Properties

Label 91.8.a.e.1.5
Level $91$
Weight $8$
Character 91.1
Self dual yes
Analytic conductor $28.427$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.4270373191\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 6 x^{11} - 1243 x^{10} + 5598 x^{9} + 567554 x^{8} - 1739560 x^{7} - 117081910 x^{6} + 186018392 x^{5} + 10752389517 x^{4} + 491049966 x^{3} + \cdots + 59402280000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(10.5927\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.59274 q^{2} +50.3591 q^{3} -35.9794 q^{4} +255.731 q^{5} -483.082 q^{6} +343.000 q^{7} +1573.01 q^{8} +349.039 q^{9} +O(q^{10})\) \(q-9.59274 q^{2} +50.3591 q^{3} -35.9794 q^{4} +255.731 q^{5} -483.082 q^{6} +343.000 q^{7} +1573.01 q^{8} +349.039 q^{9} -2453.16 q^{10} +2581.16 q^{11} -1811.89 q^{12} +2197.00 q^{13} -3290.31 q^{14} +12878.4 q^{15} -10484.1 q^{16} +2930.15 q^{17} -3348.23 q^{18} +21147.7 q^{19} -9201.06 q^{20} +17273.2 q^{21} -24760.4 q^{22} -17170.0 q^{23} +79215.4 q^{24} -12726.5 q^{25} -21075.2 q^{26} -92558.1 q^{27} -12340.9 q^{28} -93079.5 q^{29} -123539. q^{30} +203907. q^{31} -100774. q^{32} +129985. q^{33} -28108.2 q^{34} +87715.8 q^{35} -12558.2 q^{36} +108445. q^{37} -202864. q^{38} +110639. q^{39} +402268. q^{40} +568659. q^{41} -165697. q^{42} +788479. q^{43} -92868.6 q^{44} +89260.1 q^{45} +164707. q^{46} +640403. q^{47} -527971. q^{48} +117649. q^{49} +122082. q^{50} +147560. q^{51} -79046.8 q^{52} +82954.4 q^{53} +887885. q^{54} +660084. q^{55} +539543. q^{56} +1.06498e6 q^{57} +892887. q^{58} +192889. q^{59} -463357. q^{60} +2.44511e6 q^{61} -1.95603e6 q^{62} +119720. q^{63} +2.30867e6 q^{64} +561842. q^{65} -1.24691e6 q^{66} +4.16532e6 q^{67} -105425. q^{68} -864664. q^{69} -841435. q^{70} +3.47378e6 q^{71} +549042. q^{72} -4.77837e6 q^{73} -1.04028e6 q^{74} -640896. q^{75} -760881. q^{76} +885338. q^{77} -1.06133e6 q^{78} -7.65148e6 q^{79} -2.68112e6 q^{80} -5.42449e6 q^{81} -5.45500e6 q^{82} -4.92570e6 q^{83} -621478. q^{84} +749332. q^{85} -7.56367e6 q^{86} -4.68740e6 q^{87} +4.06020e6 q^{88} -5.29816e6 q^{89} -856248. q^{90} +753571. q^{91} +617765. q^{92} +1.02686e7 q^{93} -6.14322e6 q^{94} +5.40812e6 q^{95} -5.07489e6 q^{96} +5.37546e6 q^{97} -1.12858e6 q^{98} +900925. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 82 q^{3} + 986 q^{4} + 1026 q^{5} + 309 q^{6} + 4116 q^{7} + 228 q^{8} + 10902 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} + 82 q^{3} + 986 q^{4} + 1026 q^{5} + 309 q^{6} + 4116 q^{7} + 228 q^{8} + 10902 q^{9} + 6668 q^{10} + 12168 q^{11} - 183 q^{12} + 26364 q^{13} + 2058 q^{14} - 28790 q^{15} + 85914 q^{16} + 82710 q^{17} - 44965 q^{18} - 10302 q^{19} + 141318 q^{20} + 28126 q^{21} - 97457 q^{22} + 98376 q^{23} - 519981 q^{24} + 272736 q^{25} + 13182 q^{26} + 306652 q^{27} + 338198 q^{28} + 350592 q^{29} + 231528 q^{30} + 55092 q^{31} + 114420 q^{32} + 609912 q^{33} + 812002 q^{34} + 351918 q^{35} + 1472143 q^{36} + 376310 q^{37} + 2825424 q^{38} + 180154 q^{39} + 2169290 q^{40} + 1387272 q^{41} + 105987 q^{42} + 568708 q^{43} + 3392031 q^{44} + 3556226 q^{45} - 1736829 q^{46} + 1359444 q^{47} + 4151249 q^{48} + 1411788 q^{49} + 3983712 q^{50} + 2709260 q^{51} + 2166242 q^{52} + 2061780 q^{53} + 2196651 q^{54} - 2112846 q^{55} + 78204 q^{56} + 2359902 q^{57} + 670268 q^{58} + 395964 q^{59} - 1052376 q^{60} + 444006 q^{61} + 2854353 q^{62} + 3739386 q^{63} + 12026858 q^{64} + 2254122 q^{65} - 4605681 q^{66} - 3094010 q^{67} + 4668954 q^{68} + 3839892 q^{69} + 2287124 q^{70} + 5694366 q^{71} - 9780585 q^{72} + 7052346 q^{73} - 4436259 q^{74} - 16288696 q^{75} - 3051830 q^{76} + 4173624 q^{77} + 678873 q^{78} + 4304160 q^{79} + 3807018 q^{80} - 6689556 q^{81} - 4733665 q^{82} + 2704554 q^{83} - 62769 q^{84} + 9301878 q^{85} + 1510998 q^{86} + 16231802 q^{87} - 70453923 q^{88} - 10986042 q^{89} - 12851300 q^{90} + 9042852 q^{91} - 16505451 q^{92} - 47230934 q^{93} - 24306151 q^{94} - 21839424 q^{95} - 86512741 q^{96} - 24462382 q^{97} + 705894 q^{98} + 11555078 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.59274 −0.847886 −0.423943 0.905689i \(-0.639354\pi\)
−0.423943 + 0.905689i \(0.639354\pi\)
\(3\) 50.3591 1.07685 0.538423 0.842675i \(-0.319020\pi\)
0.538423 + 0.842675i \(0.319020\pi\)
\(4\) −35.9794 −0.281089
\(5\) 255.731 0.914932 0.457466 0.889227i \(-0.348757\pi\)
0.457466 + 0.889227i \(0.348757\pi\)
\(6\) −483.082 −0.913043
\(7\) 343.000 0.377964
\(8\) 1573.01 1.08622
\(9\) 349.039 0.159597
\(10\) −2453.16 −0.775758
\(11\) 2581.16 0.584710 0.292355 0.956310i \(-0.405561\pi\)
0.292355 + 0.956310i \(0.405561\pi\)
\(12\) −1811.89 −0.302690
\(13\) 2197.00 0.277350
\(14\) −3290.31 −0.320471
\(15\) 12878.4 0.985241
\(16\) −10484.1 −0.639900
\(17\) 2930.15 0.144650 0.0723251 0.997381i \(-0.476958\pi\)
0.0723251 + 0.997381i \(0.476958\pi\)
\(18\) −3348.23 −0.135320
\(19\) 21147.7 0.707335 0.353667 0.935371i \(-0.384935\pi\)
0.353667 + 0.935371i \(0.384935\pi\)
\(20\) −9201.06 −0.257177
\(21\) 17273.2 0.407009
\(22\) −24760.4 −0.495768
\(23\) −17170.0 −0.294254 −0.147127 0.989118i \(-0.547003\pi\)
−0.147127 + 0.989118i \(0.547003\pi\)
\(24\) 79215.4 1.16969
\(25\) −12726.5 −0.162900
\(26\) −21075.2 −0.235161
\(27\) −92558.1 −0.904985
\(28\) −12340.9 −0.106242
\(29\) −93079.5 −0.708698 −0.354349 0.935113i \(-0.615298\pi\)
−0.354349 + 0.935113i \(0.615298\pi\)
\(30\) −123539. −0.835372
\(31\) 203907. 1.22933 0.614663 0.788790i \(-0.289292\pi\)
0.614663 + 0.788790i \(0.289292\pi\)
\(32\) −100774. −0.543655
\(33\) 129985. 0.629643
\(34\) −28108.2 −0.122647
\(35\) 87715.8 0.345812
\(36\) −12558.2 −0.0448610
\(37\) 108445. 0.351967 0.175984 0.984393i \(-0.443689\pi\)
0.175984 + 0.984393i \(0.443689\pi\)
\(38\) −202864. −0.599739
\(39\) 110639. 0.298663
\(40\) 402268. 0.993815
\(41\) 568659. 1.28857 0.644286 0.764785i \(-0.277155\pi\)
0.644286 + 0.764785i \(0.277155\pi\)
\(42\) −165697. −0.345098
\(43\) 788479. 1.51234 0.756172 0.654373i \(-0.227067\pi\)
0.756172 + 0.654373i \(0.227067\pi\)
\(44\) −92868.6 −0.164356
\(45\) 89260.1 0.146020
\(46\) 164707. 0.249494
\(47\) 640403. 0.899727 0.449863 0.893097i \(-0.351473\pi\)
0.449863 + 0.893097i \(0.351473\pi\)
\(48\) −527971. −0.689073
\(49\) 117649. 0.142857
\(50\) 122082. 0.138120
\(51\) 147560. 0.155766
\(52\) −79046.8 −0.0779601
\(53\) 82954.4 0.0765374 0.0382687 0.999267i \(-0.487816\pi\)
0.0382687 + 0.999267i \(0.487816\pi\)
\(54\) 887885. 0.767324
\(55\) 660084. 0.534970
\(56\) 539543. 0.410552
\(57\) 1.06498e6 0.761690
\(58\) 892887. 0.600895
\(59\) 192889. 0.122272 0.0611359 0.998129i \(-0.480528\pi\)
0.0611359 + 0.998129i \(0.480528\pi\)
\(60\) −463357. −0.276940
\(61\) 2.44511e6 1.37925 0.689627 0.724165i \(-0.257774\pi\)
0.689627 + 0.724165i \(0.257774\pi\)
\(62\) −1.95603e6 −1.04233
\(63\) 119720. 0.0603220
\(64\) 2.30867e6 1.10086
\(65\) 561842. 0.253756
\(66\) −1.24691e6 −0.533865
\(67\) 4.16532e6 1.69195 0.845973 0.533226i \(-0.179021\pi\)
0.845973 + 0.533226i \(0.179021\pi\)
\(68\) −105425. −0.0406596
\(69\) −864664. −0.316866
\(70\) −841435. −0.293209
\(71\) 3.47378e6 1.15186 0.575928 0.817500i \(-0.304641\pi\)
0.575928 + 0.817500i \(0.304641\pi\)
\(72\) 549042. 0.173357
\(73\) −4.77837e6 −1.43764 −0.718819 0.695197i \(-0.755317\pi\)
−0.718819 + 0.695197i \(0.755317\pi\)
\(74\) −1.04028e6 −0.298428
\(75\) −640896. −0.175418
\(76\) −760881. −0.198824
\(77\) 885338. 0.221000
\(78\) −1.06133e6 −0.253232
\(79\) −7.65148e6 −1.74603 −0.873013 0.487696i \(-0.837837\pi\)
−0.873013 + 0.487696i \(0.837837\pi\)
\(80\) −2.68112e6 −0.585465
\(81\) −5.42449e6 −1.13413
\(82\) −5.45500e6 −1.09256
\(83\) −4.92570e6 −0.945573 −0.472786 0.881177i \(-0.656752\pi\)
−0.472786 + 0.881177i \(0.656752\pi\)
\(84\) −621478. −0.114406
\(85\) 749332. 0.132345
\(86\) −7.56367e6 −1.28230
\(87\) −4.68740e6 −0.763158
\(88\) 4.06020e6 0.635123
\(89\) −5.29816e6 −0.796636 −0.398318 0.917247i \(-0.630406\pi\)
−0.398318 + 0.917247i \(0.630406\pi\)
\(90\) −856248. −0.123809
\(91\) 753571. 0.104828
\(92\) 617765. 0.0827115
\(93\) 1.02686e7 1.32380
\(94\) −6.14322e6 −0.762866
\(95\) 5.40812e6 0.647163
\(96\) −5.07489e6 −0.585433
\(97\) 5.37546e6 0.598018 0.299009 0.954250i \(-0.403344\pi\)
0.299009 + 0.954250i \(0.403344\pi\)
\(98\) −1.12858e6 −0.121127
\(99\) 900925. 0.0933180
\(100\) 457893. 0.0457893
\(101\) 2.66268e6 0.257154 0.128577 0.991700i \(-0.458959\pi\)
0.128577 + 0.991700i \(0.458959\pi\)
\(102\) −1.41550e6 −0.132072
\(103\) −3.37971e6 −0.304754 −0.152377 0.988322i \(-0.548693\pi\)
−0.152377 + 0.988322i \(0.548693\pi\)
\(104\) 3.45591e6 0.301263
\(105\) 4.41729e6 0.372386
\(106\) −795760. −0.0648950
\(107\) 1.12679e7 0.889199 0.444599 0.895729i \(-0.353346\pi\)
0.444599 + 0.895729i \(0.353346\pi\)
\(108\) 3.33018e6 0.254381
\(109\) −1.27866e6 −0.0945716 −0.0472858 0.998881i \(-0.515057\pi\)
−0.0472858 + 0.998881i \(0.515057\pi\)
\(110\) −6.33201e6 −0.453594
\(111\) 5.46117e6 0.379014
\(112\) −3.59605e6 −0.241859
\(113\) 1.33359e7 0.869458 0.434729 0.900561i \(-0.356844\pi\)
0.434729 + 0.900561i \(0.356844\pi\)
\(114\) −1.02160e7 −0.645827
\(115\) −4.39090e6 −0.269222
\(116\) 3.34895e6 0.199207
\(117\) 766838. 0.0442642
\(118\) −1.85034e6 −0.103673
\(119\) 1.00504e6 0.0546727
\(120\) 2.02579e7 1.07019
\(121\) −1.28248e7 −0.658114
\(122\) −2.34553e7 −1.16945
\(123\) 2.86372e7 1.38759
\(124\) −7.33647e6 −0.345550
\(125\) −2.32336e7 −1.06397
\(126\) −1.14844e6 −0.0511462
\(127\) −542790. −0.0235136 −0.0117568 0.999931i \(-0.503742\pi\)
−0.0117568 + 0.999931i \(0.503742\pi\)
\(128\) −9.24735e6 −0.389747
\(129\) 3.97071e7 1.62856
\(130\) −5.38960e6 −0.215157
\(131\) −2.56069e7 −0.995195 −0.497598 0.867408i \(-0.665784\pi\)
−0.497598 + 0.867408i \(0.665784\pi\)
\(132\) −4.67678e6 −0.176986
\(133\) 7.25365e6 0.267347
\(134\) −3.99568e7 −1.43458
\(135\) −2.36700e7 −0.827999
\(136\) 4.60917e6 0.157122
\(137\) 2.79367e7 0.928223 0.464112 0.885777i \(-0.346374\pi\)
0.464112 + 0.885777i \(0.346374\pi\)
\(138\) 8.29449e6 0.268666
\(139\) −1.40247e7 −0.442936 −0.221468 0.975168i \(-0.571085\pi\)
−0.221468 + 0.975168i \(0.571085\pi\)
\(140\) −3.15596e6 −0.0972039
\(141\) 3.22501e7 0.968867
\(142\) −3.33231e7 −0.976643
\(143\) 5.67081e6 0.162169
\(144\) −3.65936e6 −0.102126
\(145\) −2.38033e7 −0.648410
\(146\) 4.58376e7 1.21895
\(147\) 5.92470e6 0.153835
\(148\) −3.90177e6 −0.0989341
\(149\) −5.24708e7 −1.29947 −0.649734 0.760162i \(-0.725120\pi\)
−0.649734 + 0.760162i \(0.725120\pi\)
\(150\) 6.14795e6 0.148734
\(151\) 2.15427e7 0.509191 0.254596 0.967048i \(-0.418058\pi\)
0.254596 + 0.967048i \(0.418058\pi\)
\(152\) 3.32655e7 0.768319
\(153\) 1.02274e6 0.0230857
\(154\) −8.49282e6 −0.187383
\(155\) 5.21455e7 1.12475
\(156\) −3.98072e6 −0.0839510
\(157\) 2.98651e7 0.615906 0.307953 0.951402i \(-0.400356\pi\)
0.307953 + 0.951402i \(0.400356\pi\)
\(158\) 7.33987e7 1.48043
\(159\) 4.17751e6 0.0824190
\(160\) −2.57711e7 −0.497408
\(161\) −5.88930e6 −0.111217
\(162\) 5.20357e7 0.961610
\(163\) −5.04574e7 −0.912575 −0.456288 0.889832i \(-0.650821\pi\)
−0.456288 + 0.889832i \(0.650821\pi\)
\(164\) −2.04600e7 −0.362203
\(165\) 3.32412e7 0.576080
\(166\) 4.72510e7 0.801738
\(167\) −3.22889e6 −0.0536470 −0.0268235 0.999640i \(-0.508539\pi\)
−0.0268235 + 0.999640i \(0.508539\pi\)
\(168\) 2.71709e7 0.442101
\(169\) 4.82681e6 0.0769231
\(170\) −7.18814e6 −0.112214
\(171\) 7.38135e6 0.112888
\(172\) −2.83690e7 −0.425103
\(173\) −8.18996e7 −1.20260 −0.601299 0.799024i \(-0.705350\pi\)
−0.601299 + 0.799024i \(0.705350\pi\)
\(174\) 4.49650e7 0.647071
\(175\) −4.36520e6 −0.0615702
\(176\) −2.70612e7 −0.374156
\(177\) 9.71373e6 0.131668
\(178\) 5.08238e7 0.675456
\(179\) 9.06789e7 1.18174 0.590868 0.806768i \(-0.298785\pi\)
0.590868 + 0.806768i \(0.298785\pi\)
\(180\) −3.21152e6 −0.0410447
\(181\) −4.70416e7 −0.589668 −0.294834 0.955549i \(-0.595264\pi\)
−0.294834 + 0.955549i \(0.595264\pi\)
\(182\) −7.22881e6 −0.0888826
\(183\) 1.23134e8 1.48524
\(184\) −2.70086e7 −0.319624
\(185\) 2.77327e7 0.322026
\(186\) −9.85039e7 −1.12243
\(187\) 7.56320e6 0.0845785
\(188\) −2.30413e7 −0.252903
\(189\) −3.17474e7 −0.342052
\(190\) −5.18787e7 −0.548721
\(191\) 3.45353e7 0.358630 0.179315 0.983792i \(-0.442612\pi\)
0.179315 + 0.983792i \(0.442612\pi\)
\(192\) 1.16262e8 1.18545
\(193\) −1.73463e8 −1.73682 −0.868412 0.495843i \(-0.834859\pi\)
−0.868412 + 0.495843i \(0.834859\pi\)
\(194\) −5.15654e7 −0.507051
\(195\) 2.82938e7 0.273257
\(196\) −4.23294e6 −0.0401556
\(197\) 9.64533e7 0.898846 0.449423 0.893319i \(-0.351630\pi\)
0.449423 + 0.893319i \(0.351630\pi\)
\(198\) −8.64233e6 −0.0791230
\(199\) −8.24829e7 −0.741956 −0.370978 0.928642i \(-0.620978\pi\)
−0.370978 + 0.928642i \(0.620978\pi\)
\(200\) −2.00190e7 −0.176944
\(201\) 2.09762e8 1.82196
\(202\) −2.55424e7 −0.218037
\(203\) −3.19263e7 −0.267863
\(204\) −5.30912e6 −0.0437841
\(205\) 1.45424e8 1.17896
\(206\) 3.24207e7 0.258397
\(207\) −5.99298e6 −0.0469620
\(208\) −2.30336e7 −0.177476
\(209\) 5.45855e7 0.413586
\(210\) −4.23739e7 −0.315741
\(211\) −8.55662e6 −0.0627067 −0.0313533 0.999508i \(-0.509982\pi\)
−0.0313533 + 0.999508i \(0.509982\pi\)
\(212\) −2.98465e6 −0.0215138
\(213\) 1.74936e8 1.24037
\(214\) −1.08090e8 −0.753939
\(215\) 2.01639e8 1.38369
\(216\) −1.45595e8 −0.983010
\(217\) 6.99403e7 0.464642
\(218\) 1.22658e7 0.0801860
\(219\) −2.40634e8 −1.54812
\(220\) −2.37494e7 −0.150374
\(221\) 6.43755e6 0.0401188
\(222\) −5.23876e7 −0.321361
\(223\) −2.46144e8 −1.48635 −0.743177 0.669094i \(-0.766682\pi\)
−0.743177 + 0.669094i \(0.766682\pi\)
\(224\) −3.45655e7 −0.205482
\(225\) −4.44205e6 −0.0259983
\(226\) −1.27928e8 −0.737201
\(227\) 1.78066e8 1.01039 0.505197 0.863004i \(-0.331420\pi\)
0.505197 + 0.863004i \(0.331420\pi\)
\(228\) −3.83173e7 −0.214103
\(229\) −2.76317e8 −1.52049 −0.760246 0.649636i \(-0.774921\pi\)
−0.760246 + 0.649636i \(0.774921\pi\)
\(230\) 4.21207e7 0.228270
\(231\) 4.45848e7 0.237983
\(232\) −1.46415e8 −0.769800
\(233\) 5.58863e7 0.289440 0.144720 0.989473i \(-0.453772\pi\)
0.144720 + 0.989473i \(0.453772\pi\)
\(234\) −7.35607e6 −0.0375310
\(235\) 1.63771e8 0.823189
\(236\) −6.94004e6 −0.0343692
\(237\) −3.85322e8 −1.88020
\(238\) −9.64111e6 −0.0463562
\(239\) 1.28066e8 0.606794 0.303397 0.952864i \(-0.401879\pi\)
0.303397 + 0.952864i \(0.401879\pi\)
\(240\) −1.35019e8 −0.630455
\(241\) −6.44744e7 −0.296707 −0.148353 0.988934i \(-0.547397\pi\)
−0.148353 + 0.988934i \(0.547397\pi\)
\(242\) 1.23025e8 0.558006
\(243\) −7.07478e7 −0.316294
\(244\) −8.79737e7 −0.387693
\(245\) 3.00865e7 0.130705
\(246\) −2.74709e8 −1.17652
\(247\) 4.64614e7 0.196179
\(248\) 3.20749e8 1.33532
\(249\) −2.48054e8 −1.01824
\(250\) 2.22874e8 0.902129
\(251\) −6.87480e7 −0.274411 −0.137206 0.990543i \(-0.543812\pi\)
−0.137206 + 0.990543i \(0.543812\pi\)
\(252\) −4.30746e6 −0.0169559
\(253\) −4.43185e7 −0.172053
\(254\) 5.20684e6 0.0199368
\(255\) 3.77357e7 0.142515
\(256\) −2.06802e8 −0.770397
\(257\) 1.12481e8 0.413344 0.206672 0.978410i \(-0.433737\pi\)
0.206672 + 0.978410i \(0.433737\pi\)
\(258\) −3.80900e8 −1.38083
\(259\) 3.71965e7 0.133031
\(260\) −2.02147e7 −0.0713282
\(261\) −3.24883e7 −0.113106
\(262\) 2.45641e8 0.843812
\(263\) 1.08966e8 0.369357 0.184678 0.982799i \(-0.440876\pi\)
0.184678 + 0.982799i \(0.440876\pi\)
\(264\) 2.04468e8 0.683929
\(265\) 2.12140e7 0.0700266
\(266\) −6.95824e7 −0.226680
\(267\) −2.66810e8 −0.857854
\(268\) −1.49866e8 −0.475587
\(269\) −6.77506e7 −0.212217 −0.106108 0.994355i \(-0.533839\pi\)
−0.106108 + 0.994355i \(0.533839\pi\)
\(270\) 2.27060e8 0.702049
\(271\) 5.74959e7 0.175487 0.0877434 0.996143i \(-0.472034\pi\)
0.0877434 + 0.996143i \(0.472034\pi\)
\(272\) −3.07201e7 −0.0925617
\(273\) 3.79492e7 0.112884
\(274\) −2.67989e8 −0.787028
\(275\) −3.28492e7 −0.0952490
\(276\) 3.11101e7 0.0890676
\(277\) 1.27673e8 0.360927 0.180464 0.983582i \(-0.442240\pi\)
0.180464 + 0.983582i \(0.442240\pi\)
\(278\) 1.34535e8 0.375559
\(279\) 7.11716e7 0.196197
\(280\) 1.37978e8 0.375627
\(281\) −4.20286e8 −1.12999 −0.564993 0.825096i \(-0.691121\pi\)
−0.564993 + 0.825096i \(0.691121\pi\)
\(282\) −3.09367e8 −0.821489
\(283\) 4.95902e8 1.30060 0.650300 0.759677i \(-0.274643\pi\)
0.650300 + 0.759677i \(0.274643\pi\)
\(284\) −1.24985e8 −0.323774
\(285\) 2.72348e8 0.696895
\(286\) −5.43986e7 −0.137501
\(287\) 1.95050e8 0.487034
\(288\) −3.51740e7 −0.0867658
\(289\) −4.01753e8 −0.979076
\(290\) 2.28339e8 0.549778
\(291\) 2.70703e8 0.643973
\(292\) 1.71923e8 0.404105
\(293\) 1.95439e7 0.0453915 0.0226957 0.999742i \(-0.492775\pi\)
0.0226957 + 0.999742i \(0.492775\pi\)
\(294\) −5.68341e7 −0.130435
\(295\) 4.93278e7 0.111870
\(296\) 1.70585e8 0.382313
\(297\) −2.38907e8 −0.529154
\(298\) 5.03338e8 1.10180
\(299\) −3.77224e7 −0.0816113
\(300\) 2.30591e7 0.0493080
\(301\) 2.70448e8 0.571612
\(302\) −2.06653e8 −0.431736
\(303\) 1.34090e8 0.276915
\(304\) −2.21715e8 −0.452623
\(305\) 6.25292e8 1.26192
\(306\) −9.81084e6 −0.0195741
\(307\) −2.49035e8 −0.491220 −0.245610 0.969369i \(-0.578988\pi\)
−0.245610 + 0.969369i \(0.578988\pi\)
\(308\) −3.18539e7 −0.0621206
\(309\) −1.70199e8 −0.328173
\(310\) −5.00218e8 −0.953660
\(311\) −8.93074e7 −0.168355 −0.0841775 0.996451i \(-0.526826\pi\)
−0.0841775 + 0.996451i \(0.526826\pi\)
\(312\) 1.74036e8 0.324413
\(313\) −4.27203e8 −0.787461 −0.393731 0.919226i \(-0.628816\pi\)
−0.393731 + 0.919226i \(0.628816\pi\)
\(314\) −2.86488e8 −0.522218
\(315\) 3.06162e7 0.0551905
\(316\) 2.75296e8 0.490789
\(317\) −9.12027e8 −1.60805 −0.804026 0.594594i \(-0.797313\pi\)
−0.804026 + 0.594594i \(0.797313\pi\)
\(318\) −4.00737e7 −0.0698819
\(319\) −2.40253e8 −0.414383
\(320\) 5.90398e8 1.00721
\(321\) 5.67440e8 0.957530
\(322\) 5.64945e7 0.0942998
\(323\) 6.19659e7 0.102316
\(324\) 1.95170e8 0.318790
\(325\) −2.79602e7 −0.0451802
\(326\) 4.84025e8 0.773760
\(327\) −6.43919e7 −0.101839
\(328\) 8.94507e8 1.39967
\(329\) 2.19658e8 0.340065
\(330\) −3.18874e8 −0.488450
\(331\) 4.33628e8 0.657233 0.328616 0.944463i \(-0.393418\pi\)
0.328616 + 0.944463i \(0.393418\pi\)
\(332\) 1.77224e8 0.265790
\(333\) 3.78514e7 0.0561729
\(334\) 3.09739e7 0.0454865
\(335\) 1.06520e9 1.54801
\(336\) −1.81094e8 −0.260445
\(337\) 1.02946e8 0.146522 0.0732612 0.997313i \(-0.476659\pi\)
0.0732612 + 0.997313i \(0.476659\pi\)
\(338\) −4.63023e7 −0.0652220
\(339\) 6.71585e8 0.936272
\(340\) −2.69605e7 −0.0372008
\(341\) 5.26318e8 0.718800
\(342\) −7.08074e7 −0.0957166
\(343\) 4.03536e7 0.0539949
\(344\) 1.24029e9 1.64274
\(345\) −2.21122e8 −0.289911
\(346\) 7.85641e8 1.01967
\(347\) 2.95663e8 0.379877 0.189939 0.981796i \(-0.439171\pi\)
0.189939 + 0.981796i \(0.439171\pi\)
\(348\) 1.68650e8 0.214516
\(349\) 5.28198e8 0.665132 0.332566 0.943080i \(-0.392086\pi\)
0.332566 + 0.943080i \(0.392086\pi\)
\(350\) 4.18742e7 0.0522045
\(351\) −2.03350e8 −0.250998
\(352\) −2.60114e8 −0.317881
\(353\) 2.36776e8 0.286501 0.143250 0.989686i \(-0.454245\pi\)
0.143250 + 0.989686i \(0.454245\pi\)
\(354\) −9.31812e7 −0.111639
\(355\) 8.88354e8 1.05387
\(356\) 1.90624e8 0.223926
\(357\) 5.06131e7 0.0588740
\(358\) −8.69859e8 −1.00198
\(359\) −1.09643e9 −1.25069 −0.625347 0.780347i \(-0.715042\pi\)
−0.625347 + 0.780347i \(0.715042\pi\)
\(360\) 1.40407e8 0.158610
\(361\) −4.46648e8 −0.499678
\(362\) 4.51258e8 0.499971
\(363\) −6.45844e8 −0.708687
\(364\) −2.71130e7 −0.0294661
\(365\) −1.22198e9 −1.31534
\(366\) −1.18119e9 −1.25932
\(367\) −1.38313e9 −1.46060 −0.730302 0.683124i \(-0.760621\pi\)
−0.730302 + 0.683124i \(0.760621\pi\)
\(368\) 1.80012e8 0.188293
\(369\) 1.98484e8 0.205652
\(370\) −2.66032e8 −0.273041
\(371\) 2.84534e7 0.0289284
\(372\) −3.69458e8 −0.372104
\(373\) 3.87393e8 0.386520 0.193260 0.981148i \(-0.438094\pi\)
0.193260 + 0.981148i \(0.438094\pi\)
\(374\) −7.25518e7 −0.0717129
\(375\) −1.17002e9 −1.14574
\(376\) 1.00736e9 0.977299
\(377\) −2.04496e8 −0.196557
\(378\) 3.04545e8 0.290021
\(379\) −8.69794e8 −0.820690 −0.410345 0.911930i \(-0.634592\pi\)
−0.410345 + 0.911930i \(0.634592\pi\)
\(380\) −1.94581e8 −0.181910
\(381\) −2.73344e7 −0.0253205
\(382\) −3.31288e8 −0.304077
\(383\) 1.44762e9 1.31661 0.658306 0.752750i \(-0.271273\pi\)
0.658306 + 0.752750i \(0.271273\pi\)
\(384\) −4.65688e8 −0.419697
\(385\) 2.26409e8 0.202200
\(386\) 1.66398e9 1.47263
\(387\) 2.75210e8 0.241366
\(388\) −1.93406e8 −0.168096
\(389\) 1.20585e9 1.03866 0.519328 0.854575i \(-0.326182\pi\)
0.519328 + 0.854575i \(0.326182\pi\)
\(390\) −2.71415e8 −0.231690
\(391\) −5.03107e7 −0.0425639
\(392\) 1.85063e8 0.155174
\(393\) −1.28954e9 −1.07167
\(394\) −9.25251e8 −0.762119
\(395\) −1.95672e9 −1.59750
\(396\) −3.24147e7 −0.0262307
\(397\) −2.38600e9 −1.91383 −0.956914 0.290371i \(-0.906221\pi\)
−0.956914 + 0.290371i \(0.906221\pi\)
\(398\) 7.91237e8 0.629094
\(399\) 3.65287e8 0.287892
\(400\) 1.33426e8 0.104239
\(401\) −1.01868e9 −0.788919 −0.394460 0.918913i \(-0.629068\pi\)
−0.394460 + 0.918913i \(0.629068\pi\)
\(402\) −2.01219e9 −1.54482
\(403\) 4.47985e8 0.340954
\(404\) −9.58015e7 −0.0722832
\(405\) −1.38721e9 −1.03765
\(406\) 3.06260e8 0.227117
\(407\) 2.79913e8 0.205799
\(408\) 2.32113e8 0.169196
\(409\) 1.06801e9 0.771871 0.385936 0.922526i \(-0.373879\pi\)
0.385936 + 0.922526i \(0.373879\pi\)
\(410\) −1.39501e9 −0.999620
\(411\) 1.40686e9 0.999553
\(412\) 1.21600e8 0.0856631
\(413\) 6.61610e7 0.0462144
\(414\) 5.74891e7 0.0398184
\(415\) −1.25966e9 −0.865135
\(416\) −2.21401e8 −0.150783
\(417\) −7.06270e8 −0.476974
\(418\) −5.23625e8 −0.350674
\(419\) −9.84794e8 −0.654028 −0.327014 0.945020i \(-0.606042\pi\)
−0.327014 + 0.945020i \(0.606042\pi\)
\(420\) −1.58931e8 −0.104674
\(421\) −4.87733e7 −0.0318562 −0.0159281 0.999873i \(-0.505070\pi\)
−0.0159281 + 0.999873i \(0.505070\pi\)
\(422\) 8.20814e7 0.0531681
\(423\) 2.23525e8 0.143594
\(424\) 1.30488e8 0.0831363
\(425\) −3.72907e7 −0.0235635
\(426\) −1.67812e9 −1.05169
\(427\) 8.38673e8 0.521309
\(428\) −4.05412e8 −0.249944
\(429\) 2.85577e8 0.174631
\(430\) −1.93427e9 −1.17321
\(431\) 1.73942e9 1.04649 0.523243 0.852183i \(-0.324722\pi\)
0.523243 + 0.852183i \(0.324722\pi\)
\(432\) 9.70390e8 0.579099
\(433\) 2.68764e9 1.59097 0.795487 0.605971i \(-0.207215\pi\)
0.795487 + 0.605971i \(0.207215\pi\)
\(434\) −6.70919e8 −0.393963
\(435\) −1.19871e9 −0.698238
\(436\) 4.60053e7 0.0265830
\(437\) −3.63105e8 −0.208136
\(438\) 2.30834e9 1.31263
\(439\) −3.53346e9 −1.99331 −0.996654 0.0817358i \(-0.973954\pi\)
−0.996654 + 0.0817358i \(0.973954\pi\)
\(440\) 1.03832e9 0.581094
\(441\) 4.10640e7 0.0227996
\(442\) −6.17537e7 −0.0340161
\(443\) −3.08727e9 −1.68718 −0.843591 0.536987i \(-0.819562\pi\)
−0.843591 + 0.536987i \(0.819562\pi\)
\(444\) −1.96490e8 −0.106537
\(445\) −1.35490e9 −0.728867
\(446\) 2.36120e9 1.26026
\(447\) −2.64238e9 −1.39933
\(448\) 7.91872e8 0.416085
\(449\) 5.75588e8 0.300089 0.150044 0.988679i \(-0.452058\pi\)
0.150044 + 0.988679i \(0.452058\pi\)
\(450\) 4.26114e7 0.0220436
\(451\) 1.46780e9 0.753441
\(452\) −4.79818e8 −0.244395
\(453\) 1.08487e9 0.548320
\(454\) −1.70814e9 −0.856699
\(455\) 1.92712e8 0.0959109
\(456\) 1.67522e9 0.827361
\(457\) 4.07165e8 0.199556 0.0997779 0.995010i \(-0.468187\pi\)
0.0997779 + 0.995010i \(0.468187\pi\)
\(458\) 2.65064e9 1.28920
\(459\) −2.71209e8 −0.130906
\(460\) 1.57982e8 0.0756754
\(461\) 2.06313e9 0.980785 0.490393 0.871502i \(-0.336853\pi\)
0.490393 + 0.871502i \(0.336853\pi\)
\(462\) −4.27691e8 −0.201782
\(463\) 1.86283e9 0.872247 0.436124 0.899887i \(-0.356351\pi\)
0.436124 + 0.899887i \(0.356351\pi\)
\(464\) 9.75857e8 0.453496
\(465\) 2.62600e9 1.21118
\(466\) −5.36102e8 −0.245413
\(467\) −3.92325e9 −1.78253 −0.891266 0.453482i \(-0.850182\pi\)
−0.891266 + 0.453482i \(0.850182\pi\)
\(468\) −2.75904e7 −0.0124422
\(469\) 1.42870e9 0.639495
\(470\) −1.57101e9 −0.697970
\(471\) 1.50398e9 0.663236
\(472\) 3.03417e8 0.132814
\(473\) 2.03519e9 0.884283
\(474\) 3.69629e9 1.59420
\(475\) −2.69136e8 −0.115224
\(476\) −3.61608e7 −0.0153679
\(477\) 2.89543e7 0.0122151
\(478\) −1.22850e9 −0.514492
\(479\) 1.38329e9 0.575095 0.287547 0.957766i \(-0.407160\pi\)
0.287547 + 0.957766i \(0.407160\pi\)
\(480\) −1.29781e9 −0.535632
\(481\) 2.38253e8 0.0976181
\(482\) 6.18486e8 0.251574
\(483\) −2.96580e8 −0.119764
\(484\) 4.61428e8 0.184989
\(485\) 1.37467e9 0.547146
\(486\) 6.78665e8 0.268181
\(487\) 2.73305e9 1.07225 0.536125 0.844139i \(-0.319888\pi\)
0.536125 + 0.844139i \(0.319888\pi\)
\(488\) 3.84619e9 1.49817
\(489\) −2.54099e9 −0.982703
\(490\) −2.88612e8 −0.110823
\(491\) −1.52223e9 −0.580359 −0.290179 0.956972i \(-0.593715\pi\)
−0.290179 + 0.956972i \(0.593715\pi\)
\(492\) −1.03035e9 −0.390037
\(493\) −2.72737e8 −0.102513
\(494\) −4.45692e8 −0.166338
\(495\) 2.30395e8 0.0853796
\(496\) −2.13779e9 −0.786646
\(497\) 1.19151e9 0.435361
\(498\) 2.37952e9 0.863348
\(499\) −1.68770e9 −0.608055 −0.304028 0.952663i \(-0.598332\pi\)
−0.304028 + 0.952663i \(0.598332\pi\)
\(500\) 8.35930e8 0.299071
\(501\) −1.62604e8 −0.0577695
\(502\) 6.59481e8 0.232670
\(503\) −2.30687e9 −0.808230 −0.404115 0.914708i \(-0.632420\pi\)
−0.404115 + 0.914708i \(0.632420\pi\)
\(504\) 1.88321e8 0.0655228
\(505\) 6.80930e8 0.235279
\(506\) 4.25135e8 0.145882
\(507\) 2.43074e8 0.0828343
\(508\) 1.95292e7 0.00660941
\(509\) −4.07761e9 −1.37055 −0.685273 0.728287i \(-0.740317\pi\)
−0.685273 + 0.728287i \(0.740317\pi\)
\(510\) −3.61988e8 −0.120837
\(511\) −1.63898e9 −0.543376
\(512\) 3.16746e9 1.04296
\(513\) −1.95739e9 −0.640127
\(514\) −1.07900e9 −0.350469
\(515\) −8.64299e8 −0.278829
\(516\) −1.42864e9 −0.457771
\(517\) 1.65298e9 0.526080
\(518\) −3.56816e8 −0.112795
\(519\) −4.12439e9 −1.29501
\(520\) 8.83783e8 0.275635
\(521\) 5.21852e9 1.61665 0.808324 0.588737i \(-0.200375\pi\)
0.808324 + 0.588737i \(0.200375\pi\)
\(522\) 3.11652e8 0.0959010
\(523\) −6.18898e9 −1.89175 −0.945874 0.324533i \(-0.894793\pi\)
−0.945874 + 0.324533i \(0.894793\pi\)
\(524\) 9.21323e8 0.279738
\(525\) −2.19827e8 −0.0663016
\(526\) −1.04528e9 −0.313172
\(527\) 5.97480e8 0.177822
\(528\) −1.36278e9 −0.402908
\(529\) −3.11002e9 −0.913415
\(530\) −2.03501e8 −0.0593745
\(531\) 6.73258e7 0.0195142
\(532\) −2.60982e8 −0.0751484
\(533\) 1.24934e9 0.357386
\(534\) 2.55944e9 0.727362
\(535\) 2.88155e9 0.813556
\(536\) 6.55209e9 1.83782
\(537\) 4.56651e9 1.27255
\(538\) 6.49913e8 0.179936
\(539\) 3.03671e8 0.0835300
\(540\) 8.51632e8 0.232742
\(541\) 4.69741e9 1.27546 0.637732 0.770258i \(-0.279873\pi\)
0.637732 + 0.770258i \(0.279873\pi\)
\(542\) −5.51543e8 −0.148793
\(543\) −2.36897e9 −0.634981
\(544\) −2.95283e8 −0.0786399
\(545\) −3.26992e8 −0.0865266
\(546\) −3.64036e8 −0.0957129
\(547\) −3.92149e9 −1.02446 −0.512230 0.858848i \(-0.671181\pi\)
−0.512230 + 0.858848i \(0.671181\pi\)
\(548\) −1.00514e9 −0.260913
\(549\) 8.53438e8 0.220125
\(550\) 3.15114e8 0.0807603
\(551\) −1.96842e9 −0.501287
\(552\) −1.36013e9 −0.344185
\(553\) −2.62446e9 −0.659936
\(554\) −1.22473e9 −0.306025
\(555\) 1.39659e9 0.346772
\(556\) 5.04599e8 0.124504
\(557\) −6.01806e9 −1.47558 −0.737791 0.675030i \(-0.764131\pi\)
−0.737791 + 0.675030i \(0.764131\pi\)
\(558\) −6.82730e8 −0.166353
\(559\) 1.73229e9 0.419449
\(560\) −9.19623e8 −0.221285
\(561\) 3.80876e8 0.0910780
\(562\) 4.03170e9 0.958099
\(563\) 5.67516e8 0.134029 0.0670145 0.997752i \(-0.478653\pi\)
0.0670145 + 0.997752i \(0.478653\pi\)
\(564\) −1.16034e9 −0.272338
\(565\) 3.41041e9 0.795495
\(566\) −4.75706e9 −1.10276
\(567\) −1.86060e9 −0.428659
\(568\) 5.46430e9 1.25117
\(569\) 7.83634e9 1.78328 0.891641 0.452742i \(-0.149554\pi\)
0.891641 + 0.452742i \(0.149554\pi\)
\(570\) −2.61256e9 −0.590887
\(571\) −1.26314e9 −0.283938 −0.141969 0.989871i \(-0.545343\pi\)
−0.141969 + 0.989871i \(0.545343\pi\)
\(572\) −2.04032e8 −0.0455841
\(573\) 1.73917e9 0.386189
\(574\) −1.87106e9 −0.412950
\(575\) 2.18514e8 0.0479338
\(576\) 8.05813e8 0.175694
\(577\) 4.57842e9 0.992203 0.496102 0.868265i \(-0.334764\pi\)
0.496102 + 0.868265i \(0.334764\pi\)
\(578\) 3.85391e9 0.830145
\(579\) −8.73543e9 −1.87029
\(580\) 8.56430e8 0.182261
\(581\) −1.68952e9 −0.357393
\(582\) −2.59678e9 −0.546016
\(583\) 2.14119e8 0.0447522
\(584\) −7.51643e9 −1.56159
\(585\) 1.96104e8 0.0404988
\(586\) −1.87479e8 −0.0384868
\(587\) 2.22324e9 0.453683 0.226841 0.973932i \(-0.427160\pi\)
0.226841 + 0.973932i \(0.427160\pi\)
\(588\) −2.13167e8 −0.0432414
\(589\) 4.31217e9 0.869545
\(590\) −4.73189e8 −0.0948533
\(591\) 4.85730e9 0.967919
\(592\) −1.13695e9 −0.225224
\(593\) 7.87244e9 1.55031 0.775154 0.631772i \(-0.217672\pi\)
0.775154 + 0.631772i \(0.217672\pi\)
\(594\) 2.29177e9 0.448662
\(595\) 2.57021e8 0.0500218
\(596\) 1.88787e9 0.365266
\(597\) −4.15377e9 −0.798972
\(598\) 3.61861e8 0.0691971
\(599\) −4.20449e9 −0.799318 −0.399659 0.916664i \(-0.630871\pi\)
−0.399659 + 0.916664i \(0.630871\pi\)
\(600\) −1.00814e9 −0.190542
\(601\) 6.72387e9 1.26345 0.631726 0.775192i \(-0.282347\pi\)
0.631726 + 0.775192i \(0.282347\pi\)
\(602\) −2.59434e9 −0.484662
\(603\) 1.45386e9 0.270029
\(604\) −7.75094e8 −0.143128
\(605\) −3.27970e9 −0.602129
\(606\) −1.28629e9 −0.234793
\(607\) −6.31449e9 −1.14598 −0.572991 0.819562i \(-0.694217\pi\)
−0.572991 + 0.819562i \(0.694217\pi\)
\(608\) −2.13114e9 −0.384546
\(609\) −1.60778e9 −0.288447
\(610\) −5.99826e9 −1.06997
\(611\) 1.40696e9 0.249539
\(612\) −3.67975e7 −0.00648915
\(613\) 9.32575e9 1.63520 0.817602 0.575784i \(-0.195303\pi\)
0.817602 + 0.575784i \(0.195303\pi\)
\(614\) 2.38893e9 0.416499
\(615\) 7.32342e9 1.26955
\(616\) 1.39265e9 0.240054
\(617\) 2.95846e9 0.507069 0.253535 0.967326i \(-0.418407\pi\)
0.253535 + 0.967326i \(0.418407\pi\)
\(618\) 1.63268e9 0.278254
\(619\) 1.96622e9 0.333207 0.166603 0.986024i \(-0.446720\pi\)
0.166603 + 0.986024i \(0.446720\pi\)
\(620\) −1.87616e9 −0.316155
\(621\) 1.58922e9 0.266295
\(622\) 8.56702e8 0.142746
\(623\) −1.81727e9 −0.301100
\(624\) −1.15995e9 −0.191115
\(625\) −4.94729e9 −0.810564
\(626\) 4.09805e9 0.667677
\(627\) 2.74888e9 0.445368
\(628\) −1.07453e9 −0.173125
\(629\) 3.17759e8 0.0509121
\(630\) −2.93693e8 −0.0467953
\(631\) −2.82147e9 −0.447067 −0.223533 0.974696i \(-0.571759\pi\)
−0.223533 + 0.974696i \(0.571759\pi\)
\(632\) −1.20359e10 −1.89657
\(633\) −4.30904e8 −0.0675254
\(634\) 8.74884e9 1.36345
\(635\) −1.38808e8 −0.0215133
\(636\) −1.50304e8 −0.0231671
\(637\) 2.58475e8 0.0396214
\(638\) 2.30469e9 0.351350
\(639\) 1.21248e9 0.183833
\(640\) −2.36484e9 −0.356592
\(641\) 1.43892e9 0.215792 0.107896 0.994162i \(-0.465589\pi\)
0.107896 + 0.994162i \(0.465589\pi\)
\(642\) −5.44330e9 −0.811877
\(643\) −1.19689e10 −1.77547 −0.887737 0.460350i \(-0.847724\pi\)
−0.887737 + 0.460350i \(0.847724\pi\)
\(644\) 2.11893e8 0.0312620
\(645\) 1.01543e10 1.49002
\(646\) −5.94423e8 −0.0867524
\(647\) −9.94077e9 −1.44296 −0.721481 0.692434i \(-0.756538\pi\)
−0.721481 + 0.692434i \(0.756538\pi\)
\(648\) −8.53278e9 −1.23191
\(649\) 4.97878e8 0.0714935
\(650\) 2.68215e8 0.0383077
\(651\) 3.52213e9 0.500348
\(652\) 1.81543e9 0.256515
\(653\) 9.63667e9 1.35435 0.677175 0.735822i \(-0.263204\pi\)
0.677175 + 0.735822i \(0.263204\pi\)
\(654\) 6.17695e8 0.0863479
\(655\) −6.54850e9 −0.910536
\(656\) −5.96189e9 −0.824557
\(657\) −1.66784e9 −0.229443
\(658\) −2.10712e9 −0.288336
\(659\) 1.14767e10 1.56213 0.781065 0.624450i \(-0.214677\pi\)
0.781065 + 0.624450i \(0.214677\pi\)
\(660\) −1.19600e9 −0.161930
\(661\) −9.41385e9 −1.26783 −0.633917 0.773401i \(-0.718554\pi\)
−0.633917 + 0.773401i \(0.718554\pi\)
\(662\) −4.15968e9 −0.557259
\(663\) 3.24189e8 0.0432017
\(664\) −7.74819e9 −1.02710
\(665\) 1.85499e9 0.244605
\(666\) −3.63098e8 −0.0476282
\(667\) 1.59817e9 0.208537
\(668\) 1.16173e8 0.0150796
\(669\) −1.23956e10 −1.60058
\(670\) −1.02182e10 −1.31254
\(671\) 6.31123e9 0.806464
\(672\) −1.74069e9 −0.221273
\(673\) −1.98375e9 −0.250861 −0.125431 0.992102i \(-0.540031\pi\)
−0.125431 + 0.992102i \(0.540031\pi\)
\(674\) −9.87532e8 −0.124234
\(675\) 1.17794e9 0.147422
\(676\) −1.73666e8 −0.0216222
\(677\) 4.99973e9 0.619279 0.309639 0.950854i \(-0.399792\pi\)
0.309639 + 0.950854i \(0.399792\pi\)
\(678\) −6.44234e9 −0.793852
\(679\) 1.84378e9 0.226030
\(680\) 1.17871e9 0.143756
\(681\) 8.96724e9 1.08804
\(682\) −5.04883e9 −0.609460
\(683\) −1.02974e10 −1.23667 −0.618334 0.785916i \(-0.712192\pi\)
−0.618334 + 0.785916i \(0.712192\pi\)
\(684\) −2.65577e8 −0.0317317
\(685\) 7.14428e9 0.849261
\(686\) −3.87102e8 −0.0457815
\(687\) −1.39151e10 −1.63733
\(688\) −8.26651e9 −0.967749
\(689\) 1.82251e8 0.0212277
\(690\) 2.12116e9 0.245811
\(691\) 4.38076e9 0.505099 0.252550 0.967584i \(-0.418731\pi\)
0.252550 + 0.967584i \(0.418731\pi\)
\(692\) 2.94670e9 0.338037
\(693\) 3.09017e8 0.0352709
\(694\) −2.83621e9 −0.322093
\(695\) −3.58655e9 −0.405256
\(696\) −7.37333e9 −0.828956
\(697\) 1.66626e9 0.186392
\(698\) −5.06686e9 −0.563956
\(699\) 2.81438e9 0.311683
\(700\) 1.57057e8 0.0173067
\(701\) −5.61326e9 −0.615463 −0.307732 0.951473i \(-0.599570\pi\)
−0.307732 + 0.951473i \(0.599570\pi\)
\(702\) 1.95068e9 0.212817
\(703\) 2.29335e9 0.248958
\(704\) 5.95904e9 0.643683
\(705\) 8.24736e9 0.886448
\(706\) −2.27133e9 −0.242920
\(707\) 9.13298e8 0.0971952
\(708\) −3.49494e8 −0.0370104
\(709\) 5.90549e9 0.622292 0.311146 0.950362i \(-0.399287\pi\)
0.311146 + 0.950362i \(0.399287\pi\)
\(710\) −8.52175e9 −0.893562
\(711\) −2.67066e9 −0.278661
\(712\) −8.33406e9 −0.865320
\(713\) −3.50108e9 −0.361734
\(714\) −4.85518e8 −0.0499185
\(715\) 1.45020e9 0.148374
\(716\) −3.26257e9 −0.332173
\(717\) 6.44929e9 0.653424
\(718\) 1.05178e10 1.06045
\(719\) 1.46168e9 0.146656 0.0733282 0.997308i \(-0.476638\pi\)
0.0733282 + 0.997308i \(0.476638\pi\)
\(720\) −9.35813e8 −0.0934384
\(721\) −1.15924e9 −0.115186
\(722\) 4.28458e9 0.423670
\(723\) −3.24687e9 −0.319508
\(724\) 1.69253e9 0.165749
\(725\) 1.18458e9 0.115447
\(726\) 6.19541e9 0.600886
\(727\) 2.64041e9 0.254859 0.127430 0.991848i \(-0.459327\pi\)
0.127430 + 0.991848i \(0.459327\pi\)
\(728\) 1.18538e9 0.113867
\(729\) 8.30056e9 0.793526
\(730\) 1.17221e10 1.11526
\(731\) 2.31037e9 0.218761
\(732\) −4.43027e9 −0.417486
\(733\) −2.02561e10 −1.89973 −0.949865 0.312660i \(-0.898780\pi\)
−0.949865 + 0.312660i \(0.898780\pi\)
\(734\) 1.32680e10 1.23843
\(735\) 1.51513e9 0.140749
\(736\) 1.73029e9 0.159973
\(737\) 1.07514e10 0.989298
\(738\) −1.90400e9 −0.174370
\(739\) 3.39005e9 0.308995 0.154497 0.987993i \(-0.450624\pi\)
0.154497 + 0.987993i \(0.450624\pi\)
\(740\) −9.97805e8 −0.0905180
\(741\) 2.33976e9 0.211255
\(742\) −2.72946e8 −0.0245280
\(743\) 1.06366e10 0.951351 0.475676 0.879621i \(-0.342204\pi\)
0.475676 + 0.879621i \(0.342204\pi\)
\(744\) 1.61526e10 1.43793
\(745\) −1.34184e10 −1.18892
\(746\) −3.71616e9 −0.327725
\(747\) −1.71926e9 −0.150911
\(748\) −2.72119e8 −0.0237741
\(749\) 3.86488e9 0.336086
\(750\) 1.12237e10 0.971454
\(751\) 2.13371e10 1.83821 0.919105 0.394012i \(-0.128913\pi\)
0.919105 + 0.394012i \(0.128913\pi\)
\(752\) −6.71406e9 −0.575735
\(753\) −3.46209e9 −0.295499
\(754\) 1.96167e9 0.166658
\(755\) 5.50914e9 0.465875
\(756\) 1.14225e9 0.0961471
\(757\) −2.55448e9 −0.214026 −0.107013 0.994258i \(-0.534129\pi\)
−0.107013 + 0.994258i \(0.534129\pi\)
\(758\) 8.34370e9 0.695851
\(759\) −2.23184e9 −0.185275
\(760\) 8.50703e9 0.702960
\(761\) −2.93006e9 −0.241007 −0.120504 0.992713i \(-0.538451\pi\)
−0.120504 + 0.992713i \(0.538451\pi\)
\(762\) 2.62212e8 0.0214689
\(763\) −4.38579e8 −0.0357447
\(764\) −1.24256e9 −0.100807
\(765\) 2.61546e8 0.0211219
\(766\) −1.38866e10 −1.11634
\(767\) 4.23778e8 0.0339121
\(768\) −1.04144e10 −0.829599
\(769\) −1.78443e9 −0.141500 −0.0707500 0.997494i \(-0.522539\pi\)
−0.0707500 + 0.997494i \(0.522539\pi\)
\(770\) −2.17188e9 −0.171442
\(771\) 5.66442e9 0.445108
\(772\) 6.24109e9 0.488202
\(773\) 4.06390e9 0.316457 0.158229 0.987402i \(-0.449422\pi\)
0.158229 + 0.987402i \(0.449422\pi\)
\(774\) −2.64001e9 −0.204650
\(775\) −2.59503e9 −0.200257
\(776\) 8.45566e9 0.649578
\(777\) 1.87318e9 0.143254
\(778\) −1.15674e10 −0.880661
\(779\) 1.20258e10 0.911451
\(780\) −1.01800e9 −0.0768094
\(781\) 8.96639e9 0.673502
\(782\) 4.82617e8 0.0360893
\(783\) 8.61526e9 0.641361
\(784\) −1.23345e9 −0.0914143
\(785\) 7.63743e9 0.563512
\(786\) 1.23702e10 0.908656
\(787\) −1.31725e10 −0.963293 −0.481646 0.876366i \(-0.659961\pi\)
−0.481646 + 0.876366i \(0.659961\pi\)
\(788\) −3.47033e9 −0.252656
\(789\) 5.48743e9 0.397740
\(790\) 1.87703e10 1.35449
\(791\) 4.57422e9 0.328624
\(792\) 1.41716e9 0.101364
\(793\) 5.37191e9 0.382536
\(794\) 2.28882e10 1.62271
\(795\) 1.06832e9 0.0754078
\(796\) 2.96769e9 0.208556
\(797\) −6.91244e8 −0.0483646 −0.0241823 0.999708i \(-0.507698\pi\)
−0.0241823 + 0.999708i \(0.507698\pi\)
\(798\) −3.50410e9 −0.244100
\(799\) 1.87648e9 0.130146
\(800\) 1.28250e9 0.0885612
\(801\) −1.84926e9 −0.127141
\(802\) 9.77193e9 0.668914
\(803\) −1.23337e10 −0.840602
\(804\) −7.54710e9 −0.512134
\(805\) −1.50608e9 −0.101756
\(806\) −4.29740e9 −0.289090
\(807\) −3.41186e9 −0.228525
\(808\) 4.18842e9 0.279325
\(809\) −1.23420e10 −0.819530 −0.409765 0.912191i \(-0.634389\pi\)
−0.409765 + 0.912191i \(0.634389\pi\)
\(810\) 1.33072e10 0.879807
\(811\) −2.28996e9 −0.150749 −0.0753747 0.997155i \(-0.524015\pi\)
−0.0753747 + 0.997155i \(0.524015\pi\)
\(812\) 1.14869e9 0.0752933
\(813\) 2.89544e9 0.188972
\(814\) −2.68513e9 −0.174494
\(815\) −1.29035e10 −0.834944
\(816\) −1.54704e9 −0.0996747
\(817\) 1.66745e10 1.06973
\(818\) −1.02452e10 −0.654459
\(819\) 2.63025e8 0.0167303
\(820\) −5.23227e9 −0.331392
\(821\) −2.74582e10 −1.73169 −0.865847 0.500308i \(-0.833220\pi\)
−0.865847 + 0.500308i \(0.833220\pi\)
\(822\) −1.34957e10 −0.847508
\(823\) 1.25243e9 0.0783165 0.0391582 0.999233i \(-0.487532\pi\)
0.0391582 + 0.999233i \(0.487532\pi\)
\(824\) −5.31633e9 −0.331029
\(825\) −1.65426e9 −0.102569
\(826\) −6.34665e8 −0.0391845
\(827\) −1.18078e10 −0.725939 −0.362969 0.931801i \(-0.618237\pi\)
−0.362969 + 0.931801i \(0.618237\pi\)
\(828\) 2.15624e8 0.0132005
\(829\) −2.13259e10 −1.30007 −0.650036 0.759904i \(-0.725246\pi\)
−0.650036 + 0.759904i \(0.725246\pi\)
\(830\) 1.20836e10 0.733536
\(831\) 6.42949e9 0.388663
\(832\) 5.07214e9 0.305323
\(833\) 3.44730e8 0.0206643
\(834\) 6.77506e9 0.404419
\(835\) −8.25728e8 −0.0490833
\(836\) −1.96396e9 −0.116254
\(837\) −1.88733e10 −1.11252
\(838\) 9.44687e9 0.554541
\(839\) 2.79317e10 1.63279 0.816395 0.577494i \(-0.195969\pi\)
0.816395 + 0.577494i \(0.195969\pi\)
\(840\) 6.94845e9 0.404492
\(841\) −8.58608e9 −0.497747
\(842\) 4.67869e8 0.0270105
\(843\) −2.11652e10 −1.21682
\(844\) 3.07862e8 0.0176262
\(845\) 1.23437e9 0.0703794
\(846\) −2.14422e9 −0.121751
\(847\) −4.39890e9 −0.248744
\(848\) −8.69704e8 −0.0489763
\(849\) 2.49732e10 1.40055
\(850\) 3.57720e8 0.0199791
\(851\) −1.86199e9 −0.103568
\(852\) −6.29411e9 −0.348655
\(853\) 2.71120e10 1.49569 0.747843 0.663876i \(-0.231090\pi\)
0.747843 + 0.663876i \(0.231090\pi\)
\(854\) −8.04517e9 −0.442011
\(855\) 1.88764e9 0.103285
\(856\) 1.77245e10 0.965864
\(857\) −1.89625e10 −1.02911 −0.514556 0.857457i \(-0.672043\pi\)
−0.514556 + 0.857457i \(0.672043\pi\)
\(858\) −2.73946e9 −0.148068
\(859\) 5.82829e9 0.313736 0.156868 0.987620i \(-0.449860\pi\)
0.156868 + 0.987620i \(0.449860\pi\)
\(860\) −7.25484e9 −0.388941
\(861\) 9.82255e9 0.524461
\(862\) −1.66858e10 −0.887301
\(863\) −3.20815e10 −1.69909 −0.849545 0.527516i \(-0.823124\pi\)
−0.849545 + 0.527516i \(0.823124\pi\)
\(864\) 9.32745e9 0.492000
\(865\) −2.09443e10 −1.10030
\(866\) −2.57818e10 −1.34896
\(867\) −2.02319e10 −1.05431
\(868\) −2.51641e9 −0.130606
\(869\) −1.97497e10 −1.02092
\(870\) 1.14990e10 0.592026
\(871\) 9.15120e9 0.469261
\(872\) −2.01134e9 −0.102725
\(873\) 1.87624e9 0.0954419
\(874\) 3.48317e9 0.176476
\(875\) −7.96912e9 −0.402144
\(876\) 8.65788e9 0.435158
\(877\) −1.24151e10 −0.621515 −0.310757 0.950489i \(-0.600583\pi\)
−0.310757 + 0.950489i \(0.600583\pi\)
\(878\) 3.38956e10 1.69010
\(879\) 9.84212e8 0.0488796
\(880\) −6.92039e9 −0.342327
\(881\) 8.84588e9 0.435838 0.217919 0.975967i \(-0.430073\pi\)
0.217919 + 0.975967i \(0.430073\pi\)
\(882\) −3.93916e8 −0.0193314
\(883\) −2.84140e10 −1.38890 −0.694449 0.719542i \(-0.744352\pi\)
−0.694449 + 0.719542i \(0.744352\pi\)
\(884\) −2.31619e8 −0.0112769
\(885\) 2.48410e9 0.120467
\(886\) 2.96154e10 1.43054
\(887\) −8.27315e8 −0.0398051 −0.0199025 0.999802i \(-0.506336\pi\)
−0.0199025 + 0.999802i \(0.506336\pi\)
\(888\) 8.59049e9 0.411692
\(889\) −1.86177e8 −0.00888729
\(890\) 1.29972e10 0.617997
\(891\) −1.40015e10 −0.663135
\(892\) 8.85612e9 0.417798
\(893\) 1.35430e10 0.636408
\(894\) 2.53477e10 1.18647
\(895\) 2.31894e10 1.08121
\(896\) −3.17184e9 −0.147310
\(897\) −1.89967e9 −0.0878828
\(898\) −5.52147e9 −0.254441
\(899\) −1.89796e10 −0.871221
\(900\) 1.59822e8 0.00730783
\(901\) 2.43069e8 0.0110712
\(902\) −1.40802e10 −0.638832
\(903\) 1.36195e10 0.615538
\(904\) 2.09776e10 0.944420
\(905\) −1.20300e10 −0.539506
\(906\) −1.04069e10 −0.464913
\(907\) −4.17403e10 −1.85751 −0.928753 0.370699i \(-0.879118\pi\)
−0.928753 + 0.370699i \(0.879118\pi\)
\(908\) −6.40671e9 −0.284011
\(909\) 9.29377e8 0.0410410
\(910\) −1.84863e9 −0.0813215
\(911\) −3.21554e10 −1.40909 −0.704547 0.709657i \(-0.748850\pi\)
−0.704547 + 0.709657i \(0.748850\pi\)
\(912\) −1.11654e10 −0.487405
\(913\) −1.27140e10 −0.552886
\(914\) −3.90583e9 −0.169201
\(915\) 3.14891e10 1.35890
\(916\) 9.94173e9 0.427394
\(917\) −8.78318e9 −0.376148
\(918\) 2.60164e9 0.110994
\(919\) 3.25496e10 1.38338 0.691690 0.722195i \(-0.256867\pi\)
0.691690 + 0.722195i \(0.256867\pi\)
\(920\) −6.90693e9 −0.292434
\(921\) −1.25412e10 −0.528968
\(922\) −1.97911e10 −0.831594
\(923\) 7.63190e9 0.319467
\(924\) −1.60414e9 −0.0668943
\(925\) −1.38012e9 −0.0573353
\(926\) −1.78696e10 −0.739567
\(927\) −1.17965e9 −0.0486379
\(928\) 9.38000e9 0.385288
\(929\) 2.95871e10 1.21073 0.605365 0.795948i \(-0.293027\pi\)
0.605365 + 0.795948i \(0.293027\pi\)
\(930\) −2.51905e10 −1.02694
\(931\) 2.48800e9 0.101048
\(932\) −2.01075e9 −0.0813586
\(933\) −4.49744e9 −0.181292
\(934\) 3.76347e10 1.51138
\(935\) 1.93415e9 0.0773836
\(936\) 1.20624e9 0.0480806
\(937\) 1.81202e10 0.719573 0.359786 0.933035i \(-0.382850\pi\)
0.359786 + 0.933035i \(0.382850\pi\)
\(938\) −1.37052e10 −0.542219
\(939\) −2.15136e10 −0.847974
\(940\) −5.89238e9 −0.231389
\(941\) 1.03402e10 0.404544 0.202272 0.979329i \(-0.435167\pi\)
0.202272 + 0.979329i \(0.435167\pi\)
\(942\) −1.44273e10 −0.562349
\(943\) −9.76386e9 −0.379167
\(944\) −2.02227e9 −0.0782417
\(945\) −8.11881e9 −0.312954
\(946\) −1.95231e10 −0.749771
\(947\) −3.10070e10 −1.18641 −0.593205 0.805051i \(-0.702138\pi\)
−0.593205 + 0.805051i \(0.702138\pi\)
\(948\) 1.38637e10 0.528504
\(949\) −1.04981e10 −0.398729
\(950\) 2.58175e9 0.0976972
\(951\) −4.59289e10 −1.73163
\(952\) 1.58094e9 0.0593864
\(953\) −3.14767e10 −1.17805 −0.589026 0.808114i \(-0.700489\pi\)
−0.589026 + 0.808114i \(0.700489\pi\)
\(954\) −2.77751e8 −0.0103571
\(955\) 8.83175e9 0.328122
\(956\) −4.60774e9 −0.170563
\(957\) −1.20989e10 −0.446227
\(958\) −1.32695e10 −0.487615
\(959\) 9.58227e9 0.350835
\(960\) 2.97319e10 1.08461
\(961\) 1.40656e10 0.511244
\(962\) −2.28550e9 −0.0827690
\(963\) 3.93292e9 0.141913
\(964\) 2.31975e9 0.0834011
\(965\) −4.43599e10 −1.58908
\(966\) 2.84501e9 0.101546
\(967\) 1.15875e10 0.412094 0.206047 0.978542i \(-0.433940\pi\)
0.206047 + 0.978542i \(0.433940\pi\)
\(968\) −2.01735e10 −0.714855
\(969\) 3.12055e9 0.110179
\(970\) −1.31869e10 −0.463918
\(971\) −2.13089e10 −0.746955 −0.373477 0.927639i \(-0.621835\pi\)
−0.373477 + 0.927639i \(0.621835\pi\)
\(972\) 2.54546e9 0.0889068
\(973\) −4.81046e9 −0.167414
\(974\) −2.62174e10 −0.909146
\(975\) −1.40805e9 −0.0486521
\(976\) −2.56348e10 −0.882585
\(977\) −1.26958e10 −0.435541 −0.217770 0.976000i \(-0.569878\pi\)
−0.217770 + 0.976000i \(0.569878\pi\)
\(978\) 2.43751e10 0.833220
\(979\) −1.36754e10 −0.465801
\(980\) −1.08250e9 −0.0367396
\(981\) −4.46300e8 −0.0150933
\(982\) 1.46024e10 0.492078
\(983\) 4.72623e10 1.58700 0.793500 0.608570i \(-0.208256\pi\)
0.793500 + 0.608570i \(0.208256\pi\)
\(984\) 4.50466e10 1.50723
\(985\) 2.46661e10 0.822383
\(986\) 2.61630e9 0.0869196
\(987\) 1.10618e10 0.366197
\(988\) −1.67165e9 −0.0551439
\(989\) −1.35382e10 −0.445013
\(990\) −2.21011e9 −0.0723922
\(991\) −1.40624e10 −0.458989 −0.229495 0.973310i \(-0.573707\pi\)
−0.229495 + 0.973310i \(0.573707\pi\)
\(992\) −2.05486e10 −0.668330
\(993\) 2.18371e10 0.707739
\(994\) −1.14298e10 −0.369136
\(995\) −2.10935e10 −0.678839
\(996\) 8.92483e9 0.286215
\(997\) 2.64671e10 0.845809 0.422905 0.906174i \(-0.361010\pi\)
0.422905 + 0.906174i \(0.361010\pi\)
\(998\) 1.61896e10 0.515562
\(999\) −1.00374e10 −0.318525
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 91.8.a.e.1.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.e.1.5 12 1.1 even 1 trivial