Properties

Label 22.8.a.d.1.2
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 1.2
Root \(-60.4939\) of defining polynomial
Character \(\chi\) \(=\) 22.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} +49.4939 q^{3} +64.0000 q^{4} +104.506 q^{5} +395.951 q^{6} +43.0861 q^{7} +512.000 q^{8} +262.641 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +49.4939 q^{3} +64.0000 q^{4} +104.506 q^{5} +395.951 q^{6} +43.0861 q^{7} +512.000 q^{8} +262.641 q^{9} +836.049 q^{10} +1331.00 q^{11} +3167.61 q^{12} +4494.27 q^{13} +344.689 q^{14} +5172.41 q^{15} +4096.00 q^{16} -6878.55 q^{17} +2101.13 q^{18} -21063.0 q^{19} +6688.39 q^{20} +2132.50 q^{21} +10648.0 q^{22} -60259.0 q^{23} +25340.9 q^{24} -67203.5 q^{25} +35954.2 q^{26} -95243.9 q^{27} +2757.51 q^{28} -59039.5 q^{29} +41379.3 q^{30} -88522.5 q^{31} +32768.0 q^{32} +65876.3 q^{33} -55028.4 q^{34} +4502.76 q^{35} +16809.0 q^{36} +382131. q^{37} -168504. q^{38} +222439. q^{39} +53507.1 q^{40} +550122. q^{41} +17060.0 q^{42} +693613. q^{43} +85184.0 q^{44} +27447.6 q^{45} -482072. q^{46} -126233. q^{47} +202727. q^{48} -821687. q^{49} -537628. q^{50} -340446. q^{51} +287634. q^{52} -1.19815e6 q^{53} -761951. q^{54} +139098. q^{55} +22060.1 q^{56} -1.04249e6 q^{57} -472316. q^{58} +1.31641e6 q^{59} +331034. q^{60} +440165. q^{61} -708180. q^{62} +11316.2 q^{63} +262144. q^{64} +469679. q^{65} +527011. q^{66} +3.56366e6 q^{67} -440227. q^{68} -2.98245e6 q^{69} +36022.1 q^{70} +3.13943e6 q^{71} +134472. q^{72} -4.07747e6 q^{73} +3.05705e6 q^{74} -3.32616e6 q^{75} -1.34803e6 q^{76} +57347.6 q^{77} +1.77951e6 q^{78} +1.89723e6 q^{79} +428057. q^{80} -5.28839e6 q^{81} +4.40097e6 q^{82} +6.49257e6 q^{83} +136480. q^{84} -718851. q^{85} +5.54890e6 q^{86} -2.92209e6 q^{87} +681472. q^{88} +9.23033e6 q^{89} +219581. q^{90} +193641. q^{91} -3.85658e6 q^{92} -4.38132e6 q^{93} -1.00986e6 q^{94} -2.20122e6 q^{95} +1.62181e6 q^{96} -1.38062e7 q^{97} -6.57349e6 q^{98} +349576. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{2} - 23 q^{3} + 128 q^{4} + 331 q^{5} - 184 q^{6} + 1794 q^{7} + 1024 q^{8} + 3331 q^{9}+O(q^{10}) \) 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
\(3\) 49.4939 1.05834 0.529172 0.848515i \(-0.322503\pi\)
0.529172 + 0.848515i \(0.322503\pi\)
\(4\) 64.0000 0.500000
\(5\) 104.506 0.373893 0.186946 0.982370i \(-0.440141\pi\)
0.186946 + 0.982370i \(0.440141\pi\)
\(6\) 395.951 0.748362
\(7\) 43.0861 0.0474781 0.0237391 0.999718i \(-0.492443\pi\)
0.0237391 + 0.999718i \(0.492443\pi\)
\(8\) 512.000 0.353553
\(9\) 262.641 0.120092
\(10\) 836.049 0.264382
\(11\) 1331.00 0.301511
\(12\) 3167.61 0.529172
\(13\) 4494.27 0.567359 0.283679 0.958919i \(-0.408445\pi\)
0.283679 + 0.958919i \(0.408445\pi\)
\(14\) 344.689 0.0335721
\(15\) 5172.41 0.395707
\(16\) 4096.00 0.250000
\(17\) −6878.55 −0.339567 −0.169784 0.985481i \(-0.554307\pi\)
−0.169784 + 0.985481i \(0.554307\pi\)
\(18\) 2101.13 0.0849179
\(19\) −21063.0 −0.704503 −0.352252 0.935905i \(-0.614584\pi\)
−0.352252 + 0.935905i \(0.614584\pi\)
\(20\) 6688.39 0.186946
\(21\) 2132.50 0.0502482
\(22\) 10648.0 0.213201
\(23\) −60259.0 −1.03270 −0.516350 0.856377i \(-0.672710\pi\)
−0.516350 + 0.856377i \(0.672710\pi\)
\(24\) 25340.9 0.374181
\(25\) −67203.5 −0.860204
\(26\) 35954.2 0.401183
\(27\) −95243.9 −0.931245
\(28\) 2757.51 0.0237391
\(29\) −59039.5 −0.449521 −0.224760 0.974414i \(-0.572160\pi\)
−0.224760 + 0.974414i \(0.572160\pi\)
\(30\) 41379.3 0.279807
\(31\) −88522.5 −0.533689 −0.266844 0.963740i \(-0.585981\pi\)
−0.266844 + 0.963740i \(0.585981\pi\)
\(32\) 32768.0 0.176777
\(33\) 65876.3 0.319103
\(34\) −55028.4 −0.240110
\(35\) 4502.76 0.0177517
\(36\) 16809.0 0.0600460
\(37\) 382131. 1.24024 0.620120 0.784507i \(-0.287084\pi\)
0.620120 + 0.784507i \(0.287084\pi\)
\(38\) −168504. −0.498159
\(39\) 222439. 0.600461
\(40\) 53507.1 0.132191
\(41\) 550122. 1.24657 0.623283 0.781996i \(-0.285798\pi\)
0.623283 + 0.781996i \(0.285798\pi\)
\(42\) 17060.0 0.0355309
\(43\) 693613. 1.33039 0.665193 0.746671i \(-0.268349\pi\)
0.665193 + 0.746671i \(0.268349\pi\)
\(44\) 85184.0 0.150756
\(45\) 27447.6 0.0449015
\(46\) −482072. −0.730230
\(47\) −126233. −0.177349 −0.0886745 0.996061i \(-0.528263\pi\)
−0.0886745 + 0.996061i \(0.528263\pi\)
\(48\) 202727. 0.264586
\(49\) −821687. −0.997746
\(50\) −537628. −0.608256
\(51\) −340446. −0.359379
\(52\) 287634. 0.283679
\(53\) −1.19815e6 −1.10546 −0.552732 0.833359i \(-0.686415\pi\)
−0.552732 + 0.833359i \(0.686415\pi\)
\(54\) −761951. −0.658490
\(55\) 139098. 0.112733
\(56\) 22060.1 0.0167861
\(57\) −1.04249e6 −0.745607
\(58\) −472316. −0.317859
\(59\) 1.31641e6 0.834469 0.417234 0.908799i \(-0.362999\pi\)
0.417234 + 0.908799i \(0.362999\pi\)
\(60\) 331034. 0.197853
\(61\) 440165. 0.248291 0.124145 0.992264i \(-0.460381\pi\)
0.124145 + 0.992264i \(0.460381\pi\)
\(62\) −708180. −0.377375
\(63\) 11316.2 0.00570175
\(64\) 262144. 0.125000
\(65\) 469679. 0.212131
\(66\) 527011. 0.225640
\(67\) 3.56366e6 1.44755 0.723777 0.690034i \(-0.242404\pi\)
0.723777 + 0.690034i \(0.242404\pi\)
\(68\) −440227. −0.169784
\(69\) −2.98245e6 −1.09295
\(70\) 36022.1 0.0125524
\(71\) 3.13943e6 1.04099 0.520494 0.853865i \(-0.325748\pi\)
0.520494 + 0.853865i \(0.325748\pi\)
\(72\) 134472. 0.0424590
\(73\) −4.07747e6 −1.22676 −0.613382 0.789787i \(-0.710191\pi\)
−0.613382 + 0.789787i \(0.710191\pi\)
\(74\) 3.05705e6 0.876982
\(75\) −3.32616e6 −0.910392
\(76\) −1.34803e6 −0.352252
\(77\) 57347.6 0.0143152
\(78\) 1.77951e6 0.424590
\(79\) 1.89723e6 0.432937 0.216468 0.976290i \(-0.430546\pi\)
0.216468 + 0.976290i \(0.430546\pi\)
\(80\) 428057. 0.0934731
\(81\) −5.28839e6 −1.10567
\(82\) 4.40097e6 0.881455
\(83\) 6.49257e6 1.24636 0.623179 0.782079i \(-0.285841\pi\)
0.623179 + 0.782079i \(0.285841\pi\)
\(84\) 136480. 0.0251241
\(85\) −718851. −0.126962
\(86\) 5.54890e6 0.940725
\(87\) −2.92209e6 −0.475747
\(88\) 681472. 0.106600
\(89\) 9.23033e6 1.38788 0.693940 0.720032i \(-0.255873\pi\)
0.693940 + 0.720032i \(0.255873\pi\)
\(90\) 219581. 0.0317502
\(91\) 193641. 0.0269371
\(92\) −3.85658e6 −0.516350
\(93\) −4.38132e6 −0.564826
\(94\) −1.00986e6 −0.125405
\(95\) −2.20122e6 −0.263409
\(96\) 1.62181e6 0.187091
\(97\) −1.38062e7 −1.53593 −0.767967 0.640489i \(-0.778732\pi\)
−0.767967 + 0.640489i \(0.778732\pi\)
\(98\) −6.57349e6 −0.705513
\(99\) 349576. 0.0362091
\(100\) −4.30102e6 −0.430102
\(101\) 6.59478e6 0.636906 0.318453 0.947939i \(-0.396837\pi\)
0.318453 + 0.947939i \(0.396837\pi\)
\(102\) −2.72357e6 −0.254119
\(103\) −1.98787e7 −1.79250 −0.896249 0.443551i \(-0.853718\pi\)
−0.896249 + 0.443551i \(0.853718\pi\)
\(104\) 2.30107e6 0.200592
\(105\) 222859. 0.0187874
\(106\) −9.58517e6 −0.781681
\(107\) 2.09533e7 1.65352 0.826758 0.562558i \(-0.190183\pi\)
0.826758 + 0.562558i \(0.190183\pi\)
\(108\) −6.09561e6 −0.465623
\(109\) −598244. −0.0442472 −0.0221236 0.999755i \(-0.507043\pi\)
−0.0221236 + 0.999755i \(0.507043\pi\)
\(110\) 1.11278e6 0.0797142
\(111\) 1.89131e7 1.31260
\(112\) 176481. 0.0118695
\(113\) −1.41807e6 −0.0924535 −0.0462267 0.998931i \(-0.514720\pi\)
−0.0462267 + 0.998931i \(0.514720\pi\)
\(114\) −8.33992e6 −0.527224
\(115\) −6.29744e6 −0.386119
\(116\) −3.77853e6 −0.224760
\(117\) 1.18038e6 0.0681353
\(118\) 1.05313e7 0.590058
\(119\) −296370. −0.0161220
\(120\) 2.64827e6 0.139904
\(121\) 1.77156e6 0.0909091
\(122\) 3.52132e6 0.175568
\(123\) 2.72276e7 1.31930
\(124\) −5.66544e6 −0.266844
\(125\) −1.51877e7 −0.695517
\(126\) 90529.5 0.00403175
\(127\) −1.92988e7 −0.836023 −0.418012 0.908442i \(-0.637273\pi\)
−0.418012 + 0.908442i \(0.637273\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) 3.43296e7 1.40801
\(130\) 3.75743e6 0.149999
\(131\) 4.22451e7 1.64182 0.820912 0.571055i \(-0.193466\pi\)
0.820912 + 0.571055i \(0.193466\pi\)
\(132\) 4.21608e6 0.159551
\(133\) −907523. −0.0334485
\(134\) 2.85093e7 1.02358
\(135\) −9.95358e6 −0.348186
\(136\) −3.52182e6 −0.120055
\(137\) −7.65888e6 −0.254474 −0.127237 0.991872i \(-0.540611\pi\)
−0.127237 + 0.991872i \(0.540611\pi\)
\(138\) −2.38596e7 −0.772834
\(139\) −1.46018e7 −0.461164 −0.230582 0.973053i \(-0.574063\pi\)
−0.230582 + 0.973053i \(0.574063\pi\)
\(140\) 288177. 0.00887586
\(141\) −6.24773e6 −0.187696
\(142\) 2.51154e7 0.736090
\(143\) 5.98188e6 0.171065
\(144\) 1.07578e6 0.0300230
\(145\) −6.16999e6 −0.168072
\(146\) −3.26198e7 −0.867453
\(147\) −4.06684e7 −1.05596
\(148\) 2.44564e7 0.620120
\(149\) −3.54730e7 −0.878507 −0.439254 0.898363i \(-0.644757\pi\)
−0.439254 + 0.898363i \(0.644757\pi\)
\(150\) −2.66093e7 −0.643744
\(151\) 4.10281e7 0.969754 0.484877 0.874582i \(-0.338864\pi\)
0.484877 + 0.874582i \(0.338864\pi\)
\(152\) −1.07843e7 −0.249080
\(153\) −1.80659e6 −0.0407793
\(154\) 458780. 0.0101224
\(155\) −9.25115e6 −0.199542
\(156\) 1.42361e7 0.300230
\(157\) −6.66977e7 −1.37550 −0.687752 0.725946i \(-0.741402\pi\)
−0.687752 + 0.725946i \(0.741402\pi\)
\(158\) 1.51778e7 0.306133
\(159\) −5.93009e7 −1.16996
\(160\) 3.42446e6 0.0660955
\(161\) −2.59632e6 −0.0490307
\(162\) −4.23071e7 −0.781827
\(163\) 9.29744e6 0.168154 0.0840769 0.996459i \(-0.473206\pi\)
0.0840769 + 0.996459i \(0.473206\pi\)
\(164\) 3.52078e7 0.623283
\(165\) 6.88448e6 0.119310
\(166\) 5.19405e7 0.881309
\(167\) −5.93268e7 −0.985696 −0.492848 0.870115i \(-0.664044\pi\)
−0.492848 + 0.870115i \(0.664044\pi\)
\(168\) 1.09184e6 0.0177654
\(169\) −4.25500e7 −0.678104
\(170\) −5.75081e6 −0.0897754
\(171\) −5.53202e6 −0.0846053
\(172\) 4.43912e7 0.665193
\(173\) 1.19997e8 1.76201 0.881004 0.473108i \(-0.156868\pi\)
0.881004 + 0.473108i \(0.156868\pi\)
\(174\) −2.33767e7 −0.336404
\(175\) −2.89553e6 −0.0408409
\(176\) 5.45178e6 0.0753778
\(177\) 6.51543e7 0.883155
\(178\) 7.38426e7 0.981380
\(179\) 3.18894e7 0.415586 0.207793 0.978173i \(-0.433372\pi\)
0.207793 + 0.978173i \(0.433372\pi\)
\(180\) 1.75665e6 0.0224508
\(181\) 2.14661e7 0.269079 0.134539 0.990908i \(-0.457045\pi\)
0.134539 + 0.990908i \(0.457045\pi\)
\(182\) 1.54913e6 0.0190474
\(183\) 2.17854e7 0.262777
\(184\) −3.08526e7 −0.365115
\(185\) 3.99350e7 0.463717
\(186\) −3.50506e7 −0.399392
\(187\) −9.15535e6 −0.102383
\(188\) −8.07888e6 −0.0886745
\(189\) −4.10369e6 −0.0442138
\(190\) −1.76097e7 −0.186258
\(191\) −5.37382e7 −0.558042 −0.279021 0.960285i \(-0.590010\pi\)
−0.279021 + 0.960285i \(0.590010\pi\)
\(192\) 1.29745e7 0.132293
\(193\) 1.72101e8 1.72319 0.861593 0.507600i \(-0.169467\pi\)
0.861593 + 0.507600i \(0.169467\pi\)
\(194\) −1.10450e8 −1.08607
\(195\) 2.32462e7 0.224508
\(196\) −5.25879e7 −0.498873
\(197\) −1.14526e8 −1.06726 −0.533631 0.845718i \(-0.679173\pi\)
−0.533631 + 0.845718i \(0.679173\pi\)
\(198\) 2.79661e6 0.0256037
\(199\) −3.40644e7 −0.306418 −0.153209 0.988194i \(-0.548961\pi\)
−0.153209 + 0.988194i \(0.548961\pi\)
\(200\) −3.44082e7 −0.304128
\(201\) 1.76379e8 1.53201
\(202\) 5.27583e7 0.450361
\(203\) −2.54378e6 −0.0213424
\(204\) −2.17885e7 −0.179689
\(205\) 5.74911e7 0.466082
\(206\) −1.59030e8 −1.26749
\(207\) −1.58265e7 −0.124019
\(208\) 1.84085e7 0.141840
\(209\) −2.80349e7 −0.212416
\(210\) 1.78287e6 0.0132847
\(211\) −1.74288e8 −1.27726 −0.638628 0.769516i \(-0.720498\pi\)
−0.638628 + 0.769516i \(0.720498\pi\)
\(212\) −7.66814e7 −0.552732
\(213\) 1.55382e8 1.10172
\(214\) 1.67626e8 1.16921
\(215\) 7.24868e7 0.497421
\(216\) −4.87649e7 −0.329245
\(217\) −3.81409e6 −0.0253385
\(218\) −4.78596e6 −0.0312875
\(219\) −2.01810e8 −1.29834
\(220\) 8.90225e6 0.0563664
\(221\) −3.09141e7 −0.192656
\(222\) 1.51305e8 0.928149
\(223\) 1.45730e8 0.879998 0.439999 0.897998i \(-0.354979\pi\)
0.439999 + 0.897998i \(0.354979\pi\)
\(224\) 1.41184e6 0.00839303
\(225\) −1.76504e7 −0.103304
\(226\) −1.13446e7 −0.0653745
\(227\) −1.27102e8 −0.721211 −0.360606 0.932718i \(-0.617430\pi\)
−0.360606 + 0.932718i \(0.617430\pi\)
\(228\) −6.67194e7 −0.372804
\(229\) −1.29750e8 −0.713975 −0.356987 0.934109i \(-0.616196\pi\)
−0.356987 + 0.934109i \(0.616196\pi\)
\(230\) −5.03795e7 −0.273028
\(231\) 2.83835e6 0.0151504
\(232\) −3.02282e7 −0.158930
\(233\) −1.52210e8 −0.788310 −0.394155 0.919044i \(-0.628963\pi\)
−0.394155 + 0.919044i \(0.628963\pi\)
\(234\) 9.44306e6 0.0481789
\(235\) −1.31921e7 −0.0663095
\(236\) 8.42504e7 0.417234
\(237\) 9.39011e7 0.458196
\(238\) −2.37096e6 −0.0114000
\(239\) −2.43061e8 −1.15165 −0.575827 0.817571i \(-0.695320\pi\)
−0.575827 + 0.817571i \(0.695320\pi\)
\(240\) 2.11862e7 0.0989267
\(241\) −6.84592e7 −0.315045 −0.157522 0.987515i \(-0.550351\pi\)
−0.157522 + 0.987515i \(0.550351\pi\)
\(242\) 1.41725e7 0.0642824
\(243\) −5.34441e7 −0.238934
\(244\) 2.81705e7 0.124145
\(245\) −8.58713e7 −0.373050
\(246\) 2.17821e8 0.932883
\(247\) −9.46630e7 −0.399706
\(248\) −4.53235e7 −0.188687
\(249\) 3.21342e8 1.31908
\(250\) −1.21502e8 −0.491804
\(251\) 1.67292e8 0.667755 0.333878 0.942616i \(-0.391643\pi\)
0.333878 + 0.942616i \(0.391643\pi\)
\(252\) 724236. 0.00285087
\(253\) −8.02047e7 −0.311371
\(254\) −1.54391e8 −0.591158
\(255\) −3.55787e7 −0.134369
\(256\) 1.67772e7 0.0625000
\(257\) 3.84789e8 1.41402 0.707012 0.707202i \(-0.250043\pi\)
0.707012 + 0.707202i \(0.250043\pi\)
\(258\) 2.74637e8 0.995611
\(259\) 1.64645e7 0.0588843
\(260\) 3.00595e7 0.106066
\(261\) −1.55062e7 −0.0539839
\(262\) 3.37960e8 1.16094
\(263\) 5.09459e8 1.72689 0.863444 0.504445i \(-0.168303\pi\)
0.863444 + 0.504445i \(0.168303\pi\)
\(264\) 3.37287e7 0.112820
\(265\) −1.25214e8 −0.413325
\(266\) −7.26018e6 −0.0236517
\(267\) 4.56845e8 1.46886
\(268\) 2.28074e8 0.723777
\(269\) −5.51297e8 −1.72684 −0.863421 0.504484i \(-0.831683\pi\)
−0.863421 + 0.504484i \(0.831683\pi\)
\(270\) −7.96286e7 −0.246204
\(271\) 3.42271e8 1.04467 0.522333 0.852742i \(-0.325062\pi\)
0.522333 + 0.852742i \(0.325062\pi\)
\(272\) −2.81745e7 −0.0848918
\(273\) 9.58402e6 0.0285088
\(274\) −6.12710e7 −0.179940
\(275\) −8.94478e7 −0.259361
\(276\) −1.90877e8 −0.546476
\(277\) −5.10974e8 −1.44451 −0.722253 0.691629i \(-0.756893\pi\)
−0.722253 + 0.691629i \(0.756893\pi\)
\(278\) −1.16815e8 −0.326092
\(279\) −2.32497e7 −0.0640918
\(280\) 2.30541e6 0.00627618
\(281\) −1.51750e7 −0.0407998 −0.0203999 0.999792i \(-0.506494\pi\)
−0.0203999 + 0.999792i \(0.506494\pi\)
\(282\) −4.99819e7 −0.132721
\(283\) −2.25341e8 −0.590999 −0.295500 0.955343i \(-0.595486\pi\)
−0.295500 + 0.955343i \(0.595486\pi\)
\(284\) 2.00923e8 0.520494
\(285\) −1.08947e8 −0.278777
\(286\) 4.78550e7 0.120961
\(287\) 2.37026e7 0.0591846
\(288\) 8.60623e6 0.0212295
\(289\) −3.63024e8 −0.884694
\(290\) −4.93599e7 −0.118845
\(291\) −6.83322e8 −1.62555
\(292\) −2.60958e8 −0.613382
\(293\) 3.84004e8 0.891864 0.445932 0.895067i \(-0.352872\pi\)
0.445932 + 0.895067i \(0.352872\pi\)
\(294\) −3.25347e8 −0.746675
\(295\) 1.37573e8 0.312002
\(296\) 1.95651e8 0.438491
\(297\) −1.26770e8 −0.280781
\(298\) −2.83784e8 −0.621199
\(299\) −2.70821e8 −0.585912
\(300\) −2.12874e8 −0.455196
\(301\) 2.98851e7 0.0631643
\(302\) 3.28225e8 0.685720
\(303\) 3.26401e8 0.674066
\(304\) −8.62741e7 −0.176126
\(305\) 4.59999e7 0.0928341
\(306\) −1.44527e7 −0.0288353
\(307\) 7.46289e8 1.47205 0.736026 0.676954i \(-0.236700\pi\)
0.736026 + 0.676954i \(0.236700\pi\)
\(308\) 3.67024e6 0.00715760
\(309\) −9.83876e8 −1.89708
\(310\) −7.40092e7 −0.141098
\(311\) 6.40000e8 1.20648 0.603238 0.797561i \(-0.293877\pi\)
0.603238 + 0.797561i \(0.293877\pi\)
\(312\) 1.13889e8 0.212295
\(313\) 1.35452e8 0.249677 0.124839 0.992177i \(-0.460159\pi\)
0.124839 + 0.992177i \(0.460159\pi\)
\(314\) −5.33581e8 −0.972628
\(315\) 1.18261e6 0.00213184
\(316\) 1.21423e8 0.216468
\(317\) 9.75198e7 0.171943 0.0859717 0.996298i \(-0.472601\pi\)
0.0859717 + 0.996298i \(0.472601\pi\)
\(318\) −4.74407e8 −0.827287
\(319\) −7.85816e7 −0.135536
\(320\) 2.73957e7 0.0467366
\(321\) 1.03706e9 1.74999
\(322\) −2.07706e7 −0.0346700
\(323\) 1.44883e8 0.239226
\(324\) −3.38457e8 −0.552835
\(325\) −3.02031e8 −0.488045
\(326\) 7.43795e7 0.118903
\(327\) −2.96094e7 −0.0468288
\(328\) 2.81662e8 0.440728
\(329\) −5.43886e6 −0.00842020
\(330\) 5.50758e7 0.0843650
\(331\) 3.49842e8 0.530242 0.265121 0.964215i \(-0.414588\pi\)
0.265121 + 0.964215i \(0.414588\pi\)
\(332\) 4.15524e8 0.623179
\(333\) 1.00363e8 0.148943
\(334\) −4.74614e8 −0.696992
\(335\) 3.72425e8 0.541230
\(336\) 8.73470e6 0.0125621
\(337\) 7.20870e8 1.02601 0.513006 0.858385i \(-0.328532\pi\)
0.513006 + 0.858385i \(0.328532\pi\)
\(338\) −3.40400e8 −0.479492
\(339\) −7.01858e7 −0.0978476
\(340\) −4.60064e7 −0.0634808
\(341\) −1.17823e8 −0.160913
\(342\) −4.42562e7 −0.0598250
\(343\) −7.08865e7 −0.0948493
\(344\) 3.55130e8 0.470362
\(345\) −3.11684e8 −0.408647
\(346\) 9.59974e8 1.24593
\(347\) 1.28079e8 0.164560 0.0822798 0.996609i \(-0.473780\pi\)
0.0822798 + 0.996609i \(0.473780\pi\)
\(348\) −1.87014e8 −0.237874
\(349\) −1.00379e8 −0.126402 −0.0632012 0.998001i \(-0.520131\pi\)
−0.0632012 + 0.998001i \(0.520131\pi\)
\(350\) −2.31643e7 −0.0288789
\(351\) −4.28052e8 −0.528350
\(352\) 4.36142e7 0.0533002
\(353\) 7.51413e8 0.909216 0.454608 0.890692i \(-0.349779\pi\)
0.454608 + 0.890692i \(0.349779\pi\)
\(354\) 5.21235e8 0.624485
\(355\) 3.28089e8 0.389218
\(356\) 5.90741e8 0.693940
\(357\) −1.46685e7 −0.0170626
\(358\) 2.55115e8 0.293864
\(359\) −9.17064e8 −1.04609 −0.523045 0.852305i \(-0.675204\pi\)
−0.523045 + 0.852305i \(0.675204\pi\)
\(360\) 1.40532e7 0.0158751
\(361\) −4.50221e8 −0.503675
\(362\) 1.71729e8 0.190267
\(363\) 8.76814e7 0.0962131
\(364\) 1.23930e7 0.0134686
\(365\) −4.26121e8 −0.458678
\(366\) 1.74283e8 0.185811
\(367\) −9.16435e8 −0.967766 −0.483883 0.875133i \(-0.660774\pi\)
−0.483883 + 0.875133i \(0.660774\pi\)
\(368\) −2.46821e8 −0.258175
\(369\) 1.44485e8 0.149703
\(370\) 3.19480e8 0.327897
\(371\) −5.16234e7 −0.0524854
\(372\) −2.80405e8 −0.282413
\(373\) −1.56099e9 −1.55747 −0.778736 0.627352i \(-0.784139\pi\)
−0.778736 + 0.627352i \(0.784139\pi\)
\(374\) −7.32428e7 −0.0723960
\(375\) −7.51699e8 −0.736096
\(376\) −6.46311e7 −0.0627024
\(377\) −2.65340e8 −0.255040
\(378\) −3.28295e7 −0.0312639
\(379\) 4.34986e7 0.0410429 0.0205214 0.999789i \(-0.493467\pi\)
0.0205214 + 0.999789i \(0.493467\pi\)
\(380\) −1.40878e8 −0.131704
\(381\) −9.55174e8 −0.884800
\(382\) −4.29906e8 −0.394595
\(383\) 1.95422e8 0.177737 0.0888685 0.996043i \(-0.471675\pi\)
0.0888685 + 0.996043i \(0.471675\pi\)
\(384\) 1.03796e8 0.0935453
\(385\) 5.99317e6 0.00535235
\(386\) 1.37681e9 1.21848
\(387\) 1.82171e8 0.159769
\(388\) −8.83596e8 −0.767967
\(389\) −1.60846e9 −1.38543 −0.692716 0.721210i \(-0.743586\pi\)
−0.692716 + 0.721210i \(0.743586\pi\)
\(390\) 1.85970e8 0.158751
\(391\) 4.14495e8 0.350671
\(392\) −4.20704e8 −0.352756
\(393\) 2.09087e9 1.73761
\(394\) −9.16205e8 −0.754668
\(395\) 1.98272e8 0.161872
\(396\) 2.23728e7 0.0181046
\(397\) 1.20591e9 0.967268 0.483634 0.875270i \(-0.339317\pi\)
0.483634 + 0.875270i \(0.339317\pi\)
\(398\) −2.72515e8 −0.216670
\(399\) −4.49168e7 −0.0354000
\(400\) −2.75265e8 −0.215051
\(401\) 7.40689e8 0.573628 0.286814 0.957986i \(-0.407404\pi\)
0.286814 + 0.957986i \(0.407404\pi\)
\(402\) 1.41104e9 1.08330
\(403\) −3.97845e8 −0.302793
\(404\) 4.22066e8 0.318453
\(405\) −5.52669e8 −0.413402
\(406\) −2.03502e7 −0.0150914
\(407\) 5.08616e8 0.373947
\(408\) −1.74308e8 −0.127060
\(409\) 1.37187e7 0.00991476 0.00495738 0.999988i \(-0.498422\pi\)
0.00495738 + 0.999988i \(0.498422\pi\)
\(410\) 4.59929e8 0.329570
\(411\) −3.79067e8 −0.269321
\(412\) −1.27224e9 −0.896249
\(413\) 5.67190e7 0.0396190
\(414\) −1.26612e8 −0.0876948
\(415\) 6.78513e8 0.466004
\(416\) 1.47268e8 0.100296
\(417\) −7.22700e8 −0.488070
\(418\) −2.24279e8 −0.150201
\(419\) 1.95596e9 1.29901 0.649503 0.760359i \(-0.274977\pi\)
0.649503 + 0.760359i \(0.274977\pi\)
\(420\) 1.42630e7 0.00939372
\(421\) −2.31110e9 −1.50950 −0.754748 0.656015i \(-0.772241\pi\)
−0.754748 + 0.656015i \(0.772241\pi\)
\(422\) −1.39430e9 −0.903156
\(423\) −3.31539e7 −0.0212982
\(424\) −6.13451e8 −0.390840
\(425\) 4.62262e8 0.292097
\(426\) 1.24306e9 0.779037
\(427\) 1.89650e7 0.0117884
\(428\) 1.34101e9 0.826758
\(429\) 2.96066e8 0.181046
\(430\) 5.79895e8 0.351730
\(431\) −2.03521e9 −1.22444 −0.612222 0.790686i \(-0.709724\pi\)
−0.612222 + 0.790686i \(0.709724\pi\)
\(432\) −3.90119e8 −0.232811
\(433\) 2.81028e9 1.66357 0.831786 0.555096i \(-0.187318\pi\)
0.831786 + 0.555096i \(0.187318\pi\)
\(434\) −3.05127e7 −0.0179171
\(435\) −3.05377e8 −0.177878
\(436\) −3.82876e7 −0.0221236
\(437\) 1.26924e9 0.727541
\(438\) −1.61448e9 −0.918063
\(439\) 1.58225e9 0.892587 0.446293 0.894887i \(-0.352744\pi\)
0.446293 + 0.894887i \(0.352744\pi\)
\(440\) 7.12180e7 0.0398571
\(441\) −2.15809e8 −0.119821
\(442\) −2.47313e8 −0.136229
\(443\) −4.88989e8 −0.267231 −0.133615 0.991033i \(-0.542659\pi\)
−0.133615 + 0.991033i \(0.542659\pi\)
\(444\) 1.21044e9 0.656301
\(445\) 9.64626e8 0.518918
\(446\) 1.16584e9 0.622253
\(447\) −1.75569e9 −0.929763
\(448\) 1.12948e7 0.00593477
\(449\) 1.81072e9 0.944036 0.472018 0.881589i \(-0.343526\pi\)
0.472018 + 0.881589i \(0.343526\pi\)
\(450\) −1.41203e8 −0.0730468
\(451\) 7.32212e8 0.375854
\(452\) −9.07565e7 −0.0462267
\(453\) 2.03064e9 1.02633
\(454\) −1.01682e9 −0.509973
\(455\) 2.02366e7 0.0100716
\(456\) −5.33755e8 −0.263612
\(457\) −8.64311e8 −0.423607 −0.211804 0.977312i \(-0.567934\pi\)
−0.211804 + 0.977312i \(0.567934\pi\)
\(458\) −1.03800e9 −0.504857
\(459\) 6.55140e8 0.316220
\(460\) −4.03036e8 −0.193060
\(461\) 4.85430e8 0.230767 0.115383 0.993321i \(-0.463190\pi\)
0.115383 + 0.993321i \(0.463190\pi\)
\(462\) 2.27068e7 0.0107130
\(463\) 1.85376e9 0.867999 0.434000 0.900913i \(-0.357102\pi\)
0.434000 + 0.900913i \(0.357102\pi\)
\(464\) −2.41826e8 −0.112380
\(465\) −4.57875e8 −0.211184
\(466\) −1.21768e9 −0.557419
\(467\) 3.16192e7 0.0143662 0.00718310 0.999974i \(-0.497714\pi\)
0.00718310 + 0.999974i \(0.497714\pi\)
\(468\) 7.55445e7 0.0340677
\(469\) 1.53544e8 0.0687272
\(470\) −1.05537e8 −0.0468879
\(471\) −3.30112e9 −1.45576
\(472\) 6.74003e8 0.295029
\(473\) 9.23199e8 0.401126
\(474\) 7.51209e8 0.323994
\(475\) 1.41551e9 0.606017
\(476\) −1.89677e7 −0.00806101
\(477\) −3.14683e8 −0.132757
\(478\) −1.94449e9 −0.814343
\(479\) −3.11838e9 −1.29645 −0.648223 0.761451i \(-0.724487\pi\)
−0.648223 + 0.761451i \(0.724487\pi\)
\(480\) 1.69490e8 0.0699518
\(481\) 1.71740e9 0.703661
\(482\) −5.47674e8 −0.222770
\(483\) −1.28502e8 −0.0518914
\(484\) 1.13380e8 0.0454545
\(485\) −1.44283e9 −0.574275
\(486\) −4.27553e8 −0.168952
\(487\) −3.37608e9 −1.32453 −0.662265 0.749270i \(-0.730404\pi\)
−0.662265 + 0.749270i \(0.730404\pi\)
\(488\) 2.25364e8 0.0877840
\(489\) 4.60166e8 0.177965
\(490\) −6.86970e8 −0.263786
\(491\) −4.55043e9 −1.73487 −0.867436 0.497548i \(-0.834234\pi\)
−0.867436 + 0.497548i \(0.834234\pi\)
\(492\) 1.74257e9 0.659648
\(493\) 4.06106e8 0.152642
\(494\) −7.57304e8 −0.282635
\(495\) 3.65328e7 0.0135383
\(496\) −3.62588e8 −0.133422
\(497\) 1.35266e8 0.0494242
\(498\) 2.57074e9 0.932728
\(499\) −6.25733e8 −0.225443 −0.112722 0.993627i \(-0.535957\pi\)
−0.112722 + 0.993627i \(0.535957\pi\)
\(500\) −9.72014e8 −0.347758
\(501\) −2.93631e9 −1.04321
\(502\) 1.33834e9 0.472174
\(503\) −3.43240e9 −1.20257 −0.601285 0.799035i \(-0.705344\pi\)
−0.601285 + 0.799035i \(0.705344\pi\)
\(504\) 5.79389e6 0.00201587
\(505\) 6.89195e8 0.238135
\(506\) −6.41638e8 −0.220173
\(507\) −2.10596e9 −0.717667
\(508\) −1.23513e9 −0.418012
\(509\) 1.36035e9 0.457234 0.228617 0.973516i \(-0.426580\pi\)
0.228617 + 0.973516i \(0.426580\pi\)
\(510\) −2.84630e8 −0.0950133
\(511\) −1.75682e8 −0.0582445
\(512\) 1.34218e8 0.0441942
\(513\) 2.00613e9 0.656066
\(514\) 3.07831e9 0.999866
\(515\) −2.07745e9 −0.670202
\(516\) 2.19709e9 0.704003
\(517\) −1.68016e8 −0.0534727
\(518\) 1.31716e8 0.0416375
\(519\) 5.93910e9 1.86481
\(520\) 2.40476e8 0.0749997
\(521\) −2.51109e9 −0.777912 −0.388956 0.921256i \(-0.627164\pi\)
−0.388956 + 0.921256i \(0.627164\pi\)
\(522\) −1.24050e8 −0.0381724
\(523\) 4.25045e9 1.29921 0.649605 0.760272i \(-0.274934\pi\)
0.649605 + 0.760272i \(0.274934\pi\)
\(524\) 2.70368e9 0.820912
\(525\) −1.43311e8 −0.0432237
\(526\) 4.07567e9 1.22109
\(527\) 6.08907e8 0.181223
\(528\) 2.69829e8 0.0797757
\(529\) 2.26323e8 0.0664713
\(530\) −1.00171e9 −0.292265
\(531\) 3.45744e8 0.100213
\(532\) −5.80815e7 −0.0167243
\(533\) 2.47240e9 0.707250
\(534\) 3.65476e9 1.03864
\(535\) 2.18974e9 0.618237
\(536\) 1.82460e9 0.511788
\(537\) 1.57833e9 0.439833
\(538\) −4.41038e9 −1.22106
\(539\) −1.09366e9 −0.300832
\(540\) −6.37029e8 −0.174093
\(541\) 1.06279e9 0.288574 0.144287 0.989536i \(-0.453911\pi\)
0.144287 + 0.989536i \(0.453911\pi\)
\(542\) 2.73817e9 0.738690
\(543\) 1.06244e9 0.284778
\(544\) −2.25396e8 −0.0600276
\(545\) −6.25202e7 −0.0165437
\(546\) 7.66722e7 0.0201587
\(547\) −3.28564e9 −0.858350 −0.429175 0.903221i \(-0.641196\pi\)
−0.429175 + 0.903221i \(0.641196\pi\)
\(548\) −4.90168e8 −0.127237
\(549\) 1.15605e8 0.0298178
\(550\) −7.15582e8 −0.183396
\(551\) 1.24355e9 0.316689
\(552\) −1.52701e9 −0.386417
\(553\) 8.17441e7 0.0205550
\(554\) −4.08779e9 −1.02142
\(555\) 1.97654e9 0.490772
\(556\) −9.34516e8 −0.230582
\(557\) −4.04436e9 −0.991645 −0.495823 0.868424i \(-0.665133\pi\)
−0.495823 + 0.868424i \(0.665133\pi\)
\(558\) −1.85997e8 −0.0453197
\(559\) 3.11729e9 0.754806
\(560\) 1.84433e7 0.00443793
\(561\) −4.53133e8 −0.108357
\(562\) −1.21400e8 −0.0288498
\(563\) −1.60197e9 −0.378333 −0.189167 0.981945i \(-0.560579\pi\)
−0.189167 + 0.981945i \(0.560579\pi\)
\(564\) −3.99855e8 −0.0938481
\(565\) −1.48197e8 −0.0345677
\(566\) −1.80272e9 −0.417899
\(567\) −2.27856e8 −0.0524952
\(568\) 1.60739e9 0.368045
\(569\) −5.94461e9 −1.35279 −0.676396 0.736539i \(-0.736459\pi\)
−0.676396 + 0.736539i \(0.736459\pi\)
\(570\) −8.71573e8 −0.197125
\(571\) −1.01942e9 −0.229154 −0.114577 0.993414i \(-0.536551\pi\)
−0.114577 + 0.993414i \(0.536551\pi\)
\(572\) 3.82840e8 0.0855326
\(573\) −2.65971e9 −0.590600
\(574\) 1.89621e8 0.0418499
\(575\) 4.04961e9 0.888334
\(576\) 6.88499e7 0.0150115
\(577\) −5.05194e8 −0.109482 −0.0547410 0.998501i \(-0.517433\pi\)
−0.0547410 + 0.998501i \(0.517433\pi\)
\(578\) −2.90419e9 −0.625573
\(579\) 8.51792e9 1.82372
\(580\) −3.94879e8 −0.0840362
\(581\) 2.79739e8 0.0591748
\(582\) −5.46657e9 −1.14944
\(583\) −1.59473e9 −0.333310
\(584\) −2.08767e9 −0.433726
\(585\) 1.23357e8 0.0254753
\(586\) 3.07203e9 0.630643
\(587\) 6.05441e8 0.123549 0.0617744 0.998090i \(-0.480324\pi\)
0.0617744 + 0.998090i \(0.480324\pi\)
\(588\) −2.60278e9 −0.527979
\(589\) 1.86455e9 0.375985
\(590\) 1.10059e9 0.220618
\(591\) −5.66831e9 −1.12953
\(592\) 1.56521e9 0.310060
\(593\) −2.44359e8 −0.0481212 −0.0240606 0.999711i \(-0.507659\pi\)
−0.0240606 + 0.999711i \(0.507659\pi\)
\(594\) −1.01416e9 −0.198542
\(595\) −3.09725e7 −0.00602790
\(596\) −2.27027e9 −0.439254
\(597\) −1.68598e9 −0.324296
\(598\) −2.16656e9 −0.414302
\(599\) 4.76209e9 0.905324 0.452662 0.891682i \(-0.350474\pi\)
0.452662 + 0.891682i \(0.350474\pi\)
\(600\) −1.70299e9 −0.321872
\(601\) −4.27387e9 −0.803083 −0.401542 0.915841i \(-0.631525\pi\)
−0.401542 + 0.915841i \(0.631525\pi\)
\(602\) 2.39080e8 0.0446639
\(603\) 9.35966e8 0.173840
\(604\) 2.62580e9 0.484877
\(605\) 1.85139e8 0.0339902
\(606\) 2.61121e9 0.476637
\(607\) 6.40384e9 1.16220 0.581099 0.813833i \(-0.302623\pi\)
0.581099 + 0.813833i \(0.302623\pi\)
\(608\) −6.90193e8 −0.124540
\(609\) −1.25901e8 −0.0225876
\(610\) 3.67999e8 0.0656436
\(611\) −5.67324e8 −0.100621
\(612\) −1.15622e8 −0.0203897
\(613\) −8.98899e9 −1.57616 −0.788078 0.615575i \(-0.788924\pi\)
−0.788078 + 0.615575i \(0.788924\pi\)
\(614\) 5.97032e9 1.04090
\(615\) 2.84546e9 0.493275
\(616\) 2.93620e7 0.00506119
\(617\) −2.11391e9 −0.362316 −0.181158 0.983454i \(-0.557985\pi\)
−0.181158 + 0.983454i \(0.557985\pi\)
\(618\) −7.87100e9 −1.34144
\(619\) −4.25875e9 −0.721713 −0.360856 0.932621i \(-0.617516\pi\)
−0.360856 + 0.932621i \(0.617516\pi\)
\(620\) −5.92074e8 −0.0997711
\(621\) 5.73930e9 0.961698
\(622\) 5.12000e9 0.853107
\(623\) 3.97699e8 0.0658940
\(624\) 9.11110e8 0.150115
\(625\) 3.66306e9 0.600156
\(626\) 1.08361e9 0.176548
\(627\) −1.38755e9 −0.224809
\(628\) −4.26865e9 −0.687752
\(629\) −2.62850e9 −0.421145
\(630\) 9.46089e6 0.00150744
\(631\) 9.55885e9 1.51462 0.757309 0.653057i \(-0.226514\pi\)
0.757309 + 0.653057i \(0.226514\pi\)
\(632\) 9.71380e8 0.153066
\(633\) −8.62617e9 −1.35178
\(634\) 7.80159e8 0.121582
\(635\) −2.01685e9 −0.312583
\(636\) −3.79526e9 −0.584980
\(637\) −3.69289e9 −0.566080
\(638\) −6.28652e8 −0.0958381
\(639\) 8.24543e8 0.125015
\(640\) 2.19165e8 0.0330477
\(641\) 7.93190e9 1.18953 0.594763 0.803901i \(-0.297246\pi\)
0.594763 + 0.803901i \(0.297246\pi\)
\(642\) 8.29646e9 1.23743
\(643\) −3.67790e9 −0.545583 −0.272792 0.962073i \(-0.587947\pi\)
−0.272792 + 0.962073i \(0.587947\pi\)
\(644\) −1.66165e8 −0.0245154
\(645\) 3.58765e9 0.526443
\(646\) 1.15906e9 0.169158
\(647\) −3.34066e9 −0.484917 −0.242459 0.970162i \(-0.577954\pi\)
−0.242459 + 0.970162i \(0.577954\pi\)
\(648\) −2.70765e9 −0.390913
\(649\) 1.75215e9 0.251602
\(650\) −2.41625e9 −0.345100
\(651\) −1.88774e8 −0.0268169
\(652\) 5.95036e8 0.0840769
\(653\) 1.23649e10 1.73778 0.868891 0.495004i \(-0.164833\pi\)
0.868891 + 0.495004i \(0.164833\pi\)
\(654\) −2.36875e8 −0.0331129
\(655\) 4.41487e9 0.613865
\(656\) 2.25330e9 0.311642
\(657\) −1.07091e9 −0.147325
\(658\) −4.35109e7 −0.00595398
\(659\) 8.58087e9 1.16797 0.583986 0.811764i \(-0.301492\pi\)
0.583986 + 0.811764i \(0.301492\pi\)
\(660\) 4.40607e8 0.0596551
\(661\) −8.09007e9 −1.08955 −0.544775 0.838582i \(-0.683385\pi\)
−0.544775 + 0.838582i \(0.683385\pi\)
\(662\) 2.79873e9 0.374937
\(663\) −1.53006e9 −0.203897
\(664\) 3.32419e9 0.440654
\(665\) −9.48417e7 −0.0125062
\(666\) 8.02907e8 0.105319
\(667\) 3.55766e9 0.464220
\(668\) −3.79691e9 −0.492848
\(669\) 7.21274e9 0.931341
\(670\) 2.97940e9 0.382707
\(671\) 5.85859e8 0.0748625
\(672\) 6.98776e7 0.00888271
\(673\) −1.38929e10 −1.75687 −0.878436 0.477860i \(-0.841413\pi\)
−0.878436 + 0.477860i \(0.841413\pi\)
\(674\) 5.76696e9 0.725500
\(675\) 6.40072e9 0.801061
\(676\) −2.72320e9 −0.339052
\(677\) 6.36018e9 0.787788 0.393894 0.919156i \(-0.371128\pi\)
0.393894 + 0.919156i \(0.371128\pi\)
\(678\) −5.61486e8 −0.0691887
\(679\) −5.94855e8 −0.0729233
\(680\) −3.68052e8 −0.0448877
\(681\) −6.29077e9 −0.763289
\(682\) −9.42588e8 −0.113783
\(683\) −7.46187e9 −0.896139 −0.448069 0.893999i \(-0.647888\pi\)
−0.448069 + 0.893999i \(0.647888\pi\)
\(684\) −3.54049e8 −0.0423026
\(685\) −8.00400e8 −0.0951459
\(686\) −5.67092e8 −0.0670686
\(687\) −6.42182e9 −0.755631
\(688\) 2.84104e9 0.332596
\(689\) −5.38480e9 −0.627195
\(690\) −2.49348e9 −0.288957
\(691\) −1.01338e10 −1.16842 −0.584212 0.811601i \(-0.698596\pi\)
−0.584212 + 0.811601i \(0.698596\pi\)
\(692\) 7.67979e9 0.881004
\(693\) 1.50618e7 0.00171914
\(694\) 1.02463e9 0.116361
\(695\) −1.52598e9 −0.172426
\(696\) −1.49611e9 −0.168202
\(697\) −3.78404e9 −0.423293
\(698\) −8.03034e8 −0.0893800
\(699\) −7.53345e9 −0.834303
\(700\) −1.85314e8 −0.0204205
\(701\) 1.31006e10 1.43641 0.718206 0.695831i \(-0.244964\pi\)
0.718206 + 0.695831i \(0.244964\pi\)
\(702\) −3.42442e9 −0.373600
\(703\) −8.04883e9 −0.873754
\(704\) 3.48914e8 0.0376889
\(705\) −6.52927e8 −0.0701782
\(706\) 6.01130e9 0.642913
\(707\) 2.84143e8 0.0302391
\(708\) 4.16988e9 0.441577
\(709\) −1.04601e10 −1.10223 −0.551116 0.834429i \(-0.685798\pi\)
−0.551116 + 0.834429i \(0.685798\pi\)
\(710\) 2.62471e9 0.275219
\(711\) 4.98290e8 0.0519923
\(712\) 4.72593e9 0.490690
\(713\) 5.33428e9 0.551141
\(714\) −1.17348e8 −0.0120651
\(715\) 6.25143e8 0.0639600
\(716\) 2.04092e9 0.207793
\(717\) −1.20300e10 −1.21885
\(718\) −7.33651e9 −0.739697
\(719\) 6.14452e9 0.616505 0.308253 0.951305i \(-0.400256\pi\)
0.308253 + 0.951305i \(0.400256\pi\)
\(720\) 1.12426e8 0.0112254
\(721\) −8.56497e8 −0.0851045
\(722\) −3.60177e9 −0.356152
\(723\) −3.38831e9 −0.333426
\(724\) 1.37383e9 0.134539
\(725\) 3.96766e9 0.386680
\(726\) 7.01451e8 0.0680329
\(727\) −1.47022e9 −0.141910 −0.0709550 0.997480i \(-0.522605\pi\)
−0.0709550 + 0.997480i \(0.522605\pi\)
\(728\) 9.91440e7 0.00952372
\(729\) 8.92054e9 0.852796
\(730\) −3.40897e9 −0.324334
\(731\) −4.77105e9 −0.451755
\(732\) 1.39427e9 0.131389
\(733\) 6.30989e8 0.0591777 0.0295888 0.999562i \(-0.490580\pi\)
0.0295888 + 0.999562i \(0.490580\pi\)
\(734\) −7.33148e9 −0.684314
\(735\) −4.25010e9 −0.394815
\(736\) −1.97457e9 −0.182557
\(737\) 4.74324e9 0.436454
\(738\) 1.15588e9 0.105856
\(739\) 7.22368e9 0.658420 0.329210 0.944257i \(-0.393218\pi\)
0.329210 + 0.944257i \(0.393218\pi\)
\(740\) 2.55584e9 0.231858
\(741\) −4.68524e9 −0.423027
\(742\) −4.12987e8 −0.0371128
\(743\) 7.72789e9 0.691194 0.345597 0.938383i \(-0.387676\pi\)
0.345597 + 0.938383i \(0.387676\pi\)
\(744\) −2.24324e9 −0.199696
\(745\) −3.70714e9 −0.328467
\(746\) −1.24880e10 −1.10130
\(747\) 1.70522e9 0.149678
\(748\) −5.85942e8 −0.0511917
\(749\) 9.02793e8 0.0785059
\(750\) −6.01359e9 −0.520498
\(751\) −1.54417e10 −1.33032 −0.665158 0.746702i \(-0.731636\pi\)
−0.665158 + 0.746702i \(0.731636\pi\)
\(752\) −5.17048e8 −0.0443373
\(753\) 8.27993e9 0.706715
\(754\) −2.12272e9 −0.180340
\(755\) 4.28769e9 0.362584
\(756\) −2.62636e8 −0.0221069
\(757\) 1.94159e10 1.62676 0.813378 0.581735i \(-0.197626\pi\)
0.813378 + 0.581735i \(0.197626\pi\)
\(758\) 3.47989e8 0.0290217
\(759\) −3.96964e9 −0.329538
\(760\) −1.12702e9 −0.0931290
\(761\) 5.03230e8 0.0413924 0.0206962 0.999786i \(-0.493412\pi\)
0.0206962 + 0.999786i \(0.493412\pi\)
\(762\) −7.64139e9 −0.625648
\(763\) −2.57760e7 −0.00210078
\(764\) −3.43925e9 −0.279021
\(765\) −1.88800e8 −0.0152471
\(766\) 1.56338e9 0.125679
\(767\) 5.91632e9 0.473443
\(768\) 8.30369e8 0.0661465
\(769\) −1.87701e10 −1.48842 −0.744208 0.667948i \(-0.767173\pi\)
−0.744208 + 0.667948i \(0.767173\pi\)
\(770\) 4.79454e7 0.00378468
\(771\) 1.90447e10 1.49652
\(772\) 1.10144e10 0.861593
\(773\) 2.39166e10 1.86239 0.931196 0.364518i \(-0.118766\pi\)
0.931196 + 0.364518i \(0.118766\pi\)
\(774\) 1.45737e9 0.112974
\(775\) 5.94902e9 0.459081
\(776\) −7.06877e9 −0.543035
\(777\) 8.14892e8 0.0623199
\(778\) −1.28676e10 −0.979649
\(779\) −1.15872e10 −0.878210
\(780\) 1.48776e9 0.112254
\(781\) 4.17858e9 0.313870
\(782\) 3.31596e9 0.247962
\(783\) 5.62315e9 0.418614
\(784\) −3.36563e9 −0.249436
\(785\) −6.97032e9 −0.514291
\(786\) 1.67270e10 1.22868
\(787\) 2.09777e10 1.53408 0.767038 0.641602i \(-0.221730\pi\)
0.767038 + 0.641602i \(0.221730\pi\)
\(788\) −7.32964e9 −0.533631
\(789\) 2.52151e10 1.82764
\(790\) 1.58618e9 0.114461
\(791\) −6.10991e7 −0.00438952
\(792\) 1.78983e8 0.0128019
\(793\) 1.97822e9 0.140870
\(794\) 9.64725e9 0.683962
\(795\) −6.19731e9 −0.437440
\(796\) −2.18012e9 −0.153209
\(797\) 1.55513e10 1.08809 0.544043 0.839057i \(-0.316893\pi\)
0.544043 + 0.839057i \(0.316893\pi\)
\(798\) −3.59334e8 −0.0250316
\(799\) 8.68297e8 0.0602219
\(800\) −2.20212e9 −0.152064
\(801\) 2.42427e9 0.166673
\(802\) 5.92551e9 0.405617
\(803\) −5.42711e9 −0.369883
\(804\) 1.12883e10 0.766005
\(805\) −2.71332e8 −0.0183322
\(806\) −3.18276e9 −0.214107
\(807\) −2.72858e10 −1.82759
\(808\) 3.37653e9 0.225180
\(809\) −1.19959e10 −0.796548 −0.398274 0.917266i \(-0.630391\pi\)
−0.398274 + 0.917266i \(0.630391\pi\)
\(810\) −4.42135e9 −0.292319
\(811\) 1.18093e10 0.777412 0.388706 0.921362i \(-0.372922\pi\)
0.388706 + 0.921362i \(0.372922\pi\)
\(812\) −1.62802e8 −0.0106712
\(813\) 1.69403e10 1.10562
\(814\) 4.06893e9 0.264420
\(815\) 9.71640e8 0.0628715
\(816\) −1.39447e9 −0.0898447
\(817\) −1.46096e10 −0.937262
\(818\) 1.09750e8 0.00701079
\(819\) 5.08580e7 0.00323494
\(820\) 3.67943e9 0.233041
\(821\) −1.96308e10 −1.23805 −0.619024 0.785372i \(-0.712472\pi\)
−0.619024 + 0.785372i \(0.712472\pi\)
\(822\) −3.03254e9 −0.190439
\(823\) −2.67601e9 −0.167336 −0.0836679 0.996494i \(-0.526663\pi\)
−0.0836679 + 0.996494i \(0.526663\pi\)
\(824\) −1.01779e10 −0.633744
\(825\) −4.42712e9 −0.274494
\(826\) 4.53752e8 0.0280149
\(827\) 9.26722e8 0.0569745 0.0284872 0.999594i \(-0.490931\pi\)
0.0284872 + 0.999594i \(0.490931\pi\)
\(828\) −1.01290e9 −0.0620096
\(829\) −2.94467e10 −1.79513 −0.897565 0.440882i \(-0.854666\pi\)
−0.897565 + 0.440882i \(0.854666\pi\)
\(830\) 5.42811e9 0.329515
\(831\) −2.52900e10 −1.52878
\(832\) 1.17815e9 0.0709199
\(833\) 5.65201e9 0.338802
\(834\) −5.78160e9 −0.345117
\(835\) −6.20001e9 −0.368544
\(836\) −1.79423e9 −0.106208
\(837\) 8.43123e9 0.496995
\(838\) 1.56477e10 0.918536
\(839\) −8.93988e9 −0.522595 −0.261297 0.965258i \(-0.584150\pi\)
−0.261297 + 0.965258i \(0.584150\pi\)
\(840\) 1.14104e8 0.00664236
\(841\) −1.37642e10 −0.797931
\(842\) −1.84888e10 −1.06737
\(843\) −7.51071e8 −0.0431802
\(844\) −1.11544e10 −0.638628
\(845\) −4.44674e9 −0.253538
\(846\) −2.65231e8 −0.0150601
\(847\) 7.63296e7 0.00431619
\(848\) −4.90761e9 −0.276366
\(849\) −1.11530e10 −0.625480
\(850\) 3.69810e9 0.206544
\(851\) −2.30268e10 −1.28080
\(852\) 9.94447e9 0.550862
\(853\) 1.71708e10 0.947257 0.473628 0.880725i \(-0.342944\pi\)
0.473628 + 0.880725i \(0.342944\pi\)
\(854\) 1.51720e8 0.00833565
\(855\) −5.78130e8 −0.0316333
\(856\) 1.07281e10 0.584606
\(857\) −7.71458e9 −0.418677 −0.209339 0.977843i \(-0.567131\pi\)
−0.209339 + 0.977843i \(0.567131\pi\)
\(858\) 2.36853e9 0.128019
\(859\) 1.68606e10 0.907606 0.453803 0.891102i \(-0.350067\pi\)
0.453803 + 0.891102i \(0.350067\pi\)
\(860\) 4.63916e9 0.248711
\(861\) 1.17313e9 0.0626377
\(862\) −1.62817e10 −0.865813
\(863\) −2.42922e10 −1.28656 −0.643278 0.765633i \(-0.722426\pi\)
−0.643278 + 0.765633i \(0.722426\pi\)
\(864\) −3.12095e9 −0.164622
\(865\) 1.25404e10 0.658802
\(866\) 2.24822e10 1.17632
\(867\) −1.79675e10 −0.936311
\(868\) −2.44102e8 −0.0126693
\(869\) 2.52521e9 0.130535
\(870\) −2.44301e9 −0.125779
\(871\) 1.60161e10 0.821283
\(872\) −3.06301e8 −0.0156438
\(873\) −3.62608e9 −0.184454
\(874\) 1.01539e10 0.514449
\(875\) −6.54379e8 −0.0330218
\(876\) −1.29158e10 −0.649169
\(877\) 2.67740e10 1.34034 0.670169 0.742209i \(-0.266222\pi\)
0.670169 + 0.742209i \(0.266222\pi\)
\(878\) 1.26580e10 0.631154
\(879\) 1.90058e10 0.943899
\(880\) 5.69744e8 0.0281832
\(881\) 1.10654e10 0.545197 0.272598 0.962128i \(-0.412117\pi\)
0.272598 + 0.962128i \(0.412117\pi\)
\(882\) −1.72647e9 −0.0847265
\(883\) 2.06264e10 1.00824 0.504118 0.863635i \(-0.331818\pi\)
0.504118 + 0.863635i \(0.331818\pi\)
\(884\) −1.97850e9 −0.0963282
\(885\) 6.80903e9 0.330205
\(886\) −3.91191e9 −0.188961
\(887\) −7.62436e9 −0.366835 −0.183417 0.983035i \(-0.558716\pi\)
−0.183417 + 0.983035i \(0.558716\pi\)
\(888\) 9.68352e9 0.464075
\(889\) −8.31512e8 −0.0396928
\(890\) 7.71701e9 0.366931
\(891\) −7.03884e9 −0.333372
\(892\) 9.32672e9 0.439999
\(893\) 2.65884e9 0.124943
\(894\) −1.40455e10 −0.657442
\(895\) 3.33264e9 0.155385
\(896\) 9.03580e7 0.00419651
\(897\) −1.34040e10 −0.620097
\(898\) 1.44857e10 0.667534
\(899\) 5.22633e9 0.239904
\(900\) −1.12963e9 −0.0516519
\(901\) 8.24151e9 0.375379
\(902\) 5.85770e9 0.265769
\(903\) 1.47913e9 0.0668495
\(904\) −7.26052e8 −0.0326872
\(905\) 2.24334e9 0.100606
\(906\) 1.62451e10 0.725728
\(907\) −3.36091e10 −1.49566 −0.747828 0.663893i \(-0.768903\pi\)
−0.747828 + 0.663893i \(0.768903\pi\)
\(908\) −8.13454e9 −0.360606
\(909\) 1.73206e9 0.0764874
\(910\) 1.61893e8 0.00712170
\(911\) −3.89170e10 −1.70540 −0.852698 0.522404i \(-0.825035\pi\)
−0.852698 + 0.522404i \(0.825035\pi\)
\(912\) −4.27004e9 −0.186402
\(913\) 8.64161e9 0.375791
\(914\) −6.91449e9 −0.299536
\(915\) 2.27671e9 0.0982504
\(916\) −8.30400e9 −0.356987
\(917\) 1.82017e9 0.0779507
\(918\) 5.24112e9 0.223601
\(919\) 4.35993e10 1.85300 0.926499 0.376296i \(-0.122803\pi\)
0.926499 + 0.376296i \(0.122803\pi\)
\(920\) −3.22429e9 −0.136514
\(921\) 3.69367e10 1.55794
\(922\) 3.88344e9 0.163177
\(923\) 1.41094e10 0.590614
\(924\) 1.81655e8 0.00757520
\(925\) −2.56805e10 −1.06686
\(926\) 1.48301e10 0.613768
\(927\) −5.22098e9 −0.215265
\(928\) −1.93461e9 −0.0794648
\(929\) −3.06911e10 −1.25591 −0.627954 0.778250i \(-0.716108\pi\)
−0.627954 + 0.778250i \(0.716108\pi\)
\(930\) −3.66300e9 −0.149330
\(931\) 1.73072e10 0.702915
\(932\) −9.74143e9 −0.394155
\(933\) 3.16761e10 1.27687
\(934\) 2.52954e8 0.0101584
\(935\) −9.56790e8 −0.0382804
\(936\) 6.04356e8 0.0240895
\(937\) 3.51153e9 0.139446 0.0697232 0.997566i \(-0.477788\pi\)
0.0697232 + 0.997566i \(0.477788\pi\)
\(938\) 1.22835e9 0.0485975
\(939\) 6.70402e9 0.264244
\(940\) −8.44293e8 −0.0331547
\(941\) −1.84266e10 −0.720911 −0.360455 0.932776i \(-0.617379\pi\)
−0.360455 + 0.932776i \(0.617379\pi\)
\(942\) −2.64090e10 −1.02938
\(943\) −3.31498e10 −1.28733
\(944\) 5.39203e9 0.208617
\(945\) −4.28860e8 −0.0165312
\(946\) 7.38559e9 0.283639
\(947\) 8.78798e9 0.336251 0.168126 0.985766i \(-0.446229\pi\)
0.168126 + 0.985766i \(0.446229\pi\)
\(948\) 6.00967e9 0.229098
\(949\) −1.83253e10 −0.696015
\(950\) 1.13241e10 0.428519
\(951\) 4.82663e9 0.181975
\(952\) −1.51741e8 −0.00569999
\(953\) 1.85304e10 0.693521 0.346761 0.937954i \(-0.387282\pi\)
0.346761 + 0.937954i \(0.387282\pi\)
\(954\) −2.51746e9 −0.0938737
\(955\) −5.61598e9 −0.208648
\(956\) −1.55559e10 −0.575827
\(957\) −3.88930e9 −0.143443
\(958\) −2.49470e10 −0.916725
\(959\) −3.29991e8 −0.0120819
\(960\) 1.35592e9 0.0494634
\(961\) −1.96764e10 −0.715176
\(962\) 1.37392e10 0.497564
\(963\) 5.50319e9 0.198574
\(964\) −4.38139e9 −0.157522
\(965\) 1.79856e10 0.644286
\(966\) −1.02802e9 −0.0366927
\(967\) 4.71498e10 1.67682 0.838412 0.545037i \(-0.183484\pi\)
0.838412 + 0.545037i \(0.183484\pi\)
\(968\) 9.07039e8 0.0321412
\(969\) 7.17082e9 0.253184
\(970\) −1.15427e10 −0.406073
\(971\) −8.76215e9 −0.307145 −0.153572 0.988137i \(-0.549078\pi\)
−0.153572 + 0.988137i \(0.549078\pi\)
\(972\) −3.42042e9 −0.119467
\(973\) −6.29135e8 −0.0218952
\(974\) −2.70086e10 −0.936583
\(975\) −1.49487e10 −0.516519
\(976\) 1.80291e9 0.0620727
\(977\) −6.33661e9 −0.217383 −0.108692 0.994076i \(-0.534666\pi\)
−0.108692 + 0.994076i \(0.534666\pi\)
\(978\) 3.68133e9 0.125840
\(979\) 1.22856e10 0.418462
\(980\) −5.49576e9 −0.186525
\(981\) −1.57124e8 −0.00531374
\(982\) −3.64035e10 −1.22674
\(983\) 4.47312e10 1.50201 0.751006 0.660295i \(-0.229569\pi\)
0.751006 + 0.660295i \(0.229569\pi\)
\(984\) 1.39406e10 0.466442
\(985\) −1.19686e10 −0.399041
\(986\) 3.24885e9 0.107934
\(987\) −2.69190e8 −0.00891147
\(988\) −6.05843e9 −0.199853
\(989\) −4.17964e10 −1.37389
\(990\) 2.92262e8 0.00957304
\(991\) 4.62109e10 1.50830 0.754148 0.656705i \(-0.228050\pi\)
0.754148 + 0.656705i \(0.228050\pi\)
\(992\) −2.90071e9 −0.0943437
\(993\) 1.73150e10 0.561178
\(994\) 1.08212e9 0.0349482
\(995\) −3.55994e9 −0.114567
\(996\) 2.05659e10 0.659538
\(997\) −1.52392e10 −0.487001 −0.243501 0.969901i \(-0.578296\pi\)
−0.243501 + 0.969901i \(0.578296\pi\)
\(998\) −5.00587e9 −0.159413
\(999\) −3.63956e10 −1.15497
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.1.2 2
3.2 odd 2 198.8.a.f.1.2 2
4.3 odd 2 176.8.a.e.1.1 2
5.4 even 2 550.8.a.d.1.1 2
11.10 odd 2 242.8.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
22.8.a.d.1.2 2 1.1 even 1 trivial
176.8.a.e.1.1 2 4.3 odd 2
198.8.a.f.1.2 2 3.2 odd 2
242.8.a.h.1.2 2 11.10 odd 2
550.8.a.d.1.1 2 5.4 even 2