Properties

Label 405.4.a.c.1.1
Level $405$
Weight $4$
Character 405.1
Self dual yes
Analytic conductor $23.896$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
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 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.73205 q^{2} -0.535898 q^{4} -5.00000 q^{5} +6.92820 q^{7} +23.3205 q^{8} +O(q^{10})\) \(q-2.73205 q^{2} -0.535898 q^{4} -5.00000 q^{5} +6.92820 q^{7} +23.3205 q^{8} +13.6603 q^{10} +59.4974 q^{11} -25.0718 q^{13} -18.9282 q^{14} -59.4256 q^{16} -112.995 q^{17} -122.779 q^{19} +2.67949 q^{20} -162.550 q^{22} +97.4256 q^{23} +25.0000 q^{25} +68.4974 q^{26} -3.71281 q^{28} +126.569 q^{29} -108.215 q^{31} -24.2102 q^{32} +308.708 q^{34} -34.6410 q^{35} +294.344 q^{37} +335.440 q^{38} -116.603 q^{40} +205.862 q^{41} -80.1999 q^{43} -31.8846 q^{44} -266.172 q^{46} +269.713 q^{47} -295.000 q^{49} -68.3013 q^{50} +13.4359 q^{52} +98.7949 q^{53} -297.487 q^{55} +161.569 q^{56} -345.794 q^{58} +304.503 q^{59} +666.554 q^{61} +295.650 q^{62} +541.549 q^{64} +125.359 q^{65} -635.933 q^{67} +60.5538 q^{68} +94.6410 q^{70} +826.482 q^{71} +751.349 q^{73} -804.161 q^{74} +65.7973 q^{76} +412.210 q^{77} +23.8667 q^{79} +297.128 q^{80} -562.424 q^{82} -817.230 q^{83} +564.974 q^{85} +219.110 q^{86} +1387.51 q^{88} +513.000 q^{89} -173.703 q^{91} -52.2102 q^{92} -736.869 q^{94} +613.897 q^{95} +428.102 q^{97} +805.955 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 8 q^{4} - 10 q^{5} + 12 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} - 8 q^{4} - 10 q^{5} + 12 q^{8} + 10 q^{10} + 22 q^{11} - 64 q^{13} - 24 q^{14} - 8 q^{16} - 32 q^{17} - 10 q^{19} + 40 q^{20} - 190 q^{22} + 84 q^{23} + 50 q^{25} + 40 q^{26} + 48 q^{28} + 170 q^{29} - 258 q^{31} + 104 q^{32} + 368 q^{34} + 76 q^{37} + 418 q^{38} - 60 q^{40} + 578 q^{41} + 380 q^{43} + 248 q^{44} - 276 q^{46} + 484 q^{47} - 590 q^{49} - 50 q^{50} + 304 q^{52} + 544 q^{53} - 110 q^{55} + 240 q^{56} - 314 q^{58} + 706 q^{59} + 668 q^{61} + 186 q^{62} + 224 q^{64} + 320 q^{65} - 1452 q^{67} - 544 q^{68} + 120 q^{70} + 974 q^{71} + 1184 q^{73} - 964 q^{74} - 776 q^{76} + 672 q^{77} + 408 q^{79} + 40 q^{80} - 290 q^{82} + 444 q^{83} + 160 q^{85} + 556 q^{86} + 1812 q^{88} + 1026 q^{89} + 96 q^{91} + 48 q^{92} - 580 q^{94} + 50 q^{95} - 668 q^{97} + 590 q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.73205 −0.965926 −0.482963 0.875641i \(-0.660439\pi\)
−0.482963 + 0.875641i \(0.660439\pi\)
\(3\) 0 0
\(4\) −0.535898 −0.0669873
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 6.92820 0.374088 0.187044 0.982352i \(-0.440109\pi\)
0.187044 + 0.982352i \(0.440109\pi\)
\(8\) 23.3205 1.03063
\(9\) 0 0
\(10\) 13.6603 0.431975
\(11\) 59.4974 1.63083 0.815416 0.578876i \(-0.196508\pi\)
0.815416 + 0.578876i \(0.196508\pi\)
\(12\) 0 0
\(13\) −25.0718 −0.534897 −0.267449 0.963572i \(-0.586181\pi\)
−0.267449 + 0.963572i \(0.586181\pi\)
\(14\) −18.9282 −0.361341
\(15\) 0 0
\(16\) −59.4256 −0.928525
\(17\) −112.995 −1.61208 −0.806038 0.591864i \(-0.798392\pi\)
−0.806038 + 0.591864i \(0.798392\pi\)
\(18\) 0 0
\(19\) −122.779 −1.48250 −0.741251 0.671228i \(-0.765767\pi\)
−0.741251 + 0.671228i \(0.765767\pi\)
\(20\) 2.67949 0.0299576
\(21\) 0 0
\(22\) −162.550 −1.57526
\(23\) 97.4256 0.883246 0.441623 0.897201i \(-0.354403\pi\)
0.441623 + 0.897201i \(0.354403\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 68.4974 0.516671
\(27\) 0 0
\(28\) −3.71281 −0.0250591
\(29\) 126.569 0.810459 0.405230 0.914215i \(-0.367192\pi\)
0.405230 + 0.914215i \(0.367192\pi\)
\(30\) 0 0
\(31\) −108.215 −0.626970 −0.313485 0.949593i \(-0.601496\pi\)
−0.313485 + 0.949593i \(0.601496\pi\)
\(32\) −24.2102 −0.133744
\(33\) 0 0
\(34\) 308.708 1.55714
\(35\) −34.6410 −0.167297
\(36\) 0 0
\(37\) 294.344 1.30783 0.653916 0.756567i \(-0.273125\pi\)
0.653916 + 0.756567i \(0.273125\pi\)
\(38\) 335.440 1.43199
\(39\) 0 0
\(40\) −116.603 −0.460912
\(41\) 205.862 0.784151 0.392075 0.919933i \(-0.371757\pi\)
0.392075 + 0.919933i \(0.371757\pi\)
\(42\) 0 0
\(43\) −80.1999 −0.284427 −0.142214 0.989836i \(-0.545422\pi\)
−0.142214 + 0.989836i \(0.545422\pi\)
\(44\) −31.8846 −0.109245
\(45\) 0 0
\(46\) −266.172 −0.853150
\(47\) 269.713 0.837057 0.418528 0.908204i \(-0.362546\pi\)
0.418528 + 0.908204i \(0.362546\pi\)
\(48\) 0 0
\(49\) −295.000 −0.860058
\(50\) −68.3013 −0.193185
\(51\) 0 0
\(52\) 13.4359 0.0358313
\(53\) 98.7949 0.256048 0.128024 0.991771i \(-0.459137\pi\)
0.128024 + 0.991771i \(0.459137\pi\)
\(54\) 0 0
\(55\) −297.487 −0.729330
\(56\) 161.569 0.385546
\(57\) 0 0
\(58\) −345.794 −0.782843
\(59\) 304.503 0.671913 0.335956 0.941878i \(-0.390941\pi\)
0.335956 + 0.941878i \(0.390941\pi\)
\(60\) 0 0
\(61\) 666.554 1.39907 0.699537 0.714597i \(-0.253390\pi\)
0.699537 + 0.714597i \(0.253390\pi\)
\(62\) 295.650 0.605606
\(63\) 0 0
\(64\) 541.549 1.05771
\(65\) 125.359 0.239213
\(66\) 0 0
\(67\) −635.933 −1.15958 −0.579788 0.814767i \(-0.696865\pi\)
−0.579788 + 0.814767i \(0.696865\pi\)
\(68\) 60.5538 0.107989
\(69\) 0 0
\(70\) 94.6410 0.161597
\(71\) 826.482 1.38148 0.690742 0.723101i \(-0.257284\pi\)
0.690742 + 0.723101i \(0.257284\pi\)
\(72\) 0 0
\(73\) 751.349 1.20464 0.602320 0.798255i \(-0.294243\pi\)
0.602320 + 0.798255i \(0.294243\pi\)
\(74\) −804.161 −1.26327
\(75\) 0 0
\(76\) 65.7973 0.0993088
\(77\) 412.210 0.610074
\(78\) 0 0
\(79\) 23.8667 0.0339901 0.0169950 0.999856i \(-0.494590\pi\)
0.0169950 + 0.999856i \(0.494590\pi\)
\(80\) 297.128 0.415249
\(81\) 0 0
\(82\) −562.424 −0.757431
\(83\) −817.230 −1.08076 −0.540378 0.841423i \(-0.681719\pi\)
−0.540378 + 0.841423i \(0.681719\pi\)
\(84\) 0 0
\(85\) 564.974 0.720942
\(86\) 219.110 0.274736
\(87\) 0 0
\(88\) 1387.51 1.68078
\(89\) 513.000 0.610988 0.305494 0.952194i \(-0.401178\pi\)
0.305494 + 0.952194i \(0.401178\pi\)
\(90\) 0 0
\(91\) −173.703 −0.200099
\(92\) −52.2102 −0.0591662
\(93\) 0 0
\(94\) −736.869 −0.808535
\(95\) 613.897 0.662995
\(96\) 0 0
\(97\) 428.102 0.448116 0.224058 0.974576i \(-0.428070\pi\)
0.224058 + 0.974576i \(0.428070\pi\)
\(98\) 805.955 0.830753
\(99\) 0 0
\(100\) −13.3975 −0.0133975
\(101\) 912.087 0.898575 0.449287 0.893387i \(-0.351678\pi\)
0.449287 + 0.893387i \(0.351678\pi\)
\(102\) 0 0
\(103\) 278.497 0.266419 0.133210 0.991088i \(-0.457472\pi\)
0.133210 + 0.991088i \(0.457472\pi\)
\(104\) −584.687 −0.551282
\(105\) 0 0
\(106\) −269.913 −0.247323
\(107\) −1512.33 −1.36638 −0.683191 0.730240i \(-0.739408\pi\)
−0.683191 + 0.730240i \(0.739408\pi\)
\(108\) 0 0
\(109\) 1991.53 1.75004 0.875020 0.484087i \(-0.160848\pi\)
0.875020 + 0.484087i \(0.160848\pi\)
\(110\) 812.750 0.704479
\(111\) 0 0
\(112\) −411.713 −0.347350
\(113\) −507.902 −0.422827 −0.211413 0.977397i \(-0.567807\pi\)
−0.211413 + 0.977397i \(0.567807\pi\)
\(114\) 0 0
\(115\) −487.128 −0.394999
\(116\) −67.8282 −0.0542905
\(117\) 0 0
\(118\) −831.917 −0.649018
\(119\) −782.851 −0.603058
\(120\) 0 0
\(121\) 2208.94 1.65961
\(122\) −1821.06 −1.35140
\(123\) 0 0
\(124\) 57.9925 0.0419990
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1845.91 1.28975 0.644873 0.764290i \(-0.276910\pi\)
0.644873 + 0.764290i \(0.276910\pi\)
\(128\) −1285.86 −0.887928
\(129\) 0 0
\(130\) −342.487 −0.231062
\(131\) 2564.71 1.71053 0.855265 0.518190i \(-0.173394\pi\)
0.855265 + 0.518190i \(0.173394\pi\)
\(132\) 0 0
\(133\) −850.641 −0.554586
\(134\) 1737.40 1.12006
\(135\) 0 0
\(136\) −2635.10 −1.66145
\(137\) 506.780 0.316037 0.158019 0.987436i \(-0.449489\pi\)
0.158019 + 0.987436i \(0.449489\pi\)
\(138\) 0 0
\(139\) 2222.65 1.35628 0.678138 0.734935i \(-0.262787\pi\)
0.678138 + 0.734935i \(0.262787\pi\)
\(140\) 18.5641 0.0112068
\(141\) 0 0
\(142\) −2257.99 −1.33441
\(143\) −1491.71 −0.872327
\(144\) 0 0
\(145\) −632.846 −0.362448
\(146\) −2052.72 −1.16359
\(147\) 0 0
\(148\) −157.738 −0.0876081
\(149\) 510.092 0.280459 0.140229 0.990119i \(-0.455216\pi\)
0.140229 + 0.990119i \(0.455216\pi\)
\(150\) 0 0
\(151\) 675.261 0.363920 0.181960 0.983306i \(-0.441756\pi\)
0.181960 + 0.983306i \(0.441756\pi\)
\(152\) −2863.28 −1.52791
\(153\) 0 0
\(154\) −1126.18 −0.589286
\(155\) 541.077 0.280389
\(156\) 0 0
\(157\) −3627.98 −1.84423 −0.922115 0.386915i \(-0.873541\pi\)
−0.922115 + 0.386915i \(0.873541\pi\)
\(158\) −65.2051 −0.0328319
\(159\) 0 0
\(160\) 121.051 0.0598121
\(161\) 674.985 0.330411
\(162\) 0 0
\(163\) −119.559 −0.0574514 −0.0287257 0.999587i \(-0.509145\pi\)
−0.0287257 + 0.999587i \(0.509145\pi\)
\(164\) −110.321 −0.0525281
\(165\) 0 0
\(166\) 2232.72 1.04393
\(167\) −3068.99 −1.42207 −0.711035 0.703157i \(-0.751773\pi\)
−0.711035 + 0.703157i \(0.751773\pi\)
\(168\) 0 0
\(169\) −1568.41 −0.713885
\(170\) −1543.54 −0.696376
\(171\) 0 0
\(172\) 42.9790 0.0190530
\(173\) 2112.61 0.928432 0.464216 0.885722i \(-0.346336\pi\)
0.464216 + 0.885722i \(0.346336\pi\)
\(174\) 0 0
\(175\) 173.205 0.0748176
\(176\) −3535.67 −1.51427
\(177\) 0 0
\(178\) −1401.54 −0.590169
\(179\) −1624.79 −0.678449 −0.339225 0.940705i \(-0.610165\pi\)
−0.339225 + 0.940705i \(0.610165\pi\)
\(180\) 0 0
\(181\) −2545.41 −1.04530 −0.522649 0.852548i \(-0.675056\pi\)
−0.522649 + 0.852548i \(0.675056\pi\)
\(182\) 474.564 0.193280
\(183\) 0 0
\(184\) 2272.02 0.910300
\(185\) −1471.72 −0.584880
\(186\) 0 0
\(187\) −6722.90 −2.62902
\(188\) −144.539 −0.0560722
\(189\) 0 0
\(190\) −1677.20 −0.640404
\(191\) 4805.65 1.82055 0.910274 0.414006i \(-0.135871\pi\)
0.910274 + 0.414006i \(0.135871\pi\)
\(192\) 0 0
\(193\) 3577.84 1.33440 0.667198 0.744880i \(-0.267493\pi\)
0.667198 + 0.744880i \(0.267493\pi\)
\(194\) −1169.60 −0.432846
\(195\) 0 0
\(196\) 158.090 0.0576130
\(197\) −3788.88 −1.37029 −0.685144 0.728408i \(-0.740261\pi\)
−0.685144 + 0.728408i \(0.740261\pi\)
\(198\) 0 0
\(199\) 2048.31 0.729652 0.364826 0.931076i \(-0.381129\pi\)
0.364826 + 0.931076i \(0.381129\pi\)
\(200\) 583.013 0.206126
\(201\) 0 0
\(202\) −2491.87 −0.867956
\(203\) 876.897 0.303183
\(204\) 0 0
\(205\) −1029.31 −0.350683
\(206\) −760.869 −0.257341
\(207\) 0 0
\(208\) 1489.91 0.496666
\(209\) −7305.06 −2.41771
\(210\) 0 0
\(211\) 2381.79 0.777107 0.388553 0.921426i \(-0.372975\pi\)
0.388553 + 0.921426i \(0.372975\pi\)
\(212\) −52.9440 −0.0171519
\(213\) 0 0
\(214\) 4131.77 1.31982
\(215\) 401.000 0.127200
\(216\) 0 0
\(217\) −749.738 −0.234542
\(218\) −5440.97 −1.69041
\(219\) 0 0
\(220\) 159.423 0.0488559
\(221\) 2832.98 0.862295
\(222\) 0 0
\(223\) −557.154 −0.167309 −0.0836543 0.996495i \(-0.526659\pi\)
−0.0836543 + 0.996495i \(0.526659\pi\)
\(224\) −167.733 −0.0500320
\(225\) 0 0
\(226\) 1387.62 0.408419
\(227\) 5178.09 1.51402 0.757008 0.653405i \(-0.226660\pi\)
0.757008 + 0.653405i \(0.226660\pi\)
\(228\) 0 0
\(229\) −918.276 −0.264984 −0.132492 0.991184i \(-0.542298\pi\)
−0.132492 + 0.991184i \(0.542298\pi\)
\(230\) 1330.86 0.381540
\(231\) 0 0
\(232\) 2951.66 0.835284
\(233\) 5896.92 1.65803 0.829013 0.559230i \(-0.188903\pi\)
0.829013 + 0.559230i \(0.188903\pi\)
\(234\) 0 0
\(235\) −1348.56 −0.374343
\(236\) −163.182 −0.0450096
\(237\) 0 0
\(238\) 2138.79 0.582509
\(239\) −526.840 −0.142588 −0.0712938 0.997455i \(-0.522713\pi\)
−0.0712938 + 0.997455i \(0.522713\pi\)
\(240\) 0 0
\(241\) −5823.19 −1.55645 −0.778226 0.627984i \(-0.783880\pi\)
−0.778226 + 0.627984i \(0.783880\pi\)
\(242\) −6034.95 −1.60306
\(243\) 0 0
\(244\) −357.205 −0.0937201
\(245\) 1475.00 0.384630
\(246\) 0 0
\(247\) 3078.30 0.792986
\(248\) −2523.64 −0.646174
\(249\) 0 0
\(250\) 341.506 0.0863950
\(251\) 3244.98 0.816022 0.408011 0.912977i \(-0.366222\pi\)
0.408011 + 0.912977i \(0.366222\pi\)
\(252\) 0 0
\(253\) 5796.57 1.44042
\(254\) −5043.11 −1.24580
\(255\) 0 0
\(256\) −819.364 −0.200040
\(257\) 1261.40 0.306163 0.153082 0.988214i \(-0.451080\pi\)
0.153082 + 0.988214i \(0.451080\pi\)
\(258\) 0 0
\(259\) 2039.27 0.489244
\(260\) −67.1797 −0.0160243
\(261\) 0 0
\(262\) −7006.91 −1.65225
\(263\) −1924.28 −0.451165 −0.225582 0.974224i \(-0.572428\pi\)
−0.225582 + 0.974224i \(0.572428\pi\)
\(264\) 0 0
\(265\) −493.975 −0.114508
\(266\) 2323.99 0.535689
\(267\) 0 0
\(268\) 340.796 0.0776769
\(269\) 4762.53 1.07947 0.539733 0.841836i \(-0.318525\pi\)
0.539733 + 0.841836i \(0.318525\pi\)
\(270\) 0 0
\(271\) 1595.10 0.357547 0.178774 0.983890i \(-0.442787\pi\)
0.178774 + 0.983890i \(0.442787\pi\)
\(272\) 6714.79 1.49685
\(273\) 0 0
\(274\) −1384.55 −0.305269
\(275\) 1487.44 0.326166
\(276\) 0 0
\(277\) 1049.58 0.227666 0.113833 0.993500i \(-0.463687\pi\)
0.113833 + 0.993500i \(0.463687\pi\)
\(278\) −6072.38 −1.31006
\(279\) 0 0
\(280\) −807.846 −0.172422
\(281\) −7576.04 −1.60836 −0.804179 0.594387i \(-0.797395\pi\)
−0.804179 + 0.594387i \(0.797395\pi\)
\(282\) 0 0
\(283\) −5852.93 −1.22940 −0.614701 0.788760i \(-0.710723\pi\)
−0.614701 + 0.788760i \(0.710723\pi\)
\(284\) −442.910 −0.0925419
\(285\) 0 0
\(286\) 4075.42 0.842604
\(287\) 1426.25 0.293341
\(288\) 0 0
\(289\) 7854.84 1.59879
\(290\) 1728.97 0.350098
\(291\) 0 0
\(292\) −402.647 −0.0806956
\(293\) 3539.60 0.705752 0.352876 0.935670i \(-0.385204\pi\)
0.352876 + 0.935670i \(0.385204\pi\)
\(294\) 0 0
\(295\) −1522.51 −0.300489
\(296\) 6864.24 1.34789
\(297\) 0 0
\(298\) −1393.60 −0.270902
\(299\) −2442.64 −0.472446
\(300\) 0 0
\(301\) −555.641 −0.106401
\(302\) −1844.85 −0.351520
\(303\) 0 0
\(304\) 7296.25 1.37654
\(305\) −3332.77 −0.625685
\(306\) 0 0
\(307\) −293.303 −0.0545267 −0.0272634 0.999628i \(-0.508679\pi\)
−0.0272634 + 0.999628i \(0.508679\pi\)
\(308\) −220.903 −0.0408672
\(309\) 0 0
\(310\) −1478.25 −0.270835
\(311\) −4591.90 −0.837243 −0.418621 0.908161i \(-0.637487\pi\)
−0.418621 + 0.908161i \(0.637487\pi\)
\(312\) 0 0
\(313\) 7741.06 1.39792 0.698962 0.715159i \(-0.253646\pi\)
0.698962 + 0.715159i \(0.253646\pi\)
\(314\) 9911.82 1.78139
\(315\) 0 0
\(316\) −12.7901 −0.00227690
\(317\) 1917.32 0.339709 0.169854 0.985469i \(-0.445670\pi\)
0.169854 + 0.985469i \(0.445670\pi\)
\(318\) 0 0
\(319\) 7530.54 1.32172
\(320\) −2707.74 −0.473023
\(321\) 0 0
\(322\) −1844.09 −0.319153
\(323\) 13873.4 2.38990
\(324\) 0 0
\(325\) −626.795 −0.106979
\(326\) 326.641 0.0554938
\(327\) 0 0
\(328\) 4800.80 0.808170
\(329\) 1868.63 0.313133
\(330\) 0 0
\(331\) 8256.53 1.37106 0.685529 0.728046i \(-0.259571\pi\)
0.685529 + 0.728046i \(0.259571\pi\)
\(332\) 437.952 0.0723969
\(333\) 0 0
\(334\) 8384.63 1.37361
\(335\) 3179.67 0.518578
\(336\) 0 0
\(337\) −8518.68 −1.37698 −0.688490 0.725246i \(-0.741726\pi\)
−0.688490 + 0.725246i \(0.741726\pi\)
\(338\) 4284.96 0.689560
\(339\) 0 0
\(340\) −302.769 −0.0482939
\(341\) −6438.54 −1.02248
\(342\) 0 0
\(343\) −4420.19 −0.695825
\(344\) −1870.30 −0.293139
\(345\) 0 0
\(346\) −5771.76 −0.896796
\(347\) 9627.63 1.48945 0.744724 0.667373i \(-0.232581\pi\)
0.744724 + 0.667373i \(0.232581\pi\)
\(348\) 0 0
\(349\) −7591.79 −1.16441 −0.582205 0.813042i \(-0.697810\pi\)
−0.582205 + 0.813042i \(0.697810\pi\)
\(350\) −473.205 −0.0722682
\(351\) 0 0
\(352\) −1440.45 −0.218114
\(353\) 1387.23 0.209163 0.104582 0.994516i \(-0.466650\pi\)
0.104582 + 0.994516i \(0.466650\pi\)
\(354\) 0 0
\(355\) −4132.41 −0.617818
\(356\) −274.916 −0.0409284
\(357\) 0 0
\(358\) 4439.01 0.655332
\(359\) 6335.93 0.931469 0.465735 0.884924i \(-0.345790\pi\)
0.465735 + 0.884924i \(0.345790\pi\)
\(360\) 0 0
\(361\) 8215.79 1.19781
\(362\) 6954.19 1.00968
\(363\) 0 0
\(364\) 93.0869 0.0134041
\(365\) −3756.74 −0.538731
\(366\) 0 0
\(367\) −9190.27 −1.30716 −0.653581 0.756857i \(-0.726734\pi\)
−0.653581 + 0.756857i \(0.726734\pi\)
\(368\) −5789.58 −0.820116
\(369\) 0 0
\(370\) 4020.81 0.564951
\(371\) 684.471 0.0957843
\(372\) 0 0
\(373\) −12685.4 −1.76092 −0.880462 0.474116i \(-0.842768\pi\)
−0.880462 + 0.474116i \(0.842768\pi\)
\(374\) 18367.3 2.53944
\(375\) 0 0
\(376\) 6289.84 0.862696
\(377\) −3173.32 −0.433512
\(378\) 0 0
\(379\) −13119.0 −1.77804 −0.889022 0.457865i \(-0.848614\pi\)
−0.889022 + 0.457865i \(0.848614\pi\)
\(380\) −328.987 −0.0444122
\(381\) 0 0
\(382\) −13129.3 −1.75851
\(383\) −1734.25 −0.231374 −0.115687 0.993286i \(-0.536907\pi\)
−0.115687 + 0.993286i \(0.536907\pi\)
\(384\) 0 0
\(385\) −2061.05 −0.272833
\(386\) −9774.84 −1.28893
\(387\) 0 0
\(388\) −229.419 −0.0300181
\(389\) 9363.99 1.22050 0.610248 0.792210i \(-0.291070\pi\)
0.610248 + 0.792210i \(0.291070\pi\)
\(390\) 0 0
\(391\) −11008.6 −1.42386
\(392\) −6879.55 −0.886402
\(393\) 0 0
\(394\) 10351.4 1.32360
\(395\) −119.334 −0.0152008
\(396\) 0 0
\(397\) −7661.94 −0.968619 −0.484309 0.874897i \(-0.660929\pi\)
−0.484309 + 0.874897i \(0.660929\pi\)
\(398\) −5596.08 −0.704789
\(399\) 0 0
\(400\) −1485.64 −0.185705
\(401\) −145.649 −0.0181381 −0.00906906 0.999959i \(-0.502887\pi\)
−0.00906906 + 0.999959i \(0.502887\pi\)
\(402\) 0 0
\(403\) 2713.15 0.335364
\(404\) −488.786 −0.0601931
\(405\) 0 0
\(406\) −2395.73 −0.292852
\(407\) 17512.7 2.13285
\(408\) 0 0
\(409\) 8890.72 1.07486 0.537430 0.843308i \(-0.319395\pi\)
0.537430 + 0.843308i \(0.319395\pi\)
\(410\) 2812.12 0.338734
\(411\) 0 0
\(412\) −149.246 −0.0178467
\(413\) 2109.66 0.251354
\(414\) 0 0
\(415\) 4086.15 0.483329
\(416\) 606.994 0.0715393
\(417\) 0 0
\(418\) 19957.8 2.33533
\(419\) −3168.40 −0.369419 −0.184709 0.982793i \(-0.559134\pi\)
−0.184709 + 0.982793i \(0.559134\pi\)
\(420\) 0 0
\(421\) −8495.03 −0.983426 −0.491713 0.870757i \(-0.663629\pi\)
−0.491713 + 0.870757i \(0.663629\pi\)
\(422\) −6507.18 −0.750627
\(423\) 0 0
\(424\) 2303.95 0.263891
\(425\) −2824.87 −0.322415
\(426\) 0 0
\(427\) 4618.02 0.523376
\(428\) 810.457 0.0915302
\(429\) 0 0
\(430\) −1095.55 −0.122866
\(431\) 2072.30 0.231599 0.115799 0.993273i \(-0.463057\pi\)
0.115799 + 0.993273i \(0.463057\pi\)
\(432\) 0 0
\(433\) 2906.44 0.322574 0.161287 0.986908i \(-0.448435\pi\)
0.161287 + 0.986908i \(0.448435\pi\)
\(434\) 2048.32 0.226550
\(435\) 0 0
\(436\) −1067.26 −0.117230
\(437\) −11961.9 −1.30941
\(438\) 0 0
\(439\) 1931.08 0.209944 0.104972 0.994475i \(-0.466525\pi\)
0.104972 + 0.994475i \(0.466525\pi\)
\(440\) −6937.55 −0.751670
\(441\) 0 0
\(442\) −7739.86 −0.832913
\(443\) −244.624 −0.0262358 −0.0131179 0.999914i \(-0.504176\pi\)
−0.0131179 + 0.999914i \(0.504176\pi\)
\(444\) 0 0
\(445\) −2565.00 −0.273242
\(446\) 1522.17 0.161608
\(447\) 0 0
\(448\) 3751.96 0.395677
\(449\) 13276.7 1.39547 0.697734 0.716357i \(-0.254192\pi\)
0.697734 + 0.716357i \(0.254192\pi\)
\(450\) 0 0
\(451\) 12248.2 1.27882
\(452\) 272.184 0.0283240
\(453\) 0 0
\(454\) −14146.8 −1.46243
\(455\) 868.513 0.0894868
\(456\) 0 0
\(457\) 12514.2 1.28094 0.640470 0.767984i \(-0.278740\pi\)
0.640470 + 0.767984i \(0.278740\pi\)
\(458\) 2508.78 0.255955
\(459\) 0 0
\(460\) 261.051 0.0264599
\(461\) 4490.89 0.453712 0.226856 0.973928i \(-0.427155\pi\)
0.226856 + 0.973928i \(0.427155\pi\)
\(462\) 0 0
\(463\) −6092.89 −0.611578 −0.305789 0.952099i \(-0.598920\pi\)
−0.305789 + 0.952099i \(0.598920\pi\)
\(464\) −7521.46 −0.752532
\(465\) 0 0
\(466\) −16110.7 −1.60153
\(467\) −5193.68 −0.514635 −0.257318 0.966327i \(-0.582839\pi\)
−0.257318 + 0.966327i \(0.582839\pi\)
\(468\) 0 0
\(469\) −4405.88 −0.433783
\(470\) 3684.35 0.361588
\(471\) 0 0
\(472\) 7101.15 0.692494
\(473\) −4771.69 −0.463853
\(474\) 0 0
\(475\) −3069.49 −0.296500
\(476\) 419.529 0.0403972
\(477\) 0 0
\(478\) 1439.35 0.137729
\(479\) −18882.5 −1.80118 −0.900590 0.434669i \(-0.856865\pi\)
−0.900590 + 0.434669i \(0.856865\pi\)
\(480\) 0 0
\(481\) −7379.72 −0.699556
\(482\) 15909.3 1.50342
\(483\) 0 0
\(484\) −1183.77 −0.111173
\(485\) −2140.51 −0.200403
\(486\) 0 0
\(487\) −17619.1 −1.63942 −0.819708 0.572781i \(-0.805864\pi\)
−0.819708 + 0.572781i \(0.805864\pi\)
\(488\) 15544.4 1.44193
\(489\) 0 0
\(490\) −4029.77 −0.371524
\(491\) −916.267 −0.0842170 −0.0421085 0.999113i \(-0.513408\pi\)
−0.0421085 + 0.999113i \(0.513408\pi\)
\(492\) 0 0
\(493\) −14301.7 −1.30652
\(494\) −8410.08 −0.765966
\(495\) 0 0
\(496\) 6430.77 0.582157
\(497\) 5726.03 0.516796
\(498\) 0 0
\(499\) 12463.4 1.11812 0.559058 0.829129i \(-0.311163\pi\)
0.559058 + 0.829129i \(0.311163\pi\)
\(500\) 66.9873 0.00599153
\(501\) 0 0
\(502\) −8865.46 −0.788217
\(503\) 49.4842 0.00438646 0.00219323 0.999998i \(-0.499302\pi\)
0.00219323 + 0.999998i \(0.499302\pi\)
\(504\) 0 0
\(505\) −4560.43 −0.401855
\(506\) −15836.5 −1.39134
\(507\) 0 0
\(508\) −989.219 −0.0863966
\(509\) −6155.01 −0.535985 −0.267992 0.963421i \(-0.586360\pi\)
−0.267992 + 0.963421i \(0.586360\pi\)
\(510\) 0 0
\(511\) 5205.50 0.450641
\(512\) 12525.4 1.08115
\(513\) 0 0
\(514\) −3446.21 −0.295731
\(515\) −1392.49 −0.119146
\(516\) 0 0
\(517\) 16047.2 1.36510
\(518\) −5571.39 −0.472573
\(519\) 0 0
\(520\) 2923.44 0.246541
\(521\) −14718.1 −1.23764 −0.618821 0.785532i \(-0.712390\pi\)
−0.618821 + 0.785532i \(0.712390\pi\)
\(522\) 0 0
\(523\) −8318.46 −0.695489 −0.347744 0.937589i \(-0.613052\pi\)
−0.347744 + 0.937589i \(0.613052\pi\)
\(524\) −1374.42 −0.114584
\(525\) 0 0
\(526\) 5257.24 0.435792
\(527\) 12227.8 1.01072
\(528\) 0 0
\(529\) −2675.25 −0.219877
\(530\) 1349.56 0.110606
\(531\) 0 0
\(532\) 455.857 0.0371502
\(533\) −5161.32 −0.419440
\(534\) 0 0
\(535\) 7561.67 0.611064
\(536\) −14830.3 −1.19510
\(537\) 0 0
\(538\) −13011.5 −1.04268
\(539\) −17551.7 −1.40261
\(540\) 0 0
\(541\) 19372.0 1.53949 0.769746 0.638350i \(-0.220383\pi\)
0.769746 + 0.638350i \(0.220383\pi\)
\(542\) −4357.89 −0.345364
\(543\) 0 0
\(544\) 2735.63 0.215605
\(545\) −9957.67 −0.782641
\(546\) 0 0
\(547\) −989.727 −0.0773632 −0.0386816 0.999252i \(-0.512316\pi\)
−0.0386816 + 0.999252i \(0.512316\pi\)
\(548\) −271.582 −0.0211705
\(549\) 0 0
\(550\) −4063.75 −0.315052
\(551\) −15540.1 −1.20151
\(552\) 0 0
\(553\) 165.353 0.0127153
\(554\) −2867.52 −0.219908
\(555\) 0 0
\(556\) −1191.11 −0.0908533
\(557\) 4616.60 0.351188 0.175594 0.984463i \(-0.443815\pi\)
0.175594 + 0.984463i \(0.443815\pi\)
\(558\) 0 0
\(559\) 2010.76 0.152139
\(560\) 2058.56 0.155340
\(561\) 0 0
\(562\) 20698.1 1.55355
\(563\) 13509.6 1.01130 0.505649 0.862739i \(-0.331253\pi\)
0.505649 + 0.862739i \(0.331253\pi\)
\(564\) 0 0
\(565\) 2539.51 0.189094
\(566\) 15990.5 1.18751
\(567\) 0 0
\(568\) 19274.0 1.42380
\(569\) −11388.1 −0.839039 −0.419520 0.907746i \(-0.637801\pi\)
−0.419520 + 0.907746i \(0.637801\pi\)
\(570\) 0 0
\(571\) 9417.39 0.690202 0.345101 0.938566i \(-0.387845\pi\)
0.345101 + 0.938566i \(0.387845\pi\)
\(572\) 799.404 0.0584349
\(573\) 0 0
\(574\) −3896.59 −0.283346
\(575\) 2435.64 0.176649
\(576\) 0 0
\(577\) −3751.39 −0.270663 −0.135332 0.990800i \(-0.543210\pi\)
−0.135332 + 0.990800i \(0.543210\pi\)
\(578\) −21459.8 −1.54431
\(579\) 0 0
\(580\) 339.141 0.0242794
\(581\) −5661.94 −0.404297
\(582\) 0 0
\(583\) 5878.04 0.417571
\(584\) 17521.8 1.24154
\(585\) 0 0
\(586\) −9670.35 −0.681704
\(587\) −9293.55 −0.653468 −0.326734 0.945116i \(-0.605948\pi\)
−0.326734 + 0.945116i \(0.605948\pi\)
\(588\) 0 0
\(589\) 13286.6 0.929484
\(590\) 4159.58 0.290250
\(591\) 0 0
\(592\) −17491.5 −1.21436
\(593\) 1527.88 0.105805 0.0529027 0.998600i \(-0.483153\pi\)
0.0529027 + 0.998600i \(0.483153\pi\)
\(594\) 0 0
\(595\) 3914.26 0.269696
\(596\) −273.358 −0.0187872
\(597\) 0 0
\(598\) 6673.40 0.456347
\(599\) 1670.40 0.113941 0.0569704 0.998376i \(-0.481856\pi\)
0.0569704 + 0.998376i \(0.481856\pi\)
\(600\) 0 0
\(601\) 17393.9 1.18055 0.590276 0.807201i \(-0.299019\pi\)
0.590276 + 0.807201i \(0.299019\pi\)
\(602\) 1518.04 0.102775
\(603\) 0 0
\(604\) −361.871 −0.0243780
\(605\) −11044.7 −0.742201
\(606\) 0 0
\(607\) 4081.36 0.272911 0.136456 0.990646i \(-0.456429\pi\)
0.136456 + 0.990646i \(0.456429\pi\)
\(608\) 2972.52 0.198276
\(609\) 0 0
\(610\) 9105.29 0.604365
\(611\) −6762.18 −0.447739
\(612\) 0 0
\(613\) 2300.45 0.151573 0.0757866 0.997124i \(-0.475853\pi\)
0.0757866 + 0.997124i \(0.475853\pi\)
\(614\) 801.320 0.0526688
\(615\) 0 0
\(616\) 9612.95 0.628761
\(617\) −2905.84 −0.189602 −0.0948012 0.995496i \(-0.530222\pi\)
−0.0948012 + 0.995496i \(0.530222\pi\)
\(618\) 0 0
\(619\) 17855.5 1.15941 0.579703 0.814828i \(-0.303169\pi\)
0.579703 + 0.814828i \(0.303169\pi\)
\(620\) −289.962 −0.0187825
\(621\) 0 0
\(622\) 12545.3 0.808714
\(623\) 3554.17 0.228563
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −21149.0 −1.35029
\(627\) 0 0
\(628\) 1944.23 0.123540
\(629\) −33259.3 −2.10832
\(630\) 0 0
\(631\) −24250.9 −1.52998 −0.764988 0.644045i \(-0.777255\pi\)
−0.764988 + 0.644045i \(0.777255\pi\)
\(632\) 556.584 0.0350312
\(633\) 0 0
\(634\) −5238.23 −0.328133
\(635\) −9229.54 −0.576792
\(636\) 0 0
\(637\) 7396.18 0.460043
\(638\) −20573.8 −1.27669
\(639\) 0 0
\(640\) 6429.28 0.397093
\(641\) 1499.84 0.0924181 0.0462091 0.998932i \(-0.485286\pi\)
0.0462091 + 0.998932i \(0.485286\pi\)
\(642\) 0 0
\(643\) 23120.1 1.41799 0.708996 0.705213i \(-0.249149\pi\)
0.708996 + 0.705213i \(0.249149\pi\)
\(644\) −361.723 −0.0221334
\(645\) 0 0
\(646\) −37903.0 −2.30847
\(647\) 10703.6 0.650391 0.325195 0.945647i \(-0.394570\pi\)
0.325195 + 0.945647i \(0.394570\pi\)
\(648\) 0 0
\(649\) 18117.1 1.09578
\(650\) 1712.44 0.103334
\(651\) 0 0
\(652\) 64.0714 0.00384851
\(653\) −18109.1 −1.08525 −0.542623 0.839976i \(-0.682569\pi\)
−0.542623 + 0.839976i \(0.682569\pi\)
\(654\) 0 0
\(655\) −12823.5 −0.764973
\(656\) −12233.5 −0.728104
\(657\) 0 0
\(658\) −5105.18 −0.302463
\(659\) 25113.4 1.48449 0.742244 0.670130i \(-0.233762\pi\)
0.742244 + 0.670130i \(0.233762\pi\)
\(660\) 0 0
\(661\) −18593.6 −1.09411 −0.547055 0.837096i \(-0.684251\pi\)
−0.547055 + 0.837096i \(0.684251\pi\)
\(662\) −22557.3 −1.32434
\(663\) 0 0
\(664\) −19058.2 −1.11386
\(665\) 4253.21 0.248018
\(666\) 0 0
\(667\) 12331.1 0.715834
\(668\) 1644.67 0.0952606
\(669\) 0 0
\(670\) −8687.01 −0.500908
\(671\) 39658.2 2.28165
\(672\) 0 0
\(673\) 4290.38 0.245739 0.122869 0.992423i \(-0.460790\pi\)
0.122869 + 0.992423i \(0.460790\pi\)
\(674\) 23273.5 1.33006
\(675\) 0 0
\(676\) 840.506 0.0478212
\(677\) −26950.5 −1.52998 −0.764988 0.644045i \(-0.777255\pi\)
−0.764988 + 0.644045i \(0.777255\pi\)
\(678\) 0 0
\(679\) 2965.98 0.167635
\(680\) 13175.5 0.743025
\(681\) 0 0
\(682\) 17590.4 0.987642
\(683\) 18870.9 1.05721 0.528607 0.848867i \(-0.322715\pi\)
0.528607 + 0.848867i \(0.322715\pi\)
\(684\) 0 0
\(685\) −2533.90 −0.141336
\(686\) 12076.2 0.672115
\(687\) 0 0
\(688\) 4765.93 0.264098
\(689\) −2476.97 −0.136959
\(690\) 0 0
\(691\) 3696.05 0.203480 0.101740 0.994811i \(-0.467559\pi\)
0.101740 + 0.994811i \(0.467559\pi\)
\(692\) −1132.14 −0.0621931
\(693\) 0 0
\(694\) −26303.2 −1.43870
\(695\) −11113.2 −0.606545
\(696\) 0 0
\(697\) −23261.3 −1.26411
\(698\) 20741.1 1.12473
\(699\) 0 0
\(700\) −92.8203 −0.00501183
\(701\) −1123.53 −0.0605349 −0.0302674 0.999542i \(-0.509636\pi\)
−0.0302674 + 0.999542i \(0.509636\pi\)
\(702\) 0 0
\(703\) −36139.3 −1.93886
\(704\) 32220.7 1.72495
\(705\) 0 0
\(706\) −3789.97 −0.202036
\(707\) 6319.12 0.336146
\(708\) 0 0
\(709\) −26467.7 −1.40200 −0.700998 0.713163i \(-0.747262\pi\)
−0.700998 + 0.713163i \(0.747262\pi\)
\(710\) 11290.0 0.596767
\(711\) 0 0
\(712\) 11963.4 0.629702
\(713\) −10543.0 −0.553768
\(714\) 0 0
\(715\) 7458.54 0.390117
\(716\) 870.722 0.0454475
\(717\) 0 0
\(718\) −17310.1 −0.899730
\(719\) −7882.34 −0.408848 −0.204424 0.978882i \(-0.565532\pi\)
−0.204424 + 0.978882i \(0.565532\pi\)
\(720\) 0 0
\(721\) 1929.49 0.0996641
\(722\) −22446.0 −1.15700
\(723\) 0 0
\(724\) 1364.08 0.0700216
\(725\) 3164.23 0.162092
\(726\) 0 0
\(727\) 20008.9 1.02076 0.510379 0.859950i \(-0.329505\pi\)
0.510379 + 0.859950i \(0.329505\pi\)
\(728\) −4050.83 −0.206228
\(729\) 0 0
\(730\) 10263.6 0.520375
\(731\) 9062.18 0.458518
\(732\) 0 0
\(733\) −36293.5 −1.82883 −0.914414 0.404781i \(-0.867348\pi\)
−0.914414 + 0.404781i \(0.867348\pi\)
\(734\) 25108.3 1.26262
\(735\) 0 0
\(736\) −2358.70 −0.118129
\(737\) −37836.4 −1.89107
\(738\) 0 0
\(739\) −27475.5 −1.36766 −0.683832 0.729639i \(-0.739688\pi\)
−0.683832 + 0.729639i \(0.739688\pi\)
\(740\) 788.691 0.0391795
\(741\) 0 0
\(742\) −1870.01 −0.0925205
\(743\) −38148.9 −1.88364 −0.941822 0.336113i \(-0.890888\pi\)
−0.941822 + 0.336113i \(0.890888\pi\)
\(744\) 0 0
\(745\) −2550.46 −0.125425
\(746\) 34657.1 1.70092
\(747\) 0 0
\(748\) 3602.79 0.176111
\(749\) −10477.8 −0.511146
\(750\) 0 0
\(751\) 28223.1 1.37134 0.685669 0.727914i \(-0.259510\pi\)
0.685669 + 0.727914i \(0.259510\pi\)
\(752\) −16027.9 −0.777228
\(753\) 0 0
\(754\) 8669.67 0.418741
\(755\) −3376.31 −0.162750
\(756\) 0 0
\(757\) 34786.4 1.67019 0.835095 0.550106i \(-0.185413\pi\)
0.835095 + 0.550106i \(0.185413\pi\)
\(758\) 35841.8 1.71746
\(759\) 0 0
\(760\) 14316.4 0.683303
\(761\) −1165.96 −0.0555403 −0.0277702 0.999614i \(-0.508841\pi\)
−0.0277702 + 0.999614i \(0.508841\pi\)
\(762\) 0 0
\(763\) 13797.7 0.654668
\(764\) −2575.34 −0.121954
\(765\) 0 0
\(766\) 4738.07 0.223490
\(767\) −7634.43 −0.359404
\(768\) 0 0
\(769\) −26345.3 −1.23542 −0.617710 0.786406i \(-0.711939\pi\)
−0.617710 + 0.786406i \(0.711939\pi\)
\(770\) 5630.90 0.263537
\(771\) 0 0
\(772\) −1917.36 −0.0893876
\(773\) −25867.0 −1.20358 −0.601792 0.798653i \(-0.705546\pi\)
−0.601792 + 0.798653i \(0.705546\pi\)
\(774\) 0 0
\(775\) −2705.38 −0.125394
\(776\) 9983.56 0.461842
\(777\) 0 0
\(778\) −25582.9 −1.17891
\(779\) −25275.6 −1.16250
\(780\) 0 0
\(781\) 49173.5 2.25297
\(782\) 30076.0 1.37534
\(783\) 0 0
\(784\) 17530.6 0.798586
\(785\) 18139.9 0.824765
\(786\) 0 0
\(787\) −8070.97 −0.365564 −0.182782 0.983153i \(-0.558510\pi\)
−0.182782 + 0.983153i \(0.558510\pi\)
\(788\) 2030.46 0.0917919
\(789\) 0 0
\(790\) 326.025 0.0146829
\(791\) −3518.85 −0.158174
\(792\) 0 0
\(793\) −16711.7 −0.748361
\(794\) 20932.8 0.935614
\(795\) 0 0
\(796\) −1097.68 −0.0488774
\(797\) 12901.1 0.573375 0.286688 0.958024i \(-0.407446\pi\)
0.286688 + 0.958024i \(0.407446\pi\)
\(798\) 0 0
\(799\) −30476.2 −1.34940
\(800\) −605.256 −0.0267488
\(801\) 0 0
\(802\) 397.922 0.0175201
\(803\) 44703.3 1.96456
\(804\) 0 0
\(805\) −3374.92 −0.147764
\(806\) −7412.48 −0.323937
\(807\) 0 0
\(808\) 21270.3 0.926098
\(809\) −26916.2 −1.16974 −0.584872 0.811126i \(-0.698855\pi\)
−0.584872 + 0.811126i \(0.698855\pi\)
\(810\) 0 0
\(811\) −28082.5 −1.21592 −0.607959 0.793969i \(-0.708011\pi\)
−0.607959 + 0.793969i \(0.708011\pi\)
\(812\) −469.928 −0.0203094
\(813\) 0 0
\(814\) −47845.5 −2.06018
\(815\) 597.795 0.0256930
\(816\) 0 0
\(817\) 9846.90 0.421664
\(818\) −24289.9 −1.03823
\(819\) 0 0
\(820\) 551.604 0.0234913
\(821\) 46373.7 1.97132 0.985659 0.168750i \(-0.0539731\pi\)
0.985659 + 0.168750i \(0.0539731\pi\)
\(822\) 0 0
\(823\) −7804.55 −0.330558 −0.165279 0.986247i \(-0.552852\pi\)
−0.165279 + 0.986247i \(0.552852\pi\)
\(824\) 6494.70 0.274580
\(825\) 0 0
\(826\) −5763.69 −0.242790
\(827\) 23964.1 1.00764 0.503818 0.863810i \(-0.331928\pi\)
0.503818 + 0.863810i \(0.331928\pi\)
\(828\) 0 0
\(829\) −4583.86 −0.192044 −0.0960218 0.995379i \(-0.530612\pi\)
−0.0960218 + 0.995379i \(0.530612\pi\)
\(830\) −11163.6 −0.466859
\(831\) 0 0
\(832\) −13577.6 −0.565767
\(833\) 33333.5 1.38648
\(834\) 0 0
\(835\) 15344.9 0.635969
\(836\) 3914.77 0.161956
\(837\) 0 0
\(838\) 8656.23 0.356831
\(839\) 26928.3 1.10807 0.554034 0.832494i \(-0.313088\pi\)
0.554034 + 0.832494i \(0.313088\pi\)
\(840\) 0 0
\(841\) −8369.23 −0.343156
\(842\) 23208.9 0.949917
\(843\) 0 0
\(844\) −1276.40 −0.0520563
\(845\) 7842.03 0.319259
\(846\) 0 0
\(847\) 15304.0 0.620841
\(848\) −5870.95 −0.237747
\(849\) 0 0
\(850\) 7717.69 0.311429
\(851\) 28676.6 1.15514
\(852\) 0 0
\(853\) −7013.39 −0.281517 −0.140759 0.990044i \(-0.544954\pi\)
−0.140759 + 0.990044i \(0.544954\pi\)
\(854\) −12616.7 −0.505543
\(855\) 0 0
\(856\) −35268.4 −1.40823
\(857\) −20665.0 −0.823692 −0.411846 0.911253i \(-0.635116\pi\)
−0.411846 + 0.911253i \(0.635116\pi\)
\(858\) 0 0
\(859\) −20255.9 −0.804565 −0.402283 0.915516i \(-0.631783\pi\)
−0.402283 + 0.915516i \(0.631783\pi\)
\(860\) −214.895 −0.00852077
\(861\) 0 0
\(862\) −5661.62 −0.223707
\(863\) 16293.0 0.642666 0.321333 0.946966i \(-0.395869\pi\)
0.321333 + 0.946966i \(0.395869\pi\)
\(864\) 0 0
\(865\) −10563.0 −0.415207
\(866\) −7940.54 −0.311583
\(867\) 0 0
\(868\) 401.783 0.0157113
\(869\) 1420.01 0.0554321
\(870\) 0 0
\(871\) 15944.0 0.620254
\(872\) 46443.6 1.80364
\(873\) 0 0
\(874\) 32680.4 1.26480
\(875\) −866.025 −0.0334594
\(876\) 0 0
\(877\) 28264.1 1.08827 0.544134 0.838998i \(-0.316858\pi\)
0.544134 + 0.838998i \(0.316858\pi\)
\(878\) −5275.80 −0.202790
\(879\) 0 0
\(880\) 17678.4 0.677202
\(881\) −24644.6 −0.942447 −0.471224 0.882014i \(-0.656188\pi\)
−0.471224 + 0.882014i \(0.656188\pi\)
\(882\) 0 0
\(883\) 36026.4 1.37303 0.686515 0.727116i \(-0.259140\pi\)
0.686515 + 0.727116i \(0.259140\pi\)
\(884\) −1518.19 −0.0577628
\(885\) 0 0
\(886\) 668.326 0.0253418
\(887\) 14174.7 0.536572 0.268286 0.963339i \(-0.413543\pi\)
0.268286 + 0.963339i \(0.413543\pi\)
\(888\) 0 0
\(889\) 12788.8 0.482478
\(890\) 7007.71 0.263931
\(891\) 0 0
\(892\) 298.578 0.0112075
\(893\) −33115.2 −1.24094
\(894\) 0 0
\(895\) 8123.95 0.303412
\(896\) −8908.67 −0.332163
\(897\) 0 0
\(898\) −36272.5 −1.34792
\(899\) −13696.7 −0.508133
\(900\) 0 0
\(901\) −11163.3 −0.412768
\(902\) −33462.8 −1.23524
\(903\) 0 0
\(904\) −11844.5 −0.435778
\(905\) 12727.0 0.467471
\(906\) 0 0
\(907\) 36005.1 1.31812 0.659058 0.752092i \(-0.270955\pi\)
0.659058 + 0.752092i \(0.270955\pi\)
\(908\) −2774.93 −0.101420
\(909\) 0 0
\(910\) −2372.82 −0.0864376
\(911\) −1202.49 −0.0437325 −0.0218662 0.999761i \(-0.506961\pi\)
−0.0218662 + 0.999761i \(0.506961\pi\)
\(912\) 0 0
\(913\) −48623.1 −1.76253
\(914\) −34189.4 −1.23729
\(915\) 0 0
\(916\) 492.103 0.0177506
\(917\) 17768.8 0.639889
\(918\) 0 0
\(919\) 1151.63 0.0413370 0.0206685 0.999786i \(-0.493421\pi\)
0.0206685 + 0.999786i \(0.493421\pi\)
\(920\) −11360.1 −0.407098
\(921\) 0 0
\(922\) −12269.3 −0.438253
\(923\) −20721.4 −0.738952
\(924\) 0 0
\(925\) 7358.59 0.261566
\(926\) 16646.1 0.590739
\(927\) 0 0
\(928\) −3064.27 −0.108394
\(929\) −5397.06 −0.190605 −0.0953023 0.995448i \(-0.530382\pi\)
−0.0953023 + 0.995448i \(0.530382\pi\)
\(930\) 0 0
\(931\) 36219.9 1.27504
\(932\) −3160.15 −0.111067
\(933\) 0 0
\(934\) 14189.4 0.497100
\(935\) 33614.5 1.17573
\(936\) 0 0
\(937\) 45594.6 1.58966 0.794830 0.606832i \(-0.207560\pi\)
0.794830 + 0.606832i \(0.207560\pi\)
\(938\) 12037.1 0.419003
\(939\) 0 0
\(940\) 722.693 0.0250762
\(941\) 37245.5 1.29030 0.645149 0.764057i \(-0.276795\pi\)
0.645149 + 0.764057i \(0.276795\pi\)
\(942\) 0 0
\(943\) 20056.2 0.692598
\(944\) −18095.3 −0.623888
\(945\) 0 0
\(946\) 13036.5 0.448048
\(947\) −4777.54 −0.163938 −0.0819689 0.996635i \(-0.526121\pi\)
−0.0819689 + 0.996635i \(0.526121\pi\)
\(948\) 0 0
\(949\) −18837.7 −0.644359
\(950\) 8385.99 0.286397
\(951\) 0 0
\(952\) −18256.5 −0.621530
\(953\) 56883.2 1.93350 0.966752 0.255717i \(-0.0823114\pi\)
0.966752 + 0.255717i \(0.0823114\pi\)
\(954\) 0 0
\(955\) −24028.3 −0.814174
\(956\) 282.333 0.00955156
\(957\) 0 0
\(958\) 51588.1 1.73981
\(959\) 3511.07 0.118226
\(960\) 0 0
\(961\) −18080.4 −0.606909
\(962\) 20161.8 0.675719
\(963\) 0 0
\(964\) 3120.64 0.104262
\(965\) −17889.2 −0.596760
\(966\) 0 0
\(967\) 14050.0 0.467236 0.233618 0.972329i \(-0.424944\pi\)
0.233618 + 0.972329i \(0.424944\pi\)
\(968\) 51513.7 1.71045
\(969\) 0 0
\(970\) 5847.99 0.193575
\(971\) −779.492 −0.0257622 −0.0128811 0.999917i \(-0.504100\pi\)
−0.0128811 + 0.999917i \(0.504100\pi\)
\(972\) 0 0
\(973\) 15398.9 0.507366
\(974\) 48136.2 1.58355
\(975\) 0 0
\(976\) −39610.4 −1.29907
\(977\) −46754.9 −1.53103 −0.765517 0.643415i \(-0.777517\pi\)
−0.765517 + 0.643415i \(0.777517\pi\)
\(978\) 0 0
\(979\) 30522.2 0.996418
\(980\) −790.450 −0.0257653
\(981\) 0 0
\(982\) 2503.29 0.0813474
\(983\) 3596.10 0.116681 0.0583406 0.998297i \(-0.481419\pi\)
0.0583406 + 0.998297i \(0.481419\pi\)
\(984\) 0 0
\(985\) 18944.4 0.612811
\(986\) 39072.9 1.26200
\(987\) 0 0
\(988\) −1649.66 −0.0531200
\(989\) −7813.53 −0.251219
\(990\) 0 0
\(991\) −15143.8 −0.485427 −0.242713 0.970098i \(-0.578037\pi\)
−0.242713 + 0.970098i \(0.578037\pi\)
\(992\) 2619.92 0.0838534
\(993\) 0 0
\(994\) −15643.8 −0.499187
\(995\) −10241.5 −0.326310
\(996\) 0 0
\(997\) 22008.9 0.699126 0.349563 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(998\) −34050.7 −1.08002
\(999\) 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.c.1.1 2
3.2 odd 2 405.4.a.f.1.2 yes 2
5.4 even 2 2025.4.a.m.1.2 2
9.2 odd 6 405.4.e.o.271.1 4
9.4 even 3 405.4.e.p.136.2 4
9.5 odd 6 405.4.e.o.136.1 4
9.7 even 3 405.4.e.p.271.2 4
15.14 odd 2 2025.4.a.i.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.4.a.c.1.1 2 1.1 even 1 trivial
405.4.a.f.1.2 yes 2 3.2 odd 2
405.4.e.o.136.1 4 9.5 odd 6
405.4.e.o.271.1 4 9.2 odd 6
405.4.e.p.136.2 4 9.4 even 3
405.4.e.p.271.2 4 9.7 even 3
2025.4.a.i.1.1 2 15.14 odd 2
2025.4.a.m.1.2 2 5.4 even 2