Properties

Label 43.8.a.a.1.6
Level $43$
Weight $8$
Character 43.1
Self dual yes
Analytic conductor $13.433$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.4325560958\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + 209758592 x^{4} + 3410917248 x^{3} - 14180732672 x^{2} - 60918607872 x + 238240894976\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(3.52766\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.52766 q^{2} -57.7031 q^{3} -97.4450 q^{4} +304.619 q^{5} +318.963 q^{6} +1421.71 q^{7} +1246.18 q^{8} +1142.65 q^{9} +O(q^{10})\) \(q-5.52766 q^{2} -57.7031 q^{3} -97.4450 q^{4} +304.619 q^{5} +318.963 q^{6} +1421.71 q^{7} +1246.18 q^{8} +1142.65 q^{9} -1683.83 q^{10} -7143.27 q^{11} +5622.88 q^{12} -1897.38 q^{13} -7858.73 q^{14} -17577.5 q^{15} +5584.49 q^{16} +1140.96 q^{17} -6316.19 q^{18} -16176.5 q^{19} -29683.6 q^{20} -82037.1 q^{21} +39485.6 q^{22} +91608.2 q^{23} -71908.7 q^{24} +14667.6 q^{25} +10488.0 q^{26} +60262.2 q^{27} -138539. q^{28} -229144. q^{29} +97162.2 q^{30} -212418. q^{31} -190381. q^{32} +412189. q^{33} -6306.85 q^{34} +433079. q^{35} -111346. q^{36} -482724. q^{37} +89418.0 q^{38} +109485. q^{39} +379611. q^{40} -384844. q^{41} +453473. q^{42} +79507.0 q^{43} +696076. q^{44} +348073. q^{45} -506379. q^{46} +380758. q^{47} -322242. q^{48} +1.19772e6 q^{49} -81077.5 q^{50} -65837.1 q^{51} +184890. q^{52} -361687. q^{53} -333109. q^{54} -2.17598e6 q^{55} +1.77171e6 q^{56} +933433. q^{57} +1.26663e6 q^{58} -1.95582e6 q^{59} +1.71284e6 q^{60} -479312. q^{61} +1.17417e6 q^{62} +1.62452e6 q^{63} +337544. q^{64} -577976. q^{65} -2.27844e6 q^{66} -2.46363e6 q^{67} -111181. q^{68} -5.28608e6 q^{69} -2.39392e6 q^{70} -1.56017e6 q^{71} +1.42395e6 q^{72} +2.98243e6 q^{73} +2.66833e6 q^{74} -846366. q^{75} +1.57632e6 q^{76} -1.01557e7 q^{77} -605193. q^{78} +2.64376e6 q^{79} +1.70114e6 q^{80} -5.97630e6 q^{81} +2.12728e6 q^{82} +5.54958e6 q^{83} +7.99411e6 q^{84} +347559. q^{85} -439488. q^{86} +1.32223e7 q^{87} -8.90183e6 q^{88} -9.15035e6 q^{89} -1.92403e6 q^{90} -2.69752e6 q^{91} -8.92676e6 q^{92} +1.22572e7 q^{93} -2.10470e6 q^{94} -4.92766e6 q^{95} +1.09856e7 q^{96} +4.33147e6 q^{97} -6.62056e6 q^{98} -8.16227e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} + O(q^{10}) \) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} - 1333q^{10} + 1333q^{11} + 5089q^{12} - 17967q^{13} - 22352q^{14} - 49504q^{15} - 34406q^{16} - 63095q^{17} - 165931q^{18} - 54524q^{19} - 280995q^{20} - 139788q^{21} - 289358q^{22} - 138139q^{23} - 429583q^{24} + 3455q^{25} - 132946q^{26} - 240356q^{27} - 12704q^{28} - 308658q^{29} + 421284q^{30} - 209523q^{31} - 644934q^{32} + 96814q^{33} + 762435q^{34} - 578892q^{35} + 426161q^{36} - 298472q^{37} - 369707q^{38} + 292298q^{39} + 2633173q^{40} - 1346735q^{41} + 1173266q^{42} + 874577q^{43} + 3134292q^{44} + 1893784q^{45} + 3588111q^{46} + 499284q^{47} + 5647533q^{48} + 2544563q^{49} + 3049745q^{50} + 1258424q^{51} + 983088q^{52} - 2210495q^{53} + 6789698q^{54} - 1855072q^{55} - 469976q^{56} - 1238444q^{57} + 4397067q^{58} - 5824216q^{59} - 2889372q^{60} - 4453034q^{61} + 1002789q^{62} - 6240564q^{63} + 4757538q^{64} - 2162872q^{65} - 258940q^{66} - 6859513q^{67} - 9397005q^{68} - 10040030q^{69} + 845078q^{70} - 10726554q^{71} + 1199517q^{72} - 4456898q^{73} + 1046637q^{74} - 3349114q^{75} + 5861267q^{76} - 17019816q^{77} + 1999122q^{78} - 15541320q^{79} - 15680911q^{80} - 12976697q^{81} + 20233655q^{82} - 11146767q^{83} + 12348278q^{84} - 12471976q^{85} - 1908168q^{86} - 18648900q^{87} - 24463544q^{88} - 13531356q^{89} + 20858990q^{90} - 19746448q^{91} - 26023161q^{92} - 21903110q^{93} + 20288857q^{94} - 12291624q^{95} - 13954503q^{96} - 10999901q^{97} + 29909168q^{98} + 29396057q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.52766 −0.488581 −0.244290 0.969702i \(-0.578555\pi\)
−0.244290 + 0.969702i \(0.578555\pi\)
\(3\) −57.7031 −1.23389 −0.616943 0.787008i \(-0.711629\pi\)
−0.616943 + 0.787008i \(0.711629\pi\)
\(4\) −97.4450 −0.761289
\(5\) 304.619 1.08984 0.544919 0.838489i \(-0.316561\pi\)
0.544919 + 0.838489i \(0.316561\pi\)
\(6\) 318.963 0.602853
\(7\) 1421.71 1.56663 0.783317 0.621622i \(-0.213526\pi\)
0.783317 + 0.621622i \(0.213526\pi\)
\(8\) 1246.18 0.860532
\(9\) 1142.65 0.522474
\(10\) −1683.83 −0.532473
\(11\) −7143.27 −1.61817 −0.809083 0.587695i \(-0.800036\pi\)
−0.809083 + 0.587695i \(0.800036\pi\)
\(12\) 5622.88 0.939344
\(13\) −1897.38 −0.239525 −0.119763 0.992803i \(-0.538213\pi\)
−0.119763 + 0.992803i \(0.538213\pi\)
\(14\) −7858.73 −0.765427
\(15\) −17577.5 −1.34473
\(16\) 5584.49 0.340850
\(17\) 1140.96 0.0563249 0.0281624 0.999603i \(-0.491034\pi\)
0.0281624 + 0.999603i \(0.491034\pi\)
\(18\) −6316.19 −0.255271
\(19\) −16176.5 −0.541061 −0.270531 0.962711i \(-0.587199\pi\)
−0.270531 + 0.962711i \(0.587199\pi\)
\(20\) −29683.6 −0.829681
\(21\) −82037.1 −1.93305
\(22\) 39485.6 0.790604
\(23\) 91608.2 1.56995 0.784977 0.619525i \(-0.212675\pi\)
0.784977 + 0.619525i \(0.212675\pi\)
\(24\) −71908.7 −1.06180
\(25\) 14667.6 0.187745
\(26\) 10488.0 0.117027
\(27\) 60262.2 0.589212
\(28\) −138539. −1.19266
\(29\) −229144. −1.74468 −0.872341 0.488898i \(-0.837399\pi\)
−0.872341 + 0.488898i \(0.837399\pi\)
\(30\) 97162.2 0.657011
\(31\) −212418. −1.28064 −0.640318 0.768110i \(-0.721197\pi\)
−0.640318 + 0.768110i \(0.721197\pi\)
\(32\) −190381. −1.02706
\(33\) 412189. 1.99663
\(34\) −6306.85 −0.0275192
\(35\) 433079. 1.70738
\(36\) −111346. −0.397754
\(37\) −482724. −1.56672 −0.783362 0.621566i \(-0.786497\pi\)
−0.783362 + 0.621566i \(0.786497\pi\)
\(38\) 89418.0 0.264352
\(39\) 109485. 0.295547
\(40\) 379611. 0.937839
\(41\) −384844. −0.872049 −0.436025 0.899935i \(-0.643614\pi\)
−0.436025 + 0.899935i \(0.643614\pi\)
\(42\) 453473. 0.944450
\(43\) 79507.0 0.152499
\(44\) 696076. 1.23189
\(45\) 348073. 0.569412
\(46\) −506379. −0.767049
\(47\) 380758. 0.534941 0.267471 0.963566i \(-0.413812\pi\)
0.267471 + 0.963566i \(0.413812\pi\)
\(48\) −322242. −0.420570
\(49\) 1.19772e6 1.45435
\(50\) −81077.5 −0.0917287
\(51\) −65837.1 −0.0694985
\(52\) 184890. 0.182348
\(53\) −361687. −0.333709 −0.166854 0.985982i \(-0.553361\pi\)
−0.166854 + 0.985982i \(0.553361\pi\)
\(54\) −333109. −0.287878
\(55\) −2.17598e6 −1.76354
\(56\) 1.77171e6 1.34814
\(57\) 933433. 0.667608
\(58\) 1.26663e6 0.852418
\(59\) −1.95582e6 −1.23979 −0.619894 0.784685i \(-0.712824\pi\)
−0.619894 + 0.784685i \(0.712824\pi\)
\(60\) 1.71284e6 1.02373
\(61\) −479312. −0.270374 −0.135187 0.990820i \(-0.543163\pi\)
−0.135187 + 0.990820i \(0.543163\pi\)
\(62\) 1.17417e6 0.625694
\(63\) 1.62452e6 0.818526
\(64\) 337544. 0.160954
\(65\) −577976. −0.261044
\(66\) −2.27844e6 −0.975515
\(67\) −2.46363e6 −1.00072 −0.500362 0.865816i \(-0.666800\pi\)
−0.500362 + 0.865816i \(0.666800\pi\)
\(68\) −111181. −0.0428795
\(69\) −5.28608e6 −1.93714
\(70\) −2.39392e6 −0.834191
\(71\) −1.56017e6 −0.517332 −0.258666 0.965967i \(-0.583283\pi\)
−0.258666 + 0.965967i \(0.583283\pi\)
\(72\) 1.42395e6 0.449606
\(73\) 2.98243e6 0.897305 0.448652 0.893706i \(-0.351904\pi\)
0.448652 + 0.893706i \(0.351904\pi\)
\(74\) 2.66833e6 0.765471
\(75\) −846366. −0.231656
\(76\) 1.57632e6 0.411904
\(77\) −1.01557e7 −2.53507
\(78\) −605193. −0.144399
\(79\) 2.64376e6 0.603292 0.301646 0.953420i \(-0.402464\pi\)
0.301646 + 0.953420i \(0.402464\pi\)
\(80\) 1.70114e6 0.371471
\(81\) −5.97630e6 −1.24949
\(82\) 2.12728e6 0.426066
\(83\) 5.54958e6 1.06534 0.532668 0.846324i \(-0.321190\pi\)
0.532668 + 0.846324i \(0.321190\pi\)
\(84\) 7.99411e6 1.47161
\(85\) 347559. 0.0613850
\(86\) −439488. −0.0745078
\(87\) 1.32223e7 2.15274
\(88\) −8.90183e6 −1.39248
\(89\) −9.15035e6 −1.37585 −0.687927 0.725779i \(-0.741479\pi\)
−0.687927 + 0.725779i \(0.741479\pi\)
\(90\) −1.92403e6 −0.278204
\(91\) −2.69752e6 −0.375249
\(92\) −8.92676e6 −1.19519
\(93\) 1.22572e7 1.58016
\(94\) −2.10470e6 −0.261362
\(95\) −4.92766e6 −0.589669
\(96\) 1.09856e7 1.26728
\(97\) 4.33147e6 0.481875 0.240937 0.970541i \(-0.422545\pi\)
0.240937 + 0.970541i \(0.422545\pi\)
\(98\) −6.62056e6 −0.710565
\(99\) −8.16227e6 −0.845450
\(100\) −1.42928e6 −0.142928
\(101\) −5.03114e6 −0.485894 −0.242947 0.970040i \(-0.578114\pi\)
−0.242947 + 0.970040i \(0.578114\pi\)
\(102\) 363925. 0.0339556
\(103\) 1.58973e7 1.43348 0.716741 0.697339i \(-0.245633\pi\)
0.716741 + 0.697339i \(0.245633\pi\)
\(104\) −2.36448e6 −0.206119
\(105\) −2.49900e7 −2.10671
\(106\) 1.99928e6 0.163044
\(107\) −1.56322e7 −1.23361 −0.616805 0.787116i \(-0.711573\pi\)
−0.616805 + 0.787116i \(0.711573\pi\)
\(108\) −5.87225e6 −0.448561
\(109\) 1.27583e7 0.943624 0.471812 0.881699i \(-0.343600\pi\)
0.471812 + 0.881699i \(0.343600\pi\)
\(110\) 1.20280e7 0.861630
\(111\) 2.78547e7 1.93316
\(112\) 7.93952e6 0.533987
\(113\) −1.58622e7 −1.03416 −0.517081 0.855937i \(-0.672981\pi\)
−0.517081 + 0.855937i \(0.672981\pi\)
\(114\) −5.15970e6 −0.326180
\(115\) 2.79056e7 1.71099
\(116\) 2.23290e7 1.32821
\(117\) −2.16804e6 −0.125146
\(118\) 1.08111e7 0.605737
\(119\) 1.62212e6 0.0882405
\(120\) −2.19047e7 −1.15719
\(121\) 3.15392e7 1.61846
\(122\) 2.64948e6 0.132099
\(123\) 2.22067e7 1.07601
\(124\) 2.06991e7 0.974934
\(125\) −1.93303e7 −0.885225
\(126\) −8.97978e6 −0.399916
\(127\) −2.44940e7 −1.06108 −0.530539 0.847660i \(-0.678011\pi\)
−0.530539 + 0.847660i \(0.678011\pi\)
\(128\) 2.25029e7 0.948425
\(129\) −4.58780e6 −0.188166
\(130\) 3.19486e6 0.127541
\(131\) 3.59703e7 1.39796 0.698979 0.715142i \(-0.253638\pi\)
0.698979 + 0.715142i \(0.253638\pi\)
\(132\) −4.01658e7 −1.52001
\(133\) −2.29983e7 −0.847645
\(134\) 1.36181e7 0.488934
\(135\) 1.83570e7 0.642145
\(136\) 1.42185e6 0.0484693
\(137\) −1.88193e7 −0.625289 −0.312644 0.949870i \(-0.601215\pi\)
−0.312644 + 0.949870i \(0.601215\pi\)
\(138\) 2.92196e7 0.946451
\(139\) −3.72014e7 −1.17492 −0.587459 0.809254i \(-0.699872\pi\)
−0.587459 + 0.809254i \(0.699872\pi\)
\(140\) −4.22014e7 −1.29981
\(141\) −2.19709e7 −0.660056
\(142\) 8.62411e6 0.252758
\(143\) 1.35535e7 0.387592
\(144\) 6.38112e6 0.178085
\(145\) −6.98017e7 −1.90142
\(146\) −1.64858e7 −0.438406
\(147\) −6.91119e7 −1.79450
\(148\) 4.70390e7 1.19273
\(149\) −4.39692e7 −1.08892 −0.544461 0.838786i \(-0.683266\pi\)
−0.544461 + 0.838786i \(0.683266\pi\)
\(150\) 4.67842e6 0.113183
\(151\) −3.09651e7 −0.731902 −0.365951 0.930634i \(-0.619256\pi\)
−0.365951 + 0.930634i \(0.619256\pi\)
\(152\) −2.01588e7 −0.465600
\(153\) 1.30372e6 0.0294283
\(154\) 5.61370e7 1.23859
\(155\) −6.47065e7 −1.39568
\(156\) −1.06687e7 −0.224997
\(157\) −1.67885e6 −0.0346228 −0.0173114 0.999850i \(-0.505511\pi\)
−0.0173114 + 0.999850i \(0.505511\pi\)
\(158\) −1.46138e7 −0.294757
\(159\) 2.08705e7 0.411759
\(160\) −5.79935e7 −1.11933
\(161\) 1.30240e8 2.45954
\(162\) 3.30349e7 0.610479
\(163\) 8.30485e6 0.150202 0.0751009 0.997176i \(-0.476072\pi\)
0.0751009 + 0.997176i \(0.476072\pi\)
\(164\) 3.75011e7 0.663881
\(165\) 1.25561e8 2.17600
\(166\) −3.06762e7 −0.520502
\(167\) 4.62254e7 0.768021 0.384010 0.923329i \(-0.374543\pi\)
0.384010 + 0.923329i \(0.374543\pi\)
\(168\) −1.02233e8 −1.66345
\(169\) −5.91485e7 −0.942628
\(170\) −1.92119e6 −0.0299915
\(171\) −1.84841e7 −0.282691
\(172\) −7.74756e6 −0.116095
\(173\) −4.56672e7 −0.670568 −0.335284 0.942117i \(-0.608832\pi\)
−0.335284 + 0.942117i \(0.608832\pi\)
\(174\) −7.30886e7 −1.05179
\(175\) 2.08531e7 0.294128
\(176\) −3.98915e7 −0.551552
\(177\) 1.12857e8 1.52976
\(178\) 5.05800e7 0.672216
\(179\) 7.68660e7 1.00173 0.500863 0.865527i \(-0.333016\pi\)
0.500863 + 0.865527i \(0.333016\pi\)
\(180\) −3.39180e7 −0.433487
\(181\) 7.38362e7 0.925538 0.462769 0.886479i \(-0.346856\pi\)
0.462769 + 0.886479i \(0.346856\pi\)
\(182\) 1.49110e7 0.183339
\(183\) 2.76578e7 0.333610
\(184\) 1.14161e8 1.35100
\(185\) −1.47047e8 −1.70747
\(186\) −6.77535e7 −0.772034
\(187\) −8.15021e6 −0.0911430
\(188\) −3.71029e7 −0.407245
\(189\) 8.56754e7 0.923081
\(190\) 2.72384e7 0.288101
\(191\) −4.61537e7 −0.479280 −0.239640 0.970862i \(-0.577029\pi\)
−0.239640 + 0.970862i \(0.577029\pi\)
\(192\) −1.94774e7 −0.198598
\(193\) 1.04470e8 1.04603 0.523013 0.852325i \(-0.324808\pi\)
0.523013 + 0.852325i \(0.324808\pi\)
\(194\) −2.39429e7 −0.235435
\(195\) 3.33510e7 0.322098
\(196\) −1.16711e8 −1.10718
\(197\) 3.24556e7 0.302453 0.151226 0.988499i \(-0.451678\pi\)
0.151226 + 0.988499i \(0.451678\pi\)
\(198\) 4.51182e7 0.413070
\(199\) 1.90706e8 1.71545 0.857725 0.514109i \(-0.171877\pi\)
0.857725 + 0.514109i \(0.171877\pi\)
\(200\) 1.82785e7 0.161561
\(201\) 1.42159e8 1.23478
\(202\) 2.78104e7 0.237398
\(203\) −3.25777e8 −2.73328
\(204\) 6.41550e6 0.0529084
\(205\) −1.17231e8 −0.950392
\(206\) −8.78747e7 −0.700372
\(207\) 1.04676e8 0.820261
\(208\) −1.05959e7 −0.0816422
\(209\) 1.15553e8 0.875526
\(210\) 1.38136e8 1.02930
\(211\) −4.93604e6 −0.0361735 −0.0180867 0.999836i \(-0.505758\pi\)
−0.0180867 + 0.999836i \(0.505758\pi\)
\(212\) 3.52446e7 0.254049
\(213\) 9.00270e7 0.638328
\(214\) 8.64097e7 0.602718
\(215\) 2.42193e7 0.166199
\(216\) 7.50977e7 0.507036
\(217\) −3.01997e8 −2.00629
\(218\) −7.05233e7 −0.461036
\(219\) −1.72095e8 −1.10717
\(220\) 2.12038e8 1.34256
\(221\) −2.16484e6 −0.0134912
\(222\) −1.53971e8 −0.944504
\(223\) 2.77562e8 1.67608 0.838038 0.545612i \(-0.183703\pi\)
0.838038 + 0.545612i \(0.183703\pi\)
\(224\) −2.70666e8 −1.60903
\(225\) 1.67599e7 0.0980921
\(226\) 8.76807e7 0.505271
\(227\) −1.27335e8 −0.722535 −0.361267 0.932462i \(-0.617656\pi\)
−0.361267 + 0.932462i \(0.617656\pi\)
\(228\) −9.09584e7 −0.508242
\(229\) 1.66075e8 0.913861 0.456931 0.889502i \(-0.348949\pi\)
0.456931 + 0.889502i \(0.348949\pi\)
\(230\) −1.54252e8 −0.835959
\(231\) 5.86014e8 3.12799
\(232\) −2.85556e8 −1.50135
\(233\) −1.97183e8 −1.02123 −0.510616 0.859809i \(-0.670583\pi\)
−0.510616 + 0.859809i \(0.670583\pi\)
\(234\) 1.19842e7 0.0611438
\(235\) 1.15986e8 0.582999
\(236\) 1.90585e8 0.943837
\(237\) −1.52553e8 −0.744393
\(238\) −8.96652e6 −0.0431126
\(239\) 7.22966e7 0.342551 0.171275 0.985223i \(-0.445211\pi\)
0.171275 + 0.985223i \(0.445211\pi\)
\(240\) −9.81611e7 −0.458353
\(241\) −3.09906e8 −1.42617 −0.713084 0.701079i \(-0.752702\pi\)
−0.713084 + 0.701079i \(0.752702\pi\)
\(242\) −1.74338e8 −0.790748
\(243\) 2.13058e8 0.952522
\(244\) 4.67066e7 0.205832
\(245\) 3.64847e8 1.58500
\(246\) −1.22751e8 −0.525717
\(247\) 3.06929e7 0.129598
\(248\) −2.64712e8 −1.10203
\(249\) −3.20228e8 −1.31450
\(250\) 1.06851e8 0.432504
\(251\) 1.11482e8 0.444988 0.222494 0.974934i \(-0.428580\pi\)
0.222494 + 0.974934i \(0.428580\pi\)
\(252\) −1.58301e8 −0.623135
\(253\) −6.54383e8 −2.54045
\(254\) 1.35395e8 0.518422
\(255\) −2.00552e7 −0.0757420
\(256\) −1.67594e8 −0.624336
\(257\) 8.26622e7 0.303767 0.151884 0.988398i \(-0.451466\pi\)
0.151884 + 0.988398i \(0.451466\pi\)
\(258\) 2.53598e7 0.0919342
\(259\) −6.86293e8 −2.45448
\(260\) 5.63209e7 0.198730
\(261\) −2.61832e8 −0.911551
\(262\) −1.98831e8 −0.683015
\(263\) −2.74508e8 −0.930486 −0.465243 0.885183i \(-0.654033\pi\)
−0.465243 + 0.885183i \(0.654033\pi\)
\(264\) 5.13663e8 1.71816
\(265\) −1.10177e8 −0.363688
\(266\) 1.27126e8 0.414143
\(267\) 5.28004e8 1.69765
\(268\) 2.40069e8 0.761840
\(269\) −9.06481e7 −0.283940 −0.141970 0.989871i \(-0.545344\pi\)
−0.141970 + 0.989871i \(0.545344\pi\)
\(270\) −1.01471e8 −0.313740
\(271\) 2.46059e8 0.751011 0.375505 0.926820i \(-0.377469\pi\)
0.375505 + 0.926820i \(0.377469\pi\)
\(272\) 6.37169e6 0.0191983
\(273\) 1.55655e8 0.463014
\(274\) 1.04026e8 0.305504
\(275\) −1.04775e8 −0.303803
\(276\) 5.15102e8 1.47473
\(277\) 6.60621e8 1.86755 0.933777 0.357857i \(-0.116492\pi\)
0.933777 + 0.357857i \(0.116492\pi\)
\(278\) 2.05637e8 0.574042
\(279\) −2.42720e8 −0.669099
\(280\) 5.39696e8 1.46925
\(281\) 2.01108e7 0.0540700 0.0270350 0.999634i \(-0.491393\pi\)
0.0270350 + 0.999634i \(0.491393\pi\)
\(282\) 1.21448e8 0.322491
\(283\) 2.51117e8 0.658602 0.329301 0.944225i \(-0.393187\pi\)
0.329301 + 0.944225i \(0.393187\pi\)
\(284\) 1.52031e8 0.393839
\(285\) 2.84341e8 0.727584
\(286\) −7.49190e7 −0.189370
\(287\) −5.47136e8 −1.36618
\(288\) −2.17539e8 −0.536615
\(289\) −4.09037e8 −0.996828
\(290\) 3.85840e8 0.928997
\(291\) −2.49939e8 −0.594578
\(292\) −2.90623e8 −0.683108
\(293\) −3.74733e8 −0.870333 −0.435166 0.900350i \(-0.643310\pi\)
−0.435166 + 0.900350i \(0.643310\pi\)
\(294\) 3.82027e8 0.876756
\(295\) −5.95780e8 −1.35117
\(296\) −6.01562e8 −1.34822
\(297\) −4.30469e8 −0.953443
\(298\) 2.43047e8 0.532027
\(299\) −1.73815e8 −0.376044
\(300\) 8.24742e7 0.176357
\(301\) 1.13036e8 0.238910
\(302\) 1.71164e8 0.357593
\(303\) 2.90312e8 0.599537
\(304\) −9.03373e7 −0.184421
\(305\) −1.46008e8 −0.294663
\(306\) −7.20653e6 −0.0143781
\(307\) 3.69155e8 0.728156 0.364078 0.931369i \(-0.381384\pi\)
0.364078 + 0.931369i \(0.381384\pi\)
\(308\) 9.89619e8 1.92992
\(309\) −9.17322e8 −1.76875
\(310\) 3.57676e8 0.681904
\(311\) −7.67471e8 −1.44677 −0.723387 0.690443i \(-0.757416\pi\)
−0.723387 + 0.690443i \(0.757416\pi\)
\(312\) 1.36438e8 0.254328
\(313\) 3.94815e8 0.727761 0.363881 0.931446i \(-0.381452\pi\)
0.363881 + 0.931446i \(0.381452\pi\)
\(314\) 9.28008e6 0.0169160
\(315\) 4.94859e8 0.892061
\(316\) −2.57621e8 −0.459279
\(317\) 5.25976e8 0.927381 0.463691 0.885997i \(-0.346525\pi\)
0.463691 + 0.885997i \(0.346525\pi\)
\(318\) −1.15365e8 −0.201177
\(319\) 1.63684e9 2.82318
\(320\) 1.02822e8 0.175413
\(321\) 9.02029e8 1.52213
\(322\) −7.19924e8 −1.20169
\(323\) −1.84568e7 −0.0304752
\(324\) 5.82360e8 0.951227
\(325\) −2.78299e7 −0.0449698
\(326\) −4.59064e7 −0.0733857
\(327\) −7.36192e8 −1.16432
\(328\) −4.79586e8 −0.750426
\(329\) 5.41327e8 0.838058
\(330\) −6.94056e8 −1.06315
\(331\) 1.17755e9 1.78477 0.892386 0.451273i \(-0.149030\pi\)
0.892386 + 0.451273i \(0.149030\pi\)
\(332\) −5.40778e8 −0.811028
\(333\) −5.51585e8 −0.818573
\(334\) −2.55518e8 −0.375240
\(335\) −7.50469e8 −1.09063
\(336\) −4.58135e8 −0.658880
\(337\) −4.76656e8 −0.678423 −0.339212 0.940710i \(-0.610160\pi\)
−0.339212 + 0.940710i \(0.610160\pi\)
\(338\) 3.26953e8 0.460550
\(339\) 9.15297e8 1.27604
\(340\) −3.38679e7 −0.0467317
\(341\) 1.51736e9 2.07228
\(342\) 1.02174e8 0.138117
\(343\) 5.31965e8 0.711793
\(344\) 9.90803e7 0.131230
\(345\) −1.61024e9 −2.11117
\(346\) 2.52433e8 0.327626
\(347\) 1.08505e8 0.139411 0.0697057 0.997568i \(-0.477794\pi\)
0.0697057 + 0.997568i \(0.477794\pi\)
\(348\) −1.28845e9 −1.63886
\(349\) −7.46559e8 −0.940102 −0.470051 0.882639i \(-0.655765\pi\)
−0.470051 + 0.882639i \(0.655765\pi\)
\(350\) −1.15269e8 −0.143705
\(351\) −1.14340e8 −0.141131
\(352\) 1.35994e9 1.66196
\(353\) −4.13749e8 −0.500640 −0.250320 0.968163i \(-0.580536\pi\)
−0.250320 + 0.968163i \(0.580536\pi\)
\(354\) −6.23836e8 −0.747410
\(355\) −4.75259e8 −0.563807
\(356\) 8.91656e8 1.04742
\(357\) −9.36013e7 −0.108879
\(358\) −4.24889e8 −0.489424
\(359\) −1.58431e8 −0.180721 −0.0903607 0.995909i \(-0.528802\pi\)
−0.0903607 + 0.995909i \(0.528802\pi\)
\(360\) 4.33763e8 0.489997
\(361\) −6.32193e8 −0.707253
\(362\) −4.08141e8 −0.452200
\(363\) −1.81991e9 −1.99699
\(364\) 2.62860e8 0.285673
\(365\) 9.08504e8 0.977916
\(366\) −1.52883e8 −0.162995
\(367\) 5.61609e7 0.0593065 0.0296533 0.999560i \(-0.490560\pi\)
0.0296533 + 0.999560i \(0.490560\pi\)
\(368\) 5.11585e8 0.535119
\(369\) −4.39742e8 −0.455623
\(370\) 8.12824e8 0.834239
\(371\) −5.14214e8 −0.522800
\(372\) −1.19440e9 −1.20296
\(373\) 1.21691e9 1.21416 0.607080 0.794641i \(-0.292341\pi\)
0.607080 + 0.794641i \(0.292341\pi\)
\(374\) 4.50516e7 0.0445307
\(375\) 1.11542e9 1.09227
\(376\) 4.74494e8 0.460334
\(377\) 4.34773e8 0.417896
\(378\) −4.73584e8 −0.450999
\(379\) −1.99936e9 −1.88649 −0.943245 0.332098i \(-0.892244\pi\)
−0.943245 + 0.332098i \(0.892244\pi\)
\(380\) 4.80176e8 0.448908
\(381\) 1.41338e9 1.30925
\(382\) 2.55122e8 0.234167
\(383\) −7.65586e8 −0.696303 −0.348151 0.937438i \(-0.613191\pi\)
−0.348151 + 0.937438i \(0.613191\pi\)
\(384\) −1.29849e9 −1.17025
\(385\) −3.09361e9 −2.76282
\(386\) −5.77476e8 −0.511068
\(387\) 9.08488e7 0.0796766
\(388\) −4.22080e8 −0.366846
\(389\) −6.56361e8 −0.565352 −0.282676 0.959215i \(-0.591222\pi\)
−0.282676 + 0.959215i \(0.591222\pi\)
\(390\) −1.84353e8 −0.157371
\(391\) 1.04522e8 0.0884275
\(392\) 1.49257e9 1.25151
\(393\) −2.07560e9 −1.72492
\(394\) −1.79403e8 −0.147773
\(395\) 8.05339e8 0.657490
\(396\) 7.95372e8 0.643632
\(397\) 9.12150e8 0.731644 0.365822 0.930685i \(-0.380788\pi\)
0.365822 + 0.930685i \(0.380788\pi\)
\(398\) −1.05416e9 −0.838136
\(399\) 1.32707e9 1.04590
\(400\) 8.19110e7 0.0639930
\(401\) −1.15178e8 −0.0891997 −0.0445999 0.999005i \(-0.514201\pi\)
−0.0445999 + 0.999005i \(0.514201\pi\)
\(402\) −7.85809e8 −0.603289
\(403\) 4.03037e8 0.306745
\(404\) 4.90259e8 0.369906
\(405\) −1.82049e9 −1.36175
\(406\) 1.80078e9 1.33543
\(407\) 3.44823e9 2.53522
\(408\) −8.20451e7 −0.0598056
\(409\) −8.39721e8 −0.606881 −0.303440 0.952850i \(-0.598135\pi\)
−0.303440 + 0.952850i \(0.598135\pi\)
\(410\) 6.48011e8 0.464343
\(411\) 1.08593e9 0.771535
\(412\) −1.54911e9 −1.09129
\(413\) −2.78061e9 −1.94230
\(414\) −5.78614e8 −0.400763
\(415\) 1.69051e9 1.16104
\(416\) 3.61223e8 0.246008
\(417\) 2.14664e9 1.44971
\(418\) −6.38738e8 −0.427765
\(419\) 9.18009e8 0.609674 0.304837 0.952404i \(-0.401398\pi\)
0.304837 + 0.952404i \(0.401398\pi\)
\(420\) 2.43515e9 1.60381
\(421\) 9.86735e7 0.0644486 0.0322243 0.999481i \(-0.489741\pi\)
0.0322243 + 0.999481i \(0.489741\pi\)
\(422\) 2.72847e7 0.0176737
\(423\) 4.35073e8 0.279493
\(424\) −4.50728e8 −0.287167
\(425\) 1.67352e7 0.0105747
\(426\) −4.97638e8 −0.311875
\(427\) −6.81443e8 −0.423577
\(428\) 1.52328e9 0.939134
\(429\) −7.82078e8 −0.478244
\(430\) −1.33876e8 −0.0812014
\(431\) 2.42488e9 1.45888 0.729440 0.684045i \(-0.239781\pi\)
0.729440 + 0.684045i \(0.239781\pi\)
\(432\) 3.36533e8 0.200833
\(433\) 1.63675e9 0.968888 0.484444 0.874822i \(-0.339022\pi\)
0.484444 + 0.874822i \(0.339022\pi\)
\(434\) 1.66934e9 0.980233
\(435\) 4.02778e9 2.34613
\(436\) −1.24323e9 −0.718370
\(437\) −1.48190e9 −0.849441
\(438\) 9.51285e8 0.540943
\(439\) −1.40300e9 −0.791465 −0.395733 0.918366i \(-0.629509\pi\)
−0.395733 + 0.918366i \(0.629509\pi\)
\(440\) −2.71166e9 −1.51758
\(441\) 1.36857e9 0.759858
\(442\) 1.19665e7 0.00659156
\(443\) 4.39909e8 0.240409 0.120204 0.992749i \(-0.461645\pi\)
0.120204 + 0.992749i \(0.461645\pi\)
\(444\) −2.71430e9 −1.47169
\(445\) −2.78737e9 −1.49946
\(446\) −1.53427e9 −0.818898
\(447\) 2.53716e9 1.34361
\(448\) 4.79890e8 0.252156
\(449\) 2.11015e9 1.10015 0.550073 0.835116i \(-0.314600\pi\)
0.550073 + 0.835116i \(0.314600\pi\)
\(450\) −9.26433e7 −0.0479259
\(451\) 2.74904e9 1.41112
\(452\) 1.54569e9 0.787295
\(453\) 1.78678e9 0.903083
\(454\) 7.03866e8 0.353016
\(455\) −8.21715e8 −0.408960
\(456\) 1.16323e9 0.574498
\(457\) −2.13547e9 −1.04661 −0.523307 0.852144i \(-0.675302\pi\)
−0.523307 + 0.852144i \(0.675302\pi\)
\(458\) −9.18006e8 −0.446495
\(459\) 6.87569e7 0.0331873
\(460\) −2.71926e9 −1.30256
\(461\) 7.44367e8 0.353862 0.176931 0.984223i \(-0.443383\pi\)
0.176931 + 0.984223i \(0.443383\pi\)
\(462\) −3.23928e9 −1.52828
\(463\) 1.70021e9 0.796105 0.398052 0.917363i \(-0.369686\pi\)
0.398052 + 0.917363i \(0.369686\pi\)
\(464\) −1.27965e9 −0.594675
\(465\) 3.73377e9 1.72211
\(466\) 1.08996e9 0.498954
\(467\) −3.93170e9 −1.78637 −0.893185 0.449689i \(-0.851535\pi\)
−0.893185 + 0.449689i \(0.851535\pi\)
\(468\) 2.11265e8 0.0952722
\(469\) −3.50257e9 −1.56777
\(470\) −6.41131e8 −0.284842
\(471\) 9.68746e7 0.0427206
\(472\) −2.43731e9 −1.06688
\(473\) −5.67940e8 −0.246768
\(474\) 8.43263e8 0.363696
\(475\) −2.37270e8 −0.101582
\(476\) −1.58067e8 −0.0671766
\(477\) −4.13282e8 −0.174354
\(478\) −3.99631e8 −0.167364
\(479\) −1.45058e9 −0.603071 −0.301536 0.953455i \(-0.597499\pi\)
−0.301536 + 0.953455i \(0.597499\pi\)
\(480\) 3.34641e9 1.38113
\(481\) 9.15908e8 0.375270
\(482\) 1.71305e9 0.696798
\(483\) −7.51527e9 −3.03480
\(484\) −3.07334e9 −1.23212
\(485\) 1.31945e9 0.525165
\(486\) −1.17771e9 −0.465384
\(487\) 4.29131e9 1.68360 0.841800 0.539790i \(-0.181496\pi\)
0.841800 + 0.539790i \(0.181496\pi\)
\(488\) −5.97311e8 −0.232665
\(489\) −4.79216e8 −0.185332
\(490\) −2.01675e9 −0.774400
\(491\) −3.50889e8 −0.133778 −0.0668889 0.997760i \(-0.521307\pi\)
−0.0668889 + 0.997760i \(0.521307\pi\)
\(492\) −2.16393e9 −0.819154
\(493\) −2.61445e8 −0.0982690
\(494\) −1.69660e8 −0.0633190
\(495\) −2.48638e9 −0.921403
\(496\) −1.18625e9 −0.436505
\(497\) −2.21812e9 −0.810470
\(498\) 1.77011e9 0.642240
\(499\) −3.72276e9 −1.34126 −0.670630 0.741792i \(-0.733976\pi\)
−0.670630 + 0.741792i \(0.733976\pi\)
\(500\) 1.88364e9 0.673912
\(501\) −2.66735e9 −0.947650
\(502\) −6.16236e8 −0.217412
\(503\) 5.48803e9 1.92277 0.961387 0.275199i \(-0.0887439\pi\)
0.961387 + 0.275199i \(0.0887439\pi\)
\(504\) 2.02445e9 0.704368
\(505\) −1.53258e9 −0.529545
\(506\) 3.61720e9 1.24121
\(507\) 3.41305e9 1.16309
\(508\) 2.38682e9 0.807788
\(509\) 3.85451e8 0.129556 0.0647778 0.997900i \(-0.479366\pi\)
0.0647778 + 0.997900i \(0.479366\pi\)
\(510\) 1.10858e8 0.0370061
\(511\) 4.24015e9 1.40575
\(512\) −1.95397e9 −0.643387
\(513\) −9.74830e8 −0.318800
\(514\) −4.56928e8 −0.148415
\(515\) 4.84261e9 1.56226
\(516\) 4.47058e8 0.143249
\(517\) −2.71986e9 −0.865623
\(518\) 3.79359e9 1.19921
\(519\) 2.63514e9 0.827404
\(520\) −7.20264e8 −0.224636
\(521\) −3.74846e9 −1.16124 −0.580618 0.814176i \(-0.697189\pi\)
−0.580618 + 0.814176i \(0.697189\pi\)
\(522\) 1.44732e9 0.445366
\(523\) 2.64980e9 0.809949 0.404974 0.914328i \(-0.367280\pi\)
0.404974 + 0.914328i \(0.367280\pi\)
\(524\) −3.50512e9 −1.06425
\(525\) −1.20329e9 −0.362921
\(526\) 1.51739e9 0.454617
\(527\) −2.42361e8 −0.0721316
\(528\) 2.30187e9 0.680552
\(529\) 4.98724e9 1.46476
\(530\) 6.09019e8 0.177691
\(531\) −2.23482e9 −0.647758
\(532\) 2.24106e9 0.645303
\(533\) 7.30193e8 0.208878
\(534\) −2.91862e9 −0.829438
\(535\) −4.76187e9 −1.34443
\(536\) −3.07014e9 −0.861155
\(537\) −4.43541e9 −1.23602
\(538\) 5.01072e8 0.138727
\(539\) −8.55561e9 −2.35337
\(540\) −1.78880e9 −0.488858
\(541\) −5.07052e9 −1.37677 −0.688386 0.725344i \(-0.741681\pi\)
−0.688386 + 0.725344i \(0.741681\pi\)
\(542\) −1.36013e9 −0.366929
\(543\) −4.26058e9 −1.14201
\(544\) −2.17217e8 −0.0578493
\(545\) 3.88641e9 1.02840
\(546\) −8.60409e8 −0.226220
\(547\) 3.00582e9 0.785249 0.392624 0.919699i \(-0.371567\pi\)
0.392624 + 0.919699i \(0.371567\pi\)
\(548\) 1.83384e9 0.476026
\(549\) −5.47687e8 −0.141263
\(550\) 5.79159e8 0.148432
\(551\) 3.70675e9 0.943980
\(552\) −6.58742e9 −1.66697
\(553\) 3.75866e9 0.945138
\(554\) −3.65169e9 −0.912450
\(555\) 8.48505e9 2.10683
\(556\) 3.62509e9 0.894452
\(557\) 5.43732e9 1.33319 0.666594 0.745421i \(-0.267751\pi\)
0.666594 + 0.745421i \(0.267751\pi\)
\(558\) 1.34167e9 0.326909
\(559\) −1.50855e8 −0.0365273
\(560\) 2.41853e9 0.581959
\(561\) 4.70293e8 0.112460
\(562\) −1.11166e8 −0.0264176
\(563\) −4.80656e9 −1.13516 −0.567578 0.823320i \(-0.692119\pi\)
−0.567578 + 0.823320i \(0.692119\pi\)
\(564\) 2.14095e9 0.502494
\(565\) −4.83192e9 −1.12707
\(566\) −1.38809e9 −0.321780
\(567\) −8.49656e9 −1.95750
\(568\) −1.94426e9 −0.445180
\(569\) 7.07656e9 1.61038 0.805191 0.593015i \(-0.202063\pi\)
0.805191 + 0.593015i \(0.202063\pi\)
\(570\) −1.57174e9 −0.355483
\(571\) −3.80378e9 −0.855045 −0.427523 0.904005i \(-0.640614\pi\)
−0.427523 + 0.904005i \(0.640614\pi\)
\(572\) −1.32072e9 −0.295069
\(573\) 2.66321e9 0.591377
\(574\) 3.02438e9 0.667490
\(575\) 1.34367e9 0.294751
\(576\) 3.85695e8 0.0840942
\(577\) −2.81170e8 −0.0609332 −0.0304666 0.999536i \(-0.509699\pi\)
−0.0304666 + 0.999536i \(0.509699\pi\)
\(578\) 2.26102e9 0.487031
\(579\) −6.02826e9 −1.29068
\(580\) 6.80182e9 1.44753
\(581\) 7.88989e9 1.66899
\(582\) 1.38158e9 0.290499
\(583\) 2.58363e9 0.539996
\(584\) 3.71665e9 0.772159
\(585\) −6.60425e8 −0.136389
\(586\) 2.07140e9 0.425228
\(587\) 8.79495e8 0.179473 0.0897367 0.995966i \(-0.471397\pi\)
0.0897367 + 0.995966i \(0.471397\pi\)
\(588\) 6.73461e9 1.36613
\(589\) 3.43618e9 0.692902
\(590\) 3.29327e9 0.660154
\(591\) −1.87279e9 −0.373192
\(592\) −2.69576e9 −0.534018
\(593\) −6.78081e8 −0.133534 −0.0667668 0.997769i \(-0.521268\pi\)
−0.0667668 + 0.997769i \(0.521268\pi\)
\(594\) 2.37949e9 0.465834
\(595\) 4.94128e8 0.0961678
\(596\) 4.28458e9 0.828985
\(597\) −1.10043e10 −2.11667
\(598\) 9.60791e8 0.183728
\(599\) −6.13578e9 −1.16648 −0.583238 0.812301i \(-0.698215\pi\)
−0.583238 + 0.812301i \(0.698215\pi\)
\(600\) −1.05473e9 −0.199347
\(601\) −4.78315e8 −0.0898780 −0.0449390 0.998990i \(-0.514309\pi\)
−0.0449390 + 0.998990i \(0.514309\pi\)
\(602\) −6.24824e8 −0.116727
\(603\) −2.81507e9 −0.522853
\(604\) 3.01739e9 0.557189
\(605\) 9.60743e9 1.76386
\(606\) −1.60475e9 −0.292922
\(607\) −8.97283e9 −1.62843 −0.814215 0.580563i \(-0.802832\pi\)
−0.814215 + 0.580563i \(0.802832\pi\)
\(608\) 3.07969e9 0.555705
\(609\) 1.87983e10 3.37255
\(610\) 8.07080e8 0.143967
\(611\) −7.22440e8 −0.128132
\(612\) −1.27041e8 −0.0224034
\(613\) 7.77928e9 1.36404 0.682021 0.731333i \(-0.261101\pi\)
0.682021 + 0.731333i \(0.261101\pi\)
\(614\) −2.04056e9 −0.355763
\(615\) 6.76457e9 1.17267
\(616\) −1.26558e10 −2.18151
\(617\) −8.81774e9 −1.51133 −0.755665 0.654958i \(-0.772686\pi\)
−0.755665 + 0.654958i \(0.772686\pi\)
\(618\) 5.07064e9 0.864179
\(619\) −4.50857e9 −0.764050 −0.382025 0.924152i \(-0.624773\pi\)
−0.382025 + 0.924152i \(0.624773\pi\)
\(620\) 6.30533e9 1.06252
\(621\) 5.52051e9 0.925036
\(622\) 4.24232e9 0.706865
\(623\) −1.30091e10 −2.15546
\(624\) 6.11415e8 0.100737
\(625\) −7.03428e9 −1.15250
\(626\) −2.18240e9 −0.355570
\(627\) −6.66777e9 −1.08030
\(628\) 1.63595e8 0.0263579
\(629\) −5.50770e8 −0.0882456
\(630\) −2.73541e9 −0.435843
\(631\) −1.47926e9 −0.234391 −0.117195 0.993109i \(-0.537390\pi\)
−0.117195 + 0.993109i \(0.537390\pi\)
\(632\) 3.29461e9 0.519152
\(633\) 2.84825e8 0.0446339
\(634\) −2.90741e9 −0.453101
\(635\) −7.46135e9 −1.15640
\(636\) −2.03372e9 −0.313467
\(637\) −2.27252e9 −0.348353
\(638\) −9.04790e9 −1.37935
\(639\) −1.78274e9 −0.270292
\(640\) 6.85480e9 1.03363
\(641\) 6.56349e9 0.984309 0.492155 0.870508i \(-0.336209\pi\)
0.492155 + 0.870508i \(0.336209\pi\)
\(642\) −4.98611e9 −0.743685
\(643\) 6.43261e9 0.954221 0.477110 0.878843i \(-0.341684\pi\)
0.477110 + 0.878843i \(0.341684\pi\)
\(644\) −1.26913e10 −1.87242
\(645\) −1.39753e9 −0.205070
\(646\) 1.02023e8 0.0148896
\(647\) 1.74903e8 0.0253882 0.0126941 0.999919i \(-0.495959\pi\)
0.0126941 + 0.999919i \(0.495959\pi\)
\(648\) −7.44756e9 −1.07523
\(649\) 1.39710e10 2.00618
\(650\) 1.53834e8 0.0219714
\(651\) 1.74262e10 2.47553
\(652\) −8.09266e8 −0.114347
\(653\) −1.10036e10 −1.54646 −0.773230 0.634126i \(-0.781360\pi\)
−0.773230 + 0.634126i \(0.781360\pi\)
\(654\) 4.06942e9 0.568866
\(655\) 1.09572e10 1.52355
\(656\) −2.14915e9 −0.297238
\(657\) 3.40788e9 0.468819
\(658\) −2.99227e9 −0.409459
\(659\) −1.09969e10 −1.49683 −0.748416 0.663230i \(-0.769185\pi\)
−0.748416 + 0.663230i \(0.769185\pi\)
\(660\) −1.22353e10 −1.65657
\(661\) 1.20713e10 1.62574 0.812869 0.582447i \(-0.197905\pi\)
0.812869 + 0.582447i \(0.197905\pi\)
\(662\) −6.50911e9 −0.872005
\(663\) 1.24918e8 0.0166467
\(664\) 6.91579e9 0.916755
\(665\) −7.00570e9 −0.923795
\(666\) 3.04897e9 0.399939
\(667\) −2.09915e10 −2.73907
\(668\) −4.50443e9 −0.584686
\(669\) −1.60162e10 −2.06809
\(670\) 4.14834e9 0.532859
\(671\) 3.42386e9 0.437509
\(672\) 1.56183e10 1.98537
\(673\) 4.08129e9 0.516113 0.258057 0.966130i \(-0.416918\pi\)
0.258057 + 0.966130i \(0.416918\pi\)
\(674\) 2.63479e9 0.331464
\(675\) 8.83902e8 0.110622
\(676\) 5.76372e9 0.717612
\(677\) −7.05725e9 −0.874128 −0.437064 0.899430i \(-0.643982\pi\)
−0.437064 + 0.899430i \(0.643982\pi\)
\(678\) −5.05945e9 −0.623447
\(679\) 6.15809e9 0.754922
\(680\) 4.33122e8 0.0528237
\(681\) 7.34765e9 0.891525
\(682\) −8.38745e9 −1.01248
\(683\) 7.13819e9 0.857266 0.428633 0.903479i \(-0.358995\pi\)
0.428633 + 0.903479i \(0.358995\pi\)
\(684\) 1.80118e9 0.215209
\(685\) −5.73270e9 −0.681463
\(686\) −2.94052e9 −0.347768
\(687\) −9.58305e9 −1.12760
\(688\) 4.44006e8 0.0519791
\(689\) 6.86257e8 0.0799317
\(690\) 8.90085e9 1.03148
\(691\) −1.09452e10 −1.26197 −0.630987 0.775793i \(-0.717350\pi\)
−0.630987 + 0.775793i \(0.717350\pi\)
\(692\) 4.45004e9 0.510496
\(693\) −1.16044e10 −1.32451
\(694\) −5.99781e8 −0.0681137
\(695\) −1.13322e10 −1.28047
\(696\) 1.64775e10 1.85250
\(697\) −4.39093e8 −0.0491181
\(698\) 4.12672e9 0.459316
\(699\) 1.13781e10 1.26008
\(700\) −2.03203e9 −0.223917
\(701\) 1.00558e10 1.10256 0.551282 0.834319i \(-0.314139\pi\)
0.551282 + 0.834319i \(0.314139\pi\)
\(702\) 6.32033e8 0.0689540
\(703\) 7.80877e9 0.847694
\(704\) −2.41117e9 −0.260450
\(705\) −6.69275e9 −0.719354
\(706\) 2.28706e9 0.244603
\(707\) −7.15282e9 −0.761218
\(708\) −1.09974e10 −1.16459
\(709\) 1.17573e10 1.23892 0.619462 0.785027i \(-0.287351\pi\)
0.619462 + 0.785027i \(0.287351\pi\)
\(710\) 2.62707e9 0.275465
\(711\) 3.02090e9 0.315204
\(712\) −1.14030e10 −1.18397
\(713\) −1.94592e10 −2.01054
\(714\) 5.17396e8 0.0531960
\(715\) 4.12864e9 0.422412
\(716\) −7.49021e9 −0.762603
\(717\) −4.17174e9 −0.422669
\(718\) 8.75753e8 0.0882970
\(719\) −1.29980e9 −0.130414 −0.0652072 0.997872i \(-0.520771\pi\)
−0.0652072 + 0.997872i \(0.520771\pi\)
\(720\) 1.94381e9 0.194084
\(721\) 2.26013e10 2.24574
\(722\) 3.49455e9 0.345550
\(723\) 1.78825e10 1.75973
\(724\) −7.19497e9 −0.704602
\(725\) −3.36100e9 −0.327556
\(726\) 1.00598e10 0.975693
\(727\) 1.55479e10 1.50073 0.750363 0.661026i \(-0.229879\pi\)
0.750363 + 0.661026i \(0.229879\pi\)
\(728\) −3.36160e9 −0.322914
\(729\) 7.76072e8 0.0741918
\(730\) −5.02190e9 −0.477791
\(731\) 9.07145e7 0.00858946
\(732\) −2.69512e9 −0.253974
\(733\) 2.52604e9 0.236906 0.118453 0.992960i \(-0.462206\pi\)
0.118453 + 0.992960i \(0.462206\pi\)
\(734\) −3.10438e8 −0.0289760
\(735\) −2.10528e10 −1.95571
\(736\) −1.74404e10 −1.61244
\(737\) 1.75984e10 1.61934
\(738\) 2.43074e9 0.222609
\(739\) −8.96231e9 −0.816892 −0.408446 0.912783i \(-0.633929\pi\)
−0.408446 + 0.912783i \(0.633929\pi\)
\(740\) 1.43290e10 1.29988
\(741\) −1.77107e9 −0.159909
\(742\) 2.84240e9 0.255430
\(743\) 9.22267e9 0.824890 0.412445 0.910983i \(-0.364675\pi\)
0.412445 + 0.910983i \(0.364675\pi\)
\(744\) 1.52747e10 1.35978
\(745\) −1.33939e10 −1.18675
\(746\) −6.72664e9 −0.593215
\(747\) 6.34123e9 0.556610
\(748\) 7.94197e8 0.0693862
\(749\) −2.22245e10 −1.93262
\(750\) −6.16566e9 −0.533661
\(751\) 1.49684e10 1.28954 0.644770 0.764377i \(-0.276953\pi\)
0.644770 + 0.764377i \(0.276953\pi\)
\(752\) 2.12634e9 0.182335
\(753\) −6.43288e9 −0.549064
\(754\) −2.40328e9 −0.204176
\(755\) −9.43255e9 −0.797654
\(756\) −8.34863e9 −0.702731
\(757\) −1.31799e10 −1.10427 −0.552136 0.833754i \(-0.686187\pi\)
−0.552136 + 0.833754i \(0.686187\pi\)
\(758\) 1.10518e10 0.921702
\(759\) 3.77599e10 3.13462
\(760\) −6.14076e9 −0.507428
\(761\) −5.65319e9 −0.464994 −0.232497 0.972597i \(-0.574690\pi\)
−0.232497 + 0.972597i \(0.574690\pi\)
\(762\) −7.81270e9 −0.639674
\(763\) 1.81385e10 1.47831
\(764\) 4.49745e9 0.364871
\(765\) 3.97138e8 0.0320721
\(766\) 4.23190e9 0.340200
\(767\) 3.71093e9 0.296961
\(768\) 9.67069e9 0.770359
\(769\) 2.55751e9 0.202803 0.101402 0.994846i \(-0.467667\pi\)
0.101402 + 0.994846i \(0.467667\pi\)
\(770\) 1.71004e10 1.34986
\(771\) −4.76987e9 −0.374814
\(772\) −1.01801e10 −0.796328
\(773\) 6.24612e9 0.486387 0.243194 0.969978i \(-0.421805\pi\)
0.243194 + 0.969978i \(0.421805\pi\)
\(774\) −5.02181e8 −0.0389284
\(775\) −3.11566e9 −0.240433
\(776\) 5.39780e9 0.414668
\(777\) 3.96013e10 3.02855
\(778\) 3.62814e9 0.276220
\(779\) 6.22542e9 0.471832
\(780\) −3.24989e9 −0.245210
\(781\) 1.11448e10 0.837128
\(782\) −5.77760e8 −0.0432040
\(783\) −1.38087e10 −1.02799
\(784\) 6.68863e9 0.495713
\(785\) −5.11408e8 −0.0377332
\(786\) 1.14732e10 0.842763
\(787\) −2.24074e10 −1.63863 −0.819314 0.573345i \(-0.805646\pi\)
−0.819314 + 0.573345i \(0.805646\pi\)
\(788\) −3.16263e9 −0.230254
\(789\) 1.58400e10 1.14811
\(790\) −4.45164e9 −0.321237
\(791\) −2.25514e10 −1.62015
\(792\) −1.01717e10 −0.727536
\(793\) 9.09436e8 0.0647613
\(794\) −5.04206e9 −0.357467
\(795\) 6.35754e9 0.448750
\(796\) −1.85833e10 −1.30595
\(797\) 7.81380e9 0.546712 0.273356 0.961913i \(-0.411866\pi\)
0.273356 + 0.961913i \(0.411866\pi\)
\(798\) −7.33560e9 −0.511005
\(799\) 4.34430e8 0.0301305
\(800\) −2.79243e9 −0.192826
\(801\) −1.04557e10 −0.718849
\(802\) 6.36664e8 0.0435813
\(803\) −2.13043e10 −1.45199
\(804\) −1.38527e10 −0.940024
\(805\) 3.96736e10 2.68050
\(806\) −2.22785e9 −0.149870
\(807\) 5.23068e9 0.350349
\(808\) −6.26972e9 −0.418127
\(809\) 1.22193e9 0.0811386 0.0405693 0.999177i \(-0.487083\pi\)
0.0405693 + 0.999177i \(0.487083\pi\)
\(810\) 1.00631e10 0.665323
\(811\) −1.05088e10 −0.691797 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(812\) 3.17453e10 2.08082
\(813\) −1.41984e10 −0.926662
\(814\) −1.90606e10 −1.23866
\(815\) 2.52981e9 0.163695
\(816\) −3.67667e8 −0.0236886
\(817\) −1.28614e9 −0.0825111
\(818\) 4.64169e9 0.296510
\(819\) −3.08232e9 −0.196058
\(820\) 1.14235e10 0.723523
\(821\) −1.87258e10 −1.18097 −0.590486 0.807048i \(-0.701064\pi\)
−0.590486 + 0.807048i \(0.701064\pi\)
\(822\) −6.00265e9 −0.376957
\(823\) −3.81070e9 −0.238290 −0.119145 0.992877i \(-0.538015\pi\)
−0.119145 + 0.992877i \(0.538015\pi\)
\(824\) 1.98109e10 1.23356
\(825\) 6.04583e9 0.374858
\(826\) 1.53703e10 0.948968
\(827\) 1.63714e10 1.00651 0.503254 0.864139i \(-0.332136\pi\)
0.503254 + 0.864139i \(0.332136\pi\)
\(828\) −1.02002e10 −0.624455
\(829\) −6.71162e9 −0.409153 −0.204577 0.978851i \(-0.565582\pi\)
−0.204577 + 0.978851i \(0.565582\pi\)
\(830\) −9.34453e9 −0.567263
\(831\) −3.81199e10 −2.30435
\(832\) −6.40448e8 −0.0385525
\(833\) 1.36655e9 0.0819158
\(834\) −1.18659e10 −0.708303
\(835\) 1.40811e10 0.837018
\(836\) −1.12601e10 −0.666529
\(837\) −1.28008e10 −0.754566
\(838\) −5.07444e9 −0.297875
\(839\) −1.35926e10 −0.794577 −0.397289 0.917694i \(-0.630049\pi\)
−0.397289 + 0.917694i \(0.630049\pi\)
\(840\) −3.11422e10 −1.81289
\(841\) 3.52573e10 2.04391
\(842\) −5.45434e8 −0.0314883
\(843\) −1.16045e9 −0.0667163
\(844\) 4.80993e8 0.0275385
\(845\) −1.80177e10 −1.02731
\(846\) −2.40494e9 −0.136555
\(847\) 4.48396e10 2.53554
\(848\) −2.01984e9 −0.113745
\(849\) −1.44902e10 −0.812640
\(850\) −9.25064e7 −0.00516661
\(851\) −4.42214e10 −2.45968
\(852\) −8.77268e9 −0.485952
\(853\) −8.31755e9 −0.458853 −0.229427 0.973326i \(-0.573685\pi\)
−0.229427 + 0.973326i \(0.573685\pi\)
\(854\) 3.76679e9 0.206951
\(855\) −5.63059e9 −0.308087
\(856\) −1.94806e10 −1.06156
\(857\) −5.32901e8 −0.0289210 −0.0144605 0.999895i \(-0.504603\pi\)
−0.0144605 + 0.999895i \(0.504603\pi\)
\(858\) 4.32306e9 0.233661
\(859\) −3.23176e10 −1.73965 −0.869827 0.493357i \(-0.835770\pi\)
−0.869827 + 0.493357i \(0.835770\pi\)
\(860\) −2.36005e9 −0.126525
\(861\) 3.15715e10 1.68571
\(862\) −1.34039e10 −0.712780
\(863\) 5.97266e9 0.316323 0.158161 0.987413i \(-0.449443\pi\)
0.158161 + 0.987413i \(0.449443\pi\)
\(864\) −1.14727e10 −0.605159
\(865\) −1.39111e10 −0.730810
\(866\) −9.04737e9 −0.473380
\(867\) 2.36027e10 1.22997
\(868\) 2.94281e10 1.52737
\(869\) −1.88851e10 −0.976226
\(870\) −2.22642e10 −1.14628
\(871\) 4.67444e9 0.239699
\(872\) 1.58991e10 0.812018
\(873\) 4.94936e9 0.251767
\(874\) 8.19143e9 0.415020
\(875\) −2.74821e10 −1.38683
\(876\) 1.67698e10 0.842878
\(877\) −7.05767e9 −0.353315 −0.176658 0.984272i \(-0.556529\pi\)
−0.176658 + 0.984272i \(0.556529\pi\)
\(878\) 7.75530e9 0.386695
\(879\) 2.16233e10 1.07389
\(880\) −1.21517e10 −0.601102
\(881\) 2.62054e10 1.29114 0.645572 0.763700i \(-0.276619\pi\)
0.645572 + 0.763700i \(0.276619\pi\)
\(882\) −7.56499e9 −0.371252
\(883\) −5.36619e8 −0.0262303 −0.0131152 0.999914i \(-0.504175\pi\)
−0.0131152 + 0.999914i \(0.504175\pi\)
\(884\) 2.10952e8 0.0102707
\(885\) 3.43784e10 1.66719
\(886\) −2.43167e9 −0.117459
\(887\) −2.81727e10 −1.35549 −0.677744 0.735298i \(-0.737042\pi\)
−0.677744 + 0.735298i \(0.737042\pi\)
\(888\) 3.47120e10 1.66354
\(889\) −3.48234e10 −1.66232
\(890\) 1.54076e10 0.732606
\(891\) 4.26903e10 2.02189
\(892\) −2.70471e10 −1.27598
\(893\) −6.15932e9 −0.289436
\(894\) −1.40246e10 −0.656460
\(895\) 2.34148e10 1.09172
\(896\) 3.19926e10 1.48584
\(897\) 1.00297e10 0.463995
\(898\) −1.16642e10 −0.537510
\(899\) 4.86744e10 2.23430
\(900\) −1.63317e9 −0.0746764
\(901\) −4.12672e8 −0.0187961
\(902\) −1.51958e10 −0.689446
\(903\) −6.52252e9 −0.294787
\(904\) −1.97672e10 −0.889928
\(905\) 2.24919e10 1.00869
\(906\) −9.87672e9 −0.441229
\(907\) 3.54442e9 0.157732 0.0788659 0.996885i \(-0.474870\pi\)
0.0788659 + 0.996885i \(0.474870\pi\)
\(908\) 1.24082e10 0.550058
\(909\) −5.74883e9 −0.253867
\(910\) 4.54216e9 0.199810
\(911\) 9.71330e9 0.425650 0.212825 0.977090i \(-0.431734\pi\)
0.212825 + 0.977090i \(0.431734\pi\)
\(912\) 5.21275e9 0.227554
\(913\) −3.96421e10 −1.72389
\(914\) 1.18041e10 0.511355
\(915\) 8.42509e9 0.363581
\(916\) −1.61832e10 −0.695713
\(917\) 5.11393e10 2.19009
\(918\) −3.80065e8 −0.0162147
\(919\) 1.30593e10 0.555029 0.277515 0.960721i \(-0.410489\pi\)
0.277515 + 0.960721i \(0.410489\pi\)
\(920\) 3.47755e10 1.47236
\(921\) −2.13014e10 −0.898461
\(922\) −4.11461e9 −0.172890
\(923\) 2.96024e9 0.123914
\(924\) −5.71041e10 −2.38131
\(925\) −7.08040e9 −0.294145
\(926\) −9.39820e9 −0.388961
\(927\) 1.81650e10 0.748957
\(928\) 4.36246e10 1.79190
\(929\) −1.76183e10 −0.720955 −0.360478 0.932768i \(-0.617386\pi\)
−0.360478 + 0.932768i \(0.617386\pi\)
\(930\) −2.06390e10 −0.841392
\(931\) −1.93748e10 −0.786890
\(932\) 1.92145e10 0.777452
\(933\) 4.42855e10 1.78515
\(934\) 2.17331e10 0.872786
\(935\) −2.48271e9 −0.0993310
\(936\) −2.70177e9 −0.107692
\(937\) 2.93955e10 1.16733 0.583664 0.811996i \(-0.301619\pi\)
0.583664 + 0.811996i \(0.301619\pi\)
\(938\) 1.93610e10 0.765982
\(939\) −2.27821e10 −0.897974
\(940\) −1.13022e10 −0.443831
\(941\) 5.66655e9 0.221694 0.110847 0.993837i \(-0.464644\pi\)
0.110847 + 0.993837i \(0.464644\pi\)
\(942\) −5.35490e8 −0.0208724
\(943\) −3.52548e10 −1.36908
\(944\) −1.09223e10 −0.422582
\(945\) 2.60983e10 1.00601
\(946\) 3.13938e9 0.120566
\(947\) 2.82326e10 1.08025 0.540126 0.841584i \(-0.318376\pi\)
0.540126 + 0.841584i \(0.318376\pi\)
\(948\) 1.48656e10 0.566698
\(949\) −5.65879e9 −0.214927
\(950\) 1.31155e9 0.0496308
\(951\) −3.03505e10 −1.14428
\(952\) 2.02146e9 0.0759338
\(953\) −2.48053e10 −0.928366 −0.464183 0.885739i \(-0.653652\pi\)
−0.464183 + 0.885739i \(0.653652\pi\)
\(954\) 2.28448e9 0.0851861
\(955\) −1.40593e10 −0.522338
\(956\) −7.04494e9 −0.260780
\(957\) −9.44509e10 −3.48349
\(958\) 8.01833e9 0.294649
\(959\) −2.67555e10 −0.979599
\(960\) −5.93317e9 −0.216440
\(961\) 1.76088e10 0.640027
\(962\) −5.06283e9 −0.183350
\(963\) −1.78622e10 −0.644530
\(964\) 3.01988e10 1.08573
\(965\) 3.18236e10 1.14000
\(966\) 4.15419e10 1.48274
\(967\) −3.08573e10 −1.09740 −0.548700 0.836019i \(-0.684877\pi\)
−0.548700 + 0.836019i \(0.684877\pi\)
\(968\) 3.93036e10 1.39274
\(969\) 1.06501e9 0.0376029
\(970\) −7.29345e9 −0.256585
\(971\) −3.26691e9 −0.114517 −0.0572585 0.998359i \(-0.518236\pi\)
−0.0572585 + 0.998359i \(0.518236\pi\)
\(972\) −2.07614e10 −0.725144
\(973\) −5.28896e10 −1.84067
\(974\) −2.37209e10 −0.822574
\(975\) 1.60587e9 0.0554876
\(976\) −2.67671e9 −0.0921568
\(977\) 1.92954e10 0.661948 0.330974 0.943640i \(-0.392623\pi\)
0.330974 + 0.943640i \(0.392623\pi\)
\(978\) 2.64894e9 0.0905495
\(979\) 6.53635e10 2.22636
\(980\) −3.55525e10 −1.20664
\(981\) 1.45782e10 0.493019
\(982\) 1.93959e9 0.0653612
\(983\) 4.50199e10 1.51170 0.755852 0.654742i \(-0.227223\pi\)
0.755852 + 0.654742i \(0.227223\pi\)
\(984\) 2.76736e10 0.925940
\(985\) 9.88658e9 0.329624
\(986\) 1.44518e9 0.0480123
\(987\) −3.12363e10 −1.03407
\(988\) −2.99087e9 −0.0986615
\(989\) 7.28349e9 0.239416
\(990\) 1.37439e10 0.450179
\(991\) −4.47544e10 −1.46076 −0.730379 0.683042i \(-0.760657\pi\)
−0.730379 + 0.683042i \(0.760657\pi\)
\(992\) 4.04403e10 1.31529
\(993\) −6.79485e10 −2.20220
\(994\) 1.22610e10 0.395980
\(995\) 5.80926e10 1.86956
\(996\) 3.12046e10 1.00072
\(997\) 4.11948e10 1.31647 0.658233 0.752815i \(-0.271304\pi\)
0.658233 + 0.752815i \(0.271304\pi\)
\(998\) 2.05781e10 0.655314
\(999\) −2.90900e10 −0.923133
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 43.8.a.a.1.6 11
3.2 odd 2 387.8.a.b.1.6 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.a.1.6 11 1.1 even 1 trivial
387.8.a.b.1.6 11 3.2 odd 2