Properties

Label 33.6.a.c.1.1
Level $33$
Weight $6$
Character 33.1
Self dual yes
Analytic conductor $5.293$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.29266605383\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
Defining polynomial: \(x^{2} - x - 44\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(7.15207\) of defining polynomial
Character \(\chi\) \(=\) 33.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.15207 q^{2} -9.00000 q^{3} +51.7603 q^{4} +95.5207 q^{5} +82.3686 q^{6} -209.521 q^{7} -180.848 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-9.15207 q^{2} -9.00000 q^{3} +51.7603 q^{4} +95.5207 q^{5} +82.3686 q^{6} -209.521 q^{7} -180.848 q^{8} +81.0000 q^{9} -874.212 q^{10} -121.000 q^{11} -465.843 q^{12} -335.779 q^{13} +1917.55 q^{14} -859.686 q^{15} -1.19832 q^{16} -799.124 q^{17} -741.317 q^{18} -658.175 q^{19} +4944.18 q^{20} +1885.69 q^{21} +1107.40 q^{22} -4119.66 q^{23} +1627.63 q^{24} +5999.20 q^{25} +3073.07 q^{26} -729.000 q^{27} -10844.9 q^{28} +559.348 q^{29} +7867.90 q^{30} -6052.31 q^{31} +5798.10 q^{32} +1089.00 q^{33} +7313.64 q^{34} -20013.6 q^{35} +4192.59 q^{36} -14053.3 q^{37} +6023.66 q^{38} +3022.01 q^{39} -17274.7 q^{40} +1846.27 q^{41} -17257.9 q^{42} +1623.49 q^{43} -6263.00 q^{44} +7737.17 q^{45} +37703.4 q^{46} +20728.1 q^{47} +10.7848 q^{48} +27091.9 q^{49} -54905.1 q^{50} +7192.12 q^{51} -17380.0 q^{52} -7585.46 q^{53} +6671.86 q^{54} -11558.0 q^{55} +37891.4 q^{56} +5923.58 q^{57} -5119.19 q^{58} +18468.5 q^{59} -44497.6 q^{60} -16972.3 q^{61} +55391.2 q^{62} -16971.2 q^{63} -53026.3 q^{64} -32073.8 q^{65} -9966.60 q^{66} -5618.62 q^{67} -41362.9 q^{68} +37076.9 q^{69} +183165. q^{70} +3703.10 q^{71} -14648.7 q^{72} -19808.0 q^{73} +128617. q^{74} -53992.8 q^{75} -34067.4 q^{76} +25352.0 q^{77} -27657.6 q^{78} +64009.9 q^{79} -114.464 q^{80} +6561.00 q^{81} -16897.2 q^{82} -46390.2 q^{83} +97603.7 q^{84} -76332.9 q^{85} -14858.3 q^{86} -5034.13 q^{87} +21882.6 q^{88} -53959.5 q^{89} -70811.1 q^{90} +70352.5 q^{91} -213235. q^{92} +54470.8 q^{93} -189705. q^{94} -62869.3 q^{95} -52182.9 q^{96} +145249. q^{97} -247947. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 5q^{2} - 18q^{3} + 37q^{4} + 58q^{5} + 45q^{6} - 286q^{7} - 375q^{8} + 162q^{9} + O(q^{10}) \) \( 2q - 5q^{2} - 18q^{3} + 37q^{4} + 58q^{5} + 45q^{6} - 286q^{7} - 375q^{8} + 162q^{9} - 1030q^{10} - 242q^{11} - 333q^{12} - 166q^{13} + 1600q^{14} - 522q^{15} - 335q^{16} - 800q^{17} - 405q^{18} - 1476q^{19} + 5498q^{20} + 2574q^{21} + 605q^{22} - 3370q^{23} + 3375q^{24} + 4282q^{25} + 3778q^{26} - 1458q^{27} - 9716q^{28} + 6600q^{29} + 9270q^{30} - 7528q^{31} + 10625q^{32} + 2178q^{33} + 7310q^{34} - 17144q^{35} + 2997q^{36} - 29916q^{37} + 2628q^{38} + 1494q^{39} - 9990q^{40} - 5780q^{41} - 14400q^{42} - 16656q^{43} - 4477q^{44} + 4698q^{45} + 40816q^{46} + 7850q^{47} + 3015q^{48} + 16134q^{49} - 62035q^{50} + 7200q^{51} - 19886q^{52} + 14178q^{53} + 3645q^{54} - 7018q^{55} + 52740q^{56} + 13284q^{57} + 19962q^{58} + 17300q^{59} - 49482q^{60} - 2946q^{61} + 49264q^{62} - 23166q^{63} - 22303q^{64} - 38444q^{65} - 5445q^{66} + 31336q^{67} - 41350q^{68} + 30330q^{69} + 195080q^{70} - 33810q^{71} - 30375q^{72} + 60644q^{73} + 62754q^{74} - 38538q^{75} - 21996q^{76} + 34606q^{77} - 34002q^{78} + 1870q^{79} + 12410q^{80} + 13122q^{81} - 48562q^{82} - 58296q^{83} + 87444q^{84} - 76300q^{85} - 90756q^{86} - 59400q^{87} + 45375q^{88} + 92388q^{89} - 83430q^{90} + 57368q^{91} - 224300q^{92} + 67752q^{93} - 243176q^{94} - 32184q^{95} - 95625q^{96} + 7120q^{97} - 293445q^{98} - 19602q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.15207 −1.61787 −0.808936 0.587897i \(-0.799956\pi\)
−0.808936 + 0.587897i \(0.799956\pi\)
\(3\) −9.00000 −0.577350
\(4\) 51.7603 1.61751
\(5\) 95.5207 1.70873 0.854363 0.519677i \(-0.173948\pi\)
0.854363 + 0.519677i \(0.173948\pi\)
\(6\) 82.3686 0.934079
\(7\) −209.521 −1.61615 −0.808075 0.589079i \(-0.799491\pi\)
−0.808075 + 0.589079i \(0.799491\pi\)
\(8\) −180.848 −0.999053
\(9\) 81.0000 0.333333
\(10\) −874.212 −2.76450
\(11\) −121.000 −0.301511
\(12\) −465.843 −0.933870
\(13\) −335.779 −0.551055 −0.275527 0.961293i \(-0.588852\pi\)
−0.275527 + 0.961293i \(0.588852\pi\)
\(14\) 1917.55 2.61472
\(15\) −859.686 −0.986533
\(16\) −1.19832 −0.00117023
\(17\) −799.124 −0.670644 −0.335322 0.942104i \(-0.608845\pi\)
−0.335322 + 0.942104i \(0.608845\pi\)
\(18\) −741.317 −0.539291
\(19\) −658.175 −0.418271 −0.209135 0.977887i \(-0.567065\pi\)
−0.209135 + 0.977887i \(0.567065\pi\)
\(20\) 4944.18 2.76388
\(21\) 1885.69 0.933085
\(22\) 1107.40 0.487807
\(23\) −4119.66 −1.62383 −0.811917 0.583773i \(-0.801576\pi\)
−0.811917 + 0.583773i \(0.801576\pi\)
\(24\) 1627.63 0.576804
\(25\) 5999.20 1.91974
\(26\) 3073.07 0.891536
\(27\) −729.000 −0.192450
\(28\) −10844.9 −2.61414
\(29\) 559.348 0.123506 0.0617529 0.998091i \(-0.480331\pi\)
0.0617529 + 0.998091i \(0.480331\pi\)
\(30\) 7867.90 1.59608
\(31\) −6052.31 −1.13114 −0.565571 0.824700i \(-0.691344\pi\)
−0.565571 + 0.824700i \(0.691344\pi\)
\(32\) 5798.10 1.00095
\(33\) 1089.00 0.174078
\(34\) 7313.64 1.08502
\(35\) −20013.6 −2.76156
\(36\) 4192.59 0.539170
\(37\) −14053.3 −1.68762 −0.843810 0.536642i \(-0.819692\pi\)
−0.843810 + 0.536642i \(0.819692\pi\)
\(38\) 6023.66 0.676709
\(39\) 3022.01 0.318151
\(40\) −17274.7 −1.70711
\(41\) 1846.27 0.171528 0.0857642 0.996315i \(-0.472667\pi\)
0.0857642 + 0.996315i \(0.472667\pi\)
\(42\) −17257.9 −1.50961
\(43\) 1623.49 0.133900 0.0669498 0.997756i \(-0.478673\pi\)
0.0669498 + 0.997756i \(0.478673\pi\)
\(44\) −6263.00 −0.487698
\(45\) 7737.17 0.569575
\(46\) 37703.4 2.62715
\(47\) 20728.1 1.36872 0.684361 0.729143i \(-0.260081\pi\)
0.684361 + 0.729143i \(0.260081\pi\)
\(48\) 10.7848 0.000675633 0
\(49\) 27091.9 1.61194
\(50\) −54905.1 −3.10590
\(51\) 7192.12 0.387196
\(52\) −17380.0 −0.891337
\(53\) −7585.46 −0.370930 −0.185465 0.982651i \(-0.559379\pi\)
−0.185465 + 0.982651i \(0.559379\pi\)
\(54\) 6671.86 0.311360
\(55\) −11558.0 −0.515200
\(56\) 37891.4 1.61462
\(57\) 5923.58 0.241489
\(58\) −5119.19 −0.199817
\(59\) 18468.5 0.690717 0.345359 0.938471i \(-0.387757\pi\)
0.345359 + 0.938471i \(0.387757\pi\)
\(60\) −44497.6 −1.59573
\(61\) −16972.3 −0.584005 −0.292002 0.956418i \(-0.594322\pi\)
−0.292002 + 0.956418i \(0.594322\pi\)
\(62\) 55391.2 1.83004
\(63\) −16971.2 −0.538717
\(64\) −53026.3 −1.61823
\(65\) −32073.8 −0.941601
\(66\) −9966.60 −0.281635
\(67\) −5618.62 −0.152912 −0.0764561 0.997073i \(-0.524361\pi\)
−0.0764561 + 0.997073i \(0.524361\pi\)
\(68\) −41362.9 −1.08477
\(69\) 37076.9 0.937521
\(70\) 183165. 4.46785
\(71\) 3703.10 0.0871807 0.0435903 0.999049i \(-0.486120\pi\)
0.0435903 + 0.999049i \(0.486120\pi\)
\(72\) −14648.7 −0.333018
\(73\) −19808.0 −0.435044 −0.217522 0.976055i \(-0.569797\pi\)
−0.217522 + 0.976055i \(0.569797\pi\)
\(74\) 128617. 2.73035
\(75\) −53992.8 −1.10836
\(76\) −34067.4 −0.676557
\(77\) 25352.0 0.487288
\(78\) −27657.6 −0.514728
\(79\) 64009.9 1.15393 0.576965 0.816769i \(-0.304237\pi\)
0.576965 + 0.816769i \(0.304237\pi\)
\(80\) −114.464 −0.00199960
\(81\) 6561.00 0.111111
\(82\) −16897.2 −0.277511
\(83\) −46390.2 −0.739147 −0.369573 0.929202i \(-0.620496\pi\)
−0.369573 + 0.929202i \(0.620496\pi\)
\(84\) 97603.7 1.50927
\(85\) −76332.9 −1.14595
\(86\) −14858.3 −0.216632
\(87\) −5034.13 −0.0713061
\(88\) 21882.6 0.301226
\(89\) −53959.5 −0.722093 −0.361046 0.932548i \(-0.617580\pi\)
−0.361046 + 0.932548i \(0.617580\pi\)
\(90\) −70811.1 −0.921500
\(91\) 70352.5 0.890587
\(92\) −213235. −2.62657
\(93\) 54470.8 0.653065
\(94\) −189705. −2.21442
\(95\) −62869.3 −0.714710
\(96\) −52182.9 −0.577897
\(97\) 145249. 1.56741 0.783707 0.621130i \(-0.213326\pi\)
0.783707 + 0.621130i \(0.213326\pi\)
\(98\) −247947. −2.60792
\(99\) −9801.00 −0.100504
\(100\) 310521. 3.10521
\(101\) −77215.1 −0.753180 −0.376590 0.926380i \(-0.622903\pi\)
−0.376590 + 0.926380i \(0.622903\pi\)
\(102\) −65822.7 −0.626434
\(103\) −106203. −0.986374 −0.493187 0.869923i \(-0.664168\pi\)
−0.493187 + 0.869923i \(0.664168\pi\)
\(104\) 60724.9 0.550533
\(105\) 180122. 1.59439
\(106\) 69422.6 0.600118
\(107\) 33750.8 0.284987 0.142493 0.989796i \(-0.454488\pi\)
0.142493 + 0.989796i \(0.454488\pi\)
\(108\) −37733.3 −0.311290
\(109\) −37977.1 −0.306165 −0.153083 0.988213i \(-0.548920\pi\)
−0.153083 + 0.988213i \(0.548920\pi\)
\(110\) 105780. 0.833528
\(111\) 126480. 0.974348
\(112\) 251.072 0.00189127
\(113\) 239549. 1.76481 0.882407 0.470486i \(-0.155922\pi\)
0.882407 + 0.470486i \(0.155922\pi\)
\(114\) −54213.0 −0.390698
\(115\) −393512. −2.77469
\(116\) 28952.1 0.199772
\(117\) −27198.1 −0.183685
\(118\) −169025. −1.11749
\(119\) 167433. 1.08386
\(120\) 155472. 0.985599
\(121\) 14641.0 0.0909091
\(122\) 155332. 0.944845
\(123\) −16616.4 −0.0990320
\(124\) −313270. −1.82963
\(125\) 274545. 1.57159
\(126\) 155321. 0.871575
\(127\) 88793.2 0.488507 0.244253 0.969711i \(-0.421457\pi\)
0.244253 + 0.969711i \(0.421457\pi\)
\(128\) 299761. 1.61715
\(129\) −14611.4 −0.0773070
\(130\) 293542. 1.52339
\(131\) 333640. 1.69864 0.849318 0.527881i \(-0.177013\pi\)
0.849318 + 0.527881i \(0.177013\pi\)
\(132\) 56367.0 0.281572
\(133\) 137901. 0.675988
\(134\) 51421.9 0.247392
\(135\) −69634.6 −0.328844
\(136\) 144520. 0.670009
\(137\) −377709. −1.71932 −0.859659 0.510869i \(-0.829324\pi\)
−0.859659 + 0.510869i \(0.829324\pi\)
\(138\) −339330. −1.51679
\(139\) −365308. −1.60370 −0.801849 0.597527i \(-0.796150\pi\)
−0.801849 + 0.597527i \(0.796150\pi\)
\(140\) −1.03591e6 −4.46685
\(141\) −186553. −0.790232
\(142\) −33891.1 −0.141047
\(143\) 40629.2 0.166149
\(144\) −97.0636 −0.000390077 0
\(145\) 53429.3 0.211038
\(146\) 181284. 0.703845
\(147\) −243827. −0.930655
\(148\) −727405. −2.72974
\(149\) 297568. 1.09805 0.549023 0.835807i \(-0.315000\pi\)
0.549023 + 0.835807i \(0.315000\pi\)
\(150\) 494146. 1.79319
\(151\) 318093. 1.13530 0.567651 0.823269i \(-0.307852\pi\)
0.567651 + 0.823269i \(0.307852\pi\)
\(152\) 119030. 0.417875
\(153\) −64729.0 −0.223548
\(154\) −232023. −0.788369
\(155\) −578121. −1.93281
\(156\) 156420. 0.514613
\(157\) −34489.6 −0.111671 −0.0558353 0.998440i \(-0.517782\pi\)
−0.0558353 + 0.998440i \(0.517782\pi\)
\(158\) −585823. −1.86691
\(159\) 68269.1 0.214157
\(160\) 553839. 1.71034
\(161\) 863153. 2.62436
\(162\) −60046.7 −0.179764
\(163\) −45865.7 −0.135213 −0.0676067 0.997712i \(-0.521536\pi\)
−0.0676067 + 0.997712i \(0.521536\pi\)
\(164\) 95563.7 0.277449
\(165\) 104022. 0.297451
\(166\) 424566. 1.19584
\(167\) 290943. 0.807265 0.403633 0.914921i \(-0.367747\pi\)
0.403633 + 0.914921i \(0.367747\pi\)
\(168\) −341022. −0.932201
\(169\) −258546. −0.696339
\(170\) 698604. 1.85399
\(171\) −53312.2 −0.139424
\(172\) 84032.5 0.216584
\(173\) −139360. −0.354016 −0.177008 0.984209i \(-0.556642\pi\)
−0.177008 + 0.984209i \(0.556642\pi\)
\(174\) 46072.7 0.115364
\(175\) −1.25696e6 −3.10259
\(176\) 144.996 0.000352838 0
\(177\) −166216. −0.398786
\(178\) 493841. 1.16825
\(179\) −599556. −1.39861 −0.699306 0.714823i \(-0.746507\pi\)
−0.699306 + 0.714823i \(0.746507\pi\)
\(180\) 400479. 0.921294
\(181\) 130631. 0.296380 0.148190 0.988959i \(-0.452655\pi\)
0.148190 + 0.988959i \(0.452655\pi\)
\(182\) −643871. −1.44086
\(183\) 152751. 0.337175
\(184\) 745031. 1.62230
\(185\) −1.34238e6 −2.88368
\(186\) −498520. −1.05658
\(187\) 96694.0 0.202207
\(188\) 1.07289e6 2.21392
\(189\) 152741. 0.311028
\(190\) 575384. 1.15631
\(191\) 338243. 0.670882 0.335441 0.942061i \(-0.391115\pi\)
0.335441 + 0.942061i \(0.391115\pi\)
\(192\) 477236. 0.934287
\(193\) −329094. −0.635955 −0.317978 0.948098i \(-0.603004\pi\)
−0.317978 + 0.948098i \(0.603004\pi\)
\(194\) −1.32933e6 −2.53588
\(195\) 288664. 0.543634
\(196\) 1.40229e6 2.60733
\(197\) 517397. 0.949858 0.474929 0.880024i \(-0.342474\pi\)
0.474929 + 0.880024i \(0.342474\pi\)
\(198\) 89699.4 0.162602
\(199\) −531436. −0.951302 −0.475651 0.879634i \(-0.657788\pi\)
−0.475651 + 0.879634i \(0.657788\pi\)
\(200\) −1.08494e6 −1.91793
\(201\) 50567.5 0.0882839
\(202\) 706678. 1.21855
\(203\) −117195. −0.199604
\(204\) 372266. 0.626294
\(205\) 176357. 0.293095
\(206\) 971973. 1.59583
\(207\) −333692. −0.541278
\(208\) 402.369 0.000644861 0
\(209\) 79639.2 0.126113
\(210\) −1.64849e6 −2.57951
\(211\) 409913. 0.633848 0.316924 0.948451i \(-0.397350\pi\)
0.316924 + 0.948451i \(0.397350\pi\)
\(212\) −392626. −0.599984
\(213\) −33327.9 −0.0503338
\(214\) −308890. −0.461072
\(215\) 155077. 0.228798
\(216\) 131838. 0.192268
\(217\) 1.26808e6 1.82810
\(218\) 347569. 0.495336
\(219\) 178272. 0.251173
\(220\) −598246. −0.833342
\(221\) 268329. 0.369561
\(222\) −1.15755e6 −1.57637
\(223\) 269898. 0.363444 0.181722 0.983350i \(-0.441833\pi\)
0.181722 + 0.983350i \(0.441833\pi\)
\(224\) −1.21482e6 −1.61768
\(225\) 485935. 0.639915
\(226\) −2.19237e6 −2.85524
\(227\) −1.02146e6 −1.31570 −0.657852 0.753147i \(-0.728535\pi\)
−0.657852 + 0.753147i \(0.728535\pi\)
\(228\) 306606. 0.390611
\(229\) 169276. 0.213307 0.106654 0.994296i \(-0.465986\pi\)
0.106654 + 0.994296i \(0.465986\pi\)
\(230\) 3.60145e6 4.48909
\(231\) −228168. −0.281336
\(232\) −101157. −0.123389
\(233\) −20445.6 −0.0246723 −0.0123361 0.999924i \(-0.503927\pi\)
−0.0123361 + 0.999924i \(0.503927\pi\)
\(234\) 248919. 0.297179
\(235\) 1.97996e6 2.33877
\(236\) 955933. 1.11724
\(237\) −576089. −0.666222
\(238\) −1.53236e6 −1.75355
\(239\) −933552. −1.05717 −0.528583 0.848881i \(-0.677277\pi\)
−0.528583 + 0.848881i \(0.677277\pi\)
\(240\) 1030.18 0.00115447
\(241\) −1.30259e6 −1.44466 −0.722331 0.691547i \(-0.756929\pi\)
−0.722331 + 0.691547i \(0.756929\pi\)
\(242\) −133995. −0.147079
\(243\) −59049.0 −0.0641500
\(244\) −878493. −0.944634
\(245\) 2.58784e6 2.75437
\(246\) 152075. 0.160221
\(247\) 221001. 0.230490
\(248\) 1.09455e6 1.13007
\(249\) 417511. 0.426747
\(250\) −2.51266e6 −2.54263
\(251\) −930872. −0.932622 −0.466311 0.884621i \(-0.654417\pi\)
−0.466311 + 0.884621i \(0.654417\pi\)
\(252\) −878434. −0.871380
\(253\) 498478. 0.489604
\(254\) −812642. −0.790342
\(255\) 686996. 0.661612
\(256\) −1.04659e6 −0.998106
\(257\) −25313.6 −0.0239067 −0.0119534 0.999929i \(-0.503805\pi\)
−0.0119534 + 0.999929i \(0.503805\pi\)
\(258\) 133725. 0.125073
\(259\) 2.94446e6 2.72745
\(260\) −1.66015e6 −1.52305
\(261\) 45307.2 0.0411686
\(262\) −3.05350e6 −2.74818
\(263\) −1.69322e6 −1.50947 −0.754735 0.656030i \(-0.772235\pi\)
−0.754735 + 0.656030i \(0.772235\pi\)
\(264\) −196943. −0.173913
\(265\) −724568. −0.633818
\(266\) −1.26208e6 −1.09366
\(267\) 485636. 0.416901
\(268\) −290821. −0.247337
\(269\) 469365. 0.395485 0.197742 0.980254i \(-0.436639\pi\)
0.197742 + 0.980254i \(0.436639\pi\)
\(270\) 637300. 0.532028
\(271\) −2.18341e6 −1.80598 −0.902988 0.429666i \(-0.858631\pi\)
−0.902988 + 0.429666i \(0.858631\pi\)
\(272\) 957.603 0.000784808 0
\(273\) −633173. −0.514181
\(274\) 3.45682e6 2.78164
\(275\) −725903. −0.578825
\(276\) 1.91911e6 1.51645
\(277\) −84280.4 −0.0659974 −0.0329987 0.999455i \(-0.510506\pi\)
−0.0329987 + 0.999455i \(0.510506\pi\)
\(278\) 3.34333e6 2.59458
\(279\) −490237. −0.377047
\(280\) 3.61941e6 2.75894
\(281\) 649515. 0.490708 0.245354 0.969433i \(-0.421096\pi\)
0.245354 + 0.969433i \(0.421096\pi\)
\(282\) 1.70735e6 1.27849
\(283\) 1.54893e6 1.14965 0.574826 0.818276i \(-0.305070\pi\)
0.574826 + 0.818276i \(0.305070\pi\)
\(284\) 191674. 0.141016
\(285\) 565824. 0.412638
\(286\) −371841. −0.268808
\(287\) −386832. −0.277216
\(288\) 469646. 0.333649
\(289\) −781258. −0.550237
\(290\) −488989. −0.341432
\(291\) −1.30724e6 −0.904947
\(292\) −1.02527e6 −0.703688
\(293\) −70276.6 −0.0478236 −0.0239118 0.999714i \(-0.507612\pi\)
−0.0239118 + 0.999714i \(0.507612\pi\)
\(294\) 2.23152e6 1.50568
\(295\) 1.76412e6 1.18025
\(296\) 2.54151e6 1.68602
\(297\) 88209.0 0.0580259
\(298\) −2.72336e6 −1.77650
\(299\) 1.38329e6 0.894821
\(300\) −2.79469e6 −1.79279
\(301\) −340155. −0.216402
\(302\) −2.91121e6 −1.83678
\(303\) 694936. 0.434849
\(304\) 788.702 0.000489473 0
\(305\) −1.62121e6 −0.997904
\(306\) 592405. 0.361672
\(307\) 1.43343e6 0.868022 0.434011 0.900908i \(-0.357098\pi\)
0.434011 + 0.900908i \(0.357098\pi\)
\(308\) 1.31223e6 0.788193
\(309\) 955823. 0.569484
\(310\) 5.29100e6 3.12704
\(311\) −227879. −0.133599 −0.0667996 0.997766i \(-0.521279\pi\)
−0.0667996 + 0.997766i \(0.521279\pi\)
\(312\) −546524. −0.317850
\(313\) −723760. −0.417574 −0.208787 0.977961i \(-0.566952\pi\)
−0.208787 + 0.977961i \(0.566952\pi\)
\(314\) 315651. 0.180669
\(315\) −1.62110e6 −0.920519
\(316\) 3.31317e6 1.86649
\(317\) 1.26086e6 0.704723 0.352362 0.935864i \(-0.385379\pi\)
0.352362 + 0.935864i \(0.385379\pi\)
\(318\) −624804. −0.346478
\(319\) −67681.1 −0.0372384
\(320\) −5.06510e6 −2.76512
\(321\) −303757. −0.164537
\(322\) −7.89964e6 −4.24588
\(323\) 525964. 0.280511
\(324\) 339600. 0.179723
\(325\) −2.01440e6 −1.05788
\(326\) 419766. 0.218758
\(327\) 341794. 0.176765
\(328\) −333894. −0.171366
\(329\) −4.34297e6 −2.21206
\(330\) −952016. −0.481238
\(331\) 433444. 0.217452 0.108726 0.994072i \(-0.465323\pi\)
0.108726 + 0.994072i \(0.465323\pi\)
\(332\) −2.40117e6 −1.19558
\(333\) −1.13832e6 −0.562540
\(334\) −2.66273e6 −1.30605
\(335\) −536694. −0.261285
\(336\) −2259.65 −0.00109192
\(337\) 3.80554e6 1.82533 0.912666 0.408706i \(-0.134020\pi\)
0.912666 + 0.408706i \(0.134020\pi\)
\(338\) 2.36623e6 1.12659
\(339\) −2.15594e6 −1.01892
\(340\) −3.95101e6 −1.85358
\(341\) 732330. 0.341052
\(342\) 487917. 0.225570
\(343\) −2.15490e6 −0.988991
\(344\) −293605. −0.133773
\(345\) 3.54161e6 1.60197
\(346\) 1.27543e6 0.572753
\(347\) 2.91029e6 1.29752 0.648759 0.760994i \(-0.275288\pi\)
0.648759 + 0.760994i \(0.275288\pi\)
\(348\) −260568. −0.115338
\(349\) −4.13500e6 −1.81724 −0.908620 0.417625i \(-0.862863\pi\)
−0.908620 + 0.417625i \(0.862863\pi\)
\(350\) 1.15037e7 5.01960
\(351\) 244783. 0.106050
\(352\) −701570. −0.301797
\(353\) 499935. 0.213539 0.106769 0.994284i \(-0.465949\pi\)
0.106769 + 0.994284i \(0.465949\pi\)
\(354\) 1.52122e6 0.645185
\(355\) 353723. 0.148968
\(356\) −2.79296e6 −1.16799
\(357\) −1.50690e6 −0.625768
\(358\) 5.48718e6 2.26278
\(359\) −2.96365e6 −1.21364 −0.606822 0.794838i \(-0.707556\pi\)
−0.606822 + 0.794838i \(0.707556\pi\)
\(360\) −1.39925e6 −0.569036
\(361\) −2.04290e6 −0.825050
\(362\) −1.19554e6 −0.479504
\(363\) −131769. −0.0524864
\(364\) 3.64147e6 1.44053
\(365\) −1.89207e6 −0.743371
\(366\) −1.39799e6 −0.545507
\(367\) 257505. 0.0997976 0.0498988 0.998754i \(-0.484110\pi\)
0.0498988 + 0.998754i \(0.484110\pi\)
\(368\) 4936.65 0.00190026
\(369\) 149548. 0.0571761
\(370\) 1.22856e7 4.66542
\(371\) 1.58931e6 0.599479
\(372\) 2.81943e6 1.05634
\(373\) −1.28217e6 −0.477169 −0.238584 0.971122i \(-0.576683\pi\)
−0.238584 + 0.971122i \(0.576683\pi\)
\(374\) −884950. −0.327145
\(375\) −2.47091e6 −0.907358
\(376\) −3.74864e6 −1.36743
\(377\) −187817. −0.0680584
\(378\) −1.39789e6 −0.503204
\(379\) 3.13440e6 1.12087 0.560436 0.828198i \(-0.310634\pi\)
0.560436 + 0.828198i \(0.310634\pi\)
\(380\) −3.25414e6 −1.15605
\(381\) −799139. −0.282040
\(382\) −3.09563e6 −1.08540
\(383\) 875781. 0.305069 0.152535 0.988298i \(-0.451256\pi\)
0.152535 + 0.988298i \(0.451256\pi\)
\(384\) −2.69785e6 −0.933661
\(385\) 2.42164e6 0.832641
\(386\) 3.01189e6 1.02889
\(387\) 131503. 0.0446332
\(388\) 7.51814e6 2.53531
\(389\) −1.95280e6 −0.654312 −0.327156 0.944970i \(-0.606090\pi\)
−0.327156 + 0.944970i \(0.606090\pi\)
\(390\) −2.64187e6 −0.879530
\(391\) 3.29212e6 1.08901
\(392\) −4.89952e6 −1.61042
\(393\) −3.00276e6 −0.980708
\(394\) −4.73526e6 −1.53675
\(395\) 6.11427e6 1.97175
\(396\) −507303. −0.162566
\(397\) −1.70718e6 −0.543629 −0.271814 0.962350i \(-0.587624\pi\)
−0.271814 + 0.962350i \(0.587624\pi\)
\(398\) 4.86374e6 1.53909
\(399\) −1.24111e6 −0.390282
\(400\) −7188.94 −0.00224654
\(401\) 4.51630e6 1.40256 0.701280 0.712886i \(-0.252612\pi\)
0.701280 + 0.712886i \(0.252612\pi\)
\(402\) −462798. −0.142832
\(403\) 2.03224e6 0.623321
\(404\) −3.99668e6 −1.21828
\(405\) 626711. 0.189858
\(406\) 1.07258e6 0.322934
\(407\) 1.70045e6 0.508836
\(408\) −1.30068e6 −0.386830
\(409\) −721013. −0.213125 −0.106563 0.994306i \(-0.533984\pi\)
−0.106563 + 0.994306i \(0.533984\pi\)
\(410\) −1.61403e6 −0.474190
\(411\) 3.39938e6 0.992648
\(412\) −5.49708e6 −1.59547
\(413\) −3.86952e6 −1.11630
\(414\) 3.05397e6 0.875718
\(415\) −4.43122e6 −1.26300
\(416\) −1.94688e6 −0.551576
\(417\) 3.28777e6 0.925895
\(418\) −728863. −0.204035
\(419\) 3.89904e6 1.08498 0.542491 0.840061i \(-0.317481\pi\)
0.542491 + 0.840061i \(0.317481\pi\)
\(420\) 9.32318e6 2.57894
\(421\) −3.39665e6 −0.933996 −0.466998 0.884258i \(-0.654665\pi\)
−0.466998 + 0.884258i \(0.654665\pi\)
\(422\) −3.75155e6 −1.02549
\(423\) 1.67898e6 0.456241
\(424\) 1.37181e6 0.370579
\(425\) −4.79410e6 −1.28746
\(426\) 305020. 0.0814336
\(427\) 3.55605e6 0.943840
\(428\) 1.74695e6 0.460969
\(429\) −365663. −0.0959263
\(430\) −1.41928e6 −0.370165
\(431\) −5.15196e6 −1.33592 −0.667958 0.744199i \(-0.732832\pi\)
−0.667958 + 0.744199i \(0.732832\pi\)
\(432\) 873.573 0.000225211 0
\(433\) −3.23450e6 −0.829063 −0.414531 0.910035i \(-0.636055\pi\)
−0.414531 + 0.910035i \(0.636055\pi\)
\(434\) −1.16056e7 −2.95762
\(435\) −480864. −0.121843
\(436\) −1.96571e6 −0.495226
\(437\) 2.71146e6 0.679202
\(438\) −1.63156e6 −0.406365
\(439\) 6.31060e6 1.56282 0.781411 0.624017i \(-0.214500\pi\)
0.781411 + 0.624017i \(0.214500\pi\)
\(440\) 2.09024e6 0.514712
\(441\) 2.19444e6 0.537314
\(442\) −2.45576e6 −0.597903
\(443\) 2.87512e6 0.696061 0.348030 0.937483i \(-0.386851\pi\)
0.348030 + 0.937483i \(0.386851\pi\)
\(444\) 6.54664e6 1.57602
\(445\) −5.15425e6 −1.23386
\(446\) −2.47013e6 −0.588006
\(447\) −2.67811e6 −0.633957
\(448\) 1.11101e7 2.61531
\(449\) −4.54361e6 −1.06362 −0.531809 0.846865i \(-0.678487\pi\)
−0.531809 + 0.846865i \(0.678487\pi\)
\(450\) −4.44731e6 −1.03530
\(451\) −223399. −0.0517178
\(452\) 1.23992e7 2.85461
\(453\) −2.86284e6 −0.655467
\(454\) 9.34850e6 2.12864
\(455\) 6.72012e6 1.52177
\(456\) −1.07127e6 −0.241260
\(457\) 2.78485e6 0.623751 0.311876 0.950123i \(-0.399043\pi\)
0.311876 + 0.950123i \(0.399043\pi\)
\(458\) −1.54922e6 −0.345104
\(459\) 582561. 0.129065
\(460\) −2.03683e7 −4.48808
\(461\) −2.00495e6 −0.439391 −0.219695 0.975569i \(-0.570506\pi\)
−0.219695 + 0.975569i \(0.570506\pi\)
\(462\) 2.08821e6 0.455165
\(463\) −4.67402e6 −1.01330 −0.506650 0.862152i \(-0.669116\pi\)
−0.506650 + 0.862152i \(0.669116\pi\)
\(464\) −670.276 −0.000144530 0
\(465\) 5.20309e6 1.11591
\(466\) 187119. 0.0399166
\(467\) −1.41948e6 −0.301188 −0.150594 0.988596i \(-0.548119\pi\)
−0.150594 + 0.988596i \(0.548119\pi\)
\(468\) −1.40778e6 −0.297112
\(469\) 1.17722e6 0.247129
\(470\) −1.81208e7 −3.78383
\(471\) 310406. 0.0644731
\(472\) −3.33998e6 −0.690063
\(473\) −196443. −0.0403722
\(474\) 5.27241e6 1.07786
\(475\) −3.94852e6 −0.802973
\(476\) 8.66639e6 1.75316
\(477\) −614422. −0.123643
\(478\) 8.54393e6 1.71036
\(479\) −3.60077e6 −0.717062 −0.358531 0.933518i \(-0.616722\pi\)
−0.358531 + 0.933518i \(0.616722\pi\)
\(480\) −4.98455e6 −0.987467
\(481\) 4.71880e6 0.929970
\(482\) 1.19214e7 2.33728
\(483\) −7.76838e6 −1.51517
\(484\) 757823. 0.147046
\(485\) 1.38743e7 2.67828
\(486\) 540420. 0.103787
\(487\) −5.57519e6 −1.06521 −0.532607 0.846363i \(-0.678788\pi\)
−0.532607 + 0.846363i \(0.678788\pi\)
\(488\) 3.06941e6 0.583452
\(489\) 412792. 0.0780654
\(490\) −2.36841e7 −4.45621
\(491\) −1.80948e6 −0.338728 −0.169364 0.985554i \(-0.554171\pi\)
−0.169364 + 0.985554i \(0.554171\pi\)
\(492\) −860073. −0.160185
\(493\) −446989. −0.0828284
\(494\) −2.02262e6 −0.372903
\(495\) −936198. −0.171733
\(496\) 7252.58 0.00132370
\(497\) −775877. −0.140897
\(498\) −3.82109e6 −0.690421
\(499\) −5.94504e6 −1.06882 −0.534409 0.845226i \(-0.679466\pi\)
−0.534409 + 0.845226i \(0.679466\pi\)
\(500\) 1.42106e7 2.54206
\(501\) −2.61848e6 −0.466075
\(502\) 8.51940e6 1.50886
\(503\) 6.57296e6 1.15835 0.579176 0.815202i \(-0.303374\pi\)
0.579176 + 0.815202i \(0.303374\pi\)
\(504\) 3.06920e6 0.538207
\(505\) −7.37564e6 −1.28698
\(506\) −4.56211e6 −0.792117
\(507\) 2.32691e6 0.402031
\(508\) 4.59597e6 0.790165
\(509\) 1.76650e6 0.302218 0.151109 0.988517i \(-0.451716\pi\)
0.151109 + 0.988517i \(0.451716\pi\)
\(510\) −6.28743e6 −1.07040
\(511\) 4.15018e6 0.703096
\(512\) −13882.8 −0.00234046
\(513\) 479810. 0.0804962
\(514\) 231671. 0.0386780
\(515\) −1.01445e7 −1.68544
\(516\) −756293. −0.125045
\(517\) −2.50810e6 −0.412685
\(518\) −2.69479e7 −4.41266
\(519\) 1.25424e6 0.204391
\(520\) 5.80048e6 0.940709
\(521\) 9.74164e6 1.57231 0.786154 0.618031i \(-0.212069\pi\)
0.786154 + 0.618031i \(0.212069\pi\)
\(522\) −414655. −0.0666055
\(523\) 6.30069e6 1.00724 0.503621 0.863925i \(-0.332001\pi\)
0.503621 + 0.863925i \(0.332001\pi\)
\(524\) 1.72693e7 2.74756
\(525\) 1.13126e7 1.79128
\(526\) 1.54965e7 2.44213
\(527\) 4.83655e6 0.758593
\(528\) −1304.97 −0.000203711 0
\(529\) 1.05352e7 1.63683
\(530\) 6.63130e6 1.02544
\(531\) 1.49594e6 0.230239
\(532\) 7.13782e6 1.09342
\(533\) −619939. −0.0945215
\(534\) −4.44457e6 −0.674492
\(535\) 3.22390e6 0.486964
\(536\) 1.01611e6 0.152767
\(537\) 5.39600e6 0.807489
\(538\) −4.29566e6 −0.639844
\(539\) −3.27812e6 −0.486019
\(540\) −3.60431e6 −0.531909
\(541\) −1.12441e7 −1.65169 −0.825847 0.563894i \(-0.809303\pi\)
−0.825847 + 0.563894i \(0.809303\pi\)
\(542\) 1.99827e7 2.92184
\(543\) −1.17567e6 −0.171115
\(544\) −4.63340e6 −0.671278
\(545\) −3.62760e6 −0.523153
\(546\) 5.79484e6 0.831879
\(547\) 60699.0 0.00867388 0.00433694 0.999991i \(-0.498620\pi\)
0.00433694 + 0.999991i \(0.498620\pi\)
\(548\) −1.95504e7 −2.78101
\(549\) −1.37476e6 −0.194668
\(550\) 6.64351e6 0.936464
\(551\) −368149. −0.0516589
\(552\) −6.70528e6 −0.936633
\(553\) −1.34114e7 −1.86492
\(554\) 771340. 0.106775
\(555\) 1.20814e7 1.66489
\(556\) −1.89085e7 −2.59400
\(557\) 2.28778e6 0.312447 0.156224 0.987722i \(-0.450068\pi\)
0.156224 + 0.987722i \(0.450068\pi\)
\(558\) 4.48668e6 0.610014
\(559\) −545134. −0.0737860
\(560\) 23982.6 0.00323166
\(561\) −870246. −0.116744
\(562\) −5.94441e6 −0.793904
\(563\) −5.37938e6 −0.715256 −0.357628 0.933864i \(-0.616414\pi\)
−0.357628 + 0.933864i \(0.616414\pi\)
\(564\) −9.65605e6 −1.27821
\(565\) 2.28819e7 3.01558
\(566\) −1.41759e7 −1.85999
\(567\) −1.37467e6 −0.179572
\(568\) −669699. −0.0870981
\(569\) −6.95488e6 −0.900552 −0.450276 0.892889i \(-0.648674\pi\)
−0.450276 + 0.892889i \(0.648674\pi\)
\(570\) −5.17846e6 −0.667596
\(571\) −3.19590e6 −0.410206 −0.205103 0.978740i \(-0.565753\pi\)
−0.205103 + 0.978740i \(0.565753\pi\)
\(572\) 2.10298e6 0.268748
\(573\) −3.04419e6 −0.387334
\(574\) 3.54031e6 0.448500
\(575\) −2.47146e7 −3.11734
\(576\) −4.29513e6 −0.539411
\(577\) 2.54992e6 0.318851 0.159425 0.987210i \(-0.449036\pi\)
0.159425 + 0.987210i \(0.449036\pi\)
\(578\) 7.15012e6 0.890213
\(579\) 2.96184e6 0.367169
\(580\) 2.76552e6 0.341355
\(581\) 9.71970e6 1.19457
\(582\) 1.19640e7 1.46409
\(583\) 917841. 0.111840
\(584\) 3.58223e6 0.434632
\(585\) −2.59798e6 −0.313867
\(586\) 643177. 0.0773724
\(587\) −1.96757e6 −0.235687 −0.117843 0.993032i \(-0.537598\pi\)
−0.117843 + 0.993032i \(0.537598\pi\)
\(588\) −1.26206e7 −1.50534
\(589\) 3.98348e6 0.473124
\(590\) −1.61453e7 −1.90949
\(591\) −4.65658e6 −0.548401
\(592\) 16840.3 0.00197490
\(593\) −8.41413e6 −0.982590 −0.491295 0.870993i \(-0.663476\pi\)
−0.491295 + 0.870993i \(0.663476\pi\)
\(594\) −807295. −0.0938785
\(595\) 1.59933e7 1.85202
\(596\) 1.54022e7 1.77610
\(597\) 4.78293e6 0.549235
\(598\) −1.26600e7 −1.44771
\(599\) −1.00298e7 −1.14216 −0.571078 0.820896i \(-0.693475\pi\)
−0.571078 + 0.820896i \(0.693475\pi\)
\(600\) 9.76448e6 1.10732
\(601\) −1.42206e7 −1.60595 −0.802977 0.596010i \(-0.796752\pi\)
−0.802977 + 0.596010i \(0.796752\pi\)
\(602\) 3.11312e6 0.350111
\(603\) −455108. −0.0509708
\(604\) 1.64646e7 1.83636
\(605\) 1.39852e6 0.155339
\(606\) −6.36010e6 −0.703530
\(607\) 1.51717e7 1.67133 0.835665 0.549240i \(-0.185083\pi\)
0.835665 + 0.549240i \(0.185083\pi\)
\(608\) −3.81617e6 −0.418667
\(609\) 1.05476e6 0.115241
\(610\) 1.48374e7 1.61448
\(611\) −6.96006e6 −0.754241
\(612\) −3.35040e6 −0.361591
\(613\) −1.52884e7 −1.64328 −0.821638 0.570010i \(-0.806939\pi\)
−0.821638 + 0.570010i \(0.806939\pi\)
\(614\) −1.31189e7 −1.40435
\(615\) −1.58721e6 −0.169218
\(616\) −4.58486e6 −0.486826
\(617\) −1.81989e6 −0.192457 −0.0962284 0.995359i \(-0.530678\pi\)
−0.0962284 + 0.995359i \(0.530678\pi\)
\(618\) −8.74775e6 −0.921352
\(619\) 1.39678e7 1.46521 0.732606 0.680653i \(-0.238304\pi\)
0.732606 + 0.680653i \(0.238304\pi\)
\(620\) −2.99237e7 −3.12634
\(621\) 3.00323e6 0.312507
\(622\) 2.08557e6 0.216147
\(623\) 1.13056e7 1.16701
\(624\) −3621.32 −0.000372311 0
\(625\) 7.47727e6 0.765672
\(626\) 6.62390e6 0.675582
\(627\) −716753. −0.0728116
\(628\) −1.78519e6 −0.180628
\(629\) 1.12303e7 1.13179
\(630\) 1.48364e7 1.48928
\(631\) −1.64990e7 −1.64962 −0.824808 0.565413i \(-0.808717\pi\)
−0.824808 + 0.565413i \(0.808717\pi\)
\(632\) −1.15761e7 −1.15284
\(633\) −3.68921e6 −0.365952
\(634\) −1.15395e7 −1.14015
\(635\) 8.48159e6 0.834724
\(636\) 3.53363e6 0.346401
\(637\) −9.09688e6 −0.888268
\(638\) 619422. 0.0602470
\(639\) 299951. 0.0290602
\(640\) 2.86333e7 2.76326
\(641\) 1.14852e7 1.10406 0.552030 0.833824i \(-0.313853\pi\)
0.552030 + 0.833824i \(0.313853\pi\)
\(642\) 2.78001e6 0.266200
\(643\) −5.83065e6 −0.556147 −0.278073 0.960560i \(-0.589696\pi\)
−0.278073 + 0.960560i \(0.589696\pi\)
\(644\) 4.46771e7 4.24493
\(645\) −1.39569e6 −0.132096
\(646\) −4.81365e6 −0.453830
\(647\) −1.27091e7 −1.19359 −0.596795 0.802394i \(-0.703559\pi\)
−0.596795 + 0.802394i \(0.703559\pi\)
\(648\) −1.18654e6 −0.111006
\(649\) −2.23468e6 −0.208259
\(650\) 1.84359e7 1.71152
\(651\) −1.14128e7 −1.05545
\(652\) −2.37403e6 −0.218709
\(653\) −8.01118e6 −0.735214 −0.367607 0.929981i \(-0.619823\pi\)
−0.367607 + 0.929981i \(0.619823\pi\)
\(654\) −3.12812e6 −0.285983
\(655\) 3.18696e7 2.90250
\(656\) −2212.42 −0.000200728 0
\(657\) −1.60445e6 −0.145015
\(658\) 3.97472e7 3.57883
\(659\) 1.48896e7 1.33558 0.667790 0.744350i \(-0.267241\pi\)
0.667790 + 0.744350i \(0.267241\pi\)
\(660\) 5.38421e6 0.481130
\(661\) −1.15205e7 −1.02558 −0.512788 0.858515i \(-0.671387\pi\)
−0.512788 + 0.858515i \(0.671387\pi\)
\(662\) −3.96691e6 −0.351809
\(663\) −2.41496e6 −0.213366
\(664\) 8.38956e6 0.738447
\(665\) 1.31724e7 1.15508
\(666\) 1.04180e7 0.910118
\(667\) −2.30432e6 −0.200553
\(668\) 1.50593e7 1.30576
\(669\) −2.42908e6 −0.209835
\(670\) 4.91186e6 0.422726
\(671\) 2.05365e6 0.176084
\(672\) 1.09334e7 0.933968
\(673\) −1.68386e7 −1.43308 −0.716538 0.697548i \(-0.754274\pi\)
−0.716538 + 0.697548i \(0.754274\pi\)
\(674\) −3.48286e7 −2.95315
\(675\) −4.37342e6 −0.369455
\(676\) −1.33824e7 −1.12634
\(677\) −6.56811e6 −0.550768 −0.275384 0.961334i \(-0.588805\pi\)
−0.275384 + 0.961334i \(0.588805\pi\)
\(678\) 1.97314e7 1.64848
\(679\) −3.04327e7 −2.53318
\(680\) 1.38046e7 1.14486
\(681\) 9.19317e6 0.759622
\(682\) −6.70233e6 −0.551779
\(683\) −1.01525e7 −0.832761 −0.416381 0.909190i \(-0.636702\pi\)
−0.416381 + 0.909190i \(0.636702\pi\)
\(684\) −2.75946e6 −0.225519
\(685\) −3.60790e7 −2.93784
\(686\) 1.97218e7 1.60006
\(687\) −1.52348e6 −0.123153
\(688\) −1945.46 −0.000156693 0
\(689\) 2.54704e6 0.204403
\(690\) −3.24131e7 −2.59178
\(691\) −1.01356e7 −0.807521 −0.403760 0.914865i \(-0.632297\pi\)
−0.403760 + 0.914865i \(0.632297\pi\)
\(692\) −7.21332e6 −0.572625
\(693\) 2.05351e6 0.162429
\(694\) −2.66352e7 −2.09922
\(695\) −3.48945e7 −2.74028
\(696\) 910413. 0.0712386
\(697\) −1.47540e6 −0.115034
\(698\) 3.78438e7 2.94006
\(699\) 184010. 0.0142445
\(700\) −6.50605e7 −5.01848
\(701\) 9.16091e6 0.704115 0.352058 0.935978i \(-0.385482\pi\)
0.352058 + 0.935978i \(0.385482\pi\)
\(702\) −2.24027e6 −0.171576
\(703\) 9.24955e6 0.705882
\(704\) 6.41618e6 0.487916
\(705\) −1.78197e7 −1.35029
\(706\) −4.57544e6 −0.345478
\(707\) 1.61782e7 1.21725
\(708\) −8.60340e6 −0.645040
\(709\) −8.97668e6 −0.670656 −0.335328 0.942101i \(-0.608847\pi\)
−0.335328 + 0.942101i \(0.608847\pi\)
\(710\) −3.23730e6 −0.241011
\(711\) 5.18480e6 0.384643
\(712\) 9.75847e6 0.721409
\(713\) 2.49334e7 1.83679
\(714\) 1.37912e7 1.01241
\(715\) 3.88093e6 0.283903
\(716\) −3.10332e7 −2.26227
\(717\) 8.40196e6 0.610356
\(718\) 2.71235e7 1.96352
\(719\) 7.92637e6 0.571811 0.285905 0.958258i \(-0.407706\pi\)
0.285905 + 0.958258i \(0.407706\pi\)
\(720\) −9271.58 −0.000666534 0
\(721\) 2.22516e7 1.59413
\(722\) 1.86968e7 1.33482
\(723\) 1.17233e7 0.834076
\(724\) 6.76148e6 0.479397
\(725\) 3.35564e6 0.237099
\(726\) 1.20596e6 0.0849163
\(727\) 1.78172e7 1.25027 0.625135 0.780516i \(-0.285044\pi\)
0.625135 + 0.780516i \(0.285044\pi\)
\(728\) −1.27231e7 −0.889744
\(729\) 531441. 0.0370370
\(730\) 1.73164e7 1.20268
\(731\) −1.29737e6 −0.0897989
\(732\) 7.90644e6 0.545385
\(733\) 1.81797e7 1.24976 0.624882 0.780719i \(-0.285147\pi\)
0.624882 + 0.780719i \(0.285147\pi\)
\(734\) −2.35670e6 −0.161460
\(735\) −2.32905e7 −1.59023
\(736\) −2.38862e7 −1.62537
\(737\) 679852. 0.0461048
\(738\) −1.36867e6 −0.0925037
\(739\) −1.38291e7 −0.931502 −0.465751 0.884916i \(-0.654216\pi\)
−0.465751 + 0.884916i \(0.654216\pi\)
\(740\) −6.94822e7 −4.66438
\(741\) −1.98901e6 −0.133073
\(742\) −1.45455e7 −0.969881
\(743\) −7.07226e6 −0.469987 −0.234994 0.971997i \(-0.575507\pi\)
−0.234994 + 0.971997i \(0.575507\pi\)
\(744\) −9.85093e6 −0.652447
\(745\) 2.84239e7 1.87626
\(746\) 1.17345e7 0.771998
\(747\) −3.75760e6 −0.246382
\(748\) 5.00491e6 0.327071
\(749\) −7.07149e6 −0.460582
\(750\) 2.26139e7 1.46799
\(751\) −653678. −0.0422926 −0.0211463 0.999776i \(-0.506732\pi\)
−0.0211463 + 0.999776i \(0.506732\pi\)
\(752\) −24838.8 −0.00160172
\(753\) 8.37785e6 0.538449
\(754\) 1.71892e6 0.110110
\(755\) 3.03845e7 1.93992
\(756\) 7.90590e6 0.503092
\(757\) 2.72250e7 1.72674 0.863372 0.504569i \(-0.168348\pi\)
0.863372 + 0.504569i \(0.168348\pi\)
\(758\) −2.86862e7 −1.81343
\(759\) −4.48631e6 −0.282673
\(760\) 1.13698e7 0.714033
\(761\) −5.41780e6 −0.339126 −0.169563 0.985519i \(-0.554236\pi\)
−0.169563 + 0.985519i \(0.554236\pi\)
\(762\) 7.31378e6 0.456304
\(763\) 7.95700e6 0.494809
\(764\) 1.75076e7 1.08516
\(765\) −6.18296e6 −0.381982
\(766\) −8.01521e6 −0.493563
\(767\) −6.20131e6 −0.380623
\(768\) 9.41931e6 0.576257
\(769\) 2.96407e7 1.80748 0.903739 0.428084i \(-0.140811\pi\)
0.903739 + 0.428084i \(0.140811\pi\)
\(770\) −2.21630e7 −1.34711
\(771\) 227822. 0.0138026
\(772\) −1.70340e7 −1.02866
\(773\) 1.87755e7 1.13017 0.565083 0.825034i \(-0.308844\pi\)
0.565083 + 0.825034i \(0.308844\pi\)
\(774\) −1.20352e6 −0.0722108
\(775\) −3.63090e7 −2.17150
\(776\) −2.62680e7 −1.56593
\(777\) −2.65001e7 −1.57469
\(778\) 1.78722e7 1.05859
\(779\) −1.21517e6 −0.0717453
\(780\) 1.49414e7 0.879333
\(781\) −448076. −0.0262860
\(782\) −3.01297e7 −1.76188
\(783\) −407765. −0.0237687
\(784\) −32464.7 −0.00188634
\(785\) −3.29447e6 −0.190814
\(786\) 2.74815e7 1.58666
\(787\) −1.54112e7 −0.886951 −0.443475 0.896286i \(-0.646255\pi\)
−0.443475 + 0.896286i \(0.646255\pi\)
\(788\) 2.67807e7 1.53641
\(789\) 1.52390e7 0.871493
\(790\) −5.59582e7 −3.19004
\(791\) −5.01906e7 −2.85221
\(792\) 1.77249e6 0.100409
\(793\) 5.69894e6 0.321819
\(794\) 1.56242e7 0.879522
\(795\) 6.52111e6 0.365935
\(796\) −2.75073e7 −1.53874
\(797\) 2.10597e7 1.17437 0.587186 0.809452i \(-0.300236\pi\)
0.587186 + 0.809452i \(0.300236\pi\)
\(798\) 1.13587e7 0.631427
\(799\) −1.65643e7 −0.917925
\(800\) 3.47840e7 1.92156
\(801\) −4.37072e6 −0.240698
\(802\) −4.13334e7 −2.26916
\(803\) 2.39677e6 0.131171
\(804\) 2.61739e6 0.142800
\(805\) 8.24490e7 4.48431
\(806\) −1.85992e7 −1.00845
\(807\) −4.22428e6 −0.228333
\(808\) 1.39642e7 0.752467
\(809\) 1.95659e7 1.05106 0.525532 0.850774i \(-0.323866\pi\)
0.525532 + 0.850774i \(0.323866\pi\)
\(810\) −5.73570e6 −0.307167
\(811\) 1.29945e7 0.693758 0.346879 0.937910i \(-0.387241\pi\)
0.346879 + 0.937910i \(0.387241\pi\)
\(812\) −6.06605e6 −0.322861
\(813\) 1.96507e7 1.04268
\(814\) −1.55626e7 −0.823232
\(815\) −4.38113e6 −0.231042
\(816\) −8618.43 −0.000453109 0
\(817\) −1.06854e6 −0.0560063
\(818\) 6.59876e6 0.344809
\(819\) 5.69856e6 0.296862
\(820\) 9.12830e6 0.474084
\(821\) −3.50872e7 −1.81673 −0.908367 0.418175i \(-0.862670\pi\)
−0.908367 + 0.418175i \(0.862670\pi\)
\(822\) −3.11114e7 −1.60598
\(823\) 1.38494e7 0.712742 0.356371 0.934345i \(-0.384014\pi\)
0.356371 + 0.934345i \(0.384014\pi\)
\(824\) 1.92065e7 0.985440
\(825\) 6.53313e6 0.334184
\(826\) 3.54141e7 1.80604
\(827\) 2.31031e7 1.17464 0.587322 0.809353i \(-0.300182\pi\)
0.587322 + 0.809353i \(0.300182\pi\)
\(828\) −1.72720e7 −0.875522
\(829\) −1.28103e6 −0.0647401 −0.0323701 0.999476i \(-0.510306\pi\)
−0.0323701 + 0.999476i \(0.510306\pi\)
\(830\) 4.05548e7 2.04337
\(831\) 758523. 0.0381036
\(832\) 1.78051e7 0.891735
\(833\) −2.16498e7 −1.08104
\(834\) −3.00899e7 −1.49798
\(835\) 2.77910e7 1.37939
\(836\) 4.12215e6 0.203990
\(837\) 4.41213e6 0.217688
\(838\) −3.56843e7 −1.75536
\(839\) 1.98332e7 0.972721 0.486361 0.873758i \(-0.338324\pi\)
0.486361 + 0.873758i \(0.338324\pi\)
\(840\) −3.25747e7 −1.59288
\(841\) −2.01983e7 −0.984746
\(842\) 3.10863e7 1.51109
\(843\) −5.84564e6 −0.283311
\(844\) 2.12172e7 1.02526
\(845\) −2.46965e7 −1.18985
\(846\) −1.53661e7 −0.738139
\(847\) −3.06759e6 −0.146923
\(848\) 9089.78 0.000434074 0
\(849\) −1.39404e7 −0.663751
\(850\) 4.38760e7 2.08295
\(851\) 5.78948e7 2.74041
\(852\) −1.72507e6 −0.0814154
\(853\) −2.65097e7 −1.24747 −0.623737 0.781634i \(-0.714386\pi\)
−0.623737 + 0.781634i \(0.714386\pi\)
\(854\) −3.25452e7 −1.52701
\(855\) −5.09242e6 −0.238237
\(856\) −6.10376e6 −0.284717
\(857\) −1.96831e7 −0.915465 −0.457733 0.889090i \(-0.651338\pi\)
−0.457733 + 0.889090i \(0.651338\pi\)
\(858\) 3.34657e6 0.155196
\(859\) −1.74273e7 −0.805835 −0.402917 0.915236i \(-0.632004\pi\)
−0.402917 + 0.915236i \(0.632004\pi\)
\(860\) 8.02684e6 0.370083
\(861\) 3.48149e6 0.160051
\(862\) 4.71511e7 2.16134
\(863\) −6.02685e6 −0.275463 −0.137732 0.990470i \(-0.543981\pi\)
−0.137732 + 0.990470i \(0.543981\pi\)
\(864\) −4.22682e6 −0.192632
\(865\) −1.33118e7 −0.604916
\(866\) 2.96024e7 1.34132
\(867\) 7.03132e6 0.317679
\(868\) 6.56365e7 2.95696
\(869\) −7.74520e6 −0.347923
\(870\) 4.40090e6 0.197126
\(871\) 1.88661e6 0.0842630
\(872\) 6.86809e6 0.305875
\(873\) 1.17652e7 0.522472
\(874\) −2.48154e7 −1.09886
\(875\) −5.75229e7 −2.53993
\(876\) 9.22741e6 0.406274
\(877\) −5.71518e6 −0.250917 −0.125459 0.992099i \(-0.540040\pi\)
−0.125459 + 0.992099i \(0.540040\pi\)
\(878\) −5.77551e7 −2.52845
\(879\) 632490. 0.0276109
\(880\) 13850.1 0.000602903 0
\(881\) 1.76619e7 0.766649 0.383324 0.923614i \(-0.374779\pi\)
0.383324 + 0.923614i \(0.374779\pi\)
\(882\) −2.00837e7 −0.869306
\(883\) 2.70117e7 1.16587 0.582936 0.812518i \(-0.301904\pi\)
0.582936 + 0.812518i \(0.301904\pi\)
\(884\) 1.38888e7 0.597769
\(885\) −1.58771e7 −0.681416
\(886\) −2.63133e7 −1.12614
\(887\) 1.30141e7 0.555398 0.277699 0.960668i \(-0.410428\pi\)
0.277699 + 0.960668i \(0.410428\pi\)
\(888\) −2.28736e7 −0.973425
\(889\) −1.86040e7 −0.789500
\(890\) 4.71720e7 1.99623
\(891\) −793881. −0.0335013
\(892\) 1.39700e7 0.587875
\(893\) −1.36427e7 −0.572496
\(894\) 2.45103e7 1.02566
\(895\) −5.72700e7 −2.38984
\(896\) −6.28061e7 −2.61355
\(897\) −1.24496e7 −0.516625
\(898\) 4.15834e7 1.72080
\(899\) −3.38535e6 −0.139703
\(900\) 2.51522e7 1.03507
\(901\) 6.06172e6 0.248762
\(902\) 2.04456e6 0.0836727
\(903\) 3.06140e6 0.124940
\(904\) −4.33220e7 −1.76314
\(905\) 1.24779e7 0.506431
\(906\) 2.62009e7 1.06046
\(907\) 9.84123e6 0.397220 0.198610 0.980079i \(-0.436357\pi\)
0.198610 + 0.980079i \(0.436357\pi\)
\(908\) −5.28713e7 −2.12816
\(909\) −6.25442e6 −0.251060
\(910\) −6.15030e7 −2.46203
\(911\) 7.41735e6 0.296110 0.148055 0.988979i \(-0.452699\pi\)
0.148055 + 0.988979i \(0.452699\pi\)
\(912\) −7098.32 −0.000282598 0
\(913\) 5.61321e6 0.222861
\(914\) −2.54871e7 −1.00915
\(915\) 1.45909e7 0.576140
\(916\) 8.76177e6 0.345027
\(917\) −6.99046e7 −2.74525
\(918\) −5.33164e6 −0.208811
\(919\) 1.15896e7 0.452667 0.226333 0.974050i \(-0.427326\pi\)
0.226333 + 0.974050i \(0.427326\pi\)
\(920\) 7.11659e7 2.77206
\(921\) −1.29009e7 −0.501153
\(922\) 1.83494e7 0.710878
\(923\) −1.24342e6 −0.0480413
\(924\) −1.18101e7 −0.455063
\(925\) −8.43087e7 −3.23980
\(926\) 4.27769e7 1.63939
\(927\) −8.60240e6 −0.328791
\(928\) 3.24316e6 0.123623
\(929\) 1.17151e7 0.445357 0.222679 0.974892i \(-0.428520\pi\)
0.222679 + 0.974892i \(0.428520\pi\)
\(930\) −4.76190e7 −1.80540
\(931\) −1.78312e7 −0.674228
\(932\) −1.05827e6 −0.0399077
\(933\) 2.05091e6 0.0771336
\(934\) 1.29912e7 0.487284
\(935\) 9.23628e6 0.345516
\(936\) 4.91871e6 0.183511
\(937\) 9.18288e6 0.341688 0.170844 0.985298i \(-0.445351\pi\)
0.170844 + 0.985298i \(0.445351\pi\)
\(938\) −1.07740e7 −0.399823
\(939\) 6.51384e6 0.241087
\(940\) 1.02484e8 3.78299
\(941\) 3.24899e6 0.119612 0.0598059 0.998210i \(-0.480952\pi\)
0.0598059 + 0.998210i \(0.480952\pi\)
\(942\) −2.84086e6 −0.104309
\(943\) −7.60601e6 −0.278534
\(944\) −22131.0 −0.000808299 0
\(945\) 1.45899e7 0.531462
\(946\) 1.79786e6 0.0653171
\(947\) 3.50906e7 1.27150 0.635750 0.771895i \(-0.280691\pi\)
0.635750 + 0.771895i \(0.280691\pi\)
\(948\) −2.98186e7 −1.07762
\(949\) 6.65109e6 0.239733
\(950\) 3.61372e7 1.29911
\(951\) −1.13477e7 −0.406872
\(952\) −3.02799e7 −1.08283
\(953\) −3.27415e7 −1.16780 −0.583898 0.811827i \(-0.698473\pi\)
−0.583898 + 0.811827i \(0.698473\pi\)
\(954\) 5.62323e6 0.200039
\(955\) 3.23092e7 1.14635
\(956\) −4.83209e7 −1.70998
\(957\) 609130. 0.0214996
\(958\) 3.29545e7 1.16011
\(959\) 7.91379e7 2.77868
\(960\) 4.55859e7 1.59644
\(961\) 8.00132e6 0.279482
\(962\) −4.31868e7 −1.50457
\(963\) 2.73382e6 0.0949956
\(964\) −6.74227e7 −2.33676
\(965\) −3.14353e7 −1.08667
\(966\) 7.10967e7 2.45136
\(967\) −3.94890e7 −1.35803 −0.679016 0.734124i \(-0.737593\pi\)
−0.679016 + 0.734124i \(0.737593\pi\)
\(968\) −2.64779e6 −0.0908230
\(969\) −4.73367e6 −0.161953
\(970\) −1.26978e8 −4.33312
\(971\) 1.43971e7 0.490033 0.245017 0.969519i \(-0.421207\pi\)
0.245017 + 0.969519i \(0.421207\pi\)
\(972\) −3.05640e6 −0.103763
\(973\) 7.65396e7 2.59182
\(974\) 5.10245e7 1.72338
\(975\) 1.81296e7 0.610769
\(976\) 20338.2 0.000683421 0
\(977\) −4.55108e7 −1.52538 −0.762690 0.646764i \(-0.776122\pi\)
−0.762690 + 0.646764i \(0.776122\pi\)
\(978\) −3.77790e6 −0.126300
\(979\) 6.52910e6 0.217719
\(980\) 1.33947e8 4.45522
\(981\) −3.07615e6 −0.102055
\(982\) 1.65605e7 0.548018
\(983\) −3.34028e7 −1.10255 −0.551276 0.834323i \(-0.685859\pi\)
−0.551276 + 0.834323i \(0.685859\pi\)
\(984\) 3.00505e6 0.0989382
\(985\) 4.94222e7 1.62305
\(986\) 4.09087e6 0.134006
\(987\) 3.90867e7 1.27713
\(988\) 1.14391e7 0.372820
\(989\) −6.68823e6 −0.217431
\(990\) 8.56815e6 0.277843
\(991\) 4.23892e7 1.37111 0.685553 0.728023i \(-0.259560\pi\)
0.685553 + 0.728023i \(0.259560\pi\)
\(992\) −3.50919e7 −1.13221
\(993\) −3.90100e6 −0.125546
\(994\) 7.10088e6 0.227953
\(995\) −5.07632e7 −1.62551
\(996\) 2.16105e7 0.690267
\(997\) 3.39103e7 1.08042 0.540212 0.841529i \(-0.318344\pi\)
0.540212 + 0.841529i \(0.318344\pi\)
\(998\) 5.44094e7 1.72921
\(999\) 1.02449e7 0.324783
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 33.6.a.c.1.1 2
3.2 odd 2 99.6.a.f.1.2 2
4.3 odd 2 528.6.a.s.1.2 2
5.4 even 2 825.6.a.e.1.2 2
11.10 odd 2 363.6.a.j.1.2 2
33.32 even 2 1089.6.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.6.a.c.1.1 2 1.1 even 1 trivial
99.6.a.f.1.2 2 3.2 odd 2
363.6.a.j.1.2 2 11.10 odd 2
528.6.a.s.1.2 2 4.3 odd 2
825.6.a.e.1.2 2 5.4 even 2
1089.6.a.j.1.1 2 33.32 even 2