Properties

Label 1815.4.a.k.1.2
Level $1815$
Weight $4$
Character 1815.1
Self dual yes
Analytic conductor $107.088$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(107.088466660\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.46410 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +10.3923 q^{6} +27.7128 q^{7} -13.8564 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.46410 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +10.3923 q^{6} +27.7128 q^{7} -13.8564 q^{8} +9.00000 q^{9} -17.3205 q^{10} +12.0000 q^{12} -58.8897 q^{13} +96.0000 q^{14} -15.0000 q^{15} -80.0000 q^{16} +19.0526 q^{17} +31.1769 q^{18} -45.0333 q^{19} -20.0000 q^{20} +83.1384 q^{21} -75.0000 q^{23} -41.5692 q^{24} +25.0000 q^{25} -204.000 q^{26} +27.0000 q^{27} +110.851 q^{28} -128.172 q^{29} -51.9615 q^{30} -263.000 q^{31} -166.277 q^{32} +66.0000 q^{34} -138.564 q^{35} +36.0000 q^{36} -308.000 q^{37} -156.000 q^{38} -176.669 q^{39} +69.2820 q^{40} +162.813 q^{41} +288.000 q^{42} -38.1051 q^{43} -45.0000 q^{45} -259.808 q^{46} +93.0000 q^{47} -240.000 q^{48} +425.000 q^{49} +86.6025 q^{50} +57.1577 q^{51} -235.559 q^{52} +525.000 q^{53} +93.5307 q^{54} -384.000 q^{56} -135.100 q^{57} -444.000 q^{58} +498.000 q^{59} -60.0000 q^{60} -441.673 q^{61} -911.059 q^{62} +249.415 q^{63} +64.0000 q^{64} +294.449 q^{65} +316.000 q^{67} +76.2102 q^{68} -225.000 q^{69} -480.000 q^{70} -288.000 q^{71} -124.708 q^{72} -928.379 q^{73} -1066.94 q^{74} +75.0000 q^{75} -180.133 q^{76} -612.000 q^{78} -1137.96 q^{79} +400.000 q^{80} +81.0000 q^{81} +564.000 q^{82} +571.577 q^{83} +332.554 q^{84} -95.2628 q^{85} -132.000 q^{86} -384.515 q^{87} -180.000 q^{89} -155.885 q^{90} -1632.00 q^{91} -300.000 q^{92} -789.000 q^{93} +322.161 q^{94} +225.167 q^{95} -498.831 q^{96} -904.000 q^{97} +1472.24 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} + 8q^{4} - 10q^{5} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} + 8q^{4} - 10q^{5} + 18q^{9} + 24q^{12} + 192q^{14} - 30q^{15} - 160q^{16} - 40q^{20} - 150q^{23} + 50q^{25} - 408q^{26} + 54q^{27} - 526q^{31} + 132q^{34} + 72q^{36} - 616q^{37} - 312q^{38} + 576q^{42} - 90q^{45} + 186q^{47} - 480q^{48} + 850q^{49} + 1050q^{53} - 768q^{56} - 888q^{58} + 996q^{59} - 120q^{60} + 128q^{64} + 632q^{67} - 450q^{69} - 960q^{70} - 576q^{71} + 150q^{75} - 1224q^{78} + 800q^{80} + 162q^{81} + 1128q^{82} - 264q^{86} - 360q^{89} - 3264q^{91} - 600q^{92} - 1578q^{93} - 1808q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.46410 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) 3.00000 0.577350
\(4\) 4.00000 0.500000
\(5\) −5.00000 −0.447214
\(6\) 10.3923 0.707107
\(7\) 27.7128 1.49635 0.748176 0.663501i \(-0.230930\pi\)
0.748176 + 0.663501i \(0.230930\pi\)
\(8\) −13.8564 −0.612372
\(9\) 9.00000 0.333333
\(10\) −17.3205 −0.547723
\(11\) 0 0
\(12\) 12.0000 0.288675
\(13\) −58.8897 −1.25639 −0.628195 0.778056i \(-0.716206\pi\)
−0.628195 + 0.778056i \(0.716206\pi\)
\(14\) 96.0000 1.83265
\(15\) −15.0000 −0.258199
\(16\) −80.0000 −1.25000
\(17\) 19.0526 0.271819 0.135910 0.990721i \(-0.456604\pi\)
0.135910 + 0.990721i \(0.456604\pi\)
\(18\) 31.1769 0.408248
\(19\) −45.0333 −0.543755 −0.271878 0.962332i \(-0.587645\pi\)
−0.271878 + 0.962332i \(0.587645\pi\)
\(20\) −20.0000 −0.223607
\(21\) 83.1384 0.863919
\(22\) 0 0
\(23\) −75.0000 −0.679938 −0.339969 0.940437i \(-0.610417\pi\)
−0.339969 + 0.940437i \(0.610417\pi\)
\(24\) −41.5692 −0.353553
\(25\) 25.0000 0.200000
\(26\) −204.000 −1.53876
\(27\) 27.0000 0.192450
\(28\) 110.851 0.748176
\(29\) −128.172 −0.820721 −0.410360 0.911923i \(-0.634597\pi\)
−0.410360 + 0.911923i \(0.634597\pi\)
\(30\) −51.9615 −0.316228
\(31\) −263.000 −1.52375 −0.761874 0.647725i \(-0.775721\pi\)
−0.761874 + 0.647725i \(0.775721\pi\)
\(32\) −166.277 −0.918559
\(33\) 0 0
\(34\) 66.0000 0.332909
\(35\) −138.564 −0.669189
\(36\) 36.0000 0.166667
\(37\) −308.000 −1.36851 −0.684255 0.729243i \(-0.739873\pi\)
−0.684255 + 0.729243i \(0.739873\pi\)
\(38\) −156.000 −0.665962
\(39\) −176.669 −0.725377
\(40\) 69.2820 0.273861
\(41\) 162.813 0.620173 0.310086 0.950708i \(-0.399642\pi\)
0.310086 + 0.950708i \(0.399642\pi\)
\(42\) 288.000 1.05808
\(43\) −38.1051 −0.135139 −0.0675695 0.997715i \(-0.521524\pi\)
−0.0675695 + 0.997715i \(0.521524\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) −259.808 −0.832751
\(47\) 93.0000 0.288626 0.144313 0.989532i \(-0.453903\pi\)
0.144313 + 0.989532i \(0.453903\pi\)
\(48\) −240.000 −0.721688
\(49\) 425.000 1.23907
\(50\) 86.6025 0.244949
\(51\) 57.1577 0.156935
\(52\) −235.559 −0.628195
\(53\) 525.000 1.36065 0.680324 0.732912i \(-0.261839\pi\)
0.680324 + 0.732912i \(0.261839\pi\)
\(54\) 93.5307 0.235702
\(55\) 0 0
\(56\) −384.000 −0.916324
\(57\) −135.100 −0.313937
\(58\) −444.000 −1.00517
\(59\) 498.000 1.09888 0.549441 0.835532i \(-0.314841\pi\)
0.549441 + 0.835532i \(0.314841\pi\)
\(60\) −60.0000 −0.129099
\(61\) −441.673 −0.927056 −0.463528 0.886082i \(-0.653417\pi\)
−0.463528 + 0.886082i \(0.653417\pi\)
\(62\) −911.059 −1.86620
\(63\) 249.415 0.498784
\(64\) 64.0000 0.125000
\(65\) 294.449 0.561875
\(66\) 0 0
\(67\) 316.000 0.576202 0.288101 0.957600i \(-0.406976\pi\)
0.288101 + 0.957600i \(0.406976\pi\)
\(68\) 76.2102 0.135910
\(69\) −225.000 −0.392563
\(70\) −480.000 −0.819585
\(71\) −288.000 −0.481399 −0.240699 0.970600i \(-0.577377\pi\)
−0.240699 + 0.970600i \(0.577377\pi\)
\(72\) −124.708 −0.204124
\(73\) −928.379 −1.48847 −0.744237 0.667916i \(-0.767187\pi\)
−0.744237 + 0.667916i \(0.767187\pi\)
\(74\) −1066.94 −1.67608
\(75\) 75.0000 0.115470
\(76\) −180.133 −0.271878
\(77\) 0 0
\(78\) −612.000 −0.888402
\(79\) −1137.96 −1.62064 −0.810318 0.585991i \(-0.800706\pi\)
−0.810318 + 0.585991i \(0.800706\pi\)
\(80\) 400.000 0.559017
\(81\) 81.0000 0.111111
\(82\) 564.000 0.759553
\(83\) 571.577 0.755888 0.377944 0.925828i \(-0.376631\pi\)
0.377944 + 0.925828i \(0.376631\pi\)
\(84\) 332.554 0.431959
\(85\) −95.2628 −0.121561
\(86\) −132.000 −0.165511
\(87\) −384.515 −0.473843
\(88\) 0 0
\(89\) −180.000 −0.214382 −0.107191 0.994238i \(-0.534186\pi\)
−0.107191 + 0.994238i \(0.534186\pi\)
\(90\) −155.885 −0.182574
\(91\) −1632.00 −1.88000
\(92\) −300.000 −0.339969
\(93\) −789.000 −0.879736
\(94\) 322.161 0.353494
\(95\) 225.167 0.243175
\(96\) −498.831 −0.530330
\(97\) −904.000 −0.946261 −0.473130 0.880992i \(-0.656876\pi\)
−0.473130 + 0.880992i \(0.656876\pi\)
\(98\) 1472.24 1.51754
\(99\) 0 0
\(100\) 100.000 0.100000
\(101\) −1046.16 −1.03066 −0.515330 0.856992i \(-0.672331\pi\)
−0.515330 + 0.856992i \(0.672331\pi\)
\(102\) 198.000 0.192205
\(103\) 1352.00 1.29336 0.646682 0.762760i \(-0.276156\pi\)
0.646682 + 0.762760i \(0.276156\pi\)
\(104\) 816.000 0.769379
\(105\) −415.692 −0.386356
\(106\) 1818.65 1.66645
\(107\) −1047.89 −0.946761 −0.473380 0.880858i \(-0.656966\pi\)
−0.473380 + 0.880858i \(0.656966\pi\)
\(108\) 108.000 0.0962250
\(109\) 1870.61 1.64378 0.821892 0.569644i \(-0.192919\pi\)
0.821892 + 0.569644i \(0.192919\pi\)
\(110\) 0 0
\(111\) −924.000 −0.790110
\(112\) −2217.03 −1.87044
\(113\) −1971.00 −1.64085 −0.820425 0.571754i \(-0.806263\pi\)
−0.820425 + 0.571754i \(0.806263\pi\)
\(114\) −468.000 −0.384493
\(115\) 375.000 0.304078
\(116\) −512.687 −0.410360
\(117\) −530.008 −0.418797
\(118\) 1725.12 1.34585
\(119\) 528.000 0.406737
\(120\) 207.846 0.158114
\(121\) 0 0
\(122\) −1530.00 −1.13541
\(123\) 488.438 0.358057
\(124\) −1052.00 −0.761874
\(125\) −125.000 −0.0894427
\(126\) 864.000 0.610883
\(127\) 945.700 0.660766 0.330383 0.943847i \(-0.392822\pi\)
0.330383 + 0.943847i \(0.392822\pi\)
\(128\) 1551.92 1.07165
\(129\) −114.315 −0.0780225
\(130\) 1020.00 0.688153
\(131\) −980.341 −0.653838 −0.326919 0.945052i \(-0.606010\pi\)
−0.326919 + 0.945052i \(0.606010\pi\)
\(132\) 0 0
\(133\) −1248.00 −0.813649
\(134\) 1094.66 0.705701
\(135\) −135.000 −0.0860663
\(136\) −264.000 −0.166455
\(137\) −2169.00 −1.35263 −0.676315 0.736613i \(-0.736424\pi\)
−0.676315 + 0.736613i \(0.736424\pi\)
\(138\) −779.423 −0.480789
\(139\) 1002.86 0.611951 0.305976 0.952039i \(-0.401017\pi\)
0.305976 + 0.952039i \(0.401017\pi\)
\(140\) −554.256 −0.334594
\(141\) 279.000 0.166639
\(142\) −997.661 −0.589591
\(143\) 0 0
\(144\) −720.000 −0.416667
\(145\) 640.859 0.367037
\(146\) −3216.00 −1.82300
\(147\) 1275.00 0.715376
\(148\) −1232.00 −0.684255
\(149\) 536.936 0.295218 0.147609 0.989046i \(-0.452842\pi\)
0.147609 + 0.989046i \(0.452842\pi\)
\(150\) 259.808 0.141421
\(151\) 226.899 0.122283 0.0611416 0.998129i \(-0.480526\pi\)
0.0611416 + 0.998129i \(0.480526\pi\)
\(152\) 624.000 0.332981
\(153\) 171.473 0.0906064
\(154\) 0 0
\(155\) 1315.00 0.681441
\(156\) −706.677 −0.362689
\(157\) −2576.00 −1.30947 −0.654736 0.755857i \(-0.727220\pi\)
−0.654736 + 0.755857i \(0.727220\pi\)
\(158\) −3942.00 −1.98487
\(159\) 1575.00 0.785570
\(160\) 831.384 0.410792
\(161\) −2078.46 −1.01743
\(162\) 280.592 0.136083
\(163\) 1694.00 0.814014 0.407007 0.913425i \(-0.366572\pi\)
0.407007 + 0.913425i \(0.366572\pi\)
\(164\) 651.251 0.310086
\(165\) 0 0
\(166\) 1980.00 0.925770
\(167\) −874.686 −0.405301 −0.202650 0.979251i \(-0.564955\pi\)
−0.202650 + 0.979251i \(0.564955\pi\)
\(168\) −1152.00 −0.529040
\(169\) 1271.00 0.578516
\(170\) −330.000 −0.148881
\(171\) −405.300 −0.181252
\(172\) −152.420 −0.0675695
\(173\) −2868.28 −1.26053 −0.630263 0.776382i \(-0.717053\pi\)
−0.630263 + 0.776382i \(0.717053\pi\)
\(174\) −1332.00 −0.580337
\(175\) 692.820 0.299270
\(176\) 0 0
\(177\) 1494.00 0.634440
\(178\) −623.538 −0.262563
\(179\) −4416.00 −1.84395 −0.921976 0.387247i \(-0.873426\pi\)
−0.921976 + 0.387247i \(0.873426\pi\)
\(180\) −180.000 −0.0745356
\(181\) 4430.00 1.81922 0.909611 0.415460i \(-0.136379\pi\)
0.909611 + 0.415460i \(0.136379\pi\)
\(182\) −5653.41 −2.30252
\(183\) −1325.02 −0.535236
\(184\) 1039.23 0.416375
\(185\) 1540.00 0.612016
\(186\) −2733.18 −1.07745
\(187\) 0 0
\(188\) 372.000 0.144313
\(189\) 748.246 0.287973
\(190\) 780.000 0.297827
\(191\) −4632.00 −1.75476 −0.877382 0.479793i \(-0.840712\pi\)
−0.877382 + 0.479793i \(0.840712\pi\)
\(192\) 192.000 0.0721688
\(193\) 3076.12 1.14728 0.573638 0.819109i \(-0.305532\pi\)
0.573638 + 0.819109i \(0.305532\pi\)
\(194\) −3131.55 −1.15893
\(195\) 883.346 0.324399
\(196\) 1700.00 0.619534
\(197\) 1406.43 0.508648 0.254324 0.967119i \(-0.418147\pi\)
0.254324 + 0.967119i \(0.418147\pi\)
\(198\) 0 0
\(199\) 4753.00 1.69312 0.846561 0.532292i \(-0.178669\pi\)
0.846561 + 0.532292i \(0.178669\pi\)
\(200\) −346.410 −0.122474
\(201\) 948.000 0.332670
\(202\) −3624.00 −1.26230
\(203\) −3552.00 −1.22809
\(204\) 228.631 0.0784674
\(205\) −814.064 −0.277350
\(206\) 4683.47 1.58404
\(207\) −675.000 −0.226646
\(208\) 4711.18 1.57049
\(209\) 0 0
\(210\) −1440.00 −0.473188
\(211\) −3670.22 −1.19748 −0.598739 0.800944i \(-0.704332\pi\)
−0.598739 + 0.800944i \(0.704332\pi\)
\(212\) 2100.00 0.680324
\(213\) −864.000 −0.277936
\(214\) −3630.00 −1.15954
\(215\) 190.526 0.0604360
\(216\) −374.123 −0.117851
\(217\) −7288.47 −2.28006
\(218\) 6480.00 2.01322
\(219\) −2785.14 −0.859371
\(220\) 0 0
\(221\) −1122.00 −0.341511
\(222\) −3200.83 −0.967683
\(223\) −1606.00 −0.482268 −0.241134 0.970492i \(-0.577519\pi\)
−0.241134 + 0.970492i \(0.577519\pi\)
\(224\) −4608.00 −1.37449
\(225\) 225.000 0.0666667
\(226\) −6827.74 −2.00962
\(227\) 4581.27 1.33951 0.669757 0.742580i \(-0.266398\pi\)
0.669757 + 0.742580i \(0.266398\pi\)
\(228\) −540.400 −0.156969
\(229\) 4021.00 1.16033 0.580164 0.814500i \(-0.302988\pi\)
0.580164 + 0.814500i \(0.302988\pi\)
\(230\) 1299.04 0.372418
\(231\) 0 0
\(232\) 1776.00 0.502587
\(233\) 1498.22 0.421253 0.210626 0.977567i \(-0.432450\pi\)
0.210626 + 0.977567i \(0.432450\pi\)
\(234\) −1836.00 −0.512919
\(235\) −465.000 −0.129078
\(236\) 1992.00 0.549441
\(237\) −3413.87 −0.935674
\(238\) 1829.05 0.498149
\(239\) −543.864 −0.147195 −0.0735976 0.997288i \(-0.523448\pi\)
−0.0735976 + 0.997288i \(0.523448\pi\)
\(240\) 1200.00 0.322749
\(241\) −2270.72 −0.606929 −0.303464 0.952843i \(-0.598143\pi\)
−0.303464 + 0.952843i \(0.598143\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) −1766.69 −0.463528
\(245\) −2125.00 −0.554128
\(246\) 1692.00 0.438528
\(247\) 2652.00 0.683169
\(248\) 3644.23 0.933101
\(249\) 1714.73 0.436412
\(250\) −433.013 −0.109545
\(251\) −4746.00 −1.19349 −0.596743 0.802433i \(-0.703539\pi\)
−0.596743 + 0.802433i \(0.703539\pi\)
\(252\) 997.661 0.249392
\(253\) 0 0
\(254\) 3276.00 0.809270
\(255\) −285.788 −0.0701834
\(256\) 4864.00 1.18750
\(257\) −2691.00 −0.653152 −0.326576 0.945171i \(-0.605895\pi\)
−0.326576 + 0.945171i \(0.605895\pi\)
\(258\) −396.000 −0.0955577
\(259\) −8535.55 −2.04777
\(260\) 1177.79 0.280937
\(261\) −1153.55 −0.273574
\(262\) −3396.00 −0.800785
\(263\) 3909.24 0.916555 0.458278 0.888809i \(-0.348467\pi\)
0.458278 + 0.888809i \(0.348467\pi\)
\(264\) 0 0
\(265\) −2625.00 −0.608500
\(266\) −4323.20 −0.996513
\(267\) −540.000 −0.123773
\(268\) 1264.00 0.288101
\(269\) −7848.00 −1.77881 −0.889407 0.457116i \(-0.848882\pi\)
−0.889407 + 0.457116i \(0.848882\pi\)
\(270\) −467.654 −0.105409
\(271\) 4823.76 1.08126 0.540632 0.841259i \(-0.318185\pi\)
0.540632 + 0.841259i \(0.318185\pi\)
\(272\) −1524.20 −0.339774
\(273\) −4896.00 −1.08542
\(274\) −7513.64 −1.65663
\(275\) 0 0
\(276\) −900.000 −0.196281
\(277\) 4842.81 1.05046 0.525228 0.850961i \(-0.323980\pi\)
0.525228 + 0.850961i \(0.323980\pi\)
\(278\) 3474.00 0.749484
\(279\) −2367.00 −0.507916
\(280\) 1920.00 0.409793
\(281\) 7676.45 1.62967 0.814837 0.579690i \(-0.196826\pi\)
0.814837 + 0.579690i \(0.196826\pi\)
\(282\) 966.484 0.204090
\(283\) −7430.50 −1.56077 −0.780384 0.625301i \(-0.784976\pi\)
−0.780384 + 0.625301i \(0.784976\pi\)
\(284\) −1152.00 −0.240699
\(285\) 675.500 0.140397
\(286\) 0 0
\(287\) 4512.00 0.927996
\(288\) −1496.49 −0.306186
\(289\) −4550.00 −0.926114
\(290\) 2220.00 0.449527
\(291\) −2712.00 −0.546324
\(292\) −3713.52 −0.744237
\(293\) 4460.03 0.889276 0.444638 0.895710i \(-0.353332\pi\)
0.444638 + 0.895710i \(0.353332\pi\)
\(294\) 4416.73 0.876153
\(295\) −2490.00 −0.491435
\(296\) 4267.77 0.838038
\(297\) 0 0
\(298\) 1860.00 0.361567
\(299\) 4416.73 0.854268
\(300\) 300.000 0.0577350
\(301\) −1056.00 −0.202215
\(302\) 786.000 0.149766
\(303\) −3138.48 −0.595052
\(304\) 3602.67 0.679694
\(305\) 2208.36 0.414592
\(306\) 594.000 0.110970
\(307\) −827.920 −0.153915 −0.0769575 0.997034i \(-0.524521\pi\)
−0.0769575 + 0.997034i \(0.524521\pi\)
\(308\) 0 0
\(309\) 4056.00 0.746724
\(310\) 4555.29 0.834591
\(311\) 6300.00 1.14868 0.574341 0.818616i \(-0.305258\pi\)
0.574341 + 0.818616i \(0.305258\pi\)
\(312\) 2448.00 0.444201
\(313\) −2318.00 −0.418598 −0.209299 0.977852i \(-0.567118\pi\)
−0.209299 + 0.977852i \(0.567118\pi\)
\(314\) −8923.53 −1.60377
\(315\) −1247.08 −0.223063
\(316\) −4551.83 −0.810318
\(317\) 1767.00 0.313074 0.156537 0.987672i \(-0.449967\pi\)
0.156537 + 0.987672i \(0.449967\pi\)
\(318\) 5455.96 0.962123
\(319\) 0 0
\(320\) −320.000 −0.0559017
\(321\) −3143.67 −0.546613
\(322\) −7200.00 −1.24609
\(323\) −858.000 −0.147803
\(324\) 324.000 0.0555556
\(325\) −1472.24 −0.251278
\(326\) 5868.19 0.996960
\(327\) 5611.84 0.949039
\(328\) −2256.00 −0.379777
\(329\) 2577.29 0.431887
\(330\) 0 0
\(331\) −1309.00 −0.217369 −0.108685 0.994076i \(-0.534664\pi\)
−0.108685 + 0.994076i \(0.534664\pi\)
\(332\) 2286.31 0.377944
\(333\) −2772.00 −0.456170
\(334\) −3030.00 −0.496390
\(335\) −1580.00 −0.257685
\(336\) −6651.08 −1.07990
\(337\) 7278.08 1.17645 0.588223 0.808699i \(-0.299828\pi\)
0.588223 + 0.808699i \(0.299828\pi\)
\(338\) 4402.87 0.708535
\(339\) −5913.00 −0.947345
\(340\) −381.051 −0.0607806
\(341\) 0 0
\(342\) −1404.00 −0.221987
\(343\) 2272.45 0.357728
\(344\) 528.000 0.0827554
\(345\) 1125.00 0.175559
\(346\) −9936.00 −1.54382
\(347\) 1536.33 0.237679 0.118839 0.992914i \(-0.462083\pi\)
0.118839 + 0.992914i \(0.462083\pi\)
\(348\) −1538.06 −0.236922
\(349\) 10806.3 1.65744 0.828719 0.559664i \(-0.189070\pi\)
0.828719 + 0.559664i \(0.189070\pi\)
\(350\) 2400.00 0.366530
\(351\) −1590.02 −0.241792
\(352\) 0 0
\(353\) −8319.00 −1.25432 −0.627161 0.778890i \(-0.715783\pi\)
−0.627161 + 0.778890i \(0.715783\pi\)
\(354\) 5175.37 0.777027
\(355\) 1440.00 0.215288
\(356\) −720.000 −0.107191
\(357\) 1584.00 0.234830
\(358\) −15297.5 −2.25837
\(359\) −10142.9 −1.49115 −0.745573 0.666424i \(-0.767824\pi\)
−0.745573 + 0.666424i \(0.767824\pi\)
\(360\) 623.538 0.0912871
\(361\) −4831.00 −0.704330
\(362\) 15346.0 2.22808
\(363\) 0 0
\(364\) −6528.00 −0.940000
\(365\) 4641.90 0.665666
\(366\) −4590.00 −0.655528
\(367\) 1136.00 0.161577 0.0807884 0.996731i \(-0.474256\pi\)
0.0807884 + 0.996731i \(0.474256\pi\)
\(368\) 6000.00 0.849923
\(369\) 1465.31 0.206724
\(370\) 5334.72 0.749564
\(371\) 14549.2 2.03601
\(372\) −3156.00 −0.439868
\(373\) 9976.61 1.38490 0.692452 0.721464i \(-0.256530\pi\)
0.692452 + 0.721464i \(0.256530\pi\)
\(374\) 0 0
\(375\) −375.000 −0.0516398
\(376\) −1288.65 −0.176747
\(377\) 7548.00 1.03115
\(378\) 2592.00 0.352693
\(379\) 12929.0 1.75229 0.876145 0.482047i \(-0.160107\pi\)
0.876145 + 0.482047i \(0.160107\pi\)
\(380\) 900.666 0.121587
\(381\) 2837.10 0.381493
\(382\) −16045.7 −2.14914
\(383\) 360.000 0.0480291 0.0240145 0.999712i \(-0.492355\pi\)
0.0240145 + 0.999712i \(0.492355\pi\)
\(384\) 4655.75 0.618718
\(385\) 0 0
\(386\) 10656.0 1.40512
\(387\) −342.946 −0.0450463
\(388\) −3616.00 −0.473130
\(389\) −3918.00 −0.510670 −0.255335 0.966853i \(-0.582186\pi\)
−0.255335 + 0.966853i \(0.582186\pi\)
\(390\) 3060.00 0.397305
\(391\) −1428.94 −0.184820
\(392\) −5888.97 −0.758771
\(393\) −2941.02 −0.377494
\(394\) 4872.00 0.622964
\(395\) 5689.79 0.724770
\(396\) 0 0
\(397\) 5546.00 0.701123 0.350561 0.936540i \(-0.385991\pi\)
0.350561 + 0.936540i \(0.385991\pi\)
\(398\) 16464.9 2.07364
\(399\) −3744.00 −0.469761
\(400\) −2000.00 −0.250000
\(401\) −672.000 −0.0836860 −0.0418430 0.999124i \(-0.513323\pi\)
−0.0418430 + 0.999124i \(0.513323\pi\)
\(402\) 3283.97 0.407436
\(403\) 15488.0 1.91442
\(404\) −4184.63 −0.515330
\(405\) −405.000 −0.0496904
\(406\) −12304.5 −1.50409
\(407\) 0 0
\(408\) −792.000 −0.0961026
\(409\) −226.899 −0.0274313 −0.0137157 0.999906i \(-0.504366\pi\)
−0.0137157 + 0.999906i \(0.504366\pi\)
\(410\) −2820.00 −0.339683
\(411\) −6507.00 −0.780941
\(412\) 5408.00 0.646682
\(413\) 13801.0 1.64431
\(414\) −2338.27 −0.277584
\(415\) −2857.88 −0.338043
\(416\) 9792.00 1.15407
\(417\) 3008.57 0.353310
\(418\) 0 0
\(419\) 4470.00 0.521178 0.260589 0.965450i \(-0.416083\pi\)
0.260589 + 0.965450i \(0.416083\pi\)
\(420\) −1662.77 −0.193178
\(421\) 5539.00 0.641222 0.320611 0.947211i \(-0.396112\pi\)
0.320611 + 0.947211i \(0.396112\pi\)
\(422\) −12714.0 −1.46661
\(423\) 837.000 0.0962088
\(424\) −7274.61 −0.833223
\(425\) 476.314 0.0543638
\(426\) −2992.98 −0.340400
\(427\) −12240.0 −1.38720
\(428\) −4191.56 −0.473380
\(429\) 0 0
\(430\) 660.000 0.0740187
\(431\) −9980.08 −1.11537 −0.557684 0.830054i \(-0.688310\pi\)
−0.557684 + 0.830054i \(0.688310\pi\)
\(432\) −2160.00 −0.240563
\(433\) 13526.0 1.50120 0.750598 0.660759i \(-0.229765\pi\)
0.750598 + 0.660759i \(0.229765\pi\)
\(434\) −25248.0 −2.79249
\(435\) 1922.58 0.211909
\(436\) 7482.46 0.821892
\(437\) 3377.50 0.369720
\(438\) −9648.00 −1.05251
\(439\) −11869.7 −1.29046 −0.645230 0.763988i \(-0.723238\pi\)
−0.645230 + 0.763988i \(0.723238\pi\)
\(440\) 0 0
\(441\) 3825.00 0.413022
\(442\) −3886.72 −0.418264
\(443\) −12108.0 −1.29857 −0.649287 0.760543i \(-0.724933\pi\)
−0.649287 + 0.760543i \(0.724933\pi\)
\(444\) −3696.00 −0.395055
\(445\) 900.000 0.0958744
\(446\) −5563.35 −0.590655
\(447\) 1610.81 0.170444
\(448\) 1773.62 0.187044
\(449\) −5586.00 −0.587126 −0.293563 0.955940i \(-0.594841\pi\)
−0.293563 + 0.955940i \(0.594841\pi\)
\(450\) 779.423 0.0816497
\(451\) 0 0
\(452\) −7884.00 −0.820425
\(453\) 680.696 0.0706002
\(454\) 15870.0 1.64056
\(455\) 8160.00 0.840762
\(456\) 1872.00 0.192247
\(457\) 17767.4 1.81865 0.909325 0.416087i \(-0.136599\pi\)
0.909325 + 0.416087i \(0.136599\pi\)
\(458\) 13929.2 1.42111
\(459\) 514.419 0.0523116
\(460\) 1500.00 0.152039
\(461\) 17594.2 1.77753 0.888766 0.458361i \(-0.151563\pi\)
0.888766 + 0.458361i \(0.151563\pi\)
\(462\) 0 0
\(463\) −622.000 −0.0624337 −0.0312168 0.999513i \(-0.509938\pi\)
−0.0312168 + 0.999513i \(0.509938\pi\)
\(464\) 10253.7 1.02590
\(465\) 3945.00 0.393430
\(466\) 5190.00 0.515927
\(467\) −14019.0 −1.38913 −0.694563 0.719432i \(-0.744402\pi\)
−0.694563 + 0.719432i \(0.744402\pi\)
\(468\) −2120.03 −0.209398
\(469\) 8757.25 0.862201
\(470\) −1610.81 −0.158087
\(471\) −7728.00 −0.756024
\(472\) −6900.49 −0.672925
\(473\) 0 0
\(474\) −11826.0 −1.14596
\(475\) −1125.83 −0.108751
\(476\) 2112.00 0.203368
\(477\) 4725.00 0.453549
\(478\) −1884.00 −0.180276
\(479\) 13735.2 1.31018 0.655089 0.755551i \(-0.272631\pi\)
0.655089 + 0.755551i \(0.272631\pi\)
\(480\) 2494.15 0.237171
\(481\) 18138.0 1.71938
\(482\) −7866.00 −0.743333
\(483\) −6235.38 −0.587411
\(484\) 0 0
\(485\) 4520.00 0.423181
\(486\) 841.777 0.0785674
\(487\) 6982.00 0.649660 0.324830 0.945772i \(-0.394693\pi\)
0.324830 + 0.945772i \(0.394693\pi\)
\(488\) 6120.00 0.567704
\(489\) 5082.00 0.469971
\(490\) −7361.22 −0.678665
\(491\) −6609.51 −0.607501 −0.303750 0.952752i \(-0.598239\pi\)
−0.303750 + 0.952752i \(0.598239\pi\)
\(492\) 1953.75 0.179028
\(493\) −2442.00 −0.223088
\(494\) 9186.80 0.836708
\(495\) 0 0
\(496\) 21040.0 1.90469
\(497\) −7981.29 −0.720342
\(498\) 5940.00 0.534494
\(499\) −6104.00 −0.547600 −0.273800 0.961787i \(-0.588281\pi\)
−0.273800 + 0.961787i \(0.588281\pi\)
\(500\) −500.000 −0.0447214
\(501\) −2624.06 −0.234000
\(502\) −16440.6 −1.46172
\(503\) 12496.7 1.10776 0.553879 0.832597i \(-0.313147\pi\)
0.553879 + 0.832597i \(0.313147\pi\)
\(504\) −3456.00 −0.305441
\(505\) 5230.79 0.460925
\(506\) 0 0
\(507\) 3813.00 0.334006
\(508\) 3782.80 0.330383
\(509\) 5904.00 0.514126 0.257063 0.966395i \(-0.417245\pi\)
0.257063 + 0.966395i \(0.417245\pi\)
\(510\) −990.000 −0.0859567
\(511\) −25728.0 −2.22728
\(512\) 4434.05 0.382733
\(513\) −1215.90 −0.104646
\(514\) −9321.90 −0.799944
\(515\) −6760.00 −0.578410
\(516\) −457.261 −0.0390113
\(517\) 0 0
\(518\) −29568.0 −2.50800
\(519\) −8604.83 −0.727765
\(520\) −4080.00 −0.344077
\(521\) −22878.0 −1.92381 −0.961903 0.273389i \(-0.911855\pi\)
−0.961903 + 0.273389i \(0.911855\pi\)
\(522\) −3996.00 −0.335058
\(523\) 6858.92 0.573460 0.286730 0.958011i \(-0.407432\pi\)
0.286730 + 0.958011i \(0.407432\pi\)
\(524\) −3921.36 −0.326919
\(525\) 2078.46 0.172784
\(526\) 13542.0 1.12255
\(527\) −5010.82 −0.414184
\(528\) 0 0
\(529\) −6542.00 −0.537684
\(530\) −9093.27 −0.745257
\(531\) 4482.00 0.366294
\(532\) −4992.00 −0.406825
\(533\) −9588.00 −0.779179
\(534\) −1870.61 −0.151591
\(535\) 5239.45 0.423404
\(536\) −4378.62 −0.352850
\(537\) −13248.0 −1.06461
\(538\) −27186.3 −2.17859
\(539\) 0 0
\(540\) −540.000 −0.0430331
\(541\) −4240.06 −0.336958 −0.168479 0.985705i \(-0.553886\pi\)
−0.168479 + 0.985705i \(0.553886\pi\)
\(542\) 16710.0 1.32427
\(543\) 13290.0 1.05033
\(544\) −3168.00 −0.249682
\(545\) −9353.07 −0.735122
\(546\) −16960.2 −1.32936
\(547\) 11261.8 0.880292 0.440146 0.897926i \(-0.354927\pi\)
0.440146 + 0.897926i \(0.354927\pi\)
\(548\) −8676.00 −0.676315
\(549\) −3975.06 −0.309019
\(550\) 0 0
\(551\) 5772.00 0.446271
\(552\) 3117.69 0.240394
\(553\) −31536.0 −2.42504
\(554\) 16776.0 1.28654
\(555\) 4620.00 0.353348
\(556\) 4011.43 0.305976
\(557\) 14138.7 1.07554 0.537771 0.843091i \(-0.319266\pi\)
0.537771 + 0.843091i \(0.319266\pi\)
\(558\) −8199.53 −0.622068
\(559\) 2244.00 0.169787
\(560\) 11085.1 0.836486
\(561\) 0 0
\(562\) 26592.0 1.99594
\(563\) −6460.55 −0.483623 −0.241811 0.970323i \(-0.577742\pi\)
−0.241811 + 0.970323i \(0.577742\pi\)
\(564\) 1116.00 0.0833193
\(565\) 9855.00 0.733811
\(566\) −25740.0 −1.91154
\(567\) 2244.74 0.166261
\(568\) 3990.65 0.294795
\(569\) −12037.8 −0.886905 −0.443452 0.896298i \(-0.646246\pi\)
−0.443452 + 0.896298i \(0.646246\pi\)
\(570\) 2340.00 0.171951
\(571\) −17013.9 −1.24695 −0.623477 0.781841i \(-0.714281\pi\)
−0.623477 + 0.781841i \(0.714281\pi\)
\(572\) 0 0
\(573\) −13896.0 −1.01311
\(574\) 15630.0 1.13656
\(575\) −1875.00 −0.135988
\(576\) 576.000 0.0416667
\(577\) −9788.00 −0.706204 −0.353102 0.935585i \(-0.614873\pi\)
−0.353102 + 0.935585i \(0.614873\pi\)
\(578\) −15761.7 −1.13425
\(579\) 9228.37 0.662380
\(580\) 2563.44 0.183519
\(581\) 15840.0 1.13107
\(582\) −9394.64 −0.669107
\(583\) 0 0
\(584\) 12864.0 0.911500
\(585\) 2650.04 0.187292
\(586\) 15450.0 1.08914
\(587\) −27633.0 −1.94299 −0.971496 0.237057i \(-0.923817\pi\)
−0.971496 + 0.237057i \(0.923817\pi\)
\(588\) 5100.00 0.357688
\(589\) 11843.8 0.828546
\(590\) −8625.61 −0.601883
\(591\) 4219.28 0.293668
\(592\) 24640.0 1.71064
\(593\) 1960.68 0.135777 0.0678883 0.997693i \(-0.478374\pi\)
0.0678883 + 0.997693i \(0.478374\pi\)
\(594\) 0 0
\(595\) −2640.00 −0.181898
\(596\) 2147.74 0.147609
\(597\) 14259.0 0.977524
\(598\) 15300.0 1.04626
\(599\) 20496.0 1.39807 0.699035 0.715088i \(-0.253613\pi\)
0.699035 + 0.715088i \(0.253613\pi\)
\(600\) −1039.23 −0.0707107
\(601\) −10399.2 −0.705813 −0.352906 0.935659i \(-0.614807\pi\)
−0.352906 + 0.935659i \(0.614807\pi\)
\(602\) −3658.09 −0.247662
\(603\) 2844.00 0.192067
\(604\) 907.595 0.0611416
\(605\) 0 0
\(606\) −10872.0 −0.728787
\(607\) −1953.75 −0.130643 −0.0653216 0.997864i \(-0.520807\pi\)
−0.0653216 + 0.997864i \(0.520807\pi\)
\(608\) 7488.00 0.499471
\(609\) −10656.0 −0.709036
\(610\) 7650.00 0.507770
\(611\) −5476.74 −0.362627
\(612\) 685.892 0.0453032
\(613\) −5203.08 −0.342823 −0.171411 0.985200i \(-0.554833\pi\)
−0.171411 + 0.985200i \(0.554833\pi\)
\(614\) −2868.00 −0.188507
\(615\) −2442.19 −0.160128
\(616\) 0 0
\(617\) 23046.0 1.50372 0.751861 0.659321i \(-0.229156\pi\)
0.751861 + 0.659321i \(0.229156\pi\)
\(618\) 14050.4 0.914547
\(619\) −17584.0 −1.14178 −0.570889 0.821027i \(-0.693401\pi\)
−0.570889 + 0.821027i \(0.693401\pi\)
\(620\) 5260.00 0.340720
\(621\) −2025.00 −0.130854
\(622\) 21823.8 1.40684
\(623\) −4988.31 −0.320790
\(624\) 14133.5 0.906721
\(625\) 625.000 0.0400000
\(626\) −8029.79 −0.512675
\(627\) 0 0
\(628\) −10304.0 −0.654736
\(629\) −5868.19 −0.371987
\(630\) −4320.00 −0.273195
\(631\) 6925.00 0.436894 0.218447 0.975849i \(-0.429901\pi\)
0.218447 + 0.975849i \(0.429901\pi\)
\(632\) 15768.0 0.992433
\(633\) −11010.6 −0.691365
\(634\) 6121.07 0.383436
\(635\) −4728.50 −0.295504
\(636\) 6300.00 0.392785
\(637\) −25028.1 −1.55675
\(638\) 0 0
\(639\) −2592.00 −0.160466
\(640\) −7759.59 −0.479257
\(641\) 5388.00 0.332002 0.166001 0.986126i \(-0.446915\pi\)
0.166001 + 0.986126i \(0.446915\pi\)
\(642\) −10890.0 −0.669461
\(643\) 17882.0 1.09673 0.548365 0.836239i \(-0.315251\pi\)
0.548365 + 0.836239i \(0.315251\pi\)
\(644\) −8313.84 −0.508713
\(645\) 571.577 0.0348927
\(646\) −2972.20 −0.181021
\(647\) 4989.00 0.303150 0.151575 0.988446i \(-0.451566\pi\)
0.151575 + 0.988446i \(0.451566\pi\)
\(648\) −1122.37 −0.0680414
\(649\) 0 0
\(650\) −5100.00 −0.307751
\(651\) −21865.4 −1.31639
\(652\) 6776.00 0.407007
\(653\) 9258.00 0.554814 0.277407 0.960752i \(-0.410525\pi\)
0.277407 + 0.960752i \(0.410525\pi\)
\(654\) 19440.0 1.16233
\(655\) 4901.70 0.292405
\(656\) −13025.0 −0.775216
\(657\) −8355.41 −0.496158
\(658\) 8928.00 0.528951
\(659\) −595.825 −0.0352201 −0.0176101 0.999845i \(-0.505606\pi\)
−0.0176101 + 0.999845i \(0.505606\pi\)
\(660\) 0 0
\(661\) 11410.0 0.671403 0.335702 0.941968i \(-0.391027\pi\)
0.335702 + 0.941968i \(0.391027\pi\)
\(662\) −4534.51 −0.266222
\(663\) −3366.00 −0.197171
\(664\) −7920.00 −0.462885
\(665\) 6240.00 0.363875
\(666\) −9602.49 −0.558692
\(667\) 9612.88 0.558039
\(668\) −3498.74 −0.202650
\(669\) −4818.00 −0.278437
\(670\) −5473.28 −0.315599
\(671\) 0 0
\(672\) −13824.0 −0.793560
\(673\) 24858.4 1.42380 0.711902 0.702278i \(-0.247834\pi\)
0.711902 + 0.702278i \(0.247834\pi\)
\(674\) 25212.0 1.44085
\(675\) 675.000 0.0384900
\(676\) 5084.00 0.289258
\(677\) 17805.5 1.01081 0.505406 0.862881i \(-0.331343\pi\)
0.505406 + 0.862881i \(0.331343\pi\)
\(678\) −20483.2 −1.16026
\(679\) −25052.4 −1.41594
\(680\) 1320.00 0.0744407
\(681\) 13743.8 0.773369
\(682\) 0 0
\(683\) −25164.0 −1.40977 −0.704886 0.709321i \(-0.749002\pi\)
−0.704886 + 0.709321i \(0.749002\pi\)
\(684\) −1621.20 −0.0906259
\(685\) 10845.0 0.604914
\(686\) 7872.00 0.438126
\(687\) 12063.0 0.669916
\(688\) 3048.41 0.168924
\(689\) −30917.1 −1.70950
\(690\) 3897.11 0.215015
\(691\) −2707.00 −0.149029 −0.0745146 0.997220i \(-0.523741\pi\)
−0.0745146 + 0.997220i \(0.523741\pi\)
\(692\) −11473.1 −0.630263
\(693\) 0 0
\(694\) 5322.00 0.291096
\(695\) −5014.29 −0.273673
\(696\) 5328.00 0.290169
\(697\) 3102.00 0.168575
\(698\) 37434.0 2.02994
\(699\) 4494.67 0.243210
\(700\) 2771.28 0.149635
\(701\) 3775.87 0.203442 0.101721 0.994813i \(-0.467565\pi\)
0.101721 + 0.994813i \(0.467565\pi\)
\(702\) −5508.00 −0.296134
\(703\) 13870.3 0.744135
\(704\) 0 0
\(705\) −1395.00 −0.0745230
\(706\) −28817.9 −1.53622
\(707\) −28992.0 −1.54223
\(708\) 5976.00 0.317220
\(709\) 8119.00 0.430064 0.215032 0.976607i \(-0.431014\pi\)
0.215032 + 0.976607i \(0.431014\pi\)
\(710\) 4988.31 0.263673
\(711\) −10241.6 −0.540212
\(712\) 2494.15 0.131281
\(713\) 19725.0 1.03605
\(714\) 5487.14 0.287606
\(715\) 0 0
\(716\) −17664.0 −0.921976
\(717\) −1631.59 −0.0849831
\(718\) −35136.0 −1.82627
\(719\) 27042.0 1.40264 0.701319 0.712848i \(-0.252595\pi\)
0.701319 + 0.712848i \(0.252595\pi\)
\(720\) 3600.00 0.186339
\(721\) 37467.7 1.93533
\(722\) −16735.1 −0.862625
\(723\) −6812.16 −0.350411
\(724\) 17720.0 0.909611
\(725\) −3204.29 −0.164144
\(726\) 0 0
\(727\) −12290.0 −0.626975 −0.313488 0.949592i \(-0.601497\pi\)
−0.313488 + 0.949592i \(0.601497\pi\)
\(728\) 22613.7 1.15126
\(729\) 729.000 0.0370370
\(730\) 16080.0 0.815271
\(731\) −726.000 −0.0367334
\(732\) −5300.08 −0.267618
\(733\) −1326.75 −0.0668549 −0.0334275 0.999441i \(-0.510642\pi\)
−0.0334275 + 0.999441i \(0.510642\pi\)
\(734\) 3935.22 0.197890
\(735\) −6375.00 −0.319926
\(736\) 12470.8 0.624563
\(737\) 0 0
\(738\) 5076.00 0.253184
\(739\) 4830.69 0.240460 0.120230 0.992746i \(-0.461637\pi\)
0.120230 + 0.992746i \(0.461637\pi\)
\(740\) 6160.00 0.306008
\(741\) 7956.00 0.394428
\(742\) 50400.0 2.49359
\(743\) −743.050 −0.0366889 −0.0183445 0.999832i \(-0.505840\pi\)
−0.0183445 + 0.999832i \(0.505840\pi\)
\(744\) 10932.7 0.538726
\(745\) −2684.68 −0.132026
\(746\) 34560.0 1.69615
\(747\) 5144.19 0.251963
\(748\) 0 0
\(749\) −29040.0 −1.41669
\(750\) −1299.04 −0.0632456
\(751\) −10835.0 −0.526464 −0.263232 0.964733i \(-0.584789\pi\)
−0.263232 + 0.964733i \(0.584789\pi\)
\(752\) −7440.00 −0.360783
\(753\) −14238.0 −0.689059
\(754\) 26147.0 1.26289
\(755\) −1134.49 −0.0546867
\(756\) 2992.98 0.143986
\(757\) −21050.0 −1.01067 −0.505334 0.862924i \(-0.668631\pi\)
−0.505334 + 0.862924i \(0.668631\pi\)
\(758\) 44787.4 2.14611
\(759\) 0 0
\(760\) −3120.00 −0.148914
\(761\) −26621.6 −1.26811 −0.634056 0.773287i \(-0.718611\pi\)
−0.634056 + 0.773287i \(0.718611\pi\)
\(762\) 9828.00 0.467232
\(763\) 51840.0 2.45968
\(764\) −18528.0 −0.877382
\(765\) −857.365 −0.0405204
\(766\) 1247.08 0.0588234
\(767\) −29327.1 −1.38063
\(768\) 14592.0 0.685603
\(769\) −20654.7 −0.968567 −0.484283 0.874911i \(-0.660920\pi\)
−0.484283 + 0.874911i \(0.660920\pi\)
\(770\) 0 0
\(771\) −8073.00 −0.377097
\(772\) 12304.5 0.573638
\(773\) 15435.0 0.718187 0.359093 0.933302i \(-0.383086\pi\)
0.359093 + 0.933302i \(0.383086\pi\)
\(774\) −1188.00 −0.0551703
\(775\) −6575.00 −0.304750
\(776\) 12526.2 0.579464
\(777\) −25606.6 −1.18228
\(778\) −13572.4 −0.625440
\(779\) −7332.00 −0.337222
\(780\) 3533.38 0.162199
\(781\) 0 0
\(782\) −4950.00 −0.226358
\(783\) −3460.64 −0.157948
\(784\) −34000.0 −1.54883
\(785\) 12880.0 0.585614
\(786\) −10188.0 −0.462333
\(787\) −10149.8 −0.459723 −0.229861 0.973223i \(-0.573827\pi\)
−0.229861 + 0.973223i \(0.573827\pi\)
\(788\) 5625.70 0.254324
\(789\) 11727.7 0.529173
\(790\) 19710.0 0.887659
\(791\) −54622.0 −2.45529
\(792\) 0 0
\(793\) 26010.0 1.16474
\(794\) 19211.9 0.858697
\(795\) −7875.00 −0.351318
\(796\) 19012.0 0.846561
\(797\) −954.000 −0.0423995 −0.0211998 0.999775i \(-0.506749\pi\)
−0.0211998 + 0.999775i \(0.506749\pi\)
\(798\) −12969.6 −0.575337
\(799\) 1771.89 0.0784542
\(800\) −4156.92 −0.183712
\(801\) −1620.00 −0.0714605
\(802\) −2327.88 −0.102494
\(803\) 0 0
\(804\) 3792.00 0.166335
\(805\) 10392.3 0.455007
\(806\) 53652.0 2.34468
\(807\) −23544.0 −1.02700
\(808\) 14496.0 0.631148
\(809\) 33421.7 1.45246 0.726232 0.687450i \(-0.241270\pi\)
0.726232 + 0.687450i \(0.241270\pi\)
\(810\) −1402.96 −0.0608581
\(811\) 3386.16 0.146614 0.0733071 0.997309i \(-0.476645\pi\)
0.0733071 + 0.997309i \(0.476645\pi\)
\(812\) −14208.0 −0.614043
\(813\) 14471.3 0.624268
\(814\) 0 0
\(815\) −8470.00 −0.364038
\(816\) −4572.61 −0.196169
\(817\) 1716.00 0.0734825
\(818\) −786.000 −0.0335964
\(819\) −14688.0 −0.626667
\(820\) −3256.26 −0.138675
\(821\) −35462.0 −1.50747 −0.753735 0.657179i \(-0.771750\pi\)
−0.753735 + 0.657179i \(0.771750\pi\)
\(822\) −22540.9 −0.956453
\(823\) −5668.00 −0.240066 −0.120033 0.992770i \(-0.538300\pi\)
−0.120033 + 0.992770i \(0.538300\pi\)
\(824\) −18733.9 −0.792021
\(825\) 0 0
\(826\) 47808.0 2.01387
\(827\) 24397.7 1.02586 0.512932 0.858429i \(-0.328559\pi\)
0.512932 + 0.858429i \(0.328559\pi\)
\(828\) −2700.00 −0.113323
\(829\) −27889.0 −1.16843 −0.584213 0.811600i \(-0.698597\pi\)
−0.584213 + 0.811600i \(0.698597\pi\)
\(830\) −9900.00 −0.414017
\(831\) 14528.4 0.606481
\(832\) −3768.94 −0.157049
\(833\) 8097.34 0.336802
\(834\) 10422.0 0.432715
\(835\) 4373.43 0.181256
\(836\) 0 0
\(837\) −7101.00 −0.293245
\(838\) 15484.5 0.638311
\(839\) −38220.0 −1.57271 −0.786353 0.617777i \(-0.788033\pi\)
−0.786353 + 0.617777i \(0.788033\pi\)
\(840\) 5760.00 0.236594
\(841\) −7961.00 −0.326418
\(842\) 19187.7 0.785333
\(843\) 23029.3 0.940893
\(844\) −14680.9 −0.598739
\(845\) −6355.00 −0.258720
\(846\) 2899.45 0.117831
\(847\) 0 0
\(848\) −42000.0 −1.70081
\(849\) −22291.5 −0.901110
\(850\) 1650.00 0.0665818
\(851\) 23100.0 0.930503
\(852\) −3456.00 −0.138968
\(853\) 45459.4 1.82474 0.912368 0.409370i \(-0.134252\pi\)
0.912368 + 0.409370i \(0.134252\pi\)
\(854\) −42400.6 −1.69897
\(855\) 2026.50 0.0810583
\(856\) 14520.0 0.579770
\(857\) 24728.5 0.985658 0.492829 0.870126i \(-0.335963\pi\)
0.492829 + 0.870126i \(0.335963\pi\)
\(858\) 0 0
\(859\) −18556.0 −0.737046 −0.368523 0.929619i \(-0.620136\pi\)
−0.368523 + 0.929619i \(0.620136\pi\)
\(860\) 762.102 0.0302180
\(861\) 13536.0 0.535779
\(862\) −34572.0 −1.36604
\(863\) 41088.0 1.62069 0.810343 0.585956i \(-0.199281\pi\)
0.810343 + 0.585956i \(0.199281\pi\)
\(864\) −4489.48 −0.176777
\(865\) 14341.4 0.563724
\(866\) 46855.4 1.83858
\(867\) −13650.0 −0.534692
\(868\) −29153.9 −1.14003
\(869\) 0 0
\(870\) 6660.00 0.259535
\(871\) −18609.2 −0.723935
\(872\) −25920.0 −1.00661
\(873\) −8136.00 −0.315420
\(874\) 11700.0 0.452813
\(875\) −3464.10 −0.133838
\(876\) −11140.6 −0.429685
\(877\) 7520.56 0.289568 0.144784 0.989463i \(-0.453751\pi\)
0.144784 + 0.989463i \(0.453751\pi\)
\(878\) −41118.0 −1.58048
\(879\) 13380.1 0.513424
\(880\) 0 0
\(881\) 19320.0 0.738828 0.369414 0.929265i \(-0.379558\pi\)
0.369414 + 0.929265i \(0.379558\pi\)
\(882\) 13250.2 0.505847
\(883\) 5818.00 0.221734 0.110867 0.993835i \(-0.464637\pi\)
0.110867 + 0.993835i \(0.464637\pi\)
\(884\) −4488.00 −0.170755
\(885\) −7470.00 −0.283730
\(886\) −41943.3 −1.59042
\(887\) −30667.7 −1.16090 −0.580451 0.814295i \(-0.697124\pi\)
−0.580451 + 0.814295i \(0.697124\pi\)
\(888\) 12803.3 0.483842
\(889\) 26208.0 0.988738
\(890\) 3117.69 0.117422
\(891\) 0 0
\(892\) −6424.00 −0.241134
\(893\) −4188.10 −0.156942
\(894\) 5580.00 0.208751
\(895\) 22080.0 0.824640
\(896\) 43008.0 1.60357
\(897\) 13250.2 0.493212
\(898\) −19350.5 −0.719080
\(899\) 33709.2 1.25057
\(900\) 900.000 0.0333333
\(901\) 10002.6 0.369850
\(902\) 0 0
\(903\) −3168.00 −0.116749
\(904\) 27311.0 1.00481
\(905\) −22150.0 −0.813581
\(906\) 2358.00 0.0864672
\(907\) −31474.0 −1.15223 −0.576117 0.817367i \(-0.695433\pi\)
−0.576117 + 0.817367i \(0.695433\pi\)
\(908\) 18325.1 0.669757
\(909\) −9415.43 −0.343553
\(910\) 28267.1 1.02972
\(911\) 16560.0 0.602258 0.301129 0.953583i \(-0.402636\pi\)
0.301129 + 0.953583i \(0.402636\pi\)
\(912\) 10808.0 0.392422
\(913\) 0 0
\(914\) 61548.0 2.22738
\(915\) 6625.09 0.239365
\(916\) 16084.0 0.580164
\(917\) −27168.0 −0.978371
\(918\) 1782.00 0.0640684
\(919\) −13617.4 −0.488788 −0.244394 0.969676i \(-0.578589\pi\)
−0.244394 + 0.969676i \(0.578589\pi\)
\(920\) −5196.15 −0.186209
\(921\) −2483.76 −0.0888629
\(922\) 60948.0 2.17702
\(923\) 16960.2 0.604825
\(924\) 0 0
\(925\) −7700.00 −0.273702
\(926\) −2154.67 −0.0764653
\(927\) 12168.0 0.431121
\(928\) 21312.0 0.753880
\(929\) −2634.00 −0.0930234 −0.0465117 0.998918i \(-0.514810\pi\)
−0.0465117 + 0.998918i \(0.514810\pi\)
\(930\) 13665.9 0.481851
\(931\) −19139.2 −0.673749
\(932\) 5992.90 0.210626
\(933\) 18900.0 0.663192
\(934\) −48563.2 −1.70133
\(935\) 0 0
\(936\) 7344.00 0.256460
\(937\) −48455.9 −1.68942 −0.844709 0.535227i \(-0.820226\pi\)
−0.844709 + 0.535227i \(0.820226\pi\)
\(938\) 30336.0 1.05598
\(939\) −6954.00 −0.241678
\(940\) −1860.00 −0.0645388
\(941\) −33861.6 −1.17307 −0.586534 0.809925i \(-0.699508\pi\)
−0.586534 + 0.809925i \(0.699508\pi\)
\(942\) −26770.6 −0.925937
\(943\) −12211.0 −0.421679
\(944\) −39840.0 −1.37360
\(945\) −3741.23 −0.128785
\(946\) 0 0
\(947\) −4557.00 −0.156370 −0.0781851 0.996939i \(-0.524913\pi\)
−0.0781851 + 0.996939i \(0.524913\pi\)
\(948\) −13655.5 −0.467837
\(949\) 54672.0 1.87010
\(950\) −3900.00 −0.133192
\(951\) 5301.00 0.180754
\(952\) −7316.18 −0.249074
\(953\) −36601.7 −1.24412 −0.622059 0.782970i \(-0.713704\pi\)
−0.622059 + 0.782970i \(0.713704\pi\)
\(954\) 16367.9 0.555482
\(955\) 23160.0 0.784754
\(956\) −2175.46 −0.0735976
\(957\) 0 0
\(958\) 47580.0 1.60463
\(959\) −60109.1 −2.02401
\(960\) −960.000 −0.0322749
\(961\) 39378.0 1.32181
\(962\) 62832.0 2.10581
\(963\) −9431.02 −0.315587
\(964\) −9082.87 −0.303464
\(965\) −15380.6 −0.513077
\(966\) −21600.0 −0.719429
\(967\) −12242.1 −0.407115 −0.203558 0.979063i \(-0.565250\pi\)
−0.203558 + 0.979063i \(0.565250\pi\)
\(968\) 0 0
\(969\) −2574.00 −0.0853342
\(970\) 15657.7 0.518288
\(971\) 31398.0 1.03770 0.518852 0.854864i \(-0.326360\pi\)
0.518852 + 0.854864i \(0.326360\pi\)
\(972\) 972.000 0.0320750
\(973\) 27792.0 0.915694
\(974\) 24186.4 0.795668
\(975\) −4416.73 −0.145075
\(976\) 35333.8 1.15882
\(977\) −54783.0 −1.79392 −0.896962 0.442108i \(-0.854231\pi\)
−0.896962 + 0.442108i \(0.854231\pi\)
\(978\) 17604.6 0.575595
\(979\) 0 0
\(980\) −8500.00 −0.277064
\(981\) 16835.5 0.547928
\(982\) −22896.0 −0.744033
\(983\) −14325.0 −0.464798 −0.232399 0.972621i \(-0.574658\pi\)
−0.232399 + 0.972621i \(0.574658\pi\)
\(984\) −6768.00 −0.219264
\(985\) −7032.13 −0.227474
\(986\) −8459.34 −0.273225
\(987\) 7731.87 0.249350
\(988\) 10608.0 0.341584
\(989\) 2857.88 0.0918862
\(990\) 0 0
\(991\) 17017.0 0.545472 0.272736 0.962089i \(-0.412071\pi\)
0.272736 + 0.962089i \(0.412071\pi\)
\(992\) 43730.8 1.39965
\(993\) −3927.00 −0.125498
\(994\) −27648.0 −0.882235
\(995\) −23765.0 −0.757187
\(996\) 6858.92 0.218206
\(997\) −4964.06 −0.157686 −0.0788432 0.996887i \(-0.525123\pi\)
−0.0788432 + 0.996887i \(0.525123\pi\)
\(998\) −21144.9 −0.670671
\(999\) −8316.00 −0.263370
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 1815.4.a.k.1.2 yes 2
11.10 odd 2 inner 1815.4.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.4.a.k.1.1 2 11.10 odd 2 inner
1815.4.a.k.1.2 yes 2 1.1 even 1 trivial