Properties

Label 400.6.n.g
Level $400$
Weight $6$
Character orbit 400.n
Analytic conductor $64.154$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.1535279252\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 271 x^{18} + 109637 x^{16} + 25993614 x^{14} + 5522961902 x^{12} + 881545050522 x^{10} + 133816049059481 x^{8} + 14779507781220031 x^{6} + 824105698447750789 x^{4} + 12044868290803250652 x^{2} + 579398322543528055824\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{67}\cdot 3^{4}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{6} q^{3} + ( -\beta_{5} + \beta_{15} ) q^{7} + ( 103 \beta_{4} + \beta_{18} ) q^{9} +O(q^{10})\) \( q -\beta_{6} q^{3} + ( -\beta_{5} + \beta_{15} ) q^{7} + ( 103 \beta_{4} + \beta_{18} ) q^{9} + ( -\beta_{2} + \beta_{5} + \beta_{6} + \beta_{15} - \beta_{16} ) q^{11} + ( -40 - \beta_{1} + 40 \beta_{4} - 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{18} ) q^{13} + ( 112 - \beta_{1} + 112 \beta_{4} + 4 \beta_{7} + 5 \beta_{9} - 2 \beta_{10} + \beta_{12} + 2 \beta_{18} ) q^{17} + ( 5 \beta_{5} - 5 \beta_{6} - 2 \beta_{13} - 2 \beta_{14} - 3 \beta_{15} - 3 \beta_{16} - \beta_{17} - \beta_{19} ) q^{19} + ( -224 - 13 \beta_{1} - 12 \beta_{8} - 10 \beta_{9} + 2 \beta_{10} - 5 \beta_{11} + 5 \beta_{12} ) q^{21} + ( 3 \beta_{2} + \beta_{3} - 59 \beta_{6} + \beta_{14} - 5 \beta_{16} - 3 \beta_{17} + \beta_{19} ) q^{23} + ( 7 \beta_{2} - \beta_{3} - 142 \beta_{5} + 6 \beta_{13} + 7 \beta_{17} + \beta_{19} ) q^{27} + ( 19 \beta_{1} + 960 \beta_{4} - 16 \beta_{7} + 19 \beta_{8} - 19 \beta_{9} + 10 \beta_{11} + 10 \beta_{12} - 2 \beta_{18} ) q^{29} + ( -8 \beta_{2} - \beta_{3} - 96 \beta_{5} - 96 \beta_{6} + 4 \beta_{13} - 4 \beta_{14} - 23 \beta_{15} + 23 \beta_{16} ) q^{31} + ( 559 - 12 \beta_{1} - 559 \beta_{4} + 20 \beta_{7} - 31 \beta_{8} - 8 \beta_{9} - 8 \beta_{10} - 21 \beta_{11} - 8 \beta_{18} ) q^{33} + ( -2209 - 18 \beta_{1} - 2209 \beta_{4} - 25 \beta_{7} - 7 \beta_{9} + 14 \beta_{10} + 20 \beta_{12} - 14 \beta_{18} ) q^{37} + ( -225 \beta_{5} + 225 \beta_{6} - 11 \beta_{13} - 11 \beta_{14} + 5 \beta_{15} + 5 \beta_{16} ) q^{39} + ( -324 - 26 \beta_{1} - 9 \beta_{8} - 20 \beta_{9} - 11 \beta_{10} - 35 \beta_{11} + 35 \beta_{12} ) q^{41} + ( -8 \beta_{2} - \beta_{3} - 323 \beta_{6} - 36 \beta_{14} - 10 \beta_{16} + 8 \beta_{17} - \beta_{19} ) q^{43} + ( -19 \beta_{2} + 2 \beta_{3} - 97 \beta_{5} + 23 \beta_{13} + 11 \beta_{15} - 19 \beta_{17} - 2 \beta_{19} ) q^{47} + ( 35 \beta_{1} - 89 \beta_{4} - 120 \beta_{7} + 35 \beta_{8} - 35 \beta_{9} + 25 \beta_{11} + 25 \beta_{12} - 23 \beta_{18} ) q^{49} + ( 30 \beta_{2} + 3 \beta_{3} - 568 \beta_{5} - 568 \beta_{6} + 16 \beta_{13} - 16 \beta_{14} + 68 \beta_{15} - 68 \beta_{16} ) q^{51} + ( -9114 - 42 \beta_{1} + 9114 \beta_{4} + 17 \beta_{7} + 50 \beta_{8} + 25 \beta_{9} + 25 \beta_{10} - 43 \beta_{11} + 25 \beta_{18} ) q^{53} + ( 1719 + 110 \beta_{1} + 1719 \beta_{4} - 46 \beta_{7} - 156 \beta_{9} - 27 \beta_{10} + 23 \beta_{12} + 27 \beta_{18} ) q^{57} + ( -59 \beta_{5} + 59 \beta_{6} - 42 \beta_{13} - 42 \beta_{14} + 59 \beta_{15} + 59 \beta_{16} + 15 \beta_{17} + 11 \beta_{19} ) q^{59} + ( -2044 + 72 \beta_{1} - 55 \beta_{8} - 47 \beta_{9} + 8 \beta_{10} - 25 \beta_{11} + 25 \beta_{12} ) q^{61} + ( -5 \beta_{2} - 10 \beta_{3} + 163 \beta_{6} - 185 \beta_{14} + 179 \beta_{16} + 5 \beta_{17} - 10 \beta_{19} ) q^{63} + ( -26 \beta_{2} + 8 \beta_{3} + 1099 \beta_{5} + 92 \beta_{13} - 14 \beta_{15} - 26 \beta_{17} - 8 \beta_{19} ) q^{67} + ( -209 \beta_{1} + 20062 \beta_{4} + 321 \beta_{7} - 209 \beta_{8} + 209 \beta_{9} - 10 \beta_{11} - 10 \beta_{12} + 90 \beta_{18} ) q^{69} + ( 40 \beta_{2} + 5 \beta_{3} + 530 \beta_{5} + 530 \beta_{6} + 67 \beta_{13} - 67 \beta_{14} + 5 \beta_{15} - 5 \beta_{16} ) q^{71} + ( -13230 - 143 \beta_{1} + 13230 \beta_{4} + 182 \beta_{7} - 229 \beta_{8} - 39 \beta_{9} - 39 \beta_{10} + 57 \beta_{11} - 39 \beta_{18} ) q^{73} + ( -19981 - 197 \beta_{1} - 19981 \beta_{4} - 41 \beta_{7} + 156 \beta_{9} - 19 \beta_{10} - 79 \beta_{12} + 19 \beta_{18} ) q^{77} + ( -163 \beta_{5} + 163 \beta_{6} - 166 \beta_{13} - 166 \beta_{14} - 224 \beta_{15} - 224 \beta_{16} - 24 \beta_{17} + 3 \beta_{19} ) q^{79} + ( -26065 - 66 \beta_{1} + 259 \beta_{8} + 292 \beta_{9} + 33 \beta_{10} + 135 \beta_{11} - 135 \beta_{12} ) q^{81} + ( 9 \beta_{2} + 8 \beta_{3} + 163 \beta_{6} - 372 \beta_{14} - 416 \beta_{16} - 9 \beta_{17} + 8 \beta_{19} ) q^{83} + ( 74 \beta_{2} - 17 \beta_{3} - 1750 \beta_{5} + 342 \beta_{13} - 354 \beta_{15} + 74 \beta_{17} + 17 \beta_{19} ) q^{87} + ( 128 \beta_{1} + 28740 \beta_{4} - 182 \beta_{7} + 128 \beta_{8} - 128 \beta_{9} - 130 \beta_{11} - 130 \beta_{12} - 46 \beta_{18} ) q^{89} + ( -112 \beta_{2} - 23 \beta_{3} + 570 \beta_{5} + 570 \beta_{6} + 198 \beta_{13} - 198 \beta_{14} - 46 \beta_{15} + 46 \beta_{16} ) q^{91} + ( -35718 + 683 \beta_{1} + 35718 \beta_{4} - 703 \beta_{7} + 722 \beta_{8} + 20 \beta_{9} + 20 \beta_{10} + 216 \beta_{11} + 20 \beta_{18} ) q^{93} + ( -18762 + 218 \beta_{1} - 18762 \beta_{4} + 634 \beta_{7} + 416 \beta_{9} + 99 \beta_{10} - 117 \beta_{12} - 99 \beta_{18} ) q^{97} + ( 3562 \beta_{5} - 3562 \beta_{6} - 414 \beta_{13} - 414 \beta_{14} + 171 \beta_{15} + 171 \beta_{16} + \beta_{17} - 47 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + O(q^{10}) \) \( 20q - 804q^{13} + 2236q^{17} - 4520q^{21} + 11096q^{33} - 44260q^{37} - 6760q^{41} - 182452q^{53} + 34288q^{57} - 41080q^{61} - 264372q^{73} - 399304q^{77} - 520220q^{81} - 713496q^{93} - 374772q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 271 x^{18} + 109637 x^{16} + 25993614 x^{14} + 5522961902 x^{12} + 881545050522 x^{10} + 133816049059481 x^{8} + 14779507781220031 x^{6} + 824105698447750789 x^{4} + 12044868290803250652 x^{2} + 579398322543528055824\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(21\!\cdots\!17\)\( \nu^{18} - \)\(27\!\cdots\!66\)\( \nu^{16} - \)\(92\!\cdots\!48\)\( \nu^{14} - \)\(14\!\cdots\!30\)\( \nu^{12} - \)\(22\!\cdots\!52\)\( \nu^{10} - \)\(31\!\cdots\!96\)\( \nu^{8} - \)\(40\!\cdots\!67\)\( \nu^{6} - \)\(23\!\cdots\!06\)\( \nu^{4} - \)\(33\!\cdots\!40\)\( \nu^{2} + \)\(21\!\cdots\!84\)\(\)\()/ \)\(11\!\cdots\!08\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(12\!\cdots\!45\)\( \nu^{18} - \)\(82\!\cdots\!16\)\( \nu^{16} + \)\(12\!\cdots\!04\)\( \nu^{14} - \)\(77\!\cdots\!26\)\( \nu^{12} - \)\(92\!\cdots\!08\)\( \nu^{10} - \)\(32\!\cdots\!56\)\( \nu^{8} - \)\(33\!\cdots\!39\)\( \nu^{6} - \)\(73\!\cdots\!00\)\( \nu^{4} - \)\(46\!\cdots\!32\)\( \nu^{2} - \)\(59\!\cdots\!24\)\(\)\()/ \)\(11\!\cdots\!60\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(24\!\cdots\!39\)\( \nu^{18} - \)\(75\!\cdots\!68\)\( \nu^{16} + \)\(28\!\cdots\!84\)\( \nu^{14} - \)\(57\!\cdots\!62\)\( \nu^{12} - \)\(91\!\cdots\!96\)\( \nu^{10} - \)\(26\!\cdots\!32\)\( \nu^{8} - \)\(27\!\cdots\!09\)\( \nu^{6} - \)\(74\!\cdots\!52\)\( \nu^{4} - \)\(52\!\cdots\!88\)\( \nu^{2} - \)\(60\!\cdots\!64\)\(\)\()/ \)\(22\!\cdots\!20\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(10\!\cdots\!57\)\( \nu^{19} + \)\(48\!\cdots\!81\)\( \nu^{17} + \)\(74\!\cdots\!38\)\( \nu^{15} + \)\(49\!\cdots\!50\)\( \nu^{13} + \)\(11\!\cdots\!18\)\( \nu^{11} + \)\(26\!\cdots\!44\)\( \nu^{9} + \)\(39\!\cdots\!11\)\( \nu^{7} + \)\(65\!\cdots\!81\)\( \nu^{5} + \)\(77\!\cdots\!98\)\( \nu^{3} + \)\(27\!\cdots\!16\)\( \nu\)\()/ \)\(16\!\cdots\!20\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(42\!\cdots\!31\)\( \nu^{19} + \)\(18\!\cdots\!83\)\( \nu^{18} + \)\(11\!\cdots\!49\)\( \nu^{17} + \)\(53\!\cdots\!64\)\( \nu^{16} + \)\(45\!\cdots\!44\)\( \nu^{15} + \)\(21\!\cdots\!68\)\( \nu^{14} + \)\(10\!\cdots\!86\)\( \nu^{13} + \)\(52\!\cdots\!46\)\( \nu^{12} + \)\(22\!\cdots\!30\)\( \nu^{11} + \)\(11\!\cdots\!08\)\( \nu^{10} + \)\(35\!\cdots\!92\)\( \nu^{9} + \)\(18\!\cdots\!56\)\( \nu^{8} + \)\(52\!\cdots\!07\)\( \nu^{7} + \)\(28\!\cdots\!07\)\( \nu^{6} + \)\(55\!\cdots\!57\)\( \nu^{5} + \)\(31\!\cdots\!36\)\( \nu^{4} + \)\(26\!\cdots\!00\)\( \nu^{3} + \)\(18\!\cdots\!44\)\( \nu^{2} - \)\(17\!\cdots\!08\)\( \nu + \)\(25\!\cdots\!12\)\(\)\()/ \)\(26\!\cdots\!80\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(42\!\cdots\!31\)\( \nu^{19} + \)\(18\!\cdots\!83\)\( \nu^{18} - \)\(11\!\cdots\!49\)\( \nu^{17} + \)\(53\!\cdots\!64\)\( \nu^{16} - \)\(45\!\cdots\!44\)\( \nu^{15} + \)\(21\!\cdots\!68\)\( \nu^{14} - \)\(10\!\cdots\!86\)\( \nu^{13} + \)\(52\!\cdots\!46\)\( \nu^{12} - \)\(22\!\cdots\!30\)\( \nu^{11} + \)\(11\!\cdots\!08\)\( \nu^{10} - \)\(35\!\cdots\!92\)\( \nu^{9} + \)\(18\!\cdots\!56\)\( \nu^{8} - \)\(52\!\cdots\!07\)\( \nu^{7} + \)\(28\!\cdots\!07\)\( \nu^{6} - \)\(55\!\cdots\!57\)\( \nu^{5} + \)\(31\!\cdots\!36\)\( \nu^{4} - \)\(26\!\cdots\!00\)\( \nu^{3} + \)\(18\!\cdots\!44\)\( \nu^{2} + \)\(17\!\cdots\!08\)\( \nu + \)\(25\!\cdots\!12\)\(\)\()/ \)\(26\!\cdots\!80\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(46\!\cdots\!17\)\( \nu^{19} + \)\(16\!\cdots\!91\)\( \nu^{17} + \)\(14\!\cdots\!48\)\( \nu^{15} + \)\(16\!\cdots\!70\)\( \nu^{13} + \)\(36\!\cdots\!58\)\( \nu^{11} + \)\(87\!\cdots\!44\)\( \nu^{9} + \)\(13\!\cdots\!91\)\( \nu^{7} + \)\(21\!\cdots\!51\)\( \nu^{5} + \)\(25\!\cdots\!08\)\( \nu^{3} + \)\(88\!\cdots\!96\)\( \nu\)\()/ \)\(28\!\cdots\!20\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(65\!\cdots\!21\)\( \nu^{19} - \)\(68\!\cdots\!45\)\( \nu^{18} - \)\(82\!\cdots\!47\)\( \nu^{17} - \)\(10\!\cdots\!40\)\( \nu^{16} - \)\(54\!\cdots\!76\)\( \nu^{15} - \)\(63\!\cdots\!20\)\( \nu^{14} - \)\(85\!\cdots\!70\)\( \nu^{13} - \)\(11\!\cdots\!50\)\( \nu^{12} - \)\(19\!\cdots\!06\)\( \nu^{11} - \)\(24\!\cdots\!80\)\( \nu^{10} - \)\(24\!\cdots\!28\)\( \nu^{9} - \)\(36\!\cdots\!40\)\( \nu^{8} - \)\(37\!\cdots\!77\)\( \nu^{7} - \)\(50\!\cdots\!05\)\( \nu^{6} - \)\(27\!\cdots\!47\)\( \nu^{5} - \)\(46\!\cdots\!40\)\( \nu^{4} + \)\(14\!\cdots\!04\)\( \nu^{3} - \)\(43\!\cdots\!00\)\( \nu^{2} + \)\(30\!\cdots\!88\)\( \nu - \)\(73\!\cdots\!20\)\(\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(65\!\cdots\!21\)\( \nu^{19} - \)\(64\!\cdots\!10\)\( \nu^{18} + \)\(82\!\cdots\!47\)\( \nu^{17} - \)\(16\!\cdots\!70\)\( \nu^{16} + \)\(54\!\cdots\!76\)\( \nu^{15} - \)\(65\!\cdots\!60\)\( \nu^{14} + \)\(85\!\cdots\!70\)\( \nu^{13} - \)\(14\!\cdots\!00\)\( \nu^{12} + \)\(19\!\cdots\!06\)\( \nu^{11} - \)\(29\!\cdots\!40\)\( \nu^{10} + \)\(24\!\cdots\!28\)\( \nu^{9} - \)\(42\!\cdots\!20\)\( \nu^{8} + \)\(37\!\cdots\!77\)\( \nu^{7} - \)\(58\!\cdots\!90\)\( \nu^{6} + \)\(27\!\cdots\!47\)\( \nu^{5} - \)\(50\!\cdots\!70\)\( \nu^{4} - \)\(14\!\cdots\!04\)\( \nu^{3} - \)\(49\!\cdots\!00\)\( \nu^{2} - \)\(30\!\cdots\!88\)\( \nu + \)\(32\!\cdots\!00\)\(\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(65\!\cdots\!21\)\( \nu^{19} + \)\(40\!\cdots\!66\)\( \nu^{18} - \)\(82\!\cdots\!47\)\( \nu^{17} + \)\(80\!\cdots\!94\)\( \nu^{16} - \)\(54\!\cdots\!76\)\( \nu^{15} + \)\(37\!\cdots\!04\)\( \nu^{14} - \)\(85\!\cdots\!70\)\( \nu^{13} + \)\(75\!\cdots\!76\)\( \nu^{12} - \)\(19\!\cdots\!06\)\( \nu^{11} + \)\(15\!\cdots\!40\)\( \nu^{10} - \)\(24\!\cdots\!28\)\( \nu^{9} + \)\(22\!\cdots\!72\)\( \nu^{8} - \)\(37\!\cdots\!77\)\( \nu^{7} + \)\(33\!\cdots\!42\)\( \nu^{6} - \)\(27\!\cdots\!47\)\( \nu^{5} + \)\(31\!\cdots\!42\)\( \nu^{4} + \)\(14\!\cdots\!04\)\( \nu^{3} + \)\(29\!\cdots\!40\)\( \nu^{2} + \)\(30\!\cdots\!88\)\( \nu - \)\(27\!\cdots\!88\)\(\)\()/ \)\(21\!\cdots\!40\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(10\!\cdots\!61\)\( \nu^{19} - \)\(10\!\cdots\!06\)\( \nu^{18} - \)\(28\!\cdots\!01\)\( \nu^{17} - \)\(23\!\cdots\!74\)\( \nu^{16} - \)\(11\!\cdots\!10\)\( \nu^{15} - \)\(10\!\cdots\!84\)\( \nu^{14} - \)\(27\!\cdots\!94\)\( \nu^{13} - \)\(23\!\cdots\!96\)\( \nu^{12} - \)\(59\!\cdots\!18\)\( \nu^{11} - \)\(46\!\cdots\!20\)\( \nu^{10} - \)\(98\!\cdots\!32\)\( \nu^{9} - \)\(67\!\cdots\!32\)\( \nu^{8} - \)\(15\!\cdots\!53\)\( \nu^{7} - \)\(10\!\cdots\!42\)\( \nu^{6} - \)\(17\!\cdots\!17\)\( \nu^{5} - \)\(93\!\cdots\!22\)\( \nu^{4} - \)\(11\!\cdots\!94\)\( \nu^{3} - \)\(87\!\cdots\!40\)\( \nu^{2} - \)\(43\!\cdots\!28\)\( \nu + \)\(14\!\cdots\!48\)\(\)\()/ \)\(28\!\cdots\!20\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(10\!\cdots\!61\)\( \nu^{19} + \)\(10\!\cdots\!06\)\( \nu^{18} - \)\(28\!\cdots\!01\)\( \nu^{17} + \)\(23\!\cdots\!74\)\( \nu^{16} - \)\(11\!\cdots\!10\)\( \nu^{15} + \)\(10\!\cdots\!84\)\( \nu^{14} - \)\(27\!\cdots\!94\)\( \nu^{13} + \)\(23\!\cdots\!96\)\( \nu^{12} - \)\(59\!\cdots\!18\)\( \nu^{11} + \)\(46\!\cdots\!20\)\( \nu^{10} - \)\(98\!\cdots\!32\)\( \nu^{9} + \)\(67\!\cdots\!32\)\( \nu^{8} - \)\(15\!\cdots\!53\)\( \nu^{7} + \)\(10\!\cdots\!42\)\( \nu^{6} - \)\(17\!\cdots\!17\)\( \nu^{5} + \)\(93\!\cdots\!22\)\( \nu^{4} - \)\(11\!\cdots\!94\)\( \nu^{3} + \)\(87\!\cdots\!40\)\( \nu^{2} - \)\(43\!\cdots\!28\)\( \nu - \)\(14\!\cdots\!48\)\(\)\()/ \)\(28\!\cdots\!20\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(22\!\cdots\!60\)\( \nu^{19} + \)\(72\!\cdots\!68\)\( \nu^{18} + \)\(54\!\cdots\!60\)\( \nu^{17} + \)\(27\!\cdots\!16\)\( \nu^{16} + \)\(23\!\cdots\!40\)\( \nu^{15} + \)\(89\!\cdots\!92\)\( \nu^{14} + \)\(51\!\cdots\!20\)\( \nu^{13} + \)\(25\!\cdots\!36\)\( \nu^{12} + \)\(11\!\cdots\!80\)\( \nu^{11} + \)\(48\!\cdots\!40\)\( \nu^{10} + \)\(16\!\cdots\!20\)\( \nu^{9} + \)\(86\!\cdots\!92\)\( \nu^{8} + \)\(26\!\cdots\!20\)\( \nu^{7} + \)\(12\!\cdots\!44\)\( \nu^{6} + \)\(26\!\cdots\!00\)\( \nu^{5} + \)\(15\!\cdots\!16\)\( \nu^{4} + \)\(12\!\cdots\!40\)\( \nu^{3} + \)\(90\!\cdots\!28\)\( \nu^{2} - \)\(88\!\cdots\!40\)\( \nu + \)\(12\!\cdots\!64\)\(\)\()/ \)\(20\!\cdots\!05\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(22\!\cdots\!60\)\( \nu^{19} - \)\(72\!\cdots\!68\)\( \nu^{18} + \)\(54\!\cdots\!60\)\( \nu^{17} - \)\(27\!\cdots\!16\)\( \nu^{16} + \)\(23\!\cdots\!40\)\( \nu^{15} - \)\(89\!\cdots\!92\)\( \nu^{14} + \)\(51\!\cdots\!20\)\( \nu^{13} - \)\(25\!\cdots\!36\)\( \nu^{12} + \)\(11\!\cdots\!80\)\( \nu^{11} - \)\(48\!\cdots\!40\)\( \nu^{10} + \)\(16\!\cdots\!20\)\( \nu^{9} - \)\(86\!\cdots\!92\)\( \nu^{8} + \)\(26\!\cdots\!20\)\( \nu^{7} - \)\(12\!\cdots\!44\)\( \nu^{6} + \)\(26\!\cdots\!00\)\( \nu^{5} - \)\(15\!\cdots\!16\)\( \nu^{4} + \)\(12\!\cdots\!40\)\( \nu^{3} - \)\(90\!\cdots\!28\)\( \nu^{2} - \)\(88\!\cdots\!40\)\( \nu - \)\(12\!\cdots\!64\)\(\)\()/ \)\(20\!\cdots\!05\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(25\!\cdots\!11\)\( \nu^{19} - \)\(16\!\cdots\!12\)\( \nu^{18} - \)\(71\!\cdots\!04\)\( \nu^{17} - \)\(41\!\cdots\!63\)\( \nu^{16} - \)\(27\!\cdots\!84\)\( \nu^{15} - \)\(17\!\cdots\!68\)\( \nu^{14} - \)\(67\!\cdots\!46\)\( \nu^{13} - \)\(39\!\cdots\!72\)\( \nu^{12} - \)\(13\!\cdots\!00\)\( \nu^{11} - \)\(86\!\cdots\!50\)\( \nu^{10} - \)\(22\!\cdots\!12\)\( \nu^{9} - \)\(13\!\cdots\!84\)\( \nu^{8} - \)\(32\!\cdots\!27\)\( \nu^{7} - \)\(20\!\cdots\!24\)\( \nu^{6} - \)\(36\!\cdots\!52\)\( \nu^{5} - \)\(21\!\cdots\!39\)\( \nu^{4} - \)\(17\!\cdots\!00\)\( \nu^{3} - \)\(12\!\cdots\!80\)\( \nu^{2} + \)\(11\!\cdots\!68\)\( \nu - \)\(17\!\cdots\!64\)\(\)\()/ \)\(22\!\cdots\!40\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(25\!\cdots\!11\)\( \nu^{19} + \)\(16\!\cdots\!12\)\( \nu^{18} - \)\(71\!\cdots\!04\)\( \nu^{17} + \)\(41\!\cdots\!63\)\( \nu^{16} - \)\(27\!\cdots\!84\)\( \nu^{15} + \)\(17\!\cdots\!68\)\( \nu^{14} - \)\(67\!\cdots\!46\)\( \nu^{13} + \)\(39\!\cdots\!72\)\( \nu^{12} - \)\(13\!\cdots\!00\)\( \nu^{11} + \)\(86\!\cdots\!50\)\( \nu^{10} - \)\(22\!\cdots\!12\)\( \nu^{9} + \)\(13\!\cdots\!84\)\( \nu^{8} - \)\(32\!\cdots\!27\)\( \nu^{7} + \)\(20\!\cdots\!24\)\( \nu^{6} - \)\(36\!\cdots\!52\)\( \nu^{5} + \)\(21\!\cdots\!39\)\( \nu^{4} - \)\(17\!\cdots\!00\)\( \nu^{3} + \)\(12\!\cdots\!80\)\( \nu^{2} + \)\(11\!\cdots\!68\)\( \nu + \)\(17\!\cdots\!64\)\(\)\()/ \)\(22\!\cdots\!40\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(39\!\cdots\!47\)\( \nu^{19} - \)\(80\!\cdots\!13\)\( \nu^{17} - \)\(40\!\cdots\!28\)\( \nu^{15} - \)\(80\!\cdots\!22\)\( \nu^{13} - \)\(19\!\cdots\!10\)\( \nu^{11} - \)\(26\!\cdots\!24\)\( \nu^{9} - \)\(43\!\cdots\!59\)\( \nu^{7} - \)\(39\!\cdots\!69\)\( \nu^{5} - \)\(19\!\cdots\!00\)\( \nu^{3} + \)\(14\!\cdots\!56\)\( \nu\)\()/ \)\(30\!\cdots\!40\)\( \)
\(\beta_{18}\)\(=\)\((\)\(\)\(41\!\cdots\!27\)\( \nu^{19} + \)\(11\!\cdots\!63\)\( \nu^{17} + \)\(46\!\cdots\!88\)\( \nu^{15} + \)\(10\!\cdots\!42\)\( \nu^{13} + \)\(23\!\cdots\!30\)\( \nu^{11} + \)\(37\!\cdots\!44\)\( \nu^{9} + \)\(56\!\cdots\!59\)\( \nu^{7} + \)\(63\!\cdots\!99\)\( \nu^{5} + \)\(37\!\cdots\!40\)\( \nu^{3} + \)\(14\!\cdots\!84\)\( \nu\)\()/ \)\(28\!\cdots\!20\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(15\!\cdots\!99\)\( \nu^{19} + \)\(30\!\cdots\!01\)\( \nu^{17} + \)\(14\!\cdots\!56\)\( \nu^{15} + \)\(30\!\cdots\!14\)\( \nu^{13} + \)\(66\!\cdots\!10\)\( \nu^{11} + \)\(95\!\cdots\!48\)\( \nu^{9} + \)\(15\!\cdots\!43\)\( \nu^{7} + \)\(13\!\cdots\!73\)\( \nu^{5} + \)\(67\!\cdots\!80\)\( \nu^{3} - \)\(44\!\cdots\!52\)\( \nu\)\()/ \)\(18\!\cdots\!40\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-12 \beta_{19} + 80 \beta_{18} + 16 \beta_{17} + 96 \beta_{16} + 96 \beta_{15} + 19 \beta_{14} + 19 \beta_{13} - 60 \beta_{12} - 60 \beta_{11} - 52 \beta_{9} + 52 \beta_{8} - 88 \beta_{7} - 492 \beta_{6} + 492 \beta_{5} - 408 \beta_{4} + 52 \beta_{1}\)\()/3840\)
\(\nu^{2}\)\(=\)\((\)\(1968 \beta_{16} - 1968 \beta_{15} + 77 \beta_{14} - 77 \beta_{13} - 240 \beta_{12} + 240 \beta_{11} - 720 \beta_{10} - 1200 \beta_{9} - 480 \beta_{8} - 13416 \beta_{6} - 13416 \beta_{5} + 24 \beta_{3} - 608 \beta_{2} + 480 \beta_{1} - 104160\)\()/3840\)
\(\nu^{3}\)\(=\)\((\)\(339 \beta_{19} + 120 \beta_{18} - 472 \beta_{17} - 1032 \beta_{16} - 1032 \beta_{15} - 1588 \beta_{14} - 1588 \beta_{13} + 90 \beta_{12} + 90 \beta_{11} - 2262 \beta_{9} + 2262 \beta_{8} - 3828 \beta_{7} + 4179 \beta_{6} - 4179 \beta_{5} + 662052 \beta_{4} + 2262 \beta_{1}\)\()/480\)
\(\nu^{4}\)\(=\)\((\)\(54912 \beta_{16} - 54912 \beta_{15} + 36633 \beta_{14} - 36633 \beta_{13} + 79020 \beta_{12} - 79020 \beta_{11} + 77840 \beta_{10} + 240564 \beta_{9} + 162724 \beta_{8} - 744924 \beta_{6} - 744924 \beta_{5} - 38844 \beta_{3} + 52368 \beta_{2} + 249412 \beta_{1} - 55968264\)\()/3840\)
\(\nu^{5}\)\(=\)\((\)\(-50448 \beta_{19} + 648560 \beta_{18} - 1483456 \beta_{17} - 4880496 \beta_{16} - 4880496 \beta_{15} - 362479 \beta_{14} - 362479 \beta_{13} + 1941240 \beta_{12} + 1941240 \beta_{11} + 2169368 \beta_{9} - 2169368 \beta_{8} + 2057792 \beta_{7} + 39809712 \beta_{6} - 39809712 \beta_{5} - 179150928 \beta_{4} - 2169368 \beta_{1}\)\()/3840\)
\(\nu^{6}\)\(=\)\((\)\(-6352440 \beta_{16} + 6352440 \beta_{15} - 5280055 \beta_{14} + 5280055 \beta_{13} + 139395 \beta_{12} - 139395 \beta_{11} + 2214600 \beta_{10} + 9725613 \beta_{9} + 7511013 \beta_{8} + 40920630 \beta_{6} + 40920630 \beta_{5} + 726150 \beta_{3} + 3580120 \beta_{2} - 2893011 \beta_{1} - 419870898\)\()/480\)
\(\nu^{7}\)\(=\)\((\)\(16934100 \beta_{19} - 97402960 \beta_{18} + 357716240 \beta_{17} + 322186560 \beta_{16} + 322186560 \beta_{15} + 532797245 \beta_{14} + 532797245 \beta_{13} + 34112340 \beta_{12} + 34112340 \beta_{11} + 308735708 \beta_{9} - 308735708 \beta_{8} + 1422146792 \beta_{7} + 34175220 \beta_{6} - 34175220 \beta_{5} - 221957110008 \beta_{4} - 308735708 \beta_{1}\)\()/3840\)
\(\nu^{8}\)\(=\)\((\)\(-7550452272 \beta_{16} + 7550452272 \beta_{15} - 575458893 \beta_{14} + 575458893 \beta_{13} - 8852128560 \beta_{12} + 8852128560 \beta_{11} + 657386000 \beta_{10} - 9965868912 \beta_{9} - 10623254912 \beta_{8} + 65813060904 \beta_{6} + 65813060904 \beta_{5} + 728816424 \beta_{3} - 1110229728 \beta_{2} - 5439480896 \beta_{1} + 1268248071072\)\()/3840\)
\(\nu^{9}\)\(=\)\((\)\(-425670693 \beta_{19} - 7813909080 \beta_{18} + 12704120624 \beta_{17} + 21259349184 \beta_{16} + 21259349184 \beta_{15} + 7781297711 \beta_{14} + 7781297711 \beta_{13} - 8239378860 \beta_{12} - 8239378860 \beta_{11} + 9096824328 \beta_{9} - 9096824328 \beta_{8} - 1525798488 \beta_{7} - 167008810773 \beta_{6} + 167008810773 \beta_{5} + 1288055722632 \beta_{4} - 9096824328 \beta_{1}\)\()/480\)
\(\nu^{10}\)\(=\)\((\)\(2132499788352 \beta_{16} - 2132499788352 \beta_{15} + 3016806827423 \beta_{14} - 3016806827423 \beta_{13} + 305046798660 \beta_{12} - 305046798660 \beta_{11} - 510876137040 \beta_{10} - 1672259724900 \beta_{9} - 1161383587860 \beta_{8} - 5686955744244 \beta_{6} - 5686955744244 \beta_{5} - 172680569364 \beta_{3} - 1549752741392 \beta_{2} - 948932108340 \beta_{1} + 361597278178920\)\()/3840\)
\(\nu^{11}\)\(=\)\((\)\(-2307104256480 \beta_{19} - 1703697317680 \beta_{18} - 21570378063680 \beta_{17} - 5096896541520 \beta_{16} - 5096896541520 \beta_{15} - 15593726881505 \beta_{14} - 15593726881505 \beta_{13} - 18496362809160 \beta_{12} - 18496362809160 \beta_{11} - 20558837655112 \beta_{9} + 20558837655112 \beta_{8} - 51807147285088 \beta_{7} - 32373269611680 \beta_{6} + 32373269611680 \beta_{5} + 6400568462744112 \beta_{4} + 20558837655112 \beta_{1}\)\()/3840\)
\(\nu^{12}\)\(=\)\((\)\(12410747120880 \beta_{16} - 12410747120880 \beta_{15} - 1628446716390 \beta_{14} + 1628446716390 \beta_{13} + 18156210673345 \beta_{12} - 18156210673345 \beta_{11} - 6741746542040 \beta_{10} + 1106879061279 \beta_{9} + 7848625603319 \beta_{8} - 124121284715660 \beta_{6} - 124121284715660 \beta_{5} - 13367968300 \beta_{3} - 4734521643440 \beta_{2} + 19348158111407 \beta_{1} - 2382801661432934\)\()/160\)
\(\nu^{13}\)\(=\)\((\)\(440456221542876 \beta_{19} + 2622145591363600 \beta_{18} - 5054280616403728 \beta_{17} - 7163056313422848 \beta_{16} - 7163056313422848 \beta_{15} - 7963850873682997 \beta_{14} - 7963850873682997 \beta_{13} + 3119829047575740 \beta_{12} + 3119829047575740 \beta_{11} - 3053392107546092 \beta_{9} + 3053392107546092 \beta_{8} - 108773241188168 \beta_{7} + 29972870644361916 \beta_{6} - 29972870644361916 \beta_{5} - 262370419783884648 \beta_{4} + 3053392107546092 \beta_{1}\)\()/3840\)
\(\nu^{14}\)\(=\)\((\)\(-84205040731630992 \beta_{16} + 84205040731630992 \beta_{15} - 120547493546657123 \beta_{14} + 120547493546657123 \beta_{13} + 7030156399557600 \beta_{12} - 7030156399557600 \beta_{11} + 3043671214528560 \beta_{10} + 21702646541893632 \beta_{9} + 18658975327365072 \beta_{8} + 209968187928002184 \beta_{6} + 209968187928002184 \beta_{5} + 133874082937224 \beta_{3} + 97456376417952992 \beta_{2} + 73374875750432016 \beta_{1} - 15023226309335919552\)\()/3840\)
\(\nu^{15}\)\(=\)\((\)\(3487660655181351 \beta_{19} + 47398577306303160 \beta_{18} + 86133028754561272 \beta_{17} + 34977364257281352 \beta_{16} + 34977364257281352 \beta_{15} + 110700788835411238 \beta_{14} + 110700788835411238 \beta_{13} + 173080483479383190 \beta_{12} + 173080483479383190 \beta_{11} + 117038481840317358 \beta_{9} - 117038481840317358 \beta_{8} + 239597181295702452 \beta_{7} + 219161334218589831 \beta_{6} - 219161334218589831 \beta_{5} - 25226294182487728068 \beta_{4} - 117038481840317358 \beta_{1}\)\()/480\)
\(\nu^{16}\)\(=\)\((\)\(-12251622931276270848 \beta_{16} + 12251622931276270848 \beta_{15} - 8118701567081188407 \beta_{14} + 8118701567081188407 \beta_{13} - 18517924754091579060 \beta_{12} + 18517924754091579060 \beta_{11} + 9380517320458882960 \beta_{10} + 9153526402022194068 \beta_{9} - 226990918436688892 \beta_{8} + 75078278917671317316 \beta_{6} + 75078278917671317316 \beta_{5} + 113158507977357156 \beta_{3} + 8727103804645430928 \beta_{2} - 30369662179190557276 \beta_{1} + 3395023935851246219832\)\()/3840\)
\(\nu^{17}\)\(=\)\((\)\(-15236107113943399536 \beta_{19} - 63969575176698141200 \beta_{18} + 293392227530733222208 \beta_{17} + 317220726629521897488 \beta_{16} + 317220726629521897488 \beta_{15} + 441689693809038574417 \beta_{14} + 441689693809038574417 \beta_{13} - 137634910590601483560 \beta_{12} - 137634910590601483560 \beta_{11} - 18832414496397422792 \beta_{9} + 18832414496397422792 \beta_{8} - 59209619368296750848 \beta_{7} - 902856014155155059376 \beta_{6} + 902856014155155059376 \beta_{5} + 5040531846945107486832 \beta_{4} + 18832414496397422792 \beta_{1}\)\()/3840\)
\(\nu^{18}\)\(=\)\((\)\(526163551596594027960 \beta_{16} - 526163551596594027960 \beta_{15} + 659755447141189572475 \beta_{14} - 659755447141189572475 \beta_{13} - 259147863864720893985 \beta_{12} + 259147863864720893985 \beta_{11} + 83532478440403840200 \beta_{10} - 60828125318245908111 \beta_{9} - 144360603758649748311 \beta_{8} - 1708504331750013376470 \beta_{6} - 1708504331750013376470 \beta_{5} + 6152838546166078170 \beta_{3} - 585750067488518263000 \beta_{2} - 354408868139717865903 \beta_{1} + 52416086050617997130406\)\()/480\)
\(\nu^{19}\)\(=\)\((\)\(1804110465624400481844 \beta_{19} - 25647352402046987464400 \beta_{18} - 17259974752170871319792 \beta_{17} - 18206281846999046970432 \beta_{16} - 18206281846999046970432 \beta_{15} - 34101527580668764092083 \beta_{14} - 34101527580668764092083 \beta_{13} - 68841569159995305234060 \beta_{12} - 68841569159995305234060 \beta_{11} - 26295881740381091576132 \beta_{9} + 26295881740381091576132 \beta_{8} - 88330204944137178981848 \beta_{7} + 14103553879394172364884 \beta_{6} - 14103553879394172364884 \beta_{5} + 11160908773514238066455112 \beta_{4} + 26295881740381091576132 \beta_{1}\)\()/3840\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(\beta_{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
−11.4741 7.80740i
3.75557 3.81117i
−10.8505 10.2794i
5.50401 + 11.9953i
−1.99079 + 10.4027i
1.99079 + 10.4027i
−5.50401 + 11.9953i
10.8505 10.2794i
−3.75557 3.81117i
11.4741 7.80740i
−11.4741 + 7.80740i
3.75557 + 3.81117i
−10.8505 + 10.2794i
5.50401 11.9953i
−1.99079 10.4027i
1.99079 10.4027i
−5.50401 11.9953i
10.8505 + 10.2794i
−3.75557 + 3.81117i
11.4741 + 7.80740i
0 −20.3843 + 20.3843i 0 0 0 −76.9082 76.9082i 0 588.037i 0
143.2 0 −17.2921 + 17.2921i 0 0 0 154.079 + 154.079i 0 355.037i 0
143.3 0 −9.68301 + 9.68301i 0 0 0 −48.6629 48.6629i 0 55.4787i 0
143.4 0 −7.48311 + 7.48311i 0 0 0 −19.2260 19.2260i 0 131.006i 0
143.5 0 −0.839817 + 0.839817i 0 0 0 99.3589 + 99.3589i 0 241.589i 0
143.6 0 0.839817 0.839817i 0 0 0 −99.3589 99.3589i 0 241.589i 0
143.7 0 7.48311 7.48311i 0 0 0 19.2260 + 19.2260i 0 131.006i 0
143.8 0 9.68301 9.68301i 0 0 0 48.6629 + 48.6629i 0 55.4787i 0
143.9 0 17.2921 17.2921i 0 0 0 −154.079 154.079i 0 355.037i 0
143.10 0 20.3843 20.3843i 0 0 0 76.9082 + 76.9082i 0 588.037i 0
207.1 0 −20.3843 20.3843i 0 0 0 −76.9082 + 76.9082i 0 588.037i 0
207.2 0 −17.2921 17.2921i 0 0 0 154.079 154.079i 0 355.037i 0
207.3 0 −9.68301 9.68301i 0 0 0 −48.6629 + 48.6629i 0 55.4787i 0
207.4 0 −7.48311 7.48311i 0 0 0 −19.2260 + 19.2260i 0 131.006i 0
207.5 0 −0.839817 0.839817i 0 0 0 99.3589 99.3589i 0 241.589i 0
207.6 0 0.839817 + 0.839817i 0 0 0 −99.3589 + 99.3589i 0 241.589i 0
207.7 0 7.48311 + 7.48311i 0 0 0 19.2260 19.2260i 0 131.006i 0
207.8 0 9.68301 + 9.68301i 0 0 0 48.6629 48.6629i 0 55.4787i 0
207.9 0 17.2921 + 17.2921i 0 0 0 −154.079 + 154.079i 0 355.037i 0
207.10 0 20.3843 + 20.3843i 0 0 0 76.9082 76.9082i 0 588.037i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 207.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.6.n.g 20
4.b odd 2 1 inner 400.6.n.g 20
5.b even 2 1 80.6.n.d 20
5.c odd 4 1 80.6.n.d 20
5.c odd 4 1 inner 400.6.n.g 20
20.d odd 2 1 80.6.n.d 20
20.e even 4 1 80.6.n.d 20
20.e even 4 1 inner 400.6.n.g 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.6.n.d 20 5.b even 2 1
80.6.n.d 20 5.c odd 4 1
80.6.n.d 20 20.d odd 2 1
80.6.n.d 20 20.e even 4 1
400.6.n.g 20 1.a even 1 1 trivial
400.6.n.g 20 4.b odd 2 1 inner
400.6.n.g 20 5.c odd 4 1 inner
400.6.n.g 20 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 1095980 T_{3}^{16} + 297452922160 T_{3}^{12} + \)12246534533136960

'>\(12\!\cdots\!60\)\( T_{3}^{8} + \)108964362492176302080
'>\(10\!\cdots\!80\)\( T_{3}^{4} + \)216763542247808434176'>\(21\!\cdots\!76\)\( \) acting on \(S_{6}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 4910 T^{4} + 4404762445 T^{8} + 138021311000520 T^{12} - 6849972091873491390 T^{16} + \)\(85\!\cdots\!52\)\( T^{20} - \)\(23\!\cdots\!90\)\( T^{24} + \)\(16\!\cdots\!20\)\( T^{28} + \)\(18\!\cdots\!45\)\( T^{32} + \)\(72\!\cdots\!10\)\( T^{36} + \)\(51\!\cdots\!01\)\( T^{40} \)
$5$ 1
$7$ \( 1 - 50878370 T^{4} + 88119764714863965 T^{8} + \)\(17\!\cdots\!80\)\( T^{12} + \)\(80\!\cdots\!70\)\( T^{16} + \)\(16\!\cdots\!76\)\( T^{20} + \)\(63\!\cdots\!70\)\( T^{24} + \)\(10\!\cdots\!80\)\( T^{28} + \)\(44\!\cdots\!65\)\( T^{32} - \)\(20\!\cdots\!70\)\( T^{36} + \)\(32\!\cdots\!01\)\( T^{40} \)
$11$ \( ( 1 - 630250 T^{2} + 230184799605 T^{4} - 63113737617167000 T^{6} + \)\(13\!\cdots\!10\)\( T^{8} - \)\(24\!\cdots\!00\)\( T^{10} + \)\(35\!\cdots\!10\)\( T^{12} - \)\(42\!\cdots\!00\)\( T^{14} + \)\(40\!\cdots\!05\)\( T^{16} - \)\(28\!\cdots\!50\)\( T^{18} + \)\(11\!\cdots\!01\)\( T^{20} )^{2} \)
$13$ \( ( 1 + 402 T + 80802 T^{2} + 412195466 T^{3} + 274076716325 T^{4} - 98802411059208 T^{5} + 23088035017184312 T^{6} + 33522742681396790296 T^{7} - \)\(55\!\cdots\!50\)\( T^{8} - \)\(34\!\cdots\!68\)\( T^{9} + \)\(30\!\cdots\!32\)\( T^{10} - \)\(12\!\cdots\!24\)\( T^{11} - \)\(75\!\cdots\!50\)\( T^{12} + \)\(17\!\cdots\!72\)\( T^{13} + \)\(43\!\cdots\!12\)\( T^{14} - \)\(69\!\cdots\!44\)\( T^{15} + \)\(71\!\cdots\!25\)\( T^{16} + \)\(40\!\cdots\!62\)\( T^{17} + \)\(29\!\cdots\!02\)\( T^{18} + \)\(53\!\cdots\!86\)\( T^{19} + \)\(49\!\cdots\!49\)\( T^{20} )^{2} \)
$17$ \( ( 1 - 1118 T + 624962 T^{2} - 1188702526 T^{3} - 1739300659939 T^{4} + 3459207301051384 T^{5} - 2073890095879259656 T^{6} + \)\(50\!\cdots\!88\)\( T^{7} + \)\(54\!\cdots\!26\)\( T^{8} - \)\(82\!\cdots\!84\)\( T^{9} + \)\(52\!\cdots\!56\)\( T^{10} - \)\(11\!\cdots\!88\)\( T^{11} + \)\(10\!\cdots\!74\)\( T^{12} + \)\(14\!\cdots\!84\)\( T^{13} - \)\(84\!\cdots\!56\)\( T^{14} + \)\(19\!\cdots\!88\)\( T^{15} - \)\(14\!\cdots\!11\)\( T^{16} - \)\(13\!\cdots\!18\)\( T^{17} + \)\(10\!\cdots\!62\)\( T^{18} - \)\(26\!\cdots\!26\)\( T^{19} + \)\(33\!\cdots\!49\)\( T^{20} )^{2} \)
$19$ \( ( 1 + 8249550 T^{2} + 36376777568405 T^{4} + \)\(13\!\cdots\!00\)\( T^{6} + \)\(42\!\cdots\!10\)\( T^{8} + \)\(11\!\cdots\!00\)\( T^{10} + \)\(26\!\cdots\!10\)\( T^{12} + \)\(49\!\cdots\!00\)\( T^{14} + \)\(83\!\cdots\!05\)\( T^{16} + \)\(11\!\cdots\!50\)\( T^{18} + \)\(86\!\cdots\!01\)\( T^{20} )^{2} \)
$23$ \( 1 - 207859623483490 T^{4} + \)\(17\!\cdots\!45\)\( T^{8} - \)\(53\!\cdots\!80\)\( T^{12} - \)\(11\!\cdots\!90\)\( T^{16} + \)\(13\!\cdots\!52\)\( T^{20} - \)\(19\!\cdots\!90\)\( T^{24} - \)\(15\!\cdots\!80\)\( T^{28} + \)\(86\!\cdots\!45\)\( T^{32} - \)\(18\!\cdots\!90\)\( T^{36} + \)\(14\!\cdots\!01\)\( T^{40} \)
$29$ \( ( 1 - 99799410 T^{2} + 5177263888867445 T^{4} - \)\(18\!\cdots\!20\)\( T^{6} + \)\(49\!\cdots\!10\)\( T^{8} - \)\(10\!\cdots\!52\)\( T^{10} + \)\(20\!\cdots\!10\)\( T^{12} - \)\(32\!\cdots\!20\)\( T^{14} + \)\(38\!\cdots\!45\)\( T^{16} - \)\(31\!\cdots\!10\)\( T^{18} + \)\(13\!\cdots\!01\)\( T^{20} )^{2} \)
$31$ \( ( 1 - 101180050 T^{2} + 5842226806223805 T^{4} - \)\(25\!\cdots\!00\)\( T^{6} + \)\(95\!\cdots\!10\)\( T^{8} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(77\!\cdots\!10\)\( T^{12} - \)\(17\!\cdots\!00\)\( T^{14} + \)\(32\!\cdots\!05\)\( T^{16} - \)\(45\!\cdots\!50\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \)
$37$ \( ( 1 + 22130 T + 244868450 T^{2} + 1308308732410 T^{3} + 13809729158765845 T^{4} + \)\(24\!\cdots\!80\)\( T^{5} + \)\(28\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!10\)\( T^{8} + \)\(15\!\cdots\!80\)\( T^{9} + \)\(18\!\cdots\!00\)\( T^{10} + \)\(10\!\cdots\!60\)\( T^{11} + \)\(50\!\cdots\!90\)\( T^{12} + \)\(51\!\cdots\!80\)\( T^{13} + \)\(64\!\cdots\!00\)\( T^{14} + \)\(38\!\cdots\!60\)\( T^{15} + \)\(15\!\cdots\!05\)\( T^{16} + \)\(10\!\cdots\!30\)\( T^{17} + \)\(13\!\cdots\!50\)\( T^{18} + \)\(82\!\cdots\!10\)\( T^{19} + \)\(25\!\cdots\!49\)\( T^{20} )^{2} \)
$41$ \( ( 1 + 1690 T + 164119545 T^{2} - 1028151329120 T^{3} + 25291870422416110 T^{4} - 93379183076463398452 T^{5} + \)\(29\!\cdots\!10\)\( T^{6} - \)\(13\!\cdots\!20\)\( T^{7} + \)\(25\!\cdots\!45\)\( T^{8} + \)\(30\!\cdots\!90\)\( T^{9} + \)\(20\!\cdots\!01\)\( T^{10} )^{4} \)
$43$ \( 1 + 22292551011039310 T^{4} + \)\(40\!\cdots\!45\)\( T^{8} + \)\(23\!\cdots\!20\)\( T^{12} + \)\(67\!\cdots\!10\)\( T^{16} + \)\(21\!\cdots\!52\)\( T^{20} + \)\(31\!\cdots\!10\)\( T^{24} + \)\(51\!\cdots\!20\)\( T^{28} + \)\(41\!\cdots\!45\)\( T^{32} + \)\(10\!\cdots\!10\)\( T^{36} + \)\(22\!\cdots\!01\)\( T^{40} \)
$47$ \( 1 + 74279974827999230 T^{4} + \)\(48\!\cdots\!65\)\( T^{8} + \)\(19\!\cdots\!80\)\( T^{12} + \)\(14\!\cdots\!70\)\( T^{16} - \)\(25\!\cdots\!24\)\( T^{20} + \)\(39\!\cdots\!70\)\( T^{24} + \)\(15\!\cdots\!80\)\( T^{28} + \)\(10\!\cdots\!65\)\( T^{32} + \)\(43\!\cdots\!30\)\( T^{36} + \)\(16\!\cdots\!01\)\( T^{40} \)
$53$ \( ( 1 + 91226 T + 4161091538 T^{2} + 141622957033378 T^{3} + 3893612180777232821 T^{4} + \)\(81\!\cdots\!12\)\( T^{5} + \)\(12\!\cdots\!56\)\( T^{6} + \)\(10\!\cdots\!76\)\( T^{7} - \)\(19\!\cdots\!54\)\( T^{8} - \)\(10\!\cdots\!72\)\( T^{9} - \)\(26\!\cdots\!96\)\( T^{10} - \)\(45\!\cdots\!96\)\( T^{11} - \)\(34\!\cdots\!46\)\( T^{12} + \)\(80\!\cdots\!32\)\( T^{13} + \)\(39\!\cdots\!56\)\( T^{14} + \)\(10\!\cdots\!16\)\( T^{15} + \)\(20\!\cdots\!29\)\( T^{16} + \)\(31\!\cdots\!46\)\( T^{17} + \)\(38\!\cdots\!38\)\( T^{18} + \)\(35\!\cdots\!18\)\( T^{19} + \)\(16\!\cdots\!49\)\( T^{20} )^{2} \)
$59$ \( ( 1 + 4265422750 T^{2} + 8278744468566020805 T^{4} + \)\(98\!\cdots\!00\)\( T^{6} + \)\(85\!\cdots\!10\)\( T^{8} + \)\(63\!\cdots\!00\)\( T^{10} + \)\(43\!\cdots\!10\)\( T^{12} + \)\(25\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!05\)\( T^{16} + \)\(29\!\cdots\!50\)\( T^{18} + \)\(34\!\cdots\!01\)\( T^{20} )^{2} \)
$61$ \( ( 1 + 10270 T + 3306972765 T^{2} + 29698468018720 T^{3} + 4981733552475787870 T^{4} + \)\(35\!\cdots\!24\)\( T^{5} + \)\(42\!\cdots\!70\)\( T^{6} + \)\(21\!\cdots\!20\)\( T^{7} + \)\(19\!\cdots\!65\)\( T^{8} + \)\(52\!\cdots\!70\)\( T^{9} + \)\(42\!\cdots\!01\)\( T^{10} )^{4} \)
$67$ \( 1 - 2038648142805486290 T^{4} - \)\(25\!\cdots\!55\)\( T^{8} + \)\(14\!\cdots\!20\)\( T^{12} + \)\(73\!\cdots\!10\)\( T^{16} - \)\(57\!\cdots\!48\)\( T^{20} + \)\(24\!\cdots\!10\)\( T^{24} + \)\(15\!\cdots\!20\)\( T^{28} - \)\(93\!\cdots\!55\)\( T^{32} - \)\(24\!\cdots\!90\)\( T^{36} + \)\(40\!\cdots\!01\)\( T^{40} \)
$71$ \( ( 1 - 14421599170 T^{2} + 96751044929093627565 T^{4} - \)\(40\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!70\)\( T^{8} - \)\(24\!\cdots\!24\)\( T^{10} + \)\(37\!\cdots\!70\)\( T^{12} - \)\(42\!\cdots\!20\)\( T^{14} + \)\(33\!\cdots\!65\)\( T^{16} - \)\(16\!\cdots\!70\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \)
$73$ \( ( 1 + 132186 T + 8736569298 T^{2} + 431249337673738 T^{3} + 10750357541220477581 T^{4} - \)\(12\!\cdots\!88\)\( T^{5} - \)\(17\!\cdots\!84\)\( T^{6} - \)\(58\!\cdots\!44\)\( T^{7} + \)\(47\!\cdots\!06\)\( T^{8} + \)\(60\!\cdots\!08\)\( T^{9} + \)\(32\!\cdots\!04\)\( T^{10} + \)\(12\!\cdots\!44\)\( T^{11} + \)\(20\!\cdots\!94\)\( T^{12} - \)\(52\!\cdots\!08\)\( T^{13} - \)\(32\!\cdots\!84\)\( T^{14} - \)\(47\!\cdots\!84\)\( T^{15} + \)\(85\!\cdots\!69\)\( T^{16} + \)\(70\!\cdots\!66\)\( T^{17} + \)\(29\!\cdots\!98\)\( T^{18} + \)\(93\!\cdots\!98\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} )^{2} \)
$79$ \( ( 1 + 18614864790 T^{2} + \)\(16\!\cdots\!45\)\( T^{4} + \)\(10\!\cdots\!80\)\( T^{6} + \)\(44\!\cdots\!10\)\( T^{8} + \)\(15\!\cdots\!48\)\( T^{10} + \)\(42\!\cdots\!10\)\( T^{12} + \)\(90\!\cdots\!80\)\( T^{14} + \)\(14\!\cdots\!45\)\( T^{16} + \)\(14\!\cdots\!90\)\( T^{18} + \)\(76\!\cdots\!01\)\( T^{20} )^{2} \)
$83$ \( 1 - 18609684090616711570 T^{4} - \)\(46\!\cdots\!35\)\( T^{8} + \)\(34\!\cdots\!80\)\( T^{12} + \)\(25\!\cdots\!70\)\( T^{16} - \)\(17\!\cdots\!24\)\( T^{20} + \)\(60\!\cdots\!70\)\( T^{24} + \)\(20\!\cdots\!80\)\( T^{28} - \)\(64\!\cdots\!35\)\( T^{32} - \)\(62\!\cdots\!70\)\( T^{36} + \)\(80\!\cdots\!01\)\( T^{40} \)
$89$ \( ( 1 - 33120686970 T^{2} + \)\(41\!\cdots\!65\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} - \)\(32\!\cdots\!30\)\( T^{8} + \)\(73\!\cdots\!76\)\( T^{10} - \)\(10\!\cdots\!30\)\( T^{12} - \)\(20\!\cdots\!20\)\( T^{14} + \)\(12\!\cdots\!65\)\( T^{16} - \)\(31\!\cdots\!70\)\( T^{18} + \)\(29\!\cdots\!01\)\( T^{20} )^{2} \)
$97$ \( ( 1 + 187386 T + 17556756498 T^{2} + 497237634071002 T^{3} - \)\(23\!\cdots\!35\)\( T^{4} - \)\(42\!\cdots\!24\)\( T^{5} - \)\(37\!\cdots\!32\)\( T^{6} - \)\(18\!\cdots\!68\)\( T^{7} + \)\(11\!\cdots\!50\)\( T^{8} + \)\(33\!\cdots\!76\)\( T^{9} + \)\(36\!\cdots\!68\)\( T^{10} + \)\(29\!\cdots\!32\)\( T^{11} + \)\(87\!\cdots\!50\)\( T^{12} - \)\(11\!\cdots\!24\)\( T^{13} - \)\(20\!\cdots\!32\)\( T^{14} - \)\(19\!\cdots\!68\)\( T^{15} - \)\(93\!\cdots\!15\)\( T^{16} + \)\(17\!\cdots\!86\)\( T^{17} + \)\(51\!\cdots\!98\)\( T^{18} + \)\(47\!\cdots\!02\)\( T^{19} + \)\(21\!\cdots\!49\)\( T^{20} )^{2} \)
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