Properties

Label 4730.2.a.bf.1.8
Level 4730
Weight 2
Character 4730.1
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.921464\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.921464 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.921464 q^{6} -3.84192 q^{7} +1.00000 q^{8} -2.15090 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.921464 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.921464 q^{6} -3.84192 q^{7} +1.00000 q^{8} -2.15090 q^{9} -1.00000 q^{10} +1.00000 q^{11} +0.921464 q^{12} +3.36077 q^{13} -3.84192 q^{14} -0.921464 q^{15} +1.00000 q^{16} +4.48968 q^{17} -2.15090 q^{18} -6.54987 q^{19} -1.00000 q^{20} -3.54019 q^{21} +1.00000 q^{22} +2.73904 q^{23} +0.921464 q^{24} +1.00000 q^{25} +3.36077 q^{26} -4.74637 q^{27} -3.84192 q^{28} +2.61423 q^{29} -0.921464 q^{30} -0.588567 q^{31} +1.00000 q^{32} +0.921464 q^{33} +4.48968 q^{34} +3.84192 q^{35} -2.15090 q^{36} -0.240321 q^{37} -6.54987 q^{38} +3.09682 q^{39} -1.00000 q^{40} +10.0856 q^{41} -3.54019 q^{42} +1.00000 q^{43} +1.00000 q^{44} +2.15090 q^{45} +2.73904 q^{46} +9.27274 q^{47} +0.921464 q^{48} +7.76033 q^{49} +1.00000 q^{50} +4.13708 q^{51} +3.36077 q^{52} +12.6401 q^{53} -4.74637 q^{54} -1.00000 q^{55} -3.84192 q^{56} -6.03547 q^{57} +2.61423 q^{58} +12.5429 q^{59} -0.921464 q^{60} -9.61487 q^{61} -0.588567 q^{62} +8.26360 q^{63} +1.00000 q^{64} -3.36077 q^{65} +0.921464 q^{66} -8.11130 q^{67} +4.48968 q^{68} +2.52392 q^{69} +3.84192 q^{70} -10.4421 q^{71} -2.15090 q^{72} +11.8631 q^{73} -0.240321 q^{74} +0.921464 q^{75} -6.54987 q^{76} -3.84192 q^{77} +3.09682 q^{78} +5.98496 q^{79} -1.00000 q^{80} +2.07910 q^{81} +10.0856 q^{82} +8.56754 q^{83} -3.54019 q^{84} -4.48968 q^{85} +1.00000 q^{86} +2.40892 q^{87} +1.00000 q^{88} +13.9559 q^{89} +2.15090 q^{90} -12.9118 q^{91} +2.73904 q^{92} -0.542343 q^{93} +9.27274 q^{94} +6.54987 q^{95} +0.921464 q^{96} -8.94728 q^{97} +7.76033 q^{98} -2.15090 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 13q^{2} + 13q^{4} - 13q^{5} + 7q^{7} + 13q^{8} + 25q^{9} + O(q^{10}) \) \( 13q + 13q^{2} + 13q^{4} - 13q^{5} + 7q^{7} + 13q^{8} + 25q^{9} - 13q^{10} + 13q^{11} + 11q^{13} + 7q^{14} + 13q^{16} - 2q^{17} + 25q^{18} + 7q^{19} - 13q^{20} + 12q^{21} + 13q^{22} + 12q^{23} + 13q^{25} + 11q^{26} - 15q^{27} + 7q^{28} + 14q^{29} + 20q^{31} + 13q^{32} - 2q^{34} - 7q^{35} + 25q^{36} + 17q^{37} + 7q^{38} - 4q^{39} - 13q^{40} + 9q^{41} + 12q^{42} + 13q^{43} + 13q^{44} - 25q^{45} + 12q^{46} - 9q^{47} + 30q^{49} + 13q^{50} - 3q^{51} + 11q^{52} + 22q^{53} - 15q^{54} - 13q^{55} + 7q^{56} + 17q^{57} + 14q^{58} + 19q^{59} + 2q^{61} + 20q^{62} + 12q^{63} + 13q^{64} - 11q^{65} + 9q^{67} - 2q^{68} - 6q^{69} - 7q^{70} + 6q^{71} + 25q^{72} + 7q^{73} + 17q^{74} + 7q^{76} + 7q^{77} - 4q^{78} + 50q^{79} - 13q^{80} + 85q^{81} + 9q^{82} + q^{83} + 12q^{84} + 2q^{85} + 13q^{86} - 21q^{87} + 13q^{88} + 5q^{89} - 25q^{90} + 5q^{91} + 12q^{92} + 3q^{93} - 9q^{94} - 7q^{95} + 20q^{97} + 30q^{98} + 25q^{99} + O(q^{100}) \)

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
\(3\) 0.921464 0.532007 0.266004 0.963972i \(-0.414297\pi\)
0.266004 + 0.963972i \(0.414297\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0.921464 0.376186
\(7\) −3.84192 −1.45211 −0.726054 0.687638i \(-0.758648\pi\)
−0.726054 + 0.687638i \(0.758648\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.15090 −0.716968
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 0.921464 0.266004
\(13\) 3.36077 0.932109 0.466054 0.884756i \(-0.345675\pi\)
0.466054 + 0.884756i \(0.345675\pi\)
\(14\) −3.84192 −1.02680
\(15\) −0.921464 −0.237921
\(16\) 1.00000 0.250000
\(17\) 4.48968 1.08891 0.544453 0.838791i \(-0.316737\pi\)
0.544453 + 0.838791i \(0.316737\pi\)
\(18\) −2.15090 −0.506973
\(19\) −6.54987 −1.50264 −0.751322 0.659936i \(-0.770583\pi\)
−0.751322 + 0.659936i \(0.770583\pi\)
\(20\) −1.00000 −0.223607
\(21\) −3.54019 −0.772532
\(22\) 1.00000 0.213201
\(23\) 2.73904 0.571128 0.285564 0.958360i \(-0.407819\pi\)
0.285564 + 0.958360i \(0.407819\pi\)
\(24\) 0.921464 0.188093
\(25\) 1.00000 0.200000
\(26\) 3.36077 0.659100
\(27\) −4.74637 −0.913440
\(28\) −3.84192 −0.726054
\(29\) 2.61423 0.485451 0.242726 0.970095i \(-0.421959\pi\)
0.242726 + 0.970095i \(0.421959\pi\)
\(30\) −0.921464 −0.168235
\(31\) −0.588567 −0.105710 −0.0528549 0.998602i \(-0.516832\pi\)
−0.0528549 + 0.998602i \(0.516832\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.921464 0.160406
\(34\) 4.48968 0.769973
\(35\) 3.84192 0.649402
\(36\) −2.15090 −0.358484
\(37\) −0.240321 −0.0395085 −0.0197543 0.999805i \(-0.506288\pi\)
−0.0197543 + 0.999805i \(0.506288\pi\)
\(38\) −6.54987 −1.06253
\(39\) 3.09682 0.495889
\(40\) −1.00000 −0.158114
\(41\) 10.0856 1.57511 0.787556 0.616244i \(-0.211346\pi\)
0.787556 + 0.616244i \(0.211346\pi\)
\(42\) −3.54019 −0.546263
\(43\) 1.00000 0.152499
\(44\) 1.00000 0.150756
\(45\) 2.15090 0.320638
\(46\) 2.73904 0.403849
\(47\) 9.27274 1.35257 0.676284 0.736641i \(-0.263589\pi\)
0.676284 + 0.736641i \(0.263589\pi\)
\(48\) 0.921464 0.133002
\(49\) 7.76033 1.10862
\(50\) 1.00000 0.141421
\(51\) 4.13708 0.579306
\(52\) 3.36077 0.466054
\(53\) 12.6401 1.73625 0.868127 0.496343i \(-0.165324\pi\)
0.868127 + 0.496343i \(0.165324\pi\)
\(54\) −4.74637 −0.645899
\(55\) −1.00000 −0.134840
\(56\) −3.84192 −0.513398
\(57\) −6.03547 −0.799417
\(58\) 2.61423 0.343266
\(59\) 12.5429 1.63294 0.816470 0.577388i \(-0.195928\pi\)
0.816470 + 0.577388i \(0.195928\pi\)
\(60\) −0.921464 −0.118960
\(61\) −9.61487 −1.23106 −0.615529 0.788114i \(-0.711057\pi\)
−0.615529 + 0.788114i \(0.711057\pi\)
\(62\) −0.588567 −0.0747481
\(63\) 8.26360 1.04112
\(64\) 1.00000 0.125000
\(65\) −3.36077 −0.416852
\(66\) 0.921464 0.113424
\(67\) −8.11130 −0.990953 −0.495477 0.868621i \(-0.665007\pi\)
−0.495477 + 0.868621i \(0.665007\pi\)
\(68\) 4.48968 0.544453
\(69\) 2.52392 0.303845
\(70\) 3.84192 0.459197
\(71\) −10.4421 −1.23925 −0.619624 0.784898i \(-0.712715\pi\)
−0.619624 + 0.784898i \(0.712715\pi\)
\(72\) −2.15090 −0.253487
\(73\) 11.8631 1.38848 0.694238 0.719746i \(-0.255742\pi\)
0.694238 + 0.719746i \(0.255742\pi\)
\(74\) −0.240321 −0.0279367
\(75\) 0.921464 0.106401
\(76\) −6.54987 −0.751322
\(77\) −3.84192 −0.437827
\(78\) 3.09682 0.350646
\(79\) 5.98496 0.673360 0.336680 0.941619i \(-0.390696\pi\)
0.336680 + 0.941619i \(0.390696\pi\)
\(80\) −1.00000 −0.111803
\(81\) 2.07910 0.231012
\(82\) 10.0856 1.11377
\(83\) 8.56754 0.940410 0.470205 0.882557i \(-0.344180\pi\)
0.470205 + 0.882557i \(0.344180\pi\)
\(84\) −3.54019 −0.386266
\(85\) −4.48968 −0.486974
\(86\) 1.00000 0.107833
\(87\) 2.40892 0.258264
\(88\) 1.00000 0.106600
\(89\) 13.9559 1.47933 0.739664 0.672977i \(-0.234985\pi\)
0.739664 + 0.672977i \(0.234985\pi\)
\(90\) 2.15090 0.226725
\(91\) −12.9118 −1.35352
\(92\) 2.73904 0.285564
\(93\) −0.542343 −0.0562384
\(94\) 9.27274 0.956410
\(95\) 6.54987 0.672003
\(96\) 0.921464 0.0940465
\(97\) −8.94728 −0.908458 −0.454229 0.890885i \(-0.650085\pi\)
−0.454229 + 0.890885i \(0.650085\pi\)
\(98\) 7.76033 0.783911
\(99\) −2.15090 −0.216174
\(100\) 1.00000 0.100000
\(101\) 6.76897 0.673538 0.336769 0.941587i \(-0.390666\pi\)
0.336769 + 0.941587i \(0.390666\pi\)
\(102\) 4.13708 0.409632
\(103\) −11.8563 −1.16824 −0.584119 0.811668i \(-0.698560\pi\)
−0.584119 + 0.811668i \(0.698560\pi\)
\(104\) 3.36077 0.329550
\(105\) 3.54019 0.345487
\(106\) 12.6401 1.22772
\(107\) −15.8462 −1.53191 −0.765956 0.642893i \(-0.777734\pi\)
−0.765956 + 0.642893i \(0.777734\pi\)
\(108\) −4.74637 −0.456720
\(109\) −9.86228 −0.944635 −0.472318 0.881428i \(-0.656582\pi\)
−0.472318 + 0.881428i \(0.656582\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −0.221447 −0.0210188
\(112\) −3.84192 −0.363027
\(113\) 11.1604 1.04988 0.524942 0.851138i \(-0.324087\pi\)
0.524942 + 0.851138i \(0.324087\pi\)
\(114\) −6.03547 −0.565273
\(115\) −2.73904 −0.255416
\(116\) 2.61423 0.242726
\(117\) −7.22869 −0.668292
\(118\) 12.5429 1.15466
\(119\) −17.2490 −1.58121
\(120\) −0.921464 −0.0841177
\(121\) 1.00000 0.0909091
\(122\) −9.61487 −0.870489
\(123\) 9.29355 0.837971
\(124\) −0.588567 −0.0528549
\(125\) −1.00000 −0.0894427
\(126\) 8.26360 0.736180
\(127\) 20.5175 1.82064 0.910318 0.413909i \(-0.135837\pi\)
0.910318 + 0.413909i \(0.135837\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0.921464 0.0811304
\(130\) −3.36077 −0.294759
\(131\) −9.26132 −0.809166 −0.404583 0.914501i \(-0.632583\pi\)
−0.404583 + 0.914501i \(0.632583\pi\)
\(132\) 0.921464 0.0802031
\(133\) 25.1641 2.18200
\(134\) −8.11130 −0.700710
\(135\) 4.74637 0.408503
\(136\) 4.48968 0.384987
\(137\) 20.5662 1.75709 0.878545 0.477659i \(-0.158515\pi\)
0.878545 + 0.477659i \(0.158515\pi\)
\(138\) 2.52392 0.214851
\(139\) −9.86057 −0.836362 −0.418181 0.908364i \(-0.637332\pi\)
−0.418181 + 0.908364i \(0.637332\pi\)
\(140\) 3.84192 0.324701
\(141\) 8.54449 0.719576
\(142\) −10.4421 −0.876281
\(143\) 3.36077 0.281041
\(144\) −2.15090 −0.179242
\(145\) −2.61423 −0.217100
\(146\) 11.8631 0.981800
\(147\) 7.15086 0.589793
\(148\) −0.240321 −0.0197543
\(149\) 4.43344 0.363201 0.181601 0.983372i \(-0.441872\pi\)
0.181601 + 0.983372i \(0.441872\pi\)
\(150\) 0.921464 0.0752372
\(151\) 18.0595 1.46966 0.734832 0.678250i \(-0.237261\pi\)
0.734832 + 0.678250i \(0.237261\pi\)
\(152\) −6.54987 −0.531265
\(153\) −9.65687 −0.780712
\(154\) −3.84192 −0.309590
\(155\) 0.588567 0.0472749
\(156\) 3.09682 0.247944
\(157\) 19.0697 1.52193 0.760965 0.648793i \(-0.224726\pi\)
0.760965 + 0.648793i \(0.224726\pi\)
\(158\) 5.98496 0.476138
\(159\) 11.6474 0.923699
\(160\) −1.00000 −0.0790569
\(161\) −10.5231 −0.829340
\(162\) 2.07910 0.163350
\(163\) −11.5905 −0.907835 −0.453917 0.891044i \(-0.649974\pi\)
−0.453917 + 0.891044i \(0.649974\pi\)
\(164\) 10.0856 0.787556
\(165\) −0.921464 −0.0717359
\(166\) 8.56754 0.664970
\(167\) 6.33426 0.490160 0.245080 0.969503i \(-0.421186\pi\)
0.245080 + 0.969503i \(0.421186\pi\)
\(168\) −3.54019 −0.273131
\(169\) −1.70526 −0.131174
\(170\) −4.48968 −0.344343
\(171\) 14.0881 1.07735
\(172\) 1.00000 0.0762493
\(173\) −2.54982 −0.193860 −0.0969298 0.995291i \(-0.530902\pi\)
−0.0969298 + 0.995291i \(0.530902\pi\)
\(174\) 2.40892 0.182620
\(175\) −3.84192 −0.290422
\(176\) 1.00000 0.0753778
\(177\) 11.5578 0.868736
\(178\) 13.9559 1.04604
\(179\) −6.35898 −0.475293 −0.237646 0.971352i \(-0.576376\pi\)
−0.237646 + 0.971352i \(0.576376\pi\)
\(180\) 2.15090 0.160319
\(181\) 4.96116 0.368760 0.184380 0.982855i \(-0.440972\pi\)
0.184380 + 0.982855i \(0.440972\pi\)
\(182\) −12.9118 −0.957085
\(183\) −8.85975 −0.654932
\(184\) 2.73904 0.201924
\(185\) 0.240321 0.0176687
\(186\) −0.542343 −0.0397665
\(187\) 4.48968 0.328318
\(188\) 9.27274 0.676284
\(189\) 18.2352 1.32641
\(190\) 6.54987 0.475178
\(191\) 2.82460 0.204381 0.102190 0.994765i \(-0.467415\pi\)
0.102190 + 0.994765i \(0.467415\pi\)
\(192\) 0.921464 0.0665009
\(193\) −0.714821 −0.0514540 −0.0257270 0.999669i \(-0.508190\pi\)
−0.0257270 + 0.999669i \(0.508190\pi\)
\(194\) −8.94728 −0.642377
\(195\) −3.09682 −0.221768
\(196\) 7.76033 0.554309
\(197\) −8.23931 −0.587026 −0.293513 0.955955i \(-0.594824\pi\)
−0.293513 + 0.955955i \(0.594824\pi\)
\(198\) −2.15090 −0.152858
\(199\) −9.90281 −0.701991 −0.350996 0.936377i \(-0.614157\pi\)
−0.350996 + 0.936377i \(0.614157\pi\)
\(200\) 1.00000 0.0707107
\(201\) −7.47427 −0.527194
\(202\) 6.76897 0.476263
\(203\) −10.0437 −0.704928
\(204\) 4.13708 0.289653
\(205\) −10.0856 −0.704411
\(206\) −11.8563 −0.826069
\(207\) −5.89140 −0.409481
\(208\) 3.36077 0.233027
\(209\) −6.54987 −0.453064
\(210\) 3.54019 0.244296
\(211\) −14.6306 −1.00722 −0.503608 0.863933i \(-0.667994\pi\)
−0.503608 + 0.863933i \(0.667994\pi\)
\(212\) 12.6401 0.868127
\(213\) −9.62201 −0.659289
\(214\) −15.8462 −1.08322
\(215\) −1.00000 −0.0681994
\(216\) −4.74637 −0.322950
\(217\) 2.26123 0.153502
\(218\) −9.86228 −0.667958
\(219\) 10.9315 0.738679
\(220\) −1.00000 −0.0674200
\(221\) 15.0888 1.01498
\(222\) −0.221447 −0.0148626
\(223\) 16.2067 1.08528 0.542639 0.839966i \(-0.317425\pi\)
0.542639 + 0.839966i \(0.317425\pi\)
\(224\) −3.84192 −0.256699
\(225\) −2.15090 −0.143394
\(226\) 11.1604 0.742381
\(227\) −7.28678 −0.483640 −0.241820 0.970321i \(-0.577744\pi\)
−0.241820 + 0.970321i \(0.577744\pi\)
\(228\) −6.03547 −0.399709
\(229\) −7.26471 −0.480065 −0.240033 0.970765i \(-0.577158\pi\)
−0.240033 + 0.970765i \(0.577158\pi\)
\(230\) −2.73904 −0.180607
\(231\) −3.54019 −0.232927
\(232\) 2.61423 0.171633
\(233\) 19.0465 1.24778 0.623890 0.781512i \(-0.285551\pi\)
0.623890 + 0.781512i \(0.285551\pi\)
\(234\) −7.22869 −0.472554
\(235\) −9.27274 −0.604887
\(236\) 12.5429 0.816470
\(237\) 5.51492 0.358233
\(238\) −17.2490 −1.11808
\(239\) 8.88667 0.574831 0.287416 0.957806i \(-0.407204\pi\)
0.287416 + 0.957806i \(0.407204\pi\)
\(240\) −0.921464 −0.0594802
\(241\) 2.76143 0.177880 0.0889398 0.996037i \(-0.471652\pi\)
0.0889398 + 0.996037i \(0.471652\pi\)
\(242\) 1.00000 0.0642824
\(243\) 16.1549 1.03634
\(244\) −9.61487 −0.615529
\(245\) −7.76033 −0.495789
\(246\) 9.29355 0.592535
\(247\) −22.0126 −1.40063
\(248\) −0.588567 −0.0373741
\(249\) 7.89468 0.500305
\(250\) −1.00000 −0.0632456
\(251\) 2.07731 0.131118 0.0655592 0.997849i \(-0.479117\pi\)
0.0655592 + 0.997849i \(0.479117\pi\)
\(252\) 8.26360 0.520558
\(253\) 2.73904 0.172202
\(254\) 20.5175 1.28738
\(255\) −4.13708 −0.259074
\(256\) 1.00000 0.0625000
\(257\) −18.3451 −1.14434 −0.572168 0.820137i \(-0.693897\pi\)
−0.572168 + 0.820137i \(0.693897\pi\)
\(258\) 0.921464 0.0573678
\(259\) 0.923293 0.0573706
\(260\) −3.36077 −0.208426
\(261\) −5.62297 −0.348053
\(262\) −9.26132 −0.572167
\(263\) 12.4911 0.770237 0.385119 0.922867i \(-0.374161\pi\)
0.385119 + 0.922867i \(0.374161\pi\)
\(264\) 0.921464 0.0567122
\(265\) −12.6401 −0.776476
\(266\) 25.1641 1.54291
\(267\) 12.8599 0.787013
\(268\) −8.11130 −0.495477
\(269\) 27.5872 1.68202 0.841010 0.541020i \(-0.181962\pi\)
0.841010 + 0.541020i \(0.181962\pi\)
\(270\) 4.74637 0.288855
\(271\) 16.8218 1.02185 0.510927 0.859624i \(-0.329302\pi\)
0.510927 + 0.859624i \(0.329302\pi\)
\(272\) 4.48968 0.272227
\(273\) −11.8977 −0.720084
\(274\) 20.5662 1.24245
\(275\) 1.00000 0.0603023
\(276\) 2.52392 0.151922
\(277\) −25.2847 −1.51921 −0.759607 0.650383i \(-0.774608\pi\)
−0.759607 + 0.650383i \(0.774608\pi\)
\(278\) −9.86057 −0.591397
\(279\) 1.26595 0.0757906
\(280\) 3.84192 0.229598
\(281\) −6.07659 −0.362499 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(282\) 8.54449 0.508817
\(283\) 25.4783 1.51453 0.757265 0.653108i \(-0.226535\pi\)
0.757265 + 0.653108i \(0.226535\pi\)
\(284\) −10.4421 −0.619624
\(285\) 6.03547 0.357510
\(286\) 3.36077 0.198726
\(287\) −38.7482 −2.28723
\(288\) −2.15090 −0.126743
\(289\) 3.15721 0.185718
\(290\) −2.61423 −0.153513
\(291\) −8.24459 −0.483306
\(292\) 11.8631 0.694238
\(293\) −5.56061 −0.324854 −0.162427 0.986721i \(-0.551932\pi\)
−0.162427 + 0.986721i \(0.551932\pi\)
\(294\) 7.15086 0.417047
\(295\) −12.5429 −0.730273
\(296\) −0.240321 −0.0139684
\(297\) −4.74637 −0.275412
\(298\) 4.43344 0.256822
\(299\) 9.20526 0.532354
\(300\) 0.921464 0.0532007
\(301\) −3.84192 −0.221444
\(302\) 18.0595 1.03921
\(303\) 6.23736 0.358327
\(304\) −6.54987 −0.375661
\(305\) 9.61487 0.550546
\(306\) −9.65687 −0.552046
\(307\) −21.8905 −1.24936 −0.624680 0.780881i \(-0.714770\pi\)
−0.624680 + 0.780881i \(0.714770\pi\)
\(308\) −3.84192 −0.218914
\(309\) −10.9252 −0.621511
\(310\) 0.588567 0.0334284
\(311\) −3.49295 −0.198067 −0.0990337 0.995084i \(-0.531575\pi\)
−0.0990337 + 0.995084i \(0.531575\pi\)
\(312\) 3.09682 0.175323
\(313\) −33.6717 −1.90324 −0.951618 0.307282i \(-0.900580\pi\)
−0.951618 + 0.307282i \(0.900580\pi\)
\(314\) 19.0697 1.07617
\(315\) −8.26360 −0.465601
\(316\) 5.98496 0.336680
\(317\) −18.5042 −1.03930 −0.519650 0.854379i \(-0.673938\pi\)
−0.519650 + 0.854379i \(0.673938\pi\)
\(318\) 11.6474 0.653154
\(319\) 2.61423 0.146369
\(320\) −1.00000 −0.0559017
\(321\) −14.6017 −0.814988
\(322\) −10.5231 −0.586432
\(323\) −29.4068 −1.63624
\(324\) 2.07910 0.115506
\(325\) 3.36077 0.186422
\(326\) −11.5905 −0.641936
\(327\) −9.08773 −0.502553
\(328\) 10.0856 0.556886
\(329\) −35.6251 −1.96408
\(330\) −0.921464 −0.0507249
\(331\) 13.8933 0.763646 0.381823 0.924235i \(-0.375296\pi\)
0.381823 + 0.924235i \(0.375296\pi\)
\(332\) 8.56754 0.470205
\(333\) 0.516907 0.0283264
\(334\) 6.33426 0.346595
\(335\) 8.11130 0.443168
\(336\) −3.54019 −0.193133
\(337\) 5.09506 0.277546 0.138773 0.990324i \(-0.455684\pi\)
0.138773 + 0.990324i \(0.455684\pi\)
\(338\) −1.70526 −0.0927537
\(339\) 10.2839 0.558546
\(340\) −4.48968 −0.243487
\(341\) −0.588567 −0.0318727
\(342\) 14.0881 0.761800
\(343\) −2.92111 −0.157725
\(344\) 1.00000 0.0539164
\(345\) −2.52392 −0.135883
\(346\) −2.54982 −0.137079
\(347\) −27.6747 −1.48565 −0.742827 0.669483i \(-0.766516\pi\)
−0.742827 + 0.669483i \(0.766516\pi\)
\(348\) 2.40892 0.129132
\(349\) 5.36885 0.287388 0.143694 0.989622i \(-0.454102\pi\)
0.143694 + 0.989622i \(0.454102\pi\)
\(350\) −3.84192 −0.205359
\(351\) −15.9514 −0.851425
\(352\) 1.00000 0.0533002
\(353\) −24.2743 −1.29199 −0.645996 0.763341i \(-0.723558\pi\)
−0.645996 + 0.763341i \(0.723558\pi\)
\(354\) 11.5578 0.614289
\(355\) 10.4421 0.554209
\(356\) 13.9559 0.739664
\(357\) −15.8943 −0.841216
\(358\) −6.35898 −0.336083
\(359\) 11.0758 0.584559 0.292280 0.956333i \(-0.405586\pi\)
0.292280 + 0.956333i \(0.405586\pi\)
\(360\) 2.15090 0.113363
\(361\) 23.9008 1.25794
\(362\) 4.96116 0.260753
\(363\) 0.921464 0.0483643
\(364\) −12.9118 −0.676761
\(365\) −11.8631 −0.620945
\(366\) −8.85975 −0.463107
\(367\) −0.627963 −0.0327794 −0.0163897 0.999866i \(-0.505217\pi\)
−0.0163897 + 0.999866i \(0.505217\pi\)
\(368\) 2.73904 0.142782
\(369\) −21.6932 −1.12930
\(370\) 0.240321 0.0124937
\(371\) −48.5623 −2.52123
\(372\) −0.542343 −0.0281192
\(373\) −34.8731 −1.80566 −0.902830 0.429999i \(-0.858514\pi\)
−0.902830 + 0.429999i \(0.858514\pi\)
\(374\) 4.48968 0.232156
\(375\) −0.921464 −0.0475842
\(376\) 9.27274 0.478205
\(377\) 8.78583 0.452493
\(378\) 18.2352 0.937916
\(379\) 19.7139 1.01264 0.506318 0.862347i \(-0.331006\pi\)
0.506318 + 0.862347i \(0.331006\pi\)
\(380\) 6.54987 0.336001
\(381\) 18.9062 0.968592
\(382\) 2.82460 0.144519
\(383\) −32.6656 −1.66914 −0.834568 0.550906i \(-0.814282\pi\)
−0.834568 + 0.550906i \(0.814282\pi\)
\(384\) 0.921464 0.0470232
\(385\) 3.84192 0.195802
\(386\) −0.714821 −0.0363834
\(387\) −2.15090 −0.109337
\(388\) −8.94728 −0.454229
\(389\) 13.0008 0.659164 0.329582 0.944127i \(-0.393092\pi\)
0.329582 + 0.944127i \(0.393092\pi\)
\(390\) −3.09682 −0.156814
\(391\) 12.2974 0.621906
\(392\) 7.76033 0.391956
\(393\) −8.53397 −0.430482
\(394\) −8.23931 −0.415090
\(395\) −5.98496 −0.301136
\(396\) −2.15090 −0.108087
\(397\) 7.23835 0.363282 0.181641 0.983365i \(-0.441859\pi\)
0.181641 + 0.983365i \(0.441859\pi\)
\(398\) −9.90281 −0.496383
\(399\) 23.1878 1.16084
\(400\) 1.00000 0.0500000
\(401\) 25.5243 1.27462 0.637312 0.770606i \(-0.280046\pi\)
0.637312 + 0.770606i \(0.280046\pi\)
\(402\) −7.47427 −0.372783
\(403\) −1.97804 −0.0985330
\(404\) 6.76897 0.336769
\(405\) −2.07910 −0.103312
\(406\) −10.0437 −0.498459
\(407\) −0.240321 −0.0119123
\(408\) 4.13708 0.204816
\(409\) −8.29621 −0.410221 −0.205111 0.978739i \(-0.565755\pi\)
−0.205111 + 0.978739i \(0.565755\pi\)
\(410\) −10.0856 −0.498094
\(411\) 18.9510 0.934785
\(412\) −11.8563 −0.584119
\(413\) −48.1886 −2.37121
\(414\) −5.89140 −0.289547
\(415\) −8.56754 −0.420564
\(416\) 3.36077 0.164775
\(417\) −9.08615 −0.444951
\(418\) −6.54987 −0.320365
\(419\) 3.75774 0.183578 0.0917888 0.995778i \(-0.470742\pi\)
0.0917888 + 0.995778i \(0.470742\pi\)
\(420\) 3.54019 0.172743
\(421\) 0.464546 0.0226406 0.0113203 0.999936i \(-0.496397\pi\)
0.0113203 + 0.999936i \(0.496397\pi\)
\(422\) −14.6306 −0.712209
\(423\) −19.9448 −0.969748
\(424\) 12.6401 0.613858
\(425\) 4.48968 0.217781
\(426\) −9.62201 −0.466188
\(427\) 36.9395 1.78763
\(428\) −15.8462 −0.765956
\(429\) 3.09682 0.149516
\(430\) −1.00000 −0.0482243
\(431\) −23.9618 −1.15420 −0.577099 0.816674i \(-0.695815\pi\)
−0.577099 + 0.816674i \(0.695815\pi\)
\(432\) −4.74637 −0.228360
\(433\) −29.5327 −1.41925 −0.709626 0.704578i \(-0.751136\pi\)
−0.709626 + 0.704578i \(0.751136\pi\)
\(434\) 2.26123 0.108542
\(435\) −2.40892 −0.115499
\(436\) −9.86228 −0.472318
\(437\) −17.9403 −0.858202
\(438\) 10.9315 0.522325
\(439\) −22.1469 −1.05701 −0.528507 0.848929i \(-0.677248\pi\)
−0.528507 + 0.848929i \(0.677248\pi\)
\(440\) −1.00000 −0.0476731
\(441\) −16.6917 −0.794844
\(442\) 15.0888 0.717699
\(443\) 37.4638 1.77996 0.889979 0.456002i \(-0.150719\pi\)
0.889979 + 0.456002i \(0.150719\pi\)
\(444\) −0.221447 −0.0105094
\(445\) −13.9559 −0.661575
\(446\) 16.2067 0.767408
\(447\) 4.08525 0.193226
\(448\) −3.84192 −0.181514
\(449\) 39.1108 1.84575 0.922875 0.385099i \(-0.125833\pi\)
0.922875 + 0.385099i \(0.125833\pi\)
\(450\) −2.15090 −0.101395
\(451\) 10.0856 0.474914
\(452\) 11.1604 0.524942
\(453\) 16.6412 0.781872
\(454\) −7.28678 −0.341985
\(455\) 12.9118 0.605314
\(456\) −6.03547 −0.282637
\(457\) 7.77236 0.363576 0.181788 0.983338i \(-0.441812\pi\)
0.181788 + 0.983338i \(0.441812\pi\)
\(458\) −7.26471 −0.339458
\(459\) −21.3097 −0.994651
\(460\) −2.73904 −0.127708
\(461\) 8.64592 0.402680 0.201340 0.979521i \(-0.435470\pi\)
0.201340 + 0.979521i \(0.435470\pi\)
\(462\) −3.54019 −0.164704
\(463\) 22.3778 1.03999 0.519993 0.854171i \(-0.325935\pi\)
0.519993 + 0.854171i \(0.325935\pi\)
\(464\) 2.61423 0.121363
\(465\) 0.542343 0.0251506
\(466\) 19.0465 0.882314
\(467\) 25.2601 1.16890 0.584450 0.811430i \(-0.301310\pi\)
0.584450 + 0.811430i \(0.301310\pi\)
\(468\) −7.22869 −0.334146
\(469\) 31.1629 1.43897
\(470\) −9.27274 −0.427720
\(471\) 17.5721 0.809678
\(472\) 12.5429 0.577331
\(473\) 1.00000 0.0459800
\(474\) 5.51492 0.253309
\(475\) −6.54987 −0.300529
\(476\) −17.2490 −0.790605
\(477\) −27.1877 −1.24484
\(478\) 8.88667 0.406467
\(479\) 24.8707 1.13637 0.568186 0.822900i \(-0.307646\pi\)
0.568186 + 0.822900i \(0.307646\pi\)
\(480\) −0.921464 −0.0420589
\(481\) −0.807662 −0.0368262
\(482\) 2.76143 0.125780
\(483\) −9.69670 −0.441215
\(484\) 1.00000 0.0454545
\(485\) 8.94728 0.406275
\(486\) 16.1549 0.732803
\(487\) −15.1078 −0.684598 −0.342299 0.939591i \(-0.611206\pi\)
−0.342299 + 0.939591i \(0.611206\pi\)
\(488\) −9.61487 −0.435245
\(489\) −10.6802 −0.482975
\(490\) −7.76033 −0.350576
\(491\) −20.9423 −0.945114 −0.472557 0.881300i \(-0.656669\pi\)
−0.472557 + 0.881300i \(0.656669\pi\)
\(492\) 9.29355 0.418985
\(493\) 11.7371 0.528611
\(494\) −22.0126 −0.990393
\(495\) 2.15090 0.0966760
\(496\) −0.588567 −0.0264274
\(497\) 40.1177 1.79952
\(498\) 7.89468 0.353769
\(499\) −25.5069 −1.14184 −0.570922 0.821004i \(-0.693414\pi\)
−0.570922 + 0.821004i \(0.693414\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 5.83679 0.260769
\(502\) 2.07731 0.0927147
\(503\) −10.1492 −0.452532 −0.226266 0.974066i \(-0.572652\pi\)
−0.226266 + 0.974066i \(0.572652\pi\)
\(504\) 8.26360 0.368090
\(505\) −6.76897 −0.301215
\(506\) 2.73904 0.121765
\(507\) −1.57133 −0.0697853
\(508\) 20.5175 0.910318
\(509\) −31.3909 −1.39138 −0.695689 0.718343i \(-0.744901\pi\)
−0.695689 + 0.718343i \(0.744901\pi\)
\(510\) −4.13708 −0.183193
\(511\) −45.5772 −2.01622
\(512\) 1.00000 0.0441942
\(513\) 31.0881 1.37257
\(514\) −18.3451 −0.809167
\(515\) 11.8563 0.522452
\(516\) 0.921464 0.0405652
\(517\) 9.27274 0.407815
\(518\) 0.923293 0.0405672
\(519\) −2.34957 −0.103135
\(520\) −3.36077 −0.147379
\(521\) 6.33199 0.277409 0.138705 0.990334i \(-0.455706\pi\)
0.138705 + 0.990334i \(0.455706\pi\)
\(522\) −5.62297 −0.246111
\(523\) 21.1781 0.926055 0.463027 0.886344i \(-0.346763\pi\)
0.463027 + 0.886344i \(0.346763\pi\)
\(524\) −9.26132 −0.404583
\(525\) −3.54019 −0.154506
\(526\) 12.4911 0.544640
\(527\) −2.64248 −0.115108
\(528\) 0.921464 0.0401016
\(529\) −15.4977 −0.673812
\(530\) −12.6401 −0.549051
\(531\) −26.9785 −1.17077
\(532\) 25.1641 1.09100
\(533\) 33.8954 1.46817
\(534\) 12.8599 0.556502
\(535\) 15.8462 0.685092
\(536\) −8.11130 −0.350355
\(537\) −5.85957 −0.252859
\(538\) 27.5872 1.18937
\(539\) 7.76033 0.334261
\(540\) 4.74637 0.204251
\(541\) 22.5374 0.968958 0.484479 0.874803i \(-0.339009\pi\)
0.484479 + 0.874803i \(0.339009\pi\)
\(542\) 16.8218 0.722560
\(543\) 4.57153 0.196183
\(544\) 4.48968 0.192493
\(545\) 9.86228 0.422454
\(546\) −11.8977 −0.509176
\(547\) −3.63414 −0.155384 −0.0776922 0.996977i \(-0.524755\pi\)
−0.0776922 + 0.996977i \(0.524755\pi\)
\(548\) 20.5662 0.878545
\(549\) 20.6807 0.882629
\(550\) 1.00000 0.0426401
\(551\) −17.1229 −0.729460
\(552\) 2.52392 0.107425
\(553\) −22.9937 −0.977792
\(554\) −25.2847 −1.07425
\(555\) 0.221447 0.00939990
\(556\) −9.86057 −0.418181
\(557\) 18.0972 0.766804 0.383402 0.923582i \(-0.374752\pi\)
0.383402 + 0.923582i \(0.374752\pi\)
\(558\) 1.26595 0.0535920
\(559\) 3.36077 0.142145
\(560\) 3.84192 0.162351
\(561\) 4.13708 0.174667
\(562\) −6.07659 −0.256326
\(563\) −19.0052 −0.800975 −0.400488 0.916302i \(-0.631159\pi\)
−0.400488 + 0.916302i \(0.631159\pi\)
\(564\) 8.54449 0.359788
\(565\) −11.1604 −0.469523
\(566\) 25.4783 1.07093
\(567\) −7.98775 −0.335454
\(568\) −10.4421 −0.438141
\(569\) −21.5947 −0.905297 −0.452649 0.891689i \(-0.649521\pi\)
−0.452649 + 0.891689i \(0.649521\pi\)
\(570\) 6.03547 0.252798
\(571\) −45.5796 −1.90745 −0.953723 0.300686i \(-0.902784\pi\)
−0.953723 + 0.300686i \(0.902784\pi\)
\(572\) 3.36077 0.140521
\(573\) 2.60277 0.108732
\(574\) −38.7482 −1.61732
\(575\) 2.73904 0.114226
\(576\) −2.15090 −0.0896210
\(577\) −4.24414 −0.176686 −0.0883430 0.996090i \(-0.528157\pi\)
−0.0883430 + 0.996090i \(0.528157\pi\)
\(578\) 3.15721 0.131323
\(579\) −0.658682 −0.0273739
\(580\) −2.61423 −0.108550
\(581\) −32.9158 −1.36558
\(582\) −8.24459 −0.341749
\(583\) 12.6401 0.523500
\(584\) 11.8631 0.490900
\(585\) 7.22869 0.298869
\(586\) −5.56061 −0.229707
\(587\) 0.406717 0.0167870 0.00839351 0.999965i \(-0.497328\pi\)
0.00839351 + 0.999965i \(0.497328\pi\)
\(588\) 7.15086 0.294896
\(589\) 3.85504 0.158844
\(590\) −12.5429 −0.516381
\(591\) −7.59222 −0.312302
\(592\) −0.240321 −0.00987713
\(593\) 24.2297 0.994993 0.497497 0.867466i \(-0.334253\pi\)
0.497497 + 0.867466i \(0.334253\pi\)
\(594\) −4.74637 −0.194746
\(595\) 17.2490 0.707139
\(596\) 4.43344 0.181601
\(597\) −9.12508 −0.373465
\(598\) 9.20526 0.376431
\(599\) 31.8343 1.30072 0.650358 0.759628i \(-0.274619\pi\)
0.650358 + 0.759628i \(0.274619\pi\)
\(600\) 0.921464 0.0376186
\(601\) 23.6792 0.965894 0.482947 0.875650i \(-0.339566\pi\)
0.482947 + 0.875650i \(0.339566\pi\)
\(602\) −3.84192 −0.156585
\(603\) 17.4466 0.710482
\(604\) 18.0595 0.734832
\(605\) −1.00000 −0.0406558
\(606\) 6.23736 0.253375
\(607\) 17.8004 0.722497 0.361249 0.932470i \(-0.382351\pi\)
0.361249 + 0.932470i \(0.382351\pi\)
\(608\) −6.54987 −0.265632
\(609\) −9.25488 −0.375027
\(610\) 9.61487 0.389295
\(611\) 31.1635 1.26074
\(612\) −9.65687 −0.390356
\(613\) 1.56785 0.0633247 0.0316624 0.999499i \(-0.489920\pi\)
0.0316624 + 0.999499i \(0.489920\pi\)
\(614\) −21.8905 −0.883431
\(615\) −9.29355 −0.374752
\(616\) −3.84192 −0.154795
\(617\) −11.8883 −0.478605 −0.239302 0.970945i \(-0.576919\pi\)
−0.239302 + 0.970945i \(0.576919\pi\)
\(618\) −10.9252 −0.439475
\(619\) −22.1870 −0.891772 −0.445886 0.895090i \(-0.647111\pi\)
−0.445886 + 0.895090i \(0.647111\pi\)
\(620\) 0.588567 0.0236374
\(621\) −13.0005 −0.521691
\(622\) −3.49295 −0.140055
\(623\) −53.6176 −2.14814
\(624\) 3.09682 0.123972
\(625\) 1.00000 0.0400000
\(626\) −33.6717 −1.34579
\(627\) −6.03547 −0.241033
\(628\) 19.0697 0.760965
\(629\) −1.07896 −0.0430211
\(630\) −8.26360 −0.329230
\(631\) 29.8463 1.18816 0.594081 0.804405i \(-0.297516\pi\)
0.594081 + 0.804405i \(0.297516\pi\)
\(632\) 5.98496 0.238069
\(633\) −13.4816 −0.535846
\(634\) −18.5042 −0.734896
\(635\) −20.5175 −0.814213
\(636\) 11.6474 0.461850
\(637\) 26.0806 1.03335
\(638\) 2.61423 0.103499
\(639\) 22.4600 0.888502
\(640\) −1.00000 −0.0395285
\(641\) 0.667023 0.0263458 0.0131729 0.999913i \(-0.495807\pi\)
0.0131729 + 0.999913i \(0.495807\pi\)
\(642\) −14.6017 −0.576284
\(643\) 46.1975 1.82185 0.910926 0.412570i \(-0.135369\pi\)
0.910926 + 0.412570i \(0.135369\pi\)
\(644\) −10.5231 −0.414670
\(645\) −0.921464 −0.0362826
\(646\) −29.4068 −1.15700
\(647\) 11.9147 0.468415 0.234207 0.972187i \(-0.424751\pi\)
0.234207 + 0.972187i \(0.424751\pi\)
\(648\) 2.07910 0.0816749
\(649\) 12.5429 0.492350
\(650\) 3.36077 0.131820
\(651\) 2.08364 0.0816642
\(652\) −11.5905 −0.453917
\(653\) 29.0362 1.13627 0.568137 0.822934i \(-0.307664\pi\)
0.568137 + 0.822934i \(0.307664\pi\)
\(654\) −9.08773 −0.355358
\(655\) 9.26132 0.361870
\(656\) 10.0856 0.393778
\(657\) −25.5165 −0.995493
\(658\) −35.6251 −1.38881
\(659\) 19.8186 0.772023 0.386011 0.922494i \(-0.373853\pi\)
0.386011 + 0.922494i \(0.373853\pi\)
\(660\) −0.921464 −0.0358679
\(661\) −13.0120 −0.506108 −0.253054 0.967452i \(-0.581435\pi\)
−0.253054 + 0.967452i \(0.581435\pi\)
\(662\) 13.8933 0.539979
\(663\) 13.9037 0.539977
\(664\) 8.56754 0.332485
\(665\) −25.1641 −0.975820
\(666\) 0.516907 0.0200298
\(667\) 7.16048 0.277255
\(668\) 6.33426 0.245080
\(669\) 14.9339 0.577376
\(670\) 8.11130 0.313367
\(671\) −9.61487 −0.371178
\(672\) −3.54019 −0.136566
\(673\) −26.7712 −1.03195 −0.515977 0.856602i \(-0.672571\pi\)
−0.515977 + 0.856602i \(0.672571\pi\)
\(674\) 5.09506 0.196254
\(675\) −4.74637 −0.182688
\(676\) −1.70526 −0.0655868
\(677\) −9.86767 −0.379245 −0.189623 0.981857i \(-0.560726\pi\)
−0.189623 + 0.981857i \(0.560726\pi\)
\(678\) 10.2839 0.394952
\(679\) 34.3747 1.31918
\(680\) −4.48968 −0.172171
\(681\) −6.71450 −0.257300
\(682\) −0.588567 −0.0225374
\(683\) 18.8993 0.723163 0.361581 0.932341i \(-0.382237\pi\)
0.361581 + 0.932341i \(0.382237\pi\)
\(684\) 14.0881 0.538674
\(685\) −20.5662 −0.785795
\(686\) −2.92111 −0.111528
\(687\) −6.69417 −0.255398
\(688\) 1.00000 0.0381246
\(689\) 42.4805 1.61838
\(690\) −2.52392 −0.0960841
\(691\) −18.8426 −0.716805 −0.358402 0.933567i \(-0.616678\pi\)
−0.358402 + 0.933567i \(0.616678\pi\)
\(692\) −2.54982 −0.0969298
\(693\) 8.26360 0.313908
\(694\) −27.6747 −1.05052
\(695\) 9.86057 0.374033
\(696\) 2.40892 0.0913100
\(697\) 45.2812 1.71515
\(698\) 5.36885 0.203214
\(699\) 17.5507 0.663829
\(700\) −3.84192 −0.145211
\(701\) −28.3200 −1.06963 −0.534816 0.844969i \(-0.679619\pi\)
−0.534816 + 0.844969i \(0.679619\pi\)
\(702\) −15.9514 −0.602048
\(703\) 1.57407 0.0593672
\(704\) 1.00000 0.0376889
\(705\) −8.54449 −0.321804
\(706\) −24.2743 −0.913576
\(707\) −26.0058 −0.978050
\(708\) 11.5578 0.434368
\(709\) 30.6657 1.15167 0.575837 0.817564i \(-0.304676\pi\)
0.575837 + 0.817564i \(0.304676\pi\)
\(710\) 10.4421 0.391885
\(711\) −12.8731 −0.482778
\(712\) 13.9559 0.523021
\(713\) −1.61211 −0.0603739
\(714\) −15.8943 −0.594829
\(715\) −3.36077 −0.125685
\(716\) −6.35898 −0.237646
\(717\) 8.18875 0.305814
\(718\) 11.0758 0.413346
\(719\) 8.13201 0.303273 0.151636 0.988436i \(-0.451546\pi\)
0.151636 + 0.988436i \(0.451546\pi\)
\(720\) 2.15090 0.0801595
\(721\) 45.5510 1.69641
\(722\) 23.9008 0.889496
\(723\) 2.54456 0.0946332
\(724\) 4.96116 0.184380
\(725\) 2.61423 0.0970902
\(726\) 0.921464 0.0341987
\(727\) −51.9357 −1.92619 −0.963094 0.269166i \(-0.913252\pi\)
−0.963094 + 0.269166i \(0.913252\pi\)
\(728\) −12.9118 −0.478542
\(729\) 8.64887 0.320329
\(730\) −11.8631 −0.439074
\(731\) 4.48968 0.166057
\(732\) −8.85975 −0.327466
\(733\) −25.6325 −0.946758 −0.473379 0.880859i \(-0.656966\pi\)
−0.473379 + 0.880859i \(0.656966\pi\)
\(734\) −0.627963 −0.0231785
\(735\) −7.15086 −0.263763
\(736\) 2.73904 0.100962
\(737\) −8.11130 −0.298784
\(738\) −21.6932 −0.798539
\(739\) 25.0386 0.921060 0.460530 0.887644i \(-0.347659\pi\)
0.460530 + 0.887644i \(0.347659\pi\)
\(740\) 0.240321 0.00883437
\(741\) −20.2838 −0.745144
\(742\) −48.5623 −1.78278
\(743\) −44.8111 −1.64396 −0.821980 0.569517i \(-0.807130\pi\)
−0.821980 + 0.569517i \(0.807130\pi\)
\(744\) −0.542343 −0.0198833
\(745\) −4.43344 −0.162429
\(746\) −34.8731 −1.27679
\(747\) −18.4280 −0.674244
\(748\) 4.48968 0.164159
\(749\) 60.8798 2.22450
\(750\) −0.921464 −0.0336471
\(751\) −11.6909 −0.426605 −0.213303 0.976986i \(-0.568422\pi\)
−0.213303 + 0.976986i \(0.568422\pi\)
\(752\) 9.27274 0.338142
\(753\) 1.91416 0.0697559
\(754\) 8.78583 0.319961
\(755\) −18.0595 −0.657253
\(756\) 18.2352 0.663207
\(757\) −16.0482 −0.583281 −0.291640 0.956528i \(-0.594201\pi\)
−0.291640 + 0.956528i \(0.594201\pi\)
\(758\) 19.7139 0.716042
\(759\) 2.52392 0.0916126
\(760\) 6.54987 0.237589
\(761\) 11.6821 0.423474 0.211737 0.977327i \(-0.432088\pi\)
0.211737 + 0.977327i \(0.432088\pi\)
\(762\) 18.9062 0.684898
\(763\) 37.8901 1.37171
\(764\) 2.82460 0.102190
\(765\) 9.65687 0.349145
\(766\) −32.6656 −1.18026
\(767\) 42.1536 1.52208
\(768\) 0.921464 0.0332505
\(769\) −52.4264 −1.89055 −0.945273 0.326281i \(-0.894204\pi\)
−0.945273 + 0.326281i \(0.894204\pi\)
\(770\) 3.84192 0.138453
\(771\) −16.9043 −0.608795
\(772\) −0.714821 −0.0257270
\(773\) −23.2049 −0.834622 −0.417311 0.908764i \(-0.637027\pi\)
−0.417311 + 0.908764i \(0.637027\pi\)
\(774\) −2.15090 −0.0773127
\(775\) −0.588567 −0.0211420
\(776\) −8.94728 −0.321189
\(777\) 0.850781 0.0305216
\(778\) 13.0008 0.466100
\(779\) −66.0596 −2.36683
\(780\) −3.09682 −0.110884
\(781\) −10.4421 −0.373648
\(782\) 12.2974 0.439754
\(783\) −12.4081 −0.443430
\(784\) 7.76033 0.277154
\(785\) −19.0697 −0.680628
\(786\) −8.53397 −0.304397
\(787\) 22.4183 0.799126 0.399563 0.916706i \(-0.369162\pi\)
0.399563 + 0.916706i \(0.369162\pi\)
\(788\) −8.23931 −0.293513
\(789\) 11.5101 0.409772
\(790\) −5.98496 −0.212935
\(791\) −42.8774 −1.52455
\(792\) −2.15090 −0.0764291
\(793\) −32.3133 −1.14748
\(794\) 7.23835 0.256879
\(795\) −11.6474 −0.413091
\(796\) −9.90281 −0.350996
\(797\) 30.7866 1.09052 0.545259 0.838267i \(-0.316431\pi\)
0.545259 + 0.838267i \(0.316431\pi\)
\(798\) 23.1878 0.820838
\(799\) 41.6316 1.47282
\(800\) 1.00000 0.0353553
\(801\) −30.0179 −1.06063
\(802\) 25.5243 0.901295
\(803\) 11.8631 0.418641
\(804\) −7.47427 −0.263597
\(805\) 10.5231 0.370892
\(806\) −1.97804 −0.0696733
\(807\) 25.4206 0.894847
\(808\) 6.76897 0.238132
\(809\) 3.76887 0.132506 0.0662532 0.997803i \(-0.478895\pi\)
0.0662532 + 0.997803i \(0.478895\pi\)
\(810\) −2.07910 −0.0730523
\(811\) 2.67036 0.0937691 0.0468846 0.998900i \(-0.485071\pi\)
0.0468846 + 0.998900i \(0.485071\pi\)
\(812\) −10.0437 −0.352464
\(813\) 15.5007 0.543634
\(814\) −0.240321 −0.00842324
\(815\) 11.5905 0.405996
\(816\) 4.13708 0.144827
\(817\) −6.54987 −0.229151
\(818\) −8.29621 −0.290070
\(819\) 27.7720 0.970433
\(820\) −10.0856 −0.352206
\(821\) 2.90152 0.101264 0.0506320 0.998717i \(-0.483876\pi\)
0.0506320 + 0.998717i \(0.483876\pi\)
\(822\) 18.9510 0.660993
\(823\) 39.2886 1.36952 0.684758 0.728771i \(-0.259908\pi\)
0.684758 + 0.728771i \(0.259908\pi\)
\(824\) −11.8563 −0.413034
\(825\) 0.921464 0.0320812
\(826\) −48.1886 −1.67670
\(827\) −24.0062 −0.834776 −0.417388 0.908728i \(-0.637054\pi\)
−0.417388 + 0.908728i \(0.637054\pi\)
\(828\) −5.89140 −0.204740
\(829\) 37.6561 1.30785 0.653925 0.756559i \(-0.273121\pi\)
0.653925 + 0.756559i \(0.273121\pi\)
\(830\) −8.56754 −0.297384
\(831\) −23.2990 −0.808232
\(832\) 3.36077 0.116514
\(833\) 34.8414 1.20718
\(834\) −9.08615 −0.314628
\(835\) −6.33426 −0.219206
\(836\) −6.54987 −0.226532
\(837\) 2.79356 0.0965595
\(838\) 3.75774 0.129809
\(839\) 27.5505 0.951150 0.475575 0.879675i \(-0.342240\pi\)
0.475575 + 0.879675i \(0.342240\pi\)
\(840\) 3.54019 0.122148
\(841\) −22.1658 −0.764337
\(842\) 0.464546 0.0160093
\(843\) −5.59936 −0.192852
\(844\) −14.6306 −0.503608
\(845\) 1.70526 0.0586626
\(846\) −19.9448 −0.685716
\(847\) −3.84192 −0.132010
\(848\) 12.6401 0.434063
\(849\) 23.4774 0.805741
\(850\) 4.48968 0.153995
\(851\) −0.658248 −0.0225644
\(852\) −9.62201 −0.329645
\(853\) −15.3388 −0.525189 −0.262595 0.964906i \(-0.584578\pi\)
−0.262595 + 0.964906i \(0.584578\pi\)
\(854\) 36.9395 1.26404
\(855\) −14.0881 −0.481804
\(856\) −15.8462 −0.541612
\(857\) −25.9966 −0.888026 −0.444013 0.896020i \(-0.646446\pi\)
−0.444013 + 0.896020i \(0.646446\pi\)
\(858\) 3.09682 0.105724
\(859\) −31.7364 −1.08283 −0.541416 0.840755i \(-0.682112\pi\)
−0.541416 + 0.840755i \(0.682112\pi\)
\(860\) −1.00000 −0.0340997
\(861\) −35.7050 −1.21682
\(862\) −23.9618 −0.816142
\(863\) 9.44518 0.321518 0.160759 0.986994i \(-0.448606\pi\)
0.160759 + 0.986994i \(0.448606\pi\)
\(864\) −4.74637 −0.161475
\(865\) 2.54982 0.0866966
\(866\) −29.5327 −1.00356
\(867\) 2.90925 0.0988035
\(868\) 2.26123 0.0767510
\(869\) 5.98496 0.203026
\(870\) −2.40892 −0.0816701
\(871\) −27.2602 −0.923676
\(872\) −9.86228 −0.333979
\(873\) 19.2447 0.651336
\(874\) −17.9403 −0.606841
\(875\) 3.84192 0.129880
\(876\) 10.9315 0.369340
\(877\) −24.3574 −0.822492 −0.411246 0.911524i \(-0.634906\pi\)
−0.411246 + 0.911524i \(0.634906\pi\)
\(878\) −22.1469 −0.747422
\(879\) −5.12390 −0.172825
\(880\) −1.00000 −0.0337100
\(881\) 9.70156 0.326854 0.163427 0.986555i \(-0.447745\pi\)
0.163427 + 0.986555i \(0.447745\pi\)
\(882\) −16.6917 −0.562039
\(883\) 2.39138 0.0804763 0.0402382 0.999190i \(-0.487188\pi\)
0.0402382 + 0.999190i \(0.487188\pi\)
\(884\) 15.0888 0.507490
\(885\) −11.5578 −0.388511
\(886\) 37.4638 1.25862
\(887\) −55.8144 −1.87407 −0.937033 0.349241i \(-0.886439\pi\)
−0.937033 + 0.349241i \(0.886439\pi\)
\(888\) −0.221447 −0.00743128
\(889\) −78.8267 −2.64376
\(890\) −13.9559 −0.467804
\(891\) 2.07910 0.0696526
\(892\) 16.2067 0.542639
\(893\) −60.7352 −2.03243
\(894\) 4.08525 0.136631
\(895\) 6.35898 0.212557
\(896\) −3.84192 −0.128349
\(897\) 8.48231 0.283216
\(898\) 39.1108 1.30514
\(899\) −1.53865 −0.0513169
\(900\) −2.15090 −0.0716968
\(901\) 56.7500 1.89062
\(902\) 10.0856 0.335815
\(903\) −3.54019 −0.117810
\(904\) 11.1604 0.371190
\(905\) −4.96116 −0.164914
\(906\) 16.6412 0.552867
\(907\) −6.13589 −0.203739 −0.101869 0.994798i \(-0.532482\pi\)
−0.101869 + 0.994798i \(0.532482\pi\)
\(908\) −7.28678 −0.241820
\(909\) −14.5594 −0.482905
\(910\) 12.9118 0.428021
\(911\) 41.6942 1.38139 0.690696 0.723146i \(-0.257305\pi\)
0.690696 + 0.723146i \(0.257305\pi\)
\(912\) −6.03547 −0.199854
\(913\) 8.56754 0.283544
\(914\) 7.77236 0.257087
\(915\) 8.85975 0.292894
\(916\) −7.26471 −0.240033
\(917\) 35.5812 1.17500
\(918\) −21.3097 −0.703324
\(919\) −11.6575 −0.384546 −0.192273 0.981341i \(-0.561586\pi\)
−0.192273 + 0.981341i \(0.561586\pi\)
\(920\) −2.73904 −0.0903033
\(921\) −20.1713 −0.664668
\(922\) 8.64592 0.284738
\(923\) −35.0934 −1.15511
\(924\) −3.54019 −0.116464
\(925\) −0.240321 −0.00790170
\(926\) 22.3778 0.735381
\(927\) 25.5018 0.837589
\(928\) 2.61423 0.0858164
\(929\) 20.8673 0.684634 0.342317 0.939585i \(-0.388788\pi\)
0.342317 + 0.939585i \(0.388788\pi\)
\(930\) 0.542343 0.0177841
\(931\) −50.8291 −1.66586
\(932\) 19.0465 0.623890
\(933\) −3.21863 −0.105373
\(934\) 25.2601 0.826537
\(935\) −4.48968 −0.146828
\(936\) −7.22869 −0.236277
\(937\) −3.29248 −0.107560 −0.0537802 0.998553i \(-0.517127\pi\)
−0.0537802 + 0.998553i \(0.517127\pi\)
\(938\) 31.1629 1.01751
\(939\) −31.0272 −1.01254
\(940\) −9.27274 −0.302443
\(941\) 34.4049 1.12157 0.560784 0.827962i \(-0.310500\pi\)
0.560784 + 0.827962i \(0.310500\pi\)
\(942\) 17.5721 0.572529
\(943\) 27.6249 0.899591
\(944\) 12.5429 0.408235
\(945\) −18.2352 −0.593190
\(946\) 1.00000 0.0325128
\(947\) −30.8053 −1.00104 −0.500519 0.865725i \(-0.666858\pi\)
−0.500519 + 0.865725i \(0.666858\pi\)
\(948\) 5.51492 0.179116
\(949\) 39.8692 1.29421
\(950\) −6.54987 −0.212506
\(951\) −17.0510 −0.552915
\(952\) −17.2490 −0.559042
\(953\) −7.40881 −0.239995 −0.119997 0.992774i \(-0.538289\pi\)
−0.119997 + 0.992774i \(0.538289\pi\)
\(954\) −27.1877 −0.880234
\(955\) −2.82460 −0.0914019
\(956\) 8.88667 0.287416
\(957\) 2.40892 0.0778694
\(958\) 24.8707 0.803536
\(959\) −79.0137 −2.55149
\(960\) −0.921464 −0.0297401
\(961\) −30.6536 −0.988825
\(962\) −0.807662 −0.0260401
\(963\) 34.0837 1.09833
\(964\) 2.76143 0.0889398
\(965\) 0.714821 0.0230109
\(966\) −9.69670 −0.311986
\(967\) 48.0204 1.54423 0.772116 0.635482i \(-0.219198\pi\)
0.772116 + 0.635482i \(0.219198\pi\)
\(968\) 1.00000 0.0321412
\(969\) −27.0973 −0.870491
\(970\) 8.94728 0.287280
\(971\) −33.0553 −1.06079 −0.530397 0.847749i \(-0.677957\pi\)
−0.530397 + 0.847749i \(0.677957\pi\)
\(972\) 16.1549 0.518170
\(973\) 37.8835 1.21449
\(974\) −15.1078 −0.484084
\(975\) 3.09682 0.0991777
\(976\) −9.61487 −0.307764
\(977\) −4.57570 −0.146389 −0.0731947 0.997318i \(-0.523319\pi\)
−0.0731947 + 0.997318i \(0.523319\pi\)
\(978\) −10.6802 −0.341515
\(979\) 13.9559 0.446034
\(980\) −7.76033 −0.247895
\(981\) 21.2128 0.677273
\(982\) −20.9423 −0.668296
\(983\) 23.2444 0.741382 0.370691 0.928756i \(-0.379121\pi\)
0.370691 + 0.928756i \(0.379121\pi\)
\(984\) 9.29355 0.296267
\(985\) 8.23931 0.262526
\(986\) 11.7371 0.373784
\(987\) −32.8272 −1.04490
\(988\) −22.0126 −0.700313
\(989\) 2.73904 0.0870963
\(990\) 2.15090 0.0683602
\(991\) 31.6459 1.00526 0.502632 0.864500i \(-0.332365\pi\)
0.502632 + 0.864500i \(0.332365\pi\)
\(992\) −0.588567 −0.0186870
\(993\) 12.8022 0.406265
\(994\) 40.1177 1.27246
\(995\) 9.90281 0.313940
\(996\) 7.89468 0.250152
\(997\) 49.3229 1.56207 0.781036 0.624486i \(-0.214691\pi\)
0.781036 + 0.624486i \(0.214691\pi\)
\(998\) −25.5069 −0.807405
\(999\) 1.14065 0.0360886
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 4730.2.a.bf.1.8 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.bf.1.8 13 1.1 even 1 trivial