Properties

Label 546.8.a.j.1.2
Level $546$
Weight $8$
Character 546.1
Self dual yes
Analytic conductor $170.562$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(170.562223914\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 5672 x^{3} - 117684 x^{2} + 1695035 x + 39011360\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-20.7333\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -251.667 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -251.667 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +2013.33 q^{10} +6935.03 q^{11} +1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} -6795.00 q^{15} +4096.00 q^{16} +3533.29 q^{17} -5832.00 q^{18} +5190.95 q^{19} -16106.7 q^{20} -9261.00 q^{21} -55480.2 q^{22} -40997.6 q^{23} -13824.0 q^{24} -14788.9 q^{25} +17576.0 q^{26} +19683.0 q^{27} -21952.0 q^{28} -70252.8 q^{29} +54360.0 q^{30} -288221. q^{31} -32768.0 q^{32} +187246. q^{33} -28266.3 q^{34} +86321.7 q^{35} +46656.0 q^{36} +288649. q^{37} -41527.6 q^{38} -59319.0 q^{39} +128853. q^{40} +8236.51 q^{41} +74088.0 q^{42} +284453. q^{43} +443842. q^{44} -183465. q^{45} +327981. q^{46} +254392. q^{47} +110592. q^{48} +117649. q^{49} +118311. q^{50} +95398.8 q^{51} -140608. q^{52} +679737. q^{53} -157464. q^{54} -1.74532e6 q^{55} +175616. q^{56} +140156. q^{57} +562022. q^{58} +1.06690e6 q^{59} -434880. q^{60} +2.93017e6 q^{61} +2.30577e6 q^{62} -250047. q^{63} +262144. q^{64} +552912. q^{65} -1.49797e6 q^{66} +1.20866e6 q^{67} +226131. q^{68} -1.10694e6 q^{69} -690573. q^{70} -2.99805e6 q^{71} -373248. q^{72} -1.35994e6 q^{73} -2.30919e6 q^{74} -399300. q^{75} +332221. q^{76} -2.37872e6 q^{77} +474552. q^{78} -565403. q^{79} -1.03083e6 q^{80} +531441. q^{81} -65892.1 q^{82} +2.10144e6 q^{83} -592704. q^{84} -889211. q^{85} -2.27563e6 q^{86} -1.89682e6 q^{87} -3.55074e6 q^{88} +6.47234e6 q^{89} +1.46772e6 q^{90} +753571. q^{91} -2.62385e6 q^{92} -7.78196e6 q^{93} -2.03513e6 q^{94} -1.30639e6 q^{95} -884736. q^{96} -4.68279e6 q^{97} -941192. q^{98} +5.05564e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 56 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + O(q^{10}) \) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 56 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} - 448 q^{10} - 3679 q^{11} + 8640 q^{12} - 10985 q^{13} + 13720 q^{14} + 1512 q^{15} + 20480 q^{16} + 409 q^{17} - 29160 q^{18} + 33730 q^{19} + 3584 q^{20} - 46305 q^{21} + 29432 q^{22} - 142142 q^{23} - 69120 q^{24} + 153981 q^{25} + 87880 q^{26} + 98415 q^{27} - 109760 q^{28} + 88028 q^{29} - 12096 q^{30} + 244543 q^{31} - 163840 q^{32} - 99333 q^{33} - 3272 q^{34} - 19208 q^{35} + 233280 q^{36} + 730963 q^{37} - 269840 q^{38} - 296595 q^{39} - 28672 q^{40} + 479512 q^{41} + 370440 q^{42} - 406536 q^{43} - 235456 q^{44} + 40824 q^{45} + 1137136 q^{46} + 1138945 q^{47} + 552960 q^{48} + 588245 q^{49} - 1231848 q^{50} + 11043 q^{51} - 703040 q^{52} + 297595 q^{53} - 787320 q^{54} - 1834423 q^{55} + 878080 q^{56} + 910710 q^{57} - 704224 q^{58} + 941652 q^{59} + 96768 q^{60} - 2985259 q^{61} - 1956344 q^{62} - 1250235 q^{63} + 1310720 q^{64} - 123032 q^{65} + 794664 q^{66} - 2333504 q^{67} + 26176 q^{68} - 3837834 q^{69} + 153664 q^{70} - 11322272 q^{71} - 1866240 q^{72} - 6631604 q^{73} - 5847704 q^{74} + 4157487 q^{75} + 2158720 q^{76} + 1261897 q^{77} + 2372760 q^{78} - 10600265 q^{79} + 229376 q^{80} + 2657205 q^{81} - 3836096 q^{82} - 2425229 q^{83} - 2963520 q^{84} - 12267705 q^{85} + 3252288 q^{86} + 2376756 q^{87} + 1883648 q^{88} - 1581837 q^{89} - 326592 q^{90} + 3767855 q^{91} - 9097088 q^{92} + 6602661 q^{93} - 9111560 q^{94} - 11507718 q^{95} - 4423680 q^{96} + 5298407 q^{97} - 4705960 q^{98} - 2681991 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.00000 −0.707107
\(3\) 27.0000 0.577350
\(4\) 64.0000 0.500000
\(5\) −251.667 −0.900390 −0.450195 0.892930i \(-0.648646\pi\)
−0.450195 + 0.892930i \(0.648646\pi\)
\(6\) −216.000 −0.408248
\(7\) −343.000 −0.377964
\(8\) −512.000 −0.353553
\(9\) 729.000 0.333333
\(10\) 2013.33 0.636672
\(11\) 6935.03 1.57099 0.785496 0.618867i \(-0.212408\pi\)
0.785496 + 0.618867i \(0.212408\pi\)
\(12\) 1728.00 0.288675
\(13\) −2197.00 −0.277350
\(14\) 2744.00 0.267261
\(15\) −6795.00 −0.519840
\(16\) 4096.00 0.250000
\(17\) 3533.29 0.174425 0.0872124 0.996190i \(-0.472204\pi\)
0.0872124 + 0.996190i \(0.472204\pi\)
\(18\) −5832.00 −0.235702
\(19\) 5190.95 0.173624 0.0868118 0.996225i \(-0.472332\pi\)
0.0868118 + 0.996225i \(0.472332\pi\)
\(20\) −16106.7 −0.450195
\(21\) −9261.00 −0.218218
\(22\) −55480.2 −1.11086
\(23\) −40997.6 −0.702605 −0.351303 0.936262i \(-0.614261\pi\)
−0.351303 + 0.936262i \(0.614261\pi\)
\(24\) −13824.0 −0.204124
\(25\) −14788.9 −0.189298
\(26\) 17576.0 0.196116
\(27\) 19683.0 0.192450
\(28\) −21952.0 −0.188982
\(29\) −70252.8 −0.534897 −0.267449 0.963572i \(-0.586181\pi\)
−0.267449 + 0.963572i \(0.586181\pi\)
\(30\) 54360.0 0.367583
\(31\) −288221. −1.73764 −0.868819 0.495130i \(-0.835120\pi\)
−0.868819 + 0.495130i \(0.835120\pi\)
\(32\) −32768.0 −0.176777
\(33\) 187246. 0.907013
\(34\) −28266.3 −0.123337
\(35\) 86321.7 0.340315
\(36\) 46656.0 0.166667
\(37\) 288649. 0.936837 0.468418 0.883507i \(-0.344824\pi\)
0.468418 + 0.883507i \(0.344824\pi\)
\(38\) −41527.6 −0.122770
\(39\) −59319.0 −0.160128
\(40\) 128853. 0.318336
\(41\) 8236.51 0.0186638 0.00933190 0.999956i \(-0.497030\pi\)
0.00933190 + 0.999956i \(0.497030\pi\)
\(42\) 74088.0 0.154303
\(43\) 284453. 0.545597 0.272798 0.962071i \(-0.412051\pi\)
0.272798 + 0.962071i \(0.412051\pi\)
\(44\) 443842. 0.785496
\(45\) −183465. −0.300130
\(46\) 327981. 0.496817
\(47\) 254392. 0.357405 0.178702 0.983903i \(-0.442810\pi\)
0.178702 + 0.983903i \(0.442810\pi\)
\(48\) 110592. 0.144338
\(49\) 117649. 0.142857
\(50\) 118311. 0.133854
\(51\) 95398.8 0.100704
\(52\) −140608. −0.138675
\(53\) 679737. 0.627155 0.313578 0.949563i \(-0.398472\pi\)
0.313578 + 0.949563i \(0.398472\pi\)
\(54\) −157464. −0.136083
\(55\) −1.74532e6 −1.41451
\(56\) 175616. 0.133631
\(57\) 140156. 0.100242
\(58\) 562022. 0.378229
\(59\) 1.06690e6 0.676304 0.338152 0.941092i \(-0.390198\pi\)
0.338152 + 0.941092i \(0.390198\pi\)
\(60\) −434880. −0.259920
\(61\) 2.93017e6 1.65287 0.826435 0.563032i \(-0.190365\pi\)
0.826435 + 0.563032i \(0.190365\pi\)
\(62\) 2.30577e6 1.22870
\(63\) −250047. −0.125988
\(64\) 262144. 0.125000
\(65\) 552912. 0.249723
\(66\) −1.49797e6 −0.641355
\(67\) 1.20866e6 0.490957 0.245479 0.969402i \(-0.421055\pi\)
0.245479 + 0.969402i \(0.421055\pi\)
\(68\) 226131. 0.0872124
\(69\) −1.10694e6 −0.405649
\(70\) −690573. −0.240639
\(71\) −2.99805e6 −0.994111 −0.497056 0.867719i \(-0.665585\pi\)
−0.497056 + 0.867719i \(0.665585\pi\)
\(72\) −373248. −0.117851
\(73\) −1.35994e6 −0.409156 −0.204578 0.978850i \(-0.565582\pi\)
−0.204578 + 0.978850i \(0.565582\pi\)
\(74\) −2.30919e6 −0.662444
\(75\) −399300. −0.109291
\(76\) 332221. 0.0868118
\(77\) −2.37872e6 −0.593779
\(78\) 474552. 0.113228
\(79\) −565403. −0.129022 −0.0645109 0.997917i \(-0.520549\pi\)
−0.0645109 + 0.997917i \(0.520549\pi\)
\(80\) −1.03083e6 −0.225098
\(81\) 531441. 0.111111
\(82\) −65892.1 −0.0131973
\(83\) 2.10144e6 0.403408 0.201704 0.979447i \(-0.435352\pi\)
0.201704 + 0.979447i \(0.435352\pi\)
\(84\) −592704. −0.109109
\(85\) −889211. −0.157050
\(86\) −2.27563e6 −0.385795
\(87\) −1.89682e6 −0.308823
\(88\) −3.55074e6 −0.555429
\(89\) 6.47234e6 0.973188 0.486594 0.873628i \(-0.338239\pi\)
0.486594 + 0.873628i \(0.338239\pi\)
\(90\) 1.46772e6 0.212224
\(91\) 753571. 0.104828
\(92\) −2.62385e6 −0.351303
\(93\) −7.78196e6 −1.00323
\(94\) −2.03513e6 −0.252723
\(95\) −1.30639e6 −0.156329
\(96\) −884736. −0.102062
\(97\) −4.68279e6 −0.520959 −0.260479 0.965479i \(-0.583881\pi\)
−0.260479 + 0.965479i \(0.583881\pi\)
\(98\) −941192. −0.101015
\(99\) 5.05564e6 0.523664
\(100\) −946489. −0.0946489
\(101\) −1.10597e7 −1.06812 −0.534060 0.845446i \(-0.679334\pi\)
−0.534060 + 0.845446i \(0.679334\pi\)
\(102\) −763191. −0.0712086
\(103\) −6.69767e6 −0.603940 −0.301970 0.953318i \(-0.597644\pi\)
−0.301970 + 0.953318i \(0.597644\pi\)
\(104\) 1.12486e6 0.0980581
\(105\) 2.33069e6 0.196481
\(106\) −5.43789e6 −0.443466
\(107\) −1.95155e7 −1.54006 −0.770029 0.638009i \(-0.779758\pi\)
−0.770029 + 0.638009i \(0.779758\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) −1.70235e7 −1.25909 −0.629545 0.776964i \(-0.716758\pi\)
−0.629545 + 0.776964i \(0.716758\pi\)
\(110\) 1.39625e7 1.00021
\(111\) 7.79352e6 0.540883
\(112\) −1.40493e6 −0.0944911
\(113\) 1.11219e7 0.725110 0.362555 0.931962i \(-0.381904\pi\)
0.362555 + 0.931962i \(0.381904\pi\)
\(114\) −1.12124e6 −0.0708816
\(115\) 1.03177e7 0.632619
\(116\) −4.49618e6 −0.267449
\(117\) −1.60161e6 −0.0924500
\(118\) −8.53520e6 −0.478219
\(119\) −1.21192e6 −0.0659264
\(120\) 3.47904e6 0.183791
\(121\) 2.86075e7 1.46802
\(122\) −2.34414e7 −1.16876
\(123\) 222386. 0.0107755
\(124\) −1.84461e7 −0.868819
\(125\) 2.33833e7 1.07083
\(126\) 2.00038e6 0.0890871
\(127\) 2.81859e7 1.22101 0.610504 0.792013i \(-0.290967\pi\)
0.610504 + 0.792013i \(0.290967\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 7.68024e6 0.315000
\(130\) −4.42329e6 −0.176581
\(131\) −3.21524e7 −1.24958 −0.624789 0.780793i \(-0.714815\pi\)
−0.624789 + 0.780793i \(0.714815\pi\)
\(132\) 1.19837e7 0.453506
\(133\) −1.78049e6 −0.0656236
\(134\) −9.66931e6 −0.347159
\(135\) −4.95356e6 −0.173280
\(136\) −1.80904e6 −0.0616685
\(137\) −1.69595e7 −0.563496 −0.281748 0.959488i \(-0.590914\pi\)
−0.281748 + 0.959488i \(0.590914\pi\)
\(138\) 8.85549e6 0.286837
\(139\) 5.15015e7 1.62655 0.813276 0.581878i \(-0.197682\pi\)
0.813276 + 0.581878i \(0.197682\pi\)
\(140\) 5.52459e6 0.170158
\(141\) 6.86858e6 0.206348
\(142\) 2.39844e7 0.702943
\(143\) −1.52363e7 −0.435715
\(144\) 2.98598e6 0.0833333
\(145\) 1.76803e7 0.481616
\(146\) 1.08795e7 0.289317
\(147\) 3.17652e6 0.0824786
\(148\) 1.84735e7 0.468418
\(149\) −1.98903e7 −0.492594 −0.246297 0.969194i \(-0.579214\pi\)
−0.246297 + 0.969194i \(0.579214\pi\)
\(150\) 3.19440e6 0.0772805
\(151\) 6.76658e7 1.59937 0.799687 0.600418i \(-0.204999\pi\)
0.799687 + 0.600418i \(0.204999\pi\)
\(152\) −2.65776e6 −0.0613852
\(153\) 2.57577e6 0.0581416
\(154\) 1.90297e7 0.419865
\(155\) 7.25355e7 1.56455
\(156\) −3.79642e6 −0.0800641
\(157\) −7.01512e6 −0.144673 −0.0723363 0.997380i \(-0.523045\pi\)
−0.0723363 + 0.997380i \(0.523045\pi\)
\(158\) 4.52322e6 0.0912322
\(159\) 1.83529e7 0.362088
\(160\) 8.24661e6 0.159168
\(161\) 1.40622e7 0.265560
\(162\) −4.25153e6 −0.0785674
\(163\) −3.60262e7 −0.651571 −0.325786 0.945444i \(-0.605629\pi\)
−0.325786 + 0.945444i \(0.605629\pi\)
\(164\) 527137. 0.00933190
\(165\) −4.71235e7 −0.816665
\(166\) −1.68116e7 −0.285253
\(167\) −7.32229e7 −1.21658 −0.608288 0.793717i \(-0.708143\pi\)
−0.608288 + 0.793717i \(0.708143\pi\)
\(168\) 4.74163e6 0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) 7.11369e6 0.111051
\(171\) 3.78420e6 0.0578746
\(172\) 1.82050e7 0.272798
\(173\) −8.04650e7 −1.18153 −0.590766 0.806843i \(-0.701174\pi\)
−0.590766 + 0.806843i \(0.701174\pi\)
\(174\) 1.51746e7 0.218371
\(175\) 5.07259e6 0.0715478
\(176\) 2.84059e7 0.392748
\(177\) 2.88063e7 0.390464
\(178\) −5.17788e7 −0.688148
\(179\) −1.43295e8 −1.86743 −0.933715 0.358016i \(-0.883453\pi\)
−0.933715 + 0.358016i \(0.883453\pi\)
\(180\) −1.17418e7 −0.150065
\(181\) −1.37748e8 −1.72668 −0.863339 0.504624i \(-0.831631\pi\)
−0.863339 + 0.504624i \(0.831631\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) 7.91147e7 0.954285
\(184\) 2.09908e7 0.248408
\(185\) −7.26433e7 −0.843518
\(186\) 6.22557e7 0.709387
\(187\) 2.45035e7 0.274020
\(188\) 1.62811e7 0.178702
\(189\) −6.75127e6 −0.0727393
\(190\) 1.04511e7 0.110541
\(191\) 3.36027e7 0.348945 0.174473 0.984662i \(-0.444178\pi\)
0.174473 + 0.984662i \(0.444178\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) 3.17594e7 0.317996 0.158998 0.987279i \(-0.449174\pi\)
0.158998 + 0.987279i \(0.449174\pi\)
\(194\) 3.74623e7 0.368373
\(195\) 1.49286e7 0.144178
\(196\) 7.52954e6 0.0714286
\(197\) −1.15390e8 −1.07532 −0.537659 0.843163i \(-0.680691\pi\)
−0.537659 + 0.843163i \(0.680691\pi\)
\(198\) −4.04451e7 −0.370286
\(199\) 1.08761e6 0.00978338 0.00489169 0.999988i \(-0.498443\pi\)
0.00489169 + 0.999988i \(0.498443\pi\)
\(200\) 7.57191e6 0.0669269
\(201\) 3.26339e7 0.283454
\(202\) 8.84780e7 0.755275
\(203\) 2.40967e7 0.202172
\(204\) 6.10553e6 0.0503521
\(205\) −2.07286e6 −0.0168047
\(206\) 5.35814e7 0.427050
\(207\) −2.98873e7 −0.234202
\(208\) −8.99891e6 −0.0693375
\(209\) 3.59994e7 0.272761
\(210\) −1.86455e7 −0.138933
\(211\) −1.70584e6 −0.0125011 −0.00625056 0.999980i \(-0.501990\pi\)
−0.00625056 + 0.999980i \(0.501990\pi\)
\(212\) 4.35031e7 0.313578
\(213\) −8.09474e7 −0.573950
\(214\) 1.56124e8 1.08899
\(215\) −7.15875e7 −0.491250
\(216\) −1.00777e7 −0.0680414
\(217\) 9.88597e7 0.656765
\(218\) 1.36188e8 0.890310
\(219\) −3.67183e7 −0.236227
\(220\) −1.11700e8 −0.707253
\(221\) −7.76264e6 −0.0483767
\(222\) −6.23482e7 −0.382462
\(223\) 6.10969e7 0.368937 0.184469 0.982838i \(-0.440944\pi\)
0.184469 + 0.982838i \(0.440944\pi\)
\(224\) 1.12394e7 0.0668153
\(225\) −1.07811e7 −0.0630992
\(226\) −8.89751e7 −0.512730
\(227\) −1.32246e8 −0.750396 −0.375198 0.926945i \(-0.622425\pi\)
−0.375198 + 0.926945i \(0.622425\pi\)
\(228\) 8.96996e6 0.0501208
\(229\) 1.01733e8 0.559809 0.279904 0.960028i \(-0.409697\pi\)
0.279904 + 0.960028i \(0.409697\pi\)
\(230\) −8.25419e7 −0.447329
\(231\) −6.42253e7 −0.342819
\(232\) 3.59694e7 0.189115
\(233\) −7.05503e7 −0.365387 −0.182694 0.983170i \(-0.558482\pi\)
−0.182694 + 0.983170i \(0.558482\pi\)
\(234\) 1.28129e7 0.0653720
\(235\) −6.40219e7 −0.321804
\(236\) 6.82816e7 0.338152
\(237\) −1.52659e7 −0.0744908
\(238\) 9.69535e6 0.0466170
\(239\) 1.77323e7 0.0840180 0.0420090 0.999117i \(-0.486624\pi\)
0.0420090 + 0.999117i \(0.486624\pi\)
\(240\) −2.78323e7 −0.129960
\(241\) −2.54626e8 −1.17177 −0.585886 0.810394i \(-0.699253\pi\)
−0.585886 + 0.810394i \(0.699253\pi\)
\(242\) −2.28860e8 −1.03804
\(243\) 1.43489e7 0.0641500
\(244\) 1.87531e8 0.826435
\(245\) −2.96083e7 −0.128627
\(246\) −1.77909e6 −0.00761946
\(247\) −1.14045e7 −0.0481545
\(248\) 1.47569e8 0.614348
\(249\) 5.67390e7 0.232908
\(250\) −1.87067e8 −0.757192
\(251\) −2.53930e8 −1.01358 −0.506788 0.862071i \(-0.669167\pi\)
−0.506788 + 0.862071i \(0.669167\pi\)
\(252\) −1.60030e7 −0.0629941
\(253\) −2.84320e8 −1.10379
\(254\) −2.25487e8 −0.863382
\(255\) −2.40087e7 −0.0906730
\(256\) 1.67772e7 0.0625000
\(257\) −1.29185e8 −0.474729 −0.237364 0.971421i \(-0.576284\pi\)
−0.237364 + 0.971421i \(0.576284\pi\)
\(258\) −6.14419e7 −0.222739
\(259\) −9.90066e7 −0.354091
\(260\) 3.53863e7 0.124862
\(261\) −5.12143e7 −0.178299
\(262\) 2.57219e8 0.883586
\(263\) −3.61265e6 −0.0122456 −0.00612282 0.999981i \(-0.501949\pi\)
−0.00612282 + 0.999981i \(0.501949\pi\)
\(264\) −9.58699e7 −0.320677
\(265\) −1.71067e8 −0.564684
\(266\) 1.42440e7 0.0464029
\(267\) 1.74753e8 0.561870
\(268\) 7.73545e7 0.245479
\(269\) −5.05729e8 −1.58411 −0.792055 0.610450i \(-0.790988\pi\)
−0.792055 + 0.610450i \(0.790988\pi\)
\(270\) 3.96284e7 0.122528
\(271\) −4.25958e8 −1.30009 −0.650047 0.759894i \(-0.725251\pi\)
−0.650047 + 0.759894i \(0.725251\pi\)
\(272\) 1.44724e7 0.0436062
\(273\) 2.03464e7 0.0605228
\(274\) 1.35676e8 0.398452
\(275\) −1.02561e8 −0.297385
\(276\) −7.08439e7 −0.202825
\(277\) −1.25232e8 −0.354026 −0.177013 0.984209i \(-0.556643\pi\)
−0.177013 + 0.984209i \(0.556643\pi\)
\(278\) −4.12012e8 −1.15015
\(279\) −2.10113e8 −0.579212
\(280\) −4.41967e7 −0.120320
\(281\) 5.86467e8 1.57678 0.788390 0.615176i \(-0.210915\pi\)
0.788390 + 0.615176i \(0.210915\pi\)
\(282\) −5.49486e7 −0.145910
\(283\) −3.31438e7 −0.0869260 −0.0434630 0.999055i \(-0.513839\pi\)
−0.0434630 + 0.999055i \(0.513839\pi\)
\(284\) −1.91875e8 −0.497056
\(285\) −3.52725e7 −0.0902566
\(286\) 1.21890e8 0.308097
\(287\) −2.82512e6 −0.00705425
\(288\) −2.38879e7 −0.0589256
\(289\) −3.97855e8 −0.969576
\(290\) −1.41442e8 −0.340554
\(291\) −1.26435e8 −0.300776
\(292\) −8.70361e7 −0.204578
\(293\) 2.85209e8 0.662409 0.331204 0.943559i \(-0.392545\pi\)
0.331204 + 0.943559i \(0.392545\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) −2.68503e8 −0.608937
\(296\) −1.47788e8 −0.331222
\(297\) 1.36502e8 0.302338
\(298\) 1.59122e8 0.348317
\(299\) 9.00718e7 0.194868
\(300\) −2.55552e7 −0.0546455
\(301\) −9.75675e7 −0.206216
\(302\) −5.41326e8 −1.13093
\(303\) −2.98613e8 −0.616680
\(304\) 2.12621e7 0.0434059
\(305\) −7.37427e8 −1.48823
\(306\) −2.06061e7 −0.0411123
\(307\) −1.61743e8 −0.319037 −0.159518 0.987195i \(-0.550994\pi\)
−0.159518 + 0.987195i \(0.550994\pi\)
\(308\) −1.52238e8 −0.296890
\(309\) −1.80837e8 −0.348685
\(310\) −5.80284e8 −1.10630
\(311\) −2.35974e8 −0.444839 −0.222419 0.974951i \(-0.571395\pi\)
−0.222419 + 0.974951i \(0.571395\pi\)
\(312\) 3.03713e7 0.0566139
\(313\) −4.68223e7 −0.0863074 −0.0431537 0.999068i \(-0.513741\pi\)
−0.0431537 + 0.999068i \(0.513741\pi\)
\(314\) 5.61210e7 0.102299
\(315\) 6.29285e7 0.113438
\(316\) −3.61858e7 −0.0645109
\(317\) −5.38537e8 −0.949529 −0.474764 0.880113i \(-0.657467\pi\)
−0.474764 + 0.880113i \(0.657467\pi\)
\(318\) −1.46823e8 −0.256035
\(319\) −4.87205e8 −0.840319
\(320\) −6.59729e7 −0.112549
\(321\) −5.26919e8 −0.889153
\(322\) −1.12497e8 −0.187779
\(323\) 1.83411e7 0.0302843
\(324\) 3.40122e7 0.0555556
\(325\) 3.24912e7 0.0525017
\(326\) 2.88210e8 0.460730
\(327\) −4.59635e8 −0.726935
\(328\) −4.21709e6 −0.00659865
\(329\) −8.72564e7 −0.135086
\(330\) 3.76988e8 0.577469
\(331\) 9.09442e8 1.37841 0.689203 0.724569i \(-0.257961\pi\)
0.689203 + 0.724569i \(0.257961\pi\)
\(332\) 1.34492e8 0.201704
\(333\) 2.10425e8 0.312279
\(334\) 5.85783e8 0.860249
\(335\) −3.04180e8 −0.442053
\(336\) −3.79331e7 −0.0545545
\(337\) 1.79739e8 0.255822 0.127911 0.991786i \(-0.459173\pi\)
0.127911 + 0.991786i \(0.459173\pi\)
\(338\) −3.86145e7 −0.0543928
\(339\) 3.00291e8 0.418642
\(340\) −5.69095e7 −0.0785252
\(341\) −1.99882e9 −2.72981
\(342\) −3.02736e7 −0.0409235
\(343\) −4.03536e7 −0.0539949
\(344\) −1.45640e8 −0.192898
\(345\) 2.78579e8 0.365243
\(346\) 6.43720e8 0.835470
\(347\) 1.45690e7 0.0187187 0.00935934 0.999956i \(-0.497021\pi\)
0.00935934 + 0.999956i \(0.497021\pi\)
\(348\) −1.21397e8 −0.154412
\(349\) −1.24811e9 −1.57168 −0.785839 0.618431i \(-0.787769\pi\)
−0.785839 + 0.618431i \(0.787769\pi\)
\(350\) −4.05807e7 −0.0505919
\(351\) −4.32436e7 −0.0533761
\(352\) −2.27247e8 −0.277715
\(353\) 2.12266e8 0.256843 0.128422 0.991720i \(-0.459009\pi\)
0.128422 + 0.991720i \(0.459009\pi\)
\(354\) −2.30450e8 −0.276100
\(355\) 7.54510e8 0.895088
\(356\) 4.14230e8 0.486594
\(357\) −3.27218e7 −0.0380626
\(358\) 1.14636e9 1.32047
\(359\) 5.08137e8 0.579630 0.289815 0.957083i \(-0.406406\pi\)
0.289815 + 0.957083i \(0.406406\pi\)
\(360\) 9.39341e7 0.106112
\(361\) −8.66926e8 −0.969855
\(362\) 1.10199e9 1.22095
\(363\) 7.72402e8 0.847559
\(364\) 4.82285e7 0.0524142
\(365\) 3.42251e8 0.368400
\(366\) −6.32917e8 −0.674781
\(367\) −1.57790e8 −0.166628 −0.0833139 0.996523i \(-0.526550\pi\)
−0.0833139 + 0.996523i \(0.526550\pi\)
\(368\) −1.67926e8 −0.175651
\(369\) 6.00442e6 0.00622126
\(370\) 5.81147e8 0.596458
\(371\) −2.33150e8 −0.237042
\(372\) −4.98045e8 −0.501613
\(373\) −3.67655e8 −0.366825 −0.183413 0.983036i \(-0.558714\pi\)
−0.183413 + 0.983036i \(0.558714\pi\)
\(374\) −1.96028e8 −0.193761
\(375\) 6.31350e8 0.618245
\(376\) −1.30249e8 −0.126362
\(377\) 1.54345e8 0.148354
\(378\) 5.40102e7 0.0514344
\(379\) −9.04041e8 −0.853004 −0.426502 0.904487i \(-0.640254\pi\)
−0.426502 + 0.904487i \(0.640254\pi\)
\(380\) −8.36088e7 −0.0781645
\(381\) 7.61018e8 0.704949
\(382\) −2.68822e8 −0.246742
\(383\) −2.12799e9 −1.93541 −0.967707 0.252079i \(-0.918886\pi\)
−0.967707 + 0.252079i \(0.918886\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) 5.98643e8 0.534633
\(386\) −2.54075e8 −0.224857
\(387\) 2.07367e8 0.181866
\(388\) −2.99698e8 −0.260479
\(389\) −1.73532e9 −1.49471 −0.747353 0.664428i \(-0.768675\pi\)
−0.747353 + 0.664428i \(0.768675\pi\)
\(390\) −1.19429e8 −0.101949
\(391\) −1.44857e8 −0.122552
\(392\) −6.02363e7 −0.0505076
\(393\) −8.68114e8 −0.721445
\(394\) 9.23120e8 0.760364
\(395\) 1.42293e8 0.116170
\(396\) 3.23561e8 0.261832
\(397\) 1.90708e9 1.52969 0.764843 0.644217i \(-0.222816\pi\)
0.764843 + 0.644217i \(0.222816\pi\)
\(398\) −8.70091e6 −0.00691789
\(399\) −4.80734e7 −0.0378878
\(400\) −6.05753e7 −0.0473244
\(401\) 2.12546e9 1.64607 0.823034 0.567992i \(-0.192280\pi\)
0.823034 + 0.567992i \(0.192280\pi\)
\(402\) −2.61071e8 −0.200433
\(403\) 6.33221e8 0.481934
\(404\) −7.07824e8 −0.534060
\(405\) −1.33746e8 −0.100043
\(406\) −1.92774e8 −0.142957
\(407\) 2.00179e9 1.47176
\(408\) −4.88442e7 −0.0356043
\(409\) 3.09769e8 0.223875 0.111938 0.993715i \(-0.464294\pi\)
0.111938 + 0.993715i \(0.464294\pi\)
\(410\) 1.65828e7 0.0118827
\(411\) −4.57906e8 −0.325334
\(412\) −4.28651e8 −0.301970
\(413\) −3.65947e8 −0.255619
\(414\) 2.39098e8 0.165606
\(415\) −5.28864e8 −0.363225
\(416\) 7.19913e7 0.0490290
\(417\) 1.39054e9 0.939090
\(418\) −2.87995e8 −0.192871
\(419\) −1.81985e9 −1.20861 −0.604305 0.796753i \(-0.706549\pi\)
−0.604305 + 0.796753i \(0.706549\pi\)
\(420\) 1.49164e8 0.0982406
\(421\) 4.07380e8 0.266080 0.133040 0.991111i \(-0.457526\pi\)
0.133040 + 0.991111i \(0.457526\pi\)
\(422\) 1.36467e7 0.00883962
\(423\) 1.85452e8 0.119135
\(424\) −3.48025e8 −0.221733
\(425\) −5.22534e7 −0.0330182
\(426\) 6.47579e8 0.405844
\(427\) −1.00505e9 −0.624726
\(428\) −1.24899e9 −0.770029
\(429\) −4.11379e8 −0.251560
\(430\) 5.72700e8 0.347366
\(431\) 1.22921e9 0.739527 0.369764 0.929126i \(-0.379439\pi\)
0.369764 + 0.929126i \(0.379439\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) 2.38026e8 0.140902 0.0704509 0.997515i \(-0.477556\pi\)
0.0704509 + 0.997515i \(0.477556\pi\)
\(434\) −7.90877e8 −0.464403
\(435\) 4.77367e8 0.278061
\(436\) −1.08950e9 −0.629545
\(437\) −2.12816e8 −0.121989
\(438\) 2.93747e8 0.167037
\(439\) −2.18758e9 −1.23407 −0.617033 0.786937i \(-0.711665\pi\)
−0.617033 + 0.786937i \(0.711665\pi\)
\(440\) 8.93602e8 0.500103
\(441\) 8.57661e7 0.0476190
\(442\) 6.21011e7 0.0342075
\(443\) 7.00372e8 0.382751 0.191375 0.981517i \(-0.438705\pi\)
0.191375 + 0.981517i \(0.438705\pi\)
\(444\) 4.98785e8 0.270441
\(445\) −1.62887e9 −0.876248
\(446\) −4.88775e8 −0.260878
\(447\) −5.37038e8 −0.284399
\(448\) −8.99154e7 −0.0472456
\(449\) 2.41556e9 1.25937 0.629687 0.776849i \(-0.283183\pi\)
0.629687 + 0.776849i \(0.283183\pi\)
\(450\) 8.62488e7 0.0446179
\(451\) 5.71205e7 0.0293207
\(452\) 7.11801e8 0.362555
\(453\) 1.82698e9 0.923398
\(454\) 1.05796e9 0.530610
\(455\) −1.89649e8 −0.0943865
\(456\) −7.17596e7 −0.0354408
\(457\) −7.67101e8 −0.375964 −0.187982 0.982173i \(-0.560195\pi\)
−0.187982 + 0.982173i \(0.560195\pi\)
\(458\) −8.13868e8 −0.395844
\(459\) 6.95458e7 0.0335681
\(460\) 6.60335e8 0.316309
\(461\) −2.40905e9 −1.14523 −0.572614 0.819825i \(-0.694071\pi\)
−0.572614 + 0.819825i \(0.694071\pi\)
\(462\) 5.13802e8 0.242409
\(463\) 1.53195e9 0.717315 0.358658 0.933469i \(-0.383235\pi\)
0.358658 + 0.933469i \(0.383235\pi\)
\(464\) −2.87755e8 −0.133724
\(465\) 1.95846e9 0.903294
\(466\) 5.64403e8 0.258368
\(467\) 4.71604e8 0.214273 0.107137 0.994244i \(-0.465832\pi\)
0.107137 + 0.994244i \(0.465832\pi\)
\(468\) −1.02503e8 −0.0462250
\(469\) −4.14572e8 −0.185564
\(470\) 5.12175e8 0.227550
\(471\) −1.89408e8 −0.0835268
\(472\) −5.46253e8 −0.239109
\(473\) 1.97269e9 0.857128
\(474\) 1.22127e8 0.0526729
\(475\) −7.67683e7 −0.0328666
\(476\) −7.75628e7 −0.0329632
\(477\) 4.95528e8 0.209052
\(478\) −1.41858e8 −0.0594097
\(479\) −1.55082e9 −0.644744 −0.322372 0.946613i \(-0.604480\pi\)
−0.322372 + 0.946613i \(0.604480\pi\)
\(480\) 2.22659e8 0.0918957
\(481\) −6.34162e8 −0.259832
\(482\) 2.03701e9 0.828568
\(483\) 3.79679e8 0.153321
\(484\) 1.83088e9 0.734008
\(485\) 1.17850e9 0.469066
\(486\) −1.14791e8 −0.0453609
\(487\) 4.23914e9 1.66313 0.831566 0.555426i \(-0.187445\pi\)
0.831566 + 0.555426i \(0.187445\pi\)
\(488\) −1.50025e9 −0.584378
\(489\) −9.72707e8 −0.376185
\(490\) 2.36867e8 0.0909531
\(491\) −4.19200e9 −1.59822 −0.799108 0.601187i \(-0.794695\pi\)
−0.799108 + 0.601187i \(0.794695\pi\)
\(492\) 1.42327e7 0.00538777
\(493\) −2.48223e8 −0.0932993
\(494\) 9.12361e7 0.0340504
\(495\) −1.27234e9 −0.471502
\(496\) −1.18055e9 −0.434409
\(497\) 1.02833e9 0.375739
\(498\) −4.53912e8 −0.164691
\(499\) −2.55636e9 −0.921022 −0.460511 0.887654i \(-0.652334\pi\)
−0.460511 + 0.887654i \(0.652334\pi\)
\(500\) 1.49653e9 0.535416
\(501\) −1.97702e9 −0.702390
\(502\) 2.03144e9 0.716707
\(503\) 3.40746e9 1.19383 0.596916 0.802304i \(-0.296393\pi\)
0.596916 + 0.802304i \(0.296393\pi\)
\(504\) 1.28024e8 0.0445435
\(505\) 2.78337e9 0.961725
\(506\) 2.27456e9 0.780495
\(507\) 1.30324e8 0.0444116
\(508\) 1.80389e9 0.610504
\(509\) −1.85676e9 −0.624086 −0.312043 0.950068i \(-0.601013\pi\)
−0.312043 + 0.950068i \(0.601013\pi\)
\(510\) 1.92070e8 0.0641155
\(511\) 4.66459e8 0.154647
\(512\) −1.34218e8 −0.0441942
\(513\) 1.02173e8 0.0334139
\(514\) 1.03348e9 0.335684
\(515\) 1.68558e9 0.543781
\(516\) 4.91536e8 0.157500
\(517\) 1.76421e9 0.561480
\(518\) 7.92053e8 0.250380
\(519\) −2.17256e9 −0.682158
\(520\) −2.83091e8 −0.0882905
\(521\) −4.88865e9 −1.51446 −0.757229 0.653150i \(-0.773447\pi\)
−0.757229 + 0.653150i \(0.773447\pi\)
\(522\) 4.09714e8 0.126076
\(523\) 2.83241e9 0.865767 0.432883 0.901450i \(-0.357496\pi\)
0.432883 + 0.901450i \(0.357496\pi\)
\(524\) −2.05775e9 −0.624789
\(525\) 1.36960e8 0.0413082
\(526\) 2.89012e7 0.00865897
\(527\) −1.01837e9 −0.303087
\(528\) 7.66959e8 0.226753
\(529\) −1.72402e9 −0.506346
\(530\) 1.36854e9 0.399292
\(531\) 7.77770e8 0.225435
\(532\) −1.13952e8 −0.0328118
\(533\) −1.80956e7 −0.00517640
\(534\) −1.39803e9 −0.397302
\(535\) 4.91141e9 1.38665
\(536\) −6.18836e8 −0.173580
\(537\) −3.86896e9 −1.07816
\(538\) 4.04583e9 1.12013
\(539\) 8.15899e8 0.224427
\(540\) −3.17028e8 −0.0866401
\(541\) 1.15617e9 0.313929 0.156965 0.987604i \(-0.449829\pi\)
0.156965 + 0.987604i \(0.449829\pi\)
\(542\) 3.40767e9 0.919305
\(543\) −3.71921e9 −0.996898
\(544\) −1.15779e8 −0.0308342
\(545\) 4.28425e9 1.13367
\(546\) −1.62771e8 −0.0427960
\(547\) 4.72294e9 1.23383 0.616917 0.787028i \(-0.288381\pi\)
0.616917 + 0.787028i \(0.288381\pi\)
\(548\) −1.08541e9 −0.281748
\(549\) 2.13610e9 0.550957
\(550\) 8.20491e8 0.210283
\(551\) −3.64678e8 −0.0928708
\(552\) 5.66751e8 0.143419
\(553\) 1.93933e8 0.0487657
\(554\) 1.00185e9 0.250334
\(555\) −1.96137e9 −0.487006
\(556\) 3.29609e9 0.813276
\(557\) −3.75863e9 −0.921587 −0.460794 0.887507i \(-0.652435\pi\)
−0.460794 + 0.887507i \(0.652435\pi\)
\(558\) 1.68090e9 0.409565
\(559\) −6.24944e8 −0.151321
\(560\) 3.53574e8 0.0850789
\(561\) 6.61594e8 0.158205
\(562\) −4.69173e9 −1.11495
\(563\) −7.13132e8 −0.168419 −0.0842094 0.996448i \(-0.526836\pi\)
−0.0842094 + 0.996448i \(0.526836\pi\)
\(564\) 4.39589e8 0.103174
\(565\) −2.79901e9 −0.652882
\(566\) 2.65150e8 0.0614659
\(567\) −1.82284e8 −0.0419961
\(568\) 1.53500e9 0.351471
\(569\) −4.30301e9 −0.979217 −0.489609 0.871942i \(-0.662860\pi\)
−0.489609 + 0.871942i \(0.662860\pi\)
\(570\) 2.82180e8 0.0638211
\(571\) 4.59909e9 1.03382 0.516911 0.856039i \(-0.327082\pi\)
0.516911 + 0.856039i \(0.327082\pi\)
\(572\) −9.75121e8 −0.217857
\(573\) 9.07273e8 0.201464
\(574\) 2.26010e7 0.00498811
\(575\) 6.06309e8 0.133002
\(576\) 1.91103e8 0.0416667
\(577\) −9.60905e8 −0.208240 −0.104120 0.994565i \(-0.533203\pi\)
−0.104120 + 0.994565i \(0.533203\pi\)
\(578\) 3.18284e9 0.685594
\(579\) 8.57504e8 0.183595
\(580\) 1.13154e9 0.240808
\(581\) −7.20795e8 −0.152474
\(582\) 1.01148e9 0.212681
\(583\) 4.71399e9 0.985256
\(584\) 6.96289e8 0.144659
\(585\) 4.03073e8 0.0832411
\(586\) −2.28167e9 −0.468394
\(587\) 7.63413e9 1.55785 0.778926 0.627116i \(-0.215765\pi\)
0.778926 + 0.627116i \(0.215765\pi\)
\(588\) 2.03297e8 0.0412393
\(589\) −1.49614e9 −0.301695
\(590\) 2.14803e9 0.430584
\(591\) −3.11553e9 −0.620835
\(592\) 1.18231e9 0.234209
\(593\) −8.34954e9 −1.64426 −0.822132 0.569297i \(-0.807215\pi\)
−0.822132 + 0.569297i \(0.807215\pi\)
\(594\) −1.09202e9 −0.213785
\(595\) 3.05000e8 0.0593594
\(596\) −1.27298e9 −0.246297
\(597\) 2.93656e7 0.00564844
\(598\) −7.20574e8 −0.137792
\(599\) 1.70616e8 0.0324358 0.0162179 0.999868i \(-0.494837\pi\)
0.0162179 + 0.999868i \(0.494837\pi\)
\(600\) 2.04442e8 0.0386402
\(601\) 6.88290e9 1.29333 0.646667 0.762772i \(-0.276162\pi\)
0.646667 + 0.762772i \(0.276162\pi\)
\(602\) 7.80540e8 0.145817
\(603\) 8.81116e8 0.163652
\(604\) 4.33061e9 0.799687
\(605\) −7.19955e9 −1.32179
\(606\) 2.38890e9 0.436058
\(607\) −2.87762e9 −0.522243 −0.261122 0.965306i \(-0.584092\pi\)
−0.261122 + 0.965306i \(0.584092\pi\)
\(608\) −1.70097e8 −0.0306926
\(609\) 6.50611e8 0.116724
\(610\) 5.89941e9 1.05234
\(611\) −5.58899e8 −0.0991263
\(612\) 1.64849e8 0.0290708
\(613\) −4.83038e9 −0.846974 −0.423487 0.905902i \(-0.639194\pi\)
−0.423487 + 0.905902i \(0.639194\pi\)
\(614\) 1.29394e9 0.225593
\(615\) −5.59671e7 −0.00970219
\(616\) 1.21790e9 0.209933
\(617\) 7.92314e9 1.35800 0.678999 0.734139i \(-0.262414\pi\)
0.678999 + 0.734139i \(0.262414\pi\)
\(618\) 1.44670e9 0.246557
\(619\) 1.07147e9 0.181578 0.0907892 0.995870i \(-0.471061\pi\)
0.0907892 + 0.995870i \(0.471061\pi\)
\(620\) 4.64227e9 0.782276
\(621\) −8.06956e8 −0.135216
\(622\) 1.88779e9 0.314549
\(623\) −2.22001e9 −0.367830
\(624\) −2.42971e8 −0.0400320
\(625\) −4.72942e9 −0.774869
\(626\) 3.74579e8 0.0610285
\(627\) 9.71983e8 0.157479
\(628\) −4.48968e8 −0.0723363
\(629\) 1.01988e9 0.163408
\(630\) −5.03428e8 −0.0802131
\(631\) 8.28876e9 1.31337 0.656685 0.754165i \(-0.271958\pi\)
0.656685 + 0.754165i \(0.271958\pi\)
\(632\) 2.89486e8 0.0456161
\(633\) −4.60576e7 −0.00721752
\(634\) 4.30829e9 0.671418
\(635\) −7.09344e9 −1.09938
\(636\) 1.17458e9 0.181044
\(637\) −2.58475e8 −0.0396214
\(638\) 3.89764e9 0.594195
\(639\) −2.18558e9 −0.331370
\(640\) 5.27783e8 0.0795840
\(641\) −6.03889e9 −0.905636 −0.452818 0.891603i \(-0.649581\pi\)
−0.452818 + 0.891603i \(0.649581\pi\)
\(642\) 4.21536e9 0.628726
\(643\) 4.04616e9 0.600212 0.300106 0.953906i \(-0.402978\pi\)
0.300106 + 0.953906i \(0.402978\pi\)
\(644\) 8.99980e8 0.132780
\(645\) −1.93286e9 −0.283623
\(646\) −1.46729e8 −0.0214142
\(647\) −1.01485e10 −1.47311 −0.736556 0.676377i \(-0.763549\pi\)
−0.736556 + 0.676377i \(0.763549\pi\)
\(648\) −2.72098e8 −0.0392837
\(649\) 7.39898e9 1.06247
\(650\) −2.59929e8 −0.0371243
\(651\) 2.66921e9 0.379184
\(652\) −2.30568e9 −0.325786
\(653\) 2.38202e9 0.334772 0.167386 0.985891i \(-0.446467\pi\)
0.167386 + 0.985891i \(0.446467\pi\)
\(654\) 3.67708e9 0.514021
\(655\) 8.09168e9 1.12511
\(656\) 3.37368e7 0.00466595
\(657\) −9.91395e8 −0.136385
\(658\) 6.98051e8 0.0955205
\(659\) −1.40084e10 −1.90674 −0.953368 0.301812i \(-0.902409\pi\)
−0.953368 + 0.301812i \(0.902409\pi\)
\(660\) −3.01591e9 −0.408333
\(661\) −3.36924e8 −0.0453760 −0.0226880 0.999743i \(-0.507222\pi\)
−0.0226880 + 0.999743i \(0.507222\pi\)
\(662\) −7.27554e9 −0.974680
\(663\) −2.09591e8 −0.0279303
\(664\) −1.07594e9 −0.142626
\(665\) 4.48091e8 0.0590868
\(666\) −1.68340e9 −0.220815
\(667\) 2.88020e9 0.375821
\(668\) −4.68626e9 −0.608288
\(669\) 1.64962e9 0.213006
\(670\) 2.43344e9 0.312579
\(671\) 2.03208e10 2.59665
\(672\) 3.03464e8 0.0385758
\(673\) −6.58816e9 −0.833127 −0.416563 0.909107i \(-0.636766\pi\)
−0.416563 + 0.909107i \(0.636766\pi\)
\(674\) −1.43791e9 −0.180893
\(675\) −2.91090e8 −0.0364304
\(676\) 3.08916e8 0.0384615
\(677\) −6.29588e9 −0.779823 −0.389911 0.920852i \(-0.627494\pi\)
−0.389911 + 0.920852i \(0.627494\pi\)
\(678\) −2.40233e9 −0.296025
\(679\) 1.60620e9 0.196904
\(680\) 4.55276e8 0.0555257
\(681\) −3.57063e9 −0.433241
\(682\) 1.59905e10 1.93027
\(683\) 4.25942e8 0.0511538 0.0255769 0.999673i \(-0.491858\pi\)
0.0255769 + 0.999673i \(0.491858\pi\)
\(684\) 2.42189e8 0.0289373
\(685\) 4.26814e9 0.507366
\(686\) 3.22829e8 0.0381802
\(687\) 2.74680e9 0.323206
\(688\) 1.16512e9 0.136399
\(689\) −1.49338e9 −0.173942
\(690\) −2.22863e9 −0.258265
\(691\) 1.11504e10 1.28563 0.642815 0.766021i \(-0.277766\pi\)
0.642815 + 0.766021i \(0.277766\pi\)
\(692\) −5.14976e9 −0.590766
\(693\) −1.73408e9 −0.197926
\(694\) −1.16552e8 −0.0132361
\(695\) −1.29612e10 −1.46453
\(696\) 9.71174e8 0.109185
\(697\) 2.91020e7 0.00325543
\(698\) 9.98488e9 1.11134
\(699\) −1.90486e9 −0.210956
\(700\) 3.24646e8 0.0357739
\(701\) −1.31586e10 −1.44277 −0.721385 0.692534i \(-0.756494\pi\)
−0.721385 + 0.692534i \(0.756494\pi\)
\(702\) 3.45948e8 0.0377426
\(703\) 1.49836e9 0.162657
\(704\) 1.81798e9 0.196374
\(705\) −1.72859e9 −0.185794
\(706\) −1.69813e9 −0.181616
\(707\) 3.79349e9 0.403712
\(708\) 1.84360e9 0.195232
\(709\) 1.43701e9 0.151425 0.0757124 0.997130i \(-0.475877\pi\)
0.0757124 + 0.997130i \(0.475877\pi\)
\(710\) −6.03608e9 −0.632923
\(711\) −4.12179e8 −0.0430073
\(712\) −3.31384e9 −0.344074
\(713\) 1.18164e10 1.22087
\(714\) 2.61774e8 0.0269143
\(715\) 3.83446e9 0.392313
\(716\) −9.17086e9 −0.933715
\(717\) 4.78772e8 0.0485078
\(718\) −4.06510e9 −0.409860
\(719\) 7.52557e8 0.0755072 0.0377536 0.999287i \(-0.487980\pi\)
0.0377536 + 0.999287i \(0.487980\pi\)
\(720\) −7.51473e8 −0.0750325
\(721\) 2.29730e9 0.228268
\(722\) 6.93541e9 0.685791
\(723\) −6.87490e9 −0.676523
\(724\) −8.81589e9 −0.863339
\(725\) 1.03896e9 0.101255
\(726\) −6.17921e9 −0.599315
\(727\) −4.19798e9 −0.405200 −0.202600 0.979262i \(-0.564939\pi\)
−0.202600 + 0.979262i \(0.564939\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) −2.73801e9 −0.260498
\(731\) 1.00506e9 0.0951655
\(732\) 5.06334e9 0.477143
\(733\) −1.61340e10 −1.51314 −0.756570 0.653913i \(-0.773126\pi\)
−0.756570 + 0.653913i \(0.773126\pi\)
\(734\) 1.26232e9 0.117824
\(735\) −7.99425e8 −0.0742629
\(736\) 1.34341e9 0.124204
\(737\) 8.38212e9 0.771290
\(738\) −4.80353e7 −0.00439910
\(739\) 1.59667e10 1.45532 0.727661 0.685937i \(-0.240608\pi\)
0.727661 + 0.685937i \(0.240608\pi\)
\(740\) −4.64917e9 −0.421759
\(741\) −3.07922e8 −0.0278020
\(742\) 1.86520e9 0.167614
\(743\) −1.18458e10 −1.05951 −0.529755 0.848151i \(-0.677716\pi\)
−0.529755 + 0.848151i \(0.677716\pi\)
\(744\) 3.98436e9 0.354694
\(745\) 5.00573e9 0.443527
\(746\) 2.94124e9 0.259385
\(747\) 1.53195e9 0.134469
\(748\) 1.56822e9 0.137010
\(749\) 6.69383e9 0.582087
\(750\) −5.05080e9 −0.437165
\(751\) 2.43069e9 0.209406 0.104703 0.994504i \(-0.466611\pi\)
0.104703 + 0.994504i \(0.466611\pi\)
\(752\) 1.04199e9 0.0893512
\(753\) −6.85612e9 −0.585189
\(754\) −1.23476e9 −0.104902
\(755\) −1.70292e10 −1.44006
\(756\) −4.32081e8 −0.0363696
\(757\) −1.34156e10 −1.12402 −0.562011 0.827130i \(-0.689972\pi\)
−0.562011 + 0.827130i \(0.689972\pi\)
\(758\) 7.23233e9 0.603165
\(759\) −7.67663e9 −0.637272
\(760\) 6.68871e8 0.0552707
\(761\) −7.21735e9 −0.593651 −0.296826 0.954932i \(-0.595928\pi\)
−0.296826 + 0.954932i \(0.595928\pi\)
\(762\) −6.08815e9 −0.498474
\(763\) 5.83906e9 0.475891
\(764\) 2.15057e9 0.174473
\(765\) −6.48235e8 −0.0523501
\(766\) 1.70239e10 1.36854
\(767\) −2.34398e9 −0.187573
\(768\) 4.52985e8 0.0360844
\(769\) −1.38066e10 −1.09483 −0.547414 0.836862i \(-0.684388\pi\)
−0.547414 + 0.836862i \(0.684388\pi\)
\(770\) −4.78915e9 −0.378042
\(771\) −3.48799e9 −0.274085
\(772\) 2.03260e9 0.158998
\(773\) 9.76421e9 0.760342 0.380171 0.924916i \(-0.375865\pi\)
0.380171 + 0.924916i \(0.375865\pi\)
\(774\) −1.65893e9 −0.128598
\(775\) 4.26246e9 0.328931
\(776\) 2.39759e9 0.184187
\(777\) −2.67318e9 −0.204435
\(778\) 1.38825e10 1.05692
\(779\) 4.27553e7 0.00324048
\(780\) 9.55431e8 0.0720889
\(781\) −2.07916e10 −1.56174
\(782\) 1.15885e9 0.0866571
\(783\) −1.38278e9 −0.102941
\(784\) 4.81890e8 0.0357143
\(785\) 1.76547e9 0.130262
\(786\) 6.94492e9 0.510138
\(787\) −8.95294e9 −0.654717 −0.327359 0.944900i \(-0.606159\pi\)
−0.327359 + 0.944900i \(0.606159\pi\)
\(788\) −7.38496e9 −0.537659
\(789\) −9.75416e7 −0.00707002
\(790\) −1.13834e9 −0.0821445
\(791\) −3.81481e9 −0.274066
\(792\) −2.58849e9 −0.185143
\(793\) −6.43759e9 −0.458424
\(794\) −1.52566e10 −1.08165
\(795\) −4.61881e9 −0.326021
\(796\) 6.96073e7 0.00489169
\(797\) 1.29583e10 0.906660 0.453330 0.891343i \(-0.350236\pi\)
0.453330 + 0.891343i \(0.350236\pi\)
\(798\) 3.84587e8 0.0267907
\(799\) 8.98840e8 0.0623403
\(800\) 4.84602e8 0.0334634
\(801\) 4.71834e9 0.324396
\(802\) −1.70037e10 −1.16395
\(803\) −9.43122e9 −0.642781
\(804\) 2.08857e9 0.141727
\(805\) −3.53898e9 −0.239107
\(806\) −5.06577e9 −0.340779
\(807\) −1.36547e10 −0.914586
\(808\) 5.66259e9 0.377638
\(809\) −3.31421e9 −0.220070 −0.110035 0.993928i \(-0.535096\pi\)
−0.110035 + 0.993928i \(0.535096\pi\)
\(810\) 1.06997e9 0.0707413
\(811\) 1.73663e10 1.14323 0.571615 0.820522i \(-0.306317\pi\)
0.571615 + 0.820522i \(0.306317\pi\)
\(812\) 1.54219e9 0.101086
\(813\) −1.15009e10 −0.750609
\(814\) −1.60143e10 −1.04069
\(815\) 9.06659e9 0.586668
\(816\) 3.90754e8 0.0251760
\(817\) 1.47658e9 0.0947285
\(818\) −2.47815e9 −0.158304
\(819\) 5.49353e8 0.0349428
\(820\) −1.32663e8 −0.00840235
\(821\) 9.16526e9 0.578021 0.289011 0.957326i \(-0.406674\pi\)
0.289011 + 0.957326i \(0.406674\pi\)
\(822\) 3.66325e9 0.230046
\(823\) −1.23495e10 −0.772239 −0.386119 0.922449i \(-0.626185\pi\)
−0.386119 + 0.922449i \(0.626185\pi\)
\(824\) 3.42921e9 0.213525
\(825\) −2.76916e9 −0.171695
\(826\) 2.92757e9 0.180750
\(827\) 1.08370e10 0.666255 0.333127 0.942882i \(-0.391896\pi\)
0.333127 + 0.942882i \(0.391896\pi\)
\(828\) −1.91279e9 −0.117101
\(829\) 1.78237e9 0.108657 0.0543283 0.998523i \(-0.482698\pi\)
0.0543283 + 0.998523i \(0.482698\pi\)
\(830\) 4.23091e9 0.256839
\(831\) −3.38126e9 −0.204397
\(832\) −5.75930e8 −0.0346688
\(833\) 4.15688e8 0.0249178
\(834\) −1.11243e10 −0.664037
\(835\) 1.84278e10 1.09539
\(836\) 2.30396e9 0.136381
\(837\) −5.67305e9 −0.334408
\(838\) 1.45588e10 0.854616
\(839\) 2.24227e10 1.31075 0.655377 0.755302i \(-0.272510\pi\)
0.655377 + 0.755302i \(0.272510\pi\)
\(840\) −1.19331e9 −0.0694666
\(841\) −1.23144e10 −0.713885
\(842\) −3.25904e9 −0.188147
\(843\) 1.58346e10 0.910354
\(844\) −1.09173e8 −0.00625056
\(845\) −1.21475e9 −0.0692608
\(846\) −1.48361e9 −0.0842411
\(847\) −9.81236e9 −0.554858
\(848\) 2.78420e9 0.156789
\(849\) −8.94882e8 −0.0501867
\(850\) 4.18027e8 0.0233474
\(851\) −1.18339e10 −0.658226
\(852\) −5.18063e9 −0.286975
\(853\) 2.99743e10 1.65359 0.826796 0.562502i \(-0.190161\pi\)
0.826796 + 0.562502i \(0.190161\pi\)
\(854\) 8.04039e9 0.441748
\(855\) −9.52357e8 −0.0521097
\(856\) 9.99195e9 0.544493
\(857\) 2.58131e10 1.40090 0.700450 0.713701i \(-0.252982\pi\)
0.700450 + 0.713701i \(0.252982\pi\)
\(858\) 3.29103e9 0.177880
\(859\) −2.65860e9 −0.143112 −0.0715562 0.997437i \(-0.522797\pi\)
−0.0715562 + 0.997437i \(0.522797\pi\)
\(860\) −4.58160e9 −0.245625
\(861\) −7.62783e7 −0.00407277
\(862\) −9.83364e9 −0.522925
\(863\) −3.64194e9 −0.192884 −0.0964418 0.995339i \(-0.530746\pi\)
−0.0964418 + 0.995339i \(0.530746\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) 2.02504e10 1.06384
\(866\) −1.90421e9 −0.0996327
\(867\) −1.07421e10 −0.559785
\(868\) 6.32702e9 0.328383
\(869\) −3.92108e9 −0.202692
\(870\) −3.81894e9 −0.196619
\(871\) −2.65543e9 −0.136167
\(872\) 8.71604e9 0.445155
\(873\) −3.41375e9 −0.173653
\(874\) 1.70253e9 0.0862592
\(875\) −8.02048e9 −0.404736
\(876\) −2.34997e9 −0.118113
\(877\) −8.39113e9 −0.420070 −0.210035 0.977694i \(-0.567358\pi\)
−0.210035 + 0.977694i \(0.567358\pi\)
\(878\) 1.75006e10 0.872616
\(879\) 7.70064e9 0.382442
\(880\) −7.14881e9 −0.353626
\(881\) −2.31554e10 −1.14087 −0.570435 0.821343i \(-0.693225\pi\)
−0.570435 + 0.821343i \(0.693225\pi\)
\(882\) −6.86129e8 −0.0336718
\(883\) 2.66795e7 0.00130411 0.000652055 1.00000i \(-0.499792\pi\)
0.000652055 1.00000i \(0.499792\pi\)
\(884\) −4.96809e8 −0.0241884
\(885\) −7.24959e9 −0.351570
\(886\) −5.60298e9 −0.270646
\(887\) 5.10941e9 0.245832 0.122916 0.992417i \(-0.460775\pi\)
0.122916 + 0.992417i \(0.460775\pi\)
\(888\) −3.99028e9 −0.191231
\(889\) −9.66775e9 −0.461497
\(890\) 1.30310e10 0.619601
\(891\) 3.68556e9 0.174555
\(892\) 3.91020e9 0.184469
\(893\) 1.32053e9 0.0620540
\(894\) 4.29630e9 0.201101
\(895\) 3.60625e10 1.68142
\(896\) 7.19323e8 0.0334077
\(897\) 2.43194e9 0.112507
\(898\) −1.93245e10 −0.890513
\(899\) 2.02483e10 0.929457
\(900\) −6.89990e8 −0.0315496
\(901\) 2.40171e9 0.109391
\(902\) −4.56964e8 −0.0207328
\(903\) −2.63432e9 −0.119059
\(904\) −5.69441e9 −0.256365
\(905\) 3.46667e10 1.55468
\(906\) −1.46158e10 −0.652941
\(907\) 4.00769e9 0.178348 0.0891741 0.996016i \(-0.471577\pi\)
0.0891741 + 0.996016i \(0.471577\pi\)
\(908\) −8.46371e9 −0.375198
\(909\) −8.06255e9 −0.356040
\(910\) 1.51719e9 0.0667414
\(911\) −1.71929e10 −0.753415 −0.376707 0.926332i \(-0.622944\pi\)
−0.376707 + 0.926332i \(0.622944\pi\)
\(912\) 5.74077e8 0.0250604
\(913\) 1.45736e10 0.633751
\(914\) 6.13681e9 0.265846
\(915\) −1.99105e10 −0.859229
\(916\) 6.51094e9 0.279904
\(917\) 1.10283e10 0.472296
\(918\) −5.56366e8 −0.0237362
\(919\) −1.75106e10 −0.744214 −0.372107 0.928190i \(-0.621365\pi\)
−0.372107 + 0.928190i \(0.621365\pi\)
\(920\) −5.28268e9 −0.223664
\(921\) −4.36706e9 −0.184196
\(922\) 1.92724e10 0.809799
\(923\) 6.58672e9 0.275717
\(924\) −4.11042e9 −0.171409
\(925\) −4.26880e9 −0.177341
\(926\) −1.22556e10 −0.507218
\(927\) −4.88260e9 −0.201313
\(928\) 2.30204e9 0.0945574
\(929\) 3.67318e10 1.50310 0.751549 0.659678i \(-0.229307\pi\)
0.751549 + 0.659678i \(0.229307\pi\)
\(930\) −1.56677e10 −0.638725
\(931\) 6.10710e8 0.0248034
\(932\) −4.51522e9 −0.182694
\(933\) −6.37130e9 −0.256828
\(934\) −3.77283e9 −0.151514
\(935\) −6.16671e9 −0.246725
\(936\) 8.20026e8 0.0326860
\(937\) −1.19505e10 −0.474567 −0.237283 0.971440i \(-0.576257\pi\)
−0.237283 + 0.971440i \(0.576257\pi\)
\(938\) 3.31657e9 0.131214
\(939\) −1.26420e9 −0.0498296
\(940\) −4.09740e9 −0.160902
\(941\) 1.33467e9 0.0522167 0.0261083 0.999659i \(-0.491689\pi\)
0.0261083 + 0.999659i \(0.491689\pi\)
\(942\) 1.51527e9 0.0590624
\(943\) −3.37677e8 −0.0131133
\(944\) 4.37002e9 0.169076
\(945\) 1.69907e9 0.0654937
\(946\) −1.57815e10 −0.606081
\(947\) −2.67182e10 −1.02231 −0.511155 0.859489i \(-0.670782\pi\)
−0.511155 + 0.859489i \(0.670782\pi\)
\(948\) −9.77016e8 −0.0372454
\(949\) 2.98779e9 0.113480
\(950\) 6.14147e8 0.0232402
\(951\) −1.45405e10 −0.548211
\(952\) 6.20502e8 0.0233085
\(953\) −2.98289e10 −1.11638 −0.558191 0.829713i \(-0.688504\pi\)
−0.558191 + 0.829713i \(0.688504\pi\)
\(954\) −3.96422e9 −0.147822
\(955\) −8.45668e9 −0.314187
\(956\) 1.13487e9 0.0420090
\(957\) −1.31545e10 −0.485159
\(958\) 1.24066e10 0.455903
\(959\) 5.81710e9 0.212981
\(960\) −1.78127e9 −0.0649801
\(961\) 5.55585e10 2.01938
\(962\) 5.07329e9 0.183729
\(963\) −1.42268e10 −0.513353
\(964\) −1.62961e10 −0.585886
\(965\) −7.99279e9 −0.286321
\(966\) −3.03743e9 −0.108414
\(967\) −1.52547e10 −0.542515 −0.271257 0.962507i \(-0.587439\pi\)
−0.271257 + 0.962507i \(0.587439\pi\)
\(968\) −1.46470e10 −0.519022
\(969\) 4.95210e8 0.0174846
\(970\) −9.42801e9 −0.331680
\(971\) 1.08089e10 0.378893 0.189446 0.981891i \(-0.439331\pi\)
0.189446 + 0.981891i \(0.439331\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) −1.76650e10 −0.614779
\(974\) −3.39131e10 −1.17601
\(975\) 8.77262e8 0.0303119
\(976\) 1.20020e10 0.413218
\(977\) 3.59918e10 1.23473 0.617366 0.786676i \(-0.288200\pi\)
0.617366 + 0.786676i \(0.288200\pi\)
\(978\) 7.78166e9 0.266003
\(979\) 4.48859e10 1.52887
\(980\) −1.89493e9 −0.0643136
\(981\) −1.24101e10 −0.419696
\(982\) 3.35360e10 1.13011
\(983\) −2.16979e10 −0.728585 −0.364293 0.931285i \(-0.618689\pi\)
−0.364293 + 0.931285i \(0.618689\pi\)
\(984\) −1.13862e8 −0.00380973
\(985\) 2.90398e10 0.968205
\(986\) 1.98579e9 0.0659726
\(987\) −2.35592e9 −0.0779921
\(988\) −7.29889e8 −0.0240773
\(989\) −1.16619e10 −0.383339
\(990\) 1.01787e10 0.333402
\(991\) −3.26123e10 −1.06445 −0.532224 0.846604i \(-0.678643\pi\)
−0.532224 + 0.846604i \(0.678643\pi\)
\(992\) 9.44441e9 0.307174
\(993\) 2.45549e10 0.795823
\(994\) −8.22665e9 −0.265687
\(995\) −2.73716e8 −0.00880886
\(996\) 3.63130e9 0.116454
\(997\) 4.15307e10 1.32720 0.663599 0.748088i \(-0.269028\pi\)
0.663599 + 0.748088i \(0.269028\pi\)
\(998\) 2.04509e10 0.651261
\(999\) 5.68148e9 0.180294
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 546.8.a.j.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.j.1.2 5 1.1 even 1 trivial