Properties

Label 3450.2.a.bt.1.1
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: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.96239 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.96239 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.35026 q^{11} -1.00000 q^{12} +4.96239 q^{13} -2.96239 q^{14} +1.00000 q^{16} -1.35026 q^{17} +1.00000 q^{18} -4.96239 q^{19} +2.96239 q^{21} -3.35026 q^{22} +1.00000 q^{23} -1.00000 q^{24} +4.96239 q^{26} -1.00000 q^{27} -2.96239 q^{28} +7.73813 q^{29} -4.00000 q^{31} +1.00000 q^{32} +3.35026 q^{33} -1.35026 q^{34} +1.00000 q^{36} +7.61213 q^{37} -4.96239 q^{38} -4.96239 q^{39} +4.70052 q^{41} +2.96239 q^{42} +10.3127 q^{43} -3.35026 q^{44} +1.00000 q^{46} -3.22425 q^{47} -1.00000 q^{48} +1.77575 q^{49} +1.35026 q^{51} +4.96239 q^{52} +6.96239 q^{53} -1.00000 q^{54} -2.96239 q^{56} +4.96239 q^{57} +7.73813 q^{58} +1.22425 q^{59} -11.0884 q^{61} -4.00000 q^{62} -2.96239 q^{63} +1.00000 q^{64} +3.35026 q^{66} +7.61213 q^{67} -1.35026 q^{68} -1.00000 q^{69} -2.18664 q^{71} +1.00000 q^{72} +9.92478 q^{73} +7.61213 q^{74} -4.96239 q^{76} +9.92478 q^{77} -4.96239 q^{78} -4.12601 q^{79} +1.00000 q^{81} +4.70052 q^{82} +6.38787 q^{83} +2.96239 q^{84} +10.3127 q^{86} -7.73813 q^{87} -3.35026 q^{88} +9.92478 q^{89} -14.7005 q^{91} +1.00000 q^{92} +4.00000 q^{93} -3.22425 q^{94} -1.00000 q^{96} -12.8872 q^{97} +1.77575 q^{98} -3.35026 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 2q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 2q^{7} + 3q^{8} + 3q^{9} - 3q^{12} + 4q^{13} + 2q^{14} + 3q^{16} + 6q^{17} + 3q^{18} - 4q^{19} - 2q^{21} + 3q^{23} - 3q^{24} + 4q^{26} - 3q^{27} + 2q^{28} + 14q^{29} - 12q^{31} + 3q^{32} + 6q^{34} + 3q^{36} + 22q^{37} - 4q^{38} - 4q^{39} - 6q^{41} - 2q^{42} + 10q^{43} + 3q^{46} - 8q^{47} - 3q^{48} + 7q^{49} - 6q^{51} + 4q^{52} + 10q^{53} - 3q^{54} + 2q^{56} + 4q^{57} + 14q^{58} + 2q^{59} - 14q^{61} - 12q^{62} + 2q^{63} + 3q^{64} + 22q^{67} + 6q^{68} - 3q^{69} + 6q^{71} + 3q^{72} + 8q^{73} + 22q^{74} - 4q^{76} + 8q^{77} - 4q^{78} - 4q^{79} + 3q^{81} - 6q^{82} + 20q^{83} - 2q^{84} + 10q^{86} - 14q^{87} + 8q^{89} - 24q^{91} + 3q^{92} + 12q^{93} - 8q^{94} - 3q^{96} - 6q^{97} + 7q^{98} + O(q^{100}) \)

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
\(7\) −2.96239 −1.11968 −0.559839 0.828602i \(-0.689137\pi\)
−0.559839 + 0.828602i \(0.689137\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.35026 −1.01014 −0.505071 0.863078i \(-0.668534\pi\)
−0.505071 + 0.863078i \(0.668534\pi\)
\(12\) −1.00000 −0.288675
\(13\) 4.96239 1.37632 0.688159 0.725559i \(-0.258419\pi\)
0.688159 + 0.725559i \(0.258419\pi\)
\(14\) −2.96239 −0.791732
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.35026 −0.327487 −0.163743 0.986503i \(-0.552357\pi\)
−0.163743 + 0.986503i \(0.552357\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.96239 −1.13845 −0.569225 0.822182i \(-0.692757\pi\)
−0.569225 + 0.822182i \(0.692757\pi\)
\(20\) 0 0
\(21\) 2.96239 0.646446
\(22\) −3.35026 −0.714278
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 4.96239 0.973204
\(27\) −1.00000 −0.192450
\(28\) −2.96239 −0.559839
\(29\) 7.73813 1.43694 0.718468 0.695560i \(-0.244844\pi\)
0.718468 + 0.695560i \(0.244844\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.35026 0.583206
\(34\) −1.35026 −0.231568
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 7.61213 1.25143 0.625713 0.780053i \(-0.284808\pi\)
0.625713 + 0.780053i \(0.284808\pi\)
\(38\) −4.96239 −0.805006
\(39\) −4.96239 −0.794618
\(40\) 0 0
\(41\) 4.70052 0.734098 0.367049 0.930202i \(-0.380368\pi\)
0.367049 + 0.930202i \(0.380368\pi\)
\(42\) 2.96239 0.457106
\(43\) 10.3127 1.57266 0.786332 0.617804i \(-0.211977\pi\)
0.786332 + 0.617804i \(0.211977\pi\)
\(44\) −3.35026 −0.505071
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −3.22425 −0.470306 −0.235153 0.971958i \(-0.575559\pi\)
−0.235153 + 0.971958i \(0.575559\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.77575 0.253678
\(50\) 0 0
\(51\) 1.35026 0.189074
\(52\) 4.96239 0.688159
\(53\) 6.96239 0.956358 0.478179 0.878263i \(-0.341297\pi\)
0.478179 + 0.878263i \(0.341297\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −2.96239 −0.395866
\(57\) 4.96239 0.657284
\(58\) 7.73813 1.01607
\(59\) 1.22425 0.159384 0.0796921 0.996820i \(-0.474606\pi\)
0.0796921 + 0.996820i \(0.474606\pi\)
\(60\) 0 0
\(61\) −11.0884 −1.41972 −0.709862 0.704341i \(-0.751243\pi\)
−0.709862 + 0.704341i \(0.751243\pi\)
\(62\) −4.00000 −0.508001
\(63\) −2.96239 −0.373226
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.35026 0.412389
\(67\) 7.61213 0.929969 0.464985 0.885319i \(-0.346060\pi\)
0.464985 + 0.885319i \(0.346060\pi\)
\(68\) −1.35026 −0.163743
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −2.18664 −0.259507 −0.129753 0.991546i \(-0.541419\pi\)
−0.129753 + 0.991546i \(0.541419\pi\)
\(72\) 1.00000 0.117851
\(73\) 9.92478 1.16161 0.580804 0.814044i \(-0.302738\pi\)
0.580804 + 0.814044i \(0.302738\pi\)
\(74\) 7.61213 0.884892
\(75\) 0 0
\(76\) −4.96239 −0.569225
\(77\) 9.92478 1.13103
\(78\) −4.96239 −0.561880
\(79\) −4.12601 −0.464212 −0.232106 0.972690i \(-0.574562\pi\)
−0.232106 + 0.972690i \(0.574562\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 4.70052 0.519086
\(83\) 6.38787 0.701160 0.350580 0.936533i \(-0.385984\pi\)
0.350580 + 0.936533i \(0.385984\pi\)
\(84\) 2.96239 0.323223
\(85\) 0 0
\(86\) 10.3127 1.11204
\(87\) −7.73813 −0.829615
\(88\) −3.35026 −0.357139
\(89\) 9.92478 1.05202 0.526012 0.850477i \(-0.323687\pi\)
0.526012 + 0.850477i \(0.323687\pi\)
\(90\) 0 0
\(91\) −14.7005 −1.54103
\(92\) 1.00000 0.104257
\(93\) 4.00000 0.414781
\(94\) −3.22425 −0.332556
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −12.8872 −1.30849 −0.654247 0.756281i \(-0.727014\pi\)
−0.654247 + 0.756281i \(0.727014\pi\)
\(98\) 1.77575 0.179377
\(99\) −3.35026 −0.336714
\(100\) 0 0
\(101\) 4.26187 0.424071 0.212036 0.977262i \(-0.431991\pi\)
0.212036 + 0.977262i \(0.431991\pi\)
\(102\) 1.35026 0.133696
\(103\) 0.261865 0.0258023 0.0129012 0.999917i \(-0.495893\pi\)
0.0129012 + 0.999917i \(0.495893\pi\)
\(104\) 4.96239 0.486602
\(105\) 0 0
\(106\) 6.96239 0.676247
\(107\) 9.08840 0.878608 0.439304 0.898338i \(-0.355225\pi\)
0.439304 + 0.898338i \(0.355225\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 18.9380 1.81393 0.906963 0.421210i \(-0.138394\pi\)
0.906963 + 0.421210i \(0.138394\pi\)
\(110\) 0 0
\(111\) −7.61213 −0.722511
\(112\) −2.96239 −0.279919
\(113\) 6.64974 0.625555 0.312777 0.949826i \(-0.398741\pi\)
0.312777 + 0.949826i \(0.398741\pi\)
\(114\) 4.96239 0.464770
\(115\) 0 0
\(116\) 7.73813 0.718468
\(117\) 4.96239 0.458773
\(118\) 1.22425 0.112702
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 0.224254 0.0203867
\(122\) −11.0884 −1.00390
\(123\) −4.70052 −0.423832
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) −2.96239 −0.263911
\(127\) −0.186642 −0.0165618 −0.00828091 0.999966i \(-0.502636\pi\)
−0.00828091 + 0.999966i \(0.502636\pi\)
\(128\) 1.00000 0.0883883
\(129\) −10.3127 −0.907978
\(130\) 0 0
\(131\) 7.92478 0.692391 0.346196 0.938162i \(-0.387473\pi\)
0.346196 + 0.938162i \(0.387473\pi\)
\(132\) 3.35026 0.291603
\(133\) 14.7005 1.27470
\(134\) 7.61213 0.657588
\(135\) 0 0
\(136\) −1.35026 −0.115784
\(137\) −9.19982 −0.785993 −0.392997 0.919540i \(-0.628562\pi\)
−0.392997 + 0.919540i \(0.628562\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 8.62530 0.731588 0.365794 0.930696i \(-0.380797\pi\)
0.365794 + 0.930696i \(0.380797\pi\)
\(140\) 0 0
\(141\) 3.22425 0.271531
\(142\) −2.18664 −0.183499
\(143\) −16.6253 −1.39028
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 9.92478 0.821380
\(147\) −1.77575 −0.146461
\(148\) 7.61213 0.625713
\(149\) 4.64974 0.380921 0.190461 0.981695i \(-0.439002\pi\)
0.190461 + 0.981695i \(0.439002\pi\)
\(150\) 0 0
\(151\) 10.7005 0.870796 0.435398 0.900238i \(-0.356608\pi\)
0.435398 + 0.900238i \(0.356608\pi\)
\(152\) −4.96239 −0.402503
\(153\) −1.35026 −0.109162
\(154\) 9.92478 0.799761
\(155\) 0 0
\(156\) −4.96239 −0.397309
\(157\) 17.0132 1.35780 0.678900 0.734231i \(-0.262457\pi\)
0.678900 + 0.734231i \(0.262457\pi\)
\(158\) −4.12601 −0.328248
\(159\) −6.96239 −0.552153
\(160\) 0 0
\(161\) −2.96239 −0.233469
\(162\) 1.00000 0.0785674
\(163\) −12.6253 −0.988890 −0.494445 0.869209i \(-0.664629\pi\)
−0.494445 + 0.869209i \(0.664629\pi\)
\(164\) 4.70052 0.367049
\(165\) 0 0
\(166\) 6.38787 0.495795
\(167\) 24.6253 1.90556 0.952781 0.303657i \(-0.0982076\pi\)
0.952781 + 0.303657i \(0.0982076\pi\)
\(168\) 2.96239 0.228553
\(169\) 11.6253 0.894254
\(170\) 0 0
\(171\) −4.96239 −0.379483
\(172\) 10.3127 0.786332
\(173\) 4.44851 0.338214 0.169107 0.985598i \(-0.445912\pi\)
0.169107 + 0.985598i \(0.445912\pi\)
\(174\) −7.73813 −0.586626
\(175\) 0 0
\(176\) −3.35026 −0.252535
\(177\) −1.22425 −0.0920205
\(178\) 9.92478 0.743894
\(179\) −13.8496 −1.03516 −0.517582 0.855634i \(-0.673168\pi\)
−0.517582 + 0.855634i \(0.673168\pi\)
\(180\) 0 0
\(181\) −22.6859 −1.68623 −0.843116 0.537732i \(-0.819281\pi\)
−0.843116 + 0.537732i \(0.819281\pi\)
\(182\) −14.7005 −1.08968
\(183\) 11.0884 0.819678
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) 4.52373 0.330808
\(188\) −3.22425 −0.235153
\(189\) 2.96239 0.215482
\(190\) 0 0
\(191\) −19.3258 −1.39837 −0.699184 0.714942i \(-0.746453\pi\)
−0.699184 + 0.714942i \(0.746453\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 5.92478 0.426475 0.213237 0.977000i \(-0.431599\pi\)
0.213237 + 0.977000i \(0.431599\pi\)
\(194\) −12.8872 −0.925245
\(195\) 0 0
\(196\) 1.77575 0.126839
\(197\) −8.07522 −0.575336 −0.287668 0.957730i \(-0.592880\pi\)
−0.287668 + 0.957730i \(0.592880\pi\)
\(198\) −3.35026 −0.238093
\(199\) 15.9756 1.13248 0.566239 0.824241i \(-0.308398\pi\)
0.566239 + 0.824241i \(0.308398\pi\)
\(200\) 0 0
\(201\) −7.61213 −0.536918
\(202\) 4.26187 0.299864
\(203\) −22.9234 −1.60890
\(204\) 1.35026 0.0945372
\(205\) 0 0
\(206\) 0.261865 0.0182450
\(207\) 1.00000 0.0695048
\(208\) 4.96239 0.344080
\(209\) 16.6253 1.15000
\(210\) 0 0
\(211\) 21.7743 1.49901 0.749503 0.662000i \(-0.230292\pi\)
0.749503 + 0.662000i \(0.230292\pi\)
\(212\) 6.96239 0.478179
\(213\) 2.18664 0.149826
\(214\) 9.08840 0.621270
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 11.8496 0.804400
\(218\) 18.9380 1.28264
\(219\) −9.92478 −0.670654
\(220\) 0 0
\(221\) −6.70052 −0.450726
\(222\) −7.61213 −0.510893
\(223\) 3.81336 0.255361 0.127681 0.991815i \(-0.459247\pi\)
0.127681 + 0.991815i \(0.459247\pi\)
\(224\) −2.96239 −0.197933
\(225\) 0 0
\(226\) 6.64974 0.442334
\(227\) 16.9380 1.12421 0.562106 0.827065i \(-0.309991\pi\)
0.562106 + 0.827065i \(0.309991\pi\)
\(228\) 4.96239 0.328642
\(229\) −26.9380 −1.78011 −0.890055 0.455853i \(-0.849334\pi\)
−0.890055 + 0.455853i \(0.849334\pi\)
\(230\) 0 0
\(231\) −9.92478 −0.653002
\(232\) 7.73813 0.508033
\(233\) 27.4010 1.79510 0.897551 0.440910i \(-0.145344\pi\)
0.897551 + 0.440910i \(0.145344\pi\)
\(234\) 4.96239 0.324401
\(235\) 0 0
\(236\) 1.22425 0.0796921
\(237\) 4.12601 0.268013
\(238\) 4.00000 0.259281
\(239\) 7.48612 0.484237 0.242118 0.970247i \(-0.422158\pi\)
0.242118 + 0.970247i \(0.422158\pi\)
\(240\) 0 0
\(241\) −13.0738 −0.842158 −0.421079 0.907024i \(-0.638348\pi\)
−0.421079 + 0.907024i \(0.638348\pi\)
\(242\) 0.224254 0.0144156
\(243\) −1.00000 −0.0641500
\(244\) −11.0884 −0.709862
\(245\) 0 0
\(246\) −4.70052 −0.299694
\(247\) −24.6253 −1.56687
\(248\) −4.00000 −0.254000
\(249\) −6.38787 −0.404815
\(250\) 0 0
\(251\) −5.02302 −0.317050 −0.158525 0.987355i \(-0.550674\pi\)
−0.158525 + 0.987355i \(0.550674\pi\)
\(252\) −2.96239 −0.186613
\(253\) −3.35026 −0.210629
\(254\) −0.186642 −0.0117110
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −19.4010 −1.21020 −0.605102 0.796148i \(-0.706868\pi\)
−0.605102 + 0.796148i \(0.706868\pi\)
\(258\) −10.3127 −0.642038
\(259\) −22.5501 −1.40119
\(260\) 0 0
\(261\) 7.73813 0.478979
\(262\) 7.92478 0.489594
\(263\) −15.4763 −0.954308 −0.477154 0.878820i \(-0.658332\pi\)
−0.477154 + 0.878820i \(0.658332\pi\)
\(264\) 3.35026 0.206194
\(265\) 0 0
\(266\) 14.7005 0.901347
\(267\) −9.92478 −0.607387
\(268\) 7.61213 0.464985
\(269\) −20.2130 −1.23241 −0.616204 0.787587i \(-0.711330\pi\)
−0.616204 + 0.787587i \(0.711330\pi\)
\(270\) 0 0
\(271\) 30.3996 1.84665 0.923323 0.384024i \(-0.125462\pi\)
0.923323 + 0.384024i \(0.125462\pi\)
\(272\) −1.35026 −0.0818716
\(273\) 14.7005 0.889716
\(274\) −9.19982 −0.555781
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −20.8119 −1.25047 −0.625234 0.780437i \(-0.714997\pi\)
−0.625234 + 0.780437i \(0.714997\pi\)
\(278\) 8.62530 0.517311
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 19.8496 1.18413 0.592063 0.805892i \(-0.298314\pi\)
0.592063 + 0.805892i \(0.298314\pi\)
\(282\) 3.22425 0.192002
\(283\) −12.3879 −0.736383 −0.368191 0.929750i \(-0.620023\pi\)
−0.368191 + 0.929750i \(0.620023\pi\)
\(284\) −2.18664 −0.129753
\(285\) 0 0
\(286\) −16.6253 −0.983075
\(287\) −13.9248 −0.821954
\(288\) 1.00000 0.0589256
\(289\) −15.1768 −0.892753
\(290\) 0 0
\(291\) 12.8872 0.755459
\(292\) 9.92478 0.580804
\(293\) −9.03761 −0.527983 −0.263991 0.964525i \(-0.585039\pi\)
−0.263991 + 0.964525i \(0.585039\pi\)
\(294\) −1.77575 −0.103564
\(295\) 0 0
\(296\) 7.61213 0.442446
\(297\) 3.35026 0.194402
\(298\) 4.64974 0.269352
\(299\) 4.96239 0.286982
\(300\) 0 0
\(301\) −30.5501 −1.76088
\(302\) 10.7005 0.615746
\(303\) −4.26187 −0.244838
\(304\) −4.96239 −0.284613
\(305\) 0 0
\(306\) −1.35026 −0.0771893
\(307\) 30.5501 1.74359 0.871793 0.489875i \(-0.162958\pi\)
0.871793 + 0.489875i \(0.162958\pi\)
\(308\) 9.92478 0.565517
\(309\) −0.261865 −0.0148970
\(310\) 0 0
\(311\) 14.4387 0.818741 0.409371 0.912368i \(-0.365748\pi\)
0.409371 + 0.912368i \(0.365748\pi\)
\(312\) −4.96239 −0.280940
\(313\) −27.9610 −1.58045 −0.790224 0.612818i \(-0.790036\pi\)
−0.790224 + 0.612818i \(0.790036\pi\)
\(314\) 17.0132 0.960109
\(315\) 0 0
\(316\) −4.12601 −0.232106
\(317\) −1.47627 −0.0829156 −0.0414578 0.999140i \(-0.513200\pi\)
−0.0414578 + 0.999140i \(0.513200\pi\)
\(318\) −6.96239 −0.390431
\(319\) −25.9248 −1.45151
\(320\) 0 0
\(321\) −9.08840 −0.507265
\(322\) −2.96239 −0.165087
\(323\) 6.70052 0.372827
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −12.6253 −0.699251
\(327\) −18.9380 −1.04727
\(328\) 4.70052 0.259543
\(329\) 9.55149 0.526591
\(330\) 0 0
\(331\) 23.1754 1.27383 0.636917 0.770932i \(-0.280209\pi\)
0.636917 + 0.770932i \(0.280209\pi\)
\(332\) 6.38787 0.350580
\(333\) 7.61213 0.417142
\(334\) 24.6253 1.34744
\(335\) 0 0
\(336\) 2.96239 0.161612
\(337\) 21.0376 1.14599 0.572996 0.819558i \(-0.305781\pi\)
0.572996 + 0.819558i \(0.305781\pi\)
\(338\) 11.6253 0.632333
\(339\) −6.64974 −0.361164
\(340\) 0 0
\(341\) 13.4010 0.725707
\(342\) −4.96239 −0.268335
\(343\) 15.4763 0.835640
\(344\) 10.3127 0.556021
\(345\) 0 0
\(346\) 4.44851 0.239153
\(347\) −3.37470 −0.181163 −0.0905817 0.995889i \(-0.528873\pi\)
−0.0905817 + 0.995889i \(0.528873\pi\)
\(348\) −7.73813 −0.414808
\(349\) −4.44851 −0.238123 −0.119062 0.992887i \(-0.537989\pi\)
−0.119062 + 0.992887i \(0.537989\pi\)
\(350\) 0 0
\(351\) −4.96239 −0.264873
\(352\) −3.35026 −0.178570
\(353\) 35.4010 1.88421 0.942104 0.335321i \(-0.108845\pi\)
0.942104 + 0.335321i \(0.108845\pi\)
\(354\) −1.22425 −0.0650684
\(355\) 0 0
\(356\) 9.92478 0.526012
\(357\) −4.00000 −0.211702
\(358\) −13.8496 −0.731972
\(359\) −34.5501 −1.82348 −0.911742 0.410764i \(-0.865262\pi\)
−0.911742 + 0.410764i \(0.865262\pi\)
\(360\) 0 0
\(361\) 5.62530 0.296068
\(362\) −22.6859 −1.19235
\(363\) −0.224254 −0.0117703
\(364\) −14.7005 −0.770517
\(365\) 0 0
\(366\) 11.0884 0.579600
\(367\) −22.8119 −1.19077 −0.595387 0.803439i \(-0.703001\pi\)
−0.595387 + 0.803439i \(0.703001\pi\)
\(368\) 1.00000 0.0521286
\(369\) 4.70052 0.244699
\(370\) 0 0
\(371\) −20.6253 −1.07081
\(372\) 4.00000 0.207390
\(373\) −10.8364 −0.561087 −0.280543 0.959841i \(-0.590515\pi\)
−0.280543 + 0.959841i \(0.590515\pi\)
\(374\) 4.52373 0.233917
\(375\) 0 0
\(376\) −3.22425 −0.166278
\(377\) 38.3996 1.97768
\(378\) 2.96239 0.152369
\(379\) −32.4387 −1.66626 −0.833131 0.553076i \(-0.813454\pi\)
−0.833131 + 0.553076i \(0.813454\pi\)
\(380\) 0 0
\(381\) 0.186642 0.00956198
\(382\) −19.3258 −0.988795
\(383\) 32.9986 1.68615 0.843074 0.537797i \(-0.180743\pi\)
0.843074 + 0.537797i \(0.180743\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 5.92478 0.301563
\(387\) 10.3127 0.524221
\(388\) −12.8872 −0.654247
\(389\) 29.1246 1.47668 0.738338 0.674431i \(-0.235611\pi\)
0.738338 + 0.674431i \(0.235611\pi\)
\(390\) 0 0
\(391\) −1.35026 −0.0682857
\(392\) 1.77575 0.0896887
\(393\) −7.92478 −0.399752
\(394\) −8.07522 −0.406824
\(395\) 0 0
\(396\) −3.35026 −0.168357
\(397\) −13.7381 −0.689497 −0.344749 0.938695i \(-0.612036\pi\)
−0.344749 + 0.938695i \(0.612036\pi\)
\(398\) 15.9756 0.800783
\(399\) −14.7005 −0.735947
\(400\) 0 0
\(401\) −1.44992 −0.0724057 −0.0362028 0.999344i \(-0.511526\pi\)
−0.0362028 + 0.999344i \(0.511526\pi\)
\(402\) −7.61213 −0.379658
\(403\) −19.8496 −0.988777
\(404\) 4.26187 0.212036
\(405\) 0 0
\(406\) −22.9234 −1.13767
\(407\) −25.5026 −1.26412
\(408\) 1.35026 0.0668479
\(409\) −17.8496 −0.882604 −0.441302 0.897359i \(-0.645483\pi\)
−0.441302 + 0.897359i \(0.645483\pi\)
\(410\) 0 0
\(411\) 9.19982 0.453793
\(412\) 0.261865 0.0129012
\(413\) −3.62672 −0.178459
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 4.96239 0.243301
\(417\) −8.62530 −0.422383
\(418\) 16.6253 0.813170
\(419\) 17.9003 0.874489 0.437244 0.899343i \(-0.355954\pi\)
0.437244 + 0.899343i \(0.355954\pi\)
\(420\) 0 0
\(421\) 7.61213 0.370992 0.185496 0.982645i \(-0.440611\pi\)
0.185496 + 0.982645i \(0.440611\pi\)
\(422\) 21.7743 1.05996
\(423\) −3.22425 −0.156769
\(424\) 6.96239 0.338123
\(425\) 0 0
\(426\) 2.18664 0.105943
\(427\) 32.8481 1.58963
\(428\) 9.08840 0.439304
\(429\) 16.6253 0.802677
\(430\) 0 0
\(431\) −22.9525 −1.10558 −0.552792 0.833319i \(-0.686438\pi\)
−0.552792 + 0.833319i \(0.686438\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −33.7645 −1.62262 −0.811309 0.584618i \(-0.801244\pi\)
−0.811309 + 0.584618i \(0.801244\pi\)
\(434\) 11.8496 0.568797
\(435\) 0 0
\(436\) 18.9380 0.906963
\(437\) −4.96239 −0.237383
\(438\) −9.92478 −0.474224
\(439\) −23.4763 −1.12046 −0.560231 0.828337i \(-0.689287\pi\)
−0.560231 + 0.828337i \(0.689287\pi\)
\(440\) 0 0
\(441\) 1.77575 0.0845593
\(442\) −6.70052 −0.318711
\(443\) 30.5501 1.45148 0.725739 0.687970i \(-0.241498\pi\)
0.725739 + 0.687970i \(0.241498\pi\)
\(444\) −7.61213 −0.361256
\(445\) 0 0
\(446\) 3.81336 0.180568
\(447\) −4.64974 −0.219925
\(448\) −2.96239 −0.139960
\(449\) 32.7005 1.54323 0.771617 0.636088i \(-0.219448\pi\)
0.771617 + 0.636088i \(0.219448\pi\)
\(450\) 0 0
\(451\) −15.7480 −0.741544
\(452\) 6.64974 0.312777
\(453\) −10.7005 −0.502754
\(454\) 16.9380 0.794937
\(455\) 0 0
\(456\) 4.96239 0.232385
\(457\) −5.81336 −0.271937 −0.135969 0.990713i \(-0.543415\pi\)
−0.135969 + 0.990713i \(0.543415\pi\)
\(458\) −26.9380 −1.25873
\(459\) 1.35026 0.0630248
\(460\) 0 0
\(461\) −23.9902 −1.11733 −0.558666 0.829392i \(-0.688687\pi\)
−0.558666 + 0.829392i \(0.688687\pi\)
\(462\) −9.92478 −0.461742
\(463\) 35.3620 1.64341 0.821706 0.569911i \(-0.193022\pi\)
0.821706 + 0.569911i \(0.193022\pi\)
\(464\) 7.73813 0.359234
\(465\) 0 0
\(466\) 27.4010 1.26933
\(467\) 1.98541 0.0918739 0.0459369 0.998944i \(-0.485373\pi\)
0.0459369 + 0.998944i \(0.485373\pi\)
\(468\) 4.96239 0.229386
\(469\) −22.5501 −1.04127
\(470\) 0 0
\(471\) −17.0132 −0.783926
\(472\) 1.22425 0.0563508
\(473\) −34.5501 −1.58861
\(474\) 4.12601 0.189514
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 6.96239 0.318786
\(478\) 7.48612 0.342407
\(479\) 11.0738 0.505975 0.252988 0.967470i \(-0.418587\pi\)
0.252988 + 0.967470i \(0.418587\pi\)
\(480\) 0 0
\(481\) 37.7743 1.72236
\(482\) −13.0738 −0.595496
\(483\) 2.96239 0.134793
\(484\) 0.224254 0.0101934
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −17.4372 −0.790157 −0.395078 0.918647i \(-0.629283\pi\)
−0.395078 + 0.918647i \(0.629283\pi\)
\(488\) −11.0884 −0.501948
\(489\) 12.6253 0.570936
\(490\) 0 0
\(491\) 14.8773 0.671404 0.335702 0.941968i \(-0.391027\pi\)
0.335702 + 0.941968i \(0.391027\pi\)
\(492\) −4.70052 −0.211916
\(493\) −10.4485 −0.470577
\(494\) −24.6253 −1.10794
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 6.47768 0.290564
\(498\) −6.38787 −0.286247
\(499\) −16.1016 −0.720805 −0.360403 0.932797i \(-0.617361\pi\)
−0.360403 + 0.932797i \(0.617361\pi\)
\(500\) 0 0
\(501\) −24.6253 −1.10018
\(502\) −5.02302 −0.224188
\(503\) −38.8021 −1.73010 −0.865050 0.501686i \(-0.832713\pi\)
−0.865050 + 0.501686i \(0.832713\pi\)
\(504\) −2.96239 −0.131955
\(505\) 0 0
\(506\) −3.35026 −0.148937
\(507\) −11.6253 −0.516298
\(508\) −0.186642 −0.00828091
\(509\) 23.4372 1.03884 0.519419 0.854520i \(-0.326148\pi\)
0.519419 + 0.854520i \(0.326148\pi\)
\(510\) 0 0
\(511\) −29.4010 −1.30063
\(512\) 1.00000 0.0441942
\(513\) 4.96239 0.219095
\(514\) −19.4010 −0.855743
\(515\) 0 0
\(516\) −10.3127 −0.453989
\(517\) 10.8021 0.475076
\(518\) −22.5501 −0.990794
\(519\) −4.44851 −0.195268
\(520\) 0 0
\(521\) 21.1490 0.926556 0.463278 0.886213i \(-0.346673\pi\)
0.463278 + 0.886213i \(0.346673\pi\)
\(522\) 7.73813 0.338689
\(523\) −11.7626 −0.514341 −0.257171 0.966366i \(-0.582790\pi\)
−0.257171 + 0.966366i \(0.582790\pi\)
\(524\) 7.92478 0.346196
\(525\) 0 0
\(526\) −15.4763 −0.674797
\(527\) 5.40105 0.235273
\(528\) 3.35026 0.145801
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 1.22425 0.0531281
\(532\) 14.7005 0.637349
\(533\) 23.3258 1.01035
\(534\) −9.92478 −0.429487
\(535\) 0 0
\(536\) 7.61213 0.328794
\(537\) 13.8496 0.597652
\(538\) −20.2130 −0.871444
\(539\) −5.94921 −0.256251
\(540\) 0 0
\(541\) −21.1754 −0.910401 −0.455200 0.890389i \(-0.650432\pi\)
−0.455200 + 0.890389i \(0.650432\pi\)
\(542\) 30.3996 1.30578
\(543\) 22.6859 0.973547
\(544\) −1.35026 −0.0578920
\(545\) 0 0
\(546\) 14.7005 0.629124
\(547\) −5.29948 −0.226589 −0.113295 0.993561i \(-0.536140\pi\)
−0.113295 + 0.993561i \(0.536140\pi\)
\(548\) −9.19982 −0.392997
\(549\) −11.0884 −0.473241
\(550\) 0 0
\(551\) −38.3996 −1.63588
\(552\) −1.00000 −0.0425628
\(553\) 12.2228 0.519768
\(554\) −20.8119 −0.884215
\(555\) 0 0
\(556\) 8.62530 0.365794
\(557\) 7.99015 0.338554 0.169277 0.985569i \(-0.445857\pi\)
0.169277 + 0.985569i \(0.445857\pi\)
\(558\) −4.00000 −0.169334
\(559\) 51.1754 2.16449
\(560\) 0 0
\(561\) −4.52373 −0.190992
\(562\) 19.8496 0.837303
\(563\) −15.1636 −0.639070 −0.319535 0.947574i \(-0.603527\pi\)
−0.319535 + 0.947574i \(0.603527\pi\)
\(564\) 3.22425 0.135766
\(565\) 0 0
\(566\) −12.3879 −0.520701
\(567\) −2.96239 −0.124409
\(568\) −2.18664 −0.0917495
\(569\) −25.5223 −1.06995 −0.534976 0.844868i \(-0.679679\pi\)
−0.534976 + 0.844868i \(0.679679\pi\)
\(570\) 0 0
\(571\) 6.76590 0.283144 0.141572 0.989928i \(-0.454784\pi\)
0.141572 + 0.989928i \(0.454784\pi\)
\(572\) −16.6253 −0.695139
\(573\) 19.3258 0.807348
\(574\) −13.9248 −0.581209
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) −15.1768 −0.631271
\(579\) −5.92478 −0.246225
\(580\) 0 0
\(581\) −18.9234 −0.785073
\(582\) 12.8872 0.534190
\(583\) −23.3258 −0.966057
\(584\) 9.92478 0.410690
\(585\) 0 0
\(586\) −9.03761 −0.373340
\(587\) 12.8773 0.531504 0.265752 0.964041i \(-0.414380\pi\)
0.265752 + 0.964041i \(0.414380\pi\)
\(588\) −1.77575 −0.0732305
\(589\) 19.8496 0.817887
\(590\) 0 0
\(591\) 8.07522 0.332170
\(592\) 7.61213 0.312856
\(593\) −16.2981 −0.669281 −0.334641 0.942346i \(-0.608615\pi\)
−0.334641 + 0.942346i \(0.608615\pi\)
\(594\) 3.35026 0.137463
\(595\) 0 0
\(596\) 4.64974 0.190461
\(597\) −15.9756 −0.653836
\(598\) 4.96239 0.202927
\(599\) 9.91493 0.405113 0.202556 0.979271i \(-0.435075\pi\)
0.202556 + 0.979271i \(0.435075\pi\)
\(600\) 0 0
\(601\) −7.40105 −0.301895 −0.150948 0.988542i \(-0.548232\pi\)
−0.150948 + 0.988542i \(0.548232\pi\)
\(602\) −30.5501 −1.24513
\(603\) 7.61213 0.309990
\(604\) 10.7005 0.435398
\(605\) 0 0
\(606\) −4.26187 −0.173126
\(607\) 13.9610 0.566658 0.283329 0.959023i \(-0.408561\pi\)
0.283329 + 0.959023i \(0.408561\pi\)
\(608\) −4.96239 −0.201251
\(609\) 22.9234 0.928902
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) −1.35026 −0.0545811
\(613\) 41.3865 1.67158 0.835792 0.549047i \(-0.185009\pi\)
0.835792 + 0.549047i \(0.185009\pi\)
\(614\) 30.5501 1.23290
\(615\) 0 0
\(616\) 9.92478 0.399881
\(617\) −29.3014 −1.17963 −0.589815 0.807539i \(-0.700799\pi\)
−0.589815 + 0.807539i \(0.700799\pi\)
\(618\) −0.261865 −0.0105338
\(619\) −19.0376 −0.765186 −0.382593 0.923917i \(-0.624969\pi\)
−0.382593 + 0.923917i \(0.624969\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 14.4387 0.578937
\(623\) −29.4010 −1.17793
\(624\) −4.96239 −0.198655
\(625\) 0 0
\(626\) −27.9610 −1.11755
\(627\) −16.6253 −0.663951
\(628\) 17.0132 0.678900
\(629\) −10.2784 −0.409825
\(630\) 0 0
\(631\) 12.4993 0.497589 0.248794 0.968556i \(-0.419966\pi\)
0.248794 + 0.968556i \(0.419966\pi\)
\(632\) −4.12601 −0.164124
\(633\) −21.7743 −0.865452
\(634\) −1.47627 −0.0586302
\(635\) 0 0
\(636\) −6.96239 −0.276077
\(637\) 8.81194 0.349142
\(638\) −25.9248 −1.02637
\(639\) −2.18664 −0.0865022
\(640\) 0 0
\(641\) −34.0263 −1.34396 −0.671980 0.740569i \(-0.734556\pi\)
−0.671980 + 0.740569i \(0.734556\pi\)
\(642\) −9.08840 −0.358690
\(643\) 7.34041 0.289478 0.144739 0.989470i \(-0.453766\pi\)
0.144739 + 0.989470i \(0.453766\pi\)
\(644\) −2.96239 −0.116734
\(645\) 0 0
\(646\) 6.70052 0.263629
\(647\) 14.9525 0.587845 0.293922 0.955829i \(-0.405039\pi\)
0.293922 + 0.955829i \(0.405039\pi\)
\(648\) 1.00000 0.0392837
\(649\) −4.10157 −0.161001
\(650\) 0 0
\(651\) −11.8496 −0.464421
\(652\) −12.6253 −0.494445
\(653\) −20.4485 −0.800212 −0.400106 0.916469i \(-0.631027\pi\)
−0.400106 + 0.916469i \(0.631027\pi\)
\(654\) −18.9380 −0.740532
\(655\) 0 0
\(656\) 4.70052 0.183525
\(657\) 9.92478 0.387202
\(658\) 9.55149 0.372356
\(659\) 38.4241 1.49679 0.748395 0.663254i \(-0.230825\pi\)
0.748395 + 0.663254i \(0.230825\pi\)
\(660\) 0 0
\(661\) −39.3112 −1.52903 −0.764515 0.644606i \(-0.777021\pi\)
−0.764515 + 0.644606i \(0.777021\pi\)
\(662\) 23.1754 0.900737
\(663\) 6.70052 0.260227
\(664\) 6.38787 0.247898
\(665\) 0 0
\(666\) 7.61213 0.294964
\(667\) 7.73813 0.299622
\(668\) 24.6253 0.952781
\(669\) −3.81336 −0.147433
\(670\) 0 0
\(671\) 37.1490 1.43412
\(672\) 2.96239 0.114277
\(673\) −25.5515 −0.984938 −0.492469 0.870330i \(-0.663905\pi\)
−0.492469 + 0.870330i \(0.663905\pi\)
\(674\) 21.0376 0.810339
\(675\) 0 0
\(676\) 11.6253 0.447127
\(677\) −36.2130 −1.39178 −0.695889 0.718149i \(-0.744990\pi\)
−0.695889 + 0.718149i \(0.744990\pi\)
\(678\) −6.64974 −0.255382
\(679\) 38.1768 1.46509
\(680\) 0 0
\(681\) −16.9380 −0.649064
\(682\) 13.4010 0.513153
\(683\) 15.4763 0.592183 0.296092 0.955160i \(-0.404317\pi\)
0.296092 + 0.955160i \(0.404317\pi\)
\(684\) −4.96239 −0.189742
\(685\) 0 0
\(686\) 15.4763 0.590887
\(687\) 26.9380 1.02775
\(688\) 10.3127 0.393166
\(689\) 34.5501 1.31625
\(690\) 0 0
\(691\) −37.6531 −1.43239 −0.716195 0.697900i \(-0.754118\pi\)
−0.716195 + 0.697900i \(0.754118\pi\)
\(692\) 4.44851 0.169107
\(693\) 9.92478 0.377011
\(694\) −3.37470 −0.128102
\(695\) 0 0
\(696\) −7.73813 −0.293313
\(697\) −6.34694 −0.240407
\(698\) −4.44851 −0.168378
\(699\) −27.4010 −1.03640
\(700\) 0 0
\(701\) 28.3488 1.07072 0.535361 0.844624i \(-0.320176\pi\)
0.535361 + 0.844624i \(0.320176\pi\)
\(702\) −4.96239 −0.187293
\(703\) −37.7743 −1.42469
\(704\) −3.35026 −0.126268
\(705\) 0 0
\(706\) 35.4010 1.33234
\(707\) −12.6253 −0.474823
\(708\) −1.22425 −0.0460103
\(709\) −43.5633 −1.63605 −0.818026 0.575181i \(-0.804932\pi\)
−0.818026 + 0.575181i \(0.804932\pi\)
\(710\) 0 0
\(711\) −4.12601 −0.154737
\(712\) 9.92478 0.371947
\(713\) −4.00000 −0.149801
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) −13.8496 −0.517582
\(717\) −7.48612 −0.279574
\(718\) −34.5501 −1.28940
\(719\) 32.3634 1.20695 0.603476 0.797381i \(-0.293782\pi\)
0.603476 + 0.797381i \(0.293782\pi\)
\(720\) 0 0
\(721\) −0.775746 −0.0288903
\(722\) 5.62530 0.209352
\(723\) 13.0738 0.486220
\(724\) −22.6859 −0.843116
\(725\) 0 0
\(726\) −0.224254 −0.00832284
\(727\) 6.96239 0.258221 0.129110 0.991630i \(-0.458788\pi\)
0.129110 + 0.991630i \(0.458788\pi\)
\(728\) −14.7005 −0.544838
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −13.9248 −0.515026
\(732\) 11.0884 0.409839
\(733\) −9.68735 −0.357810 −0.178905 0.983866i \(-0.557256\pi\)
−0.178905 + 0.983866i \(0.557256\pi\)
\(734\) −22.8119 −0.842004
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −25.5026 −0.939401
\(738\) 4.70052 0.173029
\(739\) −42.9234 −1.57896 −0.789481 0.613775i \(-0.789650\pi\)
−0.789481 + 0.613775i \(0.789650\pi\)
\(740\) 0 0
\(741\) 24.6253 0.904633
\(742\) −20.6253 −0.757179
\(743\) 16.9986 0.623618 0.311809 0.950145i \(-0.399065\pi\)
0.311809 + 0.950145i \(0.399065\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) −10.8364 −0.396748
\(747\) 6.38787 0.233720
\(748\) 4.52373 0.165404
\(749\) −26.9234 −0.983758
\(750\) 0 0
\(751\) −17.9003 −0.653193 −0.326596 0.945164i \(-0.605902\pi\)
−0.326596 + 0.945164i \(0.605902\pi\)
\(752\) −3.22425 −0.117576
\(753\) 5.02302 0.183049
\(754\) 38.3996 1.39843
\(755\) 0 0
\(756\) 2.96239 0.107741
\(757\) −12.3879 −0.450245 −0.225122 0.974330i \(-0.572278\pi\)
−0.225122 + 0.974330i \(0.572278\pi\)
\(758\) −32.4387 −1.17823
\(759\) 3.35026 0.121607
\(760\) 0 0
\(761\) 38.5764 1.39839 0.699197 0.714929i \(-0.253541\pi\)
0.699197 + 0.714929i \(0.253541\pi\)
\(762\) 0.186642 0.00676134
\(763\) −56.1016 −2.03101
\(764\) −19.3258 −0.699184
\(765\) 0 0
\(766\) 32.9986 1.19229
\(767\) 6.07522 0.219364
\(768\) −1.00000 −0.0360844
\(769\) 34.2228 1.23411 0.617054 0.786921i \(-0.288326\pi\)
0.617054 + 0.786921i \(0.288326\pi\)
\(770\) 0 0
\(771\) 19.4010 0.698712
\(772\) 5.92478 0.213237
\(773\) 13.5613 0.487768 0.243884 0.969804i \(-0.421578\pi\)
0.243884 + 0.969804i \(0.421578\pi\)
\(774\) 10.3127 0.370681
\(775\) 0 0
\(776\) −12.8872 −0.462622
\(777\) 22.5501 0.808980
\(778\) 29.1246 1.04417
\(779\) −23.3258 −0.835734
\(780\) 0 0
\(781\) 7.32582 0.262139
\(782\) −1.35026 −0.0482853
\(783\) −7.73813 −0.276538
\(784\) 1.77575 0.0634195
\(785\) 0 0
\(786\) −7.92478 −0.282667
\(787\) −37.0132 −1.31938 −0.659689 0.751539i \(-0.729312\pi\)
−0.659689 + 0.751539i \(0.729312\pi\)
\(788\) −8.07522 −0.287668
\(789\) 15.4763 0.550970
\(790\) 0 0
\(791\) −19.6991 −0.700420
\(792\) −3.35026 −0.119046
\(793\) −55.0249 −1.95399
\(794\) −13.7381 −0.487548
\(795\) 0 0
\(796\) 15.9756 0.566239
\(797\) −47.6893 −1.68924 −0.844620 0.535366i \(-0.820174\pi\)
−0.844620 + 0.535366i \(0.820174\pi\)
\(798\) −14.7005 −0.520393
\(799\) 4.35359 0.154019
\(800\) 0 0
\(801\) 9.92478 0.350675
\(802\) −1.44992 −0.0511985
\(803\) −33.2506 −1.17339
\(804\) −7.61213 −0.268459
\(805\) 0 0
\(806\) −19.8496 −0.699171
\(807\) 20.2130 0.711531
\(808\) 4.26187 0.149932
\(809\) −9.17538 −0.322589 −0.161295 0.986906i \(-0.551567\pi\)
−0.161295 + 0.986906i \(0.551567\pi\)
\(810\) 0 0
\(811\) −38.5501 −1.35368 −0.676838 0.736132i \(-0.736650\pi\)
−0.676838 + 0.736132i \(0.736650\pi\)
\(812\) −22.9234 −0.804452
\(813\) −30.3996 −1.06616
\(814\) −25.5026 −0.893866
\(815\) 0 0
\(816\) 1.35026 0.0472686
\(817\) −51.1754 −1.79040
\(818\) −17.8496 −0.624095
\(819\) −14.7005 −0.513678
\(820\) 0 0
\(821\) 43.4372 1.51597 0.757985 0.652272i \(-0.226184\pi\)
0.757985 + 0.652272i \(0.226184\pi\)
\(822\) 9.19982 0.320880
\(823\) 3.51247 0.122437 0.0612184 0.998124i \(-0.480501\pi\)
0.0612184 + 0.998124i \(0.480501\pi\)
\(824\) 0.261865 0.00912250
\(825\) 0 0
\(826\) −3.62672 −0.126190
\(827\) 9.56325 0.332547 0.166273 0.986080i \(-0.446827\pi\)
0.166273 + 0.986080i \(0.446827\pi\)
\(828\) 1.00000 0.0347524
\(829\) 30.5764 1.06196 0.530982 0.847383i \(-0.321823\pi\)
0.530982 + 0.847383i \(0.321823\pi\)
\(830\) 0 0
\(831\) 20.8119 0.721958
\(832\) 4.96239 0.172040
\(833\) −2.39772 −0.0830762
\(834\) −8.62530 −0.298670
\(835\) 0 0
\(836\) 16.6253 0.574998
\(837\) 4.00000 0.138260
\(838\) 17.9003 0.618357
\(839\) 22.1768 0.765628 0.382814 0.923825i \(-0.374955\pi\)
0.382814 + 0.923825i \(0.374955\pi\)
\(840\) 0 0
\(841\) 30.8787 1.06478
\(842\) 7.61213 0.262331
\(843\) −19.8496 −0.683655
\(844\) 21.7743 0.749503
\(845\) 0 0
\(846\) −3.22425 −0.110852
\(847\) −0.664327 −0.0228265
\(848\) 6.96239 0.239089
\(849\) 12.3879 0.425151
\(850\) 0 0
\(851\) 7.61213 0.260940
\(852\) 2.18664 0.0749131
\(853\) −29.7381 −1.01821 −0.509107 0.860703i \(-0.670024\pi\)
−0.509107 + 0.860703i \(0.670024\pi\)
\(854\) 32.8481 1.12404
\(855\) 0 0
\(856\) 9.08840 0.310635
\(857\) 22.2520 0.760114 0.380057 0.924963i \(-0.375904\pi\)
0.380057 + 0.924963i \(0.375904\pi\)
\(858\) 16.6253 0.567578
\(859\) 11.0738 0.377833 0.188917 0.981993i \(-0.439502\pi\)
0.188917 + 0.981993i \(0.439502\pi\)
\(860\) 0 0
\(861\) 13.9248 0.474555
\(862\) −22.9525 −0.781767
\(863\) −22.8218 −0.776863 −0.388431 0.921478i \(-0.626983\pi\)
−0.388431 + 0.921478i \(0.626983\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −33.7645 −1.14736
\(867\) 15.1768 0.515431
\(868\) 11.8496 0.402200
\(869\) 13.8232 0.468920
\(870\) 0 0
\(871\) 37.7743 1.27993
\(872\) 18.9380 0.641320
\(873\) −12.8872 −0.436164
\(874\) −4.96239 −0.167855
\(875\) 0 0
\(876\) −9.92478 −0.335327
\(877\) −8.81194 −0.297558 −0.148779 0.988870i \(-0.547534\pi\)
−0.148779 + 0.988870i \(0.547534\pi\)
\(878\) −23.4763 −0.792286
\(879\) 9.03761 0.304831
\(880\) 0 0
\(881\) 21.4010 0.721020 0.360510 0.932755i \(-0.382603\pi\)
0.360510 + 0.932755i \(0.382603\pi\)
\(882\) 1.77575 0.0597925
\(883\) 6.17679 0.207866 0.103933 0.994584i \(-0.466857\pi\)
0.103933 + 0.994584i \(0.466857\pi\)
\(884\) −6.70052 −0.225363
\(885\) 0 0
\(886\) 30.5501 1.02635
\(887\) −35.5975 −1.19525 −0.597624 0.801776i \(-0.703889\pi\)
−0.597624 + 0.801776i \(0.703889\pi\)
\(888\) −7.61213 −0.255446
\(889\) 0.552907 0.0185439
\(890\) 0 0
\(891\) −3.35026 −0.112238
\(892\) 3.81336 0.127681
\(893\) 16.0000 0.535420
\(894\) −4.64974 −0.155511
\(895\) 0 0
\(896\) −2.96239 −0.0989665
\(897\) −4.96239 −0.165689
\(898\) 32.7005 1.09123
\(899\) −30.9525 −1.03232
\(900\) 0 0
\(901\) −9.40105 −0.313194
\(902\) −15.7480 −0.524351
\(903\) 30.5501 1.01664
\(904\) 6.64974 0.221167
\(905\) 0 0
\(906\) −10.7005 −0.355501
\(907\) 5.53690 0.183850 0.0919249 0.995766i \(-0.470698\pi\)
0.0919249 + 0.995766i \(0.470698\pi\)
\(908\) 16.9380 0.562106
\(909\) 4.26187 0.141357
\(910\) 0 0
\(911\) −15.4763 −0.512752 −0.256376 0.966577i \(-0.582528\pi\)
−0.256376 + 0.966577i \(0.582528\pi\)
\(912\) 4.96239 0.164321
\(913\) −21.4010 −0.708271
\(914\) −5.81336 −0.192289
\(915\) 0 0
\(916\) −26.9380 −0.890055
\(917\) −23.4763 −0.775255
\(918\) 1.35026 0.0445653
\(919\) −5.17347 −0.170657 −0.0853285 0.996353i \(-0.527194\pi\)
−0.0853285 + 0.996353i \(0.527194\pi\)
\(920\) 0 0
\(921\) −30.5501 −1.00666
\(922\) −23.9902 −0.790074
\(923\) −10.8510 −0.357164
\(924\) −9.92478 −0.326501
\(925\) 0 0
\(926\) 35.3620 1.16207
\(927\) 0.261865 0.00860078
\(928\) 7.73813 0.254017
\(929\) −44.3996 −1.45670 −0.728352 0.685203i \(-0.759714\pi\)
−0.728352 + 0.685203i \(0.759714\pi\)
\(930\) 0 0
\(931\) −8.81194 −0.288800
\(932\) 27.4010 0.897551
\(933\) −14.4387 −0.472700
\(934\) 1.98541 0.0649647
\(935\) 0 0
\(936\) 4.96239 0.162201
\(937\) 3.99015 0.130353 0.0651763 0.997874i \(-0.479239\pi\)
0.0651763 + 0.997874i \(0.479239\pi\)
\(938\) −22.5501 −0.736286
\(939\) 27.9610 0.912472
\(940\) 0 0
\(941\) −2.30280 −0.0750692 −0.0375346 0.999295i \(-0.511950\pi\)
−0.0375346 + 0.999295i \(0.511950\pi\)
\(942\) −17.0132 −0.554319
\(943\) 4.70052 0.153070
\(944\) 1.22425 0.0398461
\(945\) 0 0
\(946\) −34.5501 −1.12332
\(947\) −24.7269 −0.803515 −0.401758 0.915746i \(-0.631601\pi\)
−0.401758 + 0.915746i \(0.631601\pi\)
\(948\) 4.12601 0.134007
\(949\) 49.2506 1.59874
\(950\) 0 0
\(951\) 1.47627 0.0478713
\(952\) 4.00000 0.129641
\(953\) 11.6775 0.378271 0.189136 0.981951i \(-0.439431\pi\)
0.189136 + 0.981951i \(0.439431\pi\)
\(954\) 6.96239 0.225416
\(955\) 0 0
\(956\) 7.48612 0.242118
\(957\) 25.9248 0.838029
\(958\) 11.0738 0.357779
\(959\) 27.2534 0.880059
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 37.7743 1.21789
\(963\) 9.08840 0.292869
\(964\) −13.0738 −0.421079
\(965\) 0 0
\(966\) 2.96239 0.0953133
\(967\) −51.2116 −1.64685 −0.823427 0.567423i \(-0.807941\pi\)
−0.823427 + 0.567423i \(0.807941\pi\)
\(968\) 0.224254 0.00720779
\(969\) −6.70052 −0.215252
\(970\) 0 0
\(971\) 14.8265 0.475806 0.237903 0.971289i \(-0.423540\pi\)
0.237903 + 0.971289i \(0.423540\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −25.5515 −0.819143
\(974\) −17.4372 −0.558725
\(975\) 0 0
\(976\) −11.0884 −0.354931
\(977\) 16.6958 0.534145 0.267073 0.963676i \(-0.413944\pi\)
0.267073 + 0.963676i \(0.413944\pi\)
\(978\) 12.6253 0.403713
\(979\) −33.2506 −1.06269
\(980\) 0 0
\(981\) 18.9380 0.604642
\(982\) 14.8773 0.474754
\(983\) 30.0263 0.957692 0.478846 0.877899i \(-0.341055\pi\)
0.478846 + 0.877899i \(0.341055\pi\)
\(984\) −4.70052 −0.149847
\(985\) 0 0
\(986\) −10.4485 −0.332748
\(987\) −9.55149 −0.304027
\(988\) −24.6253 −0.783435
\(989\) 10.3127 0.327923
\(990\) 0 0
\(991\) −1.40105 −0.0445057 −0.0222529 0.999752i \(-0.507084\pi\)
−0.0222529 + 0.999752i \(0.507084\pi\)
\(992\) −4.00000 −0.127000
\(993\) −23.1754 −0.735448
\(994\) 6.47768 0.205460
\(995\) 0 0
\(996\) −6.38787 −0.202408
\(997\) −13.2144 −0.418504 −0.209252 0.977862i \(-0.567103\pi\)
−0.209252 + 0.977862i \(0.567103\pi\)
\(998\) −16.1016 −0.509686
\(999\) −7.61213 −0.240837
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.bt.1.1 3
5.2 odd 4 690.2.d.c.139.4 yes 6
5.3 odd 4 690.2.d.c.139.1 6
5.4 even 2 3450.2.a.bo.1.3 3
15.2 even 4 2070.2.d.e.829.3 6
15.8 even 4 2070.2.d.e.829.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.d.c.139.1 6 5.3 odd 4
690.2.d.c.139.4 yes 6 5.2 odd 4
2070.2.d.e.829.3 6 15.2 even 4
2070.2.d.e.829.6 6 15.8 even 4
3450.2.a.bo.1.3 3 5.4 even 2
3450.2.a.bt.1.1 3 1.1 even 1 trivial