Properties

Label 503.2.a.e.1.4
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 1.4
Root \(1.07636\) of defining polynomial
Character \(\chi\) \(=\) 503.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.37178 q^{2} +0.0763625 q^{3} -0.118218 q^{4} +1.17276 q^{5} -0.104753 q^{6} +0.469303 q^{7} +2.90573 q^{8} -2.99417 q^{9} +O(q^{10})\) \(q-1.37178 q^{2} +0.0763625 q^{3} -0.118218 q^{4} +1.17276 q^{5} -0.104753 q^{6} +0.469303 q^{7} +2.90573 q^{8} -2.99417 q^{9} -1.60876 q^{10} -5.74596 q^{11} -0.00902740 q^{12} -1.85873 q^{13} -0.643780 q^{14} +0.0895546 q^{15} -3.74959 q^{16} +5.22916 q^{17} +4.10734 q^{18} +2.12602 q^{19} -0.138641 q^{20} +0.0358371 q^{21} +7.88220 q^{22} +0.171951 q^{23} +0.221889 q^{24} -3.62464 q^{25} +2.54977 q^{26} -0.457730 q^{27} -0.0554799 q^{28} -6.19149 q^{29} -0.122849 q^{30} -0.396234 q^{31} -0.667846 q^{32} -0.438776 q^{33} -7.17326 q^{34} +0.550377 q^{35} +0.353964 q^{36} -8.17999 q^{37} -2.91644 q^{38} -0.141937 q^{39} +3.40771 q^{40} -12.4282 q^{41} -0.0491607 q^{42} -4.97920 q^{43} +0.679275 q^{44} -3.51143 q^{45} -0.235879 q^{46} -0.521599 q^{47} -0.286328 q^{48} -6.77976 q^{49} +4.97222 q^{50} +0.399312 q^{51} +0.219735 q^{52} +8.76106 q^{53} +0.627905 q^{54} -6.73861 q^{55} +1.36367 q^{56} +0.162349 q^{57} +8.49336 q^{58} +3.35297 q^{59} -0.0105869 q^{60} -5.38243 q^{61} +0.543546 q^{62} -1.40517 q^{63} +8.41532 q^{64} -2.17983 q^{65} +0.601905 q^{66} +8.42823 q^{67} -0.618179 q^{68} +0.0131306 q^{69} -0.754997 q^{70} +7.47643 q^{71} -8.70025 q^{72} -4.60009 q^{73} +11.2212 q^{74} -0.276787 q^{75} -0.251334 q^{76} -2.69660 q^{77} +0.194707 q^{78} -17.1992 q^{79} -4.39735 q^{80} +8.94755 q^{81} +17.0487 q^{82} +5.97721 q^{83} -0.00423658 q^{84} +6.13253 q^{85} +6.83038 q^{86} -0.472798 q^{87} -16.6962 q^{88} -4.25595 q^{89} +4.81691 q^{90} -0.872306 q^{91} -0.0203277 q^{92} -0.0302574 q^{93} +0.715520 q^{94} +2.49331 q^{95} -0.0509984 q^{96} -2.64532 q^{97} +9.30034 q^{98} +17.2044 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} + O(q^{10}) \) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} - 4q^{10} - 3q^{11} - 7q^{12} - 18q^{13} + q^{14} - 2q^{15} - 4q^{16} - 11q^{17} - q^{18} - 3q^{20} + q^{21} - 18q^{22} - 2q^{23} + 10q^{24} - 27q^{25} + 11q^{26} - 2q^{27} - 22q^{28} - 9q^{29} + 12q^{30} - 22q^{31} - 10q^{32} - 10q^{33} - 10q^{34} - 6q^{35} + 2q^{36} - 35q^{37} + 2q^{38} + 8q^{39} - 19q^{40} - 4q^{41} + 4q^{42} - 20q^{43} + 9q^{44} + 2q^{45} - q^{46} + 7q^{47} - 27q^{49} + 16q^{50} + 9q^{51} - 7q^{52} - 24q^{53} + 17q^{54} - 11q^{55} + 12q^{56} - 23q^{57} + 2q^{58} + 17q^{59} - 4q^{61} + 8q^{62} + 10q^{63} + 3q^{64} - 16q^{65} + 46q^{66} - 6q^{67} + 28q^{68} - 2q^{69} + 26q^{70} - q^{71} - q^{72} - 31q^{73} + 11q^{74} + 30q^{75} + 20q^{76} + 3q^{77} + 11q^{78} - 10q^{79} + 24q^{80} - 6q^{81} - 9q^{82} + 22q^{83} + 22q^{84} - 6q^{85} + 38q^{86} + 25q^{87} - 3q^{88} + q^{89} + 2q^{90} + 10q^{91} + 27q^{92} - 6q^{93} + 33q^{94} + 39q^{95} + 46q^{96} - 57q^{97} + 40q^{98} + 35q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.37178 −0.969995 −0.484998 0.874515i \(-0.661180\pi\)
−0.484998 + 0.874515i \(0.661180\pi\)
\(3\) 0.0763625 0.0440879 0.0220440 0.999757i \(-0.492983\pi\)
0.0220440 + 0.999757i \(0.492983\pi\)
\(4\) −0.118218 −0.0591089
\(5\) 1.17276 0.524472 0.262236 0.965004i \(-0.415540\pi\)
0.262236 + 0.965004i \(0.415540\pi\)
\(6\) −0.104753 −0.0427651
\(7\) 0.469303 0.177380 0.0886899 0.996059i \(-0.471732\pi\)
0.0886899 + 0.996059i \(0.471732\pi\)
\(8\) 2.90573 1.02733
\(9\) −2.99417 −0.998056
\(10\) −1.60876 −0.508736
\(11\) −5.74596 −1.73247 −0.866237 0.499634i \(-0.833468\pi\)
−0.866237 + 0.499634i \(0.833468\pi\)
\(12\) −0.00902740 −0.00260599
\(13\) −1.85873 −0.515518 −0.257759 0.966209i \(-0.582984\pi\)
−0.257759 + 0.966209i \(0.582984\pi\)
\(14\) −0.643780 −0.172058
\(15\) 0.0895546 0.0231229
\(16\) −3.74959 −0.937397
\(17\) 5.22916 1.26826 0.634129 0.773228i \(-0.281359\pi\)
0.634129 + 0.773228i \(0.281359\pi\)
\(18\) 4.10734 0.968110
\(19\) 2.12602 0.487743 0.243872 0.969808i \(-0.421582\pi\)
0.243872 + 0.969808i \(0.421582\pi\)
\(20\) −0.138641 −0.0310010
\(21\) 0.0358371 0.00782030
\(22\) 7.88220 1.68049
\(23\) 0.171951 0.0358543 0.0179271 0.999839i \(-0.494293\pi\)
0.0179271 + 0.999839i \(0.494293\pi\)
\(24\) 0.221889 0.0452929
\(25\) −3.62464 −0.724929
\(26\) 2.54977 0.500050
\(27\) −0.457730 −0.0880902
\(28\) −0.0554799 −0.0104847
\(29\) −6.19149 −1.14973 −0.574865 0.818248i \(-0.694946\pi\)
−0.574865 + 0.818248i \(0.694946\pi\)
\(30\) −0.122849 −0.0224291
\(31\) −0.396234 −0.0711658 −0.0355829 0.999367i \(-0.511329\pi\)
−0.0355829 + 0.999367i \(0.511329\pi\)
\(32\) −0.667846 −0.118060
\(33\) −0.438776 −0.0763812
\(34\) −7.17326 −1.23020
\(35\) 0.550377 0.0930307
\(36\) 0.353964 0.0589940
\(37\) −8.17999 −1.34478 −0.672391 0.740196i \(-0.734733\pi\)
−0.672391 + 0.740196i \(0.734733\pi\)
\(38\) −2.91644 −0.473109
\(39\) −0.141937 −0.0227281
\(40\) 3.40771 0.538807
\(41\) −12.4282 −1.94095 −0.970477 0.241193i \(-0.922461\pi\)
−0.970477 + 0.241193i \(0.922461\pi\)
\(42\) −0.0491607 −0.00758566
\(43\) −4.97920 −0.759321 −0.379661 0.925126i \(-0.623959\pi\)
−0.379661 + 0.925126i \(0.623959\pi\)
\(44\) 0.679275 0.102405
\(45\) −3.51143 −0.523453
\(46\) −0.235879 −0.0347785
\(47\) −0.521599 −0.0760831 −0.0380415 0.999276i \(-0.512112\pi\)
−0.0380415 + 0.999276i \(0.512112\pi\)
\(48\) −0.286328 −0.0413279
\(49\) −6.77976 −0.968536
\(50\) 4.97222 0.703178
\(51\) 0.399312 0.0559148
\(52\) 0.219735 0.0304717
\(53\) 8.76106 1.20342 0.601712 0.798713i \(-0.294486\pi\)
0.601712 + 0.798713i \(0.294486\pi\)
\(54\) 0.627905 0.0854470
\(55\) −6.73861 −0.908634
\(56\) 1.36367 0.182228
\(57\) 0.162349 0.0215036
\(58\) 8.49336 1.11523
\(59\) 3.35297 0.436519 0.218260 0.975891i \(-0.429962\pi\)
0.218260 + 0.975891i \(0.429962\pi\)
\(60\) −0.0105869 −0.00136677
\(61\) −5.38243 −0.689150 −0.344575 0.938759i \(-0.611977\pi\)
−0.344575 + 0.938759i \(0.611977\pi\)
\(62\) 0.543546 0.0690305
\(63\) −1.40517 −0.177035
\(64\) 8.41532 1.05191
\(65\) −2.17983 −0.270375
\(66\) 0.601905 0.0740894
\(67\) 8.42823 1.02967 0.514836 0.857289i \(-0.327853\pi\)
0.514836 + 0.857289i \(0.327853\pi\)
\(68\) −0.618179 −0.0749653
\(69\) 0.0131306 0.00158074
\(70\) −0.754997 −0.0902394
\(71\) 7.47643 0.887289 0.443644 0.896203i \(-0.353685\pi\)
0.443644 + 0.896203i \(0.353685\pi\)
\(72\) −8.70025 −1.02533
\(73\) −4.60009 −0.538400 −0.269200 0.963084i \(-0.586759\pi\)
−0.269200 + 0.963084i \(0.586759\pi\)
\(74\) 11.2212 1.30443
\(75\) −0.276787 −0.0319606
\(76\) −0.251334 −0.0288299
\(77\) −2.69660 −0.307306
\(78\) 0.194707 0.0220462
\(79\) −17.1992 −1.93506 −0.967530 0.252758i \(-0.918662\pi\)
−0.967530 + 0.252758i \(0.918662\pi\)
\(80\) −4.39735 −0.491639
\(81\) 8.94755 0.994173
\(82\) 17.0487 1.88272
\(83\) 5.97721 0.656084 0.328042 0.944663i \(-0.393611\pi\)
0.328042 + 0.944663i \(0.393611\pi\)
\(84\) −0.00423658 −0.000462249 0
\(85\) 6.13253 0.665166
\(86\) 6.83038 0.736538
\(87\) −0.472798 −0.0506892
\(88\) −16.6962 −1.77982
\(89\) −4.25595 −0.451130 −0.225565 0.974228i \(-0.572423\pi\)
−0.225565 + 0.974228i \(0.572423\pi\)
\(90\) 4.81691 0.507747
\(91\) −0.872306 −0.0914425
\(92\) −0.0203277 −0.00211930
\(93\) −0.0302574 −0.00313755
\(94\) 0.715520 0.0738002
\(95\) 2.49331 0.255808
\(96\) −0.0509984 −0.00520501
\(97\) −2.64532 −0.268592 −0.134296 0.990941i \(-0.542877\pi\)
−0.134296 + 0.990941i \(0.542877\pi\)
\(98\) 9.30034 0.939476
\(99\) 17.2044 1.72911
\(100\) 0.428497 0.0428497
\(101\) 6.39045 0.635873 0.317937 0.948112i \(-0.397010\pi\)
0.317937 + 0.948112i \(0.397010\pi\)
\(102\) −0.547768 −0.0542371
\(103\) −16.6014 −1.63578 −0.817890 0.575375i \(-0.804856\pi\)
−0.817890 + 0.575375i \(0.804856\pi\)
\(104\) −5.40096 −0.529608
\(105\) 0.0420282 0.00410153
\(106\) −12.0182 −1.16732
\(107\) 9.15760 0.885299 0.442649 0.896695i \(-0.354039\pi\)
0.442649 + 0.896695i \(0.354039\pi\)
\(108\) 0.0541118 0.00520691
\(109\) 6.51891 0.624398 0.312199 0.950017i \(-0.398934\pi\)
0.312199 + 0.950017i \(0.398934\pi\)
\(110\) 9.24390 0.881371
\(111\) −0.624645 −0.0592887
\(112\) −1.75969 −0.166275
\(113\) 1.36694 0.128591 0.0642954 0.997931i \(-0.479520\pi\)
0.0642954 + 0.997931i \(0.479520\pi\)
\(114\) −0.222707 −0.0208584
\(115\) 0.201656 0.0188046
\(116\) 0.731944 0.0679593
\(117\) 5.56535 0.514516
\(118\) −4.59954 −0.423422
\(119\) 2.45406 0.224963
\(120\) 0.260221 0.0237549
\(121\) 22.0161 2.00146
\(122\) 7.38352 0.668473
\(123\) −0.949047 −0.0855726
\(124\) 0.0468419 0.00420653
\(125\) −10.1146 −0.904677
\(126\) 1.92759 0.171723
\(127\) 6.77469 0.601157 0.300578 0.953757i \(-0.402820\pi\)
0.300578 + 0.953757i \(0.402820\pi\)
\(128\) −10.2083 −0.902293
\(129\) −0.380225 −0.0334769
\(130\) 2.99025 0.262263
\(131\) 16.6839 1.45768 0.728838 0.684687i \(-0.240061\pi\)
0.728838 + 0.684687i \(0.240061\pi\)
\(132\) 0.0518711 0.00451480
\(133\) 0.997748 0.0865158
\(134\) −11.5617 −0.998777
\(135\) −0.536805 −0.0462008
\(136\) 15.1945 1.30292
\(137\) 13.0662 1.11632 0.558162 0.829732i \(-0.311507\pi\)
0.558162 + 0.829732i \(0.311507\pi\)
\(138\) −0.0180123 −0.00153331
\(139\) −10.8468 −0.920017 −0.460008 0.887915i \(-0.652154\pi\)
−0.460008 + 0.887915i \(0.652154\pi\)
\(140\) −0.0650644 −0.00549894
\(141\) −0.0398306 −0.00335434
\(142\) −10.2560 −0.860666
\(143\) 10.6802 0.893122
\(144\) 11.2269 0.935575
\(145\) −7.26110 −0.603002
\(146\) 6.31032 0.522245
\(147\) −0.517719 −0.0427008
\(148\) 0.967020 0.0794886
\(149\) 0.0210826 0.00172716 0.000863578 1.00000i \(-0.499725\pi\)
0.000863578 1.00000i \(0.499725\pi\)
\(150\) 0.379691 0.0310016
\(151\) 15.1704 1.23455 0.617276 0.786746i \(-0.288236\pi\)
0.617276 + 0.786746i \(0.288236\pi\)
\(152\) 6.17765 0.501074
\(153\) −15.6570 −1.26579
\(154\) 3.69914 0.298085
\(155\) −0.464686 −0.0373245
\(156\) 0.0167795 0.00134343
\(157\) 11.1437 0.889367 0.444684 0.895688i \(-0.353316\pi\)
0.444684 + 0.895688i \(0.353316\pi\)
\(158\) 23.5935 1.87700
\(159\) 0.669016 0.0530565
\(160\) −0.783221 −0.0619190
\(161\) 0.0806970 0.00635982
\(162\) −12.2741 −0.964343
\(163\) −10.3388 −0.809797 −0.404899 0.914362i \(-0.632693\pi\)
−0.404899 + 0.914362i \(0.632693\pi\)
\(164\) 1.46923 0.114728
\(165\) −0.514578 −0.0400598
\(166\) −8.19942 −0.636398
\(167\) −15.6444 −1.21060 −0.605300 0.795998i \(-0.706947\pi\)
−0.605300 + 0.795998i \(0.706947\pi\)
\(168\) 0.104133 0.00803404
\(169\) −9.54513 −0.734241
\(170\) −8.41248 −0.645208
\(171\) −6.36567 −0.486795
\(172\) 0.588630 0.0448826
\(173\) −5.69316 −0.432843 −0.216422 0.976300i \(-0.569439\pi\)
−0.216422 + 0.976300i \(0.569439\pi\)
\(174\) 0.648575 0.0491683
\(175\) −1.70105 −0.128588
\(176\) 21.5450 1.62402
\(177\) 0.256041 0.0192452
\(178\) 5.83823 0.437594
\(179\) 10.8571 0.811495 0.405748 0.913985i \(-0.367011\pi\)
0.405748 + 0.913985i \(0.367011\pi\)
\(180\) 0.415113 0.0309407
\(181\) 6.26595 0.465744 0.232872 0.972507i \(-0.425188\pi\)
0.232872 + 0.972507i \(0.425188\pi\)
\(182\) 1.19661 0.0886988
\(183\) −0.411016 −0.0303832
\(184\) 0.499643 0.0368342
\(185\) −9.59313 −0.705301
\(186\) 0.0415066 0.00304341
\(187\) −30.0466 −2.19722
\(188\) 0.0616623 0.00449718
\(189\) −0.214814 −0.0156254
\(190\) −3.42027 −0.248132
\(191\) 8.26296 0.597887 0.298943 0.954271i \(-0.403366\pi\)
0.298943 + 0.954271i \(0.403366\pi\)
\(192\) 0.642615 0.0463767
\(193\) −19.8469 −1.42861 −0.714306 0.699833i \(-0.753258\pi\)
−0.714306 + 0.699833i \(0.753258\pi\)
\(194\) 3.62880 0.260533
\(195\) −0.166458 −0.0119203
\(196\) 0.801487 0.0572491
\(197\) −4.24866 −0.302704 −0.151352 0.988480i \(-0.548363\pi\)
−0.151352 + 0.988480i \(0.548363\pi\)
\(198\) −23.6006 −1.67723
\(199\) 4.63286 0.328415 0.164207 0.986426i \(-0.447493\pi\)
0.164207 + 0.986426i \(0.447493\pi\)
\(200\) −10.5322 −0.744742
\(201\) 0.643601 0.0453961
\(202\) −8.76630 −0.616794
\(203\) −2.90568 −0.203939
\(204\) −0.0472057 −0.00330506
\(205\) −14.5752 −1.01798
\(206\) 22.7734 1.58670
\(207\) −0.514850 −0.0357846
\(208\) 6.96947 0.483246
\(209\) −12.2161 −0.845002
\(210\) −0.0576535 −0.00397847
\(211\) 10.2699 0.707007 0.353503 0.935433i \(-0.384990\pi\)
0.353503 + 0.935433i \(0.384990\pi\)
\(212\) −1.03571 −0.0711330
\(213\) 0.570919 0.0391187
\(214\) −12.5622 −0.858736
\(215\) −5.83939 −0.398243
\(216\) −1.33004 −0.0904977
\(217\) −0.185954 −0.0126234
\(218\) −8.94251 −0.605663
\(219\) −0.351275 −0.0237369
\(220\) 0.796624 0.0537083
\(221\) −9.71958 −0.653810
\(222\) 0.856876 0.0575097
\(223\) 14.7887 0.990326 0.495163 0.868800i \(-0.335108\pi\)
0.495163 + 0.868800i \(0.335108\pi\)
\(224\) −0.313422 −0.0209414
\(225\) 10.8528 0.723520
\(226\) −1.87514 −0.124732
\(227\) 14.9154 0.989969 0.494984 0.868902i \(-0.335174\pi\)
0.494984 + 0.868902i \(0.335174\pi\)
\(228\) −0.0191925 −0.00127105
\(229\) −18.2488 −1.20592 −0.602959 0.797772i \(-0.706012\pi\)
−0.602959 + 0.797772i \(0.706012\pi\)
\(230\) −0.276628 −0.0182403
\(231\) −0.205919 −0.0135485
\(232\) −17.9908 −1.18115
\(233\) −16.6741 −1.09235 −0.546177 0.837670i \(-0.683918\pi\)
−0.546177 + 0.837670i \(0.683918\pi\)
\(234\) −7.63443 −0.499079
\(235\) −0.611708 −0.0399035
\(236\) −0.396380 −0.0258022
\(237\) −1.31337 −0.0853127
\(238\) −3.36643 −0.218213
\(239\) 24.3691 1.57631 0.788153 0.615479i \(-0.211037\pi\)
0.788153 + 0.615479i \(0.211037\pi\)
\(240\) −0.335793 −0.0216753
\(241\) 18.1071 1.16638 0.583190 0.812336i \(-0.301804\pi\)
0.583190 + 0.812336i \(0.301804\pi\)
\(242\) −30.2013 −1.94141
\(243\) 2.05645 0.131921
\(244\) 0.636299 0.0407349
\(245\) −7.95100 −0.507971
\(246\) 1.30188 0.0830051
\(247\) −3.95170 −0.251441
\(248\) −1.15135 −0.0731108
\(249\) 0.456435 0.0289254
\(250\) 13.8750 0.877533
\(251\) 26.9687 1.70225 0.851124 0.524965i \(-0.175922\pi\)
0.851124 + 0.524965i \(0.175922\pi\)
\(252\) 0.166116 0.0104643
\(253\) −0.988024 −0.0621165
\(254\) −9.29340 −0.583119
\(255\) 0.468295 0.0293258
\(256\) −2.82712 −0.176695
\(257\) 20.3022 1.26642 0.633208 0.773982i \(-0.281738\pi\)
0.633208 + 0.773982i \(0.281738\pi\)
\(258\) 0.521585 0.0324724
\(259\) −3.83889 −0.238537
\(260\) 0.257695 0.0159816
\(261\) 18.5384 1.14750
\(262\) −22.8866 −1.41394
\(263\) −12.0595 −0.743622 −0.371811 0.928309i \(-0.621263\pi\)
−0.371811 + 0.928309i \(0.621263\pi\)
\(264\) −1.27497 −0.0784687
\(265\) 10.2746 0.631162
\(266\) −1.36869 −0.0839199
\(267\) −0.324995 −0.0198894
\(268\) −0.996366 −0.0608628
\(269\) −25.0120 −1.52501 −0.762506 0.646982i \(-0.776031\pi\)
−0.762506 + 0.646982i \(0.776031\pi\)
\(270\) 0.736379 0.0448146
\(271\) −18.0354 −1.09557 −0.547787 0.836618i \(-0.684530\pi\)
−0.547787 + 0.836618i \(0.684530\pi\)
\(272\) −19.6072 −1.18886
\(273\) −0.0666115 −0.00403151
\(274\) −17.9240 −1.08283
\(275\) 20.8271 1.25592
\(276\) −0.00155227 −9.34357e−5 0
\(277\) −29.3688 −1.76460 −0.882301 0.470686i \(-0.844006\pi\)
−0.882301 + 0.470686i \(0.844006\pi\)
\(278\) 14.8795 0.892412
\(279\) 1.18639 0.0710274
\(280\) 1.59925 0.0955733
\(281\) 12.0977 0.721686 0.360843 0.932626i \(-0.382489\pi\)
0.360843 + 0.932626i \(0.382489\pi\)
\(282\) 0.0546389 0.00325370
\(283\) −8.04149 −0.478017 −0.239009 0.971017i \(-0.576822\pi\)
−0.239009 + 0.971017i \(0.576822\pi\)
\(284\) −0.883846 −0.0524466
\(285\) 0.190395 0.0112780
\(286\) −14.6509 −0.866324
\(287\) −5.83257 −0.344286
\(288\) 1.99964 0.117830
\(289\) 10.3441 0.608477
\(290\) 9.96064 0.584909
\(291\) −0.202003 −0.0118416
\(292\) 0.543812 0.0318242
\(293\) −13.5236 −0.790056 −0.395028 0.918669i \(-0.629265\pi\)
−0.395028 + 0.918669i \(0.629265\pi\)
\(294\) 0.710197 0.0414195
\(295\) 3.93221 0.228942
\(296\) −23.7688 −1.38154
\(297\) 2.63010 0.152614
\(298\) −0.0289207 −0.00167533
\(299\) −0.319610 −0.0184835
\(300\) 0.0327211 0.00188916
\(301\) −2.33675 −0.134688
\(302\) −20.8105 −1.19751
\(303\) 0.487991 0.0280343
\(304\) −7.97171 −0.457209
\(305\) −6.31228 −0.361440
\(306\) 21.4780 1.22781
\(307\) 3.60632 0.205824 0.102912 0.994690i \(-0.467184\pi\)
0.102912 + 0.994690i \(0.467184\pi\)
\(308\) 0.318785 0.0181645
\(309\) −1.26772 −0.0721181
\(310\) 0.637447 0.0362046
\(311\) −27.5438 −1.56187 −0.780934 0.624613i \(-0.785257\pi\)
−0.780934 + 0.624613i \(0.785257\pi\)
\(312\) −0.412431 −0.0233493
\(313\) −30.2190 −1.70808 −0.854039 0.520209i \(-0.825854\pi\)
−0.854039 + 0.520209i \(0.825854\pi\)
\(314\) −15.2868 −0.862682
\(315\) −1.64792 −0.0928499
\(316\) 2.03325 0.114379
\(317\) −5.23152 −0.293831 −0.146916 0.989149i \(-0.546935\pi\)
−0.146916 + 0.989149i \(0.546935\pi\)
\(318\) −0.917744 −0.0514645
\(319\) 35.5761 1.99188
\(320\) 9.86911 0.551700
\(321\) 0.699298 0.0390310
\(322\) −0.110699 −0.00616899
\(323\) 11.1173 0.618584
\(324\) −1.05776 −0.0587644
\(325\) 6.73723 0.373714
\(326\) 14.1826 0.785500
\(327\) 0.497800 0.0275284
\(328\) −36.1129 −1.99400
\(329\) −0.244788 −0.0134956
\(330\) 0.705888 0.0388578
\(331\) −5.56111 −0.305666 −0.152833 0.988252i \(-0.548840\pi\)
−0.152833 + 0.988252i \(0.548840\pi\)
\(332\) −0.706612 −0.0387804
\(333\) 24.4923 1.34217
\(334\) 21.4607 1.17428
\(335\) 9.88426 0.540035
\(336\) −0.134375 −0.00733073
\(337\) 1.20314 0.0655392 0.0327696 0.999463i \(-0.489567\pi\)
0.0327696 + 0.999463i \(0.489567\pi\)
\(338\) 13.0938 0.712210
\(339\) 0.104383 0.00566930
\(340\) −0.724973 −0.0393172
\(341\) 2.27675 0.123293
\(342\) 8.73231 0.472189
\(343\) −6.46688 −0.349178
\(344\) −14.4682 −0.780074
\(345\) 0.0153990 0.000829054 0
\(346\) 7.80977 0.419856
\(347\) 18.8760 1.01332 0.506659 0.862147i \(-0.330880\pi\)
0.506659 + 0.862147i \(0.330880\pi\)
\(348\) 0.0558931 0.00299618
\(349\) 18.3633 0.982966 0.491483 0.870887i \(-0.336455\pi\)
0.491483 + 0.870887i \(0.336455\pi\)
\(350\) 2.33347 0.124729
\(351\) 0.850795 0.0454121
\(352\) 3.83742 0.204535
\(353\) −2.06519 −0.109919 −0.0549596 0.998489i \(-0.517503\pi\)
−0.0549596 + 0.998489i \(0.517503\pi\)
\(354\) −0.351232 −0.0186678
\(355\) 8.76802 0.465358
\(356\) 0.503129 0.0266658
\(357\) 0.187398 0.00991816
\(358\) −14.8935 −0.787147
\(359\) −7.83605 −0.413571 −0.206786 0.978386i \(-0.566300\pi\)
−0.206786 + 0.978386i \(0.566300\pi\)
\(360\) −10.2033 −0.537759
\(361\) −14.4800 −0.762107
\(362\) −8.59550 −0.451770
\(363\) 1.68121 0.0882404
\(364\) 0.103122 0.00540506
\(365\) −5.39478 −0.282376
\(366\) 0.563824 0.0294716
\(367\) 27.3479 1.42755 0.713775 0.700375i \(-0.246984\pi\)
0.713775 + 0.700375i \(0.246984\pi\)
\(368\) −0.644745 −0.0336097
\(369\) 37.2120 1.93718
\(370\) 13.1597 0.684139
\(371\) 4.11159 0.213463
\(372\) 0.00357697 0.000185457 0
\(373\) 31.3381 1.62263 0.811313 0.584612i \(-0.198753\pi\)
0.811313 + 0.584612i \(0.198753\pi\)
\(374\) 41.2173 2.13130
\(375\) −0.772376 −0.0398853
\(376\) −1.51563 −0.0781625
\(377\) 11.5083 0.592707
\(378\) 0.294677 0.0151566
\(379\) −1.97192 −0.101291 −0.0506454 0.998717i \(-0.516128\pi\)
−0.0506454 + 0.998717i \(0.516128\pi\)
\(380\) −0.294753 −0.0151205
\(381\) 0.517333 0.0265038
\(382\) −11.3350 −0.579948
\(383\) −24.9455 −1.27466 −0.637328 0.770593i \(-0.719960\pi\)
−0.637328 + 0.770593i \(0.719960\pi\)
\(384\) −0.779530 −0.0397802
\(385\) −3.16245 −0.161173
\(386\) 27.2256 1.38575
\(387\) 14.9086 0.757845
\(388\) 0.312724 0.0158761
\(389\) −8.98152 −0.455381 −0.227691 0.973734i \(-0.573117\pi\)
−0.227691 + 0.973734i \(0.573117\pi\)
\(390\) 0.228343 0.0115626
\(391\) 0.899159 0.0454724
\(392\) −19.7001 −0.995007
\(393\) 1.27402 0.0642659
\(394\) 5.82822 0.293622
\(395\) −20.1704 −1.01488
\(396\) −2.03386 −0.102205
\(397\) −10.3125 −0.517569 −0.258784 0.965935i \(-0.583322\pi\)
−0.258784 + 0.965935i \(0.583322\pi\)
\(398\) −6.35527 −0.318561
\(399\) 0.0761906 0.00381430
\(400\) 13.5909 0.679546
\(401\) −22.2691 −1.11207 −0.556033 0.831160i \(-0.687677\pi\)
−0.556033 + 0.831160i \(0.687677\pi\)
\(402\) −0.882879 −0.0440340
\(403\) 0.736492 0.0366873
\(404\) −0.755464 −0.0375858
\(405\) 10.4933 0.521416
\(406\) 3.98596 0.197820
\(407\) 47.0019 2.32980
\(408\) 1.16029 0.0574430
\(409\) −9.36108 −0.462876 −0.231438 0.972850i \(-0.574343\pi\)
−0.231438 + 0.972850i \(0.574343\pi\)
\(410\) 19.9940 0.987433
\(411\) 0.997771 0.0492164
\(412\) 1.96257 0.0966891
\(413\) 1.57356 0.0774297
\(414\) 0.706262 0.0347109
\(415\) 7.00980 0.344098
\(416\) 1.24134 0.0608619
\(417\) −0.828292 −0.0405616
\(418\) 16.7578 0.819648
\(419\) −34.9797 −1.70887 −0.854436 0.519557i \(-0.826097\pi\)
−0.854436 + 0.519557i \(0.826097\pi\)
\(420\) −0.00496848 −0.000242437 0
\(421\) 15.9244 0.776107 0.388054 0.921637i \(-0.373147\pi\)
0.388054 + 0.921637i \(0.373147\pi\)
\(422\) −14.0880 −0.685793
\(423\) 1.56176 0.0759352
\(424\) 25.4573 1.23631
\(425\) −18.9538 −0.919396
\(426\) −0.783175 −0.0379450
\(427\) −2.52599 −0.122241
\(428\) −1.08259 −0.0523290
\(429\) 0.815566 0.0393759
\(430\) 8.01036 0.386294
\(431\) −37.5257 −1.80755 −0.903775 0.428008i \(-0.859215\pi\)
−0.903775 + 0.428008i \(0.859215\pi\)
\(432\) 1.71630 0.0825755
\(433\) 27.7776 1.33491 0.667454 0.744651i \(-0.267384\pi\)
0.667454 + 0.744651i \(0.267384\pi\)
\(434\) 0.255088 0.0122446
\(435\) −0.554476 −0.0265851
\(436\) −0.770651 −0.0369075
\(437\) 0.365572 0.0174877
\(438\) 0.481872 0.0230247
\(439\) 37.3507 1.78265 0.891327 0.453362i \(-0.149775\pi\)
0.891327 + 0.453362i \(0.149775\pi\)
\(440\) −19.5806 −0.933468
\(441\) 20.2997 0.966654
\(442\) 13.3331 0.634193
\(443\) 15.8245 0.751844 0.375922 0.926651i \(-0.377326\pi\)
0.375922 + 0.926651i \(0.377326\pi\)
\(444\) 0.0738441 0.00350449
\(445\) −4.99119 −0.236605
\(446\) −20.2869 −0.960612
\(447\) 0.00160992 7.61467e−5 0
\(448\) 3.94933 0.186588
\(449\) −33.0395 −1.55923 −0.779616 0.626258i \(-0.784586\pi\)
−0.779616 + 0.626258i \(0.784586\pi\)
\(450\) −14.8877 −0.701811
\(451\) 71.4118 3.36265
\(452\) −0.161596 −0.00760086
\(453\) 1.15845 0.0544289
\(454\) −20.4606 −0.960265
\(455\) −1.02300 −0.0479591
\(456\) 0.471741 0.0220913
\(457\) −6.38585 −0.298717 −0.149359 0.988783i \(-0.547721\pi\)
−0.149359 + 0.988783i \(0.547721\pi\)
\(458\) 25.0334 1.16973
\(459\) −2.39354 −0.111721
\(460\) −0.0238394 −0.00111152
\(461\) 4.10510 0.191193 0.0955967 0.995420i \(-0.469524\pi\)
0.0955967 + 0.995420i \(0.469524\pi\)
\(462\) 0.282476 0.0131420
\(463\) −21.9482 −1.02002 −0.510011 0.860168i \(-0.670359\pi\)
−0.510011 + 0.860168i \(0.670359\pi\)
\(464\) 23.2155 1.07775
\(465\) −0.0354846 −0.00164556
\(466\) 22.8732 1.05958
\(467\) −20.9539 −0.969633 −0.484816 0.874616i \(-0.661114\pi\)
−0.484816 + 0.874616i \(0.661114\pi\)
\(468\) −0.657922 −0.0304125
\(469\) 3.95539 0.182643
\(470\) 0.839130 0.0387062
\(471\) 0.850964 0.0392104
\(472\) 9.74282 0.448450
\(473\) 28.6103 1.31550
\(474\) 1.80166 0.0827530
\(475\) −7.70608 −0.353579
\(476\) −0.290113 −0.0132973
\(477\) −26.2321 −1.20108
\(478\) −33.4291 −1.52901
\(479\) 40.1617 1.83503 0.917517 0.397698i \(-0.130191\pi\)
0.917517 + 0.397698i \(0.130191\pi\)
\(480\) −0.0598087 −0.00272988
\(481\) 15.2044 0.693260
\(482\) −24.8389 −1.13138
\(483\) 0.00616223 0.000280391 0
\(484\) −2.60269 −0.118304
\(485\) −3.10232 −0.140869
\(486\) −2.82099 −0.127963
\(487\) −1.56804 −0.0710547 −0.0355273 0.999369i \(-0.511311\pi\)
−0.0355273 + 0.999369i \(0.511311\pi\)
\(488\) −15.6399 −0.707985
\(489\) −0.789497 −0.0357023
\(490\) 10.9070 0.492729
\(491\) 9.66243 0.436059 0.218030 0.975942i \(-0.430037\pi\)
0.218030 + 0.975942i \(0.430037\pi\)
\(492\) 0.112194 0.00505810
\(493\) −32.3763 −1.45815
\(494\) 5.42086 0.243896
\(495\) 20.1765 0.906868
\(496\) 1.48572 0.0667106
\(497\) 3.50871 0.157387
\(498\) −0.626128 −0.0280575
\(499\) −20.6363 −0.923810 −0.461905 0.886930i \(-0.652834\pi\)
−0.461905 + 0.886930i \(0.652834\pi\)
\(500\) 1.19573 0.0534745
\(501\) −1.19465 −0.0533728
\(502\) −36.9951 −1.65117
\(503\) −1.00000 −0.0445878
\(504\) −4.08305 −0.181873
\(505\) 7.49444 0.333498
\(506\) 1.35535 0.0602528
\(507\) −0.728890 −0.0323712
\(508\) −0.800889 −0.0355337
\(509\) −23.3284 −1.03401 −0.517006 0.855982i \(-0.672953\pi\)
−0.517006 + 0.855982i \(0.672953\pi\)
\(510\) −0.642398 −0.0284459
\(511\) −2.15883 −0.0955012
\(512\) 24.2947 1.07369
\(513\) −0.973144 −0.0429654
\(514\) −27.8501 −1.22842
\(515\) −19.4693 −0.857921
\(516\) 0.0449493 0.00197878
\(517\) 2.99709 0.131812
\(518\) 5.26612 0.231380
\(519\) −0.434744 −0.0190832
\(520\) −6.33401 −0.277765
\(521\) −43.2647 −1.89546 −0.947730 0.319073i \(-0.896629\pi\)
−0.947730 + 0.319073i \(0.896629\pi\)
\(522\) −25.4306 −1.11307
\(523\) −43.4082 −1.89811 −0.949054 0.315112i \(-0.897958\pi\)
−0.949054 + 0.315112i \(0.897958\pi\)
\(524\) −1.97233 −0.0861615
\(525\) −0.129897 −0.00566916
\(526\) 16.5430 0.721310
\(527\) −2.07197 −0.0902565
\(528\) 1.64523 0.0715995
\(529\) −22.9704 −0.998714
\(530\) −14.0945 −0.612225
\(531\) −10.0394 −0.435671
\(532\) −0.117952 −0.00511385
\(533\) 23.1006 1.00060
\(534\) 0.445822 0.0192926
\(535\) 10.7396 0.464315
\(536\) 24.4902 1.05781
\(537\) 0.829073 0.0357771
\(538\) 34.3110 1.47925
\(539\) 38.9562 1.67796
\(540\) 0.0634599 0.00273088
\(541\) 32.5657 1.40011 0.700054 0.714090i \(-0.253159\pi\)
0.700054 + 0.714090i \(0.253159\pi\)
\(542\) 24.7406 1.06270
\(543\) 0.478484 0.0205337
\(544\) −3.49228 −0.149730
\(545\) 7.64509 0.327480
\(546\) 0.0913763 0.00391055
\(547\) 27.8377 1.19025 0.595127 0.803631i \(-0.297102\pi\)
0.595127 + 0.803631i \(0.297102\pi\)
\(548\) −1.54466 −0.0659847
\(549\) 16.1159 0.687811
\(550\) −28.5702 −1.21824
\(551\) −13.1633 −0.560773
\(552\) 0.0381540 0.00162394
\(553\) −8.07162 −0.343240
\(554\) 40.2876 1.71166
\(555\) −0.732556 −0.0310953
\(556\) 1.28229 0.0543811
\(557\) 15.7225 0.666182 0.333091 0.942895i \(-0.391908\pi\)
0.333091 + 0.942895i \(0.391908\pi\)
\(558\) −1.62747 −0.0688963
\(559\) 9.25498 0.391444
\(560\) −2.06369 −0.0872068
\(561\) −2.29443 −0.0968710
\(562\) −16.5953 −0.700033
\(563\) −24.7817 −1.04442 −0.522212 0.852816i \(-0.674893\pi\)
−0.522212 + 0.852816i \(0.674893\pi\)
\(564\) 0.00470869 0.000198271 0
\(565\) 1.60309 0.0674423
\(566\) 11.0312 0.463674
\(567\) 4.19911 0.176346
\(568\) 21.7245 0.911539
\(569\) 2.95346 0.123816 0.0619078 0.998082i \(-0.480282\pi\)
0.0619078 + 0.998082i \(0.480282\pi\)
\(570\) −0.261180 −0.0109396
\(571\) −10.7926 −0.451656 −0.225828 0.974167i \(-0.572509\pi\)
−0.225828 + 0.974167i \(0.572509\pi\)
\(572\) −1.26259 −0.0527914
\(573\) 0.630981 0.0263596
\(574\) 8.00101 0.333956
\(575\) −0.623261 −0.0259918
\(576\) −25.1969 −1.04987
\(577\) −5.92593 −0.246700 −0.123350 0.992363i \(-0.539364\pi\)
−0.123350 + 0.992363i \(0.539364\pi\)
\(578\) −14.1898 −0.590220
\(579\) −1.51556 −0.0629845
\(580\) 0.858391 0.0356428
\(581\) 2.80512 0.116376
\(582\) 0.277104 0.0114863
\(583\) −50.3407 −2.08490
\(584\) −13.3666 −0.553115
\(585\) 6.52679 0.269850
\(586\) 18.5514 0.766350
\(587\) −14.8090 −0.611234 −0.305617 0.952155i \(-0.598863\pi\)
−0.305617 + 0.952155i \(0.598863\pi\)
\(588\) 0.0612036 0.00252399
\(589\) −0.842403 −0.0347106
\(590\) −5.39414 −0.222073
\(591\) −0.324438 −0.0133456
\(592\) 30.6716 1.26060
\(593\) 33.4465 1.37348 0.686741 0.726902i \(-0.259041\pi\)
0.686741 + 0.726902i \(0.259041\pi\)
\(594\) −3.60792 −0.148035
\(595\) 2.87801 0.117987
\(596\) −0.00249234 −0.000102090 0
\(597\) 0.353777 0.0144791
\(598\) 0.438435 0.0179289
\(599\) 9.35823 0.382367 0.191183 0.981554i \(-0.438767\pi\)
0.191183 + 0.981554i \(0.438767\pi\)
\(600\) −0.804268 −0.0328341
\(601\) −6.16182 −0.251346 −0.125673 0.992072i \(-0.540109\pi\)
−0.125673 + 0.992072i \(0.540109\pi\)
\(602\) 3.20551 0.130647
\(603\) −25.2355 −1.02767
\(604\) −1.79341 −0.0729730
\(605\) 25.8195 1.04971
\(606\) −0.669416 −0.0271932
\(607\) 35.7361 1.45048 0.725242 0.688494i \(-0.241728\pi\)
0.725242 + 0.688494i \(0.241728\pi\)
\(608\) −1.41986 −0.0575828
\(609\) −0.221885 −0.00899124
\(610\) 8.65907 0.350595
\(611\) 0.969511 0.0392222
\(612\) 1.85093 0.0748195
\(613\) −1.96528 −0.0793771 −0.0396885 0.999212i \(-0.512637\pi\)
−0.0396885 + 0.999212i \(0.512637\pi\)
\(614\) −4.94708 −0.199648
\(615\) −1.11300 −0.0448805
\(616\) −7.83558 −0.315705
\(617\) −35.4461 −1.42701 −0.713503 0.700652i \(-0.752893\pi\)
−0.713503 + 0.700652i \(0.752893\pi\)
\(618\) 1.73904 0.0699543
\(619\) −34.4643 −1.38524 −0.692618 0.721304i \(-0.743543\pi\)
−0.692618 + 0.721304i \(0.743543\pi\)
\(620\) 0.0549341 0.00220621
\(621\) −0.0787071 −0.00315841
\(622\) 37.7841 1.51501
\(623\) −1.99733 −0.0800213
\(624\) 0.532206 0.0213053
\(625\) 6.26126 0.250451
\(626\) 41.4538 1.65683
\(627\) −0.932849 −0.0372544
\(628\) −1.31739 −0.0525695
\(629\) −42.7745 −1.70553
\(630\) 2.26059 0.0900640
\(631\) −29.7560 −1.18457 −0.592284 0.805729i \(-0.701774\pi\)
−0.592284 + 0.805729i \(0.701774\pi\)
\(632\) −49.9762 −1.98795
\(633\) 0.784233 0.0311705
\(634\) 7.17650 0.285015
\(635\) 7.94506 0.315290
\(636\) −0.0790896 −0.00313611
\(637\) 12.6017 0.499298
\(638\) −48.8026 −1.93211
\(639\) −22.3857 −0.885564
\(640\) −11.9718 −0.473228
\(641\) −32.9327 −1.30077 −0.650383 0.759607i \(-0.725391\pi\)
−0.650383 + 0.759607i \(0.725391\pi\)
\(642\) −0.959283 −0.0378599
\(643\) −0.492253 −0.0194126 −0.00970628 0.999953i \(-0.503090\pi\)
−0.00970628 + 0.999953i \(0.503090\pi\)
\(644\) −0.00953982 −0.000375922 0
\(645\) −0.445911 −0.0175577
\(646\) −15.2505 −0.600024
\(647\) 3.76376 0.147969 0.0739844 0.997259i \(-0.476429\pi\)
0.0739844 + 0.997259i \(0.476429\pi\)
\(648\) 25.9992 1.02134
\(649\) −19.2660 −0.756258
\(650\) −9.24200 −0.362501
\(651\) −0.0141999 −0.000556538 0
\(652\) 1.22223 0.0478662
\(653\) 18.5728 0.726809 0.363404 0.931631i \(-0.381614\pi\)
0.363404 + 0.931631i \(0.381614\pi\)
\(654\) −0.682873 −0.0267024
\(655\) 19.5661 0.764510
\(656\) 46.6005 1.81945
\(657\) 13.7734 0.537353
\(658\) 0.335795 0.0130907
\(659\) −34.1930 −1.33197 −0.665985 0.745965i \(-0.731988\pi\)
−0.665985 + 0.745965i \(0.731988\pi\)
\(660\) 0.0608322 0.00236789
\(661\) 29.5439 1.14912 0.574562 0.818461i \(-0.305173\pi\)
0.574562 + 0.818461i \(0.305173\pi\)
\(662\) 7.62862 0.296495
\(663\) −0.742212 −0.0288251
\(664\) 17.3682 0.674015
\(665\) 1.17012 0.0453751
\(666\) −33.5980 −1.30190
\(667\) −1.06463 −0.0412227
\(668\) 1.84944 0.0715572
\(669\) 1.12930 0.0436614
\(670\) −13.5590 −0.523831
\(671\) 30.9273 1.19393
\(672\) −0.0239337 −0.000923263 0
\(673\) −9.15185 −0.352778 −0.176389 0.984321i \(-0.556442\pi\)
−0.176389 + 0.984321i \(0.556442\pi\)
\(674\) −1.65045 −0.0635728
\(675\) 1.65911 0.0638591
\(676\) 1.12840 0.0434001
\(677\) −33.1256 −1.27312 −0.636561 0.771227i \(-0.719644\pi\)
−0.636561 + 0.771227i \(0.719644\pi\)
\(678\) −0.143190 −0.00549920
\(679\) −1.24146 −0.0476427
\(680\) 17.8195 0.683345
\(681\) 1.13898 0.0436457
\(682\) −3.12320 −0.119593
\(683\) 49.1452 1.88049 0.940245 0.340498i \(-0.110596\pi\)
0.940245 + 0.340498i \(0.110596\pi\)
\(684\) 0.752535 0.0287739
\(685\) 15.3235 0.585481
\(686\) 8.87113 0.338701
\(687\) −1.39353 −0.0531664
\(688\) 18.6700 0.711786
\(689\) −16.2844 −0.620387
\(690\) −0.0211240 −0.000804179 0
\(691\) 34.1013 1.29727 0.648637 0.761098i \(-0.275339\pi\)
0.648637 + 0.761098i \(0.275339\pi\)
\(692\) 0.673033 0.0255849
\(693\) 8.07406 0.306708
\(694\) −25.8938 −0.982914
\(695\) −12.7207 −0.482523
\(696\) −1.37382 −0.0520746
\(697\) −64.9889 −2.46163
\(698\) −25.1904 −0.953473
\(699\) −1.27327 −0.0481596
\(700\) 0.201095 0.00760067
\(701\) 26.2590 0.991787 0.495894 0.868383i \(-0.334841\pi\)
0.495894 + 0.868383i \(0.334841\pi\)
\(702\) −1.16710 −0.0440495
\(703\) −17.3909 −0.655909
\(704\) −48.3541 −1.82241
\(705\) −0.0467116 −0.00175926
\(706\) 2.83299 0.106621
\(707\) 2.99905 0.112791
\(708\) −0.0302686 −0.00113756
\(709\) 28.4811 1.06963 0.534815 0.844969i \(-0.320381\pi\)
0.534815 + 0.844969i \(0.320381\pi\)
\(710\) −12.0278 −0.451395
\(711\) 51.4973 1.93130
\(712\) −12.3666 −0.463460
\(713\) −0.0681329 −0.00255160
\(714\) −0.257069 −0.00962057
\(715\) 12.5252 0.468418
\(716\) −1.28350 −0.0479666
\(717\) 1.86089 0.0694961
\(718\) 10.7493 0.401162
\(719\) 13.8368 0.516024 0.258012 0.966142i \(-0.416933\pi\)
0.258012 + 0.966142i \(0.416933\pi\)
\(720\) 13.1664 0.490683
\(721\) −7.79106 −0.290154
\(722\) 19.8634 0.739240
\(723\) 1.38270 0.0514233
\(724\) −0.740746 −0.0275296
\(725\) 22.4419 0.833473
\(726\) −2.30625 −0.0855928
\(727\) 9.61759 0.356697 0.178348 0.983967i \(-0.442925\pi\)
0.178348 + 0.983967i \(0.442925\pi\)
\(728\) −2.53469 −0.0939417
\(729\) −26.6856 −0.988356
\(730\) 7.40046 0.273903
\(731\) −26.0370 −0.963015
\(732\) 0.0485894 0.00179592
\(733\) −16.9530 −0.626175 −0.313087 0.949724i \(-0.601363\pi\)
−0.313087 + 0.949724i \(0.601363\pi\)
\(734\) −37.5153 −1.38472
\(735\) −0.607158 −0.0223954
\(736\) −0.114837 −0.00423294
\(737\) −48.4283 −1.78388
\(738\) −51.0468 −1.87906
\(739\) 9.82452 0.361401 0.180700 0.983538i \(-0.442164\pi\)
0.180700 + 0.983538i \(0.442164\pi\)
\(740\) 1.13408 0.0416895
\(741\) −0.301762 −0.0110855
\(742\) −5.64019 −0.207058
\(743\) 28.8650 1.05896 0.529478 0.848324i \(-0.322388\pi\)
0.529478 + 0.848324i \(0.322388\pi\)
\(744\) −0.0879200 −0.00322330
\(745\) 0.0247248 0.000905845 0
\(746\) −42.9890 −1.57394
\(747\) −17.8968 −0.654808
\(748\) 3.55204 0.129875
\(749\) 4.29769 0.157034
\(750\) 1.05953 0.0386886
\(751\) −9.70834 −0.354262 −0.177131 0.984187i \(-0.556682\pi\)
−0.177131 + 0.984187i \(0.556682\pi\)
\(752\) 1.95578 0.0713200
\(753\) 2.05940 0.0750486
\(754\) −15.7869 −0.574923
\(755\) 17.7912 0.647489
\(756\) 0.0253948 0.000923600 0
\(757\) −5.99476 −0.217883 −0.108942 0.994048i \(-0.534746\pi\)
−0.108942 + 0.994048i \(0.534746\pi\)
\(758\) 2.70505 0.0982517
\(759\) −0.0754480 −0.00273859
\(760\) 7.24488 0.262799
\(761\) 15.4793 0.561123 0.280562 0.959836i \(-0.409479\pi\)
0.280562 + 0.959836i \(0.409479\pi\)
\(762\) −0.709667 −0.0257085
\(763\) 3.05934 0.110756
\(764\) −0.976829 −0.0353404
\(765\) −18.3618 −0.663873
\(766\) 34.2198 1.23641
\(767\) −6.23226 −0.225034
\(768\) −0.215886 −0.00779011
\(769\) −27.9740 −1.00877 −0.504384 0.863480i \(-0.668280\pi\)
−0.504384 + 0.863480i \(0.668280\pi\)
\(770\) 4.33819 0.156337
\(771\) 1.55033 0.0558336
\(772\) 2.34626 0.0844436
\(773\) 31.7330 1.14136 0.570679 0.821173i \(-0.306680\pi\)
0.570679 + 0.821173i \(0.306680\pi\)
\(774\) −20.4513 −0.735107
\(775\) 1.43621 0.0515901
\(776\) −7.68659 −0.275932
\(777\) −0.293147 −0.0105166
\(778\) 12.3207 0.441718
\(779\) −26.4226 −0.946687
\(780\) 0.0196782 0.000704594 0
\(781\) −42.9593 −1.53720
\(782\) −1.23345 −0.0441080
\(783\) 2.83403 0.101280
\(784\) 25.4213 0.907903
\(785\) 13.0689 0.466448
\(786\) −1.74768 −0.0623376
\(787\) 19.1339 0.682051 0.341025 0.940054i \(-0.389226\pi\)
0.341025 + 0.940054i \(0.389226\pi\)
\(788\) 0.502266 0.0178925
\(789\) −0.920895 −0.0327847
\(790\) 27.6694 0.984434
\(791\) 0.641508 0.0228094
\(792\) 49.9913 1.77636
\(793\) 10.0045 0.355270
\(794\) 14.1465 0.502039
\(795\) 0.784593 0.0278266
\(796\) −0.547686 −0.0194122
\(797\) −2.45993 −0.0871352 −0.0435676 0.999050i \(-0.513872\pi\)
−0.0435676 + 0.999050i \(0.513872\pi\)
\(798\) −0.104517 −0.00369985
\(799\) −2.72752 −0.0964929
\(800\) 2.42071 0.0855849
\(801\) 12.7430 0.450253
\(802\) 30.5483 1.07870
\(803\) 26.4320 0.932764
\(804\) −0.0760850 −0.00268331
\(805\) 0.0946379 0.00333555
\(806\) −1.01031 −0.0355865
\(807\) −1.90998 −0.0672346
\(808\) 18.5689 0.653252
\(809\) 16.9350 0.595404 0.297702 0.954659i \(-0.403780\pi\)
0.297702 + 0.954659i \(0.403780\pi\)
\(810\) −14.3945 −0.505771
\(811\) −23.7584 −0.834269 −0.417135 0.908845i \(-0.636966\pi\)
−0.417135 + 0.908845i \(0.636966\pi\)
\(812\) 0.343503 0.0120546
\(813\) −1.37723 −0.0483016
\(814\) −64.4764 −2.25990
\(815\) −12.1249 −0.424716
\(816\) −1.49726 −0.0524144
\(817\) −10.5859 −0.370354
\(818\) 12.8413 0.448987
\(819\) 2.61183 0.0912648
\(820\) 1.72305 0.0601714
\(821\) 19.2765 0.672755 0.336378 0.941727i \(-0.390798\pi\)
0.336378 + 0.941727i \(0.390798\pi\)
\(822\) −1.36872 −0.0477397
\(823\) 52.8192 1.84116 0.920580 0.390553i \(-0.127716\pi\)
0.920580 + 0.390553i \(0.127716\pi\)
\(824\) −48.2390 −1.68049
\(825\) 1.59041 0.0553709
\(826\) −2.15858 −0.0751064
\(827\) −25.7144 −0.894176 −0.447088 0.894490i \(-0.647539\pi\)
−0.447088 + 0.894490i \(0.647539\pi\)
\(828\) 0.0608644 0.00211518
\(829\) 23.3380 0.810562 0.405281 0.914192i \(-0.367174\pi\)
0.405281 + 0.914192i \(0.367174\pi\)
\(830\) −9.61591 −0.333773
\(831\) −2.24268 −0.0777976
\(832\) −15.6418 −0.542281
\(833\) −35.4524 −1.22835
\(834\) 1.13623 0.0393446
\(835\) −18.3471 −0.634926
\(836\) 1.44415 0.0499471
\(837\) 0.181368 0.00626900
\(838\) 47.9845 1.65760
\(839\) 18.8028 0.649144 0.324572 0.945861i \(-0.394780\pi\)
0.324572 + 0.945861i \(0.394780\pi\)
\(840\) 0.122123 0.00421363
\(841\) 9.33453 0.321880
\(842\) −21.8448 −0.752821
\(843\) 0.923809 0.0318177
\(844\) −1.21408 −0.0417904
\(845\) −11.1941 −0.385089
\(846\) −2.14239 −0.0736568
\(847\) 10.3322 0.355019
\(848\) −32.8504 −1.12809
\(849\) −0.614069 −0.0210748
\(850\) 26.0005 0.891810
\(851\) −1.40656 −0.0482162
\(852\) −0.0674927 −0.00231226
\(853\) −45.0553 −1.54266 −0.771332 0.636433i \(-0.780409\pi\)
−0.771332 + 0.636433i \(0.780409\pi\)
\(854\) 3.46511 0.118573
\(855\) −7.46538 −0.255311
\(856\) 26.6095 0.909495
\(857\) −33.4362 −1.14216 −0.571079 0.820895i \(-0.693475\pi\)
−0.571079 + 0.820895i \(0.693475\pi\)
\(858\) −1.11878 −0.0381944
\(859\) 19.4483 0.663567 0.331784 0.943355i \(-0.392350\pi\)
0.331784 + 0.943355i \(0.392350\pi\)
\(860\) 0.690319 0.0235397
\(861\) −0.445390 −0.0151789
\(862\) 51.4770 1.75332
\(863\) −16.1628 −0.550186 −0.275093 0.961418i \(-0.588709\pi\)
−0.275093 + 0.961418i \(0.588709\pi\)
\(864\) 0.305693 0.0103999
\(865\) −6.67669 −0.227014
\(866\) −38.1048 −1.29485
\(867\) 0.789902 0.0268265
\(868\) 0.0219830 0.000746153 0
\(869\) 98.8259 3.35244
\(870\) 0.760620 0.0257874
\(871\) −15.6658 −0.530815
\(872\) 18.9422 0.641463
\(873\) 7.92054 0.268070
\(874\) −0.501484 −0.0169630
\(875\) −4.74681 −0.160471
\(876\) 0.0415269 0.00140306
\(877\) −22.3959 −0.756255 −0.378127 0.925754i \(-0.623432\pi\)
−0.378127 + 0.925754i \(0.623432\pi\)
\(878\) −51.2370 −1.72917
\(879\) −1.03269 −0.0348319
\(880\) 25.2670 0.851751
\(881\) 10.4074 0.350633 0.175316 0.984512i \(-0.443905\pi\)
0.175316 + 0.984512i \(0.443905\pi\)
\(882\) −27.8468 −0.937650
\(883\) −18.3582 −0.617803 −0.308902 0.951094i \(-0.599961\pi\)
−0.308902 + 0.951094i \(0.599961\pi\)
\(884\) 1.14903 0.0386460
\(885\) 0.300274 0.0100936
\(886\) −21.7077 −0.729286
\(887\) −22.9758 −0.771453 −0.385726 0.922613i \(-0.626049\pi\)
−0.385726 + 0.922613i \(0.626049\pi\)
\(888\) −1.81505 −0.0609091
\(889\) 3.17938 0.106633
\(890\) 6.84682 0.229506
\(891\) −51.4123 −1.72238
\(892\) −1.74829 −0.0585370
\(893\) −1.10893 −0.0371090
\(894\) −0.00220846 −7.38619e−5 0
\(895\) 12.7327 0.425607
\(896\) −4.79077 −0.160048
\(897\) −0.0244062 −0.000814900 0
\(898\) 45.3230 1.51245
\(899\) 2.45328 0.0818215
\(900\) −1.28299 −0.0427664
\(901\) 45.8130 1.52625
\(902\) −97.9614 −3.26176
\(903\) −0.178440 −0.00593812
\(904\) 3.97196 0.132105
\(905\) 7.34843 0.244270
\(906\) −1.58914 −0.0527957
\(907\) 0.415835 0.0138076 0.00690378 0.999976i \(-0.497802\pi\)
0.00690378 + 0.999976i \(0.497802\pi\)
\(908\) −1.76326 −0.0585159
\(909\) −19.1341 −0.634638
\(910\) 1.40333 0.0465201
\(911\) −13.6089 −0.450881 −0.225441 0.974257i \(-0.572382\pi\)
−0.225441 + 0.974257i \(0.572382\pi\)
\(912\) −0.608740 −0.0201574
\(913\) −34.3448 −1.13665
\(914\) 8.75998 0.289754
\(915\) −0.482022 −0.0159351
\(916\) 2.15734 0.0712804
\(917\) 7.82978 0.258562
\(918\) 3.28342 0.108369
\(919\) −14.8913 −0.491218 −0.245609 0.969369i \(-0.578988\pi\)
−0.245609 + 0.969369i \(0.578988\pi\)
\(920\) 0.585959 0.0193185
\(921\) 0.275388 0.00907433
\(922\) −5.63129 −0.185457
\(923\) −13.8966 −0.457414
\(924\) 0.0243433 0.000800835 0
\(925\) 29.6496 0.974871
\(926\) 30.1082 0.989416
\(927\) 49.7072 1.63260
\(928\) 4.13496 0.135737
\(929\) −19.4977 −0.639699 −0.319850 0.947468i \(-0.603632\pi\)
−0.319850 + 0.947468i \(0.603632\pi\)
\(930\) 0.0486771 0.00159618
\(931\) −14.4139 −0.472397
\(932\) 1.97117 0.0645678
\(933\) −2.10332 −0.0688595
\(934\) 28.7442 0.940539
\(935\) −35.2373 −1.15238
\(936\) 16.1714 0.528578
\(937\) 12.5125 0.408764 0.204382 0.978891i \(-0.434481\pi\)
0.204382 + 0.978891i \(0.434481\pi\)
\(938\) −5.42593 −0.177163
\(939\) −2.30760 −0.0753056
\(940\) 0.0723148 0.00235865
\(941\) 4.08256 0.133088 0.0665439 0.997783i \(-0.478803\pi\)
0.0665439 + 0.997783i \(0.478803\pi\)
\(942\) −1.16734 −0.0380339
\(943\) −2.13704 −0.0695915
\(944\) −12.5723 −0.409192
\(945\) −0.251924 −0.00819509
\(946\) −39.2471 −1.27603
\(947\) −0.698574 −0.0227006 −0.0113503 0.999936i \(-0.503613\pi\)
−0.0113503 + 0.999936i \(0.503613\pi\)
\(948\) 0.155264 0.00504274
\(949\) 8.55032 0.277555
\(950\) 10.5710 0.342970
\(951\) −0.399492 −0.0129544
\(952\) 7.13083 0.231112
\(953\) 7.31223 0.236866 0.118433 0.992962i \(-0.462213\pi\)
0.118433 + 0.992962i \(0.462213\pi\)
\(954\) 35.9847 1.16505
\(955\) 9.69044 0.313575
\(956\) −2.88086 −0.0931737
\(957\) 2.71668 0.0878178
\(958\) −55.0930 −1.77997
\(959\) 6.13202 0.198013
\(960\) 0.753630 0.0243233
\(961\) −30.8430 −0.994935
\(962\) −20.8571 −0.672459
\(963\) −27.4194 −0.883578
\(964\) −2.14058 −0.0689434
\(965\) −23.2756 −0.749267
\(966\) −0.00845323 −0.000271978 0
\(967\) 38.1107 1.22556 0.612778 0.790255i \(-0.290052\pi\)
0.612778 + 0.790255i \(0.290052\pi\)
\(968\) 63.9729 2.05617
\(969\) 0.848946 0.0272721
\(970\) 4.25570 0.136642
\(971\) −51.0424 −1.63803 −0.819014 0.573773i \(-0.805479\pi\)
−0.819014 + 0.573773i \(0.805479\pi\)
\(972\) −0.243109 −0.00779771
\(973\) −5.09045 −0.163192
\(974\) 2.15101 0.0689227
\(975\) 0.514472 0.0164763
\(976\) 20.1819 0.646007
\(977\) 53.4134 1.70885 0.854424 0.519577i \(-0.173910\pi\)
0.854424 + 0.519577i \(0.173910\pi\)
\(978\) 1.08302 0.0346310
\(979\) 24.4546 0.781571
\(980\) 0.939949 0.0300256
\(981\) −19.5187 −0.623185
\(982\) −13.2547 −0.422976
\(983\) 1.20932 0.0385714 0.0192857 0.999814i \(-0.493861\pi\)
0.0192857 + 0.999814i \(0.493861\pi\)
\(984\) −2.75767 −0.0879114
\(985\) −4.98264 −0.158760
\(986\) 44.4132 1.41440
\(987\) −0.0186926 −0.000594993 0
\(988\) 0.467161 0.0148624
\(989\) −0.856179 −0.0272249
\(990\) −27.6778 −0.879658
\(991\) −13.3298 −0.423433 −0.211717 0.977331i \(-0.567905\pi\)
−0.211717 + 0.977331i \(0.567905\pi\)
\(992\) 0.264624 0.00840181
\(993\) −0.424660 −0.0134762
\(994\) −4.81318 −0.152665
\(995\) 5.43321 0.172244
\(996\) −0.0539587 −0.00170975
\(997\) −39.2475 −1.24298 −0.621490 0.783422i \(-0.713472\pi\)
−0.621490 + 0.783422i \(0.713472\pi\)
\(998\) 28.3085 0.896091
\(999\) 3.74423 0.118462
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.1.4 10
3.2 odd 2 4527.2.a.k.1.7 10
4.3 odd 2 8048.2.a.p.1.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.e.1.4 10 1.1 even 1 trivial
4527.2.a.k.1.7 10 3.2 odd 2
8048.2.a.p.1.4 10 4.3 odd 2