Properties

Label 165.6.a.a.1.3
Level $165$
Weight $6$
Character 165.1
Self dual yes
Analytic conductor $26.463$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 165.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.4633302691\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.34253.1
Defining polynomial: \( x^{3} - x^{2} - 52x + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-7.17710\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.17710 q^{2} +9.00000 q^{3} -5.19759 q^{4} +25.0000 q^{5} +46.5939 q^{6} -123.437 q^{7} -192.576 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+5.17710 q^{2} +9.00000 q^{3} -5.19759 q^{4} +25.0000 q^{5} +46.5939 q^{6} -123.437 q^{7} -192.576 q^{8} +81.0000 q^{9} +129.428 q^{10} -121.000 q^{11} -46.7783 q^{12} -500.053 q^{13} -639.048 q^{14} +225.000 q^{15} -830.662 q^{16} -422.631 q^{17} +419.345 q^{18} -932.948 q^{19} -129.940 q^{20} -1110.94 q^{21} -626.430 q^{22} -1225.18 q^{23} -1733.18 q^{24} +625.000 q^{25} -2588.83 q^{26} +729.000 q^{27} +641.576 q^{28} -2111.62 q^{29} +1164.85 q^{30} -159.612 q^{31} +1862.00 q^{32} -1089.00 q^{33} -2188.00 q^{34} -3085.93 q^{35} -421.005 q^{36} +5414.46 q^{37} -4829.97 q^{38} -4500.47 q^{39} -4814.40 q^{40} -18066.7 q^{41} -5751.43 q^{42} +6815.47 q^{43} +628.908 q^{44} +2025.00 q^{45} -6342.88 q^{46} -15098.9 q^{47} -7475.96 q^{48} -1570.23 q^{49} +3235.69 q^{50} -3803.67 q^{51} +2599.07 q^{52} +15367.7 q^{53} +3774.11 q^{54} -3025.00 q^{55} +23771.0 q^{56} -8396.53 q^{57} -10932.1 q^{58} +23400.5 q^{59} -1169.46 q^{60} +10768.4 q^{61} -826.328 q^{62} -9998.42 q^{63} +36221.0 q^{64} -12501.3 q^{65} -5637.87 q^{66} +14507.1 q^{67} +2196.66 q^{68} -11026.6 q^{69} -15976.2 q^{70} -28114.0 q^{71} -15598.6 q^{72} -28836.7 q^{73} +28031.2 q^{74} +5625.00 q^{75} +4849.08 q^{76} +14935.9 q^{77} -23299.4 q^{78} -8150.52 q^{79} -20766.6 q^{80} +6561.00 q^{81} -93533.0 q^{82} -109864. q^{83} +5774.19 q^{84} -10565.8 q^{85} +35284.4 q^{86} -19004.5 q^{87} +23301.7 q^{88} +69673.6 q^{89} +10483.6 q^{90} +61725.2 q^{91} +6367.98 q^{92} -1436.51 q^{93} -78168.5 q^{94} -23323.7 q^{95} +16758.0 q^{96} +91551.4 q^{97} -8129.23 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 7 q^{2} + 27 q^{3} + 25 q^{4} + 75 q^{5} - 63 q^{6} - 172 q^{7} - 231 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 7 q^{2} + 27 q^{3} + 25 q^{4} + 75 q^{5} - 63 q^{6} - 172 q^{7} - 231 q^{8} + 243 q^{9} - 175 q^{10} - 363 q^{11} + 225 q^{12} - 654 q^{13} - 728 q^{14} + 675 q^{15} - 415 q^{16} - 2366 q^{17} - 567 q^{18} - 2872 q^{19} + 625 q^{20} - 1548 q^{21} + 847 q^{22} + 2272 q^{23} - 2079 q^{24} + 1875 q^{25} + 3422 q^{26} + 2187 q^{27} + 4592 q^{28} - 7738 q^{29} - 1575 q^{30} + 568 q^{31} + 1001 q^{32} - 3267 q^{33} + 2506 q^{34} - 4300 q^{35} + 2025 q^{36} - 9126 q^{37} + 13076 q^{38} - 5886 q^{39} - 5775 q^{40} - 8758 q^{41} - 6552 q^{42} - 14672 q^{43} - 3025 q^{44} + 6075 q^{45} - 28768 q^{46} - 19392 q^{47} - 3735 q^{48} - 26629 q^{49} - 4375 q^{50} - 21294 q^{51} - 61506 q^{52} - 4598 q^{53} - 5103 q^{54} - 9075 q^{55} + 2688 q^{56} - 25848 q^{57} + 8550 q^{58} - 9348 q^{59} + 5625 q^{60} - 60078 q^{61} - 14096 q^{62} - 13932 q^{63} - 7087 q^{64} - 16350 q^{65} + 7623 q^{66} - 38468 q^{67} + 59778 q^{68} + 20448 q^{69} - 18200 q^{70} - 74032 q^{71} - 18711 q^{72} - 44442 q^{73} + 82542 q^{74} + 16875 q^{75} - 98708 q^{76} + 20812 q^{77} + 30798 q^{78} - 108116 q^{79} - 10375 q^{80} + 19683 q^{81} - 92230 q^{82} - 81892 q^{83} + 41328 q^{84} - 59150 q^{85} + 126412 q^{86} - 69642 q^{87} + 27951 q^{88} + 167342 q^{89} - 14175 q^{90} - 31832 q^{91} + 72960 q^{92} + 5112 q^{93} + 12728 q^{94} - 71800 q^{95} + 9009 q^{96} + 159702 q^{97} + 163121 q^{98} - 29403 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.17710 0.915191 0.457596 0.889160i \(-0.348711\pi\)
0.457596 + 0.889160i \(0.348711\pi\)
\(3\) 9.00000 0.577350
\(4\) −5.19759 −0.162425
\(5\) 25.0000 0.447214
\(6\) 46.5939 0.528386
\(7\) −123.437 −0.952141 −0.476071 0.879407i \(-0.657939\pi\)
−0.476071 + 0.879407i \(0.657939\pi\)
\(8\) −192.576 −1.06384
\(9\) 81.0000 0.333333
\(10\) 129.428 0.409286
\(11\) −121.000 −0.301511
\(12\) −46.7783 −0.0937759
\(13\) −500.053 −0.820649 −0.410324 0.911940i \(-0.634585\pi\)
−0.410324 + 0.911940i \(0.634585\pi\)
\(14\) −639.048 −0.871392
\(15\) 225.000 0.258199
\(16\) −830.662 −0.811194
\(17\) −422.631 −0.354682 −0.177341 0.984150i \(-0.556749\pi\)
−0.177341 + 0.984150i \(0.556749\pi\)
\(18\) 419.345 0.305064
\(19\) −932.948 −0.592889 −0.296445 0.955050i \(-0.595801\pi\)
−0.296445 + 0.955050i \(0.595801\pi\)
\(20\) −129.940 −0.0726385
\(21\) −1110.94 −0.549719
\(22\) −626.430 −0.275941
\(23\) −1225.18 −0.482926 −0.241463 0.970410i \(-0.577627\pi\)
−0.241463 + 0.970410i \(0.577627\pi\)
\(24\) −1733.18 −0.614209
\(25\) 625.000 0.200000
\(26\) −2588.83 −0.751051
\(27\) 729.000 0.192450
\(28\) 641.576 0.154651
\(29\) −2111.62 −0.466251 −0.233126 0.972447i \(-0.574895\pi\)
−0.233126 + 0.972447i \(0.574895\pi\)
\(30\) 1164.85 0.236301
\(31\) −159.612 −0.0298306 −0.0149153 0.999889i \(-0.504748\pi\)
−0.0149153 + 0.999889i \(0.504748\pi\)
\(32\) 1862.00 0.321444
\(33\) −1089.00 −0.174078
\(34\) −2188.00 −0.324601
\(35\) −3085.93 −0.425811
\(36\) −421.005 −0.0541415
\(37\) 5414.46 0.650205 0.325103 0.945679i \(-0.394601\pi\)
0.325103 + 0.945679i \(0.394601\pi\)
\(38\) −4829.97 −0.542607
\(39\) −4500.47 −0.473802
\(40\) −4814.40 −0.475764
\(41\) −18066.7 −1.67849 −0.839244 0.543756i \(-0.817002\pi\)
−0.839244 + 0.543756i \(0.817002\pi\)
\(42\) −5751.43 −0.503098
\(43\) 6815.47 0.562114 0.281057 0.959691i \(-0.409315\pi\)
0.281057 + 0.959691i \(0.409315\pi\)
\(44\) 628.908 0.0489729
\(45\) 2025.00 0.149071
\(46\) −6342.88 −0.441969
\(47\) −15098.9 −0.997012 −0.498506 0.866886i \(-0.666118\pi\)
−0.498506 + 0.866886i \(0.666118\pi\)
\(48\) −7475.96 −0.468343
\(49\) −1570.23 −0.0934270
\(50\) 3235.69 0.183038
\(51\) −3803.67 −0.204775
\(52\) 2599.07 0.133294
\(53\) 15367.7 0.751484 0.375742 0.926724i \(-0.377388\pi\)
0.375742 + 0.926724i \(0.377388\pi\)
\(54\) 3774.11 0.176129
\(55\) −3025.00 −0.134840
\(56\) 23771.0 1.01293
\(57\) −8396.53 −0.342305
\(58\) −10932.1 −0.426709
\(59\) 23400.5 0.875177 0.437588 0.899175i \(-0.355833\pi\)
0.437588 + 0.899175i \(0.355833\pi\)
\(60\) −1169.46 −0.0419379
\(61\) 10768.4 0.370532 0.185266 0.982688i \(-0.440685\pi\)
0.185266 + 0.982688i \(0.440685\pi\)
\(62\) −826.328 −0.0273007
\(63\) −9998.42 −0.317380
\(64\) 36221.0 1.10538
\(65\) −12501.3 −0.367005
\(66\) −5637.87 −0.159314
\(67\) 14507.1 0.394814 0.197407 0.980322i \(-0.436748\pi\)
0.197407 + 0.980322i \(0.436748\pi\)
\(68\) 2196.66 0.0576090
\(69\) −11026.6 −0.278817
\(70\) −15976.2 −0.389698
\(71\) −28114.0 −0.661876 −0.330938 0.943653i \(-0.607365\pi\)
−0.330938 + 0.943653i \(0.607365\pi\)
\(72\) −15598.6 −0.354614
\(73\) −28836.7 −0.633342 −0.316671 0.948536i \(-0.602565\pi\)
−0.316671 + 0.948536i \(0.602565\pi\)
\(74\) 28031.2 0.595062
\(75\) 5625.00 0.115470
\(76\) 4849.08 0.0962998
\(77\) 14935.9 0.287081
\(78\) −23299.4 −0.433619
\(79\) −8150.52 −0.146932 −0.0734662 0.997298i \(-0.523406\pi\)
−0.0734662 + 0.997298i \(0.523406\pi\)
\(80\) −20766.6 −0.362777
\(81\) 6561.00 0.111111
\(82\) −93533.0 −1.53614
\(83\) −109864. −1.75049 −0.875246 0.483679i \(-0.839300\pi\)
−0.875246 + 0.483679i \(0.839300\pi\)
\(84\) 5774.19 0.0892879
\(85\) −10565.8 −0.158618
\(86\) 35284.4 0.514442
\(87\) −19004.5 −0.269190
\(88\) 23301.7 0.320760
\(89\) 69673.6 0.932380 0.466190 0.884685i \(-0.345626\pi\)
0.466190 + 0.884685i \(0.345626\pi\)
\(90\) 10483.6 0.136429
\(91\) 61725.2 0.781374
\(92\) 6367.98 0.0784390
\(93\) −1436.51 −0.0172227
\(94\) −78168.5 −0.912457
\(95\) −23323.7 −0.265148
\(96\) 16758.0 0.185586
\(97\) 91551.4 0.987952 0.493976 0.869476i \(-0.335543\pi\)
0.493976 + 0.869476i \(0.335543\pi\)
\(98\) −8129.23 −0.0855036
\(99\) −9801.00 −0.100504
\(100\) −3248.49 −0.0324849
\(101\) 21299.3 0.207760 0.103880 0.994590i \(-0.466874\pi\)
0.103880 + 0.994590i \(0.466874\pi\)
\(102\) −19692.0 −0.187409
\(103\) −6548.06 −0.0608163 −0.0304081 0.999538i \(-0.509681\pi\)
−0.0304081 + 0.999538i \(0.509681\pi\)
\(104\) 96298.1 0.873040
\(105\) −27773.4 −0.245842
\(106\) 79560.3 0.687752
\(107\) 127171. 1.07381 0.536905 0.843643i \(-0.319593\pi\)
0.536905 + 0.843643i \(0.319593\pi\)
\(108\) −3789.04 −0.0312586
\(109\) 57285.2 0.461823 0.230912 0.972975i \(-0.425829\pi\)
0.230912 + 0.972975i \(0.425829\pi\)
\(110\) −15660.7 −0.123404
\(111\) 48730.1 0.375396
\(112\) 102535. 0.772371
\(113\) −67774.2 −0.499308 −0.249654 0.968335i \(-0.580317\pi\)
−0.249654 + 0.968335i \(0.580317\pi\)
\(114\) −43469.7 −0.313274
\(115\) −30629.5 −0.215971
\(116\) 10975.3 0.0757307
\(117\) −40504.3 −0.273550
\(118\) 121147. 0.800954
\(119\) 52168.4 0.337707
\(120\) −43329.6 −0.274683
\(121\) 14641.0 0.0909091
\(122\) 55749.1 0.339108
\(123\) −162600. −0.969075
\(124\) 829.598 0.00484522
\(125\) 15625.0 0.0894427
\(126\) −51762.9 −0.290464
\(127\) −115352. −0.634622 −0.317311 0.948322i \(-0.602780\pi\)
−0.317311 + 0.948322i \(0.602780\pi\)
\(128\) 127936. 0.690187
\(129\) 61339.2 0.324537
\(130\) −64720.6 −0.335880
\(131\) 43276.4 0.220329 0.110165 0.993913i \(-0.464862\pi\)
0.110165 + 0.993913i \(0.464862\pi\)
\(132\) 5660.17 0.0282745
\(133\) 115161. 0.564514
\(134\) 75104.6 0.361331
\(135\) 18225.0 0.0860663
\(136\) 81388.4 0.377325
\(137\) 401439. 1.82734 0.913668 0.406461i \(-0.133237\pi\)
0.913668 + 0.406461i \(0.133237\pi\)
\(138\) −57085.9 −0.255171
\(139\) −302018. −1.32586 −0.662928 0.748683i \(-0.730686\pi\)
−0.662928 + 0.748683i \(0.730686\pi\)
\(140\) 16039.4 0.0691621
\(141\) −135890. −0.575625
\(142\) −145549. −0.605743
\(143\) 60506.4 0.247435
\(144\) −67283.6 −0.270398
\(145\) −52790.4 −0.208514
\(146\) −149290. −0.579629
\(147\) −14132.1 −0.0539401
\(148\) −28142.1 −0.105609
\(149\) 326942. 1.20644 0.603218 0.797576i \(-0.293885\pi\)
0.603218 + 0.797576i \(0.293885\pi\)
\(150\) 29121.2 0.105677
\(151\) 164681. 0.587761 0.293881 0.955842i \(-0.405053\pi\)
0.293881 + 0.955842i \(0.405053\pi\)
\(152\) 179663. 0.630740
\(153\) −34233.1 −0.118227
\(154\) 77324.8 0.262734
\(155\) −3990.30 −0.0133406
\(156\) 23391.6 0.0769571
\(157\) 248620. 0.804982 0.402491 0.915424i \(-0.368144\pi\)
0.402491 + 0.915424i \(0.368144\pi\)
\(158\) −42196.1 −0.134471
\(159\) 138309. 0.433870
\(160\) 46550.0 0.143754
\(161\) 151233. 0.459813
\(162\) 33967.0 0.101688
\(163\) −417656. −1.23126 −0.615629 0.788036i \(-0.711098\pi\)
−0.615629 + 0.788036i \(0.711098\pi\)
\(164\) 93903.0 0.272628
\(165\) −27225.0 −0.0778499
\(166\) −568777. −1.60203
\(167\) 704955. 1.95601 0.978004 0.208588i \(-0.0668869\pi\)
0.978004 + 0.208588i \(0.0668869\pi\)
\(168\) 213939. 0.584814
\(169\) −121240. −0.326535
\(170\) −54700.1 −0.145166
\(171\) −75568.8 −0.197630
\(172\) −35424.0 −0.0913012
\(173\) −171060. −0.434545 −0.217272 0.976111i \(-0.569716\pi\)
−0.217272 + 0.976111i \(0.569716\pi\)
\(174\) −98388.5 −0.246361
\(175\) −77148.3 −0.190428
\(176\) 100510. 0.244584
\(177\) 210605. 0.505284
\(178\) 360707. 0.853306
\(179\) −166327. −0.387999 −0.193999 0.981002i \(-0.562146\pi\)
−0.193999 + 0.981002i \(0.562146\pi\)
\(180\) −10525.1 −0.0242128
\(181\) −584391. −1.32589 −0.662945 0.748668i \(-0.730694\pi\)
−0.662945 + 0.748668i \(0.730694\pi\)
\(182\) 319558. 0.715107
\(183\) 96915.5 0.213927
\(184\) 235940. 0.513756
\(185\) 135361. 0.290781
\(186\) −7436.96 −0.0157621
\(187\) 51138.3 0.106941
\(188\) 78477.8 0.161939
\(189\) −89985.8 −0.183240
\(190\) −120749. −0.242661
\(191\) −715510. −1.41916 −0.709581 0.704624i \(-0.751116\pi\)
−0.709581 + 0.704624i \(0.751116\pi\)
\(192\) 325989. 0.638189
\(193\) −922088. −1.78188 −0.890941 0.454118i \(-0.849954\pi\)
−0.890941 + 0.454118i \(0.849954\pi\)
\(194\) 473971. 0.904165
\(195\) −112512. −0.211891
\(196\) 8161.40 0.0151749
\(197\) −613632. −1.12653 −0.563265 0.826276i \(-0.690455\pi\)
−0.563265 + 0.826276i \(0.690455\pi\)
\(198\) −50740.8 −0.0919802
\(199\) −378985. −0.678406 −0.339203 0.940713i \(-0.610157\pi\)
−0.339203 + 0.940713i \(0.610157\pi\)
\(200\) −120360. −0.212768
\(201\) 130564. 0.227946
\(202\) 110269. 0.190140
\(203\) 260652. 0.443937
\(204\) 19769.9 0.0332606
\(205\) −451666. −0.750642
\(206\) −33900.0 −0.0556585
\(207\) −99239.5 −0.160975
\(208\) 415375. 0.665705
\(209\) 112887. 0.178763
\(210\) −143786. −0.224992
\(211\) −473721. −0.732515 −0.366257 0.930514i \(-0.619361\pi\)
−0.366257 + 0.930514i \(0.619361\pi\)
\(212\) −79875.1 −0.122060
\(213\) −253026. −0.382134
\(214\) 658376. 0.982741
\(215\) 170387. 0.251385
\(216\) −140388. −0.204736
\(217\) 19702.1 0.0284029
\(218\) 296571. 0.422657
\(219\) −259530. −0.365660
\(220\) 15722.7 0.0219013
\(221\) 211338. 0.291069
\(222\) 252281. 0.343559
\(223\) −822747. −1.10791 −0.553954 0.832547i \(-0.686882\pi\)
−0.553954 + 0.832547i \(0.686882\pi\)
\(224\) −229840. −0.306060
\(225\) 50625.0 0.0666667
\(226\) −350874. −0.456962
\(227\) −556097. −0.716285 −0.358143 0.933667i \(-0.616590\pi\)
−0.358143 + 0.933667i \(0.616590\pi\)
\(228\) 43641.7 0.0555987
\(229\) −634919. −0.800073 −0.400036 0.916499i \(-0.631003\pi\)
−0.400036 + 0.916499i \(0.631003\pi\)
\(230\) −158572. −0.197655
\(231\) 134423. 0.165747
\(232\) 406646. 0.496017
\(233\) −906561. −1.09397 −0.546987 0.837141i \(-0.684225\pi\)
−0.546987 + 0.837141i \(0.684225\pi\)
\(234\) −209695. −0.250350
\(235\) −377472. −0.445877
\(236\) −121626. −0.142150
\(237\) −73354.7 −0.0848314
\(238\) 270081. 0.309066
\(239\) 662586. 0.750322 0.375161 0.926960i \(-0.377587\pi\)
0.375161 + 0.926960i \(0.377587\pi\)
\(240\) −186899. −0.209449
\(241\) −1.31823e6 −1.46200 −0.731001 0.682376i \(-0.760947\pi\)
−0.731001 + 0.682376i \(0.760947\pi\)
\(242\) 75798.0 0.0831992
\(243\) 59049.0 0.0641500
\(244\) −55969.6 −0.0601836
\(245\) −39255.7 −0.0417818
\(246\) −841797. −0.886889
\(247\) 466523. 0.486554
\(248\) 30737.4 0.0317350
\(249\) −988775. −1.01065
\(250\) 80892.3 0.0818572
\(251\) −11073.9 −0.0110947 −0.00554737 0.999985i \(-0.501766\pi\)
−0.00554737 + 0.999985i \(0.501766\pi\)
\(252\) 51967.7 0.0515504
\(253\) 148247. 0.145608
\(254\) −597188. −0.580800
\(255\) −95091.9 −0.0915784
\(256\) −496734. −0.473723
\(257\) −788596. −0.744769 −0.372384 0.928079i \(-0.621460\pi\)
−0.372384 + 0.928079i \(0.621460\pi\)
\(258\) 317560. 0.297013
\(259\) −668346. −0.619087
\(260\) 64976.7 0.0596107
\(261\) −171041. −0.155417
\(262\) 224046. 0.201644
\(263\) 211325. 0.188391 0.0941956 0.995554i \(-0.469972\pi\)
0.0941956 + 0.995554i \(0.469972\pi\)
\(264\) 209715. 0.185191
\(265\) 384193. 0.336074
\(266\) 596198. 0.516639
\(267\) 627062. 0.538310
\(268\) −75401.8 −0.0641276
\(269\) 552342. 0.465401 0.232701 0.972548i \(-0.425244\pi\)
0.232701 + 0.972548i \(0.425244\pi\)
\(270\) 94352.7 0.0787671
\(271\) −1.49255e6 −1.23454 −0.617271 0.786751i \(-0.711762\pi\)
−0.617271 + 0.786751i \(0.711762\pi\)
\(272\) 351063. 0.287715
\(273\) 555527. 0.451126
\(274\) 2.07829e6 1.67236
\(275\) −75625.0 −0.0603023
\(276\) 57311.8 0.0452868
\(277\) 1.01431e6 0.794275 0.397138 0.917759i \(-0.370004\pi\)
0.397138 + 0.917759i \(0.370004\pi\)
\(278\) −1.56358e6 −1.21341
\(279\) −12928.6 −0.00994352
\(280\) 594276. 0.452995
\(281\) −658216. −0.497282 −0.248641 0.968596i \(-0.579984\pi\)
−0.248641 + 0.968596i \(0.579984\pi\)
\(282\) −703517. −0.526807
\(283\) 1.65981e6 1.23195 0.615974 0.787767i \(-0.288763\pi\)
0.615974 + 0.787767i \(0.288763\pi\)
\(284\) 146125. 0.107505
\(285\) −209913. −0.153083
\(286\) 313248. 0.226450
\(287\) 2.23010e6 1.59816
\(288\) 150822. 0.107148
\(289\) −1.24124e6 −0.874201
\(290\) −273301. −0.190830
\(291\) 823963. 0.570394
\(292\) 149881. 0.102870
\(293\) −410864. −0.279594 −0.139797 0.990180i \(-0.544645\pi\)
−0.139797 + 0.990180i \(0.544645\pi\)
\(294\) −73163.1 −0.0493655
\(295\) 585013. 0.391391
\(296\) −1.04269e6 −0.691715
\(297\) −88209.0 −0.0580259
\(298\) 1.69261e6 1.10412
\(299\) 612654. 0.396312
\(300\) −29236.4 −0.0187552
\(301\) −841283. −0.535212
\(302\) 852570. 0.537914
\(303\) 191694. 0.119950
\(304\) 774965. 0.480948
\(305\) 269210. 0.165707
\(306\) −177228. −0.108200
\(307\) 1.34831e6 0.816477 0.408238 0.912875i \(-0.366143\pi\)
0.408238 + 0.912875i \(0.366143\pi\)
\(308\) −77630.7 −0.0466291
\(309\) −58932.6 −0.0351123
\(310\) −20658.2 −0.0122092
\(311\) −2.52585e6 −1.48083 −0.740417 0.672148i \(-0.765372\pi\)
−0.740417 + 0.672148i \(0.765372\pi\)
\(312\) 866682. 0.504050
\(313\) 2.23061e6 1.28695 0.643476 0.765466i \(-0.277491\pi\)
0.643476 + 0.765466i \(0.277491\pi\)
\(314\) 1.28713e6 0.736713
\(315\) −249961. −0.141937
\(316\) 42363.0 0.0238654
\(317\) −120908. −0.0675780 −0.0337890 0.999429i \(-0.510757\pi\)
−0.0337890 + 0.999429i \(0.510757\pi\)
\(318\) 716043. 0.397074
\(319\) 255506. 0.140580
\(320\) 905524. 0.494339
\(321\) 1.14454e6 0.619964
\(322\) 782948. 0.420817
\(323\) 394292. 0.210287
\(324\) −34101.4 −0.0180472
\(325\) −312533. −0.164130
\(326\) −2.16225e6 −1.12684
\(327\) 515566. 0.266634
\(328\) 3.47920e6 1.78564
\(329\) 1.86377e6 0.949296
\(330\) −140947. −0.0712476
\(331\) 578480. 0.290214 0.145107 0.989416i \(-0.453647\pi\)
0.145107 + 0.989416i \(0.453647\pi\)
\(332\) 571028. 0.284323
\(333\) 438571. 0.216735
\(334\) 3.64963e6 1.79012
\(335\) 362677. 0.176566
\(336\) 922812. 0.445929
\(337\) −1.78710e6 −0.857186 −0.428593 0.903498i \(-0.640991\pi\)
−0.428593 + 0.903498i \(0.640991\pi\)
\(338\) −627674. −0.298842
\(339\) −609968. −0.288276
\(340\) 54916.5 0.0257635
\(341\) 19313.1 0.00899425
\(342\) −391228. −0.180869
\(343\) 2.26844e6 1.04110
\(344\) −1.31249e6 −0.598000
\(345\) −275665. −0.124691
\(346\) −885598. −0.397692
\(347\) −1.46282e6 −0.652179 −0.326089 0.945339i \(-0.605731\pi\)
−0.326089 + 0.945339i \(0.605731\pi\)
\(348\) 98777.8 0.0437231
\(349\) 1.48501e6 0.652627 0.326314 0.945262i \(-0.394193\pi\)
0.326314 + 0.945262i \(0.394193\pi\)
\(350\) −399405. −0.174278
\(351\) −364538. −0.157934
\(352\) −225302. −0.0969189
\(353\) 1.34528e6 0.574616 0.287308 0.957838i \(-0.407240\pi\)
0.287308 + 0.957838i \(0.407240\pi\)
\(354\) 1.09032e6 0.462431
\(355\) −702850. −0.296000
\(356\) −362135. −0.151442
\(357\) 469515. 0.194975
\(358\) −861092. −0.355093
\(359\) −1.83956e6 −0.753316 −0.376658 0.926352i \(-0.622927\pi\)
−0.376658 + 0.926352i \(0.622927\pi\)
\(360\) −389966. −0.158588
\(361\) −1.60571e6 −0.648483
\(362\) −3.02546e6 −1.21344
\(363\) 131769. 0.0524864
\(364\) −320822. −0.126914
\(365\) −720917. −0.283239
\(366\) 501742. 0.195784
\(367\) −312079. −0.120948 −0.0604741 0.998170i \(-0.519261\pi\)
−0.0604741 + 0.998170i \(0.519261\pi\)
\(368\) 1.01771e6 0.391746
\(369\) −1.46340e6 −0.559496
\(370\) 700780. 0.266120
\(371\) −1.89695e6 −0.715519
\(372\) 7466.38 0.00279739
\(373\) 3.38658e6 1.26034 0.630172 0.776455i \(-0.282984\pi\)
0.630172 + 0.776455i \(0.282984\pi\)
\(374\) 264748. 0.0978710
\(375\) 140625. 0.0516398
\(376\) 2.90768e6 1.06066
\(377\) 1.05592e6 0.382629
\(378\) −465866. −0.167699
\(379\) −1.62370e6 −0.580641 −0.290321 0.956929i \(-0.593762\pi\)
−0.290321 + 0.956929i \(0.593762\pi\)
\(380\) 121227. 0.0430666
\(381\) −1.03817e6 −0.366399
\(382\) −3.70427e6 −1.29881
\(383\) −3.06187e6 −1.06657 −0.533285 0.845935i \(-0.679043\pi\)
−0.533285 + 0.845935i \(0.679043\pi\)
\(384\) 1.15142e6 0.398480
\(385\) 373398. 0.128387
\(386\) −4.77375e6 −1.63076
\(387\) 552053. 0.187371
\(388\) −475847. −0.160468
\(389\) 4.21840e6 1.41343 0.706715 0.707499i \(-0.250176\pi\)
0.706715 + 0.707499i \(0.250176\pi\)
\(390\) −582486. −0.193920
\(391\) 517798. 0.171285
\(392\) 302388. 0.0993915
\(393\) 389487. 0.127207
\(394\) −3.17684e6 −1.03099
\(395\) −203763. −0.0657101
\(396\) 50941.6 0.0163243
\(397\) 513185. 0.163417 0.0817085 0.996656i \(-0.473962\pi\)
0.0817085 + 0.996656i \(0.473962\pi\)
\(398\) −1.96205e6 −0.620871
\(399\) 1.03645e6 0.325922
\(400\) −519164. −0.162239
\(401\) −2.18458e6 −0.678432 −0.339216 0.940709i \(-0.610162\pi\)
−0.339216 + 0.940709i \(0.610162\pi\)
\(402\) 675942. 0.208614
\(403\) 79814.5 0.0244804
\(404\) −110705. −0.0337453
\(405\) 164025. 0.0496904
\(406\) 1.34942e6 0.406287
\(407\) −655149. −0.196044
\(408\) 732496. 0.217849
\(409\) 1.22307e6 0.361529 0.180764 0.983526i \(-0.442143\pi\)
0.180764 + 0.983526i \(0.442143\pi\)
\(410\) −2.33832e6 −0.686981
\(411\) 3.61295e6 1.05501
\(412\) 34034.1 0.00987806
\(413\) −2.88850e6 −0.833292
\(414\) −513774. −0.147323
\(415\) −2.74660e6 −0.782844
\(416\) −931098. −0.263792
\(417\) −2.71816e6 −0.765483
\(418\) 584426. 0.163602
\(419\) 2.42879e6 0.675858 0.337929 0.941172i \(-0.390274\pi\)
0.337929 + 0.941172i \(0.390274\pi\)
\(420\) 144355. 0.0399308
\(421\) −727467. −0.200036 −0.100018 0.994986i \(-0.531890\pi\)
−0.100018 + 0.994986i \(0.531890\pi\)
\(422\) −2.45250e6 −0.670391
\(423\) −1.22301e6 −0.332337
\(424\) −2.95945e6 −0.799460
\(425\) −264144. −0.0709363
\(426\) −1.30994e6 −0.349726
\(427\) −1.32922e6 −0.352799
\(428\) −660981. −0.174413
\(429\) 544557. 0.142857
\(430\) 882110. 0.230066
\(431\) 6.18223e6 1.60307 0.801534 0.597949i \(-0.204018\pi\)
0.801534 + 0.597949i \(0.204018\pi\)
\(432\) −605553. −0.156114
\(433\) 2.42337e6 0.621154 0.310577 0.950548i \(-0.399478\pi\)
0.310577 + 0.950548i \(0.399478\pi\)
\(434\) 102000. 0.0259941
\(435\) −475114. −0.120386
\(436\) −297745. −0.0750115
\(437\) 1.14303e6 0.286321
\(438\) −1.34361e6 −0.334649
\(439\) −3.79993e6 −0.941054 −0.470527 0.882385i \(-0.655936\pi\)
−0.470527 + 0.882385i \(0.655936\pi\)
\(440\) 582542. 0.143448
\(441\) −127188. −0.0311423
\(442\) 1.09412e6 0.266384
\(443\) −5.58044e6 −1.35101 −0.675506 0.737354i \(-0.736075\pi\)
−0.675506 + 0.737354i \(0.736075\pi\)
\(444\) −253279. −0.0609736
\(445\) 1.74184e6 0.416973
\(446\) −4.25945e6 −1.01395
\(447\) 2.94247e6 0.696537
\(448\) −4.47102e6 −1.05247
\(449\) 2.69609e6 0.631130 0.315565 0.948904i \(-0.397806\pi\)
0.315565 + 0.948904i \(0.397806\pi\)
\(450\) 262091. 0.0610128
\(451\) 2.18607e6 0.506083
\(452\) 352263. 0.0810999
\(453\) 1.48213e6 0.339344
\(454\) −2.87897e6 −0.655538
\(455\) 1.54313e6 0.349441
\(456\) 1.61697e6 0.364158
\(457\) 1.14257e6 0.255912 0.127956 0.991780i \(-0.459158\pi\)
0.127956 + 0.991780i \(0.459158\pi\)
\(458\) −3.28704e6 −0.732220
\(459\) −308098. −0.0682585
\(460\) 159199. 0.0350790
\(461\) 5.37091e6 1.17705 0.588526 0.808478i \(-0.299708\pi\)
0.588526 + 0.808478i \(0.299708\pi\)
\(462\) 695923. 0.151690
\(463\) −3.80227e6 −0.824311 −0.412155 0.911114i \(-0.635224\pi\)
−0.412155 + 0.911114i \(0.635224\pi\)
\(464\) 1.75404e6 0.378220
\(465\) −35912.7 −0.00770222
\(466\) −4.69336e6 −1.00120
\(467\) 9.41694e6 1.99810 0.999051 0.0435609i \(-0.0138703\pi\)
0.999051 + 0.0435609i \(0.0138703\pi\)
\(468\) 210525. 0.0444312
\(469\) −1.79071e6 −0.375919
\(470\) −1.95421e6 −0.408063
\(471\) 2.23758e6 0.464757
\(472\) −4.50638e6 −0.931049
\(473\) −824672. −0.169484
\(474\) −379765. −0.0776370
\(475\) −583093. −0.118578
\(476\) −271150. −0.0548519
\(477\) 1.24479e6 0.250495
\(478\) 3.43028e6 0.686688
\(479\) −4.69047e6 −0.934065 −0.467033 0.884240i \(-0.654677\pi\)
−0.467033 + 0.884240i \(0.654677\pi\)
\(480\) 418950. 0.0829964
\(481\) −2.70751e6 −0.533590
\(482\) −6.82461e6 −1.33801
\(483\) 1.36110e6 0.265473
\(484\) −76097.9 −0.0147659
\(485\) 2.28879e6 0.441825
\(486\) 305703. 0.0587096
\(487\) 7.07177e6 1.35116 0.675579 0.737288i \(-0.263894\pi\)
0.675579 + 0.737288i \(0.263894\pi\)
\(488\) −2.07373e6 −0.394187
\(489\) −3.75890e6 −0.710867
\(490\) −203231. −0.0382384
\(491\) 906051. 0.169609 0.0848045 0.996398i \(-0.472973\pi\)
0.0848045 + 0.996398i \(0.472973\pi\)
\(492\) 845127. 0.157402
\(493\) 892434. 0.165371
\(494\) 2.41524e6 0.445290
\(495\) −245025. −0.0449467
\(496\) 132584. 0.0241984
\(497\) 3.47031e6 0.630199
\(498\) −5.11899e6 −0.924935
\(499\) −7.13323e6 −1.28243 −0.641217 0.767360i \(-0.721570\pi\)
−0.641217 + 0.767360i \(0.721570\pi\)
\(500\) −81212.3 −0.0145277
\(501\) 6.34460e6 1.12930
\(502\) −57330.9 −0.0101538
\(503\) −2.34927e6 −0.414013 −0.207006 0.978340i \(-0.566372\pi\)
−0.207006 + 0.978340i \(0.566372\pi\)
\(504\) 1.92545e6 0.337642
\(505\) 532482. 0.0929131
\(506\) 767489. 0.133259
\(507\) −1.09116e6 −0.188525
\(508\) 599551. 0.103078
\(509\) 1.97148e6 0.337286 0.168643 0.985677i \(-0.446062\pi\)
0.168643 + 0.985677i \(0.446062\pi\)
\(510\) −492301. −0.0838117
\(511\) 3.55952e6 0.603031
\(512\) −6.66559e6 −1.12373
\(513\) −680119. −0.114102
\(514\) −4.08264e6 −0.681606
\(515\) −163702. −0.0271979
\(516\) −318816. −0.0527128
\(517\) 1.82697e6 0.300610
\(518\) −3.46010e6 −0.566583
\(519\) −1.53954e6 −0.250884
\(520\) 2.40745e6 0.390435
\(521\) 1.20074e7 1.93801 0.969004 0.247046i \(-0.0794599\pi\)
0.969004 + 0.247046i \(0.0794599\pi\)
\(522\) −885497. −0.142236
\(523\) 4.16772e6 0.666260 0.333130 0.942881i \(-0.391895\pi\)
0.333130 + 0.942881i \(0.391895\pi\)
\(524\) −224933. −0.0357869
\(525\) −694335. −0.109944
\(526\) 1.09405e6 0.172414
\(527\) 67456.9 0.0105804
\(528\) 904591. 0.141211
\(529\) −4.93528e6 −0.766783
\(530\) 1.98901e6 0.307572
\(531\) 1.89544e6 0.291726
\(532\) −598557. −0.0916910
\(533\) 9.03428e6 1.37745
\(534\) 3.24637e6 0.492657
\(535\) 3.17927e6 0.480222
\(536\) −2.79371e6 −0.420020
\(537\) −1.49694e6 −0.224011
\(538\) 2.85953e6 0.425931
\(539\) 189998. 0.0281693
\(540\) −94726.1 −0.0139793
\(541\) 1.01106e7 1.48519 0.742594 0.669741i \(-0.233595\pi\)
0.742594 + 0.669741i \(0.233595\pi\)
\(542\) −7.72709e6 −1.12984
\(543\) −5.25952e6 −0.765503
\(544\) −786938. −0.114010
\(545\) 1.43213e6 0.206534
\(546\) 2.87602e6 0.412867
\(547\) −7.39087e6 −1.05615 −0.528077 0.849196i \(-0.677087\pi\)
−0.528077 + 0.849196i \(0.677087\pi\)
\(548\) −2.08652e6 −0.296805
\(549\) 872239. 0.123511
\(550\) −391519. −0.0551881
\(551\) 1.97003e6 0.276435
\(552\) 2.12346e6 0.296617
\(553\) 1.00608e6 0.139900
\(554\) 5.25119e6 0.726914
\(555\) 1.21825e6 0.167882
\(556\) 1.56977e6 0.215352
\(557\) 1.52459e6 0.208217 0.104108 0.994566i \(-0.466801\pi\)
0.104108 + 0.994566i \(0.466801\pi\)
\(558\) −66932.6 −0.00910023
\(559\) −3.40809e6 −0.461299
\(560\) 2.56337e6 0.345415
\(561\) 460245. 0.0617421
\(562\) −3.40765e6 −0.455108
\(563\) −1.40845e7 −1.87271 −0.936353 0.351059i \(-0.885822\pi\)
−0.936353 + 0.351059i \(0.885822\pi\)
\(564\) 706300. 0.0934957
\(565\) −1.69436e6 −0.223297
\(566\) 8.59301e6 1.12747
\(567\) −809872. −0.105793
\(568\) 5.41407e6 0.704131
\(569\) −1.30473e7 −1.68942 −0.844712 0.535221i \(-0.820228\pi\)
−0.844712 + 0.535221i \(0.820228\pi\)
\(570\) −1.08674e6 −0.140101
\(571\) −1.19873e7 −1.53862 −0.769312 0.638873i \(-0.779401\pi\)
−0.769312 + 0.638873i \(0.779401\pi\)
\(572\) −314487. −0.0401895
\(573\) −6.43959e6 −0.819354
\(574\) 1.15455e7 1.46262
\(575\) −765737. −0.0965851
\(576\) 2.93390e6 0.368459
\(577\) 568904. 0.0711376 0.0355688 0.999367i \(-0.488676\pi\)
0.0355688 + 0.999367i \(0.488676\pi\)
\(578\) −6.42603e6 −0.800061
\(579\) −8.29879e6 −1.02877
\(580\) 274383. 0.0338678
\(581\) 1.35613e7 1.66671
\(582\) 4.26574e6 0.522020
\(583\) −1.85949e6 −0.226581
\(584\) 5.55325e6 0.673775
\(585\) −1.01261e6 −0.122335
\(586\) −2.12708e6 −0.255882
\(587\) 2.93323e6 0.351358 0.175679 0.984447i \(-0.443788\pi\)
0.175679 + 0.984447i \(0.443788\pi\)
\(588\) 73452.6 0.00876120
\(589\) 148910. 0.0176862
\(590\) 3.02868e6 0.358198
\(591\) −5.52269e6 −0.650402
\(592\) −4.49758e6 −0.527442
\(593\) 5.90557e6 0.689644 0.344822 0.938668i \(-0.387939\pi\)
0.344822 + 0.938668i \(0.387939\pi\)
\(594\) −456667. −0.0531048
\(595\) 1.30421e6 0.151027
\(596\) −1.69931e6 −0.195955
\(597\) −3.41087e6 −0.391678
\(598\) 3.17178e6 0.362702
\(599\) 1.07665e7 1.22605 0.613025 0.790064i \(-0.289953\pi\)
0.613025 + 0.790064i \(0.289953\pi\)
\(600\) −1.08324e6 −0.122842
\(601\) −1.19746e7 −1.35231 −0.676154 0.736760i \(-0.736355\pi\)
−0.676154 + 0.736760i \(0.736355\pi\)
\(602\) −4.35541e6 −0.489822
\(603\) 1.17507e6 0.131605
\(604\) −855944. −0.0954669
\(605\) 366025. 0.0406558
\(606\) 992418. 0.109777
\(607\) −3.45506e6 −0.380613 −0.190307 0.981725i \(-0.560948\pi\)
−0.190307 + 0.981725i \(0.560948\pi\)
\(608\) −1.73715e6 −0.190580
\(609\) 2.34587e6 0.256307
\(610\) 1.39373e6 0.151654
\(611\) 7.55024e6 0.818197
\(612\) 177929. 0.0192030
\(613\) −8.22419e6 −0.883979 −0.441990 0.897020i \(-0.645727\pi\)
−0.441990 + 0.897020i \(0.645727\pi\)
\(614\) 6.98034e6 0.747233
\(615\) −4.06500e6 −0.433384
\(616\) −2.87630e6 −0.305409
\(617\) 9.83567e6 1.04014 0.520069 0.854124i \(-0.325906\pi\)
0.520069 + 0.854124i \(0.325906\pi\)
\(618\) −305100. −0.0321345
\(619\) 267648. 0.0280762 0.0140381 0.999901i \(-0.495531\pi\)
0.0140381 + 0.999901i \(0.495531\pi\)
\(620\) 20739.9 0.00216685
\(621\) −893156. −0.0929391
\(622\) −1.30766e7 −1.35525
\(623\) −8.60032e6 −0.887758
\(624\) 3.73837e6 0.384345
\(625\) 390625. 0.0400000
\(626\) 1.15481e7 1.17781
\(627\) 1.01598e6 0.103209
\(628\) −1.29222e6 −0.130749
\(629\) −2.28831e6 −0.230616
\(630\) −1.29407e6 −0.129899
\(631\) −4.17135e6 −0.417065 −0.208532 0.978015i \(-0.566869\pi\)
−0.208532 + 0.978015i \(0.566869\pi\)
\(632\) 1.56959e6 0.156313
\(633\) −4.26349e6 −0.422918
\(634\) −625951. −0.0618468
\(635\) −2.88379e6 −0.283811
\(636\) −718876. −0.0704711
\(637\) 785197. 0.0766708
\(638\) 1.32278e6 0.128658
\(639\) −2.27723e6 −0.220625
\(640\) 3.19839e6 0.308661
\(641\) 8.24673e6 0.792751 0.396376 0.918088i \(-0.370268\pi\)
0.396376 + 0.918088i \(0.370268\pi\)
\(642\) 5.92538e6 0.567386
\(643\) 1.07773e7 1.02797 0.513986 0.857799i \(-0.328168\pi\)
0.513986 + 0.857799i \(0.328168\pi\)
\(644\) −786046. −0.0746850
\(645\) 1.53348e6 0.145137
\(646\) 2.04129e6 0.192453
\(647\) 7.89194e6 0.741179 0.370590 0.928797i \(-0.379156\pi\)
0.370590 + 0.928797i \(0.379156\pi\)
\(648\) −1.26349e6 −0.118205
\(649\) −2.83147e6 −0.263876
\(650\) −1.61802e6 −0.150210
\(651\) 177319. 0.0163984
\(652\) 2.17080e6 0.199987
\(653\) 1.47858e7 1.35695 0.678473 0.734625i \(-0.262642\pi\)
0.678473 + 0.734625i \(0.262642\pi\)
\(654\) 2.66914e6 0.244021
\(655\) 1.08191e6 0.0985343
\(656\) 1.50073e7 1.36158
\(657\) −2.33577e6 −0.211114
\(658\) 9.64891e6 0.868788
\(659\) −475088. −0.0426148 −0.0213074 0.999773i \(-0.506783\pi\)
−0.0213074 + 0.999773i \(0.506783\pi\)
\(660\) 141504. 0.0126447
\(661\) 2.73604e6 0.243567 0.121784 0.992557i \(-0.461139\pi\)
0.121784 + 0.992557i \(0.461139\pi\)
\(662\) 2.99485e6 0.265601
\(663\) 1.90204e6 0.168049
\(664\) 2.11571e7 1.86224
\(665\) 2.87902e6 0.252458
\(666\) 2.27053e6 0.198354
\(667\) 2.58711e6 0.225165
\(668\) −3.66407e6 −0.317704
\(669\) −7.40472e6 −0.639651
\(670\) 1.87762e6 0.161592
\(671\) −1.30297e6 −0.111720
\(672\) −2.06856e6 −0.176704
\(673\) 1.52861e7 1.30094 0.650471 0.759531i \(-0.274572\pi\)
0.650471 + 0.759531i \(0.274572\pi\)
\(674\) −9.25202e6 −0.784489
\(675\) 455625. 0.0384900
\(676\) 630157. 0.0530374
\(677\) −3.93225e6 −0.329739 −0.164869 0.986315i \(-0.552720\pi\)
−0.164869 + 0.986315i \(0.552720\pi\)
\(678\) −3.15787e6 −0.263827
\(679\) −1.13009e7 −0.940670
\(680\) 2.03471e6 0.168745
\(681\) −5.00487e6 −0.413547
\(682\) 99985.7 0.00823146
\(683\) 3.42591e6 0.281011 0.140506 0.990080i \(-0.455127\pi\)
0.140506 + 0.990080i \(0.455127\pi\)
\(684\) 392776. 0.0320999
\(685\) 1.00360e7 0.817210
\(686\) 1.17439e7 0.952803
\(687\) −5.71427e6 −0.461922
\(688\) −5.66135e6 −0.455984
\(689\) −7.68467e6 −0.616705
\(690\) −1.42715e6 −0.114116
\(691\) 1.81896e7 1.44920 0.724599 0.689171i \(-0.242025\pi\)
0.724599 + 0.689171i \(0.242025\pi\)
\(692\) 889102. 0.0705808
\(693\) 1.20981e6 0.0956938
\(694\) −7.57316e6 −0.596868
\(695\) −7.55046e6 −0.592941
\(696\) 3.65982e6 0.286376
\(697\) 7.63552e6 0.595328
\(698\) 7.68804e6 0.597279
\(699\) −8.15905e6 −0.631607
\(700\) 400985. 0.0309302
\(701\) −1.15598e6 −0.0888494 −0.0444247 0.999013i \(-0.514145\pi\)
−0.0444247 + 0.999013i \(0.514145\pi\)
\(702\) −1.88725e6 −0.144540
\(703\) −5.05141e6 −0.385500
\(704\) −4.38274e6 −0.333283
\(705\) −3.39725e6 −0.257427
\(706\) 6.96468e6 0.525883
\(707\) −2.62913e6 −0.197817
\(708\) −1.09464e6 −0.0820705
\(709\) 2.42664e7 1.81296 0.906481 0.422246i \(-0.138758\pi\)
0.906481 + 0.422246i \(0.138758\pi\)
\(710\) −3.63873e6 −0.270897
\(711\) −660192. −0.0489775
\(712\) −1.34174e7 −0.991904
\(713\) 195553. 0.0144059
\(714\) 2.43073e6 0.178440
\(715\) 1.51266e6 0.110656
\(716\) 864499. 0.0630205
\(717\) 5.96328e6 0.433199
\(718\) −9.52359e6 −0.689429
\(719\) −2.25503e7 −1.62679 −0.813394 0.581714i \(-0.802382\pi\)
−0.813394 + 0.581714i \(0.802382\pi\)
\(720\) −1.68209e6 −0.120926
\(721\) 808276. 0.0579057
\(722\) −8.31291e6 −0.593486
\(723\) −1.18641e7 −0.844088
\(724\) 3.03743e6 0.215357
\(725\) −1.31976e6 −0.0932503
\(726\) 682182. 0.0480351
\(727\) −2.17941e7 −1.52933 −0.764667 0.644425i \(-0.777097\pi\)
−0.764667 + 0.644425i \(0.777097\pi\)
\(728\) −1.18868e7 −0.831257
\(729\) 531441. 0.0370370
\(730\) −3.73226e6 −0.259218
\(731\) −2.88043e6 −0.199372
\(732\) −503727. −0.0347470
\(733\) 1.73565e7 1.19317 0.596583 0.802551i \(-0.296524\pi\)
0.596583 + 0.802551i \(0.296524\pi\)
\(734\) −1.61567e6 −0.110691
\(735\) −353301. −0.0241228
\(736\) −2.28129e6 −0.155233
\(737\) −1.75536e6 −0.119041
\(738\) −7.57617e6 −0.512046
\(739\) 1.16961e6 0.0787825 0.0393912 0.999224i \(-0.487458\pi\)
0.0393912 + 0.999224i \(0.487458\pi\)
\(740\) −703553. −0.0472300
\(741\) 4.19871e6 0.280912
\(742\) −9.82071e6 −0.654837
\(743\) 2.98047e6 0.198067 0.0990336 0.995084i \(-0.468425\pi\)
0.0990336 + 0.995084i \(0.468425\pi\)
\(744\) 276637. 0.0183222
\(745\) 8.17354e6 0.539535
\(746\) 1.75327e7 1.15346
\(747\) −8.89898e6 −0.583497
\(748\) −265796. −0.0173698
\(749\) −1.56976e7 −1.02242
\(750\) 728030. 0.0472603
\(751\) −1.83578e7 −1.18774 −0.593868 0.804563i \(-0.702400\pi\)
−0.593868 + 0.804563i \(0.702400\pi\)
\(752\) 1.25421e7 0.808770
\(753\) −99665.3 −0.00640555
\(754\) 5.46661e6 0.350178
\(755\) 4.11702e6 0.262855
\(756\) 467709. 0.0297626
\(757\) 1.60050e7 1.01511 0.507557 0.861618i \(-0.330549\pi\)
0.507557 + 0.861618i \(0.330549\pi\)
\(758\) −8.40607e6 −0.531398
\(759\) 1.33422e6 0.0840665
\(760\) 4.49158e6 0.282075
\(761\) −1.67436e7 −1.04807 −0.524033 0.851698i \(-0.675573\pi\)
−0.524033 + 0.851698i \(0.675573\pi\)
\(762\) −5.37469e6 −0.335325
\(763\) −7.07113e6 −0.439721
\(764\) 3.71892e6 0.230507
\(765\) −855827. −0.0528728
\(766\) −1.58516e7 −0.976116
\(767\) −1.17015e7 −0.718213
\(768\) −4.47061e6 −0.273504
\(769\) −1.73863e7 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(770\) 1.93312e6 0.117498
\(771\) −7.09736e6 −0.429993
\(772\) 4.79263e6 0.289422
\(773\) 1.79095e7 1.07804 0.539021 0.842292i \(-0.318794\pi\)
0.539021 + 0.842292i \(0.318794\pi\)
\(774\) 2.85804e6 0.171481
\(775\) −99757.5 −0.00596611
\(776\) −1.76306e7 −1.05102
\(777\) −6.01511e6 −0.357430
\(778\) 2.18391e7 1.29356
\(779\) 1.68552e7 0.995157
\(780\) 584790. 0.0344163
\(781\) 3.40179e6 0.199563
\(782\) 2.68070e6 0.156758
\(783\) −1.53937e6 −0.0897301
\(784\) 1.30433e6 0.0757874
\(785\) 6.21549e6 0.359999
\(786\) 2.01642e6 0.116419
\(787\) −3.01291e7 −1.73400 −0.867000 0.498308i \(-0.833955\pi\)
−0.867000 + 0.498308i \(0.833955\pi\)
\(788\) 3.18941e6 0.182976
\(789\) 1.90192e6 0.108768
\(790\) −1.05490e6 −0.0601374
\(791\) 8.36587e6 0.475412
\(792\) 1.88744e6 0.106920
\(793\) −5.38476e6 −0.304077
\(794\) 2.65681e6 0.149558
\(795\) 3.45774e6 0.194032
\(796\) 1.96981e6 0.110190
\(797\) −5.51251e6 −0.307400 −0.153700 0.988118i \(-0.549119\pi\)
−0.153700 + 0.988118i \(0.549119\pi\)
\(798\) 5.36579e6 0.298281
\(799\) 6.38125e6 0.353622
\(800\) 1.16375e6 0.0642887
\(801\) 5.64356e6 0.310793
\(802\) −1.13098e7 −0.620895
\(803\) 3.48924e6 0.190960
\(804\) −678616. −0.0370241
\(805\) 3.78082e6 0.205635
\(806\) 413208. 0.0224043
\(807\) 4.97108e6 0.268700
\(808\) −4.10173e6 −0.221024
\(809\) 3.11564e7 1.67369 0.836847 0.547436i \(-0.184396\pi\)
0.836847 + 0.547436i \(0.184396\pi\)
\(810\) 849175. 0.0454762
\(811\) −1.11336e7 −0.594406 −0.297203 0.954814i \(-0.596054\pi\)
−0.297203 + 0.954814i \(0.596054\pi\)
\(812\) −1.35476e6 −0.0721063
\(813\) −1.34330e7 −0.712763
\(814\) −3.39178e6 −0.179418
\(815\) −1.04414e7 −0.550636
\(816\) 3.15957e6 0.166113
\(817\) −6.35848e6 −0.333271
\(818\) 6.33195e6 0.330868
\(819\) 4.99974e6 0.260458
\(820\) 2.34758e6 0.121923
\(821\) −3.44603e7 −1.78427 −0.892136 0.451767i \(-0.850794\pi\)
−0.892136 + 0.451767i \(0.850794\pi\)
\(822\) 1.87046e7 0.965539
\(823\) −9.74417e6 −0.501470 −0.250735 0.968056i \(-0.580672\pi\)
−0.250735 + 0.968056i \(0.580672\pi\)
\(824\) 1.26100e6 0.0646989
\(825\) −680625. −0.0348155
\(826\) −1.49541e7 −0.762622
\(827\) −9.63214e6 −0.489733 −0.244866 0.969557i \(-0.578744\pi\)
−0.244866 + 0.969557i \(0.578744\pi\)
\(828\) 515806. 0.0261463
\(829\) −2.79310e7 −1.41156 −0.705780 0.708431i \(-0.749403\pi\)
−0.705780 + 0.708431i \(0.749403\pi\)
\(830\) −1.42194e7 −0.716452
\(831\) 9.12879e6 0.458575
\(832\) −1.81124e7 −0.907126
\(833\) 663626. 0.0331368
\(834\) −1.40722e7 −0.700564
\(835\) 1.76239e7 0.874753
\(836\) −586739. −0.0290355
\(837\) −116357. −0.00574090
\(838\) 1.25741e7 0.618540
\(839\) −3.51475e7 −1.72381 −0.861906 0.507068i \(-0.830729\pi\)
−0.861906 + 0.507068i \(0.830729\pi\)
\(840\) 5.34848e6 0.261537
\(841\) −1.60522e7 −0.782610
\(842\) −3.76617e6 −0.183071
\(843\) −5.92394e6 −0.287106
\(844\) 2.46221e6 0.118978
\(845\) −3.03101e6 −0.146031
\(846\) −6.33165e6 −0.304152
\(847\) −1.80725e6 −0.0865583
\(848\) −1.27654e7 −0.609599
\(849\) 1.49383e7 0.711265
\(850\) −1.36750e6 −0.0649203
\(851\) −6.63368e6 −0.314001
\(852\) 1.31512e6 0.0620680
\(853\) 2.50498e7 1.17878 0.589389 0.807849i \(-0.299369\pi\)
0.589389 + 0.807849i \(0.299369\pi\)
\(854\) −6.88152e6 −0.322879
\(855\) −1.88922e6 −0.0883827
\(856\) −2.44900e7 −1.14236
\(857\) −1.89631e7 −0.881975 −0.440987 0.897513i \(-0.645372\pi\)
−0.440987 + 0.897513i \(0.645372\pi\)
\(858\) 2.81923e6 0.130741
\(859\) −3.04605e7 −1.40849 −0.704245 0.709957i \(-0.748714\pi\)
−0.704245 + 0.709957i \(0.748714\pi\)
\(860\) −885600. −0.0408312
\(861\) 2.00709e7 0.922696
\(862\) 3.20060e7 1.46711
\(863\) 3.46176e7 1.58223 0.791115 0.611667i \(-0.209501\pi\)
0.791115 + 0.611667i \(0.209501\pi\)
\(864\) 1.35740e6 0.0618619
\(865\) −4.27651e6 −0.194334
\(866\) 1.25460e7 0.568475
\(867\) −1.11712e7 −0.504720
\(868\) −102403. −0.00461333
\(869\) 986213. 0.0443018
\(870\) −2.45971e6 −0.110176
\(871\) −7.25430e6 −0.324004
\(872\) −1.10317e7 −0.491307
\(873\) 7.41566e6 0.329317
\(874\) 5.91758e6 0.262039
\(875\) −1.92871e6 −0.0851621
\(876\) 1.34893e6 0.0593922
\(877\) −3.61979e7 −1.58922 −0.794611 0.607119i \(-0.792325\pi\)
−0.794611 + 0.607119i \(0.792325\pi\)
\(878\) −1.96727e7 −0.861245
\(879\) −3.69777e6 −0.161424
\(880\) 2.51275e6 0.109381
\(881\) 2.40371e7 1.04338 0.521690 0.853135i \(-0.325302\pi\)
0.521690 + 0.853135i \(0.325302\pi\)
\(882\) −658468. −0.0285012
\(883\) 3.64938e7 1.57513 0.787567 0.616229i \(-0.211341\pi\)
0.787567 + 0.616229i \(0.211341\pi\)
\(884\) −1.09845e6 −0.0472768
\(885\) 5.26512e6 0.225970
\(886\) −2.88905e7 −1.23644
\(887\) 2.29197e7 0.978137 0.489069 0.872245i \(-0.337337\pi\)
0.489069 + 0.872245i \(0.337337\pi\)
\(888\) −9.38424e6 −0.399362
\(889\) 1.42387e7 0.604250
\(890\) 9.01768e6 0.381610
\(891\) −793881. −0.0335013
\(892\) 4.27630e6 0.179952
\(893\) 1.40865e7 0.591117
\(894\) 1.52335e7 0.637464
\(895\) −4.15817e6 −0.173518
\(896\) −1.57920e7 −0.657156
\(897\) 5.51389e6 0.228811
\(898\) 1.39579e7 0.577605
\(899\) 337039. 0.0139085
\(900\) −263128. −0.0108283
\(901\) −6.49487e6 −0.266538
\(902\) 1.13175e7 0.463163
\(903\) −7.57155e6 −0.309005
\(904\) 1.30517e7 0.531184
\(905\) −1.46098e7 −0.592956
\(906\) 7.67313e6 0.310565
\(907\) −2.66664e7 −1.07633 −0.538167 0.842838i \(-0.680883\pi\)
−0.538167 + 0.842838i \(0.680883\pi\)
\(908\) 2.89036e6 0.116342
\(909\) 1.72524e6 0.0692533
\(910\) 7.98894e6 0.319805
\(911\) 1.73286e7 0.691778 0.345889 0.938276i \(-0.387577\pi\)
0.345889 + 0.938276i \(0.387577\pi\)
\(912\) 6.97468e6 0.277675
\(913\) 1.32935e7 0.527793
\(914\) 5.91518e6 0.234208
\(915\) 2.42289e6 0.0956710
\(916\) 3.30005e6 0.129952
\(917\) −5.34192e6 −0.209785
\(918\) −1.59505e6 −0.0624696
\(919\) −1.72198e7 −0.672573 −0.336286 0.941760i \(-0.609171\pi\)
−0.336286 + 0.941760i \(0.609171\pi\)
\(920\) 5.89850e6 0.229759
\(921\) 1.21348e7 0.471393
\(922\) 2.78058e7 1.07723
\(923\) 1.40585e7 0.543168
\(924\) −698677. −0.0269213
\(925\) 3.38404e6 0.130041
\(926\) −1.96848e7 −0.754402
\(927\) −530393. −0.0202721
\(928\) −3.93183e6 −0.149874
\(929\) 5.47137e6 0.207997 0.103998 0.994577i \(-0.466836\pi\)
0.103998 + 0.994577i \(0.466836\pi\)
\(930\) −185924. −0.00704901
\(931\) 1.46494e6 0.0553919
\(932\) 4.71193e6 0.177688
\(933\) −2.27326e7 −0.854960
\(934\) 4.87525e7 1.82865
\(935\) 1.27846e6 0.0478252
\(936\) 7.80014e6 0.291013
\(937\) 8.04892e6 0.299494 0.149747 0.988724i \(-0.452154\pi\)
0.149747 + 0.988724i \(0.452154\pi\)
\(938\) −9.27072e6 −0.344038
\(939\) 2.00755e7 0.743022
\(940\) 1.96195e6 0.0724215
\(941\) −1.13721e7 −0.418665 −0.209333 0.977844i \(-0.567129\pi\)
−0.209333 + 0.977844i \(0.567129\pi\)
\(942\) 1.15842e7 0.425341
\(943\) 2.21349e7 0.810584
\(944\) −1.94379e7 −0.709938
\(945\) −2.24965e6 −0.0819473
\(946\) −4.26941e6 −0.155110
\(947\) 4.97017e7 1.80093 0.900463 0.434932i \(-0.143228\pi\)
0.900463 + 0.434932i \(0.143228\pi\)
\(948\) 381267. 0.0137787
\(949\) 1.44199e7 0.519751
\(950\) −3.01873e6 −0.108521
\(951\) −1.08817e6 −0.0390162
\(952\) −1.00464e7 −0.359266
\(953\) 2.95809e7 1.05507 0.527533 0.849534i \(-0.323117\pi\)
0.527533 + 0.849534i \(0.323117\pi\)
\(954\) 6.44438e6 0.229251
\(955\) −1.78877e7 −0.634669
\(956\) −3.44385e6 −0.121871
\(957\) 2.29955e6 0.0811639
\(958\) −2.42830e7 −0.854849
\(959\) −4.95526e7 −1.73988
\(960\) 8.14972e6 0.285407
\(961\) −2.86037e7 −0.999110
\(962\) −1.40171e7 −0.488337
\(963\) 1.03008e7 0.357937
\(964\) 6.85161e6 0.237465
\(965\) −2.30522e7 −0.796882
\(966\) 7.04654e6 0.242959
\(967\) 3.10475e7 1.06773 0.533863 0.845571i \(-0.320740\pi\)
0.533863 + 0.845571i \(0.320740\pi\)
\(968\) −2.81950e6 −0.0967128
\(969\) 3.54863e6 0.121409
\(970\) 1.18493e7 0.404355
\(971\) −1.89426e7 −0.644751 −0.322376 0.946612i \(-0.604481\pi\)
−0.322376 + 0.946612i \(0.604481\pi\)
\(972\) −306912. −0.0104195
\(973\) 3.72803e7 1.26240
\(974\) 3.66113e7 1.23657
\(975\) −2.81280e6 −0.0947604
\(976\) −8.94489e6 −0.300573
\(977\) 5.49466e7 1.84164 0.920819 0.389989i \(-0.127521\pi\)
0.920819 + 0.389989i \(0.127521\pi\)
\(978\) −1.94602e7 −0.650580
\(979\) −8.43050e6 −0.281123
\(980\) 204035. 0.00678640
\(981\) 4.64010e6 0.153941
\(982\) 4.69072e6 0.155225
\(983\) −4.72203e7 −1.55864 −0.779318 0.626628i \(-0.784435\pi\)
−0.779318 + 0.626628i \(0.784435\pi\)
\(984\) 3.13128e7 1.03094
\(985\) −1.53408e7 −0.503799
\(986\) 4.62022e6 0.151346
\(987\) 1.67739e7 0.548076
\(988\) −2.42480e6 −0.0790283
\(989\) −8.35018e6 −0.271459
\(990\) −1.26852e6 −0.0411348
\(991\) 4.48844e7 1.45182 0.725908 0.687792i \(-0.241420\pi\)
0.725908 + 0.687792i \(0.241420\pi\)
\(992\) −297198. −0.00958885
\(993\) 5.20632e6 0.167555
\(994\) 1.79662e7 0.576753
\(995\) −9.47463e6 −0.303392
\(996\) 5.13925e6 0.164154
\(997\) 5.08797e7 1.62109 0.810543 0.585679i \(-0.199172\pi\)
0.810543 + 0.585679i \(0.199172\pi\)
\(998\) −3.69295e7 −1.17367
\(999\) 3.94714e6 0.125132
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 165.6.a.a.1.3 3
3.2 odd 2 495.6.a.e.1.1 3
5.4 even 2 825.6.a.j.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.a.1.3 3 1.1 even 1 trivial
495.6.a.e.1.1 3 3.2 odd 2
825.6.a.j.1.1 3 5.4 even 2