Properties

Label 5070.2.a.bw.1.2
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.35690 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.35690 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.69202 q^{11} +1.00000 q^{12} +3.35690 q^{14} -1.00000 q^{15} +1.00000 q^{16} +0.939001 q^{17} +1.00000 q^{18} -4.85086 q^{19} -1.00000 q^{20} +3.35690 q^{21} +1.69202 q^{22} +4.04892 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +3.35690 q^{28} -8.12498 q^{29} -1.00000 q^{30} +4.08815 q^{31} +1.00000 q^{32} +1.69202 q^{33} +0.939001 q^{34} -3.35690 q^{35} +1.00000 q^{36} +11.6310 q^{37} -4.85086 q^{38} -1.00000 q^{40} +3.86294 q^{41} +3.35690 q^{42} +4.02177 q^{43} +1.69202 q^{44} -1.00000 q^{45} +4.04892 q^{46} -1.27413 q^{47} +1.00000 q^{48} +4.26875 q^{49} +1.00000 q^{50} +0.939001 q^{51} +5.74094 q^{53} +1.00000 q^{54} -1.69202 q^{55} +3.35690 q^{56} -4.85086 q^{57} -8.12498 q^{58} +0.417895 q^{59} -1.00000 q^{60} +0.198062 q^{61} +4.08815 q^{62} +3.35690 q^{63} +1.00000 q^{64} +1.69202 q^{66} +8.93900 q^{67} +0.939001 q^{68} +4.04892 q^{69} -3.35690 q^{70} -5.15883 q^{71} +1.00000 q^{72} -11.5308 q^{73} +11.6310 q^{74} +1.00000 q^{75} -4.85086 q^{76} +5.67994 q^{77} -4.94869 q^{79} -1.00000 q^{80} +1.00000 q^{81} +3.86294 q^{82} +13.1739 q^{83} +3.35690 q^{84} -0.939001 q^{85} +4.02177 q^{86} -8.12498 q^{87} +1.69202 q^{88} -9.47650 q^{89} -1.00000 q^{90} +4.04892 q^{92} +4.08815 q^{93} -1.27413 q^{94} +4.85086 q^{95} +1.00000 q^{96} +15.4547 q^{97} +4.26875 q^{98} +1.69202 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} + 3q^{6} + 6q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} + 3q^{6} + 6q^{7} + 3q^{8} + 3q^{9} - 3q^{10} + 3q^{12} + 6q^{14} - 3q^{15} + 3q^{16} - 7q^{17} + 3q^{18} - q^{19} - 3q^{20} + 6q^{21} + 3q^{23} + 3q^{24} + 3q^{25} + 3q^{27} + 6q^{28} - 3q^{30} + 16q^{31} + 3q^{32} - 7q^{34} - 6q^{35} + 3q^{36} + 20q^{37} - q^{38} - 3q^{40} + 17q^{41} + 6q^{42} + 9q^{43} - 3q^{45} + 3q^{46} + 7q^{47} + 3q^{48} + 5q^{49} + 3q^{50} - 7q^{51} + 3q^{53} + 3q^{54} + 6q^{56} - q^{57} + 7q^{59} - 3q^{60} + 5q^{61} + 16q^{62} + 6q^{63} + 3q^{64} + 17q^{67} - 7q^{68} + 3q^{69} - 6q^{70} - 7q^{71} + 3q^{72} + 3q^{73} + 20q^{74} + 3q^{75} - q^{76} - 7q^{77} + 17q^{79} - 3q^{80} + 3q^{81} + 17q^{82} + 6q^{83} + 6q^{84} + 7q^{85} + 9q^{86} - 3q^{89} - 3q^{90} + 3q^{92} + 16q^{93} + 7q^{94} + q^{95} + 3q^{96} + 24q^{97} + 5q^{98} + O(q^{100}) \)

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
\(7\) 3.35690 1.26879 0.634394 0.773010i \(-0.281250\pi\)
0.634394 + 0.773010i \(0.281250\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 1.69202 0.510164 0.255082 0.966919i \(-0.417898\pi\)
0.255082 + 0.966919i \(0.417898\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 3.35690 0.897168
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 0.939001 0.227741 0.113871 0.993496i \(-0.463675\pi\)
0.113871 + 0.993496i \(0.463675\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.85086 −1.11286 −0.556431 0.830894i \(-0.687830\pi\)
−0.556431 + 0.830894i \(0.687830\pi\)
\(20\) −1.00000 −0.223607
\(21\) 3.35690 0.732535
\(22\) 1.69202 0.360740
\(23\) 4.04892 0.844258 0.422129 0.906536i \(-0.361283\pi\)
0.422129 + 0.906536i \(0.361283\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 3.35690 0.634394
\(29\) −8.12498 −1.50877 −0.754386 0.656432i \(-0.772065\pi\)
−0.754386 + 0.656432i \(0.772065\pi\)
\(30\) −1.00000 −0.182574
\(31\) 4.08815 0.734253 0.367126 0.930171i \(-0.380342\pi\)
0.367126 + 0.930171i \(0.380342\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.69202 0.294543
\(34\) 0.939001 0.161037
\(35\) −3.35690 −0.567419
\(36\) 1.00000 0.166667
\(37\) 11.6310 1.91213 0.956064 0.293157i \(-0.0947060\pi\)
0.956064 + 0.293157i \(0.0947060\pi\)
\(38\) −4.85086 −0.786913
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 3.86294 0.603289 0.301645 0.953420i \(-0.402464\pi\)
0.301645 + 0.953420i \(0.402464\pi\)
\(42\) 3.35690 0.517980
\(43\) 4.02177 0.613314 0.306657 0.951820i \(-0.400790\pi\)
0.306657 + 0.951820i \(0.400790\pi\)
\(44\) 1.69202 0.255082
\(45\) −1.00000 −0.149071
\(46\) 4.04892 0.596980
\(47\) −1.27413 −0.185850 −0.0929252 0.995673i \(-0.529622\pi\)
−0.0929252 + 0.995673i \(0.529622\pi\)
\(48\) 1.00000 0.144338
\(49\) 4.26875 0.609821
\(50\) 1.00000 0.141421
\(51\) 0.939001 0.131486
\(52\) 0 0
\(53\) 5.74094 0.788579 0.394289 0.918986i \(-0.370991\pi\)
0.394289 + 0.918986i \(0.370991\pi\)
\(54\) 1.00000 0.136083
\(55\) −1.69202 −0.228152
\(56\) 3.35690 0.448584
\(57\) −4.85086 −0.642511
\(58\) −8.12498 −1.06686
\(59\) 0.417895 0.0544053 0.0272026 0.999630i \(-0.491340\pi\)
0.0272026 + 0.999630i \(0.491340\pi\)
\(60\) −1.00000 −0.129099
\(61\) 0.198062 0.0253593 0.0126796 0.999920i \(-0.495964\pi\)
0.0126796 + 0.999920i \(0.495964\pi\)
\(62\) 4.08815 0.519195
\(63\) 3.35690 0.422929
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.69202 0.208273
\(67\) 8.93900 1.09207 0.546036 0.837761i \(-0.316136\pi\)
0.546036 + 0.837761i \(0.316136\pi\)
\(68\) 0.939001 0.113871
\(69\) 4.04892 0.487432
\(70\) −3.35690 −0.401226
\(71\) −5.15883 −0.612241 −0.306120 0.951993i \(-0.599031\pi\)
−0.306120 + 0.951993i \(0.599031\pi\)
\(72\) 1.00000 0.117851
\(73\) −11.5308 −1.34958 −0.674789 0.738011i \(-0.735765\pi\)
−0.674789 + 0.738011i \(0.735765\pi\)
\(74\) 11.6310 1.35208
\(75\) 1.00000 0.115470
\(76\) −4.85086 −0.556431
\(77\) 5.67994 0.647289
\(78\) 0 0
\(79\) −4.94869 −0.556771 −0.278386 0.960469i \(-0.589799\pi\)
−0.278386 + 0.960469i \(0.589799\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 3.86294 0.426590
\(83\) 13.1739 1.44602 0.723012 0.690836i \(-0.242757\pi\)
0.723012 + 0.690836i \(0.242757\pi\)
\(84\) 3.35690 0.366267
\(85\) −0.939001 −0.101849
\(86\) 4.02177 0.433679
\(87\) −8.12498 −0.871089
\(88\) 1.69202 0.180370
\(89\) −9.47650 −1.00451 −0.502254 0.864720i \(-0.667496\pi\)
−0.502254 + 0.864720i \(0.667496\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 4.04892 0.422129
\(93\) 4.08815 0.423921
\(94\) −1.27413 −0.131416
\(95\) 4.85086 0.497687
\(96\) 1.00000 0.102062
\(97\) 15.4547 1.56919 0.784595 0.620009i \(-0.212871\pi\)
0.784595 + 0.620009i \(0.212871\pi\)
\(98\) 4.26875 0.431209
\(99\) 1.69202 0.170055
\(100\) 1.00000 0.100000
\(101\) 5.38404 0.535732 0.267866 0.963456i \(-0.413682\pi\)
0.267866 + 0.963456i \(0.413682\pi\)
\(102\) 0.939001 0.0929750
\(103\) 4.08575 0.402581 0.201291 0.979532i \(-0.435486\pi\)
0.201291 + 0.979532i \(0.435486\pi\)
\(104\) 0 0
\(105\) −3.35690 −0.327599
\(106\) 5.74094 0.557609
\(107\) −16.1196 −1.55834 −0.779171 0.626812i \(-0.784359\pi\)
−0.779171 + 0.626812i \(0.784359\pi\)
\(108\) 1.00000 0.0962250
\(109\) −14.2325 −1.36323 −0.681613 0.731713i \(-0.738721\pi\)
−0.681613 + 0.731713i \(0.738721\pi\)
\(110\) −1.69202 −0.161328
\(111\) 11.6310 1.10397
\(112\) 3.35690 0.317197
\(113\) −13.3274 −1.25373 −0.626866 0.779127i \(-0.715663\pi\)
−0.626866 + 0.779127i \(0.715663\pi\)
\(114\) −4.85086 −0.454324
\(115\) −4.04892 −0.377563
\(116\) −8.12498 −0.754386
\(117\) 0 0
\(118\) 0.417895 0.0384703
\(119\) 3.15213 0.288955
\(120\) −1.00000 −0.0912871
\(121\) −8.13706 −0.739733
\(122\) 0.198062 0.0179317
\(123\) 3.86294 0.348309
\(124\) 4.08815 0.367126
\(125\) −1.00000 −0.0894427
\(126\) 3.35690 0.299056
\(127\) −12.8509 −1.14033 −0.570164 0.821531i \(-0.693121\pi\)
−0.570164 + 0.821531i \(0.693121\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.02177 0.354097
\(130\) 0 0
\(131\) −0.570024 −0.0498032 −0.0249016 0.999690i \(-0.507927\pi\)
−0.0249016 + 0.999690i \(0.507927\pi\)
\(132\) 1.69202 0.147272
\(133\) −16.2838 −1.41199
\(134\) 8.93900 0.772212
\(135\) −1.00000 −0.0860663
\(136\) 0.939001 0.0805187
\(137\) 21.5405 1.84033 0.920164 0.391533i \(-0.128055\pi\)
0.920164 + 0.391533i \(0.128055\pi\)
\(138\) 4.04892 0.344667
\(139\) 7.09783 0.602030 0.301015 0.953619i \(-0.402674\pi\)
0.301015 + 0.953619i \(0.402674\pi\)
\(140\) −3.35690 −0.283709
\(141\) −1.27413 −0.107301
\(142\) −5.15883 −0.432920
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 8.12498 0.674743
\(146\) −11.5308 −0.954295
\(147\) 4.26875 0.352081
\(148\) 11.6310 0.956064
\(149\) −1.14914 −0.0941416 −0.0470708 0.998892i \(-0.514989\pi\)
−0.0470708 + 0.998892i \(0.514989\pi\)
\(150\) 1.00000 0.0816497
\(151\) 0.0217703 0.00177164 0.000885819 1.00000i \(-0.499718\pi\)
0.000885819 1.00000i \(0.499718\pi\)
\(152\) −4.85086 −0.393456
\(153\) 0.939001 0.0759137
\(154\) 5.67994 0.457703
\(155\) −4.08815 −0.328368
\(156\) 0 0
\(157\) −10.1588 −0.810763 −0.405382 0.914148i \(-0.632861\pi\)
−0.405382 + 0.914148i \(0.632861\pi\)
\(158\) −4.94869 −0.393697
\(159\) 5.74094 0.455286
\(160\) −1.00000 −0.0790569
\(161\) 13.5918 1.07118
\(162\) 1.00000 0.0785674
\(163\) 19.1511 1.50003 0.750014 0.661422i \(-0.230047\pi\)
0.750014 + 0.661422i \(0.230047\pi\)
\(164\) 3.86294 0.301645
\(165\) −1.69202 −0.131724
\(166\) 13.1739 1.02249
\(167\) −15.2174 −1.17756 −0.588780 0.808293i \(-0.700392\pi\)
−0.588780 + 0.808293i \(0.700392\pi\)
\(168\) 3.35690 0.258990
\(169\) 0 0
\(170\) −0.939001 −0.0720181
\(171\) −4.85086 −0.370954
\(172\) 4.02177 0.306657
\(173\) 21.8213 1.65904 0.829522 0.558474i \(-0.188613\pi\)
0.829522 + 0.558474i \(0.188613\pi\)
\(174\) −8.12498 −0.615953
\(175\) 3.35690 0.253757
\(176\) 1.69202 0.127541
\(177\) 0.417895 0.0314109
\(178\) −9.47650 −0.710294
\(179\) −20.2513 −1.51365 −0.756826 0.653616i \(-0.773251\pi\)
−0.756826 + 0.653616i \(0.773251\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 14.4969 1.07755 0.538775 0.842450i \(-0.318887\pi\)
0.538775 + 0.842450i \(0.318887\pi\)
\(182\) 0 0
\(183\) 0.198062 0.0146412
\(184\) 4.04892 0.298490
\(185\) −11.6310 −0.855130
\(186\) 4.08815 0.299757
\(187\) 1.58881 0.116185
\(188\) −1.27413 −0.0929252
\(189\) 3.35690 0.244178
\(190\) 4.85086 0.351918
\(191\) −10.1806 −0.736643 −0.368321 0.929699i \(-0.620067\pi\)
−0.368321 + 0.929699i \(0.620067\pi\)
\(192\) 1.00000 0.0721688
\(193\) 27.6407 1.98962 0.994811 0.101741i \(-0.0324412\pi\)
0.994811 + 0.101741i \(0.0324412\pi\)
\(194\) 15.4547 1.10958
\(195\) 0 0
\(196\) 4.26875 0.304911
\(197\) −13.0261 −0.928070 −0.464035 0.885817i \(-0.653599\pi\)
−0.464035 + 0.885817i \(0.653599\pi\)
\(198\) 1.69202 0.120247
\(199\) 6.67563 0.473223 0.236611 0.971604i \(-0.423963\pi\)
0.236611 + 0.971604i \(0.423963\pi\)
\(200\) 1.00000 0.0707107
\(201\) 8.93900 0.630509
\(202\) 5.38404 0.378820
\(203\) −27.2747 −1.91431
\(204\) 0.939001 0.0657432
\(205\) −3.86294 −0.269799
\(206\) 4.08575 0.284668
\(207\) 4.04892 0.281419
\(208\) 0 0
\(209\) −8.20775 −0.567742
\(210\) −3.35690 −0.231648
\(211\) −11.3056 −0.778309 −0.389154 0.921173i \(-0.627233\pi\)
−0.389154 + 0.921173i \(0.627233\pi\)
\(212\) 5.74094 0.394289
\(213\) −5.15883 −0.353477
\(214\) −16.1196 −1.10191
\(215\) −4.02177 −0.274282
\(216\) 1.00000 0.0680414
\(217\) 13.7235 0.931611
\(218\) −14.2325 −0.963947
\(219\) −11.5308 −0.779179
\(220\) −1.69202 −0.114076
\(221\) 0 0
\(222\) 11.6310 0.780623
\(223\) 18.0911 1.21147 0.605736 0.795666i \(-0.292879\pi\)
0.605736 + 0.795666i \(0.292879\pi\)
\(224\) 3.35690 0.224292
\(225\) 1.00000 0.0666667
\(226\) −13.3274 −0.886523
\(227\) −9.37196 −0.622039 −0.311019 0.950404i \(-0.600670\pi\)
−0.311019 + 0.950404i \(0.600670\pi\)
\(228\) −4.85086 −0.321256
\(229\) 21.2664 1.40532 0.702660 0.711526i \(-0.251995\pi\)
0.702660 + 0.711526i \(0.251995\pi\)
\(230\) −4.04892 −0.266978
\(231\) 5.67994 0.373713
\(232\) −8.12498 −0.533431
\(233\) 2.03146 0.133085 0.0665427 0.997784i \(-0.478803\pi\)
0.0665427 + 0.997784i \(0.478803\pi\)
\(234\) 0 0
\(235\) 1.27413 0.0831149
\(236\) 0.417895 0.0272026
\(237\) −4.94869 −0.321452
\(238\) 3.15213 0.204322
\(239\) −11.9903 −0.775589 −0.387794 0.921746i \(-0.626763\pi\)
−0.387794 + 0.921746i \(0.626763\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 21.7627 1.40186 0.700930 0.713230i \(-0.252769\pi\)
0.700930 + 0.713230i \(0.252769\pi\)
\(242\) −8.13706 −0.523070
\(243\) 1.00000 0.0641500
\(244\) 0.198062 0.0126796
\(245\) −4.26875 −0.272720
\(246\) 3.86294 0.246292
\(247\) 0 0
\(248\) 4.08815 0.259598
\(249\) 13.1739 0.834862
\(250\) −1.00000 −0.0632456
\(251\) 13.7506 0.867932 0.433966 0.900929i \(-0.357114\pi\)
0.433966 + 0.900929i \(0.357114\pi\)
\(252\) 3.35690 0.211465
\(253\) 6.85086 0.430710
\(254\) −12.8509 −0.806334
\(255\) −0.939001 −0.0588025
\(256\) 1.00000 0.0625000
\(257\) −24.1782 −1.50820 −0.754098 0.656762i \(-0.771925\pi\)
−0.754098 + 0.656762i \(0.771925\pi\)
\(258\) 4.02177 0.250384
\(259\) 39.0441 2.42608
\(260\) 0 0
\(261\) −8.12498 −0.502924
\(262\) −0.570024 −0.0352162
\(263\) 0.960771 0.0592437 0.0296218 0.999561i \(-0.490570\pi\)
0.0296218 + 0.999561i \(0.490570\pi\)
\(264\) 1.69202 0.104137
\(265\) −5.74094 −0.352663
\(266\) −16.2838 −0.998425
\(267\) −9.47650 −0.579952
\(268\) 8.93900 0.546036
\(269\) 10.0761 0.614348 0.307174 0.951653i \(-0.400617\pi\)
0.307174 + 0.951653i \(0.400617\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −14.9584 −0.908657 −0.454328 0.890834i \(-0.650121\pi\)
−0.454328 + 0.890834i \(0.650121\pi\)
\(272\) 0.939001 0.0569353
\(273\) 0 0
\(274\) 21.5405 1.30131
\(275\) 1.69202 0.102033
\(276\) 4.04892 0.243716
\(277\) −13.0465 −0.783890 −0.391945 0.919989i \(-0.628198\pi\)
−0.391945 + 0.919989i \(0.628198\pi\)
\(278\) 7.09783 0.425700
\(279\) 4.08815 0.244751
\(280\) −3.35690 −0.200613
\(281\) 16.2881 0.971668 0.485834 0.874051i \(-0.338516\pi\)
0.485834 + 0.874051i \(0.338516\pi\)
\(282\) −1.27413 −0.0758731
\(283\) −12.3773 −0.735756 −0.367878 0.929874i \(-0.619916\pi\)
−0.367878 + 0.929874i \(0.619916\pi\)
\(284\) −5.15883 −0.306120
\(285\) 4.85086 0.287340
\(286\) 0 0
\(287\) 12.9675 0.765446
\(288\) 1.00000 0.0589256
\(289\) −16.1183 −0.948134
\(290\) 8.12498 0.477115
\(291\) 15.4547 0.905972
\(292\) −11.5308 −0.674789
\(293\) −1.73663 −0.101455 −0.0507274 0.998713i \(-0.516154\pi\)
−0.0507274 + 0.998713i \(0.516154\pi\)
\(294\) 4.26875 0.248959
\(295\) −0.417895 −0.0243308
\(296\) 11.6310 0.676039
\(297\) 1.69202 0.0981810
\(298\) −1.14914 −0.0665682
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 13.5007 0.778165
\(302\) 0.0217703 0.00125274
\(303\) 5.38404 0.309305
\(304\) −4.85086 −0.278216
\(305\) −0.198062 −0.0113410
\(306\) 0.939001 0.0536791
\(307\) 25.4403 1.45195 0.725976 0.687720i \(-0.241388\pi\)
0.725976 + 0.687720i \(0.241388\pi\)
\(308\) 5.67994 0.323645
\(309\) 4.08575 0.232430
\(310\) −4.08815 −0.232191
\(311\) −10.0325 −0.568892 −0.284446 0.958692i \(-0.591810\pi\)
−0.284446 + 0.958692i \(0.591810\pi\)
\(312\) 0 0
\(313\) −22.3274 −1.26202 −0.631008 0.775776i \(-0.717359\pi\)
−0.631008 + 0.775776i \(0.717359\pi\)
\(314\) −10.1588 −0.573296
\(315\) −3.35690 −0.189140
\(316\) −4.94869 −0.278386
\(317\) −5.33811 −0.299818 −0.149909 0.988700i \(-0.547898\pi\)
−0.149909 + 0.988700i \(0.547898\pi\)
\(318\) 5.74094 0.321936
\(319\) −13.7476 −0.769720
\(320\) −1.00000 −0.0559017
\(321\) −16.1196 −0.899709
\(322\) 13.5918 0.757441
\(323\) −4.55496 −0.253445
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 19.1511 1.06068
\(327\) −14.2325 −0.787059
\(328\) 3.86294 0.213295
\(329\) −4.27711 −0.235805
\(330\) −1.69202 −0.0931427
\(331\) −4.49827 −0.247247 −0.123624 0.992329i \(-0.539452\pi\)
−0.123624 + 0.992329i \(0.539452\pi\)
\(332\) 13.1739 0.723012
\(333\) 11.6310 0.637376
\(334\) −15.2174 −0.832661
\(335\) −8.93900 −0.488390
\(336\) 3.35690 0.183134
\(337\) −11.5676 −0.630129 −0.315064 0.949070i \(-0.602026\pi\)
−0.315064 + 0.949070i \(0.602026\pi\)
\(338\) 0 0
\(339\) −13.3274 −0.723843
\(340\) −0.939001 −0.0509245
\(341\) 6.91723 0.374589
\(342\) −4.85086 −0.262304
\(343\) −9.16852 −0.495054
\(344\) 4.02177 0.216839
\(345\) −4.04892 −0.217986
\(346\) 21.8213 1.17312
\(347\) 27.1226 1.45602 0.728008 0.685568i \(-0.240446\pi\)
0.728008 + 0.685568i \(0.240446\pi\)
\(348\) −8.12498 −0.435545
\(349\) 12.8562 0.688178 0.344089 0.938937i \(-0.388188\pi\)
0.344089 + 0.938937i \(0.388188\pi\)
\(350\) 3.35690 0.179434
\(351\) 0 0
\(352\) 1.69202 0.0901850
\(353\) 24.6262 1.31072 0.655361 0.755316i \(-0.272516\pi\)
0.655361 + 0.755316i \(0.272516\pi\)
\(354\) 0.417895 0.0222109
\(355\) 5.15883 0.273802
\(356\) −9.47650 −0.502254
\(357\) 3.15213 0.166828
\(358\) −20.2513 −1.07031
\(359\) −23.8629 −1.25944 −0.629719 0.776823i \(-0.716830\pi\)
−0.629719 + 0.776823i \(0.716830\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 4.53079 0.238463
\(362\) 14.4969 0.761942
\(363\) −8.13706 −0.427085
\(364\) 0 0
\(365\) 11.5308 0.603549
\(366\) 0.198062 0.0103529
\(367\) 33.1933 1.73267 0.866337 0.499459i \(-0.166468\pi\)
0.866337 + 0.499459i \(0.166468\pi\)
\(368\) 4.04892 0.211064
\(369\) 3.86294 0.201096
\(370\) −11.6310 −0.604668
\(371\) 19.2717 1.00054
\(372\) 4.08815 0.211960
\(373\) −10.8465 −0.561613 −0.280806 0.959764i \(-0.590602\pi\)
−0.280806 + 0.959764i \(0.590602\pi\)
\(374\) 1.58881 0.0821554
\(375\) −1.00000 −0.0516398
\(376\) −1.27413 −0.0657081
\(377\) 0 0
\(378\) 3.35690 0.172660
\(379\) −6.09485 −0.313071 −0.156536 0.987672i \(-0.550033\pi\)
−0.156536 + 0.987672i \(0.550033\pi\)
\(380\) 4.85086 0.248844
\(381\) −12.8509 −0.658369
\(382\) −10.1806 −0.520885
\(383\) −7.95407 −0.406434 −0.203217 0.979134i \(-0.565140\pi\)
−0.203217 + 0.979134i \(0.565140\pi\)
\(384\) 1.00000 0.0510310
\(385\) −5.67994 −0.289477
\(386\) 27.6407 1.40688
\(387\) 4.02177 0.204438
\(388\) 15.4547 0.784595
\(389\) 1.50365 0.0762380 0.0381190 0.999273i \(-0.487863\pi\)
0.0381190 + 0.999273i \(0.487863\pi\)
\(390\) 0 0
\(391\) 3.80194 0.192272
\(392\) 4.26875 0.215604
\(393\) −0.570024 −0.0287539
\(394\) −13.0261 −0.656245
\(395\) 4.94869 0.248996
\(396\) 1.69202 0.0850273
\(397\) 34.9627 1.75473 0.877364 0.479826i \(-0.159300\pi\)
0.877364 + 0.479826i \(0.159300\pi\)
\(398\) 6.67563 0.334619
\(399\) −16.2838 −0.815210
\(400\) 1.00000 0.0500000
\(401\) −31.1148 −1.55380 −0.776900 0.629624i \(-0.783209\pi\)
−0.776900 + 0.629624i \(0.783209\pi\)
\(402\) 8.93900 0.445837
\(403\) 0 0
\(404\) 5.38404 0.267866
\(405\) −1.00000 −0.0496904
\(406\) −27.2747 −1.35362
\(407\) 19.6799 0.975498
\(408\) 0.939001 0.0464875
\(409\) −32.7318 −1.61849 −0.809243 0.587474i \(-0.800122\pi\)
−0.809243 + 0.587474i \(0.800122\pi\)
\(410\) −3.86294 −0.190777
\(411\) 21.5405 1.06251
\(412\) 4.08575 0.201291
\(413\) 1.40283 0.0690287
\(414\) 4.04892 0.198993
\(415\) −13.1739 −0.646681
\(416\) 0 0
\(417\) 7.09783 0.347582
\(418\) −8.20775 −0.401454
\(419\) 13.0694 0.638480 0.319240 0.947674i \(-0.396572\pi\)
0.319240 + 0.947674i \(0.396572\pi\)
\(420\) −3.35690 −0.163800
\(421\) 16.4789 0.803132 0.401566 0.915830i \(-0.368466\pi\)
0.401566 + 0.915830i \(0.368466\pi\)
\(422\) −11.3056 −0.550347
\(423\) −1.27413 −0.0619502
\(424\) 5.74094 0.278805
\(425\) 0.939001 0.0455482
\(426\) −5.15883 −0.249946
\(427\) 0.664874 0.0321755
\(428\) −16.1196 −0.779171
\(429\) 0 0
\(430\) −4.02177 −0.193947
\(431\) −41.0640 −1.97798 −0.988991 0.147974i \(-0.952725\pi\)
−0.988991 + 0.147974i \(0.952725\pi\)
\(432\) 1.00000 0.0481125
\(433\) −6.04593 −0.290549 −0.145275 0.989391i \(-0.546407\pi\)
−0.145275 + 0.989391i \(0.546407\pi\)
\(434\) 13.7235 0.658748
\(435\) 8.12498 0.389563
\(436\) −14.2325 −0.681613
\(437\) −19.6407 −0.939543
\(438\) −11.5308 −0.550963
\(439\) −26.2543 −1.25305 −0.626524 0.779402i \(-0.715523\pi\)
−0.626524 + 0.779402i \(0.715523\pi\)
\(440\) −1.69202 −0.0806640
\(441\) 4.26875 0.203274
\(442\) 0 0
\(443\) −39.3846 −1.87122 −0.935610 0.353035i \(-0.885150\pi\)
−0.935610 + 0.353035i \(0.885150\pi\)
\(444\) 11.6310 0.551984
\(445\) 9.47650 0.449229
\(446\) 18.0911 0.856640
\(447\) −1.14914 −0.0543527
\(448\) 3.35690 0.158598
\(449\) −9.39075 −0.443177 −0.221588 0.975140i \(-0.571124\pi\)
−0.221588 + 0.975140i \(0.571124\pi\)
\(450\) 1.00000 0.0471405
\(451\) 6.53617 0.307776
\(452\) −13.3274 −0.626866
\(453\) 0.0217703 0.00102286
\(454\) −9.37196 −0.439848
\(455\) 0 0
\(456\) −4.85086 −0.227162
\(457\) −2.66487 −0.124658 −0.0623288 0.998056i \(-0.519853\pi\)
−0.0623288 + 0.998056i \(0.519853\pi\)
\(458\) 21.2664 0.993712
\(459\) 0.939001 0.0438288
\(460\) −4.04892 −0.188782
\(461\) −24.5284 −1.14240 −0.571201 0.820810i \(-0.693522\pi\)
−0.571201 + 0.820810i \(0.693522\pi\)
\(462\) 5.67994 0.264255
\(463\) −4.50498 −0.209364 −0.104682 0.994506i \(-0.533382\pi\)
−0.104682 + 0.994506i \(0.533382\pi\)
\(464\) −8.12498 −0.377193
\(465\) −4.08815 −0.189583
\(466\) 2.03146 0.0941055
\(467\) 21.3317 0.987112 0.493556 0.869714i \(-0.335697\pi\)
0.493556 + 0.869714i \(0.335697\pi\)
\(468\) 0 0
\(469\) 30.0073 1.38561
\(470\) 1.27413 0.0587711
\(471\) −10.1588 −0.468094
\(472\) 0.417895 0.0192352
\(473\) 6.80492 0.312891
\(474\) −4.94869 −0.227301
\(475\) −4.85086 −0.222572
\(476\) 3.15213 0.144478
\(477\) 5.74094 0.262860
\(478\) −11.9903 −0.548424
\(479\) −32.9517 −1.50560 −0.752800 0.658249i \(-0.771297\pi\)
−0.752800 + 0.658249i \(0.771297\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 21.7627 0.991264
\(483\) 13.5918 0.618448
\(484\) −8.13706 −0.369867
\(485\) −15.4547 −0.701763
\(486\) 1.00000 0.0453609
\(487\) −9.33034 −0.422798 −0.211399 0.977400i \(-0.567802\pi\)
−0.211399 + 0.977400i \(0.567802\pi\)
\(488\) 0.198062 0.00896586
\(489\) 19.1511 0.866041
\(490\) −4.26875 −0.192842
\(491\) −20.4349 −0.922213 −0.461107 0.887345i \(-0.652547\pi\)
−0.461107 + 0.887345i \(0.652547\pi\)
\(492\) 3.86294 0.174155
\(493\) −7.62937 −0.343609
\(494\) 0 0
\(495\) −1.69202 −0.0760507
\(496\) 4.08815 0.183563
\(497\) −17.3177 −0.776804
\(498\) 13.1739 0.590337
\(499\) 3.54958 0.158901 0.0794505 0.996839i \(-0.474683\pi\)
0.0794505 + 0.996839i \(0.474683\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −15.2174 −0.679865
\(502\) 13.7506 0.613721
\(503\) −19.8146 −0.883490 −0.441745 0.897141i \(-0.645640\pi\)
−0.441745 + 0.897141i \(0.645640\pi\)
\(504\) 3.35690 0.149528
\(505\) −5.38404 −0.239587
\(506\) 6.85086 0.304558
\(507\) 0 0
\(508\) −12.8509 −0.570164
\(509\) 32.4325 1.43754 0.718772 0.695246i \(-0.244704\pi\)
0.718772 + 0.695246i \(0.244704\pi\)
\(510\) −0.939001 −0.0415797
\(511\) −38.7077 −1.71233
\(512\) 1.00000 0.0441942
\(513\) −4.85086 −0.214170
\(514\) −24.1782 −1.06646
\(515\) −4.08575 −0.180040
\(516\) 4.02177 0.177049
\(517\) −2.15585 −0.0948142
\(518\) 39.0441 1.71550
\(519\) 21.8213 0.957849
\(520\) 0 0
\(521\) 7.87694 0.345095 0.172547 0.985001i \(-0.444800\pi\)
0.172547 + 0.985001i \(0.444800\pi\)
\(522\) −8.12498 −0.355621
\(523\) −23.4276 −1.02442 −0.512208 0.858861i \(-0.671172\pi\)
−0.512208 + 0.858861i \(0.671172\pi\)
\(524\) −0.570024 −0.0249016
\(525\) 3.35690 0.146507
\(526\) 0.960771 0.0418916
\(527\) 3.83877 0.167220
\(528\) 1.69202 0.0736358
\(529\) −6.60627 −0.287229
\(530\) −5.74094 −0.249370
\(531\) 0.417895 0.0181351
\(532\) −16.2838 −0.705993
\(533\) 0 0
\(534\) −9.47650 −0.410088
\(535\) 16.1196 0.696911
\(536\) 8.93900 0.386106
\(537\) −20.2513 −0.873908
\(538\) 10.0761 0.434410
\(539\) 7.22282 0.311109
\(540\) −1.00000 −0.0430331
\(541\) −1.21552 −0.0522593 −0.0261297 0.999659i \(-0.508318\pi\)
−0.0261297 + 0.999659i \(0.508318\pi\)
\(542\) −14.9584 −0.642517
\(543\) 14.4969 0.622123
\(544\) 0.939001 0.0402593
\(545\) 14.2325 0.609654
\(546\) 0 0
\(547\) −15.1371 −0.647214 −0.323607 0.946192i \(-0.604896\pi\)
−0.323607 + 0.946192i \(0.604896\pi\)
\(548\) 21.5405 0.920164
\(549\) 0.198062 0.00845309
\(550\) 1.69202 0.0721480
\(551\) 39.4131 1.67905
\(552\) 4.04892 0.172333
\(553\) −16.6122 −0.706424
\(554\) −13.0465 −0.554294
\(555\) −11.6310 −0.493709
\(556\) 7.09783 0.301015
\(557\) 0.147817 0.00626321 0.00313160 0.999995i \(-0.499003\pi\)
0.00313160 + 0.999995i \(0.499003\pi\)
\(558\) 4.08815 0.173065
\(559\) 0 0
\(560\) −3.35690 −0.141855
\(561\) 1.58881 0.0670796
\(562\) 16.2881 0.687073
\(563\) 40.6945 1.71507 0.857535 0.514426i \(-0.171995\pi\)
0.857535 + 0.514426i \(0.171995\pi\)
\(564\) −1.27413 −0.0536504
\(565\) 13.3274 0.560686
\(566\) −12.3773 −0.520258
\(567\) 3.35690 0.140976
\(568\) −5.15883 −0.216460
\(569\) −26.2433 −1.10017 −0.550087 0.835107i \(-0.685406\pi\)
−0.550087 + 0.835107i \(0.685406\pi\)
\(570\) 4.85086 0.203180
\(571\) 2.31575 0.0969110 0.0484555 0.998825i \(-0.484570\pi\)
0.0484555 + 0.998825i \(0.484570\pi\)
\(572\) 0 0
\(573\) −10.1806 −0.425301
\(574\) 12.9675 0.541252
\(575\) 4.04892 0.168852
\(576\) 1.00000 0.0416667
\(577\) −14.5241 −0.604646 −0.302323 0.953206i \(-0.597762\pi\)
−0.302323 + 0.953206i \(0.597762\pi\)
\(578\) −16.1183 −0.670432
\(579\) 27.6407 1.14871
\(580\) 8.12498 0.337372
\(581\) 44.2234 1.83470
\(582\) 15.4547 0.640619
\(583\) 9.71379 0.402304
\(584\) −11.5308 −0.477148
\(585\) 0 0
\(586\) −1.73663 −0.0717394
\(587\) −32.9028 −1.35804 −0.679021 0.734119i \(-0.737596\pi\)
−0.679021 + 0.734119i \(0.737596\pi\)
\(588\) 4.26875 0.176040
\(589\) −19.8310 −0.817122
\(590\) −0.417895 −0.0172045
\(591\) −13.0261 −0.535821
\(592\) 11.6310 0.478032
\(593\) −36.0146 −1.47894 −0.739471 0.673188i \(-0.764924\pi\)
−0.739471 + 0.673188i \(0.764924\pi\)
\(594\) 1.69202 0.0694245
\(595\) −3.15213 −0.129225
\(596\) −1.14914 −0.0470708
\(597\) 6.67563 0.273215
\(598\) 0 0
\(599\) −29.4644 −1.20388 −0.601942 0.798540i \(-0.705606\pi\)
−0.601942 + 0.798540i \(0.705606\pi\)
\(600\) 1.00000 0.0408248
\(601\) −28.4004 −1.15848 −0.579239 0.815158i \(-0.696650\pi\)
−0.579239 + 0.815158i \(0.696650\pi\)
\(602\) 13.5007 0.550246
\(603\) 8.93900 0.364024
\(604\) 0.0217703 0.000885819 0
\(605\) 8.13706 0.330819
\(606\) 5.38404 0.218712
\(607\) 15.6450 0.635012 0.317506 0.948256i \(-0.397155\pi\)
0.317506 + 0.948256i \(0.397155\pi\)
\(608\) −4.85086 −0.196728
\(609\) −27.2747 −1.10523
\(610\) −0.198062 −0.00801931
\(611\) 0 0
\(612\) 0.939001 0.0379569
\(613\) −17.6329 −0.712188 −0.356094 0.934450i \(-0.615892\pi\)
−0.356094 + 0.934450i \(0.615892\pi\)
\(614\) 25.4403 1.02669
\(615\) −3.86294 −0.155769
\(616\) 5.67994 0.228851
\(617\) 10.8582 0.437133 0.218566 0.975822i \(-0.429862\pi\)
0.218566 + 0.975822i \(0.429862\pi\)
\(618\) 4.08575 0.164353
\(619\) −34.2083 −1.37495 −0.687475 0.726208i \(-0.741281\pi\)
−0.687475 + 0.726208i \(0.741281\pi\)
\(620\) −4.08815 −0.164184
\(621\) 4.04892 0.162477
\(622\) −10.0325 −0.402268
\(623\) −31.8116 −1.27451
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −22.3274 −0.892381
\(627\) −8.20775 −0.327786
\(628\) −10.1588 −0.405382
\(629\) 10.9215 0.435470
\(630\) −3.35690 −0.133742
\(631\) 31.9571 1.27219 0.636095 0.771611i \(-0.280549\pi\)
0.636095 + 0.771611i \(0.280549\pi\)
\(632\) −4.94869 −0.196848
\(633\) −11.3056 −0.449357
\(634\) −5.33811 −0.212003
\(635\) 12.8509 0.509971
\(636\) 5.74094 0.227643
\(637\) 0 0
\(638\) −13.7476 −0.544274
\(639\) −5.15883 −0.204080
\(640\) −1.00000 −0.0395285
\(641\) −45.8297 −1.81016 −0.905082 0.425238i \(-0.860190\pi\)
−0.905082 + 0.425238i \(0.860190\pi\)
\(642\) −16.1196 −0.636190
\(643\) 20.6418 0.814032 0.407016 0.913421i \(-0.366569\pi\)
0.407016 + 0.913421i \(0.366569\pi\)
\(644\) 13.5918 0.535592
\(645\) −4.02177 −0.158357
\(646\) −4.55496 −0.179212
\(647\) 1.40389 0.0551928 0.0275964 0.999619i \(-0.491215\pi\)
0.0275964 + 0.999619i \(0.491215\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0.707087 0.0277556
\(650\) 0 0
\(651\) 13.7235 0.537866
\(652\) 19.1511 0.750014
\(653\) −4.54229 −0.177753 −0.0888767 0.996043i \(-0.528328\pi\)
−0.0888767 + 0.996043i \(0.528328\pi\)
\(654\) −14.2325 −0.556535
\(655\) 0.570024 0.0222727
\(656\) 3.86294 0.150822
\(657\) −11.5308 −0.449859
\(658\) −4.27711 −0.166739
\(659\) 17.4161 0.678435 0.339217 0.940708i \(-0.389838\pi\)
0.339217 + 0.940708i \(0.389838\pi\)
\(660\) −1.69202 −0.0658618
\(661\) 43.3682 1.68683 0.843415 0.537263i \(-0.180542\pi\)
0.843415 + 0.537263i \(0.180542\pi\)
\(662\) −4.49827 −0.174830
\(663\) 0 0
\(664\) 13.1739 0.511246
\(665\) 16.2838 0.631459
\(666\) 11.6310 0.450693
\(667\) −32.8974 −1.27379
\(668\) −15.2174 −0.588780
\(669\) 18.0911 0.699443
\(670\) −8.93900 −0.345344
\(671\) 0.335126 0.0129374
\(672\) 3.35690 0.129495
\(673\) −3.49875 −0.134867 −0.0674334 0.997724i \(-0.521481\pi\)
−0.0674334 + 0.997724i \(0.521481\pi\)
\(674\) −11.5676 −0.445568
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −13.3284 −0.512253 −0.256126 0.966643i \(-0.582446\pi\)
−0.256126 + 0.966643i \(0.582446\pi\)
\(678\) −13.3274 −0.511834
\(679\) 51.8799 1.99097
\(680\) −0.939001 −0.0360090
\(681\) −9.37196 −0.359134
\(682\) 6.91723 0.264874
\(683\) −24.1739 −0.924989 −0.462494 0.886622i \(-0.653045\pi\)
−0.462494 + 0.886622i \(0.653045\pi\)
\(684\) −4.85086 −0.185477
\(685\) −21.5405 −0.823020
\(686\) −9.16852 −0.350056
\(687\) 21.2664 0.811362
\(688\) 4.02177 0.153329
\(689\) 0 0
\(690\) −4.04892 −0.154140
\(691\) −37.8810 −1.44106 −0.720530 0.693423i \(-0.756102\pi\)
−0.720530 + 0.693423i \(0.756102\pi\)
\(692\) 21.8213 0.829522
\(693\) 5.67994 0.215763
\(694\) 27.1226 1.02956
\(695\) −7.09783 −0.269236
\(696\) −8.12498 −0.307977
\(697\) 3.62730 0.137394
\(698\) 12.8562 0.486616
\(699\) 2.03146 0.0768368
\(700\) 3.35690 0.126879
\(701\) −21.5967 −0.815696 −0.407848 0.913050i \(-0.633721\pi\)
−0.407848 + 0.913050i \(0.633721\pi\)
\(702\) 0 0
\(703\) −56.4204 −2.12794
\(704\) 1.69202 0.0637705
\(705\) 1.27413 0.0479864
\(706\) 24.6262 0.926821
\(707\) 18.0737 0.679730
\(708\) 0.417895 0.0157054
\(709\) 41.3062 1.55129 0.775643 0.631172i \(-0.217426\pi\)
0.775643 + 0.631172i \(0.217426\pi\)
\(710\) 5.15883 0.193608
\(711\) −4.94869 −0.185590
\(712\) −9.47650 −0.355147
\(713\) 16.5526 0.619898
\(714\) 3.15213 0.117965
\(715\) 0 0
\(716\) −20.2513 −0.756826
\(717\) −11.9903 −0.447786
\(718\) −23.8629 −0.890557
\(719\) −35.5338 −1.32519 −0.662593 0.748980i \(-0.730544\pi\)
−0.662593 + 0.748980i \(0.730544\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 13.7154 0.510790
\(722\) 4.53079 0.168619
\(723\) 21.7627 0.809364
\(724\) 14.4969 0.538775
\(725\) −8.12498 −0.301754
\(726\) −8.13706 −0.301995
\(727\) 19.6635 0.729281 0.364640 0.931148i \(-0.381192\pi\)
0.364640 + 0.931148i \(0.381192\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 11.5308 0.426774
\(731\) 3.77645 0.139677
\(732\) 0.198062 0.00732059
\(733\) −46.8702 −1.73119 −0.865596 0.500743i \(-0.833060\pi\)
−0.865596 + 0.500743i \(0.833060\pi\)
\(734\) 33.1933 1.22519
\(735\) −4.26875 −0.157455
\(736\) 4.04892 0.149245
\(737\) 15.1250 0.557136
\(738\) 3.86294 0.142197
\(739\) −8.57779 −0.315539 −0.157770 0.987476i \(-0.550430\pi\)
−0.157770 + 0.987476i \(0.550430\pi\)
\(740\) −11.6310 −0.427565
\(741\) 0 0
\(742\) 19.2717 0.707488
\(743\) −44.9506 −1.64908 −0.824539 0.565805i \(-0.808565\pi\)
−0.824539 + 0.565805i \(0.808565\pi\)
\(744\) 4.08815 0.149879
\(745\) 1.14914 0.0421014
\(746\) −10.8465 −0.397120
\(747\) 13.1739 0.482008
\(748\) 1.58881 0.0580926
\(749\) −54.1118 −1.97720
\(750\) −1.00000 −0.0365148
\(751\) 44.0374 1.60695 0.803474 0.595339i \(-0.202982\pi\)
0.803474 + 0.595339i \(0.202982\pi\)
\(752\) −1.27413 −0.0464626
\(753\) 13.7506 0.501101
\(754\) 0 0
\(755\) −0.0217703 −0.000792301 0
\(756\) 3.35690 0.122089
\(757\) −13.2798 −0.482661 −0.241331 0.970443i \(-0.577584\pi\)
−0.241331 + 0.970443i \(0.577584\pi\)
\(758\) −6.09485 −0.221375
\(759\) 6.85086 0.248670
\(760\) 4.85086 0.175959
\(761\) 9.11231 0.330321 0.165160 0.986267i \(-0.447186\pi\)
0.165160 + 0.986267i \(0.447186\pi\)
\(762\) −12.8509 −0.465537
\(763\) −47.7770 −1.72964
\(764\) −10.1806 −0.368321
\(765\) −0.939001 −0.0339497
\(766\) −7.95407 −0.287392
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −31.7784 −1.14596 −0.572979 0.819570i \(-0.694212\pi\)
−0.572979 + 0.819570i \(0.694212\pi\)
\(770\) −5.67994 −0.204691
\(771\) −24.1782 −0.870757
\(772\) 27.6407 0.994811
\(773\) 7.11960 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(774\) 4.02177 0.144560
\(775\) 4.08815 0.146851
\(776\) 15.4547 0.554792
\(777\) 39.0441 1.40070
\(778\) 1.50365 0.0539084
\(779\) −18.7385 −0.671378
\(780\) 0 0
\(781\) −8.72886 −0.312343
\(782\) 3.80194 0.135957
\(783\) −8.12498 −0.290363
\(784\) 4.26875 0.152455
\(785\) 10.1588 0.362584
\(786\) −0.570024 −0.0203321
\(787\) −44.2121 −1.57599 −0.787995 0.615682i \(-0.788881\pi\)
−0.787995 + 0.615682i \(0.788881\pi\)
\(788\) −13.0261 −0.464035
\(789\) 0.960771 0.0342044
\(790\) 4.94869 0.176066
\(791\) −44.7385 −1.59072
\(792\) 1.69202 0.0601234
\(793\) 0 0
\(794\) 34.9627 1.24078
\(795\) −5.74094 −0.203610
\(796\) 6.67563 0.236611
\(797\) 19.2241 0.680954 0.340477 0.940253i \(-0.389411\pi\)
0.340477 + 0.940253i \(0.389411\pi\)
\(798\) −16.2838 −0.576441
\(799\) −1.19641 −0.0423258
\(800\) 1.00000 0.0353553
\(801\) −9.47650 −0.334836
\(802\) −31.1148 −1.09870
\(803\) −19.5104 −0.688505
\(804\) 8.93900 0.315254
\(805\) −13.5918 −0.479048
\(806\) 0 0
\(807\) 10.0761 0.354694
\(808\) 5.38404 0.189410
\(809\) 0.0988996 0.00347713 0.00173856 0.999998i \(-0.499447\pi\)
0.00173856 + 0.999998i \(0.499447\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 6.70304 0.235376 0.117688 0.993051i \(-0.462452\pi\)
0.117688 + 0.993051i \(0.462452\pi\)
\(812\) −27.2747 −0.957155
\(813\) −14.9584 −0.524613
\(814\) 19.6799 0.689782
\(815\) −19.1511 −0.670833
\(816\) 0.939001 0.0328716
\(817\) −19.5090 −0.682534
\(818\) −32.7318 −1.14444
\(819\) 0 0
\(820\) −3.86294 −0.134900
\(821\) −23.7450 −0.828706 −0.414353 0.910116i \(-0.635992\pi\)
−0.414353 + 0.910116i \(0.635992\pi\)
\(822\) 21.5405 0.751311
\(823\) 18.0049 0.627611 0.313806 0.949487i \(-0.398396\pi\)
0.313806 + 0.949487i \(0.398396\pi\)
\(824\) 4.08575 0.142334
\(825\) 1.69202 0.0589086
\(826\) 1.40283 0.0488107
\(827\) 40.4566 1.40682 0.703408 0.710787i \(-0.251661\pi\)
0.703408 + 0.710787i \(0.251661\pi\)
\(828\) 4.04892 0.140710
\(829\) 8.56704 0.297546 0.148773 0.988871i \(-0.452468\pi\)
0.148773 + 0.988871i \(0.452468\pi\)
\(830\) −13.1739 −0.457273
\(831\) −13.0465 −0.452579
\(832\) 0 0
\(833\) 4.00836 0.138881
\(834\) 7.09783 0.245778
\(835\) 15.2174 0.526621
\(836\) −8.20775 −0.283871
\(837\) 4.08815 0.141307
\(838\) 13.0694 0.451474
\(839\) −6.75600 −0.233243 −0.116622 0.993176i \(-0.537206\pi\)
−0.116622 + 0.993176i \(0.537206\pi\)
\(840\) −3.35690 −0.115824
\(841\) 37.0153 1.27639
\(842\) 16.4789 0.567900
\(843\) 16.2881 0.560993
\(844\) −11.3056 −0.389154
\(845\) 0 0
\(846\) −1.27413 −0.0438054
\(847\) −27.3153 −0.938564
\(848\) 5.74094 0.197145
\(849\) −12.3773 −0.424789
\(850\) 0.939001 0.0322075
\(851\) 47.0930 1.61433
\(852\) −5.15883 −0.176739
\(853\) −37.1269 −1.27120 −0.635600 0.772018i \(-0.719247\pi\)
−0.635600 + 0.772018i \(0.719247\pi\)
\(854\) 0.664874 0.0227515
\(855\) 4.85086 0.165896
\(856\) −16.1196 −0.550957
\(857\) −24.3937 −0.833274 −0.416637 0.909073i \(-0.636791\pi\)
−0.416637 + 0.909073i \(0.636791\pi\)
\(858\) 0 0
\(859\) 43.9366 1.49910 0.749549 0.661949i \(-0.230270\pi\)
0.749549 + 0.661949i \(0.230270\pi\)
\(860\) −4.02177 −0.137141
\(861\) 12.9675 0.441930
\(862\) −41.0640 −1.39864
\(863\) 16.3846 0.557739 0.278870 0.960329i \(-0.410040\pi\)
0.278870 + 0.960329i \(0.410040\pi\)
\(864\) 1.00000 0.0340207
\(865\) −21.8213 −0.741947
\(866\) −6.04593 −0.205449
\(867\) −16.1183 −0.547405
\(868\) 13.7235 0.465805
\(869\) −8.37329 −0.284044
\(870\) 8.12498 0.275463
\(871\) 0 0
\(872\) −14.2325 −0.481973
\(873\) 15.4547 0.523063
\(874\) −19.6407 −0.664357
\(875\) −3.35690 −0.113484
\(876\) −11.5308 −0.389589
\(877\) 28.1347 0.950040 0.475020 0.879975i \(-0.342441\pi\)
0.475020 + 0.879975i \(0.342441\pi\)
\(878\) −26.2543 −0.886039
\(879\) −1.73663 −0.0585750
\(880\) −1.69202 −0.0570380
\(881\) 39.1454 1.31884 0.659421 0.751773i \(-0.270801\pi\)
0.659421 + 0.751773i \(0.270801\pi\)
\(882\) 4.26875 0.143736
\(883\) 46.2833 1.55756 0.778779 0.627298i \(-0.215839\pi\)
0.778779 + 0.627298i \(0.215839\pi\)
\(884\) 0 0
\(885\) −0.417895 −0.0140474
\(886\) −39.3846 −1.32315
\(887\) −3.08097 −0.103449 −0.0517244 0.998661i \(-0.516472\pi\)
−0.0517244 + 0.998661i \(0.516472\pi\)
\(888\) 11.6310 0.390312
\(889\) −43.1390 −1.44684
\(890\) 9.47650 0.317653
\(891\) 1.69202 0.0566849
\(892\) 18.0911 0.605736
\(893\) 6.18060 0.206826
\(894\) −1.14914 −0.0384332
\(895\) 20.2513 0.676926
\(896\) 3.35690 0.112146
\(897\) 0 0
\(898\) −9.39075 −0.313373
\(899\) −33.2161 −1.10782
\(900\) 1.00000 0.0333333
\(901\) 5.39075 0.179592
\(902\) 6.53617 0.217631
\(903\) 13.5007 0.449274
\(904\) −13.3274 −0.443261
\(905\) −14.4969 −0.481895
\(906\) 0.0217703 0.000723269 0
\(907\) −2.61224 −0.0867379 −0.0433689 0.999059i \(-0.513809\pi\)
−0.0433689 + 0.999059i \(0.513809\pi\)
\(908\) −9.37196 −0.311019
\(909\) 5.38404 0.178577
\(910\) 0 0
\(911\) 24.9302 0.825973 0.412987 0.910737i \(-0.364486\pi\)
0.412987 + 0.910737i \(0.364486\pi\)
\(912\) −4.85086 −0.160628
\(913\) 22.2905 0.737709
\(914\) −2.66487 −0.0881462
\(915\) −0.198062 −0.00654774
\(916\) 21.2664 0.702660
\(917\) −1.91351 −0.0631897
\(918\) 0.939001 0.0309917
\(919\) −54.4462 −1.79602 −0.898008 0.439980i \(-0.854986\pi\)
−0.898008 + 0.439980i \(0.854986\pi\)
\(920\) −4.04892 −0.133489
\(921\) 25.4403 0.838285
\(922\) −24.5284 −0.807800
\(923\) 0 0
\(924\) 5.67994 0.186856
\(925\) 11.6310 0.382426
\(926\) −4.50498 −0.148043
\(927\) 4.08575 0.134194
\(928\) −8.12498 −0.266716
\(929\) −40.8267 −1.33948 −0.669740 0.742596i \(-0.733595\pi\)
−0.669740 + 0.742596i \(0.733595\pi\)
\(930\) −4.08815 −0.134056
\(931\) −20.7071 −0.678647
\(932\) 2.03146 0.0665427
\(933\) −10.0325 −0.328450
\(934\) 21.3317 0.697993
\(935\) −1.58881 −0.0519596
\(936\) 0 0
\(937\) −46.9571 −1.53402 −0.767010 0.641635i \(-0.778256\pi\)
−0.767010 + 0.641635i \(0.778256\pi\)
\(938\) 30.0073 0.979773
\(939\) −22.3274 −0.728626
\(940\) 1.27413 0.0415574
\(941\) 35.8525 1.16876 0.584379 0.811481i \(-0.301338\pi\)
0.584379 + 0.811481i \(0.301338\pi\)
\(942\) −10.1588 −0.330993
\(943\) 15.6407 0.509332
\(944\) 0.417895 0.0136013
\(945\) −3.35690 −0.109200
\(946\) 6.80492 0.221247
\(947\) −8.73019 −0.283693 −0.141846 0.989889i \(-0.545304\pi\)
−0.141846 + 0.989889i \(0.545304\pi\)
\(948\) −4.94869 −0.160726
\(949\) 0 0
\(950\) −4.85086 −0.157383
\(951\) −5.33811 −0.173100
\(952\) 3.15213 0.102161
\(953\) 10.2107 0.330758 0.165379 0.986230i \(-0.447115\pi\)
0.165379 + 0.986230i \(0.447115\pi\)
\(954\) 5.74094 0.185870
\(955\) 10.1806 0.329437
\(956\) −11.9903 −0.387794
\(957\) −13.7476 −0.444398
\(958\) −32.9517 −1.06462
\(959\) 72.3092 2.33498
\(960\) −1.00000 −0.0322749
\(961\) −14.2871 −0.460873
\(962\) 0 0
\(963\) −16.1196 −0.519447
\(964\) 21.7627 0.700930
\(965\) −27.6407 −0.889786
\(966\) 13.5918 0.437309
\(967\) −34.6907 −1.11558 −0.557789 0.829983i \(-0.688350\pi\)
−0.557789 + 0.829983i \(0.688350\pi\)
\(968\) −8.13706 −0.261535
\(969\) −4.55496 −0.146326
\(970\) −15.4547 −0.496221
\(971\) 34.8039 1.11691 0.558454 0.829535i \(-0.311395\pi\)
0.558454 + 0.829535i \(0.311395\pi\)
\(972\) 1.00000 0.0320750
\(973\) 23.8267 0.763849
\(974\) −9.33034 −0.298963
\(975\) 0 0
\(976\) 0.198062 0.00633982
\(977\) 23.3086 0.745707 0.372854 0.927890i \(-0.378379\pi\)
0.372854 + 0.927890i \(0.378379\pi\)
\(978\) 19.1511 0.612383
\(979\) −16.0344 −0.512463
\(980\) −4.26875 −0.136360
\(981\) −14.2325 −0.454409
\(982\) −20.4349 −0.652103
\(983\) 41.3639 1.31930 0.659652 0.751571i \(-0.270704\pi\)
0.659652 + 0.751571i \(0.270704\pi\)
\(984\) 3.86294 0.123146
\(985\) 13.0261 0.415045
\(986\) −7.62937 −0.242969
\(987\) −4.27711 −0.136142
\(988\) 0 0
\(989\) 16.2838 0.517795
\(990\) −1.69202 −0.0537760
\(991\) 49.4209 1.56991 0.784953 0.619555i \(-0.212687\pi\)
0.784953 + 0.619555i \(0.212687\pi\)
\(992\) 4.08815 0.129799
\(993\) −4.49827 −0.142748
\(994\) −17.3177 −0.549283
\(995\) −6.67563 −0.211632
\(996\) 13.1739 0.417431
\(997\) 44.1879 1.39944 0.699722 0.714415i \(-0.253307\pi\)
0.699722 + 0.714415i \(0.253307\pi\)
\(998\) 3.54958 0.112360
\(999\) 11.6310 0.367989
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 5070.2.a.bw.1.2 yes 3
13.5 odd 4 5070.2.b.z.1351.2 6
13.8 odd 4 5070.2.b.z.1351.5 6
13.12 even 2 5070.2.a.bp.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bp.1.2 3 13.12 even 2
5070.2.a.bw.1.2 yes 3 1.1 even 1 trivial
5070.2.b.z.1351.2 6 13.5 odd 4
5070.2.b.z.1351.5 6 13.8 odd 4