Properties

Label 1150.4.b.n.599.7
Level $1150$
Weight $4$
Character 1150.599
Analytic conductor $67.852$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 136 x^{6} + 5308 x^{4} + 58833 x^{2} + 116964\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.7
Root \(1.58997i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.4.b.n.599.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{2} +0.589969i q^{3} -4.00000 q^{4} -1.17994 q^{6} +18.5077i q^{7} -8.00000i q^{8} +26.6519 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} +0.589969i q^{3} -4.00000 q^{4} -1.17994 q^{6} +18.5077i q^{7} -8.00000i q^{8} +26.6519 q^{9} +47.9296 q^{11} -2.35988i q^{12} +42.3717i q^{13} -37.0155 q^{14} +16.0000 q^{16} -1.70534i q^{17} +53.3039i q^{18} -21.4208 q^{19} -10.9190 q^{21} +95.8592i q^{22} -23.0000i q^{23} +4.71976 q^{24} -84.7434 q^{26} +31.6530i q^{27} -74.0310i q^{28} -57.6332 q^{29} +295.699 q^{31} +32.0000i q^{32} +28.2770i q^{33} +3.41069 q^{34} -106.608 q^{36} +7.85184i q^{37} -42.8416i q^{38} -24.9980 q^{39} +465.929 q^{41} -21.8380i q^{42} +182.374i q^{43} -191.718 q^{44} +46.0000 q^{46} -449.193i q^{47} +9.43951i q^{48} +0.463605 q^{49} +1.00610 q^{51} -169.487i q^{52} -368.316i q^{53} -63.3060 q^{54} +148.062 q^{56} -12.6376i q^{57} -115.266i q^{58} +377.032 q^{59} +849.042 q^{61} +591.398i q^{62} +493.267i q^{63} -64.0000 q^{64} -56.5540 q^{66} -92.3424i q^{67} +6.82138i q^{68} +13.5693 q^{69} -626.854 q^{71} -213.215i q^{72} +439.227i q^{73} -15.7037 q^{74} +85.6831 q^{76} +887.068i q^{77} -49.9960i q^{78} -641.707 q^{79} +700.928 q^{81} +931.859i q^{82} -609.932i q^{83} +43.6760 q^{84} -364.747 q^{86} -34.0018i q^{87} -383.437i q^{88} -1122.87 q^{89} -784.204 q^{91} +92.0000i q^{92} +174.453i q^{93} +898.386 q^{94} -18.8790 q^{96} +1428.80i q^{97} +0.927209i q^{98} +1277.42 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} + 16 q^{6} - 64 q^{9} + O(q^{10}) \) \( 8 q - 32 q^{4} + 16 q^{6} - 64 q^{9} - 78 q^{11} - 4 q^{14} + 128 q^{16} - 106 q^{19} + 600 q^{21} - 64 q^{24} + 80 q^{26} - 322 q^{29} + 776 q^{31} - 92 q^{34} + 256 q^{36} - 2094 q^{39} + 968 q^{41} + 312 q^{44} + 368 q^{46} - 3286 q^{49} + 3650 q^{51} + 548 q^{54} + 16 q^{56} + 188 q^{59} + 2306 q^{61} - 512 q^{64} - 348 q^{66} - 184 q^{69} + 400 q^{71} + 1864 q^{74} + 424 q^{76} + 1816 q^{79} - 2112 q^{81} - 2400 q^{84} - 3576 q^{86} + 3568 q^{89} + 4658 q^{91} - 1060 q^{94} + 256 q^{96} + 5330 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) 0.589969i 0.113540i 0.998387 + 0.0567698i \(0.0180801\pi\)
−0.998387 + 0.0567698i \(0.981920\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) −1.17994 −0.0802847
\(7\) 18.5077i 0.999324i 0.866220 + 0.499662i \(0.166542\pi\)
−0.866220 + 0.499662i \(0.833458\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) 26.6519 0.987109
\(10\) 0 0
\(11\) 47.9296 1.31376 0.656878 0.753997i \(-0.271877\pi\)
0.656878 + 0.753997i \(0.271877\pi\)
\(12\) − 2.35988i − 0.0567698i
\(13\) 42.3717i 0.903984i 0.892022 + 0.451992i \(0.149286\pi\)
−0.892022 + 0.451992i \(0.850714\pi\)
\(14\) −37.0155 −0.706629
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 1.70534i − 0.0243298i −0.999926 0.0121649i \(-0.996128\pi\)
0.999926 0.0121649i \(-0.00387230\pi\)
\(18\) 53.3039i 0.697991i
\(19\) −21.4208 −0.258645 −0.129323 0.991603i \(-0.541280\pi\)
−0.129323 + 0.991603i \(0.541280\pi\)
\(20\) 0 0
\(21\) −10.9190 −0.113463
\(22\) 95.8592i 0.928966i
\(23\) − 23.0000i − 0.208514i
\(24\) 4.71976 0.0401423
\(25\) 0 0
\(26\) −84.7434 −0.639213
\(27\) 31.6530i 0.225616i
\(28\) − 74.0310i − 0.499662i
\(29\) −57.6332 −0.369042 −0.184521 0.982829i \(-0.559073\pi\)
−0.184521 + 0.982829i \(0.559073\pi\)
\(30\) 0 0
\(31\) 295.699 1.71320 0.856599 0.515983i \(-0.172573\pi\)
0.856599 + 0.515983i \(0.172573\pi\)
\(32\) 32.0000i 0.176777i
\(33\) 28.2770i 0.149163i
\(34\) 3.41069 0.0172038
\(35\) 0 0
\(36\) −106.608 −0.493554
\(37\) 7.85184i 0.0348874i 0.999848 + 0.0174437i \(0.00555279\pi\)
−0.999848 + 0.0174437i \(0.994447\pi\)
\(38\) − 42.8416i − 0.182890i
\(39\) −24.9980 −0.102638
\(40\) 0 0
\(41\) 465.929 1.77478 0.887390 0.461020i \(-0.152516\pi\)
0.887390 + 0.461020i \(0.152516\pi\)
\(42\) − 21.8380i − 0.0802304i
\(43\) 182.374i 0.646784i 0.946265 + 0.323392i \(0.104823\pi\)
−0.946265 + 0.323392i \(0.895177\pi\)
\(44\) −191.718 −0.656878
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) − 449.193i − 1.39408i −0.717035 0.697038i \(-0.754501\pi\)
0.717035 0.697038i \(-0.245499\pi\)
\(48\) 9.43951i 0.0283849i
\(49\) 0.463605 0.00135162
\(50\) 0 0
\(51\) 1.00610 0.00276240
\(52\) − 169.487i − 0.451992i
\(53\) − 368.316i − 0.954567i −0.878749 0.477283i \(-0.841621\pi\)
0.878749 0.477283i \(-0.158379\pi\)
\(54\) −63.3060 −0.159534
\(55\) 0 0
\(56\) 148.062 0.353314
\(57\) − 12.6376i − 0.0293665i
\(58\) − 115.266i − 0.260952i
\(59\) 377.032 0.831955 0.415977 0.909375i \(-0.363440\pi\)
0.415977 + 0.909375i \(0.363440\pi\)
\(60\) 0 0
\(61\) 849.042 1.78211 0.891055 0.453896i \(-0.149966\pi\)
0.891055 + 0.453896i \(0.149966\pi\)
\(62\) 591.398i 1.21141i
\(63\) 493.267i 0.986441i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) −56.5540 −0.105474
\(67\) − 92.3424i − 0.168379i −0.996450 0.0841897i \(-0.973170\pi\)
0.996450 0.0841897i \(-0.0268302\pi\)
\(68\) 6.82138i 0.0121649i
\(69\) 13.5693 0.0236747
\(70\) 0 0
\(71\) −626.854 −1.04780 −0.523901 0.851779i \(-0.675524\pi\)
−0.523901 + 0.851779i \(0.675524\pi\)
\(72\) − 213.215i − 0.348996i
\(73\) 439.227i 0.704214i 0.935960 + 0.352107i \(0.114535\pi\)
−0.935960 + 0.352107i \(0.885465\pi\)
\(74\) −15.7037 −0.0246691
\(75\) 0 0
\(76\) 85.6831 0.129323
\(77\) 887.068i 1.31287i
\(78\) − 49.9960i − 0.0725761i
\(79\) −641.707 −0.913894 −0.456947 0.889494i \(-0.651057\pi\)
−0.456947 + 0.889494i \(0.651057\pi\)
\(80\) 0 0
\(81\) 700.928 0.961492
\(82\) 931.859i 1.25496i
\(83\) − 609.932i − 0.806611i −0.915065 0.403306i \(-0.867861\pi\)
0.915065 0.403306i \(-0.132139\pi\)
\(84\) 43.6760 0.0567315
\(85\) 0 0
\(86\) −364.747 −0.457346
\(87\) − 34.0018i − 0.0419009i
\(88\) − 383.437i − 0.464483i
\(89\) −1122.87 −1.33735 −0.668673 0.743557i \(-0.733137\pi\)
−0.668673 + 0.743557i \(0.733137\pi\)
\(90\) 0 0
\(91\) −784.204 −0.903373
\(92\) 92.0000i 0.104257i
\(93\) 174.453i 0.194516i
\(94\) 898.386 0.985760
\(95\) 0 0
\(96\) −18.8790 −0.0200712
\(97\) 1428.80i 1.49559i 0.663930 + 0.747795i \(0.268887\pi\)
−0.663930 + 0.747795i \(0.731113\pi\)
\(98\) 0.927209i 0 0.000955737i
\(99\) 1277.42 1.29682
\(100\) 0 0
\(101\) −1512.15 −1.48975 −0.744875 0.667204i \(-0.767491\pi\)
−0.744875 + 0.667204i \(0.767491\pi\)
\(102\) 2.01220i 0.00195331i
\(103\) 957.279i 0.915762i 0.889013 + 0.457881i \(0.151391\pi\)
−0.889013 + 0.457881i \(0.848609\pi\)
\(104\) 338.974 0.319607
\(105\) 0 0
\(106\) 736.631 0.674981
\(107\) 1742.16i 1.57403i 0.616936 + 0.787013i \(0.288374\pi\)
−0.616936 + 0.787013i \(0.711626\pi\)
\(108\) − 126.612i − 0.112808i
\(109\) −1166.77 −1.02529 −0.512644 0.858601i \(-0.671334\pi\)
−0.512644 + 0.858601i \(0.671334\pi\)
\(110\) 0 0
\(111\) −4.63234 −0.00396111
\(112\) 296.124i 0.249831i
\(113\) − 393.287i − 0.327410i −0.986509 0.163705i \(-0.947655\pi\)
0.986509 0.163705i \(-0.0523445\pi\)
\(114\) 25.2752 0.0207653
\(115\) 0 0
\(116\) 230.533 0.184521
\(117\) 1129.29i 0.892331i
\(118\) 754.063i 0.588281i
\(119\) 31.5621 0.0243134
\(120\) 0 0
\(121\) 966.245 0.725954
\(122\) 1698.08i 1.26014i
\(123\) 274.884i 0.201508i
\(124\) −1182.80 −0.856599
\(125\) 0 0
\(126\) −986.534 −0.697519
\(127\) 1067.87i 0.746127i 0.927806 + 0.373063i \(0.121693\pi\)
−0.927806 + 0.373063i \(0.878307\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) −107.595 −0.0734357
\(130\) 0 0
\(131\) −175.497 −0.117047 −0.0585237 0.998286i \(-0.518639\pi\)
−0.0585237 + 0.998286i \(0.518639\pi\)
\(132\) − 113.108i − 0.0745817i
\(133\) − 396.450i − 0.258471i
\(134\) 184.685 0.119062
\(135\) 0 0
\(136\) −13.6428 −0.00860189
\(137\) − 475.898i − 0.296779i −0.988929 0.148389i \(-0.952591\pi\)
0.988929 0.148389i \(-0.0474089\pi\)
\(138\) 27.1386i 0.0167405i
\(139\) −153.167 −0.0934638 −0.0467319 0.998907i \(-0.514881\pi\)
−0.0467319 + 0.998907i \(0.514881\pi\)
\(140\) 0 0
\(141\) 265.010 0.158283
\(142\) − 1253.71i − 0.740907i
\(143\) 2030.86i 1.18761i
\(144\) 426.431 0.246777
\(145\) 0 0
\(146\) −878.454 −0.497954
\(147\) 0.273513i 0 0.000153462i
\(148\) − 31.4074i − 0.0174437i
\(149\) −506.906 −0.278707 −0.139353 0.990243i \(-0.544502\pi\)
−0.139353 + 0.990243i \(0.544502\pi\)
\(150\) 0 0
\(151\) 2437.84 1.31383 0.656916 0.753964i \(-0.271861\pi\)
0.656916 + 0.753964i \(0.271861\pi\)
\(152\) 171.366i 0.0914450i
\(153\) − 45.4507i − 0.0240162i
\(154\) −1774.14 −0.928338
\(155\) 0 0
\(156\) 99.9920 0.0513190
\(157\) − 255.966i − 0.130117i −0.997881 0.0650584i \(-0.979277\pi\)
0.997881 0.0650584i \(-0.0207234\pi\)
\(158\) − 1283.41i − 0.646221i
\(159\) 217.295 0.108381
\(160\) 0 0
\(161\) 425.678 0.208373
\(162\) 1401.86i 0.679878i
\(163\) 321.632i 0.154553i 0.997010 + 0.0772767i \(0.0246225\pi\)
−0.997010 + 0.0772767i \(0.975378\pi\)
\(164\) −1863.72 −0.887390
\(165\) 0 0
\(166\) 1219.86 0.570360
\(167\) 2926.22i 1.35591i 0.735102 + 0.677957i \(0.237134\pi\)
−0.735102 + 0.677957i \(0.762866\pi\)
\(168\) 87.3520i 0.0401152i
\(169\) 401.640 0.182813
\(170\) 0 0
\(171\) −570.905 −0.255311
\(172\) − 729.495i − 0.323392i
\(173\) 1811.84i 0.796254i 0.917330 + 0.398127i \(0.130340\pi\)
−0.917330 + 0.398127i \(0.869660\pi\)
\(174\) 68.0037 0.0296284
\(175\) 0 0
\(176\) 766.873 0.328439
\(177\) 222.437i 0.0944599i
\(178\) − 2245.74i − 0.945646i
\(179\) −912.664 −0.381093 −0.190547 0.981678i \(-0.561026\pi\)
−0.190547 + 0.981678i \(0.561026\pi\)
\(180\) 0 0
\(181\) 3670.55 1.50735 0.753673 0.657249i \(-0.228280\pi\)
0.753673 + 0.657249i \(0.228280\pi\)
\(182\) − 1568.41i − 0.638781i
\(183\) 500.909i 0.202340i
\(184\) −184.000 −0.0737210
\(185\) 0 0
\(186\) −348.907 −0.137544
\(187\) − 81.7364i − 0.0319634i
\(188\) 1796.77i 0.697038i
\(189\) −585.826 −0.225463
\(190\) 0 0
\(191\) −1840.96 −0.697419 −0.348710 0.937231i \(-0.613380\pi\)
−0.348710 + 0.937231i \(0.613380\pi\)
\(192\) − 37.7580i − 0.0141925i
\(193\) − 611.817i − 0.228184i −0.993470 0.114092i \(-0.963604\pi\)
0.993470 0.114092i \(-0.0363959\pi\)
\(194\) −2857.59 −1.05754
\(195\) 0 0
\(196\) −1.85442 −0.000675808 0
\(197\) − 2830.26i − 1.02359i −0.859107 0.511796i \(-0.828980\pi\)
0.859107 0.511796i \(-0.171020\pi\)
\(198\) 2554.83i 0.916990i
\(199\) 1162.74 0.414195 0.207097 0.978320i \(-0.433598\pi\)
0.207097 + 0.978320i \(0.433598\pi\)
\(200\) 0 0
\(201\) 54.4792 0.0191177
\(202\) − 3024.31i − 1.05341i
\(203\) − 1066.66i − 0.368792i
\(204\) −4.02440 −0.00138120
\(205\) 0 0
\(206\) −1914.56 −0.647542
\(207\) − 612.995i − 0.205826i
\(208\) 677.947i 0.225996i
\(209\) −1026.69 −0.339797
\(210\) 0 0
\(211\) 1399.58 0.456641 0.228320 0.973586i \(-0.426677\pi\)
0.228320 + 0.973586i \(0.426677\pi\)
\(212\) 1473.26i 0.477283i
\(213\) − 369.825i − 0.118967i
\(214\) −3484.32 −1.11300
\(215\) 0 0
\(216\) 253.224 0.0797672
\(217\) 5472.72i 1.71204i
\(218\) − 2333.54i − 0.724988i
\(219\) −259.130 −0.0799562
\(220\) 0 0
\(221\) 72.2583 0.0219938
\(222\) − 9.26469i − 0.00280092i
\(223\) 4257.98i 1.27863i 0.768943 + 0.639317i \(0.220783\pi\)
−0.768943 + 0.639317i \(0.779217\pi\)
\(224\) −592.248 −0.176657
\(225\) 0 0
\(226\) 786.574 0.231514
\(227\) 4025.03i 1.17688i 0.808542 + 0.588438i \(0.200257\pi\)
−0.808542 + 0.588438i \(0.799743\pi\)
\(228\) 50.5504i 0.0146833i
\(229\) −3623.04 −1.04549 −0.522745 0.852489i \(-0.675092\pi\)
−0.522745 + 0.852489i \(0.675092\pi\)
\(230\) 0 0
\(231\) −523.343 −0.149063
\(232\) 461.066i 0.130476i
\(233\) 6502.81i 1.82838i 0.405282 + 0.914192i \(0.367173\pi\)
−0.405282 + 0.914192i \(0.632827\pi\)
\(234\) −2258.58 −0.630973
\(235\) 0 0
\(236\) −1508.13 −0.415977
\(237\) − 378.587i − 0.103763i
\(238\) 63.1241i 0.0171921i
\(239\) 2690.07 0.728059 0.364030 0.931387i \(-0.381401\pi\)
0.364030 + 0.931387i \(0.381401\pi\)
\(240\) 0 0
\(241\) −44.6958 −0.0119465 −0.00597326 0.999982i \(-0.501901\pi\)
−0.00597326 + 0.999982i \(0.501901\pi\)
\(242\) 1932.49i 0.513327i
\(243\) 1268.16i 0.334783i
\(244\) −3396.17 −0.891055
\(245\) 0 0
\(246\) −549.768 −0.142488
\(247\) − 907.635i − 0.233811i
\(248\) − 2365.59i − 0.605707i
\(249\) 359.841 0.0915824
\(250\) 0 0
\(251\) −6801.41 −1.71036 −0.855181 0.518329i \(-0.826554\pi\)
−0.855181 + 0.518329i \(0.826554\pi\)
\(252\) − 1973.07i − 0.493221i
\(253\) − 1102.38i − 0.273937i
\(254\) −2135.74 −0.527591
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 5576.00i − 1.35339i −0.736263 0.676695i \(-0.763411\pi\)
0.736263 0.676695i \(-0.236589\pi\)
\(258\) − 215.190i − 0.0519269i
\(259\) −145.320 −0.0348638
\(260\) 0 0
\(261\) −1536.04 −0.364285
\(262\) − 350.993i − 0.0827651i
\(263\) − 5669.58i − 1.32928i −0.747163 0.664641i \(-0.768585\pi\)
0.747163 0.664641i \(-0.231415\pi\)
\(264\) 226.216 0.0527372
\(265\) 0 0
\(266\) 792.900 0.182766
\(267\) − 662.458i − 0.151842i
\(268\) 369.370i 0.0841897i
\(269\) −6040.21 −1.36906 −0.684532 0.728983i \(-0.739994\pi\)
−0.684532 + 0.728983i \(0.739994\pi\)
\(270\) 0 0
\(271\) −6899.26 −1.54650 −0.773248 0.634104i \(-0.781369\pi\)
−0.773248 + 0.634104i \(0.781369\pi\)
\(272\) − 27.2855i − 0.00608245i
\(273\) − 462.657i − 0.102569i
\(274\) 951.795 0.209854
\(275\) 0 0
\(276\) −54.2772 −0.0118373
\(277\) − 5617.68i − 1.21853i −0.792965 0.609267i \(-0.791464\pi\)
0.792965 0.609267i \(-0.208536\pi\)
\(278\) − 306.334i − 0.0660889i
\(279\) 7880.96 1.69111
\(280\) 0 0
\(281\) 3069.89 0.651723 0.325861 0.945418i \(-0.394346\pi\)
0.325861 + 0.945418i \(0.394346\pi\)
\(282\) 530.020i 0.111923i
\(283\) 1404.86i 0.295089i 0.989055 + 0.147544i \(0.0471369\pi\)
−0.989055 + 0.147544i \(0.952863\pi\)
\(284\) 2507.42 0.523901
\(285\) 0 0
\(286\) −4061.72 −0.839770
\(287\) 8623.30i 1.77358i
\(288\) 852.862i 0.174498i
\(289\) 4910.09 0.999408
\(290\) 0 0
\(291\) −842.946 −0.169809
\(292\) − 1756.91i − 0.352107i
\(293\) − 2407.05i − 0.479936i −0.970781 0.239968i \(-0.922863\pi\)
0.970781 0.239968i \(-0.0771369\pi\)
\(294\) −0.547025 −0.000108514 0
\(295\) 0 0
\(296\) 62.8147 0.0123346
\(297\) 1517.12i 0.296404i
\(298\) − 1013.81i − 0.197076i
\(299\) 974.549 0.188494
\(300\) 0 0
\(301\) −3375.33 −0.646347
\(302\) 4875.68i 0.929020i
\(303\) − 892.124i − 0.169146i
\(304\) −342.732 −0.0646614
\(305\) 0 0
\(306\) 90.9015 0.0169820
\(307\) 459.743i 0.0854689i 0.999086 + 0.0427344i \(0.0136069\pi\)
−0.999086 + 0.0427344i \(0.986393\pi\)
\(308\) − 3548.27i − 0.656434i
\(309\) −564.765 −0.103975
\(310\) 0 0
\(311\) 4119.48 0.751107 0.375553 0.926801i \(-0.377453\pi\)
0.375553 + 0.926801i \(0.377453\pi\)
\(312\) 199.984i 0.0362880i
\(313\) 1684.15i 0.304133i 0.988370 + 0.152066i \(0.0485927\pi\)
−0.988370 + 0.152066i \(0.951407\pi\)
\(314\) 511.933 0.0920065
\(315\) 0 0
\(316\) 2566.83 0.456947
\(317\) − 8686.47i − 1.53906i −0.638613 0.769528i \(-0.720492\pi\)
0.638613 0.769528i \(-0.279508\pi\)
\(318\) 434.590i 0.0766371i
\(319\) −2762.34 −0.484831
\(320\) 0 0
\(321\) −1027.82 −0.178714
\(322\) 851.356i 0.147342i
\(323\) 36.5298i 0.00629279i
\(324\) −2803.71 −0.480746
\(325\) 0 0
\(326\) −643.265 −0.109286
\(327\) − 688.360i − 0.116411i
\(328\) − 3727.44i − 0.627479i
\(329\) 8313.55 1.39313
\(330\) 0 0
\(331\) 4307.91 0.715359 0.357680 0.933844i \(-0.383568\pi\)
0.357680 + 0.933844i \(0.383568\pi\)
\(332\) 2439.73i 0.403306i
\(333\) 209.267i 0.0344377i
\(334\) −5852.44 −0.958776
\(335\) 0 0
\(336\) −174.704 −0.0283657
\(337\) 290.156i 0.0469015i 0.999725 + 0.0234507i \(0.00746529\pi\)
−0.999725 + 0.0234507i \(0.992535\pi\)
\(338\) 803.280i 0.129268i
\(339\) 232.027 0.0371740
\(340\) 0 0
\(341\) 14172.7 2.25072
\(342\) − 1141.81i − 0.180532i
\(343\) 6356.73i 1.00067i
\(344\) 1458.99 0.228673
\(345\) 0 0
\(346\) −3623.69 −0.563037
\(347\) − 1042.94i − 0.161349i −0.996741 0.0806743i \(-0.974293\pi\)
0.996741 0.0806743i \(-0.0257074\pi\)
\(348\) 136.007i 0.0209505i
\(349\) 1819.56 0.279080 0.139540 0.990216i \(-0.455438\pi\)
0.139540 + 0.990216i \(0.455438\pi\)
\(350\) 0 0
\(351\) −1341.19 −0.203953
\(352\) 1533.75i 0.232241i
\(353\) − 4514.29i − 0.680656i −0.940307 0.340328i \(-0.889462\pi\)
0.940307 0.340328i \(-0.110538\pi\)
\(354\) −444.874 −0.0667932
\(355\) 0 0
\(356\) 4491.47 0.668673
\(357\) 18.6207i 0.00276053i
\(358\) − 1825.33i − 0.269474i
\(359\) −11527.9 −1.69476 −0.847379 0.530988i \(-0.821821\pi\)
−0.847379 + 0.530988i \(0.821821\pi\)
\(360\) 0 0
\(361\) −6400.15 −0.933103
\(362\) 7341.10i 1.06585i
\(363\) 570.055i 0.0824246i
\(364\) 3136.82 0.451686
\(365\) 0 0
\(366\) −1001.82 −0.143076
\(367\) − 6894.09i − 0.980568i −0.871563 0.490284i \(-0.836893\pi\)
0.871563 0.490284i \(-0.163107\pi\)
\(368\) − 368.000i − 0.0521286i
\(369\) 12417.9 1.75190
\(370\) 0 0
\(371\) 6816.69 0.953922
\(372\) − 697.814i − 0.0972580i
\(373\) 7733.25i 1.07349i 0.843744 + 0.536746i \(0.180347\pi\)
−0.843744 + 0.536746i \(0.819653\pi\)
\(374\) 163.473 0.0226016
\(375\) 0 0
\(376\) −3593.54 −0.492880
\(377\) − 2442.02i − 0.333608i
\(378\) − 1171.65i − 0.159427i
\(379\) 9495.72 1.28697 0.643486 0.765458i \(-0.277488\pi\)
0.643486 + 0.765458i \(0.277488\pi\)
\(380\) 0 0
\(381\) −630.011 −0.0847150
\(382\) − 3681.92i − 0.493150i
\(383\) − 12877.0i − 1.71797i −0.511998 0.858987i \(-0.671094\pi\)
0.511998 0.858987i \(-0.328906\pi\)
\(384\) 75.5161 0.0100356
\(385\) 0 0
\(386\) 1223.63 0.161351
\(387\) 4860.61i 0.638447i
\(388\) − 5715.18i − 0.747795i
\(389\) 8200.40 1.06884 0.534418 0.845221i \(-0.320531\pi\)
0.534418 + 0.845221i \(0.320531\pi\)
\(390\) 0 0
\(391\) −39.2229 −0.00507312
\(392\) − 3.70884i 0 0.000477869i
\(393\) − 103.538i − 0.0132895i
\(394\) 5660.52 0.723789
\(395\) 0 0
\(396\) −5109.66 −0.648410
\(397\) − 7842.68i − 0.991468i −0.868475 0.495734i \(-0.834899\pi\)
0.868475 0.495734i \(-0.165101\pi\)
\(398\) 2325.49i 0.292880i
\(399\) 233.893 0.0293467
\(400\) 0 0
\(401\) −14534.2 −1.80999 −0.904993 0.425427i \(-0.860124\pi\)
−0.904993 + 0.425427i \(0.860124\pi\)
\(402\) 108.958i 0.0135183i
\(403\) 12529.3i 1.54870i
\(404\) 6048.61 0.744875
\(405\) 0 0
\(406\) 2133.32 0.260776
\(407\) 376.335i 0.0458335i
\(408\) − 8.04881i 0 0.000976655i
\(409\) −13388.0 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(410\) 0 0
\(411\) 280.765 0.0336962
\(412\) − 3829.12i − 0.457881i
\(413\) 6978.00i 0.831393i
\(414\) 1225.99 0.145541
\(415\) 0 0
\(416\) −1355.89 −0.159803
\(417\) − 90.3640i − 0.0106119i
\(418\) − 2053.38i − 0.240273i
\(419\) 3103.10 0.361805 0.180903 0.983501i \(-0.442098\pi\)
0.180903 + 0.983501i \(0.442098\pi\)
\(420\) 0 0
\(421\) −6075.82 −0.703367 −0.351684 0.936119i \(-0.614391\pi\)
−0.351684 + 0.936119i \(0.614391\pi\)
\(422\) 2799.16i 0.322894i
\(423\) − 11971.9i − 1.37610i
\(424\) −2946.53 −0.337490
\(425\) 0 0
\(426\) 739.649 0.0841224
\(427\) 15713.8i 1.78090i
\(428\) − 6968.63i − 0.787013i
\(429\) −1198.14 −0.134841
\(430\) 0 0
\(431\) 14120.3 1.57808 0.789040 0.614341i \(-0.210578\pi\)
0.789040 + 0.614341i \(0.210578\pi\)
\(432\) 506.448i 0.0564039i
\(433\) 8555.96i 0.949592i 0.880096 + 0.474796i \(0.157478\pi\)
−0.880096 + 0.474796i \(0.842522\pi\)
\(434\) −10945.4 −1.21060
\(435\) 0 0
\(436\) 4667.09 0.512644
\(437\) 492.678i 0.0539313i
\(438\) − 518.261i − 0.0565376i
\(439\) 10894.1 1.18439 0.592194 0.805796i \(-0.298262\pi\)
0.592194 + 0.805796i \(0.298262\pi\)
\(440\) 0 0
\(441\) 12.3560 0.00133419
\(442\) 144.517i 0.0155519i
\(443\) 16120.5i 1.72891i 0.502708 + 0.864456i \(0.332337\pi\)
−0.502708 + 0.864456i \(0.667663\pi\)
\(444\) 18.5294 0.00198055
\(445\) 0 0
\(446\) −8515.96 −0.904130
\(447\) − 299.059i − 0.0316443i
\(448\) − 1184.50i − 0.124915i
\(449\) −4811.29 −0.505699 −0.252849 0.967506i \(-0.581368\pi\)
−0.252849 + 0.967506i \(0.581368\pi\)
\(450\) 0 0
\(451\) 22331.8 2.33163
\(452\) 1573.15i 0.163705i
\(453\) 1438.25i 0.149172i
\(454\) −8050.06 −0.832177
\(455\) 0 0
\(456\) −101.101 −0.0103826
\(457\) 226.329i 0.0231667i 0.999933 + 0.0115834i \(0.00368718\pi\)
−0.999933 + 0.0115834i \(0.996313\pi\)
\(458\) − 7246.08i − 0.739273i
\(459\) 53.9793 0.00548919
\(460\) 0 0
\(461\) −4349.17 −0.439395 −0.219697 0.975568i \(-0.570507\pi\)
−0.219697 + 0.975568i \(0.570507\pi\)
\(462\) − 1046.69i − 0.105403i
\(463\) − 989.313i − 0.0993030i −0.998767 0.0496515i \(-0.984189\pi\)
0.998767 0.0496515i \(-0.0158111\pi\)
\(464\) −922.131 −0.0922605
\(465\) 0 0
\(466\) −13005.6 −1.29286
\(467\) − 8512.91i − 0.843535i −0.906704 0.421767i \(-0.861410\pi\)
0.906704 0.421767i \(-0.138590\pi\)
\(468\) − 4517.15i − 0.446165i
\(469\) 1709.05 0.168266
\(470\) 0 0
\(471\) 151.012 0.0147734
\(472\) − 3016.25i − 0.294140i
\(473\) 8741.10i 0.849717i
\(474\) 757.175 0.0733717
\(475\) 0 0
\(476\) −126.248 −0.0121567
\(477\) − 9816.33i − 0.942261i
\(478\) 5380.14i 0.514816i
\(479\) 12058.6 1.15025 0.575125 0.818065i \(-0.304953\pi\)
0.575125 + 0.818065i \(0.304953\pi\)
\(480\) 0 0
\(481\) −332.696 −0.0315377
\(482\) − 89.3916i − 0.00844746i
\(483\) 251.137i 0.0236587i
\(484\) −3864.98 −0.362977
\(485\) 0 0
\(486\) −2536.31 −0.236727
\(487\) − 11605.3i − 1.07985i −0.841714 0.539924i \(-0.818453\pi\)
0.841714 0.539924i \(-0.181547\pi\)
\(488\) − 6792.34i − 0.630071i
\(489\) −189.753 −0.0175479
\(490\) 0 0
\(491\) −4651.88 −0.427569 −0.213785 0.976881i \(-0.568579\pi\)
−0.213785 + 0.976881i \(0.568579\pi\)
\(492\) − 1099.54i − 0.100754i
\(493\) 98.2845i 0.00897872i
\(494\) 1815.27 0.165330
\(495\) 0 0
\(496\) 4731.19 0.428300
\(497\) − 11601.6i − 1.04709i
\(498\) 719.682i 0.0647585i
\(499\) 7953.03 0.713480 0.356740 0.934204i \(-0.383888\pi\)
0.356740 + 0.934204i \(0.383888\pi\)
\(500\) 0 0
\(501\) −1726.38 −0.153950
\(502\) − 13602.8i − 1.20941i
\(503\) − 11805.6i − 1.04649i −0.852182 0.523245i \(-0.824721\pi\)
0.852182 0.523245i \(-0.175279\pi\)
\(504\) 3946.14 0.348760
\(505\) 0 0
\(506\) 2204.76 0.193703
\(507\) 236.955i 0.0207565i
\(508\) − 4271.48i − 0.373063i
\(509\) 2732.89 0.237983 0.118991 0.992895i \(-0.462034\pi\)
0.118991 + 0.992895i \(0.462034\pi\)
\(510\) 0 0
\(511\) −8129.09 −0.703738
\(512\) 512.000i 0.0441942i
\(513\) − 678.032i − 0.0583545i
\(514\) 11152.0 0.956992
\(515\) 0 0
\(516\) 430.380 0.0367178
\(517\) − 21529.6i − 1.83147i
\(518\) − 290.640i − 0.0246525i
\(519\) −1068.93 −0.0904065
\(520\) 0 0
\(521\) 14710.9 1.23704 0.618518 0.785771i \(-0.287733\pi\)
0.618518 + 0.785771i \(0.287733\pi\)
\(522\) − 3072.07i − 0.257588i
\(523\) − 7034.35i − 0.588127i −0.955786 0.294064i \(-0.904992\pi\)
0.955786 0.294064i \(-0.0950078\pi\)
\(524\) 701.987 0.0585237
\(525\) 0 0
\(526\) 11339.2 0.939944
\(527\) − 504.269i − 0.0416818i
\(528\) 452.432i 0.0372908i
\(529\) −529.000 −0.0434783
\(530\) 0 0
\(531\) 10048.6 0.821230
\(532\) 1585.80i 0.129235i
\(533\) 19742.2i 1.60437i
\(534\) 1324.92 0.107368
\(535\) 0 0
\(536\) −738.739 −0.0595311
\(537\) − 538.444i − 0.0432692i
\(538\) − 12080.4i − 0.968075i
\(539\) 22.2204 0.00177569
\(540\) 0 0
\(541\) 1552.90 0.123409 0.0617045 0.998094i \(-0.480346\pi\)
0.0617045 + 0.998094i \(0.480346\pi\)
\(542\) − 13798.5i − 1.09354i
\(543\) 2165.51i 0.171144i
\(544\) 54.5710 0.00430094
\(545\) 0 0
\(546\) 925.313 0.0725270
\(547\) − 174.657i − 0.0136523i −0.999977 0.00682614i \(-0.997827\pi\)
0.999977 0.00682614i \(-0.00217284\pi\)
\(548\) 1903.59i 0.148389i
\(549\) 22628.6 1.75914
\(550\) 0 0
\(551\) 1234.55 0.0954510
\(552\) − 108.554i − 0.00837026i
\(553\) − 11876.5i − 0.913276i
\(554\) 11235.4 0.861633
\(555\) 0 0
\(556\) 612.669 0.0467319
\(557\) 1990.63i 0.151429i 0.997130 + 0.0757143i \(0.0241237\pi\)
−0.997130 + 0.0757143i \(0.975876\pi\)
\(558\) 15761.9i 1.19580i
\(559\) −7727.48 −0.584683
\(560\) 0 0
\(561\) 48.2220 0.00362912
\(562\) 6139.78i 0.460838i
\(563\) 208.006i 0.0155709i 0.999970 + 0.00778543i \(0.00247820\pi\)
−0.999970 + 0.00778543i \(0.997522\pi\)
\(564\) −1060.04 −0.0791414
\(565\) 0 0
\(566\) −2809.71 −0.208659
\(567\) 12972.6i 0.960842i
\(568\) 5014.83i 0.370454i
\(569\) 3003.05 0.221256 0.110628 0.993862i \(-0.464714\pi\)
0.110628 + 0.993862i \(0.464714\pi\)
\(570\) 0 0
\(571\) −10796.1 −0.791249 −0.395624 0.918412i \(-0.629472\pi\)
−0.395624 + 0.918412i \(0.629472\pi\)
\(572\) − 8123.43i − 0.593807i
\(573\) − 1086.11i − 0.0791847i
\(574\) −17246.6 −1.25411
\(575\) 0 0
\(576\) −1705.72 −0.123389
\(577\) 26011.3i 1.87671i 0.345667 + 0.938357i \(0.387653\pi\)
−0.345667 + 0.938357i \(0.612347\pi\)
\(578\) 9820.18i 0.706688i
\(579\) 360.953 0.0259079
\(580\) 0 0
\(581\) 11288.5 0.806066
\(582\) − 1685.89i − 0.120073i
\(583\) − 17653.2i − 1.25407i
\(584\) 3513.81 0.248977
\(585\) 0 0
\(586\) 4814.09 0.339366
\(587\) 2774.97i 0.195120i 0.995230 + 0.0975598i \(0.0311037\pi\)
−0.995230 + 0.0975598i \(0.968896\pi\)
\(588\) − 1.09405i 0 7.67311e-5i
\(589\) −6334.11 −0.443111
\(590\) 0 0
\(591\) 1669.77 0.116218
\(592\) 125.629i 0.00872185i
\(593\) − 2384.07i − 0.165096i −0.996587 0.0825481i \(-0.973694\pi\)
0.996587 0.0825481i \(-0.0263058\pi\)
\(594\) −3034.23 −0.209589
\(595\) 0 0
\(596\) 2027.62 0.139353
\(597\) 685.984i 0.0470276i
\(598\) 1949.10i 0.133285i
\(599\) 1090.87 0.0744102 0.0372051 0.999308i \(-0.488155\pi\)
0.0372051 + 0.999308i \(0.488155\pi\)
\(600\) 0 0
\(601\) −5192.37 −0.352414 −0.176207 0.984353i \(-0.556383\pi\)
−0.176207 + 0.984353i \(0.556383\pi\)
\(602\) − 6750.65i − 0.457036i
\(603\) − 2461.10i − 0.166209i
\(604\) −9751.36 −0.656916
\(605\) 0 0
\(606\) 1784.25 0.119604
\(607\) − 1205.44i − 0.0806054i −0.999188 0.0403027i \(-0.987168\pi\)
0.999188 0.0403027i \(-0.0128322\pi\)
\(608\) − 685.465i − 0.0457225i
\(609\) 629.297 0.0418726
\(610\) 0 0
\(611\) 19033.1 1.26022
\(612\) 181.803i 0.0120081i
\(613\) − 14243.2i − 0.938466i −0.883075 0.469233i \(-0.844531\pi\)
0.883075 0.469233i \(-0.155469\pi\)
\(614\) −919.487 −0.0604356
\(615\) 0 0
\(616\) 7096.55 0.464169
\(617\) 422.492i 0.0275671i 0.999905 + 0.0137835i \(0.00438757\pi\)
−0.999905 + 0.0137835i \(0.995612\pi\)
\(618\) − 1129.53i − 0.0735217i
\(619\) 8826.35 0.573119 0.286560 0.958062i \(-0.407488\pi\)
0.286560 + 0.958062i \(0.407488\pi\)
\(620\) 0 0
\(621\) 728.019 0.0470441
\(622\) 8238.96i 0.531113i
\(623\) − 20781.7i − 1.33644i
\(624\) −399.968 −0.0256595
\(625\) 0 0
\(626\) −3368.29 −0.215054
\(627\) − 605.715i − 0.0385804i
\(628\) 1023.87i 0.0650584i
\(629\) 13.3901 0.000848804 0
\(630\) 0 0
\(631\) −19684.1 −1.24186 −0.620929 0.783867i \(-0.713244\pi\)
−0.620929 + 0.783867i \(0.713244\pi\)
\(632\) 5133.65i 0.323110i
\(633\) 825.710i 0.0518468i
\(634\) 17372.9 1.08828
\(635\) 0 0
\(636\) −869.180 −0.0541906
\(637\) 19.6437i 0.00122184i
\(638\) − 5524.67i − 0.342827i
\(639\) −16706.9 −1.03429
\(640\) 0 0
\(641\) 15113.1 0.931253 0.465626 0.884981i \(-0.345829\pi\)
0.465626 + 0.884981i \(0.345829\pi\)
\(642\) − 2055.64i − 0.126370i
\(643\) − 16917.8i − 1.03760i −0.854897 0.518798i \(-0.826380\pi\)
0.854897 0.518798i \(-0.173620\pi\)
\(644\) −1702.71 −0.104187
\(645\) 0 0
\(646\) −73.0596 −0.00444968
\(647\) − 19564.5i − 1.18881i −0.804166 0.594405i \(-0.797388\pi\)
0.804166 0.594405i \(-0.202612\pi\)
\(648\) − 5607.42i − 0.339939i
\(649\) 18071.0 1.09299
\(650\) 0 0
\(651\) −3228.74 −0.194384
\(652\) − 1286.53i − 0.0772767i
\(653\) − 22735.0i − 1.36247i −0.732067 0.681233i \(-0.761444\pi\)
0.732067 0.681233i \(-0.238556\pi\)
\(654\) 1376.72 0.0823149
\(655\) 0 0
\(656\) 7454.87 0.443695
\(657\) 11706.2i 0.695136i
\(658\) 16627.1i 0.985094i
\(659\) 30730.7 1.81654 0.908269 0.418388i \(-0.137405\pi\)
0.908269 + 0.418388i \(0.137405\pi\)
\(660\) 0 0
\(661\) −29204.3 −1.71848 −0.859240 0.511573i \(-0.829063\pi\)
−0.859240 + 0.511573i \(0.829063\pi\)
\(662\) 8615.81i 0.505835i
\(663\) 42.6302i 0.00249716i
\(664\) −4879.45 −0.285180
\(665\) 0 0
\(666\) −418.533 −0.0243511
\(667\) 1325.56i 0.0769506i
\(668\) − 11704.9i − 0.677957i
\(669\) −2512.08 −0.145176
\(670\) 0 0
\(671\) 40694.2 2.34126
\(672\) − 349.408i − 0.0200576i
\(673\) − 17530.1i − 1.00406i −0.864849 0.502032i \(-0.832586\pi\)
0.864849 0.502032i \(-0.167414\pi\)
\(674\) −580.312 −0.0331644
\(675\) 0 0
\(676\) −1606.56 −0.0914064
\(677\) 20553.4i 1.16681i 0.812180 + 0.583407i \(0.198281\pi\)
−0.812180 + 0.583407i \(0.801719\pi\)
\(678\) 464.055i 0.0262860i
\(679\) −26443.8 −1.49458
\(680\) 0 0
\(681\) −2374.65 −0.133622
\(682\) 28345.5i 1.59150i
\(683\) 13174.8i 0.738098i 0.929410 + 0.369049i \(0.120316\pi\)
−0.929410 + 0.369049i \(0.879684\pi\)
\(684\) 2283.62 0.127656
\(685\) 0 0
\(686\) −12713.5 −0.707584
\(687\) − 2137.48i − 0.118705i
\(688\) 2917.98i 0.161696i
\(689\) 15606.2 0.862913
\(690\) 0 0
\(691\) −14800.8 −0.814831 −0.407415 0.913243i \(-0.633570\pi\)
−0.407415 + 0.913243i \(0.633570\pi\)
\(692\) − 7247.38i − 0.398127i
\(693\) 23642.1i 1.29594i
\(694\) 2085.88 0.114091
\(695\) 0 0
\(696\) −272.015 −0.0148142
\(697\) − 794.570i − 0.0431800i
\(698\) 3639.13i 0.197339i
\(699\) −3836.46 −0.207594
\(700\) 0 0
\(701\) −12311.5 −0.663336 −0.331668 0.943396i \(-0.607611\pi\)
−0.331668 + 0.943396i \(0.607611\pi\)
\(702\) − 2682.38i − 0.144217i
\(703\) − 168.192i − 0.00902347i
\(704\) −3067.49 −0.164219
\(705\) 0 0
\(706\) 9028.59 0.481297
\(707\) − 27986.5i − 1.48874i
\(708\) − 889.749i − 0.0472299i
\(709\) −3893.89 −0.206260 −0.103130 0.994668i \(-0.532886\pi\)
−0.103130 + 0.994668i \(0.532886\pi\)
\(710\) 0 0
\(711\) −17102.7 −0.902113
\(712\) 8982.94i 0.472823i
\(713\) − 6801.08i − 0.357227i
\(714\) −37.2413 −0.00195199
\(715\) 0 0
\(716\) 3650.65 0.190547
\(717\) 1587.06i 0.0826636i
\(718\) − 23055.8i − 1.19838i
\(719\) 19942.5 1.03440 0.517198 0.855866i \(-0.326975\pi\)
0.517198 + 0.855866i \(0.326975\pi\)
\(720\) 0 0
\(721\) −17717.1 −0.915143
\(722\) − 12800.3i − 0.659803i
\(723\) − 26.3692i − 0.00135640i
\(724\) −14682.2 −0.753673
\(725\) 0 0
\(726\) −1140.11 −0.0582830
\(727\) − 36167.0i − 1.84506i −0.385926 0.922530i \(-0.626118\pi\)
0.385926 0.922530i \(-0.373882\pi\)
\(728\) 6273.63i 0.319391i
\(729\) 18176.9 0.923481
\(730\) 0 0
\(731\) 311.010 0.0157361
\(732\) − 2003.64i − 0.101170i
\(733\) − 11669.5i − 0.588025i −0.955802 0.294013i \(-0.905009\pi\)
0.955802 0.294013i \(-0.0949908\pi\)
\(734\) 13788.2 0.693366
\(735\) 0 0
\(736\) 736.000 0.0368605
\(737\) − 4425.93i − 0.221209i
\(738\) 24835.8i 1.23878i
\(739\) −75.0536 −0.00373598 −0.00186799 0.999998i \(-0.500595\pi\)
−0.00186799 + 0.999998i \(0.500595\pi\)
\(740\) 0 0
\(741\) 535.477 0.0265469
\(742\) 13633.4i 0.674524i
\(743\) 28279.8i 1.39635i 0.715929 + 0.698173i \(0.246003\pi\)
−0.715929 + 0.698173i \(0.753997\pi\)
\(744\) 1395.63 0.0687718
\(745\) 0 0
\(746\) −15466.5 −0.759074
\(747\) − 16255.9i − 0.796213i
\(748\) 326.946i 0.0159817i
\(749\) −32243.4 −1.57296
\(750\) 0 0
\(751\) −17268.1 −0.839044 −0.419522 0.907745i \(-0.637802\pi\)
−0.419522 + 0.907745i \(0.637802\pi\)
\(752\) − 7187.09i − 0.348519i
\(753\) − 4012.62i − 0.194194i
\(754\) 4884.03 0.235897
\(755\) 0 0
\(756\) 2343.30 0.112732
\(757\) 12525.8i 0.601400i 0.953719 + 0.300700i \(0.0972202\pi\)
−0.953719 + 0.300700i \(0.902780\pi\)
\(758\) 18991.4i 0.910026i
\(759\) 650.371 0.0311027
\(760\) 0 0
\(761\) −18670.1 −0.889343 −0.444672 0.895694i \(-0.646680\pi\)
−0.444672 + 0.895694i \(0.646680\pi\)
\(762\) − 1260.02i − 0.0599026i
\(763\) − 21594.3i − 1.02460i
\(764\) 7363.83 0.348710
\(765\) 0 0
\(766\) 25754.0 1.21479
\(767\) 15975.5i 0.752074i
\(768\) 151.032i 0.00709623i
\(769\) 32969.6 1.54605 0.773027 0.634373i \(-0.218742\pi\)
0.773027 + 0.634373i \(0.218742\pi\)
\(770\) 0 0
\(771\) 3289.67 0.153664
\(772\) 2447.27i 0.114092i
\(773\) − 23251.3i − 1.08188i −0.841062 0.540938i \(-0.818069\pi\)
0.841062 0.540938i \(-0.181931\pi\)
\(774\) −9721.23 −0.451450
\(775\) 0 0
\(776\) 11430.4 0.528771
\(777\) − 85.7342i − 0.00395843i
\(778\) 16400.8i 0.755781i
\(779\) −9980.57 −0.459039
\(780\) 0 0
\(781\) −30044.8 −1.37655
\(782\) − 78.4458i − 0.00358723i
\(783\) − 1824.26i − 0.0832617i
\(784\) 7.41767 0.000337904 0
\(785\) 0 0
\(786\) 207.075 0.00939712
\(787\) 29782.8i 1.34897i 0.738288 + 0.674486i \(0.235635\pi\)
−0.738288 + 0.674486i \(0.764365\pi\)
\(788\) 11321.0i 0.511796i
\(789\) 3344.88 0.150926
\(790\) 0 0
\(791\) 7278.86 0.327189
\(792\) − 10219.3i − 0.458495i
\(793\) 35975.3i 1.61100i
\(794\) 15685.4 0.701073
\(795\) 0 0
\(796\) −4650.98 −0.207097
\(797\) − 35307.2i − 1.56919i −0.620008 0.784596i \(-0.712871\pi\)
0.620008 0.784596i \(-0.287129\pi\)
\(798\) 467.787i 0.0207512i
\(799\) −766.029 −0.0339176
\(800\) 0 0
\(801\) −29926.6 −1.32011
\(802\) − 29068.4i − 1.27985i
\(803\) 21052.0i 0.925165i
\(804\) −217.917 −0.00955887
\(805\) 0 0
\(806\) −25058.5 −1.09510
\(807\) − 3563.54i − 0.155443i
\(808\) 12097.2i 0.526706i
\(809\) −7752.46 −0.336912 −0.168456 0.985709i \(-0.553878\pi\)
−0.168456 + 0.985709i \(0.553878\pi\)
\(810\) 0 0
\(811\) 23471.6 1.01627 0.508137 0.861276i \(-0.330334\pi\)
0.508137 + 0.861276i \(0.330334\pi\)
\(812\) 4266.64i 0.184396i
\(813\) − 4070.35i − 0.175589i
\(814\) −752.671 −0.0324092
\(815\) 0 0
\(816\) 16.0976 0.000690600 0
\(817\) − 3906.59i − 0.167288i
\(818\) − 26776.0i − 1.14450i
\(819\) −20900.6 −0.891727
\(820\) 0 0
\(821\) 2467.55 0.104894 0.0524470 0.998624i \(-0.483298\pi\)
0.0524470 + 0.998624i \(0.483298\pi\)
\(822\) 561.530i 0.0238268i
\(823\) 21984.5i 0.931144i 0.885010 + 0.465572i \(0.154151\pi\)
−0.885010 + 0.465572i \(0.845849\pi\)
\(824\) 7658.23 0.323771
\(825\) 0 0
\(826\) −13956.0 −0.587883
\(827\) 587.293i 0.0246943i 0.999924 + 0.0123472i \(0.00393032\pi\)
−0.999924 + 0.0123472i \(0.996070\pi\)
\(828\) 2451.98i 0.102913i
\(829\) −41822.8 −1.75219 −0.876096 0.482137i \(-0.839861\pi\)
−0.876096 + 0.482137i \(0.839861\pi\)
\(830\) 0 0
\(831\) 3314.26 0.138352
\(832\) − 2711.79i − 0.112998i
\(833\) − 0.790605i 0 3.28846e-5i
\(834\) 180.728 0.00750371
\(835\) 0 0
\(836\) 4106.76 0.169898
\(837\) 9359.77i 0.386524i
\(838\) 6206.20i 0.255835i
\(839\) 10126.5 0.416693 0.208346 0.978055i \(-0.433192\pi\)
0.208346 + 0.978055i \(0.433192\pi\)
\(840\) 0 0
\(841\) −21067.4 −0.863808
\(842\) − 12151.6i − 0.497356i
\(843\) 1811.14i 0.0739964i
\(844\) −5598.33 −0.228320
\(845\) 0 0
\(846\) 23943.7 0.973052
\(847\) 17883.0i 0.725463i
\(848\) − 5893.05i − 0.238642i
\(849\) −828.822 −0.0335043
\(850\) 0 0
\(851\) 180.592 0.00727453
\(852\) 1479.30i 0.0594835i
\(853\) 11584.4i 0.464995i 0.972597 + 0.232498i \(0.0746898\pi\)
−0.972597 + 0.232498i \(0.925310\pi\)
\(854\) −31427.7 −1.25929
\(855\) 0 0
\(856\) 13937.3 0.556502
\(857\) 9221.47i 0.367561i 0.982967 + 0.183780i \(0.0588335\pi\)
−0.982967 + 0.183780i \(0.941166\pi\)
\(858\) − 2396.29i − 0.0953472i
\(859\) −34654.2 −1.37647 −0.688235 0.725488i \(-0.741614\pi\)
−0.688235 + 0.725488i \(0.741614\pi\)
\(860\) 0 0
\(861\) −5087.48 −0.201372
\(862\) 28240.7i 1.11587i
\(863\) 42887.7i 1.69167i 0.533441 + 0.845837i \(0.320899\pi\)
−0.533441 + 0.845837i \(0.679101\pi\)
\(864\) −1012.90 −0.0398836
\(865\) 0 0
\(866\) −17111.9 −0.671463
\(867\) 2896.80i 0.113472i
\(868\) − 21890.9i − 0.856020i
\(869\) −30756.7 −1.20063
\(870\) 0 0
\(871\) 3912.70 0.152212
\(872\) 9334.17i 0.362494i
\(873\) 38080.2i 1.47631i
\(874\) −985.356 −0.0381352
\(875\) 0 0
\(876\) 1036.52 0.0399781
\(877\) − 45935.5i − 1.76868i −0.466846 0.884339i \(-0.654610\pi\)
0.466846 0.884339i \(-0.345390\pi\)
\(878\) 21788.2i 0.837488i
\(879\) 1420.08 0.0544917
\(880\) 0 0
\(881\) −71.9309 −0.00275075 −0.00137538 0.999999i \(-0.500438\pi\)
−0.00137538 + 0.999999i \(0.500438\pi\)
\(882\) 24.7119i 0 0.000943417i
\(883\) 20003.8i 0.762379i 0.924497 + 0.381189i \(0.124485\pi\)
−0.924497 + 0.381189i \(0.875515\pi\)
\(884\) −289.033 −0.0109969
\(885\) 0 0
\(886\) −32241.0 −1.22253
\(887\) − 26357.3i − 0.997737i −0.866678 0.498869i \(-0.833749\pi\)
0.866678 0.498869i \(-0.166251\pi\)
\(888\) 37.0588i 0.00140046i
\(889\) −19763.9 −0.745623
\(890\) 0 0
\(891\) 33595.2 1.26317
\(892\) − 17031.9i − 0.639317i
\(893\) 9622.06i 0.360571i
\(894\) 598.118 0.0223759
\(895\) 0 0
\(896\) 2368.99 0.0883286
\(897\) 574.954i 0.0214015i
\(898\) − 9622.57i − 0.357583i
\(899\) −17042.1 −0.632242
\(900\) 0 0
\(901\) −628.105 −0.0232244
\(902\) 44663.6i 1.64871i
\(903\) − 1991.34i − 0.0733860i
\(904\) −3146.30 −0.115757
\(905\) 0 0
\(906\) −2876.50 −0.105481
\(907\) 39300.4i 1.43875i 0.694620 + 0.719377i \(0.255572\pi\)
−0.694620 + 0.719377i \(0.744428\pi\)
\(908\) − 16100.1i − 0.588438i
\(909\) −40301.8 −1.47055
\(910\) 0 0
\(911\) 31099.2 1.13103 0.565513 0.824740i \(-0.308678\pi\)
0.565513 + 0.824740i \(0.308678\pi\)
\(912\) − 202.202i − 0.00734163i
\(913\) − 29233.8i − 1.05969i
\(914\) −452.657 −0.0163814
\(915\) 0 0
\(916\) 14492.2 0.522745
\(917\) − 3248.05i − 0.116968i
\(918\) 107.959i 0.00388144i
\(919\) −18935.8 −0.679690 −0.339845 0.940481i \(-0.610375\pi\)
−0.339845 + 0.940481i \(0.610375\pi\)
\(920\) 0 0
\(921\) −271.235 −0.00970411
\(922\) − 8698.33i − 0.310699i
\(923\) − 26560.9i − 0.947195i
\(924\) 2093.37 0.0745313
\(925\) 0 0
\(926\) 1978.63 0.0702178
\(927\) 25513.3i 0.903957i
\(928\) − 1844.26i − 0.0652380i
\(929\) −40156.1 −1.41817 −0.709085 0.705123i \(-0.750892\pi\)
−0.709085 + 0.705123i \(0.750892\pi\)
\(930\) 0 0
\(931\) −9.93077 −0.000349590 0
\(932\) − 26011.2i − 0.914192i
\(933\) 2430.37i 0.0852804i
\(934\) 17025.8 0.596469
\(935\) 0 0
\(936\) 9034.30 0.315486
\(937\) − 126.898i − 0.00442432i −0.999998 0.00221216i \(-0.999296\pi\)
0.999998 0.00221216i \(-0.000704153\pi\)
\(938\) 3418.10i 0.118982i
\(939\) −993.594 −0.0345311
\(940\) 0 0
\(941\) −19266.6 −0.667452 −0.333726 0.942670i \(-0.608306\pi\)
−0.333726 + 0.942670i \(0.608306\pi\)
\(942\) 302.025i 0.0104464i
\(943\) − 10716.4i − 0.370067i
\(944\) 6032.51 0.207989
\(945\) 0 0
\(946\) −17482.2 −0.600840
\(947\) 54560.4i 1.87220i 0.351732 + 0.936101i \(0.385593\pi\)
−0.351732 + 0.936101i \(0.614407\pi\)
\(948\) 1514.35i 0.0518816i
\(949\) −18610.8 −0.636598
\(950\) 0 0
\(951\) 5124.75 0.174744
\(952\) − 252.497i − 0.00859607i
\(953\) − 44976.2i − 1.52877i −0.644758 0.764387i \(-0.723042\pi\)
0.644758 0.764387i \(-0.276958\pi\)
\(954\) 19632.7 0.666279
\(955\) 0 0
\(956\) −10760.3 −0.364030
\(957\) − 1629.69i − 0.0550476i
\(958\) 24117.1i 0.813350i
\(959\) 8807.79 0.296578
\(960\) 0 0
\(961\) 57647.0 1.93505
\(962\) − 665.391i − 0.0223005i
\(963\) 46431.9i 1.55374i
\(964\) 178.783 0.00597326
\(965\) 0 0
\(966\) −502.274 −0.0167292
\(967\) 39431.7i 1.31131i 0.755059 + 0.655656i \(0.227608\pi\)
−0.755059 + 0.655656i \(0.772392\pi\)
\(968\) − 7729.96i − 0.256664i
\(969\) −21.5515 −0.000714482 0
\(970\) 0 0
\(971\) 20249.9 0.669258 0.334629 0.942350i \(-0.391389\pi\)
0.334629 + 0.942350i \(0.391389\pi\)
\(972\) − 5072.63i − 0.167392i
\(973\) − 2834.78i − 0.0934006i
\(974\) 23210.6 0.763567
\(975\) 0 0
\(976\) 13584.7 0.445527
\(977\) 9673.55i 0.316770i 0.987377 + 0.158385i \(0.0506287\pi\)
−0.987377 + 0.158385i \(0.949371\pi\)
\(978\) − 379.507i − 0.0124083i
\(979\) −53818.6 −1.75695
\(980\) 0 0
\(981\) −31096.7 −1.01207
\(982\) − 9303.77i − 0.302337i
\(983\) − 17264.3i − 0.560168i −0.959975 0.280084i \(-0.909638\pi\)
0.959975 0.280084i \(-0.0903623\pi\)
\(984\) 2199.07 0.0712438
\(985\) 0 0
\(986\) −196.569 −0.00634891
\(987\) 4904.74i 0.158176i
\(988\) 3630.54i 0.116906i
\(989\) 4194.60 0.134864
\(990\) 0 0
\(991\) 23892.9 0.765876 0.382938 0.923774i \(-0.374912\pi\)
0.382938 + 0.923774i \(0.374912\pi\)
\(992\) 9462.37i 0.302854i
\(993\) 2541.53i 0.0812217i
\(994\) 23203.3 0.740406
\(995\) 0 0
\(996\) −1439.36 −0.0457912
\(997\) 35830.2i 1.13817i 0.822280 + 0.569083i \(0.192702\pi\)
−0.822280 + 0.569083i \(0.807298\pi\)
\(998\) 15906.1i 0.504507i
\(999\) −248.534 −0.00787115
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 1150.4.b.n.599.7 8
5.2 odd 4 230.4.a.h.1.3 4
5.3 odd 4 1150.4.a.p.1.2 4
5.4 even 2 inner 1150.4.b.n.599.2 8
15.2 even 4 2070.4.a.bj.1.2 4
20.7 even 4 1840.4.a.m.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.h.1.3 4 5.2 odd 4
1150.4.a.p.1.2 4 5.3 odd 4
1150.4.b.n.599.2 8 5.4 even 2 inner
1150.4.b.n.599.7 8 1.1 even 1 trivial
1840.4.a.m.1.2 4 20.7 even 4
2070.4.a.bj.1.2 4 15.2 even 4