Properties

Label 637.2.a.l.1.5
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.746052.1
Defining polynomial: \(x^{5} - x^{4} - 7 x^{3} + 8 x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.72525\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.72525 q^{2} -1.34642 q^{3} +5.42699 q^{4} +2.18716 q^{5} -3.66932 q^{6} +9.33940 q^{8} -1.18716 q^{9} +O(q^{10})\) \(q+2.72525 q^{2} -1.34642 q^{3} +5.42699 q^{4} +2.18716 q^{5} -3.66932 q^{6} +9.33940 q^{8} -1.18716 q^{9} +5.96057 q^{10} -1.04815 q^{11} -7.30699 q^{12} +1.00000 q^{13} -2.94483 q^{15} +14.5982 q^{16} -5.29125 q^{17} -3.23532 q^{18} +0.756906 q^{19} +11.8697 q^{20} -2.85648 q^{22} +0.653584 q^{23} -12.5747 q^{24} -0.216314 q^{25} +2.72525 q^{26} +5.63766 q^{27} -3.10408 q^{29} -8.02541 q^{30} +1.02791 q^{31} +21.1050 q^{32} +1.41125 q^{33} -14.4200 q^{34} -6.44273 q^{36} -10.8932 q^{37} +2.06276 q^{38} -1.34642 q^{39} +20.4268 q^{40} +7.32040 q^{41} +0.887771 q^{43} -5.68833 q^{44} -2.59652 q^{45} +1.78118 q^{46} +2.33751 q^{47} -19.6553 q^{48} -0.589510 q^{50} +7.12422 q^{51} +5.42699 q^{52} +4.88814 q^{53} +15.3640 q^{54} -2.29249 q^{55} -1.01911 q^{57} -8.45941 q^{58} -1.04815 q^{59} -15.9816 q^{60} -12.4998 q^{61} +2.80132 q^{62} +28.3200 q^{64} +2.18716 q^{65} +3.84602 q^{66} +4.47889 q^{67} -28.7155 q^{68} -0.879996 q^{69} -6.60274 q^{71} -11.0874 q^{72} -8.28347 q^{73} -29.6868 q^{74} +0.291249 q^{75} +4.10772 q^{76} -3.66932 q^{78} +2.14014 q^{79} +31.9287 q^{80} -4.02915 q^{81} +19.9499 q^{82} -6.66558 q^{83} -11.5728 q^{85} +2.41940 q^{86} +4.17939 q^{87} -9.78914 q^{88} -5.76777 q^{89} -7.07617 q^{90} +3.54699 q^{92} -1.38400 q^{93} +6.37030 q^{94} +1.65548 q^{95} -28.4162 q^{96} -2.88777 q^{97} +1.24433 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 4q^{2} + 8q^{4} + 2q^{5} - 5q^{6} + 9q^{8} + 3q^{9} + O(q^{10}) \) \( 5q + 4q^{2} + 8q^{4} + 2q^{5} - 5q^{6} + 9q^{8} + 3q^{9} - 5q^{10} + 11q^{11} + 5q^{12} + 5q^{13} + 10q^{16} - 5q^{17} + 9q^{18} + 9q^{19} + q^{20} + 8q^{22} + 10q^{23} + 9q^{25} + 4q^{26} - 3q^{29} - 13q^{30} - 6q^{31} + 22q^{32} + 8q^{33} - 22q^{34} + 7q^{36} + 4q^{37} - 10q^{38} + 28q^{40} + 14q^{41} + 2q^{43} - 32q^{45} + 3q^{46} + q^{47} - 23q^{48} + 9q^{50} - 8q^{51} + 8q^{52} + 17q^{53} + 23q^{54} - 16q^{57} - 27q^{58} + 11q^{59} - 29q^{60} - 11q^{61} - 23q^{62} + 9q^{64} + 2q^{65} + 21q^{66} + 13q^{67} - 32q^{68} + 18q^{69} + 15q^{71} - 19q^{72} - 33q^{74} - 20q^{75} + 8q^{76} - 5q^{78} + 2q^{79} + 55q^{80} - 19q^{81} + 34q^{82} + 6q^{83} - 22q^{85} + 28q^{86} - 8q^{87} - 3q^{88} - 4q^{89} - 34q^{90} + 21q^{92} + 18q^{93} + 20q^{94} - 12q^{95} - 37q^{96} - 12q^{97} + 11q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.72525 1.92704 0.963521 0.267631i \(-0.0862408\pi\)
0.963521 + 0.267631i \(0.0862408\pi\)
\(3\) −1.34642 −0.777354 −0.388677 0.921374i \(-0.627068\pi\)
−0.388677 + 0.921374i \(0.627068\pi\)
\(4\) 5.42699 2.71349
\(5\) 2.18716 0.978129 0.489065 0.872247i \(-0.337338\pi\)
0.489065 + 0.872247i \(0.337338\pi\)
\(6\) −3.66932 −1.49799
\(7\) 0 0
\(8\) 9.33940 3.30198
\(9\) −1.18716 −0.395721
\(10\) 5.96057 1.88490
\(11\) −1.04815 −0.316031 −0.158015 0.987437i \(-0.550510\pi\)
−0.158015 + 0.987437i \(0.550510\pi\)
\(12\) −7.30699 −2.10934
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −2.94483 −0.760352
\(16\) 14.5982 3.64956
\(17\) −5.29125 −1.28332 −0.641658 0.766991i \(-0.721753\pi\)
−0.641658 + 0.766991i \(0.721753\pi\)
\(18\) −3.23532 −0.762572
\(19\) 0.756906 0.173646 0.0868231 0.996224i \(-0.472329\pi\)
0.0868231 + 0.996224i \(0.472329\pi\)
\(20\) 11.8697 2.65415
\(21\) 0 0
\(22\) −2.85648 −0.609005
\(23\) 0.653584 0.136282 0.0681408 0.997676i \(-0.478293\pi\)
0.0681408 + 0.997676i \(0.478293\pi\)
\(24\) −12.5747 −2.56680
\(25\) −0.216314 −0.0432628
\(26\) 2.72525 0.534466
\(27\) 5.63766 1.08497
\(28\) 0 0
\(29\) −3.10408 −0.576414 −0.288207 0.957568i \(-0.593059\pi\)
−0.288207 + 0.957568i \(0.593059\pi\)
\(30\) −8.02541 −1.46523
\(31\) 1.02791 0.184618 0.0923092 0.995730i \(-0.470575\pi\)
0.0923092 + 0.995730i \(0.470575\pi\)
\(32\) 21.1050 3.73088
\(33\) 1.41125 0.245668
\(34\) −14.4200 −2.47301
\(35\) 0 0
\(36\) −6.44273 −1.07379
\(37\) −10.8932 −1.79084 −0.895418 0.445227i \(-0.853123\pi\)
−0.895418 + 0.445227i \(0.853123\pi\)
\(38\) 2.06276 0.334624
\(39\) −1.34642 −0.215599
\(40\) 20.4268 3.22976
\(41\) 7.32040 1.14325 0.571627 0.820514i \(-0.306312\pi\)
0.571627 + 0.820514i \(0.306312\pi\)
\(42\) 0 0
\(43\) 0.887771 0.135384 0.0676919 0.997706i \(-0.478437\pi\)
0.0676919 + 0.997706i \(0.478437\pi\)
\(44\) −5.68833 −0.857547
\(45\) −2.59652 −0.387067
\(46\) 1.78118 0.262621
\(47\) 2.33751 0.340961 0.170480 0.985361i \(-0.445468\pi\)
0.170480 + 0.985361i \(0.445468\pi\)
\(48\) −19.6553 −2.83700
\(49\) 0 0
\(50\) −0.589510 −0.0833692
\(51\) 7.12422 0.997591
\(52\) 5.42699 0.752588
\(53\) 4.88814 0.671438 0.335719 0.941962i \(-0.391021\pi\)
0.335719 + 0.941962i \(0.391021\pi\)
\(54\) 15.3640 2.09078
\(55\) −2.29249 −0.309119
\(56\) 0 0
\(57\) −1.01911 −0.134985
\(58\) −8.45941 −1.11077
\(59\) −1.04815 −0.136458 −0.0682291 0.997670i \(-0.521735\pi\)
−0.0682291 + 0.997670i \(0.521735\pi\)
\(60\) −15.9816 −2.06321
\(61\) −12.4998 −1.60043 −0.800217 0.599711i \(-0.795282\pi\)
−0.800217 + 0.599711i \(0.795282\pi\)
\(62\) 2.80132 0.355768
\(63\) 0 0
\(64\) 28.3200 3.54000
\(65\) 2.18716 0.271284
\(66\) 3.84602 0.473412
\(67\) 4.47889 0.547183 0.273592 0.961846i \(-0.411788\pi\)
0.273592 + 0.961846i \(0.411788\pi\)
\(68\) −28.7155 −3.48227
\(69\) −0.879996 −0.105939
\(70\) 0 0
\(71\) −6.60274 −0.783601 −0.391801 0.920050i \(-0.628148\pi\)
−0.391801 + 0.920050i \(0.628148\pi\)
\(72\) −11.0874 −1.30666
\(73\) −8.28347 −0.969507 −0.484754 0.874651i \(-0.661091\pi\)
−0.484754 + 0.874651i \(0.661091\pi\)
\(74\) −29.6868 −3.45102
\(75\) 0.291249 0.0336305
\(76\) 4.10772 0.471188
\(77\) 0 0
\(78\) −3.66932 −0.415469
\(79\) 2.14014 0.240785 0.120392 0.992726i \(-0.461585\pi\)
0.120392 + 0.992726i \(0.461585\pi\)
\(80\) 31.9287 3.56974
\(81\) −4.02915 −0.447683
\(82\) 19.9499 2.20310
\(83\) −6.66558 −0.731642 −0.365821 0.930685i \(-0.619212\pi\)
−0.365821 + 0.930685i \(0.619212\pi\)
\(84\) 0 0
\(85\) −11.5728 −1.25525
\(86\) 2.41940 0.260890
\(87\) 4.17939 0.448078
\(88\) −9.78914 −1.04353
\(89\) −5.76777 −0.611382 −0.305691 0.952131i \(-0.598887\pi\)
−0.305691 + 0.952131i \(0.598887\pi\)
\(90\) −7.07617 −0.745894
\(91\) 0 0
\(92\) 3.54699 0.369800
\(93\) −1.38400 −0.143514
\(94\) 6.37030 0.657046
\(95\) 1.65548 0.169848
\(96\) −28.4162 −2.90021
\(97\) −2.88777 −0.293209 −0.146604 0.989195i \(-0.546834\pi\)
−0.146604 + 0.989195i \(0.546834\pi\)
\(98\) 0 0
\(99\) 1.24433 0.125060
\(100\) −1.17393 −0.117393
\(101\) −11.2543 −1.11985 −0.559924 0.828544i \(-0.689170\pi\)
−0.559924 + 0.828544i \(0.689170\pi\)
\(102\) 19.4153 1.92240
\(103\) 20.2334 1.99366 0.996828 0.0795900i \(-0.0253611\pi\)
0.996828 + 0.0795900i \(0.0253611\pi\)
\(104\) 9.33940 0.915804
\(105\) 0 0
\(106\) 13.3214 1.29389
\(107\) 9.05517 0.875396 0.437698 0.899122i \(-0.355794\pi\)
0.437698 + 0.899122i \(0.355794\pi\)
\(108\) 30.5955 2.94406
\(109\) 15.1014 1.44645 0.723226 0.690612i \(-0.242659\pi\)
0.723226 + 0.690612i \(0.242659\pi\)
\(110\) −6.24760 −0.595685
\(111\) 14.6668 1.39211
\(112\) 0 0
\(113\) 3.10408 0.292008 0.146004 0.989284i \(-0.453359\pi\)
0.146004 + 0.989284i \(0.453359\pi\)
\(114\) −2.77733 −0.260121
\(115\) 1.42950 0.133301
\(116\) −16.8458 −1.56410
\(117\) −1.18716 −0.109753
\(118\) −2.85648 −0.262961
\(119\) 0 0
\(120\) −27.5030 −2.51067
\(121\) −9.90137 −0.900125
\(122\) −34.0651 −3.08410
\(123\) −9.85630 −0.888713
\(124\) 5.57847 0.500961
\(125\) −11.4089 −1.02045
\(126\) 0 0
\(127\) 8.78914 0.779910 0.389955 0.920834i \(-0.372491\pi\)
0.389955 + 0.920834i \(0.372491\pi\)
\(128\) 34.9691 3.09086
\(129\) −1.19531 −0.105241
\(130\) 5.96057 0.522776
\(131\) −10.5145 −0.918653 −0.459327 0.888267i \(-0.651909\pi\)
−0.459327 + 0.888267i \(0.651909\pi\)
\(132\) 7.65885 0.666618
\(133\) 0 0
\(134\) 12.2061 1.05445
\(135\) 12.3305 1.06124
\(136\) −49.4171 −4.23748
\(137\) 8.73165 0.745995 0.372998 0.927832i \(-0.378330\pi\)
0.372998 + 0.927832i \(0.378330\pi\)
\(138\) −2.39821 −0.204149
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −3.14726 −0.265047
\(142\) −17.9941 −1.51003
\(143\) −1.04815 −0.0876511
\(144\) −17.3305 −1.44421
\(145\) −6.78914 −0.563808
\(146\) −22.5745 −1.86828
\(147\) 0 0
\(148\) −59.1174 −4.85942
\(149\) 15.3926 1.26101 0.630507 0.776183i \(-0.282847\pi\)
0.630507 + 0.776183i \(0.282847\pi\)
\(150\) 0.793725 0.0648074
\(151\) −13.6757 −1.11291 −0.556457 0.830876i \(-0.687840\pi\)
−0.556457 + 0.830876i \(0.687840\pi\)
\(152\) 7.06905 0.573376
\(153\) 6.28158 0.507836
\(154\) 0 0
\(155\) 2.24821 0.180581
\(156\) −7.30699 −0.585027
\(157\) 3.38756 0.270357 0.135178 0.990821i \(-0.456839\pi\)
0.135178 + 0.990821i \(0.456839\pi\)
\(158\) 5.83242 0.464002
\(159\) −6.58147 −0.521945
\(160\) 46.1602 3.64928
\(161\) 0 0
\(162\) −10.9804 −0.862705
\(163\) −13.8100 −1.08169 −0.540843 0.841124i \(-0.681895\pi\)
−0.540843 + 0.841124i \(0.681895\pi\)
\(164\) 39.7277 3.10221
\(165\) 3.08664 0.240295
\(166\) −18.1654 −1.40991
\(167\) 16.3783 1.26739 0.633695 0.773583i \(-0.281538\pi\)
0.633695 + 0.773583i \(0.281538\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −31.5389 −2.41892
\(171\) −0.898572 −0.0687155
\(172\) 4.81792 0.367363
\(173\) 4.12546 0.313653 0.156826 0.987626i \(-0.449874\pi\)
0.156826 + 0.987626i \(0.449874\pi\)
\(174\) 11.3899 0.863465
\(175\) 0 0
\(176\) −15.3012 −1.15337
\(177\) 1.41125 0.106076
\(178\) −15.7186 −1.17816
\(179\) 14.4136 1.07732 0.538661 0.842523i \(-0.318931\pi\)
0.538661 + 0.842523i \(0.318931\pi\)
\(180\) −14.0913 −1.05030
\(181\) 18.1014 1.34547 0.672733 0.739885i \(-0.265120\pi\)
0.672733 + 0.739885i \(0.265120\pi\)
\(182\) 0 0
\(183\) 16.8299 1.24410
\(184\) 6.10408 0.449999
\(185\) −23.8253 −1.75167
\(186\) −3.77174 −0.276557
\(187\) 5.54605 0.405567
\(188\) 12.6856 0.925195
\(189\) 0 0
\(190\) 4.51159 0.327305
\(191\) 5.54135 0.400958 0.200479 0.979698i \(-0.435750\pi\)
0.200479 + 0.979698i \(0.435750\pi\)
\(192\) −38.1305 −2.75184
\(193\) −8.74088 −0.629182 −0.314591 0.949227i \(-0.601867\pi\)
−0.314591 + 0.949227i \(0.601867\pi\)
\(194\) −7.86990 −0.565026
\(195\) −2.94483 −0.210884
\(196\) 0 0
\(197\) −5.46874 −0.389632 −0.194816 0.980840i \(-0.562411\pi\)
−0.194816 + 0.980840i \(0.562411\pi\)
\(198\) 3.39112 0.240996
\(199\) 19.5368 1.38493 0.692463 0.721454i \(-0.256526\pi\)
0.692463 + 0.721454i \(0.256526\pi\)
\(200\) −2.02024 −0.142853
\(201\) −6.03045 −0.425355
\(202\) −30.6708 −2.15799
\(203\) 0 0
\(204\) 38.6631 2.70696
\(205\) 16.0109 1.11825
\(206\) 55.1411 3.84186
\(207\) −0.775911 −0.0539296
\(208\) 14.5982 1.01221
\(209\) −0.793355 −0.0548775
\(210\) 0 0
\(211\) 16.6905 1.14902 0.574511 0.818497i \(-0.305192\pi\)
0.574511 + 0.818497i \(0.305192\pi\)
\(212\) 26.5279 1.82194
\(213\) 8.89004 0.609135
\(214\) 24.6776 1.68693
\(215\) 1.94170 0.132423
\(216\) 52.6524 3.58254
\(217\) 0 0
\(218\) 41.1551 2.78737
\(219\) 11.1530 0.753650
\(220\) −12.4413 −0.838792
\(221\) −5.29125 −0.355928
\(222\) 39.9707 2.68266
\(223\) −5.34217 −0.357738 −0.178869 0.983873i \(-0.557244\pi\)
−0.178869 + 0.983873i \(0.557244\pi\)
\(224\) 0 0
\(225\) 0.256800 0.0171200
\(226\) 8.45941 0.562711
\(227\) 20.1215 1.33551 0.667757 0.744380i \(-0.267255\pi\)
0.667757 + 0.744380i \(0.267255\pi\)
\(228\) −5.53070 −0.366280
\(229\) 25.2497 1.66855 0.834275 0.551349i \(-0.185887\pi\)
0.834275 + 0.551349i \(0.185887\pi\)
\(230\) 3.89573 0.256877
\(231\) 0 0
\(232\) −28.9903 −1.90331
\(233\) −0.793355 −0.0519744 −0.0259872 0.999662i \(-0.508273\pi\)
−0.0259872 + 0.999662i \(0.508273\pi\)
\(234\) −3.23532 −0.211499
\(235\) 5.11252 0.333504
\(236\) −5.68833 −0.370278
\(237\) −2.88152 −0.187175
\(238\) 0 0
\(239\) 20.0488 1.29685 0.648425 0.761279i \(-0.275428\pi\)
0.648425 + 0.761279i \(0.275428\pi\)
\(240\) −42.9894 −2.77495
\(241\) 13.8120 0.889712 0.444856 0.895602i \(-0.353255\pi\)
0.444856 + 0.895602i \(0.353255\pi\)
\(242\) −26.9837 −1.73458
\(243\) −11.4881 −0.736961
\(244\) −67.8362 −4.34277
\(245\) 0 0
\(246\) −26.8609 −1.71259
\(247\) 0.756906 0.0481608
\(248\) 9.60008 0.609606
\(249\) 8.97464 0.568745
\(250\) −31.0922 −1.96644
\(251\) −26.1095 −1.64802 −0.824010 0.566576i \(-0.808268\pi\)
−0.824010 + 0.566576i \(0.808268\pi\)
\(252\) 0 0
\(253\) −0.685057 −0.0430692
\(254\) 23.9526 1.50292
\(255\) 15.5818 0.975773
\(256\) 38.6595 2.41622
\(257\) 10.6198 0.662444 0.331222 0.943553i \(-0.392539\pi\)
0.331222 + 0.943553i \(0.392539\pi\)
\(258\) −3.25752 −0.202804
\(259\) 0 0
\(260\) 11.8697 0.736128
\(261\) 3.68506 0.228099
\(262\) −28.6546 −1.77028
\(263\) 10.3578 0.638686 0.319343 0.947639i \(-0.396538\pi\)
0.319343 + 0.947639i \(0.396538\pi\)
\(264\) 13.1803 0.811189
\(265\) 10.6912 0.656753
\(266\) 0 0
\(267\) 7.76581 0.475260
\(268\) 24.3069 1.48478
\(269\) 11.9701 0.729827 0.364914 0.931041i \(-0.381099\pi\)
0.364914 + 0.931041i \(0.381099\pi\)
\(270\) 33.6037 2.04506
\(271\) −2.75691 −0.167470 −0.0837351 0.996488i \(-0.526685\pi\)
−0.0837351 + 0.996488i \(0.526685\pi\)
\(272\) −77.2429 −4.68354
\(273\) 0 0
\(274\) 23.7959 1.43757
\(275\) 0.226731 0.0136724
\(276\) −4.77573 −0.287465
\(277\) −23.9275 −1.43766 −0.718831 0.695185i \(-0.755322\pi\)
−0.718831 + 0.695185i \(0.755322\pi\)
\(278\) −10.9010 −0.653799
\(279\) −1.22030 −0.0730574
\(280\) 0 0
\(281\) −3.87870 −0.231384 −0.115692 0.993285i \(-0.536909\pi\)
−0.115692 + 0.993285i \(0.536909\pi\)
\(282\) −8.57707 −0.510757
\(283\) −6.20999 −0.369146 −0.184573 0.982819i \(-0.559090\pi\)
−0.184573 + 0.982819i \(0.559090\pi\)
\(284\) −35.8330 −2.12630
\(285\) −2.22896 −0.132032
\(286\) −2.85648 −0.168907
\(287\) 0 0
\(288\) −25.0551 −1.47639
\(289\) 10.9973 0.646901
\(290\) −18.5021 −1.08648
\(291\) 3.88814 0.227927
\(292\) −44.9543 −2.63075
\(293\) 16.5754 0.968347 0.484174 0.874972i \(-0.339120\pi\)
0.484174 + 0.874972i \(0.339120\pi\)
\(294\) 0 0
\(295\) −2.29249 −0.133474
\(296\) −101.736 −5.91330
\(297\) −5.90915 −0.342883
\(298\) 41.9488 2.43003
\(299\) 0.653584 0.0377977
\(300\) 1.58060 0.0912561
\(301\) 0 0
\(302\) −37.2698 −2.14463
\(303\) 15.1530 0.870517
\(304\) 11.0495 0.633732
\(305\) −27.3391 −1.56543
\(306\) 17.1189 0.978621
\(307\) −7.05788 −0.402815 −0.201407 0.979508i \(-0.564551\pi\)
−0.201407 + 0.979508i \(0.564551\pi\)
\(308\) 0 0
\(309\) −27.2426 −1.54978
\(310\) 6.12694 0.347987
\(311\) 21.1102 1.19705 0.598525 0.801104i \(-0.295754\pi\)
0.598525 + 0.801104i \(0.295754\pi\)
\(312\) −12.5747 −0.711903
\(313\) 1.98052 0.111946 0.0559728 0.998432i \(-0.482174\pi\)
0.0559728 + 0.998432i \(0.482174\pi\)
\(314\) 9.23194 0.520989
\(315\) 0 0
\(316\) 11.6145 0.653368
\(317\) −18.0459 −1.01356 −0.506781 0.862075i \(-0.669165\pi\)
−0.506781 + 0.862075i \(0.669165\pi\)
\(318\) −17.9362 −1.00581
\(319\) 3.25356 0.182164
\(320\) 61.9406 3.46258
\(321\) −12.1920 −0.680492
\(322\) 0 0
\(323\) −4.00498 −0.222843
\(324\) −21.8662 −1.21479
\(325\) −0.216314 −0.0119989
\(326\) −37.6358 −2.08446
\(327\) −20.3328 −1.12440
\(328\) 68.3682 3.77500
\(329\) 0 0
\(330\) 8.41187 0.463058
\(331\) −14.6738 −0.806544 −0.403272 0.915080i \(-0.632127\pi\)
−0.403272 + 0.915080i \(0.632127\pi\)
\(332\) −36.1740 −1.98531
\(333\) 12.9320 0.708672
\(334\) 44.6349 2.44231
\(335\) 9.79606 0.535216
\(336\) 0 0
\(337\) 12.8080 0.697698 0.348849 0.937179i \(-0.386573\pi\)
0.348849 + 0.937179i \(0.386573\pi\)
\(338\) 2.72525 0.148234
\(339\) −4.17939 −0.226993
\(340\) −62.8056 −3.40611
\(341\) −1.07741 −0.0583451
\(342\) −2.44883 −0.132418
\(343\) 0 0
\(344\) 8.29125 0.447034
\(345\) −1.92470 −0.103622
\(346\) 11.2429 0.604423
\(347\) 20.2054 1.08468 0.542342 0.840158i \(-0.317538\pi\)
0.542342 + 0.840158i \(0.317538\pi\)
\(348\) 22.6815 1.21586
\(349\) −18.4434 −0.987252 −0.493626 0.869674i \(-0.664329\pi\)
−0.493626 + 0.869674i \(0.664329\pi\)
\(350\) 0 0
\(351\) 5.63766 0.300916
\(352\) −22.1213 −1.17907
\(353\) −8.14436 −0.433480 −0.216740 0.976229i \(-0.569542\pi\)
−0.216740 + 0.976229i \(0.569542\pi\)
\(354\) 3.84602 0.204413
\(355\) −14.4413 −0.766463
\(356\) −31.3016 −1.65898
\(357\) 0 0
\(358\) 39.2806 2.07604
\(359\) −32.6100 −1.72109 −0.860545 0.509375i \(-0.829877\pi\)
−0.860545 + 0.509375i \(0.829877\pi\)
\(360\) −24.2500 −1.27809
\(361\) −18.4271 −0.969847
\(362\) 49.3308 2.59277
\(363\) 13.3314 0.699715
\(364\) 0 0
\(365\) −18.1173 −0.948304
\(366\) 45.8657 2.39744
\(367\) −3.16012 −0.164957 −0.0824786 0.996593i \(-0.526284\pi\)
−0.0824786 + 0.996593i \(0.526284\pi\)
\(368\) 9.54117 0.497368
\(369\) −8.69051 −0.452410
\(370\) −64.9298 −3.37554
\(371\) 0 0
\(372\) −7.51094 −0.389424
\(373\) −1.47770 −0.0765123 −0.0382561 0.999268i \(-0.512180\pi\)
−0.0382561 + 0.999268i \(0.512180\pi\)
\(374\) 15.1144 0.781545
\(375\) 15.3612 0.793247
\(376\) 21.8309 1.12584
\(377\) −3.10408 −0.159868
\(378\) 0 0
\(379\) 10.7254 0.550927 0.275463 0.961312i \(-0.411169\pi\)
0.275463 + 0.961312i \(0.411169\pi\)
\(380\) 8.98426 0.460883
\(381\) −11.8338 −0.606266
\(382\) 15.1016 0.772664
\(383\) −21.4109 −1.09405 −0.547023 0.837118i \(-0.684239\pi\)
−0.547023 + 0.837118i \(0.684239\pi\)
\(384\) −47.0830 −2.40269
\(385\) 0 0
\(386\) −23.8211 −1.21246
\(387\) −1.05393 −0.0535742
\(388\) −15.6719 −0.795620
\(389\) 34.7819 1.76351 0.881755 0.471707i \(-0.156362\pi\)
0.881755 + 0.471707i \(0.156362\pi\)
\(390\) −8.02541 −0.406382
\(391\) −3.45828 −0.174893
\(392\) 0 0
\(393\) 14.1568 0.714119
\(394\) −14.9037 −0.750837
\(395\) 4.68084 0.235519
\(396\) 6.75297 0.339350
\(397\) 4.45211 0.223445 0.111722 0.993739i \(-0.464363\pi\)
0.111722 + 0.993739i \(0.464363\pi\)
\(398\) 53.2426 2.66881
\(399\) 0 0
\(400\) −3.15780 −0.157890
\(401\) −13.7537 −0.686829 −0.343415 0.939184i \(-0.611584\pi\)
−0.343415 + 0.939184i \(0.611584\pi\)
\(402\) −16.4345 −0.819677
\(403\) 1.02791 0.0512039
\(404\) −61.0771 −3.03870
\(405\) −8.81241 −0.437892
\(406\) 0 0
\(407\) 11.4178 0.565959
\(408\) 66.5360 3.29402
\(409\) −3.49207 −0.172672 −0.0863358 0.996266i \(-0.527516\pi\)
−0.0863358 + 0.996266i \(0.527516\pi\)
\(410\) 43.6337 2.15492
\(411\) −11.7564 −0.579902
\(412\) 109.806 5.40977
\(413\) 0 0
\(414\) −2.11455 −0.103925
\(415\) −14.5787 −0.715641
\(416\) 21.1050 1.03476
\(417\) 5.38566 0.263737
\(418\) −2.16209 −0.105751
\(419\) −3.56737 −0.174278 −0.0871388 0.996196i \(-0.527772\pi\)
−0.0871388 + 0.996196i \(0.527772\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 45.4858 2.21422
\(423\) −2.77501 −0.134925
\(424\) 45.6523 2.21707
\(425\) 1.14457 0.0555198
\(426\) 24.2276 1.17383
\(427\) 0 0
\(428\) 49.1423 2.37538
\(429\) 1.41125 0.0681359
\(430\) 5.29162 0.255185
\(431\) −11.3642 −0.547396 −0.273698 0.961816i \(-0.588247\pi\)
−0.273698 + 0.961816i \(0.588247\pi\)
\(432\) 82.2999 3.95966
\(433\) −21.2136 −1.01946 −0.509731 0.860334i \(-0.670255\pi\)
−0.509731 + 0.860334i \(0.670255\pi\)
\(434\) 0 0
\(435\) 9.14101 0.438278
\(436\) 81.9551 3.92494
\(437\) 0.494702 0.0236648
\(438\) 30.3947 1.45232
\(439\) −24.5007 −1.16935 −0.584676 0.811267i \(-0.698778\pi\)
−0.584676 + 0.811267i \(0.698778\pi\)
\(440\) −21.4105 −1.02070
\(441\) 0 0
\(442\) −14.4200 −0.685888
\(443\) 40.4688 1.92273 0.961366 0.275274i \(-0.0887686\pi\)
0.961366 + 0.275274i \(0.0887686\pi\)
\(444\) 79.5966 3.77749
\(445\) −12.6151 −0.598011
\(446\) −14.5587 −0.689376
\(447\) −20.7249 −0.980254
\(448\) 0 0
\(449\) −27.7638 −1.31025 −0.655127 0.755519i \(-0.727385\pi\)
−0.655127 + 0.755519i \(0.727385\pi\)
\(450\) 0.699845 0.0329910
\(451\) −7.67291 −0.361303
\(452\) 16.8458 0.792361
\(453\) 18.4132 0.865128
\(454\) 54.8362 2.57359
\(455\) 0 0
\(456\) −9.51789 −0.445716
\(457\) −11.1939 −0.523629 −0.261815 0.965118i \(-0.584321\pi\)
−0.261815 + 0.965118i \(0.584321\pi\)
\(458\) 68.8119 3.21537
\(459\) −29.8303 −1.39236
\(460\) 7.75786 0.361712
\(461\) −9.29773 −0.433038 −0.216519 0.976278i \(-0.569470\pi\)
−0.216519 + 0.976278i \(0.569470\pi\)
\(462\) 0 0
\(463\) 28.2439 1.31260 0.656302 0.754499i \(-0.272120\pi\)
0.656302 + 0.754499i \(0.272120\pi\)
\(464\) −45.3142 −2.10366
\(465\) −3.02703 −0.140375
\(466\) −2.16209 −0.100157
\(467\) 22.2606 1.03010 0.515050 0.857160i \(-0.327773\pi\)
0.515050 + 0.857160i \(0.327773\pi\)
\(468\) −6.44273 −0.297815
\(469\) 0 0
\(470\) 13.9329 0.642676
\(471\) −4.56106 −0.210163
\(472\) −9.78914 −0.450582
\(473\) −0.930521 −0.0427854
\(474\) −7.85286 −0.360694
\(475\) −0.163729 −0.00751242
\(476\) 0 0
\(477\) −5.80302 −0.265702
\(478\) 54.6380 2.49909
\(479\) −32.8764 −1.50216 −0.751081 0.660210i \(-0.770467\pi\)
−0.751081 + 0.660210i \(0.770467\pi\)
\(480\) −62.1508 −2.83678
\(481\) −10.8932 −0.496688
\(482\) 37.6413 1.71451
\(483\) 0 0
\(484\) −53.7346 −2.44248
\(485\) −6.31603 −0.286796
\(486\) −31.3079 −1.42016
\(487\) −27.8924 −1.26392 −0.631962 0.774999i \(-0.717750\pi\)
−0.631962 + 0.774999i \(0.717750\pi\)
\(488\) −116.741 −5.28460
\(489\) 18.5941 0.840852
\(490\) 0 0
\(491\) 10.6571 0.480948 0.240474 0.970656i \(-0.422697\pi\)
0.240474 + 0.970656i \(0.422697\pi\)
\(492\) −53.4900 −2.41152
\(493\) 16.4245 0.739722
\(494\) 2.06276 0.0928079
\(495\) 2.72156 0.122325
\(496\) 15.0057 0.673776
\(497\) 0 0
\(498\) 24.4581 1.09600
\(499\) −24.5114 −1.09728 −0.548641 0.836058i \(-0.684854\pi\)
−0.548641 + 0.836058i \(0.684854\pi\)
\(500\) −61.9162 −2.76897
\(501\) −22.0520 −0.985210
\(502\) −71.1550 −3.17580
\(503\) −38.0054 −1.69458 −0.847288 0.531134i \(-0.821766\pi\)
−0.847288 + 0.531134i \(0.821766\pi\)
\(504\) 0 0
\(505\) −24.6151 −1.09536
\(506\) −1.86695 −0.0829962
\(507\) −1.34642 −0.0597964
\(508\) 47.6986 2.11628
\(509\) 39.8501 1.76632 0.883161 0.469070i \(-0.155411\pi\)
0.883161 + 0.469070i \(0.155411\pi\)
\(510\) 42.4644 1.88036
\(511\) 0 0
\(512\) 35.4186 1.56530
\(513\) 4.26718 0.188401
\(514\) 28.9416 1.27656
\(515\) 44.2537 1.95005
\(516\) −6.48693 −0.285571
\(517\) −2.45007 −0.107754
\(518\) 0 0
\(519\) −5.55459 −0.243819
\(520\) 20.4268 0.895775
\(521\) −19.6334 −0.860155 −0.430077 0.902792i \(-0.641514\pi\)
−0.430077 + 0.902792i \(0.641514\pi\)
\(522\) 10.0427 0.439557
\(523\) 22.8324 0.998391 0.499195 0.866489i \(-0.333629\pi\)
0.499195 + 0.866489i \(0.333629\pi\)
\(524\) −57.0619 −2.49276
\(525\) 0 0
\(526\) 28.2275 1.23078
\(527\) −5.43894 −0.236924
\(528\) 20.6018 0.896578
\(529\) −22.5728 −0.981427
\(530\) 29.1361 1.26559
\(531\) 1.24433 0.0539994
\(532\) 0 0
\(533\) 7.32040 0.317082
\(534\) 21.1638 0.915847
\(535\) 19.8051 0.856251
\(536\) 41.8301 1.80679
\(537\) −19.4067 −0.837460
\(538\) 32.6214 1.40641
\(539\) 0 0
\(540\) 66.9175 2.87967
\(541\) −9.64668 −0.414743 −0.207372 0.978262i \(-0.566491\pi\)
−0.207372 + 0.978262i \(0.566491\pi\)
\(542\) −7.51326 −0.322722
\(543\) −24.3720 −1.04590
\(544\) −111.672 −4.78790
\(545\) 33.0292 1.41482
\(546\) 0 0
\(547\) −43.8570 −1.87519 −0.937596 0.347728i \(-0.886953\pi\)
−0.937596 + 0.347728i \(0.886953\pi\)
\(548\) 47.3866 2.02425
\(549\) 14.8393 0.633326
\(550\) 0.617897 0.0263472
\(551\) −2.34950 −0.100092
\(552\) −8.21864 −0.349808
\(553\) 0 0
\(554\) −65.2083 −2.77044
\(555\) 32.0787 1.36167
\(556\) −21.7080 −0.920622
\(557\) 14.9195 0.632161 0.316080 0.948732i \(-0.397633\pi\)
0.316080 + 0.948732i \(0.397633\pi\)
\(558\) −3.32562 −0.140785
\(559\) 0.887771 0.0375487
\(560\) 0 0
\(561\) −7.46729 −0.315269
\(562\) −10.5704 −0.445886
\(563\) −17.2697 −0.727832 −0.363916 0.931432i \(-0.618560\pi\)
−0.363916 + 0.931432i \(0.618560\pi\)
\(564\) −17.0801 −0.719204
\(565\) 6.78914 0.285621
\(566\) −16.9238 −0.711359
\(567\) 0 0
\(568\) −61.6657 −2.58743
\(569\) 26.5324 1.11230 0.556148 0.831083i \(-0.312279\pi\)
0.556148 + 0.831083i \(0.312279\pi\)
\(570\) −6.07448 −0.254432
\(571\) −1.98569 −0.0830985 −0.0415492 0.999136i \(-0.513229\pi\)
−0.0415492 + 0.999136i \(0.513229\pi\)
\(572\) −5.68833 −0.237841
\(573\) −7.46097 −0.311686
\(574\) 0 0
\(575\) −0.141379 −0.00589593
\(576\) −33.6205 −1.40086
\(577\) 11.8983 0.495333 0.247666 0.968845i \(-0.420336\pi\)
0.247666 + 0.968845i \(0.420336\pi\)
\(578\) 29.9704 1.24661
\(579\) 11.7689 0.489097
\(580\) −36.8446 −1.52989
\(581\) 0 0
\(582\) 10.5962 0.439225
\(583\) −5.12353 −0.212195
\(584\) −77.3627 −3.20129
\(585\) −2.59652 −0.107353
\(586\) 45.1722 1.86605
\(587\) 33.5122 1.38320 0.691598 0.722283i \(-0.256907\pi\)
0.691598 + 0.722283i \(0.256907\pi\)
\(588\) 0 0
\(589\) 0.778033 0.0320583
\(590\) −6.24760 −0.257210
\(591\) 7.36320 0.302882
\(592\) −159.022 −6.53576
\(593\) 35.2815 1.44884 0.724419 0.689360i \(-0.242108\pi\)
0.724419 + 0.689360i \(0.242108\pi\)
\(594\) −16.1039 −0.660751
\(595\) 0 0
\(596\) 83.5357 3.42176
\(597\) −26.3046 −1.07658
\(598\) 1.78118 0.0728379
\(599\) −25.0068 −1.02175 −0.510876 0.859655i \(-0.670679\pi\)
−0.510876 + 0.859655i \(0.670679\pi\)
\(600\) 2.72009 0.111047
\(601\) −28.4688 −1.16127 −0.580634 0.814165i \(-0.697195\pi\)
−0.580634 + 0.814165i \(0.697195\pi\)
\(602\) 0 0
\(603\) −5.31717 −0.216532
\(604\) −74.2180 −3.01989
\(605\) −21.6559 −0.880438
\(606\) 41.2957 1.67752
\(607\) 36.0469 1.46310 0.731549 0.681789i \(-0.238798\pi\)
0.731549 + 0.681789i \(0.238798\pi\)
\(608\) 15.9745 0.647853
\(609\) 0 0
\(610\) −74.5058 −3.01665
\(611\) 2.33751 0.0945655
\(612\) 34.0901 1.37801
\(613\) 18.3253 0.740151 0.370075 0.929002i \(-0.379332\pi\)
0.370075 + 0.929002i \(0.379332\pi\)
\(614\) −19.2345 −0.776241
\(615\) −21.5573 −0.869276
\(616\) 0 0
\(617\) 44.3782 1.78660 0.893299 0.449463i \(-0.148385\pi\)
0.893299 + 0.449463i \(0.148385\pi\)
\(618\) −74.2428 −2.98648
\(619\) −25.0085 −1.00518 −0.502588 0.864526i \(-0.667619\pi\)
−0.502588 + 0.864526i \(0.667619\pi\)
\(620\) 12.2010 0.490005
\(621\) 3.68469 0.147861
\(622\) 57.5306 2.30677
\(623\) 0 0
\(624\) −19.6553 −0.786842
\(625\) −23.8716 −0.954866
\(626\) 5.39741 0.215724
\(627\) 1.06819 0.0426592
\(628\) 18.3842 0.733611
\(629\) 57.6388 2.29821
\(630\) 0 0
\(631\) −18.4638 −0.735032 −0.367516 0.930017i \(-0.619792\pi\)
−0.367516 + 0.930017i \(0.619792\pi\)
\(632\) 19.9876 0.795066
\(633\) −22.4724 −0.893197
\(634\) −49.1797 −1.95318
\(635\) 19.2233 0.762853
\(636\) −35.7176 −1.41629
\(637\) 0 0
\(638\) 8.86677 0.351039
\(639\) 7.83853 0.310088
\(640\) 76.4832 3.02326
\(641\) −21.2567 −0.839589 −0.419795 0.907619i \(-0.637898\pi\)
−0.419795 + 0.907619i \(0.637898\pi\)
\(642\) −33.2263 −1.31134
\(643\) −36.0554 −1.42188 −0.710942 0.703251i \(-0.751731\pi\)
−0.710942 + 0.703251i \(0.751731\pi\)
\(644\) 0 0
\(645\) −2.61434 −0.102939
\(646\) −10.9146 −0.429428
\(647\) 39.8234 1.56562 0.782809 0.622262i \(-0.213786\pi\)
0.782809 + 0.622262i \(0.213786\pi\)
\(648\) −37.6299 −1.47824
\(649\) 1.09863 0.0431250
\(650\) −0.589510 −0.0231225
\(651\) 0 0
\(652\) −74.9469 −2.93515
\(653\) −32.4669 −1.27053 −0.635265 0.772295i \(-0.719109\pi\)
−0.635265 + 0.772295i \(0.719109\pi\)
\(654\) −55.4119 −2.16678
\(655\) −22.9969 −0.898562
\(656\) 106.865 4.17237
\(657\) 9.83384 0.383655
\(658\) 0 0
\(659\) 23.5230 0.916327 0.458164 0.888868i \(-0.348507\pi\)
0.458164 + 0.888868i \(0.348507\pi\)
\(660\) 16.7512 0.652038
\(661\) 14.0389 0.546049 0.273025 0.962007i \(-0.411976\pi\)
0.273025 + 0.962007i \(0.411976\pi\)
\(662\) −39.9897 −1.55424
\(663\) 7.12422 0.276682
\(664\) −62.2525 −2.41587
\(665\) 0 0
\(666\) 35.2431 1.36564
\(667\) −2.02878 −0.0785547
\(668\) 88.8848 3.43905
\(669\) 7.19278 0.278089
\(670\) 26.6967 1.03138
\(671\) 13.1017 0.505786
\(672\) 0 0
\(673\) −47.1937 −1.81918 −0.909592 0.415502i \(-0.863606\pi\)
−0.909592 + 0.415502i \(0.863606\pi\)
\(674\) 34.9051 1.34449
\(675\) −1.21951 −0.0469388
\(676\) 5.42699 0.208730
\(677\) −9.58876 −0.368526 −0.184263 0.982877i \(-0.558990\pi\)
−0.184263 + 0.982877i \(0.558990\pi\)
\(678\) −11.3899 −0.437426
\(679\) 0 0
\(680\) −108.083 −4.14481
\(681\) −27.0920 −1.03817
\(682\) −2.93621 −0.112433
\(683\) 47.3161 1.81050 0.905250 0.424879i \(-0.139683\pi\)
0.905250 + 0.424879i \(0.139683\pi\)
\(684\) −4.87654 −0.186459
\(685\) 19.0976 0.729680
\(686\) 0 0
\(687\) −33.9967 −1.29705
\(688\) 12.9599 0.494091
\(689\) 4.88814 0.186223
\(690\) −5.24528 −0.199684
\(691\) −27.1119 −1.03138 −0.515692 0.856774i \(-0.672465\pi\)
−0.515692 + 0.856774i \(0.672465\pi\)
\(692\) 22.3888 0.851095
\(693\) 0 0
\(694\) 55.0648 2.09023
\(695\) −8.74866 −0.331855
\(696\) 39.0330 1.47954
\(697\) −38.7340 −1.46716
\(698\) −50.2628 −1.90248
\(699\) 1.06819 0.0404025
\(700\) 0 0
\(701\) −1.79821 −0.0679176 −0.0339588 0.999423i \(-0.510811\pi\)
−0.0339588 + 0.999423i \(0.510811\pi\)
\(702\) 15.3640 0.579879
\(703\) −8.24515 −0.310972
\(704\) −29.6838 −1.11875
\(705\) −6.88357 −0.259250
\(706\) −22.1954 −0.835336
\(707\) 0 0
\(708\) 7.65885 0.287837
\(709\) −28.3230 −1.06369 −0.531846 0.846841i \(-0.678502\pi\)
−0.531846 + 0.846841i \(0.678502\pi\)
\(710\) −39.3561 −1.47701
\(711\) −2.54070 −0.0952836
\(712\) −53.8675 −2.01877
\(713\) 0.671827 0.0251601
\(714\) 0 0
\(715\) −2.29249 −0.0857341
\(716\) 78.2223 2.92331
\(717\) −26.9940 −1.00811
\(718\) −88.8704 −3.31661
\(719\) −41.8971 −1.56250 −0.781249 0.624220i \(-0.785417\pi\)
−0.781249 + 0.624220i \(0.785417\pi\)
\(720\) −37.9046 −1.41262
\(721\) 0 0
\(722\) −50.2184 −1.86894
\(723\) −18.5968 −0.691621
\(724\) 98.2361 3.65092
\(725\) 0.671457 0.0249373
\(726\) 36.3313 1.34838
\(727\) 19.5123 0.723670 0.361835 0.932242i \(-0.382150\pi\)
0.361835 + 0.932242i \(0.382150\pi\)
\(728\) 0 0
\(729\) 27.5552 1.02056
\(730\) −49.3742 −1.82742
\(731\) −4.69742 −0.173740
\(732\) 91.3358 3.37587
\(733\) 17.7540 0.655757 0.327879 0.944720i \(-0.393666\pi\)
0.327879 + 0.944720i \(0.393666\pi\)
\(734\) −8.61213 −0.317880
\(735\) 0 0
\(736\) 13.7939 0.508450
\(737\) −4.69457 −0.172927
\(738\) −23.6838 −0.871814
\(739\) 44.3142 1.63012 0.815061 0.579375i \(-0.196703\pi\)
0.815061 + 0.579375i \(0.196703\pi\)
\(740\) −129.299 −4.75314
\(741\) −1.01911 −0.0374380
\(742\) 0 0
\(743\) −7.16727 −0.262941 −0.131471 0.991320i \(-0.541970\pi\)
−0.131471 + 0.991320i \(0.541970\pi\)
\(744\) −12.9257 −0.473879
\(745\) 33.6662 1.23344
\(746\) −4.02710 −0.147442
\(747\) 7.91313 0.289526
\(748\) 30.0983 1.10050
\(749\) 0 0
\(750\) 41.8630 1.52862
\(751\) −33.9065 −1.23726 −0.618632 0.785681i \(-0.712313\pi\)
−0.618632 + 0.785681i \(0.712313\pi\)
\(752\) 34.1235 1.24436
\(753\) 35.1543 1.28109
\(754\) −8.45941 −0.308073
\(755\) −29.9110 −1.08857
\(756\) 0 0
\(757\) −0.906670 −0.0329535 −0.0164767 0.999864i \(-0.505245\pi\)
−0.0164767 + 0.999864i \(0.505245\pi\)
\(758\) 29.2294 1.06166
\(759\) 0.922372 0.0334800
\(760\) 15.4612 0.560836
\(761\) 20.2494 0.734039 0.367020 0.930213i \(-0.380378\pi\)
0.367020 + 0.930213i \(0.380378\pi\)
\(762\) −32.2502 −1.16830
\(763\) 0 0
\(764\) 30.0729 1.08800
\(765\) 13.7388 0.496729
\(766\) −58.3500 −2.10827
\(767\) −1.04815 −0.0378467
\(768\) −52.0518 −1.87826
\(769\) −36.9094 −1.33099 −0.665494 0.746403i \(-0.731779\pi\)
−0.665494 + 0.746403i \(0.731779\pi\)
\(770\) 0 0
\(771\) −14.2987 −0.514954
\(772\) −47.4367 −1.70728
\(773\) −9.88037 −0.355372 −0.177686 0.984087i \(-0.556861\pi\)
−0.177686 + 0.984087i \(0.556861\pi\)
\(774\) −2.87222 −0.103240
\(775\) −0.222352 −0.00798711
\(776\) −26.9701 −0.968169
\(777\) 0 0
\(778\) 94.7893 3.39836
\(779\) 5.54086 0.198522
\(780\) −15.9816 −0.572232
\(781\) 6.92069 0.247642
\(782\) −9.42467 −0.337025
\(783\) −17.4998 −0.625391
\(784\) 0 0
\(785\) 7.40915 0.264444
\(786\) 38.5810 1.37614
\(787\) 37.6821 1.34322 0.671611 0.740904i \(-0.265603\pi\)
0.671611 + 0.740904i \(0.265603\pi\)
\(788\) −29.6788 −1.05726
\(789\) −13.9458 −0.496485
\(790\) 12.7565 0.453854
\(791\) 0 0
\(792\) 11.6213 0.412945
\(793\) −12.4998 −0.443880
\(794\) 12.1331 0.430588
\(795\) −14.3948 −0.510529
\(796\) 106.026 3.75799
\(797\) 28.3837 1.00540 0.502701 0.864460i \(-0.332340\pi\)
0.502701 + 0.864460i \(0.332340\pi\)
\(798\) 0 0
\(799\) −12.3683 −0.437560
\(800\) −4.56531 −0.161408
\(801\) 6.84728 0.241937
\(802\) −37.4824 −1.32355
\(803\) 8.68236 0.306394
\(804\) −32.7272 −1.15420
\(805\) 0 0
\(806\) 2.80132 0.0986722
\(807\) −16.1167 −0.567334
\(808\) −105.109 −3.69771
\(809\) −11.7465 −0.412987 −0.206493 0.978448i \(-0.566205\pi\)
−0.206493 + 0.978448i \(0.566205\pi\)
\(810\) −24.0160 −0.843837
\(811\) 2.01940 0.0709108 0.0354554 0.999371i \(-0.488712\pi\)
0.0354554 + 0.999371i \(0.488712\pi\)
\(812\) 0 0
\(813\) 3.71194 0.130184
\(814\) 31.1163 1.09063
\(815\) −30.2048 −1.05803
\(816\) 104.001 3.64077
\(817\) 0.671959 0.0235089
\(818\) −9.51676 −0.332746
\(819\) 0 0
\(820\) 86.8910 3.03437
\(821\) −15.0842 −0.526441 −0.263220 0.964736i \(-0.584785\pi\)
−0.263220 + 0.964736i \(0.584785\pi\)
\(822\) −32.0392 −1.11750
\(823\) −14.7766 −0.515079 −0.257539 0.966268i \(-0.582912\pi\)
−0.257539 + 0.966268i \(0.582912\pi\)
\(824\) 188.968 6.58301
\(825\) −0.305274 −0.0106283
\(826\) 0 0
\(827\) 13.0407 0.453471 0.226736 0.973956i \(-0.427195\pi\)
0.226736 + 0.973956i \(0.427195\pi\)
\(828\) −4.21086 −0.146338
\(829\) −25.4581 −0.884198 −0.442099 0.896966i \(-0.645766\pi\)
−0.442099 + 0.896966i \(0.645766\pi\)
\(830\) −39.7306 −1.37907
\(831\) 32.2163 1.11757
\(832\) 28.3200 0.981820
\(833\) 0 0
\(834\) 14.6773 0.508233
\(835\) 35.8220 1.23967
\(836\) −4.30553 −0.148910
\(837\) 5.79502 0.200305
\(838\) −9.72198 −0.335840
\(839\) 32.1703 1.11064 0.555321 0.831636i \(-0.312596\pi\)
0.555321 + 0.831636i \(0.312596\pi\)
\(840\) 0 0
\(841\) −19.3647 −0.667747
\(842\) −27.2525 −0.939183
\(843\) 5.22234 0.179867
\(844\) 90.5792 3.11787
\(845\) 2.18716 0.0752407
\(846\) −7.56259 −0.260007
\(847\) 0 0
\(848\) 71.3582 2.45045
\(849\) 8.36123 0.286957
\(850\) 3.11924 0.106989
\(851\) −7.11964 −0.244058
\(852\) 48.2461 1.65288
\(853\) −19.3910 −0.663934 −0.331967 0.943291i \(-0.607712\pi\)
−0.331967 + 0.943291i \(0.607712\pi\)
\(854\) 0 0
\(855\) −1.96532 −0.0672127
\(856\) 84.5699 2.89054
\(857\) 17.4242 0.595199 0.297600 0.954691i \(-0.403814\pi\)
0.297600 + 0.954691i \(0.403814\pi\)
\(858\) 3.84602 0.131301
\(859\) −35.4917 −1.21096 −0.605481 0.795860i \(-0.707019\pi\)
−0.605481 + 0.795860i \(0.707019\pi\)
\(860\) 10.5376 0.359329
\(861\) 0 0
\(862\) −30.9704 −1.05486
\(863\) 56.0019 1.90633 0.953164 0.302455i \(-0.0978062\pi\)
0.953164 + 0.302455i \(0.0978062\pi\)
\(864\) 118.983 4.04789
\(865\) 9.02306 0.306793
\(866\) −57.8124 −1.96455
\(867\) −14.8070 −0.502871
\(868\) 0 0
\(869\) −2.24320 −0.0760953
\(870\) 24.9115 0.844580
\(871\) 4.47889 0.151761
\(872\) 141.038 4.77615
\(873\) 3.42826 0.116029
\(874\) 1.34819 0.0456031
\(875\) 0 0
\(876\) 60.5272 2.04503
\(877\) 25.2062 0.851153 0.425577 0.904922i \(-0.360071\pi\)
0.425577 + 0.904922i \(0.360071\pi\)
\(878\) −66.7704 −2.25339
\(879\) −22.3174 −0.752748
\(880\) −33.4663 −1.12815
\(881\) −18.6082 −0.626925 −0.313463 0.949601i \(-0.601489\pi\)
−0.313463 + 0.949601i \(0.601489\pi\)
\(882\) 0 0
\(883\) −11.2552 −0.378768 −0.189384 0.981903i \(-0.560649\pi\)
−0.189384 + 0.981903i \(0.560649\pi\)
\(884\) −28.7155 −0.965808
\(885\) 3.08664 0.103756
\(886\) 110.288 3.70519
\(887\) −39.2112 −1.31658 −0.658292 0.752762i \(-0.728721\pi\)
−0.658292 + 0.752762i \(0.728721\pi\)
\(888\) 136.979 4.59672
\(889\) 0 0
\(890\) −34.3792 −1.15239
\(891\) 4.22317 0.141482
\(892\) −28.9919 −0.970720
\(893\) 1.76928 0.0592066
\(894\) −56.4805 −1.88899
\(895\) 31.5249 1.05376
\(896\) 0 0
\(897\) −0.879996 −0.0293822
\(898\) −75.6633 −2.52492
\(899\) −3.19073 −0.106417
\(900\) 1.39365 0.0464550
\(901\) −25.8644 −0.861667
\(902\) −20.9106 −0.696247
\(903\) 0 0
\(904\) 28.9903 0.964203
\(905\) 39.5907 1.31604
\(906\) 50.1806 1.66714
\(907\) 21.5970 0.717116 0.358558 0.933508i \(-0.383269\pi\)
0.358558 + 0.933508i \(0.383269\pi\)
\(908\) 109.199 3.62391
\(909\) 13.3607 0.443147
\(910\) 0 0
\(911\) −32.4434 −1.07490 −0.537449 0.843297i \(-0.680612\pi\)
−0.537449 + 0.843297i \(0.680612\pi\)
\(912\) −14.8772 −0.492634
\(913\) 6.98656 0.231221
\(914\) −30.5062 −1.00906
\(915\) 36.8098 1.21689
\(916\) 137.030 4.52760
\(917\) 0 0
\(918\) −81.2950 −2.68313
\(919\) 35.7372 1.17886 0.589430 0.807819i \(-0.299352\pi\)
0.589430 + 0.807819i \(0.299352\pi\)
\(920\) 13.3506 0.440157
\(921\) 9.50285 0.313129
\(922\) −25.3386 −0.834483
\(923\) −6.60274 −0.217332
\(924\) 0 0
\(925\) 2.35636 0.0774765
\(926\) 76.9716 2.52944
\(927\) −24.0204 −0.788932
\(928\) −65.5118 −2.15053
\(929\) 11.7769 0.386389 0.193194 0.981161i \(-0.438115\pi\)
0.193194 + 0.981161i \(0.438115\pi\)
\(930\) −8.24941 −0.270509
\(931\) 0 0
\(932\) −4.30553 −0.141032
\(933\) −28.4231 −0.930531
\(934\) 60.6658 1.98505
\(935\) 12.1301 0.396697
\(936\) −11.0874 −0.362403
\(937\) 18.9937 0.620497 0.310248 0.950655i \(-0.399588\pi\)
0.310248 + 0.950655i \(0.399588\pi\)
\(938\) 0 0
\(939\) −2.66660 −0.0870213
\(940\) 27.7456 0.904961
\(941\) 6.81864 0.222281 0.111141 0.993805i \(-0.464550\pi\)
0.111141 + 0.993805i \(0.464550\pi\)
\(942\) −12.4300 −0.404993
\(943\) 4.78450 0.155805
\(944\) −15.3012 −0.498012
\(945\) 0 0
\(946\) −2.53590 −0.0824493
\(947\) 1.05992 0.0344426 0.0172213 0.999852i \(-0.494518\pi\)
0.0172213 + 0.999852i \(0.494518\pi\)
\(948\) −15.6380 −0.507898
\(949\) −8.28347 −0.268893
\(950\) −0.446204 −0.0144768
\(951\) 24.2973 0.787895
\(952\) 0 0
\(953\) −40.4127 −1.30910 −0.654548 0.756020i \(-0.727141\pi\)
−0.654548 + 0.756020i \(0.727141\pi\)
\(954\) −15.8147 −0.512020
\(955\) 12.1199 0.392189
\(956\) 108.805 3.51899
\(957\) −4.38065 −0.141606
\(958\) −89.5964 −2.89473
\(959\) 0 0
\(960\) −83.3978 −2.69165
\(961\) −29.9434 −0.965916
\(962\) −29.6868 −0.957140
\(963\) −10.7500 −0.346413
\(964\) 74.9578 2.41423
\(965\) −19.1177 −0.615422
\(966\) 0 0
\(967\) −36.2949 −1.16717 −0.583583 0.812053i \(-0.698350\pi\)
−0.583583 + 0.812053i \(0.698350\pi\)
\(968\) −92.4729 −2.97219
\(969\) 5.39237 0.173228
\(970\) −17.2128 −0.552668
\(971\) −21.4437 −0.688160 −0.344080 0.938940i \(-0.611809\pi\)
−0.344080 + 0.938940i \(0.611809\pi\)
\(972\) −62.3457 −1.99974
\(973\) 0 0
\(974\) −76.0137 −2.43564
\(975\) 0.291249 0.00932742
\(976\) −182.475 −5.84088
\(977\) −39.8277 −1.27420 −0.637100 0.770781i \(-0.719866\pi\)
−0.637100 + 0.770781i \(0.719866\pi\)
\(978\) 50.6735 1.62036
\(979\) 6.04551 0.193215
\(980\) 0 0
\(981\) −17.9278 −0.572392
\(982\) 29.0432 0.926807
\(983\) 15.8814 0.506538 0.253269 0.967396i \(-0.418494\pi\)
0.253269 + 0.967396i \(0.418494\pi\)
\(984\) −92.0520 −2.93451
\(985\) −11.9610 −0.381110
\(986\) 44.7608 1.42548
\(987\) 0 0
\(988\) 4.10772 0.130684
\(989\) 0.580233 0.0184503
\(990\) 7.41693 0.235725
\(991\) −17.6687 −0.561265 −0.280633 0.959815i \(-0.590544\pi\)
−0.280633 + 0.959815i \(0.590544\pi\)
\(992\) 21.6941 0.688789
\(993\) 19.7570 0.626970
\(994\) 0 0
\(995\) 42.7301 1.35464
\(996\) 48.7053 1.54329
\(997\) −24.8608 −0.787350 −0.393675 0.919250i \(-0.628797\pi\)
−0.393675 + 0.919250i \(0.628797\pi\)
\(998\) −66.7997 −2.11451
\(999\) −61.4124 −1.94300
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 637.2.a.l.1.5 5
3.2 odd 2 5733.2.a.bl.1.1 5
7.2 even 3 91.2.e.c.53.1 10
7.3 odd 6 637.2.e.m.79.1 10
7.4 even 3 91.2.e.c.79.1 yes 10
7.5 odd 6 637.2.e.m.508.1 10
7.6 odd 2 637.2.a.k.1.5 5
13.12 even 2 8281.2.a.bw.1.1 5
21.2 odd 6 819.2.j.h.235.5 10
21.11 odd 6 819.2.j.h.352.5 10
21.20 even 2 5733.2.a.bm.1.1 5
28.11 odd 6 1456.2.r.p.625.2 10
28.23 odd 6 1456.2.r.p.417.2 10
91.25 even 6 1183.2.e.f.170.5 10
91.51 even 6 1183.2.e.f.508.5 10
91.90 odd 2 8281.2.a.bx.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.1 10 7.2 even 3
91.2.e.c.79.1 yes 10 7.4 even 3
637.2.a.k.1.5 5 7.6 odd 2
637.2.a.l.1.5 5 1.1 even 1 trivial
637.2.e.m.79.1 10 7.3 odd 6
637.2.e.m.508.1 10 7.5 odd 6
819.2.j.h.235.5 10 21.2 odd 6
819.2.j.h.352.5 10 21.11 odd 6
1183.2.e.f.170.5 10 91.25 even 6
1183.2.e.f.508.5 10 91.51 even 6
1456.2.r.p.417.2 10 28.23 odd 6
1456.2.r.p.625.2 10 28.11 odd 6
5733.2.a.bl.1.1 5 3.2 odd 2
5733.2.a.bm.1.1 5 21.20 even 2
8281.2.a.bw.1.1 5 13.12 even 2
8281.2.a.bx.1.1 5 91.90 odd 2