Properties

Label 91.10.a.a.1.2
Level $91$
Weight $10$
Character 91.1
Self dual yes
Analytic conductor $46.868$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.8682610909\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 4522 x^{10} + 11094 x^{9} + 7471016 x^{8} - 18339296 x^{7} - 5497728352 x^{6} + 13724467264 x^{5} + 1698856105344 x^{4} - 3404524011264 x^{3} - 154369782114304 x^{2} + 70325953652224 x + 170905444356096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-35.8265\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-37.8265 q^{2} +32.9997 q^{3} +918.841 q^{4} -2427.98 q^{5} -1248.26 q^{6} +2401.00 q^{7} -15389.3 q^{8} -18594.0 q^{9} +O(q^{10})\) \(q-37.8265 q^{2} +32.9997 q^{3} +918.841 q^{4} -2427.98 q^{5} -1248.26 q^{6} +2401.00 q^{7} -15389.3 q^{8} -18594.0 q^{9} +91841.7 q^{10} +38536.0 q^{11} +30321.5 q^{12} +28561.0 q^{13} -90821.3 q^{14} -80122.4 q^{15} +111678. q^{16} -187269. q^{17} +703346. q^{18} -109656. q^{19} -2.23092e6 q^{20} +79232.2 q^{21} -1.45768e6 q^{22} +599698. q^{23} -507844. q^{24} +3.94194e6 q^{25} -1.08036e6 q^{26} -1.26313e6 q^{27} +2.20614e6 q^{28} +1.91020e6 q^{29} +3.03075e6 q^{30} +9.83738e6 q^{31} +3.65496e6 q^{32} +1.27167e6 q^{33} +7.08374e6 q^{34} -5.82957e6 q^{35} -1.70849e7 q^{36} -6.01821e6 q^{37} +4.14790e6 q^{38} +942504. q^{39} +3.73650e7 q^{40} -2.93333e6 q^{41} -2.99707e6 q^{42} +3.42326e7 q^{43} +3.54084e7 q^{44} +4.51458e7 q^{45} -2.26844e7 q^{46} +4.06316e7 q^{47} +3.68534e6 q^{48} +5.76480e6 q^{49} -1.49110e8 q^{50} -6.17983e6 q^{51} +2.62430e7 q^{52} -1.10955e8 q^{53} +4.77797e7 q^{54} -9.35644e7 q^{55} -3.69498e7 q^{56} -3.61861e6 q^{57} -7.22562e7 q^{58} -3.51298e7 q^{59} -7.36197e7 q^{60} -4.03761e7 q^{61} -3.72113e8 q^{62} -4.46442e7 q^{63} -1.95433e8 q^{64} -6.93454e7 q^{65} -4.81029e7 q^{66} +9.71041e7 q^{67} -1.72071e8 q^{68} +1.97898e7 q^{69} +2.20512e8 q^{70} -3.17659e8 q^{71} +2.86150e8 q^{72} +1.93918e6 q^{73} +2.27648e8 q^{74} +1.30083e8 q^{75} -1.00756e8 q^{76} +9.25248e7 q^{77} -3.56516e7 q^{78} +2.55882e8 q^{79} -2.71151e8 q^{80} +3.24303e8 q^{81} +1.10957e8 q^{82} -6.62526e8 q^{83} +7.28018e7 q^{84} +4.54685e8 q^{85} -1.29490e9 q^{86} +6.30361e7 q^{87} -5.93043e8 q^{88} -1.50214e7 q^{89} -1.70771e9 q^{90} +6.85750e7 q^{91} +5.51027e8 q^{92} +3.24630e8 q^{93} -1.53695e9 q^{94} +2.66242e8 q^{95} +1.20613e8 q^{96} +7.63478e8 q^{97} -2.18062e8 q^{98} -7.16539e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} + O(q^{10}) \) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} - 65812 q^{10} - 80061 q^{11} - 184395 q^{12} + 342732 q^{13} - 50421 q^{14} + 160096 q^{15} + 385497 q^{16} - 1493598 q^{17} + 1520858 q^{18} - 109038 q^{19} - 622260 q^{20} - 775523 q^{21} + 4636975 q^{22} - 3367443 q^{23} - 5963895 q^{24} - 51480 q^{25} - 599781 q^{26} - 8158937 q^{27} + 7070945 q^{28} - 13333098 q^{29} + 2915424 q^{30} - 3954765 q^{31} + 4389297 q^{32} - 5790219 q^{33} + 14879968 q^{34} - 12490002 q^{35} + 80697058 q^{36} + 580535 q^{37} - 19134246 q^{38} - 9225203 q^{39} + 12365024 q^{40} - 27018171 q^{41} - 23762697 q^{42} + 31237588 q^{43} - 125053839 q^{44} - 62765470 q^{45} - 114008121 q^{46} - 21983709 q^{47} - 309724207 q^{48} + 69177612 q^{49} - 131331747 q^{50} - 176522692 q^{51} + 84112145 q^{52} - 196548234 q^{53} - 456152547 q^{54} - 309055872 q^{55} - 61059831 q^{56} - 274411494 q^{57} - 521980612 q^{58} - 215907906 q^{59} - 177006648 q^{60} - 218340705 q^{61} - 673289997 q^{62} + 188524119 q^{63} - 386667247 q^{64} - 148574322 q^{65} - 777397365 q^{66} + 14544775 q^{67} - 1246637448 q^{68} - 65252625 q^{69} - 158014612 q^{70} - 552451776 q^{71} + 369379470 q^{72} - 349395159 q^{73} + 73591023 q^{74} + 329300747 q^{75} - 1036299002 q^{76} - 192226461 q^{77} - 282668217 q^{78} + 962249727 q^{79} - 1494536184 q^{80} + 874458108 q^{81} - 1417698067 q^{82} - 2032575912 q^{83} - 442732395 q^{84} - 411671064 q^{85} - 2139249420 q^{86} - 759642172 q^{87} + 558651957 q^{88} - 280821684 q^{89} - 5764700804 q^{90} + 822899532 q^{91} - 4491569571 q^{92} - 1729557923 q^{93} - 1591372165 q^{94} - 1282463328 q^{95} - 2148993055 q^{96} - 2115165937 q^{97} - 121060821 q^{98} - 3595669198 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −37.8265 −1.67171 −0.835855 0.548951i \(-0.815027\pi\)
−0.835855 + 0.548951i \(0.815027\pi\)
\(3\) 32.9997 0.235214 0.117607 0.993060i \(-0.462478\pi\)
0.117607 + 0.993060i \(0.462478\pi\)
\(4\) 918.841 1.79461
\(5\) −2427.98 −1.73732 −0.868659 0.495411i \(-0.835018\pi\)
−0.868659 + 0.495411i \(0.835018\pi\)
\(6\) −1248.26 −0.393210
\(7\) 2401.00 0.377964
\(8\) −15389.3 −1.32836
\(9\) −18594.0 −0.944674
\(10\) 91841.7 2.90429
\(11\) 38536.0 0.793595 0.396798 0.917906i \(-0.370121\pi\)
0.396798 + 0.917906i \(0.370121\pi\)
\(12\) 30321.5 0.422118
\(13\) 28561.0 0.277350
\(14\) −90821.3 −0.631847
\(15\) −80122.4 −0.408642
\(16\) 111678. 0.426018
\(17\) −187269. −0.543809 −0.271905 0.962324i \(-0.587654\pi\)
−0.271905 + 0.962324i \(0.587654\pi\)
\(18\) 703346. 1.57922
\(19\) −109656. −0.193037 −0.0965186 0.995331i \(-0.530771\pi\)
−0.0965186 + 0.995331i \(0.530771\pi\)
\(20\) −2.23092e6 −3.11781
\(21\) 79232.2 0.0889027
\(22\) −1.45768e6 −1.32666
\(23\) 599698. 0.446845 0.223423 0.974722i \(-0.428277\pi\)
0.223423 + 0.974722i \(0.428277\pi\)
\(24\) −507844. −0.312449
\(25\) 3.94194e6 2.01827
\(26\) −1.08036e6 −0.463649
\(27\) −1.26313e6 −0.457416
\(28\) 2.20614e6 0.678299
\(29\) 1.91020e6 0.501520 0.250760 0.968049i \(-0.419319\pi\)
0.250760 + 0.968049i \(0.419319\pi\)
\(30\) 3.03075e6 0.683131
\(31\) 9.83738e6 1.91316 0.956582 0.291465i \(-0.0941427\pi\)
0.956582 + 0.291465i \(0.0941427\pi\)
\(32\) 3.65496e6 0.616181
\(33\) 1.27167e6 0.186665
\(34\) 7.08374e6 0.909091
\(35\) −5.82957e6 −0.656644
\(36\) −1.70849e7 −1.69532
\(37\) −6.01821e6 −0.527910 −0.263955 0.964535i \(-0.585027\pi\)
−0.263955 + 0.964535i \(0.585027\pi\)
\(38\) 4.14790e6 0.322702
\(39\) 942504. 0.0652368
\(40\) 3.73650e7 2.30778
\(41\) −2.93333e6 −0.162119 −0.0810593 0.996709i \(-0.525830\pi\)
−0.0810593 + 0.996709i \(0.525830\pi\)
\(42\) −2.99707e6 −0.148619
\(43\) 3.42326e7 1.52697 0.763487 0.645824i \(-0.223486\pi\)
0.763487 + 0.645824i \(0.223486\pi\)
\(44\) 3.54084e7 1.42420
\(45\) 4.51458e7 1.64120
\(46\) −2.26844e7 −0.746996
\(47\) 4.06316e7 1.21457 0.607287 0.794483i \(-0.292258\pi\)
0.607287 + 0.794483i \(0.292258\pi\)
\(48\) 3.68534e6 0.100206
\(49\) 5.76480e6 0.142857
\(50\) −1.49110e8 −3.37397
\(51\) −6.17983e6 −0.127912
\(52\) 2.62430e7 0.497736
\(53\) −1.10955e8 −1.93155 −0.965776 0.259376i \(-0.916483\pi\)
−0.965776 + 0.259376i \(0.916483\pi\)
\(54\) 4.77797e7 0.764666
\(55\) −9.35644e7 −1.37873
\(56\) −3.69498e7 −0.502072
\(57\) −3.61861e6 −0.0454051
\(58\) −7.22562e7 −0.838396
\(59\) −3.51298e7 −0.377435 −0.188717 0.982031i \(-0.560433\pi\)
−0.188717 + 0.982031i \(0.560433\pi\)
\(60\) −7.36197e7 −0.733354
\(61\) −4.03761e7 −0.373371 −0.186685 0.982420i \(-0.559774\pi\)
−0.186685 + 0.982420i \(0.559774\pi\)
\(62\) −3.72113e8 −3.19825
\(63\) −4.46442e7 −0.357053
\(64\) −1.95433e8 −1.45609
\(65\) −6.93454e7 −0.481845
\(66\) −4.81029e7 −0.312050
\(67\) 9.71041e7 0.588710 0.294355 0.955696i \(-0.404895\pi\)
0.294355 + 0.955696i \(0.404895\pi\)
\(68\) −1.72071e8 −0.975926
\(69\) 1.97898e7 0.105105
\(70\) 2.20512e8 1.09772
\(71\) −3.17659e8 −1.48354 −0.741768 0.670657i \(-0.766012\pi\)
−0.741768 + 0.670657i \(0.766012\pi\)
\(72\) 2.86150e8 1.25487
\(73\) 1.93918e6 0.00799220 0.00399610 0.999992i \(-0.498728\pi\)
0.00399610 + 0.999992i \(0.498728\pi\)
\(74\) 2.27648e8 0.882512
\(75\) 1.30083e8 0.474727
\(76\) −1.00756e8 −0.346427
\(77\) 9.25248e7 0.299951
\(78\) −3.56516e7 −0.109057
\(79\) 2.55882e8 0.739124 0.369562 0.929206i \(-0.379508\pi\)
0.369562 + 0.929206i \(0.379508\pi\)
\(80\) −2.71151e8 −0.740128
\(81\) 3.24303e8 0.837083
\(82\) 1.10957e8 0.271015
\(83\) −6.62526e8 −1.53233 −0.766164 0.642645i \(-0.777837\pi\)
−0.766164 + 0.642645i \(0.777837\pi\)
\(84\) 7.28018e7 0.159546
\(85\) 4.54685e8 0.944770
\(86\) −1.29490e9 −2.55266
\(87\) 6.30361e7 0.117965
\(88\) −5.93043e8 −1.05418
\(89\) −1.50214e7 −0.0253779 −0.0126889 0.999919i \(-0.504039\pi\)
−0.0126889 + 0.999919i \(0.504039\pi\)
\(90\) −1.70771e9 −2.74361
\(91\) 6.85750e7 0.104828
\(92\) 5.51027e8 0.801914
\(93\) 3.24630e8 0.450004
\(94\) −1.53695e9 −2.03041
\(95\) 2.66242e8 0.335367
\(96\) 1.20613e8 0.144935
\(97\) 7.63478e8 0.875636 0.437818 0.899064i \(-0.355751\pi\)
0.437818 + 0.899064i \(0.355751\pi\)
\(98\) −2.18062e8 −0.238816
\(99\) −7.16539e8 −0.749689
\(100\) 3.62202e9 3.62202
\(101\) −1.81478e9 −1.73531 −0.867655 0.497167i \(-0.834374\pi\)
−0.867655 + 0.497167i \(0.834374\pi\)
\(102\) 2.33761e8 0.213831
\(103\) −1.54586e9 −1.35332 −0.676662 0.736294i \(-0.736574\pi\)
−0.676662 + 0.736294i \(0.736574\pi\)
\(104\) −4.39535e8 −0.368420
\(105\) −1.92374e8 −0.154452
\(106\) 4.19705e9 3.22899
\(107\) −1.24455e9 −0.917881 −0.458941 0.888467i \(-0.651771\pi\)
−0.458941 + 0.888467i \(0.651771\pi\)
\(108\) −1.16061e9 −0.820883
\(109\) 2.12104e9 1.43923 0.719615 0.694374i \(-0.244318\pi\)
0.719615 + 0.694374i \(0.244318\pi\)
\(110\) 3.53921e9 2.30483
\(111\) −1.98599e8 −0.124172
\(112\) 2.68139e8 0.161020
\(113\) −2.03847e8 −0.117612 −0.0588060 0.998269i \(-0.518729\pi\)
−0.0588060 + 0.998269i \(0.518729\pi\)
\(114\) 1.36879e8 0.0759042
\(115\) −1.45605e9 −0.776313
\(116\) 1.75517e9 0.900034
\(117\) −5.31064e8 −0.262005
\(118\) 1.32884e9 0.630961
\(119\) −4.49634e8 −0.205541
\(120\) 1.23303e9 0.542823
\(121\) −8.72927e8 −0.370206
\(122\) 1.52728e9 0.624167
\(123\) −9.67988e7 −0.0381327
\(124\) 9.03899e9 3.43338
\(125\) −4.82879e9 −1.76906
\(126\) 1.68873e9 0.596889
\(127\) −4.46639e9 −1.52349 −0.761746 0.647876i \(-0.775657\pi\)
−0.761746 + 0.647876i \(0.775657\pi\)
\(128\) 5.52121e9 1.81798
\(129\) 1.12966e9 0.359166
\(130\) 2.62309e9 0.805505
\(131\) −1.88424e9 −0.559005 −0.279502 0.960145i \(-0.590170\pi\)
−0.279502 + 0.960145i \(0.590170\pi\)
\(132\) 1.16847e9 0.334991
\(133\) −2.63284e8 −0.0729612
\(134\) −3.67311e9 −0.984151
\(135\) 3.06685e9 0.794676
\(136\) 2.88195e9 0.722374
\(137\) 1.19271e9 0.289263 0.144631 0.989486i \(-0.453800\pi\)
0.144631 + 0.989486i \(0.453800\pi\)
\(138\) −7.48580e8 −0.175704
\(139\) −5.91993e9 −1.34509 −0.672543 0.740058i \(-0.734798\pi\)
−0.672543 + 0.740058i \(0.734798\pi\)
\(140\) −5.35645e9 −1.17842
\(141\) 1.34083e9 0.285685
\(142\) 1.20159e10 2.48004
\(143\) 1.10063e9 0.220104
\(144\) −2.07654e9 −0.402448
\(145\) −4.63793e9 −0.871300
\(146\) −7.33525e7 −0.0133606
\(147\) 1.90237e8 0.0336021
\(148\) −5.52978e9 −0.947393
\(149\) 6.45915e9 1.07359 0.536794 0.843714i \(-0.319635\pi\)
0.536794 + 0.843714i \(0.319635\pi\)
\(150\) −4.92057e9 −0.793605
\(151\) 2.95041e9 0.461835 0.230917 0.972973i \(-0.425827\pi\)
0.230917 + 0.972973i \(0.425827\pi\)
\(152\) 1.68753e9 0.256423
\(153\) 3.48209e9 0.513723
\(154\) −3.49989e9 −0.501431
\(155\) −2.38849e10 −3.32377
\(156\) 8.66011e8 0.117075
\(157\) −8.18461e9 −1.07510 −0.537551 0.843231i \(-0.680650\pi\)
−0.537551 + 0.843231i \(0.680650\pi\)
\(158\) −9.67910e9 −1.23560
\(159\) −3.66149e9 −0.454329
\(160\) −8.87416e9 −1.07050
\(161\) 1.43987e9 0.168892
\(162\) −1.22672e10 −1.39936
\(163\) −2.70691e9 −0.300351 −0.150175 0.988659i \(-0.547984\pi\)
−0.150175 + 0.988659i \(0.547984\pi\)
\(164\) −2.69526e9 −0.290940
\(165\) −3.08759e9 −0.324297
\(166\) 2.50610e10 2.56161
\(167\) 1.00308e10 0.997959 0.498979 0.866614i \(-0.333708\pi\)
0.498979 + 0.866614i \(0.333708\pi\)
\(168\) −1.21933e9 −0.118095
\(169\) 8.15731e8 0.0769231
\(170\) −1.71991e10 −1.57938
\(171\) 2.03894e9 0.182357
\(172\) 3.14543e10 2.74032
\(173\) 1.40569e10 1.19311 0.596556 0.802571i \(-0.296535\pi\)
0.596556 + 0.802571i \(0.296535\pi\)
\(174\) −2.38443e9 −0.197203
\(175\) 9.46460e9 0.762836
\(176\) 4.30362e9 0.338086
\(177\) −1.15927e9 −0.0887782
\(178\) 5.68206e8 0.0424244
\(179\) −8.96605e9 −0.652773 −0.326387 0.945236i \(-0.605831\pi\)
−0.326387 + 0.945236i \(0.605831\pi\)
\(180\) 4.14818e10 2.94531
\(181\) 1.33482e10 0.924422 0.462211 0.886770i \(-0.347056\pi\)
0.462211 + 0.886770i \(0.347056\pi\)
\(182\) −2.59395e9 −0.175243
\(183\) −1.33240e9 −0.0878222
\(184\) −9.22896e9 −0.593571
\(185\) 1.46121e10 0.917147
\(186\) −1.22796e10 −0.752275
\(187\) −7.21661e9 −0.431565
\(188\) 3.73340e10 2.17969
\(189\) −3.03277e9 −0.172887
\(190\) −1.00710e10 −0.560636
\(191\) −1.17556e10 −0.639139 −0.319569 0.947563i \(-0.603538\pi\)
−0.319569 + 0.947563i \(0.603538\pi\)
\(192\) −6.44924e9 −0.342494
\(193\) 2.07058e10 1.07420 0.537099 0.843520i \(-0.319520\pi\)
0.537099 + 0.843520i \(0.319520\pi\)
\(194\) −2.88797e10 −1.46381
\(195\) −2.28838e9 −0.113337
\(196\) 5.29693e9 0.256373
\(197\) −7.32118e9 −0.346324 −0.173162 0.984893i \(-0.555398\pi\)
−0.173162 + 0.984893i \(0.555398\pi\)
\(198\) 2.71041e10 1.25326
\(199\) −2.27663e10 −1.02909 −0.514546 0.857463i \(-0.672040\pi\)
−0.514546 + 0.857463i \(0.672040\pi\)
\(200\) −6.06639e10 −2.68099
\(201\) 3.20440e9 0.138473
\(202\) 6.86466e10 2.90093
\(203\) 4.58640e9 0.189557
\(204\) −5.67828e9 −0.229552
\(205\) 7.12204e9 0.281652
\(206\) 5.84743e10 2.26236
\(207\) −1.11508e10 −0.422123
\(208\) 3.18964e9 0.118156
\(209\) −4.22570e9 −0.153193
\(210\) 7.27682e9 0.258199
\(211\) 1.63637e10 0.568343 0.284171 0.958773i \(-0.408282\pi\)
0.284171 + 0.958773i \(0.408282\pi\)
\(212\) −1.01950e11 −3.46639
\(213\) −1.04826e10 −0.348949
\(214\) 4.70770e10 1.53443
\(215\) −8.31158e10 −2.65284
\(216\) 1.94387e10 0.607612
\(217\) 2.36196e10 0.723108
\(218\) −8.02315e10 −2.40597
\(219\) 6.39925e7 0.00187988
\(220\) −8.59708e10 −2.47428
\(221\) −5.34860e9 −0.150826
\(222\) 7.51230e9 0.207579
\(223\) 1.35777e10 0.367667 0.183834 0.982957i \(-0.441149\pi\)
0.183834 + 0.982957i \(0.441149\pi\)
\(224\) 8.77557e9 0.232894
\(225\) −7.32965e10 −1.90661
\(226\) 7.71081e9 0.196613
\(227\) 7.52071e9 0.187993 0.0939966 0.995573i \(-0.470036\pi\)
0.0939966 + 0.995573i \(0.470036\pi\)
\(228\) −3.32493e9 −0.0814846
\(229\) −4.84088e10 −1.16323 −0.581614 0.813465i \(-0.697578\pi\)
−0.581614 + 0.813465i \(0.697578\pi\)
\(230\) 5.50773e10 1.29777
\(231\) 3.05329e9 0.0705528
\(232\) −2.93968e10 −0.666199
\(233\) 8.15887e10 1.81355 0.906773 0.421620i \(-0.138538\pi\)
0.906773 + 0.421620i \(0.138538\pi\)
\(234\) 2.00883e10 0.437997
\(235\) −9.86526e10 −2.11010
\(236\) −3.22787e10 −0.677349
\(237\) 8.44401e9 0.173853
\(238\) 1.70081e10 0.343604
\(239\) 4.53407e10 0.898871 0.449436 0.893313i \(-0.351625\pi\)
0.449436 + 0.893313i \(0.351625\pi\)
\(240\) −8.94791e9 −0.174089
\(241\) −7.77958e10 −1.48552 −0.742761 0.669556i \(-0.766484\pi\)
−0.742761 + 0.669556i \(0.766484\pi\)
\(242\) 3.30197e10 0.618877
\(243\) 3.55641e10 0.654310
\(244\) −3.70992e10 −0.670055
\(245\) −1.39968e10 −0.248188
\(246\) 3.66156e9 0.0637467
\(247\) −3.13188e9 −0.0535389
\(248\) −1.51391e11 −2.54137
\(249\) −2.18631e10 −0.360426
\(250\) 1.82656e11 2.95736
\(251\) 2.00266e10 0.318475 0.159238 0.987240i \(-0.449096\pi\)
0.159238 + 0.987240i \(0.449096\pi\)
\(252\) −4.10210e10 −0.640772
\(253\) 2.31099e10 0.354614
\(254\) 1.68948e11 2.54683
\(255\) 1.50045e10 0.222223
\(256\) −1.08786e11 −1.58305
\(257\) −8.67825e10 −1.24089 −0.620445 0.784250i \(-0.713048\pi\)
−0.620445 + 0.784250i \(0.713048\pi\)
\(258\) −4.27312e10 −0.600421
\(259\) −1.44497e10 −0.199531
\(260\) −6.37174e10 −0.864725
\(261\) −3.55184e10 −0.473773
\(262\) 7.12742e10 0.934494
\(263\) 2.98699e10 0.384975 0.192487 0.981299i \(-0.438345\pi\)
0.192487 + 0.981299i \(0.438345\pi\)
\(264\) −1.95702e10 −0.247958
\(265\) 2.69397e11 3.35572
\(266\) 9.95910e9 0.121970
\(267\) −4.95701e8 −0.00596924
\(268\) 8.92232e10 1.05650
\(269\) 6.90189e10 0.803679 0.401839 0.915710i \(-0.368371\pi\)
0.401839 + 0.915710i \(0.368371\pi\)
\(270\) −1.16008e11 −1.32847
\(271\) −5.45891e10 −0.614814 −0.307407 0.951578i \(-0.599461\pi\)
−0.307407 + 0.951578i \(0.599461\pi\)
\(272\) −2.09139e10 −0.231672
\(273\) 2.26295e9 0.0246572
\(274\) −4.51160e10 −0.483563
\(275\) 1.51906e11 1.60169
\(276\) 1.81837e10 0.188622
\(277\) −3.93625e10 −0.401720 −0.200860 0.979620i \(-0.564374\pi\)
−0.200860 + 0.979620i \(0.564374\pi\)
\(278\) 2.23930e11 2.24859
\(279\) −1.82917e11 −1.80732
\(280\) 8.97133e10 0.872259
\(281\) −1.41844e11 −1.35716 −0.678582 0.734525i \(-0.737405\pi\)
−0.678582 + 0.734525i \(0.737405\pi\)
\(282\) −5.07189e10 −0.477583
\(283\) 1.78445e9 0.0165374 0.00826869 0.999966i \(-0.497368\pi\)
0.00826869 + 0.999966i \(0.497368\pi\)
\(284\) −2.91878e11 −2.66237
\(285\) 8.78590e9 0.0788832
\(286\) −4.16328e10 −0.367949
\(287\) −7.04292e9 −0.0612751
\(288\) −6.79605e10 −0.582090
\(289\) −8.35181e10 −0.704271
\(290\) 1.75436e11 1.45656
\(291\) 2.51945e10 0.205962
\(292\) 1.78180e9 0.0143429
\(293\) 1.72454e11 1.36700 0.683502 0.729948i \(-0.260456\pi\)
0.683502 + 0.729948i \(0.260456\pi\)
\(294\) −7.19598e9 −0.0561729
\(295\) 8.52944e10 0.655725
\(296\) 9.26164e10 0.701253
\(297\) −4.86759e10 −0.363003
\(298\) −2.44327e11 −1.79473
\(299\) 1.71280e10 0.123933
\(300\) 1.19525e11 0.851950
\(301\) 8.21924e10 0.577142
\(302\) −1.11604e11 −0.772054
\(303\) −5.98870e10 −0.408170
\(304\) −1.22462e10 −0.0822373
\(305\) 9.80322e10 0.648663
\(306\) −1.31715e11 −0.858795
\(307\) −1.82140e11 −1.17026 −0.585129 0.810940i \(-0.698956\pi\)
−0.585129 + 0.810940i \(0.698956\pi\)
\(308\) 8.50156e10 0.538295
\(309\) −5.10128e10 −0.318321
\(310\) 9.03482e11 5.55638
\(311\) −1.87477e11 −1.13639 −0.568194 0.822895i \(-0.692358\pi\)
−0.568194 + 0.822895i \(0.692358\pi\)
\(312\) −1.45045e10 −0.0866578
\(313\) 2.91361e10 0.171586 0.0857930 0.996313i \(-0.472658\pi\)
0.0857930 + 0.996313i \(0.472658\pi\)
\(314\) 3.09595e11 1.79726
\(315\) 1.08395e11 0.620315
\(316\) 2.35115e11 1.32644
\(317\) 7.68072e10 0.427204 0.213602 0.976921i \(-0.431480\pi\)
0.213602 + 0.976921i \(0.431480\pi\)
\(318\) 1.38501e11 0.759506
\(319\) 7.36115e10 0.398004
\(320\) 4.74508e11 2.52970
\(321\) −4.10699e10 −0.215899
\(322\) −5.44654e10 −0.282338
\(323\) 2.05352e10 0.104975
\(324\) 2.97983e11 1.50224
\(325\) 1.12586e11 0.559768
\(326\) 1.02393e11 0.502099
\(327\) 6.99937e10 0.338528
\(328\) 4.51420e10 0.215352
\(329\) 9.75565e10 0.459066
\(330\) 1.16793e11 0.542130
\(331\) −4.32748e11 −1.98157 −0.990784 0.135452i \(-0.956751\pi\)
−0.990784 + 0.135452i \(0.956751\pi\)
\(332\) −6.08756e11 −2.74993
\(333\) 1.11903e11 0.498703
\(334\) −3.79431e11 −1.66830
\(335\) −2.35766e11 −1.02278
\(336\) 8.84850e9 0.0378741
\(337\) 2.09548e10 0.0885010 0.0442505 0.999020i \(-0.485910\pi\)
0.0442505 + 0.999020i \(0.485910\pi\)
\(338\) −3.08562e10 −0.128593
\(339\) −6.72689e9 −0.0276640
\(340\) 4.17784e11 1.69549
\(341\) 3.79093e11 1.51828
\(342\) −7.71261e10 −0.304848
\(343\) 1.38413e10 0.0539949
\(344\) −5.26817e11 −2.02837
\(345\) −4.80492e10 −0.182600
\(346\) −5.31722e11 −1.99454
\(347\) 4.16346e11 1.54160 0.770801 0.637076i \(-0.219856\pi\)
0.770801 + 0.637076i \(0.219856\pi\)
\(348\) 5.79201e10 0.211701
\(349\) 3.93666e11 1.42041 0.710204 0.703996i \(-0.248603\pi\)
0.710204 + 0.703996i \(0.248603\pi\)
\(350\) −3.58012e11 −1.27524
\(351\) −3.60762e10 −0.126864
\(352\) 1.40848e11 0.488998
\(353\) −5.47569e11 −1.87695 −0.938474 0.345349i \(-0.887761\pi\)
−0.938474 + 0.345349i \(0.887761\pi\)
\(354\) 4.38512e10 0.148411
\(355\) 7.71267e11 2.57737
\(356\) −1.38023e10 −0.0455434
\(357\) −1.48378e10 −0.0483461
\(358\) 3.39154e11 1.09125
\(359\) −4.62717e11 −1.47025 −0.735124 0.677932i \(-0.762876\pi\)
−0.735124 + 0.677932i \(0.762876\pi\)
\(360\) −6.94765e11 −2.18010
\(361\) −3.10663e11 −0.962737
\(362\) −5.04917e11 −1.54537
\(363\) −2.88063e10 −0.0870779
\(364\) 6.30095e10 0.188126
\(365\) −4.70829e9 −0.0138850
\(366\) 5.03999e10 0.146813
\(367\) −1.23136e10 −0.0354313 −0.0177156 0.999843i \(-0.505639\pi\)
−0.0177156 + 0.999843i \(0.505639\pi\)
\(368\) 6.69731e10 0.190364
\(369\) 5.45423e10 0.153149
\(370\) −5.52723e11 −1.53320
\(371\) −2.66404e11 −0.730058
\(372\) 2.98284e11 0.807581
\(373\) −4.88330e11 −1.30624 −0.653122 0.757253i \(-0.726541\pi\)
−0.653122 + 0.757253i \(0.726541\pi\)
\(374\) 2.72979e11 0.721450
\(375\) −1.59349e11 −0.416109
\(376\) −6.25294e11 −1.61339
\(377\) 5.45573e10 0.139097
\(378\) 1.14719e11 0.289016
\(379\) 4.62229e11 1.15075 0.575374 0.817890i \(-0.304856\pi\)
0.575374 + 0.817890i \(0.304856\pi\)
\(380\) 2.44634e11 0.601853
\(381\) −1.47389e11 −0.358347
\(382\) 4.44673e11 1.06845
\(383\) −2.34025e11 −0.555734 −0.277867 0.960620i \(-0.589627\pi\)
−0.277867 + 0.960620i \(0.589627\pi\)
\(384\) 1.82198e11 0.427616
\(385\) −2.24648e11 −0.521110
\(386\) −7.83227e11 −1.79574
\(387\) −6.36521e11 −1.44249
\(388\) 7.01515e11 1.57143
\(389\) 2.33383e11 0.516768 0.258384 0.966042i \(-0.416810\pi\)
0.258384 + 0.966042i \(0.416810\pi\)
\(390\) 8.65612e10 0.189466
\(391\) −1.12305e11 −0.242999
\(392\) −8.87165e10 −0.189765
\(393\) −6.21793e10 −0.131486
\(394\) 2.76934e11 0.578953
\(395\) −6.21274e11 −1.28409
\(396\) −6.58385e11 −1.34540
\(397\) −2.06380e11 −0.416976 −0.208488 0.978025i \(-0.566854\pi\)
−0.208488 + 0.978025i \(0.566854\pi\)
\(398\) 8.61170e11 1.72034
\(399\) −8.68828e9 −0.0171615
\(400\) 4.40228e11 0.859820
\(401\) −4.59933e11 −0.888269 −0.444135 0.895960i \(-0.646489\pi\)
−0.444135 + 0.895960i \(0.646489\pi\)
\(402\) −1.21211e11 −0.231487
\(403\) 2.80966e11 0.530616
\(404\) −1.66749e12 −3.11421
\(405\) −7.87400e11 −1.45428
\(406\) −1.73487e11 −0.316884
\(407\) −2.31918e11 −0.418947
\(408\) 9.51035e10 0.169913
\(409\) −5.62917e11 −0.994695 −0.497347 0.867552i \(-0.665693\pi\)
−0.497347 + 0.867552i \(0.665693\pi\)
\(410\) −2.69402e11 −0.470840
\(411\) 3.93590e10 0.0680387
\(412\) −1.42040e12 −2.42869
\(413\) −8.43468e10 −0.142657
\(414\) 4.21795e11 0.705667
\(415\) 1.60860e12 2.66214
\(416\) 1.04389e11 0.170898
\(417\) −1.95356e11 −0.316384
\(418\) 1.59843e11 0.256095
\(419\) 1.87181e11 0.296687 0.148343 0.988936i \(-0.452606\pi\)
0.148343 + 0.988936i \(0.452606\pi\)
\(420\) −1.76761e11 −0.277182
\(421\) −2.66341e11 −0.413208 −0.206604 0.978425i \(-0.566241\pi\)
−0.206604 + 0.978425i \(0.566241\pi\)
\(422\) −6.18981e11 −0.950104
\(423\) −7.55505e11 −1.14738
\(424\) 1.70753e12 2.56579
\(425\) −7.38205e11 −1.09756
\(426\) 3.96521e11 0.583341
\(427\) −9.69430e10 −0.141121
\(428\) −1.14355e12 −1.64724
\(429\) 3.63203e10 0.0517716
\(430\) 3.14398e12 4.43477
\(431\) 1.65717e10 0.0231324 0.0115662 0.999933i \(-0.496318\pi\)
0.0115662 + 0.999933i \(0.496318\pi\)
\(432\) −1.41064e11 −0.194867
\(433\) 1.23308e12 1.68576 0.842878 0.538105i \(-0.180860\pi\)
0.842878 + 0.538105i \(0.180860\pi\)
\(434\) −8.93444e11 −1.20883
\(435\) −1.53050e11 −0.204942
\(436\) 1.94890e12 2.58286
\(437\) −6.57604e10 −0.0862578
\(438\) −2.42061e9 −0.00314261
\(439\) 1.00390e11 0.129004 0.0645018 0.997918i \(-0.479454\pi\)
0.0645018 + 0.997918i \(0.479454\pi\)
\(440\) 1.43989e12 1.83144
\(441\) −1.07191e11 −0.134953
\(442\) 2.02319e11 0.252136
\(443\) 5.39889e11 0.666021 0.333010 0.942923i \(-0.391936\pi\)
0.333010 + 0.942923i \(0.391936\pi\)
\(444\) −1.82481e11 −0.222840
\(445\) 3.64715e10 0.0440894
\(446\) −5.13597e11 −0.614633
\(447\) 2.13150e11 0.252523
\(448\) −4.69236e11 −0.550351
\(449\) −3.83225e11 −0.444985 −0.222493 0.974934i \(-0.571419\pi\)
−0.222493 + 0.974934i \(0.571419\pi\)
\(450\) 2.77255e12 3.18730
\(451\) −1.13039e11 −0.128657
\(452\) −1.87303e11 −0.211068
\(453\) 9.73627e10 0.108630
\(454\) −2.84482e11 −0.314270
\(455\) −1.66498e11 −0.182120
\(456\) 5.56881e10 0.0603143
\(457\) 8.64714e11 0.927363 0.463681 0.886002i \(-0.346528\pi\)
0.463681 + 0.886002i \(0.346528\pi\)
\(458\) 1.83113e12 1.94458
\(459\) 2.36545e11 0.248747
\(460\) −1.33788e12 −1.39318
\(461\) −1.69137e12 −1.74415 −0.872075 0.489373i \(-0.837226\pi\)
−0.872075 + 0.489373i \(0.837226\pi\)
\(462\) −1.15495e11 −0.117944
\(463\) 1.16394e12 1.17710 0.588552 0.808459i \(-0.299698\pi\)
0.588552 + 0.808459i \(0.299698\pi\)
\(464\) 2.13328e11 0.213657
\(465\) −7.88195e11 −0.781799
\(466\) −3.08621e12 −3.03172
\(467\) −2.92923e11 −0.284988 −0.142494 0.989796i \(-0.545512\pi\)
−0.142494 + 0.989796i \(0.545512\pi\)
\(468\) −4.87963e11 −0.470198
\(469\) 2.33147e11 0.222511
\(470\) 3.73168e12 3.52747
\(471\) −2.70090e11 −0.252880
\(472\) 5.40625e11 0.501369
\(473\) 1.31919e12 1.21180
\(474\) −3.19407e11 −0.290631
\(475\) −4.32257e11 −0.389602
\(476\) −4.13142e11 −0.368865
\(477\) 2.06310e12 1.82469
\(478\) −1.71508e12 −1.50265
\(479\) 9.83590e11 0.853698 0.426849 0.904323i \(-0.359624\pi\)
0.426849 + 0.904323i \(0.359624\pi\)
\(480\) −2.92844e11 −0.251797
\(481\) −1.71886e11 −0.146416
\(482\) 2.94274e12 2.48336
\(483\) 4.75154e10 0.0397258
\(484\) −8.02081e11 −0.664376
\(485\) −1.85371e12 −1.52126
\(486\) −1.34526e12 −1.09382
\(487\) −1.98477e12 −1.59893 −0.799464 0.600714i \(-0.794883\pi\)
−0.799464 + 0.600714i \(0.794883\pi\)
\(488\) 6.21362e11 0.495970
\(489\) −8.93270e10 −0.0706469
\(490\) 5.29449e11 0.414899
\(491\) −1.34044e12 −1.04083 −0.520416 0.853913i \(-0.674223\pi\)
−0.520416 + 0.853913i \(0.674223\pi\)
\(492\) −8.89427e10 −0.0684333
\(493\) −3.57723e11 −0.272731
\(494\) 1.18468e11 0.0895014
\(495\) 1.73974e12 1.30245
\(496\) 1.09862e12 0.815041
\(497\) −7.62698e11 −0.560724
\(498\) 8.27005e11 0.602527
\(499\) 8.16247e11 0.589344 0.294672 0.955598i \(-0.404790\pi\)
0.294672 + 0.955598i \(0.404790\pi\)
\(500\) −4.43689e12 −3.17478
\(501\) 3.31014e11 0.234734
\(502\) −7.57536e11 −0.532398
\(503\) 2.58243e12 1.79876 0.899380 0.437168i \(-0.144019\pi\)
0.899380 + 0.437168i \(0.144019\pi\)
\(504\) 6.87046e11 0.474295
\(505\) 4.40623e12 3.01478
\(506\) −8.74167e11 −0.592812
\(507\) 2.69188e10 0.0180934
\(508\) −4.10390e12 −2.73407
\(509\) −1.50530e12 −0.994018 −0.497009 0.867745i \(-0.665568\pi\)
−0.497009 + 0.867745i \(0.665568\pi\)
\(510\) −5.67566e11 −0.371493
\(511\) 4.65598e9 0.00302077
\(512\) 1.28813e12 0.828408
\(513\) 1.38510e11 0.0882982
\(514\) 3.28267e12 2.07441
\(515\) 3.75330e12 2.35115
\(516\) 1.03798e12 0.644564
\(517\) 1.56578e12 0.963880
\(518\) 5.46582e11 0.333558
\(519\) 4.63873e11 0.280637
\(520\) 1.06718e12 0.640063
\(521\) −1.92587e12 −1.14513 −0.572567 0.819858i \(-0.694053\pi\)
−0.572567 + 0.819858i \(0.694053\pi\)
\(522\) 1.34353e12 0.792011
\(523\) −3.01269e12 −1.76075 −0.880374 0.474280i \(-0.842708\pi\)
−0.880374 + 0.474280i \(0.842708\pi\)
\(524\) −1.73132e12 −1.00320
\(525\) 3.12329e11 0.179430
\(526\) −1.12987e12 −0.643566
\(527\) −1.84224e12 −1.04040
\(528\) 1.42018e11 0.0795227
\(529\) −1.44151e12 −0.800329
\(530\) −1.01903e13 −5.60979
\(531\) 6.53205e11 0.356553
\(532\) −2.41916e11 −0.130937
\(533\) −8.37787e10 −0.0449636
\(534\) 1.87506e10 0.00997883
\(535\) 3.02174e12 1.59465
\(536\) −1.49437e12 −0.782017
\(537\) −2.95877e11 −0.153542
\(538\) −2.61074e12 −1.34352
\(539\) 2.22152e11 0.113371
\(540\) 2.81794e12 1.42613
\(541\) 6.80680e10 0.0341630 0.0170815 0.999854i \(-0.494563\pi\)
0.0170815 + 0.999854i \(0.494563\pi\)
\(542\) 2.06491e12 1.02779
\(543\) 4.40488e11 0.217437
\(544\) −6.84463e11 −0.335085
\(545\) −5.14984e12 −2.50040
\(546\) −8.55994e10 −0.0412196
\(547\) 7.82289e11 0.373615 0.186807 0.982397i \(-0.440186\pi\)
0.186807 + 0.982397i \(0.440186\pi\)
\(548\) 1.09591e12 0.519114
\(549\) 7.50754e11 0.352714
\(550\) −5.74608e12 −2.67756
\(551\) −2.09465e11 −0.0968121
\(552\) −3.04553e11 −0.139616
\(553\) 6.14372e11 0.279363
\(554\) 1.48894e12 0.671559
\(555\) 4.82194e11 0.215726
\(556\) −5.43947e12 −2.41391
\(557\) −2.05884e12 −0.906305 −0.453152 0.891433i \(-0.649701\pi\)
−0.453152 + 0.891433i \(0.649701\pi\)
\(558\) 6.91908e12 3.02131
\(559\) 9.77717e11 0.423506
\(560\) −6.51035e11 −0.279742
\(561\) −2.38146e11 −0.101510
\(562\) 5.36545e12 2.26878
\(563\) 1.66481e11 0.0698356 0.0349178 0.999390i \(-0.488883\pi\)
0.0349178 + 0.999390i \(0.488883\pi\)
\(564\) 1.23201e12 0.512694
\(565\) 4.94936e11 0.204329
\(566\) −6.74996e10 −0.0276457
\(567\) 7.78652e11 0.316388
\(568\) 4.88856e12 1.97067
\(569\) 2.32174e12 0.928557 0.464279 0.885689i \(-0.346314\pi\)
0.464279 + 0.885689i \(0.346314\pi\)
\(570\) −3.32339e11 −0.131870
\(571\) 4.50517e12 1.77357 0.886786 0.462180i \(-0.152933\pi\)
0.886786 + 0.462180i \(0.152933\pi\)
\(572\) 1.01130e12 0.395001
\(573\) −3.87931e11 −0.150335
\(574\) 2.66409e11 0.102434
\(575\) 2.36397e12 0.901856
\(576\) 3.63389e12 1.37553
\(577\) 2.63619e12 0.990115 0.495058 0.868860i \(-0.335147\pi\)
0.495058 + 0.868860i \(0.335147\pi\)
\(578\) 3.15919e12 1.17734
\(579\) 6.83284e11 0.252667
\(580\) −4.26152e12 −1.56365
\(581\) −1.59073e12 −0.579165
\(582\) −9.53020e11 −0.344309
\(583\) −4.27577e12 −1.53287
\(584\) −2.98428e10 −0.0106165
\(585\) 1.28941e12 0.455187
\(586\) −6.52334e12 −2.28523
\(587\) 9.93841e11 0.345498 0.172749 0.984966i \(-0.444735\pi\)
0.172749 + 0.984966i \(0.444735\pi\)
\(588\) 1.74797e11 0.0603026
\(589\) −1.07873e12 −0.369312
\(590\) −3.22638e12 −1.09618
\(591\) −2.41597e11 −0.0814605
\(592\) −6.72102e11 −0.224899
\(593\) 3.68590e12 1.22405 0.612023 0.790840i \(-0.290356\pi\)
0.612023 + 0.790840i \(0.290356\pi\)
\(594\) 1.84124e12 0.606835
\(595\) 1.09170e12 0.357089
\(596\) 5.93493e12 1.92667
\(597\) −7.51282e11 −0.242057
\(598\) −6.47891e11 −0.207179
\(599\) −4.70229e12 −1.49241 −0.746206 0.665716i \(-0.768126\pi\)
−0.746206 + 0.665716i \(0.768126\pi\)
\(600\) −2.00189e12 −0.630608
\(601\) 4.24671e11 0.132775 0.0663877 0.997794i \(-0.478853\pi\)
0.0663877 + 0.997794i \(0.478853\pi\)
\(602\) −3.10905e12 −0.964813
\(603\) −1.80556e12 −0.556139
\(604\) 2.71096e12 0.828814
\(605\) 2.11945e12 0.643166
\(606\) 2.26531e12 0.682341
\(607\) −3.16983e12 −0.947735 −0.473868 0.880596i \(-0.657142\pi\)
−0.473868 + 0.880596i \(0.657142\pi\)
\(608\) −4.00788e11 −0.118946
\(609\) 1.51350e11 0.0445865
\(610\) −3.70821e12 −1.08438
\(611\) 1.16048e12 0.336862
\(612\) 3.19949e12 0.921932
\(613\) 4.79240e12 1.37082 0.685410 0.728157i \(-0.259623\pi\)
0.685410 + 0.728157i \(0.259623\pi\)
\(614\) 6.88970e12 1.95633
\(615\) 2.35025e11 0.0662485
\(616\) −1.42390e12 −0.398442
\(617\) 2.79264e12 0.775767 0.387884 0.921708i \(-0.373206\pi\)
0.387884 + 0.921708i \(0.373206\pi\)
\(618\) 1.92963e12 0.532141
\(619\) −6.02914e12 −1.65062 −0.825310 0.564680i \(-0.809000\pi\)
−0.825310 + 0.564680i \(0.809000\pi\)
\(620\) −2.19464e13 −5.96488
\(621\) −7.57496e11 −0.204394
\(622\) 7.09159e12 1.89971
\(623\) −3.60663e10 −0.00959193
\(624\) 1.05257e11 0.0277920
\(625\) 4.02509e12 1.05515
\(626\) −1.10212e12 −0.286842
\(627\) −1.39447e11 −0.0360333
\(628\) −7.52036e12 −1.92939
\(629\) 1.12703e12 0.287082
\(630\) −4.10020e12 −1.03699
\(631\) −7.23665e12 −1.81721 −0.908606 0.417654i \(-0.862853\pi\)
−0.908606 + 0.417654i \(0.862853\pi\)
\(632\) −3.93785e12 −0.981822
\(633\) 5.39997e11 0.133682
\(634\) −2.90534e12 −0.714161
\(635\) 1.08443e13 2.64679
\(636\) −3.36433e12 −0.815344
\(637\) 1.64648e11 0.0396214
\(638\) −2.78446e12 −0.665347
\(639\) 5.90655e12 1.40146
\(640\) −1.34054e13 −3.15841
\(641\) 5.22035e11 0.122135 0.0610673 0.998134i \(-0.480550\pi\)
0.0610673 + 0.998134i \(0.480550\pi\)
\(642\) 1.55353e12 0.360920
\(643\) −7.19567e12 −1.66005 −0.830026 0.557724i \(-0.811675\pi\)
−0.830026 + 0.557724i \(0.811675\pi\)
\(644\) 1.32302e12 0.303095
\(645\) −2.74280e12 −0.623986
\(646\) −7.76774e11 −0.175488
\(647\) 6.24024e12 1.40001 0.700007 0.714136i \(-0.253180\pi\)
0.700007 + 0.714136i \(0.253180\pi\)
\(648\) −4.99082e12 −1.11195
\(649\) −1.35376e12 −0.299531
\(650\) −4.25872e12 −0.935770
\(651\) 7.79438e11 0.170085
\(652\) −2.48722e12 −0.539013
\(653\) 7.84804e12 1.68909 0.844543 0.535488i \(-0.179872\pi\)
0.844543 + 0.535488i \(0.179872\pi\)
\(654\) −2.64761e12 −0.565920
\(655\) 4.57489e12 0.971169
\(656\) −3.27588e11 −0.0690654
\(657\) −3.60572e10 −0.00755002
\(658\) −3.69022e12 −0.767424
\(659\) −1.05598e12 −0.218108 −0.109054 0.994036i \(-0.534782\pi\)
−0.109054 + 0.994036i \(0.534782\pi\)
\(660\) −2.83701e12 −0.581986
\(661\) 5.88556e12 1.19917 0.599586 0.800311i \(-0.295332\pi\)
0.599586 + 0.800311i \(0.295332\pi\)
\(662\) 1.63693e13 3.31260
\(663\) −1.76502e11 −0.0354764
\(664\) 1.01958e13 2.03548
\(665\) 6.39247e11 0.126757
\(666\) −4.23289e12 −0.833686
\(667\) 1.14555e12 0.224102
\(668\) 9.21673e12 1.79095
\(669\) 4.48060e11 0.0864807
\(670\) 8.91821e12 1.70978
\(671\) −1.55593e12 −0.296305
\(672\) 2.89591e11 0.0547801
\(673\) 7.23246e12 1.35900 0.679498 0.733677i \(-0.262198\pi\)
0.679498 + 0.733677i \(0.262198\pi\)
\(674\) −7.92644e11 −0.147948
\(675\) −4.97918e12 −0.923189
\(676\) 7.49527e11 0.138047
\(677\) −9.20487e12 −1.68410 −0.842051 0.539398i \(-0.818652\pi\)
−0.842051 + 0.539398i \(0.818652\pi\)
\(678\) 2.54454e11 0.0462462
\(679\) 1.83311e12 0.330959
\(680\) −6.99731e12 −1.25499
\(681\) 2.48181e11 0.0442187
\(682\) −1.43397e13 −2.53812
\(683\) 5.85392e12 1.02933 0.514664 0.857392i \(-0.327917\pi\)
0.514664 + 0.857392i \(0.327917\pi\)
\(684\) 1.87347e12 0.327260
\(685\) −2.89587e12 −0.502541
\(686\) −5.23567e11 −0.0902638
\(687\) −1.59747e12 −0.273608
\(688\) 3.82303e12 0.650518
\(689\) −3.16899e12 −0.535716
\(690\) 1.81753e12 0.305254
\(691\) −5.99081e12 −0.999619 −0.499810 0.866135i \(-0.666597\pi\)
−0.499810 + 0.866135i \(0.666597\pi\)
\(692\) 1.29160e13 2.14117
\(693\) −1.72041e12 −0.283356
\(694\) −1.57489e13 −2.57711
\(695\) 1.43734e13 2.33684
\(696\) −9.70084e11 −0.156700
\(697\) 5.49322e11 0.0881616
\(698\) −1.48910e13 −2.37451
\(699\) 2.69240e12 0.426572
\(700\) 8.69646e12 1.36899
\(701\) −8.32407e12 −1.30198 −0.650990 0.759086i \(-0.725646\pi\)
−0.650990 + 0.759086i \(0.725646\pi\)
\(702\) 1.36464e12 0.212080
\(703\) 6.59933e11 0.101906
\(704\) −7.53122e12 −1.15555
\(705\) −3.25550e12 −0.496326
\(706\) 2.07126e13 3.13771
\(707\) −4.35728e12 −0.655885
\(708\) −1.06519e12 −0.159322
\(709\) −1.01539e13 −1.50913 −0.754563 0.656227i \(-0.772151\pi\)
−0.754563 + 0.656227i \(0.772151\pi\)
\(710\) −2.91743e13 −4.30862
\(711\) −4.75787e12 −0.698231
\(712\) 2.31169e11 0.0337109
\(713\) 5.89946e12 0.854888
\(714\) 5.61260e11 0.0808206
\(715\) −2.67229e12 −0.382390
\(716\) −8.23837e12 −1.17147
\(717\) 1.49623e12 0.211427
\(718\) 1.75030e13 2.45783
\(719\) 1.65891e12 0.231495 0.115748 0.993279i \(-0.463074\pi\)
0.115748 + 0.993279i \(0.463074\pi\)
\(720\) 5.04180e12 0.699180
\(721\) −3.71160e12 −0.511508
\(722\) 1.17513e13 1.60942
\(723\) −2.56723e12 −0.349416
\(724\) 1.22649e13 1.65898
\(725\) 7.52991e12 1.01221
\(726\) 1.08964e12 0.145569
\(727\) −3.35563e12 −0.445522 −0.222761 0.974873i \(-0.571507\pi\)
−0.222761 + 0.974873i \(0.571507\pi\)
\(728\) −1.05532e12 −0.139250
\(729\) −5.20966e12 −0.683180
\(730\) 1.78098e11 0.0232117
\(731\) −6.41071e12 −0.830382
\(732\) −1.22426e12 −0.157607
\(733\) 3.50033e12 0.447858 0.223929 0.974605i \(-0.428112\pi\)
0.223929 + 0.974605i \(0.428112\pi\)
\(734\) 4.65779e11 0.0592308
\(735\) −4.61890e11 −0.0583775
\(736\) 2.19187e12 0.275338
\(737\) 3.74200e12 0.467197
\(738\) −2.06314e12 −0.256021
\(739\) −2.83109e12 −0.349183 −0.174592 0.984641i \(-0.555861\pi\)
−0.174592 + 0.984641i \(0.555861\pi\)
\(740\) 1.34262e13 1.64592
\(741\) −1.03351e11 −0.0125931
\(742\) 1.00771e13 1.22045
\(743\) −1.13531e13 −1.36667 −0.683335 0.730105i \(-0.739471\pi\)
−0.683335 + 0.730105i \(0.739471\pi\)
\(744\) −4.99585e12 −0.597766
\(745\) −1.56827e13 −1.86516
\(746\) 1.84718e13 2.18366
\(747\) 1.23190e13 1.44755
\(748\) −6.63091e12 −0.774491
\(749\) −2.98817e12 −0.346926
\(750\) 6.02759e12 0.695614
\(751\) 4.74099e11 0.0543863 0.0271931 0.999630i \(-0.491343\pi\)
0.0271931 + 0.999630i \(0.491343\pi\)
\(752\) 4.53766e12 0.517430
\(753\) 6.60872e11 0.0749100
\(754\) −2.06371e12 −0.232529
\(755\) −7.16353e12 −0.802354
\(756\) −2.78664e12 −0.310265
\(757\) −9.72131e12 −1.07595 −0.537976 0.842960i \(-0.680811\pi\)
−0.537976 + 0.842960i \(0.680811\pi\)
\(758\) −1.74845e13 −1.92372
\(759\) 7.62621e11 0.0834105
\(760\) −4.09729e12 −0.445487
\(761\) 1.21142e13 1.30937 0.654685 0.755902i \(-0.272801\pi\)
0.654685 + 0.755902i \(0.272801\pi\)
\(762\) 5.57522e12 0.599052
\(763\) 5.09262e12 0.543978
\(764\) −1.08015e13 −1.14701
\(765\) −8.45443e12 −0.892499
\(766\) 8.85232e12 0.929026
\(767\) −1.00334e12 −0.104682
\(768\) −3.58990e12 −0.372355
\(769\) 3.73032e12 0.384660 0.192330 0.981330i \(-0.438396\pi\)
0.192330 + 0.981330i \(0.438396\pi\)
\(770\) 8.49764e12 0.871144
\(771\) −2.86379e12 −0.291875
\(772\) 1.90253e13 1.92777
\(773\) −2.49728e12 −0.251570 −0.125785 0.992058i \(-0.540145\pi\)
−0.125785 + 0.992058i \(0.540145\pi\)
\(774\) 2.40773e13 2.41143
\(775\) 3.87784e13 3.86129
\(776\) −1.17494e13 −1.16316
\(777\) −4.76836e11 −0.0469326
\(778\) −8.82804e12 −0.863885
\(779\) 3.21657e11 0.0312949
\(780\) −2.10265e12 −0.203396
\(781\) −1.22413e13 −1.17733
\(782\) 4.24810e12 0.406223
\(783\) −2.41283e12 −0.229403
\(784\) 6.43801e11 0.0608597
\(785\) 1.98720e13 1.86779
\(786\) 2.35202e12 0.219806
\(787\) −4.16341e11 −0.0386868 −0.0193434 0.999813i \(-0.506158\pi\)
−0.0193434 + 0.999813i \(0.506158\pi\)
\(788\) −6.72700e12 −0.621517
\(789\) 9.85695e11 0.0905516
\(790\) 2.35006e13 2.14663
\(791\) −4.89437e11 −0.0444531
\(792\) 1.10271e13 0.995856
\(793\) −1.15318e12 −0.103554
\(794\) 7.80664e12 0.697063
\(795\) 8.89000e12 0.789314
\(796\) −2.09186e13 −1.84682
\(797\) −1.41842e13 −1.24521 −0.622607 0.782535i \(-0.713926\pi\)
−0.622607 + 0.782535i \(0.713926\pi\)
\(798\) 3.28647e11 0.0286891
\(799\) −7.60906e12 −0.660496
\(800\) 1.44076e13 1.24362
\(801\) 2.79308e11 0.0239738
\(802\) 1.73976e13 1.48493
\(803\) 7.47284e10 0.00634257
\(804\) 2.94434e12 0.248505
\(805\) −3.49598e12 −0.293419
\(806\) −1.06279e13 −0.887035
\(807\) 2.27760e12 0.189037
\(808\) 2.79282e13 2.30511
\(809\) 7.88321e12 0.647045 0.323523 0.946220i \(-0.395133\pi\)
0.323523 + 0.946220i \(0.395133\pi\)
\(810\) 2.97846e13 2.43113
\(811\) −6.02618e12 −0.489157 −0.244579 0.969629i \(-0.578650\pi\)
−0.244579 + 0.969629i \(0.578650\pi\)
\(812\) 4.21417e12 0.340181
\(813\) −1.80142e12 −0.144613
\(814\) 8.77262e12 0.700357
\(815\) 6.57230e12 0.521805
\(816\) −6.90151e11 −0.0544927
\(817\) −3.75380e12 −0.294763
\(818\) 2.12932e13 1.66284
\(819\) −1.27508e12 −0.0990288
\(820\) 6.54402e12 0.505455
\(821\) −4.85583e12 −0.373009 −0.186504 0.982454i \(-0.559716\pi\)
−0.186504 + 0.982454i \(0.559716\pi\)
\(822\) −1.48881e12 −0.113741
\(823\) −1.33705e13 −1.01589 −0.507946 0.861389i \(-0.669595\pi\)
−0.507946 + 0.861389i \(0.669595\pi\)
\(824\) 2.37897e13 1.79770
\(825\) 5.01286e12 0.376741
\(826\) 3.19054e12 0.238481
\(827\) 2.10835e12 0.156735 0.0783677 0.996925i \(-0.475029\pi\)
0.0783677 + 0.996925i \(0.475029\pi\)
\(828\) −1.02458e13 −0.757547
\(829\) 1.09881e13 0.808032 0.404016 0.914752i \(-0.367614\pi\)
0.404016 + 0.914752i \(0.367614\pi\)
\(830\) −6.08475e13 −4.45032
\(831\) −1.29895e12 −0.0944904
\(832\) −5.58177e12 −0.403847
\(833\) −1.07957e12 −0.0776870
\(834\) 7.38962e12 0.528901
\(835\) −2.43546e13 −1.73377
\(836\) −3.88274e12 −0.274923
\(837\) −1.24259e13 −0.875110
\(838\) −7.08038e12 −0.495974
\(839\) −1.80392e13 −1.25686 −0.628431 0.777866i \(-0.716302\pi\)
−0.628431 + 0.777866i \(0.716302\pi\)
\(840\) 2.96051e12 0.205168
\(841\) −1.08583e13 −0.748477
\(842\) 1.00747e13 0.690764
\(843\) −4.68080e12 −0.319224
\(844\) 1.50356e13 1.01995
\(845\) −1.98057e12 −0.133640
\(846\) 2.85781e13 1.91808
\(847\) −2.09590e12 −0.139925
\(848\) −1.23913e13 −0.822876
\(849\) 5.88864e10 0.00388983
\(850\) 2.79237e13 1.83479
\(851\) −3.60911e12 −0.235894
\(852\) −9.63187e12 −0.626228
\(853\) −1.67337e13 −1.08223 −0.541116 0.840948i \(-0.681998\pi\)
−0.541116 + 0.840948i \(0.681998\pi\)
\(854\) 3.66701e12 0.235913
\(855\) −4.95051e12 −0.316812
\(856\) 1.91529e13 1.21928
\(857\) −1.39355e13 −0.882487 −0.441243 0.897387i \(-0.645462\pi\)
−0.441243 + 0.897387i \(0.645462\pi\)
\(858\) −1.37387e12 −0.0865470
\(859\) −7.59876e12 −0.476183 −0.238091 0.971243i \(-0.576522\pi\)
−0.238091 + 0.971243i \(0.576522\pi\)
\(860\) −7.63702e13 −4.76081
\(861\) −2.32414e11 −0.0144128
\(862\) −6.26850e11 −0.0386706
\(863\) −2.41467e12 −0.148187 −0.0740934 0.997251i \(-0.523606\pi\)
−0.0740934 + 0.997251i \(0.523606\pi\)
\(864\) −4.61669e12 −0.281851
\(865\) −3.41298e13 −2.07282
\(866\) −4.66429e13 −2.81809
\(867\) −2.75607e12 −0.165655
\(868\) 2.17026e13 1.29770
\(869\) 9.86065e12 0.586565
\(870\) 5.78934e12 0.342604
\(871\) 2.77339e12 0.163279
\(872\) −3.26415e13 −1.91181
\(873\) −1.41961e13 −0.827191
\(874\) 2.48748e12 0.144198
\(875\) −1.15939e13 −0.668643
\(876\) 5.87989e10 0.00337366
\(877\) −1.98971e13 −1.13578 −0.567888 0.823106i \(-0.692239\pi\)
−0.567888 + 0.823106i \(0.692239\pi\)
\(878\) −3.79741e12 −0.215656
\(879\) 5.69094e12 0.321539
\(880\) −1.04491e13 −0.587362
\(881\) −9.20036e12 −0.514533 −0.257266 0.966340i \(-0.582822\pi\)
−0.257266 + 0.966340i \(0.582822\pi\)
\(882\) 4.05465e12 0.225603
\(883\) −2.39231e13 −1.32432 −0.662162 0.749361i \(-0.730361\pi\)
−0.662162 + 0.749361i \(0.730361\pi\)
\(884\) −4.91451e12 −0.270673
\(885\) 2.81469e12 0.154236
\(886\) −2.04221e13 −1.11339
\(887\) −1.94162e13 −1.05320 −0.526598 0.850115i \(-0.676533\pi\)
−0.526598 + 0.850115i \(0.676533\pi\)
\(888\) 3.05631e12 0.164945
\(889\) −1.07238e13 −0.575826
\(890\) −1.37959e12 −0.0737046
\(891\) 1.24973e13 0.664306
\(892\) 1.24758e13 0.659820
\(893\) −4.45550e12 −0.234458
\(894\) −8.06271e12 −0.422145
\(895\) 2.17693e13 1.13407
\(896\) 1.32564e13 0.687133
\(897\) 5.65218e11 0.0291507
\(898\) 1.44961e13 0.743886
\(899\) 1.87914e13 0.959490
\(900\) −6.73478e13 −3.42162
\(901\) 2.07785e13 1.05040
\(902\) 4.27585e12 0.215076
\(903\) 2.71232e12 0.135752
\(904\) 3.13707e12 0.156231
\(905\) −3.24092e13 −1.60602
\(906\) −3.68289e12 −0.181598
\(907\) 2.75418e13 1.35133 0.675663 0.737211i \(-0.263857\pi\)
0.675663 + 0.737211i \(0.263857\pi\)
\(908\) 6.91033e12 0.337375
\(909\) 3.37440e13 1.63930
\(910\) 6.29804e12 0.304452
\(911\) 2.26747e13 1.09071 0.545354 0.838206i \(-0.316395\pi\)
0.545354 + 0.838206i \(0.316395\pi\)
\(912\) −4.04119e11 −0.0193434
\(913\) −2.55311e13 −1.21605
\(914\) −3.27091e13 −1.55028
\(915\) 3.23503e12 0.152575
\(916\) −4.44800e13 −2.08754
\(917\) −4.52406e12 −0.211284
\(918\) −8.94768e12 −0.415832
\(919\) 1.07689e13 0.498024 0.249012 0.968500i \(-0.419894\pi\)
0.249012 + 0.968500i \(0.419894\pi\)
\(920\) 2.24077e13 1.03122
\(921\) −6.01055e12 −0.275262
\(922\) 6.39784e13 2.91571
\(923\) −9.07265e12 −0.411459
\(924\) 2.80549e12 0.126615
\(925\) −2.37234e13 −1.06547
\(926\) −4.40276e13 −1.96778
\(927\) 2.87437e13 1.27845
\(928\) 6.98172e12 0.309027
\(929\) 3.44561e13 1.51773 0.758867 0.651246i \(-0.225753\pi\)
0.758867 + 0.651246i \(0.225753\pi\)
\(930\) 2.98146e13 1.30694
\(931\) −6.32145e11 −0.0275767
\(932\) 7.49670e13 3.25461
\(933\) −6.18668e12 −0.267295
\(934\) 1.10802e13 0.476417
\(935\) 1.75217e13 0.749765
\(936\) 8.17273e12 0.348037
\(937\) 1.12966e13 0.478764 0.239382 0.970925i \(-0.423055\pi\)
0.239382 + 0.970925i \(0.423055\pi\)
\(938\) −8.81913e12 −0.371974
\(939\) 9.61482e11 0.0403595
\(940\) −9.06460e13 −3.78681
\(941\) −4.40804e13 −1.83270 −0.916352 0.400373i \(-0.868881\pi\)
−0.916352 + 0.400373i \(0.868881\pi\)
\(942\) 1.02165e13 0.422741
\(943\) −1.75911e12 −0.0724420
\(944\) −3.92323e12 −0.160794
\(945\) 7.36350e12 0.300359
\(946\) −4.99001e13 −2.02578
\(947\) 2.80960e13 1.13519 0.567597 0.823307i \(-0.307873\pi\)
0.567597 + 0.823307i \(0.307873\pi\)
\(948\) 7.75870e12 0.311998
\(949\) 5.53851e10 0.00221664
\(950\) 1.63508e13 0.651301
\(951\) 2.53461e12 0.100485
\(952\) 6.91957e12 0.273032
\(953\) 1.22439e13 0.480841 0.240420 0.970669i \(-0.422715\pi\)
0.240420 + 0.970669i \(0.422715\pi\)
\(954\) −7.80399e13 −3.05035
\(955\) 2.85423e13 1.11039
\(956\) 4.16609e13 1.61312
\(957\) 2.42916e12 0.0936164
\(958\) −3.72057e13 −1.42714
\(959\) 2.86370e12 0.109331
\(960\) 1.56586e13 0.595021
\(961\) 7.03345e13 2.66019
\(962\) 6.50184e12 0.244765
\(963\) 2.31413e13 0.867099
\(964\) −7.14819e13 −2.66594
\(965\) −5.02732e13 −1.86622
\(966\) −1.79734e12 −0.0664099
\(967\) 3.18431e12 0.117111 0.0585553 0.998284i \(-0.481351\pi\)
0.0585553 + 0.998284i \(0.481351\pi\)
\(968\) 1.34338e13 0.491767
\(969\) 6.77655e11 0.0246917
\(970\) 7.01191e13 2.54310
\(971\) 1.16608e13 0.420961 0.210480 0.977598i \(-0.432497\pi\)
0.210480 + 0.977598i \(0.432497\pi\)
\(972\) 3.26777e13 1.17423
\(973\) −1.42138e13 −0.508395
\(974\) 7.50766e13 2.67294
\(975\) 3.71529e12 0.131666
\(976\) −4.50912e12 −0.159063
\(977\) 4.88625e13 1.71574 0.857868 0.513870i \(-0.171789\pi\)
0.857868 + 0.513870i \(0.171789\pi\)
\(978\) 3.37893e12 0.118101
\(979\) −5.78863e11 −0.0201397
\(980\) −1.28608e13 −0.445401
\(981\) −3.94387e13 −1.35960
\(982\) 5.07041e13 1.73997
\(983\) −2.12652e13 −0.726404 −0.363202 0.931710i \(-0.618317\pi\)
−0.363202 + 0.931710i \(0.618317\pi\)
\(984\) 1.48967e12 0.0506538
\(985\) 1.77756e13 0.601675
\(986\) 1.35314e13 0.455928
\(987\) 3.21933e12 0.107979
\(988\) −2.87770e12 −0.0960815
\(989\) 2.05292e13 0.682321
\(990\) −6.58081e13 −2.17731
\(991\) 3.63511e13 1.19725 0.598627 0.801028i \(-0.295713\pi\)
0.598627 + 0.801028i \(0.295713\pi\)
\(992\) 3.59553e13 1.17885
\(993\) −1.42805e13 −0.466093
\(994\) 2.88502e13 0.937367
\(995\) 5.52761e13 1.78786
\(996\) −2.00888e13 −0.646824
\(997\) −4.50181e13 −1.44297 −0.721487 0.692428i \(-0.756541\pi\)
−0.721487 + 0.692428i \(0.756541\pi\)
\(998\) −3.08757e13 −0.985212
\(999\) 7.60178e12 0.241474
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 91.10.a.a.1.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.10.a.a.1.2 12 1.1 even 1 trivial