Properties

Label 105.4.a.e.1.1
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: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
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 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 105.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.82843 q^{2} -3.00000 q^{3} +6.65685 q^{4} +5.00000 q^{5} +11.4853 q^{6} -7.00000 q^{7} +5.14214 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.82843 q^{2} -3.00000 q^{3} +6.65685 q^{4} +5.00000 q^{5} +11.4853 q^{6} -7.00000 q^{7} +5.14214 q^{8} +9.00000 q^{9} -19.1421 q^{10} +48.5685 q^{11} -19.9706 q^{12} -43.6569 q^{13} +26.7990 q^{14} -15.0000 q^{15} -72.9411 q^{16} -67.6569 q^{17} -34.4558 q^{18} -93.2548 q^{19} +33.2843 q^{20} +21.0000 q^{21} -185.941 q^{22} -104.167 q^{23} -15.4264 q^{24} +25.0000 q^{25} +167.137 q^{26} -27.0000 q^{27} -46.5980 q^{28} -58.7351 q^{29} +57.4264 q^{30} -9.08831 q^{31} +238.113 q^{32} -145.706 q^{33} +259.019 q^{34} -35.0000 q^{35} +59.9117 q^{36} -252.676 q^{37} +357.019 q^{38} +130.971 q^{39} +25.7107 q^{40} +276.274 q^{41} -80.3970 q^{42} -92.6375 q^{43} +323.314 q^{44} +45.0000 q^{45} +398.794 q^{46} -582.794 q^{47} +218.823 q^{48} +49.0000 q^{49} -95.7107 q^{50} +202.971 q^{51} -290.617 q^{52} +623.019 q^{53} +103.368 q^{54} +242.843 q^{55} -35.9949 q^{56} +279.765 q^{57} +224.863 q^{58} -524.999 q^{59} -99.8528 q^{60} -352.794 q^{61} +34.7939 q^{62} -63.0000 q^{63} -328.068 q^{64} -218.284 q^{65} +557.823 q^{66} -736.520 q^{67} -450.382 q^{68} +312.500 q^{69} +133.995 q^{70} -492.264 q^{71} +46.2792 q^{72} +1164.75 q^{73} +967.352 q^{74} -75.0000 q^{75} -620.784 q^{76} -339.980 q^{77} -501.411 q^{78} -872.195 q^{79} -364.706 q^{80} +81.0000 q^{81} -1057.70 q^{82} -529.588 q^{83} +139.794 q^{84} -338.284 q^{85} +354.656 q^{86} +176.205 q^{87} +249.746 q^{88} -385.216 q^{89} -172.279 q^{90} +305.598 q^{91} -693.421 q^{92} +27.2649 q^{93} +2231.18 q^{94} -466.274 q^{95} -714.338 q^{96} -463.892 q^{97} -187.593 q^{98} +437.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 6q^{3} + 2q^{4} + 10q^{5} + 6q^{6} - 14q^{7} - 18q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 6q^{3} + 2q^{4} + 10q^{5} + 6q^{6} - 14q^{7} - 18q^{8} + 18q^{9} - 10q^{10} - 16q^{11} - 6q^{12} - 76q^{13} + 14q^{14} - 30q^{15} - 78q^{16} - 124q^{17} - 18q^{18} - 96q^{19} + 10q^{20} + 42q^{21} - 304q^{22} - 16q^{23} + 54q^{24} + 50q^{25} + 108q^{26} - 54q^{27} - 14q^{28} + 188q^{29} + 30q^{30} - 120q^{31} + 414q^{32} + 48q^{33} + 156q^{34} - 70q^{35} + 18q^{36} - 132q^{37} + 352q^{38} + 228q^{39} - 90q^{40} + 100q^{41} - 42q^{42} - 536q^{43} + 624q^{44} + 90q^{45} + 560q^{46} - 928q^{47} + 234q^{48} + 98q^{49} - 50q^{50} + 372q^{51} - 140q^{52} + 884q^{53} + 54q^{54} - 80q^{55} + 126q^{56} + 288q^{57} + 676q^{58} + 104q^{59} - 30q^{60} - 468q^{61} - 168q^{62} - 126q^{63} + 34q^{64} - 380q^{65} + 912q^{66} - 1688q^{67} - 188q^{68} + 48q^{69} + 70q^{70} - 136q^{71} - 162q^{72} + 508q^{73} + 1188q^{74} - 150q^{75} - 608q^{76} + 112q^{77} - 324q^{78} - 432q^{79} - 390q^{80} + 162q^{81} - 1380q^{82} - 584q^{83} + 42q^{84} - 620q^{85} - 456q^{86} - 564q^{87} + 1744q^{88} - 1404q^{89} - 90q^{90} + 532q^{91} - 1104q^{92} + 360q^{93} + 1600q^{94} - 480q^{95} - 1242q^{96} - 1188q^{97} - 98q^{98} - 144q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.82843 −1.35355 −0.676777 0.736188i \(-0.736624\pi\)
−0.676777 + 0.736188i \(0.736624\pi\)
\(3\) −3.00000 −0.577350
\(4\) 6.65685 0.832107
\(5\) 5.00000 0.447214
\(6\) 11.4853 0.781474
\(7\) −7.00000 −0.377964
\(8\) 5.14214 0.227252
\(9\) 9.00000 0.333333
\(10\) −19.1421 −0.605327
\(11\) 48.5685 1.33127 0.665635 0.746278i \(-0.268161\pi\)
0.665635 + 0.746278i \(0.268161\pi\)
\(12\) −19.9706 −0.480417
\(13\) −43.6569 −0.931403 −0.465701 0.884942i \(-0.654198\pi\)
−0.465701 + 0.884942i \(0.654198\pi\)
\(14\) 26.7990 0.511595
\(15\) −15.0000 −0.258199
\(16\) −72.9411 −1.13971
\(17\) −67.6569 −0.965247 −0.482623 0.875828i \(-0.660316\pi\)
−0.482623 + 0.875828i \(0.660316\pi\)
\(18\) −34.4558 −0.451184
\(19\) −93.2548 −1.12601 −0.563003 0.826455i \(-0.690354\pi\)
−0.563003 + 0.826455i \(0.690354\pi\)
\(20\) 33.2843 0.372129
\(21\) 21.0000 0.218218
\(22\) −185.941 −1.80194
\(23\) −104.167 −0.944357 −0.472179 0.881503i \(-0.656532\pi\)
−0.472179 + 0.881503i \(0.656532\pi\)
\(24\) −15.4264 −0.131204
\(25\) 25.0000 0.200000
\(26\) 167.137 1.26070
\(27\) −27.0000 −0.192450
\(28\) −46.5980 −0.314507
\(29\) −58.7351 −0.376098 −0.188049 0.982160i \(-0.560216\pi\)
−0.188049 + 0.982160i \(0.560216\pi\)
\(30\) 57.4264 0.349486
\(31\) −9.08831 −0.0526551 −0.0263276 0.999653i \(-0.508381\pi\)
−0.0263276 + 0.999653i \(0.508381\pi\)
\(32\) 238.113 1.31540
\(33\) −145.706 −0.768609
\(34\) 259.019 1.30651
\(35\) −35.0000 −0.169031
\(36\) 59.9117 0.277369
\(37\) −252.676 −1.12269 −0.561347 0.827580i \(-0.689717\pi\)
−0.561347 + 0.827580i \(0.689717\pi\)
\(38\) 357.019 1.52411
\(39\) 130.971 0.537745
\(40\) 25.7107 0.101630
\(41\) 276.274 1.05236 0.526180 0.850373i \(-0.323624\pi\)
0.526180 + 0.850373i \(0.323624\pi\)
\(42\) −80.3970 −0.295370
\(43\) −92.6375 −0.328537 −0.164268 0.986416i \(-0.552526\pi\)
−0.164268 + 0.986416i \(0.552526\pi\)
\(44\) 323.314 1.10776
\(45\) 45.0000 0.149071
\(46\) 398.794 1.27824
\(47\) −582.794 −1.80871 −0.904354 0.426784i \(-0.859646\pi\)
−0.904354 + 0.426784i \(0.859646\pi\)
\(48\) 218.823 0.658009
\(49\) 49.0000 0.142857
\(50\) −95.7107 −0.270711
\(51\) 202.971 0.557286
\(52\) −290.617 −0.775026
\(53\) 623.019 1.61468 0.807342 0.590083i \(-0.200905\pi\)
0.807342 + 0.590083i \(0.200905\pi\)
\(54\) 103.368 0.260491
\(55\) 242.843 0.595362
\(56\) −35.9949 −0.0858933
\(57\) 279.765 0.650100
\(58\) 224.863 0.509068
\(59\) −524.999 −1.15846 −0.579229 0.815165i \(-0.696646\pi\)
−0.579229 + 0.815165i \(0.696646\pi\)
\(60\) −99.8528 −0.214849
\(61\) −352.794 −0.740502 −0.370251 0.928932i \(-0.620728\pi\)
−0.370251 + 0.928932i \(0.620728\pi\)
\(62\) 34.7939 0.0712715
\(63\) −63.0000 −0.125988
\(64\) −328.068 −0.640758
\(65\) −218.284 −0.416536
\(66\) 557.823 1.04035
\(67\) −736.520 −1.34299 −0.671494 0.741010i \(-0.734347\pi\)
−0.671494 + 0.741010i \(0.734347\pi\)
\(68\) −450.382 −0.803188
\(69\) 312.500 0.545225
\(70\) 133.995 0.228792
\(71\) −492.264 −0.822831 −0.411415 0.911448i \(-0.634965\pi\)
−0.411415 + 0.911448i \(0.634965\pi\)
\(72\) 46.2792 0.0757508
\(73\) 1164.75 1.86745 0.933727 0.357987i \(-0.116537\pi\)
0.933727 + 0.357987i \(0.116537\pi\)
\(74\) 967.352 1.51963
\(75\) −75.0000 −0.115470
\(76\) −620.784 −0.936958
\(77\) −339.980 −0.503173
\(78\) −501.411 −0.727867
\(79\) −872.195 −1.24215 −0.621074 0.783752i \(-0.713303\pi\)
−0.621074 + 0.783752i \(0.713303\pi\)
\(80\) −364.706 −0.509692
\(81\) 81.0000 0.111111
\(82\) −1057.70 −1.42443
\(83\) −529.588 −0.700359 −0.350180 0.936683i \(-0.613879\pi\)
−0.350180 + 0.936683i \(0.613879\pi\)
\(84\) 139.794 0.181581
\(85\) −338.284 −0.431672
\(86\) 354.656 0.444692
\(87\) 176.205 0.217140
\(88\) 249.746 0.302534
\(89\) −385.216 −0.458796 −0.229398 0.973333i \(-0.573676\pi\)
−0.229398 + 0.973333i \(0.573676\pi\)
\(90\) −172.279 −0.201776
\(91\) 305.598 0.352037
\(92\) −693.421 −0.785806
\(93\) 27.2649 0.0304005
\(94\) 2231.18 2.44818
\(95\) −466.274 −0.503565
\(96\) −714.338 −0.759446
\(97\) −463.892 −0.485579 −0.242789 0.970079i \(-0.578062\pi\)
−0.242789 + 0.970079i \(0.578062\pi\)
\(98\) −187.593 −0.193365
\(99\) 437.117 0.443757
\(100\) 166.421 0.166421
\(101\) −432.725 −0.426314 −0.213157 0.977018i \(-0.568375\pi\)
−0.213157 + 0.977018i \(0.568375\pi\)
\(102\) −777.058 −0.754316
\(103\) 512.626 0.490393 0.245197 0.969473i \(-0.421147\pi\)
0.245197 + 0.969473i \(0.421147\pi\)
\(104\) −224.489 −0.211663
\(105\) 105.000 0.0975900
\(106\) −2385.18 −2.18556
\(107\) 1963.09 1.77363 0.886817 0.462122i \(-0.152912\pi\)
0.886817 + 0.462122i \(0.152912\pi\)
\(108\) −179.735 −0.160139
\(109\) 545.176 0.479068 0.239534 0.970888i \(-0.423005\pi\)
0.239534 + 0.970888i \(0.423005\pi\)
\(110\) −929.706 −0.805854
\(111\) 758.029 0.648188
\(112\) 510.588 0.430768
\(113\) −231.823 −0.192992 −0.0964961 0.995333i \(-0.530764\pi\)
−0.0964961 + 0.995333i \(0.530764\pi\)
\(114\) −1071.06 −0.879945
\(115\) −520.833 −0.422329
\(116\) −390.991 −0.312953
\(117\) −392.912 −0.310468
\(118\) 2009.92 1.56804
\(119\) 473.598 0.364829
\(120\) −77.1320 −0.0586763
\(121\) 1027.90 0.772279
\(122\) 1350.65 1.00231
\(123\) −828.823 −0.607581
\(124\) −60.4996 −0.0438147
\(125\) 125.000 0.0894427
\(126\) 241.191 0.170532
\(127\) 2372.90 1.65796 0.828979 0.559280i \(-0.188922\pi\)
0.828979 + 0.559280i \(0.188922\pi\)
\(128\) −648.917 −0.448099
\(129\) 277.913 0.189681
\(130\) 835.685 0.563804
\(131\) 1200.04 0.800364 0.400182 0.916436i \(-0.368947\pi\)
0.400182 + 0.916436i \(0.368947\pi\)
\(132\) −969.941 −0.639565
\(133\) 652.784 0.425591
\(134\) 2819.71 1.81781
\(135\) −135.000 −0.0860663
\(136\) −347.901 −0.219355
\(137\) 2781.25 1.73444 0.867221 0.497924i \(-0.165904\pi\)
0.867221 + 0.497924i \(0.165904\pi\)
\(138\) −1196.38 −0.737991
\(139\) −1245.60 −0.760078 −0.380039 0.924971i \(-0.624089\pi\)
−0.380039 + 0.924971i \(0.624089\pi\)
\(140\) −232.990 −0.140652
\(141\) 1748.38 1.04426
\(142\) 1884.60 1.11375
\(143\) −2120.35 −1.23995
\(144\) −656.470 −0.379902
\(145\) −293.675 −0.168196
\(146\) −4459.17 −2.52770
\(147\) −147.000 −0.0824786
\(148\) −1682.03 −0.934202
\(149\) 19.4046 0.0106690 0.00533452 0.999986i \(-0.498302\pi\)
0.00533452 + 0.999986i \(0.498302\pi\)
\(150\) 287.132 0.156295
\(151\) −2349.80 −1.26638 −0.633192 0.773995i \(-0.718256\pi\)
−0.633192 + 0.773995i \(0.718256\pi\)
\(152\) −479.529 −0.255888
\(153\) −608.912 −0.321749
\(154\) 1301.59 0.681071
\(155\) −45.4416 −0.0235481
\(156\) 871.852 0.447462
\(157\) −3898.46 −1.98172 −0.990862 0.134880i \(-0.956935\pi\)
−0.990862 + 0.134880i \(0.956935\pi\)
\(158\) 3339.14 1.68131
\(159\) −1869.06 −0.932239
\(160\) 1190.56 0.588264
\(161\) 729.166 0.356934
\(162\) −310.103 −0.150395
\(163\) 1527.54 0.734024 0.367012 0.930216i \(-0.380381\pi\)
0.367012 + 0.930216i \(0.380381\pi\)
\(164\) 1839.12 0.875676
\(165\) −728.528 −0.343732
\(166\) 2027.49 0.947974
\(167\) −998.518 −0.462681 −0.231340 0.972873i \(-0.574311\pi\)
−0.231340 + 0.972873i \(0.574311\pi\)
\(168\) 107.985 0.0495905
\(169\) −291.079 −0.132489
\(170\) 1295.10 0.584290
\(171\) −839.294 −0.375336
\(172\) −616.674 −0.273378
\(173\) 685.253 0.301149 0.150575 0.988599i \(-0.451888\pi\)
0.150575 + 0.988599i \(0.451888\pi\)
\(174\) −674.589 −0.293911
\(175\) −175.000 −0.0755929
\(176\) −3542.64 −1.51725
\(177\) 1575.00 0.668836
\(178\) 1474.77 0.621005
\(179\) 1025.58 0.428245 0.214122 0.976807i \(-0.431311\pi\)
0.214122 + 0.976807i \(0.431311\pi\)
\(180\) 299.558 0.124043
\(181\) 2899.40 1.19067 0.595333 0.803479i \(-0.297020\pi\)
0.595333 + 0.803479i \(0.297020\pi\)
\(182\) −1169.96 −0.476501
\(183\) 1058.38 0.427529
\(184\) −535.638 −0.214608
\(185\) −1263.38 −0.502084
\(186\) −104.382 −0.0411486
\(187\) −3285.99 −1.28500
\(188\) −3879.57 −1.50504
\(189\) 189.000 0.0727393
\(190\) 1785.10 0.681603
\(191\) 1074.18 0.406939 0.203469 0.979081i \(-0.434778\pi\)
0.203469 + 0.979081i \(0.434778\pi\)
\(192\) 984.204 0.369942
\(193\) 898.999 0.335292 0.167646 0.985847i \(-0.446383\pi\)
0.167646 + 0.985847i \(0.446383\pi\)
\(194\) 1775.98 0.657257
\(195\) 654.853 0.240487
\(196\) 326.186 0.118872
\(197\) 3063.63 1.10799 0.553996 0.832519i \(-0.313102\pi\)
0.553996 + 0.832519i \(0.313102\pi\)
\(198\) −1673.47 −0.600648
\(199\) −949.522 −0.338240 −0.169120 0.985595i \(-0.554093\pi\)
−0.169120 + 0.985595i \(0.554093\pi\)
\(200\) 128.553 0.0454505
\(201\) 2209.56 0.775375
\(202\) 1656.66 0.577039
\(203\) 411.145 0.142151
\(204\) 1351.15 0.463721
\(205\) 1381.37 0.470630
\(206\) −1962.55 −0.663774
\(207\) −937.499 −0.314786
\(208\) 3184.38 1.06152
\(209\) −4529.25 −1.49902
\(210\) −401.985 −0.132093
\(211\) 2306.64 0.752587 0.376294 0.926500i \(-0.377198\pi\)
0.376294 + 0.926500i \(0.377198\pi\)
\(212\) 4147.35 1.34359
\(213\) 1476.79 0.475062
\(214\) −7515.53 −2.40071
\(215\) −463.188 −0.146926
\(216\) −138.838 −0.0437348
\(217\) 63.6182 0.0199018
\(218\) −2087.17 −0.648444
\(219\) −3494.26 −1.07817
\(220\) 1616.57 0.495405
\(221\) 2953.69 0.899033
\(222\) −2902.06 −0.877357
\(223\) −3227.61 −0.969222 −0.484611 0.874730i \(-0.661039\pi\)
−0.484611 + 0.874730i \(0.661039\pi\)
\(224\) −1666.79 −0.497174
\(225\) 225.000 0.0666667
\(226\) 887.519 0.261225
\(227\) −637.820 −0.186492 −0.0932458 0.995643i \(-0.529724\pi\)
−0.0932458 + 0.995643i \(0.529724\pi\)
\(228\) 1862.35 0.540953
\(229\) 544.774 0.157204 0.0786019 0.996906i \(-0.474954\pi\)
0.0786019 + 0.996906i \(0.474954\pi\)
\(230\) 1993.97 0.571646
\(231\) 1019.94 0.290507
\(232\) −302.024 −0.0854691
\(233\) −5748.54 −1.61631 −0.808154 0.588972i \(-0.799533\pi\)
−0.808154 + 0.588972i \(0.799533\pi\)
\(234\) 1504.23 0.420234
\(235\) −2913.97 −0.808878
\(236\) −3494.84 −0.963961
\(237\) 2616.59 0.717154
\(238\) −1813.14 −0.493816
\(239\) −2678.10 −0.724820 −0.362410 0.932019i \(-0.618046\pi\)
−0.362410 + 0.932019i \(0.618046\pi\)
\(240\) 1094.12 0.294271
\(241\) −2202.16 −0.588604 −0.294302 0.955713i \(-0.595087\pi\)
−0.294302 + 0.955713i \(0.595087\pi\)
\(242\) −3935.25 −1.04532
\(243\) −243.000 −0.0641500
\(244\) −2348.50 −0.616177
\(245\) 245.000 0.0638877
\(246\) 3173.09 0.822393
\(247\) 4071.21 1.04877
\(248\) −46.7333 −0.0119660
\(249\) 1588.76 0.404353
\(250\) −478.553 −0.121065
\(251\) −5716.90 −1.43764 −0.718820 0.695196i \(-0.755317\pi\)
−0.718820 + 0.695196i \(0.755317\pi\)
\(252\) −419.382 −0.104836
\(253\) −5059.22 −1.25719
\(254\) −9084.47 −2.24413
\(255\) 1014.85 0.249226
\(256\) 5108.88 1.24728
\(257\) 4724.29 1.14666 0.573332 0.819323i \(-0.305650\pi\)
0.573332 + 0.819323i \(0.305650\pi\)
\(258\) −1063.97 −0.256743
\(259\) 1768.73 0.424339
\(260\) −1453.09 −0.346602
\(261\) −528.616 −0.125366
\(262\) −4594.25 −1.08334
\(263\) 5975.36 1.40097 0.700487 0.713665i \(-0.252966\pi\)
0.700487 + 0.713665i \(0.252966\pi\)
\(264\) −749.238 −0.174668
\(265\) 3115.10 0.722109
\(266\) −2499.14 −0.576059
\(267\) 1155.65 0.264886
\(268\) −4902.90 −1.11751
\(269\) 4486.11 1.01681 0.508407 0.861117i \(-0.330234\pi\)
0.508407 + 0.861117i \(0.330234\pi\)
\(270\) 516.838 0.116495
\(271\) 3827.68 0.857989 0.428994 0.903307i \(-0.358868\pi\)
0.428994 + 0.903307i \(0.358868\pi\)
\(272\) 4934.97 1.10010
\(273\) −916.794 −0.203249
\(274\) −10647.8 −2.34766
\(275\) 1214.21 0.266254
\(276\) 2080.26 0.453685
\(277\) −3420.54 −0.741950 −0.370975 0.928643i \(-0.620976\pi\)
−0.370975 + 0.928643i \(0.620976\pi\)
\(278\) 4768.71 1.02881
\(279\) −81.7948 −0.0175517
\(280\) −179.975 −0.0384127
\(281\) 5235.92 1.11156 0.555781 0.831329i \(-0.312419\pi\)
0.555781 + 0.831329i \(0.312419\pi\)
\(282\) −6693.55 −1.41346
\(283\) −6985.88 −1.46738 −0.733688 0.679486i \(-0.762203\pi\)
−0.733688 + 0.679486i \(0.762203\pi\)
\(284\) −3276.93 −0.684683
\(285\) 1398.82 0.290734
\(286\) 8117.60 1.67834
\(287\) −1933.92 −0.397755
\(288\) 2143.01 0.438466
\(289\) −335.550 −0.0682984
\(290\) 1124.31 0.227662
\(291\) 1391.68 0.280349
\(292\) 7753.59 1.55392
\(293\) 7399.70 1.47541 0.737705 0.675123i \(-0.235910\pi\)
0.737705 + 0.675123i \(0.235910\pi\)
\(294\) 562.779 0.111639
\(295\) −2625.00 −0.518078
\(296\) −1299.30 −0.255135
\(297\) −1311.35 −0.256203
\(298\) −74.2892 −0.0144411
\(299\) 4547.58 0.879577
\(300\) −499.264 −0.0960834
\(301\) 648.463 0.124175
\(302\) 8996.04 1.71412
\(303\) 1298.17 0.246133
\(304\) 6802.11 1.28332
\(305\) −1763.97 −0.331163
\(306\) 2331.17 0.435504
\(307\) 2668.64 0.496116 0.248058 0.968745i \(-0.420208\pi\)
0.248058 + 0.968745i \(0.420208\pi\)
\(308\) −2263.20 −0.418693
\(309\) −1537.88 −0.283129
\(310\) 173.970 0.0318736
\(311\) 6189.25 1.12849 0.564244 0.825608i \(-0.309168\pi\)
0.564244 + 0.825608i \(0.309168\pi\)
\(312\) 673.468 0.122204
\(313\) −2921.59 −0.527598 −0.263799 0.964578i \(-0.584976\pi\)
−0.263799 + 0.964578i \(0.584976\pi\)
\(314\) 14925.0 2.68237
\(315\) −315.000 −0.0563436
\(316\) −5806.08 −1.03360
\(317\) 9825.56 1.74088 0.870439 0.492276i \(-0.163835\pi\)
0.870439 + 0.492276i \(0.163835\pi\)
\(318\) 7155.55 1.26183
\(319\) −2852.68 −0.500687
\(320\) −1640.34 −0.286556
\(321\) −5889.26 −1.02401
\(322\) −2791.56 −0.483129
\(323\) 6309.33 1.08687
\(324\) 539.205 0.0924563
\(325\) −1091.42 −0.186281
\(326\) −5848.07 −0.993541
\(327\) −1635.53 −0.276590
\(328\) 1420.64 0.239151
\(329\) 4079.56 0.683627
\(330\) 2789.12 0.465260
\(331\) −9258.17 −1.53739 −0.768693 0.639618i \(-0.779093\pi\)
−0.768693 + 0.639618i \(0.779093\pi\)
\(332\) −3525.39 −0.582774
\(333\) −2274.09 −0.374232
\(334\) 3822.75 0.626263
\(335\) −3682.60 −0.600603
\(336\) −1531.76 −0.248704
\(337\) −3693.98 −0.597103 −0.298552 0.954394i \(-0.596503\pi\)
−0.298552 + 0.954394i \(0.596503\pi\)
\(338\) 1114.38 0.179331
\(339\) 695.470 0.111424
\(340\) −2251.91 −0.359197
\(341\) −441.406 −0.0700982
\(342\) 3213.17 0.508037
\(343\) −343.000 −0.0539949
\(344\) −476.355 −0.0746608
\(345\) 1562.50 0.243832
\(346\) −2623.44 −0.407622
\(347\) 3832.83 0.592960 0.296480 0.955039i \(-0.404187\pi\)
0.296480 + 0.955039i \(0.404187\pi\)
\(348\) 1172.97 0.180684
\(349\) 8325.22 1.27690 0.638451 0.769662i \(-0.279575\pi\)
0.638451 + 0.769662i \(0.279575\pi\)
\(350\) 669.975 0.102319
\(351\) 1178.74 0.179248
\(352\) 11564.8 1.75115
\(353\) −8991.52 −1.35572 −0.677862 0.735189i \(-0.737093\pi\)
−0.677862 + 0.735189i \(0.737093\pi\)
\(354\) −6029.76 −0.905306
\(355\) −2461.32 −0.367981
\(356\) −2564.33 −0.381767
\(357\) −1420.79 −0.210634
\(358\) −3926.38 −0.579652
\(359\) −12893.8 −1.89557 −0.947783 0.318917i \(-0.896681\pi\)
−0.947783 + 0.318917i \(0.896681\pi\)
\(360\) 231.396 0.0338768
\(361\) 1837.46 0.267891
\(362\) −11100.1 −1.61163
\(363\) −3083.71 −0.445875
\(364\) 2034.32 0.292932
\(365\) 5823.77 0.835150
\(366\) −4051.94 −0.578684
\(367\) −7480.17 −1.06393 −0.531964 0.846767i \(-0.678546\pi\)
−0.531964 + 0.846767i \(0.678546\pi\)
\(368\) 7598.02 1.07629
\(369\) 2486.47 0.350787
\(370\) 4836.76 0.679598
\(371\) −4361.14 −0.610293
\(372\) 181.499 0.0252964
\(373\) −3523.32 −0.489090 −0.244545 0.969638i \(-0.578639\pi\)
−0.244545 + 0.969638i \(0.578639\pi\)
\(374\) 12580.2 1.73932
\(375\) −375.000 −0.0516398
\(376\) −2996.81 −0.411033
\(377\) 2564.19 0.350298
\(378\) −723.573 −0.0984565
\(379\) 13515.4 1.83177 0.915886 0.401438i \(-0.131490\pi\)
0.915886 + 0.401438i \(0.131490\pi\)
\(380\) −3103.92 −0.419020
\(381\) −7118.69 −0.957222
\(382\) −4112.44 −0.550813
\(383\) −657.182 −0.0876774 −0.0438387 0.999039i \(-0.513959\pi\)
−0.0438387 + 0.999039i \(0.513959\pi\)
\(384\) 1946.75 0.258710
\(385\) −1699.90 −0.225026
\(386\) −3441.75 −0.453836
\(387\) −833.738 −0.109512
\(388\) −3088.06 −0.404053
\(389\) −9741.87 −1.26975 −0.634875 0.772615i \(-0.718948\pi\)
−0.634875 + 0.772615i \(0.718948\pi\)
\(390\) −2507.06 −0.325512
\(391\) 7047.58 0.911538
\(392\) 251.965 0.0324646
\(393\) −3600.11 −0.462091
\(394\) −11728.9 −1.49973
\(395\) −4360.98 −0.555505
\(396\) 2909.82 0.369253
\(397\) 4407.42 0.557184 0.278592 0.960410i \(-0.410132\pi\)
0.278592 + 0.960410i \(0.410132\pi\)
\(398\) 3635.18 0.457827
\(399\) −1958.35 −0.245715
\(400\) −1823.53 −0.227941
\(401\) −11569.5 −1.44078 −0.720391 0.693568i \(-0.756037\pi\)
−0.720391 + 0.693568i \(0.756037\pi\)
\(402\) −8459.14 −1.04951
\(403\) 396.767 0.0490431
\(404\) −2880.59 −0.354739
\(405\) 405.000 0.0496904
\(406\) −1574.04 −0.192410
\(407\) −12272.1 −1.49461
\(408\) 1043.70 0.126645
\(409\) −3083.03 −0.372729 −0.186364 0.982481i \(-0.559670\pi\)
−0.186364 + 0.982481i \(0.559670\pi\)
\(410\) −5288.48 −0.637023
\(411\) −8343.76 −1.00138
\(412\) 3412.47 0.408060
\(413\) 3674.99 0.437856
\(414\) 3589.15 0.426079
\(415\) −2647.94 −0.313210
\(416\) −10395.3 −1.22517
\(417\) 3736.81 0.438831
\(418\) 17339.9 2.02900
\(419\) −5415.21 −0.631385 −0.315692 0.948862i \(-0.602237\pi\)
−0.315692 + 0.948862i \(0.602237\pi\)
\(420\) 698.970 0.0812053
\(421\) 4188.34 0.484863 0.242432 0.970168i \(-0.422055\pi\)
0.242432 + 0.970168i \(0.422055\pi\)
\(422\) −8830.82 −1.01867
\(423\) −5245.15 −0.602902
\(424\) 3203.65 0.366941
\(425\) −1691.42 −0.193049
\(426\) −5653.79 −0.643021
\(427\) 2469.56 0.279884
\(428\) 13068.0 1.47585
\(429\) 6361.05 0.715884
\(430\) 1773.28 0.198872
\(431\) −9108.41 −1.01795 −0.508975 0.860781i \(-0.669976\pi\)
−0.508975 + 0.860781i \(0.669976\pi\)
\(432\) 1969.41 0.219336
\(433\) −16847.0 −1.86979 −0.934893 0.354930i \(-0.884505\pi\)
−0.934893 + 0.354930i \(0.884505\pi\)
\(434\) −243.558 −0.0269381
\(435\) 881.026 0.0971080
\(436\) 3629.16 0.398635
\(437\) 9714.03 1.06335
\(438\) 13377.5 1.45937
\(439\) 8434.14 0.916946 0.458473 0.888708i \(-0.348397\pi\)
0.458473 + 0.888708i \(0.348397\pi\)
\(440\) 1248.73 0.135297
\(441\) 441.000 0.0476190
\(442\) −11308.0 −1.21689
\(443\) −4298.49 −0.461010 −0.230505 0.973071i \(-0.574038\pi\)
−0.230505 + 0.973071i \(0.574038\pi\)
\(444\) 5046.09 0.539362
\(445\) −1926.08 −0.205180
\(446\) 12356.7 1.31189
\(447\) −58.2139 −0.00615978
\(448\) 2296.48 0.242184
\(449\) 10545.0 1.10835 0.554173 0.832402i \(-0.313035\pi\)
0.554173 + 0.832402i \(0.313035\pi\)
\(450\) −861.396 −0.0902369
\(451\) 13418.2 1.40098
\(452\) −1543.21 −0.160590
\(453\) 7049.40 0.731147
\(454\) 2441.85 0.252426
\(455\) 1527.99 0.157436
\(456\) 1438.59 0.147737
\(457\) 11952.4 1.22344 0.611719 0.791075i \(-0.290478\pi\)
0.611719 + 0.791075i \(0.290478\pi\)
\(458\) −2085.63 −0.212784
\(459\) 1826.74 0.185762
\(460\) −3467.11 −0.351423
\(461\) −17200.9 −1.73780 −0.868900 0.494988i \(-0.835173\pi\)
−0.868900 + 0.494988i \(0.835173\pi\)
\(462\) −3904.76 −0.393217
\(463\) −10368.7 −1.04076 −0.520381 0.853934i \(-0.674210\pi\)
−0.520381 + 0.853934i \(0.674210\pi\)
\(464\) 4284.20 0.428640
\(465\) 136.325 0.0135955
\(466\) 22007.9 2.18776
\(467\) −16879.5 −1.67257 −0.836284 0.548296i \(-0.815277\pi\)
−0.836284 + 0.548296i \(0.815277\pi\)
\(468\) −2615.56 −0.258342
\(469\) 5155.64 0.507602
\(470\) 11155.9 1.09486
\(471\) 11695.4 1.14415
\(472\) −2699.62 −0.263263
\(473\) −4499.27 −0.437371
\(474\) −10017.4 −0.970706
\(475\) −2331.37 −0.225201
\(476\) 3152.67 0.303577
\(477\) 5607.17 0.538228
\(478\) 10252.9 0.981082
\(479\) 7329.12 0.699115 0.349558 0.936915i \(-0.386332\pi\)
0.349558 + 0.936915i \(0.386332\pi\)
\(480\) −3571.69 −0.339635
\(481\) 11031.0 1.04568
\(482\) 8430.80 0.796706
\(483\) −2187.50 −0.206076
\(484\) 6842.60 0.642619
\(485\) −2319.46 −0.217157
\(486\) 930.308 0.0868305
\(487\) −17209.5 −1.60131 −0.800655 0.599125i \(-0.795515\pi\)
−0.800655 + 0.599125i \(0.795515\pi\)
\(488\) −1814.11 −0.168281
\(489\) −4582.61 −0.423789
\(490\) −937.965 −0.0864754
\(491\) 11392.5 1.04712 0.523560 0.851989i \(-0.324604\pi\)
0.523560 + 0.851989i \(0.324604\pi\)
\(492\) −5517.35 −0.505572
\(493\) 3973.83 0.363027
\(494\) −15586.3 −1.41956
\(495\) 2185.58 0.198454
\(496\) 662.912 0.0600113
\(497\) 3445.85 0.311001
\(498\) −6082.47 −0.547313
\(499\) 19079.4 1.71164 0.855822 0.517271i \(-0.173052\pi\)
0.855822 + 0.517271i \(0.173052\pi\)
\(500\) 832.107 0.0744259
\(501\) 2995.55 0.267129
\(502\) 21886.7 1.94592
\(503\) −13499.4 −1.19663 −0.598317 0.801259i \(-0.704164\pi\)
−0.598317 + 0.801259i \(0.704164\pi\)
\(504\) −323.955 −0.0286311
\(505\) −2163.62 −0.190654
\(506\) 19368.8 1.70168
\(507\) 873.237 0.0764928
\(508\) 15796.0 1.37960
\(509\) −4328.85 −0.376960 −0.188480 0.982077i \(-0.560356\pi\)
−0.188480 + 0.982077i \(0.560356\pi\)
\(510\) −3885.29 −0.337340
\(511\) −8153.27 −0.705831
\(512\) −14367.6 −1.24017
\(513\) 2517.88 0.216700
\(514\) −18086.6 −1.55207
\(515\) 2563.13 0.219311
\(516\) 1850.02 0.157835
\(517\) −28305.5 −2.40788
\(518\) −6771.47 −0.574365
\(519\) −2055.76 −0.173869
\(520\) −1122.45 −0.0946588
\(521\) 19395.7 1.63098 0.815490 0.578771i \(-0.196468\pi\)
0.815490 + 0.578771i \(0.196468\pi\)
\(522\) 2023.77 0.169689
\(523\) 20413.8 1.70675 0.853377 0.521294i \(-0.174550\pi\)
0.853377 + 0.521294i \(0.174550\pi\)
\(524\) 7988.47 0.665989
\(525\) 525.000 0.0436436
\(526\) −22876.2 −1.89629
\(527\) 614.887 0.0508252
\(528\) 10627.9 0.875987
\(529\) −1316.34 −0.108189
\(530\) −11925.9 −0.977413
\(531\) −4724.99 −0.386153
\(532\) 4345.49 0.354137
\(533\) −12061.3 −0.980171
\(534\) −4424.32 −0.358537
\(535\) 9815.43 0.793193
\(536\) −3787.28 −0.305197
\(537\) −3076.75 −0.247247
\(538\) −17174.8 −1.37631
\(539\) 2379.86 0.190181
\(540\) −898.675 −0.0716163
\(541\) −4919.18 −0.390928 −0.195464 0.980711i \(-0.562621\pi\)
−0.195464 + 0.980711i \(0.562621\pi\)
\(542\) −14654.0 −1.16133
\(543\) −8698.19 −0.687431
\(544\) −16110.0 −1.26969
\(545\) 2725.88 0.214246
\(546\) 3509.88 0.275108
\(547\) 15334.2 1.19862 0.599308 0.800518i \(-0.295442\pi\)
0.599308 + 0.800518i \(0.295442\pi\)
\(548\) 18514.4 1.44324
\(549\) −3175.15 −0.246834
\(550\) −4648.53 −0.360389
\(551\) 5477.33 0.423488
\(552\) 1606.92 0.123904
\(553\) 6105.37 0.469487
\(554\) 13095.3 1.00427
\(555\) 3790.14 0.289879
\(556\) −8291.81 −0.632466
\(557\) −8613.78 −0.655256 −0.327628 0.944807i \(-0.606249\pi\)
−0.327628 + 0.944807i \(0.606249\pi\)
\(558\) 313.145 0.0237572
\(559\) 4044.26 0.306000
\(560\) 2552.94 0.192645
\(561\) 9857.98 0.741897
\(562\) −20045.3 −1.50456
\(563\) −2320.81 −0.173731 −0.0868654 0.996220i \(-0.527685\pi\)
−0.0868654 + 0.996220i \(0.527685\pi\)
\(564\) 11638.7 0.868934
\(565\) −1159.12 −0.0863087
\(566\) 26744.9 1.98617
\(567\) −567.000 −0.0419961
\(568\) −2531.29 −0.186990
\(569\) −1736.04 −0.127906 −0.0639529 0.997953i \(-0.520371\pi\)
−0.0639529 + 0.997953i \(0.520371\pi\)
\(570\) −5355.29 −0.393524
\(571\) 23897.8 1.75148 0.875738 0.482786i \(-0.160375\pi\)
0.875738 + 0.482786i \(0.160375\pi\)
\(572\) −14114.9 −1.03177
\(573\) −3222.55 −0.234946
\(574\) 7403.87 0.538382
\(575\) −2604.16 −0.188871
\(576\) −2952.61 −0.213586
\(577\) 8029.26 0.579311 0.289655 0.957131i \(-0.406459\pi\)
0.289655 + 0.957131i \(0.406459\pi\)
\(578\) 1284.63 0.0924455
\(579\) −2697.00 −0.193581
\(580\) −1954.95 −0.139957
\(581\) 3707.12 0.264711
\(582\) −5327.93 −0.379467
\(583\) 30259.1 2.14958
\(584\) 5989.32 0.424383
\(585\) −1964.56 −0.138845
\(586\) −28329.2 −1.99705
\(587\) −8015.14 −0.563578 −0.281789 0.959476i \(-0.590928\pi\)
−0.281789 + 0.959476i \(0.590928\pi\)
\(588\) −978.558 −0.0686310
\(589\) 847.529 0.0592900
\(590\) 10049.6 0.701247
\(591\) −9190.88 −0.639700
\(592\) 18430.5 1.27954
\(593\) 12820.3 0.887801 0.443901 0.896076i \(-0.353594\pi\)
0.443901 + 0.896076i \(0.353594\pi\)
\(594\) 5020.41 0.346784
\(595\) 2367.99 0.163157
\(596\) 129.174 0.00887779
\(597\) 2848.57 0.195283
\(598\) −17410.1 −1.19055
\(599\) −15330.5 −1.04572 −0.522860 0.852418i \(-0.675135\pi\)
−0.522860 + 0.852418i \(0.675135\pi\)
\(600\) −385.660 −0.0262409
\(601\) 107.658 0.00730694 0.00365347 0.999993i \(-0.498837\pi\)
0.00365347 + 0.999993i \(0.498837\pi\)
\(602\) −2482.59 −0.168078
\(603\) −6628.68 −0.447663
\(604\) −15642.3 −1.05377
\(605\) 5139.52 0.345374
\(606\) −4969.97 −0.333154
\(607\) −8213.06 −0.549189 −0.274595 0.961560i \(-0.588544\pi\)
−0.274595 + 0.961560i \(0.588544\pi\)
\(608\) −22205.2 −1.48115
\(609\) −1233.44 −0.0820712
\(610\) 6753.23 0.448246
\(611\) 25443.0 1.68463
\(612\) −4053.44 −0.267729
\(613\) 1242.60 0.0818732 0.0409366 0.999162i \(-0.486966\pi\)
0.0409366 + 0.999162i \(0.486966\pi\)
\(614\) −10216.7 −0.671519
\(615\) −4144.11 −0.271718
\(616\) −1748.22 −0.114347
\(617\) −13170.6 −0.859367 −0.429683 0.902980i \(-0.641375\pi\)
−0.429683 + 0.902980i \(0.641375\pi\)
\(618\) 5887.65 0.383230
\(619\) 19774.1 1.28399 0.641993 0.766711i \(-0.278108\pi\)
0.641993 + 0.766711i \(0.278108\pi\)
\(620\) −302.498 −0.0195945
\(621\) 2812.50 0.181742
\(622\) −23695.1 −1.52747
\(623\) 2696.51 0.173409
\(624\) −9553.14 −0.612871
\(625\) 625.000 0.0400000
\(626\) 11185.1 0.714132
\(627\) 13587.8 0.865459
\(628\) −25951.5 −1.64901
\(629\) 17095.3 1.08368
\(630\) 1205.95 0.0762641
\(631\) −14308.8 −0.902735 −0.451367 0.892338i \(-0.649064\pi\)
−0.451367 + 0.892338i \(0.649064\pi\)
\(632\) −4484.95 −0.282281
\(633\) −6919.93 −0.434507
\(634\) −37616.4 −2.35637
\(635\) 11864.5 0.741461
\(636\) −12442.0 −0.775722
\(637\) −2139.19 −0.133058
\(638\) 10921.3 0.677707
\(639\) −4430.38 −0.274277
\(640\) −3244.58 −0.200396
\(641\) 11537.5 0.710925 0.355463 0.934691i \(-0.384323\pi\)
0.355463 + 0.934691i \(0.384323\pi\)
\(642\) 22546.6 1.38605
\(643\) 19603.0 1.20228 0.601139 0.799144i \(-0.294714\pi\)
0.601139 + 0.799144i \(0.294714\pi\)
\(644\) 4853.95 0.297007
\(645\) 1389.56 0.0848279
\(646\) −24154.8 −1.47114
\(647\) 21650.0 1.31553 0.657765 0.753223i \(-0.271502\pi\)
0.657765 + 0.753223i \(0.271502\pi\)
\(648\) 416.513 0.0252503
\(649\) −25498.4 −1.54222
\(650\) 4178.43 0.252141
\(651\) −190.855 −0.0114903
\(652\) 10168.6 0.610787
\(653\) −2927.33 −0.175429 −0.0877145 0.996146i \(-0.527956\pi\)
−0.0877145 + 0.996146i \(0.527956\pi\)
\(654\) 6261.50 0.374379
\(655\) 6000.18 0.357934
\(656\) −20151.7 −1.19938
\(657\) 10482.8 0.622484
\(658\) −15618.3 −0.925326
\(659\) −4778.76 −0.282480 −0.141240 0.989975i \(-0.545109\pi\)
−0.141240 + 0.989975i \(0.545109\pi\)
\(660\) −4849.71 −0.286022
\(661\) −31510.3 −1.85417 −0.927086 0.374849i \(-0.877695\pi\)
−0.927086 + 0.374849i \(0.877695\pi\)
\(662\) 35444.2 2.08093
\(663\) −8861.06 −0.519057
\(664\) −2723.21 −0.159158
\(665\) 3263.92 0.190330
\(666\) 8706.17 0.506542
\(667\) 6118.23 0.355170
\(668\) −6646.99 −0.385000
\(669\) 9682.82 0.559581
\(670\) 14098.6 0.812948
\(671\) −17134.7 −0.985808
\(672\) 5000.37 0.287044
\(673\) −8992.15 −0.515040 −0.257520 0.966273i \(-0.582905\pi\)
−0.257520 + 0.966273i \(0.582905\pi\)
\(674\) 14142.1 0.808211
\(675\) −675.000 −0.0384900
\(676\) −1937.67 −0.110245
\(677\) 19340.8 1.09797 0.548985 0.835832i \(-0.315014\pi\)
0.548985 + 0.835832i \(0.315014\pi\)
\(678\) −2662.56 −0.150818
\(679\) 3247.25 0.183531
\(680\) −1739.50 −0.0980984
\(681\) 1913.46 0.107671
\(682\) 1689.89 0.0948816
\(683\) −4255.14 −0.238387 −0.119194 0.992871i \(-0.538031\pi\)
−0.119194 + 0.992871i \(0.538031\pi\)
\(684\) −5587.05 −0.312319
\(685\) 13906.3 0.775666
\(686\) 1313.15 0.0730850
\(687\) −1634.32 −0.0907616
\(688\) 6757.08 0.374435
\(689\) −27199.1 −1.50392
\(690\) −5981.91 −0.330040
\(691\) −17505.5 −0.963733 −0.481867 0.876245i \(-0.660041\pi\)
−0.481867 + 0.876245i \(0.660041\pi\)
\(692\) 4561.63 0.250588
\(693\) −3059.82 −0.167724
\(694\) −14673.7 −0.802603
\(695\) −6228.02 −0.339917
\(696\) 906.071 0.0493456
\(697\) −18691.8 −1.01579
\(698\) −31872.5 −1.72836
\(699\) 17245.6 0.933175
\(700\) −1164.95 −0.0629014
\(701\) −3240.77 −0.174611 −0.0873054 0.996182i \(-0.527826\pi\)
−0.0873054 + 0.996182i \(0.527826\pi\)
\(702\) −4512.70 −0.242622
\(703\) 23563.3 1.26416
\(704\) −15933.8 −0.853022
\(705\) 8741.91 0.467006
\(706\) 34423.4 1.83504
\(707\) 3029.07 0.161132
\(708\) 10484.5 0.556543
\(709\) 19949.3 1.05672 0.528358 0.849022i \(-0.322808\pi\)
0.528358 + 0.849022i \(0.322808\pi\)
\(710\) 9422.99 0.498082
\(711\) −7849.76 −0.414049
\(712\) −1980.83 −0.104262
\(713\) 946.698 0.0497253
\(714\) 5439.41 0.285105
\(715\) −10601.7 −0.554522
\(716\) 6827.17 0.356345
\(717\) 8034.30 0.418475
\(718\) 49362.9 2.56575
\(719\) −11259.4 −0.584011 −0.292006 0.956417i \(-0.594323\pi\)
−0.292006 + 0.956417i \(0.594323\pi\)
\(720\) −3282.35 −0.169897
\(721\) −3588.38 −0.185351
\(722\) −7034.60 −0.362605
\(723\) 6606.47 0.339830
\(724\) 19300.9 0.990761
\(725\) −1468.38 −0.0752195
\(726\) 11805.8 0.603516
\(727\) −12228.5 −0.623840 −0.311920 0.950108i \(-0.600972\pi\)
−0.311920 + 0.950108i \(0.600972\pi\)
\(728\) 1571.43 0.0800013
\(729\) 729.000 0.0370370
\(730\) −22295.9 −1.13042
\(731\) 6267.56 0.317119
\(732\) 7045.49 0.355750
\(733\) −26635.1 −1.34214 −0.671072 0.741392i \(-0.734166\pi\)
−0.671072 + 0.741392i \(0.734166\pi\)
\(734\) 28637.3 1.44008
\(735\) −735.000 −0.0368856
\(736\) −24803.4 −1.24221
\(737\) −35771.7 −1.78788
\(738\) −9519.26 −0.474809
\(739\) −6074.00 −0.302349 −0.151174 0.988507i \(-0.548306\pi\)
−0.151174 + 0.988507i \(0.548306\pi\)
\(740\) −8410.14 −0.417788
\(741\) −12213.6 −0.605505
\(742\) 16696.3 0.826065
\(743\) −4016.87 −0.198337 −0.0991686 0.995071i \(-0.531618\pi\)
−0.0991686 + 0.995071i \(0.531618\pi\)
\(744\) 140.200 0.00690858
\(745\) 97.0231 0.00477134
\(746\) 13488.8 0.662010
\(747\) −4766.29 −0.233453
\(748\) −21874.4 −1.06926
\(749\) −13741.6 −0.670370
\(750\) 1435.66 0.0698972
\(751\) −23913.2 −1.16192 −0.580962 0.813931i \(-0.697324\pi\)
−0.580962 + 0.813931i \(0.697324\pi\)
\(752\) 42509.6 2.06139
\(753\) 17150.7 0.830022
\(754\) −9816.81 −0.474147
\(755\) −11749.0 −0.566344
\(756\) 1258.15 0.0605269
\(757\) 31044.9 1.49055 0.745275 0.666758i \(-0.232318\pi\)
0.745275 + 0.666758i \(0.232318\pi\)
\(758\) −51742.9 −2.47940
\(759\) 15177.6 0.725842
\(760\) −2397.65 −0.114436
\(761\) 14011.8 0.667446 0.333723 0.942671i \(-0.391695\pi\)
0.333723 + 0.942671i \(0.391695\pi\)
\(762\) 27253.4 1.29565
\(763\) −3816.23 −0.181071
\(764\) 7150.69 0.338616
\(765\) −3044.56 −0.143891
\(766\) 2515.97 0.118676
\(767\) 22919.8 1.07899
\(768\) −15326.6 −0.720120
\(769\) 3342.49 0.156740 0.0783701 0.996924i \(-0.475028\pi\)
0.0783701 + 0.996924i \(0.475028\pi\)
\(770\) 6507.94 0.304584
\(771\) −14172.9 −0.662027
\(772\) 5984.51 0.278999
\(773\) −21074.6 −0.980594 −0.490297 0.871555i \(-0.663112\pi\)
−0.490297 + 0.871555i \(0.663112\pi\)
\(774\) 3191.90 0.148231
\(775\) −227.208 −0.0105310
\(776\) −2385.40 −0.110349
\(777\) −5306.20 −0.244992
\(778\) 37296.0 1.71867
\(779\) −25763.9 −1.18496
\(780\) 4359.26 0.200111
\(781\) −23908.5 −1.09541
\(782\) −26981.1 −1.23382
\(783\) 1585.85 0.0723800
\(784\) −3574.12 −0.162815
\(785\) −19492.3 −0.886254
\(786\) 13782.8 0.625464
\(787\) 21394.8 0.969048 0.484524 0.874778i \(-0.338993\pi\)
0.484524 + 0.874778i \(0.338993\pi\)
\(788\) 20394.1 0.921968
\(789\) −17926.1 −0.808853
\(790\) 16695.7 0.751906
\(791\) 1622.76 0.0729442
\(792\) 2247.71 0.100845
\(793\) 15401.9 0.689706
\(794\) −16873.5 −0.754179
\(795\) −9345.29 −0.416910
\(796\) −6320.83 −0.281452
\(797\) 20645.0 0.917547 0.458773 0.888553i \(-0.348289\pi\)
0.458773 + 0.888553i \(0.348289\pi\)
\(798\) 7497.41 0.332588
\(799\) 39430.0 1.74585
\(800\) 5952.82 0.263080
\(801\) −3466.95 −0.152932
\(802\) 44293.0 1.95017
\(803\) 56570.4 2.48608
\(804\) 14708.7 0.645194
\(805\) 3645.83 0.159626
\(806\) −1518.99 −0.0663825
\(807\) −13458.3 −0.587058
\(808\) −2225.13 −0.0968810
\(809\) −15939.0 −0.692688 −0.346344 0.938108i \(-0.612577\pi\)
−0.346344 + 0.938108i \(0.612577\pi\)
\(810\) −1550.51 −0.0672586
\(811\) 22829.2 0.988460 0.494230 0.869331i \(-0.335450\pi\)
0.494230 + 0.869331i \(0.335450\pi\)
\(812\) 2736.94 0.118285
\(813\) −11483.0 −0.495360
\(814\) 46982.9 2.02303
\(815\) 7637.69 0.328266
\(816\) −14804.9 −0.635141
\(817\) 8638.90 0.369935
\(818\) 11803.2 0.504508
\(819\) 2750.38 0.117346
\(820\) 9195.58 0.391614
\(821\) −5700.22 −0.242313 −0.121157 0.992633i \(-0.538660\pi\)
−0.121157 + 0.992633i \(0.538660\pi\)
\(822\) 31943.5 1.35542
\(823\) −32438.0 −1.37390 −0.686948 0.726707i \(-0.741050\pi\)
−0.686948 + 0.726707i \(0.741050\pi\)
\(824\) 2635.99 0.111443
\(825\) −3642.64 −0.153722
\(826\) −14069.4 −0.592662
\(827\) −12762.6 −0.536638 −0.268319 0.963330i \(-0.586468\pi\)
−0.268319 + 0.963330i \(0.586468\pi\)
\(828\) −6240.79 −0.261935
\(829\) −30766.7 −1.28899 −0.644494 0.764609i \(-0.722932\pi\)
−0.644494 + 0.764609i \(0.722932\pi\)
\(830\) 10137.4 0.423947
\(831\) 10261.6 0.428365
\(832\) 14322.4 0.596804
\(833\) −3315.19 −0.137892
\(834\) −14306.1 −0.593981
\(835\) −4992.59 −0.206917
\(836\) −30150.6 −1.24734
\(837\) 245.384 0.0101335
\(838\) 20731.7 0.854613
\(839\) −9779.71 −0.402423 −0.201212 0.979548i \(-0.564488\pi\)
−0.201212 + 0.979548i \(0.564488\pi\)
\(840\) 539.924 0.0221776
\(841\) −20939.2 −0.858551
\(842\) −16034.8 −0.656288
\(843\) −15707.8 −0.641760
\(844\) 15355.0 0.626233
\(845\) −1455.40 −0.0592510
\(846\) 20080.7 0.816061
\(847\) −7195.32 −0.291894
\(848\) −45443.7 −1.84026
\(849\) 20957.6 0.847190
\(850\) 6475.48 0.261303
\(851\) 26320.4 1.06023
\(852\) 9830.79 0.395302
\(853\) 24201.5 0.971445 0.485723 0.874113i \(-0.338556\pi\)
0.485723 + 0.874113i \(0.338556\pi\)
\(854\) −9454.52 −0.378837
\(855\) −4196.47 −0.167855
\(856\) 10094.5 0.403062
\(857\) 21036.7 0.838507 0.419254 0.907869i \(-0.362292\pi\)
0.419254 + 0.907869i \(0.362292\pi\)
\(858\) −24352.8 −0.968988
\(859\) −6179.19 −0.245438 −0.122719 0.992441i \(-0.539161\pi\)
−0.122719 + 0.992441i \(0.539161\pi\)
\(860\) −3083.37 −0.122258
\(861\) 5801.76 0.229644
\(862\) 34870.9 1.37785
\(863\) 50256.2 1.98232 0.991160 0.132671i \(-0.0423555\pi\)
0.991160 + 0.132671i \(0.0423555\pi\)
\(864\) −6429.04 −0.253149
\(865\) 3426.27 0.134678
\(866\) 64497.7 2.53085
\(867\) 1006.65 0.0394321
\(868\) 423.497 0.0165604
\(869\) −42361.2 −1.65363
\(870\) −3372.94 −0.131441
\(871\) 32154.1 1.25086
\(872\) 2803.37 0.108869
\(873\) −4175.03 −0.161860
\(874\) −37189.5 −1.43930
\(875\) −875.000 −0.0338062
\(876\) −23260.8 −0.897156
\(877\) −9175.95 −0.353306 −0.176653 0.984273i \(-0.556527\pi\)
−0.176653 + 0.984273i \(0.556527\pi\)
\(878\) −32289.5 −1.24114
\(879\) −22199.1 −0.851828
\(880\) −17713.2 −0.678537
\(881\) −26172.7 −1.00089 −0.500444 0.865769i \(-0.666830\pi\)
−0.500444 + 0.865769i \(0.666830\pi\)
\(882\) −1688.34 −0.0644549
\(883\) 18615.5 0.709471 0.354736 0.934967i \(-0.384571\pi\)
0.354736 + 0.934967i \(0.384571\pi\)
\(884\) 19662.3 0.748092
\(885\) 7874.99 0.299113
\(886\) 16456.4 0.624001
\(887\) −12837.8 −0.485964 −0.242982 0.970031i \(-0.578126\pi\)
−0.242982 + 0.970031i \(0.578126\pi\)
\(888\) 3897.89 0.147302
\(889\) −16610.3 −0.626649
\(890\) 7373.86 0.277722
\(891\) 3934.05 0.147919
\(892\) −21485.7 −0.806496
\(893\) 54348.4 2.03662
\(894\) 222.868 0.00833759
\(895\) 5127.92 0.191517
\(896\) 4542.42 0.169366
\(897\) −13642.7 −0.507824
\(898\) −40370.6 −1.50020
\(899\) 533.803 0.0198035
\(900\) 1497.79 0.0554738
\(901\) −42151.5 −1.55857
\(902\) −51370.7 −1.89630
\(903\) −1945.39 −0.0716926
\(904\) −1192.07 −0.0438579
\(905\) 14497.0 0.532482
\(906\) −26988.1 −0.989647
\(907\) −26766.1 −0.979885 −0.489942 0.871755i \(-0.662982\pi\)
−0.489942 + 0.871755i \(0.662982\pi\)
\(908\) −4245.87 −0.155181
\(909\) −3894.52 −0.142105
\(910\) −5849.80 −0.213098
\(911\) 5022.67 0.182666 0.0913328 0.995820i \(-0.470887\pi\)
0.0913328 + 0.995820i \(0.470887\pi\)
\(912\) −20406.3 −0.740923
\(913\) −25721.3 −0.932367
\(914\) −45759.0 −1.65599
\(915\) 5291.91 0.191197
\(916\) 3626.48 0.130810
\(917\) −8400.26 −0.302509
\(918\) −6993.52 −0.251439
\(919\) 9541.79 0.342497 0.171248 0.985228i \(-0.445220\pi\)
0.171248 + 0.985228i \(0.445220\pi\)
\(920\) −2678.19 −0.0959754
\(921\) −8005.93 −0.286432
\(922\) 65852.4 2.35220
\(923\) 21490.7 0.766387
\(924\) 6789.59 0.241733
\(925\) −6316.90 −0.224539
\(926\) 39695.7 1.40873
\(927\) 4613.63 0.163464
\(928\) −13985.6 −0.494718
\(929\) −25479.6 −0.899846 −0.449923 0.893067i \(-0.648549\pi\)
−0.449923 + 0.893067i \(0.648549\pi\)
\(930\) −521.909 −0.0184022
\(931\) −4569.49 −0.160858
\(932\) −38267.2 −1.34494
\(933\) −18567.7 −0.651533
\(934\) 64621.9 2.26391
\(935\) −16430.0 −0.574671
\(936\) −2020.41 −0.0705545
\(937\) −33608.3 −1.17176 −0.585878 0.810399i \(-0.699250\pi\)
−0.585878 + 0.810399i \(0.699250\pi\)
\(938\) −19738.0 −0.687066
\(939\) 8764.77 0.304609
\(940\) −19397.9 −0.673073
\(941\) −19173.6 −0.664232 −0.332116 0.943239i \(-0.607762\pi\)
−0.332116 + 0.943239i \(0.607762\pi\)
\(942\) −44774.9 −1.54867
\(943\) −28778.5 −0.993804
\(944\) 38294.0 1.32030
\(945\) 945.000 0.0325300
\(946\) 17225.1 0.592005
\(947\) −979.315 −0.0336045 −0.0168023 0.999859i \(-0.505349\pi\)
−0.0168023 + 0.999859i \(0.505349\pi\)
\(948\) 17418.2 0.596749
\(949\) −50849.5 −1.73935
\(950\) 8925.48 0.304822
\(951\) −29476.7 −1.00510
\(952\) 2435.31 0.0829083
\(953\) 3048.61 0.103624 0.0518122 0.998657i \(-0.483500\pi\)
0.0518122 + 0.998657i \(0.483500\pi\)
\(954\) −21466.7 −0.728521
\(955\) 5370.92 0.181989
\(956\) −17827.7 −0.603127
\(957\) 8558.03 0.289072
\(958\) −28059.0 −0.946290
\(959\) −19468.8 −0.655557
\(960\) 4921.02 0.165443
\(961\) −29708.4 −0.997227
\(962\) −42231.6 −1.41538
\(963\) 17667.8 0.591211
\(964\) −14659.4 −0.489781
\(965\) 4495.00 0.149947
\(966\) 8374.67 0.278934
\(967\) −14467.9 −0.481133 −0.240567 0.970633i \(-0.577333\pi\)
−0.240567 + 0.970633i \(0.577333\pi\)
\(968\) 5285.62 0.175502
\(969\) −18928.0 −0.627507
\(970\) 8879.89 0.293934
\(971\) 12952.8 0.428090 0.214045 0.976824i \(-0.431336\pi\)
0.214045 + 0.976824i \(0.431336\pi\)
\(972\) −1617.62 −0.0533797
\(973\) 8719.23 0.287282
\(974\) 65885.4 2.16746
\(975\) 3274.26 0.107549
\(976\) 25733.2 0.843954
\(977\) 47244.2 1.54706 0.773529 0.633760i \(-0.218489\pi\)
0.773529 + 0.633760i \(0.218489\pi\)
\(978\) 17544.2 0.573621
\(979\) −18709.4 −0.610781
\(980\) 1630.93 0.0531614
\(981\) 4906.58 0.159689
\(982\) −43615.3 −1.41733
\(983\) −1536.84 −0.0498651 −0.0249326 0.999689i \(-0.507937\pi\)
−0.0249326 + 0.999689i \(0.507937\pi\)
\(984\) −4261.92 −0.138074
\(985\) 15318.1 0.495509
\(986\) −15213.5 −0.491376
\(987\) −12238.7 −0.394692
\(988\) 27101.5 0.872685
\(989\) 9649.73 0.310256
\(990\) −8367.35 −0.268618
\(991\) 3785.22 0.121334 0.0606668 0.998158i \(-0.480677\pi\)
0.0606668 + 0.998158i \(0.480677\pi\)
\(992\) −2164.04 −0.0692625
\(993\) 27774.5 0.887610
\(994\) −13192.2 −0.420956
\(995\) −4747.61 −0.151266
\(996\) 10576.2 0.336465
\(997\) −25894.9 −0.822566 −0.411283 0.911508i \(-0.634919\pi\)
−0.411283 + 0.911508i \(0.634919\pi\)
\(998\) −73044.0 −2.31680
\(999\) 6822.26 0.216063
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.1.1 2
3.2 odd 2 315.4.a.k.1.2 2
4.3 odd 2 1680.4.a.bo.1.1 2
5.2 odd 4 525.4.d.l.274.1 4
5.3 odd 4 525.4.d.l.274.4 4
5.4 even 2 525.4.a.l.1.2 2
7.6 odd 2 735.4.a.o.1.1 2
15.14 odd 2 1575.4.a.q.1.1 2
21.20 even 2 2205.4.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.e.1.1 2 1.1 even 1 trivial
315.4.a.k.1.2 2 3.2 odd 2
525.4.a.l.1.2 2 5.4 even 2
525.4.d.l.274.1 4 5.2 odd 4
525.4.d.l.274.4 4 5.3 odd 4
735.4.a.o.1.1 2 7.6 odd 2
1575.4.a.q.1.1 2 15.14 odd 2
1680.4.a.bo.1.1 2 4.3 odd 2
2205.4.a.bb.1.2 2 21.20 even 2