Properties

Label 1805.2.a.m.1.2
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4129975648\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.7168.1
Defining polynomial: \( x^{4} - 6x^{2} + 7 \) Copy content Toggle raw display
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 1.2
Root \(-1.25928\) of defining polynomial
Character \(\chi\) \(=\) 1805.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.25928 q^{2} -1.78089 q^{3} -0.414214 q^{4} +1.00000 q^{5} +2.24264 q^{6} -0.828427 q^{7} +3.04017 q^{8} +0.171573 q^{9} +O(q^{10})\) \(q-1.25928 q^{2} -1.78089 q^{3} -0.414214 q^{4} +1.00000 q^{5} +2.24264 q^{6} -0.828427 q^{7} +3.04017 q^{8} +0.171573 q^{9} -1.25928 q^{10} +2.00000 q^{11} +0.737669 q^{12} -4.29945 q^{13} +1.04322 q^{14} -1.78089 q^{15} -3.00000 q^{16} +7.65685 q^{17} -0.216058 q^{18} -0.414214 q^{20} +1.47534 q^{21} -2.51856 q^{22} -0.828427 q^{23} -5.41421 q^{24} +1.00000 q^{25} +5.41421 q^{26} +5.03712 q^{27} +0.343146 q^{28} -8.59890 q^{29} +2.24264 q^{30} -3.56178 q^{31} -2.30250 q^{32} -3.56178 q^{33} -9.64212 q^{34} -0.828427 q^{35} -0.0710678 q^{36} -0.737669 q^{37} +7.65685 q^{39} +3.04017 q^{40} -1.85786 q^{42} -4.82843 q^{43} -0.828427 q^{44} +0.171573 q^{45} +1.04322 q^{46} +10.4853 q^{47} +5.34267 q^{48} -6.31371 q^{49} -1.25928 q^{50} -13.6360 q^{51} +1.78089 q^{52} +7.86123 q^{53} -6.34315 q^{54} +2.00000 q^{55} -2.51856 q^{56} +10.8284 q^{58} +7.12356 q^{59} +0.737669 q^{60} -2.82843 q^{61} +4.48528 q^{62} -0.142136 q^{63} +8.89949 q^{64} -4.29945 q^{65} +4.48528 q^{66} -6.81801 q^{67} -3.17157 q^{68} +1.47534 q^{69} +1.04322 q^{70} -13.6360 q^{71} +0.521611 q^{72} -11.6569 q^{73} +0.928932 q^{74} -1.78089 q^{75} -1.65685 q^{77} -9.64212 q^{78} -7.12356 q^{79} -3.00000 q^{80} -9.48528 q^{81} +8.82843 q^{83} -0.611105 q^{84} +7.65685 q^{85} +6.08034 q^{86} +15.3137 q^{87} +6.08034 q^{88} +15.7225 q^{89} -0.216058 q^{90} +3.56178 q^{91} +0.343146 q^{92} +6.34315 q^{93} -13.2039 q^{94} +4.10051 q^{96} -4.29945 q^{97} +7.95073 q^{98} +0.343146 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 4 q^{5} - 8 q^{6} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 4 q^{5} - 8 q^{6} + 8 q^{7} + 12 q^{9} + 8 q^{11} - 12 q^{16} + 8 q^{17} + 4 q^{20} + 8 q^{23} - 16 q^{24} + 4 q^{25} + 16 q^{26} + 24 q^{28} - 8 q^{30} + 8 q^{35} + 28 q^{36} + 8 q^{39} - 64 q^{42} - 8 q^{43} + 8 q^{44} + 12 q^{45} + 8 q^{47} + 20 q^{49} - 48 q^{54} + 8 q^{55} + 32 q^{58} - 16 q^{62} + 56 q^{63} - 4 q^{64} - 16 q^{66} - 24 q^{68} - 24 q^{73} + 32 q^{74} + 16 q^{77} - 12 q^{80} - 4 q^{81} + 24 q^{83} + 8 q^{85} + 16 q^{87} + 24 q^{92} + 48 q^{93} + 56 q^{96} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.25928 −0.890446 −0.445223 0.895420i \(-0.646876\pi\)
−0.445223 + 0.895420i \(0.646876\pi\)
\(3\) −1.78089 −1.02820 −0.514099 0.857731i \(-0.671874\pi\)
−0.514099 + 0.857731i \(0.671874\pi\)
\(4\) −0.414214 −0.207107
\(5\) 1.00000 0.447214
\(6\) 2.24264 0.915554
\(7\) −0.828427 −0.313116 −0.156558 0.987669i \(-0.550040\pi\)
−0.156558 + 0.987669i \(0.550040\pi\)
\(8\) 3.04017 1.07486
\(9\) 0.171573 0.0571910
\(10\) −1.25928 −0.398219
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0.737669 0.212947
\(13\) −4.29945 −1.19245 −0.596227 0.802816i \(-0.703334\pi\)
−0.596227 + 0.802816i \(0.703334\pi\)
\(14\) 1.04322 0.278813
\(15\) −1.78089 −0.459824
\(16\) −3.00000 −0.750000
\(17\) 7.65685 1.85706 0.928530 0.371257i \(-0.121073\pi\)
0.928530 + 0.371257i \(0.121073\pi\)
\(18\) −0.216058 −0.0509254
\(19\) 0 0
\(20\) −0.414214 −0.0926210
\(21\) 1.47534 0.321945
\(22\) −2.51856 −0.536959
\(23\) −0.828427 −0.172739 −0.0863695 0.996263i \(-0.527527\pi\)
−0.0863695 + 0.996263i \(0.527527\pi\)
\(24\) −5.41421 −1.10517
\(25\) 1.00000 0.200000
\(26\) 5.41421 1.06181
\(27\) 5.03712 0.969394
\(28\) 0.343146 0.0648485
\(29\) −8.59890 −1.59678 −0.798388 0.602143i \(-0.794314\pi\)
−0.798388 + 0.602143i \(0.794314\pi\)
\(30\) 2.24264 0.409448
\(31\) −3.56178 −0.639715 −0.319857 0.947466i \(-0.603635\pi\)
−0.319857 + 0.947466i \(0.603635\pi\)
\(32\) −2.30250 −0.407029
\(33\) −3.56178 −0.620027
\(34\) −9.64212 −1.65361
\(35\) −0.828427 −0.140030
\(36\) −0.0710678 −0.0118446
\(37\) −0.737669 −0.121272 −0.0606360 0.998160i \(-0.519313\pi\)
−0.0606360 + 0.998160i \(0.519313\pi\)
\(38\) 0 0
\(39\) 7.65685 1.22608
\(40\) 3.04017 0.480693
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −1.85786 −0.286675
\(43\) −4.82843 −0.736328 −0.368164 0.929761i \(-0.620014\pi\)
−0.368164 + 0.929761i \(0.620014\pi\)
\(44\) −0.828427 −0.124890
\(45\) 0.171573 0.0255766
\(46\) 1.04322 0.153815
\(47\) 10.4853 1.52944 0.764718 0.644365i \(-0.222878\pi\)
0.764718 + 0.644365i \(0.222878\pi\)
\(48\) 5.34267 0.771148
\(49\) −6.31371 −0.901958
\(50\) −1.25928 −0.178089
\(51\) −13.6360 −1.90943
\(52\) 1.78089 0.246965
\(53\) 7.86123 1.07982 0.539912 0.841722i \(-0.318458\pi\)
0.539912 + 0.841722i \(0.318458\pi\)
\(54\) −6.34315 −0.863193
\(55\) 2.00000 0.269680
\(56\) −2.51856 −0.336557
\(57\) 0 0
\(58\) 10.8284 1.42184
\(59\) 7.12356 0.927409 0.463705 0.885990i \(-0.346520\pi\)
0.463705 + 0.885990i \(0.346520\pi\)
\(60\) 0.737669 0.0952327
\(61\) −2.82843 −0.362143 −0.181071 0.983470i \(-0.557957\pi\)
−0.181071 + 0.983470i \(0.557957\pi\)
\(62\) 4.48528 0.569631
\(63\) −0.142136 −0.0179074
\(64\) 8.89949 1.11244
\(65\) −4.29945 −0.533281
\(66\) 4.48528 0.552100
\(67\) −6.81801 −0.832953 −0.416476 0.909147i \(-0.636735\pi\)
−0.416476 + 0.909147i \(0.636735\pi\)
\(68\) −3.17157 −0.384610
\(69\) 1.47534 0.177610
\(70\) 1.04322 0.124689
\(71\) −13.6360 −1.61830 −0.809149 0.587603i \(-0.800072\pi\)
−0.809149 + 0.587603i \(0.800072\pi\)
\(72\) 0.521611 0.0614724
\(73\) −11.6569 −1.36433 −0.682166 0.731198i \(-0.738962\pi\)
−0.682166 + 0.731198i \(0.738962\pi\)
\(74\) 0.928932 0.107986
\(75\) −1.78089 −0.205640
\(76\) 0 0
\(77\) −1.65685 −0.188816
\(78\) −9.64212 −1.09176
\(79\) −7.12356 −0.801464 −0.400732 0.916195i \(-0.631244\pi\)
−0.400732 + 0.916195i \(0.631244\pi\)
\(80\) −3.00000 −0.335410
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) 8.82843 0.969046 0.484523 0.874779i \(-0.338993\pi\)
0.484523 + 0.874779i \(0.338993\pi\)
\(84\) −0.611105 −0.0666770
\(85\) 7.65685 0.830502
\(86\) 6.08034 0.655660
\(87\) 15.3137 1.64180
\(88\) 6.08034 0.648167
\(89\) 15.7225 1.66658 0.833289 0.552838i \(-0.186455\pi\)
0.833289 + 0.552838i \(0.186455\pi\)
\(90\) −0.216058 −0.0227745
\(91\) 3.56178 0.373376
\(92\) 0.343146 0.0357754
\(93\) 6.34315 0.657754
\(94\) −13.2039 −1.36188
\(95\) 0 0
\(96\) 4.10051 0.418506
\(97\) −4.29945 −0.436543 −0.218272 0.975888i \(-0.570042\pi\)
−0.218272 + 0.975888i \(0.570042\pi\)
\(98\) 7.95073 0.803145
\(99\) 0.343146 0.0344874
\(100\) −0.414214 −0.0414214
\(101\) 0.485281 0.0482873 0.0241437 0.999708i \(-0.492314\pi\)
0.0241437 + 0.999708i \(0.492314\pi\)
\(102\) 17.1716 1.70024
\(103\) 5.34267 0.526429 0.263215 0.964737i \(-0.415217\pi\)
0.263215 + 0.964737i \(0.415217\pi\)
\(104\) −13.0711 −1.28172
\(105\) 1.47534 0.143978
\(106\) −9.89949 −0.961524
\(107\) 13.9416 1.34778 0.673891 0.738830i \(-0.264622\pi\)
0.673891 + 0.738830i \(0.264622\pi\)
\(108\) −2.08644 −0.200768
\(109\) 18.6731 1.78856 0.894281 0.447505i \(-0.147687\pi\)
0.894281 + 0.447505i \(0.147687\pi\)
\(110\) −2.51856 −0.240135
\(111\) 1.31371 0.124692
\(112\) 2.48528 0.234837
\(113\) 9.33657 0.878311 0.439155 0.898411i \(-0.355278\pi\)
0.439155 + 0.898411i \(0.355278\pi\)
\(114\) 0 0
\(115\) −0.828427 −0.0772512
\(116\) 3.56178 0.330703
\(117\) −0.737669 −0.0681975
\(118\) −8.97056 −0.825807
\(119\) −6.34315 −0.581475
\(120\) −5.41421 −0.494248
\(121\) −7.00000 −0.636364
\(122\) 3.56178 0.322469
\(123\) 0 0
\(124\) 1.47534 0.132489
\(125\) 1.00000 0.0894427
\(126\) 0.178989 0.0159456
\(127\) 17.5034 1.55317 0.776586 0.630011i \(-0.216950\pi\)
0.776586 + 0.630011i \(0.216950\pi\)
\(128\) −6.60195 −0.583536
\(129\) 8.59890 0.757091
\(130\) 5.41421 0.474858
\(131\) 15.3137 1.33796 0.668982 0.743278i \(-0.266730\pi\)
0.668982 + 0.743278i \(0.266730\pi\)
\(132\) 1.47534 0.128412
\(133\) 0 0
\(134\) 8.58579 0.741699
\(135\) 5.03712 0.433526
\(136\) 23.2781 1.99608
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) −1.85786 −0.158152
\(139\) 19.6569 1.66727 0.833636 0.552314i \(-0.186255\pi\)
0.833636 + 0.552314i \(0.186255\pi\)
\(140\) 0.343146 0.0290011
\(141\) −18.6731 −1.57256
\(142\) 17.1716 1.44101
\(143\) −8.59890 −0.719076
\(144\) −0.514719 −0.0428932
\(145\) −8.59890 −0.714100
\(146\) 14.6792 1.21486
\(147\) 11.2440 0.927392
\(148\) 0.305553 0.0251163
\(149\) −14.8284 −1.21479 −0.607396 0.794399i \(-0.707786\pi\)
−0.607396 + 0.794399i \(0.707786\pi\)
\(150\) 2.24264 0.183111
\(151\) 10.6853 0.869561 0.434781 0.900536i \(-0.356826\pi\)
0.434781 + 0.900536i \(0.356826\pi\)
\(152\) 0 0
\(153\) 1.31371 0.106207
\(154\) 2.08644 0.168130
\(155\) −3.56178 −0.286089
\(156\) −3.17157 −0.253929
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 8.97056 0.713660
\(159\) −14.0000 −1.11027
\(160\) −2.30250 −0.182029
\(161\) 0.686292 0.0540873
\(162\) 11.9446 0.938458
\(163\) 7.17157 0.561721 0.280860 0.959749i \(-0.409380\pi\)
0.280860 + 0.959749i \(0.409380\pi\)
\(164\) 0 0
\(165\) −3.56178 −0.277284
\(166\) −11.1175 −0.862882
\(167\) 16.8923 1.30716 0.653581 0.756857i \(-0.273266\pi\)
0.653581 + 0.756857i \(0.273266\pi\)
\(168\) 4.48528 0.346047
\(169\) 5.48528 0.421945
\(170\) −9.64212 −0.739517
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) −2.82411 −0.214713 −0.107357 0.994221i \(-0.534239\pi\)
−0.107357 + 0.994221i \(0.534239\pi\)
\(174\) −19.2842 −1.46194
\(175\) −0.828427 −0.0626232
\(176\) −6.00000 −0.452267
\(177\) −12.6863 −0.953560
\(178\) −19.7990 −1.48400
\(179\) −17.1978 −1.28542 −0.642712 0.766108i \(-0.722191\pi\)
−0.642712 + 0.766108i \(0.722191\pi\)
\(180\) −0.0710678 −0.00529708
\(181\) 10.0742 0.748812 0.374406 0.927265i \(-0.377847\pi\)
0.374406 + 0.927265i \(0.377847\pi\)
\(182\) −4.48528 −0.332471
\(183\) 5.03712 0.372355
\(184\) −2.51856 −0.185671
\(185\) −0.737669 −0.0542345
\(186\) −7.98780 −0.585694
\(187\) 15.3137 1.11985
\(188\) −4.34315 −0.316756
\(189\) −4.17289 −0.303533
\(190\) 0 0
\(191\) −2.34315 −0.169544 −0.0847720 0.996400i \(-0.527016\pi\)
−0.0847720 + 0.996400i \(0.527016\pi\)
\(192\) −15.8490 −1.14381
\(193\) 20.0219 1.44121 0.720605 0.693346i \(-0.243864\pi\)
0.720605 + 0.693346i \(0.243864\pi\)
\(194\) 5.41421 0.388718
\(195\) 7.65685 0.548319
\(196\) 2.61522 0.186802
\(197\) 9.31371 0.663574 0.331787 0.943354i \(-0.392348\pi\)
0.331787 + 0.943354i \(0.392348\pi\)
\(198\) −0.432117 −0.0307092
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 3.04017 0.214973
\(201\) 12.1421 0.856440
\(202\) −0.611105 −0.0429972
\(203\) 7.12356 0.499976
\(204\) 5.64823 0.395455
\(205\) 0 0
\(206\) −6.72792 −0.468757
\(207\) −0.142136 −0.00987911
\(208\) 12.8984 0.894340
\(209\) 0 0
\(210\) −1.85786 −0.128205
\(211\) 13.6360 0.938743 0.469371 0.883001i \(-0.344481\pi\)
0.469371 + 0.883001i \(0.344481\pi\)
\(212\) −3.25623 −0.223639
\(213\) 24.2843 1.66393
\(214\) −17.5563 −1.20013
\(215\) −4.82843 −0.329296
\(216\) 15.3137 1.04197
\(217\) 2.95068 0.200305
\(218\) −23.5147 −1.59262
\(219\) 20.7596 1.40280
\(220\) −0.828427 −0.0558525
\(221\) −32.9203 −2.21446
\(222\) −1.65433 −0.111031
\(223\) −0.305553 −0.0204613 −0.0102307 0.999948i \(-0.503257\pi\)
−0.0102307 + 0.999948i \(0.503257\pi\)
\(224\) 1.90746 0.127447
\(225\) 0.171573 0.0114382
\(226\) −11.7574 −0.782088
\(227\) 16.8923 1.12118 0.560589 0.828094i \(-0.310575\pi\)
0.560589 + 0.828094i \(0.310575\pi\)
\(228\) 0 0
\(229\) −4.48528 −0.296396 −0.148198 0.988958i \(-0.547347\pi\)
−0.148198 + 0.988958i \(0.547347\pi\)
\(230\) 1.04322 0.0687880
\(231\) 2.95068 0.194140
\(232\) −26.1421 −1.71632
\(233\) −9.31371 −0.610161 −0.305081 0.952327i \(-0.598683\pi\)
−0.305081 + 0.952327i \(0.598683\pi\)
\(234\) 0.928932 0.0607262
\(235\) 10.4853 0.683984
\(236\) −2.95068 −0.192073
\(237\) 12.6863 0.824063
\(238\) 7.98780 0.517772
\(239\) 9.65685 0.624650 0.312325 0.949975i \(-0.398892\pi\)
0.312325 + 0.949975i \(0.398892\pi\)
\(240\) 5.34267 0.344868
\(241\) −24.3214 −1.56668 −0.783339 0.621595i \(-0.786485\pi\)
−0.783339 + 0.621595i \(0.786485\pi\)
\(242\) 8.81496 0.566647
\(243\) 1.78089 0.114244
\(244\) 1.17157 0.0750023
\(245\) −6.31371 −0.403368
\(246\) 0 0
\(247\) 0 0
\(248\) −10.8284 −0.687606
\(249\) −15.7225 −0.996371
\(250\) −1.25928 −0.0796439
\(251\) 20.9706 1.32365 0.661825 0.749658i \(-0.269782\pi\)
0.661825 + 0.749658i \(0.269782\pi\)
\(252\) 0.0588745 0.00370875
\(253\) −1.65685 −0.104166
\(254\) −22.0416 −1.38301
\(255\) −13.6360 −0.853921
\(256\) −9.48528 −0.592830
\(257\) −20.0219 −1.24893 −0.624466 0.781052i \(-0.714684\pi\)
−0.624466 + 0.781052i \(0.714684\pi\)
\(258\) −10.8284 −0.674148
\(259\) 0.611105 0.0379722
\(260\) 1.78089 0.110446
\(261\) −1.47534 −0.0913212
\(262\) −19.2842 −1.19138
\(263\) −16.1421 −0.995367 −0.497683 0.867359i \(-0.665816\pi\)
−0.497683 + 0.867359i \(0.665816\pi\)
\(264\) −10.8284 −0.666444
\(265\) 7.86123 0.482912
\(266\) 0 0
\(267\) −28.0000 −1.71357
\(268\) 2.82411 0.172510
\(269\) −8.59890 −0.524284 −0.262142 0.965029i \(-0.584429\pi\)
−0.262142 + 0.965029i \(0.584429\pi\)
\(270\) −6.34315 −0.386032
\(271\) 30.9706 1.88133 0.940664 0.339340i \(-0.110204\pi\)
0.940664 + 0.339340i \(0.110204\pi\)
\(272\) −22.9706 −1.39279
\(273\) −6.34315 −0.383905
\(274\) 2.51856 0.152152
\(275\) 2.00000 0.120605
\(276\) −0.611105 −0.0367842
\(277\) −17.3137 −1.04028 −0.520140 0.854081i \(-0.674120\pi\)
−0.520140 + 0.854081i \(0.674120\pi\)
\(278\) −24.7535 −1.48462
\(279\) −0.611105 −0.0365859
\(280\) −2.51856 −0.150513
\(281\) −14.2471 −0.849912 −0.424956 0.905214i \(-0.639711\pi\)
−0.424956 + 0.905214i \(0.639711\pi\)
\(282\) 23.5147 1.40028
\(283\) −9.79899 −0.582489 −0.291245 0.956649i \(-0.594069\pi\)
−0.291245 + 0.956649i \(0.594069\pi\)
\(284\) 5.64823 0.335161
\(285\) 0 0
\(286\) 10.8284 0.640298
\(287\) 0 0
\(288\) −0.395047 −0.0232784
\(289\) 41.6274 2.44867
\(290\) 10.8284 0.635867
\(291\) 7.65685 0.448853
\(292\) 4.82843 0.282562
\(293\) 17.9355 1.04780 0.523901 0.851779i \(-0.324476\pi\)
0.523901 + 0.851779i \(0.324476\pi\)
\(294\) −14.1594 −0.825792
\(295\) 7.12356 0.414750
\(296\) −2.24264 −0.130351
\(297\) 10.0742 0.584567
\(298\) 18.6731 1.08171
\(299\) 3.56178 0.205983
\(300\) 0.737669 0.0425894
\(301\) 4.00000 0.230556
\(302\) −13.4558 −0.774297
\(303\) −0.864233 −0.0496489
\(304\) 0 0
\(305\) −2.82843 −0.161955
\(306\) −1.65433 −0.0945716
\(307\) −3.25623 −0.185843 −0.0929214 0.995673i \(-0.529621\pi\)
−0.0929214 + 0.995673i \(0.529621\pi\)
\(308\) 0.686292 0.0391051
\(309\) −9.51472 −0.541273
\(310\) 4.48528 0.254747
\(311\) −20.3431 −1.15355 −0.576777 0.816902i \(-0.695690\pi\)
−0.576777 + 0.816902i \(0.695690\pi\)
\(312\) 23.2781 1.31787
\(313\) 24.6274 1.39202 0.696012 0.718030i \(-0.254956\pi\)
0.696012 + 0.718030i \(0.254956\pi\)
\(314\) −22.6670 −1.27918
\(315\) −0.142136 −0.00800844
\(316\) 2.95068 0.165989
\(317\) 5.77479 0.324345 0.162172 0.986762i \(-0.448150\pi\)
0.162172 + 0.986762i \(0.448150\pi\)
\(318\) 17.6299 0.988637
\(319\) −17.1978 −0.962892
\(320\) 8.89949 0.497497
\(321\) −24.8284 −1.38579
\(322\) −0.864233 −0.0481618
\(323\) 0 0
\(324\) 3.92893 0.218274
\(325\) −4.29945 −0.238491
\(326\) −9.03102 −0.500182
\(327\) −33.2548 −1.83900
\(328\) 0 0
\(329\) −8.68629 −0.478891
\(330\) 4.48528 0.246907
\(331\) 17.8089 0.978866 0.489433 0.872041i \(-0.337204\pi\)
0.489433 + 0.872041i \(0.337204\pi\)
\(332\) −3.65685 −0.200696
\(333\) −0.126564 −0.00693567
\(334\) −21.2721 −1.16396
\(335\) −6.81801 −0.372508
\(336\) −4.42602 −0.241459
\(337\) 13.5095 0.735907 0.367954 0.929844i \(-0.380059\pi\)
0.367954 + 0.929844i \(0.380059\pi\)
\(338\) −6.90751 −0.375719
\(339\) −16.6274 −0.903077
\(340\) −3.17157 −0.172003
\(341\) −7.12356 −0.385763
\(342\) 0 0
\(343\) 11.0294 0.595534
\(344\) −14.6792 −0.791452
\(345\) 1.47534 0.0794296
\(346\) 3.55635 0.191191
\(347\) 19.1716 1.02918 0.514592 0.857435i \(-0.327943\pi\)
0.514592 + 0.857435i \(0.327943\pi\)
\(348\) −6.34315 −0.340028
\(349\) −29.3137 −1.56913 −0.784563 0.620049i \(-0.787113\pi\)
−0.784563 + 0.620049i \(0.787113\pi\)
\(350\) 1.04322 0.0557626
\(351\) −21.6569 −1.15596
\(352\) −4.60500 −0.245448
\(353\) 7.65685 0.407533 0.203767 0.979019i \(-0.434682\pi\)
0.203767 + 0.979019i \(0.434682\pi\)
\(354\) 15.9756 0.849093
\(355\) −13.6360 −0.723725
\(356\) −6.51246 −0.345160
\(357\) 11.2965 0.597872
\(358\) 21.6569 1.14460
\(359\) −2.68629 −0.141777 −0.0708885 0.997484i \(-0.522583\pi\)
−0.0708885 + 0.997484i \(0.522583\pi\)
\(360\) 0.521611 0.0274913
\(361\) 0 0
\(362\) −12.6863 −0.666777
\(363\) 12.4662 0.654308
\(364\) −1.47534 −0.0773287
\(365\) −11.6569 −0.610148
\(366\) −6.34315 −0.331562
\(367\) 3.17157 0.165555 0.0827774 0.996568i \(-0.473621\pi\)
0.0827774 + 0.996568i \(0.473621\pi\)
\(368\) 2.48528 0.129554
\(369\) 0 0
\(370\) 0.928932 0.0482929
\(371\) −6.51246 −0.338110
\(372\) −2.62742 −0.136225
\(373\) 10.8119 0.559819 0.279910 0.960026i \(-0.409695\pi\)
0.279910 + 0.960026i \(0.409695\pi\)
\(374\) −19.2842 −0.997165
\(375\) −1.78089 −0.0919648
\(376\) 31.8771 1.64393
\(377\) 36.9706 1.90408
\(378\) 5.25483 0.270279
\(379\) 17.1978 0.883392 0.441696 0.897165i \(-0.354377\pi\)
0.441696 + 0.897165i \(0.354377\pi\)
\(380\) 0 0
\(381\) −31.1716 −1.59697
\(382\) 2.95068 0.150970
\(383\) −32.6147 −1.66653 −0.833267 0.552871i \(-0.813532\pi\)
−0.833267 + 0.552871i \(0.813532\pi\)
\(384\) 11.7574 0.599990
\(385\) −1.65685 −0.0844411
\(386\) −25.2132 −1.28332
\(387\) −0.828427 −0.0421113
\(388\) 1.78089 0.0904110
\(389\) −20.6274 −1.04585 −0.522926 0.852378i \(-0.675160\pi\)
−0.522926 + 0.852378i \(0.675160\pi\)
\(390\) −9.64212 −0.488248
\(391\) −6.34315 −0.320787
\(392\) −19.1948 −0.969482
\(393\) −27.2720 −1.37569
\(394\) −11.7286 −0.590877
\(395\) −7.12356 −0.358425
\(396\) −0.142136 −0.00714258
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 5.03712 0.252488
\(399\) 0 0
\(400\) −3.00000 −0.150000
\(401\) 10.0742 0.503084 0.251542 0.967846i \(-0.419062\pi\)
0.251542 + 0.967846i \(0.419062\pi\)
\(402\) −15.2904 −0.762613
\(403\) 15.3137 0.762830
\(404\) −0.201010 −0.0100006
\(405\) −9.48528 −0.471327
\(406\) −8.97056 −0.445202
\(407\) −1.47534 −0.0731298
\(408\) −41.4558 −2.05237
\(409\) −12.7718 −0.631524 −0.315762 0.948838i \(-0.602260\pi\)
−0.315762 + 0.948838i \(0.602260\pi\)
\(410\) 0 0
\(411\) 3.56178 0.175690
\(412\) −2.21301 −0.109027
\(413\) −5.90135 −0.290387
\(414\) 0.178989 0.00879681
\(415\) 8.82843 0.433370
\(416\) 9.89949 0.485363
\(417\) −35.0067 −1.71429
\(418\) 0 0
\(419\) −24.9706 −1.21989 −0.609946 0.792443i \(-0.708809\pi\)
−0.609946 + 0.792443i \(0.708809\pi\)
\(420\) −0.611105 −0.0298189
\(421\) 10.0742 0.490988 0.245494 0.969398i \(-0.421050\pi\)
0.245494 + 0.969398i \(0.421050\pi\)
\(422\) −17.1716 −0.835899
\(423\) 1.79899 0.0874699
\(424\) 23.8995 1.16066
\(425\) 7.65685 0.371412
\(426\) −30.5807 −1.48164
\(427\) 2.34315 0.113393
\(428\) −5.77479 −0.279135
\(429\) 15.3137 0.739353
\(430\) 6.08034 0.293220
\(431\) −3.56178 −0.171565 −0.0857825 0.996314i \(-0.527339\pi\)
−0.0857825 + 0.996314i \(0.527339\pi\)
\(432\) −15.1114 −0.727046
\(433\) −4.91056 −0.235986 −0.117993 0.993014i \(-0.537646\pi\)
−0.117993 + 0.993014i \(0.537646\pi\)
\(434\) −3.71573 −0.178361
\(435\) 15.3137 0.734236
\(436\) −7.73467 −0.370423
\(437\) 0 0
\(438\) −26.1421 −1.24912
\(439\) −2.95068 −0.140828 −0.0704141 0.997518i \(-0.522432\pi\)
−0.0704141 + 0.997518i \(0.522432\pi\)
\(440\) 6.08034 0.289869
\(441\) −1.08326 −0.0515839
\(442\) 41.4558 1.97185
\(443\) 20.1421 0.956982 0.478491 0.878093i \(-0.341184\pi\)
0.478491 + 0.878093i \(0.341184\pi\)
\(444\) −0.544156 −0.0258245
\(445\) 15.7225 0.745316
\(446\) 0.384776 0.0182197
\(447\) 26.4078 1.24905
\(448\) −7.37258 −0.348322
\(449\) 8.59890 0.405807 0.202904 0.979199i \(-0.434962\pi\)
0.202904 + 0.979199i \(0.434962\pi\)
\(450\) −0.216058 −0.0101851
\(451\) 0 0
\(452\) −3.86733 −0.181904
\(453\) −19.0294 −0.894081
\(454\) −21.2721 −0.998348
\(455\) 3.56178 0.166979
\(456\) 0 0
\(457\) 5.31371 0.248565 0.124282 0.992247i \(-0.460337\pi\)
0.124282 + 0.992247i \(0.460337\pi\)
\(458\) 5.64823 0.263924
\(459\) 38.5685 1.80022
\(460\) 0.343146 0.0159993
\(461\) −10.6863 −0.497710 −0.248855 0.968541i \(-0.580054\pi\)
−0.248855 + 0.968541i \(0.580054\pi\)
\(462\) −3.71573 −0.172871
\(463\) −32.8284 −1.52567 −0.762833 0.646595i \(-0.776192\pi\)
−0.762833 + 0.646595i \(0.776192\pi\)
\(464\) 25.7967 1.19758
\(465\) 6.34315 0.294156
\(466\) 11.7286 0.543315
\(467\) 38.4853 1.78089 0.890443 0.455094i \(-0.150394\pi\)
0.890443 + 0.455094i \(0.150394\pi\)
\(468\) 0.305553 0.0141242
\(469\) 5.64823 0.260811
\(470\) −13.2039 −0.609051
\(471\) −32.0560 −1.47706
\(472\) 21.6569 0.996838
\(473\) −9.65685 −0.444023
\(474\) −15.9756 −0.733783
\(475\) 0 0
\(476\) 2.62742 0.120427
\(477\) 1.34877 0.0617561
\(478\) −12.1607 −0.556217
\(479\) 10.0000 0.456912 0.228456 0.973554i \(-0.426632\pi\)
0.228456 + 0.973554i \(0.426632\pi\)
\(480\) 4.10051 0.187162
\(481\) 3.17157 0.144611
\(482\) 30.6274 1.39504
\(483\) −1.22221 −0.0556125
\(484\) 2.89949 0.131795
\(485\) −4.29945 −0.195228
\(486\) −2.24264 −0.101728
\(487\) 29.6640 1.34421 0.672103 0.740458i \(-0.265391\pi\)
0.672103 + 0.740458i \(0.265391\pi\)
\(488\) −8.59890 −0.389254
\(489\) −12.7718 −0.577560
\(490\) 7.95073 0.359177
\(491\) 32.2843 1.45697 0.728484 0.685062i \(-0.240225\pi\)
0.728484 + 0.685062i \(0.240225\pi\)
\(492\) 0 0
\(493\) −65.8405 −2.96531
\(494\) 0 0
\(495\) 0.343146 0.0154233
\(496\) 10.6853 0.479786
\(497\) 11.2965 0.506715
\(498\) 19.7990 0.887214
\(499\) 1.02944 0.0460839 0.0230420 0.999734i \(-0.492665\pi\)
0.0230420 + 0.999734i \(0.492665\pi\)
\(500\) −0.414214 −0.0185242
\(501\) −30.0833 −1.34402
\(502\) −26.4078 −1.17864
\(503\) 2.48528 0.110813 0.0554066 0.998464i \(-0.482354\pi\)
0.0554066 + 0.998464i \(0.482354\pi\)
\(504\) −0.432117 −0.0192480
\(505\) 0.485281 0.0215947
\(506\) 2.08644 0.0927537
\(507\) −9.76869 −0.433843
\(508\) −7.25013 −0.321672
\(509\) 32.9203 1.45917 0.729583 0.683893i \(-0.239714\pi\)
0.729583 + 0.683893i \(0.239714\pi\)
\(510\) 17.1716 0.760370
\(511\) 9.65685 0.427194
\(512\) 25.1485 1.11142
\(513\) 0 0
\(514\) 25.2132 1.11211
\(515\) 5.34267 0.235426
\(516\) −3.56178 −0.156799
\(517\) 20.9706 0.922284
\(518\) −0.769553 −0.0338122
\(519\) 5.02944 0.220768
\(520\) −13.0711 −0.573204
\(521\) 17.1978 0.753450 0.376725 0.926325i \(-0.377050\pi\)
0.376725 + 0.926325i \(0.377050\pi\)
\(522\) 1.85786 0.0813165
\(523\) −15.4169 −0.674135 −0.337067 0.941481i \(-0.609435\pi\)
−0.337067 + 0.941481i \(0.609435\pi\)
\(524\) −6.34315 −0.277102
\(525\) 1.47534 0.0643890
\(526\) 20.3275 0.886320
\(527\) −27.2720 −1.18799
\(528\) 10.6853 0.465020
\(529\) −22.3137 −0.970161
\(530\) −9.89949 −0.430007
\(531\) 1.22221 0.0530394
\(532\) 0 0
\(533\) 0 0
\(534\) 35.2598 1.52584
\(535\) 13.9416 0.602747
\(536\) −20.7279 −0.895310
\(537\) 30.6274 1.32167
\(538\) 10.8284 0.466847
\(539\) −12.6274 −0.543901
\(540\) −2.08644 −0.0897862
\(541\) 22.1421 0.951965 0.475982 0.879455i \(-0.342093\pi\)
0.475982 + 0.879455i \(0.342093\pi\)
\(542\) −39.0006 −1.67522
\(543\) −17.9411 −0.769927
\(544\) −17.6299 −0.755877
\(545\) 18.6731 0.799870
\(546\) 7.98780 0.341846
\(547\) −11.8551 −0.506889 −0.253444 0.967350i \(-0.581563\pi\)
−0.253444 + 0.967350i \(0.581563\pi\)
\(548\) 0.828427 0.0353887
\(549\) −0.485281 −0.0207113
\(550\) −2.51856 −0.107392
\(551\) 0 0
\(552\) 4.48528 0.190906
\(553\) 5.90135 0.250951
\(554\) 21.8028 0.926313
\(555\) 1.31371 0.0557638
\(556\) −8.14214 −0.345303
\(557\) −28.6274 −1.21298 −0.606491 0.795090i \(-0.707424\pi\)
−0.606491 + 0.795090i \(0.707424\pi\)
\(558\) 0.769553 0.0325778
\(559\) 20.7596 0.878037
\(560\) 2.48528 0.105022
\(561\) −27.2720 −1.15143
\(562\) 17.9411 0.756801
\(563\) 26.1023 1.10008 0.550040 0.835139i \(-0.314613\pi\)
0.550040 + 0.835139i \(0.314613\pi\)
\(564\) 7.73467 0.325688
\(565\) 9.33657 0.392793
\(566\) 12.3397 0.518675
\(567\) 7.85786 0.329999
\(568\) −41.4558 −1.73945
\(569\) 15.7225 0.659120 0.329560 0.944135i \(-0.393100\pi\)
0.329560 + 0.944135i \(0.393100\pi\)
\(570\) 0 0
\(571\) 13.3137 0.557161 0.278581 0.960413i \(-0.410136\pi\)
0.278581 + 0.960413i \(0.410136\pi\)
\(572\) 3.56178 0.148926
\(573\) 4.17289 0.174325
\(574\) 0 0
\(575\) −0.828427 −0.0345478
\(576\) 1.52691 0.0636213
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) −52.4206 −2.18041
\(579\) −35.6569 −1.48185
\(580\) 3.56178 0.147895
\(581\) −7.31371 −0.303424
\(582\) −9.64212 −0.399679
\(583\) 15.7225 0.651158
\(584\) −35.4388 −1.46647
\(585\) −0.737669 −0.0304989
\(586\) −22.5858 −0.933010
\(587\) 18.4853 0.762969 0.381485 0.924375i \(-0.375413\pi\)
0.381485 + 0.924375i \(0.375413\pi\)
\(588\) −4.65743 −0.192069
\(589\) 0 0
\(590\) −8.97056 −0.369312
\(591\) −16.5867 −0.682286
\(592\) 2.21301 0.0909541
\(593\) −9.31371 −0.382468 −0.191234 0.981544i \(-0.561249\pi\)
−0.191234 + 0.981544i \(0.561249\pi\)
\(594\) −12.6863 −0.520525
\(595\) −6.34315 −0.260044
\(596\) 6.14214 0.251592
\(597\) 7.12356 0.291548
\(598\) −4.48528 −0.183417
\(599\) 34.3956 1.40537 0.702683 0.711503i \(-0.251985\pi\)
0.702683 + 0.711503i \(0.251985\pi\)
\(600\) −5.41421 −0.221034
\(601\) 17.1978 0.701513 0.350757 0.936467i \(-0.385924\pi\)
0.350757 + 0.936467i \(0.385924\pi\)
\(602\) −5.03712 −0.205298
\(603\) −1.16979 −0.0476374
\(604\) −4.42602 −0.180092
\(605\) −7.00000 −0.284590
\(606\) 1.08831 0.0442096
\(607\) −33.2258 −1.34859 −0.674297 0.738460i \(-0.735553\pi\)
−0.674297 + 0.738460i \(0.735553\pi\)
\(608\) 0 0
\(609\) −12.6863 −0.514074
\(610\) 3.56178 0.144212
\(611\) −45.0810 −1.82378
\(612\) −0.544156 −0.0219962
\(613\) −14.2843 −0.576936 −0.288468 0.957489i \(-0.593146\pi\)
−0.288468 + 0.957489i \(0.593146\pi\)
\(614\) 4.10051 0.165483
\(615\) 0 0
\(616\) −5.03712 −0.202951
\(617\) 30.2843 1.21920 0.609599 0.792710i \(-0.291330\pi\)
0.609599 + 0.792710i \(0.291330\pi\)
\(618\) 11.9817 0.481975
\(619\) −18.9706 −0.762491 −0.381246 0.924474i \(-0.624505\pi\)
−0.381246 + 0.924474i \(0.624505\pi\)
\(620\) 1.47534 0.0592510
\(621\) −4.17289 −0.167452
\(622\) 25.6177 1.02718
\(623\) −13.0249 −0.521832
\(624\) −22.9706 −0.919558
\(625\) 1.00000 0.0400000
\(626\) −31.0128 −1.23952
\(627\) 0 0
\(628\) −7.45584 −0.297521
\(629\) −5.64823 −0.225210
\(630\) 0.178989 0.00713108
\(631\) 18.2843 0.727885 0.363943 0.931421i \(-0.381430\pi\)
0.363943 + 0.931421i \(0.381430\pi\)
\(632\) −21.6569 −0.861463
\(633\) −24.2843 −0.965213
\(634\) −7.27208 −0.288811
\(635\) 17.5034 0.694600
\(636\) 5.79899 0.229945
\(637\) 27.1455 1.07554
\(638\) 21.6569 0.857403
\(639\) −2.33957 −0.0925520
\(640\) −6.60195 −0.260965
\(641\) 21.3707 0.844092 0.422046 0.906575i \(-0.361312\pi\)
0.422046 + 0.906575i \(0.361312\pi\)
\(642\) 31.2659 1.23397
\(643\) 9.51472 0.375224 0.187612 0.982243i \(-0.439925\pi\)
0.187612 + 0.982243i \(0.439925\pi\)
\(644\) −0.284271 −0.0112019
\(645\) 8.59890 0.338581
\(646\) 0 0
\(647\) −27.1716 −1.06822 −0.534112 0.845413i \(-0.679354\pi\)
−0.534112 + 0.845413i \(0.679354\pi\)
\(648\) −28.8369 −1.13282
\(649\) 14.2471 0.559249
\(650\) 5.41421 0.212363
\(651\) −5.25483 −0.205953
\(652\) −2.97056 −0.116336
\(653\) 27.6569 1.08230 0.541148 0.840927i \(-0.317990\pi\)
0.541148 + 0.840927i \(0.317990\pi\)
\(654\) 41.8772 1.63753
\(655\) 15.3137 0.598356
\(656\) 0 0
\(657\) −2.00000 −0.0780274
\(658\) 10.9385 0.426426
\(659\) 20.1485 0.784873 0.392437 0.919779i \(-0.371632\pi\)
0.392437 + 0.919779i \(0.371632\pi\)
\(660\) 1.47534 0.0574275
\(661\) −2.95068 −0.114768 −0.0573840 0.998352i \(-0.518276\pi\)
−0.0573840 + 0.998352i \(0.518276\pi\)
\(662\) −22.4264 −0.871627
\(663\) 58.6274 2.27690
\(664\) 26.8399 1.04159
\(665\) 0 0
\(666\) 0.159380 0.00617583
\(667\) 7.12356 0.275826
\(668\) −6.99700 −0.270722
\(669\) 0.544156 0.0210383
\(670\) 8.58579 0.331698
\(671\) −5.65685 −0.218380
\(672\) −3.39697 −0.131041
\(673\) −50.8557 −1.96034 −0.980172 0.198146i \(-0.936508\pi\)
−0.980172 + 0.198146i \(0.936508\pi\)
\(674\) −17.0122 −0.655285
\(675\) 5.03712 0.193879
\(676\) −2.27208 −0.0873876
\(677\) −2.21301 −0.0850528 −0.0425264 0.999095i \(-0.513541\pi\)
−0.0425264 + 0.999095i \(0.513541\pi\)
\(678\) 20.9386 0.804141
\(679\) 3.56178 0.136689
\(680\) 23.2781 0.892676
\(681\) −30.0833 −1.15279
\(682\) 8.97056 0.343501
\(683\) 11.8551 0.453624 0.226812 0.973939i \(-0.427170\pi\)
0.226812 + 0.973939i \(0.427170\pi\)
\(684\) 0 0
\(685\) −2.00000 −0.0764161
\(686\) −13.8892 −0.530290
\(687\) 7.98780 0.304753
\(688\) 14.4853 0.552246
\(689\) −33.7990 −1.28764
\(690\) −1.85786 −0.0707277
\(691\) 16.6274 0.632537 0.316268 0.948670i \(-0.397570\pi\)
0.316268 + 0.948670i \(0.397570\pi\)
\(692\) 1.16979 0.0444686
\(693\) −0.284271 −0.0107986
\(694\) −24.1424 −0.916432
\(695\) 19.6569 0.745627
\(696\) 46.5563 1.76471
\(697\) 0 0
\(698\) 36.9142 1.39722
\(699\) 16.5867 0.627367
\(700\) 0.343146 0.0129697
\(701\) −24.4853 −0.924796 −0.462398 0.886672i \(-0.653011\pi\)
−0.462398 + 0.886672i \(0.653011\pi\)
\(702\) 27.2720 1.02932
\(703\) 0 0
\(704\) 17.7990 0.670825
\(705\) −18.6731 −0.703271
\(706\) −9.64212 −0.362886
\(707\) −0.402020 −0.0151195
\(708\) 5.25483 0.197489
\(709\) −5.31371 −0.199561 −0.0997803 0.995009i \(-0.531814\pi\)
−0.0997803 + 0.995009i \(0.531814\pi\)
\(710\) 17.1716 0.644438
\(711\) −1.22221 −0.0458365
\(712\) 47.7990 1.79134
\(713\) 2.95068 0.110504
\(714\) −14.2254 −0.532372
\(715\) −8.59890 −0.321581
\(716\) 7.12356 0.266220
\(717\) −17.1978 −0.642264
\(718\) 3.38279 0.126245
\(719\) 31.9411 1.19120 0.595601 0.803280i \(-0.296914\pi\)
0.595601 + 0.803280i \(0.296914\pi\)
\(720\) −0.514719 −0.0191824
\(721\) −4.42602 −0.164833
\(722\) 0 0
\(723\) 43.3137 1.61085
\(724\) −4.17289 −0.155084
\(725\) −8.59890 −0.319355
\(726\) −15.6985 −0.582625
\(727\) 6.48528 0.240526 0.120263 0.992742i \(-0.461626\pi\)
0.120263 + 0.992742i \(0.461626\pi\)
\(728\) 10.8284 0.401328
\(729\) 25.2843 0.936454
\(730\) 14.6792 0.543303
\(731\) −36.9706 −1.36741
\(732\) −2.08644 −0.0771172
\(733\) −35.6569 −1.31702 −0.658508 0.752574i \(-0.728812\pi\)
−0.658508 + 0.752574i \(0.728812\pi\)
\(734\) −3.99390 −0.147417
\(735\) 11.2440 0.414742
\(736\) 1.90746 0.0703097
\(737\) −13.6360 −0.502289
\(738\) 0 0
\(739\) 1.37258 0.0504913 0.0252456 0.999681i \(-0.491963\pi\)
0.0252456 + 0.999681i \(0.491963\pi\)
\(740\) 0.305553 0.0112323
\(741\) 0 0
\(742\) 8.20101 0.301069
\(743\) −6.81801 −0.250129 −0.125064 0.992149i \(-0.539914\pi\)
−0.125064 + 0.992149i \(0.539914\pi\)
\(744\) 19.2842 0.706995
\(745\) −14.8284 −0.543272
\(746\) −13.6152 −0.498489
\(747\) 1.51472 0.0554207
\(748\) −6.34315 −0.231928
\(749\) −11.5496 −0.422012
\(750\) 2.24264 0.0818897
\(751\) 6.51246 0.237643 0.118822 0.992916i \(-0.462088\pi\)
0.118822 + 0.992916i \(0.462088\pi\)
\(752\) −31.4558 −1.14708
\(753\) −37.3463 −1.36097
\(754\) −46.5563 −1.69548
\(755\) 10.6853 0.388880
\(756\) 1.72847 0.0628637
\(757\) −6.68629 −0.243017 −0.121509 0.992590i \(-0.538773\pi\)
−0.121509 + 0.992590i \(0.538773\pi\)
\(758\) −21.6569 −0.786612
\(759\) 2.95068 0.107103
\(760\) 0 0
\(761\) −9.17157 −0.332469 −0.166235 0.986086i \(-0.553161\pi\)
−0.166235 + 0.986086i \(0.553161\pi\)
\(762\) 39.2537 1.42201
\(763\) −15.4693 −0.560028
\(764\) 0.970563 0.0351137
\(765\) 1.31371 0.0474972
\(766\) 41.0711 1.48396
\(767\) −30.6274 −1.10589
\(768\) 16.8923 0.609547
\(769\) 14.1421 0.509978 0.254989 0.966944i \(-0.417928\pi\)
0.254989 + 0.966944i \(0.417928\pi\)
\(770\) 2.08644 0.0751902
\(771\) 35.6569 1.28415
\(772\) −8.29335 −0.298484
\(773\) −18.5466 −0.667074 −0.333537 0.942737i \(-0.608242\pi\)
−0.333537 + 0.942737i \(0.608242\pi\)
\(774\) 1.04322 0.0374978
\(775\) −3.56178 −0.127943
\(776\) −13.0711 −0.469224
\(777\) −1.08831 −0.0390430
\(778\) 25.9757 0.931274
\(779\) 0 0
\(780\) −3.17157 −0.113561
\(781\) −27.2720 −0.975871
\(782\) 7.98780 0.285643
\(783\) −43.3137 −1.54791
\(784\) 18.9411 0.676469
\(785\) 18.0000 0.642448
\(786\) 34.3431 1.22498
\(787\) −46.2507 −1.64866 −0.824330 0.566109i \(-0.808448\pi\)
−0.824330 + 0.566109i \(0.808448\pi\)
\(788\) −3.85786 −0.137431
\(789\) 28.7474 1.02343
\(790\) 8.97056 0.319158
\(791\) −7.73467 −0.275013
\(792\) 1.04322 0.0370693
\(793\) 12.1607 0.431839
\(794\) 17.6299 0.625663
\(795\) −14.0000 −0.496529
\(796\) 1.65685 0.0587256
\(797\) 0.737669 0.0261296 0.0130648 0.999915i \(-0.495841\pi\)
0.0130648 + 0.999915i \(0.495841\pi\)
\(798\) 0 0
\(799\) 80.2843 2.84025
\(800\) −2.30250 −0.0814057
\(801\) 2.69755 0.0953132
\(802\) −12.6863 −0.447969
\(803\) −23.3137 −0.822723
\(804\) −5.02944 −0.177375
\(805\) 0.686292 0.0241886
\(806\) −19.2842 −0.679259
\(807\) 15.3137 0.539068
\(808\) 1.47534 0.0519022
\(809\) −13.3137 −0.468085 −0.234043 0.972226i \(-0.575196\pi\)
−0.234043 + 0.972226i \(0.575196\pi\)
\(810\) 11.9446 0.419691
\(811\) 33.7845 1.18633 0.593167 0.805079i \(-0.297877\pi\)
0.593167 + 0.805079i \(0.297877\pi\)
\(812\) −2.95068 −0.103548
\(813\) −55.1552 −1.93438
\(814\) 1.85786 0.0651181
\(815\) 7.17157 0.251209
\(816\) 40.9081 1.43207
\(817\) 0 0
\(818\) 16.0833 0.562338
\(819\) 0.611105 0.0213537
\(820\) 0 0
\(821\) 2.68629 0.0937522 0.0468761 0.998901i \(-0.485073\pi\)
0.0468761 + 0.998901i \(0.485073\pi\)
\(822\) −4.48528 −0.156442
\(823\) 42.0833 1.46693 0.733465 0.679727i \(-0.237902\pi\)
0.733465 + 0.679727i \(0.237902\pi\)
\(824\) 16.2426 0.565839
\(825\) −3.56178 −0.124005
\(826\) 7.43146 0.258573
\(827\) 18.9787 0.659954 0.329977 0.943989i \(-0.392959\pi\)
0.329977 + 0.943989i \(0.392959\pi\)
\(828\) 0.0588745 0.00204603
\(829\) −42.9945 −1.49326 −0.746631 0.665239i \(-0.768330\pi\)
−0.746631 + 0.665239i \(0.768330\pi\)
\(830\) −11.1175 −0.385893
\(831\) 30.8338 1.06961
\(832\) −38.2629 −1.32653
\(833\) −48.3431 −1.67499
\(834\) 44.0833 1.52648
\(835\) 16.8923 0.584581
\(836\) 0 0
\(837\) −17.9411 −0.620136
\(838\) 31.4449 1.08625
\(839\) 31.4449 1.08560 0.542800 0.839862i \(-0.317364\pi\)
0.542800 + 0.839862i \(0.317364\pi\)
\(840\) 4.48528 0.154757
\(841\) 44.9411 1.54969
\(842\) −12.6863 −0.437198
\(843\) 25.3726 0.873878
\(844\) −5.64823 −0.194420
\(845\) 5.48528 0.188699
\(846\) −2.26543 −0.0778872
\(847\) 5.79899 0.199256
\(848\) −23.5837 −0.809868
\(849\) 17.4509 0.598914
\(850\) −9.64212 −0.330722
\(851\) 0.611105 0.0209484
\(852\) −10.0589 −0.344611
\(853\) 36.3431 1.24437 0.622183 0.782872i \(-0.286246\pi\)
0.622183 + 0.782872i \(0.286246\pi\)
\(854\) −2.95068 −0.100970
\(855\) 0 0
\(856\) 42.3848 1.44868
\(857\) −17.0712 −0.583142 −0.291571 0.956549i \(-0.594178\pi\)
−0.291571 + 0.956549i \(0.594178\pi\)
\(858\) −19.2842 −0.658353
\(859\) −44.2843 −1.51096 −0.755480 0.655172i \(-0.772596\pi\)
−0.755480 + 0.655172i \(0.772596\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) 4.48528 0.152769
\(863\) 24.6269 0.838310 0.419155 0.907915i \(-0.362326\pi\)
0.419155 + 0.907915i \(0.362326\pi\)
\(864\) −11.5980 −0.394571
\(865\) −2.82411 −0.0960227
\(866\) 6.18377 0.210133
\(867\) −74.1339 −2.51772
\(868\) −1.22221 −0.0414845
\(869\) −14.2471 −0.483301
\(870\) −19.2842 −0.653797
\(871\) 29.3137 0.993257
\(872\) 56.7696 1.92246
\(873\) −0.737669 −0.0249663
\(874\) 0 0
\(875\) −0.828427 −0.0280059
\(876\) −8.59890 −0.290530
\(877\) 10.8119 0.365092 0.182546 0.983197i \(-0.441566\pi\)
0.182546 + 0.983197i \(0.441566\pi\)
\(878\) 3.71573 0.125400
\(879\) −31.9411 −1.07735
\(880\) −6.00000 −0.202260
\(881\) 4.48528 0.151113 0.0755565 0.997142i \(-0.475927\pi\)
0.0755565 + 0.997142i \(0.475927\pi\)
\(882\) 1.36413 0.0459326
\(883\) 3.85786 0.129827 0.0649137 0.997891i \(-0.479323\pi\)
0.0649137 + 0.997891i \(0.479323\pi\)
\(884\) 13.6360 0.458629
\(885\) −12.6863 −0.426445
\(886\) −25.3646 −0.852140
\(887\) −1.16979 −0.0392776 −0.0196388 0.999807i \(-0.506252\pi\)
−0.0196388 + 0.999807i \(0.506252\pi\)
\(888\) 3.99390 0.134026
\(889\) −14.5003 −0.486323
\(890\) −19.7990 −0.663664
\(891\) −18.9706 −0.635538
\(892\) 0.126564 0.00423768
\(893\) 0 0
\(894\) −33.2548 −1.11221
\(895\) −17.1978 −0.574859
\(896\) 5.46924 0.182714
\(897\) −6.34315 −0.211791
\(898\) −10.8284 −0.361349
\(899\) 30.6274 1.02148
\(900\) −0.0710678 −0.00236893
\(901\) 60.1923 2.00530
\(902\) 0 0
\(903\) −7.12356 −0.237057
\(904\) 28.3848 0.944064
\(905\) 10.0742 0.334879
\(906\) 23.9634 0.796130
\(907\) −30.5283 −1.01367 −0.506837 0.862042i \(-0.669186\pi\)
−0.506837 + 0.862042i \(0.669186\pi\)
\(908\) −6.99700 −0.232204
\(909\) 0.0832611 0.00276160
\(910\) −4.48528 −0.148686
\(911\) 26.6609 0.883316 0.441658 0.897183i \(-0.354390\pi\)
0.441658 + 0.897183i \(0.354390\pi\)
\(912\) 0 0
\(913\) 17.6569 0.584357
\(914\) −6.69145 −0.221333
\(915\) 5.03712 0.166522
\(916\) 1.85786 0.0613856
\(917\) −12.6863 −0.418938
\(918\) −48.5685 −1.60300
\(919\) 0.284271 0.00937724 0.00468862 0.999989i \(-0.498508\pi\)
0.00468862 + 0.999989i \(0.498508\pi\)
\(920\) −2.51856 −0.0830345
\(921\) 5.79899 0.191083
\(922\) 13.4570 0.443184
\(923\) 58.6274 1.92974
\(924\) −1.22221 −0.0402078
\(925\) −0.737669 −0.0242544
\(926\) 41.3402 1.35852
\(927\) 0.916658 0.0301070
\(928\) 19.7990 0.649934
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) −7.98780 −0.261930
\(931\) 0 0
\(932\) 3.85786 0.126369
\(933\) 36.2289 1.18608
\(934\) −48.4638 −1.58578
\(935\) 15.3137 0.500812
\(936\) −2.24264 −0.0733030
\(937\) 10.9706 0.358393 0.179196 0.983813i \(-0.442650\pi\)
0.179196 + 0.983813i \(0.442650\pi\)
\(938\) −7.11270 −0.232238
\(939\) −43.8587 −1.43128
\(940\) −4.34315 −0.141658
\(941\) −2.95068 −0.0961893 −0.0480947 0.998843i \(-0.515315\pi\)
−0.0480947 + 0.998843i \(0.515315\pi\)
\(942\) 40.3675 1.31525
\(943\) 0 0
\(944\) −21.3707 −0.695557
\(945\) −4.17289 −0.135744
\(946\) 12.1607 0.395378
\(947\) −4.14214 −0.134601 −0.0673007 0.997733i \(-0.521439\pi\)
−0.0673007 + 0.997733i \(0.521439\pi\)
\(948\) −5.25483 −0.170669
\(949\) 50.1181 1.62690
\(950\) 0 0
\(951\) −10.2843 −0.333490
\(952\) −19.2842 −0.625006
\(953\) 1.34877 0.0436911 0.0218455 0.999761i \(-0.493046\pi\)
0.0218455 + 0.999761i \(0.493046\pi\)
\(954\) −1.69848 −0.0549905
\(955\) −2.34315 −0.0758224
\(956\) −4.00000 −0.129369
\(957\) 30.6274 0.990044
\(958\) −12.5928 −0.406855
\(959\) 1.65685 0.0535026
\(960\) −15.8490 −0.511525
\(961\) −18.3137 −0.590765
\(962\) −3.99390 −0.128768
\(963\) 2.39200 0.0770810
\(964\) 10.0742 0.324469
\(965\) 20.0219 0.644528
\(966\) 1.53911 0.0495199
\(967\) 21.5147 0.691867 0.345933 0.938259i \(-0.387562\pi\)
0.345933 + 0.938259i \(0.387562\pi\)
\(968\) −21.2812 −0.684004
\(969\) 0 0
\(970\) 5.41421 0.173840
\(971\) −7.73467 −0.248217 −0.124109 0.992269i \(-0.539607\pi\)
−0.124109 + 0.992269i \(0.539607\pi\)
\(972\) −0.737669 −0.0236608
\(973\) −16.2843 −0.522050
\(974\) −37.3553 −1.19694
\(975\) 7.65685 0.245216
\(976\) 8.48528 0.271607
\(977\) −47.9051 −1.53262 −0.766309 0.642472i \(-0.777909\pi\)
−0.766309 + 0.642472i \(0.777909\pi\)
\(978\) 16.0833 0.514286
\(979\) 31.4449 1.00498
\(980\) 2.61522 0.0835403
\(981\) 3.20380 0.102290
\(982\) −40.6549 −1.29735
\(983\) −33.8369 −1.07923 −0.539615 0.841912i \(-0.681430\pi\)
−0.539615 + 0.841912i \(0.681430\pi\)
\(984\) 0 0
\(985\) 9.31371 0.296759
\(986\) 82.9117 2.64045
\(987\) 15.4693 0.492394
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) −0.432117 −0.0137336
\(991\) −27.8832 −0.885737 −0.442869 0.896586i \(-0.646039\pi\)
−0.442869 + 0.896586i \(0.646039\pi\)
\(992\) 8.20101 0.260382
\(993\) −31.7157 −1.00647
\(994\) −14.2254 −0.451202
\(995\) −4.00000 −0.126809
\(996\) 6.51246 0.206355
\(997\) −7.65685 −0.242495 −0.121248 0.992622i \(-0.538689\pi\)
−0.121248 + 0.992622i \(0.538689\pi\)
\(998\) −1.29635 −0.0410352
\(999\) −3.71573 −0.117560
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 1805.2.a.m.1.2 4
5.4 even 2 9025.2.a.bn.1.3 4
19.18 odd 2 inner 1805.2.a.m.1.3 yes 4
95.94 odd 2 9025.2.a.bn.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.m.1.2 4 1.1 even 1 trivial
1805.2.a.m.1.3 yes 4 19.18 odd 2 inner
9025.2.a.bn.1.2 4 95.94 odd 2
9025.2.a.bn.1.3 4 5.4 even 2