Properties

Label 1521.4.a.bf.1.8
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(2.37150\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.17344 q^{2} +2.07074 q^{4} -6.74147 q^{5} -14.1726 q^{7} -18.8162 q^{8} +O(q^{10})\) \(q+3.17344 q^{2} +2.07074 q^{4} -6.74147 q^{5} -14.1726 q^{7} -18.8162 q^{8} -21.3937 q^{10} -62.4956 q^{11} -44.9761 q^{14} -76.2779 q^{16} +58.6172 q^{17} +64.1652 q^{19} -13.9598 q^{20} -198.326 q^{22} -10.9221 q^{23} -79.5526 q^{25} -29.3478 q^{28} -216.316 q^{29} -38.6271 q^{31} -91.5342 q^{32} +186.018 q^{34} +95.5445 q^{35} +423.770 q^{37} +203.625 q^{38} +126.849 q^{40} +366.126 q^{41} -128.297 q^{43} -129.412 q^{44} -34.6605 q^{46} +93.1169 q^{47} -142.136 q^{49} -252.455 q^{50} -131.909 q^{53} +421.313 q^{55} +266.675 q^{56} -686.467 q^{58} -386.729 q^{59} -621.077 q^{61} -122.581 q^{62} +319.745 q^{64} +865.273 q^{67} +121.381 q^{68} +303.205 q^{70} +607.506 q^{71} +980.958 q^{73} +1344.81 q^{74} +132.869 q^{76} +885.728 q^{77} +1331.91 q^{79} +514.226 q^{80} +1161.88 q^{82} -907.633 q^{83} -395.166 q^{85} -407.142 q^{86} +1175.93 q^{88} +1033.67 q^{89} -22.6167 q^{92} +295.501 q^{94} -432.568 q^{95} +1046.17 q^{97} -451.061 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8} - 54 q^{10} - 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} + 352 q^{19} - 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} + 940 q^{28} + 97 q^{29} + 717 q^{31} - 707 q^{32} + 632 q^{34} + 418 q^{35} + 1108 q^{37} + 660 q^{38} - 1506 q^{40} - 334 q^{41} + 242 q^{43} + 307 q^{44} + 979 q^{46} + 184 q^{47} - 38 q^{49} + 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} + 1161 q^{58} - 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} + 2308 q^{67} - 2785 q^{68} - 1420 q^{70} - 96 q^{71} + 2505 q^{73} + 1191 q^{74} + 2409 q^{76} + 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 1517 q^{82} - 1539 q^{83} + 4296 q^{85} + 3763 q^{86} - 3716 q^{88} + 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} + 1445 q^{97} - 1457 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.17344 1.12198 0.560991 0.827822i \(-0.310420\pi\)
0.560991 + 0.827822i \(0.310420\pi\)
\(3\) 0 0
\(4\) 2.07074 0.258842
\(5\) −6.74147 −0.602975 −0.301488 0.953470i \(-0.597483\pi\)
−0.301488 + 0.953470i \(0.597483\pi\)
\(6\) 0 0
\(7\) −14.1726 −0.765251 −0.382625 0.923904i \(-0.624980\pi\)
−0.382625 + 0.923904i \(0.624980\pi\)
\(8\) −18.8162 −0.831565
\(9\) 0 0
\(10\) −21.3937 −0.676527
\(11\) −62.4956 −1.71301 −0.856507 0.516136i \(-0.827370\pi\)
−0.856507 + 0.516136i \(0.827370\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −44.9761 −0.858597
\(15\) 0 0
\(16\) −76.2779 −1.19184
\(17\) 58.6172 0.836280 0.418140 0.908383i \(-0.362682\pi\)
0.418140 + 0.908383i \(0.362682\pi\)
\(18\) 0 0
\(19\) 64.1652 0.774764 0.387382 0.921919i \(-0.373380\pi\)
0.387382 + 0.921919i \(0.373380\pi\)
\(20\) −13.9598 −0.156075
\(21\) 0 0
\(22\) −198.326 −1.92197
\(23\) −10.9221 −0.0990177 −0.0495088 0.998774i \(-0.515766\pi\)
−0.0495088 + 0.998774i \(0.515766\pi\)
\(24\) 0 0
\(25\) −79.5526 −0.636421
\(26\) 0 0
\(27\) 0 0
\(28\) −29.3478 −0.198079
\(29\) −216.316 −1.38514 −0.692568 0.721353i \(-0.743521\pi\)
−0.692568 + 0.721353i \(0.743521\pi\)
\(30\) 0 0
\(31\) −38.6271 −0.223795 −0.111897 0.993720i \(-0.535693\pi\)
−0.111897 + 0.993720i \(0.535693\pi\)
\(32\) −91.5342 −0.505660
\(33\) 0 0
\(34\) 186.018 0.938290
\(35\) 95.5445 0.461427
\(36\) 0 0
\(37\) 423.770 1.88290 0.941452 0.337147i \(-0.109462\pi\)
0.941452 + 0.337147i \(0.109462\pi\)
\(38\) 203.625 0.869270
\(39\) 0 0
\(40\) 126.849 0.501414
\(41\) 366.126 1.39461 0.697307 0.716772i \(-0.254381\pi\)
0.697307 + 0.716772i \(0.254381\pi\)
\(42\) 0 0
\(43\) −128.297 −0.455001 −0.227501 0.973778i \(-0.573055\pi\)
−0.227501 + 0.973778i \(0.573055\pi\)
\(44\) −129.412 −0.443400
\(45\) 0 0
\(46\) −34.6605 −0.111096
\(47\) 93.1169 0.288989 0.144495 0.989506i \(-0.453844\pi\)
0.144495 + 0.989506i \(0.453844\pi\)
\(48\) 0 0
\(49\) −142.136 −0.414391
\(50\) −252.455 −0.714052
\(51\) 0 0
\(52\) 0 0
\(53\) −131.909 −0.341869 −0.170934 0.985282i \(-0.554679\pi\)
−0.170934 + 0.985282i \(0.554679\pi\)
\(54\) 0 0
\(55\) 421.313 1.03290
\(56\) 266.675 0.636356
\(57\) 0 0
\(58\) −686.467 −1.55410
\(59\) −386.729 −0.853353 −0.426677 0.904404i \(-0.640316\pi\)
−0.426677 + 0.904404i \(0.640316\pi\)
\(60\) 0 0
\(61\) −621.077 −1.30362 −0.651810 0.758382i \(-0.725990\pi\)
−0.651810 + 0.758382i \(0.725990\pi\)
\(62\) −122.581 −0.251094
\(63\) 0 0
\(64\) 319.745 0.624502
\(65\) 0 0
\(66\) 0 0
\(67\) 865.273 1.57776 0.788880 0.614547i \(-0.210661\pi\)
0.788880 + 0.614547i \(0.210661\pi\)
\(68\) 121.381 0.216464
\(69\) 0 0
\(70\) 303.205 0.517713
\(71\) 607.506 1.01546 0.507730 0.861516i \(-0.330485\pi\)
0.507730 + 0.861516i \(0.330485\pi\)
\(72\) 0 0
\(73\) 980.958 1.57277 0.786387 0.617735i \(-0.211949\pi\)
0.786387 + 0.617735i \(0.211949\pi\)
\(74\) 1344.81 2.11258
\(75\) 0 0
\(76\) 132.869 0.200541
\(77\) 885.728 1.31088
\(78\) 0 0
\(79\) 1331.91 1.89685 0.948425 0.317003i \(-0.102676\pi\)
0.948425 + 0.317003i \(0.102676\pi\)
\(80\) 514.226 0.718652
\(81\) 0 0
\(82\) 1161.88 1.56473
\(83\) −907.633 −1.20031 −0.600155 0.799884i \(-0.704894\pi\)
−0.600155 + 0.799884i \(0.704894\pi\)
\(84\) 0 0
\(85\) −395.166 −0.504256
\(86\) −407.142 −0.510503
\(87\) 0 0
\(88\) 1175.93 1.42448
\(89\) 1033.67 1.23110 0.615552 0.788096i \(-0.288933\pi\)
0.615552 + 0.788096i \(0.288933\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −22.6167 −0.0256299
\(93\) 0 0
\(94\) 295.501 0.324240
\(95\) −432.568 −0.467163
\(96\) 0 0
\(97\) 1046.17 1.09508 0.547538 0.836781i \(-0.315565\pi\)
0.547538 + 0.836781i \(0.315565\pi\)
\(98\) −451.061 −0.464939
\(99\) 0 0
\(100\) −164.732 −0.164732
\(101\) −1416.64 −1.39566 −0.697828 0.716265i \(-0.745850\pi\)
−0.697828 + 0.716265i \(0.745850\pi\)
\(102\) 0 0
\(103\) 387.629 0.370818 0.185409 0.982661i \(-0.440639\pi\)
0.185409 + 0.982661i \(0.440639\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −418.605 −0.383570
\(107\) −86.4526 −0.0781092 −0.0390546 0.999237i \(-0.512435\pi\)
−0.0390546 + 0.999237i \(0.512435\pi\)
\(108\) 0 0
\(109\) −940.072 −0.826079 −0.413039 0.910713i \(-0.635533\pi\)
−0.413039 + 0.910713i \(0.635533\pi\)
\(110\) 1337.01 1.15890
\(111\) 0 0
\(112\) 1081.06 0.912059
\(113\) −960.499 −0.799612 −0.399806 0.916600i \(-0.630922\pi\)
−0.399806 + 0.916600i \(0.630922\pi\)
\(114\) 0 0
\(115\) 73.6307 0.0597052
\(116\) −447.934 −0.358531
\(117\) 0 0
\(118\) −1227.26 −0.957447
\(119\) −830.760 −0.639964
\(120\) 0 0
\(121\) 2574.71 1.93441
\(122\) −1970.95 −1.46264
\(123\) 0 0
\(124\) −79.9866 −0.0579275
\(125\) 1378.99 0.986721
\(126\) 0 0
\(127\) 2022.18 1.41291 0.706456 0.707757i \(-0.250293\pi\)
0.706456 + 0.707757i \(0.250293\pi\)
\(128\) 1746.97 1.20634
\(129\) 0 0
\(130\) 0 0
\(131\) −1857.90 −1.23912 −0.619561 0.784948i \(-0.712690\pi\)
−0.619561 + 0.784948i \(0.712690\pi\)
\(132\) 0 0
\(133\) −909.390 −0.592888
\(134\) 2745.90 1.77022
\(135\) 0 0
\(136\) −1102.95 −0.695421
\(137\) −1894.12 −1.18121 −0.590604 0.806961i \(-0.701110\pi\)
−0.590604 + 0.806961i \(0.701110\pi\)
\(138\) 0 0
\(139\) −1226.08 −0.748165 −0.374082 0.927395i \(-0.622042\pi\)
−0.374082 + 0.927395i \(0.622042\pi\)
\(140\) 197.847 0.119437
\(141\) 0 0
\(142\) 1927.89 1.13933
\(143\) 0 0
\(144\) 0 0
\(145\) 1458.29 0.835203
\(146\) 3113.01 1.76462
\(147\) 0 0
\(148\) 877.517 0.487375
\(149\) −3195.65 −1.75703 −0.878517 0.477711i \(-0.841467\pi\)
−0.878517 + 0.477711i \(0.841467\pi\)
\(150\) 0 0
\(151\) 508.232 0.273903 0.136951 0.990578i \(-0.456270\pi\)
0.136951 + 0.990578i \(0.456270\pi\)
\(152\) −1207.34 −0.644267
\(153\) 0 0
\(154\) 2810.81 1.47079
\(155\) 260.404 0.134943
\(156\) 0 0
\(157\) −1243.08 −0.631900 −0.315950 0.948776i \(-0.602323\pi\)
−0.315950 + 0.948776i \(0.602323\pi\)
\(158\) 4226.73 2.12823
\(159\) 0 0
\(160\) 617.075 0.304901
\(161\) 154.794 0.0757734
\(162\) 0 0
\(163\) −33.9996 −0.0163378 −0.00816888 0.999967i \(-0.502600\pi\)
−0.00816888 + 0.999967i \(0.502600\pi\)
\(164\) 758.149 0.360985
\(165\) 0 0
\(166\) −2880.32 −1.34673
\(167\) 2210.67 1.02435 0.512176 0.858880i \(-0.328839\pi\)
0.512176 + 0.858880i \(0.328839\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −1254.04 −0.565766
\(171\) 0 0
\(172\) −265.668 −0.117773
\(173\) 661.307 0.290626 0.145313 0.989386i \(-0.453581\pi\)
0.145313 + 0.989386i \(0.453581\pi\)
\(174\) 0 0
\(175\) 1127.47 0.487021
\(176\) 4767.04 2.04164
\(177\) 0 0
\(178\) 3280.28 1.38128
\(179\) 2325.05 0.970850 0.485425 0.874278i \(-0.338665\pi\)
0.485425 + 0.874278i \(0.338665\pi\)
\(180\) 0 0
\(181\) 2122.20 0.871503 0.435752 0.900067i \(-0.356483\pi\)
0.435752 + 0.900067i \(0.356483\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 205.511 0.0823397
\(185\) −2856.84 −1.13534
\(186\) 0 0
\(187\) −3663.32 −1.43256
\(188\) 192.820 0.0748025
\(189\) 0 0
\(190\) −1372.73 −0.524149
\(191\) 2484.37 0.941166 0.470583 0.882356i \(-0.344044\pi\)
0.470583 + 0.882356i \(0.344044\pi\)
\(192\) 0 0
\(193\) −266.771 −0.0994955 −0.0497478 0.998762i \(-0.515842\pi\)
−0.0497478 + 0.998762i \(0.515842\pi\)
\(194\) 3319.95 1.22865
\(195\) 0 0
\(196\) −294.327 −0.107262
\(197\) 1231.03 0.445216 0.222608 0.974908i \(-0.428543\pi\)
0.222608 + 0.974908i \(0.428543\pi\)
\(198\) 0 0
\(199\) −3246.14 −1.15635 −0.578173 0.815914i \(-0.696234\pi\)
−0.578173 + 0.815914i \(0.696234\pi\)
\(200\) 1496.88 0.529225
\(201\) 0 0
\(202\) −4495.63 −1.56590
\(203\) 3065.77 1.05998
\(204\) 0 0
\(205\) −2468.23 −0.840919
\(206\) 1230.12 0.416051
\(207\) 0 0
\(208\) 0 0
\(209\) −4010.05 −1.32718
\(210\) 0 0
\(211\) 330.708 0.107900 0.0539500 0.998544i \(-0.482819\pi\)
0.0539500 + 0.998544i \(0.482819\pi\)
\(212\) −273.148 −0.0884900
\(213\) 0 0
\(214\) −274.352 −0.0876371
\(215\) 864.908 0.274355
\(216\) 0 0
\(217\) 547.449 0.171259
\(218\) −2983.26 −0.926845
\(219\) 0 0
\(220\) 872.427 0.267359
\(221\) 0 0
\(222\) 0 0
\(223\) 5785.86 1.73744 0.868722 0.495300i \(-0.164942\pi\)
0.868722 + 0.495300i \(0.164942\pi\)
\(224\) 1297.28 0.386957
\(225\) 0 0
\(226\) −3048.09 −0.897149
\(227\) −2945.35 −0.861189 −0.430595 0.902545i \(-0.641696\pi\)
−0.430595 + 0.902545i \(0.641696\pi\)
\(228\) 0 0
\(229\) −3541.26 −1.02189 −0.510945 0.859613i \(-0.670704\pi\)
−0.510945 + 0.859613i \(0.670704\pi\)
\(230\) 233.663 0.0669882
\(231\) 0 0
\(232\) 4070.25 1.15183
\(233\) −2340.76 −0.658148 −0.329074 0.944304i \(-0.606737\pi\)
−0.329074 + 0.944304i \(0.606737\pi\)
\(234\) 0 0
\(235\) −627.745 −0.174253
\(236\) −800.814 −0.220884
\(237\) 0 0
\(238\) −2636.37 −0.718027
\(239\) 1515.70 0.410218 0.205109 0.978739i \(-0.434245\pi\)
0.205109 + 0.978739i \(0.434245\pi\)
\(240\) 0 0
\(241\) 2392.47 0.639472 0.319736 0.947507i \(-0.396406\pi\)
0.319736 + 0.947507i \(0.396406\pi\)
\(242\) 8170.68 2.17038
\(243\) 0 0
\(244\) −1286.09 −0.337432
\(245\) 958.207 0.249868
\(246\) 0 0
\(247\) 0 0
\(248\) 726.815 0.186100
\(249\) 0 0
\(250\) 4376.13 1.10708
\(251\) 2198.78 0.552931 0.276465 0.961024i \(-0.410837\pi\)
0.276465 + 0.961024i \(0.410837\pi\)
\(252\) 0 0
\(253\) 682.581 0.169619
\(254\) 6417.29 1.58526
\(255\) 0 0
\(256\) 2985.94 0.728988
\(257\) 6194.26 1.50345 0.751727 0.659475i \(-0.229221\pi\)
0.751727 + 0.659475i \(0.229221\pi\)
\(258\) 0 0
\(259\) −6005.95 −1.44089
\(260\) 0 0
\(261\) 0 0
\(262\) −5895.92 −1.39027
\(263\) 4181.74 0.980445 0.490222 0.871597i \(-0.336916\pi\)
0.490222 + 0.871597i \(0.336916\pi\)
\(264\) 0 0
\(265\) 889.258 0.206139
\(266\) −2885.90 −0.665210
\(267\) 0 0
\(268\) 1791.75 0.408391
\(269\) −2767.69 −0.627320 −0.313660 0.949535i \(-0.601555\pi\)
−0.313660 + 0.949535i \(0.601555\pi\)
\(270\) 0 0
\(271\) −7191.36 −1.61197 −0.805986 0.591935i \(-0.798364\pi\)
−0.805986 + 0.591935i \(0.798364\pi\)
\(272\) −4471.20 −0.996714
\(273\) 0 0
\(274\) −6010.88 −1.32529
\(275\) 4971.69 1.09020
\(276\) 0 0
\(277\) 1317.27 0.285729 0.142864 0.989742i \(-0.454369\pi\)
0.142864 + 0.989742i \(0.454369\pi\)
\(278\) −3890.90 −0.839427
\(279\) 0 0
\(280\) −1797.78 −0.383707
\(281\) 3948.92 0.838338 0.419169 0.907908i \(-0.362321\pi\)
0.419169 + 0.907908i \(0.362321\pi\)
\(282\) 0 0
\(283\) −4981.52 −1.04636 −0.523181 0.852221i \(-0.675255\pi\)
−0.523181 + 0.852221i \(0.675255\pi\)
\(284\) 1257.99 0.262844
\(285\) 0 0
\(286\) 0 0
\(287\) −5188.97 −1.06723
\(288\) 0 0
\(289\) −1477.03 −0.300636
\(290\) 4627.80 0.937082
\(291\) 0 0
\(292\) 2031.31 0.407100
\(293\) 3203.02 0.638644 0.319322 0.947646i \(-0.396545\pi\)
0.319322 + 0.947646i \(0.396545\pi\)
\(294\) 0 0
\(295\) 2607.12 0.514551
\(296\) −7973.74 −1.56576
\(297\) 0 0
\(298\) −10141.2 −1.97136
\(299\) 0 0
\(300\) 0 0
\(301\) 1818.30 0.348190
\(302\) 1612.84 0.307314
\(303\) 0 0
\(304\) −4894.39 −0.923396
\(305\) 4186.97 0.786051
\(306\) 0 0
\(307\) 4795.67 0.891542 0.445771 0.895147i \(-0.352930\pi\)
0.445771 + 0.895147i \(0.352930\pi\)
\(308\) 1834.11 0.339312
\(309\) 0 0
\(310\) 826.376 0.151403
\(311\) −630.213 −0.114907 −0.0574535 0.998348i \(-0.518298\pi\)
−0.0574535 + 0.998348i \(0.518298\pi\)
\(312\) 0 0
\(313\) −9314.73 −1.68211 −0.841054 0.540952i \(-0.818064\pi\)
−0.841054 + 0.540952i \(0.818064\pi\)
\(314\) −3944.83 −0.708980
\(315\) 0 0
\(316\) 2758.02 0.490984
\(317\) −576.333 −0.102114 −0.0510569 0.998696i \(-0.516259\pi\)
−0.0510569 + 0.998696i \(0.516259\pi\)
\(318\) 0 0
\(319\) 13518.8 2.37275
\(320\) −2155.55 −0.376559
\(321\) 0 0
\(322\) 491.231 0.0850163
\(323\) 3761.18 0.647919
\(324\) 0 0
\(325\) 0 0
\(326\) −107.896 −0.0183307
\(327\) 0 0
\(328\) −6889.08 −1.15971
\(329\) −1319.71 −0.221149
\(330\) 0 0
\(331\) 1575.95 0.261699 0.130849 0.991402i \(-0.458230\pi\)
0.130849 + 0.991402i \(0.458230\pi\)
\(332\) −1879.47 −0.310691
\(333\) 0 0
\(334\) 7015.44 1.14930
\(335\) −5833.22 −0.951351
\(336\) 0 0
\(337\) 9289.32 1.50155 0.750774 0.660559i \(-0.229681\pi\)
0.750774 + 0.660559i \(0.229681\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −818.284 −0.130523
\(341\) 2414.03 0.383363
\(342\) 0 0
\(343\) 6875.66 1.08236
\(344\) 2414.05 0.378363
\(345\) 0 0
\(346\) 2098.62 0.326077
\(347\) 7701.82 1.19151 0.595757 0.803164i \(-0.296852\pi\)
0.595757 + 0.803164i \(0.296852\pi\)
\(348\) 0 0
\(349\) −4972.89 −0.762730 −0.381365 0.924425i \(-0.624546\pi\)
−0.381365 + 0.924425i \(0.624546\pi\)
\(350\) 3577.96 0.546429
\(351\) 0 0
\(352\) 5720.49 0.866202
\(353\) −1575.34 −0.237526 −0.118763 0.992923i \(-0.537893\pi\)
−0.118763 + 0.992923i \(0.537893\pi\)
\(354\) 0 0
\(355\) −4095.49 −0.612298
\(356\) 2140.45 0.318661
\(357\) 0 0
\(358\) 7378.41 1.08928
\(359\) −7567.42 −1.11252 −0.556258 0.831010i \(-0.687763\pi\)
−0.556258 + 0.831010i \(0.687763\pi\)
\(360\) 0 0
\(361\) −2741.83 −0.399741
\(362\) 6734.69 0.977810
\(363\) 0 0
\(364\) 0 0
\(365\) −6613.10 −0.948344
\(366\) 0 0
\(367\) 4368.25 0.621310 0.310655 0.950523i \(-0.399452\pi\)
0.310655 + 0.950523i \(0.399452\pi\)
\(368\) 833.112 0.118014
\(369\) 0 0
\(370\) −9066.01 −1.27384
\(371\) 1869.49 0.261615
\(372\) 0 0
\(373\) −801.944 −0.111322 −0.0556610 0.998450i \(-0.517727\pi\)
−0.0556610 + 0.998450i \(0.517727\pi\)
\(374\) −11625.3 −1.60730
\(375\) 0 0
\(376\) −1752.10 −0.240313
\(377\) 0 0
\(378\) 0 0
\(379\) 68.0819 0.00922727 0.00461363 0.999989i \(-0.498531\pi\)
0.00461363 + 0.999989i \(0.498531\pi\)
\(380\) −895.734 −0.120922
\(381\) 0 0
\(382\) 7884.01 1.05597
\(383\) −1549.01 −0.206659 −0.103330 0.994647i \(-0.532950\pi\)
−0.103330 + 0.994647i \(0.532950\pi\)
\(384\) 0 0
\(385\) −5971.11 −0.790431
\(386\) −846.584 −0.111632
\(387\) 0 0
\(388\) 2166.34 0.283451
\(389\) −7300.51 −0.951544 −0.475772 0.879569i \(-0.657831\pi\)
−0.475772 + 0.879569i \(0.657831\pi\)
\(390\) 0 0
\(391\) −640.220 −0.0828065
\(392\) 2674.46 0.344594
\(393\) 0 0
\(394\) 3906.61 0.499524
\(395\) −8979.00 −1.14375
\(396\) 0 0
\(397\) 6096.27 0.770688 0.385344 0.922773i \(-0.374083\pi\)
0.385344 + 0.922773i \(0.374083\pi\)
\(398\) −10301.4 −1.29740
\(399\) 0 0
\(400\) 6068.11 0.758513
\(401\) −7592.37 −0.945498 −0.472749 0.881197i \(-0.656738\pi\)
−0.472749 + 0.881197i \(0.656738\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −2933.49 −0.361254
\(405\) 0 0
\(406\) 9729.05 1.18927
\(407\) −26483.8 −3.22544
\(408\) 0 0
\(409\) −7233.86 −0.874551 −0.437275 0.899328i \(-0.644057\pi\)
−0.437275 + 0.899328i \(0.644057\pi\)
\(410\) −7832.77 −0.943495
\(411\) 0 0
\(412\) 802.678 0.0959832
\(413\) 5480.97 0.653029
\(414\) 0 0
\(415\) 6118.78 0.723757
\(416\) 0 0
\(417\) 0 0
\(418\) −12725.6 −1.48907
\(419\) 5312.55 0.619416 0.309708 0.950832i \(-0.399769\pi\)
0.309708 + 0.950832i \(0.399769\pi\)
\(420\) 0 0
\(421\) 15028.1 1.73973 0.869865 0.493290i \(-0.164206\pi\)
0.869865 + 0.493290i \(0.164206\pi\)
\(422\) 1049.48 0.121062
\(423\) 0 0
\(424\) 2482.02 0.284286
\(425\) −4663.15 −0.532226
\(426\) 0 0
\(427\) 8802.31 0.997596
\(428\) −179.020 −0.0202179
\(429\) 0 0
\(430\) 2744.73 0.307821
\(431\) 7154.66 0.799600 0.399800 0.916602i \(-0.369080\pi\)
0.399800 + 0.916602i \(0.369080\pi\)
\(432\) 0 0
\(433\) −9542.58 −1.05909 −0.529546 0.848281i \(-0.677638\pi\)
−0.529546 + 0.848281i \(0.677638\pi\)
\(434\) 1737.30 0.192150
\(435\) 0 0
\(436\) −1946.64 −0.213824
\(437\) −700.816 −0.0767153
\(438\) 0 0
\(439\) 7070.70 0.768715 0.384358 0.923184i \(-0.374423\pi\)
0.384358 + 0.923184i \(0.374423\pi\)
\(440\) −7927.49 −0.858928
\(441\) 0 0
\(442\) 0 0
\(443\) −2092.58 −0.224428 −0.112214 0.993684i \(-0.535794\pi\)
−0.112214 + 0.993684i \(0.535794\pi\)
\(444\) 0 0
\(445\) −6968.42 −0.742326
\(446\) 18361.1 1.94938
\(447\) 0 0
\(448\) −4531.63 −0.477901
\(449\) −5842.05 −0.614038 −0.307019 0.951703i \(-0.599332\pi\)
−0.307019 + 0.951703i \(0.599332\pi\)
\(450\) 0 0
\(451\) −22881.3 −2.38899
\(452\) −1988.94 −0.206973
\(453\) 0 0
\(454\) −9346.91 −0.966238
\(455\) 0 0
\(456\) 0 0
\(457\) 5954.40 0.609486 0.304743 0.952435i \(-0.401429\pi\)
0.304743 + 0.952435i \(0.401429\pi\)
\(458\) −11238.0 −1.14654
\(459\) 0 0
\(460\) 152.470 0.0154542
\(461\) 1865.94 0.188515 0.0942576 0.995548i \(-0.469952\pi\)
0.0942576 + 0.995548i \(0.469952\pi\)
\(462\) 0 0
\(463\) −6700.05 −0.672522 −0.336261 0.941769i \(-0.609162\pi\)
−0.336261 + 0.941769i \(0.609162\pi\)
\(464\) 16500.2 1.65086
\(465\) 0 0
\(466\) −7428.28 −0.738430
\(467\) 16585.8 1.64347 0.821734 0.569871i \(-0.193007\pi\)
0.821734 + 0.569871i \(0.193007\pi\)
\(468\) 0 0
\(469\) −12263.2 −1.20738
\(470\) −1992.11 −0.195509
\(471\) 0 0
\(472\) 7276.77 0.709619
\(473\) 8017.98 0.779423
\(474\) 0 0
\(475\) −5104.51 −0.493075
\(476\) −1720.29 −0.165649
\(477\) 0 0
\(478\) 4809.97 0.460257
\(479\) 6166.88 0.588250 0.294125 0.955767i \(-0.404972\pi\)
0.294125 + 0.955767i \(0.404972\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 7592.38 0.717476
\(483\) 0 0
\(484\) 5331.53 0.500708
\(485\) −7052.71 −0.660303
\(486\) 0 0
\(487\) 5718.51 0.532095 0.266047 0.963960i \(-0.414282\pi\)
0.266047 + 0.963960i \(0.414282\pi\)
\(488\) 11686.3 1.08405
\(489\) 0 0
\(490\) 3040.82 0.280347
\(491\) 21060.9 1.93578 0.967888 0.251383i \(-0.0808854\pi\)
0.967888 + 0.251383i \(0.0808854\pi\)
\(492\) 0 0
\(493\) −12679.8 −1.15836
\(494\) 0 0
\(495\) 0 0
\(496\) 2946.40 0.266728
\(497\) −8609.97 −0.777082
\(498\) 0 0
\(499\) −7863.87 −0.705481 −0.352741 0.935721i \(-0.614750\pi\)
−0.352741 + 0.935721i \(0.614750\pi\)
\(500\) 2855.51 0.255405
\(501\) 0 0
\(502\) 6977.70 0.620378
\(503\) 6504.06 0.576544 0.288272 0.957549i \(-0.406919\pi\)
0.288272 + 0.957549i \(0.406919\pi\)
\(504\) 0 0
\(505\) 9550.26 0.841546
\(506\) 2166.13 0.190309
\(507\) 0 0
\(508\) 4187.41 0.365721
\(509\) −14799.3 −1.28873 −0.644367 0.764717i \(-0.722879\pi\)
−0.644367 + 0.764717i \(0.722879\pi\)
\(510\) 0 0
\(511\) −13902.8 −1.20357
\(512\) −4500.03 −0.388428
\(513\) 0 0
\(514\) 19657.1 1.68685
\(515\) −2613.19 −0.223594
\(516\) 0 0
\(517\) −5819.40 −0.495042
\(518\) −19059.5 −1.61666
\(519\) 0 0
\(520\) 0 0
\(521\) 6633.65 0.557822 0.278911 0.960317i \(-0.410026\pi\)
0.278911 + 0.960317i \(0.410026\pi\)
\(522\) 0 0
\(523\) 4527.41 0.378527 0.189264 0.981926i \(-0.439390\pi\)
0.189264 + 0.981926i \(0.439390\pi\)
\(524\) −3847.21 −0.320737
\(525\) 0 0
\(526\) 13270.5 1.10004
\(527\) −2264.21 −0.187155
\(528\) 0 0
\(529\) −12047.7 −0.990195
\(530\) 2822.01 0.231284
\(531\) 0 0
\(532\) −1883.11 −0.153464
\(533\) 0 0
\(534\) 0 0
\(535\) 582.818 0.0470980
\(536\) −16281.1 −1.31201
\(537\) 0 0
\(538\) −8783.11 −0.703841
\(539\) 8882.89 0.709858
\(540\) 0 0
\(541\) 8685.42 0.690232 0.345116 0.938560i \(-0.387840\pi\)
0.345116 + 0.938560i \(0.387840\pi\)
\(542\) −22821.4 −1.80860
\(543\) 0 0
\(544\) −5365.48 −0.422873
\(545\) 6337.47 0.498105
\(546\) 0 0
\(547\) 24656.6 1.92731 0.963657 0.267142i \(-0.0860793\pi\)
0.963657 + 0.267142i \(0.0860793\pi\)
\(548\) −3922.22 −0.305746
\(549\) 0 0
\(550\) 15777.4 1.22318
\(551\) −13880.0 −1.07315
\(552\) 0 0
\(553\) −18876.6 −1.45157
\(554\) 4180.27 0.320582
\(555\) 0 0
\(556\) −2538.89 −0.193656
\(557\) −9215.90 −0.701059 −0.350530 0.936552i \(-0.613998\pi\)
−0.350530 + 0.936552i \(0.613998\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −7287.93 −0.549949
\(561\) 0 0
\(562\) 12531.7 0.940600
\(563\) 19686.7 1.47370 0.736850 0.676056i \(-0.236312\pi\)
0.736850 + 0.676056i \(0.236312\pi\)
\(564\) 0 0
\(565\) 6475.17 0.482146
\(566\) −15808.6 −1.17400
\(567\) 0 0
\(568\) −11430.9 −0.844422
\(569\) 3559.36 0.262243 0.131121 0.991366i \(-0.458142\pi\)
0.131121 + 0.991366i \(0.458142\pi\)
\(570\) 0 0
\(571\) −710.968 −0.0521070 −0.0260535 0.999661i \(-0.508294\pi\)
−0.0260535 + 0.999661i \(0.508294\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −16466.9 −1.19741
\(575\) 868.878 0.0630169
\(576\) 0 0
\(577\) −8041.67 −0.580206 −0.290103 0.956995i \(-0.593690\pi\)
−0.290103 + 0.956995i \(0.593690\pi\)
\(578\) −4687.26 −0.337308
\(579\) 0 0
\(580\) 3019.73 0.216186
\(581\) 12863.6 0.918538
\(582\) 0 0
\(583\) 8243.72 0.585626
\(584\) −18457.9 −1.30786
\(585\) 0 0
\(586\) 10164.6 0.716546
\(587\) 14641.3 1.02949 0.514744 0.857344i \(-0.327887\pi\)
0.514744 + 0.857344i \(0.327887\pi\)
\(588\) 0 0
\(589\) −2478.52 −0.173388
\(590\) 8273.56 0.577317
\(591\) 0 0
\(592\) −32324.3 −2.24413
\(593\) −5735.76 −0.397200 −0.198600 0.980081i \(-0.563639\pi\)
−0.198600 + 0.980081i \(0.563639\pi\)
\(594\) 0 0
\(595\) 5600.55 0.385882
\(596\) −6617.35 −0.454794
\(597\) 0 0
\(598\) 0 0
\(599\) 16109.3 1.09884 0.549422 0.835545i \(-0.314848\pi\)
0.549422 + 0.835545i \(0.314848\pi\)
\(600\) 0 0
\(601\) −21005.4 −1.42567 −0.712837 0.701330i \(-0.752590\pi\)
−0.712837 + 0.701330i \(0.752590\pi\)
\(602\) 5770.28 0.390663
\(603\) 0 0
\(604\) 1052.41 0.0708975
\(605\) −17357.3 −1.16640
\(606\) 0 0
\(607\) 7478.16 0.500048 0.250024 0.968240i \(-0.419561\pi\)
0.250024 + 0.968240i \(0.419561\pi\)
\(608\) −5873.31 −0.391767
\(609\) 0 0
\(610\) 13287.1 0.881934
\(611\) 0 0
\(612\) 0 0
\(613\) 16435.5 1.08291 0.541455 0.840730i \(-0.317874\pi\)
0.541455 + 0.840730i \(0.317874\pi\)
\(614\) 15218.8 1.00029
\(615\) 0 0
\(616\) −16666.0 −1.09009
\(617\) −1290.89 −0.0842289 −0.0421145 0.999113i \(-0.513409\pi\)
−0.0421145 + 0.999113i \(0.513409\pi\)
\(618\) 0 0
\(619\) −26719.0 −1.73494 −0.867470 0.497490i \(-0.834255\pi\)
−0.867470 + 0.497490i \(0.834255\pi\)
\(620\) 539.227 0.0349289
\(621\) 0 0
\(622\) −1999.94 −0.128924
\(623\) −14649.8 −0.942103
\(624\) 0 0
\(625\) 647.683 0.0414517
\(626\) −29559.8 −1.88729
\(627\) 0 0
\(628\) −2574.08 −0.163562
\(629\) 24840.2 1.57463
\(630\) 0 0
\(631\) −10697.4 −0.674893 −0.337447 0.941345i \(-0.609563\pi\)
−0.337447 + 0.941345i \(0.609563\pi\)
\(632\) −25061.4 −1.57735
\(633\) 0 0
\(634\) −1828.96 −0.114570
\(635\) −13632.5 −0.851951
\(636\) 0 0
\(637\) 0 0
\(638\) 42901.2 2.66219
\(639\) 0 0
\(640\) −11777.1 −0.727393
\(641\) −23572.8 −1.45253 −0.726264 0.687416i \(-0.758745\pi\)
−0.726264 + 0.687416i \(0.758745\pi\)
\(642\) 0 0
\(643\) −14000.3 −0.858661 −0.429331 0.903147i \(-0.641250\pi\)
−0.429331 + 0.903147i \(0.641250\pi\)
\(644\) 320.538 0.0196133
\(645\) 0 0
\(646\) 11935.9 0.726953
\(647\) −614.196 −0.0373207 −0.0186604 0.999826i \(-0.505940\pi\)
−0.0186604 + 0.999826i \(0.505940\pi\)
\(648\) 0 0
\(649\) 24168.9 1.46181
\(650\) 0 0
\(651\) 0 0
\(652\) −70.4042 −0.00422890
\(653\) 5333.42 0.319622 0.159811 0.987148i \(-0.448912\pi\)
0.159811 + 0.987148i \(0.448912\pi\)
\(654\) 0 0
\(655\) 12524.9 0.747161
\(656\) −27927.3 −1.66216
\(657\) 0 0
\(658\) −4188.03 −0.248125
\(659\) −21396.8 −1.26480 −0.632398 0.774644i \(-0.717929\pi\)
−0.632398 + 0.774644i \(0.717929\pi\)
\(660\) 0 0
\(661\) −16107.9 −0.947841 −0.473921 0.880568i \(-0.657162\pi\)
−0.473921 + 0.880568i \(0.657162\pi\)
\(662\) 5001.20 0.293621
\(663\) 0 0
\(664\) 17078.2 0.998136
\(665\) 6130.63 0.357497
\(666\) 0 0
\(667\) 2362.62 0.137153
\(668\) 4577.72 0.265145
\(669\) 0 0
\(670\) −18511.4 −1.06740
\(671\) 38814.6 2.23312
\(672\) 0 0
\(673\) 20329.9 1.16443 0.582213 0.813036i \(-0.302187\pi\)
0.582213 + 0.813036i \(0.302187\pi\)
\(674\) 29479.1 1.68471
\(675\) 0 0
\(676\) 0 0
\(677\) 24883.5 1.41263 0.706314 0.707899i \(-0.250357\pi\)
0.706314 + 0.707899i \(0.250357\pi\)
\(678\) 0 0
\(679\) −14827.0 −0.838007
\(680\) 7435.51 0.419322
\(681\) 0 0
\(682\) 7660.78 0.430127
\(683\) 258.953 0.0145074 0.00725369 0.999974i \(-0.497691\pi\)
0.00725369 + 0.999974i \(0.497691\pi\)
\(684\) 0 0
\(685\) 12769.1 0.712240
\(686\) 21819.5 1.21439
\(687\) 0 0
\(688\) 9786.20 0.542290
\(689\) 0 0
\(690\) 0 0
\(691\) −658.193 −0.0362357 −0.0181178 0.999836i \(-0.505767\pi\)
−0.0181178 + 0.999836i \(0.505767\pi\)
\(692\) 1369.39 0.0752262
\(693\) 0 0
\(694\) 24441.3 1.33686
\(695\) 8265.60 0.451125
\(696\) 0 0
\(697\) 21461.2 1.16629
\(698\) −15781.2 −0.855768
\(699\) 0 0
\(700\) 2334.69 0.126062
\(701\) 8222.16 0.443005 0.221503 0.975160i \(-0.428904\pi\)
0.221503 + 0.975160i \(0.428904\pi\)
\(702\) 0 0
\(703\) 27191.3 1.45881
\(704\) −19982.7 −1.06978
\(705\) 0 0
\(706\) −4999.25 −0.266500
\(707\) 20077.6 1.06803
\(708\) 0 0
\(709\) 6817.51 0.361124 0.180562 0.983564i \(-0.442208\pi\)
0.180562 + 0.983564i \(0.442208\pi\)
\(710\) −12996.8 −0.686987
\(711\) 0 0
\(712\) −19449.6 −1.02374
\(713\) 421.888 0.0221596
\(714\) 0 0
\(715\) 0 0
\(716\) 4814.56 0.251297
\(717\) 0 0
\(718\) −24014.8 −1.24822
\(719\) −23385.7 −1.21299 −0.606496 0.795087i \(-0.707425\pi\)
−0.606496 + 0.795087i \(0.707425\pi\)
\(720\) 0 0
\(721\) −5493.73 −0.283769
\(722\) −8701.03 −0.448502
\(723\) 0 0
\(724\) 4394.52 0.225582
\(725\) 17208.5 0.881529
\(726\) 0 0
\(727\) 20488.6 1.04523 0.522613 0.852570i \(-0.324957\pi\)
0.522613 + 0.852570i \(0.324957\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −20986.3 −1.06402
\(731\) −7520.38 −0.380508
\(732\) 0 0
\(733\) −16993.4 −0.856299 −0.428149 0.903708i \(-0.640834\pi\)
−0.428149 + 0.903708i \(0.640834\pi\)
\(734\) 13862.4 0.697099
\(735\) 0 0
\(736\) 999.742 0.0500693
\(737\) −54075.8 −2.70272
\(738\) 0 0
\(739\) −14014.3 −0.697600 −0.348800 0.937197i \(-0.613411\pi\)
−0.348800 + 0.937197i \(0.613411\pi\)
\(740\) −5915.75 −0.293875
\(741\) 0 0
\(742\) 5932.73 0.293528
\(743\) 141.560 0.00698971 0.00349485 0.999994i \(-0.498888\pi\)
0.00349485 + 0.999994i \(0.498888\pi\)
\(744\) 0 0
\(745\) 21543.4 1.05945
\(746\) −2544.92 −0.124901
\(747\) 0 0
\(748\) −7585.76 −0.370806
\(749\) 1225.26 0.0597731
\(750\) 0 0
\(751\) 20734.3 1.00746 0.503732 0.863860i \(-0.331960\pi\)
0.503732 + 0.863860i \(0.331960\pi\)
\(752\) −7102.76 −0.344430
\(753\) 0 0
\(754\) 0 0
\(755\) −3426.23 −0.165157
\(756\) 0 0
\(757\) 17449.8 0.837812 0.418906 0.908030i \(-0.362414\pi\)
0.418906 + 0.908030i \(0.362414\pi\)
\(758\) 216.054 0.0103528
\(759\) 0 0
\(760\) 8139.27 0.388477
\(761\) 424.121 0.0202028 0.0101014 0.999949i \(-0.496785\pi\)
0.0101014 + 0.999949i \(0.496785\pi\)
\(762\) 0 0
\(763\) 13323.3 0.632157
\(764\) 5144.48 0.243613
\(765\) 0 0
\(766\) −4915.68 −0.231868
\(767\) 0 0
\(768\) 0 0
\(769\) 38060.7 1.78479 0.892396 0.451252i \(-0.149023\pi\)
0.892396 + 0.451252i \(0.149023\pi\)
\(770\) −18949.0 −0.886849
\(771\) 0 0
\(772\) −552.413 −0.0257536
\(773\) 16683.3 0.776268 0.388134 0.921603i \(-0.373120\pi\)
0.388134 + 0.921603i \(0.373120\pi\)
\(774\) 0 0
\(775\) 3072.89 0.142428
\(776\) −19684.9 −0.910627
\(777\) 0 0
\(778\) −23167.7 −1.06761
\(779\) 23492.5 1.08050
\(780\) 0 0
\(781\) −37966.5 −1.73950
\(782\) −2031.70 −0.0929073
\(783\) 0 0
\(784\) 10841.9 0.493889
\(785\) 8380.17 0.381020
\(786\) 0 0
\(787\) −39105.2 −1.77122 −0.885609 0.464432i \(-0.846259\pi\)
−0.885609 + 0.464432i \(0.846259\pi\)
\(788\) 2549.14 0.115240
\(789\) 0 0
\(790\) −28494.3 −1.28327
\(791\) 13612.8 0.611903
\(792\) 0 0
\(793\) 0 0
\(794\) 19346.2 0.864697
\(795\) 0 0
\(796\) −6721.90 −0.299311
\(797\) 4346.93 0.193195 0.0965974 0.995324i \(-0.469204\pi\)
0.0965974 + 0.995324i \(0.469204\pi\)
\(798\) 0 0
\(799\) 5458.25 0.241676
\(800\) 7281.78 0.321812
\(801\) 0 0
\(802\) −24093.9 −1.06083
\(803\) −61305.6 −2.69418
\(804\) 0 0
\(805\) −1043.54 −0.0456895
\(806\) 0 0
\(807\) 0 0
\(808\) 26655.8 1.16058
\(809\) 23030.2 1.00086 0.500432 0.865776i \(-0.333174\pi\)
0.500432 + 0.865776i \(0.333174\pi\)
\(810\) 0 0
\(811\) 7898.11 0.341973 0.170987 0.985273i \(-0.445305\pi\)
0.170987 + 0.985273i \(0.445305\pi\)
\(812\) 6348.41 0.274366
\(813\) 0 0
\(814\) −84044.8 −3.61888
\(815\) 229.207 0.00985127
\(816\) 0 0
\(817\) −8232.18 −0.352518
\(818\) −22956.2 −0.981230
\(819\) 0 0
\(820\) −5111.04 −0.217665
\(821\) 3939.61 0.167471 0.0837353 0.996488i \(-0.473315\pi\)
0.0837353 + 0.996488i \(0.473315\pi\)
\(822\) 0 0
\(823\) −17599.8 −0.745430 −0.372715 0.927946i \(-0.621573\pi\)
−0.372715 + 0.927946i \(0.621573\pi\)
\(824\) −7293.70 −0.308359
\(825\) 0 0
\(826\) 17393.6 0.732687
\(827\) −12510.6 −0.526042 −0.263021 0.964790i \(-0.584719\pi\)
−0.263021 + 0.964790i \(0.584719\pi\)
\(828\) 0 0
\(829\) 28630.8 1.19950 0.599752 0.800186i \(-0.295266\pi\)
0.599752 + 0.800186i \(0.295266\pi\)
\(830\) 19417.6 0.812042
\(831\) 0 0
\(832\) 0 0
\(833\) −8331.62 −0.346547
\(834\) 0 0
\(835\) −14903.2 −0.617660
\(836\) −8303.75 −0.343530
\(837\) 0 0
\(838\) 16859.1 0.694973
\(839\) −21250.3 −0.874426 −0.437213 0.899358i \(-0.644034\pi\)
−0.437213 + 0.899358i \(0.644034\pi\)
\(840\) 0 0
\(841\) 22403.7 0.918600
\(842\) 47690.9 1.95194
\(843\) 0 0
\(844\) 684.810 0.0279291
\(845\) 0 0
\(846\) 0 0
\(847\) −36490.4 −1.48031
\(848\) 10061.7 0.407454
\(849\) 0 0
\(850\) −14798.2 −0.597147
\(851\) −4628.45 −0.186441
\(852\) 0 0
\(853\) 34721.1 1.39370 0.696852 0.717215i \(-0.254584\pi\)
0.696852 + 0.717215i \(0.254584\pi\)
\(854\) 27933.6 1.11928
\(855\) 0 0
\(856\) 1626.71 0.0649529
\(857\) −4898.06 −0.195233 −0.0976163 0.995224i \(-0.531122\pi\)
−0.0976163 + 0.995224i \(0.531122\pi\)
\(858\) 0 0
\(859\) −12564.2 −0.499051 −0.249526 0.968368i \(-0.580275\pi\)
−0.249526 + 0.968368i \(0.580275\pi\)
\(860\) 1791.00 0.0710145
\(861\) 0 0
\(862\) 22704.9 0.897137
\(863\) 22826.6 0.900380 0.450190 0.892933i \(-0.351356\pi\)
0.450190 + 0.892933i \(0.351356\pi\)
\(864\) 0 0
\(865\) −4458.18 −0.175240
\(866\) −30282.8 −1.18828
\(867\) 0 0
\(868\) 1133.62 0.0443291
\(869\) −83238.3 −3.24933
\(870\) 0 0
\(871\) 0 0
\(872\) 17688.6 0.686939
\(873\) 0 0
\(874\) −2224.00 −0.0860731
\(875\) −19543.9 −0.755089
\(876\) 0 0
\(877\) 25212.7 0.970776 0.485388 0.874299i \(-0.338678\pi\)
0.485388 + 0.874299i \(0.338678\pi\)
\(878\) 22438.5 0.862484
\(879\) 0 0
\(880\) −32136.9 −1.23106
\(881\) 18026.2 0.689352 0.344676 0.938722i \(-0.387989\pi\)
0.344676 + 0.938722i \(0.387989\pi\)
\(882\) 0 0
\(883\) −18833.1 −0.717764 −0.358882 0.933383i \(-0.616842\pi\)
−0.358882 + 0.933383i \(0.616842\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −6640.68 −0.251803
\(887\) −38451.4 −1.45555 −0.727775 0.685816i \(-0.759445\pi\)
−0.727775 + 0.685816i \(0.759445\pi\)
\(888\) 0 0
\(889\) −28659.7 −1.08123
\(890\) −22113.9 −0.832876
\(891\) 0 0
\(892\) 11981.0 0.449723
\(893\) 5974.86 0.223898
\(894\) 0 0
\(895\) −15674.2 −0.585399
\(896\) −24759.1 −0.923152
\(897\) 0 0
\(898\) −18539.4 −0.688940
\(899\) 8355.68 0.309986
\(900\) 0 0
\(901\) −7732.11 −0.285898
\(902\) −72612.3 −2.68041
\(903\) 0 0
\(904\) 18072.9 0.664929
\(905\) −14306.8 −0.525495
\(906\) 0 0
\(907\) −5531.31 −0.202496 −0.101248 0.994861i \(-0.532284\pi\)
−0.101248 + 0.994861i \(0.532284\pi\)
\(908\) −6099.05 −0.222912
\(909\) 0 0
\(910\) 0 0
\(911\) −15695.2 −0.570806 −0.285403 0.958408i \(-0.592128\pi\)
−0.285403 + 0.958408i \(0.592128\pi\)
\(912\) 0 0
\(913\) 56723.1 2.05615
\(914\) 18895.9 0.683832
\(915\) 0 0
\(916\) −7333.01 −0.264508
\(917\) 26331.3 0.948240
\(918\) 0 0
\(919\) −51503.5 −1.84869 −0.924344 0.381561i \(-0.875386\pi\)
−0.924344 + 0.381561i \(0.875386\pi\)
\(920\) −1385.45 −0.0496488
\(921\) 0 0
\(922\) 5921.45 0.211510
\(923\) 0 0
\(924\) 0 0
\(925\) −33712.0 −1.19832
\(926\) −21262.2 −0.754557
\(927\) 0 0
\(928\) 19800.3 0.700407
\(929\) 42927.0 1.51603 0.758014 0.652238i \(-0.226170\pi\)
0.758014 + 0.652238i \(0.226170\pi\)
\(930\) 0 0
\(931\) −9120.20 −0.321055
\(932\) −4847.10 −0.170356
\(933\) 0 0
\(934\) 52634.1 1.84394
\(935\) 24696.2 0.863797
\(936\) 0 0
\(937\) 44206.6 1.54127 0.770633 0.637280i \(-0.219940\pi\)
0.770633 + 0.637280i \(0.219940\pi\)
\(938\) −38916.6 −1.35466
\(939\) 0 0
\(940\) −1299.89 −0.0451041
\(941\) 44067.7 1.52664 0.763319 0.646022i \(-0.223568\pi\)
0.763319 + 0.646022i \(0.223568\pi\)
\(942\) 0 0
\(943\) −3998.85 −0.138092
\(944\) 29498.9 1.01706
\(945\) 0 0
\(946\) 25444.6 0.874498
\(947\) 44402.5 1.52364 0.761820 0.647789i \(-0.224306\pi\)
0.761820 + 0.647789i \(0.224306\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −16198.9 −0.553221
\(951\) 0 0
\(952\) 15631.7 0.532172
\(953\) 10361.7 0.352202 0.176101 0.984372i \(-0.443651\pi\)
0.176101 + 0.984372i \(0.443651\pi\)
\(954\) 0 0
\(955\) −16748.3 −0.567500
\(956\) 3138.60 0.106182
\(957\) 0 0
\(958\) 19570.2 0.660006
\(959\) 26844.7 0.903920
\(960\) 0 0
\(961\) −28298.9 −0.949916
\(962\) 0 0
\(963\) 0 0
\(964\) 4954.18 0.165522
\(965\) 1798.43 0.0599933
\(966\) 0 0
\(967\) 8432.54 0.280426 0.140213 0.990121i \(-0.455221\pi\)
0.140213 + 0.990121i \(0.455221\pi\)
\(968\) −48446.1 −1.60859
\(969\) 0 0
\(970\) −22381.4 −0.740848
\(971\) 36917.1 1.22011 0.610055 0.792359i \(-0.291147\pi\)
0.610055 + 0.792359i \(0.291147\pi\)
\(972\) 0 0
\(973\) 17376.8 0.572534
\(974\) 18147.4 0.597001
\(975\) 0 0
\(976\) 47374.5 1.55371
\(977\) 38306.9 1.25440 0.627199 0.778859i \(-0.284201\pi\)
0.627199 + 0.778859i \(0.284201\pi\)
\(978\) 0 0
\(979\) −64599.6 −2.10890
\(980\) 1984.19 0.0646763
\(981\) 0 0
\(982\) 66835.6 2.17190
\(983\) 18810.9 0.610350 0.305175 0.952296i \(-0.401285\pi\)
0.305175 + 0.952296i \(0.401285\pi\)
\(984\) 0 0
\(985\) −8298.97 −0.268454
\(986\) −40238.8 −1.29966
\(987\) 0 0
\(988\) 0 0
\(989\) 1401.26 0.0450532
\(990\) 0 0
\(991\) −6492.09 −0.208101 −0.104051 0.994572i \(-0.533180\pi\)
−0.104051 + 0.994572i \(0.533180\pi\)
\(992\) 3535.71 0.113164
\(993\) 0 0
\(994\) −27323.2 −0.871872
\(995\) 21883.8 0.697249
\(996\) 0 0
\(997\) 48552.1 1.54229 0.771143 0.636662i \(-0.219685\pi\)
0.771143 + 0.636662i \(0.219685\pi\)
\(998\) −24955.5 −0.791537
\(999\) 0 0
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 1521.4.a.bf.1.8 9
3.2 odd 2 507.4.a.p.1.2 yes 9
13.12 even 2 1521.4.a.bi.1.2 9
39.5 even 4 507.4.b.k.337.14 18
39.8 even 4 507.4.b.k.337.5 18
39.38 odd 2 507.4.a.o.1.8 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.8 9 39.38 odd 2
507.4.a.p.1.2 yes 9 3.2 odd 2
507.4.b.k.337.5 18 39.8 even 4
507.4.b.k.337.14 18 39.5 even 4
1521.4.a.bf.1.8 9 1.1 even 1 trivial
1521.4.a.bi.1.2 9 13.12 even 2