Properties

Label 3450.2.a.bp.1.3
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3368.1
Defining polynomial: \(x^{3} - x^{2} - 15 x + 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.03932\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.03932 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.03932 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.69736 q^{11} +1.00000 q^{12} +0.381275 q^{13} -4.03932 q^{14} +1.00000 q^{16} -3.65804 q^{17} -1.00000 q^{18} +0.381275 q^{19} +4.03932 q^{21} +5.69736 q^{22} -1.00000 q^{23} -1.00000 q^{24} -0.381275 q^{26} +1.00000 q^{27} +4.03932 q^{28} -1.00000 q^{29} +6.00000 q^{31} -1.00000 q^{32} -5.69736 q^{33} +3.65804 q^{34} +1.00000 q^{36} +3.65804 q^{37} -0.381275 q^{38} +0.381275 q^{39} +5.61873 q^{41} -4.03932 q^{42} +3.61873 q^{43} -5.69736 q^{44} +1.00000 q^{46} -3.38127 q^{47} +1.00000 q^{48} +9.31608 q^{49} -3.65804 q^{51} +0.381275 q^{52} +9.31608 q^{53} -1.00000 q^{54} -4.03932 q^{56} +0.381275 q^{57} +1.00000 q^{58} +13.3947 q^{59} +14.0786 q^{61} -6.00000 q^{62} +4.03932 q^{63} +1.00000 q^{64} +5.69736 q^{66} -2.76255 q^{67} -3.65804 q^{68} -1.00000 q^{69} +10.6974 q^{71} -1.00000 q^{72} -7.07863 q^{73} -3.65804 q^{74} +0.381275 q^{76} -23.0134 q^{77} -0.381275 q^{78} +5.69736 q^{79} +1.00000 q^{81} -5.61873 q^{82} +16.1180 q^{83} +4.03932 q^{84} -3.61873 q^{86} -1.00000 q^{87} +5.69736 q^{88} +5.65804 q^{89} +1.54009 q^{91} -1.00000 q^{92} +6.00000 q^{93} +3.38127 q^{94} -1.00000 q^{96} -4.07863 q^{97} -9.31608 q^{98} -5.69736 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} + q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} + q^{7} - 3q^{8} + 3q^{9} + 3q^{11} + 3q^{12} - q^{13} - q^{14} + 3q^{16} - 2q^{17} - 3q^{18} - q^{19} + q^{21} - 3q^{22} - 3q^{23} - 3q^{24} + q^{26} + 3q^{27} + q^{28} - 3q^{29} + 18q^{31} - 3q^{32} + 3q^{33} + 2q^{34} + 3q^{36} + 2q^{37} + q^{38} - q^{39} + 19q^{41} - q^{42} + 13q^{43} + 3q^{44} + 3q^{46} - 8q^{47} + 3q^{48} + 10q^{49} - 2q^{51} - q^{52} + 10q^{53} - 3q^{54} - q^{56} - q^{57} + 3q^{58} + 20q^{61} - 18q^{62} + q^{63} + 3q^{64} - 3q^{66} - 4q^{67} - 2q^{68} - 3q^{69} + 12q^{71} - 3q^{72} + q^{73} - 2q^{74} - q^{76} - 31q^{77} + q^{78} - 3q^{79} + 3q^{81} - 19q^{82} + 15q^{83} + q^{84} - 13q^{86} - 3q^{87} - 3q^{88} + 8q^{89} + 29q^{91} - 3q^{92} + 18q^{93} + 8q^{94} - 3q^{96} + 10q^{97} - 10q^{98} + 3q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 4.03932 1.52672 0.763359 0.645974i \(-0.223549\pi\)
0.763359 + 0.645974i \(0.223549\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.69736 −1.71782 −0.858909 0.512128i \(-0.828857\pi\)
−0.858909 + 0.512128i \(0.828857\pi\)
\(12\) 1.00000 0.288675
\(13\) 0.381275 0.105747 0.0528733 0.998601i \(-0.483162\pi\)
0.0528733 + 0.998601i \(0.483162\pi\)
\(14\) −4.03932 −1.07955
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.65804 −0.887206 −0.443603 0.896223i \(-0.646300\pi\)
−0.443603 + 0.896223i \(0.646300\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0.381275 0.0874705 0.0437352 0.999043i \(-0.486074\pi\)
0.0437352 + 0.999043i \(0.486074\pi\)
\(20\) 0 0
\(21\) 4.03932 0.881451
\(22\) 5.69736 1.21468
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −0.381275 −0.0747742
\(27\) 1.00000 0.192450
\(28\) 4.03932 0.763359
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.69736 −0.991783
\(34\) 3.65804 0.627349
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 3.65804 0.601378 0.300689 0.953722i \(-0.402783\pi\)
0.300689 + 0.953722i \(0.402783\pi\)
\(38\) −0.381275 −0.0618510
\(39\) 0.381275 0.0610528
\(40\) 0 0
\(41\) 5.61873 0.877497 0.438749 0.898610i \(-0.355422\pi\)
0.438749 + 0.898610i \(0.355422\pi\)
\(42\) −4.03932 −0.623280
\(43\) 3.61873 0.551850 0.275925 0.961179i \(-0.411016\pi\)
0.275925 + 0.961179i \(0.411016\pi\)
\(44\) −5.69736 −0.858909
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −3.38127 −0.493210 −0.246605 0.969116i \(-0.579315\pi\)
−0.246605 + 0.969116i \(0.579315\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.31608 1.33087
\(50\) 0 0
\(51\) −3.65804 −0.512228
\(52\) 0.381275 0.0528733
\(53\) 9.31608 1.27966 0.639831 0.768515i \(-0.279004\pi\)
0.639831 + 0.768515i \(0.279004\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −4.03932 −0.539776
\(57\) 0.381275 0.0505011
\(58\) 1.00000 0.131306
\(59\) 13.3947 1.74384 0.871922 0.489645i \(-0.162874\pi\)
0.871922 + 0.489645i \(0.162874\pi\)
\(60\) 0 0
\(61\) 14.0786 1.80258 0.901292 0.433212i \(-0.142620\pi\)
0.901292 + 0.433212i \(0.142620\pi\)
\(62\) −6.00000 −0.762001
\(63\) 4.03932 0.508906
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 5.69736 0.701297
\(67\) −2.76255 −0.337499 −0.168750 0.985659i \(-0.553973\pi\)
−0.168750 + 0.985659i \(0.553973\pi\)
\(68\) −3.65804 −0.443603
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 10.6974 1.26954 0.634772 0.772700i \(-0.281094\pi\)
0.634772 + 0.772700i \(0.281094\pi\)
\(72\) −1.00000 −0.117851
\(73\) −7.07863 −0.828492 −0.414246 0.910165i \(-0.635955\pi\)
−0.414246 + 0.910165i \(0.635955\pi\)
\(74\) −3.65804 −0.425239
\(75\) 0 0
\(76\) 0.381275 0.0437352
\(77\) −23.0134 −2.62263
\(78\) −0.381275 −0.0431709
\(79\) 5.69736 0.641003 0.320502 0.947248i \(-0.396149\pi\)
0.320502 + 0.947248i \(0.396149\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −5.61873 −0.620484
\(83\) 16.1180 1.76918 0.884588 0.466374i \(-0.154440\pi\)
0.884588 + 0.466374i \(0.154440\pi\)
\(84\) 4.03932 0.440726
\(85\) 0 0
\(86\) −3.61873 −0.390217
\(87\) −1.00000 −0.107211
\(88\) 5.69736 0.607341
\(89\) 5.65804 0.599751 0.299876 0.953978i \(-0.403055\pi\)
0.299876 + 0.953978i \(0.403055\pi\)
\(90\) 0 0
\(91\) 1.54009 0.161445
\(92\) −1.00000 −0.104257
\(93\) 6.00000 0.622171
\(94\) 3.38127 0.348752
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −4.07863 −0.414123 −0.207061 0.978328i \(-0.566390\pi\)
−0.207061 + 0.978328i \(0.566390\pi\)
\(98\) −9.31608 −0.941067
\(99\) −5.69736 −0.572606
\(100\) 0 0
\(101\) 16.7760 1.66927 0.834637 0.550801i \(-0.185677\pi\)
0.834637 + 0.550801i \(0.185677\pi\)
\(102\) 3.65804 0.362200
\(103\) 13.3554 1.31595 0.657973 0.753041i \(-0.271414\pi\)
0.657973 + 0.753041i \(0.271414\pi\)
\(104\) −0.381275 −0.0373871
\(105\) 0 0
\(106\) −9.31608 −0.904858
\(107\) −6.76255 −0.653760 −0.326880 0.945066i \(-0.605997\pi\)
−0.326880 + 0.945066i \(0.605997\pi\)
\(108\) 1.00000 0.0962250
\(109\) −13.0528 −1.25023 −0.625114 0.780534i \(-0.714948\pi\)
−0.625114 + 0.780534i \(0.714948\pi\)
\(110\) 0 0
\(111\) 3.65804 0.347206
\(112\) 4.03932 0.381680
\(113\) −1.65804 −0.155976 −0.0779878 0.996954i \(-0.524850\pi\)
−0.0779878 + 0.996954i \(0.524850\pi\)
\(114\) −0.381275 −0.0357097
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) 0.381275 0.0352489
\(118\) −13.3947 −1.23308
\(119\) −14.7760 −1.35451
\(120\) 0 0
\(121\) 21.4599 1.95090
\(122\) −14.0786 −1.27462
\(123\) 5.61873 0.506623
\(124\) 6.00000 0.538816
\(125\) 0 0
\(126\) −4.03932 −0.359851
\(127\) −18.6322 −1.65334 −0.826669 0.562689i \(-0.809767\pi\)
−0.826669 + 0.562689i \(0.809767\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 3.61873 0.318611
\(130\) 0 0
\(131\) −17.4734 −1.52665 −0.763327 0.646012i \(-0.776435\pi\)
−0.763327 + 0.646012i \(0.776435\pi\)
\(132\) −5.69736 −0.495892
\(133\) 1.54009 0.133543
\(134\) 2.76255 0.238648
\(135\) 0 0
\(136\) 3.65804 0.313675
\(137\) −14.3420 −1.22532 −0.612658 0.790348i \(-0.709900\pi\)
−0.612658 + 0.790348i \(0.709900\pi\)
\(138\) 1.00000 0.0851257
\(139\) −8.01344 −0.679692 −0.339846 0.940481i \(-0.610375\pi\)
−0.339846 + 0.940481i \(0.610375\pi\)
\(140\) 0 0
\(141\) −3.38127 −0.284755
\(142\) −10.6974 −0.897702
\(143\) −2.17226 −0.181654
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 7.07863 0.585832
\(147\) 9.31608 0.768378
\(148\) 3.65804 0.300689
\(149\) 3.23745 0.265222 0.132611 0.991168i \(-0.457664\pi\)
0.132611 + 0.991168i \(0.457664\pi\)
\(150\) 0 0
\(151\) −9.31608 −0.758132 −0.379066 0.925370i \(-0.623755\pi\)
−0.379066 + 0.925370i \(0.623755\pi\)
\(152\) −0.381275 −0.0309255
\(153\) −3.65804 −0.295735
\(154\) 23.0134 1.85448
\(155\) 0 0
\(156\) 0.381275 0.0305264
\(157\) −6.84118 −0.545986 −0.272993 0.962016i \(-0.588014\pi\)
−0.272993 + 0.962016i \(0.588014\pi\)
\(158\) −5.69736 −0.453258
\(159\) 9.31608 0.738814
\(160\) 0 0
\(161\) −4.03932 −0.318343
\(162\) −1.00000 −0.0785674
\(163\) 4.76255 0.373032 0.186516 0.982452i \(-0.440280\pi\)
0.186516 + 0.982452i \(0.440280\pi\)
\(164\) 5.61873 0.438749
\(165\) 0 0
\(166\) −16.1180 −1.25100
\(167\) 14.8546 1.14949 0.574743 0.818334i \(-0.305102\pi\)
0.574743 + 0.818334i \(0.305102\pi\)
\(168\) −4.03932 −0.311640
\(169\) −12.8546 −0.988818
\(170\) 0 0
\(171\) 0.381275 0.0291568
\(172\) 3.61873 0.275925
\(173\) 9.69736 0.737277 0.368638 0.929573i \(-0.379824\pi\)
0.368638 + 0.929573i \(0.379824\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) −5.69736 −0.429455
\(177\) 13.3947 1.00681
\(178\) −5.65804 −0.424088
\(179\) −5.31608 −0.397343 −0.198671 0.980066i \(-0.563663\pi\)
−0.198671 + 0.980066i \(0.563663\pi\)
\(180\) 0 0
\(181\) 24.3688 1.81132 0.905661 0.424002i \(-0.139375\pi\)
0.905661 + 0.424002i \(0.139375\pi\)
\(182\) −1.54009 −0.114159
\(183\) 14.0786 1.04072
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −6.00000 −0.439941
\(187\) 20.8412 1.52406
\(188\) −3.38127 −0.246605
\(189\) 4.03932 0.293817
\(190\) 0 0
\(191\) 10.4599 0.756853 0.378426 0.925631i \(-0.376465\pi\)
0.378426 + 0.925631i \(0.376465\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0134 −1.00871 −0.504355 0.863496i \(-0.668270\pi\)
−0.504355 + 0.863496i \(0.668270\pi\)
\(194\) 4.07863 0.292829
\(195\) 0 0
\(196\) 9.31608 0.665435
\(197\) 8.47335 0.603702 0.301851 0.953355i \(-0.402396\pi\)
0.301851 + 0.953355i \(0.402396\pi\)
\(198\) 5.69736 0.404894
\(199\) 1.19813 0.0849334 0.0424667 0.999098i \(-0.486478\pi\)
0.0424667 + 0.999098i \(0.486478\pi\)
\(200\) 0 0
\(201\) −2.76255 −0.194855
\(202\) −16.7760 −1.18035
\(203\) −4.03932 −0.283505
\(204\) −3.65804 −0.256114
\(205\) 0 0
\(206\) −13.3554 −0.930515
\(207\) −1.00000 −0.0695048
\(208\) 0.381275 0.0264367
\(209\) −2.17226 −0.150258
\(210\) 0 0
\(211\) −16.2225 −1.11680 −0.558400 0.829572i \(-0.688585\pi\)
−0.558400 + 0.829572i \(0.688585\pi\)
\(212\) 9.31608 0.639831
\(213\) 10.6974 0.732971
\(214\) 6.76255 0.462278
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 24.2359 1.64524
\(218\) 13.0528 0.884045
\(219\) −7.07863 −0.478330
\(220\) 0 0
\(221\) −1.39472 −0.0938190
\(222\) −3.65804 −0.245512
\(223\) −13.6037 −0.910973 −0.455487 0.890243i \(-0.650535\pi\)
−0.455487 + 0.890243i \(0.650535\pi\)
\(224\) −4.03932 −0.269888
\(225\) 0 0
\(226\) 1.65804 0.110291
\(227\) −12.4992 −0.829603 −0.414801 0.909912i \(-0.636149\pi\)
−0.414801 + 0.909912i \(0.636149\pi\)
\(228\) 0.381275 0.0252505
\(229\) 21.4734 1.41900 0.709500 0.704706i \(-0.248921\pi\)
0.709500 + 0.704706i \(0.248921\pi\)
\(230\) 0 0
\(231\) −23.0134 −1.51417
\(232\) 1.00000 0.0656532
\(233\) −11.1438 −0.730056 −0.365028 0.930996i \(-0.618941\pi\)
−0.365028 + 0.930996i \(0.618941\pi\)
\(234\) −0.381275 −0.0249247
\(235\) 0 0
\(236\) 13.3947 0.871922
\(237\) 5.69736 0.370083
\(238\) 14.7760 0.957785
\(239\) 22.1707 1.43410 0.717052 0.697020i \(-0.245491\pi\)
0.717052 + 0.697020i \(0.245491\pi\)
\(240\) 0 0
\(241\) −13.8696 −0.893421 −0.446710 0.894679i \(-0.647405\pi\)
−0.446710 + 0.894679i \(0.647405\pi\)
\(242\) −21.4599 −1.37950
\(243\) 1.00000 0.0641500
\(244\) 14.0786 0.901292
\(245\) 0 0
\(246\) −5.61873 −0.358237
\(247\) 0.145371 0.00924971
\(248\) −6.00000 −0.381000
\(249\) 16.1180 1.02143
\(250\) 0 0
\(251\) 5.73668 0.362096 0.181048 0.983474i \(-0.442051\pi\)
0.181048 + 0.983474i \(0.442051\pi\)
\(252\) 4.03932 0.254453
\(253\) 5.69736 0.358190
\(254\) 18.6322 1.16909
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −20.6322 −1.28700 −0.643500 0.765446i \(-0.722518\pi\)
−0.643500 + 0.765446i \(0.722518\pi\)
\(258\) −3.61873 −0.225292
\(259\) 14.7760 0.918136
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 17.4734 1.07951
\(263\) 19.8696 1.22521 0.612607 0.790388i \(-0.290121\pi\)
0.612607 + 0.790388i \(0.290121\pi\)
\(264\) 5.69736 0.350648
\(265\) 0 0
\(266\) −1.54009 −0.0944290
\(267\) 5.65804 0.346267
\(268\) −2.76255 −0.168750
\(269\) −8.17226 −0.498272 −0.249136 0.968469i \(-0.580147\pi\)
−0.249136 + 0.968469i \(0.580147\pi\)
\(270\) 0 0
\(271\) −11.3947 −0.692180 −0.346090 0.938201i \(-0.612491\pi\)
−0.346090 + 0.938201i \(0.612491\pi\)
\(272\) −3.65804 −0.221801
\(273\) 1.54009 0.0932105
\(274\) 14.3420 0.866429
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −4.93481 −0.296504 −0.148252 0.988950i \(-0.547365\pi\)
−0.148252 + 0.988950i \(0.547365\pi\)
\(278\) 8.01344 0.480614
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 14.8955 0.888591 0.444295 0.895880i \(-0.353454\pi\)
0.444295 + 0.895880i \(0.353454\pi\)
\(282\) 3.38127 0.201352
\(283\) −14.6322 −0.869792 −0.434896 0.900481i \(-0.643215\pi\)
−0.434896 + 0.900481i \(0.643215\pi\)
\(284\) 10.6974 0.634772
\(285\) 0 0
\(286\) 2.17226 0.128448
\(287\) 22.6958 1.33969
\(288\) −1.00000 −0.0589256
\(289\) −3.61873 −0.212866
\(290\) 0 0
\(291\) −4.07863 −0.239094
\(292\) −7.07863 −0.414246
\(293\) −9.39472 −0.548845 −0.274423 0.961609i \(-0.588487\pi\)
−0.274423 + 0.961609i \(0.588487\pi\)
\(294\) −9.31608 −0.543325
\(295\) 0 0
\(296\) −3.65804 −0.212619
\(297\) −5.69736 −0.330594
\(298\) −3.23745 −0.187540
\(299\) −0.381275 −0.0220497
\(300\) 0 0
\(301\) 14.6172 0.842520
\(302\) 9.31608 0.536080
\(303\) 16.7760 0.963756
\(304\) 0.381275 0.0218676
\(305\) 0 0
\(306\) 3.65804 0.209116
\(307\) 9.22401 0.526442 0.263221 0.964736i \(-0.415215\pi\)
0.263221 + 0.964736i \(0.415215\pi\)
\(308\) −23.0134 −1.31131
\(309\) 13.3554 0.759762
\(310\) 0 0
\(311\) 11.3295 0.642439 0.321219 0.947005i \(-0.395907\pi\)
0.321219 + 0.947005i \(0.395907\pi\)
\(312\) −0.381275 −0.0215854
\(313\) 15.6037 0.881975 0.440988 0.897513i \(-0.354628\pi\)
0.440988 + 0.897513i \(0.354628\pi\)
\(314\) 6.84118 0.386070
\(315\) 0 0
\(316\) 5.69736 0.320502
\(317\) −11.0000 −0.617822 −0.308911 0.951091i \(-0.599964\pi\)
−0.308911 + 0.951091i \(0.599964\pi\)
\(318\) −9.31608 −0.522420
\(319\) 5.69736 0.318991
\(320\) 0 0
\(321\) −6.76255 −0.377449
\(322\) 4.03932 0.225102
\(323\) −1.39472 −0.0776043
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.76255 −0.263773
\(327\) −13.0528 −0.721819
\(328\) −5.61873 −0.310242
\(329\) −13.6580 −0.752992
\(330\) 0 0
\(331\) 2.69736 0.148260 0.0741302 0.997249i \(-0.476382\pi\)
0.0741302 + 0.997249i \(0.476382\pi\)
\(332\) 16.1180 0.884588
\(333\) 3.65804 0.200459
\(334\) −14.8546 −0.812809
\(335\) 0 0
\(336\) 4.03932 0.220363
\(337\) −17.3161 −0.943267 −0.471634 0.881795i \(-0.656336\pi\)
−0.471634 + 0.881795i \(0.656336\pi\)
\(338\) 12.8546 0.699200
\(339\) −1.65804 −0.0900525
\(340\) 0 0
\(341\) −34.1842 −1.85118
\(342\) −0.381275 −0.0206170
\(343\) 9.35540 0.505144
\(344\) −3.61873 −0.195109
\(345\) 0 0
\(346\) −9.69736 −0.521333
\(347\) −3.47335 −0.186459 −0.0932297 0.995645i \(-0.529719\pi\)
−0.0932297 + 0.995645i \(0.529719\pi\)
\(348\) −1.00000 −0.0536056
\(349\) −13.1438 −0.703573 −0.351786 0.936080i \(-0.614426\pi\)
−0.351786 + 0.936080i \(0.614426\pi\)
\(350\) 0 0
\(351\) 0.381275 0.0203509
\(352\) 5.69736 0.303670
\(353\) −13.8546 −0.737408 −0.368704 0.929547i \(-0.620198\pi\)
−0.368704 + 0.929547i \(0.620198\pi\)
\(354\) −13.3947 −0.711921
\(355\) 0 0
\(356\) 5.65804 0.299876
\(357\) −14.7760 −0.782029
\(358\) 5.31608 0.280964
\(359\) 14.9348 0.788229 0.394115 0.919061i \(-0.371051\pi\)
0.394115 + 0.919061i \(0.371051\pi\)
\(360\) 0 0
\(361\) −18.8546 −0.992349
\(362\) −24.3688 −1.28080
\(363\) 21.4599 1.12635
\(364\) 1.54009 0.0807227
\(365\) 0 0
\(366\) −14.0786 −0.735902
\(367\) 28.9348 1.51038 0.755192 0.655503i \(-0.227543\pi\)
0.755192 + 0.655503i \(0.227543\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 5.61873 0.292499
\(370\) 0 0
\(371\) 37.6306 1.95368
\(372\) 6.00000 0.311086
\(373\) 26.4475 1.36940 0.684699 0.728826i \(-0.259934\pi\)
0.684699 + 0.728826i \(0.259934\pi\)
\(374\) −20.8412 −1.07767
\(375\) 0 0
\(376\) 3.38127 0.174376
\(377\) −0.381275 −0.0196367
\(378\) −4.03932 −0.207760
\(379\) −26.2359 −1.34765 −0.673824 0.738892i \(-0.735349\pi\)
−0.673824 + 0.738892i \(0.735349\pi\)
\(380\) 0 0
\(381\) −18.6322 −0.954555
\(382\) −10.4599 −0.535176
\(383\) −29.7242 −1.51884 −0.759419 0.650602i \(-0.774517\pi\)
−0.759419 + 0.650602i \(0.774517\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 14.0134 0.713266
\(387\) 3.61873 0.183950
\(388\) −4.07863 −0.207061
\(389\) 8.47490 0.429695 0.214847 0.976648i \(-0.431075\pi\)
0.214847 + 0.976648i \(0.431075\pi\)
\(390\) 0 0
\(391\) 3.65804 0.184995
\(392\) −9.31608 −0.470533
\(393\) −17.4734 −0.881414
\(394\) −8.47335 −0.426881
\(395\) 0 0
\(396\) −5.69736 −0.286303
\(397\) 21.8696 1.09760 0.548802 0.835952i \(-0.315084\pi\)
0.548802 + 0.835952i \(0.315084\pi\)
\(398\) −1.19813 −0.0600570
\(399\) 1.54009 0.0771010
\(400\) 0 0
\(401\) −4.71080 −0.235246 −0.117623 0.993058i \(-0.537527\pi\)
−0.117623 + 0.993058i \(0.537527\pi\)
\(402\) 2.76255 0.137783
\(403\) 2.28765 0.113956
\(404\) 16.7760 0.834637
\(405\) 0 0
\(406\) 4.03932 0.200468
\(407\) −20.8412 −1.03306
\(408\) 3.65804 0.181100
\(409\) 35.1055 1.73586 0.867928 0.496690i \(-0.165451\pi\)
0.867928 + 0.496690i \(0.165451\pi\)
\(410\) 0 0
\(411\) −14.3420 −0.707437
\(412\) 13.3554 0.657973
\(413\) 54.1055 2.66236
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −0.381275 −0.0186935
\(417\) −8.01344 −0.392420
\(418\) 2.17226 0.106249
\(419\) −24.7232 −1.20781 −0.603904 0.797057i \(-0.706389\pi\)
−0.603904 + 0.797057i \(0.706389\pi\)
\(420\) 0 0
\(421\) 17.4734 0.851599 0.425800 0.904817i \(-0.359993\pi\)
0.425800 + 0.904817i \(0.359993\pi\)
\(422\) 16.2225 0.789697
\(423\) −3.38127 −0.164403
\(424\) −9.31608 −0.452429
\(425\) 0 0
\(426\) −10.6974 −0.518289
\(427\) 56.8681 2.75204
\(428\) −6.76255 −0.326880
\(429\) −2.17226 −0.104878
\(430\) 0 0
\(431\) 4.84118 0.233192 0.116596 0.993179i \(-0.462802\pi\)
0.116596 + 0.993179i \(0.462802\pi\)
\(432\) 1.00000 0.0481125
\(433\) 16.8412 0.809336 0.404668 0.914464i \(-0.367387\pi\)
0.404668 + 0.914464i \(0.367387\pi\)
\(434\) −24.2359 −1.16336
\(435\) 0 0
\(436\) −13.0528 −0.625114
\(437\) −0.381275 −0.0182389
\(438\) 7.07863 0.338230
\(439\) −20.7108 −0.988473 −0.494236 0.869328i \(-0.664552\pi\)
−0.494236 + 0.869328i \(0.664552\pi\)
\(440\) 0 0
\(441\) 9.31608 0.443623
\(442\) 1.39472 0.0663401
\(443\) −14.8412 −0.705126 −0.352563 0.935788i \(-0.614690\pi\)
−0.352563 + 0.935788i \(0.614690\pi\)
\(444\) 3.65804 0.173603
\(445\) 0 0
\(446\) 13.6037 0.644155
\(447\) 3.23745 0.153126
\(448\) 4.03932 0.190840
\(449\) −30.1573 −1.42321 −0.711605 0.702580i \(-0.752031\pi\)
−0.711605 + 0.702580i \(0.752031\pi\)
\(450\) 0 0
\(451\) −32.0119 −1.50738
\(452\) −1.65804 −0.0779878
\(453\) −9.31608 −0.437708
\(454\) 12.4992 0.586618
\(455\) 0 0
\(456\) −0.381275 −0.0178548
\(457\) 12.7625 0.597007 0.298503 0.954409i \(-0.403513\pi\)
0.298503 + 0.954409i \(0.403513\pi\)
\(458\) −21.4734 −1.00338
\(459\) −3.65804 −0.170743
\(460\) 0 0
\(461\) 17.2359 0.802756 0.401378 0.915912i \(-0.368531\pi\)
0.401378 + 0.915912i \(0.368531\pi\)
\(462\) 23.0134 1.07068
\(463\) −12.0000 −0.557687 −0.278844 0.960337i \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) 11.1438 0.516228
\(467\) −5.88205 −0.272189 −0.136094 0.990696i \(-0.543455\pi\)
−0.136094 + 0.990696i \(0.543455\pi\)
\(468\) 0.381275 0.0176244
\(469\) −11.1588 −0.515266
\(470\) 0 0
\(471\) −6.84118 −0.315225
\(472\) −13.3947 −0.616542
\(473\) −20.6172 −0.947979
\(474\) −5.69736 −0.261688
\(475\) 0 0
\(476\) −14.7760 −0.677257
\(477\) 9.31608 0.426554
\(478\) −22.1707 −1.01406
\(479\) −28.4868 −1.30160 −0.650798 0.759251i \(-0.725565\pi\)
−0.650798 + 0.759251i \(0.725565\pi\)
\(480\) 0 0
\(481\) 1.39472 0.0635937
\(482\) 13.8696 0.631744
\(483\) −4.03932 −0.183795
\(484\) 21.4599 0.975450
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −25.6306 −1.16143 −0.580717 0.814105i \(-0.697228\pi\)
−0.580717 + 0.814105i \(0.697228\pi\)
\(488\) −14.0786 −0.637310
\(489\) 4.76255 0.215370
\(490\) 0 0
\(491\) 33.9483 1.53206 0.766032 0.642803i \(-0.222229\pi\)
0.766032 + 0.642803i \(0.222229\pi\)
\(492\) 5.61873 0.253312
\(493\) 3.65804 0.164750
\(494\) −0.145371 −0.00654053
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) 43.2100 1.93823
\(498\) −16.1180 −0.722263
\(499\) 18.0134 0.806393 0.403196 0.915114i \(-0.367899\pi\)
0.403196 + 0.915114i \(0.367899\pi\)
\(500\) 0 0
\(501\) 14.8546 0.663656
\(502\) −5.73668 −0.256040
\(503\) 14.3813 0.641229 0.320615 0.947210i \(-0.396111\pi\)
0.320615 + 0.947210i \(0.396111\pi\)
\(504\) −4.03932 −0.179925
\(505\) 0 0
\(506\) −5.69736 −0.253279
\(507\) −12.8546 −0.570894
\(508\) −18.6322 −0.826669
\(509\) 32.3797 1.43521 0.717603 0.696452i \(-0.245239\pi\)
0.717603 + 0.696452i \(0.245239\pi\)
\(510\) 0 0
\(511\) −28.5929 −1.26487
\(512\) −1.00000 −0.0441942
\(513\) 0.381275 0.0168337
\(514\) 20.6322 0.910046
\(515\) 0 0
\(516\) 3.61873 0.159305
\(517\) 19.2643 0.847245
\(518\) −14.7760 −0.649220
\(519\) 9.69736 0.425667
\(520\) 0 0
\(521\) 24.3420 1.06644 0.533220 0.845976i \(-0.320982\pi\)
0.533220 + 0.845976i \(0.320982\pi\)
\(522\) 1.00000 0.0437688
\(523\) 3.54009 0.154797 0.0773987 0.997000i \(-0.475339\pi\)
0.0773987 + 0.997000i \(0.475339\pi\)
\(524\) −17.4734 −0.763327
\(525\) 0 0
\(526\) −19.8696 −0.866357
\(527\) −21.9483 −0.956081
\(528\) −5.69736 −0.247946
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 13.3947 0.581281
\(532\) 1.54009 0.0667714
\(533\) 2.14228 0.0927924
\(534\) −5.65804 −0.244847
\(535\) 0 0
\(536\) 2.76255 0.119324
\(537\) −5.31608 −0.229406
\(538\) 8.17226 0.352331
\(539\) −53.0771 −2.28619
\(540\) 0 0
\(541\) −23.2493 −0.999568 −0.499784 0.866150i \(-0.666587\pi\)
−0.499784 + 0.866150i \(0.666587\pi\)
\(542\) 11.3947 0.489445
\(543\) 24.3688 1.04577
\(544\) 3.65804 0.156837
\(545\) 0 0
\(546\) −1.54009 −0.0659098
\(547\) 32.2225 1.37773 0.688866 0.724888i \(-0.258109\pi\)
0.688866 + 0.724888i \(0.258109\pi\)
\(548\) −14.3420 −0.612658
\(549\) 14.0786 0.600861
\(550\) 0 0
\(551\) −0.381275 −0.0162429
\(552\) 1.00000 0.0425628
\(553\) 23.0134 0.978631
\(554\) 4.93481 0.209660
\(555\) 0 0
\(556\) −8.01344 −0.339846
\(557\) −44.9985 −1.90665 −0.953323 0.301953i \(-0.902361\pi\)
−0.953323 + 0.301953i \(0.902361\pi\)
\(558\) −6.00000 −0.254000
\(559\) 1.37973 0.0583563
\(560\) 0 0
\(561\) 20.8412 0.879916
\(562\) −14.8955 −0.628328
\(563\) −13.0921 −0.551765 −0.275883 0.961191i \(-0.588970\pi\)
−0.275883 + 0.961191i \(0.588970\pi\)
\(564\) −3.38127 −0.142377
\(565\) 0 0
\(566\) 14.6322 0.615036
\(567\) 4.03932 0.169635
\(568\) −10.6974 −0.448851
\(569\) −34.0786 −1.42865 −0.714325 0.699814i \(-0.753266\pi\)
−0.714325 + 0.699814i \(0.753266\pi\)
\(570\) 0 0
\(571\) 23.4734 0.982329 0.491165 0.871067i \(-0.336571\pi\)
0.491165 + 0.871067i \(0.336571\pi\)
\(572\) −2.17226 −0.0908268
\(573\) 10.4599 0.436969
\(574\) −22.6958 −0.947305
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 24.9348 1.03805 0.519025 0.854759i \(-0.326295\pi\)
0.519025 + 0.854759i \(0.326295\pi\)
\(578\) 3.61873 0.150519
\(579\) −14.0134 −0.582379
\(580\) 0 0
\(581\) 65.1055 2.70103
\(582\) 4.07863 0.169065
\(583\) −53.0771 −2.19823
\(584\) 7.07863 0.292916
\(585\) 0 0
\(586\) 9.39472 0.388092
\(587\) −30.0786 −1.24148 −0.620739 0.784017i \(-0.713167\pi\)
−0.620739 + 0.784017i \(0.713167\pi\)
\(588\) 9.31608 0.384189
\(589\) 2.28765 0.0942610
\(590\) 0 0
\(591\) 8.47335 0.348547
\(592\) 3.65804 0.150345
\(593\) 16.4599 0.675927 0.337964 0.941159i \(-0.390262\pi\)
0.337964 + 0.941159i \(0.390262\pi\)
\(594\) 5.69736 0.233766
\(595\) 0 0
\(596\) 3.23745 0.132611
\(597\) 1.19813 0.0490363
\(598\) 0.381275 0.0155915
\(599\) −26.0269 −1.06343 −0.531715 0.846923i \(-0.678452\pi\)
−0.531715 + 0.846923i \(0.678452\pi\)
\(600\) 0 0
\(601\) −30.8546 −1.25859 −0.629293 0.777168i \(-0.716656\pi\)
−0.629293 + 0.777168i \(0.716656\pi\)
\(602\) −14.6172 −0.595752
\(603\) −2.76255 −0.112500
\(604\) −9.31608 −0.379066
\(605\) 0 0
\(606\) −16.7760 −0.681478
\(607\) 5.05020 0.204981 0.102491 0.994734i \(-0.467319\pi\)
0.102491 + 0.994734i \(0.467319\pi\)
\(608\) −0.381275 −0.0154627
\(609\) −4.03932 −0.163681
\(610\) 0 0
\(611\) −1.28920 −0.0521553
\(612\) −3.65804 −0.147868
\(613\) 48.5261 1.95995 0.979976 0.199117i \(-0.0638074\pi\)
0.979976 + 0.199117i \(0.0638074\pi\)
\(614\) −9.22401 −0.372251
\(615\) 0 0
\(616\) 23.0134 0.927238
\(617\) −20.7108 −0.833786 −0.416893 0.908956i \(-0.636881\pi\)
−0.416893 + 0.908956i \(0.636881\pi\)
\(618\) −13.3554 −0.537233
\(619\) 42.2359 1.69760 0.848802 0.528711i \(-0.177324\pi\)
0.848802 + 0.528711i \(0.177324\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −11.3295 −0.454273
\(623\) 22.8546 0.915651
\(624\) 0.381275 0.0152632
\(625\) 0 0
\(626\) −15.6037 −0.623651
\(627\) −2.17226 −0.0867517
\(628\) −6.84118 −0.272993
\(629\) −13.3813 −0.533546
\(630\) 0 0
\(631\) 25.5913 1.01877 0.509387 0.860538i \(-0.329872\pi\)
0.509387 + 0.860538i \(0.329872\pi\)
\(632\) −5.69736 −0.226629
\(633\) −16.2225 −0.644785
\(634\) 11.0000 0.436866
\(635\) 0 0
\(636\) 9.31608 0.369407
\(637\) 3.55199 0.140735
\(638\) −5.69736 −0.225561
\(639\) 10.6974 0.423181
\(640\) 0 0
\(641\) −36.6834 −1.44891 −0.724453 0.689324i \(-0.757908\pi\)
−0.724453 + 0.689324i \(0.757908\pi\)
\(642\) 6.76255 0.266897
\(643\) 13.6974 0.540171 0.270086 0.962836i \(-0.412948\pi\)
0.270086 + 0.962836i \(0.412948\pi\)
\(644\) −4.03932 −0.159171
\(645\) 0 0
\(646\) 1.39472 0.0548745
\(647\) 24.0652 0.946100 0.473050 0.881036i \(-0.343153\pi\)
0.473050 + 0.881036i \(0.343153\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −76.3145 −2.99561
\(650\) 0 0
\(651\) 24.2359 0.949880
\(652\) 4.76255 0.186516
\(653\) −14.9214 −0.583918 −0.291959 0.956431i \(-0.594307\pi\)
−0.291959 + 0.956431i \(0.594307\pi\)
\(654\) 13.0528 0.510403
\(655\) 0 0
\(656\) 5.61873 0.219374
\(657\) −7.07863 −0.276164
\(658\) 13.6580 0.532446
\(659\) 40.8288 1.59046 0.795231 0.606306i \(-0.207349\pi\)
0.795231 + 0.606306i \(0.207349\pi\)
\(660\) 0 0
\(661\) −29.1045 −1.13203 −0.566017 0.824394i \(-0.691516\pi\)
−0.566017 + 0.824394i \(0.691516\pi\)
\(662\) −2.69736 −0.104836
\(663\) −1.39472 −0.0541664
\(664\) −16.1180 −0.625498
\(665\) 0 0
\(666\) −3.65804 −0.141746
\(667\) 1.00000 0.0387202
\(668\) 14.8546 0.574743
\(669\) −13.6037 −0.525951
\(670\) 0 0
\(671\) −80.2110 −3.09651
\(672\) −4.03932 −0.155820
\(673\) −41.9198 −1.61589 −0.807945 0.589258i \(-0.799420\pi\)
−0.807945 + 0.589258i \(0.799420\pi\)
\(674\) 17.3161 0.666991
\(675\) 0 0
\(676\) −12.8546 −0.494409
\(677\) −24.3145 −0.934484 −0.467242 0.884130i \(-0.654752\pi\)
−0.467242 + 0.884130i \(0.654752\pi\)
\(678\) 1.65804 0.0636767
\(679\) −16.4749 −0.632249
\(680\) 0 0
\(681\) −12.4992 −0.478971
\(682\) 34.1842 1.30898
\(683\) 44.2628 1.69367 0.846834 0.531857i \(-0.178506\pi\)
0.846834 + 0.531857i \(0.178506\pi\)
\(684\) 0.381275 0.0145784
\(685\) 0 0
\(686\) −9.35540 −0.357191
\(687\) 21.4734 0.819260
\(688\) 3.61873 0.137963
\(689\) 3.55199 0.135320
\(690\) 0 0
\(691\) −37.1423 −1.41296 −0.706479 0.707734i \(-0.749718\pi\)
−0.706479 + 0.707734i \(0.749718\pi\)
\(692\) 9.69736 0.368638
\(693\) −23.0134 −0.874208
\(694\) 3.47335 0.131847
\(695\) 0 0
\(696\) 1.00000 0.0379049
\(697\) −20.5535 −0.778521
\(698\) 13.1438 0.497501
\(699\) −11.1438 −0.421498
\(700\) 0 0
\(701\) −0.130380 −0.00492438 −0.00246219 0.999997i \(-0.500784\pi\)
−0.00246219 + 0.999997i \(0.500784\pi\)
\(702\) −0.381275 −0.0143903
\(703\) 1.39472 0.0526029
\(704\) −5.69736 −0.214727
\(705\) 0 0
\(706\) 13.8546 0.521426
\(707\) 67.7636 2.54851
\(708\) 13.3947 0.503404
\(709\) −21.2618 −0.798503 −0.399251 0.916841i \(-0.630730\pi\)
−0.399251 + 0.916841i \(0.630730\pi\)
\(710\) 0 0
\(711\) 5.69736 0.213668
\(712\) −5.65804 −0.212044
\(713\) −6.00000 −0.224702
\(714\) 14.7760 0.552978
\(715\) 0 0
\(716\) −5.31608 −0.198671
\(717\) 22.1707 0.827981
\(718\) −14.9348 −0.557362
\(719\) −23.3026 −0.869042 −0.434521 0.900662i \(-0.643082\pi\)
−0.434521 + 0.900662i \(0.643082\pi\)
\(720\) 0 0
\(721\) 53.9467 2.00908
\(722\) 18.8546 0.701697
\(723\) −13.8696 −0.515817
\(724\) 24.3688 0.905661
\(725\) 0 0
\(726\) −21.4599 −0.796452
\(727\) 14.6053 0.541680 0.270840 0.962624i \(-0.412699\pi\)
0.270840 + 0.962624i \(0.412699\pi\)
\(728\) −1.54009 −0.0570795
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −13.2375 −0.489605
\(732\) 14.0786 0.520361
\(733\) 26.6565 0.984580 0.492290 0.870431i \(-0.336160\pi\)
0.492290 + 0.870431i \(0.336160\pi\)
\(734\) −28.9348 −1.06800
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 15.7392 0.579762
\(738\) −5.61873 −0.206828
\(739\) 49.1992 1.80982 0.904910 0.425603i \(-0.139938\pi\)
0.904910 + 0.425603i \(0.139938\pi\)
\(740\) 0 0
\(741\) 0.145371 0.00534032
\(742\) −37.6306 −1.38146
\(743\) −17.1438 −0.628946 −0.314473 0.949266i \(-0.601828\pi\)
−0.314473 + 0.949266i \(0.601828\pi\)
\(744\) −6.00000 −0.219971
\(745\) 0 0
\(746\) −26.4475 −0.968311
\(747\) 16.1180 0.589725
\(748\) 20.8412 0.762029
\(749\) −27.3161 −0.998108
\(750\) 0 0
\(751\) 42.2752 1.54264 0.771322 0.636445i \(-0.219596\pi\)
0.771322 + 0.636445i \(0.219596\pi\)
\(752\) −3.38127 −0.123302
\(753\) 5.73668 0.209056
\(754\) 0.381275 0.0138852
\(755\) 0 0
\(756\) 4.03932 0.146909
\(757\) −53.8939 −1.95881 −0.979404 0.201908i \(-0.935286\pi\)
−0.979404 + 0.201908i \(0.935286\pi\)
\(758\) 26.2359 0.952931
\(759\) 5.69736 0.206801
\(760\) 0 0
\(761\) −41.6456 −1.50965 −0.754826 0.655925i \(-0.772279\pi\)
−0.754826 + 0.655925i \(0.772279\pi\)
\(762\) 18.6322 0.674972
\(763\) −52.7242 −1.90875
\(764\) 10.4599 0.378426
\(765\) 0 0
\(766\) 29.7242 1.07398
\(767\) 5.10707 0.184406
\(768\) 1.00000 0.0360844
\(769\) −12.5266 −0.451722 −0.225861 0.974159i \(-0.572520\pi\)
−0.225861 + 0.974159i \(0.572520\pi\)
\(770\) 0 0
\(771\) −20.6322 −0.743049
\(772\) −14.0134 −0.504355
\(773\) 18.3663 0.660589 0.330295 0.943878i \(-0.392852\pi\)
0.330295 + 0.943878i \(0.392852\pi\)
\(774\) −3.61873 −0.130072
\(775\) 0 0
\(776\) 4.07863 0.146414
\(777\) 14.7760 0.530086
\(778\) −8.47490 −0.303840
\(779\) 2.14228 0.0767551
\(780\) 0 0
\(781\) −60.9467 −2.18084
\(782\) −3.65804 −0.130811
\(783\) −1.00000 −0.0357371
\(784\) 9.31608 0.332717
\(785\) 0 0
\(786\) 17.4734 0.623254
\(787\) −32.6172 −1.16268 −0.581338 0.813662i \(-0.697471\pi\)
−0.581338 + 0.813662i \(0.697471\pi\)
\(788\) 8.47335 0.301851
\(789\) 19.8696 0.707377
\(790\) 0 0
\(791\) −6.69736 −0.238131
\(792\) 5.69736 0.202447
\(793\) 5.36783 0.190617
\(794\) −21.8696 −0.776124
\(795\) 0 0
\(796\) 1.19813 0.0424667
\(797\) −21.7910 −0.771876 −0.385938 0.922525i \(-0.626122\pi\)
−0.385938 + 0.922525i \(0.626122\pi\)
\(798\) −1.54009 −0.0545186
\(799\) 12.3688 0.437578
\(800\) 0 0
\(801\) 5.65804 0.199917
\(802\) 4.71080 0.166344
\(803\) 40.3295 1.42320
\(804\) −2.76255 −0.0974276
\(805\) 0 0
\(806\) −2.28765 −0.0805790
\(807\) −8.17226 −0.287677
\(808\) −16.7760 −0.590177
\(809\) 8.17226 0.287321 0.143661 0.989627i \(-0.454113\pi\)
0.143661 + 0.989627i \(0.454113\pi\)
\(810\) 0 0
\(811\) −29.4216 −1.03313 −0.516566 0.856247i \(-0.672790\pi\)
−0.516566 + 0.856247i \(0.672790\pi\)
\(812\) −4.03932 −0.141752
\(813\) −11.3947 −0.399630
\(814\) 20.8412 0.730483
\(815\) 0 0
\(816\) −3.65804 −0.128057
\(817\) 1.37973 0.0482706
\(818\) −35.1055 −1.22744
\(819\) 1.54009 0.0538151
\(820\) 0 0
\(821\) 41.6974 1.45525 0.727624 0.685976i \(-0.240625\pi\)
0.727624 + 0.685976i \(0.240625\pi\)
\(822\) 14.3420 0.500233
\(823\) −26.2359 −0.914526 −0.457263 0.889331i \(-0.651170\pi\)
−0.457263 + 0.889331i \(0.651170\pi\)
\(824\) −13.3554 −0.465257
\(825\) 0 0
\(826\) −54.1055 −1.88257
\(827\) −41.9358 −1.45825 −0.729126 0.684380i \(-0.760073\pi\)
−0.729126 + 0.684380i \(0.760073\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −27.7760 −0.964700 −0.482350 0.875979i \(-0.660217\pi\)
−0.482350 + 0.875979i \(0.660217\pi\)
\(830\) 0 0
\(831\) −4.93481 −0.171187
\(832\) 0.381275 0.0132183
\(833\) −34.0786 −1.18075
\(834\) 8.01344 0.277483
\(835\) 0 0
\(836\) −2.17226 −0.0751292
\(837\) 6.00000 0.207390
\(838\) 24.7232 0.854050
\(839\) 31.3011 1.08063 0.540317 0.841462i \(-0.318304\pi\)
0.540317 + 0.841462i \(0.318304\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −17.4734 −0.602172
\(843\) 14.8955 0.513028
\(844\) −16.2225 −0.558400
\(845\) 0 0
\(846\) 3.38127 0.116251
\(847\) 86.6834 2.97848
\(848\) 9.31608 0.319916
\(849\) −14.6322 −0.502175
\(850\) 0 0
\(851\) −3.65804 −0.125396
\(852\) 10.6974 0.366486
\(853\) −20.6172 −0.705919 −0.352959 0.935639i \(-0.614825\pi\)
−0.352959 + 0.935639i \(0.614825\pi\)
\(854\) −56.8681 −1.94599
\(855\) 0 0
\(856\) 6.76255 0.231139
\(857\) 0.235904 0.00805834 0.00402917 0.999992i \(-0.498717\pi\)
0.00402917 + 0.999992i \(0.498717\pi\)
\(858\) 2.17226 0.0741597
\(859\) −23.7392 −0.809972 −0.404986 0.914323i \(-0.632724\pi\)
−0.404986 + 0.914323i \(0.632724\pi\)
\(860\) 0 0
\(861\) 22.6958 0.773471
\(862\) −4.84118 −0.164891
\(863\) −20.2225 −0.688381 −0.344190 0.938900i \(-0.611847\pi\)
−0.344190 + 0.938900i \(0.611847\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −16.8412 −0.572287
\(867\) −3.61873 −0.122898
\(868\) 24.2359 0.822620
\(869\) −32.4599 −1.10113
\(870\) 0 0
\(871\) −1.05329 −0.0356894
\(872\) 13.0528 0.442022
\(873\) −4.07863 −0.138041
\(874\) 0.381275 0.0128968
\(875\) 0 0
\(876\) −7.07863 −0.239165
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 20.7108 0.698956
\(879\) −9.39472 −0.316876
\(880\) 0 0
\(881\) 26.0786 0.878612 0.439306 0.898338i \(-0.355224\pi\)
0.439306 + 0.898338i \(0.355224\pi\)
\(882\) −9.31608 −0.313689
\(883\) 4.40971 0.148399 0.0741993 0.997243i \(-0.476360\pi\)
0.0741993 + 0.997243i \(0.476360\pi\)
\(884\) −1.39472 −0.0469095
\(885\) 0 0
\(886\) 14.8412 0.498599
\(887\) 22.6456 0.760365 0.380183 0.924911i \(-0.375861\pi\)
0.380183 + 0.924911i \(0.375861\pi\)
\(888\) −3.65804 −0.122756
\(889\) −75.2612 −2.52418
\(890\) 0 0
\(891\) −5.69736 −0.190869
\(892\) −13.6037 −0.455487
\(893\) −1.28920 −0.0431413
\(894\) −3.23745 −0.108277
\(895\) 0 0
\(896\) −4.03932 −0.134944
\(897\) −0.381275 −0.0127304
\(898\) 30.1573 1.00636
\(899\) −6.00000 −0.200111
\(900\) 0 0
\(901\) −34.0786 −1.13532
\(902\) 32.0119 1.06588
\(903\) 14.6172 0.486429
\(904\) 1.65804 0.0551457
\(905\) 0 0
\(906\) 9.31608 0.309506
\(907\) 16.2240 0.538709 0.269355 0.963041i \(-0.413190\pi\)
0.269355 + 0.963041i \(0.413190\pi\)
\(908\) −12.4992 −0.414801
\(909\) 16.7760 0.556425
\(910\) 0 0
\(911\) −1.77599 −0.0588413 −0.0294207 0.999567i \(-0.509366\pi\)
−0.0294207 + 0.999567i \(0.509366\pi\)
\(912\) 0.381275 0.0126253
\(913\) −91.8298 −3.03912
\(914\) −12.7625 −0.422148
\(915\) 0 0
\(916\) 21.4734 0.709500
\(917\) −70.5804 −2.33077
\(918\) 3.65804 0.120733
\(919\) −27.6580 −0.912355 −0.456177 0.889889i \(-0.650782\pi\)
−0.456177 + 0.889889i \(0.650782\pi\)
\(920\) 0 0
\(921\) 9.22401 0.303941
\(922\) −17.2359 −0.567634
\(923\) 4.07863 0.134250
\(924\) −23.0134 −0.757087
\(925\) 0 0
\(926\) 12.0000 0.394344
\(927\) 13.3554 0.438649
\(928\) 1.00000 0.0328266
\(929\) 0.777541 0.0255103 0.0127551 0.999919i \(-0.495940\pi\)
0.0127551 + 0.999919i \(0.495940\pi\)
\(930\) 0 0
\(931\) 3.55199 0.116412
\(932\) −11.1438 −0.365028
\(933\) 11.3295 0.370912
\(934\) 5.88205 0.192466
\(935\) 0 0
\(936\) −0.381275 −0.0124624
\(937\) 7.52510 0.245834 0.122917 0.992417i \(-0.460775\pi\)
0.122917 + 0.992417i \(0.460775\pi\)
\(938\) 11.1588 0.364348
\(939\) 15.6037 0.509209
\(940\) 0 0
\(941\) −40.6322 −1.32457 −0.662285 0.749252i \(-0.730413\pi\)
−0.662285 + 0.749252i \(0.730413\pi\)
\(942\) 6.84118 0.222898
\(943\) −5.61873 −0.182971
\(944\) 13.3947 0.435961
\(945\) 0 0
\(946\) 20.6172 0.670322
\(947\) 23.3947 0.760226 0.380113 0.924940i \(-0.375885\pi\)
0.380113 + 0.924940i \(0.375885\pi\)
\(948\) 5.69736 0.185042
\(949\) −2.69891 −0.0876102
\(950\) 0 0
\(951\) −11.0000 −0.356699
\(952\) 14.7760 0.478893
\(953\) −34.2902 −1.11077 −0.555384 0.831594i \(-0.687429\pi\)
−0.555384 + 0.831594i \(0.687429\pi\)
\(954\) −9.31608 −0.301619
\(955\) 0 0
\(956\) 22.1707 0.717052
\(957\) 5.69736 0.184169
\(958\) 28.4868 0.920367
\(959\) −57.9317 −1.87071
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −1.39472 −0.0449676
\(963\) −6.76255 −0.217920
\(964\) −13.8696 −0.446710
\(965\) 0 0
\(966\) 4.03932 0.129963
\(967\) −5.39472 −0.173482 −0.0867412 0.996231i \(-0.527645\pi\)
−0.0867412 + 0.996231i \(0.527645\pi\)
\(968\) −21.4599 −0.689748
\(969\) −1.39472 −0.0448049
\(970\) 0 0
\(971\) 58.3807 1.87353 0.936764 0.349963i \(-0.113806\pi\)
0.936764 + 0.349963i \(0.113806\pi\)
\(972\) 1.00000 0.0320750
\(973\) −32.3688 −1.03770
\(974\) 25.6306 0.821258
\(975\) 0 0
\(976\) 14.0786 0.450646
\(977\) −33.4190 −1.06917 −0.534585 0.845115i \(-0.679532\pi\)
−0.534585 + 0.845115i \(0.679532\pi\)
\(978\) −4.76255 −0.152290
\(979\) −32.2359 −1.03026
\(980\) 0 0
\(981\) −13.0528 −0.416743
\(982\) −33.9483 −1.08333
\(983\) −42.4868 −1.35512 −0.677559 0.735468i \(-0.736962\pi\)
−0.677559 + 0.735468i \(0.736962\pi\)
\(984\) −5.61873 −0.179118
\(985\) 0 0
\(986\) −3.65804 −0.116496
\(987\) −13.6580 −0.434740
\(988\) 0.145371 0.00462485
\(989\) −3.61873 −0.115069
\(990\) 0 0
\(991\) 16.7625 0.532480 0.266240 0.963907i \(-0.414219\pi\)
0.266240 + 0.963907i \(0.414219\pi\)
\(992\) −6.00000 −0.190500
\(993\) 2.69736 0.0855981
\(994\) −43.2100 −1.37054
\(995\) 0 0
\(996\) 16.1180 0.510717
\(997\) 15.7760 0.499631 0.249815 0.968293i \(-0.419630\pi\)
0.249815 + 0.968293i \(0.419630\pi\)
\(998\) −18.0134 −0.570206
\(999\) 3.65804 0.115735
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 3450.2.a.bp.1.3 3
5.2 odd 4 3450.2.d.ba.2899.3 6
5.3 odd 4 3450.2.d.ba.2899.4 6
5.4 even 2 3450.2.a.bs.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.bp.1.3 3 1.1 even 1 trivial
3450.2.a.bs.1.1 yes 3 5.4 even 2
3450.2.d.ba.2899.3 6 5.2 odd 4
3450.2.d.ba.2899.4 6 5.3 odd 4