Properties

Label 503.2.a.e.1.8
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 1.8
Root \(0.208270\) of defining polynomial
Character \(\chi\) \(=\) 503.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.17266 q^{2} -0.791730 q^{3} -0.624870 q^{4} +0.178789 q^{5} -0.928430 q^{6} -0.0809018 q^{7} -3.07808 q^{8} -2.37316 q^{9} +O(q^{10})\) \(q+1.17266 q^{2} -0.791730 q^{3} -0.624870 q^{4} +0.178789 q^{5} -0.928430 q^{6} -0.0809018 q^{7} -3.07808 q^{8} -2.37316 q^{9} +0.209658 q^{10} -4.30391 q^{11} +0.494728 q^{12} +1.10019 q^{13} -0.0948703 q^{14} -0.141552 q^{15} -2.35980 q^{16} -3.06504 q^{17} -2.78291 q^{18} -2.15853 q^{19} -0.111720 q^{20} +0.0640524 q^{21} -5.04702 q^{22} +9.15620 q^{23} +2.43701 q^{24} -4.96803 q^{25} +1.29015 q^{26} +4.25410 q^{27} +0.0505531 q^{28} -6.17849 q^{29} -0.165993 q^{30} -9.82812 q^{31} +3.38892 q^{32} +3.40754 q^{33} -3.59425 q^{34} -0.0144643 q^{35} +1.48292 q^{36} +4.08321 q^{37} -2.53122 q^{38} -0.871056 q^{39} -0.550326 q^{40} +4.58598 q^{41} +0.0751117 q^{42} +2.03126 q^{43} +2.68938 q^{44} -0.424295 q^{45} +10.7371 q^{46} -5.12836 q^{47} +1.86832 q^{48} -6.99345 q^{49} -5.82581 q^{50} +2.42668 q^{51} -0.687477 q^{52} +2.69017 q^{53} +4.98861 q^{54} -0.769491 q^{55} +0.249022 q^{56} +1.70897 q^{57} -7.24527 q^{58} +9.21598 q^{59} +0.0884518 q^{60} -14.0554 q^{61} -11.5250 q^{62} +0.191993 q^{63} +8.69364 q^{64} +0.196702 q^{65} +3.99588 q^{66} +5.62801 q^{67} +1.91525 q^{68} -7.24924 q^{69} -0.0169617 q^{70} -4.88191 q^{71} +7.30478 q^{72} +10.9754 q^{73} +4.78821 q^{74} +3.93334 q^{75} +1.34880 q^{76} +0.348194 q^{77} -1.02145 q^{78} +17.4356 q^{79} -0.421905 q^{80} +3.75139 q^{81} +5.37780 q^{82} -2.48372 q^{83} -0.0400244 q^{84} -0.547994 q^{85} +2.38198 q^{86} +4.89170 q^{87} +13.2478 q^{88} +5.57230 q^{89} -0.497553 q^{90} -0.0890076 q^{91} -5.72143 q^{92} +7.78122 q^{93} -6.01382 q^{94} -0.385920 q^{95} -2.68311 q^{96} -13.9731 q^{97} -8.20094 q^{98} +10.2139 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} + O(q^{10}) \) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} - 4q^{10} - 3q^{11} - 7q^{12} - 18q^{13} + q^{14} - 2q^{15} - 4q^{16} - 11q^{17} - q^{18} - 3q^{20} + q^{21} - 18q^{22} - 2q^{23} + 10q^{24} - 27q^{25} + 11q^{26} - 2q^{27} - 22q^{28} - 9q^{29} + 12q^{30} - 22q^{31} - 10q^{32} - 10q^{33} - 10q^{34} - 6q^{35} + 2q^{36} - 35q^{37} + 2q^{38} + 8q^{39} - 19q^{40} - 4q^{41} + 4q^{42} - 20q^{43} + 9q^{44} + 2q^{45} - q^{46} + 7q^{47} - 27q^{49} + 16q^{50} + 9q^{51} - 7q^{52} - 24q^{53} + 17q^{54} - 11q^{55} + 12q^{56} - 23q^{57} + 2q^{58} + 17q^{59} - 4q^{61} + 8q^{62} + 10q^{63} + 3q^{64} - 16q^{65} + 46q^{66} - 6q^{67} + 28q^{68} - 2q^{69} + 26q^{70} - q^{71} - q^{72} - 31q^{73} + 11q^{74} + 30q^{75} + 20q^{76} + 3q^{77} + 11q^{78} - 10q^{79} + 24q^{80} - 6q^{81} - 9q^{82} + 22q^{83} + 22q^{84} - 6q^{85} + 38q^{86} + 25q^{87} - 3q^{88} + q^{89} + 2q^{90} + 10q^{91} + 27q^{92} - 6q^{93} + 33q^{94} + 39q^{95} + 46q^{96} - 57q^{97} + 40q^{98} + 35q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.17266 0.829195 0.414598 0.910005i \(-0.363922\pi\)
0.414598 + 0.910005i \(0.363922\pi\)
\(3\) −0.791730 −0.457106 −0.228553 0.973532i \(-0.573399\pi\)
−0.228553 + 0.973532i \(0.573399\pi\)
\(4\) −0.624870 −0.312435
\(5\) 0.178789 0.0799568 0.0399784 0.999201i \(-0.487271\pi\)
0.0399784 + 0.999201i \(0.487271\pi\)
\(6\) −0.928430 −0.379030
\(7\) −0.0809018 −0.0305780 −0.0152890 0.999883i \(-0.504867\pi\)
−0.0152890 + 0.999883i \(0.504867\pi\)
\(8\) −3.07808 −1.08827
\(9\) −2.37316 −0.791054
\(10\) 0.209658 0.0662998
\(11\) −4.30391 −1.29768 −0.648839 0.760926i \(-0.724745\pi\)
−0.648839 + 0.760926i \(0.724745\pi\)
\(12\) 0.494728 0.142816
\(13\) 1.10019 0.305139 0.152569 0.988293i \(-0.451245\pi\)
0.152569 + 0.988293i \(0.451245\pi\)
\(14\) −0.0948703 −0.0253551
\(15\) −0.141552 −0.0365487
\(16\) −2.35980 −0.589950
\(17\) −3.06504 −0.743381 −0.371690 0.928357i \(-0.621222\pi\)
−0.371690 + 0.928357i \(0.621222\pi\)
\(18\) −2.78291 −0.655939
\(19\) −2.15853 −0.495200 −0.247600 0.968862i \(-0.579642\pi\)
−0.247600 + 0.968862i \(0.579642\pi\)
\(20\) −0.111720 −0.0249813
\(21\) 0.0640524 0.0139774
\(22\) −5.04702 −1.07603
\(23\) 9.15620 1.90920 0.954600 0.297892i \(-0.0962835\pi\)
0.954600 + 0.297892i \(0.0962835\pi\)
\(24\) 2.43701 0.497452
\(25\) −4.96803 −0.993607
\(26\) 1.29015 0.253020
\(27\) 4.25410 0.818701
\(28\) 0.0505531 0.00955364
\(29\) −6.17849 −1.14732 −0.573658 0.819095i \(-0.694476\pi\)
−0.573658 + 0.819095i \(0.694476\pi\)
\(30\) −0.165993 −0.0303060
\(31\) −9.82812 −1.76518 −0.882591 0.470141i \(-0.844203\pi\)
−0.882591 + 0.470141i \(0.844203\pi\)
\(32\) 3.38892 0.599082
\(33\) 3.40754 0.593176
\(34\) −3.59425 −0.616408
\(35\) −0.0144643 −0.00244492
\(36\) 1.48292 0.247153
\(37\) 4.08321 0.671275 0.335637 0.941991i \(-0.391048\pi\)
0.335637 + 0.941991i \(0.391048\pi\)
\(38\) −2.53122 −0.410618
\(39\) −0.871056 −0.139481
\(40\) −0.550326 −0.0870141
\(41\) 4.58598 0.716210 0.358105 0.933681i \(-0.383423\pi\)
0.358105 + 0.933681i \(0.383423\pi\)
\(42\) 0.0751117 0.0115900
\(43\) 2.03126 0.309764 0.154882 0.987933i \(-0.450500\pi\)
0.154882 + 0.987933i \(0.450500\pi\)
\(44\) 2.68938 0.405440
\(45\) −0.424295 −0.0632501
\(46\) 10.7371 1.58310
\(47\) −5.12836 −0.748049 −0.374024 0.927419i \(-0.622022\pi\)
−0.374024 + 0.927419i \(0.622022\pi\)
\(48\) 1.86832 0.269669
\(49\) −6.99345 −0.999065
\(50\) −5.82581 −0.823894
\(51\) 2.42668 0.339804
\(52\) −0.687477 −0.0953359
\(53\) 2.69017 0.369523 0.184761 0.982783i \(-0.440849\pi\)
0.184761 + 0.982783i \(0.440849\pi\)
\(54\) 4.98861 0.678863
\(55\) −0.769491 −0.103758
\(56\) 0.249022 0.0332770
\(57\) 1.70897 0.226359
\(58\) −7.24527 −0.951350
\(59\) 9.21598 1.19982 0.599909 0.800068i \(-0.295203\pi\)
0.599909 + 0.800068i \(0.295203\pi\)
\(60\) 0.0884518 0.0114191
\(61\) −14.0554 −1.79961 −0.899805 0.436291i \(-0.856292\pi\)
−0.899805 + 0.436291i \(0.856292\pi\)
\(62\) −11.5250 −1.46368
\(63\) 0.191993 0.0241889
\(64\) 8.69364 1.08671
\(65\) 0.196702 0.0243979
\(66\) 3.99588 0.491859
\(67\) 5.62801 0.687570 0.343785 0.939048i \(-0.388291\pi\)
0.343785 + 0.939048i \(0.388291\pi\)
\(68\) 1.91525 0.232258
\(69\) −7.24924 −0.872706
\(70\) −0.0169617 −0.00202732
\(71\) −4.88191 −0.579376 −0.289688 0.957121i \(-0.593552\pi\)
−0.289688 + 0.957121i \(0.593552\pi\)
\(72\) 7.30478 0.860877
\(73\) 10.9754 1.28457 0.642286 0.766465i \(-0.277986\pi\)
0.642286 + 0.766465i \(0.277986\pi\)
\(74\) 4.78821 0.556618
\(75\) 3.93334 0.454183
\(76\) 1.34880 0.154718
\(77\) 0.348194 0.0396804
\(78\) −1.02145 −0.115657
\(79\) 17.4356 1.96166 0.980828 0.194874i \(-0.0624299\pi\)
0.980828 + 0.194874i \(0.0624299\pi\)
\(80\) −0.421905 −0.0471705
\(81\) 3.75139 0.416822
\(82\) 5.37780 0.593878
\(83\) −2.48372 −0.272624 −0.136312 0.990666i \(-0.543525\pi\)
−0.136312 + 0.990666i \(0.543525\pi\)
\(84\) −0.0400244 −0.00436702
\(85\) −0.547994 −0.0594383
\(86\) 2.38198 0.256855
\(87\) 4.89170 0.524445
\(88\) 13.2478 1.41222
\(89\) 5.57230 0.590662 0.295331 0.955395i \(-0.404570\pi\)
0.295331 + 0.955395i \(0.404570\pi\)
\(90\) −0.497553 −0.0524467
\(91\) −0.0890076 −0.00933053
\(92\) −5.72143 −0.596500
\(93\) 7.78122 0.806875
\(94\) −6.01382 −0.620279
\(95\) −0.385920 −0.0395946
\(96\) −2.68311 −0.273844
\(97\) −13.9731 −1.41876 −0.709379 0.704828i \(-0.751024\pi\)
−0.709379 + 0.704828i \(0.751024\pi\)
\(98\) −8.20094 −0.828420
\(99\) 10.2139 1.02653
\(100\) 3.10437 0.310437
\(101\) −7.98379 −0.794417 −0.397208 0.917728i \(-0.630021\pi\)
−0.397208 + 0.917728i \(0.630021\pi\)
\(102\) 2.84567 0.281764
\(103\) 10.5314 1.03769 0.518846 0.854868i \(-0.326362\pi\)
0.518846 + 0.854868i \(0.326362\pi\)
\(104\) −3.38648 −0.332072
\(105\) 0.0114518 0.00111759
\(106\) 3.15465 0.306407
\(107\) −3.31671 −0.320638 −0.160319 0.987065i \(-0.551252\pi\)
−0.160319 + 0.987065i \(0.551252\pi\)
\(108\) −2.65826 −0.255791
\(109\) −12.7956 −1.22559 −0.612797 0.790240i \(-0.709956\pi\)
−0.612797 + 0.790240i \(0.709956\pi\)
\(110\) −0.902350 −0.0860357
\(111\) −3.23280 −0.306844
\(112\) 0.190912 0.0180395
\(113\) −4.42086 −0.415880 −0.207940 0.978142i \(-0.566676\pi\)
−0.207940 + 0.978142i \(0.566676\pi\)
\(114\) 2.00404 0.187696
\(115\) 1.63702 0.152653
\(116\) 3.86075 0.358462
\(117\) −2.61094 −0.241381
\(118\) 10.8072 0.994884
\(119\) 0.247967 0.0227311
\(120\) 0.435709 0.0397747
\(121\) 7.52364 0.683967
\(122\) −16.4822 −1.49223
\(123\) −3.63086 −0.327384
\(124\) 6.14130 0.551505
\(125\) −1.78217 −0.159402
\(126\) 0.225143 0.0200573
\(127\) −17.5071 −1.55351 −0.776753 0.629806i \(-0.783135\pi\)
−0.776753 + 0.629806i \(0.783135\pi\)
\(128\) 3.41685 0.302010
\(129\) −1.60821 −0.141595
\(130\) 0.230665 0.0202306
\(131\) 11.8336 1.03390 0.516952 0.856014i \(-0.327066\pi\)
0.516952 + 0.856014i \(0.327066\pi\)
\(132\) −2.12927 −0.185329
\(133\) 0.174629 0.0151422
\(134\) 6.59973 0.570130
\(135\) 0.760584 0.0654607
\(136\) 9.43443 0.808995
\(137\) −2.19499 −0.187530 −0.0937651 0.995594i \(-0.529890\pi\)
−0.0937651 + 0.995594i \(0.529890\pi\)
\(138\) −8.50089 −0.723644
\(139\) 6.11773 0.518899 0.259450 0.965757i \(-0.416459\pi\)
0.259450 + 0.965757i \(0.416459\pi\)
\(140\) 0.00903832 0.000763878 0
\(141\) 4.06028 0.341937
\(142\) −5.72482 −0.480416
\(143\) −4.73513 −0.395972
\(144\) 5.60019 0.466682
\(145\) −1.10464 −0.0917357
\(146\) 12.8704 1.06516
\(147\) 5.53693 0.456678
\(148\) −2.55147 −0.209730
\(149\) −17.3950 −1.42505 −0.712526 0.701646i \(-0.752449\pi\)
−0.712526 + 0.701646i \(0.752449\pi\)
\(150\) 4.61247 0.376607
\(151\) −21.9595 −1.78704 −0.893520 0.449024i \(-0.851772\pi\)
−0.893520 + 0.449024i \(0.851772\pi\)
\(152\) 6.64411 0.538909
\(153\) 7.27383 0.588055
\(154\) 0.408313 0.0329028
\(155\) −1.75716 −0.141138
\(156\) 0.544296 0.0435786
\(157\) −19.5736 −1.56214 −0.781070 0.624444i \(-0.785326\pi\)
−0.781070 + 0.624444i \(0.785326\pi\)
\(158\) 20.4460 1.62660
\(159\) −2.12989 −0.168911
\(160\) 0.605900 0.0479006
\(161\) −0.740753 −0.0583795
\(162\) 4.39911 0.345627
\(163\) −14.2549 −1.11653 −0.558265 0.829663i \(-0.688533\pi\)
−0.558265 + 0.829663i \(0.688533\pi\)
\(164\) −2.86564 −0.223769
\(165\) 0.609229 0.0474284
\(166\) −2.91256 −0.226058
\(167\) −19.3555 −1.49777 −0.748886 0.662698i \(-0.769411\pi\)
−0.748886 + 0.662698i \(0.769411\pi\)
\(168\) −0.197158 −0.0152111
\(169\) −11.7896 −0.906890
\(170\) −0.642611 −0.0492860
\(171\) 5.12254 0.391730
\(172\) −1.26927 −0.0967811
\(173\) 24.1751 1.83800 0.919000 0.394258i \(-0.128998\pi\)
0.919000 + 0.394258i \(0.128998\pi\)
\(174\) 5.73629 0.434867
\(175\) 0.401923 0.0303825
\(176\) 10.1564 0.765564
\(177\) −7.29657 −0.548444
\(178\) 6.53441 0.489775
\(179\) 11.7844 0.880807 0.440403 0.897800i \(-0.354835\pi\)
0.440403 + 0.897800i \(0.354835\pi\)
\(180\) 0.265129 0.0197616
\(181\) 0.193747 0.0144011 0.00720054 0.999974i \(-0.497708\pi\)
0.00720054 + 0.999974i \(0.497708\pi\)
\(182\) −0.104376 −0.00773683
\(183\) 11.1281 0.822612
\(184\) −28.1835 −2.07771
\(185\) 0.730031 0.0536730
\(186\) 9.12472 0.669057
\(187\) 13.1916 0.964669
\(188\) 3.20456 0.233716
\(189\) −0.344164 −0.0250342
\(190\) −0.452553 −0.0328317
\(191\) −20.0106 −1.44791 −0.723957 0.689845i \(-0.757679\pi\)
−0.723957 + 0.689845i \(0.757679\pi\)
\(192\) −6.88302 −0.496739
\(193\) −16.4429 −1.18358 −0.591792 0.806091i \(-0.701579\pi\)
−0.591792 + 0.806091i \(0.701579\pi\)
\(194\) −16.3857 −1.17643
\(195\) −0.155735 −0.0111524
\(196\) 4.37000 0.312143
\(197\) 24.3311 1.73352 0.866761 0.498723i \(-0.166198\pi\)
0.866761 + 0.498723i \(0.166198\pi\)
\(198\) 11.9774 0.851197
\(199\) −16.2467 −1.15170 −0.575851 0.817555i \(-0.695329\pi\)
−0.575851 + 0.817555i \(0.695329\pi\)
\(200\) 15.2920 1.08131
\(201\) −4.45586 −0.314292
\(202\) −9.36226 −0.658727
\(203\) 0.499851 0.0350827
\(204\) −1.51636 −0.106166
\(205\) 0.819922 0.0572659
\(206\) 12.3498 0.860449
\(207\) −21.7292 −1.51028
\(208\) −2.59623 −0.180016
\(209\) 9.29010 0.642610
\(210\) 0.0134291 0.000926697 0
\(211\) 13.5844 0.935189 0.467594 0.883943i \(-0.345121\pi\)
0.467594 + 0.883943i \(0.345121\pi\)
\(212\) −1.68100 −0.115452
\(213\) 3.86515 0.264836
\(214\) −3.88937 −0.265872
\(215\) 0.363166 0.0247677
\(216\) −13.0944 −0.890964
\(217\) 0.795113 0.0539758
\(218\) −15.0049 −1.01626
\(219\) −8.68955 −0.587185
\(220\) 0.480831 0.0324176
\(221\) −3.37213 −0.226834
\(222\) −3.79097 −0.254433
\(223\) 24.6725 1.65219 0.826096 0.563530i \(-0.190557\pi\)
0.826096 + 0.563530i \(0.190557\pi\)
\(224\) −0.274169 −0.0183187
\(225\) 11.7900 0.785997
\(226\) −5.18417 −0.344846
\(227\) 17.9854 1.19373 0.596866 0.802341i \(-0.296412\pi\)
0.596866 + 0.802341i \(0.296412\pi\)
\(228\) −1.06788 −0.0707224
\(229\) 15.8558 1.04778 0.523891 0.851786i \(-0.324480\pi\)
0.523891 + 0.851786i \(0.324480\pi\)
\(230\) 1.91967 0.126579
\(231\) −0.275676 −0.0181381
\(232\) 19.0179 1.24858
\(233\) 2.33607 0.153041 0.0765205 0.997068i \(-0.475619\pi\)
0.0765205 + 0.997068i \(0.475619\pi\)
\(234\) −3.06174 −0.200152
\(235\) −0.916894 −0.0598115
\(236\) −5.75879 −0.374865
\(237\) −13.8043 −0.896684
\(238\) 0.290781 0.0188485
\(239\) 4.84852 0.313625 0.156812 0.987628i \(-0.449878\pi\)
0.156812 + 0.987628i \(0.449878\pi\)
\(240\) 0.334035 0.0215619
\(241\) −3.42679 −0.220739 −0.110369 0.993891i \(-0.535203\pi\)
−0.110369 + 0.993891i \(0.535203\pi\)
\(242\) 8.82267 0.567143
\(243\) −15.7324 −1.00923
\(244\) 8.78280 0.562261
\(245\) −1.25035 −0.0798820
\(246\) −4.25776 −0.271465
\(247\) −2.37480 −0.151105
\(248\) 30.2517 1.92099
\(249\) 1.96644 0.124618
\(250\) −2.08988 −0.132176
\(251\) −25.7166 −1.62321 −0.811607 0.584204i \(-0.801407\pi\)
−0.811607 + 0.584204i \(0.801407\pi\)
\(252\) −0.119971 −0.00755745
\(253\) −39.4074 −2.47753
\(254\) −20.5299 −1.28816
\(255\) 0.433864 0.0271696
\(256\) −13.3805 −0.836280
\(257\) −24.1897 −1.50891 −0.754457 0.656350i \(-0.772100\pi\)
−0.754457 + 0.656350i \(0.772100\pi\)
\(258\) −1.88588 −0.117410
\(259\) −0.330339 −0.0205262
\(260\) −0.122913 −0.00762275
\(261\) 14.6626 0.907590
\(262\) 13.8768 0.857309
\(263\) 18.1779 1.12089 0.560447 0.828190i \(-0.310629\pi\)
0.560447 + 0.828190i \(0.310629\pi\)
\(264\) −10.4887 −0.645532
\(265\) 0.480972 0.0295458
\(266\) 0.204780 0.0125559
\(267\) −4.41176 −0.269995
\(268\) −3.51677 −0.214821
\(269\) −5.69010 −0.346931 −0.173466 0.984840i \(-0.555497\pi\)
−0.173466 + 0.984840i \(0.555497\pi\)
\(270\) 0.891906 0.0542797
\(271\) 16.4138 0.997069 0.498535 0.866870i \(-0.333872\pi\)
0.498535 + 0.866870i \(0.333872\pi\)
\(272\) 7.23287 0.438557
\(273\) 0.0704700 0.00426504
\(274\) −2.57397 −0.155499
\(275\) 21.3820 1.28938
\(276\) 4.52983 0.272664
\(277\) −26.4497 −1.58921 −0.794605 0.607127i \(-0.792322\pi\)
−0.794605 + 0.607127i \(0.792322\pi\)
\(278\) 7.17402 0.430269
\(279\) 23.3237 1.39636
\(280\) 0.0445223 0.00266072
\(281\) 3.32607 0.198417 0.0992084 0.995067i \(-0.468369\pi\)
0.0992084 + 0.995067i \(0.468369\pi\)
\(282\) 4.76133 0.283533
\(283\) −21.9817 −1.30668 −0.653339 0.757065i \(-0.726633\pi\)
−0.653339 + 0.757065i \(0.726633\pi\)
\(284\) 3.05056 0.181017
\(285\) 0.305545 0.0180989
\(286\) −5.55270 −0.328338
\(287\) −0.371014 −0.0219003
\(288\) −8.04245 −0.473906
\(289\) −7.60555 −0.447385
\(290\) −1.29537 −0.0760669
\(291\) 11.0630 0.648522
\(292\) −6.85819 −0.401345
\(293\) −6.54118 −0.382140 −0.191070 0.981576i \(-0.561196\pi\)
−0.191070 + 0.981576i \(0.561196\pi\)
\(294\) 6.49293 0.378676
\(295\) 1.64771 0.0959336
\(296\) −12.5684 −0.730525
\(297\) −18.3092 −1.06241
\(298\) −20.3984 −1.18165
\(299\) 10.0736 0.582570
\(300\) −2.45783 −0.141903
\(301\) −0.164333 −0.00947197
\(302\) −25.7510 −1.48181
\(303\) 6.32101 0.363132
\(304\) 5.09369 0.292143
\(305\) −2.51295 −0.143891
\(306\) 8.52973 0.487612
\(307\) 15.6985 0.895959 0.447980 0.894044i \(-0.352144\pi\)
0.447980 + 0.894044i \(0.352144\pi\)
\(308\) −0.217576 −0.0123975
\(309\) −8.33804 −0.474335
\(310\) −2.06055 −0.117031
\(311\) 25.6166 1.45258 0.726292 0.687386i \(-0.241242\pi\)
0.726292 + 0.687386i \(0.241242\pi\)
\(312\) 2.68118 0.151792
\(313\) 3.69793 0.209019 0.104510 0.994524i \(-0.466673\pi\)
0.104510 + 0.994524i \(0.466673\pi\)
\(314\) −22.9531 −1.29532
\(315\) 0.0343262 0.00193406
\(316\) −10.8950 −0.612890
\(317\) 19.0196 1.06825 0.534124 0.845406i \(-0.320641\pi\)
0.534124 + 0.845406i \(0.320641\pi\)
\(318\) −2.49763 −0.140060
\(319\) 26.5917 1.48885
\(320\) 1.55433 0.0868894
\(321\) 2.62594 0.146566
\(322\) −0.868651 −0.0484080
\(323\) 6.61597 0.368122
\(324\) −2.34413 −0.130230
\(325\) −5.46580 −0.303188
\(326\) −16.7161 −0.925822
\(327\) 10.1306 0.560226
\(328\) −14.1160 −0.779427
\(329\) 0.414894 0.0228738
\(330\) 0.714418 0.0393274
\(331\) −4.74934 −0.261048 −0.130524 0.991445i \(-0.541666\pi\)
−0.130524 + 0.991445i \(0.541666\pi\)
\(332\) 1.55200 0.0851771
\(333\) −9.69011 −0.531015
\(334\) −22.6974 −1.24195
\(335\) 1.00622 0.0549759
\(336\) −0.151151 −0.00824595
\(337\) −29.8451 −1.62576 −0.812882 0.582428i \(-0.802103\pi\)
−0.812882 + 0.582428i \(0.802103\pi\)
\(338\) −13.8252 −0.751989
\(339\) 3.50013 0.190101
\(340\) 0.342425 0.0185706
\(341\) 42.2993 2.29064
\(342\) 6.00699 0.324821
\(343\) 1.13210 0.0611274
\(344\) −6.25238 −0.337105
\(345\) −1.29608 −0.0697787
\(346\) 28.3492 1.52406
\(347\) 15.3781 0.825540 0.412770 0.910835i \(-0.364561\pi\)
0.412770 + 0.910835i \(0.364561\pi\)
\(348\) −3.05667 −0.163855
\(349\) 6.64130 0.355501 0.177750 0.984076i \(-0.443118\pi\)
0.177750 + 0.984076i \(0.443118\pi\)
\(350\) 0.471319 0.0251930
\(351\) 4.68033 0.249817
\(352\) −14.5856 −0.777415
\(353\) −33.7272 −1.79512 −0.897558 0.440896i \(-0.854661\pi\)
−0.897558 + 0.440896i \(0.854661\pi\)
\(354\) −8.55639 −0.454767
\(355\) −0.872830 −0.0463250
\(356\) −3.48196 −0.184544
\(357\) −0.196323 −0.0103905
\(358\) 13.8191 0.730361
\(359\) −27.2110 −1.43614 −0.718071 0.695970i \(-0.754975\pi\)
−0.718071 + 0.695970i \(0.754975\pi\)
\(360\) 1.30601 0.0688329
\(361\) −14.3408 −0.754777
\(362\) 0.227199 0.0119413
\(363\) −5.95669 −0.312645
\(364\) 0.0556181 0.00291518
\(365\) 1.96228 0.102710
\(366\) 13.0495 0.682106
\(367\) 14.0523 0.733525 0.366763 0.930315i \(-0.380466\pi\)
0.366763 + 0.930315i \(0.380466\pi\)
\(368\) −21.6068 −1.12633
\(369\) −10.8833 −0.566561
\(370\) 0.856078 0.0445054
\(371\) −0.217639 −0.0112993
\(372\) −4.86225 −0.252096
\(373\) 19.8673 1.02869 0.514345 0.857584i \(-0.328035\pi\)
0.514345 + 0.857584i \(0.328035\pi\)
\(374\) 15.4693 0.799899
\(375\) 1.41100 0.0728637
\(376\) 15.7855 0.814075
\(377\) −6.79753 −0.350091
\(378\) −0.403587 −0.0207583
\(379\) −16.4863 −0.846844 −0.423422 0.905933i \(-0.639171\pi\)
−0.423422 + 0.905933i \(0.639171\pi\)
\(380\) 0.241150 0.0123707
\(381\) 13.8609 0.710116
\(382\) −23.4656 −1.20060
\(383\) 22.2392 1.13637 0.568186 0.822900i \(-0.307645\pi\)
0.568186 + 0.822900i \(0.307645\pi\)
\(384\) −2.70522 −0.138050
\(385\) 0.0622532 0.00317272
\(386\) −19.2819 −0.981422
\(387\) −4.82051 −0.245040
\(388\) 8.73139 0.443269
\(389\) −10.3650 −0.525528 −0.262764 0.964860i \(-0.584634\pi\)
−0.262764 + 0.964860i \(0.584634\pi\)
\(390\) −0.182624 −0.00924753
\(391\) −28.0641 −1.41926
\(392\) 21.5264 1.08725
\(393\) −9.36900 −0.472604
\(394\) 28.5322 1.43743
\(395\) 3.11729 0.156848
\(396\) −6.38235 −0.320725
\(397\) −2.54363 −0.127661 −0.0638306 0.997961i \(-0.520332\pi\)
−0.0638306 + 0.997961i \(0.520332\pi\)
\(398\) −19.0519 −0.954985
\(399\) −0.138259 −0.00692160
\(400\) 11.7236 0.586178
\(401\) 12.4246 0.620455 0.310227 0.950662i \(-0.399595\pi\)
0.310227 + 0.950662i \(0.399595\pi\)
\(402\) −5.22521 −0.260610
\(403\) −10.8128 −0.538625
\(404\) 4.98883 0.248203
\(405\) 0.670707 0.0333277
\(406\) 0.586155 0.0290904
\(407\) −17.5738 −0.871098
\(408\) −7.46952 −0.369796
\(409\) 14.3575 0.709932 0.354966 0.934879i \(-0.384492\pi\)
0.354966 + 0.934879i \(0.384492\pi\)
\(410\) 0.961490 0.0474846
\(411\) 1.73784 0.0857211
\(412\) −6.58076 −0.324211
\(413\) −0.745589 −0.0366880
\(414\) −25.4809 −1.25232
\(415\) −0.444061 −0.0217981
\(416\) 3.72846 0.182803
\(417\) −4.84359 −0.237192
\(418\) 10.8941 0.532849
\(419\) 15.6860 0.766309 0.383155 0.923684i \(-0.374838\pi\)
0.383155 + 0.923684i \(0.374838\pi\)
\(420\) −0.00715591 −0.000349173 0
\(421\) 0.689920 0.0336247 0.0168123 0.999859i \(-0.494648\pi\)
0.0168123 + 0.999859i \(0.494648\pi\)
\(422\) 15.9299 0.775454
\(423\) 12.1704 0.591747
\(424\) −8.28054 −0.402139
\(425\) 15.2272 0.738628
\(426\) 4.53251 0.219601
\(427\) 1.13711 0.0550285
\(428\) 2.07251 0.100179
\(429\) 3.74895 0.181001
\(430\) 0.425870 0.0205373
\(431\) 29.0858 1.40102 0.700508 0.713645i \(-0.252957\pi\)
0.700508 + 0.713645i \(0.252957\pi\)
\(432\) −10.0388 −0.482992
\(433\) 18.1169 0.870643 0.435321 0.900275i \(-0.356635\pi\)
0.435321 + 0.900275i \(0.356635\pi\)
\(434\) 0.932396 0.0447565
\(435\) 0.874580 0.0419329
\(436\) 7.99557 0.382918
\(437\) −19.7639 −0.945435
\(438\) −10.1899 −0.486891
\(439\) 3.25712 0.155454 0.0777270 0.996975i \(-0.475234\pi\)
0.0777270 + 0.996975i \(0.475234\pi\)
\(440\) 2.36855 0.112916
\(441\) 16.5966 0.790315
\(442\) −3.95436 −0.188090
\(443\) −21.5717 −1.02490 −0.512451 0.858717i \(-0.671262\pi\)
−0.512451 + 0.858717i \(0.671262\pi\)
\(444\) 2.02008 0.0958686
\(445\) 0.996264 0.0472274
\(446\) 28.9324 1.36999
\(447\) 13.7721 0.651399
\(448\) −0.703331 −0.0332293
\(449\) 18.9612 0.894834 0.447417 0.894325i \(-0.352344\pi\)
0.447417 + 0.894325i \(0.352344\pi\)
\(450\) 13.8256 0.651745
\(451\) −19.7377 −0.929410
\(452\) 2.76246 0.129935
\(453\) 17.3860 0.816866
\(454\) 21.0907 0.989837
\(455\) −0.0159136 −0.000746039 0
\(456\) −5.26035 −0.246338
\(457\) 4.18789 0.195901 0.0979506 0.995191i \(-0.468771\pi\)
0.0979506 + 0.995191i \(0.468771\pi\)
\(458\) 18.5935 0.868815
\(459\) −13.0390 −0.608607
\(460\) −1.02293 −0.0476942
\(461\) 13.7101 0.638543 0.319271 0.947663i \(-0.396562\pi\)
0.319271 + 0.947663i \(0.396562\pi\)
\(462\) −0.323274 −0.0150401
\(463\) 4.87808 0.226703 0.113352 0.993555i \(-0.463841\pi\)
0.113352 + 0.993555i \(0.463841\pi\)
\(464\) 14.5800 0.676859
\(465\) 1.39119 0.0645151
\(466\) 2.73941 0.126901
\(467\) 12.3373 0.570904 0.285452 0.958393i \(-0.407856\pi\)
0.285452 + 0.958393i \(0.407856\pi\)
\(468\) 1.63150 0.0754159
\(469\) −0.455316 −0.0210245
\(470\) −1.07520 −0.0495955
\(471\) 15.4970 0.714063
\(472\) −28.3675 −1.30572
\(473\) −8.74236 −0.401974
\(474\) −16.1877 −0.743526
\(475\) 10.7236 0.492034
\(476\) −0.154947 −0.00710199
\(477\) −6.38421 −0.292313
\(478\) 5.68566 0.260056
\(479\) −3.56361 −0.162826 −0.0814128 0.996680i \(-0.525943\pi\)
−0.0814128 + 0.996680i \(0.525943\pi\)
\(480\) −0.479709 −0.0218956
\(481\) 4.49231 0.204832
\(482\) −4.01846 −0.183036
\(483\) 0.586476 0.0266856
\(484\) −4.70130 −0.213695
\(485\) −2.49824 −0.113439
\(486\) −18.4487 −0.836851
\(487\) −5.09262 −0.230769 −0.115384 0.993321i \(-0.536810\pi\)
−0.115384 + 0.993321i \(0.536810\pi\)
\(488\) 43.2637 1.95845
\(489\) 11.2860 0.510372
\(490\) −1.46624 −0.0662378
\(491\) 27.1049 1.22323 0.611614 0.791156i \(-0.290521\pi\)
0.611614 + 0.791156i \(0.290521\pi\)
\(492\) 2.26882 0.102286
\(493\) 18.9373 0.852893
\(494\) −2.78483 −0.125295
\(495\) 1.82613 0.0820783
\(496\) 23.1924 1.04137
\(497\) 0.394955 0.0177162
\(498\) 2.30596 0.103333
\(499\) −4.49778 −0.201348 −0.100674 0.994919i \(-0.532100\pi\)
−0.100674 + 0.994919i \(0.532100\pi\)
\(500\) 1.11363 0.0498028
\(501\) 15.3243 0.684640
\(502\) −30.1568 −1.34596
\(503\) −1.00000 −0.0445878
\(504\) −0.590970 −0.0263239
\(505\) −1.42741 −0.0635190
\(506\) −46.2115 −2.05435
\(507\) 9.33416 0.414545
\(508\) 10.9397 0.485369
\(509\) −9.92656 −0.439987 −0.219993 0.975501i \(-0.570604\pi\)
−0.219993 + 0.975501i \(0.570604\pi\)
\(510\) 0.508774 0.0225289
\(511\) −0.887929 −0.0392797
\(512\) −22.5244 −0.995449
\(513\) −9.18258 −0.405421
\(514\) −28.3663 −1.25118
\(515\) 1.88290 0.0829704
\(516\) 1.00492 0.0442392
\(517\) 22.0720 0.970726
\(518\) −0.387375 −0.0170203
\(519\) −19.1402 −0.840160
\(520\) −0.605464 −0.0265514
\(521\) −8.65790 −0.379309 −0.189655 0.981851i \(-0.560737\pi\)
−0.189655 + 0.981851i \(0.560737\pi\)
\(522\) 17.1942 0.752570
\(523\) 17.7111 0.774451 0.387226 0.921985i \(-0.373433\pi\)
0.387226 + 0.921985i \(0.373433\pi\)
\(524\) −7.39445 −0.323028
\(525\) −0.318215 −0.0138880
\(526\) 21.3164 0.929441
\(527\) 30.1236 1.31220
\(528\) −8.04110 −0.349944
\(529\) 60.8360 2.64504
\(530\) 0.564016 0.0244993
\(531\) −21.8710 −0.949122
\(532\) −0.109120 −0.00473096
\(533\) 5.04547 0.218543
\(534\) −5.17349 −0.223879
\(535\) −0.592990 −0.0256372
\(536\) −17.3234 −0.748259
\(537\) −9.33006 −0.402622
\(538\) −6.67255 −0.287674
\(539\) 30.0992 1.29646
\(540\) −0.475266 −0.0204522
\(541\) −10.8187 −0.465131 −0.232565 0.972581i \(-0.574712\pi\)
−0.232565 + 0.972581i \(0.574712\pi\)
\(542\) 19.2478 0.826765
\(543\) −0.153395 −0.00658282
\(544\) −10.3872 −0.445346
\(545\) −2.28771 −0.0979945
\(546\) 0.0826373 0.00353655
\(547\) −33.1188 −1.41606 −0.708028 0.706184i \(-0.750415\pi\)
−0.708028 + 0.706184i \(0.750415\pi\)
\(548\) 1.37158 0.0585910
\(549\) 33.3558 1.42359
\(550\) 25.0738 1.06915
\(551\) 13.3364 0.568151
\(552\) 22.3137 0.949735
\(553\) −1.41057 −0.0599835
\(554\) −31.0165 −1.31777
\(555\) −0.577988 −0.0245342
\(556\) −3.82279 −0.162122
\(557\) 12.3324 0.522541 0.261270 0.965266i \(-0.415859\pi\)
0.261270 + 0.965266i \(0.415859\pi\)
\(558\) 27.3508 1.15785
\(559\) 2.23478 0.0945210
\(560\) 0.0341329 0.00144238
\(561\) −10.4442 −0.440955
\(562\) 3.90035 0.164526
\(563\) 29.3356 1.23635 0.618174 0.786041i \(-0.287873\pi\)
0.618174 + 0.786041i \(0.287873\pi\)
\(564\) −2.53715 −0.106833
\(565\) −0.790400 −0.0332524
\(566\) −25.7771 −1.08349
\(567\) −0.303495 −0.0127456
\(568\) 15.0269 0.630514
\(569\) 0.339405 0.0142286 0.00711431 0.999975i \(-0.497735\pi\)
0.00711431 + 0.999975i \(0.497735\pi\)
\(570\) 0.358300 0.0150075
\(571\) −16.5993 −0.694658 −0.347329 0.937743i \(-0.612911\pi\)
−0.347329 + 0.937743i \(0.612911\pi\)
\(572\) 2.95884 0.123715
\(573\) 15.8430 0.661850
\(574\) −0.435073 −0.0181596
\(575\) −45.4883 −1.89699
\(576\) −20.6314 −0.859643
\(577\) −19.0207 −0.791840 −0.395920 0.918285i \(-0.629574\pi\)
−0.395920 + 0.918285i \(0.629574\pi\)
\(578\) −8.91872 −0.370970
\(579\) 13.0183 0.541023
\(580\) 0.690259 0.0286614
\(581\) 0.200937 0.00833629
\(582\) 12.9731 0.537751
\(583\) −11.5782 −0.479522
\(584\) −33.7831 −1.39796
\(585\) −0.466806 −0.0193001
\(586\) −7.67058 −0.316869
\(587\) 29.3939 1.21322 0.606609 0.795000i \(-0.292529\pi\)
0.606609 + 0.795000i \(0.292529\pi\)
\(588\) −3.45986 −0.142682
\(589\) 21.2143 0.874118
\(590\) 1.93221 0.0795477
\(591\) −19.2637 −0.792403
\(592\) −9.63554 −0.396018
\(593\) −21.6941 −0.890871 −0.445436 0.895314i \(-0.646951\pi\)
−0.445436 + 0.895314i \(0.646951\pi\)
\(594\) −21.4705 −0.880946
\(595\) 0.0443337 0.00181751
\(596\) 10.8696 0.445236
\(597\) 12.8630 0.526449
\(598\) 11.8129 0.483065
\(599\) −7.41054 −0.302786 −0.151393 0.988474i \(-0.548376\pi\)
−0.151393 + 0.988474i \(0.548376\pi\)
\(600\) −12.1071 −0.494272
\(601\) −17.7208 −0.722848 −0.361424 0.932402i \(-0.617709\pi\)
−0.361424 + 0.932402i \(0.617709\pi\)
\(602\) −0.192706 −0.00785411
\(603\) −13.3562 −0.543906
\(604\) 13.7218 0.558333
\(605\) 1.34514 0.0546878
\(606\) 7.41239 0.301108
\(607\) 37.0318 1.50307 0.751536 0.659692i \(-0.229313\pi\)
0.751536 + 0.659692i \(0.229313\pi\)
\(608\) −7.31507 −0.296665
\(609\) −0.395747 −0.0160365
\(610\) −2.94683 −0.119314
\(611\) −5.64219 −0.228259
\(612\) −4.54520 −0.183729
\(613\) 37.6348 1.52005 0.760027 0.649891i \(-0.225185\pi\)
0.760027 + 0.649891i \(0.225185\pi\)
\(614\) 18.4090 0.742925
\(615\) −0.649157 −0.0261765
\(616\) −1.07177 −0.0431828
\(617\) −8.73601 −0.351699 −0.175849 0.984417i \(-0.556267\pi\)
−0.175849 + 0.984417i \(0.556267\pi\)
\(618\) −9.77768 −0.393316
\(619\) 17.0846 0.686687 0.343343 0.939210i \(-0.388441\pi\)
0.343343 + 0.939210i \(0.388441\pi\)
\(620\) 1.09799 0.0440965
\(621\) 38.9513 1.56306
\(622\) 30.0396 1.20448
\(623\) −0.450809 −0.0180613
\(624\) 2.05552 0.0822865
\(625\) 24.5215 0.980862
\(626\) 4.33641 0.173318
\(627\) −7.35526 −0.293741
\(628\) 12.2309 0.488067
\(629\) −12.5152 −0.499013
\(630\) 0.0402530 0.00160372
\(631\) −38.0472 −1.51464 −0.757318 0.653046i \(-0.773491\pi\)
−0.757318 + 0.653046i \(0.773491\pi\)
\(632\) −53.6681 −2.13480
\(633\) −10.7552 −0.427480
\(634\) 22.3035 0.885787
\(635\) −3.13008 −0.124213
\(636\) 1.33090 0.0527737
\(637\) −7.69415 −0.304853
\(638\) 31.1830 1.23455
\(639\) 11.5856 0.458318
\(640\) 0.610894 0.0241477
\(641\) 27.5905 1.08976 0.544879 0.838514i \(-0.316575\pi\)
0.544879 + 0.838514i \(0.316575\pi\)
\(642\) 3.07933 0.121532
\(643\) 6.89268 0.271821 0.135910 0.990721i \(-0.456604\pi\)
0.135910 + 0.990721i \(0.456604\pi\)
\(644\) 0.462874 0.0182398
\(645\) −0.287530 −0.0113215
\(646\) 7.75827 0.305245
\(647\) 8.60822 0.338424 0.169212 0.985580i \(-0.445878\pi\)
0.169212 + 0.985580i \(0.445878\pi\)
\(648\) −11.5471 −0.453612
\(649\) −39.6647 −1.55698
\(650\) −6.40952 −0.251402
\(651\) −0.629515 −0.0246726
\(652\) 8.90746 0.348843
\(653\) 3.73378 0.146114 0.0730571 0.997328i \(-0.476724\pi\)
0.0730571 + 0.997328i \(0.476724\pi\)
\(654\) 11.8798 0.464537
\(655\) 2.11571 0.0826677
\(656\) −10.8220 −0.422528
\(657\) −26.0464 −1.01617
\(658\) 0.486529 0.0189669
\(659\) 10.1373 0.394893 0.197447 0.980314i \(-0.436735\pi\)
0.197447 + 0.980314i \(0.436735\pi\)
\(660\) −0.380689 −0.0148183
\(661\) −11.0003 −0.427863 −0.213932 0.976849i \(-0.568627\pi\)
−0.213932 + 0.976849i \(0.568627\pi\)
\(662\) −5.56936 −0.216459
\(663\) 2.66982 0.103687
\(664\) 7.64508 0.296687
\(665\) 0.0312216 0.00121072
\(666\) −11.3632 −0.440315
\(667\) −56.5715 −2.19046
\(668\) 12.0947 0.467957
\(669\) −19.5339 −0.755226
\(670\) 1.17996 0.0455858
\(671\) 60.4932 2.33531
\(672\) 0.217068 0.00837359
\(673\) −44.9940 −1.73439 −0.867196 0.497967i \(-0.834080\pi\)
−0.867196 + 0.497967i \(0.834080\pi\)
\(674\) −34.9981 −1.34808
\(675\) −21.1345 −0.813467
\(676\) 7.36695 0.283344
\(677\) 15.4870 0.595213 0.297607 0.954689i \(-0.403812\pi\)
0.297607 + 0.954689i \(0.403812\pi\)
\(678\) 4.10446 0.157631
\(679\) 1.13045 0.0433828
\(680\) 1.68677 0.0646846
\(681\) −14.2396 −0.545661
\(682\) 49.6027 1.89939
\(683\) 33.7255 1.29047 0.645235 0.763984i \(-0.276759\pi\)
0.645235 + 0.763984i \(0.276759\pi\)
\(684\) −3.20092 −0.122390
\(685\) −0.392439 −0.0149943
\(686\) 1.32756 0.0506866
\(687\) −12.5535 −0.478947
\(688\) −4.79336 −0.182745
\(689\) 2.95970 0.112756
\(690\) −1.51986 −0.0578602
\(691\) 13.5728 0.516333 0.258166 0.966100i \(-0.416882\pi\)
0.258166 + 0.966100i \(0.416882\pi\)
\(692\) −15.1063 −0.574255
\(693\) −0.826321 −0.0313894
\(694\) 18.0333 0.684534
\(695\) 1.09378 0.0414895
\(696\) −15.0570 −0.570735
\(697\) −14.0562 −0.532417
\(698\) 7.78799 0.294780
\(699\) −1.84954 −0.0699559
\(700\) −0.251149 −0.00949256
\(701\) −34.7789 −1.31358 −0.656790 0.754074i \(-0.728086\pi\)
−0.656790 + 0.754074i \(0.728086\pi\)
\(702\) 5.48843 0.207147
\(703\) −8.81371 −0.332415
\(704\) −37.4167 −1.41019
\(705\) 0.725932 0.0273402
\(706\) −39.5505 −1.48850
\(707\) 0.645903 0.0242917
\(708\) 4.55940 0.171353
\(709\) −23.1752 −0.870362 −0.435181 0.900343i \(-0.643316\pi\)
−0.435181 + 0.900343i \(0.643316\pi\)
\(710\) −1.02353 −0.0384125
\(711\) −41.3775 −1.55178
\(712\) −17.1520 −0.642797
\(713\) −89.9882 −3.37009
\(714\) −0.230220 −0.00861577
\(715\) −0.846588 −0.0316606
\(716\) −7.36371 −0.275195
\(717\) −3.83872 −0.143360
\(718\) −31.9092 −1.19084
\(719\) −39.5082 −1.47341 −0.736704 0.676215i \(-0.763619\pi\)
−0.736704 + 0.676215i \(0.763619\pi\)
\(720\) 1.00125 0.0373144
\(721\) −0.852011 −0.0317305
\(722\) −16.8168 −0.625858
\(723\) 2.71309 0.100901
\(724\) −0.121067 −0.00449940
\(725\) 30.6950 1.13998
\(726\) −6.98517 −0.259244
\(727\) −5.75881 −0.213583 −0.106791 0.994281i \(-0.534058\pi\)
−0.106791 + 0.994281i \(0.534058\pi\)
\(728\) 0.273972 0.0101541
\(729\) 1.20162 0.0445043
\(730\) 2.30108 0.0851669
\(731\) −6.22589 −0.230273
\(732\) −6.95361 −0.257013
\(733\) −41.9036 −1.54774 −0.773872 0.633343i \(-0.781682\pi\)
−0.773872 + 0.633343i \(0.781682\pi\)
\(734\) 16.4786 0.608236
\(735\) 0.989941 0.0365145
\(736\) 31.0296 1.14377
\(737\) −24.2224 −0.892245
\(738\) −12.7624 −0.469790
\(739\) −14.5650 −0.535780 −0.267890 0.963449i \(-0.586326\pi\)
−0.267890 + 0.963449i \(0.586326\pi\)
\(740\) −0.456174 −0.0167693
\(741\) 1.88020 0.0690708
\(742\) −0.255217 −0.00936931
\(743\) −22.9633 −0.842440 −0.421220 0.906958i \(-0.638398\pi\)
−0.421220 + 0.906958i \(0.638398\pi\)
\(744\) −23.9512 −0.878094
\(745\) −3.11003 −0.113943
\(746\) 23.2976 0.852984
\(747\) 5.89427 0.215660
\(748\) −8.24306 −0.301396
\(749\) 0.268328 0.00980448
\(750\) 1.65462 0.0604183
\(751\) −8.52609 −0.311121 −0.155561 0.987826i \(-0.549718\pi\)
−0.155561 + 0.987826i \(0.549718\pi\)
\(752\) 12.1019 0.441311
\(753\) 20.3606 0.741980
\(754\) −7.97119 −0.290294
\(755\) −3.92611 −0.142886
\(756\) 0.215058 0.00782157
\(757\) −19.7364 −0.717331 −0.358665 0.933466i \(-0.616768\pi\)
−0.358665 + 0.933466i \(0.616768\pi\)
\(758\) −19.3328 −0.702199
\(759\) 31.2001 1.13249
\(760\) 1.18789 0.0430894
\(761\) 44.7253 1.62129 0.810646 0.585537i \(-0.199116\pi\)
0.810646 + 0.585537i \(0.199116\pi\)
\(762\) 16.2541 0.588825
\(763\) 1.03519 0.0374762
\(764\) 12.5040 0.452379
\(765\) 1.30048 0.0470189
\(766\) 26.0791 0.942275
\(767\) 10.1394 0.366111
\(768\) 10.5937 0.382268
\(769\) −28.1170 −1.01393 −0.506963 0.861968i \(-0.669232\pi\)
−0.506963 + 0.861968i \(0.669232\pi\)
\(770\) 0.0730018 0.00263080
\(771\) 19.1517 0.689733
\(772\) 10.2746 0.369793
\(773\) 34.5978 1.24440 0.622199 0.782859i \(-0.286240\pi\)
0.622199 + 0.782859i \(0.286240\pi\)
\(774\) −5.65282 −0.203186
\(775\) 48.8264 1.75390
\(776\) 43.0104 1.54398
\(777\) 0.261539 0.00938266
\(778\) −12.1546 −0.435765
\(779\) −9.89897 −0.354667
\(780\) 0.0973141 0.00348440
\(781\) 21.0113 0.751843
\(782\) −32.9096 −1.17685
\(783\) −26.2839 −0.939310
\(784\) 16.5031 0.589398
\(785\) −3.49953 −0.124904
\(786\) −10.9867 −0.391881
\(787\) −46.1869 −1.64639 −0.823193 0.567762i \(-0.807810\pi\)
−0.823193 + 0.567762i \(0.807810\pi\)
\(788\) −15.2038 −0.541613
\(789\) −14.3920 −0.512367
\(790\) 3.65552 0.130057
\(791\) 0.357656 0.0127168
\(792\) −31.4391 −1.11714
\(793\) −15.4637 −0.549131
\(794\) −2.98282 −0.105856
\(795\) −0.380800 −0.0135056
\(796\) 10.1521 0.359832
\(797\) 31.0936 1.10139 0.550695 0.834707i \(-0.314363\pi\)
0.550695 + 0.834707i \(0.314363\pi\)
\(798\) −0.162131 −0.00573936
\(799\) 15.7186 0.556085
\(800\) −16.8363 −0.595252
\(801\) −13.2240 −0.467246
\(802\) 14.5698 0.514478
\(803\) −47.2371 −1.66696
\(804\) 2.78433 0.0981959
\(805\) −0.132438 −0.00466784
\(806\) −12.6798 −0.446626
\(807\) 4.50502 0.158584
\(808\) 24.5747 0.864536
\(809\) −22.7451 −0.799676 −0.399838 0.916586i \(-0.630934\pi\)
−0.399838 + 0.916586i \(0.630934\pi\)
\(810\) 0.786511 0.0276352
\(811\) −20.2514 −0.711121 −0.355561 0.934653i \(-0.615710\pi\)
−0.355561 + 0.934653i \(0.615710\pi\)
\(812\) −0.312342 −0.0109610
\(813\) −12.9953 −0.455766
\(814\) −20.6080 −0.722311
\(815\) −2.54862 −0.0892741
\(816\) −5.72648 −0.200467
\(817\) −4.38453 −0.153395
\(818\) 16.8364 0.588673
\(819\) 0.211230 0.00738096
\(820\) −0.512345 −0.0178918
\(821\) 47.0787 1.64306 0.821529 0.570167i \(-0.193121\pi\)
0.821529 + 0.570167i \(0.193121\pi\)
\(822\) 2.03789 0.0710796
\(823\) −3.97277 −0.138482 −0.0692411 0.997600i \(-0.522058\pi\)
−0.0692411 + 0.997600i \(0.522058\pi\)
\(824\) −32.4165 −1.12928
\(825\) −16.9288 −0.589384
\(826\) −0.874322 −0.0304216
\(827\) −12.1175 −0.421366 −0.210683 0.977554i \(-0.567569\pi\)
−0.210683 + 0.977554i \(0.567569\pi\)
\(828\) 13.5779 0.471864
\(829\) 27.0304 0.938804 0.469402 0.882985i \(-0.344470\pi\)
0.469402 + 0.882985i \(0.344470\pi\)
\(830\) −0.520732 −0.0180749
\(831\) 20.9410 0.726437
\(832\) 9.56468 0.331596
\(833\) 21.4352 0.742686
\(834\) −5.67989 −0.196678
\(835\) −3.46054 −0.119757
\(836\) −5.80511 −0.200774
\(837\) −41.8098 −1.44516
\(838\) 18.3943 0.635420
\(839\) −21.4624 −0.740964 −0.370482 0.928840i \(-0.620807\pi\)
−0.370482 + 0.928840i \(0.620807\pi\)
\(840\) −0.0352497 −0.00121623
\(841\) 9.17374 0.316336
\(842\) 0.809042 0.0278814
\(843\) −2.63335 −0.0906974
\(844\) −8.48848 −0.292186
\(845\) −2.10784 −0.0725120
\(846\) 14.2718 0.490674
\(847\) −0.608676 −0.0209144
\(848\) −6.34825 −0.218000
\(849\) 17.4036 0.597290
\(850\) 17.8563 0.612467
\(851\) 37.3866 1.28160
\(852\) −2.41522 −0.0827440
\(853\) −20.5093 −0.702226 −0.351113 0.936333i \(-0.614197\pi\)
−0.351113 + 0.936333i \(0.614197\pi\)
\(854\) 1.33344 0.0456294
\(855\) 0.915852 0.0313215
\(856\) 10.2091 0.348939
\(857\) −23.2772 −0.795135 −0.397567 0.917573i \(-0.630146\pi\)
−0.397567 + 0.917573i \(0.630146\pi\)
\(858\) 4.39624 0.150085
\(859\) −3.80551 −0.129842 −0.0649212 0.997890i \(-0.520680\pi\)
−0.0649212 + 0.997890i \(0.520680\pi\)
\(860\) −0.226932 −0.00773830
\(861\) 0.293743 0.0100107
\(862\) 34.1078 1.16172
\(863\) −35.9633 −1.22420 −0.612102 0.790779i \(-0.709676\pi\)
−0.612102 + 0.790779i \(0.709676\pi\)
\(864\) 14.4168 0.490469
\(865\) 4.32224 0.146960
\(866\) 21.2450 0.721933
\(867\) 6.02154 0.204502
\(868\) −0.496842 −0.0168639
\(869\) −75.0412 −2.54560
\(870\) 1.02558 0.0347706
\(871\) 6.19189 0.209804
\(872\) 39.3858 1.33377
\(873\) 33.1605 1.12231
\(874\) −23.1763 −0.783951
\(875\) 0.144181 0.00487421
\(876\) 5.42984 0.183457
\(877\) 14.2388 0.480810 0.240405 0.970673i \(-0.422720\pi\)
0.240405 + 0.970673i \(0.422720\pi\)
\(878\) 3.81949 0.128902
\(879\) 5.17885 0.174678
\(880\) 1.81584 0.0612120
\(881\) 27.4845 0.925976 0.462988 0.886365i \(-0.346777\pi\)
0.462988 + 0.886365i \(0.346777\pi\)
\(882\) 19.4622 0.655325
\(883\) 24.3360 0.818973 0.409487 0.912316i \(-0.365708\pi\)
0.409487 + 0.912316i \(0.365708\pi\)
\(884\) 2.10714 0.0708709
\(885\) −1.30454 −0.0438518
\(886\) −25.2962 −0.849844
\(887\) −26.2023 −0.879787 −0.439893 0.898050i \(-0.644984\pi\)
−0.439893 + 0.898050i \(0.644984\pi\)
\(888\) 9.95080 0.333927
\(889\) 1.41636 0.0475031
\(890\) 1.16828 0.0391608
\(891\) −16.1457 −0.540900
\(892\) −15.4171 −0.516202
\(893\) 11.0697 0.370434
\(894\) 16.1500 0.540137
\(895\) 2.10692 0.0704265
\(896\) −0.276429 −0.00923485
\(897\) −7.97556 −0.266296
\(898\) 22.2350 0.741992
\(899\) 60.7230 2.02522
\(900\) −7.36719 −0.245573
\(901\) −8.24546 −0.274696
\(902\) −23.1456 −0.770663
\(903\) 0.130107 0.00432969
\(904\) 13.6078 0.452587
\(905\) 0.0346397 0.00115146
\(906\) 20.3879 0.677341
\(907\) 58.2340 1.93363 0.966814 0.255482i \(-0.0822342\pi\)
0.966814 + 0.255482i \(0.0822342\pi\)
\(908\) −11.2385 −0.372963
\(909\) 18.9468 0.628427
\(910\) −0.0186612 −0.000618612 0
\(911\) −30.4851 −1.01002 −0.505008 0.863114i \(-0.668511\pi\)
−0.505008 + 0.863114i \(0.668511\pi\)
\(912\) −4.03283 −0.133540
\(913\) 10.6897 0.353778
\(914\) 4.91097 0.162440
\(915\) 1.98958 0.0657734
\(916\) −9.90781 −0.327363
\(917\) −0.957358 −0.0316148
\(918\) −15.2903 −0.504654
\(919\) −2.94915 −0.0972833 −0.0486417 0.998816i \(-0.515489\pi\)
−0.0486417 + 0.998816i \(0.515489\pi\)
\(920\) −5.03889 −0.166127
\(921\) −12.4290 −0.409548
\(922\) 16.0773 0.529477
\(923\) −5.37104 −0.176790
\(924\) 0.172261 0.00566699
\(925\) −20.2855 −0.666983
\(926\) 5.72032 0.187982
\(927\) −24.9928 −0.820870
\(928\) −20.9384 −0.687336
\(929\) 28.9881 0.951070 0.475535 0.879697i \(-0.342255\pi\)
0.475535 + 0.879697i \(0.342255\pi\)
\(930\) 1.63140 0.0534956
\(931\) 15.0956 0.494737
\(932\) −1.45974 −0.0478153
\(933\) −20.2814 −0.663984
\(934\) 14.4675 0.473391
\(935\) 2.35852 0.0771318
\(936\) 8.03667 0.262687
\(937\) −27.5952 −0.901497 −0.450748 0.892651i \(-0.648843\pi\)
−0.450748 + 0.892651i \(0.648843\pi\)
\(938\) −0.533930 −0.0174334
\(939\) −2.92776 −0.0955439
\(940\) 0.572939 0.0186872
\(941\) 19.2064 0.626112 0.313056 0.949735i \(-0.398647\pi\)
0.313056 + 0.949735i \(0.398647\pi\)
\(942\) 18.1727 0.592098
\(943\) 41.9902 1.36739
\(944\) −21.7478 −0.707832
\(945\) −0.0615326 −0.00200166
\(946\) −10.2518 −0.333315
\(947\) −41.2922 −1.34182 −0.670908 0.741541i \(-0.734095\pi\)
−0.670908 + 0.741541i \(0.734095\pi\)
\(948\) 8.62588 0.280155
\(949\) 12.0750 0.391973
\(950\) 12.5752 0.407992
\(951\) −15.0584 −0.488302
\(952\) −0.763262 −0.0247375
\(953\) −24.0272 −0.778317 −0.389158 0.921171i \(-0.627234\pi\)
−0.389158 + 0.921171i \(0.627234\pi\)
\(954\) −7.48650 −0.242384
\(955\) −3.57767 −0.115771
\(956\) −3.02969 −0.0979872
\(957\) −21.0534 −0.680561
\(958\) −4.17890 −0.135014
\(959\) 0.177578 0.00573430
\(960\) −1.23061 −0.0397176
\(961\) 65.5920 2.11587
\(962\) 5.26795 0.169846
\(963\) 7.87109 0.253642
\(964\) 2.14130 0.0689665
\(965\) −2.93980 −0.0946355
\(966\) 0.687737 0.0221276
\(967\) −10.8553 −0.349084 −0.174542 0.984650i \(-0.555845\pi\)
−0.174542 + 0.984650i \(0.555845\pi\)
\(968\) −23.1584 −0.744338
\(969\) −5.23806 −0.168271
\(970\) −2.92958 −0.0940633
\(971\) −17.1111 −0.549122 −0.274561 0.961570i \(-0.588533\pi\)
−0.274561 + 0.961570i \(0.588533\pi\)
\(972\) 9.83069 0.315319
\(973\) −0.494936 −0.0158669
\(974\) −5.97191 −0.191352
\(975\) 4.32744 0.138589
\(976\) 33.1679 1.06168
\(977\) 22.5277 0.720726 0.360363 0.932812i \(-0.382653\pi\)
0.360363 + 0.932812i \(0.382653\pi\)
\(978\) 13.2347 0.423198
\(979\) −23.9827 −0.766489
\(980\) 0.781306 0.0249579
\(981\) 30.3660 0.969512
\(982\) 31.7849 1.01430
\(983\) −27.0618 −0.863138 −0.431569 0.902080i \(-0.642040\pi\)
−0.431569 + 0.902080i \(0.642040\pi\)
\(984\) 11.1761 0.356280
\(985\) 4.35014 0.138607
\(986\) 22.2070 0.707215
\(987\) −0.328484 −0.0104558
\(988\) 1.48394 0.0472104
\(989\) 18.5986 0.591401
\(990\) 2.14143 0.0680590
\(991\) −19.8216 −0.629655 −0.314827 0.949149i \(-0.601947\pi\)
−0.314827 + 0.949149i \(0.601947\pi\)
\(992\) −33.3067 −1.05749
\(993\) 3.76020 0.119326
\(994\) 0.463148 0.0146902
\(995\) −2.90473 −0.0920863
\(996\) −1.22877 −0.0389349
\(997\) −23.1120 −0.731966 −0.365983 0.930622i \(-0.619267\pi\)
−0.365983 + 0.930622i \(0.619267\pi\)
\(998\) −5.27436 −0.166957
\(999\) 17.3703 0.549573
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.1.8 10
3.2 odd 2 4527.2.a.k.1.3 10
4.3 odd 2 8048.2.a.p.1.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.e.1.8 10 1.1 even 1 trivial
4527.2.a.k.1.3 10 3.2 odd 2
8048.2.a.p.1.5 10 4.3 odd 2