Properties

Label 147.4.a.l.1.1
Level $147$
Weight $4$
Character 147.1
Self dual yes
Analytic conductor $8.673$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.57516.1
Defining polynomial: \(x^{3} - x^{2} - 24 x + 6\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.55637\) of defining polynomial
Character \(\chi\) \(=\) 147.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.55637 q^{2} -3.00000 q^{3} +12.7605 q^{4} +17.8732 q^{5} +13.6691 q^{6} -21.6905 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.55637 q^{2} -3.00000 q^{3} +12.7605 q^{4} +17.8732 q^{5} +13.6691 q^{6} -21.6905 q^{8} +9.00000 q^{9} -81.4369 q^{10} -11.3942 q^{11} -38.2814 q^{12} -13.0987 q^{13} -53.6196 q^{15} -3.25412 q^{16} +53.2674 q^{17} -41.0073 q^{18} -42.4223 q^{19} +228.071 q^{20} +51.9159 q^{22} +152.085 q^{23} +65.0714 q^{24} +194.451 q^{25} +59.6823 q^{26} -27.0000 q^{27} +186.493 q^{29} +244.311 q^{30} -157.874 q^{31} +188.351 q^{32} +34.1825 q^{33} -242.706 q^{34} +114.844 q^{36} +3.74588 q^{37} +193.291 q^{38} +39.2960 q^{39} -387.678 q^{40} -39.3230 q^{41} +429.439 q^{43} -145.395 q^{44} +160.859 q^{45} -692.957 q^{46} +21.1869 q^{47} +9.76236 q^{48} -885.992 q^{50} -159.802 q^{51} -167.145 q^{52} +365.904 q^{53} +123.022 q^{54} -203.650 q^{55} +127.267 q^{57} -849.732 q^{58} -226.578 q^{59} -684.212 q^{60} +651.973 q^{61} +719.331 q^{62} -832.161 q^{64} -234.115 q^{65} -155.748 q^{66} +145.433 q^{67} +679.717 q^{68} -456.256 q^{69} -368.962 q^{71} -195.214 q^{72} +608.906 q^{73} -17.0676 q^{74} -583.354 q^{75} -541.328 q^{76} -179.047 q^{78} +910.237 q^{79} -58.1615 q^{80} +81.0000 q^{81} +179.170 q^{82} -327.929 q^{83} +952.058 q^{85} -1956.68 q^{86} -559.480 q^{87} +247.144 q^{88} -37.6118 q^{89} -732.932 q^{90} +1940.68 q^{92} +473.621 q^{93} -96.5352 q^{94} -758.222 q^{95} -565.052 q^{96} +722.013 q^{97} -102.547 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} - 9q^{3} + 25q^{4} + 11q^{5} - 3q^{6} + 39q^{8} + 27q^{9} + O(q^{10}) \) \( 3q + q^{2} - 9q^{3} + 25q^{4} + 11q^{5} - 3q^{6} + 39q^{8} + 27q^{9} - 55q^{10} + 35q^{11} - 75q^{12} + 62q^{13} - 33q^{15} + 241q^{16} + 48q^{17} + 9q^{18} - 202q^{19} + 439q^{20} - 7q^{22} + 216q^{23} - 117q^{24} + 130q^{25} + 274q^{26} - 81q^{27} + 53q^{29} + 165q^{30} - 95q^{31} + 683q^{32} - 105q^{33} - 24q^{34} + 225q^{36} + 262q^{37} - 398q^{38} - 186q^{39} + 21q^{40} + 244q^{41} + 360q^{43} - 905q^{44} + 99q^{45} - 1056q^{46} - 210q^{47} - 723q^{48} - 1378q^{50} - 144q^{51} + 324q^{52} + 393q^{53} - 27q^{54} - 1031q^{55} + 606q^{57} - 1249q^{58} + 1143q^{59} - 1317q^{60} - 70q^{61} + 1059q^{62} - 399q^{64} - 472q^{65} + 21q^{66} - 628q^{67} + 1944q^{68} - 648q^{69} + 318q^{71} + 351q^{72} + 988q^{73} + 1002q^{74} - 390q^{75} - 2340q^{76} - 822q^{78} + 861q^{79} + 175q^{80} + 243q^{81} + 124q^{82} + 519q^{83} + 1800q^{85} - 3208q^{86} - 159q^{87} - 891q^{88} + 1766q^{89} - 495q^{90} - 672q^{92} + 285q^{93} - 3294q^{94} - 736q^{95} - 2049q^{96} + 19q^{97} + 315q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.55637 −1.61092 −0.805459 0.592651i \(-0.798081\pi\)
−0.805459 + 0.592651i \(0.798081\pi\)
\(3\) −3.00000 −0.577350
\(4\) 12.7605 1.59506
\(5\) 17.8732 1.59863 0.799314 0.600914i \(-0.205196\pi\)
0.799314 + 0.600914i \(0.205196\pi\)
\(6\) 13.6691 0.930064
\(7\) 0 0
\(8\) −21.6905 −0.958592
\(9\) 9.00000 0.333333
\(10\) −81.4369 −2.57526
\(11\) −11.3942 −0.312315 −0.156158 0.987732i \(-0.549911\pi\)
−0.156158 + 0.987732i \(0.549911\pi\)
\(12\) −38.2814 −0.920908
\(13\) −13.0987 −0.279455 −0.139728 0.990190i \(-0.544623\pi\)
−0.139728 + 0.990190i \(0.544623\pi\)
\(14\) 0 0
\(15\) −53.6196 −0.922968
\(16\) −3.25412 −0.0508456
\(17\) 53.2674 0.759955 0.379977 0.924996i \(-0.375932\pi\)
0.379977 + 0.924996i \(0.375932\pi\)
\(18\) −41.0073 −0.536973
\(19\) −42.4223 −0.512228 −0.256114 0.966647i \(-0.582442\pi\)
−0.256114 + 0.966647i \(0.582442\pi\)
\(20\) 228.071 2.54991
\(21\) 0 0
\(22\) 51.9159 0.503114
\(23\) 152.085 1.37878 0.689391 0.724389i \(-0.257878\pi\)
0.689391 + 0.724389i \(0.257878\pi\)
\(24\) 65.0714 0.553443
\(25\) 194.451 1.55561
\(26\) 59.6823 0.450180
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 186.493 1.19417 0.597085 0.802178i \(-0.296325\pi\)
0.597085 + 0.802178i \(0.296325\pi\)
\(30\) 244.311 1.48683
\(31\) −157.874 −0.914676 −0.457338 0.889293i \(-0.651197\pi\)
−0.457338 + 0.889293i \(0.651197\pi\)
\(32\) 188.351 1.04050
\(33\) 34.1825 0.180315
\(34\) −242.706 −1.22423
\(35\) 0 0
\(36\) 114.844 0.531686
\(37\) 3.74588 0.0166438 0.00832188 0.999965i \(-0.497351\pi\)
0.00832188 + 0.999965i \(0.497351\pi\)
\(38\) 193.291 0.825158
\(39\) 39.2960 0.161344
\(40\) −387.678 −1.53243
\(41\) −39.3230 −0.149786 −0.0748930 0.997192i \(-0.523862\pi\)
−0.0748930 + 0.997192i \(0.523862\pi\)
\(42\) 0 0
\(43\) 429.439 1.52300 0.761498 0.648168i \(-0.224464\pi\)
0.761498 + 0.648168i \(0.224464\pi\)
\(44\) −145.395 −0.498161
\(45\) 160.859 0.532876
\(46\) −692.957 −2.22111
\(47\) 21.1869 0.0657537 0.0328768 0.999459i \(-0.489533\pi\)
0.0328768 + 0.999459i \(0.489533\pi\)
\(48\) 9.76236 0.0293557
\(49\) 0 0
\(50\) −885.992 −2.50596
\(51\) −159.802 −0.438760
\(52\) −167.145 −0.445748
\(53\) 365.904 0.948317 0.474158 0.880440i \(-0.342752\pi\)
0.474158 + 0.880440i \(0.342752\pi\)
\(54\) 123.022 0.310021
\(55\) −203.650 −0.499276
\(56\) 0 0
\(57\) 127.267 0.295735
\(58\) −849.732 −1.92371
\(59\) −226.578 −0.499964 −0.249982 0.968250i \(-0.580425\pi\)
−0.249982 + 0.968250i \(0.580425\pi\)
\(60\) −684.212 −1.47219
\(61\) 651.973 1.36847 0.684235 0.729262i \(-0.260136\pi\)
0.684235 + 0.729262i \(0.260136\pi\)
\(62\) 719.331 1.47347
\(63\) 0 0
\(64\) −832.161 −1.62532
\(65\) −234.115 −0.446745
\(66\) −155.748 −0.290473
\(67\) 145.433 0.265186 0.132593 0.991171i \(-0.457670\pi\)
0.132593 + 0.991171i \(0.457670\pi\)
\(68\) 679.717 1.21217
\(69\) −456.256 −0.796041
\(70\) 0 0
\(71\) −368.962 −0.616728 −0.308364 0.951268i \(-0.599782\pi\)
−0.308364 + 0.951268i \(0.599782\pi\)
\(72\) −195.214 −0.319531
\(73\) 608.906 0.976261 0.488130 0.872771i \(-0.337679\pi\)
0.488130 + 0.872771i \(0.337679\pi\)
\(74\) −17.0676 −0.0268117
\(75\) −583.354 −0.898133
\(76\) −541.328 −0.817034
\(77\) 0 0
\(78\) −179.047 −0.259911
\(79\) 910.237 1.29633 0.648163 0.761502i \(-0.275538\pi\)
0.648163 + 0.761502i \(0.275538\pi\)
\(80\) −58.1615 −0.0812832
\(81\) 81.0000 0.111111
\(82\) 179.170 0.241293
\(83\) −327.929 −0.433674 −0.216837 0.976208i \(-0.569574\pi\)
−0.216837 + 0.976208i \(0.569574\pi\)
\(84\) 0 0
\(85\) 952.058 1.21489
\(86\) −1956.68 −2.45342
\(87\) −559.480 −0.689455
\(88\) 247.144 0.299383
\(89\) −37.6118 −0.0447960 −0.0223980 0.999749i \(-0.507130\pi\)
−0.0223980 + 0.999749i \(0.507130\pi\)
\(90\) −732.932 −0.858420
\(91\) 0 0
\(92\) 1940.68 2.19924
\(93\) 473.621 0.528089
\(94\) −96.5352 −0.105924
\(95\) −758.222 −0.818863
\(96\) −565.052 −0.600733
\(97\) 722.013 0.755766 0.377883 0.925853i \(-0.376652\pi\)
0.377883 + 0.925853i \(0.376652\pi\)
\(98\) 0 0
\(99\) −102.547 −0.104105
\(100\) 2481.29 2.48129
\(101\) 1518.67 1.49617 0.748087 0.663601i \(-0.230973\pi\)
0.748087 + 0.663601i \(0.230973\pi\)
\(102\) 728.117 0.706807
\(103\) −1051.88 −1.00626 −0.503132 0.864210i \(-0.667819\pi\)
−0.503132 + 0.864210i \(0.667819\pi\)
\(104\) 284.116 0.267883
\(105\) 0 0
\(106\) −1667.19 −1.52766
\(107\) 766.520 0.692545 0.346273 0.938134i \(-0.387447\pi\)
0.346273 + 0.938134i \(0.387447\pi\)
\(108\) −344.533 −0.306969
\(109\) −1427.05 −1.25400 −0.627002 0.779018i \(-0.715718\pi\)
−0.627002 + 0.779018i \(0.715718\pi\)
\(110\) 927.904 0.804292
\(111\) −11.2376 −0.00960928
\(112\) 0 0
\(113\) 362.564 0.301833 0.150917 0.988546i \(-0.451778\pi\)
0.150917 + 0.988546i \(0.451778\pi\)
\(114\) −579.874 −0.476405
\(115\) 2718.25 2.20416
\(116\) 2379.74 1.90477
\(117\) −117.888 −0.0931517
\(118\) 1032.37 0.805402
\(119\) 0 0
\(120\) 1163.03 0.884750
\(121\) −1201.17 −0.902459
\(122\) −2970.63 −2.20449
\(123\) 117.969 0.0864789
\(124\) −2014.54 −1.45896
\(125\) 1241.32 0.888216
\(126\) 0 0
\(127\) 974.777 0.681082 0.340541 0.940230i \(-0.389390\pi\)
0.340541 + 0.940230i \(0.389390\pi\)
\(128\) 2284.83 1.57775
\(129\) −1288.32 −0.879302
\(130\) 1066.71 0.719670
\(131\) −1792.70 −1.19564 −0.597821 0.801629i \(-0.703967\pi\)
−0.597821 + 0.801629i \(0.703967\pi\)
\(132\) 436.184 0.287613
\(133\) 0 0
\(134\) −662.647 −0.427194
\(135\) −482.577 −0.307656
\(136\) −1155.39 −0.728486
\(137\) 1684.42 1.05043 0.525217 0.850969i \(-0.323984\pi\)
0.525217 + 0.850969i \(0.323984\pi\)
\(138\) 2078.87 1.28236
\(139\) 315.089 0.192270 0.0961350 0.995368i \(-0.469352\pi\)
0.0961350 + 0.995368i \(0.469352\pi\)
\(140\) 0 0
\(141\) −63.5606 −0.0379629
\(142\) 1681.13 0.993499
\(143\) 149.248 0.0872781
\(144\) −29.2871 −0.0169485
\(145\) 3333.23 1.90903
\(146\) −2774.40 −1.57268
\(147\) 0 0
\(148\) 47.7992 0.0265478
\(149\) 1893.77 1.04123 0.520617 0.853790i \(-0.325702\pi\)
0.520617 + 0.853790i \(0.325702\pi\)
\(150\) 2657.98 1.44682
\(151\) −2011.84 −1.08425 −0.542124 0.840299i \(-0.682380\pi\)
−0.542124 + 0.840299i \(0.682380\pi\)
\(152\) 920.159 0.491018
\(153\) 479.406 0.253318
\(154\) 0 0
\(155\) −2821.71 −1.46223
\(156\) 501.436 0.257352
\(157\) −3828.50 −1.94616 −0.973082 0.230460i \(-0.925977\pi\)
−0.973082 + 0.230460i \(0.925977\pi\)
\(158\) −4147.38 −2.08827
\(159\) −1097.71 −0.547511
\(160\) 3366.43 1.66337
\(161\) 0 0
\(162\) −369.066 −0.178991
\(163\) −3509.26 −1.68630 −0.843148 0.537682i \(-0.819300\pi\)
−0.843148 + 0.537682i \(0.819300\pi\)
\(164\) −501.780 −0.238917
\(165\) 610.950 0.288257
\(166\) 1494.17 0.698613
\(167\) −343.008 −0.158939 −0.0794694 0.996837i \(-0.525323\pi\)
−0.0794694 + 0.996837i \(0.525323\pi\)
\(168\) 0 0
\(169\) −2025.42 −0.921905
\(170\) −4337.93 −1.95708
\(171\) −381.800 −0.170743
\(172\) 5479.84 2.42927
\(173\) −4187.21 −1.84016 −0.920081 0.391729i \(-0.871877\pi\)
−0.920081 + 0.391729i \(0.871877\pi\)
\(174\) 2549.20 1.11066
\(175\) 0 0
\(176\) 37.0779 0.0158798
\(177\) 679.733 0.288654
\(178\) 171.373 0.0721627
\(179\) −1970.29 −0.822716 −0.411358 0.911474i \(-0.634945\pi\)
−0.411358 + 0.911474i \(0.634945\pi\)
\(180\) 2052.63 0.849969
\(181\) −3613.10 −1.48376 −0.741878 0.670535i \(-0.766065\pi\)
−0.741878 + 0.670535i \(0.766065\pi\)
\(182\) 0 0
\(183\) −1955.92 −0.790086
\(184\) −3298.80 −1.32169
\(185\) 66.9509 0.0266072
\(186\) −2157.99 −0.850708
\(187\) −606.936 −0.237345
\(188\) 270.355 0.104881
\(189\) 0 0
\(190\) 3454.74 1.31912
\(191\) 1907.77 0.722729 0.361365 0.932425i \(-0.382311\pi\)
0.361365 + 0.932425i \(0.382311\pi\)
\(192\) 2496.48 0.938376
\(193\) 2399.93 0.895080 0.447540 0.894264i \(-0.352300\pi\)
0.447540 + 0.894264i \(0.352300\pi\)
\(194\) −3289.75 −1.21748
\(195\) 702.346 0.257928
\(196\) 0 0
\(197\) 1514.32 0.547668 0.273834 0.961777i \(-0.411708\pi\)
0.273834 + 0.961777i \(0.411708\pi\)
\(198\) 467.243 0.167705
\(199\) 1367.78 0.487232 0.243616 0.969872i \(-0.421666\pi\)
0.243616 + 0.969872i \(0.421666\pi\)
\(200\) −4217.74 −1.49120
\(201\) −436.300 −0.153105
\(202\) −6919.63 −2.41021
\(203\) 0 0
\(204\) −2039.15 −0.699848
\(205\) −702.828 −0.239452
\(206\) 4792.76 1.62101
\(207\) 1368.77 0.459594
\(208\) 42.6246 0.0142091
\(209\) 483.366 0.159977
\(210\) 0 0
\(211\) 4302.52 1.40378 0.701891 0.712285i \(-0.252339\pi\)
0.701891 + 0.712285i \(0.252339\pi\)
\(212\) 4669.11 1.51262
\(213\) 1106.89 0.356068
\(214\) −3492.55 −1.11563
\(215\) 7675.45 2.43470
\(216\) 585.642 0.184481
\(217\) 0 0
\(218\) 6502.16 2.02010
\(219\) −1826.72 −0.563644
\(220\) −2598.67 −0.796374
\(221\) −697.731 −0.212373
\(222\) 51.2028 0.0154798
\(223\) −1497.19 −0.449592 −0.224796 0.974406i \(-0.572172\pi\)
−0.224796 + 0.974406i \(0.572172\pi\)
\(224\) 0 0
\(225\) 1750.06 0.518537
\(226\) −1651.97 −0.486229
\(227\) 1603.32 0.468795 0.234397 0.972141i \(-0.424688\pi\)
0.234397 + 0.972141i \(0.424688\pi\)
\(228\) 1623.98 0.471715
\(229\) 1010.52 0.291603 0.145802 0.989314i \(-0.453424\pi\)
0.145802 + 0.989314i \(0.453424\pi\)
\(230\) −12385.4 −3.55072
\(231\) 0 0
\(232\) −4045.13 −1.14472
\(233\) 198.217 0.0557323 0.0278661 0.999612i \(-0.491129\pi\)
0.0278661 + 0.999612i \(0.491129\pi\)
\(234\) 537.141 0.150060
\(235\) 378.677 0.105116
\(236\) −2891.24 −0.797472
\(237\) −2730.71 −0.748434
\(238\) 0 0
\(239\) −1201.19 −0.325098 −0.162549 0.986700i \(-0.551972\pi\)
−0.162549 + 0.986700i \(0.551972\pi\)
\(240\) 174.485 0.0469289
\(241\) −2732.69 −0.730407 −0.365204 0.930928i \(-0.619001\pi\)
−0.365204 + 0.930928i \(0.619001\pi\)
\(242\) 5472.99 1.45379
\(243\) −243.000 −0.0641500
\(244\) 8319.49 2.18279
\(245\) 0 0
\(246\) −537.510 −0.139311
\(247\) 555.675 0.143145
\(248\) 3424.35 0.876801
\(249\) 983.788 0.250382
\(250\) −5655.91 −1.43084
\(251\) 7565.82 1.90259 0.951295 0.308281i \(-0.0997537\pi\)
0.951295 + 0.308281i \(0.0997537\pi\)
\(252\) 0 0
\(253\) −1732.88 −0.430615
\(254\) −4441.44 −1.09717
\(255\) −2856.18 −0.701414
\(256\) −3753.22 −0.916313
\(257\) 5008.68 1.21569 0.607846 0.794055i \(-0.292034\pi\)
0.607846 + 0.794055i \(0.292034\pi\)
\(258\) 5870.04 1.41648
\(259\) 0 0
\(260\) −2987.42 −0.712584
\(261\) 1678.44 0.398057
\(262\) 8168.21 1.92608
\(263\) −6248.81 −1.46509 −0.732544 0.680720i \(-0.761667\pi\)
−0.732544 + 0.680720i \(0.761667\pi\)
\(264\) −741.433 −0.172849
\(265\) 6539.88 1.51601
\(266\) 0 0
\(267\) 112.835 0.0258630
\(268\) 1855.80 0.422988
\(269\) −3588.45 −0.813351 −0.406676 0.913573i \(-0.633312\pi\)
−0.406676 + 0.913573i \(0.633312\pi\)
\(270\) 2198.80 0.495609
\(271\) −1983.14 −0.444529 −0.222264 0.974986i \(-0.571345\pi\)
−0.222264 + 0.974986i \(0.571345\pi\)
\(272\) −173.338 −0.0386404
\(273\) 0 0
\(274\) −7674.81 −1.69216
\(275\) −2215.61 −0.485841
\(276\) −5822.05 −1.26973
\(277\) 7363.91 1.59731 0.798654 0.601790i \(-0.205546\pi\)
0.798654 + 0.601790i \(0.205546\pi\)
\(278\) −1435.66 −0.309731
\(279\) −1420.86 −0.304892
\(280\) 0 0
\(281\) −5312.05 −1.12772 −0.563861 0.825869i \(-0.690685\pi\)
−0.563861 + 0.825869i \(0.690685\pi\)
\(282\) 289.606 0.0611552
\(283\) −1091.76 −0.229324 −0.114662 0.993405i \(-0.536578\pi\)
−0.114662 + 0.993405i \(0.536578\pi\)
\(284\) −4708.13 −0.983718
\(285\) 2274.67 0.472770
\(286\) −680.030 −0.140598
\(287\) 0 0
\(288\) 1695.16 0.346833
\(289\) −2075.59 −0.422469
\(290\) −15187.4 −3.07530
\(291\) −2166.04 −0.436341
\(292\) 7769.92 1.55719
\(293\) −7191.86 −1.43397 −0.716985 0.697089i \(-0.754478\pi\)
−0.716985 + 0.697089i \(0.754478\pi\)
\(294\) 0 0
\(295\) −4049.67 −0.799257
\(296\) −81.2499 −0.0159546
\(297\) 307.642 0.0601051
\(298\) −8628.73 −1.67734
\(299\) −1992.12 −0.385308
\(300\) −7443.88 −1.43257
\(301\) 0 0
\(302\) 9166.69 1.74663
\(303\) −4556.02 −0.863816
\(304\) 138.047 0.0260446
\(305\) 11652.9 2.18767
\(306\) −2184.35 −0.408075
\(307\) 541.355 0.100641 0.0503204 0.998733i \(-0.483976\pi\)
0.0503204 + 0.998733i \(0.483976\pi\)
\(308\) 0 0
\(309\) 3155.65 0.580966
\(310\) 12856.7 2.35553
\(311\) 54.0168 0.00984892 0.00492446 0.999988i \(-0.498432\pi\)
0.00492446 + 0.999988i \(0.498432\pi\)
\(312\) −852.348 −0.154663
\(313\) 3772.94 0.681340 0.340670 0.940183i \(-0.389346\pi\)
0.340670 + 0.940183i \(0.389346\pi\)
\(314\) 17444.1 3.13511
\(315\) 0 0
\(316\) 11615.1 2.06772
\(317\) 1719.24 0.304612 0.152306 0.988333i \(-0.451330\pi\)
0.152306 + 0.988333i \(0.451330\pi\)
\(318\) 5001.58 0.881996
\(319\) −2124.93 −0.372957
\(320\) −14873.4 −2.59827
\(321\) −2299.56 −0.399841
\(322\) 0 0
\(323\) −2259.72 −0.389270
\(324\) 1033.60 0.177229
\(325\) −2547.06 −0.434724
\(326\) 15989.5 2.71649
\(327\) 4281.15 0.724000
\(328\) 852.934 0.143584
\(329\) 0 0
\(330\) −2783.71 −0.464358
\(331\) 8408.21 1.39625 0.698123 0.715978i \(-0.254019\pi\)
0.698123 + 0.715978i \(0.254019\pi\)
\(332\) −4184.53 −0.691735
\(333\) 33.7129 0.00554792
\(334\) 1562.87 0.256037
\(335\) 2599.36 0.423935
\(336\) 0 0
\(337\) 2789.46 0.450894 0.225447 0.974255i \(-0.427616\pi\)
0.225447 + 0.974255i \(0.427616\pi\)
\(338\) 9228.58 1.48511
\(339\) −1087.69 −0.174263
\(340\) 12148.7 1.93781
\(341\) 1798.84 0.285667
\(342\) 1739.62 0.275053
\(343\) 0 0
\(344\) −9314.72 −1.45993
\(345\) −8154.76 −1.27257
\(346\) 19078.5 2.96435
\(347\) −3471.96 −0.537132 −0.268566 0.963261i \(-0.586550\pi\)
−0.268566 + 0.963261i \(0.586550\pi\)
\(348\) −7139.23 −1.09972
\(349\) −6626.12 −1.01630 −0.508149 0.861269i \(-0.669670\pi\)
−0.508149 + 0.861269i \(0.669670\pi\)
\(350\) 0 0
\(351\) 353.664 0.0537812
\(352\) −2146.10 −0.324964
\(353\) 9468.40 1.42763 0.713813 0.700337i \(-0.246967\pi\)
0.713813 + 0.700337i \(0.246967\pi\)
\(354\) −3097.11 −0.464999
\(355\) −6594.53 −0.985919
\(356\) −479.944 −0.0714522
\(357\) 0 0
\(358\) 8977.35 1.32533
\(359\) −6279.55 −0.923182 −0.461591 0.887093i \(-0.652721\pi\)
−0.461591 + 0.887093i \(0.652721\pi\)
\(360\) −3489.10 −0.510811
\(361\) −5059.35 −0.737622
\(362\) 16462.6 2.39021
\(363\) 3603.52 0.521035
\(364\) 0 0
\(365\) 10883.1 1.56068
\(366\) 8911.89 1.27276
\(367\) −10827.8 −1.54008 −0.770038 0.637998i \(-0.779763\pi\)
−0.770038 + 0.637998i \(0.779763\pi\)
\(368\) −494.904 −0.0701050
\(369\) −353.907 −0.0499286
\(370\) −305.053 −0.0428620
\(371\) 0 0
\(372\) 6043.63 0.842332
\(373\) 5239.23 0.727284 0.363642 0.931539i \(-0.381533\pi\)
0.363642 + 0.931539i \(0.381533\pi\)
\(374\) 2765.42 0.382344
\(375\) −3723.96 −0.512812
\(376\) −459.553 −0.0630310
\(377\) −2442.81 −0.333717
\(378\) 0 0
\(379\) −11050.4 −1.49768 −0.748839 0.662751i \(-0.769389\pi\)
−0.748839 + 0.662751i \(0.769389\pi\)
\(380\) −9675.27 −1.30613
\(381\) −2924.33 −0.393223
\(382\) −8692.49 −1.16426
\(383\) −10468.0 −1.39658 −0.698292 0.715813i \(-0.746056\pi\)
−0.698292 + 0.715813i \(0.746056\pi\)
\(384\) −6854.48 −0.910915
\(385\) 0 0
\(386\) −10934.9 −1.44190
\(387\) 3864.95 0.507665
\(388\) 9213.22 1.20549
\(389\) −11614.0 −1.51377 −0.756884 0.653550i \(-0.773279\pi\)
−0.756884 + 0.653550i \(0.773279\pi\)
\(390\) −3200.14 −0.415501
\(391\) 8101.19 1.04781
\(392\) 0 0
\(393\) 5378.11 0.690304
\(394\) −6899.78 −0.882248
\(395\) 16268.9 2.07234
\(396\) −1308.55 −0.166054
\(397\) −6707.30 −0.847934 −0.423967 0.905678i \(-0.639363\pi\)
−0.423967 + 0.905678i \(0.639363\pi\)
\(398\) −6232.10 −0.784892
\(399\) 0 0
\(400\) −632.768 −0.0790960
\(401\) 5526.38 0.688215 0.344107 0.938930i \(-0.388182\pi\)
0.344107 + 0.938930i \(0.388182\pi\)
\(402\) 1987.94 0.246640
\(403\) 2067.94 0.255611
\(404\) 19379.0 2.38649
\(405\) 1447.73 0.177625
\(406\) 0 0
\(407\) −42.6811 −0.00519810
\(408\) 3466.18 0.420592
\(409\) −1318.91 −0.159452 −0.0797258 0.996817i \(-0.525404\pi\)
−0.0797258 + 0.996817i \(0.525404\pi\)
\(410\) 3202.34 0.385738
\(411\) −5053.25 −0.606468
\(412\) −13422.5 −1.60505
\(413\) 0 0
\(414\) −6236.61 −0.740369
\(415\) −5861.15 −0.693283
\(416\) −2467.14 −0.290773
\(417\) −945.268 −0.111007
\(418\) −2202.39 −0.257709
\(419\) 3656.13 0.426286 0.213143 0.977021i \(-0.431630\pi\)
0.213143 + 0.977021i \(0.431630\pi\)
\(420\) 0 0
\(421\) −135.389 −0.0156733 −0.00783663 0.999969i \(-0.502495\pi\)
−0.00783663 + 0.999969i \(0.502495\pi\)
\(422\) −19603.9 −2.26138
\(423\) 190.682 0.0219179
\(424\) −7936.63 −0.909049
\(425\) 10357.9 1.18219
\(426\) −5043.38 −0.573597
\(427\) 0 0
\(428\) 9781.16 1.10465
\(429\) −447.745 −0.0503900
\(430\) −34972.1 −3.92211
\(431\) 8389.16 0.937568 0.468784 0.883313i \(-0.344692\pi\)
0.468784 + 0.883313i \(0.344692\pi\)
\(432\) 87.8612 0.00978524
\(433\) −8243.02 −0.914859 −0.457430 0.889246i \(-0.651230\pi\)
−0.457430 + 0.889246i \(0.651230\pi\)
\(434\) 0 0
\(435\) −9999.70 −1.10218
\(436\) −18209.8 −2.00021
\(437\) −6451.81 −0.706252
\(438\) 8323.19 0.907985
\(439\) −18283.2 −1.98772 −0.993859 0.110649i \(-0.964707\pi\)
−0.993859 + 0.110649i \(0.964707\pi\)
\(440\) 4417.26 0.478602
\(441\) 0 0
\(442\) 3179.12 0.342116
\(443\) 1210.44 0.129818 0.0649092 0.997891i \(-0.479324\pi\)
0.0649092 + 0.997891i \(0.479324\pi\)
\(444\) −143.398 −0.0153274
\(445\) −672.243 −0.0716121
\(446\) 6821.72 0.724256
\(447\) −5681.32 −0.601157
\(448\) 0 0
\(449\) −8301.16 −0.872508 −0.436254 0.899824i \(-0.643695\pi\)
−0.436254 + 0.899824i \(0.643695\pi\)
\(450\) −7973.93 −0.835321
\(451\) 448.052 0.0467804
\(452\) 4626.49 0.481442
\(453\) 6035.52 0.625990
\(454\) −7305.34 −0.755190
\(455\) 0 0
\(456\) −2760.48 −0.283489
\(457\) −12293.8 −1.25838 −0.629188 0.777253i \(-0.716612\pi\)
−0.629188 + 0.777253i \(0.716612\pi\)
\(458\) −4604.31 −0.469749
\(459\) −1438.22 −0.146253
\(460\) 34686.2 3.51577
\(461\) 19434.2 1.96343 0.981717 0.190346i \(-0.0609609\pi\)
0.981717 + 0.190346i \(0.0609609\pi\)
\(462\) 0 0
\(463\) −12491.1 −1.25380 −0.626902 0.779098i \(-0.715678\pi\)
−0.626902 + 0.779098i \(0.715678\pi\)
\(464\) −606.871 −0.0607183
\(465\) 8465.13 0.844217
\(466\) −903.149 −0.0897802
\(467\) 3385.17 0.335433 0.167716 0.985835i \(-0.446361\pi\)
0.167716 + 0.985835i \(0.446361\pi\)
\(468\) −1504.31 −0.148583
\(469\) 0 0
\(470\) −1725.39 −0.169333
\(471\) 11485.5 1.12362
\(472\) 4914.57 0.479262
\(473\) −4893.09 −0.475654
\(474\) 12442.1 1.20567
\(475\) −8249.07 −0.796828
\(476\) 0 0
\(477\) 3293.14 0.316106
\(478\) 5473.05 0.523706
\(479\) 5979.41 0.570368 0.285184 0.958473i \(-0.407945\pi\)
0.285184 + 0.958473i \(0.407945\pi\)
\(480\) −10099.3 −0.960349
\(481\) −49.0661 −0.00465119
\(482\) 12451.1 1.17663
\(483\) 0 0
\(484\) −15327.5 −1.43948
\(485\) 12904.7 1.20819
\(486\) 1107.20 0.103340
\(487\) −1114.96 −0.103745 −0.0518725 0.998654i \(-0.516519\pi\)
−0.0518725 + 0.998654i \(0.516519\pi\)
\(488\) −14141.6 −1.31180
\(489\) 10527.8 0.973583
\(490\) 0 0
\(491\) 1086.23 0.0998387 0.0499194 0.998753i \(-0.484104\pi\)
0.0499194 + 0.998753i \(0.484104\pi\)
\(492\) 1505.34 0.137939
\(493\) 9934.01 0.907516
\(494\) −2531.86 −0.230595
\(495\) −1832.85 −0.166425
\(496\) 513.740 0.0465073
\(497\) 0 0
\(498\) −4482.50 −0.403344
\(499\) −2213.50 −0.198577 −0.0992884 0.995059i \(-0.531657\pi\)
−0.0992884 + 0.995059i \(0.531657\pi\)
\(500\) 15839.8 1.41676
\(501\) 1029.02 0.0917634
\(502\) −34472.6 −3.06492
\(503\) −2643.32 −0.234314 −0.117157 0.993113i \(-0.537378\pi\)
−0.117157 + 0.993113i \(0.537378\pi\)
\(504\) 0 0
\(505\) 27143.5 2.39183
\(506\) 7895.66 0.693685
\(507\) 6076.27 0.532262
\(508\) 12438.6 1.08637
\(509\) −665.169 −0.0579236 −0.0289618 0.999581i \(-0.509220\pi\)
−0.0289618 + 0.999581i \(0.509220\pi\)
\(510\) 13013.8 1.12992
\(511\) 0 0
\(512\) −1177.58 −0.101645
\(513\) 1145.40 0.0985784
\(514\) −22821.4 −1.95838
\(515\) −18800.5 −1.60864
\(516\) −16439.5 −1.40254
\(517\) −241.406 −0.0205359
\(518\) 0 0
\(519\) 12561.6 1.06242
\(520\) 5078.07 0.428246
\(521\) −11762.0 −0.989063 −0.494531 0.869160i \(-0.664660\pi\)
−0.494531 + 0.869160i \(0.664660\pi\)
\(522\) −7647.59 −0.641237
\(523\) 10122.6 0.846330 0.423165 0.906053i \(-0.360919\pi\)
0.423165 + 0.906053i \(0.360919\pi\)
\(524\) −22875.7 −1.90712
\(525\) 0 0
\(526\) 28471.9 2.36014
\(527\) −8409.52 −0.695112
\(528\) −111.234 −0.00916823
\(529\) 10963.0 0.901042
\(530\) −29798.1 −2.44216
\(531\) −2039.20 −0.166655
\(532\) 0 0
\(533\) 515.079 0.0418585
\(534\) −514.119 −0.0416631
\(535\) 13700.2 1.10712
\(536\) −3154.51 −0.254206
\(537\) 5910.86 0.474995
\(538\) 16350.3 1.31024
\(539\) 0 0
\(540\) −6157.90 −0.490730
\(541\) 16118.0 1.28090 0.640449 0.768001i \(-0.278748\pi\)
0.640449 + 0.768001i \(0.278748\pi\)
\(542\) 9035.92 0.716100
\(543\) 10839.3 0.856647
\(544\) 10032.9 0.790733
\(545\) −25505.9 −2.00469
\(546\) 0 0
\(547\) −626.100 −0.0489399 −0.0244699 0.999701i \(-0.507790\pi\)
−0.0244699 + 0.999701i \(0.507790\pi\)
\(548\) 21493.9 1.67550
\(549\) 5867.76 0.456156
\(550\) 10095.1 0.782650
\(551\) −7911.47 −0.611688
\(552\) 9896.41 0.763078
\(553\) 0 0
\(554\) −33552.7 −2.57313
\(555\) −200.853 −0.0153617
\(556\) 4020.69 0.306682
\(557\) 20771.2 1.58008 0.790039 0.613057i \(-0.210060\pi\)
0.790039 + 0.613057i \(0.210060\pi\)
\(558\) 6473.97 0.491156
\(559\) −5625.08 −0.425609
\(560\) 0 0
\(561\) 1820.81 0.137031
\(562\) 24203.6 1.81667
\(563\) −5521.72 −0.413344 −0.206672 0.978410i \(-0.566263\pi\)
−0.206672 + 0.978410i \(0.566263\pi\)
\(564\) −811.064 −0.0605531
\(565\) 6480.18 0.482519
\(566\) 4974.48 0.369422
\(567\) 0 0
\(568\) 8002.95 0.591191
\(569\) 7574.81 0.558089 0.279044 0.960278i \(-0.409982\pi\)
0.279044 + 0.960278i \(0.409982\pi\)
\(570\) −10364.2 −0.761595
\(571\) −331.248 −0.0242772 −0.0121386 0.999926i \(-0.503864\pi\)
−0.0121386 + 0.999926i \(0.503864\pi\)
\(572\) 1904.48 0.139214
\(573\) −5723.31 −0.417268
\(574\) 0 0
\(575\) 29573.2 2.14485
\(576\) −7489.45 −0.541772
\(577\) 2038.11 0.147050 0.0735248 0.997293i \(-0.476575\pi\)
0.0735248 + 0.997293i \(0.476575\pi\)
\(578\) 9457.14 0.680563
\(579\) −7199.78 −0.516775
\(580\) 42533.6 3.04502
\(581\) 0 0
\(582\) 9869.26 0.702911
\(583\) −4169.17 −0.296174
\(584\) −13207.4 −0.935835
\(585\) −2107.04 −0.148915
\(586\) 32768.8 2.31001
\(587\) 5232.90 0.367947 0.183973 0.982931i \(-0.441104\pi\)
0.183973 + 0.982931i \(0.441104\pi\)
\(588\) 0 0
\(589\) 6697.36 0.468523
\(590\) 18451.8 1.28754
\(591\) −4542.95 −0.316196
\(592\) −12.1895 −0.000846262 0
\(593\) −5720.24 −0.396125 −0.198062 0.980189i \(-0.563465\pi\)
−0.198062 + 0.980189i \(0.563465\pi\)
\(594\) −1401.73 −0.0968244
\(595\) 0 0
\(596\) 24165.5 1.66083
\(597\) −4103.33 −0.281304
\(598\) 9076.81 0.620700
\(599\) 18088.4 1.23384 0.616922 0.787024i \(-0.288379\pi\)
0.616922 + 0.787024i \(0.288379\pi\)
\(600\) 12653.2 0.860943
\(601\) −1821.43 −0.123623 −0.0618117 0.998088i \(-0.519688\pi\)
−0.0618117 + 0.998088i \(0.519688\pi\)
\(602\) 0 0
\(603\) 1308.90 0.0883955
\(604\) −25672.0 −1.72944
\(605\) −21468.8 −1.44270
\(606\) 20758.9 1.39154
\(607\) 2372.20 0.158624 0.0793120 0.996850i \(-0.474728\pi\)
0.0793120 + 0.996850i \(0.474728\pi\)
\(608\) −7990.26 −0.532974
\(609\) 0 0
\(610\) −53094.7 −3.52416
\(611\) −277.520 −0.0183752
\(612\) 6117.45 0.404058
\(613\) 9725.08 0.640770 0.320385 0.947287i \(-0.396188\pi\)
0.320385 + 0.947287i \(0.396188\pi\)
\(614\) −2466.61 −0.162124
\(615\) 2108.48 0.138248
\(616\) 0 0
\(617\) −5329.51 −0.347744 −0.173872 0.984768i \(-0.555628\pi\)
−0.173872 + 0.984768i \(0.555628\pi\)
\(618\) −14378.3 −0.935890
\(619\) −15976.6 −1.03740 −0.518702 0.854955i \(-0.673585\pi\)
−0.518702 + 0.854955i \(0.673585\pi\)
\(620\) −36006.3 −2.33234
\(621\) −4106.31 −0.265347
\(622\) −246.120 −0.0158658
\(623\) 0 0
\(624\) −127.874 −0.00820361
\(625\) −2120.06 −0.135684
\(626\) −17190.9 −1.09758
\(627\) −1450.10 −0.0923625
\(628\) −48853.5 −3.10425
\(629\) 199.533 0.0126485
\(630\) 0 0
\(631\) −4199.98 −0.264974 −0.132487 0.991185i \(-0.542296\pi\)
−0.132487 + 0.991185i \(0.542296\pi\)
\(632\) −19743.5 −1.24265
\(633\) −12907.6 −0.810474
\(634\) −7833.47 −0.490705
\(635\) 17422.4 1.08880
\(636\) −14007.3 −0.873312
\(637\) 0 0
\(638\) 9681.97 0.600804
\(639\) −3320.66 −0.205576
\(640\) 40837.2 2.52224
\(641\) 2648.51 0.163198 0.0815988 0.996665i \(-0.473997\pi\)
0.0815988 + 0.996665i \(0.473997\pi\)
\(642\) 10477.6 0.644112
\(643\) 13.4305 0.000823715 0 0.000411857 1.00000i \(-0.499869\pi\)
0.000411857 1.00000i \(0.499869\pi\)
\(644\) 0 0
\(645\) −23026.3 −1.40568
\(646\) 10296.1 0.627083
\(647\) 11624.1 0.706324 0.353162 0.935562i \(-0.385106\pi\)
0.353162 + 0.935562i \(0.385106\pi\)
\(648\) −1756.93 −0.106510
\(649\) 2581.66 0.156146
\(650\) 11605.3 0.700305
\(651\) 0 0
\(652\) −44779.8 −2.68974
\(653\) −28516.6 −1.70894 −0.854471 0.519499i \(-0.826119\pi\)
−0.854471 + 0.519499i \(0.826119\pi\)
\(654\) −19506.5 −1.16630
\(655\) −32041.3 −1.91139
\(656\) 127.962 0.00761595
\(657\) 5480.15 0.325420
\(658\) 0 0
\(659\) 18048.6 1.06688 0.533440 0.845838i \(-0.320899\pi\)
0.533440 + 0.845838i \(0.320899\pi\)
\(660\) 7796.01 0.459787
\(661\) 17841.4 1.04985 0.524926 0.851148i \(-0.324093\pi\)
0.524926 + 0.851148i \(0.324093\pi\)
\(662\) −38310.9 −2.24924
\(663\) 2093.19 0.122614
\(664\) 7112.94 0.415716
\(665\) 0 0
\(666\) −153.608 −0.00893725
\(667\) 28362.9 1.64650
\(668\) −4376.95 −0.253517
\(669\) 4491.56 0.259572
\(670\) −11843.6 −0.682924
\(671\) −7428.68 −0.427394
\(672\) 0 0
\(673\) −6826.13 −0.390978 −0.195489 0.980706i \(-0.562629\pi\)
−0.195489 + 0.980706i \(0.562629\pi\)
\(674\) −12709.8 −0.726354
\(675\) −5250.19 −0.299378
\(676\) −25845.4 −1.47049
\(677\) −21286.9 −1.20845 −0.604225 0.796814i \(-0.706517\pi\)
−0.604225 + 0.796814i \(0.706517\pi\)
\(678\) 4955.92 0.280724
\(679\) 0 0
\(680\) −20650.6 −1.16458
\(681\) −4809.97 −0.270659
\(682\) −8196.16 −0.460187
\(683\) −20696.8 −1.15951 −0.579753 0.814793i \(-0.696851\pi\)
−0.579753 + 0.814793i \(0.696851\pi\)
\(684\) −4871.95 −0.272345
\(685\) 30105.9 1.67925
\(686\) 0 0
\(687\) −3031.56 −0.168357
\(688\) −1397.44 −0.0774376
\(689\) −4792.86 −0.265012
\(690\) 37156.1 2.05001
\(691\) 31341.9 1.72548 0.862738 0.505652i \(-0.168748\pi\)
0.862738 + 0.505652i \(0.168748\pi\)
\(692\) −53430.8 −2.93517
\(693\) 0 0
\(694\) 15819.5 0.865276
\(695\) 5631.66 0.307368
\(696\) 12135.4 0.660906
\(697\) −2094.63 −0.113831
\(698\) 30191.0 1.63717
\(699\) −594.651 −0.0321770
\(700\) 0 0
\(701\) −9213.32 −0.496408 −0.248204 0.968708i \(-0.579840\pi\)
−0.248204 + 0.968708i \(0.579840\pi\)
\(702\) −1611.42 −0.0866371
\(703\) −158.909 −0.00852541
\(704\) 9481.77 0.507610
\(705\) −1136.03 −0.0606886
\(706\) −43141.5 −2.29979
\(707\) 0 0
\(708\) 8673.71 0.460421
\(709\) 14516.5 0.768942 0.384471 0.923137i \(-0.374384\pi\)
0.384471 + 0.923137i \(0.374384\pi\)
\(710\) 30047.1 1.58824
\(711\) 8192.14 0.432108
\(712\) 815.817 0.0429411
\(713\) −24010.3 −1.26114
\(714\) 0 0
\(715\) 2667.54 0.139525
\(716\) −25141.8 −1.31228
\(717\) 3603.56 0.187695
\(718\) 28611.9 1.48717
\(719\) 25882.4 1.34249 0.671246 0.741235i \(-0.265760\pi\)
0.671246 + 0.741235i \(0.265760\pi\)
\(720\) −523.454 −0.0270944
\(721\) 0 0
\(722\) 23052.3 1.18825
\(723\) 8198.08 0.421701
\(724\) −46104.9 −2.36668
\(725\) 36263.9 1.85767
\(726\) −16419.0 −0.839345
\(727\) 32181.2 1.64172 0.820862 0.571127i \(-0.193494\pi\)
0.820862 + 0.571127i \(0.193494\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −49587.4 −2.51412
\(731\) 22875.1 1.15741
\(732\) −24958.5 −1.26023
\(733\) 20836.1 1.04993 0.524966 0.851123i \(-0.324078\pi\)
0.524966 + 0.851123i \(0.324078\pi\)
\(734\) 49335.5 2.48094
\(735\) 0 0
\(736\) 28645.4 1.43462
\(737\) −1657.09 −0.0828217
\(738\) 1612.53 0.0804310
\(739\) 26434.9 1.31586 0.657931 0.753078i \(-0.271432\pi\)
0.657931 + 0.753078i \(0.271432\pi\)
\(740\) 854.325 0.0424400
\(741\) −1667.03 −0.0826447
\(742\) 0 0
\(743\) 9954.69 0.491524 0.245762 0.969330i \(-0.420962\pi\)
0.245762 + 0.969330i \(0.420962\pi\)
\(744\) −10273.1 −0.506221
\(745\) 33847.8 1.66455
\(746\) −23871.8 −1.17160
\(747\) −2951.36 −0.144558
\(748\) −7744.79 −0.378580
\(749\) 0 0
\(750\) 16967.7 0.826098
\(751\) −33204.5 −1.61338 −0.806692 0.590973i \(-0.798744\pi\)
−0.806692 + 0.590973i \(0.798744\pi\)
\(752\) −68.9446 −0.00334329
\(753\) −22697.5 −1.09846
\(754\) 11130.4 0.537591
\(755\) −35958.1 −1.73331
\(756\) 0 0
\(757\) 1964.06 0.0942998 0.0471499 0.998888i \(-0.484986\pi\)
0.0471499 + 0.998888i \(0.484986\pi\)
\(758\) 50349.6 2.41264
\(759\) 5198.65 0.248615
\(760\) 16446.2 0.784955
\(761\) −38553.8 −1.83650 −0.918248 0.396005i \(-0.870396\pi\)
−0.918248 + 0.396005i \(0.870396\pi\)
\(762\) 13324.3 0.633450
\(763\) 0 0
\(764\) 24344.0 1.15280
\(765\) 8568.53 0.404962
\(766\) 47696.2 2.24978
\(767\) 2967.87 0.139718
\(768\) 11259.7 0.529034
\(769\) −19715.0 −0.924501 −0.462251 0.886749i \(-0.652958\pi\)
−0.462251 + 0.886749i \(0.652958\pi\)
\(770\) 0 0
\(771\) −15026.0 −0.701880
\(772\) 30624.2 1.42771
\(773\) −14700.7 −0.684019 −0.342010 0.939696i \(-0.611108\pi\)
−0.342010 + 0.939696i \(0.611108\pi\)
\(774\) −17610.1 −0.817807
\(775\) −30698.8 −1.42288
\(776\) −15660.8 −0.724471
\(777\) 0 0
\(778\) 52917.8 2.43856
\(779\) 1668.17 0.0767246
\(780\) 8962.26 0.411411
\(781\) 4204.01 0.192614
\(782\) −36912.0 −1.68794
\(783\) −5035.32 −0.229818
\(784\) 0 0
\(785\) −68427.6 −3.11119
\(786\) −24504.6 −1.11202
\(787\) −23918.6 −1.08336 −0.541681 0.840584i \(-0.682212\pi\)
−0.541681 + 0.840584i \(0.682212\pi\)
\(788\) 19323.4 0.873562
\(789\) 18746.4 0.845869
\(790\) −74126.9 −3.33837
\(791\) 0 0
\(792\) 2224.30 0.0997942
\(793\) −8539.98 −0.382426
\(794\) 30560.9 1.36595
\(795\) −19619.6 −0.875266
\(796\) 17453.5 0.777164
\(797\) 38252.7 1.70010 0.850051 0.526700i \(-0.176571\pi\)
0.850051 + 0.526700i \(0.176571\pi\)
\(798\) 0 0
\(799\) 1128.57 0.0499698
\(800\) 36625.1 1.61861
\(801\) −338.506 −0.0149320
\(802\) −25180.2 −1.10866
\(803\) −6937.96 −0.304901
\(804\) −5567.39 −0.244212
\(805\) 0 0
\(806\) −9422.27 −0.411769
\(807\) 10765.3 0.469589
\(808\) −32940.7 −1.43422
\(809\) −31435.6 −1.36615 −0.683075 0.730348i \(-0.739358\pi\)
−0.683075 + 0.730348i \(0.739358\pi\)
\(810\) −6596.39 −0.286140
\(811\) 11467.0 0.496501 0.248250 0.968696i \(-0.420144\pi\)
0.248250 + 0.968696i \(0.420144\pi\)
\(812\) 0 0
\(813\) 5949.43 0.256649
\(814\) 194.471 0.00837371
\(815\) −62721.7 −2.69576
\(816\) 520.015 0.0223090
\(817\) −18217.8 −0.780121
\(818\) 6009.42 0.256864
\(819\) 0 0
\(820\) −8968.42 −0.381940
\(821\) −5030.57 −0.213847 −0.106923 0.994267i \(-0.534100\pi\)
−0.106923 + 0.994267i \(0.534100\pi\)
\(822\) 23024.4 0.976970
\(823\) −13985.2 −0.592336 −0.296168 0.955136i \(-0.595709\pi\)
−0.296168 + 0.955136i \(0.595709\pi\)
\(824\) 22815.8 0.964596
\(825\) 6646.83 0.280500
\(826\) 0 0
\(827\) −13939.5 −0.586125 −0.293063 0.956093i \(-0.594674\pi\)
−0.293063 + 0.956093i \(0.594674\pi\)
\(828\) 17466.1 0.733080
\(829\) −20104.4 −0.842286 −0.421143 0.906994i \(-0.638371\pi\)
−0.421143 + 0.906994i \(0.638371\pi\)
\(830\) 26705.5 1.11682
\(831\) −22091.7 −0.922207
\(832\) 10900.2 0.454203
\(833\) 0 0
\(834\) 4306.99 0.178823
\(835\) −6130.66 −0.254084
\(836\) 6167.98 0.255172
\(837\) 4262.59 0.176030
\(838\) −16658.7 −0.686711
\(839\) 15949.5 0.656302 0.328151 0.944625i \(-0.393575\pi\)
0.328151 + 0.944625i \(0.393575\pi\)
\(840\) 0 0
\(841\) 10390.8 0.426043
\(842\) 616.881 0.0252484
\(843\) 15936.1 0.651091
\(844\) 54902.2 2.23911
\(845\) −36200.8 −1.47378
\(846\) −868.817 −0.0353080
\(847\) 0 0
\(848\) −1190.70 −0.0482177
\(849\) 3275.29 0.132400
\(850\) −47194.5 −1.90442
\(851\) 569.694 0.0229481
\(852\) 14124.4 0.567950
\(853\) 11802.0 0.473730 0.236865 0.971543i \(-0.423880\pi\)
0.236865 + 0.971543i \(0.423880\pi\)
\(854\) 0 0
\(855\) −6824.00 −0.272954
\(856\) −16626.2 −0.663868
\(857\) 9595.28 0.382460 0.191230 0.981545i \(-0.438752\pi\)
0.191230 + 0.981545i \(0.438752\pi\)
\(858\) 2040.09 0.0811742
\(859\) 21840.9 0.867521 0.433760 0.901028i \(-0.357186\pi\)
0.433760 + 0.901028i \(0.357186\pi\)
\(860\) 97942.3 3.88350
\(861\) 0 0
\(862\) −38224.1 −1.51035
\(863\) −26531.6 −1.04652 −0.523260 0.852173i \(-0.675284\pi\)
−0.523260 + 0.852173i \(0.675284\pi\)
\(864\) −5085.47 −0.200244
\(865\) −74838.9 −2.94173
\(866\) 37558.2 1.47376
\(867\) 6226.77 0.243912
\(868\) 0 0
\(869\) −10371.4 −0.404862
\(870\) 45562.3 1.77552
\(871\) −1904.98 −0.0741077
\(872\) 30953.3 1.20208
\(873\) 6498.11 0.251922
\(874\) 29396.8 1.13771
\(875\) 0 0
\(876\) −23309.8 −0.899046
\(877\) −6832.46 −0.263074 −0.131537 0.991311i \(-0.541991\pi\)
−0.131537 + 0.991311i \(0.541991\pi\)
\(878\) 83304.9 3.20205
\(879\) 21575.6 0.827903
\(880\) 662.701 0.0253860
\(881\) −3994.77 −0.152766 −0.0763832 0.997079i \(-0.524337\pi\)
−0.0763832 + 0.997079i \(0.524337\pi\)
\(882\) 0 0
\(883\) 13727.0 0.523161 0.261580 0.965182i \(-0.415756\pi\)
0.261580 + 0.965182i \(0.415756\pi\)
\(884\) −8903.38 −0.338748
\(885\) 12149.0 0.461451
\(886\) −5515.19 −0.209127
\(887\) 44119.4 1.67011 0.835054 0.550169i \(-0.185437\pi\)
0.835054 + 0.550169i \(0.185437\pi\)
\(888\) 243.750 0.00921138
\(889\) 0 0
\(890\) 3062.99 0.115361
\(891\) −922.926 −0.0347017
\(892\) −19104.8 −0.717125
\(893\) −898.796 −0.0336809
\(894\) 25886.2 0.968415
\(895\) −35215.3 −1.31522
\(896\) 0 0
\(897\) 5976.35 0.222458
\(898\) 37823.1 1.40554
\(899\) −29442.4 −1.09228
\(900\) 22331.6 0.827098
\(901\) 19490.7 0.720678
\(902\) −2041.49 −0.0753594
\(903\) 0 0
\(904\) −7864.18 −0.289335
\(905\) −64577.7 −2.37197
\(906\) −27500.1 −1.00842
\(907\) 36905.8 1.35109 0.675545 0.737319i \(-0.263908\pi\)
0.675545 + 0.737319i \(0.263908\pi\)
\(908\) 20459.2 0.747755
\(909\) 13668.1 0.498725
\(910\) 0 0
\(911\) −3169.56 −0.115271 −0.0576356 0.998338i \(-0.518356\pi\)
−0.0576356 + 0.998338i \(0.518356\pi\)
\(912\) −414.141 −0.0150368
\(913\) 3736.48 0.135443
\(914\) 56014.9 2.02714
\(915\) −34958.6 −1.26305
\(916\) 12894.7 0.465124
\(917\) 0 0
\(918\) 6553.05 0.235602
\(919\) 8727.00 0.313250 0.156625 0.987658i \(-0.449939\pi\)
0.156625 + 0.987658i \(0.449939\pi\)
\(920\) −58960.2 −2.11289
\(921\) −1624.06 −0.0581050
\(922\) −88549.5 −3.16293
\(923\) 4832.91 0.172348
\(924\) 0 0
\(925\) 728.392 0.0258912
\(926\) 56914.1 2.01978
\(927\) −9466.95 −0.335421
\(928\) 35126.1 1.24253
\(929\) 19405.1 0.685317 0.342659 0.939460i \(-0.388673\pi\)
0.342659 + 0.939460i \(0.388673\pi\)
\(930\) −38570.2 −1.35997
\(931\) 0 0
\(932\) 2529.34 0.0888963
\(933\) −162.050 −0.00568628
\(934\) −15424.1 −0.540355
\(935\) −10847.9 −0.379427
\(936\) 2557.05 0.0892945
\(937\) −615.692 −0.0214662 −0.0107331 0.999942i \(-0.503417\pi\)
−0.0107331 + 0.999942i \(0.503417\pi\)
\(938\) 0 0
\(939\) −11318.8 −0.393372
\(940\) 4832.10 0.167666
\(941\) −29602.0 −1.02550 −0.512751 0.858537i \(-0.671374\pi\)
−0.512751 + 0.858537i \(0.671374\pi\)
\(942\) −52332.2 −1.81006
\(943\) −5980.46 −0.206522
\(944\) 737.310 0.0254210
\(945\) 0 0
\(946\) 22294.7 0.766240
\(947\) 13537.4 0.464527 0.232264 0.972653i \(-0.425387\pi\)
0.232264 + 0.972653i \(0.425387\pi\)
\(948\) −34845.2 −1.19380
\(949\) −7975.85 −0.272821
\(950\) 37585.8 1.28363
\(951\) −5157.71 −0.175868
\(952\) 0 0
\(953\) 33468.5 1.13762 0.568810 0.822469i \(-0.307404\pi\)
0.568810 + 0.822469i \(0.307404\pi\)
\(954\) −15004.7 −0.509220
\(955\) 34097.9 1.15538
\(956\) −15327.7 −0.518550
\(957\) 6374.80 0.215327
\(958\) −27244.4 −0.918817
\(959\) 0 0
\(960\) 44620.2 1.50011
\(961\) −4866.88 −0.163368
\(962\) 223.563 0.00749268
\(963\) 6898.68 0.230848
\(964\) −34870.4 −1.16504
\(965\) 42894.4 1.43090
\(966\) 0 0
\(967\) −55733.5 −1.85343 −0.926715 0.375764i \(-0.877380\pi\)
−0.926715 + 0.375764i \(0.877380\pi\)
\(968\) 26054.0 0.865090
\(969\) 6779.17 0.224745
\(970\) −58798.4 −1.94629
\(971\) −19491.3 −0.644187 −0.322094 0.946708i \(-0.604387\pi\)
−0.322094 + 0.946708i \(0.604387\pi\)
\(972\) −3100.79 −0.102323
\(973\) 0 0
\(974\) 5080.18 0.167125
\(975\) 7641.17 0.250988
\(976\) −2121.60 −0.0695807
\(977\) −6241.56 −0.204386 −0.102193 0.994765i \(-0.532586\pi\)
−0.102193 + 0.994765i \(0.532586\pi\)
\(978\) −47968.4 −1.56836
\(979\) 428.554 0.0139905
\(980\) 0 0
\(981\) −12843.4 −0.418001
\(982\) −4949.25 −0.160832
\(983\) 59694.6 1.93689 0.968444 0.249231i \(-0.0801778\pi\)
0.968444 + 0.249231i \(0.0801778\pi\)
\(984\) −2558.80 −0.0828980
\(985\) 27065.7 0.875517
\(986\) −45263.0 −1.46193
\(987\) 0 0
\(988\) 7090.68 0.228324
\(989\) 65311.4 2.09988
\(990\) 8351.14 0.268097
\(991\) 15561.6 0.498821 0.249411 0.968398i \(-0.419763\pi\)
0.249411 + 0.968398i \(0.419763\pi\)
\(992\) −29735.6 −0.951720
\(993\) −25224.6 −0.806123
\(994\) 0 0
\(995\) 24446.6 0.778903
\(996\) 12553.6 0.399374
\(997\) 19884.9 0.631657 0.315829 0.948816i \(-0.397718\pi\)
0.315829 + 0.948816i \(0.397718\pi\)
\(998\) 10085.5 0.319891
\(999\) −101.139 −0.00320309
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 147.4.a.l.1.1 3
3.2 odd 2 441.4.a.s.1.3 3
4.3 odd 2 2352.4.a.ci.1.3 3
7.2 even 3 21.4.e.b.4.3 6
7.3 odd 6 147.4.e.n.79.3 6
7.4 even 3 21.4.e.b.16.3 yes 6
7.5 odd 6 147.4.e.n.67.3 6
7.6 odd 2 147.4.a.m.1.1 3
21.2 odd 6 63.4.e.c.46.1 6
21.5 even 6 441.4.e.w.361.1 6
21.11 odd 6 63.4.e.c.37.1 6
21.17 even 6 441.4.e.w.226.1 6
21.20 even 2 441.4.a.t.1.3 3
28.11 odd 6 336.4.q.k.289.1 6
28.23 odd 6 336.4.q.k.193.1 6
28.27 even 2 2352.4.a.cg.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.b.4.3 6 7.2 even 3
21.4.e.b.16.3 yes 6 7.4 even 3
63.4.e.c.37.1 6 21.11 odd 6
63.4.e.c.46.1 6 21.2 odd 6
147.4.a.l.1.1 3 1.1 even 1 trivial
147.4.a.m.1.1 3 7.6 odd 2
147.4.e.n.67.3 6 7.5 odd 6
147.4.e.n.79.3 6 7.3 odd 6
336.4.q.k.193.1 6 28.23 odd 6
336.4.q.k.289.1 6 28.11 odd 6
441.4.a.s.1.3 3 3.2 odd 2
441.4.a.t.1.3 3 21.20 even 2
441.4.e.w.226.1 6 21.17 even 6
441.4.e.w.361.1 6 21.5 even 6
2352.4.a.cg.1.1 3 28.27 even 2
2352.4.a.ci.1.3 3 4.3 odd 2