Properties

Label 1045.4.a.c.1.5
Level $1045$
Weight $4$
Character 1045.1
Self dual yes
Analytic conductor $61.657$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.6569959560\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + 1317797 x^{12} - 1160501 x^{11} - 9914845 x^{10} + 7570653 x^{9} + 42786958 x^{8} - 23777633 x^{7} - 102801526 x^{6} + 28436356 x^{5} + 122325928 x^{4} + 411232 x^{3} - 47350496 x^{2} - 4782848 x + 150528\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.81607\) of defining polynomial
Character \(\chi\) \(=\) 1045.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.81607 q^{2} +3.33662 q^{3} -0.0697562 q^{4} -5.00000 q^{5} -9.39614 q^{6} +35.3821 q^{7} +22.7250 q^{8} -15.8670 q^{9} +O(q^{10})\) \(q-2.81607 q^{2} +3.33662 q^{3} -0.0697562 q^{4} -5.00000 q^{5} -9.39614 q^{6} +35.3821 q^{7} +22.7250 q^{8} -15.8670 q^{9} +14.0803 q^{10} -11.0000 q^{11} -0.232750 q^{12} +16.5610 q^{13} -99.6385 q^{14} -16.6831 q^{15} -63.4371 q^{16} -35.7214 q^{17} +44.6825 q^{18} +19.0000 q^{19} +0.348781 q^{20} +118.057 q^{21} +30.9768 q^{22} -96.3766 q^{23} +75.8246 q^{24} +25.0000 q^{25} -46.6369 q^{26} -143.031 q^{27} -2.46812 q^{28} -163.372 q^{29} +46.9807 q^{30} +130.338 q^{31} -3.15672 q^{32} -36.7028 q^{33} +100.594 q^{34} -176.911 q^{35} +1.10682 q^{36} +7.26988 q^{37} -53.5053 q^{38} +55.2577 q^{39} -113.625 q^{40} -315.838 q^{41} -332.456 q^{42} -250.799 q^{43} +0.767318 q^{44} +79.3350 q^{45} +271.403 q^{46} -280.397 q^{47} -211.665 q^{48} +908.895 q^{49} -70.4017 q^{50} -119.189 q^{51} -1.15523 q^{52} +504.085 q^{53} +402.784 q^{54} +55.0000 q^{55} +804.059 q^{56} +63.3957 q^{57} +460.066 q^{58} +49.5715 q^{59} +1.16375 q^{60} +360.508 q^{61} -367.039 q^{62} -561.408 q^{63} +516.386 q^{64} -82.8050 q^{65} +103.358 q^{66} -370.276 q^{67} +2.49179 q^{68} -321.572 q^{69} +498.193 q^{70} +309.651 q^{71} -360.577 q^{72} +42.6753 q^{73} -20.4725 q^{74} +83.4154 q^{75} -1.32537 q^{76} -389.203 q^{77} -155.610 q^{78} -771.511 q^{79} +317.185 q^{80} -48.8298 q^{81} +889.423 q^{82} -363.907 q^{83} -8.23518 q^{84} +178.607 q^{85} +706.268 q^{86} -545.109 q^{87} -249.975 q^{88} +0.664243 q^{89} -223.413 q^{90} +585.964 q^{91} +6.72286 q^{92} +434.886 q^{93} +789.617 q^{94} -95.0000 q^{95} -10.5327 q^{96} -573.501 q^{97} -2559.51 q^{98} +174.537 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + O(q^{10}) \) \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.81607 −0.995631 −0.497815 0.867283i \(-0.665864\pi\)
−0.497815 + 0.867283i \(0.665864\pi\)
\(3\) 3.33662 0.642132 0.321066 0.947057i \(-0.395959\pi\)
0.321066 + 0.947057i \(0.395959\pi\)
\(4\) −0.0697562 −0.00871953
\(5\) −5.00000 −0.447214
\(6\) −9.39614 −0.639326
\(7\) 35.3821 1.91046 0.955228 0.295871i \(-0.0956099\pi\)
0.955228 + 0.295871i \(0.0956099\pi\)
\(8\) 22.7250 1.00431
\(9\) −15.8670 −0.587666
\(10\) 14.0803 0.445260
\(11\) −11.0000 −0.301511
\(12\) −0.232750 −0.00559909
\(13\) 16.5610 0.353323 0.176661 0.984272i \(-0.443470\pi\)
0.176661 + 0.984272i \(0.443470\pi\)
\(14\) −99.6385 −1.90211
\(15\) −16.6831 −0.287170
\(16\) −63.4371 −0.991204
\(17\) −35.7214 −0.509631 −0.254815 0.966990i \(-0.582015\pi\)
−0.254815 + 0.966990i \(0.582015\pi\)
\(18\) 44.6825 0.585099
\(19\) 19.0000 0.229416
\(20\) 0.348781 0.00389949
\(21\) 118.057 1.22676
\(22\) 30.9768 0.300194
\(23\) −96.3766 −0.873735 −0.436868 0.899526i \(-0.643912\pi\)
−0.436868 + 0.899526i \(0.643912\pi\)
\(24\) 75.8246 0.644901
\(25\) 25.0000 0.200000
\(26\) −46.6369 −0.351779
\(27\) −143.031 −1.01949
\(28\) −2.46812 −0.0166583
\(29\) −163.372 −1.04612 −0.523058 0.852297i \(-0.675209\pi\)
−0.523058 + 0.852297i \(0.675209\pi\)
\(30\) 46.9807 0.285915
\(31\) 130.338 0.755139 0.377570 0.925981i \(-0.376760\pi\)
0.377570 + 0.925981i \(0.376760\pi\)
\(32\) −3.15672 −0.0174386
\(33\) −36.7028 −0.193610
\(34\) 100.594 0.507404
\(35\) −176.911 −0.854382
\(36\) 1.10682 0.00512417
\(37\) 7.26988 0.0323016 0.0161508 0.999870i \(-0.494859\pi\)
0.0161508 + 0.999870i \(0.494859\pi\)
\(38\) −53.5053 −0.228413
\(39\) 55.2577 0.226880
\(40\) −113.625 −0.449142
\(41\) −315.838 −1.20307 −0.601533 0.798848i \(-0.705443\pi\)
−0.601533 + 0.798848i \(0.705443\pi\)
\(42\) −332.456 −1.22140
\(43\) −250.799 −0.889454 −0.444727 0.895666i \(-0.646699\pi\)
−0.444727 + 0.895666i \(0.646699\pi\)
\(44\) 0.767318 0.00262904
\(45\) 79.3350 0.262812
\(46\) 271.403 0.869917
\(47\) −280.397 −0.870214 −0.435107 0.900379i \(-0.643290\pi\)
−0.435107 + 0.900379i \(0.643290\pi\)
\(48\) −211.665 −0.636484
\(49\) 908.895 2.64984
\(50\) −70.4017 −0.199126
\(51\) −119.189 −0.327250
\(52\) −1.15523 −0.00308081
\(53\) 504.085 1.30644 0.653221 0.757167i \(-0.273417\pi\)
0.653221 + 0.757167i \(0.273417\pi\)
\(54\) 402.784 1.01504
\(55\) 55.0000 0.134840
\(56\) 804.059 1.91869
\(57\) 63.3957 0.147315
\(58\) 460.066 1.04154
\(59\) 49.5715 0.109384 0.0546921 0.998503i \(-0.482582\pi\)
0.0546921 + 0.998503i \(0.482582\pi\)
\(60\) 1.16375 0.00250399
\(61\) 360.508 0.756695 0.378347 0.925664i \(-0.376493\pi\)
0.378347 + 0.925664i \(0.376493\pi\)
\(62\) −367.039 −0.751840
\(63\) −561.408 −1.12271
\(64\) 516.386 1.00857
\(65\) −82.8050 −0.158011
\(66\) 103.358 0.192764
\(67\) −370.276 −0.675170 −0.337585 0.941295i \(-0.609610\pi\)
−0.337585 + 0.941295i \(0.609610\pi\)
\(68\) 2.49179 0.00444374
\(69\) −321.572 −0.561053
\(70\) 498.193 0.850649
\(71\) 309.651 0.517590 0.258795 0.965932i \(-0.416675\pi\)
0.258795 + 0.965932i \(0.416675\pi\)
\(72\) −360.577 −0.590200
\(73\) 42.6753 0.0684214 0.0342107 0.999415i \(-0.489108\pi\)
0.0342107 + 0.999415i \(0.489108\pi\)
\(74\) −20.4725 −0.0321605
\(75\) 83.4154 0.128426
\(76\) −1.32537 −0.00200040
\(77\) −389.203 −0.576024
\(78\) −155.610 −0.225889
\(79\) −771.511 −1.09876 −0.549378 0.835574i \(-0.685135\pi\)
−0.549378 + 0.835574i \(0.685135\pi\)
\(80\) 317.185 0.443280
\(81\) −48.8298 −0.0669819
\(82\) 889.423 1.19781
\(83\) −363.907 −0.481253 −0.240626 0.970618i \(-0.577353\pi\)
−0.240626 + 0.970618i \(0.577353\pi\)
\(84\) −8.23518 −0.0106968
\(85\) 178.607 0.227914
\(86\) 706.268 0.885568
\(87\) −545.109 −0.671744
\(88\) −249.975 −0.302811
\(89\) 0.664243 0.000791119 0 0.000395559 1.00000i \(-0.499874\pi\)
0.000395559 1.00000i \(0.499874\pi\)
\(90\) −223.413 −0.261664
\(91\) 585.964 0.675008
\(92\) 6.72286 0.00761855
\(93\) 434.886 0.484899
\(94\) 789.617 0.866412
\(95\) −95.0000 −0.102598
\(96\) −10.5327 −0.0111979
\(97\) −573.501 −0.600311 −0.300156 0.953890i \(-0.597039\pi\)
−0.300156 + 0.953890i \(0.597039\pi\)
\(98\) −2559.51 −2.63826
\(99\) 174.537 0.177188
\(100\) −1.74391 −0.00174391
\(101\) −970.755 −0.956374 −0.478187 0.878258i \(-0.658706\pi\)
−0.478187 + 0.878258i \(0.658706\pi\)
\(102\) 335.644 0.325820
\(103\) 425.291 0.406846 0.203423 0.979091i \(-0.434793\pi\)
0.203423 + 0.979091i \(0.434793\pi\)
\(104\) 376.349 0.354846
\(105\) −590.283 −0.548626
\(106\) −1419.54 −1.30073
\(107\) 905.406 0.818027 0.409014 0.912528i \(-0.365873\pi\)
0.409014 + 0.912528i \(0.365873\pi\)
\(108\) 9.97728 0.00888948
\(109\) −827.160 −0.726858 −0.363429 0.931622i \(-0.618394\pi\)
−0.363429 + 0.931622i \(0.618394\pi\)
\(110\) −154.884 −0.134251
\(111\) 24.2568 0.0207419
\(112\) −2244.54 −1.89365
\(113\) −2239.00 −1.86396 −0.931980 0.362510i \(-0.881920\pi\)
−0.931980 + 0.362510i \(0.881920\pi\)
\(114\) −178.527 −0.146672
\(115\) 481.883 0.390746
\(116\) 11.3962 0.00912163
\(117\) −262.773 −0.207636
\(118\) −139.597 −0.108906
\(119\) −1263.90 −0.973627
\(120\) −379.123 −0.288409
\(121\) 121.000 0.0909091
\(122\) −1015.22 −0.753388
\(123\) −1053.83 −0.772527
\(124\) −9.09185 −0.00658445
\(125\) −125.000 −0.0894427
\(126\) 1580.96 1.11781
\(127\) −26.8256 −0.0187432 −0.00937161 0.999956i \(-0.502983\pi\)
−0.00937161 + 0.999956i \(0.502983\pi\)
\(128\) −1428.93 −0.986722
\(129\) −836.821 −0.571147
\(130\) 233.185 0.157320
\(131\) 1073.61 0.716047 0.358023 0.933713i \(-0.383451\pi\)
0.358023 + 0.933713i \(0.383451\pi\)
\(132\) 2.56025 0.00168819
\(133\) 672.261 0.438289
\(134\) 1042.72 0.672220
\(135\) 715.154 0.455930
\(136\) −811.769 −0.511828
\(137\) 720.676 0.449427 0.224713 0.974425i \(-0.427855\pi\)
0.224713 + 0.974425i \(0.427855\pi\)
\(138\) 905.568 0.558602
\(139\) −107.451 −0.0655674 −0.0327837 0.999462i \(-0.510437\pi\)
−0.0327837 + 0.999462i \(0.510437\pi\)
\(140\) 12.3406 0.00744980
\(141\) −935.577 −0.558793
\(142\) −872.000 −0.515328
\(143\) −182.171 −0.106531
\(144\) 1006.56 0.582497
\(145\) 816.858 0.467837
\(146\) −120.177 −0.0681225
\(147\) 3032.64 1.70155
\(148\) −0.507119 −0.000281655 0
\(149\) 1961.09 1.07825 0.539123 0.842227i \(-0.318756\pi\)
0.539123 + 0.842227i \(0.318756\pi\)
\(150\) −234.904 −0.127865
\(151\) 902.642 0.486463 0.243232 0.969968i \(-0.421792\pi\)
0.243232 + 0.969968i \(0.421792\pi\)
\(152\) 431.775 0.230405
\(153\) 566.792 0.299493
\(154\) 1096.02 0.573507
\(155\) −651.688 −0.337708
\(156\) −3.85457 −0.00197829
\(157\) −2620.69 −1.33219 −0.666096 0.745866i \(-0.732036\pi\)
−0.666096 + 0.745866i \(0.732036\pi\)
\(158\) 2172.63 1.09396
\(159\) 1681.94 0.838908
\(160\) 15.7836 0.00779876
\(161\) −3410.01 −1.66923
\(162\) 137.508 0.0666892
\(163\) 887.061 0.426258 0.213129 0.977024i \(-0.431635\pi\)
0.213129 + 0.977024i \(0.431635\pi\)
\(164\) 22.0317 0.0104902
\(165\) 183.514 0.0865851
\(166\) 1024.79 0.479150
\(167\) −2935.44 −1.36019 −0.680093 0.733125i \(-0.738061\pi\)
−0.680093 + 0.733125i \(0.738061\pi\)
\(168\) 2682.84 1.23205
\(169\) −1922.73 −0.875163
\(170\) −502.970 −0.226918
\(171\) −301.473 −0.134820
\(172\) 17.4948 0.00775562
\(173\) −2867.13 −1.26002 −0.630010 0.776587i \(-0.716949\pi\)
−0.630010 + 0.776587i \(0.716949\pi\)
\(174\) 1535.06 0.668809
\(175\) 884.553 0.382091
\(176\) 697.808 0.298859
\(177\) 165.401 0.0702390
\(178\) −1.87055 −0.000787662 0
\(179\) −4338.27 −1.81150 −0.905748 0.423816i \(-0.860690\pi\)
−0.905748 + 0.423816i \(0.860690\pi\)
\(180\) −5.53411 −0.00229160
\(181\) 275.587 0.113172 0.0565862 0.998398i \(-0.481978\pi\)
0.0565862 + 0.998398i \(0.481978\pi\)
\(182\) −1650.11 −0.672058
\(183\) 1202.88 0.485898
\(184\) −2190.16 −0.877503
\(185\) −36.3494 −0.0144457
\(186\) −1224.67 −0.482780
\(187\) 392.936 0.153659
\(188\) 19.5594 0.00758786
\(189\) −5060.73 −1.94769
\(190\) 267.527 0.102150
\(191\) 731.067 0.276954 0.138477 0.990366i \(-0.455779\pi\)
0.138477 + 0.990366i \(0.455779\pi\)
\(192\) 1722.98 0.647633
\(193\) 1773.81 0.661564 0.330782 0.943707i \(-0.392688\pi\)
0.330782 + 0.943707i \(0.392688\pi\)
\(194\) 1615.02 0.597688
\(195\) −276.289 −0.101464
\(196\) −63.4011 −0.0231054
\(197\) 1060.34 0.383484 0.191742 0.981445i \(-0.438586\pi\)
0.191742 + 0.981445i \(0.438586\pi\)
\(198\) −491.508 −0.176414
\(199\) −4735.68 −1.68695 −0.843476 0.537167i \(-0.819494\pi\)
−0.843476 + 0.537167i \(0.819494\pi\)
\(200\) 568.125 0.200862
\(201\) −1235.47 −0.433549
\(202\) 2733.71 0.952195
\(203\) −5780.44 −1.99856
\(204\) 8.31415 0.00285347
\(205\) 1579.19 0.538027
\(206\) −1197.65 −0.405068
\(207\) 1529.21 0.513465
\(208\) −1050.58 −0.350215
\(209\) −209.000 −0.0691714
\(210\) 1662.28 0.546229
\(211\) −131.852 −0.0430192 −0.0215096 0.999769i \(-0.506847\pi\)
−0.0215096 + 0.999769i \(0.506847\pi\)
\(212\) −35.1631 −0.0113916
\(213\) 1033.19 0.332361
\(214\) −2549.69 −0.814453
\(215\) 1254.00 0.397776
\(216\) −3250.37 −1.02389
\(217\) 4611.62 1.44266
\(218\) 2329.34 0.723682
\(219\) 142.391 0.0439356
\(220\) −3.83659 −0.00117574
\(221\) −591.583 −0.180064
\(222\) −68.3088 −0.0206513
\(223\) 15.1854 0.00456006 0.00228003 0.999997i \(-0.499274\pi\)
0.00228003 + 0.999997i \(0.499274\pi\)
\(224\) −111.691 −0.0333156
\(225\) −396.675 −0.117533
\(226\) 6305.18 1.85582
\(227\) −1126.42 −0.329352 −0.164676 0.986348i \(-0.552658\pi\)
−0.164676 + 0.986348i \(0.552658\pi\)
\(228\) −4.42224 −0.00128452
\(229\) −3232.93 −0.932918 −0.466459 0.884543i \(-0.654470\pi\)
−0.466459 + 0.884543i \(0.654470\pi\)
\(230\) −1357.02 −0.389039
\(231\) −1298.62 −0.369884
\(232\) −3712.62 −1.05063
\(233\) −3006.01 −0.845195 −0.422598 0.906317i \(-0.638882\pi\)
−0.422598 + 0.906317i \(0.638882\pi\)
\(234\) 739.988 0.206729
\(235\) 1401.98 0.389172
\(236\) −3.45792 −0.000953777 0
\(237\) −2574.24 −0.705547
\(238\) 3559.23 0.969373
\(239\) −923.389 −0.249912 −0.124956 0.992162i \(-0.539879\pi\)
−0.124956 + 0.992162i \(0.539879\pi\)
\(240\) 1058.33 0.284644
\(241\) −5944.82 −1.58896 −0.794480 0.607290i \(-0.792257\pi\)
−0.794480 + 0.607290i \(0.792257\pi\)
\(242\) −340.744 −0.0905119
\(243\) 3698.90 0.976480
\(244\) −25.1477 −0.00659802
\(245\) −4544.48 −1.18504
\(246\) 2967.66 0.769152
\(247\) 314.659 0.0810578
\(248\) 2961.92 0.758395
\(249\) −1214.22 −0.309028
\(250\) 352.009 0.0890519
\(251\) −3644.44 −0.916474 −0.458237 0.888830i \(-0.651519\pi\)
−0.458237 + 0.888830i \(0.651519\pi\)
\(252\) 39.1617 0.00978950
\(253\) 1060.14 0.263441
\(254\) 75.5428 0.0186613
\(255\) 595.944 0.146351
\(256\) −107.137 −0.0261566
\(257\) 3492.93 0.847794 0.423897 0.905710i \(-0.360662\pi\)
0.423897 + 0.905710i \(0.360662\pi\)
\(258\) 2356.55 0.568652
\(259\) 257.224 0.0617108
\(260\) 5.77617 0.00137778
\(261\) 2592.22 0.614767
\(262\) −3023.37 −0.712918
\(263\) −4129.23 −0.968134 −0.484067 0.875031i \(-0.660841\pi\)
−0.484067 + 0.875031i \(0.660841\pi\)
\(264\) −834.070 −0.194445
\(265\) −2520.43 −0.584259
\(266\) −1893.13 −0.436374
\(267\) 2.21632 0.000508003 0
\(268\) 25.8291 0.00588717
\(269\) −6512.16 −1.47603 −0.738017 0.674782i \(-0.764238\pi\)
−0.738017 + 0.674782i \(0.764238\pi\)
\(270\) −2013.92 −0.453938
\(271\) 2674.98 0.599607 0.299803 0.954001i \(-0.403079\pi\)
0.299803 + 0.954001i \(0.403079\pi\)
\(272\) 2266.06 0.505148
\(273\) 1955.14 0.433444
\(274\) −2029.47 −0.447463
\(275\) −275.000 −0.0603023
\(276\) 22.4316 0.00489212
\(277\) −5144.41 −1.11588 −0.557938 0.829883i \(-0.688407\pi\)
−0.557938 + 0.829883i \(0.688407\pi\)
\(278\) 302.589 0.0652809
\(279\) −2068.06 −0.443770
\(280\) −4020.29 −0.858066
\(281\) 6593.16 1.39970 0.699849 0.714291i \(-0.253251\pi\)
0.699849 + 0.714291i \(0.253251\pi\)
\(282\) 2634.65 0.556351
\(283\) 6742.17 1.41619 0.708093 0.706119i \(-0.249556\pi\)
0.708093 + 0.706119i \(0.249556\pi\)
\(284\) −21.6001 −0.00451314
\(285\) −316.979 −0.0658814
\(286\) 513.006 0.106065
\(287\) −11175.0 −2.29840
\(288\) 50.0876 0.0102480
\(289\) −3636.98 −0.740277
\(290\) −2300.33 −0.465793
\(291\) −1913.55 −0.385479
\(292\) −2.97687 −0.000596602 0
\(293\) 514.102 0.102506 0.0512529 0.998686i \(-0.483679\pi\)
0.0512529 + 0.998686i \(0.483679\pi\)
\(294\) −8540.11 −1.69411
\(295\) −247.858 −0.0489181
\(296\) 165.208 0.0324409
\(297\) 1573.34 0.307388
\(298\) −5522.56 −1.07353
\(299\) −1596.09 −0.308711
\(300\) −5.81874 −0.00111982
\(301\) −8873.81 −1.69926
\(302\) −2541.90 −0.484338
\(303\) −3239.04 −0.614118
\(304\) −1205.30 −0.227398
\(305\) −1802.54 −0.338404
\(306\) −1596.12 −0.298184
\(307\) −2887.23 −0.536751 −0.268376 0.963314i \(-0.586487\pi\)
−0.268376 + 0.963314i \(0.586487\pi\)
\(308\) 27.1494 0.00502266
\(309\) 1419.03 0.261249
\(310\) 1835.20 0.336233
\(311\) 6666.03 1.21542 0.607711 0.794158i \(-0.292088\pi\)
0.607711 + 0.794158i \(0.292088\pi\)
\(312\) 1255.73 0.227858
\(313\) 6935.87 1.25252 0.626260 0.779614i \(-0.284585\pi\)
0.626260 + 0.779614i \(0.284585\pi\)
\(314\) 7380.06 1.32637
\(315\) 2807.04 0.502091
\(316\) 53.8177 0.00958064
\(317\) −2548.21 −0.451488 −0.225744 0.974187i \(-0.572481\pi\)
−0.225744 + 0.974187i \(0.572481\pi\)
\(318\) −4736.46 −0.835243
\(319\) 1797.09 0.315416
\(320\) −2581.93 −0.451045
\(321\) 3020.99 0.525282
\(322\) 9602.82 1.66194
\(323\) −678.707 −0.116917
\(324\) 3.40618 0.000584050 0
\(325\) 414.025 0.0706646
\(326\) −2498.02 −0.424395
\(327\) −2759.91 −0.466739
\(328\) −7177.43 −1.20825
\(329\) −9921.04 −1.66251
\(330\) −516.788 −0.0862068
\(331\) 2780.92 0.461792 0.230896 0.972978i \(-0.425834\pi\)
0.230896 + 0.972978i \(0.425834\pi\)
\(332\) 25.3848 0.00419630
\(333\) −115.351 −0.0189826
\(334\) 8266.40 1.35424
\(335\) 1851.38 0.301945
\(336\) −7489.17 −1.21597
\(337\) −10910.3 −1.76357 −0.881783 0.471655i \(-0.843657\pi\)
−0.881783 + 0.471655i \(0.843657\pi\)
\(338\) 5414.55 0.871339
\(339\) −7470.69 −1.19691
\(340\) −12.4590 −0.00198730
\(341\) −1433.71 −0.227683
\(342\) 848.968 0.134231
\(343\) 20022.6 3.15195
\(344\) −5699.41 −0.893290
\(345\) 1607.86 0.250911
\(346\) 8074.02 1.25452
\(347\) −6518.77 −1.00849 −0.504245 0.863561i \(-0.668229\pi\)
−0.504245 + 0.863561i \(0.668229\pi\)
\(348\) 38.0247 0.00585729
\(349\) −4855.11 −0.744665 −0.372333 0.928099i \(-0.621442\pi\)
−0.372333 + 0.928099i \(0.621442\pi\)
\(350\) −2490.96 −0.380422
\(351\) −2368.73 −0.360210
\(352\) 34.7239 0.00525792
\(353\) 4649.09 0.700981 0.350491 0.936566i \(-0.386015\pi\)
0.350491 + 0.936566i \(0.386015\pi\)
\(354\) −465.781 −0.0699321
\(355\) −1548.26 −0.231473
\(356\) −0.0463350 −6.89818e−6 0
\(357\) −4217.15 −0.625197
\(358\) 12216.9 1.80358
\(359\) −2432.15 −0.357559 −0.178780 0.983889i \(-0.557215\pi\)
−0.178780 + 0.983889i \(0.557215\pi\)
\(360\) 1802.89 0.263946
\(361\) 361.000 0.0526316
\(362\) −776.071 −0.112678
\(363\) 403.731 0.0583756
\(364\) −40.8746 −0.00588575
\(365\) −213.376 −0.0305990
\(366\) −3387.39 −0.483775
\(367\) −7051.07 −1.00290 −0.501448 0.865188i \(-0.667199\pi\)
−0.501448 + 0.865188i \(0.667199\pi\)
\(368\) 6113.85 0.866050
\(369\) 5011.41 0.707001
\(370\) 102.362 0.0143826
\(371\) 17835.6 2.49590
\(372\) −30.3360 −0.00422809
\(373\) −6658.95 −0.924363 −0.462182 0.886785i \(-0.652933\pi\)
−0.462182 + 0.886785i \(0.652933\pi\)
\(374\) −1106.53 −0.152988
\(375\) −417.077 −0.0574340
\(376\) −6372.01 −0.873967
\(377\) −2705.60 −0.369617
\(378\) 14251.4 1.93918
\(379\) 9402.70 1.27437 0.637183 0.770713i \(-0.280100\pi\)
0.637183 + 0.770713i \(0.280100\pi\)
\(380\) 6.62684 0.000894604 0
\(381\) −89.5068 −0.0120356
\(382\) −2058.73 −0.275743
\(383\) 8799.50 1.17398 0.586989 0.809595i \(-0.300313\pi\)
0.586989 + 0.809595i \(0.300313\pi\)
\(384\) −4767.78 −0.633606
\(385\) 1946.02 0.257606
\(386\) −4995.18 −0.658673
\(387\) 3979.43 0.522702
\(388\) 40.0052 0.00523443
\(389\) −7090.90 −0.924223 −0.462112 0.886822i \(-0.652908\pi\)
−0.462112 + 0.886822i \(0.652908\pi\)
\(390\) 778.048 0.101020
\(391\) 3442.71 0.445282
\(392\) 20654.6 2.66127
\(393\) 3582.24 0.459796
\(394\) −2986.00 −0.381809
\(395\) 3857.56 0.491379
\(396\) −12.1750 −0.00154500
\(397\) −1343.40 −0.169832 −0.0849159 0.996388i \(-0.527062\pi\)
−0.0849159 + 0.996388i \(0.527062\pi\)
\(398\) 13336.0 1.67958
\(399\) 2243.08 0.281439
\(400\) −1585.93 −0.198241
\(401\) −1765.83 −0.219904 −0.109952 0.993937i \(-0.535070\pi\)
−0.109952 + 0.993937i \(0.535070\pi\)
\(402\) 3479.17 0.431654
\(403\) 2158.52 0.266808
\(404\) 67.7162 0.00833913
\(405\) 244.149 0.0299552
\(406\) 16278.1 1.98983
\(407\) −79.9687 −0.00973931
\(408\) −2708.56 −0.328661
\(409\) 11987.7 1.44927 0.724636 0.689131i \(-0.242008\pi\)
0.724636 + 0.689131i \(0.242008\pi\)
\(410\) −4447.11 −0.535676
\(411\) 2404.62 0.288591
\(412\) −29.6667 −0.00354750
\(413\) 1753.95 0.208973
\(414\) −4306.35 −0.511221
\(415\) 1819.53 0.215223
\(416\) −52.2784 −0.00616144
\(417\) −358.523 −0.0421029
\(418\) 588.558 0.0688692
\(419\) 14960.5 1.74432 0.872160 0.489220i \(-0.162719\pi\)
0.872160 + 0.489220i \(0.162719\pi\)
\(420\) 41.1759 0.00478376
\(421\) 9875.94 1.14329 0.571643 0.820502i \(-0.306306\pi\)
0.571643 + 0.820502i \(0.306306\pi\)
\(422\) 371.304 0.0428312
\(423\) 4449.05 0.511396
\(424\) 11455.3 1.31208
\(425\) −893.036 −0.101926
\(426\) −2909.53 −0.330909
\(427\) 12755.6 1.44563
\(428\) −63.1577 −0.00713281
\(429\) −607.835 −0.0684069
\(430\) −3531.34 −0.396038
\(431\) 11892.2 1.32907 0.664535 0.747257i \(-0.268630\pi\)
0.664535 + 0.747257i \(0.268630\pi\)
\(432\) 9073.45 1.01052
\(433\) −2273.84 −0.252364 −0.126182 0.992007i \(-0.540272\pi\)
−0.126182 + 0.992007i \(0.540272\pi\)
\(434\) −12986.6 −1.43636
\(435\) 2725.54 0.300413
\(436\) 57.6995 0.00633786
\(437\) −1831.15 −0.200449
\(438\) −400.983 −0.0437436
\(439\) −10428.2 −1.13374 −0.566870 0.823808i \(-0.691846\pi\)
−0.566870 + 0.823808i \(0.691846\pi\)
\(440\) 1249.87 0.135421
\(441\) −14421.4 −1.55722
\(442\) 1665.94 0.179277
\(443\) 3826.67 0.410407 0.205204 0.978719i \(-0.434214\pi\)
0.205204 + 0.978719i \(0.434214\pi\)
\(444\) −1.69206 −0.000180860 0
\(445\) −3.32121 −0.000353799 0
\(446\) −42.7633 −0.00454013
\(447\) 6543.40 0.692376
\(448\) 18270.8 1.92682
\(449\) −6096.21 −0.640753 −0.320376 0.947290i \(-0.603809\pi\)
−0.320376 + 0.947290i \(0.603809\pi\)
\(450\) 1117.06 0.117020
\(451\) 3474.22 0.362738
\(452\) 156.184 0.0162528
\(453\) 3011.77 0.312374
\(454\) 3172.07 0.327913
\(455\) −2929.82 −0.301873
\(456\) 1440.67 0.147950
\(457\) −199.059 −0.0203755 −0.0101877 0.999948i \(-0.503243\pi\)
−0.0101877 + 0.999948i \(0.503243\pi\)
\(458\) 9104.16 0.928842
\(459\) 5109.26 0.519564
\(460\) −33.6143 −0.00340712
\(461\) 5245.95 0.529996 0.264998 0.964249i \(-0.414629\pi\)
0.264998 + 0.964249i \(0.414629\pi\)
\(462\) 3657.01 0.368267
\(463\) 14613.6 1.46685 0.733427 0.679769i \(-0.237920\pi\)
0.733427 + 0.679769i \(0.237920\pi\)
\(464\) 10363.8 1.03691
\(465\) −2174.43 −0.216853
\(466\) 8465.14 0.841503
\(467\) −1392.53 −0.137984 −0.0689920 0.997617i \(-0.521978\pi\)
−0.0689920 + 0.997617i \(0.521978\pi\)
\(468\) 18.3301 0.00181049
\(469\) −13101.2 −1.28988
\(470\) −3948.08 −0.387471
\(471\) −8744.25 −0.855443
\(472\) 1126.51 0.109856
\(473\) 2758.79 0.268181
\(474\) 7249.23 0.702464
\(475\) 475.000 0.0458831
\(476\) 88.1649 0.00848956
\(477\) −7998.31 −0.767752
\(478\) 2600.33 0.248820
\(479\) −6679.96 −0.637192 −0.318596 0.947891i \(-0.603211\pi\)
−0.318596 + 0.947891i \(0.603211\pi\)
\(480\) 52.6637 0.00500783
\(481\) 120.396 0.0114129
\(482\) 16741.0 1.58202
\(483\) −11377.9 −1.07187
\(484\) −8.44050 −0.000792684 0
\(485\) 2867.50 0.268467
\(486\) −10416.4 −0.972214
\(487\) −2320.34 −0.215903 −0.107951 0.994156i \(-0.534429\pi\)
−0.107951 + 0.994156i \(0.534429\pi\)
\(488\) 8192.55 0.759957
\(489\) 2959.78 0.273714
\(490\) 12797.6 1.17987
\(491\) −8455.23 −0.777147 −0.388573 0.921418i \(-0.627032\pi\)
−0.388573 + 0.921418i \(0.627032\pi\)
\(492\) 73.5113 0.00673607
\(493\) 5835.87 0.533133
\(494\) −886.102 −0.0807037
\(495\) −872.685 −0.0792409
\(496\) −8268.23 −0.748497
\(497\) 10956.1 0.988832
\(498\) 3419.32 0.307678
\(499\) 13473.2 1.20870 0.604352 0.796718i \(-0.293432\pi\)
0.604352 + 0.796718i \(0.293432\pi\)
\(500\) 8.71953 0.000779898 0
\(501\) −9794.44 −0.873420
\(502\) 10263.0 0.912470
\(503\) 4812.36 0.426585 0.213293 0.976988i \(-0.431581\pi\)
0.213293 + 0.976988i \(0.431581\pi\)
\(504\) −12758.0 −1.12755
\(505\) 4853.78 0.427703
\(506\) −2985.43 −0.262290
\(507\) −6415.42 −0.561970
\(508\) 1.87125 0.000163432 0
\(509\) 8693.16 0.757009 0.378505 0.925599i \(-0.376438\pi\)
0.378505 + 0.925599i \(0.376438\pi\)
\(510\) −1678.22 −0.145711
\(511\) 1509.94 0.130716
\(512\) 11733.1 1.01276
\(513\) −2717.58 −0.233887
\(514\) −9836.33 −0.844090
\(515\) −2126.45 −0.181947
\(516\) 58.3735 0.00498013
\(517\) 3084.37 0.262380
\(518\) −724.360 −0.0614412
\(519\) −9566.50 −0.809100
\(520\) −1881.74 −0.158692
\(521\) −4186.69 −0.352058 −0.176029 0.984385i \(-0.556325\pi\)
−0.176029 + 0.984385i \(0.556325\pi\)
\(522\) −7299.86 −0.612081
\(523\) 3761.19 0.314465 0.157233 0.987562i \(-0.449743\pi\)
0.157233 + 0.987562i \(0.449743\pi\)
\(524\) −74.8912 −0.00624359
\(525\) 2951.42 0.245353
\(526\) 11628.2 0.963904
\(527\) −4655.84 −0.384842
\(528\) 2328.32 0.191907
\(529\) −2878.56 −0.236587
\(530\) 7097.69 0.581706
\(531\) −786.551 −0.0642814
\(532\) −46.8943 −0.00382167
\(533\) −5230.60 −0.425071
\(534\) −6.24132 −0.000505783 0
\(535\) −4527.03 −0.365833
\(536\) −8414.52 −0.678082
\(537\) −14475.2 −1.16322
\(538\) 18338.7 1.46959
\(539\) −9997.85 −0.798957
\(540\) −49.8864 −0.00397550
\(541\) −1941.09 −0.154258 −0.0771292 0.997021i \(-0.524575\pi\)
−0.0771292 + 0.997021i \(0.524575\pi\)
\(542\) −7532.92 −0.596987
\(543\) 919.527 0.0726716
\(544\) 112.762 0.00888722
\(545\) 4135.80 0.325061
\(546\) −5505.80 −0.431550
\(547\) −11815.2 −0.923551 −0.461776 0.886997i \(-0.652787\pi\)
−0.461776 + 0.886997i \(0.652787\pi\)
\(548\) −50.2716 −0.00391879
\(549\) −5720.18 −0.444684
\(550\) 774.419 0.0600388
\(551\) −3104.06 −0.239995
\(552\) −7307.71 −0.563473
\(553\) −27297.7 −2.09913
\(554\) 14487.0 1.11100
\(555\) −121.284 −0.00927607
\(556\) 7.49537 0.000571717 0
\(557\) −6769.46 −0.514957 −0.257479 0.966284i \(-0.582892\pi\)
−0.257479 + 0.966284i \(0.582892\pi\)
\(558\) 5823.81 0.441831
\(559\) −4153.49 −0.314265
\(560\) 11222.7 0.846867
\(561\) 1311.08 0.0986697
\(562\) −18566.8 −1.39358
\(563\) −6761.89 −0.506180 −0.253090 0.967443i \(-0.581447\pi\)
−0.253090 + 0.967443i \(0.581447\pi\)
\(564\) 65.2623 0.00487241
\(565\) 11195.0 0.833588
\(566\) −18986.4 −1.41000
\(567\) −1727.70 −0.127966
\(568\) 7036.83 0.519822
\(569\) −11385.7 −0.838860 −0.419430 0.907788i \(-0.637770\pi\)
−0.419430 + 0.907788i \(0.637770\pi\)
\(570\) 892.633 0.0655935
\(571\) 17478.8 1.28102 0.640512 0.767948i \(-0.278722\pi\)
0.640512 + 0.767948i \(0.278722\pi\)
\(572\) 12.7076 0.000928898 0
\(573\) 2439.29 0.177841
\(574\) 31469.7 2.28836
\(575\) −2409.41 −0.174747
\(576\) −8193.50 −0.592701
\(577\) −5534.31 −0.399300 −0.199650 0.979867i \(-0.563981\pi\)
−0.199650 + 0.979867i \(0.563981\pi\)
\(578\) 10242.0 0.737042
\(579\) 5918.53 0.424811
\(580\) −56.9809 −0.00407932
\(581\) −12875.8 −0.919412
\(582\) 5388.69 0.383795
\(583\) −5544.94 −0.393907
\(584\) 969.796 0.0687165
\(585\) 1313.87 0.0928576
\(586\) −1447.75 −0.102058
\(587\) 25907.1 1.82163 0.910817 0.412810i \(-0.135453\pi\)
0.910817 + 0.412810i \(0.135453\pi\)
\(588\) −211.545 −0.0148367
\(589\) 2476.41 0.173241
\(590\) 697.984 0.0487043
\(591\) 3537.96 0.246248
\(592\) −461.180 −0.0320175
\(593\) −25291.4 −1.75142 −0.875712 0.482835i \(-0.839607\pi\)
−0.875712 + 0.482835i \(0.839607\pi\)
\(594\) −4430.63 −0.306045
\(595\) 6319.50 0.435419
\(596\) −136.798 −0.00940179
\(597\) −15801.1 −1.08325
\(598\) 4494.71 0.307362
\(599\) 21055.0 1.43620 0.718099 0.695941i \(-0.245012\pi\)
0.718099 + 0.695941i \(0.245012\pi\)
\(600\) 1895.61 0.128980
\(601\) 28962.6 1.96574 0.982870 0.184300i \(-0.0590017\pi\)
0.982870 + 0.184300i \(0.0590017\pi\)
\(602\) 24989.3 1.69184
\(603\) 5875.17 0.396775
\(604\) −62.9649 −0.00424173
\(605\) −605.000 −0.0406558
\(606\) 9121.35 0.611435
\(607\) −7890.64 −0.527629 −0.263815 0.964573i \(-0.584981\pi\)
−0.263815 + 0.964573i \(0.584981\pi\)
\(608\) −59.9776 −0.00400068
\(609\) −19287.1 −1.28334
\(610\) 5076.08 0.336925
\(611\) −4643.65 −0.307467
\(612\) −39.5372 −0.00261144
\(613\) −5050.70 −0.332783 −0.166391 0.986060i \(-0.553212\pi\)
−0.166391 + 0.986060i \(0.553212\pi\)
\(614\) 8130.63 0.534406
\(615\) 5269.16 0.345485
\(616\) −8844.64 −0.578508
\(617\) 23340.9 1.52297 0.761484 0.648184i \(-0.224471\pi\)
0.761484 + 0.648184i \(0.224471\pi\)
\(618\) −3996.09 −0.260107
\(619\) −5647.72 −0.366722 −0.183361 0.983046i \(-0.558698\pi\)
−0.183361 + 0.983046i \(0.558698\pi\)
\(620\) 45.4593 0.00294466
\(621\) 13784.8 0.890765
\(622\) −18772.0 −1.21011
\(623\) 23.5023 0.00151140
\(624\) −3505.39 −0.224884
\(625\) 625.000 0.0400000
\(626\) −19531.9 −1.24705
\(627\) −697.353 −0.0444172
\(628\) 182.810 0.0116161
\(629\) −259.690 −0.0164619
\(630\) −7904.82 −0.499898
\(631\) −9535.77 −0.601605 −0.300803 0.953686i \(-0.597255\pi\)
−0.300803 + 0.953686i \(0.597255\pi\)
\(632\) −17532.6 −1.10349
\(633\) −439.939 −0.0276240
\(634\) 7175.94 0.449516
\(635\) 134.128 0.00838223
\(636\) −117.326 −0.00731488
\(637\) 15052.2 0.936249
\(638\) −5060.72 −0.314038
\(639\) −4913.24 −0.304170
\(640\) 7144.63 0.441275
\(641\) −8928.98 −0.550193 −0.275096 0.961417i \(-0.588710\pi\)
−0.275096 + 0.961417i \(0.588710\pi\)
\(642\) −8507.32 −0.522986
\(643\) 14311.6 0.877753 0.438877 0.898547i \(-0.355377\pi\)
0.438877 + 0.898547i \(0.355377\pi\)
\(644\) 237.869 0.0145549
\(645\) 4184.11 0.255425
\(646\) 1911.29 0.116406
\(647\) −25100.4 −1.52519 −0.762595 0.646876i \(-0.776075\pi\)
−0.762595 + 0.646876i \(0.776075\pi\)
\(648\) −1109.66 −0.0672707
\(649\) −545.287 −0.0329805
\(650\) −1165.92 −0.0703558
\(651\) 15387.2 0.926378
\(652\) −61.8780 −0.00371676
\(653\) −25852.1 −1.54927 −0.774635 0.632409i \(-0.782066\pi\)
−0.774635 + 0.632409i \(0.782066\pi\)
\(654\) 7772.11 0.464699
\(655\) −5368.07 −0.320226
\(656\) 20035.9 1.19248
\(657\) −677.129 −0.0402090
\(658\) 27938.3 1.65524
\(659\) −31004.7 −1.83274 −0.916368 0.400336i \(-0.868893\pi\)
−0.916368 + 0.400336i \(0.868893\pi\)
\(660\) −12.8012 −0.000754981 0
\(661\) 6075.32 0.357493 0.178746 0.983895i \(-0.442796\pi\)
0.178746 + 0.983895i \(0.442796\pi\)
\(662\) −7831.26 −0.459774
\(663\) −1973.89 −0.115625
\(664\) −8269.78 −0.483328
\(665\) −3361.30 −0.196009
\(666\) 324.837 0.0188996
\(667\) 15745.2 0.914028
\(668\) 204.765 0.0118602
\(669\) 50.6680 0.00292816
\(670\) −5213.61 −0.300626
\(671\) −3965.59 −0.228152
\(672\) −372.671 −0.0213930
\(673\) 3403.29 0.194929 0.0974644 0.995239i \(-0.468927\pi\)
0.0974644 + 0.995239i \(0.468927\pi\)
\(674\) 30724.1 1.75586
\(675\) −3575.77 −0.203898
\(676\) 134.123 0.00763101
\(677\) 21049.5 1.19497 0.597486 0.801879i \(-0.296166\pi\)
0.597486 + 0.801879i \(0.296166\pi\)
\(678\) 21038.0 1.19168
\(679\) −20291.7 −1.14687
\(680\) 4058.85 0.228897
\(681\) −3758.42 −0.211487
\(682\) 4037.43 0.226688
\(683\) −18806.7 −1.05361 −0.526806 0.849985i \(-0.676611\pi\)
−0.526806 + 0.849985i \(0.676611\pi\)
\(684\) 21.0296 0.00117557
\(685\) −3603.38 −0.200990
\(686\) −56385.0 −3.13818
\(687\) −10787.1 −0.599057
\(688\) 15910.0 0.881631
\(689\) 8348.16 0.461596
\(690\) −4527.84 −0.249814
\(691\) −24254.6 −1.33529 −0.667647 0.744478i \(-0.732698\pi\)
−0.667647 + 0.744478i \(0.732698\pi\)
\(692\) 200.000 0.0109868
\(693\) 6175.49 0.338510
\(694\) 18357.3 1.00408
\(695\) 537.255 0.0293226
\(696\) −12387.6 −0.674641
\(697\) 11282.2 0.613119
\(698\) 13672.3 0.741412
\(699\) −10029.9 −0.542727
\(700\) −61.7031 −0.00333165
\(701\) 2422.32 0.130513 0.0652565 0.997869i \(-0.479213\pi\)
0.0652565 + 0.997869i \(0.479213\pi\)
\(702\) 6670.51 0.358636
\(703\) 138.128 0.00741050
\(704\) −5680.25 −0.304094
\(705\) 4677.88 0.249900
\(706\) −13092.2 −0.697918
\(707\) −34347.4 −1.82711
\(708\) −11.5378 −0.000612451 0
\(709\) −25061.8 −1.32753 −0.663764 0.747942i \(-0.731042\pi\)
−0.663764 + 0.747942i \(0.731042\pi\)
\(710\) 4360.00 0.230462
\(711\) 12241.6 0.645702
\(712\) 15.0949 0.000794530 0
\(713\) −12561.5 −0.659792
\(714\) 11875.8 0.622465
\(715\) 910.855 0.0476420
\(716\) 302.622 0.0157954
\(717\) −3080.99 −0.160477
\(718\) 6849.10 0.355997
\(719\) −12118.3 −0.628562 −0.314281 0.949330i \(-0.601763\pi\)
−0.314281 + 0.949330i \(0.601763\pi\)
\(720\) −5032.78 −0.260501
\(721\) 15047.7 0.777261
\(722\) −1016.60 −0.0524016
\(723\) −19835.6 −1.02032
\(724\) −19.2239 −0.000986809 0
\(725\) −4084.29 −0.209223
\(726\) −1136.93 −0.0581206
\(727\) 3705.97 0.189060 0.0945300 0.995522i \(-0.469865\pi\)
0.0945300 + 0.995522i \(0.469865\pi\)
\(728\) 13316.0 0.677918
\(729\) 13660.2 0.694011
\(730\) 600.883 0.0304653
\(731\) 8958.91 0.453293
\(732\) −83.9082 −0.00423680
\(733\) 24264.3 1.22268 0.611338 0.791370i \(-0.290632\pi\)
0.611338 + 0.791370i \(0.290632\pi\)
\(734\) 19856.3 0.998515
\(735\) −15163.2 −0.760955
\(736\) 304.233 0.0152367
\(737\) 4073.04 0.203572
\(738\) −14112.5 −0.703912
\(739\) 26472.4 1.31773 0.658864 0.752262i \(-0.271037\pi\)
0.658864 + 0.752262i \(0.271037\pi\)
\(740\) 2.53560 0.000125960 0
\(741\) 1049.90 0.0520498
\(742\) −50226.3 −2.48499
\(743\) 9403.97 0.464331 0.232166 0.972676i \(-0.425419\pi\)
0.232166 + 0.972676i \(0.425419\pi\)
\(744\) 9882.79 0.486990
\(745\) −9805.44 −0.482206
\(746\) 18752.1 0.920325
\(747\) 5774.11 0.282816
\(748\) −27.4097 −0.00133984
\(749\) 32035.2 1.56280
\(750\) 1174.52 0.0571831
\(751\) −6747.81 −0.327871 −0.163935 0.986471i \(-0.552419\pi\)
−0.163935 + 0.986471i \(0.552419\pi\)
\(752\) 17787.6 0.862560
\(753\) −12160.1 −0.588498
\(754\) 7619.15 0.368002
\(755\) −4513.21 −0.217553
\(756\) 353.017 0.0169830
\(757\) 22692.4 1.08952 0.544762 0.838591i \(-0.316620\pi\)
0.544762 + 0.838591i \(0.316620\pi\)
\(758\) −26478.7 −1.26880
\(759\) 3537.29 0.169164
\(760\) −2158.87 −0.103040
\(761\) 36689.5 1.74769 0.873846 0.486203i \(-0.161618\pi\)
0.873846 + 0.486203i \(0.161618\pi\)
\(762\) 252.057 0.0119830
\(763\) −29266.7 −1.38863
\(764\) −50.9964 −0.00241490
\(765\) −2833.96 −0.133937
\(766\) −24780.0 −1.16885
\(767\) 820.954 0.0386479
\(768\) −357.476 −0.0167960
\(769\) 30205.0 1.41641 0.708206 0.706006i \(-0.249505\pi\)
0.708206 + 0.706006i \(0.249505\pi\)
\(770\) −5480.12 −0.256480
\(771\) 11654.6 0.544396
\(772\) −123.734 −0.00576852
\(773\) −29371.6 −1.36665 −0.683327 0.730113i \(-0.739468\pi\)
−0.683327 + 0.730113i \(0.739468\pi\)
\(774\) −11206.3 −0.520418
\(775\) 3258.44 0.151028
\(776\) −13032.8 −0.602900
\(777\) 858.257 0.0396265
\(778\) 19968.5 0.920185
\(779\) −6000.93 −0.276002
\(780\) 19.2728 0.000884716 0
\(781\) −3406.17 −0.156059
\(782\) −9694.91 −0.443337
\(783\) 23367.2 1.06651
\(784\) −57657.7 −2.62653
\(785\) 13103.5 0.595774
\(786\) −10087.8 −0.457787
\(787\) −33921.9 −1.53645 −0.768224 0.640181i \(-0.778859\pi\)
−0.768224 + 0.640181i \(0.778859\pi\)
\(788\) −73.9656 −0.00334380
\(789\) −13777.7 −0.621670
\(790\) −10863.1 −0.489232
\(791\) −79220.6 −3.56101
\(792\) 3966.35 0.177952
\(793\) 5970.38 0.267357
\(794\) 3783.10 0.169090
\(795\) −8409.69 −0.375171
\(796\) 330.343 0.0147094
\(797\) −24073.9 −1.06994 −0.534969 0.844872i \(-0.679677\pi\)
−0.534969 + 0.844872i \(0.679677\pi\)
\(798\) −6316.66 −0.280209
\(799\) 10016.2 0.443488
\(800\) −78.9179 −0.00348771
\(801\) −10.5395 −0.000464914 0
\(802\) 4972.71 0.218943
\(803\) −469.428 −0.0206298
\(804\) 86.1816 0.00378034
\(805\) 17050.0 0.746503
\(806\) −6078.54 −0.265642
\(807\) −21728.6 −0.947809
\(808\) −22060.4 −0.960498
\(809\) −670.425 −0.0291358 −0.0145679 0.999894i \(-0.504637\pi\)
−0.0145679 + 0.999894i \(0.504637\pi\)
\(810\) −687.541 −0.0298243
\(811\) 10775.0 0.466536 0.233268 0.972412i \(-0.425058\pi\)
0.233268 + 0.972412i \(0.425058\pi\)
\(812\) 403.221 0.0174265
\(813\) 8925.38 0.385027
\(814\) 225.197 0.00969676
\(815\) −4435.31 −0.190628
\(816\) 7560.99 0.324372
\(817\) −4765.19 −0.204055
\(818\) −33758.1 −1.44294
\(819\) −9297.48 −0.396679
\(820\) −110.158 −0.00469134
\(821\) −28550.6 −1.21367 −0.606835 0.794828i \(-0.707561\pi\)
−0.606835 + 0.794828i \(0.707561\pi\)
\(822\) −6771.57 −0.287330
\(823\) 31286.3 1.32512 0.662558 0.749011i \(-0.269471\pi\)
0.662558 + 0.749011i \(0.269471\pi\)
\(824\) 9664.72 0.408600
\(825\) −917.569 −0.0387220
\(826\) −4939.23 −0.208060
\(827\) 24268.3 1.02043 0.510213 0.860048i \(-0.329566\pi\)
0.510213 + 0.860048i \(0.329566\pi\)
\(828\) −106.672 −0.00447717
\(829\) −2245.33 −0.0940696 −0.0470348 0.998893i \(-0.514977\pi\)
−0.0470348 + 0.998893i \(0.514977\pi\)
\(830\) −5123.94 −0.214282
\(831\) −17164.9 −0.716540
\(832\) 8551.88 0.356350
\(833\) −32467.1 −1.35044
\(834\) 1009.62 0.0419190
\(835\) 14677.2 0.608294
\(836\) 14.5790 0.000603142 0
\(837\) −18642.3 −0.769858
\(838\) −42129.9 −1.73670
\(839\) −1461.78 −0.0601506 −0.0300753 0.999548i \(-0.509575\pi\)
−0.0300753 + 0.999548i \(0.509575\pi\)
\(840\) −13414.2 −0.550992
\(841\) 2301.29 0.0943579
\(842\) −27811.3 −1.13829
\(843\) 21998.8 0.898791
\(844\) 9.19748 0.000375107 0
\(845\) 9613.67 0.391385
\(846\) −12528.8 −0.509161
\(847\) 4281.24 0.173678
\(848\) −31977.7 −1.29495
\(849\) 22496.0 0.909378
\(850\) 2514.85 0.101481
\(851\) −700.646 −0.0282231
\(852\) −72.0713 −0.00289803
\(853\) −6468.16 −0.259631 −0.129816 0.991538i \(-0.541439\pi\)
−0.129816 + 0.991538i \(0.541439\pi\)
\(854\) −35920.5 −1.43932
\(855\) 1507.36 0.0602933
\(856\) 20575.3 0.821555
\(857\) 22875.3 0.911790 0.455895 0.890034i \(-0.349319\pi\)
0.455895 + 0.890034i \(0.349319\pi\)
\(858\) 1711.71 0.0681080
\(859\) 29145.0 1.15764 0.578821 0.815455i \(-0.303513\pi\)
0.578821 + 0.815455i \(0.303513\pi\)
\(860\) −87.4740 −0.00346842
\(861\) −37286.8 −1.47588
\(862\) −33489.4 −1.32326
\(863\) 43299.0 1.70790 0.853949 0.520357i \(-0.174201\pi\)
0.853949 + 0.520357i \(0.174201\pi\)
\(864\) 451.507 0.0177785
\(865\) 14335.6 0.563498
\(866\) 6403.28 0.251261
\(867\) −12135.2 −0.475355
\(868\) −321.689 −0.0125793
\(869\) 8486.62 0.331288
\(870\) −7675.32 −0.299101
\(871\) −6132.15 −0.238553
\(872\) −18797.2 −0.729992
\(873\) 9099.73 0.352783
\(874\) 5156.66 0.199573
\(875\) −4422.77 −0.170876
\(876\) −9.93266 −0.000383098 0
\(877\) 28197.3 1.08569 0.542847 0.839832i \(-0.317346\pi\)
0.542847 + 0.839832i \(0.317346\pi\)
\(878\) 29366.6 1.12879
\(879\) 1715.36 0.0658222
\(880\) −3489.04 −0.133654
\(881\) −36316.5 −1.38880 −0.694402 0.719588i \(-0.744331\pi\)
−0.694402 + 0.719588i \(0.744331\pi\)
\(882\) 40611.8 1.55042
\(883\) −16385.4 −0.624478 −0.312239 0.950004i \(-0.601079\pi\)
−0.312239 + 0.950004i \(0.601079\pi\)
\(884\) 41.2666 0.00157007
\(885\) −827.006 −0.0314119
\(886\) −10776.2 −0.408614
\(887\) 19539.7 0.739661 0.369830 0.929099i \(-0.379416\pi\)
0.369830 + 0.929099i \(0.379416\pi\)
\(888\) 551.235 0.0208314
\(889\) −949.148 −0.0358081
\(890\) 9.35277 0.000352253 0
\(891\) 537.128 0.0201958
\(892\) −1.05928 −3.97615e−5 0
\(893\) −5327.54 −0.199641
\(894\) −18426.7 −0.689351
\(895\) 21691.4 0.810126
\(896\) −50558.4 −1.88509
\(897\) −5325.55 −0.198233
\(898\) 17167.4 0.637953
\(899\) −21293.5 −0.789963
\(900\) 27.6705 0.00102483
\(901\) −18006.6 −0.665803
\(902\) −9783.65 −0.361153
\(903\) −29608.5 −1.09115
\(904\) −50881.3 −1.87200
\(905\) −1377.93 −0.0506122
\(906\) −8481.35 −0.311009
\(907\) 44433.5 1.62667 0.813336 0.581794i \(-0.197649\pi\)
0.813336 + 0.581794i \(0.197649\pi\)
\(908\) 78.5745 0.00287179
\(909\) 15403.0 0.562029
\(910\) 8250.57 0.300554
\(911\) −7675.35 −0.279139 −0.139570 0.990212i \(-0.544572\pi\)
−0.139570 + 0.990212i \(0.544572\pi\)
\(912\) −4021.64 −0.146019
\(913\) 4002.98 0.145103
\(914\) 560.564 0.0202865
\(915\) −6014.39 −0.217300
\(916\) 225.517 0.00813460
\(917\) 37986.8 1.36798
\(918\) −14388.0 −0.517294
\(919\) −19102.2 −0.685661 −0.342831 0.939397i \(-0.611386\pi\)
−0.342831 + 0.939397i \(0.611386\pi\)
\(920\) 10950.8 0.392431
\(921\) −9633.56 −0.344665
\(922\) −14772.9 −0.527680
\(923\) 5128.14 0.182876
\(924\) 90.5870 0.00322521
\(925\) 181.747 0.00646033
\(926\) −41153.0 −1.46044
\(927\) −6748.08 −0.239090
\(928\) 515.718 0.0182427
\(929\) −37268.3 −1.31618 −0.658091 0.752939i \(-0.728636\pi\)
−0.658091 + 0.752939i \(0.728636\pi\)
\(930\) 6123.35 0.215906
\(931\) 17269.0 0.607915
\(932\) 209.688 0.00736970
\(933\) 22242.0 0.780461
\(934\) 3921.45 0.137381
\(935\) −1964.68 −0.0687186
\(936\) −5971.52 −0.208531
\(937\) −21507.6 −0.749863 −0.374931 0.927053i \(-0.622334\pi\)
−0.374931 + 0.927053i \(0.622334\pi\)
\(938\) 36893.8 1.28425
\(939\) 23142.3 0.804283
\(940\) −97.7971 −0.00339339
\(941\) 11451.4 0.396710 0.198355 0.980130i \(-0.436440\pi\)
0.198355 + 0.980130i \(0.436440\pi\)
\(942\) 24624.4 0.851706
\(943\) 30439.4 1.05116
\(944\) −3144.67 −0.108422
\(945\) 25303.7 0.871035
\(946\) −7768.95 −0.267009
\(947\) 16012.3 0.549450 0.274725 0.961523i \(-0.411413\pi\)
0.274725 + 0.961523i \(0.411413\pi\)
\(948\) 179.569 0.00615204
\(949\) 706.746 0.0241749
\(950\) −1337.63 −0.0456827
\(951\) −8502.40 −0.289915
\(952\) −28722.1 −0.977825
\(953\) −24428.9 −0.830357 −0.415179 0.909740i \(-0.636281\pi\)
−0.415179 + 0.909740i \(0.636281\pi\)
\(954\) 22523.8 0.764397
\(955\) −3655.33 −0.123857
\(956\) 64.4121 0.00217912
\(957\) 5996.19 0.202539
\(958\) 18811.2 0.634408
\(959\) 25499.0 0.858610
\(960\) −8614.91 −0.289630
\(961\) −12803.1 −0.429765
\(962\) −339.045 −0.0113630
\(963\) −14366.1 −0.480727
\(964\) 414.688 0.0138550
\(965\) −8869.06 −0.295860
\(966\) 32040.9 1.06718
\(967\) 22318.7 0.742213 0.371107 0.928590i \(-0.378978\pi\)
0.371107 + 0.928590i \(0.378978\pi\)
\(968\) 2749.72 0.0913011
\(969\) −2264.59 −0.0750764
\(970\) −8075.09 −0.267294
\(971\) −19197.0 −0.634459 −0.317230 0.948349i \(-0.602753\pi\)
−0.317230 + 0.948349i \(0.602753\pi\)
\(972\) −258.021 −0.00851444
\(973\) −3801.84 −0.125264
\(974\) 6534.24 0.214960
\(975\) 1381.44 0.0453760
\(976\) −22869.6 −0.750039
\(977\) −32172.6 −1.05353 −0.526763 0.850012i \(-0.676594\pi\)
−0.526763 + 0.850012i \(0.676594\pi\)
\(978\) −8334.95 −0.272518
\(979\) −7.30667 −0.000238531 0
\(980\) 317.005 0.0103330
\(981\) 13124.5 0.427150
\(982\) 23810.5 0.773751
\(983\) 44722.4 1.45109 0.725546 0.688173i \(-0.241587\pi\)
0.725546 + 0.688173i \(0.241587\pi\)
\(984\) −23948.3 −0.775858
\(985\) −5301.72 −0.171499
\(986\) −16434.2 −0.530803
\(987\) −33102.7 −1.06755
\(988\) −21.9494 −0.000706786 0
\(989\) 24171.2 0.777147
\(990\) 2457.54 0.0788947
\(991\) −39887.1 −1.27856 −0.639282 0.768973i \(-0.720768\pi\)
−0.639282 + 0.768973i \(0.720768\pi\)
\(992\) −411.438 −0.0131685
\(993\) 9278.86 0.296531
\(994\) −30853.2 −0.984512
\(995\) 23678.4 0.754428
\(996\) 84.6992 0.00269458
\(997\) −268.045 −0.00851462 −0.00425731 0.999991i \(-0.501355\pi\)
−0.00425731 + 0.999991i \(0.501355\pi\)
\(998\) −37941.4 −1.20342
\(999\) −1039.82 −0.0329312
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 1045.4.a.c.1.5 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.4.a.c.1.5 20 1.1 even 1 trivial