Properties

Label 825.2.a.k.1.2
Level $825$
Weight $2$
Character 825.1
Self dual yes
Analytic conductor $6.588$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.193937 q^{2} -1.00000 q^{3} -1.96239 q^{4} -0.193937 q^{6} -3.35026 q^{7} -0.768452 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.193937 q^{2} -1.00000 q^{3} -1.96239 q^{4} -0.193937 q^{6} -3.35026 q^{7} -0.768452 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.96239 q^{12} -2.96239 q^{13} -0.649738 q^{14} +3.77575 q^{16} +4.57452 q^{17} +0.193937 q^{18} -4.31265 q^{19} +3.35026 q^{21} +0.193937 q^{22} +6.70052 q^{23} +0.768452 q^{24} -0.574515 q^{26} -1.00000 q^{27} +6.57452 q^{28} -3.61213 q^{29} +9.92478 q^{31} +2.26916 q^{32} -1.00000 q^{33} +0.887166 q^{34} -1.96239 q^{36} +2.00000 q^{37} -0.836381 q^{38} +2.96239 q^{39} -4.38787 q^{41} +0.649738 q^{42} +9.27504 q^{43} -1.96239 q^{44} +1.29948 q^{46} +9.92478 q^{47} -3.77575 q^{48} +4.22425 q^{49} -4.57452 q^{51} +5.81336 q^{52} -4.70052 q^{53} -0.193937 q^{54} +2.57452 q^{56} +4.31265 q^{57} -0.700523 q^{58} +10.7005 q^{59} -8.70052 q^{61} +1.92478 q^{62} -3.35026 q^{63} -7.11142 q^{64} -0.193937 q^{66} -5.92478 q^{67} -8.97698 q^{68} -6.70052 q^{69} +9.92478 q^{71} -0.768452 q^{72} +7.73813 q^{73} +0.387873 q^{74} +8.46310 q^{76} -3.35026 q^{77} +0.574515 q^{78} +11.5369 q^{79} +1.00000 q^{81} -0.850969 q^{82} -10.8872 q^{83} -6.57452 q^{84} +1.79877 q^{86} +3.61213 q^{87} -0.768452 q^{88} -2.77575 q^{89} +9.92478 q^{91} -13.1490 q^{92} -9.92478 q^{93} +1.92478 q^{94} -2.26916 q^{96} -0.0752228 q^{97} +0.819237 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} + 9 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} + 9 q^{8} + 3 q^{9} + 3 q^{11} - 5 q^{12} + 2 q^{13} - 12 q^{14} + 13 q^{16} + 2 q^{17} + q^{18} + 8 q^{19} + q^{22} - 9 q^{24} + 10 q^{26} - 3 q^{27} + 8 q^{28} - 10 q^{29} + 8 q^{31} + 29 q^{32} - 3 q^{33} - 30 q^{34} + 5 q^{36} + 6 q^{37} - 2 q^{39} - 14 q^{41} + 12 q^{42} - 4 q^{43} + 5 q^{44} + 24 q^{46} + 8 q^{47} - 13 q^{48} + 11 q^{49} - 2 q^{51} + 30 q^{52} + 6 q^{53} - q^{54} - 4 q^{56} - 8 q^{57} + 18 q^{58} + 12 q^{59} - 6 q^{61} - 16 q^{62} + 13 q^{64} - q^{66} + 4 q^{67} - 42 q^{68} + 8 q^{71} + 9 q^{72} + 14 q^{73} + 2 q^{74} + 48 q^{76} - 10 q^{78} + 12 q^{79} + 3 q^{81} - 26 q^{82} - 8 q^{84} - 8 q^{86} + 10 q^{87} + 9 q^{88} - 10 q^{89} + 8 q^{91} - 16 q^{92} - 8 q^{93} - 16 q^{94} - 29 q^{96} - 22 q^{97} - 39 q^{98} + 3 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.193937 0.137134 0.0685669 0.997647i \(-0.478157\pi\)
0.0685669 + 0.997647i \(0.478157\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.96239 −0.981194
\(5\) 0 0
\(6\) −0.193937 −0.0791743
\(7\) −3.35026 −1.26628 −0.633140 0.774037i \(-0.718234\pi\)
−0.633140 + 0.774037i \(0.718234\pi\)
\(8\) −0.768452 −0.271689
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 1.96239 0.566493
\(13\) −2.96239 −0.821619 −0.410809 0.911721i \(-0.634754\pi\)
−0.410809 + 0.911721i \(0.634754\pi\)
\(14\) −0.649738 −0.173650
\(15\) 0 0
\(16\) 3.77575 0.943937
\(17\) 4.57452 1.10948 0.554741 0.832023i \(-0.312817\pi\)
0.554741 + 0.832023i \(0.312817\pi\)
\(18\) 0.193937 0.0457113
\(19\) −4.31265 −0.989390 −0.494695 0.869067i \(-0.664720\pi\)
−0.494695 + 0.869067i \(0.664720\pi\)
\(20\) 0 0
\(21\) 3.35026 0.731087
\(22\) 0.193937 0.0413474
\(23\) 6.70052 1.39716 0.698578 0.715534i \(-0.253817\pi\)
0.698578 + 0.715534i \(0.253817\pi\)
\(24\) 0.768452 0.156860
\(25\) 0 0
\(26\) −0.574515 −0.112672
\(27\) −1.00000 −0.192450
\(28\) 6.57452 1.24247
\(29\) −3.61213 −0.670755 −0.335378 0.942084i \(-0.608864\pi\)
−0.335378 + 0.942084i \(0.608864\pi\)
\(30\) 0 0
\(31\) 9.92478 1.78254 0.891271 0.453470i \(-0.149814\pi\)
0.891271 + 0.453470i \(0.149814\pi\)
\(32\) 2.26916 0.401134
\(33\) −1.00000 −0.174078
\(34\) 0.887166 0.152148
\(35\) 0 0
\(36\) −1.96239 −0.327065
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −0.836381 −0.135679
\(39\) 2.96239 0.474362
\(40\) 0 0
\(41\) −4.38787 −0.685271 −0.342635 0.939468i \(-0.611320\pi\)
−0.342635 + 0.939468i \(0.611320\pi\)
\(42\) 0.649738 0.100257
\(43\) 9.27504 1.41443 0.707215 0.706998i \(-0.249951\pi\)
0.707215 + 0.706998i \(0.249951\pi\)
\(44\) −1.96239 −0.295841
\(45\) 0 0
\(46\) 1.29948 0.191597
\(47\) 9.92478 1.44768 0.723839 0.689969i \(-0.242376\pi\)
0.723839 + 0.689969i \(0.242376\pi\)
\(48\) −3.77575 −0.544982
\(49\) 4.22425 0.603465
\(50\) 0 0
\(51\) −4.57452 −0.640560
\(52\) 5.81336 0.806168
\(53\) −4.70052 −0.645667 −0.322833 0.946456i \(-0.604635\pi\)
−0.322833 + 0.946456i \(0.604635\pi\)
\(54\) −0.193937 −0.0263914
\(55\) 0 0
\(56\) 2.57452 0.344034
\(57\) 4.31265 0.571224
\(58\) −0.700523 −0.0919832
\(59\) 10.7005 1.39309 0.696545 0.717513i \(-0.254720\pi\)
0.696545 + 0.717513i \(0.254720\pi\)
\(60\) 0 0
\(61\) −8.70052 −1.11399 −0.556994 0.830517i \(-0.688045\pi\)
−0.556994 + 0.830517i \(0.688045\pi\)
\(62\) 1.92478 0.244447
\(63\) −3.35026 −0.422093
\(64\) −7.11142 −0.888927
\(65\) 0 0
\(66\) −0.193937 −0.0238719
\(67\) −5.92478 −0.723827 −0.361913 0.932212i \(-0.617876\pi\)
−0.361913 + 0.932212i \(0.617876\pi\)
\(68\) −8.97698 −1.08862
\(69\) −6.70052 −0.806648
\(70\) 0 0
\(71\) 9.92478 1.17785 0.588927 0.808186i \(-0.299550\pi\)
0.588927 + 0.808186i \(0.299550\pi\)
\(72\) −0.768452 −0.0905629
\(73\) 7.73813 0.905680 0.452840 0.891592i \(-0.350411\pi\)
0.452840 + 0.891592i \(0.350411\pi\)
\(74\) 0.387873 0.0450893
\(75\) 0 0
\(76\) 8.46310 0.970784
\(77\) −3.35026 −0.381798
\(78\) 0.574515 0.0650511
\(79\) 11.5369 1.29800 0.649002 0.760787i \(-0.275187\pi\)
0.649002 + 0.760787i \(0.275187\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −0.850969 −0.0939738
\(83\) −10.8872 −1.19502 −0.597511 0.801861i \(-0.703844\pi\)
−0.597511 + 0.801861i \(0.703844\pi\)
\(84\) −6.57452 −0.717338
\(85\) 0 0
\(86\) 1.79877 0.193966
\(87\) 3.61213 0.387261
\(88\) −0.768452 −0.0819173
\(89\) −2.77575 −0.294229 −0.147114 0.989120i \(-0.546999\pi\)
−0.147114 + 0.989120i \(0.546999\pi\)
\(90\) 0 0
\(91\) 9.92478 1.04040
\(92\) −13.1490 −1.37088
\(93\) −9.92478 −1.02915
\(94\) 1.92478 0.198526
\(95\) 0 0
\(96\) −2.26916 −0.231595
\(97\) −0.0752228 −0.00763772 −0.00381886 0.999993i \(-0.501216\pi\)
−0.00381886 + 0.999993i \(0.501216\pi\)
\(98\) 0.819237 0.0827555
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −15.0884 −1.50135 −0.750676 0.660671i \(-0.770272\pi\)
−0.750676 + 0.660671i \(0.770272\pi\)
\(102\) −0.887166 −0.0878425
\(103\) 3.22425 0.317695 0.158848 0.987303i \(-0.449222\pi\)
0.158848 + 0.987303i \(0.449222\pi\)
\(104\) 2.27645 0.223225
\(105\) 0 0
\(106\) −0.911603 −0.0885427
\(107\) 0.962389 0.0930376 0.0465188 0.998917i \(-0.485187\pi\)
0.0465188 + 0.998917i \(0.485187\pi\)
\(108\) 1.96239 0.188831
\(109\) 11.4010 1.09202 0.546011 0.837778i \(-0.316146\pi\)
0.546011 + 0.837778i \(0.316146\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) −12.6497 −1.19529
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0.836381 0.0783342
\(115\) 0 0
\(116\) 7.08840 0.658141
\(117\) −2.96239 −0.273873
\(118\) 2.07522 0.191040
\(119\) −15.3258 −1.40492
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −1.68735 −0.152765
\(123\) 4.38787 0.395641
\(124\) −19.4763 −1.74902
\(125\) 0 0
\(126\) −0.649738 −0.0578833
\(127\) 14.5745 1.29328 0.646640 0.762796i \(-0.276174\pi\)
0.646640 + 0.762796i \(0.276174\pi\)
\(128\) −5.91748 −0.523037
\(129\) −9.27504 −0.816622
\(130\) 0 0
\(131\) −5.92478 −0.517650 −0.258825 0.965924i \(-0.583335\pi\)
−0.258825 + 0.965924i \(0.583335\pi\)
\(132\) 1.96239 0.170804
\(133\) 14.4485 1.25284
\(134\) −1.14903 −0.0992612
\(135\) 0 0
\(136\) −3.51530 −0.301434
\(137\) −13.8496 −1.18325 −0.591624 0.806214i \(-0.701513\pi\)
−0.591624 + 0.806214i \(0.701513\pi\)
\(138\) −1.29948 −0.110619
\(139\) 13.6121 1.15457 0.577283 0.816544i \(-0.304113\pi\)
0.577283 + 0.816544i \(0.304113\pi\)
\(140\) 0 0
\(141\) −9.92478 −0.835817
\(142\) 1.92478 0.161524
\(143\) −2.96239 −0.247727
\(144\) 3.77575 0.314646
\(145\) 0 0
\(146\) 1.50071 0.124199
\(147\) −4.22425 −0.348411
\(148\) −3.92478 −0.322615
\(149\) 1.53690 0.125908 0.0629540 0.998016i \(-0.479948\pi\)
0.0629540 + 0.998016i \(0.479948\pi\)
\(150\) 0 0
\(151\) −6.76116 −0.550215 −0.275108 0.961413i \(-0.588713\pi\)
−0.275108 + 0.961413i \(0.588713\pi\)
\(152\) 3.31406 0.268806
\(153\) 4.57452 0.369828
\(154\) −0.649738 −0.0523574
\(155\) 0 0
\(156\) −5.81336 −0.465441
\(157\) 5.47627 0.437054 0.218527 0.975831i \(-0.429875\pi\)
0.218527 + 0.975831i \(0.429875\pi\)
\(158\) 2.23743 0.178000
\(159\) 4.70052 0.372776
\(160\) 0 0
\(161\) −22.4485 −1.76919
\(162\) 0.193937 0.0152371
\(163\) −12.6253 −0.988890 −0.494445 0.869209i \(-0.664629\pi\)
−0.494445 + 0.869209i \(0.664629\pi\)
\(164\) 8.61071 0.672384
\(165\) 0 0
\(166\) −2.11142 −0.163878
\(167\) −18.3634 −1.42101 −0.710503 0.703695i \(-0.751532\pi\)
−0.710503 + 0.703695i \(0.751532\pi\)
\(168\) −2.57452 −0.198628
\(169\) −4.22425 −0.324943
\(170\) 0 0
\(171\) −4.31265 −0.329797
\(172\) −18.2012 −1.38783
\(173\) 8.57452 0.651908 0.325954 0.945386i \(-0.394314\pi\)
0.325954 + 0.945386i \(0.394314\pi\)
\(174\) 0.700523 0.0531065
\(175\) 0 0
\(176\) 3.77575 0.284608
\(177\) −10.7005 −0.804301
\(178\) −0.538319 −0.0403487
\(179\) 14.1768 1.05962 0.529812 0.848115i \(-0.322263\pi\)
0.529812 + 0.848115i \(0.322263\pi\)
\(180\) 0 0
\(181\) −5.22425 −0.388316 −0.194158 0.980970i \(-0.562197\pi\)
−0.194158 + 0.980970i \(0.562197\pi\)
\(182\) 1.92478 0.142674
\(183\) 8.70052 0.643161
\(184\) −5.14903 −0.379592
\(185\) 0 0
\(186\) −1.92478 −0.141132
\(187\) 4.57452 0.334522
\(188\) −19.4763 −1.42045
\(189\) 3.35026 0.243696
\(190\) 0 0
\(191\) −16.6253 −1.20296 −0.601482 0.798886i \(-0.705423\pi\)
−0.601482 + 0.798886i \(0.705423\pi\)
\(192\) 7.11142 0.513222
\(193\) 16.3634 1.17787 0.588933 0.808182i \(-0.299548\pi\)
0.588933 + 0.808182i \(0.299548\pi\)
\(194\) −0.0145884 −0.00104739
\(195\) 0 0
\(196\) −8.28963 −0.592116
\(197\) 20.4241 1.45515 0.727577 0.686026i \(-0.240646\pi\)
0.727577 + 0.686026i \(0.240646\pi\)
\(198\) 0.193937 0.0137825
\(199\) −8.62530 −0.611431 −0.305716 0.952123i \(-0.598896\pi\)
−0.305716 + 0.952123i \(0.598896\pi\)
\(200\) 0 0
\(201\) 5.92478 0.417902
\(202\) −2.92619 −0.205886
\(203\) 12.1016 0.849364
\(204\) 8.97698 0.628514
\(205\) 0 0
\(206\) 0.625301 0.0435668
\(207\) 6.70052 0.465719
\(208\) −11.1852 −0.775556
\(209\) −4.31265 −0.298312
\(210\) 0 0
\(211\) 9.08840 0.625671 0.312836 0.949807i \(-0.398721\pi\)
0.312836 + 0.949807i \(0.398721\pi\)
\(212\) 9.22425 0.633524
\(213\) −9.92478 −0.680035
\(214\) 0.186642 0.0127586
\(215\) 0 0
\(216\) 0.768452 0.0522865
\(217\) −33.2506 −2.25720
\(218\) 2.21108 0.149753
\(219\) −7.73813 −0.522895
\(220\) 0 0
\(221\) −13.5515 −0.911572
\(222\) −0.387873 −0.0260323
\(223\) 6.70052 0.448700 0.224350 0.974509i \(-0.427974\pi\)
0.224350 + 0.974509i \(0.427974\pi\)
\(224\) −7.60228 −0.507949
\(225\) 0 0
\(226\) 1.16362 0.0774028
\(227\) −16.9624 −1.12583 −0.562917 0.826514i \(-0.690321\pi\)
−0.562917 + 0.826514i \(0.690321\pi\)
\(228\) −8.46310 −0.560482
\(229\) 25.8496 1.70819 0.854093 0.520120i \(-0.174113\pi\)
0.854093 + 0.520120i \(0.174113\pi\)
\(230\) 0 0
\(231\) 3.35026 0.220431
\(232\) 2.77575 0.182237
\(233\) 19.2750 1.26275 0.631375 0.775478i \(-0.282491\pi\)
0.631375 + 0.775478i \(0.282491\pi\)
\(234\) −0.574515 −0.0375573
\(235\) 0 0
\(236\) −20.9986 −1.36689
\(237\) −11.5369 −0.749402
\(238\) −2.97224 −0.192662
\(239\) 26.5501 1.71738 0.858691 0.512494i \(-0.171278\pi\)
0.858691 + 0.512494i \(0.171278\pi\)
\(240\) 0 0
\(241\) 28.5501 1.83907 0.919536 0.393006i \(-0.128565\pi\)
0.919536 + 0.393006i \(0.128565\pi\)
\(242\) 0.193937 0.0124667
\(243\) −1.00000 −0.0641500
\(244\) 17.0738 1.09304
\(245\) 0 0
\(246\) 0.850969 0.0542558
\(247\) 12.7757 0.812901
\(248\) −7.62672 −0.484297
\(249\) 10.8872 0.689946
\(250\) 0 0
\(251\) 29.9248 1.88884 0.944418 0.328748i \(-0.106627\pi\)
0.944418 + 0.328748i \(0.106627\pi\)
\(252\) 6.57452 0.414156
\(253\) 6.70052 0.421258
\(254\) 2.82653 0.177352
\(255\) 0 0
\(256\) 13.0752 0.817201
\(257\) −8.70052 −0.542724 −0.271362 0.962477i \(-0.587474\pi\)
−0.271362 + 0.962477i \(0.587474\pi\)
\(258\) −1.79877 −0.111986
\(259\) −6.70052 −0.416350
\(260\) 0 0
\(261\) −3.61213 −0.223585
\(262\) −1.14903 −0.0709874
\(263\) −12.2882 −0.757724 −0.378862 0.925453i \(-0.623684\pi\)
−0.378862 + 0.925453i \(0.623684\pi\)
\(264\) 0.768452 0.0472950
\(265\) 0 0
\(266\) 2.80209 0.171807
\(267\) 2.77575 0.169873
\(268\) 11.6267 0.710215
\(269\) −5.84955 −0.356654 −0.178327 0.983971i \(-0.557068\pi\)
−0.178327 + 0.983971i \(0.557068\pi\)
\(270\) 0 0
\(271\) −5.08840 −0.309098 −0.154549 0.987985i \(-0.549392\pi\)
−0.154549 + 0.987985i \(0.549392\pi\)
\(272\) 17.2722 1.04728
\(273\) −9.92478 −0.600675
\(274\) −2.68594 −0.162263
\(275\) 0 0
\(276\) 13.1490 0.791479
\(277\) −1.41090 −0.0847725 −0.0423863 0.999101i \(-0.513496\pi\)
−0.0423863 + 0.999101i \(0.513496\pi\)
\(278\) 2.63989 0.158330
\(279\) 9.92478 0.594181
\(280\) 0 0
\(281\) −4.38787 −0.261759 −0.130879 0.991398i \(-0.541780\pi\)
−0.130879 + 0.991398i \(0.541780\pi\)
\(282\) −1.92478 −0.114619
\(283\) −26.5745 −1.57969 −0.789845 0.613306i \(-0.789839\pi\)
−0.789845 + 0.613306i \(0.789839\pi\)
\(284\) −19.4763 −1.15570
\(285\) 0 0
\(286\) −0.574515 −0.0339718
\(287\) 14.7005 0.867744
\(288\) 2.26916 0.133711
\(289\) 3.92619 0.230952
\(290\) 0 0
\(291\) 0.0752228 0.00440964
\(292\) −15.1852 −0.888648
\(293\) 3.42548 0.200119 0.100059 0.994981i \(-0.468097\pi\)
0.100059 + 0.994981i \(0.468097\pi\)
\(294\) −0.819237 −0.0477789
\(295\) 0 0
\(296\) −1.53690 −0.0893307
\(297\) −1.00000 −0.0580259
\(298\) 0.298062 0.0172663
\(299\) −19.8496 −1.14793
\(300\) 0 0
\(301\) −31.0738 −1.79106
\(302\) −1.31124 −0.0754531
\(303\) 15.0884 0.866806
\(304\) −16.2835 −0.933921
\(305\) 0 0
\(306\) 0.887166 0.0507159
\(307\) 16.6497 0.950251 0.475125 0.879918i \(-0.342403\pi\)
0.475125 + 0.879918i \(0.342403\pi\)
\(308\) 6.57452 0.374618
\(309\) −3.22425 −0.183421
\(310\) 0 0
\(311\) 32.9986 1.87118 0.935589 0.353091i \(-0.114869\pi\)
0.935589 + 0.353091i \(0.114869\pi\)
\(312\) −2.27645 −0.128879
\(313\) −15.4010 −0.870519 −0.435259 0.900305i \(-0.643343\pi\)
−0.435259 + 0.900305i \(0.643343\pi\)
\(314\) 1.06205 0.0599349
\(315\) 0 0
\(316\) −22.6399 −1.27359
\(317\) −2.15045 −0.120781 −0.0603905 0.998175i \(-0.519235\pi\)
−0.0603905 + 0.998175i \(0.519235\pi\)
\(318\) 0.911603 0.0511202
\(319\) −3.61213 −0.202240
\(320\) 0 0
\(321\) −0.962389 −0.0537153
\(322\) −4.35359 −0.242616
\(323\) −19.7283 −1.09771
\(324\) −1.96239 −0.109022
\(325\) 0 0
\(326\) −2.44851 −0.135610
\(327\) −11.4010 −0.630479
\(328\) 3.37187 0.186180
\(329\) −33.2506 −1.83316
\(330\) 0 0
\(331\) −14.5501 −0.799745 −0.399872 0.916571i \(-0.630946\pi\)
−0.399872 + 0.916571i \(0.630946\pi\)
\(332\) 21.3649 1.17255
\(333\) 2.00000 0.109599
\(334\) −3.56134 −0.194868
\(335\) 0 0
\(336\) 12.6497 0.690100
\(337\) −16.2619 −0.885840 −0.442920 0.896561i \(-0.646057\pi\)
−0.442920 + 0.896561i \(0.646057\pi\)
\(338\) −0.819237 −0.0445606
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 9.92478 0.537457
\(342\) −0.836381 −0.0452263
\(343\) 9.29948 0.502125
\(344\) −7.12742 −0.384285
\(345\) 0 0
\(346\) 1.66291 0.0893987
\(347\) 0.962389 0.0516637 0.0258319 0.999666i \(-0.491777\pi\)
0.0258319 + 0.999666i \(0.491777\pi\)
\(348\) −7.08840 −0.379978
\(349\) 20.7005 1.10807 0.554037 0.832492i \(-0.313087\pi\)
0.554037 + 0.832492i \(0.313087\pi\)
\(350\) 0 0
\(351\) 2.96239 0.158121
\(352\) 2.26916 0.120947
\(353\) −20.5501 −1.09377 −0.546885 0.837208i \(-0.684187\pi\)
−0.546885 + 0.837208i \(0.684187\pi\)
\(354\) −2.07522 −0.110297
\(355\) 0 0
\(356\) 5.44709 0.288695
\(357\) 15.3258 0.811129
\(358\) 2.74940 0.145310
\(359\) 17.9248 0.946034 0.473017 0.881053i \(-0.343165\pi\)
0.473017 + 0.881053i \(0.343165\pi\)
\(360\) 0 0
\(361\) −0.401047 −0.0211077
\(362\) −1.01317 −0.0532512
\(363\) −1.00000 −0.0524864
\(364\) −19.4763 −1.02083
\(365\) 0 0
\(366\) 1.68735 0.0881992
\(367\) 29.6531 1.54788 0.773939 0.633261i \(-0.218284\pi\)
0.773939 + 0.633261i \(0.218284\pi\)
\(368\) 25.2995 1.31883
\(369\) −4.38787 −0.228424
\(370\) 0 0
\(371\) 15.7480 0.817595
\(372\) 19.4763 1.00980
\(373\) 9.13918 0.473209 0.236604 0.971606i \(-0.423965\pi\)
0.236604 + 0.971606i \(0.423965\pi\)
\(374\) 0.887166 0.0458743
\(375\) 0 0
\(376\) −7.62672 −0.393318
\(377\) 10.7005 0.551105
\(378\) 0.649738 0.0334189
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −14.5745 −0.746675
\(382\) −3.22425 −0.164967
\(383\) 34.9234 1.78450 0.892250 0.451541i \(-0.149126\pi\)
0.892250 + 0.451541i \(0.149126\pi\)
\(384\) 5.91748 0.301975
\(385\) 0 0
\(386\) 3.17347 0.161525
\(387\) 9.27504 0.471477
\(388\) 0.147616 0.00749408
\(389\) 2.77575 0.140736 0.0703680 0.997521i \(-0.477583\pi\)
0.0703680 + 0.997521i \(0.477583\pi\)
\(390\) 0 0
\(391\) 30.6516 1.55012
\(392\) −3.24614 −0.163955
\(393\) 5.92478 0.298865
\(394\) 3.96097 0.199551
\(395\) 0 0
\(396\) −1.96239 −0.0986137
\(397\) 19.9248 0.999996 0.499998 0.866027i \(-0.333334\pi\)
0.499998 + 0.866027i \(0.333334\pi\)
\(398\) −1.67276 −0.0838479
\(399\) −14.4485 −0.723330
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 1.14903 0.0573085
\(403\) −29.4010 −1.46457
\(404\) 29.6093 1.47312
\(405\) 0 0
\(406\) 2.34694 0.116477
\(407\) 2.00000 0.0991363
\(408\) 3.51530 0.174033
\(409\) −13.0738 −0.646458 −0.323229 0.946321i \(-0.604768\pi\)
−0.323229 + 0.946321i \(0.604768\pi\)
\(410\) 0 0
\(411\) 13.8496 0.683148
\(412\) −6.32724 −0.311721
\(413\) −35.8496 −1.76404
\(414\) 1.29948 0.0638658
\(415\) 0 0
\(416\) −6.72213 −0.329580
\(417\) −13.6121 −0.666589
\(418\) −0.836381 −0.0409087
\(419\) 7.22425 0.352928 0.176464 0.984307i \(-0.443534\pi\)
0.176464 + 0.984307i \(0.443534\pi\)
\(420\) 0 0
\(421\) 30.6253 1.49259 0.746293 0.665618i \(-0.231832\pi\)
0.746293 + 0.665618i \(0.231832\pi\)
\(422\) 1.76257 0.0858007
\(423\) 9.92478 0.482559
\(424\) 3.61213 0.175420
\(425\) 0 0
\(426\) −1.92478 −0.0932558
\(427\) 29.1490 1.41062
\(428\) −1.88858 −0.0912880
\(429\) 2.96239 0.143025
\(430\) 0 0
\(431\) −33.8759 −1.63174 −0.815872 0.578232i \(-0.803743\pi\)
−0.815872 + 0.578232i \(0.803743\pi\)
\(432\) −3.77575 −0.181661
\(433\) 9.47627 0.455400 0.227700 0.973731i \(-0.426879\pi\)
0.227700 + 0.973731i \(0.426879\pi\)
\(434\) −6.44851 −0.309538
\(435\) 0 0
\(436\) −22.3733 −1.07149
\(437\) −28.8970 −1.38233
\(438\) −1.50071 −0.0717066
\(439\) −29.4617 −1.40613 −0.703065 0.711126i \(-0.748186\pi\)
−0.703065 + 0.711126i \(0.748186\pi\)
\(440\) 0 0
\(441\) 4.22425 0.201155
\(442\) −2.62813 −0.125007
\(443\) 19.0738 0.906224 0.453112 0.891454i \(-0.350314\pi\)
0.453112 + 0.891454i \(0.350314\pi\)
\(444\) 3.92478 0.186262
\(445\) 0 0
\(446\) 1.29948 0.0615320
\(447\) −1.53690 −0.0726931
\(448\) 23.8251 1.12563
\(449\) 35.8759 1.69309 0.846544 0.532318i \(-0.178679\pi\)
0.846544 + 0.532318i \(0.178679\pi\)
\(450\) 0 0
\(451\) −4.38787 −0.206617
\(452\) −11.7743 −0.553818
\(453\) 6.76116 0.317667
\(454\) −3.28963 −0.154390
\(455\) 0 0
\(456\) −3.31406 −0.155195
\(457\) −5.28963 −0.247438 −0.123719 0.992317i \(-0.539482\pi\)
−0.123719 + 0.992317i \(0.539482\pi\)
\(458\) 5.01317 0.234250
\(459\) −4.57452 −0.213520
\(460\) 0 0
\(461\) 36.3390 1.69248 0.846238 0.532805i \(-0.178862\pi\)
0.846238 + 0.532805i \(0.178862\pi\)
\(462\) 0.649738 0.0302286
\(463\) −10.5501 −0.490304 −0.245152 0.969485i \(-0.578838\pi\)
−0.245152 + 0.969485i \(0.578838\pi\)
\(464\) −13.6385 −0.633150
\(465\) 0 0
\(466\) 3.73813 0.173166
\(467\) −18.7005 −0.865357 −0.432679 0.901548i \(-0.642431\pi\)
−0.432679 + 0.901548i \(0.642431\pi\)
\(468\) 5.81336 0.268723
\(469\) 19.8496 0.916567
\(470\) 0 0
\(471\) −5.47627 −0.252333
\(472\) −8.22284 −0.378487
\(473\) 9.27504 0.426467
\(474\) −2.23743 −0.102768
\(475\) 0 0
\(476\) 30.0752 1.37850
\(477\) −4.70052 −0.215222
\(478\) 5.14903 0.235511
\(479\) −9.29948 −0.424904 −0.212452 0.977172i \(-0.568145\pi\)
−0.212452 + 0.977172i \(0.568145\pi\)
\(480\) 0 0
\(481\) −5.92478 −0.270147
\(482\) 5.53690 0.252199
\(483\) 22.4485 1.02144
\(484\) −1.96239 −0.0891995
\(485\) 0 0
\(486\) −0.193937 −0.00879714
\(487\) 35.4763 1.60758 0.803792 0.594911i \(-0.202813\pi\)
0.803792 + 0.594911i \(0.202813\pi\)
\(488\) 6.68594 0.302658
\(489\) 12.6253 0.570936
\(490\) 0 0
\(491\) 24.7757 1.11811 0.559057 0.829129i \(-0.311163\pi\)
0.559057 + 0.829129i \(0.311163\pi\)
\(492\) −8.61071 −0.388201
\(493\) −16.5237 −0.744191
\(494\) 2.47768 0.111476
\(495\) 0 0
\(496\) 37.4734 1.68261
\(497\) −33.2506 −1.49149
\(498\) 2.11142 0.0946150
\(499\) 14.1768 0.634640 0.317320 0.948318i \(-0.397217\pi\)
0.317320 + 0.948318i \(0.397217\pi\)
\(500\) 0 0
\(501\) 18.3634 0.820418
\(502\) 5.80351 0.259023
\(503\) 8.43866 0.376261 0.188131 0.982144i \(-0.439757\pi\)
0.188131 + 0.982144i \(0.439757\pi\)
\(504\) 2.57452 0.114678
\(505\) 0 0
\(506\) 1.29948 0.0577688
\(507\) 4.22425 0.187606
\(508\) −28.6009 −1.26896
\(509\) 1.10299 0.0488890 0.0244445 0.999701i \(-0.492218\pi\)
0.0244445 + 0.999701i \(0.492218\pi\)
\(510\) 0 0
\(511\) −25.9248 −1.14684
\(512\) 14.3707 0.635103
\(513\) 4.31265 0.190408
\(514\) −1.68735 −0.0744258
\(515\) 0 0
\(516\) 18.2012 0.801265
\(517\) 9.92478 0.436491
\(518\) −1.29948 −0.0570957
\(519\) −8.57452 −0.376379
\(520\) 0 0
\(521\) −12.4485 −0.545379 −0.272690 0.962102i \(-0.587913\pi\)
−0.272690 + 0.962102i \(0.587913\pi\)
\(522\) −0.700523 −0.0306611
\(523\) −30.0508 −1.31403 −0.657015 0.753878i \(-0.728181\pi\)
−0.657015 + 0.753878i \(0.728181\pi\)
\(524\) 11.6267 0.507915
\(525\) 0 0
\(526\) −2.38313 −0.103910
\(527\) 45.4010 1.97770
\(528\) −3.77575 −0.164318
\(529\) 21.8970 0.952044
\(530\) 0 0
\(531\) 10.7005 0.464363
\(532\) −28.3536 −1.22928
\(533\) 12.9986 0.563031
\(534\) 0.538319 0.0232953
\(535\) 0 0
\(536\) 4.55291 0.196656
\(537\) −14.1768 −0.611774
\(538\) −1.13444 −0.0489093
\(539\) 4.22425 0.181951
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −0.986826 −0.0423878
\(543\) 5.22425 0.224194
\(544\) 10.3803 0.445052
\(545\) 0 0
\(546\) −1.92478 −0.0823729
\(547\) 14.3028 0.611544 0.305772 0.952105i \(-0.401086\pi\)
0.305772 + 0.952105i \(0.401086\pi\)
\(548\) 27.1782 1.16100
\(549\) −8.70052 −0.371329
\(550\) 0 0
\(551\) 15.5778 0.663638
\(552\) 5.14903 0.219157
\(553\) −38.6516 −1.64364
\(554\) −0.273624 −0.0116252
\(555\) 0 0
\(556\) −26.7123 −1.13285
\(557\) 11.7988 0.499930 0.249965 0.968255i \(-0.419581\pi\)
0.249965 + 0.968255i \(0.419581\pi\)
\(558\) 1.92478 0.0814823
\(559\) −27.4763 −1.16212
\(560\) 0 0
\(561\) −4.57452 −0.193136
\(562\) −0.850969 −0.0358960
\(563\) −30.4847 −1.28478 −0.642389 0.766379i \(-0.722056\pi\)
−0.642389 + 0.766379i \(0.722056\pi\)
\(564\) 19.4763 0.820099
\(565\) 0 0
\(566\) −5.15377 −0.216629
\(567\) −3.35026 −0.140698
\(568\) −7.62672 −0.320010
\(569\) −27.0884 −1.13560 −0.567802 0.823165i \(-0.692206\pi\)
−0.567802 + 0.823165i \(0.692206\pi\)
\(570\) 0 0
\(571\) 7.28489 0.304863 0.152432 0.988314i \(-0.451290\pi\)
0.152432 + 0.988314i \(0.451290\pi\)
\(572\) 5.81336 0.243069
\(573\) 16.6253 0.694532
\(574\) 2.85097 0.118997
\(575\) 0 0
\(576\) −7.11142 −0.296309
\(577\) 31.6239 1.31652 0.658260 0.752791i \(-0.271293\pi\)
0.658260 + 0.752791i \(0.271293\pi\)
\(578\) 0.761432 0.0316714
\(579\) −16.3634 −0.680041
\(580\) 0 0
\(581\) 36.4749 1.51323
\(582\) 0.0145884 0.000604711 0
\(583\) −4.70052 −0.194676
\(584\) −5.94639 −0.246063
\(585\) 0 0
\(586\) 0.664327 0.0274431
\(587\) −33.1490 −1.36821 −0.684103 0.729385i \(-0.739806\pi\)
−0.684103 + 0.729385i \(0.739806\pi\)
\(588\) 8.28963 0.341858
\(589\) −42.8021 −1.76363
\(590\) 0 0
\(591\) −20.4241 −0.840134
\(592\) 7.55149 0.310364
\(593\) −34.4993 −1.41672 −0.708358 0.705853i \(-0.750564\pi\)
−0.708358 + 0.705853i \(0.750564\pi\)
\(594\) −0.193937 −0.00795731
\(595\) 0 0
\(596\) −3.01600 −0.123540
\(597\) 8.62530 0.353010
\(598\) −3.84955 −0.157420
\(599\) −14.4485 −0.590350 −0.295175 0.955443i \(-0.595378\pi\)
−0.295175 + 0.955443i \(0.595378\pi\)
\(600\) 0 0
\(601\) −15.9248 −0.649585 −0.324793 0.945785i \(-0.605295\pi\)
−0.324793 + 0.945785i \(0.605295\pi\)
\(602\) −6.02635 −0.245616
\(603\) −5.92478 −0.241276
\(604\) 13.2680 0.539868
\(605\) 0 0
\(606\) 2.92619 0.118868
\(607\) 14.5745 0.591561 0.295781 0.955256i \(-0.404420\pi\)
0.295781 + 0.955256i \(0.404420\pi\)
\(608\) −9.78609 −0.396878
\(609\) −12.1016 −0.490380
\(610\) 0 0
\(611\) −29.4010 −1.18944
\(612\) −8.97698 −0.362873
\(613\) −16.4123 −0.662887 −0.331443 0.943475i \(-0.607536\pi\)
−0.331443 + 0.943475i \(0.607536\pi\)
\(614\) 3.22899 0.130312
\(615\) 0 0
\(616\) 2.57452 0.103730
\(617\) 17.8496 0.718596 0.359298 0.933223i \(-0.383016\pi\)
0.359298 + 0.933223i \(0.383016\pi\)
\(618\) −0.625301 −0.0251533
\(619\) −0.402462 −0.0161763 −0.00808815 0.999967i \(-0.502575\pi\)
−0.00808815 + 0.999967i \(0.502575\pi\)
\(620\) 0 0
\(621\) −6.70052 −0.268883
\(622\) 6.39963 0.256602
\(623\) 9.29948 0.372576
\(624\) 11.1852 0.447767
\(625\) 0 0
\(626\) −2.98683 −0.119378
\(627\) 4.31265 0.172231
\(628\) −10.7466 −0.428835
\(629\) 9.14903 0.364796
\(630\) 0 0
\(631\) −38.0263 −1.51380 −0.756902 0.653528i \(-0.773288\pi\)
−0.756902 + 0.653528i \(0.773288\pi\)
\(632\) −8.86556 −0.352653
\(633\) −9.08840 −0.361231
\(634\) −0.417050 −0.0165632
\(635\) 0 0
\(636\) −9.22425 −0.365765
\(637\) −12.5139 −0.495818
\(638\) −0.700523 −0.0277340
\(639\) 9.92478 0.392618
\(640\) 0 0
\(641\) −28.0263 −1.10697 −0.553487 0.832858i \(-0.686703\pi\)
−0.553487 + 0.832858i \(0.686703\pi\)
\(642\) −0.186642 −0.00736619
\(643\) 4.62530 0.182404 0.0912020 0.995832i \(-0.470929\pi\)
0.0912020 + 0.995832i \(0.470929\pi\)
\(644\) 44.0527 1.73592
\(645\) 0 0
\(646\) −3.82604 −0.150533
\(647\) −23.5778 −0.926941 −0.463470 0.886112i \(-0.653396\pi\)
−0.463470 + 0.886112i \(0.653396\pi\)
\(648\) −0.768452 −0.0301876
\(649\) 10.7005 0.420032
\(650\) 0 0
\(651\) 33.2506 1.30319
\(652\) 24.7757 0.970293
\(653\) 2.25202 0.0881282 0.0440641 0.999029i \(-0.485969\pi\)
0.0440641 + 0.999029i \(0.485969\pi\)
\(654\) −2.21108 −0.0864601
\(655\) 0 0
\(656\) −16.5675 −0.646852
\(657\) 7.73813 0.301893
\(658\) −6.44851 −0.251389
\(659\) −41.4010 −1.61276 −0.806378 0.591401i \(-0.798575\pi\)
−0.806378 + 0.591401i \(0.798575\pi\)
\(660\) 0 0
\(661\) 3.40105 0.132285 0.0661427 0.997810i \(-0.478931\pi\)
0.0661427 + 0.997810i \(0.478931\pi\)
\(662\) −2.82179 −0.109672
\(663\) 13.5515 0.526296
\(664\) 8.36626 0.324674
\(665\) 0 0
\(666\) 0.387873 0.0150298
\(667\) −24.2031 −0.937149
\(668\) 36.0362 1.39428
\(669\) −6.70052 −0.259057
\(670\) 0 0
\(671\) −8.70052 −0.335880
\(672\) 7.60228 0.293264
\(673\) −0.887166 −0.0341977 −0.0170989 0.999854i \(-0.505443\pi\)
−0.0170989 + 0.999854i \(0.505443\pi\)
\(674\) −3.15377 −0.121479
\(675\) 0 0
\(676\) 8.28963 0.318832
\(677\) −18.9018 −0.726453 −0.363227 0.931701i \(-0.618325\pi\)
−0.363227 + 0.931701i \(0.618325\pi\)
\(678\) −1.16362 −0.0446885
\(679\) 0.252016 0.00967149
\(680\) 0 0
\(681\) 16.9624 0.650000
\(682\) 1.92478 0.0737035
\(683\) 20.8773 0.798848 0.399424 0.916766i \(-0.369210\pi\)
0.399424 + 0.916766i \(0.369210\pi\)
\(684\) 8.46310 0.323595
\(685\) 0 0
\(686\) 1.80351 0.0688583
\(687\) −25.8496 −0.986222
\(688\) 35.0202 1.33513
\(689\) 13.9248 0.530492
\(690\) 0 0
\(691\) −2.44851 −0.0931456 −0.0465728 0.998915i \(-0.514830\pi\)
−0.0465728 + 0.998915i \(0.514830\pi\)
\(692\) −16.8265 −0.639649
\(693\) −3.35026 −0.127266
\(694\) 0.186642 0.00708485
\(695\) 0 0
\(696\) −2.77575 −0.105214
\(697\) −20.0724 −0.760296
\(698\) 4.01459 0.151954
\(699\) −19.2750 −0.729049
\(700\) 0 0
\(701\) −2.98683 −0.112811 −0.0564054 0.998408i \(-0.517964\pi\)
−0.0564054 + 0.998408i \(0.517964\pi\)
\(702\) 0.574515 0.0216837
\(703\) −8.62530 −0.325309
\(704\) −7.11142 −0.268022
\(705\) 0 0
\(706\) −3.98541 −0.149993
\(707\) 50.5501 1.90113
\(708\) 20.9986 0.789175
\(709\) 24.1768 0.907979 0.453989 0.891007i \(-0.350000\pi\)
0.453989 + 0.891007i \(0.350000\pi\)
\(710\) 0 0
\(711\) 11.5369 0.432668
\(712\) 2.13303 0.0799386
\(713\) 66.5012 2.49049
\(714\) 2.97224 0.111233
\(715\) 0 0
\(716\) −27.8204 −1.03970
\(717\) −26.5501 −0.991531
\(718\) 3.47627 0.129733
\(719\) −30.0263 −1.11979 −0.559897 0.828562i \(-0.689159\pi\)
−0.559897 + 0.828562i \(0.689159\pi\)
\(720\) 0 0
\(721\) −10.8021 −0.402291
\(722\) −0.0777777 −0.00289459
\(723\) −28.5501 −1.06179
\(724\) 10.2520 0.381013
\(725\) 0 0
\(726\) −0.193937 −0.00719766
\(727\) −14.9525 −0.554559 −0.277279 0.960789i \(-0.589433\pi\)
−0.277279 + 0.960789i \(0.589433\pi\)
\(728\) −7.62672 −0.282665
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 42.4288 1.56929
\(732\) −17.0738 −0.631066
\(733\) 19.1128 0.705949 0.352974 0.935633i \(-0.385170\pi\)
0.352974 + 0.935633i \(0.385170\pi\)
\(734\) 5.75081 0.212266
\(735\) 0 0
\(736\) 15.2046 0.560447
\(737\) −5.92478 −0.218242
\(738\) −0.850969 −0.0313246
\(739\) −3.31406 −0.121910 −0.0609549 0.998141i \(-0.519415\pi\)
−0.0609549 + 0.998141i \(0.519415\pi\)
\(740\) 0 0
\(741\) −12.7757 −0.469329
\(742\) 3.05411 0.112120
\(743\) 34.9887 1.28361 0.641806 0.766867i \(-0.278185\pi\)
0.641806 + 0.766867i \(0.278185\pi\)
\(744\) 7.62672 0.279609
\(745\) 0 0
\(746\) 1.77242 0.0648930
\(747\) −10.8872 −0.398341
\(748\) −8.97698 −0.328231
\(749\) −3.22425 −0.117812
\(750\) 0 0
\(751\) −26.9234 −0.982447 −0.491224 0.871033i \(-0.663450\pi\)
−0.491224 + 0.871033i \(0.663450\pi\)
\(752\) 37.4734 1.36652
\(753\) −29.9248 −1.09052
\(754\) 2.07522 0.0755752
\(755\) 0 0
\(756\) −6.57452 −0.239113
\(757\) −15.9248 −0.578796 −0.289398 0.957209i \(-0.593455\pi\)
−0.289398 + 0.957209i \(0.593455\pi\)
\(758\) −3.87873 −0.140882
\(759\) −6.70052 −0.243214
\(760\) 0 0
\(761\) −30.9380 −1.12150 −0.560750 0.827985i \(-0.689487\pi\)
−0.560750 + 0.827985i \(0.689487\pi\)
\(762\) −2.82653 −0.102394
\(763\) −38.1965 −1.38281
\(764\) 32.6253 1.18034
\(765\) 0 0
\(766\) 6.77292 0.244715
\(767\) −31.6991 −1.14459
\(768\) −13.0752 −0.471811
\(769\) 9.32582 0.336298 0.168149 0.985762i \(-0.446221\pi\)
0.168149 + 0.985762i \(0.446221\pi\)
\(770\) 0 0
\(771\) 8.70052 0.313342
\(772\) −32.1114 −1.15572
\(773\) −44.7005 −1.60777 −0.803883 0.594787i \(-0.797236\pi\)
−0.803883 + 0.594787i \(0.797236\pi\)
\(774\) 1.79877 0.0646554
\(775\) 0 0
\(776\) 0.0578051 0.00207508
\(777\) 6.70052 0.240380
\(778\) 0.538319 0.0192997
\(779\) 18.9234 0.678000
\(780\) 0 0
\(781\) 9.92478 0.355136
\(782\) 5.94448 0.212574
\(783\) 3.61213 0.129087
\(784\) 15.9497 0.569633
\(785\) 0 0
\(786\) 1.14903 0.0409846
\(787\) −21.6775 −0.772719 −0.386360 0.922348i \(-0.626268\pi\)
−0.386360 + 0.922348i \(0.626268\pi\)
\(788\) −40.0800 −1.42779
\(789\) 12.2882 0.437472
\(790\) 0 0
\(791\) −20.1016 −0.714730
\(792\) −0.768452 −0.0273058
\(793\) 25.7743 0.915273
\(794\) 3.86414 0.137133
\(795\) 0 0
\(796\) 16.9262 0.599933
\(797\) −22.7466 −0.805725 −0.402862 0.915261i \(-0.631985\pi\)
−0.402862 + 0.915261i \(0.631985\pi\)
\(798\) −2.80209 −0.0991930
\(799\) 45.4010 1.60617
\(800\) 0 0
\(801\) −2.77575 −0.0980762
\(802\) 0.387873 0.0136963
\(803\) 7.73813 0.273073
\(804\) −11.6267 −0.410043
\(805\) 0 0
\(806\) −5.70194 −0.200842
\(807\) 5.84955 0.205914
\(808\) 11.5947 0.407900
\(809\) −23.6121 −0.830158 −0.415079 0.909785i \(-0.636246\pi\)
−0.415079 + 0.909785i \(0.636246\pi\)
\(810\) 0 0
\(811\) −26.0870 −0.916038 −0.458019 0.888942i \(-0.651441\pi\)
−0.458019 + 0.888942i \(0.651441\pi\)
\(812\) −23.7480 −0.833391
\(813\) 5.08840 0.178458
\(814\) 0.387873 0.0135949
\(815\) 0 0
\(816\) −17.2722 −0.604648
\(817\) −40.0000 −1.39942
\(818\) −2.53549 −0.0886513
\(819\) 9.92478 0.346800
\(820\) 0 0
\(821\) −54.4142 −1.89907 −0.949535 0.313662i \(-0.898444\pi\)
−0.949535 + 0.313662i \(0.898444\pi\)
\(822\) 2.68594 0.0936827
\(823\) −0.121269 −0.00422716 −0.00211358 0.999998i \(-0.500673\pi\)
−0.00211358 + 0.999998i \(0.500673\pi\)
\(824\) −2.47768 −0.0863142
\(825\) 0 0
\(826\) −6.95254 −0.241910
\(827\) 18.2130 0.633328 0.316664 0.948538i \(-0.397437\pi\)
0.316664 + 0.948538i \(0.397437\pi\)
\(828\) −13.1490 −0.456960
\(829\) 13.0738 0.454072 0.227036 0.973886i \(-0.427096\pi\)
0.227036 + 0.973886i \(0.427096\pi\)
\(830\) 0 0
\(831\) 1.41090 0.0489434
\(832\) 21.0668 0.730359
\(833\) 19.3239 0.669534
\(834\) −2.63989 −0.0914119
\(835\) 0 0
\(836\) 8.46310 0.292702
\(837\) −9.92478 −0.343050
\(838\) 1.40105 0.0483984
\(839\) −26.5501 −0.916610 −0.458305 0.888795i \(-0.651543\pi\)
−0.458305 + 0.888795i \(0.651543\pi\)
\(840\) 0 0
\(841\) −15.9525 −0.550088
\(842\) 5.93937 0.204684
\(843\) 4.38787 0.151126
\(844\) −17.8350 −0.613905
\(845\) 0 0
\(846\) 1.92478 0.0661752
\(847\) −3.35026 −0.115116
\(848\) −17.7480 −0.609468
\(849\) 26.5745 0.912035
\(850\) 0 0
\(851\) 13.4010 0.459382
\(852\) 19.4763 0.667246
\(853\) −40.6155 −1.39065 −0.695323 0.718697i \(-0.744739\pi\)
−0.695323 + 0.718697i \(0.744739\pi\)
\(854\) 5.65306 0.193444
\(855\) 0 0
\(856\) −0.739549 −0.0252773
\(857\) −20.1721 −0.689064 −0.344532 0.938775i \(-0.611962\pi\)
−0.344532 + 0.938775i \(0.611962\pi\)
\(858\) 0.574515 0.0196136
\(859\) 21.8035 0.743926 0.371963 0.928248i \(-0.378685\pi\)
0.371963 + 0.928248i \(0.378685\pi\)
\(860\) 0 0
\(861\) −14.7005 −0.500993
\(862\) −6.56978 −0.223767
\(863\) −35.4274 −1.20596 −0.602981 0.797755i \(-0.706021\pi\)
−0.602981 + 0.797755i \(0.706021\pi\)
\(864\) −2.26916 −0.0771984
\(865\) 0 0
\(866\) 1.83780 0.0624508
\(867\) −3.92619 −0.133340
\(868\) 65.2506 2.21475
\(869\) 11.5369 0.391363
\(870\) 0 0
\(871\) 17.5515 0.594710
\(872\) −8.76116 −0.296690
\(873\) −0.0752228 −0.00254591
\(874\) −5.60419 −0.189564
\(875\) 0 0
\(876\) 15.1852 0.513061
\(877\) 14.0362 0.473969 0.236984 0.971513i \(-0.423841\pi\)
0.236984 + 0.971513i \(0.423841\pi\)
\(878\) −5.71370 −0.192828
\(879\) −3.42548 −0.115539
\(880\) 0 0
\(881\) −21.0738 −0.709995 −0.354997 0.934867i \(-0.615518\pi\)
−0.354997 + 0.934867i \(0.615518\pi\)
\(882\) 0.819237 0.0275852
\(883\) 42.1476 1.41838 0.709190 0.705017i \(-0.249061\pi\)
0.709190 + 0.705017i \(0.249061\pi\)
\(884\) 26.5933 0.894429
\(885\) 0 0
\(886\) 3.69911 0.124274
\(887\) −6.93604 −0.232889 −0.116445 0.993197i \(-0.537150\pi\)
−0.116445 + 0.993197i \(0.537150\pi\)
\(888\) 1.53690 0.0515751
\(889\) −48.8284 −1.63765
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) −13.1490 −0.440262
\(893\) −42.8021 −1.43232
\(894\) −0.298062 −0.00996868
\(895\) 0 0
\(896\) 19.8251 0.662311
\(897\) 19.8496 0.662757
\(898\) 6.95765 0.232180
\(899\) −35.8496 −1.19565
\(900\) 0 0
\(901\) −21.5026 −0.716356
\(902\) −0.850969 −0.0283342
\(903\) 31.0738 1.03407
\(904\) −4.61071 −0.153350
\(905\) 0 0
\(906\) 1.31124 0.0435629
\(907\) −53.2017 −1.76653 −0.883267 0.468870i \(-0.844661\pi\)
−0.883267 + 0.468870i \(0.844661\pi\)
\(908\) 33.2868 1.10466
\(909\) −15.0884 −0.500451
\(910\) 0 0
\(911\) 36.4749 1.20847 0.604233 0.796808i \(-0.293480\pi\)
0.604233 + 0.796808i \(0.293480\pi\)
\(912\) 16.2835 0.539200
\(913\) −10.8872 −0.360313
\(914\) −1.02585 −0.0339322
\(915\) 0 0
\(916\) −50.7269 −1.67606
\(917\) 19.8496 0.655490
\(918\) −0.887166 −0.0292808
\(919\) 9.73340 0.321075 0.160538 0.987030i \(-0.448677\pi\)
0.160538 + 0.987030i \(0.448677\pi\)
\(920\) 0 0
\(921\) −16.6497 −0.548628
\(922\) 7.04746 0.232096
\(923\) −29.4010 −0.967747
\(924\) −6.57452 −0.216286
\(925\) 0 0
\(926\) −2.04605 −0.0672372
\(927\) 3.22425 0.105898
\(928\) −8.19649 −0.269063
\(929\) −24.1768 −0.793215 −0.396607 0.917988i \(-0.629813\pi\)
−0.396607 + 0.917988i \(0.629813\pi\)
\(930\) 0 0
\(931\) −18.2177 −0.597062
\(932\) −37.8251 −1.23900
\(933\) −32.9986 −1.08033
\(934\) −3.62672 −0.118670
\(935\) 0 0
\(936\) 2.27645 0.0744082
\(937\) 7.48612 0.244561 0.122280 0.992496i \(-0.460979\pi\)
0.122280 + 0.992496i \(0.460979\pi\)
\(938\) 3.84955 0.125692
\(939\) 15.4010 0.502594
\(940\) 0 0
\(941\) −21.2360 −0.692274 −0.346137 0.938184i \(-0.612507\pi\)
−0.346137 + 0.938184i \(0.612507\pi\)
\(942\) −1.06205 −0.0346034
\(943\) −29.4010 −0.957430
\(944\) 40.4025 1.31499
\(945\) 0 0
\(946\) 1.79877 0.0584830
\(947\) 15.4763 0.502911 0.251456 0.967869i \(-0.419091\pi\)
0.251456 + 0.967869i \(0.419091\pi\)
\(948\) 22.6399 0.735309
\(949\) −22.9234 −0.744124
\(950\) 0 0
\(951\) 2.15045 0.0697330
\(952\) 11.7772 0.381700
\(953\) 32.0508 1.03823 0.519113 0.854705i \(-0.326262\pi\)
0.519113 + 0.854705i \(0.326262\pi\)
\(954\) −0.911603 −0.0295142
\(955\) 0 0
\(956\) −52.1016 −1.68509
\(957\) 3.61213 0.116763
\(958\) −1.80351 −0.0582687
\(959\) 46.3996 1.49832
\(960\) 0 0
\(961\) 67.5012 2.17746
\(962\) −1.14903 −0.0370462
\(963\) 0.962389 0.0310125
\(964\) −56.0263 −1.80449
\(965\) 0 0
\(966\) 4.35359 0.140074
\(967\) 17.3766 0.558794 0.279397 0.960176i \(-0.409865\pi\)
0.279397 + 0.960176i \(0.409865\pi\)
\(968\) −0.768452 −0.0246990
\(969\) 19.7283 0.633764
\(970\) 0 0
\(971\) 36.2031 1.16181 0.580907 0.813970i \(-0.302698\pi\)
0.580907 + 0.813970i \(0.302698\pi\)
\(972\) 1.96239 0.0629436
\(973\) −45.6042 −1.46200
\(974\) 6.88015 0.220454
\(975\) 0 0
\(976\) −32.8510 −1.05153
\(977\) 28.1476 0.900522 0.450261 0.892897i \(-0.351331\pi\)
0.450261 + 0.892897i \(0.351331\pi\)
\(978\) 2.44851 0.0782946
\(979\) −2.77575 −0.0887132
\(980\) 0 0
\(981\) 11.4010 0.364007
\(982\) 4.80492 0.153331
\(983\) −7.07381 −0.225619 −0.112810 0.993617i \(-0.535985\pi\)
−0.112810 + 0.993617i \(0.535985\pi\)
\(984\) −3.37187 −0.107491
\(985\) 0 0
\(986\) −3.20456 −0.102054
\(987\) 33.2506 1.05838
\(988\) −25.0710 −0.797614
\(989\) 62.1476 1.97618
\(990\) 0 0
\(991\) 44.4260 1.41124 0.705619 0.708592i \(-0.250669\pi\)
0.705619 + 0.708592i \(0.250669\pi\)
\(992\) 22.5209 0.715039
\(993\) 14.5501 0.461733
\(994\) −6.44851 −0.204534
\(995\) 0 0
\(996\) −21.3649 −0.676971
\(997\) 28.4847 0.902120 0.451060 0.892494i \(-0.351046\pi\)
0.451060 + 0.892494i \(0.351046\pi\)
\(998\) 2.74940 0.0870307
\(999\) −2.00000 −0.0632772
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 825.2.a.k.1.2 3
3.2 odd 2 2475.2.a.bb.1.2 3
5.2 odd 4 825.2.c.g.199.4 6
5.3 odd 4 825.2.c.g.199.3 6
5.4 even 2 165.2.a.c.1.2 3
11.10 odd 2 9075.2.a.cf.1.2 3
15.2 even 4 2475.2.c.r.199.3 6
15.8 even 4 2475.2.c.r.199.4 6
15.14 odd 2 495.2.a.e.1.2 3
20.19 odd 2 2640.2.a.be.1.1 3
35.34 odd 2 8085.2.a.bk.1.2 3
55.54 odd 2 1815.2.a.m.1.2 3
60.59 even 2 7920.2.a.cj.1.1 3
165.164 even 2 5445.2.a.z.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.2 3 5.4 even 2
495.2.a.e.1.2 3 15.14 odd 2
825.2.a.k.1.2 3 1.1 even 1 trivial
825.2.c.g.199.3 6 5.3 odd 4
825.2.c.g.199.4 6 5.2 odd 4
1815.2.a.m.1.2 3 55.54 odd 2
2475.2.a.bb.1.2 3 3.2 odd 2
2475.2.c.r.199.3 6 15.2 even 4
2475.2.c.r.199.4 6 15.8 even 4
2640.2.a.be.1.1 3 20.19 odd 2
5445.2.a.z.1.2 3 165.164 even 2
7920.2.a.cj.1.1 3 60.59 even 2
8085.2.a.bk.1.2 3 35.34 odd 2
9075.2.a.cf.1.2 3 11.10 odd 2