Properties

Label 22.8.a.d.1.1
Level $22$
Weight $8$
Character 22.1
Self dual yes
Analytic conductor $6.872$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 22.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.87247056065\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14881}) \)
Defining polynomial: \( x^{2} - x - 3720 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(61.4939\) of defining polynomial
Character \(\chi\) \(=\) 22.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} -72.4939 q^{3} +64.0000 q^{4} +226.494 q^{5} -579.951 q^{6} +1750.91 q^{7} +512.000 q^{8} +3068.36 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} -72.4939 q^{3} +64.0000 q^{4} +226.494 q^{5} -579.951 q^{6} +1750.91 q^{7} +512.000 q^{8} +3068.36 q^{9} +1811.95 q^{10} +1331.00 q^{11} -4639.61 q^{12} -9900.27 q^{13} +14007.3 q^{14} -16419.4 q^{15} +4096.00 q^{16} +21910.5 q^{17} +24546.9 q^{18} +37979.0 q^{19} +14495.6 q^{20} -126930. q^{21} +10648.0 q^{22} +8908.01 q^{23} -37116.9 q^{24} -26825.5 q^{25} -79202.2 q^{26} -63893.1 q^{27} +112058. q^{28} -148091. q^{29} -131355. q^{30} +69451.5 q^{31} +32768.0 q^{32} -96489.3 q^{33} +175284. q^{34} +396571. q^{35} +196375. q^{36} -30797.7 q^{37} +303832. q^{38} +717709. q^{39} +115965. q^{40} -426512. q^{41} -1.01544e6 q^{42} -853435. q^{43} +85184.0 q^{44} +694964. q^{45} +71264.1 q^{46} +577393. q^{47} -296935. q^{48} +2.24216e6 q^{49} -214604. q^{50} -1.58838e6 q^{51} -633618. q^{52} -62685.2 q^{53} -511145. q^{54} +301463. q^{55} +896468. q^{56} -2.75325e6 q^{57} -1.18472e6 q^{58} -428866. q^{59} -1.05084e6 q^{60} -1.03808e6 q^{61} +555612. q^{62} +5.37243e6 q^{63} +262144. q^{64} -2.24235e6 q^{65} -771915. q^{66} -698953. q^{67} +1.40228e6 q^{68} -645776. q^{69} +3.17257e6 q^{70} -1.83316e6 q^{71} +1.57100e6 q^{72} -500059. q^{73} -246382. q^{74} +1.94469e6 q^{75} +2.43066e6 q^{76} +2.33047e6 q^{77} +5.74167e6 q^{78} -4.84357e6 q^{79} +927719. q^{80} -2.07864e6 q^{81} -3.41209e6 q^{82} +3.47288e6 q^{83} -8.12355e6 q^{84} +4.96260e6 q^{85} -6.82748e6 q^{86} +1.07357e7 q^{87} +681472. q^{88} +955049. q^{89} +5.55971e6 q^{90} -1.73345e7 q^{91} +570113. q^{92} -5.03481e6 q^{93} +4.61914e6 q^{94} +8.60202e6 q^{95} -2.37548e6 q^{96} -1.39593e7 q^{97} +1.79373e7 q^{98} +4.08399e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{2} - 23 q^{3} + 128 q^{4} + 331 q^{5} - 184 q^{6} + 1794 q^{7} + 1024 q^{8} + 3331 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{2} - 23 q^{3} + 128 q^{4} + 331 q^{5} - 184 q^{6} + 1794 q^{7} + 1024 q^{8} + 3331 q^{9} + 2648 q^{10} + 2662 q^{11} - 1472 q^{12} - 5406 q^{13} + 14352 q^{14} - 11247 q^{15} + 8192 q^{16} + 15032 q^{17} + 26648 q^{18} + 16916 q^{19} + 21184 q^{20} - 124798 q^{21} + 21296 q^{22} - 51351 q^{23} - 11776 q^{24} - 94029 q^{25} - 43248 q^{26} - 159137 q^{27} + 114816 q^{28} - 207130 q^{29} - 89976 q^{30} - 19071 q^{31} + 65536 q^{32} - 30613 q^{33} + 120256 q^{34} + 401074 q^{35} + 213184 q^{36} + 351333 q^{37} + 135328 q^{38} + 940148 q^{39} + 169472 q^{40} + 123610 q^{41} - 998384 q^{42} - 159822 q^{43} + 170368 q^{44} + 722412 q^{45} - 410808 q^{46} + 451160 q^{47} - 94208 q^{48} + 1420470 q^{49} - 752232 q^{50} - 1928826 q^{51} - 345984 q^{52} - 1260832 q^{53} - 1273096 q^{54} + 440561 q^{55} + 918528 q^{56} - 3795736 q^{57} - 1657040 q^{58} + 887547 q^{59} - 719808 q^{60} - 597918 q^{61} - 152568 q^{62} + 5383748 q^{63} + 524288 q^{64} - 1772672 q^{65} - 244904 q^{66} + 2864711 q^{67} + 962048 q^{68} - 3628227 q^{69} + 3208592 q^{70} + 1306267 q^{71} + 1705472 q^{72} - 4577530 q^{73} + 2810664 q^{74} - 1381472 q^{75} + 1082624 q^{76} + 2387814 q^{77} + 7521184 q^{78} - 2946342 q^{79} + 1355776 q^{80} - 7367030 q^{81} + 988880 q^{82} + 9965450 q^{83} - 7987072 q^{84} + 4243754 q^{85} - 1278576 q^{86} + 7813560 q^{87} + 1362944 q^{88} + 10185377 q^{89} + 5779296 q^{90} - 17140888 q^{91} - 3286464 q^{92} - 9416131 q^{93} + 3609280 q^{94} + 6400800 q^{95} - 753664 q^{96} - 27765477 q^{97} + 11363760 q^{98} + 4433561 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −72.4939 −1.55016 −0.775080 0.631863i \(-0.782291\pi\)
−0.775080 + 0.631863i \(0.782291\pi\)
\(4\) 64.0000 0.500000
\(5\) 226.494 0.810329 0.405165 0.914244i \(-0.367214\pi\)
0.405165 + 0.914244i \(0.367214\pi\)
\(6\) −579.951 −1.09613
\(7\) 1750.91 1.92940 0.964699 0.263356i \(-0.0848295\pi\)
0.964699 + 0.263356i \(0.0848295\pi\)
\(8\) 512.000 0.353553
\(9\) 3068.36 1.40300
\(10\) 1811.95 0.572989
\(11\) 1331.00 0.301511
\(12\) −4639.61 −0.775080
\(13\) −9900.27 −1.24981 −0.624907 0.780699i \(-0.714863\pi\)
−0.624907 + 0.780699i \(0.714863\pi\)
\(14\) 14007.3 1.36429
\(15\) −16419.4 −1.25614
\(16\) 4096.00 0.250000
\(17\) 21910.5 1.08164 0.540819 0.841139i \(-0.318114\pi\)
0.540819 + 0.841139i \(0.318114\pi\)
\(18\) 24546.9 0.992070
\(19\) 37979.0 1.27030 0.635150 0.772389i \(-0.280938\pi\)
0.635150 + 0.772389i \(0.280938\pi\)
\(20\) 14495.6 0.405165
\(21\) −126930. −2.99088
\(22\) 10648.0 0.213201
\(23\) 8908.01 0.152663 0.0763314 0.997082i \(-0.475679\pi\)
0.0763314 + 0.997082i \(0.475679\pi\)
\(24\) −37116.9 −0.548065
\(25\) −26825.5 −0.343367
\(26\) −79202.2 −0.883752
\(27\) −63893.1 −0.624713
\(28\) 112058. 0.964699
\(29\) −148091. −1.12755 −0.563773 0.825930i \(-0.690651\pi\)
−0.563773 + 0.825930i \(0.690651\pi\)
\(30\) −131355. −0.888225
\(31\) 69451.5 0.418713 0.209356 0.977839i \(-0.432863\pi\)
0.209356 + 0.977839i \(0.432863\pi\)
\(32\) 32768.0 0.176777
\(33\) −96489.3 −0.467391
\(34\) 175284. 0.764834
\(35\) 396571. 1.56345
\(36\) 196375. 0.701499
\(37\) −30797.7 −0.0999567 −0.0499784 0.998750i \(-0.515915\pi\)
−0.0499784 + 0.998750i \(0.515915\pi\)
\(38\) 303832. 0.898238
\(39\) 717709. 1.93741
\(40\) 115965. 0.286495
\(41\) −426512. −0.966468 −0.483234 0.875491i \(-0.660538\pi\)
−0.483234 + 0.875491i \(0.660538\pi\)
\(42\) −1.01544e6 −2.11487
\(43\) −853435. −1.63693 −0.818466 0.574554i \(-0.805175\pi\)
−0.818466 + 0.574554i \(0.805175\pi\)
\(44\) 85184.0 0.150756
\(45\) 694964. 1.13689
\(46\) 71264.1 0.107949
\(47\) 577393. 0.811201 0.405601 0.914050i \(-0.367062\pi\)
0.405601 + 0.914050i \(0.367062\pi\)
\(48\) −296935. −0.387540
\(49\) 2.24216e6 2.72257
\(50\) −214604. −0.242797
\(51\) −1.58838e6 −1.67671
\(52\) −633618. −0.624907
\(53\) −62685.2 −0.0578362 −0.0289181 0.999582i \(-0.509206\pi\)
−0.0289181 + 0.999582i \(0.509206\pi\)
\(54\) −511145. −0.441739
\(55\) 301463. 0.244323
\(56\) 896468. 0.682145
\(57\) −2.75325e6 −1.96917
\(58\) −1.18472e6 −0.797295
\(59\) −428866. −0.271856 −0.135928 0.990719i \(-0.543402\pi\)
−0.135928 + 0.990719i \(0.543402\pi\)
\(60\) −1.05084e6 −0.628070
\(61\) −1.03808e6 −0.585568 −0.292784 0.956179i \(-0.594582\pi\)
−0.292784 + 0.956179i \(0.594582\pi\)
\(62\) 555612. 0.296074
\(63\) 5.37243e6 2.70694
\(64\) 262144. 0.125000
\(65\) −2.24235e6 −1.01276
\(66\) −771915. −0.330495
\(67\) −698953. −0.283913 −0.141957 0.989873i \(-0.545339\pi\)
−0.141957 + 0.989873i \(0.545339\pi\)
\(68\) 1.40228e6 0.540819
\(69\) −645776. −0.236652
\(70\) 3.17257e6 1.10552
\(71\) −1.83316e6 −0.607849 −0.303925 0.952696i \(-0.598297\pi\)
−0.303925 + 0.952696i \(0.598297\pi\)
\(72\) 1.57100e6 0.496035
\(73\) −500059. −0.150450 −0.0752249 0.997167i \(-0.523967\pi\)
−0.0752249 + 0.997167i \(0.523967\pi\)
\(74\) −246382. −0.0706801
\(75\) 1.94469e6 0.532274
\(76\) 2.43066e6 0.635150
\(77\) 2.33047e6 0.581735
\(78\) 5.74167e6 1.36996
\(79\) −4.84357e6 −1.10528 −0.552638 0.833421i \(-0.686379\pi\)
−0.552638 + 0.833421i \(0.686379\pi\)
\(80\) 927719. 0.202582
\(81\) −2.07864e6 −0.434593
\(82\) −3.41209e6 −0.683396
\(83\) 3.47288e6 0.666679 0.333340 0.942807i \(-0.391824\pi\)
0.333340 + 0.942807i \(0.391824\pi\)
\(84\) −8.12355e6 −1.49544
\(85\) 4.96260e6 0.876483
\(86\) −6.82748e6 −1.15749
\(87\) 1.07357e7 1.74788
\(88\) 681472. 0.106600
\(89\) 955049. 0.143602 0.0718010 0.997419i \(-0.477125\pi\)
0.0718010 + 0.997419i \(0.477125\pi\)
\(90\) 5.55971e6 0.803903
\(91\) −1.73345e7 −2.41139
\(92\) 570113. 0.0763314
\(93\) −5.03481e6 −0.649072
\(94\) 4.61914e6 0.573606
\(95\) 8.60202e6 1.02936
\(96\) −2.37548e6 −0.274032
\(97\) −1.39593e7 −1.55297 −0.776483 0.630138i \(-0.782998\pi\)
−0.776483 + 0.630138i \(0.782998\pi\)
\(98\) 1.79373e7 1.92515
\(99\) 4.08399e6 0.423020
\(100\) −1.71683e6 −0.171683
\(101\) 9.16872e6 0.885491 0.442745 0.896647i \(-0.354005\pi\)
0.442745 + 0.896647i \(0.354005\pi\)
\(102\) −1.27070e7 −1.18562
\(103\) −1.36882e6 −0.123428 −0.0617142 0.998094i \(-0.519657\pi\)
−0.0617142 + 0.998094i \(0.519657\pi\)
\(104\) −5.06894e6 −0.441876
\(105\) −2.87490e7 −2.42359
\(106\) −501482. −0.0408964
\(107\) 2.15337e7 1.69932 0.849659 0.527332i \(-0.176807\pi\)
0.849659 + 0.527332i \(0.176807\pi\)
\(108\) −4.08916e6 −0.312357
\(109\) −2.11639e7 −1.56532 −0.782660 0.622450i \(-0.786138\pi\)
−0.782660 + 0.622450i \(0.786138\pi\)
\(110\) 2.41171e6 0.172763
\(111\) 2.23264e6 0.154949
\(112\) 7.17174e6 0.482349
\(113\) 2.05115e7 1.33728 0.668642 0.743584i \(-0.266876\pi\)
0.668642 + 0.743584i \(0.266876\pi\)
\(114\) −2.20260e7 −1.39241
\(115\) 2.01761e6 0.123707
\(116\) −9.47779e6 −0.563773
\(117\) −3.03776e7 −1.75349
\(118\) −3.43092e6 −0.192231
\(119\) 3.83635e7 2.08691
\(120\) −8.40674e6 −0.444113
\(121\) 1.77156e6 0.0909091
\(122\) −8.30466e6 −0.414059
\(123\) 3.09195e7 1.49818
\(124\) 4.44490e6 0.209356
\(125\) −2.37707e7 −1.08857
\(126\) 4.29795e7 1.91410
\(127\) 1.78362e7 0.772660 0.386330 0.922361i \(-0.373743\pi\)
0.386330 + 0.922361i \(0.373743\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) 6.18688e7 2.53751
\(130\) −1.79388e7 −0.716130
\(131\) −7.31001e6 −0.284098 −0.142049 0.989860i \(-0.545369\pi\)
−0.142049 + 0.989860i \(0.545369\pi\)
\(132\) −6.17532e6 −0.233696
\(133\) 6.64980e7 2.45091
\(134\) −5.59162e6 −0.200757
\(135\) −1.44714e7 −0.506223
\(136\) 1.12182e7 0.382417
\(137\) −1.66829e7 −0.554307 −0.277153 0.960826i \(-0.589391\pi\)
−0.277153 + 0.960826i \(0.589391\pi\)
\(138\) −5.16621e6 −0.167338
\(139\) −1.29918e7 −0.410316 −0.205158 0.978729i \(-0.565771\pi\)
−0.205158 + 0.978729i \(0.565771\pi\)
\(140\) 2.53806e7 0.781723
\(141\) −4.18574e7 −1.25749
\(142\) −1.46653e7 −0.429814
\(143\) −1.31773e7 −0.376833
\(144\) 1.25680e7 0.350750
\(145\) −3.35416e7 −0.913683
\(146\) −4.00047e6 −0.106384
\(147\) −1.62543e8 −4.22043
\(148\) −1.97105e6 −0.0499784
\(149\) −1.12735e7 −0.279195 −0.139598 0.990208i \(-0.544581\pi\)
−0.139598 + 0.990208i \(0.544581\pi\)
\(150\) 1.55575e7 0.376374
\(151\) −4.16803e7 −0.985171 −0.492586 0.870264i \(-0.663948\pi\)
−0.492586 + 0.870264i \(0.663948\pi\)
\(152\) 1.94453e7 0.449119
\(153\) 6.72294e7 1.51754
\(154\) 1.86437e7 0.411349
\(155\) 1.57303e7 0.339295
\(156\) 4.59334e7 0.968707
\(157\) 3.55935e7 0.734045 0.367022 0.930212i \(-0.380377\pi\)
0.367022 + 0.930212i \(0.380377\pi\)
\(158\) −3.87486e7 −0.781548
\(159\) 4.54429e6 0.0896554
\(160\) 7.42175e6 0.143247
\(161\) 1.55972e7 0.294547
\(162\) −1.66292e7 −0.307304
\(163\) −9.91643e7 −1.79349 −0.896744 0.442549i \(-0.854074\pi\)
−0.896744 + 0.442549i \(0.854074\pi\)
\(164\) −2.72968e7 −0.483234
\(165\) −2.18542e7 −0.378741
\(166\) 2.77831e7 0.471413
\(167\) 8.13890e7 1.35225 0.676126 0.736786i \(-0.263657\pi\)
0.676126 + 0.736786i \(0.263657\pi\)
\(168\) −6.49884e7 −1.05743
\(169\) 3.52669e7 0.562036
\(170\) 3.97008e7 0.619767
\(171\) 1.16533e8 1.78223
\(172\) −5.46198e7 −0.818466
\(173\) −2.47680e7 −0.363689 −0.181844 0.983327i \(-0.558207\pi\)
−0.181844 + 0.983327i \(0.558207\pi\)
\(174\) 8.58852e7 1.23594
\(175\) −4.69692e7 −0.662491
\(176\) 5.45178e6 0.0753778
\(177\) 3.10901e7 0.421421
\(178\) 7.64039e6 0.101542
\(179\) −1.33876e8 −1.74469 −0.872343 0.488894i \(-0.837401\pi\)
−0.872343 + 0.488894i \(0.837401\pi\)
\(180\) 4.44777e7 0.568445
\(181\) 2.75561e7 0.345417 0.172708 0.984973i \(-0.444748\pi\)
0.172708 + 0.984973i \(0.444748\pi\)
\(182\) −1.38676e8 −1.70511
\(183\) 7.52546e7 0.907725
\(184\) 4.56090e6 0.0539745
\(185\) −6.97549e6 −0.0809979
\(186\) −4.02785e7 −0.458963
\(187\) 2.91629e7 0.326126
\(188\) 3.69531e7 0.405601
\(189\) −1.11871e8 −1.20532
\(190\) 6.88161e7 0.727868
\(191\) 1.10570e8 1.14820 0.574102 0.818784i \(-0.305351\pi\)
0.574102 + 0.818784i \(0.305351\pi\)
\(192\) −1.90038e7 −0.193770
\(193\) 6.81474e7 0.682336 0.341168 0.940002i \(-0.389177\pi\)
0.341168 + 0.940002i \(0.389177\pi\)
\(194\) −1.11674e8 −1.09811
\(195\) 1.62557e8 1.56994
\(196\) 1.43498e8 1.36129
\(197\) 1.58884e8 1.48064 0.740318 0.672257i \(-0.234675\pi\)
0.740318 + 0.672257i \(0.234675\pi\)
\(198\) 3.26719e7 0.299120
\(199\) −5.17682e7 −0.465669 −0.232834 0.972516i \(-0.574800\pi\)
−0.232834 + 0.972516i \(0.574800\pi\)
\(200\) −1.37347e7 −0.121399
\(201\) 5.06698e7 0.440112
\(202\) 7.33498e7 0.626136
\(203\) −2.59294e8 −2.17548
\(204\) −1.01656e8 −0.838357
\(205\) −9.66023e7 −0.783157
\(206\) −1.09505e7 −0.0872770
\(207\) 2.73330e7 0.214186
\(208\) −4.05515e7 −0.312454
\(209\) 5.05501e7 0.383010
\(210\) −2.29992e8 −1.71374
\(211\) −6.61157e7 −0.484525 −0.242263 0.970211i \(-0.577890\pi\)
−0.242263 + 0.970211i \(0.577890\pi\)
\(212\) −4.01185e6 −0.0289181
\(213\) 1.32893e8 0.942264
\(214\) 1.72269e8 1.20160
\(215\) −1.93298e8 −1.32645
\(216\) −3.27133e7 −0.220869
\(217\) 1.21604e8 0.807863
\(218\) −1.69311e8 −1.10685
\(219\) 3.62512e7 0.233221
\(220\) 1.92937e7 0.122162
\(221\) −2.16920e8 −1.35185
\(222\) 1.78611e7 0.109566
\(223\) 1.13417e8 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(224\) 5.73739e7 0.341072
\(225\) −8.23104e7 −0.481743
\(226\) 1.64092e8 0.945603
\(227\) 2.69424e8 1.52878 0.764391 0.644752i \(-0.223040\pi\)
0.764391 + 0.644752i \(0.223040\pi\)
\(228\) −1.76208e8 −0.984584
\(229\) −2.58093e8 −1.42021 −0.710105 0.704096i \(-0.751353\pi\)
−0.710105 + 0.704096i \(0.751353\pi\)
\(230\) 1.61409e7 0.0874742
\(231\) −1.68944e8 −0.901783
\(232\) −7.58223e7 −0.398648
\(233\) −1.97294e8 −1.02181 −0.510903 0.859638i \(-0.670689\pi\)
−0.510903 + 0.859638i \(0.670689\pi\)
\(234\) −2.43021e8 −1.23990
\(235\) 1.30776e8 0.657340
\(236\) −2.74474e7 −0.135928
\(237\) 3.51129e8 1.71336
\(238\) 3.06908e8 1.47567
\(239\) −1.76075e8 −0.834265 −0.417133 0.908846i \(-0.636965\pi\)
−0.417133 + 0.908846i \(0.636965\pi\)
\(240\) −6.72539e7 −0.314035
\(241\) −4.22960e7 −0.194644 −0.0973218 0.995253i \(-0.531028\pi\)
−0.0973218 + 0.995253i \(0.531028\pi\)
\(242\) 1.41725e7 0.0642824
\(243\) 2.90423e8 1.29840
\(244\) −6.64373e7 −0.292784
\(245\) 5.07835e8 2.20618
\(246\) 2.47356e8 1.05937
\(247\) −3.76003e8 −1.58764
\(248\) 3.55592e7 0.148037
\(249\) −2.51763e8 −1.03346
\(250\) −1.90165e8 −0.769735
\(251\) 1.00835e8 0.402489 0.201245 0.979541i \(-0.435501\pi\)
0.201245 + 0.979541i \(0.435501\pi\)
\(252\) 3.43836e8 1.35347
\(253\) 1.18566e7 0.0460296
\(254\) 1.42689e8 0.546353
\(255\) −3.59758e8 −1.35869
\(256\) 1.67772e7 0.0625000
\(257\) 8.85623e7 0.325449 0.162725 0.986672i \(-0.447972\pi\)
0.162725 + 0.986672i \(0.447972\pi\)
\(258\) 4.94950e8 1.79429
\(259\) −5.39241e7 −0.192856
\(260\) −1.43510e8 −0.506380
\(261\) −4.54395e8 −1.58195
\(262\) −5.84800e7 −0.200888
\(263\) −4.15591e8 −1.40871 −0.704353 0.709850i \(-0.748763\pi\)
−0.704353 + 0.709850i \(0.748763\pi\)
\(264\) −4.94025e7 −0.165248
\(265\) −1.41978e7 −0.0468663
\(266\) 5.31984e8 1.73306
\(267\) −6.92351e7 −0.222606
\(268\) −4.47330e7 −0.141957
\(269\) −1.84945e8 −0.579308 −0.289654 0.957131i \(-0.593540\pi\)
−0.289654 + 0.957131i \(0.593540\pi\)
\(270\) −1.15771e8 −0.357954
\(271\) 5.17862e8 1.58060 0.790300 0.612721i \(-0.209925\pi\)
0.790300 + 0.612721i \(0.209925\pi\)
\(272\) 8.97456e7 0.270410
\(273\) 1.25665e9 3.73804
\(274\) −1.33463e8 −0.391954
\(275\) −3.57048e7 −0.103529
\(276\) −4.13297e7 −0.118326
\(277\) −3.23904e7 −0.0915667 −0.0457833 0.998951i \(-0.514578\pi\)
−0.0457833 + 0.998951i \(0.514578\pi\)
\(278\) −1.03935e8 −0.290137
\(279\) 2.13102e8 0.587453
\(280\) 2.03044e8 0.552762
\(281\) 3.54172e8 0.952231 0.476115 0.879383i \(-0.342044\pi\)
0.476115 + 0.879383i \(0.342044\pi\)
\(282\) −3.34859e8 −0.889182
\(283\) 1.82414e8 0.478415 0.239207 0.970968i \(-0.423112\pi\)
0.239207 + 0.970968i \(0.423112\pi\)
\(284\) −1.17322e8 −0.303925
\(285\) −6.23593e8 −1.59567
\(286\) −1.05418e8 −0.266461
\(287\) −7.46785e8 −1.86470
\(288\) 1.00544e8 0.248018
\(289\) 6.97335e7 0.169941
\(290\) −2.68333e8 −0.646072
\(291\) 1.01196e9 2.40735
\(292\) −3.20038e7 −0.0752249
\(293\) 4.36215e6 0.0101313 0.00506563 0.999987i \(-0.498388\pi\)
0.00506563 + 0.999987i \(0.498388\pi\)
\(294\) −1.30034e9 −2.98429
\(295\) −9.71354e7 −0.220293
\(296\) −1.57684e7 −0.0353400
\(297\) −8.50417e7 −0.188358
\(298\) −9.01882e7 −0.197421
\(299\) −8.81918e7 −0.190800
\(300\) 1.24460e8 0.266137
\(301\) −1.49429e9 −3.15829
\(302\) −3.33443e8 −0.696621
\(303\) −6.64676e8 −1.37265
\(304\) 1.55562e8 0.317575
\(305\) −2.35119e8 −0.474503
\(306\) 5.37835e8 1.07306
\(307\) −4.08466e8 −0.805697 −0.402849 0.915267i \(-0.631980\pi\)
−0.402849 + 0.915267i \(0.631980\pi\)
\(308\) 1.49150e8 0.290868
\(309\) 9.92308e7 0.191334
\(310\) 1.25843e8 0.239918
\(311\) 8.06482e8 1.52031 0.760157 0.649740i \(-0.225122\pi\)
0.760157 + 0.649740i \(0.225122\pi\)
\(312\) 3.67467e8 0.684979
\(313\) 2.34539e8 0.432325 0.216162 0.976357i \(-0.430646\pi\)
0.216162 + 0.976357i \(0.430646\pi\)
\(314\) 2.84748e8 0.519048
\(315\) 1.21682e9 2.19351
\(316\) −3.09988e8 −0.552638
\(317\) 2.10448e8 0.371054 0.185527 0.982639i \(-0.440601\pi\)
0.185527 + 0.982639i \(0.440601\pi\)
\(318\) 3.63543e7 0.0633959
\(319\) −1.97108e8 −0.339968
\(320\) 5.93740e7 0.101291
\(321\) −1.56106e9 −2.63422
\(322\) 1.24777e8 0.208276
\(323\) 8.32141e8 1.37400
\(324\) −1.33033e8 −0.217296
\(325\) 2.65580e8 0.429145
\(326\) −7.93314e8 −1.26819
\(327\) 1.53425e9 2.42650
\(328\) −2.18374e8 −0.341698
\(329\) 1.01096e9 1.56513
\(330\) −1.74834e8 −0.267810
\(331\) −9.04546e8 −1.37099 −0.685493 0.728080i \(-0.740413\pi\)
−0.685493 + 0.728080i \(0.740413\pi\)
\(332\) 2.22265e8 0.333340
\(333\) −9.44984e7 −0.140239
\(334\) 6.51112e8 0.956187
\(335\) −1.58308e8 −0.230063
\(336\) −5.19907e8 −0.747719
\(337\) 6.56273e8 0.934071 0.467036 0.884238i \(-0.345322\pi\)
0.467036 + 0.884238i \(0.345322\pi\)
\(338\) 2.82135e8 0.397419
\(339\) −1.48696e9 −2.07301
\(340\) 3.17607e8 0.438241
\(341\) 9.24400e7 0.126247
\(342\) 9.32266e8 1.26023
\(343\) 2.48387e9 3.32353
\(344\) −4.36959e8 −0.578743
\(345\) −1.46264e8 −0.191766
\(346\) −1.98144e8 −0.257167
\(347\) 8.23306e8 1.05781 0.528906 0.848681i \(-0.322603\pi\)
0.528906 + 0.848681i \(0.322603\pi\)
\(348\) 6.87082e8 0.873939
\(349\) −5.24128e8 −0.660007 −0.330004 0.943980i \(-0.607050\pi\)
−0.330004 + 0.943980i \(0.607050\pi\)
\(350\) −3.75754e8 −0.468452
\(351\) 6.32559e8 0.780775
\(352\) 4.36142e7 0.0533002
\(353\) −3.31476e8 −0.401089 −0.200544 0.979685i \(-0.564271\pi\)
−0.200544 + 0.979685i \(0.564271\pi\)
\(354\) 2.48721e8 0.297989
\(355\) −4.15199e8 −0.492558
\(356\) 6.11231e7 0.0718010
\(357\) −2.78112e9 −3.23505
\(358\) −1.07101e9 −1.23368
\(359\) −1.60697e9 −1.83306 −0.916529 0.399968i \(-0.869021\pi\)
−0.916529 + 0.399968i \(0.869021\pi\)
\(360\) 3.55822e8 0.401952
\(361\) 5.48535e8 0.613661
\(362\) 2.20449e8 0.244246
\(363\) −1.28427e8 −0.140924
\(364\) −1.10941e9 −1.20569
\(365\) −1.13260e8 −0.121914
\(366\) 6.02037e8 0.641858
\(367\) −2.48331e7 −0.0262240 −0.0131120 0.999914i \(-0.504174\pi\)
−0.0131120 + 0.999914i \(0.504174\pi\)
\(368\) 3.64872e7 0.0381657
\(369\) −1.30869e9 −1.35595
\(370\) −5.58039e7 −0.0572741
\(371\) −1.09756e8 −0.111589
\(372\) −3.22228e8 −0.324536
\(373\) 1.33542e9 1.33241 0.666204 0.745770i \(-0.267918\pi\)
0.666204 + 0.745770i \(0.267918\pi\)
\(374\) 2.33304e8 0.230606
\(375\) 1.72323e9 1.68746
\(376\) 2.95625e8 0.286803
\(377\) 1.46614e9 1.40922
\(378\) −8.94970e8 −0.852290
\(379\) −5.06235e8 −0.477656 −0.238828 0.971062i \(-0.576763\pi\)
−0.238828 + 0.971062i \(0.576763\pi\)
\(380\) 5.50529e8 0.514680
\(381\) −1.29301e9 −1.19775
\(382\) 8.84556e8 0.811902
\(383\) 4.48094e8 0.407543 0.203771 0.979019i \(-0.434680\pi\)
0.203771 + 0.979019i \(0.434680\pi\)
\(384\) −1.52031e8 −0.137016
\(385\) 5.27836e8 0.471397
\(386\) 5.45179e8 0.482485
\(387\) −2.61864e9 −2.29662
\(388\) −8.93394e8 −0.776483
\(389\) 7.22169e8 0.622036 0.311018 0.950404i \(-0.399330\pi\)
0.311018 + 0.950404i \(0.399330\pi\)
\(390\) 1.30045e9 1.11012
\(391\) 1.95179e8 0.165126
\(392\) 1.14798e9 0.962575
\(393\) 5.29930e8 0.440398
\(394\) 1.27107e9 1.04697
\(395\) −1.09704e9 −0.895637
\(396\) 2.61375e8 0.211510
\(397\) −9.71374e8 −0.779147 −0.389574 0.920995i \(-0.627378\pi\)
−0.389574 + 0.920995i \(0.627378\pi\)
\(398\) −4.14146e8 −0.329278
\(399\) −4.82070e9 −3.79931
\(400\) −1.09877e8 −0.0858417
\(401\) 1.22326e9 0.947354 0.473677 0.880699i \(-0.342926\pi\)
0.473677 + 0.880699i \(0.342926\pi\)
\(402\) 4.05358e8 0.311206
\(403\) −6.87589e8 −0.523313
\(404\) 5.86798e8 0.442745
\(405\) −4.70800e8 −0.352163
\(406\) −2.07435e9 −1.53830
\(407\) −4.09917e7 −0.0301381
\(408\) −8.13251e8 −0.592808
\(409\) 1.75949e9 1.27161 0.635806 0.771849i \(-0.280668\pi\)
0.635806 + 0.771849i \(0.280668\pi\)
\(410\) −7.72818e8 −0.553776
\(411\) 1.20941e9 0.859264
\(412\) −8.76043e7 −0.0617142
\(413\) −7.50907e8 −0.524518
\(414\) 2.18664e8 0.151452
\(415\) 7.86587e8 0.540229
\(416\) −3.24412e8 −0.220938
\(417\) 9.41827e8 0.636056
\(418\) 4.04401e8 0.270829
\(419\) 1.02214e9 0.678830 0.339415 0.940637i \(-0.389771\pi\)
0.339415 + 0.940637i \(0.389771\pi\)
\(420\) −1.83993e9 −1.21180
\(421\) 8.07976e8 0.527729 0.263865 0.964560i \(-0.415003\pi\)
0.263865 + 0.964560i \(0.415003\pi\)
\(422\) −5.28926e8 −0.342611
\(423\) 1.77165e9 1.13811
\(424\) −3.20948e7 −0.0204482
\(425\) −5.87762e8 −0.371399
\(426\) 1.06314e9 0.666281
\(427\) −1.81759e9 −1.12979
\(428\) 1.37816e9 0.849659
\(429\) 9.55271e8 0.584152
\(430\) −1.54638e9 −0.937945
\(431\) 2.11098e9 1.27003 0.635015 0.772499i \(-0.280994\pi\)
0.635015 + 0.772499i \(0.280994\pi\)
\(432\) −2.61706e8 −0.156178
\(433\) −2.05374e9 −1.21573 −0.607867 0.794039i \(-0.707975\pi\)
−0.607867 + 0.794039i \(0.707975\pi\)
\(434\) 9.72829e8 0.571245
\(435\) 2.43156e9 1.41636
\(436\) −1.35449e9 −0.782660
\(437\) 3.38318e8 0.193928
\(438\) 2.90010e8 0.164912
\(439\) −2.72396e8 −0.153665 −0.0768324 0.997044i \(-0.524481\pi\)
−0.0768324 + 0.997044i \(0.524481\pi\)
\(440\) 1.54349e8 0.0863814
\(441\) 6.87974e9 3.81977
\(442\) −1.73536e9 −0.955900
\(443\) 2.53670e8 0.138629 0.0693147 0.997595i \(-0.477919\pi\)
0.0693147 + 0.997595i \(0.477919\pi\)
\(444\) 1.42889e8 0.0774745
\(445\) 2.16313e8 0.116365
\(446\) 9.07338e8 0.484280
\(447\) 8.17261e8 0.432797
\(448\) 4.58992e8 0.241175
\(449\) −1.18063e9 −0.615533 −0.307766 0.951462i \(-0.599581\pi\)
−0.307766 + 0.951462i \(0.599581\pi\)
\(450\) −6.58483e8 −0.340644
\(451\) −5.67687e8 −0.291401
\(452\) 1.31274e9 0.668642
\(453\) 3.02157e9 1.52717
\(454\) 2.15539e9 1.08101
\(455\) −3.92616e9 −1.95402
\(456\) −1.40966e9 −0.696206
\(457\) 1.44893e9 0.710133 0.355066 0.934841i \(-0.384458\pi\)
0.355066 + 0.934841i \(0.384458\pi\)
\(458\) −2.06475e9 −1.00424
\(459\) −1.39993e9 −0.675714
\(460\) 1.29127e8 0.0618536
\(461\) −2.23879e9 −1.06429 −0.532145 0.846653i \(-0.678614\pi\)
−0.532145 + 0.846653i \(0.678614\pi\)
\(462\) −1.35156e9 −0.637657
\(463\) 2.75405e9 1.28955 0.644776 0.764372i \(-0.276951\pi\)
0.644776 + 0.764372i \(0.276951\pi\)
\(464\) −6.06579e8 −0.281886
\(465\) −1.14035e9 −0.525962
\(466\) −1.57835e9 −0.722526
\(467\) −2.08168e9 −0.945811 −0.472905 0.881113i \(-0.656795\pi\)
−0.472905 + 0.881113i \(0.656795\pi\)
\(468\) −1.94417e9 −0.876744
\(469\) −1.22381e9 −0.547782
\(470\) 1.04621e9 0.464810
\(471\) −2.58031e9 −1.13789
\(472\) −2.19579e8 −0.0961157
\(473\) −1.13592e9 −0.493554
\(474\) 2.80903e9 1.21153
\(475\) −1.01881e9 −0.436179
\(476\) 2.45526e9 1.04345
\(477\) −1.92341e8 −0.0811441
\(478\) −1.40860e9 −0.589915
\(479\) 4.51168e8 0.187570 0.0937852 0.995592i \(-0.470103\pi\)
0.0937852 + 0.995592i \(0.470103\pi\)
\(480\) −5.38031e8 −0.222056
\(481\) 3.04906e8 0.124927
\(482\) −3.38368e8 −0.137634
\(483\) −1.13070e9 −0.456596
\(484\) 1.13380e8 0.0454545
\(485\) −3.16169e9 −1.25841
\(486\) 2.32339e9 0.918109
\(487\) −1.48099e9 −0.581033 −0.290516 0.956870i \(-0.593827\pi\)
−0.290516 + 0.956870i \(0.593827\pi\)
\(488\) −5.31498e8 −0.207030
\(489\) 7.18880e9 2.78020
\(490\) 4.06268e9 1.56001
\(491\) −6.07557e8 −0.231634 −0.115817 0.993271i \(-0.536949\pi\)
−0.115817 + 0.993271i \(0.536949\pi\)
\(492\) 1.97885e9 0.749091
\(493\) −3.24474e9 −1.21960
\(494\) −3.00802e9 −1.12263
\(495\) 9.24998e8 0.342785
\(496\) 2.84474e8 0.104678
\(497\) −3.20970e9 −1.17278
\(498\) −2.01410e9 −0.730767
\(499\) 4.69266e9 1.69070 0.845351 0.534212i \(-0.179391\pi\)
0.845351 + 0.534212i \(0.179391\pi\)
\(500\) −1.52132e9 −0.544285
\(501\) −5.90020e9 −2.09621
\(502\) 8.06682e8 0.284603
\(503\) −2.56555e9 −0.898860 −0.449430 0.893315i \(-0.648373\pi\)
−0.449430 + 0.893315i \(0.648373\pi\)
\(504\) 2.75069e9 0.957049
\(505\) 2.07666e9 0.717539
\(506\) 9.48525e7 0.0325478
\(507\) −2.55663e9 −0.871246
\(508\) 1.14151e9 0.386330
\(509\) 5.55142e9 1.86591 0.932957 0.359987i \(-0.117219\pi\)
0.932957 + 0.359987i \(0.117219\pi\)
\(510\) −2.87807e9 −0.960739
\(511\) −8.75561e8 −0.290277
\(512\) 1.34218e8 0.0441942
\(513\) −2.42660e9 −0.793573
\(514\) 7.08499e8 0.230127
\(515\) −3.10029e8 −0.100018
\(516\) 3.95960e9 1.26875
\(517\) 7.68509e8 0.244586
\(518\) −4.31393e8 −0.136370
\(519\) 1.79553e9 0.563776
\(520\) −1.14808e9 −0.358065
\(521\) −3.93859e9 −1.22014 −0.610069 0.792348i \(-0.708858\pi\)
−0.610069 + 0.792348i \(0.708858\pi\)
\(522\) −3.63516e9 −1.11860
\(523\) 1.28298e9 0.392160 0.196080 0.980588i \(-0.437179\pi\)
0.196080 + 0.980588i \(0.437179\pi\)
\(524\) −4.67840e8 −0.142049
\(525\) 3.40498e9 1.02697
\(526\) −3.32472e9 −0.996106
\(527\) 1.52172e9 0.452895
\(528\) −3.95220e8 −0.116848
\(529\) −3.32547e9 −0.976694
\(530\) −1.13583e8 −0.0331395
\(531\) −1.31591e9 −0.381414
\(532\) 4.25587e9 1.22546
\(533\) 4.22258e9 1.20791
\(534\) −5.53881e8 −0.157406
\(535\) 4.87724e9 1.37701
\(536\) −3.57864e8 −0.100379
\(537\) 9.70520e9 2.70455
\(538\) −1.47956e9 −0.409633
\(539\) 2.98431e9 0.820887
\(540\) −9.26169e8 −0.253112
\(541\) 3.43581e9 0.932908 0.466454 0.884546i \(-0.345531\pi\)
0.466454 + 0.884546i \(0.345531\pi\)
\(542\) 4.14290e9 1.11765
\(543\) −1.99765e9 −0.535451
\(544\) 7.17965e8 0.191208
\(545\) −4.79349e9 −1.26842
\(546\) 1.00532e10 2.64319
\(547\) −1.20501e8 −0.0314800 −0.0157400 0.999876i \(-0.505010\pi\)
−0.0157400 + 0.999876i \(0.505010\pi\)
\(548\) −1.06771e9 −0.277153
\(549\) −3.18521e9 −0.821551
\(550\) −2.85638e8 −0.0732061
\(551\) −5.62433e9 −1.43232
\(552\) −3.30637e8 −0.0836691
\(553\) −8.48067e9 −2.13252
\(554\) −2.59123e8 −0.0647474
\(555\) 5.05680e8 0.125560
\(556\) −8.31477e8 −0.205158
\(557\) −1.81976e9 −0.446191 −0.223096 0.974797i \(-0.571616\pi\)
−0.223096 + 0.974797i \(0.571616\pi\)
\(558\) 1.70482e9 0.415392
\(559\) 8.44924e9 2.04586
\(560\) 1.62436e9 0.390862
\(561\) −2.11413e9 −0.505548
\(562\) 2.83338e9 0.673329
\(563\) 6.87134e9 1.62279 0.811394 0.584499i \(-0.198709\pi\)
0.811394 + 0.584499i \(0.198709\pi\)
\(564\) −2.67887e9 −0.628746
\(565\) 4.64574e9 1.08364
\(566\) 1.45931e9 0.338290
\(567\) −3.63953e9 −0.838503
\(568\) −9.38577e8 −0.214907
\(569\) −6.30670e8 −0.143519 −0.0717595 0.997422i \(-0.522861\pi\)
−0.0717595 + 0.997422i \(0.522861\pi\)
\(570\) −4.98875e9 −1.12831
\(571\) 2.00096e9 0.449793 0.224896 0.974383i \(-0.427796\pi\)
0.224896 + 0.974383i \(0.427796\pi\)
\(572\) −8.43345e8 −0.188417
\(573\) −8.01561e9 −1.77990
\(574\) −5.97428e9 −1.31854
\(575\) −2.38962e8 −0.0524194
\(576\) 8.04352e8 0.175375
\(577\) −5.92670e9 −1.28439 −0.642196 0.766541i \(-0.721976\pi\)
−0.642196 + 0.766541i \(0.721976\pi\)
\(578\) 5.57868e8 0.120167
\(579\) −4.94027e9 −1.05773
\(580\) −2.14666e9 −0.456842
\(581\) 6.08072e9 1.28629
\(582\) 8.09570e9 1.70225
\(583\) −8.34340e7 −0.0174383
\(584\) −2.56030e8 −0.0531920
\(585\) −6.88034e9 −1.42090
\(586\) 3.48972e7 0.00716388
\(587\) −1.54400e9 −0.315075 −0.157538 0.987513i \(-0.550356\pi\)
−0.157538 + 0.987513i \(0.550356\pi\)
\(588\) −1.04027e10 −2.11021
\(589\) 2.63770e9 0.531890
\(590\) −7.77083e8 −0.155771
\(591\) −1.15181e10 −2.29522
\(592\) −1.26147e8 −0.0249892
\(593\) 6.29215e9 1.23910 0.619552 0.784956i \(-0.287314\pi\)
0.619552 + 0.784956i \(0.287314\pi\)
\(594\) −6.80334e8 −0.133189
\(595\) 8.68909e9 1.69108
\(596\) −7.21505e8 −0.139598
\(597\) 3.75288e9 0.721862
\(598\) −7.05534e8 −0.134916
\(599\) −3.20572e9 −0.609441 −0.304721 0.952442i \(-0.598563\pi\)
−0.304721 + 0.952442i \(0.598563\pi\)
\(600\) 9.95679e8 0.188187
\(601\) 1.65231e9 0.310477 0.155239 0.987877i \(-0.450385\pi\)
0.155239 + 0.987877i \(0.450385\pi\)
\(602\) −1.19543e10 −2.23325
\(603\) −2.14464e9 −0.398330
\(604\) −2.66754e9 −0.492586
\(605\) 4.01248e8 0.0736663
\(606\) −5.31741e9 −0.970612
\(607\) 1.08264e9 0.196482 0.0982410 0.995163i \(-0.468678\pi\)
0.0982410 + 0.995163i \(0.468678\pi\)
\(608\) 1.24450e9 0.224559
\(609\) 1.87972e10 3.37235
\(610\) −1.88095e9 −0.335524
\(611\) −5.71634e9 −1.01385
\(612\) 4.30268e9 0.758769
\(613\) 1.86372e9 0.326790 0.163395 0.986561i \(-0.447755\pi\)
0.163395 + 0.986561i \(0.447755\pi\)
\(614\) −3.26773e9 −0.569714
\(615\) 7.00307e9 1.21402
\(616\) 1.19320e9 0.205674
\(617\) −4.23169e8 −0.0725297 −0.0362649 0.999342i \(-0.511546\pi\)
−0.0362649 + 0.999342i \(0.511546\pi\)
\(618\) 7.93846e8 0.135293
\(619\) 2.37963e9 0.403266 0.201633 0.979461i \(-0.435375\pi\)
0.201633 + 0.979461i \(0.435375\pi\)
\(620\) 1.00674e9 0.169647
\(621\) −5.69160e8 −0.0953705
\(622\) 6.45185e9 1.07502
\(623\) 1.67221e9 0.277065
\(624\) 2.93974e9 0.484353
\(625\) −3.28816e9 −0.538732
\(626\) 1.87631e9 0.305700
\(627\) −3.66457e9 −0.593727
\(628\) 2.27799e9 0.367022
\(629\) −6.74794e8 −0.108117
\(630\) 9.73458e9 1.55105
\(631\) 7.34037e9 1.16309 0.581547 0.813513i \(-0.302447\pi\)
0.581547 + 0.813513i \(0.302447\pi\)
\(632\) −2.47991e9 −0.390774
\(633\) 4.79299e9 0.751092
\(634\) 1.68358e9 0.262375
\(635\) 4.03978e9 0.626109
\(636\) 2.90835e8 0.0448277
\(637\) −2.21980e10 −3.40271
\(638\) −1.57687e9 −0.240394
\(639\) −5.62479e9 −0.852812
\(640\) 4.74992e8 0.0716236
\(641\) −4.78647e9 −0.717814 −0.358907 0.933373i \(-0.616850\pi\)
−0.358907 + 0.933373i \(0.616850\pi\)
\(642\) −1.24885e10 −1.86267
\(643\) 4.12710e9 0.612219 0.306110 0.951996i \(-0.400973\pi\)
0.306110 + 0.951996i \(0.400973\pi\)
\(644\) 9.98219e8 0.147274
\(645\) 1.40129e10 2.05622
\(646\) 6.65713e9 0.971568
\(647\) 2.07770e9 0.301591 0.150796 0.988565i \(-0.451816\pi\)
0.150796 + 0.988565i \(0.451816\pi\)
\(648\) −1.06427e9 −0.153652
\(649\) −5.70820e8 −0.0819677
\(650\) 2.12464e9 0.303451
\(651\) −8.81552e9 −1.25232
\(652\) −6.34651e9 −0.896744
\(653\) −1.03614e10 −1.45621 −0.728104 0.685467i \(-0.759598\pi\)
−0.728104 + 0.685467i \(0.759598\pi\)
\(654\) 1.22740e10 1.71579
\(655\) −1.65567e9 −0.230213
\(656\) −1.74699e9 −0.241617
\(657\) −1.53436e9 −0.211081
\(658\) 8.08772e9 1.10671
\(659\) 6.72335e9 0.915139 0.457569 0.889174i \(-0.348720\pi\)
0.457569 + 0.889174i \(0.348720\pi\)
\(660\) −1.39867e9 −0.189370
\(661\) −1.14081e10 −1.53641 −0.768204 0.640206i \(-0.778849\pi\)
−0.768204 + 0.640206i \(0.778849\pi\)
\(662\) −7.23637e9 −0.969433
\(663\) 1.57254e10 2.09558
\(664\) 1.77812e9 0.235707
\(665\) 1.50614e10 1.98605
\(666\) −7.55987e8 −0.0991641
\(667\) −1.31919e9 −0.172134
\(668\) 5.20889e9 0.676126
\(669\) −8.22205e9 −1.06167
\(670\) −1.26647e9 −0.162679
\(671\) −1.38169e9 −0.176555
\(672\) −4.15926e9 −0.528717
\(673\) −4.18258e9 −0.528922 −0.264461 0.964396i \(-0.585194\pi\)
−0.264461 + 0.964396i \(0.585194\pi\)
\(674\) 5.25019e9 0.660488
\(675\) 1.71397e9 0.214506
\(676\) 2.25708e9 0.281018
\(677\) 3.43078e9 0.424945 0.212472 0.977167i \(-0.431848\pi\)
0.212472 + 0.977167i \(0.431848\pi\)
\(678\) −1.18957e10 −1.46584
\(679\) −2.44415e10 −2.99629
\(680\) 2.54085e9 0.309883
\(681\) −1.95316e10 −2.36986
\(682\) 7.39520e8 0.0892698
\(683\) 1.62297e10 1.94911 0.974556 0.224144i \(-0.0719585\pi\)
0.974556 + 0.224144i \(0.0719585\pi\)
\(684\) 7.45813e9 0.891115
\(685\) −3.77858e9 −0.449171
\(686\) 1.98710e10 2.35009
\(687\) 1.87102e10 2.20155
\(688\) −3.49567e9 −0.409233
\(689\) 6.20601e8 0.0722845
\(690\) −1.17011e9 −0.135599
\(691\) 6.75599e9 0.778962 0.389481 0.921035i \(-0.372654\pi\)
0.389481 + 0.921035i \(0.372654\pi\)
\(692\) −1.58515e9 −0.181844
\(693\) 7.15071e9 0.816174
\(694\) 6.58645e9 0.747985
\(695\) −2.94257e9 −0.332491
\(696\) 5.49665e9 0.617968
\(697\) −9.34511e9 −1.04537
\(698\) −4.19303e9 −0.466696
\(699\) 1.43026e10 1.58396
\(700\) −3.00603e9 −0.331246
\(701\) −7.32837e9 −0.803516 −0.401758 0.915746i \(-0.631601\pi\)
−0.401758 + 0.915746i \(0.631601\pi\)
\(702\) 5.06047e9 0.552092
\(703\) −1.16967e9 −0.126975
\(704\) 3.48914e8 0.0376889
\(705\) −9.48045e9 −1.01898
\(706\) −2.65181e9 −0.283613
\(707\) 1.60536e10 1.70846
\(708\) 1.98977e9 0.210710
\(709\) −1.38412e10 −1.45852 −0.729260 0.684236i \(-0.760136\pi\)
−0.729260 + 0.684236i \(0.760136\pi\)
\(710\) −3.32159e9 −0.348291
\(711\) −1.48618e10 −1.55070
\(712\) 4.88985e8 0.0507710
\(713\) 6.18675e8 0.0639219
\(714\) −2.22489e10 −2.28752
\(715\) −2.98457e9 −0.305359
\(716\) −8.56807e9 −0.872343
\(717\) 1.27643e10 1.29325
\(718\) −1.28557e10 −1.29617
\(719\) 1.31832e9 0.132273 0.0661365 0.997811i \(-0.478933\pi\)
0.0661365 + 0.997811i \(0.478933\pi\)
\(720\) 2.84657e9 0.284223
\(721\) −2.39668e9 −0.238142
\(722\) 4.38828e9 0.433924
\(723\) 3.06620e9 0.301729
\(724\) 1.76359e9 0.172708
\(725\) 3.97261e9 0.387162
\(726\) −1.02742e9 −0.0996481
\(727\) −1.99337e9 −0.192405 −0.0962027 0.995362i \(-0.530670\pi\)
−0.0962027 + 0.995362i \(0.530670\pi\)
\(728\) −8.87528e9 −0.852555
\(729\) −1.65079e10 −1.57814
\(730\) −9.06083e8 −0.0862061
\(731\) −1.86992e10 −1.77057
\(732\) 4.81629e9 0.453862
\(733\) 8.58436e9 0.805089 0.402545 0.915400i \(-0.368126\pi\)
0.402545 + 0.915400i \(0.368126\pi\)
\(734\) −1.98664e8 −0.0185432
\(735\) −3.68149e10 −3.41993
\(736\) 2.91898e8 0.0269872
\(737\) −9.30306e8 −0.0856031
\(738\) −1.04695e10 −0.958804
\(739\) 4.87753e9 0.444575 0.222287 0.974981i \(-0.428648\pi\)
0.222287 + 0.974981i \(0.428648\pi\)
\(740\) −4.46431e8 −0.0404989
\(741\) 2.72579e10 2.46110
\(742\) −8.78051e8 −0.0789053
\(743\) −1.46925e9 −0.131412 −0.0657059 0.997839i \(-0.520930\pi\)
−0.0657059 + 0.997839i \(0.520930\pi\)
\(744\) −2.57782e9 −0.229482
\(745\) −2.55338e9 −0.226240
\(746\) 1.06834e10 0.942155
\(747\) 1.06561e10 0.935350
\(748\) 1.86643e9 0.163063
\(749\) 3.77036e10 3.27866
\(750\) 1.37858e10 1.19321
\(751\) 7.26899e9 0.626230 0.313115 0.949715i \(-0.398627\pi\)
0.313115 + 0.949715i \(0.398627\pi\)
\(752\) 2.36500e9 0.202800
\(753\) −7.30993e9 −0.623923
\(754\) 1.17291e10 0.996471
\(755\) −9.44034e9 −0.798313
\(756\) −7.15976e9 −0.602660
\(757\) 2.05022e10 1.71777 0.858884 0.512170i \(-0.171158\pi\)
0.858884 + 0.512170i \(0.171158\pi\)
\(758\) −4.04988e9 −0.337754
\(759\) −8.59528e8 −0.0713533
\(760\) 4.40423e9 0.363934
\(761\) −1.30953e10 −1.07713 −0.538566 0.842583i \(-0.681034\pi\)
−0.538566 + 0.842583i \(0.681034\pi\)
\(762\) −1.03441e10 −0.846935
\(763\) −3.70562e10 −3.02012
\(764\) 7.07645e9 0.574102
\(765\) 1.52271e10 1.22970
\(766\) 3.58475e9 0.288176
\(767\) 4.24589e9 0.339770
\(768\) −1.21625e9 −0.0968851
\(769\) 2.32970e10 1.84739 0.923694 0.383131i \(-0.125154\pi\)
0.923694 + 0.383131i \(0.125154\pi\)
\(770\) 4.22269e9 0.333328
\(771\) −6.42022e9 −0.504498
\(772\) 4.36143e9 0.341168
\(773\) −7.01207e9 −0.546032 −0.273016 0.962009i \(-0.588021\pi\)
−0.273016 + 0.962009i \(0.588021\pi\)
\(774\) −2.09492e10 −1.62395
\(775\) −1.86307e9 −0.143772
\(776\) −7.14715e9 −0.549057
\(777\) 3.90917e9 0.298958
\(778\) 5.77735e9 0.439846
\(779\) −1.61985e10 −1.22770
\(780\) 1.04036e10 0.784971
\(781\) −2.43993e9 −0.183273
\(782\) 1.56144e9 0.116762
\(783\) 9.46196e9 0.704393
\(784\) 9.18387e9 0.680643
\(785\) 8.06172e9 0.594818
\(786\) 4.23944e9 0.311408
\(787\) 9.15470e9 0.669472 0.334736 0.942312i \(-0.391353\pi\)
0.334736 + 0.942312i \(0.391353\pi\)
\(788\) 1.01686e10 0.740318
\(789\) 3.01278e10 2.18372
\(790\) −8.77631e9 −0.633311
\(791\) 3.59139e10 2.58015
\(792\) 2.09100e9 0.149560
\(793\) 1.02773e10 0.731851
\(794\) −7.77099e9 −0.550940
\(795\) 1.02925e9 0.0726504
\(796\) −3.31316e9 −0.232834
\(797\) 3.58195e9 0.250620 0.125310 0.992118i \(-0.460007\pi\)
0.125310 + 0.992118i \(0.460007\pi\)
\(798\) −3.85656e10 −2.68652
\(799\) 1.26510e10 0.877426
\(800\) −8.79019e8 −0.0606993
\(801\) 2.93043e9 0.201473
\(802\) 9.78606e9 0.669881
\(803\) −6.65579e8 −0.0453623
\(804\) 3.24287e9 0.220056
\(805\) 3.53266e9 0.238680
\(806\) −5.50071e9 −0.370038
\(807\) 1.34074e10 0.898021
\(808\) 4.69439e9 0.313068
\(809\) −1.39506e10 −0.926345 −0.463173 0.886268i \(-0.653289\pi\)
−0.463173 + 0.886268i \(0.653289\pi\)
\(810\) −3.76640e9 −0.249017
\(811\) 3.44260e8 0.0226628 0.0113314 0.999936i \(-0.496393\pi\)
0.0113314 + 0.999936i \(0.496393\pi\)
\(812\) −1.65948e10 −1.08774
\(813\) −3.75418e10 −2.45018
\(814\) −3.27934e8 −0.0213108
\(815\) −2.24601e10 −1.45332
\(816\) −6.50600e9 −0.419178
\(817\) −3.24126e10 −2.07940
\(818\) 1.40759e10 0.899166
\(819\) −5.31885e10 −3.38318
\(820\) −6.18255e9 −0.391579
\(821\) 7.16819e9 0.452073 0.226036 0.974119i \(-0.427423\pi\)
0.226036 + 0.974119i \(0.427423\pi\)
\(822\) 9.67527e9 0.607592
\(823\) 9.52943e9 0.595892 0.297946 0.954583i \(-0.403698\pi\)
0.297946 + 0.954583i \(0.403698\pi\)
\(824\) −7.00834e8 −0.0436385
\(825\) 2.58838e9 0.160487
\(826\) −6.00725e9 −0.370891
\(827\) −7.58166e9 −0.466117 −0.233059 0.972463i \(-0.574873\pi\)
−0.233059 + 0.972463i \(0.574873\pi\)
\(828\) 1.74931e9 0.107093
\(829\) −1.26823e10 −0.773139 −0.386569 0.922260i \(-0.626340\pi\)
−0.386569 + 0.922260i \(0.626340\pi\)
\(830\) 6.29269e9 0.382000
\(831\) 2.34811e9 0.141943
\(832\) −2.59530e9 −0.156227
\(833\) 4.91269e10 2.94484
\(834\) 7.53462e9 0.449759
\(835\) 1.84341e10 1.09577
\(836\) 3.23521e9 0.191505
\(837\) −4.43747e9 −0.261575
\(838\) 8.17712e9 0.480005
\(839\) −1.50145e10 −0.877695 −0.438847 0.898562i \(-0.644613\pi\)
−0.438847 + 0.898562i \(0.644613\pi\)
\(840\) −1.47195e10 −0.856870
\(841\) 4.68092e9 0.271360
\(842\) 6.46381e9 0.373161
\(843\) −2.56753e10 −1.47611
\(844\) −4.23141e9 −0.242263
\(845\) 7.98774e9 0.455434
\(846\) 1.41732e10 0.804769
\(847\) 3.10185e9 0.175400
\(848\) −2.56759e8 −0.0144590
\(849\) −1.32239e10 −0.741620
\(850\) −4.70210e9 −0.262619
\(851\) −2.74346e8 −0.0152597
\(852\) 8.50514e9 0.471132
\(853\) −2.88665e10 −1.59247 −0.796236 0.604986i \(-0.793179\pi\)
−0.796236 + 0.604986i \(0.793179\pi\)
\(854\) −1.45407e10 −0.798885
\(855\) 2.63941e10 1.44419
\(856\) 1.10252e10 0.600800
\(857\) −2.27563e10 −1.23500 −0.617502 0.786569i \(-0.711855\pi\)
−0.617502 + 0.786569i \(0.711855\pi\)
\(858\) 7.64217e9 0.413058
\(859\) 7.60631e8 0.0409447 0.0204723 0.999790i \(-0.493483\pi\)
0.0204723 + 0.999790i \(0.493483\pi\)
\(860\) −1.23711e10 −0.663227
\(861\) 5.41374e10 2.89059
\(862\) 1.68879e10 0.898047
\(863\) 1.03239e10 0.546772 0.273386 0.961904i \(-0.411856\pi\)
0.273386 + 0.961904i \(0.411856\pi\)
\(864\) −2.09365e9 −0.110435
\(865\) −5.60981e9 −0.294708
\(866\) −1.64300e10 −0.859654
\(867\) −5.05525e9 −0.263436
\(868\) 7.78263e9 0.403931
\(869\) −6.44679e9 −0.333253
\(870\) 1.94525e10 1.00151
\(871\) 6.91982e9 0.354839
\(872\) −1.08359e10 −0.553424
\(873\) −4.28321e10 −2.17881
\(874\) 2.70654e9 0.137128
\(875\) −4.16204e10 −2.10028
\(876\) 2.32008e9 0.116611
\(877\) −3.48516e9 −0.174471 −0.0872357 0.996188i \(-0.527803\pi\)
−0.0872357 + 0.996188i \(0.527803\pi\)
\(878\) −2.17917e9 −0.108657
\(879\) −3.16229e8 −0.0157051
\(880\) 1.23479e9 0.0610808
\(881\) −3.74696e10 −1.84613 −0.923067 0.384638i \(-0.874326\pi\)
−0.923067 + 0.384638i \(0.874326\pi\)
\(882\) 5.50379e10 2.70098
\(883\) −2.65271e10 −1.29667 −0.648333 0.761357i \(-0.724533\pi\)
−0.648333 + 0.761357i \(0.724533\pi\)
\(884\) −1.38829e10 −0.675924
\(885\) 7.04172e9 0.341489
\(886\) 2.02936e9 0.0980258
\(887\) −3.93196e10 −1.89181 −0.945903 0.324450i \(-0.894821\pi\)
−0.945903 + 0.324450i \(0.894821\pi\)
\(888\) 1.14311e9 0.0547828
\(889\) 3.12296e10 1.49077
\(890\) 1.73050e9 0.0822824
\(891\) −2.76668e9 −0.131035
\(892\) 7.25870e9 0.342438
\(893\) 2.19288e10 1.03047
\(894\) 6.53809e9 0.306034
\(895\) −3.03221e10 −1.41377
\(896\) 3.67193e9 0.170536
\(897\) 6.39336e9 0.295771
\(898\) −9.44503e9 −0.435247
\(899\) −1.02851e10 −0.472118
\(900\) −5.26786e9 −0.240872
\(901\) −1.37347e9 −0.0625578
\(902\) −4.54150e9 −0.206052
\(903\) 1.08327e11 4.89586
\(904\) 1.05019e10 0.472801
\(905\) 6.24130e9 0.279901
\(906\) 2.41725e10 1.07988
\(907\) 2.63565e10 1.17290 0.586451 0.809984i \(-0.300524\pi\)
0.586451 + 0.809984i \(0.300524\pi\)
\(908\) 1.72431e10 0.764391
\(909\) 2.81329e10 1.24234
\(910\) −3.14093e10 −1.38170
\(911\) −2.82134e10 −1.23635 −0.618175 0.786041i \(-0.712128\pi\)
−0.618175 + 0.786041i \(0.712128\pi\)
\(912\) −1.12773e10 −0.492292
\(913\) 4.62241e9 0.201011
\(914\) 1.15914e10 0.502140
\(915\) 1.70447e10 0.735556
\(916\) −1.65180e10 −0.710105
\(917\) −1.27992e10 −0.548138
\(918\) −1.11995e10 −0.477802
\(919\) 3.73720e10 1.58834 0.794168 0.607698i \(-0.207907\pi\)
0.794168 + 0.607698i \(0.207907\pi\)
\(920\) 1.03302e9 0.0437371
\(921\) 2.96113e10 1.24896
\(922\) −1.79103e10 −0.752567
\(923\) 1.81488e10 0.759699
\(924\) −1.08124e10 −0.450892
\(925\) 8.26164e8 0.0343218
\(926\) 2.20324e10 0.911851
\(927\) −4.20002e9 −0.173170
\(928\) −4.85263e9 −0.199324
\(929\) 1.01264e10 0.414380 0.207190 0.978301i \(-0.433568\pi\)
0.207190 + 0.978301i \(0.433568\pi\)
\(930\) −9.12283e9 −0.371911
\(931\) 8.51549e10 3.45848
\(932\) −1.26268e10 −0.510903
\(933\) −5.84650e10 −2.35673
\(934\) −1.66534e10 −0.668789
\(935\) 6.60523e9 0.264270
\(936\) −1.55533e10 −0.619952
\(937\) −1.52255e10 −0.604622 −0.302311 0.953209i \(-0.597758\pi\)
−0.302311 + 0.953209i \(0.597758\pi\)
\(938\) −9.79045e9 −0.387340
\(939\) −1.70026e10 −0.670173
\(940\) 8.36966e9 0.328670
\(941\) 1.50765e10 0.589844 0.294922 0.955521i \(-0.404706\pi\)
0.294922 + 0.955521i \(0.404706\pi\)
\(942\) −2.06425e10 −0.804608
\(943\) −3.79937e9 −0.147544
\(944\) −1.75663e9 −0.0679640
\(945\) −2.53382e10 −0.976706
\(946\) −9.08738e9 −0.348995
\(947\) −2.57203e10 −0.984125 −0.492063 0.870560i \(-0.663757\pi\)
−0.492063 + 0.870560i \(0.663757\pi\)
\(948\) 2.24723e10 0.856678
\(949\) 4.95072e9 0.188034
\(950\) −8.15046e9 −0.308425
\(951\) −1.52562e10 −0.575193
\(952\) 1.96421e10 0.737834
\(953\) −3.41315e10 −1.27741 −0.638704 0.769452i \(-0.720529\pi\)
−0.638704 + 0.769452i \(0.720529\pi\)
\(954\) −1.53873e9 −0.0573775
\(955\) 2.50433e10 0.930422
\(956\) −1.12688e10 −0.417133
\(957\) 1.42892e10 0.527005
\(958\) 3.60935e9 0.132632
\(959\) −2.92104e10 −1.06948
\(960\) −4.30425e9 −0.157018
\(961\) −2.26891e10 −0.824680
\(962\) 2.43924e9 0.0883370
\(963\) 6.60730e10 2.38414
\(964\) −2.70695e9 −0.0973218
\(965\) 1.54350e10 0.552917
\(966\) −9.04559e9 −0.322862
\(967\) −3.81102e10 −1.35534 −0.677670 0.735366i \(-0.737010\pi\)
−0.677670 + 0.735366i \(0.737010\pi\)
\(968\) 9.07039e8 0.0321412
\(969\) −6.03251e10 −2.12993
\(970\) −2.52935e10 −0.889833
\(971\) 4.36002e10 1.52835 0.764173 0.645011i \(-0.223147\pi\)
0.764173 + 0.645011i \(0.223147\pi\)
\(972\) 1.85871e10 0.649201
\(973\) −2.27476e10 −0.791662
\(974\) −1.18479e10 −0.410852
\(975\) −1.92529e10 −0.665244
\(976\) −4.25199e9 −0.146392
\(977\) 2.46833e10 0.846785 0.423392 0.905946i \(-0.360839\pi\)
0.423392 + 0.905946i \(0.360839\pi\)
\(978\) 5.75104e10 1.96590
\(979\) 1.27117e9 0.0432976
\(980\) 3.25014e10 1.10309
\(981\) −6.49385e10 −2.19614
\(982\) −4.86046e9 −0.163790
\(983\) 1.78371e10 0.598946 0.299473 0.954105i \(-0.403189\pi\)
0.299473 + 0.954105i \(0.403189\pi\)
\(984\) 1.58308e10 0.529687
\(985\) 3.59862e10 1.19980
\(986\) −2.59580e10 −0.862385
\(987\) −7.32887e10 −2.42620
\(988\) −2.40642e10 −0.793819
\(989\) −7.60241e9 −0.249899
\(990\) 7.39998e9 0.242386
\(991\) 3.07472e10 1.00357 0.501785 0.864993i \(-0.332677\pi\)
0.501785 + 0.864993i \(0.332677\pi\)
\(992\) 2.27579e9 0.0740186
\(993\) 6.55740e10 2.12525
\(994\) −2.56776e10 −0.829283
\(995\) −1.17252e10 −0.377345
\(996\) −1.61128e10 −0.516730
\(997\) −4.48450e9 −0.143312 −0.0716558 0.997429i \(-0.522828\pi\)
−0.0716558 + 0.997429i \(0.522828\pi\)
\(998\) 3.75413e10 1.19551
\(999\) 1.96776e9 0.0624443
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 22.8.a.d.1.1 2
3.2 odd 2 198.8.a.f.1.1 2
4.3 odd 2 176.8.a.e.1.2 2
5.4 even 2 550.8.a.d.1.2 2
11.10 odd 2 242.8.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
22.8.a.d.1.1 2 1.1 even 1 trivial
176.8.a.e.1.2 2 4.3 odd 2
198.8.a.f.1.1 2 3.2 odd 2
242.8.a.h.1.1 2 11.10 odd 2
550.8.a.d.1.2 2 5.4 even 2