Properties

Label 3630.2.a.bj.1.2
Level $3630$
Weight $2$
Character 3630.1
Self dual yes
Analytic conductor $28.986$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9856959337\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 330)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 3630.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.23607 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.23607 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +5.61803 q^{13} +1.23607 q^{14} +1.00000 q^{15} +1.00000 q^{16} +5.09017 q^{17} +1.00000 q^{18} -3.23607 q^{19} -1.00000 q^{20} -1.23607 q^{21} +0.145898 q^{23} -1.00000 q^{24} +1.00000 q^{25} +5.61803 q^{26} -1.00000 q^{27} +1.23607 q^{28} -0.381966 q^{29} +1.00000 q^{30} -3.61803 q^{31} +1.00000 q^{32} +5.09017 q^{34} -1.23607 q^{35} +1.00000 q^{36} -8.61803 q^{37} -3.23607 q^{38} -5.61803 q^{39} -1.00000 q^{40} -3.23607 q^{41} -1.23607 q^{42} +11.0902 q^{43} -1.00000 q^{45} +0.145898 q^{46} +11.3262 q^{47} -1.00000 q^{48} -5.47214 q^{49} +1.00000 q^{50} -5.09017 q^{51} +5.61803 q^{52} -0.472136 q^{53} -1.00000 q^{54} +1.23607 q^{56} +3.23607 q^{57} -0.381966 q^{58} +2.90983 q^{59} +1.00000 q^{60} -3.61803 q^{62} +1.23607 q^{63} +1.00000 q^{64} -5.61803 q^{65} +6.09017 q^{67} +5.09017 q^{68} -0.145898 q^{69} -1.23607 q^{70} +1.52786 q^{71} +1.00000 q^{72} -13.4164 q^{73} -8.61803 q^{74} -1.00000 q^{75} -3.23607 q^{76} -5.61803 q^{78} +7.09017 q^{79} -1.00000 q^{80} +1.00000 q^{81} -3.23607 q^{82} +4.00000 q^{83} -1.23607 q^{84} -5.09017 q^{85} +11.0902 q^{86} +0.381966 q^{87} +12.7639 q^{89} -1.00000 q^{90} +6.94427 q^{91} +0.145898 q^{92} +3.61803 q^{93} +11.3262 q^{94} +3.23607 q^{95} -1.00000 q^{96} -12.6525 q^{97} -5.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} - 2q^{10} - 2q^{12} + 9q^{13} - 2q^{14} + 2q^{15} + 2q^{16} - q^{17} + 2q^{18} - 2q^{19} - 2q^{20} + 2q^{21} + 7q^{23} - 2q^{24} + 2q^{25} + 9q^{26} - 2q^{27} - 2q^{28} - 3q^{29} + 2q^{30} - 5q^{31} + 2q^{32} - q^{34} + 2q^{35} + 2q^{36} - 15q^{37} - 2q^{38} - 9q^{39} - 2q^{40} - 2q^{41} + 2q^{42} + 11q^{43} - 2q^{45} + 7q^{46} + 7q^{47} - 2q^{48} - 2q^{49} + 2q^{50} + q^{51} + 9q^{52} + 8q^{53} - 2q^{54} - 2q^{56} + 2q^{57} - 3q^{58} + 17q^{59} + 2q^{60} - 5q^{62} - 2q^{63} + 2q^{64} - 9q^{65} + q^{67} - q^{68} - 7q^{69} + 2q^{70} + 12q^{71} + 2q^{72} - 15q^{74} - 2q^{75} - 2q^{76} - 9q^{78} + 3q^{79} - 2q^{80} + 2q^{81} - 2q^{82} + 8q^{83} + 2q^{84} + q^{85} + 11q^{86} + 3q^{87} + 30q^{89} - 2q^{90} - 4q^{91} + 7q^{92} + 5q^{93} + 7q^{94} + 2q^{95} - 2q^{96} + 6q^{97} - 2q^{98} + O(q^{100}) \)

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
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) 1.23607 0.467190 0.233595 0.972334i \(-0.424951\pi\)
0.233595 + 0.972334i \(0.424951\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −1.00000 −0.288675
\(13\) 5.61803 1.55816 0.779081 0.626923i \(-0.215686\pi\)
0.779081 + 0.626923i \(0.215686\pi\)
\(14\) 1.23607 0.330353
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 5.09017 1.23455 0.617274 0.786748i \(-0.288237\pi\)
0.617274 + 0.786748i \(0.288237\pi\)
\(18\) 1.00000 0.235702
\(19\) −3.23607 −0.742405 −0.371202 0.928552i \(-0.621054\pi\)
−0.371202 + 0.928552i \(0.621054\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.23607 −0.269732
\(22\) 0 0
\(23\) 0.145898 0.0304218 0.0152109 0.999884i \(-0.495158\pi\)
0.0152109 + 0.999884i \(0.495158\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 5.61803 1.10179
\(27\) −1.00000 −0.192450
\(28\) 1.23607 0.233595
\(29\) −0.381966 −0.0709293 −0.0354647 0.999371i \(-0.511291\pi\)
−0.0354647 + 0.999371i \(0.511291\pi\)
\(30\) 1.00000 0.182574
\(31\) −3.61803 −0.649818 −0.324909 0.945745i \(-0.605334\pi\)
−0.324909 + 0.945745i \(0.605334\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 5.09017 0.872957
\(35\) −1.23607 −0.208934
\(36\) 1.00000 0.166667
\(37\) −8.61803 −1.41680 −0.708398 0.705813i \(-0.750582\pi\)
−0.708398 + 0.705813i \(0.750582\pi\)
\(38\) −3.23607 −0.524960
\(39\) −5.61803 −0.899605
\(40\) −1.00000 −0.158114
\(41\) −3.23607 −0.505389 −0.252694 0.967546i \(-0.581317\pi\)
−0.252694 + 0.967546i \(0.581317\pi\)
\(42\) −1.23607 −0.190729
\(43\) 11.0902 1.69124 0.845618 0.533789i \(-0.179232\pi\)
0.845618 + 0.533789i \(0.179232\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0.145898 0.0215115
\(47\) 11.3262 1.65210 0.826051 0.563596i \(-0.190582\pi\)
0.826051 + 0.563596i \(0.190582\pi\)
\(48\) −1.00000 −0.144338
\(49\) −5.47214 −0.781734
\(50\) 1.00000 0.141421
\(51\) −5.09017 −0.712766
\(52\) 5.61803 0.779081
\(53\) −0.472136 −0.0648529 −0.0324264 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.23607 0.165177
\(57\) 3.23607 0.428628
\(58\) −0.381966 −0.0501546
\(59\) 2.90983 0.378828 0.189414 0.981897i \(-0.439341\pi\)
0.189414 + 0.981897i \(0.439341\pi\)
\(60\) 1.00000 0.129099
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −3.61803 −0.459491
\(63\) 1.23607 0.155730
\(64\) 1.00000 0.125000
\(65\) −5.61803 −0.696831
\(66\) 0 0
\(67\) 6.09017 0.744033 0.372016 0.928226i \(-0.378667\pi\)
0.372016 + 0.928226i \(0.378667\pi\)
\(68\) 5.09017 0.617274
\(69\) −0.145898 −0.0175641
\(70\) −1.23607 −0.147738
\(71\) 1.52786 0.181324 0.0906621 0.995882i \(-0.471102\pi\)
0.0906621 + 0.995882i \(0.471102\pi\)
\(72\) 1.00000 0.117851
\(73\) −13.4164 −1.57027 −0.785136 0.619324i \(-0.787407\pi\)
−0.785136 + 0.619324i \(0.787407\pi\)
\(74\) −8.61803 −1.00183
\(75\) −1.00000 −0.115470
\(76\) −3.23607 −0.371202
\(77\) 0 0
\(78\) −5.61803 −0.636117
\(79\) 7.09017 0.797706 0.398853 0.917015i \(-0.369408\pi\)
0.398853 + 0.917015i \(0.369408\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −3.23607 −0.357364
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −1.23607 −0.134866
\(85\) −5.09017 −0.552106
\(86\) 11.0902 1.19588
\(87\) 0.381966 0.0409511
\(88\) 0 0
\(89\) 12.7639 1.35297 0.676487 0.736455i \(-0.263501\pi\)
0.676487 + 0.736455i \(0.263501\pi\)
\(90\) −1.00000 −0.105409
\(91\) 6.94427 0.727957
\(92\) 0.145898 0.0152109
\(93\) 3.61803 0.375173
\(94\) 11.3262 1.16821
\(95\) 3.23607 0.332014
\(96\) −1.00000 −0.102062
\(97\) −12.6525 −1.28466 −0.642332 0.766426i \(-0.722033\pi\)
−0.642332 + 0.766426i \(0.722033\pi\)
\(98\) −5.47214 −0.552769
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 13.3820 1.33156 0.665778 0.746150i \(-0.268100\pi\)
0.665778 + 0.746150i \(0.268100\pi\)
\(102\) −5.09017 −0.504002
\(103\) 11.2361 1.10712 0.553561 0.832808i \(-0.313268\pi\)
0.553561 + 0.832808i \(0.313268\pi\)
\(104\) 5.61803 0.550894
\(105\) 1.23607 0.120628
\(106\) −0.472136 −0.0458579
\(107\) 19.7082 1.90526 0.952632 0.304125i \(-0.0983642\pi\)
0.952632 + 0.304125i \(0.0983642\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.47214 0.619918 0.309959 0.950750i \(-0.399685\pi\)
0.309959 + 0.950750i \(0.399685\pi\)
\(110\) 0 0
\(111\) 8.61803 0.817988
\(112\) 1.23607 0.116797
\(113\) 0.562306 0.0528973 0.0264486 0.999650i \(-0.491580\pi\)
0.0264486 + 0.999650i \(0.491580\pi\)
\(114\) 3.23607 0.303086
\(115\) −0.145898 −0.0136051
\(116\) −0.381966 −0.0354647
\(117\) 5.61803 0.519387
\(118\) 2.90983 0.267872
\(119\) 6.29180 0.576768
\(120\) 1.00000 0.0912871
\(121\) 0 0
\(122\) 0 0
\(123\) 3.23607 0.291786
\(124\) −3.61803 −0.324909
\(125\) −1.00000 −0.0894427
\(126\) 1.23607 0.110118
\(127\) −16.6525 −1.47767 −0.738834 0.673887i \(-0.764623\pi\)
−0.738834 + 0.673887i \(0.764623\pi\)
\(128\) 1.00000 0.0883883
\(129\) −11.0902 −0.976435
\(130\) −5.61803 −0.492734
\(131\) −5.61803 −0.490850 −0.245425 0.969416i \(-0.578927\pi\)
−0.245425 + 0.969416i \(0.578927\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 6.09017 0.526111
\(135\) 1.00000 0.0860663
\(136\) 5.09017 0.436478
\(137\) 11.7984 1.00800 0.504002 0.863703i \(-0.331861\pi\)
0.504002 + 0.863703i \(0.331861\pi\)
\(138\) −0.145898 −0.0124197
\(139\) 13.2361 1.12267 0.561334 0.827589i \(-0.310288\pi\)
0.561334 + 0.827589i \(0.310288\pi\)
\(140\) −1.23607 −0.104467
\(141\) −11.3262 −0.953841
\(142\) 1.52786 0.128216
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0.381966 0.0317206
\(146\) −13.4164 −1.11035
\(147\) 5.47214 0.451334
\(148\) −8.61803 −0.708398
\(149\) −5.38197 −0.440908 −0.220454 0.975397i \(-0.570754\pi\)
−0.220454 + 0.975397i \(0.570754\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 3.05573 0.248672 0.124336 0.992240i \(-0.460320\pi\)
0.124336 + 0.992240i \(0.460320\pi\)
\(152\) −3.23607 −0.262480
\(153\) 5.09017 0.411516
\(154\) 0 0
\(155\) 3.61803 0.290607
\(156\) −5.61803 −0.449803
\(157\) 24.2705 1.93700 0.968499 0.249018i \(-0.0801079\pi\)
0.968499 + 0.249018i \(0.0801079\pi\)
\(158\) 7.09017 0.564064
\(159\) 0.472136 0.0374428
\(160\) −1.00000 −0.0790569
\(161\) 0.180340 0.0142128
\(162\) 1.00000 0.0785674
\(163\) −5.85410 −0.458529 −0.229264 0.973364i \(-0.573632\pi\)
−0.229264 + 0.973364i \(0.573632\pi\)
\(164\) −3.23607 −0.252694
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 20.2705 1.56858 0.784290 0.620395i \(-0.213028\pi\)
0.784290 + 0.620395i \(0.213028\pi\)
\(168\) −1.23607 −0.0953647
\(169\) 18.5623 1.42787
\(170\) −5.09017 −0.390398
\(171\) −3.23607 −0.247468
\(172\) 11.0902 0.845618
\(173\) −18.4721 −1.40441 −0.702205 0.711975i \(-0.747801\pi\)
−0.702205 + 0.711975i \(0.747801\pi\)
\(174\) 0.381966 0.0289568
\(175\) 1.23607 0.0934380
\(176\) 0 0
\(177\) −2.90983 −0.218716
\(178\) 12.7639 0.956697
\(179\) 24.8541 1.85768 0.928841 0.370478i \(-0.120806\pi\)
0.928841 + 0.370478i \(0.120806\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −13.8885 −1.03233 −0.516164 0.856490i \(-0.672640\pi\)
−0.516164 + 0.856490i \(0.672640\pi\)
\(182\) 6.94427 0.514744
\(183\) 0 0
\(184\) 0.145898 0.0107557
\(185\) 8.61803 0.633610
\(186\) 3.61803 0.265287
\(187\) 0 0
\(188\) 11.3262 0.826051
\(189\) −1.23607 −0.0899107
\(190\) 3.23607 0.234769
\(191\) 4.76393 0.344706 0.172353 0.985035i \(-0.444863\pi\)
0.172353 + 0.985035i \(0.444863\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −3.05573 −0.219956 −0.109978 0.993934i \(-0.535078\pi\)
−0.109978 + 0.993934i \(0.535078\pi\)
\(194\) −12.6525 −0.908395
\(195\) 5.61803 0.402316
\(196\) −5.47214 −0.390867
\(197\) −6.94427 −0.494759 −0.247379 0.968919i \(-0.579569\pi\)
−0.247379 + 0.968919i \(0.579569\pi\)
\(198\) 0 0
\(199\) −20.9787 −1.48714 −0.743571 0.668657i \(-0.766869\pi\)
−0.743571 + 0.668657i \(0.766869\pi\)
\(200\) 1.00000 0.0707107
\(201\) −6.09017 −0.429567
\(202\) 13.3820 0.941552
\(203\) −0.472136 −0.0331374
\(204\) −5.09017 −0.356383
\(205\) 3.23607 0.226017
\(206\) 11.2361 0.782854
\(207\) 0.145898 0.0101406
\(208\) 5.61803 0.389541
\(209\) 0 0
\(210\) 1.23607 0.0852968
\(211\) −20.7639 −1.42945 −0.714724 0.699407i \(-0.753448\pi\)
−0.714724 + 0.699407i \(0.753448\pi\)
\(212\) −0.472136 −0.0324264
\(213\) −1.52786 −0.104688
\(214\) 19.7082 1.34723
\(215\) −11.0902 −0.756343
\(216\) −1.00000 −0.0680414
\(217\) −4.47214 −0.303588
\(218\) 6.47214 0.438348
\(219\) 13.4164 0.906597
\(220\) 0 0
\(221\) 28.5967 1.92363
\(222\) 8.61803 0.578405
\(223\) 2.76393 0.185087 0.0925433 0.995709i \(-0.470500\pi\)
0.0925433 + 0.995709i \(0.470500\pi\)
\(224\) 1.23607 0.0825883
\(225\) 1.00000 0.0666667
\(226\) 0.562306 0.0374040
\(227\) −24.3607 −1.61688 −0.808438 0.588582i \(-0.799686\pi\)
−0.808438 + 0.588582i \(0.799686\pi\)
\(228\) 3.23607 0.214314
\(229\) 5.23607 0.346009 0.173005 0.984921i \(-0.444652\pi\)
0.173005 + 0.984921i \(0.444652\pi\)
\(230\) −0.145898 −0.00962023
\(231\) 0 0
\(232\) −0.381966 −0.0250773
\(233\) −24.5623 −1.60913 −0.804565 0.593864i \(-0.797602\pi\)
−0.804565 + 0.593864i \(0.797602\pi\)
\(234\) 5.61803 0.367262
\(235\) −11.3262 −0.738842
\(236\) 2.90983 0.189414
\(237\) −7.09017 −0.460556
\(238\) 6.29180 0.407837
\(239\) 1.23607 0.0799546 0.0399773 0.999201i \(-0.487271\pi\)
0.0399773 + 0.999201i \(0.487271\pi\)
\(240\) 1.00000 0.0645497
\(241\) 11.8885 0.765808 0.382904 0.923788i \(-0.374924\pi\)
0.382904 + 0.923788i \(0.374924\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 5.47214 0.349602
\(246\) 3.23607 0.206324
\(247\) −18.1803 −1.15679
\(248\) −3.61803 −0.229745
\(249\) −4.00000 −0.253490
\(250\) −1.00000 −0.0632456
\(251\) 10.6180 0.670204 0.335102 0.942182i \(-0.391229\pi\)
0.335102 + 0.942182i \(0.391229\pi\)
\(252\) 1.23607 0.0778650
\(253\) 0 0
\(254\) −16.6525 −1.04487
\(255\) 5.09017 0.318759
\(256\) 1.00000 0.0625000
\(257\) −14.3607 −0.895795 −0.447897 0.894085i \(-0.647827\pi\)
−0.447897 + 0.894085i \(0.647827\pi\)
\(258\) −11.0902 −0.690444
\(259\) −10.6525 −0.661913
\(260\) −5.61803 −0.348416
\(261\) −0.381966 −0.0236431
\(262\) −5.61803 −0.347083
\(263\) 0.145898 0.00899646 0.00449823 0.999990i \(-0.498568\pi\)
0.00449823 + 0.999990i \(0.498568\pi\)
\(264\) 0 0
\(265\) 0.472136 0.0290031
\(266\) −4.00000 −0.245256
\(267\) −12.7639 −0.781140
\(268\) 6.09017 0.372016
\(269\) 27.7984 1.69490 0.847448 0.530878i \(-0.178138\pi\)
0.847448 + 0.530878i \(0.178138\pi\)
\(270\) 1.00000 0.0608581
\(271\) −8.61803 −0.523508 −0.261754 0.965135i \(-0.584301\pi\)
−0.261754 + 0.965135i \(0.584301\pi\)
\(272\) 5.09017 0.308637
\(273\) −6.94427 −0.420286
\(274\) 11.7984 0.712766
\(275\) 0 0
\(276\) −0.145898 −0.00878203
\(277\) 14.5066 0.871616 0.435808 0.900040i \(-0.356463\pi\)
0.435808 + 0.900040i \(0.356463\pi\)
\(278\) 13.2361 0.793847
\(279\) −3.61803 −0.216606
\(280\) −1.23607 −0.0738692
\(281\) −16.4721 −0.982645 −0.491323 0.870978i \(-0.663486\pi\)
−0.491323 + 0.870978i \(0.663486\pi\)
\(282\) −11.3262 −0.674468
\(283\) −6.67376 −0.396714 −0.198357 0.980130i \(-0.563561\pi\)
−0.198357 + 0.980130i \(0.563561\pi\)
\(284\) 1.52786 0.0906621
\(285\) −3.23607 −0.191688
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 1.00000 0.0589256
\(289\) 8.90983 0.524108
\(290\) 0.381966 0.0224298
\(291\) 12.6525 0.741701
\(292\) −13.4164 −0.785136
\(293\) −20.0000 −1.16841 −0.584206 0.811605i \(-0.698594\pi\)
−0.584206 + 0.811605i \(0.698594\pi\)
\(294\) 5.47214 0.319141
\(295\) −2.90983 −0.169417
\(296\) −8.61803 −0.500913
\(297\) 0 0
\(298\) −5.38197 −0.311769
\(299\) 0.819660 0.0474022
\(300\) −1.00000 −0.0577350
\(301\) 13.7082 0.790128
\(302\) 3.05573 0.175837
\(303\) −13.3820 −0.768774
\(304\) −3.23607 −0.185601
\(305\) 0 0
\(306\) 5.09017 0.290986
\(307\) 22.4508 1.28134 0.640669 0.767817i \(-0.278657\pi\)
0.640669 + 0.767817i \(0.278657\pi\)
\(308\) 0 0
\(309\) −11.2361 −0.639198
\(310\) 3.61803 0.205491
\(311\) 4.76393 0.270138 0.135069 0.990836i \(-0.456874\pi\)
0.135069 + 0.990836i \(0.456874\pi\)
\(312\) −5.61803 −0.318059
\(313\) −0.472136 −0.0266867 −0.0133434 0.999911i \(-0.504247\pi\)
−0.0133434 + 0.999911i \(0.504247\pi\)
\(314\) 24.2705 1.36966
\(315\) −1.23607 −0.0696445
\(316\) 7.09017 0.398853
\(317\) −0.583592 −0.0327778 −0.0163889 0.999866i \(-0.505217\pi\)
−0.0163889 + 0.999866i \(0.505217\pi\)
\(318\) 0.472136 0.0264761
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −19.7082 −1.10000
\(322\) 0.180340 0.0100499
\(323\) −16.4721 −0.916534
\(324\) 1.00000 0.0555556
\(325\) 5.61803 0.311632
\(326\) −5.85410 −0.324229
\(327\) −6.47214 −0.357910
\(328\) −3.23607 −0.178682
\(329\) 14.0000 0.771845
\(330\) 0 0
\(331\) 4.76393 0.261849 0.130925 0.991392i \(-0.458205\pi\)
0.130925 + 0.991392i \(0.458205\pi\)
\(332\) 4.00000 0.219529
\(333\) −8.61803 −0.472265
\(334\) 20.2705 1.10915
\(335\) −6.09017 −0.332742
\(336\) −1.23607 −0.0674330
\(337\) −10.7639 −0.586349 −0.293174 0.956059i \(-0.594712\pi\)
−0.293174 + 0.956059i \(0.594712\pi\)
\(338\) 18.5623 1.00966
\(339\) −0.562306 −0.0305403
\(340\) −5.09017 −0.276053
\(341\) 0 0
\(342\) −3.23607 −0.174987
\(343\) −15.4164 −0.832408
\(344\) 11.0902 0.597942
\(345\) 0.145898 0.00785489
\(346\) −18.4721 −0.993068
\(347\) 0.111456 0.00598328 0.00299164 0.999996i \(-0.499048\pi\)
0.00299164 + 0.999996i \(0.499048\pi\)
\(348\) 0.381966 0.0204755
\(349\) 15.4164 0.825221 0.412611 0.910907i \(-0.364617\pi\)
0.412611 + 0.910907i \(0.364617\pi\)
\(350\) 1.23607 0.0660706
\(351\) −5.61803 −0.299868
\(352\) 0 0
\(353\) 21.3820 1.13805 0.569024 0.822321i \(-0.307321\pi\)
0.569024 + 0.822321i \(0.307321\pi\)
\(354\) −2.90983 −0.154656
\(355\) −1.52786 −0.0810906
\(356\) 12.7639 0.676487
\(357\) −6.29180 −0.332997
\(358\) 24.8541 1.31358
\(359\) 10.1803 0.537298 0.268649 0.963238i \(-0.413423\pi\)
0.268649 + 0.963238i \(0.413423\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −8.52786 −0.448835
\(362\) −13.8885 −0.729966
\(363\) 0 0
\(364\) 6.94427 0.363979
\(365\) 13.4164 0.702247
\(366\) 0 0
\(367\) −31.2361 −1.63051 −0.815255 0.579103i \(-0.803403\pi\)
−0.815255 + 0.579103i \(0.803403\pi\)
\(368\) 0.145898 0.00760546
\(369\) −3.23607 −0.168463
\(370\) 8.61803 0.448030
\(371\) −0.583592 −0.0302986
\(372\) 3.61803 0.187586
\(373\) −20.4721 −1.06001 −0.530004 0.847995i \(-0.677809\pi\)
−0.530004 + 0.847995i \(0.677809\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 11.3262 0.584106
\(377\) −2.14590 −0.110519
\(378\) −1.23607 −0.0635765
\(379\) 32.3607 1.66226 0.831128 0.556081i \(-0.187696\pi\)
0.831128 + 0.556081i \(0.187696\pi\)
\(380\) 3.23607 0.166007
\(381\) 16.6525 0.853132
\(382\) 4.76393 0.243744
\(383\) 15.3262 0.783134 0.391567 0.920150i \(-0.371933\pi\)
0.391567 + 0.920150i \(0.371933\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −3.05573 −0.155532
\(387\) 11.0902 0.563745
\(388\) −12.6525 −0.642332
\(389\) 0.145898 0.00739732 0.00369866 0.999993i \(-0.498823\pi\)
0.00369866 + 0.999993i \(0.498823\pi\)
\(390\) 5.61803 0.284480
\(391\) 0.742646 0.0375572
\(392\) −5.47214 −0.276385
\(393\) 5.61803 0.283392
\(394\) −6.94427 −0.349847
\(395\) −7.09017 −0.356745
\(396\) 0 0
\(397\) 26.7984 1.34497 0.672486 0.740110i \(-0.265227\pi\)
0.672486 + 0.740110i \(0.265227\pi\)
\(398\) −20.9787 −1.05157
\(399\) 4.00000 0.200250
\(400\) 1.00000 0.0500000
\(401\) −36.0689 −1.80119 −0.900597 0.434655i \(-0.856870\pi\)
−0.900597 + 0.434655i \(0.856870\pi\)
\(402\) −6.09017 −0.303750
\(403\) −20.3262 −1.01252
\(404\) 13.3820 0.665778
\(405\) −1.00000 −0.0496904
\(406\) −0.472136 −0.0234317
\(407\) 0 0
\(408\) −5.09017 −0.252001
\(409\) −33.9787 −1.68014 −0.840070 0.542479i \(-0.817486\pi\)
−0.840070 + 0.542479i \(0.817486\pi\)
\(410\) 3.23607 0.159818
\(411\) −11.7984 −0.581971
\(412\) 11.2361 0.553561
\(413\) 3.59675 0.176984
\(414\) 0.145898 0.00717050
\(415\) −4.00000 −0.196352
\(416\) 5.61803 0.275447
\(417\) −13.2361 −0.648173
\(418\) 0 0
\(419\) 16.1459 0.788779 0.394389 0.918943i \(-0.370956\pi\)
0.394389 + 0.918943i \(0.370956\pi\)
\(420\) 1.23607 0.0603139
\(421\) 6.47214 0.315433 0.157716 0.987484i \(-0.449587\pi\)
0.157716 + 0.987484i \(0.449587\pi\)
\(422\) −20.7639 −1.01077
\(423\) 11.3262 0.550701
\(424\) −0.472136 −0.0229289
\(425\) 5.09017 0.246910
\(426\) −1.52786 −0.0740253
\(427\) 0 0
\(428\) 19.7082 0.952632
\(429\) 0 0
\(430\) −11.0902 −0.534815
\(431\) −22.3607 −1.07708 −0.538538 0.842601i \(-0.681023\pi\)
−0.538538 + 0.842601i \(0.681023\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 30.8328 1.48173 0.740865 0.671654i \(-0.234416\pi\)
0.740865 + 0.671654i \(0.234416\pi\)
\(434\) −4.47214 −0.214669
\(435\) −0.381966 −0.0183139
\(436\) 6.47214 0.309959
\(437\) −0.472136 −0.0225853
\(438\) 13.4164 0.641061
\(439\) 17.3262 0.826936 0.413468 0.910519i \(-0.364317\pi\)
0.413468 + 0.910519i \(0.364317\pi\)
\(440\) 0 0
\(441\) −5.47214 −0.260578
\(442\) 28.5967 1.36021
\(443\) 12.6525 0.601137 0.300569 0.953760i \(-0.402824\pi\)
0.300569 + 0.953760i \(0.402824\pi\)
\(444\) 8.61803 0.408994
\(445\) −12.7639 −0.605068
\(446\) 2.76393 0.130876
\(447\) 5.38197 0.254558
\(448\) 1.23607 0.0583987
\(449\) 29.1246 1.37448 0.687238 0.726433i \(-0.258823\pi\)
0.687238 + 0.726433i \(0.258823\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 0.562306 0.0264486
\(453\) −3.05573 −0.143571
\(454\) −24.3607 −1.14330
\(455\) −6.94427 −0.325552
\(456\) 3.23607 0.151543
\(457\) −26.6525 −1.24675 −0.623375 0.781923i \(-0.714239\pi\)
−0.623375 + 0.781923i \(0.714239\pi\)
\(458\) 5.23607 0.244665
\(459\) −5.09017 −0.237589
\(460\) −0.145898 −0.00680253
\(461\) −28.2148 −1.31409 −0.657047 0.753850i \(-0.728195\pi\)
−0.657047 + 0.753850i \(0.728195\pi\)
\(462\) 0 0
\(463\) −23.8885 −1.11019 −0.555097 0.831785i \(-0.687319\pi\)
−0.555097 + 0.831785i \(0.687319\pi\)
\(464\) −0.381966 −0.0177323
\(465\) −3.61803 −0.167782
\(466\) −24.5623 −1.13783
\(467\) −38.5410 −1.78347 −0.891733 0.452562i \(-0.850510\pi\)
−0.891733 + 0.452562i \(0.850510\pi\)
\(468\) 5.61803 0.259694
\(469\) 7.52786 0.347604
\(470\) −11.3262 −0.522440
\(471\) −24.2705 −1.11833
\(472\) 2.90983 0.133936
\(473\) 0 0
\(474\) −7.09017 −0.325662
\(475\) −3.23607 −0.148481
\(476\) 6.29180 0.288384
\(477\) −0.472136 −0.0216176
\(478\) 1.23607 0.0565364
\(479\) 17.2361 0.787536 0.393768 0.919210i \(-0.371171\pi\)
0.393768 + 0.919210i \(0.371171\pi\)
\(480\) 1.00000 0.0456435
\(481\) −48.4164 −2.20760
\(482\) 11.8885 0.541508
\(483\) −0.180340 −0.00820575
\(484\) 0 0
\(485\) 12.6525 0.574519
\(486\) −1.00000 −0.0453609
\(487\) −15.4164 −0.698584 −0.349292 0.937014i \(-0.613578\pi\)
−0.349292 + 0.937014i \(0.613578\pi\)
\(488\) 0 0
\(489\) 5.85410 0.264732
\(490\) 5.47214 0.247206
\(491\) −1.03444 −0.0466837 −0.0233419 0.999728i \(-0.507431\pi\)
−0.0233419 + 0.999728i \(0.507431\pi\)
\(492\) 3.23607 0.145893
\(493\) −1.94427 −0.0875656
\(494\) −18.1803 −0.817972
\(495\) 0 0
\(496\) −3.61803 −0.162455
\(497\) 1.88854 0.0847128
\(498\) −4.00000 −0.179244
\(499\) −23.5279 −1.05325 −0.526626 0.850097i \(-0.676543\pi\)
−0.526626 + 0.850097i \(0.676543\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −20.2705 −0.905620
\(502\) 10.6180 0.473906
\(503\) −8.67376 −0.386744 −0.193372 0.981126i \(-0.561942\pi\)
−0.193372 + 0.981126i \(0.561942\pi\)
\(504\) 1.23607 0.0550588
\(505\) −13.3820 −0.595490
\(506\) 0 0
\(507\) −18.5623 −0.824381
\(508\) −16.6525 −0.738834
\(509\) −15.7984 −0.700251 −0.350125 0.936703i \(-0.613861\pi\)
−0.350125 + 0.936703i \(0.613861\pi\)
\(510\) 5.09017 0.225397
\(511\) −16.5836 −0.733615
\(512\) 1.00000 0.0441942
\(513\) 3.23607 0.142876
\(514\) −14.3607 −0.633422
\(515\) −11.2361 −0.495120
\(516\) −11.0902 −0.488218
\(517\) 0 0
\(518\) −10.6525 −0.468043
\(519\) 18.4721 0.810837
\(520\) −5.61803 −0.246367
\(521\) 42.3607 1.85586 0.927928 0.372761i \(-0.121589\pi\)
0.927928 + 0.372761i \(0.121589\pi\)
\(522\) −0.381966 −0.0167182
\(523\) −14.8328 −0.648594 −0.324297 0.945955i \(-0.605128\pi\)
−0.324297 + 0.945955i \(0.605128\pi\)
\(524\) −5.61803 −0.245425
\(525\) −1.23607 −0.0539464
\(526\) 0.145898 0.00636146
\(527\) −18.4164 −0.802231
\(528\) 0 0
\(529\) −22.9787 −0.999075
\(530\) 0.472136 0.0205083
\(531\) 2.90983 0.126276
\(532\) −4.00000 −0.173422
\(533\) −18.1803 −0.787478
\(534\) −12.7639 −0.552349
\(535\) −19.7082 −0.852060
\(536\) 6.09017 0.263055
\(537\) −24.8541 −1.07253
\(538\) 27.7984 1.19847
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) −32.0689 −1.37875 −0.689374 0.724405i \(-0.742115\pi\)
−0.689374 + 0.724405i \(0.742115\pi\)
\(542\) −8.61803 −0.370176
\(543\) 13.8885 0.596014
\(544\) 5.09017 0.218239
\(545\) −6.47214 −0.277236
\(546\) −6.94427 −0.297187
\(547\) 18.6738 0.798432 0.399216 0.916857i \(-0.369282\pi\)
0.399216 + 0.916857i \(0.369282\pi\)
\(548\) 11.7984 0.504002
\(549\) 0 0
\(550\) 0 0
\(551\) 1.23607 0.0526583
\(552\) −0.145898 −0.00620983
\(553\) 8.76393 0.372680
\(554\) 14.5066 0.616325
\(555\) −8.61803 −0.365815
\(556\) 13.2361 0.561334
\(557\) 5.12461 0.217137 0.108568 0.994089i \(-0.465373\pi\)
0.108568 + 0.994089i \(0.465373\pi\)
\(558\) −3.61803 −0.153164
\(559\) 62.3050 2.63522
\(560\) −1.23607 −0.0522334
\(561\) 0 0
\(562\) −16.4721 −0.694835
\(563\) 34.9443 1.47273 0.736363 0.676587i \(-0.236542\pi\)
0.736363 + 0.676587i \(0.236542\pi\)
\(564\) −11.3262 −0.476921
\(565\) −0.562306 −0.0236564
\(566\) −6.67376 −0.280519
\(567\) 1.23607 0.0519100
\(568\) 1.52786 0.0641078
\(569\) 1.41641 0.0593789 0.0296895 0.999559i \(-0.490548\pi\)
0.0296895 + 0.999559i \(0.490548\pi\)
\(570\) −3.23607 −0.135544
\(571\) 16.3607 0.684673 0.342337 0.939577i \(-0.388782\pi\)
0.342337 + 0.939577i \(0.388782\pi\)
\(572\) 0 0
\(573\) −4.76393 −0.199016
\(574\) −4.00000 −0.166957
\(575\) 0.145898 0.00608437
\(576\) 1.00000 0.0416667
\(577\) 34.9443 1.45475 0.727375 0.686241i \(-0.240740\pi\)
0.727375 + 0.686241i \(0.240740\pi\)
\(578\) 8.90983 0.370600
\(579\) 3.05573 0.126992
\(580\) 0.381966 0.0158603
\(581\) 4.94427 0.205123
\(582\) 12.6525 0.524462
\(583\) 0 0
\(584\) −13.4164 −0.555175
\(585\) −5.61803 −0.232277
\(586\) −20.0000 −0.826192
\(587\) −39.7082 −1.63893 −0.819466 0.573127i \(-0.805730\pi\)
−0.819466 + 0.573127i \(0.805730\pi\)
\(588\) 5.47214 0.225667
\(589\) 11.7082 0.482428
\(590\) −2.90983 −0.119796
\(591\) 6.94427 0.285649
\(592\) −8.61803 −0.354199
\(593\) −33.3262 −1.36854 −0.684272 0.729227i \(-0.739880\pi\)
−0.684272 + 0.729227i \(0.739880\pi\)
\(594\) 0 0
\(595\) −6.29180 −0.257938
\(596\) −5.38197 −0.220454
\(597\) 20.9787 0.858602
\(598\) 0.819660 0.0335184
\(599\) −4.18034 −0.170804 −0.0854020 0.996347i \(-0.527217\pi\)
−0.0854020 + 0.996347i \(0.527217\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −1.05573 −0.0430640 −0.0215320 0.999768i \(-0.506854\pi\)
−0.0215320 + 0.999768i \(0.506854\pi\)
\(602\) 13.7082 0.558705
\(603\) 6.09017 0.248011
\(604\) 3.05573 0.124336
\(605\) 0 0
\(606\) −13.3820 −0.543605
\(607\) 4.47214 0.181518 0.0907592 0.995873i \(-0.471071\pi\)
0.0907592 + 0.995873i \(0.471071\pi\)
\(608\) −3.23607 −0.131240
\(609\) 0.472136 0.0191319
\(610\) 0 0
\(611\) 63.6312 2.57424
\(612\) 5.09017 0.205758
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 22.4508 0.906043
\(615\) −3.23607 −0.130491
\(616\) 0 0
\(617\) −18.3607 −0.739173 −0.369587 0.929196i \(-0.620501\pi\)
−0.369587 + 0.929196i \(0.620501\pi\)
\(618\) −11.2361 −0.451981
\(619\) 18.4721 0.742458 0.371229 0.928541i \(-0.378937\pi\)
0.371229 + 0.928541i \(0.378937\pi\)
\(620\) 3.61803 0.145304
\(621\) −0.145898 −0.00585469
\(622\) 4.76393 0.191016
\(623\) 15.7771 0.632096
\(624\) −5.61803 −0.224901
\(625\) 1.00000 0.0400000
\(626\) −0.472136 −0.0188703
\(627\) 0 0
\(628\) 24.2705 0.968499
\(629\) −43.8673 −1.74910
\(630\) −1.23607 −0.0492461
\(631\) −21.3262 −0.848984 −0.424492 0.905432i \(-0.639547\pi\)
−0.424492 + 0.905432i \(0.639547\pi\)
\(632\) 7.09017 0.282032
\(633\) 20.7639 0.825292
\(634\) −0.583592 −0.0231774
\(635\) 16.6525 0.660833
\(636\) 0.472136 0.0187214
\(637\) −30.7426 −1.21807
\(638\) 0 0
\(639\) 1.52786 0.0604414
\(640\) −1.00000 −0.0395285
\(641\) −10.1803 −0.402099 −0.201050 0.979581i \(-0.564435\pi\)
−0.201050 + 0.979581i \(0.564435\pi\)
\(642\) −19.7082 −0.777821
\(643\) 19.7984 0.780772 0.390386 0.920651i \(-0.372342\pi\)
0.390386 + 0.920651i \(0.372342\pi\)
\(644\) 0.180340 0.00710639
\(645\) 11.0902 0.436675
\(646\) −16.4721 −0.648088
\(647\) −14.6180 −0.574694 −0.287347 0.957827i \(-0.592773\pi\)
−0.287347 + 0.957827i \(0.592773\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 5.61803 0.220357
\(651\) 4.47214 0.175277
\(652\) −5.85410 −0.229264
\(653\) −42.5410 −1.66476 −0.832379 0.554206i \(-0.813022\pi\)
−0.832379 + 0.554206i \(0.813022\pi\)
\(654\) −6.47214 −0.253081
\(655\) 5.61803 0.219515
\(656\) −3.23607 −0.126347
\(657\) −13.4164 −0.523424
\(658\) 14.0000 0.545777
\(659\) 11.4164 0.444720 0.222360 0.974965i \(-0.428624\pi\)
0.222360 + 0.974965i \(0.428624\pi\)
\(660\) 0 0
\(661\) −36.3607 −1.41427 −0.707133 0.707080i \(-0.750012\pi\)
−0.707133 + 0.707080i \(0.750012\pi\)
\(662\) 4.76393 0.185155
\(663\) −28.5967 −1.11061
\(664\) 4.00000 0.155230
\(665\) 4.00000 0.155113
\(666\) −8.61803 −0.333942
\(667\) −0.0557281 −0.00215780
\(668\) 20.2705 0.784290
\(669\) −2.76393 −0.106860
\(670\) −6.09017 −0.235284
\(671\) 0 0
\(672\) −1.23607 −0.0476824
\(673\) −2.29180 −0.0883422 −0.0441711 0.999024i \(-0.514065\pi\)
−0.0441711 + 0.999024i \(0.514065\pi\)
\(674\) −10.7639 −0.414611
\(675\) −1.00000 −0.0384900
\(676\) 18.5623 0.713935
\(677\) 22.7639 0.874889 0.437444 0.899245i \(-0.355884\pi\)
0.437444 + 0.899245i \(0.355884\pi\)
\(678\) −0.562306 −0.0215952
\(679\) −15.6393 −0.600182
\(680\) −5.09017 −0.195199
\(681\) 24.3607 0.933503
\(682\) 0 0
\(683\) −19.1246 −0.731783 −0.365891 0.930658i \(-0.619236\pi\)
−0.365891 + 0.930658i \(0.619236\pi\)
\(684\) −3.23607 −0.123734
\(685\) −11.7984 −0.450793
\(686\) −15.4164 −0.588601
\(687\) −5.23607 −0.199768
\(688\) 11.0902 0.422809
\(689\) −2.65248 −0.101051
\(690\) 0.145898 0.00555424
\(691\) 10.6525 0.405239 0.202620 0.979258i \(-0.435054\pi\)
0.202620 + 0.979258i \(0.435054\pi\)
\(692\) −18.4721 −0.702205
\(693\) 0 0
\(694\) 0.111456 0.00423082
\(695\) −13.2361 −0.502073
\(696\) 0.381966 0.0144784
\(697\) −16.4721 −0.623927
\(698\) 15.4164 0.583520
\(699\) 24.5623 0.929032
\(700\) 1.23607 0.0467190
\(701\) −12.4721 −0.471066 −0.235533 0.971866i \(-0.575684\pi\)
−0.235533 + 0.971866i \(0.575684\pi\)
\(702\) −5.61803 −0.212039
\(703\) 27.8885 1.05184
\(704\) 0 0
\(705\) 11.3262 0.426571
\(706\) 21.3820 0.804721
\(707\) 16.5410 0.622089
\(708\) −2.90983 −0.109358
\(709\) −37.4164 −1.40520 −0.702601 0.711584i \(-0.747978\pi\)
−0.702601 + 0.711584i \(0.747978\pi\)
\(710\) −1.52786 −0.0573397
\(711\) 7.09017 0.265902
\(712\) 12.7639 0.478349
\(713\) −0.527864 −0.0197687
\(714\) −6.29180 −0.235465
\(715\) 0 0
\(716\) 24.8541 0.928841
\(717\) −1.23607 −0.0461618
\(718\) 10.1803 0.379927
\(719\) 9.34752 0.348604 0.174302 0.984692i \(-0.444233\pi\)
0.174302 + 0.984692i \(0.444233\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 13.8885 0.517236
\(722\) −8.52786 −0.317374
\(723\) −11.8885 −0.442140
\(724\) −13.8885 −0.516164
\(725\) −0.381966 −0.0141859
\(726\) 0 0
\(727\) −27.8197 −1.03177 −0.515887 0.856657i \(-0.672538\pi\)
−0.515887 + 0.856657i \(0.672538\pi\)
\(728\) 6.94427 0.257372
\(729\) 1.00000 0.0370370
\(730\) 13.4164 0.496564
\(731\) 56.4508 2.08791
\(732\) 0 0
\(733\) 35.6738 1.31764 0.658820 0.752300i \(-0.271056\pi\)
0.658820 + 0.752300i \(0.271056\pi\)
\(734\) −31.2361 −1.15294
\(735\) −5.47214 −0.201843
\(736\) 0.145898 0.00537787
\(737\) 0 0
\(738\) −3.23607 −0.119121
\(739\) −49.0132 −1.80298 −0.901489 0.432802i \(-0.857525\pi\)
−0.901489 + 0.432802i \(0.857525\pi\)
\(740\) 8.61803 0.316805
\(741\) 18.1803 0.667871
\(742\) −0.583592 −0.0214243
\(743\) −13.7984 −0.506213 −0.253107 0.967438i \(-0.581452\pi\)
−0.253107 + 0.967438i \(0.581452\pi\)
\(744\) 3.61803 0.132644
\(745\) 5.38197 0.197180
\(746\) −20.4721 −0.749538
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 24.3607 0.890120
\(750\) 1.00000 0.0365148
\(751\) 0.0344419 0.00125680 0.000628401 1.00000i \(-0.499800\pi\)
0.000628401 1.00000i \(0.499800\pi\)
\(752\) 11.3262 0.413025
\(753\) −10.6180 −0.386943
\(754\) −2.14590 −0.0781490
\(755\) −3.05573 −0.111209
\(756\) −1.23607 −0.0449554
\(757\) −47.1459 −1.71355 −0.856773 0.515693i \(-0.827534\pi\)
−0.856773 + 0.515693i \(0.827534\pi\)
\(758\) 32.3607 1.17539
\(759\) 0 0
\(760\) 3.23607 0.117385
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 16.6525 0.603256
\(763\) 8.00000 0.289619
\(764\) 4.76393 0.172353
\(765\) −5.09017 −0.184035
\(766\) 15.3262 0.553759
\(767\) 16.3475 0.590275
\(768\) −1.00000 −0.0360844
\(769\) 31.4508 1.13415 0.567073 0.823667i \(-0.308076\pi\)
0.567073 + 0.823667i \(0.308076\pi\)
\(770\) 0 0
\(771\) 14.3607 0.517187
\(772\) −3.05573 −0.109978
\(773\) −29.5967 −1.06452 −0.532261 0.846581i \(-0.678657\pi\)
−0.532261 + 0.846581i \(0.678657\pi\)
\(774\) 11.0902 0.398628
\(775\) −3.61803 −0.129964
\(776\) −12.6525 −0.454197
\(777\) 10.6525 0.382155
\(778\) 0.145898 0.00523070
\(779\) 10.4721 0.375203
\(780\) 5.61803 0.201158
\(781\) 0 0
\(782\) 0.742646 0.0265570
\(783\) 0.381966 0.0136504
\(784\) −5.47214 −0.195433
\(785\) −24.2705 −0.866252
\(786\) 5.61803 0.200389
\(787\) −8.03444 −0.286397 −0.143198 0.989694i \(-0.545739\pi\)
−0.143198 + 0.989694i \(0.545739\pi\)
\(788\) −6.94427 −0.247379
\(789\) −0.145898 −0.00519411
\(790\) −7.09017 −0.252257
\(791\) 0.695048 0.0247131
\(792\) 0 0
\(793\) 0 0
\(794\) 26.7984 0.951039
\(795\) −0.472136 −0.0167449
\(796\) −20.9787 −0.743571
\(797\) −8.29180 −0.293710 −0.146855 0.989158i \(-0.546915\pi\)
−0.146855 + 0.989158i \(0.546915\pi\)
\(798\) 4.00000 0.141598
\(799\) 57.6525 2.03960
\(800\) 1.00000 0.0353553
\(801\) 12.7639 0.450991
\(802\) −36.0689 −1.27364
\(803\) 0 0
\(804\) −6.09017 −0.214784
\(805\) −0.180340 −0.00635615
\(806\) −20.3262 −0.715961
\(807\) −27.7984 −0.978549
\(808\) 13.3820 0.470776
\(809\) 28.6525 1.00737 0.503684 0.863888i \(-0.331978\pi\)
0.503684 + 0.863888i \(0.331978\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −31.4164 −1.10318 −0.551590 0.834116i \(-0.685979\pi\)
−0.551590 + 0.834116i \(0.685979\pi\)
\(812\) −0.472136 −0.0165687
\(813\) 8.61803 0.302248
\(814\) 0 0
\(815\) 5.85410 0.205060
\(816\) −5.09017 −0.178192
\(817\) −35.8885 −1.25558
\(818\) −33.9787 −1.18804
\(819\) 6.94427 0.242652
\(820\) 3.23607 0.113008
\(821\) 19.3050 0.673747 0.336874 0.941550i \(-0.390630\pi\)
0.336874 + 0.941550i \(0.390630\pi\)
\(822\) −11.7984 −0.411516
\(823\) −17.2361 −0.600812 −0.300406 0.953811i \(-0.597122\pi\)
−0.300406 + 0.953811i \(0.597122\pi\)
\(824\) 11.2361 0.391427
\(825\) 0 0
\(826\) 3.59675 0.125147
\(827\) 43.7082 1.51988 0.759942 0.649991i \(-0.225228\pi\)
0.759942 + 0.649991i \(0.225228\pi\)
\(828\) 0.145898 0.00507031
\(829\) −13.5967 −0.472235 −0.236117 0.971725i \(-0.575875\pi\)
−0.236117 + 0.971725i \(0.575875\pi\)
\(830\) −4.00000 −0.138842
\(831\) −14.5066 −0.503228
\(832\) 5.61803 0.194770
\(833\) −27.8541 −0.965087
\(834\) −13.2361 −0.458328
\(835\) −20.2705 −0.701490
\(836\) 0 0
\(837\) 3.61803 0.125058
\(838\) 16.1459 0.557751
\(839\) −54.6525 −1.88681 −0.943407 0.331639i \(-0.892399\pi\)
−0.943407 + 0.331639i \(0.892399\pi\)
\(840\) 1.23607 0.0426484
\(841\) −28.8541 −0.994969
\(842\) 6.47214 0.223045
\(843\) 16.4721 0.567330
\(844\) −20.7639 −0.714724
\(845\) −18.5623 −0.638563
\(846\) 11.3262 0.389404
\(847\) 0 0
\(848\) −0.472136 −0.0162132
\(849\) 6.67376 0.229043
\(850\) 5.09017 0.174591
\(851\) −1.25735 −0.0431015
\(852\) −1.52786 −0.0523438
\(853\) −36.4721 −1.24878 −0.624391 0.781112i \(-0.714653\pi\)
−0.624391 + 0.781112i \(0.714653\pi\)
\(854\) 0 0
\(855\) 3.23607 0.110671
\(856\) 19.7082 0.673613
\(857\) −45.1591 −1.54260 −0.771302 0.636469i \(-0.780394\pi\)
−0.771302 + 0.636469i \(0.780394\pi\)
\(858\) 0 0
\(859\) 21.1246 0.720762 0.360381 0.932805i \(-0.382647\pi\)
0.360381 + 0.932805i \(0.382647\pi\)
\(860\) −11.0902 −0.378172
\(861\) 4.00000 0.136320
\(862\) −22.3607 −0.761608
\(863\) 47.3394 1.61145 0.805726 0.592289i \(-0.201775\pi\)
0.805726 + 0.592289i \(0.201775\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 18.4721 0.628071
\(866\) 30.8328 1.04774
\(867\) −8.90983 −0.302594
\(868\) −4.47214 −0.151794
\(869\) 0 0
\(870\) −0.381966 −0.0129499
\(871\) 34.2148 1.15932
\(872\) 6.47214 0.219174
\(873\) −12.6525 −0.428221
\(874\) −0.472136 −0.0159702
\(875\) −1.23607 −0.0417867
\(876\) 13.4164 0.453298
\(877\) 48.0344 1.62201 0.811004 0.585041i \(-0.198921\pi\)
0.811004 + 0.585041i \(0.198921\pi\)
\(878\) 17.3262 0.584732
\(879\) 20.0000 0.674583
\(880\) 0 0
\(881\) 8.00000 0.269527 0.134763 0.990878i \(-0.456973\pi\)
0.134763 + 0.990878i \(0.456973\pi\)
\(882\) −5.47214 −0.184256
\(883\) −20.3262 −0.684032 −0.342016 0.939694i \(-0.611110\pi\)
−0.342016 + 0.939694i \(0.611110\pi\)
\(884\) 28.5967 0.961813
\(885\) 2.90983 0.0978129
\(886\) 12.6525 0.425068
\(887\) 13.4508 0.451635 0.225818 0.974170i \(-0.427495\pi\)
0.225818 + 0.974170i \(0.427495\pi\)
\(888\) 8.61803 0.289202
\(889\) −20.5836 −0.690352
\(890\) −12.7639 −0.427848
\(891\) 0 0
\(892\) 2.76393 0.0925433
\(893\) −36.6525 −1.22653
\(894\) 5.38197 0.180000
\(895\) −24.8541 −0.830781
\(896\) 1.23607 0.0412941
\(897\) −0.819660 −0.0273677
\(898\) 29.1246 0.971901
\(899\) 1.38197 0.0460911
\(900\) 1.00000 0.0333333
\(901\) −2.40325 −0.0800639
\(902\) 0 0
\(903\) −13.7082 −0.456180
\(904\) 0.562306 0.0187020
\(905\) 13.8885 0.461671
\(906\) −3.05573 −0.101520
\(907\) 14.5066 0.481683 0.240842 0.970564i \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(908\) −24.3607 −0.808438
\(909\) 13.3820 0.443852
\(910\) −6.94427 −0.230200
\(911\) 12.4721 0.413220 0.206610 0.978423i \(-0.433757\pi\)
0.206610 + 0.978423i \(0.433757\pi\)
\(912\) 3.23607 0.107157
\(913\) 0 0
\(914\) −26.6525 −0.881586
\(915\) 0 0
\(916\) 5.23607 0.173005
\(917\) −6.94427 −0.229320
\(918\) −5.09017 −0.168001
\(919\) −45.3394 −1.49561 −0.747804 0.663919i \(-0.768892\pi\)
−0.747804 + 0.663919i \(0.768892\pi\)
\(920\) −0.145898 −0.00481012
\(921\) −22.4508 −0.739781
\(922\) −28.2148 −0.929204
\(923\) 8.58359 0.282532
\(924\) 0 0
\(925\) −8.61803 −0.283359
\(926\) −23.8885 −0.785026
\(927\) 11.2361 0.369041
\(928\) −0.381966 −0.0125386
\(929\) −21.5279 −0.706306 −0.353153 0.935566i \(-0.614891\pi\)
−0.353153 + 0.935566i \(0.614891\pi\)
\(930\) −3.61803 −0.118640
\(931\) 17.7082 0.580363
\(932\) −24.5623 −0.804565
\(933\) −4.76393 −0.155964
\(934\) −38.5410 −1.26110
\(935\) 0 0
\(936\) 5.61803 0.183631
\(937\) −11.3475 −0.370707 −0.185354 0.982672i \(-0.559343\pi\)
−0.185354 + 0.982672i \(0.559343\pi\)
\(938\) 7.52786 0.245793
\(939\) 0.472136 0.0154076
\(940\) −11.3262 −0.369421
\(941\) 47.9787 1.56406 0.782031 0.623240i \(-0.214184\pi\)
0.782031 + 0.623240i \(0.214184\pi\)
\(942\) −24.2705 −0.790776
\(943\) −0.472136 −0.0153749
\(944\) 2.90983 0.0947069
\(945\) 1.23607 0.0402093
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) −7.09017 −0.230278
\(949\) −75.3738 −2.44674
\(950\) −3.23607 −0.104992
\(951\) 0.583592 0.0189243
\(952\) 6.29180 0.203918
\(953\) −7.78522 −0.252188 −0.126094 0.992018i \(-0.540244\pi\)
−0.126094 + 0.992018i \(0.540244\pi\)
\(954\) −0.472136 −0.0152860
\(955\) −4.76393 −0.154157
\(956\) 1.23607 0.0399773
\(957\) 0 0
\(958\) 17.2361 0.556872
\(959\) 14.5836 0.470929
\(960\) 1.00000 0.0322749
\(961\) −17.9098 −0.577736
\(962\) −48.4164 −1.56101
\(963\) 19.7082 0.635088
\(964\) 11.8885 0.382904
\(965\) 3.05573 0.0983674
\(966\) −0.180340 −0.00580234
\(967\) −9.30495 −0.299227 −0.149614 0.988745i \(-0.547803\pi\)
−0.149614 + 0.988745i \(0.547803\pi\)
\(968\) 0 0
\(969\) 16.4721 0.529161
\(970\) 12.6525 0.406247
\(971\) −36.3607 −1.16687 −0.583435 0.812160i \(-0.698292\pi\)
−0.583435 + 0.812160i \(0.698292\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 16.3607 0.524499
\(974\) −15.4164 −0.493974
\(975\) −5.61803 −0.179921
\(976\) 0 0
\(977\) −24.8328 −0.794472 −0.397236 0.917716i \(-0.630031\pi\)
−0.397236 + 0.917716i \(0.630031\pi\)
\(978\) 5.85410 0.187194
\(979\) 0 0
\(980\) 5.47214 0.174801
\(981\) 6.47214 0.206639
\(982\) −1.03444 −0.0330104
\(983\) 47.1935 1.50524 0.752619 0.658456i \(-0.228790\pi\)
0.752619 + 0.658456i \(0.228790\pi\)
\(984\) 3.23607 0.103162
\(985\) 6.94427 0.221263
\(986\) −1.94427 −0.0619182
\(987\) −14.0000 −0.445625
\(988\) −18.1803 −0.578394
\(989\) 1.61803 0.0514505
\(990\) 0 0
\(991\) −59.5197 −1.89071 −0.945353 0.326048i \(-0.894283\pi\)
−0.945353 + 0.326048i \(0.894283\pi\)
\(992\) −3.61803 −0.114873
\(993\) −4.76393 −0.151179
\(994\) 1.88854 0.0599010
\(995\) 20.9787 0.665070
\(996\) −4.00000 −0.126745
\(997\) 4.03444 0.127772 0.0638860 0.997957i \(-0.479651\pi\)
0.0638860 + 0.997957i \(0.479651\pi\)
\(998\) −23.5279 −0.744762
\(999\) 8.61803 0.272663
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 3630.2.a.bj.1.2 2
11.3 even 5 330.2.m.b.31.1 4
11.4 even 5 330.2.m.b.181.1 yes 4
11.10 odd 2 3630.2.a.bb.1.1 2
33.14 odd 10 990.2.n.e.361.1 4
33.26 odd 10 990.2.n.e.181.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
330.2.m.b.31.1 4 11.3 even 5
330.2.m.b.181.1 yes 4 11.4 even 5
990.2.n.e.181.1 4 33.26 odd 10
990.2.n.e.361.1 4 33.14 odd 10
3630.2.a.bb.1.1 2 11.10 odd 2
3630.2.a.bj.1.2 2 1.1 even 1 trivial