Properties

Label 1521.4.a.r.1.2
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{2} -1.43845 q^{4} +0.561553 q^{5} -18.1771 q^{7} -24.1771 q^{8} +O(q^{10})\) \(q+2.56155 q^{2} -1.43845 q^{4} +0.561553 q^{5} -18.1771 q^{7} -24.1771 q^{8} +1.43845 q^{10} +64.7386 q^{11} -46.5616 q^{14} -50.4233 q^{16} +25.5464 q^{17} +107.970 q^{19} -0.807764 q^{20} +165.831 q^{22} -73.2614 q^{23} -124.685 q^{25} +26.1468 q^{28} -175.909 q^{29} +113.093 q^{31} +64.2547 q^{32} +65.4384 q^{34} -10.2074 q^{35} -114.808 q^{37} +276.570 q^{38} -13.5767 q^{40} -69.6458 q^{41} +438.302 q^{43} -93.1231 q^{44} -187.663 q^{46} -31.9479 q^{47} -12.5937 q^{49} -319.386 q^{50} -2.84658 q^{53} +36.3542 q^{55} +439.469 q^{56} -450.600 q^{58} +71.6325 q^{59} -920.695 q^{61} +289.693 q^{62} +567.978 q^{64} +444.280 q^{67} -36.7471 q^{68} -26.1468 q^{70} -541.719 q^{71} -764.004 q^{73} -294.086 q^{74} -155.309 q^{76} -1176.76 q^{77} -421.538 q^{79} -28.3153 q^{80} -178.401 q^{82} +603.797 q^{83} +14.3457 q^{85} +1122.73 q^{86} -1565.19 q^{88} -1159.88 q^{89} +105.383 q^{92} -81.8362 q^{94} +60.6307 q^{95} -583.269 q^{97} -32.2595 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8} + 7 q^{10} + 80 q^{11} - 89 q^{14} - 39 q^{16} - 19 q^{17} + 84 q^{19} + 19 q^{20} + 142 q^{22} - 196 q^{23} - 237 q^{25} - 125 q^{28} + 44 q^{29} + 86 q^{31} - 123 q^{32} + 135 q^{34} - 107 q^{35} - 209 q^{37} + 314 q^{38} - 89 q^{40} - 230 q^{41} + 287 q^{43} - 178 q^{44} + 4 q^{46} + 435 q^{47} + 383 q^{49} - 144 q^{50} + 118 q^{53} - 18 q^{55} + 1015 q^{56} - 794 q^{58} - 368 q^{59} - 1058 q^{61} + 332 q^{62} + 769 q^{64} - 68 q^{67} + 211 q^{68} + 125 q^{70} - 131 q^{71} - 456 q^{73} - 147 q^{74} - 22 q^{76} - 762 q^{77} - 1008 q^{79} - 69 q^{80} + 72 q^{82} + 1958 q^{83} + 173 q^{85} + 1359 q^{86} - 1242 q^{88} - 720 q^{89} + 788 q^{92} - 811 q^{94} + 146 q^{95} + 928 q^{97} - 650 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56155 0.905646 0.452823 0.891601i \(-0.350417\pi\)
0.452823 + 0.891601i \(0.350417\pi\)
\(3\) 0 0
\(4\) −1.43845 −0.179806
\(5\) 0.561553 0.0502268 0.0251134 0.999685i \(-0.492005\pi\)
0.0251134 + 0.999685i \(0.492005\pi\)
\(6\) 0 0
\(7\) −18.1771 −0.981470 −0.490735 0.871309i \(-0.663272\pi\)
−0.490735 + 0.871309i \(0.663272\pi\)
\(8\) −24.1771 −1.06849
\(9\) 0 0
\(10\) 1.43845 0.0454877
\(11\) 64.7386 1.77449 0.887247 0.461295i \(-0.152615\pi\)
0.887247 + 0.461295i \(0.152615\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −46.5616 −0.888864
\(15\) 0 0
\(16\) −50.4233 −0.787864
\(17\) 25.5464 0.364465 0.182233 0.983255i \(-0.441668\pi\)
0.182233 + 0.983255i \(0.441668\pi\)
\(18\) 0 0
\(19\) 107.970 1.30368 0.651841 0.758356i \(-0.273997\pi\)
0.651841 + 0.758356i \(0.273997\pi\)
\(20\) −0.807764 −0.00903108
\(21\) 0 0
\(22\) 165.831 1.60706
\(23\) −73.2614 −0.664176 −0.332088 0.943248i \(-0.607753\pi\)
−0.332088 + 0.943248i \(0.607753\pi\)
\(24\) 0 0
\(25\) −124.685 −0.997477
\(26\) 0 0
\(27\) 0 0
\(28\) 26.1468 0.176474
\(29\) −175.909 −1.12640 −0.563198 0.826322i \(-0.690429\pi\)
−0.563198 + 0.826322i \(0.690429\pi\)
\(30\) 0 0
\(31\) 113.093 0.655228 0.327614 0.944812i \(-0.393755\pi\)
0.327614 + 0.944812i \(0.393755\pi\)
\(32\) 64.2547 0.354961
\(33\) 0 0
\(34\) 65.4384 0.330077
\(35\) −10.2074 −0.0492961
\(36\) 0 0
\(37\) −114.808 −0.510116 −0.255058 0.966926i \(-0.582095\pi\)
−0.255058 + 0.966926i \(0.582095\pi\)
\(38\) 276.570 1.18067
\(39\) 0 0
\(40\) −13.5767 −0.0536666
\(41\) −69.6458 −0.265289 −0.132645 0.991164i \(-0.542347\pi\)
−0.132645 + 0.991164i \(0.542347\pi\)
\(42\) 0 0
\(43\) 438.302 1.55443 0.777214 0.629236i \(-0.216632\pi\)
0.777214 + 0.629236i \(0.216632\pi\)
\(44\) −93.1231 −0.319064
\(45\) 0 0
\(46\) −187.663 −0.601508
\(47\) −31.9479 −0.0991506 −0.0495753 0.998770i \(-0.515787\pi\)
−0.0495753 + 0.998770i \(0.515787\pi\)
\(48\) 0 0
\(49\) −12.5937 −0.0367164
\(50\) −319.386 −0.903361
\(51\) 0 0
\(52\) 0 0
\(53\) −2.84658 −0.00737752 −0.00368876 0.999993i \(-0.501174\pi\)
−0.00368876 + 0.999993i \(0.501174\pi\)
\(54\) 0 0
\(55\) 36.3542 0.0891272
\(56\) 439.469 1.04869
\(57\) 0 0
\(58\) −450.600 −1.02012
\(59\) 71.6325 0.158064 0.0790319 0.996872i \(-0.474817\pi\)
0.0790319 + 0.996872i \(0.474817\pi\)
\(60\) 0 0
\(61\) −920.695 −1.93251 −0.966253 0.257593i \(-0.917071\pi\)
−0.966253 + 0.257593i \(0.917071\pi\)
\(62\) 289.693 0.593404
\(63\) 0 0
\(64\) 567.978 1.10933
\(65\) 0 0
\(66\) 0 0
\(67\) 444.280 0.810112 0.405056 0.914292i \(-0.367252\pi\)
0.405056 + 0.914292i \(0.367252\pi\)
\(68\) −36.7471 −0.0655330
\(69\) 0 0
\(70\) −26.1468 −0.0446448
\(71\) −541.719 −0.905496 −0.452748 0.891639i \(-0.649556\pi\)
−0.452748 + 0.891639i \(0.649556\pi\)
\(72\) 0 0
\(73\) −764.004 −1.22493 −0.612465 0.790498i \(-0.709822\pi\)
−0.612465 + 0.790498i \(0.709822\pi\)
\(74\) −294.086 −0.461984
\(75\) 0 0
\(76\) −155.309 −0.234410
\(77\) −1176.76 −1.74161
\(78\) 0 0
\(79\) −421.538 −0.600338 −0.300169 0.953886i \(-0.597043\pi\)
−0.300169 + 0.953886i \(0.597043\pi\)
\(80\) −28.3153 −0.0395719
\(81\) 0 0
\(82\) −178.401 −0.240258
\(83\) 603.797 0.798498 0.399249 0.916842i \(-0.369271\pi\)
0.399249 + 0.916842i \(0.369271\pi\)
\(84\) 0 0
\(85\) 14.3457 0.0183059
\(86\) 1122.73 1.40776
\(87\) 0 0
\(88\) −1565.19 −1.89602
\(89\) −1159.88 −1.38143 −0.690715 0.723127i \(-0.742704\pi\)
−0.690715 + 0.723127i \(0.742704\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 105.383 0.119423
\(93\) 0 0
\(94\) −81.8362 −0.0897953
\(95\) 60.6307 0.0654798
\(96\) 0 0
\(97\) −583.269 −0.610536 −0.305268 0.952267i \(-0.598746\pi\)
−0.305268 + 0.952267i \(0.598746\pi\)
\(98\) −32.2595 −0.0332521
\(99\) 0 0
\(100\) 179.352 0.179352
\(101\) −921.740 −0.908085 −0.454043 0.890980i \(-0.650019\pi\)
−0.454043 + 0.890980i \(0.650019\pi\)
\(102\) 0 0
\(103\) −930.712 −0.890347 −0.445174 0.895444i \(-0.646858\pi\)
−0.445174 + 0.895444i \(0.646858\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −7.29168 −0.00668142
\(107\) −857.383 −0.774638 −0.387319 0.921946i \(-0.626599\pi\)
−0.387319 + 0.921946i \(0.626599\pi\)
\(108\) 0 0
\(109\) −671.853 −0.590384 −0.295192 0.955438i \(-0.595384\pi\)
−0.295192 + 0.955438i \(0.595384\pi\)
\(110\) 93.1231 0.0807176
\(111\) 0 0
\(112\) 916.548 0.773265
\(113\) −641.474 −0.534024 −0.267012 0.963693i \(-0.586036\pi\)
−0.267012 + 0.963693i \(0.586036\pi\)
\(114\) 0 0
\(115\) −41.1401 −0.0333594
\(116\) 253.036 0.202533
\(117\) 0 0
\(118\) 183.491 0.143150
\(119\) −464.359 −0.357712
\(120\) 0 0
\(121\) 2860.09 2.14883
\(122\) −2358.41 −1.75017
\(123\) 0 0
\(124\) −162.678 −0.117814
\(125\) −140.211 −0.100327
\(126\) 0 0
\(127\) −553.174 −0.386506 −0.193253 0.981149i \(-0.561904\pi\)
−0.193253 + 0.981149i \(0.561904\pi\)
\(128\) 940.868 0.649702
\(129\) 0 0
\(130\) 0 0
\(131\) −2056.40 −1.37152 −0.685758 0.727830i \(-0.740529\pi\)
−0.685758 + 0.727830i \(0.740529\pi\)
\(132\) 0 0
\(133\) −1962.57 −1.27952
\(134\) 1138.05 0.733674
\(135\) 0 0
\(136\) −617.637 −0.389426
\(137\) −1808.57 −1.12786 −0.563928 0.825824i \(-0.690710\pi\)
−0.563928 + 0.825824i \(0.690710\pi\)
\(138\) 0 0
\(139\) 1493.64 0.911428 0.455714 0.890126i \(-0.349384\pi\)
0.455714 + 0.890126i \(0.349384\pi\)
\(140\) 14.6828 0.00886373
\(141\) 0 0
\(142\) −1387.64 −0.820058
\(143\) 0 0
\(144\) 0 0
\(145\) −98.7822 −0.0565753
\(146\) −1957.04 −1.10935
\(147\) 0 0
\(148\) 165.145 0.0917218
\(149\) −2759.02 −1.51696 −0.758482 0.651694i \(-0.774059\pi\)
−0.758482 + 0.651694i \(0.774059\pi\)
\(150\) 0 0
\(151\) 976.355 0.526190 0.263095 0.964770i \(-0.415257\pi\)
0.263095 + 0.964770i \(0.415257\pi\)
\(152\) −2610.39 −1.39297
\(153\) 0 0
\(154\) −3014.33 −1.57728
\(155\) 63.5076 0.0329100
\(156\) 0 0
\(157\) −564.875 −0.287146 −0.143573 0.989640i \(-0.545859\pi\)
−0.143573 + 0.989640i \(0.545859\pi\)
\(158\) −1079.79 −0.543694
\(159\) 0 0
\(160\) 36.0824 0.0178285
\(161\) 1331.68 0.651869
\(162\) 0 0
\(163\) −1508.53 −0.724892 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(164\) 100.182 0.0477005
\(165\) 0 0
\(166\) 1546.66 0.723157
\(167\) 592.521 0.274555 0.137277 0.990533i \(-0.456165\pi\)
0.137277 + 0.990533i \(0.456165\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 36.7471 0.0165787
\(171\) 0 0
\(172\) −630.474 −0.279495
\(173\) 4495.57 1.97568 0.987838 0.155488i \(-0.0496952\pi\)
0.987838 + 0.155488i \(0.0496952\pi\)
\(174\) 0 0
\(175\) 2266.40 0.978994
\(176\) −3264.34 −1.39806
\(177\) 0 0
\(178\) −2971.10 −1.25109
\(179\) 154.285 0.0644235 0.0322117 0.999481i \(-0.489745\pi\)
0.0322117 + 0.999481i \(0.489745\pi\)
\(180\) 0 0
\(181\) 1071.35 0.439959 0.219979 0.975505i \(-0.429401\pi\)
0.219979 + 0.975505i \(0.429401\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1771.25 0.709663
\(185\) −64.4706 −0.0256215
\(186\) 0 0
\(187\) 1653.84 0.646742
\(188\) 45.9554 0.0178279
\(189\) 0 0
\(190\) 155.309 0.0593015
\(191\) −677.203 −0.256548 −0.128274 0.991739i \(-0.540944\pi\)
−0.128274 + 0.991739i \(0.540944\pi\)
\(192\) 0 0
\(193\) −1321.68 −0.492936 −0.246468 0.969151i \(-0.579270\pi\)
−0.246468 + 0.969151i \(0.579270\pi\)
\(194\) −1494.07 −0.552929
\(195\) 0 0
\(196\) 18.1154 0.00660183
\(197\) 1267.37 0.458356 0.229178 0.973385i \(-0.426396\pi\)
0.229178 + 0.973385i \(0.426396\pi\)
\(198\) 0 0
\(199\) 2396.24 0.853593 0.426796 0.904348i \(-0.359642\pi\)
0.426796 + 0.904348i \(0.359642\pi\)
\(200\) 3014.51 1.06579
\(201\) 0 0
\(202\) −2361.09 −0.822403
\(203\) 3197.51 1.10552
\(204\) 0 0
\(205\) −39.1098 −0.0133246
\(206\) −2384.07 −0.806339
\(207\) 0 0
\(208\) 0 0
\(209\) 6989.81 2.31337
\(210\) 0 0
\(211\) −91.5539 −0.0298712 −0.0149356 0.999888i \(-0.504754\pi\)
−0.0149356 + 0.999888i \(0.504754\pi\)
\(212\) 4.09466 0.00132652
\(213\) 0 0
\(214\) −2196.23 −0.701548
\(215\) 246.130 0.0780740
\(216\) 0 0
\(217\) −2055.70 −0.643087
\(218\) −1720.99 −0.534679
\(219\) 0 0
\(220\) −52.2935 −0.0160256
\(221\) 0 0
\(222\) 0 0
\(223\) −1235.42 −0.370985 −0.185493 0.982646i \(-0.559388\pi\)
−0.185493 + 0.982646i \(0.559388\pi\)
\(224\) −1167.96 −0.348383
\(225\) 0 0
\(226\) −1643.17 −0.483637
\(227\) 3301.66 0.965370 0.482685 0.875794i \(-0.339662\pi\)
0.482685 + 0.875794i \(0.339662\pi\)
\(228\) 0 0
\(229\) −211.283 −0.0609694 −0.0304847 0.999535i \(-0.509705\pi\)
−0.0304847 + 0.999535i \(0.509705\pi\)
\(230\) −105.383 −0.0302118
\(231\) 0 0
\(232\) 4252.97 1.20354
\(233\) 256.724 0.0721827 0.0360913 0.999348i \(-0.488509\pi\)
0.0360913 + 0.999348i \(0.488509\pi\)
\(234\) 0 0
\(235\) −17.9404 −0.00498002
\(236\) −103.040 −0.0284208
\(237\) 0 0
\(238\) −1189.48 −0.323960
\(239\) −3549.62 −0.960694 −0.480347 0.877078i \(-0.659489\pi\)
−0.480347 + 0.877078i \(0.659489\pi\)
\(240\) 0 0
\(241\) 5030.10 1.34447 0.672235 0.740338i \(-0.265335\pi\)
0.672235 + 0.740338i \(0.265335\pi\)
\(242\) 7326.27 1.94608
\(243\) 0 0
\(244\) 1324.37 0.347476
\(245\) −7.07204 −0.00184415
\(246\) 0 0
\(247\) 0 0
\(248\) −2734.25 −0.700102
\(249\) 0 0
\(250\) −359.158 −0.0908606
\(251\) 718.784 0.180754 0.0903770 0.995908i \(-0.471193\pi\)
0.0903770 + 0.995908i \(0.471193\pi\)
\(252\) 0 0
\(253\) −4742.84 −1.17858
\(254\) −1416.98 −0.350038
\(255\) 0 0
\(256\) −2133.74 −0.520933
\(257\) −1280.79 −0.310871 −0.155435 0.987846i \(-0.549678\pi\)
−0.155435 + 0.987846i \(0.549678\pi\)
\(258\) 0 0
\(259\) 2086.87 0.500663
\(260\) 0 0
\(261\) 0 0
\(262\) −5267.58 −1.24211
\(263\) −5225.55 −1.22517 −0.612587 0.790403i \(-0.709871\pi\)
−0.612587 + 0.790403i \(0.709871\pi\)
\(264\) 0 0
\(265\) −1.59851 −0.000370549 0
\(266\) −5027.24 −1.15880
\(267\) 0 0
\(268\) −639.074 −0.145663
\(269\) −6443.80 −1.46054 −0.730270 0.683158i \(-0.760606\pi\)
−0.730270 + 0.683158i \(0.760606\pi\)
\(270\) 0 0
\(271\) 3929.93 0.880909 0.440455 0.897775i \(-0.354817\pi\)
0.440455 + 0.897775i \(0.354817\pi\)
\(272\) −1288.13 −0.287149
\(273\) 0 0
\(274\) −4632.74 −1.02144
\(275\) −8071.91 −1.77002
\(276\) 0 0
\(277\) −5884.40 −1.27639 −0.638194 0.769876i \(-0.720318\pi\)
−0.638194 + 0.769876i \(0.720318\pi\)
\(278\) 3826.03 0.825431
\(279\) 0 0
\(280\) 246.785 0.0526722
\(281\) 3529.79 0.749358 0.374679 0.927155i \(-0.377753\pi\)
0.374679 + 0.927155i \(0.377753\pi\)
\(282\) 0 0
\(283\) −2611.00 −0.548438 −0.274219 0.961667i \(-0.588419\pi\)
−0.274219 + 0.961667i \(0.588419\pi\)
\(284\) 779.234 0.162813
\(285\) 0 0
\(286\) 0 0
\(287\) 1265.96 0.260373
\(288\) 0 0
\(289\) −4260.38 −0.867165
\(290\) −253.036 −0.0512372
\(291\) 0 0
\(292\) 1098.98 0.220250
\(293\) −5491.03 −1.09484 −0.547422 0.836857i \(-0.684391\pi\)
−0.547422 + 0.836857i \(0.684391\pi\)
\(294\) 0 0
\(295\) 40.2255 0.00793904
\(296\) 2775.72 0.545052
\(297\) 0 0
\(298\) −7067.37 −1.37383
\(299\) 0 0
\(300\) 0 0
\(301\) −7967.05 −1.52563
\(302\) 2500.99 0.476542
\(303\) 0 0
\(304\) −5444.19 −1.02712
\(305\) −517.019 −0.0970637
\(306\) 0 0
\(307\) 7307.59 1.35852 0.679261 0.733897i \(-0.262300\pi\)
0.679261 + 0.733897i \(0.262300\pi\)
\(308\) 1692.71 0.313152
\(309\) 0 0
\(310\) 162.678 0.0298048
\(311\) −7904.92 −1.44131 −0.720654 0.693295i \(-0.756158\pi\)
−0.720654 + 0.693295i \(0.756158\pi\)
\(312\) 0 0
\(313\) 10002.4 1.80629 0.903145 0.429336i \(-0.141252\pi\)
0.903145 + 0.429336i \(0.141252\pi\)
\(314\) −1446.96 −0.260053
\(315\) 0 0
\(316\) 606.360 0.107944
\(317\) −6230.81 −1.10397 −0.551983 0.833856i \(-0.686129\pi\)
−0.551983 + 0.833856i \(0.686129\pi\)
\(318\) 0 0
\(319\) −11388.1 −1.99878
\(320\) 318.950 0.0557182
\(321\) 0 0
\(322\) 3411.16 0.590362
\(323\) 2758.24 0.475147
\(324\) 0 0
\(325\) 0 0
\(326\) −3864.19 −0.656495
\(327\) 0 0
\(328\) 1683.83 0.283458
\(329\) 580.719 0.0973134
\(330\) 0 0
\(331\) 4634.51 0.769594 0.384797 0.923001i \(-0.374271\pi\)
0.384797 + 0.923001i \(0.374271\pi\)
\(332\) −868.531 −0.143575
\(333\) 0 0
\(334\) 1517.77 0.248649
\(335\) 249.487 0.0406893
\(336\) 0 0
\(337\) 3029.82 0.489747 0.244874 0.969555i \(-0.421254\pi\)
0.244874 + 0.969555i \(0.421254\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −20.6355 −0.00329151
\(341\) 7321.47 1.16270
\(342\) 0 0
\(343\) 6463.66 1.01751
\(344\) −10596.9 −1.66089
\(345\) 0 0
\(346\) 11515.6 1.78926
\(347\) −2841.60 −0.439611 −0.219805 0.975544i \(-0.570542\pi\)
−0.219805 + 0.975544i \(0.570542\pi\)
\(348\) 0 0
\(349\) −7565.68 −1.16040 −0.580202 0.814472i \(-0.697027\pi\)
−0.580202 + 0.814472i \(0.697027\pi\)
\(350\) 5805.51 0.886622
\(351\) 0 0
\(352\) 4159.76 0.629875
\(353\) −2339.44 −0.352736 −0.176368 0.984324i \(-0.556435\pi\)
−0.176368 + 0.984324i \(0.556435\pi\)
\(354\) 0 0
\(355\) −304.204 −0.0454802
\(356\) 1668.43 0.248389
\(357\) 0 0
\(358\) 395.209 0.0583449
\(359\) −2531.68 −0.372192 −0.186096 0.982532i \(-0.559583\pi\)
−0.186096 + 0.982532i \(0.559583\pi\)
\(360\) 0 0
\(361\) 4798.45 0.699585
\(362\) 2744.31 0.398447
\(363\) 0 0
\(364\) 0 0
\(365\) −429.028 −0.0615243
\(366\) 0 0
\(367\) 6577.81 0.935583 0.467792 0.883839i \(-0.345050\pi\)
0.467792 + 0.883839i \(0.345050\pi\)
\(368\) 3694.08 0.523280
\(369\) 0 0
\(370\) −165.145 −0.0232040
\(371\) 51.7426 0.00724081
\(372\) 0 0
\(373\) 2902.72 0.402942 0.201471 0.979495i \(-0.435428\pi\)
0.201471 + 0.979495i \(0.435428\pi\)
\(374\) 4236.40 0.585719
\(375\) 0 0
\(376\) 772.407 0.105941
\(377\) 0 0
\(378\) 0 0
\(379\) 1865.73 0.252866 0.126433 0.991975i \(-0.459647\pi\)
0.126433 + 0.991975i \(0.459647\pi\)
\(380\) −87.2140 −0.0117736
\(381\) 0 0
\(382\) −1734.69 −0.232342
\(383\) 10836.0 1.44567 0.722837 0.691019i \(-0.242838\pi\)
0.722837 + 0.691019i \(0.242838\pi\)
\(384\) 0 0
\(385\) −660.813 −0.0874757
\(386\) −3385.55 −0.446425
\(387\) 0 0
\(388\) 839.001 0.109778
\(389\) 9520.34 1.24088 0.620438 0.784256i \(-0.286955\pi\)
0.620438 + 0.784256i \(0.286955\pi\)
\(390\) 0 0
\(391\) −1871.56 −0.242069
\(392\) 304.480 0.0392310
\(393\) 0 0
\(394\) 3246.43 0.415108
\(395\) −236.716 −0.0301531
\(396\) 0 0
\(397\) 10108.8 1.27796 0.638978 0.769225i \(-0.279358\pi\)
0.638978 + 0.769225i \(0.279358\pi\)
\(398\) 6138.10 0.773053
\(399\) 0 0
\(400\) 6287.01 0.785876
\(401\) 2084.38 0.259573 0.129787 0.991542i \(-0.458571\pi\)
0.129787 + 0.991542i \(0.458571\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1325.88 0.163279
\(405\) 0 0
\(406\) 8190.60 1.00121
\(407\) −7432.50 −0.905197
\(408\) 0 0
\(409\) 9716.53 1.17470 0.587349 0.809334i \(-0.300172\pi\)
0.587349 + 0.809334i \(0.300172\pi\)
\(410\) −100.182 −0.0120674
\(411\) 0 0
\(412\) 1338.78 0.160090
\(413\) −1302.07 −0.155135
\(414\) 0 0
\(415\) 339.064 0.0401060
\(416\) 0 0
\(417\) 0 0
\(418\) 17904.8 2.09510
\(419\) −13381.9 −1.56026 −0.780129 0.625619i \(-0.784847\pi\)
−0.780129 + 0.625619i \(0.784847\pi\)
\(420\) 0 0
\(421\) 9463.37 1.09553 0.547763 0.836633i \(-0.315479\pi\)
0.547763 + 0.836633i \(0.315479\pi\)
\(422\) −234.520 −0.0270527
\(423\) 0 0
\(424\) 68.8221 0.00788278
\(425\) −3185.24 −0.363546
\(426\) 0 0
\(427\) 16735.5 1.89670
\(428\) 1233.30 0.139285
\(429\) 0 0
\(430\) 630.474 0.0707074
\(431\) 4852.28 0.542288 0.271144 0.962539i \(-0.412598\pi\)
0.271144 + 0.962539i \(0.412598\pi\)
\(432\) 0 0
\(433\) −8208.00 −0.910973 −0.455486 0.890243i \(-0.650535\pi\)
−0.455486 + 0.890243i \(0.650535\pi\)
\(434\) −5265.78 −0.582409
\(435\) 0 0
\(436\) 966.425 0.106155
\(437\) −7910.01 −0.865874
\(438\) 0 0
\(439\) −2993.80 −0.325481 −0.162741 0.986669i \(-0.552033\pi\)
−0.162741 + 0.986669i \(0.552033\pi\)
\(440\) −878.938 −0.0952311
\(441\) 0 0
\(442\) 0 0
\(443\) −9743.67 −1.04500 −0.522501 0.852639i \(-0.675001\pi\)
−0.522501 + 0.852639i \(0.675001\pi\)
\(444\) 0 0
\(445\) −651.335 −0.0693848
\(446\) −3164.59 −0.335981
\(447\) 0 0
\(448\) −10324.2 −1.08878
\(449\) −561.459 −0.0590131 −0.0295065 0.999565i \(-0.509394\pi\)
−0.0295065 + 0.999565i \(0.509394\pi\)
\(450\) 0 0
\(451\) −4508.78 −0.470754
\(452\) 922.726 0.0960207
\(453\) 0 0
\(454\) 8457.38 0.874283
\(455\) 0 0
\(456\) 0 0
\(457\) −13758.4 −1.40830 −0.704148 0.710054i \(-0.748671\pi\)
−0.704148 + 0.710054i \(0.748671\pi\)
\(458\) −541.213 −0.0552166
\(459\) 0 0
\(460\) 59.1779 0.00599823
\(461\) 12009.2 1.21329 0.606644 0.794974i \(-0.292515\pi\)
0.606644 + 0.794974i \(0.292515\pi\)
\(462\) 0 0
\(463\) −13635.7 −1.36870 −0.684348 0.729156i \(-0.739913\pi\)
−0.684348 + 0.729156i \(0.739913\pi\)
\(464\) 8869.91 0.887447
\(465\) 0 0
\(466\) 657.613 0.0653719
\(467\) −8821.95 −0.874157 −0.437079 0.899423i \(-0.643987\pi\)
−0.437079 + 0.899423i \(0.643987\pi\)
\(468\) 0 0
\(469\) −8075.72 −0.795100
\(470\) −45.9554 −0.00451013
\(471\) 0 0
\(472\) −1731.87 −0.168889
\(473\) 28375.1 2.75832
\(474\) 0 0
\(475\) −13462.2 −1.30039
\(476\) 667.956 0.0643187
\(477\) 0 0
\(478\) −9092.54 −0.870049
\(479\) −14620.0 −1.39459 −0.697293 0.716786i \(-0.745612\pi\)
−0.697293 + 0.716786i \(0.745612\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 12884.9 1.21761
\(483\) 0 0
\(484\) −4114.09 −0.386372
\(485\) −327.536 −0.0306653
\(486\) 0 0
\(487\) 9798.86 0.911763 0.455882 0.890040i \(-0.349324\pi\)
0.455882 + 0.890040i \(0.349324\pi\)
\(488\) 22259.7 2.06486
\(489\) 0 0
\(490\) −18.1154 −0.00167014
\(491\) 10836.1 0.995977 0.497989 0.867184i \(-0.334072\pi\)
0.497989 + 0.867184i \(0.334072\pi\)
\(492\) 0 0
\(493\) −4493.84 −0.410532
\(494\) 0 0
\(495\) 0 0
\(496\) −5702.51 −0.516230
\(497\) 9846.86 0.888717
\(498\) 0 0
\(499\) −2589.96 −0.232349 −0.116175 0.993229i \(-0.537063\pi\)
−0.116175 + 0.993229i \(0.537063\pi\)
\(500\) 201.686 0.0180394
\(501\) 0 0
\(502\) 1841.20 0.163699
\(503\) 17067.5 1.51292 0.756462 0.654038i \(-0.226926\pi\)
0.756462 + 0.654038i \(0.226926\pi\)
\(504\) 0 0
\(505\) −517.606 −0.0456102
\(506\) −12149.0 −1.06737
\(507\) 0 0
\(508\) 795.712 0.0694961
\(509\) −1012.89 −0.0882038 −0.0441019 0.999027i \(-0.514043\pi\)
−0.0441019 + 0.999027i \(0.514043\pi\)
\(510\) 0 0
\(511\) 13887.4 1.20223
\(512\) −12992.6 −1.12148
\(513\) 0 0
\(514\) −3280.82 −0.281539
\(515\) −522.644 −0.0447193
\(516\) 0 0
\(517\) −2068.26 −0.175942
\(518\) 5345.63 0.453424
\(519\) 0 0
\(520\) 0 0
\(521\) 14367.7 1.20818 0.604089 0.796917i \(-0.293537\pi\)
0.604089 + 0.796917i \(0.293537\pi\)
\(522\) 0 0
\(523\) −16219.9 −1.35611 −0.678057 0.735010i \(-0.737178\pi\)
−0.678057 + 0.735010i \(0.737178\pi\)
\(524\) 2958.02 0.246607
\(525\) 0 0
\(526\) −13385.5 −1.10957
\(527\) 2889.11 0.238808
\(528\) 0 0
\(529\) −6799.77 −0.558870
\(530\) −4.09466 −0.000335586 0
\(531\) 0 0
\(532\) 2823.06 0.230066
\(533\) 0 0
\(534\) 0 0
\(535\) −481.466 −0.0389076
\(536\) −10741.4 −0.865593
\(537\) 0 0
\(538\) −16506.1 −1.32273
\(539\) −815.301 −0.0651530
\(540\) 0 0
\(541\) −17592.2 −1.39806 −0.699029 0.715094i \(-0.746384\pi\)
−0.699029 + 0.715094i \(0.746384\pi\)
\(542\) 10066.7 0.797792
\(543\) 0 0
\(544\) 1641.48 0.129371
\(545\) −377.281 −0.0296531
\(546\) 0 0
\(547\) 10504.6 0.821103 0.410552 0.911837i \(-0.365336\pi\)
0.410552 + 0.911837i \(0.365336\pi\)
\(548\) 2601.53 0.202795
\(549\) 0 0
\(550\) −20676.6 −1.60301
\(551\) −18992.8 −1.46846
\(552\) 0 0
\(553\) 7662.33 0.589214
\(554\) −15073.2 −1.15596
\(555\) 0 0
\(556\) −2148.52 −0.163880
\(557\) −507.558 −0.0386102 −0.0193051 0.999814i \(-0.506145\pi\)
−0.0193051 + 0.999814i \(0.506145\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 514.690 0.0388386
\(561\) 0 0
\(562\) 9041.75 0.678653
\(563\) 3443.14 0.257746 0.128873 0.991661i \(-0.458864\pi\)
0.128873 + 0.991661i \(0.458864\pi\)
\(564\) 0 0
\(565\) −360.221 −0.0268223
\(566\) −6688.21 −0.496690
\(567\) 0 0
\(568\) 13097.2 0.967509
\(569\) −23972.2 −1.76620 −0.883098 0.469189i \(-0.844546\pi\)
−0.883098 + 0.469189i \(0.844546\pi\)
\(570\) 0 0
\(571\) −7458.32 −0.546622 −0.273311 0.961926i \(-0.588119\pi\)
−0.273311 + 0.961926i \(0.588119\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 3242.82 0.235806
\(575\) 9134.57 0.662501
\(576\) 0 0
\(577\) −5669.57 −0.409059 −0.204530 0.978860i \(-0.565566\pi\)
−0.204530 + 0.978860i \(0.565566\pi\)
\(578\) −10913.2 −0.785344
\(579\) 0 0
\(580\) 142.093 0.0101726
\(581\) −10975.3 −0.783702
\(582\) 0 0
\(583\) −184.284 −0.0130914
\(584\) 18471.4 1.30882
\(585\) 0 0
\(586\) −14065.6 −0.991541
\(587\) 1017.39 0.0715371 0.0357685 0.999360i \(-0.488612\pi\)
0.0357685 + 0.999360i \(0.488612\pi\)
\(588\) 0 0
\(589\) 12210.6 0.854208
\(590\) 103.040 0.00718996
\(591\) 0 0
\(592\) 5788.99 0.401902
\(593\) −10198.2 −0.706221 −0.353111 0.935582i \(-0.614876\pi\)
−0.353111 + 0.935582i \(0.614876\pi\)
\(594\) 0 0
\(595\) −260.762 −0.0179667
\(596\) 3968.70 0.272759
\(597\) 0 0
\(598\) 0 0
\(599\) −12516.3 −0.853763 −0.426881 0.904308i \(-0.640388\pi\)
−0.426881 + 0.904308i \(0.640388\pi\)
\(600\) 0 0
\(601\) 9627.46 0.653431 0.326716 0.945123i \(-0.394058\pi\)
0.326716 + 0.945123i \(0.394058\pi\)
\(602\) −20408.0 −1.38168
\(603\) 0 0
\(604\) −1404.44 −0.0946120
\(605\) 1606.09 0.107929
\(606\) 0 0
\(607\) 6667.20 0.445821 0.222910 0.974839i \(-0.428444\pi\)
0.222910 + 0.974839i \(0.428444\pi\)
\(608\) 6937.56 0.462755
\(609\) 0 0
\(610\) −1324.37 −0.0879053
\(611\) 0 0
\(612\) 0 0
\(613\) 23085.4 1.52106 0.760530 0.649302i \(-0.224939\pi\)
0.760530 + 0.649302i \(0.224939\pi\)
\(614\) 18718.8 1.23034
\(615\) 0 0
\(616\) 28450.6 1.86089
\(617\) 3049.24 0.198959 0.0994796 0.995040i \(-0.468282\pi\)
0.0994796 + 0.995040i \(0.468282\pi\)
\(618\) 0 0
\(619\) −7296.58 −0.473787 −0.236894 0.971536i \(-0.576129\pi\)
−0.236894 + 0.971536i \(0.576129\pi\)
\(620\) −91.3523 −0.00591741
\(621\) 0 0
\(622\) −20248.9 −1.30531
\(623\) 21083.3 1.35583
\(624\) 0 0
\(625\) 15506.8 0.992438
\(626\) 25621.7 1.63586
\(627\) 0 0
\(628\) 812.543 0.0516305
\(629\) −2932.92 −0.185920
\(630\) 0 0
\(631\) 23829.5 1.50339 0.751694 0.659512i \(-0.229237\pi\)
0.751694 + 0.659512i \(0.229237\pi\)
\(632\) 10191.6 0.641453
\(633\) 0 0
\(634\) −15960.5 −0.999801
\(635\) −310.637 −0.0194130
\(636\) 0 0
\(637\) 0 0
\(638\) −29171.3 −1.81019
\(639\) 0 0
\(640\) 528.347 0.0326324
\(641\) −13405.3 −0.826016 −0.413008 0.910727i \(-0.635522\pi\)
−0.413008 + 0.910727i \(0.635522\pi\)
\(642\) 0 0
\(643\) −5251.51 −0.322083 −0.161042 0.986948i \(-0.551485\pi\)
−0.161042 + 0.986948i \(0.551485\pi\)
\(644\) −1915.55 −0.117210
\(645\) 0 0
\(646\) 7065.37 0.430315
\(647\) −21611.4 −1.31319 −0.656595 0.754244i \(-0.728004\pi\)
−0.656595 + 0.754244i \(0.728004\pi\)
\(648\) 0 0
\(649\) 4637.39 0.280483
\(650\) 0 0
\(651\) 0 0
\(652\) 2169.94 0.130340
\(653\) 21595.8 1.29420 0.647099 0.762406i \(-0.275982\pi\)
0.647099 + 0.762406i \(0.275982\pi\)
\(654\) 0 0
\(655\) −1154.78 −0.0688869
\(656\) 3511.77 0.209012
\(657\) 0 0
\(658\) 1487.54 0.0881314
\(659\) 16642.6 0.983768 0.491884 0.870661i \(-0.336308\pi\)
0.491884 + 0.870661i \(0.336308\pi\)
\(660\) 0 0
\(661\) −26981.1 −1.58766 −0.793831 0.608139i \(-0.791916\pi\)
−0.793831 + 0.608139i \(0.791916\pi\)
\(662\) 11871.5 0.696980
\(663\) 0 0
\(664\) −14598.1 −0.853185
\(665\) −1102.09 −0.0642664
\(666\) 0 0
\(667\) 12887.3 0.748126
\(668\) −852.310 −0.0493665
\(669\) 0 0
\(670\) 639.074 0.0368501
\(671\) −59604.5 −3.42922
\(672\) 0 0
\(673\) 11149.2 0.638591 0.319296 0.947655i \(-0.396554\pi\)
0.319296 + 0.947655i \(0.396554\pi\)
\(674\) 7761.04 0.443537
\(675\) 0 0
\(676\) 0 0
\(677\) −3314.33 −0.188154 −0.0940769 0.995565i \(-0.529990\pi\)
−0.0940769 + 0.995565i \(0.529990\pi\)
\(678\) 0 0
\(679\) 10602.1 0.599223
\(680\) −346.836 −0.0195596
\(681\) 0 0
\(682\) 18754.3 1.05299
\(683\) 24505.2 1.37287 0.686433 0.727193i \(-0.259176\pi\)
0.686433 + 0.727193i \(0.259176\pi\)
\(684\) 0 0
\(685\) −1015.61 −0.0566486
\(686\) 16557.0 0.921500
\(687\) 0 0
\(688\) −22100.6 −1.22468
\(689\) 0 0
\(690\) 0 0
\(691\) 21752.8 1.19756 0.598782 0.800912i \(-0.295652\pi\)
0.598782 + 0.800912i \(0.295652\pi\)
\(692\) −6466.64 −0.355238
\(693\) 0 0
\(694\) −7278.90 −0.398132
\(695\) 838.755 0.0457781
\(696\) 0 0
\(697\) −1779.20 −0.0966887
\(698\) −19379.9 −1.05092
\(699\) 0 0
\(700\) −3260.10 −0.176029
\(701\) −34250.9 −1.84542 −0.922709 0.385496i \(-0.874030\pi\)
−0.922709 + 0.385496i \(0.874030\pi\)
\(702\) 0 0
\(703\) −12395.8 −0.665028
\(704\) 36770.1 1.96850
\(705\) 0 0
\(706\) −5992.59 −0.319454
\(707\) 16754.6 0.891259
\(708\) 0 0
\(709\) 5527.11 0.292771 0.146386 0.989228i \(-0.453236\pi\)
0.146386 + 0.989228i \(0.453236\pi\)
\(710\) −779.234 −0.0411889
\(711\) 0 0
\(712\) 28042.6 1.47604
\(713\) −8285.33 −0.435187
\(714\) 0 0
\(715\) 0 0
\(716\) −221.931 −0.0115837
\(717\) 0 0
\(718\) −6485.02 −0.337074
\(719\) 3777.78 0.195949 0.0979745 0.995189i \(-0.468764\pi\)
0.0979745 + 0.995189i \(0.468764\pi\)
\(720\) 0 0
\(721\) 16917.6 0.873849
\(722\) 12291.5 0.633576
\(723\) 0 0
\(724\) −1541.08 −0.0791072
\(725\) 21933.2 1.12355
\(726\) 0 0
\(727\) 19076.8 0.973204 0.486602 0.873624i \(-0.338236\pi\)
0.486602 + 0.873624i \(0.338236\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −1098.98 −0.0557192
\(731\) 11197.0 0.566535
\(732\) 0 0
\(733\) −7997.30 −0.402984 −0.201492 0.979490i \(-0.564579\pi\)
−0.201492 + 0.979490i \(0.564579\pi\)
\(734\) 16849.4 0.847307
\(735\) 0 0
\(736\) −4707.39 −0.235756
\(737\) 28762.1 1.43754
\(738\) 0 0
\(739\) −28983.6 −1.44273 −0.721367 0.692553i \(-0.756486\pi\)
−0.721367 + 0.692553i \(0.756486\pi\)
\(740\) 92.7376 0.00460689
\(741\) 0 0
\(742\) 132.541 0.00655761
\(743\) −19145.4 −0.945324 −0.472662 0.881244i \(-0.656707\pi\)
−0.472662 + 0.881244i \(0.656707\pi\)
\(744\) 0 0
\(745\) −1549.33 −0.0761923
\(746\) 7435.47 0.364922
\(747\) 0 0
\(748\) −2378.96 −0.116288
\(749\) 15584.7 0.760284
\(750\) 0 0
\(751\) −25516.9 −1.23985 −0.619923 0.784663i \(-0.712836\pi\)
−0.619923 + 0.784663i \(0.712836\pi\)
\(752\) 1610.92 0.0781172
\(753\) 0 0
\(754\) 0 0
\(755\) 548.275 0.0264288
\(756\) 0 0
\(757\) −17230.6 −0.827289 −0.413645 0.910438i \(-0.635744\pi\)
−0.413645 + 0.910438i \(0.635744\pi\)
\(758\) 4779.17 0.229007
\(759\) 0 0
\(760\) −1465.87 −0.0699642
\(761\) −2343.06 −0.111611 −0.0558053 0.998442i \(-0.517773\pi\)
−0.0558053 + 0.998442i \(0.517773\pi\)
\(762\) 0 0
\(763\) 12212.3 0.579444
\(764\) 974.121 0.0461289
\(765\) 0 0
\(766\) 27756.9 1.30927
\(767\) 0 0
\(768\) 0 0
\(769\) 7100.18 0.332950 0.166475 0.986046i \(-0.446761\pi\)
0.166475 + 0.986046i \(0.446761\pi\)
\(770\) −1692.71 −0.0792219
\(771\) 0 0
\(772\) 1901.17 0.0886328
\(773\) 12270.4 0.570940 0.285470 0.958388i \(-0.407850\pi\)
0.285470 + 0.958388i \(0.407850\pi\)
\(774\) 0 0
\(775\) −14100.9 −0.653575
\(776\) 14101.7 0.652349
\(777\) 0 0
\(778\) 24386.9 1.12379
\(779\) −7519.64 −0.345852
\(780\) 0 0
\(781\) −35070.1 −1.60680
\(782\) −4794.11 −0.219229
\(783\) 0 0
\(784\) 635.017 0.0289275
\(785\) −317.207 −0.0144224
\(786\) 0 0
\(787\) −3425.04 −0.155133 −0.0775663 0.996987i \(-0.524715\pi\)
−0.0775663 + 0.996987i \(0.524715\pi\)
\(788\) −1823.04 −0.0824151
\(789\) 0 0
\(790\) −606.360 −0.0273080
\(791\) 11660.1 0.524129
\(792\) 0 0
\(793\) 0 0
\(794\) 25894.3 1.15737
\(795\) 0 0
\(796\) −3446.87 −0.153481
\(797\) 11781.1 0.523600 0.261800 0.965122i \(-0.415684\pi\)
0.261800 + 0.965122i \(0.415684\pi\)
\(798\) 0 0
\(799\) −816.154 −0.0361370
\(800\) −8011.58 −0.354065
\(801\) 0 0
\(802\) 5339.24 0.235081
\(803\) −49460.6 −2.17363
\(804\) 0 0
\(805\) 747.807 0.0327413
\(806\) 0 0
\(807\) 0 0
\(808\) 22285.0 0.970276
\(809\) −18910.1 −0.821810 −0.410905 0.911678i \(-0.634787\pi\)
−0.410905 + 0.911678i \(0.634787\pi\)
\(810\) 0 0
\(811\) −12803.3 −0.554359 −0.277180 0.960818i \(-0.589400\pi\)
−0.277180 + 0.960818i \(0.589400\pi\)
\(812\) −4599.45 −0.198780
\(813\) 0 0
\(814\) −19038.7 −0.819788
\(815\) −847.121 −0.0364090
\(816\) 0 0
\(817\) 47323.3 2.02648
\(818\) 24889.4 1.06386
\(819\) 0 0
\(820\) 56.2574 0.00239585
\(821\) 19335.1 0.821923 0.410962 0.911653i \(-0.365193\pi\)
0.410962 + 0.911653i \(0.365193\pi\)
\(822\) 0 0
\(823\) −2125.90 −0.0900417 −0.0450209 0.998986i \(-0.514335\pi\)
−0.0450209 + 0.998986i \(0.514335\pi\)
\(824\) 22501.9 0.951324
\(825\) 0 0
\(826\) −3335.32 −0.140497
\(827\) −6989.24 −0.293881 −0.146941 0.989145i \(-0.546943\pi\)
−0.146941 + 0.989145i \(0.546943\pi\)
\(828\) 0 0
\(829\) −32649.7 −1.36788 −0.683938 0.729540i \(-0.739734\pi\)
−0.683938 + 0.729540i \(0.739734\pi\)
\(830\) 868.531 0.0363219
\(831\) 0 0
\(832\) 0 0
\(833\) −321.724 −0.0133819
\(834\) 0 0
\(835\) 332.732 0.0137900
\(836\) −10054.5 −0.415958
\(837\) 0 0
\(838\) −34278.4 −1.41304
\(839\) −4038.23 −0.166168 −0.0830841 0.996543i \(-0.526477\pi\)
−0.0830841 + 0.996543i \(0.526477\pi\)
\(840\) 0 0
\(841\) 6555.00 0.268769
\(842\) 24240.9 0.992159
\(843\) 0 0
\(844\) 131.695 0.00537102
\(845\) 0 0
\(846\) 0 0
\(847\) −51988.1 −2.10901
\(848\) 143.534 0.00581248
\(849\) 0 0
\(850\) −8159.17 −0.329244
\(851\) 8410.97 0.338807
\(852\) 0 0
\(853\) −8114.12 −0.325700 −0.162850 0.986651i \(-0.552069\pi\)
−0.162850 + 0.986651i \(0.552069\pi\)
\(854\) 42869.0 1.71774
\(855\) 0 0
\(856\) 20729.0 0.827690
\(857\) 22298.1 0.888786 0.444393 0.895832i \(-0.353419\pi\)
0.444393 + 0.895832i \(0.353419\pi\)
\(858\) 0 0
\(859\) 33550.5 1.33263 0.666315 0.745670i \(-0.267870\pi\)
0.666315 + 0.745670i \(0.267870\pi\)
\(860\) −354.045 −0.0140382
\(861\) 0 0
\(862\) 12429.4 0.491121
\(863\) −14120.5 −0.556972 −0.278486 0.960440i \(-0.589833\pi\)
−0.278486 + 0.960440i \(0.589833\pi\)
\(864\) 0 0
\(865\) 2524.50 0.0992319
\(866\) −21025.2 −0.825018
\(867\) 0 0
\(868\) 2957.01 0.115631
\(869\) −27289.8 −1.06530
\(870\) 0 0
\(871\) 0 0
\(872\) 16243.4 0.630817
\(873\) 0 0
\(874\) −20261.9 −0.784175
\(875\) 2548.63 0.0984679
\(876\) 0 0
\(877\) 1941.69 0.0747619 0.0373809 0.999301i \(-0.488099\pi\)
0.0373809 + 0.999301i \(0.488099\pi\)
\(878\) −7668.78 −0.294771
\(879\) 0 0
\(880\) −1833.10 −0.0702201
\(881\) 790.231 0.0302197 0.0151099 0.999886i \(-0.495190\pi\)
0.0151099 + 0.999886i \(0.495190\pi\)
\(882\) 0 0
\(883\) −36638.6 −1.39636 −0.698180 0.715922i \(-0.746007\pi\)
−0.698180 + 0.715922i \(0.746007\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −24958.9 −0.946401
\(887\) 40686.3 1.54015 0.770075 0.637954i \(-0.220219\pi\)
0.770075 + 0.637954i \(0.220219\pi\)
\(888\) 0 0
\(889\) 10055.1 0.379344
\(890\) −1668.43 −0.0628381
\(891\) 0 0
\(892\) 1777.08 0.0667053
\(893\) −3449.40 −0.129261
\(894\) 0 0
\(895\) 86.6392 0.00323579
\(896\) −17102.2 −0.637663
\(897\) 0 0
\(898\) −1438.21 −0.0534449
\(899\) −19894.0 −0.738046
\(900\) 0 0
\(901\) −72.7200 −0.00268885
\(902\) −11549.5 −0.426336
\(903\) 0 0
\(904\) 15509.0 0.570598
\(905\) 601.618 0.0220977
\(906\) 0 0
\(907\) −10464.4 −0.383093 −0.191547 0.981484i \(-0.561350\pi\)
−0.191547 + 0.981484i \(0.561350\pi\)
\(908\) −4749.26 −0.173579
\(909\) 0 0
\(910\) 0 0
\(911\) 35611.5 1.29513 0.647563 0.762011i \(-0.275788\pi\)
0.647563 + 0.762011i \(0.275788\pi\)
\(912\) 0 0
\(913\) 39089.0 1.41693
\(914\) −35242.9 −1.27542
\(915\) 0 0
\(916\) 303.920 0.0109627
\(917\) 37379.4 1.34610
\(918\) 0 0
\(919\) 1077.25 0.0386674 0.0193337 0.999813i \(-0.493846\pi\)
0.0193337 + 0.999813i \(0.493846\pi\)
\(920\) 994.648 0.0356441
\(921\) 0 0
\(922\) 30762.3 1.09881
\(923\) 0 0
\(924\) 0 0
\(925\) 14314.8 0.508829
\(926\) −34928.6 −1.23955
\(927\) 0 0
\(928\) −11303.0 −0.399826
\(929\) 55733.8 1.96832 0.984159 0.177290i \(-0.0567330\pi\)
0.984159 + 0.177290i \(0.0567330\pi\)
\(930\) 0 0
\(931\) −1359.74 −0.0478665
\(932\) −369.284 −0.0129789
\(933\) 0 0
\(934\) −22597.9 −0.791677
\(935\) 928.718 0.0324838
\(936\) 0 0
\(937\) −3198.60 −0.111519 −0.0557596 0.998444i \(-0.517758\pi\)
−0.0557596 + 0.998444i \(0.517758\pi\)
\(938\) −20686.4 −0.720079
\(939\) 0 0
\(940\) 25.8064 0.000895437 0
\(941\) 8823.35 0.305667 0.152834 0.988252i \(-0.451160\pi\)
0.152834 + 0.988252i \(0.451160\pi\)
\(942\) 0 0
\(943\) 5102.35 0.176199
\(944\) −3611.95 −0.124533
\(945\) 0 0
\(946\) 72684.3 2.49806
\(947\) 28290.4 0.970766 0.485383 0.874301i \(-0.338680\pi\)
0.485383 + 0.874301i \(0.338680\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −34484.0 −1.17769
\(951\) 0 0
\(952\) 11226.8 0.382210
\(953\) 12399.0 0.421452 0.210726 0.977545i \(-0.432417\pi\)
0.210726 + 0.977545i \(0.432417\pi\)
\(954\) 0 0
\(955\) −380.285 −0.0128856
\(956\) 5105.94 0.172739
\(957\) 0 0
\(958\) −37450.0 −1.26300
\(959\) 32874.5 1.10696
\(960\) 0 0
\(961\) −17001.0 −0.570676
\(962\) 0 0
\(963\) 0 0
\(964\) −7235.53 −0.241744
\(965\) −742.193 −0.0247586
\(966\) 0 0
\(967\) 26667.1 0.886820 0.443410 0.896319i \(-0.353769\pi\)
0.443410 + 0.896319i \(0.353769\pi\)
\(968\) −69148.6 −2.29599
\(969\) 0 0
\(970\) −839.001 −0.0277719
\(971\) −49420.7 −1.63335 −0.816676 0.577096i \(-0.804186\pi\)
−0.816676 + 0.577096i \(0.804186\pi\)
\(972\) 0 0
\(973\) −27149.9 −0.894539
\(974\) 25100.3 0.825735
\(975\) 0 0
\(976\) 46424.5 1.52255
\(977\) 778.759 0.0255012 0.0127506 0.999919i \(-0.495941\pi\)
0.0127506 + 0.999919i \(0.495941\pi\)
\(978\) 0 0
\(979\) −75089.2 −2.45134
\(980\) 10.1728 0.000331589 0
\(981\) 0 0
\(982\) 27757.2 0.902003
\(983\) 5997.90 0.194612 0.0973059 0.995255i \(-0.468977\pi\)
0.0973059 + 0.995255i \(0.468977\pi\)
\(984\) 0 0
\(985\) 711.693 0.0230218
\(986\) −11511.2 −0.371797
\(987\) 0 0
\(988\) 0 0
\(989\) −32110.6 −1.03241
\(990\) 0 0
\(991\) 8974.94 0.287688 0.143844 0.989600i \(-0.454054\pi\)
0.143844 + 0.989600i \(0.454054\pi\)
\(992\) 7266.75 0.232580
\(993\) 0 0
\(994\) 25223.3 0.804863
\(995\) 1345.62 0.0428732
\(996\) 0 0
\(997\) 28530.2 0.906280 0.453140 0.891439i \(-0.350304\pi\)
0.453140 + 0.891439i \(0.350304\pi\)
\(998\) −6634.31 −0.210426
\(999\) 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.r.1.2 2
3.2 odd 2 169.4.a.g.1.1 2
13.12 even 2 117.4.a.d.1.1 2
39.2 even 12 169.4.e.f.147.1 8
39.5 even 4 169.4.b.f.168.4 4
39.8 even 4 169.4.b.f.168.1 4
39.11 even 12 169.4.e.f.147.4 8
39.17 odd 6 169.4.c.g.146.1 4
39.20 even 12 169.4.e.f.23.1 8
39.23 odd 6 169.4.c.g.22.1 4
39.29 odd 6 169.4.c.j.22.2 4
39.32 even 12 169.4.e.f.23.4 8
39.35 odd 6 169.4.c.j.146.2 4
39.38 odd 2 13.4.a.b.1.2 2
52.51 odd 2 1872.4.a.bb.1.1 2
156.155 even 2 208.4.a.h.1.2 2
195.38 even 4 325.4.b.e.274.1 4
195.77 even 4 325.4.b.e.274.4 4
195.194 odd 2 325.4.a.f.1.1 2
273.272 even 2 637.4.a.b.1.2 2
312.77 odd 2 832.4.a.s.1.2 2
312.155 even 2 832.4.a.z.1.1 2
429.428 even 2 1573.4.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.a.b.1.2 2 39.38 odd 2
117.4.a.d.1.1 2 13.12 even 2
169.4.a.g.1.1 2 3.2 odd 2
169.4.b.f.168.1 4 39.8 even 4
169.4.b.f.168.4 4 39.5 even 4
169.4.c.g.22.1 4 39.23 odd 6
169.4.c.g.146.1 4 39.17 odd 6
169.4.c.j.22.2 4 39.29 odd 6
169.4.c.j.146.2 4 39.35 odd 6
169.4.e.f.23.1 8 39.20 even 12
169.4.e.f.23.4 8 39.32 even 12
169.4.e.f.147.1 8 39.2 even 12
169.4.e.f.147.4 8 39.11 even 12
208.4.a.h.1.2 2 156.155 even 2
325.4.a.f.1.1 2 195.194 odd 2
325.4.b.e.274.1 4 195.38 even 4
325.4.b.e.274.4 4 195.77 even 4
637.4.a.b.1.2 2 273.272 even 2
832.4.a.s.1.2 2 312.77 odd 2
832.4.a.z.1.1 2 312.155 even 2
1521.4.a.r.1.2 2 1.1 even 1 trivial
1573.4.a.b.1.1 2 429.428 even 2
1872.4.a.bb.1.1 2 52.51 odd 2