Properties

Label 91.10.a.a.1.6
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 1.6
Root \(-0.857959\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.85796 q^{2} +143.668 q^{3} -503.832 q^{4} -15.3125 q^{5} -410.598 q^{6} +2401.00 q^{7} +2903.21 q^{8} +957.575 q^{9} +O(q^{10})\) \(q-2.85796 q^{2} +143.668 q^{3} -503.832 q^{4} -15.3125 q^{5} -410.598 q^{6} +2401.00 q^{7} +2903.21 q^{8} +957.575 q^{9} +43.7625 q^{10} -2617.92 q^{11} -72384.7 q^{12} +28561.0 q^{13} -6861.96 q^{14} -2199.92 q^{15} +249665. q^{16} +86313.3 q^{17} -2736.71 q^{18} -90449.5 q^{19} +7714.92 q^{20} +344948. q^{21} +7481.92 q^{22} -855540. q^{23} +417099. q^{24} -1.95289e6 q^{25} -81626.2 q^{26} -2.69025e6 q^{27} -1.20970e6 q^{28} +1.60148e6 q^{29} +6287.28 q^{30} -2.47079e6 q^{31} -2.19997e6 q^{32} -376112. q^{33} -246680. q^{34} -36765.3 q^{35} -482457. q^{36} +1.00763e6 q^{37} +258501. q^{38} +4.10331e6 q^{39} -44455.3 q^{40} -3.12371e7 q^{41} -985846. q^{42} +1.84833e7 q^{43} +1.31899e6 q^{44} -14662.9 q^{45} +2.44510e6 q^{46} +2.65637e6 q^{47} +3.58689e7 q^{48} +5.76480e6 q^{49} +5.58128e6 q^{50} +1.24005e7 q^{51} -1.43899e7 q^{52} -6.38851e7 q^{53} +7.68862e6 q^{54} +40086.9 q^{55} +6.97060e6 q^{56} -1.29947e7 q^{57} -4.57696e6 q^{58} -1.84110e7 q^{59} +1.10839e6 q^{60} -9.45649e7 q^{61} +7.06143e6 q^{62} +2.29914e6 q^{63} -1.21541e8 q^{64} -437340. q^{65} +1.07491e6 q^{66} -1.40600e8 q^{67} -4.34874e7 q^{68} -1.22914e8 q^{69} +105074. q^{70} -2.69191e8 q^{71} +2.78004e6 q^{72} -1.26381e8 q^{73} -2.87978e6 q^{74} -2.80568e8 q^{75} +4.55713e7 q^{76} -6.28563e6 q^{77} -1.17271e7 q^{78} -4.38026e8 q^{79} -3.82299e6 q^{80} -4.05351e8 q^{81} +8.92745e7 q^{82} -3.83977e7 q^{83} -1.73796e8 q^{84} -1.32167e6 q^{85} -5.28244e7 q^{86} +2.30082e8 q^{87} -7.60037e6 q^{88} -2.45390e8 q^{89} +41905.8 q^{90} +6.85750e7 q^{91} +4.31048e8 q^{92} -3.54975e8 q^{93} -7.59181e6 q^{94} +1.38501e6 q^{95} -3.16066e8 q^{96} +4.47528e8 q^{97} -1.64756e7 q^{98} -2.50686e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} + O(q^{10}) \) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} - 65812 q^{10} - 80061 q^{11} - 184395 q^{12} + 342732 q^{13} - 50421 q^{14} + 160096 q^{15} + 385497 q^{16} - 1493598 q^{17} + 1520858 q^{18} - 109038 q^{19} - 622260 q^{20} - 775523 q^{21} + 4636975 q^{22} - 3367443 q^{23} - 5963895 q^{24} - 51480 q^{25} - 599781 q^{26} - 8158937 q^{27} + 7070945 q^{28} - 13333098 q^{29} + 2915424 q^{30} - 3954765 q^{31} + 4389297 q^{32} - 5790219 q^{33} + 14879968 q^{34} - 12490002 q^{35} + 80697058 q^{36} + 580535 q^{37} - 19134246 q^{38} - 9225203 q^{39} + 12365024 q^{40} - 27018171 q^{41} - 23762697 q^{42} + 31237588 q^{43} - 125053839 q^{44} - 62765470 q^{45} - 114008121 q^{46} - 21983709 q^{47} - 309724207 q^{48} + 69177612 q^{49} - 131331747 q^{50} - 176522692 q^{51} + 84112145 q^{52} - 196548234 q^{53} - 456152547 q^{54} - 309055872 q^{55} - 61059831 q^{56} - 274411494 q^{57} - 521980612 q^{58} - 215907906 q^{59} - 177006648 q^{60} - 218340705 q^{61} - 673289997 q^{62} + 188524119 q^{63} - 386667247 q^{64} - 148574322 q^{65} - 777397365 q^{66} + 14544775 q^{67} - 1246637448 q^{68} - 65252625 q^{69} - 158014612 q^{70} - 552451776 q^{71} + 369379470 q^{72} - 349395159 q^{73} + 73591023 q^{74} + 329300747 q^{75} - 1036299002 q^{76} - 192226461 q^{77} - 282668217 q^{78} + 962249727 q^{79} - 1494536184 q^{80} + 874458108 q^{81} - 1417698067 q^{82} - 2032575912 q^{83} - 442732395 q^{84} - 411671064 q^{85} - 2139249420 q^{86} - 759642172 q^{87} + 558651957 q^{88} - 280821684 q^{89} - 5764700804 q^{90} + 822899532 q^{91} - 4491569571 q^{92} - 1729557923 q^{93} - 1591372165 q^{94} - 1282463328 q^{95} - 2148993055 q^{96} - 2115165937 q^{97} - 121060821 q^{98} - 3595669198 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.85796 −0.126305 −0.0631526 0.998004i \(-0.520115\pi\)
−0.0631526 + 0.998004i \(0.520115\pi\)
\(3\) 143.668 1.02404 0.512018 0.858975i \(-0.328898\pi\)
0.512018 + 0.858975i \(0.328898\pi\)
\(4\) −503.832 −0.984047
\(5\) −15.3125 −0.0109567 −0.00547836 0.999985i \(-0.501744\pi\)
−0.00547836 + 0.999985i \(0.501744\pi\)
\(6\) −410.598 −0.129341
\(7\) 2401.00 0.377964
\(8\) 2903.21 0.250595
\(9\) 957.575 0.0486498
\(10\) 43.7625 0.00138389
\(11\) −2617.92 −0.0539125 −0.0269563 0.999637i \(-0.508581\pi\)
−0.0269563 + 0.999637i \(0.508581\pi\)
\(12\) −72384.7 −1.00770
\(13\) 28561.0 0.277350
\(14\) −6861.96 −0.0477389
\(15\) −2199.92 −0.0112201
\(16\) 249665. 0.952396
\(17\) 86313.3 0.250644 0.125322 0.992116i \(-0.460004\pi\)
0.125322 + 0.992116i \(0.460004\pi\)
\(18\) −2736.71 −0.00614473
\(19\) −90449.5 −0.159226 −0.0796131 0.996826i \(-0.525368\pi\)
−0.0796131 + 0.996826i \(0.525368\pi\)
\(20\) 7714.92 0.0107819
\(21\) 344948. 0.387049
\(22\) 7481.92 0.00680943
\(23\) −855540. −0.637477 −0.318739 0.947843i \(-0.603259\pi\)
−0.318739 + 0.947843i \(0.603259\pi\)
\(24\) 417099. 0.256619
\(25\) −1.95289e6 −0.999880
\(26\) −81626.2 −0.0350307
\(27\) −2.69025e6 −0.974217
\(28\) −1.20970e6 −0.371935
\(29\) 1.60148e6 0.420466 0.210233 0.977651i \(-0.432578\pi\)
0.210233 + 0.977651i \(0.432578\pi\)
\(30\) 6287.28 0.00141715
\(31\) −2.47079e6 −0.480517 −0.240259 0.970709i \(-0.577232\pi\)
−0.240259 + 0.970709i \(0.577232\pi\)
\(32\) −2.19997e6 −0.370888
\(33\) −376112. −0.0552084
\(34\) −246680. −0.0316576
\(35\) −36765.3 −0.00414125
\(36\) −482457. −0.0478737
\(37\) 1.00763e6 0.0883884 0.0441942 0.999023i \(-0.485928\pi\)
0.0441942 + 0.999023i \(0.485928\pi\)
\(38\) 258501. 0.0201111
\(39\) 4.10331e6 0.284017
\(40\) −44455.3 −0.00274570
\(41\) −3.12371e7 −1.72641 −0.863205 0.504854i \(-0.831546\pi\)
−0.863205 + 0.504854i \(0.831546\pi\)
\(42\) −985846. −0.0488863
\(43\) 1.84833e7 0.824462 0.412231 0.911079i \(-0.364750\pi\)
0.412231 + 0.911079i \(0.364750\pi\)
\(44\) 1.31899e6 0.0530525
\(45\) −14662.9 −0.000533043 0
\(46\) 2.44510e6 0.0805167
\(47\) 2.65637e6 0.0794052 0.0397026 0.999212i \(-0.487359\pi\)
0.0397026 + 0.999212i \(0.487359\pi\)
\(48\) 3.58689e7 0.975287
\(49\) 5.76480e6 0.142857
\(50\) 5.58128e6 0.126290
\(51\) 1.24005e7 0.256669
\(52\) −1.43899e7 −0.272926
\(53\) −6.38851e7 −1.11214 −0.556068 0.831137i \(-0.687691\pi\)
−0.556068 + 0.831137i \(0.687691\pi\)
\(54\) 7.68862e6 0.123049
\(55\) 40086.9 0.000590705 0
\(56\) 6.97060e6 0.0947161
\(57\) −1.29947e7 −0.163053
\(58\) −4.57696e6 −0.0531070
\(59\) −1.84110e7 −0.197808 −0.0989040 0.995097i \(-0.531534\pi\)
−0.0989040 + 0.995097i \(0.531534\pi\)
\(60\) 1.10839e6 0.0110411
\(61\) −9.45649e7 −0.874471 −0.437236 0.899347i \(-0.644042\pi\)
−0.437236 + 0.899347i \(0.644042\pi\)
\(62\) 7.06143e6 0.0606918
\(63\) 2.29914e6 0.0183879
\(64\) −1.21541e8 −0.905550
\(65\) −437340. −0.00303885
\(66\) 1.07491e6 0.00697310
\(67\) −1.40600e8 −0.852408 −0.426204 0.904627i \(-0.640149\pi\)
−0.426204 + 0.904627i \(0.640149\pi\)
\(68\) −4.34874e7 −0.246645
\(69\) −1.22914e8 −0.652800
\(70\) 105074. 0.000523061 0
\(71\) −2.69191e8 −1.25718 −0.628590 0.777737i \(-0.716368\pi\)
−0.628590 + 0.777737i \(0.716368\pi\)
\(72\) 2.78004e6 0.0121914
\(73\) −1.26381e8 −0.520869 −0.260435 0.965492i \(-0.583866\pi\)
−0.260435 + 0.965492i \(0.583866\pi\)
\(74\) −2.87978e6 −0.0111639
\(75\) −2.80568e8 −1.02391
\(76\) 4.55713e7 0.156686
\(77\) −6.28563e6 −0.0203770
\(78\) −1.17271e7 −0.0358727
\(79\) −4.38026e8 −1.26526 −0.632628 0.774456i \(-0.718024\pi\)
−0.632628 + 0.774456i \(0.718024\pi\)
\(80\) −3.82299e6 −0.0104351
\(81\) −4.05351e8 −1.04628
\(82\) 8.92745e7 0.218054
\(83\) −3.83977e7 −0.0888084 −0.0444042 0.999014i \(-0.514139\pi\)
−0.0444042 + 0.999014i \(0.514139\pi\)
\(84\) −1.73796e8 −0.380875
\(85\) −1.32167e6 −0.00274624
\(86\) −5.28244e7 −0.104134
\(87\) 2.30082e8 0.430572
\(88\) −7.60037e6 −0.0135102
\(89\) −2.45390e8 −0.414574 −0.207287 0.978280i \(-0.566463\pi\)
−0.207287 + 0.978280i \(0.566463\pi\)
\(90\) 41905.8 6.73261e−5 0
\(91\) 6.85750e7 0.104828
\(92\) 4.31048e8 0.627308
\(93\) −3.54975e8 −0.492067
\(94\) −7.59181e6 −0.0100293
\(95\) 1.38501e6 0.00174460
\(96\) −3.16066e8 −0.379802
\(97\) 4.47528e8 0.513272 0.256636 0.966508i \(-0.417386\pi\)
0.256636 + 0.966508i \(0.417386\pi\)
\(98\) −1.64756e7 −0.0180436
\(99\) −2.50686e6 −0.00262284
\(100\) 9.83929e8 0.983929
\(101\) 1.39090e9 1.32999 0.664995 0.746848i \(-0.268434\pi\)
0.664995 + 0.746848i \(0.268434\pi\)
\(102\) −3.54401e7 −0.0324186
\(103\) −1.82795e8 −0.160028 −0.0800140 0.996794i \(-0.525497\pi\)
−0.0800140 + 0.996794i \(0.525497\pi\)
\(104\) 8.29185e7 0.0695026
\(105\) −5.28200e6 −0.00424079
\(106\) 1.82581e8 0.140468
\(107\) 9.74520e8 0.718727 0.359363 0.933198i \(-0.382994\pi\)
0.359363 + 0.933198i \(0.382994\pi\)
\(108\) 1.35543e9 0.958675
\(109\) −5.98386e8 −0.406034 −0.203017 0.979175i \(-0.565075\pi\)
−0.203017 + 0.979175i \(0.565075\pi\)
\(110\) −114567. −7.46091e−5 0
\(111\) 1.44765e8 0.0905129
\(112\) 5.99445e8 0.359972
\(113\) −4.67968e8 −0.270000 −0.135000 0.990846i \(-0.543103\pi\)
−0.135000 + 0.990846i \(0.543103\pi\)
\(114\) 3.71384e7 0.0205945
\(115\) 1.31004e7 0.00698466
\(116\) −8.06877e8 −0.413758
\(117\) 2.73493e7 0.0134930
\(118\) 5.26179e7 0.0249842
\(119\) 2.07238e8 0.0947345
\(120\) −6.38682e6 −0.00281170
\(121\) −2.35109e9 −0.997093
\(122\) 2.70263e8 0.110450
\(123\) −4.48779e9 −1.76791
\(124\) 1.24487e9 0.472851
\(125\) 5.98108e7 0.0219121
\(126\) −6.57084e6 −0.00232249
\(127\) −6.49754e8 −0.221632 −0.110816 0.993841i \(-0.535346\pi\)
−0.110816 + 0.993841i \(0.535346\pi\)
\(128\) 1.47375e9 0.485263
\(129\) 2.65546e9 0.844279
\(130\) 1.24990e6 0.000383822 0
\(131\) 3.34637e9 0.992782 0.496391 0.868099i \(-0.334658\pi\)
0.496391 + 0.868099i \(0.334658\pi\)
\(132\) 1.89498e8 0.0543276
\(133\) −2.17169e8 −0.0601819
\(134\) 4.01828e8 0.107664
\(135\) 4.11944e7 0.0106742
\(136\) 2.50585e8 0.0628102
\(137\) −4.28524e9 −1.03928 −0.519640 0.854386i \(-0.673934\pi\)
−0.519640 + 0.854386i \(0.673934\pi\)
\(138\) 3.51283e8 0.0824520
\(139\) 9.12782e8 0.207396 0.103698 0.994609i \(-0.466932\pi\)
0.103698 + 0.994609i \(0.466932\pi\)
\(140\) 1.85235e7 0.00407519
\(141\) 3.81637e8 0.0813138
\(142\) 7.69336e8 0.158788
\(143\) −7.47705e7 −0.0149526
\(144\) 2.39073e8 0.0463339
\(145\) −2.45226e7 −0.00460692
\(146\) 3.61191e8 0.0657884
\(147\) 8.28219e8 0.146291
\(148\) −5.07678e8 −0.0869783
\(149\) 1.02262e10 1.69972 0.849859 0.527010i \(-0.176687\pi\)
0.849859 + 0.527010i \(0.176687\pi\)
\(150\) 8.01853e8 0.129325
\(151\) 2.84718e9 0.445675 0.222838 0.974856i \(-0.428468\pi\)
0.222838 + 0.974856i \(0.428468\pi\)
\(152\) −2.62593e8 −0.0399014
\(153\) 8.26514e7 0.0121938
\(154\) 1.79641e7 0.00257372
\(155\) 3.78340e7 0.00526489
\(156\) −2.06738e9 −0.279486
\(157\) 1.42852e10 1.87645 0.938224 0.346029i \(-0.112470\pi\)
0.938224 + 0.346029i \(0.112470\pi\)
\(158\) 1.25186e9 0.159808
\(159\) −9.17826e9 −1.13887
\(160\) 3.36871e7 0.00406372
\(161\) −2.05415e9 −0.240944
\(162\) 1.15848e9 0.132151
\(163\) 2.93151e9 0.325272 0.162636 0.986686i \(-0.448000\pi\)
0.162636 + 0.986686i \(0.448000\pi\)
\(164\) 1.57383e10 1.69887
\(165\) 5.75922e6 0.000604903 0
\(166\) 1.09739e8 0.0112170
\(167\) 1.14737e10 1.14151 0.570753 0.821121i \(-0.306651\pi\)
0.570753 + 0.821121i \(0.306651\pi\)
\(168\) 1.00145e9 0.0969927
\(169\) 8.15731e8 0.0769231
\(170\) 3.77728e6 0.000346864 0
\(171\) −8.66121e7 −0.00774633
\(172\) −9.31246e9 −0.811310
\(173\) 1.26545e10 1.07408 0.537042 0.843555i \(-0.319542\pi\)
0.537042 + 0.843555i \(0.319542\pi\)
\(174\) −6.57564e8 −0.0543834
\(175\) −4.68889e9 −0.377919
\(176\) −6.53603e8 −0.0513461
\(177\) −2.64508e9 −0.202563
\(178\) 7.01315e8 0.0523628
\(179\) −1.03113e10 −0.750715 −0.375358 0.926880i \(-0.622480\pi\)
−0.375358 + 0.926880i \(0.622480\pi\)
\(180\) 7.38762e6 0.000524539 0
\(181\) −8.02667e9 −0.555881 −0.277940 0.960598i \(-0.589652\pi\)
−0.277940 + 0.960598i \(0.589652\pi\)
\(182\) −1.95984e8 −0.0132404
\(183\) −1.35860e10 −0.895490
\(184\) −2.48381e9 −0.159749
\(185\) −1.54294e7 −0.000968447 0
\(186\) 1.01450e9 0.0621506
\(187\) −2.25961e8 −0.0135129
\(188\) −1.33837e9 −0.0781385
\(189\) −6.45929e9 −0.368219
\(190\) −3.95829e6 −0.000220352 0
\(191\) 3.21002e9 0.174525 0.0872626 0.996185i \(-0.472188\pi\)
0.0872626 + 0.996185i \(0.472188\pi\)
\(192\) −1.74616e10 −0.927316
\(193\) −4.45146e9 −0.230938 −0.115469 0.993311i \(-0.536837\pi\)
−0.115469 + 0.993311i \(0.536837\pi\)
\(194\) −1.27902e9 −0.0648289
\(195\) −6.28319e7 −0.00311189
\(196\) −2.90449e9 −0.140578
\(197\) −1.25795e10 −0.595064 −0.297532 0.954712i \(-0.596164\pi\)
−0.297532 + 0.954712i \(0.596164\pi\)
\(198\) 7.16450e6 0.000331278 0
\(199\) 1.73373e10 0.783685 0.391843 0.920032i \(-0.371838\pi\)
0.391843 + 0.920032i \(0.371838\pi\)
\(200\) −5.66964e9 −0.250565
\(201\) −2.01997e10 −0.872897
\(202\) −3.97512e9 −0.167985
\(203\) 3.84515e9 0.158921
\(204\) −6.24776e9 −0.252574
\(205\) 4.78318e8 0.0189158
\(206\) 5.22420e8 0.0202124
\(207\) −8.19243e8 −0.0310132
\(208\) 7.13068e9 0.264147
\(209\) 2.36790e8 0.00858429
\(210\) 1.50958e7 0.000535634 0
\(211\) −3.20449e10 −1.11298 −0.556490 0.830854i \(-0.687852\pi\)
−0.556490 + 0.830854i \(0.687852\pi\)
\(212\) 3.21873e10 1.09439
\(213\) −3.86742e10 −1.28740
\(214\) −2.78514e9 −0.0907789
\(215\) −2.83025e8 −0.00903341
\(216\) −7.81035e9 −0.244134
\(217\) −5.93238e9 −0.181618
\(218\) 1.71016e9 0.0512842
\(219\) −1.81569e10 −0.533389
\(220\) −2.01971e7 −0.000581281 0
\(221\) 2.46519e9 0.0695161
\(222\) −4.13733e8 −0.0114322
\(223\) −4.67025e10 −1.26464 −0.632322 0.774705i \(-0.717898\pi\)
−0.632322 + 0.774705i \(0.717898\pi\)
\(224\) −5.28214e9 −0.140182
\(225\) −1.87004e9 −0.0486440
\(226\) 1.33743e9 0.0341023
\(227\) −3.63837e10 −0.909475 −0.454737 0.890626i \(-0.650267\pi\)
−0.454737 + 0.890626i \(0.650267\pi\)
\(228\) 6.54716e9 0.160452
\(229\) 4.58774e10 1.10240 0.551200 0.834373i \(-0.314170\pi\)
0.551200 + 0.834373i \(0.314170\pi\)
\(230\) −3.74405e7 −0.000882199 0
\(231\) −9.03046e8 −0.0208668
\(232\) 4.64943e9 0.105367
\(233\) 4.92733e10 1.09524 0.547621 0.836727i \(-0.315534\pi\)
0.547621 + 0.836727i \(0.315534\pi\)
\(234\) −7.81632e7 −0.00170424
\(235\) −4.06757e7 −0.000870021 0
\(236\) 9.27606e9 0.194652
\(237\) −6.29305e10 −1.29567
\(238\) −5.92278e8 −0.0119655
\(239\) −2.13419e10 −0.423100 −0.211550 0.977367i \(-0.567851\pi\)
−0.211550 + 0.977367i \(0.567851\pi\)
\(240\) −5.49242e8 −0.0106860
\(241\) 3.43411e10 0.655748 0.327874 0.944721i \(-0.393668\pi\)
0.327874 + 0.944721i \(0.393668\pi\)
\(242\) 6.71933e9 0.125938
\(243\) −5.28397e9 −0.0972147
\(244\) 4.76448e10 0.860521
\(245\) −8.82734e7 −0.00156525
\(246\) 1.28259e10 0.223296
\(247\) −2.58333e9 −0.0441614
\(248\) −7.17322e9 −0.120415
\(249\) −5.51653e9 −0.0909430
\(250\) −1.70937e8 −0.00276762
\(251\) 2.49984e10 0.397539 0.198769 0.980046i \(-0.436305\pi\)
0.198769 + 0.980046i \(0.436305\pi\)
\(252\) −1.15838e9 −0.0180946
\(253\) 2.23974e9 0.0343680
\(254\) 1.85697e9 0.0279933
\(255\) −1.89882e8 −0.00281225
\(256\) 5.80171e10 0.844259
\(257\) −7.08375e10 −1.01289 −0.506447 0.862271i \(-0.669042\pi\)
−0.506447 + 0.862271i \(0.669042\pi\)
\(258\) −7.58920e9 −0.106637
\(259\) 2.41933e9 0.0334077
\(260\) 2.20346e8 0.00299037
\(261\) 1.53354e9 0.0204556
\(262\) −9.56380e9 −0.125393
\(263\) −7.08775e10 −0.913498 −0.456749 0.889596i \(-0.650986\pi\)
−0.456749 + 0.889596i \(0.650986\pi\)
\(264\) −1.09193e9 −0.0138350
\(265\) 9.78239e8 0.0121854
\(266\) 6.20661e8 0.00760128
\(267\) −3.52548e10 −0.424538
\(268\) 7.08386e10 0.838810
\(269\) −6.78221e10 −0.789743 −0.394871 0.918736i \(-0.629211\pi\)
−0.394871 + 0.918736i \(0.629211\pi\)
\(270\) −1.17732e8 −0.00134821
\(271\) 2.64639e10 0.298052 0.149026 0.988833i \(-0.452386\pi\)
0.149026 + 0.988833i \(0.452386\pi\)
\(272\) 2.15494e10 0.238712
\(273\) 9.85205e9 0.107348
\(274\) 1.22470e10 0.131266
\(275\) 5.11252e9 0.0539061
\(276\) 6.19280e10 0.642386
\(277\) 1.07161e11 1.09365 0.546826 0.837247i \(-0.315836\pi\)
0.546826 + 0.837247i \(0.315836\pi\)
\(278\) −2.60869e9 −0.0261952
\(279\) −2.36597e9 −0.0233771
\(280\) −1.06737e8 −0.00103778
\(281\) 9.81760e10 0.939349 0.469674 0.882840i \(-0.344371\pi\)
0.469674 + 0.882840i \(0.344371\pi\)
\(282\) −1.09070e9 −0.0102704
\(283\) −1.67457e11 −1.55191 −0.775953 0.630791i \(-0.782731\pi\)
−0.775953 + 0.630791i \(0.782731\pi\)
\(284\) 1.35627e11 1.23712
\(285\) 1.98981e8 0.00178653
\(286\) 2.13691e8 0.00188860
\(287\) −7.50004e10 −0.652522
\(288\) −2.10664e9 −0.0180436
\(289\) −1.11138e11 −0.937178
\(290\) 7.00847e7 0.000581878 0
\(291\) 6.42956e10 0.525609
\(292\) 6.36747e10 0.512560
\(293\) −1.39998e10 −0.110973 −0.0554866 0.998459i \(-0.517671\pi\)
−0.0554866 + 0.998459i \(0.517671\pi\)
\(294\) −2.36702e9 −0.0184773
\(295\) 2.81919e8 0.00216733
\(296\) 2.92537e9 0.0221497
\(297\) 7.04287e9 0.0525225
\(298\) −2.92261e10 −0.214683
\(299\) −2.44351e10 −0.176804
\(300\) 1.41359e11 1.00758
\(301\) 4.43783e10 0.311617
\(302\) −8.13712e9 −0.0562911
\(303\) 1.99828e11 1.36196
\(304\) −2.25820e10 −0.151646
\(305\) 1.44802e9 0.00958134
\(306\) −2.36214e8 −0.00154014
\(307\) 2.48086e11 1.59397 0.796985 0.603999i \(-0.206427\pi\)
0.796985 + 0.603999i \(0.206427\pi\)
\(308\) 3.16690e9 0.0200519
\(309\) −2.62618e10 −0.163874
\(310\) −1.08128e8 −0.000664983 0
\(311\) −1.67737e11 −1.01673 −0.508367 0.861141i \(-0.669751\pi\)
−0.508367 + 0.861141i \(0.669751\pi\)
\(312\) 1.19128e10 0.0711732
\(313\) −1.10406e11 −0.650196 −0.325098 0.945680i \(-0.605397\pi\)
−0.325098 + 0.945680i \(0.605397\pi\)
\(314\) −4.08264e10 −0.237005
\(315\) −3.52055e7 −0.000201471 0
\(316\) 2.20692e11 1.24507
\(317\) −7.89060e10 −0.438877 −0.219439 0.975626i \(-0.570423\pi\)
−0.219439 + 0.975626i \(0.570423\pi\)
\(318\) 2.62311e10 0.143845
\(319\) −4.19255e9 −0.0226684
\(320\) 1.86109e9 0.00992187
\(321\) 1.40008e11 0.736002
\(322\) 5.87068e9 0.0304324
\(323\) −7.80699e9 −0.0399091
\(324\) 2.04229e11 1.02959
\(325\) −5.57765e10 −0.277317
\(326\) −8.37813e9 −0.0410836
\(327\) −8.59691e10 −0.415793
\(328\) −9.06879e10 −0.432630
\(329\) 6.37796e9 0.0300123
\(330\) −1.64596e7 −7.64024e−5 0
\(331\) 8.20817e10 0.375855 0.187928 0.982183i \(-0.439823\pi\)
0.187928 + 0.982183i \(0.439823\pi\)
\(332\) 1.93460e10 0.0873916
\(333\) 9.64885e8 0.00430008
\(334\) −3.27913e10 −0.144178
\(335\) 2.15293e9 0.00933960
\(336\) 8.61212e10 0.368624
\(337\) 2.83816e11 1.19868 0.599338 0.800496i \(-0.295431\pi\)
0.599338 + 0.800496i \(0.295431\pi\)
\(338\) −2.33133e9 −0.00971578
\(339\) −6.72321e10 −0.276489
\(340\) 6.65900e8 0.00270243
\(341\) 6.46835e9 0.0259059
\(342\) 2.47534e8 0.000978402 0
\(343\) 1.38413e10 0.0539949
\(344\) 5.36608e10 0.206606
\(345\) 1.88212e9 0.00715255
\(346\) −3.61661e10 −0.135662
\(347\) −3.38226e11 −1.25234 −0.626172 0.779685i \(-0.715379\pi\)
−0.626172 + 0.779685i \(0.715379\pi\)
\(348\) −1.15923e11 −0.423703
\(349\) 5.12708e11 1.84993 0.924966 0.380049i \(-0.124093\pi\)
0.924966 + 0.380049i \(0.124093\pi\)
\(350\) 1.34007e10 0.0477331
\(351\) −7.68362e10 −0.270199
\(352\) 5.75936e9 0.0199955
\(353\) −4.34925e11 −1.49083 −0.745414 0.666601i \(-0.767748\pi\)
−0.745414 + 0.666601i \(0.767748\pi\)
\(354\) 7.55953e9 0.0255847
\(355\) 4.12198e9 0.0137746
\(356\) 1.23635e11 0.407960
\(357\) 2.97735e10 0.0970116
\(358\) 2.94693e10 0.0948192
\(359\) 7.29872e10 0.231911 0.115956 0.993254i \(-0.463007\pi\)
0.115956 + 0.993254i \(0.463007\pi\)
\(360\) −4.25693e7 −0.000133578 0
\(361\) −3.14507e11 −0.974647
\(362\) 2.29399e10 0.0702106
\(363\) −3.37778e11 −1.02106
\(364\) −3.45503e10 −0.103156
\(365\) 1.93521e9 0.00570702
\(366\) 3.88282e10 0.113105
\(367\) −1.94125e11 −0.558577 −0.279289 0.960207i \(-0.590099\pi\)
−0.279289 + 0.960207i \(0.590099\pi\)
\(368\) −2.13598e11 −0.607131
\(369\) −2.99119e10 −0.0839896
\(370\) 4.40965e7 0.000122320 0
\(371\) −1.53388e11 −0.420348
\(372\) 1.78848e11 0.484217
\(373\) 9.78363e10 0.261704 0.130852 0.991402i \(-0.458229\pi\)
0.130852 + 0.991402i \(0.458229\pi\)
\(374\) 6.45789e8 0.00170674
\(375\) 8.59292e9 0.0224388
\(376\) 7.71200e9 0.0198986
\(377\) 4.57399e10 0.116616
\(378\) 1.84604e10 0.0465080
\(379\) 2.10996e11 0.525287 0.262644 0.964893i \(-0.415406\pi\)
0.262644 + 0.964893i \(0.415406\pi\)
\(380\) −6.97811e8 −0.00171677
\(381\) −9.33491e10 −0.226959
\(382\) −9.17411e9 −0.0220434
\(383\) −3.68029e11 −0.873951 −0.436976 0.899473i \(-0.643950\pi\)
−0.436976 + 0.899473i \(0.643950\pi\)
\(384\) 2.11730e11 0.496927
\(385\) 9.62487e7 0.000223265 0
\(386\) 1.27221e10 0.0291686
\(387\) 1.76991e10 0.0401100
\(388\) −2.25479e11 −0.505084
\(389\) −3.34352e11 −0.740339 −0.370170 0.928964i \(-0.620700\pi\)
−0.370170 + 0.928964i \(0.620700\pi\)
\(390\) 1.79571e8 0.000393048 0
\(391\) −7.38444e10 −0.159780
\(392\) 1.67364e10 0.0357993
\(393\) 4.80768e11 1.01664
\(394\) 3.59516e10 0.0751597
\(395\) 6.70727e9 0.0138631
\(396\) 1.26303e9 0.00258099
\(397\) −4.55124e11 −0.919544 −0.459772 0.888037i \(-0.652069\pi\)
−0.459772 + 0.888037i \(0.652069\pi\)
\(398\) −4.95492e10 −0.0989835
\(399\) −3.12003e10 −0.0616284
\(400\) −4.87568e11 −0.952281
\(401\) 4.86374e11 0.939335 0.469668 0.882843i \(-0.344374\pi\)
0.469668 + 0.882843i \(0.344374\pi\)
\(402\) 5.77299e10 0.110251
\(403\) −7.05683e10 −0.133271
\(404\) −7.00778e11 −1.30877
\(405\) 6.20694e9 0.0114638
\(406\) −1.09893e10 −0.0200725
\(407\) −2.63791e9 −0.00476524
\(408\) 3.60011e10 0.0643199
\(409\) −2.38979e11 −0.422285 −0.211142 0.977455i \(-0.567718\pi\)
−0.211142 + 0.977455i \(0.567718\pi\)
\(410\) −1.36701e9 −0.00238916
\(411\) −6.15653e11 −1.06426
\(412\) 9.20978e10 0.157475
\(413\) −4.42049e10 −0.0747644
\(414\) 2.34136e9 0.00391712
\(415\) 5.87964e8 0.000973049 0
\(416\) −6.28334e10 −0.102866
\(417\) 1.31138e11 0.212381
\(418\) −6.76735e8 −0.00108424
\(419\) 1.39300e11 0.220794 0.110397 0.993888i \(-0.464788\pi\)
0.110397 + 0.993888i \(0.464788\pi\)
\(420\) 2.66124e9 0.00417314
\(421\) −1.04165e11 −0.161604 −0.0808019 0.996730i \(-0.525748\pi\)
−0.0808019 + 0.996730i \(0.525748\pi\)
\(422\) 9.15830e10 0.140575
\(423\) 2.54368e9 0.00386305
\(424\) −1.85472e11 −0.278696
\(425\) −1.68560e11 −0.250614
\(426\) 1.10529e11 0.162605
\(427\) −2.27050e11 −0.330519
\(428\) −4.90994e11 −0.707261
\(429\) −1.07421e10 −0.0153121
\(430\) 8.08873e8 0.00114097
\(431\) −1.10468e12 −1.54202 −0.771010 0.636823i \(-0.780248\pi\)
−0.771010 + 0.636823i \(0.780248\pi\)
\(432\) −6.71661e11 −0.927840
\(433\) −3.53626e11 −0.483447 −0.241724 0.970345i \(-0.577713\pi\)
−0.241724 + 0.970345i \(0.577713\pi\)
\(434\) 1.69545e10 0.0229393
\(435\) −3.52312e9 −0.00471766
\(436\) 3.01486e11 0.399556
\(437\) 7.73831e10 0.101503
\(438\) 5.18918e10 0.0673697
\(439\) 6.43978e11 0.827524 0.413762 0.910385i \(-0.364214\pi\)
0.413762 + 0.910385i \(0.364214\pi\)
\(440\) 1.16381e8 0.000148028 0
\(441\) 5.52023e9 0.00694998
\(442\) −7.04542e9 −0.00878025
\(443\) −6.41849e11 −0.791802 −0.395901 0.918293i \(-0.629568\pi\)
−0.395901 + 0.918293i \(0.629568\pi\)
\(444\) −7.29373e10 −0.0890689
\(445\) 3.75753e9 0.00454237
\(446\) 1.33474e11 0.159731
\(447\) 1.46918e12 1.74057
\(448\) −2.91820e11 −0.342266
\(449\) 1.25397e12 1.45606 0.728029 0.685546i \(-0.240437\pi\)
0.728029 + 0.685546i \(0.240437\pi\)
\(450\) 5.34449e9 0.00614399
\(451\) 8.17764e10 0.0930751
\(452\) 2.35777e11 0.265692
\(453\) 4.09049e11 0.456388
\(454\) 1.03983e11 0.114871
\(455\) −1.05005e9 −0.00114858
\(456\) −3.77264e10 −0.0408604
\(457\) 1.25906e12 1.35028 0.675141 0.737689i \(-0.264083\pi\)
0.675141 + 0.737689i \(0.264083\pi\)
\(458\) −1.31116e11 −0.139239
\(459\) −2.32204e11 −0.244182
\(460\) −6.60042e9 −0.00687324
\(461\) 4.43884e11 0.457737 0.228868 0.973457i \(-0.426497\pi\)
0.228868 + 0.973457i \(0.426497\pi\)
\(462\) 2.58087e9 0.00263559
\(463\) −1.26115e12 −1.27542 −0.637710 0.770277i \(-0.720118\pi\)
−0.637710 + 0.770277i \(0.720118\pi\)
\(464\) 3.99833e11 0.400450
\(465\) 5.43555e9 0.00539144
\(466\) −1.40821e11 −0.138335
\(467\) 1.85885e11 0.180850 0.0904249 0.995903i \(-0.471178\pi\)
0.0904249 + 0.995903i \(0.471178\pi\)
\(468\) −1.37795e10 −0.0132778
\(469\) −3.37580e11 −0.322180
\(470\) 1.16250e8 0.000109888 0
\(471\) 2.05232e12 1.92155
\(472\) −5.34510e10 −0.0495698
\(473\) −4.83878e10 −0.0444489
\(474\) 1.79853e11 0.163649
\(475\) 1.76638e11 0.159207
\(476\) −1.04413e11 −0.0932232
\(477\) −6.11747e10 −0.0541052
\(478\) 6.09943e10 0.0534396
\(479\) 1.51780e11 0.131736 0.0658680 0.997828i \(-0.479018\pi\)
0.0658680 + 0.997828i \(0.479018\pi\)
\(480\) 4.83976e9 0.00416139
\(481\) 2.87790e10 0.0245145
\(482\) −9.81453e10 −0.0828243
\(483\) −2.95116e11 −0.246735
\(484\) 1.18456e12 0.981187
\(485\) −6.85277e9 −0.00562378
\(486\) 1.51014e10 0.0122787
\(487\) −7.60064e10 −0.0612308 −0.0306154 0.999531i \(-0.509747\pi\)
−0.0306154 + 0.999531i \(0.509747\pi\)
\(488\) −2.74541e11 −0.219138
\(489\) 4.21165e11 0.333091
\(490\) 2.52282e8 0.000197699 0
\(491\) −7.25712e11 −0.563505 −0.281752 0.959487i \(-0.590916\pi\)
−0.281752 + 0.959487i \(0.590916\pi\)
\(492\) 2.26109e12 1.73970
\(493\) 1.38229e11 0.105387
\(494\) 7.38304e9 0.00557782
\(495\) 3.83862e7 2.87377e−5 0
\(496\) −6.16870e11 −0.457642
\(497\) −6.46327e11 −0.475170
\(498\) 1.57660e10 0.0114866
\(499\) 2.70447e12 1.95268 0.976338 0.216251i \(-0.0693828\pi\)
0.976338 + 0.216251i \(0.0693828\pi\)
\(500\) −3.01346e10 −0.0215626
\(501\) 1.64840e12 1.16894
\(502\) −7.14443e10 −0.0502112
\(503\) −7.93395e11 −0.552629 −0.276314 0.961067i \(-0.589113\pi\)
−0.276314 + 0.961067i \(0.589113\pi\)
\(504\) 6.67487e9 0.00460793
\(505\) −2.12981e10 −0.0145723
\(506\) −6.40108e9 −0.00434086
\(507\) 1.17195e11 0.0787720
\(508\) 3.27367e11 0.218096
\(509\) 1.61881e12 1.06897 0.534487 0.845177i \(-0.320505\pi\)
0.534487 + 0.845177i \(0.320505\pi\)
\(510\) 5.42675e8 0.000355201 0
\(511\) −3.03441e11 −0.196870
\(512\) −9.20368e11 −0.591898
\(513\) 2.43332e11 0.155121
\(514\) 2.02451e11 0.127934
\(515\) 2.79904e9 0.00175338
\(516\) −1.33791e12 −0.830810
\(517\) −6.95418e9 −0.00428094
\(518\) −6.91434e9 −0.00421956
\(519\) 1.81805e12 1.09990
\(520\) −1.26969e9 −0.000761521 0
\(521\) −3.78680e11 −0.225166 −0.112583 0.993642i \(-0.535912\pi\)
−0.112583 + 0.993642i \(0.535912\pi\)
\(522\) −4.38278e9 −0.00258365
\(523\) −6.27715e11 −0.366864 −0.183432 0.983032i \(-0.558721\pi\)
−0.183432 + 0.983032i \(0.558721\pi\)
\(524\) −1.68601e12 −0.976944
\(525\) −6.73645e11 −0.387003
\(526\) 2.02565e11 0.115380
\(527\) −2.13262e11 −0.120439
\(528\) −9.39020e10 −0.0525802
\(529\) −1.06920e12 −0.593623
\(530\) −2.79577e9 −0.00153907
\(531\) −1.76299e10 −0.00962333
\(532\) 1.09417e11 0.0592218
\(533\) −8.92164e11 −0.478820
\(534\) 1.00757e11 0.0536214
\(535\) −1.49223e10 −0.00787489
\(536\) −4.08190e11 −0.213610
\(537\) −1.48141e12 −0.768760
\(538\) 1.93833e11 0.0997486
\(539\) −1.50918e10 −0.00770179
\(540\) −2.07551e10 −0.0105039
\(541\) 1.45230e12 0.728902 0.364451 0.931223i \(-0.381257\pi\)
0.364451 + 0.931223i \(0.381257\pi\)
\(542\) −7.56328e10 −0.0376456
\(543\) −1.15318e12 −0.569242
\(544\) −1.89887e11 −0.0929608
\(545\) 9.16278e9 0.00444880
\(546\) −2.81567e10 −0.0135586
\(547\) 2.61898e12 1.25080 0.625402 0.780302i \(-0.284935\pi\)
0.625402 + 0.780302i \(0.284935\pi\)
\(548\) 2.15904e12 1.02270
\(549\) −9.05529e10 −0.0425429
\(550\) −1.46114e10 −0.00680861
\(551\) −1.44853e11 −0.0669492
\(552\) −3.56844e11 −0.163589
\(553\) −1.05170e12 −0.478222
\(554\) −3.06263e11 −0.138134
\(555\) −2.21671e9 −0.000991724 0
\(556\) −4.59889e11 −0.204088
\(557\) −1.64364e12 −0.723533 −0.361767 0.932269i \(-0.617826\pi\)
−0.361767 + 0.932269i \(0.617826\pi\)
\(558\) 6.76185e9 0.00295265
\(559\) 5.27901e11 0.228665
\(560\) −9.17900e9 −0.00394411
\(561\) −3.24635e10 −0.0138377
\(562\) −2.80583e11 −0.118645
\(563\) −3.36157e12 −1.41011 −0.705056 0.709151i \(-0.749078\pi\)
−0.705056 + 0.709151i \(0.749078\pi\)
\(564\) −1.92281e11 −0.0800166
\(565\) 7.16575e9 0.00295831
\(566\) 4.78587e11 0.196014
\(567\) −9.73249e11 −0.395458
\(568\) −7.81517e11 −0.315044
\(569\) 3.54649e12 1.41838 0.709192 0.705015i \(-0.249060\pi\)
0.709192 + 0.705015i \(0.249060\pi\)
\(570\) −5.68681e8 −0.000225648 0
\(571\) 1.25130e12 0.492604 0.246302 0.969193i \(-0.420785\pi\)
0.246302 + 0.969193i \(0.420785\pi\)
\(572\) 3.76718e10 0.0147141
\(573\) 4.61178e11 0.178720
\(574\) 2.14348e11 0.0824168
\(575\) 1.67078e12 0.637401
\(576\) −1.16385e11 −0.0440549
\(577\) 9.21167e11 0.345977 0.172989 0.984924i \(-0.444658\pi\)
0.172989 + 0.984924i \(0.444658\pi\)
\(578\) 3.17628e11 0.118370
\(579\) −6.39534e11 −0.236488
\(580\) 1.23553e10 0.00453343
\(581\) −9.21929e10 −0.0335664
\(582\) −1.83754e11 −0.0663872
\(583\) 1.67246e11 0.0599581
\(584\) −3.66910e11 −0.130527
\(585\) −4.18786e8 −0.000147839 0
\(586\) 4.00109e10 0.0140165
\(587\) 3.24374e12 1.12765 0.563826 0.825894i \(-0.309329\pi\)
0.563826 + 0.825894i \(0.309329\pi\)
\(588\) −4.17283e11 −0.143957
\(589\) 2.23482e11 0.0765110
\(590\) −8.05712e8 −0.000273745 0
\(591\) −1.80727e12 −0.609367
\(592\) 2.51571e11 0.0841807
\(593\) −1.00916e12 −0.335130 −0.167565 0.985861i \(-0.553590\pi\)
−0.167565 + 0.985861i \(0.553590\pi\)
\(594\) −2.01282e10 −0.00663386
\(595\) −3.17333e9 −0.00103798
\(596\) −5.15230e12 −1.67260
\(597\) 2.49081e12 0.802522
\(598\) 6.98344e10 0.0223313
\(599\) −1.04975e12 −0.333170 −0.166585 0.986027i \(-0.553274\pi\)
−0.166585 + 0.986027i \(0.553274\pi\)
\(600\) −8.14548e11 −0.256588
\(601\) 8.80068e11 0.275157 0.137579 0.990491i \(-0.456068\pi\)
0.137579 + 0.990491i \(0.456068\pi\)
\(602\) −1.26831e11 −0.0393589
\(603\) −1.34635e11 −0.0414695
\(604\) −1.43450e12 −0.438565
\(605\) 3.60011e10 0.0109249
\(606\) −5.71099e11 −0.172022
\(607\) 2.70896e12 0.809942 0.404971 0.914329i \(-0.367282\pi\)
0.404971 + 0.914329i \(0.367282\pi\)
\(608\) 1.98986e11 0.0590551
\(609\) 5.52426e11 0.162741
\(610\) −4.13839e9 −0.00121017
\(611\) 7.58687e10 0.0220230
\(612\) −4.16424e10 −0.0119993
\(613\) 1.95586e12 0.559455 0.279727 0.960079i \(-0.409756\pi\)
0.279727 + 0.960079i \(0.409756\pi\)
\(614\) −7.09021e11 −0.201327
\(615\) 6.87192e10 0.0193705
\(616\) −1.82485e10 −0.00510639
\(617\) −4.94329e12 −1.37320 −0.686599 0.727037i \(-0.740897\pi\)
−0.686599 + 0.727037i \(0.740897\pi\)
\(618\) 7.50552e10 0.0206982
\(619\) −1.57550e12 −0.431332 −0.215666 0.976467i \(-0.569192\pi\)
−0.215666 + 0.976467i \(0.569192\pi\)
\(620\) −1.90620e10 −0.00518090
\(621\) 2.30161e12 0.621041
\(622\) 4.79385e11 0.128419
\(623\) −5.89181e11 −0.156694
\(624\) 1.02445e12 0.270496
\(625\) 3.81332e12 0.999640
\(626\) 3.15537e11 0.0821231
\(627\) 3.40192e10 0.00879063
\(628\) −7.19732e12 −1.84651
\(629\) 8.69722e10 0.0221540
\(630\) 1.00616e8 2.54469e−5 0
\(631\) 7.29862e12 1.83277 0.916386 0.400295i \(-0.131092\pi\)
0.916386 + 0.400295i \(0.131092\pi\)
\(632\) −1.27168e12 −0.317067
\(633\) −4.60383e12 −1.13973
\(634\) 2.25510e11 0.0554325
\(635\) 9.94935e9 0.00242836
\(636\) 4.62430e12 1.12070
\(637\) 1.64648e11 0.0396214
\(638\) 1.19821e10 0.00286313
\(639\) −2.57770e11 −0.0611616
\(640\) −2.25667e10 −0.00531690
\(641\) −5.11451e12 −1.19658 −0.598292 0.801278i \(-0.704154\pi\)
−0.598292 + 0.801278i \(0.704154\pi\)
\(642\) −4.00136e11 −0.0929608
\(643\) −2.13356e12 −0.492216 −0.246108 0.969242i \(-0.579152\pi\)
−0.246108 + 0.969242i \(0.579152\pi\)
\(644\) 1.03495e12 0.237100
\(645\) −4.06617e10 −0.00925053
\(646\) 2.23120e10 0.00504073
\(647\) −1.64637e12 −0.369368 −0.184684 0.982798i \(-0.559126\pi\)
−0.184684 + 0.982798i \(0.559126\pi\)
\(648\) −1.17682e12 −0.262194
\(649\) 4.81986e10 0.0106643
\(650\) 1.59407e11 0.0350265
\(651\) −8.52294e11 −0.185984
\(652\) −1.47699e12 −0.320083
\(653\) −7.65775e12 −1.64813 −0.824066 0.566494i \(-0.808299\pi\)
−0.824066 + 0.566494i \(0.808299\pi\)
\(654\) 2.45696e11 0.0525168
\(655\) −5.12413e10 −0.0108776
\(656\) −7.79881e12 −1.64422
\(657\) −1.21019e11 −0.0253402
\(658\) −1.82279e10 −0.00379071
\(659\) 7.62072e11 0.157402 0.0787012 0.996898i \(-0.474923\pi\)
0.0787012 + 0.996898i \(0.474923\pi\)
\(660\) −2.90168e9 −0.000595253 0
\(661\) −3.40869e12 −0.694513 −0.347257 0.937770i \(-0.612887\pi\)
−0.347257 + 0.937770i \(0.612887\pi\)
\(662\) −2.34586e11 −0.0474724
\(663\) 3.54170e11 0.0711870
\(664\) −1.11476e11 −0.0222550
\(665\) 3.32540e9 0.000659396 0
\(666\) −2.75760e9 −0.000543122 0
\(667\) −1.37013e12 −0.268037
\(668\) −5.78081e12 −1.12330
\(669\) −6.70967e12 −1.29504
\(670\) −6.15299e9 −0.00117964
\(671\) 2.47564e11 0.0471450
\(672\) −7.58875e11 −0.143552
\(673\) 1.78459e12 0.335328 0.167664 0.985844i \(-0.446378\pi\)
0.167664 + 0.985844i \(0.446378\pi\)
\(674\) −8.11133e11 −0.151399
\(675\) 5.25376e12 0.974100
\(676\) −4.10991e11 −0.0756959
\(677\) −2.42571e12 −0.443803 −0.221902 0.975069i \(-0.571226\pi\)
−0.221902 + 0.975069i \(0.571226\pi\)
\(678\) 1.92147e11 0.0349220
\(679\) 1.07452e12 0.193999
\(680\) −3.83708e9 −0.000688194 0
\(681\) −5.22719e12 −0.931335
\(682\) −1.84863e10 −0.00327205
\(683\) 2.67882e12 0.471033 0.235516 0.971870i \(-0.424322\pi\)
0.235516 + 0.971870i \(0.424322\pi\)
\(684\) 4.36380e10 0.00762276
\(685\) 6.56176e10 0.0113871
\(686\) −3.95578e10 −0.00681984
\(687\) 6.59113e12 1.12890
\(688\) 4.61462e12 0.785214
\(689\) −1.82462e12 −0.308451
\(690\) −5.37901e9 −0.000903404 0
\(691\) 5.77069e11 0.0962890 0.0481445 0.998840i \(-0.484669\pi\)
0.0481445 + 0.998840i \(0.484669\pi\)
\(692\) −6.37576e12 −1.05695
\(693\) −6.01896e9 −0.000991339 0
\(694\) 9.66635e11 0.158178
\(695\) −1.39770e10 −0.00227238
\(696\) 6.67975e11 0.107899
\(697\) −2.69618e12 −0.432714
\(698\) −1.46530e12 −0.233656
\(699\) 7.07901e12 1.12157
\(700\) 2.36241e12 0.371890
\(701\) 4.67001e12 0.730443 0.365222 0.930921i \(-0.380993\pi\)
0.365222 + 0.930921i \(0.380993\pi\)
\(702\) 2.19595e11 0.0341275
\(703\) −9.11400e10 −0.0140738
\(704\) 3.18185e11 0.0488205
\(705\) −5.84381e9 −0.000890933 0
\(706\) 1.24300e12 0.188299
\(707\) 3.33954e12 0.502689
\(708\) 1.33268e12 0.199331
\(709\) 3.32880e12 0.494743 0.247372 0.968921i \(-0.420433\pi\)
0.247372 + 0.968921i \(0.420433\pi\)
\(710\) −1.17805e10 −0.00173980
\(711\) −4.19443e11 −0.0615545
\(712\) −7.12418e11 −0.103890
\(713\) 2.11386e12 0.306319
\(714\) −8.50916e10 −0.0122531
\(715\) 1.14492e9 0.000163832 0
\(716\) 5.19517e12 0.738739
\(717\) −3.06615e12 −0.433269
\(718\) −2.08594e11 −0.0292916
\(719\) −7.15618e11 −0.0998623 −0.0499311 0.998753i \(-0.515900\pi\)
−0.0499311 + 0.998753i \(0.515900\pi\)
\(720\) −3.66080e9 −0.000507668 0
\(721\) −4.38890e11 −0.0604849
\(722\) 8.98847e11 0.123103
\(723\) 4.93372e12 0.671510
\(724\) 4.04409e12 0.547013
\(725\) −3.12751e12 −0.420415
\(726\) 9.65355e11 0.128965
\(727\) 8.28829e12 1.10042 0.550212 0.835025i \(-0.314547\pi\)
0.550212 + 0.835025i \(0.314547\pi\)
\(728\) 1.99087e11 0.0262695
\(729\) 7.21939e12 0.946732
\(730\) −5.53074e9 −0.000720826 0
\(731\) 1.59535e12 0.206647
\(732\) 6.84505e12 0.881205
\(733\) 7.87930e12 1.00814 0.504069 0.863664i \(-0.331836\pi\)
0.504069 + 0.863664i \(0.331836\pi\)
\(734\) 5.54800e11 0.0705512
\(735\) −1.26821e10 −0.00160287
\(736\) 1.88216e12 0.236433
\(737\) 3.68079e11 0.0459555
\(738\) 8.54870e10 0.0106083
\(739\) 9.31945e12 1.14945 0.574725 0.818346i \(-0.305109\pi\)
0.574725 + 0.818346i \(0.305109\pi\)
\(740\) 7.77382e9 0.000952997 0
\(741\) −3.71142e11 −0.0452229
\(742\) 4.38377e11 0.0530921
\(743\) 3.17339e12 0.382009 0.191004 0.981589i \(-0.438826\pi\)
0.191004 + 0.981589i \(0.438826\pi\)
\(744\) −1.03056e12 −0.123310
\(745\) −1.56589e11 −0.0186233
\(746\) −2.79612e11 −0.0330546
\(747\) −3.67687e10 −0.00432051
\(748\) 1.13847e11 0.0132973
\(749\) 2.33982e12 0.271653
\(750\) −2.45582e10 −0.00283414
\(751\) −3.64334e12 −0.417946 −0.208973 0.977921i \(-0.567012\pi\)
−0.208973 + 0.977921i \(0.567012\pi\)
\(752\) 6.63203e11 0.0756252
\(753\) 3.59147e12 0.407094
\(754\) −1.30723e11 −0.0147292
\(755\) −4.35974e10 −0.00488314
\(756\) 3.25440e12 0.362345
\(757\) −6.03038e12 −0.667441 −0.333721 0.942672i \(-0.608304\pi\)
−0.333721 + 0.942672i \(0.608304\pi\)
\(758\) −6.03017e11 −0.0663465
\(759\) 3.21779e11 0.0351941
\(760\) 4.02096e9 0.000437188 0
\(761\) −2.97715e11 −0.0321788 −0.0160894 0.999871i \(-0.505122\pi\)
−0.0160894 + 0.999871i \(0.505122\pi\)
\(762\) 2.66788e11 0.0286661
\(763\) −1.43672e12 −0.153466
\(764\) −1.61731e12 −0.171741
\(765\) −1.26560e9 −0.000133604 0
\(766\) 1.05181e12 0.110385
\(767\) −5.25837e11 −0.0548621
\(768\) 8.33521e12 0.864552
\(769\) 1.20527e13 1.24284 0.621418 0.783479i \(-0.286557\pi\)
0.621418 + 0.783479i \(0.286557\pi\)
\(770\) −2.75075e8 −2.81996e−5 0
\(771\) −1.01771e13 −1.03724
\(772\) 2.24279e12 0.227253
\(773\) −1.72502e13 −1.73775 −0.868876 0.495030i \(-0.835157\pi\)
−0.868876 + 0.495030i \(0.835157\pi\)
\(774\) −5.05833e10 −0.00506609
\(775\) 4.82519e12 0.480459
\(776\) 1.29927e12 0.128624
\(777\) 3.47581e11 0.0342106
\(778\) 9.55564e11 0.0935087
\(779\) 2.82538e12 0.274890
\(780\) 3.16567e10 0.00306225
\(781\) 7.04721e11 0.0677778
\(782\) 2.11044e11 0.0201810
\(783\) −4.30838e12 −0.409625
\(784\) 1.43927e12 0.136057
\(785\) −2.18741e11 −0.0205597
\(786\) −1.37402e12 −0.128407
\(787\) 1.33731e13 1.24264 0.621320 0.783557i \(-0.286597\pi\)
0.621320 + 0.783557i \(0.286597\pi\)
\(788\) 6.33793e12 0.585571
\(789\) −1.01829e13 −0.935455
\(790\) −1.91691e10 −0.00175098
\(791\) −1.12359e12 −0.102050
\(792\) −7.27792e9 −0.000657271 0
\(793\) −2.70087e12 −0.242535
\(794\) 1.30073e12 0.116143
\(795\) 1.40542e11 0.0124783
\(796\) −8.73507e12 −0.771183
\(797\) −4.33914e12 −0.380926 −0.190463 0.981694i \(-0.560999\pi\)
−0.190463 + 0.981694i \(0.560999\pi\)
\(798\) 8.91692e10 0.00778399
\(799\) 2.29280e11 0.0199024
\(800\) 4.29631e12 0.370843
\(801\) −2.34979e11 −0.0201689
\(802\) −1.39004e12 −0.118643
\(803\) 3.30855e11 0.0280814
\(804\) 1.01773e13 0.858971
\(805\) 3.14542e10 0.00263995
\(806\) 2.01681e11 0.0168329
\(807\) −9.74388e12 −0.808725
\(808\) 4.03806e12 0.333289
\(809\) −1.12260e13 −0.921421 −0.460710 0.887551i \(-0.652405\pi\)
−0.460710 + 0.887551i \(0.652405\pi\)
\(810\) −1.77392e10 −0.00144794
\(811\) −1.10314e13 −0.895439 −0.447720 0.894174i \(-0.647764\pi\)
−0.447720 + 0.894174i \(0.647764\pi\)
\(812\) −1.93731e12 −0.156386
\(813\) 3.80203e12 0.305216
\(814\) 7.53903e9 0.000601874 0
\(815\) −4.48887e10 −0.00356392
\(816\) 3.09596e12 0.244450
\(817\) −1.67180e12 −0.131276
\(818\) 6.82993e11 0.0533367
\(819\) 6.56657e10 0.00509989
\(820\) −2.40992e11 −0.0186140
\(821\) −2.22237e13 −1.70715 −0.853576 0.520969i \(-0.825571\pi\)
−0.853576 + 0.520969i \(0.825571\pi\)
\(822\) 1.75951e12 0.134421
\(823\) −3.71507e12 −0.282272 −0.141136 0.989990i \(-0.545076\pi\)
−0.141136 + 0.989990i \(0.545076\pi\)
\(824\) −5.30691e11 −0.0401023
\(825\) 7.34507e11 0.0552018
\(826\) 1.26336e11 0.00944313
\(827\) −8.46104e12 −0.628997 −0.314499 0.949258i \(-0.601836\pi\)
−0.314499 + 0.949258i \(0.601836\pi\)
\(828\) 4.12761e11 0.0305184
\(829\) −7.73805e12 −0.569031 −0.284516 0.958671i \(-0.591833\pi\)
−0.284516 + 0.958671i \(0.591833\pi\)
\(830\) −1.68038e9 −0.000122901 0
\(831\) 1.53957e13 1.11994
\(832\) −3.47133e12 −0.251155
\(833\) 4.97579e11 0.0358063
\(834\) −3.74787e11 −0.0268248
\(835\) −1.75691e11 −0.0125072
\(836\) −1.19302e11 −0.00844735
\(837\) 6.64705e12 0.468128
\(838\) −3.98113e11 −0.0278874
\(839\) 3.66210e12 0.255153 0.127577 0.991829i \(-0.459280\pi\)
0.127577 + 0.991829i \(0.459280\pi\)
\(840\) −1.53348e10 −0.00106272
\(841\) −1.19424e13 −0.823209
\(842\) 2.97699e11 0.0204114
\(843\) 1.41048e13 0.961927
\(844\) 1.61452e13 1.09523
\(845\) −1.24909e10 −0.000842825 0
\(846\) −7.26973e9 −0.000487923 0
\(847\) −5.64498e12 −0.376866
\(848\) −1.59498e13 −1.05919
\(849\) −2.40583e13 −1.58921
\(850\) 4.81739e11 0.0316538
\(851\) −8.62071e11 −0.0563456
\(852\) 1.94853e13 1.26686
\(853\) −2.33183e13 −1.50809 −0.754043 0.656825i \(-0.771899\pi\)
−0.754043 + 0.656825i \(0.771899\pi\)
\(854\) 6.48900e11 0.0417463
\(855\) 1.32625e9 8.48744e−5 0
\(856\) 2.82923e12 0.180110
\(857\) 4.14264e12 0.262340 0.131170 0.991360i \(-0.458127\pi\)
0.131170 + 0.991360i \(0.458127\pi\)
\(858\) 3.07006e10 0.00193399
\(859\) 7.00855e12 0.439197 0.219598 0.975590i \(-0.429525\pi\)
0.219598 + 0.975590i \(0.429525\pi\)
\(860\) 1.42597e11 0.00888930
\(861\) −1.07752e13 −0.668206
\(862\) 3.15714e12 0.194765
\(863\) 2.00687e13 1.23160 0.615800 0.787902i \(-0.288833\pi\)
0.615800 + 0.787902i \(0.288833\pi\)
\(864\) 5.91848e12 0.361325
\(865\) −1.93772e11 −0.0117684
\(866\) 1.01065e12 0.0610619
\(867\) −1.59670e13 −0.959704
\(868\) 2.98892e12 0.178721
\(869\) 1.14672e12 0.0682131
\(870\) 1.00689e10 0.000595864 0
\(871\) −4.01567e12 −0.236415
\(872\) −1.73724e12 −0.101750
\(873\) 4.28542e11 0.0249706
\(874\) −2.21158e11 −0.0128204
\(875\) 1.43606e11 0.00828201
\(876\) 9.14804e12 0.524880
\(877\) −1.46130e12 −0.0834142 −0.0417071 0.999130i \(-0.513280\pi\)
−0.0417071 + 0.999130i \(0.513280\pi\)
\(878\) −1.84046e12 −0.104521
\(879\) −2.01133e12 −0.113641
\(880\) 1.00083e10 0.000562585 0
\(881\) −2.41994e13 −1.35336 −0.676681 0.736276i \(-0.736582\pi\)
−0.676681 + 0.736276i \(0.736582\pi\)
\(882\) −1.57766e10 −0.000877818 0
\(883\) 4.59585e12 0.254415 0.127207 0.991876i \(-0.459399\pi\)
0.127207 + 0.991876i \(0.459399\pi\)
\(884\) −1.24204e12 −0.0684071
\(885\) 4.05027e10 0.00221942
\(886\) 1.83438e12 0.100009
\(887\) 1.20937e13 0.655999 0.327999 0.944678i \(-0.393626\pi\)
0.327999 + 0.944678i \(0.393626\pi\)
\(888\) 4.20283e11 0.0226821
\(889\) −1.56006e12 −0.0837690
\(890\) −1.07389e10 −0.000573725 0
\(891\) 1.06118e12 0.0564078
\(892\) 2.35302e13 1.24447
\(893\) −2.40268e11 −0.0126434
\(894\) −4.19887e12 −0.219843
\(895\) 1.57892e11 0.00822538
\(896\) 3.53846e12 0.183412
\(897\) −3.51054e12 −0.181054
\(898\) −3.58380e12 −0.183908
\(899\) −3.95693e12 −0.202041
\(900\) 9.42186e11 0.0478680
\(901\) −5.51413e12 −0.278750
\(902\) −2.33714e11 −0.0117559
\(903\) 6.37576e12 0.319108
\(904\) −1.35861e12 −0.0676606
\(905\) 1.22908e11 0.00609063
\(906\) −1.16905e12 −0.0576441
\(907\) 2.56987e11 0.0126089 0.00630446 0.999980i \(-0.497993\pi\)
0.00630446 + 0.999980i \(0.497993\pi\)
\(908\) 1.83313e13 0.894966
\(909\) 1.33189e12 0.0647038
\(910\) 3.00101e9 0.000145071 0
\(911\) −3.43373e13 −1.65171 −0.825853 0.563885i \(-0.809306\pi\)
−0.825853 + 0.563885i \(0.809306\pi\)
\(912\) −3.24432e12 −0.155291
\(913\) 1.00522e11 0.00478788
\(914\) −3.59835e12 −0.170547
\(915\) 2.08035e11 0.00981164
\(916\) −2.31145e13 −1.08481
\(917\) 8.03465e12 0.375236
\(918\) 6.63630e11 0.0308414
\(919\) −2.98053e13 −1.37840 −0.689198 0.724573i \(-0.742037\pi\)
−0.689198 + 0.724573i \(0.742037\pi\)
\(920\) 3.80333e10 0.00175032
\(921\) 3.56421e13 1.63228
\(922\) −1.26860e12 −0.0578145
\(923\) −7.68836e12 −0.348679
\(924\) 4.54984e11 0.0205339
\(925\) −1.96780e12 −0.0883777
\(926\) 3.60432e12 0.161092
\(927\) −1.75040e11 −0.00778534
\(928\) −3.52321e12 −0.155946
\(929\) −4.65926e12 −0.205232 −0.102616 0.994721i \(-0.532721\pi\)
−0.102616 + 0.994721i \(0.532721\pi\)
\(930\) −1.55346e10 −0.000680967 0
\(931\) −5.21423e11 −0.0227466
\(932\) −2.48255e13 −1.07777
\(933\) −2.40985e13 −1.04117
\(934\) −5.31251e11 −0.0228423
\(935\) 3.46003e9 0.000148057 0
\(936\) 7.94007e10 0.00338129
\(937\) −4.08252e13 −1.73022 −0.865108 0.501585i \(-0.832750\pi\)
−0.865108 + 0.501585i \(0.832750\pi\)
\(938\) 9.64789e11 0.0406930
\(939\) −1.58619e13 −0.665824
\(940\) 2.04937e10 0.000856141 0
\(941\) 2.07456e13 0.862526 0.431263 0.902226i \(-0.358068\pi\)
0.431263 + 0.902226i \(0.358068\pi\)
\(942\) −5.86546e12 −0.242702
\(943\) 2.67246e13 1.10055
\(944\) −4.59658e12 −0.188391
\(945\) 9.89078e10 0.00403448
\(946\) 1.38290e11 0.00561412
\(947\) 1.68721e13 0.681700 0.340850 0.940118i \(-0.389285\pi\)
0.340850 + 0.940118i \(0.389285\pi\)
\(948\) 3.17064e13 1.27500
\(949\) −3.60956e12 −0.144463
\(950\) −5.04824e11 −0.0201087
\(951\) −1.13363e13 −0.449426
\(952\) 6.01655e11 0.0237400
\(953\) −1.38338e13 −0.543280 −0.271640 0.962399i \(-0.587566\pi\)
−0.271640 + 0.962399i \(0.587566\pi\)
\(954\) 1.74835e11 0.00683377
\(955\) −4.91534e10 −0.00191222
\(956\) 1.07527e13 0.416350
\(957\) −6.02336e11 −0.0232132
\(958\) −4.33780e11 −0.0166389
\(959\) −1.02889e13 −0.392811
\(960\) 2.67380e11 0.0101603
\(961\) −2.03348e13 −0.769103
\(962\) −8.22493e10 −0.00309631
\(963\) 9.33176e11 0.0349659
\(964\) −1.73021e13 −0.645287
\(965\) 6.81629e10 0.00253032
\(966\) 8.43430e11 0.0311639
\(967\) 2.30154e13 0.846448 0.423224 0.906025i \(-0.360898\pi\)
0.423224 + 0.906025i \(0.360898\pi\)
\(968\) −6.82571e12 −0.249867
\(969\) −1.12162e12 −0.0408684
\(970\) 1.95849e10 0.000710313 0
\(971\) 2.99062e13 1.07963 0.539814 0.841784i \(-0.318495\pi\)
0.539814 + 0.841784i \(0.318495\pi\)
\(972\) 2.66223e12 0.0956638
\(973\) 2.19159e12 0.0783884
\(974\) 2.17223e11 0.00773377
\(975\) −8.01331e12 −0.283982
\(976\) −2.36095e13 −0.832843
\(977\) −3.16873e13 −1.11265 −0.556326 0.830964i \(-0.687789\pi\)
−0.556326 + 0.830964i \(0.687789\pi\)
\(978\) −1.20367e12 −0.0420711
\(979\) 6.42412e11 0.0223507
\(980\) 4.44750e10 0.00154028
\(981\) −5.72999e11 −0.0197535
\(982\) 2.07406e12 0.0711736
\(983\) 1.11886e13 0.382194 0.191097 0.981571i \(-0.438795\pi\)
0.191097 + 0.981571i \(0.438795\pi\)
\(984\) −1.30290e13 −0.443029
\(985\) 1.92623e11 0.00651995
\(986\) −3.95053e11 −0.0133109
\(987\) 9.16310e11 0.0307337
\(988\) 1.30156e12 0.0434569
\(989\) −1.58132e13 −0.525576
\(990\) −1.09706e8 −3.62972e−6 0
\(991\) −2.37751e13 −0.783052 −0.391526 0.920167i \(-0.628053\pi\)
−0.391526 + 0.920167i \(0.628053\pi\)
\(992\) 5.43568e12 0.178218
\(993\) 1.17925e13 0.384889
\(994\) 1.84718e12 0.0600164
\(995\) −2.65477e11 −0.00858662
\(996\) 2.77941e12 0.0894922
\(997\) 4.78551e13 1.53391 0.766956 0.641700i \(-0.221770\pi\)
0.766956 + 0.641700i \(0.221770\pi\)
\(998\) −7.72927e12 −0.246633
\(999\) −2.71079e12 −0.0861094
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.1.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.10.a.a.1.6 12 1.1 even 1 trivial