Properties

Label 1849.4.a.j.1.7
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $50$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.094531601\)
Analytic rank: \(1\)
Dimension: \(50\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.20684 q^{2} -4.82832 q^{3} +9.69750 q^{4} +7.01067 q^{5} +20.3120 q^{6} -27.8882 q^{7} -7.14112 q^{8} -3.68734 q^{9} +O(q^{10})\) \(q-4.20684 q^{2} -4.82832 q^{3} +9.69750 q^{4} +7.01067 q^{5} +20.3120 q^{6} -27.8882 q^{7} -7.14112 q^{8} -3.68734 q^{9} -29.4928 q^{10} -14.8584 q^{11} -46.8226 q^{12} -60.1508 q^{13} +117.321 q^{14} -33.8498 q^{15} -47.5385 q^{16} -53.5705 q^{17} +15.5121 q^{18} +77.3824 q^{19} +67.9860 q^{20} +134.653 q^{21} +62.5069 q^{22} +61.7129 q^{23} +34.4796 q^{24} -75.8505 q^{25} +253.045 q^{26} +148.168 q^{27} -270.446 q^{28} -53.9349 q^{29} +142.401 q^{30} -191.801 q^{31} +257.116 q^{32} +71.7411 q^{33} +225.363 q^{34} -195.515 q^{35} -35.7580 q^{36} +366.299 q^{37} -325.536 q^{38} +290.427 q^{39} -50.0641 q^{40} +5.32390 q^{41} -566.465 q^{42} -144.089 q^{44} -25.8508 q^{45} -259.616 q^{46} +559.264 q^{47} +229.531 q^{48} +434.753 q^{49} +319.091 q^{50} +258.655 q^{51} -583.312 q^{52} +97.6988 q^{53} -623.320 q^{54} -104.167 q^{55} +199.153 q^{56} -373.627 q^{57} +226.896 q^{58} +873.184 q^{59} -328.258 q^{60} -723.179 q^{61} +806.878 q^{62} +102.833 q^{63} -701.337 q^{64} -421.697 q^{65} -301.803 q^{66} -982.993 q^{67} -519.500 q^{68} -297.970 q^{69} +822.501 q^{70} -417.795 q^{71} +26.3318 q^{72} -219.278 q^{73} -1540.96 q^{74} +366.230 q^{75} +750.416 q^{76} +414.374 q^{77} -1221.78 q^{78} -801.309 q^{79} -333.277 q^{80} -615.845 q^{81} -22.3968 q^{82} -792.865 q^{83} +1305.80 q^{84} -375.565 q^{85} +260.415 q^{87} +106.106 q^{88} +1205.01 q^{89} +108.750 q^{90} +1677.50 q^{91} +598.461 q^{92} +926.078 q^{93} -2352.73 q^{94} +542.503 q^{95} -1241.44 q^{96} +577.561 q^{97} -1828.94 q^{98} +54.7880 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.20684 −1.48734 −0.743671 0.668545i \(-0.766917\pi\)
−0.743671 + 0.668545i \(0.766917\pi\)
\(3\) −4.82832 −0.929210 −0.464605 0.885518i \(-0.653804\pi\)
−0.464605 + 0.885518i \(0.653804\pi\)
\(4\) 9.69750 1.21219
\(5\) 7.01067 0.627054 0.313527 0.949579i \(-0.398489\pi\)
0.313527 + 0.949579i \(0.398489\pi\)
\(6\) 20.3120 1.38205
\(7\) −27.8882 −1.50582 −0.752911 0.658122i \(-0.771351\pi\)
−0.752911 + 0.658122i \(0.771351\pi\)
\(8\) −7.14112 −0.315596
\(9\) −3.68734 −0.136568
\(10\) −29.4928 −0.932644
\(11\) −14.8584 −0.407270 −0.203635 0.979047i \(-0.565276\pi\)
−0.203635 + 0.979047i \(0.565276\pi\)
\(12\) −46.8226 −1.12638
\(13\) −60.1508 −1.28329 −0.641647 0.767000i \(-0.721749\pi\)
−0.641647 + 0.767000i \(0.721749\pi\)
\(14\) 117.321 2.23967
\(15\) −33.8498 −0.582665
\(16\) −47.5385 −0.742789
\(17\) −53.5705 −0.764280 −0.382140 0.924104i \(-0.624813\pi\)
−0.382140 + 0.924104i \(0.624813\pi\)
\(18\) 15.5121 0.203124
\(19\) 77.3824 0.934355 0.467178 0.884163i \(-0.345271\pi\)
0.467178 + 0.884163i \(0.345271\pi\)
\(20\) 67.9860 0.760107
\(21\) 134.653 1.39923
\(22\) 62.5069 0.605751
\(23\) 61.7129 0.559480 0.279740 0.960076i \(-0.409752\pi\)
0.279740 + 0.960076i \(0.409752\pi\)
\(24\) 34.4796 0.293255
\(25\) −75.8505 −0.606804
\(26\) 253.045 1.90870
\(27\) 148.168 1.05611
\(28\) −270.446 −1.82534
\(29\) −53.9349 −0.345361 −0.172680 0.984978i \(-0.555243\pi\)
−0.172680 + 0.984978i \(0.555243\pi\)
\(30\) 142.401 0.866622
\(31\) −191.801 −1.11124 −0.555622 0.831435i \(-0.687520\pi\)
−0.555622 + 0.831435i \(0.687520\pi\)
\(32\) 257.116 1.42038
\(33\) 71.7411 0.378440
\(34\) 225.363 1.13675
\(35\) −195.515 −0.944231
\(36\) −35.7580 −0.165546
\(37\) 366.299 1.62755 0.813774 0.581182i \(-0.197410\pi\)
0.813774 + 0.581182i \(0.197410\pi\)
\(38\) −325.536 −1.38971
\(39\) 290.427 1.19245
\(40\) −50.0641 −0.197896
\(41\) 5.32390 0.0202793 0.0101397 0.999949i \(-0.496772\pi\)
0.0101397 + 0.999949i \(0.496772\pi\)
\(42\) −566.465 −2.08113
\(43\) 0 0
\(44\) −144.089 −0.493688
\(45\) −25.8508 −0.0856356
\(46\) −259.616 −0.832138
\(47\) 559.264 1.73568 0.867841 0.496843i \(-0.165507\pi\)
0.867841 + 0.496843i \(0.165507\pi\)
\(48\) 229.531 0.690207
\(49\) 434.753 1.26750
\(50\) 319.091 0.902525
\(51\) 258.655 0.710177
\(52\) −583.312 −1.55559
\(53\) 97.6988 0.253207 0.126603 0.991953i \(-0.459592\pi\)
0.126603 + 0.991953i \(0.459592\pi\)
\(54\) −623.320 −1.57080
\(55\) −104.167 −0.255380
\(56\) 199.153 0.475231
\(57\) −373.627 −0.868212
\(58\) 226.896 0.513670
\(59\) 873.184 1.92676 0.963380 0.268139i \(-0.0864086\pi\)
0.963380 + 0.268139i \(0.0864086\pi\)
\(60\) −328.258 −0.706299
\(61\) −723.179 −1.51793 −0.758964 0.651133i \(-0.774294\pi\)
−0.758964 + 0.651133i \(0.774294\pi\)
\(62\) 806.878 1.65280
\(63\) 102.833 0.205648
\(64\) −701.337 −1.36980
\(65\) −421.697 −0.804694
\(66\) −301.803 −0.562870
\(67\) −982.993 −1.79241 −0.896207 0.443637i \(-0.853688\pi\)
−0.896207 + 0.443637i \(0.853688\pi\)
\(68\) −519.500 −0.926451
\(69\) −297.970 −0.519874
\(70\) 822.501 1.40440
\(71\) −417.795 −0.698354 −0.349177 0.937057i \(-0.613539\pi\)
−0.349177 + 0.937057i \(0.613539\pi\)
\(72\) 26.3318 0.0431004
\(73\) −219.278 −0.351569 −0.175784 0.984429i \(-0.556246\pi\)
−0.175784 + 0.984429i \(0.556246\pi\)
\(74\) −1540.96 −2.42072
\(75\) 366.230 0.563848
\(76\) 750.416 1.13261
\(77\) 414.374 0.613277
\(78\) −1221.78 −1.77358
\(79\) −801.309 −1.14119 −0.570597 0.821230i \(-0.693288\pi\)
−0.570597 + 0.821230i \(0.693288\pi\)
\(80\) −333.277 −0.465768
\(81\) −615.845 −0.844781
\(82\) −22.3968 −0.0301623
\(83\) −792.865 −1.04853 −0.524266 0.851554i \(-0.675660\pi\)
−0.524266 + 0.851554i \(0.675660\pi\)
\(84\) 1305.80 1.69612
\(85\) −375.565 −0.479244
\(86\) 0 0
\(87\) 260.415 0.320913
\(88\) 106.106 0.128533
\(89\) 1205.01 1.43518 0.717589 0.696467i \(-0.245246\pi\)
0.717589 + 0.696467i \(0.245246\pi\)
\(90\) 108.750 0.127370
\(91\) 1677.50 1.93241
\(92\) 598.461 0.678194
\(93\) 926.078 1.03258
\(94\) −2352.73 −2.58155
\(95\) 542.503 0.585891
\(96\) −1241.44 −1.31983
\(97\) 577.561 0.604561 0.302281 0.953219i \(-0.402252\pi\)
0.302281 + 0.953219i \(0.402252\pi\)
\(98\) −1828.94 −1.88521
\(99\) 54.7880 0.0556202
\(100\) −735.560 −0.735560
\(101\) 1590.84 1.56727 0.783635 0.621221i \(-0.213363\pi\)
0.783635 + 0.621221i \(0.213363\pi\)
\(102\) −1088.12 −1.05628
\(103\) 1356.26 1.29744 0.648718 0.761029i \(-0.275305\pi\)
0.648718 + 0.761029i \(0.275305\pi\)
\(104\) 429.544 0.405002
\(105\) 944.010 0.877390
\(106\) −411.003 −0.376605
\(107\) 615.442 0.556047 0.278023 0.960574i \(-0.410321\pi\)
0.278023 + 0.960574i \(0.410321\pi\)
\(108\) 1436.86 1.28020
\(109\) 1400.49 1.23067 0.615333 0.788267i \(-0.289022\pi\)
0.615333 + 0.788267i \(0.289022\pi\)
\(110\) 438.215 0.379838
\(111\) −1768.61 −1.51233
\(112\) 1325.76 1.11851
\(113\) 1104.14 0.919193 0.459597 0.888128i \(-0.347994\pi\)
0.459597 + 0.888128i \(0.347994\pi\)
\(114\) 1571.79 1.29133
\(115\) 432.649 0.350824
\(116\) −523.034 −0.418642
\(117\) 221.797 0.175257
\(118\) −3673.35 −2.86575
\(119\) 1493.99 1.15087
\(120\) 241.725 0.183887
\(121\) −1110.23 −0.834131
\(122\) 3042.30 2.25768
\(123\) −25.7055 −0.0188438
\(124\) −1859.99 −1.34704
\(125\) −1408.10 −1.00755
\(126\) −432.604 −0.305868
\(127\) 561.803 0.392535 0.196268 0.980550i \(-0.437118\pi\)
0.196268 + 0.980550i \(0.437118\pi\)
\(128\) 893.486 0.616983
\(129\) 0 0
\(130\) 1774.01 1.19686
\(131\) 255.265 0.170249 0.0851245 0.996370i \(-0.472871\pi\)
0.0851245 + 0.996370i \(0.472871\pi\)
\(132\) 695.709 0.458740
\(133\) −2158.06 −1.40697
\(134\) 4135.29 2.66593
\(135\) 1038.76 0.662238
\(136\) 382.553 0.241204
\(137\) 1067.47 0.665692 0.332846 0.942981i \(-0.391991\pi\)
0.332846 + 0.942981i \(0.391991\pi\)
\(138\) 1253.51 0.773231
\(139\) −476.139 −0.290544 −0.145272 0.989392i \(-0.546406\pi\)
−0.145272 + 0.989392i \(0.546406\pi\)
\(140\) −1896.01 −1.14459
\(141\) −2700.30 −1.61281
\(142\) 1757.60 1.03869
\(143\) 893.744 0.522648
\(144\) 175.291 0.101441
\(145\) −378.120 −0.216560
\(146\) 922.466 0.522903
\(147\) −2099.13 −1.17778
\(148\) 3552.19 1.97289
\(149\) 517.779 0.284685 0.142343 0.989817i \(-0.454537\pi\)
0.142343 + 0.989817i \(0.454537\pi\)
\(150\) −1540.67 −0.838636
\(151\) −1364.45 −0.735347 −0.367674 0.929955i \(-0.619846\pi\)
−0.367674 + 0.929955i \(0.619846\pi\)
\(152\) −552.597 −0.294879
\(153\) 197.533 0.104376
\(154\) −1743.21 −0.912153
\(155\) −1344.66 −0.696809
\(156\) 2816.42 1.44547
\(157\) 2848.68 1.44808 0.724042 0.689756i \(-0.242282\pi\)
0.724042 + 0.689756i \(0.242282\pi\)
\(158\) 3370.98 1.69735
\(159\) −471.721 −0.235282
\(160\) 1802.55 0.890653
\(161\) −1721.06 −0.842477
\(162\) 2590.76 1.25648
\(163\) 2047.22 0.983745 0.491873 0.870667i \(-0.336313\pi\)
0.491873 + 0.870667i \(0.336313\pi\)
\(164\) 51.6285 0.0245824
\(165\) 502.953 0.237302
\(166\) 3335.45 1.55953
\(167\) −900.176 −0.417112 −0.208556 0.978010i \(-0.566876\pi\)
−0.208556 + 0.978010i \(0.566876\pi\)
\(168\) −961.575 −0.441590
\(169\) 1421.12 0.646845
\(170\) 1579.94 0.712801
\(171\) −285.336 −0.127603
\(172\) 0 0
\(173\) 1843.61 0.810215 0.405108 0.914269i \(-0.367234\pi\)
0.405108 + 0.914269i \(0.367234\pi\)
\(174\) −1095.52 −0.477307
\(175\) 2115.33 0.913739
\(176\) 706.345 0.302516
\(177\) −4216.01 −1.79037
\(178\) −5069.29 −2.13460
\(179\) 4436.88 1.85267 0.926335 0.376701i \(-0.122942\pi\)
0.926335 + 0.376701i \(0.122942\pi\)
\(180\) −250.688 −0.103806
\(181\) −3254.36 −1.33643 −0.668217 0.743966i \(-0.732942\pi\)
−0.668217 + 0.743966i \(0.732942\pi\)
\(182\) −7056.97 −2.87416
\(183\) 3491.74 1.41047
\(184\) −440.699 −0.176569
\(185\) 2568.00 1.02056
\(186\) −3895.86 −1.53580
\(187\) 795.972 0.311269
\(188\) 5423.46 2.10397
\(189\) −4132.15 −1.59032
\(190\) −2282.22 −0.871420
\(191\) −2246.48 −0.851043 −0.425522 0.904948i \(-0.639909\pi\)
−0.425522 + 0.904948i \(0.639909\pi\)
\(192\) 3386.28 1.27283
\(193\) −1020.49 −0.380605 −0.190302 0.981726i \(-0.560947\pi\)
−0.190302 + 0.981726i \(0.560947\pi\)
\(194\) −2429.71 −0.899190
\(195\) 2036.09 0.747730
\(196\) 4216.02 1.53645
\(197\) 2025.70 0.732614 0.366307 0.930494i \(-0.380622\pi\)
0.366307 + 0.930494i \(0.380622\pi\)
\(198\) −230.484 −0.0827263
\(199\) 2340.03 0.833569 0.416784 0.909005i \(-0.363157\pi\)
0.416784 + 0.909005i \(0.363157\pi\)
\(200\) 541.657 0.191505
\(201\) 4746.20 1.66553
\(202\) −6692.40 −2.33107
\(203\) 1504.15 0.520052
\(204\) 2508.31 0.860868
\(205\) 37.3241 0.0127162
\(206\) −5705.56 −1.92973
\(207\) −227.557 −0.0764072
\(208\) 2859.48 0.953216
\(209\) −1149.78 −0.380535
\(210\) −3971.30 −1.30498
\(211\) −3911.57 −1.27622 −0.638112 0.769943i \(-0.720284\pi\)
−0.638112 + 0.769943i \(0.720284\pi\)
\(212\) 947.435 0.306934
\(213\) 2017.25 0.648918
\(214\) −2589.07 −0.827032
\(215\) 0 0
\(216\) −1058.09 −0.333304
\(217\) 5349.00 1.67333
\(218\) −5891.64 −1.83042
\(219\) 1058.74 0.326681
\(220\) −1010.16 −0.309569
\(221\) 3222.31 0.980796
\(222\) 7440.26 2.24936
\(223\) −1.96080 −0.000588811 0 −0.000294406 1.00000i \(-0.500094\pi\)
−0.000294406 1.00000i \(0.500094\pi\)
\(224\) −7170.50 −2.13884
\(225\) 279.687 0.0828701
\(226\) −4644.94 −1.36715
\(227\) −49.4069 −0.0144461 −0.00722303 0.999974i \(-0.502299\pi\)
−0.00722303 + 0.999974i \(0.502299\pi\)
\(228\) −3623.25 −1.05244
\(229\) 1697.91 0.489960 0.244980 0.969528i \(-0.421219\pi\)
0.244980 + 0.969528i \(0.421219\pi\)
\(230\) −1820.09 −0.521795
\(231\) −2000.73 −0.569863
\(232\) 385.156 0.108995
\(233\) −3795.60 −1.06720 −0.533601 0.845736i \(-0.679162\pi\)
−0.533601 + 0.845736i \(0.679162\pi\)
\(234\) −933.063 −0.260668
\(235\) 3920.82 1.08837
\(236\) 8467.70 2.33560
\(237\) 3868.97 1.06041
\(238\) −6284.96 −1.71174
\(239\) −2950.09 −0.798433 −0.399216 0.916857i \(-0.630718\pi\)
−0.399216 + 0.916857i \(0.630718\pi\)
\(240\) 1609.17 0.432797
\(241\) 4078.25 1.09005 0.545027 0.838418i \(-0.316519\pi\)
0.545027 + 0.838418i \(0.316519\pi\)
\(242\) 4670.55 1.24064
\(243\) −1027.05 −0.271132
\(244\) −7013.03 −1.84001
\(245\) 3047.91 0.794791
\(246\) 108.139 0.0280271
\(247\) −4654.61 −1.19905
\(248\) 1369.68 0.350704
\(249\) 3828.20 0.974307
\(250\) 5923.64 1.49858
\(251\) −3528.89 −0.887417 −0.443708 0.896171i \(-0.646337\pi\)
−0.443708 + 0.896171i \(0.646337\pi\)
\(252\) 997.228 0.249284
\(253\) −916.955 −0.227860
\(254\) −2363.42 −0.583835
\(255\) 1813.35 0.445319
\(256\) 1851.94 0.452134
\(257\) 6455.09 1.56676 0.783380 0.621543i \(-0.213494\pi\)
0.783380 + 0.621543i \(0.213494\pi\)
\(258\) 0 0
\(259\) −10215.4 −2.45080
\(260\) −4089.41 −0.975441
\(261\) 198.877 0.0471654
\(262\) −1073.86 −0.253219
\(263\) 3618.65 0.848424 0.424212 0.905563i \(-0.360551\pi\)
0.424212 + 0.905563i \(0.360551\pi\)
\(264\) −512.312 −0.119434
\(265\) 684.935 0.158774
\(266\) 9078.61 2.09265
\(267\) −5818.17 −1.33358
\(268\) −9532.58 −2.17274
\(269\) 1258.66 0.285286 0.142643 0.989774i \(-0.454440\pi\)
0.142643 + 0.989774i \(0.454440\pi\)
\(270\) −4369.89 −0.984975
\(271\) −4808.76 −1.07790 −0.538951 0.842337i \(-0.681179\pi\)
−0.538951 + 0.842337i \(0.681179\pi\)
\(272\) 2546.66 0.567698
\(273\) −8099.50 −1.79562
\(274\) −4490.66 −0.990112
\(275\) 1127.02 0.247133
\(276\) −2889.56 −0.630185
\(277\) −6487.98 −1.40731 −0.703655 0.710542i \(-0.748450\pi\)
−0.703655 + 0.710542i \(0.748450\pi\)
\(278\) 2003.04 0.432138
\(279\) 707.237 0.151761
\(280\) 1396.20 0.297996
\(281\) −1896.75 −0.402672 −0.201336 0.979522i \(-0.564528\pi\)
−0.201336 + 0.979522i \(0.564528\pi\)
\(282\) 11359.7 2.39880
\(283\) 448.018 0.0941057 0.0470528 0.998892i \(-0.485017\pi\)
0.0470528 + 0.998892i \(0.485017\pi\)
\(284\) −4051.57 −0.846536
\(285\) −2619.38 −0.544416
\(286\) −3759.84 −0.777357
\(287\) −148.474 −0.0305371
\(288\) −948.074 −0.193978
\(289\) −2043.20 −0.415876
\(290\) 1590.69 0.322099
\(291\) −2788.65 −0.561765
\(292\) −2126.44 −0.426167
\(293\) −2086.41 −0.416005 −0.208002 0.978128i \(-0.566696\pi\)
−0.208002 + 0.978128i \(0.566696\pi\)
\(294\) 8830.69 1.75176
\(295\) 6121.61 1.20818
\(296\) −2615.79 −0.513647
\(297\) −2201.54 −0.430123
\(298\) −2178.21 −0.423425
\(299\) −3712.08 −0.717977
\(300\) 3551.52 0.683490
\(301\) 0 0
\(302\) 5740.03 1.09371
\(303\) −7681.08 −1.45632
\(304\) −3678.64 −0.694028
\(305\) −5069.97 −0.951822
\(306\) −830.989 −0.155243
\(307\) 393.588 0.0731702 0.0365851 0.999331i \(-0.488352\pi\)
0.0365851 + 0.999331i \(0.488352\pi\)
\(308\) 4018.40 0.743407
\(309\) −6548.44 −1.20559
\(310\) 5656.75 1.03639
\(311\) 2153.14 0.392583 0.196292 0.980546i \(-0.437110\pi\)
0.196292 + 0.980546i \(0.437110\pi\)
\(312\) −2073.98 −0.376332
\(313\) 6023.57 1.08777 0.543886 0.839159i \(-0.316953\pi\)
0.543886 + 0.839159i \(0.316953\pi\)
\(314\) −11983.9 −2.15380
\(315\) 720.932 0.128952
\(316\) −7770.69 −1.38334
\(317\) −6359.98 −1.12685 −0.563426 0.826166i \(-0.690517\pi\)
−0.563426 + 0.826166i \(0.690517\pi\)
\(318\) 1984.46 0.349946
\(319\) 801.387 0.140655
\(320\) −4916.84 −0.858937
\(321\) −2971.55 −0.516684
\(322\) 7240.24 1.25305
\(323\) −4145.42 −0.714109
\(324\) −5972.16 −1.02403
\(325\) 4562.47 0.778708
\(326\) −8612.32 −1.46317
\(327\) −6762.01 −1.14355
\(328\) −38.0186 −0.00640008
\(329\) −15596.9 −2.61363
\(330\) −2115.84 −0.352950
\(331\) −6984.33 −1.15980 −0.579900 0.814688i \(-0.696908\pi\)
−0.579900 + 0.814688i \(0.696908\pi\)
\(332\) −7688.81 −1.27102
\(333\) −1350.67 −0.222271
\(334\) 3786.90 0.620389
\(335\) −6891.44 −1.12394
\(336\) −6401.21 −1.03933
\(337\) 428.803 0.0693128 0.0346564 0.999399i \(-0.488966\pi\)
0.0346564 + 0.999399i \(0.488966\pi\)
\(338\) −5978.41 −0.962079
\(339\) −5331.14 −0.854124
\(340\) −3642.05 −0.580934
\(341\) 2849.86 0.452577
\(342\) 1200.36 0.189790
\(343\) −2558.83 −0.402810
\(344\) 0 0
\(345\) −2088.97 −0.325989
\(346\) −7755.78 −1.20507
\(347\) −871.352 −0.134803 −0.0674015 0.997726i \(-0.521471\pi\)
−0.0674015 + 0.997726i \(0.521471\pi\)
\(348\) 2525.38 0.389007
\(349\) −2954.05 −0.453085 −0.226543 0.974001i \(-0.572742\pi\)
−0.226543 + 0.974001i \(0.572742\pi\)
\(350\) −8898.87 −1.35904
\(351\) −8912.44 −1.35530
\(352\) −3820.33 −0.578478
\(353\) −1760.46 −0.265438 −0.132719 0.991154i \(-0.542371\pi\)
−0.132719 + 0.991154i \(0.542371\pi\)
\(354\) 17736.1 2.66289
\(355\) −2929.02 −0.437905
\(356\) 11685.6 1.73971
\(357\) −7213.44 −1.06940
\(358\) −18665.2 −2.75555
\(359\) 9960.60 1.46435 0.732173 0.681119i \(-0.238506\pi\)
0.732173 + 0.681119i \(0.238506\pi\)
\(360\) 184.603 0.0270263
\(361\) −870.958 −0.126980
\(362\) 13690.6 1.98774
\(363\) 5360.53 0.775083
\(364\) 16267.5 2.34245
\(365\) −1537.28 −0.220452
\(366\) −14689.2 −2.09786
\(367\) 1291.52 0.183697 0.0918485 0.995773i \(-0.470722\pi\)
0.0918485 + 0.995773i \(0.470722\pi\)
\(368\) −2933.74 −0.415575
\(369\) −19.6310 −0.00276951
\(370\) −10803.2 −1.51792
\(371\) −2724.65 −0.381285
\(372\) 8980.64 1.25168
\(373\) 7010.88 0.973216 0.486608 0.873620i \(-0.338234\pi\)
0.486608 + 0.873620i \(0.338234\pi\)
\(374\) −3348.53 −0.462963
\(375\) 6798.74 0.936228
\(376\) −3993.77 −0.547774
\(377\) 3244.23 0.443200
\(378\) 17383.3 2.36534
\(379\) 2688.90 0.364431 0.182216 0.983259i \(-0.441673\pi\)
0.182216 + 0.983259i \(0.441673\pi\)
\(380\) 5260.92 0.710210
\(381\) −2712.57 −0.364748
\(382\) 9450.56 1.26579
\(383\) 5518.96 0.736307 0.368153 0.929765i \(-0.379990\pi\)
0.368153 + 0.929765i \(0.379990\pi\)
\(384\) −4314.04 −0.573307
\(385\) 2905.04 0.384558
\(386\) 4293.05 0.566089
\(387\) 0 0
\(388\) 5600.90 0.732842
\(389\) −6817.59 −0.888601 −0.444300 0.895878i \(-0.646548\pi\)
−0.444300 + 0.895878i \(0.646548\pi\)
\(390\) −8565.50 −1.11213
\(391\) −3305.99 −0.427599
\(392\) −3104.62 −0.400018
\(393\) −1232.50 −0.158197
\(394\) −8521.78 −1.08965
\(395\) −5617.71 −0.715589
\(396\) 531.307 0.0674222
\(397\) 5351.01 0.676472 0.338236 0.941061i \(-0.390170\pi\)
0.338236 + 0.941061i \(0.390170\pi\)
\(398\) −9844.12 −1.23980
\(399\) 10419.8 1.30737
\(400\) 3605.82 0.450727
\(401\) −6590.42 −0.820723 −0.410362 0.911923i \(-0.634597\pi\)
−0.410362 + 0.911923i \(0.634597\pi\)
\(402\) −19966.5 −2.47721
\(403\) 11537.0 1.42605
\(404\) 15427.2 1.89983
\(405\) −4317.49 −0.529723
\(406\) −6327.72 −0.773496
\(407\) −5442.62 −0.662852
\(408\) −1847.09 −0.224129
\(409\) −4132.15 −0.499564 −0.249782 0.968302i \(-0.580359\pi\)
−0.249782 + 0.968302i \(0.580359\pi\)
\(410\) −157.017 −0.0189134
\(411\) −5154.07 −0.618568
\(412\) 13152.3 1.57274
\(413\) −24351.6 −2.90136
\(414\) 957.295 0.113644
\(415\) −5558.51 −0.657486
\(416\) −15465.7 −1.82276
\(417\) 2298.95 0.269976
\(418\) 4836.94 0.565986
\(419\) −5101.36 −0.594792 −0.297396 0.954754i \(-0.596118\pi\)
−0.297396 + 0.954754i \(0.596118\pi\)
\(420\) 9154.54 1.06356
\(421\) 3542.08 0.410048 0.205024 0.978757i \(-0.434273\pi\)
0.205024 + 0.978757i \(0.434273\pi\)
\(422\) 16455.3 1.89818
\(423\) −2062.20 −0.237039
\(424\) −697.679 −0.0799111
\(425\) 4063.35 0.463768
\(426\) −8486.23 −0.965163
\(427\) 20168.2 2.28573
\(428\) 5968.25 0.674033
\(429\) −4315.28 −0.485650
\(430\) 0 0
\(431\) −8499.53 −0.949903 −0.474951 0.880012i \(-0.657534\pi\)
−0.474951 + 0.880012i \(0.657534\pi\)
\(432\) −7043.69 −0.784467
\(433\) 7919.12 0.878911 0.439456 0.898264i \(-0.355171\pi\)
0.439456 + 0.898264i \(0.355171\pi\)
\(434\) −22502.4 −2.48882
\(435\) 1825.68 0.201230
\(436\) 13581.3 1.49180
\(437\) 4775.50 0.522753
\(438\) −4453.96 −0.485887
\(439\) 13630.2 1.48186 0.740929 0.671583i \(-0.234385\pi\)
0.740929 + 0.671583i \(0.234385\pi\)
\(440\) 743.872 0.0805970
\(441\) −1603.08 −0.173101
\(442\) −13555.7 −1.45878
\(443\) −15066.5 −1.61587 −0.807936 0.589270i \(-0.799416\pi\)
−0.807936 + 0.589270i \(0.799416\pi\)
\(444\) −17151.1 −1.83323
\(445\) 8447.94 0.899934
\(446\) 8.24877 0.000875764 0
\(447\) −2500.00 −0.264533
\(448\) 19559.0 2.06267
\(449\) 4920.86 0.517215 0.258608 0.965982i \(-0.416736\pi\)
0.258608 + 0.965982i \(0.416736\pi\)
\(450\) −1176.60 −0.123256
\(451\) −79.1046 −0.00825918
\(452\) 10707.4 1.11423
\(453\) 6588.00 0.683292
\(454\) 207.847 0.0214862
\(455\) 11760.4 1.21173
\(456\) 2668.12 0.274004
\(457\) −8415.53 −0.861404 −0.430702 0.902494i \(-0.641734\pi\)
−0.430702 + 0.902494i \(0.641734\pi\)
\(458\) −7142.82 −0.728738
\(459\) −7937.45 −0.807164
\(460\) 4195.62 0.425264
\(461\) −17706.4 −1.78887 −0.894437 0.447194i \(-0.852423\pi\)
−0.894437 + 0.447194i \(0.852423\pi\)
\(462\) 8416.76 0.847582
\(463\) −9891.44 −0.992860 −0.496430 0.868077i \(-0.665356\pi\)
−0.496430 + 0.868077i \(0.665356\pi\)
\(464\) 2563.98 0.256530
\(465\) 6492.43 0.647482
\(466\) 15967.5 1.58730
\(467\) −5637.96 −0.558659 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(468\) 2150.87 0.212445
\(469\) 27413.9 2.69906
\(470\) −16494.2 −1.61877
\(471\) −13754.3 −1.34557
\(472\) −6235.51 −0.608078
\(473\) 0 0
\(474\) −16276.1 −1.57719
\(475\) −5869.49 −0.566970
\(476\) 14487.9 1.39507
\(477\) −360.249 −0.0345800
\(478\) 12410.6 1.18754
\(479\) −16670.3 −1.59015 −0.795077 0.606508i \(-0.792570\pi\)
−0.795077 + 0.606508i \(0.792570\pi\)
\(480\) −8703.30 −0.827603
\(481\) −22033.2 −2.08862
\(482\) −17156.5 −1.62128
\(483\) 8309.84 0.782838
\(484\) −10766.4 −1.01112
\(485\) 4049.09 0.379092
\(486\) 4320.62 0.403266
\(487\) 16410.3 1.52694 0.763470 0.645843i \(-0.223494\pi\)
0.763470 + 0.645843i \(0.223494\pi\)
\(488\) 5164.31 0.479052
\(489\) −9884.62 −0.914106
\(490\) −12822.1 −1.18213
\(491\) −3171.88 −0.291538 −0.145769 0.989319i \(-0.546566\pi\)
−0.145769 + 0.989319i \(0.546566\pi\)
\(492\) −249.279 −0.0228422
\(493\) 2889.32 0.263952
\(494\) 19581.2 1.78340
\(495\) 384.101 0.0348769
\(496\) 9117.94 0.825419
\(497\) 11651.6 1.05160
\(498\) −16104.6 −1.44913
\(499\) −10898.7 −0.977737 −0.488869 0.872357i \(-0.662590\pi\)
−0.488869 + 0.872357i \(0.662590\pi\)
\(500\) −13655.0 −1.22134
\(501\) 4346.34 0.387585
\(502\) 14845.5 1.31989
\(503\) −932.683 −0.0826765 −0.0413383 0.999145i \(-0.513162\pi\)
−0.0413383 + 0.999145i \(0.513162\pi\)
\(504\) −734.346 −0.0649015
\(505\) 11152.8 0.982763
\(506\) 3857.48 0.338905
\(507\) −6861.61 −0.601055
\(508\) 5448.09 0.475827
\(509\) −18569.3 −1.61703 −0.808517 0.588473i \(-0.799729\pi\)
−0.808517 + 0.588473i \(0.799729\pi\)
\(510\) −7628.47 −0.662342
\(511\) 6115.26 0.529400
\(512\) −14938.7 −1.28946
\(513\) 11465.6 0.986783
\(514\) −27155.5 −2.33031
\(515\) 9508.27 0.813562
\(516\) 0 0
\(517\) −8309.76 −0.706892
\(518\) 42974.7 3.64517
\(519\) −8901.54 −0.752860
\(520\) 3011.39 0.253958
\(521\) 21616.7 1.81774 0.908872 0.417075i \(-0.136945\pi\)
0.908872 + 0.417075i \(0.136945\pi\)
\(522\) −836.642 −0.0701510
\(523\) −7553.08 −0.631498 −0.315749 0.948843i \(-0.602256\pi\)
−0.315749 + 0.948843i \(0.602256\pi\)
\(524\) 2475.43 0.206374
\(525\) −10213.5 −0.849055
\(526\) −15223.1 −1.26190
\(527\) 10274.9 0.849301
\(528\) −3410.46 −0.281101
\(529\) −8358.52 −0.686983
\(530\) −2881.41 −0.236152
\(531\) −3219.73 −0.263134
\(532\) −20927.8 −1.70552
\(533\) −320.237 −0.0260244
\(534\) 24476.1 1.98349
\(535\) 4314.66 0.348671
\(536\) 7019.67 0.565678
\(537\) −21422.7 −1.72152
\(538\) −5294.98 −0.424317
\(539\) −6459.73 −0.516216
\(540\) 10073.4 0.802757
\(541\) 17662.3 1.40363 0.701814 0.712360i \(-0.252374\pi\)
0.701814 + 0.712360i \(0.252374\pi\)
\(542\) 20229.7 1.60321
\(543\) 15713.1 1.24183
\(544\) −13773.8 −1.08557
\(545\) 9818.38 0.771694
\(546\) 34073.3 2.67070
\(547\) 19855.5 1.55203 0.776016 0.630714i \(-0.217238\pi\)
0.776016 + 0.630714i \(0.217238\pi\)
\(548\) 10351.8 0.806944
\(549\) 2666.61 0.207301
\(550\) −4741.18 −0.367572
\(551\) −4173.62 −0.322690
\(552\) 2127.84 0.164070
\(553\) 22347.1 1.71843
\(554\) 27293.9 2.09315
\(555\) −12399.1 −0.948314
\(556\) −4617.36 −0.352194
\(557\) 4237.59 0.322356 0.161178 0.986925i \(-0.448471\pi\)
0.161178 + 0.986925i \(0.448471\pi\)
\(558\) −2975.23 −0.225720
\(559\) 0 0
\(560\) 9294.49 0.701364
\(561\) −3843.21 −0.289234
\(562\) 7979.33 0.598911
\(563\) 12500.8 0.935781 0.467890 0.883787i \(-0.345014\pi\)
0.467890 + 0.883787i \(0.345014\pi\)
\(564\) −26186.2 −1.95503
\(565\) 7740.77 0.576383
\(566\) −1884.74 −0.139967
\(567\) 17174.8 1.27209
\(568\) 2983.52 0.220398
\(569\) 102.851 0.00757777 0.00378889 0.999993i \(-0.498794\pi\)
0.00378889 + 0.999993i \(0.498794\pi\)
\(570\) 11019.3 0.809733
\(571\) −16482.6 −1.20801 −0.604006 0.796979i \(-0.706430\pi\)
−0.604006 + 0.796979i \(0.706430\pi\)
\(572\) 8667.09 0.633547
\(573\) 10846.7 0.790798
\(574\) 624.606 0.0454191
\(575\) −4680.95 −0.339494
\(576\) 2586.07 0.187071
\(577\) −12994.4 −0.937547 −0.468774 0.883318i \(-0.655304\pi\)
−0.468774 + 0.883318i \(0.655304\pi\)
\(578\) 8595.42 0.618550
\(579\) 4927.26 0.353662
\(580\) −3666.82 −0.262511
\(581\) 22111.6 1.57890
\(582\) 11731.4 0.835536
\(583\) −1451.65 −0.103124
\(584\) 1565.89 0.110954
\(585\) 1554.94 0.109896
\(586\) 8777.20 0.618742
\(587\) −3247.27 −0.228329 −0.114165 0.993462i \(-0.536419\pi\)
−0.114165 + 0.993462i \(0.536419\pi\)
\(588\) −20356.3 −1.42768
\(589\) −14842.1 −1.03830
\(590\) −25752.6 −1.79698
\(591\) −9780.71 −0.680752
\(592\) −17413.3 −1.20892
\(593\) −6784.90 −0.469852 −0.234926 0.972013i \(-0.575485\pi\)
−0.234926 + 0.972013i \(0.575485\pi\)
\(594\) 9261.54 0.639740
\(595\) 10473.8 0.721657
\(596\) 5021.16 0.345092
\(597\) −11298.4 −0.774560
\(598\) 15616.1 1.06788
\(599\) −14622.2 −0.997408 −0.498704 0.866772i \(-0.666190\pi\)
−0.498704 + 0.866772i \(0.666190\pi\)
\(600\) −2615.29 −0.177948
\(601\) 18858.5 1.27995 0.639977 0.768394i \(-0.278944\pi\)
0.639977 + 0.768394i \(0.278944\pi\)
\(602\) 0 0
\(603\) 3624.63 0.244787
\(604\) −13231.8 −0.891379
\(605\) −7783.45 −0.523045
\(606\) 32313.1 2.16605
\(607\) −22932.4 −1.53344 −0.766718 0.641984i \(-0.778112\pi\)
−0.766718 + 0.641984i \(0.778112\pi\)
\(608\) 19896.2 1.32714
\(609\) −7262.51 −0.483238
\(610\) 21328.6 1.41569
\(611\) −33640.2 −2.22739
\(612\) 1915.58 0.126524
\(613\) −15582.9 −1.02673 −0.513367 0.858169i \(-0.671602\pi\)
−0.513367 + 0.858169i \(0.671602\pi\)
\(614\) −1655.76 −0.108829
\(615\) −180.213 −0.0118161
\(616\) −2959.10 −0.193548
\(617\) 9064.42 0.591442 0.295721 0.955274i \(-0.404440\pi\)
0.295721 + 0.955274i \(0.404440\pi\)
\(618\) 27548.2 1.79313
\(619\) −8539.07 −0.554466 −0.277233 0.960803i \(-0.589417\pi\)
−0.277233 + 0.960803i \(0.589417\pi\)
\(620\) −13039.8 −0.844663
\(621\) 9143.90 0.590873
\(622\) −9057.92 −0.583906
\(623\) −33605.6 −2.16112
\(624\) −13806.5 −0.885738
\(625\) −390.398 −0.0249855
\(626\) −25340.2 −1.61789
\(627\) 5551.50 0.353597
\(628\) 27625.0 1.75535
\(629\) −19622.8 −1.24390
\(630\) −3032.84 −0.191796
\(631\) 15078.5 0.951294 0.475647 0.879636i \(-0.342214\pi\)
0.475647 + 0.879636i \(0.342214\pi\)
\(632\) 5722.24 0.360156
\(633\) 18886.3 1.18588
\(634\) 26755.4 1.67602
\(635\) 3938.62 0.246141
\(636\) −4574.52 −0.285207
\(637\) −26150.7 −1.62658
\(638\) −3371.31 −0.209203
\(639\) 1540.55 0.0953730
\(640\) 6263.94 0.386881
\(641\) 22253.5 1.37123 0.685616 0.727963i \(-0.259533\pi\)
0.685616 + 0.727963i \(0.259533\pi\)
\(642\) 12500.8 0.768487
\(643\) 6114.37 0.375003 0.187502 0.982264i \(-0.439961\pi\)
0.187502 + 0.982264i \(0.439961\pi\)
\(644\) −16690.0 −1.02124
\(645\) 0 0
\(646\) 17439.1 1.06212
\(647\) 4947.98 0.300657 0.150329 0.988636i \(-0.451967\pi\)
0.150329 + 0.988636i \(0.451967\pi\)
\(648\) 4397.82 0.266609
\(649\) −12974.1 −0.784713
\(650\) −19193.6 −1.15821
\(651\) −25826.7 −1.55488
\(652\) 19852.9 1.19248
\(653\) 21153.5 1.26769 0.633845 0.773460i \(-0.281476\pi\)
0.633845 + 0.773460i \(0.281476\pi\)
\(654\) 28446.7 1.70085
\(655\) 1789.58 0.106755
\(656\) −253.090 −0.0150633
\(657\) 808.552 0.0480131
\(658\) 65613.6 3.88736
\(659\) 15358.2 0.907843 0.453922 0.891042i \(-0.350025\pi\)
0.453922 + 0.891042i \(0.350025\pi\)
\(660\) 4877.39 0.287655
\(661\) −11989.6 −0.705511 −0.352755 0.935716i \(-0.614755\pi\)
−0.352755 + 0.935716i \(0.614755\pi\)
\(662\) 29382.0 1.72502
\(663\) −15558.3 −0.911366
\(664\) 5661.94 0.330913
\(665\) −15129.4 −0.882248
\(666\) 5682.06 0.330594
\(667\) −3328.48 −0.193222
\(668\) −8729.46 −0.505618
\(669\) 9.46737 0.000547129 0
\(670\) 28991.2 1.67168
\(671\) 10745.3 0.618207
\(672\) 34621.5 1.98743
\(673\) 19868.3 1.13799 0.568994 0.822342i \(-0.307333\pi\)
0.568994 + 0.822342i \(0.307333\pi\)
\(674\) −1803.91 −0.103092
\(675\) −11238.6 −0.640852
\(676\) 13781.3 0.784097
\(677\) 11011.1 0.625098 0.312549 0.949902i \(-0.398817\pi\)
0.312549 + 0.949902i \(0.398817\pi\)
\(678\) 22427.3 1.27037
\(679\) −16107.2 −0.910362
\(680\) 2681.96 0.151248
\(681\) 238.552 0.0134234
\(682\) −11988.9 −0.673136
\(683\) −21704.9 −1.21598 −0.607989 0.793945i \(-0.708024\pi\)
−0.607989 + 0.793945i \(0.708024\pi\)
\(684\) −2767.04 −0.154679
\(685\) 7483.66 0.417425
\(686\) 10764.6 0.599116
\(687\) −8198.03 −0.455276
\(688\) 0 0
\(689\) −5876.66 −0.324939
\(690\) 8787.95 0.484857
\(691\) 4124.20 0.227051 0.113525 0.993535i \(-0.463786\pi\)
0.113525 + 0.993535i \(0.463786\pi\)
\(692\) 17878.4 0.982133
\(693\) −1527.94 −0.0837542
\(694\) 3665.64 0.200498
\(695\) −3338.06 −0.182187
\(696\) −1859.66 −0.101279
\(697\) −285.204 −0.0154991
\(698\) 12427.2 0.673893
\(699\) 18326.4 0.991655
\(700\) 20513.5 1.10762
\(701\) 27363.8 1.47435 0.737173 0.675704i \(-0.236160\pi\)
0.737173 + 0.675704i \(0.236160\pi\)
\(702\) 37493.2 2.01580
\(703\) 28345.1 1.52071
\(704\) 10420.7 0.557878
\(705\) −18930.9 −1.01132
\(706\) 7405.96 0.394797
\(707\) −44365.7 −2.36003
\(708\) −40884.8 −2.17026
\(709\) 17078.7 0.904658 0.452329 0.891851i \(-0.350593\pi\)
0.452329 + 0.891851i \(0.350593\pi\)
\(710\) 12321.9 0.651315
\(711\) 2954.70 0.155851
\(712\) −8605.13 −0.452936
\(713\) −11836.6 −0.621718
\(714\) 30345.8 1.59056
\(715\) 6265.75 0.327728
\(716\) 43026.6 2.24578
\(717\) 14244.0 0.741912
\(718\) −41902.6 −2.17798
\(719\) −14080.8 −0.730356 −0.365178 0.930938i \(-0.618992\pi\)
−0.365178 + 0.930938i \(0.618992\pi\)
\(720\) 1228.91 0.0636092
\(721\) −37823.6 −1.95371
\(722\) 3663.98 0.188863
\(723\) −19691.1 −1.01289
\(724\) −31559.2 −1.62001
\(725\) 4090.99 0.209566
\(726\) −22550.9 −1.15281
\(727\) −29786.3 −1.51955 −0.759775 0.650186i \(-0.774691\pi\)
−0.759775 + 0.650186i \(0.774691\pi\)
\(728\) −11979.2 −0.609862
\(729\) 21586.7 1.09672
\(730\) 6467.11 0.327888
\(731\) 0 0
\(732\) 33861.1 1.70976
\(733\) 15924.8 0.802452 0.401226 0.915979i \(-0.368584\pi\)
0.401226 + 0.915979i \(0.368584\pi\)
\(734\) −5433.22 −0.273220
\(735\) −14716.3 −0.738528
\(736\) 15867.4 0.794672
\(737\) 14605.7 0.729997
\(738\) 82.5846 0.00411922
\(739\) −37584.1 −1.87084 −0.935421 0.353536i \(-0.884979\pi\)
−0.935421 + 0.353536i \(0.884979\pi\)
\(740\) 24903.2 1.23711
\(741\) 22474.0 1.11417
\(742\) 11462.2 0.567101
\(743\) −17948.3 −0.886219 −0.443110 0.896467i \(-0.646125\pi\)
−0.443110 + 0.896467i \(0.646125\pi\)
\(744\) −6613.23 −0.325878
\(745\) 3629.98 0.178513
\(746\) −29493.7 −1.44751
\(747\) 2923.56 0.143196
\(748\) 7718.94 0.377316
\(749\) −17163.6 −0.837308
\(750\) −28601.2 −1.39249
\(751\) −2386.72 −0.115969 −0.0579844 0.998317i \(-0.518467\pi\)
−0.0579844 + 0.998317i \(0.518467\pi\)
\(752\) −26586.5 −1.28924
\(753\) 17038.6 0.824597
\(754\) −13648.0 −0.659190
\(755\) −9565.72 −0.461102
\(756\) −40071.5 −1.92776
\(757\) 20527.4 0.985574 0.492787 0.870150i \(-0.335978\pi\)
0.492787 + 0.870150i \(0.335978\pi\)
\(758\) −11311.8 −0.542034
\(759\) 4427.35 0.211729
\(760\) −3874.08 −0.184905
\(761\) 21708.9 1.03410 0.517048 0.855957i \(-0.327031\pi\)
0.517048 + 0.855957i \(0.327031\pi\)
\(762\) 11411.3 0.542505
\(763\) −39057.2 −1.85316
\(764\) −21785.2 −1.03162
\(765\) 1384.84 0.0654496
\(766\) −23217.4 −1.09514
\(767\) −52522.7 −2.47260
\(768\) −8941.76 −0.420128
\(769\) −3194.65 −0.149807 −0.0749037 0.997191i \(-0.523865\pi\)
−0.0749037 + 0.997191i \(0.523865\pi\)
\(770\) −12221.0 −0.571969
\(771\) −31167.2 −1.45585
\(772\) −9896.23 −0.461364
\(773\) −9743.41 −0.453359 −0.226679 0.973969i \(-0.572787\pi\)
−0.226679 + 0.973969i \(0.572787\pi\)
\(774\) 0 0
\(775\) 14548.2 0.674306
\(776\) −4124.43 −0.190797
\(777\) 49323.4 2.27731
\(778\) 28680.5 1.32165
\(779\) 411.976 0.0189481
\(780\) 19745.0 0.906390
\(781\) 6207.76 0.284419
\(782\) 13907.8 0.635986
\(783\) −7991.45 −0.364739
\(784\) −20667.5 −0.941486
\(785\) 19971.1 0.908026
\(786\) 5184.93 0.235293
\(787\) −10035.8 −0.454561 −0.227280 0.973829i \(-0.572983\pi\)
−0.227280 + 0.973829i \(0.572983\pi\)
\(788\) 19644.2 0.888066
\(789\) −17472.0 −0.788364
\(790\) 23632.8 1.06433
\(791\) −30792.5 −1.38414
\(792\) −391.248 −0.0175535
\(793\) 43499.8 1.94795
\(794\) −22510.8 −1.00615
\(795\) −3307.08 −0.147535
\(796\) 22692.4 1.01044
\(797\) 23138.7 1.02837 0.514187 0.857678i \(-0.328094\pi\)
0.514187 + 0.857678i \(0.328094\pi\)
\(798\) −43834.4 −1.94451
\(799\) −29960.1 −1.32655
\(800\) −19502.3 −0.861890
\(801\) −4443.29 −0.196000
\(802\) 27724.9 1.22070
\(803\) 3258.11 0.143183
\(804\) 46026.3 2.01893
\(805\) −12065.8 −0.528278
\(806\) −48534.3 −2.12103
\(807\) −6077.21 −0.265090
\(808\) −11360.4 −0.494624
\(809\) −2919.61 −0.126883 −0.0634413 0.997986i \(-0.520208\pi\)
−0.0634413 + 0.997986i \(0.520208\pi\)
\(810\) 18163.0 0.787879
\(811\) 16208.2 0.701784 0.350892 0.936416i \(-0.385878\pi\)
0.350892 + 0.936416i \(0.385878\pi\)
\(812\) 14586.5 0.630401
\(813\) 23218.2 1.00160
\(814\) 22896.2 0.985888
\(815\) 14352.4 0.616861
\(816\) −12296.1 −0.527511
\(817\) 0 0
\(818\) 17383.3 0.743022
\(819\) −6185.51 −0.263906
\(820\) 361.951 0.0154145
\(821\) −17404.8 −0.739867 −0.369934 0.929058i \(-0.620620\pi\)
−0.369934 + 0.929058i \(0.620620\pi\)
\(822\) 21682.3 0.920022
\(823\) 9581.39 0.405816 0.202908 0.979198i \(-0.434961\pi\)
0.202908 + 0.979198i \(0.434961\pi\)
\(824\) −9685.20 −0.409466
\(825\) −5441.59 −0.229639
\(826\) 102443. 4.31532
\(827\) −36359.4 −1.52883 −0.764414 0.644726i \(-0.776972\pi\)
−0.764414 + 0.644726i \(0.776972\pi\)
\(828\) −2206.73 −0.0926198
\(829\) −21898.1 −0.917435 −0.458717 0.888582i \(-0.651691\pi\)
−0.458717 + 0.888582i \(0.651691\pi\)
\(830\) 23383.8 0.977907
\(831\) 31326.0 1.30769
\(832\) 42186.0 1.75785
\(833\) −23289.9 −0.968726
\(834\) −9671.32 −0.401547
\(835\) −6310.84 −0.261552
\(836\) −11150.0 −0.461280
\(837\) −28418.9 −1.17360
\(838\) 21460.6 0.884659
\(839\) 5384.09 0.221549 0.110774 0.993846i \(-0.464667\pi\)
0.110774 + 0.993846i \(0.464667\pi\)
\(840\) −6741.29 −0.276901
\(841\) −21480.0 −0.880726
\(842\) −14900.9 −0.609882
\(843\) 9158.12 0.374167
\(844\) −37932.4 −1.54702
\(845\) 9962.99 0.405606
\(846\) 8675.34 0.352558
\(847\) 30962.3 1.25605
\(848\) −4644.45 −0.188079
\(849\) −2163.17 −0.0874440
\(850\) −17093.9 −0.689782
\(851\) 22605.4 0.910579
\(852\) 19562.3 0.786610
\(853\) 3558.83 0.142851 0.0714255 0.997446i \(-0.477245\pi\)
0.0714255 + 0.997446i \(0.477245\pi\)
\(854\) −84844.3 −3.39966
\(855\) −2000.39 −0.0800141
\(856\) −4394.94 −0.175486
\(857\) −38994.7 −1.55430 −0.777149 0.629317i \(-0.783335\pi\)
−0.777149 + 0.629317i \(0.783335\pi\)
\(858\) 18153.7 0.722328
\(859\) 10894.1 0.432715 0.216358 0.976314i \(-0.430582\pi\)
0.216358 + 0.976314i \(0.430582\pi\)
\(860\) 0 0
\(861\) 716.880 0.0283754
\(862\) 35756.2 1.41283
\(863\) −48209.7 −1.90160 −0.950798 0.309811i \(-0.899734\pi\)
−0.950798 + 0.309811i \(0.899734\pi\)
\(864\) 38096.4 1.50008
\(865\) 12925.0 0.508048
\(866\) −33314.5 −1.30724
\(867\) 9865.22 0.386436
\(868\) 51871.9 2.02840
\(869\) 11906.2 0.464774
\(870\) −7680.36 −0.299297
\(871\) 59127.8 2.30019
\(872\) −10001.1 −0.388393
\(873\) −2129.67 −0.0825639
\(874\) −20089.7 −0.777512
\(875\) 39269.3 1.51719
\(876\) 10267.2 0.395999
\(877\) 14792.4 0.569558 0.284779 0.958593i \(-0.408080\pi\)
0.284779 + 0.958593i \(0.408080\pi\)
\(878\) −57340.2 −2.20403
\(879\) 10073.9 0.386556
\(880\) 4951.96 0.189694
\(881\) 9682.56 0.370277 0.185138 0.982712i \(-0.440727\pi\)
0.185138 + 0.982712i \(0.440727\pi\)
\(882\) 6743.92 0.257460
\(883\) −19920.4 −0.759201 −0.379601 0.925150i \(-0.623939\pi\)
−0.379601 + 0.925150i \(0.623939\pi\)
\(884\) 31248.3 1.18891
\(885\) −29557.1 −1.12266
\(886\) 63382.4 2.40336
\(887\) −41838.2 −1.58375 −0.791876 0.610682i \(-0.790895\pi\)
−0.791876 + 0.610682i \(0.790895\pi\)
\(888\) 12629.9 0.477286
\(889\) −15667.7 −0.591089
\(890\) −35539.1 −1.33851
\(891\) 9150.47 0.344054
\(892\) −19.0149 −0.000713750 0
\(893\) 43277.2 1.62174
\(894\) 10517.1 0.393451
\(895\) 31105.5 1.16172
\(896\) −24917.7 −0.929066
\(897\) 17923.1 0.667152
\(898\) −20701.3 −0.769276
\(899\) 10344.8 0.383780
\(900\) 2712.26 0.100454
\(901\) −5233.78 −0.193521
\(902\) 332.780 0.0122842
\(903\) 0 0
\(904\) −7884.80 −0.290094
\(905\) −22815.2 −0.838016
\(906\) −27714.7 −1.01629
\(907\) −35318.6 −1.29298 −0.646492 0.762921i \(-0.723765\pi\)
−0.646492 + 0.762921i \(0.723765\pi\)
\(908\) −479.124 −0.0175113
\(909\) −5865.97 −0.214040
\(910\) −49474.1 −1.80225
\(911\) −1355.47 −0.0492959 −0.0246480 0.999696i \(-0.507846\pi\)
−0.0246480 + 0.999696i \(0.507846\pi\)
\(912\) 17761.7 0.644898
\(913\) 11780.7 0.427036
\(914\) 35402.8 1.28120
\(915\) 24479.4 0.884443
\(916\) 16465.5 0.593924
\(917\) −7118.89 −0.256365
\(918\) 33391.6 1.20053
\(919\) 10499.0 0.376854 0.188427 0.982087i \(-0.439661\pi\)
0.188427 + 0.982087i \(0.439661\pi\)
\(920\) −3089.60 −0.110719
\(921\) −1900.37 −0.0679905
\(922\) 74488.2 2.66067
\(923\) 25130.7 0.896194
\(924\) −19402.1 −0.690781
\(925\) −27784.0 −0.987602
\(926\) 41611.7 1.47672
\(927\) −5000.99 −0.177189
\(928\) −13867.5 −0.490543
\(929\) −31511.5 −1.11287 −0.556436 0.830890i \(-0.687832\pi\)
−0.556436 + 0.830890i \(0.687832\pi\)
\(930\) −27312.6 −0.963028
\(931\) 33642.2 1.18430
\(932\) −36807.9 −1.29365
\(933\) −10396.1 −0.364793
\(934\) 23718.0 0.830918
\(935\) 5580.30 0.195182
\(936\) −1583.88 −0.0553105
\(937\) 5299.66 0.184773 0.0923864 0.995723i \(-0.470550\pi\)
0.0923864 + 0.995723i \(0.470550\pi\)
\(938\) −115326. −4.01442
\(939\) −29083.7 −1.01077
\(940\) 38022.1 1.31930
\(941\) 34588.6 1.19825 0.599126 0.800655i \(-0.295515\pi\)
0.599126 + 0.800655i \(0.295515\pi\)
\(942\) 57862.2 2.00133
\(943\) 328.553 0.0113459
\(944\) −41509.8 −1.43118
\(945\) −28969.1 −0.997213
\(946\) 0 0
\(947\) −17342.1 −0.595083 −0.297541 0.954709i \(-0.596167\pi\)
−0.297541 + 0.954709i \(0.596167\pi\)
\(948\) 37519.4 1.28541
\(949\) 13189.7 0.451166
\(950\) 24692.0 0.843279
\(951\) 30708.0 1.04708
\(952\) −10668.7 −0.363210
\(953\) 53043.0 1.80297 0.901485 0.432810i \(-0.142478\pi\)
0.901485 + 0.432810i \(0.142478\pi\)
\(954\) 1515.51 0.0514324
\(955\) −15749.3 −0.533650
\(956\) −28608.5 −0.967850
\(957\) −3869.35 −0.130698
\(958\) 70129.1 2.36510
\(959\) −29769.7 −1.00241
\(960\) 23740.1 0.798133
\(961\) 6996.75 0.234861
\(962\) 92690.1 3.10650
\(963\) −2269.35 −0.0759384
\(964\) 39548.8 1.32135
\(965\) −7154.34 −0.238659
\(966\) −34958.2 −1.16435
\(967\) 4560.36 0.151656 0.0758280 0.997121i \(-0.475840\pi\)
0.0758280 + 0.997121i \(0.475840\pi\)
\(968\) 7928.27 0.263248
\(969\) 20015.4 0.663557
\(970\) −17033.9 −0.563840
\(971\) −27229.5 −0.899934 −0.449967 0.893045i \(-0.648564\pi\)
−0.449967 + 0.893045i \(0.648564\pi\)
\(972\) −9959.78 −0.328663
\(973\) 13278.7 0.437508
\(974\) −69035.3 −2.27108
\(975\) −22029.0 −0.723583
\(976\) 34378.8 1.12750
\(977\) 31182.6 1.02110 0.510552 0.859847i \(-0.329441\pi\)
0.510552 + 0.859847i \(0.329441\pi\)
\(978\) 41583.0 1.35959
\(979\) −17904.5 −0.584506
\(980\) 29557.1 0.963436
\(981\) −5164.09 −0.168070
\(982\) 13343.6 0.433617
\(983\) 28454.7 0.923259 0.461630 0.887073i \(-0.347265\pi\)
0.461630 + 0.887073i \(0.347265\pi\)
\(984\) 183.566 0.00594702
\(985\) 14201.5 0.459388
\(986\) −12154.9 −0.392588
\(987\) 75306.7 2.42861
\(988\) −45138.1 −1.45348
\(989\) 0 0
\(990\) −1615.85 −0.0518739
\(991\) 40487.4 1.29781 0.648903 0.760871i \(-0.275228\pi\)
0.648903 + 0.760871i \(0.275228\pi\)
\(992\) −49315.1 −1.57838
\(993\) 33722.6 1.07770
\(994\) −49016.2 −1.56409
\(995\) 16405.2 0.522692
\(996\) 37124.0 1.18104
\(997\) 3912.73 0.124290 0.0621451 0.998067i \(-0.480206\pi\)
0.0621451 + 0.998067i \(0.480206\pi\)
\(998\) 45848.9 1.45423
\(999\) 54273.9 1.71887
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 1849.4.a.j.1.7 yes 50
43.42 odd 2 1849.4.a.i.1.44 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.44 50 43.42 odd 2
1849.4.a.j.1.7 yes 50 1.1 even 1 trivial