Properties

Label 1110.2.a.s.1.2
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.54764.1
Defining polynomial: \(x^{4} - x^{3} - 9 x^{2} + 3 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.36007\) of defining polynomial
Character \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +0.487359 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +0.487359 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.20750 q^{11} +1.00000 q^{12} +4.12492 q^{13} +0.487359 q^{14} +1.00000 q^{15} +1.00000 q^{16} +3.67781 q^{17} +1.00000 q^{18} +0.430057 q^{19} +1.00000 q^{20} +0.487359 q^{21} -5.20750 q^{22} +7.31537 q^{23} +1.00000 q^{24} +1.00000 q^{25} +4.12492 q^{26} +1.00000 q^{27} +0.487359 q^{28} -6.77745 q^{29} +1.00000 q^{30} +7.80273 q^{31} +1.00000 q^{32} -5.20750 q^{33} +3.67781 q^{34} +0.487359 q^{35} +1.00000 q^{36} -1.00000 q^{37} +0.430057 q^{38} +4.12492 q^{39} +1.00000 q^{40} -9.80273 q^{41} +0.487359 q^{42} +4.77745 q^{43} -5.20750 q^{44} +1.00000 q^{45} +7.31537 q^{46} +3.52969 q^{47} +1.00000 q^{48} -6.76248 q^{49} +1.00000 q^{50} +3.67781 q^{51} +4.12492 q^{52} -7.67781 q^{53} +1.00000 q^{54} -5.20750 q^{55} +0.487359 q^{56} +0.430057 q^{57} -6.77745 q^{58} -0.974718 q^{59} +1.00000 q^{60} -14.5229 q^{61} +7.80273 q^{62} +0.487359 q^{63} +1.00000 q^{64} +4.12492 q^{65} -5.20750 q^{66} -1.19045 q^{67} +3.67781 q^{68} +7.31537 q^{69} +0.487359 q^{70} +8.58018 q^{71} +1.00000 q^{72} -7.15020 q^{73} -1.00000 q^{74} +1.00000 q^{75} +0.430057 q^{76} -2.53792 q^{77} +4.12492 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -9.80273 q^{82} -16.5399 q^{83} +0.487359 q^{84} +3.67781 q^{85} +4.77745 q^{86} -6.77745 q^{87} -5.20750 q^{88} +16.7051 q^{89} +1.00000 q^{90} +2.01032 q^{91} +7.31537 q^{92} +7.80273 q^{93} +3.52969 q^{94} +0.430057 q^{95} +1.00000 q^{96} -7.16726 q^{97} -6.76248 q^{98} -5.20750 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} + 4q^{6} + 4q^{7} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} + 4q^{6} + 4q^{7} + 4q^{8} + 4q^{9} + 4q^{10} + 2q^{11} + 4q^{12} - 3q^{13} + 4q^{14} + 4q^{15} + 4q^{16} + 6q^{17} + 4q^{18} + 3q^{19} + 4q^{20} + 4q^{21} + 2q^{22} - q^{23} + 4q^{24} + 4q^{25} - 3q^{26} + 4q^{27} + 4q^{28} - 3q^{29} + 4q^{30} + 3q^{31} + 4q^{32} + 2q^{33} + 6q^{34} + 4q^{35} + 4q^{36} - 4q^{37} + 3q^{38} - 3q^{39} + 4q^{40} - 11q^{41} + 4q^{42} - 5q^{43} + 2q^{44} + 4q^{45} - q^{46} + 4q^{48} + 14q^{49} + 4q^{50} + 6q^{51} - 3q^{52} - 22q^{53} + 4q^{54} + 2q^{55} + 4q^{56} + 3q^{57} - 3q^{58} - 8q^{59} + 4q^{60} - 5q^{61} + 3q^{62} + 4q^{63} + 4q^{64} - 3q^{65} + 2q^{66} + 6q^{67} + 6q^{68} - q^{69} + 4q^{70} - 18q^{71} + 4q^{72} - 5q^{73} - 4q^{74} + 4q^{75} + 3q^{76} - 4q^{77} - 3q^{78} + 16q^{79} + 4q^{80} + 4q^{81} - 11q^{82} - q^{83} + 4q^{84} + 6q^{85} - 5q^{86} - 3q^{87} + 2q^{88} - 5q^{89} + 4q^{90} - 13q^{91} - q^{92} + 3q^{93} + 3q^{95} + 4q^{96} + 7q^{97} + 14q^{98} + 2q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 0.487359 0.184204 0.0921022 0.995750i \(-0.470641\pi\)
0.0921022 + 0.995750i \(0.470641\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −5.20750 −1.57012 −0.785061 0.619419i \(-0.787368\pi\)
−0.785061 + 0.619419i \(0.787368\pi\)
\(12\) 1.00000 0.288675
\(13\) 4.12492 1.14405 0.572023 0.820237i \(-0.306159\pi\)
0.572023 + 0.820237i \(0.306159\pi\)
\(14\) 0.487359 0.130252
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 3.67781 0.892000 0.446000 0.895033i \(-0.352848\pi\)
0.446000 + 0.895033i \(0.352848\pi\)
\(18\) 1.00000 0.235702
\(19\) 0.430057 0.0986618 0.0493309 0.998782i \(-0.484291\pi\)
0.0493309 + 0.998782i \(0.484291\pi\)
\(20\) 1.00000 0.223607
\(21\) 0.487359 0.106350
\(22\) −5.20750 −1.11024
\(23\) 7.31537 1.52536 0.762680 0.646776i \(-0.223883\pi\)
0.762680 + 0.646776i \(0.223883\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 4.12492 0.808963
\(27\) 1.00000 0.192450
\(28\) 0.487359 0.0921022
\(29\) −6.77745 −1.25854 −0.629270 0.777187i \(-0.716646\pi\)
−0.629270 + 0.777187i \(0.716646\pi\)
\(30\) 1.00000 0.182574
\(31\) 7.80273 1.40141 0.700706 0.713450i \(-0.252869\pi\)
0.700706 + 0.713450i \(0.252869\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.20750 −0.906510
\(34\) 3.67781 0.630739
\(35\) 0.487359 0.0823787
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) 0.430057 0.0697645
\(39\) 4.12492 0.660516
\(40\) 1.00000 0.158114
\(41\) −9.80273 −1.53093 −0.765465 0.643478i \(-0.777491\pi\)
−0.765465 + 0.643478i \(0.777491\pi\)
\(42\) 0.487359 0.0752011
\(43\) 4.77745 0.728554 0.364277 0.931291i \(-0.381316\pi\)
0.364277 + 0.931291i \(0.381316\pi\)
\(44\) −5.20750 −0.785061
\(45\) 1.00000 0.149071
\(46\) 7.31537 1.07859
\(47\) 3.52969 0.514859 0.257429 0.966297i \(-0.417125\pi\)
0.257429 + 0.966297i \(0.417125\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.76248 −0.966069
\(50\) 1.00000 0.141421
\(51\) 3.67781 0.514996
\(52\) 4.12492 0.572023
\(53\) −7.67781 −1.05463 −0.527314 0.849670i \(-0.676801\pi\)
−0.527314 + 0.849670i \(0.676801\pi\)
\(54\) 1.00000 0.136083
\(55\) −5.20750 −0.702180
\(56\) 0.487359 0.0651261
\(57\) 0.430057 0.0569624
\(58\) −6.77745 −0.889922
\(59\) −0.974718 −0.126897 −0.0634487 0.997985i \(-0.520210\pi\)
−0.0634487 + 0.997985i \(0.520210\pi\)
\(60\) 1.00000 0.129099
\(61\) −14.5229 −1.85946 −0.929732 0.368237i \(-0.879961\pi\)
−0.929732 + 0.368237i \(0.879961\pi\)
\(62\) 7.80273 0.990948
\(63\) 0.487359 0.0614014
\(64\) 1.00000 0.125000
\(65\) 4.12492 0.511633
\(66\) −5.20750 −0.640999
\(67\) −1.19045 −0.145437 −0.0727183 0.997353i \(-0.523167\pi\)
−0.0727183 + 0.997353i \(0.523167\pi\)
\(68\) 3.67781 0.446000
\(69\) 7.31537 0.880667
\(70\) 0.487359 0.0582505
\(71\) 8.58018 1.01828 0.509140 0.860684i \(-0.329964\pi\)
0.509140 + 0.860684i \(0.329964\pi\)
\(72\) 1.00000 0.117851
\(73\) −7.15020 −0.836868 −0.418434 0.908247i \(-0.637421\pi\)
−0.418434 + 0.908247i \(0.637421\pi\)
\(74\) −1.00000 −0.116248
\(75\) 1.00000 0.115470
\(76\) 0.430057 0.0493309
\(77\) −2.53792 −0.289223
\(78\) 4.12492 0.467055
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −9.80273 −1.08253
\(83\) −16.5399 −1.81549 −0.907747 0.419519i \(-0.862199\pi\)
−0.907747 + 0.419519i \(0.862199\pi\)
\(84\) 0.487359 0.0531752
\(85\) 3.67781 0.398914
\(86\) 4.77745 0.515165
\(87\) −6.77745 −0.726619
\(88\) −5.20750 −0.555122
\(89\) 16.7051 1.77074 0.885368 0.464890i \(-0.153906\pi\)
0.885368 + 0.464890i \(0.153906\pi\)
\(90\) 1.00000 0.105409
\(91\) 2.01032 0.210738
\(92\) 7.31537 0.762680
\(93\) 7.80273 0.809105
\(94\) 3.52969 0.364060
\(95\) 0.430057 0.0441229
\(96\) 1.00000 0.102062
\(97\) −7.16726 −0.727725 −0.363862 0.931453i \(-0.618542\pi\)
−0.363862 + 0.931453i \(0.618542\pi\)
\(98\) −6.76248 −0.683114
\(99\) −5.20750 −0.523374
\(100\) 1.00000 0.100000
\(101\) −13.7454 −1.36772 −0.683861 0.729613i \(-0.739700\pi\)
−0.683861 + 0.729613i \(0.739700\pi\)
\(102\) 3.67781 0.364157
\(103\) −3.52969 −0.347791 −0.173896 0.984764i \(-0.555636\pi\)
−0.173896 + 0.984764i \(0.555636\pi\)
\(104\) 4.12492 0.404482
\(105\) 0.487359 0.0475614
\(106\) −7.67781 −0.745735
\(107\) 7.09964 0.686348 0.343174 0.939272i \(-0.388498\pi\)
0.343174 + 0.939272i \(0.388498\pi\)
\(108\) 1.00000 0.0962250
\(109\) 16.4321 1.57391 0.786953 0.617013i \(-0.211657\pi\)
0.786953 + 0.617013i \(0.211657\pi\)
\(110\) −5.20750 −0.496516
\(111\) −1.00000 −0.0949158
\(112\) 0.487359 0.0460511
\(113\) −1.22255 −0.115008 −0.0575040 0.998345i \(-0.518314\pi\)
−0.0575040 + 0.998345i \(0.518314\pi\)
\(114\) 0.430057 0.0402785
\(115\) 7.31537 0.682162
\(116\) −6.77745 −0.629270
\(117\) 4.12492 0.381349
\(118\) −0.974718 −0.0897300
\(119\) 1.79241 0.164310
\(120\) 1.00000 0.0912871
\(121\) 16.1181 1.46528
\(122\) −14.5229 −1.31484
\(123\) −9.80273 −0.883882
\(124\) 7.80273 0.700706
\(125\) 1.00000 0.0894427
\(126\) 0.487359 0.0434174
\(127\) 15.5652 1.38119 0.690595 0.723242i \(-0.257349\pi\)
0.690595 + 0.723242i \(0.257349\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.77745 0.420631
\(130\) 4.12492 0.361779
\(131\) −17.5549 −1.53378 −0.766889 0.641780i \(-0.778196\pi\)
−0.766889 + 0.641780i \(0.778196\pi\)
\(132\) −5.20750 −0.453255
\(133\) 0.209592 0.0181739
\(134\) −1.19045 −0.102839
\(135\) 1.00000 0.0860663
\(136\) 3.67781 0.315370
\(137\) −21.2498 −1.81549 −0.907745 0.419523i \(-0.862197\pi\)
−0.907745 + 0.419523i \(0.862197\pi\)
\(138\) 7.31537 0.622726
\(139\) −15.4082 −1.30691 −0.653453 0.756967i \(-0.726680\pi\)
−0.653453 + 0.756967i \(0.726680\pi\)
\(140\) 0.487359 0.0411893
\(141\) 3.52969 0.297254
\(142\) 8.58018 0.720032
\(143\) −21.4805 −1.79629
\(144\) 1.00000 0.0833333
\(145\) −6.77745 −0.562836
\(146\) −7.15020 −0.591755
\(147\) −6.76248 −0.557760
\(148\) −1.00000 −0.0821995
\(149\) −3.47913 −0.285021 −0.142511 0.989793i \(-0.545518\pi\)
−0.142511 + 0.989793i \(0.545518\pi\)
\(150\) 1.00000 0.0816497
\(151\) −17.7304 −1.44288 −0.721439 0.692478i \(-0.756519\pi\)
−0.721439 + 0.692478i \(0.756519\pi\)
\(152\) 0.430057 0.0348822
\(153\) 3.67781 0.297333
\(154\) −2.53792 −0.204512
\(155\) 7.80273 0.626730
\(156\) 4.12492 0.330258
\(157\) 11.8533 0.945996 0.472998 0.881064i \(-0.343172\pi\)
0.472998 + 0.881064i \(0.343172\pi\)
\(158\) 4.00000 0.318223
\(159\) −7.67781 −0.608890
\(160\) 1.00000 0.0790569
\(161\) 3.56521 0.280978
\(162\) 1.00000 0.0785674
\(163\) 5.89354 0.461618 0.230809 0.972999i \(-0.425863\pi\)
0.230809 + 0.972999i \(0.425863\pi\)
\(164\) −9.80273 −0.765465
\(165\) −5.20750 −0.405404
\(166\) −16.5399 −1.28375
\(167\) 24.8362 1.92188 0.960940 0.276758i \(-0.0892599\pi\)
0.960940 + 0.276758i \(0.0892599\pi\)
\(168\) 0.487359 0.0376006
\(169\) 4.01497 0.308844
\(170\) 3.67781 0.282075
\(171\) 0.430057 0.0328873
\(172\) 4.77745 0.364277
\(173\) −19.9276 −1.51507 −0.757536 0.652794i \(-0.773597\pi\)
−0.757536 + 0.652794i \(0.773597\pi\)
\(174\) −6.77745 −0.513797
\(175\) 0.487359 0.0368409
\(176\) −5.20750 −0.392530
\(177\) −0.974718 −0.0732643
\(178\) 16.7051 1.25210
\(179\) −4.33034 −0.323665 −0.161832 0.986818i \(-0.551740\pi\)
−0.161832 + 0.986818i \(0.551740\pi\)
\(180\) 1.00000 0.0745356
\(181\) −0.0505645 −0.00375843 −0.00187922 0.999998i \(-0.500598\pi\)
−0.00187922 + 0.999998i \(0.500598\pi\)
\(182\) 2.01032 0.149015
\(183\) −14.5229 −1.07356
\(184\) 7.31537 0.539296
\(185\) −1.00000 −0.0735215
\(186\) 7.80273 0.572124
\(187\) −19.1522 −1.40055
\(188\) 3.52969 0.257429
\(189\) 0.487359 0.0354501
\(190\) 0.430057 0.0311996
\(191\) −4.37276 −0.316401 −0.158201 0.987407i \(-0.550569\pi\)
−0.158201 + 0.987407i \(0.550569\pi\)
\(192\) 1.00000 0.0721688
\(193\) −2.47031 −0.177816 −0.0889082 0.996040i \(-0.528338\pi\)
−0.0889082 + 0.996040i \(0.528338\pi\)
\(194\) −7.16726 −0.514579
\(195\) 4.12492 0.295392
\(196\) −6.76248 −0.483034
\(197\) −13.3154 −0.948681 −0.474340 0.880341i \(-0.657313\pi\)
−0.474340 + 0.880341i \(0.657313\pi\)
\(198\) −5.20750 −0.370081
\(199\) 21.4403 1.51986 0.759931 0.650004i \(-0.225233\pi\)
0.759931 + 0.650004i \(0.225233\pi\)
\(200\) 1.00000 0.0707107
\(201\) −1.19045 −0.0839679
\(202\) −13.7454 −0.967125
\(203\) −3.30305 −0.231829
\(204\) 3.67781 0.257498
\(205\) −9.80273 −0.684652
\(206\) −3.52969 −0.245925
\(207\) 7.31537 0.508453
\(208\) 4.12492 0.286012
\(209\) −2.23952 −0.154911
\(210\) 0.487359 0.0336310
\(211\) 13.9679 0.961590 0.480795 0.876833i \(-0.340348\pi\)
0.480795 + 0.876833i \(0.340348\pi\)
\(212\) −7.67781 −0.527314
\(213\) 8.58018 0.587904
\(214\) 7.09964 0.485321
\(215\) 4.77745 0.325819
\(216\) 1.00000 0.0680414
\(217\) 3.80273 0.258146
\(218\) 16.4321 1.11292
\(219\) −7.15020 −0.483166
\(220\) −5.20750 −0.351090
\(221\) 15.1707 1.02049
\(222\) −1.00000 −0.0671156
\(223\) −22.4335 −1.50226 −0.751128 0.660156i \(-0.770490\pi\)
−0.751128 + 0.660156i \(0.770490\pi\)
\(224\) 0.487359 0.0325630
\(225\) 1.00000 0.0666667
\(226\) −1.22255 −0.0813230
\(227\) 23.8368 1.58211 0.791053 0.611747i \(-0.209533\pi\)
0.791053 + 0.611747i \(0.209533\pi\)
\(228\) 0.430057 0.0284812
\(229\) 10.2157 0.675075 0.337537 0.941312i \(-0.390406\pi\)
0.337537 + 0.941312i \(0.390406\pi\)
\(230\) 7.31537 0.482361
\(231\) −2.53792 −0.166983
\(232\) −6.77745 −0.444961
\(233\) 2.07576 0.135988 0.0679939 0.997686i \(-0.478340\pi\)
0.0679939 + 0.997686i \(0.478340\pi\)
\(234\) 4.12492 0.269654
\(235\) 3.52969 0.230252
\(236\) −0.974718 −0.0634487
\(237\) 4.00000 0.259828
\(238\) 1.79241 0.116185
\(239\) −28.2683 −1.82852 −0.914262 0.405123i \(-0.867229\pi\)
−0.914262 + 0.405123i \(0.867229\pi\)
\(240\) 1.00000 0.0645497
\(241\) 20.4491 1.31724 0.658622 0.752474i \(-0.271140\pi\)
0.658622 + 0.752474i \(0.271140\pi\)
\(242\) 16.1181 1.03611
\(243\) 1.00000 0.0641500
\(244\) −14.5229 −0.929732
\(245\) −6.76248 −0.432039
\(246\) −9.80273 −0.624999
\(247\) 1.77395 0.112874
\(248\) 7.80273 0.495474
\(249\) −16.5399 −1.04818
\(250\) 1.00000 0.0632456
\(251\) −19.3092 −1.21879 −0.609394 0.792868i \(-0.708587\pi\)
−0.609394 + 0.792868i \(0.708587\pi\)
\(252\) 0.487359 0.0307007
\(253\) −38.0948 −2.39500
\(254\) 15.5652 0.976648
\(255\) 3.67781 0.230313
\(256\) 1.00000 0.0625000
\(257\) 0.735194 0.0458601 0.0229301 0.999737i \(-0.492700\pi\)
0.0229301 + 0.999737i \(0.492700\pi\)
\(258\) 4.77745 0.297431
\(259\) −0.487359 −0.0302830
\(260\) 4.12492 0.255817
\(261\) −6.77745 −0.419513
\(262\) −17.5549 −1.08454
\(263\) 6.47231 0.399100 0.199550 0.979888i \(-0.436052\pi\)
0.199550 + 0.979888i \(0.436052\pi\)
\(264\) −5.20750 −0.320500
\(265\) −7.67781 −0.471644
\(266\) 0.209592 0.0128509
\(267\) 16.7051 1.02234
\(268\) −1.19045 −0.0727183
\(269\) −0.810958 −0.0494450 −0.0247225 0.999694i \(-0.507870\pi\)
−0.0247225 + 0.999694i \(0.507870\pi\)
\(270\) 1.00000 0.0608581
\(271\) 17.4403 1.05942 0.529711 0.848178i \(-0.322300\pi\)
0.529711 + 0.848178i \(0.322300\pi\)
\(272\) 3.67781 0.223000
\(273\) 2.01032 0.121670
\(274\) −21.2498 −1.28374
\(275\) −5.20750 −0.314024
\(276\) 7.31537 0.440334
\(277\) −1.46007 −0.0877272 −0.0438636 0.999038i \(-0.513967\pi\)
−0.0438636 + 0.999038i \(0.513967\pi\)
\(278\) −15.4082 −0.924122
\(279\) 7.80273 0.467137
\(280\) 0.487359 0.0291253
\(281\) −12.1249 −0.723312 −0.361656 0.932312i \(-0.617789\pi\)
−0.361656 + 0.932312i \(0.617789\pi\)
\(282\) 3.52969 0.210190
\(283\) 19.5652 1.16303 0.581516 0.813535i \(-0.302460\pi\)
0.581516 + 0.813535i \(0.302460\pi\)
\(284\) 8.58018 0.509140
\(285\) 0.430057 0.0254744
\(286\) −21.4805 −1.27017
\(287\) −4.77745 −0.282004
\(288\) 1.00000 0.0589256
\(289\) −3.47372 −0.204336
\(290\) −6.77745 −0.397985
\(291\) −7.16726 −0.420152
\(292\) −7.15020 −0.418434
\(293\) −17.5126 −1.02310 −0.511550 0.859254i \(-0.670928\pi\)
−0.511550 + 0.859254i \(0.670928\pi\)
\(294\) −6.76248 −0.394396
\(295\) −0.974718 −0.0567503
\(296\) −1.00000 −0.0581238
\(297\) −5.20750 −0.302170
\(298\) −3.47913 −0.201541
\(299\) 30.1753 1.74508
\(300\) 1.00000 0.0577350
\(301\) 2.32833 0.134203
\(302\) −17.7304 −1.02027
\(303\) −13.7454 −0.789654
\(304\) 0.430057 0.0246655
\(305\) −14.5229 −0.831577
\(306\) 3.67781 0.210246
\(307\) −0.364444 −0.0207999 −0.0104000 0.999946i \(-0.503310\pi\)
−0.0104000 + 0.999946i \(0.503310\pi\)
\(308\) −2.53792 −0.144612
\(309\) −3.52969 −0.200797
\(310\) 7.80273 0.443165
\(311\) −15.1583 −0.859551 −0.429776 0.902936i \(-0.641407\pi\)
−0.429776 + 0.902936i \(0.641407\pi\)
\(312\) 4.12492 0.233528
\(313\) −14.7201 −0.832032 −0.416016 0.909357i \(-0.636574\pi\)
−0.416016 + 0.909357i \(0.636574\pi\)
\(314\) 11.8533 0.668920
\(315\) 0.487359 0.0274596
\(316\) 4.00000 0.225018
\(317\) −0.942615 −0.0529426 −0.0264713 0.999650i \(-0.508427\pi\)
−0.0264713 + 0.999650i \(0.508427\pi\)
\(318\) −7.67781 −0.430550
\(319\) 35.2936 1.97606
\(320\) 1.00000 0.0559017
\(321\) 7.09964 0.396263
\(322\) 3.56521 0.198681
\(323\) 1.58167 0.0880063
\(324\) 1.00000 0.0555556
\(325\) 4.12492 0.228809
\(326\) 5.89354 0.326413
\(327\) 16.4321 0.908695
\(328\) −9.80273 −0.541265
\(329\) 1.72023 0.0948392
\(330\) −5.20750 −0.286664
\(331\) 15.8600 0.871746 0.435873 0.900008i \(-0.356440\pi\)
0.435873 + 0.900008i \(0.356440\pi\)
\(332\) −16.5399 −0.907747
\(333\) −1.00000 −0.0547997
\(334\) 24.8362 1.35897
\(335\) −1.19045 −0.0650413
\(336\) 0.487359 0.0265876
\(337\) 16.9549 0.923594 0.461797 0.886986i \(-0.347205\pi\)
0.461797 + 0.886986i \(0.347205\pi\)
\(338\) 4.01497 0.218385
\(339\) −1.22255 −0.0664000
\(340\) 3.67781 0.199457
\(341\) −40.6327 −2.20039
\(342\) 0.430057 0.0232548
\(343\) −6.70727 −0.362158
\(344\) 4.77745 0.257583
\(345\) 7.31537 0.393846
\(346\) −19.9276 −1.07132
\(347\) −5.44029 −0.292050 −0.146025 0.989281i \(-0.546648\pi\)
−0.146025 + 0.989281i \(0.546648\pi\)
\(348\) −6.77745 −0.363309
\(349\) 23.1262 1.23792 0.618960 0.785423i \(-0.287554\pi\)
0.618960 + 0.785423i \(0.287554\pi\)
\(350\) 0.487359 0.0260504
\(351\) 4.12492 0.220172
\(352\) −5.20750 −0.277561
\(353\) 24.7638 1.31804 0.659022 0.752123i \(-0.270970\pi\)
0.659022 + 0.752123i \(0.270970\pi\)
\(354\) −0.974718 −0.0518057
\(355\) 8.58018 0.455388
\(356\) 16.7051 0.885368
\(357\) 1.79241 0.0948646
\(358\) −4.33034 −0.228865
\(359\) −17.5549 −0.926512 −0.463256 0.886225i \(-0.653319\pi\)
−0.463256 + 0.886225i \(0.653319\pi\)
\(360\) 1.00000 0.0527046
\(361\) −18.8151 −0.990266
\(362\) −0.0505645 −0.00265761
\(363\) 16.1181 0.845981
\(364\) 2.01032 0.105369
\(365\) −7.15020 −0.374259
\(366\) −14.5229 −0.759123
\(367\) −8.17331 −0.426644 −0.213322 0.976982i \(-0.568428\pi\)
−0.213322 + 0.976982i \(0.568428\pi\)
\(368\) 7.31537 0.381340
\(369\) −9.80273 −0.510310
\(370\) −1.00000 −0.0519875
\(371\) −3.74185 −0.194267
\(372\) 7.80273 0.404553
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) −19.1522 −0.990337
\(375\) 1.00000 0.0516398
\(376\) 3.52969 0.182030
\(377\) −27.9564 −1.43983
\(378\) 0.487359 0.0250670
\(379\) 26.1993 1.34577 0.672883 0.739749i \(-0.265056\pi\)
0.672883 + 0.739749i \(0.265056\pi\)
\(380\) 0.430057 0.0220615
\(381\) 15.5652 0.797430
\(382\) −4.37276 −0.223730
\(383\) −5.87508 −0.300203 −0.150101 0.988671i \(-0.547960\pi\)
−0.150101 + 0.988671i \(0.547960\pi\)
\(384\) 1.00000 0.0510310
\(385\) −2.53792 −0.129345
\(386\) −2.47031 −0.125735
\(387\) 4.77745 0.242851
\(388\) −7.16726 −0.363862
\(389\) 33.8368 1.71560 0.857798 0.513987i \(-0.171832\pi\)
0.857798 + 0.513987i \(0.171832\pi\)
\(390\) 4.12492 0.208873
\(391\) 26.9045 1.36062
\(392\) −6.76248 −0.341557
\(393\) −17.5549 −0.885527
\(394\) −13.3154 −0.670819
\(395\) 4.00000 0.201262
\(396\) −5.20750 −0.261687
\(397\) −1.75016 −0.0878380 −0.0439190 0.999035i \(-0.513984\pi\)
−0.0439190 + 0.999035i \(0.513984\pi\)
\(398\) 21.4403 1.07470
\(399\) 0.209592 0.0104927
\(400\) 1.00000 0.0500000
\(401\) 24.7051 1.23371 0.616857 0.787075i \(-0.288406\pi\)
0.616857 + 0.787075i \(0.288406\pi\)
\(402\) −1.19045 −0.0593743
\(403\) 32.1856 1.60328
\(404\) −13.7454 −0.683861
\(405\) 1.00000 0.0496904
\(406\) −3.30305 −0.163928
\(407\) 5.20750 0.258126
\(408\) 3.67781 0.182079
\(409\) 6.75899 0.334210 0.167105 0.985939i \(-0.446558\pi\)
0.167105 + 0.985939i \(0.446558\pi\)
\(410\) −9.80273 −0.484122
\(411\) −21.2498 −1.04817
\(412\) −3.52969 −0.173896
\(413\) −0.475037 −0.0233751
\(414\) 7.31537 0.359531
\(415\) −16.5399 −0.811913
\(416\) 4.12492 0.202241
\(417\) −15.4082 −0.754542
\(418\) −2.23952 −0.109539
\(419\) 12.2143 0.596709 0.298354 0.954455i \(-0.403562\pi\)
0.298354 + 0.954455i \(0.403562\pi\)
\(420\) 0.487359 0.0237807
\(421\) −8.91942 −0.434706 −0.217353 0.976093i \(-0.569742\pi\)
−0.217353 + 0.976093i \(0.569742\pi\)
\(422\) 13.9679 0.679947
\(423\) 3.52969 0.171620
\(424\) −7.67781 −0.372867
\(425\) 3.67781 0.178400
\(426\) 8.58018 0.415711
\(427\) −7.07785 −0.342521
\(428\) 7.09964 0.343174
\(429\) −21.4805 −1.03709
\(430\) 4.77745 0.230389
\(431\) 4.98704 0.240217 0.120109 0.992761i \(-0.461676\pi\)
0.120109 + 0.992761i \(0.461676\pi\)
\(432\) 1.00000 0.0481125
\(433\) 15.4805 0.743947 0.371974 0.928243i \(-0.378681\pi\)
0.371974 + 0.928243i \(0.378681\pi\)
\(434\) 3.80273 0.182537
\(435\) −6.77745 −0.324954
\(436\) 16.4321 0.786953
\(437\) 3.14603 0.150495
\(438\) −7.15020 −0.341650
\(439\) −21.4929 −1.02580 −0.512899 0.858449i \(-0.671429\pi\)
−0.512899 + 0.858449i \(0.671429\pi\)
\(440\) −5.20750 −0.248258
\(441\) −6.76248 −0.322023
\(442\) 15.1707 0.721595
\(443\) −32.4355 −1.54106 −0.770528 0.637406i \(-0.780007\pi\)
−0.770528 + 0.637406i \(0.780007\pi\)
\(444\) −1.00000 −0.0474579
\(445\) 16.7051 0.791898
\(446\) −22.4335 −1.06226
\(447\) −3.47913 −0.164557
\(448\) 0.487359 0.0230255
\(449\) 15.8348 0.747292 0.373646 0.927571i \(-0.378108\pi\)
0.373646 + 0.927571i \(0.378108\pi\)
\(450\) 1.00000 0.0471405
\(451\) 51.0478 2.40374
\(452\) −1.22255 −0.0575040
\(453\) −17.7304 −0.833046
\(454\) 23.8368 1.11872
\(455\) 2.01032 0.0942451
\(456\) 0.430057 0.0201393
\(457\) −29.4470 −1.37747 −0.688737 0.725011i \(-0.741834\pi\)
−0.688737 + 0.725011i \(0.741834\pi\)
\(458\) 10.2157 0.477350
\(459\) 3.67781 0.171665
\(460\) 7.31537 0.341081
\(461\) 18.4975 0.861515 0.430757 0.902468i \(-0.358246\pi\)
0.430757 + 0.902468i \(0.358246\pi\)
\(462\) −2.53792 −0.118075
\(463\) −7.46549 −0.346951 −0.173475 0.984838i \(-0.555500\pi\)
−0.173475 + 0.984838i \(0.555500\pi\)
\(464\) −6.77745 −0.314635
\(465\) 7.80273 0.361843
\(466\) 2.07576 0.0961579
\(467\) 33.3917 1.54519 0.772593 0.634902i \(-0.218960\pi\)
0.772593 + 0.634902i \(0.218960\pi\)
\(468\) 4.12492 0.190674
\(469\) −0.580177 −0.0267901
\(470\) 3.52969 0.162813
\(471\) 11.8533 0.546171
\(472\) −0.974718 −0.0448650
\(473\) −24.8786 −1.14392
\(474\) 4.00000 0.183726
\(475\) 0.430057 0.0197324
\(476\) 1.79241 0.0821551
\(477\) −7.67781 −0.351543
\(478\) −28.2683 −1.29296
\(479\) 26.7051 1.22019 0.610094 0.792329i \(-0.291132\pi\)
0.610094 + 0.792329i \(0.291132\pi\)
\(480\) 1.00000 0.0456435
\(481\) −4.12492 −0.188080
\(482\) 20.4491 0.931432
\(483\) 3.56521 0.162223
\(484\) 16.1181 0.732641
\(485\) −7.16726 −0.325448
\(486\) 1.00000 0.0453609
\(487\) −1.61437 −0.0731539 −0.0365770 0.999331i \(-0.511645\pi\)
−0.0365770 + 0.999331i \(0.511645\pi\)
\(488\) −14.5229 −0.657420
\(489\) 5.89354 0.266515
\(490\) −6.76248 −0.305498
\(491\) 1.40477 0.0633966 0.0316983 0.999497i \(-0.489908\pi\)
0.0316983 + 0.999497i \(0.489908\pi\)
\(492\) −9.80273 −0.441941
\(493\) −24.9262 −1.12262
\(494\) 1.77395 0.0798138
\(495\) −5.20750 −0.234060
\(496\) 7.80273 0.350353
\(497\) 4.18163 0.187572
\(498\) −16.5399 −0.741172
\(499\) −9.98495 −0.446988 −0.223494 0.974705i \(-0.571746\pi\)
−0.223494 + 0.974705i \(0.571746\pi\)
\(500\) 1.00000 0.0447214
\(501\) 24.8362 1.10960
\(502\) −19.3092 −0.861813
\(503\) −20.8300 −0.928765 −0.464382 0.885635i \(-0.653724\pi\)
−0.464382 + 0.885635i \(0.653724\pi\)
\(504\) 0.487359 0.0217087
\(505\) −13.7454 −0.611663
\(506\) −38.0948 −1.69352
\(507\) 4.01497 0.178311
\(508\) 15.5652 0.690595
\(509\) 26.5816 1.17821 0.589104 0.808057i \(-0.299481\pi\)
0.589104 + 0.808057i \(0.299481\pi\)
\(510\) 3.67781 0.162856
\(511\) −3.48471 −0.154155
\(512\) 1.00000 0.0441942
\(513\) 0.430057 0.0189875
\(514\) 0.735194 0.0324280
\(515\) −3.52969 −0.155537
\(516\) 4.77745 0.210315
\(517\) −18.3809 −0.808391
\(518\) −0.487359 −0.0214133
\(519\) −19.9276 −0.874727
\(520\) 4.12492 0.180890
\(521\) −9.91733 −0.434486 −0.217243 0.976118i \(-0.569706\pi\)
−0.217243 + 0.976118i \(0.569706\pi\)
\(522\) −6.77745 −0.296641
\(523\) 5.61910 0.245706 0.122853 0.992425i \(-0.460796\pi\)
0.122853 + 0.992425i \(0.460796\pi\)
\(524\) −17.5549 −0.766889
\(525\) 0.487359 0.0212701
\(526\) 6.47231 0.282206
\(527\) 28.6970 1.25006
\(528\) −5.20750 −0.226628
\(529\) 30.5146 1.32672
\(530\) −7.67781 −0.333503
\(531\) −0.974718 −0.0422991
\(532\) 0.209592 0.00908697
\(533\) −40.4355 −1.75145
\(534\) 16.7051 0.722900
\(535\) 7.09964 0.306944
\(536\) −1.19045 −0.0514196
\(537\) −4.33034 −0.186868
\(538\) −0.810958 −0.0349629
\(539\) 35.2156 1.51685
\(540\) 1.00000 0.0430331
\(541\) −0.315288 −0.0135553 −0.00677764 0.999977i \(-0.502157\pi\)
−0.00677764 + 0.999977i \(0.502157\pi\)
\(542\) 17.4403 0.749125
\(543\) −0.0505645 −0.00216993
\(544\) 3.67781 0.157685
\(545\) 16.4321 0.703872
\(546\) 2.01032 0.0860336
\(547\) 22.1133 0.945496 0.472748 0.881198i \(-0.343262\pi\)
0.472748 + 0.881198i \(0.343262\pi\)
\(548\) −21.2498 −0.907745
\(549\) −14.5229 −0.619821
\(550\) −5.20750 −0.222049
\(551\) −2.91469 −0.124170
\(552\) 7.31537 0.311363
\(553\) 1.94944 0.0828984
\(554\) −1.46007 −0.0620325
\(555\) −1.00000 −0.0424476
\(556\) −15.4082 −0.653453
\(557\) 24.9611 1.05763 0.528817 0.848736i \(-0.322636\pi\)
0.528817 + 0.848736i \(0.322636\pi\)
\(558\) 7.80273 0.330316
\(559\) 19.7066 0.833500
\(560\) 0.487359 0.0205947
\(561\) −19.1522 −0.808607
\(562\) −12.1249 −0.511459
\(563\) −29.0642 −1.22491 −0.612455 0.790505i \(-0.709818\pi\)
−0.612455 + 0.790505i \(0.709818\pi\)
\(564\) 3.52969 0.148627
\(565\) −1.22255 −0.0514332
\(566\) 19.5652 0.822387
\(567\) 0.487359 0.0204671
\(568\) 8.58018 0.360016
\(569\) 34.5740 1.44942 0.724709 0.689055i \(-0.241974\pi\)
0.724709 + 0.689055i \(0.241974\pi\)
\(570\) 0.430057 0.0180131
\(571\) −40.4539 −1.69294 −0.846472 0.532433i \(-0.821278\pi\)
−0.846472 + 0.532433i \(0.821278\pi\)
\(572\) −21.4805 −0.898146
\(573\) −4.37276 −0.182674
\(574\) −4.77745 −0.199407
\(575\) 7.31537 0.305072
\(576\) 1.00000 0.0416667
\(577\) −15.5803 −0.648615 −0.324307 0.945952i \(-0.605131\pi\)
−0.324307 + 0.945952i \(0.605131\pi\)
\(578\) −3.47372 −0.144488
\(579\) −2.47031 −0.102662
\(580\) −6.77745 −0.281418
\(581\) −8.06088 −0.334422
\(582\) −7.16726 −0.297092
\(583\) 39.9822 1.65589
\(584\) −7.15020 −0.295877
\(585\) 4.12492 0.170544
\(586\) −17.5126 −0.723441
\(587\) 25.6717 1.05958 0.529792 0.848128i \(-0.322270\pi\)
0.529792 + 0.848128i \(0.322270\pi\)
\(588\) −6.76248 −0.278880
\(589\) 3.35562 0.138266
\(590\) −0.974718 −0.0401285
\(591\) −13.3154 −0.547721
\(592\) −1.00000 −0.0410997
\(593\) 22.9399 0.942028 0.471014 0.882126i \(-0.343888\pi\)
0.471014 + 0.882126i \(0.343888\pi\)
\(594\) −5.20750 −0.213666
\(595\) 1.79241 0.0734818
\(596\) −3.47913 −0.142511
\(597\) 21.4403 0.877493
\(598\) 30.1753 1.23396
\(599\) −10.9583 −0.447742 −0.223871 0.974619i \(-0.571869\pi\)
−0.223871 + 0.974619i \(0.571869\pi\)
\(600\) 1.00000 0.0408248
\(601\) 38.0123 1.55055 0.775277 0.631621i \(-0.217610\pi\)
0.775277 + 0.631621i \(0.217610\pi\)
\(602\) 2.32833 0.0948957
\(603\) −1.19045 −0.0484789
\(604\) −17.7304 −0.721439
\(605\) 16.1181 0.655294
\(606\) −13.7454 −0.558370
\(607\) −5.16509 −0.209644 −0.104822 0.994491i \(-0.533427\pi\)
−0.104822 + 0.994491i \(0.533427\pi\)
\(608\) 0.430057 0.0174411
\(609\) −3.30305 −0.133846
\(610\) −14.5229 −0.588014
\(611\) 14.5597 0.589023
\(612\) 3.67781 0.148667
\(613\) 35.6075 1.43817 0.719086 0.694921i \(-0.244561\pi\)
0.719086 + 0.694921i \(0.244561\pi\)
\(614\) −0.364444 −0.0147078
\(615\) −9.80273 −0.395284
\(616\) −2.53792 −0.102256
\(617\) 32.0799 1.29149 0.645745 0.763553i \(-0.276547\pi\)
0.645745 + 0.763553i \(0.276547\pi\)
\(618\) −3.52969 −0.141985
\(619\) 24.6628 0.991283 0.495642 0.868527i \(-0.334933\pi\)
0.495642 + 0.868527i \(0.334933\pi\)
\(620\) 7.80273 0.313365
\(621\) 7.31537 0.293556
\(622\) −15.1583 −0.607794
\(623\) 8.14138 0.326177
\(624\) 4.12492 0.165129
\(625\) 1.00000 0.0400000
\(626\) −14.7201 −0.588335
\(627\) −2.23952 −0.0894380
\(628\) 11.8533 0.472998
\(629\) −3.67781 −0.146644
\(630\) 0.487359 0.0194168
\(631\) 8.89205 0.353987 0.176993 0.984212i \(-0.443363\pi\)
0.176993 + 0.984212i \(0.443363\pi\)
\(632\) 4.00000 0.159111
\(633\) 13.9679 0.555174
\(634\) −0.942615 −0.0374360
\(635\) 15.5652 0.617687
\(636\) −7.67781 −0.304445
\(637\) −27.8947 −1.10523
\(638\) 35.2936 1.39729
\(639\) 8.58018 0.339427
\(640\) 1.00000 0.0395285
\(641\) 41.5775 1.64221 0.821107 0.570774i \(-0.193357\pi\)
0.821107 + 0.570774i \(0.193357\pi\)
\(642\) 7.09964 0.280200
\(643\) 43.1522 1.70176 0.850878 0.525363i \(-0.176070\pi\)
0.850878 + 0.525363i \(0.176070\pi\)
\(644\) 3.56521 0.140489
\(645\) 4.77745 0.188112
\(646\) 1.58167 0.0622299
\(647\) −15.4641 −0.607956 −0.303978 0.952679i \(-0.598315\pi\)
−0.303978 + 0.952679i \(0.598315\pi\)
\(648\) 1.00000 0.0392837
\(649\) 5.07585 0.199244
\(650\) 4.12492 0.161793
\(651\) 3.80273 0.149041
\(652\) 5.89354 0.230809
\(653\) 17.3556 0.679178 0.339589 0.940574i \(-0.389712\pi\)
0.339589 + 0.940574i \(0.389712\pi\)
\(654\) 16.4321 0.642544
\(655\) −17.5549 −0.685926
\(656\) −9.80273 −0.382732
\(657\) −7.15020 −0.278956
\(658\) 1.72023 0.0670615
\(659\) −15.3509 −0.597986 −0.298993 0.954255i \(-0.596651\pi\)
−0.298993 + 0.954255i \(0.596651\pi\)
\(660\) −5.20750 −0.202702
\(661\) −23.8218 −0.926560 −0.463280 0.886212i \(-0.653328\pi\)
−0.463280 + 0.886212i \(0.653328\pi\)
\(662\) 15.8600 0.616418
\(663\) 15.1707 0.589180
\(664\) −16.5399 −0.641874
\(665\) 0.209592 0.00812763
\(666\) −1.00000 −0.0387492
\(667\) −49.5795 −1.91973
\(668\) 24.8362 0.960940
\(669\) −22.4335 −0.867328
\(670\) −1.19045 −0.0459911
\(671\) 75.6279 2.91958
\(672\) 0.487359 0.0188003
\(673\) −37.3523 −1.43983 −0.719913 0.694065i \(-0.755818\pi\)
−0.719913 + 0.694065i \(0.755818\pi\)
\(674\) 16.9549 0.653080
\(675\) 1.00000 0.0384900
\(676\) 4.01497 0.154422
\(677\) 40.0143 1.53788 0.768938 0.639324i \(-0.220786\pi\)
0.768938 + 0.639324i \(0.220786\pi\)
\(678\) −1.22255 −0.0469519
\(679\) −3.49303 −0.134050
\(680\) 3.67781 0.141038
\(681\) 23.8368 0.913430
\(682\) −40.6327 −1.55591
\(683\) 7.25948 0.277776 0.138888 0.990308i \(-0.455647\pi\)
0.138888 + 0.990308i \(0.455647\pi\)
\(684\) 0.430057 0.0164436
\(685\) −21.2498 −0.811911
\(686\) −6.70727 −0.256085
\(687\) 10.2157 0.389755
\(688\) 4.77745 0.182138
\(689\) −31.6703 −1.20654
\(690\) 7.31537 0.278491
\(691\) 7.91733 0.301190 0.150595 0.988596i \(-0.451881\pi\)
0.150595 + 0.988596i \(0.451881\pi\)
\(692\) −19.9276 −0.757536
\(693\) −2.53792 −0.0964077
\(694\) −5.44029 −0.206511
\(695\) −15.4082 −0.584466
\(696\) −6.77745 −0.256898
\(697\) −36.0526 −1.36559
\(698\) 23.1262 0.875341
\(699\) 2.07576 0.0785126
\(700\) 0.487359 0.0184204
\(701\) −39.1303 −1.47793 −0.738965 0.673744i \(-0.764685\pi\)
−0.738965 + 0.673744i \(0.764685\pi\)
\(702\) 4.12492 0.155685
\(703\) −0.430057 −0.0162199
\(704\) −5.20750 −0.196265
\(705\) 3.52969 0.132936
\(706\) 24.7638 0.931998
\(707\) −6.69896 −0.251940
\(708\) −0.974718 −0.0366321
\(709\) −40.6478 −1.52656 −0.763280 0.646068i \(-0.776412\pi\)
−0.763280 + 0.646068i \(0.776412\pi\)
\(710\) 8.58018 0.322008
\(711\) 4.00000 0.150012
\(712\) 16.7051 0.626050
\(713\) 57.0799 2.13766
\(714\) 1.79241 0.0670794
\(715\) −21.4805 −0.803327
\(716\) −4.33034 −0.161832
\(717\) −28.2683 −1.05570
\(718\) −17.5549 −0.655143
\(719\) −34.9147 −1.30210 −0.651049 0.759036i \(-0.725671\pi\)
−0.651049 + 0.759036i \(0.725671\pi\)
\(720\) 1.00000 0.0372678
\(721\) −1.72023 −0.0640646
\(722\) −18.8151 −0.700224
\(723\) 20.4491 0.760511
\(724\) −0.0505645 −0.00187922
\(725\) −6.77745 −0.251708
\(726\) 16.1181 0.598199
\(727\) −16.1741 −0.599863 −0.299932 0.953961i \(-0.596964\pi\)
−0.299932 + 0.953961i \(0.596964\pi\)
\(728\) 2.01032 0.0745073
\(729\) 1.00000 0.0370370
\(730\) −7.15020 −0.264641
\(731\) 17.5705 0.649870
\(732\) −14.5229 −0.536781
\(733\) 50.0895 1.85010 0.925049 0.379848i \(-0.124024\pi\)
0.925049 + 0.379848i \(0.124024\pi\)
\(734\) −8.17331 −0.301683
\(735\) −6.76248 −0.249438
\(736\) 7.31537 0.269648
\(737\) 6.19928 0.228353
\(738\) −9.80273 −0.360843
\(739\) −4.01846 −0.147822 −0.0739108 0.997265i \(-0.523548\pi\)
−0.0739108 + 0.997265i \(0.523548\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 1.77395 0.0651677
\(742\) −3.74185 −0.137368
\(743\) −14.4382 −0.529686 −0.264843 0.964292i \(-0.585320\pi\)
−0.264843 + 0.964292i \(0.585320\pi\)
\(744\) 7.80273 0.286062
\(745\) −3.47913 −0.127465
\(746\) −6.00000 −0.219676
\(747\) −16.5399 −0.605164
\(748\) −19.1522 −0.700274
\(749\) 3.46007 0.126428
\(750\) 1.00000 0.0365148
\(751\) 25.0088 0.912585 0.456292 0.889830i \(-0.349177\pi\)
0.456292 + 0.889830i \(0.349177\pi\)
\(752\) 3.52969 0.128715
\(753\) −19.3092 −0.703667
\(754\) −27.9564 −1.01811
\(755\) −17.7304 −0.645275
\(756\) 0.487359 0.0177251
\(757\) −16.4049 −0.596245 −0.298122 0.954528i \(-0.596360\pi\)
−0.298122 + 0.954528i \(0.596360\pi\)
\(758\) 26.1993 0.951601
\(759\) −38.0948 −1.38275
\(760\) 0.430057 0.0155998
\(761\) 19.3877 0.702804 0.351402 0.936225i \(-0.385705\pi\)
0.351402 + 0.936225i \(0.385705\pi\)
\(762\) 15.5652 0.563868
\(763\) 8.00831 0.289920
\(764\) −4.37276 −0.158201
\(765\) 3.67781 0.132971
\(766\) −5.87508 −0.212275
\(767\) −4.02063 −0.145177
\(768\) 1.00000 0.0360844
\(769\) 14.1693 0.510960 0.255480 0.966814i \(-0.417767\pi\)
0.255480 + 0.966814i \(0.417767\pi\)
\(770\) −2.53792 −0.0914604
\(771\) 0.735194 0.0264774
\(772\) −2.47031 −0.0889082
\(773\) −9.58367 −0.344701 −0.172350 0.985036i \(-0.555136\pi\)
−0.172350 + 0.985036i \(0.555136\pi\)
\(774\) 4.77745 0.171722
\(775\) 7.80273 0.280282
\(776\) −7.16726 −0.257289
\(777\) −0.487359 −0.0174839
\(778\) 33.8368 1.21311
\(779\) −4.21573 −0.151044
\(780\) 4.12492 0.147696
\(781\) −44.6813 −1.59882
\(782\) 26.9045 0.962104
\(783\) −6.77745 −0.242206
\(784\) −6.76248 −0.241517
\(785\) 11.8533 0.423062
\(786\) −17.5549 −0.626162
\(787\) 2.47921 0.0883744 0.0441872 0.999023i \(-0.485930\pi\)
0.0441872 + 0.999023i \(0.485930\pi\)
\(788\) −13.3154 −0.474340
\(789\) 6.47231 0.230420
\(790\) 4.00000 0.142314
\(791\) −0.595822 −0.0211850
\(792\) −5.20750 −0.185041
\(793\) −59.9057 −2.12731
\(794\) −1.75016 −0.0621108
\(795\) −7.67781 −0.272304
\(796\) 21.4403 0.759931
\(797\) −47.0961 −1.66823 −0.834116 0.551590i \(-0.814021\pi\)
−0.834116 + 0.551590i \(0.814021\pi\)
\(798\) 0.209592 0.00741948
\(799\) 12.9815 0.459254
\(800\) 1.00000 0.0353553
\(801\) 16.7051 0.590246
\(802\) 24.7051 0.872367
\(803\) 37.2347 1.31398
\(804\) −1.19045 −0.0419840
\(805\) 3.56521 0.125657
\(806\) 32.1856 1.13369
\(807\) −0.810958 −0.0285471
\(808\) −13.7454 −0.483562
\(809\) 36.6409 1.28823 0.644113 0.764931i \(-0.277227\pi\)
0.644113 + 0.764931i \(0.277227\pi\)
\(810\) 1.00000 0.0351364
\(811\) 12.4997 0.438923 0.219462 0.975621i \(-0.429570\pi\)
0.219462 + 0.975621i \(0.429570\pi\)
\(812\) −3.30305 −0.115914
\(813\) 17.4403 0.611658
\(814\) 5.20750 0.182523
\(815\) 5.89354 0.206442
\(816\) 3.67781 0.128749
\(817\) 2.05457 0.0718805
\(818\) 6.75899 0.236322
\(819\) 2.01032 0.0702461
\(820\) −9.80273 −0.342326
\(821\) −11.7351 −0.409558 −0.204779 0.978808i \(-0.565648\pi\)
−0.204779 + 0.978808i \(0.565648\pi\)
\(822\) −21.2498 −0.741170
\(823\) 40.5058 1.41194 0.705972 0.708240i \(-0.250510\pi\)
0.705972 + 0.708240i \(0.250510\pi\)
\(824\) −3.52969 −0.122963
\(825\) −5.20750 −0.181302
\(826\) −0.475037 −0.0165287
\(827\) −6.51115 −0.226415 −0.113207 0.993571i \(-0.536112\pi\)
−0.113207 + 0.993571i \(0.536112\pi\)
\(828\) 7.31537 0.254227
\(829\) 11.0422 0.383510 0.191755 0.981443i \(-0.438582\pi\)
0.191755 + 0.981443i \(0.438582\pi\)
\(830\) −16.5399 −0.574109
\(831\) −1.46007 −0.0506493
\(832\) 4.12492 0.143006
\(833\) −24.8711 −0.861733
\(834\) −15.4082 −0.533542
\(835\) 24.8362 0.859491
\(836\) −2.23952 −0.0774555
\(837\) 7.80273 0.269702
\(838\) 12.2143 0.421937
\(839\) 3.12641 0.107936 0.0539679 0.998543i \(-0.482813\pi\)
0.0539679 + 0.998543i \(0.482813\pi\)
\(840\) 0.487359 0.0168155
\(841\) 16.9338 0.583924
\(842\) −8.91942 −0.307384
\(843\) −12.1249 −0.417604
\(844\) 13.9679 0.480795
\(845\) 4.01497 0.138119
\(846\) 3.52969 0.121353
\(847\) 7.85530 0.269911
\(848\) −7.67781 −0.263657
\(849\) 19.5652 0.671476
\(850\) 3.67781 0.126148
\(851\) −7.31537 −0.250768
\(852\) 8.58018 0.293952
\(853\) −8.98503 −0.307642 −0.153821 0.988099i \(-0.549158\pi\)
−0.153821 + 0.988099i \(0.549158\pi\)
\(854\) −7.07785 −0.242199
\(855\) 0.430057 0.0147076
\(856\) 7.09964 0.242661
\(857\) 5.59314 0.191058 0.0955290 0.995427i \(-0.469546\pi\)
0.0955290 + 0.995427i \(0.469546\pi\)
\(858\) −21.4805 −0.733334
\(859\) −18.8110 −0.641822 −0.320911 0.947109i \(-0.603989\pi\)
−0.320911 + 0.947109i \(0.603989\pi\)
\(860\) 4.77745 0.162910
\(861\) −4.77745 −0.162815
\(862\) 4.98704 0.169859
\(863\) −29.8116 −1.01480 −0.507400 0.861711i \(-0.669393\pi\)
−0.507400 + 0.861711i \(0.669393\pi\)
\(864\) 1.00000 0.0340207
\(865\) −19.9276 −0.677560
\(866\) 15.4805 0.526050
\(867\) −3.47372 −0.117974
\(868\) 3.80273 0.129073
\(869\) −20.8300 −0.706610
\(870\) −6.77745 −0.229777
\(871\) −4.91051 −0.166386
\(872\) 16.4321 0.556460
\(873\) −7.16726 −0.242575
\(874\) 3.14603 0.106416
\(875\) 0.487359 0.0164757
\(876\) −7.15020 −0.241583
\(877\) −9.96806 −0.336598 −0.168299 0.985736i \(-0.553827\pi\)
−0.168299 + 0.985736i \(0.553827\pi\)
\(878\) −21.4929 −0.725349
\(879\) −17.5126 −0.590687
\(880\) −5.20750 −0.175545
\(881\) 27.7387 0.934540 0.467270 0.884115i \(-0.345238\pi\)
0.467270 + 0.884115i \(0.345238\pi\)
\(882\) −6.76248 −0.227705
\(883\) 7.79659 0.262376 0.131188 0.991358i \(-0.458121\pi\)
0.131188 + 0.991358i \(0.458121\pi\)
\(884\) 15.1707 0.510245
\(885\) −0.974718 −0.0327648
\(886\) −32.4355 −1.08969
\(887\) 19.1536 0.643115 0.321558 0.946890i \(-0.395794\pi\)
0.321558 + 0.946890i \(0.395794\pi\)
\(888\) −1.00000 −0.0335578
\(889\) 7.58584 0.254421
\(890\) 16.7051 0.559956
\(891\) −5.20750 −0.174458
\(892\) −22.4335 −0.751128
\(893\) 1.51797 0.0507969
\(894\) −3.47913 −0.116360
\(895\) −4.33034 −0.144747
\(896\) 0.487359 0.0162815
\(897\) 30.1753 1.00752
\(898\) 15.8348 0.528415
\(899\) −52.8826 −1.76373
\(900\) 1.00000 0.0333333
\(901\) −28.2375 −0.940728
\(902\) 51.0478 1.69970
\(903\) 2.32833 0.0774820
\(904\) −1.22255 −0.0406615
\(905\) −0.0505645 −0.00168082
\(906\) −17.7304 −0.589052
\(907\) 1.34530 0.0446700 0.0223350 0.999751i \(-0.492890\pi\)
0.0223350 + 0.999751i \(0.492890\pi\)
\(908\) 23.8368 0.791053
\(909\) −13.7454 −0.455907
\(910\) 2.01032 0.0666413
\(911\) 2.95826 0.0980116 0.0490058 0.998798i \(-0.484395\pi\)
0.0490058 + 0.998798i \(0.484395\pi\)
\(912\) 0.430057 0.0142406
\(913\) 86.1317 2.85054
\(914\) −29.4470 −0.974021
\(915\) −14.5229 −0.480111
\(916\) 10.2157 0.337537
\(917\) −8.55553 −0.282529
\(918\) 3.67781 0.121386
\(919\) 16.0504 0.529454 0.264727 0.964323i \(-0.414718\pi\)
0.264727 + 0.964323i \(0.414718\pi\)
\(920\) 7.31537 0.241181
\(921\) −0.364444 −0.0120088
\(922\) 18.4975 0.609183
\(923\) 35.3925 1.16496
\(924\) −2.53792 −0.0834915
\(925\) −1.00000 −0.0328798
\(926\) −7.46549 −0.245331
\(927\) −3.52969 −0.115930
\(928\) −6.77745 −0.222481
\(929\) −2.20009 −0.0721825 −0.0360913 0.999348i \(-0.511491\pi\)
−0.0360913 + 0.999348i \(0.511491\pi\)
\(930\) 7.80273 0.255862
\(931\) −2.90825 −0.0953141
\(932\) 2.07576 0.0679939
\(933\) −15.1583 −0.496262
\(934\) 33.3917 1.09261
\(935\) −19.1522 −0.626344
\(936\) 4.12492 0.134827
\(937\) −15.9030 −0.519530 −0.259765 0.965672i \(-0.583645\pi\)
−0.259765 + 0.965672i \(0.583645\pi\)
\(938\) −0.580177 −0.0189434
\(939\) −14.7201 −0.480374
\(940\) 3.52969 0.115126
\(941\) 50.0021 1.63002 0.815011 0.579446i \(-0.196731\pi\)
0.815011 + 0.579446i \(0.196731\pi\)
\(942\) 11.8533 0.386201
\(943\) −71.7106 −2.33522
\(944\) −0.974718 −0.0317244
\(945\) 0.487359 0.0158538
\(946\) −24.8786 −0.808872
\(947\) 8.11677 0.263760 0.131880 0.991266i \(-0.457899\pi\)
0.131880 + 0.991266i \(0.457899\pi\)
\(948\) 4.00000 0.129914
\(949\) −29.4940 −0.957416
\(950\) 0.430057 0.0139529
\(951\) −0.942615 −0.0305664
\(952\) 1.79241 0.0580924
\(953\) −34.2451 −1.10931 −0.554654 0.832081i \(-0.687149\pi\)
−0.554654 + 0.832081i \(0.687149\pi\)
\(954\) −7.67781 −0.248578
\(955\) −4.37276 −0.141499
\(956\) −28.2683 −0.914262
\(957\) 35.2936 1.14088
\(958\) 26.7051 0.862803
\(959\) −10.3563 −0.334421
\(960\) 1.00000 0.0322749
\(961\) 29.8826 0.963954
\(962\) −4.12492 −0.132993
\(963\) 7.09964 0.228783
\(964\) 20.4491 0.658622
\(965\) −2.47031 −0.0795219
\(966\) 3.56521 0.114709
\(967\) −7.49960 −0.241171 −0.120585 0.992703i \(-0.538477\pi\)
−0.120585 + 0.992703i \(0.538477\pi\)
\(968\) 16.1181 0.518055
\(969\) 1.58167 0.0508105
\(970\) −7.16726 −0.230127
\(971\) 55.6327 1.78534 0.892669 0.450714i \(-0.148830\pi\)
0.892669 + 0.450714i \(0.148830\pi\)
\(972\) 1.00000 0.0320750
\(973\) −7.50932 −0.240738
\(974\) −1.61437 −0.0517277
\(975\) 4.12492 0.132103
\(976\) −14.5229 −0.464866
\(977\) 33.1685 1.06115 0.530577 0.847637i \(-0.321975\pi\)
0.530577 + 0.847637i \(0.321975\pi\)
\(978\) 5.89354 0.188455
\(979\) −86.9919 −2.78027
\(980\) −6.76248 −0.216020
\(981\) 16.4321 0.524635
\(982\) 1.40477 0.0448282
\(983\) 55.2178 1.76117 0.880587 0.473884i \(-0.157148\pi\)
0.880587 + 0.473884i \(0.157148\pi\)
\(984\) −9.80273 −0.312500
\(985\) −13.3154 −0.424263
\(986\) −24.9262 −0.793811
\(987\) 1.72023 0.0547555
\(988\) 1.77395 0.0564369
\(989\) 34.9488 1.11131
\(990\) −5.20750 −0.165505
\(991\) −36.7815 −1.16840 −0.584201 0.811609i \(-0.698592\pi\)
−0.584201 + 0.811609i \(0.698592\pi\)
\(992\) 7.80273 0.247737
\(993\) 15.8600 0.503303
\(994\) 4.18163 0.132633
\(995\) 21.4403 0.679703
\(996\) −16.5399 −0.524088
\(997\) −42.5058 −1.34617 −0.673086 0.739564i \(-0.735032\pi\)
−0.673086 + 0.739564i \(0.735032\pi\)
\(998\) −9.98495 −0.316068
\(999\) −1.00000 −0.0316386
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 1110.2.a.s.1.2 4
3.2 odd 2 3330.2.a.bj.1.2 4
4.3 odd 2 8880.2.a.cg.1.3 4
5.4 even 2 5550.2.a.cj.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.s.1.2 4 1.1 even 1 trivial
3330.2.a.bj.1.2 4 3.2 odd 2
5550.2.a.cj.1.3 4 5.4 even 2
8880.2.a.cg.1.3 4 4.3 odd 2