Properties

Label 525.4.a.i.1.1
Level $525$
Weight $4$
Character 525.1
Self dual yes
Analytic conductor $30.976$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.70156\) of defining polynomial
Character \(\chi\) \(=\) 525.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.70156 q^{2} -3.00000 q^{3} +14.1047 q^{4} +14.1047 q^{6} -7.00000 q^{7} -28.7016 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.70156 q^{2} -3.00000 q^{3} +14.1047 q^{4} +14.1047 q^{6} -7.00000 q^{7} -28.7016 q^{8} +9.00000 q^{9} +24.5969 q^{11} -42.3141 q^{12} +35.0156 q^{13} +32.9109 q^{14} +22.1047 q^{16} +18.4187 q^{17} -42.3141 q^{18} -67.4031 q^{19} +21.0000 q^{21} -115.644 q^{22} +145.675 q^{23} +86.1047 q^{24} -164.628 q^{26} -27.0000 q^{27} -98.7328 q^{28} +214.419 q^{29} -88.6594 q^{31} +125.686 q^{32} -73.7906 q^{33} -86.5969 q^{34} +126.942 q^{36} -162.125 q^{37} +316.900 q^{38} -105.047 q^{39} -337.769 q^{41} -98.7328 q^{42} -122.156 q^{43} +346.931 q^{44} -684.900 q^{46} -354.219 q^{47} -66.3141 q^{48} +49.0000 q^{49} -55.2562 q^{51} +493.884 q^{52} -676.691 q^{53} +126.942 q^{54} +200.911 q^{56} +202.209 q^{57} -1008.10 q^{58} +501.319 q^{59} -708.931 q^{61} +416.837 q^{62} -63.0000 q^{63} -767.758 q^{64} +346.931 q^{66} +907.956 q^{67} +259.791 q^{68} -437.025 q^{69} +430.334 q^{71} -258.314 q^{72} -41.3406 q^{73} +762.241 q^{74} -950.700 q^{76} -172.178 q^{77} +493.884 q^{78} +890.388 q^{79} +81.0000 q^{81} +1588.04 q^{82} +1057.15 q^{83} +296.198 q^{84} +574.325 q^{86} -643.256 q^{87} -705.969 q^{88} +1473.72 q^{89} -245.109 q^{91} +2054.70 q^{92} +265.978 q^{93} +1665.38 q^{94} -377.058 q^{96} -555.034 q^{97} -230.377 q^{98} +221.372 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} - 6q^{3} + 9q^{4} + 9q^{6} - 14q^{7} - 51q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 3q^{2} - 6q^{3} + 9q^{4} + 9q^{6} - 14q^{7} - 51q^{8} + 18q^{9} + 62q^{11} - 27q^{12} + 6q^{13} + 21q^{14} + 25q^{16} - 40q^{17} - 27q^{18} - 122q^{19} + 42q^{21} - 52q^{22} - 16q^{23} + 153q^{24} - 214q^{26} - 54q^{27} - 63q^{28} + 352q^{29} + 66q^{31} + 309q^{32} - 186q^{33} - 186q^{34} + 81q^{36} + 188q^{37} + 224q^{38} - 18q^{39} + 16q^{41} - 63q^{42} + 396q^{43} + 156q^{44} - 960q^{46} + 188q^{47} - 75q^{48} + 98q^{49} + 120q^{51} + 642q^{52} - 982q^{53} + 81q^{54} + 357q^{56} + 366q^{57} - 774q^{58} + 516q^{59} - 880q^{61} + 680q^{62} - 126q^{63} - 479q^{64} + 156q^{66} + 356q^{67} + 558q^{68} + 48q^{69} + 310q^{71} - 459q^{72} - 326q^{73} + 1358q^{74} - 672q^{76} - 434q^{77} + 642q^{78} + 1832q^{79} + 162q^{81} + 2190q^{82} + 680q^{83} + 189q^{84} + 1456q^{86} - 1056q^{87} - 1540q^{88} + 796q^{89} - 42q^{91} + 2880q^{92} - 198q^{93} + 2588q^{94} - 927q^{96} + 670q^{97} - 147q^{98} + 558q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.70156 −1.66225 −0.831127 0.556083i \(-0.812304\pi\)
−0.831127 + 0.556083i \(0.812304\pi\)
\(3\) −3.00000 −0.577350
\(4\) 14.1047 1.76309
\(5\) 0 0
\(6\) 14.1047 0.959702
\(7\) −7.00000 −0.377964
\(8\) −28.7016 −1.26844
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 24.5969 0.674203 0.337102 0.941468i \(-0.390553\pi\)
0.337102 + 0.941468i \(0.390553\pi\)
\(12\) −42.3141 −1.01792
\(13\) 35.0156 0.747045 0.373523 0.927621i \(-0.378150\pi\)
0.373523 + 0.927621i \(0.378150\pi\)
\(14\) 32.9109 0.628273
\(15\) 0 0
\(16\) 22.1047 0.345386
\(17\) 18.4187 0.262777 0.131388 0.991331i \(-0.458057\pi\)
0.131388 + 0.991331i \(0.458057\pi\)
\(18\) −42.3141 −0.554084
\(19\) −67.4031 −0.813860 −0.406930 0.913459i \(-0.633401\pi\)
−0.406930 + 0.913459i \(0.633401\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) −115.644 −1.12070
\(23\) 145.675 1.32067 0.660333 0.750973i \(-0.270415\pi\)
0.660333 + 0.750973i \(0.270415\pi\)
\(24\) 86.1047 0.732335
\(25\) 0 0
\(26\) −164.628 −1.24178
\(27\) −27.0000 −0.192450
\(28\) −98.7328 −0.666384
\(29\) 214.419 1.37298 0.686492 0.727137i \(-0.259149\pi\)
0.686492 + 0.727137i \(0.259149\pi\)
\(30\) 0 0
\(31\) −88.6594 −0.513667 −0.256834 0.966456i \(-0.582679\pi\)
−0.256834 + 0.966456i \(0.582679\pi\)
\(32\) 125.686 0.694323
\(33\) −73.7906 −0.389251
\(34\) −86.5969 −0.436801
\(35\) 0 0
\(36\) 126.942 0.587695
\(37\) −162.125 −0.720356 −0.360178 0.932884i \(-0.617284\pi\)
−0.360178 + 0.932884i \(0.617284\pi\)
\(38\) 316.900 1.35284
\(39\) −105.047 −0.431307
\(40\) 0 0
\(41\) −337.769 −1.28660 −0.643300 0.765614i \(-0.722435\pi\)
−0.643300 + 0.765614i \(0.722435\pi\)
\(42\) −98.7328 −0.362733
\(43\) −122.156 −0.433224 −0.216612 0.976258i \(-0.569501\pi\)
−0.216612 + 0.976258i \(0.569501\pi\)
\(44\) 346.931 1.18868
\(45\) 0 0
\(46\) −684.900 −2.19528
\(47\) −354.219 −1.09932 −0.549661 0.835388i \(-0.685243\pi\)
−0.549661 + 0.835388i \(0.685243\pi\)
\(48\) −66.3141 −0.199409
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −55.2562 −0.151714
\(52\) 493.884 1.31710
\(53\) −676.691 −1.75378 −0.876892 0.480687i \(-0.840387\pi\)
−0.876892 + 0.480687i \(0.840387\pi\)
\(54\) 126.942 0.319901
\(55\) 0 0
\(56\) 200.911 0.479426
\(57\) 202.209 0.469882
\(58\) −1008.10 −2.28225
\(59\) 501.319 1.10621 0.553103 0.833113i \(-0.313444\pi\)
0.553103 + 0.833113i \(0.313444\pi\)
\(60\) 0 0
\(61\) −708.931 −1.48802 −0.744011 0.668167i \(-0.767079\pi\)
−0.744011 + 0.668167i \(0.767079\pi\)
\(62\) 416.837 0.853845
\(63\) −63.0000 −0.125988
\(64\) −767.758 −1.49953
\(65\) 0 0
\(66\) 346.931 0.647035
\(67\) 907.956 1.65559 0.827795 0.561031i \(-0.189595\pi\)
0.827795 + 0.561031i \(0.189595\pi\)
\(68\) 259.791 0.463298
\(69\) −437.025 −0.762487
\(70\) 0 0
\(71\) 430.334 0.719314 0.359657 0.933085i \(-0.382894\pi\)
0.359657 + 0.933085i \(0.382894\pi\)
\(72\) −258.314 −0.422814
\(73\) −41.3406 −0.0662816 −0.0331408 0.999451i \(-0.510551\pi\)
−0.0331408 + 0.999451i \(0.510551\pi\)
\(74\) 762.241 1.19741
\(75\) 0 0
\(76\) −950.700 −1.43490
\(77\) −172.178 −0.254825
\(78\) 493.884 0.716941
\(79\) 890.388 1.26806 0.634028 0.773310i \(-0.281400\pi\)
0.634028 + 0.773310i \(0.281400\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 1588.04 2.13866
\(83\) 1057.15 1.39804 0.699020 0.715102i \(-0.253620\pi\)
0.699020 + 0.715102i \(0.253620\pi\)
\(84\) 296.198 0.384737
\(85\) 0 0
\(86\) 574.325 0.720129
\(87\) −643.256 −0.792693
\(88\) −705.969 −0.855188
\(89\) 1473.72 1.75522 0.877610 0.479376i \(-0.159137\pi\)
0.877610 + 0.479376i \(0.159137\pi\)
\(90\) 0 0
\(91\) −245.109 −0.282356
\(92\) 2054.70 2.32845
\(93\) 265.978 0.296566
\(94\) 1665.38 1.82735
\(95\) 0 0
\(96\) −377.058 −0.400868
\(97\) −555.034 −0.580981 −0.290491 0.956878i \(-0.593819\pi\)
−0.290491 + 0.956878i \(0.593819\pi\)
\(98\) −230.377 −0.237465
\(99\) 221.372 0.224734
\(100\) 0 0
\(101\) 1890.14 1.86214 0.931071 0.364838i \(-0.118876\pi\)
0.931071 + 0.364838i \(0.118876\pi\)
\(102\) 259.791 0.252187
\(103\) −662.700 −0.633959 −0.316979 0.948432i \(-0.602669\pi\)
−0.316979 + 0.948432i \(0.602669\pi\)
\(104\) −1005.00 −0.947583
\(105\) 0 0
\(106\) 3181.50 2.91523
\(107\) −1614.53 −1.45872 −0.729358 0.684132i \(-0.760181\pi\)
−0.729358 + 0.684132i \(0.760181\pi\)
\(108\) −380.827 −0.339306
\(109\) 217.206 0.190868 0.0954339 0.995436i \(-0.469576\pi\)
0.0954339 + 0.995436i \(0.469576\pi\)
\(110\) 0 0
\(111\) 486.375 0.415898
\(112\) −154.733 −0.130544
\(113\) 1658.20 1.38044 0.690221 0.723598i \(-0.257513\pi\)
0.690221 + 0.723598i \(0.257513\pi\)
\(114\) −950.700 −0.781063
\(115\) 0 0
\(116\) 3024.31 2.42069
\(117\) 315.141 0.249015
\(118\) −2356.98 −1.83879
\(119\) −128.931 −0.0993202
\(120\) 0 0
\(121\) −725.994 −0.545450
\(122\) 3333.08 2.47347
\(123\) 1013.31 0.742819
\(124\) −1250.51 −0.905640
\(125\) 0 0
\(126\) 296.198 0.209424
\(127\) 1108.81 0.774734 0.387367 0.921926i \(-0.373385\pi\)
0.387367 + 0.921926i \(0.373385\pi\)
\(128\) 2604.17 1.79827
\(129\) 366.469 0.250122
\(130\) 0 0
\(131\) 185.488 0.123711 0.0618554 0.998085i \(-0.480298\pi\)
0.0618554 + 0.998085i \(0.480298\pi\)
\(132\) −1040.79 −0.686284
\(133\) 471.822 0.307610
\(134\) −4268.81 −2.75201
\(135\) 0 0
\(136\) −528.647 −0.333317
\(137\) 37.9907 0.0236917 0.0118458 0.999930i \(-0.496229\pi\)
0.0118458 + 0.999930i \(0.496229\pi\)
\(138\) 2054.70 1.26745
\(139\) 183.609 0.112040 0.0560199 0.998430i \(-0.482159\pi\)
0.0560199 + 0.998430i \(0.482159\pi\)
\(140\) 0 0
\(141\) 1062.66 0.634694
\(142\) −2023.24 −1.19568
\(143\) 861.275 0.503660
\(144\) 198.942 0.115129
\(145\) 0 0
\(146\) 194.366 0.110177
\(147\) −147.000 −0.0824786
\(148\) −2286.72 −1.27005
\(149\) −1383.34 −0.760587 −0.380293 0.924866i \(-0.624177\pi\)
−0.380293 + 0.924866i \(0.624177\pi\)
\(150\) 0 0
\(151\) 765.256 0.412422 0.206211 0.978508i \(-0.433887\pi\)
0.206211 + 0.978508i \(0.433887\pi\)
\(152\) 1934.57 1.03233
\(153\) 165.769 0.0875922
\(154\) 809.506 0.423584
\(155\) 0 0
\(156\) −1481.65 −0.760431
\(157\) 2366.76 1.20311 0.601554 0.798832i \(-0.294548\pi\)
0.601554 + 0.798832i \(0.294548\pi\)
\(158\) −4186.21 −2.10783
\(159\) 2030.07 1.01255
\(160\) 0 0
\(161\) −1019.72 −0.499165
\(162\) −380.827 −0.184695
\(163\) 3137.69 1.50775 0.753875 0.657018i \(-0.228183\pi\)
0.753875 + 0.657018i \(0.228183\pi\)
\(164\) −4764.12 −2.26839
\(165\) 0 0
\(166\) −4970.26 −2.32390
\(167\) −146.469 −0.0678688 −0.0339344 0.999424i \(-0.510804\pi\)
−0.0339344 + 0.999424i \(0.510804\pi\)
\(168\) −602.733 −0.276797
\(169\) −970.906 −0.441924
\(170\) 0 0
\(171\) −606.628 −0.271287
\(172\) −1722.98 −0.763812
\(173\) 1424.12 0.625860 0.312930 0.949776i \(-0.398689\pi\)
0.312930 + 0.949776i \(0.398689\pi\)
\(174\) 3024.31 1.31766
\(175\) 0 0
\(176\) 543.706 0.232860
\(177\) −1503.96 −0.638668
\(178\) −6928.81 −2.91762
\(179\) 1244.70 0.519737 0.259869 0.965644i \(-0.416321\pi\)
0.259869 + 0.965644i \(0.416321\pi\)
\(180\) 0 0
\(181\) −3879.09 −1.59299 −0.796493 0.604648i \(-0.793314\pi\)
−0.796493 + 0.604648i \(0.793314\pi\)
\(182\) 1152.40 0.469348
\(183\) 2126.79 0.859110
\(184\) −4181.10 −1.67519
\(185\) 0 0
\(186\) −1250.51 −0.492968
\(187\) 453.044 0.177165
\(188\) −4996.14 −1.93820
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) 1574.90 0.596628 0.298314 0.954468i \(-0.403576\pi\)
0.298314 + 0.954468i \(0.403576\pi\)
\(192\) 2303.27 0.865752
\(193\) 4775.67 1.78114 0.890572 0.454843i \(-0.150305\pi\)
0.890572 + 0.454843i \(0.150305\pi\)
\(194\) 2609.53 0.965738
\(195\) 0 0
\(196\) 691.130 0.251869
\(197\) 2803.58 1.01394 0.506971 0.861963i \(-0.330765\pi\)
0.506971 + 0.861963i \(0.330765\pi\)
\(198\) −1040.79 −0.373566
\(199\) 4102.92 1.46155 0.730774 0.682620i \(-0.239159\pi\)
0.730774 + 0.682620i \(0.239159\pi\)
\(200\) 0 0
\(201\) −2723.87 −0.955855
\(202\) −8886.63 −3.09535
\(203\) −1500.93 −0.518940
\(204\) −779.372 −0.267485
\(205\) 0 0
\(206\) 3115.72 1.05380
\(207\) 1311.07 0.440222
\(208\) 774.009 0.258019
\(209\) −1657.91 −0.548707
\(210\) 0 0
\(211\) −823.512 −0.268687 −0.134343 0.990935i \(-0.542893\pi\)
−0.134343 + 0.990935i \(0.542893\pi\)
\(212\) −9544.51 −3.09207
\(213\) −1291.00 −0.415296
\(214\) 7590.82 2.42476
\(215\) 0 0
\(216\) 774.942 0.244112
\(217\) 620.616 0.194148
\(218\) −1021.21 −0.317271
\(219\) 124.022 0.0382677
\(220\) 0 0
\(221\) 644.944 0.196306
\(222\) −2286.72 −0.691328
\(223\) −817.194 −0.245396 −0.122698 0.992444i \(-0.539155\pi\)
−0.122698 + 0.992444i \(0.539155\pi\)
\(224\) −879.802 −0.262430
\(225\) 0 0
\(226\) −7796.12 −2.29465
\(227\) −3655.85 −1.06893 −0.534465 0.845190i \(-0.679487\pi\)
−0.534465 + 0.845190i \(0.679487\pi\)
\(228\) 2852.10 0.828443
\(229\) 939.393 0.271078 0.135539 0.990772i \(-0.456723\pi\)
0.135539 + 0.990772i \(0.456723\pi\)
\(230\) 0 0
\(231\) 516.534 0.147123
\(232\) −6154.15 −1.74155
\(233\) 7.64701 0.00215010 0.00107505 0.999999i \(-0.499658\pi\)
0.00107505 + 0.999999i \(0.499658\pi\)
\(234\) −1481.65 −0.413926
\(235\) 0 0
\(236\) 7070.94 1.95034
\(237\) −2671.16 −0.732112
\(238\) 606.178 0.165095
\(239\) −889.115 −0.240636 −0.120318 0.992735i \(-0.538391\pi\)
−0.120318 + 0.992735i \(0.538391\pi\)
\(240\) 0 0
\(241\) 2140.23 0.572051 0.286026 0.958222i \(-0.407666\pi\)
0.286026 + 0.958222i \(0.407666\pi\)
\(242\) 3413.30 0.906676
\(243\) −243.000 −0.0641500
\(244\) −9999.25 −2.62351
\(245\) 0 0
\(246\) −4764.12 −1.23475
\(247\) −2360.16 −0.607990
\(248\) 2544.66 0.651557
\(249\) −3171.45 −0.807158
\(250\) 0 0
\(251\) −6749.81 −1.69739 −0.848693 0.528886i \(-0.822610\pi\)
−0.848693 + 0.528886i \(0.822610\pi\)
\(252\) −888.595 −0.222128
\(253\) 3583.15 0.890398
\(254\) −5213.15 −1.28780
\(255\) 0 0
\(256\) −6101.62 −1.48965
\(257\) −3068.64 −0.744811 −0.372405 0.928070i \(-0.621467\pi\)
−0.372405 + 0.928070i \(0.621467\pi\)
\(258\) −1722.98 −0.415766
\(259\) 1134.87 0.272269
\(260\) 0 0
\(261\) 1929.77 0.457662
\(262\) −872.081 −0.205639
\(263\) 4674.12 1.09589 0.547944 0.836515i \(-0.315411\pi\)
0.547944 + 0.836515i \(0.315411\pi\)
\(264\) 2117.91 0.493743
\(265\) 0 0
\(266\) −2218.30 −0.511326
\(267\) −4421.17 −1.01338
\(268\) 12806.4 2.91895
\(269\) 2417.38 0.547919 0.273960 0.961741i \(-0.411667\pi\)
0.273960 + 0.961741i \(0.411667\pi\)
\(270\) 0 0
\(271\) 7724.30 1.73143 0.865715 0.500537i \(-0.166864\pi\)
0.865715 + 0.500537i \(0.166864\pi\)
\(272\) 407.141 0.0907593
\(273\) 735.328 0.163019
\(274\) −178.616 −0.0393816
\(275\) 0 0
\(276\) −6164.10 −1.34433
\(277\) 4576.17 0.992620 0.496310 0.868145i \(-0.334688\pi\)
0.496310 + 0.868145i \(0.334688\pi\)
\(278\) −863.250 −0.186239
\(279\) −797.934 −0.171222
\(280\) 0 0
\(281\) −1358.56 −0.288415 −0.144208 0.989547i \(-0.546063\pi\)
−0.144208 + 0.989547i \(0.546063\pi\)
\(282\) −4996.14 −1.05502
\(283\) −3885.04 −0.816048 −0.408024 0.912971i \(-0.633782\pi\)
−0.408024 + 0.912971i \(0.633782\pi\)
\(284\) 6069.73 1.26821
\(285\) 0 0
\(286\) −4049.34 −0.837211
\(287\) 2364.38 0.486289
\(288\) 1131.17 0.231441
\(289\) −4573.75 −0.930948
\(290\) 0 0
\(291\) 1665.10 0.335430
\(292\) −583.097 −0.116860
\(293\) 4033.91 0.804312 0.402156 0.915571i \(-0.368261\pi\)
0.402156 + 0.915571i \(0.368261\pi\)
\(294\) 691.130 0.137100
\(295\) 0 0
\(296\) 4653.24 0.913730
\(297\) −664.116 −0.129750
\(298\) 6503.85 1.26429
\(299\) 5100.90 0.986598
\(300\) 0 0
\(301\) 855.093 0.163743
\(302\) −3597.90 −0.685549
\(303\) −5670.43 −1.07511
\(304\) −1489.92 −0.281096
\(305\) 0 0
\(306\) −779.372 −0.145600
\(307\) 4620.36 0.858950 0.429475 0.903079i \(-0.358699\pi\)
0.429475 + 0.903079i \(0.358699\pi\)
\(308\) −2428.52 −0.449278
\(309\) 1988.10 0.366016
\(310\) 0 0
\(311\) 6675.89 1.21722 0.608609 0.793470i \(-0.291728\pi\)
0.608609 + 0.793470i \(0.291728\pi\)
\(312\) 3015.01 0.547087
\(313\) −2836.78 −0.512283 −0.256141 0.966639i \(-0.582451\pi\)
−0.256141 + 0.966639i \(0.582451\pi\)
\(314\) −11127.5 −1.99987
\(315\) 0 0
\(316\) 12558.6 2.23569
\(317\) −4010.63 −0.710597 −0.355299 0.934753i \(-0.615621\pi\)
−0.355299 + 0.934753i \(0.615621\pi\)
\(318\) −9544.51 −1.68311
\(319\) 5274.03 0.925671
\(320\) 0 0
\(321\) 4843.59 0.842190
\(322\) 4794.30 0.829739
\(323\) −1241.48 −0.213863
\(324\) 1142.48 0.195898
\(325\) 0 0
\(326\) −14752.1 −2.50626
\(327\) −651.619 −0.110198
\(328\) 9694.49 1.63198
\(329\) 2479.53 0.415504
\(330\) 0 0
\(331\) 11087.5 1.84117 0.920583 0.390546i \(-0.127714\pi\)
0.920583 + 0.390546i \(0.127714\pi\)
\(332\) 14910.8 2.46486
\(333\) −1459.12 −0.240119
\(334\) 688.631 0.112815
\(335\) 0 0
\(336\) 464.198 0.0753693
\(337\) −12118.7 −1.95890 −0.979450 0.201689i \(-0.935357\pi\)
−0.979450 + 0.201689i \(0.935357\pi\)
\(338\) 4564.78 0.734589
\(339\) −4974.59 −0.796999
\(340\) 0 0
\(341\) −2180.74 −0.346316
\(342\) 2852.10 0.450947
\(343\) −343.000 −0.0539949
\(344\) 3506.07 0.549520
\(345\) 0 0
\(346\) −6695.58 −1.04034
\(347\) 6361.22 0.984116 0.492058 0.870562i \(-0.336245\pi\)
0.492058 + 0.870562i \(0.336245\pi\)
\(348\) −9072.93 −1.39759
\(349\) −3115.18 −0.477799 −0.238899 0.971044i \(-0.576787\pi\)
−0.238899 + 0.971044i \(0.576787\pi\)
\(350\) 0 0
\(351\) −945.422 −0.143769
\(352\) 3091.48 0.468115
\(353\) 11927.4 1.79839 0.899194 0.437550i \(-0.144154\pi\)
0.899194 + 0.437550i \(0.144154\pi\)
\(354\) 7070.94 1.06163
\(355\) 0 0
\(356\) 20786.4 3.09460
\(357\) 386.794 0.0573426
\(358\) −5852.02 −0.863935
\(359\) −6143.95 −0.903245 −0.451623 0.892209i \(-0.649155\pi\)
−0.451623 + 0.892209i \(0.649155\pi\)
\(360\) 0 0
\(361\) −2315.82 −0.337632
\(362\) 18237.8 2.64794
\(363\) 2177.98 0.314916
\(364\) −3457.19 −0.497819
\(365\) 0 0
\(366\) −9999.25 −1.42806
\(367\) 1927.67 0.274178 0.137089 0.990559i \(-0.456225\pi\)
0.137089 + 0.990559i \(0.456225\pi\)
\(368\) 3220.10 0.456139
\(369\) −3039.92 −0.428867
\(370\) 0 0
\(371\) 4736.83 0.662868
\(372\) 3751.54 0.522871
\(373\) −10452.0 −1.45090 −0.725449 0.688276i \(-0.758368\pi\)
−0.725449 + 0.688276i \(0.758368\pi\)
\(374\) −2130.01 −0.294493
\(375\) 0 0
\(376\) 10166.6 1.39443
\(377\) 7508.01 1.02568
\(378\) −888.595 −0.120911
\(379\) 7066.43 0.957726 0.478863 0.877890i \(-0.341049\pi\)
0.478863 + 0.877890i \(0.341049\pi\)
\(380\) 0 0
\(381\) −3326.44 −0.447293
\(382\) −7404.51 −0.991747
\(383\) −7168.04 −0.956318 −0.478159 0.878273i \(-0.658696\pi\)
−0.478159 + 0.878273i \(0.658696\pi\)
\(384\) −7812.52 −1.03823
\(385\) 0 0
\(386\) −22453.1 −2.96071
\(387\) −1099.41 −0.144408
\(388\) −7828.58 −1.02432
\(389\) −7414.06 −0.966344 −0.483172 0.875525i \(-0.660515\pi\)
−0.483172 + 0.875525i \(0.660515\pi\)
\(390\) 0 0
\(391\) 2683.15 0.347040
\(392\) −1406.38 −0.181206
\(393\) −556.463 −0.0714245
\(394\) −13181.2 −1.68543
\(395\) 0 0
\(396\) 3122.38 0.396226
\(397\) 8936.01 1.12969 0.564843 0.825198i \(-0.308937\pi\)
0.564843 + 0.825198i \(0.308937\pi\)
\(398\) −19290.1 −2.42946
\(399\) −1415.47 −0.177599
\(400\) 0 0
\(401\) 1782.91 0.222031 0.111015 0.993819i \(-0.464590\pi\)
0.111015 + 0.993819i \(0.464590\pi\)
\(402\) 12806.4 1.58887
\(403\) −3104.46 −0.383733
\(404\) 26659.9 3.28312
\(405\) 0 0
\(406\) 7056.72 0.862609
\(407\) −3987.77 −0.485667
\(408\) 1585.94 0.192441
\(409\) −8759.92 −1.05905 −0.529524 0.848295i \(-0.677629\pi\)
−0.529524 + 0.848295i \(0.677629\pi\)
\(410\) 0 0
\(411\) −113.972 −0.0136784
\(412\) −9347.17 −1.11772
\(413\) −3509.23 −0.418106
\(414\) −6164.10 −0.731761
\(415\) 0 0
\(416\) 4400.97 0.518691
\(417\) −550.828 −0.0646862
\(418\) 7794.75 0.912090
\(419\) −3212.74 −0.374588 −0.187294 0.982304i \(-0.559972\pi\)
−0.187294 + 0.982304i \(0.559972\pi\)
\(420\) 0 0
\(421\) 15757.8 1.82420 0.912101 0.409965i \(-0.134459\pi\)
0.912101 + 0.409965i \(0.134459\pi\)
\(422\) 3871.79 0.446626
\(423\) −3187.97 −0.366440
\(424\) 19422.1 2.22457
\(425\) 0 0
\(426\) 6069.73 0.690327
\(427\) 4962.52 0.562419
\(428\) −22772.5 −2.57184
\(429\) −2583.82 −0.290788
\(430\) 0 0
\(431\) −405.917 −0.0453650 −0.0226825 0.999743i \(-0.507221\pi\)
−0.0226825 + 0.999743i \(0.507221\pi\)
\(432\) −596.827 −0.0664695
\(433\) 7845.25 0.870713 0.435357 0.900258i \(-0.356622\pi\)
0.435357 + 0.900258i \(0.356622\pi\)
\(434\) −2917.86 −0.322723
\(435\) 0 0
\(436\) 3063.63 0.336516
\(437\) −9818.95 −1.07484
\(438\) −583.097 −0.0636106
\(439\) 423.029 0.0459911 0.0229955 0.999736i \(-0.492680\pi\)
0.0229955 + 0.999736i \(0.492680\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) −3032.24 −0.326310
\(443\) 16058.7 1.72229 0.861143 0.508362i \(-0.169749\pi\)
0.861143 + 0.508362i \(0.169749\pi\)
\(444\) 6860.17 0.733264
\(445\) 0 0
\(446\) 3842.09 0.407911
\(447\) 4150.01 0.439125
\(448\) 5374.30 0.566768
\(449\) 2186.75 0.229842 0.114921 0.993375i \(-0.463338\pi\)
0.114921 + 0.993375i \(0.463338\pi\)
\(450\) 0 0
\(451\) −8308.05 −0.867430
\(452\) 23388.3 2.43384
\(453\) −2295.77 −0.238112
\(454\) 17188.2 1.77683
\(455\) 0 0
\(456\) −5803.72 −0.596018
\(457\) 5799.22 0.593602 0.296801 0.954939i \(-0.404080\pi\)
0.296801 + 0.954939i \(0.404080\pi\)
\(458\) −4416.62 −0.450600
\(459\) −497.306 −0.0505714
\(460\) 0 0
\(461\) 9873.35 0.997500 0.498750 0.866746i \(-0.333793\pi\)
0.498750 + 0.866746i \(0.333793\pi\)
\(462\) −2428.52 −0.244556
\(463\) 6181.84 0.620506 0.310253 0.950654i \(-0.399586\pi\)
0.310253 + 0.950654i \(0.399586\pi\)
\(464\) 4739.66 0.474209
\(465\) 0 0
\(466\) −35.9529 −0.00357400
\(467\) −6145.50 −0.608950 −0.304475 0.952520i \(-0.598481\pi\)
−0.304475 + 0.952520i \(0.598481\pi\)
\(468\) 4444.96 0.439035
\(469\) −6355.69 −0.625754
\(470\) 0 0
\(471\) −7100.28 −0.694615
\(472\) −14388.6 −1.40316
\(473\) −3004.66 −0.292081
\(474\) 12558.6 1.21696
\(475\) 0 0
\(476\) −1818.53 −0.175110
\(477\) −6090.22 −0.584595
\(478\) 4180.23 0.399999
\(479\) 10879.4 1.03777 0.518887 0.854843i \(-0.326347\pi\)
0.518887 + 0.854843i \(0.326347\pi\)
\(480\) 0 0
\(481\) −5676.91 −0.538139
\(482\) −10062.4 −0.950894
\(483\) 3059.17 0.288193
\(484\) −10239.9 −0.961675
\(485\) 0 0
\(486\) 1142.48 0.106634
\(487\) −8087.51 −0.752526 −0.376263 0.926513i \(-0.622791\pi\)
−0.376263 + 0.926513i \(0.622791\pi\)
\(488\) 20347.4 1.88747
\(489\) −9413.08 −0.870499
\(490\) 0 0
\(491\) −6959.90 −0.639707 −0.319853 0.947467i \(-0.603634\pi\)
−0.319853 + 0.947467i \(0.603634\pi\)
\(492\) 14292.4 1.30965
\(493\) 3949.32 0.360788
\(494\) 11096.4 1.01063
\(495\) 0 0
\(496\) −1959.79 −0.177413
\(497\) −3012.34 −0.271875
\(498\) 14910.8 1.34170
\(499\) 18632.0 1.67151 0.835756 0.549101i \(-0.185030\pi\)
0.835756 + 0.549101i \(0.185030\pi\)
\(500\) 0 0
\(501\) 439.406 0.0391840
\(502\) 31734.6 2.82149
\(503\) −4627.62 −0.410209 −0.205105 0.978740i \(-0.565753\pi\)
−0.205105 + 0.978740i \(0.565753\pi\)
\(504\) 1808.20 0.159809
\(505\) 0 0
\(506\) −16846.4 −1.48007
\(507\) 2912.72 0.255145
\(508\) 15639.4 1.36592
\(509\) −11351.8 −0.988528 −0.494264 0.869312i \(-0.664562\pi\)
−0.494264 + 0.869312i \(0.664562\pi\)
\(510\) 0 0
\(511\) 289.384 0.0250521
\(512\) 7853.76 0.677911
\(513\) 1819.88 0.156627
\(514\) 14427.4 1.23806
\(515\) 0 0
\(516\) 5168.93 0.440987
\(517\) −8712.67 −0.741166
\(518\) −5335.68 −0.452580
\(519\) −4272.36 −0.361340
\(520\) 0 0
\(521\) 19096.1 1.60579 0.802893 0.596123i \(-0.203293\pi\)
0.802893 + 0.596123i \(0.203293\pi\)
\(522\) −9072.93 −0.760750
\(523\) 3145.11 0.262956 0.131478 0.991319i \(-0.458028\pi\)
0.131478 + 0.991319i \(0.458028\pi\)
\(524\) 2616.24 0.218113
\(525\) 0 0
\(526\) −21975.7 −1.82164
\(527\) −1632.99 −0.134980
\(528\) −1631.12 −0.134442
\(529\) 9054.20 0.744160
\(530\) 0 0
\(531\) 4511.87 0.368735
\(532\) 6654.90 0.542343
\(533\) −11827.2 −0.961148
\(534\) 20786.4 1.68449
\(535\) 0 0
\(536\) −26059.8 −2.10002
\(537\) −3734.09 −0.300071
\(538\) −11365.5 −0.910781
\(539\) 1205.25 0.0963148
\(540\) 0 0
\(541\) 8776.12 0.697440 0.348720 0.937227i \(-0.386616\pi\)
0.348720 + 0.937227i \(0.386616\pi\)
\(542\) −36316.3 −2.87808
\(543\) 11637.3 0.919710
\(544\) 2314.98 0.182452
\(545\) 0 0
\(546\) −3457.19 −0.270978
\(547\) 13695.1 1.07049 0.535247 0.844696i \(-0.320219\pi\)
0.535247 + 0.844696i \(0.320219\pi\)
\(548\) 535.847 0.0417705
\(549\) −6380.38 −0.496007
\(550\) 0 0
\(551\) −14452.5 −1.11742
\(552\) 12543.3 0.967171
\(553\) −6232.71 −0.479280
\(554\) −21515.2 −1.64999
\(555\) 0 0
\(556\) 2589.75 0.197536
\(557\) −7850.44 −0.597188 −0.298594 0.954380i \(-0.596518\pi\)
−0.298594 + 0.954380i \(0.596518\pi\)
\(558\) 3751.54 0.284615
\(559\) −4277.38 −0.323638
\(560\) 0 0
\(561\) −1359.13 −0.102286
\(562\) 6387.33 0.479419
\(563\) 4948.81 0.370457 0.185229 0.982695i \(-0.440697\pi\)
0.185229 + 0.982695i \(0.440697\pi\)
\(564\) 14988.4 1.11902
\(565\) 0 0
\(566\) 18265.7 1.35648
\(567\) −567.000 −0.0419961
\(568\) −12351.3 −0.912408
\(569\) −8115.76 −0.597945 −0.298972 0.954262i \(-0.596644\pi\)
−0.298972 + 0.954262i \(0.596644\pi\)
\(570\) 0 0
\(571\) 5656.42 0.414560 0.207280 0.978282i \(-0.433539\pi\)
0.207280 + 0.978282i \(0.433539\pi\)
\(572\) 12148.0 0.887996
\(573\) −4724.71 −0.344463
\(574\) −11116.3 −0.808336
\(575\) 0 0
\(576\) −6909.82 −0.499842
\(577\) 9536.77 0.688078 0.344039 0.938955i \(-0.388205\pi\)
0.344039 + 0.938955i \(0.388205\pi\)
\(578\) 21503.8 1.54747
\(579\) −14327.0 −1.02834
\(580\) 0 0
\(581\) −7400.05 −0.528409
\(582\) −7828.58 −0.557569
\(583\) −16644.5 −1.18241
\(584\) 1186.54 0.0840743
\(585\) 0 0
\(586\) −18965.7 −1.33697
\(587\) 13089.6 0.920383 0.460191 0.887820i \(-0.347781\pi\)
0.460191 + 0.887820i \(0.347781\pi\)
\(588\) −2073.39 −0.145417
\(589\) 5975.92 0.418053
\(590\) 0 0
\(591\) −8410.73 −0.585400
\(592\) −3583.72 −0.248801
\(593\) −4281.96 −0.296524 −0.148262 0.988948i \(-0.547368\pi\)
−0.148262 + 0.988948i \(0.547368\pi\)
\(594\) 3122.38 0.215678
\(595\) 0 0
\(596\) −19511.5 −1.34098
\(597\) −12308.7 −0.843825
\(598\) −23982.2 −1.63997
\(599\) 3699.92 0.252378 0.126189 0.992006i \(-0.459725\pi\)
0.126189 + 0.992006i \(0.459725\pi\)
\(600\) 0 0
\(601\) −17286.1 −1.17323 −0.586616 0.809865i \(-0.699540\pi\)
−0.586616 + 0.809865i \(0.699540\pi\)
\(602\) −4020.28 −0.272183
\(603\) 8171.61 0.551863
\(604\) 10793.7 0.727135
\(605\) 0 0
\(606\) 26659.9 1.78710
\(607\) −14456.7 −0.966689 −0.483344 0.875430i \(-0.660578\pi\)
−0.483344 + 0.875430i \(0.660578\pi\)
\(608\) −8471.63 −0.565082
\(609\) 4502.79 0.299610
\(610\) 0 0
\(611\) −12403.2 −0.821243
\(612\) 2338.12 0.154433
\(613\) −17981.9 −1.18480 −0.592400 0.805644i \(-0.701819\pi\)
−0.592400 + 0.805644i \(0.701819\pi\)
\(614\) −21722.9 −1.42779
\(615\) 0 0
\(616\) 4941.78 0.323231
\(617\) −19614.7 −1.27983 −0.639916 0.768445i \(-0.721031\pi\)
−0.639916 + 0.768445i \(0.721031\pi\)
\(618\) −9347.17 −0.608412
\(619\) −10462.9 −0.679385 −0.339692 0.940537i \(-0.610323\pi\)
−0.339692 + 0.940537i \(0.610323\pi\)
\(620\) 0 0
\(621\) −3933.22 −0.254162
\(622\) −31387.1 −2.02332
\(623\) −10316.1 −0.663411
\(624\) −2322.03 −0.148967
\(625\) 0 0
\(626\) 13337.3 0.851544
\(627\) 4973.72 0.316796
\(628\) 33382.4 2.12118
\(629\) −2986.14 −0.189293
\(630\) 0 0
\(631\) 24481.9 1.54454 0.772272 0.635292i \(-0.219120\pi\)
0.772272 + 0.635292i \(0.219120\pi\)
\(632\) −25555.5 −1.60846
\(633\) 2470.54 0.155126
\(634\) 18856.2 1.18119
\(635\) 0 0
\(636\) 28633.5 1.78521
\(637\) 1715.77 0.106721
\(638\) −24796.2 −1.53870
\(639\) 3873.01 0.239771
\(640\) 0 0
\(641\) −1109.39 −0.0683595 −0.0341797 0.999416i \(-0.510882\pi\)
−0.0341797 + 0.999416i \(0.510882\pi\)
\(642\) −22772.5 −1.39993
\(643\) −30112.5 −1.84684 −0.923422 0.383787i \(-0.874620\pi\)
−0.923422 + 0.383787i \(0.874620\pi\)
\(644\) −14382.9 −0.880071
\(645\) 0 0
\(646\) 5836.90 0.355495
\(647\) 4260.27 0.258869 0.129435 0.991588i \(-0.458684\pi\)
0.129435 + 0.991588i \(0.458684\pi\)
\(648\) −2324.83 −0.140938
\(649\) 12330.9 0.745808
\(650\) 0 0
\(651\) −1861.85 −0.112091
\(652\) 44256.2 2.65829
\(653\) 10576.8 0.633844 0.316922 0.948452i \(-0.397351\pi\)
0.316922 + 0.948452i \(0.397351\pi\)
\(654\) 3063.63 0.183176
\(655\) 0 0
\(656\) −7466.27 −0.444373
\(657\) −372.066 −0.0220939
\(658\) −11657.7 −0.690674
\(659\) 3394.70 0.200666 0.100333 0.994954i \(-0.468009\pi\)
0.100333 + 0.994954i \(0.468009\pi\)
\(660\) 0 0
\(661\) −33174.4 −1.95210 −0.976048 0.217554i \(-0.930192\pi\)
−0.976048 + 0.217554i \(0.930192\pi\)
\(662\) −52128.7 −3.06048
\(663\) −1934.83 −0.113337
\(664\) −30341.9 −1.77333
\(665\) 0 0
\(666\) 6860.17 0.399138
\(667\) 31235.4 1.81326
\(668\) −2065.89 −0.119658
\(669\) 2451.58 0.141680
\(670\) 0 0
\(671\) −17437.5 −1.00323
\(672\) 2639.40 0.151514
\(673\) −753.881 −0.0431797 −0.0215899 0.999767i \(-0.506873\pi\)
−0.0215899 + 0.999767i \(0.506873\pi\)
\(674\) 56976.9 3.25619
\(675\) 0 0
\(676\) −13694.3 −0.779149
\(677\) −15668.8 −0.889511 −0.444756 0.895652i \(-0.646709\pi\)
−0.444756 + 0.895652i \(0.646709\pi\)
\(678\) 23388.3 1.32481
\(679\) 3885.24 0.219590
\(680\) 0 0
\(681\) 10967.5 0.617147
\(682\) 10252.9 0.575665
\(683\) 11557.4 0.647485 0.323742 0.946145i \(-0.395059\pi\)
0.323742 + 0.946145i \(0.395059\pi\)
\(684\) −8556.30 −0.478302
\(685\) 0 0
\(686\) 1612.64 0.0897532
\(687\) −2818.18 −0.156507
\(688\) −2700.22 −0.149630
\(689\) −23694.7 −1.31016
\(690\) 0 0
\(691\) −18503.1 −1.01866 −0.509328 0.860572i \(-0.670106\pi\)
−0.509328 + 0.860572i \(0.670106\pi\)
\(692\) 20086.8 1.10344
\(693\) −1549.60 −0.0849416
\(694\) −29907.7 −1.63585
\(695\) 0 0
\(696\) 18462.5 1.00549
\(697\) −6221.28 −0.338088
\(698\) 14646.2 0.794223
\(699\) −22.9410 −0.00124136
\(700\) 0 0
\(701\) 22580.4 1.21662 0.608311 0.793699i \(-0.291847\pi\)
0.608311 + 0.793699i \(0.291847\pi\)
\(702\) 4444.96 0.238980
\(703\) 10927.7 0.586269
\(704\) −18884.4 −1.01099
\(705\) 0 0
\(706\) −56077.4 −2.98938
\(707\) −13231.0 −0.703823
\(708\) −21212.8 −1.12603
\(709\) −27426.6 −1.45279 −0.726394 0.687278i \(-0.758805\pi\)
−0.726394 + 0.687278i \(0.758805\pi\)
\(710\) 0 0
\(711\) 8013.49 0.422685
\(712\) −42298.2 −2.22639
\(713\) −12915.5 −0.678383
\(714\) −1818.53 −0.0953178
\(715\) 0 0
\(716\) 17556.1 0.916342
\(717\) 2667.35 0.138931
\(718\) 28886.1 1.50142
\(719\) 19383.0 1.00538 0.502688 0.864468i \(-0.332344\pi\)
0.502688 + 0.864468i \(0.332344\pi\)
\(720\) 0 0
\(721\) 4638.90 0.239614
\(722\) 10888.0 0.561230
\(723\) −6420.69 −0.330274
\(724\) −54713.3 −2.80857
\(725\) 0 0
\(726\) −10239.9 −0.523469
\(727\) 12317.3 0.628368 0.314184 0.949362i \(-0.398269\pi\)
0.314184 + 0.949362i \(0.398269\pi\)
\(728\) 7035.02 0.358153
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −2249.96 −0.113841
\(732\) 29997.8 1.51468
\(733\) −1234.02 −0.0621822 −0.0310911 0.999517i \(-0.509898\pi\)
−0.0310911 + 0.999517i \(0.509898\pi\)
\(734\) −9063.05 −0.455754
\(735\) 0 0
\(736\) 18309.3 0.916970
\(737\) 22332.9 1.11620
\(738\) 14292.4 0.712885
\(739\) −15257.3 −0.759473 −0.379736 0.925095i \(-0.623985\pi\)
−0.379736 + 0.925095i \(0.623985\pi\)
\(740\) 0 0
\(741\) 7080.49 0.351023
\(742\) −22270.5 −1.10186
\(743\) 35565.1 1.75606 0.878032 0.478602i \(-0.158856\pi\)
0.878032 + 0.478602i \(0.158856\pi\)
\(744\) −7633.99 −0.376177
\(745\) 0 0
\(746\) 49140.8 2.41176
\(747\) 9514.35 0.466013
\(748\) 6390.04 0.312357
\(749\) 11301.7 0.551343
\(750\) 0 0
\(751\) 14266.7 0.693209 0.346605 0.938011i \(-0.387335\pi\)
0.346605 + 0.938011i \(0.387335\pi\)
\(752\) −7829.89 −0.379690
\(753\) 20249.4 0.979986
\(754\) −35299.4 −1.70494
\(755\) 0 0
\(756\) 2665.79 0.128246
\(757\) 15927.9 0.764744 0.382372 0.924009i \(-0.375107\pi\)
0.382372 + 0.924009i \(0.375107\pi\)
\(758\) −33223.3 −1.59198
\(759\) −10749.4 −0.514071
\(760\) 0 0
\(761\) −2566.48 −0.122253 −0.0611266 0.998130i \(-0.519469\pi\)
−0.0611266 + 0.998130i \(0.519469\pi\)
\(762\) 15639.4 0.743514
\(763\) −1520.44 −0.0721413
\(764\) 22213.5 1.05191
\(765\) 0 0
\(766\) 33701.0 1.58964
\(767\) 17554.0 0.826386
\(768\) 18304.9 0.860052
\(769\) 14433.1 0.676816 0.338408 0.940999i \(-0.390112\pi\)
0.338408 + 0.940999i \(0.390112\pi\)
\(770\) 0 0
\(771\) 9205.91 0.430017
\(772\) 67359.4 3.14031
\(773\) 29443.2 1.36999 0.684993 0.728550i \(-0.259805\pi\)
0.684993 + 0.728550i \(0.259805\pi\)
\(774\) 5168.93 0.240043
\(775\) 0 0
\(776\) 15930.4 0.736941
\(777\) −3404.62 −0.157195
\(778\) 34857.7 1.60631
\(779\) 22766.7 1.04711
\(780\) 0 0
\(781\) 10584.9 0.484964
\(782\) −12615.0 −0.576869
\(783\) −5789.31 −0.264231
\(784\) 1083.13 0.0493408
\(785\) 0 0
\(786\) 2616.24 0.118726
\(787\) −26390.6 −1.19533 −0.597664 0.801747i \(-0.703904\pi\)
−0.597664 + 0.801747i \(0.703904\pi\)
\(788\) 39543.6 1.78767
\(789\) −14022.4 −0.632711
\(790\) 0 0
\(791\) −11607.4 −0.521758
\(792\) −6353.72 −0.285063
\(793\) −24823.7 −1.11162
\(794\) −42013.2 −1.87783
\(795\) 0 0
\(796\) 57870.3 2.57683
\(797\) 3738.33 0.166146 0.0830730 0.996543i \(-0.473527\pi\)
0.0830730 + 0.996543i \(0.473527\pi\)
\(798\) 6654.90 0.295214
\(799\) −6524.26 −0.288876
\(800\) 0 0
\(801\) 13263.5 0.585073
\(802\) −8382.48 −0.369072
\(803\) −1016.85 −0.0446873
\(804\) −38419.3 −1.68525
\(805\) 0 0
\(806\) 14595.8 0.637861
\(807\) −7252.14 −0.316341
\(808\) −54250.1 −2.36202
\(809\) 43204.1 1.87760 0.938798 0.344468i \(-0.111941\pi\)
0.938798 + 0.344468i \(0.111941\pi\)
\(810\) 0 0
\(811\) −30192.4 −1.30727 −0.653637 0.756809i \(-0.726758\pi\)
−0.653637 + 0.756809i \(0.726758\pi\)
\(812\) −21170.2 −0.914935
\(813\) −23172.9 −0.999642
\(814\) 18748.7 0.807301
\(815\) 0 0
\(816\) −1221.42 −0.0523999
\(817\) 8233.71 0.352584
\(818\) 41185.3 1.76041
\(819\) −2205.98 −0.0941188
\(820\) 0 0
\(821\) −40274.7 −1.71206 −0.856028 0.516929i \(-0.827075\pi\)
−0.856028 + 0.516929i \(0.827075\pi\)
\(822\) 535.847 0.0227370
\(823\) −25184.2 −1.06667 −0.533334 0.845905i \(-0.679061\pi\)
−0.533334 + 0.845905i \(0.679061\pi\)
\(824\) 19020.5 0.804140
\(825\) 0 0
\(826\) 16498.9 0.694999
\(827\) 38941.7 1.63741 0.818703 0.574218i \(-0.194694\pi\)
0.818703 + 0.574218i \(0.194694\pi\)
\(828\) 18492.3 0.776150
\(829\) −8327.05 −0.348867 −0.174433 0.984669i \(-0.555809\pi\)
−0.174433 + 0.984669i \(0.555809\pi\)
\(830\) 0 0
\(831\) −13728.5 −0.573089
\(832\) −26883.5 −1.12021
\(833\) 902.519 0.0375395
\(834\) 2589.75 0.107525
\(835\) 0 0
\(836\) −23384.2 −0.967418
\(837\) 2393.80 0.0988553
\(838\) 15104.9 0.622660
\(839\) 8784.41 0.361468 0.180734 0.983532i \(-0.442153\pi\)
0.180734 + 0.983532i \(0.442153\pi\)
\(840\) 0 0
\(841\) 21586.4 0.885087
\(842\) −74086.4 −3.03229
\(843\) 4075.67 0.166517
\(844\) −11615.4 −0.473718
\(845\) 0 0
\(846\) 14988.4 0.609117
\(847\) 5081.96 0.206161
\(848\) −14958.0 −0.605732
\(849\) 11655.1 0.471145
\(850\) 0 0
\(851\) −23617.6 −0.951350
\(852\) −18209.2 −0.732203
\(853\) 9076.15 0.364316 0.182158 0.983269i \(-0.441692\pi\)
0.182158 + 0.983269i \(0.441692\pi\)
\(854\) −23331.6 −0.934884
\(855\) 0 0
\(856\) 46339.6 1.85030
\(857\) −36396.7 −1.45074 −0.725372 0.688357i \(-0.758332\pi\)
−0.725372 + 0.688357i \(0.758332\pi\)
\(858\) 12148.0 0.483364
\(859\) 8915.27 0.354115 0.177058 0.984200i \(-0.443342\pi\)
0.177058 + 0.984200i \(0.443342\pi\)
\(860\) 0 0
\(861\) −7093.14 −0.280759
\(862\) 1908.44 0.0754081
\(863\) 6148.26 0.242514 0.121257 0.992621i \(-0.461308\pi\)
0.121257 + 0.992621i \(0.461308\pi\)
\(864\) −3393.52 −0.133623
\(865\) 0 0
\(866\) −36884.9 −1.44735
\(867\) 13721.2 0.537483
\(868\) 8753.59 0.342300
\(869\) 21900.8 0.854928
\(870\) 0 0
\(871\) 31792.6 1.23680
\(872\) −6234.16 −0.242105
\(873\) −4995.31 −0.193660
\(874\) 46164.4 1.78665
\(875\) 0 0
\(876\) 1749.29 0.0674692
\(877\) 14287.0 0.550101 0.275050 0.961430i \(-0.411306\pi\)
0.275050 + 0.961430i \(0.411306\pi\)
\(878\) −1988.90 −0.0764488
\(879\) −12101.7 −0.464370
\(880\) 0 0
\(881\) −13315.9 −0.509221 −0.254610 0.967044i \(-0.581947\pi\)
−0.254610 + 0.967044i \(0.581947\pi\)
\(882\) −2073.39 −0.0791549
\(883\) 5271.78 0.200917 0.100458 0.994941i \(-0.467969\pi\)
0.100458 + 0.994941i \(0.467969\pi\)
\(884\) 9096.73 0.346104
\(885\) 0 0
\(886\) −75501.1 −2.86288
\(887\) −2606.07 −0.0986507 −0.0493253 0.998783i \(-0.515707\pi\)
−0.0493253 + 0.998783i \(0.515707\pi\)
\(888\) −13959.7 −0.527542
\(889\) −7761.69 −0.292822
\(890\) 0 0
\(891\) 1992.35 0.0749115
\(892\) −11526.3 −0.432654
\(893\) 23875.4 0.894694
\(894\) −19511.5 −0.729937
\(895\) 0 0
\(896\) −18229.2 −0.679682
\(897\) −15302.7 −0.569612
\(898\) −10281.1 −0.382056
\(899\) −19010.2 −0.705258
\(900\) 0 0
\(901\) −12463.8 −0.460854
\(902\) 39060.8 1.44189
\(903\) −2565.28 −0.0945373
\(904\) −47592.8 −1.75101
\(905\) 0 0
\(906\) 10793.7 0.395802
\(907\) −18610.6 −0.681317 −0.340659 0.940187i \(-0.610650\pi\)
−0.340659 + 0.940187i \(0.610650\pi\)
\(908\) −51564.6 −1.88462
\(909\) 17011.3 0.620714
\(910\) 0 0
\(911\) 41091.7 1.49443 0.747216 0.664581i \(-0.231390\pi\)
0.747216 + 0.664581i \(0.231390\pi\)
\(912\) 4469.77 0.162291
\(913\) 26002.6 0.942563
\(914\) −27265.4 −0.986716
\(915\) 0 0
\(916\) 13249.8 0.477934
\(917\) −1298.41 −0.0467583
\(918\) 2338.12 0.0840624
\(919\) 38891.3 1.39598 0.697990 0.716107i \(-0.254078\pi\)
0.697990 + 0.716107i \(0.254078\pi\)
\(920\) 0 0
\(921\) −13861.1 −0.495915
\(922\) −46420.2 −1.65810
\(923\) 15068.4 0.537360
\(924\) 7285.56 0.259391
\(925\) 0 0
\(926\) −29064.3 −1.03144
\(927\) −5964.30 −0.211320
\(928\) 26949.4 0.953295
\(929\) 18699.4 0.660396 0.330198 0.943912i \(-0.392885\pi\)
0.330198 + 0.943912i \(0.392885\pi\)
\(930\) 0 0
\(931\) −3302.75 −0.116266
\(932\) 107.859 0.00379080
\(933\) −20027.7 −0.702761
\(934\) 28893.4 1.01223
\(935\) 0 0
\(936\) −9045.03 −0.315861
\(937\) −21509.6 −0.749933 −0.374967 0.927038i \(-0.622346\pi\)
−0.374967 + 0.927038i \(0.622346\pi\)
\(938\) 29881.7 1.04016
\(939\) 8510.35 0.295767
\(940\) 0 0
\(941\) −11241.7 −0.389448 −0.194724 0.980858i \(-0.562381\pi\)
−0.194724 + 0.980858i \(0.562381\pi\)
\(942\) 33382.4 1.15463
\(943\) −49204.5 −1.69917
\(944\) 11081.5 0.382068
\(945\) 0 0
\(946\) 14126.6 0.485513
\(947\) 36556.3 1.25441 0.627203 0.778856i \(-0.284200\pi\)
0.627203 + 0.778856i \(0.284200\pi\)
\(948\) −37675.9 −1.29078
\(949\) −1447.57 −0.0495153
\(950\) 0 0
\(951\) 12031.9 0.410263
\(952\) 3700.53 0.125982
\(953\) 36633.4 1.24520 0.622598 0.782542i \(-0.286077\pi\)
0.622598 + 0.782542i \(0.286077\pi\)
\(954\) 28633.5 0.971745
\(955\) 0 0
\(956\) −12540.7 −0.424263
\(957\) −15822.1 −0.534436
\(958\) −51150.3 −1.72504
\(959\) −265.935 −0.00895462
\(960\) 0 0
\(961\) −21930.5 −0.736146
\(962\) 26690.3 0.894523
\(963\) −14530.8 −0.486239
\(964\) 30187.3 1.00858
\(965\) 0 0
\(966\) −14382.9 −0.479050
\(967\) 35515.8 1.18109 0.590544 0.807006i \(-0.298913\pi\)
0.590544 + 0.807006i \(0.298913\pi\)
\(968\) 20837.2 0.691871
\(969\) 3724.44 0.123474
\(970\) 0 0
\(971\) −39661.0 −1.31080 −0.655398 0.755283i \(-0.727499\pi\)
−0.655398 + 0.755283i \(0.727499\pi\)
\(972\) −3427.44 −0.113102
\(973\) −1285.26 −0.0423471
\(974\) 38023.9 1.25089
\(975\) 0 0
\(976\) −15670.7 −0.513942
\(977\) −50325.3 −1.64795 −0.823977 0.566624i \(-0.808249\pi\)
−0.823977 + 0.566624i \(0.808249\pi\)
\(978\) 44256.2 1.44699
\(979\) 36249.0 1.18337
\(980\) 0 0
\(981\) 1954.86 0.0636226
\(982\) 32722.4 1.06335
\(983\) −51189.0 −1.66091 −0.830456 0.557084i \(-0.811920\pi\)
−0.830456 + 0.557084i \(0.811920\pi\)
\(984\) −29083.5 −0.942223
\(985\) 0 0
\(986\) −18568.0 −0.599721
\(987\) −7438.59 −0.239892
\(988\) −33289.3 −1.07194
\(989\) −17795.1 −0.572145
\(990\) 0 0
\(991\) −55137.3 −1.76740 −0.883700 0.468054i \(-0.844955\pi\)
−0.883700 + 0.468054i \(0.844955\pi\)
\(992\) −11143.2 −0.356651
\(993\) −33262.6 −1.06300
\(994\) 14162.7 0.451925
\(995\) 0 0
\(996\) −44732.3 −1.42309
\(997\) −41606.5 −1.32166 −0.660828 0.750537i \(-0.729795\pi\)
−0.660828 + 0.750537i \(0.729795\pi\)
\(998\) −87599.6 −2.77848
\(999\) 4377.37 0.138633
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 525.4.a.i.1.1 2
3.2 odd 2 1575.4.a.y.1.2 2
5.2 odd 4 525.4.d.j.274.1 4
5.3 odd 4 525.4.d.j.274.4 4
5.4 even 2 105.4.a.g.1.2 2
15.14 odd 2 315.4.a.g.1.1 2
20.19 odd 2 1680.4.a.y.1.2 2
35.34 odd 2 735.4.a.q.1.2 2
105.104 even 2 2205.4.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.g.1.2 2 5.4 even 2
315.4.a.g.1.1 2 15.14 odd 2
525.4.a.i.1.1 2 1.1 even 1 trivial
525.4.d.j.274.1 4 5.2 odd 4
525.4.d.j.274.4 4 5.3 odd 4
735.4.a.q.1.2 2 35.34 odd 2
1575.4.a.y.1.2 2 3.2 odd 2
1680.4.a.y.1.2 2 20.19 odd 2
2205.4.a.v.1.1 2 105.104 even 2