Properties

Label 17.10.a.a.1.2
Level $17$
Weight $10$
Character 17.1
Self dual yes
Analytic conductor $8.756$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 17.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.75560921479\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 1596 x^{3} + 5754 x^{2} + 488987 x - 2711704\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-21.1654\) of defining polynomial
Character \(\chi\) \(=\) 17.1

$q$-expansion

\(f(q)\) \(=\) \(q-28.1654 q^{2} -3.02373 q^{3} +281.287 q^{4} +762.851 q^{5} +85.1644 q^{6} +5573.11 q^{7} +6498.11 q^{8} -19673.9 q^{9} +O(q^{10})\) \(q-28.1654 q^{2} -3.02373 q^{3} +281.287 q^{4} +762.851 q^{5} +85.1644 q^{6} +5573.11 q^{7} +6498.11 q^{8} -19673.9 q^{9} -21486.0 q^{10} -47641.7 q^{11} -850.536 q^{12} -92260.9 q^{13} -156969. q^{14} -2306.65 q^{15} -327041. q^{16} -83521.0 q^{17} +554121. q^{18} -8373.93 q^{19} +214580. q^{20} -16851.6 q^{21} +1.34184e6 q^{22} +364592. q^{23} -19648.5 q^{24} -1.37118e6 q^{25} +2.59856e6 q^{26} +119004. q^{27} +1.56764e6 q^{28} -3.50595e6 q^{29} +64967.7 q^{30} -5.20629e6 q^{31} +5.88418e6 q^{32} +144055. q^{33} +2.35240e6 q^{34} +4.25145e6 q^{35} -5.53400e6 q^{36} -499530. q^{37} +235855. q^{38} +278972. q^{39} +4.95709e6 q^{40} -5.43648e6 q^{41} +474631. q^{42} -3.54411e7 q^{43} -1.34010e7 q^{44} -1.50082e7 q^{45} -1.02689e7 q^{46} +1.21753e7 q^{47} +988882. q^{48} -9.29403e6 q^{49} +3.86199e7 q^{50} +252545. q^{51} -2.59518e7 q^{52} +1.04471e8 q^{53} -3.35180e6 q^{54} -3.63435e7 q^{55} +3.62147e7 q^{56} +25320.5 q^{57} +9.87464e7 q^{58} +4.16714e7 q^{59} -648832. q^{60} +5.67537e7 q^{61} +1.46637e8 q^{62} -1.09645e8 q^{63} +1.71477e6 q^{64} -7.03813e7 q^{65} -4.05737e6 q^{66} -1.74621e8 q^{67} -2.34934e7 q^{68} -1.10243e6 q^{69} -1.19744e8 q^{70} +3.46330e7 q^{71} -1.27843e8 q^{72} -3.93220e8 q^{73} +1.40694e7 q^{74} +4.14609e6 q^{75} -2.35548e6 q^{76} -2.65512e8 q^{77} -7.85734e6 q^{78} +1.85772e8 q^{79} -2.49483e8 q^{80} +3.86881e8 q^{81} +1.53120e8 q^{82} +3.62239e8 q^{83} -4.74013e6 q^{84} -6.37141e7 q^{85} +9.98211e8 q^{86} +1.06010e7 q^{87} -3.09581e8 q^{88} -5.04798e7 q^{89} +4.22712e8 q^{90} -5.14180e8 q^{91} +1.02555e8 q^{92} +1.57424e7 q^{93} -3.42920e8 q^{94} -6.38806e6 q^{95} -1.77922e7 q^{96} -9.67620e8 q^{97} +2.61770e8 q^{98} +9.37295e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 33q^{2} - 236q^{3} + 853q^{4} + 1480q^{5} + 7578q^{6} - 13202q^{7} - 42423q^{8} + 10981q^{9} + O(q^{10}) \) \( 5q - 33q^{2} - 236q^{3} + 853q^{4} + 1480q^{5} + 7578q^{6} - 13202q^{7} - 42423q^{8} + 10981q^{9} - 89328q^{10} - 68036q^{11} - 406010q^{12} - 158862q^{13} - 84700q^{14} - 687324q^{15} + 350225q^{16} - 417605q^{17} - 1911585q^{18} - 370992q^{19} + 1632640q^{20} + 1783880q^{21} + 122290q^{22} + 1645870q^{23} + 9678702q^{24} + 3270239q^{25} + 734846q^{26} - 2998268q^{27} + 183372q^{28} + 3668616q^{29} + 17048544q^{30} - 7262362q^{31} - 5605919q^{32} - 11334900q^{33} + 2756193q^{34} - 26503988q^{35} + 49782133q^{36} - 31420708q^{37} + 18513700q^{38} - 42449884q^{39} - 53930464q^{40} - 7996938q^{41} - 44519496q^{42} - 56908268q^{43} + 43323054q^{44} + 12799536q^{45} - 32063472q^{46} - 16903336q^{47} - 102794498q^{48} - 11784059q^{49} + 85921093q^{50} + 19710956q^{51} + 173619082q^{52} - 83362982q^{53} + 386329164q^{54} + 6363364q^{55} + 317409372q^{56} + 136615904q^{57} + 64577488q^{58} - 37946604q^{59} - 223158912q^{60} - 77685452q^{61} + 324855300q^{62} - 191945278q^{63} + 131623105q^{64} - 40321288q^{65} + 298037676q^{66} - 304503600q^{67} - 71243413q^{68} - 333409272q^{69} - 122787392q^{70} - 476602922q^{71} - 1301701911q^{72} - 289980486q^{73} + 262289012q^{74} - 153685772q^{75} - 1031276084q^{76} - 143385648q^{77} + 691646196q^{78} - 828240610q^{79} + 912750944q^{80} + 891328609q^{81} - 1109615654q^{82} + 194681148q^{83} + 1541719592q^{84} - 123611080q^{85} + 1164707144q^{86} + 158149884q^{87} - 1017979978q^{88} + 376848106q^{89} - 2240087472q^{90} + 194543664q^{91} + 2506713088q^{92} + 3494835920q^{93} - 2244811104q^{94} + 1498679864q^{95} + 2935047582q^{96} + 692035246q^{97} + 871744055q^{98} + 2027106408q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −28.1654 −1.24474 −0.622372 0.782721i \(-0.713831\pi\)
−0.622372 + 0.782721i \(0.713831\pi\)
\(3\) −3.02373 −0.0215525 −0.0107762 0.999942i \(-0.503430\pi\)
−0.0107762 + 0.999942i \(0.503430\pi\)
\(4\) 281.287 0.549389
\(5\) 762.851 0.545852 0.272926 0.962035i \(-0.412009\pi\)
0.272926 + 0.962035i \(0.412009\pi\)
\(6\) 85.1644 0.0268273
\(7\) 5573.11 0.877317 0.438659 0.898654i \(-0.355454\pi\)
0.438659 + 0.898654i \(0.355454\pi\)
\(8\) 6498.11 0.560896
\(9\) −19673.9 −0.999535
\(10\) −21486.0 −0.679446
\(11\) −47641.7 −0.981115 −0.490558 0.871409i \(-0.663207\pi\)
−0.490558 + 0.871409i \(0.663207\pi\)
\(12\) −850.536 −0.0118407
\(13\) −92260.9 −0.895927 −0.447963 0.894052i \(-0.647851\pi\)
−0.447963 + 0.894052i \(0.647851\pi\)
\(14\) −156969. −1.09204
\(15\) −2306.65 −0.0117645
\(16\) −327041. −1.24756
\(17\) −83521.0 −0.242536
\(18\) 554121. 1.24417
\(19\) −8373.93 −0.0147414 −0.00737069 0.999973i \(-0.502346\pi\)
−0.00737069 + 0.999973i \(0.502346\pi\)
\(20\) 214580. 0.299885
\(21\) −16851.6 −0.0189084
\(22\) 1.34184e6 1.22124
\(23\) 364592. 0.271664 0.135832 0.990732i \(-0.456629\pi\)
0.135832 + 0.990732i \(0.456629\pi\)
\(24\) −19648.5 −0.0120887
\(25\) −1.37118e6 −0.702046
\(26\) 2.59856e6 1.11520
\(27\) 119004. 0.0430949
\(28\) 1.56764e6 0.481988
\(29\) −3.50595e6 −0.920482 −0.460241 0.887794i \(-0.652237\pi\)
−0.460241 + 0.887794i \(0.652237\pi\)
\(30\) 64967.7 0.0146437
\(31\) −5.20629e6 −1.01251 −0.506257 0.862383i \(-0.668971\pi\)
−0.506257 + 0.862383i \(0.668971\pi\)
\(32\) 5.88418e6 0.991999
\(33\) 144055. 0.0211455
\(34\) 2.35240e6 0.301895
\(35\) 4.25145e6 0.478885
\(36\) −5.53400e6 −0.549134
\(37\) −499530. −0.0438182 −0.0219091 0.999760i \(-0.506974\pi\)
−0.0219091 + 0.999760i \(0.506974\pi\)
\(38\) 235855. 0.0183493
\(39\) 278972. 0.0193094
\(40\) 4.95709e6 0.306166
\(41\) −5.43648e6 −0.300463 −0.150231 0.988651i \(-0.548002\pi\)
−0.150231 + 0.988651i \(0.548002\pi\)
\(42\) 474631. 0.0235361
\(43\) −3.54411e7 −1.58088 −0.790440 0.612539i \(-0.790148\pi\)
−0.790440 + 0.612539i \(0.790148\pi\)
\(44\) −1.34010e7 −0.539014
\(45\) −1.50082e7 −0.545598
\(46\) −1.02689e7 −0.338152
\(47\) 1.21753e7 0.363947 0.181973 0.983303i \(-0.441752\pi\)
0.181973 + 0.983303i \(0.441752\pi\)
\(48\) 988882. 0.0268880
\(49\) −9.29403e6 −0.230315
\(50\) 3.86199e7 0.873868
\(51\) 252545. 0.00522724
\(52\) −2.59518e7 −0.492212
\(53\) 1.04471e8 1.81867 0.909336 0.416063i \(-0.136590\pi\)
0.909336 + 0.416063i \(0.136590\pi\)
\(54\) −3.35180e6 −0.0536422
\(55\) −3.63435e7 −0.535543
\(56\) 3.62147e7 0.492083
\(57\) 25320.5 0.000317713 0
\(58\) 9.87464e7 1.14576
\(59\) 4.16714e7 0.447718 0.223859 0.974622i \(-0.428135\pi\)
0.223859 + 0.974622i \(0.428135\pi\)
\(60\) −648832. −0.00646326
\(61\) 5.67537e7 0.524819 0.262410 0.964957i \(-0.415483\pi\)
0.262410 + 0.964957i \(0.415483\pi\)
\(62\) 1.46637e8 1.26032
\(63\) −1.09645e8 −0.876910
\(64\) 1.71477e6 0.0127760
\(65\) −7.03813e7 −0.489043
\(66\) −4.05737e6 −0.0263207
\(67\) −1.74621e8 −1.05867 −0.529333 0.848414i \(-0.677558\pi\)
−0.529333 + 0.848414i \(0.677558\pi\)
\(68\) −2.34934e7 −0.133246
\(69\) −1.10243e6 −0.00585503
\(70\) −1.19744e8 −0.596089
\(71\) 3.46330e7 0.161744 0.0808719 0.996725i \(-0.474230\pi\)
0.0808719 + 0.996725i \(0.474230\pi\)
\(72\) −1.27843e8 −0.560635
\(73\) −3.93220e8 −1.62063 −0.810314 0.585996i \(-0.800703\pi\)
−0.810314 + 0.585996i \(0.800703\pi\)
\(74\) 1.40694e7 0.0545424
\(75\) 4.14609e6 0.0151308
\(76\) −2.35548e6 −0.00809875
\(77\) −2.65512e8 −0.860749
\(78\) −7.85734e6 −0.0240353
\(79\) 1.85772e8 0.536609 0.268305 0.963334i \(-0.413537\pi\)
0.268305 + 0.963334i \(0.413537\pi\)
\(80\) −2.49483e8 −0.680983
\(81\) 3.86881e8 0.998607
\(82\) 1.53120e8 0.373999
\(83\) 3.62239e8 0.837807 0.418903 0.908031i \(-0.362415\pi\)
0.418903 + 0.908031i \(0.362415\pi\)
\(84\) −4.74013e6 −0.0103880
\(85\) −6.37141e7 −0.132388
\(86\) 9.98211e8 1.96779
\(87\) 1.06010e7 0.0198387
\(88\) −3.09581e8 −0.550303
\(89\) −5.04798e7 −0.0852831 −0.0426416 0.999090i \(-0.513577\pi\)
−0.0426416 + 0.999090i \(0.513577\pi\)
\(90\) 4.22712e8 0.679130
\(91\) −5.14180e8 −0.786012
\(92\) 1.02555e8 0.149249
\(93\) 1.57424e7 0.0218222
\(94\) −3.42920e8 −0.453020
\(95\) −6.38806e6 −0.00804661
\(96\) −1.77922e7 −0.0213800
\(97\) −9.67620e8 −1.10977 −0.554884 0.831928i \(-0.687237\pi\)
−0.554884 + 0.831928i \(0.687237\pi\)
\(98\) 2.61770e8 0.286683
\(99\) 9.37295e8 0.980659
\(100\) −3.85696e8 −0.385696
\(101\) 1.60604e9 1.53572 0.767858 0.640621i \(-0.221323\pi\)
0.767858 + 0.640621i \(0.221323\pi\)
\(102\) −7.11301e6 −0.00650658
\(103\) 1.76819e9 1.54796 0.773982 0.633208i \(-0.218262\pi\)
0.773982 + 0.633208i \(0.218262\pi\)
\(104\) −5.99522e8 −0.502522
\(105\) −1.28552e7 −0.0103212
\(106\) −2.94246e9 −2.26378
\(107\) −2.59627e9 −1.91479 −0.957397 0.288774i \(-0.906752\pi\)
−0.957397 + 0.288774i \(0.906752\pi\)
\(108\) 3.34744e7 0.0236759
\(109\) 2.64333e9 1.79363 0.896814 0.442408i \(-0.145876\pi\)
0.896814 + 0.442408i \(0.145876\pi\)
\(110\) 1.02363e9 0.666614
\(111\) 1.51044e6 0.000944390 0
\(112\) −1.82263e9 −1.09451
\(113\) 9.53189e8 0.549954 0.274977 0.961451i \(-0.411330\pi\)
0.274977 + 0.961451i \(0.411330\pi\)
\(114\) −713161. −0.000395472 0
\(115\) 2.78129e8 0.148288
\(116\) −9.86179e8 −0.505702
\(117\) 1.81513e9 0.895511
\(118\) −1.17369e9 −0.557294
\(119\) −4.65472e8 −0.212781
\(120\) −1.49889e7 −0.00659863
\(121\) −8.82187e7 −0.0374133
\(122\) −1.59849e9 −0.653266
\(123\) 1.64384e7 0.00647571
\(124\) −1.46446e9 −0.556264
\(125\) −2.53595e9 −0.929065
\(126\) 3.08818e9 1.09153
\(127\) 9.64764e8 0.329082 0.164541 0.986370i \(-0.447386\pi\)
0.164541 + 0.986370i \(0.447386\pi\)
\(128\) −3.06100e9 −1.00790
\(129\) 1.07164e8 0.0340719
\(130\) 1.98231e9 0.608734
\(131\) −5.24089e9 −1.55483 −0.777417 0.628985i \(-0.783471\pi\)
−0.777417 + 0.628985i \(0.783471\pi\)
\(132\) 4.05209e7 0.0116171
\(133\) −4.66689e7 −0.0129329
\(134\) 4.91825e9 1.31777
\(135\) 9.07826e7 0.0235234
\(136\) −5.42729e8 −0.136037
\(137\) 3.09452e9 0.750500 0.375250 0.926924i \(-0.377557\pi\)
0.375250 + 0.926924i \(0.377557\pi\)
\(138\) 3.10503e7 0.00728802
\(139\) −6.10981e8 −0.138823 −0.0694115 0.997588i \(-0.522112\pi\)
−0.0694115 + 0.997588i \(0.522112\pi\)
\(140\) 1.19588e9 0.263094
\(141\) −3.68147e7 −0.00784395
\(142\) −9.75450e8 −0.201330
\(143\) 4.39546e9 0.879007
\(144\) 6.43415e9 1.24698
\(145\) −2.67452e9 −0.502446
\(146\) 1.10752e10 2.01727
\(147\) 2.81026e7 0.00496385
\(148\) −1.40511e8 −0.0240732
\(149\) 2.02582e9 0.336716 0.168358 0.985726i \(-0.446154\pi\)
0.168358 + 0.985726i \(0.446154\pi\)
\(150\) −1.16776e8 −0.0188340
\(151\) −7.09269e9 −1.11024 −0.555118 0.831772i \(-0.687327\pi\)
−0.555118 + 0.831772i \(0.687327\pi\)
\(152\) −5.44147e7 −0.00826838
\(153\) 1.64318e9 0.242423
\(154\) 7.47825e9 1.07141
\(155\) −3.97162e9 −0.552682
\(156\) 7.84712e7 0.0106084
\(157\) −8.25715e9 −1.08463 −0.542315 0.840175i \(-0.682452\pi\)
−0.542315 + 0.840175i \(0.682452\pi\)
\(158\) −5.23233e9 −0.667941
\(159\) −3.15892e8 −0.0391969
\(160\) 4.48875e9 0.541484
\(161\) 2.03191e9 0.238335
\(162\) −1.08966e10 −1.24301
\(163\) −6.13273e9 −0.680471 −0.340235 0.940340i \(-0.610507\pi\)
−0.340235 + 0.940340i \(0.610507\pi\)
\(164\) −1.52921e9 −0.165071
\(165\) 1.09893e8 0.0115423
\(166\) −1.02026e10 −1.04286
\(167\) −1.53224e10 −1.52442 −0.762208 0.647332i \(-0.775885\pi\)
−0.762208 + 0.647332i \(0.775885\pi\)
\(168\) −1.09503e8 −0.0106056
\(169\) −2.09243e9 −0.197315
\(170\) 1.79453e9 0.164790
\(171\) 1.64748e8 0.0147345
\(172\) −9.96912e9 −0.868518
\(173\) 1.45473e10 1.23474 0.617370 0.786673i \(-0.288198\pi\)
0.617370 + 0.786673i \(0.288198\pi\)
\(174\) −2.98582e8 −0.0246941
\(175\) −7.64176e9 −0.615917
\(176\) 1.55808e10 1.22400
\(177\) −1.26003e8 −0.00964942
\(178\) 1.42178e9 0.106156
\(179\) 4.64898e9 0.338469 0.169235 0.985576i \(-0.445870\pi\)
0.169235 + 0.985576i \(0.445870\pi\)
\(180\) −4.22162e9 −0.299745
\(181\) 1.08585e9 0.0751997 0.0375998 0.999293i \(-0.488029\pi\)
0.0375998 + 0.999293i \(0.488029\pi\)
\(182\) 1.44821e10 0.978384
\(183\) −1.71608e8 −0.0113112
\(184\) 2.36916e9 0.152375
\(185\) −3.81067e8 −0.0239182
\(186\) −4.43391e8 −0.0271630
\(187\) 3.97908e9 0.237955
\(188\) 3.42474e9 0.199948
\(189\) 6.63225e8 0.0378079
\(190\) 1.79922e8 0.0100160
\(191\) −3.78328e9 −0.205693 −0.102846 0.994697i \(-0.532795\pi\)
−0.102846 + 0.994697i \(0.532795\pi\)
\(192\) −5.18500e6 −0.000275355 0
\(193\) 3.16352e10 1.64121 0.820603 0.571499i \(-0.193638\pi\)
0.820603 + 0.571499i \(0.193638\pi\)
\(194\) 2.72534e10 1.38138
\(195\) 2.12814e8 0.0105401
\(196\) −2.61429e9 −0.126532
\(197\) 1.24212e10 0.587576 0.293788 0.955871i \(-0.405084\pi\)
0.293788 + 0.955871i \(0.405084\pi\)
\(198\) −2.63993e10 −1.22067
\(199\) 4.01627e10 1.81545 0.907724 0.419568i \(-0.137818\pi\)
0.907724 + 0.419568i \(0.137818\pi\)
\(200\) −8.91010e9 −0.393775
\(201\) 5.28005e8 0.0228169
\(202\) −4.52347e10 −1.91157
\(203\) −1.95391e10 −0.807554
\(204\) 7.10376e7 0.00287179
\(205\) −4.14722e9 −0.164008
\(206\) −4.98016e10 −1.92682
\(207\) −7.17293e9 −0.271538
\(208\) 3.01731e10 1.11772
\(209\) 3.98948e8 0.0144630
\(210\) 3.62072e8 0.0128472
\(211\) 1.34292e10 0.466422 0.233211 0.972426i \(-0.425077\pi\)
0.233211 + 0.972426i \(0.425077\pi\)
\(212\) 2.93863e10 0.999158
\(213\) −1.04721e8 −0.00348598
\(214\) 7.31247e10 2.38343
\(215\) −2.70363e10 −0.862926
\(216\) 7.73304e8 0.0241718
\(217\) −2.90153e10 −0.888296
\(218\) −7.44503e10 −2.23261
\(219\) 1.18899e9 0.0349285
\(220\) −1.02230e10 −0.294221
\(221\) 7.70572e9 0.217294
\(222\) −4.25422e7 −0.00117552
\(223\) −4.41977e10 −1.19682 −0.598409 0.801191i \(-0.704200\pi\)
−0.598409 + 0.801191i \(0.704200\pi\)
\(224\) 3.27932e10 0.870297
\(225\) 2.69765e10 0.701720
\(226\) −2.68469e10 −0.684552
\(227\) −3.42932e10 −0.857218 −0.428609 0.903490i \(-0.640996\pi\)
−0.428609 + 0.903490i \(0.640996\pi\)
\(228\) 7.12233e6 0.000174548 0
\(229\) 5.87161e10 1.41090 0.705452 0.708757i \(-0.250744\pi\)
0.705452 + 0.708757i \(0.250744\pi\)
\(230\) −7.83361e9 −0.184581
\(231\) 8.02837e8 0.0185513
\(232\) −2.27821e10 −0.516294
\(233\) 8.16442e10 1.81478 0.907390 0.420290i \(-0.138072\pi\)
0.907390 + 0.420290i \(0.138072\pi\)
\(234\) −5.11237e10 −1.11468
\(235\) 9.28790e9 0.198661
\(236\) 1.17216e10 0.245971
\(237\) −5.61724e8 −0.0115653
\(238\) 1.31102e10 0.264858
\(239\) −6.99278e10 −1.38631 −0.693153 0.720790i \(-0.743779\pi\)
−0.693153 + 0.720790i \(0.743779\pi\)
\(240\) 7.54369e8 0.0146769
\(241\) 2.20186e10 0.420448 0.210224 0.977653i \(-0.432581\pi\)
0.210224 + 0.977653i \(0.432581\pi\)
\(242\) 2.48471e9 0.0465700
\(243\) −3.51219e9 −0.0646174
\(244\) 1.59641e10 0.288330
\(245\) −7.08996e9 −0.125718
\(246\) −4.62995e8 −0.00806061
\(247\) 7.72586e8 0.0132072
\(248\) −3.38311e10 −0.567915
\(249\) −1.09531e9 −0.0180568
\(250\) 7.14260e10 1.15645
\(251\) −7.51995e10 −1.19587 −0.597934 0.801545i \(-0.704012\pi\)
−0.597934 + 0.801545i \(0.704012\pi\)
\(252\) −3.08416e10 −0.481764
\(253\) −1.73698e10 −0.266534
\(254\) −2.71729e10 −0.409623
\(255\) 1.92654e8 0.00285330
\(256\) 8.53361e10 1.24180
\(257\) −4.88179e10 −0.698039 −0.349020 0.937115i \(-0.613485\pi\)
−0.349020 + 0.937115i \(0.613485\pi\)
\(258\) −3.01832e9 −0.0424108
\(259\) −2.78394e9 −0.0384424
\(260\) −1.97973e10 −0.268675
\(261\) 6.89756e10 0.920054
\(262\) 1.47612e11 1.93537
\(263\) 1.51056e10 0.194687 0.0973437 0.995251i \(-0.468965\pi\)
0.0973437 + 0.995251i \(0.468965\pi\)
\(264\) 9.36088e8 0.0118604
\(265\) 7.96958e10 0.992725
\(266\) 1.31444e9 0.0160981
\(267\) 1.52637e8 0.00183806
\(268\) −4.91185e10 −0.581619
\(269\) 7.34701e9 0.0855511 0.0427755 0.999085i \(-0.486380\pi\)
0.0427755 + 0.999085i \(0.486380\pi\)
\(270\) −2.55692e9 −0.0292807
\(271\) 9.26149e10 1.04308 0.521542 0.853226i \(-0.325357\pi\)
0.521542 + 0.853226i \(0.325357\pi\)
\(272\) 2.73148e10 0.302578
\(273\) 1.55474e9 0.0169405
\(274\) −8.71583e10 −0.934181
\(275\) 6.53255e10 0.688788
\(276\) −3.10099e8 −0.00321669
\(277\) 2.09446e10 0.213753 0.106877 0.994272i \(-0.465915\pi\)
0.106877 + 0.994272i \(0.465915\pi\)
\(278\) 1.72085e10 0.172799
\(279\) 1.02428e11 1.01204
\(280\) 2.76264e10 0.268605
\(281\) −7.54010e10 −0.721438 −0.360719 0.932675i \(-0.617469\pi\)
−0.360719 + 0.932675i \(0.617469\pi\)
\(282\) 1.03690e9 0.00976371
\(283\) −1.56859e11 −1.45368 −0.726842 0.686805i \(-0.759013\pi\)
−0.726842 + 0.686805i \(0.759013\pi\)
\(284\) 9.74181e9 0.0888602
\(285\) 1.93158e7 0.000173424 0
\(286\) −1.23800e11 −1.09414
\(287\) −3.02981e10 −0.263601
\(288\) −1.15765e11 −0.991538
\(289\) 6.97576e9 0.0588235
\(290\) 7.53288e10 0.625417
\(291\) 2.92582e9 0.0239182
\(292\) −1.10608e11 −0.890355
\(293\) −1.90997e11 −1.51399 −0.756994 0.653422i \(-0.773333\pi\)
−0.756994 + 0.653422i \(0.773333\pi\)
\(294\) −7.91520e8 −0.00617873
\(295\) 3.17891e10 0.244387
\(296\) −3.24600e9 −0.0245774
\(297\) −5.66957e9 −0.0422811
\(298\) −5.70580e10 −0.419125
\(299\) −3.36376e10 −0.243391
\(300\) 1.16624e9 0.00831271
\(301\) −1.97517e11 −1.38693
\(302\) 1.99768e11 1.38196
\(303\) −4.85623e9 −0.0330985
\(304\) 2.73862e9 0.0183908
\(305\) 4.32946e10 0.286473
\(306\) −4.62807e10 −0.301755
\(307\) 7.83989e10 0.503718 0.251859 0.967764i \(-0.418958\pi\)
0.251859 + 0.967764i \(0.418958\pi\)
\(308\) −7.46852e10 −0.472886
\(309\) −5.34652e9 −0.0333624
\(310\) 1.11862e11 0.687948
\(311\) −2.81000e11 −1.70328 −0.851639 0.524130i \(-0.824391\pi\)
−0.851639 + 0.524130i \(0.824391\pi\)
\(312\) 1.81279e9 0.0108306
\(313\) −8.83831e10 −0.520499 −0.260249 0.965541i \(-0.583805\pi\)
−0.260249 + 0.965541i \(0.583805\pi\)
\(314\) 2.32566e11 1.35009
\(315\) −8.36425e10 −0.478662
\(316\) 5.22552e10 0.294807
\(317\) 2.32664e11 1.29409 0.647043 0.762454i \(-0.276006\pi\)
0.647043 + 0.762454i \(0.276006\pi\)
\(318\) 8.89721e9 0.0487901
\(319\) 1.67029e11 0.903098
\(320\) 1.30811e9 0.00697382
\(321\) 7.85040e9 0.0412686
\(322\) −5.72295e10 −0.296667
\(323\) 6.99399e8 0.00357531
\(324\) 1.08825e11 0.548623
\(325\) 1.26507e11 0.628982
\(326\) 1.72730e11 0.847012
\(327\) −7.99271e9 −0.0386571
\(328\) −3.53269e10 −0.168528
\(329\) 6.78540e10 0.319297
\(330\) −3.09517e9 −0.0143672
\(331\) −1.08788e11 −0.498144 −0.249072 0.968485i \(-0.580126\pi\)
−0.249072 + 0.968485i \(0.580126\pi\)
\(332\) 1.01893e11 0.460282
\(333\) 9.82769e9 0.0437978
\(334\) 4.31562e11 1.89751
\(335\) −1.33209e11 −0.577874
\(336\) 5.51115e9 0.0235893
\(337\) 1.67127e11 0.705849 0.352925 0.935652i \(-0.385187\pi\)
0.352925 + 0.935652i \(0.385187\pi\)
\(338\) 5.89339e10 0.245607
\(339\) −2.88218e9 −0.0118529
\(340\) −1.79219e10 −0.0727327
\(341\) 2.48037e11 0.993393
\(342\) −4.64017e9 −0.0183407
\(343\) −2.76692e11 −1.07938
\(344\) −2.30300e11 −0.886709
\(345\) −8.40988e8 −0.00319598
\(346\) −4.09731e11 −1.53694
\(347\) 4.77989e11 1.76985 0.884923 0.465737i \(-0.154211\pi\)
0.884923 + 0.465737i \(0.154211\pi\)
\(348\) 2.98194e9 0.0108991
\(349\) −4.80098e11 −1.73227 −0.866134 0.499811i \(-0.833403\pi\)
−0.866134 + 0.499811i \(0.833403\pi\)
\(350\) 2.15233e11 0.766659
\(351\) −1.09795e10 −0.0386099
\(352\) −2.80332e11 −0.973265
\(353\) −2.98926e11 −1.02466 −0.512328 0.858790i \(-0.671217\pi\)
−0.512328 + 0.858790i \(0.671217\pi\)
\(354\) 3.54892e9 0.0120111
\(355\) 2.64198e10 0.0882881
\(356\) −1.41993e10 −0.0468536
\(357\) 1.40746e9 0.00458595
\(358\) −1.30940e11 −0.421308
\(359\) 3.68878e11 1.17208 0.586041 0.810282i \(-0.300686\pi\)
0.586041 + 0.810282i \(0.300686\pi\)
\(360\) −9.75251e10 −0.306024
\(361\) −3.22618e11 −0.999783
\(362\) −3.05833e10 −0.0936044
\(363\) 2.66749e8 0.000806350 0
\(364\) −1.44632e11 −0.431826
\(365\) −2.99969e11 −0.884622
\(366\) 4.83339e9 0.0140795
\(367\) −6.01112e11 −1.72965 −0.864825 0.502074i \(-0.832571\pi\)
−0.864825 + 0.502074i \(0.832571\pi\)
\(368\) −1.19236e11 −0.338917
\(369\) 1.06957e11 0.300323
\(370\) 1.07329e10 0.0297721
\(371\) 5.82229e11 1.59555
\(372\) 4.42814e9 0.0119889
\(373\) 3.28495e11 0.878697 0.439349 0.898317i \(-0.355209\pi\)
0.439349 + 0.898317i \(0.355209\pi\)
\(374\) −1.12072e11 −0.296194
\(375\) 7.66803e9 0.0200236
\(376\) 7.91161e10 0.204136
\(377\) 3.23462e11 0.824684
\(378\) −1.86800e10 −0.0470612
\(379\) 1.10170e11 0.274276 0.137138 0.990552i \(-0.456210\pi\)
0.137138 + 0.990552i \(0.456210\pi\)
\(380\) −1.79688e9 −0.00442072
\(381\) −2.91718e9 −0.00709253
\(382\) 1.06557e11 0.256035
\(383\) 4.46966e11 1.06140 0.530702 0.847559i \(-0.321928\pi\)
0.530702 + 0.847559i \(0.321928\pi\)
\(384\) 9.25562e9 0.0217228
\(385\) −2.02546e11 −0.469841
\(386\) −8.91017e11 −2.04288
\(387\) 6.97263e11 1.58015
\(388\) −2.72179e11 −0.609694
\(389\) 5.35885e11 1.18658 0.593292 0.804987i \(-0.297828\pi\)
0.593292 + 0.804987i \(0.297828\pi\)
\(390\) −5.99398e9 −0.0131197
\(391\) −3.04511e10 −0.0658882
\(392\) −6.03936e10 −0.129183
\(393\) 1.58470e10 0.0335105
\(394\) −3.49846e11 −0.731382
\(395\) 1.41716e11 0.292909
\(396\) 2.63649e11 0.538763
\(397\) −5.15293e11 −1.04111 −0.520555 0.853828i \(-0.674275\pi\)
−0.520555 + 0.853828i \(0.674275\pi\)
\(398\) −1.13120e12 −2.25977
\(399\) 1.41114e8 0.000278735 0
\(400\) 4.48433e11 0.875845
\(401\) −4.93193e11 −0.952504 −0.476252 0.879309i \(-0.658005\pi\)
−0.476252 + 0.879309i \(0.658005\pi\)
\(402\) −1.48714e10 −0.0284012
\(403\) 4.80337e11 0.907138
\(404\) 4.51759e11 0.843705
\(405\) 2.95132e11 0.545091
\(406\) 5.50325e11 1.00520
\(407\) 2.37985e10 0.0429907
\(408\) 1.64106e9 0.00293194
\(409\) −2.86274e11 −0.505857 −0.252928 0.967485i \(-0.581394\pi\)
−0.252928 + 0.967485i \(0.581394\pi\)
\(410\) 1.16808e11 0.204148
\(411\) −9.35699e9 −0.0161751
\(412\) 4.97368e11 0.850434
\(413\) 2.32239e11 0.392790
\(414\) 2.02028e11 0.337995
\(415\) 2.76334e11 0.457318
\(416\) −5.42880e11 −0.888758
\(417\) 1.84744e9 0.00299198
\(418\) −1.12365e10 −0.0180027
\(419\) −1.66697e11 −0.264219 −0.132110 0.991235i \(-0.542175\pi\)
−0.132110 + 0.991235i \(0.542175\pi\)
\(420\) −3.61601e9 −0.00567033
\(421\) −5.15187e11 −0.799274 −0.399637 0.916674i \(-0.630864\pi\)
−0.399637 + 0.916674i \(0.630864\pi\)
\(422\) −3.78238e11 −0.580576
\(423\) −2.39534e11 −0.363777
\(424\) 6.78864e11 1.02009
\(425\) 1.14523e11 0.170271
\(426\) 2.94950e9 0.00433915
\(427\) 3.16295e11 0.460433
\(428\) −7.30296e11 −1.05197
\(429\) −1.32907e10 −0.0189448
\(430\) 7.61486e11 1.07412
\(431\) 5.42449e10 0.0757202 0.0378601 0.999283i \(-0.487946\pi\)
0.0378601 + 0.999283i \(0.487946\pi\)
\(432\) −3.89193e10 −0.0537635
\(433\) −2.49025e11 −0.340445 −0.170222 0.985406i \(-0.554449\pi\)
−0.170222 + 0.985406i \(0.554449\pi\)
\(434\) 8.17225e11 1.10570
\(435\) 8.08702e9 0.0108290
\(436\) 7.43535e11 0.985399
\(437\) −3.05307e9 −0.00400470
\(438\) −3.34884e10 −0.0434771
\(439\) 2.46288e11 0.316485 0.158242 0.987400i \(-0.449417\pi\)
0.158242 + 0.987400i \(0.449417\pi\)
\(440\) −2.36164e11 −0.300384
\(441\) 1.82849e11 0.230208
\(442\) −2.17034e11 −0.270476
\(443\) −1.20603e12 −1.48779 −0.743895 0.668297i \(-0.767024\pi\)
−0.743895 + 0.668297i \(0.767024\pi\)
\(444\) 4.24868e8 0.000518837 0
\(445\) −3.85086e10 −0.0465519
\(446\) 1.24484e12 1.48973
\(447\) −6.12554e9 −0.00725706
\(448\) 9.55660e9 0.0112086
\(449\) −1.43303e12 −1.66398 −0.831988 0.554794i \(-0.812797\pi\)
−0.831988 + 0.554794i \(0.812797\pi\)
\(450\) −7.59802e11 −0.873462
\(451\) 2.59003e11 0.294788
\(452\) 2.68120e11 0.302138
\(453\) 2.14464e10 0.0239283
\(454\) 9.65879e11 1.06702
\(455\) −3.92243e11 −0.429046
\(456\) 1.64535e8 0.000178204 0
\(457\) −1.45072e12 −1.55582 −0.777911 0.628375i \(-0.783721\pi\)
−0.777911 + 0.628375i \(0.783721\pi\)
\(458\) −1.65376e12 −1.75622
\(459\) −9.93937e9 −0.0104521
\(460\) 7.82342e10 0.0814679
\(461\) 4.79650e11 0.494619 0.247309 0.968937i \(-0.420454\pi\)
0.247309 + 0.968937i \(0.420454\pi\)
\(462\) −2.26122e10 −0.0230916
\(463\) −1.02183e11 −0.103339 −0.0516697 0.998664i \(-0.516454\pi\)
−0.0516697 + 0.998664i \(0.516454\pi\)
\(464\) 1.14659e12 1.14836
\(465\) 1.20091e10 0.0119117
\(466\) −2.29954e12 −2.25894
\(467\) −9.48806e11 −0.923106 −0.461553 0.887113i \(-0.652708\pi\)
−0.461553 + 0.887113i \(0.652708\pi\)
\(468\) 5.10572e11 0.491984
\(469\) −9.73180e11 −0.928785
\(470\) −2.61597e11 −0.247282
\(471\) 2.49674e10 0.0233765
\(472\) 2.70785e11 0.251123
\(473\) 1.68847e12 1.55103
\(474\) 1.58211e10 0.0143958
\(475\) 1.14822e10 0.0103491
\(476\) −1.30931e11 −0.116899
\(477\) −2.05535e12 −1.81783
\(478\) 1.96954e12 1.72560
\(479\) 1.54645e12 1.34223 0.671114 0.741354i \(-0.265816\pi\)
0.671114 + 0.741354i \(0.265816\pi\)
\(480\) −1.35728e10 −0.0116703
\(481\) 4.60871e10 0.0392579
\(482\) −6.20161e11 −0.523350
\(483\) −6.14395e9 −0.00513672
\(484\) −2.48148e10 −0.0205545
\(485\) −7.38150e11 −0.605768
\(486\) 9.89220e10 0.0804321
\(487\) −8.16740e11 −0.657966 −0.328983 0.944336i \(-0.606706\pi\)
−0.328983 + 0.944336i \(0.606706\pi\)
\(488\) 3.68792e11 0.294369
\(489\) 1.85437e10 0.0146658
\(490\) 1.99691e11 0.156486
\(491\) 2.36995e11 0.184023 0.0920116 0.995758i \(-0.470670\pi\)
0.0920116 + 0.995758i \(0.470670\pi\)
\(492\) 4.62392e9 0.00355768
\(493\) 2.92821e11 0.223250
\(494\) −2.17602e10 −0.0164396
\(495\) 7.15017e11 0.535294
\(496\) 1.70267e12 1.26317
\(497\) 1.93014e11 0.141901
\(498\) 3.08499e10 0.0224761
\(499\) 1.95695e12 1.41295 0.706477 0.707736i \(-0.250283\pi\)
0.706477 + 0.707736i \(0.250283\pi\)
\(500\) −7.13330e11 −0.510418
\(501\) 4.63309e10 0.0328549
\(502\) 2.11802e12 1.48855
\(503\) 1.25952e12 0.877302 0.438651 0.898658i \(-0.355456\pi\)
0.438651 + 0.898658i \(0.355456\pi\)
\(504\) −7.12483e11 −0.491855
\(505\) 1.22517e12 0.838273
\(506\) 4.89226e11 0.331766
\(507\) 6.32693e9 0.00425263
\(508\) 2.71376e11 0.180794
\(509\) 8.63543e11 0.570235 0.285118 0.958493i \(-0.407967\pi\)
0.285118 + 0.958493i \(0.407967\pi\)
\(510\) −5.42617e9 −0.00355163
\(511\) −2.19146e12 −1.42180
\(512\) −8.36291e11 −0.537827
\(513\) −9.96535e8 −0.000635279 0
\(514\) 1.37497e12 0.868881
\(515\) 1.34886e12 0.844958
\(516\) 3.01439e10 0.0187187
\(517\) −5.80049e11 −0.357073
\(518\) 7.84106e10 0.0478510
\(519\) −4.39872e10 −0.0266117
\(520\) −4.57345e11 −0.274302
\(521\) 8.44577e11 0.502192 0.251096 0.967962i \(-0.419209\pi\)
0.251096 + 0.967962i \(0.419209\pi\)
\(522\) −1.94272e12 −1.14523
\(523\) 9.06283e11 0.529671 0.264835 0.964294i \(-0.414682\pi\)
0.264835 + 0.964294i \(0.414682\pi\)
\(524\) −1.47419e12 −0.854209
\(525\) 2.31066e10 0.0132745
\(526\) −4.25455e11 −0.242336
\(527\) 4.34835e11 0.245571
\(528\) −4.71120e10 −0.0263802
\(529\) −1.66823e12 −0.926199
\(530\) −2.24466e12 −1.23569
\(531\) −8.19837e11 −0.447510
\(532\) −1.31273e10 −0.00710517
\(533\) 5.01575e11 0.269193
\(534\) −4.29908e9 −0.00228792
\(535\) −1.98056e12 −1.04519
\(536\) −1.13470e12 −0.593801
\(537\) −1.40573e10 −0.00729485
\(538\) −2.06931e11 −0.106489
\(539\) 4.42783e11 0.225965
\(540\) 2.55360e10 0.0129235
\(541\) 1.55099e12 0.778435 0.389217 0.921146i \(-0.372745\pi\)
0.389217 + 0.921146i \(0.372745\pi\)
\(542\) −2.60853e12 −1.29837
\(543\) −3.28331e9 −0.00162074
\(544\) −4.91453e11 −0.240595
\(545\) 2.01647e12 0.979054
\(546\) −4.37898e10 −0.0210866
\(547\) −2.93476e12 −1.40162 −0.700808 0.713350i \(-0.747177\pi\)
−0.700808 + 0.713350i \(0.747177\pi\)
\(548\) 8.70449e11 0.412317
\(549\) −1.11656e12 −0.524576
\(550\) −1.83992e12 −0.857365
\(551\) 2.93586e10 0.0135692
\(552\) −7.16370e9 −0.00328406
\(553\) 1.03533e12 0.470776
\(554\) −5.89912e11 −0.266068
\(555\) 1.15224e9 0.000515497 0
\(556\) −1.71861e11 −0.0762678
\(557\) −2.35783e12 −1.03792 −0.518961 0.854798i \(-0.673681\pi\)
−0.518961 + 0.854798i \(0.673681\pi\)
\(558\) −2.88492e12 −1.25974
\(559\) 3.26983e12 1.41635
\(560\) −1.39040e12 −0.597438
\(561\) −1.20317e10 −0.00512853
\(562\) 2.12370e12 0.898006
\(563\) 2.31243e12 0.970020 0.485010 0.874509i \(-0.338816\pi\)
0.485010 + 0.874509i \(0.338816\pi\)
\(564\) −1.03555e10 −0.00430938
\(565\) 7.27141e11 0.300193
\(566\) 4.41798e12 1.80946
\(567\) 2.15613e12 0.876095
\(568\) 2.25049e11 0.0907214
\(569\) −8.35674e11 −0.334220 −0.167110 0.985938i \(-0.553443\pi\)
−0.167110 + 0.985938i \(0.553443\pi\)
\(570\) −5.44035e8 −0.000215869 0
\(571\) −2.79849e12 −1.10169 −0.550846 0.834607i \(-0.685695\pi\)
−0.550846 + 0.834607i \(0.685695\pi\)
\(572\) 1.23639e12 0.482917
\(573\) 1.14396e10 0.00443318
\(574\) 8.53357e11 0.328116
\(575\) −4.99923e11 −0.190721
\(576\) −3.37361e10 −0.0127701
\(577\) 4.23010e12 1.58876 0.794382 0.607419i \(-0.207795\pi\)
0.794382 + 0.607419i \(0.207795\pi\)
\(578\) −1.96475e11 −0.0732203
\(579\) −9.56563e10 −0.0353720
\(580\) −7.52308e11 −0.276038
\(581\) 2.01880e12 0.735022
\(582\) −8.24068e10 −0.0297721
\(583\) −4.97717e12 −1.78433
\(584\) −2.55519e12 −0.909003
\(585\) 1.38467e12 0.488816
\(586\) 5.37950e12 1.88453
\(587\) −9.29095e11 −0.322990 −0.161495 0.986874i \(-0.551632\pi\)
−0.161495 + 0.986874i \(0.551632\pi\)
\(588\) 7.90490e9 0.00272708
\(589\) 4.35971e10 0.0149259
\(590\) −8.95350e11 −0.304200
\(591\) −3.75582e10 −0.0126637
\(592\) 1.63367e11 0.0546658
\(593\) 3.56436e12 1.18368 0.591841 0.806055i \(-0.298401\pi\)
0.591841 + 0.806055i \(0.298401\pi\)
\(594\) 1.59685e11 0.0526291
\(595\) −3.55086e11 −0.116147
\(596\) 5.69838e11 0.184988
\(597\) −1.21441e11 −0.0391274
\(598\) 9.47415e11 0.302960
\(599\) −5.69493e12 −1.80746 −0.903728 0.428106i \(-0.859181\pi\)
−0.903728 + 0.428106i \(0.859181\pi\)
\(600\) 2.69417e10 0.00848682
\(601\) −3.23003e12 −1.00989 −0.504943 0.863153i \(-0.668486\pi\)
−0.504943 + 0.863153i \(0.668486\pi\)
\(602\) 5.56314e12 1.72638
\(603\) 3.43546e12 1.05817
\(604\) −1.99508e12 −0.609951
\(605\) −6.72977e10 −0.0204221
\(606\) 1.36778e11 0.0411991
\(607\) −1.62163e12 −0.484846 −0.242423 0.970171i \(-0.577942\pi\)
−0.242423 + 0.970171i \(0.577942\pi\)
\(608\) −4.92737e10 −0.0146234
\(609\) 5.90808e10 0.0174048
\(610\) −1.21941e12 −0.356586
\(611\) −1.12330e12 −0.326070
\(612\) 4.62205e11 0.133184
\(613\) 3.22227e12 0.921700 0.460850 0.887478i \(-0.347545\pi\)
0.460850 + 0.887478i \(0.347545\pi\)
\(614\) −2.20813e12 −0.627000
\(615\) 1.25401e10 0.00353478
\(616\) −1.72533e12 −0.482790
\(617\) −3.76562e12 −1.04605 −0.523026 0.852317i \(-0.675197\pi\)
−0.523026 + 0.852317i \(0.675197\pi\)
\(618\) 1.50587e11 0.0415277
\(619\) 3.75883e12 1.02907 0.514535 0.857470i \(-0.327965\pi\)
0.514535 + 0.857470i \(0.327965\pi\)
\(620\) −1.11717e12 −0.303637
\(621\) 4.33881e10 0.0117073
\(622\) 7.91448e12 2.12014
\(623\) −2.81330e11 −0.0748203
\(624\) −9.12351e10 −0.0240897
\(625\) 7.43541e11 0.194915
\(626\) 2.48934e12 0.647888
\(627\) −1.20631e9 −0.000311713 0
\(628\) −2.32263e12 −0.595884
\(629\) 4.17213e10 0.0106275
\(630\) 2.35582e12 0.595812
\(631\) −5.72141e12 −1.43672 −0.718358 0.695673i \(-0.755106\pi\)
−0.718358 + 0.695673i \(0.755106\pi\)
\(632\) 1.20717e12 0.300982
\(633\) −4.06062e10 −0.0100525
\(634\) −6.55307e12 −1.61081
\(635\) 7.35971e11 0.179630
\(636\) −8.88563e10 −0.0215343
\(637\) 8.57475e11 0.206345
\(638\) −4.70444e12 −1.12413
\(639\) −6.81365e11 −0.161669
\(640\) −2.33508e12 −0.550165
\(641\) −2.45043e12 −0.573299 −0.286650 0.958035i \(-0.592542\pi\)
−0.286650 + 0.958035i \(0.592542\pi\)
\(642\) −2.21109e11 −0.0513688
\(643\) 5.20042e12 1.19975 0.599873 0.800095i \(-0.295218\pi\)
0.599873 + 0.800095i \(0.295218\pi\)
\(644\) 5.71551e11 0.130939
\(645\) 8.17503e10 0.0185982
\(646\) −1.96988e10 −0.00445035
\(647\) 9.06811e11 0.203445 0.101723 0.994813i \(-0.467565\pi\)
0.101723 + 0.994813i \(0.467565\pi\)
\(648\) 2.51399e12 0.560114
\(649\) −1.98530e12 −0.439262
\(650\) −3.56310e12 −0.782922
\(651\) 8.77342e10 0.0191450
\(652\) −1.72506e12 −0.373843
\(653\) 5.74944e12 1.23742 0.618708 0.785621i \(-0.287656\pi\)
0.618708 + 0.785621i \(0.287656\pi\)
\(654\) 2.25118e11 0.0481182
\(655\) −3.99802e12 −0.848709
\(656\) 1.77795e12 0.374845
\(657\) 7.73616e12 1.61987
\(658\) −1.91113e12 −0.397443
\(659\) −1.14504e12 −0.236503 −0.118252 0.992984i \(-0.537729\pi\)
−0.118252 + 0.992984i \(0.537729\pi\)
\(660\) 3.09114e10 0.00634120
\(661\) −2.18103e12 −0.444380 −0.222190 0.975003i \(-0.571321\pi\)
−0.222190 + 0.975003i \(0.571321\pi\)
\(662\) 3.06405e12 0.620062
\(663\) −2.33000e10 −0.00468323
\(664\) 2.35387e12 0.469922
\(665\) −3.56014e10 −0.00705943
\(666\) −2.76800e11 −0.0545171
\(667\) −1.27824e12 −0.250062
\(668\) −4.31000e12 −0.837497
\(669\) 1.33642e11 0.0257944
\(670\) 3.75189e12 0.719306
\(671\) −2.70384e12 −0.514908
\(672\) −9.91577e10 −0.0187571
\(673\) −5.45288e12 −1.02461 −0.512304 0.858804i \(-0.671208\pi\)
−0.512304 + 0.858804i \(0.671208\pi\)
\(674\) −4.70719e12 −0.878602
\(675\) −1.63177e11 −0.0302546
\(676\) −5.88573e11 −0.108403
\(677\) −4.84898e12 −0.887159 −0.443579 0.896235i \(-0.646292\pi\)
−0.443579 + 0.896235i \(0.646292\pi\)
\(678\) 8.11777e10 0.0147538
\(679\) −5.39266e12 −0.973618
\(680\) −4.14021e11 −0.0742561
\(681\) 1.03693e11 0.0184752
\(682\) −6.98604e12 −1.23652
\(683\) −1.30756e11 −0.0229916 −0.0114958 0.999934i \(-0.503659\pi\)
−0.0114958 + 0.999934i \(0.503659\pi\)
\(684\) 4.63414e10 0.00809499
\(685\) 2.36066e12 0.409662
\(686\) 7.79312e12 1.34355
\(687\) −1.77542e11 −0.0304085
\(688\) 1.15907e13 1.97224
\(689\) −9.63859e12 −1.62940
\(690\) 2.36867e10 0.00397818
\(691\) 3.22565e12 0.538227 0.269113 0.963108i \(-0.413269\pi\)
0.269113 + 0.963108i \(0.413269\pi\)
\(692\) 4.09198e12 0.678353
\(693\) 5.22365e12 0.860349
\(694\) −1.34627e13 −2.20301
\(695\) −4.66088e11 −0.0757767
\(696\) 6.88868e10 0.0111274
\(697\) 4.54060e11 0.0728729
\(698\) 1.35221e13 2.15623
\(699\) −2.46870e11 −0.0391130
\(700\) −2.14953e12 −0.338378
\(701\) 8.43820e12 1.31983 0.659916 0.751339i \(-0.270592\pi\)
0.659916 + 0.751339i \(0.270592\pi\)
\(702\) 3.09240e11 0.0480595
\(703\) 4.18303e9 0.000645940 0
\(704\) −8.16945e10 −0.0125348
\(705\) −2.80841e10 −0.00428163
\(706\) 8.41936e12 1.27543
\(707\) 8.95065e12 1.34731
\(708\) −3.54430e10 −0.00530128
\(709\) 5.72297e12 0.850576 0.425288 0.905058i \(-0.360173\pi\)
0.425288 + 0.905058i \(0.360173\pi\)
\(710\) −7.44123e11 −0.109896
\(711\) −3.65485e12 −0.536360
\(712\) −3.28024e11 −0.0478349
\(713\) −1.89817e12 −0.275064
\(714\) −3.96416e10 −0.00570833
\(715\) 3.35308e12 0.479808
\(716\) 1.30770e12 0.185951
\(717\) 2.11443e11 0.0298783
\(718\) −1.03896e13 −1.45894
\(719\) −6.14814e12 −0.857954 −0.428977 0.903316i \(-0.641126\pi\)
−0.428977 + 0.903316i \(0.641126\pi\)
\(720\) 4.90830e12 0.680667
\(721\) 9.85431e12 1.35805
\(722\) 9.08664e12 1.24447
\(723\) −6.65782e10 −0.00906170
\(724\) 3.05435e11 0.0413138
\(725\) 4.80731e12 0.646220
\(726\) −7.51309e9 −0.00100370
\(727\) −9.21269e12 −1.22316 −0.611578 0.791184i \(-0.709465\pi\)
−0.611578 + 0.791184i \(0.709465\pi\)
\(728\) −3.34120e12 −0.440871
\(729\) −7.60435e12 −0.997214
\(730\) 8.44872e12 1.10113
\(731\) 2.96008e12 0.383420
\(732\) −4.82710e10 −0.00621422
\(733\) −8.69962e12 −1.11310 −0.556548 0.830816i \(-0.687874\pi\)
−0.556548 + 0.830816i \(0.687874\pi\)
\(734\) 1.69305e13 2.15297
\(735\) 2.14381e10 0.00270953
\(736\) 2.14533e12 0.269490
\(737\) 8.31921e12 1.03867
\(738\) −3.01247e12 −0.373826
\(739\) −8.74888e12 −1.07908 −0.539539 0.841961i \(-0.681401\pi\)
−0.539539 + 0.841961i \(0.681401\pi\)
\(740\) −1.07189e11 −0.0131404
\(741\) −2.33609e9 −0.000284648 0
\(742\) −1.63987e13 −1.98605
\(743\) −4.65215e11 −0.0560021 −0.0280010 0.999608i \(-0.508914\pi\)
−0.0280010 + 0.999608i \(0.508914\pi\)
\(744\) 1.02296e11 0.0122400
\(745\) 1.54540e12 0.183797
\(746\) −9.25219e12 −1.09375
\(747\) −7.12664e12 −0.837418
\(748\) 1.11926e12 0.130730
\(749\) −1.44693e13 −1.67988
\(750\) −2.15973e11 −0.0249243
\(751\) −8.93867e12 −1.02540 −0.512700 0.858568i \(-0.671355\pi\)
−0.512700 + 0.858568i \(0.671355\pi\)
\(752\) −3.98180e12 −0.454045
\(753\) 2.27383e11 0.0257739
\(754\) −9.11043e12 −1.02652
\(755\) −5.41067e12 −0.606024
\(756\) 1.86557e11 0.0207712
\(757\) 6.47580e12 0.716741 0.358371 0.933579i \(-0.383332\pi\)
0.358371 + 0.933579i \(0.383332\pi\)
\(758\) −3.10298e12 −0.341403
\(759\) 5.25215e10 0.00574446
\(760\) −4.15103e10 −0.00451331
\(761\) 1.50384e12 0.162544 0.0812720 0.996692i \(-0.474102\pi\)
0.0812720 + 0.996692i \(0.474102\pi\)
\(762\) 8.21635e10 0.00882839
\(763\) 1.47316e13 1.57358
\(764\) −1.06419e12 −0.113005
\(765\) 1.25350e12 0.132327
\(766\) −1.25890e13 −1.32118
\(767\) −3.84464e12 −0.401122
\(768\) −2.58033e11 −0.0267639
\(769\) 1.49083e13 1.53731 0.768653 0.639666i \(-0.220927\pi\)
0.768653 + 0.639666i \(0.220927\pi\)
\(770\) 5.70479e12 0.584832
\(771\) 1.47612e11 0.0150445
\(772\) 8.89858e12 0.901660
\(773\) −2.45775e11 −0.0247588 −0.0123794 0.999923i \(-0.503941\pi\)
−0.0123794 + 0.999923i \(0.503941\pi\)
\(774\) −1.96387e13 −1.96688
\(775\) 7.13878e12 0.710831
\(776\) −6.28770e12 −0.622464
\(777\) 8.41787e9 0.000828529 0
\(778\) −1.50934e13 −1.47699
\(779\) 4.55247e10 0.00442924
\(780\) 5.98618e10 0.00579061
\(781\) −1.64997e12 −0.158689
\(782\) 8.57666e11 0.0820140
\(783\) −4.17224e11 −0.0396681
\(784\) 3.03952e12 0.287332
\(785\) −6.29898e12 −0.592047
\(786\) −4.46337e11 −0.0417120
\(787\) 8.04995e12 0.748009 0.374004 0.927427i \(-0.377984\pi\)
0.374004 + 0.927427i \(0.377984\pi\)
\(788\) 3.49391e12 0.322808
\(789\) −4.56753e10 −0.00419600
\(790\) −3.99149e12 −0.364597
\(791\) 5.31223e12 0.482484
\(792\) 6.09065e12 0.550048
\(793\) −5.23615e12 −0.470200
\(794\) 1.45134e13 1.29592
\(795\) −2.40978e11 −0.0213957
\(796\) 1.12972e13 0.997387
\(797\) 2.27278e12 0.199524 0.0997618 0.995011i \(-0.468192\pi\)
0.0997618 + 0.995011i \(0.468192\pi\)
\(798\) −3.97452e9 −0.000346954 0
\(799\) −1.01689e12 −0.0882700
\(800\) −8.06829e12 −0.696429
\(801\) 9.93133e11 0.0852435
\(802\) 1.38909e13 1.18562
\(803\) 1.87337e13 1.59002
\(804\) 1.48521e11 0.0125353
\(805\) 1.55005e12 0.130096
\(806\) −1.35289e13 −1.12916
\(807\) −2.22154e10 −0.00184384
\(808\) 1.04362e13 0.861376
\(809\) 2.47519e12 0.203161 0.101581 0.994827i \(-0.467610\pi\)
0.101581 + 0.994827i \(0.467610\pi\)
\(810\) −8.31250e12 −0.678499
\(811\) 2.45293e13 1.99110 0.995548 0.0942611i \(-0.0300488\pi\)
0.995548 + 0.0942611i \(0.0300488\pi\)
\(812\) −5.49609e12 −0.443661
\(813\) −2.80042e11 −0.0224810
\(814\) −6.70292e11 −0.0535124
\(815\) −4.67836e12 −0.371436
\(816\) −8.25924e10 −0.00652130
\(817\) 2.96781e11 0.0233044
\(818\) 8.06302e12 0.629662
\(819\) 1.01159e13 0.785647
\(820\) −1.16656e12 −0.0901042
\(821\) −8.21852e12 −0.631319 −0.315660 0.948872i \(-0.602226\pi\)
−0.315660 + 0.948872i \(0.602226\pi\)
\(822\) 2.63543e11 0.0201339
\(823\) 1.55798e12 0.118376 0.0591878 0.998247i \(-0.481149\pi\)
0.0591878 + 0.998247i \(0.481149\pi\)
\(824\) 1.14899e13 0.868246
\(825\) −1.97527e11 −0.0148451
\(826\) −6.54110e12 −0.488923
\(827\) 1.18962e13 0.884367 0.442184 0.896924i \(-0.354204\pi\)
0.442184 + 0.896924i \(0.354204\pi\)
\(828\) −2.01765e12 −0.149180
\(829\) −8.80333e12 −0.647368 −0.323684 0.946165i \(-0.604922\pi\)
−0.323684 + 0.946165i \(0.604922\pi\)
\(830\) −7.78305e12 −0.569244
\(831\) −6.33308e10 −0.00460691
\(832\) −1.58206e11 −0.0114464
\(833\) 7.76247e11 0.0558595
\(834\) −5.20338e10 −0.00372425
\(835\) −1.16887e13 −0.832105
\(836\) 1.12219e11 0.00794581
\(837\) −6.19572e11 −0.0436342
\(838\) 4.69508e12 0.328886
\(839\) −6.42832e12 −0.447887 −0.223944 0.974602i \(-0.571893\pi\)
−0.223944 + 0.974602i \(0.571893\pi\)
\(840\) −8.35348e10 −0.00578909
\(841\) −2.21544e12 −0.152714
\(842\) 1.45104e13 0.994892
\(843\) 2.27992e11 0.0155488
\(844\) 3.77746e12 0.256247
\(845\) −1.59621e12 −0.107705
\(846\) 6.74656e12 0.452810
\(847\) −4.91653e11 −0.0328234
\(848\) −3.41663e13 −2.26890
\(849\) 4.74298e11 0.0313305
\(850\) −3.22557e12 −0.211944
\(851\) −1.82125e11 −0.0119038
\(852\) −2.94566e10 −0.00191516
\(853\) −1.34906e13 −0.872492 −0.436246 0.899827i \(-0.643692\pi\)
−0.436246 + 0.899827i \(0.643692\pi\)
\(854\) −8.90855e12 −0.573121
\(855\) 1.25678e11 0.00804287
\(856\) −1.68708e13 −1.07400
\(857\) −2.46136e13 −1.55870 −0.779348 0.626592i \(-0.784449\pi\)
−0.779348 + 0.626592i \(0.784449\pi\)
\(858\) 3.74337e11 0.0235814
\(859\) 2.45934e13 1.54117 0.770583 0.637340i \(-0.219965\pi\)
0.770583 + 0.637340i \(0.219965\pi\)
\(860\) −7.60495e12 −0.474082
\(861\) 9.16133e10 0.00568125
\(862\) −1.52783e12 −0.0942523
\(863\) 1.57068e12 0.0963918 0.0481959 0.998838i \(-0.484653\pi\)
0.0481959 + 0.998838i \(0.484653\pi\)
\(864\) 7.00244e11 0.0427501
\(865\) 1.10974e13 0.673985
\(866\) 7.01386e12 0.423767
\(867\) −2.10928e10 −0.00126779
\(868\) −8.16162e12 −0.488020
\(869\) −8.85048e12 −0.526475
\(870\) −2.27774e11 −0.0134793
\(871\) 1.61106e13 0.948487
\(872\) 1.71767e13 1.00604
\(873\) 1.90368e13 1.10925
\(874\) 8.59908e10 0.00498483
\(875\) −1.41331e13 −0.815084
\(876\) 3.34448e11 0.0191893
\(877\) −2.26638e12 −0.129370 −0.0646852 0.997906i \(-0.520604\pi\)
−0.0646852 + 0.997906i \(0.520604\pi\)
\(878\) −6.93678e12 −0.393942
\(879\) 5.77523e11 0.0326302
\(880\) 1.18858e13 0.668123
\(881\) −2.20342e13 −1.23227 −0.616134 0.787641i \(-0.711302\pi\)
−0.616134 + 0.787641i \(0.711302\pi\)
\(882\) −5.15002e12 −0.286550
\(883\) 7.71424e12 0.427041 0.213521 0.976939i \(-0.431507\pi\)
0.213521 + 0.976939i \(0.431507\pi\)
\(884\) 2.16752e12 0.119379
\(885\) −9.61215e10 −0.00526715
\(886\) 3.39683e13 1.85192
\(887\) −3.10912e13 −1.68648 −0.843239 0.537539i \(-0.819354\pi\)
−0.843239 + 0.537539i \(0.819354\pi\)
\(888\) 9.81503e9 0.000529704 0
\(889\) 5.37674e12 0.288709
\(890\) 1.08461e12 0.0579452
\(891\) −1.84316e13 −0.979748
\(892\) −1.24322e13 −0.657518
\(893\) −1.01955e11 −0.00536508
\(894\) 1.72528e11 0.00903318
\(895\) 3.54648e12 0.184754
\(896\) −1.70593e13 −0.884249
\(897\) 1.01711e11 0.00524568
\(898\) 4.03618e13 2.07122
\(899\) 1.82530e13 0.932000
\(900\) 7.58813e12 0.385517
\(901\) −8.72552e12 −0.441093
\(902\) −7.29491e12 −0.366936
\(903\) 5.97238e11 0.0298918
\(904\) 6.19393e12 0.308467
\(905\) 8.28341e11 0.0410479
\(906\) −6.04045e11 −0.0297846
\(907\) −9.61906e12 −0.471954 −0.235977 0.971759i \(-0.575829\pi\)
−0.235977 + 0.971759i \(0.575829\pi\)
\(908\) −9.64623e12 −0.470946
\(909\) −3.15970e13 −1.53500
\(910\) 1.10477e13 0.534052
\(911\) −1.52076e13 −0.731525 −0.365762 0.930708i \(-0.619192\pi\)
−0.365762 + 0.930708i \(0.619192\pi\)
\(912\) −8.28083e9 −0.000396367 0
\(913\) −1.72577e13 −0.821985
\(914\) 4.08600e13 1.93660
\(915\) −1.30911e11 −0.00617421
\(916\) 1.65161e13 0.775135
\(917\) −2.92081e13 −1.36408
\(918\) 2.79946e11 0.0130101
\(919\) 1.96701e13 0.909675 0.454838 0.890574i \(-0.349697\pi\)
0.454838 + 0.890574i \(0.349697\pi\)
\(920\) 1.80732e12 0.0831742
\(921\) −2.37057e11 −0.0108564
\(922\) −1.35095e13 −0.615674
\(923\) −3.19527e12 −0.144911
\(924\) 2.25828e11 0.0101919
\(925\) 6.84948e11 0.0307624
\(926\) 2.87803e12 0.128631
\(927\) −3.47871e13 −1.54724
\(928\) −2.06297e13 −0.913116
\(929\) 3.18903e13 1.40471 0.702357 0.711825i \(-0.252131\pi\)
0.702357 + 0.711825i \(0.252131\pi\)
\(930\) −3.38241e11 −0.0148270
\(931\) 7.78276e10 0.00339516
\(932\) 2.29655e13 0.997020
\(933\) 8.49669e11 0.0367098
\(934\) 2.67235e13 1.14903
\(935\) 3.03544e12 0.129888
\(936\) 1.17949e13 0.502288
\(937\) −2.61645e13 −1.10888 −0.554439 0.832224i \(-0.687067\pi\)
−0.554439 + 0.832224i \(0.687067\pi\)
\(938\) 2.74099e13 1.15610
\(939\) 2.67246e11 0.0112180
\(940\) 2.61257e12 0.109142
\(941\) 3.93723e13 1.63696 0.818479 0.574536i \(-0.194817\pi\)
0.818479 + 0.574536i \(0.194817\pi\)
\(942\) −7.03215e11 −0.0290977
\(943\) −1.98210e12 −0.0816249
\(944\) −1.36282e13 −0.558555
\(945\) 5.05942e11 0.0206375
\(946\) −4.75564e13 −1.93063
\(947\) −2.26031e13 −0.913258 −0.456629 0.889657i \(-0.650943\pi\)
−0.456629 + 0.889657i \(0.650943\pi\)
\(948\) −1.58006e11 −0.00635382
\(949\) 3.62789e13 1.45196
\(950\) −3.23400e11 −0.0128820
\(951\) −7.03513e11 −0.0278907
\(952\) −3.02469e12 −0.119348
\(953\) −3.27556e13 −1.28637 −0.643187 0.765709i \(-0.722388\pi\)
−0.643187 + 0.765709i \(0.722388\pi\)
\(954\) 5.78896e13 2.26273
\(955\) −2.88608e12 −0.112278
\(956\) −1.96698e13 −0.761621
\(957\) −5.05052e11 −0.0194640
\(958\) −4.35563e13 −1.67073
\(959\) 1.72461e13 0.658427
\(960\) −3.95538e9 −0.000150303 0
\(961\) 6.65864e11 0.0251843
\(962\) −1.29806e12 −0.0488660
\(963\) 5.10786e13 1.91390
\(964\) 6.19354e12 0.230989
\(965\) 2.41329e13 0.895854
\(966\) 1.73047e11 0.00639390
\(967\) 2.07251e13 0.762217 0.381108 0.924530i \(-0.375543\pi\)
0.381108 + 0.924530i \(0.375543\pi\)
\(968\) −5.73255e11 −0.0209850
\(969\) −2.11479e9 −7.70568e−5 0
\(970\) 2.07902e13 0.754027
\(971\) 5.19068e13 1.87386 0.936932 0.349512i \(-0.113652\pi\)
0.936932 + 0.349512i \(0.113652\pi\)
\(972\) −9.87933e11 −0.0355001
\(973\) −3.40507e12 −0.121792
\(974\) 2.30038e13 0.819000
\(975\) −3.82522e11 −0.0135561
\(976\) −1.85608e13 −0.654744
\(977\) 2.18772e12 0.0768185 0.0384093 0.999262i \(-0.487771\pi\)
0.0384093 + 0.999262i \(0.487771\pi\)
\(978\) −5.22290e11 −0.0182552
\(979\) 2.40494e12 0.0836725
\(980\) −1.99431e12 −0.0690679
\(981\) −5.20045e13 −1.79279
\(982\) −6.67505e12 −0.229062
\(983\) −1.84743e13 −0.631068 −0.315534 0.948914i \(-0.602184\pi\)
−0.315534 + 0.948914i \(0.602184\pi\)
\(984\) 1.06819e11 0.00363220
\(985\) 9.47549e12 0.320729
\(986\) −8.24740e12 −0.277889
\(987\) −2.05172e11 −0.00688163
\(988\) 2.17319e11 0.00725589
\(989\) −1.29215e13 −0.429468
\(990\) −2.01387e13 −0.666305
\(991\) −3.55303e13 −1.17022 −0.585110 0.810954i \(-0.698949\pi\)
−0.585110 + 0.810954i \(0.698949\pi\)
\(992\) −3.06348e13 −1.00441
\(993\) 3.28945e11 0.0107362
\(994\) −5.43629e12 −0.176630
\(995\) 3.06381e13 0.990965
\(996\) −3.08097e11 −0.00992021
\(997\) 5.54350e13 1.77687 0.888435 0.459003i \(-0.151793\pi\)
0.888435 + 0.459003i \(0.151793\pi\)
\(998\) −5.51183e13 −1.75877
\(999\) −5.94463e10 −0.00188834
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 17.10.a.a.1.2 5
3.2 odd 2 153.10.a.c.1.4 5
4.3 odd 2 272.10.a.f.1.2 5
17.16 even 2 289.10.a.a.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.10.a.a.1.2 5 1.1 even 1 trivial
153.10.a.c.1.4 5 3.2 odd 2
272.10.a.f.1.2 5 4.3 odd 2
289.10.a.a.1.2 5 17.16 even 2