Properties

Label 4235.2.a.u.1.4
Level $4235$
Weight $2$
Character 4235.1
Self dual yes
Analytic conductor $33.817$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.8166452560\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.15952.1
Defining polynomial: \(x^{4} - 6 x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.21161\) of defining polynomial
Character \(\chi\) \(=\) 4235.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.21161 q^{2} -1.45216 q^{3} +2.89124 q^{4} +1.00000 q^{5} -3.21161 q^{6} +1.00000 q^{7} +1.97107 q^{8} -0.891236 q^{9} +O(q^{10})\) \(q+2.21161 q^{2} -1.45216 q^{3} +2.89124 q^{4} +1.00000 q^{5} -3.21161 q^{6} +1.00000 q^{7} +1.97107 q^{8} -0.891236 q^{9} +2.21161 q^{10} -4.19853 q^{12} -6.19853 q^{13} +2.21161 q^{14} -1.45216 q^{15} -1.42323 q^{16} +6.75354 q^{17} -1.97107 q^{18} -6.55501 q^{19} +2.89124 q^{20} -1.45216 q^{21} -1.75946 q^{23} -2.86230 q^{24} +1.00000 q^{25} -13.7088 q^{26} +5.65069 q^{27} +2.89124 q^{28} +3.32754 q^{29} -3.21161 q^{30} +5.76662 q^{31} -7.08977 q^{32} +14.9362 q^{34} +1.00000 q^{35} -2.57677 q^{36} -9.70159 q^{37} -14.4971 q^{38} +9.00125 q^{39} +1.97107 q^{40} -7.98415 q^{41} -3.21161 q^{42} -5.27837 q^{43} -0.891236 q^{45} -3.89124 q^{46} -11.9493 q^{47} +2.06675 q^{48} +1.00000 q^{49} +2.21161 q^{50} -9.80721 q^{51} -17.9214 q^{52} +5.33623 q^{53} +12.4971 q^{54} +1.97107 q^{56} +9.51891 q^{57} +7.35924 q^{58} -14.1637 q^{59} -4.19853 q^{60} -10.9362 q^{61} +12.7535 q^{62} -0.891236 q^{63} -12.8334 q^{64} -6.19853 q^{65} +7.27837 q^{67} +19.5261 q^{68} +2.55501 q^{69} +2.21161 q^{70} +8.82029 q^{71} -1.75669 q^{72} -0.811402 q^{73} -21.4562 q^{74} -1.45216 q^{75} -18.9521 q^{76} +19.9073 q^{78} +4.53199 q^{79} -1.42323 q^{80} -5.53199 q^{81} -17.6579 q^{82} +0.639032 q^{83} -4.19853 q^{84} +6.75354 q^{85} -11.6737 q^{86} -4.83212 q^{87} +10.1001 q^{89} -1.97107 q^{90} -6.19853 q^{91} -5.08700 q^{92} -8.37405 q^{93} -26.4272 q^{94} -6.55501 q^{95} +10.2955 q^{96} -7.25363 q^{97} +2.21161 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9} - 8 q^{12} - 16 q^{13} - 2 q^{15} + 12 q^{16} - 2 q^{17} + 6 q^{18} - 6 q^{19} + 4 q^{20} - 2 q^{21} - 2 q^{23} + 10 q^{24} + 4 q^{25} + 2 q^{26} + 10 q^{27} + 4 q^{28} - 12 q^{29} - 4 q^{30} - 6 q^{31} - 12 q^{32} + 8 q^{34} + 4 q^{35} - 28 q^{36} - 6 q^{37} - 10 q^{38} - 12 q^{39} - 6 q^{40} - 18 q^{41} - 4 q^{42} - 6 q^{43} + 4 q^{45} - 8 q^{46} + 4 q^{47} + 2 q^{48} + 4 q^{49} - 4 q^{51} - 30 q^{52} + 34 q^{53} + 2 q^{54} - 6 q^{56} + 28 q^{57} + 32 q^{58} - 10 q^{59} - 8 q^{60} + 8 q^{61} + 22 q^{62} + 4 q^{63} - 16 q^{64} - 16 q^{65} + 14 q^{67} + 44 q^{68} - 10 q^{69} + 12 q^{72} - 2 q^{73} - 48 q^{74} - 2 q^{75} - 38 q^{76} + 14 q^{78} + 8 q^{79} + 12 q^{80} - 12 q^{81} - 34 q^{82} + 10 q^{83} - 8 q^{84} - 2 q^{85} - 24 q^{86} - 32 q^{87} + 10 q^{89} + 6 q^{90} - 16 q^{91} + 10 q^{92} - 26 q^{93} - 54 q^{94} - 6 q^{95} - 8 q^{96} - 34 q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.21161 1.56385 0.781924 0.623374i \(-0.214239\pi\)
0.781924 + 0.623374i \(0.214239\pi\)
\(3\) −1.45216 −0.838404 −0.419202 0.907893i \(-0.637690\pi\)
−0.419202 + 0.907893i \(0.637690\pi\)
\(4\) 2.89124 1.44562
\(5\) 1.00000 0.447214
\(6\) −3.21161 −1.31114
\(7\) 1.00000 0.377964
\(8\) 1.97107 0.696878
\(9\) −0.891236 −0.297079
\(10\) 2.21161 0.699374
\(11\) 0 0
\(12\) −4.19853 −1.21201
\(13\) −6.19853 −1.71916 −0.859582 0.510998i \(-0.829276\pi\)
−0.859582 + 0.510998i \(0.829276\pi\)
\(14\) 2.21161 0.591079
\(15\) −1.45216 −0.374946
\(16\) −1.42323 −0.355807
\(17\) 6.75354 1.63797 0.818987 0.573812i \(-0.194536\pi\)
0.818987 + 0.573812i \(0.194536\pi\)
\(18\) −1.97107 −0.464585
\(19\) −6.55501 −1.50382 −0.751911 0.659265i \(-0.770868\pi\)
−0.751911 + 0.659265i \(0.770868\pi\)
\(20\) 2.89124 0.646500
\(21\) −1.45216 −0.316887
\(22\) 0 0
\(23\) −1.75946 −0.366872 −0.183436 0.983032i \(-0.558722\pi\)
−0.183436 + 0.983032i \(0.558722\pi\)
\(24\) −2.86230 −0.584266
\(25\) 1.00000 0.200000
\(26\) −13.7088 −2.68851
\(27\) 5.65069 1.08748
\(28\) 2.89124 0.546392
\(29\) 3.32754 0.617910 0.308955 0.951077i \(-0.400021\pi\)
0.308955 + 0.951077i \(0.400021\pi\)
\(30\) −3.21161 −0.586358
\(31\) 5.76662 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(32\) −7.08977 −1.25331
\(33\) 0 0
\(34\) 14.9362 2.56154
\(35\) 1.00000 0.169031
\(36\) −2.57677 −0.429462
\(37\) −9.70159 −1.59493 −0.797466 0.603364i \(-0.793827\pi\)
−0.797466 + 0.603364i \(0.793827\pi\)
\(38\) −14.4971 −2.35175
\(39\) 9.00125 1.44135
\(40\) 1.97107 0.311653
\(41\) −7.98415 −1.24691 −0.623457 0.781857i \(-0.714273\pi\)
−0.623457 + 0.781857i \(0.714273\pi\)
\(42\) −3.21161 −0.495563
\(43\) −5.27837 −0.804943 −0.402472 0.915432i \(-0.631849\pi\)
−0.402472 + 0.915432i \(0.631849\pi\)
\(44\) 0 0
\(45\) −0.891236 −0.132858
\(46\) −3.89124 −0.573731
\(47\) −11.9493 −1.74299 −0.871493 0.490409i \(-0.836848\pi\)
−0.871493 + 0.490409i \(0.836848\pi\)
\(48\) 2.06675 0.298310
\(49\) 1.00000 0.142857
\(50\) 2.21161 0.312769
\(51\) −9.80721 −1.37328
\(52\) −17.9214 −2.48525
\(53\) 5.33623 0.732987 0.366494 0.930421i \(-0.380558\pi\)
0.366494 + 0.930421i \(0.380558\pi\)
\(54\) 12.4971 1.70065
\(55\) 0 0
\(56\) 1.97107 0.263395
\(57\) 9.51891 1.26081
\(58\) 7.35924 0.966316
\(59\) −14.1637 −1.84395 −0.921977 0.387244i \(-0.873427\pi\)
−0.921977 + 0.387244i \(0.873427\pi\)
\(60\) −4.19853 −0.542028
\(61\) −10.9362 −1.40024 −0.700120 0.714025i \(-0.746870\pi\)
−0.700120 + 0.714025i \(0.746870\pi\)
\(62\) 12.7535 1.61970
\(63\) −0.891236 −0.112285
\(64\) −12.8334 −1.60417
\(65\) −6.19853 −0.768833
\(66\) 0 0
\(67\) 7.27837 0.889194 0.444597 0.895731i \(-0.353347\pi\)
0.444597 + 0.895731i \(0.353347\pi\)
\(68\) 19.5261 2.36788
\(69\) 2.55501 0.307587
\(70\) 2.21161 0.264338
\(71\) 8.82029 1.04678 0.523388 0.852094i \(-0.324668\pi\)
0.523388 + 0.852094i \(0.324668\pi\)
\(72\) −1.75669 −0.207028
\(73\) −0.811402 −0.0949674 −0.0474837 0.998872i \(-0.515120\pi\)
−0.0474837 + 0.998872i \(0.515120\pi\)
\(74\) −21.4562 −2.49423
\(75\) −1.45216 −0.167681
\(76\) −18.9521 −2.17395
\(77\) 0 0
\(78\) 19.9073 2.25406
\(79\) 4.53199 0.509889 0.254944 0.966956i \(-0.417943\pi\)
0.254944 + 0.966956i \(0.417943\pi\)
\(80\) −1.42323 −0.159122
\(81\) −5.53199 −0.614666
\(82\) −17.6579 −1.94998
\(83\) 0.639032 0.0701429 0.0350715 0.999385i \(-0.488834\pi\)
0.0350715 + 0.999385i \(0.488834\pi\)
\(84\) −4.19853 −0.458097
\(85\) 6.75354 0.732524
\(86\) −11.6737 −1.25881
\(87\) −4.83212 −0.518058
\(88\) 0 0
\(89\) 10.1001 1.07061 0.535303 0.844660i \(-0.320197\pi\)
0.535303 + 0.844660i \(0.320197\pi\)
\(90\) −1.97107 −0.207769
\(91\) −6.19853 −0.649783
\(92\) −5.08700 −0.530356
\(93\) −8.37405 −0.868348
\(94\) −26.4272 −2.72576
\(95\) −6.55501 −0.672530
\(96\) 10.2955 1.05078
\(97\) −7.25363 −0.736494 −0.368247 0.929728i \(-0.620042\pi\)
−0.368247 + 0.929728i \(0.620042\pi\)
\(98\) 2.21161 0.223407
\(99\) 0 0
\(100\) 2.89124 0.289124
\(101\) −18.1898 −1.80996 −0.904979 0.425457i \(-0.860114\pi\)
−0.904979 + 0.425457i \(0.860114\pi\)
\(102\) −21.6898 −2.14761
\(103\) 3.79136 0.373574 0.186787 0.982400i \(-0.440193\pi\)
0.186787 + 0.982400i \(0.440193\pi\)
\(104\) −12.2177 −1.19805
\(105\) −1.45216 −0.141716
\(106\) 11.8017 1.14628
\(107\) 3.73644 0.361215 0.180608 0.983555i \(-0.442194\pi\)
0.180608 + 0.983555i \(0.442194\pi\)
\(108\) 16.3375 1.57207
\(109\) 9.50708 0.910613 0.455307 0.890335i \(-0.349530\pi\)
0.455307 + 0.890335i \(0.349530\pi\)
\(110\) 0 0
\(111\) 14.0883 1.33720
\(112\) −1.42323 −0.134482
\(113\) −12.8465 −1.20849 −0.604246 0.796797i \(-0.706526\pi\)
−0.604246 + 0.796797i \(0.706526\pi\)
\(114\) 21.0522 1.97171
\(115\) −1.75946 −0.164070
\(116\) 9.62072 0.893261
\(117\) 5.52435 0.510727
\(118\) −31.3246 −2.88366
\(119\) 6.75354 0.619096
\(120\) −2.86230 −0.261291
\(121\) 0 0
\(122\) −24.1867 −2.18976
\(123\) 11.5943 1.04542
\(124\) 16.6727 1.49725
\(125\) 1.00000 0.0894427
\(126\) −1.97107 −0.175597
\(127\) 10.1478 0.900475 0.450237 0.892909i \(-0.351339\pi\)
0.450237 + 0.892909i \(0.351339\pi\)
\(128\) −14.2029 −1.25537
\(129\) 7.66502 0.674868
\(130\) −13.7088 −1.20234
\(131\) −4.96218 −0.433548 −0.216774 0.976222i \(-0.569553\pi\)
−0.216774 + 0.976222i \(0.569553\pi\)
\(132\) 0 0
\(133\) −6.55501 −0.568391
\(134\) 16.0969 1.39056
\(135\) 5.65069 0.486334
\(136\) 13.3117 1.14147
\(137\) 2.61774 0.223649 0.111824 0.993728i \(-0.464331\pi\)
0.111824 + 0.993728i \(0.464331\pi\)
\(138\) 5.65069 0.481019
\(139\) −5.94214 −0.504006 −0.252003 0.967727i \(-0.581089\pi\)
−0.252003 + 0.967727i \(0.581089\pi\)
\(140\) 2.89124 0.244354
\(141\) 17.3523 1.46133
\(142\) 19.5071 1.63700
\(143\) 0 0
\(144\) 1.26843 0.105703
\(145\) 3.32754 0.276338
\(146\) −1.79451 −0.148515
\(147\) −1.45216 −0.119772
\(148\) −28.0496 −2.30566
\(149\) 0.686788 0.0562639 0.0281319 0.999604i \(-0.491044\pi\)
0.0281319 + 0.999604i \(0.491044\pi\)
\(150\) −3.21161 −0.262227
\(151\) 14.9751 1.21866 0.609328 0.792918i \(-0.291439\pi\)
0.609328 + 0.792918i \(0.291439\pi\)
\(152\) −12.9204 −1.04798
\(153\) −6.01900 −0.486607
\(154\) 0 0
\(155\) 5.76662 0.463186
\(156\) 26.0247 2.08365
\(157\) 0.746374 0.0595671 0.0297836 0.999556i \(-0.490518\pi\)
0.0297836 + 0.999556i \(0.490518\pi\)
\(158\) 10.0230 0.797388
\(159\) −7.74905 −0.614540
\(160\) −7.08977 −0.560495
\(161\) −1.75946 −0.138665
\(162\) −12.2346 −0.961243
\(163\) −22.1447 −1.73451 −0.867253 0.497868i \(-0.834117\pi\)
−0.867253 + 0.497868i \(0.834117\pi\)
\(164\) −23.0841 −1.80256
\(165\) 0 0
\(166\) 1.41329 0.109693
\(167\) −16.4393 −1.27211 −0.636055 0.771644i \(-0.719435\pi\)
−0.636055 + 0.771644i \(0.719435\pi\)
\(168\) −2.86230 −0.220832
\(169\) 25.4218 1.95552
\(170\) 14.9362 1.14556
\(171\) 5.84206 0.446753
\(172\) −15.2610 −1.16364
\(173\) 7.23463 0.550039 0.275019 0.961439i \(-0.411316\pi\)
0.275019 + 0.961439i \(0.411316\pi\)
\(174\) −10.6868 −0.810163
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 20.5679 1.54598
\(178\) 22.3375 1.67426
\(179\) 1.80863 0.135184 0.0675918 0.997713i \(-0.478468\pi\)
0.0675918 + 0.997713i \(0.478468\pi\)
\(180\) −2.57677 −0.192061
\(181\) −9.25363 −0.687817 −0.343908 0.939003i \(-0.611751\pi\)
−0.343908 + 0.939003i \(0.611751\pi\)
\(182\) −13.7088 −1.01616
\(183\) 15.8811 1.17397
\(184\) −3.46801 −0.255665
\(185\) −9.70159 −0.713275
\(186\) −18.5202 −1.35796
\(187\) 0 0
\(188\) −34.5483 −2.51969
\(189\) 5.65069 0.411027
\(190\) −14.4971 −1.05173
\(191\) −16.7886 −1.21478 −0.607390 0.794404i \(-0.707783\pi\)
−0.607390 + 0.794404i \(0.707783\pi\)
\(192\) 18.6361 1.34494
\(193\) 1.33069 0.0957853 0.0478926 0.998852i \(-0.484749\pi\)
0.0478926 + 0.998852i \(0.484749\pi\)
\(194\) −16.0422 −1.15176
\(195\) 9.00125 0.644593
\(196\) 2.89124 0.206517
\(197\) −5.59425 −0.398574 −0.199287 0.979941i \(-0.563863\pi\)
−0.199287 + 0.979941i \(0.563863\pi\)
\(198\) 0 0
\(199\) 8.05807 0.571221 0.285611 0.958346i \(-0.407804\pi\)
0.285611 + 0.958346i \(0.407804\pi\)
\(200\) 1.97107 0.139376
\(201\) −10.5693 −0.745504
\(202\) −40.2289 −2.83050
\(203\) 3.32754 0.233548
\(204\) −28.3550 −1.98524
\(205\) −7.98415 −0.557637
\(206\) 8.38503 0.584213
\(207\) 1.56809 0.108990
\(208\) 8.82192 0.611690
\(209\) 0 0
\(210\) −3.21161 −0.221622
\(211\) 2.13350 0.146876 0.0734382 0.997300i \(-0.476603\pi\)
0.0734382 + 0.997300i \(0.476603\pi\)
\(212\) 15.4283 1.05962
\(213\) −12.8085 −0.877621
\(214\) 8.26356 0.564885
\(215\) −5.27837 −0.359982
\(216\) 11.1379 0.757838
\(217\) 5.76662 0.391464
\(218\) 21.0260 1.42406
\(219\) 1.17828 0.0796211
\(220\) 0 0
\(221\) −41.8620 −2.81595
\(222\) 31.1578 2.09117
\(223\) 15.5421 1.04078 0.520389 0.853929i \(-0.325787\pi\)
0.520389 + 0.853929i \(0.325787\pi\)
\(224\) −7.08977 −0.473705
\(225\) −0.891236 −0.0594157
\(226\) −28.4114 −1.88990
\(227\) 20.5592 1.36456 0.682282 0.731089i \(-0.260988\pi\)
0.682282 + 0.731089i \(0.260988\pi\)
\(228\) 27.5214 1.82265
\(229\) 26.3274 1.73976 0.869881 0.493262i \(-0.164196\pi\)
0.869881 + 0.493262i \(0.164196\pi\)
\(230\) −3.89124 −0.256580
\(231\) 0 0
\(232\) 6.55882 0.430608
\(233\) 0.0808778 0.00529848 0.00264924 0.999996i \(-0.499157\pi\)
0.00264924 + 0.999996i \(0.499157\pi\)
\(234\) 12.2177 0.798698
\(235\) −11.9493 −0.779487
\(236\) −40.9506 −2.66565
\(237\) −6.58117 −0.427493
\(238\) 14.9362 0.968172
\(239\) 6.87815 0.444911 0.222455 0.974943i \(-0.428593\pi\)
0.222455 + 0.974943i \(0.428593\pi\)
\(240\) 2.06675 0.133408
\(241\) −9.65928 −0.622209 −0.311104 0.950376i \(-0.600699\pi\)
−0.311104 + 0.950376i \(0.600699\pi\)
\(242\) 0 0
\(243\) −8.91874 −0.572138
\(244\) −31.6192 −2.02421
\(245\) 1.00000 0.0638877
\(246\) 25.6420 1.63487
\(247\) 40.6314 2.58532
\(248\) 11.3664 0.721768
\(249\) −0.927976 −0.0588081
\(250\) 2.21161 0.139875
\(251\) −9.31723 −0.588098 −0.294049 0.955790i \(-0.595003\pi\)
−0.294049 + 0.955790i \(0.595003\pi\)
\(252\) −2.57677 −0.162321
\(253\) 0 0
\(254\) 22.4431 1.40821
\(255\) −9.80721 −0.614151
\(256\) −5.74465 −0.359041
\(257\) −6.12624 −0.382145 −0.191072 0.981576i \(-0.561196\pi\)
−0.191072 + 0.981576i \(0.561196\pi\)
\(258\) 16.9521 1.05539
\(259\) −9.70159 −0.602828
\(260\) −17.9214 −1.11144
\(261\) −2.96563 −0.183568
\(262\) −10.9744 −0.678002
\(263\) −7.97384 −0.491688 −0.245844 0.969309i \(-0.579065\pi\)
−0.245844 + 0.969309i \(0.579065\pi\)
\(264\) 0 0
\(265\) 5.33623 0.327802
\(266\) −14.4971 −0.888877
\(267\) −14.6669 −0.897601
\(268\) 21.0435 1.28543
\(269\) −26.7084 −1.62844 −0.814219 0.580557i \(-0.802835\pi\)
−0.814219 + 0.580557i \(0.802835\pi\)
\(270\) 12.4971 0.760552
\(271\) −17.7706 −1.07949 −0.539745 0.841829i \(-0.681479\pi\)
−0.539745 + 0.841829i \(0.681479\pi\)
\(272\) −9.61183 −0.582803
\(273\) 9.00125 0.544781
\(274\) 5.78943 0.349752
\(275\) 0 0
\(276\) 7.38713 0.444653
\(277\) 30.2465 1.81734 0.908668 0.417520i \(-0.137101\pi\)
0.908668 + 0.417520i \(0.137101\pi\)
\(278\) −13.1417 −0.788188
\(279\) −5.13942 −0.307689
\(280\) 1.97107 0.117794
\(281\) −26.8925 −1.60427 −0.802136 0.597142i \(-0.796303\pi\)
−0.802136 + 0.597142i \(0.796303\pi\)
\(282\) 38.3766 2.28529
\(283\) −12.1795 −0.723998 −0.361999 0.932178i \(-0.617906\pi\)
−0.361999 + 0.932178i \(0.617906\pi\)
\(284\) 25.5015 1.51324
\(285\) 9.51891 0.563852
\(286\) 0 0
\(287\) −7.98415 −0.471290
\(288\) 6.31865 0.372330
\(289\) 28.6103 1.68296
\(290\) 7.35924 0.432150
\(291\) 10.5334 0.617480
\(292\) −2.34596 −0.137287
\(293\) −7.02339 −0.410311 −0.205156 0.978729i \(-0.565770\pi\)
−0.205156 + 0.978729i \(0.565770\pi\)
\(294\) −3.21161 −0.187305
\(295\) −14.1637 −0.824642
\(296\) −19.1225 −1.11147
\(297\) 0 0
\(298\) 1.51891 0.0879881
\(299\) 10.9060 0.630713
\(300\) −4.19853 −0.242402
\(301\) −5.27837 −0.304240
\(302\) 33.1191 1.90579
\(303\) 26.4145 1.51748
\(304\) 9.32927 0.535070
\(305\) −10.9362 −0.626206
\(306\) −13.3117 −0.760979
\(307\) 4.02159 0.229524 0.114762 0.993393i \(-0.463389\pi\)
0.114762 + 0.993393i \(0.463389\pi\)
\(308\) 0 0
\(309\) −5.50566 −0.313206
\(310\) 12.7535 0.724352
\(311\) 4.14209 0.234877 0.117438 0.993080i \(-0.462532\pi\)
0.117438 + 0.993080i \(0.462532\pi\)
\(312\) 17.7421 1.00445
\(313\) 2.35103 0.132888 0.0664441 0.997790i \(-0.478835\pi\)
0.0664441 + 0.997790i \(0.478835\pi\)
\(314\) 1.65069 0.0931539
\(315\) −0.891236 −0.0502154
\(316\) 13.1031 0.737105
\(317\) −0.0198694 −0.00111598 −0.000557988 1.00000i \(-0.500178\pi\)
−0.000557988 1.00000i \(0.500178\pi\)
\(318\) −17.1379 −0.961046
\(319\) 0 0
\(320\) −12.8334 −0.717407
\(321\) −5.42590 −0.302844
\(322\) −3.89124 −0.216850
\(323\) −44.2695 −2.46322
\(324\) −15.9943 −0.888572
\(325\) −6.19853 −0.343833
\(326\) −48.9755 −2.71250
\(327\) −13.8058 −0.763462
\(328\) −15.7373 −0.868948
\(329\) −11.9493 −0.658786
\(330\) 0 0
\(331\) 9.11555 0.501036 0.250518 0.968112i \(-0.419399\pi\)
0.250518 + 0.968112i \(0.419399\pi\)
\(332\) 1.84759 0.101400
\(333\) 8.64641 0.473820
\(334\) −36.3573 −1.98939
\(335\) 7.27837 0.397660
\(336\) 2.06675 0.112751
\(337\) −0.637610 −0.0347328 −0.0173664 0.999849i \(-0.505528\pi\)
−0.0173664 + 0.999849i \(0.505528\pi\)
\(338\) 56.2232 3.05814
\(339\) 18.6551 1.01321
\(340\) 19.5261 1.05895
\(341\) 0 0
\(342\) 12.9204 0.698654
\(343\) 1.00000 0.0539949
\(344\) −10.4040 −0.560947
\(345\) 2.55501 0.137557
\(346\) 16.0002 0.860176
\(347\) −5.42620 −0.291294 −0.145647 0.989337i \(-0.546526\pi\)
−0.145647 + 0.989337i \(0.546526\pi\)
\(348\) −13.9708 −0.748914
\(349\) −20.0842 −1.07508 −0.537542 0.843237i \(-0.680647\pi\)
−0.537542 + 0.843237i \(0.680647\pi\)
\(350\) 2.21161 0.118216
\(351\) −35.0260 −1.86955
\(352\) 0 0
\(353\) 15.0117 0.798990 0.399495 0.916735i \(-0.369185\pi\)
0.399495 + 0.916735i \(0.369185\pi\)
\(354\) 45.4883 2.41768
\(355\) 8.82029 0.468133
\(356\) 29.2017 1.54769
\(357\) −9.80721 −0.519053
\(358\) 4.00000 0.211407
\(359\) 5.58968 0.295012 0.147506 0.989061i \(-0.452875\pi\)
0.147506 + 0.989061i \(0.452875\pi\)
\(360\) −1.75669 −0.0925855
\(361\) 23.9681 1.26148
\(362\) −20.4654 −1.07564
\(363\) 0 0
\(364\) −17.9214 −0.939338
\(365\) −0.811402 −0.0424707
\(366\) 35.1229 1.83590
\(367\) 26.4503 1.38069 0.690346 0.723479i \(-0.257458\pi\)
0.690346 + 0.723479i \(0.257458\pi\)
\(368\) 2.50411 0.130536
\(369\) 7.11576 0.370432
\(370\) −21.4562 −1.11545
\(371\) 5.33623 0.277043
\(372\) −24.2113 −1.25530
\(373\) −21.7595 −1.12666 −0.563331 0.826231i \(-0.690480\pi\)
−0.563331 + 0.826231i \(0.690480\pi\)
\(374\) 0 0
\(375\) −1.45216 −0.0749891
\(376\) −23.5529 −1.21465
\(377\) −20.6259 −1.06229
\(378\) 12.4971 0.642784
\(379\) 4.28972 0.220348 0.110174 0.993912i \(-0.464859\pi\)
0.110174 + 0.993912i \(0.464859\pi\)
\(380\) −18.9521 −0.972221
\(381\) −14.7363 −0.754962
\(382\) −37.1299 −1.89973
\(383\) 17.8319 0.911165 0.455583 0.890193i \(-0.349431\pi\)
0.455583 + 0.890193i \(0.349431\pi\)
\(384\) 20.6249 1.05251
\(385\) 0 0
\(386\) 2.94298 0.149793
\(387\) 4.70427 0.239131
\(388\) −20.9719 −1.06469
\(389\) 26.6694 1.35219 0.676097 0.736813i \(-0.263670\pi\)
0.676097 + 0.736813i \(0.263670\pi\)
\(390\) 19.9073 1.00804
\(391\) −11.8826 −0.600927
\(392\) 1.97107 0.0995540
\(393\) 7.20587 0.363488
\(394\) −12.3723 −0.623309
\(395\) 4.53199 0.228029
\(396\) 0 0
\(397\) −0.514512 −0.0258226 −0.0129113 0.999917i \(-0.504110\pi\)
−0.0129113 + 0.999917i \(0.504110\pi\)
\(398\) 17.8213 0.893303
\(399\) 9.51891 0.476542
\(400\) −1.42323 −0.0711614
\(401\) 5.03437 0.251405 0.125702 0.992068i \(-0.459882\pi\)
0.125702 + 0.992068i \(0.459882\pi\)
\(402\) −23.3753 −1.16585
\(403\) −35.7446 −1.78056
\(404\) −52.5911 −2.61651
\(405\) −5.53199 −0.274887
\(406\) 7.35924 0.365233
\(407\) 0 0
\(408\) −19.3307 −0.957012
\(409\) −25.4435 −1.25810 −0.629049 0.777365i \(-0.716556\pi\)
−0.629049 + 0.777365i \(0.716556\pi\)
\(410\) −17.6579 −0.872059
\(411\) −3.80137 −0.187508
\(412\) 10.9617 0.540045
\(413\) −14.1637 −0.696949
\(414\) 3.46801 0.170443
\(415\) 0.639032 0.0313689
\(416\) 43.9462 2.15464
\(417\) 8.62893 0.422560
\(418\) 0 0
\(419\) −3.24961 −0.158754 −0.0793768 0.996845i \(-0.525293\pi\)
−0.0793768 + 0.996845i \(0.525293\pi\)
\(420\) −4.19853 −0.204867
\(421\) 36.8737 1.79711 0.898557 0.438857i \(-0.144616\pi\)
0.898557 + 0.438857i \(0.144616\pi\)
\(422\) 4.71849 0.229692
\(423\) 10.6496 0.517804
\(424\) 10.5181 0.510803
\(425\) 6.75354 0.327595
\(426\) −28.3274 −1.37247
\(427\) −10.9362 −0.529241
\(428\) 10.8029 0.522179
\(429\) 0 0
\(430\) −11.6737 −0.562956
\(431\) 2.29459 0.110527 0.0552633 0.998472i \(-0.482400\pi\)
0.0552633 + 0.998472i \(0.482400\pi\)
\(432\) −8.04222 −0.386931
\(433\) 19.3213 0.928520 0.464260 0.885699i \(-0.346320\pi\)
0.464260 + 0.885699i \(0.346320\pi\)
\(434\) 12.7535 0.612190
\(435\) −4.83212 −0.231683
\(436\) 27.4872 1.31640
\(437\) 11.5332 0.551710
\(438\) 2.60591 0.124515
\(439\) −21.4610 −1.02428 −0.512140 0.858902i \(-0.671147\pi\)
−0.512140 + 0.858902i \(0.671147\pi\)
\(440\) 0 0
\(441\) −0.891236 −0.0424398
\(442\) −92.5827 −4.40371
\(443\) 25.8000 1.22579 0.612896 0.790163i \(-0.290004\pi\)
0.612896 + 0.790163i \(0.290004\pi\)
\(444\) 40.7325 1.93308
\(445\) 10.1001 0.478790
\(446\) 34.3732 1.62762
\(447\) −0.997326 −0.0471719
\(448\) −12.8334 −0.606320
\(449\) 18.5408 0.874994 0.437497 0.899220i \(-0.355865\pi\)
0.437497 + 0.899220i \(0.355865\pi\)
\(450\) −1.97107 −0.0929171
\(451\) 0 0
\(452\) −37.1421 −1.74702
\(453\) −21.7462 −1.02173
\(454\) 45.4691 2.13397
\(455\) −6.19853 −0.290592
\(456\) 18.7624 0.878631
\(457\) −10.9155 −0.510606 −0.255303 0.966861i \(-0.582175\pi\)
−0.255303 + 0.966861i \(0.582175\pi\)
\(458\) 58.2260 2.72072
\(459\) 38.1622 1.78126
\(460\) −5.08700 −0.237183
\(461\) −20.2160 −0.941554 −0.470777 0.882252i \(-0.656026\pi\)
−0.470777 + 0.882252i \(0.656026\pi\)
\(462\) 0 0
\(463\) 32.5425 1.51238 0.756189 0.654353i \(-0.227059\pi\)
0.756189 + 0.654353i \(0.227059\pi\)
\(464\) −4.73585 −0.219856
\(465\) −8.37405 −0.388337
\(466\) 0.178870 0.00828601
\(467\) 30.5276 1.41265 0.706325 0.707888i \(-0.250352\pi\)
0.706325 + 0.707888i \(0.250352\pi\)
\(468\) 15.9722 0.738316
\(469\) 7.27837 0.336084
\(470\) −26.4272 −1.21900
\(471\) −1.08385 −0.0499413
\(472\) −27.9176 −1.28501
\(473\) 0 0
\(474\) −14.5550 −0.668534
\(475\) −6.55501 −0.300764
\(476\) 19.5261 0.894976
\(477\) −4.75584 −0.217755
\(478\) 15.2118 0.695773
\(479\) 13.0560 0.596542 0.298271 0.954481i \(-0.403590\pi\)
0.298271 + 0.954481i \(0.403590\pi\)
\(480\) 10.2955 0.469922
\(481\) 60.1356 2.74195
\(482\) −21.3626 −0.973040
\(483\) 2.55501 0.116257
\(484\) 0 0
\(485\) −7.25363 −0.329370
\(486\) −19.7248 −0.894736
\(487\) 2.07534 0.0940427 0.0470213 0.998894i \(-0.485027\pi\)
0.0470213 + 0.998894i \(0.485027\pi\)
\(488\) −21.5561 −0.975797
\(489\) 32.1576 1.45422
\(490\) 2.21161 0.0999105
\(491\) 24.4163 1.10189 0.550945 0.834541i \(-0.314267\pi\)
0.550945 + 0.834541i \(0.314267\pi\)
\(492\) 33.5217 1.51128
\(493\) 22.4727 1.01212
\(494\) 89.8610 4.04304
\(495\) 0 0
\(496\) −8.20722 −0.368515
\(497\) 8.82029 0.395644
\(498\) −2.05233 −0.0919669
\(499\) 4.60906 0.206330 0.103165 0.994664i \(-0.467103\pi\)
0.103165 + 0.994664i \(0.467103\pi\)
\(500\) 2.89124 0.129300
\(501\) 23.8724 1.06654
\(502\) −20.6061 −0.919696
\(503\) 30.2653 1.34946 0.674731 0.738064i \(-0.264260\pi\)
0.674731 + 0.738064i \(0.264260\pi\)
\(504\) −1.75669 −0.0782491
\(505\) −18.1898 −0.809438
\(506\) 0 0
\(507\) −36.9165 −1.63952
\(508\) 29.3398 1.30174
\(509\) −5.34741 −0.237020 −0.118510 0.992953i \(-0.537812\pi\)
−0.118510 + 0.992953i \(0.537812\pi\)
\(510\) −21.6898 −0.960439
\(511\) −0.811402 −0.0358943
\(512\) 15.7009 0.693889
\(513\) −37.0403 −1.63537
\(514\) −13.5489 −0.597616
\(515\) 3.79136 0.167067
\(516\) 22.1614 0.975601
\(517\) 0 0
\(518\) −21.4562 −0.942730
\(519\) −10.5058 −0.461155
\(520\) −12.2177 −0.535783
\(521\) 15.8247 0.693292 0.346646 0.937996i \(-0.387320\pi\)
0.346646 + 0.937996i \(0.387320\pi\)
\(522\) −6.55882 −0.287072
\(523\) 36.1406 1.58032 0.790159 0.612902i \(-0.209998\pi\)
0.790159 + 0.612902i \(0.209998\pi\)
\(524\) −14.3468 −0.626744
\(525\) −1.45216 −0.0633774
\(526\) −17.6350 −0.768925
\(527\) 38.9451 1.69648
\(528\) 0 0
\(529\) −19.9043 −0.865405
\(530\) 11.8017 0.512632
\(531\) 12.6232 0.547799
\(532\) −18.9521 −0.821677
\(533\) 49.4900 2.14365
\(534\) −32.4376 −1.40371
\(535\) 3.73644 0.161540
\(536\) 14.3462 0.619660
\(537\) −2.62642 −0.113339
\(538\) −59.0686 −2.54663
\(539\) 0 0
\(540\) 16.3375 0.703053
\(541\) −19.5594 −0.840925 −0.420462 0.907310i \(-0.638132\pi\)
−0.420462 + 0.907310i \(0.638132\pi\)
\(542\) −39.3018 −1.68816
\(543\) 13.4377 0.576668
\(544\) −47.8810 −2.05288
\(545\) 9.50708 0.407239
\(546\) 19.9073 0.851953
\(547\) −24.1043 −1.03063 −0.515313 0.857002i \(-0.672324\pi\)
−0.515313 + 0.857002i \(0.672324\pi\)
\(548\) 7.56850 0.323310
\(549\) 9.74675 0.415981
\(550\) 0 0
\(551\) −21.8121 −0.929226
\(552\) 5.03610 0.214351
\(553\) 4.53199 0.192720
\(554\) 66.8936 2.84204
\(555\) 14.0883 0.598013
\(556\) −17.1801 −0.728599
\(557\) −33.6176 −1.42442 −0.712211 0.701965i \(-0.752306\pi\)
−0.712211 + 0.701965i \(0.752306\pi\)
\(558\) −11.3664 −0.481179
\(559\) 32.7181 1.38383
\(560\) −1.42323 −0.0601423
\(561\) 0 0
\(562\) −59.4758 −2.50884
\(563\) 10.5711 0.445517 0.222759 0.974874i \(-0.428494\pi\)
0.222759 + 0.974874i \(0.428494\pi\)
\(564\) 50.1695 2.11252
\(565\) −12.8465 −0.540454
\(566\) −26.9364 −1.13222
\(567\) −5.53199 −0.232322
\(568\) 17.3854 0.729475
\(569\) −31.6316 −1.32607 −0.663033 0.748590i \(-0.730731\pi\)
−0.663033 + 0.748590i \(0.730731\pi\)
\(570\) 21.0522 0.881778
\(571\) 13.9160 0.582365 0.291183 0.956667i \(-0.405951\pi\)
0.291183 + 0.956667i \(0.405951\pi\)
\(572\) 0 0
\(573\) 24.3797 1.01848
\(574\) −17.6579 −0.737025
\(575\) −1.75946 −0.0733744
\(576\) 11.4376 0.476565
\(577\) 45.7055 1.90275 0.951373 0.308042i \(-0.0996736\pi\)
0.951373 + 0.308042i \(0.0996736\pi\)
\(578\) 63.2750 2.63189
\(579\) −1.93237 −0.0803068
\(580\) 9.62072 0.399478
\(581\) 0.639032 0.0265115
\(582\) 23.2958 0.965644
\(583\) 0 0
\(584\) −1.59933 −0.0661807
\(585\) 5.52435 0.228404
\(586\) −15.5330 −0.641664
\(587\) 12.8553 0.530597 0.265298 0.964166i \(-0.414530\pi\)
0.265298 + 0.964166i \(0.414530\pi\)
\(588\) −4.19853 −0.173145
\(589\) −37.8003 −1.55753
\(590\) −31.3246 −1.28961
\(591\) 8.12374 0.334166
\(592\) 13.8076 0.567488
\(593\) −15.4564 −0.634718 −0.317359 0.948306i \(-0.602796\pi\)
−0.317359 + 0.948306i \(0.602796\pi\)
\(594\) 0 0
\(595\) 6.75354 0.276868
\(596\) 1.98567 0.0813361
\(597\) −11.7016 −0.478914
\(598\) 24.1200 0.986338
\(599\) −34.2496 −1.39940 −0.699701 0.714436i \(-0.746684\pi\)
−0.699701 + 0.714436i \(0.746684\pi\)
\(600\) −2.86230 −0.116853
\(601\) 25.5119 1.04065 0.520325 0.853968i \(-0.325811\pi\)
0.520325 + 0.853968i \(0.325811\pi\)
\(602\) −11.6737 −0.475785
\(603\) −6.48674 −0.264160
\(604\) 43.2965 1.76171
\(605\) 0 0
\(606\) 58.4188 2.37310
\(607\) −1.52063 −0.0617206 −0.0308603 0.999524i \(-0.509825\pi\)
−0.0308603 + 0.999524i \(0.509825\pi\)
\(608\) 46.4735 1.88475
\(609\) −4.83212 −0.195807
\(610\) −24.1867 −0.979291
\(611\) 74.0682 2.99648
\(612\) −17.4023 −0.703448
\(613\) −41.1735 −1.66298 −0.831492 0.555537i \(-0.812513\pi\)
−0.831492 + 0.555537i \(0.812513\pi\)
\(614\) 8.89421 0.358941
\(615\) 11.5943 0.467525
\(616\) 0 0
\(617\) 42.9037 1.72724 0.863618 0.504146i \(-0.168193\pi\)
0.863618 + 0.504146i \(0.168193\pi\)
\(618\) −12.1764 −0.489806
\(619\) −41.9939 −1.68788 −0.843939 0.536439i \(-0.819769\pi\)
−0.843939 + 0.536439i \(0.819769\pi\)
\(620\) 16.6727 0.669590
\(621\) −9.94214 −0.398964
\(622\) 9.16071 0.367311
\(623\) 10.1001 0.404651
\(624\) −12.8108 −0.512844
\(625\) 1.00000 0.0400000
\(626\) 5.19958 0.207817
\(627\) 0 0
\(628\) 2.15794 0.0861113
\(629\) −65.5201 −2.61246
\(630\) −1.97107 −0.0785293
\(631\) 0.0696914 0.00277437 0.00138719 0.999999i \(-0.499558\pi\)
0.00138719 + 0.999999i \(0.499558\pi\)
\(632\) 8.93287 0.355330
\(633\) −3.09819 −0.123142
\(634\) −0.0439434 −0.00174522
\(635\) 10.1478 0.402705
\(636\) −22.4043 −0.888389
\(637\) −6.19853 −0.245595
\(638\) 0 0
\(639\) −7.86096 −0.310975
\(640\) −14.2029 −0.561420
\(641\) −9.52913 −0.376378 −0.188189 0.982133i \(-0.560262\pi\)
−0.188189 + 0.982133i \(0.560262\pi\)
\(642\) −12.0000 −0.473602
\(643\) −33.6596 −1.32740 −0.663702 0.747997i \(-0.731016\pi\)
−0.663702 + 0.747997i \(0.731016\pi\)
\(644\) −5.08700 −0.200456
\(645\) 7.66502 0.301810
\(646\) −97.9071 −3.85210
\(647\) 45.7694 1.79938 0.899691 0.436528i \(-0.143792\pi\)
0.899691 + 0.436528i \(0.143792\pi\)
\(648\) −10.9039 −0.428347
\(649\) 0 0
\(650\) −13.7088 −0.537702
\(651\) −8.37405 −0.328205
\(652\) −64.0255 −2.50743
\(653\) 10.7743 0.421629 0.210815 0.977526i \(-0.432388\pi\)
0.210815 + 0.977526i \(0.432388\pi\)
\(654\) −30.5331 −1.19394
\(655\) −4.96218 −0.193888
\(656\) 11.3633 0.443661
\(657\) 0.723151 0.0282128
\(658\) −26.4272 −1.03024
\(659\) −16.8298 −0.655594 −0.327797 0.944748i \(-0.606306\pi\)
−0.327797 + 0.944748i \(0.606306\pi\)
\(660\) 0 0
\(661\) −30.3814 −1.18170 −0.590850 0.806781i \(-0.701208\pi\)
−0.590850 + 0.806781i \(0.701208\pi\)
\(662\) 20.1601 0.783544
\(663\) 60.7903 2.36090
\(664\) 1.25958 0.0488811
\(665\) −6.55501 −0.254192
\(666\) 19.1225 0.740982
\(667\) −5.85467 −0.226694
\(668\) −47.5298 −1.83898
\(669\) −22.5696 −0.872593
\(670\) 16.0969 0.621879
\(671\) 0 0
\(672\) 10.2955 0.397156
\(673\) −17.0093 −0.655659 −0.327830 0.944737i \(-0.606317\pi\)
−0.327830 + 0.944737i \(0.606317\pi\)
\(674\) −1.41015 −0.0543168
\(675\) 5.65069 0.217495
\(676\) 73.5004 2.82694
\(677\) 11.5295 0.443116 0.221558 0.975147i \(-0.428886\pi\)
0.221558 + 0.975147i \(0.428886\pi\)
\(678\) 41.2579 1.58450
\(679\) −7.25363 −0.278369
\(680\) 13.3117 0.510480
\(681\) −29.8553 −1.14406
\(682\) 0 0
\(683\) −4.61429 −0.176561 −0.0882805 0.996096i \(-0.528137\pi\)
−0.0882805 + 0.996096i \(0.528137\pi\)
\(684\) 16.8908 0.645834
\(685\) 2.61774 0.100019
\(686\) 2.21161 0.0844398
\(687\) −38.2315 −1.45862
\(688\) 7.51232 0.286404
\(689\) −33.0768 −1.26012
\(690\) 5.65069 0.215118
\(691\) −16.3453 −0.621804 −0.310902 0.950442i \(-0.600631\pi\)
−0.310902 + 0.950442i \(0.600631\pi\)
\(692\) 20.9170 0.795146
\(693\) 0 0
\(694\) −12.0007 −0.455539
\(695\) −5.94214 −0.225398
\(696\) −9.52445 −0.361023
\(697\) −53.9213 −2.04241
\(698\) −44.4186 −1.68127
\(699\) −0.117447 −0.00444227
\(700\) 2.89124 0.109278
\(701\) −21.3419 −0.806072 −0.403036 0.915184i \(-0.632045\pi\)
−0.403036 + 0.915184i \(0.632045\pi\)
\(702\) −77.4640 −2.92369
\(703\) 63.5940 2.39849
\(704\) 0 0
\(705\) 17.3523 0.653525
\(706\) 33.2000 1.24950
\(707\) −18.1898 −0.684100
\(708\) 59.4667 2.23490
\(709\) 17.1857 0.645421 0.322710 0.946498i \(-0.395406\pi\)
0.322710 + 0.946498i \(0.395406\pi\)
\(710\) 19.5071 0.732088
\(711\) −4.03907 −0.151477
\(712\) 19.9080 0.746082
\(713\) −10.1461 −0.379975
\(714\) −21.6898 −0.811719
\(715\) 0 0
\(716\) 5.22919 0.195424
\(717\) −9.98817 −0.373015
\(718\) 12.3622 0.461354
\(719\) −19.5770 −0.730098 −0.365049 0.930988i \(-0.618948\pi\)
−0.365049 + 0.930988i \(0.618948\pi\)
\(720\) 1.26843 0.0472716
\(721\) 3.79136 0.141198
\(722\) 53.0082 1.97276
\(723\) 14.0268 0.521663
\(724\) −26.7544 −0.994320
\(725\) 3.32754 0.123582
\(726\) 0 0
\(727\) −20.7261 −0.768687 −0.384344 0.923190i \(-0.625572\pi\)
−0.384344 + 0.923190i \(0.625572\pi\)
\(728\) −12.2177 −0.452819
\(729\) 29.5474 1.09435
\(730\) −1.79451 −0.0664177
\(731\) −35.6477 −1.31848
\(732\) 45.9161 1.69711
\(733\) 10.2931 0.380183 0.190092 0.981766i \(-0.439121\pi\)
0.190092 + 0.981766i \(0.439121\pi\)
\(734\) 58.4978 2.15919
\(735\) −1.45216 −0.0535637
\(736\) 12.4741 0.459803
\(737\) 0 0
\(738\) 15.7373 0.579298
\(739\) 14.2200 0.523092 0.261546 0.965191i \(-0.415768\pi\)
0.261546 + 0.965191i \(0.415768\pi\)
\(740\) −28.0496 −1.03112
\(741\) −59.0033 −2.16754
\(742\) 11.8017 0.433253
\(743\) −1.42306 −0.0522069 −0.0261034 0.999659i \(-0.508310\pi\)
−0.0261034 + 0.999659i \(0.508310\pi\)
\(744\) −16.5058 −0.605133
\(745\) 0.686788 0.0251620
\(746\) −48.1235 −1.76193
\(747\) −0.569528 −0.0208380
\(748\) 0 0
\(749\) 3.73644 0.136527
\(750\) −3.21161 −0.117272
\(751\) −17.8201 −0.650265 −0.325133 0.945668i \(-0.605409\pi\)
−0.325133 + 0.945668i \(0.605409\pi\)
\(752\) 17.0066 0.620166
\(753\) 13.5301 0.493064
\(754\) −45.6165 −1.66126
\(755\) 14.9751 0.544999
\(756\) 16.3375 0.594188
\(757\) −22.5791 −0.820652 −0.410326 0.911939i \(-0.634585\pi\)
−0.410326 + 0.911939i \(0.634585\pi\)
\(758\) 9.48721 0.344591
\(759\) 0 0
\(760\) −12.9204 −0.468671
\(761\) 13.6408 0.494478 0.247239 0.968954i \(-0.420477\pi\)
0.247239 + 0.968954i \(0.420477\pi\)
\(762\) −32.5909 −1.18064
\(763\) 9.50708 0.344179
\(764\) −48.5398 −1.75611
\(765\) −6.01900 −0.217617
\(766\) 39.4372 1.42492
\(767\) 87.7941 3.17006
\(768\) 8.34214 0.301021
\(769\) 20.8163 0.750654 0.375327 0.926892i \(-0.377530\pi\)
0.375327 + 0.926892i \(0.377530\pi\)
\(770\) 0 0
\(771\) 8.89628 0.320392
\(772\) 3.84734 0.138469
\(773\) 9.30561 0.334699 0.167350 0.985898i \(-0.446479\pi\)
0.167350 + 0.985898i \(0.446479\pi\)
\(774\) 10.4040 0.373965
\(775\) 5.76662 0.207143
\(776\) −14.2974 −0.513247
\(777\) 14.0883 0.505413
\(778\) 58.9825 2.11462
\(779\) 52.3362 1.87514
\(780\) 26.0247 0.931835
\(781\) 0 0
\(782\) −26.2796 −0.939757
\(783\) 18.8029 0.671962
\(784\) −1.42323 −0.0508296
\(785\) 0.746374 0.0266392
\(786\) 15.9366 0.568440
\(787\) −15.0205 −0.535421 −0.267711 0.963499i \(-0.586267\pi\)
−0.267711 + 0.963499i \(0.586267\pi\)
\(788\) −16.1743 −0.576186
\(789\) 11.5793 0.412233
\(790\) 10.0230 0.356603
\(791\) −12.8465 −0.456767
\(792\) 0 0
\(793\) 67.7885 2.40724
\(794\) −1.13790 −0.0403826
\(795\) −7.74905 −0.274830
\(796\) 23.2978 0.825768
\(797\) −40.8706 −1.44771 −0.723855 0.689953i \(-0.757631\pi\)
−0.723855 + 0.689953i \(0.757631\pi\)
\(798\) 21.0522 0.745238
\(799\) −80.7001 −2.85496
\(800\) −7.08977 −0.250661
\(801\) −9.00155 −0.318054
\(802\) 11.1341 0.393158
\(803\) 0 0
\(804\) −30.5585 −1.07771
\(805\) −1.75946 −0.0620127
\(806\) −79.0532 −2.78453
\(807\) 38.7848 1.36529
\(808\) −35.8535 −1.26132
\(809\) −16.2670 −0.571918 −0.285959 0.958242i \(-0.592312\pi\)
−0.285959 + 0.958242i \(0.592312\pi\)
\(810\) −12.2346 −0.429881
\(811\) −25.1784 −0.884133 −0.442067 0.896982i \(-0.645755\pi\)
−0.442067 + 0.896982i \(0.645755\pi\)
\(812\) 9.62072 0.337621
\(813\) 25.8058 0.905049
\(814\) 0 0
\(815\) −22.1447 −0.775695
\(816\) 13.9579 0.488624
\(817\) 34.5997 1.21049
\(818\) −56.2711 −1.96747
\(819\) 5.52435 0.193037
\(820\) −23.0841 −0.806130
\(821\) 24.5153 0.855590 0.427795 0.903876i \(-0.359291\pi\)
0.427795 + 0.903876i \(0.359291\pi\)
\(822\) −8.40717 −0.293234
\(823\) −9.15460 −0.319109 −0.159555 0.987189i \(-0.551006\pi\)
−0.159555 + 0.987189i \(0.551006\pi\)
\(824\) 7.47304 0.260336
\(825\) 0 0
\(826\) −31.3246 −1.08992
\(827\) 24.3588 0.847037 0.423519 0.905887i \(-0.360795\pi\)
0.423519 + 0.905887i \(0.360795\pi\)
\(828\) 4.53372 0.157558
\(829\) −15.1100 −0.524793 −0.262396 0.964960i \(-0.584513\pi\)
−0.262396 + 0.964960i \(0.584513\pi\)
\(830\) 1.41329 0.0490561
\(831\) −43.9227 −1.52366
\(832\) 79.5481 2.75783
\(833\) 6.75354 0.233996
\(834\) 19.0839 0.660820
\(835\) −16.4393 −0.568905
\(836\) 0 0
\(837\) 32.5854 1.12632
\(838\) −7.18687 −0.248266
\(839\) 0.170947 0.00590175 0.00295087 0.999996i \(-0.499061\pi\)
0.00295087 + 0.999996i \(0.499061\pi\)
\(840\) −2.86230 −0.0987589
\(841\) −17.9274 −0.618188
\(842\) 81.5504 2.81041
\(843\) 39.0522 1.34503
\(844\) 6.16846 0.212327
\(845\) 25.4218 0.874537
\(846\) 23.5529 0.809766
\(847\) 0 0
\(848\) −7.59467 −0.260802
\(849\) 17.6866 0.607003
\(850\) 14.9362 0.512308
\(851\) 17.0695 0.585136
\(852\) −37.0323 −1.26871
\(853\) −39.4944 −1.35226 −0.676131 0.736782i \(-0.736345\pi\)
−0.676131 + 0.736782i \(0.736345\pi\)
\(854\) −24.1867 −0.827652
\(855\) 5.84206 0.199794
\(856\) 7.36478 0.251723
\(857\) −32.2687 −1.10228 −0.551138 0.834414i \(-0.685806\pi\)
−0.551138 + 0.834414i \(0.685806\pi\)
\(858\) 0 0
\(859\) 17.6791 0.603205 0.301602 0.953434i \(-0.402478\pi\)
0.301602 + 0.953434i \(0.402478\pi\)
\(860\) −15.2610 −0.520396
\(861\) 11.5943 0.395131
\(862\) 5.07476 0.172847
\(863\) −27.6057 −0.939710 −0.469855 0.882744i \(-0.655694\pi\)
−0.469855 + 0.882744i \(0.655694\pi\)
\(864\) −40.0621 −1.36294
\(865\) 7.23463 0.245985
\(866\) 42.7311 1.45206
\(867\) −41.5467 −1.41100
\(868\) 16.6727 0.565907
\(869\) 0 0
\(870\) −10.6868 −0.362316
\(871\) −45.1152 −1.52867
\(872\) 18.7391 0.634586
\(873\) 6.46469 0.218797
\(874\) 25.5071 0.862790
\(875\) 1.00000 0.0338062
\(876\) 3.40670 0.115102
\(877\) 3.64582 0.123111 0.0615553 0.998104i \(-0.480394\pi\)
0.0615553 + 0.998104i \(0.480394\pi\)
\(878\) −47.4636 −1.60182
\(879\) 10.1991 0.344006
\(880\) 0 0
\(881\) −24.0883 −0.811554 −0.405777 0.913972i \(-0.632999\pi\)
−0.405777 + 0.913972i \(0.632999\pi\)
\(882\) −1.97107 −0.0663693
\(883\) 8.40021 0.282690 0.141345 0.989960i \(-0.454857\pi\)
0.141345 + 0.989960i \(0.454857\pi\)
\(884\) −121.033 −4.07078
\(885\) 20.5679 0.691383
\(886\) 57.0595 1.91695
\(887\) −48.8895 −1.64155 −0.820774 0.571254i \(-0.806457\pi\)
−0.820774 + 0.571254i \(0.806457\pi\)
\(888\) 27.7689 0.931864
\(889\) 10.1478 0.340348
\(890\) 22.3375 0.748754
\(891\) 0 0
\(892\) 44.9360 1.50457
\(893\) 78.3278 2.62114
\(894\) −2.20570 −0.0737696
\(895\) 1.80863 0.0604560
\(896\) −14.2029 −0.474487
\(897\) −15.8373 −0.528792
\(898\) 41.0051 1.36836
\(899\) 19.1887 0.639979
\(900\) −2.57677 −0.0858924
\(901\) 36.0384 1.20061
\(902\) 0 0
\(903\) 7.66502 0.255076
\(904\) −25.3213 −0.842172
\(905\) −9.25363 −0.307601
\(906\) −48.0942 −1.59782
\(907\) 9.06429 0.300975 0.150487 0.988612i \(-0.451916\pi\)
0.150487 + 0.988612i \(0.451916\pi\)
\(908\) 59.4416 1.97264
\(909\) 16.2114 0.537700
\(910\) −13.7088 −0.454441
\(911\) 38.1305 1.26332 0.631659 0.775246i \(-0.282374\pi\)
0.631659 + 0.775246i \(0.282374\pi\)
\(912\) −13.5476 −0.448605
\(913\) 0 0
\(914\) −24.1409 −0.798509
\(915\) 15.8811 0.525014
\(916\) 76.1186 2.51503
\(917\) −4.96218 −0.163866
\(918\) 84.4000 2.78561
\(919\) 28.5711 0.942472 0.471236 0.882007i \(-0.343808\pi\)
0.471236 + 0.882007i \(0.343808\pi\)
\(920\) −3.46801 −0.114337
\(921\) −5.83999 −0.192434
\(922\) −44.7100 −1.47245
\(923\) −54.6729 −1.79958
\(924\) 0 0
\(925\) −9.70159 −0.318986
\(926\) 71.9715 2.36513
\(927\) −3.37900 −0.110981
\(928\) −23.5915 −0.774430
\(929\) −7.79299 −0.255680 −0.127840 0.991795i \(-0.540804\pi\)
−0.127840 + 0.991795i \(0.540804\pi\)
\(930\) −18.5202 −0.607300
\(931\) −6.55501 −0.214832
\(932\) 0.233837 0.00765958
\(933\) −6.01498 −0.196921
\(934\) 67.5153 2.20917
\(935\) 0 0
\(936\) 10.8889 0.355914
\(937\) −0.427891 −0.0139786 −0.00698929 0.999976i \(-0.502225\pi\)
−0.00698929 + 0.999976i \(0.502225\pi\)
\(938\) 16.0969 0.525584
\(939\) −3.41407 −0.111414
\(940\) −34.5483 −1.12684
\(941\) −50.7330 −1.65385 −0.826925 0.562313i \(-0.809912\pi\)
−0.826925 + 0.562313i \(0.809912\pi\)
\(942\) −2.39706 −0.0781006
\(943\) 14.0478 0.457458
\(944\) 20.1582 0.656092
\(945\) 5.65069 0.183817
\(946\) 0 0
\(947\) −23.7388 −0.771408 −0.385704 0.922623i \(-0.626041\pi\)
−0.385704 + 0.922623i \(0.626041\pi\)
\(948\) −19.0277 −0.617991
\(949\) 5.02950 0.163265
\(950\) −14.4971 −0.470350
\(951\) 0.0288535 0.000935639 0
\(952\) 13.3117 0.431435
\(953\) −4.55359 −0.147505 −0.0737525 0.997277i \(-0.523497\pi\)
−0.0737525 + 0.997277i \(0.523497\pi\)
\(954\) −10.5181 −0.340535
\(955\) −16.7886 −0.543266
\(956\) 19.8864 0.643171
\(957\) 0 0
\(958\) 28.8748 0.932901
\(959\) 2.61774 0.0845312
\(960\) 18.6361 0.601477
\(961\) 2.25393 0.0727073
\(962\) 132.997 4.28799
\(963\) −3.33005 −0.107309
\(964\) −27.9273 −0.899476
\(965\) 1.33069 0.0428365
\(966\) 5.65069 0.181808
\(967\) −38.9252 −1.25175 −0.625876 0.779923i \(-0.715258\pi\)
−0.625876 + 0.779923i \(0.715258\pi\)
\(968\) 0 0
\(969\) 64.2863 2.06518
\(970\) −16.0422 −0.515085
\(971\) 6.67820 0.214314 0.107157 0.994242i \(-0.465825\pi\)
0.107157 + 0.994242i \(0.465825\pi\)
\(972\) −25.7862 −0.827092
\(973\) −5.94214 −0.190496
\(974\) 4.58985 0.147068
\(975\) 9.00125 0.288271
\(976\) 15.5647 0.498215
\(977\) 34.5917 1.10669 0.553343 0.832953i \(-0.313352\pi\)
0.553343 + 0.832953i \(0.313352\pi\)
\(978\) 71.1202 2.27417
\(979\) 0 0
\(980\) 2.89124 0.0923571
\(981\) −8.47305 −0.270524
\(982\) 53.9994 1.72319
\(983\) −6.03257 −0.192409 −0.0962046 0.995362i \(-0.530670\pi\)
−0.0962046 + 0.995362i \(0.530670\pi\)
\(984\) 22.8531 0.728529
\(985\) −5.59425 −0.178248
\(986\) 49.7010 1.58280
\(987\) 17.3523 0.552329
\(988\) 117.475 3.73738
\(989\) 9.28705 0.295311
\(990\) 0 0
\(991\) 13.1912 0.419032 0.209516 0.977805i \(-0.432811\pi\)
0.209516 + 0.977805i \(0.432811\pi\)
\(992\) −40.8840 −1.29807
\(993\) −13.2372 −0.420071
\(994\) 19.5071 0.618727
\(995\) 8.05807 0.255458
\(996\) −2.68300 −0.0850141
\(997\) −2.69777 −0.0854392 −0.0427196 0.999087i \(-0.513602\pi\)
−0.0427196 + 0.999087i \(0.513602\pi\)
\(998\) 10.1935 0.322668
\(999\) −54.8207 −1.73445
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 4235.2.a.u.1.4 4
11.10 odd 2 4235.2.a.v.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4235.2.a.u.1.4 4 1.1 even 1 trivial
4235.2.a.v.1.1 yes 4 11.10 odd 2