Properties

Label 1521.4.a.bi.1.2
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
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: \(1\)
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 1.2
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 \(-0.689580\pi\)
−0.560991 + 0.827822i \(0.689580\pi\)
\(3\) 0 0
\(4\) 2.07074 0.258842
\(5\) 6.74147 0.602975 0.301488 0.953470i \(-0.402517\pi\)
0.301488 + 0.953470i \(0.402517\pi\)
\(6\) 0 0
\(7\) 14.1726 0.765251 0.382625 0.923904i \(-0.375020\pi\)
0.382625 + 0.923904i \(0.375020\pi\)
\(8\) 18.8162 0.831565
\(9\) 0 0
\(10\) −21.3937 −0.676527
\(11\) 62.4956 1.71301 0.856507 0.516136i \(-0.172630\pi\)
0.856507 + 0.516136i \(0.172630\pi\)
\(12\) 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 \(-0.626620\pi\)
−0.387382 + 0.921919i \(0.626620\pi\)
\(20\) 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 \(-0.464307\pi\)
0.111897 + 0.993720i \(0.464307\pi\)
\(32\) 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 \(-0.890538\pi\)
−0.941452 + 0.337147i \(0.890538\pi\)
\(38\) 203.625 0.869270
\(39\) 0 0
\(40\) 126.849 0.501414
\(41\) −366.126 −1.39461 −0.697307 0.716772i \(-0.745619\pi\)
−0.697307 + 0.716772i \(0.745619\pi\)
\(42\) 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 \(-0.546156\pi\)
−0.144495 + 0.989506i \(0.546156\pi\)
\(48\) 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 \(-0.359684\pi\)
0.426677 + 0.904404i \(0.359684\pi\)
\(60\) 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 \(-0.789339\pi\)
−0.788880 + 0.614547i \(0.789339\pi\)
\(68\) 121.381 0.216464
\(69\) 0 0
\(70\) −303.205 −0.517713
\(71\) −607.506 −1.01546 −0.507730 0.861516i \(-0.669515\pi\)
−0.507730 + 0.861516i \(0.669515\pi\)
\(72\) 0 0
\(73\) −980.958 −1.57277 −0.786387 0.617735i \(-0.788051\pi\)
−0.786387 + 0.617735i \(0.788051\pi\)
\(74\) 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 \(-0.295106\pi\)
0.600155 + 0.799884i \(0.295106\pi\)
\(84\) 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 \(-0.711067\pi\)
−0.615552 + 0.788096i \(0.711067\pi\)
\(90\) 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 \(-0.684435\pi\)
−0.547538 + 0.836781i \(0.684435\pi\)
\(98\) 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 \(-0.364467\pi\)
0.413039 + 0.910713i \(0.364467\pi\)
\(110\) −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 \(-0.298890\pi\)
0.590604 + 0.806961i \(0.298890\pi\)
\(138\) 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 \(-0.158533\pi\)
0.878517 + 0.477711i \(0.158533\pi\)
\(150\) 0 0
\(151\) −508.232 −0.273903 −0.136951 0.990578i \(-0.543730\pi\)
−0.136951 + 0.990578i \(0.543730\pi\)
\(152\) −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 \(-0.497400\pi\)
0.00816888 + 0.999967i \(0.497400\pi\)
\(164\) −758.149 −0.360985
\(165\) 0 0
\(166\) −2880.32 −1.34673
\(167\) −2210.67 −1.02435 −0.512176 0.858880i \(-0.671161\pi\)
−0.512176 + 0.858880i \(0.671161\pi\)
\(168\) 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 \(-0.484158\pi\)
0.0497478 + 0.998762i \(0.484158\pi\)
\(194\) 3319.95 1.22865
\(195\) 0 0
\(196\) −294.327 −0.107262
\(197\) −1231.03 −0.445216 −0.222608 0.974908i \(-0.571457\pi\)
−0.222608 + 0.974908i \(0.571457\pi\)
\(198\) 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 \(-0.835058\pi\)
−0.868722 + 0.495300i \(0.835058\pi\)
\(224\) 1297.28 0.386957
\(225\) 0 0
\(226\) 3048.09 0.897149
\(227\) 2945.35 0.861189 0.430595 0.902545i \(-0.358304\pi\)
0.430595 + 0.902545i \(0.358304\pi\)
\(228\) 0 0
\(229\) 3541.26 1.02189 0.510945 0.859613i \(-0.329296\pi\)
0.510945 + 0.859613i \(0.329296\pi\)
\(230\) 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 \(-0.565755\pi\)
−0.205109 + 0.978739i \(0.565755\pi\)
\(240\) 0 0
\(241\) −2392.47 −0.639472 −0.319736 0.947507i \(-0.603594\pi\)
−0.319736 + 0.947507i \(0.603594\pi\)
\(242\) −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 \(-0.201636\pi\)
0.805986 + 0.591935i \(0.201636\pi\)
\(272\) −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 \(-0.637679\pi\)
−0.419169 + 0.907908i \(0.637679\pi\)
\(282\) 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 \(-0.603455\pi\)
−0.319322 + 0.947646i \(0.603455\pi\)
\(294\) 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 \(-0.647070\pi\)
−0.445771 + 0.895147i \(0.647070\pi\)
\(308\) 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 \(-0.483741\pi\)
0.0510569 + 0.998696i \(0.483741\pi\)
\(318\) 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 \(-0.541770\pi\)
−0.130849 + 0.991402i \(0.541770\pi\)
\(332\) 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 \(-0.375454\pi\)
0.381365 + 0.924425i \(0.375454\pi\)
\(350\) 3577.96 0.546429
\(351\) 0 0
\(352\) 5720.49 0.866202
\(353\) 1575.34 0.237526 0.118763 0.992923i \(-0.462107\pi\)
0.118763 + 0.992923i \(0.462107\pi\)
\(354\) 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 \(-0.312237\pi\)
0.556258 + 0.831010i \(0.312237\pi\)
\(360\) 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 \(-0.501469\pi\)
−0.00461363 + 0.999989i \(0.501469\pi\)
\(380\) −895.734 −0.120922
\(381\) 0 0
\(382\) −7884.01 −1.05597
\(383\) 1549.01 0.206659 0.103330 0.994647i \(-0.467050\pi\)
0.103330 + 0.994647i \(0.467050\pi\)
\(384\) 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 \(-0.625917\pi\)
−0.385344 + 0.922773i \(0.625917\pi\)
\(398\) 10301.4 1.29740
\(399\) 0 0
\(400\) 6068.11 0.758513
\(401\) 7592.37 0.945498 0.472749 0.881197i \(-0.343262\pi\)
0.472749 + 0.881197i \(0.343262\pi\)
\(402\) 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 \(-0.355943\pi\)
0.437275 + 0.899328i \(0.355943\pi\)
\(410\) 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 \(-0.835794\pi\)
−0.869865 + 0.493290i \(0.835794\pi\)
\(422\) −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 \(-0.630920\pi\)
−0.399800 + 0.916602i \(0.630920\pi\)
\(432\) 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 \(-0.400668\pi\)
0.307019 + 0.951703i \(0.400668\pi\)
\(450\) 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 \(-0.598571\pi\)
−0.304743 + 0.952435i \(0.598571\pi\)
\(458\) −11238.0 −1.14654
\(459\) 0 0
\(460\) −152.470 −0.0154542
\(461\) −1865.94 −0.188515 −0.0942576 0.995548i \(-0.530048\pi\)
−0.0942576 + 0.995548i \(0.530048\pi\)
\(462\) 0 0
\(463\) 6700.05 0.672522 0.336261 0.941769i \(-0.390838\pi\)
0.336261 + 0.941769i \(0.390838\pi\)
\(464\) 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 \(-0.595028\pi\)
−0.294125 + 0.955767i \(0.595028\pi\)
\(480\) 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 \(-0.585718\pi\)
−0.266047 + 0.963960i \(0.585718\pi\)
\(488\) −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 \(-0.385250\pi\)
0.352741 + 0.935721i \(0.385250\pi\)
\(500\) −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 \(-0.277121\pi\)
0.644367 + 0.764717i \(0.277121\pi\)
\(510\) 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 \(-0.612160\pi\)
−0.345116 + 0.938560i \(0.612160\pi\)
\(542\) −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 \(-0.386002\pi\)
0.350530 + 0.936552i \(0.386002\pi\)
\(558\) 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 \(-0.406310\pi\)
0.290103 + 0.956995i \(0.406310\pi\)
\(578\) 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 \(-0.672113\pi\)
−0.514744 + 0.857344i \(0.672113\pi\)
\(588\) 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 \(-0.436361\pi\)
0.198600 + 0.980081i \(0.436361\pi\)
\(594\) 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 \(-0.682126\pi\)
−0.541455 + 0.840730i \(0.682126\pi\)
\(614\) 15218.8 1.00029
\(615\) 0 0
\(616\) 16666.0 1.09009
\(617\) 1290.89 0.0842289 0.0421145 0.999113i \(-0.486591\pi\)
0.0421145 + 0.999113i \(0.486591\pi\)
\(618\) 0 0
\(619\) 26719.0 1.73494 0.867470 0.497490i \(-0.165745\pi\)
0.867470 + 0.497490i \(0.165745\pi\)
\(620\) 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 \(-0.390437\pi\)
0.337447 + 0.941345i \(0.390437\pi\)
\(632\) 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 \(-0.358750\pi\)
0.429331 + 0.903147i \(0.358750\pi\)
\(644\) −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 \(-0.342838\pi\)
0.473921 + 0.880568i \(0.342838\pi\)
\(662\) 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 \(-0.502309\pi\)
−0.00725369 + 0.999974i \(0.502309\pi\)
\(684\) 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 \(-0.494233\pi\)
0.0181178 + 0.999836i \(0.494233\pi\)
\(692\) 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 \(-0.557792\pi\)
−0.180562 + 0.983564i \(0.557792\pi\)
\(710\) 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 \(-0.359166\pi\)
0.428149 + 0.903708i \(0.359166\pi\)
\(734\) −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 \(-0.386589\pi\)
0.348800 + 0.937197i \(0.386589\pi\)
\(740\) −5915.75 −0.293875
\(741\) 0 0
\(742\) 5932.73 0.293528
\(743\) −141.560 −0.00698971 −0.00349485 0.999994i \(-0.501112\pi\)
−0.00349485 + 0.999994i \(0.501112\pi\)
\(744\) 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 \(-0.503215\pi\)
−0.0101014 + 0.999949i \(0.503215\pi\)
\(762\) 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 \(-0.850977\pi\)
−0.892396 + 0.451252i \(0.850977\pi\)
\(770\) −18949.0 −0.886849
\(771\) 0 0
\(772\) 552.413 0.0257536
\(773\) −16683.3 −0.776268 −0.388134 0.921603i \(-0.626880\pi\)
−0.388134 + 0.921603i \(0.626880\pi\)
\(774\) 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 \(-0.153741\pi\)
0.885609 + 0.464432i \(0.153741\pi\)
\(788\) −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 \(-0.554695\pi\)
−0.170987 + 0.985273i \(0.554695\pi\)
\(812\) −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 \(-0.526685\pi\)
−0.0837353 + 0.996488i \(0.526685\pi\)
\(822\) 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 \(-0.415281\pi\)
0.263021 + 0.964790i \(0.415281\pi\)
\(828\) 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 \(-0.355966\pi\)
0.437213 + 0.899358i \(0.355966\pi\)
\(840\) 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 \(-0.745416\pi\)
−0.696852 + 0.717215i \(0.745416\pi\)
\(854\) 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 \(-0.648644\pi\)
−0.450190 + 0.892933i \(0.648644\pi\)
\(864\) 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 \(-0.661322\pi\)
−0.485388 + 0.874299i \(0.661322\pi\)
\(878\) −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 \(-0.773830\pi\)
−0.758014 + 0.652238i \(0.773830\pi\)
\(930\) 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 \(-0.776432\pi\)
−0.763319 + 0.646022i \(0.776432\pi\)
\(942\) 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 \(-0.775694\pi\)
−0.761820 + 0.647789i \(0.775694\pi\)
\(948\) 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 \(-0.544779\pi\)
−0.140213 + 0.990121i \(0.544779\pi\)
\(968\) 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 \(-0.715799\pi\)
−0.627199 + 0.778859i \(0.715799\pi\)
\(978\) 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 \(-0.598715\pi\)
−0.305175 + 0.952296i \(0.598715\pi\)
\(984\) 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.bi.1.2 9
3.2 odd 2 507.4.a.o.1.8 9
13.12 even 2 1521.4.a.bf.1.8 9
39.5 even 4 507.4.b.k.337.5 18
39.8 even 4 507.4.b.k.337.14 18
39.38 odd 2 507.4.a.p.1.2 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.8 9 3.2 odd 2
507.4.a.p.1.2 yes 9 39.38 odd 2
507.4.b.k.337.5 18 39.5 even 4
507.4.b.k.337.14 18 39.8 even 4
1521.4.a.bf.1.8 9 13.12 even 2
1521.4.a.bi.1.2 9 1.1 even 1 trivial