Properties

Label 570.8.a.c.1.3
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 1.3
Root \(20.3008\) of defining polynomial
Character \(\chi\) \(=\) 570.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} +216.000 q^{6} -332.718 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} +216.000 q^{6} -332.718 q^{7} +512.000 q^{8} +729.000 q^{9} +1000.00 q^{10} -3927.61 q^{11} +1728.00 q^{12} -14291.3 q^{13} -2661.75 q^{14} +3375.00 q^{15} +4096.00 q^{16} +30583.4 q^{17} +5832.00 q^{18} -6859.00 q^{19} +8000.00 q^{20} -8983.39 q^{21} -31420.9 q^{22} +21908.3 q^{23} +13824.0 q^{24} +15625.0 q^{25} -114331. q^{26} +19683.0 q^{27} -21294.0 q^{28} +99781.4 q^{29} +27000.0 q^{30} -43129.8 q^{31} +32768.0 q^{32} -106046. q^{33} +244667. q^{34} -41589.8 q^{35} +46656.0 q^{36} +18702.6 q^{37} -54872.0 q^{38} -385866. q^{39} +64000.0 q^{40} -550617. q^{41} -71867.1 q^{42} +695496. q^{43} -251367. q^{44} +91125.0 q^{45} +175266. q^{46} -1.03557e6 q^{47} +110592. q^{48} -712842. q^{49} +125000. q^{50} +825751. q^{51} -914645. q^{52} -975206. q^{53} +157464. q^{54} -490952. q^{55} -170352. q^{56} -185193. q^{57} +798251. q^{58} -2.85563e6 q^{59} +216000. q^{60} -1.48398e6 q^{61} -345038. q^{62} -242552. q^{63} +262144. q^{64} -1.78642e6 q^{65} -848364. q^{66} -3.77653e6 q^{67} +1.95734e6 q^{68} +591523. q^{69} -332718. q^{70} -1.17658e6 q^{71} +373248. q^{72} +3.93268e6 q^{73} +149621. q^{74} +421875. q^{75} -438976. q^{76} +1.30679e6 q^{77} -3.08693e6 q^{78} +1.87113e6 q^{79} +512000. q^{80} +531441. q^{81} -4.40494e6 q^{82} +2.44065e6 q^{83} -574937. q^{84} +3.82292e6 q^{85} +5.56397e6 q^{86} +2.69410e6 q^{87} -2.01094e6 q^{88} -5.65647e6 q^{89} +729000. q^{90} +4.75499e6 q^{91} +1.40213e6 q^{92} -1.16450e6 q^{93} -8.28456e6 q^{94} -857375. q^{95} +884736. q^{96} -2.38713e6 q^{97} -5.70273e6 q^{98} -2.86323e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} + 500q^{5} + 864q^{6} - 1316q^{7} + 2048q^{8} + 2916q^{9} + O(q^{10}) \) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} + 500q^{5} + 864q^{6} - 1316q^{7} + 2048q^{8} + 2916q^{9} + 4000q^{10} - 6570q^{11} + 6912q^{12} - 17578q^{13} - 10528q^{14} + 13500q^{15} + 16384q^{16} - 28466q^{17} + 23328q^{18} - 27436q^{19} + 32000q^{20} - 35532q^{21} - 52560q^{22} - 32136q^{23} + 55296q^{24} + 62500q^{25} - 140624q^{26} + 78732q^{27} - 84224q^{28} - 159122q^{29} + 108000q^{30} - 67974q^{31} + 131072q^{32} - 177390q^{33} - 227728q^{34} - 164500q^{35} + 186624q^{36} - 823702q^{37} - 219488q^{38} - 474606q^{39} + 256000q^{40} - 781924q^{41} - 284256q^{42} - 1115638q^{43} - 420480q^{44} + 364500q^{45} - 257088q^{46} - 209160q^{47} + 442368q^{48} - 1159308q^{49} + 500000q^{50} - 768582q^{51} - 1124992q^{52} - 848424q^{53} + 629856q^{54} - 821250q^{55} - 673792q^{56} - 740772q^{57} - 1272976q^{58} - 3677830q^{59} + 864000q^{60} - 1161072q^{61} - 543792q^{62} - 959364q^{63} + 1048576q^{64} - 2197250q^{65} - 1419120q^{66} - 6154740q^{67} - 1821824q^{68} - 867672q^{69} - 1316000q^{70} - 4456224q^{71} + 1492992q^{72} + 1057792q^{73} - 6589616q^{74} + 1687500q^{75} - 1755904q^{76} + 2402388q^{77} - 3796848q^{78} + 2910090q^{79} + 2048000q^{80} + 2125764q^{81} - 6255392q^{82} - 1767198q^{83} - 2274048q^{84} - 3558250q^{85} - 8925104q^{86} - 4296294q^{87} - 3363840q^{88} - 3677360q^{89} + 2916000q^{90} - 6727732q^{91} - 2056704q^{92} - 1835298q^{93} - 1673280q^{94} - 3429500q^{95} + 3538944q^{96} - 10419094q^{97} - 9274464q^{98} - 4789530q^{99} + O(q^{100}) \)

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
\(7\) −332.718 −0.366635 −0.183317 0.983054i \(-0.558684\pi\)
−0.183317 + 0.983054i \(0.558684\pi\)
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) 1000.00 0.316228
\(11\) −3927.61 −0.889722 −0.444861 0.895600i \(-0.646747\pi\)
−0.444861 + 0.895600i \(0.646747\pi\)
\(12\) 1728.00 0.288675
\(13\) −14291.3 −1.80414 −0.902071 0.431587i \(-0.857954\pi\)
−0.902071 + 0.431587i \(0.857954\pi\)
\(14\) −2661.75 −0.259250
\(15\) 3375.00 0.258199
\(16\) 4096.00 0.250000
\(17\) 30583.4 1.50978 0.754891 0.655850i \(-0.227690\pi\)
0.754891 + 0.655850i \(0.227690\pi\)
\(18\) 5832.00 0.235702
\(19\) −6859.00 −0.229416
\(20\) 8000.00 0.223607
\(21\) −8983.39 −0.211677
\(22\) −31420.9 −0.629128
\(23\) 21908.3 0.375457 0.187729 0.982221i \(-0.439887\pi\)
0.187729 + 0.982221i \(0.439887\pi\)
\(24\) 13824.0 0.204124
\(25\) 15625.0 0.200000
\(26\) −114331. −1.27572
\(27\) 19683.0 0.192450
\(28\) −21294.0 −0.183317
\(29\) 99781.4 0.759725 0.379863 0.925043i \(-0.375971\pi\)
0.379863 + 0.925043i \(0.375971\pi\)
\(30\) 27000.0 0.182574
\(31\) −43129.8 −0.260023 −0.130011 0.991513i \(-0.541501\pi\)
−0.130011 + 0.991513i \(0.541501\pi\)
\(32\) 32768.0 0.176777
\(33\) −106046. −0.513681
\(34\) 244667. 1.06758
\(35\) −41589.8 −0.163964
\(36\) 46656.0 0.166667
\(37\) 18702.6 0.0607011 0.0303506 0.999539i \(-0.490338\pi\)
0.0303506 + 0.999539i \(0.490338\pi\)
\(38\) −54872.0 −0.162221
\(39\) −385866. −1.04162
\(40\) 64000.0 0.158114
\(41\) −550617. −1.24769 −0.623844 0.781549i \(-0.714430\pi\)
−0.623844 + 0.781549i \(0.714430\pi\)
\(42\) −71867.1 −0.149678
\(43\) 695496. 1.33400 0.666999 0.745058i \(-0.267578\pi\)
0.666999 + 0.745058i \(0.267578\pi\)
\(44\) −251367. −0.444861
\(45\) 91125.0 0.149071
\(46\) 175266. 0.265488
\(47\) −1.03557e6 −1.45491 −0.727456 0.686154i \(-0.759298\pi\)
−0.727456 + 0.686154i \(0.759298\pi\)
\(48\) 110592. 0.144338
\(49\) −712842. −0.865579
\(50\) 125000. 0.141421
\(51\) 825751. 0.871673
\(52\) −914645. −0.902071
\(53\) −975206. −0.899768 −0.449884 0.893087i \(-0.648535\pi\)
−0.449884 + 0.893087i \(0.648535\pi\)
\(54\) 157464. 0.136083
\(55\) −490952. −0.397896
\(56\) −170352. −0.129625
\(57\) −185193. −0.132453
\(58\) 798251. 0.537207
\(59\) −2.85563e6 −1.81017 −0.905087 0.425227i \(-0.860194\pi\)
−0.905087 + 0.425227i \(0.860194\pi\)
\(60\) 216000. 0.129099
\(61\) −1.48398e6 −0.837092 −0.418546 0.908196i \(-0.637460\pi\)
−0.418546 + 0.908196i \(0.637460\pi\)
\(62\) −345038. −0.183864
\(63\) −242552. −0.122212
\(64\) 262144. 0.125000
\(65\) −1.78642e6 −0.806837
\(66\) −848364. −0.363227
\(67\) −3.77653e6 −1.53402 −0.767010 0.641636i \(-0.778256\pi\)
−0.767010 + 0.641636i \(0.778256\pi\)
\(68\) 1.95734e6 0.754891
\(69\) 591523. 0.216770
\(70\) −332718. −0.115940
\(71\) −1.17658e6 −0.390138 −0.195069 0.980790i \(-0.562493\pi\)
−0.195069 + 0.980790i \(0.562493\pi\)
\(72\) 373248. 0.117851
\(73\) 3.93268e6 1.18320 0.591601 0.806231i \(-0.298496\pi\)
0.591601 + 0.806231i \(0.298496\pi\)
\(74\) 149621. 0.0429222
\(75\) 421875. 0.115470
\(76\) −438976. −0.114708
\(77\) 1.30679e6 0.326203
\(78\) −3.08693e6 −0.736538
\(79\) 1.87113e6 0.426981 0.213491 0.976945i \(-0.431517\pi\)
0.213491 + 0.976945i \(0.431517\pi\)
\(80\) 512000. 0.111803
\(81\) 531441. 0.111111
\(82\) −4.40494e6 −0.882249
\(83\) 2.44065e6 0.468525 0.234263 0.972173i \(-0.424732\pi\)
0.234263 + 0.972173i \(0.424732\pi\)
\(84\) −574937. −0.105838
\(85\) 3.82292e6 0.675195
\(86\) 5.56397e6 0.943279
\(87\) 2.69410e6 0.438628
\(88\) −2.01094e6 −0.314564
\(89\) −5.65647e6 −0.850512 −0.425256 0.905073i \(-0.639816\pi\)
−0.425256 + 0.905073i \(0.639816\pi\)
\(90\) 729000. 0.105409
\(91\) 4.75499e6 0.661461
\(92\) 1.40213e6 0.187729
\(93\) −1.16450e6 −0.150124
\(94\) −8.28456e6 −1.02878
\(95\) −857375. −0.102598
\(96\) 884736. 0.102062
\(97\) −2.38713e6 −0.265567 −0.132784 0.991145i \(-0.542392\pi\)
−0.132784 + 0.991145i \(0.542392\pi\)
\(98\) −5.70273e6 −0.612057
\(99\) −2.86323e6 −0.296574
\(100\) 1.00000e6 0.100000
\(101\) −6.61014e6 −0.638390 −0.319195 0.947689i \(-0.603412\pi\)
−0.319195 + 0.947689i \(0.603412\pi\)
\(102\) 6.60601e6 0.616366
\(103\) −1.75261e6 −0.158036 −0.0790180 0.996873i \(-0.525178\pi\)
−0.0790180 + 0.996873i \(0.525178\pi\)
\(104\) −7.31716e6 −0.637861
\(105\) −1.12292e6 −0.0946647
\(106\) −7.80165e6 −0.636232
\(107\) −1.39199e6 −0.109848 −0.0549242 0.998491i \(-0.517492\pi\)
−0.0549242 + 0.998491i \(0.517492\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) −1.22583e7 −0.906646 −0.453323 0.891346i \(-0.649762\pi\)
−0.453323 + 0.891346i \(0.649762\pi\)
\(110\) −3.92761e6 −0.281355
\(111\) 504971. 0.0350458
\(112\) −1.36281e6 −0.0916587
\(113\) −6.44735e6 −0.420346 −0.210173 0.977664i \(-0.567403\pi\)
−0.210173 + 0.977664i \(0.567403\pi\)
\(114\) −1.48154e6 −0.0936586
\(115\) 2.73853e6 0.167910
\(116\) 6.38601e6 0.379863
\(117\) −1.04184e7 −0.601381
\(118\) −2.28451e7 −1.27999
\(119\) −1.01757e7 −0.553539
\(120\) 1.72800e6 0.0912871
\(121\) −4.06103e6 −0.208395
\(122\) −1.18718e7 −0.591914
\(123\) −1.48667e7 −0.720353
\(124\) −2.76031e6 −0.130011
\(125\) 1.95312e6 0.0894427
\(126\) −1.94041e6 −0.0864166
\(127\) 8.64444e6 0.374476 0.187238 0.982315i \(-0.440046\pi\)
0.187238 + 0.982315i \(0.440046\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) 1.87784e7 0.770184
\(130\) −1.42913e7 −0.570520
\(131\) 1.04588e7 0.406474 0.203237 0.979130i \(-0.434854\pi\)
0.203237 + 0.979130i \(0.434854\pi\)
\(132\) −6.78691e6 −0.256841
\(133\) 2.28211e6 0.0841118
\(134\) −3.02122e7 −1.08472
\(135\) 2.46038e6 0.0860663
\(136\) 1.56587e7 0.533789
\(137\) −2.79751e7 −0.929499 −0.464749 0.885442i \(-0.653856\pi\)
−0.464749 + 0.885442i \(0.653856\pi\)
\(138\) 4.73218e6 0.153280
\(139\) 1.99331e6 0.0629541 0.0314770 0.999504i \(-0.489979\pi\)
0.0314770 + 0.999504i \(0.489979\pi\)
\(140\) −2.66175e6 −0.0819820
\(141\) −2.79604e7 −0.839994
\(142\) −9.41266e6 −0.275869
\(143\) 5.61308e7 1.60518
\(144\) 2.98598e6 0.0833333
\(145\) 1.24727e7 0.339760
\(146\) 3.14615e7 0.836650
\(147\) −1.92467e7 −0.499742
\(148\) 1.19697e6 0.0303506
\(149\) 2.05022e7 0.507748 0.253874 0.967237i \(-0.418295\pi\)
0.253874 + 0.967237i \(0.418295\pi\)
\(150\) 3.37500e6 0.0816497
\(151\) −1.90022e7 −0.449143 −0.224572 0.974458i \(-0.572098\pi\)
−0.224572 + 0.974458i \(0.572098\pi\)
\(152\) −3.51181e6 −0.0811107
\(153\) 2.22953e7 0.503261
\(154\) 1.04543e7 0.230660
\(155\) −5.39122e6 −0.116286
\(156\) −2.46954e7 −0.520811
\(157\) −4.41247e7 −0.909982 −0.454991 0.890496i \(-0.650357\pi\)
−0.454991 + 0.890496i \(0.650357\pi\)
\(158\) 1.49690e7 0.301921
\(159\) −2.63306e7 −0.519482
\(160\) 4.09600e6 0.0790569
\(161\) −7.28928e6 −0.137656
\(162\) 4.25153e6 0.0785674
\(163\) 2.34216e7 0.423603 0.211802 0.977313i \(-0.432067\pi\)
0.211802 + 0.977313i \(0.432067\pi\)
\(164\) −3.52395e7 −0.623844
\(165\) −1.32557e7 −0.229725
\(166\) 1.95252e7 0.331297
\(167\) 1.13995e7 0.189399 0.0946994 0.995506i \(-0.469811\pi\)
0.0946994 + 0.995506i \(0.469811\pi\)
\(168\) −4.59950e6 −0.0748390
\(169\) 1.41494e8 2.25493
\(170\) 3.05834e7 0.477435
\(171\) −5.00021e6 −0.0764719
\(172\) 4.45118e7 0.666999
\(173\) −5.77373e6 −0.0847803 −0.0423901 0.999101i \(-0.513497\pi\)
−0.0423901 + 0.999101i \(0.513497\pi\)
\(174\) 2.15528e7 0.310157
\(175\) −5.19872e6 −0.0733269
\(176\) −1.60875e7 −0.222430
\(177\) −7.71021e7 −1.04510
\(178\) −4.52517e7 −0.601403
\(179\) −1.01319e8 −1.32040 −0.660200 0.751090i \(-0.729529\pi\)
−0.660200 + 0.751090i \(0.729529\pi\)
\(180\) 5.83200e6 0.0745356
\(181\) 1.10505e8 1.38519 0.692594 0.721328i \(-0.256468\pi\)
0.692594 + 0.721328i \(0.256468\pi\)
\(182\) 3.80399e7 0.467724
\(183\) −4.00674e7 −0.483295
\(184\) 1.12170e7 0.132744
\(185\) 2.33783e6 0.0271464
\(186\) −9.31603e6 −0.106154
\(187\) −1.20120e8 −1.34329
\(188\) −6.62765e7 −0.727456
\(189\) −6.54889e6 −0.0705589
\(190\) −6.85900e6 −0.0725476
\(191\) −8.93938e7 −0.928305 −0.464153 0.885755i \(-0.653641\pi\)
−0.464153 + 0.885755i \(0.653641\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) −5.21763e7 −0.522424 −0.261212 0.965281i \(-0.584122\pi\)
−0.261212 + 0.965281i \(0.584122\pi\)
\(194\) −1.90970e7 −0.187784
\(195\) −4.82332e7 −0.465828
\(196\) −4.56219e7 −0.432790
\(197\) −1.03049e8 −0.960310 −0.480155 0.877184i \(-0.659420\pi\)
−0.480155 + 0.877184i \(0.659420\pi\)
\(198\) −2.29058e7 −0.209709
\(199\) 7.32063e7 0.658510 0.329255 0.944241i \(-0.393202\pi\)
0.329255 + 0.944241i \(0.393202\pi\)
\(200\) 8.00000e6 0.0707107
\(201\) −1.01966e8 −0.885666
\(202\) −5.28811e7 −0.451410
\(203\) −3.31991e7 −0.278542
\(204\) 5.28481e7 0.435837
\(205\) −6.88271e7 −0.557983
\(206\) −1.40209e7 −0.111748
\(207\) 1.59711e7 0.125152
\(208\) −5.85373e7 −0.451036
\(209\) 2.69395e7 0.204116
\(210\) −8.98339e6 −0.0669380
\(211\) 2.38061e6 0.0174462 0.00872309 0.999962i \(-0.497223\pi\)
0.00872309 + 0.999962i \(0.497223\pi\)
\(212\) −6.24132e7 −0.449884
\(213\) −3.17677e7 −0.225246
\(214\) −1.11359e7 −0.0776745
\(215\) 8.69370e7 0.596582
\(216\) 1.00777e7 0.0680414
\(217\) 1.43501e7 0.0953333
\(218\) −9.80665e7 −0.641096
\(219\) 1.06182e8 0.683122
\(220\) −3.14209e7 −0.198948
\(221\) −4.37077e8 −2.72386
\(222\) 4.03977e6 0.0247811
\(223\) −1.26275e8 −0.762516 −0.381258 0.924469i \(-0.624509\pi\)
−0.381258 + 0.924469i \(0.624509\pi\)
\(224\) −1.09025e7 −0.0648125
\(225\) 1.13906e7 0.0666667
\(226\) −5.15788e7 −0.297229
\(227\) 2.82547e8 1.60324 0.801622 0.597831i \(-0.203971\pi\)
0.801622 + 0.597831i \(0.203971\pi\)
\(228\) −1.18524e7 −0.0662266
\(229\) 3.07407e8 1.69157 0.845783 0.533527i \(-0.179134\pi\)
0.845783 + 0.533527i \(0.179134\pi\)
\(230\) 2.19083e7 0.118730
\(231\) 3.52833e7 0.188333
\(232\) 5.10881e7 0.268604
\(233\) 3.54739e8 1.83723 0.918616 0.395152i \(-0.129308\pi\)
0.918616 + 0.395152i \(0.129308\pi\)
\(234\) −8.33470e7 −0.425240
\(235\) −1.29446e8 −0.650657
\(236\) −1.82760e8 −0.905087
\(237\) 5.05205e7 0.246518
\(238\) −8.14052e7 −0.391411
\(239\) 6.74968e7 0.319809 0.159904 0.987132i \(-0.448881\pi\)
0.159904 + 0.987132i \(0.448881\pi\)
\(240\) 1.38240e7 0.0645497
\(241\) −4.23322e8 −1.94810 −0.974051 0.226330i \(-0.927327\pi\)
−0.974051 + 0.226330i \(0.927327\pi\)
\(242\) −3.24883e7 −0.147358
\(243\) 1.43489e7 0.0641500
\(244\) −9.49746e7 −0.418546
\(245\) −8.91052e7 −0.387099
\(246\) −1.18933e8 −0.509367
\(247\) 9.80242e7 0.413899
\(248\) −2.20824e7 −0.0919319
\(249\) 6.58977e7 0.270503
\(250\) 1.56250e7 0.0632456
\(251\) 2.11433e8 0.843945 0.421973 0.906609i \(-0.361338\pi\)
0.421973 + 0.906609i \(0.361338\pi\)
\(252\) −1.55233e7 −0.0611058
\(253\) −8.60471e7 −0.334052
\(254\) 6.91555e7 0.264794
\(255\) 1.03219e8 0.389824
\(256\) 1.67772e7 0.0625000
\(257\) 4.84998e8 1.78227 0.891135 0.453737i \(-0.149910\pi\)
0.891135 + 0.453737i \(0.149910\pi\)
\(258\) 1.50227e8 0.544603
\(259\) −6.22271e6 −0.0222551
\(260\) −1.14331e8 −0.403419
\(261\) 7.27407e7 0.253242
\(262\) 8.36704e7 0.287420
\(263\) −2.87065e8 −0.973050 −0.486525 0.873667i \(-0.661736\pi\)
−0.486525 + 0.873667i \(0.661736\pi\)
\(264\) −5.42953e7 −0.181614
\(265\) −1.21901e8 −0.402389
\(266\) 1.82569e7 0.0594760
\(267\) −1.52725e8 −0.491043
\(268\) −2.41698e8 −0.767010
\(269\) −2.67462e8 −0.837778 −0.418889 0.908038i \(-0.637580\pi\)
−0.418889 + 0.908038i \(0.637580\pi\)
\(270\) 1.96830e7 0.0608581
\(271\) 2.45103e8 0.748095 0.374048 0.927410i \(-0.377970\pi\)
0.374048 + 0.927410i \(0.377970\pi\)
\(272\) 1.25270e8 0.377446
\(273\) 1.28385e8 0.381895
\(274\) −2.23800e8 −0.657255
\(275\) −6.13689e7 −0.177944
\(276\) 3.78575e7 0.108385
\(277\) −2.79593e8 −0.790400 −0.395200 0.918595i \(-0.629325\pi\)
−0.395200 + 0.918595i \(0.629325\pi\)
\(278\) 1.59465e7 0.0445153
\(279\) −3.14416e7 −0.0866742
\(280\) −2.12940e7 −0.0579700
\(281\) −2.64641e8 −0.711516 −0.355758 0.934578i \(-0.615777\pi\)
−0.355758 + 0.934578i \(0.615777\pi\)
\(282\) −2.23683e8 −0.593966
\(283\) −2.06594e8 −0.541831 −0.270916 0.962603i \(-0.587326\pi\)
−0.270916 + 0.962603i \(0.587326\pi\)
\(284\) −7.53013e7 −0.195069
\(285\) −2.31491e7 −0.0592349
\(286\) 4.49046e8 1.13504
\(287\) 1.83200e8 0.457446
\(288\) 2.38879e7 0.0589256
\(289\) 5.25005e8 1.27944
\(290\) 9.97814e7 0.240246
\(291\) −6.44524e7 −0.153325
\(292\) 2.51692e8 0.591601
\(293\) −1.74393e8 −0.405036 −0.202518 0.979279i \(-0.564912\pi\)
−0.202518 + 0.979279i \(0.564912\pi\)
\(294\) −1.53974e8 −0.353371
\(295\) −3.56954e8 −0.809534
\(296\) 9.57575e6 0.0214611
\(297\) −7.73072e7 −0.171227
\(298\) 1.64017e8 0.359032
\(299\) −3.13098e8 −0.677378
\(300\) 2.70000e7 0.0577350
\(301\) −2.31404e8 −0.489090
\(302\) −1.52018e8 −0.317592
\(303\) −1.78474e8 −0.368574
\(304\) −2.80945e7 −0.0573539
\(305\) −1.85497e8 −0.374359
\(306\) 1.78362e8 0.355859
\(307\) 3.91587e7 0.0772403 0.0386202 0.999254i \(-0.487704\pi\)
0.0386202 + 0.999254i \(0.487704\pi\)
\(308\) 8.36345e7 0.163101
\(309\) −4.73206e7 −0.0912421
\(310\) −4.31298e7 −0.0822264
\(311\) 7.03665e8 1.32649 0.663246 0.748402i \(-0.269178\pi\)
0.663246 + 0.748402i \(0.269178\pi\)
\(312\) −1.97563e8 −0.368269
\(313\) −5.57911e8 −1.02840 −0.514198 0.857672i \(-0.671910\pi\)
−0.514198 + 0.857672i \(0.671910\pi\)
\(314\) −3.52997e8 −0.643454
\(315\) −3.03190e7 −0.0546547
\(316\) 1.19752e8 0.213491
\(317\) −4.42988e8 −0.781061 −0.390530 0.920590i \(-0.627708\pi\)
−0.390530 + 0.920590i \(0.627708\pi\)
\(318\) −2.10644e8 −0.367329
\(319\) −3.91903e8 −0.675944
\(320\) 3.27680e7 0.0559017
\(321\) −3.75838e7 −0.0634210
\(322\) −5.83142e7 −0.0973372
\(323\) −2.09771e8 −0.346368
\(324\) 3.40122e7 0.0555556
\(325\) −2.23302e8 −0.360829
\(326\) 1.87372e8 0.299533
\(327\) −3.30974e8 −0.523453
\(328\) −2.81916e8 −0.441125
\(329\) 3.44553e8 0.533421
\(330\) −1.06046e8 −0.162440
\(331\) 5.11746e8 0.775633 0.387817 0.921737i \(-0.373229\pi\)
0.387817 + 0.921737i \(0.373229\pi\)
\(332\) 1.56202e8 0.234263
\(333\) 1.36342e7 0.0202337
\(334\) 9.11958e7 0.133925
\(335\) −4.72066e8 −0.686034
\(336\) −3.67960e7 −0.0529192
\(337\) −6.53704e7 −0.0930415 −0.0465207 0.998917i \(-0.514813\pi\)
−0.0465207 + 0.998917i \(0.514813\pi\)
\(338\) 1.13195e9 1.59448
\(339\) −1.74078e8 −0.242687
\(340\) 2.44667e8 0.337598
\(341\) 1.69397e8 0.231348
\(342\) −4.00017e7 −0.0540738
\(343\) 5.11183e8 0.683986
\(344\) 3.56094e8 0.471640
\(345\) 7.39403e7 0.0969426
\(346\) −4.61898e7 −0.0599487
\(347\) −9.43498e7 −0.121224 −0.0606119 0.998161i \(-0.519305\pi\)
−0.0606119 + 0.998161i \(0.519305\pi\)
\(348\) 1.72422e8 0.219314
\(349\) 4.23470e8 0.533253 0.266627 0.963800i \(-0.414091\pi\)
0.266627 + 0.963800i \(0.414091\pi\)
\(350\) −4.15898e7 −0.0518500
\(351\) −2.81296e8 −0.347207
\(352\) −1.28700e8 −0.157282
\(353\) 1.22062e8 0.147696 0.0738479 0.997270i \(-0.476472\pi\)
0.0738479 + 0.997270i \(0.476472\pi\)
\(354\) −6.16816e8 −0.739000
\(355\) −1.47073e8 −0.174475
\(356\) −3.62014e8 −0.425256
\(357\) −2.74743e8 −0.319586
\(358\) −8.10553e8 −0.933664
\(359\) 2.98888e8 0.340940 0.170470 0.985363i \(-0.445471\pi\)
0.170470 + 0.985363i \(0.445471\pi\)
\(360\) 4.66560e7 0.0527046
\(361\) 4.70459e7 0.0526316
\(362\) 8.84043e8 0.979475
\(363\) −1.09648e8 −0.120317
\(364\) 3.04319e8 0.330731
\(365\) 4.91585e8 0.529144
\(366\) −3.20539e8 −0.341741
\(367\) −9.24854e6 −0.00976657 −0.00488329 0.999988i \(-0.501554\pi\)
−0.00488329 + 0.999988i \(0.501554\pi\)
\(368\) 8.97362e7 0.0938643
\(369\) −4.01400e8 −0.415896
\(370\) 1.87026e7 0.0191954
\(371\) 3.24469e8 0.329886
\(372\) −7.45282e7 −0.0750621
\(373\) −6.28142e8 −0.626725 −0.313363 0.949634i \(-0.601456\pi\)
−0.313363 + 0.949634i \(0.601456\pi\)
\(374\) −9.60957e8 −0.949847
\(375\) 5.27344e7 0.0516398
\(376\) −5.30212e8 −0.514389
\(377\) −1.42601e9 −1.37065
\(378\) −5.23911e7 −0.0498927
\(379\) 3.11661e8 0.294066 0.147033 0.989132i \(-0.453028\pi\)
0.147033 + 0.989132i \(0.453028\pi\)
\(380\) −5.48720e7 −0.0512989
\(381\) 2.33400e8 0.216204
\(382\) −7.15151e8 −0.656411
\(383\) −6.12353e8 −0.556937 −0.278469 0.960445i \(-0.589827\pi\)
−0.278469 + 0.960445i \(0.589827\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) 1.63349e8 0.145882
\(386\) −4.17411e8 −0.369409
\(387\) 5.07017e8 0.444666
\(388\) −1.52776e8 −0.132784
\(389\) −1.71480e9 −1.47703 −0.738517 0.674234i \(-0.764474\pi\)
−0.738517 + 0.674234i \(0.764474\pi\)
\(390\) −3.85866e8 −0.329390
\(391\) 6.70029e8 0.566858
\(392\) −3.64975e8 −0.306028
\(393\) 2.82388e8 0.234678
\(394\) −8.24391e8 −0.679041
\(395\) 2.33891e8 0.190952
\(396\) −1.83247e8 −0.148287
\(397\) 1.19433e9 0.957985 0.478992 0.877819i \(-0.341002\pi\)
0.478992 + 0.877819i \(0.341002\pi\)
\(398\) 5.85650e8 0.465637
\(399\) 6.16171e7 0.0485619
\(400\) 6.40000e7 0.0500000
\(401\) −2.38886e9 −1.85006 −0.925030 0.379895i \(-0.875960\pi\)
−0.925030 + 0.379895i \(0.875960\pi\)
\(402\) −8.15730e8 −0.626261
\(403\) 6.16382e8 0.469118
\(404\) −4.23049e8 −0.319195
\(405\) 6.64301e7 0.0496904
\(406\) −2.65593e8 −0.196959
\(407\) −7.34567e7 −0.0540071
\(408\) 4.22785e8 0.308183
\(409\) −3.89477e8 −0.281482 −0.140741 0.990046i \(-0.544948\pi\)
−0.140741 + 0.990046i \(0.544948\pi\)
\(410\) −5.50617e8 −0.394554
\(411\) −7.55326e8 −0.536646
\(412\) −1.12167e8 −0.0790180
\(413\) 9.50121e8 0.663672
\(414\) 1.27769e8 0.0884961
\(415\) 3.05082e8 0.209531
\(416\) −4.68298e8 −0.318930
\(417\) 5.38195e7 0.0363466
\(418\) 2.15516e8 0.144332
\(419\) 1.19601e9 0.794304 0.397152 0.917753i \(-0.369999\pi\)
0.397152 + 0.917753i \(0.369999\pi\)
\(420\) −7.18671e7 −0.0473323
\(421\) 2.04720e9 1.33713 0.668565 0.743653i \(-0.266909\pi\)
0.668565 + 0.743653i \(0.266909\pi\)
\(422\) 1.90449e7 0.0123363
\(423\) −7.54931e8 −0.484971
\(424\) −4.99305e8 −0.318116
\(425\) 4.77865e8 0.301956
\(426\) −2.54142e8 −0.159273
\(427\) 4.93747e8 0.306907
\(428\) −8.90875e7 −0.0549242
\(429\) 1.51553e9 0.926754
\(430\) 6.95496e8 0.421847
\(431\) 1.02239e9 0.615101 0.307551 0.951532i \(-0.400491\pi\)
0.307551 + 0.951532i \(0.400491\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) 1.93712e9 1.14670 0.573348 0.819312i \(-0.305644\pi\)
0.573348 + 0.819312i \(0.305644\pi\)
\(434\) 1.14801e8 0.0674108
\(435\) 3.36762e8 0.196160
\(436\) −7.84532e8 −0.453323
\(437\) −1.50269e8 −0.0861358
\(438\) 8.49459e8 0.483040
\(439\) −1.94872e9 −1.09932 −0.549659 0.835389i \(-0.685242\pi\)
−0.549659 + 0.835389i \(0.685242\pi\)
\(440\) −2.51367e8 −0.140677
\(441\) −5.19661e8 −0.288526
\(442\) −3.49662e9 −1.92606
\(443\) 1.91815e9 1.04826 0.524132 0.851637i \(-0.324390\pi\)
0.524132 + 0.851637i \(0.324390\pi\)
\(444\) 3.23181e7 0.0175229
\(445\) −7.07059e8 −0.380360
\(446\) −1.01020e9 −0.539180
\(447\) 5.53559e8 0.293148
\(448\) −8.72201e7 −0.0458293
\(449\) −3.82247e9 −1.99288 −0.996441 0.0842908i \(-0.973138\pi\)
−0.996441 + 0.0842908i \(0.973138\pi\)
\(450\) 9.11250e7 0.0471405
\(451\) 2.16261e9 1.11010
\(452\) −4.12630e8 −0.210173
\(453\) −5.13060e8 −0.259313
\(454\) 2.26037e9 1.13367
\(455\) 5.94373e8 0.295814
\(456\) −9.48188e7 −0.0468293
\(457\) 1.63777e9 0.802688 0.401344 0.915927i \(-0.368543\pi\)
0.401344 + 0.915927i \(0.368543\pi\)
\(458\) 2.45925e9 1.19612
\(459\) 6.01973e8 0.290558
\(460\) 1.75266e8 0.0839548
\(461\) 2.29804e8 0.109246 0.0546229 0.998507i \(-0.482604\pi\)
0.0546229 + 0.998507i \(0.482604\pi\)
\(462\) 2.82266e8 0.133172
\(463\) −4.03314e9 −1.88847 −0.944235 0.329274i \(-0.893196\pi\)
−0.944235 + 0.329274i \(0.893196\pi\)
\(464\) 4.08705e8 0.189931
\(465\) −1.45563e8 −0.0671376
\(466\) 2.83792e9 1.29912
\(467\) 1.02273e9 0.464677 0.232338 0.972635i \(-0.425362\pi\)
0.232338 + 0.972635i \(0.425362\pi\)
\(468\) −6.66776e8 −0.300690
\(469\) 1.25652e9 0.562425
\(470\) −1.03557e9 −0.460084
\(471\) −1.19137e9 −0.525378
\(472\) −1.46208e9 −0.639993
\(473\) −2.73164e9 −1.18689
\(474\) 4.04164e8 0.174314
\(475\) −1.07172e8 −0.0458831
\(476\) −6.51242e8 −0.276769
\(477\) −7.10925e8 −0.299923
\(478\) 5.39975e8 0.226139
\(479\) 1.53415e9 0.637814 0.318907 0.947786i \(-0.396684\pi\)
0.318907 + 0.947786i \(0.396684\pi\)
\(480\) 1.10592e8 0.0456435
\(481\) −2.67285e8 −0.109513
\(482\) −3.38658e9 −1.37752
\(483\) −1.96810e8 −0.0794755
\(484\) −2.59906e8 −0.104198
\(485\) −2.98391e8 −0.118765
\(486\) 1.14791e8 0.0453609
\(487\) 5.28220e8 0.207235 0.103618 0.994617i \(-0.466958\pi\)
0.103618 + 0.994617i \(0.466958\pi\)
\(488\) −7.59797e8 −0.295957
\(489\) 6.32382e8 0.244567
\(490\) −7.12842e8 −0.273720
\(491\) 8.59835e8 0.327816 0.163908 0.986476i \(-0.447590\pi\)
0.163908 + 0.986476i \(0.447590\pi\)
\(492\) −9.51466e8 −0.360177
\(493\) 3.05165e9 1.14702
\(494\) 7.84194e8 0.292671
\(495\) −3.57904e8 −0.132632
\(496\) −1.76660e8 −0.0650057
\(497\) 3.91471e8 0.143038
\(498\) 5.27181e8 0.191275
\(499\) 1.71016e9 0.616147 0.308074 0.951363i \(-0.400316\pi\)
0.308074 + 0.951363i \(0.400316\pi\)
\(500\) 1.25000e8 0.0447214
\(501\) 3.07786e8 0.109349
\(502\) 1.69146e9 0.596759
\(503\) −2.41522e9 −0.846192 −0.423096 0.906085i \(-0.639057\pi\)
−0.423096 + 0.906085i \(0.639057\pi\)
\(504\) −1.24186e8 −0.0432083
\(505\) −8.26268e8 −0.285497
\(506\) −6.88377e8 −0.236211
\(507\) 3.82033e9 1.30188
\(508\) 5.53244e8 0.187238
\(509\) −5.62309e8 −0.189001 −0.0945003 0.995525i \(-0.530125\pi\)
−0.0945003 + 0.995525i \(0.530125\pi\)
\(510\) 8.25751e8 0.275647
\(511\) −1.30848e9 −0.433803
\(512\) 1.34218e8 0.0441942
\(513\) −1.35006e8 −0.0441511
\(514\) 3.87998e9 1.26026
\(515\) −2.19077e8 −0.0706759
\(516\) 1.20182e9 0.385092
\(517\) 4.06732e9 1.29447
\(518\) −4.97817e7 −0.0157368
\(519\) −1.55891e8 −0.0489479
\(520\) −9.14645e8 −0.285260
\(521\) −2.15163e9 −0.666553 −0.333277 0.942829i \(-0.608154\pi\)
−0.333277 + 0.942829i \(0.608154\pi\)
\(522\) 5.81925e8 0.179069
\(523\) 2.44070e9 0.746034 0.373017 0.927824i \(-0.378323\pi\)
0.373017 + 0.927824i \(0.378323\pi\)
\(524\) 6.69364e8 0.203237
\(525\) −1.40366e8 −0.0423353
\(526\) −2.29652e9 −0.688050
\(527\) −1.31905e9 −0.392578
\(528\) −4.34362e8 −0.128420
\(529\) −2.92485e9 −0.859032
\(530\) −9.75206e8 −0.284532
\(531\) −2.08176e9 −0.603391
\(532\) 1.46055e8 0.0420559
\(533\) 7.86905e9 2.25101
\(534\) −1.22180e9 −0.347220
\(535\) −1.73999e8 −0.0491257
\(536\) −1.93358e9 −0.542358
\(537\) −2.73561e9 −0.762333
\(538\) −2.13969e9 −0.592398
\(539\) 2.79977e9 0.770125
\(540\) 1.57464e8 0.0430331
\(541\) 2.96766e9 0.805793 0.402896 0.915246i \(-0.368003\pi\)
0.402896 + 0.915246i \(0.368003\pi\)
\(542\) 1.96083e9 0.528983
\(543\) 2.98365e9 0.799738
\(544\) 1.00216e9 0.266894
\(545\) −1.53229e9 −0.405465
\(546\) 1.02708e9 0.270040
\(547\) −5.15902e8 −0.134776 −0.0673879 0.997727i \(-0.521467\pi\)
−0.0673879 + 0.997727i \(0.521467\pi\)
\(548\) −1.79040e9 −0.464749
\(549\) −1.08182e9 −0.279031
\(550\) −4.90952e8 −0.125826
\(551\) −6.84401e8 −0.174293
\(552\) 3.02860e8 0.0766399
\(553\) −6.22558e8 −0.156546
\(554\) −2.23674e9 −0.558897
\(555\) 6.31214e7 0.0156730
\(556\) 1.27572e8 0.0314770
\(557\) 6.30702e8 0.154643 0.0773216 0.997006i \(-0.475363\pi\)
0.0773216 + 0.997006i \(0.475363\pi\)
\(558\) −2.51533e8 −0.0612879
\(559\) −9.93957e9 −2.40672
\(560\) −1.70352e8 −0.0409910
\(561\) −3.24323e9 −0.775547
\(562\) −2.11713e9 −0.503118
\(563\) 5.81575e9 1.37349 0.686746 0.726897i \(-0.259038\pi\)
0.686746 + 0.726897i \(0.259038\pi\)
\(564\) −1.78947e9 −0.419997
\(565\) −8.05918e8 −0.187984
\(566\) −1.65275e9 −0.383133
\(567\) −1.76820e8 −0.0407372
\(568\) −6.02410e8 −0.137935
\(569\) 3.38553e9 0.770431 0.385216 0.922827i \(-0.374127\pi\)
0.385216 + 0.922827i \(0.374127\pi\)
\(570\) −1.85193e8 −0.0418854
\(571\) −3.74462e9 −0.841747 −0.420873 0.907119i \(-0.638276\pi\)
−0.420873 + 0.907119i \(0.638276\pi\)
\(572\) 3.59237e9 0.802592
\(573\) −2.41363e9 −0.535957
\(574\) 1.46560e9 0.323463
\(575\) 3.42316e8 0.0750914
\(576\) 1.91103e8 0.0416667
\(577\) 1.94442e9 0.421380 0.210690 0.977553i \(-0.432429\pi\)
0.210690 + 0.977553i \(0.432429\pi\)
\(578\) 4.20004e9 0.904703
\(579\) −1.40876e9 −0.301622
\(580\) 7.98251e8 0.169880
\(581\) −8.12050e8 −0.171778
\(582\) −5.15619e8 −0.108417
\(583\) 3.83023e9 0.800543
\(584\) 2.01353e9 0.418325
\(585\) −1.30230e9 −0.268946
\(586\) −1.39515e9 −0.286403
\(587\) 7.71193e9 1.57373 0.786864 0.617126i \(-0.211703\pi\)
0.786864 + 0.617126i \(0.211703\pi\)
\(588\) −1.23179e9 −0.249871
\(589\) 2.95827e8 0.0596533
\(590\) −2.85563e9 −0.572427
\(591\) −2.78232e9 −0.554435
\(592\) 7.66060e7 0.0151753
\(593\) 9.15574e9 1.80303 0.901513 0.432752i \(-0.142457\pi\)
0.901513 + 0.432752i \(0.142457\pi\)
\(594\) −6.18458e8 −0.121076
\(595\) −1.27196e9 −0.247550
\(596\) 1.31214e9 0.253874
\(597\) 1.97657e9 0.380191
\(598\) −2.50478e9 −0.478979
\(599\) −6.00767e9 −1.14212 −0.571061 0.820908i \(-0.693468\pi\)
−0.571061 + 0.820908i \(0.693468\pi\)
\(600\) 2.16000e8 0.0408248
\(601\) −2.97355e9 −0.558747 −0.279373 0.960183i \(-0.590127\pi\)
−0.279373 + 0.960183i \(0.590127\pi\)
\(602\) −1.85123e9 −0.345839
\(603\) −2.75309e9 −0.511340
\(604\) −1.21614e9 −0.224572
\(605\) −5.07629e8 −0.0931972
\(606\) −1.42779e9 −0.260622
\(607\) −2.14526e9 −0.389331 −0.194665 0.980870i \(-0.562362\pi\)
−0.194665 + 0.980870i \(0.562362\pi\)
\(608\) −2.24756e8 −0.0405554
\(609\) −8.96376e8 −0.160816
\(610\) −1.48398e9 −0.264712
\(611\) 1.47997e10 2.62487
\(612\) 1.42690e9 0.251630
\(613\) 1.03064e10 1.80716 0.903579 0.428422i \(-0.140930\pi\)
0.903579 + 0.428422i \(0.140930\pi\)
\(614\) 3.13270e8 0.0546172
\(615\) −1.85833e9 −0.322152
\(616\) 6.69076e8 0.115330
\(617\) 3.87942e9 0.664919 0.332460 0.943118i \(-0.392122\pi\)
0.332460 + 0.943118i \(0.392122\pi\)
\(618\) −3.78565e8 −0.0645179
\(619\) 2.21402e9 0.375201 0.187600 0.982245i \(-0.439929\pi\)
0.187600 + 0.982245i \(0.439929\pi\)
\(620\) −3.45038e8 −0.0581428
\(621\) 4.31220e8 0.0722567
\(622\) 5.62932e9 0.937971
\(623\) 1.88201e9 0.311827
\(624\) −1.58051e9 −0.260406
\(625\) 2.44141e8 0.0400000
\(626\) −4.46329e9 −0.727185
\(627\) 7.27366e8 0.117847
\(628\) −2.82398e9 −0.454991
\(629\) 5.71990e8 0.0916455
\(630\) −2.42552e8 −0.0386467
\(631\) −3.23646e9 −0.512823 −0.256412 0.966568i \(-0.582540\pi\)
−0.256412 + 0.966568i \(0.582540\pi\)
\(632\) 9.58017e8 0.150961
\(633\) 6.42766e7 0.0100726
\(634\) −3.54391e9 −0.552293
\(635\) 1.08056e9 0.167471
\(636\) −1.68516e9 −0.259741
\(637\) 1.01875e10 1.56163
\(638\) −3.13522e9 −0.477965
\(639\) −8.57729e8 −0.130046
\(640\) 2.62144e8 0.0395285
\(641\) 1.16659e10 1.74950 0.874750 0.484574i \(-0.161025\pi\)
0.874750 + 0.484574i \(0.161025\pi\)
\(642\) −3.00670e8 −0.0448454
\(643\) −1.40342e9 −0.208184 −0.104092 0.994568i \(-0.533194\pi\)
−0.104092 + 0.994568i \(0.533194\pi\)
\(644\) −4.66514e8 −0.0688278
\(645\) 2.34730e9 0.344437
\(646\) −1.67817e9 −0.244919
\(647\) −3.61471e9 −0.524696 −0.262348 0.964973i \(-0.584497\pi\)
−0.262348 + 0.964973i \(0.584497\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) 1.12158e10 1.61055
\(650\) −1.78642e9 −0.255144
\(651\) 3.87452e8 0.0550407
\(652\) 1.49898e9 0.211802
\(653\) −1.95439e9 −0.274672 −0.137336 0.990525i \(-0.543854\pi\)
−0.137336 + 0.990525i \(0.543854\pi\)
\(654\) −2.64780e9 −0.370137
\(655\) 1.30735e9 0.181781
\(656\) −2.25533e9 −0.311922
\(657\) 2.86693e9 0.394401
\(658\) 2.75642e9 0.377186
\(659\) 7.53677e9 1.02586 0.512928 0.858432i \(-0.328561\pi\)
0.512928 + 0.858432i \(0.328561\pi\)
\(660\) −8.48364e8 −0.114863
\(661\) 5.24752e9 0.706723 0.353361 0.935487i \(-0.385039\pi\)
0.353361 + 0.935487i \(0.385039\pi\)
\(662\) 4.09397e9 0.548456
\(663\) −1.18011e10 −1.57262
\(664\) 1.24962e9 0.165649
\(665\) 2.85264e8 0.0376159
\(666\) 1.09074e8 0.0143074
\(667\) 2.18604e9 0.285244
\(668\) 7.29566e8 0.0946994
\(669\) −3.40941e9 −0.440239
\(670\) −3.77653e9 −0.485100
\(671\) 5.82849e9 0.744779
\(672\) −2.94368e8 −0.0374195
\(673\) 8.75932e9 1.10769 0.553845 0.832620i \(-0.313160\pi\)
0.553845 + 0.832620i \(0.313160\pi\)
\(674\) −5.22963e8 −0.0657903
\(675\) 3.07547e8 0.0384900
\(676\) 9.05559e9 1.12747
\(677\) −4.39888e9 −0.544856 −0.272428 0.962176i \(-0.587827\pi\)
−0.272428 + 0.962176i \(0.587827\pi\)
\(678\) −1.39263e9 −0.171605
\(679\) 7.94241e8 0.0973661
\(680\) 1.95734e9 0.238718
\(681\) 7.62876e9 0.925634
\(682\) 1.35518e9 0.163588
\(683\) −3.81565e9 −0.458243 −0.229122 0.973398i \(-0.573585\pi\)
−0.229122 + 0.973398i \(0.573585\pi\)
\(684\) −3.20014e8 −0.0382360
\(685\) −3.49688e9 −0.415685
\(686\) 4.08947e9 0.483651
\(687\) 8.29998e9 0.976626
\(688\) 2.84875e9 0.333500
\(689\) 1.39370e10 1.62331
\(690\) 5.91523e8 0.0685488
\(691\) −1.29278e10 −1.49057 −0.745285 0.666746i \(-0.767687\pi\)
−0.745285 + 0.666746i \(0.767687\pi\)
\(692\) −3.69519e8 −0.0423901
\(693\) 9.52649e8 0.108734
\(694\) −7.54799e8 −0.0857182
\(695\) 2.49164e8 0.0281539
\(696\) 1.37938e9 0.155078
\(697\) −1.68397e10 −1.88374
\(698\) 3.38776e9 0.377067
\(699\) 9.57797e9 1.06073
\(700\) −3.32718e8 −0.0366635
\(701\) 7.56473e9 0.829432 0.414716 0.909951i \(-0.363881\pi\)
0.414716 + 0.909951i \(0.363881\pi\)
\(702\) −2.25037e9 −0.245513
\(703\) −1.28281e8 −0.0139258
\(704\) −1.02960e9 −0.111215
\(705\) −3.49505e9 −0.375657
\(706\) 9.76494e8 0.104437
\(707\) 2.19931e9 0.234056
\(708\) −4.93453e9 −0.522552
\(709\) −6.65401e9 −0.701168 −0.350584 0.936531i \(-0.614017\pi\)
−0.350584 + 0.936531i \(0.614017\pi\)
\(710\) −1.17658e9 −0.123372
\(711\) 1.36405e9 0.142327
\(712\) −2.89611e9 −0.300701
\(713\) −9.44898e8 −0.0976274
\(714\) −2.19794e9 −0.225981
\(715\) 7.01635e9 0.717860
\(716\) −6.48442e9 −0.660200
\(717\) 1.82241e9 0.184642
\(718\) 2.39110e9 0.241081
\(719\) −1.84578e10 −1.85195 −0.925975 0.377584i \(-0.876755\pi\)
−0.925975 + 0.377584i \(0.876755\pi\)
\(720\) 3.73248e8 0.0372678
\(721\) 5.83127e8 0.0579415
\(722\) 3.76367e8 0.0372161
\(723\) −1.14297e10 −1.12474
\(724\) 7.07235e9 0.692594
\(725\) 1.55908e9 0.151945
\(726\) −8.77183e8 −0.0850770
\(727\) 1.41775e9 0.136845 0.0684223 0.997656i \(-0.478203\pi\)
0.0684223 + 0.997656i \(0.478203\pi\)
\(728\) 2.43455e9 0.233862
\(729\) 3.87420e8 0.0370370
\(730\) 3.93268e9 0.374161
\(731\) 2.12706e10 2.01405
\(732\) −2.56432e9 −0.241648
\(733\) −1.27897e10 −1.19949 −0.599744 0.800192i \(-0.704731\pi\)
−0.599744 + 0.800192i \(0.704731\pi\)
\(734\) −7.39883e7 −0.00690601
\(735\) −2.40584e9 −0.223492
\(736\) 7.17890e8 0.0663721
\(737\) 1.48327e10 1.36485
\(738\) −3.21120e9 −0.294083
\(739\) 1.05992e10 0.966094 0.483047 0.875594i \(-0.339530\pi\)
0.483047 + 0.875594i \(0.339530\pi\)
\(740\) 1.49621e8 0.0135732
\(741\) 2.64665e9 0.238965
\(742\) 2.59575e9 0.233265
\(743\) −8.98168e9 −0.803335 −0.401668 0.915785i \(-0.631569\pi\)
−0.401668 + 0.915785i \(0.631569\pi\)
\(744\) −5.96226e8 −0.0530769
\(745\) 2.56277e9 0.227072
\(746\) −5.02514e9 −0.443162
\(747\) 1.77924e9 0.156175
\(748\) −7.68766e9 −0.671643
\(749\) 4.63141e8 0.0402742
\(750\) 4.21875e8 0.0365148
\(751\) 7.96423e9 0.686126 0.343063 0.939312i \(-0.388536\pi\)
0.343063 + 0.939312i \(0.388536\pi\)
\(752\) −4.24170e9 −0.363728
\(753\) 5.70868e9 0.487252
\(754\) −1.14081e10 −0.969198
\(755\) −2.37528e9 −0.200863
\(756\) −4.19129e8 −0.0352794
\(757\) 1.06811e10 0.894911 0.447455 0.894306i \(-0.352330\pi\)
0.447455 + 0.894306i \(0.352330\pi\)
\(758\) 2.49329e9 0.207936
\(759\) −2.32327e9 −0.192865
\(760\) −4.38976e8 −0.0362738
\(761\) −3.49539e9 −0.287508 −0.143754 0.989613i \(-0.545917\pi\)
−0.143754 + 0.989613i \(0.545917\pi\)
\(762\) 1.86720e9 0.152879
\(763\) 4.07856e9 0.332408
\(764\) −5.72121e9 −0.464153
\(765\) 2.78691e9 0.225065
\(766\) −4.89882e9 −0.393814
\(767\) 4.08108e10 3.26581
\(768\) 4.52985e8 0.0360844
\(769\) 1.20094e10 0.952309 0.476155 0.879362i \(-0.342030\pi\)
0.476155 + 0.879362i \(0.342030\pi\)
\(770\) 1.30679e9 0.103154
\(771\) 1.30949e10 1.02899
\(772\) −3.33929e9 −0.261212
\(773\) 2.03787e10 1.58690 0.793448 0.608638i \(-0.208284\pi\)
0.793448 + 0.608638i \(0.208284\pi\)
\(774\) 4.05613e9 0.314426
\(775\) −6.73903e8 −0.0520045
\(776\) −1.22221e9 −0.0938922
\(777\) −1.68013e8 −0.0128490
\(778\) −1.37184e10 −1.04442
\(779\) 3.77668e9 0.286239
\(780\) −3.08693e9 −0.232914
\(781\) 4.62116e9 0.347114
\(782\) 5.36023e9 0.400829
\(783\) 1.96400e9 0.146209
\(784\) −2.91980e9 −0.216395
\(785\) −5.51558e9 −0.406956
\(786\) 2.25910e9 0.165942
\(787\) −1.61839e10 −1.18351 −0.591754 0.806119i \(-0.701564\pi\)
−0.591754 + 0.806119i \(0.701564\pi\)
\(788\) −6.59513e9 −0.480155
\(789\) −7.75076e9 −0.561791
\(790\) 1.87113e9 0.135023
\(791\) 2.14515e9 0.154113
\(792\) −1.46597e9 −0.104855
\(793\) 2.12080e10 1.51023
\(794\) 9.55466e9 0.677398
\(795\) −3.29132e9 −0.232319
\(796\) 4.68520e9 0.329255
\(797\) −4.25959e9 −0.298032 −0.149016 0.988835i \(-0.547611\pi\)
−0.149016 + 0.988835i \(0.547611\pi\)
\(798\) 4.92937e8 0.0343385
\(799\) −3.16712e10 −2.19660
\(800\) 5.12000e8 0.0353553
\(801\) −4.12357e9 −0.283504
\(802\) −1.91109e10 −1.30819
\(803\) −1.54461e10 −1.05272
\(804\) −6.52584e9 −0.442833
\(805\) −9.11159e8 −0.0615614
\(806\) 4.93105e9 0.331717
\(807\) −7.22147e9 −0.483691
\(808\) −3.38439e9 −0.225705
\(809\) 1.45169e10 0.963951 0.481975 0.876185i \(-0.339919\pi\)
0.481975 + 0.876185i \(0.339919\pi\)
\(810\) 5.31441e8 0.0351364
\(811\) 4.37265e9 0.287853 0.143927 0.989588i \(-0.454027\pi\)
0.143927 + 0.989588i \(0.454027\pi\)
\(812\) −2.12474e9 −0.139271
\(813\) 6.61779e9 0.431913
\(814\) −5.87653e8 −0.0381888
\(815\) 2.92769e9 0.189441
\(816\) 3.38228e9 0.217918
\(817\) −4.77041e9 −0.306040
\(818\) −3.11581e9 −0.199038
\(819\) 3.46638e9 0.220487
\(820\) −4.40494e9 −0.278992
\(821\) −4.64314e9 −0.292827 −0.146413 0.989223i \(-0.546773\pi\)
−0.146413 + 0.989223i \(0.546773\pi\)
\(822\) −6.04261e9 −0.379466
\(823\) 4.32686e9 0.270566 0.135283 0.990807i \(-0.456806\pi\)
0.135283 + 0.990807i \(0.456806\pi\)
\(824\) −8.97339e8 −0.0558742
\(825\) −1.65696e9 −0.102736
\(826\) 7.60097e9 0.469287
\(827\) 9.51024e9 0.584685 0.292343 0.956314i \(-0.405565\pi\)
0.292343 + 0.956314i \(0.405565\pi\)
\(828\) 1.02215e9 0.0625762
\(829\) −1.48006e10 −0.902277 −0.451138 0.892454i \(-0.648982\pi\)
−0.451138 + 0.892454i \(0.648982\pi\)
\(830\) 2.44065e9 0.148161
\(831\) −7.54900e9 −0.456337
\(832\) −3.74639e9 −0.225518
\(833\) −2.18011e10 −1.30684
\(834\) 4.30556e8 0.0257009
\(835\) 1.42493e9 0.0847017
\(836\) 1.72413e9 0.102058
\(837\) −8.48923e8 −0.0500414
\(838\) 9.56810e9 0.561658
\(839\) 1.70007e10 0.993800 0.496900 0.867808i \(-0.334472\pi\)
0.496900 + 0.867808i \(0.334472\pi\)
\(840\) −5.74937e8 −0.0334690
\(841\) −7.29355e9 −0.422817
\(842\) 1.63776e10 0.945494
\(843\) −7.14530e9 −0.410794
\(844\) 1.52359e8 0.00872309
\(845\) 1.76867e10 1.00844
\(846\) −6.03944e9 −0.342926
\(847\) 1.35118e9 0.0764049
\(848\) −3.99444e9 −0.224942
\(849\) −5.57803e9 −0.312827
\(850\) 3.82292e9 0.213515
\(851\) 4.09742e8 0.0227907
\(852\) −2.03314e9 −0.112623
\(853\) −5.47273e9 −0.301913 −0.150957 0.988540i \(-0.548235\pi\)
−0.150957 + 0.988540i \(0.548235\pi\)
\(854\) 3.94997e9 0.217016
\(855\) −6.25026e8 −0.0341993
\(856\) −7.12700e8 −0.0388372
\(857\) −1.00479e10 −0.545307 −0.272654 0.962112i \(-0.587901\pi\)
−0.272654 + 0.962112i \(0.587901\pi\)
\(858\) 1.21243e10 0.655314
\(859\) −1.99055e10 −1.07151 −0.535755 0.844374i \(-0.679973\pi\)
−0.535755 + 0.844374i \(0.679973\pi\)
\(860\) 5.56397e9 0.298291
\(861\) 4.94641e9 0.264107
\(862\) 8.17913e9 0.434942
\(863\) −8.33403e9 −0.441385 −0.220692 0.975343i \(-0.570832\pi\)
−0.220692 + 0.975343i \(0.570832\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) −7.21716e8 −0.0379149
\(866\) 1.54969e10 0.810836
\(867\) 1.41751e10 0.738687
\(868\) 9.18404e8 0.0476667
\(869\) −7.34906e9 −0.379894
\(870\) 2.69410e9 0.138706
\(871\) 5.39716e10 2.76759
\(872\) −6.27626e9 −0.320548
\(873\) −1.74022e9 −0.0885224
\(874\) −1.20215e9 −0.0609072
\(875\) −6.49840e8 −0.0327928
\(876\) 6.79567e9 0.341561
\(877\) −2.25386e10 −1.12831 −0.564155 0.825669i \(-0.690798\pi\)
−0.564155 + 0.825669i \(0.690798\pi\)
\(878\) −1.55897e10 −0.777335
\(879\) −4.70862e9 −0.233847
\(880\) −2.01094e9 −0.0994739
\(881\) 1.58194e10 0.779427 0.389713 0.920936i \(-0.372574\pi\)
0.389713 + 0.920936i \(0.372574\pi\)
\(882\) −4.15729e9 −0.204019
\(883\) 1.97832e10 0.967015 0.483507 0.875340i \(-0.339363\pi\)
0.483507 + 0.875340i \(0.339363\pi\)
\(884\) −2.79729e10 −1.36193
\(885\) −9.63776e9 −0.467385
\(886\) 1.53452e10 0.741234
\(887\) 2.83958e10 1.36622 0.683111 0.730314i \(-0.260626\pi\)
0.683111 + 0.730314i \(0.260626\pi\)
\(888\) 2.58545e8 0.0123906
\(889\) −2.87616e9 −0.137296
\(890\) −5.65647e9 −0.268955
\(891\) −2.08729e9 −0.0988580
\(892\) −8.08157e9 −0.381258
\(893\) 7.10298e9 0.333780
\(894\) 4.42847e9 0.207287
\(895\) −1.26649e10 −0.590501
\(896\) −6.97761e8 −0.0324062
\(897\) −8.45365e9 −0.391084
\(898\) −3.05797e10 −1.40918
\(899\) −4.30355e9 −0.197546
\(900\) 7.29000e8 0.0333333
\(901\) −2.98251e10 −1.35845
\(902\) 1.73009e10 0.784956
\(903\) −6.24792e9 −0.282376
\(904\) −3.30104e9 −0.148615
\(905\) 1.38132e10 0.619475
\(906\) −4.10448e9 −0.183362
\(907\) −7.32028e9 −0.325763 −0.162882 0.986646i \(-0.552079\pi\)
−0.162882 + 0.986646i \(0.552079\pi\)
\(908\) 1.80830e10 0.801622
\(909\) −4.81879e9 −0.212797
\(910\) 4.75499e9 0.209172
\(911\) 9.56317e8 0.0419071 0.0209536 0.999780i \(-0.493330\pi\)
0.0209536 + 0.999780i \(0.493330\pi\)
\(912\) −7.58551e8 −0.0331133
\(913\) −9.58595e9 −0.416857
\(914\) 1.31022e10 0.567586
\(915\) −5.00843e9 −0.216136
\(916\) 1.96740e10 0.845783
\(917\) −3.47984e9 −0.149027
\(918\) 4.81578e9 0.205455
\(919\) −2.86457e10 −1.21746 −0.608730 0.793378i \(-0.708321\pi\)
−0.608730 + 0.793378i \(0.708321\pi\)
\(920\) 1.40213e9 0.0593650
\(921\) 1.05729e9 0.0445947
\(922\) 1.83843e9 0.0772484
\(923\) 1.68149e10 0.703865
\(924\) 2.25813e9 0.0941666
\(925\) 2.92229e8 0.0121402
\(926\) −3.22651e10 −1.33535
\(927\) −1.27766e9 −0.0526787
\(928\) 3.26964e9 0.134302
\(929\) −1.99379e10 −0.815876 −0.407938 0.913010i \(-0.633752\pi\)
−0.407938 + 0.913010i \(0.633752\pi\)
\(930\) −1.16450e9 −0.0474734
\(931\) 4.88938e9 0.198577
\(932\) 2.27033e10 0.918616
\(933\) 1.89989e10 0.765850
\(934\) 8.18182e9 0.328576
\(935\) −1.50150e10 −0.600736
\(936\) −5.33421e9 −0.212620
\(937\) −2.50542e10 −0.994927 −0.497464 0.867485i \(-0.665735\pi\)
−0.497464 + 0.867485i \(0.665735\pi\)
\(938\) 1.00522e10 0.397694
\(939\) −1.50636e10 −0.593744
\(940\) −8.28456e9 −0.325328
\(941\) 3.64762e10 1.42707 0.713536 0.700618i \(-0.247092\pi\)
0.713536 + 0.700618i \(0.247092\pi\)
\(942\) −9.53093e9 −0.371498
\(943\) −1.20631e10 −0.468454
\(944\) −1.16967e10 −0.452543
\(945\) −8.18612e8 −0.0315549
\(946\) −2.18531e10 −0.839256
\(947\) 3.75110e10 1.43527 0.717636 0.696419i \(-0.245224\pi\)
0.717636 + 0.696419i \(0.245224\pi\)
\(948\) 3.23331e9 0.123259
\(949\) −5.62033e10 −2.13466
\(950\) −8.57375e8 −0.0324443
\(951\) −1.19607e10 −0.450946
\(952\) −5.20993e9 −0.195705
\(953\) −1.31985e10 −0.493969 −0.246984 0.969019i \(-0.579440\pi\)
−0.246984 + 0.969019i \(0.579440\pi\)
\(954\) −5.68740e9 −0.212077
\(955\) −1.11742e10 −0.415151
\(956\) 4.31980e9 0.159904
\(957\) −1.05814e10 −0.390257
\(958\) 1.22732e10 0.451003
\(959\) 9.30781e9 0.340787
\(960\) 8.84736e8 0.0322749
\(961\) −2.56524e10 −0.932388
\(962\) −2.13828e9 −0.0774377
\(963\) −1.01476e9 −0.0366161
\(964\) −2.70926e10 −0.974051
\(965\) −6.52204e9 −0.233635
\(966\) −1.57448e9 −0.0561977
\(967\) −4.10453e9 −0.145973 −0.0729863 0.997333i \(-0.523253\pi\)
−0.0729863 + 0.997333i \(0.523253\pi\)
\(968\) −2.07925e9 −0.0736788
\(969\) −5.66383e9 −0.199976
\(970\) −2.38713e9 −0.0839797
\(971\) 2.68931e10 0.942700 0.471350 0.881946i \(-0.343767\pi\)
0.471350 + 0.881946i \(0.343767\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) −6.63212e8 −0.0230811
\(974\) 4.22576e9 0.146537
\(975\) −6.02915e9 −0.208324
\(976\) −6.07838e9 −0.209273
\(977\) 1.19451e10 0.409789 0.204894 0.978784i \(-0.434315\pi\)
0.204894 + 0.978784i \(0.434315\pi\)
\(978\) 5.05906e9 0.172935
\(979\) 2.22164e10 0.756719
\(980\) −5.70273e9 −0.193549
\(981\) −8.93631e9 −0.302215
\(982\) 6.87868e9 0.231801
\(983\) −5.34291e10 −1.79408 −0.897038 0.441954i \(-0.854285\pi\)
−0.897038 + 0.441954i \(0.854285\pi\)
\(984\) −7.61173e9 −0.254683
\(985\) −1.28811e10 −0.429464
\(986\) 2.44132e10 0.811066
\(987\) 9.30293e9 0.307971
\(988\) 6.27355e9 0.206949
\(989\) 1.52371e10 0.500859
\(990\) −2.86323e9 −0.0937849
\(991\) −5.53048e10 −1.80511 −0.902557 0.430570i \(-0.858313\pi\)
−0.902557 + 0.430570i \(0.858313\pi\)
\(992\) −1.41328e9 −0.0459660
\(993\) 1.38171e10 0.447812
\(994\) 3.13177e9 0.101143
\(995\) 9.15078e9 0.294495
\(996\) 4.21745e9 0.135252
\(997\) −7.17326e9 −0.229236 −0.114618 0.993410i \(-0.536564\pi\)
−0.114618 + 0.993410i \(0.536564\pi\)
\(998\) 1.36813e10 0.435682
\(999\) 3.68124e8 0.0116819
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.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.8.a.c.1.3 4 1.1 even 1 trivial