Properties

Label 546.8.a.q.1.1
Level $546$
Weight $8$
Character 546.1
Self dual yes
Analytic conductor $170.562$
Analytic rank $0$
Dimension $6$
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: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - x^{5} - 367021 x^{4} - 17702143 x^{3} + 34815194576 x^{2} + 1422988371620 x - 933871993059968\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-419.580\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -417.580 q^{5} -216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -417.580 q^{5} -216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} -3340.64 q^{10} +5214.78 q^{11} -1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} +11274.7 q^{15} +4096.00 q^{16} -9289.46 q^{17} +5832.00 q^{18} -28641.6 q^{19} -26725.1 q^{20} -9261.00 q^{21} +41718.2 q^{22} -70760.2 q^{23} -13824.0 q^{24} +96248.1 q^{25} -17576.0 q^{26} -19683.0 q^{27} +21952.0 q^{28} +40590.0 q^{29} +90197.3 q^{30} +239911. q^{31} +32768.0 q^{32} -140799. q^{33} -74315.7 q^{34} -143230. q^{35} +46656.0 q^{36} +334019. q^{37} -229132. q^{38} +59319.0 q^{39} -213801. q^{40} -347373. q^{41} -74088.0 q^{42} -454134. q^{43} +333746. q^{44} -304416. q^{45} -566082. q^{46} -87902.0 q^{47} -110592. q^{48} +117649. q^{49} +769985. q^{50} +250815. q^{51} -140608. q^{52} -2.03091e6 q^{53} -157464. q^{54} -2.17759e6 q^{55} +175616. q^{56} +773322. q^{57} +324720. q^{58} -385876. q^{59} +721578. q^{60} -2.16065e6 q^{61} +1.91928e6 q^{62} +250047. q^{63} +262144. q^{64} +917423. q^{65} -1.12639e6 q^{66} +1.94749e6 q^{67} -594525. q^{68} +1.91053e6 q^{69} -1.14584e6 q^{70} +2.74370e6 q^{71} +373248. q^{72} -801961. q^{73} +2.67215e6 q^{74} -2.59870e6 q^{75} -1.83306e6 q^{76} +1.78867e6 q^{77} +474552. q^{78} -7.49135e6 q^{79} -1.71041e6 q^{80} +531441. q^{81} -2.77899e6 q^{82} +2.59240e6 q^{83} -592704. q^{84} +3.87909e6 q^{85} -3.63307e6 q^{86} -1.09593e6 q^{87} +2.66997e6 q^{88} +712280. q^{89} -2.43533e6 q^{90} -753571. q^{91} -4.52865e6 q^{92} -6.47759e6 q^{93} -703216. q^{94} +1.19601e7 q^{95} -884736. q^{96} +1.71130e7 q^{97} +941192. q^{98} +3.80157e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + O(q^{10}) \) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + 104 q^{10} + 10054 q^{11} - 10368 q^{12} - 13182 q^{13} + 16464 q^{14} - 351 q^{15} + 24576 q^{16} + 21222 q^{17} + 34992 q^{18} + 9527 q^{19} + 832 q^{20} - 55566 q^{21} + 80432 q^{22} + 33229 q^{23} - 82944 q^{24} + 265321 q^{25} - 105456 q^{26} - 118098 q^{27} + 131712 q^{28} + 174185 q^{29} - 2808 q^{30} + 119045 q^{31} + 196608 q^{32} - 271458 q^{33} + 169776 q^{34} + 4459 q^{35} + 279936 q^{36} + 56562 q^{37} + 76216 q^{38} + 355914 q^{39} + 6656 q^{40} + 101632 q^{41} - 444528 q^{42} + 441323 q^{43} + 643456 q^{44} + 9477 q^{45} + 265832 q^{46} - 892849 q^{47} - 663552 q^{48} + 705894 q^{49} + 2122568 q^{50} - 572994 q^{51} - 843648 q^{52} + 2093965 q^{53} - 944784 q^{54} - 331222 q^{55} + 1053696 q^{56} - 257229 q^{57} + 1393480 q^{58} - 136204 q^{59} - 22464 q^{60} - 3989946 q^{61} + 952360 q^{62} + 1500282 q^{63} + 1572864 q^{64} - 28561 q^{65} - 2171664 q^{66} - 2218250 q^{67} + 1358208 q^{68} - 897183 q^{69} + 35672 q^{70} + 2045928 q^{71} + 2239488 q^{72} - 8557479 q^{73} + 452496 q^{74} - 7163667 q^{75} + 609728 q^{76} + 3448522 q^{77} + 2847312 q^{78} - 8559709 q^{79} + 53248 q^{80} + 3188646 q^{81} + 813056 q^{82} + 2496351 q^{83} - 3556224 q^{84} + 5335304 q^{85} + 3530584 q^{86} - 4702995 q^{87} + 5147648 q^{88} - 2446683 q^{89} + 75816 q^{90} - 4521426 q^{91} + 2126656 q^{92} - 3214215 q^{93} - 7142792 q^{94} + 16410211 q^{95} - 5308416 q^{96} + 5786889 q^{97} + 5647152 q^{98} + 7329366 q^{99} + O(q^{100}) \)

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
\(5\) −417.580 −1.49398 −0.746990 0.664835i \(-0.768502\pi\)
−0.746990 + 0.664835i \(0.768502\pi\)
\(6\) −216.000 −0.408248
\(7\) 343.000 0.377964
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) −3340.64 −1.05640
\(11\) 5214.78 1.18130 0.590651 0.806927i \(-0.298871\pi\)
0.590651 + 0.806927i \(0.298871\pi\)
\(12\) −1728.00 −0.288675
\(13\) −2197.00 −0.277350
\(14\) 2744.00 0.267261
\(15\) 11274.7 0.862550
\(16\) 4096.00 0.250000
\(17\) −9289.46 −0.458584 −0.229292 0.973358i \(-0.573641\pi\)
−0.229292 + 0.973358i \(0.573641\pi\)
\(18\) 5832.00 0.235702
\(19\) −28641.6 −0.957986 −0.478993 0.877819i \(-0.658998\pi\)
−0.478993 + 0.877819i \(0.658998\pi\)
\(20\) −26725.1 −0.746990
\(21\) −9261.00 −0.218218
\(22\) 41718.2 0.835307
\(23\) −70760.2 −1.21267 −0.606334 0.795210i \(-0.707360\pi\)
−0.606334 + 0.795210i \(0.707360\pi\)
\(24\) −13824.0 −0.204124
\(25\) 96248.1 1.23198
\(26\) −17576.0 −0.196116
\(27\) −19683.0 −0.192450
\(28\) 21952.0 0.188982
\(29\) 40590.0 0.309048 0.154524 0.987989i \(-0.450616\pi\)
0.154524 + 0.987989i \(0.450616\pi\)
\(30\) 90197.3 0.609915
\(31\) 239911. 1.44638 0.723192 0.690647i \(-0.242674\pi\)
0.723192 + 0.690647i \(0.242674\pi\)
\(32\) 32768.0 0.176777
\(33\) −140799. −0.682026
\(34\) −74315.7 −0.324268
\(35\) −143230. −0.564671
\(36\) 46656.0 0.166667
\(37\) 334019. 1.08409 0.542045 0.840350i \(-0.317650\pi\)
0.542045 + 0.840350i \(0.317650\pi\)
\(38\) −229132. −0.677398
\(39\) 59319.0 0.160128
\(40\) −213801. −0.528202
\(41\) −347373. −0.787141 −0.393571 0.919294i \(-0.628760\pi\)
−0.393571 + 0.919294i \(0.628760\pi\)
\(42\) −74088.0 −0.154303
\(43\) −454134. −0.871052 −0.435526 0.900176i \(-0.643438\pi\)
−0.435526 + 0.900176i \(0.643438\pi\)
\(44\) 333746. 0.590651
\(45\) −304416. −0.497993
\(46\) −566082. −0.857485
\(47\) −87902.0 −0.123497 −0.0617485 0.998092i \(-0.519668\pi\)
−0.0617485 + 0.998092i \(0.519668\pi\)
\(48\) −110592. −0.144338
\(49\) 117649. 0.142857
\(50\) 769985. 0.871138
\(51\) 250815. 0.264764
\(52\) −140608. −0.138675
\(53\) −2.03091e6 −1.87380 −0.936902 0.349592i \(-0.886320\pi\)
−0.936902 + 0.349592i \(0.886320\pi\)
\(54\) −157464. −0.136083
\(55\) −2.17759e6 −1.76484
\(56\) 175616. 0.133631
\(57\) 773322. 0.553093
\(58\) 324720. 0.218530
\(59\) −385876. −0.244605 −0.122303 0.992493i \(-0.539028\pi\)
−0.122303 + 0.992493i \(0.539028\pi\)
\(60\) 721578. 0.431275
\(61\) −2.16065e6 −1.21879 −0.609397 0.792865i \(-0.708589\pi\)
−0.609397 + 0.792865i \(0.708589\pi\)
\(62\) 1.91928e6 1.02275
\(63\) 250047. 0.125988
\(64\) 262144. 0.125000
\(65\) 917423. 0.414355
\(66\) −1.12639e6 −0.482265
\(67\) 1.94749e6 0.791066 0.395533 0.918452i \(-0.370560\pi\)
0.395533 + 0.918452i \(0.370560\pi\)
\(68\) −594525. −0.229292
\(69\) 1.91053e6 0.700134
\(70\) −1.14584e6 −0.399283
\(71\) 2.74370e6 0.909771 0.454885 0.890550i \(-0.349680\pi\)
0.454885 + 0.890550i \(0.349680\pi\)
\(72\) 373248. 0.117851
\(73\) −801961. −0.241281 −0.120641 0.992696i \(-0.538495\pi\)
−0.120641 + 0.992696i \(0.538495\pi\)
\(74\) 2.67215e6 0.766567
\(75\) −2.59870e6 −0.711281
\(76\) −1.83306e6 −0.478993
\(77\) 1.78867e6 0.446491
\(78\) 474552. 0.113228
\(79\) −7.49135e6 −1.70948 −0.854742 0.519053i \(-0.826285\pi\)
−0.854742 + 0.519053i \(0.826285\pi\)
\(80\) −1.71041e6 −0.373495
\(81\) 531441. 0.111111
\(82\) −2.77899e6 −0.556593
\(83\) 2.59240e6 0.497656 0.248828 0.968548i \(-0.419955\pi\)
0.248828 + 0.968548i \(0.419955\pi\)
\(84\) −592704. −0.109109
\(85\) 3.87909e6 0.685116
\(86\) −3.63307e6 −0.615927
\(87\) −1.09593e6 −0.178429
\(88\) 2.66997e6 0.417654
\(89\) 712280. 0.107099 0.0535495 0.998565i \(-0.482946\pi\)
0.0535495 + 0.998565i \(0.482946\pi\)
\(90\) −2.43533e6 −0.352134
\(91\) −753571. −0.104828
\(92\) −4.52865e6 −0.606334
\(93\) −6.47759e6 −0.835070
\(94\) −703216. −0.0873255
\(95\) 1.19601e7 1.43121
\(96\) −884736. −0.102062
\(97\) 1.71130e7 1.90381 0.951907 0.306387i \(-0.0991202\pi\)
0.951907 + 0.306387i \(0.0991202\pi\)
\(98\) 941192. 0.101015
\(99\) 3.80157e6 0.393768
\(100\) 6.15988e6 0.615988
\(101\) −8.28018e6 −0.799678 −0.399839 0.916585i \(-0.630934\pi\)
−0.399839 + 0.916585i \(0.630934\pi\)
\(102\) 2.00652e6 0.187216
\(103\) −1.79135e7 −1.61529 −0.807645 0.589669i \(-0.799258\pi\)
−0.807645 + 0.589669i \(0.799258\pi\)
\(104\) −1.12486e6 −0.0980581
\(105\) 3.86721e6 0.326013
\(106\) −1.62472e7 −1.32498
\(107\) 2.05021e7 1.61791 0.808957 0.587868i \(-0.200033\pi\)
0.808957 + 0.587868i \(0.200033\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) 2.18594e7 1.61676 0.808380 0.588661i \(-0.200345\pi\)
0.808380 + 0.588661i \(0.200345\pi\)
\(110\) −1.74207e7 −1.24793
\(111\) −9.01851e6 −0.625899
\(112\) 1.40493e6 0.0944911
\(113\) 1.74187e7 1.13564 0.567822 0.823151i \(-0.307786\pi\)
0.567822 + 0.823151i \(0.307786\pi\)
\(114\) 6.18658e6 0.391096
\(115\) 2.95481e7 1.81170
\(116\) 2.59776e6 0.154524
\(117\) −1.60161e6 −0.0924500
\(118\) −3.08701e6 −0.172962
\(119\) −3.18628e6 −0.173329
\(120\) 5.77263e6 0.304957
\(121\) 7.70672e6 0.395476
\(122\) −1.72852e7 −0.861818
\(123\) 9.37908e6 0.454456
\(124\) 1.53543e7 0.723192
\(125\) −7.56784e6 −0.346567
\(126\) 2.00038e6 0.0890871
\(127\) 2.64428e7 1.14550 0.572748 0.819731i \(-0.305877\pi\)
0.572748 + 0.819731i \(0.305877\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) 1.22616e7 0.502902
\(130\) 7.33939e6 0.292994
\(131\) 2.78528e7 1.08248 0.541238 0.840869i \(-0.317956\pi\)
0.541238 + 0.840869i \(0.317956\pi\)
\(132\) −9.01113e6 −0.341013
\(133\) −9.82406e6 −0.362085
\(134\) 1.55799e7 0.559368
\(135\) 8.21923e6 0.287517
\(136\) −4.75620e6 −0.162134
\(137\) 1.34607e7 0.447244 0.223622 0.974676i \(-0.428212\pi\)
0.223622 + 0.974676i \(0.428212\pi\)
\(138\) 1.52842e7 0.495069
\(139\) 1.64036e7 0.518068 0.259034 0.965868i \(-0.416596\pi\)
0.259034 + 0.965868i \(0.416596\pi\)
\(140\) −9.16672e6 −0.282336
\(141\) 2.37335e6 0.0713010
\(142\) 2.19496e7 0.643305
\(143\) −1.14569e7 −0.327634
\(144\) 2.98598e6 0.0833333
\(145\) −1.69496e7 −0.461711
\(146\) −6.41569e6 −0.170612
\(147\) −3.17652e6 −0.0824786
\(148\) 2.13772e7 0.542045
\(149\) 6.94840e6 0.172081 0.0860405 0.996292i \(-0.472579\pi\)
0.0860405 + 0.996292i \(0.472579\pi\)
\(150\) −2.07896e7 −0.502952
\(151\) 7.14181e7 1.68806 0.844032 0.536293i \(-0.180176\pi\)
0.844032 + 0.536293i \(0.180176\pi\)
\(152\) −1.46645e7 −0.338699
\(153\) −6.77201e6 −0.152861
\(154\) 1.43093e7 0.315716
\(155\) −1.00182e8 −2.16087
\(156\) 3.79642e6 0.0800641
\(157\) 3.30011e7 0.680580 0.340290 0.940321i \(-0.389475\pi\)
0.340290 + 0.940321i \(0.389475\pi\)
\(158\) −5.99308e7 −1.20879
\(159\) 5.48344e7 1.08184
\(160\) −1.36833e7 −0.264101
\(161\) −2.42708e7 −0.458345
\(162\) 4.25153e6 0.0785674
\(163\) 4.27115e7 0.772481 0.386241 0.922398i \(-0.373773\pi\)
0.386241 + 0.922398i \(0.373773\pi\)
\(164\) −2.22319e7 −0.393571
\(165\) 5.87948e7 1.01893
\(166\) 2.07392e7 0.351896
\(167\) −7.77575e6 −0.129192 −0.0645958 0.997912i \(-0.520576\pi\)
−0.0645958 + 0.997912i \(0.520576\pi\)
\(168\) −4.74163e6 −0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) 3.10327e7 0.484450
\(171\) −2.08797e7 −0.319329
\(172\) −2.90645e7 −0.435526
\(173\) −6.03299e7 −0.885873 −0.442937 0.896553i \(-0.646063\pi\)
−0.442937 + 0.896553i \(0.646063\pi\)
\(174\) −8.76743e6 −0.126168
\(175\) 3.30131e7 0.465643
\(176\) 2.13597e7 0.295326
\(177\) 1.04187e7 0.141223
\(178\) 5.69824e6 0.0757305
\(179\) 1.35910e8 1.77119 0.885594 0.464459i \(-0.153751\pi\)
0.885594 + 0.464459i \(0.153751\pi\)
\(180\) −1.94826e7 −0.248997
\(181\) −2.57944e7 −0.323334 −0.161667 0.986845i \(-0.551687\pi\)
−0.161667 + 0.986845i \(0.551687\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) 5.83376e7 0.703672
\(184\) −3.62292e7 −0.428743
\(185\) −1.39480e8 −1.61961
\(186\) −5.18207e7 −0.590484
\(187\) −4.84424e7 −0.541727
\(188\) −5.62573e6 −0.0617485
\(189\) −6.75127e6 −0.0727393
\(190\) 9.56812e7 1.01202
\(191\) −8.97084e7 −0.931572 −0.465786 0.884897i \(-0.654228\pi\)
−0.465786 + 0.884897i \(0.654228\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) −5.89536e7 −0.590282 −0.295141 0.955454i \(-0.595367\pi\)
−0.295141 + 0.955454i \(0.595367\pi\)
\(194\) 1.36904e8 1.34620
\(195\) −2.47704e7 −0.239228
\(196\) 7.52954e6 0.0714286
\(197\) 8.91164e7 0.830474 0.415237 0.909713i \(-0.363699\pi\)
0.415237 + 0.909713i \(0.363699\pi\)
\(198\) 3.04126e7 0.278436
\(199\) 1.64764e8 1.48210 0.741048 0.671453i \(-0.234329\pi\)
0.741048 + 0.671453i \(0.234329\pi\)
\(200\) 4.92790e7 0.435569
\(201\) −5.25821e7 −0.456722
\(202\) −6.62415e7 −0.565458
\(203\) 1.39224e7 0.116809
\(204\) 1.60522e7 0.132382
\(205\) 1.45056e8 1.17597
\(206\) −1.43308e8 −1.14218
\(207\) −5.15842e7 −0.404222
\(208\) −8.99891e6 −0.0693375
\(209\) −1.49359e8 −1.13167
\(210\) 3.09377e7 0.230526
\(211\) 2.06727e8 1.51499 0.757494 0.652842i \(-0.226424\pi\)
0.757494 + 0.652842i \(0.226424\pi\)
\(212\) −1.29978e8 −0.936902
\(213\) −7.40798e7 −0.525256
\(214\) 1.64017e8 1.14404
\(215\) 1.89637e8 1.30133
\(216\) −1.00777e7 −0.0680414
\(217\) 8.22893e7 0.546682
\(218\) 1.74875e8 1.14322
\(219\) 2.16530e7 0.139304
\(220\) −1.39366e8 −0.882421
\(221\) 2.04089e7 0.127188
\(222\) −7.21481e7 −0.442578
\(223\) −8.56828e7 −0.517400 −0.258700 0.965958i \(-0.583294\pi\)
−0.258700 + 0.965958i \(0.583294\pi\)
\(224\) 1.12394e7 0.0668153
\(225\) 7.01649e7 0.410659
\(226\) 1.39350e8 0.803021
\(227\) 3.19553e8 1.81323 0.906613 0.421964i \(-0.138659\pi\)
0.906613 + 0.421964i \(0.138659\pi\)
\(228\) 4.94926e7 0.276547
\(229\) 8.92189e7 0.490945 0.245473 0.969404i \(-0.421057\pi\)
0.245473 + 0.969404i \(0.421057\pi\)
\(230\) 2.36384e8 1.28107
\(231\) −4.82940e7 −0.257781
\(232\) 2.07821e7 0.109265
\(233\) −2.98130e8 −1.54405 −0.772023 0.635594i \(-0.780755\pi\)
−0.772023 + 0.635594i \(0.780755\pi\)
\(234\) −1.28129e7 −0.0653720
\(235\) 3.67061e7 0.184502
\(236\) −2.46961e7 −0.122303
\(237\) 2.02266e8 0.986971
\(238\) −2.54903e7 −0.122562
\(239\) −1.47766e8 −0.700135 −0.350067 0.936724i \(-0.613841\pi\)
−0.350067 + 0.936724i \(0.613841\pi\)
\(240\) 4.61810e7 0.215637
\(241\) −6.66547e7 −0.306741 −0.153370 0.988169i \(-0.549013\pi\)
−0.153370 + 0.988169i \(0.549013\pi\)
\(242\) 6.16537e7 0.279644
\(243\) −1.43489e7 −0.0641500
\(244\) −1.38282e8 −0.609397
\(245\) −4.91279e7 −0.213426
\(246\) 7.50326e7 0.321349
\(247\) 6.29255e7 0.265697
\(248\) 1.22834e8 0.511374
\(249\) −6.99949e7 −0.287322
\(250\) −6.05427e7 −0.245060
\(251\) −1.62351e8 −0.648034 −0.324017 0.946051i \(-0.605033\pi\)
−0.324017 + 0.946051i \(0.605033\pi\)
\(252\) 1.60030e7 0.0629941
\(253\) −3.68999e8 −1.43253
\(254\) 2.11542e8 0.809988
\(255\) −1.04735e8 −0.395552
\(256\) 1.67772e7 0.0625000
\(257\) −4.07344e8 −1.49691 −0.748454 0.663186i \(-0.769204\pi\)
−0.748454 + 0.663186i \(0.769204\pi\)
\(258\) 9.80928e7 0.355605
\(259\) 1.14569e8 0.409747
\(260\) 5.87151e7 0.207178
\(261\) 2.95901e7 0.103016
\(262\) 2.22822e8 0.765427
\(263\) 5.04618e8 1.71048 0.855240 0.518232i \(-0.173410\pi\)
0.855240 + 0.518232i \(0.173410\pi\)
\(264\) −7.20891e7 −0.241132
\(265\) 8.48065e8 2.79942
\(266\) −7.85924e7 −0.256032
\(267\) −1.92316e7 −0.0618337
\(268\) 1.24639e8 0.395533
\(269\) 1.19771e8 0.375161 0.187581 0.982249i \(-0.439935\pi\)
0.187581 + 0.982249i \(0.439935\pi\)
\(270\) 6.57538e7 0.203305
\(271\) 4.65730e8 1.42148 0.710741 0.703454i \(-0.248360\pi\)
0.710741 + 0.703454i \(0.248360\pi\)
\(272\) −3.80496e7 −0.114646
\(273\) 2.03464e7 0.0605228
\(274\) 1.07685e8 0.316249
\(275\) 5.01912e8 1.45534
\(276\) 1.22274e8 0.350067
\(277\) −1.74789e8 −0.494122 −0.247061 0.969000i \(-0.579465\pi\)
−0.247061 + 0.969000i \(0.579465\pi\)
\(278\) 1.31229e8 0.366330
\(279\) 1.74895e8 0.482128
\(280\) −7.33337e7 −0.199641
\(281\) 4.02412e8 1.08193 0.540965 0.841045i \(-0.318059\pi\)
0.540965 + 0.841045i \(0.318059\pi\)
\(282\) 1.89868e7 0.0504174
\(283\) 1.34858e8 0.353690 0.176845 0.984239i \(-0.443411\pi\)
0.176845 + 0.984239i \(0.443411\pi\)
\(284\) 1.75597e8 0.454885
\(285\) −3.22924e8 −0.826310
\(286\) −9.16549e7 −0.231673
\(287\) −1.19149e8 −0.297512
\(288\) 2.38879e7 0.0589256
\(289\) −3.24045e8 −0.789701
\(290\) −1.35596e8 −0.326479
\(291\) −4.62050e8 −1.09917
\(292\) −5.13255e7 −0.120641
\(293\) 1.37180e8 0.318605 0.159303 0.987230i \(-0.449075\pi\)
0.159303 + 0.987230i \(0.449075\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) 1.61134e8 0.365435
\(296\) 1.71018e8 0.383284
\(297\) −1.02642e8 −0.227342
\(298\) 5.55872e7 0.121680
\(299\) 1.55460e8 0.336333
\(300\) −1.66317e8 −0.355641
\(301\) −1.55768e8 −0.329227
\(302\) 5.71345e8 1.19364
\(303\) 2.23565e8 0.461694
\(304\) −1.17316e8 −0.239496
\(305\) 9.02246e8 1.82086
\(306\) −5.41761e7 −0.108089
\(307\) −5.62841e8 −1.11020 −0.555101 0.831783i \(-0.687320\pi\)
−0.555101 + 0.831783i \(0.687320\pi\)
\(308\) 1.14475e8 0.223245
\(309\) 4.83665e8 0.932588
\(310\) −8.01455e8 −1.52796
\(311\) −3.69566e8 −0.696676 −0.348338 0.937369i \(-0.613254\pi\)
−0.348338 + 0.937369i \(0.613254\pi\)
\(312\) 3.03713e7 0.0566139
\(313\) 1.69417e7 0.0312286 0.0156143 0.999878i \(-0.495030\pi\)
0.0156143 + 0.999878i \(0.495030\pi\)
\(314\) 2.64008e8 0.481243
\(315\) −1.04415e8 −0.188224
\(316\) −4.79446e8 −0.854742
\(317\) −2.87449e8 −0.506820 −0.253410 0.967359i \(-0.581552\pi\)
−0.253410 + 0.967359i \(0.581552\pi\)
\(318\) 4.38676e8 0.764977
\(319\) 2.11668e8 0.365079
\(320\) −1.09466e8 −0.186747
\(321\) −5.53557e8 −0.934103
\(322\) −1.94166e8 −0.324099
\(323\) 2.66065e8 0.439317
\(324\) 3.40122e7 0.0555556
\(325\) −2.11457e8 −0.341689
\(326\) 3.41692e8 0.546227
\(327\) −5.90204e8 −0.933437
\(328\) −1.77855e8 −0.278297
\(329\) −3.01504e7 −0.0466774
\(330\) 4.70359e8 0.720494
\(331\) −2.57572e8 −0.390391 −0.195196 0.980764i \(-0.562534\pi\)
−0.195196 + 0.980764i \(0.562534\pi\)
\(332\) 1.65914e8 0.248828
\(333\) 2.43500e8 0.361363
\(334\) −6.22060e7 −0.0913523
\(335\) −8.13232e8 −1.18184
\(336\) −3.79331e7 −0.0545545
\(337\) −7.89868e8 −1.12422 −0.562108 0.827064i \(-0.690010\pi\)
−0.562108 + 0.827064i \(0.690010\pi\)
\(338\) 3.86145e7 0.0543928
\(339\) −4.70306e8 −0.655664
\(340\) 2.48262e8 0.342558
\(341\) 1.25108e9 1.70862
\(342\) −1.67038e8 −0.225799
\(343\) 4.03536e7 0.0539949
\(344\) −2.32516e8 −0.307963
\(345\) −7.97797e8 −1.04599
\(346\) −4.82640e8 −0.626407
\(347\) −5.09917e8 −0.655158 −0.327579 0.944824i \(-0.606233\pi\)
−0.327579 + 0.944824i \(0.606233\pi\)
\(348\) −7.01395e7 −0.0892144
\(349\) 4.13805e8 0.521082 0.260541 0.965463i \(-0.416099\pi\)
0.260541 + 0.965463i \(0.416099\pi\)
\(350\) 2.64105e8 0.329259
\(351\) 4.32436e7 0.0533761
\(352\) 1.70878e8 0.208827
\(353\) −3.76671e7 −0.0455776 −0.0227888 0.999740i \(-0.507255\pi\)
−0.0227888 + 0.999740i \(0.507255\pi\)
\(354\) 8.33492e7 0.0998597
\(355\) −1.14571e9 −1.35918
\(356\) 4.55859e7 0.0535495
\(357\) 8.60297e7 0.100071
\(358\) 1.08728e9 1.25242
\(359\) −1.17417e8 −0.133937 −0.0669683 0.997755i \(-0.521333\pi\)
−0.0669683 + 0.997755i \(0.521333\pi\)
\(360\) −1.55861e8 −0.176067
\(361\) −7.35326e7 −0.0822631
\(362\) −2.06356e8 −0.228632
\(363\) −2.08081e8 −0.228328
\(364\) −4.82285e7 −0.0524142
\(365\) 3.34883e8 0.360469
\(366\) 4.66701e8 0.497571
\(367\) 7.55687e8 0.798015 0.399007 0.916948i \(-0.369355\pi\)
0.399007 + 0.916948i \(0.369355\pi\)
\(368\) −2.89834e8 −0.303167
\(369\) −2.53235e8 −0.262380
\(370\) −1.11584e9 −1.14524
\(371\) −6.96600e8 −0.708231
\(372\) −4.14565e8 −0.417535
\(373\) 4.77037e8 0.475961 0.237980 0.971270i \(-0.423515\pi\)
0.237980 + 0.971270i \(0.423515\pi\)
\(374\) −3.87540e8 −0.383059
\(375\) 2.04332e8 0.200090
\(376\) −4.50058e7 −0.0436627
\(377\) −8.91761e7 −0.0857144
\(378\) −5.40102e7 −0.0514344
\(379\) 1.36069e9 1.28387 0.641935 0.766759i \(-0.278132\pi\)
0.641935 + 0.766759i \(0.278132\pi\)
\(380\) 7.65449e8 0.715606
\(381\) −7.13954e8 −0.661353
\(382\) −7.17667e8 −0.658721
\(383\) −1.97066e9 −1.79233 −0.896163 0.443726i \(-0.853656\pi\)
−0.896163 + 0.443726i \(0.853656\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) −7.46912e8 −0.667048
\(386\) −4.71629e8 −0.417392
\(387\) −3.31063e8 −0.290351
\(388\) 1.09523e9 0.951907
\(389\) −1.56134e9 −1.34485 −0.672425 0.740165i \(-0.734747\pi\)
−0.672425 + 0.740165i \(0.734747\pi\)
\(390\) −1.98163e8 −0.169160
\(391\) 6.57324e8 0.556110
\(392\) 6.02363e7 0.0505076
\(393\) −7.52024e8 −0.624968
\(394\) 7.12931e8 0.587234
\(395\) 3.12824e9 2.55394
\(396\) 2.43301e8 0.196884
\(397\) 8.83178e8 0.708405 0.354203 0.935169i \(-0.384752\pi\)
0.354203 + 0.935169i \(0.384752\pi\)
\(398\) 1.31811e9 1.04800
\(399\) 2.65250e8 0.209050
\(400\) 3.94232e8 0.307994
\(401\) −5.92174e7 −0.0458611 −0.0229305 0.999737i \(-0.507300\pi\)
−0.0229305 + 0.999737i \(0.507300\pi\)
\(402\) −4.20657e8 −0.322951
\(403\) −5.27084e8 −0.401155
\(404\) −5.29932e8 −0.399839
\(405\) −2.21919e8 −0.165998
\(406\) 1.11379e8 0.0825965
\(407\) 1.74183e9 1.28064
\(408\) 1.28417e8 0.0936081
\(409\) 4.68060e8 0.338275 0.169137 0.985592i \(-0.445902\pi\)
0.169137 + 0.985592i \(0.445902\pi\)
\(410\) 1.16045e9 0.831539
\(411\) −3.63438e8 −0.258216
\(412\) −1.14646e9 −0.807645
\(413\) −1.32355e8 −0.0924521
\(414\) −4.12674e8 −0.285828
\(415\) −1.08254e9 −0.743488
\(416\) −7.19913e7 −0.0490290
\(417\) −4.42897e8 −0.299107
\(418\) −1.19487e9 −0.800213
\(419\) −1.92159e9 −1.27618 −0.638090 0.769962i \(-0.720275\pi\)
−0.638090 + 0.769962i \(0.720275\pi\)
\(420\) 2.47501e8 0.163007
\(421\) 2.94243e9 1.92185 0.960924 0.276813i \(-0.0892784\pi\)
0.960924 + 0.276813i \(0.0892784\pi\)
\(422\) 1.65382e9 1.07126
\(423\) −6.40805e7 −0.0411656
\(424\) −1.03982e9 −0.662490
\(425\) −8.94093e8 −0.564965
\(426\) −5.92639e8 −0.371412
\(427\) −7.41104e8 −0.460661
\(428\) 1.31213e9 0.808957
\(429\) 3.09335e8 0.189160
\(430\) 1.51710e9 0.920182
\(431\) 2.27957e9 1.37146 0.685728 0.727858i \(-0.259484\pi\)
0.685728 + 0.727858i \(0.259484\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) 5.87675e8 0.347880 0.173940 0.984756i \(-0.444350\pi\)
0.173940 + 0.984756i \(0.444350\pi\)
\(434\) 6.58315e8 0.386562
\(435\) 4.57638e8 0.266569
\(436\) 1.39900e9 0.808380
\(437\) 2.02668e9 1.16172
\(438\) 1.73224e8 0.0985026
\(439\) 1.73709e9 0.979932 0.489966 0.871741i \(-0.337009\pi\)
0.489966 + 0.871741i \(0.337009\pi\)
\(440\) −1.11492e9 −0.623966
\(441\) 8.57661e7 0.0476190
\(442\) 1.63272e8 0.0899358
\(443\) 1.11318e9 0.608350 0.304175 0.952616i \(-0.401619\pi\)
0.304175 + 0.952616i \(0.401619\pi\)
\(444\) −5.77185e8 −0.312950
\(445\) −2.97434e8 −0.160004
\(446\) −6.85462e8 −0.365857
\(447\) −1.87607e8 −0.0993511
\(448\) 8.99154e7 0.0472456
\(449\) −3.34446e8 −0.174367 −0.0871833 0.996192i \(-0.527787\pi\)
−0.0871833 + 0.996192i \(0.527787\pi\)
\(450\) 5.61319e8 0.290379
\(451\) −1.81147e9 −0.929852
\(452\) 1.11480e9 0.567822
\(453\) −1.92829e9 −0.974604
\(454\) 2.55642e9 1.28214
\(455\) 3.14676e8 0.156612
\(456\) 3.95941e8 0.195548
\(457\) 2.60831e8 0.127836 0.0639179 0.997955i \(-0.479640\pi\)
0.0639179 + 0.997955i \(0.479640\pi\)
\(458\) 7.13752e8 0.347151
\(459\) 1.82844e8 0.0882546
\(460\) 1.89108e9 0.905850
\(461\) 4.44614e8 0.211363 0.105682 0.994400i \(-0.466298\pi\)
0.105682 + 0.994400i \(0.466298\pi\)
\(462\) −3.86352e8 −0.182279
\(463\) −1.06605e9 −0.499167 −0.249583 0.968353i \(-0.580294\pi\)
−0.249583 + 0.968353i \(0.580294\pi\)
\(464\) 1.66256e8 0.0772620
\(465\) 2.70491e9 1.24758
\(466\) −2.38504e9 −1.09181
\(467\) 3.94192e9 1.79101 0.895507 0.445047i \(-0.146813\pi\)
0.895507 + 0.445047i \(0.146813\pi\)
\(468\) −1.02503e8 −0.0462250
\(469\) 6.67988e8 0.298995
\(470\) 2.93649e8 0.130463
\(471\) −8.91029e8 −0.392933
\(472\) −1.97569e8 −0.0864810
\(473\) −2.36820e9 −1.02898
\(474\) 1.61813e9 0.697894
\(475\) −2.75670e9 −1.18022
\(476\) −2.03922e8 −0.0866643
\(477\) −1.48053e9 −0.624601
\(478\) −1.18213e9 −0.495070
\(479\) −1.27553e9 −0.530296 −0.265148 0.964208i \(-0.585421\pi\)
−0.265148 + 0.964208i \(0.585421\pi\)
\(480\) 3.69448e8 0.152479
\(481\) −7.33840e8 −0.300672
\(482\) −5.33238e8 −0.216898
\(483\) 6.55310e8 0.264626
\(484\) 4.93230e8 0.197738
\(485\) −7.14604e9 −2.84426
\(486\) −1.14791e8 −0.0453609
\(487\) 8.31710e7 0.0326302 0.0163151 0.999867i \(-0.494807\pi\)
0.0163151 + 0.999867i \(0.494807\pi\)
\(488\) −1.10625e9 −0.430909
\(489\) −1.15321e9 −0.445992
\(490\) −3.93023e8 −0.150915
\(491\) 3.42177e8 0.130456 0.0652282 0.997870i \(-0.479222\pi\)
0.0652282 + 0.997870i \(0.479222\pi\)
\(492\) 6.00261e8 0.227228
\(493\) −3.77059e8 −0.141724
\(494\) 5.03404e8 0.187876
\(495\) −1.58746e9 −0.588281
\(496\) 9.82674e8 0.361596
\(497\) 9.41088e8 0.343861
\(498\) −5.59959e8 −0.203167
\(499\) 4.03473e9 1.45366 0.726830 0.686818i \(-0.240993\pi\)
0.726830 + 0.686818i \(0.240993\pi\)
\(500\) −4.84342e8 −0.173283
\(501\) 2.09945e8 0.0745888
\(502\) −1.29881e9 −0.458229
\(503\) 4.57881e9 1.60422 0.802111 0.597174i \(-0.203710\pi\)
0.802111 + 0.597174i \(0.203710\pi\)
\(504\) 1.28024e8 0.0445435
\(505\) 3.45764e9 1.19470
\(506\) −2.95199e9 −1.01295
\(507\) −1.30324e8 −0.0444116
\(508\) 1.69234e9 0.572748
\(509\) −9.10655e8 −0.306085 −0.153042 0.988220i \(-0.548907\pi\)
−0.153042 + 0.988220i \(0.548907\pi\)
\(510\) −8.37884e8 −0.279697
\(511\) −2.75073e8 −0.0911957
\(512\) 1.34218e8 0.0441942
\(513\) 5.63752e8 0.184364
\(514\) −3.25875e9 −1.05847
\(515\) 7.48032e9 2.41321
\(516\) 7.84743e8 0.251451
\(517\) −4.58389e8 −0.145887
\(518\) 9.16548e8 0.289735
\(519\) 1.62891e9 0.511459
\(520\) 4.69721e8 0.146497
\(521\) 3.78933e9 1.17390 0.586949 0.809624i \(-0.300329\pi\)
0.586949 + 0.809624i \(0.300329\pi\)
\(522\) 2.36721e8 0.0728433
\(523\) 7.58145e8 0.231738 0.115869 0.993265i \(-0.463035\pi\)
0.115869 + 0.993265i \(0.463035\pi\)
\(524\) 1.78258e9 0.541238
\(525\) −8.91354e8 −0.268839
\(526\) 4.03695e9 1.20949
\(527\) −2.22864e9 −0.663289
\(528\) −5.76713e8 −0.170506
\(529\) 1.60218e9 0.470562
\(530\) 6.78452e9 1.97949
\(531\) −2.81304e8 −0.0815351
\(532\) −6.28740e8 −0.181042
\(533\) 7.63179e8 0.218314
\(534\) −1.53853e8 −0.0437230
\(535\) −8.56127e9 −2.41713
\(536\) 9.97113e8 0.279684
\(537\) −3.66956e9 −1.02260
\(538\) 9.58167e8 0.265279
\(539\) 6.13513e8 0.168758
\(540\) 5.26031e8 0.143758
\(541\) −2.69793e9 −0.732554 −0.366277 0.930506i \(-0.619368\pi\)
−0.366277 + 0.930506i \(0.619368\pi\)
\(542\) 3.72584e9 1.00514
\(543\) 6.96450e8 0.186677
\(544\) −3.04397e8 −0.0810670
\(545\) −9.12805e9 −2.41541
\(546\) 1.62771e8 0.0427960
\(547\) 4.73299e9 1.23646 0.618230 0.785997i \(-0.287850\pi\)
0.618230 + 0.785997i \(0.287850\pi\)
\(548\) 8.61483e8 0.223622
\(549\) −1.57512e9 −0.406265
\(550\) 4.01530e9 1.02908
\(551\) −1.16256e9 −0.296063
\(552\) 9.78189e8 0.247535
\(553\) −2.56953e9 −0.646124
\(554\) −1.39831e9 −0.349397
\(555\) 3.76595e9 0.935081
\(556\) 1.04983e9 0.259034
\(557\) −7.53882e9 −1.84846 −0.924231 0.381834i \(-0.875293\pi\)
−0.924231 + 0.381834i \(0.875293\pi\)
\(558\) 1.39916e9 0.340916
\(559\) 9.97731e8 0.241586
\(560\) −5.86670e8 −0.141168
\(561\) 1.30795e9 0.312766
\(562\) 3.21930e9 0.765040
\(563\) −5.85241e9 −1.38215 −0.691076 0.722782i \(-0.742863\pi\)
−0.691076 + 0.722782i \(0.742863\pi\)
\(564\) 1.51895e8 0.0356505
\(565\) −7.27372e9 −1.69663
\(566\) 1.07886e9 0.250097
\(567\) 1.82284e8 0.0419961
\(568\) 1.40477e9 0.321653
\(569\) 1.36162e8 0.0309857 0.0154929 0.999880i \(-0.495068\pi\)
0.0154929 + 0.999880i \(0.495068\pi\)
\(570\) −2.58339e9 −0.584290
\(571\) 2.69163e9 0.605046 0.302523 0.953142i \(-0.402171\pi\)
0.302523 + 0.953142i \(0.402171\pi\)
\(572\) −7.33239e8 −0.163817
\(573\) 2.42213e9 0.537843
\(574\) −9.53192e8 −0.210372
\(575\) −6.81054e9 −1.49398
\(576\) 1.91103e8 0.0416667
\(577\) −1.66238e9 −0.360259 −0.180130 0.983643i \(-0.557652\pi\)
−0.180130 + 0.983643i \(0.557652\pi\)
\(578\) −2.59236e9 −0.558403
\(579\) 1.59175e9 0.340800
\(580\) −1.08477e9 −0.230856
\(581\) 8.89195e8 0.188096
\(582\) −3.69640e9 −0.777229
\(583\) −1.05907e10 −2.21353
\(584\) −4.10604e8 −0.0853058
\(585\) 6.68802e8 0.138118
\(586\) 1.09744e9 0.225288
\(587\) −5.21278e9 −1.06374 −0.531871 0.846825i \(-0.678511\pi\)
−0.531871 + 0.846825i \(0.678511\pi\)
\(588\) −2.03297e8 −0.0412393
\(589\) −6.87141e9 −1.38561
\(590\) 1.28907e9 0.258402
\(591\) −2.40614e9 −0.479474
\(592\) 1.36814e9 0.271022
\(593\) 2.40644e9 0.473897 0.236948 0.971522i \(-0.423853\pi\)
0.236948 + 0.971522i \(0.423853\pi\)
\(594\) −8.21139e8 −0.160755
\(595\) 1.33053e9 0.258949
\(596\) 4.44698e8 0.0860405
\(597\) −4.44862e9 −0.855688
\(598\) 1.24368e9 0.237824
\(599\) 3.57730e9 0.680082 0.340041 0.940411i \(-0.389559\pi\)
0.340041 + 0.940411i \(0.389559\pi\)
\(600\) −1.33053e9 −0.251476
\(601\) 7.94216e9 1.49238 0.746188 0.665736i \(-0.231882\pi\)
0.746188 + 0.665736i \(0.231882\pi\)
\(602\) −1.24614e9 −0.232798
\(603\) 1.41972e9 0.263689
\(604\) 4.57076e9 0.844032
\(605\) −3.21817e9 −0.590834
\(606\) 1.78852e9 0.326467
\(607\) 1.49061e9 0.270523 0.135261 0.990810i \(-0.456813\pi\)
0.135261 + 0.990810i \(0.456813\pi\)
\(608\) −9.38527e8 −0.169350
\(609\) −3.75904e8 −0.0674398
\(610\) 7.21797e9 1.28754
\(611\) 1.93121e8 0.0342519
\(612\) −4.33409e8 −0.0764307
\(613\) 1.08594e10 1.90411 0.952056 0.305923i \(-0.0989650\pi\)
0.952056 + 0.305923i \(0.0989650\pi\)
\(614\) −4.50273e9 −0.785031
\(615\) −3.91651e9 −0.678949
\(616\) 9.15798e8 0.157858
\(617\) −5.50961e9 −0.944328 −0.472164 0.881511i \(-0.656527\pi\)
−0.472164 + 0.881511i \(0.656527\pi\)
\(618\) 3.86932e9 0.659439
\(619\) −3.31953e9 −0.562548 −0.281274 0.959627i \(-0.590757\pi\)
−0.281274 + 0.959627i \(0.590757\pi\)
\(620\) −6.41164e9 −1.08043
\(621\) 1.39277e9 0.233378
\(622\) −2.95653e9 −0.492625
\(623\) 2.44312e8 0.0404797
\(624\) 2.42971e8 0.0400320
\(625\) −4.35920e9 −0.714212
\(626\) 1.35534e8 0.0220819
\(627\) 4.03270e9 0.653371
\(628\) 2.11207e9 0.340290
\(629\) −3.10286e9 −0.497146
\(630\) −8.35317e8 −0.133094
\(631\) −6.04983e9 −0.958607 −0.479303 0.877649i \(-0.659111\pi\)
−0.479303 + 0.877649i \(0.659111\pi\)
\(632\) −3.83557e9 −0.604394
\(633\) −5.58163e9 −0.874678
\(634\) −2.29959e9 −0.358376
\(635\) −1.10420e10 −1.71135
\(636\) 3.50940e9 0.540921
\(637\) −2.58475e8 −0.0396214
\(638\) 1.69334e9 0.258150
\(639\) 2.00016e9 0.303257
\(640\) −8.75729e8 −0.132050
\(641\) 5.52551e8 0.0828647 0.0414323 0.999141i \(-0.486808\pi\)
0.0414323 + 0.999141i \(0.486808\pi\)
\(642\) −4.42846e9 −0.660510
\(643\) −1.22576e10 −1.81831 −0.909156 0.416456i \(-0.863272\pi\)
−0.909156 + 0.416456i \(0.863272\pi\)
\(644\) −1.55333e9 −0.229173
\(645\) −5.12020e9 −0.751325
\(646\) 2.12852e9 0.310644
\(647\) −9.44182e9 −1.37054 −0.685269 0.728290i \(-0.740315\pi\)
−0.685269 + 0.728290i \(0.740315\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) −2.01226e9 −0.288953
\(650\) −1.69166e9 −0.241610
\(651\) −2.22181e9 −0.315627
\(652\) 2.73353e9 0.386241
\(653\) −1.47622e9 −0.207470 −0.103735 0.994605i \(-0.533079\pi\)
−0.103735 + 0.994605i \(0.533079\pi\)
\(654\) −4.72163e9 −0.660040
\(655\) −1.16308e10 −1.61720
\(656\) −1.42284e9 −0.196785
\(657\) −5.84630e8 −0.0804271
\(658\) −2.41203e8 −0.0330059
\(659\) −1.04164e10 −1.41781 −0.708906 0.705303i \(-0.750811\pi\)
−0.708906 + 0.705303i \(0.750811\pi\)
\(660\) 3.76287e9 0.509466
\(661\) −1.98552e9 −0.267405 −0.133702 0.991022i \(-0.542687\pi\)
−0.133702 + 0.991022i \(0.542687\pi\)
\(662\) −2.06057e9 −0.276048
\(663\) −5.51041e8 −0.0734322
\(664\) 1.32731e9 0.175948
\(665\) 4.10233e9 0.540947
\(666\) 1.94800e9 0.255522
\(667\) −2.87215e9 −0.374772
\(668\) −4.97648e8 −0.0645958
\(669\) 2.31343e9 0.298721
\(670\) −6.50585e9 −0.835685
\(671\) −1.12673e10 −1.43977
\(672\) −3.03464e8 −0.0385758
\(673\) 4.00058e9 0.505907 0.252953 0.967478i \(-0.418598\pi\)
0.252953 + 0.967478i \(0.418598\pi\)
\(674\) −6.31895e9 −0.794941
\(675\) −1.89445e9 −0.237094
\(676\) 3.08916e8 0.0384615
\(677\) 4.72498e9 0.585247 0.292624 0.956228i \(-0.405472\pi\)
0.292624 + 0.956228i \(0.405472\pi\)
\(678\) −3.76245e9 −0.463625
\(679\) 5.86975e9 0.719574
\(680\) 1.98610e9 0.242225
\(681\) −8.62792e9 −1.04687
\(682\) 1.00086e10 1.20817
\(683\) −7.14183e9 −0.857704 −0.428852 0.903375i \(-0.641082\pi\)
−0.428852 + 0.903375i \(0.641082\pi\)
\(684\) −1.33630e9 −0.159664
\(685\) −5.62091e9 −0.668174
\(686\) 3.22829e8 0.0381802
\(687\) −2.40891e9 −0.283447
\(688\) −1.86013e9 −0.217763
\(689\) 4.46190e9 0.519700
\(690\) −6.38238e9 −0.739624
\(691\) 1.21613e10 1.40219 0.701096 0.713067i \(-0.252694\pi\)
0.701096 + 0.713067i \(0.252694\pi\)
\(692\) −3.86112e9 −0.442937
\(693\) 1.30394e9 0.148830
\(694\) −4.07933e9 −0.463267
\(695\) −6.84981e9 −0.773983
\(696\) −5.61116e8 −0.0630841
\(697\) 3.22691e9 0.360971
\(698\) 3.31044e9 0.368461
\(699\) 8.04952e9 0.891456
\(700\) 2.11284e9 0.232821
\(701\) −5.36586e9 −0.588338 −0.294169 0.955753i \(-0.595043\pi\)
−0.294169 + 0.955753i \(0.595043\pi\)
\(702\) 3.45948e8 0.0377426
\(703\) −9.56683e9 −1.03854
\(704\) 1.36702e9 0.147663
\(705\) −9.91065e8 −0.106522
\(706\) −3.01337e8 −0.0322282
\(707\) −2.84010e9 −0.302250
\(708\) 6.66794e8 0.0706115
\(709\) −1.47913e8 −0.0155864 −0.00779319 0.999970i \(-0.502481\pi\)
−0.00779319 + 0.999970i \(0.502481\pi\)
\(710\) −9.16571e9 −0.961085
\(711\) −5.46119e9 −0.569828
\(712\) 3.64687e8 0.0378652
\(713\) −1.69761e10 −1.75398
\(714\) 6.88237e8 0.0707611
\(715\) 4.78416e9 0.489479
\(716\) 8.69822e9 0.885594
\(717\) 3.98968e9 0.404223
\(718\) −9.39333e8 −0.0947075
\(719\) 1.37582e10 1.38042 0.690209 0.723610i \(-0.257518\pi\)
0.690209 + 0.723610i \(0.257518\pi\)
\(720\) −1.24689e9 −0.124498
\(721\) −6.14433e9 −0.610522
\(722\) −5.88261e8 −0.0581688
\(723\) 1.79968e9 0.177097
\(724\) −1.65084e9 −0.161667
\(725\) 3.90671e9 0.380739
\(726\) −1.66465e9 −0.161453
\(727\) 1.08983e10 1.05193 0.525966 0.850506i \(-0.323704\pi\)
0.525966 + 0.850506i \(0.323704\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) 2.67906e9 0.254890
\(731\) 4.21865e9 0.399451
\(732\) 3.73361e9 0.351836
\(733\) 1.46717e10 1.37599 0.687996 0.725714i \(-0.258490\pi\)
0.687996 + 0.725714i \(0.258490\pi\)
\(734\) 6.04550e9 0.564282
\(735\) 1.32645e9 0.123221
\(736\) −2.31867e9 −0.214371
\(737\) 1.01557e10 0.934488
\(738\) −2.02588e9 −0.185531
\(739\) 1.07386e10 0.978797 0.489398 0.872060i \(-0.337216\pi\)
0.489398 + 0.872060i \(0.337216\pi\)
\(740\) −8.92670e9 −0.809804
\(741\) −1.69899e9 −0.153401
\(742\) −5.57280e9 −0.500795
\(743\) 1.26609e10 1.13241 0.566205 0.824264i \(-0.308411\pi\)
0.566205 + 0.824264i \(0.308411\pi\)
\(744\) −3.31652e9 −0.295242
\(745\) −2.90151e9 −0.257086
\(746\) 3.81630e9 0.336555
\(747\) 1.88986e9 0.165885
\(748\) −3.10032e9 −0.270863
\(749\) 7.03222e9 0.611514
\(750\) 1.63465e9 0.141485
\(751\) −6.84407e7 −0.00589623 −0.00294812 0.999996i \(-0.500938\pi\)
−0.00294812 + 0.999996i \(0.500938\pi\)
\(752\) −3.60046e8 −0.0308742
\(753\) 4.38348e9 0.374142
\(754\) −7.13409e8 −0.0606093
\(755\) −2.98228e10 −2.52193
\(756\) −4.32081e8 −0.0363696
\(757\) 1.57268e10 1.31766 0.658830 0.752292i \(-0.271051\pi\)
0.658830 + 0.752292i \(0.271051\pi\)
\(758\) 1.08855e10 0.907833
\(759\) 9.96296e9 0.827070
\(760\) 6.12359e9 0.506010
\(761\) 1.32261e10 1.08789 0.543945 0.839121i \(-0.316930\pi\)
0.543945 + 0.839121i \(0.316930\pi\)
\(762\) −5.71164e9 −0.467647
\(763\) 7.49778e9 0.611078
\(764\) −5.74134e9 −0.465786
\(765\) 2.82786e9 0.228372
\(766\) −1.57653e10 −1.26737
\(767\) 8.47770e8 0.0678413
\(768\) −4.52985e8 −0.0360844
\(769\) 1.28832e10 1.02160 0.510801 0.859699i \(-0.329349\pi\)
0.510801 + 0.859699i \(0.329349\pi\)
\(770\) −5.97530e9 −0.471674
\(771\) 1.09983e10 0.864241
\(772\) −3.77303e9 −0.295141
\(773\) 7.09192e9 0.552249 0.276125 0.961122i \(-0.410950\pi\)
0.276125 + 0.961122i \(0.410950\pi\)
\(774\) −2.64851e9 −0.205309
\(775\) 2.30909e10 1.78191
\(776\) 8.76184e9 0.673100
\(777\) −3.09335e9 −0.236568
\(778\) −1.24907e10 −0.950952
\(779\) 9.94931e9 0.754070
\(780\) −1.58531e9 −0.119614
\(781\) 1.43078e10 1.07471
\(782\) 5.25859e9 0.393229
\(783\) −7.98932e8 −0.0594763
\(784\) 4.81890e8 0.0357143
\(785\) −1.37806e10 −1.01677
\(786\) −6.01620e9 −0.441919
\(787\) −3.46049e9 −0.253061 −0.126531 0.991963i \(-0.540384\pi\)
−0.126531 + 0.991963i \(0.540384\pi\)
\(788\) 5.70345e9 0.415237
\(789\) −1.36247e10 −0.987546
\(790\) 2.50259e10 1.80591
\(791\) 5.97463e9 0.429233
\(792\) 1.94640e9 0.139218
\(793\) 4.74696e9 0.338033
\(794\) 7.06543e9 0.500918
\(795\) −2.28978e10 −1.61625
\(796\) 1.05449e10 0.741048
\(797\) −1.22093e10 −0.854253 −0.427126 0.904192i \(-0.640474\pi\)
−0.427126 + 0.904192i \(0.640474\pi\)
\(798\) 2.12200e9 0.147820
\(799\) 8.16562e8 0.0566337
\(800\) 3.15386e9 0.217785
\(801\) 5.19252e8 0.0356997
\(802\) −4.73739e8 −0.0324287
\(803\) −4.18205e9 −0.285026
\(804\) −3.36526e9 −0.228361
\(805\) 1.01350e10 0.684759
\(806\) −4.21667e9 −0.283659
\(807\) −3.23381e9 −0.216599
\(808\) −4.23945e9 −0.282729
\(809\) −1.33753e10 −0.888142 −0.444071 0.895992i \(-0.646466\pi\)
−0.444071 + 0.895992i \(0.646466\pi\)
\(810\) −1.77535e9 −0.117378
\(811\) −2.49686e10 −1.64369 −0.821847 0.569709i \(-0.807056\pi\)
−0.821847 + 0.569709i \(0.807056\pi\)
\(812\) 8.91031e8 0.0584045
\(813\) −1.25747e10 −0.820693
\(814\) 1.39347e10 0.905548
\(815\) −1.78355e10 −1.15407
\(816\) 1.02734e9 0.0661909
\(817\) 1.30071e10 0.834455
\(818\) 3.74448e9 0.239196
\(819\) −5.49353e8 −0.0349428
\(820\) 9.28359e9 0.587987
\(821\) 2.22509e10 1.40329 0.701643 0.712529i \(-0.252450\pi\)
0.701643 + 0.712529i \(0.252450\pi\)
\(822\) −2.90750e9 −0.182587
\(823\) 5.96037e9 0.372712 0.186356 0.982482i \(-0.440332\pi\)
0.186356 + 0.982482i \(0.440332\pi\)
\(824\) −9.17172e9 −0.571091
\(825\) −1.35516e10 −0.840239
\(826\) −1.05884e9 −0.0653735
\(827\) −2.18600e10 −1.34395 −0.671973 0.740576i \(-0.734553\pi\)
−0.671973 + 0.740576i \(0.734553\pi\)
\(828\) −3.30139e9 −0.202111
\(829\) 1.31510e10 0.801714 0.400857 0.916141i \(-0.368712\pi\)
0.400857 + 0.916141i \(0.368712\pi\)
\(830\) −8.66029e9 −0.525726
\(831\) 4.71929e9 0.285281
\(832\) −5.75930e8 −0.0346688
\(833\) −1.09290e9 −0.0655120
\(834\) −3.54317e9 −0.211500
\(835\) 3.24700e9 0.193010
\(836\) −9.55900e9 −0.565836
\(837\) −4.72216e9 −0.278357
\(838\) −1.53727e10 −0.902395
\(839\) −1.98047e9 −0.115772 −0.0578858 0.998323i \(-0.518436\pi\)
−0.0578858 + 0.998323i \(0.518436\pi\)
\(840\) 1.98001e9 0.115263
\(841\) −1.56023e10 −0.904489
\(842\) 2.35394e10 1.35895
\(843\) −1.08651e10 −0.624653
\(844\) 1.32305e10 0.757494
\(845\) −2.01558e9 −0.114922
\(846\) −5.12644e8 −0.0291085
\(847\) 2.64340e9 0.149476
\(848\) −8.31859e9 −0.468451
\(849\) −3.64116e9 −0.204203
\(850\) −7.15274e9 −0.399490
\(851\) −2.36353e10 −1.31464
\(852\) −4.74111e9 −0.262628
\(853\) −1.15271e10 −0.635916 −0.317958 0.948105i \(-0.602997\pi\)
−0.317958 + 0.948105i \(0.602997\pi\)
\(854\) −5.92883e9 −0.325737
\(855\) 8.71895e9 0.477071
\(856\) 1.04971e10 0.572019
\(857\) 1.48622e10 0.806586 0.403293 0.915071i \(-0.367866\pi\)
0.403293 + 0.915071i \(0.367866\pi\)
\(858\) 2.47468e9 0.133756
\(859\) 3.35830e10 1.80777 0.903885 0.427776i \(-0.140703\pi\)
0.903885 + 0.427776i \(0.140703\pi\)
\(860\) 1.21368e10 0.650667
\(861\) 3.21702e9 0.171768
\(862\) 1.82365e10 0.969765
\(863\) 2.97813e10 1.57727 0.788635 0.614862i \(-0.210788\pi\)
0.788635 + 0.614862i \(0.210788\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) 2.51926e10 1.32348
\(866\) 4.70140e9 0.245988
\(867\) 8.74921e9 0.455934
\(868\) 5.26652e9 0.273341
\(869\) −3.90657e10 −2.01942
\(870\) 3.66110e9 0.188493
\(871\) −4.27863e9 −0.219402
\(872\) 1.11920e10 0.571611
\(873\) 1.24754e10 0.634605
\(874\) 1.62135e10 0.821459
\(875\) −2.59577e9 −0.130990
\(876\) 1.38579e9 0.0696519
\(877\) 1.84914e9 0.0925704 0.0462852 0.998928i \(-0.485262\pi\)
0.0462852 + 0.998928i \(0.485262\pi\)
\(878\) 1.38967e10 0.692917
\(879\) −3.70385e9 −0.183947
\(880\) −8.91939e9 −0.441211
\(881\) 1.54420e10 0.760831 0.380415 0.924816i \(-0.375781\pi\)
0.380415 + 0.924816i \(0.375781\pi\)
\(882\) 6.86129e8 0.0336718
\(883\) −2.81407e10 −1.37554 −0.687768 0.725931i \(-0.741409\pi\)
−0.687768 + 0.725931i \(0.741409\pi\)
\(884\) 1.30617e9 0.0635942
\(885\) −4.35062e9 −0.210984
\(886\) 8.90546e9 0.430168
\(887\) 7.47660e9 0.359726 0.179863 0.983692i \(-0.442435\pi\)
0.179863 + 0.983692i \(0.442435\pi\)
\(888\) −4.61748e9 −0.221289
\(889\) 9.06986e9 0.432957
\(890\) −2.37947e9 −0.113140
\(891\) 2.77135e9 0.131256
\(892\) −5.48370e9 −0.258700
\(893\) 2.51765e9 0.118308
\(894\) −1.50086e9 −0.0702518
\(895\) −5.67532e10 −2.64612
\(896\) 7.19323e8 0.0334077
\(897\) −4.19743e9 −0.194182
\(898\) −2.67557e9 −0.123296
\(899\) 9.73796e9 0.447002
\(900\) 4.49055e9 0.205329
\(901\) 1.88660e10 0.859297
\(902\) −1.44918e10 −0.657505
\(903\) 4.20573e9 0.190079
\(904\) 8.91839e9 0.401511
\(905\) 1.07712e10 0.483054
\(906\) −1.54263e10 −0.689149
\(907\) 1.22547e10 0.545350 0.272675 0.962106i \(-0.412092\pi\)
0.272675 + 0.962106i \(0.412092\pi\)
\(908\) 2.04514e10 0.906613
\(909\) −6.03625e9 −0.266559
\(910\) 2.51741e9 0.110741
\(911\) −1.25825e10 −0.551381 −0.275691 0.961246i \(-0.588907\pi\)
−0.275691 + 0.961246i \(0.588907\pi\)
\(912\) 3.16753e9 0.138273
\(913\) 1.35188e10 0.587883
\(914\) 2.08665e9 0.0903936
\(915\) −2.43606e10 −1.05127
\(916\) 5.71001e9 0.245473
\(917\) 9.55350e9 0.409138
\(918\) 1.46276e9 0.0624054
\(919\) −1.59682e8 −0.00678658 −0.00339329 0.999994i \(-0.501080\pi\)
−0.00339329 + 0.999994i \(0.501080\pi\)
\(920\) 1.51286e10 0.640533
\(921\) 1.51967e10 0.640975
\(922\) 3.55691e9 0.149456
\(923\) −6.02790e9 −0.252325
\(924\) −3.09082e9 −0.128891
\(925\) 3.21487e10 1.33557
\(926\) −8.52843e9 −0.352964
\(927\) −1.30589e10 −0.538430
\(928\) 1.33005e9 0.0546324
\(929\) −5.05924e9 −0.207028 −0.103514 0.994628i \(-0.533009\pi\)
−0.103514 + 0.994628i \(0.533009\pi\)
\(930\) 2.16393e10 0.882171
\(931\) −3.36965e9 −0.136855
\(932\) −1.90803e10 −0.772023
\(933\) 9.97829e9 0.402226
\(934\) 3.15354e10 1.26644
\(935\) 2.02286e10 0.809329
\(936\) −8.20026e8 −0.0326860
\(937\) 2.59671e9 0.103118 0.0515591 0.998670i \(-0.483581\pi\)
0.0515591 + 0.998670i \(0.483581\pi\)
\(938\) 5.34390e9 0.211421
\(939\) −4.57426e8 −0.0180298
\(940\) 2.34919e9 0.0922509
\(941\) 4.20695e10 1.64590 0.822950 0.568114i \(-0.192327\pi\)
0.822950 + 0.568114i \(0.192327\pi\)
\(942\) −7.12823e9 −0.277846
\(943\) 2.45802e10 0.954541
\(944\) −1.58055e9 −0.0611513
\(945\) 2.81920e9 0.108671
\(946\) −1.89456e10 −0.727596
\(947\) −3.01808e10 −1.15480 −0.577398 0.816463i \(-0.695932\pi\)
−0.577398 + 0.816463i \(0.695932\pi\)
\(948\) 1.29451e10 0.493486
\(949\) 1.76191e9 0.0669194
\(950\) −2.20536e10 −0.834538
\(951\) 7.76113e9 0.292613
\(952\) −1.63138e9 −0.0612809
\(953\) −4.11878e9 −0.154150 −0.0770751 0.997025i \(-0.524558\pi\)
−0.0770751 + 0.997025i \(0.524558\pi\)
\(954\) −1.18442e10 −0.441660
\(955\) 3.74604e10 1.39175
\(956\) −9.45702e9 −0.350067
\(957\) −5.71502e9 −0.210778
\(958\) −1.02043e10 −0.374976
\(959\) 4.61701e9 0.169042
\(960\) 2.95558e9 0.107819
\(961\) 3.00445e10 1.09203
\(962\) −5.87072e9 −0.212607
\(963\) 1.49460e10 0.539304
\(964\) −4.26590e9 −0.153370
\(965\) 2.46178e10 0.881870
\(966\) 5.24248e9 0.187119
\(967\) −5.03708e10 −1.79138 −0.895688 0.444684i \(-0.853316\pi\)
−0.895688 + 0.444684i \(0.853316\pi\)
\(968\) 3.94584e9 0.139822
\(969\) −7.18374e9 −0.253640
\(970\) −5.71683e10 −2.01120
\(971\) 2.36143e10 0.827765 0.413882 0.910330i \(-0.364172\pi\)
0.413882 + 0.910330i \(0.364172\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) 5.62643e9 0.195811
\(974\) 6.65368e8 0.0230731
\(975\) 5.70934e9 0.197274
\(976\) −8.85004e9 −0.304699
\(977\) −3.85449e9 −0.132232 −0.0661159 0.997812i \(-0.521061\pi\)
−0.0661159 + 0.997812i \(0.521061\pi\)
\(978\) −9.22568e9 −0.315364
\(979\) 3.71438e9 0.126516
\(980\) −3.14418e9 −0.106713
\(981\) 1.59355e10 0.538920
\(982\) 2.73741e9 0.0922466
\(983\) 3.36188e10 1.12887 0.564436 0.825477i \(-0.309094\pi\)
0.564436 + 0.825477i \(0.309094\pi\)
\(984\) 4.80209e9 0.160675
\(985\) −3.72132e10 −1.24071
\(986\) −3.01647e9 −0.100214
\(987\) 8.14060e8 0.0269492
\(988\) 4.02723e9 0.132849
\(989\) 3.21346e10 1.05630
\(990\) −1.26997e10 −0.415977
\(991\) −2.11989e10 −0.691918 −0.345959 0.938250i \(-0.612446\pi\)
−0.345959 + 0.938250i \(0.612446\pi\)
\(992\) 7.86139e9 0.255687
\(993\) 6.95443e9 0.225392
\(994\) 7.52871e9 0.243146
\(995\) −6.88021e10 −2.21422
\(996\) −4.47968e9 −0.143661
\(997\) −2.65730e9 −0.0849196 −0.0424598 0.999098i \(-0.513519\pi\)
−0.0424598 + 0.999098i \(0.513519\pi\)
\(998\) 3.22779e10 1.02789
\(999\) −6.57450e9 −0.208633
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.q.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.q.1.1 6 1.1 even 1 trivial