Properties

Label 5445.2.a.cd.1.7
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} - 3 x^{6} + 22 x^{5} - 3 x^{4} - 32 x^{3} + 9 x^{2} + 8 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 495)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-1.39491\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.39491 q^{2} +3.73560 q^{4} +1.00000 q^{5} +1.96984 q^{7} +4.15660 q^{8} +O(q^{10})\) \(q+2.39491 q^{2} +3.73560 q^{4} +1.00000 q^{5} +1.96984 q^{7} +4.15660 q^{8} +2.39491 q^{10} +2.92022 q^{13} +4.71760 q^{14} +2.48350 q^{16} +6.62030 q^{17} -3.56013 q^{19} +3.73560 q^{20} -2.78918 q^{23} +1.00000 q^{25} +6.99366 q^{26} +7.35855 q^{28} +3.51753 q^{29} +8.11992 q^{31} -2.36545 q^{32} +15.8550 q^{34} +1.96984 q^{35} +4.42991 q^{37} -8.52619 q^{38} +4.15660 q^{40} -7.07811 q^{41} -10.8914 q^{43} -6.67984 q^{46} +0.0271524 q^{47} -3.11971 q^{49} +2.39491 q^{50} +10.9088 q^{52} -4.10948 q^{53} +8.18786 q^{56} +8.42417 q^{58} +9.51441 q^{59} +3.10231 q^{61} +19.4465 q^{62} -10.6320 q^{64} +2.92022 q^{65} +1.58295 q^{67} +24.7308 q^{68} +4.71760 q^{70} +6.05837 q^{71} -2.37292 q^{73} +10.6092 q^{74} -13.2992 q^{76} -12.5389 q^{79} +2.48350 q^{80} -16.9515 q^{82} +5.04333 q^{83} +6.62030 q^{85} -26.0840 q^{86} -12.0573 q^{89} +5.75238 q^{91} -10.4193 q^{92} +0.0650277 q^{94} -3.56013 q^{95} +10.4097 q^{97} -7.47144 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{2} + 6q^{4} + 8q^{5} + 8q^{7} + 12q^{8} + O(q^{10}) \) \( 8q + 4q^{2} + 6q^{4} + 8q^{5} + 8q^{7} + 12q^{8} + 4q^{10} + 6q^{13} + 14q^{14} + 14q^{16} + 8q^{17} - 2q^{19} + 6q^{20} + 4q^{23} + 8q^{25} - 2q^{26} + 24q^{28} + 22q^{29} + 10q^{31} + 28q^{32} - 2q^{34} + 8q^{35} - 14q^{37} - 20q^{38} + 12q^{40} + 22q^{41} + 14q^{43} - 2q^{46} + 10q^{47} + 4q^{50} - 10q^{52} - 18q^{53} + 34q^{56} + 12q^{58} + 2q^{59} - 14q^{61} + 30q^{62} + 30q^{64} + 6q^{65} + 10q^{67} + 6q^{68} + 14q^{70} - 2q^{71} + 16q^{73} + 24q^{74} - 22q^{76} - 16q^{79} + 14q^{80} + 10q^{82} + 46q^{83} + 8q^{85} - 28q^{86} + 38q^{89} + 8q^{91} - 24q^{92} - 10q^{94} - 2q^{95} - 4q^{97} + 4q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.39491 1.69346 0.846729 0.532025i \(-0.178569\pi\)
0.846729 + 0.532025i \(0.178569\pi\)
\(3\) 0 0
\(4\) 3.73560 1.86780
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.96984 0.744531 0.372266 0.928126i \(-0.378581\pi\)
0.372266 + 0.928126i \(0.378581\pi\)
\(8\) 4.15660 1.46958
\(9\) 0 0
\(10\) 2.39491 0.757337
\(11\) 0 0
\(12\) 0 0
\(13\) 2.92022 0.809923 0.404961 0.914334i \(-0.367285\pi\)
0.404961 + 0.914334i \(0.367285\pi\)
\(14\) 4.71760 1.26083
\(15\) 0 0
\(16\) 2.48350 0.620875
\(17\) 6.62030 1.60566 0.802829 0.596210i \(-0.203327\pi\)
0.802829 + 0.596210i \(0.203327\pi\)
\(18\) 0 0
\(19\) −3.56013 −0.816750 −0.408375 0.912814i \(-0.633904\pi\)
−0.408375 + 0.912814i \(0.633904\pi\)
\(20\) 3.73560 0.835305
\(21\) 0 0
\(22\) 0 0
\(23\) −2.78918 −0.581585 −0.290792 0.956786i \(-0.593919\pi\)
−0.290792 + 0.956786i \(0.593919\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 6.99366 1.37157
\(27\) 0 0
\(28\) 7.35855 1.39063
\(29\) 3.51753 0.653188 0.326594 0.945165i \(-0.394099\pi\)
0.326594 + 0.945165i \(0.394099\pi\)
\(30\) 0 0
\(31\) 8.11992 1.45838 0.729191 0.684311i \(-0.239897\pi\)
0.729191 + 0.684311i \(0.239897\pi\)
\(32\) −2.36545 −0.418156
\(33\) 0 0
\(34\) 15.8550 2.71911
\(35\) 1.96984 0.332964
\(36\) 0 0
\(37\) 4.42991 0.728273 0.364137 0.931346i \(-0.381364\pi\)
0.364137 + 0.931346i \(0.381364\pi\)
\(38\) −8.52619 −1.38313
\(39\) 0 0
\(40\) 4.15660 0.657217
\(41\) −7.07811 −1.10542 −0.552708 0.833375i \(-0.686405\pi\)
−0.552708 + 0.833375i \(0.686405\pi\)
\(42\) 0 0
\(43\) −10.8914 −1.66092 −0.830462 0.557075i \(-0.811924\pi\)
−0.830462 + 0.557075i \(0.811924\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −6.67984 −0.984889
\(47\) 0.0271524 0.00396059 0.00198029 0.999998i \(-0.499370\pi\)
0.00198029 + 0.999998i \(0.499370\pi\)
\(48\) 0 0
\(49\) −3.11971 −0.445673
\(50\) 2.39491 0.338692
\(51\) 0 0
\(52\) 10.9088 1.51277
\(53\) −4.10948 −0.564481 −0.282240 0.959344i \(-0.591078\pi\)
−0.282240 + 0.959344i \(0.591078\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 8.18786 1.09415
\(57\) 0 0
\(58\) 8.42417 1.10615
\(59\) 9.51441 1.23867 0.619335 0.785127i \(-0.287402\pi\)
0.619335 + 0.785127i \(0.287402\pi\)
\(60\) 0 0
\(61\) 3.10231 0.397210 0.198605 0.980080i \(-0.436359\pi\)
0.198605 + 0.980080i \(0.436359\pi\)
\(62\) 19.4465 2.46971
\(63\) 0 0
\(64\) −10.6320 −1.32900
\(65\) 2.92022 0.362209
\(66\) 0 0
\(67\) 1.58295 0.193388 0.0966939 0.995314i \(-0.469173\pi\)
0.0966939 + 0.995314i \(0.469173\pi\)
\(68\) 24.7308 2.99905
\(69\) 0 0
\(70\) 4.71760 0.563861
\(71\) 6.05837 0.718997 0.359498 0.933146i \(-0.382948\pi\)
0.359498 + 0.933146i \(0.382948\pi\)
\(72\) 0 0
\(73\) −2.37292 −0.277729 −0.138864 0.990311i \(-0.544345\pi\)
−0.138864 + 0.990311i \(0.544345\pi\)
\(74\) 10.6092 1.23330
\(75\) 0 0
\(76\) −13.2992 −1.52552
\(77\) 0 0
\(78\) 0 0
\(79\) −12.5389 −1.41074 −0.705369 0.708840i \(-0.749219\pi\)
−0.705369 + 0.708840i \(0.749219\pi\)
\(80\) 2.48350 0.277664
\(81\) 0 0
\(82\) −16.9515 −1.87197
\(83\) 5.04333 0.553577 0.276789 0.960931i \(-0.410730\pi\)
0.276789 + 0.960931i \(0.410730\pi\)
\(84\) 0 0
\(85\) 6.62030 0.718072
\(86\) −26.0840 −2.81271
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0573 −1.27807 −0.639035 0.769177i \(-0.720666\pi\)
−0.639035 + 0.769177i \(0.720666\pi\)
\(90\) 0 0
\(91\) 5.75238 0.603013
\(92\) −10.4193 −1.08628
\(93\) 0 0
\(94\) 0.0650277 0.00670709
\(95\) −3.56013 −0.365262
\(96\) 0 0
\(97\) 10.4097 1.05694 0.528470 0.848952i \(-0.322766\pi\)
0.528470 + 0.848952i \(0.322766\pi\)
\(98\) −7.47144 −0.754729
\(99\) 0 0
\(100\) 3.73560 0.373560
\(101\) 17.2166 1.71311 0.856556 0.516054i \(-0.172600\pi\)
0.856556 + 0.516054i \(0.172600\pi\)
\(102\) 0 0
\(103\) 6.66999 0.657213 0.328607 0.944467i \(-0.393421\pi\)
0.328607 + 0.944467i \(0.393421\pi\)
\(104\) 12.1382 1.19025
\(105\) 0 0
\(106\) −9.84185 −0.955925
\(107\) 9.61908 0.929912 0.464956 0.885334i \(-0.346070\pi\)
0.464956 + 0.885334i \(0.346070\pi\)
\(108\) 0 0
\(109\) −19.2602 −1.84480 −0.922398 0.386240i \(-0.873773\pi\)
−0.922398 + 0.386240i \(0.873773\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.89211 0.462261
\(113\) 3.56283 0.335163 0.167582 0.985858i \(-0.446404\pi\)
0.167582 + 0.985858i \(0.446404\pi\)
\(114\) 0 0
\(115\) −2.78918 −0.260093
\(116\) 13.1401 1.22003
\(117\) 0 0
\(118\) 22.7862 2.09764
\(119\) 13.0410 1.19546
\(120\) 0 0
\(121\) 0 0
\(122\) 7.42975 0.672658
\(123\) 0 0
\(124\) 30.3328 2.72396
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 19.3179 1.71419 0.857094 0.515159i \(-0.172267\pi\)
0.857094 + 0.515159i \(0.172267\pi\)
\(128\) −20.7319 −1.83246
\(129\) 0 0
\(130\) 6.99366 0.613385
\(131\) 13.9027 1.21468 0.607341 0.794441i \(-0.292236\pi\)
0.607341 + 0.794441i \(0.292236\pi\)
\(132\) 0 0
\(133\) −7.01290 −0.608096
\(134\) 3.79102 0.327494
\(135\) 0 0
\(136\) 27.5180 2.35964
\(137\) −22.7270 −1.94169 −0.970847 0.239699i \(-0.922951\pi\)
−0.970847 + 0.239699i \(0.922951\pi\)
\(138\) 0 0
\(139\) 12.7100 1.07804 0.539022 0.842291i \(-0.318794\pi\)
0.539022 + 0.842291i \(0.318794\pi\)
\(140\) 7.35855 0.621911
\(141\) 0 0
\(142\) 14.5093 1.21759
\(143\) 0 0
\(144\) 0 0
\(145\) 3.51753 0.292115
\(146\) −5.68292 −0.470322
\(147\) 0 0
\(148\) 16.5484 1.36027
\(149\) −4.30274 −0.352494 −0.176247 0.984346i \(-0.556396\pi\)
−0.176247 + 0.984346i \(0.556396\pi\)
\(150\) 0 0
\(151\) 17.5844 1.43100 0.715499 0.698614i \(-0.246199\pi\)
0.715499 + 0.698614i \(0.246199\pi\)
\(152\) −14.7980 −1.20028
\(153\) 0 0
\(154\) 0 0
\(155\) 8.11992 0.652208
\(156\) 0 0
\(157\) −23.8112 −1.90034 −0.950172 0.311726i \(-0.899093\pi\)
−0.950172 + 0.311726i \(0.899093\pi\)
\(158\) −30.0296 −2.38903
\(159\) 0 0
\(160\) −2.36545 −0.187005
\(161\) −5.49425 −0.433008
\(162\) 0 0
\(163\) −19.8452 −1.55440 −0.777198 0.629256i \(-0.783360\pi\)
−0.777198 + 0.629256i \(0.783360\pi\)
\(164\) −26.4410 −2.06469
\(165\) 0 0
\(166\) 12.0783 0.937460
\(167\) −14.4760 −1.12019 −0.560094 0.828429i \(-0.689235\pi\)
−0.560094 + 0.828429i \(0.689235\pi\)
\(168\) 0 0
\(169\) −4.47232 −0.344025
\(170\) 15.8550 1.21602
\(171\) 0 0
\(172\) −40.6859 −3.10227
\(173\) 4.31904 0.328371 0.164185 0.986429i \(-0.447500\pi\)
0.164185 + 0.986429i \(0.447500\pi\)
\(174\) 0 0
\(175\) 1.96984 0.148906
\(176\) 0 0
\(177\) 0 0
\(178\) −28.8761 −2.16436
\(179\) −7.45911 −0.557520 −0.278760 0.960361i \(-0.589923\pi\)
−0.278760 + 0.960361i \(0.589923\pi\)
\(180\) 0 0
\(181\) −20.4634 −1.52104 −0.760518 0.649317i \(-0.775055\pi\)
−0.760518 + 0.649317i \(0.775055\pi\)
\(182\) 13.7764 1.02118
\(183\) 0 0
\(184\) −11.5935 −0.854686
\(185\) 4.42991 0.325694
\(186\) 0 0
\(187\) 0 0
\(188\) 0.101431 0.00739759
\(189\) 0 0
\(190\) −8.52619 −0.618555
\(191\) 6.55174 0.474067 0.237034 0.971501i \(-0.423825\pi\)
0.237034 + 0.971501i \(0.423825\pi\)
\(192\) 0 0
\(193\) −6.82579 −0.491331 −0.245665 0.969355i \(-0.579006\pi\)
−0.245665 + 0.969355i \(0.579006\pi\)
\(194\) 24.9302 1.78988
\(195\) 0 0
\(196\) −11.6540 −0.832428
\(197\) 3.45091 0.245867 0.122934 0.992415i \(-0.460770\pi\)
0.122934 + 0.992415i \(0.460770\pi\)
\(198\) 0 0
\(199\) −10.6742 −0.756671 −0.378336 0.925668i \(-0.623504\pi\)
−0.378336 + 0.925668i \(0.623504\pi\)
\(200\) 4.15660 0.293916
\(201\) 0 0
\(202\) 41.2321 2.90108
\(203\) 6.92898 0.486319
\(204\) 0 0
\(205\) −7.07811 −0.494357
\(206\) 15.9740 1.11296
\(207\) 0 0
\(208\) 7.25237 0.502861
\(209\) 0 0
\(210\) 0 0
\(211\) −1.37923 −0.0949503 −0.0474752 0.998872i \(-0.515117\pi\)
−0.0474752 + 0.998872i \(0.515117\pi\)
\(212\) −15.3514 −1.05434
\(213\) 0 0
\(214\) 23.0368 1.57477
\(215\) −10.8914 −0.742788
\(216\) 0 0
\(217\) 15.9950 1.08581
\(218\) −46.1266 −3.12408
\(219\) 0 0
\(220\) 0 0
\(221\) 19.3327 1.30046
\(222\) 0 0
\(223\) 12.1038 0.810531 0.405265 0.914199i \(-0.367179\pi\)
0.405265 + 0.914199i \(0.367179\pi\)
\(224\) −4.65956 −0.311330
\(225\) 0 0
\(226\) 8.53267 0.567585
\(227\) −17.3320 −1.15037 −0.575183 0.818025i \(-0.695069\pi\)
−0.575183 + 0.818025i \(0.695069\pi\)
\(228\) 0 0
\(229\) −0.837178 −0.0553223 −0.0276611 0.999617i \(-0.508806\pi\)
−0.0276611 + 0.999617i \(0.508806\pi\)
\(230\) −6.67984 −0.440456
\(231\) 0 0
\(232\) 14.6210 0.959914
\(233\) −19.0481 −1.24788 −0.623941 0.781471i \(-0.714470\pi\)
−0.623941 + 0.781471i \(0.714470\pi\)
\(234\) 0 0
\(235\) 0.0271524 0.00177123
\(236\) 35.5420 2.31359
\(237\) 0 0
\(238\) 31.2319 2.02446
\(239\) 25.7936 1.66845 0.834223 0.551427i \(-0.185916\pi\)
0.834223 + 0.551427i \(0.185916\pi\)
\(240\) 0 0
\(241\) −17.0680 −1.09944 −0.549722 0.835348i \(-0.685266\pi\)
−0.549722 + 0.835348i \(0.685266\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 11.5890 0.741908
\(245\) −3.11971 −0.199311
\(246\) 0 0
\(247\) −10.3964 −0.661504
\(248\) 33.7513 2.14321
\(249\) 0 0
\(250\) 2.39491 0.151467
\(251\) −10.3166 −0.651177 −0.325589 0.945512i \(-0.605562\pi\)
−0.325589 + 0.945512i \(0.605562\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 46.2647 2.90291
\(255\) 0 0
\(256\) −28.3869 −1.77418
\(257\) −11.2636 −0.702604 −0.351302 0.936262i \(-0.614261\pi\)
−0.351302 + 0.936262i \(0.614261\pi\)
\(258\) 0 0
\(259\) 8.72624 0.542222
\(260\) 10.9088 0.676533
\(261\) 0 0
\(262\) 33.2957 2.05701
\(263\) 5.15082 0.317613 0.158806 0.987310i \(-0.449235\pi\)
0.158806 + 0.987310i \(0.449235\pi\)
\(264\) 0 0
\(265\) −4.10948 −0.252444
\(266\) −16.7953 −1.02978
\(267\) 0 0
\(268\) 5.91326 0.361210
\(269\) 23.1522 1.41161 0.705806 0.708405i \(-0.250585\pi\)
0.705806 + 0.708405i \(0.250585\pi\)
\(270\) 0 0
\(271\) 13.1449 0.798498 0.399249 0.916843i \(-0.369271\pi\)
0.399249 + 0.916843i \(0.369271\pi\)
\(272\) 16.4415 0.996913
\(273\) 0 0
\(274\) −54.4290 −3.28818
\(275\) 0 0
\(276\) 0 0
\(277\) 10.3489 0.621807 0.310903 0.950442i \(-0.399368\pi\)
0.310903 + 0.950442i \(0.399368\pi\)
\(278\) 30.4392 1.82562
\(279\) 0 0
\(280\) 8.18786 0.489319
\(281\) 0.387020 0.0230877 0.0115438 0.999933i \(-0.496325\pi\)
0.0115438 + 0.999933i \(0.496325\pi\)
\(282\) 0 0
\(283\) −14.5253 −0.863442 −0.431721 0.902007i \(-0.642094\pi\)
−0.431721 + 0.902007i \(0.642094\pi\)
\(284\) 22.6317 1.34294
\(285\) 0 0
\(286\) 0 0
\(287\) −13.9428 −0.823016
\(288\) 0 0
\(289\) 26.8283 1.57814
\(290\) 8.42417 0.494684
\(291\) 0 0
\(292\) −8.86426 −0.518742
\(293\) −14.8176 −0.865652 −0.432826 0.901478i \(-0.642483\pi\)
−0.432826 + 0.901478i \(0.642483\pi\)
\(294\) 0 0
\(295\) 9.51441 0.553950
\(296\) 18.4134 1.07026
\(297\) 0 0
\(298\) −10.3047 −0.596934
\(299\) −8.14502 −0.471039
\(300\) 0 0
\(301\) −21.4544 −1.23661
\(302\) 42.1131 2.42334
\(303\) 0 0
\(304\) −8.84158 −0.507100
\(305\) 3.10231 0.177638
\(306\) 0 0
\(307\) 32.3539 1.84653 0.923266 0.384161i \(-0.125509\pi\)
0.923266 + 0.384161i \(0.125509\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 19.4465 1.10449
\(311\) 1.76355 0.100002 0.0500008 0.998749i \(-0.484078\pi\)
0.0500008 + 0.998749i \(0.484078\pi\)
\(312\) 0 0
\(313\) −26.1735 −1.47941 −0.739706 0.672930i \(-0.765036\pi\)
−0.739706 + 0.672930i \(0.765036\pi\)
\(314\) −57.0258 −3.21815
\(315\) 0 0
\(316\) −46.8404 −2.63498
\(317\) 10.3755 0.582744 0.291372 0.956610i \(-0.405888\pi\)
0.291372 + 0.956610i \(0.405888\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −10.6320 −0.594349
\(321\) 0 0
\(322\) −13.1582 −0.733281
\(323\) −23.5691 −1.31142
\(324\) 0 0
\(325\) 2.92022 0.161985
\(326\) −47.5275 −2.63230
\(327\) 0 0
\(328\) −29.4209 −1.62450
\(329\) 0.0534861 0.00294878
\(330\) 0 0
\(331\) 9.22380 0.506986 0.253493 0.967337i \(-0.418421\pi\)
0.253493 + 0.967337i \(0.418421\pi\)
\(332\) 18.8399 1.03397
\(333\) 0 0
\(334\) −34.6688 −1.89699
\(335\) 1.58295 0.0864857
\(336\) 0 0
\(337\) 28.5372 1.55452 0.777261 0.629178i \(-0.216608\pi\)
0.777261 + 0.629178i \(0.216608\pi\)
\(338\) −10.7108 −0.582592
\(339\) 0 0
\(340\) 24.7308 1.34121
\(341\) 0 0
\(342\) 0 0
\(343\) −19.9343 −1.07635
\(344\) −45.2713 −2.44086
\(345\) 0 0
\(346\) 10.3437 0.556082
\(347\) −10.1698 −0.545945 −0.272972 0.962022i \(-0.588007\pi\)
−0.272972 + 0.962022i \(0.588007\pi\)
\(348\) 0 0
\(349\) −23.7174 −1.26957 −0.634783 0.772691i \(-0.718910\pi\)
−0.634783 + 0.772691i \(0.718910\pi\)
\(350\) 4.71760 0.252166
\(351\) 0 0
\(352\) 0 0
\(353\) 14.1383 0.752504 0.376252 0.926517i \(-0.377213\pi\)
0.376252 + 0.926517i \(0.377213\pi\)
\(354\) 0 0
\(355\) 6.05837 0.321545
\(356\) −45.0412 −2.38718
\(357\) 0 0
\(358\) −17.8639 −0.944137
\(359\) 24.0330 1.26841 0.634207 0.773163i \(-0.281327\pi\)
0.634207 + 0.773163i \(0.281327\pi\)
\(360\) 0 0
\(361\) −6.32548 −0.332920
\(362\) −49.0081 −2.57581
\(363\) 0 0
\(364\) 21.4886 1.12631
\(365\) −2.37292 −0.124204
\(366\) 0 0
\(367\) −1.79705 −0.0938055 −0.0469027 0.998899i \(-0.514935\pi\)
−0.0469027 + 0.998899i \(0.514935\pi\)
\(368\) −6.92694 −0.361091
\(369\) 0 0
\(370\) 10.6092 0.551549
\(371\) −8.09504 −0.420274
\(372\) 0 0
\(373\) −6.87802 −0.356131 −0.178065 0.984019i \(-0.556984\pi\)
−0.178065 + 0.984019i \(0.556984\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0.112862 0.00582041
\(377\) 10.2719 0.529032
\(378\) 0 0
\(379\) −3.53329 −0.181493 −0.0907465 0.995874i \(-0.528925\pi\)
−0.0907465 + 0.995874i \(0.528925\pi\)
\(380\) −13.2992 −0.682235
\(381\) 0 0
\(382\) 15.6908 0.802813
\(383\) 1.90335 0.0972564 0.0486282 0.998817i \(-0.484515\pi\)
0.0486282 + 0.998817i \(0.484515\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −16.3472 −0.832048
\(387\) 0 0
\(388\) 38.8863 1.97415
\(389\) 7.02219 0.356039 0.178020 0.984027i \(-0.443031\pi\)
0.178020 + 0.984027i \(0.443031\pi\)
\(390\) 0 0
\(391\) −18.4652 −0.933826
\(392\) −12.9674 −0.654953
\(393\) 0 0
\(394\) 8.26463 0.416366
\(395\) −12.5389 −0.630902
\(396\) 0 0
\(397\) −12.5149 −0.628104 −0.314052 0.949406i \(-0.601687\pi\)
−0.314052 + 0.949406i \(0.601687\pi\)
\(398\) −25.5637 −1.28139
\(399\) 0 0
\(400\) 2.48350 0.124175
\(401\) −27.3284 −1.36472 −0.682358 0.731018i \(-0.739045\pi\)
−0.682358 + 0.731018i \(0.739045\pi\)
\(402\) 0 0
\(403\) 23.7120 1.18118
\(404\) 64.3142 3.19975
\(405\) 0 0
\(406\) 16.5943 0.823561
\(407\) 0 0
\(408\) 0 0
\(409\) −11.5990 −0.573531 −0.286766 0.958001i \(-0.592580\pi\)
−0.286766 + 0.958001i \(0.592580\pi\)
\(410\) −16.9515 −0.837172
\(411\) 0 0
\(412\) 24.9164 1.22754
\(413\) 18.7419 0.922229
\(414\) 0 0
\(415\) 5.04333 0.247567
\(416\) −6.90762 −0.338674
\(417\) 0 0
\(418\) 0 0
\(419\) −9.07542 −0.443363 −0.221682 0.975119i \(-0.571155\pi\)
−0.221682 + 0.975119i \(0.571155\pi\)
\(420\) 0 0
\(421\) 18.5151 0.902372 0.451186 0.892430i \(-0.351001\pi\)
0.451186 + 0.892430i \(0.351001\pi\)
\(422\) −3.30314 −0.160794
\(423\) 0 0
\(424\) −17.0815 −0.829551
\(425\) 6.62030 0.321132
\(426\) 0 0
\(427\) 6.11106 0.295735
\(428\) 35.9330 1.73689
\(429\) 0 0
\(430\) −26.0840 −1.25788
\(431\) 24.7913 1.19415 0.597077 0.802184i \(-0.296329\pi\)
0.597077 + 0.802184i \(0.296329\pi\)
\(432\) 0 0
\(433\) −35.3858 −1.70053 −0.850267 0.526352i \(-0.823559\pi\)
−0.850267 + 0.526352i \(0.823559\pi\)
\(434\) 38.3066 1.83877
\(435\) 0 0
\(436\) −71.9485 −3.44571
\(437\) 9.92985 0.475009
\(438\) 0 0
\(439\) −10.2916 −0.491192 −0.245596 0.969372i \(-0.578984\pi\)
−0.245596 + 0.969372i \(0.578984\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 46.3001 2.20227
\(443\) −35.8502 −1.70330 −0.851648 0.524114i \(-0.824397\pi\)
−0.851648 + 0.524114i \(0.824397\pi\)
\(444\) 0 0
\(445\) −12.0573 −0.571571
\(446\) 28.9875 1.37260
\(447\) 0 0
\(448\) −20.9435 −0.989485
\(449\) 26.8436 1.26683 0.633414 0.773813i \(-0.281653\pi\)
0.633414 + 0.773813i \(0.281653\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 13.3093 0.626017
\(453\) 0 0
\(454\) −41.5086 −1.94810
\(455\) 5.75238 0.269676
\(456\) 0 0
\(457\) −19.6380 −0.918627 −0.459314 0.888274i \(-0.651905\pi\)
−0.459314 + 0.888274i \(0.651905\pi\)
\(458\) −2.00497 −0.0936860
\(459\) 0 0
\(460\) −10.4193 −0.485801
\(461\) −19.5626 −0.911122 −0.455561 0.890205i \(-0.650561\pi\)
−0.455561 + 0.890205i \(0.650561\pi\)
\(462\) 0 0
\(463\) −14.7474 −0.685371 −0.342686 0.939450i \(-0.611337\pi\)
−0.342686 + 0.939450i \(0.611337\pi\)
\(464\) 8.73578 0.405549
\(465\) 0 0
\(466\) −45.6185 −2.11324
\(467\) 10.3049 0.476854 0.238427 0.971160i \(-0.423368\pi\)
0.238427 + 0.971160i \(0.423368\pi\)
\(468\) 0 0
\(469\) 3.11816 0.143983
\(470\) 0.0650277 0.00299950
\(471\) 0 0
\(472\) 39.5476 1.82033
\(473\) 0 0
\(474\) 0 0
\(475\) −3.56013 −0.163350
\(476\) 48.7158 2.23288
\(477\) 0 0
\(478\) 61.7733 2.82544
\(479\) 11.3976 0.520768 0.260384 0.965505i \(-0.416151\pi\)
0.260384 + 0.965505i \(0.416151\pi\)
\(480\) 0 0
\(481\) 12.9363 0.589845
\(482\) −40.8762 −1.86186
\(483\) 0 0
\(484\) 0 0
\(485\) 10.4097 0.472678
\(486\) 0 0
\(487\) −34.6435 −1.56984 −0.784922 0.619594i \(-0.787297\pi\)
−0.784922 + 0.619594i \(0.787297\pi\)
\(488\) 12.8951 0.583732
\(489\) 0 0
\(490\) −7.47144 −0.337525
\(491\) 36.6153 1.65242 0.826212 0.563360i \(-0.190491\pi\)
0.826212 + 0.563360i \(0.190491\pi\)
\(492\) 0 0
\(493\) 23.2871 1.04880
\(494\) −24.8983 −1.12023
\(495\) 0 0
\(496\) 20.1658 0.905473
\(497\) 11.9341 0.535315
\(498\) 0 0
\(499\) −21.3515 −0.955824 −0.477912 0.878408i \(-0.658606\pi\)
−0.477912 + 0.878408i \(0.658606\pi\)
\(500\) 3.73560 0.167061
\(501\) 0 0
\(502\) −24.7073 −1.10274
\(503\) −11.6575 −0.519783 −0.259891 0.965638i \(-0.583687\pi\)
−0.259891 + 0.965638i \(0.583687\pi\)
\(504\) 0 0
\(505\) 17.2166 0.766127
\(506\) 0 0
\(507\) 0 0
\(508\) 72.1640 3.20176
\(509\) −29.3434 −1.30062 −0.650311 0.759668i \(-0.725361\pi\)
−0.650311 + 0.759668i \(0.725361\pi\)
\(510\) 0 0
\(511\) −4.67427 −0.206778
\(512\) −26.5205 −1.17205
\(513\) 0 0
\(514\) −26.9753 −1.18983
\(515\) 6.66999 0.293915
\(516\) 0 0
\(517\) 0 0
\(518\) 20.8986 0.918230
\(519\) 0 0
\(520\) 12.1382 0.532295
\(521\) −4.00057 −0.175268 −0.0876340 0.996153i \(-0.527931\pi\)
−0.0876340 + 0.996153i \(0.527931\pi\)
\(522\) 0 0
\(523\) −4.44571 −0.194397 −0.0971986 0.995265i \(-0.530988\pi\)
−0.0971986 + 0.995265i \(0.530988\pi\)
\(524\) 51.9348 2.26878
\(525\) 0 0
\(526\) 12.3357 0.537864
\(527\) 53.7563 2.34166
\(528\) 0 0
\(529\) −15.2205 −0.661759
\(530\) −9.84185 −0.427502
\(531\) 0 0
\(532\) −26.1974 −1.13580
\(533\) −20.6696 −0.895301
\(534\) 0 0
\(535\) 9.61908 0.415869
\(536\) 6.57969 0.284199
\(537\) 0 0
\(538\) 55.4474 2.39051
\(539\) 0 0
\(540\) 0 0
\(541\) −13.1859 −0.566905 −0.283452 0.958986i \(-0.591480\pi\)
−0.283452 + 0.958986i \(0.591480\pi\)
\(542\) 31.4810 1.35222
\(543\) 0 0
\(544\) −15.6599 −0.671415
\(545\) −19.2602 −0.825018
\(546\) 0 0
\(547\) 1.86825 0.0798806 0.0399403 0.999202i \(-0.487283\pi\)
0.0399403 + 0.999202i \(0.487283\pi\)
\(548\) −84.8988 −3.62670
\(549\) 0 0
\(550\) 0 0
\(551\) −12.5229 −0.533491
\(552\) 0 0
\(553\) −24.6997 −1.05034
\(554\) 24.7848 1.05300
\(555\) 0 0
\(556\) 47.4793 2.01357
\(557\) −13.5209 −0.572900 −0.286450 0.958095i \(-0.592475\pi\)
−0.286450 + 0.958095i \(0.592475\pi\)
\(558\) 0 0
\(559\) −31.8053 −1.34522
\(560\) 4.89211 0.206729
\(561\) 0 0
\(562\) 0.926879 0.0390980
\(563\) 30.5720 1.28846 0.644228 0.764833i \(-0.277179\pi\)
0.644228 + 0.764833i \(0.277179\pi\)
\(564\) 0 0
\(565\) 3.56283 0.149890
\(566\) −34.7869 −1.46220
\(567\) 0 0
\(568\) 25.1823 1.05662
\(569\) −36.4840 −1.52949 −0.764745 0.644333i \(-0.777135\pi\)
−0.764745 + 0.644333i \(0.777135\pi\)
\(570\) 0 0
\(571\) −19.9929 −0.836679 −0.418339 0.908291i \(-0.637388\pi\)
−0.418339 + 0.908291i \(0.637388\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −33.3917 −1.39374
\(575\) −2.78918 −0.116317
\(576\) 0 0
\(577\) −31.1211 −1.29559 −0.647794 0.761816i \(-0.724308\pi\)
−0.647794 + 0.761816i \(0.724308\pi\)
\(578\) 64.2514 2.67251
\(579\) 0 0
\(580\) 13.1401 0.545612
\(581\) 9.93458 0.412156
\(582\) 0 0
\(583\) 0 0
\(584\) −9.86327 −0.408145
\(585\) 0 0
\(586\) −35.4868 −1.46594
\(587\) 31.9068 1.31693 0.658467 0.752610i \(-0.271205\pi\)
0.658467 + 0.752610i \(0.271205\pi\)
\(588\) 0 0
\(589\) −28.9080 −1.19113
\(590\) 22.7862 0.938091
\(591\) 0 0
\(592\) 11.0017 0.452167
\(593\) 34.5829 1.42015 0.710074 0.704127i \(-0.248662\pi\)
0.710074 + 0.704127i \(0.248662\pi\)
\(594\) 0 0
\(595\) 13.0410 0.534627
\(596\) −16.0733 −0.658388
\(597\) 0 0
\(598\) −19.5066 −0.797684
\(599\) 17.7818 0.726543 0.363271 0.931683i \(-0.381660\pi\)
0.363271 + 0.931683i \(0.381660\pi\)
\(600\) 0 0
\(601\) −14.8208 −0.604554 −0.302277 0.953220i \(-0.597747\pi\)
−0.302277 + 0.953220i \(0.597747\pi\)
\(602\) −51.3813 −2.09415
\(603\) 0 0
\(604\) 65.6883 2.67282
\(605\) 0 0
\(606\) 0 0
\(607\) −13.8034 −0.560264 −0.280132 0.959962i \(-0.590378\pi\)
−0.280132 + 0.959962i \(0.590378\pi\)
\(608\) 8.42129 0.341528
\(609\) 0 0
\(610\) 7.42975 0.300822
\(611\) 0.0792910 0.00320777
\(612\) 0 0
\(613\) 22.4951 0.908570 0.454285 0.890856i \(-0.349895\pi\)
0.454285 + 0.890856i \(0.349895\pi\)
\(614\) 77.4846 3.12702
\(615\) 0 0
\(616\) 0 0
\(617\) 10.6195 0.427525 0.213762 0.976886i \(-0.431428\pi\)
0.213762 + 0.976886i \(0.431428\pi\)
\(618\) 0 0
\(619\) −2.63250 −0.105809 −0.0529046 0.998600i \(-0.516848\pi\)
−0.0529046 + 0.998600i \(0.516848\pi\)
\(620\) 30.3328 1.21819
\(621\) 0 0
\(622\) 4.22354 0.169349
\(623\) −23.7510 −0.951563
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −62.6832 −2.50532
\(627\) 0 0
\(628\) −88.9493 −3.54946
\(629\) 29.3273 1.16936
\(630\) 0 0
\(631\) 24.1315 0.960660 0.480330 0.877088i \(-0.340517\pi\)
0.480330 + 0.877088i \(0.340517\pi\)
\(632\) −52.1193 −2.07320
\(633\) 0 0
\(634\) 24.8483 0.986852
\(635\) 19.3179 0.766608
\(636\) 0 0
\(637\) −9.11024 −0.360961
\(638\) 0 0
\(639\) 0 0
\(640\) −20.7319 −0.819500
\(641\) −45.7507 −1.80705 −0.903523 0.428540i \(-0.859028\pi\)
−0.903523 + 0.428540i \(0.859028\pi\)
\(642\) 0 0
\(643\) −6.60925 −0.260643 −0.130322 0.991472i \(-0.541601\pi\)
−0.130322 + 0.991472i \(0.541601\pi\)
\(644\) −20.5243 −0.808772
\(645\) 0 0
\(646\) −56.4459 −2.22083
\(647\) −1.86148 −0.0731824 −0.0365912 0.999330i \(-0.511650\pi\)
−0.0365912 + 0.999330i \(0.511650\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 6.99366 0.274314
\(651\) 0 0
\(652\) −74.1337 −2.90330
\(653\) 31.9277 1.24943 0.624713 0.780854i \(-0.285216\pi\)
0.624713 + 0.780854i \(0.285216\pi\)
\(654\) 0 0
\(655\) 13.9027 0.543223
\(656\) −17.5785 −0.686325
\(657\) 0 0
\(658\) 0.128094 0.00499364
\(659\) −8.36924 −0.326019 −0.163010 0.986624i \(-0.552120\pi\)
−0.163010 + 0.986624i \(0.552120\pi\)
\(660\) 0 0
\(661\) −34.0773 −1.32545 −0.662726 0.748862i \(-0.730601\pi\)
−0.662726 + 0.748862i \(0.730601\pi\)
\(662\) 22.0902 0.858559
\(663\) 0 0
\(664\) 20.9631 0.813527
\(665\) −7.01290 −0.271949
\(666\) 0 0
\(667\) −9.81102 −0.379884
\(668\) −54.0766 −2.09228
\(669\) 0 0
\(670\) 3.79102 0.146460
\(671\) 0 0
\(672\) 0 0
\(673\) −2.06786 −0.0797103 −0.0398551 0.999205i \(-0.512690\pi\)
−0.0398551 + 0.999205i \(0.512690\pi\)
\(674\) 68.3441 2.63252
\(675\) 0 0
\(676\) −16.7068 −0.642569
\(677\) 18.0905 0.695275 0.347637 0.937629i \(-0.386984\pi\)
0.347637 + 0.937629i \(0.386984\pi\)
\(678\) 0 0
\(679\) 20.5054 0.786925
\(680\) 27.5180 1.05527
\(681\) 0 0
\(682\) 0 0
\(683\) −47.3828 −1.81305 −0.906526 0.422149i \(-0.861276\pi\)
−0.906526 + 0.422149i \(0.861276\pi\)
\(684\) 0 0
\(685\) −22.7270 −0.868352
\(686\) −47.7408 −1.82275
\(687\) 0 0
\(688\) −27.0488 −1.03123
\(689\) −12.0006 −0.457186
\(690\) 0 0
\(691\) 15.8475 0.602869 0.301435 0.953487i \(-0.402535\pi\)
0.301435 + 0.953487i \(0.402535\pi\)
\(692\) 16.1342 0.613331
\(693\) 0 0
\(694\) −24.3558 −0.924534
\(695\) 12.7100 0.482116
\(696\) 0 0
\(697\) −46.8592 −1.77492
\(698\) −56.8012 −2.14996
\(699\) 0 0
\(700\) 7.35855 0.278127
\(701\) 26.3059 0.993560 0.496780 0.867876i \(-0.334516\pi\)
0.496780 + 0.867876i \(0.334516\pi\)
\(702\) 0 0
\(703\) −15.7711 −0.594817
\(704\) 0 0
\(705\) 0 0
\(706\) 33.8599 1.27433
\(707\) 33.9139 1.27547
\(708\) 0 0
\(709\) 30.8329 1.15795 0.578977 0.815344i \(-0.303452\pi\)
0.578977 + 0.815344i \(0.303452\pi\)
\(710\) 14.5093 0.544523
\(711\) 0 0
\(712\) −50.1174 −1.87823
\(713\) −22.6479 −0.848172
\(714\) 0 0
\(715\) 0 0
\(716\) −27.8643 −1.04134
\(717\) 0 0
\(718\) 57.5569 2.14801
\(719\) 47.2434 1.76188 0.880940 0.473228i \(-0.156911\pi\)
0.880940 + 0.473228i \(0.156911\pi\)
\(720\) 0 0
\(721\) 13.1388 0.489316
\(722\) −15.1490 −0.563786
\(723\) 0 0
\(724\) −76.4432 −2.84099
\(725\) 3.51753 0.130638
\(726\) 0 0
\(727\) −24.4299 −0.906054 −0.453027 0.891497i \(-0.649656\pi\)
−0.453027 + 0.891497i \(0.649656\pi\)
\(728\) 23.9104 0.886177
\(729\) 0 0
\(730\) −5.68292 −0.210334
\(731\) −72.1043 −2.66688
\(732\) 0 0
\(733\) 27.9875 1.03374 0.516870 0.856064i \(-0.327097\pi\)
0.516870 + 0.856064i \(0.327097\pi\)
\(734\) −4.30379 −0.158856
\(735\) 0 0
\(736\) 6.59766 0.243193
\(737\) 0 0
\(738\) 0 0
\(739\) −2.70533 −0.0995172 −0.0497586 0.998761i \(-0.515845\pi\)
−0.0497586 + 0.998761i \(0.515845\pi\)
\(740\) 16.5484 0.608331
\(741\) 0 0
\(742\) −19.3869 −0.711716
\(743\) 4.14414 0.152034 0.0760168 0.997107i \(-0.475780\pi\)
0.0760168 + 0.997107i \(0.475780\pi\)
\(744\) 0 0
\(745\) −4.30274 −0.157640
\(746\) −16.4723 −0.603092
\(747\) 0 0
\(748\) 0 0
\(749\) 18.9481 0.692348
\(750\) 0 0
\(751\) −44.4316 −1.62133 −0.810666 0.585508i \(-0.800895\pi\)
−0.810666 + 0.585508i \(0.800895\pi\)
\(752\) 0.0674331 0.00245903
\(753\) 0 0
\(754\) 24.6004 0.895894
\(755\) 17.5844 0.639962
\(756\) 0 0
\(757\) 29.9901 1.09001 0.545004 0.838433i \(-0.316528\pi\)
0.545004 + 0.838433i \(0.316528\pi\)
\(758\) −8.46192 −0.307351
\(759\) 0 0
\(760\) −14.7980 −0.536782
\(761\) 39.8420 1.44427 0.722135 0.691752i \(-0.243161\pi\)
0.722135 + 0.691752i \(0.243161\pi\)
\(762\) 0 0
\(763\) −37.9397 −1.37351
\(764\) 24.4747 0.885462
\(765\) 0 0
\(766\) 4.55834 0.164700
\(767\) 27.7841 1.00323
\(768\) 0 0
\(769\) 20.5304 0.740346 0.370173 0.928963i \(-0.379298\pi\)
0.370173 + 0.928963i \(0.379298\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −25.4984 −0.917708
\(773\) 16.3315 0.587401 0.293701 0.955897i \(-0.405113\pi\)
0.293701 + 0.955897i \(0.405113\pi\)
\(774\) 0 0
\(775\) 8.11992 0.291676
\(776\) 43.2688 1.55326
\(777\) 0 0
\(778\) 16.8175 0.602938
\(779\) 25.1990 0.902848
\(780\) 0 0
\(781\) 0 0
\(782\) −44.2225 −1.58139
\(783\) 0 0
\(784\) −7.74781 −0.276708
\(785\) −23.8112 −0.849860
\(786\) 0 0
\(787\) −7.38766 −0.263342 −0.131671 0.991293i \(-0.542034\pi\)
−0.131671 + 0.991293i \(0.542034\pi\)
\(788\) 12.8912 0.459231
\(789\) 0 0
\(790\) −30.0296 −1.06841
\(791\) 7.01823 0.249539
\(792\) 0 0
\(793\) 9.05941 0.321709
\(794\) −29.9720 −1.06367
\(795\) 0 0
\(796\) −39.8744 −1.41331
\(797\) 23.1554 0.820207 0.410104 0.912039i \(-0.365492\pi\)
0.410104 + 0.912039i \(0.365492\pi\)
\(798\) 0 0
\(799\) 0.179757 0.00635935
\(800\) −2.36545 −0.0836311
\(801\) 0 0
\(802\) −65.4491 −2.31109
\(803\) 0 0
\(804\) 0 0
\(805\) −5.49425 −0.193647
\(806\) 56.7880 2.00027
\(807\) 0 0
\(808\) 71.5624 2.51756
\(809\) −14.9484 −0.525559 −0.262779 0.964856i \(-0.584639\pi\)
−0.262779 + 0.964856i \(0.584639\pi\)
\(810\) 0 0
\(811\) −25.1135 −0.881855 −0.440927 0.897543i \(-0.645350\pi\)
−0.440927 + 0.897543i \(0.645350\pi\)
\(812\) 25.8839 0.908347
\(813\) 0 0
\(814\) 0 0
\(815\) −19.8452 −0.695147
\(816\) 0 0
\(817\) 38.7748 1.35656
\(818\) −27.7785 −0.971251
\(819\) 0 0
\(820\) −26.4410 −0.923359
\(821\) −13.1035 −0.457315 −0.228657 0.973507i \(-0.573434\pi\)
−0.228657 + 0.973507i \(0.573434\pi\)
\(822\) 0 0
\(823\) −20.5741 −0.717169 −0.358584 0.933497i \(-0.616741\pi\)
−0.358584 + 0.933497i \(0.616741\pi\)
\(824\) 27.7245 0.965828
\(825\) 0 0
\(826\) 44.8852 1.56176
\(827\) 12.6343 0.439338 0.219669 0.975574i \(-0.429502\pi\)
0.219669 + 0.975574i \(0.429502\pi\)
\(828\) 0 0
\(829\) 7.78876 0.270515 0.135257 0.990810i \(-0.456814\pi\)
0.135257 + 0.990810i \(0.456814\pi\)
\(830\) 12.0783 0.419245
\(831\) 0 0
\(832\) −31.0479 −1.07639
\(833\) −20.6534 −0.715599
\(834\) 0 0
\(835\) −14.4760 −0.500963
\(836\) 0 0
\(837\) 0 0
\(838\) −21.7348 −0.750817
\(839\) 30.5522 1.05478 0.527389 0.849624i \(-0.323171\pi\)
0.527389 + 0.849624i \(0.323171\pi\)
\(840\) 0 0
\(841\) −16.6270 −0.573345
\(842\) 44.3421 1.52813
\(843\) 0 0
\(844\) −5.15226 −0.177348
\(845\) −4.47232 −0.153853
\(846\) 0 0
\(847\) 0 0
\(848\) −10.2059 −0.350472
\(849\) 0 0
\(850\) 15.8550 0.543823
\(851\) −12.3558 −0.423553
\(852\) 0 0
\(853\) 25.6737 0.879051 0.439525 0.898230i \(-0.355147\pi\)
0.439525 + 0.898230i \(0.355147\pi\)
\(854\) 14.6354 0.500815
\(855\) 0 0
\(856\) 39.9827 1.36658
\(857\) 14.7370 0.503407 0.251703 0.967804i \(-0.419009\pi\)
0.251703 + 0.967804i \(0.419009\pi\)
\(858\) 0 0
\(859\) −1.68741 −0.0575738 −0.0287869 0.999586i \(-0.509164\pi\)
−0.0287869 + 0.999586i \(0.509164\pi\)
\(860\) −40.6859 −1.38738
\(861\) 0 0
\(862\) 59.3729 2.02225
\(863\) −13.1473 −0.447538 −0.223769 0.974642i \(-0.571836\pi\)
−0.223769 + 0.974642i \(0.571836\pi\)
\(864\) 0 0
\(865\) 4.31904 0.146852
\(866\) −84.7458 −2.87978
\(867\) 0 0
\(868\) 59.7509 2.02808
\(869\) 0 0
\(870\) 0 0
\(871\) 4.62255 0.156629
\(872\) −80.0572 −2.71108
\(873\) 0 0
\(874\) 23.7811 0.804408
\(875\) 1.96984 0.0665929
\(876\) 0 0
\(877\) −46.9492 −1.58536 −0.792681 0.609637i \(-0.791315\pi\)
−0.792681 + 0.609637i \(0.791315\pi\)
\(878\) −24.6475 −0.831813
\(879\) 0 0
\(880\) 0 0
\(881\) −14.1843 −0.477880 −0.238940 0.971034i \(-0.576800\pi\)
−0.238940 + 0.971034i \(0.576800\pi\)
\(882\) 0 0
\(883\) −10.3899 −0.349647 −0.174824 0.984600i \(-0.555936\pi\)
−0.174824 + 0.984600i \(0.555936\pi\)
\(884\) 72.2193 2.42900
\(885\) 0 0
\(886\) −85.8581 −2.88446
\(887\) 3.84764 0.129191 0.0645955 0.997912i \(-0.479424\pi\)
0.0645955 + 0.997912i \(0.479424\pi\)
\(888\) 0 0
\(889\) 38.0533 1.27627
\(890\) −28.8761 −0.967931
\(891\) 0 0
\(892\) 45.2150 1.51391
\(893\) −0.0966662 −0.00323481
\(894\) 0 0
\(895\) −7.45911 −0.249331
\(896\) −40.8386 −1.36432
\(897\) 0 0
\(898\) 64.2881 2.14532
\(899\) 28.5621 0.952598
\(900\) 0 0
\(901\) −27.2060 −0.906363
\(902\) 0 0
\(903\) 0 0
\(904\) 14.8093 0.492550
\(905\) −20.4634 −0.680228
\(906\) 0 0
\(907\) 1.20437 0.0399904 0.0199952 0.999800i \(-0.493635\pi\)
0.0199952 + 0.999800i \(0.493635\pi\)
\(908\) −64.7455 −2.14865
\(909\) 0 0
\(910\) 13.7764 0.456684
\(911\) −38.9307 −1.28983 −0.644915 0.764254i \(-0.723107\pi\)
−0.644915 + 0.764254i \(0.723107\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −47.0313 −1.55566
\(915\) 0 0
\(916\) −3.12736 −0.103331
\(917\) 27.3861 0.904369
\(918\) 0 0
\(919\) −9.56757 −0.315605 −0.157802 0.987471i \(-0.550441\pi\)
−0.157802 + 0.987471i \(0.550441\pi\)
\(920\) −11.5935 −0.382227
\(921\) 0 0
\(922\) −46.8507 −1.54295
\(923\) 17.6918 0.582332
\(924\) 0 0
\(925\) 4.42991 0.145655
\(926\) −35.3188 −1.16065
\(927\) 0 0
\(928\) −8.32052 −0.273134
\(929\) 6.45603 0.211816 0.105908 0.994376i \(-0.466225\pi\)
0.105908 + 0.994376i \(0.466225\pi\)
\(930\) 0 0
\(931\) 11.1066 0.364003
\(932\) −71.1561 −2.33079
\(933\) 0 0
\(934\) 24.6793 0.807532
\(935\) 0 0
\(936\) 0 0
\(937\) 0.478554 0.0156337 0.00781684 0.999969i \(-0.497512\pi\)
0.00781684 + 0.999969i \(0.497512\pi\)
\(938\) 7.46772 0.243830
\(939\) 0 0
\(940\) 0.101431 0.00330830
\(941\) −18.3345 −0.597689 −0.298845 0.954302i \(-0.596601\pi\)
−0.298845 + 0.954302i \(0.596601\pi\)
\(942\) 0 0
\(943\) 19.7421 0.642893
\(944\) 23.6290 0.769060
\(945\) 0 0
\(946\) 0 0
\(947\) 23.6619 0.768908 0.384454 0.923144i \(-0.374390\pi\)
0.384454 + 0.923144i \(0.374390\pi\)
\(948\) 0 0
\(949\) −6.92943 −0.224939
\(950\) −8.52619 −0.276626
\(951\) 0 0
\(952\) 54.2061 1.75683
\(953\) −19.5111 −0.632026 −0.316013 0.948755i \(-0.602344\pi\)
−0.316013 + 0.948755i \(0.602344\pi\)
\(954\) 0 0
\(955\) 6.55174 0.212009
\(956\) 96.3544 3.11632
\(957\) 0 0
\(958\) 27.2962 0.881899
\(959\) −44.7686 −1.44565
\(960\) 0 0
\(961\) 34.9332 1.12688
\(962\) 30.9813 0.998878
\(963\) 0 0
\(964\) −63.7591 −2.05354
\(965\) −6.82579 −0.219730
\(966\) 0 0
\(967\) −13.3686 −0.429905 −0.214953 0.976624i \(-0.568960\pi\)
−0.214953 + 0.976624i \(0.568960\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 24.9302 0.800461
\(971\) −55.6001 −1.78429 −0.892146 0.451747i \(-0.850801\pi\)
−0.892146 + 0.451747i \(0.850801\pi\)
\(972\) 0 0
\(973\) 25.0366 0.802638
\(974\) −82.9680 −2.65847
\(975\) 0 0
\(976\) 7.70458 0.246618
\(977\) 42.9068 1.37271 0.686355 0.727267i \(-0.259210\pi\)
0.686355 + 0.727267i \(0.259210\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −11.6540 −0.372273
\(981\) 0 0
\(982\) 87.6903 2.79831
\(983\) −34.5788 −1.10289 −0.551446 0.834211i \(-0.685924\pi\)
−0.551446 + 0.834211i \(0.685924\pi\)
\(984\) 0 0
\(985\) 3.45091 0.109955
\(986\) 55.7705 1.77609
\(987\) 0 0
\(988\) −38.8366 −1.23556
\(989\) 30.3781 0.965968
\(990\) 0 0
\(991\) 15.5839 0.495038 0.247519 0.968883i \(-0.420385\pi\)
0.247519 + 0.968883i \(0.420385\pi\)
\(992\) −19.2072 −0.609830
\(993\) 0 0
\(994\) 28.5810 0.906534
\(995\) −10.6742 −0.338394
\(996\) 0 0
\(997\) 4.04074 0.127972 0.0639858 0.997951i \(-0.479619\pi\)
0.0639858 + 0.997951i \(0.479619\pi\)
\(998\) −51.1349 −1.61865
\(999\) 0 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 5445.2.a.cd.1.7 8
3.2 odd 2 5445.2.a.ca.1.2 8
11.5 even 5 495.2.n.g.91.1 16
11.9 even 5 495.2.n.g.136.1 yes 16
11.10 odd 2 5445.2.a.cb.1.2 8
33.5 odd 10 495.2.n.h.91.4 yes 16
33.20 odd 10 495.2.n.h.136.4 yes 16
33.32 even 2 5445.2.a.cc.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
495.2.n.g.91.1 16 11.5 even 5
495.2.n.g.136.1 yes 16 11.9 even 5
495.2.n.h.91.4 yes 16 33.5 odd 10
495.2.n.h.136.4 yes 16 33.20 odd 10
5445.2.a.ca.1.2 8 3.2 odd 2
5445.2.a.cb.1.2 8 11.10 odd 2
5445.2.a.cc.1.7 8 33.32 even 2
5445.2.a.cd.1.7 8 1.1 even 1 trivial