Properties

Label 570.8.a.c.1.4
Level $570$
Weight $8$
Character 570.1
Self dual yes
Analytic conductor $178.059$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 570.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(178.059464526\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 2087 x^{2} + 44517 x - 205110\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-54.0924\) of defining polynomial
Character \(\chi\) \(=\) 570.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} +216.000 q^{6} +617.139 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} +216.000 q^{6} +617.139 q^{7} +512.000 q^{8} +729.000 q^{9} +1000.00 q^{10} -3899.34 q^{11} +1728.00 q^{12} -4970.13 q^{13} +4937.11 q^{14} +3375.00 q^{15} +4096.00 q^{16} -30312.4 q^{17} +5832.00 q^{18} -6859.00 q^{19} +8000.00 q^{20} +16662.7 q^{21} -31194.7 q^{22} +24239.7 q^{23} +13824.0 q^{24} +15625.0 q^{25} -39761.0 q^{26} +19683.0 q^{27} +39496.9 q^{28} -111451. q^{29} +27000.0 q^{30} -135412. q^{31} +32768.0 q^{32} -105282. q^{33} -242499. q^{34} +77142.3 q^{35} +46656.0 q^{36} +80372.0 q^{37} -54872.0 q^{38} -134193. q^{39} +64000.0 q^{40} -511416. q^{41} +133302. q^{42} -828701. q^{43} -249558. q^{44} +91125.0 q^{45} +193918. q^{46} -514925. q^{47} +110592. q^{48} -442683. q^{49} +125000. q^{50} -818436. q^{51} -318088. q^{52} +546977. q^{53} +157464. q^{54} -487417. q^{55} +315975. q^{56} -185193. q^{57} -891606. q^{58} +20228.9 q^{59} +216000. q^{60} -1.58093e6 q^{61} -1.08330e6 q^{62} +449894. q^{63} +262144. q^{64} -621266. q^{65} -842257. q^{66} +1.93405e6 q^{67} -1.94000e6 q^{68} +654472. q^{69} +617139. q^{70} +555676. q^{71} +373248. q^{72} -2.03651e6 q^{73} +642976. q^{74} +421875. q^{75} -438976. q^{76} -2.40643e6 q^{77} -1.07355e6 q^{78} +712815. q^{79} +512000. q^{80} +531441. q^{81} -4.09132e6 q^{82} +2.94412e6 q^{83} +1.06642e6 q^{84} -3.78905e6 q^{85} -6.62961e6 q^{86} -3.00917e6 q^{87} -1.99646e6 q^{88} +1.06933e7 q^{89} +729000. q^{90} -3.06726e6 q^{91} +1.55134e6 q^{92} -3.65613e6 q^{93} -4.11940e6 q^{94} -857375. q^{95} +884736. q^{96} +3.72622e6 q^{97} -3.54146e6 q^{98} -2.84262e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} + 500q^{5} + 864q^{6} - 1316q^{7} + 2048q^{8} + 2916q^{9} + O(q^{10}) \) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} + 500q^{5} + 864q^{6} - 1316q^{7} + 2048q^{8} + 2916q^{9} + 4000q^{10} - 6570q^{11} + 6912q^{12} - 17578q^{13} - 10528q^{14} + 13500q^{15} + 16384q^{16} - 28466q^{17} + 23328q^{18} - 27436q^{19} + 32000q^{20} - 35532q^{21} - 52560q^{22} - 32136q^{23} + 55296q^{24} + 62500q^{25} - 140624q^{26} + 78732q^{27} - 84224q^{28} - 159122q^{29} + 108000q^{30} - 67974q^{31} + 131072q^{32} - 177390q^{33} - 227728q^{34} - 164500q^{35} + 186624q^{36} - 823702q^{37} - 219488q^{38} - 474606q^{39} + 256000q^{40} - 781924q^{41} - 284256q^{42} - 1115638q^{43} - 420480q^{44} + 364500q^{45} - 257088q^{46} - 209160q^{47} + 442368q^{48} - 1159308q^{49} + 500000q^{50} - 768582q^{51} - 1124992q^{52} - 848424q^{53} + 629856q^{54} - 821250q^{55} - 673792q^{56} - 740772q^{57} - 1272976q^{58} - 3677830q^{59} + 864000q^{60} - 1161072q^{61} - 543792q^{62} - 959364q^{63} + 1048576q^{64} - 2197250q^{65} - 1419120q^{66} - 6154740q^{67} - 1821824q^{68} - 867672q^{69} - 1316000q^{70} - 4456224q^{71} + 1492992q^{72} + 1057792q^{73} - 6589616q^{74} + 1687500q^{75} - 1755904q^{76} + 2402388q^{77} - 3796848q^{78} + 2910090q^{79} + 2048000q^{80} + 2125764q^{81} - 6255392q^{82} - 1767198q^{83} - 2274048q^{84} - 3558250q^{85} - 8925104q^{86} - 4296294q^{87} - 3363840q^{88} - 3677360q^{89} + 2916000q^{90} - 6727732q^{91} - 2056704q^{92} - 1835298q^{93} - 1673280q^{94} - 3429500q^{95} + 3538944q^{96} - 10419094q^{97} - 9274464q^{98} - 4789530q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 27.0000 0.577350
\(4\) 64.0000 0.500000
\(5\) 125.000 0.447214
\(6\) 216.000 0.408248
\(7\) 617.139 0.680048 0.340024 0.940417i \(-0.389565\pi\)
0.340024 + 0.940417i \(0.389565\pi\)
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) 1000.00 0.316228
\(11\) −3899.34 −0.883317 −0.441658 0.897183i \(-0.645610\pi\)
−0.441658 + 0.897183i \(0.645610\pi\)
\(12\) 1728.00 0.288675
\(13\) −4970.13 −0.627431 −0.313715 0.949517i \(-0.601574\pi\)
−0.313715 + 0.949517i \(0.601574\pi\)
\(14\) 4937.11 0.480867
\(15\) 3375.00 0.258199
\(16\) 4096.00 0.250000
\(17\) −30312.4 −1.49641 −0.748203 0.663470i \(-0.769083\pi\)
−0.748203 + 0.663470i \(0.769083\pi\)
\(18\) 5832.00 0.235702
\(19\) −6859.00 −0.229416
\(20\) 8000.00 0.223607
\(21\) 16662.7 0.392626
\(22\) −31194.7 −0.624599
\(23\) 24239.7 0.415413 0.207706 0.978191i \(-0.433400\pi\)
0.207706 + 0.978191i \(0.433400\pi\)
\(24\) 13824.0 0.204124
\(25\) 15625.0 0.200000
\(26\) −39761.0 −0.443660
\(27\) 19683.0 0.192450
\(28\) 39496.9 0.340024
\(29\) −111451. −0.848574 −0.424287 0.905528i \(-0.639475\pi\)
−0.424287 + 0.905528i \(0.639475\pi\)
\(30\) 27000.0 0.182574
\(31\) −135412. −0.816379 −0.408190 0.912897i \(-0.633840\pi\)
−0.408190 + 0.912897i \(0.633840\pi\)
\(32\) 32768.0 0.176777
\(33\) −105282. −0.509983
\(34\) −242499. −1.05812
\(35\) 77142.3 0.304127
\(36\) 46656.0 0.166667
\(37\) 80372.0 0.260855 0.130427 0.991458i \(-0.458365\pi\)
0.130427 + 0.991458i \(0.458365\pi\)
\(38\) −54872.0 −0.162221
\(39\) −134193. −0.362247
\(40\) 64000.0 0.158114
\(41\) −511416. −1.15886 −0.579429 0.815023i \(-0.696724\pi\)
−0.579429 + 0.815023i \(0.696724\pi\)
\(42\) 133302. 0.277628
\(43\) −828701. −1.58949 −0.794746 0.606942i \(-0.792396\pi\)
−0.794746 + 0.606942i \(0.792396\pi\)
\(44\) −249558. −0.441658
\(45\) 91125.0 0.149071
\(46\) 193918. 0.293741
\(47\) −514925. −0.723439 −0.361719 0.932287i \(-0.617810\pi\)
−0.361719 + 0.932287i \(0.617810\pi\)
\(48\) 110592. 0.144338
\(49\) −442683. −0.537535
\(50\) 125000. 0.141421
\(51\) −818436. −0.863950
\(52\) −318088. −0.313715
\(53\) 546977. 0.504665 0.252333 0.967641i \(-0.418802\pi\)
0.252333 + 0.967641i \(0.418802\pi\)
\(54\) 157464. 0.136083
\(55\) −487417. −0.395031
\(56\) 315975. 0.240433
\(57\) −185193. −0.132453
\(58\) −891606. −0.600033
\(59\) 20228.9 0.0128230 0.00641151 0.999979i \(-0.497959\pi\)
0.00641151 + 0.999979i \(0.497959\pi\)
\(60\) 216000. 0.129099
\(61\) −1.58093e6 −0.891783 −0.445891 0.895087i \(-0.647113\pi\)
−0.445891 + 0.895087i \(0.647113\pi\)
\(62\) −1.08330e6 −0.577267
\(63\) 449894. 0.226683
\(64\) 262144. 0.125000
\(65\) −621266. −0.280595
\(66\) −842257. −0.360612
\(67\) 1.93405e6 0.785607 0.392804 0.919622i \(-0.371505\pi\)
0.392804 + 0.919622i \(0.371505\pi\)
\(68\) −1.94000e6 −0.748203
\(69\) 654472. 0.239839
\(70\) 617139. 0.215050
\(71\) 555676. 0.184254 0.0921270 0.995747i \(-0.470633\pi\)
0.0921270 + 0.995747i \(0.470633\pi\)
\(72\) 373248. 0.117851
\(73\) −2.03651e6 −0.612712 −0.306356 0.951917i \(-0.599110\pi\)
−0.306356 + 0.951917i \(0.599110\pi\)
\(74\) 642976. 0.184452
\(75\) 421875. 0.115470
\(76\) −438976. −0.114708
\(77\) −2.40643e6 −0.600698
\(78\) −1.07355e6 −0.256147
\(79\) 712815. 0.162661 0.0813303 0.996687i \(-0.474083\pi\)
0.0813303 + 0.996687i \(0.474083\pi\)
\(80\) 512000. 0.111803
\(81\) 531441. 0.111111
\(82\) −4.09132e6 −0.819437
\(83\) 2.94412e6 0.565174 0.282587 0.959242i \(-0.408807\pi\)
0.282587 + 0.959242i \(0.408807\pi\)
\(84\) 1.06642e6 0.196313
\(85\) −3.78905e6 −0.669213
\(86\) −6.62961e6 −1.12394
\(87\) −3.00917e6 −0.489925
\(88\) −1.99646e6 −0.312300
\(89\) 1.06933e7 1.60786 0.803930 0.594724i \(-0.202739\pi\)
0.803930 + 0.594724i \(0.202739\pi\)
\(90\) 729000. 0.105409
\(91\) −3.06726e6 −0.426683
\(92\) 1.55134e6 0.207706
\(93\) −3.65613e6 −0.471337
\(94\) −4.11940e6 −0.511548
\(95\) −857375. −0.102598
\(96\) 884736. 0.102062
\(97\) 3.72622e6 0.414541 0.207271 0.978284i \(-0.433542\pi\)
0.207271 + 0.978284i \(0.433542\pi\)
\(98\) −3.54146e6 −0.380094
\(99\) −2.84262e6 −0.294439
\(100\) 1.00000e6 0.100000
\(101\) 2.30207e6 0.222328 0.111164 0.993802i \(-0.464542\pi\)
0.111164 + 0.993802i \(0.464542\pi\)
\(102\) −6.54748e6 −0.610905
\(103\) −6.09959e6 −0.550010 −0.275005 0.961443i \(-0.588680\pi\)
−0.275005 + 0.961443i \(0.588680\pi\)
\(104\) −2.54470e6 −0.221830
\(105\) 2.08284e6 0.175588
\(106\) 4.37582e6 0.356852
\(107\) −7.37181e6 −0.581743 −0.290871 0.956762i \(-0.593945\pi\)
−0.290871 + 0.956762i \(0.593945\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) 2.38555e7 1.76439 0.882196 0.470882i \(-0.156064\pi\)
0.882196 + 0.470882i \(0.156064\pi\)
\(110\) −3.89934e6 −0.279329
\(111\) 2.17004e6 0.150605
\(112\) 2.52780e6 0.170012
\(113\) 1.50684e7 0.982410 0.491205 0.871044i \(-0.336557\pi\)
0.491205 + 0.871044i \(0.336557\pi\)
\(114\) −1.48154e6 −0.0936586
\(115\) 3.02996e6 0.185778
\(116\) −7.13285e6 −0.424287
\(117\) −3.62322e6 −0.209144
\(118\) 161831. 0.00906725
\(119\) −1.87070e7 −1.01763
\(120\) 1.72800e6 0.0912871
\(121\) −4.28234e6 −0.219752
\(122\) −1.26475e7 −0.630586
\(123\) −1.38082e7 −0.669067
\(124\) −8.66638e6 −0.408190
\(125\) 1.95312e6 0.0894427
\(126\) 3.59915e6 0.160289
\(127\) −3.51742e7 −1.52374 −0.761871 0.647729i \(-0.775719\pi\)
−0.761871 + 0.647729i \(0.775719\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −2.23749e7 −0.917694
\(130\) −4.97013e6 −0.198411
\(131\) 1.21096e7 0.470630 0.235315 0.971919i \(-0.424388\pi\)
0.235315 + 0.971919i \(0.424388\pi\)
\(132\) −6.73805e6 −0.254992
\(133\) −4.23295e6 −0.156014
\(134\) 1.54724e7 0.555508
\(135\) 2.46038e6 0.0860663
\(136\) −1.55200e7 −0.529059
\(137\) 2.39599e6 0.0796091 0.0398045 0.999207i \(-0.487326\pi\)
0.0398045 + 0.999207i \(0.487326\pi\)
\(138\) 5.23578e6 0.169592
\(139\) −3.62795e7 −1.14580 −0.572901 0.819625i \(-0.694182\pi\)
−0.572901 + 0.819625i \(0.694182\pi\)
\(140\) 4.93711e6 0.152063
\(141\) −1.39030e7 −0.417678
\(142\) 4.44541e6 0.130287
\(143\) 1.93802e7 0.554220
\(144\) 2.98598e6 0.0833333
\(145\) −1.39313e7 −0.379494
\(146\) −1.62921e7 −0.433253
\(147\) −1.19524e7 −0.310346
\(148\) 5.14381e6 0.130427
\(149\) −5.26977e7 −1.30509 −0.652544 0.757751i \(-0.726298\pi\)
−0.652544 + 0.757751i \(0.726298\pi\)
\(150\) 3.37500e6 0.0816497
\(151\) 3.44151e7 0.813448 0.406724 0.913551i \(-0.366671\pi\)
0.406724 + 0.913551i \(0.366671\pi\)
\(152\) −3.51181e6 −0.0811107
\(153\) −2.20978e7 −0.498802
\(154\) −1.92515e7 −0.424757
\(155\) −1.69265e7 −0.365096
\(156\) −8.58838e6 −0.181124
\(157\) 1.05722e7 0.218030 0.109015 0.994040i \(-0.465230\pi\)
0.109015 + 0.994040i \(0.465230\pi\)
\(158\) 5.70252e6 0.115018
\(159\) 1.47684e7 0.291369
\(160\) 4.09600e6 0.0790569
\(161\) 1.49593e7 0.282501
\(162\) 4.25153e6 0.0785674
\(163\) 3.92791e7 0.710404 0.355202 0.934790i \(-0.384412\pi\)
0.355202 + 0.934790i \(0.384412\pi\)
\(164\) −3.27306e7 −0.579429
\(165\) −1.31603e7 −0.228071
\(166\) 2.35529e7 0.399638
\(167\) −2.14163e7 −0.355825 −0.177912 0.984046i \(-0.556934\pi\)
−0.177912 + 0.984046i \(0.556934\pi\)
\(168\) 8.53132e6 0.138814
\(169\) −3.80464e7 −0.606331
\(170\) −3.03124e7 −0.473205
\(171\) −5.00021e6 −0.0764719
\(172\) −5.30369e7 −0.794746
\(173\) 1.22025e8 1.79179 0.895897 0.444261i \(-0.146534\pi\)
0.895897 + 0.444261i \(0.146534\pi\)
\(174\) −2.40734e7 −0.346429
\(175\) 9.64279e6 0.136010
\(176\) −1.59717e7 −0.220829
\(177\) 546181. 0.00740338
\(178\) 8.55467e7 1.13693
\(179\) −5.57624e7 −0.726701 −0.363350 0.931653i \(-0.618367\pi\)
−0.363350 + 0.931653i \(0.618367\pi\)
\(180\) 5.83200e6 0.0745356
\(181\) −1.10873e8 −1.38979 −0.694896 0.719110i \(-0.744550\pi\)
−0.694896 + 0.719110i \(0.744550\pi\)
\(182\) −2.45381e7 −0.301710
\(183\) −4.26852e7 −0.514871
\(184\) 1.24107e7 0.146871
\(185\) 1.00465e7 0.116658
\(186\) −2.92490e7 −0.333285
\(187\) 1.18198e8 1.32180
\(188\) −3.29552e7 −0.361719
\(189\) 1.21471e7 0.130875
\(190\) −6.85900e6 −0.0725476
\(191\) −9.58547e6 −0.0995398 −0.0497699 0.998761i \(-0.515849\pi\)
−0.0497699 + 0.998761i \(0.515849\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) −3.86980e7 −0.387470 −0.193735 0.981054i \(-0.562060\pi\)
−0.193735 + 0.981054i \(0.562060\pi\)
\(194\) 2.98098e7 0.293125
\(195\) −1.67742e7 −0.162002
\(196\) −2.83317e7 −0.268767
\(197\) 2.75631e7 0.256860 0.128430 0.991719i \(-0.459006\pi\)
0.128430 + 0.991719i \(0.459006\pi\)
\(198\) −2.27409e7 −0.208200
\(199\) 1.72448e8 1.55122 0.775609 0.631214i \(-0.217443\pi\)
0.775609 + 0.631214i \(0.217443\pi\)
\(200\) 8.00000e6 0.0707107
\(201\) 5.22193e7 0.453571
\(202\) 1.84165e7 0.157209
\(203\) −6.87805e7 −0.577071
\(204\) −5.23799e7 −0.431975
\(205\) −6.39269e7 −0.518257
\(206\) −4.87967e7 −0.388916
\(207\) 1.76707e7 0.138471
\(208\) −2.03576e7 −0.156858
\(209\) 2.67456e7 0.202647
\(210\) 1.66627e7 0.124159
\(211\) −2.17057e8 −1.59069 −0.795343 0.606159i \(-0.792709\pi\)
−0.795343 + 0.606159i \(0.792709\pi\)
\(212\) 3.50065e7 0.252333
\(213\) 1.50032e7 0.106379
\(214\) −5.89745e7 −0.411354
\(215\) −1.03588e8 −0.710843
\(216\) 1.00777e7 0.0680414
\(217\) −8.35681e7 −0.555177
\(218\) 1.90844e8 1.24761
\(219\) −5.49858e7 −0.353750
\(220\) −3.11947e7 −0.197516
\(221\) 1.50657e8 0.938891
\(222\) 1.73603e7 0.106493
\(223\) 5.20130e7 0.314084 0.157042 0.987592i \(-0.449804\pi\)
0.157042 + 0.987592i \(0.449804\pi\)
\(224\) 2.02224e7 0.120217
\(225\) 1.13906e7 0.0666667
\(226\) 1.20547e8 0.694669
\(227\) −2.01058e8 −1.14086 −0.570428 0.821348i \(-0.693223\pi\)
−0.570428 + 0.821348i \(0.693223\pi\)
\(228\) −1.18524e7 −0.0662266
\(229\) −2.50710e8 −1.37958 −0.689790 0.724010i \(-0.742297\pi\)
−0.689790 + 0.724010i \(0.742297\pi\)
\(230\) 2.42397e7 0.131365
\(231\) −6.49737e7 −0.346813
\(232\) −5.70628e7 −0.300016
\(233\) −3.51661e8 −1.82129 −0.910645 0.413190i \(-0.864415\pi\)
−0.910645 + 0.413190i \(0.864415\pi\)
\(234\) −2.89858e7 −0.147887
\(235\) −6.43657e7 −0.323532
\(236\) 1.29465e6 0.00641151
\(237\) 1.92460e7 0.0939121
\(238\) −1.49656e8 −0.719572
\(239\) 1.19631e8 0.566826 0.283413 0.958998i \(-0.408533\pi\)
0.283413 + 0.958998i \(0.408533\pi\)
\(240\) 1.38240e7 0.0645497
\(241\) 2.13046e8 0.980425 0.490212 0.871603i \(-0.336919\pi\)
0.490212 + 0.871603i \(0.336919\pi\)
\(242\) −3.42587e7 −0.155388
\(243\) 1.43489e7 0.0641500
\(244\) −1.01180e8 −0.445891
\(245\) −5.53354e7 −0.240393
\(246\) −1.10466e8 −0.473102
\(247\) 3.40901e7 0.143942
\(248\) −6.93311e7 −0.288634
\(249\) 7.94912e7 0.326303
\(250\) 1.56250e7 0.0632456
\(251\) −2.38887e8 −0.953531 −0.476765 0.879031i \(-0.658191\pi\)
−0.476765 + 0.879031i \(0.658191\pi\)
\(252\) 2.87932e7 0.113341
\(253\) −9.45188e7 −0.366941
\(254\) −2.81394e8 −1.07745
\(255\) −1.02304e8 −0.386370
\(256\) 1.67772e7 0.0625000
\(257\) −4.60534e7 −0.169237 −0.0846187 0.996413i \(-0.526967\pi\)
−0.0846187 + 0.996413i \(0.526967\pi\)
\(258\) −1.78999e8 −0.648907
\(259\) 4.96007e7 0.177394
\(260\) −3.97610e7 −0.140298
\(261\) −8.12476e7 −0.282858
\(262\) 9.68767e7 0.332786
\(263\) 3.53267e8 1.19745 0.598726 0.800954i \(-0.295674\pi\)
0.598726 + 0.800954i \(0.295674\pi\)
\(264\) −5.39044e7 −0.180306
\(265\) 6.83721e7 0.225693
\(266\) −3.38636e7 −0.110318
\(267\) 2.88720e8 0.928298
\(268\) 1.23779e8 0.392804
\(269\) 4.31453e8 1.35145 0.675726 0.737153i \(-0.263830\pi\)
0.675726 + 0.737153i \(0.263830\pi\)
\(270\) 1.96830e7 0.0608581
\(271\) 2.73481e8 0.834707 0.417353 0.908744i \(-0.362958\pi\)
0.417353 + 0.908744i \(0.362958\pi\)
\(272\) −1.24160e8 −0.374102
\(273\) −8.28159e7 −0.246345
\(274\) 1.91679e7 0.0562921
\(275\) −6.09271e7 −0.176663
\(276\) 4.18862e7 0.119919
\(277\) 1.99480e8 0.563924 0.281962 0.959426i \(-0.409015\pi\)
0.281962 + 0.959426i \(0.409015\pi\)
\(278\) −2.90236e8 −0.810204
\(279\) −9.87155e7 −0.272126
\(280\) 3.94969e7 0.107525
\(281\) −1.44168e8 −0.387612 −0.193806 0.981040i \(-0.562083\pi\)
−0.193806 + 0.981040i \(0.562083\pi\)
\(282\) −1.11224e8 −0.295343
\(283\) −2.66542e8 −0.699057 −0.349528 0.936926i \(-0.613658\pi\)
−0.349528 + 0.936926i \(0.613658\pi\)
\(284\) 3.55632e7 0.0921270
\(285\) −2.31491e7 −0.0592349
\(286\) 1.55042e8 0.391893
\(287\) −3.15614e8 −0.788079
\(288\) 2.38879e7 0.0589256
\(289\) 5.08505e8 1.23923
\(290\) −1.11451e8 −0.268343
\(291\) 1.00608e8 0.239335
\(292\) −1.30337e8 −0.306356
\(293\) 2.37563e8 0.551751 0.275875 0.961193i \(-0.411032\pi\)
0.275875 + 0.961193i \(0.411032\pi\)
\(294\) −9.56195e7 −0.219448
\(295\) 2.52861e6 0.00573463
\(296\) 4.11505e7 0.0922261
\(297\) −7.67507e7 −0.169994
\(298\) −4.21582e8 −0.922837
\(299\) −1.20474e8 −0.260643
\(300\) 2.70000e7 0.0577350
\(301\) −5.11424e8 −1.08093
\(302\) 2.75321e8 0.575194
\(303\) 6.21558e7 0.128361
\(304\) −2.80945e7 −0.0573539
\(305\) −1.97617e8 −0.398817
\(306\) −1.76782e8 −0.352706
\(307\) 6.83552e8 1.34830 0.674151 0.738593i \(-0.264510\pi\)
0.674151 + 0.738593i \(0.264510\pi\)
\(308\) −1.54012e8 −0.300349
\(309\) −1.64689e8 −0.317548
\(310\) −1.35412e8 −0.258162
\(311\) −1.59619e8 −0.300900 −0.150450 0.988618i \(-0.548072\pi\)
−0.150450 + 0.988618i \(0.548072\pi\)
\(312\) −6.87070e7 −0.128074
\(313\) −8.70033e8 −1.60373 −0.801864 0.597507i \(-0.796158\pi\)
−0.801864 + 0.597507i \(0.796158\pi\)
\(314\) 8.45776e7 0.154171
\(315\) 5.62368e7 0.101376
\(316\) 4.56202e7 0.0813303
\(317\) −5.96120e8 −1.05106 −0.525528 0.850776i \(-0.676132\pi\)
−0.525528 + 0.850776i \(0.676132\pi\)
\(318\) 1.18147e8 0.206029
\(319\) 4.34584e8 0.749560
\(320\) 3.27680e7 0.0559017
\(321\) −1.99039e8 −0.335869
\(322\) 1.19674e8 0.199758
\(323\) 2.07913e8 0.343299
\(324\) 3.40122e7 0.0555556
\(325\) −7.76582e7 −0.125486
\(326\) 3.14233e8 0.502331
\(327\) 6.44098e8 1.01867
\(328\) −2.61845e8 −0.409718
\(329\) −3.17780e8 −0.491973
\(330\) −1.05282e8 −0.161271
\(331\) −1.05032e9 −1.59192 −0.795962 0.605346i \(-0.793035\pi\)
−0.795962 + 0.605346i \(0.793035\pi\)
\(332\) 1.88424e8 0.282587
\(333\) 5.85912e7 0.0869516
\(334\) −1.71330e8 −0.251606
\(335\) 2.41756e8 0.351334
\(336\) 6.82506e7 0.0981565
\(337\) −2.63958e8 −0.375690 −0.187845 0.982199i \(-0.560150\pi\)
−0.187845 + 0.982199i \(0.560150\pi\)
\(338\) −3.04371e8 −0.428741
\(339\) 4.06847e8 0.567195
\(340\) −2.42499e8 −0.334607
\(341\) 5.28018e8 0.721121
\(342\) −4.00017e7 −0.0540738
\(343\) −7.81437e8 −1.04560
\(344\) −4.24295e8 −0.561970
\(345\) 8.18090e7 0.107259
\(346\) 9.76202e8 1.26699
\(347\) −5.36070e8 −0.688761 −0.344380 0.938830i \(-0.611911\pi\)
−0.344380 + 0.938830i \(0.611911\pi\)
\(348\) −1.92587e8 −0.244962
\(349\) −6.21040e8 −0.782043 −0.391022 0.920382i \(-0.627878\pi\)
−0.391022 + 0.920382i \(0.627878\pi\)
\(350\) 7.71423e7 0.0961733
\(351\) −9.78270e7 −0.120749
\(352\) −1.27773e8 −0.156150
\(353\) 1.14350e9 1.38365 0.691825 0.722065i \(-0.256807\pi\)
0.691825 + 0.722065i \(0.256807\pi\)
\(354\) 4.36945e6 0.00523498
\(355\) 6.94595e7 0.0824009
\(356\) 6.84373e8 0.803930
\(357\) −5.05088e8 −0.587528
\(358\) −4.46099e8 −0.513855
\(359\) −1.02998e9 −1.17489 −0.587444 0.809265i \(-0.699866\pi\)
−0.587444 + 0.809265i \(0.699866\pi\)
\(360\) 4.66560e7 0.0527046
\(361\) 4.70459e7 0.0526316
\(362\) −8.86982e8 −0.982732
\(363\) −1.15623e8 −0.126874
\(364\) −1.96304e8 −0.213341
\(365\) −2.54564e8 −0.274013
\(366\) −3.41482e8 −0.364069
\(367\) −4.12106e8 −0.435189 −0.217594 0.976039i \(-0.569821\pi\)
−0.217594 + 0.976039i \(0.569821\pi\)
\(368\) 9.92858e7 0.103853
\(369\) −3.72822e8 −0.386286
\(370\) 8.03720e7 0.0824895
\(371\) 3.37561e8 0.343197
\(372\) −2.33992e8 −0.235668
\(373\) 4.10195e8 0.409269 0.204635 0.978838i \(-0.434399\pi\)
0.204635 + 0.978838i \(0.434399\pi\)
\(374\) 9.45587e8 0.934654
\(375\) 5.27344e7 0.0516398
\(376\) −2.63642e8 −0.255774
\(377\) 5.53924e8 0.532421
\(378\) 9.71771e7 0.0925428
\(379\) 7.81318e8 0.737209 0.368604 0.929586i \(-0.379836\pi\)
0.368604 + 0.929586i \(0.379836\pi\)
\(380\) −5.48720e7 −0.0512989
\(381\) −9.49704e8 −0.879733
\(382\) −7.66838e7 −0.0703852
\(383\) 1.40717e9 1.27983 0.639913 0.768447i \(-0.278970\pi\)
0.639913 + 0.768447i \(0.278970\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) −3.00804e8 −0.268640
\(386\) −3.09584e8 −0.273983
\(387\) −6.04123e8 −0.529831
\(388\) 2.38478e8 0.207271
\(389\) −9.11166e8 −0.784827 −0.392413 0.919789i \(-0.628360\pi\)
−0.392413 + 0.919789i \(0.628360\pi\)
\(390\) −1.34193e8 −0.114553
\(391\) −7.34764e8 −0.621626
\(392\) −2.26654e8 −0.190047
\(393\) 3.26959e8 0.271719
\(394\) 2.20505e8 0.181627
\(395\) 8.91019e7 0.0727440
\(396\) −1.81927e8 −0.147219
\(397\) 1.23062e9 0.987091 0.493545 0.869720i \(-0.335701\pi\)
0.493545 + 0.869720i \(0.335701\pi\)
\(398\) 1.37959e9 1.09688
\(399\) −1.14290e8 −0.0900746
\(400\) 6.40000e7 0.0500000
\(401\) −3.29019e7 −0.0254809 −0.0127405 0.999919i \(-0.504056\pi\)
−0.0127405 + 0.999919i \(0.504056\pi\)
\(402\) 4.17754e8 0.320723
\(403\) 6.73016e8 0.512221
\(404\) 1.47332e8 0.111164
\(405\) 6.64301e7 0.0496904
\(406\) −5.50244e8 −0.408051
\(407\) −3.13397e8 −0.230417
\(408\) −4.19039e8 −0.305453
\(409\) 8.09372e8 0.584947 0.292474 0.956274i \(-0.405522\pi\)
0.292474 + 0.956274i \(0.405522\pi\)
\(410\) −5.11416e8 −0.366463
\(411\) 6.46917e7 0.0459623
\(412\) −3.90374e8 −0.275005
\(413\) 1.24840e7 0.00872028
\(414\) 1.41366e8 0.0979137
\(415\) 3.68015e8 0.252753
\(416\) −1.62861e8 −0.110915
\(417\) −9.79546e8 −0.661529
\(418\) 2.13964e8 0.143293
\(419\) 1.69058e9 1.12276 0.561378 0.827560i \(-0.310271\pi\)
0.561378 + 0.827560i \(0.310271\pi\)
\(420\) 1.33302e8 0.0877938
\(421\) 1.93291e9 1.26248 0.631239 0.775589i \(-0.282547\pi\)
0.631239 + 0.775589i \(0.282547\pi\)
\(422\) −1.73645e9 −1.12479
\(423\) −3.75381e8 −0.241146
\(424\) 2.80052e8 0.178426
\(425\) −4.73632e8 −0.299281
\(426\) 1.20026e8 0.0752214
\(427\) −9.75655e8 −0.606455
\(428\) −4.71796e8 −0.290871
\(429\) 5.23265e8 0.319979
\(430\) −8.28701e8 −0.502642
\(431\) −1.03164e9 −0.620665 −0.310333 0.950628i \(-0.600440\pi\)
−0.310333 + 0.950628i \(0.600440\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) −1.67696e8 −0.0992693 −0.0496347 0.998767i \(-0.515806\pi\)
−0.0496347 + 0.998767i \(0.515806\pi\)
\(434\) −6.68545e8 −0.392570
\(435\) −3.76146e8 −0.219101
\(436\) 1.52675e9 0.882196
\(437\) −1.66260e8 −0.0953022
\(438\) −4.39886e8 −0.250139
\(439\) −1.24889e9 −0.704526 −0.352263 0.935901i \(-0.614588\pi\)
−0.352263 + 0.935901i \(0.614588\pi\)
\(440\) −2.49558e8 −0.139665
\(441\) −3.22716e8 −0.179178
\(442\) 1.20525e9 0.663896
\(443\) −1.49377e9 −0.816340 −0.408170 0.912906i \(-0.633833\pi\)
−0.408170 + 0.912906i \(0.633833\pi\)
\(444\) 1.38883e8 0.0753023
\(445\) 1.33667e9 0.719057
\(446\) 4.16104e8 0.222091
\(447\) −1.42284e9 −0.753493
\(448\) 1.61779e8 0.0850060
\(449\) −2.86865e9 −1.49560 −0.747799 0.663925i \(-0.768889\pi\)
−0.747799 + 0.663925i \(0.768889\pi\)
\(450\) 9.11250e7 0.0471405
\(451\) 1.99418e9 1.02364
\(452\) 9.64378e8 0.491205
\(453\) 9.29208e8 0.469644
\(454\) −1.60846e9 −0.806707
\(455\) −3.83407e8 −0.190818
\(456\) −9.48188e7 −0.0468293
\(457\) 2.70676e9 1.32661 0.663306 0.748349i \(-0.269153\pi\)
0.663306 + 0.748349i \(0.269153\pi\)
\(458\) −2.00568e9 −0.975510
\(459\) −5.96639e8 −0.287983
\(460\) 1.93918e8 0.0928891
\(461\) −2.74743e9 −1.30609 −0.653046 0.757318i \(-0.726509\pi\)
−0.653046 + 0.757318i \(0.726509\pi\)
\(462\) −5.19789e8 −0.245234
\(463\) 1.06477e9 0.498567 0.249284 0.968431i \(-0.419805\pi\)
0.249284 + 0.968431i \(0.419805\pi\)
\(464\) −4.56502e8 −0.212144
\(465\) −4.57016e8 −0.210788
\(466\) −2.81329e9 −1.28785
\(467\) −8.23306e8 −0.374070 −0.187035 0.982353i \(-0.559888\pi\)
−0.187035 + 0.982353i \(0.559888\pi\)
\(468\) −2.31886e8 −0.104572
\(469\) 1.19358e9 0.534251
\(470\) −5.14925e8 −0.228771
\(471\) 2.85450e8 0.125880
\(472\) 1.03572e7 0.00453363
\(473\) 3.23139e9 1.40402
\(474\) 1.53968e8 0.0664059
\(475\) −1.07172e8 −0.0458831
\(476\) −1.19725e9 −0.508814
\(477\) 3.98746e8 0.168222
\(478\) 9.57046e8 0.400807
\(479\) 5.94152e8 0.247015 0.123507 0.992344i \(-0.460586\pi\)
0.123507 + 0.992344i \(0.460586\pi\)
\(480\) 1.10592e8 0.0456435
\(481\) −3.99459e8 −0.163668
\(482\) 1.70437e9 0.693265
\(483\) 4.03900e8 0.163102
\(484\) −2.74070e8 −0.109876
\(485\) 4.65778e8 0.185388
\(486\) 1.14791e8 0.0453609
\(487\) 9.50680e8 0.372978 0.186489 0.982457i \(-0.440289\pi\)
0.186489 + 0.982457i \(0.440289\pi\)
\(488\) −8.09438e8 −0.315293
\(489\) 1.06054e9 0.410152
\(490\) −4.42683e8 −0.169983
\(491\) 1.00970e9 0.384952 0.192476 0.981302i \(-0.438348\pi\)
0.192476 + 0.981302i \(0.438348\pi\)
\(492\) −8.83726e8 −0.334534
\(493\) 3.37834e9 1.26981
\(494\) 2.72721e8 0.101783
\(495\) −3.55327e8 −0.131677
\(496\) −5.54649e8 −0.204095
\(497\) 3.42929e8 0.125302
\(498\) 6.35929e8 0.230731
\(499\) −1.16136e9 −0.418423 −0.209211 0.977870i \(-0.567090\pi\)
−0.209211 + 0.977870i \(0.567090\pi\)
\(500\) 1.25000e8 0.0447214
\(501\) −5.78239e8 −0.205435
\(502\) −1.91110e9 −0.674248
\(503\) −1.84999e8 −0.0648158 −0.0324079 0.999475i \(-0.510318\pi\)
−0.0324079 + 0.999475i \(0.510318\pi\)
\(504\) 2.30346e8 0.0801444
\(505\) 2.87759e8 0.0994279
\(506\) −7.56150e8 −0.259466
\(507\) −1.02725e9 −0.350065
\(508\) −2.25115e9 −0.761871
\(509\) 1.19774e9 0.402579 0.201290 0.979532i \(-0.435487\pi\)
0.201290 + 0.979532i \(0.435487\pi\)
\(510\) −8.18436e8 −0.273205
\(511\) −1.25681e9 −0.416674
\(512\) 1.34218e8 0.0441942
\(513\) −1.35006e8 −0.0441511
\(514\) −3.68428e8 −0.119669
\(515\) −7.62449e8 −0.245972
\(516\) −1.43200e9 −0.458847
\(517\) 2.00787e9 0.639025
\(518\) 3.96805e8 0.125436
\(519\) 3.29468e9 1.03449
\(520\) −3.18088e8 −0.0992055
\(521\) 1.15136e9 0.356680 0.178340 0.983969i \(-0.442927\pi\)
0.178340 + 0.983969i \(0.442927\pi\)
\(522\) −6.49981e8 −0.200011
\(523\) −1.31591e9 −0.402227 −0.201113 0.979568i \(-0.564456\pi\)
−0.201113 + 0.979568i \(0.564456\pi\)
\(524\) 7.75014e8 0.235315
\(525\) 2.60355e8 0.0785252
\(526\) 2.82614e9 0.846727
\(527\) 4.10467e9 1.22164
\(528\) −4.31235e8 −0.127496
\(529\) −2.81726e9 −0.827432
\(530\) 5.46977e8 0.159589
\(531\) 1.47469e7 0.00427434
\(532\) −2.70909e8 −0.0780069
\(533\) 2.54180e9 0.727103
\(534\) 2.30976e9 0.656406
\(535\) −9.21476e8 −0.260163
\(536\) 9.90233e8 0.277754
\(537\) −1.50558e9 −0.419561
\(538\) 3.45162e9 0.955621
\(539\) 1.72617e9 0.474813
\(540\) 1.57464e8 0.0430331
\(541\) −4.60448e9 −1.25023 −0.625115 0.780532i \(-0.714948\pi\)
−0.625115 + 0.780532i \(0.714948\pi\)
\(542\) 2.18784e9 0.590227
\(543\) −2.99356e9 −0.802397
\(544\) −9.93278e8 −0.264530
\(545\) 2.98193e9 0.789060
\(546\) −6.62527e8 −0.174193
\(547\) −1.97338e8 −0.0515532 −0.0257766 0.999668i \(-0.508206\pi\)
−0.0257766 + 0.999668i \(0.508206\pi\)
\(548\) 1.53343e8 0.0398045
\(549\) −1.15250e9 −0.297261
\(550\) −4.87417e8 −0.124920
\(551\) 7.64440e8 0.194676
\(552\) 3.35090e8 0.0847958
\(553\) 4.39906e8 0.110617
\(554\) 1.59584e9 0.398754
\(555\) 2.71255e8 0.0673524
\(556\) −2.32189e9 −0.572901
\(557\) 1.13888e9 0.279245 0.139622 0.990205i \(-0.455411\pi\)
0.139622 + 0.990205i \(0.455411\pi\)
\(558\) −7.89724e8 −0.192422
\(559\) 4.11875e9 0.997296
\(560\) 3.15975e8 0.0760317
\(561\) 3.19136e9 0.763142
\(562\) −1.15335e9 −0.274083
\(563\) −5.72180e9 −1.35130 −0.675652 0.737221i \(-0.736138\pi\)
−0.675652 + 0.737221i \(0.736138\pi\)
\(564\) −8.89791e8 −0.208839
\(565\) 1.88355e9 0.439347
\(566\) −2.13233e9 −0.494308
\(567\) 3.27973e8 0.0755609
\(568\) 2.84506e8 0.0651437
\(569\) −5.71793e9 −1.30120 −0.650602 0.759419i \(-0.725484\pi\)
−0.650602 + 0.759419i \(0.725484\pi\)
\(570\) −1.85193e8 −0.0418854
\(571\) −1.89431e9 −0.425818 −0.212909 0.977072i \(-0.568294\pi\)
−0.212909 + 0.977072i \(0.568294\pi\)
\(572\) 1.24033e9 0.277110
\(573\) −2.58808e8 −0.0574693
\(574\) −2.52491e9 −0.557256
\(575\) 3.78745e8 0.0830826
\(576\) 1.91103e8 0.0416667
\(577\) 2.21556e9 0.480140 0.240070 0.970756i \(-0.422830\pi\)
0.240070 + 0.970756i \(0.422830\pi\)
\(578\) 4.06804e9 0.876269
\(579\) −1.04485e9 −0.223706
\(580\) −8.91606e8 −0.189747
\(581\) 1.81693e9 0.384345
\(582\) 8.04864e8 0.169236
\(583\) −2.13285e9 −0.445779
\(584\) −1.04269e9 −0.216627
\(585\) −4.52903e8 −0.0935318
\(586\) 1.90051e9 0.390147
\(587\) 4.62738e9 0.944281 0.472141 0.881523i \(-0.343481\pi\)
0.472141 + 0.881523i \(0.343481\pi\)
\(588\) −7.64956e8 −0.155173
\(589\) 9.28793e8 0.187290
\(590\) 2.02289e7 0.00405500
\(591\) 7.44203e8 0.148298
\(592\) 3.29204e8 0.0652137
\(593\) 6.14225e9 1.20958 0.604792 0.796384i \(-0.293256\pi\)
0.604792 + 0.796384i \(0.293256\pi\)
\(594\) −6.14005e8 −0.120204
\(595\) −2.33837e9 −0.455097
\(596\) −3.37265e9 −0.652544
\(597\) 4.65610e9 0.895596
\(598\) −9.63795e8 −0.184302
\(599\) −5.34698e8 −0.101652 −0.0508259 0.998708i \(-0.516185\pi\)
−0.0508259 + 0.998708i \(0.516185\pi\)
\(600\) 2.16000e8 0.0408248
\(601\) 1.36037e9 0.255620 0.127810 0.991799i \(-0.459205\pi\)
0.127810 + 0.991799i \(0.459205\pi\)
\(602\) −4.09139e9 −0.764334
\(603\) 1.40992e9 0.261869
\(604\) 2.20257e9 0.406724
\(605\) −5.35293e8 −0.0982760
\(606\) 4.97247e8 0.0907649
\(607\) −5.54565e9 −1.00645 −0.503225 0.864155i \(-0.667853\pi\)
−0.503225 + 0.864155i \(0.667853\pi\)
\(608\) −2.24756e8 −0.0405554
\(609\) −1.85707e9 −0.333172
\(610\) −1.58093e9 −0.282006
\(611\) 2.55924e9 0.453908
\(612\) −1.41426e9 −0.249401
\(613\) 7.38465e9 1.29485 0.647423 0.762131i \(-0.275847\pi\)
0.647423 + 0.762131i \(0.275847\pi\)
\(614\) 5.46842e9 0.953394
\(615\) −1.72603e9 −0.299216
\(616\) −1.23209e9 −0.212379
\(617\) 5.89706e9 1.01074 0.505368 0.862904i \(-0.331357\pi\)
0.505368 + 0.862904i \(0.331357\pi\)
\(618\) −1.31751e9 −0.224541
\(619\) 7.54341e9 1.27835 0.639176 0.769061i \(-0.279276\pi\)
0.639176 + 0.769061i \(0.279276\pi\)
\(620\) −1.08330e9 −0.182548
\(621\) 4.77110e8 0.0799462
\(622\) −1.27695e9 −0.212769
\(623\) 6.59927e9 1.09342
\(624\) −5.49656e8 −0.0905618
\(625\) 2.44141e8 0.0400000
\(626\) −6.96026e9 −1.13401
\(627\) 7.22130e8 0.116998
\(628\) 6.76621e8 0.109015
\(629\) −2.43627e9 −0.390345
\(630\) 4.49894e8 0.0716834
\(631\) 8.19812e9 1.29901 0.649504 0.760358i \(-0.274977\pi\)
0.649504 + 0.760358i \(0.274977\pi\)
\(632\) 3.64961e8 0.0575092
\(633\) −5.86053e9 −0.918384
\(634\) −4.76896e9 −0.743209
\(635\) −4.39678e9 −0.681438
\(636\) 9.45176e8 0.145684
\(637\) 2.20019e9 0.337266
\(638\) 3.47667e9 0.530019
\(639\) 4.05088e8 0.0614180
\(640\) 2.62144e8 0.0395285
\(641\) −9.39397e8 −0.140879 −0.0704395 0.997516i \(-0.522440\pi\)
−0.0704395 + 0.997516i \(0.522440\pi\)
\(642\) −1.59231e9 −0.237495
\(643\) 2.12166e9 0.314729 0.157364 0.987541i \(-0.449700\pi\)
0.157364 + 0.987541i \(0.449700\pi\)
\(644\) 9.57393e8 0.141250
\(645\) −2.79687e9 −0.410405
\(646\) 1.66330e9 0.242749
\(647\) 1.30389e10 1.89267 0.946334 0.323190i \(-0.104755\pi\)
0.946334 + 0.323190i \(0.104755\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) −7.88794e7 −0.0113268
\(650\) −6.21266e8 −0.0887321
\(651\) −2.25634e9 −0.320532
\(652\) 2.51386e9 0.355202
\(653\) 7.62830e9 1.07209 0.536046 0.844189i \(-0.319918\pi\)
0.536046 + 0.844189i \(0.319918\pi\)
\(654\) 5.15278e9 0.720310
\(655\) 1.51370e9 0.210472
\(656\) −2.09476e9 −0.289715
\(657\) −1.48462e9 −0.204237
\(658\) −2.54224e9 −0.347878
\(659\) 8.42652e9 1.14696 0.573481 0.819219i \(-0.305593\pi\)
0.573481 + 0.819219i \(0.305593\pi\)
\(660\) −8.42257e8 −0.114036
\(661\) 1.14011e10 1.53547 0.767735 0.640767i \(-0.221384\pi\)
0.767735 + 0.640767i \(0.221384\pi\)
\(662\) −8.40254e9 −1.12566
\(663\) 4.06773e9 0.542069
\(664\) 1.50739e9 0.199819
\(665\) −5.29119e8 −0.0697715
\(666\) 4.68729e8 0.0614840
\(667\) −2.70153e9 −0.352509
\(668\) −1.37064e9 −0.177912
\(669\) 1.40435e9 0.181336
\(670\) 1.93405e9 0.248431
\(671\) 6.16459e9 0.787727
\(672\) 5.46005e8 0.0694071
\(673\) −1.05006e10 −1.32789 −0.663946 0.747781i \(-0.731119\pi\)
−0.663946 + 0.747781i \(0.731119\pi\)
\(674\) −2.11166e9 −0.265653
\(675\) 3.07547e8 0.0384900
\(676\) −2.43497e9 −0.303165
\(677\) 1.18985e10 1.47377 0.736886 0.676017i \(-0.236295\pi\)
0.736886 + 0.676017i \(0.236295\pi\)
\(678\) 3.25477e9 0.401067
\(679\) 2.29960e9 0.281908
\(680\) −1.94000e9 −0.236603
\(681\) −5.42856e9 −0.658673
\(682\) 4.22414e9 0.509910
\(683\) −5.15003e8 −0.0618496 −0.0309248 0.999522i \(-0.509845\pi\)
−0.0309248 + 0.999522i \(0.509845\pi\)
\(684\) −3.20014e8 −0.0382360
\(685\) 2.99498e8 0.0356023
\(686\) −6.25150e9 −0.739349
\(687\) −6.76916e9 −0.796501
\(688\) −3.39436e9 −0.397373
\(689\) −2.71854e9 −0.316642
\(690\) 6.54472e8 0.0758437
\(691\) 3.67190e9 0.423368 0.211684 0.977338i \(-0.432105\pi\)
0.211684 + 0.977338i \(0.432105\pi\)
\(692\) 7.80961e9 0.895897
\(693\) −1.75429e9 −0.200233
\(694\) −4.28856e9 −0.487027
\(695\) −4.53494e9 −0.512418
\(696\) −1.54069e9 −0.173214
\(697\) 1.55022e10 1.73412
\(698\) −4.96832e9 −0.552988
\(699\) −9.49486e9 −1.05152
\(700\) 6.17139e8 0.0680048
\(701\) 5.94285e9 0.651601 0.325800 0.945439i \(-0.394366\pi\)
0.325800 + 0.945439i \(0.394366\pi\)
\(702\) −7.82616e8 −0.0853825
\(703\) −5.51271e8 −0.0598442
\(704\) −1.02219e9 −0.110415
\(705\) −1.73787e9 −0.186791
\(706\) 9.14803e9 0.978388
\(707\) 1.42070e9 0.151193
\(708\) 3.49556e7 0.00370169
\(709\) 4.94540e9 0.521122 0.260561 0.965457i \(-0.416093\pi\)
0.260561 + 0.965457i \(0.416093\pi\)
\(710\) 5.55676e8 0.0582663
\(711\) 5.19642e8 0.0542202
\(712\) 5.47499e9 0.568464
\(713\) −3.28235e9 −0.339134
\(714\) −4.04071e9 −0.415445
\(715\) 2.42252e9 0.247855
\(716\) −3.56879e9 −0.363350
\(717\) 3.23003e9 0.327257
\(718\) −8.23980e9 −0.830771
\(719\) 5.85264e9 0.587220 0.293610 0.955925i \(-0.405143\pi\)
0.293610 + 0.955925i \(0.405143\pi\)
\(720\) 3.73248e8 0.0372678
\(721\) −3.76429e9 −0.374033
\(722\) 3.76367e8 0.0372161
\(723\) 5.75225e9 0.566048
\(724\) −7.09586e9 −0.694896
\(725\) −1.74142e9 −0.169715
\(726\) −9.24986e8 −0.0897133
\(727\) −4.76651e9 −0.460076 −0.230038 0.973182i \(-0.573885\pi\)
−0.230038 + 0.973182i \(0.573885\pi\)
\(728\) −1.57044e9 −0.150855
\(729\) 3.87420e8 0.0370370
\(730\) −2.03651e9 −0.193757
\(731\) 2.51199e10 2.37853
\(732\) −2.73185e9 −0.257436
\(733\) 6.90084e9 0.647199 0.323600 0.946194i \(-0.395107\pi\)
0.323600 + 0.946194i \(0.395107\pi\)
\(734\) −3.29685e9 −0.307725
\(735\) −1.49405e9 −0.138791
\(736\) 7.94287e8 0.0734353
\(737\) −7.54151e9 −0.693940
\(738\) −2.98258e9 −0.273146
\(739\) −1.43630e10 −1.30915 −0.654577 0.755995i \(-0.727153\pi\)
−0.654577 + 0.755995i \(0.727153\pi\)
\(740\) 6.42976e8 0.0583289
\(741\) 9.20433e8 0.0831052
\(742\) 2.70048e9 0.242677
\(743\) 1.06577e10 0.953240 0.476620 0.879109i \(-0.341862\pi\)
0.476620 + 0.879109i \(0.341862\pi\)
\(744\) −1.87194e9 −0.166643
\(745\) −6.58721e9 −0.583653
\(746\) 3.28156e9 0.289397
\(747\) 2.14626e9 0.188391
\(748\) 7.56470e9 0.660900
\(749\) −4.54943e9 −0.395613
\(750\) 4.21875e8 0.0365148
\(751\) 2.86094e9 0.246473 0.123236 0.992377i \(-0.460673\pi\)
0.123236 + 0.992377i \(0.460673\pi\)
\(752\) −2.10913e9 −0.180860
\(753\) −6.44995e9 −0.550521
\(754\) 4.43139e9 0.376479
\(755\) 4.30189e9 0.363785
\(756\) 7.77417e8 0.0654377
\(757\) −7.22882e8 −0.0605664 −0.0302832 0.999541i \(-0.509641\pi\)
−0.0302832 + 0.999541i \(0.509641\pi\)
\(758\) 6.25054e9 0.521285
\(759\) −2.55201e9 −0.211854
\(760\) −4.38976e8 −0.0362738
\(761\) 1.61196e10 1.32589 0.662946 0.748668i \(-0.269306\pi\)
0.662946 + 0.748668i \(0.269306\pi\)
\(762\) −7.59763e9 −0.622065
\(763\) 1.47221e10 1.19987
\(764\) −6.13470e8 −0.0497699
\(765\) −2.76222e9 −0.223071
\(766\) 1.12574e10 0.904974
\(767\) −1.00540e8 −0.00804556
\(768\) 4.52985e8 0.0360844
\(769\) −2.48228e10 −1.96838 −0.984189 0.177124i \(-0.943321\pi\)
−0.984189 + 0.177124i \(0.943321\pi\)
\(770\) −2.40643e9 −0.189957
\(771\) −1.24344e9 −0.0977092
\(772\) −2.47667e9 −0.193735
\(773\) −1.24574e10 −0.970060 −0.485030 0.874498i \(-0.661191\pi\)
−0.485030 + 0.874498i \(0.661191\pi\)
\(774\) −4.83299e9 −0.374647
\(775\) −2.11582e9 −0.163276
\(776\) 1.90783e9 0.146562
\(777\) 1.33922e9 0.102418
\(778\) −7.28933e9 −0.554956
\(779\) 3.50780e9 0.265860
\(780\) −1.07355e9 −0.0810009
\(781\) −2.16677e9 −0.162755
\(782\) −5.87811e9 −0.439556
\(783\) −2.19368e9 −0.163308
\(784\) −1.81323e9 −0.134384
\(785\) 1.32153e9 0.0975061
\(786\) 2.61567e9 0.192134
\(787\) 3.95767e9 0.289420 0.144710 0.989474i \(-0.453775\pi\)
0.144710 + 0.989474i \(0.453775\pi\)
\(788\) 1.76404e9 0.128430
\(789\) 9.53822e9 0.691349
\(790\) 7.12815e8 0.0514378
\(791\) 9.29929e9 0.668086
\(792\) −1.45542e9 −0.104100
\(793\) 7.85744e9 0.559532
\(794\) 9.84496e9 0.697978
\(795\) 1.84605e9 0.130304
\(796\) 1.10367e10 0.775609
\(797\) 1.61800e10 1.13208 0.566038 0.824379i \(-0.308476\pi\)
0.566038 + 0.824379i \(0.308476\pi\)
\(798\) −9.14318e8 −0.0636923
\(799\) 1.56086e10 1.08256
\(800\) 5.12000e8 0.0353553
\(801\) 7.79544e9 0.535953
\(802\) −2.63215e8 −0.0180177
\(803\) 7.94104e9 0.541219
\(804\) 3.34204e9 0.226785
\(805\) 1.86991e9 0.126338
\(806\) 5.38413e9 0.362195
\(807\) 1.16492e10 0.780261
\(808\) 1.17866e9 0.0786047
\(809\) −1.54849e10 −1.02823 −0.514114 0.857722i \(-0.671879\pi\)
−0.514114 + 0.857722i \(0.671879\pi\)
\(810\) 5.31441e8 0.0351364
\(811\) −3.51012e9 −0.231073 −0.115536 0.993303i \(-0.536859\pi\)
−0.115536 + 0.993303i \(0.536859\pi\)
\(812\) −4.40195e9 −0.288536
\(813\) 7.38398e9 0.481918
\(814\) −2.50718e9 −0.162930
\(815\) 4.90989e9 0.317702
\(816\) −3.35231e9 −0.215988
\(817\) 5.68406e9 0.364655
\(818\) 6.47498e9 0.413620
\(819\) −2.23603e9 −0.142228
\(820\) −4.09132e9 −0.259129
\(821\) 1.63061e10 1.02837 0.514184 0.857680i \(-0.328095\pi\)
0.514184 + 0.857680i \(0.328095\pi\)
\(822\) 5.17533e8 0.0325003
\(823\) −2.53495e10 −1.58515 −0.792574 0.609776i \(-0.791259\pi\)
−0.792574 + 0.609776i \(0.791259\pi\)
\(824\) −3.12299e9 −0.194458
\(825\) −1.64503e9 −0.101997
\(826\) 9.98724e7 0.00616617
\(827\) 6.40564e9 0.393816 0.196908 0.980422i \(-0.436910\pi\)
0.196908 + 0.980422i \(0.436910\pi\)
\(828\) 1.13093e9 0.0692355
\(829\) −2.66102e10 −1.62221 −0.811106 0.584899i \(-0.801134\pi\)
−0.811106 + 0.584899i \(0.801134\pi\)
\(830\) 2.94412e9 0.178724
\(831\) 5.38596e9 0.325581
\(832\) −1.30289e9 −0.0784288
\(833\) 1.34188e10 0.804370
\(834\) −7.83637e9 −0.467771
\(835\) −2.67703e9 −0.159130
\(836\) 1.71172e9 0.101323
\(837\) −2.66532e9 −0.157112
\(838\) 1.35246e10 0.793908
\(839\) 1.24536e10 0.727995 0.363997 0.931400i \(-0.381412\pi\)
0.363997 + 0.931400i \(0.381412\pi\)
\(840\) 1.06642e9 0.0620796
\(841\) −4.82862e9 −0.279922
\(842\) 1.54633e10 0.892707
\(843\) −3.89254e9 −0.223788
\(844\) −1.38916e10 −0.795343
\(845\) −4.75580e9 −0.271159
\(846\) −3.00304e9 −0.170516
\(847\) −2.64280e9 −0.149442
\(848\) 2.24042e9 0.126166
\(849\) −7.19662e9 −0.403601
\(850\) −3.78905e9 −0.211624
\(851\) 1.94819e9 0.108362
\(852\) 9.60208e8 0.0531896
\(853\) 1.74849e10 0.964590 0.482295 0.876009i \(-0.339803\pi\)
0.482295 + 0.876009i \(0.339803\pi\)
\(854\) −7.80524e9 −0.428829
\(855\) −6.25026e8 −0.0341993
\(856\) −3.77437e9 −0.205677
\(857\) 1.42851e10 0.775263 0.387631 0.921814i \(-0.373293\pi\)
0.387631 + 0.921814i \(0.373293\pi\)
\(858\) 4.18612e9 0.226259
\(859\) −3.03280e10 −1.63256 −0.816278 0.577659i \(-0.803966\pi\)
−0.816278 + 0.577659i \(0.803966\pi\)
\(860\) −6.62961e9 −0.355421
\(861\) −8.52159e9 −0.454998
\(862\) −8.25311e9 −0.438876
\(863\) 2.39179e10 1.26673 0.633366 0.773852i \(-0.281673\pi\)
0.633366 + 0.773852i \(0.281673\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) 1.52532e10 0.801315
\(866\) −1.34157e9 −0.0701940
\(867\) 1.37296e10 0.715471
\(868\) −5.34836e9 −0.277589
\(869\) −2.77951e9 −0.143681
\(870\) −3.00917e9 −0.154928
\(871\) −9.61246e9 −0.492914
\(872\) 1.22140e10 0.623807
\(873\) 2.71642e9 0.138180
\(874\) −1.33008e9 −0.0673889
\(875\) 1.20535e9 0.0608253
\(876\) −3.51909e9 −0.176875
\(877\) −1.17981e10 −0.590627 −0.295314 0.955400i \(-0.595424\pi\)
−0.295314 + 0.955400i \(0.595424\pi\)
\(878\) −9.99109e9 −0.498175
\(879\) 6.41421e9 0.318554
\(880\) −1.99646e9 −0.0987578
\(881\) 2.29166e9 0.112911 0.0564553 0.998405i \(-0.482020\pi\)
0.0564553 + 0.998405i \(0.482020\pi\)
\(882\) −2.58173e9 −0.126698
\(883\) −3.40653e10 −1.66514 −0.832568 0.553922i \(-0.813130\pi\)
−0.832568 + 0.553922i \(0.813130\pi\)
\(884\) 9.64202e9 0.469445
\(885\) 6.82726e7 0.00331089
\(886\) −1.19502e10 −0.577239
\(887\) −3.23552e9 −0.155672 −0.0778361 0.996966i \(-0.524801\pi\)
−0.0778361 + 0.996966i \(0.524801\pi\)
\(888\) 1.11106e9 0.0532467
\(889\) −2.17074e10 −1.03622
\(890\) 1.06933e10 0.508450
\(891\) −2.07227e9 −0.0981463
\(892\) 3.32883e9 0.157042
\(893\) 3.53187e9 0.165968
\(894\) −1.13827e10 −0.532800
\(895\) −6.97030e9 −0.324991
\(896\) 1.29423e9 0.0601083
\(897\) −3.25281e9 −0.150482
\(898\) −2.29492e10 −1.05755
\(899\) 1.50918e10 0.692758
\(900\) 7.29000e8 0.0333333
\(901\) −1.65802e10 −0.755184
\(902\) 1.59535e10 0.723822
\(903\) −1.38084e10 −0.624076
\(904\) 7.71502e9 0.347334
\(905\) −1.38591e10 −0.621534
\(906\) 7.43366e9 0.332089
\(907\) −8.16332e8 −0.0363280 −0.0181640 0.999835i \(-0.505782\pi\)
−0.0181640 + 0.999835i \(0.505782\pi\)
\(908\) −1.28677e10 −0.570428
\(909\) 1.67821e9 0.0741092
\(910\) −3.06726e9 −0.134929
\(911\) 2.35955e10 1.03399 0.516994 0.855989i \(-0.327051\pi\)
0.516994 + 0.855989i \(0.327051\pi\)
\(912\) −7.58551e8 −0.0331133
\(913\) −1.14801e10 −0.499227
\(914\) 2.16541e10 0.938056
\(915\) −5.33565e9 −0.230257
\(916\) −1.60454e10 −0.689790
\(917\) 7.47330e9 0.320051
\(918\) −4.77312e9 −0.203635
\(919\) −3.94356e10 −1.67604 −0.838020 0.545640i \(-0.816287\pi\)
−0.838020 + 0.545640i \(0.816287\pi\)
\(920\) 1.55134e9 0.0656825
\(921\) 1.84559e10 0.778443
\(922\) −2.19795e10 −0.923546
\(923\) −2.76178e9 −0.115607
\(924\) −4.15831e9 −0.173406
\(925\) 1.25581e9 0.0521709
\(926\) 8.51819e9 0.352540
\(927\) −4.44660e9 −0.183337
\(928\) −3.65202e9 −0.150008
\(929\) 3.67740e10 1.50483 0.752413 0.658692i \(-0.228890\pi\)
0.752413 + 0.658692i \(0.228890\pi\)
\(930\) −3.65613e9 −0.149050
\(931\) 3.03636e9 0.123319
\(932\) −2.25063e10 −0.910645
\(933\) −4.30971e9 −0.173725
\(934\) −6.58645e9 −0.264507
\(935\) 1.47748e10 0.591127
\(936\) −1.85509e9 −0.0739434
\(937\) 2.25171e10 0.894179 0.447089 0.894489i \(-0.352461\pi\)
0.447089 + 0.894489i \(0.352461\pi\)
\(938\) 9.54861e9 0.377772
\(939\) −2.34909e10 −0.925912
\(940\) −4.11940e9 −0.161766
\(941\) −1.99173e10 −0.779230 −0.389615 0.920978i \(-0.627392\pi\)
−0.389615 + 0.920978i \(0.627392\pi\)
\(942\) 2.28360e9 0.0890105
\(943\) −1.23966e10 −0.481405
\(944\) 8.28576e7 0.00320576
\(945\) 1.51839e9 0.0585292
\(946\) 2.58511e10 0.992795
\(947\) 6.22928e9 0.238349 0.119174 0.992873i \(-0.461975\pi\)
0.119174 + 0.992873i \(0.461975\pi\)
\(948\) 1.23174e9 0.0469561
\(949\) 1.01217e10 0.384435
\(950\) −8.57375e8 −0.0324443
\(951\) −1.60952e10 −0.606828
\(952\) −9.57797e9 −0.359786
\(953\) −3.58488e10 −1.34168 −0.670840 0.741602i \(-0.734066\pi\)
−0.670840 + 0.741602i \(0.734066\pi\)
\(954\) 3.18997e9 0.118951
\(955\) −1.19818e9 −0.0445155
\(956\) 7.65637e9 0.283413
\(957\) 1.17338e10 0.432758
\(958\) 4.75321e9 0.174666
\(959\) 1.47866e9 0.0541380
\(960\) 8.84736e8 0.0322749
\(961\) −9.17614e9 −0.333525
\(962\) −3.19567e9 −0.115731
\(963\) −5.37405e9 −0.193914
\(964\) 1.36350e10 0.490212
\(965\) −4.83725e9 −0.173282
\(966\) 3.23120e9 0.115330
\(967\) −3.86678e10 −1.37517 −0.687586 0.726103i \(-0.741330\pi\)
−0.687586 + 0.726103i \(0.741330\pi\)
\(968\) −2.19256e9 −0.0776940
\(969\) 5.61365e9 0.198204
\(970\) 3.72622e9 0.131089
\(971\) −1.32865e10 −0.465738 −0.232869 0.972508i \(-0.574811\pi\)
−0.232869 + 0.972508i \(0.574811\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) −2.23895e10 −0.779200
\(974\) 7.60544e9 0.263735
\(975\) −2.09677e9 −0.0724494
\(976\) −6.47550e9 −0.222946
\(977\) −3.41614e10 −1.17194 −0.585969 0.810334i \(-0.699286\pi\)
−0.585969 + 0.810334i \(0.699286\pi\)
\(978\) 8.48429e9 0.290021
\(979\) −4.16969e10 −1.42025
\(980\) −3.54146e9 −0.120196
\(981\) 1.73906e10 0.588131
\(982\) 8.07759e9 0.272202
\(983\) −4.25214e10 −1.42781 −0.713905 0.700243i \(-0.753075\pi\)
−0.713905 + 0.700243i \(0.753075\pi\)
\(984\) −7.06981e9 −0.236551
\(985\) 3.44538e9 0.114871
\(986\) 2.70267e10 0.897892
\(987\) −8.58007e9 −0.284041
\(988\) 2.18177e9 0.0719712
\(989\) −2.00875e10 −0.660295
\(990\) −2.84262e9 −0.0931097
\(991\) 1.65974e10 0.541729 0.270865 0.962617i \(-0.412690\pi\)
0.270865 + 0.962617i \(0.412690\pi\)
\(992\) −4.43719e9 −0.144317
\(993\) −2.83586e10 −0.919098
\(994\) 2.74343e9 0.0886016
\(995\) 2.15560e10 0.693726
\(996\) 5.08744e9 0.163152
\(997\) −2.20018e10 −0.703114 −0.351557 0.936167i \(-0.614348\pi\)
−0.351557 + 0.936167i \(0.614348\pi\)
\(998\) −9.29088e9 −0.295869
\(999\) 1.58196e9 0.0502015
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 570.8.a.c.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.8.a.c.1.4 4 1.1 even 1 trivial