Properties

Label 405.4.a.g.1.2
Level $405$
Weight $4$
Character 405.1
Self dual yes
Analytic conductor $23.896$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.8957735523\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.7032.1
Defining polynomial: \( x^{3} - x^{2} - 14x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.32681\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.32681 q^{2} -6.23958 q^{4} +5.00000 q^{5} -24.1043 q^{7} +18.8932 q^{8} +O(q^{10})\) \(q-1.32681 q^{2} -6.23958 q^{4} +5.00000 q^{5} -24.1043 q^{7} +18.8932 q^{8} -6.63404 q^{10} +8.27619 q^{11} +87.1145 q^{13} +31.9818 q^{14} +24.8490 q^{16} +51.9166 q^{17} -88.5107 q^{19} -31.1979 q^{20} -10.9809 q^{22} -129.245 q^{23} +25.0000 q^{25} -115.584 q^{26} +150.400 q^{28} -271.109 q^{29} +224.547 q^{31} -184.115 q^{32} -68.8834 q^{34} -120.521 q^{35} -70.5268 q^{37} +117.437 q^{38} +94.4660 q^{40} +366.938 q^{41} -195.547 q^{43} -51.6399 q^{44} +171.483 q^{46} -359.192 q^{47} +238.016 q^{49} -33.1702 q^{50} -543.558 q^{52} +29.4890 q^{53} +41.3810 q^{55} -455.407 q^{56} +359.709 q^{58} -858.104 q^{59} -556.811 q^{61} -297.931 q^{62} +45.4941 q^{64} +435.573 q^{65} -41.8987 q^{67} -323.938 q^{68} +159.909 q^{70} -549.163 q^{71} -185.505 q^{73} +93.5756 q^{74} +552.269 q^{76} -199.492 q^{77} +80.4913 q^{79} +124.245 q^{80} -486.857 q^{82} -576.753 q^{83} +259.583 q^{85} +259.454 q^{86} +156.364 q^{88} +224.516 q^{89} -2099.83 q^{91} +806.433 q^{92} +476.579 q^{94} -442.553 q^{95} -555.016 q^{97} -315.801 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 5 q^{4} + 15 q^{5} - 25 q^{7} + 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 5 q^{4} + 15 q^{5} - 25 q^{7} + 27 q^{8} - 5 q^{10} - 58 q^{11} - 47 q^{13} - 159 q^{14} - 127 q^{16} - 34 q^{17} - 5 q^{19} + 25 q^{20} + 260 q^{22} + 51 q^{23} + 75 q^{25} - 253 q^{26} + 83 q^{28} - 350 q^{29} + 638 q^{31} - 245 q^{32} - 154 q^{34} - 125 q^{35} - 414 q^{37} - 397 q^{38} + 135 q^{40} - 179 q^{41} - 836 q^{43} - 332 q^{44} + 261 q^{46} - 235 q^{47} + 892 q^{49} - 25 q^{50} - 1335 q^{52} - 505 q^{53} - 290 q^{55} - 15 q^{56} + 1876 q^{58} - 535 q^{59} - 104 q^{61} - 348 q^{62} - 303 q^{64} - 235 q^{65} - 40 q^{67} - 830 q^{68} - 795 q^{70} - 452 q^{71} - 710 q^{73} - 1394 q^{74} + 849 q^{76} - 2148 q^{77} - 634 q^{79} - 635 q^{80} + 613 q^{82} - 1734 q^{83} - 170 q^{85} - 460 q^{86} - 768 q^{88} + 852 q^{89} - 1229 q^{91} + 1839 q^{92} + 1751 q^{94} - 25 q^{95} - 38 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.32681 −0.469098 −0.234549 0.972104i \(-0.575361\pi\)
−0.234549 + 0.972104i \(0.575361\pi\)
\(3\) 0 0
\(4\) −6.23958 −0.779947
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −24.1043 −1.30151 −0.650754 0.759289i \(-0.725547\pi\)
−0.650754 + 0.759289i \(0.725547\pi\)
\(8\) 18.8932 0.834969
\(9\) 0 0
\(10\) −6.63404 −0.209787
\(11\) 8.27619 0.226851 0.113426 0.993546i \(-0.463818\pi\)
0.113426 + 0.993546i \(0.463818\pi\)
\(12\) 0 0
\(13\) 87.1145 1.85856 0.929278 0.369382i \(-0.120430\pi\)
0.929278 + 0.369382i \(0.120430\pi\)
\(14\) 31.9818 0.610534
\(15\) 0 0
\(16\) 24.8490 0.388265
\(17\) 51.9166 0.740684 0.370342 0.928895i \(-0.379240\pi\)
0.370342 + 0.928895i \(0.379240\pi\)
\(18\) 0 0
\(19\) −88.5107 −1.06872 −0.534362 0.845256i \(-0.679448\pi\)
−0.534362 + 0.845256i \(0.679448\pi\)
\(20\) −31.1979 −0.348803
\(21\) 0 0
\(22\) −10.9809 −0.106415
\(23\) −129.245 −1.17171 −0.585856 0.810415i \(-0.699242\pi\)
−0.585856 + 0.810415i \(0.699242\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −115.584 −0.871844
\(27\) 0 0
\(28\) 150.400 1.01511
\(29\) −271.109 −1.73599 −0.867993 0.496576i \(-0.834590\pi\)
−0.867993 + 0.496576i \(0.834590\pi\)
\(30\) 0 0
\(31\) 224.547 1.30096 0.650481 0.759523i \(-0.274568\pi\)
0.650481 + 0.759523i \(0.274568\pi\)
\(32\) −184.115 −1.01710
\(33\) 0 0
\(34\) −68.8834 −0.347453
\(35\) −120.521 −0.582052
\(36\) 0 0
\(37\) −70.5268 −0.313366 −0.156683 0.987649i \(-0.550080\pi\)
−0.156683 + 0.987649i \(0.550080\pi\)
\(38\) 117.437 0.501336
\(39\) 0 0
\(40\) 94.4660 0.373410
\(41\) 366.938 1.39771 0.698855 0.715263i \(-0.253693\pi\)
0.698855 + 0.715263i \(0.253693\pi\)
\(42\) 0 0
\(43\) −195.547 −0.693504 −0.346752 0.937957i \(-0.612715\pi\)
−0.346752 + 0.937957i \(0.612715\pi\)
\(44\) −51.6399 −0.176932
\(45\) 0 0
\(46\) 171.483 0.549648
\(47\) −359.192 −1.11476 −0.557378 0.830259i \(-0.688193\pi\)
−0.557378 + 0.830259i \(0.688193\pi\)
\(48\) 0 0
\(49\) 238.016 0.693923
\(50\) −33.1702 −0.0938195
\(51\) 0 0
\(52\) −543.558 −1.44958
\(53\) 29.4890 0.0764270 0.0382135 0.999270i \(-0.487833\pi\)
0.0382135 + 0.999270i \(0.487833\pi\)
\(54\) 0 0
\(55\) 41.3810 0.101451
\(56\) −455.407 −1.08672
\(57\) 0 0
\(58\) 359.709 0.814347
\(59\) −858.104 −1.89349 −0.946743 0.321991i \(-0.895648\pi\)
−0.946743 + 0.321991i \(0.895648\pi\)
\(60\) 0 0
\(61\) −556.811 −1.16873 −0.584364 0.811492i \(-0.698656\pi\)
−0.584364 + 0.811492i \(0.698656\pi\)
\(62\) −297.931 −0.610278
\(63\) 0 0
\(64\) 45.4941 0.0888557
\(65\) 435.573 0.831171
\(66\) 0 0
\(67\) −41.8987 −0.0763991 −0.0381995 0.999270i \(-0.512162\pi\)
−0.0381995 + 0.999270i \(0.512162\pi\)
\(68\) −323.938 −0.577695
\(69\) 0 0
\(70\) 159.909 0.273039
\(71\) −549.163 −0.917939 −0.458970 0.888452i \(-0.651781\pi\)
−0.458970 + 0.888452i \(0.651781\pi\)
\(72\) 0 0
\(73\) −185.505 −0.297420 −0.148710 0.988881i \(-0.547512\pi\)
−0.148710 + 0.988881i \(0.547512\pi\)
\(74\) 93.5756 0.146999
\(75\) 0 0
\(76\) 552.269 0.833548
\(77\) −199.492 −0.295249
\(78\) 0 0
\(79\) 80.4913 0.114633 0.0573163 0.998356i \(-0.481746\pi\)
0.0573163 + 0.998356i \(0.481746\pi\)
\(80\) 124.245 0.173637
\(81\) 0 0
\(82\) −486.857 −0.655663
\(83\) −576.753 −0.762734 −0.381367 0.924424i \(-0.624547\pi\)
−0.381367 + 0.924424i \(0.624547\pi\)
\(84\) 0 0
\(85\) 259.583 0.331244
\(86\) 259.454 0.325321
\(87\) 0 0
\(88\) 156.364 0.189414
\(89\) 224.516 0.267401 0.133700 0.991022i \(-0.457314\pi\)
0.133700 + 0.991022i \(0.457314\pi\)
\(90\) 0 0
\(91\) −2099.83 −2.41893
\(92\) 806.433 0.913874
\(93\) 0 0
\(94\) 476.579 0.522930
\(95\) −442.553 −0.477948
\(96\) 0 0
\(97\) −555.016 −0.580963 −0.290481 0.956881i \(-0.593815\pi\)
−0.290481 + 0.956881i \(0.593815\pi\)
\(98\) −315.801 −0.325518
\(99\) 0 0
\(100\) −155.989 −0.155989
\(101\) 227.000 0.223637 0.111818 0.993729i \(-0.464333\pi\)
0.111818 + 0.993729i \(0.464333\pi\)
\(102\) 0 0
\(103\) 383.524 0.366891 0.183446 0.983030i \(-0.441275\pi\)
0.183446 + 0.983030i \(0.441275\pi\)
\(104\) 1645.87 1.55184
\(105\) 0 0
\(106\) −39.1263 −0.0358517
\(107\) −1775.32 −1.60399 −0.801993 0.597334i \(-0.796227\pi\)
−0.801993 + 0.597334i \(0.796227\pi\)
\(108\) 0 0
\(109\) 1530.50 1.34491 0.672454 0.740139i \(-0.265240\pi\)
0.672454 + 0.740139i \(0.265240\pi\)
\(110\) −54.9046 −0.0475904
\(111\) 0 0
\(112\) −598.966 −0.505330
\(113\) −840.782 −0.699948 −0.349974 0.936759i \(-0.613810\pi\)
−0.349974 + 0.936759i \(0.613810\pi\)
\(114\) 0 0
\(115\) −646.224 −0.524006
\(116\) 1691.60 1.35398
\(117\) 0 0
\(118\) 1138.54 0.888230
\(119\) −1251.41 −0.964007
\(120\) 0 0
\(121\) −1262.50 −0.948538
\(122\) 738.782 0.548247
\(123\) 0 0
\(124\) −1401.08 −1.01468
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 1038.45 0.725572 0.362786 0.931873i \(-0.381826\pi\)
0.362786 + 0.931873i \(0.381826\pi\)
\(128\) 1412.56 0.975422
\(129\) 0 0
\(130\) −577.922 −0.389901
\(131\) −803.410 −0.535834 −0.267917 0.963442i \(-0.586335\pi\)
−0.267917 + 0.963442i \(0.586335\pi\)
\(132\) 0 0
\(133\) 2133.49 1.39095
\(134\) 55.5915 0.0358386
\(135\) 0 0
\(136\) 980.871 0.618449
\(137\) 869.152 0.542019 0.271010 0.962577i \(-0.412642\pi\)
0.271010 + 0.962577i \(0.412642\pi\)
\(138\) 0 0
\(139\) −196.398 −0.119844 −0.0599218 0.998203i \(-0.519085\pi\)
−0.0599218 + 0.998203i \(0.519085\pi\)
\(140\) 752.002 0.453970
\(141\) 0 0
\(142\) 728.635 0.430603
\(143\) 720.976 0.421616
\(144\) 0 0
\(145\) −1355.54 −0.776357
\(146\) 246.129 0.139519
\(147\) 0 0
\(148\) 440.057 0.244409
\(149\) 754.283 0.414720 0.207360 0.978265i \(-0.433513\pi\)
0.207360 + 0.978265i \(0.433513\pi\)
\(150\) 0 0
\(151\) 2057.64 1.10893 0.554464 0.832208i \(-0.312923\pi\)
0.554464 + 0.832208i \(0.312923\pi\)
\(152\) −1672.25 −0.892351
\(153\) 0 0
\(154\) 264.687 0.138501
\(155\) 1122.73 0.581808
\(156\) 0 0
\(157\) −3358.58 −1.70729 −0.853644 0.520857i \(-0.825613\pi\)
−0.853644 + 0.520857i \(0.825613\pi\)
\(158\) −106.797 −0.0537739
\(159\) 0 0
\(160\) −920.577 −0.454863
\(161\) 3115.35 1.52499
\(162\) 0 0
\(163\) −710.376 −0.341356 −0.170678 0.985327i \(-0.554596\pi\)
−0.170678 + 0.985327i \(0.554596\pi\)
\(164\) −2289.54 −1.09014
\(165\) 0 0
\(166\) 765.241 0.357797
\(167\) −1148.27 −0.532070 −0.266035 0.963963i \(-0.585714\pi\)
−0.266035 + 0.963963i \(0.585714\pi\)
\(168\) 0 0
\(169\) 5391.94 2.45423
\(170\) −344.417 −0.155386
\(171\) 0 0
\(172\) 1220.13 0.540897
\(173\) −2925.29 −1.28558 −0.642790 0.766042i \(-0.722223\pi\)
−0.642790 + 0.766042i \(0.722223\pi\)
\(174\) 0 0
\(175\) −602.607 −0.260302
\(176\) 205.655 0.0880785
\(177\) 0 0
\(178\) −297.890 −0.125437
\(179\) 3422.02 1.42890 0.714452 0.699685i \(-0.246676\pi\)
0.714452 + 0.699685i \(0.246676\pi\)
\(180\) 0 0
\(181\) −1151.58 −0.472907 −0.236454 0.971643i \(-0.575985\pi\)
−0.236454 + 0.971643i \(0.575985\pi\)
\(182\) 2786.07 1.13471
\(183\) 0 0
\(184\) −2441.85 −0.978344
\(185\) −352.634 −0.140141
\(186\) 0 0
\(187\) 429.672 0.168025
\(188\) 2241.21 0.869451
\(189\) 0 0
\(190\) 587.184 0.224204
\(191\) 932.120 0.353120 0.176560 0.984290i \(-0.443503\pi\)
0.176560 + 0.984290i \(0.443503\pi\)
\(192\) 0 0
\(193\) −4272.81 −1.59359 −0.796797 0.604247i \(-0.793474\pi\)
−0.796797 + 0.604247i \(0.793474\pi\)
\(194\) 736.400 0.272528
\(195\) 0 0
\(196\) −1485.12 −0.541224
\(197\) −1924.15 −0.695888 −0.347944 0.937515i \(-0.613120\pi\)
−0.347944 + 0.937515i \(0.613120\pi\)
\(198\) 0 0
\(199\) 1738.84 0.619414 0.309707 0.950832i \(-0.399769\pi\)
0.309707 + 0.950832i \(0.399769\pi\)
\(200\) 472.330 0.166994
\(201\) 0 0
\(202\) −301.185 −0.104907
\(203\) 6534.87 2.25940
\(204\) 0 0
\(205\) 1834.69 0.625075
\(206\) −508.864 −0.172108
\(207\) 0 0
\(208\) 2164.71 0.721613
\(209\) −732.531 −0.242441
\(210\) 0 0
\(211\) 3202.35 1.04483 0.522414 0.852692i \(-0.325031\pi\)
0.522414 + 0.852692i \(0.325031\pi\)
\(212\) −183.999 −0.0596090
\(213\) 0 0
\(214\) 2355.51 0.752426
\(215\) −977.736 −0.310144
\(216\) 0 0
\(217\) −5412.54 −1.69321
\(218\) −2030.68 −0.630893
\(219\) 0 0
\(220\) −258.200 −0.0791265
\(221\) 4522.69 1.37660
\(222\) 0 0
\(223\) −1404.99 −0.421906 −0.210953 0.977496i \(-0.567657\pi\)
−0.210953 + 0.977496i \(0.567657\pi\)
\(224\) 4437.97 1.32377
\(225\) 0 0
\(226\) 1115.56 0.328344
\(227\) 6238.35 1.82402 0.912012 0.410163i \(-0.134528\pi\)
0.912012 + 0.410163i \(0.134528\pi\)
\(228\) 0 0
\(229\) 6630.31 1.91329 0.956644 0.291259i \(-0.0940742\pi\)
0.956644 + 0.291259i \(0.0940742\pi\)
\(230\) 857.415 0.245810
\(231\) 0 0
\(232\) −5122.11 −1.44949
\(233\) 2453.48 0.689840 0.344920 0.938632i \(-0.387906\pi\)
0.344920 + 0.938632i \(0.387906\pi\)
\(234\) 0 0
\(235\) −1795.96 −0.498534
\(236\) 5354.21 1.47682
\(237\) 0 0
\(238\) 1660.38 0.452213
\(239\) −6949.74 −1.88093 −0.940463 0.339895i \(-0.889608\pi\)
−0.940463 + 0.339895i \(0.889608\pi\)
\(240\) 0 0
\(241\) 6335.19 1.69330 0.846651 0.532149i \(-0.178615\pi\)
0.846651 + 0.532149i \(0.178615\pi\)
\(242\) 1675.10 0.444957
\(243\) 0 0
\(244\) 3474.27 0.911546
\(245\) 1190.08 0.310332
\(246\) 0 0
\(247\) −7710.57 −1.98628
\(248\) 4242.41 1.08626
\(249\) 0 0
\(250\) −165.851 −0.0419574
\(251\) −4022.65 −1.01158 −0.505792 0.862656i \(-0.668800\pi\)
−0.505792 + 0.862656i \(0.668800\pi\)
\(252\) 0 0
\(253\) −1069.65 −0.265805
\(254\) −1377.83 −0.340364
\(255\) 0 0
\(256\) −2238.15 −0.546424
\(257\) −4997.74 −1.21304 −0.606519 0.795069i \(-0.707435\pi\)
−0.606519 + 0.795069i \(0.707435\pi\)
\(258\) 0 0
\(259\) 1700.00 0.407848
\(260\) −2717.79 −0.648270
\(261\) 0 0
\(262\) 1065.97 0.251359
\(263\) −3992.47 −0.936069 −0.468035 0.883710i \(-0.655038\pi\)
−0.468035 + 0.883710i \(0.655038\pi\)
\(264\) 0 0
\(265\) 147.445 0.0341792
\(266\) −2830.73 −0.652493
\(267\) 0 0
\(268\) 261.430 0.0595872
\(269\) −2188.89 −0.496131 −0.248065 0.968743i \(-0.579795\pi\)
−0.248065 + 0.968743i \(0.579795\pi\)
\(270\) 0 0
\(271\) 4280.26 0.959437 0.479718 0.877423i \(-0.340739\pi\)
0.479718 + 0.877423i \(0.340739\pi\)
\(272\) 1290.08 0.287582
\(273\) 0 0
\(274\) −1153.20 −0.254260
\(275\) 206.905 0.0453703
\(276\) 0 0
\(277\) −3879.97 −0.841606 −0.420803 0.907152i \(-0.638252\pi\)
−0.420803 + 0.907152i \(0.638252\pi\)
\(278\) 260.583 0.0562184
\(279\) 0 0
\(280\) −2277.03 −0.485996
\(281\) −6397.43 −1.35814 −0.679072 0.734071i \(-0.737618\pi\)
−0.679072 + 0.734071i \(0.737618\pi\)
\(282\) 0 0
\(283\) −342.869 −0.0720193 −0.0360096 0.999351i \(-0.511465\pi\)
−0.0360096 + 0.999351i \(0.511465\pi\)
\(284\) 3426.55 0.715944
\(285\) 0 0
\(286\) −956.598 −0.197779
\(287\) −8844.78 −1.81913
\(288\) 0 0
\(289\) −2217.66 −0.451387
\(290\) 1798.55 0.364187
\(291\) 0 0
\(292\) 1157.47 0.231972
\(293\) −7333.43 −1.46220 −0.731098 0.682272i \(-0.760992\pi\)
−0.731098 + 0.682272i \(0.760992\pi\)
\(294\) 0 0
\(295\) −4290.52 −0.846793
\(296\) −1332.48 −0.261651
\(297\) 0 0
\(298\) −1000.79 −0.194544
\(299\) −11259.1 −2.17769
\(300\) 0 0
\(301\) 4713.52 0.902601
\(302\) −2730.09 −0.520195
\(303\) 0 0
\(304\) −2199.40 −0.414948
\(305\) −2784.06 −0.522671
\(306\) 0 0
\(307\) −7965.33 −1.48080 −0.740399 0.672167i \(-0.765364\pi\)
−0.740399 + 0.672167i \(0.765364\pi\)
\(308\) 1244.74 0.230279
\(309\) 0 0
\(310\) −1489.65 −0.272925
\(311\) −2186.19 −0.398609 −0.199305 0.979938i \(-0.563868\pi\)
−0.199305 + 0.979938i \(0.563868\pi\)
\(312\) 0 0
\(313\) 38.6303 0.00697609 0.00348805 0.999994i \(-0.498890\pi\)
0.00348805 + 0.999994i \(0.498890\pi\)
\(314\) 4456.20 0.800885
\(315\) 0 0
\(316\) −502.232 −0.0894074
\(317\) 9535.81 1.68954 0.844770 0.535129i \(-0.179737\pi\)
0.844770 + 0.535129i \(0.179737\pi\)
\(318\) 0 0
\(319\) −2243.75 −0.393811
\(320\) 227.471 0.0397375
\(321\) 0 0
\(322\) −4133.47 −0.715371
\(323\) −4595.18 −0.791587
\(324\) 0 0
\(325\) 2177.86 0.371711
\(326\) 942.533 0.160129
\(327\) 0 0
\(328\) 6932.63 1.16704
\(329\) 8658.06 1.45086
\(330\) 0 0
\(331\) −3010.58 −0.499929 −0.249964 0.968255i \(-0.580419\pi\)
−0.249964 + 0.968255i \(0.580419\pi\)
\(332\) 3598.70 0.594892
\(333\) 0 0
\(334\) 1523.53 0.249593
\(335\) −209.493 −0.0341667
\(336\) 0 0
\(337\) 2812.72 0.454655 0.227327 0.973818i \(-0.427001\pi\)
0.227327 + 0.973818i \(0.427001\pi\)
\(338\) −7154.07 −1.15127
\(339\) 0 0
\(340\) −1619.69 −0.258353
\(341\) 1858.39 0.295125
\(342\) 0 0
\(343\) 2530.57 0.398361
\(344\) −3694.51 −0.579054
\(345\) 0 0
\(346\) 3881.30 0.603063
\(347\) −139.354 −0.0215588 −0.0107794 0.999942i \(-0.503431\pi\)
−0.0107794 + 0.999942i \(0.503431\pi\)
\(348\) 0 0
\(349\) −5210.29 −0.799141 −0.399571 0.916702i \(-0.630841\pi\)
−0.399571 + 0.916702i \(0.630841\pi\)
\(350\) 799.544 0.122107
\(351\) 0 0
\(352\) −1523.77 −0.230731
\(353\) −3410.46 −0.514223 −0.257112 0.966382i \(-0.582771\pi\)
−0.257112 + 0.966382i \(0.582771\pi\)
\(354\) 0 0
\(355\) −2745.82 −0.410515
\(356\) −1400.89 −0.208558
\(357\) 0 0
\(358\) −4540.36 −0.670295
\(359\) 8131.85 1.19549 0.597747 0.801685i \(-0.296063\pi\)
0.597747 + 0.801685i \(0.296063\pi\)
\(360\) 0 0
\(361\) 975.143 0.142170
\(362\) 1527.93 0.221840
\(363\) 0 0
\(364\) 13102.1 1.88663
\(365\) −927.523 −0.133010
\(366\) 0 0
\(367\) 132.151 0.0187963 0.00939816 0.999956i \(-0.497008\pi\)
0.00939816 + 0.999956i \(0.497008\pi\)
\(368\) −3211.60 −0.454935
\(369\) 0 0
\(370\) 467.878 0.0657400
\(371\) −710.812 −0.0994703
\(372\) 0 0
\(373\) 9348.88 1.29777 0.648883 0.760888i \(-0.275236\pi\)
0.648883 + 0.760888i \(0.275236\pi\)
\(374\) −570.092 −0.0788203
\(375\) 0 0
\(376\) −6786.29 −0.930787
\(377\) −23617.5 −3.22643
\(378\) 0 0
\(379\) 6164.17 0.835441 0.417720 0.908576i \(-0.362829\pi\)
0.417720 + 0.908576i \(0.362829\pi\)
\(380\) 2761.35 0.372774
\(381\) 0 0
\(382\) −1236.75 −0.165648
\(383\) −2601.27 −0.347047 −0.173523 0.984830i \(-0.555515\pi\)
−0.173523 + 0.984830i \(0.555515\pi\)
\(384\) 0 0
\(385\) −997.458 −0.132039
\(386\) 5669.20 0.747552
\(387\) 0 0
\(388\) 3463.07 0.453120
\(389\) −4250.57 −0.554017 −0.277008 0.960867i \(-0.589343\pi\)
−0.277008 + 0.960867i \(0.589343\pi\)
\(390\) 0 0
\(391\) −6709.95 −0.867869
\(392\) 4496.88 0.579405
\(393\) 0 0
\(394\) 2552.98 0.326440
\(395\) 402.456 0.0512653
\(396\) 0 0
\(397\) −6088.35 −0.769687 −0.384843 0.922982i \(-0.625745\pi\)
−0.384843 + 0.922982i \(0.625745\pi\)
\(398\) −2307.11 −0.290566
\(399\) 0 0
\(400\) 621.224 0.0776530
\(401\) −7725.79 −0.962114 −0.481057 0.876689i \(-0.659747\pi\)
−0.481057 + 0.876689i \(0.659747\pi\)
\(402\) 0 0
\(403\) 19561.3 2.41791
\(404\) −1416.38 −0.174425
\(405\) 0 0
\(406\) −8670.53 −1.05988
\(407\) −583.693 −0.0710875
\(408\) 0 0
\(409\) 16321.1 1.97318 0.986588 0.163232i \(-0.0521920\pi\)
0.986588 + 0.163232i \(0.0521920\pi\)
\(410\) −2434.28 −0.293221
\(411\) 0 0
\(412\) −2393.03 −0.286156
\(413\) 20684.0 2.46439
\(414\) 0 0
\(415\) −2883.77 −0.341105
\(416\) −16039.1 −1.89034
\(417\) 0 0
\(418\) 971.929 0.113729
\(419\) −1152.46 −0.134370 −0.0671851 0.997741i \(-0.521402\pi\)
−0.0671851 + 0.997741i \(0.521402\pi\)
\(420\) 0 0
\(421\) −3531.23 −0.408792 −0.204396 0.978888i \(-0.565523\pi\)
−0.204396 + 0.978888i \(0.565523\pi\)
\(422\) −4248.91 −0.490127
\(423\) 0 0
\(424\) 557.142 0.0638142
\(425\) 1297.92 0.148137
\(426\) 0 0
\(427\) 13421.5 1.52111
\(428\) 11077.2 1.25102
\(429\) 0 0
\(430\) 1297.27 0.145488
\(431\) −10230.6 −1.14337 −0.571683 0.820475i \(-0.693709\pi\)
−0.571683 + 0.820475i \(0.693709\pi\)
\(432\) 0 0
\(433\) −7311.31 −0.811453 −0.405726 0.913995i \(-0.632981\pi\)
−0.405726 + 0.913995i \(0.632981\pi\)
\(434\) 7181.40 0.794282
\(435\) 0 0
\(436\) −9549.65 −1.04896
\(437\) 11439.5 1.25224
\(438\) 0 0
\(439\) 7053.18 0.766811 0.383405 0.923580i \(-0.374751\pi\)
0.383405 + 0.923580i \(0.374751\pi\)
\(440\) 781.818 0.0847085
\(441\) 0 0
\(442\) −6000.75 −0.645761
\(443\) 167.790 0.0179954 0.00899770 0.999960i \(-0.497136\pi\)
0.00899770 + 0.999960i \(0.497136\pi\)
\(444\) 0 0
\(445\) 1122.58 0.119585
\(446\) 1864.15 0.197915
\(447\) 0 0
\(448\) −1096.60 −0.115646
\(449\) −3949.94 −0.415165 −0.207582 0.978218i \(-0.566560\pi\)
−0.207582 + 0.978218i \(0.566560\pi\)
\(450\) 0 0
\(451\) 3036.85 0.317072
\(452\) 5246.13 0.545922
\(453\) 0 0
\(454\) −8277.09 −0.855646
\(455\) −10499.2 −1.08178
\(456\) 0 0
\(457\) −7447.60 −0.762328 −0.381164 0.924507i \(-0.624477\pi\)
−0.381164 + 0.924507i \(0.624477\pi\)
\(458\) −8797.15 −0.897519
\(459\) 0 0
\(460\) 4032.16 0.408697
\(461\) −10887.6 −1.09997 −0.549987 0.835173i \(-0.685367\pi\)
−0.549987 + 0.835173i \(0.685367\pi\)
\(462\) 0 0
\(463\) −3171.27 −0.318318 −0.159159 0.987253i \(-0.550878\pi\)
−0.159159 + 0.987253i \(0.550878\pi\)
\(464\) −6736.77 −0.674023
\(465\) 0 0
\(466\) −3255.29 −0.323602
\(467\) 16348.5 1.61996 0.809978 0.586461i \(-0.199479\pi\)
0.809978 + 0.586461i \(0.199479\pi\)
\(468\) 0 0
\(469\) 1009.94 0.0994340
\(470\) 2382.90 0.233861
\(471\) 0 0
\(472\) −16212.3 −1.58100
\(473\) −1618.39 −0.157322
\(474\) 0 0
\(475\) −2212.77 −0.213745
\(476\) 7808.29 0.751874
\(477\) 0 0
\(478\) 9220.98 0.882338
\(479\) −9413.69 −0.897959 −0.448980 0.893542i \(-0.648212\pi\)
−0.448980 + 0.893542i \(0.648212\pi\)
\(480\) 0 0
\(481\) −6143.91 −0.582408
\(482\) −8405.59 −0.794324
\(483\) 0 0
\(484\) 7877.50 0.739810
\(485\) −2775.08 −0.259814
\(486\) 0 0
\(487\) 13482.3 1.25450 0.627250 0.778818i \(-0.284180\pi\)
0.627250 + 0.778818i \(0.284180\pi\)
\(488\) −10519.9 −0.975852
\(489\) 0 0
\(490\) −1579.01 −0.145576
\(491\) 9219.74 0.847416 0.423708 0.905799i \(-0.360728\pi\)
0.423708 + 0.905799i \(0.360728\pi\)
\(492\) 0 0
\(493\) −14075.0 −1.28582
\(494\) 10230.4 0.931760
\(495\) 0 0
\(496\) 5579.76 0.505118
\(497\) 13237.2 1.19471
\(498\) 0 0
\(499\) 104.346 0.00936108 0.00468054 0.999989i \(-0.498510\pi\)
0.00468054 + 0.999989i \(0.498510\pi\)
\(500\) −779.947 −0.0697606
\(501\) 0 0
\(502\) 5337.29 0.474531
\(503\) −490.652 −0.0434933 −0.0217466 0.999764i \(-0.506923\pi\)
−0.0217466 + 0.999764i \(0.506923\pi\)
\(504\) 0 0
\(505\) 1135.00 0.100013
\(506\) 1419.23 0.124688
\(507\) 0 0
\(508\) −6479.50 −0.565908
\(509\) 3412.81 0.297191 0.148595 0.988898i \(-0.452525\pi\)
0.148595 + 0.988898i \(0.452525\pi\)
\(510\) 0 0
\(511\) 4471.45 0.387095
\(512\) −8330.89 −0.719095
\(513\) 0 0
\(514\) 6631.05 0.569033
\(515\) 1917.62 0.164079
\(516\) 0 0
\(517\) −2972.74 −0.252884
\(518\) −2255.57 −0.191321
\(519\) 0 0
\(520\) 8229.36 0.694002
\(521\) 3486.31 0.293163 0.146582 0.989199i \(-0.453173\pi\)
0.146582 + 0.989199i \(0.453173\pi\)
\(522\) 0 0
\(523\) 14465.6 1.20943 0.604717 0.796440i \(-0.293286\pi\)
0.604717 + 0.796440i \(0.293286\pi\)
\(524\) 5012.94 0.417923
\(525\) 0 0
\(526\) 5297.24 0.439108
\(527\) 11657.7 0.963602
\(528\) 0 0
\(529\) 4537.21 0.372911
\(530\) −195.632 −0.0160334
\(531\) 0 0
\(532\) −13312.1 −1.08487
\(533\) 31965.6 2.59772
\(534\) 0 0
\(535\) −8876.59 −0.717324
\(536\) −791.600 −0.0637909
\(537\) 0 0
\(538\) 2904.24 0.232734
\(539\) 1969.86 0.157417
\(540\) 0 0
\(541\) 6602.78 0.524724 0.262362 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(542\) −5679.09 −0.450070
\(543\) 0 0
\(544\) −9558.65 −0.753353
\(545\) 7652.48 0.601461
\(546\) 0 0
\(547\) 6358.56 0.497024 0.248512 0.968629i \(-0.420058\pi\)
0.248512 + 0.968629i \(0.420058\pi\)
\(548\) −5423.14 −0.422746
\(549\) 0 0
\(550\) −274.523 −0.0212831
\(551\) 23996.0 1.85529
\(552\) 0 0
\(553\) −1940.18 −0.149195
\(554\) 5147.98 0.394795
\(555\) 0 0
\(556\) 1225.44 0.0934717
\(557\) 20782.3 1.58092 0.790461 0.612513i \(-0.209841\pi\)
0.790461 + 0.612513i \(0.209841\pi\)
\(558\) 0 0
\(559\) −17035.0 −1.28892
\(560\) −2994.83 −0.225991
\(561\) 0 0
\(562\) 8488.17 0.637103
\(563\) −16470.0 −1.23291 −0.616453 0.787392i \(-0.711431\pi\)
−0.616453 + 0.787392i \(0.711431\pi\)
\(564\) 0 0
\(565\) −4203.91 −0.313026
\(566\) 454.922 0.0337841
\(567\) 0 0
\(568\) −10375.4 −0.766451
\(569\) −5425.39 −0.399726 −0.199863 0.979824i \(-0.564050\pi\)
−0.199863 + 0.979824i \(0.564050\pi\)
\(570\) 0 0
\(571\) −14689.7 −1.07661 −0.538306 0.842749i \(-0.680936\pi\)
−0.538306 + 0.842749i \(0.680936\pi\)
\(572\) −4498.59 −0.328838
\(573\) 0 0
\(574\) 11735.3 0.853350
\(575\) −3231.12 −0.234343
\(576\) 0 0
\(577\) −17933.1 −1.29387 −0.646935 0.762545i \(-0.723949\pi\)
−0.646935 + 0.762545i \(0.723949\pi\)
\(578\) 2942.41 0.211745
\(579\) 0 0
\(580\) 8458.02 0.605517
\(581\) 13902.2 0.992704
\(582\) 0 0
\(583\) 244.057 0.0173376
\(584\) −3504.78 −0.248337
\(585\) 0 0
\(586\) 9730.06 0.685913
\(587\) −4801.16 −0.337589 −0.168795 0.985651i \(-0.553987\pi\)
−0.168795 + 0.985651i \(0.553987\pi\)
\(588\) 0 0
\(589\) −19874.8 −1.39037
\(590\) 5692.70 0.397228
\(591\) 0 0
\(592\) −1752.52 −0.121669
\(593\) 16803.8 1.16366 0.581830 0.813311i \(-0.302337\pi\)
0.581830 + 0.813311i \(0.302337\pi\)
\(594\) 0 0
\(595\) −6257.06 −0.431117
\(596\) −4706.41 −0.323460
\(597\) 0 0
\(598\) 14938.7 1.02155
\(599\) 17220.2 1.17462 0.587310 0.809362i \(-0.300187\pi\)
0.587310 + 0.809362i \(0.300187\pi\)
\(600\) 0 0
\(601\) 21536.7 1.46173 0.730865 0.682522i \(-0.239117\pi\)
0.730865 + 0.682522i \(0.239117\pi\)
\(602\) −6253.94 −0.423408
\(603\) 0 0
\(604\) −12838.8 −0.864905
\(605\) −6312.52 −0.424199
\(606\) 0 0
\(607\) −24050.9 −1.60823 −0.804117 0.594472i \(-0.797361\pi\)
−0.804117 + 0.594472i \(0.797361\pi\)
\(608\) 16296.2 1.08700
\(609\) 0 0
\(610\) 3693.91 0.245184
\(611\) −31290.8 −2.07184
\(612\) 0 0
\(613\) −3554.49 −0.234200 −0.117100 0.993120i \(-0.537360\pi\)
−0.117100 + 0.993120i \(0.537360\pi\)
\(614\) 10568.5 0.694639
\(615\) 0 0
\(616\) −3769.03 −0.246524
\(617\) −9747.42 −0.636007 −0.318003 0.948090i \(-0.603012\pi\)
−0.318003 + 0.948090i \(0.603012\pi\)
\(618\) 0 0
\(619\) −20765.2 −1.34834 −0.674171 0.738576i \(-0.735499\pi\)
−0.674171 + 0.738576i \(0.735499\pi\)
\(620\) −7005.39 −0.453779
\(621\) 0 0
\(622\) 2900.66 0.186987
\(623\) −5411.80 −0.348024
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −51.2551 −0.00327247
\(627\) 0 0
\(628\) 20956.2 1.33159
\(629\) −3661.51 −0.232105
\(630\) 0 0
\(631\) 23740.6 1.49778 0.748890 0.662694i \(-0.230587\pi\)
0.748890 + 0.662694i \(0.230587\pi\)
\(632\) 1520.74 0.0957147
\(633\) 0 0
\(634\) −12652.2 −0.792560
\(635\) 5192.25 0.324486
\(636\) 0 0
\(637\) 20734.6 1.28970
\(638\) 2977.02 0.184736
\(639\) 0 0
\(640\) 7062.81 0.436222
\(641\) 26108.7 1.60879 0.804394 0.594097i \(-0.202490\pi\)
0.804394 + 0.594097i \(0.202490\pi\)
\(642\) 0 0
\(643\) −24017.9 −1.47305 −0.736526 0.676409i \(-0.763535\pi\)
−0.736526 + 0.676409i \(0.763535\pi\)
\(644\) −19438.5 −1.18941
\(645\) 0 0
\(646\) 6096.92 0.371331
\(647\) 18588.3 1.12949 0.564747 0.825264i \(-0.308974\pi\)
0.564747 + 0.825264i \(0.308974\pi\)
\(648\) 0 0
\(649\) −7101.83 −0.429540
\(650\) −2889.61 −0.174369
\(651\) 0 0
\(652\) 4432.45 0.266239
\(653\) 12171.9 0.729441 0.364721 0.931117i \(-0.381165\pi\)
0.364721 + 0.931117i \(0.381165\pi\)
\(654\) 0 0
\(655\) −4017.05 −0.239632
\(656\) 9118.04 0.542682
\(657\) 0 0
\(658\) −11487.6 −0.680597
\(659\) 8589.64 0.507746 0.253873 0.967238i \(-0.418295\pi\)
0.253873 + 0.967238i \(0.418295\pi\)
\(660\) 0 0
\(661\) 24995.3 1.47081 0.735405 0.677628i \(-0.236992\pi\)
0.735405 + 0.677628i \(0.236992\pi\)
\(662\) 3994.46 0.234515
\(663\) 0 0
\(664\) −10896.7 −0.636859
\(665\) 10667.4 0.622053
\(666\) 0 0
\(667\) 35039.4 2.03408
\(668\) 7164.71 0.414987
\(669\) 0 0
\(670\) 277.958 0.0160275
\(671\) −4608.28 −0.265127
\(672\) 0 0
\(673\) −2540.17 −0.145492 −0.0727461 0.997350i \(-0.523176\pi\)
−0.0727461 + 0.997350i \(0.523176\pi\)
\(674\) −3731.94 −0.213277
\(675\) 0 0
\(676\) −33643.4 −1.91417
\(677\) 4197.79 0.238307 0.119154 0.992876i \(-0.461982\pi\)
0.119154 + 0.992876i \(0.461982\pi\)
\(678\) 0 0
\(679\) 13378.3 0.756128
\(680\) 4904.36 0.276579
\(681\) 0 0
\(682\) −2465.73 −0.138442
\(683\) 8523.02 0.477488 0.238744 0.971083i \(-0.423264\pi\)
0.238744 + 0.971083i \(0.423264\pi\)
\(684\) 0 0
\(685\) 4345.76 0.242398
\(686\) −3357.58 −0.186870
\(687\) 0 0
\(688\) −4859.15 −0.269263
\(689\) 2568.92 0.142044
\(690\) 0 0
\(691\) −20292.0 −1.11714 −0.558571 0.829457i \(-0.688650\pi\)
−0.558571 + 0.829457i \(0.688650\pi\)
\(692\) 18252.6 1.00269
\(693\) 0 0
\(694\) 184.896 0.0101132
\(695\) −981.990 −0.0535957
\(696\) 0 0
\(697\) 19050.2 1.03526
\(698\) 6913.05 0.374875
\(699\) 0 0
\(700\) 3760.01 0.203022
\(701\) 11223.6 0.604721 0.302361 0.953194i \(-0.402225\pi\)
0.302361 + 0.953194i \(0.402225\pi\)
\(702\) 0 0
\(703\) 6242.38 0.334901
\(704\) 376.518 0.0201570
\(705\) 0 0
\(706\) 4525.03 0.241221
\(707\) −5471.66 −0.291065
\(708\) 0 0
\(709\) −19931.0 −1.05575 −0.527873 0.849324i \(-0.677010\pi\)
−0.527873 + 0.849324i \(0.677010\pi\)
\(710\) 3643.17 0.192572
\(711\) 0 0
\(712\) 4241.83 0.223271
\(713\) −29021.5 −1.52435
\(714\) 0 0
\(715\) 3604.88 0.188552
\(716\) −21352.0 −1.11447
\(717\) 0 0
\(718\) −10789.4 −0.560804
\(719\) 3186.13 0.165261 0.0826305 0.996580i \(-0.473668\pi\)
0.0826305 + 0.996580i \(0.473668\pi\)
\(720\) 0 0
\(721\) −9244.58 −0.477512
\(722\) −1293.83 −0.0666916
\(723\) 0 0
\(724\) 7185.37 0.368843
\(725\) −6777.71 −0.347197
\(726\) 0 0
\(727\) −23088.2 −1.17785 −0.588923 0.808189i \(-0.700448\pi\)
−0.588923 + 0.808189i \(0.700448\pi\)
\(728\) −39672.5 −2.01973
\(729\) 0 0
\(730\) 1230.65 0.0623949
\(731\) −10152.2 −0.513667
\(732\) 0 0
\(733\) −29801.4 −1.50169 −0.750846 0.660477i \(-0.770354\pi\)
−0.750846 + 0.660477i \(0.770354\pi\)
\(734\) −175.340 −0.00881731
\(735\) 0 0
\(736\) 23796.0 1.19175
\(737\) −346.761 −0.0173312
\(738\) 0 0
\(739\) 39741.5 1.97823 0.989117 0.147134i \(-0.0470047\pi\)
0.989117 + 0.147134i \(0.0470047\pi\)
\(740\) 2200.29 0.109303
\(741\) 0 0
\(742\) 943.111 0.0466613
\(743\) −9713.53 −0.479616 −0.239808 0.970820i \(-0.577085\pi\)
−0.239808 + 0.970820i \(0.577085\pi\)
\(744\) 0 0
\(745\) 3771.42 0.185468
\(746\) −12404.2 −0.608779
\(747\) 0 0
\(748\) −2680.97 −0.131051
\(749\) 42792.7 2.08760
\(750\) 0 0
\(751\) −2509.28 −0.121924 −0.0609619 0.998140i \(-0.519417\pi\)
−0.0609619 + 0.998140i \(0.519417\pi\)
\(752\) −8925.55 −0.432821
\(753\) 0 0
\(754\) 31335.9 1.51351
\(755\) 10288.2 0.495927
\(756\) 0 0
\(757\) 9705.73 0.465998 0.232999 0.972477i \(-0.425146\pi\)
0.232999 + 0.972477i \(0.425146\pi\)
\(758\) −8178.67 −0.391903
\(759\) 0 0
\(760\) −8361.25 −0.399072
\(761\) 7778.44 0.370523 0.185262 0.982689i \(-0.440687\pi\)
0.185262 + 0.982689i \(0.440687\pi\)
\(762\) 0 0
\(763\) −36891.5 −1.75041
\(764\) −5816.04 −0.275415
\(765\) 0 0
\(766\) 3451.39 0.162799
\(767\) −74753.3 −3.51915
\(768\) 0 0
\(769\) 1387.90 0.0650833 0.0325416 0.999470i \(-0.489640\pi\)
0.0325416 + 0.999470i \(0.489640\pi\)
\(770\) 1323.44 0.0619393
\(771\) 0 0
\(772\) 26660.5 1.24292
\(773\) −20692.0 −0.962796 −0.481398 0.876502i \(-0.659871\pi\)
−0.481398 + 0.876502i \(0.659871\pi\)
\(774\) 0 0
\(775\) 5613.67 0.260192
\(776\) −10486.0 −0.485086
\(777\) 0 0
\(778\) 5639.70 0.259888
\(779\) −32478.0 −1.49377
\(780\) 0 0
\(781\) −4544.98 −0.208236
\(782\) 8902.82 0.407115
\(783\) 0 0
\(784\) 5914.45 0.269426
\(785\) −16792.9 −0.763523
\(786\) 0 0
\(787\) 33896.8 1.53531 0.767655 0.640863i \(-0.221423\pi\)
0.767655 + 0.640863i \(0.221423\pi\)
\(788\) 12005.9 0.542756
\(789\) 0 0
\(790\) −533.983 −0.0240484
\(791\) 20266.4 0.910988
\(792\) 0 0
\(793\) −48506.4 −2.17215
\(794\) 8078.08 0.361058
\(795\) 0 0
\(796\) −10849.7 −0.483110
\(797\) 11954.5 0.531307 0.265653 0.964069i \(-0.414412\pi\)
0.265653 + 0.964069i \(0.414412\pi\)
\(798\) 0 0
\(799\) −18648.0 −0.825682
\(800\) −4602.89 −0.203421
\(801\) 0 0
\(802\) 10250.7 0.451325
\(803\) −1535.27 −0.0674702
\(804\) 0 0
\(805\) 15576.8 0.681998
\(806\) −25954.1 −1.13424
\(807\) 0 0
\(808\) 4288.75 0.186730
\(809\) −27377.3 −1.18978 −0.594891 0.803807i \(-0.702805\pi\)
−0.594891 + 0.803807i \(0.702805\pi\)
\(810\) 0 0
\(811\) −713.034 −0.0308730 −0.0154365 0.999881i \(-0.504914\pi\)
−0.0154365 + 0.999881i \(0.504914\pi\)
\(812\) −40774.9 −1.76221
\(813\) 0 0
\(814\) 774.449 0.0333470
\(815\) −3551.88 −0.152659
\(816\) 0 0
\(817\) 17308.0 0.741164
\(818\) −21655.0 −0.925612
\(819\) 0 0
\(820\) −11447.7 −0.487526
\(821\) −6384.12 −0.271385 −0.135693 0.990751i \(-0.543326\pi\)
−0.135693 + 0.990751i \(0.543326\pi\)
\(822\) 0 0
\(823\) 19636.3 0.831686 0.415843 0.909436i \(-0.363487\pi\)
0.415843 + 0.909436i \(0.363487\pi\)
\(824\) 7246.00 0.306343
\(825\) 0 0
\(826\) −27443.7 −1.15604
\(827\) −43578.4 −1.83237 −0.916185 0.400756i \(-0.868747\pi\)
−0.916185 + 0.400756i \(0.868747\pi\)
\(828\) 0 0
\(829\) −20418.8 −0.855458 −0.427729 0.903907i \(-0.640686\pi\)
−0.427729 + 0.903907i \(0.640686\pi\)
\(830\) 3826.21 0.160012
\(831\) 0 0
\(832\) 3963.20 0.165143
\(833\) 12357.0 0.513978
\(834\) 0 0
\(835\) −5741.34 −0.237949
\(836\) 4570.69 0.189092
\(837\) 0 0
\(838\) 1529.09 0.0630328
\(839\) 10962.0 0.451073 0.225536 0.974235i \(-0.427587\pi\)
0.225536 + 0.974235i \(0.427587\pi\)
\(840\) 0 0
\(841\) 49110.9 2.01365
\(842\) 4685.26 0.191763
\(843\) 0 0
\(844\) −19981.3 −0.814911
\(845\) 26959.7 1.09756
\(846\) 0 0
\(847\) 30431.8 1.23453
\(848\) 732.772 0.0296739
\(849\) 0 0
\(850\) −1722.09 −0.0694907
\(851\) 9115.22 0.367175
\(852\) 0 0
\(853\) −5246.41 −0.210590 −0.105295 0.994441i \(-0.533579\pi\)
−0.105295 + 0.994441i \(0.533579\pi\)
\(854\) −17807.8 −0.713548
\(855\) 0 0
\(856\) −33541.4 −1.33928
\(857\) 27283.1 1.08748 0.543741 0.839253i \(-0.317007\pi\)
0.543741 + 0.839253i \(0.317007\pi\)
\(858\) 0 0
\(859\) 3581.05 0.142240 0.0711198 0.997468i \(-0.477343\pi\)
0.0711198 + 0.997468i \(0.477343\pi\)
\(860\) 6100.66 0.241896
\(861\) 0 0
\(862\) 13574.0 0.536350
\(863\) 41710.9 1.64526 0.822629 0.568579i \(-0.192507\pi\)
0.822629 + 0.568579i \(0.192507\pi\)
\(864\) 0 0
\(865\) −14626.4 −0.574929
\(866\) 9700.71 0.380651
\(867\) 0 0
\(868\) 33771.9 1.32062
\(869\) 666.161 0.0260046
\(870\) 0 0
\(871\) −3649.98 −0.141992
\(872\) 28916.0 1.12296
\(873\) 0 0
\(874\) −15178.1 −0.587422
\(875\) −3013.03 −0.116410
\(876\) 0 0
\(877\) 21533.6 0.829119 0.414560 0.910022i \(-0.363936\pi\)
0.414560 + 0.910022i \(0.363936\pi\)
\(878\) −9358.22 −0.359709
\(879\) 0 0
\(880\) 1028.27 0.0393899
\(881\) 30919.1 1.18240 0.591199 0.806526i \(-0.298655\pi\)
0.591199 + 0.806526i \(0.298655\pi\)
\(882\) 0 0
\(883\) 5341.72 0.203582 0.101791 0.994806i \(-0.467543\pi\)
0.101791 + 0.994806i \(0.467543\pi\)
\(884\) −28219.7 −1.07368
\(885\) 0 0
\(886\) −222.626 −0.00844160
\(887\) 11210.3 0.424358 0.212179 0.977231i \(-0.431944\pi\)
0.212179 + 0.977231i \(0.431944\pi\)
\(888\) 0 0
\(889\) −25031.1 −0.944337
\(890\) −1489.45 −0.0560972
\(891\) 0 0
\(892\) 8766.54 0.329064
\(893\) 31792.3 1.19137
\(894\) 0 0
\(895\) 17110.1 0.639025
\(896\) −34048.8 −1.26952
\(897\) 0 0
\(898\) 5240.81 0.194753
\(899\) −60876.6 −2.25845
\(900\) 0 0
\(901\) 1530.97 0.0566083
\(902\) −4029.32 −0.148738
\(903\) 0 0
\(904\) −15885.1 −0.584435
\(905\) −5757.90 −0.211491
\(906\) 0 0
\(907\) 20690.8 0.757473 0.378737 0.925505i \(-0.376359\pi\)
0.378737 + 0.925505i \(0.376359\pi\)
\(908\) −38924.6 −1.42264
\(909\) 0 0
\(910\) 13930.4 0.507459
\(911\) 19471.6 0.708147 0.354073 0.935218i \(-0.384796\pi\)
0.354073 + 0.935218i \(0.384796\pi\)
\(912\) 0 0
\(913\) −4773.32 −0.173027
\(914\) 9881.53 0.357606
\(915\) 0 0
\(916\) −41370.3 −1.49226
\(917\) 19365.6 0.697393
\(918\) 0 0
\(919\) −30002.0 −1.07690 −0.538452 0.842656i \(-0.680991\pi\)
−0.538452 + 0.842656i \(0.680991\pi\)
\(920\) −12209.2 −0.437529
\(921\) 0 0
\(922\) 14445.8 0.515996
\(923\) −47840.1 −1.70604
\(924\) 0 0
\(925\) −1763.17 −0.0626732
\(926\) 4207.66 0.149322
\(927\) 0 0
\(928\) 49915.3 1.76568
\(929\) 22262.8 0.786241 0.393121 0.919487i \(-0.371395\pi\)
0.393121 + 0.919487i \(0.371395\pi\)
\(930\) 0 0
\(931\) −21066.9 −0.741612
\(932\) −15308.7 −0.538039
\(933\) 0 0
\(934\) −21691.4 −0.759917
\(935\) 2148.36 0.0751432
\(936\) 0 0
\(937\) 23361.5 0.814500 0.407250 0.913317i \(-0.366488\pi\)
0.407250 + 0.913317i \(0.366488\pi\)
\(938\) −1339.99 −0.0466443
\(939\) 0 0
\(940\) 11206.0 0.388830
\(941\) 39330.9 1.36254 0.681271 0.732031i \(-0.261428\pi\)
0.681271 + 0.732031i \(0.261428\pi\)
\(942\) 0 0
\(943\) −47424.8 −1.63771
\(944\) −21323.0 −0.735175
\(945\) 0 0
\(946\) 2147.29 0.0737995
\(947\) −26869.9 −0.922021 −0.461011 0.887395i \(-0.652513\pi\)
−0.461011 + 0.887395i \(0.652513\pi\)
\(948\) 0 0
\(949\) −16160.1 −0.552772
\(950\) 2935.92 0.100267
\(951\) 0 0
\(952\) −23643.2 −0.804916
\(953\) 16422.6 0.558218 0.279109 0.960259i \(-0.409961\pi\)
0.279109 + 0.960259i \(0.409961\pi\)
\(954\) 0 0
\(955\) 4660.60 0.157920
\(956\) 43363.5 1.46702
\(957\) 0 0
\(958\) 12490.2 0.421231
\(959\) −20950.3 −0.705442
\(960\) 0 0
\(961\) 20630.3 0.692500
\(962\) 8151.79 0.273206
\(963\) 0 0
\(964\) −39528.9 −1.32069
\(965\) −21364.1 −0.712677
\(966\) 0 0
\(967\) 10304.4 0.342677 0.171339 0.985212i \(-0.445191\pi\)
0.171339 + 0.985212i \(0.445191\pi\)
\(968\) −23852.7 −0.792000
\(969\) 0 0
\(970\) 3682.00 0.121878
\(971\) −44153.1 −1.45926 −0.729630 0.683842i \(-0.760308\pi\)
−0.729630 + 0.683842i \(0.760308\pi\)
\(972\) 0 0
\(973\) 4734.03 0.155977
\(974\) −17888.4 −0.588483
\(975\) 0 0
\(976\) −13836.2 −0.453776
\(977\) 16115.6 0.527722 0.263861 0.964561i \(-0.415004\pi\)
0.263861 + 0.964561i \(0.415004\pi\)
\(978\) 0 0
\(979\) 1858.14 0.0606602
\(980\) −7425.59 −0.242043
\(981\) 0 0
\(982\) −12232.8 −0.397521
\(983\) 38748.6 1.25726 0.628630 0.777704i \(-0.283616\pi\)
0.628630 + 0.777704i \(0.283616\pi\)
\(984\) 0 0
\(985\) −9620.75 −0.311211
\(986\) 18674.9 0.603174
\(987\) 0 0
\(988\) 48110.7 1.54920
\(989\) 25273.5 0.812587
\(990\) 0 0
\(991\) −42906.4 −1.37534 −0.687672 0.726022i \(-0.741367\pi\)
−0.687672 + 0.726022i \(0.741367\pi\)
\(992\) −41342.5 −1.32321
\(993\) 0 0
\(994\) −17563.2 −0.560434
\(995\) 8694.22 0.277010
\(996\) 0 0
\(997\) −19395.3 −0.616104 −0.308052 0.951369i \(-0.599677\pi\)
−0.308052 + 0.951369i \(0.599677\pi\)
\(998\) −138.447 −0.00439126
\(999\) 0 0
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 405.4.a.g.1.2 3
3.2 odd 2 405.4.a.i.1.2 yes 3
5.4 even 2 2025.4.a.r.1.2 3
9.2 odd 6 405.4.e.s.271.2 6
9.4 even 3 405.4.e.u.136.2 6
9.5 odd 6 405.4.e.s.136.2 6
9.7 even 3 405.4.e.u.271.2 6
15.14 odd 2 2025.4.a.p.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.4.a.g.1.2 3 1.1 even 1 trivial
405.4.a.i.1.2 yes 3 3.2 odd 2
405.4.e.s.136.2 6 9.5 odd 6
405.4.e.s.271.2 6 9.2 odd 6
405.4.e.u.136.2 6 9.4 even 3
405.4.e.u.271.2 6 9.7 even 3
2025.4.a.p.1.2 3 15.14 odd 2
2025.4.a.r.1.2 3 5.4 even 2