Properties

Label 1150.2.a.q.1.3
Level $1150$
Weight $2$
Character 1150.1
Self dual yes
Analytic conductor $9.183$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
Defining polynomial: \(x^{3} - x^{2} - 9 x + 12\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.11903\) of defining polynomial
Character \(\chi\) \(=\) 1150.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.11903 q^{3} +1.00000 q^{4} -3.11903 q^{6} -4.50973 q^{7} -1.00000 q^{8} +6.72833 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.11903 q^{3} +1.00000 q^{4} -3.11903 q^{6} -4.50973 q^{7} -1.00000 q^{8} +6.72833 q^{9} +4.33763 q^{11} +3.11903 q^{12} +3.72833 q^{13} +4.50973 q^{14} +1.00000 q^{16} -1.11903 q^{17} -6.72833 q^{18} +4.50973 q^{19} -14.0660 q^{21} -4.33763 q^{22} +1.00000 q^{23} -3.11903 q^{24} -3.72833 q^{26} +11.6288 q^{27} -4.50973 q^{28} -8.23805 q^{29} +1.72833 q^{31} -1.00000 q^{32} +13.5292 q^{33} +1.11903 q^{34} +6.72833 q^{36} +0.781399 q^{37} -4.50973 q^{38} +11.6288 q^{39} +3.90043 q^{41} +14.0660 q^{42} -8.00000 q^{43} +4.33763 q^{44} -1.00000 q^{46} +11.4567 q^{47} +3.11903 q^{48} +13.3376 q^{49} -3.49027 q^{51} +3.72833 q^{52} +6.00000 q^{53} -11.6288 q^{54} +4.50973 q^{56} +14.0660 q^{57} +8.23805 q^{58} -2.23805 q^{59} +3.55623 q^{61} -1.72833 q^{62} -30.3429 q^{63} +1.00000 q^{64} -13.5292 q^{66} -2.43720 q^{67} -1.11903 q^{68} +3.11903 q^{69} +7.11903 q^{71} -6.72833 q^{72} +9.45665 q^{73} -0.781399 q^{74} +4.50973 q^{76} -19.5615 q^{77} -11.6288 q^{78} -14.9133 q^{79} +16.0854 q^{81} -3.90043 q^{82} -2.78140 q^{83} -14.0660 q^{84} +8.00000 q^{86} -25.6947 q^{87} -4.33763 q^{88} -7.69471 q^{89} -16.8137 q^{91} +1.00000 q^{92} +5.39070 q^{93} -11.4567 q^{94} -3.11903 q^{96} +0.642920 q^{97} -13.3376 q^{98} +29.1850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} - q^{3} + 3q^{4} + q^{6} - 3q^{7} - 3q^{8} + 10q^{9} + O(q^{10}) \) \( 3q - 3q^{2} - q^{3} + 3q^{4} + q^{6} - 3q^{7} - 3q^{8} + 10q^{9} + 3q^{11} - q^{12} + q^{13} + 3q^{14} + 3q^{16} + 7q^{17} - 10q^{18} + 3q^{19} - 22q^{21} - 3q^{22} + 3q^{23} + q^{24} - q^{26} + 14q^{27} - 3q^{28} - 4q^{29} - 5q^{31} - 3q^{32} + 9q^{33} - 7q^{34} + 10q^{36} + 2q^{37} - 3q^{38} + 14q^{39} + q^{41} + 22q^{42} - 24q^{43} + 3q^{44} - 3q^{46} + 14q^{47} - q^{48} + 30q^{49} - 21q^{51} + q^{52} + 18q^{53} - 14q^{54} + 3q^{56} + 22q^{57} + 4q^{58} + 14q^{59} + q^{61} + 5q^{62} - 8q^{63} + 3q^{64} - 9q^{66} - 8q^{67} + 7q^{68} - q^{69} + 11q^{71} - 10q^{72} + 8q^{73} - 2q^{74} + 3q^{76} + 24q^{77} - 14q^{78} - 4q^{79} + 7q^{81} - q^{82} - 8q^{83} - 22q^{84} + 24q^{86} - 36q^{87} - 3q^{88} + 18q^{89} + q^{91} + 3q^{92} + 16q^{93} - 14q^{94} + q^{96} + 33q^{97} - 30q^{98} + 57q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 3.11903 1.80077 0.900385 0.435093i \(-0.143285\pi\)
0.900385 + 0.435093i \(0.143285\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −3.11903 −1.27334
\(7\) −4.50973 −1.70452 −0.852258 0.523122i \(-0.824767\pi\)
−0.852258 + 0.523122i \(0.824767\pi\)
\(8\) −1.00000 −0.353553
\(9\) 6.72833 2.24278
\(10\) 0 0
\(11\) 4.33763 1.30784 0.653922 0.756562i \(-0.273122\pi\)
0.653922 + 0.756562i \(0.273122\pi\)
\(12\) 3.11903 0.900385
\(13\) 3.72833 1.03405 0.517026 0.855970i \(-0.327039\pi\)
0.517026 + 0.855970i \(0.327039\pi\)
\(14\) 4.50973 1.20527
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.11903 −0.271404 −0.135702 0.990750i \(-0.543329\pi\)
−0.135702 + 0.990750i \(0.543329\pi\)
\(18\) −6.72833 −1.58588
\(19\) 4.50973 1.03460 0.517301 0.855803i \(-0.326937\pi\)
0.517301 + 0.855803i \(0.326937\pi\)
\(20\) 0 0
\(21\) −14.0660 −3.06944
\(22\) −4.33763 −0.924785
\(23\) 1.00000 0.208514
\(24\) −3.11903 −0.636669
\(25\) 0 0
\(26\) −3.72833 −0.731185
\(27\) 11.6288 2.23795
\(28\) −4.50973 −0.852258
\(29\) −8.23805 −1.52977 −0.764884 0.644168i \(-0.777204\pi\)
−0.764884 + 0.644168i \(0.777204\pi\)
\(30\) 0 0
\(31\) 1.72833 0.310417 0.155208 0.987882i \(-0.450395\pi\)
0.155208 + 0.987882i \(0.450395\pi\)
\(32\) −1.00000 −0.176777
\(33\) 13.5292 2.35513
\(34\) 1.11903 0.191911
\(35\) 0 0
\(36\) 6.72833 1.12139
\(37\) 0.781399 0.128461 0.0642306 0.997935i \(-0.479541\pi\)
0.0642306 + 0.997935i \(0.479541\pi\)
\(38\) −4.50973 −0.731574
\(39\) 11.6288 1.86209
\(40\) 0 0
\(41\) 3.90043 0.609144 0.304572 0.952489i \(-0.401487\pi\)
0.304572 + 0.952489i \(0.401487\pi\)
\(42\) 14.0660 2.17042
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 4.33763 0.653922
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 11.4567 1.67112 0.835562 0.549396i \(-0.185142\pi\)
0.835562 + 0.549396i \(0.185142\pi\)
\(48\) 3.11903 0.450193
\(49\) 13.3376 1.90538
\(50\) 0 0
\(51\) −3.49027 −0.488736
\(52\) 3.72833 0.517026
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −11.6288 −1.58247
\(55\) 0 0
\(56\) 4.50973 0.602637
\(57\) 14.0660 1.86308
\(58\) 8.23805 1.08171
\(59\) −2.23805 −0.291370 −0.145685 0.989331i \(-0.546539\pi\)
−0.145685 + 0.989331i \(0.546539\pi\)
\(60\) 0 0
\(61\) 3.55623 0.455329 0.227664 0.973740i \(-0.426891\pi\)
0.227664 + 0.973740i \(0.426891\pi\)
\(62\) −1.72833 −0.219498
\(63\) −30.3429 −3.82285
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −13.5292 −1.66533
\(67\) −2.43720 −0.297752 −0.148876 0.988856i \(-0.547565\pi\)
−0.148876 + 0.988856i \(0.547565\pi\)
\(68\) −1.11903 −0.135702
\(69\) 3.11903 0.375487
\(70\) 0 0
\(71\) 7.11903 0.844873 0.422437 0.906393i \(-0.361175\pi\)
0.422437 + 0.906393i \(0.361175\pi\)
\(72\) −6.72833 −0.792941
\(73\) 9.45665 1.10682 0.553409 0.832910i \(-0.313327\pi\)
0.553409 + 0.832910i \(0.313327\pi\)
\(74\) −0.781399 −0.0908357
\(75\) 0 0
\(76\) 4.50973 0.517301
\(77\) −19.5615 −2.22924
\(78\) −11.6288 −1.31670
\(79\) −14.9133 −1.67788 −0.838939 0.544225i \(-0.816824\pi\)
−0.838939 + 0.544225i \(0.816824\pi\)
\(80\) 0 0
\(81\) 16.0854 1.78727
\(82\) −3.90043 −0.430730
\(83\) −2.78140 −0.305298 −0.152649 0.988280i \(-0.548780\pi\)
−0.152649 + 0.988280i \(0.548780\pi\)
\(84\) −14.0660 −1.53472
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) −25.6947 −2.75476
\(88\) −4.33763 −0.462393
\(89\) −7.69471 −0.815637 −0.407819 0.913063i \(-0.633710\pi\)
−0.407819 + 0.913063i \(0.633710\pi\)
\(90\) 0 0
\(91\) −16.8137 −1.76256
\(92\) 1.00000 0.104257
\(93\) 5.39070 0.558989
\(94\) −11.4567 −1.18166
\(95\) 0 0
\(96\) −3.11903 −0.318334
\(97\) 0.642920 0.0652786 0.0326393 0.999467i \(-0.489609\pi\)
0.0326393 + 0.999467i \(0.489609\pi\)
\(98\) −13.3376 −1.34730
\(99\) 29.1850 2.93320
\(100\) 0 0
\(101\) −8.23805 −0.819717 −0.409858 0.912149i \(-0.634422\pi\)
−0.409858 + 0.912149i \(0.634422\pi\)
\(102\) 3.49027 0.345589
\(103\) −12.3376 −1.21566 −0.607831 0.794066i \(-0.707960\pi\)
−0.607831 + 0.794066i \(0.707960\pi\)
\(104\) −3.72833 −0.365593
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 15.9328 1.54028 0.770139 0.637876i \(-0.220187\pi\)
0.770139 + 0.637876i \(0.220187\pi\)
\(108\) 11.6288 1.11898
\(109\) −1.49027 −0.142742 −0.0713712 0.997450i \(-0.522737\pi\)
−0.0713712 + 0.997450i \(0.522737\pi\)
\(110\) 0 0
\(111\) 2.43720 0.231329
\(112\) −4.50973 −0.426129
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −14.0660 −1.31740
\(115\) 0 0
\(116\) −8.23805 −0.764884
\(117\) 25.0854 2.31915
\(118\) 2.23805 0.206030
\(119\) 5.04650 0.462612
\(120\) 0 0
\(121\) 7.81502 0.710456
\(122\) −3.55623 −0.321966
\(123\) 12.1655 1.09693
\(124\) 1.72833 0.155208
\(125\) 0 0
\(126\) 30.3429 2.70316
\(127\) 0.675256 0.0599193 0.0299597 0.999551i \(-0.490462\pi\)
0.0299597 + 0.999551i \(0.490462\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −24.9522 −2.19692
\(130\) 0 0
\(131\) −13.6947 −1.19651 −0.598256 0.801305i \(-0.704139\pi\)
−0.598256 + 0.801305i \(0.704139\pi\)
\(132\) 13.5292 1.17756
\(133\) −20.3376 −1.76350
\(134\) 2.43720 0.210542
\(135\) 0 0
\(136\) 1.11903 0.0959557
\(137\) −7.52918 −0.643261 −0.321631 0.946865i \(-0.604231\pi\)
−0.321631 + 0.946865i \(0.604231\pi\)
\(138\) −3.11903 −0.265509
\(139\) 4.67526 0.396550 0.198275 0.980146i \(-0.436466\pi\)
0.198275 + 0.980146i \(0.436466\pi\)
\(140\) 0 0
\(141\) 35.7336 3.00931
\(142\) −7.11903 −0.597415
\(143\) 16.1721 1.35238
\(144\) 6.72833 0.560694
\(145\) 0 0
\(146\) −9.45665 −0.782638
\(147\) 41.6004 3.43114
\(148\) 0.781399 0.0642306
\(149\) 7.52918 0.616814 0.308407 0.951254i \(-0.400204\pi\)
0.308407 + 0.951254i \(0.400204\pi\)
\(150\) 0 0
\(151\) −13.3571 −1.08698 −0.543492 0.839414i \(-0.682898\pi\)
−0.543492 + 0.839414i \(0.682898\pi\)
\(152\) −4.50973 −0.365787
\(153\) −7.52918 −0.608698
\(154\) 19.5615 1.57631
\(155\) 0 0
\(156\) 11.6288 0.931045
\(157\) −16.2381 −1.29594 −0.647969 0.761667i \(-0.724381\pi\)
−0.647969 + 0.761667i \(0.724381\pi\)
\(158\) 14.9133 1.18644
\(159\) 18.7142 1.48413
\(160\) 0 0
\(161\) −4.50973 −0.355416
\(162\) −16.0854 −1.26379
\(163\) 3.29112 0.257781 0.128890 0.991659i \(-0.458858\pi\)
0.128890 + 0.991659i \(0.458858\pi\)
\(164\) 3.90043 0.304572
\(165\) 0 0
\(166\) 2.78140 0.215878
\(167\) −22.9133 −1.77309 −0.886543 0.462647i \(-0.846900\pi\)
−0.886543 + 0.462647i \(0.846900\pi\)
\(168\) 14.0660 1.08521
\(169\) 0.900425 0.0692635
\(170\) 0 0
\(171\) 30.3429 2.32038
\(172\) −8.00000 −0.609994
\(173\) −0.575681 −0.0437683 −0.0218841 0.999761i \(-0.506966\pi\)
−0.0218841 + 0.999761i \(0.506966\pi\)
\(174\) 25.6947 1.94791
\(175\) 0 0
\(176\) 4.33763 0.326961
\(177\) −6.98055 −0.524690
\(178\) 7.69471 0.576743
\(179\) 5.01945 0.375171 0.187586 0.982248i \(-0.439934\pi\)
0.187586 + 0.982248i \(0.439934\pi\)
\(180\) 0 0
\(181\) −11.5292 −0.856957 −0.428479 0.903552i \(-0.640950\pi\)
−0.428479 + 0.903552i \(0.640950\pi\)
\(182\) 16.8137 1.24632
\(183\) 11.0920 0.819942
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −5.39070 −0.395265
\(187\) −4.85392 −0.354954
\(188\) 11.4567 0.835562
\(189\) −52.4425 −3.81463
\(190\) 0 0
\(191\) −18.7142 −1.35411 −0.677055 0.735933i \(-0.736744\pi\)
−0.677055 + 0.735933i \(0.736744\pi\)
\(192\) 3.11903 0.225096
\(193\) −23.4956 −1.69125 −0.845624 0.533780i \(-0.820771\pi\)
−0.845624 + 0.533780i \(0.820771\pi\)
\(194\) −0.642920 −0.0461590
\(195\) 0 0
\(196\) 13.3376 0.952688
\(197\) 18.1385 1.29231 0.646157 0.763205i \(-0.276375\pi\)
0.646157 + 0.763205i \(0.276375\pi\)
\(198\) −29.1850 −2.07409
\(199\) −23.2575 −1.64868 −0.824340 0.566094i \(-0.808454\pi\)
−0.824340 + 0.566094i \(0.808454\pi\)
\(200\) 0 0
\(201\) −7.60170 −0.536183
\(202\) 8.23805 0.579627
\(203\) 37.1514 2.60751
\(204\) −3.49027 −0.244368
\(205\) 0 0
\(206\) 12.3376 0.859603
\(207\) 6.72833 0.467651
\(208\) 3.72833 0.258513
\(209\) 19.5615 1.35310
\(210\) 0 0
\(211\) 4.34420 0.299067 0.149533 0.988757i \(-0.452223\pi\)
0.149533 + 0.988757i \(0.452223\pi\)
\(212\) 6.00000 0.412082
\(213\) 22.2044 1.52142
\(214\) −15.9328 −1.08914
\(215\) 0 0
\(216\) −11.6288 −0.791236
\(217\) −7.79428 −0.529110
\(218\) 1.49027 0.100934
\(219\) 29.4956 1.99313
\(220\) 0 0
\(221\) −4.17210 −0.280646
\(222\) −2.43720 −0.163574
\(223\) −12.4761 −0.835462 −0.417731 0.908571i \(-0.637175\pi\)
−0.417731 + 0.908571i \(0.637175\pi\)
\(224\) 4.50973 0.301319
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −15.9328 −1.05749 −0.528747 0.848779i \(-0.677338\pi\)
−0.528747 + 0.848779i \(0.677338\pi\)
\(228\) 14.0660 0.931541
\(229\) −3.56280 −0.235436 −0.117718 0.993047i \(-0.537558\pi\)
−0.117718 + 0.993047i \(0.537558\pi\)
\(230\) 0 0
\(231\) −61.0129 −4.01435
\(232\) 8.23805 0.540855
\(233\) 27.4956 1.80129 0.900647 0.434552i \(-0.143093\pi\)
0.900647 + 0.434552i \(0.143093\pi\)
\(234\) −25.0854 −1.63988
\(235\) 0 0
\(236\) −2.23805 −0.145685
\(237\) −46.5150 −3.02147
\(238\) −5.04650 −0.327116
\(239\) 10.0389 0.649363 0.324681 0.945823i \(-0.394743\pi\)
0.324681 + 0.945823i \(0.394743\pi\)
\(240\) 0 0
\(241\) −23.6947 −1.52631 −0.763155 0.646215i \(-0.776351\pi\)
−0.763155 + 0.646215i \(0.776351\pi\)
\(242\) −7.81502 −0.502368
\(243\) 15.2846 0.980505
\(244\) 3.55623 0.227664
\(245\) 0 0
\(246\) −12.1655 −0.775646
\(247\) 16.8137 1.06983
\(248\) −1.72833 −0.109749
\(249\) −8.67526 −0.549772
\(250\) 0 0
\(251\) 12.4425 0.785363 0.392681 0.919675i \(-0.371548\pi\)
0.392681 + 0.919675i \(0.371548\pi\)
\(252\) −30.3429 −1.91142
\(253\) 4.33763 0.272704
\(254\) −0.675256 −0.0423693
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −5.45665 −0.340377 −0.170188 0.985412i \(-0.554438\pi\)
−0.170188 + 0.985412i \(0.554438\pi\)
\(258\) 24.9522 1.55346
\(259\) −3.52389 −0.218964
\(260\) 0 0
\(261\) −55.4283 −3.43093
\(262\) 13.6947 0.846062
\(263\) −0.138479 −0.00853895 −0.00426948 0.999991i \(-0.501359\pi\)
−0.00426948 + 0.999991i \(0.501359\pi\)
\(264\) −13.5292 −0.832663
\(265\) 0 0
\(266\) 20.3376 1.24698
\(267\) −24.0000 −1.46878
\(268\) −2.43720 −0.148876
\(269\) 14.6753 0.894766 0.447383 0.894342i \(-0.352356\pi\)
0.447383 + 0.894342i \(0.352356\pi\)
\(270\) 0 0
\(271\) −8.31058 −0.504832 −0.252416 0.967619i \(-0.581225\pi\)
−0.252416 + 0.967619i \(0.581225\pi\)
\(272\) −1.11903 −0.0678510
\(273\) −52.4425 −3.17396
\(274\) 7.52918 0.454854
\(275\) 0 0
\(276\) 3.11903 0.187743
\(277\) −12.9133 −0.775886 −0.387943 0.921683i \(-0.626814\pi\)
−0.387943 + 0.921683i \(0.626814\pi\)
\(278\) −4.67526 −0.280403
\(279\) 11.6288 0.696195
\(280\) 0 0
\(281\) 2.67526 0.159592 0.0797962 0.996811i \(-0.474573\pi\)
0.0797962 + 0.996811i \(0.474573\pi\)
\(282\) −35.7336 −2.12791
\(283\) −0.742495 −0.0441367 −0.0220684 0.999756i \(-0.507025\pi\)
−0.0220684 + 0.999756i \(0.507025\pi\)
\(284\) 7.11903 0.422437
\(285\) 0 0
\(286\) −16.1721 −0.956276
\(287\) −17.5898 −1.03830
\(288\) −6.72833 −0.396470
\(289\) −15.7478 −0.926340
\(290\) 0 0
\(291\) 2.00528 0.117552
\(292\) 9.45665 0.553409
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −41.6004 −2.42619
\(295\) 0 0
\(296\) −0.781399 −0.0454179
\(297\) 50.4412 2.92690
\(298\) −7.52918 −0.436154
\(299\) 3.72833 0.215615
\(300\) 0 0
\(301\) 36.0778 2.07949
\(302\) 13.3571 0.768614
\(303\) −25.6947 −1.47612
\(304\) 4.50973 0.258651
\(305\) 0 0
\(306\) 7.52918 0.430414
\(307\) −30.5084 −1.74121 −0.870604 0.491984i \(-0.836272\pi\)
−0.870604 + 0.491984i \(0.836272\pi\)
\(308\) −19.5615 −1.11462
\(309\) −38.4814 −2.18913
\(310\) 0 0
\(311\) 5.56280 0.315437 0.157719 0.987484i \(-0.449586\pi\)
0.157719 + 0.987484i \(0.449586\pi\)
\(312\) −11.6288 −0.658348
\(313\) −4.07252 −0.230193 −0.115096 0.993354i \(-0.536718\pi\)
−0.115096 + 0.993354i \(0.536718\pi\)
\(314\) 16.2381 0.916366
\(315\) 0 0
\(316\) −14.9133 −0.838939
\(317\) 6.16553 0.346291 0.173145 0.984896i \(-0.444607\pi\)
0.173145 + 0.984896i \(0.444607\pi\)
\(318\) −18.7142 −1.04944
\(319\) −35.7336 −2.00070
\(320\) 0 0
\(321\) 49.6947 2.77369
\(322\) 4.50973 0.251317
\(323\) −5.04650 −0.280795
\(324\) 16.0854 0.893634
\(325\) 0 0
\(326\) −3.29112 −0.182279
\(327\) −4.64820 −0.257046
\(328\) −3.90043 −0.215365
\(329\) −51.6664 −2.84846
\(330\) 0 0
\(331\) 27.5886 1.51640 0.758202 0.652019i \(-0.226078\pi\)
0.758202 + 0.652019i \(0.226078\pi\)
\(332\) −2.78140 −0.152649
\(333\) 5.25751 0.288110
\(334\) 22.9133 1.25376
\(335\) 0 0
\(336\) −14.0660 −0.767361
\(337\) 17.4230 0.949093 0.474547 0.880230i \(-0.342612\pi\)
0.474547 + 0.880230i \(0.342612\pi\)
\(338\) −0.900425 −0.0489767
\(339\) 18.7142 1.01641
\(340\) 0 0
\(341\) 7.49684 0.405977
\(342\) −30.3429 −1.64076
\(343\) −28.5810 −1.54323
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) 0.575681 0.0309488
\(347\) −4.88097 −0.262024 −0.131012 0.991381i \(-0.541823\pi\)
−0.131012 + 0.991381i \(0.541823\pi\)
\(348\) −25.6947 −1.37738
\(349\) 24.0389 1.28677 0.643387 0.765542i \(-0.277529\pi\)
0.643387 + 0.765542i \(0.277529\pi\)
\(350\) 0 0
\(351\) 43.3558 2.31416
\(352\) −4.33763 −0.231196
\(353\) 14.3442 0.763464 0.381732 0.924273i \(-0.375328\pi\)
0.381732 + 0.924273i \(0.375328\pi\)
\(354\) 6.98055 0.371012
\(355\) 0 0
\(356\) −7.69471 −0.407819
\(357\) 15.7402 0.833059
\(358\) −5.01945 −0.265286
\(359\) 26.7814 1.41347 0.706734 0.707479i \(-0.250168\pi\)
0.706734 + 0.707479i \(0.250168\pi\)
\(360\) 0 0
\(361\) 1.33763 0.0704015
\(362\) 11.5292 0.605960
\(363\) 24.3752 1.27937
\(364\) −16.8137 −0.881279
\(365\) 0 0
\(366\) −11.0920 −0.579787
\(367\) 20.4761 1.06884 0.534422 0.845218i \(-0.320529\pi\)
0.534422 + 0.845218i \(0.320529\pi\)
\(368\) 1.00000 0.0521286
\(369\) 26.2433 1.36617
\(370\) 0 0
\(371\) −27.0584 −1.40480
\(372\) 5.39070 0.279495
\(373\) 3.89386 0.201616 0.100808 0.994906i \(-0.467857\pi\)
0.100808 + 0.994906i \(0.467857\pi\)
\(374\) 4.85392 0.250990
\(375\) 0 0
\(376\) −11.4567 −0.590832
\(377\) −30.7142 −1.58186
\(378\) 52.4425 2.69735
\(379\) −30.3765 −1.56034 −0.780169 0.625569i \(-0.784867\pi\)
−0.780169 + 0.625569i \(0.784867\pi\)
\(380\) 0 0
\(381\) 2.10614 0.107901
\(382\) 18.7142 0.957500
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −3.11903 −0.159167
\(385\) 0 0
\(386\) 23.4956 1.19589
\(387\) −53.8266 −2.73616
\(388\) 0.642920 0.0326393
\(389\) −18.6818 −0.947206 −0.473603 0.880738i \(-0.657047\pi\)
−0.473603 + 0.880738i \(0.657047\pi\)
\(390\) 0 0
\(391\) −1.11903 −0.0565916
\(392\) −13.3376 −0.673652
\(393\) −42.7142 −2.15464
\(394\) −18.1385 −0.913803
\(395\) 0 0
\(396\) 29.1850 1.46660
\(397\) 28.5757 1.43417 0.717086 0.696985i \(-0.245475\pi\)
0.717086 + 0.696985i \(0.245475\pi\)
\(398\) 23.2575 1.16579
\(399\) −63.4336 −3.17565
\(400\) 0 0
\(401\) 12.1061 0.604552 0.302276 0.953220i \(-0.402254\pi\)
0.302276 + 0.953220i \(0.402254\pi\)
\(402\) 7.60170 0.379138
\(403\) 6.44377 0.320987
\(404\) −8.23805 −0.409858
\(405\) 0 0
\(406\) −37.1514 −1.84379
\(407\) 3.38942 0.168007
\(408\) 3.49027 0.172794
\(409\) 25.2911 1.25057 0.625283 0.780398i \(-0.284984\pi\)
0.625283 + 0.780398i \(0.284984\pi\)
\(410\) 0 0
\(411\) −23.4837 −1.15837
\(412\) −12.3376 −0.607831
\(413\) 10.0930 0.496644
\(414\) −6.72833 −0.330679
\(415\) 0 0
\(416\) −3.72833 −0.182796
\(417\) 14.5822 0.714096
\(418\) −19.5615 −0.956785
\(419\) −17.3505 −0.847628 −0.423814 0.905749i \(-0.639309\pi\)
−0.423814 + 0.905749i \(0.639309\pi\)
\(420\) 0 0
\(421\) 21.4230 1.04409 0.522047 0.852916i \(-0.325168\pi\)
0.522047 + 0.852916i \(0.325168\pi\)
\(422\) −4.34420 −0.211472
\(423\) 77.0841 3.74796
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −22.2044 −1.07581
\(427\) −16.0376 −0.776115
\(428\) 15.9328 0.770139
\(429\) 50.4412 2.43532
\(430\) 0 0
\(431\) 22.5822 1.08775 0.543874 0.839167i \(-0.316957\pi\)
0.543874 + 0.839167i \(0.316957\pi\)
\(432\) 11.6288 0.559489
\(433\) −1.01417 −0.0487378 −0.0243689 0.999703i \(-0.507758\pi\)
−0.0243689 + 0.999703i \(0.507758\pi\)
\(434\) 7.79428 0.374138
\(435\) 0 0
\(436\) −1.49027 −0.0713712
\(437\) 4.50973 0.215729
\(438\) −29.4956 −1.40935
\(439\) −26.7478 −1.27660 −0.638301 0.769787i \(-0.720362\pi\)
−0.638301 + 0.769787i \(0.720362\pi\)
\(440\) 0 0
\(441\) 89.7399 4.27333
\(442\) 4.17210 0.198446
\(443\) −10.2044 −0.484827 −0.242414 0.970173i \(-0.577939\pi\)
−0.242414 + 0.970173i \(0.577939\pi\)
\(444\) 2.43720 0.115665
\(445\) 0 0
\(446\) 12.4761 0.590761
\(447\) 23.4837 1.11074
\(448\) −4.50973 −0.213065
\(449\) 38.7867 1.83046 0.915228 0.402936i \(-0.132010\pi\)
0.915228 + 0.402936i \(0.132010\pi\)
\(450\) 0 0
\(451\) 16.9186 0.796665
\(452\) 6.00000 0.282216
\(453\) −41.6611 −1.95741
\(454\) 15.9328 0.747762
\(455\) 0 0
\(456\) −14.0660 −0.658699
\(457\) −34.9522 −1.63500 −0.817498 0.575932i \(-0.804639\pi\)
−0.817498 + 0.575932i \(0.804639\pi\)
\(458\) 3.56280 0.166479
\(459\) −13.0129 −0.607389
\(460\) 0 0
\(461\) 16.3700 0.762425 0.381213 0.924487i \(-0.375507\pi\)
0.381213 + 0.924487i \(0.375507\pi\)
\(462\) 61.0129 2.83858
\(463\) −29.2186 −1.35790 −0.678952 0.734183i \(-0.737565\pi\)
−0.678952 + 0.734183i \(0.737565\pi\)
\(464\) −8.23805 −0.382442
\(465\) 0 0
\(466\) −27.4956 −1.27371
\(467\) −24.2770 −1.12340 −0.561702 0.827340i \(-0.689853\pi\)
−0.561702 + 0.827340i \(0.689853\pi\)
\(468\) 25.0854 1.15957
\(469\) 10.9911 0.507523
\(470\) 0 0
\(471\) −50.6469 −2.33369
\(472\) 2.23805 0.103015
\(473\) −34.7010 −1.59555
\(474\) 46.5150 2.13651
\(475\) 0 0
\(476\) 5.04650 0.231306
\(477\) 40.3700 1.84841
\(478\) −10.0389 −0.459169
\(479\) 24.6080 1.12437 0.562185 0.827012i \(-0.309961\pi\)
0.562185 + 0.827012i \(0.309961\pi\)
\(480\) 0 0
\(481\) 2.91331 0.132835
\(482\) 23.6947 1.07926
\(483\) −14.0660 −0.640023
\(484\) 7.81502 0.355228
\(485\) 0 0
\(486\) −15.2846 −0.693322
\(487\) 30.2381 1.37022 0.685108 0.728441i \(-0.259755\pi\)
0.685108 + 0.728441i \(0.259755\pi\)
\(488\) −3.55623 −0.160983
\(489\) 10.2651 0.464204
\(490\) 0 0
\(491\) 12.3311 0.556493 0.278246 0.960510i \(-0.410247\pi\)
0.278246 + 0.960510i \(0.410247\pi\)
\(492\) 12.1655 0.548464
\(493\) 9.21860 0.415185
\(494\) −16.8137 −0.756486
\(495\) 0 0
\(496\) 1.72833 0.0776042
\(497\) −32.1049 −1.44010
\(498\) 8.67526 0.388748
\(499\) −26.9133 −1.20481 −0.602403 0.798192i \(-0.705790\pi\)
−0.602403 + 0.798192i \(0.705790\pi\)
\(500\) 0 0
\(501\) −71.4672 −3.19292
\(502\) −12.4425 −0.555335
\(503\) −20.5097 −0.914483 −0.457242 0.889342i \(-0.651163\pi\)
−0.457242 + 0.889342i \(0.651163\pi\)
\(504\) 30.3429 1.35158
\(505\) 0 0
\(506\) −4.33763 −0.192831
\(507\) 2.80845 0.124728
\(508\) 0.675256 0.0299597
\(509\) −36.7142 −1.62733 −0.813663 0.581336i \(-0.802530\pi\)
−0.813663 + 0.581336i \(0.802530\pi\)
\(510\) 0 0
\(511\) −42.6469 −1.88659
\(512\) −1.00000 −0.0441942
\(513\) 52.4425 2.31539
\(514\) 5.45665 0.240683
\(515\) 0 0
\(516\) −24.9522 −1.09846
\(517\) 49.6947 2.18557
\(518\) 3.52389 0.154831
\(519\) −1.79557 −0.0788166
\(520\) 0 0
\(521\) −4.91331 −0.215256 −0.107628 0.994191i \(-0.534326\pi\)
−0.107628 + 0.994191i \(0.534326\pi\)
\(522\) 55.4283 2.42603
\(523\) 0.344196 0.0150506 0.00752531 0.999972i \(-0.497605\pi\)
0.00752531 + 0.999972i \(0.497605\pi\)
\(524\) −13.6947 −0.598256
\(525\) 0 0
\(526\) 0.138479 0.00603795
\(527\) −1.93404 −0.0842483
\(528\) 13.5292 0.588782
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −15.0584 −0.653477
\(532\) −20.3376 −0.881748
\(533\) 14.5421 0.629887
\(534\) 24.0000 1.03858
\(535\) 0 0
\(536\) 2.43720 0.105271
\(537\) 15.6558 0.675598
\(538\) −14.6753 −0.632695
\(539\) 57.8537 2.49193
\(540\) 0 0
\(541\) −6.13191 −0.263631 −0.131816 0.991274i \(-0.542081\pi\)
−0.131816 + 0.991274i \(0.542081\pi\)
\(542\) 8.31058 0.356970
\(543\) −35.9598 −1.54318
\(544\) 1.11903 0.0479779
\(545\) 0 0
\(546\) 52.4425 2.24433
\(547\) 9.18498 0.392721 0.196361 0.980532i \(-0.437088\pi\)
0.196361 + 0.980532i \(0.437088\pi\)
\(548\) −7.52918 −0.321631
\(549\) 23.9275 1.02120
\(550\) 0 0
\(551\) −37.1514 −1.58270
\(552\) −3.11903 −0.132755
\(553\) 67.2549 2.85997
\(554\) 12.9133 0.548634
\(555\) 0 0
\(556\) 4.67526 0.198275
\(557\) 4.30529 0.182421 0.0912105 0.995832i \(-0.470926\pi\)
0.0912105 + 0.995832i \(0.470926\pi\)
\(558\) −11.6288 −0.492284
\(559\) −29.8266 −1.26153
\(560\) 0 0
\(561\) −15.1395 −0.639191
\(562\) −2.67526 −0.112849
\(563\) −11.1256 −0.468888 −0.234444 0.972130i \(-0.575327\pi\)
−0.234444 + 0.972130i \(0.575327\pi\)
\(564\) 35.7336 1.50466
\(565\) 0 0
\(566\) 0.742495 0.0312094
\(567\) −72.5408 −3.04643
\(568\) −7.11903 −0.298708
\(569\) −16.0389 −0.672386 −0.336193 0.941793i \(-0.609139\pi\)
−0.336193 + 0.941793i \(0.609139\pi\)
\(570\) 0 0
\(571\) 17.9004 0.749109 0.374555 0.927205i \(-0.377796\pi\)
0.374555 + 0.927205i \(0.377796\pi\)
\(572\) 16.1721 0.676189
\(573\) −58.3700 −2.43844
\(574\) 17.5898 0.734186
\(575\) 0 0
\(576\) 6.72833 0.280347
\(577\) 9.12559 0.379903 0.189952 0.981793i \(-0.439167\pi\)
0.189952 + 0.981793i \(0.439167\pi\)
\(578\) 15.7478 0.655021
\(579\) −73.2833 −3.04555
\(580\) 0 0
\(581\) 12.5433 0.520386
\(582\) −2.00528 −0.0831217
\(583\) 26.0258 1.07788
\(584\) −9.45665 −0.391319
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 33.6340 1.38823 0.694113 0.719866i \(-0.255797\pi\)
0.694113 + 0.719866i \(0.255797\pi\)
\(588\) 41.6004 1.71557
\(589\) 7.79428 0.321158
\(590\) 0 0
\(591\) 56.5744 2.32716
\(592\) 0.781399 0.0321153
\(593\) 17.4567 0.716859 0.358429 0.933557i \(-0.383312\pi\)
0.358429 + 0.933557i \(0.383312\pi\)
\(594\) −50.4412 −2.06963
\(595\) 0 0
\(596\) 7.52918 0.308407
\(597\) −72.5408 −2.96890
\(598\) −3.72833 −0.152463
\(599\) −11.5951 −0.473764 −0.236882 0.971538i \(-0.576126\pi\)
−0.236882 + 0.971538i \(0.576126\pi\)
\(600\) 0 0
\(601\) −31.6611 −1.29148 −0.645741 0.763556i \(-0.723452\pi\)
−0.645741 + 0.763556i \(0.723452\pi\)
\(602\) −36.0778 −1.47042
\(603\) −16.3983 −0.667790
\(604\) −13.3571 −0.543492
\(605\) 0 0
\(606\) 25.6947 1.04378
\(607\) 36.0778 1.46435 0.732177 0.681115i \(-0.238505\pi\)
0.732177 + 0.681115i \(0.238505\pi\)
\(608\) −4.50973 −0.182894
\(609\) 115.876 4.69554
\(610\) 0 0
\(611\) 42.7142 1.72803
\(612\) −7.52918 −0.304349
\(613\) 32.0389 1.29404 0.647020 0.762473i \(-0.276015\pi\)
0.647020 + 0.762473i \(0.276015\pi\)
\(614\) 30.5084 1.23122
\(615\) 0 0
\(616\) 19.5615 0.788156
\(617\) 13.3960 0.539302 0.269651 0.962958i \(-0.413092\pi\)
0.269651 + 0.962958i \(0.413092\pi\)
\(618\) 38.4814 1.54795
\(619\) 37.1309 1.49242 0.746208 0.665713i \(-0.231872\pi\)
0.746208 + 0.665713i \(0.231872\pi\)
\(620\) 0 0
\(621\) 11.6288 0.466646
\(622\) −5.56280 −0.223048
\(623\) 34.7010 1.39027
\(624\) 11.6288 0.465523
\(625\) 0 0
\(626\) 4.07252 0.162771
\(627\) 61.0129 2.43662
\(628\) −16.2381 −0.647969
\(629\) −0.874406 −0.0348648
\(630\) 0 0
\(631\) 11.1125 0.442380 0.221190 0.975231i \(-0.429006\pi\)
0.221190 + 0.975231i \(0.429006\pi\)
\(632\) 14.9133 0.593220
\(633\) 13.5497 0.538551
\(634\) −6.16553 −0.244864
\(635\) 0 0
\(636\) 18.7142 0.742065
\(637\) 49.7270 1.97026
\(638\) 35.7336 1.41471
\(639\) 47.8991 1.89486
\(640\) 0 0
\(641\) 12.3831 0.489103 0.244552 0.969636i \(-0.421359\pi\)
0.244552 + 0.969636i \(0.421359\pi\)
\(642\) −49.6947 −1.96129
\(643\) 37.4956 1.47868 0.739340 0.673332i \(-0.235138\pi\)
0.739340 + 0.673332i \(0.235138\pi\)
\(644\) −4.50973 −0.177708
\(645\) 0 0
\(646\) 5.04650 0.198552
\(647\) −14.5691 −0.572771 −0.286385 0.958114i \(-0.592454\pi\)
−0.286385 + 0.958114i \(0.592454\pi\)
\(648\) −16.0854 −0.631894
\(649\) −9.70784 −0.381066
\(650\) 0 0
\(651\) −24.3106 −0.952807
\(652\) 3.29112 0.128890
\(653\) 4.41672 0.172840 0.0864198 0.996259i \(-0.472457\pi\)
0.0864198 + 0.996259i \(0.472457\pi\)
\(654\) 4.64820 0.181759
\(655\) 0 0
\(656\) 3.90043 0.152286
\(657\) 63.6275 2.48234
\(658\) 51.6664 2.01416
\(659\) −31.8655 −1.24130 −0.620652 0.784086i \(-0.713132\pi\)
−0.620652 + 0.784086i \(0.713132\pi\)
\(660\) 0 0
\(661\) 33.1190 1.28818 0.644090 0.764949i \(-0.277236\pi\)
0.644090 + 0.764949i \(0.277236\pi\)
\(662\) −27.5886 −1.07226
\(663\) −13.0129 −0.505379
\(664\) 2.78140 0.107939
\(665\) 0 0
\(666\) −5.25751 −0.203724
\(667\) −8.23805 −0.318979
\(668\) −22.9133 −0.886543
\(669\) −38.9133 −1.50448
\(670\) 0 0
\(671\) 15.4256 0.595499
\(672\) 14.0660 0.542606
\(673\) −19.3505 −0.745907 −0.372954 0.927850i \(-0.621655\pi\)
−0.372954 + 0.927850i \(0.621655\pi\)
\(674\) −17.4230 −0.671110
\(675\) 0 0
\(676\) 0.900425 0.0346317
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) −18.7142 −0.718713
\(679\) −2.89939 −0.111268
\(680\) 0 0
\(681\) −49.6947 −1.90431
\(682\) −7.49684 −0.287069
\(683\) −38.3495 −1.46740 −0.733701 0.679472i \(-0.762209\pi\)
−0.733701 + 0.679472i \(0.762209\pi\)
\(684\) 30.3429 1.16019
\(685\) 0 0
\(686\) 28.5810 1.09123
\(687\) −11.1125 −0.423967
\(688\) −8.00000 −0.304997
\(689\) 22.3700 0.852228
\(690\) 0 0
\(691\) −21.0195 −0.799618 −0.399809 0.916599i \(-0.630923\pi\)
−0.399809 + 0.916599i \(0.630923\pi\)
\(692\) −0.575681 −0.0218841
\(693\) −131.616 −4.99969
\(694\) 4.88097 0.185279
\(695\) 0 0
\(696\) 25.6947 0.973955
\(697\) −4.36468 −0.165324
\(698\) −24.0389 −0.909886
\(699\) 85.7594 3.24372
\(700\) 0 0
\(701\) 19.3169 0.729589 0.364794 0.931088i \(-0.381139\pi\)
0.364794 + 0.931088i \(0.381139\pi\)
\(702\) −43.3558 −1.63636
\(703\) 3.52389 0.132906
\(704\) 4.33763 0.163481
\(705\) 0 0
\(706\) −14.3442 −0.539851
\(707\) 37.1514 1.39722
\(708\) −6.98055 −0.262345
\(709\) 12.2315 0.459363 0.229682 0.973266i \(-0.426232\pi\)
0.229682 + 0.973266i \(0.426232\pi\)
\(710\) 0 0
\(711\) −100.342 −3.76311
\(712\) 7.69471 0.288371
\(713\) 1.72833 0.0647264
\(714\) −15.7402 −0.589061
\(715\) 0 0
\(716\) 5.01945 0.187586
\(717\) 31.3116 1.16935
\(718\) −26.7814 −0.999473
\(719\) −40.6416 −1.51568 −0.757839 0.652442i \(-0.773745\pi\)
−0.757839 + 0.652442i \(0.773745\pi\)
\(720\) 0 0
\(721\) 55.6393 2.07212
\(722\) −1.33763 −0.0497814
\(723\) −73.9044 −2.74854
\(724\) −11.5292 −0.428479
\(725\) 0 0
\(726\) −24.3752 −0.904650
\(727\) 23.4501 0.869716 0.434858 0.900499i \(-0.356799\pi\)
0.434858 + 0.900499i \(0.356799\pi\)
\(728\) 16.8137 0.623158
\(729\) −0.583281 −0.0216030
\(730\) 0 0
\(731\) 8.95221 0.331110
\(732\) 11.0920 0.409971
\(733\) 12.5150 0.462252 0.231126 0.972924i \(-0.425759\pi\)
0.231126 + 0.972924i \(0.425759\pi\)
\(734\) −20.4761 −0.755787
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −10.5717 −0.389413
\(738\) −26.2433 −0.966031
\(739\) −21.3505 −0.785391 −0.392696 0.919668i \(-0.628457\pi\)
−0.392696 + 0.919668i \(0.628457\pi\)
\(740\) 0 0
\(741\) 52.4425 1.92652
\(742\) 27.0584 0.993343
\(743\) −24.9858 −0.916641 −0.458321 0.888787i \(-0.651549\pi\)
−0.458321 + 0.888787i \(0.651549\pi\)
\(744\) −5.39070 −0.197633
\(745\) 0 0
\(746\) −3.89386 −0.142564
\(747\) −18.7142 −0.684715
\(748\) −4.85392 −0.177477
\(749\) −71.8524 −2.62543
\(750\) 0 0
\(751\) −33.6275 −1.22708 −0.613542 0.789662i \(-0.710256\pi\)
−0.613542 + 0.789662i \(0.710256\pi\)
\(752\) 11.4567 0.417781
\(753\) 38.8085 1.41426
\(754\) 30.7142 1.11854
\(755\) 0 0
\(756\) −52.4425 −1.90731
\(757\) 37.1230 1.34926 0.674630 0.738156i \(-0.264303\pi\)
0.674630 + 0.738156i \(0.264303\pi\)
\(758\) 30.3765 1.10333
\(759\) 13.5292 0.491078
\(760\) 0 0
\(761\) −3.87337 −0.140410 −0.0702048 0.997533i \(-0.522365\pi\)
−0.0702048 + 0.997533i \(0.522365\pi\)
\(762\) −2.10614 −0.0762975
\(763\) 6.72073 0.243307
\(764\) −18.7142 −0.677055
\(765\) 0 0
\(766\) 0 0
\(767\) −8.34420 −0.301291
\(768\) 3.11903 0.112548
\(769\) 23.1645 0.835333 0.417667 0.908600i \(-0.362848\pi\)
0.417667 + 0.908600i \(0.362848\pi\)
\(770\) 0 0
\(771\) −17.0195 −0.612941
\(772\) −23.4956 −0.845624
\(773\) 8.78140 0.315845 0.157922 0.987452i \(-0.449520\pi\)
0.157922 + 0.987452i \(0.449520\pi\)
\(774\) 53.8266 1.93476
\(775\) 0 0
\(776\) −0.642920 −0.0230795
\(777\) −10.9911 −0.394304
\(778\) 18.6818 0.669776
\(779\) 17.5898 0.630222
\(780\) 0 0
\(781\) 30.8797 1.10496
\(782\) 1.11903 0.0400163
\(783\) −95.7983 −3.42355
\(784\) 13.3376 0.476344
\(785\) 0 0
\(786\) 42.7142 1.52356
\(787\) −49.6275 −1.76903 −0.884514 0.466513i \(-0.845510\pi\)
−0.884514 + 0.466513i \(0.845510\pi\)
\(788\) 18.1385 0.646157
\(789\) −0.431918 −0.0153767
\(790\) 0 0
\(791\) −27.0584 −0.962084
\(792\) −29.1850 −1.03704
\(793\) 13.2588 0.470833
\(794\) −28.5757 −1.01411
\(795\) 0 0
\(796\) −23.2575 −0.824340
\(797\) −18.3311 −0.649319 −0.324660 0.945831i \(-0.605250\pi\)
−0.324660 + 0.945831i \(0.605250\pi\)
\(798\) 63.4336 2.24553
\(799\) −12.8203 −0.453550
\(800\) 0 0
\(801\) −51.7725 −1.82929
\(802\) −12.1061 −0.427483
\(803\) 41.0195 1.44755
\(804\) −7.60170 −0.268091
\(805\) 0 0
\(806\) −6.44377 −0.226972
\(807\) 45.7725 1.61127
\(808\) 8.23805 0.289814
\(809\) −1.93933 −0.0681832 −0.0340916 0.999419i \(-0.510854\pi\)
−0.0340916 + 0.999419i \(0.510854\pi\)
\(810\) 0 0
\(811\) −5.41775 −0.190243 −0.0951215 0.995466i \(-0.530324\pi\)
−0.0951215 + 0.995466i \(0.530324\pi\)
\(812\) 37.1514 1.30376
\(813\) −25.9209 −0.909086
\(814\) −3.38942 −0.118799
\(815\) 0 0
\(816\) −3.49027 −0.122184
\(817\) −36.0778 −1.26220
\(818\) −25.2911 −0.884283
\(819\) −113.128 −3.95302
\(820\) 0 0
\(821\) 37.8655 1.32152 0.660758 0.750599i \(-0.270235\pi\)
0.660758 + 0.750599i \(0.270235\pi\)
\(822\) 23.4837 0.819088
\(823\) −43.7336 −1.52446 −0.762229 0.647308i \(-0.775895\pi\)
−0.762229 + 0.647308i \(0.775895\pi\)
\(824\) 12.3376 0.429802
\(825\) 0 0
\(826\) −10.0930 −0.351181
\(827\) −28.1991 −0.980581 −0.490290 0.871559i \(-0.663109\pi\)
−0.490290 + 0.871559i \(0.663109\pi\)
\(828\) 6.72833 0.233826
\(829\) 1.12559 0.0390935 0.0195468 0.999809i \(-0.493778\pi\)
0.0195468 + 0.999809i \(0.493778\pi\)
\(830\) 0 0
\(831\) −40.2770 −1.39719
\(832\) 3.72833 0.129256
\(833\) −14.9252 −0.517126
\(834\) −14.5822 −0.504942
\(835\) 0 0
\(836\) 19.5615 0.676549
\(837\) 20.0983 0.694699
\(838\) 17.3505 0.599364
\(839\) 39.9328 1.37863 0.689316 0.724461i \(-0.257911\pi\)
0.689316 + 0.724461i \(0.257911\pi\)
\(840\) 0 0
\(841\) 38.8655 1.34019
\(842\) −21.4230 −0.738287
\(843\) 8.34420 0.287389
\(844\) 4.34420 0.149533
\(845\) 0 0
\(846\) −77.0841 −2.65021
\(847\) −35.2436 −1.21098
\(848\) 6.00000 0.206041
\(849\) −2.31586 −0.0794801
\(850\) 0 0
\(851\) 0.781399 0.0267860
\(852\) 22.2044 0.760711
\(853\) −47.9921 −1.64322 −0.821610 0.570050i \(-0.806924\pi\)
−0.821610 + 0.570050i \(0.806924\pi\)
\(854\) 16.0376 0.548796
\(855\) 0 0
\(856\) −15.9328 −0.544571
\(857\) 43.4283 1.48348 0.741742 0.670686i \(-0.234000\pi\)
0.741742 + 0.670686i \(0.234000\pi\)
\(858\) −50.4412 −1.72203
\(859\) 32.5433 1.11036 0.555182 0.831729i \(-0.312648\pi\)
0.555182 + 0.831729i \(0.312648\pi\)
\(860\) 0 0
\(861\) −54.8632 −1.86973
\(862\) −22.5822 −0.769154
\(863\) −46.1036 −1.56938 −0.784692 0.619886i \(-0.787179\pi\)
−0.784692 + 0.619886i \(0.787179\pi\)
\(864\) −11.6288 −0.395618
\(865\) 0 0
\(866\) 1.01417 0.0344628
\(867\) −49.1177 −1.66813
\(868\) −7.79428 −0.264555
\(869\) −64.6884 −2.19440
\(870\) 0 0
\(871\) −9.08669 −0.307891
\(872\) 1.49027 0.0504670
\(873\) 4.32578 0.146405
\(874\) −4.50973 −0.152544
\(875\) 0 0
\(876\) 29.4956 0.996563
\(877\) 24.0996 0.813785 0.406892 0.913476i \(-0.366612\pi\)
0.406892 + 0.913476i \(0.366612\pi\)
\(878\) 26.7478 0.902694
\(879\) 18.7142 0.631213
\(880\) 0 0
\(881\) 2.34420 0.0789780 0.0394890 0.999220i \(-0.487427\pi\)
0.0394890 + 0.999220i \(0.487427\pi\)
\(882\) −89.7399 −3.02170
\(883\) 41.0505 1.38146 0.690730 0.723113i \(-0.257289\pi\)
0.690730 + 0.723113i \(0.257289\pi\)
\(884\) −4.17210 −0.140323
\(885\) 0 0
\(886\) 10.2044 0.342825
\(887\) −54.7788 −1.83929 −0.919647 0.392747i \(-0.871525\pi\)
−0.919647 + 0.392747i \(0.871525\pi\)
\(888\) −2.43720 −0.0817872
\(889\) −3.04522 −0.102133
\(890\) 0 0
\(891\) 69.7725 2.33747
\(892\) −12.4761 −0.417731
\(893\) 51.6664 1.72895
\(894\) −23.4837 −0.785413
\(895\) 0 0
\(896\) 4.50973 0.150659
\(897\) 11.6288 0.388273
\(898\) −38.7867 −1.29433
\(899\) −14.2381 −0.474866
\(900\) 0 0
\(901\) −6.71416 −0.223681
\(902\) −16.9186 −0.563328
\(903\) 112.528 3.74469
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 41.6611 1.38410
\(907\) 10.1061 0.335569 0.167784 0.985824i \(-0.446339\pi\)
0.167784 + 0.985824i \(0.446339\pi\)
\(908\) −15.9328 −0.528747
\(909\) −55.4283 −1.83844
\(910\) 0 0
\(911\) −25.4178 −0.842128 −0.421064 0.907031i \(-0.638343\pi\)
−0.421064 + 0.907031i \(0.638343\pi\)
\(912\) 14.0660 0.465770
\(913\) −12.0647 −0.399282
\(914\) 34.9522 1.15612
\(915\) 0 0
\(916\) −3.56280 −0.117718
\(917\) 61.7594 2.03947
\(918\) 13.0129 0.429489
\(919\) 23.6017 0.778548 0.389274 0.921122i \(-0.372726\pi\)
0.389274 + 0.921122i \(0.372726\pi\)
\(920\) 0 0
\(921\) −95.1566 −3.13552
\(922\) −16.3700 −0.539116
\(923\) 26.5421 0.873643
\(924\) −61.0129 −2.00718
\(925\) 0 0
\(926\) 29.2186 0.960183
\(927\) −83.0116 −2.72646
\(928\) 8.23805 0.270427
\(929\) −7.08669 −0.232507 −0.116253 0.993220i \(-0.537088\pi\)
−0.116253 + 0.993220i \(0.537088\pi\)
\(930\) 0 0
\(931\) 60.1490 1.97131
\(932\) 27.4956 0.900647
\(933\) 17.3505 0.568030
\(934\) 24.2770 0.794366
\(935\) 0 0
\(936\) −25.0854 −0.819942
\(937\) −27.3169 −0.892404 −0.446202 0.894932i \(-0.647224\pi\)
−0.446202 + 0.894932i \(0.647224\pi\)
\(938\) −10.9911 −0.358873
\(939\) −12.7023 −0.414524
\(940\) 0 0
\(941\) 55.8979 1.82222 0.911109 0.412165i \(-0.135227\pi\)
0.911109 + 0.412165i \(0.135227\pi\)
\(942\) 50.6469 1.65017
\(943\) 3.90043 0.127015
\(944\) −2.23805 −0.0728424
\(945\) 0 0
\(946\) 34.7010 1.12823
\(947\) −37.5939 −1.22164 −0.610818 0.791771i \(-0.709159\pi\)
−0.610818 + 0.791771i \(0.709159\pi\)
\(948\) −46.5150 −1.51074
\(949\) 35.2575 1.14451
\(950\) 0 0
\(951\) 19.2305 0.623590
\(952\) −5.04650 −0.163558
\(953\) 29.3828 0.951804 0.475902 0.879498i \(-0.342122\pi\)
0.475902 + 0.879498i \(0.342122\pi\)
\(954\) −40.3700 −1.30703
\(955\) 0 0
\(956\) 10.0389 0.324681
\(957\) −111.454 −3.60280
\(958\) −24.6080 −0.795049
\(959\) 33.9545 1.09645
\(960\) 0 0
\(961\) −28.0129 −0.903641
\(962\) −2.91331 −0.0939289
\(963\) 107.201 3.45450
\(964\) −23.6947 −0.763155
\(965\) 0 0
\(966\) 14.0660 0.452565
\(967\) 49.2292 1.58310 0.791552 0.611102i \(-0.209274\pi\)
0.791552 + 0.611102i \(0.209274\pi\)
\(968\) −7.81502 −0.251184
\(969\) −15.7402 −0.505647
\(970\) 0 0
\(971\) 9.62347 0.308832 0.154416 0.988006i \(-0.450650\pi\)
0.154416 + 0.988006i \(0.450650\pi\)
\(972\) 15.2846 0.490252
\(973\) −21.0841 −0.675926
\(974\) −30.2381 −0.968890
\(975\) 0 0
\(976\) 3.55623 0.113832
\(977\) −18.9858 −0.607411 −0.303705 0.952766i \(-0.598224\pi\)
−0.303705 + 0.952766i \(0.598224\pi\)
\(978\) −10.2651 −0.328242
\(979\) −33.3768 −1.06673
\(980\) 0 0
\(981\) −10.0271 −0.320139
\(982\) −12.3311 −0.393500
\(983\) −4.33763 −0.138349 −0.0691744 0.997605i \(-0.522037\pi\)
−0.0691744 + 0.997605i \(0.522037\pi\)
\(984\) −12.1655 −0.387823
\(985\) 0 0
\(986\) −9.21860 −0.293580
\(987\) −161.149 −5.12942
\(988\) 16.8137 0.534916
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) 1.96766 0.0625049 0.0312524 0.999512i \(-0.490050\pi\)
0.0312524 + 0.999512i \(0.490050\pi\)
\(992\) −1.72833 −0.0548744
\(993\) 86.0495 2.73070
\(994\) 32.1049 1.01830
\(995\) 0 0
\(996\) −8.67526 −0.274886
\(997\) −3.96110 −0.125449 −0.0627246 0.998031i \(-0.519979\pi\)
−0.0627246 + 0.998031i \(0.519979\pi\)
\(998\) 26.9133 0.851926
\(999\) 9.08669 0.287490
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 1150.2.a.q.1.3 3
4.3 odd 2 9200.2.a.cf.1.1 3
5.2 odd 4 1150.2.b.j.599.1 6
5.3 odd 4 1150.2.b.j.599.6 6
5.4 even 2 230.2.a.d.1.1 3
15.14 odd 2 2070.2.a.z.1.3 3
20.19 odd 2 1840.2.a.r.1.3 3
40.19 odd 2 7360.2.a.ce.1.1 3
40.29 even 2 7360.2.a.bz.1.3 3
115.114 odd 2 5290.2.a.r.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.1 3 5.4 even 2
1150.2.a.q.1.3 3 1.1 even 1 trivial
1150.2.b.j.599.1 6 5.2 odd 4
1150.2.b.j.599.6 6 5.3 odd 4
1840.2.a.r.1.3 3 20.19 odd 2
2070.2.a.z.1.3 3 15.14 odd 2
5290.2.a.r.1.1 3 115.114 odd 2
7360.2.a.bz.1.3 3 40.29 even 2
7360.2.a.ce.1.1 3 40.19 odd 2
9200.2.a.cf.1.1 3 4.3 odd 2