Properties

Label 29.12.a.a.1.4
Level $29$
Weight $12$
Character 29.1
Self dual yes
Analytic conductor $22.282$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.2819522362\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - x^{10} - 15790 x^{9} + 14666 x^{8} + 87206462 x^{7} - 14008334 x^{6} - 203974096304 x^{5} - 388180519304 x^{4} + 193065378004825 x^{3} + 1279291654973975 x^{2} - 65244901875230266 x - 758324542468966858\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-20.7638\) of defining polynomial
Character \(\chi\) \(=\) 29.1

$q$-expansion

\(f(q)\) \(=\) \(q-23.7638 q^{2} -595.785 q^{3} -1483.28 q^{4} -5297.67 q^{5} +14158.1 q^{6} +80499.9 q^{7} +83916.8 q^{8} +177813. q^{9} +O(q^{10})\) \(q-23.7638 q^{2} -595.785 q^{3} -1483.28 q^{4} -5297.67 q^{5} +14158.1 q^{6} +80499.9 q^{7} +83916.8 q^{8} +177813. q^{9} +125893. q^{10} -48833.0 q^{11} +883716. q^{12} -930345. q^{13} -1.91299e6 q^{14} +3.15627e6 q^{15} +1.04357e6 q^{16} +506191. q^{17} -4.22552e6 q^{18} +1.04796e6 q^{19} +7.85793e6 q^{20} -4.79607e7 q^{21} +1.16046e6 q^{22} +2.83418e7 q^{23} -4.99964e7 q^{24} -2.07628e7 q^{25} +2.21086e7 q^{26} -396829. q^{27} -1.19404e8 q^{28} +2.05111e7 q^{29} -7.50052e7 q^{30} +1.94175e8 q^{31} -1.96661e8 q^{32} +2.90940e7 q^{33} -1.20290e7 q^{34} -4.26462e8 q^{35} -2.63747e8 q^{36} -7.56788e8 q^{37} -2.49035e7 q^{38} +5.54286e8 q^{39} -4.44563e8 q^{40} +5.07893e8 q^{41} +1.13973e9 q^{42} -1.48159e9 q^{43} +7.24330e7 q^{44} -9.41995e8 q^{45} -6.73509e8 q^{46} +1.66313e9 q^{47} -6.21745e8 q^{48} +4.50291e9 q^{49} +4.93404e8 q^{50} -3.01581e8 q^{51} +1.37996e9 q^{52} -8.34639e8 q^{53} +9.43018e6 q^{54} +2.58701e8 q^{55} +6.75530e9 q^{56} -6.24359e8 q^{57} -4.87424e8 q^{58} +6.60274e8 q^{59} -4.68164e9 q^{60} -7.10239e9 q^{61} -4.61435e9 q^{62} +1.43139e10 q^{63} +2.53618e9 q^{64} +4.92866e9 q^{65} -6.91385e8 q^{66} +8.27289e8 q^{67} -7.50823e8 q^{68} -1.68856e10 q^{69} +1.01344e10 q^{70} -2.25416e10 q^{71} +1.49215e10 q^{72} +6.51950e9 q^{73} +1.79842e10 q^{74} +1.23702e10 q^{75} -1.55442e9 q^{76} -3.93106e9 q^{77} -1.31720e10 q^{78} +2.48536e10 q^{79} -5.52851e9 q^{80} -3.12626e10 q^{81} -1.20695e10 q^{82} -4.41362e10 q^{83} +7.11391e10 q^{84} -2.68163e9 q^{85} +3.52083e10 q^{86} -1.22202e10 q^{87} -4.09791e9 q^{88} -7.45858e10 q^{89} +2.23854e10 q^{90} -7.48927e10 q^{91} -4.20388e10 q^{92} -1.15687e11 q^{93} -3.95223e10 q^{94} -5.55174e9 q^{95} +1.17168e11 q^{96} -1.19334e11 q^{97} -1.07007e11 q^{98} -8.68315e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 32q^{2} - 982q^{3} + 9146q^{4} - 2740q^{5} - 28202q^{6} - 49432q^{7} - 150054q^{8} + 330749q^{9} + O(q^{10}) \) \( 11q - 32q^{2} - 982q^{3} + 9146q^{4} - 2740q^{5} - 28202q^{6} - 49432q^{7} - 150054q^{8} + 330749q^{9} - 685834q^{10} - 612246q^{11} + 2578538q^{12} + 1510364q^{13} + 3955400q^{14} - 2462818q^{15} + 3024818q^{16} - 3291098q^{17} - 27885614q^{18} - 44121388q^{19} - 49472662q^{20} - 46916800q^{21} - 43435618q^{22} - 88684076q^{23} - 224700678q^{24} - 44195521q^{25} - 324999762q^{26} - 236304286q^{27} - 391274848q^{28} + 225622639q^{29} - 494910382q^{30} - 292235934q^{31} - 632542514q^{32} - 1079766410q^{33} - 1113307936q^{34} - 1312820120q^{35} - 2236726492q^{36} - 1380429338q^{37} - 1222857284q^{38} - 1186931090q^{39} - 2713154106q^{40} - 1062067494q^{41} + 205598960q^{42} + 74588594q^{43} + 52891466q^{44} + 4527996830q^{45} - 87670324q^{46} - 1821239394q^{47} + 2666035542q^{48} + 4692522003q^{49} + 9494259926q^{50} + 8768158380q^{51} + 3266669866q^{52} + 7818635688q^{53} + 17402728558q^{54} - 191002682q^{55} + 11263587512q^{56} + 15495358340q^{57} - 656356768q^{58} + 1230002712q^{59} + 31834046430q^{60} - 18602654230q^{61} + 22075953162q^{62} - 9964531456q^{63} + 11813658086q^{64} + 32245789334q^{65} + 42677188354q^{66} + 27481284652q^{67} + 29588811820q^{68} - 20565315068q^{69} + 42862666712q^{70} - 20347168516q^{71} + 47061083616q^{72} - 57740010478q^{73} - 2640709564q^{74} - 23544691000q^{75} - 33350650772q^{76} + 871959792q^{77} - 15384525342q^{78} - 120245016462q^{79} - 84319695274q^{80} - 48880047865q^{81} - 111495532412q^{82} - 142463983824q^{83} - 134146226376q^{84} - 181628566552q^{85} + 47870165542q^{86} - 20141948318q^{87} - 180608014462q^{88} - 96700717270q^{89} - 25522461244q^{90} - 355162031176q^{91} - 22429477796q^{92} - 172582115142q^{93} + 172608565078q^{94} - 195922150708q^{95} + 226391047758q^{96} - 303190852014q^{97} - 123776497136q^{98} - 139125462440q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −23.7638 −0.525112 −0.262556 0.964917i \(-0.584565\pi\)
−0.262556 + 0.964917i \(0.584565\pi\)
\(3\) −595.785 −1.41554 −0.707771 0.706442i \(-0.750299\pi\)
−0.707771 + 0.706442i \(0.750299\pi\)
\(4\) −1483.28 −0.724258
\(5\) −5297.67 −0.758141 −0.379070 0.925368i \(-0.623756\pi\)
−0.379070 + 0.925368i \(0.623756\pi\)
\(6\) 14158.1 0.743318
\(7\) 80499.9 1.81032 0.905162 0.425066i \(-0.139749\pi\)
0.905162 + 0.425066i \(0.139749\pi\)
\(8\) 83916.8 0.905428
\(9\) 177813. 1.00376
\(10\) 125893. 0.398109
\(11\) −48833.0 −0.0914226 −0.0457113 0.998955i \(-0.514555\pi\)
−0.0457113 + 0.998955i \(0.514555\pi\)
\(12\) 883716. 1.02522
\(13\) −930345. −0.694953 −0.347476 0.937689i \(-0.612961\pi\)
−0.347476 + 0.937689i \(0.612961\pi\)
\(14\) −1.91299e6 −0.950622
\(15\) 3.15627e6 1.07318
\(16\) 1.04357e6 0.248807
\(17\) 506191. 0.0864660 0.0432330 0.999065i \(-0.486234\pi\)
0.0432330 + 0.999065i \(0.486234\pi\)
\(18\) −4.22552e6 −0.527086
\(19\) 1.04796e6 0.0970956 0.0485478 0.998821i \(-0.484541\pi\)
0.0485478 + 0.998821i \(0.484541\pi\)
\(20\) 7.85793e6 0.549089
\(21\) −4.79607e7 −2.56259
\(22\) 1.16046e6 0.0480071
\(23\) 2.83418e7 0.918171 0.459086 0.888392i \(-0.348177\pi\)
0.459086 + 0.888392i \(0.348177\pi\)
\(24\) −4.99964e7 −1.28167
\(25\) −2.07628e7 −0.425222
\(26\) 2.21086e7 0.364928
\(27\) −396829. −0.00532234
\(28\) −1.19404e8 −1.31114
\(29\) 2.05111e7 0.185695
\(30\) −7.50052e7 −0.563539
\(31\) 1.94175e8 1.21816 0.609081 0.793108i \(-0.291539\pi\)
0.609081 + 0.793108i \(0.291539\pi\)
\(32\) −1.96661e8 −1.03608
\(33\) 2.90940e7 0.129413
\(34\) −1.20290e7 −0.0454043
\(35\) −4.26462e8 −1.37248
\(36\) −2.63747e8 −0.726981
\(37\) −7.56788e8 −1.79418 −0.897088 0.441852i \(-0.854322\pi\)
−0.897088 + 0.441852i \(0.854322\pi\)
\(38\) −2.49035e7 −0.0509860
\(39\) 5.54286e8 0.983735
\(40\) −4.44563e8 −0.686442
\(41\) 5.07893e8 0.684638 0.342319 0.939584i \(-0.388788\pi\)
0.342319 + 0.939584i \(0.388788\pi\)
\(42\) 1.13973e9 1.34565
\(43\) −1.48159e9 −1.53692 −0.768461 0.639896i \(-0.778977\pi\)
−0.768461 + 0.639896i \(0.778977\pi\)
\(44\) 7.24330e7 0.0662136
\(45\) −9.41995e8 −0.760991
\(46\) −6.73509e8 −0.482142
\(47\) 1.66313e9 1.05776 0.528880 0.848696i \(-0.322612\pi\)
0.528880 + 0.848696i \(0.322612\pi\)
\(48\) −6.21745e8 −0.352197
\(49\) 4.50291e9 2.27727
\(50\) 4.93404e8 0.223289
\(51\) −3.01581e8 −0.122396
\(52\) 1.37996e9 0.503325
\(53\) −8.34639e8 −0.274146 −0.137073 0.990561i \(-0.543769\pi\)
−0.137073 + 0.990561i \(0.543769\pi\)
\(54\) 9.43018e6 0.00279482
\(55\) 2.58701e8 0.0693112
\(56\) 6.75530e9 1.63912
\(57\) −6.24359e8 −0.137443
\(58\) −4.87424e8 −0.0975108
\(59\) 6.60274e8 0.120237 0.0601185 0.998191i \(-0.480852\pi\)
0.0601185 + 0.998191i \(0.480852\pi\)
\(60\) −4.68164e9 −0.777259
\(61\) −7.10239e9 −1.07669 −0.538345 0.842725i \(-0.680950\pi\)
−0.538345 + 0.842725i \(0.680950\pi\)
\(62\) −4.61435e9 −0.639671
\(63\) 1.43139e10 1.81713
\(64\) 2.53618e9 0.295250
\(65\) 4.92866e9 0.526872
\(66\) −6.91385e8 −0.0679561
\(67\) 8.27289e8 0.0748593 0.0374297 0.999299i \(-0.488083\pi\)
0.0374297 + 0.999299i \(0.488083\pi\)
\(68\) −7.50823e8 −0.0626237
\(69\) −1.68856e10 −1.29971
\(70\) 1.01344e10 0.720706
\(71\) −2.25416e10 −1.48273 −0.741367 0.671100i \(-0.765822\pi\)
−0.741367 + 0.671100i \(0.765822\pi\)
\(72\) 1.49215e10 0.908832
\(73\) 6.51950e9 0.368077 0.184039 0.982919i \(-0.441083\pi\)
0.184039 + 0.982919i \(0.441083\pi\)
\(74\) 1.79842e10 0.942142
\(75\) 1.23702e10 0.601920
\(76\) −1.55442e9 −0.0703223
\(77\) −3.93106e9 −0.165505
\(78\) −1.31720e10 −0.516571
\(79\) 2.48536e10 0.908742 0.454371 0.890812i \(-0.349864\pi\)
0.454371 + 0.890812i \(0.349864\pi\)
\(80\) −5.52851e9 −0.188631
\(81\) −3.12626e10 −0.996226
\(82\) −1.20695e10 −0.359511
\(83\) −4.41362e10 −1.22989 −0.614944 0.788570i \(-0.710822\pi\)
−0.614944 + 0.788570i \(0.710822\pi\)
\(84\) 7.11391e10 1.85598
\(85\) −2.68163e9 −0.0655534
\(86\) 3.52083e10 0.807056
\(87\) −1.22202e10 −0.262860
\(88\) −4.09791e9 −0.0827766
\(89\) −7.45858e10 −1.41583 −0.707915 0.706298i \(-0.750364\pi\)
−0.707915 + 0.706298i \(0.750364\pi\)
\(90\) 2.23854e10 0.399605
\(91\) −7.48927e10 −1.25809
\(92\) −4.20388e10 −0.664993
\(93\) −1.15687e11 −1.72436
\(94\) −3.95223e10 −0.555442
\(95\) −5.55174e9 −0.0736122
\(96\) 1.17168e11 1.46661
\(97\) −1.19334e11 −1.41098 −0.705490 0.708720i \(-0.749273\pi\)
−0.705490 + 0.708720i \(0.749273\pi\)
\(98\) −1.07007e11 −1.19582
\(99\) −8.68315e9 −0.0917664
\(100\) 3.07971e10 0.307971
\(101\) −2.52198e10 −0.238767 −0.119383 0.992848i \(-0.538092\pi\)
−0.119383 + 0.992848i \(0.538092\pi\)
\(102\) 7.16672e9 0.0642717
\(103\) 3.15441e10 0.268110 0.134055 0.990974i \(-0.457200\pi\)
0.134055 + 0.990974i \(0.457200\pi\)
\(104\) −7.80715e10 −0.629230
\(105\) 2.54080e11 1.94280
\(106\) 1.98342e10 0.143957
\(107\) 2.50848e11 1.72902 0.864511 0.502614i \(-0.167628\pi\)
0.864511 + 0.502614i \(0.167628\pi\)
\(108\) 5.88609e8 0.00385475
\(109\) 1.24993e11 0.778109 0.389054 0.921215i \(-0.372802\pi\)
0.389054 + 0.921215i \(0.372802\pi\)
\(110\) −6.14773e9 −0.0363961
\(111\) 4.50883e11 2.53973
\(112\) 8.40076e10 0.450422
\(113\) 6.74405e10 0.344342 0.172171 0.985067i \(-0.444922\pi\)
0.172171 + 0.985067i \(0.444922\pi\)
\(114\) 1.48372e10 0.0721729
\(115\) −1.50145e11 −0.696103
\(116\) −3.04238e10 −0.134491
\(117\) −1.65427e11 −0.697566
\(118\) −1.56906e10 −0.0631379
\(119\) 4.07483e10 0.156531
\(120\) 2.64864e11 0.971687
\(121\) −2.82927e11 −0.991642
\(122\) 1.68780e11 0.565382
\(123\) −3.02595e11 −0.969134
\(124\) −2.88016e11 −0.882263
\(125\) 3.68670e11 1.08052
\(126\) −3.40154e11 −0.954197
\(127\) −4.02221e11 −1.08030 −0.540150 0.841569i \(-0.681633\pi\)
−0.540150 + 0.841569i \(0.681633\pi\)
\(128\) 3.42492e11 0.881040
\(129\) 8.82710e11 2.17558
\(130\) −1.17124e11 −0.276667
\(131\) −6.76340e11 −1.53170 −0.765849 0.643020i \(-0.777681\pi\)
−0.765849 + 0.643020i \(0.777681\pi\)
\(132\) −4.31545e10 −0.0937281
\(133\) 8.43607e10 0.175775
\(134\) −1.96596e10 −0.0393095
\(135\) 2.10227e9 0.00403508
\(136\) 4.24779e10 0.0782887
\(137\) 2.81271e11 0.497923 0.248961 0.968513i \(-0.419911\pi\)
0.248961 + 0.968513i \(0.419911\pi\)
\(138\) 4.01267e11 0.682493
\(139\) −3.80202e11 −0.621488 −0.310744 0.950494i \(-0.600578\pi\)
−0.310744 + 0.950494i \(0.600578\pi\)
\(140\) 6.32563e11 0.994030
\(141\) −9.90867e11 −1.49730
\(142\) 5.35674e11 0.778601
\(143\) 4.54315e10 0.0635344
\(144\) 1.85561e11 0.249743
\(145\) −1.08661e11 −0.140783
\(146\) −1.54928e11 −0.193282
\(147\) −2.68277e12 −3.22358
\(148\) 1.12253e12 1.29945
\(149\) 1.74127e12 1.94241 0.971207 0.238236i \(-0.0765693\pi\)
0.971207 + 0.238236i \(0.0765693\pi\)
\(150\) −2.93963e11 −0.316075
\(151\) −1.67155e12 −1.73279 −0.866394 0.499361i \(-0.833568\pi\)
−0.866394 + 0.499361i \(0.833568\pi\)
\(152\) 8.79414e10 0.0879131
\(153\) 9.00074e10 0.0867911
\(154\) 9.34170e10 0.0869084
\(155\) −1.02868e12 −0.923538
\(156\) −8.22161e11 −0.712478
\(157\) −1.81666e12 −1.51994 −0.759969 0.649959i \(-0.774786\pi\)
−0.759969 + 0.649959i \(0.774786\pi\)
\(158\) −5.90618e11 −0.477191
\(159\) 4.97266e11 0.388065
\(160\) 1.04184e12 0.785494
\(161\) 2.28151e12 1.66219
\(162\) 7.42920e11 0.523130
\(163\) 1.07869e12 0.734284 0.367142 0.930165i \(-0.380336\pi\)
0.367142 + 0.930165i \(0.380336\pi\)
\(164\) −7.53347e11 −0.495854
\(165\) −1.54130e11 −0.0981130
\(166\) 1.04885e12 0.645829
\(167\) −2.12043e12 −1.26323 −0.631617 0.775281i \(-0.717608\pi\)
−0.631617 + 0.775281i \(0.717608\pi\)
\(168\) −4.02471e12 −2.32024
\(169\) −9.26619e11 −0.517040
\(170\) 6.37259e10 0.0344228
\(171\) 1.86341e11 0.0974607
\(172\) 2.19761e12 1.11313
\(173\) 1.04942e11 0.0514868 0.0257434 0.999669i \(-0.491805\pi\)
0.0257434 + 0.999669i \(0.491805\pi\)
\(174\) 2.90400e11 0.138031
\(175\) −1.67141e12 −0.769791
\(176\) −5.09608e10 −0.0227466
\(177\) −3.93382e11 −0.170201
\(178\) 1.77245e12 0.743469
\(179\) 3.02670e12 1.23105 0.615527 0.788116i \(-0.288943\pi\)
0.615527 + 0.788116i \(0.288943\pi\)
\(180\) 1.39724e12 0.551154
\(181\) −2.56476e12 −0.981328 −0.490664 0.871349i \(-0.663246\pi\)
−0.490664 + 0.871349i \(0.663246\pi\)
\(182\) 1.77974e12 0.660638
\(183\) 4.23150e12 1.52410
\(184\) 2.37835e12 0.831338
\(185\) 4.00921e12 1.36024
\(186\) 2.74916e12 0.905481
\(187\) −2.47188e10 −0.00790495
\(188\) −2.46688e12 −0.766091
\(189\) −3.19447e10 −0.00963516
\(190\) 1.31931e11 0.0386546
\(191\) −3.71872e12 −1.05855 −0.529273 0.848452i \(-0.677535\pi\)
−0.529273 + 0.848452i \(0.677535\pi\)
\(192\) −1.51102e12 −0.417939
\(193\) 3.83026e11 0.102959 0.0514794 0.998674i \(-0.483606\pi\)
0.0514794 + 0.998674i \(0.483606\pi\)
\(194\) 2.83584e12 0.740922
\(195\) −2.93642e12 −0.745810
\(196\) −6.67908e12 −1.64933
\(197\) 6.41435e12 1.54024 0.770120 0.637899i \(-0.220196\pi\)
0.770120 + 0.637899i \(0.220196\pi\)
\(198\) 2.06345e11 0.0481876
\(199\) −1.85317e12 −0.420944 −0.210472 0.977600i \(-0.567500\pi\)
−0.210472 + 0.977600i \(0.567500\pi\)
\(200\) −1.74235e12 −0.385008
\(201\) −4.92887e11 −0.105967
\(202\) 5.99319e11 0.125379
\(203\) 1.65115e12 0.336169
\(204\) 4.47329e11 0.0886464
\(205\) −2.69065e12 −0.519052
\(206\) −7.49609e11 −0.140788
\(207\) 5.03953e12 0.921623
\(208\) −9.70882e11 −0.172909
\(209\) −5.11750e10 −0.00887674
\(210\) −6.03791e12 −1.02019
\(211\) −3.00374e12 −0.494434 −0.247217 0.968960i \(-0.579516\pi\)
−0.247217 + 0.968960i \(0.579516\pi\)
\(212\) 1.23800e12 0.198552
\(213\) 1.34299e13 2.09887
\(214\) −5.96112e12 −0.907929
\(215\) 7.84898e12 1.16520
\(216\) −3.33006e10 −0.00481900
\(217\) 1.56311e13 2.20527
\(218\) −2.97032e12 −0.408594
\(219\) −3.88422e12 −0.521029
\(220\) −3.83726e11 −0.0501992
\(221\) −4.70932e11 −0.0600898
\(222\) −1.07147e13 −1.33364
\(223\) −7.06080e11 −0.0857388 −0.0428694 0.999081i \(-0.513650\pi\)
−0.0428694 + 0.999081i \(0.513650\pi\)
\(224\) −1.58312e13 −1.87564
\(225\) −3.69190e12 −0.426821
\(226\) −1.60265e12 −0.180818
\(227\) 3.18905e12 0.351171 0.175585 0.984464i \(-0.443818\pi\)
0.175585 + 0.984464i \(0.443818\pi\)
\(228\) 9.26099e11 0.0995441
\(229\) −1.02975e13 −1.08053 −0.540266 0.841495i \(-0.681676\pi\)
−0.540266 + 0.841495i \(0.681676\pi\)
\(230\) 3.56803e12 0.365532
\(231\) 2.34206e12 0.234279
\(232\) 1.72123e12 0.168134
\(233\) −6.19234e12 −0.590742 −0.295371 0.955383i \(-0.595443\pi\)
−0.295371 + 0.955383i \(0.595443\pi\)
\(234\) 3.93119e12 0.366300
\(235\) −8.81070e12 −0.801931
\(236\) −9.79371e11 −0.0870826
\(237\) −1.48074e13 −1.28636
\(238\) −9.68337e11 −0.0821965
\(239\) 1.38733e12 0.115078 0.0575390 0.998343i \(-0.481675\pi\)
0.0575390 + 0.998343i \(0.481675\pi\)
\(240\) 3.29380e12 0.267015
\(241\) 8.34573e12 0.661257 0.330629 0.943761i \(-0.392739\pi\)
0.330629 + 0.943761i \(0.392739\pi\)
\(242\) 6.72343e12 0.520723
\(243\) 1.86961e13 1.41552
\(244\) 1.05348e13 0.779800
\(245\) −2.38550e13 −1.72649
\(246\) 7.19082e12 0.508903
\(247\) −9.74963e11 −0.0674769
\(248\) 1.62946e13 1.10296
\(249\) 2.62957e13 1.74096
\(250\) −8.76101e12 −0.567393
\(251\) −1.99492e13 −1.26392 −0.631960 0.775001i \(-0.717749\pi\)
−0.631960 + 0.775001i \(0.717749\pi\)
\(252\) −2.12316e13 −1.31607
\(253\) −1.38401e12 −0.0839416
\(254\) 9.55832e12 0.567278
\(255\) 1.59768e12 0.0927936
\(256\) −1.33330e13 −0.757894
\(257\) −1.70759e13 −0.950060 −0.475030 0.879970i \(-0.657563\pi\)
−0.475030 + 0.879970i \(0.657563\pi\)
\(258\) −2.09766e13 −1.14242
\(259\) −6.09214e13 −3.24804
\(260\) −7.31058e12 −0.381591
\(261\) 3.64715e12 0.186394
\(262\) 1.60724e13 0.804313
\(263\) 2.47821e12 0.121446 0.0607228 0.998155i \(-0.480659\pi\)
0.0607228 + 0.998155i \(0.480659\pi\)
\(264\) 2.44147e12 0.117174
\(265\) 4.42164e12 0.207841
\(266\) −2.00473e12 −0.0923013
\(267\) 4.44371e13 2.00417
\(268\) −1.22710e12 −0.0542174
\(269\) 3.70849e13 1.60531 0.802657 0.596441i \(-0.203419\pi\)
0.802657 + 0.596441i \(0.203419\pi\)
\(270\) −4.99580e10 −0.00211887
\(271\) 1.89763e13 0.788642 0.394321 0.918973i \(-0.370980\pi\)
0.394321 + 0.918973i \(0.370980\pi\)
\(272\) 5.28247e11 0.0215134
\(273\) 4.46200e13 1.78088
\(274\) −6.68408e12 −0.261465
\(275\) 1.01391e12 0.0388750
\(276\) 2.50461e13 0.941325
\(277\) 3.21867e13 1.18587 0.592935 0.805250i \(-0.297969\pi\)
0.592935 + 0.805250i \(0.297969\pi\)
\(278\) 9.03506e12 0.326351
\(279\) 3.45269e13 1.22274
\(280\) −3.57873e13 −1.24268
\(281\) 7.67486e12 0.261328 0.130664 0.991427i \(-0.458289\pi\)
0.130664 + 0.991427i \(0.458289\pi\)
\(282\) 2.35468e13 0.786252
\(283\) −5.46880e13 −1.79088 −0.895440 0.445183i \(-0.853139\pi\)
−0.895440 + 0.445183i \(0.853139\pi\)
\(284\) 3.34355e13 1.07388
\(285\) 3.30765e12 0.104201
\(286\) −1.07963e12 −0.0333627
\(287\) 4.08853e13 1.23942
\(288\) −3.49689e13 −1.03997
\(289\) −3.40157e13 −0.992524
\(290\) 2.58221e12 0.0739269
\(291\) 7.10976e13 1.99730
\(292\) −9.67025e12 −0.266583
\(293\) −4.17914e12 −0.113062 −0.0565308 0.998401i \(-0.518004\pi\)
−0.0565308 + 0.998401i \(0.518004\pi\)
\(294\) 6.37529e13 1.69274
\(295\) −3.49791e12 −0.0911566
\(296\) −6.35072e13 −1.62450
\(297\) 1.93784e10 0.000486583 0
\(298\) −4.13793e13 −1.01998
\(299\) −2.63676e13 −0.638086
\(300\) −1.83484e13 −0.435946
\(301\) −1.19268e14 −2.78233
\(302\) 3.97224e13 0.909907
\(303\) 1.50256e13 0.337984
\(304\) 1.09362e12 0.0241581
\(305\) 3.76261e13 0.816282
\(306\) −2.13892e12 −0.0455750
\(307\) −1.85704e13 −0.388651 −0.194326 0.980937i \(-0.562252\pi\)
−0.194326 + 0.980937i \(0.562252\pi\)
\(308\) 5.83086e12 0.119868
\(309\) −1.87935e13 −0.379521
\(310\) 2.44453e13 0.484961
\(311\) −5.58068e13 −1.08769 −0.543845 0.839186i \(-0.683032\pi\)
−0.543845 + 0.839186i \(0.683032\pi\)
\(312\) 4.65139e13 0.890701
\(313\) 4.67501e13 0.879608 0.439804 0.898094i \(-0.355048\pi\)
0.439804 + 0.898094i \(0.355048\pi\)
\(314\) 4.31709e13 0.798137
\(315\) −7.58305e13 −1.37764
\(316\) −3.68649e13 −0.658164
\(317\) −7.21065e13 −1.26517 −0.632584 0.774491i \(-0.718006\pi\)
−0.632584 + 0.774491i \(0.718006\pi\)
\(318\) −1.18169e13 −0.203777
\(319\) −1.00162e12 −0.0169768
\(320\) −1.34358e13 −0.223841
\(321\) −1.49452e14 −2.44750
\(322\) −5.42174e13 −0.872834
\(323\) 5.30468e11 0.00839547
\(324\) 4.63712e13 0.721524
\(325\) 1.93166e13 0.295510
\(326\) −2.56338e13 −0.385581
\(327\) −7.44691e13 −1.10145
\(328\) 4.26207e13 0.619890
\(329\) 1.33882e14 1.91489
\(330\) 3.66273e12 0.0515203
\(331\) 7.90037e13 1.09293 0.546466 0.837481i \(-0.315973\pi\)
0.546466 + 0.837481i \(0.315973\pi\)
\(332\) 6.54664e13 0.890757
\(333\) −1.34567e14 −1.80092
\(334\) 5.03896e13 0.663339
\(335\) −4.38270e12 −0.0567539
\(336\) −5.00505e13 −0.637591
\(337\) −4.89939e13 −0.614013 −0.307006 0.951707i \(-0.599327\pi\)
−0.307006 + 0.951707i \(0.599327\pi\)
\(338\) 2.20200e13 0.271504
\(339\) −4.01801e13 −0.487430
\(340\) 3.97761e12 0.0474776
\(341\) −9.48217e12 −0.111368
\(342\) −4.42817e12 −0.0511777
\(343\) 2.03310e14 2.31228
\(344\) −1.24330e14 −1.39157
\(345\) 8.94543e13 0.985363
\(346\) −2.49383e12 −0.0270363
\(347\) 1.34018e14 1.43005 0.715027 0.699097i \(-0.246415\pi\)
0.715027 + 0.699097i \(0.246415\pi\)
\(348\) 1.81260e13 0.190378
\(349\) −1.23800e14 −1.27991 −0.639956 0.768412i \(-0.721047\pi\)
−0.639956 + 0.768412i \(0.721047\pi\)
\(350\) 3.97190e13 0.404226
\(351\) 3.69188e11 0.00369878
\(352\) 9.60354e12 0.0947211
\(353\) 8.30822e12 0.0806765 0.0403383 0.999186i \(-0.487156\pi\)
0.0403383 + 0.999186i \(0.487156\pi\)
\(354\) 9.34826e12 0.0893743
\(355\) 1.19418e14 1.12412
\(356\) 1.10632e14 1.02543
\(357\) −2.42773e13 −0.221577
\(358\) −7.19260e13 −0.646441
\(359\) −2.00482e14 −1.77442 −0.887210 0.461365i \(-0.847360\pi\)
−0.887210 + 0.461365i \(0.847360\pi\)
\(360\) −7.90492e13 −0.689023
\(361\) −1.15392e14 −0.990572
\(362\) 6.09485e13 0.515307
\(363\) 1.68564e14 1.40371
\(364\) 1.11087e14 0.911182
\(365\) −3.45382e13 −0.279054
\(366\) −1.00557e14 −0.800322
\(367\) 1.35764e13 0.106444 0.0532221 0.998583i \(-0.483051\pi\)
0.0532221 + 0.998583i \(0.483051\pi\)
\(368\) 2.95767e13 0.228448
\(369\) 9.03100e13 0.687212
\(370\) −9.52743e13 −0.714277
\(371\) −6.71884e13 −0.496293
\(372\) 1.71596e14 1.24888
\(373\) −4.27855e13 −0.306830 −0.153415 0.988162i \(-0.549027\pi\)
−0.153415 + 0.988162i \(0.549027\pi\)
\(374\) 5.87414e11 0.00415098
\(375\) −2.19648e14 −1.52952
\(376\) 1.39564e14 0.957726
\(377\) −1.90824e13 −0.129050
\(378\) 7.59129e11 0.00505954
\(379\) −8.97682e13 −0.589667 −0.294833 0.955549i \(-0.595264\pi\)
−0.294833 + 0.955549i \(0.595264\pi\)
\(380\) 8.23479e12 0.0533142
\(381\) 2.39638e14 1.52921
\(382\) 8.83710e13 0.555855
\(383\) −1.56083e14 −0.967750 −0.483875 0.875137i \(-0.660771\pi\)
−0.483875 + 0.875137i \(0.660771\pi\)
\(384\) −2.04052e14 −1.24715
\(385\) 2.08254e13 0.125476
\(386\) −9.10217e12 −0.0540648
\(387\) −2.63446e14 −1.54270
\(388\) 1.77006e14 1.02191
\(389\) 6.64866e13 0.378452 0.189226 0.981934i \(-0.439402\pi\)
0.189226 + 0.981934i \(0.439402\pi\)
\(390\) 6.97807e13 0.391633
\(391\) 1.43463e13 0.0793906
\(392\) 3.77870e14 2.06191
\(393\) 4.02954e14 2.16818
\(394\) −1.52430e14 −0.808798
\(395\) −1.31666e14 −0.688955
\(396\) 1.28795e13 0.0664625
\(397\) 7.81206e13 0.397574 0.198787 0.980043i \(-0.436300\pi\)
0.198787 + 0.980043i \(0.436300\pi\)
\(398\) 4.40385e13 0.221043
\(399\) −5.02609e13 −0.248816
\(400\) −2.16675e13 −0.105798
\(401\) 3.95314e14 1.90392 0.951958 0.306228i \(-0.0990670\pi\)
0.951958 + 0.306228i \(0.0990670\pi\)
\(402\) 1.17129e13 0.0556443
\(403\) −1.80650e14 −0.846565
\(404\) 3.74080e13 0.172929
\(405\) 1.65619e14 0.755280
\(406\) −3.92376e13 −0.176526
\(407\) 3.69563e13 0.164028
\(408\) −2.53077e13 −0.110821
\(409\) −4.19423e14 −1.81207 −0.906033 0.423207i \(-0.860904\pi\)
−0.906033 + 0.423207i \(0.860904\pi\)
\(410\) 6.39401e13 0.272560
\(411\) −1.67577e14 −0.704830
\(412\) −4.67887e13 −0.194181
\(413\) 5.31520e13 0.217668
\(414\) −1.19759e14 −0.483955
\(415\) 2.33819e14 0.932429
\(416\) 1.82962e14 0.720026
\(417\) 2.26519e14 0.879743
\(418\) 1.21612e12 0.00466128
\(419\) 1.16987e14 0.442547 0.221273 0.975212i \(-0.428979\pi\)
0.221273 + 0.975212i \(0.428979\pi\)
\(420\) −3.76872e14 −1.40709
\(421\) −4.85624e14 −1.78957 −0.894785 0.446497i \(-0.852672\pi\)
−0.894785 + 0.446497i \(0.852672\pi\)
\(422\) 7.13804e13 0.259633
\(423\) 2.95726e14 1.06174
\(424\) −7.00402e13 −0.248219
\(425\) −1.05100e13 −0.0367673
\(426\) −3.19147e14 −1.10214
\(427\) −5.71742e14 −1.94916
\(428\) −3.72078e14 −1.25226
\(429\) −2.70674e13 −0.0899357
\(430\) −1.86522e14 −0.611862
\(431\) 1.43717e14 0.465460 0.232730 0.972541i \(-0.425234\pi\)
0.232730 + 0.972541i \(0.425234\pi\)
\(432\) −4.14120e11 −0.00132424
\(433\) 3.13984e14 0.991345 0.495672 0.868510i \(-0.334922\pi\)
0.495672 + 0.868510i \(0.334922\pi\)
\(434\) −3.71455e14 −1.15801
\(435\) 6.47388e13 0.199285
\(436\) −1.85400e14 −0.563551
\(437\) 2.97010e13 0.0891504
\(438\) 9.23041e13 0.273598
\(439\) −4.69695e14 −1.37487 −0.687434 0.726247i \(-0.741263\pi\)
−0.687434 + 0.726247i \(0.741263\pi\)
\(440\) 2.17094e13 0.0627563
\(441\) 8.00677e14 2.28584
\(442\) 1.11912e13 0.0315538
\(443\) −1.86393e14 −0.519050 −0.259525 0.965736i \(-0.583566\pi\)
−0.259525 + 0.965736i \(0.583566\pi\)
\(444\) −6.68786e14 −1.83942
\(445\) 3.95131e14 1.07340
\(446\) 1.67792e13 0.0450224
\(447\) −1.03742e15 −2.74957
\(448\) 2.04162e14 0.534498
\(449\) −1.00294e14 −0.259371 −0.129685 0.991555i \(-0.541397\pi\)
−0.129685 + 0.991555i \(0.541397\pi\)
\(450\) 8.77337e13 0.224129
\(451\) −2.48019e13 −0.0625914
\(452\) −1.00033e14 −0.249392
\(453\) 9.95883e14 2.45283
\(454\) −7.57840e13 −0.184404
\(455\) 3.96757e14 0.953810
\(456\) −5.23942e13 −0.124445
\(457\) 2.45758e14 0.576726 0.288363 0.957521i \(-0.406889\pi\)
0.288363 + 0.957521i \(0.406889\pi\)
\(458\) 2.44709e14 0.567399
\(459\) −2.00871e11 −0.000460202 0
\(460\) 2.22707e14 0.504158
\(461\) −6.63539e14 −1.48426 −0.742132 0.670253i \(-0.766185\pi\)
−0.742132 + 0.670253i \(0.766185\pi\)
\(462\) −5.56564e13 −0.123023
\(463\) −1.04412e14 −0.228062 −0.114031 0.993477i \(-0.536376\pi\)
−0.114031 + 0.993477i \(0.536376\pi\)
\(464\) 2.14049e13 0.0462023
\(465\) 6.12871e14 1.30731
\(466\) 1.47154e14 0.310205
\(467\) −6.80677e14 −1.41807 −0.709036 0.705172i \(-0.750870\pi\)
−0.709036 + 0.705172i \(0.750870\pi\)
\(468\) 2.45375e14 0.505218
\(469\) 6.65967e13 0.135520
\(470\) 2.09376e14 0.421104
\(471\) 1.08234e15 2.15154
\(472\) 5.54081e13 0.108866
\(473\) 7.23506e13 0.140510
\(474\) 3.51881e14 0.675484
\(475\) −2.17586e13 −0.0412872
\(476\) −6.04412e13 −0.113369
\(477\) −1.48410e14 −0.275177
\(478\) −3.29684e13 −0.0604288
\(479\) −5.79575e14 −1.05018 −0.525090 0.851047i \(-0.675968\pi\)
−0.525090 + 0.851047i \(0.675968\pi\)
\(480\) −6.20715e14 −1.11190
\(481\) 7.04074e14 1.24687
\(482\) −1.98326e14 −0.347234
\(483\) −1.35929e15 −2.35290
\(484\) 4.19660e14 0.718204
\(485\) 6.32194e14 1.06972
\(486\) −4.44291e14 −0.743307
\(487\) 6.09413e14 1.00810 0.504048 0.863675i \(-0.331843\pi\)
0.504048 + 0.863675i \(0.331843\pi\)
\(488\) −5.96009e14 −0.974864
\(489\) −6.42667e14 −1.03941
\(490\) 5.66885e14 0.906602
\(491\) 3.47139e14 0.548978 0.274489 0.961590i \(-0.411491\pi\)
0.274489 + 0.961590i \(0.411491\pi\)
\(492\) 4.48833e14 0.701903
\(493\) 1.03826e13 0.0160563
\(494\) 2.31689e13 0.0354329
\(495\) 4.60005e13 0.0695718
\(496\) 2.02636e14 0.303087
\(497\) −1.81460e15 −2.68423
\(498\) −6.24887e14 −0.914198
\(499\) 8.73757e14 1.26426 0.632132 0.774861i \(-0.282180\pi\)
0.632132 + 0.774861i \(0.282180\pi\)
\(500\) −5.46841e14 −0.782575
\(501\) 1.26332e15 1.78816
\(502\) 4.74069e14 0.663699
\(503\) −1.16225e15 −1.60944 −0.804721 0.593653i \(-0.797685\pi\)
−0.804721 + 0.593653i \(0.797685\pi\)
\(504\) 1.20118e15 1.64528
\(505\) 1.33606e14 0.181019
\(506\) 3.28895e13 0.0440787
\(507\) 5.52066e14 0.731893
\(508\) 5.96607e14 0.782416
\(509\) 9.89379e13 0.128356 0.0641778 0.997938i \(-0.479558\pi\)
0.0641778 + 0.997938i \(0.479558\pi\)
\(510\) −3.79669e13 −0.0487270
\(511\) 5.24820e14 0.666339
\(512\) −3.84580e14 −0.483061
\(513\) −4.15861e11 −0.000516776 0
\(514\) 4.05789e14 0.498888
\(515\) −1.67110e14 −0.203265
\(516\) −1.30931e15 −1.57568
\(517\) −8.12155e13 −0.0967033
\(518\) 1.44773e15 1.70558
\(519\) −6.25229e13 −0.0728818
\(520\) 4.13597e14 0.477045
\(521\) 7.97660e14 0.910354 0.455177 0.890401i \(-0.349576\pi\)
0.455177 + 0.890401i \(0.349576\pi\)
\(522\) −8.66703e13 −0.0978774
\(523\) 4.00131e14 0.447140 0.223570 0.974688i \(-0.428229\pi\)
0.223570 + 0.974688i \(0.428229\pi\)
\(524\) 1.00320e15 1.10934
\(525\) 9.95799e14 1.08967
\(526\) −5.88919e13 −0.0637725
\(527\) 9.82898e13 0.105330
\(528\) 3.03617e13 0.0321988
\(529\) −1.49555e14 −0.156962
\(530\) −1.05075e14 −0.109140
\(531\) 1.17405e14 0.120689
\(532\) −1.25131e14 −0.127306
\(533\) −4.72515e14 −0.475791
\(534\) −1.05600e15 −1.05241
\(535\) −1.32891e15 −1.31084
\(536\) 6.94234e13 0.0677797
\(537\) −1.80326e15 −1.74261
\(538\) −8.81280e14 −0.842969
\(539\) −2.19891e14 −0.208194
\(540\) −3.11825e12 −0.00292244
\(541\) 4.56696e14 0.423684 0.211842 0.977304i \(-0.432054\pi\)
0.211842 + 0.977304i \(0.432054\pi\)
\(542\) −4.50949e14 −0.414125
\(543\) 1.52805e15 1.38911
\(544\) −9.95479e13 −0.0895856
\(545\) −6.62172e14 −0.589916
\(546\) −1.06034e15 −0.935161
\(547\) −6.82824e13 −0.0596182 −0.0298091 0.999556i \(-0.509490\pi\)
−0.0298091 + 0.999556i \(0.509490\pi\)
\(548\) −4.17204e14 −0.360624
\(549\) −1.26290e15 −1.08074
\(550\) −2.40944e13 −0.0204137
\(551\) 2.14949e13 0.0180302
\(552\) −1.41698e15 −1.17679
\(553\) 2.00072e15 1.64512
\(554\) −7.64879e14 −0.622714
\(555\) −2.38863e15 −1.92547
\(556\) 5.63946e14 0.450118
\(557\) 3.28321e14 0.259475 0.129737 0.991548i \(-0.458587\pi\)
0.129737 + 0.991548i \(0.458587\pi\)
\(558\) −8.20492e14 −0.642076
\(559\) 1.37839e15 1.06809
\(560\) −4.45044e14 −0.341483
\(561\) 1.47271e13 0.0111898
\(562\) −1.82384e14 −0.137226
\(563\) −7.71171e14 −0.574585 −0.287293 0.957843i \(-0.592755\pi\)
−0.287293 + 0.957843i \(0.592755\pi\)
\(564\) 1.46973e15 1.08443
\(565\) −3.57278e14 −0.261059
\(566\) 1.29960e15 0.940412
\(567\) −2.51664e15 −1.80349
\(568\) −1.89162e15 −1.34251
\(569\) 2.18238e15 1.53396 0.766978 0.641674i \(-0.221760\pi\)
0.766978 + 0.641674i \(0.221760\pi\)
\(570\) −7.86024e13 −0.0547172
\(571\) 8.46547e14 0.583650 0.291825 0.956472i \(-0.405738\pi\)
0.291825 + 0.956472i \(0.405738\pi\)
\(572\) −6.73877e13 −0.0460153
\(573\) 2.21556e15 1.49842
\(574\) −9.71593e14 −0.650832
\(575\) −5.88455e14 −0.390427
\(576\) 4.50966e14 0.296360
\(577\) −1.61690e15 −1.05249 −0.526244 0.850334i \(-0.676400\pi\)
−0.526244 + 0.850334i \(0.676400\pi\)
\(578\) 8.08343e14 0.521186
\(579\) −2.28201e14 −0.145742
\(580\) 1.61175e14 0.101963
\(581\) −3.55297e15 −2.22650
\(582\) −1.68955e15 −1.04881
\(583\) 4.07580e13 0.0250631
\(584\) 5.47096e14 0.333267
\(585\) 8.76380e14 0.528853
\(586\) 9.93124e13 0.0593700
\(587\) 8.67880e14 0.513985 0.256992 0.966413i \(-0.417268\pi\)
0.256992 + 0.966413i \(0.417268\pi\)
\(588\) 3.97930e15 2.33470
\(589\) 2.03488e14 0.118278
\(590\) 8.31239e13 0.0478674
\(591\) −3.82157e15 −2.18027
\(592\) −7.89764e14 −0.446404
\(593\) 2.67501e15 1.49804 0.749022 0.662546i \(-0.230524\pi\)
0.749022 + 0.662546i \(0.230524\pi\)
\(594\) −4.60504e11 −0.000255510 0
\(595\) −2.15871e14 −0.118673
\(596\) −2.58279e15 −1.40681
\(597\) 1.10409e15 0.595864
\(598\) 6.26595e14 0.335066
\(599\) 4.64308e14 0.246013 0.123007 0.992406i \(-0.460746\pi\)
0.123007 + 0.992406i \(0.460746\pi\)
\(600\) 1.03807e15 0.544995
\(601\) 4.46744e14 0.232407 0.116203 0.993225i \(-0.462928\pi\)
0.116203 + 0.993225i \(0.462928\pi\)
\(602\) 2.83427e15 1.46103
\(603\) 1.47103e14 0.0751408
\(604\) 2.47937e15 1.25499
\(605\) 1.49885e15 0.751804
\(606\) −3.57065e14 −0.177480
\(607\) 3.59594e15 1.77123 0.885614 0.464423i \(-0.153738\pi\)
0.885614 + 0.464423i \(0.153738\pi\)
\(608\) −2.06093e14 −0.100599
\(609\) −9.83729e14 −0.475861
\(610\) −8.94141e14 −0.428639
\(611\) −1.54728e15 −0.735094
\(612\) −1.33506e14 −0.0628591
\(613\) 1.78311e15 0.832044 0.416022 0.909355i \(-0.363424\pi\)
0.416022 + 0.909355i \(0.363424\pi\)
\(614\) 4.41304e14 0.204085
\(615\) 1.60305e15 0.734740
\(616\) −3.29881e14 −0.149852
\(617\) −4.04190e14 −0.181977 −0.0909885 0.995852i \(-0.529003\pi\)
−0.0909885 + 0.995852i \(0.529003\pi\)
\(618\) 4.46606e14 0.199291
\(619\) 4.73653e14 0.209489 0.104745 0.994499i \(-0.466597\pi\)
0.104745 + 0.994499i \(0.466597\pi\)
\(620\) 1.52582e15 0.668880
\(621\) −1.12468e13 −0.00488682
\(622\) 1.32618e15 0.571158
\(623\) −6.00415e15 −2.56311
\(624\) 5.78437e14 0.244760
\(625\) −9.39282e14 −0.393963
\(626\) −1.11096e15 −0.461892
\(627\) 3.04893e13 0.0125654
\(628\) 2.69462e15 1.10083
\(629\) −3.83079e14 −0.155135
\(630\) 1.80202e15 0.723415
\(631\) 4.25063e13 0.0169158 0.00845789 0.999964i \(-0.497308\pi\)
0.00845789 + 0.999964i \(0.497308\pi\)
\(632\) 2.08564e15 0.822801
\(633\) 1.78958e15 0.699893
\(634\) 1.71353e15 0.664355
\(635\) 2.13084e15 0.819020
\(636\) −7.37584e14 −0.281059
\(637\) −4.18926e15 −1.58260
\(638\) 2.38024e13 0.00891469
\(639\) −4.00819e15 −1.48831
\(640\) −1.81441e15 −0.667952
\(641\) 4.20206e15 1.53371 0.766855 0.641820i \(-0.221820\pi\)
0.766855 + 0.641820i \(0.221820\pi\)
\(642\) 3.55155e15 1.28521
\(643\) 2.82129e15 1.01225 0.506125 0.862460i \(-0.331077\pi\)
0.506125 + 0.862460i \(0.331077\pi\)
\(644\) −3.38412e15 −1.20385
\(645\) −4.67631e15 −1.64939
\(646\) −1.26059e13 −0.00440856
\(647\) 8.24863e14 0.286028 0.143014 0.989721i \(-0.454321\pi\)
0.143014 + 0.989721i \(0.454321\pi\)
\(648\) −2.62346e15 −0.902011
\(649\) −3.22432e13 −0.0109924
\(650\) −4.59036e14 −0.155176
\(651\) −9.31278e15 −3.12165
\(652\) −1.60000e15 −0.531811
\(653\) 3.50926e14 0.115663 0.0578313 0.998326i \(-0.481581\pi\)
0.0578313 + 0.998326i \(0.481581\pi\)
\(654\) 1.76967e15 0.578382
\(655\) 3.58303e15 1.16124
\(656\) 5.30023e14 0.170343
\(657\) 1.15925e15 0.369461
\(658\) −3.18154e15 −1.00553
\(659\) 1.10967e15 0.347796 0.173898 0.984764i \(-0.444364\pi\)
0.173898 + 0.984764i \(0.444364\pi\)
\(660\) 2.28618e14 0.0710591
\(661\) −2.41663e15 −0.744908 −0.372454 0.928051i \(-0.621483\pi\)
−0.372454 + 0.928051i \(0.621483\pi\)
\(662\) −1.87743e15 −0.573911
\(663\) 2.80574e14 0.0850596
\(664\) −3.70377e15 −1.11358
\(665\) −4.46915e14 −0.133262
\(666\) 3.19782e15 0.945685
\(667\) 5.81322e14 0.170500
\(668\) 3.14519e15 0.914907
\(669\) 4.20672e14 0.121367
\(670\) 1.04150e14 0.0298021
\(671\) 3.46831e14 0.0984338
\(672\) 9.43199e15 2.65505
\(673\) 4.93288e15 1.37727 0.688633 0.725110i \(-0.258211\pi\)
0.688633 + 0.725110i \(0.258211\pi\)
\(674\) 1.16428e15 0.322425
\(675\) 8.23929e12 0.00226318
\(676\) 1.37444e15 0.374471
\(677\) −5.77963e15 −1.56193 −0.780967 0.624573i \(-0.785273\pi\)
−0.780967 + 0.624573i \(0.785273\pi\)
\(678\) 9.54833e14 0.255955
\(679\) −9.60640e15 −2.55433
\(680\) −2.25034e14 −0.0593539
\(681\) −1.89999e15 −0.497097
\(682\) 2.25333e14 0.0584804
\(683\) 2.53598e15 0.652879 0.326439 0.945218i \(-0.394151\pi\)
0.326439 + 0.945218i \(0.394151\pi\)
\(684\) −2.76396e14 −0.0705867
\(685\) −1.49008e15 −0.377495
\(686\) −4.83142e15 −1.21421
\(687\) 6.13511e15 1.52954
\(688\) −1.54615e15 −0.382397
\(689\) 7.76502e14 0.190518
\(690\) −2.12578e15 −0.517426
\(691\) −3.34143e15 −0.806870 −0.403435 0.915008i \(-0.632184\pi\)
−0.403435 + 0.915008i \(0.632184\pi\)
\(692\) −1.55658e14 −0.0372897
\(693\) −6.98993e14 −0.166127
\(694\) −3.18479e15 −0.750937
\(695\) 2.01418e15 0.471176
\(696\) −1.02548e15 −0.238000
\(697\) 2.57091e14 0.0591979
\(698\) 2.94196e15 0.672096
\(699\) 3.68931e15 0.836220
\(700\) 2.47916e15 0.557527
\(701\) −6.59658e15 −1.47187 −0.735935 0.677052i \(-0.763257\pi\)
−0.735935 + 0.677052i \(0.763257\pi\)
\(702\) −8.77332e12 −0.00194227
\(703\) −7.93084e14 −0.174207
\(704\) −1.23849e14 −0.0269925
\(705\) 5.24929e15 1.13517
\(706\) −1.97435e14 −0.0423642
\(707\) −2.03019e15 −0.432245
\(708\) 5.83495e14 0.123269
\(709\) −4.27161e15 −0.895442 −0.447721 0.894173i \(-0.647764\pi\)
−0.447721 + 0.894173i \(0.647764\pi\)
\(710\) −2.83783e15 −0.590289
\(711\) 4.41930e15 0.912159
\(712\) −6.25900e15 −1.28193
\(713\) 5.50327e15 1.11848
\(714\) 5.76921e14 0.116353
\(715\) −2.40681e14 −0.0481681
\(716\) −4.48944e15 −0.891601
\(717\) −8.26553e14 −0.162898
\(718\) 4.76423e15 0.931769
\(719\) 8.99730e15 1.74624 0.873118 0.487508i \(-0.162094\pi\)
0.873118 + 0.487508i \(0.162094\pi\)
\(720\) −9.83040e14 −0.189340
\(721\) 2.53930e15 0.485367
\(722\) 2.74216e15 0.520161
\(723\) −4.97226e15 −0.936038
\(724\) 3.80426e15 0.710735
\(725\) −4.25869e14 −0.0789618
\(726\) −4.00572e15 −0.737105
\(727\) −2.23564e14 −0.0408284 −0.0204142 0.999792i \(-0.506499\pi\)
−0.0204142 + 0.999792i \(0.506499\pi\)
\(728\) −6.28475e15 −1.13911
\(729\) −5.60079e15 −1.00751
\(730\) 8.20760e14 0.146535
\(731\) −7.49968e14 −0.132892
\(732\) −6.27650e15 −1.10384
\(733\) −3.48936e15 −0.609079 −0.304539 0.952500i \(-0.598503\pi\)
−0.304539 + 0.952500i \(0.598503\pi\)
\(734\) −3.22628e14 −0.0558951
\(735\) 1.42124e16 2.44393
\(736\) −5.57371e15 −0.951298
\(737\) −4.03990e13 −0.00684384
\(738\) −2.14611e15 −0.360863
\(739\) −1.54307e15 −0.257539 −0.128769 0.991675i \(-0.541103\pi\)
−0.128769 + 0.991675i \(0.541103\pi\)
\(740\) −5.94679e15 −0.985163
\(741\) 5.80869e14 0.0955164
\(742\) 1.59665e15 0.260609
\(743\) 2.64447e15 0.428450 0.214225 0.976784i \(-0.431277\pi\)
0.214225 + 0.976784i \(0.431277\pi\)
\(744\) −9.70807e15 −1.56128
\(745\) −9.22468e15 −1.47262
\(746\) 1.01675e15 0.161120
\(747\) −7.84800e15 −1.23451
\(748\) 3.66649e13 0.00572522
\(749\) 2.01933e16 3.13009
\(750\) 5.21968e15 0.803169
\(751\) −7.20106e15 −1.09996 −0.549980 0.835178i \(-0.685365\pi\)
−0.549980 + 0.835178i \(0.685365\pi\)
\(752\) 1.73560e15 0.263178
\(753\) 1.18854e16 1.78913
\(754\) 4.53472e14 0.0677654
\(755\) 8.85530e15 1.31370
\(756\) 4.73830e13 0.00697834
\(757\) −7.11377e15 −1.04009 −0.520047 0.854138i \(-0.674086\pi\)
−0.520047 + 0.854138i \(0.674086\pi\)
\(758\) 2.13324e15 0.309641
\(759\) 8.24575e14 0.118823
\(760\) −4.65884e14 −0.0666505
\(761\) −7.35766e15 −1.04502 −0.522509 0.852634i \(-0.675004\pi\)
−0.522509 + 0.852634i \(0.675004\pi\)
\(762\) −5.69471e15 −0.803007
\(763\) 1.00619e16 1.40863
\(764\) 5.51590e15 0.766660
\(765\) −4.76829e14 −0.0657999
\(766\) 3.70914e15 0.508177
\(767\) −6.14282e14 −0.0835591
\(768\) 7.94362e15 1.07283
\(769\) −5.70882e15 −0.765511 −0.382756 0.923850i \(-0.625025\pi\)
−0.382756 + 0.923850i \(0.625025\pi\)
\(770\) −4.94892e14 −0.0658888
\(771\) 1.01736e16 1.34485
\(772\) −5.68135e14 −0.0745687
\(773\) −2.55574e15 −0.333065 −0.166533 0.986036i \(-0.553257\pi\)
−0.166533 + 0.986036i \(0.553257\pi\)
\(774\) 6.26050e15 0.810090
\(775\) −4.03163e15 −0.517990
\(776\) −1.00141e16 −1.27754
\(777\) 3.62961e16 4.59774
\(778\) −1.57998e15 −0.198730
\(779\) 5.32251e14 0.0664753
\(780\) 4.35554e15 0.540159
\(781\) 1.10077e15 0.135556
\(782\) −3.40924e14 −0.0416889
\(783\) −8.13942e12 −0.000988334 0
\(784\) 4.69912e15 0.566602
\(785\) 9.62407e15 1.15233
\(786\) −9.57573e15 −1.13854
\(787\) 1.39857e16 1.65129 0.825643 0.564193i \(-0.190813\pi\)
0.825643 + 0.564193i \(0.190813\pi\)
\(788\) −9.51427e15 −1.11553
\(789\) −1.47648e15 −0.171911
\(790\) 3.12890e15 0.361778
\(791\) 5.42896e15 0.623370
\(792\) −7.28662e14 −0.0830878
\(793\) 6.60767e15 0.748248
\(794\) −1.85645e15 −0.208771
\(795\) −2.63435e15 −0.294208
\(796\) 2.74878e15 0.304872
\(797\) 1.33077e15 0.146582 0.0732912 0.997311i \(-0.476650\pi\)
0.0732912 + 0.997311i \(0.476650\pi\)
\(798\) 1.19439e15 0.130656
\(799\) 8.41860e14 0.0914603
\(800\) 4.08323e15 0.440564
\(801\) −1.32623e16 −1.42115
\(802\) −9.39417e15 −0.999769
\(803\) −3.18367e14 −0.0336506
\(804\) 7.31089e14 0.0767471
\(805\) −1.20867e16 −1.26017
\(806\) 4.29294e15 0.444541
\(807\) −2.20947e16 −2.27239
\(808\) −2.11636e15 −0.216186
\(809\) −1.93263e15 −0.196079 −0.0980396 0.995183i \(-0.531257\pi\)
−0.0980396 + 0.995183i \(0.531257\pi\)
\(810\) −3.93574e15 −0.396606
\(811\) −2.88987e15 −0.289243 −0.144622 0.989487i \(-0.546196\pi\)
−0.144622 + 0.989487i \(0.546196\pi\)
\(812\) −2.44911e15 −0.243473
\(813\) −1.13058e16 −1.11636
\(814\) −8.78223e14 −0.0861332
\(815\) −5.71453e15 −0.556691
\(816\) −3.14722e14 −0.0304531
\(817\) −1.55265e15 −0.149228
\(818\) 9.96710e15 0.951537
\(819\) −1.33169e16 −1.26282
\(820\) 3.99098e15 0.375927
\(821\) −1.15152e16 −1.07742 −0.538710 0.842491i \(-0.681088\pi\)
−0.538710 + 0.842491i \(0.681088\pi\)
\(822\) 3.98228e15 0.370115
\(823\) 2.15934e15 0.199353 0.0996763 0.995020i \(-0.468219\pi\)
0.0996763 + 0.995020i \(0.468219\pi\)
\(824\) 2.64708e15 0.242755
\(825\) −6.04073e14 −0.0550292
\(826\) −1.26310e15 −0.114300
\(827\) 9.76712e15 0.877984 0.438992 0.898491i \(-0.355336\pi\)
0.438992 + 0.898491i \(0.355336\pi\)
\(828\) −7.47504e15 −0.667493
\(829\) 1.75825e16 1.55966 0.779831 0.625991i \(-0.215305\pi\)
0.779831 + 0.625991i \(0.215305\pi\)
\(830\) −5.55644e15 −0.489629
\(831\) −1.91763e16 −1.67865
\(832\) −2.35952e15 −0.205185
\(833\) 2.27933e15 0.196907
\(834\) −5.38295e15 −0.461963
\(835\) 1.12333e16 0.957709
\(836\) 7.59069e13 0.00642905
\(837\) −7.70544e13 −0.00648347
\(838\) −2.78005e15 −0.232387
\(839\) −6.59993e15 −0.548085 −0.274043 0.961718i \(-0.588361\pi\)
−0.274043 + 0.961718i \(0.588361\pi\)
\(840\) 2.13216e16 1.75907
\(841\) 4.20707e14 0.0344828
\(842\) 1.15403e16 0.939724
\(843\) −4.57257e15 −0.369921
\(844\) 4.45539e15 0.358098
\(845\) 4.90892e15 0.391989
\(846\) −7.02758e15 −0.557531
\(847\) −2.27756e16 −1.79519
\(848\) −8.71007e14 −0.0682094
\(849\) 3.25823e16 2.53507
\(850\) 2.49757e14 0.0193069
\(851\) −2.14487e16 −1.64736
\(852\) −1.99204e16 −1.52013
\(853\) 1.50817e14 0.0114349 0.00571744 0.999984i \(-0.498180\pi\)
0.00571744 + 0.999984i \(0.498180\pi\)
\(854\) 1.35868e16 1.02352
\(855\) −9.87173e14 −0.0738889
\(856\) 2.10504e16 1.56550
\(857\) 1.33286e16 0.984894 0.492447 0.870343i \(-0.336103\pi\)
0.492447 + 0.870343i \(0.336103\pi\)
\(858\) 6.43226e14 0.0472263
\(859\) −8.76363e15 −0.639325 −0.319662 0.947532i \(-0.603569\pi\)
−0.319662 + 0.947532i \(0.603569\pi\)
\(860\) −1.16422e16 −0.843908
\(861\) −2.43589e16 −1.75445
\(862\) −3.41526e15 −0.244418
\(863\) −2.04820e16 −1.45651 −0.728256 0.685305i \(-0.759669\pi\)
−0.728256 + 0.685305i \(0.759669\pi\)
\(864\) 7.80407e13 0.00551437
\(865\) −5.55949e14 −0.0390343
\(866\) −7.46147e15 −0.520567
\(867\) 2.02660e16 1.40496
\(868\) −2.31853e16 −1.59718
\(869\) −1.21368e15 −0.0830796
\(870\) −1.53844e15 −0.104647
\(871\) −7.69664e14 −0.0520237
\(872\) 1.04890e16 0.704521
\(873\) −2.12192e16 −1.41628
\(874\) −7.05810e14 −0.0468139
\(875\) 2.96779e16 1.95609
\(876\) 5.76139e15 0.377359
\(877\) −2.58409e16 −1.68194 −0.840969 0.541083i \(-0.818014\pi\)
−0.840969 + 0.541083i \(0.818014\pi\)
\(878\) 1.11618e16 0.721959
\(879\) 2.48987e15 0.160043
\(880\) 2.69974e14 0.0172451
\(881\) −2.22834e15 −0.141453 −0.0707267 0.997496i \(-0.522532\pi\)
−0.0707267 + 0.997496i \(0.522532\pi\)
\(882\) −1.90272e16 −1.20032
\(883\) 5.86667e15 0.367797 0.183898 0.982945i \(-0.441128\pi\)
0.183898 + 0.982945i \(0.441128\pi\)
\(884\) 6.98524e14 0.0435205
\(885\) 2.08401e15 0.129036
\(886\) 4.42941e15 0.272559
\(887\) 1.54439e16 0.944443 0.472221 0.881480i \(-0.343452\pi\)
0.472221 + 0.881480i \(0.343452\pi\)
\(888\) 3.78367e16 2.29954
\(889\) −3.23788e16 −1.95569
\(890\) −9.38983e15 −0.563654
\(891\) 1.52665e15 0.0910776
\(892\) 1.04731e15 0.0620970
\(893\) 1.74289e15 0.102704
\(894\) 2.46532e16 1.44383
\(895\) −1.60344e16 −0.933313
\(896\) 2.75706e16 1.59497
\(897\) 1.57094e16 0.903237
\(898\) 2.38337e15 0.136199
\(899\) 3.98276e15 0.226207
\(900\) 5.47612e15 0.309129
\(901\) −4.22487e14 −0.0237043
\(902\) 5.89389e14 0.0328675
\(903\) 7.10581e16 3.93850
\(904\) 5.65939e15 0.311776
\(905\) 1.35872e16 0.743985
\(906\) −2.36660e16 −1.28801
\(907\) −1.11131e16 −0.601165 −0.300583 0.953756i \(-0.597181\pi\)
−0.300583 + 0.953756i \(0.597181\pi\)
\(908\) −4.73025e15 −0.254338
\(909\) −4.48441e15 −0.239664
\(910\) −9.42846e15 −0.500856
\(911\) −3.61381e16 −1.90816 −0.954078 0.299559i \(-0.903161\pi\)
−0.954078 + 0.299559i \(0.903161\pi\)
\(912\) −6.51564e14 −0.0341968
\(913\) 2.15531e15 0.112440
\(914\) −5.84016e15 −0.302845
\(915\) −2.24171e16 −1.15548
\(916\) 1.52741e16 0.782583
\(917\) −5.44454e16 −2.77287
\(918\) 4.77347e12 0.000241657 0
\(919\) 1.56063e15 0.0785351 0.0392675 0.999229i \(-0.487498\pi\)
0.0392675 + 0.999229i \(0.487498\pi\)
\(920\) −1.25997e16 −0.630271
\(921\) 1.10640e16 0.550152
\(922\) 1.57682e16 0.779405
\(923\) 2.09714e16 1.03043
\(924\) −3.47394e15 −0.169678
\(925\) 1.57131e16 0.762924
\(926\) 2.48122e15 0.119758
\(927\) 5.60895e15 0.269118
\(928\) −4.03374e15 −0.192395
\(929\) −3.93856e16 −1.86746 −0.933729 0.357982i \(-0.883465\pi\)
−0.933729 + 0.357982i \(0.883465\pi\)
\(930\) −1.45642e16 −0.686482
\(931\) 4.71887e15 0.221113
\(932\) 9.18498e15 0.427849
\(933\) 3.32489e16 1.53967
\(934\) 1.61755e16 0.744646
\(935\) 1.30952e14 0.00599306
\(936\) −1.38821e16 −0.631596
\(937\) 2.61331e16 1.18201 0.591007 0.806666i \(-0.298731\pi\)
0.591007 + 0.806666i \(0.298731\pi\)
\(938\) −1.58259e15 −0.0711629
\(939\) −2.78530e16 −1.24512
\(940\) 1.30687e16 0.580805
\(941\) 8.78306e15 0.388063 0.194032 0.980995i \(-0.437844\pi\)
0.194032 + 0.980995i \(0.437844\pi\)
\(942\) −2.57206e16 −1.12980
\(943\) 1.43946e16 0.628615
\(944\) 6.89044e14 0.0299158
\(945\) 1.69233e14 0.00730481
\(946\) −1.71933e15 −0.0737832
\(947\) 1.87878e16 0.801586 0.400793 0.916169i \(-0.368735\pi\)
0.400793 + 0.916169i \(0.368735\pi\)
\(948\) 2.19636e16 0.931659
\(949\) −6.06539e15 −0.255796
\(950\) 5.17068e14 0.0216804
\(951\) 4.29600e16 1.79090
\(952\) 3.41947e15 0.141728
\(953\) 1.66665e16 0.686807 0.343404 0.939188i \(-0.388420\pi\)
0.343404 + 0.939188i \(0.388420\pi\)
\(954\) 3.52679e15 0.144498
\(955\) 1.97005e16 0.802527
\(956\) −2.05781e15 −0.0833462
\(957\) 5.96751e14 0.0240313
\(958\) 1.37729e16 0.551462
\(959\) 2.26423e16 0.901401
\(960\) 8.00487e15 0.316857
\(961\) 1.22956e16 0.483918
\(962\) −1.67315e16 −0.654745
\(963\) 4.46041e16 1.73552
\(964\) −1.23790e16 −0.478921
\(965\) −2.02915e15 −0.0780572
\(966\) 3.23019e16 1.23553
\(967\) 1.23857e16 0.471057 0.235528 0.971867i \(-0.424318\pi\)
0.235528 + 0.971867i \(0.424318\pi\)
\(968\) −2.37423e16 −0.897860
\(969\) −3.16045e14 −0.0118841
\(970\) −1.50233e16 −0.561723
\(971\) −2.59670e16 −0.965421 −0.482710 0.875780i \(-0.660348\pi\)
−0.482710 + 0.875780i \(0.660348\pi\)
\(972\) −2.77316e16 −1.02520
\(973\) −3.06062e16 −1.12510
\(974\) −1.44820e16 −0.529363
\(975\) −1.15085e16 −0.418306
\(976\) −7.41186e15 −0.267888
\(977\) −4.38632e16 −1.57645 −0.788225 0.615387i \(-0.789000\pi\)
−0.788225 + 0.615387i \(0.789000\pi\)
\(978\) 1.52722e16 0.545806
\(979\) 3.64225e15 0.129439
\(980\) 3.53836e16 1.25043
\(981\) 2.22254e16 0.781035
\(982\) −8.24936e15 −0.288275
\(983\) 9.95512e15 0.345941 0.172971 0.984927i \(-0.444663\pi\)
0.172971 + 0.984927i \(0.444663\pi\)
\(984\) −2.53928e16 −0.877480
\(985\) −3.39811e16 −1.16772
\(986\) −2.46729e14 −0.00843137
\(987\) −7.97647e16 −2.71061
\(988\) 1.44614e15 0.0488707
\(989\) −4.19909e16 −1.41116
\(990\) −1.09315e15 −0.0365330
\(991\) 1.41215e16 0.469327 0.234663 0.972077i \(-0.424601\pi\)
0.234663 + 0.972077i \(0.424601\pi\)
\(992\) −3.81867e16 −1.26211
\(993\) −4.70692e16 −1.54709
\(994\) 4.31218e16 1.40952
\(995\) 9.81751e15 0.319135
\(996\) −3.90039e16 −1.26090
\(997\) −1.95475e16 −0.628445 −0.314223 0.949349i \(-0.601744\pi\)
−0.314223 + 0.949349i \(0.601744\pi\)
\(998\) −2.07638e16 −0.663880
\(999\) 3.00316e14 0.00954922
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 29.12.a.a.1.4 11
3.2 odd 2 261.12.a.a.1.8 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.12.a.a.1.4 11 1.1 even 1 trivial
261.12.a.a.1.8 11 3.2 odd 2