Properties

Label 1617.2.a.r.1.3
Level $1617$
Weight $2$
Character 1617.1
Self dual yes
Analytic conductor $12.912$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1617.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.9118100068\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.11491 q^{2} +1.00000 q^{3} +2.47283 q^{4} +1.64207 q^{5} +2.11491 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.11491 q^{2} +1.00000 q^{3} +2.47283 q^{4} +1.64207 q^{5} +2.11491 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.47283 q^{10} +1.00000 q^{11} +2.47283 q^{12} +4.58774 q^{13} +1.64207 q^{15} -2.83076 q^{16} +0.715853 q^{17} +2.11491 q^{18} +1.64207 q^{19} +4.06058 q^{20} +2.11491 q^{22} +1.00000 q^{24} -2.30359 q^{25} +9.70265 q^{26} +1.00000 q^{27} -5.53341 q^{29} +3.47283 q^{30} -5.17548 q^{31} -7.98680 q^{32} +1.00000 q^{33} +1.51396 q^{34} +2.47283 q^{36} -6.58774 q^{37} +3.47283 q^{38} +4.58774 q^{39} +1.64207 q^{40} +5.17548 q^{41} +5.17548 q^{43} +2.47283 q^{44} +1.64207 q^{45} +7.87189 q^{47} -2.83076 q^{48} -4.87189 q^{50} +0.715853 q^{51} +11.3447 q^{52} +2.71585 q^{53} +2.11491 q^{54} +1.64207 q^{55} +1.64207 q^{57} -11.7026 q^{58} -3.15604 q^{59} +4.06058 q^{60} +13.1755 q^{61} -10.9457 q^{62} -11.2298 q^{64} +7.53341 q^{65} +2.11491 q^{66} +4.21037 q^{67} +1.77018 q^{68} -1.85244 q^{71} +1.00000 q^{72} -10.4791 q^{73} -13.9325 q^{74} -2.30359 q^{75} +4.06058 q^{76} +9.70265 q^{78} -12.4596 q^{79} -4.64832 q^{80} +1.00000 q^{81} +10.9457 q^{82} -9.17548 q^{83} +1.17548 q^{85} +10.9457 q^{86} -5.53341 q^{87} +1.00000 q^{88} +4.71585 q^{89} +3.47283 q^{90} -5.17548 q^{93} +16.6483 q^{94} +2.69641 q^{95} -7.98680 q^{96} -13.6351 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 2 q^{4} + 4 q^{5} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 2 q^{4} + 4 q^{5} + 3 q^{8} + 3 q^{9} + 5 q^{10} + 3 q^{11} + 2 q^{12} + 2 q^{13} + 4 q^{15} - 4 q^{16} + 4 q^{17} + 4 q^{19} - 5 q^{20} + 3 q^{24} + 3 q^{25} + 11 q^{26} + 3 q^{27} + 6 q^{29} + 5 q^{30} + 8 q^{31} - 4 q^{32} + 3 q^{33} - 10 q^{34} + 2 q^{36} - 8 q^{37} + 5 q^{38} + 2 q^{39} + 4 q^{40} - 8 q^{41} - 8 q^{43} + 2 q^{44} + 4 q^{45} + 10 q^{47} - 4 q^{48} - q^{50} + 4 q^{51} + 15 q^{52} + 10 q^{53} + 4 q^{55} + 4 q^{57} - 17 q^{58} + 6 q^{59} - 5 q^{60} + 16 q^{61} - 22 q^{62} - 21 q^{64} + 8 q^{67} + 18 q^{68} + 3 q^{72} + 2 q^{73} - 11 q^{74} + 3 q^{75} - 5 q^{76} + 11 q^{78} - 12 q^{79} + 15 q^{80} + 3 q^{81} + 22 q^{82} - 4 q^{83} - 20 q^{85} + 22 q^{86} + 6 q^{87} + 3 q^{88} + 16 q^{89} + 5 q^{90} + 8 q^{93} + 21 q^{94} + 18 q^{95} - 4 q^{96} + 8 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.11491 1.49547 0.747733 0.664000i \(-0.231142\pi\)
0.747733 + 0.664000i \(0.231142\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.47283 1.23642
\(5\) 1.64207 0.734358 0.367179 0.930150i \(-0.380324\pi\)
0.367179 + 0.930150i \(0.380324\pi\)
\(6\) 2.11491 0.863407
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.47283 1.09821
\(11\) 1.00000 0.301511
\(12\) 2.47283 0.713846
\(13\) 4.58774 1.27241 0.636205 0.771520i \(-0.280503\pi\)
0.636205 + 0.771520i \(0.280503\pi\)
\(14\) 0 0
\(15\) 1.64207 0.423982
\(16\) −2.83076 −0.707690
\(17\) 0.715853 0.173620 0.0868099 0.996225i \(-0.472333\pi\)
0.0868099 + 0.996225i \(0.472333\pi\)
\(18\) 2.11491 0.498488
\(19\) 1.64207 0.376718 0.188359 0.982100i \(-0.439683\pi\)
0.188359 + 0.982100i \(0.439683\pi\)
\(20\) 4.06058 0.907972
\(21\) 0 0
\(22\) 2.11491 0.450900
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) −2.30359 −0.460719
\(26\) 9.70265 1.90285
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −5.53341 −1.02753 −0.513764 0.857931i \(-0.671749\pi\)
−0.513764 + 0.857931i \(0.671749\pi\)
\(30\) 3.47283 0.634050
\(31\) −5.17548 −0.929544 −0.464772 0.885430i \(-0.653864\pi\)
−0.464772 + 0.885430i \(0.653864\pi\)
\(32\) −7.98680 −1.41188
\(33\) 1.00000 0.174078
\(34\) 1.51396 0.259642
\(35\) 0 0
\(36\) 2.47283 0.412139
\(37\) −6.58774 −1.08302 −0.541509 0.840695i \(-0.682147\pi\)
−0.541509 + 0.840695i \(0.682147\pi\)
\(38\) 3.47283 0.563368
\(39\) 4.58774 0.734627
\(40\) 1.64207 0.259635
\(41\) 5.17548 0.808275 0.404137 0.914698i \(-0.367572\pi\)
0.404137 + 0.914698i \(0.367572\pi\)
\(42\) 0 0
\(43\) 5.17548 0.789254 0.394627 0.918841i \(-0.370874\pi\)
0.394627 + 0.918841i \(0.370874\pi\)
\(44\) 2.47283 0.372794
\(45\) 1.64207 0.244786
\(46\) 0 0
\(47\) 7.87189 1.14823 0.574116 0.818774i \(-0.305346\pi\)
0.574116 + 0.818774i \(0.305346\pi\)
\(48\) −2.83076 −0.408585
\(49\) 0 0
\(50\) −4.87189 −0.688989
\(51\) 0.715853 0.100239
\(52\) 11.3447 1.57323
\(53\) 2.71585 0.373051 0.186526 0.982450i \(-0.440277\pi\)
0.186526 + 0.982450i \(0.440277\pi\)
\(54\) 2.11491 0.287802
\(55\) 1.64207 0.221417
\(56\) 0 0
\(57\) 1.64207 0.217498
\(58\) −11.7026 −1.53663
\(59\) −3.15604 −0.410881 −0.205440 0.978670i \(-0.565863\pi\)
−0.205440 + 0.978670i \(0.565863\pi\)
\(60\) 4.06058 0.524218
\(61\) 13.1755 1.68695 0.843474 0.537170i \(-0.180507\pi\)
0.843474 + 0.537170i \(0.180507\pi\)
\(62\) −10.9457 −1.39010
\(63\) 0 0
\(64\) −11.2298 −1.40373
\(65\) 7.53341 0.934404
\(66\) 2.11491 0.260327
\(67\) 4.21037 0.514378 0.257189 0.966361i \(-0.417204\pi\)
0.257189 + 0.966361i \(0.417204\pi\)
\(68\) 1.77018 0.214666
\(69\) 0 0
\(70\) 0 0
\(71\) −1.85244 −0.219844 −0.109922 0.993940i \(-0.535060\pi\)
−0.109922 + 0.993940i \(0.535060\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.4791 −1.22648 −0.613242 0.789895i \(-0.710135\pi\)
−0.613242 + 0.789895i \(0.710135\pi\)
\(74\) −13.9325 −1.61962
\(75\) −2.30359 −0.265996
\(76\) 4.06058 0.465780
\(77\) 0 0
\(78\) 9.70265 1.09861
\(79\) −12.4596 −1.40182 −0.700909 0.713251i \(-0.747222\pi\)
−0.700909 + 0.713251i \(0.747222\pi\)
\(80\) −4.64832 −0.519698
\(81\) 1.00000 0.111111
\(82\) 10.9457 1.20875
\(83\) −9.17548 −1.00714 −0.503570 0.863954i \(-0.667980\pi\)
−0.503570 + 0.863954i \(0.667980\pi\)
\(84\) 0 0
\(85\) 1.17548 0.127499
\(86\) 10.9457 1.18030
\(87\) −5.53341 −0.593244
\(88\) 1.00000 0.106600
\(89\) 4.71585 0.499879 0.249940 0.968261i \(-0.419589\pi\)
0.249940 + 0.968261i \(0.419589\pi\)
\(90\) 3.47283 0.366069
\(91\) 0 0
\(92\) 0 0
\(93\) −5.17548 −0.536673
\(94\) 16.6483 1.71714
\(95\) 2.69641 0.276645
\(96\) −7.98680 −0.815149
\(97\) −13.6351 −1.38444 −0.692218 0.721688i \(-0.743366\pi\)
−0.692218 + 0.721688i \(0.743366\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −5.69641 −0.569641
\(101\) −14.6072 −1.45347 −0.726735 0.686918i \(-0.758963\pi\)
−0.726735 + 0.686918i \(0.758963\pi\)
\(102\) 1.51396 0.149905
\(103\) 17.8913 1.76289 0.881443 0.472291i \(-0.156573\pi\)
0.881443 + 0.472291i \(0.156573\pi\)
\(104\) 4.58774 0.449865
\(105\) 0 0
\(106\) 5.74378 0.557885
\(107\) 11.1560 1.07849 0.539247 0.842147i \(-0.318709\pi\)
0.539247 + 0.842147i \(0.318709\pi\)
\(108\) 2.47283 0.237949
\(109\) −0.108664 −0.0104082 −0.00520408 0.999986i \(-0.501657\pi\)
−0.00520408 + 0.999986i \(0.501657\pi\)
\(110\) 3.47283 0.331122
\(111\) −6.58774 −0.625281
\(112\) 0 0
\(113\) −13.0668 −1.22922 −0.614611 0.788830i \(-0.710687\pi\)
−0.614611 + 0.788830i \(0.710687\pi\)
\(114\) 3.47283 0.325261
\(115\) 0 0
\(116\) −13.6832 −1.27045
\(117\) 4.58774 0.424137
\(118\) −6.67472 −0.614458
\(119\) 0 0
\(120\) 1.64207 0.149900
\(121\) 1.00000 0.0909091
\(122\) 27.8649 2.52277
\(123\) 5.17548 0.466658
\(124\) −12.7981 −1.14930
\(125\) −11.9930 −1.07269
\(126\) 0 0
\(127\) −20.4596 −1.81550 −0.907749 0.419513i \(-0.862201\pi\)
−0.907749 + 0.419513i \(0.862201\pi\)
\(128\) −7.77643 −0.687346
\(129\) 5.17548 0.455676
\(130\) 15.9325 1.39737
\(131\) −15.0668 −1.31639 −0.658197 0.752846i \(-0.728681\pi\)
−0.658197 + 0.752846i \(0.728681\pi\)
\(132\) 2.47283 0.215233
\(133\) 0 0
\(134\) 8.90454 0.769235
\(135\) 1.64207 0.141327
\(136\) 0.715853 0.0613839
\(137\) −14.4596 −1.23537 −0.617685 0.786426i \(-0.711929\pi\)
−0.617685 + 0.786426i \(0.711929\pi\)
\(138\) 0 0
\(139\) −5.13659 −0.435680 −0.217840 0.975985i \(-0.569901\pi\)
−0.217840 + 0.975985i \(0.569901\pi\)
\(140\) 0 0
\(141\) 7.87189 0.662933
\(142\) −3.91774 −0.328770
\(143\) 4.58774 0.383646
\(144\) −2.83076 −0.235897
\(145\) −9.08627 −0.754573
\(146\) −22.1623 −1.83416
\(147\) 0 0
\(148\) −16.2904 −1.33906
\(149\) 16.8176 1.37775 0.688874 0.724881i \(-0.258105\pi\)
0.688874 + 0.724881i \(0.258105\pi\)
\(150\) −4.87189 −0.397788
\(151\) 4.71585 0.383771 0.191885 0.981417i \(-0.438540\pi\)
0.191885 + 0.981417i \(0.438540\pi\)
\(152\) 1.64207 0.133190
\(153\) 0.715853 0.0578733
\(154\) 0 0
\(155\) −8.49852 −0.682618
\(156\) 11.3447 0.908305
\(157\) −11.0279 −0.880124 −0.440062 0.897967i \(-0.645044\pi\)
−0.440062 + 0.897967i \(0.645044\pi\)
\(158\) −26.3510 −2.09637
\(159\) 2.71585 0.215381
\(160\) −13.1149 −1.03682
\(161\) 0 0
\(162\) 2.11491 0.166163
\(163\) 5.64207 0.441921 0.220961 0.975283i \(-0.429081\pi\)
0.220961 + 0.975283i \(0.429081\pi\)
\(164\) 12.7981 0.999364
\(165\) 1.64207 0.127835
\(166\) −19.4053 −1.50614
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) 8.04737 0.619029
\(170\) 2.48604 0.190670
\(171\) 1.64207 0.125573
\(172\) 12.7981 0.975847
\(173\) −15.0279 −1.14255 −0.571276 0.820758i \(-0.693551\pi\)
−0.571276 + 0.820758i \(0.693551\pi\)
\(174\) −11.7026 −0.887176
\(175\) 0 0
\(176\) −2.83076 −0.213377
\(177\) −3.15604 −0.237222
\(178\) 9.97359 0.747552
\(179\) 9.89134 0.739313 0.369657 0.929168i \(-0.379475\pi\)
0.369657 + 0.929168i \(0.379475\pi\)
\(180\) 4.06058 0.302657
\(181\) 6.82452 0.507262 0.253631 0.967301i \(-0.418375\pi\)
0.253631 + 0.967301i \(0.418375\pi\)
\(182\) 0 0
\(183\) 13.1755 0.973960
\(184\) 0 0
\(185\) −10.8176 −0.795323
\(186\) −10.9457 −0.802575
\(187\) 0.715853 0.0523483
\(188\) 19.4659 1.41969
\(189\) 0 0
\(190\) 5.70265 0.413714
\(191\) 11.0279 0.797953 0.398976 0.916961i \(-0.369366\pi\)
0.398976 + 0.916961i \(0.369366\pi\)
\(192\) −11.2298 −0.810442
\(193\) −7.85244 −0.565231 −0.282616 0.959233i \(-0.591202\pi\)
−0.282616 + 0.959233i \(0.591202\pi\)
\(194\) −28.8370 −2.07038
\(195\) 7.53341 0.539479
\(196\) 0 0
\(197\) 19.8913 1.41720 0.708599 0.705611i \(-0.249327\pi\)
0.708599 + 0.705611i \(0.249327\pi\)
\(198\) 2.11491 0.150300
\(199\) 23.5264 1.66775 0.833873 0.551957i \(-0.186119\pi\)
0.833873 + 0.551957i \(0.186119\pi\)
\(200\) −2.30359 −0.162889
\(201\) 4.21037 0.296976
\(202\) −30.8929 −2.17361
\(203\) 0 0
\(204\) 1.77018 0.123938
\(205\) 8.49852 0.593563
\(206\) 37.8385 2.63633
\(207\) 0 0
\(208\) −12.9868 −0.900472
\(209\) 1.64207 0.113585
\(210\) 0 0
\(211\) −4.25622 −0.293010 −0.146505 0.989210i \(-0.546803\pi\)
−0.146505 + 0.989210i \(0.546803\pi\)
\(212\) 6.71585 0.461247
\(213\) −1.85244 −0.126927
\(214\) 23.5940 1.61285
\(215\) 8.49852 0.579595
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −0.229815 −0.0155650
\(219\) −10.4791 −0.708110
\(220\) 4.06058 0.273764
\(221\) 3.28415 0.220916
\(222\) −13.9325 −0.935086
\(223\) −3.74378 −0.250702 −0.125351 0.992112i \(-0.540006\pi\)
−0.125351 + 0.992112i \(0.540006\pi\)
\(224\) 0 0
\(225\) −2.30359 −0.153573
\(226\) −27.6351 −1.83826
\(227\) −15.4876 −1.02795 −0.513973 0.857807i \(-0.671827\pi\)
−0.513973 + 0.857807i \(0.671827\pi\)
\(228\) 4.06058 0.268918
\(229\) −8.00000 −0.528655 −0.264327 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −5.53341 −0.363286
\(233\) −5.54037 −0.362962 −0.181481 0.983394i \(-0.558089\pi\)
−0.181481 + 0.983394i \(0.558089\pi\)
\(234\) 9.70265 0.634282
\(235\) 12.9262 0.843214
\(236\) −7.80435 −0.508020
\(237\) −12.4596 −0.809340
\(238\) 0 0
\(239\) −1.26470 −0.0818067 −0.0409033 0.999163i \(-0.513024\pi\)
−0.0409033 + 0.999163i \(0.513024\pi\)
\(240\) −4.64832 −0.300048
\(241\) −23.8719 −1.53772 −0.768862 0.639415i \(-0.779177\pi\)
−0.768862 + 0.639415i \(0.779177\pi\)
\(242\) 2.11491 0.135951
\(243\) 1.00000 0.0641500
\(244\) 32.5808 2.08577
\(245\) 0 0
\(246\) 10.9457 0.697870
\(247\) 7.53341 0.479339
\(248\) −5.17548 −0.328643
\(249\) −9.17548 −0.581473
\(250\) −25.3642 −1.60417
\(251\) 22.0194 1.38986 0.694928 0.719080i \(-0.255436\pi\)
0.694928 + 0.719080i \(0.255436\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −43.2702 −2.71502
\(255\) 1.17548 0.0736116
\(256\) 6.01320 0.375825
\(257\) 27.3161 1.70393 0.851965 0.523599i \(-0.175411\pi\)
0.851965 + 0.523599i \(0.175411\pi\)
\(258\) 10.9457 0.681448
\(259\) 0 0
\(260\) 18.6289 1.15531
\(261\) −5.53341 −0.342509
\(262\) −31.8649 −1.96862
\(263\) −11.8719 −0.732052 −0.366026 0.930605i \(-0.619282\pi\)
−0.366026 + 0.930605i \(0.619282\pi\)
\(264\) 1.00000 0.0615457
\(265\) 4.45963 0.273953
\(266\) 0 0
\(267\) 4.71585 0.288606
\(268\) 10.4115 0.635986
\(269\) 5.63511 0.343579 0.171789 0.985134i \(-0.445045\pi\)
0.171789 + 0.985134i \(0.445045\pi\)
\(270\) 3.47283 0.211350
\(271\) −5.68097 −0.345094 −0.172547 0.985001i \(-0.555200\pi\)
−0.172547 + 0.985001i \(0.555200\pi\)
\(272\) −2.02641 −0.122869
\(273\) 0 0
\(274\) −30.5808 −1.84745
\(275\) −2.30359 −0.138912
\(276\) 0 0
\(277\) −25.4876 −1.53140 −0.765699 0.643199i \(-0.777607\pi\)
−0.765699 + 0.643199i \(0.777607\pi\)
\(278\) −10.8634 −0.651544
\(279\) −5.17548 −0.309848
\(280\) 0 0
\(281\) 17.5723 1.04828 0.524138 0.851633i \(-0.324388\pi\)
0.524138 + 0.851633i \(0.324388\pi\)
\(282\) 16.6483 0.991393
\(283\) 25.1296 1.49380 0.746901 0.664936i \(-0.231541\pi\)
0.746901 + 0.664936i \(0.231541\pi\)
\(284\) −4.58078 −0.271819
\(285\) 2.69641 0.159721
\(286\) 9.70265 0.573730
\(287\) 0 0
\(288\) −7.98680 −0.470626
\(289\) −16.4876 −0.969856
\(290\) −19.2166 −1.12844
\(291\) −13.6351 −0.799304
\(292\) −25.9130 −1.51644
\(293\) −14.3510 −0.838392 −0.419196 0.907896i \(-0.637688\pi\)
−0.419196 + 0.907896i \(0.637688\pi\)
\(294\) 0 0
\(295\) −5.18244 −0.301734
\(296\) −6.58774 −0.382905
\(297\) 1.00000 0.0580259
\(298\) 35.5676 2.06037
\(299\) 0 0
\(300\) −5.69641 −0.328882
\(301\) 0 0
\(302\) 9.97359 0.573916
\(303\) −14.6072 −0.839161
\(304\) −4.64832 −0.266599
\(305\) 21.6351 1.23882
\(306\) 1.51396 0.0865475
\(307\) 13.1366 0.749745 0.374872 0.927076i \(-0.377687\pi\)
0.374872 + 0.927076i \(0.377687\pi\)
\(308\) 0 0
\(309\) 17.8913 1.01780
\(310\) −17.9736 −1.02083
\(311\) −14.3510 −0.813769 −0.406884 0.913480i \(-0.633385\pi\)
−0.406884 + 0.913480i \(0.633385\pi\)
\(312\) 4.58774 0.259730
\(313\) 23.7438 1.34208 0.671039 0.741422i \(-0.265848\pi\)
0.671039 + 0.741422i \(0.265848\pi\)
\(314\) −23.3230 −1.31620
\(315\) 0 0
\(316\) −30.8106 −1.73323
\(317\) −6.45963 −0.362809 −0.181404 0.983409i \(-0.558064\pi\)
−0.181404 + 0.983409i \(0.558064\pi\)
\(318\) 5.74378 0.322095
\(319\) −5.53341 −0.309811
\(320\) −18.4402 −1.03084
\(321\) 11.1560 0.622669
\(322\) 0 0
\(323\) 1.17548 0.0654056
\(324\) 2.47283 0.137380
\(325\) −10.5683 −0.586224
\(326\) 11.9325 0.660878
\(327\) −0.108664 −0.00600915
\(328\) 5.17548 0.285768
\(329\) 0 0
\(330\) 3.47283 0.191173
\(331\) 31.2702 1.71877 0.859384 0.511332i \(-0.170848\pi\)
0.859384 + 0.511332i \(0.170848\pi\)
\(332\) −22.6894 −1.24525
\(333\) −6.58774 −0.361006
\(334\) 33.8385 1.85156
\(335\) 6.91373 0.377738
\(336\) 0 0
\(337\) −16.3121 −0.888575 −0.444288 0.895884i \(-0.646543\pi\)
−0.444288 + 0.895884i \(0.646543\pi\)
\(338\) 17.0194 0.925736
\(339\) −13.0668 −0.709692
\(340\) 2.90677 0.157642
\(341\) −5.17548 −0.280268
\(342\) 3.47283 0.187789
\(343\) 0 0
\(344\) 5.17548 0.279043
\(345\) 0 0
\(346\) −31.7827 −1.70865
\(347\) −6.86341 −0.368447 −0.184224 0.982884i \(-0.558977\pi\)
−0.184224 + 0.982884i \(0.558977\pi\)
\(348\) −13.6832 −0.733497
\(349\) 23.8580 1.27709 0.638544 0.769585i \(-0.279537\pi\)
0.638544 + 0.769585i \(0.279537\pi\)
\(350\) 0 0
\(351\) 4.58774 0.244876
\(352\) −7.98680 −0.425698
\(353\) 11.7368 0.624688 0.312344 0.949969i \(-0.398886\pi\)
0.312344 + 0.949969i \(0.398886\pi\)
\(354\) −6.67472 −0.354758
\(355\) −3.04185 −0.161444
\(356\) 11.6615 0.618059
\(357\) 0 0
\(358\) 20.9193 1.10562
\(359\) −27.7827 −1.46631 −0.733157 0.680060i \(-0.761954\pi\)
−0.733157 + 0.680060i \(0.761954\pi\)
\(360\) 1.64207 0.0865449
\(361\) −16.3036 −0.858084
\(362\) 14.4332 0.758593
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −17.2074 −0.900677
\(366\) 27.8649 1.45652
\(367\) −26.5544 −1.38613 −0.693064 0.720877i \(-0.743739\pi\)
−0.693064 + 0.720877i \(0.743739\pi\)
\(368\) 0 0
\(369\) 5.17548 0.269425
\(370\) −22.8781 −1.18938
\(371\) 0 0
\(372\) −12.7981 −0.663551
\(373\) 5.06682 0.262350 0.131175 0.991359i \(-0.458125\pi\)
0.131175 + 0.991359i \(0.458125\pi\)
\(374\) 1.51396 0.0782851
\(375\) −11.9930 −0.619318
\(376\) 7.87189 0.405962
\(377\) −25.3859 −1.30744
\(378\) 0 0
\(379\) −30.5613 −1.56983 −0.784915 0.619603i \(-0.787294\pi\)
−0.784915 + 0.619603i \(0.787294\pi\)
\(380\) 6.66776 0.342049
\(381\) −20.4596 −1.04818
\(382\) 23.3230 1.19331
\(383\) 9.13659 0.466858 0.233429 0.972374i \(-0.425005\pi\)
0.233429 + 0.972374i \(0.425005\pi\)
\(384\) −7.77643 −0.396839
\(385\) 0 0
\(386\) −16.6072 −0.845284
\(387\) 5.17548 0.263085
\(388\) −33.7174 −1.71174
\(389\) 21.2702 1.07844 0.539222 0.842164i \(-0.318719\pi\)
0.539222 + 0.842164i \(0.318719\pi\)
\(390\) 15.9325 0.806772
\(391\) 0 0
\(392\) 0 0
\(393\) −15.0668 −0.760020
\(394\) 42.0683 2.11937
\(395\) −20.4596 −1.02944
\(396\) 2.47283 0.124265
\(397\) 28.9582 1.45337 0.726684 0.686972i \(-0.241060\pi\)
0.726684 + 0.686972i \(0.241060\pi\)
\(398\) 49.7563 2.49406
\(399\) 0 0
\(400\) 6.52092 0.326046
\(401\) 32.3510 1.61553 0.807765 0.589505i \(-0.200677\pi\)
0.807765 + 0.589505i \(0.200677\pi\)
\(402\) 8.90454 0.444118
\(403\) −23.7438 −1.18276
\(404\) −36.1212 −1.79709
\(405\) 1.64207 0.0815953
\(406\) 0 0
\(407\) −6.58774 −0.326542
\(408\) 0.715853 0.0354400
\(409\) 2.31207 0.114325 0.0571623 0.998365i \(-0.481795\pi\)
0.0571623 + 0.998365i \(0.481795\pi\)
\(410\) 17.9736 0.887652
\(411\) −14.4596 −0.713241
\(412\) 44.2423 2.17966
\(413\) 0 0
\(414\) 0 0
\(415\) −15.0668 −0.739601
\(416\) −36.6414 −1.79649
\(417\) −5.13659 −0.251540
\(418\) 3.47283 0.169862
\(419\) 38.4263 1.87725 0.938623 0.344945i \(-0.112102\pi\)
0.938623 + 0.344945i \(0.112102\pi\)
\(420\) 0 0
\(421\) 15.5070 0.755765 0.377883 0.925854i \(-0.376652\pi\)
0.377883 + 0.925854i \(0.376652\pi\)
\(422\) −9.00152 −0.438187
\(423\) 7.87189 0.382744
\(424\) 2.71585 0.131893
\(425\) −1.64903 −0.0799899
\(426\) −3.91774 −0.189815
\(427\) 0 0
\(428\) 27.5870 1.33347
\(429\) 4.58774 0.221498
\(430\) 17.9736 0.866764
\(431\) 37.9666 1.82879 0.914394 0.404825i \(-0.132668\pi\)
0.914394 + 0.404825i \(0.132668\pi\)
\(432\) −2.83076 −0.136195
\(433\) 10.1087 0.485791 0.242896 0.970052i \(-0.421903\pi\)
0.242896 + 0.970052i \(0.421903\pi\)
\(434\) 0 0
\(435\) −9.08627 −0.435653
\(436\) −0.268709 −0.0128688
\(437\) 0 0
\(438\) −22.1623 −1.05895
\(439\) −22.5224 −1.07494 −0.537469 0.843284i \(-0.680619\pi\)
−0.537469 + 0.843284i \(0.680619\pi\)
\(440\) 1.64207 0.0782828
\(441\) 0 0
\(442\) 6.94567 0.330372
\(443\) 20.6630 0.981731 0.490865 0.871235i \(-0.336681\pi\)
0.490865 + 0.871235i \(0.336681\pi\)
\(444\) −16.2904 −0.773108
\(445\) 7.74378 0.367090
\(446\) −7.91774 −0.374916
\(447\) 16.8176 0.795443
\(448\) 0 0
\(449\) 40.8106 1.92597 0.962986 0.269553i \(-0.0868759\pi\)
0.962986 + 0.269553i \(0.0868759\pi\)
\(450\) −4.87189 −0.229663
\(451\) 5.17548 0.243704
\(452\) −32.3121 −1.51983
\(453\) 4.71585 0.221570
\(454\) −32.7547 −1.53726
\(455\) 0 0
\(456\) 1.64207 0.0768971
\(457\) −28.3899 −1.32802 −0.664011 0.747723i \(-0.731147\pi\)
−0.664011 + 0.747723i \(0.731147\pi\)
\(458\) −16.9193 −0.790585
\(459\) 0.715853 0.0334131
\(460\) 0 0
\(461\) 21.4178 0.997526 0.498763 0.866739i \(-0.333788\pi\)
0.498763 + 0.866739i \(0.333788\pi\)
\(462\) 0 0
\(463\) 20.7478 0.964231 0.482116 0.876108i \(-0.339868\pi\)
0.482116 + 0.876108i \(0.339868\pi\)
\(464\) 15.6638 0.727172
\(465\) −8.49852 −0.394110
\(466\) −11.7174 −0.542797
\(467\) 26.8998 1.24477 0.622387 0.782709i \(-0.286163\pi\)
0.622387 + 0.782709i \(0.286163\pi\)
\(468\) 11.3447 0.524410
\(469\) 0 0
\(470\) 27.3378 1.26100
\(471\) −11.0279 −0.508140
\(472\) −3.15604 −0.145268
\(473\) 5.17548 0.237969
\(474\) −26.3510 −1.21034
\(475\) −3.78267 −0.173561
\(476\) 0 0
\(477\) 2.71585 0.124350
\(478\) −2.67472 −0.122339
\(479\) 30.8106 1.40777 0.703886 0.710313i \(-0.251447\pi\)
0.703886 + 0.710313i \(0.251447\pi\)
\(480\) −13.1149 −0.598611
\(481\) −30.2229 −1.37804
\(482\) −50.4868 −2.29961
\(483\) 0 0
\(484\) 2.47283 0.112402
\(485\) −22.3899 −1.01667
\(486\) 2.11491 0.0959342
\(487\) −23.4876 −1.06432 −0.532161 0.846643i \(-0.678620\pi\)
−0.532161 + 0.846643i \(0.678620\pi\)
\(488\) 13.1755 0.596426
\(489\) 5.64207 0.255143
\(490\) 0 0
\(491\) −16.6266 −0.750350 −0.375175 0.926954i \(-0.622417\pi\)
−0.375175 + 0.926954i \(0.622417\pi\)
\(492\) 12.7981 0.576983
\(493\) −3.96111 −0.178399
\(494\) 15.9325 0.716835
\(495\) 1.64207 0.0738057
\(496\) 14.6506 0.657829
\(497\) 0 0
\(498\) −19.4053 −0.869572
\(499\) −7.99304 −0.357818 −0.178909 0.983866i \(-0.557257\pi\)
−0.178909 + 0.983866i \(0.557257\pi\)
\(500\) −29.6568 −1.32629
\(501\) 16.0000 0.714827
\(502\) 46.5691 2.07848
\(503\) 24.6630 1.09967 0.549835 0.835273i \(-0.314690\pi\)
0.549835 + 0.835273i \(0.314690\pi\)
\(504\) 0 0
\(505\) −23.9861 −1.06737
\(506\) 0 0
\(507\) 8.04737 0.357396
\(508\) −50.5933 −2.24471
\(509\) 15.0668 0.667825 0.333912 0.942604i \(-0.391631\pi\)
0.333912 + 0.942604i \(0.391631\pi\)
\(510\) 2.48604 0.110084
\(511\) 0 0
\(512\) 28.2702 1.24938
\(513\) 1.64207 0.0724993
\(514\) 57.7710 2.54817
\(515\) 29.3789 1.29459
\(516\) 12.7981 0.563405
\(517\) 7.87189 0.346205
\(518\) 0 0
\(519\) −15.0279 −0.659653
\(520\) 7.53341 0.330362
\(521\) −42.3051 −1.85342 −0.926710 0.375777i \(-0.877376\pi\)
−0.926710 + 0.375777i \(0.877376\pi\)
\(522\) −11.7026 −0.512211
\(523\) 5.84548 0.255605 0.127803 0.991800i \(-0.459208\pi\)
0.127803 + 0.991800i \(0.459208\pi\)
\(524\) −37.2577 −1.62761
\(525\) 0 0
\(526\) −25.1079 −1.09476
\(527\) −3.70488 −0.161387
\(528\) −2.83076 −0.123193
\(529\) −23.0000 −1.00000
\(530\) 9.43171 0.409687
\(531\) −3.15604 −0.136960
\(532\) 0 0
\(533\) 23.7438 1.02846
\(534\) 9.97359 0.431600
\(535\) 18.3190 0.792001
\(536\) 4.21037 0.181860
\(537\) 9.89134 0.426843
\(538\) 11.9177 0.513810
\(539\) 0 0
\(540\) 4.06058 0.174739
\(541\) −40.2982 −1.73255 −0.866276 0.499565i \(-0.833493\pi\)
−0.866276 + 0.499565i \(0.833493\pi\)
\(542\) −12.0147 −0.516076
\(543\) 6.82452 0.292868
\(544\) −5.71737 −0.245130
\(545\) −0.178435 −0.00764331
\(546\) 0 0
\(547\) 2.98903 0.127802 0.0639009 0.997956i \(-0.479646\pi\)
0.0639009 + 0.997956i \(0.479646\pi\)
\(548\) −35.7563 −1.52743
\(549\) 13.1755 0.562316
\(550\) −4.87189 −0.207738
\(551\) −9.08627 −0.387088
\(552\) 0 0
\(553\) 0 0
\(554\) −53.9038 −2.29015
\(555\) −10.8176 −0.459180
\(556\) −12.7019 −0.538682
\(557\) 21.1127 0.894573 0.447286 0.894391i \(-0.352390\pi\)
0.447286 + 0.894391i \(0.352390\pi\)
\(558\) −10.9457 −0.463367
\(559\) 23.7438 1.00425
\(560\) 0 0
\(561\) 0.715853 0.0302233
\(562\) 37.1638 1.56766
\(563\) 29.8774 1.25918 0.629591 0.776926i \(-0.283222\pi\)
0.629591 + 0.776926i \(0.283222\pi\)
\(564\) 19.4659 0.819661
\(565\) −21.4567 −0.902689
\(566\) 53.1468 2.23393
\(567\) 0 0
\(568\) −1.85244 −0.0777267
\(569\) 0.186452 0.00781648 0.00390824 0.999992i \(-0.498756\pi\)
0.00390824 + 0.999992i \(0.498756\pi\)
\(570\) 5.70265 0.238858
\(571\) 3.06682 0.128342 0.0641712 0.997939i \(-0.479560\pi\)
0.0641712 + 0.997939i \(0.479560\pi\)
\(572\) 11.3447 0.474347
\(573\) 11.0279 0.460698
\(574\) 0 0
\(575\) 0 0
\(576\) −11.2298 −0.467909
\(577\) 18.8495 0.784715 0.392357 0.919813i \(-0.371660\pi\)
0.392357 + 0.919813i \(0.371660\pi\)
\(578\) −34.8697 −1.45039
\(579\) −7.85244 −0.326336
\(580\) −22.4688 −0.932967
\(581\) 0 0
\(582\) −28.8370 −1.19533
\(583\) 2.71585 0.112479
\(584\) −10.4791 −0.433627
\(585\) 7.53341 0.311468
\(586\) −30.3510 −1.25379
\(587\) −8.11419 −0.334908 −0.167454 0.985880i \(-0.553555\pi\)
−0.167454 + 0.985880i \(0.553555\pi\)
\(588\) 0 0
\(589\) −8.49852 −0.350176
\(590\) −10.9604 −0.451232
\(591\) 19.8913 0.818220
\(592\) 18.6483 0.766441
\(593\) 6.36489 0.261375 0.130687 0.991424i \(-0.458282\pi\)
0.130687 + 0.991424i \(0.458282\pi\)
\(594\) 2.11491 0.0867757
\(595\) 0 0
\(596\) 41.5870 1.70347
\(597\) 23.5264 0.962873
\(598\) 0 0
\(599\) 0.676959 0.0276598 0.0138299 0.999904i \(-0.495598\pi\)
0.0138299 + 0.999904i \(0.495598\pi\)
\(600\) −2.30359 −0.0940438
\(601\) −22.6964 −0.925806 −0.462903 0.886409i \(-0.653192\pi\)
−0.462903 + 0.886409i \(0.653192\pi\)
\(602\) 0 0
\(603\) 4.21037 0.171459
\(604\) 11.6615 0.474501
\(605\) 1.64207 0.0667598
\(606\) −30.8929 −1.25494
\(607\) 14.6141 0.593170 0.296585 0.955006i \(-0.404152\pi\)
0.296585 + 0.955006i \(0.404152\pi\)
\(608\) −13.1149 −0.531880
\(609\) 0 0
\(610\) 45.7563 1.85262
\(611\) 36.1142 1.46102
\(612\) 1.77018 0.0715555
\(613\) −28.3899 −1.14666 −0.573328 0.819326i \(-0.694348\pi\)
−0.573328 + 0.819326i \(0.694348\pi\)
\(614\) 27.7827 1.12122
\(615\) 8.49852 0.342694
\(616\) 0 0
\(617\) −21.5404 −0.867183 −0.433591 0.901110i \(-0.642754\pi\)
−0.433591 + 0.901110i \(0.642754\pi\)
\(618\) 37.8385 1.52209
\(619\) 23.7827 0.955906 0.477953 0.878385i \(-0.341379\pi\)
0.477953 + 0.878385i \(0.341379\pi\)
\(620\) −21.0154 −0.844000
\(621\) 0 0
\(622\) −30.3510 −1.21696
\(623\) 0 0
\(624\) −12.9868 −0.519888
\(625\) −8.17548 −0.327019
\(626\) 50.2159 2.00703
\(627\) 1.64207 0.0655781
\(628\) −27.2702 −1.08820
\(629\) −4.71585 −0.188033
\(630\) 0 0
\(631\) −40.4068 −1.60857 −0.804285 0.594244i \(-0.797451\pi\)
−0.804285 + 0.594244i \(0.797451\pi\)
\(632\) −12.4596 −0.495617
\(633\) −4.25622 −0.169170
\(634\) −13.6615 −0.542568
\(635\) −33.5962 −1.33323
\(636\) 6.71585 0.266301
\(637\) 0 0
\(638\) −11.7026 −0.463312
\(639\) −1.85244 −0.0732815
\(640\) −12.7695 −0.504758
\(641\) 13.4876 0.532726 0.266363 0.963873i \(-0.414178\pi\)
0.266363 + 0.963873i \(0.414178\pi\)
\(642\) 23.5940 0.931180
\(643\) 26.8245 1.05786 0.528928 0.848667i \(-0.322594\pi\)
0.528928 + 0.848667i \(0.322594\pi\)
\(644\) 0 0
\(645\) 8.49852 0.334629
\(646\) 2.48604 0.0978118
\(647\) 42.4791 1.67002 0.835012 0.550231i \(-0.185460\pi\)
0.835012 + 0.550231i \(0.185460\pi\)
\(648\) 1.00000 0.0392837
\(649\) −3.15604 −0.123885
\(650\) −22.3510 −0.876677
\(651\) 0 0
\(652\) 13.9519 0.546399
\(653\) 7.39281 0.289303 0.144652 0.989483i \(-0.453794\pi\)
0.144652 + 0.989483i \(0.453794\pi\)
\(654\) −0.229815 −0.00898648
\(655\) −24.7408 −0.966704
\(656\) −14.6506 −0.572008
\(657\) −10.4791 −0.408828
\(658\) 0 0
\(659\) 18.2368 0.710404 0.355202 0.934790i \(-0.384412\pi\)
0.355202 + 0.934790i \(0.384412\pi\)
\(660\) 4.06058 0.158058
\(661\) 8.16451 0.317563 0.158781 0.987314i \(-0.449243\pi\)
0.158781 + 0.987314i \(0.449243\pi\)
\(662\) 66.1336 2.57036
\(663\) 3.28415 0.127546
\(664\) −9.17548 −0.356078
\(665\) 0 0
\(666\) −13.9325 −0.539872
\(667\) 0 0
\(668\) 39.5653 1.53083
\(669\) −3.74378 −0.144743
\(670\) 14.6219 0.564894
\(671\) 13.1755 0.508634
\(672\) 0 0
\(673\) 43.6212 1.68147 0.840737 0.541444i \(-0.182122\pi\)
0.840737 + 0.541444i \(0.182122\pi\)
\(674\) −34.4985 −1.32883
\(675\) −2.30359 −0.0886654
\(676\) 19.8998 0.765377
\(677\) −23.1057 −0.888025 −0.444012 0.896021i \(-0.646445\pi\)
−0.444012 + 0.896021i \(0.646445\pi\)
\(678\) −27.6351 −1.06132
\(679\) 0 0
\(680\) 1.17548 0.0450777
\(681\) −15.4876 −0.593484
\(682\) −10.9457 −0.419131
\(683\) 27.5653 1.05476 0.527379 0.849630i \(-0.323175\pi\)
0.527379 + 0.849630i \(0.323175\pi\)
\(684\) 4.06058 0.155260
\(685\) −23.7438 −0.907203
\(686\) 0 0
\(687\) −8.00000 −0.305219
\(688\) −14.6506 −0.558547
\(689\) 12.4596 0.474674
\(690\) 0 0
\(691\) −5.85244 −0.222637 −0.111319 0.993785i \(-0.535507\pi\)
−0.111319 + 0.993785i \(0.535507\pi\)
\(692\) −37.1616 −1.41267
\(693\) 0 0
\(694\) −14.5155 −0.551000
\(695\) −8.43466 −0.319945
\(696\) −5.53341 −0.209743
\(697\) 3.70488 0.140332
\(698\) 50.4574 1.90984
\(699\) −5.54037 −0.209556
\(700\) 0 0
\(701\) 9.32304 0.352126 0.176063 0.984379i \(-0.443664\pi\)
0.176063 + 0.984379i \(0.443664\pi\)
\(702\) 9.70265 0.366203
\(703\) −10.8176 −0.407992
\(704\) −11.2298 −0.423240
\(705\) 12.9262 0.486830
\(706\) 24.8223 0.934199
\(707\) 0 0
\(708\) −7.80435 −0.293306
\(709\) 41.5629 1.56093 0.780463 0.625202i \(-0.214983\pi\)
0.780463 + 0.625202i \(0.214983\pi\)
\(710\) −6.43322 −0.241435
\(711\) −12.4596 −0.467273
\(712\) 4.71585 0.176734
\(713\) 0 0
\(714\) 0 0
\(715\) 7.53341 0.281734
\(716\) 24.4596 0.914099
\(717\) −1.26470 −0.0472311
\(718\) −58.7578 −2.19282
\(719\) −31.4372 −1.17241 −0.586205 0.810162i \(-0.699379\pi\)
−0.586205 + 0.810162i \(0.699379\pi\)
\(720\) −4.64832 −0.173233
\(721\) 0 0
\(722\) −34.4806 −1.28323
\(723\) −23.8719 −0.887805
\(724\) 16.8759 0.627188
\(725\) 12.7467 0.473402
\(726\) 2.11491 0.0784916
\(727\) −14.2732 −0.529363 −0.264681 0.964336i \(-0.585267\pi\)
−0.264681 + 0.964336i \(0.585267\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −36.3921 −1.34693
\(731\) 3.70488 0.137030
\(732\) 32.5808 1.20422
\(733\) −2.31207 −0.0853983 −0.0426992 0.999088i \(-0.513596\pi\)
−0.0426992 + 0.999088i \(0.513596\pi\)
\(734\) −56.1600 −2.07291
\(735\) 0 0
\(736\) 0 0
\(737\) 4.21037 0.155091
\(738\) 10.9457 0.402916
\(739\) −26.0558 −0.958480 −0.479240 0.877684i \(-0.659088\pi\)
−0.479240 + 0.877684i \(0.659088\pi\)
\(740\) −26.7500 −0.983350
\(741\) 7.53341 0.276747
\(742\) 0 0
\(743\) 5.04737 0.185170 0.0925851 0.995705i \(-0.470487\pi\)
0.0925851 + 0.995705i \(0.470487\pi\)
\(744\) −5.17548 −0.189742
\(745\) 27.6157 1.01176
\(746\) 10.7159 0.392335
\(747\) −9.17548 −0.335713
\(748\) 1.77018 0.0647244
\(749\) 0 0
\(750\) −25.3642 −0.926169
\(751\) −40.4387 −1.47563 −0.737815 0.675002i \(-0.764143\pi\)
−0.737815 + 0.675002i \(0.764143\pi\)
\(752\) −22.2834 −0.812593
\(753\) 22.0194 0.802433
\(754\) −53.6887 −1.95523
\(755\) 7.74378 0.281825
\(756\) 0 0
\(757\) −24.4263 −0.887788 −0.443894 0.896079i \(-0.646403\pi\)
−0.443894 + 0.896079i \(0.646403\pi\)
\(758\) −64.6344 −2.34763
\(759\) 0 0
\(760\) 2.69641 0.0978089
\(761\) 5.18940 0.188116 0.0940579 0.995567i \(-0.470016\pi\)
0.0940579 + 0.995567i \(0.470016\pi\)
\(762\) −43.2702 −1.56751
\(763\) 0 0
\(764\) 27.2702 0.986602
\(765\) 1.17548 0.0424997
\(766\) 19.3230 0.698170
\(767\) −14.4791 −0.522809
\(768\) 6.01320 0.216983
\(769\) −28.1670 −1.01573 −0.507864 0.861437i \(-0.669565\pi\)
−0.507864 + 0.861437i \(0.669565\pi\)
\(770\) 0 0
\(771\) 27.3161 0.983765
\(772\) −19.4178 −0.698861
\(773\) 38.2662 1.37634 0.688170 0.725549i \(-0.258414\pi\)
0.688170 + 0.725549i \(0.258414\pi\)
\(774\) 10.9457 0.393434
\(775\) 11.9222 0.428259
\(776\) −13.6351 −0.489472
\(777\) 0 0
\(778\) 44.9846 1.61277
\(779\) 8.49852 0.304491
\(780\) 18.6289 0.667021
\(781\) −1.85244 −0.0662856
\(782\) 0 0
\(783\) −5.53341 −0.197748
\(784\) 0 0
\(785\) −18.1087 −0.646326
\(786\) −31.8649 −1.13658
\(787\) −13.4247 −0.478540 −0.239270 0.970953i \(-0.576908\pi\)
−0.239270 + 0.970953i \(0.576908\pi\)
\(788\) 49.1880 1.75225
\(789\) −11.8719 −0.422650
\(790\) −43.2702 −1.53949
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) 60.4457 2.14649
\(794\) 61.2438 2.17346
\(795\) 4.45963 0.158167
\(796\) 58.1770 2.06203
\(797\) −22.3440 −0.791465 −0.395733 0.918366i \(-0.629509\pi\)
−0.395733 + 0.918366i \(0.629509\pi\)
\(798\) 0 0
\(799\) 5.63511 0.199356
\(800\) 18.3983 0.650479
\(801\) 4.71585 0.166626
\(802\) 68.4193 2.41597
\(803\) −10.4791 −0.369799
\(804\) 10.4115 0.367187
\(805\) 0 0
\(806\) −50.2159 −1.76878
\(807\) 5.63511 0.198365
\(808\) −14.6072 −0.513879
\(809\) 31.8455 1.11963 0.559814 0.828618i \(-0.310873\pi\)
0.559814 + 0.828618i \(0.310873\pi\)
\(810\) 3.47283 0.122023
\(811\) 8.70889 0.305811 0.152905 0.988241i \(-0.451137\pi\)
0.152905 + 0.988241i \(0.451137\pi\)
\(812\) 0 0
\(813\) −5.68097 −0.199240
\(814\) −13.9325 −0.488333
\(815\) 9.26470 0.324528
\(816\) −2.02641 −0.0709385
\(817\) 8.49852 0.297326
\(818\) 4.88982 0.170968
\(819\) 0 0
\(820\) 21.0154 0.733891
\(821\) 6.45267 0.225200 0.112600 0.993640i \(-0.464082\pi\)
0.112600 + 0.993640i \(0.464082\pi\)
\(822\) −30.5808 −1.06663
\(823\) 25.3719 0.884410 0.442205 0.896914i \(-0.354196\pi\)
0.442205 + 0.896914i \(0.354196\pi\)
\(824\) 17.8913 0.623274
\(825\) −2.30359 −0.0802009
\(826\) 0 0
\(827\) 14.0194 0.487504 0.243752 0.969838i \(-0.421622\pi\)
0.243752 + 0.969838i \(0.421622\pi\)
\(828\) 0 0
\(829\) −48.9970 −1.70174 −0.850869 0.525378i \(-0.823924\pi\)
−0.850869 + 0.525378i \(0.823924\pi\)
\(830\) −31.8649 −1.10605
\(831\) −25.4876 −0.884153
\(832\) −51.5195 −1.78612
\(833\) 0 0
\(834\) −10.8634 −0.376169
\(835\) 26.2732 0.909221
\(836\) 4.06058 0.140438
\(837\) −5.17548 −0.178891
\(838\) 81.2680 2.80736
\(839\) −13.2647 −0.457948 −0.228974 0.973433i \(-0.573537\pi\)
−0.228974 + 0.973433i \(0.573537\pi\)
\(840\) 0 0
\(841\) 1.61862 0.0558144
\(842\) 32.7959 1.13022
\(843\) 17.5723 0.605222
\(844\) −10.5249 −0.362283
\(845\) 13.2144 0.454588
\(846\) 16.6483 0.572381
\(847\) 0 0
\(848\) −7.68793 −0.264005
\(849\) 25.1296 0.862447
\(850\) −3.48755 −0.119622
\(851\) 0 0
\(852\) −4.58078 −0.156935
\(853\) −4.25622 −0.145730 −0.0728651 0.997342i \(-0.523214\pi\)
−0.0728651 + 0.997342i \(0.523214\pi\)
\(854\) 0 0
\(855\) 2.69641 0.0922151
\(856\) 11.1560 0.381305
\(857\) −23.5264 −0.803648 −0.401824 0.915717i \(-0.631624\pi\)
−0.401824 + 0.915717i \(0.631624\pi\)
\(858\) 9.70265 0.331243
\(859\) −31.0279 −1.05866 −0.529330 0.848416i \(-0.677556\pi\)
−0.529330 + 0.848416i \(0.677556\pi\)
\(860\) 21.0154 0.716620
\(861\) 0 0
\(862\) 80.2959 2.73489
\(863\) −46.3121 −1.57648 −0.788241 0.615367i \(-0.789008\pi\)
−0.788241 + 0.615367i \(0.789008\pi\)
\(864\) −7.98680 −0.271716
\(865\) −24.6770 −0.839042
\(866\) 21.3789 0.726484
\(867\) −16.4876 −0.559947
\(868\) 0 0
\(869\) −12.4596 −0.422664
\(870\) −19.2166 −0.651504
\(871\) 19.3161 0.654500
\(872\) −0.108664 −0.00367984
\(873\) −13.6351 −0.461479
\(874\) 0 0
\(875\) 0 0
\(876\) −25.9130 −0.875520
\(877\) −12.8245 −0.433053 −0.216527 0.976277i \(-0.569473\pi\)
−0.216527 + 0.976277i \(0.569473\pi\)
\(878\) −47.6329 −1.60753
\(879\) −14.3510 −0.484046
\(880\) −4.64832 −0.156695
\(881\) −56.9512 −1.91873 −0.959367 0.282160i \(-0.908949\pi\)
−0.959367 + 0.282160i \(0.908949\pi\)
\(882\) 0 0
\(883\) 8.00696 0.269456 0.134728 0.990883i \(-0.456984\pi\)
0.134728 + 0.990883i \(0.456984\pi\)
\(884\) 8.12115 0.273144
\(885\) −5.18244 −0.174206
\(886\) 43.7004 1.46814
\(887\) 35.7827 1.20146 0.600732 0.799450i \(-0.294876\pi\)
0.600732 + 0.799450i \(0.294876\pi\)
\(888\) −6.58774 −0.221070
\(889\) 0 0
\(890\) 16.3774 0.548971
\(891\) 1.00000 0.0335013
\(892\) −9.25774 −0.309972
\(893\) 12.9262 0.432559
\(894\) 35.5676 1.18956
\(895\) 16.2423 0.542920
\(896\) 0 0
\(897\) 0 0
\(898\) 86.3106 2.88022
\(899\) 28.6381 0.955133
\(900\) −5.69641 −0.189880
\(901\) 1.94415 0.0647690
\(902\) 10.9457 0.364451
\(903\) 0 0
\(904\) −13.0668 −0.434596
\(905\) 11.2064 0.372512
\(906\) 9.97359 0.331350
\(907\) 5.43171 0.180357 0.0901784 0.995926i \(-0.471256\pi\)
0.0901784 + 0.995926i \(0.471256\pi\)
\(908\) −38.2982 −1.27097
\(909\) −14.6072 −0.484490
\(910\) 0 0
\(911\) 16.6770 0.552532 0.276266 0.961081i \(-0.410903\pi\)
0.276266 + 0.961081i \(0.410903\pi\)
\(912\) −4.64832 −0.153921
\(913\) −9.17548 −0.303664
\(914\) −60.0419 −1.98601
\(915\) 21.6351 0.715235
\(916\) −19.7827 −0.653638
\(917\) 0 0
\(918\) 1.51396 0.0499682
\(919\) 14.4038 0.475137 0.237568 0.971371i \(-0.423650\pi\)
0.237568 + 0.971371i \(0.423650\pi\)
\(920\) 0 0
\(921\) 13.1366 0.432865
\(922\) 45.2966 1.49177
\(923\) −8.49852 −0.279732
\(924\) 0 0
\(925\) 15.1755 0.498967
\(926\) 43.8796 1.44197
\(927\) 17.8913 0.587629
\(928\) 44.1942 1.45075
\(929\) 1.89830 0.0622811 0.0311405 0.999515i \(-0.490086\pi\)
0.0311405 + 0.999515i \(0.490086\pi\)
\(930\) −17.9736 −0.589377
\(931\) 0 0
\(932\) −13.7004 −0.448772
\(933\) −14.3510 −0.469830
\(934\) 56.8906 1.86152
\(935\) 1.17548 0.0384424
\(936\) 4.58774 0.149955
\(937\) −13.1755 −0.430424 −0.215212 0.976567i \(-0.569044\pi\)
−0.215212 + 0.976567i \(0.569044\pi\)
\(938\) 0 0
\(939\) 23.7438 0.774849
\(940\) 31.9644 1.04256
\(941\) 0.381842 0.0124477 0.00622385 0.999981i \(-0.498019\pi\)
0.00622385 + 0.999981i \(0.498019\pi\)
\(942\) −23.3230 −0.759906
\(943\) 0 0
\(944\) 8.93398 0.290776
\(945\) 0 0
\(946\) 10.9457 0.355874
\(947\) −52.2284 −1.69719 −0.848597 0.529040i \(-0.822552\pi\)
−0.848597 + 0.529040i \(0.822552\pi\)
\(948\) −30.8106 −1.00068
\(949\) −48.0753 −1.56059
\(950\) −8.00000 −0.259554
\(951\) −6.45963 −0.209468
\(952\) 0 0
\(953\) −45.7757 −1.48282 −0.741410 0.671052i \(-0.765843\pi\)
−0.741410 + 0.671052i \(0.765843\pi\)
\(954\) 5.74378 0.185962
\(955\) 18.1087 0.585983
\(956\) −3.12739 −0.101147
\(957\) −5.53341 −0.178870
\(958\) 65.1616 2.10527
\(959\) 0 0
\(960\) −18.4402 −0.595154
\(961\) −4.21438 −0.135948
\(962\) −63.9185 −2.06082
\(963\) 11.1560 0.359498
\(964\) −59.0312 −1.90127
\(965\) −12.8943 −0.415082
\(966\) 0 0
\(967\) −15.3230 −0.492756 −0.246378 0.969174i \(-0.579240\pi\)
−0.246378 + 0.969174i \(0.579240\pi\)
\(968\) 1.00000 0.0321412
\(969\) 1.17548 0.0377620
\(970\) −47.3525 −1.52040
\(971\) 12.0753 0.387515 0.193757 0.981049i \(-0.437933\pi\)
0.193757 + 0.981049i \(0.437933\pi\)
\(972\) 2.47283 0.0793162
\(973\) 0 0
\(974\) −49.6740 −1.59166
\(975\) −10.5683 −0.338456
\(976\) −37.2966 −1.19384
\(977\) −39.3650 −1.25940 −0.629698 0.776840i \(-0.716822\pi\)
−0.629698 + 0.776840i \(0.716822\pi\)
\(978\) 11.9325 0.381558
\(979\) 4.71585 0.150719
\(980\) 0 0
\(981\) −0.108664 −0.00346939
\(982\) −35.1638 −1.12212
\(983\) 10.5683 0.337076 0.168538 0.985695i \(-0.446095\pi\)
0.168538 + 0.985695i \(0.446095\pi\)
\(984\) 5.17548 0.164988
\(985\) 32.6630 1.04073
\(986\) −8.37737 −0.266790
\(987\) 0 0
\(988\) 18.6289 0.592663
\(989\) 0 0
\(990\) 3.47283 0.110374
\(991\) 42.8036 1.35970 0.679851 0.733350i \(-0.262044\pi\)
0.679851 + 0.733350i \(0.262044\pi\)
\(992\) 41.3355 1.31240
\(993\) 31.2702 0.992331
\(994\) 0 0
\(995\) 38.6322 1.22472
\(996\) −22.6894 −0.718943
\(997\) −10.3121 −0.326587 −0.163293 0.986578i \(-0.552212\pi\)
−0.163293 + 0.986578i \(0.552212\pi\)
\(998\) −16.9045 −0.535104
\(999\) −6.58774 −0.208427
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 1617.2.a.r.1.3 yes 3
3.2 odd 2 4851.2.a.bl.1.1 3
7.6 odd 2 1617.2.a.q.1.3 3
21.20 even 2 4851.2.a.bm.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.2.a.q.1.3 3 7.6 odd 2
1617.2.a.r.1.3 yes 3 1.1 even 1 trivial
4851.2.a.bl.1.1 3 3.2 odd 2
4851.2.a.bm.1.1 3 21.20 even 2