Properties

Label 503.2.a.e.1.6
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 1.6
Root \(1.95007\) of defining polynomial
Character \(\chi\) \(=\) 503.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.392284 q^{2} +0.950069 q^{3} -1.84611 q^{4} -2.28693 q^{5} -0.372697 q^{6} +2.71022 q^{7} +1.50877 q^{8} -2.09737 q^{9} +O(q^{10})\) \(q-0.392284 q^{2} +0.950069 q^{3} -1.84611 q^{4} -2.28693 q^{5} -0.372697 q^{6} +2.71022 q^{7} +1.50877 q^{8} -2.09737 q^{9} +0.897127 q^{10} +1.36880 q^{11} -1.75393 q^{12} -2.93079 q^{13} -1.06318 q^{14} -2.17275 q^{15} +3.10036 q^{16} -2.61287 q^{17} +0.822764 q^{18} -7.79104 q^{19} +4.22194 q^{20} +2.57490 q^{21} -0.536960 q^{22} -2.61064 q^{23} +1.43343 q^{24} +0.230071 q^{25} +1.14970 q^{26} -4.84285 q^{27} -5.00337 q^{28} -0.314480 q^{29} +0.852333 q^{30} -7.95126 q^{31} -4.23376 q^{32} +1.30046 q^{33} +1.02499 q^{34} -6.19810 q^{35} +3.87198 q^{36} -4.17299 q^{37} +3.05630 q^{38} -2.78446 q^{39} -3.45045 q^{40} +6.16482 q^{41} -1.01009 q^{42} +0.457851 q^{43} -2.52697 q^{44} +4.79655 q^{45} +1.02411 q^{46} +7.67118 q^{47} +2.94556 q^{48} +0.345296 q^{49} -0.0902533 q^{50} -2.48241 q^{51} +5.41058 q^{52} +7.26306 q^{53} +1.89977 q^{54} -3.13037 q^{55} +4.08909 q^{56} -7.40203 q^{57} +0.123365 q^{58} +0.217166 q^{59} +4.01114 q^{60} +7.26694 q^{61} +3.11915 q^{62} -5.68433 q^{63} -4.53989 q^{64} +6.70253 q^{65} -0.510149 q^{66} +10.1022 q^{67} +4.82366 q^{68} -2.48029 q^{69} +2.43141 q^{70} -16.5088 q^{71} -3.16444 q^{72} +2.86838 q^{73} +1.63699 q^{74} +0.218584 q^{75} +14.3831 q^{76} +3.70976 q^{77} +1.09230 q^{78} -16.0222 q^{79} -7.09033 q^{80} +1.69106 q^{81} -2.41836 q^{82} +12.5312 q^{83} -4.75355 q^{84} +5.97547 q^{85} -0.179608 q^{86} -0.298777 q^{87} +2.06521 q^{88} -0.0952234 q^{89} -1.88161 q^{90} -7.94310 q^{91} +4.81954 q^{92} -7.55424 q^{93} -3.00928 q^{94} +17.8176 q^{95} -4.02236 q^{96} -9.27640 q^{97} -0.135454 q^{98} -2.87089 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} + O(q^{10}) \) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} - 4q^{10} - 3q^{11} - 7q^{12} - 18q^{13} + q^{14} - 2q^{15} - 4q^{16} - 11q^{17} - q^{18} - 3q^{20} + q^{21} - 18q^{22} - 2q^{23} + 10q^{24} - 27q^{25} + 11q^{26} - 2q^{27} - 22q^{28} - 9q^{29} + 12q^{30} - 22q^{31} - 10q^{32} - 10q^{33} - 10q^{34} - 6q^{35} + 2q^{36} - 35q^{37} + 2q^{38} + 8q^{39} - 19q^{40} - 4q^{41} + 4q^{42} - 20q^{43} + 9q^{44} + 2q^{45} - q^{46} + 7q^{47} - 27q^{49} + 16q^{50} + 9q^{51} - 7q^{52} - 24q^{53} + 17q^{54} - 11q^{55} + 12q^{56} - 23q^{57} + 2q^{58} + 17q^{59} - 4q^{61} + 8q^{62} + 10q^{63} + 3q^{64} - 16q^{65} + 46q^{66} - 6q^{67} + 28q^{68} - 2q^{69} + 26q^{70} - q^{71} - q^{72} - 31q^{73} + 11q^{74} + 30q^{75} + 20q^{76} + 3q^{77} + 11q^{78} - 10q^{79} + 24q^{80} - 6q^{81} - 9q^{82} + 22q^{83} + 22q^{84} - 6q^{85} + 38q^{86} + 25q^{87} - 3q^{88} + q^{89} + 2q^{90} + 10q^{91} + 27q^{92} - 6q^{93} + 33q^{94} + 39q^{95} + 46q^{96} - 57q^{97} + 40q^{98} + 35q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.392284 −0.277386 −0.138693 0.990335i \(-0.544290\pi\)
−0.138693 + 0.990335i \(0.544290\pi\)
\(3\) 0.950069 0.548523 0.274261 0.961655i \(-0.411567\pi\)
0.274261 + 0.961655i \(0.411567\pi\)
\(4\) −1.84611 −0.923057
\(5\) −2.28693 −1.02275 −0.511374 0.859358i \(-0.670863\pi\)
−0.511374 + 0.859358i \(0.670863\pi\)
\(6\) −0.372697 −0.152153
\(7\) 2.71022 1.02437 0.512184 0.858876i \(-0.328837\pi\)
0.512184 + 0.858876i \(0.328837\pi\)
\(8\) 1.50877 0.533430
\(9\) −2.09737 −0.699123
\(10\) 0.897127 0.283697
\(11\) 1.36880 0.412710 0.206355 0.978477i \(-0.433840\pi\)
0.206355 + 0.978477i \(0.433840\pi\)
\(12\) −1.75393 −0.506317
\(13\) −2.93079 −0.812856 −0.406428 0.913683i \(-0.633226\pi\)
−0.406428 + 0.913683i \(0.633226\pi\)
\(14\) −1.06318 −0.284146
\(15\) −2.17275 −0.561001
\(16\) 3.10036 0.775091
\(17\) −2.61287 −0.633715 −0.316857 0.948473i \(-0.602628\pi\)
−0.316857 + 0.948473i \(0.602628\pi\)
\(18\) 0.822764 0.193927
\(19\) −7.79104 −1.78739 −0.893694 0.448677i \(-0.851895\pi\)
−0.893694 + 0.448677i \(0.851895\pi\)
\(20\) 4.22194 0.944055
\(21\) 2.57490 0.561888
\(22\) −0.536960 −0.114480
\(23\) −2.61064 −0.544356 −0.272178 0.962247i \(-0.587744\pi\)
−0.272178 + 0.962247i \(0.587744\pi\)
\(24\) 1.43343 0.292598
\(25\) 0.230071 0.0460143
\(26\) 1.14970 0.225475
\(27\) −4.84285 −0.932007
\(28\) −5.00337 −0.945549
\(29\) −0.314480 −0.0583974 −0.0291987 0.999574i \(-0.509296\pi\)
−0.0291987 + 0.999574i \(0.509296\pi\)
\(30\) 0.852333 0.155614
\(31\) −7.95126 −1.42809 −0.714044 0.700101i \(-0.753138\pi\)
−0.714044 + 0.700101i \(0.753138\pi\)
\(32\) −4.23376 −0.748430
\(33\) 1.30046 0.226381
\(34\) 1.02499 0.175784
\(35\) −6.19810 −1.04767
\(36\) 3.87198 0.645330
\(37\) −4.17299 −0.686035 −0.343017 0.939329i \(-0.611449\pi\)
−0.343017 + 0.939329i \(0.611449\pi\)
\(38\) 3.05630 0.495797
\(39\) −2.78446 −0.445870
\(40\) −3.45045 −0.545565
\(41\) 6.16482 0.962783 0.481391 0.876506i \(-0.340132\pi\)
0.481391 + 0.876506i \(0.340132\pi\)
\(42\) −1.01009 −0.155860
\(43\) 0.457851 0.0698216 0.0349108 0.999390i \(-0.488885\pi\)
0.0349108 + 0.999390i \(0.488885\pi\)
\(44\) −2.52697 −0.380955
\(45\) 4.79655 0.715027
\(46\) 1.02411 0.150997
\(47\) 7.67118 1.11896 0.559478 0.828845i \(-0.311002\pi\)
0.559478 + 0.828845i \(0.311002\pi\)
\(48\) 2.94556 0.425155
\(49\) 0.345296 0.0493279
\(50\) −0.0902533 −0.0127637
\(51\) −2.48241 −0.347607
\(52\) 5.41058 0.750312
\(53\) 7.26306 0.997658 0.498829 0.866700i \(-0.333763\pi\)
0.498829 + 0.866700i \(0.333763\pi\)
\(54\) 1.89977 0.258526
\(55\) −3.13037 −0.422099
\(56\) 4.08909 0.546428
\(57\) −7.40203 −0.980423
\(58\) 0.123365 0.0161987
\(59\) 0.217166 0.0282726 0.0141363 0.999900i \(-0.495500\pi\)
0.0141363 + 0.999900i \(0.495500\pi\)
\(60\) 4.01114 0.517835
\(61\) 7.26694 0.930436 0.465218 0.885196i \(-0.345976\pi\)
0.465218 + 0.885196i \(0.345976\pi\)
\(62\) 3.11915 0.396132
\(63\) −5.68433 −0.716159
\(64\) −4.53989 −0.567486
\(65\) 6.70253 0.831347
\(66\) −0.510149 −0.0627950
\(67\) 10.1022 1.23418 0.617091 0.786892i \(-0.288311\pi\)
0.617091 + 0.786892i \(0.288311\pi\)
\(68\) 4.82366 0.584955
\(69\) −2.48029 −0.298592
\(70\) 2.43141 0.290609
\(71\) −16.5088 −1.95924 −0.979619 0.200863i \(-0.935625\pi\)
−0.979619 + 0.200863i \(0.935625\pi\)
\(72\) −3.16444 −0.372933
\(73\) 2.86838 0.335718 0.167859 0.985811i \(-0.446315\pi\)
0.167859 + 0.985811i \(0.446315\pi\)
\(74\) 1.63699 0.190297
\(75\) 0.218584 0.0252399
\(76\) 14.3831 1.64986
\(77\) 3.70976 0.422767
\(78\) 1.09230 0.123678
\(79\) −16.0222 −1.80264 −0.901320 0.433153i \(-0.857401\pi\)
−0.901320 + 0.433153i \(0.857401\pi\)
\(80\) −7.09033 −0.792723
\(81\) 1.69106 0.187896
\(82\) −2.41836 −0.267063
\(83\) 12.5312 1.37548 0.687741 0.725956i \(-0.258602\pi\)
0.687741 + 0.725956i \(0.258602\pi\)
\(84\) −4.75355 −0.518655
\(85\) 5.97547 0.648131
\(86\) −0.179608 −0.0193676
\(87\) −0.298777 −0.0320323
\(88\) 2.06521 0.220152
\(89\) −0.0952234 −0.0100937 −0.00504683 0.999987i \(-0.501606\pi\)
−0.00504683 + 0.999987i \(0.501606\pi\)
\(90\) −1.88161 −0.198339
\(91\) −7.94310 −0.832663
\(92\) 4.81954 0.502472
\(93\) −7.55424 −0.783339
\(94\) −3.00928 −0.310383
\(95\) 17.8176 1.82805
\(96\) −4.02236 −0.410530
\(97\) −9.27640 −0.941875 −0.470938 0.882167i \(-0.656084\pi\)
−0.470938 + 0.882167i \(0.656084\pi\)
\(98\) −0.135454 −0.0136829
\(99\) −2.87089 −0.288535
\(100\) −0.424738 −0.0424738
\(101\) 8.77727 0.873371 0.436686 0.899614i \(-0.356152\pi\)
0.436686 + 0.899614i \(0.356152\pi\)
\(102\) 0.973809 0.0964214
\(103\) 1.52475 0.150238 0.0751189 0.997175i \(-0.476066\pi\)
0.0751189 + 0.997175i \(0.476066\pi\)
\(104\) −4.42189 −0.433602
\(105\) −5.88862 −0.574670
\(106\) −2.84918 −0.276737
\(107\) −17.6358 −1.70492 −0.852460 0.522792i \(-0.824890\pi\)
−0.852460 + 0.522792i \(0.824890\pi\)
\(108\) 8.94045 0.860296
\(109\) −4.91427 −0.470702 −0.235351 0.971910i \(-0.575624\pi\)
−0.235351 + 0.971910i \(0.575624\pi\)
\(110\) 1.22799 0.117084
\(111\) −3.96462 −0.376305
\(112\) 8.40266 0.793977
\(113\) 0.102880 0.00967811 0.00483905 0.999988i \(-0.498460\pi\)
0.00483905 + 0.999988i \(0.498460\pi\)
\(114\) 2.90369 0.271956
\(115\) 5.97037 0.556740
\(116\) 0.580565 0.0539041
\(117\) 6.14696 0.568286
\(118\) −0.0851906 −0.00784243
\(119\) −7.08146 −0.649157
\(120\) −3.27817 −0.299254
\(121\) −9.12637 −0.829670
\(122\) −2.85070 −0.258090
\(123\) 5.85700 0.528108
\(124\) 14.6789 1.31821
\(125\) 10.9085 0.975687
\(126\) 2.22987 0.198653
\(127\) 14.7682 1.31046 0.655232 0.755428i \(-0.272571\pi\)
0.655232 + 0.755428i \(0.272571\pi\)
\(128\) 10.2484 0.905843
\(129\) 0.434990 0.0382987
\(130\) −2.62929 −0.230604
\(131\) 3.01646 0.263549 0.131775 0.991280i \(-0.457932\pi\)
0.131775 + 0.991280i \(0.457932\pi\)
\(132\) −2.40079 −0.208962
\(133\) −21.1154 −1.83094
\(134\) −3.96293 −0.342345
\(135\) 11.0753 0.953209
\(136\) −3.94222 −0.338042
\(137\) 0.370190 0.0316275 0.0158137 0.999875i \(-0.494966\pi\)
0.0158137 + 0.999875i \(0.494966\pi\)
\(138\) 0.972977 0.0828253
\(139\) 13.0944 1.11065 0.555324 0.831634i \(-0.312594\pi\)
0.555324 + 0.831634i \(0.312594\pi\)
\(140\) 11.4424 0.967059
\(141\) 7.28815 0.613773
\(142\) 6.47615 0.543466
\(143\) −4.01168 −0.335474
\(144\) −6.50260 −0.541884
\(145\) 0.719195 0.0597259
\(146\) −1.12522 −0.0931237
\(147\) 0.328055 0.0270575
\(148\) 7.70381 0.633249
\(149\) −5.61923 −0.460345 −0.230173 0.973150i \(-0.573929\pi\)
−0.230173 + 0.973150i \(0.573929\pi\)
\(150\) −0.0857468 −0.00700120
\(151\) −6.99662 −0.569377 −0.284689 0.958620i \(-0.591890\pi\)
−0.284689 + 0.958620i \(0.591890\pi\)
\(152\) −11.7549 −0.953446
\(153\) 5.48016 0.443045
\(154\) −1.45528 −0.117270
\(155\) 18.1840 1.46058
\(156\) 5.14042 0.411563
\(157\) 10.5849 0.844764 0.422382 0.906418i \(-0.361194\pi\)
0.422382 + 0.906418i \(0.361194\pi\)
\(158\) 6.28526 0.500028
\(159\) 6.90041 0.547238
\(160\) 9.68233 0.765455
\(161\) −7.07541 −0.557621
\(162\) −0.663377 −0.0521198
\(163\) −4.74396 −0.371576 −0.185788 0.982590i \(-0.559484\pi\)
−0.185788 + 0.982590i \(0.559484\pi\)
\(164\) −11.3810 −0.888703
\(165\) −2.97406 −0.231531
\(166\) −4.91580 −0.381540
\(167\) 15.4650 1.19672 0.598360 0.801227i \(-0.295819\pi\)
0.598360 + 0.801227i \(0.295819\pi\)
\(168\) 3.88492 0.299728
\(169\) −4.41045 −0.339265
\(170\) −2.34408 −0.179783
\(171\) 16.3407 1.24960
\(172\) −0.845245 −0.0644493
\(173\) −18.0030 −1.36874 −0.684372 0.729133i \(-0.739924\pi\)
−0.684372 + 0.729133i \(0.739924\pi\)
\(174\) 0.117206 0.00888533
\(175\) 0.623544 0.0471355
\(176\) 4.24379 0.319888
\(177\) 0.206322 0.0155081
\(178\) 0.0373546 0.00279985
\(179\) 22.6390 1.69212 0.846058 0.533092i \(-0.178970\pi\)
0.846058 + 0.533092i \(0.178970\pi\)
\(180\) −8.85497 −0.660010
\(181\) 2.97984 0.221490 0.110745 0.993849i \(-0.464676\pi\)
0.110745 + 0.993849i \(0.464676\pi\)
\(182\) 3.11595 0.230969
\(183\) 6.90409 0.510365
\(184\) −3.93885 −0.290376
\(185\) 9.54335 0.701641
\(186\) 2.96341 0.217288
\(187\) −3.57651 −0.261541
\(188\) −14.1619 −1.03286
\(189\) −13.1252 −0.954718
\(190\) −6.98956 −0.507076
\(191\) 4.09564 0.296350 0.148175 0.988961i \(-0.452660\pi\)
0.148175 + 0.988961i \(0.452660\pi\)
\(192\) −4.31321 −0.311279
\(193\) −21.1708 −1.52391 −0.761955 0.647630i \(-0.775760\pi\)
−0.761955 + 0.647630i \(0.775760\pi\)
\(194\) 3.63898 0.261263
\(195\) 6.36787 0.456013
\(196\) −0.637455 −0.0455325
\(197\) 18.5139 1.31906 0.659531 0.751677i \(-0.270755\pi\)
0.659531 + 0.751677i \(0.270755\pi\)
\(198\) 1.12620 0.0800358
\(199\) −1.56711 −0.111089 −0.0555447 0.998456i \(-0.517690\pi\)
−0.0555447 + 0.998456i \(0.517690\pi\)
\(200\) 0.347124 0.0245454
\(201\) 9.59779 0.676976
\(202\) −3.44318 −0.242261
\(203\) −0.852309 −0.0598204
\(204\) 4.58281 0.320861
\(205\) −14.0985 −0.984685
\(206\) −0.598133 −0.0416739
\(207\) 5.47548 0.380572
\(208\) −9.08652 −0.630037
\(209\) −10.6644 −0.737673
\(210\) 2.31001 0.159406
\(211\) −22.8488 −1.57298 −0.786489 0.617604i \(-0.788103\pi\)
−0.786489 + 0.617604i \(0.788103\pi\)
\(212\) −13.4084 −0.920895
\(213\) −15.6845 −1.07469
\(214\) 6.91825 0.472922
\(215\) −1.04708 −0.0714100
\(216\) −7.30674 −0.497161
\(217\) −21.5497 −1.46289
\(218\) 1.92779 0.130566
\(219\) 2.72516 0.184149
\(220\) 5.77901 0.389621
\(221\) 7.65779 0.515119
\(222\) 1.55526 0.104382
\(223\) 4.75557 0.318457 0.159228 0.987242i \(-0.449099\pi\)
0.159228 + 0.987242i \(0.449099\pi\)
\(224\) −11.4744 −0.766667
\(225\) −0.482545 −0.0321696
\(226\) −0.0403580 −0.00268458
\(227\) −27.4709 −1.82331 −0.911655 0.410956i \(-0.865195\pi\)
−0.911655 + 0.410956i \(0.865195\pi\)
\(228\) 13.6650 0.904986
\(229\) −4.52436 −0.298978 −0.149489 0.988763i \(-0.547763\pi\)
−0.149489 + 0.988763i \(0.547763\pi\)
\(230\) −2.34208 −0.154432
\(231\) 3.52453 0.231897
\(232\) −0.474477 −0.0311509
\(233\) −10.5567 −0.691594 −0.345797 0.938309i \(-0.612391\pi\)
−0.345797 + 0.938309i \(0.612391\pi\)
\(234\) −2.41135 −0.157635
\(235\) −17.5435 −1.14441
\(236\) −0.400913 −0.0260972
\(237\) −15.2222 −0.988789
\(238\) 2.77794 0.180067
\(239\) −28.3072 −1.83104 −0.915522 0.402269i \(-0.868222\pi\)
−0.915522 + 0.402269i \(0.868222\pi\)
\(240\) −6.73630 −0.434826
\(241\) −10.6922 −0.688748 −0.344374 0.938833i \(-0.611909\pi\)
−0.344374 + 0.938833i \(0.611909\pi\)
\(242\) 3.58013 0.230139
\(243\) 16.1352 1.03507
\(244\) −13.4156 −0.858846
\(245\) −0.789669 −0.0504501
\(246\) −2.29761 −0.146490
\(247\) 22.8339 1.45289
\(248\) −11.9966 −0.761785
\(249\) 11.9055 0.754483
\(250\) −4.27923 −0.270642
\(251\) −3.42359 −0.216095 −0.108048 0.994146i \(-0.534460\pi\)
−0.108048 + 0.994146i \(0.534460\pi\)
\(252\) 10.4939 0.661055
\(253\) −3.57346 −0.224661
\(254\) −5.79331 −0.363505
\(255\) 5.67711 0.355514
\(256\) 5.05948 0.316218
\(257\) −11.2524 −0.701906 −0.350953 0.936393i \(-0.614142\pi\)
−0.350953 + 0.936393i \(0.614142\pi\)
\(258\) −0.170640 −0.0106236
\(259\) −11.3097 −0.702751
\(260\) −12.3736 −0.767381
\(261\) 0.659580 0.0408270
\(262\) −1.18331 −0.0731050
\(263\) 0.429180 0.0264643 0.0132322 0.999912i \(-0.495788\pi\)
0.0132322 + 0.999912i \(0.495788\pi\)
\(264\) 1.96209 0.120758
\(265\) −16.6102 −1.02035
\(266\) 8.28325 0.507878
\(267\) −0.0904688 −0.00553660
\(268\) −18.6498 −1.13922
\(269\) 15.2320 0.928712 0.464356 0.885649i \(-0.346286\pi\)
0.464356 + 0.885649i \(0.346286\pi\)
\(270\) −4.34465 −0.264407
\(271\) 17.7850 1.08036 0.540182 0.841548i \(-0.318355\pi\)
0.540182 + 0.841548i \(0.318355\pi\)
\(272\) −8.10085 −0.491186
\(273\) −7.54649 −0.456734
\(274\) −0.145219 −0.00877303
\(275\) 0.314923 0.0189906
\(276\) 4.57889 0.275617
\(277\) 15.1882 0.912570 0.456285 0.889834i \(-0.349180\pi\)
0.456285 + 0.889834i \(0.349180\pi\)
\(278\) −5.13670 −0.308079
\(279\) 16.6767 0.998409
\(280\) −9.35149 −0.558858
\(281\) 20.0885 1.19838 0.599191 0.800606i \(-0.295489\pi\)
0.599191 + 0.800606i \(0.295489\pi\)
\(282\) −2.85902 −0.170252
\(283\) −1.63497 −0.0971889 −0.0485945 0.998819i \(-0.515474\pi\)
−0.0485945 + 0.998819i \(0.515474\pi\)
\(284\) 30.4772 1.80849
\(285\) 16.9280 1.00273
\(286\) 1.57372 0.0930559
\(287\) 16.7080 0.986243
\(288\) 8.87975 0.523244
\(289\) −10.1729 −0.598405
\(290\) −0.282128 −0.0165671
\(291\) −8.81321 −0.516640
\(292\) −5.29535 −0.309887
\(293\) −10.1892 −0.595262 −0.297631 0.954681i \(-0.596196\pi\)
−0.297631 + 0.954681i \(0.596196\pi\)
\(294\) −0.128690 −0.00750538
\(295\) −0.496644 −0.0289157
\(296\) −6.29607 −0.365951
\(297\) −6.62892 −0.384649
\(298\) 2.20433 0.127694
\(299\) 7.65125 0.442483
\(300\) −0.403530 −0.0232978
\(301\) 1.24088 0.0715230
\(302\) 2.74466 0.157938
\(303\) 8.33901 0.479064
\(304\) −24.1551 −1.38539
\(305\) −16.6190 −0.951602
\(306\) −2.14978 −0.122895
\(307\) −16.3682 −0.934184 −0.467092 0.884209i \(-0.654698\pi\)
−0.467092 + 0.884209i \(0.654698\pi\)
\(308\) −6.84864 −0.390238
\(309\) 1.44861 0.0824088
\(310\) −7.13329 −0.405144
\(311\) 6.98922 0.396322 0.198161 0.980169i \(-0.436503\pi\)
0.198161 + 0.980169i \(0.436503\pi\)
\(312\) −4.20110 −0.237840
\(313\) 2.22444 0.125733 0.0628665 0.998022i \(-0.479976\pi\)
0.0628665 + 0.998022i \(0.479976\pi\)
\(314\) −4.15227 −0.234326
\(315\) 12.9997 0.732450
\(316\) 29.5788 1.66394
\(317\) 3.49687 0.196404 0.0982019 0.995167i \(-0.468691\pi\)
0.0982019 + 0.995167i \(0.468691\pi\)
\(318\) −2.70692 −0.151796
\(319\) −0.430461 −0.0241012
\(320\) 10.3824 0.580396
\(321\) −16.7553 −0.935187
\(322\) 2.77557 0.154676
\(323\) 20.3570 1.13269
\(324\) −3.12190 −0.173439
\(325\) −0.674292 −0.0374030
\(326\) 1.86098 0.103070
\(327\) −4.66889 −0.258190
\(328\) 9.30128 0.513577
\(329\) 20.7906 1.14622
\(330\) 1.16668 0.0642235
\(331\) −13.8654 −0.762112 −0.381056 0.924552i \(-0.624439\pi\)
−0.381056 + 0.924552i \(0.624439\pi\)
\(332\) −23.1341 −1.26965
\(333\) 8.75229 0.479623
\(334\) −6.06668 −0.331954
\(335\) −23.1031 −1.26226
\(336\) 7.98311 0.435514
\(337\) −33.0259 −1.79903 −0.899517 0.436885i \(-0.856082\pi\)
−0.899517 + 0.436885i \(0.856082\pi\)
\(338\) 1.73015 0.0941076
\(339\) 0.0977428 0.00530866
\(340\) −11.0314 −0.598262
\(341\) −10.8837 −0.589387
\(342\) −6.41019 −0.346623
\(343\) −18.0357 −0.973837
\(344\) 0.690791 0.0372450
\(345\) 5.67226 0.305384
\(346\) 7.06229 0.379671
\(347\) −0.417829 −0.0224302 −0.0112151 0.999937i \(-0.503570\pi\)
−0.0112151 + 0.999937i \(0.503570\pi\)
\(348\) 0.551577 0.0295676
\(349\) −8.68607 −0.464955 −0.232477 0.972602i \(-0.574683\pi\)
−0.232477 + 0.972602i \(0.574683\pi\)
\(350\) −0.244606 −0.0130748
\(351\) 14.1934 0.757588
\(352\) −5.79519 −0.308884
\(353\) −9.09582 −0.484122 −0.242061 0.970261i \(-0.577823\pi\)
−0.242061 + 0.970261i \(0.577823\pi\)
\(354\) −0.0809369 −0.00430175
\(355\) 37.7546 2.00381
\(356\) 0.175793 0.00931703
\(357\) −6.72788 −0.356077
\(358\) −8.88089 −0.469370
\(359\) 5.51998 0.291333 0.145667 0.989334i \(-0.453467\pi\)
0.145667 + 0.989334i \(0.453467\pi\)
\(360\) 7.23687 0.381417
\(361\) 41.7004 2.19476
\(362\) −1.16894 −0.0614383
\(363\) −8.67068 −0.455093
\(364\) 14.6639 0.768595
\(365\) −6.55979 −0.343355
\(366\) −2.70836 −0.141568
\(367\) 16.3639 0.854188 0.427094 0.904207i \(-0.359537\pi\)
0.427094 + 0.904207i \(0.359537\pi\)
\(368\) −8.09393 −0.421925
\(369\) −12.9299 −0.673104
\(370\) −3.74370 −0.194626
\(371\) 19.6845 1.02197
\(372\) 13.9460 0.723066
\(373\) −23.5721 −1.22051 −0.610257 0.792203i \(-0.708934\pi\)
−0.610257 + 0.792203i \(0.708934\pi\)
\(374\) 1.40301 0.0725478
\(375\) 10.3638 0.535186
\(376\) 11.5740 0.596885
\(377\) 0.921675 0.0474687
\(378\) 5.14880 0.264826
\(379\) −22.8307 −1.17273 −0.586367 0.810045i \(-0.699442\pi\)
−0.586367 + 0.810045i \(0.699442\pi\)
\(380\) −32.8933 −1.68739
\(381\) 14.0308 0.718819
\(382\) −1.60665 −0.0822034
\(383\) 21.8257 1.11524 0.557622 0.830095i \(-0.311714\pi\)
0.557622 + 0.830095i \(0.311714\pi\)
\(384\) 9.73672 0.496875
\(385\) −8.48399 −0.432384
\(386\) 8.30497 0.422712
\(387\) −0.960283 −0.0488139
\(388\) 17.1253 0.869404
\(389\) 29.7462 1.50819 0.754097 0.656763i \(-0.228075\pi\)
0.754097 + 0.656763i \(0.228075\pi\)
\(390\) −2.49801 −0.126492
\(391\) 6.82127 0.344967
\(392\) 0.520971 0.0263130
\(393\) 2.86584 0.144563
\(394\) −7.26271 −0.365890
\(395\) 36.6418 1.84365
\(396\) 5.29999 0.266334
\(397\) 28.5561 1.43319 0.716595 0.697489i \(-0.245700\pi\)
0.716595 + 0.697489i \(0.245700\pi\)
\(398\) 0.614751 0.0308147
\(399\) −20.0611 −1.00431
\(400\) 0.713305 0.0356652
\(401\) 13.9970 0.698975 0.349487 0.936941i \(-0.386356\pi\)
0.349487 + 0.936941i \(0.386356\pi\)
\(402\) −3.76506 −0.187784
\(403\) 23.3035 1.16083
\(404\) −16.2038 −0.806171
\(405\) −3.86735 −0.192170
\(406\) 0.334347 0.0165934
\(407\) −5.71200 −0.283133
\(408\) −3.74538 −0.185424
\(409\) −18.2429 −0.902055 −0.451028 0.892510i \(-0.648942\pi\)
−0.451028 + 0.892510i \(0.648942\pi\)
\(410\) 5.53063 0.273138
\(411\) 0.351706 0.0173484
\(412\) −2.81486 −0.138678
\(413\) 0.588567 0.0289615
\(414\) −2.14794 −0.105566
\(415\) −28.6581 −1.40677
\(416\) 12.4083 0.608365
\(417\) 12.4405 0.609216
\(418\) 4.18348 0.204621
\(419\) 9.78645 0.478099 0.239050 0.971007i \(-0.423164\pi\)
0.239050 + 0.971007i \(0.423164\pi\)
\(420\) 10.8711 0.530453
\(421\) 2.26639 0.110457 0.0552286 0.998474i \(-0.482411\pi\)
0.0552286 + 0.998474i \(0.482411\pi\)
\(422\) 8.96322 0.436323
\(423\) −16.0893 −0.782288
\(424\) 10.9583 0.532181
\(425\) −0.601147 −0.0291599
\(426\) 6.15279 0.298104
\(427\) 19.6950 0.953109
\(428\) 32.5578 1.57374
\(429\) −3.81138 −0.184015
\(430\) 0.410751 0.0198082
\(431\) 4.29790 0.207023 0.103511 0.994628i \(-0.466992\pi\)
0.103511 + 0.994628i \(0.466992\pi\)
\(432\) −15.0146 −0.722390
\(433\) −16.6114 −0.798292 −0.399146 0.916887i \(-0.630693\pi\)
−0.399146 + 0.916887i \(0.630693\pi\)
\(434\) 8.45358 0.405785
\(435\) 0.683284 0.0327610
\(436\) 9.07230 0.434484
\(437\) 20.3396 0.972976
\(438\) −1.06903 −0.0510805
\(439\) −21.4863 −1.02548 −0.512742 0.858542i \(-0.671370\pi\)
−0.512742 + 0.858542i \(0.671370\pi\)
\(440\) −4.72300 −0.225160
\(441\) −0.724212 −0.0344863
\(442\) −3.00403 −0.142887
\(443\) −19.6047 −0.931447 −0.465724 0.884930i \(-0.654206\pi\)
−0.465724 + 0.884930i \(0.654206\pi\)
\(444\) 7.31915 0.347351
\(445\) 0.217770 0.0103233
\(446\) −1.86553 −0.0883356
\(447\) −5.33866 −0.252510
\(448\) −12.3041 −0.581314
\(449\) −35.1455 −1.65862 −0.829310 0.558789i \(-0.811266\pi\)
−0.829310 + 0.558789i \(0.811266\pi\)
\(450\) 0.189294 0.00892342
\(451\) 8.43843 0.397350
\(452\) −0.189928 −0.00893344
\(453\) −6.64728 −0.312316
\(454\) 10.7764 0.505762
\(455\) 18.1653 0.851605
\(456\) −11.1679 −0.522987
\(457\) −11.8432 −0.554002 −0.277001 0.960870i \(-0.589341\pi\)
−0.277001 + 0.960870i \(0.589341\pi\)
\(458\) 1.77483 0.0829325
\(459\) 12.6538 0.590627
\(460\) −11.0220 −0.513902
\(461\) 17.1286 0.797756 0.398878 0.917004i \(-0.369400\pi\)
0.398878 + 0.917004i \(0.369400\pi\)
\(462\) −1.38262 −0.0643251
\(463\) −6.92218 −0.321701 −0.160850 0.986979i \(-0.551424\pi\)
−0.160850 + 0.986979i \(0.551424\pi\)
\(464\) −0.975001 −0.0452633
\(465\) 17.2761 0.801158
\(466\) 4.14123 0.191839
\(467\) −32.6395 −1.51038 −0.755188 0.655508i \(-0.772455\pi\)
−0.755188 + 0.655508i \(0.772455\pi\)
\(468\) −11.3480 −0.524560
\(469\) 27.3792 1.26425
\(470\) 6.88202 0.317444
\(471\) 10.0563 0.463372
\(472\) 0.327653 0.0150814
\(473\) 0.626709 0.0288161
\(474\) 5.97143 0.274277
\(475\) −1.79250 −0.0822454
\(476\) 13.0732 0.599208
\(477\) −15.2333 −0.697486
\(478\) 11.1045 0.507907
\(479\) 8.78005 0.401171 0.200585 0.979676i \(-0.435716\pi\)
0.200585 + 0.979676i \(0.435716\pi\)
\(480\) 9.19888 0.419869
\(481\) 12.2302 0.557647
\(482\) 4.19439 0.191049
\(483\) −6.72213 −0.305868
\(484\) 16.8483 0.765833
\(485\) 21.2145 0.963301
\(486\) −6.32957 −0.287115
\(487\) −23.1617 −1.04956 −0.524779 0.851238i \(-0.675852\pi\)
−0.524779 + 0.851238i \(0.675852\pi\)
\(488\) 10.9641 0.496323
\(489\) −4.50709 −0.203818
\(490\) 0.309774 0.0139942
\(491\) 8.15363 0.367968 0.183984 0.982929i \(-0.441100\pi\)
0.183984 + 0.982929i \(0.441100\pi\)
\(492\) −10.8127 −0.487474
\(493\) 0.821696 0.0370073
\(494\) −8.95738 −0.403012
\(495\) 6.56554 0.295099
\(496\) −24.6518 −1.10690
\(497\) −44.7426 −2.00698
\(498\) −4.67035 −0.209283
\(499\) 30.6993 1.37429 0.687144 0.726521i \(-0.258864\pi\)
0.687144 + 0.726521i \(0.258864\pi\)
\(500\) −20.1384 −0.900615
\(501\) 14.6929 0.656428
\(502\) 1.34302 0.0599419
\(503\) −1.00000 −0.0445878
\(504\) −8.57634 −0.382020
\(505\) −20.0731 −0.893239
\(506\) 1.40181 0.0623180
\(507\) −4.19023 −0.186095
\(508\) −27.2637 −1.20963
\(509\) −6.39080 −0.283267 −0.141634 0.989919i \(-0.545235\pi\)
−0.141634 + 0.989919i \(0.545235\pi\)
\(510\) −2.22704 −0.0986149
\(511\) 7.77394 0.343899
\(512\) −22.4816 −0.993557
\(513\) 37.7309 1.66586
\(514\) 4.41414 0.194699
\(515\) −3.48700 −0.153655
\(516\) −0.803041 −0.0353519
\(517\) 10.5003 0.461805
\(518\) 4.43662 0.194934
\(519\) −17.1041 −0.750787
\(520\) 10.1126 0.443465
\(521\) −11.2745 −0.493944 −0.246972 0.969023i \(-0.579436\pi\)
−0.246972 + 0.969023i \(0.579436\pi\)
\(522\) −0.258743 −0.0113249
\(523\) 18.6154 0.813993 0.406996 0.913430i \(-0.366576\pi\)
0.406996 + 0.913430i \(0.366576\pi\)
\(524\) −5.56872 −0.243271
\(525\) 0.592410 0.0258549
\(526\) −0.168360 −0.00734085
\(527\) 20.7756 0.905001
\(528\) 4.03189 0.175466
\(529\) −16.1846 −0.703676
\(530\) 6.51589 0.283032
\(531\) −0.455477 −0.0197660
\(532\) 38.9815 1.69006
\(533\) −18.0678 −0.782604
\(534\) 0.0354894 0.00153578
\(535\) 40.3320 1.74370
\(536\) 15.2419 0.658349
\(537\) 21.5086 0.928163
\(538\) −5.97527 −0.257612
\(539\) 0.472642 0.0203581
\(540\) −20.4462 −0.879866
\(541\) −36.6446 −1.57547 −0.787736 0.616013i \(-0.788747\pi\)
−0.787736 + 0.616013i \(0.788747\pi\)
\(542\) −6.97678 −0.299678
\(543\) 2.83105 0.121492
\(544\) 11.0623 0.474291
\(545\) 11.2386 0.481409
\(546\) 2.96036 0.126692
\(547\) 19.4186 0.830280 0.415140 0.909758i \(-0.363733\pi\)
0.415140 + 0.909758i \(0.363733\pi\)
\(548\) −0.683413 −0.0291939
\(549\) −15.2415 −0.650490
\(550\) −0.123539 −0.00526772
\(551\) 2.45013 0.104379
\(552\) −3.74218 −0.159278
\(553\) −43.4238 −1.84657
\(554\) −5.95808 −0.253135
\(555\) 9.06684 0.384866
\(556\) −24.1737 −1.02519
\(557\) −35.6561 −1.51080 −0.755398 0.655266i \(-0.772556\pi\)
−0.755398 + 0.655266i \(0.772556\pi\)
\(558\) −6.54201 −0.276945
\(559\) −1.34187 −0.0567549
\(560\) −19.2163 −0.812039
\(561\) −3.39793 −0.143461
\(562\) −7.88041 −0.332415
\(563\) −34.0879 −1.43663 −0.718316 0.695717i \(-0.755087\pi\)
−0.718316 + 0.695717i \(0.755087\pi\)
\(564\) −13.4547 −0.566547
\(565\) −0.235279 −0.00989827
\(566\) 0.641373 0.0269589
\(567\) 4.58316 0.192475
\(568\) −24.9080 −1.04512
\(569\) 13.0323 0.546342 0.273171 0.961965i \(-0.411927\pi\)
0.273171 + 0.961965i \(0.411927\pi\)
\(570\) −6.64056 −0.278143
\(571\) 28.5710 1.19566 0.597829 0.801624i \(-0.296030\pi\)
0.597829 + 0.801624i \(0.296030\pi\)
\(572\) 7.40602 0.309661
\(573\) 3.89114 0.162555
\(574\) −6.55428 −0.273570
\(575\) −0.600634 −0.0250482
\(576\) 9.52183 0.396743
\(577\) 40.1508 1.67150 0.835750 0.549110i \(-0.185033\pi\)
0.835750 + 0.549110i \(0.185033\pi\)
\(578\) 3.99066 0.165990
\(579\) −20.1137 −0.835899
\(580\) −1.32771 −0.0551304
\(581\) 33.9624 1.40900
\(582\) 3.45728 0.143309
\(583\) 9.94172 0.411744
\(584\) 4.32772 0.179082
\(585\) −14.0577 −0.581214
\(586\) 3.99707 0.165118
\(587\) −1.26390 −0.0521666 −0.0260833 0.999660i \(-0.508304\pi\)
−0.0260833 + 0.999660i \(0.508304\pi\)
\(588\) −0.605626 −0.0249756
\(589\) 61.9486 2.55255
\(590\) 0.194825 0.00802083
\(591\) 17.5895 0.723535
\(592\) −12.9378 −0.531739
\(593\) 0.473003 0.0194239 0.00971196 0.999953i \(-0.496909\pi\)
0.00971196 + 0.999953i \(0.496909\pi\)
\(594\) 2.60042 0.106696
\(595\) 16.1948 0.663924
\(596\) 10.3737 0.424925
\(597\) −1.48886 −0.0609350
\(598\) −3.00146 −0.122739
\(599\) 28.8689 1.17955 0.589775 0.807568i \(-0.299217\pi\)
0.589775 + 0.807568i \(0.299217\pi\)
\(600\) 0.329792 0.0134637
\(601\) 8.88531 0.362439 0.181220 0.983443i \(-0.441996\pi\)
0.181220 + 0.983443i \(0.441996\pi\)
\(602\) −0.486776 −0.0198395
\(603\) −21.1881 −0.862845
\(604\) 12.9166 0.525568
\(605\) 20.8714 0.848544
\(606\) −3.27126 −0.132886
\(607\) 28.6612 1.16332 0.581662 0.813431i \(-0.302403\pi\)
0.581662 + 0.813431i \(0.302403\pi\)
\(608\) 32.9854 1.33773
\(609\) −0.809753 −0.0328128
\(610\) 6.51937 0.263962
\(611\) −22.4826 −0.909550
\(612\) −10.1170 −0.408955
\(613\) 46.9961 1.89815 0.949077 0.315043i \(-0.102019\pi\)
0.949077 + 0.315043i \(0.102019\pi\)
\(614\) 6.42099 0.259130
\(615\) −13.3946 −0.540122
\(616\) 5.59717 0.225516
\(617\) −24.3030 −0.978401 −0.489201 0.872171i \(-0.662711\pi\)
−0.489201 + 0.872171i \(0.662711\pi\)
\(618\) −0.568268 −0.0228591
\(619\) 39.4552 1.58584 0.792920 0.609326i \(-0.208560\pi\)
0.792920 + 0.609326i \(0.208560\pi\)
\(620\) −33.5698 −1.34819
\(621\) 12.6429 0.507344
\(622\) −2.74176 −0.109934
\(623\) −0.258077 −0.0103396
\(624\) −8.63282 −0.345589
\(625\) −26.0974 −1.04390
\(626\) −0.872613 −0.0348766
\(627\) −10.1319 −0.404630
\(628\) −19.5408 −0.779765
\(629\) 10.9035 0.434750
\(630\) −5.09957 −0.203172
\(631\) 34.7457 1.38320 0.691602 0.722279i \(-0.256905\pi\)
0.691602 + 0.722279i \(0.256905\pi\)
\(632\) −24.1738 −0.961583
\(633\) −21.7080 −0.862814
\(634\) −1.37177 −0.0544798
\(635\) −33.7738 −1.34027
\(636\) −12.7389 −0.505132
\(637\) −1.01199 −0.0400965
\(638\) 0.168863 0.00668535
\(639\) 34.6251 1.36975
\(640\) −23.4375 −0.926449
\(641\) −42.3173 −1.67143 −0.835716 0.549163i \(-0.814947\pi\)
−0.835716 + 0.549163i \(0.814947\pi\)
\(642\) 6.57282 0.259408
\(643\) −31.5672 −1.24489 −0.622444 0.782664i \(-0.713860\pi\)
−0.622444 + 0.782664i \(0.713860\pi\)
\(644\) 13.0620 0.514716
\(645\) −0.994794 −0.0391700
\(646\) −7.98572 −0.314194
\(647\) −45.6664 −1.79533 −0.897667 0.440675i \(-0.854739\pi\)
−0.897667 + 0.440675i \(0.854739\pi\)
\(648\) 2.55142 0.100229
\(649\) 0.297258 0.0116684
\(650\) 0.264514 0.0103751
\(651\) −20.4737 −0.802426
\(652\) 8.75790 0.342986
\(653\) 7.15705 0.280077 0.140039 0.990146i \(-0.455277\pi\)
0.140039 + 0.990146i \(0.455277\pi\)
\(654\) 1.83153 0.0716185
\(655\) −6.89844 −0.269545
\(656\) 19.1132 0.746244
\(657\) −6.01605 −0.234708
\(658\) −8.15581 −0.317946
\(659\) 12.1603 0.473696 0.236848 0.971547i \(-0.423886\pi\)
0.236848 + 0.971547i \(0.423886\pi\)
\(660\) 5.49046 0.213716
\(661\) −15.2194 −0.591968 −0.295984 0.955193i \(-0.595647\pi\)
−0.295984 + 0.955193i \(0.595647\pi\)
\(662\) 5.43917 0.211400
\(663\) 7.27543 0.282554
\(664\) 18.9067 0.733724
\(665\) 48.2896 1.87259
\(666\) −3.43338 −0.133041
\(667\) 0.820994 0.0317890
\(668\) −28.5502 −1.10464
\(669\) 4.51812 0.174681
\(670\) 9.06297 0.350133
\(671\) 9.94702 0.384001
\(672\) −10.9015 −0.420534
\(673\) 34.9359 1.34668 0.673340 0.739333i \(-0.264859\pi\)
0.673340 + 0.739333i \(0.264859\pi\)
\(674\) 12.9555 0.499028
\(675\) −1.11420 −0.0428856
\(676\) 8.14219 0.313161
\(677\) −17.6078 −0.676721 −0.338361 0.941016i \(-0.609872\pi\)
−0.338361 + 0.941016i \(0.609872\pi\)
\(678\) −0.0383429 −0.00147255
\(679\) −25.1411 −0.964826
\(680\) 9.01560 0.345732
\(681\) −26.0993 −1.00013
\(682\) 4.26951 0.163488
\(683\) −43.5075 −1.66477 −0.832385 0.554198i \(-0.813025\pi\)
−0.832385 + 0.554198i \(0.813025\pi\)
\(684\) −30.1668 −1.15346
\(685\) −0.846600 −0.0323469
\(686\) 7.07512 0.270129
\(687\) −4.29846 −0.163996
\(688\) 1.41950 0.0541181
\(689\) −21.2865 −0.810953
\(690\) −2.22513 −0.0847094
\(691\) −2.05256 −0.0780829 −0.0390415 0.999238i \(-0.512430\pi\)
−0.0390415 + 0.999238i \(0.512430\pi\)
\(692\) 33.2356 1.26343
\(693\) −7.78074 −0.295566
\(694\) 0.163907 0.00622184
\(695\) −29.9459 −1.13591
\(696\) −0.450786 −0.0170870
\(697\) −16.1079 −0.610130
\(698\) 3.40741 0.128972
\(699\) −10.0296 −0.379355
\(700\) −1.15113 −0.0435088
\(701\) −9.44349 −0.356676 −0.178338 0.983969i \(-0.557072\pi\)
−0.178338 + 0.983969i \(0.557072\pi\)
\(702\) −5.56784 −0.210145
\(703\) 32.5119 1.22621
\(704\) −6.21422 −0.234207
\(705\) −16.6675 −0.627735
\(706\) 3.56814 0.134289
\(707\) 23.7883 0.894653
\(708\) −0.380895 −0.0143149
\(709\) 15.9123 0.597598 0.298799 0.954316i \(-0.403414\pi\)
0.298799 + 0.954316i \(0.403414\pi\)
\(710\) −14.8105 −0.555829
\(711\) 33.6045 1.26027
\(712\) −0.143670 −0.00538426
\(713\) 20.7579 0.777389
\(714\) 2.63924 0.0987710
\(715\) 9.17446 0.343105
\(716\) −41.7941 −1.56192
\(717\) −26.8938 −1.00437
\(718\) −2.16540 −0.0808119
\(719\) 24.9061 0.928842 0.464421 0.885615i \(-0.346263\pi\)
0.464421 + 0.885615i \(0.346263\pi\)
\(720\) 14.8710 0.554211
\(721\) 4.13240 0.153899
\(722\) −16.3584 −0.608795
\(723\) −10.1584 −0.377794
\(724\) −5.50112 −0.204448
\(725\) −0.0723528 −0.00268712
\(726\) 3.40137 0.126237
\(727\) 11.5573 0.428636 0.214318 0.976764i \(-0.431247\pi\)
0.214318 + 0.976764i \(0.431247\pi\)
\(728\) −11.9843 −0.444167
\(729\) 10.2563 0.379865
\(730\) 2.57330 0.0952421
\(731\) −1.19631 −0.0442470
\(732\) −12.7457 −0.471096
\(733\) 11.9918 0.442926 0.221463 0.975169i \(-0.428917\pi\)
0.221463 + 0.975169i \(0.428917\pi\)
\(734\) −6.41928 −0.236940
\(735\) −0.750240 −0.0276730
\(736\) 11.0528 0.407412
\(737\) 13.8280 0.509359
\(738\) 5.07219 0.186710
\(739\) 48.3255 1.77768 0.888840 0.458217i \(-0.151512\pi\)
0.888840 + 0.458217i \(0.151512\pi\)
\(740\) −17.6181 −0.647654
\(741\) 21.6938 0.796942
\(742\) −7.72191 −0.283480
\(743\) 40.3146 1.47900 0.739499 0.673158i \(-0.235063\pi\)
0.739499 + 0.673158i \(0.235063\pi\)
\(744\) −11.3976 −0.417856
\(745\) 12.8508 0.470817
\(746\) 9.24693 0.338554
\(747\) −26.2826 −0.961632
\(748\) 6.60265 0.241417
\(749\) −47.7970 −1.74646
\(750\) −4.06557 −0.148453
\(751\) −48.6091 −1.77377 −0.886885 0.461990i \(-0.847136\pi\)
−0.886885 + 0.461990i \(0.847136\pi\)
\(752\) 23.7834 0.867292
\(753\) −3.25265 −0.118533
\(754\) −0.361558 −0.0131672
\(755\) 16.0008 0.582330
\(756\) 24.2306 0.881258
\(757\) −44.8319 −1.62944 −0.814721 0.579853i \(-0.803110\pi\)
−0.814721 + 0.579853i \(0.803110\pi\)
\(758\) 8.95611 0.325301
\(759\) −3.39503 −0.123232
\(760\) 26.8826 0.975136
\(761\) −31.4812 −1.14119 −0.570596 0.821231i \(-0.693288\pi\)
−0.570596 + 0.821231i \(0.693288\pi\)
\(762\) −5.50405 −0.199391
\(763\) −13.3188 −0.482171
\(764\) −7.56101 −0.273548
\(765\) −12.5328 −0.453123
\(766\) −8.56189 −0.309353
\(767\) −0.636468 −0.0229815
\(768\) 4.80686 0.173453
\(769\) 30.6134 1.10395 0.551974 0.833861i \(-0.313875\pi\)
0.551974 + 0.833861i \(0.313875\pi\)
\(770\) 3.32813 0.119937
\(771\) −10.6906 −0.385011
\(772\) 39.0837 1.40665
\(773\) 25.8605 0.930136 0.465068 0.885275i \(-0.346030\pi\)
0.465068 + 0.885275i \(0.346030\pi\)
\(774\) 0.376703 0.0135403
\(775\) −1.82936 −0.0657125
\(776\) −13.9959 −0.502424
\(777\) −10.7450 −0.385475
\(778\) −11.6690 −0.418352
\(779\) −48.0304 −1.72087
\(780\) −11.7558 −0.420925
\(781\) −22.5974 −0.808598
\(782\) −2.67588 −0.0956891
\(783\) 1.52298 0.0544268
\(784\) 1.07054 0.0382336
\(785\) −24.2069 −0.863981
\(786\) −1.12422 −0.0400997
\(787\) 31.7233 1.13081 0.565406 0.824813i \(-0.308719\pi\)
0.565406 + 0.824813i \(0.308719\pi\)
\(788\) −34.1788 −1.21757
\(789\) 0.407750 0.0145163
\(790\) −14.3740 −0.511403
\(791\) 0.278827 0.00991394
\(792\) −4.33150 −0.153913
\(793\) −21.2979 −0.756311
\(794\) −11.2021 −0.397548
\(795\) −15.7808 −0.559687
\(796\) 2.89306 0.102542
\(797\) −55.1753 −1.95441 −0.977205 0.212296i \(-0.931906\pi\)
−0.977205 + 0.212296i \(0.931906\pi\)
\(798\) 7.86965 0.278583
\(799\) −20.0438 −0.709099
\(800\) −0.974066 −0.0344384
\(801\) 0.199719 0.00705671
\(802\) −5.49078 −0.193886
\(803\) 3.92625 0.138554
\(804\) −17.7186 −0.624887
\(805\) 16.1810 0.570306
\(806\) −9.14158 −0.321999
\(807\) 14.4715 0.509420
\(808\) 13.2429 0.465882
\(809\) −6.04923 −0.212680 −0.106340 0.994330i \(-0.533913\pi\)
−0.106340 + 0.994330i \(0.533913\pi\)
\(810\) 1.51710 0.0533055
\(811\) −3.97693 −0.139649 −0.0698244 0.997559i \(-0.522244\pi\)
−0.0698244 + 0.997559i \(0.522244\pi\)
\(812\) 1.57346 0.0552176
\(813\) 16.8970 0.592604
\(814\) 2.24073 0.0785374
\(815\) 10.8491 0.380029
\(816\) −7.69637 −0.269427
\(817\) −3.56714 −0.124798
\(818\) 7.15641 0.250218
\(819\) 16.6596 0.582134
\(820\) 26.0275 0.908920
\(821\) 16.5562 0.577815 0.288907 0.957357i \(-0.406708\pi\)
0.288907 + 0.957357i \(0.406708\pi\)
\(822\) −0.137968 −0.00481220
\(823\) −29.9588 −1.04430 −0.522150 0.852854i \(-0.674870\pi\)
−0.522150 + 0.852854i \(0.674870\pi\)
\(824\) 2.30049 0.0801413
\(825\) 0.299198 0.0104168
\(826\) −0.230885 −0.00803353
\(827\) −39.4061 −1.37029 −0.685143 0.728409i \(-0.740260\pi\)
−0.685143 + 0.728409i \(0.740260\pi\)
\(828\) −10.1084 −0.351290
\(829\) 6.63817 0.230553 0.115277 0.993333i \(-0.463225\pi\)
0.115277 + 0.993333i \(0.463225\pi\)
\(830\) 11.2421 0.390220
\(831\) 14.4298 0.500565
\(832\) 13.3055 0.461285
\(833\) −0.902214 −0.0312599
\(834\) −4.88022 −0.168988
\(835\) −35.3675 −1.22394
\(836\) 19.6877 0.680914
\(837\) 38.5068 1.33099
\(838\) −3.83906 −0.132618
\(839\) −55.4839 −1.91552 −0.957759 0.287574i \(-0.907151\pi\)
−0.957759 + 0.287574i \(0.907151\pi\)
\(840\) −8.88456 −0.306546
\(841\) −28.9011 −0.996590
\(842\) −0.889068 −0.0306393
\(843\) 19.0855 0.657340
\(844\) 42.1815 1.45195
\(845\) 10.0864 0.346983
\(846\) 6.31157 0.216996
\(847\) −24.7345 −0.849887
\(848\) 22.5181 0.773276
\(849\) −1.55334 −0.0533103
\(850\) 0.235820 0.00808857
\(851\) 10.8942 0.373447
\(852\) 28.9554 0.991997
\(853\) −23.3264 −0.798680 −0.399340 0.916803i \(-0.630761\pi\)
−0.399340 + 0.916803i \(0.630761\pi\)
\(854\) −7.72603 −0.264379
\(855\) −37.3701 −1.27803
\(856\) −26.6084 −0.909456
\(857\) 20.5822 0.703075 0.351537 0.936174i \(-0.385659\pi\)
0.351537 + 0.936174i \(0.385659\pi\)
\(858\) 1.49514 0.0510433
\(859\) −45.7526 −1.56106 −0.780529 0.625120i \(-0.785050\pi\)
−0.780529 + 0.625120i \(0.785050\pi\)
\(860\) 1.93302 0.0659155
\(861\) 15.8738 0.540977
\(862\) −1.68600 −0.0574253
\(863\) 10.0848 0.343290 0.171645 0.985159i \(-0.445092\pi\)
0.171645 + 0.985159i \(0.445092\pi\)
\(864\) 20.5035 0.697542
\(865\) 41.1717 1.39988
\(866\) 6.51638 0.221435
\(867\) −9.66495 −0.328239
\(868\) 39.7831 1.35033
\(869\) −21.9313 −0.743968
\(870\) −0.268041 −0.00908745
\(871\) −29.6075 −1.00321
\(872\) −7.41449 −0.251086
\(873\) 19.4560 0.658487
\(874\) −7.97890 −0.269890
\(875\) 29.5645 0.999462
\(876\) −5.03095 −0.169980
\(877\) 3.30564 0.111624 0.0558118 0.998441i \(-0.482225\pi\)
0.0558118 + 0.998441i \(0.482225\pi\)
\(878\) 8.42872 0.284456
\(879\) −9.68048 −0.326515
\(880\) −9.70527 −0.327165
\(881\) 8.30907 0.279940 0.139970 0.990156i \(-0.455299\pi\)
0.139970 + 0.990156i \(0.455299\pi\)
\(882\) 0.284097 0.00956603
\(883\) 2.65895 0.0894807 0.0447403 0.998999i \(-0.485754\pi\)
0.0447403 + 0.998999i \(0.485754\pi\)
\(884\) −14.1372 −0.475484
\(885\) −0.471846 −0.0158609
\(886\) 7.69060 0.258371
\(887\) −21.5846 −0.724740 −0.362370 0.932034i \(-0.618032\pi\)
−0.362370 + 0.932034i \(0.618032\pi\)
\(888\) −5.98170 −0.200733
\(889\) 40.0250 1.34240
\(890\) −0.0854276 −0.00286354
\(891\) 2.31474 0.0775466
\(892\) −8.77932 −0.293954
\(893\) −59.7665 −2.00001
\(894\) 2.09427 0.0700428
\(895\) −51.7738 −1.73061
\(896\) 27.7755 0.927915
\(897\) 7.26921 0.242712
\(898\) 13.7870 0.460079
\(899\) 2.50051 0.0833967
\(900\) 0.890832 0.0296944
\(901\) −18.9775 −0.632231
\(902\) −3.31026 −0.110220
\(903\) 1.17892 0.0392320
\(904\) 0.155222 0.00516259
\(905\) −6.81470 −0.226528
\(906\) 2.60762 0.0866323
\(907\) −5.95616 −0.197771 −0.0988855 0.995099i \(-0.531528\pi\)
−0.0988855 + 0.995099i \(0.531528\pi\)
\(908\) 50.7145 1.68302
\(909\) −18.4092 −0.610594
\(910\) −7.12597 −0.236224
\(911\) 14.9730 0.496079 0.248040 0.968750i \(-0.420214\pi\)
0.248040 + 0.968750i \(0.420214\pi\)
\(912\) −22.9490 −0.759916
\(913\) 17.1528 0.567676
\(914\) 4.64590 0.153673
\(915\) −15.7892 −0.521975
\(916\) 8.35249 0.275974
\(917\) 8.17527 0.269971
\(918\) −4.96386 −0.163832
\(919\) −21.4164 −0.706463 −0.353232 0.935536i \(-0.614917\pi\)
−0.353232 + 0.935536i \(0.614917\pi\)
\(920\) 9.00790 0.296982
\(921\) −15.5509 −0.512421
\(922\) −6.71925 −0.221287
\(923\) 48.3840 1.59258
\(924\) −6.50668 −0.214054
\(925\) −0.960085 −0.0315674
\(926\) 2.71546 0.0892354
\(927\) −3.19796 −0.105035
\(928\) 1.33143 0.0437064
\(929\) −51.4153 −1.68688 −0.843441 0.537221i \(-0.819474\pi\)
−0.843441 + 0.537221i \(0.819474\pi\)
\(930\) −6.77712 −0.222230
\(931\) −2.69021 −0.0881682
\(932\) 19.4889 0.638381
\(933\) 6.64024 0.217392
\(934\) 12.8039 0.418958
\(935\) 8.17925 0.267490
\(936\) 9.27433 0.303141
\(937\) −41.9350 −1.36996 −0.684979 0.728563i \(-0.740188\pi\)
−0.684979 + 0.728563i \(0.740188\pi\)
\(938\) −10.7404 −0.350687
\(939\) 2.11337 0.0689674
\(940\) 32.3873 1.05636
\(941\) −36.8467 −1.20117 −0.600585 0.799561i \(-0.705065\pi\)
−0.600585 + 0.799561i \(0.705065\pi\)
\(942\) −3.94494 −0.128533
\(943\) −16.0941 −0.524097
\(944\) 0.673293 0.0219138
\(945\) 30.0165 0.976436
\(946\) −0.245848 −0.00799320
\(947\) 2.12177 0.0689481 0.0344741 0.999406i \(-0.489024\pi\)
0.0344741 + 0.999406i \(0.489024\pi\)
\(948\) 28.1019 0.912708
\(949\) −8.40663 −0.272891
\(950\) 0.703167 0.0228138
\(951\) 3.32227 0.107732
\(952\) −10.6843 −0.346280
\(953\) 13.5643 0.439391 0.219695 0.975569i \(-0.429494\pi\)
0.219695 + 0.975569i \(0.429494\pi\)
\(954\) 5.97579 0.193473
\(955\) −9.36645 −0.303091
\(956\) 52.2584 1.69016
\(957\) −0.408968 −0.0132201
\(958\) −3.44427 −0.111279
\(959\) 1.00330 0.0323981
\(960\) 9.86403 0.318360
\(961\) 32.2225 1.03944
\(962\) −4.79769 −0.154684
\(963\) 36.9889 1.19195
\(964\) 19.7391 0.635754
\(965\) 48.4163 1.55858
\(966\) 2.63698 0.0848435
\(967\) 5.95719 0.191570 0.0957851 0.995402i \(-0.469464\pi\)
0.0957851 + 0.995402i \(0.469464\pi\)
\(968\) −13.7696 −0.442571
\(969\) 19.3406 0.621308
\(970\) −8.32211 −0.267207
\(971\) 12.1772 0.390785 0.195392 0.980725i \(-0.437402\pi\)
0.195392 + 0.980725i \(0.437402\pi\)
\(972\) −29.7874 −0.955431
\(973\) 35.4886 1.13771
\(974\) 9.08597 0.291133
\(975\) −0.640624 −0.0205164
\(976\) 22.5301 0.721172
\(977\) 7.69439 0.246165 0.123083 0.992396i \(-0.460722\pi\)
0.123083 + 0.992396i \(0.460722\pi\)
\(978\) 1.76806 0.0565363
\(979\) −0.130342 −0.00416576
\(980\) 1.45782 0.0465683
\(981\) 10.3070 0.329078
\(982\) −3.19854 −0.102069
\(983\) −12.2144 −0.389578 −0.194789 0.980845i \(-0.562402\pi\)
−0.194789 + 0.980845i \(0.562402\pi\)
\(984\) 8.83686 0.281709
\(985\) −42.3401 −1.34907
\(986\) −0.322338 −0.0102653
\(987\) 19.7525 0.628729
\(988\) −42.1540 −1.34110
\(989\) −1.19529 −0.0380079
\(990\) −2.57555 −0.0818564
\(991\) 2.35228 0.0747227 0.0373613 0.999302i \(-0.488105\pi\)
0.0373613 + 0.999302i \(0.488105\pi\)
\(992\) 33.6637 1.06882
\(993\) −13.1731 −0.418036
\(994\) 17.5518 0.556709
\(995\) 3.58387 0.113616
\(996\) −21.9790 −0.696431
\(997\) 19.2415 0.609384 0.304692 0.952451i \(-0.401446\pi\)
0.304692 + 0.952451i \(0.401446\pi\)
\(998\) −12.0428 −0.381209
\(999\) 20.2092 0.639389
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.1.6 10
3.2 odd 2 4527.2.a.k.1.5 10
4.3 odd 2 8048.2.a.p.1.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.e.1.6 10 1.1 even 1 trivial
4527.2.a.k.1.5 10 3.2 odd 2
8048.2.a.p.1.2 10 4.3 odd 2