots\!64\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(14\!\cdots\!25\)\( \nu^{19} + \)\(64\!\cdots\!40\)\( \nu^{18} - \)\(11\!\cdots\!96\)\( \nu^{17} + \)\(26\!\cdots\!40\)\( \nu^{16} - \)\(60\!\cdots\!61\)\( \nu^{15} + \)\(17\!\cdots\!16\)\( \nu^{14} - \)\(92\!\cdots\!53\)\( \nu^{13} + \)\(15\!\cdots\!40\)\( \nu^{12} - \)\(69\!\cdots\!49\)\( \nu^{11} + \)\(14\!\cdots\!00\)\( \nu^{10} - \)\(27\!\cdots\!87\)\( \nu^{9} + \)\(45\!\cdots\!72\)\( \nu^{8} - \)\(57\!\cdots\!68\)\( \nu^{7} + \)\(10\!\cdots\!40\)\( \nu^{6} - \)\(58\!\cdots\!29\)\( \nu^{5} + \)\(60\!\cdots\!36\)\( \nu^{4} + \)\(52\!\cdots\!06\)\( \nu^{3} + \)\(28\!\cdots\!12\)\( \nu^{2} - \)\(30\!\cdots\!40\)\( \nu + \)\(18\!\cdots\!12\)\(\)\()/ \)\(14\!\cdots\!56\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(72\!\cdots\!84\)\( \nu^{19} + \)\(28\!\cdots\!96\)\( \nu^{18} - \)\(57\!\cdots\!60\)\( \nu^{17} + \)\(10\!\cdots\!08\)\( \nu^{16} - \)\(31\!\cdots\!34\)\( \nu^{15} + \)\(71\!\cdots\!44\)\( \nu^{14} - \)\(50\!\cdots\!14\)\( \nu^{13} + \)\(65\!\cdots\!24\)\( \nu^{12} - \)\(42\!\cdots\!08\)\( \nu^{11} + \)\(59\!\cdots\!16\)\( \nu^{10} - \)\(20\!\cdots\!54\)\( \nu^{9} + \)\(20\!\cdots\!24\)\( \nu^{8} - \)\(53\!\cdots\!32\)\( \nu^{7} + \)\(48\!\cdots\!40\)\( \nu^{6} - \)\(88\!\cdots\!14\)\( \nu^{5} + \)\(38\!\cdots\!56\)\( \nu^{4} - \)\(47\!\cdots\!46\)\( \nu^{3} - \)\(11\!\cdots\!64\)\( \nu^{2} - \)\(17\!\cdots\!56\)\( \nu + \)\(91\!\cdots\!60\)\(\)\()/ \)\(33\!\cdots\!05\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(18\!\cdots\!36\)\( \nu^{19} + \)\(74\!\cdots\!84\)\( \nu^{18} - \)\(14\!\cdots\!40\)\( \nu^{17} + \)\(28\!\cdots\!32\)\( \nu^{16} - \)\(79\!\cdots\!56\)\( \nu^{15} + \)\(18\!\cdots\!16\)\( \nu^{14} - \)\(12\!\cdots\!76\)\( \nu^{13} + \)\(17\!\cdots\!16\)\( \nu^{12} - \)\(10\!\cdots\!12\)\( \nu^{11} + \)\(15\!\cdots\!24\)\( \nu^{10} - \)\(52\!\cdots\!16\)\( \nu^{9} + \)\(53\!\cdots\!96\)\( \nu^{8} - \)\(14\!\cdots\!28\)\( \nu^{7} + \)\(12\!\cdots\!00\)\( \nu^{6} - \)\(23\!\cdots\!36\)\( \nu^{5} + \)\(10\!\cdots\!64\)\( \nu^{4} - \)\(12\!\cdots\!84\)\( \nu^{3} - \)\(62\!\cdots\!36\)\( \nu^{2} - \)\(52\!\cdots\!84\)\( \nu + \)\(27\!\cdots\!40\)\(\)\()/ \)\(72\!\cdots\!35\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(10\!\cdots\!99\)\( \nu^{19} + \)\(45\!\cdots\!26\)\( \nu^{18} - \)\(87\!\cdots\!70\)\( \nu^{17} + \)\(17\!\cdots\!43\)\( \nu^{16} - \)\(47\!\cdots\!14\)\( \nu^{15} + \)\(11\!\cdots\!94\)\( \nu^{14} - \)\(77\!\cdots\!34\)\( \nu^{13} + \)\(11\!\cdots\!39\)\( \nu^{12} - \)\(66\!\cdots\!53\)\( \nu^{11} + \)\(99\!\cdots\!66\)\( \nu^{10} - \)\(32\!\cdots\!44\)\( \nu^{9} + \)\(34\!\cdots\!99\)\( \nu^{8} - \)\(86\!\cdots\!82\)\( \nu^{7} + \)\(81\!\cdots\!70\)\( \nu^{6} - \)\(14\!\cdots\!64\)\( \nu^{5} + \)\(69\!\cdots\!36\)\( \nu^{4} - \)\(85\!\cdots\!31\)\( \nu^{3} + \)\(31\!\cdots\!16\)\( \nu^{2} - \)\(38\!\cdots\!36\)\( \nu + \)\(20\!\cdots\!60\)\(\)\()/ \)\(33\!\cdots\!05\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(10\!\cdots\!87\)\( \nu^{19} + \)\(43\!\cdots\!83\)\( \nu^{18} - \)\(87\!\cdots\!30\)\( \nu^{17} + \)\(16\!\cdots\!39\)\( \nu^{16} - \)\(47\!\cdots\!77\)\( \nu^{15} + \)\(10\!\cdots\!32\)\( \nu^{14} - \)\(76\!\cdots\!87\)\( \nu^{13} + \)\(10\!\cdots\!37\)\( \nu^{12} - \)\(64\!\cdots\!09\)\( \nu^{11} + \)\(89\!\cdots\!28\)\( \nu^{10} - \)\(30\!\cdots\!37\)\( \nu^{9} + \)\(30\!\cdots\!67\)\( \nu^{8} - \)\(81\!\cdots\!26\)\( \nu^{7} + \)\(72\!\cdots\!55\)\( \nu^{6} - \)\(13\!\cdots\!17\)\( \nu^{5} + \)\(56\!\cdots\!08\)\( \nu^{4} - \)\(72\!\cdots\!58\)\( \nu^{3} + \)\(22\!\cdots\!28\)\( \nu^{2} - \)\(29\!\cdots\!68\)\( \nu + \)\(14\!\cdots\!80\)\(\)\()/ \)\(24\!\cdots\!95\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(87\!\cdots\!19\)\( \nu^{19} - \)\(33\!\cdots\!56\)\( \nu^{18} + \)\(69\!\cdots\!00\)\( \nu^{17} - \)\(12\!\cdots\!38\)\( \nu^{16} + \)\(37\!\cdots\!99\)\( \nu^{15} - \)\(83\!\cdots\!14\)\( \nu^{14} + \)\(60\!\cdots\!19\)\( \nu^{13} - \)\(75\!\cdots\!44\)\( \nu^{12} + \)\(51\!\cdots\!43\)\( \nu^{11} - \)\(67\!\cdots\!46\)\( \nu^{10} + \)\(24\!\cdots\!49\)\( \nu^{9} - \)\(22\!\cdots\!64\)\( \nu^{8} + \)\(65\!\cdots\!12\)\( \nu^{7} - \)\(54\!\cdots\!90\)\( \nu^{6} + \)\(10\!\cdots\!59\)\( \nu^{5} - \)\(42\!\cdots\!66\)\( \nu^{4} + \)\(59\!\cdots\!26\)\( \nu^{3} + \)\(37\!\cdots\!84\)\( \nu^{2} + \)\(24\!\cdots\!56\)\( \nu - \)\(16\!\cdots\!20\)\(\)\()/ \)\(13\!\cdots\!20\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(18\!\cdots\!99\)\( \nu^{19} - \)\(71\!\cdots\!29\)\( \nu^{18} + \)\(14\!\cdots\!90\)\( \nu^{17} - \)\(26\!\cdots\!63\)\( \nu^{16} + \)\(79\!\cdots\!45\)\( \nu^{15} - \)\(17\!\cdots\!40\)\( \nu^{14} + \)\(12\!\cdots\!37\)\( \nu^{13} - \)\(15\!\cdots\!17\)\( \nu^{12} + \)\(10\!\cdots\!09\)\( \nu^{11} - \)\(14\!\cdots\!12\)\( \nu^{10} + \)\(51\!\cdots\!19\)\( \nu^{9} - \)\(47\!\cdots\!91\)\( \nu^{8} + \)\(13\!\cdots\!46\)\( \nu^{7} - \)\(11\!\cdots\!89\)\( \nu^{6} + \)\(23\!\cdots\!15\)\( \nu^{5} - \)\(96\!\cdots\!64\)\( \nu^{4} + \)\(15\!\cdots\!22\)\( \nu^{3} - \)\(16\!\cdots\!76\)\( \nu^{2} + \)\(78\!\cdots\!60\)\( \nu - \)\(41\!\cdots\!00\)\(\)\()/ \)\(27\!\cdots\!64\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(32\!\cdots\!03\)\( \nu^{19} - \)\(12\!\cdots\!02\)\( \nu^{18} + \)\(25\!\cdots\!60\)\( \nu^{17} - \)\(48\!\cdots\!16\)\( \nu^{16} + \)\(13\!\cdots\!83\)\( \nu^{15} - \)\(32\!\cdots\!38\)\( \nu^{14} + \)\(22\!\cdots\!03\)\( \nu^{13} - \)\(29\!\cdots\!38\)\( \nu^{12} + \)\(18\!\cdots\!31\)\( \nu^{11} - \)\(26\!\cdots\!02\)\( \nu^{10} + \)\(89\!\cdots\!13\)\( \nu^{9} - \)\(91\!\cdots\!98\)\( \nu^{8} + \)\(23\!\cdots\!44\)\( \nu^{7} - \)\(22\!\cdots\!20\)\( \nu^{6} + \)\(39\!\cdots\!23\)\( \nu^{5} - \)\(18\!\cdots\!22\)\( \nu^{4} + \)\(20\!\cdots\!02\)\( \nu^{3} - \)\(94\!\cdots\!72\)\( \nu^{2} + \)\(82\!\cdots\!52\)\( \nu - \)\(27\!\cdots\!00\)\(\)\()/ \)\(45\!\cdots\!40\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(28\!\cdots\!81\)\( \nu^{19} + \)\(11\!\cdots\!47\)\( \nu^{18} - \)\(22\!\cdots\!70\)\( \nu^{17} + \)\(43\!\cdots\!17\)\( \nu^{16} - \)\(12\!\cdots\!83\)\( \nu^{15} + \)\(28\!\cdots\!24\)\( \nu^{14} - \)\(20\!\cdots\!91\)\( \nu^{13} + \)\(26\!\cdots\!15\)\( \nu^{12} - \)\(17\!\cdots\!63\)\( \nu^{11} + \)\(23\!\cdots\!68\)\( \nu^{10} - \)\(80\!\cdots\!69\)\( \nu^{9} + \)\(81\!\cdots\!29\)\( \nu^{8} - \)\(21\!\cdots\!74\)\( \nu^{7} + \)\(19\!\cdots\!99\)\( \nu^{6} - \)\(36\!\cdots\!49\)\( \nu^{5} + \)\(15\!\cdots\!64\)\( \nu^{4} - \)\(19\!\cdots\!18\)\( \nu^{3} + \)\(23\!\cdots\!60\)\( \nu^{2} - \)\(83\!\cdots\!24\)\( \nu + \)\(43\!\cdots\!40\)\(\)\()/ \)\(27\!\cdots\!64\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(10\!\cdots\!85\)\( \nu^{19} + \)\(41\!\cdots\!49\)\( \nu^{18} - \)\(83\!\cdots\!26\)\( \nu^{17} + \)\(15\!\cdots\!95\)\( \nu^{16} - \)\(44\!\cdots\!01\)\( \nu^{15} + \)\(10\!\cdots\!36\)\( \nu^{14} - \)\(72\!\cdots\!17\)\( \nu^{13} + \)\(96\!\cdots\!85\)\( \nu^{12} - \)\(61\!\cdots\!67\)\( \nu^{11} + \)\(86\!\cdots\!24\)\( \nu^{10} - \)\(29\!\cdots\!55\)\( \nu^{9} + \)\(29\!\cdots\!07\)\( \nu^{8} - \)\(79\!\cdots\!74\)\( \nu^{7} + \)\(70\!\cdots\!01\)\( \nu^{6} - \)\(13\!\cdots\!87\)\( \nu^{5} + \)\(58\!\cdots\!56\)\( \nu^{4} - \)\(78\!\cdots\!08\)\( \nu^{3} + \)\(42\!\cdots\!00\)\( \nu^{2} - \)\(36\!\cdots\!24\)\( \nu + \)\(19\!\cdots\!40\)\(\)\()/ \)\(67\!\cdots\!41\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{18} - \beta_{17} - \beta_{15} + \beta_{13} - \beta_{12} - 2 \beta_{11} - \beta_{9} - 4 \beta_{4} + 4 \beta_{3} + 4\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(-10 \beta_{19} + 7 \beta_{18} + 3 \beta_{17} - 6 \beta_{16} - 9 \beta_{15} + 14 \beta_{14} + 11 \beta_{13} - 29 \beta_{12} - 10 \beta_{11} + 7 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 4 \beta_{1} - 118\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{17} - 5 \beta_{15} - 9 \beta_{10} + 6 \beta_{7} + 10 \beta_{6} + 3 \beta_{5} + 102 \beta_{4} - 75 \beta_{3} + 2 \beta_{2} - 8 \beta_{1} - 112\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(600 \beta_{19} - 333 \beta_{18} + 481 \beta_{17} + 338 \beta_{16} - 191 \beta_{15} - 910 \beta_{14} - 769 \beta_{13} + 1281 \beta_{12} + 732 \beta_{11} - 60 \beta_{10} - 469 \beta_{9} - 70 \beta_{8} + 280 \beta_{7} - 84 \beta_{6} - 108 \beta_{5} + 436 \beta_{4} - 12 \beta_{3} + 264 \beta_{2} + 164 \beta_{1} - 5780\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(1480 \beta_{19} + 867 \beta_{18} + 1379 \beta_{17} + 756 \beta_{16} + 3179 \beta_{15} - 3794 \beta_{14} - 2659 \beta_{13} + 3847 \beta_{12} + 9050 \beta_{11} + 1256 \beta_{10} - 833 \beta_{9} - 464 \beta_{8} - 1110 \beta_{7} - 1216 \beta_{6} - 292 \beta_{5} - 11762 \beta_{4} + 8012 \beta_{3} - 580 \beta_{2} + 788 \beta_{1} + 17968\)\()/16\)
\(\nu^{6}\)\(=\)\((\)\(-9863 \beta_{17} + 9863 \beta_{15} + 2644 \beta_{10} - 8802 \beta_{7} + 1613 \beta_{6} + 2813 \beta_{5} - 18710 \beta_{4} + 4499 \beta_{3} - 7962 \beta_{2} - 3966 \beta_{1} + 169367\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-123774 \beta_{19} - 28069 \beta_{18} - 209999 \beta_{17} - 63720 \beta_{16} - 65531 \beta_{15} + 281280 \beta_{14} + 204821 \beta_{13} - 292735 \beta_{12} - 572686 \beta_{11} + 79034 \beta_{10} + 83059 \beta_{9} + 29208 \beta_{8} - 84024 \beta_{7} - 66964 \beta_{6} - 11110 \beta_{5} - 705392 \beta_{4} + 459246 \beta_{3} - 51540 \beta_{2} + 36408 \beta_{1} + 1380264\)\()/16\)
\(\nu^{8}\)\(=\)\((\)\(-2157980 \beta_{19} + 963553 \beta_{18} + 483971 \beta_{17} - 1182358 \beta_{16} - 1904885 \beta_{15} + 3534834 \beta_{14} + 2940433 \beta_{13} - 4438749 \beta_{12} - 3578644 \beta_{11} - 394036 \beta_{10} + 1718841 \beta_{9} + 242946 \beta_{8} + 1097296 \beta_{7} - 105064 \beta_{6} - 303680 \beta_{5} + 2901828 \beta_{4} - 1013384 \beta_{3} + 963552 \beta_{2} + 403684 \beta_{1} - 20645344\)\()/16\)
\(\nu^{9}\)\(=\)\((\)\(2708064 \beta_{17} - 2708064 \beta_{15} - 2471014 \beta_{10} + 3002705 \beta_{7} + 1847372 \beta_{6} + 150332 \beta_{5} + 21589639 \beta_{4} - 13551088 \beta_{3} + 2004278 \beta_{2} - 810542 \beta_{1} - 50524736\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(132802650 \beta_{19} - 54027105 \beta_{18} + 123398467 \beta_{17} + 72435042 \beta_{16} - 23966185 \beta_{15} - 222678954 \beta_{14} - 183401109 \beta_{13} + 275028275 \beta_{12} + 243576330 \beta_{11} - 27925480 \beta_{10} - 105013473 \beta_{9} - 15809286 \beta_{8} + 68956556 \beta_{7} - 1783402 \beta_{6} - 16836026 \beta_{5} + 212575748 \beta_{4} - 87566438 \beta_{3} + 59120324 \beta_{2} + 21006812 \beta_{1} - 1281689150\)\()/16\)
\(\nu^{11}\)\(=\)\((\)\(682394814 \beta_{19} - 31995467 \beta_{18} + 151603019 \beta_{17} + 359353008 \beta_{16} + 929726799 \beta_{15} - 1378101024 \beta_{14} - 1050118429 \beta_{13} + 1521227143 \beta_{12} + 2338913918 \beta_{11} + 311514098 \beta_{10} - 496151483 \beta_{9} - 125437848 \beta_{8} - 417554392 \beta_{7} - 207564244 \beta_{6} + 1643810 \beta_{5} - 2683769112 \beta_{4} + 1632911606 \beta_{3} - 293332924 \beta_{2} + 68517048 \beta_{1} + 7154649976\)\()/16\)
\(\nu^{12}\)\(=\)\((\)\(-1152279790 \beta_{17} + 1152279790 \beta_{15} + 482830935 \beta_{10} - 1093657394 \beta_{7} - 32850285 \beta_{6} + 238841537 \beta_{5} - 3760559113 \beta_{4} + 1696064685 \beta_{3} - 919543950 \beta_{2} - 278279749 \beta_{1} + 20151252679\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(-47924797136 \beta_{19} + 5273905181 \beta_{18} - 61643806723 \beta_{17} - 25392750068 \beta_{16} - 7258461387 \beta_{15} + 93910481090 \beta_{14} + 72428935779 \beta_{13} - 105482954367 \beta_{12} - 150871618090 \beta_{11} + 19812434304 \beta_{10} + 35389817633 \beta_{9} + 8260470800 \beta_{8} - 28550003430 \beta_{7} - 11910359152 \beta_{6} + 1158865172 \beta_{5} - 168870297810 \beta_{4} + 100097458724 \beta_{3} - 20732124468 \beta_{2} + 2640578004 \beta_{1} + 495414629696\)\()/16\)
\(\nu^{14}\)\(=\)\((\)\(-523121794890 \beta_{19} + 181381395201 \beta_{18} + 59297990013 \beta_{17} - 283635645014 \beta_{16} - 523777450319 \beta_{15} + 907210986478 \beta_{14} + 736334691337 \beta_{13} - 1096803691215 \beta_{12} - 1098706955022 \beta_{11} - 131580448280 \beta_{10} + 408249232033 \beta_{9} + 67855720594 \beta_{8} + 279785088628 \beta_{7} + 20758937646 \beta_{6} - 55422825906 \beta_{5} + 1040979325228 \beta_{4} - 496219465294 \beta_{3} + 231584074724 \beta_{2} + 60114922620 \beta_{1} - 5120627804022\)\()/16\)
\(\nu^{15}\)\(=\)\((\)\(933244928353 \beta_{17} - 933244928353 \beta_{15} - 634835552577 \beta_{10} + 965165474152 \beta_{7} + 349047400658 \beta_{6} - 63781030873 \beta_{5} + 5365786995180 \beta_{4} - 3113042180315 \beta_{3} + 716742562090 \beta_{2} - 41243216340 \beta_{1} - 16895883956052\)\()/4\)
\(\nu^{16}\)\(=\)\((\)\(33328220141620 \beta_{19} - 10837840970449 \beta_{18} + 34234246792029 \beta_{17} + 18032892033126 \beta_{16} - 2974689881899 \beta_{15} - 58482510946706 \beta_{14} - 47227026711825 \beta_{13} + 70193224084709 \beta_{12} + 73177851692844 \beta_{11} - 8874322224740 \beta_{10} - 25882771070217 \beta_{9} - 4452292836146 \beta_{8} + 18010072703520 \beta_{7} + 1956251229936 \beta_{6} - 3282854216488 \beta_{5} + 70973740307492 \beta_{4} - 35072564726080 \beta_{3} + 14723751804656 \beta_{2} + 3314038483204 \beta_{1} - 327898917272952\)\()/16\)
\(\nu^{17}\)\(=\)\((\)\(224219509291284 \beta_{19} - 40523183971793 \beta_{18} + 16225053128667 \beta_{17} + 119626753439604 \beta_{16} + 269314701997483 \beta_{15} - 424285070302014 \beta_{14} - 331922560207367 \beta_{13} + 486505766263591 \beta_{12} + 635706224632978 \beta_{11} + 81857904551716 \beta_{10} - 168390426715957 \beta_{9} - 35788210572336 \beta_{8} - 129498953243226 \beta_{7} - 41741858763400 \beta_{6} + 10895319483952 \beta_{5} - 687371047027094 \beta_{4} + 391968826176632 \beta_{3} - 97674684856876 \beta_{2} + 971831582924 \beta_{1} + 2281052946812768\)\()/16\)
\(\nu^{18}\)\(=\)\((\)\(-597776987190383 \beta_{17} + 597776987190383 \beta_{15} + 297148023316908 \beta_{10} - 582588423789262 \beta_{7} - 78740530663111 \beta_{6} + 99093480949057 \beta_{5} - 2394743651781714 \beta_{4} + 1212603307639391 \beta_{3} - 471705293045546 \beta_{2} - 93347984280926 \beta_{1} + 10564001496204795\)\()/4\)
\(\nu^{19}\)\(=\)\((\)\(-15094518578462470 \beta_{19} + 3032799220060659 \beta_{18} - 17763359190364863 \beta_{17} - 8069217204374368 \beta_{16} - 751294811000939 \beta_{15} + 28272843473724712 \beta_{14} + 22211542722760053 \beta_{13} - 32617360927833255 \beta_{12} - 41446682352470510 \beta_{11} + 5302242325672202 \beta_{10} + 11390107820328883 \beta_{9} + 2354360786209048 \beta_{8} - 8639494229171888 \beta_{7} - 2541150614272836 \beta_{6} + 838766987275018 \beta_{5} - 44299095246252896 \beta_{4} + 24919439683115726 \beta_{3} - 6588486388780508 \beta_{2} - 153502512636632 \beta_{1} + 152854604418559640\)\()/16\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
0.0545091 0.0944125i
0.0545091 + 0.0944125i
0.735188 + 1.27338i
0.735188 1.27338i
−0.861157 + 1.49157i
−0.861157 1.49157i
−3.49479 6.05315i
−3.49479 + 6.05315i
−1.34238 + 2.32507i
−1.34238 2.32507i
1.76236 + 3.05250i
1.76236 3.05250i
0.567379 0.982730i
0.567379 + 0.982730i
0.855886 + 1.48244i
0.855886 1.48244i
−0.330374 0.572225i
−0.330374 + 0.572225i
4.05338 + 7.02066i
4.05338 7.02066i
0 0 0 −7.51170 0 7.72974 0 0 0
721.2 0 0 0 −7.51170 0 7.72974 0 0 0
721.3 0 0 0 −6.80804 0 9.74880 0 0 0
721.4 0 0 0 −6.80804 0 9.74880 0 0 0
721.5 0 0 0 −5.26532 0 1.82069 0 0 0
721.6 0 0 0 −5.26532 0 1.82069 0 0 0
721.7 0 0 0 −2.73712 0 −11.2420 0 0 0
721.8 0 0 0 −2.73712 0 −11.2420 0 0 0
721.9 0 0 0 0.0118390 0 −5.38136 0 0 0
721.10 0 0 0 0.0118390 0 −5.38136 0 0 0
721.11 0 0 0 0.816130 0 6.23331 0 0 0
721.12 0 0 0 0.816130 0 6.23331 0 0 0
721.13 0 0 0 1.70658 0 0.562937 0 0 0
721.14 0 0 0 1.70658 0 0.562937 0 0 0
721.15 0 0 0 3.78754 0 −0.363991 0 0 0
721.16 0 0 0 3.78754 0 −0.363991 0 0 0
721.17 0 0 0 7.79518 0 −9.11667 0 0 0
721.18 0 0 0 7.79518 0 −9.11667 0 0 0
721.19 0 0 0 8.20492 0 8.00860 0 0 0
721.20 0 0 0 8.20492 0 8.00860 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 721.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.o.r 20
3.b odd 2 1 912.3.o.e 20
4.b odd 2 1 1368.3.o.c 20
12.b even 2 1 456.3.o.a 20
19.b odd 2 1 inner 2736.3.o.r 20
57.d even 2 1 912.3.o.e 20
76.d even 2 1 1368.3.o.c 20
228.b odd 2 1 456.3.o.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.3.o.a 20 12.b even 2 1
456.3.o.a 20 228.b odd 2 1
912.3.o.e 20 3.b odd 2 1
912.3.o.e 20 57.d even 2 1
1368.3.o.c 20 4.b odd 2 1
1368.3.o.c 20 76.d even 2 1
2736.3.o.r 20 1.a even 1 1 trivial
2736.3.o.r 20 19.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\):

\(T_{5}^{10} - \cdots\)
\(T_{7}^{10} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( T^{20} \)
$5$ \( ( 2944 - 253056 T + 370656 T^{2} - 7072 T^{3} - 96156 T^{4} + 3716 T^{5} + 6433 T^{6} - 60 T^{7} - 142 T^{8} + T^{10} )^{2} \)
$7$ \( ( 774016 + 223616 T - 4274352 T^{2} + 2619680 T^{3} - 51568 T^{4} - 140592 T^{5} + 12441 T^{6} + 1976 T^{7} - 218 T^{8} - 8 T^{9} + T^{10} )^{2} \)
$11$ \( ( -185601120 - 699933504 T - 287334272 T^{2} + 132051440 T^{3} - 3601694 T^{4} - 2390048 T^{5} + 172845 T^{6} + 8456 T^{7} - 800 T^{8} - 8 T^{9} + T^{10} )^{2} \)
$13$ \( 929457403743698944 + 2677263240999731200 T^{2} + 1037212928405667840 T^{4} + 59286678970826752 T^{6} + 1437599420416000 T^{8} + 18318464647168 T^{10} + 133552089344 T^{12} + 573286144 T^{14} + 1422624 T^{16} + 1872 T^{18} + T^{20} \)
$17$ \( ( -166822288832 + 14309774400 T + 22789587952 T^{2} - 765334752 T^{3} - 295180996 T^{4} + 7047716 T^{5} + 1201073 T^{6} - 20204 T^{7} - 1862 T^{8} + 16 T^{9} + T^{10} )^{2} \)
$19$ \( \)\(37\!\cdots\!01\)\( + \)\(41\!\cdots\!40\)\( T + \)\(21\!\cdots\!74\)\( T^{2} + \)\(44\!\cdots\!96\)\( T^{3} + \)\(23\!\cdots\!61\)\( T^{4} + 24337488228726350336 T^{5} + 1064159489735586200 T^{6} - 28846513890886336 T^{7} - 2854992367239918 T^{8} - 52136621691248 T^{9} + 3613212800364 T^{10} - 144422774768 T^{11} - 21907385358 T^{12} - 613157056 T^{13} + 62658200 T^{14} + 3969536 T^{15} + 105501 T^{16} + 5576 T^{17} + 754 T^{18} + 40 T^{19} + T^{20} \)
$23$ \( ( -11287201510400 - 1943856834560 T + 199478919168 T^{2} + 21268517120 T^{3} - 1169020832 T^{4} - 71143168 T^{5} + 2973216 T^{6} + 83920 T^{7} - 2974 T^{8} - 32 T^{9} + T^{10} )^{2} \)
$29$ \( \)\(22\!\cdots\!76\)\( + \)\(53\!\cdots\!28\)\( T^{2} + \)\(56\!\cdots\!96\)\( T^{4} + \)\(17\!\cdots\!88\)\( T^{6} + 1671052441844580352 T^{8} + 7542876140470272 T^{10} + 18627752724736 T^{12} + 26040660288 T^{14} + 19856900 T^{16} + 7316 T^{18} + T^{20} \)
$31$ \( \)\(26\!\cdots\!00\)\( + \)\(22\!\cdots\!28\)\( T^{2} + \)\(46\!\cdots\!68\)\( T^{4} + \)\(34\!\cdots\!56\)\( T^{6} + \)\(11\!\cdots\!04\)\( T^{8} + 219321248116410368 T^{10} + 239254325717760 T^{12} + 155437815552 T^{14} + 58869936 T^{16} + 11944 T^{18} + T^{20} \)
$37$ \( \)\(48\!\cdots\!00\)\( + \)\(15\!\cdots\!92\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{4} + \)\(70\!\cdots\!96\)\( T^{6} + \)\(18\!\cdots\!80\)\( T^{8} + 299570454320160768 T^{10} + 295152936577280 T^{12} + 178817929984 T^{14} + 64400800 T^{16} + 12528 T^{18} + T^{20} \)
$41$ \( \)\(41\!\cdots\!00\)\( + \)\(53\!\cdots\!12\)\( T^{2} + \)\(77\!\cdots\!84\)\( T^{4} + \)\(43\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{8} + 214494561746026496 T^{10} + 221106343751680 T^{12} + 141002485760 T^{14} + 54066244 T^{16} + 11364 T^{18} + T^{20} \)
$43$ \( ( 896271935895552 + 276170356164096 T + 12435366172048 T^{2} - 1451010696352 T^{3} - 43832456968 T^{4} + 1237012184 T^{5} + 35253993 T^{6} - 359160 T^{7} - 10386 T^{8} + 32 T^{9} + T^{10} )^{2} \)
$47$ \( ( -152806724137280 - 94113798350976 T + 34161994420448 T^{2} + 308419077648 T^{3} - 83603692070 T^{4} - 572271248 T^{5} + 56003709 T^{6} + 228136 T^{7} - 13416 T^{8} - 24 T^{9} + T^{10} )^{2} \)
$53$ \( \)\(35\!\cdots\!56\)\( + \)\(18\!\cdots\!80\)\( T^{2} + \)\(37\!\cdots\!60\)\( T^{4} + \)\(17\!\cdots\!80\)\( T^{6} + \)\(30\!\cdots\!48\)\( T^{8} + 25822366531611983872 T^{10} + 11676407272159232 T^{12} + 2999741949184 T^{14} + 434677956 T^{16} + 32868 T^{18} + T^{20} \)
$59$ \( \)\(67\!\cdots\!00\)\( + \)\(75\!\cdots\!72\)\( T^{2} + \)\(45\!\cdots\!76\)\( T^{4} + \)\(16\!\cdots\!36\)\( T^{6} + \)\(24\!\cdots\!48\)\( T^{8} + 19482825196703842304 T^{10} + 9047781746982912 T^{12} + 2469853788160 T^{14} + 383798080 T^{16} + 30992 T^{18} + T^{20} \)
$61$ \( ( -15246226889176832 - 6050558653640576 T - 383025929840368 T^{2} + 27079855482464 T^{3} - 75596869848 T^{4} - 12105645088 T^{5} + 97029057 T^{6} + 1846052 T^{7} - 18290 T^{8} - 92 T^{9} + T^{10} )^{2} \)
$67$ \( \)\(23\!\cdots\!00\)\( + \)\(10\!\cdots\!88\)\( T^{2} + \)\(15\!\cdots\!92\)\( T^{4} + \)\(77\!\cdots\!88\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{8} + 15936092997984845824 T^{10} + 8591500025644032 T^{12} + 2537836945152 T^{14} + 405586512 T^{16} + 32424 T^{18} + T^{20} \)
$71$ \( \)\(13\!\cdots\!96\)\( + \)\(11\!\cdots\!36\)\( T^{2} + \)\(32\!\cdots\!52\)\( T^{4} + \)\(42\!\cdots\!80\)\( T^{6} + \)\(28\!\cdots\!32\)\( T^{8} + \)\(10\!\cdots\!32\)\( T^{10} + 210339924940800000 T^{12} + 25225446000640 T^{14} + 1749325632 T^{16} + 64976 T^{18} + T^{20} \)
$73$ \( ( 2760295013492892672 - 195746857917982464 T + 1903776322443280 T^{2} + 113240465682400 T^{3} - 1799773080200 T^{4} - 23304625672 T^{5} + 412103409 T^{6} + 1929124 T^{7} - 35378 T^{8} - 52 T^{9} + T^{10} )^{2} \)
$79$ \( \)\(22\!\cdots\!76\)\( + \)\(17\!\cdots\!72\)\( T^{2} + \)\(37\!\cdots\!40\)\( T^{4} + \)\(38\!\cdots\!40\)\( T^{6} + \)\(21\!\cdots\!28\)\( T^{8} + \)\(73\!\cdots\!92\)\( T^{10} + 149836276817957120 T^{12} + 18818284311808 T^{14} + 1412281312 T^{16} + 57984 T^{18} + T^{20} \)
$83$ \( ( -555706264565847552 + 86750290257460224 T - 3834496057074944 T^{2} + 35099813913856 T^{3} + 1555295094464 T^{4} - 38121749696 T^{5} + 105127168 T^{6} + 4106672 T^{7} - 28678 T^{8} - 112 T^{9} + T^{10} )^{2} \)
$89$ \( \)\(21\!\cdots\!36\)\( + \)\(12\!\cdots\!84\)\( T^{2} + \)\(26\!\cdots\!12\)\( T^{4} + \)\(25\!\cdots\!44\)\( T^{6} + \)\(12\!\cdots\!88\)\( T^{8} + \)\(33\!\cdots\!04\)\( T^{10} + 536351744205864960 T^{12} + 50788268544256 T^{14} + 2800937988 T^{16} + 82564 T^{18} + T^{20} \)
$97$ \( \)\(10\!\cdots\!96\)\( + \)\(45\!\cdots\!40\)\( T^{2} + \)\(67\!\cdots\!76\)\( T^{4} + \)\(47\!\cdots\!88\)\( T^{6} + \)\(18\!\cdots\!40\)\( T^{8} + \)\(42\!\cdots\!16\)\( T^{10} + 614997924778151936 T^{12} + 54494427877376 T^{14} + 2887229568 T^{16} + 83296 T^{18} + T^{20} \)
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