Properties

Label 503.2.a.e.1.7
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 1.7
Root \(1.31567\) of defining polynomial
Character \(\chi\) \(=\) 503.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.0830530 q^{2} +0.315672 q^{3} -1.99310 q^{4} +2.25024 q^{5} +0.0262175 q^{6} -3.20647 q^{7} -0.331639 q^{8} -2.90035 q^{9} +O(q^{10})\) \(q+0.0830530 q^{2} +0.315672 q^{3} -1.99310 q^{4} +2.25024 q^{5} +0.0262175 q^{6} -3.20647 q^{7} -0.331639 q^{8} -2.90035 q^{9} +0.186890 q^{10} -0.218022 q^{11} -0.629166 q^{12} -4.17856 q^{13} -0.266307 q^{14} +0.710339 q^{15} +3.95866 q^{16} -4.68939 q^{17} -0.240883 q^{18} +3.43926 q^{19} -4.48497 q^{20} -1.01219 q^{21} -0.0181074 q^{22} -3.99816 q^{23} -0.104689 q^{24} +0.0635992 q^{25} -0.347042 q^{26} -1.86257 q^{27} +6.39082 q^{28} -0.712153 q^{29} +0.0589958 q^{30} +1.04933 q^{31} +0.992057 q^{32} -0.0688235 q^{33} -0.389468 q^{34} -7.21534 q^{35} +5.78070 q^{36} +1.70998 q^{37} +0.285641 q^{38} -1.31905 q^{39} -0.746269 q^{40} +3.18460 q^{41} -0.0840656 q^{42} -6.35890 q^{43} +0.434541 q^{44} -6.52650 q^{45} -0.332059 q^{46} -3.87861 q^{47} +1.24964 q^{48} +3.28144 q^{49} +0.00528210 q^{50} -1.48031 q^{51} +8.32830 q^{52} -10.8877 q^{53} -0.154692 q^{54} -0.490604 q^{55} +1.06339 q^{56} +1.08568 q^{57} -0.0591465 q^{58} -2.63793 q^{59} -1.41578 q^{60} +11.1544 q^{61} +0.0871503 q^{62} +9.29988 q^{63} -7.83493 q^{64} -9.40279 q^{65} -0.00571600 q^{66} +6.09170 q^{67} +9.34643 q^{68} -1.26210 q^{69} -0.599256 q^{70} +9.40515 q^{71} +0.961870 q^{72} -2.78532 q^{73} +0.142019 q^{74} +0.0200765 q^{75} -6.85479 q^{76} +0.699082 q^{77} -0.109551 q^{78} +5.16993 q^{79} +8.90795 q^{80} +8.11309 q^{81} +0.264491 q^{82} +1.83793 q^{83} +2.01740 q^{84} -10.5523 q^{85} -0.528126 q^{86} -0.224807 q^{87} +0.0723048 q^{88} -8.08786 q^{89} -0.542046 q^{90} +13.3984 q^{91} +7.96873 q^{92} +0.331245 q^{93} -0.322130 q^{94} +7.73917 q^{95} +0.313165 q^{96} -12.9058 q^{97} +0.272533 q^{98} +0.632342 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} + O(q^{10}) \) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} - 4q^{10} - 3q^{11} - 7q^{12} - 18q^{13} + q^{14} - 2q^{15} - 4q^{16} - 11q^{17} - q^{18} - 3q^{20} + q^{21} - 18q^{22} - 2q^{23} + 10q^{24} - 27q^{25} + 11q^{26} - 2q^{27} - 22q^{28} - 9q^{29} + 12q^{30} - 22q^{31} - 10q^{32} - 10q^{33} - 10q^{34} - 6q^{35} + 2q^{36} - 35q^{37} + 2q^{38} + 8q^{39} - 19q^{40} - 4q^{41} + 4q^{42} - 20q^{43} + 9q^{44} + 2q^{45} - q^{46} + 7q^{47} - 27q^{49} + 16q^{50} + 9q^{51} - 7q^{52} - 24q^{53} + 17q^{54} - 11q^{55} + 12q^{56} - 23q^{57} + 2q^{58} + 17q^{59} - 4q^{61} + 8q^{62} + 10q^{63} + 3q^{64} - 16q^{65} + 46q^{66} - 6q^{67} + 28q^{68} - 2q^{69} + 26q^{70} - q^{71} - q^{72} - 31q^{73} + 11q^{74} + 30q^{75} + 20q^{76} + 3q^{77} + 11q^{78} - 10q^{79} + 24q^{80} - 6q^{81} - 9q^{82} + 22q^{83} + 22q^{84} - 6q^{85} + 38q^{86} + 25q^{87} - 3q^{88} + q^{89} + 2q^{90} + 10q^{91} + 27q^{92} - 6q^{93} + 33q^{94} + 39q^{95} + 46q^{96} - 57q^{97} + 40q^{98} + 35q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0830530 0.0587274 0.0293637 0.999569i \(-0.490652\pi\)
0.0293637 + 0.999569i \(0.490652\pi\)
\(3\) 0.315672 0.182253 0.0911266 0.995839i \(-0.470953\pi\)
0.0911266 + 0.995839i \(0.470953\pi\)
\(4\) −1.99310 −0.996551
\(5\) 2.25024 1.00634 0.503170 0.864188i \(-0.332167\pi\)
0.503170 + 0.864188i \(0.332167\pi\)
\(6\) 0.0262175 0.0107032
\(7\) −3.20647 −1.21193 −0.605966 0.795491i \(-0.707213\pi\)
−0.605966 + 0.795491i \(0.707213\pi\)
\(8\) −0.331639 −0.117252
\(9\) −2.90035 −0.966784
\(10\) 0.186890 0.0590997
\(11\) −0.218022 −0.0657362 −0.0328681 0.999460i \(-0.510464\pi\)
−0.0328681 + 0.999460i \(0.510464\pi\)
\(12\) −0.629166 −0.181625
\(13\) −4.17856 −1.15892 −0.579462 0.814999i \(-0.696737\pi\)
−0.579462 + 0.814999i \(0.696737\pi\)
\(14\) −0.266307 −0.0711735
\(15\) 0.710339 0.183409
\(16\) 3.95866 0.989665
\(17\) −4.68939 −1.13734 −0.568672 0.822564i \(-0.692543\pi\)
−0.568672 + 0.822564i \(0.692543\pi\)
\(18\) −0.240883 −0.0567767
\(19\) 3.43926 0.789020 0.394510 0.918892i \(-0.370914\pi\)
0.394510 + 0.918892i \(0.370914\pi\)
\(20\) −4.48497 −1.00287
\(21\) −1.01219 −0.220878
\(22\) −0.0181074 −0.00386052
\(23\) −3.99816 −0.833673 −0.416837 0.908981i \(-0.636861\pi\)
−0.416837 + 0.908981i \(0.636861\pi\)
\(24\) −0.104689 −0.0213696
\(25\) 0.0635992 0.0127198
\(26\) −0.347042 −0.0680606
\(27\) −1.86257 −0.358453
\(28\) 6.39082 1.20775
\(29\) −0.712153 −0.132243 −0.0661217 0.997812i \(-0.521063\pi\)
−0.0661217 + 0.997812i \(0.521063\pi\)
\(30\) 0.0589958 0.0107711
\(31\) 1.04933 0.188466 0.0942329 0.995550i \(-0.469960\pi\)
0.0942329 + 0.995550i \(0.469960\pi\)
\(32\) 0.992057 0.175373
\(33\) −0.0688235 −0.0119806
\(34\) −0.389468 −0.0667932
\(35\) −7.21534 −1.21961
\(36\) 5.78070 0.963449
\(37\) 1.70998 0.281119 0.140560 0.990072i \(-0.455110\pi\)
0.140560 + 0.990072i \(0.455110\pi\)
\(38\) 0.285641 0.0463371
\(39\) −1.31905 −0.211218
\(40\) −0.746269 −0.117996
\(41\) 3.18460 0.497351 0.248675 0.968587i \(-0.420005\pi\)
0.248675 + 0.968587i \(0.420005\pi\)
\(42\) −0.0840656 −0.0129716
\(43\) −6.35890 −0.969723 −0.484861 0.874591i \(-0.661130\pi\)
−0.484861 + 0.874591i \(0.661130\pi\)
\(44\) 0.434541 0.0655095
\(45\) −6.52650 −0.972913
\(46\) −0.332059 −0.0489594
\(47\) −3.87861 −0.565754 −0.282877 0.959156i \(-0.591289\pi\)
−0.282877 + 0.959156i \(0.591289\pi\)
\(48\) 1.24964 0.180370
\(49\) 3.28144 0.468777
\(50\) 0.00528210 0.000747002 0
\(51\) −1.48031 −0.207285
\(52\) 8.32830 1.15493
\(53\) −10.8877 −1.49555 −0.747773 0.663954i \(-0.768877\pi\)
−0.747773 + 0.663954i \(0.768877\pi\)
\(54\) −0.154692 −0.0210510
\(55\) −0.490604 −0.0661530
\(56\) 1.06339 0.142102
\(57\) 1.08568 0.143801
\(58\) −0.0591465 −0.00776631
\(59\) −2.63793 −0.343429 −0.171715 0.985147i \(-0.554931\pi\)
−0.171715 + 0.985147i \(0.554931\pi\)
\(60\) −1.41578 −0.182776
\(61\) 11.1544 1.42817 0.714084 0.700060i \(-0.246843\pi\)
0.714084 + 0.700060i \(0.246843\pi\)
\(62\) 0.0871503 0.0110681
\(63\) 9.29988 1.17168
\(64\) −7.83493 −0.979366
\(65\) −9.40279 −1.16627
\(66\) −0.00571600 −0.000703591 0
\(67\) 6.09170 0.744219 0.372110 0.928189i \(-0.378635\pi\)
0.372110 + 0.928189i \(0.378635\pi\)
\(68\) 9.34643 1.13342
\(69\) −1.26210 −0.151940
\(70\) −0.599256 −0.0716247
\(71\) 9.40515 1.11619 0.558093 0.829778i \(-0.311533\pi\)
0.558093 + 0.829778i \(0.311533\pi\)
\(72\) 0.961870 0.113358
\(73\) −2.78532 −0.325997 −0.162998 0.986626i \(-0.552117\pi\)
−0.162998 + 0.986626i \(0.552117\pi\)
\(74\) 0.142019 0.0165094
\(75\) 0.0200765 0.00231823
\(76\) −6.85479 −0.786299
\(77\) 0.699082 0.0796678
\(78\) −0.109551 −0.0124043
\(79\) 5.16993 0.581662 0.290831 0.956774i \(-0.406068\pi\)
0.290831 + 0.956774i \(0.406068\pi\)
\(80\) 8.90795 0.995939
\(81\) 8.11309 0.901455
\(82\) 0.264491 0.0292081
\(83\) 1.83793 0.201739 0.100870 0.994900i \(-0.467838\pi\)
0.100870 + 0.994900i \(0.467838\pi\)
\(84\) 2.01740 0.220117
\(85\) −10.5523 −1.14455
\(86\) −0.528126 −0.0569493
\(87\) −0.224807 −0.0241018
\(88\) 0.0723048 0.00770772
\(89\) −8.08786 −0.857311 −0.428656 0.903468i \(-0.641013\pi\)
−0.428656 + 0.903468i \(0.641013\pi\)
\(90\) −0.542046 −0.0571366
\(91\) 13.3984 1.40454
\(92\) 7.96873 0.830798
\(93\) 0.331245 0.0343485
\(94\) −0.322130 −0.0332252
\(95\) 7.73917 0.794022
\(96\) 0.313165 0.0319622
\(97\) −12.9058 −1.31039 −0.655193 0.755462i \(-0.727413\pi\)
−0.655193 + 0.755462i \(0.727413\pi\)
\(98\) 0.272533 0.0275300
\(99\) 0.632342 0.0635527
\(100\) −0.126760 −0.0126760
\(101\) 2.17744 0.216663 0.108331 0.994115i \(-0.465449\pi\)
0.108331 + 0.994115i \(0.465449\pi\)
\(102\) −0.122944 −0.0121733
\(103\) 14.7612 1.45447 0.727234 0.686390i \(-0.240806\pi\)
0.727234 + 0.686390i \(0.240806\pi\)
\(104\) 1.38578 0.135886
\(105\) −2.27768 −0.222279
\(106\) −0.904260 −0.0878295
\(107\) 10.4459 1.00984 0.504920 0.863166i \(-0.331522\pi\)
0.504920 + 0.863166i \(0.331522\pi\)
\(108\) 3.71230 0.357216
\(109\) −5.69014 −0.545017 −0.272508 0.962153i \(-0.587853\pi\)
−0.272508 + 0.962153i \(0.587853\pi\)
\(110\) −0.0407461 −0.00388499
\(111\) 0.539793 0.0512348
\(112\) −12.6933 −1.19941
\(113\) 20.9980 1.97533 0.987665 0.156584i \(-0.0500481\pi\)
0.987665 + 0.156584i \(0.0500481\pi\)
\(114\) 0.0901687 0.00844508
\(115\) −8.99683 −0.838958
\(116\) 1.41939 0.131787
\(117\) 12.1193 1.12043
\(118\) −0.219088 −0.0201687
\(119\) 15.0364 1.37838
\(120\) −0.235576 −0.0215051
\(121\) −10.9525 −0.995679
\(122\) 0.926403 0.0838726
\(123\) 1.00529 0.0906438
\(124\) −2.09143 −0.187816
\(125\) −11.1081 −0.993539
\(126\) 0.772384 0.0688094
\(127\) −0.0956906 −0.00849117 −0.00424558 0.999991i \(-0.501351\pi\)
−0.00424558 + 0.999991i \(0.501351\pi\)
\(128\) −2.63483 −0.232888
\(129\) −2.00732 −0.176735
\(130\) −0.780930 −0.0684921
\(131\) 14.5638 1.27244 0.636221 0.771507i \(-0.280496\pi\)
0.636221 + 0.771507i \(0.280496\pi\)
\(132\) 0.137172 0.0119393
\(133\) −11.0279 −0.956238
\(134\) 0.505934 0.0437060
\(135\) −4.19125 −0.360725
\(136\) 1.55519 0.133356
\(137\) −17.1079 −1.46162 −0.730811 0.682579i \(-0.760858\pi\)
−0.730811 + 0.682579i \(0.760858\pi\)
\(138\) −0.104822 −0.00892301
\(139\) −17.7270 −1.50358 −0.751791 0.659402i \(-0.770810\pi\)
−0.751791 + 0.659402i \(0.770810\pi\)
\(140\) 14.3809 1.21541
\(141\) −1.22437 −0.103110
\(142\) 0.781126 0.0655507
\(143\) 0.911020 0.0761833
\(144\) −11.4815 −0.956792
\(145\) −1.60252 −0.133082
\(146\) −0.231329 −0.0191449
\(147\) 1.03586 0.0854361
\(148\) −3.40817 −0.280150
\(149\) −3.37396 −0.276406 −0.138203 0.990404i \(-0.544133\pi\)
−0.138203 + 0.990404i \(0.544133\pi\)
\(150\) 0.00166741 0.000136144 0
\(151\) 2.76168 0.224742 0.112371 0.993666i \(-0.464155\pi\)
0.112371 + 0.993666i \(0.464155\pi\)
\(152\) −1.14059 −0.0925143
\(153\) 13.6009 1.09957
\(154\) 0.0580609 0.00467868
\(155\) 2.36126 0.189661
\(156\) 2.62901 0.210489
\(157\) −8.93566 −0.713144 −0.356572 0.934268i \(-0.616054\pi\)
−0.356572 + 0.934268i \(0.616054\pi\)
\(158\) 0.429378 0.0341595
\(159\) −3.43695 −0.272568
\(160\) 2.23237 0.176484
\(161\) 12.8200 1.01035
\(162\) 0.673817 0.0529401
\(163\) −8.75660 −0.685870 −0.342935 0.939359i \(-0.611421\pi\)
−0.342935 + 0.939359i \(0.611421\pi\)
\(164\) −6.34723 −0.495635
\(165\) −0.154870 −0.0120566
\(166\) 0.152646 0.0118476
\(167\) 14.3605 1.11125 0.555623 0.831434i \(-0.312480\pi\)
0.555623 + 0.831434i \(0.312480\pi\)
\(168\) 0.335682 0.0258985
\(169\) 4.46038 0.343106
\(170\) −0.876398 −0.0672167
\(171\) −9.97506 −0.762812
\(172\) 12.6739 0.966378
\(173\) −11.0498 −0.840098 −0.420049 0.907501i \(-0.637987\pi\)
−0.420049 + 0.907501i \(0.637987\pi\)
\(174\) −0.0186709 −0.00141543
\(175\) −0.203929 −0.0154156
\(176\) −0.863077 −0.0650569
\(177\) −0.832720 −0.0625910
\(178\) −0.671721 −0.0503476
\(179\) −6.55060 −0.489615 −0.244808 0.969572i \(-0.578725\pi\)
−0.244808 + 0.969572i \(0.578725\pi\)
\(180\) 13.0080 0.969558
\(181\) −13.9694 −1.03834 −0.519169 0.854672i \(-0.673758\pi\)
−0.519169 + 0.854672i \(0.673758\pi\)
\(182\) 1.11278 0.0824848
\(183\) 3.52111 0.260288
\(184\) 1.32595 0.0977500
\(185\) 3.84787 0.282901
\(186\) 0.0275109 0.00201720
\(187\) 1.02239 0.0747647
\(188\) 7.73047 0.563802
\(189\) 5.97229 0.434420
\(190\) 0.642762 0.0466308
\(191\) 2.23996 0.162078 0.0810390 0.996711i \(-0.474176\pi\)
0.0810390 + 0.996711i \(0.474176\pi\)
\(192\) −2.47327 −0.178493
\(193\) 22.1924 1.59744 0.798722 0.601700i \(-0.205510\pi\)
0.798722 + 0.601700i \(0.205510\pi\)
\(194\) −1.07187 −0.0769555
\(195\) −2.96819 −0.212557
\(196\) −6.54024 −0.467160
\(197\) −11.0264 −0.785595 −0.392797 0.919625i \(-0.628493\pi\)
−0.392797 + 0.919625i \(0.628493\pi\)
\(198\) 0.0525179 0.00373228
\(199\) 14.6196 1.03636 0.518179 0.855272i \(-0.326610\pi\)
0.518179 + 0.855272i \(0.326610\pi\)
\(200\) −0.0210920 −0.00149143
\(201\) 1.92298 0.135636
\(202\) 0.180843 0.0127240
\(203\) 2.28350 0.160270
\(204\) 2.95040 0.206570
\(205\) 7.16612 0.500504
\(206\) 1.22597 0.0854170
\(207\) 11.5961 0.805982
\(208\) −16.5415 −1.14695
\(209\) −0.749835 −0.0518672
\(210\) −0.189168 −0.0130538
\(211\) 1.54449 0.106327 0.0531635 0.998586i \(-0.483070\pi\)
0.0531635 + 0.998586i \(0.483070\pi\)
\(212\) 21.7004 1.49039
\(213\) 2.96894 0.203428
\(214\) 0.867562 0.0593053
\(215\) −14.3091 −0.975871
\(216\) 0.617703 0.0420294
\(217\) −3.36465 −0.228408
\(218\) −0.472584 −0.0320074
\(219\) −0.879246 −0.0594139
\(220\) 0.977823 0.0659248
\(221\) 19.5949 1.31810
\(222\) 0.0448314 0.00300889
\(223\) −26.6878 −1.78715 −0.893573 0.448918i \(-0.851810\pi\)
−0.893573 + 0.448918i \(0.851810\pi\)
\(224\) −3.18100 −0.212540
\(225\) −0.184460 −0.0122973
\(226\) 1.74395 0.116006
\(227\) −6.41989 −0.426103 −0.213052 0.977041i \(-0.568340\pi\)
−0.213052 + 0.977041i \(0.568340\pi\)
\(228\) −2.16386 −0.143305
\(229\) −27.8670 −1.84150 −0.920752 0.390149i \(-0.872423\pi\)
−0.920752 + 0.390149i \(0.872423\pi\)
\(230\) −0.747214 −0.0492698
\(231\) 0.220680 0.0145197
\(232\) 0.236178 0.0155058
\(233\) 0.647350 0.0424093 0.0212047 0.999775i \(-0.493250\pi\)
0.0212047 + 0.999775i \(0.493250\pi\)
\(234\) 1.00654 0.0657999
\(235\) −8.72782 −0.569340
\(236\) 5.25766 0.342245
\(237\) 1.63200 0.106010
\(238\) 1.24882 0.0809488
\(239\) −12.3635 −0.799727 −0.399864 0.916575i \(-0.630943\pi\)
−0.399864 + 0.916575i \(0.630943\pi\)
\(240\) 2.81199 0.181513
\(241\) −12.7531 −0.821497 −0.410748 0.911749i \(-0.634732\pi\)
−0.410748 + 0.911749i \(0.634732\pi\)
\(242\) −0.909636 −0.0584736
\(243\) 8.14880 0.522746
\(244\) −22.2318 −1.42324
\(245\) 7.38404 0.471749
\(246\) 0.0834922 0.00532327
\(247\) −14.3712 −0.914415
\(248\) −0.348000 −0.0220980
\(249\) 0.580183 0.0367676
\(250\) −0.922562 −0.0583480
\(251\) −13.4917 −0.851588 −0.425794 0.904820i \(-0.640005\pi\)
−0.425794 + 0.904820i \(0.640005\pi\)
\(252\) −18.5356 −1.16763
\(253\) 0.871688 0.0548025
\(254\) −0.00794739 −0.000498664 0
\(255\) −3.33105 −0.208599
\(256\) 15.4510 0.965689
\(257\) −4.56318 −0.284643 −0.142322 0.989820i \(-0.545457\pi\)
−0.142322 + 0.989820i \(0.545457\pi\)
\(258\) −0.166714 −0.0103792
\(259\) −5.48300 −0.340697
\(260\) 18.7407 1.16225
\(261\) 2.06549 0.127851
\(262\) 1.20957 0.0747272
\(263\) 16.0985 0.992674 0.496337 0.868130i \(-0.334678\pi\)
0.496337 + 0.868130i \(0.334678\pi\)
\(264\) 0.0228246 0.00140476
\(265\) −24.5001 −1.50503
\(266\) −0.915898 −0.0561573
\(267\) −2.55311 −0.156248
\(268\) −12.1414 −0.741652
\(269\) −7.49840 −0.457186 −0.228593 0.973522i \(-0.573412\pi\)
−0.228593 + 0.973522i \(0.573412\pi\)
\(270\) −0.348096 −0.0211844
\(271\) 6.47424 0.393282 0.196641 0.980476i \(-0.436997\pi\)
0.196641 + 0.980476i \(0.436997\pi\)
\(272\) −18.5637 −1.12559
\(273\) 4.22951 0.255981
\(274\) −1.42086 −0.0858373
\(275\) −0.0138660 −0.000836154 0
\(276\) 2.51550 0.151416
\(277\) 8.32050 0.499930 0.249965 0.968255i \(-0.419581\pi\)
0.249965 + 0.968255i \(0.419581\pi\)
\(278\) −1.47228 −0.0883014
\(279\) −3.04344 −0.182206
\(280\) 2.39289 0.143002
\(281\) −26.3798 −1.57369 −0.786844 0.617152i \(-0.788286\pi\)
−0.786844 + 0.617152i \(0.788286\pi\)
\(282\) −0.101687 −0.00605540
\(283\) −23.4048 −1.39127 −0.695637 0.718394i \(-0.744878\pi\)
−0.695637 + 0.718394i \(0.744878\pi\)
\(284\) −18.7454 −1.11234
\(285\) 2.44304 0.144713
\(286\) 0.0756630 0.00447405
\(287\) −10.2113 −0.602755
\(288\) −2.87731 −0.169547
\(289\) 4.99036 0.293551
\(290\) −0.133094 −0.00781555
\(291\) −4.07400 −0.238822
\(292\) 5.55142 0.324872
\(293\) −3.57088 −0.208613 −0.104307 0.994545i \(-0.533262\pi\)
−0.104307 + 0.994545i \(0.533262\pi\)
\(294\) 0.0860311 0.00501744
\(295\) −5.93598 −0.345606
\(296\) −0.567097 −0.0329618
\(297\) 0.406083 0.0235633
\(298\) −0.280218 −0.0162326
\(299\) 16.7065 0.966164
\(300\) −0.0400144 −0.00231023
\(301\) 20.3896 1.17524
\(302\) 0.229366 0.0131985
\(303\) 0.687355 0.0394875
\(304\) 13.6149 0.780866
\(305\) 25.1000 1.43722
\(306\) 1.12959 0.0645746
\(307\) −2.46828 −0.140872 −0.0704362 0.997516i \(-0.522439\pi\)
−0.0704362 + 0.997516i \(0.522439\pi\)
\(308\) −1.39334 −0.0793930
\(309\) 4.65970 0.265081
\(310\) 0.196110 0.0111383
\(311\) −19.8949 −1.12814 −0.564069 0.825727i \(-0.690765\pi\)
−0.564069 + 0.825727i \(0.690765\pi\)
\(312\) 0.437450 0.0247657
\(313\) −10.0179 −0.566246 −0.283123 0.959084i \(-0.591370\pi\)
−0.283123 + 0.959084i \(0.591370\pi\)
\(314\) −0.742134 −0.0418810
\(315\) 20.9270 1.17910
\(316\) −10.3042 −0.579656
\(317\) −11.2440 −0.631529 −0.315764 0.948838i \(-0.602261\pi\)
−0.315764 + 0.948838i \(0.602261\pi\)
\(318\) −0.285449 −0.0160072
\(319\) 0.155265 0.00869319
\(320\) −17.6305 −0.985575
\(321\) 3.29747 0.184047
\(322\) 1.06474 0.0593354
\(323\) −16.1280 −0.897387
\(324\) −16.1702 −0.898346
\(325\) −0.265753 −0.0147413
\(326\) −0.727262 −0.0402793
\(327\) −1.79622 −0.0993311
\(328\) −1.05614 −0.0583155
\(329\) 12.4366 0.685654
\(330\) −0.0128624 −0.000708052 0
\(331\) −30.0728 −1.65295 −0.826475 0.562974i \(-0.809657\pi\)
−0.826475 + 0.562974i \(0.809657\pi\)
\(332\) −3.66319 −0.201043
\(333\) −4.95954 −0.271781
\(334\) 1.19268 0.0652606
\(335\) 13.7078 0.748937
\(336\) −4.00692 −0.218596
\(337\) −17.3224 −0.943613 −0.471807 0.881702i \(-0.656398\pi\)
−0.471807 + 0.881702i \(0.656398\pi\)
\(338\) 0.370448 0.0201497
\(339\) 6.62849 0.360010
\(340\) 21.0317 1.14061
\(341\) −0.228778 −0.0123890
\(342\) −0.828459 −0.0447979
\(343\) 11.9234 0.643806
\(344\) 2.10886 0.113702
\(345\) −2.84004 −0.152903
\(346\) −0.917716 −0.0493367
\(347\) 21.2412 1.14029 0.570143 0.821545i \(-0.306888\pi\)
0.570143 + 0.821545i \(0.306888\pi\)
\(348\) 0.448062 0.0240187
\(349\) 11.6817 0.625308 0.312654 0.949867i \(-0.398782\pi\)
0.312654 + 0.949867i \(0.398782\pi\)
\(350\) −0.0169369 −0.000905315 0
\(351\) 7.78288 0.415420
\(352\) −0.216291 −0.0115283
\(353\) 14.0775 0.749271 0.374636 0.927172i \(-0.377768\pi\)
0.374636 + 0.927172i \(0.377768\pi\)
\(354\) −0.0691599 −0.00367581
\(355\) 21.1639 1.12326
\(356\) 16.1199 0.854355
\(357\) 4.74656 0.251215
\(358\) −0.544048 −0.0287538
\(359\) 7.85706 0.414680 0.207340 0.978269i \(-0.433519\pi\)
0.207340 + 0.978269i \(0.433519\pi\)
\(360\) 2.16444 0.114076
\(361\) −7.17150 −0.377448
\(362\) −1.16020 −0.0609788
\(363\) −3.45738 −0.181466
\(364\) −26.7044 −1.39969
\(365\) −6.26764 −0.328063
\(366\) 0.292439 0.0152860
\(367\) −19.7726 −1.03212 −0.516062 0.856551i \(-0.672602\pi\)
−0.516062 + 0.856551i \(0.672602\pi\)
\(368\) −15.8273 −0.825057
\(369\) −9.23646 −0.480831
\(370\) 0.319578 0.0166141
\(371\) 34.9112 1.81250
\(372\) −0.660205 −0.0342300
\(373\) 7.49088 0.387863 0.193931 0.981015i \(-0.437876\pi\)
0.193931 + 0.981015i \(0.437876\pi\)
\(374\) 0.0849127 0.00439073
\(375\) −3.50652 −0.181076
\(376\) 1.28630 0.0663358
\(377\) 2.97577 0.153260
\(378\) 0.496016 0.0255123
\(379\) 27.5064 1.41291 0.706455 0.707758i \(-0.250293\pi\)
0.706455 + 0.707758i \(0.250293\pi\)
\(380\) −15.4250 −0.791284
\(381\) −0.0302068 −0.00154754
\(382\) 0.186036 0.00951842
\(383\) 4.92412 0.251611 0.125805 0.992055i \(-0.459849\pi\)
0.125805 + 0.992055i \(0.459849\pi\)
\(384\) −0.831741 −0.0424446
\(385\) 1.57311 0.0801729
\(386\) 1.84315 0.0938137
\(387\) 18.4430 0.937512
\(388\) 25.7226 1.30587
\(389\) 38.9916 1.97696 0.988478 0.151368i \(-0.0483678\pi\)
0.988478 + 0.151368i \(0.0483678\pi\)
\(390\) −0.246518 −0.0124829
\(391\) 18.7489 0.948173
\(392\) −1.08825 −0.0549651
\(393\) 4.59737 0.231907
\(394\) −0.915772 −0.0461359
\(395\) 11.6336 0.585350
\(396\) −1.26032 −0.0633335
\(397\) −12.6661 −0.635694 −0.317847 0.948142i \(-0.602960\pi\)
−0.317847 + 0.948142i \(0.602960\pi\)
\(398\) 1.21420 0.0608626
\(399\) −3.48119 −0.174277
\(400\) 0.251767 0.0125884
\(401\) 26.7959 1.33812 0.669062 0.743207i \(-0.266696\pi\)
0.669062 + 0.743207i \(0.266696\pi\)
\(402\) 0.159709 0.00796556
\(403\) −4.38471 −0.218418
\(404\) −4.33985 −0.215916
\(405\) 18.2564 0.907170
\(406\) 0.189651 0.00941223
\(407\) −0.372814 −0.0184797
\(408\) 0.490928 0.0243046
\(409\) 35.3472 1.74781 0.873903 0.486101i \(-0.161581\pi\)
0.873903 + 0.486101i \(0.161581\pi\)
\(410\) 0.595168 0.0293933
\(411\) −5.40047 −0.266385
\(412\) −29.4206 −1.44945
\(413\) 8.45844 0.416212
\(414\) 0.963088 0.0473332
\(415\) 4.13580 0.203018
\(416\) −4.14537 −0.203244
\(417\) −5.59590 −0.274033
\(418\) −0.0622761 −0.00304602
\(419\) −2.49358 −0.121819 −0.0609096 0.998143i \(-0.519400\pi\)
−0.0609096 + 0.998143i \(0.519400\pi\)
\(420\) 4.53965 0.221512
\(421\) 17.2671 0.841547 0.420773 0.907166i \(-0.361759\pi\)
0.420773 + 0.907166i \(0.361759\pi\)
\(422\) 0.128274 0.00624430
\(423\) 11.2493 0.546961
\(424\) 3.61080 0.175356
\(425\) −0.298241 −0.0144668
\(426\) 0.246580 0.0119468
\(427\) −35.7661 −1.73084
\(428\) −20.8197 −1.00636
\(429\) 0.287583 0.0138847
\(430\) −1.18841 −0.0573103
\(431\) 31.0699 1.49659 0.748293 0.663368i \(-0.230874\pi\)
0.748293 + 0.663368i \(0.230874\pi\)
\(432\) −7.37330 −0.354748
\(433\) 16.4828 0.792112 0.396056 0.918226i \(-0.370379\pi\)
0.396056 + 0.918226i \(0.370379\pi\)
\(434\) −0.279445 −0.0134138
\(435\) −0.505870 −0.0242546
\(436\) 11.3410 0.543137
\(437\) −13.7507 −0.657785
\(438\) −0.0730241 −0.00348922
\(439\) 33.0499 1.57739 0.788693 0.614787i \(-0.210758\pi\)
0.788693 + 0.614787i \(0.210758\pi\)
\(440\) 0.162703 0.00775658
\(441\) −9.51733 −0.453206
\(442\) 1.62742 0.0774083
\(443\) −24.8330 −1.17985 −0.589925 0.807458i \(-0.700843\pi\)
−0.589925 + 0.807458i \(0.700843\pi\)
\(444\) −1.07586 −0.0510581
\(445\) −18.1997 −0.862747
\(446\) −2.21650 −0.104954
\(447\) −1.06506 −0.0503758
\(448\) 25.1224 1.18692
\(449\) −18.6217 −0.878811 −0.439405 0.898289i \(-0.644811\pi\)
−0.439405 + 0.898289i \(0.644811\pi\)
\(450\) −0.0153200 −0.000722190 0
\(451\) −0.694314 −0.0326940
\(452\) −41.8512 −1.96852
\(453\) 0.871784 0.0409600
\(454\) −0.533192 −0.0250239
\(455\) 30.1497 1.41344
\(456\) −0.360053 −0.0168610
\(457\) −34.2762 −1.60337 −0.801686 0.597745i \(-0.796063\pi\)
−0.801686 + 0.597745i \(0.796063\pi\)
\(458\) −2.31444 −0.108147
\(459\) 8.73433 0.407684
\(460\) 17.9316 0.836065
\(461\) 11.2220 0.522659 0.261329 0.965250i \(-0.415839\pi\)
0.261329 + 0.965250i \(0.415839\pi\)
\(462\) 0.0183282 0.000852704 0
\(463\) −26.9770 −1.25373 −0.626863 0.779129i \(-0.715662\pi\)
−0.626863 + 0.779129i \(0.715662\pi\)
\(464\) −2.81917 −0.130877
\(465\) 0.745382 0.0345663
\(466\) 0.0537644 0.00249059
\(467\) 11.5336 0.533711 0.266856 0.963737i \(-0.414015\pi\)
0.266856 + 0.963737i \(0.414015\pi\)
\(468\) −24.1550 −1.11657
\(469\) −19.5328 −0.901942
\(470\) −0.724872 −0.0334359
\(471\) −2.82074 −0.129973
\(472\) 0.874841 0.0402678
\(473\) 1.38638 0.0637459
\(474\) 0.135543 0.00622568
\(475\) 0.218734 0.0100362
\(476\) −29.9690 −1.37363
\(477\) 31.5783 1.44587
\(478\) −1.02682 −0.0469659
\(479\) −28.8893 −1.31998 −0.659992 0.751272i \(-0.729441\pi\)
−0.659992 + 0.751272i \(0.729441\pi\)
\(480\) 0.704697 0.0321649
\(481\) −7.14526 −0.325796
\(482\) −1.05918 −0.0482443
\(483\) 4.04690 0.184140
\(484\) 21.8294 0.992245
\(485\) −29.0412 −1.31869
\(486\) 0.676782 0.0306995
\(487\) 1.63083 0.0739001 0.0369501 0.999317i \(-0.488236\pi\)
0.0369501 + 0.999317i \(0.488236\pi\)
\(488\) −3.69922 −0.167456
\(489\) −2.76421 −0.125002
\(490\) 0.613267 0.0277046
\(491\) −6.45592 −0.291352 −0.145676 0.989332i \(-0.546536\pi\)
−0.145676 + 0.989332i \(0.546536\pi\)
\(492\) −2.00364 −0.0903311
\(493\) 3.33956 0.150406
\(494\) −1.19357 −0.0537012
\(495\) 1.42292 0.0639556
\(496\) 4.15396 0.186518
\(497\) −30.1573 −1.35274
\(498\) 0.0481860 0.00215927
\(499\) 29.7187 1.33039 0.665196 0.746669i \(-0.268348\pi\)
0.665196 + 0.746669i \(0.268348\pi\)
\(500\) 22.1396 0.990113
\(501\) 4.53320 0.202528
\(502\) −1.12053 −0.0500115
\(503\) −1.00000 −0.0445878
\(504\) −3.08421 −0.137381
\(505\) 4.89976 0.218037
\(506\) 0.0723963 0.00321841
\(507\) 1.40802 0.0625323
\(508\) 0.190721 0.00846188
\(509\) 7.29140 0.323186 0.161593 0.986858i \(-0.448337\pi\)
0.161593 + 0.986858i \(0.448337\pi\)
\(510\) −0.276654 −0.0122504
\(511\) 8.93103 0.395086
\(512\) 6.55291 0.289601
\(513\) −6.40587 −0.282826
\(514\) −0.378986 −0.0167163
\(515\) 33.2164 1.46369
\(516\) 4.00080 0.176126
\(517\) 0.845624 0.0371905
\(518\) −0.455380 −0.0200082
\(519\) −3.48810 −0.153110
\(520\) 3.11833 0.136748
\(521\) 1.73107 0.0758394 0.0379197 0.999281i \(-0.487927\pi\)
0.0379197 + 0.999281i \(0.487927\pi\)
\(522\) 0.171545 0.00750834
\(523\) 15.8653 0.693742 0.346871 0.937913i \(-0.387244\pi\)
0.346871 + 0.937913i \(0.387244\pi\)
\(524\) −29.0271 −1.26805
\(525\) −0.0643745 −0.00280953
\(526\) 1.33703 0.0582971
\(527\) −4.92073 −0.214350
\(528\) −0.272449 −0.0118568
\(529\) −7.01475 −0.304989
\(530\) −2.03481 −0.0883863
\(531\) 7.65092 0.332022
\(532\) 21.9797 0.952940
\(533\) −13.3070 −0.576392
\(534\) −0.212043 −0.00917602
\(535\) 23.5058 1.01624
\(536\) −2.02025 −0.0872613
\(537\) −2.06784 −0.0892339
\(538\) −0.622765 −0.0268493
\(539\) −0.715427 −0.0308156
\(540\) 8.35358 0.359481
\(541\) 0.384069 0.0165124 0.00825621 0.999966i \(-0.497372\pi\)
0.00825621 + 0.999966i \(0.497372\pi\)
\(542\) 0.537705 0.0230964
\(543\) −4.40975 −0.189240
\(544\) −4.65214 −0.199459
\(545\) −12.8042 −0.548472
\(546\) 0.351273 0.0150331
\(547\) 38.3367 1.63916 0.819579 0.572967i \(-0.194208\pi\)
0.819579 + 0.572967i \(0.194208\pi\)
\(548\) 34.0977 1.45658
\(549\) −32.3515 −1.38073
\(550\) −0.00115162 −4.91051e−5 0
\(551\) −2.44928 −0.104343
\(552\) 0.418564 0.0178152
\(553\) −16.5772 −0.704935
\(554\) 0.691043 0.0293596
\(555\) 1.21467 0.0515597
\(556\) 35.3317 1.49840
\(557\) 25.2935 1.07172 0.535860 0.844307i \(-0.319987\pi\)
0.535860 + 0.844307i \(0.319987\pi\)
\(558\) −0.252767 −0.0107005
\(559\) 26.5710 1.12384
\(560\) −28.5631 −1.20701
\(561\) 0.322740 0.0136261
\(562\) −2.19092 −0.0924185
\(563\) 32.5764 1.37293 0.686465 0.727163i \(-0.259161\pi\)
0.686465 + 0.727163i \(0.259161\pi\)
\(564\) 2.44029 0.102755
\(565\) 47.2507 1.98785
\(566\) −1.94384 −0.0817058
\(567\) −26.0144 −1.09250
\(568\) −3.11912 −0.130875
\(569\) −36.9548 −1.54923 −0.774614 0.632435i \(-0.782056\pi\)
−0.774614 + 0.632435i \(0.782056\pi\)
\(570\) 0.202902 0.00849862
\(571\) −32.4941 −1.35984 −0.679919 0.733288i \(-0.737985\pi\)
−0.679919 + 0.733288i \(0.737985\pi\)
\(572\) −1.81576 −0.0759206
\(573\) 0.707093 0.0295393
\(574\) −0.848081 −0.0353982
\(575\) −0.254279 −0.0106042
\(576\) 22.7240 0.946835
\(577\) −3.08889 −0.128592 −0.0642960 0.997931i \(-0.520480\pi\)
−0.0642960 + 0.997931i \(0.520480\pi\)
\(578\) 0.414465 0.0172395
\(579\) 7.00552 0.291139
\(580\) 3.19398 0.132623
\(581\) −5.89327 −0.244494
\(582\) −0.338358 −0.0140254
\(583\) 2.37377 0.0983116
\(584\) 0.923721 0.0382238
\(585\) 27.2714 1.12753
\(586\) −0.296572 −0.0122513
\(587\) −39.5581 −1.63274 −0.816369 0.577530i \(-0.804017\pi\)
−0.816369 + 0.577530i \(0.804017\pi\)
\(588\) −2.06457 −0.0851414
\(589\) 3.60893 0.148703
\(590\) −0.493002 −0.0202966
\(591\) −3.48071 −0.143177
\(592\) 6.76923 0.278214
\(593\) 37.0809 1.52273 0.761366 0.648323i \(-0.224529\pi\)
0.761366 + 0.648323i \(0.224529\pi\)
\(594\) 0.0337264 0.00138381
\(595\) 33.8355 1.38712
\(596\) 6.72465 0.275453
\(597\) 4.61501 0.188880
\(598\) 1.38753 0.0567403
\(599\) −7.21058 −0.294616 −0.147308 0.989091i \(-0.547061\pi\)
−0.147308 + 0.989091i \(0.547061\pi\)
\(600\) −0.00665814 −0.000271818 0
\(601\) −15.6443 −0.638145 −0.319072 0.947730i \(-0.603371\pi\)
−0.319072 + 0.947730i \(0.603371\pi\)
\(602\) 1.69342 0.0690186
\(603\) −17.6681 −0.719499
\(604\) −5.50431 −0.223967
\(605\) −24.6457 −1.00199
\(606\) 0.0570869 0.00231900
\(607\) −5.03996 −0.204566 −0.102283 0.994755i \(-0.532615\pi\)
−0.102283 + 0.994755i \(0.532615\pi\)
\(608\) 3.41194 0.138372
\(609\) 0.720835 0.0292097
\(610\) 2.08463 0.0844043
\(611\) 16.2070 0.655666
\(612\) −27.1079 −1.09577
\(613\) −41.4393 −1.67372 −0.836858 0.547420i \(-0.815610\pi\)
−0.836858 + 0.547420i \(0.815610\pi\)
\(614\) −0.204998 −0.00827306
\(615\) 2.26214 0.0912184
\(616\) −0.231843 −0.00934122
\(617\) 17.1685 0.691178 0.345589 0.938386i \(-0.387679\pi\)
0.345589 + 0.938386i \(0.387679\pi\)
\(618\) 0.387003 0.0155675
\(619\) −16.9228 −0.680186 −0.340093 0.940392i \(-0.610459\pi\)
−0.340093 + 0.940392i \(0.610459\pi\)
\(620\) −4.70623 −0.189007
\(621\) 7.44686 0.298832
\(622\) −1.65234 −0.0662526
\(623\) 25.9335 1.03900
\(624\) −5.22169 −0.209035
\(625\) −25.3140 −1.01256
\(626\) −0.832018 −0.0332541
\(627\) −0.236702 −0.00945296
\(628\) 17.8097 0.710684
\(629\) −8.01876 −0.319729
\(630\) 1.73805 0.0692456
\(631\) 15.9640 0.635518 0.317759 0.948171i \(-0.397070\pi\)
0.317759 + 0.948171i \(0.397070\pi\)
\(632\) −1.71455 −0.0682012
\(633\) 0.487551 0.0193784
\(634\) −0.933852 −0.0370880
\(635\) −0.215327 −0.00854500
\(636\) 6.85020 0.271628
\(637\) −13.7117 −0.543277
\(638\) 0.0128953 0.000510528 0
\(639\) −27.2782 −1.07911
\(640\) −5.92901 −0.234365
\(641\) 34.1774 1.34993 0.674964 0.737851i \(-0.264159\pi\)
0.674964 + 0.737851i \(0.264159\pi\)
\(642\) 0.273865 0.0108086
\(643\) −1.27460 −0.0502653 −0.0251326 0.999684i \(-0.508001\pi\)
−0.0251326 + 0.999684i \(0.508001\pi\)
\(644\) −25.5515 −1.00687
\(645\) −4.51697 −0.177856
\(646\) −1.33948 −0.0527012
\(647\) 16.4300 0.645930 0.322965 0.946411i \(-0.395320\pi\)
0.322965 + 0.946411i \(0.395320\pi\)
\(648\) −2.69062 −0.105698
\(649\) 0.575128 0.0225757
\(650\) −0.0220716 −0.000865719 0
\(651\) −1.06213 −0.0416280
\(652\) 17.4528 0.683504
\(653\) 46.5637 1.82218 0.911088 0.412211i \(-0.135243\pi\)
0.911088 + 0.412211i \(0.135243\pi\)
\(654\) −0.149181 −0.00583345
\(655\) 32.7720 1.28051
\(656\) 12.6067 0.492211
\(657\) 8.07840 0.315168
\(658\) 1.03290 0.0402667
\(659\) −21.2263 −0.826861 −0.413431 0.910536i \(-0.635670\pi\)
−0.413431 + 0.910536i \(0.635670\pi\)
\(660\) 0.308671 0.0120150
\(661\) 31.5049 1.22540 0.612700 0.790316i \(-0.290084\pi\)
0.612700 + 0.790316i \(0.290084\pi\)
\(662\) −2.49764 −0.0970734
\(663\) 6.18556 0.240227
\(664\) −0.609530 −0.0236544
\(665\) −24.8154 −0.962300
\(666\) −0.411905 −0.0159610
\(667\) 2.84730 0.110248
\(668\) −28.6219 −1.10741
\(669\) −8.42458 −0.325713
\(670\) 1.13847 0.0439831
\(671\) −2.43190 −0.0938824
\(672\) −1.00415 −0.0387360
\(673\) −46.0920 −1.77672 −0.888359 0.459150i \(-0.848154\pi\)
−0.888359 + 0.459150i \(0.848154\pi\)
\(674\) −1.43868 −0.0554159
\(675\) −0.118458 −0.00455946
\(676\) −8.89000 −0.341923
\(677\) 5.22605 0.200853 0.100427 0.994944i \(-0.467979\pi\)
0.100427 + 0.994944i \(0.467979\pi\)
\(678\) 0.550516 0.0211424
\(679\) 41.3820 1.58810
\(680\) 3.49955 0.134201
\(681\) −2.02658 −0.0776587
\(682\) −0.0190007 −0.000727575 0
\(683\) −26.7163 −1.02227 −0.511135 0.859500i \(-0.670775\pi\)
−0.511135 + 0.859500i \(0.670775\pi\)
\(684\) 19.8813 0.760181
\(685\) −38.4969 −1.47089
\(686\) 0.990279 0.0378090
\(687\) −8.79683 −0.335620
\(688\) −25.1727 −0.959701
\(689\) 45.4951 1.73323
\(690\) −0.235874 −0.00897958
\(691\) −20.7575 −0.789653 −0.394826 0.918756i \(-0.629195\pi\)
−0.394826 + 0.918756i \(0.629195\pi\)
\(692\) 22.0233 0.837200
\(693\) −2.02758 −0.0770215
\(694\) 1.76414 0.0669660
\(695\) −39.8900 −1.51311
\(696\) 0.0745547 0.00282599
\(697\) −14.9338 −0.565659
\(698\) 0.970202 0.0367227
\(699\) 0.204350 0.00772924
\(700\) 0.406451 0.0153624
\(701\) −48.5428 −1.83344 −0.916719 0.399533i \(-0.869172\pi\)
−0.916719 + 0.399533i \(0.869172\pi\)
\(702\) 0.646392 0.0243965
\(703\) 5.88106 0.221809
\(704\) 1.70819 0.0643798
\(705\) −2.75513 −0.103764
\(706\) 1.16918 0.0440027
\(707\) −6.98188 −0.262581
\(708\) 1.65970 0.0623752
\(709\) −21.9935 −0.825981 −0.412991 0.910735i \(-0.635516\pi\)
−0.412991 + 0.910735i \(0.635516\pi\)
\(710\) 1.75772 0.0659662
\(711\) −14.9946 −0.562342
\(712\) 2.68225 0.100522
\(713\) −4.19540 −0.157119
\(714\) 0.394216 0.0147532
\(715\) 2.05002 0.0766663
\(716\) 13.0560 0.487926
\(717\) −3.90280 −0.145753
\(718\) 0.652552 0.0243530
\(719\) 9.78228 0.364817 0.182409 0.983223i \(-0.441611\pi\)
0.182409 + 0.983223i \(0.441611\pi\)
\(720\) −25.8362 −0.962858
\(721\) −47.3314 −1.76271
\(722\) −0.595615 −0.0221665
\(723\) −4.02578 −0.149720
\(724\) 27.8425 1.03476
\(725\) −0.0452923 −0.00168211
\(726\) −0.287146 −0.0106570
\(727\) −41.0223 −1.52143 −0.760716 0.649085i \(-0.775152\pi\)
−0.760716 + 0.649085i \(0.775152\pi\)
\(728\) −4.44344 −0.164685
\(729\) −21.7669 −0.806183
\(730\) −0.520547 −0.0192663
\(731\) 29.8193 1.10291
\(732\) −7.01794 −0.259391
\(733\) 19.7580 0.729780 0.364890 0.931051i \(-0.381107\pi\)
0.364890 + 0.931051i \(0.381107\pi\)
\(734\) −1.64218 −0.0606139
\(735\) 2.33093 0.0859777
\(736\) −3.96640 −0.146203
\(737\) −1.32813 −0.0489222
\(738\) −0.767116 −0.0282379
\(739\) 8.08268 0.297326 0.148663 0.988888i \(-0.452503\pi\)
0.148663 + 0.988888i \(0.452503\pi\)
\(740\) −7.66921 −0.281926
\(741\) −4.53657 −0.166655
\(742\) 2.89948 0.106443
\(743\) 10.5995 0.388859 0.194429 0.980917i \(-0.437714\pi\)
0.194429 + 0.980917i \(0.437714\pi\)
\(744\) −0.109854 −0.00402744
\(745\) −7.59224 −0.278158
\(746\) 0.622140 0.0227782
\(747\) −5.33065 −0.195038
\(748\) −2.03773 −0.0745068
\(749\) −33.4944 −1.22386
\(750\) −0.291227 −0.0106341
\(751\) 14.0003 0.510877 0.255438 0.966825i \(-0.417780\pi\)
0.255438 + 0.966825i \(0.417780\pi\)
\(752\) −15.3541 −0.559907
\(753\) −4.25895 −0.155205
\(754\) 0.247147 0.00900057
\(755\) 6.21445 0.226167
\(756\) −11.9034 −0.432922
\(757\) −28.8413 −1.04826 −0.524128 0.851640i \(-0.675609\pi\)
−0.524128 + 0.851640i \(0.675609\pi\)
\(758\) 2.28449 0.0829765
\(759\) 0.275167 0.00998794
\(760\) −2.56661 −0.0931008
\(761\) −14.2005 −0.514767 −0.257383 0.966309i \(-0.582860\pi\)
−0.257383 + 0.966309i \(0.582860\pi\)
\(762\) −0.00250877 −9.08831e−5 0
\(763\) 18.2453 0.660523
\(764\) −4.46448 −0.161519
\(765\) 30.6053 1.10654
\(766\) 0.408963 0.0147764
\(767\) 11.0228 0.398008
\(768\) 4.87745 0.176000
\(769\) 19.1077 0.689043 0.344521 0.938778i \(-0.388041\pi\)
0.344521 + 0.938778i \(0.388041\pi\)
\(770\) 0.130651 0.00470834
\(771\) −1.44047 −0.0518771
\(772\) −44.2317 −1.59193
\(773\) −29.5204 −1.06178 −0.530888 0.847442i \(-0.678141\pi\)
−0.530888 + 0.847442i \(0.678141\pi\)
\(774\) 1.53175 0.0550576
\(775\) 0.0667367 0.00239725
\(776\) 4.28007 0.153646
\(777\) −1.73083 −0.0620931
\(778\) 3.23837 0.116101
\(779\) 10.9527 0.392420
\(780\) 5.91591 0.211824
\(781\) −2.05053 −0.0733739
\(782\) 1.55715 0.0556837
\(783\) 1.32644 0.0474030
\(784\) 12.9901 0.463932
\(785\) −20.1074 −0.717665
\(786\) 0.381826 0.0136193
\(787\) −20.3547 −0.725568 −0.362784 0.931873i \(-0.618174\pi\)
−0.362784 + 0.931873i \(0.618174\pi\)
\(788\) 21.9766 0.782886
\(789\) 5.08183 0.180918
\(790\) 0.966206 0.0343761
\(791\) −67.3295 −2.39396
\(792\) −0.209709 −0.00745170
\(793\) −46.6092 −1.65514
\(794\) −1.05196 −0.0373326
\(795\) −7.73398 −0.274296
\(796\) −29.1384 −1.03278
\(797\) 51.2687 1.81603 0.908016 0.418935i \(-0.137597\pi\)
0.908016 + 0.418935i \(0.137597\pi\)
\(798\) −0.289123 −0.0102349
\(799\) 18.1883 0.643456
\(800\) 0.0630940 0.00223071
\(801\) 23.4576 0.828835
\(802\) 2.22548 0.0785845
\(803\) 0.607262 0.0214298
\(804\) −3.83269 −0.135169
\(805\) 28.8480 1.01676
\(806\) −0.364163 −0.0128271
\(807\) −2.36703 −0.0833235
\(808\) −0.722123 −0.0254042
\(809\) 53.5568 1.88296 0.941478 0.337073i \(-0.109437\pi\)
0.941478 + 0.337073i \(0.109437\pi\)
\(810\) 1.51625 0.0532757
\(811\) 42.8963 1.50629 0.753146 0.657854i \(-0.228535\pi\)
0.753146 + 0.657854i \(0.228535\pi\)
\(812\) −4.55124 −0.159717
\(813\) 2.04373 0.0716769
\(814\) −0.0309633 −0.00108526
\(815\) −19.7045 −0.690218
\(816\) −5.86004 −0.205142
\(817\) −21.8699 −0.765130
\(818\) 2.93569 0.102644
\(819\) −38.8601 −1.35788
\(820\) −14.2828 −0.498778
\(821\) 5.52686 0.192889 0.0964444 0.995338i \(-0.469253\pi\)
0.0964444 + 0.995338i \(0.469253\pi\)
\(822\) −0.448525 −0.0156441
\(823\) −19.0939 −0.665572 −0.332786 0.943002i \(-0.607989\pi\)
−0.332786 + 0.943002i \(0.607989\pi\)
\(824\) −4.89540 −0.170539
\(825\) −0.00437712 −0.000152392 0
\(826\) 0.702499 0.0244431
\(827\) −47.8774 −1.66486 −0.832430 0.554130i \(-0.813051\pi\)
−0.832430 + 0.554130i \(0.813051\pi\)
\(828\) −23.1121 −0.803202
\(829\) 13.1461 0.456583 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(830\) 0.343490 0.0119227
\(831\) 2.62655 0.0911139
\(832\) 32.7387 1.13501
\(833\) −15.3879 −0.533161
\(834\) −0.464757 −0.0160932
\(835\) 32.3146 1.11829
\(836\) 1.49450 0.0516883
\(837\) −1.95446 −0.0675561
\(838\) −0.207099 −0.00715412
\(839\) 33.7580 1.16545 0.582727 0.812668i \(-0.301986\pi\)
0.582727 + 0.812668i \(0.301986\pi\)
\(840\) 0.755368 0.0260627
\(841\) −28.4928 −0.982512
\(842\) 1.43408 0.0494218
\(843\) −8.32736 −0.286810
\(844\) −3.07832 −0.105960
\(845\) 10.0370 0.345282
\(846\) 0.934291 0.0321216
\(847\) 35.1187 1.20669
\(848\) −43.1009 −1.48009
\(849\) −7.38825 −0.253564
\(850\) −0.0247698 −0.000849598 0
\(851\) −6.83677 −0.234361
\(852\) −5.91740 −0.202727
\(853\) 16.1494 0.552946 0.276473 0.961022i \(-0.410834\pi\)
0.276473 + 0.961022i \(0.410834\pi\)
\(854\) −2.97048 −0.101648
\(855\) −22.4463 −0.767648
\(856\) −3.46426 −0.118406
\(857\) 21.1749 0.723322 0.361661 0.932310i \(-0.382210\pi\)
0.361661 + 0.932310i \(0.382210\pi\)
\(858\) 0.0238847 0.000815409 0
\(859\) −21.2136 −0.723798 −0.361899 0.932217i \(-0.617871\pi\)
−0.361899 + 0.932217i \(0.617871\pi\)
\(860\) 28.5194 0.972505
\(861\) −3.22342 −0.109854
\(862\) 2.58045 0.0878906
\(863\) −45.3619 −1.54414 −0.772068 0.635539i \(-0.780778\pi\)
−0.772068 + 0.635539i \(0.780778\pi\)
\(864\) −1.84778 −0.0628628
\(865\) −24.8647 −0.845424
\(866\) 1.36895 0.0465186
\(867\) 1.57532 0.0535005
\(868\) 6.70610 0.227620
\(869\) −1.12716 −0.0382363
\(870\) −0.0420140 −0.00142441
\(871\) −25.4545 −0.862494
\(872\) 1.88707 0.0639044
\(873\) 37.4313 1.26686
\(874\) −1.14204 −0.0386300
\(875\) 35.6178 1.20410
\(876\) 1.75243 0.0592090
\(877\) 12.1962 0.411837 0.205918 0.978569i \(-0.433982\pi\)
0.205918 + 0.978569i \(0.433982\pi\)
\(878\) 2.74490 0.0926358
\(879\) −1.12723 −0.0380204
\(880\) −1.94213 −0.0654693
\(881\) 8.50736 0.286620 0.143310 0.989678i \(-0.454225\pi\)
0.143310 + 0.989678i \(0.454225\pi\)
\(882\) −0.790443 −0.0266156
\(883\) −26.5027 −0.891888 −0.445944 0.895061i \(-0.647132\pi\)
−0.445944 + 0.895061i \(0.647132\pi\)
\(884\) −39.0546 −1.31355
\(885\) −1.87382 −0.0629879
\(886\) −2.06245 −0.0692895
\(887\) 33.1863 1.11429 0.557144 0.830416i \(-0.311897\pi\)
0.557144 + 0.830416i \(0.311897\pi\)
\(888\) −0.179016 −0.00600740
\(889\) 0.306829 0.0102907
\(890\) −1.51154 −0.0506668
\(891\) −1.76884 −0.0592582
\(892\) 53.1915 1.78098
\(893\) −13.3395 −0.446391
\(894\) −0.0884569 −0.00295844
\(895\) −14.7405 −0.492719
\(896\) 8.44850 0.282244
\(897\) 5.27378 0.176087
\(898\) −1.54659 −0.0516102
\(899\) −0.747286 −0.0249234
\(900\) 0.367647 0.0122549
\(901\) 51.0568 1.70095
\(902\) −0.0576649 −0.00192003
\(903\) 6.43642 0.214191
\(904\) −6.96377 −0.231612
\(905\) −31.4346 −1.04492
\(906\) 0.0724043 0.00240547
\(907\) 6.31486 0.209682 0.104841 0.994489i \(-0.466567\pi\)
0.104841 + 0.994489i \(0.466567\pi\)
\(908\) 12.7955 0.424634
\(909\) −6.31533 −0.209466
\(910\) 2.50403 0.0830077
\(911\) 43.0949 1.42780 0.713898 0.700249i \(-0.246928\pi\)
0.713898 + 0.700249i \(0.246928\pi\)
\(912\) 4.29783 0.142315
\(913\) −0.400710 −0.0132616
\(914\) −2.84674 −0.0941618
\(915\) 7.92337 0.261938
\(916\) 55.5418 1.83515
\(917\) −46.6983 −1.54211
\(918\) 0.725413 0.0239422
\(919\) −19.4088 −0.640236 −0.320118 0.947378i \(-0.603723\pi\)
−0.320118 + 0.947378i \(0.603723\pi\)
\(920\) 2.98370 0.0983697
\(921\) −0.779167 −0.0256744
\(922\) 0.932018 0.0306944
\(923\) −39.3000 −1.29358
\(924\) −0.439839 −0.0144696
\(925\) 0.108753 0.00357579
\(926\) −2.24052 −0.0736281
\(927\) −42.8128 −1.40616
\(928\) −0.706496 −0.0231919
\(929\) −28.3914 −0.931493 −0.465747 0.884918i \(-0.654214\pi\)
−0.465747 + 0.884918i \(0.654214\pi\)
\(930\) 0.0619062 0.00202999
\(931\) 11.2857 0.369874
\(932\) −1.29024 −0.0422631
\(933\) −6.28027 −0.205607
\(934\) 0.957900 0.0313434
\(935\) 2.30063 0.0752387
\(936\) −4.01924 −0.131373
\(937\) −4.00280 −0.130766 −0.0653829 0.997860i \(-0.520827\pi\)
−0.0653829 + 0.997860i \(0.520827\pi\)
\(938\) −1.62226 −0.0529687
\(939\) −3.16237 −0.103200
\(940\) 17.3954 0.567377
\(941\) 13.9675 0.455328 0.227664 0.973740i \(-0.426891\pi\)
0.227664 + 0.973740i \(0.426891\pi\)
\(942\) −0.234271 −0.00763295
\(943\) −12.7325 −0.414628
\(944\) −10.4427 −0.339880
\(945\) 13.4391 0.437174
\(946\) 0.115143 0.00374363
\(947\) 57.2963 1.86188 0.930940 0.365171i \(-0.118990\pi\)
0.930940 + 0.365171i \(0.118990\pi\)
\(948\) −3.25274 −0.105644
\(949\) 11.6386 0.377806
\(950\) 0.0181665 0.000589400 0
\(951\) −3.54943 −0.115098
\(952\) −4.98665 −0.161618
\(953\) 5.29452 0.171506 0.0857532 0.996316i \(-0.472670\pi\)
0.0857532 + 0.996316i \(0.472670\pi\)
\(954\) 2.62267 0.0849121
\(955\) 5.04047 0.163106
\(956\) 24.6417 0.796969
\(957\) 0.0490129 0.00158436
\(958\) −2.39934 −0.0775192
\(959\) 54.8558 1.77139
\(960\) −5.56545 −0.179624
\(961\) −29.8989 −0.964481
\(962\) −0.593436 −0.0191331
\(963\) −30.2967 −0.976298
\(964\) 25.4181 0.818663
\(965\) 49.9383 1.60757
\(966\) 0.336107 0.0108141
\(967\) 44.3615 1.42657 0.713284 0.700875i \(-0.247207\pi\)
0.713284 + 0.700875i \(0.247207\pi\)
\(968\) 3.63227 0.116746
\(969\) −5.09116 −0.163552
\(970\) −2.41196 −0.0774434
\(971\) 51.4789 1.65204 0.826018 0.563644i \(-0.190601\pi\)
0.826018 + 0.563644i \(0.190601\pi\)
\(972\) −16.2414 −0.520943
\(973\) 56.8410 1.82224
\(974\) 0.135446 0.00433996
\(975\) −0.0838907 −0.00268665
\(976\) 44.1563 1.41341
\(977\) −0.892938 −0.0285676 −0.0142838 0.999898i \(-0.504547\pi\)
−0.0142838 + 0.999898i \(0.504547\pi\)
\(978\) −0.229576 −0.00734104
\(979\) 1.76333 0.0563564
\(980\) −14.7171 −0.470122
\(981\) 16.5034 0.526913
\(982\) −0.536184 −0.0171103
\(983\) 36.2679 1.15677 0.578384 0.815765i \(-0.303684\pi\)
0.578384 + 0.815765i \(0.303684\pi\)
\(984\) −0.333393 −0.0106282
\(985\) −24.8120 −0.790576
\(986\) 0.277361 0.00883296
\(987\) 3.92590 0.124963
\(988\) 28.6432 0.911261
\(989\) 25.4239 0.808432
\(990\) 0.118178 0.00375595
\(991\) 29.3407 0.932036 0.466018 0.884775i \(-0.345688\pi\)
0.466018 + 0.884775i \(0.345688\pi\)
\(992\) 1.04100 0.0330518
\(993\) −9.49313 −0.301255
\(994\) −2.50466 −0.0794429
\(995\) 32.8977 1.04293
\(996\) −1.15636 −0.0366408
\(997\) 18.6388 0.590296 0.295148 0.955452i \(-0.404631\pi\)
0.295148 + 0.955452i \(0.404631\pi\)
\(998\) 2.46823 0.0781304
\(999\) −3.18497 −0.100768
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.1.7 10
3.2 odd 2 4527.2.a.k.1.4 10
4.3 odd 2 8048.2.a.p.1.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.e.1.7 10 1.1 even 1 trivial
4527.2.a.k.1.4 10 3.2 odd 2
8048.2.a.p.1.3 10 4.3 odd 2