Properties

Label 503.2.a.e.1.2
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 1.2
Root \(2.78533\) of defining polynomial
Character \(\chi\) \(=\) 503.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.03947 q^{2} +1.78533 q^{3} +2.15945 q^{4} -0.701114 q^{5} -3.64112 q^{6} -2.02991 q^{7} -0.325186 q^{8} +0.187388 q^{9} +O(q^{10})\) \(q-2.03947 q^{2} +1.78533 q^{3} +2.15945 q^{4} -0.701114 q^{5} -3.64112 q^{6} -2.02991 q^{7} -0.325186 q^{8} +0.187388 q^{9} +1.42990 q^{10} -0.626970 q^{11} +3.85531 q^{12} -2.93743 q^{13} +4.13995 q^{14} -1.25172 q^{15} -3.65568 q^{16} -2.71003 q^{17} -0.382172 q^{18} -1.11114 q^{19} -1.51402 q^{20} -3.62406 q^{21} +1.27869 q^{22} -0.412395 q^{23} -0.580563 q^{24} -4.50844 q^{25} +5.99081 q^{26} -5.02143 q^{27} -4.38349 q^{28} +6.46349 q^{29} +2.55284 q^{30} -4.14074 q^{31} +8.10604 q^{32} -1.11935 q^{33} +5.52704 q^{34} +1.42320 q^{35} +0.404653 q^{36} +2.98634 q^{37} +2.26613 q^{38} -5.24427 q^{39} +0.227992 q^{40} -0.135430 q^{41} +7.39116 q^{42} -0.861393 q^{43} -1.35391 q^{44} -0.131380 q^{45} +0.841067 q^{46} +1.67307 q^{47} -6.52659 q^{48} -2.87945 q^{49} +9.19483 q^{50} -4.83829 q^{51} -6.34323 q^{52} -8.51721 q^{53} +10.2411 q^{54} +0.439578 q^{55} +0.660099 q^{56} -1.98374 q^{57} -13.1821 q^{58} -0.406685 q^{59} -2.70302 q^{60} -8.53132 q^{61} +8.44492 q^{62} -0.380381 q^{63} -9.22067 q^{64} +2.05948 q^{65} +2.28288 q^{66} -13.0293 q^{67} -5.85217 q^{68} -0.736259 q^{69} -2.90258 q^{70} +2.78095 q^{71} -0.0609358 q^{72} -2.11734 q^{73} -6.09055 q^{74} -8.04903 q^{75} -2.39944 q^{76} +1.27270 q^{77} +10.6955 q^{78} -0.317368 q^{79} +2.56305 q^{80} -9.52705 q^{81} +0.276205 q^{82} -5.05120 q^{83} -7.82595 q^{84} +1.90004 q^{85} +1.75679 q^{86} +11.5394 q^{87} +0.203882 q^{88} +16.4699 q^{89} +0.267946 q^{90} +5.96273 q^{91} -0.890544 q^{92} -7.39256 q^{93} -3.41218 q^{94} +0.779033 q^{95} +14.4719 q^{96} +7.68873 q^{97} +5.87256 q^{98} -0.117486 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} + O(q^{10}) \) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} - 4q^{10} - 3q^{11} - 7q^{12} - 18q^{13} + q^{14} - 2q^{15} - 4q^{16} - 11q^{17} - q^{18} - 3q^{20} + q^{21} - 18q^{22} - 2q^{23} + 10q^{24} - 27q^{25} + 11q^{26} - 2q^{27} - 22q^{28} - 9q^{29} + 12q^{30} - 22q^{31} - 10q^{32} - 10q^{33} - 10q^{34} - 6q^{35} + 2q^{36} - 35q^{37} + 2q^{38} + 8q^{39} - 19q^{40} - 4q^{41} + 4q^{42} - 20q^{43} + 9q^{44} + 2q^{45} - q^{46} + 7q^{47} - 27q^{49} + 16q^{50} + 9q^{51} - 7q^{52} - 24q^{53} + 17q^{54} - 11q^{55} + 12q^{56} - 23q^{57} + 2q^{58} + 17q^{59} - 4q^{61} + 8q^{62} + 10q^{63} + 3q^{64} - 16q^{65} + 46q^{66} - 6q^{67} + 28q^{68} - 2q^{69} + 26q^{70} - q^{71} - q^{72} - 31q^{73} + 11q^{74} + 30q^{75} + 20q^{76} + 3q^{77} + 11q^{78} - 10q^{79} + 24q^{80} - 6q^{81} - 9q^{82} + 22q^{83} + 22q^{84} - 6q^{85} + 38q^{86} + 25q^{87} - 3q^{88} + q^{89} + 2q^{90} + 10q^{91} + 27q^{92} - 6q^{93} + 33q^{94} + 39q^{95} + 46q^{96} - 57q^{97} + 40q^{98} + 35q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.03947 −1.44212 −0.721062 0.692870i \(-0.756346\pi\)
−0.721062 + 0.692870i \(0.756346\pi\)
\(3\) 1.78533 1.03076 0.515379 0.856962i \(-0.327651\pi\)
0.515379 + 0.856962i \(0.327651\pi\)
\(4\) 2.15945 1.07972
\(5\) −0.701114 −0.313548 −0.156774 0.987635i \(-0.550109\pi\)
−0.156774 + 0.987635i \(0.550109\pi\)
\(6\) −3.64112 −1.48648
\(7\) −2.02991 −0.767235 −0.383618 0.923492i \(-0.625322\pi\)
−0.383618 + 0.923492i \(0.625322\pi\)
\(8\) −0.325186 −0.114971
\(9\) 0.187388 0.0624625
\(10\) 1.42990 0.452175
\(11\) −0.626970 −0.189039 −0.0945194 0.995523i \(-0.530131\pi\)
−0.0945194 + 0.995523i \(0.530131\pi\)
\(12\) 3.85531 1.11293
\(13\) −2.93743 −0.814697 −0.407349 0.913273i \(-0.633547\pi\)
−0.407349 + 0.913273i \(0.633547\pi\)
\(14\) 4.13995 1.10645
\(15\) −1.25172 −0.323192
\(16\) −3.65568 −0.913921
\(17\) −2.71003 −0.657280 −0.328640 0.944455i \(-0.606590\pi\)
−0.328640 + 0.944455i \(0.606590\pi\)
\(18\) −0.382172 −0.0900787
\(19\) −1.11114 −0.254912 −0.127456 0.991844i \(-0.540681\pi\)
−0.127456 + 0.991844i \(0.540681\pi\)
\(20\) −1.51402 −0.338545
\(21\) −3.62406 −0.790834
\(22\) 1.27869 0.272617
\(23\) −0.412395 −0.0859902 −0.0429951 0.999075i \(-0.513690\pi\)
−0.0429951 + 0.999075i \(0.513690\pi\)
\(24\) −0.580563 −0.118507
\(25\) −4.50844 −0.901688
\(26\) 5.99081 1.17489
\(27\) −5.02143 −0.966374
\(28\) −4.38349 −0.828402
\(29\) 6.46349 1.20024 0.600120 0.799910i \(-0.295120\pi\)
0.600120 + 0.799910i \(0.295120\pi\)
\(30\) 2.55284 0.466083
\(31\) −4.14074 −0.743698 −0.371849 0.928293i \(-0.621276\pi\)
−0.371849 + 0.928293i \(0.621276\pi\)
\(32\) 8.10604 1.43296
\(33\) −1.11935 −0.194853
\(34\) 5.52704 0.947879
\(35\) 1.42320 0.240565
\(36\) 0.404653 0.0674422
\(37\) 2.98634 0.490951 0.245475 0.969403i \(-0.421056\pi\)
0.245475 + 0.969403i \(0.421056\pi\)
\(38\) 2.26613 0.367615
\(39\) −5.24427 −0.839756
\(40\) 0.227992 0.0360488
\(41\) −0.135430 −0.0211505 −0.0105753 0.999944i \(-0.503366\pi\)
−0.0105753 + 0.999944i \(0.503366\pi\)
\(42\) 7.39116 1.14048
\(43\) −0.861393 −0.131361 −0.0656806 0.997841i \(-0.520922\pi\)
−0.0656806 + 0.997841i \(0.520922\pi\)
\(44\) −1.35391 −0.204109
\(45\) −0.131380 −0.0195850
\(46\) 0.841067 0.124009
\(47\) 1.67307 0.244043 0.122021 0.992527i \(-0.461062\pi\)
0.122021 + 0.992527i \(0.461062\pi\)
\(48\) −6.52659 −0.942032
\(49\) −2.87945 −0.411350
\(50\) 9.19483 1.30035
\(51\) −4.83829 −0.677497
\(52\) −6.34323 −0.879647
\(53\) −8.51721 −1.16993 −0.584964 0.811059i \(-0.698892\pi\)
−0.584964 + 0.811059i \(0.698892\pi\)
\(54\) 10.2411 1.39363
\(55\) 0.439578 0.0592727
\(56\) 0.660099 0.0882095
\(57\) −1.98374 −0.262753
\(58\) −13.1821 −1.73089
\(59\) −0.406685 −0.0529459 −0.0264729 0.999650i \(-0.508428\pi\)
−0.0264729 + 0.999650i \(0.508428\pi\)
\(60\) −2.70302 −0.348958
\(61\) −8.53132 −1.09232 −0.546162 0.837680i \(-0.683912\pi\)
−0.546162 + 0.837680i \(0.683912\pi\)
\(62\) 8.44492 1.07251
\(63\) −0.380381 −0.0479234
\(64\) −9.22067 −1.15258
\(65\) 2.05948 0.255446
\(66\) 2.28288 0.281003
\(67\) −13.0293 −1.59178 −0.795889 0.605443i \(-0.792996\pi\)
−0.795889 + 0.605443i \(0.792996\pi\)
\(68\) −5.85217 −0.709680
\(69\) −0.736259 −0.0886352
\(70\) −2.90258 −0.346925
\(71\) 2.78095 0.330039 0.165019 0.986290i \(-0.447231\pi\)
0.165019 + 0.986290i \(0.447231\pi\)
\(72\) −0.0609358 −0.00718135
\(73\) −2.11734 −0.247816 −0.123908 0.992294i \(-0.539543\pi\)
−0.123908 + 0.992294i \(0.539543\pi\)
\(74\) −6.09055 −0.708012
\(75\) −8.04903 −0.929422
\(76\) −2.39944 −0.275234
\(77\) 1.27270 0.145037
\(78\) 10.6955 1.21103
\(79\) −0.317368 −0.0357067 −0.0178534 0.999841i \(-0.505683\pi\)
−0.0178534 + 0.999841i \(0.505683\pi\)
\(80\) 2.56305 0.286558
\(81\) −9.52705 −1.05856
\(82\) 0.276205 0.0305017
\(83\) −5.05120 −0.554442 −0.277221 0.960806i \(-0.589413\pi\)
−0.277221 + 0.960806i \(0.589413\pi\)
\(84\) −7.82595 −0.853882
\(85\) 1.90004 0.206089
\(86\) 1.75679 0.189439
\(87\) 11.5394 1.23716
\(88\) 0.203882 0.0217339
\(89\) 16.4699 1.74580 0.872901 0.487897i \(-0.162236\pi\)
0.872901 + 0.487897i \(0.162236\pi\)
\(90\) 0.267946 0.0282440
\(91\) 5.96273 0.625064
\(92\) −0.890544 −0.0928456
\(93\) −7.39256 −0.766573
\(94\) −3.41218 −0.351940
\(95\) 0.779033 0.0799271
\(96\) 14.4719 1.47703
\(97\) 7.68873 0.780673 0.390336 0.920672i \(-0.372359\pi\)
0.390336 + 0.920672i \(0.372359\pi\)
\(98\) 5.87256 0.593218
\(99\) −0.117486 −0.0118078
\(100\) −9.73573 −0.973573
\(101\) −0.549020 −0.0546296 −0.0273148 0.999627i \(-0.508696\pi\)
−0.0273148 + 0.999627i \(0.508696\pi\)
\(102\) 9.86757 0.977035
\(103\) 1.32758 0.130810 0.0654051 0.997859i \(-0.479166\pi\)
0.0654051 + 0.997859i \(0.479166\pi\)
\(104\) 0.955211 0.0936662
\(105\) 2.54088 0.247964
\(106\) 17.3706 1.68718
\(107\) 2.57528 0.248961 0.124481 0.992222i \(-0.460274\pi\)
0.124481 + 0.992222i \(0.460274\pi\)
\(108\) −10.8435 −1.04342
\(109\) 8.89011 0.851518 0.425759 0.904837i \(-0.360007\pi\)
0.425759 + 0.904837i \(0.360007\pi\)
\(110\) −0.896507 −0.0854786
\(111\) 5.33159 0.506052
\(112\) 7.42072 0.701192
\(113\) −1.91987 −0.180606 −0.0903032 0.995914i \(-0.528784\pi\)
−0.0903032 + 0.995914i \(0.528784\pi\)
\(114\) 4.04578 0.378922
\(115\) 0.289136 0.0269621
\(116\) 13.9576 1.29593
\(117\) −0.550438 −0.0508880
\(118\) 0.829423 0.0763546
\(119\) 5.50114 0.504288
\(120\) 0.407041 0.0371576
\(121\) −10.6069 −0.964264
\(122\) 17.3994 1.57527
\(123\) −0.241786 −0.0218011
\(124\) −8.94170 −0.802988
\(125\) 6.66650 0.596270
\(126\) 0.775776 0.0691116
\(127\) 21.8010 1.93452 0.967261 0.253782i \(-0.0816746\pi\)
0.967261 + 0.253782i \(0.0816746\pi\)
\(128\) 2.59322 0.229210
\(129\) −1.53787 −0.135402
\(130\) −4.20024 −0.368386
\(131\) 8.78145 0.767239 0.383620 0.923491i \(-0.374677\pi\)
0.383620 + 0.923491i \(0.374677\pi\)
\(132\) −2.41717 −0.210387
\(133\) 2.25551 0.195577
\(134\) 26.5728 2.29554
\(135\) 3.52060 0.303005
\(136\) 0.881265 0.0755678
\(137\) −4.26127 −0.364065 −0.182032 0.983293i \(-0.558268\pi\)
−0.182032 + 0.983293i \(0.558268\pi\)
\(138\) 1.50158 0.127823
\(139\) 13.4951 1.14464 0.572321 0.820030i \(-0.306043\pi\)
0.572321 + 0.820030i \(0.306043\pi\)
\(140\) 3.07333 0.259743
\(141\) 2.98698 0.251549
\(142\) −5.67168 −0.475957
\(143\) 1.84168 0.154009
\(144\) −0.685030 −0.0570858
\(145\) −4.53164 −0.376332
\(146\) 4.31826 0.357382
\(147\) −5.14076 −0.424003
\(148\) 6.44884 0.530091
\(149\) 11.2453 0.921253 0.460626 0.887594i \(-0.347625\pi\)
0.460626 + 0.887594i \(0.347625\pi\)
\(150\) 16.4158 1.34034
\(151\) −2.36107 −0.192141 −0.0960705 0.995375i \(-0.530627\pi\)
−0.0960705 + 0.995375i \(0.530627\pi\)
\(152\) 0.361325 0.0293074
\(153\) −0.507827 −0.0410554
\(154\) −2.59563 −0.209162
\(155\) 2.90313 0.233185
\(156\) −11.3247 −0.906704
\(157\) −17.5242 −1.39859 −0.699293 0.714835i \(-0.746502\pi\)
−0.699293 + 0.714835i \(0.746502\pi\)
\(158\) 0.647264 0.0514936
\(159\) −15.2060 −1.20591
\(160\) −5.68326 −0.449301
\(161\) 0.837126 0.0659747
\(162\) 19.4301 1.52658
\(163\) −7.12153 −0.557801 −0.278901 0.960320i \(-0.589970\pi\)
−0.278901 + 0.960320i \(0.589970\pi\)
\(164\) −0.292453 −0.0228367
\(165\) 0.784790 0.0610958
\(166\) 10.3018 0.799574
\(167\) 4.56080 0.352925 0.176463 0.984307i \(-0.443535\pi\)
0.176463 + 0.984307i \(0.443535\pi\)
\(168\) 1.17849 0.0909226
\(169\) −4.37149 −0.336269
\(170\) −3.87509 −0.297206
\(171\) −0.208213 −0.0159224
\(172\) −1.86013 −0.141834
\(173\) 2.91357 0.221514 0.110757 0.993847i \(-0.464672\pi\)
0.110757 + 0.993847i \(0.464672\pi\)
\(174\) −23.5343 −1.78413
\(175\) 9.15174 0.691807
\(176\) 2.29201 0.172766
\(177\) −0.726065 −0.0545744
\(178\) −33.5898 −2.51766
\(179\) −9.01141 −0.673544 −0.336772 0.941586i \(-0.609335\pi\)
−0.336772 + 0.941586i \(0.609335\pi\)
\(180\) −0.283708 −0.0211464
\(181\) 14.4065 1.07082 0.535412 0.844591i \(-0.320156\pi\)
0.535412 + 0.844591i \(0.320156\pi\)
\(182\) −12.1608 −0.901420
\(183\) −15.2312 −1.12592
\(184\) 0.134105 0.00988635
\(185\) −2.09376 −0.153937
\(186\) 15.0769 1.10549
\(187\) 1.69911 0.124251
\(188\) 3.61291 0.263499
\(189\) 10.1931 0.741437
\(190\) −1.58882 −0.115265
\(191\) 4.40730 0.318901 0.159450 0.987206i \(-0.449028\pi\)
0.159450 + 0.987206i \(0.449028\pi\)
\(192\) −16.4619 −1.18804
\(193\) −2.90027 −0.208766 −0.104383 0.994537i \(-0.533287\pi\)
−0.104383 + 0.994537i \(0.533287\pi\)
\(194\) −15.6810 −1.12583
\(195\) 3.67683 0.263304
\(196\) −6.21802 −0.444144
\(197\) 5.91510 0.421433 0.210717 0.977547i \(-0.432420\pi\)
0.210717 + 0.977547i \(0.432420\pi\)
\(198\) 0.239610 0.0170284
\(199\) −4.73473 −0.335636 −0.167818 0.985818i \(-0.553672\pi\)
−0.167818 + 0.985818i \(0.553672\pi\)
\(200\) 1.46608 0.103668
\(201\) −23.2615 −1.64074
\(202\) 1.11971 0.0787827
\(203\) −13.1203 −0.920866
\(204\) −10.4480 −0.731509
\(205\) 0.0949516 0.00663171
\(206\) −2.70756 −0.188644
\(207\) −0.0772776 −0.00537117
\(208\) 10.7383 0.744569
\(209\) 0.696649 0.0481882
\(210\) −5.18205 −0.357595
\(211\) −1.50671 −0.103726 −0.0518631 0.998654i \(-0.516516\pi\)
−0.0518631 + 0.998654i \(0.516516\pi\)
\(212\) −18.3925 −1.26320
\(213\) 4.96491 0.340190
\(214\) −5.25220 −0.359033
\(215\) 0.603935 0.0411880
\(216\) 1.63290 0.111105
\(217\) 8.40534 0.570591
\(218\) −18.1311 −1.22799
\(219\) −3.78014 −0.255438
\(220\) 0.949245 0.0639981
\(221\) 7.96054 0.535484
\(222\) −10.8736 −0.729790
\(223\) −9.79220 −0.655734 −0.327867 0.944724i \(-0.606330\pi\)
−0.327867 + 0.944724i \(0.606330\pi\)
\(224\) −16.4546 −1.09942
\(225\) −0.844825 −0.0563217
\(226\) 3.91552 0.260457
\(227\) 6.88512 0.456981 0.228491 0.973546i \(-0.426621\pi\)
0.228491 + 0.973546i \(0.426621\pi\)
\(228\) −4.28378 −0.283700
\(229\) 12.1880 0.805405 0.402703 0.915331i \(-0.368071\pi\)
0.402703 + 0.915331i \(0.368071\pi\)
\(230\) −0.589684 −0.0388826
\(231\) 2.27218 0.149498
\(232\) −2.10183 −0.137992
\(233\) −21.0547 −1.37934 −0.689669 0.724125i \(-0.742244\pi\)
−0.689669 + 0.724125i \(0.742244\pi\)
\(234\) 1.12260 0.0733869
\(235\) −1.17301 −0.0765190
\(236\) −0.878215 −0.0571669
\(237\) −0.566606 −0.0368050
\(238\) −11.2194 −0.727247
\(239\) 9.11944 0.589888 0.294944 0.955515i \(-0.404699\pi\)
0.294944 + 0.955515i \(0.404699\pi\)
\(240\) 4.57588 0.295372
\(241\) 22.7352 1.46450 0.732252 0.681034i \(-0.238469\pi\)
0.732252 + 0.681034i \(0.238469\pi\)
\(242\) 21.6325 1.39059
\(243\) −1.94460 −0.124746
\(244\) −18.4229 −1.17941
\(245\) 2.01882 0.128978
\(246\) 0.493116 0.0314399
\(247\) 3.26388 0.207676
\(248\) 1.34651 0.0855034
\(249\) −9.01804 −0.571495
\(250\) −13.5961 −0.859896
\(251\) −11.6824 −0.737385 −0.368692 0.929551i \(-0.620194\pi\)
−0.368692 + 0.929551i \(0.620194\pi\)
\(252\) −0.821411 −0.0517441
\(253\) 0.258559 0.0162555
\(254\) −44.4625 −2.78982
\(255\) 3.39220 0.212428
\(256\) 13.1525 0.822034
\(257\) −16.3010 −1.01683 −0.508415 0.861112i \(-0.669768\pi\)
−0.508415 + 0.861112i \(0.669768\pi\)
\(258\) 3.13644 0.195266
\(259\) −6.06201 −0.376675
\(260\) 4.44733 0.275811
\(261\) 1.21118 0.0749700
\(262\) −17.9095 −1.10645
\(263\) −15.7955 −0.973993 −0.486996 0.873404i \(-0.661907\pi\)
−0.486996 + 0.873404i \(0.661907\pi\)
\(264\) 0.363996 0.0224024
\(265\) 5.97154 0.366829
\(266\) −4.60005 −0.282047
\(267\) 29.4041 1.79950
\(268\) −28.1360 −1.71868
\(269\) 11.2401 0.685323 0.342662 0.939459i \(-0.388672\pi\)
0.342662 + 0.939459i \(0.388672\pi\)
\(270\) −7.18016 −0.436970
\(271\) 22.7114 1.37962 0.689809 0.723991i \(-0.257694\pi\)
0.689809 + 0.723991i \(0.257694\pi\)
\(272\) 9.90703 0.600702
\(273\) 10.6454 0.644290
\(274\) 8.69074 0.525027
\(275\) 2.82666 0.170454
\(276\) −1.58991 −0.0957014
\(277\) −3.42609 −0.205854 −0.102927 0.994689i \(-0.532821\pi\)
−0.102927 + 0.994689i \(0.532821\pi\)
\(278\) −27.5229 −1.65072
\(279\) −0.775923 −0.0464533
\(280\) −0.462805 −0.0276579
\(281\) −6.41168 −0.382488 −0.191244 0.981542i \(-0.561252\pi\)
−0.191244 + 0.981542i \(0.561252\pi\)
\(282\) −6.09186 −0.362765
\(283\) −22.3462 −1.32834 −0.664172 0.747580i \(-0.731216\pi\)
−0.664172 + 0.747580i \(0.731216\pi\)
\(284\) 6.00532 0.356350
\(285\) 1.39083 0.0823855
\(286\) −3.75606 −0.222101
\(287\) 0.274910 0.0162274
\(288\) 1.51897 0.0895062
\(289\) −9.65571 −0.567983
\(290\) 9.24216 0.542718
\(291\) 13.7269 0.804685
\(292\) −4.57228 −0.267573
\(293\) 4.95366 0.289396 0.144698 0.989476i \(-0.453779\pi\)
0.144698 + 0.989476i \(0.453779\pi\)
\(294\) 10.4844 0.611464
\(295\) 0.285133 0.0166011
\(296\) −0.971115 −0.0564449
\(297\) 3.14829 0.182682
\(298\) −22.9345 −1.32856
\(299\) 1.21138 0.0700560
\(300\) −17.3814 −1.00352
\(301\) 1.74855 0.100785
\(302\) 4.81533 0.277091
\(303\) −0.980180 −0.0563099
\(304\) 4.06196 0.232969
\(305\) 5.98143 0.342496
\(306\) 1.03570 0.0592069
\(307\) −1.86937 −0.106690 −0.0533452 0.998576i \(-0.516988\pi\)
−0.0533452 + 0.998576i \(0.516988\pi\)
\(308\) 2.74832 0.156600
\(309\) 2.37016 0.134834
\(310\) −5.92085 −0.336282
\(311\) −4.29662 −0.243639 −0.121819 0.992552i \(-0.538873\pi\)
−0.121819 + 0.992552i \(0.538873\pi\)
\(312\) 1.70536 0.0965472
\(313\) −24.2378 −1.37000 −0.684999 0.728544i \(-0.740198\pi\)
−0.684999 + 0.728544i \(0.740198\pi\)
\(314\) 35.7402 2.01693
\(315\) 0.266690 0.0150263
\(316\) −0.685340 −0.0385534
\(317\) −28.1902 −1.58332 −0.791661 0.610961i \(-0.790783\pi\)
−0.791661 + 0.610961i \(0.790783\pi\)
\(318\) 31.0122 1.73908
\(319\) −4.05242 −0.226892
\(320\) 6.46474 0.361390
\(321\) 4.59771 0.256619
\(322\) −1.70729 −0.0951438
\(323\) 3.01121 0.167548
\(324\) −20.5731 −1.14295
\(325\) 13.2432 0.734602
\(326\) 14.5242 0.804419
\(327\) 15.8717 0.877709
\(328\) 0.0440398 0.00243169
\(329\) −3.39619 −0.187238
\(330\) −1.60056 −0.0881077
\(331\) 13.5185 0.743046 0.371523 0.928424i \(-0.378836\pi\)
0.371523 + 0.928424i \(0.378836\pi\)
\(332\) −10.9078 −0.598644
\(333\) 0.559603 0.0306660
\(334\) −9.30162 −0.508962
\(335\) 9.13500 0.499098
\(336\) 13.2484 0.722760
\(337\) −1.73199 −0.0943475 −0.0471738 0.998887i \(-0.515021\pi\)
−0.0471738 + 0.998887i \(0.515021\pi\)
\(338\) 8.91554 0.484941
\(339\) −3.42760 −0.186161
\(340\) 4.10304 0.222519
\(341\) 2.59612 0.140588
\(342\) 0.424644 0.0229621
\(343\) 20.0544 1.08284
\(344\) 0.280113 0.0151027
\(345\) 0.516202 0.0277914
\(346\) −5.94214 −0.319451
\(347\) −27.7262 −1.48842 −0.744210 0.667946i \(-0.767174\pi\)
−0.744210 + 0.667946i \(0.767174\pi\)
\(348\) 24.9188 1.33579
\(349\) −16.0612 −0.859737 −0.429869 0.902891i \(-0.641440\pi\)
−0.429869 + 0.902891i \(0.641440\pi\)
\(350\) −18.6647 −0.997671
\(351\) 14.7501 0.787302
\(352\) −5.08225 −0.270885
\(353\) −16.0011 −0.851652 −0.425826 0.904805i \(-0.640016\pi\)
−0.425826 + 0.904805i \(0.640016\pi\)
\(354\) 1.48079 0.0787031
\(355\) −1.94977 −0.103483
\(356\) 35.5658 1.88498
\(357\) 9.82132 0.519799
\(358\) 18.3785 0.971335
\(359\) 29.2698 1.54480 0.772400 0.635136i \(-0.219056\pi\)
0.772400 + 0.635136i \(0.219056\pi\)
\(360\) 0.0427229 0.00225170
\(361\) −17.7654 −0.935020
\(362\) −29.3816 −1.54426
\(363\) −18.9368 −0.993923
\(364\) 12.8762 0.674896
\(365\) 1.48450 0.0777022
\(366\) 31.0636 1.62372
\(367\) 19.5331 1.01962 0.509810 0.860287i \(-0.329716\pi\)
0.509810 + 0.860287i \(0.329716\pi\)
\(368\) 1.50759 0.0785883
\(369\) −0.0253778 −0.00132112
\(370\) 4.27017 0.221996
\(371\) 17.2892 0.897611
\(372\) −15.9638 −0.827687
\(373\) −22.0592 −1.14218 −0.571091 0.820887i \(-0.693480\pi\)
−0.571091 + 0.820887i \(0.693480\pi\)
\(374\) −3.46529 −0.179186
\(375\) 11.9019 0.614610
\(376\) −0.544059 −0.0280577
\(377\) −18.9861 −0.977832
\(378\) −20.7885 −1.06924
\(379\) 10.7429 0.551828 0.275914 0.961182i \(-0.411020\pi\)
0.275914 + 0.961182i \(0.411020\pi\)
\(380\) 1.68228 0.0862991
\(381\) 38.9218 1.99403
\(382\) −8.98856 −0.459895
\(383\) 7.81876 0.399520 0.199760 0.979845i \(-0.435984\pi\)
0.199760 + 0.979845i \(0.435984\pi\)
\(384\) 4.62974 0.236261
\(385\) −0.892305 −0.0454761
\(386\) 5.91502 0.301067
\(387\) −0.161414 −0.00820515
\(388\) 16.6034 0.842910
\(389\) −35.0312 −1.77615 −0.888075 0.459698i \(-0.847957\pi\)
−0.888075 + 0.459698i \(0.847957\pi\)
\(390\) −7.49880 −0.379717
\(391\) 1.11760 0.0565197
\(392\) 0.936357 0.0472932
\(393\) 15.6778 0.790838
\(394\) −12.0637 −0.607759
\(395\) 0.222512 0.0111958
\(396\) −0.253706 −0.0127492
\(397\) −0.245438 −0.0123182 −0.00615909 0.999981i \(-0.501961\pi\)
−0.00615909 + 0.999981i \(0.501961\pi\)
\(398\) 9.65635 0.484029
\(399\) 4.02682 0.201593
\(400\) 16.4814 0.824072
\(401\) −16.3434 −0.816151 −0.408075 0.912948i \(-0.633800\pi\)
−0.408075 + 0.912948i \(0.633800\pi\)
\(402\) 47.4411 2.36615
\(403\) 12.1631 0.605889
\(404\) −1.18558 −0.0589848
\(405\) 6.67955 0.331909
\(406\) 26.7585 1.32800
\(407\) −1.87235 −0.0928087
\(408\) 1.57334 0.0778922
\(409\) 4.60893 0.227897 0.113948 0.993487i \(-0.463650\pi\)
0.113948 + 0.993487i \(0.463650\pi\)
\(410\) −0.193651 −0.00956375
\(411\) −7.60775 −0.375263
\(412\) 2.86683 0.141239
\(413\) 0.825536 0.0406220
\(414\) 0.157606 0.00774589
\(415\) 3.54147 0.173844
\(416\) −23.8109 −1.16743
\(417\) 24.0932 1.17985
\(418\) −1.42080 −0.0694934
\(419\) 11.4509 0.559414 0.279707 0.960085i \(-0.409763\pi\)
0.279707 + 0.960085i \(0.409763\pi\)
\(420\) 5.48689 0.267733
\(421\) −22.2626 −1.08501 −0.542505 0.840052i \(-0.682524\pi\)
−0.542505 + 0.840052i \(0.682524\pi\)
\(422\) 3.07289 0.149586
\(423\) 0.313513 0.0152435
\(424\) 2.76968 0.134507
\(425\) 12.2180 0.592661
\(426\) −10.1258 −0.490596
\(427\) 17.3178 0.838069
\(428\) 5.56117 0.268809
\(429\) 3.28800 0.158746
\(430\) −1.23171 −0.0593982
\(431\) −1.13299 −0.0545741 −0.0272871 0.999628i \(-0.508687\pi\)
−0.0272871 + 0.999628i \(0.508687\pi\)
\(432\) 18.3568 0.883190
\(433\) −2.65692 −0.127683 −0.0638417 0.997960i \(-0.520335\pi\)
−0.0638417 + 0.997960i \(0.520335\pi\)
\(434\) −17.1425 −0.822864
\(435\) −8.09046 −0.387908
\(436\) 19.1977 0.919403
\(437\) 0.458226 0.0219199
\(438\) 7.70950 0.368374
\(439\) 6.57228 0.313678 0.156839 0.987624i \(-0.449870\pi\)
0.156839 + 0.987624i \(0.449870\pi\)
\(440\) −0.142945 −0.00681461
\(441\) −0.539573 −0.0256940
\(442\) −16.2353 −0.772235
\(443\) 1.34406 0.0638583 0.0319291 0.999490i \(-0.489835\pi\)
0.0319291 + 0.999490i \(0.489835\pi\)
\(444\) 11.5133 0.546396
\(445\) −11.5473 −0.547393
\(446\) 19.9709 0.945650
\(447\) 20.0766 0.949589
\(448\) 18.7172 0.884303
\(449\) 26.3410 1.24311 0.621555 0.783371i \(-0.286501\pi\)
0.621555 + 0.783371i \(0.286501\pi\)
\(450\) 1.72300 0.0812229
\(451\) 0.0849103 0.00399827
\(452\) −4.14586 −0.195005
\(453\) −4.21528 −0.198051
\(454\) −14.0420 −0.659024
\(455\) −4.18056 −0.195988
\(456\) 0.645084 0.0302088
\(457\) −30.0693 −1.40658 −0.703290 0.710903i \(-0.748287\pi\)
−0.703290 + 0.710903i \(0.748287\pi\)
\(458\) −24.8571 −1.16149
\(459\) 13.6082 0.635179
\(460\) 0.624373 0.0291115
\(461\) −7.72477 −0.359778 −0.179889 0.983687i \(-0.557574\pi\)
−0.179889 + 0.983687i \(0.557574\pi\)
\(462\) −4.63404 −0.215595
\(463\) 7.82961 0.363873 0.181936 0.983310i \(-0.441764\pi\)
0.181936 + 0.983310i \(0.441764\pi\)
\(464\) −23.6285 −1.09692
\(465\) 5.18303 0.240357
\(466\) 42.9404 1.98918
\(467\) −7.19623 −0.333002 −0.166501 0.986041i \(-0.553247\pi\)
−0.166501 + 0.986041i \(0.553247\pi\)
\(468\) −1.18864 −0.0549450
\(469\) 26.4483 1.22127
\(470\) 2.39233 0.110350
\(471\) −31.2865 −1.44160
\(472\) 0.132248 0.00608722
\(473\) 0.540068 0.0248323
\(474\) 1.15558 0.0530774
\(475\) 5.00948 0.229851
\(476\) 11.8794 0.544492
\(477\) −1.59602 −0.0730767
\(478\) −18.5988 −0.850691
\(479\) −12.1955 −0.557229 −0.278614 0.960403i \(-0.589875\pi\)
−0.278614 + 0.960403i \(0.589875\pi\)
\(480\) −10.1465 −0.463121
\(481\) −8.77217 −0.399976
\(482\) −46.3678 −2.11200
\(483\) 1.49454 0.0680040
\(484\) −22.9050 −1.04114
\(485\) −5.39068 −0.244778
\(486\) 3.96595 0.179899
\(487\) 35.9684 1.62988 0.814942 0.579543i \(-0.196769\pi\)
0.814942 + 0.579543i \(0.196769\pi\)
\(488\) 2.77426 0.125585
\(489\) −12.7142 −0.574958
\(490\) −4.11734 −0.186002
\(491\) −17.3618 −0.783527 −0.391763 0.920066i \(-0.628135\pi\)
−0.391763 + 0.920066i \(0.628135\pi\)
\(492\) −0.522124 −0.0235392
\(493\) −17.5163 −0.788893
\(494\) −6.65660 −0.299495
\(495\) 0.0823714 0.00370232
\(496\) 15.1372 0.679682
\(497\) −5.64510 −0.253217
\(498\) 18.3921 0.824167
\(499\) −30.0817 −1.34664 −0.673320 0.739351i \(-0.735133\pi\)
−0.673320 + 0.739351i \(0.735133\pi\)
\(500\) 14.3960 0.643806
\(501\) 8.14251 0.363781
\(502\) 23.8259 1.06340
\(503\) −1.00000 −0.0445878
\(504\) 0.123694 0.00550979
\(505\) 0.384926 0.0171290
\(506\) −0.527324 −0.0234424
\(507\) −7.80454 −0.346612
\(508\) 47.0780 2.08875
\(509\) −32.5495 −1.44273 −0.721366 0.692554i \(-0.756485\pi\)
−0.721366 + 0.692554i \(0.756485\pi\)
\(510\) −6.91829 −0.306347
\(511\) 4.29802 0.190133
\(512\) −32.0107 −1.41469
\(513\) 5.57949 0.246340
\(514\) 33.2455 1.46639
\(515\) −0.930784 −0.0410152
\(516\) −3.32094 −0.146196
\(517\) −1.04897 −0.0461335
\(518\) 12.3633 0.543212
\(519\) 5.20166 0.228328
\(520\) −0.669712 −0.0293688
\(521\) −12.6474 −0.554094 −0.277047 0.960856i \(-0.589356\pi\)
−0.277047 + 0.960856i \(0.589356\pi\)
\(522\) −2.47016 −0.108116
\(523\) 14.7294 0.644073 0.322036 0.946727i \(-0.395633\pi\)
0.322036 + 0.946727i \(0.395633\pi\)
\(524\) 18.9631 0.828406
\(525\) 16.3388 0.713085
\(526\) 32.2145 1.40462
\(527\) 11.2215 0.488818
\(528\) 4.09198 0.178080
\(529\) −22.8299 −0.992606
\(530\) −12.1788 −0.529013
\(531\) −0.0762077 −0.00330713
\(532\) 4.87065 0.211169
\(533\) 0.397815 0.0172313
\(534\) −59.9688 −2.59510
\(535\) −1.80556 −0.0780613
\(536\) 4.23693 0.183008
\(537\) −16.0883 −0.694261
\(538\) −22.9239 −0.988321
\(539\) 1.80533 0.0777611
\(540\) 7.60254 0.327161
\(541\) −27.4172 −1.17876 −0.589379 0.807857i \(-0.700627\pi\)
−0.589379 + 0.807857i \(0.700627\pi\)
\(542\) −46.3192 −1.98958
\(543\) 25.7202 1.10376
\(544\) −21.9676 −0.941855
\(545\) −6.23298 −0.266992
\(546\) −21.7110 −0.929147
\(547\) −16.1884 −0.692164 −0.346082 0.938204i \(-0.612488\pi\)
−0.346082 + 0.938204i \(0.612488\pi\)
\(548\) −9.20198 −0.393089
\(549\) −1.59866 −0.0682293
\(550\) −5.76489 −0.245816
\(551\) −7.18181 −0.305955
\(552\) 0.239421 0.0101904
\(553\) 0.644231 0.0273955
\(554\) 6.98742 0.296867
\(555\) −3.73805 −0.158671
\(556\) 29.1420 1.23590
\(557\) −13.3858 −0.567176 −0.283588 0.958946i \(-0.591525\pi\)
−0.283588 + 0.958946i \(0.591525\pi\)
\(558\) 1.58247 0.0669914
\(559\) 2.53028 0.107020
\(560\) −5.20277 −0.219857
\(561\) 3.03347 0.128073
\(562\) 13.0764 0.551596
\(563\) 23.3845 0.985538 0.492769 0.870160i \(-0.335985\pi\)
0.492769 + 0.870160i \(0.335985\pi\)
\(564\) 6.45022 0.271603
\(565\) 1.34605 0.0566287
\(566\) 45.5745 1.91564
\(567\) 19.3391 0.812165
\(568\) −0.904327 −0.0379447
\(569\) 43.0171 1.80337 0.901685 0.432393i \(-0.142331\pi\)
0.901685 + 0.432393i \(0.142331\pi\)
\(570\) −2.83655 −0.118810
\(571\) −1.24920 −0.0522772 −0.0261386 0.999658i \(-0.508321\pi\)
−0.0261386 + 0.999658i \(0.508321\pi\)
\(572\) 3.97702 0.166287
\(573\) 7.86846 0.328710
\(574\) −0.560672 −0.0234020
\(575\) 1.85926 0.0775364
\(576\) −1.72784 −0.0719933
\(577\) −38.6392 −1.60857 −0.804284 0.594244i \(-0.797451\pi\)
−0.804284 + 0.594244i \(0.797451\pi\)
\(578\) 19.6926 0.819102
\(579\) −5.17793 −0.215187
\(580\) −9.78584 −0.406335
\(581\) 10.2535 0.425387
\(582\) −27.9956 −1.16046
\(583\) 5.34004 0.221162
\(584\) 0.688529 0.0284915
\(585\) 0.385920 0.0159558
\(586\) −10.1029 −0.417345
\(587\) −13.5179 −0.557943 −0.278972 0.960299i \(-0.589994\pi\)
−0.278972 + 0.960299i \(0.589994\pi\)
\(588\) −11.1012 −0.457805
\(589\) 4.60092 0.189578
\(590\) −0.581520 −0.0239408
\(591\) 10.5604 0.434396
\(592\) −10.9171 −0.448690
\(593\) −6.72274 −0.276070 −0.138035 0.990427i \(-0.544079\pi\)
−0.138035 + 0.990427i \(0.544079\pi\)
\(594\) −6.42084 −0.263450
\(595\) −3.85692 −0.158118
\(596\) 24.2837 0.994698
\(597\) −8.45303 −0.345960
\(598\) −2.47058 −0.101029
\(599\) 4.15139 0.169621 0.0848106 0.996397i \(-0.472971\pi\)
0.0848106 + 0.996397i \(0.472971\pi\)
\(600\) 2.61743 0.106856
\(601\) −39.8754 −1.62655 −0.813276 0.581878i \(-0.802318\pi\)
−0.813276 + 0.581878i \(0.802318\pi\)
\(602\) −3.56612 −0.145344
\(603\) −2.44152 −0.0994264
\(604\) −5.09860 −0.207459
\(605\) 7.43665 0.302343
\(606\) 1.99905 0.0812059
\(607\) −27.9713 −1.13532 −0.567660 0.823263i \(-0.692151\pi\)
−0.567660 + 0.823263i \(0.692151\pi\)
\(608\) −9.00690 −0.365278
\(609\) −23.4240 −0.949190
\(610\) −12.1990 −0.493921
\(611\) −4.91454 −0.198821
\(612\) −1.09662 −0.0443284
\(613\) −30.7521 −1.24207 −0.621033 0.783785i \(-0.713287\pi\)
−0.621033 + 0.783785i \(0.713287\pi\)
\(614\) 3.81252 0.153861
\(615\) 0.169520 0.00683569
\(616\) −0.413863 −0.0166750
\(617\) −24.4669 −0.984999 −0.492500 0.870313i \(-0.663917\pi\)
−0.492500 + 0.870313i \(0.663917\pi\)
\(618\) −4.83387 −0.194447
\(619\) 37.1492 1.49315 0.746575 0.665301i \(-0.231697\pi\)
0.746575 + 0.665301i \(0.231697\pi\)
\(620\) 6.26915 0.251775
\(621\) 2.07081 0.0830988
\(622\) 8.76284 0.351358
\(623\) −33.4324 −1.33944
\(624\) 19.1714 0.767471
\(625\) 17.8682 0.714729
\(626\) 49.4322 1.97571
\(627\) 1.24375 0.0496704
\(628\) −37.8426 −1.51009
\(629\) −8.09308 −0.322692
\(630\) −0.543907 −0.0216698
\(631\) −15.7642 −0.627563 −0.313781 0.949495i \(-0.601596\pi\)
−0.313781 + 0.949495i \(0.601596\pi\)
\(632\) 0.103204 0.00410522
\(633\) −2.68997 −0.106917
\(634\) 57.4932 2.28335
\(635\) −15.2850 −0.606565
\(636\) −32.8365 −1.30205
\(637\) 8.45819 0.335126
\(638\) 8.26479 0.327206
\(639\) 0.521116 0.0206150
\(640\) −1.81814 −0.0718684
\(641\) 48.3528 1.90982 0.954911 0.296892i \(-0.0959503\pi\)
0.954911 + 0.296892i \(0.0959503\pi\)
\(642\) −9.37689 −0.370076
\(643\) 3.30378 0.130288 0.0651441 0.997876i \(-0.479249\pi\)
0.0651441 + 0.997876i \(0.479249\pi\)
\(644\) 1.80773 0.0712344
\(645\) 1.07822 0.0424549
\(646\) −6.14129 −0.241626
\(647\) 34.3017 1.34854 0.674269 0.738486i \(-0.264459\pi\)
0.674269 + 0.738486i \(0.264459\pi\)
\(648\) 3.09806 0.121703
\(649\) 0.254980 0.0100088
\(650\) −27.0092 −1.05939
\(651\) 15.0063 0.588142
\(652\) −15.3786 −0.602271
\(653\) −13.5211 −0.529120 −0.264560 0.964369i \(-0.585227\pi\)
−0.264560 + 0.964369i \(0.585227\pi\)
\(654\) −32.3700 −1.26577
\(655\) −6.15680 −0.240566
\(656\) 0.495088 0.0193299
\(657\) −0.396763 −0.0154792
\(658\) 6.92644 0.270021
\(659\) 29.2605 1.13983 0.569914 0.821704i \(-0.306977\pi\)
0.569914 + 0.821704i \(0.306977\pi\)
\(660\) 1.69471 0.0659665
\(661\) 36.0006 1.40026 0.700131 0.714015i \(-0.253125\pi\)
0.700131 + 0.714015i \(0.253125\pi\)
\(662\) −27.5707 −1.07157
\(663\) 14.2122 0.551955
\(664\) 1.64258 0.0637445
\(665\) −1.58137 −0.0613229
\(666\) −1.14129 −0.0442242
\(667\) −2.66551 −0.103209
\(668\) 9.84880 0.381061
\(669\) −17.4823 −0.675903
\(670\) −18.6306 −0.719762
\(671\) 5.34889 0.206491
\(672\) −29.3767 −1.13323
\(673\) 14.3874 0.554595 0.277298 0.960784i \(-0.410561\pi\)
0.277298 + 0.960784i \(0.410561\pi\)
\(674\) 3.53235 0.136061
\(675\) 22.6388 0.871368
\(676\) −9.44000 −0.363077
\(677\) 25.3712 0.975094 0.487547 0.873097i \(-0.337892\pi\)
0.487547 + 0.873097i \(0.337892\pi\)
\(678\) 6.99049 0.268468
\(679\) −15.6075 −0.598960
\(680\) −0.617867 −0.0236941
\(681\) 12.2922 0.471037
\(682\) −5.29471 −0.202745
\(683\) 31.2549 1.19594 0.597968 0.801520i \(-0.295975\pi\)
0.597968 + 0.801520i \(0.295975\pi\)
\(684\) −0.449625 −0.0171918
\(685\) 2.98764 0.114152
\(686\) −40.9005 −1.56159
\(687\) 21.7595 0.830178
\(688\) 3.14898 0.120054
\(689\) 25.0187 0.953138
\(690\) −1.05278 −0.0400786
\(691\) −1.92143 −0.0730946 −0.0365473 0.999332i \(-0.511636\pi\)
−0.0365473 + 0.999332i \(0.511636\pi\)
\(692\) 6.29169 0.239174
\(693\) 0.238487 0.00905939
\(694\) 56.5468 2.14649
\(695\) −9.46162 −0.358900
\(696\) −3.75246 −0.142237
\(697\) 0.367019 0.0139018
\(698\) 32.7564 1.23985
\(699\) −37.5894 −1.42176
\(700\) 19.7627 0.746960
\(701\) 5.22129 0.197205 0.0986027 0.995127i \(-0.468563\pi\)
0.0986027 + 0.995127i \(0.468563\pi\)
\(702\) −30.0824 −1.13539
\(703\) −3.31823 −0.125149
\(704\) 5.78109 0.217883
\(705\) −2.09421 −0.0788726
\(706\) 32.6338 1.22819
\(707\) 1.11446 0.0419137
\(708\) −1.56790 −0.0589253
\(709\) 1.68414 0.0632493 0.0316247 0.999500i \(-0.489932\pi\)
0.0316247 + 0.999500i \(0.489932\pi\)
\(710\) 3.97649 0.149235
\(711\) −0.0594709 −0.00223033
\(712\) −5.35577 −0.200716
\(713\) 1.70762 0.0639508
\(714\) −20.0303 −0.749615
\(715\) −1.29123 −0.0482893
\(716\) −19.4596 −0.727241
\(717\) 16.2812 0.608032
\(718\) −59.6949 −2.22779
\(719\) −38.7868 −1.44651 −0.723253 0.690583i \(-0.757354\pi\)
−0.723253 + 0.690583i \(0.757354\pi\)
\(720\) 0.480284 0.0178991
\(721\) −2.69487 −0.100362
\(722\) 36.2320 1.34842
\(723\) 40.5898 1.50955
\(724\) 31.1100 1.15619
\(725\) −29.1402 −1.08224
\(726\) 38.6210 1.43336
\(727\) −18.0472 −0.669333 −0.334666 0.942337i \(-0.608624\pi\)
−0.334666 + 0.942337i \(0.608624\pi\)
\(728\) −1.93900 −0.0718640
\(729\) 25.1094 0.929978
\(730\) −3.02759 −0.112056
\(731\) 2.33440 0.0863410
\(732\) −32.8909 −1.21568
\(733\) 2.36945 0.0875175 0.0437587 0.999042i \(-0.486067\pi\)
0.0437587 + 0.999042i \(0.486067\pi\)
\(734\) −39.8373 −1.47042
\(735\) 3.60426 0.132945
\(736\) −3.34289 −0.123220
\(737\) 8.16896 0.300907
\(738\) 0.0517574 0.00190521
\(739\) 23.0868 0.849260 0.424630 0.905367i \(-0.360404\pi\)
0.424630 + 0.905367i \(0.360404\pi\)
\(740\) −4.52137 −0.166209
\(741\) 5.82710 0.214064
\(742\) −35.2608 −1.29447
\(743\) 2.24456 0.0823449 0.0411725 0.999152i \(-0.486891\pi\)
0.0411725 + 0.999152i \(0.486891\pi\)
\(744\) 2.40396 0.0881333
\(745\) −7.88426 −0.288857
\(746\) 44.9891 1.64717
\(747\) −0.946533 −0.0346318
\(748\) 3.66914 0.134157
\(749\) −5.22759 −0.191012
\(750\) −24.2735 −0.886344
\(751\) 54.7798 1.99894 0.999472 0.0324826i \(-0.0103413\pi\)
0.999472 + 0.0324826i \(0.0103413\pi\)
\(752\) −6.11622 −0.223036
\(753\) −20.8568 −0.760065
\(754\) 38.7215 1.41015
\(755\) 1.65538 0.0602454
\(756\) 22.0114 0.800546
\(757\) −5.96003 −0.216621 −0.108311 0.994117i \(-0.534544\pi\)
−0.108311 + 0.994117i \(0.534544\pi\)
\(758\) −21.9099 −0.795804
\(759\) 0.461613 0.0167555
\(760\) −0.253330 −0.00918926
\(761\) −33.5210 −1.21513 −0.607567 0.794268i \(-0.707854\pi\)
−0.607567 + 0.794268i \(0.707854\pi\)
\(762\) −79.3800 −2.87563
\(763\) −18.0461 −0.653314
\(764\) 9.51732 0.344325
\(765\) 0.356045 0.0128728
\(766\) −15.9461 −0.576158
\(767\) 1.19461 0.0431349
\(768\) 23.4816 0.847318
\(769\) −41.9889 −1.51416 −0.757080 0.653322i \(-0.773375\pi\)
−0.757080 + 0.653322i \(0.773375\pi\)
\(770\) 1.81983 0.0655822
\(771\) −29.1026 −1.04810
\(772\) −6.26298 −0.225409
\(773\) 11.9695 0.430512 0.215256 0.976558i \(-0.430941\pi\)
0.215256 + 0.976558i \(0.430941\pi\)
\(774\) 0.329200 0.0118328
\(775\) 18.6683 0.670584
\(776\) −2.50027 −0.0897544
\(777\) −10.8227 −0.388261
\(778\) 71.4451 2.56143
\(779\) 0.150481 0.00539153
\(780\) 7.93993 0.284295
\(781\) −1.74358 −0.0623901
\(782\) −2.27932 −0.0815084
\(783\) −32.4559 −1.15988
\(784\) 10.5264 0.375942
\(785\) 12.2865 0.438523
\(786\) −31.9743 −1.14049
\(787\) −14.5883 −0.520017 −0.260008 0.965606i \(-0.583725\pi\)
−0.260008 + 0.965606i \(0.583725\pi\)
\(788\) 12.7733 0.455031
\(789\) −28.2001 −1.00395
\(790\) −0.453806 −0.0161457
\(791\) 3.89717 0.138568
\(792\) 0.0382049 0.00135755
\(793\) 25.0602 0.889913
\(794\) 0.500564 0.0177643
\(795\) 10.6611 0.378112
\(796\) −10.2244 −0.362394
\(797\) 13.0225 0.461281 0.230641 0.973039i \(-0.425918\pi\)
0.230641 + 0.973039i \(0.425918\pi\)
\(798\) −8.21258 −0.290722
\(799\) −4.53408 −0.160404
\(800\) −36.5456 −1.29208
\(801\) 3.08625 0.109047
\(802\) 33.3319 1.17699
\(803\) 1.32751 0.0468468
\(804\) −50.2319 −1.77154
\(805\) −0.586921 −0.0206862
\(806\) −24.8064 −0.873767
\(807\) 20.0673 0.706403
\(808\) 0.178534 0.00628079
\(809\) 23.5857 0.829230 0.414615 0.909997i \(-0.363916\pi\)
0.414615 + 0.909997i \(0.363916\pi\)
\(810\) −13.6228 −0.478655
\(811\) 44.3230 1.55639 0.778195 0.628023i \(-0.216136\pi\)
0.778195 + 0.628023i \(0.216136\pi\)
\(812\) −28.3326 −0.994280
\(813\) 40.5472 1.42205
\(814\) 3.81860 0.133842
\(815\) 4.99300 0.174897
\(816\) 17.6873 0.619179
\(817\) 0.957124 0.0334855
\(818\) −9.39977 −0.328655
\(819\) 1.11734 0.0390431
\(820\) 0.205043 0.00716041
\(821\) −11.0481 −0.385581 −0.192791 0.981240i \(-0.561754\pi\)
−0.192791 + 0.981240i \(0.561754\pi\)
\(822\) 15.5158 0.541176
\(823\) 8.97374 0.312805 0.156403 0.987693i \(-0.450010\pi\)
0.156403 + 0.987693i \(0.450010\pi\)
\(824\) −0.431710 −0.0150393
\(825\) 5.04650 0.175697
\(826\) −1.68366 −0.0585819
\(827\) 36.0996 1.25531 0.627654 0.778493i \(-0.284015\pi\)
0.627654 + 0.778493i \(0.284015\pi\)
\(828\) −0.166877 −0.00579937
\(829\) −5.63971 −0.195875 −0.0979375 0.995193i \(-0.531225\pi\)
−0.0979375 + 0.995193i \(0.531225\pi\)
\(830\) −7.22273 −0.250705
\(831\) −6.11669 −0.212186
\(832\) 27.0851 0.939007
\(833\) 7.80341 0.270372
\(834\) −49.1374 −1.70149
\(835\) −3.19764 −0.110659
\(836\) 1.50438 0.0520299
\(837\) 20.7924 0.718691
\(838\) −23.3538 −0.806745
\(839\) −19.2580 −0.664859 −0.332429 0.943128i \(-0.607868\pi\)
−0.332429 + 0.943128i \(0.607868\pi\)
\(840\) −0.826257 −0.0285086
\(841\) 12.7767 0.440575
\(842\) 45.4039 1.56472
\(843\) −11.4469 −0.394253
\(844\) −3.25366 −0.111996
\(845\) 3.06492 0.105436
\(846\) −0.639401 −0.0219831
\(847\) 21.5311 0.739818
\(848\) 31.1362 1.06922
\(849\) −39.8953 −1.36920
\(850\) −24.9183 −0.854691
\(851\) −1.23155 −0.0422170
\(852\) 10.7215 0.367311
\(853\) 29.1126 0.996796 0.498398 0.866948i \(-0.333922\pi\)
0.498398 + 0.866948i \(0.333922\pi\)
\(854\) −35.3193 −1.20860
\(855\) 0.145981 0.00499245
\(856\) −0.837443 −0.0286232
\(857\) −18.6286 −0.636342 −0.318171 0.948033i \(-0.603069\pi\)
−0.318171 + 0.948033i \(0.603069\pi\)
\(858\) −6.70579 −0.228932
\(859\) 18.5140 0.631690 0.315845 0.948811i \(-0.397712\pi\)
0.315845 + 0.948811i \(0.397712\pi\)
\(860\) 1.30416 0.0444716
\(861\) 0.490804 0.0167266
\(862\) 2.31070 0.0787027
\(863\) −43.5150 −1.48127 −0.740635 0.671908i \(-0.765475\pi\)
−0.740635 + 0.671908i \(0.765475\pi\)
\(864\) −40.7039 −1.38477
\(865\) −2.04274 −0.0694553
\(866\) 5.41872 0.184135
\(867\) −17.2386 −0.585453
\(868\) 18.1509 0.616081
\(869\) 0.198981 0.00674996
\(870\) 16.5003 0.559411
\(871\) 38.2726 1.29682
\(872\) −2.89094 −0.0978995
\(873\) 1.44077 0.0487628
\(874\) −0.934540 −0.0316113
\(875\) −13.5324 −0.457479
\(876\) −8.16302 −0.275803
\(877\) 11.9837 0.404662 0.202331 0.979317i \(-0.435148\pi\)
0.202331 + 0.979317i \(0.435148\pi\)
\(878\) −13.4040 −0.452363
\(879\) 8.84390 0.298297
\(880\) −1.60696 −0.0541705
\(881\) −8.33770 −0.280904 −0.140452 0.990087i \(-0.544856\pi\)
−0.140452 + 0.990087i \(0.544856\pi\)
\(882\) 1.10044 0.0370539
\(883\) 14.7773 0.497296 0.248648 0.968594i \(-0.420014\pi\)
0.248648 + 0.968594i \(0.420014\pi\)
\(884\) 17.1904 0.578174
\(885\) 0.509055 0.0171117
\(886\) −2.74118 −0.0920916
\(887\) −26.6636 −0.895278 −0.447639 0.894214i \(-0.647735\pi\)
−0.447639 + 0.894214i \(0.647735\pi\)
\(888\) −1.73376 −0.0581811
\(889\) −44.2541 −1.48423
\(890\) 23.5503 0.789408
\(891\) 5.97318 0.200109
\(892\) −21.1457 −0.708011
\(893\) −1.85901 −0.0622094
\(894\) −40.9456 −1.36943
\(895\) 6.31803 0.211188
\(896\) −5.26401 −0.175858
\(897\) 2.16271 0.0722108
\(898\) −53.7218 −1.79272
\(899\) −26.7636 −0.892616
\(900\) −1.82435 −0.0608118
\(901\) 23.0819 0.768971
\(902\) −0.173172 −0.00576601
\(903\) 3.12174 0.103885
\(904\) 0.624315 0.0207644
\(905\) −10.1006 −0.335754
\(906\) 8.59694 0.285614
\(907\) −41.3358 −1.37253 −0.686266 0.727351i \(-0.740751\pi\)
−0.686266 + 0.727351i \(0.740751\pi\)
\(908\) 14.8680 0.493413
\(909\) −0.102880 −0.00341230
\(910\) 8.52613 0.282638
\(911\) −32.2613 −1.06887 −0.534433 0.845211i \(-0.679475\pi\)
−0.534433 + 0.845211i \(0.679475\pi\)
\(912\) 7.25192 0.240135
\(913\) 3.16696 0.104811
\(914\) 61.3254 2.02846
\(915\) 10.6788 0.353030
\(916\) 26.3193 0.869615
\(917\) −17.8256 −0.588653
\(918\) −27.7536 −0.916007
\(919\) −48.5538 −1.60164 −0.800821 0.598903i \(-0.795603\pi\)
−0.800821 + 0.598903i \(0.795603\pi\)
\(920\) −0.0940229 −0.00309984
\(921\) −3.33743 −0.109972
\(922\) 15.7545 0.518845
\(923\) −8.16886 −0.268881
\(924\) 4.90664 0.161417
\(925\) −13.4637 −0.442684
\(926\) −15.9683 −0.524750
\(927\) 0.248772 0.00817073
\(928\) 52.3933 1.71989
\(929\) 28.1849 0.924717 0.462358 0.886693i \(-0.347003\pi\)
0.462358 + 0.886693i \(0.347003\pi\)
\(930\) −10.5706 −0.346625
\(931\) 3.19946 0.104858
\(932\) −45.4664 −1.48930
\(933\) −7.67087 −0.251133
\(934\) 14.6765 0.480230
\(935\) −1.19127 −0.0389587
\(936\) 0.178995 0.00585063
\(937\) −59.4487 −1.94211 −0.971053 0.238865i \(-0.923225\pi\)
−0.971053 + 0.238865i \(0.923225\pi\)
\(938\) −53.9405 −1.76122
\(939\) −43.2723 −1.41214
\(940\) −2.53306 −0.0826194
\(941\) 24.0890 0.785277 0.392639 0.919693i \(-0.371562\pi\)
0.392639 + 0.919693i \(0.371562\pi\)
\(942\) 63.8078 2.07897
\(943\) 0.0558504 0.00181874
\(944\) 1.48671 0.0483884
\(945\) −7.14650 −0.232476
\(946\) −1.10145 −0.0358113
\(947\) 40.0415 1.30117 0.650586 0.759432i \(-0.274523\pi\)
0.650586 + 0.759432i \(0.274523\pi\)
\(948\) −1.22356 −0.0397392
\(949\) 6.21955 0.201895
\(950\) −10.2167 −0.331474
\(951\) −50.3288 −1.63202
\(952\) −1.78889 −0.0579783
\(953\) −24.4039 −0.790521 −0.395261 0.918569i \(-0.629346\pi\)
−0.395261 + 0.918569i \(0.629346\pi\)
\(954\) 3.25504 0.105386
\(955\) −3.09002 −0.0999907
\(956\) 19.6929 0.636915
\(957\) −7.23488 −0.233870
\(958\) 24.8725 0.803593
\(959\) 8.65001 0.279323
\(960\) 11.5417 0.372506
\(961\) −13.8543 −0.446913
\(962\) 17.8906 0.576816
\(963\) 0.482575 0.0155508
\(964\) 49.0955 1.58126
\(965\) 2.03342 0.0654581
\(966\) −3.04808 −0.0980702
\(967\) −16.2701 −0.523212 −0.261606 0.965175i \(-0.584252\pi\)
−0.261606 + 0.965175i \(0.584252\pi\)
\(968\) 3.44922 0.110862
\(969\) 5.37600 0.172702
\(970\) 10.9941 0.353001
\(971\) −53.0256 −1.70167 −0.850837 0.525430i \(-0.823904\pi\)
−0.850837 + 0.525430i \(0.823904\pi\)
\(972\) −4.19925 −0.134691
\(973\) −27.3939 −0.878209
\(974\) −73.3565 −2.35050
\(975\) 23.6435 0.757197
\(976\) 31.1878 0.998298
\(977\) −47.0435 −1.50505 −0.752527 0.658561i \(-0.771165\pi\)
−0.752527 + 0.658561i \(0.771165\pi\)
\(978\) 25.9303 0.829161
\(979\) −10.3261 −0.330024
\(980\) 4.35954 0.139260
\(981\) 1.66590 0.0531879
\(982\) 35.4089 1.12994
\(983\) −10.9007 −0.347678 −0.173839 0.984774i \(-0.555617\pi\)
−0.173839 + 0.984774i \(0.555617\pi\)
\(984\) 0.0786254 0.00250648
\(985\) −4.14716 −0.132139
\(986\) 35.7239 1.13768
\(987\) −6.06331 −0.192997
\(988\) 7.04818 0.224233
\(989\) 0.355234 0.0112958
\(990\) −0.167994 −0.00533921
\(991\) −48.5839 −1.54332 −0.771659 0.636036i \(-0.780573\pi\)
−0.771659 + 0.636036i \(0.780573\pi\)
\(992\) −33.5650 −1.06569
\(993\) 24.1350 0.765901
\(994\) 11.5130 0.365171
\(995\) 3.31959 0.105238
\(996\) −19.4740 −0.617057
\(997\) 20.3630 0.644902 0.322451 0.946586i \(-0.395493\pi\)
0.322451 + 0.946586i \(0.395493\pi\)
\(998\) 61.3507 1.94202
\(999\) −14.9957 −0.474442
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.1.2 10
3.2 odd 2 4527.2.a.k.1.9 10
4.3 odd 2 8048.2.a.p.1.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.e.1.2 10 1.1 even 1 trivial
4527.2.a.k.1.9 10 3.2 odd 2
8048.2.a.p.1.1 10 4.3 odd 2