Properties

Label 1225.4.a.q.1.1
Level $1225$
Weight $4$
Character 1225.1
Self dual yes
Analytic conductor $72.277$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.2773397570\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 245)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.31662\) of defining polynomial
Character \(\chi\) \(=\) 1225.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.31662 q^{2} +5.00000 q^{3} +10.6332 q^{4} -21.5831 q^{6} -11.3668 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-4.31662 q^{2} +5.00000 q^{3} +10.6332 q^{4} -21.5831 q^{6} -11.3668 q^{8} -2.00000 q^{9} +19.7335 q^{11} +53.1662 q^{12} +71.3325 q^{13} -36.0000 q^{16} -31.3325 q^{17} +8.63325 q^{18} -136.332 q^{19} -85.1821 q^{22} +100.865 q^{23} -56.8338 q^{24} -307.916 q^{26} -145.000 q^{27} -288.198 q^{29} +208.997 q^{31} +246.332 q^{32} +98.6675 q^{33} +135.251 q^{34} -21.2665 q^{36} -309.931 q^{37} +588.496 q^{38} +356.662 q^{39} +181.662 q^{41} +18.2005 q^{43} +209.831 q^{44} -435.398 q^{46} -147.665 q^{47} -180.000 q^{48} -156.662 q^{51} +758.496 q^{52} +127.995 q^{53} +625.911 q^{54} -681.662 q^{57} +1244.04 q^{58} -322.665 q^{59} -341.003 q^{61} -902.164 q^{62} -775.325 q^{64} -425.911 q^{66} +84.3960 q^{67} -333.166 q^{68} +504.327 q^{69} -315.736 q^{71} +22.7335 q^{72} -1093.32 q^{73} +1337.86 q^{74} -1449.66 q^{76} -1539.58 q^{78} +1233.19 q^{79} -671.000 q^{81} -784.169 q^{82} -643.325 q^{83} -78.5647 q^{86} -1440.99 q^{87} -224.306 q^{88} -1140.32 q^{89} +1072.53 q^{92} +1044.99 q^{93} +637.414 q^{94} +1231.66 q^{96} +1411.99 q^{97} -39.4670 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 10 q^{3} + 8 q^{4} - 10 q^{6} - 36 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 10 q^{3} + 8 q^{4} - 10 q^{6} - 36 q^{8} - 4 q^{9} + 66 q^{11} + 40 q^{12} + 10 q^{13} - 72 q^{16} + 70 q^{17} + 4 q^{18} - 140 q^{19} + 22 q^{22} + 16 q^{23} - 180 q^{24} - 450 q^{26} - 290 q^{27} - 258 q^{29} + 20 q^{31} + 360 q^{32} + 330 q^{33} + 370 q^{34} - 16 q^{36} - 328 q^{37} + 580 q^{38} + 50 q^{39} - 300 q^{41} + 116 q^{43} + 88 q^{44} - 632 q^{46} - 30 q^{47} - 360 q^{48} + 350 q^{51} + 920 q^{52} - 540 q^{53} + 290 q^{54} - 700 q^{57} + 1314 q^{58} - 380 q^{59} - 1080 q^{61} - 1340 q^{62} - 224 q^{64} + 110 q^{66} - 468 q^{67} - 600 q^{68} + 80 q^{69} - 1056 q^{71} + 72 q^{72} - 860 q^{73} + 1296 q^{74} - 1440 q^{76} - 2250 q^{78} + 158 q^{79} - 1342 q^{81} - 1900 q^{82} + 40 q^{83} + 148 q^{86} - 1290 q^{87} - 1364 q^{88} + 240 q^{89} + 1296 q^{92} + 100 q^{93} + 910 q^{94} + 1800 q^{96} + 1630 q^{97} - 132 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.31662 −1.52616 −0.763079 0.646306i \(-0.776313\pi\)
−0.763079 + 0.646306i \(0.776313\pi\)
\(3\) 5.00000 0.962250 0.481125 0.876652i \(-0.340228\pi\)
0.481125 + 0.876652i \(0.340228\pi\)
\(4\) 10.6332 1.32916
\(5\) 0 0
\(6\) −21.5831 −1.46855
\(7\) 0 0
\(8\) −11.3668 −0.502344
\(9\) −2.00000 −0.0740741
\(10\) 0 0
\(11\) 19.7335 0.540898 0.270449 0.962734i \(-0.412828\pi\)
0.270449 + 0.962734i \(0.412828\pi\)
\(12\) 53.1662 1.27898
\(13\) 71.3325 1.52185 0.760926 0.648839i \(-0.224745\pi\)
0.760926 + 0.648839i \(0.224745\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −36.0000 −0.562500
\(17\) −31.3325 −0.447014 −0.223507 0.974702i \(-0.571751\pi\)
−0.223507 + 0.974702i \(0.571751\pi\)
\(18\) 8.63325 0.113049
\(19\) −136.332 −1.64615 −0.823074 0.567934i \(-0.807743\pi\)
−0.823074 + 0.567934i \(0.807743\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −85.1821 −0.825495
\(23\) 100.865 0.914431 0.457215 0.889356i \(-0.348847\pi\)
0.457215 + 0.889356i \(0.348847\pi\)
\(24\) −56.8338 −0.483381
\(25\) 0 0
\(26\) −307.916 −2.32259
\(27\) −145.000 −1.03353
\(28\) 0 0
\(29\) −288.198 −1.84541 −0.922707 0.385501i \(-0.874029\pi\)
−0.922707 + 0.385501i \(0.874029\pi\)
\(30\) 0 0
\(31\) 208.997 1.21087 0.605436 0.795894i \(-0.292999\pi\)
0.605436 + 0.795894i \(0.292999\pi\)
\(32\) 246.332 1.36081
\(33\) 98.6675 0.520479
\(34\) 135.251 0.682214
\(35\) 0 0
\(36\) −21.2665 −0.0984560
\(37\) −309.931 −1.37709 −0.688546 0.725192i \(-0.741751\pi\)
−0.688546 + 0.725192i \(0.741751\pi\)
\(38\) 588.496 2.51228
\(39\) 356.662 1.46440
\(40\) 0 0
\(41\) 181.662 0.691973 0.345987 0.938239i \(-0.387544\pi\)
0.345987 + 0.938239i \(0.387544\pi\)
\(42\) 0 0
\(43\) 18.2005 0.0645477 0.0322738 0.999479i \(-0.489725\pi\)
0.0322738 + 0.999479i \(0.489725\pi\)
\(44\) 209.831 0.718937
\(45\) 0 0
\(46\) −435.398 −1.39557
\(47\) −147.665 −0.458280 −0.229140 0.973393i \(-0.573591\pi\)
−0.229140 + 0.973393i \(0.573591\pi\)
\(48\) −180.000 −0.541266
\(49\) 0 0
\(50\) 0 0
\(51\) −156.662 −0.430140
\(52\) 758.496 2.02278
\(53\) 127.995 0.331726 0.165863 0.986149i \(-0.446959\pi\)
0.165863 + 0.986149i \(0.446959\pi\)
\(54\) 625.911 1.57733
\(55\) 0 0
\(56\) 0 0
\(57\) −681.662 −1.58401
\(58\) 1244.04 2.81639
\(59\) −322.665 −0.711990 −0.355995 0.934488i \(-0.615858\pi\)
−0.355995 + 0.934488i \(0.615858\pi\)
\(60\) 0 0
\(61\) −341.003 −0.715752 −0.357876 0.933769i \(-0.616499\pi\)
−0.357876 + 0.933769i \(0.616499\pi\)
\(62\) −902.164 −1.84798
\(63\) 0 0
\(64\) −775.325 −1.51431
\(65\) 0 0
\(66\) −425.911 −0.794333
\(67\) 84.3960 0.153890 0.0769449 0.997035i \(-0.475483\pi\)
0.0769449 + 0.997035i \(0.475483\pi\)
\(68\) −333.166 −0.594152
\(69\) 504.327 0.879911
\(70\) 0 0
\(71\) −315.736 −0.527760 −0.263880 0.964555i \(-0.585002\pi\)
−0.263880 + 0.964555i \(0.585002\pi\)
\(72\) 22.7335 0.0372107
\(73\) −1093.32 −1.75293 −0.876466 0.481464i \(-0.840105\pi\)
−0.876466 + 0.481464i \(0.840105\pi\)
\(74\) 1337.86 2.10166
\(75\) 0 0
\(76\) −1449.66 −2.18799
\(77\) 0 0
\(78\) −1539.58 −2.23491
\(79\) 1233.19 1.75626 0.878128 0.478426i \(-0.158793\pi\)
0.878128 + 0.478426i \(0.158793\pi\)
\(80\) 0 0
\(81\) −671.000 −0.920439
\(82\) −784.169 −1.05606
\(83\) −643.325 −0.850772 −0.425386 0.905012i \(-0.639862\pi\)
−0.425386 + 0.905012i \(0.639862\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −78.5647 −0.0985099
\(87\) −1440.99 −1.77575
\(88\) −224.306 −0.271717
\(89\) −1140.32 −1.35813 −0.679064 0.734079i \(-0.737614\pi\)
−0.679064 + 0.734079i \(0.737614\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1072.53 1.21542
\(93\) 1044.99 1.16516
\(94\) 637.414 0.699407
\(95\) 0 0
\(96\) 1231.66 1.30944
\(97\) 1411.99 1.47800 0.739001 0.673705i \(-0.235298\pi\)
0.739001 + 0.673705i \(0.235298\pi\)
\(98\) 0 0
\(99\) −39.4670 −0.0400665
\(100\) 0 0
\(101\) 545.673 0.537589 0.268794 0.963198i \(-0.413375\pi\)
0.268794 + 0.963198i \(0.413375\pi\)
\(102\) 676.253 0.656461
\(103\) 780.990 0.747119 0.373559 0.927606i \(-0.378137\pi\)
0.373559 + 0.927606i \(0.378137\pi\)
\(104\) −810.819 −0.764493
\(105\) 0 0
\(106\) −552.506 −0.506266
\(107\) 620.660 0.560761 0.280381 0.959889i \(-0.409539\pi\)
0.280381 + 0.959889i \(0.409539\pi\)
\(108\) −1541.82 −1.37372
\(109\) 4.18794 0.00368011 0.00184005 0.999998i \(-0.499414\pi\)
0.00184005 + 0.999998i \(0.499414\pi\)
\(110\) 0 0
\(111\) −1549.66 −1.32511
\(112\) 0 0
\(113\) −1413.53 −1.17676 −0.588379 0.808585i \(-0.700234\pi\)
−0.588379 + 0.808585i \(0.700234\pi\)
\(114\) 2942.48 2.41744
\(115\) 0 0
\(116\) −3064.48 −2.45284
\(117\) −142.665 −0.112730
\(118\) 1392.82 1.08661
\(119\) 0 0
\(120\) 0 0
\(121\) −941.589 −0.707430
\(122\) 1471.98 1.09235
\(123\) 908.312 0.665852
\(124\) 2222.32 1.60944
\(125\) 0 0
\(126\) 0 0
\(127\) 2046.26 1.42974 0.714868 0.699259i \(-0.246487\pi\)
0.714868 + 0.699259i \(0.246487\pi\)
\(128\) 1376.13 0.950262
\(129\) 91.0025 0.0621110
\(130\) 0 0
\(131\) −1140.32 −0.760534 −0.380267 0.924877i \(-0.624168\pi\)
−0.380267 + 0.924877i \(0.624168\pi\)
\(132\) 1049.16 0.691798
\(133\) 0 0
\(134\) −364.306 −0.234860
\(135\) 0 0
\(136\) 356.149 0.224555
\(137\) 490.343 0.305787 0.152893 0.988243i \(-0.451141\pi\)
0.152893 + 0.988243i \(0.451141\pi\)
\(138\) −2176.99 −1.34288
\(139\) −2800.00 −1.70858 −0.854291 0.519795i \(-0.826008\pi\)
−0.854291 + 0.519795i \(0.826008\pi\)
\(140\) 0 0
\(141\) −738.325 −0.440980
\(142\) 1362.91 0.805445
\(143\) 1407.64 0.823166
\(144\) 72.0000 0.0416667
\(145\) 0 0
\(146\) 4719.47 2.67525
\(147\) 0 0
\(148\) −3295.58 −1.83037
\(149\) 1166.12 0.641154 0.320577 0.947222i \(-0.396123\pi\)
0.320577 + 0.947222i \(0.396123\pi\)
\(150\) 0 0
\(151\) 959.581 0.517150 0.258575 0.965991i \(-0.416747\pi\)
0.258575 + 0.965991i \(0.416747\pi\)
\(152\) 1549.66 0.826933
\(153\) 62.6650 0.0331122
\(154\) 0 0
\(155\) 0 0
\(156\) 3792.48 1.94642
\(157\) −1020.66 −0.518838 −0.259419 0.965765i \(-0.583531\pi\)
−0.259419 + 0.965765i \(0.583531\pi\)
\(158\) −5323.20 −2.68032
\(159\) 639.975 0.319203
\(160\) 0 0
\(161\) 0 0
\(162\) 2896.46 1.40473
\(163\) 1566.02 0.752514 0.376257 0.926515i \(-0.377211\pi\)
0.376257 + 0.926515i \(0.377211\pi\)
\(164\) 1931.66 0.919741
\(165\) 0 0
\(166\) 2776.99 1.29841
\(167\) −1130.30 −0.523746 −0.261873 0.965102i \(-0.584340\pi\)
−0.261873 + 0.965102i \(0.584340\pi\)
\(168\) 0 0
\(169\) 2891.32 1.31603
\(170\) 0 0
\(171\) 272.665 0.121937
\(172\) 193.530 0.0857940
\(173\) 2543.34 1.11772 0.558862 0.829260i \(-0.311238\pi\)
0.558862 + 0.829260i \(0.311238\pi\)
\(174\) 6220.21 2.71008
\(175\) 0 0
\(176\) −710.406 −0.304255
\(177\) −1613.32 −0.685113
\(178\) 4922.32 2.07272
\(179\) −1210.65 −0.505521 −0.252760 0.967529i \(-0.581338\pi\)
−0.252760 + 0.967529i \(0.581338\pi\)
\(180\) 0 0
\(181\) 3031.32 1.24484 0.622421 0.782683i \(-0.286149\pi\)
0.622421 + 0.782683i \(0.286149\pi\)
\(182\) 0 0
\(183\) −1705.01 −0.688733
\(184\) −1146.51 −0.459359
\(185\) 0 0
\(186\) −4510.82 −1.77822
\(187\) −618.300 −0.241789
\(188\) −1570.16 −0.609125
\(189\) 0 0
\(190\) 0 0
\(191\) −2168.64 −0.821555 −0.410778 0.911736i \(-0.634743\pi\)
−0.410778 + 0.911736i \(0.634743\pi\)
\(192\) −3876.62 −1.45714
\(193\) −1490.48 −0.555892 −0.277946 0.960597i \(-0.589654\pi\)
−0.277946 + 0.960597i \(0.589654\pi\)
\(194\) −6095.04 −2.25566
\(195\) 0 0
\(196\) 0 0
\(197\) −3380.57 −1.22262 −0.611308 0.791393i \(-0.709356\pi\)
−0.611308 + 0.791393i \(0.709356\pi\)
\(198\) 170.364 0.0611478
\(199\) −4595.33 −1.63696 −0.818478 0.574538i \(-0.805182\pi\)
−0.818478 + 0.574538i \(0.805182\pi\)
\(200\) 0 0
\(201\) 421.980 0.148080
\(202\) −2355.46 −0.820445
\(203\) 0 0
\(204\) −1665.83 −0.571723
\(205\) 0 0
\(206\) −3371.24 −1.14022
\(207\) −201.731 −0.0677356
\(208\) −2567.97 −0.856042
\(209\) −2690.32 −0.890398
\(210\) 0 0
\(211\) −4988.66 −1.62765 −0.813824 0.581112i \(-0.802618\pi\)
−0.813824 + 0.581112i \(0.802618\pi\)
\(212\) 1361.00 0.440915
\(213\) −1578.68 −0.507837
\(214\) −2679.16 −0.855810
\(215\) 0 0
\(216\) 1648.18 0.519187
\(217\) 0 0
\(218\) −18.0778 −0.00561643
\(219\) −5466.62 −1.68676
\(220\) 0 0
\(221\) −2235.03 −0.680290
\(222\) 6689.29 2.02232
\(223\) 3792.97 1.13900 0.569498 0.821993i \(-0.307138\pi\)
0.569498 + 0.821993i \(0.307138\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 6101.68 1.79592
\(227\) 3910.96 1.14352 0.571762 0.820420i \(-0.306260\pi\)
0.571762 + 0.820420i \(0.306260\pi\)
\(228\) −7248.29 −2.10539
\(229\) −354.327 −0.102247 −0.0511236 0.998692i \(-0.516280\pi\)
−0.0511236 + 0.998692i \(0.516280\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3275.87 0.927033
\(233\) −6492.48 −1.82548 −0.912739 0.408543i \(-0.866037\pi\)
−0.912739 + 0.408543i \(0.866037\pi\)
\(234\) 615.831 0.172043
\(235\) 0 0
\(236\) −3430.98 −0.946346
\(237\) 6165.93 1.68996
\(238\) 0 0
\(239\) −342.688 −0.0927474 −0.0463737 0.998924i \(-0.514766\pi\)
−0.0463737 + 0.998924i \(0.514766\pi\)
\(240\) 0 0
\(241\) −2313.67 −0.618408 −0.309204 0.950996i \(-0.600063\pi\)
−0.309204 + 0.950996i \(0.600063\pi\)
\(242\) 4064.49 1.07965
\(243\) 560.000 0.147835
\(244\) −3625.96 −0.951347
\(245\) 0 0
\(246\) −3920.84 −1.01619
\(247\) −9724.94 −2.50519
\(248\) −2375.62 −0.608275
\(249\) −3216.62 −0.818656
\(250\) 0 0
\(251\) −3989.29 −1.00319 −0.501597 0.865101i \(-0.667254\pi\)
−0.501597 + 0.865101i \(0.667254\pi\)
\(252\) 0 0
\(253\) 1990.43 0.494614
\(254\) −8832.95 −2.18200
\(255\) 0 0
\(256\) 262.376 0.0640566
\(257\) 2291.32 0.556142 0.278071 0.960560i \(-0.410305\pi\)
0.278071 + 0.960560i \(0.410305\pi\)
\(258\) −392.824 −0.0947912
\(259\) 0 0
\(260\) 0 0
\(261\) 576.396 0.136697
\(262\) 4922.32 1.16070
\(263\) −6360.47 −1.49127 −0.745634 0.666356i \(-0.767853\pi\)
−0.745634 + 0.666356i \(0.767853\pi\)
\(264\) −1121.53 −0.261460
\(265\) 0 0
\(266\) 0 0
\(267\) −5701.59 −1.30686
\(268\) 897.404 0.204543
\(269\) 991.345 0.224697 0.112348 0.993669i \(-0.464163\pi\)
0.112348 + 0.993669i \(0.464163\pi\)
\(270\) 0 0
\(271\) −730.977 −0.163851 −0.0819257 0.996638i \(-0.526107\pi\)
−0.0819257 + 0.996638i \(0.526107\pi\)
\(272\) 1127.97 0.251446
\(273\) 0 0
\(274\) −2116.62 −0.466679
\(275\) 0 0
\(276\) 5362.64 1.16954
\(277\) 3538.63 0.767566 0.383783 0.923423i \(-0.374621\pi\)
0.383783 + 0.923423i \(0.374621\pi\)
\(278\) 12086.5 2.60756
\(279\) −417.995 −0.0896943
\(280\) 0 0
\(281\) −4663.20 −0.989975 −0.494988 0.868900i \(-0.664827\pi\)
−0.494988 + 0.868900i \(0.664827\pi\)
\(282\) 3187.07 0.673005
\(283\) −2104.95 −0.442142 −0.221071 0.975258i \(-0.570955\pi\)
−0.221071 + 0.975258i \(0.570955\pi\)
\(284\) −3357.30 −0.701476
\(285\) 0 0
\(286\) −6076.25 −1.25628
\(287\) 0 0
\(288\) −492.665 −0.100801
\(289\) −3931.27 −0.800178
\(290\) 0 0
\(291\) 7059.96 1.42221
\(292\) −11625.6 −2.32992
\(293\) −6594.66 −1.31489 −0.657447 0.753501i \(-0.728364\pi\)
−0.657447 + 0.753501i \(0.728364\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3522.91 0.691774
\(297\) −2861.36 −0.559033
\(298\) −5033.69 −0.978503
\(299\) 7194.99 1.39163
\(300\) 0 0
\(301\) 0 0
\(302\) −4142.15 −0.789252
\(303\) 2728.36 0.517295
\(304\) 4907.97 0.925958
\(305\) 0 0
\(306\) −270.501 −0.0505344
\(307\) −2672.97 −0.496920 −0.248460 0.968642i \(-0.579924\pi\)
−0.248460 + 0.968642i \(0.579924\pi\)
\(308\) 0 0
\(309\) 3904.95 0.718915
\(310\) 0 0
\(311\) 855.698 0.156020 0.0780099 0.996953i \(-0.475143\pi\)
0.0780099 + 0.996953i \(0.475143\pi\)
\(312\) −4054.09 −0.735634
\(313\) −3349.99 −0.604960 −0.302480 0.953156i \(-0.597815\pi\)
−0.302480 + 0.953156i \(0.597815\pi\)
\(314\) 4405.81 0.791828
\(315\) 0 0
\(316\) 13112.8 2.33434
\(317\) −7633.05 −1.35241 −0.676206 0.736712i \(-0.736377\pi\)
−0.676206 + 0.736712i \(0.736377\pi\)
\(318\) −2762.53 −0.487154
\(319\) −5687.16 −0.998180
\(320\) 0 0
\(321\) 3103.30 0.539593
\(322\) 0 0
\(323\) 4271.64 0.735852
\(324\) −7134.91 −1.22341
\(325\) 0 0
\(326\) −6759.90 −1.14845
\(327\) 20.9397 0.00354119
\(328\) −2064.91 −0.347609
\(329\) 0 0
\(330\) 0 0
\(331\) −3321.70 −0.551593 −0.275796 0.961216i \(-0.588942\pi\)
−0.275796 + 0.961216i \(0.588942\pi\)
\(332\) −6840.63 −1.13081
\(333\) 619.863 0.102007
\(334\) 4879.10 0.799319
\(335\) 0 0
\(336\) 0 0
\(337\) 2233.98 0.361107 0.180553 0.983565i \(-0.442211\pi\)
0.180553 + 0.983565i \(0.442211\pi\)
\(338\) −12480.8 −2.00847
\(339\) −7067.65 −1.13234
\(340\) 0 0
\(341\) 4124.25 0.654958
\(342\) −1176.99 −0.186095
\(343\) 0 0
\(344\) −206.881 −0.0324252
\(345\) 0 0
\(346\) −10978.6 −1.70582
\(347\) −2528.61 −0.391190 −0.195595 0.980685i \(-0.562664\pi\)
−0.195595 + 0.980685i \(0.562664\pi\)
\(348\) −15322.4 −2.36025
\(349\) −1291.00 −0.198011 −0.0990054 0.995087i \(-0.531566\pi\)
−0.0990054 + 0.995087i \(0.531566\pi\)
\(350\) 0 0
\(351\) −10343.2 −1.57288
\(352\) 4861.00 0.736058
\(353\) −7768.64 −1.17134 −0.585670 0.810550i \(-0.699169\pi\)
−0.585670 + 0.810550i \(0.699169\pi\)
\(354\) 6964.12 1.04559
\(355\) 0 0
\(356\) −12125.3 −1.80516
\(357\) 0 0
\(358\) 5225.92 0.771504
\(359\) −2284.14 −0.335800 −0.167900 0.985804i \(-0.553699\pi\)
−0.167900 + 0.985804i \(0.553699\pi\)
\(360\) 0 0
\(361\) 11727.5 1.70980
\(362\) −13085.1 −1.89982
\(363\) −4707.94 −0.680725
\(364\) 0 0
\(365\) 0 0
\(366\) 7359.90 1.05112
\(367\) 10707.0 1.52289 0.761446 0.648229i \(-0.224490\pi\)
0.761446 + 0.648229i \(0.224490\pi\)
\(368\) −3631.16 −0.514367
\(369\) −363.325 −0.0512573
\(370\) 0 0
\(371\) 0 0
\(372\) 11111.6 1.54868
\(373\) 830.429 0.115276 0.0576381 0.998338i \(-0.481643\pi\)
0.0576381 + 0.998338i \(0.481643\pi\)
\(374\) 2668.97 0.369008
\(375\) 0 0
\(376\) 1678.47 0.230214
\(377\) −20557.9 −2.80845
\(378\) 0 0
\(379\) 5253.17 0.711972 0.355986 0.934491i \(-0.384145\pi\)
0.355986 + 0.934491i \(0.384145\pi\)
\(380\) 0 0
\(381\) 10231.3 1.37576
\(382\) 9361.19 1.25382
\(383\) −11243.9 −1.50010 −0.750048 0.661383i \(-0.769970\pi\)
−0.750048 + 0.661383i \(0.769970\pi\)
\(384\) 6880.63 0.914390
\(385\) 0 0
\(386\) 6433.84 0.848378
\(387\) −36.4010 −0.00478131
\(388\) 15014.1 1.96449
\(389\) −8506.85 −1.10878 −0.554389 0.832258i \(-0.687048\pi\)
−0.554389 + 0.832258i \(0.687048\pi\)
\(390\) 0 0
\(391\) −3160.37 −0.408764
\(392\) 0 0
\(393\) −5701.59 −0.731824
\(394\) 14592.6 1.86590
\(395\) 0 0
\(396\) −419.662 −0.0532546
\(397\) 3123.24 0.394838 0.197419 0.980319i \(-0.436744\pi\)
0.197419 + 0.980319i \(0.436744\pi\)
\(398\) 19836.3 2.49825
\(399\) 0 0
\(400\) 0 0
\(401\) −11255.3 −1.40166 −0.700828 0.713330i \(-0.747186\pi\)
−0.700828 + 0.713330i \(0.747186\pi\)
\(402\) −1821.53 −0.225994
\(403\) 14908.3 1.84277
\(404\) 5802.27 0.714539
\(405\) 0 0
\(406\) 0 0
\(407\) −6116.03 −0.744866
\(408\) 1780.74 0.216078
\(409\) 7919.92 0.957494 0.478747 0.877953i \(-0.341091\pi\)
0.478747 + 0.877953i \(0.341091\pi\)
\(410\) 0 0
\(411\) 2451.71 0.294243
\(412\) 8304.46 0.993037
\(413\) 0 0
\(414\) 870.797 0.103375
\(415\) 0 0
\(416\) 17571.5 2.07095
\(417\) −14000.0 −1.64408
\(418\) 11613.1 1.35889
\(419\) 5257.28 0.612972 0.306486 0.951875i \(-0.400847\pi\)
0.306486 + 0.951875i \(0.400847\pi\)
\(420\) 0 0
\(421\) 1457.36 0.168711 0.0843556 0.996436i \(-0.473117\pi\)
0.0843556 + 0.996436i \(0.473117\pi\)
\(422\) 21534.2 2.48405
\(423\) 295.330 0.0339467
\(424\) −1454.89 −0.166640
\(425\) 0 0
\(426\) 6814.57 0.775040
\(427\) 0 0
\(428\) 6599.63 0.745339
\(429\) 7038.20 0.792092
\(430\) 0 0
\(431\) −15291.2 −1.70893 −0.854467 0.519506i \(-0.826116\pi\)
−0.854467 + 0.519506i \(0.826116\pi\)
\(432\) 5220.00 0.581360
\(433\) 187.260 0.0207832 0.0103916 0.999946i \(-0.496692\pi\)
0.0103916 + 0.999946i \(0.496692\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 44.5314 0.00489144
\(437\) −13751.2 −1.50529
\(438\) 23597.4 2.57426
\(439\) −3587.92 −0.390073 −0.195037 0.980796i \(-0.562483\pi\)
−0.195037 + 0.980796i \(0.562483\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 9647.76 1.03823
\(443\) −4915.65 −0.527200 −0.263600 0.964632i \(-0.584910\pi\)
−0.263600 + 0.964632i \(0.584910\pi\)
\(444\) −16477.9 −1.76128
\(445\) 0 0
\(446\) −16372.8 −1.73829
\(447\) 5830.58 0.616951
\(448\) 0 0
\(449\) 7091.12 0.745324 0.372662 0.927967i \(-0.378445\pi\)
0.372662 + 0.927967i \(0.378445\pi\)
\(450\) 0 0
\(451\) 3584.84 0.374287
\(452\) −15030.4 −1.56410
\(453\) 4797.91 0.497628
\(454\) −16882.2 −1.74520
\(455\) 0 0
\(456\) 7748.29 0.795717
\(457\) −5051.81 −0.517098 −0.258549 0.965998i \(-0.583244\pi\)
−0.258549 + 0.965998i \(0.583244\pi\)
\(458\) 1529.50 0.156045
\(459\) 4543.21 0.462002
\(460\) 0 0
\(461\) 16681.3 1.68531 0.842653 0.538456i \(-0.180992\pi\)
0.842653 + 0.538456i \(0.180992\pi\)
\(462\) 0 0
\(463\) −15569.6 −1.56280 −0.781402 0.624027i \(-0.785495\pi\)
−0.781402 + 0.624027i \(0.785495\pi\)
\(464\) 10375.1 1.03805
\(465\) 0 0
\(466\) 28025.6 2.78597
\(467\) 3328.35 0.329802 0.164901 0.986310i \(-0.447269\pi\)
0.164901 + 0.986310i \(0.447269\pi\)
\(468\) −1516.99 −0.149835
\(469\) 0 0
\(470\) 0 0
\(471\) −5103.30 −0.499252
\(472\) 3667.65 0.357664
\(473\) 359.160 0.0349137
\(474\) −26616.0 −2.57914
\(475\) 0 0
\(476\) 0 0
\(477\) −255.990 −0.0245723
\(478\) 1479.25 0.141547
\(479\) 14607.5 1.39339 0.696695 0.717367i \(-0.254653\pi\)
0.696695 + 0.717367i \(0.254653\pi\)
\(480\) 0 0
\(481\) −22108.2 −2.09573
\(482\) 9987.23 0.943789
\(483\) 0 0
\(484\) −10012.2 −0.940285
\(485\) 0 0
\(486\) −2417.31 −0.225620
\(487\) 1879.49 0.174882 0.0874412 0.996170i \(-0.472131\pi\)
0.0874412 + 0.996170i \(0.472131\pi\)
\(488\) 3876.09 0.359554
\(489\) 7830.08 0.724107
\(490\) 0 0
\(491\) 3221.13 0.296064 0.148032 0.988983i \(-0.452706\pi\)
0.148032 + 0.988983i \(0.452706\pi\)
\(492\) 9658.31 0.885021
\(493\) 9029.96 0.824927
\(494\) 41978.9 3.82332
\(495\) 0 0
\(496\) −7523.91 −0.681116
\(497\) 0 0
\(498\) 13885.0 1.24940
\(499\) 9713.81 0.871443 0.435721 0.900082i \(-0.356493\pi\)
0.435721 + 0.900082i \(0.356493\pi\)
\(500\) 0 0
\(501\) −5651.52 −0.503975
\(502\) 17220.3 1.53103
\(503\) 2078.32 0.184230 0.0921152 0.995748i \(-0.470637\pi\)
0.0921152 + 0.995748i \(0.470637\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −8591.94 −0.754858
\(507\) 14456.6 1.26635
\(508\) 21758.4 1.90034
\(509\) 18974.5 1.65232 0.826158 0.563439i \(-0.190522\pi\)
0.826158 + 0.563439i \(0.190522\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −12141.6 −1.04802
\(513\) 19768.2 1.70134
\(514\) −9890.77 −0.848761
\(515\) 0 0
\(516\) 967.652 0.0825553
\(517\) −2913.95 −0.247883
\(518\) 0 0
\(519\) 12716.7 1.07553
\(520\) 0 0
\(521\) −17523.6 −1.47355 −0.736777 0.676136i \(-0.763653\pi\)
−0.736777 + 0.676136i \(0.763653\pi\)
\(522\) −2488.09 −0.208622
\(523\) 15218.6 1.27239 0.636197 0.771527i \(-0.280507\pi\)
0.636197 + 0.771527i \(0.280507\pi\)
\(524\) −12125.3 −1.01087
\(525\) 0 0
\(526\) 27455.8 2.27591
\(527\) −6548.41 −0.541278
\(528\) −3552.03 −0.292769
\(529\) −1993.15 −0.163816
\(530\) 0 0
\(531\) 645.330 0.0527400
\(532\) 0 0
\(533\) 12958.4 1.05308
\(534\) 24611.6 1.99447
\(535\) 0 0
\(536\) −959.308 −0.0773056
\(537\) −6053.25 −0.486438
\(538\) −4279.26 −0.342922
\(539\) 0 0
\(540\) 0 0
\(541\) −15559.3 −1.23650 −0.618249 0.785983i \(-0.712158\pi\)
−0.618249 + 0.785983i \(0.712158\pi\)
\(542\) 3155.36 0.250063
\(543\) 15156.6 1.19785
\(544\) −7718.21 −0.608301
\(545\) 0 0
\(546\) 0 0
\(547\) 8690.70 0.679319 0.339660 0.940548i \(-0.389688\pi\)
0.339660 + 0.940548i \(0.389688\pi\)
\(548\) 5213.93 0.406438
\(549\) 682.005 0.0530187
\(550\) 0 0
\(551\) 39290.8 3.03783
\(552\) −5732.56 −0.442018
\(553\) 0 0
\(554\) −15274.9 −1.17143
\(555\) 0 0
\(556\) −29773.1 −2.27097
\(557\) 7376.26 0.561117 0.280559 0.959837i \(-0.409480\pi\)
0.280559 + 0.959837i \(0.409480\pi\)
\(558\) 1804.33 0.136888
\(559\) 1298.29 0.0982320
\(560\) 0 0
\(561\) −3091.50 −0.232662
\(562\) 20129.3 1.51086
\(563\) 12875.4 0.963825 0.481913 0.876219i \(-0.339942\pi\)
0.481913 + 0.876219i \(0.339942\pi\)
\(564\) −7850.79 −0.586131
\(565\) 0 0
\(566\) 9086.28 0.674779
\(567\) 0 0
\(568\) 3588.89 0.265117
\(569\) −12064.6 −0.888882 −0.444441 0.895808i \(-0.646598\pi\)
−0.444441 + 0.895808i \(0.646598\pi\)
\(570\) 0 0
\(571\) 23745.6 1.74032 0.870158 0.492772i \(-0.164016\pi\)
0.870158 + 0.492772i \(0.164016\pi\)
\(572\) 14967.8 1.09412
\(573\) −10843.2 −0.790542
\(574\) 0 0
\(575\) 0 0
\(576\) 1550.65 0.112171
\(577\) −9846.08 −0.710394 −0.355197 0.934791i \(-0.615586\pi\)
−0.355197 + 0.934791i \(0.615586\pi\)
\(578\) 16969.8 1.22120
\(579\) −7452.40 −0.534907
\(580\) 0 0
\(581\) 0 0
\(582\) −30475.2 −2.17051
\(583\) 2525.79 0.179430
\(584\) 12427.6 0.880575
\(585\) 0 0
\(586\) 28466.7 2.00674
\(587\) 10074.7 0.708392 0.354196 0.935171i \(-0.384755\pi\)
0.354196 + 0.935171i \(0.384755\pi\)
\(588\) 0 0
\(589\) −28493.1 −1.99328
\(590\) 0 0
\(591\) −16902.8 −1.17646
\(592\) 11157.5 0.774615
\(593\) −7387.25 −0.511565 −0.255782 0.966734i \(-0.582333\pi\)
−0.255782 + 0.966734i \(0.582333\pi\)
\(594\) 12351.4 0.853172
\(595\) 0 0
\(596\) 12399.6 0.852194
\(597\) −22976.6 −1.57516
\(598\) −31058.1 −2.12384
\(599\) 1252.73 0.0854510 0.0427255 0.999087i \(-0.486396\pi\)
0.0427255 + 0.999087i \(0.486396\pi\)
\(600\) 0 0
\(601\) 1800.81 0.122224 0.0611120 0.998131i \(-0.480535\pi\)
0.0611120 + 0.998131i \(0.480535\pi\)
\(602\) 0 0
\(603\) −168.792 −0.0113992
\(604\) 10203.5 0.687373
\(605\) 0 0
\(606\) −11777.3 −0.789473
\(607\) 2497.06 0.166973 0.0834863 0.996509i \(-0.473395\pi\)
0.0834863 + 0.996509i \(0.473395\pi\)
\(608\) −33583.1 −2.24009
\(609\) 0 0
\(610\) 0 0
\(611\) −10533.3 −0.697434
\(612\) 666.332 0.0440113
\(613\) 19750.8 1.30135 0.650674 0.759357i \(-0.274487\pi\)
0.650674 + 0.759357i \(0.274487\pi\)
\(614\) 11538.2 0.758378
\(615\) 0 0
\(616\) 0 0
\(617\) −16797.4 −1.09601 −0.548004 0.836476i \(-0.684612\pi\)
−0.548004 + 0.836476i \(0.684612\pi\)
\(618\) −16856.2 −1.09718
\(619\) −26547.4 −1.72379 −0.861897 0.507084i \(-0.830723\pi\)
−0.861897 + 0.507084i \(0.830723\pi\)
\(620\) 0 0
\(621\) −14625.5 −0.945090
\(622\) −3693.73 −0.238111
\(623\) 0 0
\(624\) −12839.8 −0.823727
\(625\) 0 0
\(626\) 14460.6 0.923264
\(627\) −13451.6 −0.856786
\(628\) −10852.9 −0.689616
\(629\) 9710.93 0.615580
\(630\) 0 0
\(631\) 5394.86 0.340358 0.170179 0.985413i \(-0.445565\pi\)
0.170179 + 0.985413i \(0.445565\pi\)
\(632\) −14017.3 −0.882245
\(633\) −24943.3 −1.56620
\(634\) 32949.0 2.06399
\(635\) 0 0
\(636\) 6805.01 0.424271
\(637\) 0 0
\(638\) 24549.3 1.52338
\(639\) 631.472 0.0390933
\(640\) 0 0
\(641\) 2452.41 0.151114 0.0755572 0.997141i \(-0.475926\pi\)
0.0755572 + 0.997141i \(0.475926\pi\)
\(642\) −13395.8 −0.823504
\(643\) 7074.97 0.433919 0.216959 0.976181i \(-0.430386\pi\)
0.216959 + 0.976181i \(0.430386\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −18439.1 −1.12303
\(647\) 3341.37 0.203034 0.101517 0.994834i \(-0.467630\pi\)
0.101517 + 0.994834i \(0.467630\pi\)
\(648\) 7627.09 0.462377
\(649\) −6367.31 −0.385114
\(650\) 0 0
\(651\) 0 0
\(652\) 16651.8 1.00021
\(653\) 23061.6 1.38204 0.691019 0.722837i \(-0.257162\pi\)
0.691019 + 0.722837i \(0.257162\pi\)
\(654\) −90.3888 −0.00540441
\(655\) 0 0
\(656\) −6539.85 −0.389235
\(657\) 2186.65 0.129847
\(658\) 0 0
\(659\) 1742.64 0.103010 0.0515049 0.998673i \(-0.483598\pi\)
0.0515049 + 0.998673i \(0.483598\pi\)
\(660\) 0 0
\(661\) 12576.5 0.740046 0.370023 0.929023i \(-0.379350\pi\)
0.370023 + 0.929023i \(0.379350\pi\)
\(662\) 14338.5 0.841817
\(663\) −11175.1 −0.654609
\(664\) 7312.51 0.427380
\(665\) 0 0
\(666\) −2675.72 −0.155679
\(667\) −29069.2 −1.68750
\(668\) −12018.8 −0.696141
\(669\) 18964.8 1.09600
\(670\) 0 0
\(671\) −6729.17 −0.387149
\(672\) 0 0
\(673\) 10680.8 0.611760 0.305880 0.952070i \(-0.401049\pi\)
0.305880 + 0.952070i \(0.401049\pi\)
\(674\) −9643.27 −0.551105
\(675\) 0 0
\(676\) 30744.2 1.74921
\(677\) −29559.1 −1.67806 −0.839032 0.544082i \(-0.816878\pi\)
−0.839032 + 0.544082i \(0.816878\pi\)
\(678\) 30508.4 1.72812
\(679\) 0 0
\(680\) 0 0
\(681\) 19554.8 1.10036
\(682\) −17802.8 −0.999569
\(683\) −10250.4 −0.574263 −0.287132 0.957891i \(-0.592702\pi\)
−0.287132 + 0.957891i \(0.592702\pi\)
\(684\) 2899.31 0.162073
\(685\) 0 0
\(686\) 0 0
\(687\) −1771.64 −0.0983875
\(688\) −655.218 −0.0363081
\(689\) 9130.20 0.504837
\(690\) 0 0
\(691\) 8874.04 0.488544 0.244272 0.969707i \(-0.421451\pi\)
0.244272 + 0.969707i \(0.421451\pi\)
\(692\) 27043.9 1.48563
\(693\) 0 0
\(694\) 10915.1 0.597018
\(695\) 0 0
\(696\) 16379.4 0.892038
\(697\) −5691.94 −0.309322
\(698\) 5572.77 0.302196
\(699\) −32462.4 −1.75657
\(700\) 0 0
\(701\) −22086.2 −1.18999 −0.594996 0.803729i \(-0.702846\pi\)
−0.594996 + 0.803729i \(0.702846\pi\)
\(702\) 44647.8 2.40046
\(703\) 42253.7 2.26690
\(704\) −15299.9 −0.819085
\(705\) 0 0
\(706\) 33534.3 1.78765
\(707\) 0 0
\(708\) −17154.9 −0.910622
\(709\) −27878.9 −1.47675 −0.738373 0.674392i \(-0.764406\pi\)
−0.738373 + 0.674392i \(0.764406\pi\)
\(710\) 0 0
\(711\) −2466.37 −0.130093
\(712\) 12961.7 0.682248
\(713\) 21080.6 1.10726
\(714\) 0 0
\(715\) 0 0
\(716\) −12873.1 −0.671916
\(717\) −1713.44 −0.0892462
\(718\) 9859.76 0.512483
\(719\) −25863.3 −1.34150 −0.670750 0.741684i \(-0.734027\pi\)
−0.670750 + 0.741684i \(0.734027\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −50623.4 −2.60943
\(723\) −11568.3 −0.595064
\(724\) 32232.8 1.65459
\(725\) 0 0
\(726\) 20322.4 1.03889
\(727\) 29157.0 1.48744 0.743722 0.668489i \(-0.233059\pi\)
0.743722 + 0.668489i \(0.233059\pi\)
\(728\) 0 0
\(729\) 20917.0 1.06269
\(730\) 0 0
\(731\) −570.267 −0.0288538
\(732\) −18129.8 −0.915434
\(733\) 11006.1 0.554595 0.277297 0.960784i \(-0.410561\pi\)
0.277297 + 0.960784i \(0.410561\pi\)
\(734\) −46218.1 −2.32417
\(735\) 0 0
\(736\) 24846.4 1.24436
\(737\) 1665.43 0.0832386
\(738\) 1568.34 0.0782267
\(739\) −37214.4 −1.85244 −0.926221 0.376982i \(-0.876962\pi\)
−0.926221 + 0.376982i \(0.876962\pi\)
\(740\) 0 0
\(741\) −48624.7 −2.41062
\(742\) 0 0
\(743\) −11214.5 −0.553730 −0.276865 0.960909i \(-0.589296\pi\)
−0.276865 + 0.960909i \(0.589296\pi\)
\(744\) −11878.1 −0.585313
\(745\) 0 0
\(746\) −3584.65 −0.175930
\(747\) 1286.65 0.0630202
\(748\) −6574.54 −0.321375
\(749\) 0 0
\(750\) 0 0
\(751\) 6965.26 0.338437 0.169218 0.985579i \(-0.445876\pi\)
0.169218 + 0.985579i \(0.445876\pi\)
\(752\) 5315.94 0.257782
\(753\) −19946.4 −0.965324
\(754\) 88740.7 4.28613
\(755\) 0 0
\(756\) 0 0
\(757\) 19352.8 0.929180 0.464590 0.885526i \(-0.346202\pi\)
0.464590 + 0.885526i \(0.346202\pi\)
\(758\) −22676.0 −1.08658
\(759\) 9952.15 0.475942
\(760\) 0 0
\(761\) −32383.6 −1.54258 −0.771291 0.636483i \(-0.780388\pi\)
−0.771291 + 0.636483i \(0.780388\pi\)
\(762\) −44164.8 −2.09963
\(763\) 0 0
\(764\) −23059.7 −1.09198
\(765\) 0 0
\(766\) 48535.7 2.28938
\(767\) −23016.5 −1.08354
\(768\) 1311.88 0.0616385
\(769\) 25353.9 1.18893 0.594463 0.804123i \(-0.297365\pi\)
0.594463 + 0.804123i \(0.297365\pi\)
\(770\) 0 0
\(771\) 11456.6 0.535148
\(772\) −15848.6 −0.738867
\(773\) 26117.0 1.21522 0.607610 0.794236i \(-0.292128\pi\)
0.607610 + 0.794236i \(0.292128\pi\)
\(774\) 157.129 0.00729703
\(775\) 0 0
\(776\) −16049.8 −0.742465
\(777\) 0 0
\(778\) 36720.9 1.69217
\(779\) −24766.5 −1.13909
\(780\) 0 0
\(781\) −6230.58 −0.285464
\(782\) 13642.1 0.623838
\(783\) 41788.7 1.90729
\(784\) 0 0
\(785\) 0 0
\(786\) 24611.6 1.11688
\(787\) −2273.38 −0.102970 −0.0514849 0.998674i \(-0.516395\pi\)
−0.0514849 + 0.998674i \(0.516395\pi\)
\(788\) −35946.4 −1.62505
\(789\) −31802.4 −1.43497
\(790\) 0 0
\(791\) 0 0
\(792\) 448.612 0.0201272
\(793\) −24324.6 −1.08927
\(794\) −13481.8 −0.602585
\(795\) 0 0
\(796\) −48863.3 −2.17577
\(797\) 2937.42 0.130551 0.0652753 0.997867i \(-0.479207\pi\)
0.0652753 + 0.997867i \(0.479207\pi\)
\(798\) 0 0
\(799\) 4626.71 0.204858
\(800\) 0 0
\(801\) 2280.63 0.100602
\(802\) 48585.0 2.13915
\(803\) −21575.1 −0.948157
\(804\) 4487.02 0.196822
\(805\) 0 0
\(806\) −64353.6 −2.81236
\(807\) 4956.73 0.216214
\(808\) −6202.52 −0.270054
\(809\) −4317.51 −0.187634 −0.0938169 0.995589i \(-0.529907\pi\)
−0.0938169 + 0.995589i \(0.529907\pi\)
\(810\) 0 0
\(811\) −1286.12 −0.0556863 −0.0278432 0.999612i \(-0.508864\pi\)
−0.0278432 + 0.999612i \(0.508864\pi\)
\(812\) 0 0
\(813\) −3654.89 −0.157666
\(814\) 26400.6 1.13678
\(815\) 0 0
\(816\) 5639.85 0.241954
\(817\) −2481.32 −0.106255
\(818\) −34187.3 −1.46129
\(819\) 0 0
\(820\) 0 0
\(821\) 26350.2 1.12013 0.560066 0.828448i \(-0.310776\pi\)
0.560066 + 0.828448i \(0.310776\pi\)
\(822\) −10583.1 −0.449062
\(823\) −9820.05 −0.415924 −0.207962 0.978137i \(-0.566683\pi\)
−0.207962 + 0.978137i \(0.566683\pi\)
\(824\) −8877.32 −0.375311
\(825\) 0 0
\(826\) 0 0
\(827\) 30370.7 1.27702 0.638509 0.769615i \(-0.279552\pi\)
0.638509 + 0.769615i \(0.279552\pi\)
\(828\) −2145.06 −0.0900312
\(829\) 30817.7 1.29112 0.645562 0.763708i \(-0.276623\pi\)
0.645562 + 0.763708i \(0.276623\pi\)
\(830\) 0 0
\(831\) 17693.1 0.738590
\(832\) −55305.9 −2.30455
\(833\) 0 0
\(834\) 60432.7 2.50913
\(835\) 0 0
\(836\) −28606.8 −1.18348
\(837\) −30304.6 −1.25147
\(838\) −22693.7 −0.935491
\(839\) 24746.0 1.01827 0.509134 0.860688i \(-0.329966\pi\)
0.509134 + 0.860688i \(0.329966\pi\)
\(840\) 0 0
\(841\) 58669.1 2.40556
\(842\) −6290.88 −0.257480
\(843\) −23316.0 −0.952604
\(844\) −53045.7 −2.16340
\(845\) 0 0
\(846\) −1274.83 −0.0518079
\(847\) 0 0
\(848\) −4607.82 −0.186596
\(849\) −10524.7 −0.425452
\(850\) 0 0
\(851\) −31261.4 −1.25926
\(852\) −16786.5 −0.674995
\(853\) −11812.1 −0.474135 −0.237067 0.971493i \(-0.576186\pi\)
−0.237067 + 0.971493i \(0.576186\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −7054.89 −0.281695
\(857\) −23440.6 −0.934322 −0.467161 0.884172i \(-0.654723\pi\)
−0.467161 + 0.884172i \(0.654723\pi\)
\(858\) −30381.3 −1.20886
\(859\) −8945.58 −0.355319 −0.177660 0.984092i \(-0.556853\pi\)
−0.177660 + 0.984092i \(0.556853\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 66006.3 2.60810
\(863\) −19313.5 −0.761806 −0.380903 0.924615i \(-0.624387\pi\)
−0.380903 + 0.924615i \(0.624387\pi\)
\(864\) −35718.2 −1.40643
\(865\) 0 0
\(866\) −808.330 −0.0317184
\(867\) −19656.4 −0.769972
\(868\) 0 0
\(869\) 24335.1 0.949955
\(870\) 0 0
\(871\) 6020.18 0.234197
\(872\) −47.6033 −0.00184868
\(873\) −2823.98 −0.109482
\(874\) 59359.0 2.29731
\(875\) 0 0
\(876\) −58128.0 −2.24197
\(877\) 12154.8 0.468001 0.234001 0.972236i \(-0.424818\pi\)
0.234001 + 0.972236i \(0.424818\pi\)
\(878\) 15487.7 0.595313
\(879\) −32973.3 −1.26526
\(880\) 0 0
\(881\) 29390.4 1.12394 0.561968 0.827159i \(-0.310044\pi\)
0.561968 + 0.827159i \(0.310044\pi\)
\(882\) 0 0
\(883\) −4180.02 −0.159308 −0.0796540 0.996823i \(-0.525382\pi\)
−0.0796540 + 0.996823i \(0.525382\pi\)
\(884\) −23765.6 −0.904211
\(885\) 0 0
\(886\) 21219.0 0.804590
\(887\) 21825.8 0.826198 0.413099 0.910686i \(-0.364446\pi\)
0.413099 + 0.910686i \(0.364446\pi\)
\(888\) 17614.6 0.665660
\(889\) 0 0
\(890\) 0 0
\(891\) −13241.2 −0.497863
\(892\) 40331.6 1.51390
\(893\) 20131.5 0.754397
\(894\) −25168.4 −0.941565
\(895\) 0 0
\(896\) 0 0
\(897\) 35974.9 1.33909
\(898\) −30609.7 −1.13748
\(899\) −60232.7 −2.23456
\(900\) 0 0
\(901\) −4010.40 −0.148286
\(902\) −15474.4 −0.571221
\(903\) 0 0
\(904\) 16067.2 0.591138
\(905\) 0 0
\(906\) −20710.8 −0.759458
\(907\) 8356.11 0.305910 0.152955 0.988233i \(-0.451121\pi\)
0.152955 + 0.988233i \(0.451121\pi\)
\(908\) 41586.3 1.51992
\(909\) −1091.35 −0.0398214
\(910\) 0 0
\(911\) −4419.80 −0.160740 −0.0803701 0.996765i \(-0.525610\pi\)
−0.0803701 + 0.996765i \(0.525610\pi\)
\(912\) 24539.8 0.891004
\(913\) −12695.1 −0.460181
\(914\) 21806.8 0.789173
\(915\) 0 0
\(916\) −3767.65 −0.135903
\(917\) 0 0
\(918\) −19611.3 −0.705088
\(919\) −39257.6 −1.40913 −0.704563 0.709641i \(-0.748857\pi\)
−0.704563 + 0.709641i \(0.748857\pi\)
\(920\) 0 0
\(921\) −13364.8 −0.478162
\(922\) −72007.0 −2.57204
\(923\) −22522.2 −0.803173
\(924\) 0 0
\(925\) 0 0
\(926\) 67207.9 2.38509
\(927\) −1561.98 −0.0553421
\(928\) −70992.5 −2.51125
\(929\) 13399.9 0.473235 0.236618 0.971603i \(-0.423961\pi\)
0.236618 + 0.971603i \(0.423961\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −69036.1 −2.42635
\(933\) 4278.49 0.150130
\(934\) −14367.2 −0.503330
\(935\) 0 0
\(936\) 1621.64 0.0566291
\(937\) 27539.8 0.960176 0.480088 0.877220i \(-0.340605\pi\)
0.480088 + 0.877220i \(0.340605\pi\)
\(938\) 0 0
\(939\) −16749.9 −0.582123
\(940\) 0 0
\(941\) −14363.8 −0.497605 −0.248802 0.968554i \(-0.580037\pi\)
−0.248802 + 0.968554i \(0.580037\pi\)
\(942\) 22029.0 0.761937
\(943\) 18323.5 0.632762
\(944\) 11615.9 0.400494
\(945\) 0 0
\(946\) −1550.36 −0.0532838
\(947\) 6372.12 0.218655 0.109327 0.994006i \(-0.465130\pi\)
0.109327 + 0.994006i \(0.465130\pi\)
\(948\) 65563.8 2.24622
\(949\) −77989.6 −2.66770
\(950\) 0 0
\(951\) −38165.3 −1.30136
\(952\) 0 0
\(953\) −958.776 −0.0325895 −0.0162948 0.999867i \(-0.505187\pi\)
−0.0162948 + 0.999867i \(0.505187\pi\)
\(954\) 1105.01 0.0375012
\(955\) 0 0
\(956\) −3643.88 −0.123276
\(957\) −28435.8 −0.960500
\(958\) −63055.1 −2.12653
\(959\) 0 0
\(960\) 0 0
\(961\) 13888.9 0.466213
\(962\) 95432.7 3.19842
\(963\) −1241.32 −0.0415379
\(964\) −24601.8 −0.821961
\(965\) 0 0
\(966\) 0 0
\(967\) 40104.5 1.33369 0.666843 0.745198i \(-0.267645\pi\)
0.666843 + 0.745198i \(0.267645\pi\)
\(968\) 10702.8 0.355373
\(969\) 21358.2 0.708074
\(970\) 0 0
\(971\) −12397.4 −0.409732 −0.204866 0.978790i \(-0.565676\pi\)
−0.204866 + 0.978790i \(0.565676\pi\)
\(972\) 5954.62 0.196496
\(973\) 0 0
\(974\) −8113.05 −0.266898
\(975\) 0 0
\(976\) 12276.1 0.402611
\(977\) −44982.9 −1.47301 −0.736506 0.676432i \(-0.763525\pi\)
−0.736506 + 0.676432i \(0.763525\pi\)
\(978\) −33799.5 −1.10510
\(979\) −22502.5 −0.734608
\(980\) 0 0
\(981\) −8.37588 −0.000272601 0
\(982\) −13904.4 −0.451841
\(983\) −7895.76 −0.256191 −0.128095 0.991762i \(-0.540886\pi\)
−0.128095 + 0.991762i \(0.540886\pi\)
\(984\) −10324.6 −0.334487
\(985\) 0 0
\(986\) −38979.0 −1.25897
\(987\) 0 0
\(988\) −103408. −3.32979
\(989\) 1835.80 0.0590244
\(990\) 0 0
\(991\) 54534.9 1.74809 0.874046 0.485844i \(-0.161488\pi\)
0.874046 + 0.485844i \(0.161488\pi\)
\(992\) 51482.9 1.64776
\(993\) −16608.5 −0.530770
\(994\) 0 0
\(995\) 0 0
\(996\) −34203.2 −1.08812
\(997\) −6028.06 −0.191485 −0.0957425 0.995406i \(-0.530523\pi\)
−0.0957425 + 0.995406i \(0.530523\pi\)
\(998\) −41930.9 −1.32996
\(999\) 44940.1 1.42326
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 1225.4.a.q.1.1 2
5.4 even 2 245.4.a.i.1.2 2
7.6 odd 2 1225.4.a.p.1.1 2
15.14 odd 2 2205.4.a.x.1.1 2
35.4 even 6 245.4.e.k.226.1 4
35.9 even 6 245.4.e.k.116.1 4
35.19 odd 6 245.4.e.j.116.1 4
35.24 odd 6 245.4.e.j.226.1 4
35.34 odd 2 245.4.a.j.1.2 yes 2
105.104 even 2 2205.4.a.w.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.i.1.2 2 5.4 even 2
245.4.a.j.1.2 yes 2 35.34 odd 2
245.4.e.j.116.1 4 35.19 odd 6
245.4.e.j.226.1 4 35.24 odd 6
245.4.e.k.116.1 4 35.9 even 6
245.4.e.k.226.1 4 35.4 even 6
1225.4.a.p.1.1 2 7.6 odd 2
1225.4.a.q.1.1 2 1.1 even 1 trivial
2205.4.a.w.1.1 2 105.104 even 2
2205.4.a.x.1.1 2 15.14 odd 2