Properties

Label 503.2.a.e.1.10
Level $503$
Weight $2$
Character 503.1
Self dual yes
Analytic conductor $4.016$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 503.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.01647522167\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 9 x^{8} + 14 x^{7} + 27 x^{6} - 27 x^{5} - 34 x^{4} + 14 x^{3} + 17 x^{2} + x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-0.510671\) of defining polynomial
Character \(\chi\) \(=\) 503.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.15783 q^{2} -1.51067 q^{3} +2.65622 q^{4} -2.23445 q^{5} -3.25977 q^{6} -3.60329 q^{7} +1.41602 q^{8} -0.717874 q^{9} +O(q^{10})\) \(q+2.15783 q^{2} -1.51067 q^{3} +2.65622 q^{4} -2.23445 q^{5} -3.25977 q^{6} -3.60329 q^{7} +1.41602 q^{8} -0.717874 q^{9} -4.82156 q^{10} -1.50306 q^{11} -4.01268 q^{12} +0.00459548 q^{13} -7.77529 q^{14} +3.37551 q^{15} -2.25692 q^{16} +1.11927 q^{17} -1.54905 q^{18} +2.49963 q^{19} -5.93520 q^{20} +5.44339 q^{21} -3.24334 q^{22} +1.72480 q^{23} -2.13914 q^{24} -0.00724425 q^{25} +0.00991627 q^{26} +5.61648 q^{27} -9.57116 q^{28} -0.572061 q^{29} +7.28378 q^{30} +3.25375 q^{31} -7.70209 q^{32} +2.27063 q^{33} +2.41520 q^{34} +8.05137 q^{35} -1.90683 q^{36} -6.13809 q^{37} +5.39377 q^{38} -0.00694226 q^{39} -3.16402 q^{40} -5.89549 q^{41} +11.7459 q^{42} -7.66800 q^{43} -3.99246 q^{44} +1.60405 q^{45} +3.72183 q^{46} +6.44840 q^{47} +3.40946 q^{48} +5.98373 q^{49} -0.0156318 q^{50} -1.69085 q^{51} +0.0122066 q^{52} -7.82500 q^{53} +12.1194 q^{54} +3.35851 q^{55} -5.10234 q^{56} -3.77612 q^{57} -1.23441 q^{58} +0.253382 q^{59} +8.96613 q^{60} +7.08244 q^{61} +7.02103 q^{62} +2.58671 q^{63} -12.1059 q^{64} -0.0102684 q^{65} +4.89963 q^{66} +6.16046 q^{67} +2.97304 q^{68} -2.60561 q^{69} +17.3735 q^{70} -9.32311 q^{71} -1.01652 q^{72} -15.2654 q^{73} -13.2450 q^{74} +0.0109437 q^{75} +6.63958 q^{76} +5.41596 q^{77} -0.0149802 q^{78} -1.99661 q^{79} +5.04297 q^{80} -6.33104 q^{81} -12.7215 q^{82} +6.91660 q^{83} +14.4589 q^{84} -2.50096 q^{85} -16.5462 q^{86} +0.864195 q^{87} -2.12836 q^{88} +3.37422 q^{89} +3.46127 q^{90} -0.0165589 q^{91} +4.58146 q^{92} -4.91534 q^{93} +13.9146 q^{94} -5.58529 q^{95} +11.6353 q^{96} -14.8842 q^{97} +12.9119 q^{98} +1.07901 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} + O(q^{10}) \) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} - 4q^{10} - 3q^{11} - 7q^{12} - 18q^{13} + q^{14} - 2q^{15} - 4q^{16} - 11q^{17} - q^{18} - 3q^{20} + q^{21} - 18q^{22} - 2q^{23} + 10q^{24} - 27q^{25} + 11q^{26} - 2q^{27} - 22q^{28} - 9q^{29} + 12q^{30} - 22q^{31} - 10q^{32} - 10q^{33} - 10q^{34} - 6q^{35} + 2q^{36} - 35q^{37} + 2q^{38} + 8q^{39} - 19q^{40} - 4q^{41} + 4q^{42} - 20q^{43} + 9q^{44} + 2q^{45} - q^{46} + 7q^{47} - 27q^{49} + 16q^{50} + 9q^{51} - 7q^{52} - 24q^{53} + 17q^{54} - 11q^{55} + 12q^{56} - 23q^{57} + 2q^{58} + 17q^{59} - 4q^{61} + 8q^{62} + 10q^{63} + 3q^{64} - 16q^{65} + 46q^{66} - 6q^{67} + 28q^{68} - 2q^{69} + 26q^{70} - q^{71} - q^{72} - 31q^{73} + 11q^{74} + 30q^{75} + 20q^{76} + 3q^{77} + 11q^{78} - 10q^{79} + 24q^{80} - 6q^{81} - 9q^{82} + 22q^{83} + 22q^{84} - 6q^{85} + 38q^{86} + 25q^{87} - 3q^{88} + q^{89} + 2q^{90} + 10q^{91} + 27q^{92} - 6q^{93} + 33q^{94} + 39q^{95} + 46q^{96} - 57q^{97} + 40q^{98} + 35q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.15783 1.52582 0.762908 0.646507i \(-0.223771\pi\)
0.762908 + 0.646507i \(0.223771\pi\)
\(3\) −1.51067 −0.872186 −0.436093 0.899902i \(-0.643638\pi\)
−0.436093 + 0.899902i \(0.643638\pi\)
\(4\) 2.65622 1.32811
\(5\) −2.23445 −0.999275 −0.499638 0.866235i \(-0.666534\pi\)
−0.499638 + 0.866235i \(0.666534\pi\)
\(6\) −3.25977 −1.33080
\(7\) −3.60329 −1.36192 −0.680959 0.732322i \(-0.738437\pi\)
−0.680959 + 0.732322i \(0.738437\pi\)
\(8\) 1.41602 0.500639
\(9\) −0.717874 −0.239291
\(10\) −4.82156 −1.52471
\(11\) −1.50306 −0.453189 −0.226595 0.973989i \(-0.572759\pi\)
−0.226595 + 0.973989i \(0.572759\pi\)
\(12\) −4.01268 −1.15836
\(13\) 0.00459548 0.00127456 0.000637279 1.00000i \(-0.499797\pi\)
0.000637279 1.00000i \(0.499797\pi\)
\(14\) −7.77529 −2.07803
\(15\) 3.37551 0.871554
\(16\) −2.25692 −0.564230
\(17\) 1.11927 0.271464 0.135732 0.990746i \(-0.456661\pi\)
0.135732 + 0.990746i \(0.456661\pi\)
\(18\) −1.54905 −0.365114
\(19\) 2.49963 0.573454 0.286727 0.958012i \(-0.407433\pi\)
0.286727 + 0.958012i \(0.407433\pi\)
\(20\) −5.93520 −1.32715
\(21\) 5.44339 1.18785
\(22\) −3.24334 −0.691483
\(23\) 1.72480 0.359646 0.179823 0.983699i \(-0.442447\pi\)
0.179823 + 0.983699i \(0.442447\pi\)
\(24\) −2.13914 −0.436650
\(25\) −0.00724425 −0.00144885
\(26\) 0.00991627 0.00194474
\(27\) 5.61648 1.08089
\(28\) −9.57116 −1.80878
\(29\) −0.572061 −0.106229 −0.0531145 0.998588i \(-0.516915\pi\)
−0.0531145 + 0.998588i \(0.516915\pi\)
\(30\) 7.28378 1.32983
\(31\) 3.25375 0.584390 0.292195 0.956359i \(-0.405614\pi\)
0.292195 + 0.956359i \(0.405614\pi\)
\(32\) −7.70209 −1.36155
\(33\) 2.27063 0.395266
\(34\) 2.41520 0.414203
\(35\) 8.05137 1.36093
\(36\) −1.90683 −0.317806
\(37\) −6.13809 −1.00910 −0.504548 0.863384i \(-0.668341\pi\)
−0.504548 + 0.863384i \(0.668341\pi\)
\(38\) 5.39377 0.874986
\(39\) −0.00694226 −0.00111165
\(40\) −3.16402 −0.500276
\(41\) −5.89549 −0.920721 −0.460360 0.887732i \(-0.652280\pi\)
−0.460360 + 0.887732i \(0.652280\pi\)
\(42\) 11.7459 1.81243
\(43\) −7.66800 −1.16936 −0.584680 0.811264i \(-0.698780\pi\)
−0.584680 + 0.811264i \(0.698780\pi\)
\(44\) −3.99246 −0.601886
\(45\) 1.60405 0.239118
\(46\) 3.72183 0.548754
\(47\) 6.44840 0.940596 0.470298 0.882508i \(-0.344146\pi\)
0.470298 + 0.882508i \(0.344146\pi\)
\(48\) 3.40946 0.492113
\(49\) 5.98373 0.854819
\(50\) −0.0156318 −0.00221068
\(51\) −1.69085 −0.236767
\(52\) 0.0122066 0.00169276
\(53\) −7.82500 −1.07485 −0.537423 0.843313i \(-0.680602\pi\)
−0.537423 + 0.843313i \(0.680602\pi\)
\(54\) 12.1194 1.64924
\(55\) 3.35851 0.452861
\(56\) −5.10234 −0.681829
\(57\) −3.77612 −0.500159
\(58\) −1.23441 −0.162086
\(59\) 0.253382 0.0329875 0.0164938 0.999864i \(-0.494750\pi\)
0.0164938 + 0.999864i \(0.494750\pi\)
\(60\) 8.96613 1.15752
\(61\) 7.08244 0.906814 0.453407 0.891304i \(-0.350208\pi\)
0.453407 + 0.891304i \(0.350208\pi\)
\(62\) 7.02103 0.891672
\(63\) 2.58671 0.325895
\(64\) −12.1059 −1.51324
\(65\) −0.0102684 −0.00127363
\(66\) 4.89963 0.603102
\(67\) 6.16046 0.752621 0.376310 0.926494i \(-0.377193\pi\)
0.376310 + 0.926494i \(0.377193\pi\)
\(68\) 2.97304 0.360534
\(69\) −2.60561 −0.313678
\(70\) 17.3735 2.07653
\(71\) −9.32311 −1.10645 −0.553225 0.833032i \(-0.686603\pi\)
−0.553225 + 0.833032i \(0.686603\pi\)
\(72\) −1.01652 −0.119799
\(73\) −15.2654 −1.78668 −0.893338 0.449385i \(-0.851643\pi\)
−0.893338 + 0.449385i \(0.851643\pi\)
\(74\) −13.2450 −1.53969
\(75\) 0.0109437 0.00126367
\(76\) 6.63958 0.761612
\(77\) 5.41596 0.617206
\(78\) −0.0149802 −0.00169618
\(79\) −1.99661 −0.224636 −0.112318 0.993672i \(-0.535828\pi\)
−0.112318 + 0.993672i \(0.535828\pi\)
\(80\) 5.04297 0.563821
\(81\) −6.33104 −0.703448
\(82\) −12.7215 −1.40485
\(83\) 6.91660 0.759196 0.379598 0.925152i \(-0.376062\pi\)
0.379598 + 0.925152i \(0.376062\pi\)
\(84\) 14.4589 1.57759
\(85\) −2.50096 −0.271267
\(86\) −16.5462 −1.78423
\(87\) 0.864195 0.0926514
\(88\) −2.12836 −0.226884
\(89\) 3.37422 0.357666 0.178833 0.983879i \(-0.442768\pi\)
0.178833 + 0.983879i \(0.442768\pi\)
\(90\) 3.46127 0.364850
\(91\) −0.0165589 −0.00173584
\(92\) 4.58146 0.477651
\(93\) −4.91534 −0.509697
\(94\) 13.9146 1.43518
\(95\) −5.58529 −0.573039
\(96\) 11.6353 1.18752
\(97\) −14.8842 −1.51126 −0.755631 0.654997i \(-0.772670\pi\)
−0.755631 + 0.654997i \(0.772670\pi\)
\(98\) 12.9119 1.30430
\(99\) 1.07901 0.108444
\(100\) −0.0192424 −0.00192424
\(101\) 11.2591 1.12032 0.560159 0.828385i \(-0.310740\pi\)
0.560159 + 0.828385i \(0.310740\pi\)
\(102\) −3.64857 −0.361262
\(103\) 4.65004 0.458182 0.229091 0.973405i \(-0.426425\pi\)
0.229091 + 0.973405i \(0.426425\pi\)
\(104\) 0.00650730 0.000638093 0
\(105\) −12.1630 −1.18698
\(106\) −16.8850 −1.64002
\(107\) −8.08323 −0.781435 −0.390718 0.920511i \(-0.627773\pi\)
−0.390718 + 0.920511i \(0.627773\pi\)
\(108\) 14.9186 1.43555
\(109\) −1.20991 −0.115889 −0.0579444 0.998320i \(-0.518455\pi\)
−0.0579444 + 0.998320i \(0.518455\pi\)
\(110\) 7.24708 0.690982
\(111\) 9.27264 0.880120
\(112\) 8.13234 0.768434
\(113\) 7.68412 0.722861 0.361431 0.932399i \(-0.382288\pi\)
0.361431 + 0.932399i \(0.382288\pi\)
\(114\) −8.14822 −0.763150
\(115\) −3.85398 −0.359386
\(116\) −1.51952 −0.141084
\(117\) −0.00329898 −0.000304991 0
\(118\) 0.546755 0.0503329
\(119\) −4.03307 −0.369711
\(120\) 4.77980 0.436334
\(121\) −8.74081 −0.794619
\(122\) 15.2827 1.38363
\(123\) 8.90614 0.803040
\(124\) 8.64268 0.776136
\(125\) 11.1884 1.00072
\(126\) 5.58168 0.497255
\(127\) 8.83928 0.784360 0.392180 0.919889i \(-0.371721\pi\)
0.392180 + 0.919889i \(0.371721\pi\)
\(128\) −10.7184 −0.947380
\(129\) 11.5838 1.01990
\(130\) −0.0221574 −0.00194333
\(131\) −12.9102 −1.12797 −0.563984 0.825786i \(-0.690732\pi\)
−0.563984 + 0.825786i \(0.690732\pi\)
\(132\) 6.03130 0.524957
\(133\) −9.00690 −0.780998
\(134\) 13.2932 1.14836
\(135\) −12.5497 −1.08011
\(136\) 1.58491 0.135905
\(137\) 3.83616 0.327745 0.163873 0.986482i \(-0.447601\pi\)
0.163873 + 0.986482i \(0.447601\pi\)
\(138\) −5.62246 −0.478615
\(139\) 9.22921 0.782811 0.391406 0.920218i \(-0.371989\pi\)
0.391406 + 0.920218i \(0.371989\pi\)
\(140\) 21.3863 1.80747
\(141\) −9.74142 −0.820375
\(142\) −20.1177 −1.68824
\(143\) −0.00690728 −0.000577616 0
\(144\) 1.62018 0.135015
\(145\) 1.27824 0.106152
\(146\) −32.9401 −2.72614
\(147\) −9.03945 −0.745561
\(148\) −16.3042 −1.34019
\(149\) 22.6884 1.85871 0.929353 0.369191i \(-0.120365\pi\)
0.929353 + 0.369191i \(0.120365\pi\)
\(150\) 0.0236146 0.00192812
\(151\) −2.22521 −0.181085 −0.0905427 0.995893i \(-0.528860\pi\)
−0.0905427 + 0.995893i \(0.528860\pi\)
\(152\) 3.53953 0.287094
\(153\) −0.803497 −0.0649589
\(154\) 11.6867 0.941743
\(155\) −7.27033 −0.583967
\(156\) −0.0184402 −0.00147640
\(157\) −22.3177 −1.78114 −0.890571 0.454844i \(-0.849695\pi\)
−0.890571 + 0.454844i \(0.849695\pi\)
\(158\) −4.30834 −0.342753
\(159\) 11.8210 0.937466
\(160\) 17.2099 1.36056
\(161\) −6.21497 −0.489808
\(162\) −13.6613 −1.07333
\(163\) 7.22321 0.565765 0.282883 0.959155i \(-0.408709\pi\)
0.282883 + 0.959155i \(0.408709\pi\)
\(164\) −15.6597 −1.22282
\(165\) −5.07360 −0.394979
\(166\) 14.9248 1.15839
\(167\) −11.4485 −0.885915 −0.442958 0.896543i \(-0.646071\pi\)
−0.442958 + 0.896543i \(0.646071\pi\)
\(168\) 7.70796 0.594682
\(169\) −13.0000 −0.999998
\(170\) −5.39664 −0.413903
\(171\) −1.79442 −0.137223
\(172\) −20.3679 −1.55304
\(173\) −12.5821 −0.956599 −0.478300 0.878197i \(-0.658747\pi\)
−0.478300 + 0.878197i \(0.658747\pi\)
\(174\) 1.86479 0.141369
\(175\) 0.0261032 0.00197321
\(176\) 3.39228 0.255703
\(177\) −0.382777 −0.0287713
\(178\) 7.28098 0.545732
\(179\) 1.23407 0.0922389 0.0461194 0.998936i \(-0.485315\pi\)
0.0461194 + 0.998936i \(0.485315\pi\)
\(180\) 4.26072 0.317575
\(181\) −17.0030 −1.26382 −0.631911 0.775041i \(-0.717729\pi\)
−0.631911 + 0.775041i \(0.717729\pi\)
\(182\) −0.0357312 −0.00264858
\(183\) −10.6992 −0.790910
\(184\) 2.44236 0.180053
\(185\) 13.7152 1.00836
\(186\) −10.6065 −0.777704
\(187\) −1.68233 −0.123024
\(188\) 17.1284 1.24922
\(189\) −20.2378 −1.47209
\(190\) −12.0521 −0.874352
\(191\) 0.213652 0.0154593 0.00772966 0.999970i \(-0.497540\pi\)
0.00772966 + 0.999970i \(0.497540\pi\)
\(192\) 18.2881 1.31983
\(193\) −3.51336 −0.252897 −0.126449 0.991973i \(-0.540358\pi\)
−0.126449 + 0.991973i \(0.540358\pi\)
\(194\) −32.1176 −2.30591
\(195\) 0.0155121 0.00111085
\(196\) 15.8941 1.13530
\(197\) −11.5456 −0.822587 −0.411294 0.911503i \(-0.634923\pi\)
−0.411294 + 0.911503i \(0.634923\pi\)
\(198\) 2.32831 0.165466
\(199\) −4.63698 −0.328707 −0.164353 0.986402i \(-0.552554\pi\)
−0.164353 + 0.986402i \(0.552554\pi\)
\(200\) −0.0102580 −0.000725351 0
\(201\) −9.30643 −0.656425
\(202\) 24.2951 1.70940
\(203\) 2.06130 0.144675
\(204\) −4.49129 −0.314453
\(205\) 13.1732 0.920054
\(206\) 10.0340 0.699101
\(207\) −1.23819 −0.0860602
\(208\) −0.0103716 −0.000719144 0
\(209\) −3.75709 −0.259883
\(210\) −26.2456 −1.81112
\(211\) −0.622411 −0.0428485 −0.0214243 0.999770i \(-0.506820\pi\)
−0.0214243 + 0.999770i \(0.506820\pi\)
\(212\) −20.7850 −1.42752
\(213\) 14.0841 0.965030
\(214\) −17.4422 −1.19233
\(215\) 17.1338 1.16851
\(216\) 7.95306 0.541137
\(217\) −11.7242 −0.795891
\(218\) −2.61079 −0.176825
\(219\) 23.0609 1.55831
\(220\) 8.92095 0.601450
\(221\) 0.00514360 0.000345996 0
\(222\) 20.0088 1.34290
\(223\) 0.0281489 0.00188499 0.000942495 1.00000i \(-0.499700\pi\)
0.000942495 1.00000i \(0.499700\pi\)
\(224\) 27.7529 1.85432
\(225\) 0.00520046 0.000346697 0
\(226\) 16.5810 1.10295
\(227\) −8.12837 −0.539499 −0.269749 0.962931i \(-0.586941\pi\)
−0.269749 + 0.962931i \(0.586941\pi\)
\(228\) −10.0302 −0.664267
\(229\) 23.9995 1.58593 0.792966 0.609266i \(-0.208536\pi\)
0.792966 + 0.609266i \(0.208536\pi\)
\(230\) −8.31623 −0.548356
\(231\) −8.18174 −0.538319
\(232\) −0.810050 −0.0531824
\(233\) −12.5903 −0.824819 −0.412410 0.910999i \(-0.635313\pi\)
−0.412410 + 0.910999i \(0.635313\pi\)
\(234\) −0.00711863 −0.000465359 0
\(235\) −14.4086 −0.939915
\(236\) 0.673040 0.0438112
\(237\) 3.01622 0.195925
\(238\) −8.70268 −0.564111
\(239\) 24.1351 1.56117 0.780586 0.625049i \(-0.214921\pi\)
0.780586 + 0.625049i \(0.214921\pi\)
\(240\) −7.61826 −0.491757
\(241\) 25.4713 1.64075 0.820374 0.571827i \(-0.193765\pi\)
0.820374 + 0.571827i \(0.193765\pi\)
\(242\) −18.8612 −1.21244
\(243\) −7.28534 −0.467355
\(244\) 18.8126 1.20435
\(245\) −13.3703 −0.854199
\(246\) 19.2179 1.22529
\(247\) 0.0114870 0.000730901 0
\(248\) 4.60737 0.292569
\(249\) −10.4487 −0.662160
\(250\) 24.1427 1.52692
\(251\) 4.12389 0.260298 0.130149 0.991494i \(-0.458454\pi\)
0.130149 + 0.991494i \(0.458454\pi\)
\(252\) 6.87088 0.432825
\(253\) −2.59248 −0.162988
\(254\) 19.0737 1.19679
\(255\) 3.77812 0.236595
\(256\) 1.08345 0.0677158
\(257\) −16.7266 −1.04337 −0.521687 0.853137i \(-0.674697\pi\)
−0.521687 + 0.853137i \(0.674697\pi\)
\(258\) 24.9959 1.55618
\(259\) 22.1174 1.37431
\(260\) −0.0272751 −0.00169153
\(261\) 0.410667 0.0254197
\(262\) −27.8579 −1.72107
\(263\) 28.7329 1.77174 0.885872 0.463930i \(-0.153561\pi\)
0.885872 + 0.463930i \(0.153561\pi\)
\(264\) 3.21526 0.197885
\(265\) 17.4845 1.07407
\(266\) −19.4354 −1.19166
\(267\) −5.09733 −0.311951
\(268\) 16.3636 0.999565
\(269\) 28.6099 1.74438 0.872189 0.489169i \(-0.162700\pi\)
0.872189 + 0.489169i \(0.162700\pi\)
\(270\) −27.0802 −1.64805
\(271\) −13.8852 −0.843467 −0.421733 0.906720i \(-0.638578\pi\)
−0.421733 + 0.906720i \(0.638578\pi\)
\(272\) −2.52611 −0.153168
\(273\) 0.0250150 0.00151398
\(274\) 8.27778 0.500079
\(275\) 0.0108885 0.000656603 0
\(276\) −6.92108 −0.416600
\(277\) −14.4664 −0.869200 −0.434600 0.900624i \(-0.643110\pi\)
−0.434600 + 0.900624i \(0.643110\pi\)
\(278\) 19.9150 1.19443
\(279\) −2.33578 −0.139839
\(280\) 11.4009 0.681335
\(281\) −15.4967 −0.924456 −0.462228 0.886761i \(-0.652950\pi\)
−0.462228 + 0.886761i \(0.652950\pi\)
\(282\) −21.0203 −1.25174
\(283\) 10.4295 0.619970 0.309985 0.950741i \(-0.399676\pi\)
0.309985 + 0.950741i \(0.399676\pi\)
\(284\) −24.7643 −1.46949
\(285\) 8.43754 0.499797
\(286\) −0.0149047 −0.000881336 0
\(287\) 21.2432 1.25395
\(288\) 5.52913 0.325807
\(289\) −15.7472 −0.926307
\(290\) 2.75822 0.161968
\(291\) 22.4851 1.31810
\(292\) −40.5483 −2.37291
\(293\) −1.90244 −0.111142 −0.0555708 0.998455i \(-0.517698\pi\)
−0.0555708 + 0.998455i \(0.517698\pi\)
\(294\) −19.5056 −1.13759
\(295\) −0.566169 −0.0329636
\(296\) −8.69167 −0.505193
\(297\) −8.44191 −0.489849
\(298\) 48.9577 2.83604
\(299\) 0.00792630 0.000458390 0
\(300\) 0.0290689 0.00167829
\(301\) 27.6301 1.59257
\(302\) −4.80163 −0.276303
\(303\) −17.0087 −0.977127
\(304\) −5.64146 −0.323560
\(305\) −15.8253 −0.906157
\(306\) −1.73381 −0.0991152
\(307\) −17.6876 −1.00949 −0.504744 0.863269i \(-0.668413\pi\)
−0.504744 + 0.863269i \(0.668413\pi\)
\(308\) 14.3860 0.819720
\(309\) −7.02468 −0.399620
\(310\) −15.6881 −0.891025
\(311\) −28.0074 −1.58815 −0.794076 0.607818i \(-0.792045\pi\)
−0.794076 + 0.607818i \(0.792045\pi\)
\(312\) −0.00983039 −0.000556536 0
\(313\) 5.02123 0.283817 0.141908 0.989880i \(-0.454676\pi\)
0.141908 + 0.989880i \(0.454676\pi\)
\(314\) −48.1577 −2.71769
\(315\) −5.77987 −0.325659
\(316\) −5.30344 −0.298342
\(317\) 0.356974 0.0200496 0.0100248 0.999950i \(-0.496809\pi\)
0.0100248 + 0.999950i \(0.496809\pi\)
\(318\) 25.5077 1.43040
\(319\) 0.859841 0.0481418
\(320\) 27.0501 1.51215
\(321\) 12.2111 0.681557
\(322\) −13.4108 −0.747357
\(323\) 2.79777 0.155672
\(324\) −16.8167 −0.934259
\(325\) −3.32908e−5 0 −1.84664e−6 0
\(326\) 15.5864 0.863253
\(327\) 1.82778 0.101077
\(328\) −8.34814 −0.460949
\(329\) −23.2355 −1.28101
\(330\) −10.9480 −0.602665
\(331\) 16.2490 0.893123 0.446562 0.894753i \(-0.352648\pi\)
0.446562 + 0.894753i \(0.352648\pi\)
\(332\) 18.3720 1.00830
\(333\) 4.40637 0.241468
\(334\) −24.7040 −1.35174
\(335\) −13.7652 −0.752075
\(336\) −12.2853 −0.670218
\(337\) −24.0444 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(338\) −28.0517 −1.52581
\(339\) −11.6082 −0.630469
\(340\) −6.64310 −0.360273
\(341\) −4.89057 −0.264839
\(342\) −3.87205 −0.209376
\(343\) 3.66192 0.197725
\(344\) −10.8581 −0.585427
\(345\) 5.82210 0.313451
\(346\) −27.1500 −1.45959
\(347\) −10.7861 −0.579028 −0.289514 0.957174i \(-0.593494\pi\)
−0.289514 + 0.957174i \(0.593494\pi\)
\(348\) 2.29550 0.123052
\(349\) 17.2044 0.920933 0.460466 0.887677i \(-0.347682\pi\)
0.460466 + 0.887677i \(0.347682\pi\)
\(350\) 0.0563262 0.00301076
\(351\) 0.0258105 0.00137766
\(352\) 11.5767 0.617040
\(353\) −2.83276 −0.150773 −0.0753863 0.997154i \(-0.524019\pi\)
−0.0753863 + 0.997154i \(0.524019\pi\)
\(354\) −0.825967 −0.0438996
\(355\) 20.8320 1.10565
\(356\) 8.96268 0.475021
\(357\) 6.09264 0.322457
\(358\) 2.66292 0.140739
\(359\) 23.3759 1.23373 0.616867 0.787068i \(-0.288402\pi\)
0.616867 + 0.787068i \(0.288402\pi\)
\(360\) 2.27137 0.119712
\(361\) −12.7518 −0.671150
\(362\) −36.6895 −1.92836
\(363\) 13.2045 0.693056
\(364\) −0.0439841 −0.00230539
\(365\) 34.1097 1.78538
\(366\) −23.0871 −1.20678
\(367\) −21.8406 −1.14007 −0.570034 0.821621i \(-0.693070\pi\)
−0.570034 + 0.821621i \(0.693070\pi\)
\(368\) −3.89274 −0.202923
\(369\) 4.23222 0.220320
\(370\) 29.5951 1.53858
\(371\) 28.1958 1.46385
\(372\) −13.0563 −0.676935
\(373\) 0.594294 0.0307714 0.0153857 0.999882i \(-0.495102\pi\)
0.0153857 + 0.999882i \(0.495102\pi\)
\(374\) −3.63019 −0.187713
\(375\) −16.9020 −0.872817
\(376\) 9.13108 0.470899
\(377\) −0.00262890 −0.000135395 0
\(378\) −43.6698 −2.24613
\(379\) −16.0631 −0.825105 −0.412553 0.910934i \(-0.635363\pi\)
−0.412553 + 0.910934i \(0.635363\pi\)
\(380\) −14.8358 −0.761060
\(381\) −13.3532 −0.684108
\(382\) 0.461025 0.0235881
\(383\) 15.0745 0.770271 0.385135 0.922860i \(-0.374155\pi\)
0.385135 + 0.922860i \(0.374155\pi\)
\(384\) 16.1920 0.826292
\(385\) −12.1017 −0.616759
\(386\) −7.58123 −0.385875
\(387\) 5.50466 0.279818
\(388\) −39.5358 −2.00713
\(389\) 8.06793 0.409060 0.204530 0.978860i \(-0.434433\pi\)
0.204530 + 0.978860i \(0.434433\pi\)
\(390\) 0.0334725 0.00169495
\(391\) 1.93053 0.0976309
\(392\) 8.47309 0.427956
\(393\) 19.5030 0.983797
\(394\) −24.9133 −1.25512
\(395\) 4.46132 0.224473
\(396\) 2.86608 0.144026
\(397\) 7.64285 0.383583 0.191792 0.981436i \(-0.438570\pi\)
0.191792 + 0.981436i \(0.438570\pi\)
\(398\) −10.0058 −0.501546
\(399\) 13.6065 0.681175
\(400\) 0.0163497 0.000817484 0
\(401\) 37.0556 1.85047 0.925234 0.379397i \(-0.123868\pi\)
0.925234 + 0.379397i \(0.123868\pi\)
\(402\) −20.0817 −1.00158
\(403\) 0.0149525 0.000744839 0
\(404\) 29.9066 1.48791
\(405\) 14.1464 0.702939
\(406\) 4.44794 0.220747
\(407\) 9.22591 0.457312
\(408\) −2.39428 −0.118535
\(409\) −29.1516 −1.44145 −0.720726 0.693220i \(-0.756191\pi\)
−0.720726 + 0.693220i \(0.756191\pi\)
\(410\) 28.4254 1.40383
\(411\) −5.79518 −0.285855
\(412\) 12.3515 0.608517
\(413\) −0.913010 −0.0449263
\(414\) −2.67180 −0.131312
\(415\) −15.4548 −0.758645
\(416\) −0.0353948 −0.00173537
\(417\) −13.9423 −0.682757
\(418\) −8.10716 −0.396534
\(419\) 31.8690 1.55690 0.778451 0.627705i \(-0.216006\pi\)
0.778451 + 0.627705i \(0.216006\pi\)
\(420\) −32.3076 −1.57645
\(421\) −26.0459 −1.26940 −0.634699 0.772759i \(-0.718876\pi\)
−0.634699 + 0.772759i \(0.718876\pi\)
\(422\) −1.34306 −0.0653790
\(423\) −4.62914 −0.225077
\(424\) −11.0804 −0.538110
\(425\) −0.00810829 −0.000393310 0
\(426\) 30.3912 1.47246
\(427\) −25.5201 −1.23501
\(428\) −21.4709 −1.03783
\(429\) 0.0104346 0.000503789 0
\(430\) 36.9717 1.78293
\(431\) −22.6196 −1.08955 −0.544774 0.838583i \(-0.683384\pi\)
−0.544774 + 0.838583i \(0.683384\pi\)
\(432\) −12.6759 −0.609872
\(433\) −22.7272 −1.09220 −0.546099 0.837720i \(-0.683888\pi\)
−0.546099 + 0.837720i \(0.683888\pi\)
\(434\) −25.2988 −1.21438
\(435\) −1.93100 −0.0925843
\(436\) −3.21380 −0.153913
\(437\) 4.31137 0.206241
\(438\) 49.7616 2.37770
\(439\) 24.5936 1.17379 0.586895 0.809663i \(-0.300350\pi\)
0.586895 + 0.809663i \(0.300350\pi\)
\(440\) 4.75572 0.226720
\(441\) −4.29556 −0.204551
\(442\) 0.0110990 0.000527926 0
\(443\) 23.4765 1.11540 0.557702 0.830041i \(-0.311683\pi\)
0.557702 + 0.830041i \(0.311683\pi\)
\(444\) 24.6302 1.16890
\(445\) −7.53951 −0.357407
\(446\) 0.0607405 0.00287615
\(447\) −34.2747 −1.62114
\(448\) 43.6213 2.06091
\(449\) −13.6452 −0.643958 −0.321979 0.946747i \(-0.604348\pi\)
−0.321979 + 0.946747i \(0.604348\pi\)
\(450\) 0.0112217 0.000528996 0
\(451\) 8.86127 0.417261
\(452\) 20.4107 0.960041
\(453\) 3.36157 0.157940
\(454\) −17.5396 −0.823176
\(455\) 0.0370000 0.00173458
\(456\) −5.34706 −0.250399
\(457\) −2.17967 −0.101960 −0.0509802 0.998700i \(-0.516235\pi\)
−0.0509802 + 0.998700i \(0.516235\pi\)
\(458\) 51.7868 2.41984
\(459\) 6.28638 0.293423
\(460\) −10.2370 −0.477304
\(461\) 12.2229 0.569276 0.284638 0.958635i \(-0.408127\pi\)
0.284638 + 0.958635i \(0.408127\pi\)
\(462\) −17.6548 −0.821375
\(463\) −1.99436 −0.0926860 −0.0463430 0.998926i \(-0.514757\pi\)
−0.0463430 + 0.998926i \(0.514757\pi\)
\(464\) 1.29109 0.0599375
\(465\) 10.9831 0.509328
\(466\) −27.1677 −1.25852
\(467\) 41.2601 1.90929 0.954645 0.297747i \(-0.0962353\pi\)
0.954645 + 0.297747i \(0.0962353\pi\)
\(468\) −0.00876283 −0.000405062 0
\(469\) −22.1980 −1.02501
\(470\) −31.0913 −1.43414
\(471\) 33.7146 1.55349
\(472\) 0.358794 0.0165148
\(473\) 11.5255 0.529941
\(474\) 6.50849 0.298945
\(475\) −0.0181079 −0.000830849 0
\(476\) −10.7127 −0.491018
\(477\) 5.61736 0.257201
\(478\) 52.0795 2.38206
\(479\) 23.4921 1.07338 0.536691 0.843779i \(-0.319674\pi\)
0.536691 + 0.843779i \(0.319674\pi\)
\(480\) −25.9985 −1.18666
\(481\) −0.0282075 −0.00128615
\(482\) 54.9626 2.50348
\(483\) 9.38878 0.427204
\(484\) −23.2176 −1.05534
\(485\) 33.2580 1.51017
\(486\) −15.7205 −0.713097
\(487\) 6.09387 0.276139 0.138070 0.990423i \(-0.455910\pi\)
0.138070 + 0.990423i \(0.455910\pi\)
\(488\) 10.0289 0.453986
\(489\) −10.9119 −0.493453
\(490\) −28.8509 −1.30335
\(491\) −7.88983 −0.356063 −0.178032 0.984025i \(-0.556973\pi\)
−0.178032 + 0.984025i \(0.556973\pi\)
\(492\) 23.6567 1.06653
\(493\) −0.640292 −0.0288373
\(494\) 0.0247870 0.00111522
\(495\) −2.41098 −0.108366
\(496\) −7.34344 −0.329730
\(497\) 33.5939 1.50689
\(498\) −22.5465 −1.01033
\(499\) −0.289120 −0.0129428 −0.00647139 0.999979i \(-0.502060\pi\)
−0.00647139 + 0.999979i \(0.502060\pi\)
\(500\) 29.7190 1.32907
\(501\) 17.2950 0.772683
\(502\) 8.89865 0.397166
\(503\) −1.00000 −0.0445878
\(504\) 3.66284 0.163156
\(505\) −25.1578 −1.11951
\(506\) −5.59413 −0.248689
\(507\) 19.6387 0.872185
\(508\) 23.4791 1.04172
\(509\) 27.0889 1.20069 0.600346 0.799740i \(-0.295030\pi\)
0.600346 + 0.799740i \(0.295030\pi\)
\(510\) 8.15254 0.361001
\(511\) 55.0056 2.43331
\(512\) 23.7747 1.05070
\(513\) 14.0391 0.619843
\(514\) −36.0931 −1.59200
\(515\) −10.3903 −0.457850
\(516\) 30.7693 1.35454
\(517\) −9.69233 −0.426268
\(518\) 47.7255 2.09694
\(519\) 19.0074 0.834333
\(520\) −0.0145402 −0.000637631 0
\(521\) 21.1689 0.927424 0.463712 0.885986i \(-0.346517\pi\)
0.463712 + 0.885986i \(0.346517\pi\)
\(522\) 0.886150 0.0387857
\(523\) −0.868369 −0.0379711 −0.0189856 0.999820i \(-0.506044\pi\)
−0.0189856 + 0.999820i \(0.506044\pi\)
\(524\) −34.2923 −1.49807
\(525\) −0.0394333 −0.00172101
\(526\) 62.0006 2.70335
\(527\) 3.64183 0.158641
\(528\) −5.12462 −0.223021
\(529\) −20.0251 −0.870655
\(530\) 37.7287 1.63883
\(531\) −0.181896 −0.00789363
\(532\) −23.9244 −1.03725
\(533\) −0.0270926 −0.00117351
\(534\) −10.9992 −0.475980
\(535\) 18.0616 0.780869
\(536\) 8.72335 0.376791
\(537\) −1.86428 −0.0804495
\(538\) 61.7353 2.66160
\(539\) −8.99390 −0.387395
\(540\) −33.3349 −1.43451
\(541\) 14.8513 0.638508 0.319254 0.947669i \(-0.396568\pi\)
0.319254 + 0.947669i \(0.396568\pi\)
\(542\) −29.9619 −1.28697
\(543\) 25.6859 1.10229
\(544\) −8.62074 −0.369611
\(545\) 2.70349 0.115805
\(546\) 0.0539781 0.00231005
\(547\) −18.8256 −0.804925 −0.402463 0.915436i \(-0.631846\pi\)
−0.402463 + 0.915436i \(0.631846\pi\)
\(548\) 10.1897 0.435283
\(549\) −5.08430 −0.216993
\(550\) 0.0234956 0.00100186
\(551\) −1.42994 −0.0609175
\(552\) −3.68960 −0.157040
\(553\) 7.19437 0.305936
\(554\) −31.2160 −1.32624
\(555\) −20.7192 −0.879482
\(556\) 24.5149 1.03966
\(557\) 11.9201 0.505071 0.252535 0.967588i \(-0.418736\pi\)
0.252535 + 0.967588i \(0.418736\pi\)
\(558\) −5.04021 −0.213369
\(559\) −0.0352382 −0.00149042
\(560\) −18.1713 −0.767877
\(561\) 2.54145 0.107300
\(562\) −33.4392 −1.41055
\(563\) −22.7172 −0.957414 −0.478707 0.877975i \(-0.658894\pi\)
−0.478707 + 0.877975i \(0.658894\pi\)
\(564\) −25.8754 −1.08955
\(565\) −17.1698 −0.722337
\(566\) 22.5051 0.945960
\(567\) 22.8126 0.958039
\(568\) −13.2017 −0.553932
\(569\) −16.9403 −0.710176 −0.355088 0.934833i \(-0.615549\pi\)
−0.355088 + 0.934833i \(0.615549\pi\)
\(570\) 18.2068 0.762597
\(571\) 16.2940 0.681883 0.340941 0.940085i \(-0.389254\pi\)
0.340941 + 0.940085i \(0.389254\pi\)
\(572\) −0.0183473 −0.000767139 0
\(573\) −0.322758 −0.0134834
\(574\) 45.8392 1.91329
\(575\) −0.0124949 −0.000521073 0
\(576\) 8.69054 0.362106
\(577\) −44.3876 −1.84788 −0.923941 0.382535i \(-0.875051\pi\)
−0.923941 + 0.382535i \(0.875051\pi\)
\(578\) −33.9798 −1.41337
\(579\) 5.30753 0.220574
\(580\) 3.39529 0.140982
\(581\) −24.9226 −1.03396
\(582\) 48.5191 2.01118
\(583\) 11.7614 0.487109
\(584\) −21.6161 −0.894480
\(585\) 0.00737139 0.000304770 0
\(586\) −4.10513 −0.169582
\(587\) 33.6221 1.38773 0.693865 0.720105i \(-0.255906\pi\)
0.693865 + 0.720105i \(0.255906\pi\)
\(588\) −24.0108 −0.990189
\(589\) 8.13316 0.335121
\(590\) −1.22170 −0.0502964
\(591\) 17.4415 0.717449
\(592\) 13.8532 0.569362
\(593\) 14.3513 0.589336 0.294668 0.955600i \(-0.404791\pi\)
0.294668 + 0.955600i \(0.404791\pi\)
\(594\) −18.2162 −0.747419
\(595\) 9.01169 0.369443
\(596\) 60.2655 2.46857
\(597\) 7.00495 0.286694
\(598\) 0.0171036 0.000699418 0
\(599\) −4.26954 −0.174449 −0.0872243 0.996189i \(-0.527800\pi\)
−0.0872243 + 0.996189i \(0.527800\pi\)
\(600\) 0.0154965 0.000632641 0
\(601\) 18.2764 0.745510 0.372755 0.927930i \(-0.378413\pi\)
0.372755 + 0.927930i \(0.378413\pi\)
\(602\) 59.6210 2.42997
\(603\) −4.42244 −0.180096
\(604\) −5.91067 −0.240502
\(605\) 19.5309 0.794044
\(606\) −36.7020 −1.49091
\(607\) −29.2048 −1.18539 −0.592694 0.805428i \(-0.701936\pi\)
−0.592694 + 0.805428i \(0.701936\pi\)
\(608\) −19.2524 −0.780787
\(609\) −3.11395 −0.126184
\(610\) −34.1484 −1.38263
\(611\) 0.0296335 0.00119884
\(612\) −2.13427 −0.0862727
\(613\) −24.0622 −0.971861 −0.485931 0.873997i \(-0.661519\pi\)
−0.485931 + 0.873997i \(0.661519\pi\)
\(614\) −38.1669 −1.54029
\(615\) −19.9003 −0.802458
\(616\) 7.66912 0.308998
\(617\) −20.7734 −0.836307 −0.418154 0.908376i \(-0.637323\pi\)
−0.418154 + 0.908376i \(0.637323\pi\)
\(618\) −15.1580 −0.609746
\(619\) −6.43218 −0.258531 −0.129266 0.991610i \(-0.541262\pi\)
−0.129266 + 0.991610i \(0.541262\pi\)
\(620\) −19.3116 −0.775574
\(621\) 9.68732 0.388739
\(622\) −60.4351 −2.42323
\(623\) −12.1583 −0.487112
\(624\) 0.0156681 0.000627227 0
\(625\) −24.9637 −0.998549
\(626\) 10.8350 0.433052
\(627\) 5.67573 0.226667
\(628\) −59.2807 −2.36556
\(629\) −6.87020 −0.273933
\(630\) −12.4720 −0.496895
\(631\) −43.2199 −1.72056 −0.860279 0.509824i \(-0.829710\pi\)
−0.860279 + 0.509824i \(0.829710\pi\)
\(632\) −2.82724 −0.112462
\(633\) 0.940258 0.0373719
\(634\) 0.770288 0.0305920
\(635\) −19.7509 −0.783791
\(636\) 31.3992 1.24506
\(637\) 0.0274981 0.00108952
\(638\) 1.85539 0.0734556
\(639\) 6.69281 0.264764
\(640\) 23.9497 0.946694
\(641\) −49.6276 −1.96017 −0.980086 0.198574i \(-0.936369\pi\)
−0.980086 + 0.198574i \(0.936369\pi\)
\(642\) 26.3495 1.03993
\(643\) −45.1377 −1.78006 −0.890028 0.455906i \(-0.849315\pi\)
−0.890028 + 0.455906i \(0.849315\pi\)
\(644\) −16.5084 −0.650521
\(645\) −25.8835 −1.01916
\(646\) 6.03711 0.237527
\(647\) −29.7930 −1.17128 −0.585642 0.810570i \(-0.699157\pi\)
−0.585642 + 0.810570i \(0.699157\pi\)
\(648\) −8.96488 −0.352174
\(649\) −0.380848 −0.0149496
\(650\) −7.18359e−5 0 −2.81764e−6 0
\(651\) 17.7114 0.694165
\(652\) 19.1865 0.751400
\(653\) 29.1127 1.13927 0.569635 0.821898i \(-0.307085\pi\)
0.569635 + 0.821898i \(0.307085\pi\)
\(654\) 3.94404 0.154224
\(655\) 28.8471 1.12715
\(656\) 13.3056 0.519498
\(657\) 10.9586 0.427536
\(658\) −50.1382 −1.95459
\(659\) 17.1327 0.667395 0.333697 0.942680i \(-0.391704\pi\)
0.333697 + 0.942680i \(0.391704\pi\)
\(660\) −13.4766 −0.524577
\(661\) −0.266846 −0.0103791 −0.00518956 0.999987i \(-0.501652\pi\)
−0.00518956 + 0.999987i \(0.501652\pi\)
\(662\) 35.0625 1.36274
\(663\) −0.00777029 −0.000301773 0
\(664\) 9.79405 0.380083
\(665\) 20.1255 0.780432
\(666\) 9.50820 0.368435
\(667\) −0.986691 −0.0382048
\(668\) −30.4099 −1.17660
\(669\) −0.0425237 −0.00164406
\(670\) −29.7030 −1.14753
\(671\) −10.6453 −0.410958
\(672\) −41.9255 −1.61731
\(673\) 41.4814 1.59899 0.799495 0.600673i \(-0.205101\pi\)
0.799495 + 0.600673i \(0.205101\pi\)
\(674\) −51.8838 −1.99849
\(675\) −0.0406872 −0.00156605
\(676\) −34.5309 −1.32811
\(677\) 33.0242 1.26922 0.634611 0.772832i \(-0.281160\pi\)
0.634611 + 0.772832i \(0.281160\pi\)
\(678\) −25.0485 −0.961980
\(679\) 53.6322 2.05822
\(680\) −3.54141 −0.135807
\(681\) 12.2793 0.470543
\(682\) −10.5530 −0.404096
\(683\) 0.471704 0.0180492 0.00902462 0.999959i \(-0.497127\pi\)
0.00902462 + 0.999959i \(0.497127\pi\)
\(684\) −4.76638 −0.182247
\(685\) −8.57170 −0.327508
\(686\) 7.90179 0.301692
\(687\) −36.2553 −1.38323
\(688\) 17.3061 0.659788
\(689\) −0.0359597 −0.00136995
\(690\) 12.5631 0.478269
\(691\) −12.1972 −0.464003 −0.232002 0.972715i \(-0.574527\pi\)
−0.232002 + 0.972715i \(0.574527\pi\)
\(692\) −33.4209 −1.27047
\(693\) −3.88798 −0.147692
\(694\) −23.2745 −0.883489
\(695\) −20.6222 −0.782244
\(696\) 1.22372 0.0463849
\(697\) −6.59866 −0.249942
\(698\) 37.1242 1.40517
\(699\) 19.0198 0.719396
\(700\) 0.0693359 0.00262065
\(701\) −47.0220 −1.77600 −0.887999 0.459846i \(-0.847905\pi\)
−0.887999 + 0.459846i \(0.847905\pi\)
\(702\) 0.0556946 0.00210206
\(703\) −15.3430 −0.578671
\(704\) 18.1960 0.685786
\(705\) 21.7667 0.819781
\(706\) −6.11261 −0.230051
\(707\) −40.5697 −1.52578
\(708\) −1.01674 −0.0382115
\(709\) 26.6796 1.00197 0.500986 0.865455i \(-0.332971\pi\)
0.500986 + 0.865455i \(0.332971\pi\)
\(710\) 44.9519 1.68701
\(711\) 1.43331 0.0537535
\(712\) 4.77796 0.179062
\(713\) 5.61207 0.210174
\(714\) 13.1469 0.492010
\(715\) 0.0154340 0.000577198 0
\(716\) 3.27797 0.122504
\(717\) −36.4602 −1.36163
\(718\) 50.4412 1.88245
\(719\) 12.7457 0.475336 0.237668 0.971346i \(-0.423617\pi\)
0.237668 + 0.971346i \(0.423617\pi\)
\(720\) −3.62021 −0.134917
\(721\) −16.7555 −0.624006
\(722\) −27.5163 −1.02405
\(723\) −38.4787 −1.43104
\(724\) −45.1637 −1.67850
\(725\) 0.00414415 0.000153910 0
\(726\) 28.4930 1.05748
\(727\) 1.14219 0.0423614 0.0211807 0.999776i \(-0.493257\pi\)
0.0211807 + 0.999776i \(0.493257\pi\)
\(728\) −0.0234477 −0.000869031 0
\(729\) 29.9989 1.11107
\(730\) 73.6028 2.72416
\(731\) −8.58259 −0.317439
\(732\) −28.4196 −1.05042
\(733\) −20.6111 −0.761289 −0.380644 0.924721i \(-0.624298\pi\)
−0.380644 + 0.924721i \(0.624298\pi\)
\(734\) −47.1282 −1.73953
\(735\) 20.1982 0.745021
\(736\) −13.2846 −0.489676
\(737\) −9.25954 −0.341080
\(738\) 9.13240 0.336168
\(739\) −40.1054 −1.47530 −0.737651 0.675182i \(-0.764065\pi\)
−0.737651 + 0.675182i \(0.764065\pi\)
\(740\) 36.4308 1.33922
\(741\) −0.0173531 −0.000637482 0
\(742\) 60.8417 2.23357
\(743\) −31.7658 −1.16538 −0.582688 0.812696i \(-0.697999\pi\)
−0.582688 + 0.812696i \(0.697999\pi\)
\(744\) −6.96023 −0.255174
\(745\) −50.6961 −1.85736
\(746\) 1.28238 0.0469514
\(747\) −4.96525 −0.181669
\(748\) −4.46866 −0.163390
\(749\) 29.1263 1.06425
\(750\) −36.4717 −1.33176
\(751\) 3.19661 0.116646 0.0583230 0.998298i \(-0.481425\pi\)
0.0583230 + 0.998298i \(0.481425\pi\)
\(752\) −14.5535 −0.530713
\(753\) −6.22984 −0.227028
\(754\) −0.00567271 −0.000206588 0
\(755\) 4.97212 0.180954
\(756\) −53.7563 −1.95510
\(757\) 45.6783 1.66021 0.830103 0.557610i \(-0.188281\pi\)
0.830103 + 0.557610i \(0.188281\pi\)
\(758\) −34.6614 −1.25896
\(759\) 3.91638 0.142156
\(760\) −7.90889 −0.286886
\(761\) 37.7902 1.36989 0.684946 0.728594i \(-0.259826\pi\)
0.684946 + 0.728594i \(0.259826\pi\)
\(762\) −28.8140 −1.04382
\(763\) 4.35968 0.157831
\(764\) 0.567508 0.0205317
\(765\) 1.79537 0.0649118
\(766\) 32.5282 1.17529
\(767\) 0.00116441 4.20445e−5 0
\(768\) −1.63674 −0.0590608
\(769\) 11.6770 0.421084 0.210542 0.977585i \(-0.432477\pi\)
0.210542 + 0.977585i \(0.432477\pi\)
\(770\) −26.1134 −0.941061
\(771\) 25.2683 0.910017
\(772\) −9.33228 −0.335876
\(773\) −43.2250 −1.55470 −0.777348 0.629070i \(-0.783436\pi\)
−0.777348 + 0.629070i \(0.783436\pi\)
\(774\) 11.8781 0.426950
\(775\) −0.0235710 −0.000846694 0
\(776\) −21.0764 −0.756597
\(777\) −33.4120 −1.19865
\(778\) 17.4092 0.624150
\(779\) −14.7365 −0.527992
\(780\) 0.0412037 0.00147533
\(781\) 14.0132 0.501431
\(782\) 4.16574 0.148967
\(783\) −3.21297 −0.114822
\(784\) −13.5048 −0.482314
\(785\) 49.8676 1.77985
\(786\) 42.0842 1.50109
\(787\) 41.6501 1.48467 0.742334 0.670030i \(-0.233719\pi\)
0.742334 + 0.670030i \(0.233719\pi\)
\(788\) −30.6676 −1.09249
\(789\) −43.4059 −1.54529
\(790\) 9.62676 0.342505
\(791\) −27.6881 −0.984477
\(792\) 1.52790 0.0542914
\(793\) 0.0325473 0.00115579
\(794\) 16.4920 0.585278
\(795\) −26.4134 −0.936787
\(796\) −12.3169 −0.436560
\(797\) 20.0955 0.711819 0.355910 0.934520i \(-0.384171\pi\)
0.355910 + 0.934520i \(0.384171\pi\)
\(798\) 29.3604 1.03935
\(799\) 7.21753 0.255338
\(800\) 0.0557958 0.00197268
\(801\) −2.42226 −0.0855864
\(802\) 79.9596 2.82347
\(803\) 22.9448 0.809703
\(804\) −24.7200 −0.871807
\(805\) 13.8870 0.489453
\(806\) 0.0322650 0.00113649
\(807\) −43.2202 −1.52142
\(808\) 15.9431 0.560875
\(809\) −10.5903 −0.372334 −0.186167 0.982518i \(-0.559607\pi\)
−0.186167 + 0.982518i \(0.559607\pi\)
\(810\) 30.5254 1.07255
\(811\) 29.9332 1.05110 0.525549 0.850764i \(-0.323860\pi\)
0.525549 + 0.850764i \(0.323860\pi\)
\(812\) 5.47528 0.192145
\(813\) 20.9760 0.735660
\(814\) 19.9079 0.697773
\(815\) −16.1399 −0.565355
\(816\) 3.81612 0.133591
\(817\) −19.1672 −0.670575
\(818\) −62.9041 −2.19939
\(819\) 0.0118872 0.000415372 0
\(820\) 34.9909 1.22193
\(821\) 1.47970 0.0516419 0.0258209 0.999667i \(-0.491780\pi\)
0.0258209 + 0.999667i \(0.491780\pi\)
\(822\) −12.5050 −0.436162
\(823\) −20.9277 −0.729494 −0.364747 0.931107i \(-0.618845\pi\)
−0.364747 + 0.931107i \(0.618845\pi\)
\(824\) 6.58455 0.229384
\(825\) −0.0164490 −0.000572680 0
\(826\) −1.97012 −0.0685492
\(827\) 14.8863 0.517646 0.258823 0.965925i \(-0.416665\pi\)
0.258823 + 0.965925i \(0.416665\pi\)
\(828\) −3.28891 −0.114298
\(829\) 54.9732 1.90930 0.954649 0.297733i \(-0.0962304\pi\)
0.954649 + 0.297733i \(0.0962304\pi\)
\(830\) −33.3488 −1.15755
\(831\) 21.8539 0.758104
\(832\) −0.0556327 −0.00192872
\(833\) 6.69743 0.232052
\(834\) −30.0851 −1.04176
\(835\) 25.5812 0.885273
\(836\) −9.97968 −0.345154
\(837\) 18.2746 0.631663
\(838\) 68.7678 2.37554
\(839\) −40.8460 −1.41016 −0.705080 0.709128i \(-0.749089\pi\)
−0.705080 + 0.709128i \(0.749089\pi\)
\(840\) −17.2230 −0.594251
\(841\) −28.6727 −0.988715
\(842\) −56.2026 −1.93687
\(843\) 23.4104 0.806298
\(844\) −1.65326 −0.0569077
\(845\) 29.0478 0.999274
\(846\) −9.98889 −0.343425
\(847\) 31.4957 1.08221
\(848\) 17.6604 0.606460
\(849\) −15.7556 −0.540729
\(850\) −0.0174963 −0.000600118 0
\(851\) −10.5870 −0.362918
\(852\) 37.4107 1.28167
\(853\) −8.90994 −0.305071 −0.152535 0.988298i \(-0.548744\pi\)
−0.152535 + 0.988298i \(0.548744\pi\)
\(854\) −55.0681 −1.88439
\(855\) 4.00953 0.137123
\(856\) −11.4460 −0.391217
\(857\) 32.1479 1.09815 0.549076 0.835773i \(-0.314980\pi\)
0.549076 + 0.835773i \(0.314980\pi\)
\(858\) 0.0225162 0.000768689 0
\(859\) −25.4577 −0.868606 −0.434303 0.900767i \(-0.643005\pi\)
−0.434303 + 0.900767i \(0.643005\pi\)
\(860\) 45.5111 1.55192
\(861\) −32.0915 −1.09367
\(862\) −48.8092 −1.66245
\(863\) −42.3500 −1.44161 −0.720805 0.693138i \(-0.756228\pi\)
−0.720805 + 0.693138i \(0.756228\pi\)
\(864\) −43.2586 −1.47169
\(865\) 28.1140 0.955906
\(866\) −49.0414 −1.66649
\(867\) 23.7889 0.807913
\(868\) −31.1421 −1.05703
\(869\) 3.00102 0.101803
\(870\) −4.16676 −0.141267
\(871\) 0.0283103 0.000959259 0
\(872\) −1.71326 −0.0580184
\(873\) 10.6850 0.361632
\(874\) 9.30319 0.314685
\(875\) −40.3152 −1.36290
\(876\) 61.2551 2.06962
\(877\) −54.7498 −1.84877 −0.924385 0.381461i \(-0.875421\pi\)
−0.924385 + 0.381461i \(0.875421\pi\)
\(878\) 53.0688 1.79099
\(879\) 2.87396 0.0969361
\(880\) −7.57988 −0.255518
\(881\) −42.7752 −1.44113 −0.720566 0.693386i \(-0.756118\pi\)
−0.720566 + 0.693386i \(0.756118\pi\)
\(882\) −9.26909 −0.312107
\(883\) −51.0453 −1.71781 −0.858906 0.512133i \(-0.828856\pi\)
−0.858906 + 0.512133i \(0.828856\pi\)
\(884\) 0.0136626 0.000459522 0
\(885\) 0.855295 0.0287504
\(886\) 50.6584 1.70190
\(887\) −15.6713 −0.526191 −0.263096 0.964770i \(-0.584744\pi\)
−0.263096 + 0.964770i \(0.584744\pi\)
\(888\) 13.1302 0.440622
\(889\) −31.8505 −1.06823
\(890\) −16.2690 −0.545337
\(891\) 9.51592 0.318795
\(892\) 0.0747698 0.00250348
\(893\) 16.1186 0.539389
\(894\) −73.9590 −2.47356
\(895\) −2.75747 −0.0921720
\(896\) 38.6215 1.29025
\(897\) −0.0119740 −0.000399801 0
\(898\) −29.4441 −0.982561
\(899\) −1.86134 −0.0620792
\(900\) 0.0138136 0.000460453 0
\(901\) −8.75831 −0.291782
\(902\) 19.1211 0.636663
\(903\) −41.7400 −1.38902
\(904\) 10.8809 0.361892
\(905\) 37.9923 1.26291
\(906\) 7.25368 0.240988
\(907\) 50.8029 1.68688 0.843442 0.537221i \(-0.180526\pi\)
0.843442 + 0.537221i \(0.180526\pi\)
\(908\) −21.5908 −0.716515
\(909\) −8.08259 −0.268083
\(910\) 0.0798396 0.00264666
\(911\) −47.4669 −1.57265 −0.786324 0.617815i \(-0.788018\pi\)
−0.786324 + 0.617815i \(0.788018\pi\)
\(912\) 8.52239 0.282205
\(913\) −10.3961 −0.344059
\(914\) −4.70335 −0.155573
\(915\) 23.9069 0.790337
\(916\) 63.7481 2.10630
\(917\) 46.5192 1.53620
\(918\) 13.5649 0.447709
\(919\) 18.1424 0.598462 0.299231 0.954181i \(-0.403270\pi\)
0.299231 + 0.954181i \(0.403270\pi\)
\(920\) −5.45732 −0.179922
\(921\) 26.7202 0.880461
\(922\) 26.3749 0.868610
\(923\) −0.0428442 −0.00141023
\(924\) −21.7325 −0.714948
\(925\) 0.0444659 0.00146203
\(926\) −4.30350 −0.141422
\(927\) −3.33814 −0.109639
\(928\) 4.40606 0.144636
\(929\) −36.7632 −1.20616 −0.603081 0.797680i \(-0.706060\pi\)
−0.603081 + 0.797680i \(0.706060\pi\)
\(930\) 23.6996 0.777140
\(931\) 14.9571 0.490200
\(932\) −33.4427 −1.09545
\(933\) 42.3099 1.38516
\(934\) 89.0322 2.91322
\(935\) 3.75909 0.122935
\(936\) −0.00467142 −0.000152690 0
\(937\) −29.1597 −0.952607 −0.476304 0.879281i \(-0.658024\pi\)
−0.476304 + 0.879281i \(0.658024\pi\)
\(938\) −47.8994 −1.56397
\(939\) −7.58543 −0.247541
\(940\) −38.2725 −1.24831
\(941\) −11.1041 −0.361983 −0.180991 0.983485i \(-0.557931\pi\)
−0.180991 + 0.983485i \(0.557931\pi\)
\(942\) 72.7504 2.37034
\(943\) −10.1686 −0.331134
\(944\) −0.571863 −0.0186125
\(945\) 45.2204 1.47102
\(946\) 24.8700 0.808593
\(947\) 7.44473 0.241921 0.120961 0.992657i \(-0.461403\pi\)
0.120961 + 0.992657i \(0.461403\pi\)
\(948\) 8.01176 0.260210
\(949\) −0.0701518 −0.00227722
\(950\) −0.0390738 −0.00126772
\(951\) −0.539270 −0.0174870
\(952\) −5.71091 −0.185092
\(953\) −40.8632 −1.32369 −0.661844 0.749641i \(-0.730226\pi\)
−0.661844 + 0.749641i \(0.730226\pi\)
\(954\) 12.1213 0.392442
\(955\) −0.477394 −0.0154481
\(956\) 64.1083 2.07341
\(957\) −1.29894 −0.0419886
\(958\) 50.6919 1.63778
\(959\) −13.8228 −0.446362
\(960\) −40.8638 −1.31887
\(961\) −20.4131 −0.658488
\(962\) −0.0608670 −0.00196243
\(963\) 5.80274 0.186991
\(964\) 67.6574 2.17910
\(965\) 7.85042 0.252714
\(966\) 20.2594 0.651835
\(967\) 2.29088 0.0736697 0.0368349 0.999321i \(-0.488272\pi\)
0.0368349 + 0.999321i \(0.488272\pi\)
\(968\) −12.3772 −0.397817
\(969\) −4.22651 −0.135775
\(970\) 71.7651 2.30424
\(971\) −27.8481 −0.893689 −0.446845 0.894612i \(-0.647452\pi\)
−0.446845 + 0.894612i \(0.647452\pi\)
\(972\) −19.3515 −0.620700
\(973\) −33.2556 −1.06612
\(974\) 13.1495 0.421338
\(975\) 5.02915e−5 0 1.61062e−6 0
\(976\) −15.9845 −0.511651
\(977\) 2.65190 0.0848419 0.0424209 0.999100i \(-0.486493\pi\)
0.0424209 + 0.999100i \(0.486493\pi\)
\(978\) −23.5460 −0.752917
\(979\) −5.07165 −0.162090
\(980\) −35.5146 −1.13447
\(981\) 0.868565 0.0277312
\(982\) −17.0249 −0.543287
\(983\) 17.5940 0.561162 0.280581 0.959830i \(-0.409473\pi\)
0.280581 + 0.959830i \(0.409473\pi\)
\(984\) 12.6113 0.402033
\(985\) 25.7980 0.821991
\(986\) −1.38164 −0.0440004
\(987\) 35.1012 1.11728
\(988\) 0.0305121 0.000970719 0
\(989\) −13.2258 −0.420556
\(990\) −5.20249 −0.165346
\(991\) −40.1464 −1.27529 −0.637646 0.770329i \(-0.720092\pi\)
−0.637646 + 0.770329i \(0.720092\pi\)
\(992\) −25.0606 −0.795676
\(993\) −24.5468 −0.778970
\(994\) 72.4899 2.29924
\(995\) 10.3611 0.328469
\(996\) −27.7541 −0.879423
\(997\) −52.3474 −1.65786 −0.828930 0.559353i \(-0.811050\pi\)
−0.828930 + 0.559353i \(0.811050\pi\)
\(998\) −0.623871 −0.0197483
\(999\) −34.4745 −1.09072
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 503.2.a.e.1.10 10
3.2 odd 2 4527.2.a.k.1.1 10
4.3 odd 2 8048.2.a.p.1.7 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.e.1.10 10 1.1 even 1 trivial
4527.2.a.k.1.1 10 3.2 odd 2
8048.2.a.p.1.7 10 4.3 odd 2