Properties

Label 4006.2.a.g.1.19
Level 4006
Weight 2
Character 4006.1
Self dual yes
Analytic conductor 31.988
Analytic rank 1
Dimension 40
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4006 = 2 \cdot 2003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4006.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9880710497\)
Analytic rank: \(1\)
Dimension: \(40\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.19
Character \(\chi\) = 4006.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.535052 q^{3} +1.00000 q^{4} +2.84024 q^{5} +0.535052 q^{6} +2.46812 q^{7} -1.00000 q^{8} -2.71372 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.535052 q^{3} +1.00000 q^{4} +2.84024 q^{5} +0.535052 q^{6} +2.46812 q^{7} -1.00000 q^{8} -2.71372 q^{9} -2.84024 q^{10} +3.17739 q^{11} -0.535052 q^{12} -7.14791 q^{13} -2.46812 q^{14} -1.51968 q^{15} +1.00000 q^{16} +0.244682 q^{17} +2.71372 q^{18} -7.69456 q^{19} +2.84024 q^{20} -1.32057 q^{21} -3.17739 q^{22} +4.96361 q^{23} +0.535052 q^{24} +3.06699 q^{25} +7.14791 q^{26} +3.05714 q^{27} +2.46812 q^{28} +3.48771 q^{29} +1.51968 q^{30} -4.76087 q^{31} -1.00000 q^{32} -1.70007 q^{33} -0.244682 q^{34} +7.01006 q^{35} -2.71372 q^{36} -3.01634 q^{37} +7.69456 q^{38} +3.82450 q^{39} -2.84024 q^{40} +10.4380 q^{41} +1.32057 q^{42} -8.01689 q^{43} +3.17739 q^{44} -7.70763 q^{45} -4.96361 q^{46} -11.6638 q^{47} -0.535052 q^{48} -0.908392 q^{49} -3.06699 q^{50} -0.130918 q^{51} -7.14791 q^{52} -2.04870 q^{53} -3.05714 q^{54} +9.02456 q^{55} -2.46812 q^{56} +4.11699 q^{57} -3.48771 q^{58} -10.3363 q^{59} -1.51968 q^{60} -14.4000 q^{61} +4.76087 q^{62} -6.69778 q^{63} +1.00000 q^{64} -20.3018 q^{65} +1.70007 q^{66} -13.7795 q^{67} +0.244682 q^{68} -2.65579 q^{69} -7.01006 q^{70} -4.34345 q^{71} +2.71372 q^{72} +8.56207 q^{73} +3.01634 q^{74} -1.64100 q^{75} -7.69456 q^{76} +7.84217 q^{77} -3.82450 q^{78} -1.56339 q^{79} +2.84024 q^{80} +6.50543 q^{81} -10.4380 q^{82} +10.0964 q^{83} -1.32057 q^{84} +0.694957 q^{85} +8.01689 q^{86} -1.86611 q^{87} -3.17739 q^{88} +8.19194 q^{89} +7.70763 q^{90} -17.6419 q^{91} +4.96361 q^{92} +2.54731 q^{93} +11.6638 q^{94} -21.8544 q^{95} +0.535052 q^{96} +3.65276 q^{97} +0.908392 q^{98} -8.62255 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + O(q^{10}) \) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + 23q^{10} - 28q^{11} - q^{12} - 12q^{13} - 12q^{14} - 14q^{15} + 40q^{16} - 10q^{17} - 29q^{18} + 3q^{19} - 23q^{20} - 40q^{21} + 28q^{22} - 10q^{23} + q^{24} + 43q^{25} + 12q^{26} - 7q^{27} + 12q^{28} - 37q^{29} + 14q^{30} - 16q^{31} - 40q^{32} - 11q^{33} + 10q^{34} - 24q^{35} + 29q^{36} - 17q^{37} - 3q^{38} - 10q^{39} + 23q^{40} - 58q^{41} + 40q^{42} + 37q^{43} - 28q^{44} - 66q^{45} + 10q^{46} - 34q^{47} - q^{48} + 28q^{49} - 43q^{50} - 7q^{51} - 12q^{52} - 43q^{53} + 7q^{54} + 28q^{55} - 12q^{56} - 25q^{57} + 37q^{58} - 92q^{59} - 14q^{60} - 37q^{61} + 16q^{62} + 14q^{63} + 40q^{64} - 54q^{65} + 11q^{66} - 10q^{67} - 10q^{68} - 49q^{69} + 24q^{70} - 87q^{71} - 29q^{72} + 12q^{73} + 17q^{74} - 31q^{75} + 3q^{76} - 53q^{77} + 10q^{78} + 14q^{79} - 23q^{80} - 4q^{81} + 58q^{82} - 42q^{83} - 40q^{84} - 24q^{85} - 37q^{86} + 24q^{87} + 28q^{88} - 118q^{89} + 66q^{90} - 35q^{91} - 10q^{92} - 22q^{93} + 34q^{94} - 18q^{95} + q^{96} + 2q^{97} - 28q^{98} - 48q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.535052 −0.308912 −0.154456 0.988000i \(-0.549363\pi\)
−0.154456 + 0.988000i \(0.549363\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.84024 1.27020 0.635098 0.772432i \(-0.280960\pi\)
0.635098 + 0.772432i \(0.280960\pi\)
\(6\) 0.535052 0.218434
\(7\) 2.46812 0.932861 0.466431 0.884558i \(-0.345540\pi\)
0.466431 + 0.884558i \(0.345540\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.71372 −0.904573
\(10\) −2.84024 −0.898164
\(11\) 3.17739 0.958019 0.479010 0.877810i \(-0.340996\pi\)
0.479010 + 0.877810i \(0.340996\pi\)
\(12\) −0.535052 −0.154456
\(13\) −7.14791 −1.98247 −0.991237 0.132092i \(-0.957830\pi\)
−0.991237 + 0.132092i \(0.957830\pi\)
\(14\) −2.46812 −0.659632
\(15\) −1.51968 −0.392379
\(16\) 1.00000 0.250000
\(17\) 0.244682 0.0593442 0.0296721 0.999560i \(-0.490554\pi\)
0.0296721 + 0.999560i \(0.490554\pi\)
\(18\) 2.71372 0.639630
\(19\) −7.69456 −1.76525 −0.882626 0.470076i \(-0.844227\pi\)
−0.882626 + 0.470076i \(0.844227\pi\)
\(20\) 2.84024 0.635098
\(21\) −1.32057 −0.288172
\(22\) −3.17739 −0.677422
\(23\) 4.96361 1.03498 0.517492 0.855688i \(-0.326866\pi\)
0.517492 + 0.855688i \(0.326866\pi\)
\(24\) 0.535052 0.109217
\(25\) 3.06699 0.613398
\(26\) 7.14791 1.40182
\(27\) 3.05714 0.588346
\(28\) 2.46812 0.466431
\(29\) 3.48771 0.647652 0.323826 0.946117i \(-0.395031\pi\)
0.323826 + 0.946117i \(0.395031\pi\)
\(30\) 1.51968 0.277454
\(31\) −4.76087 −0.855078 −0.427539 0.903997i \(-0.640619\pi\)
−0.427539 + 0.903997i \(0.640619\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.70007 −0.295944
\(34\) −0.244682 −0.0419627
\(35\) 7.01006 1.18492
\(36\) −2.71372 −0.452287
\(37\) −3.01634 −0.495883 −0.247941 0.968775i \(-0.579754\pi\)
−0.247941 + 0.968775i \(0.579754\pi\)
\(38\) 7.69456 1.24822
\(39\) 3.82450 0.612411
\(40\) −2.84024 −0.449082
\(41\) 10.4380 1.63013 0.815067 0.579366i \(-0.196700\pi\)
0.815067 + 0.579366i \(0.196700\pi\)
\(42\) 1.32057 0.203768
\(43\) −8.01689 −1.22256 −0.611282 0.791413i \(-0.709346\pi\)
−0.611282 + 0.791413i \(0.709346\pi\)
\(44\) 3.17739 0.479010
\(45\) −7.70763 −1.14899
\(46\) −4.96361 −0.731844
\(47\) −11.6638 −1.70133 −0.850667 0.525705i \(-0.823801\pi\)
−0.850667 + 0.525705i \(0.823801\pi\)
\(48\) −0.535052 −0.0772280
\(49\) −0.908392 −0.129770
\(50\) −3.06699 −0.433738
\(51\) −0.130918 −0.0183321
\(52\) −7.14791 −0.991237
\(53\) −2.04870 −0.281410 −0.140705 0.990052i \(-0.544937\pi\)
−0.140705 + 0.990052i \(0.544937\pi\)
\(54\) −3.05714 −0.416023
\(55\) 9.02456 1.21687
\(56\) −2.46812 −0.329816
\(57\) 4.11699 0.545308
\(58\) −3.48771 −0.457959
\(59\) −10.3363 −1.34567 −0.672835 0.739792i \(-0.734924\pi\)
−0.672835 + 0.739792i \(0.734924\pi\)
\(60\) −1.51968 −0.196190
\(61\) −14.4000 −1.84373 −0.921866 0.387508i \(-0.873336\pi\)
−0.921866 + 0.387508i \(0.873336\pi\)
\(62\) 4.76087 0.604631
\(63\) −6.69778 −0.843841
\(64\) 1.00000 0.125000
\(65\) −20.3018 −2.51813
\(66\) 1.70007 0.209264
\(67\) −13.7795 −1.68343 −0.841716 0.539920i \(-0.818454\pi\)
−0.841716 + 0.539920i \(0.818454\pi\)
\(68\) 0.244682 0.0296721
\(69\) −2.65579 −0.319719
\(70\) −7.01006 −0.837862
\(71\) −4.34345 −0.515473 −0.257737 0.966215i \(-0.582977\pi\)
−0.257737 + 0.966215i \(0.582977\pi\)
\(72\) 2.71372 0.319815
\(73\) 8.56207 1.00211 0.501057 0.865414i \(-0.332945\pi\)
0.501057 + 0.865414i \(0.332945\pi\)
\(74\) 3.01634 0.350642
\(75\) −1.64100 −0.189486
\(76\) −7.69456 −0.882626
\(77\) 7.84217 0.893699
\(78\) −3.82450 −0.433040
\(79\) −1.56339 −0.175895 −0.0879473 0.996125i \(-0.528031\pi\)
−0.0879473 + 0.996125i \(0.528031\pi\)
\(80\) 2.84024 0.317549
\(81\) 6.50543 0.722826
\(82\) −10.4380 −1.15268
\(83\) 10.0964 1.10822 0.554111 0.832443i \(-0.313058\pi\)
0.554111 + 0.832443i \(0.313058\pi\)
\(84\) −1.32057 −0.144086
\(85\) 0.694957 0.0753787
\(86\) 8.01689 0.864483
\(87\) −1.86611 −0.200068
\(88\) −3.17739 −0.338711
\(89\) 8.19194 0.868344 0.434172 0.900830i \(-0.357041\pi\)
0.434172 + 0.900830i \(0.357041\pi\)
\(90\) 7.70763 0.812455
\(91\) −17.6419 −1.84937
\(92\) 4.96361 0.517492
\(93\) 2.54731 0.264144
\(94\) 11.6638 1.20302
\(95\) −21.8544 −2.24222
\(96\) 0.535052 0.0546085
\(97\) 3.65276 0.370881 0.185441 0.982655i \(-0.440629\pi\)
0.185441 + 0.982655i \(0.440629\pi\)
\(98\) 0.908392 0.0917615
\(99\) −8.62255 −0.866598
\(100\) 3.06699 0.306699
\(101\) −5.33264 −0.530618 −0.265309 0.964163i \(-0.585474\pi\)
−0.265309 + 0.964163i \(0.585474\pi\)
\(102\) 0.130918 0.0129628
\(103\) 6.12186 0.603205 0.301602 0.953434i \(-0.402478\pi\)
0.301602 + 0.953434i \(0.402478\pi\)
\(104\) 7.14791 0.700911
\(105\) −3.75074 −0.366035
\(106\) 2.04870 0.198987
\(107\) 8.44585 0.816492 0.408246 0.912872i \(-0.366141\pi\)
0.408246 + 0.912872i \(0.366141\pi\)
\(108\) 3.05714 0.294173
\(109\) −5.78322 −0.553932 −0.276966 0.960880i \(-0.589329\pi\)
−0.276966 + 0.960880i \(0.589329\pi\)
\(110\) −9.02456 −0.860458
\(111\) 1.61390 0.153184
\(112\) 2.46812 0.233215
\(113\) −8.59201 −0.808268 −0.404134 0.914700i \(-0.632427\pi\)
−0.404134 + 0.914700i \(0.632427\pi\)
\(114\) −4.11699 −0.385591
\(115\) 14.0979 1.31463
\(116\) 3.48771 0.323826
\(117\) 19.3974 1.79329
\(118\) 10.3363 0.951533
\(119\) 0.603905 0.0553599
\(120\) 1.51968 0.138727
\(121\) −0.904194 −0.0821995
\(122\) 14.4000 1.30372
\(123\) −5.58484 −0.503568
\(124\) −4.76087 −0.427539
\(125\) −5.49022 −0.491060
\(126\) 6.69778 0.596686
\(127\) 18.1164 1.60757 0.803786 0.594919i \(-0.202816\pi\)
0.803786 + 0.594919i \(0.202816\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.28945 0.377665
\(130\) 20.3018 1.78059
\(131\) 11.1235 0.971869 0.485934 0.873995i \(-0.338479\pi\)
0.485934 + 0.873995i \(0.338479\pi\)
\(132\) −1.70007 −0.147972
\(133\) −18.9911 −1.64674
\(134\) 13.7795 1.19037
\(135\) 8.68301 0.747315
\(136\) −0.244682 −0.0209813
\(137\) −0.247378 −0.0211349 −0.0105674 0.999944i \(-0.503364\pi\)
−0.0105674 + 0.999944i \(0.503364\pi\)
\(138\) 2.65579 0.226076
\(139\) −4.73479 −0.401600 −0.200800 0.979632i \(-0.564354\pi\)
−0.200800 + 0.979632i \(0.564354\pi\)
\(140\) 7.01006 0.592458
\(141\) 6.24071 0.525563
\(142\) 4.34345 0.364495
\(143\) −22.7117 −1.89925
\(144\) −2.71372 −0.226143
\(145\) 9.90596 0.822645
\(146\) −8.56207 −0.708602
\(147\) 0.486037 0.0400876
\(148\) −3.01634 −0.247941
\(149\) −13.2074 −1.08199 −0.540995 0.841026i \(-0.681952\pi\)
−0.540995 + 0.841026i \(0.681952\pi\)
\(150\) 1.64100 0.133987
\(151\) −18.5776 −1.51182 −0.755912 0.654674i \(-0.772806\pi\)
−0.755912 + 0.654674i \(0.772806\pi\)
\(152\) 7.69456 0.624111
\(153\) −0.663999 −0.0536811
\(154\) −7.84217 −0.631940
\(155\) −13.5220 −1.08612
\(156\) 3.82450 0.306205
\(157\) 5.00903 0.399764 0.199882 0.979820i \(-0.435944\pi\)
0.199882 + 0.979820i \(0.435944\pi\)
\(158\) 1.56339 0.124376
\(159\) 1.09616 0.0869311
\(160\) −2.84024 −0.224541
\(161\) 12.2508 0.965496
\(162\) −6.50543 −0.511115
\(163\) −11.0324 −0.864121 −0.432060 0.901845i \(-0.642213\pi\)
−0.432060 + 0.901845i \(0.642213\pi\)
\(164\) 10.4380 0.815067
\(165\) −4.82861 −0.375907
\(166\) −10.0964 −0.783631
\(167\) 4.23271 0.327537 0.163769 0.986499i \(-0.447635\pi\)
0.163769 + 0.986499i \(0.447635\pi\)
\(168\) 1.32057 0.101884
\(169\) 38.0927 2.93021
\(170\) −0.694957 −0.0533008
\(171\) 20.8809 1.59680
\(172\) −8.01689 −0.611282
\(173\) −6.77456 −0.515060 −0.257530 0.966270i \(-0.582909\pi\)
−0.257530 + 0.966270i \(0.582909\pi\)
\(174\) 1.86611 0.141469
\(175\) 7.56969 0.572215
\(176\) 3.17739 0.239505
\(177\) 5.53045 0.415694
\(178\) −8.19194 −0.614012
\(179\) 0.473280 0.0353746 0.0176873 0.999844i \(-0.494370\pi\)
0.0176873 + 0.999844i \(0.494370\pi\)
\(180\) −7.70763 −0.574493
\(181\) −11.1549 −0.829139 −0.414569 0.910018i \(-0.636068\pi\)
−0.414569 + 0.910018i \(0.636068\pi\)
\(182\) 17.6419 1.30770
\(183\) 7.70475 0.569551
\(184\) −4.96361 −0.365922
\(185\) −8.56713 −0.629868
\(186\) −2.54731 −0.186778
\(187\) 0.777451 0.0568528
\(188\) −11.6638 −0.850667
\(189\) 7.54537 0.548845
\(190\) 21.8544 1.58549
\(191\) 14.5246 1.05096 0.525481 0.850805i \(-0.323885\pi\)
0.525481 + 0.850805i \(0.323885\pi\)
\(192\) −0.535052 −0.0386140
\(193\) −15.2728 −1.09936 −0.549680 0.835375i \(-0.685250\pi\)
−0.549680 + 0.835375i \(0.685250\pi\)
\(194\) −3.65276 −0.262253
\(195\) 10.8625 0.777882
\(196\) −0.908392 −0.0648852
\(197\) −8.80473 −0.627311 −0.313656 0.949537i \(-0.601554\pi\)
−0.313656 + 0.949537i \(0.601554\pi\)
\(198\) 8.62255 0.612778
\(199\) 19.3037 1.36840 0.684202 0.729292i \(-0.260151\pi\)
0.684202 + 0.729292i \(0.260151\pi\)
\(200\) −3.06699 −0.216869
\(201\) 7.37274 0.520033
\(202\) 5.33264 0.375203
\(203\) 8.60809 0.604169
\(204\) −0.130918 −0.00916607
\(205\) 29.6463 2.07059
\(206\) −6.12186 −0.426530
\(207\) −13.4698 −0.936219
\(208\) −7.14791 −0.495619
\(209\) −24.4486 −1.69115
\(210\) 3.75074 0.258826
\(211\) 4.05547 0.279190 0.139595 0.990209i \(-0.455420\pi\)
0.139595 + 0.990209i \(0.455420\pi\)
\(212\) −2.04870 −0.140705
\(213\) 2.32397 0.159236
\(214\) −8.44585 −0.577347
\(215\) −22.7699 −1.55290
\(216\) −3.05714 −0.208012
\(217\) −11.7504 −0.797669
\(218\) 5.78322 0.391689
\(219\) −4.58115 −0.309565
\(220\) 9.02456 0.608436
\(221\) −1.74897 −0.117648
\(222\) −1.61390 −0.108318
\(223\) 10.0228 0.671178 0.335589 0.942008i \(-0.391065\pi\)
0.335589 + 0.942008i \(0.391065\pi\)
\(224\) −2.46812 −0.164908
\(225\) −8.32295 −0.554863
\(226\) 8.59201 0.571532
\(227\) 4.14121 0.274862 0.137431 0.990511i \(-0.456116\pi\)
0.137431 + 0.990511i \(0.456116\pi\)
\(228\) 4.11699 0.272654
\(229\) 6.73526 0.445079 0.222539 0.974924i \(-0.428565\pi\)
0.222539 + 0.974924i \(0.428565\pi\)
\(230\) −14.0979 −0.929585
\(231\) −4.19597 −0.276074
\(232\) −3.48771 −0.228980
\(233\) 2.49761 0.163624 0.0818121 0.996648i \(-0.473929\pi\)
0.0818121 + 0.996648i \(0.473929\pi\)
\(234\) −19.3974 −1.26805
\(235\) −33.1279 −2.16103
\(236\) −10.3363 −0.672835
\(237\) 0.836492 0.0543360
\(238\) −0.603905 −0.0391453
\(239\) −1.31372 −0.0849776 −0.0424888 0.999097i \(-0.513529\pi\)
−0.0424888 + 0.999097i \(0.513529\pi\)
\(240\) −1.51968 −0.0980948
\(241\) −29.9918 −1.93194 −0.965970 0.258656i \(-0.916720\pi\)
−0.965970 + 0.258656i \(0.916720\pi\)
\(242\) 0.904194 0.0581238
\(243\) −12.6521 −0.811636
\(244\) −14.4000 −0.921866
\(245\) −2.58006 −0.164834
\(246\) 5.58484 0.356077
\(247\) 55.0000 3.49957
\(248\) 4.76087 0.302316
\(249\) −5.40208 −0.342343
\(250\) 5.49022 0.347232
\(251\) −18.0047 −1.13645 −0.568224 0.822874i \(-0.692369\pi\)
−0.568224 + 0.822874i \(0.692369\pi\)
\(252\) −6.69778 −0.421921
\(253\) 15.7713 0.991534
\(254\) −18.1164 −1.13672
\(255\) −0.371838 −0.0232854
\(256\) 1.00000 0.0625000
\(257\) 7.64688 0.476999 0.238500 0.971143i \(-0.423344\pi\)
0.238500 + 0.971143i \(0.423344\pi\)
\(258\) −4.28945 −0.267049
\(259\) −7.44467 −0.462590
\(260\) −20.3018 −1.25907
\(261\) −9.46468 −0.585849
\(262\) −11.1235 −0.687215
\(263\) −28.0698 −1.73086 −0.865428 0.501033i \(-0.832954\pi\)
−0.865428 + 0.501033i \(0.832954\pi\)
\(264\) 1.70007 0.104632
\(265\) −5.81880 −0.357446
\(266\) 18.9911 1.16442
\(267\) −4.38311 −0.268242
\(268\) −13.7795 −0.841716
\(269\) 4.13235 0.251954 0.125977 0.992033i \(-0.459793\pi\)
0.125977 + 0.992033i \(0.459793\pi\)
\(270\) −8.68301 −0.528431
\(271\) 10.4250 0.633272 0.316636 0.948547i \(-0.397447\pi\)
0.316636 + 0.948547i \(0.397447\pi\)
\(272\) 0.244682 0.0148360
\(273\) 9.43933 0.571294
\(274\) 0.247378 0.0149446
\(275\) 9.74502 0.587647
\(276\) −2.65579 −0.159860
\(277\) −7.58992 −0.456034 −0.228017 0.973657i \(-0.573224\pi\)
−0.228017 + 0.973657i \(0.573224\pi\)
\(278\) 4.73479 0.283974
\(279\) 12.9197 0.773480
\(280\) −7.01006 −0.418931
\(281\) 23.5833 1.40686 0.703432 0.710762i \(-0.251650\pi\)
0.703432 + 0.710762i \(0.251650\pi\)
\(282\) −6.24071 −0.371629
\(283\) 26.0471 1.54834 0.774170 0.632978i \(-0.218168\pi\)
0.774170 + 0.632978i \(0.218168\pi\)
\(284\) −4.34345 −0.257737
\(285\) 11.6932 0.692648
\(286\) 22.7117 1.34297
\(287\) 25.7621 1.52069
\(288\) 2.71372 0.159907
\(289\) −16.9401 −0.996478
\(290\) −9.90596 −0.581698
\(291\) −1.95441 −0.114570
\(292\) 8.56207 0.501057
\(293\) −14.6448 −0.855561 −0.427780 0.903883i \(-0.640704\pi\)
−0.427780 + 0.903883i \(0.640704\pi\)
\(294\) −0.486037 −0.0283462
\(295\) −29.3576 −1.70927
\(296\) 3.01634 0.175321
\(297\) 9.71371 0.563647
\(298\) 13.2074 0.765083
\(299\) −35.4794 −2.05183
\(300\) −1.64100 −0.0947430
\(301\) −19.7866 −1.14048
\(302\) 18.5776 1.06902
\(303\) 2.85324 0.163914
\(304\) −7.69456 −0.441313
\(305\) −40.8995 −2.34190
\(306\) 0.663999 0.0379583
\(307\) −3.84873 −0.219659 −0.109829 0.993950i \(-0.535030\pi\)
−0.109829 + 0.993950i \(0.535030\pi\)
\(308\) 7.84217 0.446849
\(309\) −3.27551 −0.186337
\(310\) 13.5220 0.768000
\(311\) 8.00308 0.453813 0.226906 0.973917i \(-0.427139\pi\)
0.226906 + 0.973917i \(0.427139\pi\)
\(312\) −3.82450 −0.216520
\(313\) 31.6279 1.78772 0.893859 0.448349i \(-0.147988\pi\)
0.893859 + 0.448349i \(0.147988\pi\)
\(314\) −5.00903 −0.282676
\(315\) −19.0233 −1.07184
\(316\) −1.56339 −0.0879473
\(317\) −16.9172 −0.950166 −0.475083 0.879941i \(-0.657582\pi\)
−0.475083 + 0.879941i \(0.657582\pi\)
\(318\) −1.09616 −0.0614696
\(319\) 11.0818 0.620463
\(320\) 2.84024 0.158774
\(321\) −4.51897 −0.252224
\(322\) −12.2508 −0.682709
\(323\) −1.88272 −0.104757
\(324\) 6.50543 0.361413
\(325\) −21.9226 −1.21605
\(326\) 11.0324 0.611026
\(327\) 3.09432 0.171116
\(328\) −10.4380 −0.576340
\(329\) −28.7875 −1.58711
\(330\) 4.82861 0.265806
\(331\) 25.2684 1.38887 0.694437 0.719553i \(-0.255653\pi\)
0.694437 + 0.719553i \(0.255653\pi\)
\(332\) 10.0964 0.554111
\(333\) 8.18549 0.448562
\(334\) −4.23271 −0.231604
\(335\) −39.1371 −2.13829
\(336\) −1.32057 −0.0720430
\(337\) 12.5215 0.682092 0.341046 0.940047i \(-0.389219\pi\)
0.341046 + 0.940047i \(0.389219\pi\)
\(338\) −38.0927 −2.07197
\(339\) 4.59717 0.249684
\(340\) 0.694957 0.0376894
\(341\) −15.1271 −0.819181
\(342\) −20.8809 −1.12911
\(343\) −19.5188 −1.05392
\(344\) 8.01689 0.432242
\(345\) −7.54308 −0.406106
\(346\) 6.77456 0.364203
\(347\) 31.1396 1.67166 0.835830 0.548988i \(-0.184987\pi\)
0.835830 + 0.548988i \(0.184987\pi\)
\(348\) −1.86611 −0.100034
\(349\) −8.89742 −0.476268 −0.238134 0.971232i \(-0.576536\pi\)
−0.238134 + 0.971232i \(0.576536\pi\)
\(350\) −7.56969 −0.404617
\(351\) −21.8521 −1.16638
\(352\) −3.17739 −0.169355
\(353\) −1.90600 −0.101446 −0.0507231 0.998713i \(-0.516153\pi\)
−0.0507231 + 0.998713i \(0.516153\pi\)
\(354\) −5.53045 −0.293940
\(355\) −12.3365 −0.654752
\(356\) 8.19194 0.434172
\(357\) −0.323120 −0.0171013
\(358\) −0.473280 −0.0250136
\(359\) 8.14581 0.429919 0.214960 0.976623i \(-0.431038\pi\)
0.214960 + 0.976623i \(0.431038\pi\)
\(360\) 7.70763 0.406228
\(361\) 40.2062 2.11612
\(362\) 11.1549 0.586290
\(363\) 0.483791 0.0253924
\(364\) −17.6419 −0.924687
\(365\) 24.3184 1.27288
\(366\) −7.70475 −0.402734
\(367\) −0.469826 −0.0245247 −0.0122624 0.999925i \(-0.503903\pi\)
−0.0122624 + 0.999925i \(0.503903\pi\)
\(368\) 4.96361 0.258746
\(369\) −28.3257 −1.47458
\(370\) 8.56713 0.445384
\(371\) −5.05643 −0.262517
\(372\) 2.54731 0.132072
\(373\) −6.01461 −0.311425 −0.155712 0.987802i \(-0.549767\pi\)
−0.155712 + 0.987802i \(0.549767\pi\)
\(374\) −0.777451 −0.0402010
\(375\) 2.93755 0.151695
\(376\) 11.6638 0.601512
\(377\) −24.9299 −1.28395
\(378\) −7.54537 −0.388092
\(379\) −28.2641 −1.45183 −0.725914 0.687786i \(-0.758583\pi\)
−0.725914 + 0.687786i \(0.758583\pi\)
\(380\) −21.8544 −1.12111
\(381\) −9.69322 −0.496599
\(382\) −14.5246 −0.743143
\(383\) −20.2284 −1.03362 −0.516811 0.856100i \(-0.672881\pi\)
−0.516811 + 0.856100i \(0.672881\pi\)
\(384\) 0.535052 0.0273042
\(385\) 22.2737 1.13517
\(386\) 15.2728 0.777365
\(387\) 21.7556 1.10590
\(388\) 3.65276 0.185441
\(389\) 26.8697 1.36235 0.681174 0.732122i \(-0.261470\pi\)
0.681174 + 0.732122i \(0.261470\pi\)
\(390\) −10.8625 −0.550045
\(391\) 1.21451 0.0614202
\(392\) 0.908392 0.0458807
\(393\) −5.95167 −0.300222
\(394\) 8.80473 0.443576
\(395\) −4.44040 −0.223421
\(396\) −8.62255 −0.433299
\(397\) −1.98115 −0.0994313 −0.0497156 0.998763i \(-0.515832\pi\)
−0.0497156 + 0.998763i \(0.515832\pi\)
\(398\) −19.3037 −0.967608
\(399\) 10.1612 0.508697
\(400\) 3.06699 0.153349
\(401\) 9.52766 0.475789 0.237894 0.971291i \(-0.423543\pi\)
0.237894 + 0.971291i \(0.423543\pi\)
\(402\) −7.37274 −0.367719
\(403\) 34.0303 1.69517
\(404\) −5.33264 −0.265309
\(405\) 18.4770 0.918131
\(406\) −8.60809 −0.427212
\(407\) −9.58408 −0.475065
\(408\) 0.130918 0.00648139
\(409\) −31.4732 −1.55625 −0.778126 0.628108i \(-0.783830\pi\)
−0.778126 + 0.628108i \(0.783830\pi\)
\(410\) −29.6463 −1.46413
\(411\) 0.132360 0.00652882
\(412\) 6.12186 0.301602
\(413\) −25.5112 −1.25532
\(414\) 13.4698 0.662007
\(415\) 28.6762 1.40766
\(416\) 7.14791 0.350455
\(417\) 2.53336 0.124059
\(418\) 24.4486 1.19582
\(419\) −22.7639 −1.11209 −0.556044 0.831153i \(-0.687681\pi\)
−0.556044 + 0.831153i \(0.687681\pi\)
\(420\) −3.75074 −0.183018
\(421\) −15.8516 −0.772558 −0.386279 0.922382i \(-0.626240\pi\)
−0.386279 + 0.922382i \(0.626240\pi\)
\(422\) −4.05547 −0.197417
\(423\) 31.6522 1.53898
\(424\) 2.04870 0.0994936
\(425\) 0.750438 0.0364016
\(426\) −2.32397 −0.112597
\(427\) −35.5409 −1.71995
\(428\) 8.44585 0.408246
\(429\) 12.1519 0.586701
\(430\) 22.7699 1.09806
\(431\) 25.9137 1.24822 0.624109 0.781337i \(-0.285462\pi\)
0.624109 + 0.781337i \(0.285462\pi\)
\(432\) 3.05714 0.147086
\(433\) −27.1746 −1.30593 −0.652964 0.757389i \(-0.726475\pi\)
−0.652964 + 0.757389i \(0.726475\pi\)
\(434\) 11.7504 0.564037
\(435\) −5.30020 −0.254125
\(436\) −5.78322 −0.276966
\(437\) −38.1928 −1.82701
\(438\) 4.58115 0.218896
\(439\) −20.2089 −0.964518 −0.482259 0.876029i \(-0.660184\pi\)
−0.482259 + 0.876029i \(0.660184\pi\)
\(440\) −9.02456 −0.430229
\(441\) 2.46512 0.117387
\(442\) 1.74897 0.0831899
\(443\) −10.5329 −0.500432 −0.250216 0.968190i \(-0.580502\pi\)
−0.250216 + 0.968190i \(0.580502\pi\)
\(444\) 1.61390 0.0765921
\(445\) 23.2671 1.10297
\(446\) −10.0228 −0.474595
\(447\) 7.06663 0.334240
\(448\) 2.46812 0.116608
\(449\) −26.5774 −1.25427 −0.627133 0.778912i \(-0.715772\pi\)
−0.627133 + 0.778912i \(0.715772\pi\)
\(450\) 8.32295 0.392348
\(451\) 33.1654 1.56170
\(452\) −8.59201 −0.404134
\(453\) 9.93997 0.467021
\(454\) −4.14121 −0.194357
\(455\) −50.1073 −2.34907
\(456\) −4.11699 −0.192795
\(457\) 5.28293 0.247125 0.123563 0.992337i \(-0.460568\pi\)
0.123563 + 0.992337i \(0.460568\pi\)
\(458\) −6.73526 −0.314718
\(459\) 0.748027 0.0349149
\(460\) 14.0979 0.657316
\(461\) −17.6090 −0.820132 −0.410066 0.912056i \(-0.634494\pi\)
−0.410066 + 0.912056i \(0.634494\pi\)
\(462\) 4.19597 0.195214
\(463\) −6.25917 −0.290888 −0.145444 0.989366i \(-0.546461\pi\)
−0.145444 + 0.989366i \(0.546461\pi\)
\(464\) 3.48771 0.161913
\(465\) 7.23499 0.335515
\(466\) −2.49761 −0.115700
\(467\) 33.3466 1.54310 0.771549 0.636170i \(-0.219482\pi\)
0.771549 + 0.636170i \(0.219482\pi\)
\(468\) 19.3974 0.896647
\(469\) −34.0094 −1.57041
\(470\) 33.1279 1.52808
\(471\) −2.68009 −0.123492
\(472\) 10.3363 0.475766
\(473\) −25.4728 −1.17124
\(474\) −0.836492 −0.0384213
\(475\) −23.5991 −1.08280
\(476\) 0.603905 0.0276799
\(477\) 5.55959 0.254556
\(478\) 1.31372 0.0600882
\(479\) −27.0053 −1.23391 −0.616953 0.787000i \(-0.711633\pi\)
−0.616953 + 0.787000i \(0.711633\pi\)
\(480\) 1.51968 0.0693635
\(481\) 21.5605 0.983075
\(482\) 29.9918 1.36609
\(483\) −6.55480 −0.298254
\(484\) −0.904194 −0.0410997
\(485\) 10.3747 0.471092
\(486\) 12.6521 0.573913
\(487\) −8.12994 −0.368403 −0.184201 0.982889i \(-0.558970\pi\)
−0.184201 + 0.982889i \(0.558970\pi\)
\(488\) 14.4000 0.651858
\(489\) 5.90288 0.266937
\(490\) 2.58006 0.116555
\(491\) 33.7450 1.52289 0.761446 0.648228i \(-0.224490\pi\)
0.761446 + 0.648228i \(0.224490\pi\)
\(492\) −5.58484 −0.251784
\(493\) 0.853382 0.0384344
\(494\) −55.0000 −2.47457
\(495\) −24.4901 −1.10075
\(496\) −4.76087 −0.213769
\(497\) −10.7202 −0.480865
\(498\) 5.40208 0.242073
\(499\) −15.9315 −0.713192 −0.356596 0.934259i \(-0.616063\pi\)
−0.356596 + 0.934259i \(0.616063\pi\)
\(500\) −5.49022 −0.245530
\(501\) −2.26472 −0.101180
\(502\) 18.0047 0.803589
\(503\) 9.15376 0.408146 0.204073 0.978956i \(-0.434582\pi\)
0.204073 + 0.978956i \(0.434582\pi\)
\(504\) 6.69778 0.298343
\(505\) −15.1460 −0.673988
\(506\) −15.7713 −0.701121
\(507\) −20.3816 −0.905177
\(508\) 18.1164 0.803786
\(509\) −19.7803 −0.876746 −0.438373 0.898793i \(-0.644445\pi\)
−0.438373 + 0.898793i \(0.644445\pi\)
\(510\) 0.371838 0.0164653
\(511\) 21.1322 0.934833
\(512\) −1.00000 −0.0441942
\(513\) −23.5233 −1.03858
\(514\) −7.64688 −0.337290
\(515\) 17.3876 0.766189
\(516\) 4.28945 0.188832
\(517\) −37.0603 −1.62991
\(518\) 7.44467 0.327100
\(519\) 3.62474 0.159108
\(520\) 20.3018 0.890294
\(521\) −14.5791 −0.638721 −0.319361 0.947633i \(-0.603468\pi\)
−0.319361 + 0.947633i \(0.603468\pi\)
\(522\) 9.46468 0.414258
\(523\) −37.6334 −1.64560 −0.822798 0.568335i \(-0.807588\pi\)
−0.822798 + 0.568335i \(0.807588\pi\)
\(524\) 11.1235 0.485934
\(525\) −4.05018 −0.176764
\(526\) 28.0698 1.22390
\(527\) −1.16490 −0.0507439
\(528\) −1.70007 −0.0739859
\(529\) 1.63741 0.0711917
\(530\) 5.81880 0.252753
\(531\) 28.0498 1.21726
\(532\) −18.9911 −0.823368
\(533\) −74.6096 −3.23170
\(534\) 4.38311 0.189676
\(535\) 23.9883 1.03710
\(536\) 13.7795 0.595183
\(537\) −0.253229 −0.0109277
\(538\) −4.13235 −0.178158
\(539\) −2.88632 −0.124322
\(540\) 8.68301 0.373657
\(541\) −20.4824 −0.880608 −0.440304 0.897849i \(-0.645129\pi\)
−0.440304 + 0.897849i \(0.645129\pi\)
\(542\) −10.4250 −0.447791
\(543\) 5.96846 0.256131
\(544\) −0.244682 −0.0104907
\(545\) −16.4258 −0.703603
\(546\) −9.43933 −0.403966
\(547\) 14.6175 0.625000 0.312500 0.949918i \(-0.398834\pi\)
0.312500 + 0.949918i \(0.398834\pi\)
\(548\) −0.247378 −0.0105674
\(549\) 39.0776 1.66779
\(550\) −9.74502 −0.415529
\(551\) −26.8364 −1.14327
\(552\) 2.65579 0.113038
\(553\) −3.85862 −0.164085
\(554\) 7.58992 0.322465
\(555\) 4.58386 0.194574
\(556\) −4.73479 −0.200800
\(557\) 7.55027 0.319915 0.159958 0.987124i \(-0.448864\pi\)
0.159958 + 0.987124i \(0.448864\pi\)
\(558\) −12.9197 −0.546933
\(559\) 57.3040 2.42370
\(560\) 7.01006 0.296229
\(561\) −0.415976 −0.0175625
\(562\) −23.5833 −0.994803
\(563\) −38.3467 −1.61612 −0.808061 0.589099i \(-0.799483\pi\)
−0.808061 + 0.589099i \(0.799483\pi\)
\(564\) 6.24071 0.262781
\(565\) −24.4034 −1.02666
\(566\) −26.0471 −1.09484
\(567\) 16.0562 0.674296
\(568\) 4.34345 0.182247
\(569\) 24.4808 1.02629 0.513144 0.858302i \(-0.328480\pi\)
0.513144 + 0.858302i \(0.328480\pi\)
\(570\) −11.6932 −0.489776
\(571\) 31.4249 1.31509 0.657546 0.753414i \(-0.271594\pi\)
0.657546 + 0.753414i \(0.271594\pi\)
\(572\) −22.7117 −0.949624
\(573\) −7.77141 −0.324655
\(574\) −25.7621 −1.07529
\(575\) 15.2233 0.634857
\(576\) −2.71372 −0.113072
\(577\) −31.2796 −1.30219 −0.651094 0.758997i \(-0.725690\pi\)
−0.651094 + 0.758997i \(0.725690\pi\)
\(578\) 16.9401 0.704617
\(579\) 8.17173 0.339606
\(580\) 9.90596 0.411323
\(581\) 24.9190 1.03382
\(582\) 1.95441 0.0810131
\(583\) −6.50951 −0.269596
\(584\) −8.56207 −0.354301
\(585\) 55.0935 2.27783
\(586\) 14.6448 0.604973
\(587\) 34.8648 1.43902 0.719511 0.694481i \(-0.244366\pi\)
0.719511 + 0.694481i \(0.244366\pi\)
\(588\) 0.486037 0.0200438
\(589\) 36.6328 1.50943
\(590\) 29.3576 1.20863
\(591\) 4.71099 0.193784
\(592\) −3.01634 −0.123971
\(593\) 14.3276 0.588365 0.294183 0.955749i \(-0.404953\pi\)
0.294183 + 0.955749i \(0.404953\pi\)
\(594\) −9.71371 −0.398558
\(595\) 1.71524 0.0703179
\(596\) −13.2074 −0.540995
\(597\) −10.3285 −0.422717
\(598\) 35.4794 1.45086
\(599\) 24.0966 0.984560 0.492280 0.870437i \(-0.336164\pi\)
0.492280 + 0.870437i \(0.336164\pi\)
\(600\) 1.64100 0.0669934
\(601\) −3.60215 −0.146935 −0.0734674 0.997298i \(-0.523406\pi\)
−0.0734674 + 0.997298i \(0.523406\pi\)
\(602\) 19.7866 0.806443
\(603\) 37.3937 1.52279
\(604\) −18.5776 −0.755912
\(605\) −2.56813 −0.104409
\(606\) −2.85324 −0.115905
\(607\) 26.6291 1.08084 0.540422 0.841394i \(-0.318265\pi\)
0.540422 + 0.841394i \(0.318265\pi\)
\(608\) 7.69456 0.312055
\(609\) −4.60577 −0.186635
\(610\) 40.8995 1.65597
\(611\) 83.3715 3.37285
\(612\) −0.663999 −0.0268406
\(613\) −6.97460 −0.281701 −0.140851 0.990031i \(-0.544984\pi\)
−0.140851 + 0.990031i \(0.544984\pi\)
\(614\) 3.84873 0.155322
\(615\) −15.8623 −0.639631
\(616\) −7.84217 −0.315970
\(617\) 20.3877 0.820778 0.410389 0.911911i \(-0.365393\pi\)
0.410389 + 0.911911i \(0.365393\pi\)
\(618\) 3.27551 0.131760
\(619\) 10.6780 0.429186 0.214593 0.976704i \(-0.431157\pi\)
0.214593 + 0.976704i \(0.431157\pi\)
\(620\) −13.5220 −0.543058
\(621\) 15.1744 0.608929
\(622\) −8.00308 −0.320894
\(623\) 20.2187 0.810044
\(624\) 3.82450 0.153103
\(625\) −30.9285 −1.23714
\(626\) −31.6279 −1.26411
\(627\) 13.0813 0.522415
\(628\) 5.00903 0.199882
\(629\) −0.738044 −0.0294277
\(630\) 19.0233 0.757908
\(631\) −17.2178 −0.685431 −0.342716 0.939439i \(-0.611347\pi\)
−0.342716 + 0.939439i \(0.611347\pi\)
\(632\) 1.56339 0.0621881
\(633\) −2.16989 −0.0862453
\(634\) 16.9172 0.671869
\(635\) 51.4551 2.04193
\(636\) 1.09616 0.0434655
\(637\) 6.49311 0.257266
\(638\) −11.0818 −0.438734
\(639\) 11.7869 0.466283
\(640\) −2.84024 −0.112271
\(641\) −24.7289 −0.976732 −0.488366 0.872639i \(-0.662407\pi\)
−0.488366 + 0.872639i \(0.662407\pi\)
\(642\) 4.51897 0.178349
\(643\) −44.9752 −1.77365 −0.886824 0.462107i \(-0.847093\pi\)
−0.886824 + 0.462107i \(0.847093\pi\)
\(644\) 12.2508 0.482748
\(645\) 12.1831 0.479709
\(646\) 1.88272 0.0740747
\(647\) 31.7453 1.24804 0.624018 0.781410i \(-0.285499\pi\)
0.624018 + 0.781410i \(0.285499\pi\)
\(648\) −6.50543 −0.255558
\(649\) −32.8424 −1.28918
\(650\) 21.9226 0.859874
\(651\) 6.28707 0.246410
\(652\) −11.0324 −0.432060
\(653\) 10.0872 0.394741 0.197371 0.980329i \(-0.436760\pi\)
0.197371 + 0.980329i \(0.436760\pi\)
\(654\) −3.09432 −0.120998
\(655\) 31.5936 1.23446
\(656\) 10.4380 0.407534
\(657\) −23.2350 −0.906486
\(658\) 28.7875 1.12225
\(659\) 27.6259 1.07615 0.538077 0.842896i \(-0.319151\pi\)
0.538077 + 0.842896i \(0.319151\pi\)
\(660\) −4.82861 −0.187953
\(661\) −26.9595 −1.04860 −0.524301 0.851533i \(-0.675674\pi\)
−0.524301 + 0.851533i \(0.675674\pi\)
\(662\) −25.2684 −0.982083
\(663\) 0.935788 0.0363430
\(664\) −10.0964 −0.391815
\(665\) −53.9393 −2.09168
\(666\) −8.18549 −0.317181
\(667\) 17.3116 0.670310
\(668\) 4.23271 0.163769
\(669\) −5.36273 −0.207335
\(670\) 39.1371 1.51200
\(671\) −45.7544 −1.76633
\(672\) 1.32057 0.0509421
\(673\) 14.3848 0.554494 0.277247 0.960799i \(-0.410578\pi\)
0.277247 + 0.960799i \(0.410578\pi\)
\(674\) −12.5215 −0.482312
\(675\) 9.37620 0.360890
\(676\) 38.0927 1.46510
\(677\) 45.3629 1.74344 0.871719 0.490006i \(-0.163006\pi\)
0.871719 + 0.490006i \(0.163006\pi\)
\(678\) −4.59717 −0.176553
\(679\) 9.01544 0.345981
\(680\) −0.694957 −0.0266504
\(681\) −2.21576 −0.0849081
\(682\) 15.1271 0.579248
\(683\) −16.8562 −0.644983 −0.322492 0.946572i \(-0.604520\pi\)
−0.322492 + 0.946572i \(0.604520\pi\)
\(684\) 20.8809 0.798400
\(685\) −0.702613 −0.0268455
\(686\) 19.5188 0.745233
\(687\) −3.60371 −0.137490
\(688\) −8.01689 −0.305641
\(689\) 14.6439 0.557889
\(690\) 7.54308 0.287160
\(691\) −7.84206 −0.298326 −0.149163 0.988813i \(-0.547658\pi\)
−0.149163 + 0.988813i \(0.547658\pi\)
\(692\) −6.77456 −0.257530
\(693\) −21.2815 −0.808416
\(694\) −31.1396 −1.18204
\(695\) −13.4480 −0.510110
\(696\) 1.86611 0.0707346
\(697\) 2.55398 0.0967389
\(698\) 8.89742 0.336772
\(699\) −1.33635 −0.0505455
\(700\) 7.56969 0.286107
\(701\) 32.4126 1.22421 0.612103 0.790778i \(-0.290324\pi\)
0.612103 + 0.790778i \(0.290324\pi\)
\(702\) 21.8521 0.824756
\(703\) 23.2094 0.875358
\(704\) 3.17739 0.119752
\(705\) 17.7251 0.667568
\(706\) 1.90600 0.0717334
\(707\) −13.1616 −0.494992
\(708\) 5.53045 0.207847
\(709\) 10.5685 0.396909 0.198455 0.980110i \(-0.436408\pi\)
0.198455 + 0.980110i \(0.436408\pi\)
\(710\) 12.3365 0.462979
\(711\) 4.24259 0.159110
\(712\) −8.19194 −0.307006
\(713\) −23.6311 −0.884992
\(714\) 0.323120 0.0120925
\(715\) −64.5068 −2.41242
\(716\) 0.473280 0.0176873
\(717\) 0.702909 0.0262506
\(718\) −8.14581 −0.303999
\(719\) 20.5110 0.764932 0.382466 0.923969i \(-0.375075\pi\)
0.382466 + 0.923969i \(0.375075\pi\)
\(720\) −7.70763 −0.287246
\(721\) 15.1095 0.562706
\(722\) −40.2062 −1.49632
\(723\) 16.0471 0.596800
\(724\) −11.1549 −0.414569
\(725\) 10.6968 0.397268
\(726\) −0.483791 −0.0179552
\(727\) 43.7025 1.62083 0.810417 0.585853i \(-0.199240\pi\)
0.810417 + 0.585853i \(0.199240\pi\)
\(728\) 17.6419 0.653852
\(729\) −12.7468 −0.472102
\(730\) −24.3184 −0.900063
\(731\) −1.96159 −0.0725520
\(732\) 7.70475 0.284776
\(733\) −3.01984 −0.111540 −0.0557701 0.998444i \(-0.517761\pi\)
−0.0557701 + 0.998444i \(0.517761\pi\)
\(734\) 0.469826 0.0173416
\(735\) 1.38046 0.0509192
\(736\) −4.96361 −0.182961
\(737\) −43.7828 −1.61276
\(738\) 28.3257 1.04268
\(739\) −5.34037 −0.196449 −0.0982244 0.995164i \(-0.531316\pi\)
−0.0982244 + 0.995164i \(0.531316\pi\)
\(740\) −8.56713 −0.314934
\(741\) −29.4279 −1.08106
\(742\) 5.05643 0.185627
\(743\) 5.18726 0.190302 0.0951511 0.995463i \(-0.469667\pi\)
0.0951511 + 0.995463i \(0.469667\pi\)
\(744\) −2.54731 −0.0933890
\(745\) −37.5122 −1.37434
\(746\) 6.01461 0.220210
\(747\) −27.3987 −1.00247
\(748\) 0.777451 0.0284264
\(749\) 20.8454 0.761673
\(750\) −2.93755 −0.107264
\(751\) −47.2946 −1.72580 −0.862902 0.505371i \(-0.831356\pi\)
−0.862902 + 0.505371i \(0.831356\pi\)
\(752\) −11.6638 −0.425333
\(753\) 9.63345 0.351062
\(754\) 24.9299 0.907893
\(755\) −52.7649 −1.92031
\(756\) 7.54537 0.274422
\(757\) −5.19299 −0.188742 −0.0943712 0.995537i \(-0.530084\pi\)
−0.0943712 + 0.995537i \(0.530084\pi\)
\(758\) 28.2641 1.02660
\(759\) −8.43847 −0.306297
\(760\) 21.8544 0.792743
\(761\) 17.8016 0.645309 0.322654 0.946517i \(-0.395425\pi\)
0.322654 + 0.946517i \(0.395425\pi\)
\(762\) 9.69322 0.351148
\(763\) −14.2737 −0.516742
\(764\) 14.5246 0.525481
\(765\) −1.88592 −0.0681856
\(766\) 20.2284 0.730881
\(767\) 73.8829 2.66776
\(768\) −0.535052 −0.0193070
\(769\) −21.7271 −0.783500 −0.391750 0.920072i \(-0.628130\pi\)
−0.391750 + 0.920072i \(0.628130\pi\)
\(770\) −22.2737 −0.802688
\(771\) −4.09148 −0.147351
\(772\) −15.2728 −0.549680
\(773\) −34.9859 −1.25836 −0.629178 0.777261i \(-0.716608\pi\)
−0.629178 + 0.777261i \(0.716608\pi\)
\(774\) −21.7556 −0.781989
\(775\) −14.6015 −0.524503
\(776\) −3.65276 −0.131126
\(777\) 3.98329 0.142900
\(778\) −26.8697 −0.963325
\(779\) −80.3154 −2.87760
\(780\) 10.8625 0.388941
\(781\) −13.8008 −0.493833
\(782\) −1.21451 −0.0434307
\(783\) 10.6624 0.381044
\(784\) −0.908392 −0.0324426
\(785\) 14.2269 0.507778
\(786\) 5.95167 0.212289
\(787\) 26.4416 0.942543 0.471271 0.881988i \(-0.343795\pi\)
0.471271 + 0.881988i \(0.343795\pi\)
\(788\) −8.80473 −0.313656
\(789\) 15.0188 0.534683
\(790\) 4.44040 0.157982
\(791\) −21.2061 −0.754002
\(792\) 8.62255 0.306389
\(793\) 102.930 3.65515
\(794\) 1.98115 0.0703085
\(795\) 3.11336 0.110420
\(796\) 19.3037 0.684202
\(797\) −20.1583 −0.714042 −0.357021 0.934096i \(-0.616208\pi\)
−0.357021 + 0.934096i \(0.616208\pi\)
\(798\) −10.1612 −0.359703
\(799\) −2.85391 −0.100964
\(800\) −3.06699 −0.108434
\(801\) −22.2306 −0.785480
\(802\) −9.52766 −0.336433
\(803\) 27.2050 0.960044
\(804\) 7.37274 0.260016
\(805\) 34.7952 1.22637
\(806\) −34.0303 −1.19867
\(807\) −2.21102 −0.0778317
\(808\) 5.33264 0.187602
\(809\) 4.97927 0.175062 0.0875310 0.996162i \(-0.472102\pi\)
0.0875310 + 0.996162i \(0.472102\pi\)
\(810\) −18.4770 −0.649216
\(811\) −43.3426 −1.52197 −0.760983 0.648772i \(-0.775283\pi\)
−0.760983 + 0.648772i \(0.775283\pi\)
\(812\) 8.60809 0.302085
\(813\) −5.57790 −0.195626
\(814\) 9.58408 0.335922
\(815\) −31.3346 −1.09760
\(816\) −0.130918 −0.00458303
\(817\) 61.6864 2.15813
\(818\) 31.4732 1.10044
\(819\) 47.8752 1.67289
\(820\) 29.6463 1.03530
\(821\) 18.5870 0.648691 0.324346 0.945939i \(-0.394856\pi\)
0.324346 + 0.945939i \(0.394856\pi\)
\(822\) −0.132360 −0.00461658
\(823\) −1.06540 −0.0371376 −0.0185688 0.999828i \(-0.505911\pi\)
−0.0185688 + 0.999828i \(0.505911\pi\)
\(824\) −6.12186 −0.213265
\(825\) −5.21409 −0.181531
\(826\) 25.5112 0.887648
\(827\) 16.2334 0.564489 0.282245 0.959342i \(-0.408921\pi\)
0.282245 + 0.959342i \(0.408921\pi\)
\(828\) −13.4698 −0.468109
\(829\) 12.7407 0.442504 0.221252 0.975217i \(-0.428986\pi\)
0.221252 + 0.975217i \(0.428986\pi\)
\(830\) −28.6762 −0.995365
\(831\) 4.06100 0.140875
\(832\) −7.14791 −0.247809
\(833\) −0.222267 −0.00770111
\(834\) −2.53336 −0.0877230
\(835\) 12.0219 0.416036
\(836\) −24.4486 −0.845573
\(837\) −14.5546 −0.503081
\(838\) 22.7639 0.786365
\(839\) −28.9520 −0.999535 −0.499768 0.866160i \(-0.666581\pi\)
−0.499768 + 0.866160i \(0.666581\pi\)
\(840\) 3.75074 0.129413
\(841\) −16.8359 −0.580547
\(842\) 15.8516 0.546281
\(843\) −12.6183 −0.434598
\(844\) 4.05547 0.139595
\(845\) 108.193 3.72194
\(846\) −31.6522 −1.08822
\(847\) −2.23166 −0.0766807
\(848\) −2.04870 −0.0703526
\(849\) −13.9365 −0.478301
\(850\) −0.750438 −0.0257398
\(851\) −14.9719 −0.513231
\(852\) 2.32397 0.0796180
\(853\) 23.1007 0.790953 0.395476 0.918476i \(-0.370579\pi\)
0.395476 + 0.918476i \(0.370579\pi\)
\(854\) 35.5409 1.21619
\(855\) 59.3068 2.02825
\(856\) −8.44585 −0.288673
\(857\) −31.9681 −1.09201 −0.546005 0.837782i \(-0.683852\pi\)
−0.546005 + 0.837782i \(0.683852\pi\)
\(858\) −12.1519 −0.414860
\(859\) 21.1719 0.722377 0.361188 0.932493i \(-0.382371\pi\)
0.361188 + 0.932493i \(0.382371\pi\)
\(860\) −22.7699 −0.776448
\(861\) −13.7841 −0.469759
\(862\) −25.9137 −0.882624
\(863\) 18.7336 0.637700 0.318850 0.947805i \(-0.396703\pi\)
0.318850 + 0.947805i \(0.396703\pi\)
\(864\) −3.05714 −0.104006
\(865\) −19.2414 −0.654227
\(866\) 27.1746 0.923431
\(867\) 9.06384 0.307824
\(868\) −11.7504 −0.398834
\(869\) −4.96748 −0.168510
\(870\) 5.30020 0.179694
\(871\) 98.4946 3.33736
\(872\) 5.78322 0.195845
\(873\) −9.91256 −0.335489
\(874\) 38.1928 1.29189
\(875\) −13.5505 −0.458091
\(876\) −4.58115 −0.154783
\(877\) 51.1590 1.72752 0.863758 0.503908i \(-0.168105\pi\)
0.863758 + 0.503908i \(0.168105\pi\)
\(878\) 20.2089 0.682017
\(879\) 7.83575 0.264293
\(880\) 9.02456 0.304218
\(881\) −51.9475 −1.75016 −0.875078 0.483982i \(-0.839190\pi\)
−0.875078 + 0.483982i \(0.839190\pi\)
\(882\) −2.46512 −0.0830050
\(883\) 31.3160 1.05387 0.526934 0.849906i \(-0.323341\pi\)
0.526934 + 0.849906i \(0.323341\pi\)
\(884\) −1.74897 −0.0588242
\(885\) 15.7078 0.528013
\(886\) 10.5329 0.353859
\(887\) 26.1116 0.876742 0.438371 0.898794i \(-0.355556\pi\)
0.438371 + 0.898794i \(0.355556\pi\)
\(888\) −1.61390 −0.0541588
\(889\) 44.7135 1.49964
\(890\) −23.2671 −0.779915
\(891\) 20.6703 0.692481
\(892\) 10.0228 0.335589
\(893\) 89.7474 3.00328
\(894\) −7.06663 −0.236343
\(895\) 1.34423 0.0449327
\(896\) −2.46812 −0.0824540
\(897\) 18.9833 0.633835
\(898\) 26.5774 0.886900
\(899\) −16.6046 −0.553793
\(900\) −8.32295 −0.277432
\(901\) −0.501280 −0.0167001
\(902\) −33.1654 −1.10429
\(903\) 10.5869 0.352309
\(904\) 8.59201 0.285766
\(905\) −31.6827 −1.05317
\(906\) −9.93997 −0.330233
\(907\) 35.6314 1.18312 0.591561 0.806260i \(-0.298512\pi\)
0.591561 + 0.806260i \(0.298512\pi\)
\(908\) 4.14121 0.137431
\(909\) 14.4713 0.479983
\(910\) 50.1073 1.66104
\(911\) 30.3967 1.00709 0.503545 0.863969i \(-0.332029\pi\)
0.503545 + 0.863969i \(0.332029\pi\)
\(912\) 4.11699 0.136327
\(913\) 32.0801 1.06170
\(914\) −5.28293 −0.174744
\(915\) 21.8834 0.723442
\(916\) 6.73526 0.222539
\(917\) 27.4542 0.906619
\(918\) −0.748027 −0.0246886
\(919\) −49.8472 −1.64431 −0.822153 0.569267i \(-0.807227\pi\)
−0.822153 + 0.569267i \(0.807227\pi\)
\(920\) −14.0979 −0.464793
\(921\) 2.05927 0.0678552
\(922\) 17.6090 0.579921
\(923\) 31.0466 1.02191
\(924\) −4.19597 −0.138037
\(925\) −9.25107 −0.304173
\(926\) 6.25917 0.205689
\(927\) −16.6130 −0.545643
\(928\) −3.48771 −0.114490
\(929\) −53.2949 −1.74855 −0.874274 0.485433i \(-0.838662\pi\)
−0.874274 + 0.485433i \(0.838662\pi\)
\(930\) −7.23499 −0.237245
\(931\) 6.98968 0.229077
\(932\) 2.49761 0.0818121
\(933\) −4.28206 −0.140188
\(934\) −33.3466 −1.09113
\(935\) 2.20815 0.0722142
\(936\) −19.3974 −0.634025
\(937\) 11.5583 0.377592 0.188796 0.982016i \(-0.439541\pi\)
0.188796 + 0.982016i \(0.439541\pi\)
\(938\) 34.0094 1.11045
\(939\) −16.9226 −0.552248
\(940\) −33.1279 −1.08051
\(941\) −6.54093 −0.213228 −0.106614 0.994300i \(-0.534001\pi\)
−0.106614 + 0.994300i \(0.534001\pi\)
\(942\) 2.68009 0.0873220
\(943\) 51.8099 1.68716
\(944\) −10.3363 −0.336418
\(945\) 21.4307 0.697141
\(946\) 25.4728 0.828192
\(947\) −34.4635 −1.11991 −0.559956 0.828522i \(-0.689182\pi\)
−0.559956 + 0.828522i \(0.689182\pi\)
\(948\) 0.836492 0.0271680
\(949\) −61.2009 −1.98667
\(950\) 23.5991 0.765657
\(951\) 9.05159 0.293518
\(952\) −0.603905 −0.0195727
\(953\) −45.1326 −1.46199 −0.730995 0.682383i \(-0.760944\pi\)
−0.730995 + 0.682383i \(0.760944\pi\)
\(954\) −5.55959 −0.179998
\(955\) 41.2534 1.33493
\(956\) −1.31372 −0.0424888
\(957\) −5.92935 −0.191669
\(958\) 27.0053 0.872503
\(959\) −0.610557 −0.0197159
\(960\) −1.51968 −0.0490474
\(961\) −8.33411 −0.268842
\(962\) −21.5605 −0.695139
\(963\) −22.9197 −0.738576
\(964\) −29.9918 −0.965970
\(965\) −43.3785 −1.39640
\(966\) 6.55480 0.210897
\(967\) 54.1158 1.74025 0.870123 0.492834i \(-0.164039\pi\)
0.870123 + 0.492834i \(0.164039\pi\)
\(968\) 0.904194 0.0290619
\(969\) 1.00735 0.0323608
\(970\) −10.3747 −0.333112
\(971\) 22.6846 0.727983 0.363992 0.931402i \(-0.381414\pi\)
0.363992 + 0.931402i \(0.381414\pi\)
\(972\) −12.6521 −0.405818
\(973\) −11.6860 −0.374637
\(974\) 8.12994 0.260500
\(975\) 11.7297 0.375651
\(976\) −14.4000 −0.460933
\(977\) 20.3349 0.650573 0.325286 0.945616i \(-0.394539\pi\)
0.325286 + 0.945616i \(0.394539\pi\)
\(978\) −5.90288 −0.188753
\(979\) 26.0290 0.831890
\(980\) −2.58006 −0.0824169
\(981\) 15.6941 0.501073
\(982\) −33.7450 −1.07685
\(983\) −9.24115 −0.294747 −0.147373 0.989081i \(-0.547082\pi\)
−0.147373 + 0.989081i \(0.547082\pi\)
\(984\) 5.58484 0.178038
\(985\) −25.0076 −0.796808
\(986\) −0.853382 −0.0271772
\(987\) 15.4028 0.490277
\(988\) 55.0000 1.74978
\(989\) −39.7927 −1.26533
\(990\) 24.4901 0.778348
\(991\) −20.1435 −0.639878 −0.319939 0.947438i \(-0.603662\pi\)
−0.319939 + 0.947438i \(0.603662\pi\)
\(992\) 4.76087 0.151158
\(993\) −13.5199 −0.429040
\(994\) 10.7202 0.340023
\(995\) 54.8273 1.73814
\(996\) −5.40208 −0.171172
\(997\) −3.82037 −0.120992 −0.0604961 0.998168i \(-0.519268\pi\)
−0.0604961 + 0.998168i \(0.519268\pi\)
\(998\) 15.9315 0.504303
\(999\) −9.22135 −0.291751
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 4006.2.a.g.1.19 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4006.2.a.g.1.19 40 1.1 even 1 trivial