Properties

Label 4025.2.a.p.1.5
Level 4025
Weight 2
Character 4025.1
Self dual yes
Analytic conductor 32.140
Analytic rank 0
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.2147108.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 161)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-2.54577\)
Character \(\chi\) = 4025.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.54577 q^{2} -2.46268 q^{3} +4.48096 q^{4} -6.26943 q^{6} -1.00000 q^{7} +6.31597 q^{8} +3.06481 q^{9} +O(q^{10})\) \(q+2.54577 q^{2} -2.46268 q^{3} +4.48096 q^{4} -6.26943 q^{6} -1.00000 q^{7} +6.31597 q^{8} +3.06481 q^{9} -4.70095 q^{11} -11.0352 q^{12} +2.32579 q^{13} -2.54577 q^{14} +7.11710 q^{16} +1.82655 q^{17} +7.80231 q^{18} +7.09155 q^{19} +2.46268 q^{21} -11.9675 q^{22} +1.00000 q^{23} -15.5542 q^{24} +5.92093 q^{26} -0.159610 q^{27} -4.48096 q^{28} -9.98866 q^{29} +3.53732 q^{31} +5.48658 q^{32} +11.5769 q^{33} +4.64998 q^{34} +13.7333 q^{36} +0.166179 q^{37} +18.0535 q^{38} -5.72768 q^{39} +7.25116 q^{41} +6.26943 q^{42} +9.57695 q^{43} -21.0648 q^{44} +2.54577 q^{46} -4.66542 q^{47} -17.5272 q^{48} +1.00000 q^{49} -4.49821 q^{51} +10.4218 q^{52} +0.961924 q^{53} -0.406330 q^{54} -6.31597 q^{56} -17.4642 q^{57} -25.4289 q^{58} +13.8800 q^{59} +0.954652 q^{61} +9.00521 q^{62} -3.06481 q^{63} -0.266598 q^{64} +29.4723 q^{66} +11.9221 q^{67} +8.18469 q^{68} -2.46268 q^{69} +4.59958 q^{71} +19.3572 q^{72} +7.59806 q^{73} +0.423055 q^{74} +31.7770 q^{76} +4.70095 q^{77} -14.5814 q^{78} +5.73902 q^{79} -8.80137 q^{81} +18.4598 q^{82} +5.57695 q^{83} +11.0352 q^{84} +24.3807 q^{86} +24.5989 q^{87} -29.6910 q^{88} -11.2284 q^{89} -2.32579 q^{91} +4.48096 q^{92} -8.71129 q^{93} -11.8771 q^{94} -13.5117 q^{96} +1.68805 q^{97} +2.54577 q^{98} -14.4075 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 2q^{2} + 12q^{4} - 3q^{6} - 5q^{7} - 3q^{8} + 11q^{9} + O(q^{10}) \) \( 5q - 2q^{2} + 12q^{4} - 3q^{6} - 5q^{7} - 3q^{8} + 11q^{9} - 4q^{11} - 3q^{12} + 6q^{13} + 2q^{14} + 10q^{16} + 12q^{17} + 19q^{18} + 6q^{19} - 14q^{22} + 5q^{23} - 36q^{24} + q^{26} - 12q^{28} - 4q^{29} + 30q^{31} - 8q^{32} + 22q^{33} + 6q^{34} - q^{36} - 4q^{37} + 40q^{38} + 16q^{39} + 6q^{41} + 3q^{42} + 12q^{43} - 26q^{44} - 2q^{46} - 10q^{47} - 25q^{48} + 5q^{49} - 4q^{51} + 21q^{52} - 16q^{53} + 33q^{54} + 3q^{56} - 6q^{57} - 13q^{58} + 22q^{59} - 18q^{61} - 15q^{62} - 11q^{63} + 25q^{64} + 4q^{66} + 2q^{67} - 12q^{68} + 4q^{71} + 41q^{72} + 2q^{73} + 38q^{74} + 10q^{76} + 4q^{77} - 41q^{78} + 30q^{79} - 3q^{81} + 7q^{82} - 8q^{83} + 3q^{84} + 8q^{86} + 12q^{87} - 4q^{88} - 20q^{89} - 6q^{91} + 12q^{92} + 26q^{93} - 25q^{94} - q^{96} + 12q^{97} - 2q^{98} - 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.54577 1.80013 0.900067 0.435752i \(-0.143517\pi\)
0.900067 + 0.435752i \(0.143517\pi\)
\(3\) −2.46268 −1.42183 −0.710916 0.703277i \(-0.751719\pi\)
−0.710916 + 0.703277i \(0.751719\pi\)
\(4\) 4.48096 2.24048
\(5\) 0 0
\(6\) −6.26943 −2.55949
\(7\) −1.00000 −0.377964
\(8\) 6.31597 2.23303
\(9\) 3.06481 1.02160
\(10\) 0 0
\(11\) −4.70095 −1.41739 −0.708694 0.705516i \(-0.750715\pi\)
−0.708694 + 0.705516i \(0.750715\pi\)
\(12\) −11.0352 −3.18559
\(13\) 2.32579 0.645058 0.322529 0.946560i \(-0.395467\pi\)
0.322529 + 0.946560i \(0.395467\pi\)
\(14\) −2.54577 −0.680387
\(15\) 0 0
\(16\) 7.11710 1.77927
\(17\) 1.82655 0.443003 0.221502 0.975160i \(-0.428904\pi\)
0.221502 + 0.975160i \(0.428904\pi\)
\(18\) 7.80231 1.83902
\(19\) 7.09155 1.62691 0.813456 0.581626i \(-0.197583\pi\)
0.813456 + 0.581626i \(0.197583\pi\)
\(20\) 0 0
\(21\) 2.46268 0.537402
\(22\) −11.9675 −2.55149
\(23\) 1.00000 0.208514
\(24\) −15.5542 −3.17499
\(25\) 0 0
\(26\) 5.92093 1.16119
\(27\) −0.159610 −0.0307169
\(28\) −4.48096 −0.846822
\(29\) −9.98866 −1.85485 −0.927424 0.374012i \(-0.877982\pi\)
−0.927424 + 0.374012i \(0.877982\pi\)
\(30\) 0 0
\(31\) 3.53732 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(32\) 5.48658 0.969900
\(33\) 11.5769 2.01529
\(34\) 4.64998 0.797465
\(35\) 0 0
\(36\) 13.7333 2.28888
\(37\) 0.166179 0.0273197 0.0136599 0.999907i \(-0.495652\pi\)
0.0136599 + 0.999907i \(0.495652\pi\)
\(38\) 18.0535 2.92866
\(39\) −5.72768 −0.917163
\(40\) 0 0
\(41\) 7.25116 1.13244 0.566220 0.824254i \(-0.308405\pi\)
0.566220 + 0.824254i \(0.308405\pi\)
\(42\) 6.26943 0.967395
\(43\) 9.57695 1.46047 0.730235 0.683196i \(-0.239410\pi\)
0.730235 + 0.683196i \(0.239410\pi\)
\(44\) −21.0648 −3.17563
\(45\) 0 0
\(46\) 2.54577 0.375354
\(47\) −4.66542 −0.680521 −0.340261 0.940331i \(-0.610515\pi\)
−0.340261 + 0.940331i \(0.610515\pi\)
\(48\) −17.5272 −2.52983
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.49821 −0.629875
\(52\) 10.4218 1.44524
\(53\) 0.961924 0.132130 0.0660652 0.997815i \(-0.478955\pi\)
0.0660652 + 0.997815i \(0.478955\pi\)
\(54\) −0.406330 −0.0552945
\(55\) 0 0
\(56\) −6.31597 −0.844007
\(57\) −17.4642 −2.31319
\(58\) −25.4289 −3.33897
\(59\) 13.8800 1.80702 0.903512 0.428562i \(-0.140980\pi\)
0.903512 + 0.428562i \(0.140980\pi\)
\(60\) 0 0
\(61\) 0.954652 0.122231 0.0611153 0.998131i \(-0.480534\pi\)
0.0611153 + 0.998131i \(0.480534\pi\)
\(62\) 9.00521 1.14366
\(63\) −3.06481 −0.386130
\(64\) −0.266598 −0.0333248
\(65\) 0 0
\(66\) 29.4723 3.62779
\(67\) 11.9221 1.45652 0.728259 0.685302i \(-0.240330\pi\)
0.728259 + 0.685302i \(0.240330\pi\)
\(68\) 8.18469 0.992540
\(69\) −2.46268 −0.296472
\(70\) 0 0
\(71\) 4.59958 0.545870 0.272935 0.962033i \(-0.412006\pi\)
0.272935 + 0.962033i \(0.412006\pi\)
\(72\) 19.3572 2.28127
\(73\) 7.59806 0.889286 0.444643 0.895708i \(-0.353331\pi\)
0.444643 + 0.895708i \(0.353331\pi\)
\(74\) 0.423055 0.0491791
\(75\) 0 0
\(76\) 31.7770 3.64507
\(77\) 4.70095 0.535723
\(78\) −14.5814 −1.65102
\(79\) 5.73902 0.645690 0.322845 0.946452i \(-0.395361\pi\)
0.322845 + 0.946452i \(0.395361\pi\)
\(80\) 0 0
\(81\) −8.80137 −0.977930
\(82\) 18.4598 2.03854
\(83\) 5.57695 0.612149 0.306075 0.952008i \(-0.400984\pi\)
0.306075 + 0.952008i \(0.400984\pi\)
\(84\) 11.0352 1.20404
\(85\) 0 0
\(86\) 24.3807 2.62904
\(87\) 24.5989 2.63728
\(88\) −29.6910 −3.16507
\(89\) −11.2284 −1.19021 −0.595106 0.803647i \(-0.702890\pi\)
−0.595106 + 0.803647i \(0.702890\pi\)
\(90\) 0 0
\(91\) −2.32579 −0.243809
\(92\) 4.48096 0.467173
\(93\) −8.71129 −0.903319
\(94\) −11.8771 −1.22503
\(95\) 0 0
\(96\) −13.5117 −1.37903
\(97\) 1.68805 0.171396 0.0856979 0.996321i \(-0.472688\pi\)
0.0856979 + 0.996321i \(0.472688\pi\)
\(98\) 2.54577 0.257162
\(99\) −14.4075 −1.44801
\(100\) 0 0
\(101\) 3.12810 0.311258 0.155629 0.987816i \(-0.450260\pi\)
0.155629 + 0.987816i \(0.450260\pi\)
\(102\) −11.4514 −1.13386
\(103\) −1.03808 −0.102285 −0.0511423 0.998691i \(-0.516286\pi\)
−0.0511423 + 0.998691i \(0.516286\pi\)
\(104\) 14.6896 1.44043
\(105\) 0 0
\(106\) 2.44884 0.237853
\(107\) −8.21965 −0.794624 −0.397312 0.917684i \(-0.630057\pi\)
−0.397312 + 0.917684i \(0.630057\pi\)
\(108\) −0.715204 −0.0688206
\(109\) 6.99848 0.670333 0.335166 0.942159i \(-0.391207\pi\)
0.335166 + 0.942159i \(0.391207\pi\)
\(110\) 0 0
\(111\) −0.409247 −0.0388440
\(112\) −7.11710 −0.672503
\(113\) −2.65158 −0.249439 −0.124720 0.992192i \(-0.539803\pi\)
−0.124720 + 0.992192i \(0.539803\pi\)
\(114\) −44.4600 −4.16406
\(115\) 0 0
\(116\) −44.7588 −4.15575
\(117\) 7.12810 0.658993
\(118\) 35.3354 3.25289
\(119\) −1.82655 −0.167439
\(120\) 0 0
\(121\) 11.0989 1.00899
\(122\) 2.43033 0.220031
\(123\) −17.8573 −1.61014
\(124\) 15.8506 1.42342
\(125\) 0 0
\(126\) −7.80231 −0.695085
\(127\) 0.847744 0.0752251 0.0376126 0.999292i \(-0.488025\pi\)
0.0376126 + 0.999292i \(0.488025\pi\)
\(128\) −11.6519 −1.02989
\(129\) −23.5850 −2.07654
\(130\) 0 0
\(131\) −3.40769 −0.297732 −0.148866 0.988857i \(-0.547562\pi\)
−0.148866 + 0.988857i \(0.547562\pi\)
\(132\) 51.8759 4.51521
\(133\) −7.09155 −0.614915
\(134\) 30.3510 2.62193
\(135\) 0 0
\(136\) 11.5364 0.989240
\(137\) 14.4092 1.23107 0.615533 0.788111i \(-0.288941\pi\)
0.615533 + 0.788111i \(0.288941\pi\)
\(138\) −6.26943 −0.533690
\(139\) 2.60685 0.221110 0.110555 0.993870i \(-0.464737\pi\)
0.110555 + 0.993870i \(0.464737\pi\)
\(140\) 0 0
\(141\) 11.4895 0.967587
\(142\) 11.7095 0.982638
\(143\) −10.9334 −0.914298
\(144\) 21.8126 1.81771
\(145\) 0 0
\(146\) 19.3429 1.60083
\(147\) −2.46268 −0.203119
\(148\) 0.744643 0.0612093
\(149\) −0.00887233 −0.000726849 0 −0.000363425 1.00000i \(-0.500116\pi\)
−0.000363425 1.00000i \(0.500116\pi\)
\(150\) 0 0
\(151\) 13.1070 1.06663 0.533316 0.845916i \(-0.320946\pi\)
0.533316 + 0.845916i \(0.320946\pi\)
\(152\) 44.7900 3.63295
\(153\) 5.59803 0.452574
\(154\) 11.9675 0.964372
\(155\) 0 0
\(156\) −25.6655 −2.05489
\(157\) 0.249603 0.0199205 0.00996025 0.999950i \(-0.496830\pi\)
0.00996025 + 0.999950i \(0.496830\pi\)
\(158\) 14.6103 1.16233
\(159\) −2.36892 −0.187867
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) −22.4063 −1.76040
\(163\) 0.150737 0.0118066 0.00590332 0.999983i \(-0.498121\pi\)
0.00590332 + 0.999983i \(0.498121\pi\)
\(164\) 32.4922 2.53721
\(165\) 0 0
\(166\) 14.1976 1.10195
\(167\) −11.8235 −0.914931 −0.457465 0.889227i \(-0.651243\pi\)
−0.457465 + 0.889227i \(0.651243\pi\)
\(168\) 15.5542 1.20003
\(169\) −7.59071 −0.583900
\(170\) 0 0
\(171\) 21.7343 1.66206
\(172\) 42.9139 3.27216
\(173\) −6.78120 −0.515565 −0.257783 0.966203i \(-0.582992\pi\)
−0.257783 + 0.966203i \(0.582992\pi\)
\(174\) 62.6233 4.74746
\(175\) 0 0
\(176\) −33.4571 −2.52192
\(177\) −34.1821 −2.56928
\(178\) −28.5851 −2.14254
\(179\) 20.5624 1.53690 0.768451 0.639908i \(-0.221028\pi\)
0.768451 + 0.639908i \(0.221028\pi\)
\(180\) 0 0
\(181\) −1.69693 −0.126131 −0.0630657 0.998009i \(-0.520088\pi\)
−0.0630657 + 0.998009i \(0.520088\pi\)
\(182\) −5.92093 −0.438889
\(183\) −2.35101 −0.173791
\(184\) 6.31597 0.465619
\(185\) 0 0
\(186\) −22.1770 −1.62609
\(187\) −8.58651 −0.627907
\(188\) −20.9056 −1.52470
\(189\) 0.159610 0.0116099
\(190\) 0 0
\(191\) 15.8873 1.14956 0.574782 0.818307i \(-0.305087\pi\)
0.574782 + 0.818307i \(0.305087\pi\)
\(192\) 0.656548 0.0473822
\(193\) 21.1919 1.52543 0.762714 0.646736i \(-0.223866\pi\)
0.762714 + 0.646736i \(0.223866\pi\)
\(194\) 4.29740 0.308535
\(195\) 0 0
\(196\) 4.48096 0.320069
\(197\) −18.5146 −1.31911 −0.659554 0.751657i \(-0.729255\pi\)
−0.659554 + 0.751657i \(0.729255\pi\)
\(198\) −36.6783 −2.60661
\(199\) 12.3192 0.873286 0.436643 0.899635i \(-0.356167\pi\)
0.436643 + 0.899635i \(0.356167\pi\)
\(200\) 0 0
\(201\) −29.3604 −2.07092
\(202\) 7.96344 0.560306
\(203\) 9.98866 0.701066
\(204\) −20.1563 −1.41122
\(205\) 0 0
\(206\) −2.64271 −0.184126
\(207\) 3.06481 0.213019
\(208\) 16.5529 1.14773
\(209\) −33.3370 −2.30597
\(210\) 0 0
\(211\) −3.49259 −0.240440 −0.120220 0.992747i \(-0.538360\pi\)
−0.120220 + 0.992747i \(0.538360\pi\)
\(212\) 4.31035 0.296036
\(213\) −11.3273 −0.776134
\(214\) −20.9254 −1.43043
\(215\) 0 0
\(216\) −1.00809 −0.0685917
\(217\) −3.53732 −0.240129
\(218\) 17.8165 1.20669
\(219\) −18.7116 −1.26441
\(220\) 0 0
\(221\) 4.24817 0.285763
\(222\) −1.04185 −0.0699244
\(223\) −9.77808 −0.654789 −0.327394 0.944888i \(-0.606171\pi\)
−0.327394 + 0.944888i \(0.606171\pi\)
\(224\) −5.48658 −0.366588
\(225\) 0 0
\(226\) −6.75032 −0.449024
\(227\) −8.25924 −0.548185 −0.274093 0.961703i \(-0.588378\pi\)
−0.274093 + 0.961703i \(0.588378\pi\)
\(228\) −78.2566 −5.18267
\(229\) 3.13841 0.207392 0.103696 0.994609i \(-0.466933\pi\)
0.103696 + 0.994609i \(0.466933\pi\)
\(230\) 0 0
\(231\) −11.5769 −0.761707
\(232\) −63.0881 −4.14193
\(233\) −8.84601 −0.579521 −0.289761 0.957099i \(-0.593576\pi\)
−0.289761 + 0.957099i \(0.593576\pi\)
\(234\) 18.1465 1.18628
\(235\) 0 0
\(236\) 62.1958 4.04860
\(237\) −14.1334 −0.918063
\(238\) −4.64998 −0.301413
\(239\) −10.3793 −0.671379 −0.335689 0.941973i \(-0.608969\pi\)
−0.335689 + 0.941973i \(0.608969\pi\)
\(240\) 0 0
\(241\) −2.47813 −0.159630 −0.0798151 0.996810i \(-0.525433\pi\)
−0.0798151 + 0.996810i \(0.525433\pi\)
\(242\) 28.2553 1.81632
\(243\) 22.1538 1.42117
\(244\) 4.27776 0.273855
\(245\) 0 0
\(246\) −45.4607 −2.89847
\(247\) 16.4934 1.04945
\(248\) 22.3416 1.41869
\(249\) −13.7343 −0.870373
\(250\) 0 0
\(251\) −9.66697 −0.610174 −0.305087 0.952324i \(-0.598686\pi\)
−0.305087 + 0.952324i \(0.598686\pi\)
\(252\) −13.7333 −0.865117
\(253\) −4.70095 −0.295546
\(254\) 2.15816 0.135415
\(255\) 0 0
\(256\) −29.1298 −1.82061
\(257\) −16.2773 −1.01535 −0.507676 0.861548i \(-0.669495\pi\)
−0.507676 + 0.861548i \(0.669495\pi\)
\(258\) −60.0420 −3.73805
\(259\) −0.166179 −0.0103259
\(260\) 0 0
\(261\) −30.6134 −1.89492
\(262\) −8.67522 −0.535957
\(263\) 9.03331 0.557017 0.278509 0.960434i \(-0.410160\pi\)
0.278509 + 0.960434i \(0.410160\pi\)
\(264\) 73.1196 4.50020
\(265\) 0 0
\(266\) −18.0535 −1.10693
\(267\) 27.6521 1.69228
\(268\) 53.4226 3.26330
\(269\) −17.6238 −1.07454 −0.537272 0.843409i \(-0.680545\pi\)
−0.537272 + 0.843409i \(0.680545\pi\)
\(270\) 0 0
\(271\) 23.9935 1.45750 0.728750 0.684780i \(-0.240102\pi\)
0.728750 + 0.684780i \(0.240102\pi\)
\(272\) 12.9997 0.788224
\(273\) 5.72768 0.346655
\(274\) 36.6827 2.21608
\(275\) 0 0
\(276\) −11.0352 −0.664241
\(277\) −16.5641 −0.995239 −0.497620 0.867395i \(-0.665792\pi\)
−0.497620 + 0.867395i \(0.665792\pi\)
\(278\) 6.63645 0.398028
\(279\) 10.8412 0.649046
\(280\) 0 0
\(281\) −19.5228 −1.16463 −0.582316 0.812962i \(-0.697854\pi\)
−0.582316 + 0.812962i \(0.697854\pi\)
\(282\) 29.2495 1.74179
\(283\) 15.7324 0.935191 0.467596 0.883942i \(-0.345120\pi\)
0.467596 + 0.883942i \(0.345120\pi\)
\(284\) 20.6105 1.22301
\(285\) 0 0
\(286\) −27.8340 −1.64586
\(287\) −7.25116 −0.428022
\(288\) 16.8153 0.990853
\(289\) −13.6637 −0.803748
\(290\) 0 0
\(291\) −4.15714 −0.243696
\(292\) 34.0466 1.99243
\(293\) 8.16268 0.476869 0.238434 0.971159i \(-0.423366\pi\)
0.238434 + 0.971159i \(0.423366\pi\)
\(294\) −6.26943 −0.365641
\(295\) 0 0
\(296\) 1.04958 0.0610058
\(297\) 0.750316 0.0435377
\(298\) −0.0225869 −0.00130843
\(299\) 2.32579 0.134504
\(300\) 0 0
\(301\) −9.57695 −0.552006
\(302\) 33.3674 1.92008
\(303\) −7.70353 −0.442556
\(304\) 50.4712 2.89472
\(305\) 0 0
\(306\) 14.2513 0.814693
\(307\) −30.5542 −1.74382 −0.871910 0.489667i \(-0.837118\pi\)
−0.871910 + 0.489667i \(0.837118\pi\)
\(308\) 21.0648 1.20028
\(309\) 2.55645 0.145431
\(310\) 0 0
\(311\) 25.3573 1.43788 0.718941 0.695071i \(-0.244627\pi\)
0.718941 + 0.695071i \(0.244627\pi\)
\(312\) −36.1759 −2.04805
\(313\) 6.15654 0.347988 0.173994 0.984747i \(-0.444333\pi\)
0.173994 + 0.984747i \(0.444333\pi\)
\(314\) 0.635433 0.0358595
\(315\) 0 0
\(316\) 25.7163 1.44666
\(317\) 4.48063 0.251657 0.125829 0.992052i \(-0.459841\pi\)
0.125829 + 0.992052i \(0.459841\pi\)
\(318\) −6.03072 −0.338186
\(319\) 46.9562 2.62904
\(320\) 0 0
\(321\) 20.2424 1.12982
\(322\) −2.54577 −0.141870
\(323\) 12.9531 0.720727
\(324\) −39.4386 −2.19103
\(325\) 0 0
\(326\) 0.383743 0.0212535
\(327\) −17.2350 −0.953100
\(328\) 45.7981 2.52878
\(329\) 4.66542 0.257213
\(330\) 0 0
\(331\) −1.62894 −0.0895349 −0.0447674 0.998997i \(-0.514255\pi\)
−0.0447674 + 0.998997i \(0.514255\pi\)
\(332\) 24.9901 1.37151
\(333\) 0.509308 0.0279099
\(334\) −30.1000 −1.64700
\(335\) 0 0
\(336\) 17.5272 0.956185
\(337\) −4.91650 −0.267819 −0.133909 0.990994i \(-0.542753\pi\)
−0.133909 + 0.990994i \(0.542753\pi\)
\(338\) −19.3242 −1.05110
\(339\) 6.53000 0.354661
\(340\) 0 0
\(341\) −16.6287 −0.900497
\(342\) 55.3305 2.99193
\(343\) −1.00000 −0.0539949
\(344\) 60.4877 3.26128
\(345\) 0 0
\(346\) −17.2634 −0.928086
\(347\) −33.8004 −1.81450 −0.907249 0.420593i \(-0.861822\pi\)
−0.907249 + 0.420593i \(0.861822\pi\)
\(348\) 110.227 5.90878
\(349\) 3.26430 0.174734 0.0873669 0.996176i \(-0.472155\pi\)
0.0873669 + 0.996176i \(0.472155\pi\)
\(350\) 0 0
\(351\) −0.371218 −0.0198142
\(352\) −25.7921 −1.37473
\(353\) −11.1070 −0.591165 −0.295583 0.955317i \(-0.595514\pi\)
−0.295583 + 0.955317i \(0.595514\pi\)
\(354\) −87.0199 −4.62505
\(355\) 0 0
\(356\) −50.3142 −2.66665
\(357\) 4.49821 0.238071
\(358\) 52.3471 2.76663
\(359\) −19.0760 −1.00679 −0.503397 0.864056i \(-0.667916\pi\)
−0.503397 + 0.864056i \(0.667916\pi\)
\(360\) 0 0
\(361\) 31.2900 1.64684
\(362\) −4.31999 −0.227054
\(363\) −27.3331 −1.43461
\(364\) −10.4218 −0.546249
\(365\) 0 0
\(366\) −5.98513 −0.312848
\(367\) 31.9319 1.66683 0.833416 0.552647i \(-0.186382\pi\)
0.833416 + 0.552647i \(0.186382\pi\)
\(368\) 7.11710 0.371004
\(369\) 22.2234 1.15691
\(370\) 0 0
\(371\) −0.961924 −0.0499406
\(372\) −39.0350 −2.02387
\(373\) −20.8038 −1.07718 −0.538590 0.842568i \(-0.681043\pi\)
−0.538590 + 0.842568i \(0.681043\pi\)
\(374\) −21.8593 −1.13032
\(375\) 0 0
\(376\) −29.4666 −1.51963
\(377\) −23.2315 −1.19648
\(378\) 0.406330 0.0208993
\(379\) −26.5314 −1.36282 −0.681412 0.731900i \(-0.738634\pi\)
−0.681412 + 0.731900i \(0.738634\pi\)
\(380\) 0 0
\(381\) −2.08773 −0.106957
\(382\) 40.4454 2.06937
\(383\) 20.4224 1.04354 0.521768 0.853088i \(-0.325273\pi\)
0.521768 + 0.853088i \(0.325273\pi\)
\(384\) 28.6949 1.46433
\(385\) 0 0
\(386\) 53.9498 2.74597
\(387\) 29.3515 1.49202
\(388\) 7.56410 0.384009
\(389\) 1.10846 0.0562012 0.0281006 0.999605i \(-0.491054\pi\)
0.0281006 + 0.999605i \(0.491054\pi\)
\(390\) 0 0
\(391\) 1.82655 0.0923725
\(392\) 6.31597 0.319005
\(393\) 8.39207 0.423324
\(394\) −47.1339 −2.37457
\(395\) 0 0
\(396\) −64.5595 −3.24424
\(397\) 19.3304 0.970166 0.485083 0.874468i \(-0.338789\pi\)
0.485083 + 0.874468i \(0.338789\pi\)
\(398\) 31.3619 1.57203
\(399\) 17.4642 0.874305
\(400\) 0 0
\(401\) 34.3076 1.71324 0.856620 0.515947i \(-0.172560\pi\)
0.856620 + 0.515947i \(0.172560\pi\)
\(402\) −74.7449 −3.72794
\(403\) 8.22705 0.409819
\(404\) 14.0169 0.697368
\(405\) 0 0
\(406\) 25.4289 1.26201
\(407\) −0.781200 −0.0387227
\(408\) −28.4106 −1.40653
\(409\) −11.1492 −0.551293 −0.275647 0.961259i \(-0.588892\pi\)
−0.275647 + 0.961259i \(0.588892\pi\)
\(410\) 0 0
\(411\) −35.4854 −1.75037
\(412\) −4.65158 −0.229167
\(413\) −13.8800 −0.682991
\(414\) 7.80231 0.383463
\(415\) 0 0
\(416\) 12.7606 0.625642
\(417\) −6.41985 −0.314381
\(418\) −84.8684 −4.15105
\(419\) −36.8881 −1.80210 −0.901052 0.433711i \(-0.857204\pi\)
−0.901052 + 0.433711i \(0.857204\pi\)
\(420\) 0 0
\(421\) 21.8055 1.06273 0.531367 0.847142i \(-0.321679\pi\)
0.531367 + 0.847142i \(0.321679\pi\)
\(422\) −8.89134 −0.432824
\(423\) −14.2986 −0.695223
\(424\) 6.07548 0.295052
\(425\) 0 0
\(426\) −28.8368 −1.39715
\(427\) −0.954652 −0.0461988
\(428\) −36.8319 −1.78034
\(429\) 26.9255 1.29998
\(430\) 0 0
\(431\) 17.1450 0.825846 0.412923 0.910766i \(-0.364508\pi\)
0.412923 + 0.910766i \(0.364508\pi\)
\(432\) −1.13596 −0.0546537
\(433\) −2.68890 −0.129220 −0.0646102 0.997911i \(-0.520580\pi\)
−0.0646102 + 0.997911i \(0.520580\pi\)
\(434\) −9.00521 −0.432264
\(435\) 0 0
\(436\) 31.3599 1.50187
\(437\) 7.09155 0.339235
\(438\) −47.6355 −2.27611
\(439\) 23.9686 1.14396 0.571979 0.820268i \(-0.306176\pi\)
0.571979 + 0.820268i \(0.306176\pi\)
\(440\) 0 0
\(441\) 3.06481 0.145943
\(442\) 10.8149 0.514411
\(443\) 28.5038 1.35426 0.677128 0.735865i \(-0.263224\pi\)
0.677128 + 0.735865i \(0.263224\pi\)
\(444\) −1.83382 −0.0870293
\(445\) 0 0
\(446\) −24.8928 −1.17871
\(447\) 0.0218497 0.00103346
\(448\) 0.266598 0.0125956
\(449\) 1.18398 0.0558753 0.0279376 0.999610i \(-0.491106\pi\)
0.0279376 + 0.999610i \(0.491106\pi\)
\(450\) 0 0
\(451\) −34.0873 −1.60511
\(452\) −11.8816 −0.558864
\(453\) −32.2784 −1.51657
\(454\) −21.0262 −0.986807
\(455\) 0 0
\(456\) −110.304 −5.16544
\(457\) −33.3173 −1.55852 −0.779260 0.626701i \(-0.784405\pi\)
−0.779260 + 0.626701i \(0.784405\pi\)
\(458\) 7.98969 0.373334
\(459\) −0.291534 −0.0136077
\(460\) 0 0
\(461\) −1.59002 −0.0740545 −0.0370273 0.999314i \(-0.511789\pi\)
−0.0370273 + 0.999314i \(0.511789\pi\)
\(462\) −29.4723 −1.37117
\(463\) −30.4569 −1.41545 −0.707726 0.706487i \(-0.750279\pi\)
−0.707726 + 0.706487i \(0.750279\pi\)
\(464\) −71.0903 −3.30028
\(465\) 0 0
\(466\) −22.5199 −1.04322
\(467\) −27.6450 −1.27926 −0.639628 0.768685i \(-0.720912\pi\)
−0.639628 + 0.768685i \(0.720912\pi\)
\(468\) 31.9408 1.47646
\(469\) −11.9221 −0.550512
\(470\) 0 0
\(471\) −0.614693 −0.0283236
\(472\) 87.6658 4.03514
\(473\) −45.0207 −2.07005
\(474\) −35.9804 −1.65264
\(475\) 0 0
\(476\) −8.18469 −0.375145
\(477\) 2.94812 0.134985
\(478\) −26.4232 −1.20857
\(479\) 36.3245 1.65971 0.829855 0.557979i \(-0.188423\pi\)
0.829855 + 0.557979i \(0.188423\pi\)
\(480\) 0 0
\(481\) 0.386498 0.0176228
\(482\) −6.30875 −0.287356
\(483\) 2.46268 0.112056
\(484\) 49.7338 2.26063
\(485\) 0 0
\(486\) 56.3986 2.55829
\(487\) −31.6204 −1.43286 −0.716428 0.697661i \(-0.754224\pi\)
−0.716428 + 0.697661i \(0.754224\pi\)
\(488\) 6.02955 0.272945
\(489\) −0.371218 −0.0167871
\(490\) 0 0
\(491\) −41.6773 −1.88087 −0.940436 0.339972i \(-0.889582\pi\)
−0.940436 + 0.339972i \(0.889582\pi\)
\(492\) −80.0179 −3.60749
\(493\) −18.2448 −0.821703
\(494\) 41.9886 1.88915
\(495\) 0 0
\(496\) 25.1754 1.13041
\(497\) −4.59958 −0.206319
\(498\) −34.9643 −1.56679
\(499\) −27.7258 −1.24118 −0.620588 0.784137i \(-0.713106\pi\)
−0.620588 + 0.784137i \(0.713106\pi\)
\(500\) 0 0
\(501\) 29.1176 1.30088
\(502\) −24.6099 −1.09839
\(503\) 29.0492 1.29524 0.647620 0.761963i \(-0.275764\pi\)
0.647620 + 0.761963i \(0.275764\pi\)
\(504\) −19.3572 −0.862240
\(505\) 0 0
\(506\) −11.9675 −0.532022
\(507\) 18.6935 0.830208
\(508\) 3.79871 0.168540
\(509\) −8.20421 −0.363645 −0.181823 0.983331i \(-0.558200\pi\)
−0.181823 + 0.983331i \(0.558200\pi\)
\(510\) 0 0
\(511\) −7.59806 −0.336118
\(512\) −50.8542 −2.24746
\(513\) −1.13188 −0.0499737
\(514\) −41.4384 −1.82777
\(515\) 0 0
\(516\) −105.683 −4.65245
\(517\) 21.9319 0.964563
\(518\) −0.423055 −0.0185880
\(519\) 16.7000 0.733046
\(520\) 0 0
\(521\) −9.59168 −0.420219 −0.210110 0.977678i \(-0.567382\pi\)
−0.210110 + 0.977678i \(0.567382\pi\)
\(522\) −77.9347 −3.41111
\(523\) 6.06424 0.265171 0.132585 0.991172i \(-0.457672\pi\)
0.132585 + 0.991172i \(0.457672\pi\)
\(524\) −15.2697 −0.667062
\(525\) 0 0
\(526\) 22.9967 1.00271
\(527\) 6.46108 0.281449
\(528\) 82.3942 3.58575
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 42.5396 1.84606
\(532\) −31.7770 −1.37771
\(533\) 16.8647 0.730489
\(534\) 70.3960 3.04633
\(535\) 0 0
\(536\) 75.2997 3.25245
\(537\) −50.6386 −2.18522
\(538\) −44.8663 −1.93432
\(539\) −4.70095 −0.202484
\(540\) 0 0
\(541\) −28.6760 −1.23288 −0.616438 0.787403i \(-0.711425\pi\)
−0.616438 + 0.787403i \(0.711425\pi\)
\(542\) 61.0819 2.62369
\(543\) 4.17899 0.179338
\(544\) 10.0215 0.429669
\(545\) 0 0
\(546\) 14.5814 0.624026
\(547\) 24.5492 1.04965 0.524824 0.851211i \(-0.324131\pi\)
0.524824 + 0.851211i \(0.324131\pi\)
\(548\) 64.5673 2.75818
\(549\) 2.92583 0.124871
\(550\) 0 0
\(551\) −70.8350 −3.01767
\(552\) −15.5542 −0.662032
\(553\) −5.73902 −0.244048
\(554\) −42.1684 −1.79156
\(555\) 0 0
\(556\) 11.6812 0.495393
\(557\) 5.82872 0.246971 0.123485 0.992346i \(-0.460593\pi\)
0.123485 + 0.992346i \(0.460593\pi\)
\(558\) 27.5993 1.16837
\(559\) 22.2740 0.942088
\(560\) 0 0
\(561\) 21.1458 0.892778
\(562\) −49.7006 −2.09649
\(563\) 30.9619 1.30489 0.652445 0.757836i \(-0.273743\pi\)
0.652445 + 0.757836i \(0.273743\pi\)
\(564\) 51.4838 2.16786
\(565\) 0 0
\(566\) 40.0510 1.68347
\(567\) 8.80137 0.369623
\(568\) 29.0508 1.21894
\(569\) 6.77249 0.283918 0.141959 0.989873i \(-0.454660\pi\)
0.141959 + 0.989873i \(0.454660\pi\)
\(570\) 0 0
\(571\) −26.7220 −1.11828 −0.559140 0.829073i \(-0.688868\pi\)
−0.559140 + 0.829073i \(0.688868\pi\)
\(572\) −48.9922 −2.04847
\(573\) −39.1254 −1.63449
\(574\) −18.4598 −0.770497
\(575\) 0 0
\(576\) −0.817074 −0.0340447
\(577\) 3.13787 0.130631 0.0653157 0.997865i \(-0.479195\pi\)
0.0653157 + 0.997865i \(0.479195\pi\)
\(578\) −34.7847 −1.44685
\(579\) −52.1890 −2.16890
\(580\) 0 0
\(581\) −5.57695 −0.231371
\(582\) −10.5831 −0.438685
\(583\) −4.52196 −0.187280
\(584\) 47.9891 1.98580
\(585\) 0 0
\(586\) 20.7803 0.858428
\(587\) −31.4784 −1.29925 −0.649626 0.760254i \(-0.725075\pi\)
−0.649626 + 0.760254i \(0.725075\pi\)
\(588\) −11.0352 −0.455084
\(589\) 25.0850 1.03361
\(590\) 0 0
\(591\) 45.5955 1.87555
\(592\) 1.18271 0.0486093
\(593\) −15.4196 −0.633209 −0.316604 0.948558i \(-0.602543\pi\)
−0.316604 + 0.948558i \(0.602543\pi\)
\(594\) 1.91013 0.0783738
\(595\) 0 0
\(596\) −0.0397566 −0.00162849
\(597\) −30.3383 −1.24167
\(598\) 5.92093 0.242125
\(599\) −18.7196 −0.764862 −0.382431 0.923984i \(-0.624913\pi\)
−0.382431 + 0.923984i \(0.624913\pi\)
\(600\) 0 0
\(601\) 12.5523 0.512017 0.256009 0.966675i \(-0.417592\pi\)
0.256009 + 0.966675i \(0.417592\pi\)
\(602\) −24.3807 −0.993684
\(603\) 36.5390 1.48798
\(604\) 58.7319 2.38977
\(605\) 0 0
\(606\) −19.6114 −0.796660
\(607\) −20.5171 −0.832761 −0.416381 0.909190i \(-0.636702\pi\)
−0.416381 + 0.909190i \(0.636702\pi\)
\(608\) 38.9084 1.57794
\(609\) −24.5989 −0.996798
\(610\) 0 0
\(611\) −10.8508 −0.438976
\(612\) 25.0845 1.01398
\(613\) 24.9165 1.00637 0.503184 0.864179i \(-0.332162\pi\)
0.503184 + 0.864179i \(0.332162\pi\)
\(614\) −77.7840 −3.13911
\(615\) 0 0
\(616\) 29.6910 1.19629
\(617\) 8.77553 0.353289 0.176645 0.984275i \(-0.443476\pi\)
0.176645 + 0.984275i \(0.443476\pi\)
\(618\) 6.50815 0.261796
\(619\) −14.6150 −0.587427 −0.293714 0.955893i \(-0.594891\pi\)
−0.293714 + 0.955893i \(0.594891\pi\)
\(620\) 0 0
\(621\) −0.159610 −0.00640491
\(622\) 64.5540 2.58838
\(623\) 11.2284 0.449858
\(624\) −40.7645 −1.63189
\(625\) 0 0
\(626\) 15.6731 0.626425
\(627\) 82.0984 3.27870
\(628\) 1.11846 0.0446315
\(629\) 0.303535 0.0121027
\(630\) 0 0
\(631\) 18.5826 0.739760 0.369880 0.929079i \(-0.379399\pi\)
0.369880 + 0.929079i \(0.379399\pi\)
\(632\) 36.2475 1.44185
\(633\) 8.60114 0.341865
\(634\) 11.4067 0.453016
\(635\) 0 0
\(636\) −10.6150 −0.420913
\(637\) 2.32579 0.0921511
\(638\) 119.540 4.73262
\(639\) 14.0968 0.557662
\(640\) 0 0
\(641\) 9.57505 0.378192 0.189096 0.981959i \(-0.439444\pi\)
0.189096 + 0.981959i \(0.439444\pi\)
\(642\) 51.5326 2.03383
\(643\) −23.7204 −0.935443 −0.467722 0.883876i \(-0.654925\pi\)
−0.467722 + 0.883876i \(0.654925\pi\)
\(644\) −4.48096 −0.176575
\(645\) 0 0
\(646\) 32.9755 1.29741
\(647\) 20.1385 0.791727 0.395864 0.918309i \(-0.370445\pi\)
0.395864 + 0.918309i \(0.370445\pi\)
\(648\) −55.5891 −2.18375
\(649\) −65.2492 −2.56126
\(650\) 0 0
\(651\) 8.71129 0.341422
\(652\) 0.675448 0.0264526
\(653\) 32.6863 1.27911 0.639557 0.768744i \(-0.279118\pi\)
0.639557 + 0.768744i \(0.279118\pi\)
\(654\) −43.8765 −1.71571
\(655\) 0 0
\(656\) 51.6072 2.01492
\(657\) 23.2866 0.908498
\(658\) 11.8771 0.463018
\(659\) 37.3251 1.45398 0.726989 0.686649i \(-0.240919\pi\)
0.726989 + 0.686649i \(0.240919\pi\)
\(660\) 0 0
\(661\) −9.68312 −0.376630 −0.188315 0.982109i \(-0.560303\pi\)
−0.188315 + 0.982109i \(0.560303\pi\)
\(662\) −4.14692 −0.161175
\(663\) −10.4619 −0.406306
\(664\) 35.2238 1.36695
\(665\) 0 0
\(666\) 1.29658 0.0502416
\(667\) −9.98866 −0.386762
\(668\) −52.9807 −2.04988
\(669\) 24.0803 0.930999
\(670\) 0 0
\(671\) −4.48777 −0.173248
\(672\) 13.5117 0.521226
\(673\) −38.5622 −1.48646 −0.743232 0.669034i \(-0.766708\pi\)
−0.743232 + 0.669034i \(0.766708\pi\)
\(674\) −12.5163 −0.482109
\(675\) 0 0
\(676\) −34.0137 −1.30822
\(677\) −2.70428 −0.103934 −0.0519669 0.998649i \(-0.516549\pi\)
−0.0519669 + 0.998649i \(0.516549\pi\)
\(678\) 16.6239 0.638437
\(679\) −1.68805 −0.0647815
\(680\) 0 0
\(681\) 20.3399 0.779427
\(682\) −42.3330 −1.62101
\(683\) 10.6222 0.406446 0.203223 0.979132i \(-0.434858\pi\)
0.203223 + 0.979132i \(0.434858\pi\)
\(684\) 97.3904 3.72381
\(685\) 0 0
\(686\) −2.54577 −0.0971981
\(687\) −7.72892 −0.294877
\(688\) 68.1601 2.59858
\(689\) 2.23723 0.0852318
\(690\) 0 0
\(691\) 23.7327 0.902833 0.451416 0.892313i \(-0.350919\pi\)
0.451416 + 0.892313i \(0.350919\pi\)
\(692\) −30.3863 −1.15511
\(693\) 14.4075 0.547296
\(694\) −86.0481 −3.26634
\(695\) 0 0
\(696\) 155.366 5.88913
\(697\) 13.2446 0.501674
\(698\) 8.31016 0.314544
\(699\) 21.7849 0.823982
\(700\) 0 0
\(701\) 22.5615 0.852138 0.426069 0.904691i \(-0.359898\pi\)
0.426069 + 0.904691i \(0.359898\pi\)
\(702\) −0.945037 −0.0356681
\(703\) 1.17847 0.0444468
\(704\) 1.25327 0.0472342
\(705\) 0 0
\(706\) −28.2759 −1.06418
\(707\) −3.12810 −0.117644
\(708\) −153.169 −5.75643
\(709\) −16.7030 −0.627294 −0.313647 0.949540i \(-0.601551\pi\)
−0.313647 + 0.949540i \(0.601551\pi\)
\(710\) 0 0
\(711\) 17.5890 0.659640
\(712\) −70.9185 −2.65778
\(713\) 3.53732 0.132474
\(714\) 11.4514 0.428559
\(715\) 0 0
\(716\) 92.1391 3.44340
\(717\) 25.5608 0.954587
\(718\) −48.5632 −1.81236
\(719\) 27.7469 1.03479 0.517393 0.855748i \(-0.326903\pi\)
0.517393 + 0.855748i \(0.326903\pi\)
\(720\) 0 0
\(721\) 1.03808 0.0386600
\(722\) 79.6573 2.96454
\(723\) 6.10284 0.226967
\(724\) −7.60386 −0.282595
\(725\) 0 0
\(726\) −69.5838 −2.58250
\(727\) 41.4981 1.53908 0.769539 0.638600i \(-0.220486\pi\)
0.769539 + 0.638600i \(0.220486\pi\)
\(728\) −14.6896 −0.544433
\(729\) −28.1537 −1.04273
\(730\) 0 0
\(731\) 17.4928 0.646993
\(732\) −10.5348 −0.389376
\(733\) −35.7212 −1.31939 −0.659697 0.751532i \(-0.729315\pi\)
−0.659697 + 0.751532i \(0.729315\pi\)
\(734\) 81.2914 3.00052
\(735\) 0 0
\(736\) 5.48658 0.202238
\(737\) −56.0452 −2.06445
\(738\) 56.5758 2.08258
\(739\) −44.4520 −1.63519 −0.817597 0.575791i \(-0.804694\pi\)
−0.817597 + 0.575791i \(0.804694\pi\)
\(740\) 0 0
\(741\) −40.6181 −1.49214
\(742\) −2.44884 −0.0898998
\(743\) 17.0898 0.626964 0.313482 0.949594i \(-0.398504\pi\)
0.313482 + 0.949594i \(0.398504\pi\)
\(744\) −55.0202 −2.01714
\(745\) 0 0
\(746\) −52.9617 −1.93907
\(747\) 17.0923 0.625374
\(748\) −38.4758 −1.40681
\(749\) 8.21965 0.300339
\(750\) 0 0
\(751\) −40.9748 −1.49519 −0.747596 0.664154i \(-0.768792\pi\)
−0.747596 + 0.664154i \(0.768792\pi\)
\(752\) −33.2042 −1.21083
\(753\) 23.8067 0.867564
\(754\) −59.1422 −2.15383
\(755\) 0 0
\(756\) 0.715204 0.0260117
\(757\) 23.7095 0.861737 0.430868 0.902415i \(-0.358207\pi\)
0.430868 + 0.902415i \(0.358207\pi\)
\(758\) −67.5429 −2.45327
\(759\) 11.5769 0.420216
\(760\) 0 0
\(761\) −42.7680 −1.55034 −0.775170 0.631753i \(-0.782336\pi\)
−0.775170 + 0.631753i \(0.782336\pi\)
\(762\) −5.31488 −0.192538
\(763\) −6.99848 −0.253362
\(764\) 71.1904 2.57558
\(765\) 0 0
\(766\) 51.9908 1.87850
\(767\) 32.2820 1.16564
\(768\) 71.7375 2.58860
\(769\) 35.6281 1.28478 0.642392 0.766376i \(-0.277942\pi\)
0.642392 + 0.766376i \(0.277942\pi\)
\(770\) 0 0
\(771\) 40.0859 1.44366
\(772\) 94.9602 3.41769
\(773\) −16.9388 −0.609246 −0.304623 0.952473i \(-0.598530\pi\)
−0.304623 + 0.952473i \(0.598530\pi\)
\(774\) 74.7223 2.68584
\(775\) 0 0
\(776\) 10.6617 0.382732
\(777\) 0.409247 0.0146817
\(778\) 2.82189 0.101170
\(779\) 51.4219 1.84238
\(780\) 0 0
\(781\) −21.6224 −0.773709
\(782\) 4.64998 0.166283
\(783\) 1.59429 0.0569751
\(784\) 7.11710 0.254182
\(785\) 0 0
\(786\) 21.3643 0.762040
\(787\) 15.4748 0.551619 0.275809 0.961212i \(-0.411054\pi\)
0.275809 + 0.961212i \(0.411054\pi\)
\(788\) −82.9630 −2.95544
\(789\) −22.2462 −0.791985
\(790\) 0 0
\(791\) 2.65158 0.0942792
\(792\) −90.9974 −3.23345
\(793\) 2.22032 0.0788458
\(794\) 49.2109 1.74643
\(795\) 0 0
\(796\) 55.2020 1.95658
\(797\) −28.1935 −0.998664 −0.499332 0.866411i \(-0.666421\pi\)
−0.499332 + 0.866411i \(0.666421\pi\)
\(798\) 44.4600 1.57387
\(799\) −8.52161 −0.301473
\(800\) 0 0
\(801\) −34.4131 −1.21593
\(802\) 87.3394 3.08406
\(803\) −35.7181 −1.26046
\(804\) −131.563 −4.63986
\(805\) 0 0
\(806\) 20.9442 0.737728
\(807\) 43.4020 1.52782
\(808\) 19.7570 0.695049
\(809\) −30.7565 −1.08134 −0.540670 0.841235i \(-0.681829\pi\)
−0.540670 + 0.841235i \(0.681829\pi\)
\(810\) 0 0
\(811\) 1.27477 0.0447632 0.0223816 0.999750i \(-0.492875\pi\)
0.0223816 + 0.999750i \(0.492875\pi\)
\(812\) 44.7588 1.57073
\(813\) −59.0883 −2.07232
\(814\) −1.98876 −0.0697059
\(815\) 0 0
\(816\) −32.0142 −1.12072
\(817\) 67.9154 2.37606
\(818\) −28.3834 −0.992402
\(819\) −7.12810 −0.249076
\(820\) 0 0
\(821\) −5.88860 −0.205513 −0.102757 0.994707i \(-0.532766\pi\)
−0.102757 + 0.994707i \(0.532766\pi\)
\(822\) −90.3378 −3.15089
\(823\) −13.3985 −0.467043 −0.233522 0.972352i \(-0.575025\pi\)
−0.233522 + 0.972352i \(0.575025\pi\)
\(824\) −6.55645 −0.228405
\(825\) 0 0
\(826\) −35.3354 −1.22948
\(827\) −30.5493 −1.06230 −0.531151 0.847277i \(-0.678240\pi\)
−0.531151 + 0.847277i \(0.678240\pi\)
\(828\) 13.7333 0.477265
\(829\) −33.2608 −1.15520 −0.577598 0.816321i \(-0.696010\pi\)
−0.577598 + 0.816321i \(0.696010\pi\)
\(830\) 0 0
\(831\) 40.7921 1.41506
\(832\) −0.620052 −0.0214964
\(833\) 1.82655 0.0632861
\(834\) −16.3435 −0.565929
\(835\) 0 0
\(836\) −149.382 −5.16648
\(837\) −0.564589 −0.0195151
\(838\) −93.9089 −3.24403
\(839\) 19.4516 0.671543 0.335772 0.941943i \(-0.391003\pi\)
0.335772 + 0.941943i \(0.391003\pi\)
\(840\) 0 0
\(841\) 70.7733 2.44046
\(842\) 55.5118 1.91306
\(843\) 48.0785 1.65591
\(844\) −15.6502 −0.538701
\(845\) 0 0
\(846\) −36.4011 −1.25149
\(847\) −11.0989 −0.381363
\(848\) 6.84611 0.235096
\(849\) −38.7438 −1.32968
\(850\) 0 0
\(851\) 0.166179 0.00569655
\(852\) −50.7572 −1.73891
\(853\) 43.9900 1.50619 0.753094 0.657913i \(-0.228561\pi\)
0.753094 + 0.657913i \(0.228561\pi\)
\(854\) −2.43033 −0.0831641
\(855\) 0 0
\(856\) −51.9150 −1.77442
\(857\) 50.9265 1.73962 0.869808 0.493391i \(-0.164243\pi\)
0.869808 + 0.493391i \(0.164243\pi\)
\(858\) 68.5463 2.34013
\(859\) 19.0473 0.649885 0.324943 0.945734i \(-0.394655\pi\)
0.324943 + 0.945734i \(0.394655\pi\)
\(860\) 0 0
\(861\) 17.8573 0.608575
\(862\) 43.6473 1.48663
\(863\) 30.6481 1.04327 0.521637 0.853168i \(-0.325322\pi\)
0.521637 + 0.853168i \(0.325322\pi\)
\(864\) −0.875711 −0.0297923
\(865\) 0 0
\(866\) −6.84534 −0.232614
\(867\) 33.6494 1.14279
\(868\) −15.8506 −0.538004
\(869\) −26.9788 −0.915194
\(870\) 0 0
\(871\) 27.7283 0.939538
\(872\) 44.2022 1.49687
\(873\) 5.17357 0.175099
\(874\) 18.0535 0.610668
\(875\) 0 0
\(876\) −83.8461 −2.83290
\(877\) −15.6561 −0.528668 −0.264334 0.964431i \(-0.585152\pi\)
−0.264334 + 0.964431i \(0.585152\pi\)
\(878\) 61.0186 2.05928
\(879\) −20.1021 −0.678027
\(880\) 0 0
\(881\) −51.9258 −1.74943 −0.874713 0.484642i \(-0.838950\pi\)
−0.874713 + 0.484642i \(0.838950\pi\)
\(882\) 7.80231 0.262718
\(883\) −18.1603 −0.611144 −0.305572 0.952169i \(-0.598848\pi\)
−0.305572 + 0.952169i \(0.598848\pi\)
\(884\) 19.0359 0.640246
\(885\) 0 0
\(886\) 72.5642 2.43784
\(887\) 29.5677 0.992786 0.496393 0.868098i \(-0.334658\pi\)
0.496393 + 0.868098i \(0.334658\pi\)
\(888\) −2.58479 −0.0867399
\(889\) −0.847744 −0.0284324
\(890\) 0 0
\(891\) 41.3748 1.38611
\(892\) −43.8152 −1.46704
\(893\) −33.0850 −1.10715
\(894\) 0.0556245 0.00186036
\(895\) 0 0
\(896\) 11.6519 0.389261
\(897\) −5.72768 −0.191242
\(898\) 3.01413 0.100583
\(899\) −35.3330 −1.17842
\(900\) 0 0
\(901\) 1.75700 0.0585342
\(902\) −86.7785 −2.88941
\(903\) 23.5850 0.784859
\(904\) −16.7473 −0.557006
\(905\) 0 0
\(906\) −82.1734 −2.73003
\(907\) −30.4350 −1.01058 −0.505289 0.862950i \(-0.668614\pi\)
−0.505289 + 0.862950i \(0.668614\pi\)
\(908\) −37.0094 −1.22820
\(909\) 9.58705 0.317982
\(910\) 0 0
\(911\) −48.2859 −1.59978 −0.799892 0.600144i \(-0.795110\pi\)
−0.799892 + 0.600144i \(0.795110\pi\)
\(912\) −124.295 −4.11581
\(913\) −26.2169 −0.867653
\(914\) −84.8184 −2.80554
\(915\) 0 0
\(916\) 14.0631 0.464658
\(917\) 3.40769 0.112532
\(918\) −0.742181 −0.0244956
\(919\) −28.4097 −0.937150 −0.468575 0.883424i \(-0.655232\pi\)
−0.468575 + 0.883424i \(0.655232\pi\)
\(920\) 0 0
\(921\) 75.2453 2.47942
\(922\) −4.04783 −0.133308
\(923\) 10.6976 0.352117
\(924\) −51.8759 −1.70659
\(925\) 0 0
\(926\) −77.5363 −2.54800
\(927\) −3.18151 −0.104494
\(928\) −54.8036 −1.79902
\(929\) 27.0080 0.886104 0.443052 0.896496i \(-0.353896\pi\)
0.443052 + 0.896496i \(0.353896\pi\)
\(930\) 0 0
\(931\) 7.09155 0.232416
\(932\) −39.6386 −1.29841
\(933\) −62.4471 −2.04443
\(934\) −70.3778 −2.30283
\(935\) 0 0
\(936\) 45.0209 1.47155
\(937\) −15.3639 −0.501917 −0.250958 0.967998i \(-0.580746\pi\)
−0.250958 + 0.967998i \(0.580746\pi\)
\(938\) −30.3510 −0.990995
\(939\) −15.1616 −0.494780
\(940\) 0 0
\(941\) −46.8015 −1.52569 −0.762843 0.646584i \(-0.776197\pi\)
−0.762843 + 0.646584i \(0.776197\pi\)
\(942\) −1.56487 −0.0509862
\(943\) 7.25116 0.236130
\(944\) 98.7855 3.21519
\(945\) 0 0
\(946\) −114.613 −3.72637
\(947\) −13.1326 −0.426753 −0.213376 0.976970i \(-0.568446\pi\)
−0.213376 + 0.976970i \(0.568446\pi\)
\(948\) −63.3312 −2.05690
\(949\) 17.6715 0.573641
\(950\) 0 0
\(951\) −11.0344 −0.357814
\(952\) −11.5364 −0.373897
\(953\) −3.29495 −0.106734 −0.0533670 0.998575i \(-0.516995\pi\)
−0.0533670 + 0.998575i \(0.516995\pi\)
\(954\) 7.50524 0.242991
\(955\) 0 0
\(956\) −46.5091 −1.50421
\(957\) −115.638 −3.73805
\(958\) 92.4740 2.98770
\(959\) −14.4092 −0.465299
\(960\) 0 0
\(961\) −18.4874 −0.596368
\(962\) 0.983936 0.0317234
\(963\) −25.1917 −0.811790
\(964\) −11.1044 −0.357648
\(965\) 0 0
\(966\) 6.26943 0.201716
\(967\) −1.39392 −0.0448254 −0.0224127 0.999749i \(-0.507135\pi\)
−0.0224127 + 0.999749i \(0.507135\pi\)
\(968\) 70.1003 2.25311
\(969\) −31.8993 −1.02475
\(970\) 0 0
\(971\) −10.8178 −0.347158 −0.173579 0.984820i \(-0.555533\pi\)
−0.173579 + 0.984820i \(0.555533\pi\)
\(972\) 99.2704 3.18410
\(973\) −2.60685 −0.0835718
\(974\) −80.4983 −2.57933
\(975\) 0 0
\(976\) 6.79435 0.217482
\(977\) −23.4628 −0.750642 −0.375321 0.926895i \(-0.622468\pi\)
−0.375321 + 0.926895i \(0.622468\pi\)
\(978\) −0.945037 −0.0302190
\(979\) 52.7843 1.68699
\(980\) 0 0
\(981\) 21.4490 0.684815
\(982\) −106.101 −3.38582
\(983\) −57.3568 −1.82940 −0.914700 0.404134i \(-0.867573\pi\)
−0.914700 + 0.404134i \(0.867573\pi\)
\(984\) −112.786 −3.59549
\(985\) 0 0
\(986\) −46.4470 −1.47918
\(987\) −11.4895 −0.365713
\(988\) 73.9065 2.35128
\(989\) 9.57695 0.304529
\(990\) 0 0
\(991\) 20.1069 0.638718 0.319359 0.947634i \(-0.396532\pi\)
0.319359 + 0.947634i \(0.396532\pi\)
\(992\) 19.4078 0.616198
\(993\) 4.01157 0.127303
\(994\) −11.7095 −0.371402
\(995\) 0 0
\(996\) −61.5427 −1.95005
\(997\) 12.2818 0.388970 0.194485 0.980906i \(-0.437697\pi\)
0.194485 + 0.980906i \(0.437697\pi\)
\(998\) −70.5836 −2.23428
\(999\) −0.0265238 −0.000839176 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4025.2.a.p.1.5 5
5.4 even 2 161.2.a.d.1.1 5
15.14 odd 2 1449.2.a.r.1.5 5
20.19 odd 2 2576.2.a.bd.1.2 5
35.34 odd 2 1127.2.a.h.1.1 5
115.114 odd 2 3703.2.a.j.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
161.2.a.d.1.1 5 5.4 even 2
1127.2.a.h.1.1 5 35.34 odd 2
1449.2.a.r.1.5 5 15.14 odd 2
2576.2.a.bd.1.2 5 20.19 odd 2
3703.2.a.j.1.1 5 115.114 odd 2
4025.2.a.p.1.5 5 1.1 even 1 trivial