Properties

Label 405.4.a.n.1.5
Level $405$
Weight $4$
Character 405.1
Self dual yes
Analytic conductor $23.896$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.8957735523\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 44 x^{5} + 74 x^{4} + 479 x^{3} - 460 x^{2} - 1200 x + 288\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.19444\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.19444 q^{2} -3.18442 q^{4} +5.00000 q^{5} +2.76605 q^{7} -24.5436 q^{8} +O(q^{10})\) \(q+2.19444 q^{2} -3.18442 q^{4} +5.00000 q^{5} +2.76605 q^{7} -24.5436 q^{8} +10.9722 q^{10} +52.6590 q^{11} +20.4535 q^{13} +6.06994 q^{14} -28.3842 q^{16} -3.66084 q^{17} -95.6705 q^{19} -15.9221 q^{20} +115.557 q^{22} +89.8411 q^{23} +25.0000 q^{25} +44.8840 q^{26} -8.80825 q^{28} +227.780 q^{29} +279.139 q^{31} +134.061 q^{32} -8.03351 q^{34} +13.8302 q^{35} +273.725 q^{37} -209.944 q^{38} -122.718 q^{40} -64.8647 q^{41} +418.762 q^{43} -167.688 q^{44} +197.151 q^{46} -138.709 q^{47} -335.349 q^{49} +54.8611 q^{50} -65.1323 q^{52} +197.063 q^{53} +263.295 q^{55} -67.8887 q^{56} +499.852 q^{58} -741.103 q^{59} +488.468 q^{61} +612.554 q^{62} +521.263 q^{64} +102.267 q^{65} -411.468 q^{67} +11.6576 q^{68} +30.3497 q^{70} +310.343 q^{71} -51.0260 q^{73} +600.675 q^{74} +304.655 q^{76} +145.657 q^{77} +1208.00 q^{79} -141.921 q^{80} -142.342 q^{82} -905.221 q^{83} -18.3042 q^{85} +918.949 q^{86} -1292.44 q^{88} -663.633 q^{89} +56.5752 q^{91} -286.092 q^{92} -304.389 q^{94} -478.353 q^{95} -725.336 q^{97} -735.905 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8} + O(q^{10}) \) \( 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8} + 10 q^{10} + 23 q^{11} + 96 q^{13} - 21 q^{14} + 324 q^{16} + 161 q^{17} + 279 q^{19} + 180 q^{20} + 311 q^{22} + 96 q^{23} + 175 q^{25} - 358 q^{26} + 337 q^{28} - 296 q^{29} + 244 q^{31} - 314 q^{32} + 125 q^{34} + 110 q^{35} + 404 q^{37} + 305 q^{38} + 90 q^{40} - 47 q^{41} + 525 q^{43} + 55 q^{44} + 717 q^{46} + 164 q^{47} + 1225 q^{49} + 50 q^{50} + 1682 q^{52} + 506 q^{53} + 115 q^{55} - 981 q^{56} + 1183 q^{58} - 85 q^{59} + 828 q^{61} - 786 q^{62} + 2236 q^{64} + 480 q^{65} + 1093 q^{67} + 2473 q^{68} - 105 q^{70} + 328 q^{71} + 2085 q^{73} - 1316 q^{74} + 2789 q^{76} + 24 q^{77} + 2110 q^{79} + 1620 q^{80} - 62 q^{82} + 1290 q^{83} + 805 q^{85} - 2569 q^{86} + 2271 q^{88} - 3048 q^{89} + 3338 q^{91} + 2763 q^{92} - 517 q^{94} + 1395 q^{95} + 1787 q^{97} + 1279 q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.19444 0.775853 0.387927 0.921690i \(-0.373191\pi\)
0.387927 + 0.921690i \(0.373191\pi\)
\(3\) 0 0
\(4\) −3.18442 −0.398052
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 2.76605 0.149353 0.0746763 0.997208i \(-0.476208\pi\)
0.0746763 + 0.997208i \(0.476208\pi\)
\(8\) −24.5436 −1.08468
\(9\) 0 0
\(10\) 10.9722 0.346972
\(11\) 52.6590 1.44339 0.721694 0.692212i \(-0.243364\pi\)
0.721694 + 0.692212i \(0.243364\pi\)
\(12\) 0 0
\(13\) 20.4535 0.436367 0.218183 0.975908i \(-0.429987\pi\)
0.218183 + 0.975908i \(0.429987\pi\)
\(14\) 6.06994 0.115876
\(15\) 0 0
\(16\) −28.3842 −0.443503
\(17\) −3.66084 −0.0522285 −0.0261142 0.999659i \(-0.508313\pi\)
−0.0261142 + 0.999659i \(0.508313\pi\)
\(18\) 0 0
\(19\) −95.6705 −1.15517 −0.577587 0.816329i \(-0.696006\pi\)
−0.577587 + 0.816329i \(0.696006\pi\)
\(20\) −15.9221 −0.178014
\(21\) 0 0
\(22\) 115.557 1.11986
\(23\) 89.8411 0.814486 0.407243 0.913320i \(-0.366490\pi\)
0.407243 + 0.913320i \(0.366490\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 44.8840 0.338556
\(27\) 0 0
\(28\) −8.80825 −0.0594501
\(29\) 227.780 1.45854 0.729272 0.684224i \(-0.239859\pi\)
0.729272 + 0.684224i \(0.239859\pi\)
\(30\) 0 0
\(31\) 279.139 1.61725 0.808625 0.588324i \(-0.200212\pi\)
0.808625 + 0.588324i \(0.200212\pi\)
\(32\) 134.061 0.740590
\(33\) 0 0
\(34\) −8.03351 −0.0405216
\(35\) 13.8302 0.0667925
\(36\) 0 0
\(37\) 273.725 1.21622 0.608110 0.793852i \(-0.291928\pi\)
0.608110 + 0.793852i \(0.291928\pi\)
\(38\) −209.944 −0.896246
\(39\) 0 0
\(40\) −122.718 −0.485085
\(41\) −64.8647 −0.247077 −0.123539 0.992340i \(-0.539424\pi\)
−0.123539 + 0.992340i \(0.539424\pi\)
\(42\) 0 0
\(43\) 418.762 1.48513 0.742565 0.669774i \(-0.233609\pi\)
0.742565 + 0.669774i \(0.233609\pi\)
\(44\) −167.688 −0.574544
\(45\) 0 0
\(46\) 197.151 0.631921
\(47\) −138.709 −0.430484 −0.215242 0.976561i \(-0.569054\pi\)
−0.215242 + 0.976561i \(0.569054\pi\)
\(48\) 0 0
\(49\) −335.349 −0.977694
\(50\) 54.8611 0.155171
\(51\) 0 0
\(52\) −65.1323 −0.173697
\(53\) 197.063 0.510730 0.255365 0.966845i \(-0.417804\pi\)
0.255365 + 0.966845i \(0.417804\pi\)
\(54\) 0 0
\(55\) 263.295 0.645503
\(56\) −67.8887 −0.162000
\(57\) 0 0
\(58\) 499.852 1.13162
\(59\) −741.103 −1.63531 −0.817656 0.575707i \(-0.804727\pi\)
−0.817656 + 0.575707i \(0.804727\pi\)
\(60\) 0 0
\(61\) 488.468 1.02528 0.512639 0.858604i \(-0.328668\pi\)
0.512639 + 0.858604i \(0.328668\pi\)
\(62\) 612.554 1.25475
\(63\) 0 0
\(64\) 521.263 1.01809
\(65\) 102.267 0.195149
\(66\) 0 0
\(67\) −411.468 −0.750280 −0.375140 0.926968i \(-0.622405\pi\)
−0.375140 + 0.926968i \(0.622405\pi\)
\(68\) 11.6576 0.0207896
\(69\) 0 0
\(70\) 30.3497 0.0518212
\(71\) 310.343 0.518746 0.259373 0.965777i \(-0.416484\pi\)
0.259373 + 0.965777i \(0.416484\pi\)
\(72\) 0 0
\(73\) −51.0260 −0.0818101 −0.0409051 0.999163i \(-0.513024\pi\)
−0.0409051 + 0.999163i \(0.513024\pi\)
\(74\) 600.675 0.943609
\(75\) 0 0
\(76\) 304.655 0.459820
\(77\) 145.657 0.215574
\(78\) 0 0
\(79\) 1208.00 1.72039 0.860193 0.509969i \(-0.170343\pi\)
0.860193 + 0.509969i \(0.170343\pi\)
\(80\) −141.921 −0.198340
\(81\) 0 0
\(82\) −142.342 −0.191695
\(83\) −905.221 −1.19712 −0.598560 0.801078i \(-0.704260\pi\)
−0.598560 + 0.801078i \(0.704260\pi\)
\(84\) 0 0
\(85\) −18.3042 −0.0233573
\(86\) 918.949 1.15224
\(87\) 0 0
\(88\) −1292.44 −1.56562
\(89\) −663.633 −0.790393 −0.395197 0.918597i \(-0.629324\pi\)
−0.395197 + 0.918597i \(0.629324\pi\)
\(90\) 0 0
\(91\) 56.5752 0.0651725
\(92\) −286.092 −0.324208
\(93\) 0 0
\(94\) −304.389 −0.333992
\(95\) −478.353 −0.516610
\(96\) 0 0
\(97\) −725.336 −0.759244 −0.379622 0.925142i \(-0.623946\pi\)
−0.379622 + 0.925142i \(0.623946\pi\)
\(98\) −735.905 −0.758547
\(99\) 0 0
\(100\) −79.6104 −0.0796104
\(101\) −977.782 −0.963296 −0.481648 0.876365i \(-0.659962\pi\)
−0.481648 + 0.876365i \(0.659962\pi\)
\(102\) 0 0
\(103\) 1587.70 1.51885 0.759423 0.650597i \(-0.225481\pi\)
0.759423 + 0.650597i \(0.225481\pi\)
\(104\) −502.001 −0.473319
\(105\) 0 0
\(106\) 432.444 0.396252
\(107\) −897.731 −0.811093 −0.405546 0.914074i \(-0.632919\pi\)
−0.405546 + 0.914074i \(0.632919\pi\)
\(108\) 0 0
\(109\) 855.492 0.751754 0.375877 0.926669i \(-0.377341\pi\)
0.375877 + 0.926669i \(0.377341\pi\)
\(110\) 577.786 0.500816
\(111\) 0 0
\(112\) −78.5120 −0.0662383
\(113\) −910.241 −0.757772 −0.378886 0.925443i \(-0.623693\pi\)
−0.378886 + 0.925443i \(0.623693\pi\)
\(114\) 0 0
\(115\) 449.206 0.364249
\(116\) −725.348 −0.580576
\(117\) 0 0
\(118\) −1626.31 −1.26876
\(119\) −10.1261 −0.00780046
\(120\) 0 0
\(121\) 1441.97 1.08337
\(122\) 1071.92 0.795465
\(123\) 0 0
\(124\) −888.893 −0.643750
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −2038.25 −1.42414 −0.712068 0.702111i \(-0.752241\pi\)
−0.712068 + 0.702111i \(0.752241\pi\)
\(128\) 71.3936 0.0492997
\(129\) 0 0
\(130\) 224.420 0.151407
\(131\) −235.645 −0.157163 −0.0785817 0.996908i \(-0.525039\pi\)
−0.0785817 + 0.996908i \(0.525039\pi\)
\(132\) 0 0
\(133\) −264.629 −0.172528
\(134\) −902.942 −0.582107
\(135\) 0 0
\(136\) 89.8501 0.0566513
\(137\) 2972.89 1.85395 0.926975 0.375123i \(-0.122400\pi\)
0.926975 + 0.375123i \(0.122400\pi\)
\(138\) 0 0
\(139\) −2094.63 −1.27816 −0.639080 0.769140i \(-0.720685\pi\)
−0.639080 + 0.769140i \(0.720685\pi\)
\(140\) −44.0412 −0.0265869
\(141\) 0 0
\(142\) 681.031 0.402471
\(143\) 1077.06 0.629847
\(144\) 0 0
\(145\) 1138.90 0.652281
\(146\) −111.974 −0.0634726
\(147\) 0 0
\(148\) −871.655 −0.484119
\(149\) −273.937 −0.150616 −0.0753081 0.997160i \(-0.523994\pi\)
−0.0753081 + 0.997160i \(0.523994\pi\)
\(150\) 0 0
\(151\) −936.844 −0.504896 −0.252448 0.967610i \(-0.581236\pi\)
−0.252448 + 0.967610i \(0.581236\pi\)
\(152\) 2348.10 1.25300
\(153\) 0 0
\(154\) 319.637 0.167254
\(155\) 1395.69 0.723256
\(156\) 0 0
\(157\) 398.436 0.202539 0.101269 0.994859i \(-0.467710\pi\)
0.101269 + 0.994859i \(0.467710\pi\)
\(158\) 2650.89 1.33477
\(159\) 0 0
\(160\) 670.306 0.331202
\(161\) 248.505 0.121646
\(162\) 0 0
\(163\) 478.154 0.229766 0.114883 0.993379i \(-0.463351\pi\)
0.114883 + 0.993379i \(0.463351\pi\)
\(164\) 206.556 0.0983495
\(165\) 0 0
\(166\) −1986.46 −0.928789
\(167\) 340.362 0.157713 0.0788564 0.996886i \(-0.474873\pi\)
0.0788564 + 0.996886i \(0.474873\pi\)
\(168\) 0 0
\(169\) −1778.66 −0.809584
\(170\) −40.1675 −0.0181218
\(171\) 0 0
\(172\) −1333.51 −0.591159
\(173\) −3776.07 −1.65948 −0.829738 0.558154i \(-0.811510\pi\)
−0.829738 + 0.558154i \(0.811510\pi\)
\(174\) 0 0
\(175\) 69.1512 0.0298705
\(176\) −1494.68 −0.640147
\(177\) 0 0
\(178\) −1456.31 −0.613229
\(179\) −186.652 −0.0779385 −0.0389693 0.999240i \(-0.512407\pi\)
−0.0389693 + 0.999240i \(0.512407\pi\)
\(180\) 0 0
\(181\) 1438.75 0.590837 0.295418 0.955368i \(-0.404541\pi\)
0.295418 + 0.955368i \(0.404541\pi\)
\(182\) 124.151 0.0505643
\(183\) 0 0
\(184\) −2205.02 −0.883459
\(185\) 1368.63 0.543911
\(186\) 0 0
\(187\) −192.776 −0.0753860
\(188\) 441.706 0.171355
\(189\) 0 0
\(190\) −1049.72 −0.400813
\(191\) 390.435 0.147910 0.0739552 0.997262i \(-0.476438\pi\)
0.0739552 + 0.997262i \(0.476438\pi\)
\(192\) 0 0
\(193\) −3915.05 −1.46016 −0.730081 0.683361i \(-0.760518\pi\)
−0.730081 + 0.683361i \(0.760518\pi\)
\(194\) −1591.71 −0.589062
\(195\) 0 0
\(196\) 1067.89 0.389173
\(197\) 892.680 0.322847 0.161423 0.986885i \(-0.448392\pi\)
0.161423 + 0.986885i \(0.448392\pi\)
\(198\) 0 0
\(199\) −2770.50 −0.986913 −0.493457 0.869770i \(-0.664267\pi\)
−0.493457 + 0.869770i \(0.664267\pi\)
\(200\) −613.589 −0.216937
\(201\) 0 0
\(202\) −2145.69 −0.747376
\(203\) 630.052 0.217837
\(204\) 0 0
\(205\) −324.323 −0.110496
\(206\) 3484.13 1.17840
\(207\) 0 0
\(208\) −580.554 −0.193530
\(209\) −5037.91 −1.66737
\(210\) 0 0
\(211\) 4582.51 1.49513 0.747566 0.664187i \(-0.231222\pi\)
0.747566 + 0.664187i \(0.231222\pi\)
\(212\) −627.531 −0.203297
\(213\) 0 0
\(214\) −1970.02 −0.629289
\(215\) 2093.81 0.664170
\(216\) 0 0
\(217\) 772.111 0.241541
\(218\) 1877.33 0.583251
\(219\) 0 0
\(220\) −838.440 −0.256944
\(221\) −74.8768 −0.0227908
\(222\) 0 0
\(223\) 4095.34 1.22979 0.614897 0.788608i \(-0.289198\pi\)
0.614897 + 0.788608i \(0.289198\pi\)
\(224\) 370.820 0.110609
\(225\) 0 0
\(226\) −1997.47 −0.587920
\(227\) 3869.35 1.13136 0.565678 0.824626i \(-0.308614\pi\)
0.565678 + 0.824626i \(0.308614\pi\)
\(228\) 0 0
\(229\) 1236.01 0.356672 0.178336 0.983970i \(-0.442929\pi\)
0.178336 + 0.983970i \(0.442929\pi\)
\(230\) 985.757 0.282604
\(231\) 0 0
\(232\) −5590.55 −1.58206
\(233\) −2207.05 −0.620552 −0.310276 0.950647i \(-0.600421\pi\)
−0.310276 + 0.950647i \(0.600421\pi\)
\(234\) 0 0
\(235\) −693.544 −0.192518
\(236\) 2359.98 0.650939
\(237\) 0 0
\(238\) −22.2211 −0.00605201
\(239\) −876.648 −0.237262 −0.118631 0.992938i \(-0.537851\pi\)
−0.118631 + 0.992938i \(0.537851\pi\)
\(240\) 0 0
\(241\) −477.861 −0.127725 −0.0638626 0.997959i \(-0.520342\pi\)
−0.0638626 + 0.997959i \(0.520342\pi\)
\(242\) 3164.32 0.840537
\(243\) 0 0
\(244\) −1555.49 −0.408114
\(245\) −1676.74 −0.437238
\(246\) 0 0
\(247\) −1956.79 −0.504080
\(248\) −6851.06 −1.75420
\(249\) 0 0
\(250\) 274.306 0.0693944
\(251\) −6892.28 −1.73322 −0.866608 0.498990i \(-0.833704\pi\)
−0.866608 + 0.498990i \(0.833704\pi\)
\(252\) 0 0
\(253\) 4730.94 1.17562
\(254\) −4472.82 −1.10492
\(255\) 0 0
\(256\) −4013.43 −0.979842
\(257\) 7256.71 1.76133 0.880664 0.473742i \(-0.157097\pi\)
0.880664 + 0.473742i \(0.157097\pi\)
\(258\) 0 0
\(259\) 757.138 0.181646
\(260\) −325.661 −0.0776795
\(261\) 0 0
\(262\) −517.110 −0.121936
\(263\) 6317.55 1.48121 0.740603 0.671943i \(-0.234540\pi\)
0.740603 + 0.671943i \(0.234540\pi\)
\(264\) 0 0
\(265\) 985.316 0.228406
\(266\) −580.714 −0.133857
\(267\) 0 0
\(268\) 1310.28 0.298650
\(269\) −5746.22 −1.30243 −0.651214 0.758894i \(-0.725740\pi\)
−0.651214 + 0.758894i \(0.725740\pi\)
\(270\) 0 0
\(271\) 4925.20 1.10400 0.552001 0.833844i \(-0.313865\pi\)
0.552001 + 0.833844i \(0.313865\pi\)
\(272\) 103.910 0.0231635
\(273\) 0 0
\(274\) 6523.84 1.43839
\(275\) 1316.47 0.288678
\(276\) 0 0
\(277\) 2325.88 0.504508 0.252254 0.967661i \(-0.418828\pi\)
0.252254 + 0.967661i \(0.418828\pi\)
\(278\) −4596.55 −0.991665
\(279\) 0 0
\(280\) −339.444 −0.0724487
\(281\) −3283.42 −0.697055 −0.348527 0.937299i \(-0.613318\pi\)
−0.348527 + 0.937299i \(0.613318\pi\)
\(282\) 0 0
\(283\) 2014.21 0.423083 0.211541 0.977369i \(-0.432152\pi\)
0.211541 + 0.977369i \(0.432152\pi\)
\(284\) −988.262 −0.206488
\(285\) 0 0
\(286\) 2363.54 0.488669
\(287\) −179.419 −0.0369016
\(288\) 0 0
\(289\) −4899.60 −0.997272
\(290\) 2499.26 0.506074
\(291\) 0 0
\(292\) 162.488 0.0325647
\(293\) −480.697 −0.0958450 −0.0479225 0.998851i \(-0.515260\pi\)
−0.0479225 + 0.998851i \(0.515260\pi\)
\(294\) 0 0
\(295\) −3705.52 −0.731334
\(296\) −6718.20 −1.31921
\(297\) 0 0
\(298\) −601.140 −0.116856
\(299\) 1837.56 0.355414
\(300\) 0 0
\(301\) 1158.32 0.221808
\(302\) −2055.85 −0.391725
\(303\) 0 0
\(304\) 2715.53 0.512323
\(305\) 2442.34 0.458518
\(306\) 0 0
\(307\) 3222.21 0.599026 0.299513 0.954092i \(-0.403176\pi\)
0.299513 + 0.954092i \(0.403176\pi\)
\(308\) −463.833 −0.0858096
\(309\) 0 0
\(310\) 3062.77 0.561141
\(311\) −2414.01 −0.440147 −0.220074 0.975483i \(-0.570630\pi\)
−0.220074 + 0.975483i \(0.570630\pi\)
\(312\) 0 0
\(313\) −2506.94 −0.452717 −0.226359 0.974044i \(-0.572682\pi\)
−0.226359 + 0.974044i \(0.572682\pi\)
\(314\) 874.344 0.157140
\(315\) 0 0
\(316\) −3846.77 −0.684803
\(317\) −3707.12 −0.656822 −0.328411 0.944535i \(-0.606513\pi\)
−0.328411 + 0.944535i \(0.606513\pi\)
\(318\) 0 0
\(319\) 11994.7 2.10525
\(320\) 2606.31 0.455304
\(321\) 0 0
\(322\) 545.330 0.0943791
\(323\) 350.234 0.0603330
\(324\) 0 0
\(325\) 511.336 0.0872733
\(326\) 1049.28 0.178265
\(327\) 0 0
\(328\) 1592.01 0.268000
\(329\) −383.675 −0.0642939
\(330\) 0 0
\(331\) 2553.45 0.424018 0.212009 0.977268i \(-0.431999\pi\)
0.212009 + 0.977268i \(0.431999\pi\)
\(332\) 2882.60 0.476516
\(333\) 0 0
\(334\) 746.906 0.122362
\(335\) −2057.34 −0.335535
\(336\) 0 0
\(337\) 3552.89 0.574297 0.287148 0.957886i \(-0.407293\pi\)
0.287148 + 0.957886i \(0.407293\pi\)
\(338\) −3903.16 −0.628118
\(339\) 0 0
\(340\) 58.2882 0.00929741
\(341\) 14699.2 2.33432
\(342\) 0 0
\(343\) −1876.35 −0.295374
\(344\) −10277.9 −1.61089
\(345\) 0 0
\(346\) −8286.37 −1.28751
\(347\) 7768.02 1.20176 0.600878 0.799341i \(-0.294818\pi\)
0.600878 + 0.799341i \(0.294818\pi\)
\(348\) 0 0
\(349\) 696.007 0.106752 0.0533759 0.998574i \(-0.483002\pi\)
0.0533759 + 0.998574i \(0.483002\pi\)
\(350\) 151.748 0.0231751
\(351\) 0 0
\(352\) 7059.52 1.06896
\(353\) 5451.87 0.822022 0.411011 0.911630i \(-0.365176\pi\)
0.411011 + 0.911630i \(0.365176\pi\)
\(354\) 0 0
\(355\) 1551.72 0.231990
\(356\) 2113.28 0.314618
\(357\) 0 0
\(358\) −409.596 −0.0604688
\(359\) −4036.41 −0.593408 −0.296704 0.954969i \(-0.595887\pi\)
−0.296704 + 0.954969i \(0.595887\pi\)
\(360\) 0 0
\(361\) 2293.85 0.334429
\(362\) 3157.26 0.458403
\(363\) 0 0
\(364\) −180.159 −0.0259420
\(365\) −255.130 −0.0365866
\(366\) 0 0
\(367\) 11239.9 1.59869 0.799345 0.600872i \(-0.205180\pi\)
0.799345 + 0.600872i \(0.205180\pi\)
\(368\) −2550.07 −0.361227
\(369\) 0 0
\(370\) 3003.38 0.421995
\(371\) 545.086 0.0762789
\(372\) 0 0
\(373\) −6320.72 −0.877411 −0.438706 0.898631i \(-0.644563\pi\)
−0.438706 + 0.898631i \(0.644563\pi\)
\(374\) −423.036 −0.0584885
\(375\) 0 0
\(376\) 3404.41 0.466939
\(377\) 4658.90 0.636460
\(378\) 0 0
\(379\) 9325.49 1.26390 0.631950 0.775009i \(-0.282255\pi\)
0.631950 + 0.775009i \(0.282255\pi\)
\(380\) 1523.27 0.205638
\(381\) 0 0
\(382\) 856.788 0.114757
\(383\) −11148.0 −1.48730 −0.743652 0.668567i \(-0.766908\pi\)
−0.743652 + 0.668567i \(0.766908\pi\)
\(384\) 0 0
\(385\) 728.286 0.0964076
\(386\) −8591.35 −1.13287
\(387\) 0 0
\(388\) 2309.77 0.302219
\(389\) −6570.81 −0.856436 −0.428218 0.903676i \(-0.640858\pi\)
−0.428218 + 0.903676i \(0.640858\pi\)
\(390\) 0 0
\(391\) −328.894 −0.0425394
\(392\) 8230.66 1.06049
\(393\) 0 0
\(394\) 1958.94 0.250482
\(395\) 6039.99 0.769380
\(396\) 0 0
\(397\) 3969.33 0.501800 0.250900 0.968013i \(-0.419273\pi\)
0.250900 + 0.968013i \(0.419273\pi\)
\(398\) −6079.71 −0.765700
\(399\) 0 0
\(400\) −709.604 −0.0887005
\(401\) −4374.70 −0.544793 −0.272396 0.962185i \(-0.587816\pi\)
−0.272396 + 0.962185i \(0.587816\pi\)
\(402\) 0 0
\(403\) 5709.35 0.705714
\(404\) 3113.66 0.383442
\(405\) 0 0
\(406\) 1382.61 0.169010
\(407\) 14414.1 1.75548
\(408\) 0 0
\(409\) −8872.33 −1.07264 −0.536318 0.844016i \(-0.680185\pi\)
−0.536318 + 0.844016i \(0.680185\pi\)
\(410\) −711.709 −0.0857288
\(411\) 0 0
\(412\) −5055.91 −0.604580
\(413\) −2049.93 −0.244238
\(414\) 0 0
\(415\) −4526.11 −0.535368
\(416\) 2742.01 0.323169
\(417\) 0 0
\(418\) −11055.4 −1.29363
\(419\) 1014.17 0.118247 0.0591236 0.998251i \(-0.481169\pi\)
0.0591236 + 0.998251i \(0.481169\pi\)
\(420\) 0 0
\(421\) −14893.1 −1.72410 −0.862051 0.506822i \(-0.830820\pi\)
−0.862051 + 0.506822i \(0.830820\pi\)
\(422\) 10056.1 1.16000
\(423\) 0 0
\(424\) −4836.63 −0.553980
\(425\) −91.5210 −0.0104457
\(426\) 0 0
\(427\) 1351.13 0.153128
\(428\) 2858.75 0.322857
\(429\) 0 0
\(430\) 4594.75 0.515298
\(431\) 4363.90 0.487707 0.243853 0.969812i \(-0.421588\pi\)
0.243853 + 0.969812i \(0.421588\pi\)
\(432\) 0 0
\(433\) 9301.59 1.03235 0.516173 0.856484i \(-0.327356\pi\)
0.516173 + 0.856484i \(0.327356\pi\)
\(434\) 1694.35 0.187400
\(435\) 0 0
\(436\) −2724.24 −0.299237
\(437\) −8595.15 −0.940873
\(438\) 0 0
\(439\) 1520.15 0.165269 0.0826343 0.996580i \(-0.473667\pi\)
0.0826343 + 0.996580i \(0.473667\pi\)
\(440\) −6462.20 −0.700166
\(441\) 0 0
\(442\) −164.313 −0.0176823
\(443\) −4462.48 −0.478598 −0.239299 0.970946i \(-0.576918\pi\)
−0.239299 + 0.970946i \(0.576918\pi\)
\(444\) 0 0
\(445\) −3318.17 −0.353475
\(446\) 8986.98 0.954139
\(447\) 0 0
\(448\) 1441.84 0.152055
\(449\) 5371.66 0.564598 0.282299 0.959326i \(-0.408903\pi\)
0.282299 + 0.959326i \(0.408903\pi\)
\(450\) 0 0
\(451\) −3415.71 −0.356628
\(452\) 2898.59 0.301633
\(453\) 0 0
\(454\) 8491.08 0.877767
\(455\) 282.876 0.0291460
\(456\) 0 0
\(457\) 15525.4 1.58917 0.794583 0.607155i \(-0.207689\pi\)
0.794583 + 0.607155i \(0.207689\pi\)
\(458\) 2712.36 0.276725
\(459\) 0 0
\(460\) −1430.46 −0.144990
\(461\) 56.4179 0.00569988 0.00284994 0.999996i \(-0.499093\pi\)
0.00284994 + 0.999996i \(0.499093\pi\)
\(462\) 0 0
\(463\) 13373.8 1.34240 0.671201 0.741275i \(-0.265779\pi\)
0.671201 + 0.741275i \(0.265779\pi\)
\(464\) −6465.36 −0.646868
\(465\) 0 0
\(466\) −4843.25 −0.481457
\(467\) −2677.46 −0.265306 −0.132653 0.991163i \(-0.542350\pi\)
−0.132653 + 0.991163i \(0.542350\pi\)
\(468\) 0 0
\(469\) −1138.14 −0.112056
\(470\) −1521.94 −0.149366
\(471\) 0 0
\(472\) 18189.3 1.77380
\(473\) 22051.6 2.14362
\(474\) 0 0
\(475\) −2391.76 −0.231035
\(476\) 32.2456 0.00310499
\(477\) 0 0
\(478\) −1923.75 −0.184080
\(479\) 3469.93 0.330992 0.165496 0.986210i \(-0.447078\pi\)
0.165496 + 0.986210i \(0.447078\pi\)
\(480\) 0 0
\(481\) 5598.63 0.530718
\(482\) −1048.64 −0.0990960
\(483\) 0 0
\(484\) −4591.82 −0.431238
\(485\) −3626.68 −0.339544
\(486\) 0 0
\(487\) −14040.6 −1.30645 −0.653224 0.757165i \(-0.726584\pi\)
−0.653224 + 0.757165i \(0.726584\pi\)
\(488\) −11988.8 −1.11210
\(489\) 0 0
\(490\) −3679.52 −0.339232
\(491\) −9815.44 −0.902169 −0.451084 0.892481i \(-0.648963\pi\)
−0.451084 + 0.892481i \(0.648963\pi\)
\(492\) 0 0
\(493\) −833.868 −0.0761775
\(494\) −4294.07 −0.391092
\(495\) 0 0
\(496\) −7923.12 −0.717255
\(497\) 858.425 0.0774761
\(498\) 0 0
\(499\) −12052.0 −1.08121 −0.540603 0.841278i \(-0.681804\pi\)
−0.540603 + 0.841278i \(0.681804\pi\)
\(500\) −398.052 −0.0356028
\(501\) 0 0
\(502\) −15124.7 −1.34472
\(503\) 4695.09 0.416191 0.208095 0.978109i \(-0.433274\pi\)
0.208095 + 0.978109i \(0.433274\pi\)
\(504\) 0 0
\(505\) −4888.91 −0.430799
\(506\) 10381.8 0.912108
\(507\) 0 0
\(508\) 6490.62 0.566880
\(509\) 819.813 0.0713902 0.0356951 0.999363i \(-0.488635\pi\)
0.0356951 + 0.999363i \(0.488635\pi\)
\(510\) 0 0
\(511\) −141.140 −0.0122186
\(512\) −9378.41 −0.809514
\(513\) 0 0
\(514\) 15924.4 1.36653
\(515\) 7938.52 0.679249
\(516\) 0 0
\(517\) −7304.26 −0.621356
\(518\) 1661.50 0.140930
\(519\) 0 0
\(520\) −2510.00 −0.211675
\(521\) −3282.80 −0.276050 −0.138025 0.990429i \(-0.544075\pi\)
−0.138025 + 0.990429i \(0.544075\pi\)
\(522\) 0 0
\(523\) −10768.1 −0.900300 −0.450150 0.892953i \(-0.648629\pi\)
−0.450150 + 0.892953i \(0.648629\pi\)
\(524\) 750.392 0.0625592
\(525\) 0 0
\(526\) 13863.5 1.14920
\(527\) −1021.88 −0.0844665
\(528\) 0 0
\(529\) −4095.57 −0.336613
\(530\) 2162.22 0.177209
\(531\) 0 0
\(532\) 842.690 0.0686752
\(533\) −1326.71 −0.107816
\(534\) 0 0
\(535\) −4488.65 −0.362732
\(536\) 10098.9 0.813816
\(537\) 0 0
\(538\) −12609.8 −1.01049
\(539\) −17659.1 −1.41119
\(540\) 0 0
\(541\) −16037.9 −1.27453 −0.637266 0.770644i \(-0.719935\pi\)
−0.637266 + 0.770644i \(0.719935\pi\)
\(542\) 10808.1 0.856543
\(543\) 0 0
\(544\) −490.776 −0.0386799
\(545\) 4277.46 0.336195
\(546\) 0 0
\(547\) 2049.12 0.160172 0.0800862 0.996788i \(-0.474480\pi\)
0.0800862 + 0.996788i \(0.474480\pi\)
\(548\) −9466.92 −0.737968
\(549\) 0 0
\(550\) 2888.93 0.223972
\(551\) −21791.9 −1.68487
\(552\) 0 0
\(553\) 3341.38 0.256944
\(554\) 5104.02 0.391424
\(555\) 0 0
\(556\) 6670.18 0.508774
\(557\) −3644.07 −0.277207 −0.138603 0.990348i \(-0.544261\pi\)
−0.138603 + 0.990348i \(0.544261\pi\)
\(558\) 0 0
\(559\) 8565.12 0.648061
\(560\) −392.560 −0.0296227
\(561\) 0 0
\(562\) −7205.28 −0.540812
\(563\) −350.736 −0.0262553 −0.0131277 0.999914i \(-0.504179\pi\)
−0.0131277 + 0.999914i \(0.504179\pi\)
\(564\) 0 0
\(565\) −4551.21 −0.338886
\(566\) 4420.07 0.328250
\(567\) 0 0
\(568\) −7616.93 −0.562675
\(569\) 19448.7 1.43292 0.716460 0.697628i \(-0.245761\pi\)
0.716460 + 0.697628i \(0.245761\pi\)
\(570\) 0 0
\(571\) −15145.9 −1.11005 −0.555024 0.831834i \(-0.687291\pi\)
−0.555024 + 0.831834i \(0.687291\pi\)
\(572\) −3429.80 −0.250712
\(573\) 0 0
\(574\) −393.725 −0.0286302
\(575\) 2246.03 0.162897
\(576\) 0 0
\(577\) −6365.11 −0.459243 −0.229621 0.973280i \(-0.573749\pi\)
−0.229621 + 0.973280i \(0.573749\pi\)
\(578\) −10751.9 −0.773737
\(579\) 0 0
\(580\) −3626.74 −0.259642
\(581\) −2503.89 −0.178793
\(582\) 0 0
\(583\) 10377.1 0.737182
\(584\) 1252.36 0.0887381
\(585\) 0 0
\(586\) −1054.86 −0.0743617
\(587\) −10284.4 −0.723136 −0.361568 0.932346i \(-0.617759\pi\)
−0.361568 + 0.932346i \(0.617759\pi\)
\(588\) 0 0
\(589\) −26705.3 −1.86821
\(590\) −8131.55 −0.567408
\(591\) 0 0
\(592\) −7769.47 −0.539397
\(593\) −666.566 −0.0461595 −0.0230798 0.999734i \(-0.507347\pi\)
−0.0230798 + 0.999734i \(0.507347\pi\)
\(594\) 0 0
\(595\) −50.6303 −0.00348847
\(596\) 872.330 0.0599530
\(597\) 0 0
\(598\) 4032.43 0.275749
\(599\) −25213.3 −1.71984 −0.859922 0.510426i \(-0.829488\pi\)
−0.859922 + 0.510426i \(0.829488\pi\)
\(600\) 0 0
\(601\) −20618.7 −1.39942 −0.699712 0.714426i \(-0.746688\pi\)
−0.699712 + 0.714426i \(0.746688\pi\)
\(602\) 2541.86 0.172090
\(603\) 0 0
\(604\) 2983.30 0.200975
\(605\) 7209.84 0.484499
\(606\) 0 0
\(607\) −5083.52 −0.339923 −0.169962 0.985451i \(-0.554364\pi\)
−0.169962 + 0.985451i \(0.554364\pi\)
\(608\) −12825.7 −0.855511
\(609\) 0 0
\(610\) 5359.58 0.355743
\(611\) −2837.07 −0.187849
\(612\) 0 0
\(613\) 2625.18 0.172969 0.0864845 0.996253i \(-0.472437\pi\)
0.0864845 + 0.996253i \(0.472437\pi\)
\(614\) 7070.95 0.464756
\(615\) 0 0
\(616\) −3574.95 −0.233829
\(617\) 1631.32 0.106442 0.0532208 0.998583i \(-0.483051\pi\)
0.0532208 + 0.998583i \(0.483051\pi\)
\(618\) 0 0
\(619\) 3184.67 0.206789 0.103395 0.994640i \(-0.467030\pi\)
0.103395 + 0.994640i \(0.467030\pi\)
\(620\) −4444.47 −0.287894
\(621\) 0 0
\(622\) −5297.40 −0.341489
\(623\) −1835.64 −0.118047
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −5501.33 −0.351242
\(627\) 0 0
\(628\) −1268.78 −0.0806210
\(629\) −1002.06 −0.0635214
\(630\) 0 0
\(631\) 10436.7 0.658447 0.329223 0.944252i \(-0.393213\pi\)
0.329223 + 0.944252i \(0.393213\pi\)
\(632\) −29648.6 −1.86607
\(633\) 0 0
\(634\) −8135.07 −0.509598
\(635\) −10191.2 −0.636893
\(636\) 0 0
\(637\) −6859.04 −0.426633
\(638\) 26321.7 1.63336
\(639\) 0 0
\(640\) 356.968 0.0220475
\(641\) −9083.45 −0.559711 −0.279856 0.960042i \(-0.590287\pi\)
−0.279856 + 0.960042i \(0.590287\pi\)
\(642\) 0 0
\(643\) 17187.6 1.05414 0.527071 0.849821i \(-0.323290\pi\)
0.527071 + 0.849821i \(0.323290\pi\)
\(644\) −791.343 −0.0484213
\(645\) 0 0
\(646\) 768.570 0.0468096
\(647\) −2359.17 −0.143352 −0.0716758 0.997428i \(-0.522835\pi\)
−0.0716758 + 0.997428i \(0.522835\pi\)
\(648\) 0 0
\(649\) −39025.7 −2.36039
\(650\) 1122.10 0.0677113
\(651\) 0 0
\(652\) −1522.64 −0.0914589
\(653\) −5436.11 −0.325776 −0.162888 0.986645i \(-0.552081\pi\)
−0.162888 + 0.986645i \(0.552081\pi\)
\(654\) 0 0
\(655\) −1178.23 −0.0702856
\(656\) 1841.13 0.109579
\(657\) 0 0
\(658\) −841.954 −0.0498826
\(659\) −3088.91 −0.182590 −0.0912950 0.995824i \(-0.529101\pi\)
−0.0912950 + 0.995824i \(0.529101\pi\)
\(660\) 0 0
\(661\) −19053.9 −1.12120 −0.560598 0.828088i \(-0.689429\pi\)
−0.560598 + 0.828088i \(0.689429\pi\)
\(662\) 5603.39 0.328976
\(663\) 0 0
\(664\) 22217.4 1.29850
\(665\) −1323.15 −0.0771570
\(666\) 0 0
\(667\) 20464.1 1.18796
\(668\) −1083.86 −0.0627779
\(669\) 0 0
\(670\) −4514.71 −0.260326
\(671\) 25722.2 1.47987
\(672\) 0 0
\(673\) −29237.5 −1.67462 −0.837312 0.546725i \(-0.815874\pi\)
−0.837312 + 0.546725i \(0.815874\pi\)
\(674\) 7796.61 0.445570
\(675\) 0 0
\(676\) 5663.98 0.322257
\(677\) −14471.2 −0.821529 −0.410764 0.911742i \(-0.634738\pi\)
−0.410764 + 0.911742i \(0.634738\pi\)
\(678\) 0 0
\(679\) −2006.31 −0.113395
\(680\) 449.250 0.0253352
\(681\) 0 0
\(682\) 32256.5 1.81109
\(683\) −5782.94 −0.323980 −0.161990 0.986792i \(-0.551791\pi\)
−0.161990 + 0.986792i \(0.551791\pi\)
\(684\) 0 0
\(685\) 14864.5 0.829112
\(686\) −4117.54 −0.229167
\(687\) 0 0
\(688\) −11886.2 −0.658659
\(689\) 4030.62 0.222866
\(690\) 0 0
\(691\) 11055.1 0.608620 0.304310 0.952573i \(-0.401574\pi\)
0.304310 + 0.952573i \(0.401574\pi\)
\(692\) 12024.6 0.660557
\(693\) 0 0
\(694\) 17046.5 0.932386
\(695\) −10473.2 −0.571611
\(696\) 0 0
\(697\) 237.459 0.0129045
\(698\) 1527.35 0.0828238
\(699\) 0 0
\(700\) −220.206 −0.0118900
\(701\) 13554.2 0.730294 0.365147 0.930950i \(-0.381019\pi\)
0.365147 + 0.930950i \(0.381019\pi\)
\(702\) 0 0
\(703\) −26187.4 −1.40495
\(704\) 27449.2 1.46950
\(705\) 0 0
\(706\) 11963.8 0.637768
\(707\) −2704.59 −0.143871
\(708\) 0 0
\(709\) −9389.17 −0.497345 −0.248672 0.968588i \(-0.579994\pi\)
−0.248672 + 0.968588i \(0.579994\pi\)
\(710\) 3405.15 0.179990
\(711\) 0 0
\(712\) 16287.9 0.857326
\(713\) 25078.1 1.31723
\(714\) 0 0
\(715\) 5385.29 0.281676
\(716\) 594.376 0.0310236
\(717\) 0 0
\(718\) −8857.67 −0.460398
\(719\) 26301.8 1.36424 0.682122 0.731238i \(-0.261057\pi\)
0.682122 + 0.731238i \(0.261057\pi\)
\(720\) 0 0
\(721\) 4391.67 0.226844
\(722\) 5033.72 0.259467
\(723\) 0 0
\(724\) −4581.58 −0.235184
\(725\) 5694.51 0.291709
\(726\) 0 0
\(727\) −25538.6 −1.30285 −0.651427 0.758711i \(-0.725829\pi\)
−0.651427 + 0.758711i \(0.725829\pi\)
\(728\) −1388.56 −0.0706915
\(729\) 0 0
\(730\) −559.868 −0.0283858
\(731\) −1533.02 −0.0775660
\(732\) 0 0
\(733\) 6982.12 0.351829 0.175914 0.984405i \(-0.443712\pi\)
0.175914 + 0.984405i \(0.443712\pi\)
\(734\) 24665.4 1.24035
\(735\) 0 0
\(736\) 12044.2 0.603200
\(737\) −21667.5 −1.08295
\(738\) 0 0
\(739\) 8863.91 0.441224 0.220612 0.975362i \(-0.429195\pi\)
0.220612 + 0.975362i \(0.429195\pi\)
\(740\) −4358.28 −0.216505
\(741\) 0 0
\(742\) 1196.16 0.0591812
\(743\) 38944.9 1.92295 0.961473 0.274899i \(-0.0886443\pi\)
0.961473 + 0.274899i \(0.0886443\pi\)
\(744\) 0 0
\(745\) −1369.69 −0.0673576
\(746\) −13870.5 −0.680742
\(747\) 0 0
\(748\) 613.879 0.0300075
\(749\) −2483.17 −0.121139
\(750\) 0 0
\(751\) 17590.3 0.854700 0.427350 0.904086i \(-0.359447\pi\)
0.427350 + 0.904086i \(0.359447\pi\)
\(752\) 3937.13 0.190921
\(753\) 0 0
\(754\) 10223.7 0.493799
\(755\) −4684.22 −0.225796
\(756\) 0 0
\(757\) −4075.85 −0.195693 −0.0978463 0.995202i \(-0.531195\pi\)
−0.0978463 + 0.995202i \(0.531195\pi\)
\(758\) 20464.3 0.980601
\(759\) 0 0
\(760\) 11740.5 0.560358
\(761\) −40537.0 −1.93097 −0.965483 0.260466i \(-0.916124\pi\)
−0.965483 + 0.260466i \(0.916124\pi\)
\(762\) 0 0
\(763\) 2366.33 0.112276
\(764\) −1243.31 −0.0588761
\(765\) 0 0
\(766\) −24463.7 −1.15393
\(767\) −15158.1 −0.713596
\(768\) 0 0
\(769\) 24032.7 1.12697 0.563485 0.826126i \(-0.309460\pi\)
0.563485 + 0.826126i \(0.309460\pi\)
\(770\) 1598.18 0.0747981
\(771\) 0 0
\(772\) 12467.1 0.581220
\(773\) 20881.5 0.971611 0.485805 0.874067i \(-0.338526\pi\)
0.485805 + 0.874067i \(0.338526\pi\)
\(774\) 0 0
\(775\) 6978.46 0.323450
\(776\) 17802.3 0.823539
\(777\) 0 0
\(778\) −14419.3 −0.664468
\(779\) 6205.64 0.285417
\(780\) 0 0
\(781\) 16342.4 0.748752
\(782\) −721.740 −0.0330043
\(783\) 0 0
\(784\) 9518.60 0.433610
\(785\) 1992.18 0.0905782
\(786\) 0 0
\(787\) −5136.39 −0.232646 −0.116323 0.993211i \(-0.537111\pi\)
−0.116323 + 0.993211i \(0.537111\pi\)
\(788\) −2842.67 −0.128510
\(789\) 0 0
\(790\) 13254.4 0.596926
\(791\) −2517.77 −0.113175
\(792\) 0 0
\(793\) 9990.86 0.447397
\(794\) 8710.46 0.389323
\(795\) 0 0
\(796\) 8822.43 0.392843
\(797\) −21996.0 −0.977587 −0.488794 0.872400i \(-0.662563\pi\)
−0.488794 + 0.872400i \(0.662563\pi\)
\(798\) 0 0
\(799\) 507.791 0.0224835
\(800\) 3351.53 0.148118
\(801\) 0 0
\(802\) −9600.03 −0.422679
\(803\) −2686.98 −0.118084
\(804\) 0 0
\(805\) 1242.52 0.0544016
\(806\) 12528.8 0.547531
\(807\) 0 0
\(808\) 23998.3 1.04487
\(809\) −31094.2 −1.35132 −0.675658 0.737215i \(-0.736140\pi\)
−0.675658 + 0.737215i \(0.736140\pi\)
\(810\) 0 0
\(811\) 19130.6 0.828320 0.414160 0.910204i \(-0.364075\pi\)
0.414160 + 0.910204i \(0.364075\pi\)
\(812\) −2006.35 −0.0867106
\(813\) 0 0
\(814\) 31630.9 1.36199
\(815\) 2390.77 0.102755
\(816\) 0 0
\(817\) −40063.1 −1.71558
\(818\) −19469.8 −0.832208
\(819\) 0 0
\(820\) 1032.78 0.0439832
\(821\) 3557.62 0.151232 0.0756162 0.997137i \(-0.475908\pi\)
0.0756162 + 0.997137i \(0.475908\pi\)
\(822\) 0 0
\(823\) 6149.38 0.260454 0.130227 0.991484i \(-0.458429\pi\)
0.130227 + 0.991484i \(0.458429\pi\)
\(824\) −38967.9 −1.64747
\(825\) 0 0
\(826\) −4498.45 −0.189493
\(827\) −21152.8 −0.889425 −0.444713 0.895673i \(-0.646694\pi\)
−0.444713 + 0.895673i \(0.646694\pi\)
\(828\) 0 0
\(829\) −17402.4 −0.729083 −0.364541 0.931187i \(-0.618774\pi\)
−0.364541 + 0.931187i \(0.618774\pi\)
\(830\) −9932.29 −0.415367
\(831\) 0 0
\(832\) 10661.6 0.444261
\(833\) 1227.66 0.0510635
\(834\) 0 0
\(835\) 1701.81 0.0705313
\(836\) 16042.8 0.663698
\(837\) 0 0
\(838\) 2225.55 0.0917425
\(839\) 18074.2 0.743733 0.371867 0.928286i \(-0.378718\pi\)
0.371867 + 0.928286i \(0.378718\pi\)
\(840\) 0 0
\(841\) 27495.0 1.12735
\(842\) −32682.1 −1.33765
\(843\) 0 0
\(844\) −14592.6 −0.595140
\(845\) −8893.28 −0.362057
\(846\) 0 0
\(847\) 3988.55 0.161804
\(848\) −5593.47 −0.226510
\(849\) 0 0
\(850\) −200.838 −0.00810432
\(851\) 24591.8 0.990595
\(852\) 0 0
\(853\) 40792.3 1.63740 0.818699 0.574222i \(-0.194696\pi\)
0.818699 + 0.574222i \(0.194696\pi\)
\(854\) 2964.97 0.118805
\(855\) 0 0
\(856\) 22033.5 0.879778
\(857\) 11057.9 0.440758 0.220379 0.975414i \(-0.429271\pi\)
0.220379 + 0.975414i \(0.429271\pi\)
\(858\) 0 0
\(859\) −1753.63 −0.0696543 −0.0348272 0.999393i \(-0.511088\pi\)
−0.0348272 + 0.999393i \(0.511088\pi\)
\(860\) −6667.56 −0.264374
\(861\) 0 0
\(862\) 9576.33 0.378389
\(863\) 19186.5 0.756796 0.378398 0.925643i \(-0.376475\pi\)
0.378398 + 0.925643i \(0.376475\pi\)
\(864\) 0 0
\(865\) −18880.3 −0.742140
\(866\) 20411.8 0.800949
\(867\) 0 0
\(868\) −2458.72 −0.0961457
\(869\) 63612.0 2.48319
\(870\) 0 0
\(871\) −8415.93 −0.327397
\(872\) −20996.8 −0.815415
\(873\) 0 0
\(874\) −18861.6 −0.729980
\(875\) 345.756 0.0133585
\(876\) 0 0
\(877\) −8514.60 −0.327842 −0.163921 0.986473i \(-0.552414\pi\)
−0.163921 + 0.986473i \(0.552414\pi\)
\(878\) 3335.89 0.128224
\(879\) 0 0
\(880\) −7473.41 −0.286282
\(881\) −41177.0 −1.57467 −0.787337 0.616522i \(-0.788541\pi\)
−0.787337 + 0.616522i \(0.788541\pi\)
\(882\) 0 0
\(883\) 32540.4 1.24017 0.620086 0.784533i \(-0.287098\pi\)
0.620086 + 0.784533i \(0.287098\pi\)
\(884\) 238.439 0.00907191
\(885\) 0 0
\(886\) −9792.67 −0.371322
\(887\) −2166.10 −0.0819960 −0.0409980 0.999159i \(-0.513054\pi\)
−0.0409980 + 0.999159i \(0.513054\pi\)
\(888\) 0 0
\(889\) −5637.89 −0.212698
\(890\) −7281.53 −0.274244
\(891\) 0 0
\(892\) −13041.2 −0.489522
\(893\) 13270.3 0.497284
\(894\) 0 0
\(895\) −933.258 −0.0348552
\(896\) 197.478 0.00736304
\(897\) 0 0
\(898\) 11787.8 0.438045
\(899\) 63582.3 2.35883
\(900\) 0 0
\(901\) −721.416 −0.0266747
\(902\) −7495.58 −0.276691
\(903\) 0 0
\(904\) 22340.6 0.821943
\(905\) 7193.75 0.264230
\(906\) 0 0
\(907\) −706.062 −0.0258483 −0.0129241 0.999916i \(-0.504114\pi\)
−0.0129241 + 0.999916i \(0.504114\pi\)
\(908\) −12321.6 −0.450339
\(909\) 0 0
\(910\) 620.756 0.0226130
\(911\) 495.485 0.0180199 0.00900997 0.999959i \(-0.497132\pi\)
0.00900997 + 0.999959i \(0.497132\pi\)
\(912\) 0 0
\(913\) −47668.0 −1.72791
\(914\) 34069.7 1.23296
\(915\) 0 0
\(916\) −3935.98 −0.141974
\(917\) −651.806 −0.0234728
\(918\) 0 0
\(919\) 20473.6 0.734889 0.367445 0.930045i \(-0.380233\pi\)
0.367445 + 0.930045i \(0.380233\pi\)
\(920\) −11025.1 −0.395095
\(921\) 0 0
\(922\) 123.806 0.00442227
\(923\) 6347.59 0.226363
\(924\) 0 0
\(925\) 6843.13 0.243244
\(926\) 29348.0 1.04151
\(927\) 0 0
\(928\) 30536.5 1.08018
\(929\) −49715.3 −1.75577 −0.877883 0.478875i \(-0.841045\pi\)
−0.877883 + 0.478875i \(0.841045\pi\)
\(930\) 0 0
\(931\) 32083.0 1.12941
\(932\) 7028.16 0.247012
\(933\) 0 0
\(934\) −5875.53 −0.205839
\(935\) −963.880 −0.0337136
\(936\) 0 0
\(937\) −31524.9 −1.09912 −0.549560 0.835454i \(-0.685204\pi\)
−0.549560 + 0.835454i \(0.685204\pi\)
\(938\) −2497.58 −0.0869392
\(939\) 0 0
\(940\) 2208.53 0.0766323
\(941\) −37528.5 −1.30010 −0.650050 0.759892i \(-0.725252\pi\)
−0.650050 + 0.759892i \(0.725252\pi\)
\(942\) 0 0
\(943\) −5827.52 −0.201241
\(944\) 21035.6 0.725265
\(945\) 0 0
\(946\) 48390.9 1.66313
\(947\) −25863.4 −0.887483 −0.443741 0.896155i \(-0.646349\pi\)
−0.443741 + 0.896155i \(0.646349\pi\)
\(948\) 0 0
\(949\) −1043.66 −0.0356992
\(950\) −5248.59 −0.179249
\(951\) 0 0
\(952\) 248.530 0.00846102
\(953\) −1060.03 −0.0360312 −0.0180156 0.999838i \(-0.505735\pi\)
−0.0180156 + 0.999838i \(0.505735\pi\)
\(954\) 0 0
\(955\) 1952.18 0.0661476
\(956\) 2791.61 0.0944426
\(957\) 0 0
\(958\) 7614.56 0.256801
\(959\) 8223.16 0.276892
\(960\) 0 0
\(961\) 48127.3 1.61550
\(962\) 12285.9 0.411759
\(963\) 0 0
\(964\) 1521.71 0.0508412
\(965\) −19575.2 −0.653004
\(966\) 0 0
\(967\) −5185.01 −0.172429 −0.0862145 0.996277i \(-0.527477\pi\)
−0.0862145 + 0.996277i \(0.527477\pi\)
\(968\) −35391.0 −1.17511
\(969\) 0 0
\(970\) −7958.54 −0.263436
\(971\) −28314.9 −0.935805 −0.467903 0.883780i \(-0.654990\pi\)
−0.467903 + 0.883780i \(0.654990\pi\)
\(972\) 0 0
\(973\) −5793.85 −0.190897
\(974\) −30811.3 −1.01361
\(975\) 0 0
\(976\) −13864.8 −0.454713
\(977\) −44896.7 −1.47019 −0.735093 0.677967i \(-0.762861\pi\)
−0.735093 + 0.677967i \(0.762861\pi\)
\(978\) 0 0
\(979\) −34946.3 −1.14084
\(980\) 5339.45 0.174043
\(981\) 0 0
\(982\) −21539.4 −0.699950
\(983\) 14572.8 0.472838 0.236419 0.971651i \(-0.424026\pi\)
0.236419 + 0.971651i \(0.424026\pi\)
\(984\) 0 0
\(985\) 4463.40 0.144382
\(986\) −1829.88 −0.0591026
\(987\) 0 0
\(988\) 6231.24 0.200650
\(989\) 37622.0 1.20962
\(990\) 0 0
\(991\) 10602.1 0.339844 0.169922 0.985457i \(-0.445648\pi\)
0.169922 + 0.985457i \(0.445648\pi\)
\(992\) 37421.6 1.19772
\(993\) 0 0
\(994\) 1883.76 0.0601100
\(995\) −13852.5 −0.441361
\(996\) 0 0
\(997\) 27897.0 0.886164 0.443082 0.896481i \(-0.353885\pi\)
0.443082 + 0.896481i \(0.353885\pi\)
\(998\) −26447.4 −0.838857
\(999\) 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 405.4.a.n.1.5 7
3.2 odd 2 405.4.a.m.1.3 7
5.4 even 2 2025.4.a.ba.1.3 7
9.2 odd 6 45.4.e.c.31.5 yes 14
9.4 even 3 135.4.e.c.46.3 14
9.5 odd 6 45.4.e.c.16.5 14
9.7 even 3 135.4.e.c.91.3 14
15.14 odd 2 2025.4.a.bb.1.5 7
45.2 even 12 225.4.k.d.49.5 28
45.14 odd 6 225.4.e.d.151.3 14
45.23 even 12 225.4.k.d.124.5 28
45.29 odd 6 225.4.e.d.76.3 14
45.32 even 12 225.4.k.d.124.10 28
45.38 even 12 225.4.k.d.49.10 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.4.e.c.16.5 14 9.5 odd 6
45.4.e.c.31.5 yes 14 9.2 odd 6
135.4.e.c.46.3 14 9.4 even 3
135.4.e.c.91.3 14 9.7 even 3
225.4.e.d.76.3 14 45.29 odd 6
225.4.e.d.151.3 14 45.14 odd 6
225.4.k.d.49.5 28 45.2 even 12
225.4.k.d.49.10 28 45.38 even 12
225.4.k.d.124.5 28 45.23 even 12
225.4.k.d.124.10 28 45.32 even 12
405.4.a.m.1.3 7 3.2 odd 2
405.4.a.n.1.5 7 1.1 even 1 trivial
2025.4.a.ba.1.3 7 5.4 even 2
2025.4.a.bb.1.5 7 15.14 odd 2