Properties

Label 4730.2.a.be.1.2
Level 4730
Weight 2
Character 4730.1
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.56242\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.56242 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.56242 q^{6} +3.71438 q^{7} +1.00000 q^{8} +3.56600 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.56242 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.56242 q^{6} +3.71438 q^{7} +1.00000 q^{8} +3.56600 q^{9} +1.00000 q^{10} -1.00000 q^{11} -2.56242 q^{12} +5.57975 q^{13} +3.71438 q^{14} -2.56242 q^{15} +1.00000 q^{16} +3.56825 q^{17} +3.56600 q^{18} +7.50849 q^{19} +1.00000 q^{20} -9.51780 q^{21} -1.00000 q^{22} -3.57975 q^{23} -2.56242 q^{24} +1.00000 q^{25} +5.57975 q^{26} -1.45032 q^{27} +3.71438 q^{28} +10.4694 q^{29} -2.56242 q^{30} -8.61067 q^{31} +1.00000 q^{32} +2.56242 q^{33} +3.56825 q^{34} +3.71438 q^{35} +3.56600 q^{36} -1.18834 q^{37} +7.50849 q^{38} -14.2977 q^{39} +1.00000 q^{40} +9.18919 q^{41} -9.51780 q^{42} +1.00000 q^{43} -1.00000 q^{44} +3.56600 q^{45} -3.57975 q^{46} -8.24604 q^{47} -2.56242 q^{48} +6.79662 q^{49} +1.00000 q^{50} -9.14335 q^{51} +5.57975 q^{52} -9.37658 q^{53} -1.45032 q^{54} -1.00000 q^{55} +3.71438 q^{56} -19.2399 q^{57} +10.4694 q^{58} -4.51898 q^{59} -2.56242 q^{60} -10.7548 q^{61} -8.61067 q^{62} +13.2455 q^{63} +1.00000 q^{64} +5.57975 q^{65} +2.56242 q^{66} +6.61761 q^{67} +3.56825 q^{68} +9.17281 q^{69} +3.71438 q^{70} -0.469439 q^{71} +3.56600 q^{72} -1.64340 q^{73} -1.18834 q^{74} -2.56242 q^{75} +7.50849 q^{76} -3.71438 q^{77} -14.2977 q^{78} -4.43404 q^{79} +1.00000 q^{80} -6.98165 q^{81} +9.18919 q^{82} +10.4132 q^{83} -9.51780 q^{84} +3.56825 q^{85} +1.00000 q^{86} -26.8269 q^{87} -1.00000 q^{88} +10.8354 q^{89} +3.56600 q^{90} +20.7253 q^{91} -3.57975 q^{92} +22.0642 q^{93} -8.24604 q^{94} +7.50849 q^{95} -2.56242 q^{96} +0.511402 q^{97} +6.79662 q^{98} -3.56600 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{2} + 3q^{3} + 12q^{4} + 12q^{5} + 3q^{6} + 8q^{7} + 12q^{8} + 25q^{9} + O(q^{10}) \) \( 12q + 12q^{2} + 3q^{3} + 12q^{4} + 12q^{5} + 3q^{6} + 8q^{7} + 12q^{8} + 25q^{9} + 12q^{10} - 12q^{11} + 3q^{12} + 16q^{13} + 8q^{14} + 3q^{15} + 12q^{16} + 18q^{17} + 25q^{18} - 4q^{19} + 12q^{20} + 4q^{21} - 12q^{22} + 8q^{23} + 3q^{24} + 12q^{25} + 16q^{26} + 6q^{27} + 8q^{28} + 20q^{29} + 3q^{30} + 5q^{31} + 12q^{32} - 3q^{33} + 18q^{34} + 8q^{35} + 25q^{36} + 19q^{37} - 4q^{38} + 6q^{39} + 12q^{40} + 16q^{41} + 4q^{42} + 12q^{43} - 12q^{44} + 25q^{45} + 8q^{46} - q^{47} + 3q^{48} + 52q^{49} + 12q^{50} + q^{51} + 16q^{52} + 11q^{53} + 6q^{54} - 12q^{55} + 8q^{56} + 9q^{57} + 20q^{58} - 11q^{59} + 3q^{60} + 18q^{61} + 5q^{62} + 15q^{63} + 12q^{64} + 16q^{65} - 3q^{66} - 10q^{67} + 18q^{68} + 8q^{70} - 2q^{71} + 25q^{72} + 29q^{73} + 19q^{74} + 3q^{75} - 4q^{76} - 8q^{77} + 6q^{78} + 2q^{79} + 12q^{80} - 8q^{81} + 16q^{82} + 26q^{83} + 4q^{84} + 18q^{85} + 12q^{86} - 4q^{87} - 12q^{88} + 41q^{89} + 25q^{90} - 4q^{91} + 8q^{92} + 5q^{93} - q^{94} - 4q^{95} + 3q^{96} - 7q^{97} + 52q^{98} - 25q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.56242 −1.47941 −0.739707 0.672929i \(-0.765036\pi\)
−0.739707 + 0.672929i \(0.765036\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −2.56242 −1.04610
\(7\) 3.71438 1.40390 0.701952 0.712224i \(-0.252312\pi\)
0.701952 + 0.712224i \(0.252312\pi\)
\(8\) 1.00000 0.353553
\(9\) 3.56600 1.18867
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511
\(12\) −2.56242 −0.739707
\(13\) 5.57975 1.54754 0.773772 0.633465i \(-0.218368\pi\)
0.773772 + 0.633465i \(0.218368\pi\)
\(14\) 3.71438 0.992710
\(15\) −2.56242 −0.661614
\(16\) 1.00000 0.250000
\(17\) 3.56825 0.865427 0.432713 0.901532i \(-0.357556\pi\)
0.432713 + 0.901532i \(0.357556\pi\)
\(18\) 3.56600 0.840514
\(19\) 7.50849 1.72257 0.861283 0.508126i \(-0.169662\pi\)
0.861283 + 0.508126i \(0.169662\pi\)
\(20\) 1.00000 0.223607
\(21\) −9.51780 −2.07696
\(22\) −1.00000 −0.213201
\(23\) −3.57975 −0.746429 −0.373214 0.927745i \(-0.621744\pi\)
−0.373214 + 0.927745i \(0.621744\pi\)
\(24\) −2.56242 −0.523052
\(25\) 1.00000 0.200000
\(26\) 5.57975 1.09428
\(27\) −1.45032 −0.279115
\(28\) 3.71438 0.701952
\(29\) 10.4694 1.94411 0.972056 0.234751i \(-0.0754274\pi\)
0.972056 + 0.234751i \(0.0754274\pi\)
\(30\) −2.56242 −0.467832
\(31\) −8.61067 −1.54652 −0.773261 0.634088i \(-0.781376\pi\)
−0.773261 + 0.634088i \(0.781376\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.56242 0.446060
\(34\) 3.56825 0.611949
\(35\) 3.71438 0.627845
\(36\) 3.56600 0.594333
\(37\) −1.18834 −0.195362 −0.0976808 0.995218i \(-0.531142\pi\)
−0.0976808 + 0.995218i \(0.531142\pi\)
\(38\) 7.50849 1.21804
\(39\) −14.2977 −2.28946
\(40\) 1.00000 0.158114
\(41\) 9.18919 1.43511 0.717556 0.696501i \(-0.245261\pi\)
0.717556 + 0.696501i \(0.245261\pi\)
\(42\) −9.51780 −1.46863
\(43\) 1.00000 0.152499
\(44\) −1.00000 −0.150756
\(45\) 3.56600 0.531588
\(46\) −3.57975 −0.527805
\(47\) −8.24604 −1.20281 −0.601404 0.798945i \(-0.705392\pi\)
−0.601404 + 0.798945i \(0.705392\pi\)
\(48\) −2.56242 −0.369854
\(49\) 6.79662 0.970946
\(50\) 1.00000 0.141421
\(51\) −9.14335 −1.28032
\(52\) 5.57975 0.773772
\(53\) −9.37658 −1.28797 −0.643986 0.765037i \(-0.722721\pi\)
−0.643986 + 0.765037i \(0.722721\pi\)
\(54\) −1.45032 −0.197364
\(55\) −1.00000 −0.134840
\(56\) 3.71438 0.496355
\(57\) −19.2399 −2.54839
\(58\) 10.4694 1.37469
\(59\) −4.51898 −0.588321 −0.294160 0.955756i \(-0.595040\pi\)
−0.294160 + 0.955756i \(0.595040\pi\)
\(60\) −2.56242 −0.330807
\(61\) −10.7548 −1.37701 −0.688507 0.725229i \(-0.741734\pi\)
−0.688507 + 0.725229i \(0.741734\pi\)
\(62\) −8.61067 −1.09356
\(63\) 13.2455 1.66877
\(64\) 1.00000 0.125000
\(65\) 5.57975 0.692082
\(66\) 2.56242 0.315412
\(67\) 6.61761 0.808470 0.404235 0.914655i \(-0.367538\pi\)
0.404235 + 0.914655i \(0.367538\pi\)
\(68\) 3.56825 0.432713
\(69\) 9.17281 1.10428
\(70\) 3.71438 0.443953
\(71\) −0.469439 −0.0557122 −0.0278561 0.999612i \(-0.508868\pi\)
−0.0278561 + 0.999612i \(0.508868\pi\)
\(72\) 3.56600 0.420257
\(73\) −1.64340 −0.192345 −0.0961725 0.995365i \(-0.530660\pi\)
−0.0961725 + 0.995365i \(0.530660\pi\)
\(74\) −1.18834 −0.138142
\(75\) −2.56242 −0.295883
\(76\) 7.50849 0.861283
\(77\) −3.71438 −0.423293
\(78\) −14.2977 −1.61889
\(79\) −4.43404 −0.498868 −0.249434 0.968392i \(-0.580245\pi\)
−0.249434 + 0.968392i \(0.580245\pi\)
\(80\) 1.00000 0.111803
\(81\) −6.98165 −0.775739
\(82\) 9.18919 1.01478
\(83\) 10.4132 1.14300 0.571498 0.820603i \(-0.306362\pi\)
0.571498 + 0.820603i \(0.306362\pi\)
\(84\) −9.51780 −1.03848
\(85\) 3.56825 0.387031
\(86\) 1.00000 0.107833
\(87\) −26.8269 −2.87615
\(88\) −1.00000 −0.106600
\(89\) 10.8354 1.14855 0.574273 0.818664i \(-0.305285\pi\)
0.574273 + 0.818664i \(0.305285\pi\)
\(90\) 3.56600 0.375889
\(91\) 20.7253 2.17260
\(92\) −3.57975 −0.373214
\(93\) 22.0642 2.28795
\(94\) −8.24604 −0.850514
\(95\) 7.50849 0.770355
\(96\) −2.56242 −0.261526
\(97\) 0.511402 0.0519250 0.0259625 0.999663i \(-0.491735\pi\)
0.0259625 + 0.999663i \(0.491735\pi\)
\(98\) 6.79662 0.686562
\(99\) −3.56600 −0.358396
\(100\) 1.00000 0.100000
\(101\) 5.73226 0.570381 0.285191 0.958471i \(-0.407943\pi\)
0.285191 + 0.958471i \(0.407943\pi\)
\(102\) −9.14335 −0.905326
\(103\) 7.60241 0.749087 0.374544 0.927209i \(-0.377799\pi\)
0.374544 + 0.927209i \(0.377799\pi\)
\(104\) 5.57975 0.547139
\(105\) −9.51780 −0.928843
\(106\) −9.37658 −0.910734
\(107\) −2.53696 −0.245257 −0.122629 0.992453i \(-0.539132\pi\)
−0.122629 + 0.992453i \(0.539132\pi\)
\(108\) −1.45032 −0.139558
\(109\) −10.5718 −1.01259 −0.506295 0.862360i \(-0.668985\pi\)
−0.506295 + 0.862360i \(0.668985\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 3.04502 0.289021
\(112\) 3.71438 0.350976
\(113\) −16.3289 −1.53609 −0.768046 0.640395i \(-0.778771\pi\)
−0.768046 + 0.640395i \(0.778771\pi\)
\(114\) −19.2399 −1.80198
\(115\) −3.57975 −0.333813
\(116\) 10.4694 0.972056
\(117\) 19.8974 1.83951
\(118\) −4.51898 −0.416005
\(119\) 13.2538 1.21498
\(120\) −2.56242 −0.233916
\(121\) 1.00000 0.0909091
\(122\) −10.7548 −0.973697
\(123\) −23.5466 −2.12312
\(124\) −8.61067 −0.773261
\(125\) 1.00000 0.0894427
\(126\) 13.2455 1.18000
\(127\) −1.55955 −0.138388 −0.0691938 0.997603i \(-0.522043\pi\)
−0.0691938 + 0.997603i \(0.522043\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.56242 −0.225609
\(130\) 5.57975 0.489376
\(131\) −18.8774 −1.64933 −0.824664 0.565623i \(-0.808636\pi\)
−0.824664 + 0.565623i \(0.808636\pi\)
\(132\) 2.56242 0.223030
\(133\) 27.8894 2.41832
\(134\) 6.61761 0.571675
\(135\) −1.45032 −0.124824
\(136\) 3.56825 0.305975
\(137\) −6.16582 −0.526782 −0.263391 0.964689i \(-0.584841\pi\)
−0.263391 + 0.964689i \(0.584841\pi\)
\(138\) 9.17281 0.780842
\(139\) −20.7119 −1.75676 −0.878378 0.477966i \(-0.841374\pi\)
−0.878378 + 0.477966i \(0.841374\pi\)
\(140\) 3.71438 0.313922
\(141\) 21.1298 1.77945
\(142\) −0.469439 −0.0393944
\(143\) −5.57975 −0.466602
\(144\) 3.56600 0.297166
\(145\) 10.4694 0.869433
\(146\) −1.64340 −0.136009
\(147\) −17.4158 −1.43643
\(148\) −1.18834 −0.0976808
\(149\) 4.55818 0.373420 0.186710 0.982415i \(-0.440217\pi\)
0.186710 + 0.982415i \(0.440217\pi\)
\(150\) −2.56242 −0.209221
\(151\) −19.4030 −1.57900 −0.789498 0.613753i \(-0.789659\pi\)
−0.789498 + 0.613753i \(0.789659\pi\)
\(152\) 7.50849 0.609019
\(153\) 12.7244 1.02870
\(154\) −3.71438 −0.299313
\(155\) −8.61067 −0.691626
\(156\) −14.2977 −1.14473
\(157\) 9.75708 0.778700 0.389350 0.921090i \(-0.372700\pi\)
0.389350 + 0.921090i \(0.372700\pi\)
\(158\) −4.43404 −0.352753
\(159\) 24.0267 1.90544
\(160\) 1.00000 0.0790569
\(161\) −13.2965 −1.04791
\(162\) −6.98165 −0.548530
\(163\) −7.12889 −0.558378 −0.279189 0.960236i \(-0.590066\pi\)
−0.279189 + 0.960236i \(0.590066\pi\)
\(164\) 9.18919 0.717556
\(165\) 2.56242 0.199484
\(166\) 10.4132 0.808220
\(167\) 6.12444 0.473923 0.236962 0.971519i \(-0.423848\pi\)
0.236962 + 0.971519i \(0.423848\pi\)
\(168\) −9.51780 −0.734315
\(169\) 18.1336 1.39489
\(170\) 3.56825 0.273672
\(171\) 26.7753 2.04756
\(172\) 1.00000 0.0762493
\(173\) −18.6132 −1.41513 −0.707567 0.706646i \(-0.750207\pi\)
−0.707567 + 0.706646i \(0.750207\pi\)
\(174\) −26.8269 −2.03374
\(175\) 3.71438 0.280781
\(176\) −1.00000 −0.0753778
\(177\) 11.5795 0.870370
\(178\) 10.8354 0.812145
\(179\) −8.07746 −0.603738 −0.301869 0.953349i \(-0.597611\pi\)
−0.301869 + 0.953349i \(0.597611\pi\)
\(180\) 3.56600 0.265794
\(181\) −12.2583 −0.911153 −0.455576 0.890197i \(-0.650567\pi\)
−0.455576 + 0.890197i \(0.650567\pi\)
\(182\) 20.7253 1.53626
\(183\) 27.5584 2.03718
\(184\) −3.57975 −0.263902
\(185\) −1.18834 −0.0873684
\(186\) 22.0642 1.61782
\(187\) −3.56825 −0.260936
\(188\) −8.24604 −0.601404
\(189\) −5.38706 −0.391851
\(190\) 7.50849 0.544723
\(191\) 25.8507 1.87049 0.935244 0.354004i \(-0.115180\pi\)
0.935244 + 0.354004i \(0.115180\pi\)
\(192\) −2.56242 −0.184927
\(193\) 12.9265 0.930467 0.465233 0.885188i \(-0.345970\pi\)
0.465233 + 0.885188i \(0.345970\pi\)
\(194\) 0.511402 0.0367165
\(195\) −14.2977 −1.02388
\(196\) 6.79662 0.485473
\(197\) 9.14995 0.651907 0.325953 0.945386i \(-0.394315\pi\)
0.325953 + 0.945386i \(0.394315\pi\)
\(198\) −3.56600 −0.253424
\(199\) 17.2884 1.22554 0.612771 0.790260i \(-0.290055\pi\)
0.612771 + 0.790260i \(0.290055\pi\)
\(200\) 1.00000 0.0707107
\(201\) −16.9571 −1.19606
\(202\) 5.73226 0.403320
\(203\) 38.8872 2.72935
\(204\) −9.14335 −0.640162
\(205\) 9.18919 0.641801
\(206\) 7.60241 0.529685
\(207\) −12.7654 −0.887254
\(208\) 5.57975 0.386886
\(209\) −7.50849 −0.519373
\(210\) −9.51780 −0.656791
\(211\) 1.34818 0.0928123 0.0464062 0.998923i \(-0.485223\pi\)
0.0464062 + 0.998923i \(0.485223\pi\)
\(212\) −9.37658 −0.643986
\(213\) 1.20290 0.0824214
\(214\) −2.53696 −0.173423
\(215\) 1.00000 0.0681994
\(216\) −1.45032 −0.0986821
\(217\) −31.9833 −2.17117
\(218\) −10.5718 −0.716010
\(219\) 4.21107 0.284558
\(220\) −1.00000 −0.0674200
\(221\) 19.9099 1.33929
\(222\) 3.04502 0.204369
\(223\) 26.8745 1.79965 0.899824 0.436253i \(-0.143695\pi\)
0.899824 + 0.436253i \(0.143695\pi\)
\(224\) 3.71438 0.248177
\(225\) 3.56600 0.237733
\(226\) −16.3289 −1.08618
\(227\) −0.00729015 −0.000483864 0 −0.000241932 1.00000i \(-0.500077\pi\)
−0.000241932 1.00000i \(0.500077\pi\)
\(228\) −19.2399 −1.27419
\(229\) 27.4701 1.81528 0.907639 0.419753i \(-0.137883\pi\)
0.907639 + 0.419753i \(0.137883\pi\)
\(230\) −3.57975 −0.236041
\(231\) 9.51780 0.626226
\(232\) 10.4694 0.687347
\(233\) −3.41771 −0.223901 −0.111951 0.993714i \(-0.535710\pi\)
−0.111951 + 0.993714i \(0.535710\pi\)
\(234\) 19.8974 1.30073
\(235\) −8.24604 −0.537912
\(236\) −4.51898 −0.294160
\(237\) 11.3619 0.738033
\(238\) 13.2538 0.859118
\(239\) −20.9593 −1.35574 −0.677872 0.735180i \(-0.737098\pi\)
−0.677872 + 0.735180i \(0.737098\pi\)
\(240\) −2.56242 −0.165404
\(241\) 22.6301 1.45773 0.728865 0.684657i \(-0.240048\pi\)
0.728865 + 0.684657i \(0.240048\pi\)
\(242\) 1.00000 0.0642824
\(243\) 22.2409 1.42675
\(244\) −10.7548 −0.688507
\(245\) 6.79662 0.434220
\(246\) −23.5466 −1.50127
\(247\) 41.8955 2.66574
\(248\) −8.61067 −0.546778
\(249\) −26.6830 −1.69096
\(250\) 1.00000 0.0632456
\(251\) −17.7734 −1.12185 −0.560923 0.827868i \(-0.689554\pi\)
−0.560923 + 0.827868i \(0.689554\pi\)
\(252\) 13.2455 0.834386
\(253\) 3.57975 0.225057
\(254\) −1.55955 −0.0978548
\(255\) −9.14335 −0.572579
\(256\) 1.00000 0.0625000
\(257\) 1.56620 0.0976970 0.0488485 0.998806i \(-0.484445\pi\)
0.0488485 + 0.998806i \(0.484445\pi\)
\(258\) −2.56242 −0.159529
\(259\) −4.41394 −0.274269
\(260\) 5.57975 0.346041
\(261\) 37.3337 2.31090
\(262\) −18.8774 −1.16625
\(263\) 17.4609 1.07668 0.538342 0.842726i \(-0.319051\pi\)
0.538342 + 0.842726i \(0.319051\pi\)
\(264\) 2.56242 0.157706
\(265\) −9.37658 −0.575999
\(266\) 27.8894 1.71001
\(267\) −27.7647 −1.69918
\(268\) 6.61761 0.404235
\(269\) 9.08000 0.553618 0.276809 0.960925i \(-0.410723\pi\)
0.276809 + 0.960925i \(0.410723\pi\)
\(270\) −1.45032 −0.0882639
\(271\) 23.7248 1.44118 0.720590 0.693362i \(-0.243871\pi\)
0.720590 + 0.693362i \(0.243871\pi\)
\(272\) 3.56825 0.216357
\(273\) −53.1069 −3.21418
\(274\) −6.16582 −0.372491
\(275\) −1.00000 −0.0603023
\(276\) 9.17281 0.552139
\(277\) 24.6479 1.48095 0.740475 0.672084i \(-0.234601\pi\)
0.740475 + 0.672084i \(0.234601\pi\)
\(278\) −20.7119 −1.24221
\(279\) −30.7056 −1.83830
\(280\) 3.71438 0.221977
\(281\) 1.06479 0.0635202 0.0317601 0.999496i \(-0.489889\pi\)
0.0317601 + 0.999496i \(0.489889\pi\)
\(282\) 21.1298 1.25826
\(283\) 8.26267 0.491165 0.245582 0.969376i \(-0.421021\pi\)
0.245582 + 0.969376i \(0.421021\pi\)
\(284\) −0.469439 −0.0278561
\(285\) −19.2399 −1.13967
\(286\) −5.57975 −0.329937
\(287\) 34.1322 2.01476
\(288\) 3.56600 0.210128
\(289\) −4.26762 −0.251036
\(290\) 10.4694 0.614782
\(291\) −1.31043 −0.0768185
\(292\) −1.64340 −0.0961725
\(293\) −15.1924 −0.887551 −0.443776 0.896138i \(-0.646361\pi\)
−0.443776 + 0.896138i \(0.646361\pi\)
\(294\) −17.4158 −1.01571
\(295\) −4.51898 −0.263105
\(296\) −1.18834 −0.0690708
\(297\) 1.45032 0.0841564
\(298\) 4.55818 0.264048
\(299\) −19.9741 −1.15513
\(300\) −2.56242 −0.147941
\(301\) 3.71438 0.214093
\(302\) −19.4030 −1.11652
\(303\) −14.6885 −0.843830
\(304\) 7.50849 0.430641
\(305\) −10.7548 −0.615820
\(306\) 12.7244 0.727403
\(307\) −15.3667 −0.877026 −0.438513 0.898725i \(-0.644495\pi\)
−0.438513 + 0.898725i \(0.644495\pi\)
\(308\) −3.71438 −0.211646
\(309\) −19.4806 −1.10821
\(310\) −8.61067 −0.489053
\(311\) −25.1532 −1.42631 −0.713154 0.701007i \(-0.752734\pi\)
−0.713154 + 0.701007i \(0.752734\pi\)
\(312\) −14.2977 −0.809445
\(313\) 11.8493 0.669762 0.334881 0.942260i \(-0.391304\pi\)
0.334881 + 0.942260i \(0.391304\pi\)
\(314\) 9.75708 0.550624
\(315\) 13.2455 0.746298
\(316\) −4.43404 −0.249434
\(317\) −14.0466 −0.788933 −0.394467 0.918910i \(-0.629071\pi\)
−0.394467 + 0.918910i \(0.629071\pi\)
\(318\) 24.0267 1.34735
\(319\) −10.4694 −0.586172
\(320\) 1.00000 0.0559017
\(321\) 6.50076 0.362837
\(322\) −13.2965 −0.740987
\(323\) 26.7921 1.49075
\(324\) −6.98165 −0.387870
\(325\) 5.57975 0.309509
\(326\) −7.12889 −0.394833
\(327\) 27.0893 1.49804
\(328\) 9.18919 0.507388
\(329\) −30.6289 −1.68863
\(330\) 2.56242 0.141057
\(331\) 27.3606 1.50388 0.751938 0.659234i \(-0.229119\pi\)
0.751938 + 0.659234i \(0.229119\pi\)
\(332\) 10.4132 0.571498
\(333\) −4.23761 −0.232220
\(334\) 6.12444 0.335114
\(335\) 6.61761 0.361559
\(336\) −9.51780 −0.519239
\(337\) 25.1191 1.36832 0.684162 0.729330i \(-0.260168\pi\)
0.684162 + 0.729330i \(0.260168\pi\)
\(338\) 18.1336 0.986336
\(339\) 41.8414 2.27252
\(340\) 3.56825 0.193515
\(341\) 8.61067 0.466294
\(342\) 26.7753 1.44784
\(343\) −0.755424 −0.0407891
\(344\) 1.00000 0.0539164
\(345\) 9.17281 0.493848
\(346\) −18.6132 −1.00065
\(347\) 11.8815 0.637832 0.318916 0.947783i \(-0.396681\pi\)
0.318916 + 0.947783i \(0.396681\pi\)
\(348\) −26.8269 −1.43807
\(349\) 1.12629 0.0602888 0.0301444 0.999546i \(-0.490403\pi\)
0.0301444 + 0.999546i \(0.490403\pi\)
\(350\) 3.71438 0.198542
\(351\) −8.09244 −0.431943
\(352\) −1.00000 −0.0533002
\(353\) −33.2259 −1.76844 −0.884219 0.467072i \(-0.845309\pi\)
−0.884219 + 0.467072i \(0.845309\pi\)
\(354\) 11.5795 0.615444
\(355\) −0.469439 −0.0249152
\(356\) 10.8354 0.574273
\(357\) −33.9619 −1.79745
\(358\) −8.07746 −0.426907
\(359\) −37.3416 −1.97082 −0.985408 0.170210i \(-0.945555\pi\)
−0.985408 + 0.170210i \(0.945555\pi\)
\(360\) 3.56600 0.187945
\(361\) 37.3774 1.96723
\(362\) −12.2583 −0.644282
\(363\) −2.56242 −0.134492
\(364\) 20.7253 1.08630
\(365\) −1.64340 −0.0860193
\(366\) 27.5584 1.44050
\(367\) 6.58389 0.343676 0.171838 0.985125i \(-0.445029\pi\)
0.171838 + 0.985125i \(0.445029\pi\)
\(368\) −3.57975 −0.186607
\(369\) 32.7686 1.70587
\(370\) −1.18834 −0.0617788
\(371\) −34.8282 −1.80819
\(372\) 22.0642 1.14397
\(373\) 25.4328 1.31686 0.658429 0.752642i \(-0.271221\pi\)
0.658429 + 0.752642i \(0.271221\pi\)
\(374\) −3.56825 −0.184510
\(375\) −2.56242 −0.132323
\(376\) −8.24604 −0.425257
\(377\) 58.4164 3.00860
\(378\) −5.38706 −0.277080
\(379\) 32.8805 1.68896 0.844479 0.535589i \(-0.179910\pi\)
0.844479 + 0.535589i \(0.179910\pi\)
\(380\) 7.50849 0.385177
\(381\) 3.99622 0.204733
\(382\) 25.8507 1.32263
\(383\) 7.85906 0.401579 0.200789 0.979634i \(-0.435649\pi\)
0.200789 + 0.979634i \(0.435649\pi\)
\(384\) −2.56242 −0.130763
\(385\) −3.71438 −0.189302
\(386\) 12.9265 0.657939
\(387\) 3.56600 0.181270
\(388\) 0.511402 0.0259625
\(389\) −31.5297 −1.59862 −0.799310 0.600919i \(-0.794802\pi\)
−0.799310 + 0.600919i \(0.794802\pi\)
\(390\) −14.2977 −0.723990
\(391\) −12.7734 −0.645979
\(392\) 6.79662 0.343281
\(393\) 48.3719 2.44004
\(394\) 9.14995 0.460968
\(395\) −4.43404 −0.223101
\(396\) −3.56600 −0.179198
\(397\) −30.6110 −1.53632 −0.768160 0.640258i \(-0.778828\pi\)
−0.768160 + 0.640258i \(0.778828\pi\)
\(398\) 17.2884 0.866589
\(399\) −71.4643 −3.57769
\(400\) 1.00000 0.0500000
\(401\) 15.8986 0.793937 0.396969 0.917832i \(-0.370062\pi\)
0.396969 + 0.917832i \(0.370062\pi\)
\(402\) −16.9571 −0.845744
\(403\) −48.0454 −2.39331
\(404\) 5.73226 0.285191
\(405\) −6.98165 −0.346921
\(406\) 38.8872 1.92994
\(407\) 1.18834 0.0589037
\(408\) −9.14335 −0.452663
\(409\) −8.57948 −0.424228 −0.212114 0.977245i \(-0.568035\pi\)
−0.212114 + 0.977245i \(0.568035\pi\)
\(410\) 9.18919 0.453822
\(411\) 15.7994 0.779328
\(412\) 7.60241 0.374544
\(413\) −16.7852 −0.825945
\(414\) −12.7654 −0.627384
\(415\) 10.4132 0.511164
\(416\) 5.57975 0.273570
\(417\) 53.0725 2.59897
\(418\) −7.50849 −0.367252
\(419\) −14.1402 −0.690793 −0.345396 0.938457i \(-0.612256\pi\)
−0.345396 + 0.938457i \(0.612256\pi\)
\(420\) −9.51780 −0.464421
\(421\) −2.26240 −0.110263 −0.0551313 0.998479i \(-0.517558\pi\)
−0.0551313 + 0.998479i \(0.517558\pi\)
\(422\) 1.34818 0.0656282
\(423\) −29.4054 −1.42974
\(424\) −9.37658 −0.455367
\(425\) 3.56825 0.173085
\(426\) 1.20290 0.0582807
\(427\) −39.9475 −1.93320
\(428\) −2.53696 −0.122629
\(429\) 14.2977 0.690297
\(430\) 1.00000 0.0482243
\(431\) 15.9360 0.767612 0.383806 0.923414i \(-0.374613\pi\)
0.383806 + 0.923414i \(0.374613\pi\)
\(432\) −1.45032 −0.0697788
\(433\) 36.2384 1.74151 0.870754 0.491720i \(-0.163632\pi\)
0.870754 + 0.491720i \(0.163632\pi\)
\(434\) −31.9833 −1.53525
\(435\) −26.8269 −1.28625
\(436\) −10.5718 −0.506295
\(437\) −26.8785 −1.28577
\(438\) 4.21107 0.201213
\(439\) −31.2479 −1.49138 −0.745690 0.666293i \(-0.767880\pi\)
−0.745690 + 0.666293i \(0.767880\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 24.2367 1.15413
\(442\) 19.9099 0.947018
\(443\) −31.0221 −1.47391 −0.736953 0.675944i \(-0.763736\pi\)
−0.736953 + 0.675944i \(0.763736\pi\)
\(444\) 3.04502 0.144510
\(445\) 10.8354 0.513645
\(446\) 26.8745 1.27254
\(447\) −11.6800 −0.552443
\(448\) 3.71438 0.175488
\(449\) −30.6147 −1.44480 −0.722398 0.691478i \(-0.756960\pi\)
−0.722398 + 0.691478i \(0.756960\pi\)
\(450\) 3.56600 0.168103
\(451\) −9.18919 −0.432702
\(452\) −16.3289 −0.768046
\(453\) 49.7187 2.33599
\(454\) −0.00729015 −0.000342144 0
\(455\) 20.7253 0.971617
\(456\) −19.2399 −0.900991
\(457\) 8.31983 0.389185 0.194593 0.980884i \(-0.437661\pi\)
0.194593 + 0.980884i \(0.437661\pi\)
\(458\) 27.4701 1.28359
\(459\) −5.17512 −0.241554
\(460\) −3.57975 −0.166907
\(461\) −8.69871 −0.405140 −0.202570 0.979268i \(-0.564929\pi\)
−0.202570 + 0.979268i \(0.564929\pi\)
\(462\) 9.51780 0.442808
\(463\) 7.27816 0.338245 0.169122 0.985595i \(-0.445907\pi\)
0.169122 + 0.985595i \(0.445907\pi\)
\(464\) 10.4694 0.486028
\(465\) 22.0642 1.02320
\(466\) −3.41771 −0.158322
\(467\) −5.45479 −0.252418 −0.126209 0.992004i \(-0.540281\pi\)
−0.126209 + 0.992004i \(0.540281\pi\)
\(468\) 19.8974 0.919756
\(469\) 24.5803 1.13501
\(470\) −8.24604 −0.380361
\(471\) −25.0017 −1.15202
\(472\) −4.51898 −0.208003
\(473\) −1.00000 −0.0459800
\(474\) 11.3619 0.521868
\(475\) 7.50849 0.344513
\(476\) 13.2538 0.607488
\(477\) −33.4369 −1.53097
\(478\) −20.9593 −0.958656
\(479\) −32.5974 −1.48941 −0.744706 0.667393i \(-0.767410\pi\)
−0.744706 + 0.667393i \(0.767410\pi\)
\(480\) −2.56242 −0.116958
\(481\) −6.63063 −0.302331
\(482\) 22.6301 1.03077
\(483\) 34.0713 1.55030
\(484\) 1.00000 0.0454545
\(485\) 0.511402 0.0232215
\(486\) 22.2409 1.00887
\(487\) 3.63248 0.164603 0.0823017 0.996607i \(-0.473773\pi\)
0.0823017 + 0.996607i \(0.473773\pi\)
\(488\) −10.7548 −0.486848
\(489\) 18.2672 0.826072
\(490\) 6.79662 0.307040
\(491\) 5.31772 0.239985 0.119993 0.992775i \(-0.461713\pi\)
0.119993 + 0.992775i \(0.461713\pi\)
\(492\) −23.5466 −1.06156
\(493\) 37.3573 1.68249
\(494\) 41.8955 1.88497
\(495\) −3.56600 −0.160280
\(496\) −8.61067 −0.386630
\(497\) −1.74368 −0.0782145
\(498\) −26.6830 −1.19569
\(499\) −13.4007 −0.599899 −0.299950 0.953955i \(-0.596970\pi\)
−0.299950 + 0.953955i \(0.596970\pi\)
\(500\) 1.00000 0.0447214
\(501\) −15.6934 −0.701129
\(502\) −17.7734 −0.793265
\(503\) 39.5123 1.76177 0.880884 0.473332i \(-0.156949\pi\)
0.880884 + 0.473332i \(0.156949\pi\)
\(504\) 13.2455 0.590000
\(505\) 5.73226 0.255082
\(506\) 3.57975 0.159139
\(507\) −46.4658 −2.06362
\(508\) −1.55955 −0.0691938
\(509\) −27.7161 −1.22850 −0.614248 0.789113i \(-0.710541\pi\)
−0.614248 + 0.789113i \(0.710541\pi\)
\(510\) −9.14335 −0.404874
\(511\) −6.10420 −0.270034
\(512\) 1.00000 0.0441942
\(513\) −10.8897 −0.480794
\(514\) 1.56620 0.0690822
\(515\) 7.60241 0.335002
\(516\) −2.56242 −0.112804
\(517\) 8.24604 0.362660
\(518\) −4.41394 −0.193937
\(519\) 47.6948 2.09357
\(520\) 5.57975 0.244688
\(521\) −18.2117 −0.797867 −0.398934 0.916980i \(-0.630620\pi\)
−0.398934 + 0.916980i \(0.630620\pi\)
\(522\) 37.3337 1.63405
\(523\) −13.3213 −0.582502 −0.291251 0.956647i \(-0.594071\pi\)
−0.291251 + 0.956647i \(0.594071\pi\)
\(524\) −18.8774 −0.824664
\(525\) −9.51780 −0.415391
\(526\) 17.4609 0.761331
\(527\) −30.7250 −1.33840
\(528\) 2.56242 0.111515
\(529\) −10.1854 −0.442844
\(530\) −9.37658 −0.407293
\(531\) −16.1147 −0.699317
\(532\) 27.8894 1.20916
\(533\) 51.2734 2.22090
\(534\) −27.7647 −1.20150
\(535\) −2.53696 −0.109682
\(536\) 6.61761 0.285837
\(537\) 20.6979 0.893178
\(538\) 9.08000 0.391467
\(539\) −6.79662 −0.292751
\(540\) −1.45032 −0.0624120
\(541\) −12.5715 −0.540493 −0.270246 0.962791i \(-0.587105\pi\)
−0.270246 + 0.962791i \(0.587105\pi\)
\(542\) 23.7248 1.01907
\(543\) 31.4109 1.34797
\(544\) 3.56825 0.152987
\(545\) −10.5718 −0.452844
\(546\) −53.1069 −2.27277
\(547\) −27.8264 −1.18977 −0.594885 0.803811i \(-0.702802\pi\)
−0.594885 + 0.803811i \(0.702802\pi\)
\(548\) −6.16582 −0.263391
\(549\) −38.3517 −1.63681
\(550\) −1.00000 −0.0426401
\(551\) 78.6091 3.34886
\(552\) 9.17281 0.390421
\(553\) −16.4697 −0.700363
\(554\) 24.6479 1.04719
\(555\) 3.04502 0.129254
\(556\) −20.7119 −0.878378
\(557\) 15.1781 0.643115 0.321557 0.946890i \(-0.395794\pi\)
0.321557 + 0.946890i \(0.395794\pi\)
\(558\) −30.7056 −1.29987
\(559\) 5.57975 0.235998
\(560\) 3.71438 0.156961
\(561\) 9.14335 0.386032
\(562\) 1.06479 0.0449156
\(563\) −28.3603 −1.19524 −0.597622 0.801778i \(-0.703888\pi\)
−0.597622 + 0.801778i \(0.703888\pi\)
\(564\) 21.1298 0.889726
\(565\) −16.3289 −0.686961
\(566\) 8.26267 0.347306
\(567\) −25.9325 −1.08906
\(568\) −0.469439 −0.0196972
\(569\) 30.9562 1.29775 0.648875 0.760895i \(-0.275240\pi\)
0.648875 + 0.760895i \(0.275240\pi\)
\(570\) −19.2399 −0.805871
\(571\) −15.9479 −0.667399 −0.333700 0.942679i \(-0.608297\pi\)
−0.333700 + 0.942679i \(0.608297\pi\)
\(572\) −5.57975 −0.233301
\(573\) −66.2402 −2.76723
\(574\) 34.1322 1.42465
\(575\) −3.57975 −0.149286
\(576\) 3.56600 0.148583
\(577\) 7.63594 0.317888 0.158944 0.987288i \(-0.449191\pi\)
0.158944 + 0.987288i \(0.449191\pi\)
\(578\) −4.26762 −0.177509
\(579\) −33.1230 −1.37655
\(580\) 10.4694 0.434716
\(581\) 38.6786 1.60466
\(582\) −1.31043 −0.0543189
\(583\) 9.37658 0.388338
\(584\) −1.64340 −0.0680043
\(585\) 19.8974 0.822655
\(586\) −15.1924 −0.627594
\(587\) −38.5469 −1.59100 −0.795501 0.605952i \(-0.792792\pi\)
−0.795501 + 0.605952i \(0.792792\pi\)
\(588\) −17.4158 −0.718216
\(589\) −64.6531 −2.66399
\(590\) −4.51898 −0.186043
\(591\) −23.4460 −0.964440
\(592\) −1.18834 −0.0488404
\(593\) −42.3404 −1.73871 −0.869356 0.494187i \(-0.835466\pi\)
−0.869356 + 0.494187i \(0.835466\pi\)
\(594\) 1.45032 0.0595075
\(595\) 13.2538 0.543354
\(596\) 4.55818 0.186710
\(597\) −44.3002 −1.81308
\(598\) −19.9741 −0.816801
\(599\) −16.2466 −0.663817 −0.331908 0.943312i \(-0.607692\pi\)
−0.331908 + 0.943312i \(0.607692\pi\)
\(600\) −2.56242 −0.104610
\(601\) −29.8453 −1.21741 −0.608707 0.793395i \(-0.708312\pi\)
−0.608707 + 0.793395i \(0.708312\pi\)
\(602\) 3.71438 0.151387
\(603\) 23.5984 0.961001
\(604\) −19.4030 −0.789498
\(605\) 1.00000 0.0406558
\(606\) −14.6885 −0.596678
\(607\) 11.3631 0.461214 0.230607 0.973047i \(-0.425929\pi\)
0.230607 + 0.973047i \(0.425929\pi\)
\(608\) 7.50849 0.304509
\(609\) −99.6453 −4.03783
\(610\) −10.7548 −0.435450
\(611\) −46.0108 −1.86140
\(612\) 12.7244 0.514352
\(613\) 25.1962 1.01766 0.508832 0.860866i \(-0.330077\pi\)
0.508832 + 0.860866i \(0.330077\pi\)
\(614\) −15.3667 −0.620151
\(615\) −23.5466 −0.949490
\(616\) −3.71438 −0.149657
\(617\) 26.2533 1.05692 0.528460 0.848958i \(-0.322770\pi\)
0.528460 + 0.848958i \(0.322770\pi\)
\(618\) −19.4806 −0.783623
\(619\) −44.1153 −1.77314 −0.886572 0.462591i \(-0.846920\pi\)
−0.886572 + 0.462591i \(0.846920\pi\)
\(620\) −8.61067 −0.345813
\(621\) 5.19179 0.208339
\(622\) −25.1532 −1.00855
\(623\) 40.2467 1.61245
\(624\) −14.2977 −0.572364
\(625\) 1.00000 0.0400000
\(626\) 11.8493 0.473594
\(627\) 19.2399 0.768368
\(628\) 9.75708 0.389350
\(629\) −4.24028 −0.169071
\(630\) 13.2455 0.527712
\(631\) −33.3429 −1.32736 −0.663681 0.748016i \(-0.731007\pi\)
−0.663681 + 0.748016i \(0.731007\pi\)
\(632\) −4.43404 −0.176377
\(633\) −3.45460 −0.137308
\(634\) −14.0466 −0.557860
\(635\) −1.55955 −0.0618888
\(636\) 24.0267 0.952722
\(637\) 37.9234 1.50258
\(638\) −10.4694 −0.414486
\(639\) −1.67402 −0.0662232
\(640\) 1.00000 0.0395285
\(641\) −26.5788 −1.04980 −0.524900 0.851164i \(-0.675897\pi\)
−0.524900 + 0.851164i \(0.675897\pi\)
\(642\) 6.50076 0.256565
\(643\) −18.0000 −0.709850 −0.354925 0.934895i \(-0.615494\pi\)
−0.354925 + 0.934895i \(0.615494\pi\)
\(644\) −13.2965 −0.523957
\(645\) −2.56242 −0.100895
\(646\) 26.7921 1.05412
\(647\) −23.4896 −0.923470 −0.461735 0.887018i \(-0.652773\pi\)
−0.461735 + 0.887018i \(0.652773\pi\)
\(648\) −6.98165 −0.274265
\(649\) 4.51898 0.177385
\(650\) 5.57975 0.218856
\(651\) 81.9547 3.21206
\(652\) −7.12889 −0.279189
\(653\) 4.77348 0.186801 0.0934003 0.995629i \(-0.470226\pi\)
0.0934003 + 0.995629i \(0.470226\pi\)
\(654\) 27.0893 1.05928
\(655\) −18.8774 −0.737602
\(656\) 9.18919 0.358778
\(657\) −5.86035 −0.228634
\(658\) −30.6289 −1.19404
\(659\) 28.3906 1.10594 0.552971 0.833200i \(-0.313494\pi\)
0.552971 + 0.833200i \(0.313494\pi\)
\(660\) 2.56242 0.0997421
\(661\) −21.5499 −0.838195 −0.419097 0.907941i \(-0.637653\pi\)
−0.419097 + 0.907941i \(0.637653\pi\)
\(662\) 27.3606 1.06340
\(663\) −51.0176 −1.98136
\(664\) 10.4132 0.404110
\(665\) 27.8894 1.08150
\(666\) −4.23761 −0.164204
\(667\) −37.4776 −1.45114
\(668\) 6.12444 0.236962
\(669\) −68.8637 −2.66243
\(670\) 6.61761 0.255661
\(671\) 10.7548 0.415186
\(672\) −9.51780 −0.367157
\(673\) 12.9526 0.499285 0.249642 0.968338i \(-0.419687\pi\)
0.249642 + 0.968338i \(0.419687\pi\)
\(674\) 25.1191 0.967552
\(675\) −1.45032 −0.0558230
\(676\) 18.1336 0.697445
\(677\) 24.9006 0.957009 0.478504 0.878085i \(-0.341179\pi\)
0.478504 + 0.878085i \(0.341179\pi\)
\(678\) 41.8414 1.60691
\(679\) 1.89954 0.0728977
\(680\) 3.56825 0.136836
\(681\) 0.0186804 0.000715836 0
\(682\) 8.61067 0.329720
\(683\) 2.19027 0.0838085 0.0419043 0.999122i \(-0.486658\pi\)
0.0419043 + 0.999122i \(0.486658\pi\)
\(684\) 26.7753 1.02378
\(685\) −6.16582 −0.235584
\(686\) −0.755424 −0.0288422
\(687\) −70.3900 −2.68555
\(688\) 1.00000 0.0381246
\(689\) −52.3189 −1.99319
\(690\) 9.17281 0.349203
\(691\) −33.4753 −1.27346 −0.636730 0.771087i \(-0.719713\pi\)
−0.636730 + 0.771087i \(0.719713\pi\)
\(692\) −18.6132 −0.707567
\(693\) −13.2455 −0.503154
\(694\) 11.8815 0.451015
\(695\) −20.7119 −0.785646
\(696\) −26.8269 −1.01687
\(697\) 32.7893 1.24198
\(698\) 1.12629 0.0426306
\(699\) 8.75760 0.331243
\(700\) 3.71438 0.140390
\(701\) 20.3654 0.769191 0.384596 0.923085i \(-0.374341\pi\)
0.384596 + 0.923085i \(0.374341\pi\)
\(702\) −8.09244 −0.305430
\(703\) −8.92263 −0.336523
\(704\) −1.00000 −0.0376889
\(705\) 21.1298 0.795795
\(706\) −33.2259 −1.25048
\(707\) 21.2918 0.800760
\(708\) 11.5795 0.435185
\(709\) 34.6244 1.30035 0.650174 0.759786i \(-0.274696\pi\)
0.650174 + 0.759786i \(0.274696\pi\)
\(710\) −0.469439 −0.0176177
\(711\) −15.8118 −0.592988
\(712\) 10.8354 0.406072
\(713\) 30.8240 1.15437
\(714\) −33.9619 −1.27099
\(715\) −5.57975 −0.208671
\(716\) −8.07746 −0.301869
\(717\) 53.7065 2.00571
\(718\) −37.3416 −1.39358
\(719\) 18.7362 0.698741 0.349370 0.936985i \(-0.386396\pi\)
0.349370 + 0.936985i \(0.386396\pi\)
\(720\) 3.56600 0.132897
\(721\) 28.2382 1.05165
\(722\) 37.3774 1.39104
\(723\) −57.9877 −2.15659
\(724\) −12.2583 −0.455576
\(725\) 10.4694 0.388822
\(726\) −2.56242 −0.0951003
\(727\) −36.0943 −1.33866 −0.669331 0.742964i \(-0.733419\pi\)
−0.669331 + 0.742964i \(0.733419\pi\)
\(728\) 20.7253 0.768131
\(729\) −36.0456 −1.33502
\(730\) −1.64340 −0.0608249
\(731\) 3.56825 0.131976
\(732\) 27.5584 1.01859
\(733\) 25.9853 0.959789 0.479894 0.877326i \(-0.340675\pi\)
0.479894 + 0.877326i \(0.340675\pi\)
\(734\) 6.58389 0.243016
\(735\) −17.4158 −0.642392
\(736\) −3.57975 −0.131951
\(737\) −6.61761 −0.243763
\(738\) 32.7686 1.20623
\(739\) 12.4010 0.456179 0.228090 0.973640i \(-0.426752\pi\)
0.228090 + 0.973640i \(0.426752\pi\)
\(740\) −1.18834 −0.0436842
\(741\) −107.354 −3.94374
\(742\) −34.8282 −1.27858
\(743\) 28.0197 1.02794 0.513972 0.857807i \(-0.328173\pi\)
0.513972 + 0.857807i \(0.328173\pi\)
\(744\) 22.0642 0.808911
\(745\) 4.55818 0.166999
\(746\) 25.4328 0.931160
\(747\) 37.1334 1.35864
\(748\) −3.56825 −0.130468
\(749\) −9.42324 −0.344318
\(750\) −2.56242 −0.0935664
\(751\) −9.01153 −0.328835 −0.164418 0.986391i \(-0.552574\pi\)
−0.164418 + 0.986391i \(0.552574\pi\)
\(752\) −8.24604 −0.300702
\(753\) 45.5429 1.65968
\(754\) 58.4164 2.12740
\(755\) −19.4030 −0.706148
\(756\) −5.38706 −0.195925
\(757\) 22.1670 0.805672 0.402836 0.915272i \(-0.368025\pi\)
0.402836 + 0.915272i \(0.368025\pi\)
\(758\) 32.8805 1.19427
\(759\) −9.17281 −0.332952
\(760\) 7.50849 0.272362
\(761\) 7.77879 0.281981 0.140991 0.990011i \(-0.454971\pi\)
0.140991 + 0.990011i \(0.454971\pi\)
\(762\) 3.99622 0.144768
\(763\) −39.2675 −1.42158
\(764\) 25.8507 0.935244
\(765\) 12.7244 0.460050
\(766\) 7.85906 0.283959
\(767\) −25.2147 −0.910451
\(768\) −2.56242 −0.0924634
\(769\) 15.2820 0.551083 0.275542 0.961289i \(-0.411143\pi\)
0.275542 + 0.961289i \(0.411143\pi\)
\(770\) −3.71438 −0.133857
\(771\) −4.01327 −0.144534
\(772\) 12.9265 0.465233
\(773\) 3.28862 0.118284 0.0591418 0.998250i \(-0.481164\pi\)
0.0591418 + 0.998250i \(0.481164\pi\)
\(774\) 3.56600 0.128177
\(775\) −8.61067 −0.309304
\(776\) 0.511402 0.0183582
\(777\) 11.3104 0.405757
\(778\) −31.5297 −1.13040
\(779\) 68.9970 2.47207
\(780\) −14.2977 −0.511938
\(781\) 0.469439 0.0167978
\(782\) −12.7734 −0.456776
\(783\) −15.1840 −0.542631
\(784\) 6.79662 0.242736
\(785\) 9.75708 0.348245
\(786\) 48.3719 1.72537
\(787\) 38.8675 1.38548 0.692739 0.721188i \(-0.256404\pi\)
0.692739 + 0.721188i \(0.256404\pi\)
\(788\) 9.14995 0.325953
\(789\) −44.7421 −1.59286
\(790\) −4.43404 −0.157756
\(791\) −60.6517 −2.15652
\(792\) −3.56600 −0.126712
\(793\) −60.0092 −2.13099
\(794\) −30.6110 −1.08634
\(795\) 24.0267 0.852140
\(796\) 17.2884 0.612771
\(797\) 36.9300 1.30813 0.654063 0.756440i \(-0.273063\pi\)
0.654063 + 0.756440i \(0.273063\pi\)
\(798\) −71.4643 −2.52981
\(799\) −29.4239 −1.04094
\(800\) 1.00000 0.0353553
\(801\) 38.6389 1.36524
\(802\) 15.8986 0.561398
\(803\) 1.64340 0.0579942
\(804\) −16.9571 −0.598031
\(805\) −13.2965 −0.468641
\(806\) −48.0454 −1.69233
\(807\) −23.2668 −0.819030
\(808\) 5.73226 0.201660
\(809\) −31.6994 −1.11449 −0.557246 0.830348i \(-0.688142\pi\)
−0.557246 + 0.830348i \(0.688142\pi\)
\(810\) −6.98165 −0.245310
\(811\) −13.1924 −0.463246 −0.231623 0.972806i \(-0.574404\pi\)
−0.231623 + 0.972806i \(0.574404\pi\)
\(812\) 38.8872 1.36467
\(813\) −60.7929 −2.13210
\(814\) 1.18834 0.0416512
\(815\) −7.12889 −0.249714
\(816\) −9.14335 −0.320081
\(817\) 7.50849 0.262689
\(818\) −8.57948 −0.299975
\(819\) 73.9064 2.58250
\(820\) 9.18919 0.320901
\(821\) −19.4667 −0.679392 −0.339696 0.940535i \(-0.610324\pi\)
−0.339696 + 0.940535i \(0.610324\pi\)
\(822\) 15.7994 0.551068
\(823\) 27.8888 0.972142 0.486071 0.873919i \(-0.338430\pi\)
0.486071 + 0.873919i \(0.338430\pi\)
\(824\) 7.60241 0.264842
\(825\) 2.56242 0.0892120
\(826\) −16.7852 −0.584032
\(827\) −9.43030 −0.327924 −0.163962 0.986467i \(-0.552427\pi\)
−0.163962 + 0.986467i \(0.552427\pi\)
\(828\) −12.7654 −0.443627
\(829\) −45.8021 −1.59077 −0.795386 0.606103i \(-0.792732\pi\)
−0.795386 + 0.606103i \(0.792732\pi\)
\(830\) 10.4132 0.361447
\(831\) −63.1583 −2.19094
\(832\) 5.57975 0.193443
\(833\) 24.2520 0.840283
\(834\) 53.0725 1.83775
\(835\) 6.12444 0.211945
\(836\) −7.50849 −0.259687
\(837\) 12.4883 0.431658
\(838\) −14.1402 −0.488464
\(839\) 7.18828 0.248167 0.124083 0.992272i \(-0.460401\pi\)
0.124083 + 0.992272i \(0.460401\pi\)
\(840\) −9.51780 −0.328395
\(841\) 80.6075 2.77957
\(842\) −2.26240 −0.0779675
\(843\) −2.72845 −0.0939727
\(844\) 1.34818 0.0464062
\(845\) 18.1336 0.623814
\(846\) −29.4054 −1.01098
\(847\) 3.71438 0.127628
\(848\) −9.37658 −0.321993
\(849\) −21.1724 −0.726636
\(850\) 3.56825 0.122390
\(851\) 4.25395 0.145824
\(852\) 1.20290 0.0412107
\(853\) 3.57962 0.122564 0.0612820 0.998120i \(-0.480481\pi\)
0.0612820 + 0.998120i \(0.480481\pi\)
\(854\) −39.9475 −1.36698
\(855\) 26.7753 0.915694
\(856\) −2.53696 −0.0867116
\(857\) 18.9411 0.647014 0.323507 0.946226i \(-0.395138\pi\)
0.323507 + 0.946226i \(0.395138\pi\)
\(858\) 14.2977 0.488114
\(859\) 19.8700 0.677957 0.338978 0.940794i \(-0.389919\pi\)
0.338978 + 0.940794i \(0.389919\pi\)
\(860\) 1.00000 0.0340997
\(861\) −87.4609 −2.98066
\(862\) 15.9360 0.542783
\(863\) 18.6340 0.634308 0.317154 0.948374i \(-0.397273\pi\)
0.317154 + 0.948374i \(0.397273\pi\)
\(864\) −1.45032 −0.0493410
\(865\) −18.6132 −0.632868
\(866\) 36.2384 1.23143
\(867\) 10.9354 0.371387
\(868\) −31.9833 −1.08558
\(869\) 4.43404 0.150414
\(870\) −26.8269 −0.909517
\(871\) 36.9246 1.25114
\(872\) −10.5718 −0.358005
\(873\) 1.82366 0.0617214
\(874\) −26.8785 −0.909178
\(875\) 3.71438 0.125569
\(876\) 4.21107 0.142279
\(877\) 11.1192 0.375469 0.187735 0.982220i \(-0.439886\pi\)
0.187735 + 0.982220i \(0.439886\pi\)
\(878\) −31.2479 −1.05456
\(879\) 38.9294 1.31306
\(880\) −1.00000 −0.0337100
\(881\) −9.06242 −0.305321 −0.152660 0.988279i \(-0.548784\pi\)
−0.152660 + 0.988279i \(0.548784\pi\)
\(882\) 24.2367 0.816093
\(883\) 14.2250 0.478710 0.239355 0.970932i \(-0.423064\pi\)
0.239355 + 0.970932i \(0.423064\pi\)
\(884\) 19.9099 0.669643
\(885\) 11.5795 0.389241
\(886\) −31.0221 −1.04221
\(887\) 20.5493 0.689977 0.344988 0.938607i \(-0.387883\pi\)
0.344988 + 0.938607i \(0.387883\pi\)
\(888\) 3.04502 0.102184
\(889\) −5.79276 −0.194283
\(890\) 10.8354 0.363202
\(891\) 6.98165 0.233894
\(892\) 26.8745 0.899824
\(893\) −61.9153 −2.07192
\(894\) −11.6800 −0.390637
\(895\) −8.07746 −0.270000
\(896\) 3.71438 0.124089
\(897\) 51.1820 1.70892
\(898\) −30.6147 −1.02162
\(899\) −90.1482 −3.00661
\(900\) 3.56600 0.118867
\(901\) −33.4579 −1.11465
\(902\) −9.18919 −0.305967
\(903\) −9.51780 −0.316733
\(904\) −16.3289 −0.543090
\(905\) −12.2583 −0.407480
\(906\) 49.7187 1.65179
\(907\) 4.79684 0.159276 0.0796382 0.996824i \(-0.474623\pi\)
0.0796382 + 0.996824i \(0.474623\pi\)
\(908\) −0.00729015 −0.000241932 0
\(909\) 20.4412 0.677992
\(910\) 20.7253 0.687037
\(911\) −28.3419 −0.939010 −0.469505 0.882930i \(-0.655568\pi\)
−0.469505 + 0.882930i \(0.655568\pi\)
\(912\) −19.2399 −0.637097
\(913\) −10.4132 −0.344626
\(914\) 8.31983 0.275196
\(915\) 27.5584 0.911052
\(916\) 27.4701 0.907639
\(917\) −70.1179 −2.31550
\(918\) −5.17512 −0.170804
\(919\) 16.8779 0.556750 0.278375 0.960473i \(-0.410204\pi\)
0.278375 + 0.960473i \(0.410204\pi\)
\(920\) −3.57975 −0.118021
\(921\) 39.3760 1.29748
\(922\) −8.69871 −0.286477
\(923\) −2.61935 −0.0862170
\(924\) 9.51780 0.313113
\(925\) −1.18834 −0.0390723
\(926\) 7.27816 0.239175
\(927\) 27.1102 0.890415
\(928\) 10.4694 0.343674
\(929\) 23.7461 0.779084 0.389542 0.921009i \(-0.372633\pi\)
0.389542 + 0.921009i \(0.372633\pi\)
\(930\) 22.0642 0.723512
\(931\) 51.0324 1.67252
\(932\) −3.41771 −0.111951
\(933\) 64.4531 2.11010
\(934\) −5.45479 −0.178486
\(935\) −3.56825 −0.116694
\(936\) 19.8974 0.650366
\(937\) 24.3243 0.794641 0.397321 0.917680i \(-0.369940\pi\)
0.397321 + 0.917680i \(0.369940\pi\)
\(938\) 24.5803 0.802576
\(939\) −30.3629 −0.990856
\(940\) −8.24604 −0.268956
\(941\) −9.98359 −0.325456 −0.162728 0.986671i \(-0.552029\pi\)
−0.162728 + 0.986671i \(0.552029\pi\)
\(942\) −25.0017 −0.814601
\(943\) −32.8950 −1.07121
\(944\) −4.51898 −0.147080
\(945\) −5.38706 −0.175241
\(946\) −1.00000 −0.0325128
\(947\) 21.7353 0.706304 0.353152 0.935566i \(-0.385110\pi\)
0.353152 + 0.935566i \(0.385110\pi\)
\(948\) 11.3619 0.369016
\(949\) −9.16974 −0.297662
\(950\) 7.50849 0.243608
\(951\) 35.9932 1.16716
\(952\) 13.2538 0.429559
\(953\) 10.1753 0.329611 0.164806 0.986326i \(-0.447300\pi\)
0.164806 + 0.986326i \(0.447300\pi\)
\(954\) −33.4369 −1.08256
\(955\) 25.8507 0.836507
\(956\) −20.9593 −0.677872
\(957\) 26.8269 0.867190
\(958\) −32.5974 −1.05317
\(959\) −22.9022 −0.739551
\(960\) −2.56242 −0.0827018
\(961\) 43.1436 1.39173
\(962\) −6.63063 −0.213780
\(963\) −9.04680 −0.291529
\(964\) 22.6301 0.728865
\(965\) 12.9265 0.416117
\(966\) 34.0713 1.09623
\(967\) 54.7081 1.75929 0.879647 0.475627i \(-0.157779\pi\)
0.879647 + 0.475627i \(0.157779\pi\)
\(968\) 1.00000 0.0321412
\(969\) −68.6527 −2.20544
\(970\) 0.511402 0.0164201
\(971\) 23.0305 0.739083 0.369541 0.929214i \(-0.379515\pi\)
0.369541 + 0.929214i \(0.379515\pi\)
\(972\) 22.2409 0.713377
\(973\) −76.9317 −2.46632
\(974\) 3.63248 0.116392
\(975\) −14.2977 −0.457891
\(976\) −10.7548 −0.344254
\(977\) −28.5369 −0.912976 −0.456488 0.889730i \(-0.650893\pi\)
−0.456488 + 0.889730i \(0.650893\pi\)
\(978\) 18.2672 0.584121
\(979\) −10.8354 −0.346300
\(980\) 6.79662 0.217110
\(981\) −37.6989 −1.20363
\(982\) 5.31772 0.169695
\(983\) −15.7180 −0.501328 −0.250664 0.968074i \(-0.580649\pi\)
−0.250664 + 0.968074i \(0.580649\pi\)
\(984\) −23.5466 −0.750637
\(985\) 9.14995 0.291542
\(986\) 37.3573 1.18970
\(987\) 78.4842 2.49818
\(988\) 41.8955 1.33287
\(989\) −3.57975 −0.113829
\(990\) −3.56600 −0.113335
\(991\) −26.1846 −0.831781 −0.415890 0.909415i \(-0.636530\pi\)
−0.415890 + 0.909415i \(0.636530\pi\)
\(992\) −8.61067 −0.273389
\(993\) −70.1094 −2.22486
\(994\) −1.74368 −0.0553060
\(995\) 17.2884 0.548079
\(996\) −26.6830 −0.845482
\(997\) 31.1336 0.986012 0.493006 0.870026i \(-0.335898\pi\)
0.493006 + 0.870026i \(0.335898\pi\)
\(998\) −13.4007 −0.424193
\(999\) 1.72348 0.0545284
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 4730.2.a.be.1.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.be.1.2 12 1.1 even 1 trivial