Properties

Label 105.4.a.e.1.2
Level 105
Weight 4
Character 105.1
Self dual yes
Analytic conductor 6.195
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.19520055060\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of \(x^{2} - 2\)
Character \(\chi\) \(=\) 105.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.82843 q^{2} -3.00000 q^{3} -4.65685 q^{4} +5.00000 q^{5} -5.48528 q^{6} -7.00000 q^{7} -23.1421 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+1.82843 q^{2} -3.00000 q^{3} -4.65685 q^{4} +5.00000 q^{5} -5.48528 q^{6} -7.00000 q^{7} -23.1421 q^{8} +9.00000 q^{9} +9.14214 q^{10} -64.5685 q^{11} +13.9706 q^{12} -32.3431 q^{13} -12.7990 q^{14} -15.0000 q^{15} -5.05887 q^{16} -56.3431 q^{17} +16.4558 q^{18} -2.74517 q^{19} -23.2843 q^{20} +21.0000 q^{21} -118.059 q^{22} +88.1665 q^{23} +69.4264 q^{24} +25.0000 q^{25} -59.1371 q^{26} -27.0000 q^{27} +32.5980 q^{28} +246.735 q^{29} -27.4264 q^{30} -110.912 q^{31} +175.887 q^{32} +193.706 q^{33} -103.019 q^{34} -35.0000 q^{35} -41.9117 q^{36} +120.676 q^{37} -5.01934 q^{38} +97.0294 q^{39} -115.711 q^{40} -176.274 q^{41} +38.3970 q^{42} -443.362 q^{43} +300.686 q^{44} +45.0000 q^{45} +161.206 q^{46} -345.206 q^{47} +15.1766 q^{48} +49.0000 q^{49} +45.7107 q^{50} +169.029 q^{51} +150.617 q^{52} +260.981 q^{53} -49.3675 q^{54} -322.843 q^{55} +161.995 q^{56} +8.23550 q^{57} +451.137 q^{58} +628.999 q^{59} +69.8528 q^{60} -115.206 q^{61} -202.794 q^{62} -63.0000 q^{63} +362.068 q^{64} -161.716 q^{65} +354.177 q^{66} -951.480 q^{67} +262.382 q^{68} -264.500 q^{69} -63.9949 q^{70} +356.264 q^{71} -208.279 q^{72} -656.754 q^{73} +220.648 q^{74} -75.0000 q^{75} +12.7838 q^{76} +451.980 q^{77} +177.411 q^{78} +440.195 q^{79} -25.2944 q^{80} +81.0000 q^{81} -322.304 q^{82} -54.4121 q^{83} -97.7939 q^{84} -281.716 q^{85} -810.656 q^{86} -740.205 q^{87} +1494.25 q^{88} -1018.78 q^{89} +82.2792 q^{90} +226.402 q^{91} -410.579 q^{92} +332.735 q^{93} -631.184 q^{94} -13.7258 q^{95} -527.662 q^{96} -724.108 q^{97} +89.5929 q^{98} -581.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 6q^{3} + 2q^{4} + 10q^{5} + 6q^{6} - 14q^{7} - 18q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 6q^{3} + 2q^{4} + 10q^{5} + 6q^{6} - 14q^{7} - 18q^{8} + 18q^{9} - 10q^{10} - 16q^{11} - 6q^{12} - 76q^{13} + 14q^{14} - 30q^{15} - 78q^{16} - 124q^{17} - 18q^{18} - 96q^{19} + 10q^{20} + 42q^{21} - 304q^{22} - 16q^{23} + 54q^{24} + 50q^{25} + 108q^{26} - 54q^{27} - 14q^{28} + 188q^{29} + 30q^{30} - 120q^{31} + 414q^{32} + 48q^{33} + 156q^{34} - 70q^{35} + 18q^{36} - 132q^{37} + 352q^{38} + 228q^{39} - 90q^{40} + 100q^{41} - 42q^{42} - 536q^{43} + 624q^{44} + 90q^{45} + 560q^{46} - 928q^{47} + 234q^{48} + 98q^{49} - 50q^{50} + 372q^{51} - 140q^{52} + 884q^{53} + 54q^{54} - 80q^{55} + 126q^{56} + 288q^{57} + 676q^{58} + 104q^{59} - 30q^{60} - 468q^{61} - 168q^{62} - 126q^{63} + 34q^{64} - 380q^{65} + 912q^{66} - 1688q^{67} - 188q^{68} + 48q^{69} + 70q^{70} - 136q^{71} - 162q^{72} + 508q^{73} + 1188q^{74} - 150q^{75} - 608q^{76} + 112q^{77} - 324q^{78} - 432q^{79} - 390q^{80} + 162q^{81} - 1380q^{82} - 584q^{83} + 42q^{84} - 620q^{85} - 456q^{86} - 564q^{87} + 1744q^{88} - 1404q^{89} - 90q^{90} + 532q^{91} - 1104q^{92} + 360q^{93} + 1600q^{94} - 480q^{95} - 1242q^{96} - 1188q^{97} - 98q^{98} - 144q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.82843 0.646447 0.323223 0.946323i \(-0.395234\pi\)
0.323223 + 0.946323i \(0.395234\pi\)
\(3\) −3.00000 −0.577350
\(4\) −4.65685 −0.582107
\(5\) 5.00000 0.447214
\(6\) −5.48528 −0.373226
\(7\) −7.00000 −0.377964
\(8\) −23.1421 −1.02275
\(9\) 9.00000 0.333333
\(10\) 9.14214 0.289100
\(11\) −64.5685 −1.76983 −0.884916 0.465751i \(-0.845784\pi\)
−0.884916 + 0.465751i \(0.845784\pi\)
\(12\) 13.9706 0.336080
\(13\) −32.3431 −0.690029 −0.345014 0.938597i \(-0.612126\pi\)
−0.345014 + 0.938597i \(0.612126\pi\)
\(14\) −12.7990 −0.244334
\(15\) −15.0000 −0.258199
\(16\) −5.05887 −0.0790449
\(17\) −56.3431 −0.803836 −0.401918 0.915676i \(-0.631656\pi\)
−0.401918 + 0.915676i \(0.631656\pi\)
\(18\) 16.4558 0.215482
\(19\) −2.74517 −0.0331465 −0.0165733 0.999863i \(-0.505276\pi\)
−0.0165733 + 0.999863i \(0.505276\pi\)
\(20\) −23.2843 −0.260326
\(21\) 21.0000 0.218218
\(22\) −118.059 −1.14410
\(23\) 88.1665 0.799304 0.399652 0.916667i \(-0.369131\pi\)
0.399652 + 0.916667i \(0.369131\pi\)
\(24\) 69.4264 0.590484
\(25\) 25.0000 0.200000
\(26\) −59.1371 −0.446067
\(27\) −27.0000 −0.192450
\(28\) 32.5980 0.220016
\(29\) 246.735 1.57992 0.789958 0.613161i \(-0.210102\pi\)
0.789958 + 0.613161i \(0.210102\pi\)
\(30\) −27.4264 −0.166912
\(31\) −110.912 −0.642591 −0.321296 0.946979i \(-0.604118\pi\)
−0.321296 + 0.946979i \(0.604118\pi\)
\(32\) 175.887 0.971649
\(33\) 193.706 1.02181
\(34\) −103.019 −0.519637
\(35\) −35.0000 −0.169031
\(36\) −41.9117 −0.194036
\(37\) 120.676 0.536190 0.268095 0.963392i \(-0.413606\pi\)
0.268095 + 0.963392i \(0.413606\pi\)
\(38\) −5.01934 −0.0214275
\(39\) 97.0294 0.398388
\(40\) −115.711 −0.457387
\(41\) −176.274 −0.671449 −0.335724 0.941960i \(-0.608981\pi\)
−0.335724 + 0.941960i \(0.608981\pi\)
\(42\) 38.3970 0.141066
\(43\) −443.362 −1.57238 −0.786188 0.617988i \(-0.787948\pi\)
−0.786188 + 0.617988i \(0.787948\pi\)
\(44\) 300.686 1.03023
\(45\) 45.0000 0.149071
\(46\) 161.206 0.516707
\(47\) −345.206 −1.07135 −0.535675 0.844424i \(-0.679943\pi\)
−0.535675 + 0.844424i \(0.679943\pi\)
\(48\) 15.1766 0.0456366
\(49\) 49.0000 0.142857
\(50\) 45.7107 0.129289
\(51\) 169.029 0.464095
\(52\) 150.617 0.401670
\(53\) 260.981 0.676386 0.338193 0.941077i \(-0.390184\pi\)
0.338193 + 0.941077i \(0.390184\pi\)
\(54\) −49.3675 −0.124409
\(55\) −322.843 −0.791493
\(56\) 161.995 0.386562
\(57\) 8.23550 0.0191372
\(58\) 451.137 1.02133
\(59\) 628.999 1.38794 0.693972 0.720002i \(-0.255859\pi\)
0.693972 + 0.720002i \(0.255859\pi\)
\(60\) 69.8528 0.150299
\(61\) −115.206 −0.241814 −0.120907 0.992664i \(-0.538580\pi\)
−0.120907 + 0.992664i \(0.538580\pi\)
\(62\) −202.794 −0.415401
\(63\) −63.0000 −0.125988
\(64\) 362.068 0.707164
\(65\) −161.716 −0.308590
\(66\) 354.177 0.660547
\(67\) −951.480 −1.73495 −0.867476 0.497479i \(-0.834259\pi\)
−0.867476 + 0.497479i \(0.834259\pi\)
\(68\) 262.382 0.467919
\(69\) −264.500 −0.461478
\(70\) −63.9949 −0.109269
\(71\) 356.264 0.595504 0.297752 0.954643i \(-0.403763\pi\)
0.297752 + 0.954643i \(0.403763\pi\)
\(72\) −208.279 −0.340916
\(73\) −656.754 −1.05298 −0.526488 0.850183i \(-0.676491\pi\)
−0.526488 + 0.850183i \(0.676491\pi\)
\(74\) 220.648 0.346618
\(75\) −75.0000 −0.115470
\(76\) 12.7838 0.0192948
\(77\) 451.980 0.668933
\(78\) 177.411 0.257537
\(79\) 440.195 0.626909 0.313455 0.949603i \(-0.398514\pi\)
0.313455 + 0.949603i \(0.398514\pi\)
\(80\) −25.2944 −0.0353500
\(81\) 81.0000 0.111111
\(82\) −322.304 −0.434056
\(83\) −54.4121 −0.0719579 −0.0359790 0.999353i \(-0.511455\pi\)
−0.0359790 + 0.999353i \(0.511455\pi\)
\(84\) −97.7939 −0.127026
\(85\) −281.716 −0.359487
\(86\) −810.656 −1.01646
\(87\) −740.205 −0.912165
\(88\) 1494.25 1.81009
\(89\) −1018.78 −1.21338 −0.606690 0.794938i \(-0.707503\pi\)
−0.606690 + 0.794938i \(0.707503\pi\)
\(90\) 82.2792 0.0963666
\(91\) 226.402 0.260806
\(92\) −410.579 −0.465280
\(93\) 332.735 0.371000
\(94\) −631.184 −0.692571
\(95\) −13.7258 −0.0148236
\(96\) −527.662 −0.560982
\(97\) −724.108 −0.757959 −0.378979 0.925405i \(-0.623725\pi\)
−0.378979 + 0.925405i \(0.623725\pi\)
\(98\) 89.5929 0.0923495
\(99\) −581.117 −0.589944
\(100\) −116.421 −0.116421
\(101\) 268.725 0.264744 0.132372 0.991200i \(-0.457741\pi\)
0.132372 + 0.991200i \(0.457741\pi\)
\(102\) 309.058 0.300013
\(103\) −1840.63 −1.76080 −0.880399 0.474233i \(-0.842725\pi\)
−0.880399 + 0.474233i \(0.842725\pi\)
\(104\) 748.489 0.705725
\(105\) 105.000 0.0975900
\(106\) 477.184 0.437247
\(107\) −243.087 −0.219627 −0.109813 0.993952i \(-0.535025\pi\)
−0.109813 + 0.993952i \(0.535025\pi\)
\(108\) 125.735 0.112027
\(109\) −405.176 −0.356044 −0.178022 0.984027i \(-0.556970\pi\)
−0.178022 + 0.984027i \(0.556970\pi\)
\(110\) −590.294 −0.511658
\(111\) −362.029 −0.309570
\(112\) 35.4121 0.0298762
\(113\) −28.1766 −0.0234569 −0.0117285 0.999931i \(-0.503733\pi\)
−0.0117285 + 0.999931i \(0.503733\pi\)
\(114\) 15.0580 0.0123712
\(115\) 440.833 0.357460
\(116\) −1149.01 −0.919680
\(117\) −291.088 −0.230010
\(118\) 1150.08 0.897232
\(119\) 394.402 0.303822
\(120\) 347.132 0.264072
\(121\) 2838.10 2.13230
\(122\) −210.646 −0.156320
\(123\) 528.823 0.387661
\(124\) 516.500 0.374057
\(125\) 125.000 0.0894427
\(126\) −115.191 −0.0814446
\(127\) −2740.90 −1.91508 −0.957541 0.288298i \(-0.906911\pi\)
−0.957541 + 0.288298i \(0.906911\pi\)
\(128\) −745.083 −0.514505
\(129\) 1330.09 0.907811
\(130\) −295.685 −0.199487
\(131\) −1832.04 −1.22188 −0.610938 0.791678i \(-0.709208\pi\)
−0.610938 + 0.791678i \(0.709208\pi\)
\(132\) −902.059 −0.594804
\(133\) 19.2162 0.0125282
\(134\) −1739.71 −1.12155
\(135\) −135.000 −0.0860663
\(136\) 1303.90 0.822122
\(137\) 382.747 0.238688 0.119344 0.992853i \(-0.461921\pi\)
0.119344 + 0.992853i \(0.461921\pi\)
\(138\) −483.618 −0.298321
\(139\) 3053.60 1.86333 0.931667 0.363314i \(-0.118355\pi\)
0.931667 + 0.363314i \(0.118355\pi\)
\(140\) 162.990 0.0983940
\(141\) 1035.62 0.618545
\(142\) 651.403 0.384961
\(143\) 2088.35 1.22123
\(144\) −45.5299 −0.0263483
\(145\) 1233.68 0.706560
\(146\) −1200.83 −0.680692
\(147\) −147.000 −0.0824786
\(148\) −561.971 −0.312120
\(149\) 3560.60 1.95769 0.978843 0.204611i \(-0.0655929\pi\)
0.978843 + 0.204611i \(0.0655929\pi\)
\(150\) −137.132 −0.0746452
\(151\) 3261.80 1.75789 0.878945 0.476923i \(-0.158248\pi\)
0.878945 + 0.476923i \(0.158248\pi\)
\(152\) 63.5290 0.0339005
\(153\) −507.088 −0.267945
\(154\) 826.412 0.432430
\(155\) −554.558 −0.287376
\(156\) −451.852 −0.231905
\(157\) 2878.46 1.46322 0.731611 0.681723i \(-0.238769\pi\)
0.731611 + 0.681723i \(0.238769\pi\)
\(158\) 804.865 0.405263
\(159\) −782.942 −0.390512
\(160\) 879.437 0.434535
\(161\) −617.166 −0.302108
\(162\) 148.103 0.0718274
\(163\) −927.537 −0.445708 −0.222854 0.974852i \(-0.571537\pi\)
−0.222854 + 0.974852i \(0.571537\pi\)
\(164\) 820.883 0.390855
\(165\) 968.528 0.456969
\(166\) −99.4886 −0.0465169
\(167\) 1094.52 0.507164 0.253582 0.967314i \(-0.418391\pi\)
0.253582 + 0.967314i \(0.418391\pi\)
\(168\) −485.985 −0.223182
\(169\) −1150.92 −0.523860
\(170\) −515.097 −0.232389
\(171\) −24.7065 −0.0110488
\(172\) 2064.67 0.915290
\(173\) −1713.25 −0.752926 −0.376463 0.926432i \(-0.622860\pi\)
−0.376463 + 0.926432i \(0.622860\pi\)
\(174\) −1353.41 −0.589666
\(175\) −175.000 −0.0755929
\(176\) 326.644 0.139896
\(177\) −1887.00 −0.801330
\(178\) −1862.77 −0.784386
\(179\) −4065.58 −1.69763 −0.848816 0.528689i \(-0.822684\pi\)
−0.848816 + 0.528689i \(0.822684\pi\)
\(180\) −209.558 −0.0867754
\(181\) −2791.40 −1.14631 −0.573157 0.819445i \(-0.694282\pi\)
−0.573157 + 0.819445i \(0.694282\pi\)
\(182\) 413.960 0.168597
\(183\) 345.618 0.139611
\(184\) −2040.36 −0.817486
\(185\) 603.381 0.239792
\(186\) 608.382 0.239832
\(187\) 3637.99 1.42266
\(188\) 1607.57 0.623640
\(189\) 189.000 0.0727393
\(190\) −25.0967 −0.00958266
\(191\) −634.185 −0.240251 −0.120126 0.992759i \(-0.538330\pi\)
−0.120126 + 0.992759i \(0.538330\pi\)
\(192\) −1086.20 −0.408281
\(193\) −254.999 −0.0951049 −0.0475524 0.998869i \(-0.515142\pi\)
−0.0475524 + 0.998869i \(0.515142\pi\)
\(194\) −1323.98 −0.489980
\(195\) 485.147 0.178165
\(196\) −228.186 −0.0831581
\(197\) 4172.37 1.50898 0.754490 0.656311i \(-0.227884\pi\)
0.754490 + 0.656311i \(0.227884\pi\)
\(198\) −1062.53 −0.381367
\(199\) −4626.48 −1.64805 −0.824026 0.566552i \(-0.808277\pi\)
−0.824026 + 0.566552i \(0.808277\pi\)
\(200\) −578.553 −0.204550
\(201\) 2854.44 1.00168
\(202\) 491.344 0.171143
\(203\) −1727.15 −0.597152
\(204\) −787.145 −0.270153
\(205\) −881.371 −0.300281
\(206\) −3365.45 −1.13826
\(207\) 793.499 0.266435
\(208\) 163.620 0.0545433
\(209\) 177.251 0.0586638
\(210\) 191.985 0.0630867
\(211\) −1562.64 −0.509843 −0.254921 0.966962i \(-0.582050\pi\)
−0.254921 + 0.966962i \(0.582050\pi\)
\(212\) −1215.35 −0.393729
\(213\) −1068.79 −0.343814
\(214\) −444.466 −0.141977
\(215\) −2216.81 −0.703188
\(216\) 624.838 0.196828
\(217\) 776.382 0.242877
\(218\) −740.834 −0.230163
\(219\) 1970.26 0.607935
\(220\) 1503.43 0.460733
\(221\) 1822.31 0.554670
\(222\) −661.943 −0.200120
\(223\) −1236.39 −0.371278 −0.185639 0.982618i \(-0.559436\pi\)
−0.185639 + 0.982618i \(0.559436\pi\)
\(224\) −1231.21 −0.367249
\(225\) 225.000 0.0666667
\(226\) −51.5189 −0.0151637
\(227\) 4181.82 1.22272 0.611359 0.791353i \(-0.290623\pi\)
0.611359 + 0.791353i \(0.290623\pi\)
\(228\) −38.3515 −0.0111399
\(229\) −484.774 −0.139890 −0.0699449 0.997551i \(-0.522282\pi\)
−0.0699449 + 0.997551i \(0.522282\pi\)
\(230\) 806.030 0.231079
\(231\) −1355.94 −0.386209
\(232\) −5709.98 −1.61585
\(233\) 2080.54 0.584982 0.292491 0.956268i \(-0.405516\pi\)
0.292491 + 0.956268i \(0.405516\pi\)
\(234\) −532.234 −0.148689
\(235\) −1726.03 −0.479123
\(236\) −2929.16 −0.807932
\(237\) −1320.59 −0.361946
\(238\) 721.135 0.196404
\(239\) 6814.10 1.84422 0.922108 0.386933i \(-0.126466\pi\)
0.922108 + 0.386933i \(0.126466\pi\)
\(240\) 75.8831 0.0204093
\(241\) −3921.84 −1.04825 −0.524125 0.851642i \(-0.675607\pi\)
−0.524125 + 0.851642i \(0.675607\pi\)
\(242\) 5189.25 1.37842
\(243\) −243.000 −0.0641500
\(244\) 536.498 0.140761
\(245\) 245.000 0.0638877
\(246\) 966.913 0.250602
\(247\) 88.7873 0.0228721
\(248\) 2566.73 0.657209
\(249\) 163.236 0.0415449
\(250\) 228.553 0.0578199
\(251\) −5219.10 −1.31246 −0.656228 0.754562i \(-0.727849\pi\)
−0.656228 + 0.754562i \(0.727849\pi\)
\(252\) 293.382 0.0733386
\(253\) −5692.78 −1.41463
\(254\) −5011.53 −1.23800
\(255\) 845.147 0.207550
\(256\) −4258.88 −1.03976
\(257\) 6975.71 1.69312 0.846562 0.532289i \(-0.178668\pi\)
0.846562 + 0.532289i \(0.178668\pi\)
\(258\) 2431.97 0.586852
\(259\) −844.733 −0.202661
\(260\) 753.087 0.179632
\(261\) 2220.62 0.526639
\(262\) −3349.75 −0.789878
\(263\) −3607.36 −0.845776 −0.422888 0.906182i \(-0.638984\pi\)
−0.422888 + 0.906182i \(0.638984\pi\)
\(264\) −4482.76 −1.04506
\(265\) 1304.90 0.302489
\(266\) 35.1354 0.00809882
\(267\) 3056.35 0.700546
\(268\) 4430.90 1.00993
\(269\) 5.88572 0.00133405 0.000667023 1.00000i \(-0.499788\pi\)
0.000667023 1.00000i \(0.499788\pi\)
\(270\) −246.838 −0.0556373
\(271\) 6916.32 1.55032 0.775160 0.631765i \(-0.217669\pi\)
0.775160 + 0.631765i \(0.217669\pi\)
\(272\) 285.033 0.0635392
\(273\) −679.206 −0.150577
\(274\) 699.825 0.154299
\(275\) −1614.21 −0.353966
\(276\) 1231.74 0.268630
\(277\) −2119.46 −0.459733 −0.229867 0.973222i \(-0.573829\pi\)
−0.229867 + 0.973222i \(0.573829\pi\)
\(278\) 5583.29 1.20455
\(279\) −998.205 −0.214197
\(280\) 809.975 0.172876
\(281\) −239.917 −0.0509334 −0.0254667 0.999676i \(-0.508107\pi\)
−0.0254667 + 0.999676i \(0.508107\pi\)
\(282\) 1893.55 0.399856
\(283\) −4542.12 −0.954067 −0.477034 0.878885i \(-0.658288\pi\)
−0.477034 + 0.878885i \(0.658288\pi\)
\(284\) −1659.07 −0.346647
\(285\) 41.1775 0.00855840
\(286\) 3818.40 0.789463
\(287\) 1233.92 0.253784
\(288\) 1582.99 0.323883
\(289\) −1738.45 −0.353847
\(290\) 2255.69 0.456753
\(291\) 2172.32 0.437608
\(292\) 3058.41 0.612944
\(293\) −2171.70 −0.433010 −0.216505 0.976281i \(-0.569466\pi\)
−0.216505 + 0.976281i \(0.569466\pi\)
\(294\) −268.779 −0.0533180
\(295\) 3145.00 0.620708
\(296\) −2792.70 −0.548387
\(297\) 1743.35 0.340604
\(298\) 6510.29 1.26554
\(299\) −2851.58 −0.551543
\(300\) 349.264 0.0672159
\(301\) 3103.54 0.594302
\(302\) 5963.96 1.13638
\(303\) −806.175 −0.152850
\(304\) 13.8875 0.00262007
\(305\) −576.030 −0.108142
\(306\) −927.174 −0.173212
\(307\) −3508.64 −0.652276 −0.326138 0.945322i \(-0.605747\pi\)
−0.326138 + 0.945322i \(0.605747\pi\)
\(308\) −2104.80 −0.389391
\(309\) 5521.88 1.01660
\(310\) −1013.97 −0.185773
\(311\) −3133.25 −0.571287 −0.285643 0.958336i \(-0.592207\pi\)
−0.285643 + 0.958336i \(0.592207\pi\)
\(312\) −2245.47 −0.407451
\(313\) 6389.59 1.15387 0.576935 0.816790i \(-0.304249\pi\)
0.576935 + 0.816790i \(0.304249\pi\)
\(314\) 5263.05 0.945895
\(315\) −315.000 −0.0563436
\(316\) −2049.92 −0.364928
\(317\) 1634.44 0.289587 0.144794 0.989462i \(-0.453748\pi\)
0.144794 + 0.989462i \(0.453748\pi\)
\(318\) −1431.55 −0.252445
\(319\) −15931.3 −2.79618
\(320\) 1810.34 0.316253
\(321\) 729.260 0.126802
\(322\) −1128.44 −0.195297
\(323\) 154.671 0.0266444
\(324\) −377.205 −0.0646785
\(325\) −808.579 −0.138006
\(326\) −1695.93 −0.288126
\(327\) 1215.53 0.205562
\(328\) 4079.36 0.686723
\(329\) 2416.44 0.404932
\(330\) 1770.88 0.295406
\(331\) 4386.17 0.728355 0.364177 0.931330i \(-0.381350\pi\)
0.364177 + 0.931330i \(0.381350\pi\)
\(332\) 253.389 0.0418872
\(333\) 1086.09 0.178730
\(334\) 2001.25 0.327854
\(335\) −4757.40 −0.775894
\(336\) −106.236 −0.0172490
\(337\) 1713.98 0.277051 0.138526 0.990359i \(-0.455764\pi\)
0.138526 + 0.990359i \(0.455764\pi\)
\(338\) −2104.38 −0.338648
\(339\) 84.5299 0.0135429
\(340\) 1311.91 0.209260
\(341\) 7161.41 1.13728
\(342\) −45.1740 −0.00714249
\(343\) −343.000 −0.0539949
\(344\) 10260.4 1.60814
\(345\) −1322.50 −0.206379
\(346\) −3132.56 −0.486727
\(347\) −1744.83 −0.269935 −0.134967 0.990850i \(-0.543093\pi\)
−0.134967 + 0.990850i \(0.543093\pi\)
\(348\) 3447.03 0.530977
\(349\) 7046.78 1.08082 0.540409 0.841403i \(-0.318270\pi\)
0.540409 + 0.841403i \(0.318270\pi\)
\(350\) −319.975 −0.0488668
\(351\) 873.265 0.132796
\(352\) −11356.8 −1.71966
\(353\) −12668.5 −1.91013 −0.955064 0.296400i \(-0.904214\pi\)
−0.955064 + 0.296400i \(0.904214\pi\)
\(354\) −3450.24 −0.518017
\(355\) 1781.32 0.266317
\(356\) 4744.33 0.706317
\(357\) −1183.21 −0.175411
\(358\) −7433.62 −1.09743
\(359\) 37.7844 0.00555483 0.00277742 0.999996i \(-0.499116\pi\)
0.00277742 + 0.999996i \(0.499116\pi\)
\(360\) −1041.40 −0.152462
\(361\) −6851.46 −0.998901
\(362\) −5103.87 −0.741031
\(363\) −8514.29 −1.23109
\(364\) −1054.32 −0.151817
\(365\) −3283.77 −0.470905
\(366\) 631.938 0.0902511
\(367\) −759.829 −0.108073 −0.0540364 0.998539i \(-0.517209\pi\)
−0.0540364 + 0.998539i \(0.517209\pi\)
\(368\) −446.023 −0.0631809
\(369\) −1586.47 −0.223816
\(370\) 1103.24 0.155012
\(371\) −1826.86 −0.255650
\(372\) −1549.50 −0.215962
\(373\) 719.320 0.0998525 0.0499263 0.998753i \(-0.484101\pi\)
0.0499263 + 0.998753i \(0.484101\pi\)
\(374\) 6651.81 0.919671
\(375\) −375.000 −0.0516398
\(376\) 7988.81 1.09572
\(377\) −7980.19 −1.09019
\(378\) 345.573 0.0470221
\(379\) 572.559 0.0775999 0.0388000 0.999247i \(-0.487646\pi\)
0.0388000 + 0.999247i \(0.487646\pi\)
\(380\) 63.9192 0.00862891
\(381\) 8222.69 1.10567
\(382\) −1159.56 −0.155310
\(383\) 4513.18 0.602122 0.301061 0.953605i \(-0.402659\pi\)
0.301061 + 0.953605i \(0.402659\pi\)
\(384\) 2235.25 0.297050
\(385\) 2259.90 0.299156
\(386\) −466.247 −0.0614802
\(387\) −3990.26 −0.524125
\(388\) 3372.06 0.441213
\(389\) −6902.13 −0.899619 −0.449810 0.893124i \(-0.648508\pi\)
−0.449810 + 0.893124i \(0.648508\pi\)
\(390\) 887.056 0.115174
\(391\) −4967.58 −0.642510
\(392\) −1133.96 −0.146107
\(393\) 5496.11 0.705451
\(394\) 7628.88 0.975475
\(395\) 2200.98 0.280362
\(396\) 2706.18 0.343410
\(397\) 4124.58 0.521427 0.260714 0.965416i \(-0.416042\pi\)
0.260714 + 0.965416i \(0.416042\pi\)
\(398\) −8459.18 −1.06538
\(399\) −57.6485 −0.00723317
\(400\) −126.472 −0.0158090
\(401\) −1002.50 −0.124844 −0.0624219 0.998050i \(-0.519882\pi\)
−0.0624219 + 0.998050i \(0.519882\pi\)
\(402\) 5219.14 0.647530
\(403\) 3587.23 0.443406
\(404\) −1251.41 −0.154109
\(405\) 405.000 0.0496904
\(406\) −3157.96 −0.386027
\(407\) −7791.89 −0.948967
\(408\) −3911.70 −0.474652
\(409\) 10335.0 1.24947 0.624736 0.780836i \(-0.285206\pi\)
0.624736 + 0.780836i \(0.285206\pi\)
\(410\) −1611.52 −0.194116
\(411\) −1148.24 −0.137807
\(412\) 8571.53 1.02497
\(413\) −4402.99 −0.524594
\(414\) 1450.85 0.172236
\(415\) −272.061 −0.0321806
\(416\) −5688.75 −0.670466
\(417\) −9160.81 −1.07580
\(418\) 324.091 0.0379230
\(419\) 3183.21 0.371145 0.185573 0.982631i \(-0.440586\pi\)
0.185573 + 0.982631i \(0.440586\pi\)
\(420\) −488.970 −0.0568078
\(421\) −6944.34 −0.803911 −0.401956 0.915659i \(-0.631669\pi\)
−0.401956 + 0.915659i \(0.631669\pi\)
\(422\) −2857.18 −0.329586
\(423\) −3106.85 −0.357117
\(424\) −6039.65 −0.691772
\(425\) −1408.58 −0.160767
\(426\) −1954.21 −0.222258
\(427\) 806.442 0.0913969
\(428\) 1132.02 0.127846
\(429\) −6265.05 −0.705080
\(430\) −4053.28 −0.454573
\(431\) 3868.41 0.432331 0.216166 0.976357i \(-0.430645\pi\)
0.216166 + 0.976357i \(0.430645\pi\)
\(432\) 136.590 0.0152122
\(433\) −6132.96 −0.680673 −0.340336 0.940304i \(-0.610541\pi\)
−0.340336 + 0.940304i \(0.610541\pi\)
\(434\) 1419.56 0.157007
\(435\) −3701.03 −0.407932
\(436\) 1886.84 0.207256
\(437\) −242.032 −0.0264942
\(438\) 3602.48 0.392998
\(439\) −4090.14 −0.444673 −0.222337 0.974970i \(-0.571368\pi\)
−0.222337 + 0.974970i \(0.571368\pi\)
\(440\) 7471.27 0.809497
\(441\) 441.000 0.0476190
\(442\) 3331.97 0.358565
\(443\) 12434.5 1.33359 0.666795 0.745241i \(-0.267666\pi\)
0.666795 + 0.745241i \(0.267666\pi\)
\(444\) 1685.91 0.180203
\(445\) −5093.92 −0.542640
\(446\) −2260.66 −0.240012
\(447\) −10681.8 −1.13027
\(448\) −2534.48 −0.267283
\(449\) 883.046 0.0928141 0.0464071 0.998923i \(-0.485223\pi\)
0.0464071 + 0.998923i \(0.485223\pi\)
\(450\) 411.396 0.0430964
\(451\) 11381.8 1.18835
\(452\) 131.214 0.0136544
\(453\) −9785.40 −1.01492
\(454\) 7646.15 0.790422
\(455\) 1132.01 0.116636
\(456\) −190.587 −0.0195725
\(457\) −9068.44 −0.928235 −0.464118 0.885774i \(-0.653629\pi\)
−0.464118 + 0.885774i \(0.653629\pi\)
\(458\) −886.373 −0.0904312
\(459\) 1521.26 0.154698
\(460\) −2052.89 −0.208080
\(461\) 12508.9 1.26377 0.631885 0.775063i \(-0.282282\pi\)
0.631885 + 0.775063i \(0.282282\pi\)
\(462\) −2479.24 −0.249663
\(463\) 12688.7 1.27363 0.636817 0.771015i \(-0.280251\pi\)
0.636817 + 0.771015i \(0.280251\pi\)
\(464\) −1248.20 −0.124884
\(465\) 1663.68 0.165916
\(466\) 3804.12 0.378160
\(467\) −10136.5 −1.00442 −0.502208 0.864747i \(-0.667479\pi\)
−0.502208 + 0.864747i \(0.667479\pi\)
\(468\) 1355.56 0.133890
\(469\) 6660.36 0.655750
\(470\) −3155.92 −0.309727
\(471\) −8635.37 −0.844791
\(472\) −14556.4 −1.41952
\(473\) 28627.3 2.78284
\(474\) −2414.59 −0.233979
\(475\) −68.6292 −0.00662931
\(476\) −1836.67 −0.176857
\(477\) 2348.83 0.225462
\(478\) 12459.1 1.19219
\(479\) −11361.1 −1.08372 −0.541861 0.840468i \(-0.682280\pi\)
−0.541861 + 0.840468i \(0.682280\pi\)
\(480\) −2638.31 −0.250879
\(481\) −3903.05 −0.369987
\(482\) −7170.80 −0.677637
\(483\) 1851.50 0.174422
\(484\) −13216.6 −1.24123
\(485\) −3620.54 −0.338969
\(486\) −444.308 −0.0414696
\(487\) 7929.53 0.737826 0.368913 0.929464i \(-0.379730\pi\)
0.368913 + 0.929464i \(0.379730\pi\)
\(488\) 2666.11 0.247314
\(489\) 2782.61 0.257329
\(490\) 447.965 0.0413000
\(491\) 8111.51 0.745555 0.372777 0.927921i \(-0.378405\pi\)
0.372777 + 0.927921i \(0.378405\pi\)
\(492\) −2462.65 −0.225660
\(493\) −13901.8 −1.26999
\(494\) 162.341 0.0147856
\(495\) −2905.58 −0.263831
\(496\) 561.088 0.0507936
\(497\) −2493.85 −0.225079
\(498\) 298.466 0.0268566
\(499\) 16816.6 1.50865 0.754324 0.656502i \(-0.227965\pi\)
0.754324 + 0.656502i \(0.227965\pi\)
\(500\) −582.107 −0.0520652
\(501\) −3283.55 −0.292811
\(502\) −9542.74 −0.848433
\(503\) −17764.6 −1.57472 −0.787362 0.616491i \(-0.788554\pi\)
−0.787362 + 0.616491i \(0.788554\pi\)
\(504\) 1457.95 0.128854
\(505\) 1343.62 0.118397
\(506\) −10408.8 −0.914485
\(507\) 3452.76 0.302451
\(508\) 12764.0 1.11478
\(509\) 13908.8 1.21120 0.605598 0.795771i \(-0.292934\pi\)
0.605598 + 0.795771i \(0.292934\pi\)
\(510\) 1545.29 0.134170
\(511\) 4597.27 0.397987
\(512\) −1826.38 −0.157647
\(513\) 74.1195 0.00637905
\(514\) 12754.6 1.09451
\(515\) −9203.13 −0.787453
\(516\) −6194.02 −0.528443
\(517\) 22289.5 1.89611
\(518\) −1544.53 −0.131009
\(519\) 5139.76 0.434702
\(520\) 3742.45 0.315610
\(521\) −8639.68 −0.726510 −0.363255 0.931690i \(-0.618335\pi\)
−0.363255 + 0.931690i \(0.618335\pi\)
\(522\) 4060.23 0.340444
\(523\) 23242.2 1.94323 0.971617 0.236561i \(-0.0760203\pi\)
0.971617 + 0.236561i \(0.0760203\pi\)
\(524\) 8531.53 0.711263
\(525\) 525.000 0.0436436
\(526\) −6595.79 −0.546749
\(527\) 6249.11 0.516538
\(528\) −979.932 −0.0807691
\(529\) −4393.66 −0.361113
\(530\) 2385.92 0.195543
\(531\) 5660.99 0.462648
\(532\) −89.4869 −0.00729276
\(533\) 5701.26 0.463319
\(534\) 5588.32 0.452865
\(535\) −1215.43 −0.0982201
\(536\) 22019.3 1.77442
\(537\) 12196.8 0.980128
\(538\) 10.7616 0.000862390 0
\(539\) −3163.86 −0.252833
\(540\) 628.675 0.0500998
\(541\) 11395.2 0.905577 0.452789 0.891618i \(-0.350429\pi\)
0.452789 + 0.891618i \(0.350429\pi\)
\(542\) 12646.0 1.00220
\(543\) 8374.19 0.661825
\(544\) −9910.04 −0.781047
\(545\) −2025.88 −0.159228
\(546\) −1241.88 −0.0973398
\(547\) −7870.21 −0.615184 −0.307592 0.951518i \(-0.599523\pi\)
−0.307592 + 0.951518i \(0.599523\pi\)
\(548\) −1782.40 −0.138942
\(549\) −1036.85 −0.0806045
\(550\) −2951.47 −0.228820
\(551\) −677.329 −0.0523687
\(552\) 6121.08 0.471976
\(553\) −3081.37 −0.236949
\(554\) −3875.28 −0.297193
\(555\) −1810.14 −0.138444
\(556\) −14220.2 −1.08466
\(557\) 17769.8 1.35176 0.675880 0.737012i \(-0.263764\pi\)
0.675880 + 0.737012i \(0.263764\pi\)
\(558\) −1825.15 −0.138467
\(559\) 14339.7 1.08498
\(560\) 177.061 0.0133610
\(561\) −10914.0 −0.821370
\(562\) −438.672 −0.0329257
\(563\) 15192.8 1.13730 0.568651 0.822579i \(-0.307466\pi\)
0.568651 + 0.822579i \(0.307466\pi\)
\(564\) −4822.72 −0.360059
\(565\) −140.883 −0.0104903
\(566\) −8304.93 −0.616753
\(567\) −567.000 −0.0419961
\(568\) −8244.71 −0.609050
\(569\) −23300.0 −1.71667 −0.858335 0.513090i \(-0.828501\pi\)
−0.858335 + 0.513090i \(0.828501\pi\)
\(570\) 75.2900 0.00553255
\(571\) 10638.2 0.779673 0.389837 0.920884i \(-0.372531\pi\)
0.389837 + 0.920884i \(0.372531\pi\)
\(572\) −9725.14 −0.710889
\(573\) 1902.55 0.138709
\(574\) 2256.13 0.164058
\(575\) 2204.16 0.159861
\(576\) 3258.61 0.235721
\(577\) −897.258 −0.0647372 −0.0323686 0.999476i \(-0.510305\pi\)
−0.0323686 + 0.999476i \(0.510305\pi\)
\(578\) −3178.63 −0.228743
\(579\) 764.997 0.0549088
\(580\) −5745.05 −0.411293
\(581\) 380.885 0.0271975
\(582\) 3971.93 0.282890
\(583\) −16851.1 −1.19709
\(584\) 15198.7 1.07693
\(585\) −1455.44 −0.102863
\(586\) −3970.79 −0.279918
\(587\) −14712.9 −1.03452 −0.517261 0.855828i \(-0.673048\pi\)
−0.517261 + 0.855828i \(0.673048\pi\)
\(588\) 684.558 0.0480114
\(589\) 304.471 0.0212997
\(590\) 5750.40 0.401254
\(591\) −12517.1 −0.871210
\(592\) −610.486 −0.0423831
\(593\) −7216.29 −0.499726 −0.249863 0.968281i \(-0.580386\pi\)
−0.249863 + 0.968281i \(0.580386\pi\)
\(594\) 3187.59 0.220182
\(595\) 1972.01 0.135873
\(596\) −16581.2 −1.13958
\(597\) 13879.4 0.951503
\(598\) −5213.91 −0.356543
\(599\) −20885.5 −1.42464 −0.712320 0.701855i \(-0.752355\pi\)
−0.712320 + 0.701855i \(0.752355\pi\)
\(600\) 1735.66 0.118097
\(601\) −11047.7 −0.749823 −0.374911 0.927061i \(-0.622327\pi\)
−0.374911 + 0.927061i \(0.622327\pi\)
\(602\) 5674.59 0.384185
\(603\) −8563.32 −0.578317
\(604\) −15189.7 −1.02328
\(605\) 14190.5 0.953595
\(606\) −1474.03 −0.0988093
\(607\) −9434.94 −0.630894 −0.315447 0.948943i \(-0.602154\pi\)
−0.315447 + 0.948943i \(0.602154\pi\)
\(608\) −482.840 −0.0322068
\(609\) 5181.44 0.344766
\(610\) −1053.23 −0.0699082
\(611\) 11165.0 0.739263
\(612\) 2361.44 0.155973
\(613\) −17662.6 −1.16376 −0.581881 0.813274i \(-0.697683\pi\)
−0.581881 + 0.813274i \(0.697683\pi\)
\(614\) −6415.30 −0.421662
\(615\) 2644.11 0.173367
\(616\) −10459.8 −0.684150
\(617\) −10817.4 −0.705820 −0.352910 0.935657i \(-0.614808\pi\)
−0.352910 + 0.935657i \(0.614808\pi\)
\(618\) 10096.3 0.657176
\(619\) 29073.9 1.88785 0.943926 0.330158i \(-0.107102\pi\)
0.943926 + 0.330158i \(0.107102\pi\)
\(620\) 2582.50 0.167283
\(621\) −2380.50 −0.153826
\(622\) −5728.92 −0.369306
\(623\) 7131.49 0.458615
\(624\) −490.860 −0.0314906
\(625\) 625.000 0.0400000
\(626\) 11682.9 0.745915
\(627\) −531.754 −0.0338696
\(628\) −13404.5 −0.851751
\(629\) −6799.28 −0.431009
\(630\) −575.955 −0.0364231
\(631\) −2203.17 −0.138996 −0.0694981 0.997582i \(-0.522140\pi\)
−0.0694981 + 0.997582i \(0.522140\pi\)
\(632\) −10187.1 −0.641170
\(633\) 4687.93 0.294358
\(634\) 2988.45 0.187203
\(635\) −13704.5 −0.856451
\(636\) 3646.05 0.227319
\(637\) −1584.81 −0.0985755
\(638\) −29129.3 −1.80758
\(639\) 3206.38 0.198501
\(640\) −3725.42 −0.230094
\(641\) 22466.5 1.38436 0.692180 0.721725i \(-0.256651\pi\)
0.692180 + 0.721725i \(0.256651\pi\)
\(642\) 1333.40 0.0819705
\(643\) −12347.0 −0.757257 −0.378629 0.925549i \(-0.623604\pi\)
−0.378629 + 0.925549i \(0.623604\pi\)
\(644\) 2874.05 0.175859
\(645\) 6650.44 0.405986
\(646\) 282.805 0.0172242
\(647\) −24114.0 −1.46525 −0.732626 0.680631i \(-0.761706\pi\)
−0.732626 + 0.680631i \(0.761706\pi\)
\(648\) −1874.51 −0.113639
\(649\) −40613.6 −2.45643
\(650\) −1478.43 −0.0892134
\(651\) −2329.15 −0.140225
\(652\) 4319.41 0.259449
\(653\) 7843.33 0.470035 0.235018 0.971991i \(-0.424485\pi\)
0.235018 + 0.971991i \(0.424485\pi\)
\(654\) 2222.50 0.132885
\(655\) −9160.18 −0.546440
\(656\) 891.749 0.0530746
\(657\) −5910.78 −0.350992
\(658\) 4418.29 0.261767
\(659\) 21242.8 1.25569 0.627846 0.778338i \(-0.283937\pi\)
0.627846 + 0.778338i \(0.283937\pi\)
\(660\) −4510.29 −0.266005
\(661\) −22221.7 −1.30760 −0.653801 0.756667i \(-0.726827\pi\)
−0.653801 + 0.756667i \(0.726827\pi\)
\(662\) 8019.79 0.470843
\(663\) −5466.94 −0.320239
\(664\) 1259.21 0.0735948
\(665\) 96.0808 0.00560279
\(666\) 1985.83 0.115539
\(667\) 21753.8 1.26283
\(668\) −5097.01 −0.295223
\(669\) 3709.18 0.214358
\(670\) −8698.56 −0.501574
\(671\) 7438.69 0.427969
\(672\) 3693.63 0.212031
\(673\) −3787.85 −0.216955 −0.108478 0.994099i \(-0.534598\pi\)
−0.108478 + 0.994099i \(0.534598\pi\)
\(674\) 3133.88 0.179099
\(675\) −675.000 −0.0384900
\(676\) 5359.67 0.304943
\(677\) −11296.8 −0.641314 −0.320657 0.947195i \(-0.603904\pi\)
−0.320657 + 0.947195i \(0.603904\pi\)
\(678\) 154.557 0.00875474
\(679\) 5068.75 0.286481
\(680\) 6519.50 0.367664
\(681\) −12545.5 −0.705937
\(682\) 13094.1 0.735190
\(683\) 4807.14 0.269312 0.134656 0.990892i \(-0.457007\pi\)
0.134656 + 0.990892i \(0.457007\pi\)
\(684\) 115.055 0.00643161
\(685\) 1913.73 0.106745
\(686\) −627.151 −0.0349048
\(687\) 1454.32 0.0807654
\(688\) 2242.92 0.124288
\(689\) −8440.94 −0.466726
\(690\) −2418.09 −0.133413
\(691\) 5393.47 0.296928 0.148464 0.988918i \(-0.452567\pi\)
0.148464 + 0.988918i \(0.452567\pi\)
\(692\) 7978.37 0.438283
\(693\) 4067.82 0.222978
\(694\) −3190.29 −0.174498
\(695\) 15268.0 0.833308
\(696\) 17129.9 0.932914
\(697\) 9931.84 0.539735
\(698\) 12884.5 0.698691
\(699\) −6241.63 −0.337740
\(700\) 814.949 0.0440031
\(701\) 2404.77 0.129568 0.0647838 0.997899i \(-0.479364\pi\)
0.0647838 + 0.997899i \(0.479364\pi\)
\(702\) 1596.70 0.0858456
\(703\) −331.276 −0.0177729
\(704\) −23378.2 −1.25156
\(705\) 5178.09 0.276622
\(706\) −23163.4 −1.23480
\(707\) −1881.07 −0.100064
\(708\) 8787.47 0.466460
\(709\) −21617.3 −1.14507 −0.572535 0.819881i \(-0.694040\pi\)
−0.572535 + 0.819881i \(0.694040\pi\)
\(710\) 3257.01 0.172160
\(711\) 3961.76 0.208970
\(712\) 23576.8 1.24098
\(713\) −9778.70 −0.513626
\(714\) −2163.41 −0.113394
\(715\) 10441.7 0.546153
\(716\) 18932.8 0.988203
\(717\) −20442.3 −1.06476
\(718\) 69.0860 0.00359090
\(719\) −18228.6 −0.945498 −0.472749 0.881197i \(-0.656738\pi\)
−0.472749 + 0.881197i \(0.656738\pi\)
\(720\) −227.649 −0.0117833
\(721\) 12884.4 0.665519
\(722\) −12527.4 −0.645736
\(723\) 11765.5 0.605207
\(724\) 12999.1 0.667278
\(725\) 6168.38 0.315983
\(726\) −15567.8 −0.795832
\(727\) 20196.5 1.03033 0.515164 0.857092i \(-0.327731\pi\)
0.515164 + 0.857092i \(0.327731\pi\)
\(728\) −5239.43 −0.266739
\(729\) 729.000 0.0370370
\(730\) −6004.13 −0.304415
\(731\) 24980.4 1.26393
\(732\) −1609.49 −0.0812686
\(733\) −15264.9 −0.769196 −0.384598 0.923084i \(-0.625660\pi\)
−0.384598 + 0.923084i \(0.625660\pi\)
\(734\) −1389.29 −0.0698633
\(735\) −735.000 −0.0368856
\(736\) 15507.4 0.776643
\(737\) 61435.7 3.07057
\(738\) −2900.74 −0.144685
\(739\) 13906.0 0.692207 0.346103 0.938196i \(-0.387505\pi\)
0.346103 + 0.938196i \(0.387505\pi\)
\(740\) −2809.86 −0.139584
\(741\) −266.362 −0.0132052
\(742\) −3340.29 −0.165264
\(743\) 4592.87 0.226778 0.113389 0.993551i \(-0.463829\pi\)
0.113389 + 0.993551i \(0.463829\pi\)
\(744\) −7700.20 −0.379440
\(745\) 17803.0 0.875504
\(746\) 1315.22 0.0645493
\(747\) −489.709 −0.0239860
\(748\) −16941.6 −0.828137
\(749\) 1701.61 0.0830111
\(750\) −685.660 −0.0333824
\(751\) −8390.80 −0.407702 −0.203851 0.979002i \(-0.565346\pi\)
−0.203851 + 0.979002i \(0.565346\pi\)
\(752\) 1746.35 0.0846848
\(753\) 15657.3 0.757747
\(754\) −14591.2 −0.704748
\(755\) 16309.0 0.786153
\(756\) −880.145 −0.0423420
\(757\) −1368.89 −0.0657240 −0.0328620 0.999460i \(-0.510462\pi\)
−0.0328620 + 0.999460i \(0.510462\pi\)
\(758\) 1046.88 0.0501642
\(759\) 17078.4 0.816739
\(760\) 317.645 0.0151608
\(761\) −1623.77 −0.0773478 −0.0386739 0.999252i \(-0.512313\pi\)
−0.0386739 + 0.999252i \(0.512313\pi\)
\(762\) 15034.6 0.714759
\(763\) 2836.23 0.134572
\(764\) 2953.31 0.139852
\(765\) −2535.44 −0.119829
\(766\) 8252.03 0.389240
\(767\) −20343.8 −0.957722
\(768\) 12776.6 0.600308
\(769\) −26842.5 −1.25873 −0.629366 0.777109i \(-0.716685\pi\)
−0.629366 + 0.777109i \(0.716685\pi\)
\(770\) 4132.06 0.193388
\(771\) −20927.1 −0.977526
\(772\) 1187.49 0.0553612
\(773\) −20961.4 −0.975330 −0.487665 0.873031i \(-0.662151\pi\)
−0.487665 + 0.873031i \(0.662151\pi\)
\(774\) −7295.90 −0.338819
\(775\) −2772.79 −0.128518
\(776\) 16757.4 0.775200
\(777\) 2534.20 0.117006
\(778\) −12620.0 −0.581556
\(779\) 483.902 0.0222562
\(780\) −2259.26 −0.103711
\(781\) −23003.5 −1.05394
\(782\) −9082.86 −0.415348
\(783\) −6661.85 −0.304055
\(784\) −247.885 −0.0112921
\(785\) 14392.3 0.654373
\(786\) 10049.2 0.456036
\(787\) 35333.2 1.60037 0.800187 0.599751i \(-0.204734\pi\)
0.800187 + 0.599751i \(0.204734\pi\)
\(788\) −19430.1 −0.878388
\(789\) 10822.1 0.488309
\(790\) 4024.32 0.181239
\(791\) 197.236 0.00886589
\(792\) 13448.3 0.603364
\(793\) 3726.13 0.166858
\(794\) 7541.49 0.337075
\(795\) −3914.71 −0.174642
\(796\) 21544.8 0.959342
\(797\) −8137.04 −0.361642 −0.180821 0.983516i \(-0.557875\pi\)
−0.180821 + 0.983516i \(0.557875\pi\)
\(798\) −105.406 −0.00467586
\(799\) 19450.0 0.861191
\(800\) 4397.18 0.194330
\(801\) −9169.05 −0.404460
\(802\) −1832.99 −0.0807049
\(803\) 42405.6 1.86359
\(804\) −13292.7 −0.583082
\(805\) −3085.83 −0.135107
\(806\) 6558.99 0.286639
\(807\) −17.6572 −0.000770212 0
\(808\) −6218.87 −0.270766
\(809\) −36281.0 −1.57673 −0.788364 0.615209i \(-0.789072\pi\)
−0.788364 + 0.615209i \(0.789072\pi\)
\(810\) 740.513 0.0321222
\(811\) −34237.2 −1.48240 −0.741202 0.671282i \(-0.765744\pi\)
−0.741202 + 0.671282i \(0.765744\pi\)
\(812\) 8043.06 0.347606
\(813\) −20749.0 −0.895077
\(814\) −14246.9 −0.613456
\(815\) −4637.69 −0.199326
\(816\) −855.099 −0.0366844
\(817\) 1217.10 0.0521188
\(818\) 18896.8 0.807717
\(819\) 2037.62 0.0869355
\(820\) 4104.42 0.174796
\(821\) −23247.8 −0.988250 −0.494125 0.869391i \(-0.664511\pi\)
−0.494125 + 0.869391i \(0.664511\pi\)
\(822\) −2099.47 −0.0890846
\(823\) 42934.0 1.81845 0.909225 0.416306i \(-0.136675\pi\)
0.909225 + 0.416306i \(0.136675\pi\)
\(824\) 42596.0 1.80085
\(825\) 4842.64 0.204363
\(826\) −8050.55 −0.339122
\(827\) −781.391 −0.0328557 −0.0164278 0.999865i \(-0.505229\pi\)
−0.0164278 + 0.999865i \(0.505229\pi\)
\(828\) −3695.21 −0.155093
\(829\) −33493.3 −1.40322 −0.701611 0.712561i \(-0.747535\pi\)
−0.701611 + 0.712561i \(0.747535\pi\)
\(830\) −497.443 −0.0208030
\(831\) 6358.39 0.265427
\(832\) −11710.4 −0.487964
\(833\) −2760.81 −0.114834
\(834\) −16749.9 −0.695445
\(835\) 5472.59 0.226811
\(836\) −825.434 −0.0341486
\(837\) 2994.62 0.123667
\(838\) 5820.27 0.239926
\(839\) 15155.7 0.623639 0.311819 0.950141i \(-0.399062\pi\)
0.311819 + 0.950141i \(0.399062\pi\)
\(840\) −2429.92 −0.0998099
\(841\) 36489.2 1.49613
\(842\) −12697.2 −0.519686
\(843\) 719.752 0.0294064
\(844\) 7277.01 0.296783
\(845\) −5754.60 −0.234277
\(846\) −5680.66 −0.230857
\(847\) −19866.7 −0.805935
\(848\) −1320.27 −0.0534649
\(849\) 13626.4 0.550831
\(850\) −2575.48 −0.103927
\(851\) 10639.6 0.428579
\(852\) 4977.21 0.200137
\(853\) −2917.48 −0.117107 −0.0585537 0.998284i \(-0.518649\pi\)
−0.0585537 + 0.998284i \(0.518649\pi\)
\(854\) 1474.52 0.0590832
\(855\) −123.532 −0.00494119
\(856\) 5625.54 0.224623
\(857\) −31560.7 −1.25799 −0.628993 0.777411i \(-0.716532\pi\)
−0.628993 + 0.777411i \(0.716532\pi\)
\(858\) −11455.2 −0.455797
\(859\) −1404.81 −0.0557991 −0.0278995 0.999611i \(-0.508882\pi\)
−0.0278995 + 0.999611i \(0.508882\pi\)
\(860\) 10323.4 0.409330
\(861\) −3701.76 −0.146522
\(862\) 7073.11 0.279479
\(863\) −9808.24 −0.386879 −0.193439 0.981112i \(-0.561964\pi\)
−0.193439 + 0.981112i \(0.561964\pi\)
\(864\) −4748.96 −0.186994
\(865\) −8566.27 −0.336719
\(866\) −11213.7 −0.440018
\(867\) 5215.35 0.204294
\(868\) −3615.50 −0.141380
\(869\) −28422.8 −1.10952
\(870\) −6767.06 −0.263707
\(871\) 30773.9 1.19717
\(872\) 9376.63 0.364143
\(873\) −6516.97 −0.252653
\(874\) −442.537 −0.0171271
\(875\) −875.000 −0.0338062
\(876\) −9175.22 −0.353883
\(877\) −7196.05 −0.277073 −0.138537 0.990357i \(-0.544240\pi\)
−0.138537 + 0.990357i \(0.544240\pi\)
\(878\) −7478.52 −0.287458
\(879\) 6515.10 0.249999
\(880\) 1633.22 0.0625635
\(881\) −3183.27 −0.121733 −0.0608667 0.998146i \(-0.519386\pi\)
−0.0608667 + 0.998146i \(0.519386\pi\)
\(882\) 806.336 0.0307832
\(883\) 25392.5 0.967751 0.483876 0.875137i \(-0.339229\pi\)
0.483876 + 0.875137i \(0.339229\pi\)
\(884\) −8486.25 −0.322877
\(885\) −9434.99 −0.358366
\(886\) 22735.6 0.862095
\(887\) −30634.2 −1.15964 −0.579818 0.814746i \(-0.696876\pi\)
−0.579818 + 0.814746i \(0.696876\pi\)
\(888\) 8378.11 0.316612
\(889\) 19186.3 0.723833
\(890\) −9313.86 −0.350788
\(891\) −5230.05 −0.196648
\(892\) 5757.71 0.216124
\(893\) 947.648 0.0355116
\(894\) −19530.9 −0.730660
\(895\) −20327.9 −0.759204
\(896\) 5215.58 0.194465
\(897\) 8554.75 0.318433
\(898\) 1614.59 0.0599994
\(899\) −27365.8 −1.01524
\(900\) −1047.79 −0.0388071
\(901\) −14704.5 −0.543704
\(902\) 20810.7 0.768206
\(903\) −9310.61 −0.343120
\(904\) 652.067 0.0239905
\(905\) −13957.0 −0.512648
\(906\) −17891.9 −0.656091
\(907\) −28089.9 −1.02834 −0.514172 0.857687i \(-0.671901\pi\)
−0.514172 + 0.857687i \(0.671901\pi\)
\(908\) −19474.1 −0.711753
\(909\) 2418.52 0.0882480
\(910\) 2069.80 0.0753990
\(911\) −36102.7 −1.31299 −0.656495 0.754330i \(-0.727962\pi\)
−0.656495 + 0.754330i \(0.727962\pi\)
\(912\) −41.6624 −0.00151270
\(913\) 3513.31 0.127353
\(914\) −16581.0 −0.600055
\(915\) 1728.09 0.0624360
\(916\) 2257.52 0.0814308
\(917\) 12824.3 0.461826
\(918\) 2781.52 0.100004
\(919\) −14533.8 −0.521682 −0.260841 0.965382i \(-0.584000\pi\)
−0.260841 + 0.965382i \(0.584000\pi\)
\(920\) −10201.8 −0.365591
\(921\) 10525.9 0.376592
\(922\) 22871.6 0.816959
\(923\) −11522.7 −0.410915
\(924\) 6314.41 0.224815
\(925\) 3016.90 0.107238
\(926\) 23200.3 0.823336
\(927\) −16565.6 −0.586933
\(928\) 43397.6 1.53512
\(929\) 16539.6 0.584118 0.292059 0.956400i \(-0.405660\pi\)
0.292059 + 0.956400i \(0.405660\pi\)
\(930\) 3041.91 0.107256
\(931\) −134.513 −0.00473522
\(932\) −9688.79 −0.340522
\(933\) 9399.74 0.329833
\(934\) −18533.9 −0.649301
\(935\) 18190.0 0.636231
\(936\) 6736.41 0.235242
\(937\) 30212.3 1.05335 0.526677 0.850065i \(-0.323438\pi\)
0.526677 + 0.850065i \(0.323438\pi\)
\(938\) 12178.0 0.423908
\(939\) −19168.8 −0.666187
\(940\) 8037.87 0.278900
\(941\) −26414.4 −0.915074 −0.457537 0.889191i \(-0.651268\pi\)
−0.457537 + 0.889191i \(0.651268\pi\)
\(942\) −15789.1 −0.546112
\(943\) −15541.5 −0.536692
\(944\) −3182.03 −0.109710
\(945\) 945.000 0.0325300
\(946\) 52342.9 1.79896
\(947\) 10187.3 0.349570 0.174785 0.984607i \(-0.444077\pi\)
0.174785 + 0.984607i \(0.444077\pi\)
\(948\) 6149.77 0.210691
\(949\) 21241.5 0.726583
\(950\) −125.483 −0.00428549
\(951\) −4903.31 −0.167193
\(952\) −9127.31 −0.310733
\(953\) 2211.39 0.0751669 0.0375834 0.999293i \(-0.488034\pi\)
0.0375834 + 0.999293i \(0.488034\pi\)
\(954\) 4294.66 0.145749
\(955\) −3170.92 −0.107444
\(956\) −31732.3 −1.07353
\(957\) 47794.0 1.61438
\(958\) −20773.0 −0.700569
\(959\) −2679.23 −0.0902156
\(960\) −5431.02 −0.182589
\(961\) −17489.6 −0.587077
\(962\) −7136.44 −0.239177
\(963\) −2187.78 −0.0732089
\(964\) 18263.4 0.610193
\(965\) −1275.00 −0.0425322
\(966\) 3385.33 0.112755
\(967\) 7955.89 0.264575 0.132287 0.991211i \(-0.457768\pi\)
0.132287 + 0.991211i \(0.457768\pi\)
\(968\) −65679.6 −2.18081
\(969\) −464.014 −0.0153832
\(970\) −6619.89 −0.219126
\(971\) 53071.2 1.75400 0.877001 0.480488i \(-0.159541\pi\)
0.877001 + 0.480488i \(0.159541\pi\)
\(972\) 1131.62 0.0373422
\(973\) −21375.2 −0.704274
\(974\) 14498.6 0.476965
\(975\) 2425.74 0.0796777
\(976\) 582.813 0.0191141
\(977\) −22448.2 −0.735089 −0.367545 0.930006i \(-0.619801\pi\)
−0.367545 + 0.930006i \(0.619801\pi\)
\(978\) 5087.80 0.166350
\(979\) 65781.4 2.14748
\(980\) −1140.93 −0.0371894
\(981\) −3646.58 −0.118681
\(982\) 14831.3 0.481961
\(983\) 21712.8 0.704509 0.352254 0.935904i \(-0.385415\pi\)
0.352254 + 0.935904i \(0.385415\pi\)
\(984\) −12238.1 −0.396479
\(985\) 20861.9 0.674837
\(986\) −25418.5 −0.820983
\(987\) −7249.33 −0.233788
\(988\) −413.470 −0.0133140
\(989\) −39089.7 −1.25681
\(990\) −5312.65 −0.170553
\(991\) −37849.2 −1.21324 −0.606620 0.794992i \(-0.707475\pi\)
−0.606620 + 0.794992i \(0.707475\pi\)
\(992\) −19508.0 −0.624373
\(993\) −13158.5 −0.420516
\(994\) −4559.82 −0.145502
\(995\) −23132.4 −0.737031
\(996\) −760.168 −0.0241836
\(997\) −39573.1 −1.25707 −0.628533 0.777783i \(-0.716344\pi\)
−0.628533 + 0.777783i \(0.716344\pi\)
\(998\) 30748.0 0.975261
\(999\) −3258.26 −0.103190
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 105.4.a.e.1.2 2
3.2 odd 2 315.4.a.k.1.1 2
4.3 odd 2 1680.4.a.bo.1.2 2
5.2 odd 4 525.4.d.l.274.3 4
5.3 odd 4 525.4.d.l.274.2 4
5.4 even 2 525.4.a.l.1.1 2
7.6 odd 2 735.4.a.o.1.2 2
15.14 odd 2 1575.4.a.q.1.2 2
21.20 even 2 2205.4.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.e.1.2 2 1.1 even 1 trivial
315.4.a.k.1.1 2 3.2 odd 2
525.4.a.l.1.1 2 5.4 even 2
525.4.d.l.274.2 4 5.3 odd 4
525.4.d.l.274.3 4 5.2 odd 4
735.4.a.o.1.2 2 7.6 odd 2
1575.4.a.q.1.2 2 15.14 odd 2
1680.4.a.bo.1.2 2 4.3 odd 2
2205.4.a.bb.1.1 2 21.20 even 2