Properties

Label 1911.4.a.k.1.3
Level $1911$
Weight $4$
Character 1911.1
Self dual yes
Analytic conductor $112.753$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1911.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(112.752650021\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
Defining polynomial: \( x^{3} - x^{2} - 16x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.20905\) of defining polynomial
Character \(\chi\) \(=\) 1911.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.20905 q^{2} -3.00000 q^{3} +9.71610 q^{4} +11.4322 q^{5} -12.6271 q^{6} +7.22315 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+4.20905 q^{2} -3.00000 q^{3} +9.71610 q^{4} +11.4322 q^{5} -12.6271 q^{6} +7.22315 q^{8} +9.00000 q^{9} +48.1187 q^{10} +25.8785 q^{11} -29.1483 q^{12} -13.0000 q^{13} -34.2966 q^{15} -47.3262 q^{16} +20.3276 q^{17} +37.8814 q^{18} -154.712 q^{19} +111.076 q^{20} +108.924 q^{22} -180.418 q^{23} -21.6695 q^{24} +5.69520 q^{25} -54.7176 q^{26} -27.0000 q^{27} -20.4522 q^{29} -144.356 q^{30} -266.424 q^{31} -256.984 q^{32} -77.6355 q^{33} +85.5599 q^{34} +87.4449 q^{36} +115.984 q^{37} -651.190 q^{38} +39.0000 q^{39} +82.5765 q^{40} -391.184 q^{41} +151.407 q^{43} +251.438 q^{44} +102.890 q^{45} -759.390 q^{46} +467.365 q^{47} +141.979 q^{48} +23.9714 q^{50} -60.9828 q^{51} -126.309 q^{52} +79.9842 q^{53} -113.644 q^{54} +295.848 q^{55} +464.136 q^{57} -86.0843 q^{58} +873.710 q^{59} -333.229 q^{60} +187.068 q^{61} -1121.39 q^{62} -703.047 q^{64} -148.619 q^{65} -326.772 q^{66} -609.204 q^{67} +197.505 q^{68} +541.255 q^{69} +248.038 q^{71} +65.0084 q^{72} -852.765 q^{73} +488.181 q^{74} -17.0856 q^{75} -1503.20 q^{76} +164.153 q^{78} -331.221 q^{79} -541.043 q^{80} +81.0000 q^{81} -1646.51 q^{82} +435.432 q^{83} +232.389 q^{85} +637.281 q^{86} +61.3566 q^{87} +186.924 q^{88} -259.233 q^{89} +433.068 q^{90} -1752.96 q^{92} +799.273 q^{93} +1967.16 q^{94} -1768.70 q^{95} +770.951 q^{96} -1225.17 q^{97} +232.907 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - 9 q^{3} + 10 q^{4} - 4 q^{5} - 6 q^{6} - 6 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} - 9 q^{3} + 10 q^{4} - 4 q^{5} - 6 q^{6} - 6 q^{8} + 27 q^{9} + 4 q^{10} - 16 q^{11} - 30 q^{12} - 39 q^{13} + 12 q^{15} - 110 q^{16} + 146 q^{17} + 18 q^{18} - 94 q^{19} + 244 q^{20} - 56 q^{22} - 48 q^{23} + 18 q^{24} + 145 q^{25} - 26 q^{26} - 81 q^{27} - 2 q^{29} - 12 q^{30} - 302 q^{31} + 154 q^{32} + 48 q^{33} - 164 q^{34} + 90 q^{36} + 374 q^{37} - 312 q^{38} + 117 q^{39} + 516 q^{40} - 480 q^{41} - 260 q^{43} + 712 q^{44} - 36 q^{45} - 1104 q^{46} + 24 q^{47} + 330 q^{48} + 814 q^{50} - 438 q^{51} - 130 q^{52} - 678 q^{53} - 54 q^{54} + 1552 q^{55} + 282 q^{57} - 628 q^{58} + 1788 q^{59} - 732 q^{60} - 230 q^{61} - 1952 q^{62} - 750 q^{64} + 52 q^{65} + 168 q^{66} + 74 q^{67} + 460 q^{68} + 144 q^{69} - 948 q^{71} - 54 q^{72} + 222 q^{73} + 1724 q^{74} - 435 q^{75} - 2392 q^{76} + 78 q^{78} - 24 q^{79} - 1100 q^{80} + 243 q^{81} - 564 q^{82} + 796 q^{83} - 248 q^{85} + 1800 q^{86} + 6 q^{87} + 1608 q^{88} - 1436 q^{89} + 36 q^{90} - 1296 q^{92} + 906 q^{93} + 1920 q^{94} - 4032 q^{95} - 462 q^{96} - 3242 q^{97} - 144 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.20905 1.48812 0.744062 0.668111i \(-0.232897\pi\)
0.744062 + 0.668111i \(0.232897\pi\)
\(3\) −3.00000 −0.577350
\(4\) 9.71610 1.21451
\(5\) 11.4322 1.02253 0.511264 0.859424i \(-0.329178\pi\)
0.511264 + 0.859424i \(0.329178\pi\)
\(6\) −12.6271 −0.859169
\(7\) 0 0
\(8\) 7.22315 0.319221
\(9\) 9.00000 0.333333
\(10\) 48.1187 1.52165
\(11\) 25.8785 0.709333 0.354666 0.934993i \(-0.384594\pi\)
0.354666 + 0.934993i \(0.384594\pi\)
\(12\) −29.1483 −0.701199
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) −34.2966 −0.590356
\(16\) −47.3262 −0.739472
\(17\) 20.3276 0.290010 0.145005 0.989431i \(-0.453680\pi\)
0.145005 + 0.989431i \(0.453680\pi\)
\(18\) 37.8814 0.496041
\(19\) −154.712 −1.86807 −0.934035 0.357181i \(-0.883738\pi\)
−0.934035 + 0.357181i \(0.883738\pi\)
\(20\) 111.076 1.24187
\(21\) 0 0
\(22\) 108.924 1.05558
\(23\) −180.418 −1.63565 −0.817823 0.575471i \(-0.804819\pi\)
−0.817823 + 0.575471i \(0.804819\pi\)
\(24\) −21.6695 −0.184302
\(25\) 5.69520 0.0455616
\(26\) −54.7176 −0.412731
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −20.4522 −0.130961 −0.0654806 0.997854i \(-0.520858\pi\)
−0.0654806 + 0.997854i \(0.520858\pi\)
\(30\) −144.356 −0.878523
\(31\) −266.424 −1.54359 −0.771794 0.635873i \(-0.780640\pi\)
−0.771794 + 0.635873i \(0.780640\pi\)
\(32\) −256.984 −1.41965
\(33\) −77.6355 −0.409534
\(34\) 85.5599 0.431571
\(35\) 0 0
\(36\) 87.4449 0.404837
\(37\) 115.984 0.515340 0.257670 0.966233i \(-0.417045\pi\)
0.257670 + 0.966233i \(0.417045\pi\)
\(38\) −651.190 −2.77992
\(39\) 39.0000 0.160128
\(40\) 82.5765 0.326412
\(41\) −391.184 −1.49006 −0.745032 0.667029i \(-0.767566\pi\)
−0.745032 + 0.667029i \(0.767566\pi\)
\(42\) 0 0
\(43\) 151.407 0.536963 0.268482 0.963285i \(-0.413478\pi\)
0.268482 + 0.963285i \(0.413478\pi\)
\(44\) 251.438 0.861494
\(45\) 102.890 0.340842
\(46\) −759.390 −2.43404
\(47\) 467.365 1.45047 0.725236 0.688500i \(-0.241731\pi\)
0.725236 + 0.688500i \(0.241731\pi\)
\(48\) 141.979 0.426934
\(49\) 0 0
\(50\) 23.9714 0.0678012
\(51\) −60.9828 −0.167437
\(52\) −126.309 −0.336845
\(53\) 79.9842 0.207296 0.103648 0.994614i \(-0.466949\pi\)
0.103648 + 0.994614i \(0.466949\pi\)
\(54\) −113.644 −0.286390
\(55\) 295.848 0.725312
\(56\) 0 0
\(57\) 464.136 1.07853
\(58\) −86.0843 −0.194887
\(59\) 873.710 1.92792 0.963960 0.266045i \(-0.0857171\pi\)
0.963960 + 0.266045i \(0.0857171\pi\)
\(60\) −333.229 −0.716995
\(61\) 187.068 0.392649 0.196325 0.980539i \(-0.437099\pi\)
0.196325 + 0.980539i \(0.437099\pi\)
\(62\) −1121.39 −2.29705
\(63\) 0 0
\(64\) −703.047 −1.37314
\(65\) −148.619 −0.283598
\(66\) −326.772 −0.609437
\(67\) −609.204 −1.11084 −0.555418 0.831571i \(-0.687442\pi\)
−0.555418 + 0.831571i \(0.687442\pi\)
\(68\) 197.505 0.352221
\(69\) 541.255 0.944340
\(70\) 0 0
\(71\) 248.038 0.414601 0.207301 0.978277i \(-0.433532\pi\)
0.207301 + 0.978277i \(0.433532\pi\)
\(72\) 65.0084 0.106407
\(73\) −852.765 −1.36724 −0.683621 0.729838i \(-0.739596\pi\)
−0.683621 + 0.729838i \(0.739596\pi\)
\(74\) 488.181 0.766890
\(75\) −17.0856 −0.0263050
\(76\) −1503.20 −2.26880
\(77\) 0 0
\(78\) 164.153 0.238291
\(79\) −331.221 −0.471712 −0.235856 0.971788i \(-0.575789\pi\)
−0.235856 + 0.971788i \(0.575789\pi\)
\(80\) −541.043 −0.756130
\(81\) 81.0000 0.111111
\(82\) −1646.51 −2.21740
\(83\) 435.432 0.575842 0.287921 0.957654i \(-0.407036\pi\)
0.287921 + 0.957654i \(0.407036\pi\)
\(84\) 0 0
\(85\) 232.389 0.296543
\(86\) 637.281 0.799067
\(87\) 61.3566 0.0756105
\(88\) 186.924 0.226434
\(89\) −259.233 −0.308749 −0.154375 0.988012i \(-0.549336\pi\)
−0.154375 + 0.988012i \(0.549336\pi\)
\(90\) 433.068 0.507216
\(91\) 0 0
\(92\) −1752.96 −1.98651
\(93\) 799.273 0.891191
\(94\) 1967.16 2.15848
\(95\) −1768.70 −1.91015
\(96\) 770.951 0.819634
\(97\) −1225.17 −1.28245 −0.641223 0.767355i \(-0.721572\pi\)
−0.641223 + 0.767355i \(0.721572\pi\)
\(98\) 0 0
\(99\) 232.907 0.236444
\(100\) 55.3351 0.0553351
\(101\) −645.416 −0.635855 −0.317927 0.948115i \(-0.602987\pi\)
−0.317927 + 0.948115i \(0.602987\pi\)
\(102\) −256.680 −0.249167
\(103\) 511.137 0.488969 0.244484 0.969653i \(-0.421381\pi\)
0.244484 + 0.969653i \(0.421381\pi\)
\(104\) −93.9010 −0.0885360
\(105\) 0 0
\(106\) 336.657 0.308482
\(107\) 608.195 0.549499 0.274750 0.961516i \(-0.411405\pi\)
0.274750 + 0.961516i \(0.411405\pi\)
\(108\) −262.335 −0.233733
\(109\) −1300.04 −1.14239 −0.571197 0.820813i \(-0.693521\pi\)
−0.571197 + 0.820813i \(0.693521\pi\)
\(110\) 1245.24 1.07935
\(111\) −347.951 −0.297532
\(112\) 0 0
\(113\) 42.1953 0.0351274 0.0175637 0.999846i \(-0.494409\pi\)
0.0175637 + 0.999846i \(0.494409\pi\)
\(114\) 1953.57 1.60499
\(115\) −2062.58 −1.67249
\(116\) −198.716 −0.159054
\(117\) −117.000 −0.0924500
\(118\) 3677.49 2.86899
\(119\) 0 0
\(120\) −247.729 −0.188454
\(121\) −661.303 −0.496847
\(122\) 787.378 0.584311
\(123\) 1173.55 0.860289
\(124\) −2588.61 −1.87471
\(125\) −1363.92 −0.975939
\(126\) 0 0
\(127\) −311.018 −0.217310 −0.108655 0.994080i \(-0.534654\pi\)
−0.108655 + 0.994080i \(0.534654\pi\)
\(128\) −903.291 −0.623753
\(129\) −454.222 −0.310016
\(130\) −625.543 −0.422029
\(131\) −2000.98 −1.33456 −0.667278 0.744809i \(-0.732541\pi\)
−0.667278 + 0.744809i \(0.732541\pi\)
\(132\) −754.314 −0.497384
\(133\) 0 0
\(134\) −2564.17 −1.65306
\(135\) −308.669 −0.196785
\(136\) 146.829 0.0925773
\(137\) 1038.53 0.647644 0.323822 0.946118i \(-0.395032\pi\)
0.323822 + 0.946118i \(0.395032\pi\)
\(138\) 2278.17 1.40529
\(139\) 2858.46 1.74426 0.872128 0.489277i \(-0.162739\pi\)
0.872128 + 0.489277i \(0.162739\pi\)
\(140\) 0 0
\(141\) −1402.09 −0.837430
\(142\) 1044.00 0.616978
\(143\) −336.421 −0.196734
\(144\) −425.936 −0.246491
\(145\) −233.814 −0.133911
\(146\) −3589.33 −2.03462
\(147\) 0 0
\(148\) 1126.91 0.625887
\(149\) 743.479 0.408780 0.204390 0.978890i \(-0.434479\pi\)
0.204390 + 0.978890i \(0.434479\pi\)
\(150\) −71.9141 −0.0391451
\(151\) 2277.24 1.22728 0.613640 0.789586i \(-0.289705\pi\)
0.613640 + 0.789586i \(0.289705\pi\)
\(152\) −1117.51 −0.596328
\(153\) 182.948 0.0966700
\(154\) 0 0
\(155\) −3045.82 −1.57836
\(156\) 378.928 0.194478
\(157\) −3173.51 −1.61321 −0.806605 0.591091i \(-0.798697\pi\)
−0.806605 + 0.591091i \(0.798697\pi\)
\(158\) −1394.12 −0.701966
\(159\) −239.953 −0.119682
\(160\) −2937.89 −1.45163
\(161\) 0 0
\(162\) 340.933 0.165347
\(163\) −2314.65 −1.11225 −0.556126 0.831098i \(-0.687713\pi\)
−0.556126 + 0.831098i \(0.687713\pi\)
\(164\) −3800.78 −1.80970
\(165\) −887.545 −0.418759
\(166\) 1832.76 0.856925
\(167\) 2665.65 1.23517 0.617587 0.786502i \(-0.288110\pi\)
0.617587 + 0.786502i \(0.288110\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 978.138 0.441293
\(171\) −1392.41 −0.622690
\(172\) 1471.09 0.652148
\(173\) 165.243 0.0726198 0.0363099 0.999341i \(-0.488440\pi\)
0.0363099 + 0.999341i \(0.488440\pi\)
\(174\) 258.253 0.112518
\(175\) 0 0
\(176\) −1224.73 −0.524532
\(177\) −2621.13 −1.11309
\(178\) −1091.13 −0.459457
\(179\) 712.339 0.297446 0.148723 0.988879i \(-0.452484\pi\)
0.148723 + 0.988879i \(0.452484\pi\)
\(180\) 999.688 0.413957
\(181\) −2206.53 −0.906133 −0.453066 0.891477i \(-0.649670\pi\)
−0.453066 + 0.891477i \(0.649670\pi\)
\(182\) 0 0
\(183\) −561.204 −0.226696
\(184\) −1303.19 −0.522132
\(185\) 1325.95 0.526949
\(186\) 3364.18 1.32620
\(187\) 526.048 0.205714
\(188\) 4540.96 1.76162
\(189\) 0 0
\(190\) −7444.54 −2.84254
\(191\) 1470.64 0.557129 0.278565 0.960417i \(-0.410141\pi\)
0.278565 + 0.960417i \(0.410141\pi\)
\(192\) 2109.14 0.792782
\(193\) 369.560 0.137832 0.0689158 0.997622i \(-0.478046\pi\)
0.0689158 + 0.997622i \(0.478046\pi\)
\(194\) −5156.80 −1.90844
\(195\) 445.856 0.163735
\(196\) 0 0
\(197\) −4273.41 −1.54552 −0.772761 0.634697i \(-0.781125\pi\)
−0.772761 + 0.634697i \(0.781125\pi\)
\(198\) 980.315 0.351858
\(199\) −4154.31 −1.47985 −0.739927 0.672687i \(-0.765140\pi\)
−0.739927 + 0.672687i \(0.765140\pi\)
\(200\) 41.1373 0.0145442
\(201\) 1827.61 0.641342
\(202\) −2716.59 −0.946230
\(203\) 0 0
\(204\) −592.515 −0.203355
\(205\) −4472.09 −1.52363
\(206\) 2151.40 0.727646
\(207\) −1623.77 −0.545215
\(208\) 615.241 0.205093
\(209\) −4003.71 −1.32508
\(210\) 0 0
\(211\) 1231.59 0.401830 0.200915 0.979609i \(-0.435608\pi\)
0.200915 + 0.979609i \(0.435608\pi\)
\(212\) 777.134 0.251763
\(213\) −744.114 −0.239370
\(214\) 2559.92 0.817723
\(215\) 1730.92 0.549059
\(216\) −195.025 −0.0614341
\(217\) 0 0
\(218\) −5471.92 −1.70002
\(219\) 2558.30 0.789377
\(220\) 2874.49 0.880901
\(221\) −264.259 −0.0804343
\(222\) −1464.54 −0.442764
\(223\) 2187.24 0.656809 0.328404 0.944537i \(-0.393489\pi\)
0.328404 + 0.944537i \(0.393489\pi\)
\(224\) 0 0
\(225\) 51.2568 0.0151872
\(226\) 177.602 0.0522739
\(227\) −4138.67 −1.21010 −0.605051 0.796187i \(-0.706847\pi\)
−0.605051 + 0.796187i \(0.706847\pi\)
\(228\) 4509.59 1.30989
\(229\) 835.354 0.241056 0.120528 0.992710i \(-0.461541\pi\)
0.120528 + 0.992710i \(0.461541\pi\)
\(230\) −8681.50 −2.48887
\(231\) 0 0
\(232\) −147.729 −0.0418056
\(233\) 3685.51 1.03625 0.518124 0.855305i \(-0.326630\pi\)
0.518124 + 0.855305i \(0.326630\pi\)
\(234\) −492.459 −0.137577
\(235\) 5343.01 1.48315
\(236\) 8489.05 2.34148
\(237\) 993.662 0.272343
\(238\) 0 0
\(239\) 3026.21 0.819034 0.409517 0.912303i \(-0.365697\pi\)
0.409517 + 0.912303i \(0.365697\pi\)
\(240\) 1623.13 0.436552
\(241\) −3265.58 −0.872839 −0.436420 0.899743i \(-0.643754\pi\)
−0.436420 + 0.899743i \(0.643754\pi\)
\(242\) −2783.46 −0.739370
\(243\) −243.000 −0.0641500
\(244\) 1817.57 0.476877
\(245\) 0 0
\(246\) 4939.53 1.28022
\(247\) 2011.25 0.518110
\(248\) −1924.42 −0.492746
\(249\) −1306.30 −0.332463
\(250\) −5740.79 −1.45232
\(251\) 6363.16 1.60016 0.800078 0.599897i \(-0.204792\pi\)
0.800078 + 0.599897i \(0.204792\pi\)
\(252\) 0 0
\(253\) −4668.96 −1.16022
\(254\) −1309.09 −0.323385
\(255\) −697.168 −0.171209
\(256\) 1822.38 0.444917
\(257\) 6085.36 1.47702 0.738511 0.674242i \(-0.235529\pi\)
0.738511 + 0.674242i \(0.235529\pi\)
\(258\) −1911.84 −0.461342
\(259\) 0 0
\(260\) −1443.99 −0.344433
\(261\) −184.070 −0.0436538
\(262\) −8422.24 −1.98598
\(263\) 123.227 0.0288916 0.0144458 0.999896i \(-0.495402\pi\)
0.0144458 + 0.999896i \(0.495402\pi\)
\(264\) −560.773 −0.130732
\(265\) 914.395 0.211965
\(266\) 0 0
\(267\) 777.700 0.178256
\(268\) −5919.08 −1.34913
\(269\) 1935.79 0.438763 0.219381 0.975639i \(-0.429596\pi\)
0.219381 + 0.975639i \(0.429596\pi\)
\(270\) −1299.20 −0.292841
\(271\) 4612.69 1.03395 0.516976 0.856000i \(-0.327058\pi\)
0.516976 + 0.856000i \(0.327058\pi\)
\(272\) −962.028 −0.214454
\(273\) 0 0
\(274\) 4371.20 0.963774
\(275\) 147.383 0.0323183
\(276\) 5258.89 1.14691
\(277\) −5834.30 −1.26552 −0.632761 0.774347i \(-0.718078\pi\)
−0.632761 + 0.774347i \(0.718078\pi\)
\(278\) 12031.4 2.59567
\(279\) −2397.82 −0.514529
\(280\) 0 0
\(281\) 4691.91 0.996071 0.498036 0.867157i \(-0.334055\pi\)
0.498036 + 0.867157i \(0.334055\pi\)
\(282\) −5901.49 −1.24620
\(283\) −3465.60 −0.727945 −0.363973 0.931410i \(-0.618580\pi\)
−0.363973 + 0.931410i \(0.618580\pi\)
\(284\) 2409.96 0.503539
\(285\) 5306.09 1.10283
\(286\) −1416.01 −0.292764
\(287\) 0 0
\(288\) −2312.85 −0.473216
\(289\) −4499.79 −0.915894
\(290\) −984.133 −0.199277
\(291\) 3675.51 0.740420
\(292\) −8285.55 −1.66053
\(293\) −2677.31 −0.533822 −0.266911 0.963721i \(-0.586003\pi\)
−0.266911 + 0.963721i \(0.586003\pi\)
\(294\) 0 0
\(295\) 9988.43 1.97135
\(296\) 837.767 0.164507
\(297\) −698.720 −0.136511
\(298\) 3129.34 0.608315
\(299\) 2345.44 0.453646
\(300\) −166.005 −0.0319477
\(301\) 0 0
\(302\) 9585.02 1.82634
\(303\) 1936.25 0.367111
\(304\) 7321.93 1.38139
\(305\) 2138.60 0.401494
\(306\) 770.039 0.143857
\(307\) −471.915 −0.0877316 −0.0438658 0.999037i \(-0.513967\pi\)
−0.0438658 + 0.999037i \(0.513967\pi\)
\(308\) 0 0
\(309\) −1533.41 −0.282306
\(310\) −12820.0 −2.34880
\(311\) 1518.52 0.276872 0.138436 0.990371i \(-0.455793\pi\)
0.138436 + 0.990371i \(0.455793\pi\)
\(312\) 281.703 0.0511163
\(313\) −4049.86 −0.731348 −0.365674 0.930743i \(-0.619161\pi\)
−0.365674 + 0.930743i \(0.619161\pi\)
\(314\) −13357.5 −2.40066
\(315\) 0 0
\(316\) −3218.17 −0.572900
\(317\) 3253.96 0.576532 0.288266 0.957550i \(-0.406921\pi\)
0.288266 + 0.957550i \(0.406921\pi\)
\(318\) −1009.97 −0.178102
\(319\) −529.272 −0.0928951
\(320\) −8037.37 −1.40407
\(321\) −1824.59 −0.317254
\(322\) 0 0
\(323\) −3144.92 −0.541759
\(324\) 787.004 0.134946
\(325\) −74.0375 −0.0126365
\(326\) −9742.46 −1.65517
\(327\) 3900.11 0.659561
\(328\) −2825.58 −0.475660
\(329\) 0 0
\(330\) −3735.72 −0.623165
\(331\) −3422.45 −0.568322 −0.284161 0.958777i \(-0.591715\pi\)
−0.284161 + 0.958777i \(0.591715\pi\)
\(332\) 4230.71 0.699368
\(333\) 1043.85 0.171780
\(334\) 11219.8 1.83809
\(335\) −6964.54 −1.13586
\(336\) 0 0
\(337\) −9301.67 −1.50354 −0.751772 0.659423i \(-0.770801\pi\)
−0.751772 + 0.659423i \(0.770801\pi\)
\(338\) 711.329 0.114471
\(339\) −126.586 −0.0202808
\(340\) 2257.92 0.360155
\(341\) −6894.66 −1.09492
\(342\) −5860.71 −0.926640
\(343\) 0 0
\(344\) 1093.64 0.171410
\(345\) 6187.74 0.965613
\(346\) 695.518 0.108067
\(347\) 216.898 0.0335554 0.0167777 0.999859i \(-0.494659\pi\)
0.0167777 + 0.999859i \(0.494659\pi\)
\(348\) 596.147 0.0918299
\(349\) 4809.84 0.737721 0.368861 0.929485i \(-0.379748\pi\)
0.368861 + 0.929485i \(0.379748\pi\)
\(350\) 0 0
\(351\) 351.000 0.0533761
\(352\) −6650.35 −1.00700
\(353\) 2859.64 0.431170 0.215585 0.976485i \(-0.430834\pi\)
0.215585 + 0.976485i \(0.430834\pi\)
\(354\) −11032.5 −1.65641
\(355\) 2835.62 0.423941
\(356\) −2518.74 −0.374980
\(357\) 0 0
\(358\) 2998.27 0.442636
\(359\) 3686.04 0.541899 0.270949 0.962594i \(-0.412662\pi\)
0.270949 + 0.962594i \(0.412662\pi\)
\(360\) 743.188 0.108804
\(361\) 17076.8 2.48969
\(362\) −9287.39 −1.34844
\(363\) 1983.91 0.286855
\(364\) 0 0
\(365\) −9748.98 −1.39804
\(366\) −2362.14 −0.337352
\(367\) 3470.59 0.493633 0.246816 0.969062i \(-0.420616\pi\)
0.246816 + 0.969062i \(0.420616\pi\)
\(368\) 8538.52 1.20951
\(369\) −3520.65 −0.496688
\(370\) 5580.98 0.784166
\(371\) 0 0
\(372\) 7765.82 1.08236
\(373\) −11963.4 −1.66070 −0.830352 0.557240i \(-0.811860\pi\)
−0.830352 + 0.557240i \(0.811860\pi\)
\(374\) 2214.16 0.306127
\(375\) 4091.75 0.563459
\(376\) 3375.85 0.463021
\(377\) 265.879 0.0363221
\(378\) 0 0
\(379\) 345.604 0.0468403 0.0234202 0.999726i \(-0.492544\pi\)
0.0234202 + 0.999726i \(0.492544\pi\)
\(380\) −17184.8 −2.31990
\(381\) 933.055 0.125464
\(382\) 6189.99 0.829078
\(383\) 3386.40 0.451793 0.225897 0.974151i \(-0.427469\pi\)
0.225897 + 0.974151i \(0.427469\pi\)
\(384\) 2709.87 0.360124
\(385\) 0 0
\(386\) 1555.49 0.205110
\(387\) 1362.67 0.178988
\(388\) −11903.9 −1.55755
\(389\) −1629.88 −0.212438 −0.106219 0.994343i \(-0.533874\pi\)
−0.106219 + 0.994343i \(0.533874\pi\)
\(390\) 1876.63 0.243659
\(391\) −3667.47 −0.474353
\(392\) 0 0
\(393\) 6002.95 0.770506
\(394\) −17987.0 −2.29993
\(395\) −3786.58 −0.482338
\(396\) 2262.94 0.287165
\(397\) −7938.94 −1.00364 −0.501819 0.864973i \(-0.667336\pi\)
−0.501819 + 0.864973i \(0.667336\pi\)
\(398\) −17485.7 −2.20221
\(399\) 0 0
\(400\) −269.532 −0.0336915
\(401\) 214.402 0.0267001 0.0133500 0.999911i \(-0.495750\pi\)
0.0133500 + 0.999911i \(0.495750\pi\)
\(402\) 7692.51 0.954396
\(403\) 3463.52 0.428114
\(404\) −6270.93 −0.772253
\(405\) 926.008 0.113614
\(406\) 0 0
\(407\) 3001.48 0.365548
\(408\) −440.488 −0.0534495
\(409\) 4783.73 0.578338 0.289169 0.957278i \(-0.406621\pi\)
0.289169 + 0.957278i \(0.406621\pi\)
\(410\) −18823.2 −2.26735
\(411\) −3115.58 −0.373917
\(412\) 4966.25 0.593859
\(413\) 0 0
\(414\) −6834.51 −0.811347
\(415\) 4977.95 0.588815
\(416\) 3340.79 0.393739
\(417\) −8575.39 −1.00705
\(418\) −16851.8 −1.97189
\(419\) 9903.67 1.15472 0.577358 0.816491i \(-0.304084\pi\)
0.577358 + 0.816491i \(0.304084\pi\)
\(420\) 0 0
\(421\) −12120.6 −1.40314 −0.701572 0.712598i \(-0.747518\pi\)
−0.701572 + 0.712598i \(0.747518\pi\)
\(422\) 5183.82 0.597973
\(423\) 4206.28 0.483491
\(424\) 577.738 0.0661732
\(425\) 115.770 0.0132133
\(426\) −3132.01 −0.356213
\(427\) 0 0
\(428\) 5909.28 0.667374
\(429\) 1009.26 0.113584
\(430\) 7285.53 0.817068
\(431\) −13672.6 −1.52805 −0.764023 0.645189i \(-0.776779\pi\)
−0.764023 + 0.645189i \(0.776779\pi\)
\(432\) 1277.81 0.142311
\(433\) −7113.10 −0.789455 −0.394727 0.918798i \(-0.629161\pi\)
−0.394727 + 0.918798i \(0.629161\pi\)
\(434\) 0 0
\(435\) 701.441 0.0773138
\(436\) −12631.3 −1.38745
\(437\) 27912.9 3.05550
\(438\) 10768.0 1.17469
\(439\) 6022.04 0.654707 0.327353 0.944902i \(-0.393843\pi\)
0.327353 + 0.944902i \(0.393843\pi\)
\(440\) 2136.96 0.231535
\(441\) 0 0
\(442\) −1112.28 −0.119696
\(443\) −12994.4 −1.39364 −0.696821 0.717245i \(-0.745403\pi\)
−0.696821 + 0.717245i \(0.745403\pi\)
\(444\) −3380.72 −0.361356
\(445\) −2963.61 −0.315704
\(446\) 9206.20 0.977413
\(447\) −2230.44 −0.236009
\(448\) 0 0
\(449\) 10984.3 1.15452 0.577260 0.816560i \(-0.304122\pi\)
0.577260 + 0.816560i \(0.304122\pi\)
\(450\) 215.742 0.0226004
\(451\) −10123.2 −1.05695
\(452\) 409.973 0.0426627
\(453\) −6831.72 −0.708570
\(454\) −17419.9 −1.80078
\(455\) 0 0
\(456\) 3352.52 0.344290
\(457\) 9834.10 1.00661 0.503304 0.864109i \(-0.332118\pi\)
0.503304 + 0.864109i \(0.332118\pi\)
\(458\) 3516.05 0.358721
\(459\) −548.845 −0.0558124
\(460\) −20040.2 −2.03126
\(461\) −3401.42 −0.343644 −0.171822 0.985128i \(-0.554965\pi\)
−0.171822 + 0.985128i \(0.554965\pi\)
\(462\) 0 0
\(463\) 1739.42 0.174596 0.0872979 0.996182i \(-0.472177\pi\)
0.0872979 + 0.996182i \(0.472177\pi\)
\(464\) 967.925 0.0968422
\(465\) 9137.45 0.911267
\(466\) 15512.5 1.54207
\(467\) 7958.82 0.788630 0.394315 0.918975i \(-0.370982\pi\)
0.394315 + 0.918975i \(0.370982\pi\)
\(468\) −1136.78 −0.112282
\(469\) 0 0
\(470\) 22489.0 2.20711
\(471\) 9520.54 0.931387
\(472\) 6310.94 0.615433
\(473\) 3918.20 0.380886
\(474\) 4182.37 0.405280
\(475\) −881.114 −0.0851122
\(476\) 0 0
\(477\) 719.858 0.0690986
\(478\) 12737.5 1.21882
\(479\) −8431.98 −0.804315 −0.402158 0.915570i \(-0.631740\pi\)
−0.402158 + 0.915570i \(0.631740\pi\)
\(480\) 8813.66 0.838097
\(481\) −1507.79 −0.142930
\(482\) −13745.0 −1.29889
\(483\) 0 0
\(484\) −6425.29 −0.603427
\(485\) −14006.4 −1.31133
\(486\) −1022.80 −0.0954632
\(487\) −11684.7 −1.08723 −0.543617 0.839334i \(-0.682945\pi\)
−0.543617 + 0.839334i \(0.682945\pi\)
\(488\) 1351.22 0.125342
\(489\) 6943.94 0.642159
\(490\) 0 0
\(491\) −3954.70 −0.363489 −0.181745 0.983346i \(-0.558174\pi\)
−0.181745 + 0.983346i \(0.558174\pi\)
\(492\) 11402.3 1.04483
\(493\) −415.744 −0.0379801
\(494\) 8465.47 0.771011
\(495\) 2662.63 0.241771
\(496\) 12608.8 1.14144
\(497\) 0 0
\(498\) −5498.27 −0.494746
\(499\) −5690.37 −0.510493 −0.255246 0.966876i \(-0.582157\pi\)
−0.255246 + 0.966876i \(0.582157\pi\)
\(500\) −13251.9 −1.18529
\(501\) −7996.95 −0.713128
\(502\) 26782.8 2.38123
\(503\) −10859.1 −0.962595 −0.481298 0.876557i \(-0.659834\pi\)
−0.481298 + 0.876557i \(0.659834\pi\)
\(504\) 0 0
\(505\) −7378.53 −0.650178
\(506\) −19651.9 −1.72655
\(507\) −507.000 −0.0444116
\(508\) −3021.88 −0.263926
\(509\) 18558.6 1.61610 0.808049 0.589115i \(-0.200524\pi\)
0.808049 + 0.589115i \(0.200524\pi\)
\(510\) −2934.41 −0.254780
\(511\) 0 0
\(512\) 14896.8 1.28584
\(513\) 4177.22 0.359510
\(514\) 25613.6 2.19799
\(515\) 5843.42 0.499984
\(516\) −4413.27 −0.376518
\(517\) 12094.7 1.02887
\(518\) 0 0
\(519\) −495.730 −0.0419271
\(520\) −1073.49 −0.0905305
\(521\) −17297.5 −1.45454 −0.727271 0.686350i \(-0.759212\pi\)
−0.727271 + 0.686350i \(0.759212\pi\)
\(522\) −774.759 −0.0649622
\(523\) 5016.11 0.419386 0.209693 0.977767i \(-0.432753\pi\)
0.209693 + 0.977767i \(0.432753\pi\)
\(524\) −19441.8 −1.62084
\(525\) 0 0
\(526\) 518.667 0.0429942
\(527\) −5415.77 −0.447656
\(528\) 3674.19 0.302839
\(529\) 20383.8 1.67533
\(530\) 3848.73 0.315431
\(531\) 7863.39 0.642640
\(532\) 0 0
\(533\) 5085.39 0.413269
\(534\) 3273.38 0.265268
\(535\) 6953.01 0.561878
\(536\) −4400.37 −0.354603
\(537\) −2137.02 −0.171730
\(538\) 8147.83 0.652933
\(539\) 0 0
\(540\) −2999.06 −0.238998
\(541\) 17642.3 1.40204 0.701018 0.713144i \(-0.252729\pi\)
0.701018 + 0.713144i \(0.252729\pi\)
\(542\) 19415.0 1.53865
\(543\) 6619.59 0.523156
\(544\) −5223.86 −0.411712
\(545\) −14862.3 −1.16813
\(546\) 0 0
\(547\) −18414.9 −1.43943 −0.719713 0.694271i \(-0.755727\pi\)
−0.719713 + 0.694271i \(0.755727\pi\)
\(548\) 10090.4 0.786571
\(549\) 1683.61 0.130883
\(550\) 620.343 0.0480937
\(551\) 3164.20 0.244645
\(552\) 3909.57 0.301453
\(553\) 0 0
\(554\) −24556.9 −1.88325
\(555\) −3977.84 −0.304234
\(556\) 27773.1 2.11842
\(557\) 8179.15 0.622193 0.311096 0.950378i \(-0.399304\pi\)
0.311096 + 0.950378i \(0.399304\pi\)
\(558\) −10092.5 −0.765683
\(559\) −1968.30 −0.148927
\(560\) 0 0
\(561\) −1578.14 −0.118769
\(562\) 19748.5 1.48228
\(563\) 1880.07 0.140738 0.0703690 0.997521i \(-0.477582\pi\)
0.0703690 + 0.997521i \(0.477582\pi\)
\(564\) −13622.9 −1.01707
\(565\) 482.385 0.0359187
\(566\) −14586.9 −1.08327
\(567\) 0 0
\(568\) 1791.62 0.132350
\(569\) 10118.3 0.745485 0.372743 0.927935i \(-0.378417\pi\)
0.372743 + 0.927935i \(0.378417\pi\)
\(570\) 22333.6 1.64114
\(571\) 23428.9 1.71711 0.858555 0.512721i \(-0.171362\pi\)
0.858555 + 0.512721i \(0.171362\pi\)
\(572\) −3268.70 −0.238935
\(573\) −4411.92 −0.321659
\(574\) 0 0
\(575\) −1027.52 −0.0745225
\(576\) −6327.42 −0.457713
\(577\) −20508.1 −1.47966 −0.739831 0.672793i \(-0.765094\pi\)
−0.739831 + 0.672793i \(0.765094\pi\)
\(578\) −18939.8 −1.36296
\(579\) −1108.68 −0.0795771
\(580\) −2271.76 −0.162637
\(581\) 0 0
\(582\) 15470.4 1.10184
\(583\) 2069.87 0.147042
\(584\) −6159.65 −0.436452
\(585\) −1337.57 −0.0945327
\(586\) −11268.9 −0.794394
\(587\) 5968.43 0.419665 0.209833 0.977737i \(-0.432708\pi\)
0.209833 + 0.977737i \(0.432708\pi\)
\(588\) 0 0
\(589\) 41219.0 2.88353
\(590\) 42041.8 2.93361
\(591\) 12820.2 0.892308
\(592\) −5489.06 −0.381080
\(593\) 14659.5 1.01517 0.507584 0.861602i \(-0.330539\pi\)
0.507584 + 0.861602i \(0.330539\pi\)
\(594\) −2940.95 −0.203146
\(595\) 0 0
\(596\) 7223.72 0.496468
\(597\) 12462.9 0.854394
\(598\) 9872.07 0.675082
\(599\) 23635.9 1.61225 0.806125 0.591746i \(-0.201561\pi\)
0.806125 + 0.591746i \(0.201561\pi\)
\(600\) −123.412 −0.00839711
\(601\) 11527.0 0.782356 0.391178 0.920315i \(-0.372068\pi\)
0.391178 + 0.920315i \(0.372068\pi\)
\(602\) 0 0
\(603\) −5482.83 −0.370279
\(604\) 22125.9 1.49055
\(605\) −7560.15 −0.508039
\(606\) 8149.77 0.546306
\(607\) 5098.56 0.340930 0.170465 0.985364i \(-0.445473\pi\)
0.170465 + 0.985364i \(0.445473\pi\)
\(608\) 39758.4 2.65200
\(609\) 0 0
\(610\) 9001.47 0.597473
\(611\) −6075.74 −0.402288
\(612\) 1777.55 0.117407
\(613\) 1516.39 0.0999128 0.0499564 0.998751i \(-0.484092\pi\)
0.0499564 + 0.998751i \(0.484092\pi\)
\(614\) −1986.31 −0.130556
\(615\) 13416.3 0.879668
\(616\) 0 0
\(617\) 18539.3 1.20966 0.604832 0.796353i \(-0.293240\pi\)
0.604832 + 0.796353i \(0.293240\pi\)
\(618\) −6454.20 −0.420107
\(619\) −25684.9 −1.66779 −0.833897 0.551920i \(-0.813895\pi\)
−0.833897 + 0.551920i \(0.813895\pi\)
\(620\) −29593.5 −1.91694
\(621\) 4871.30 0.314780
\(622\) 6391.51 0.412020
\(623\) 0 0
\(624\) −1845.72 −0.118410
\(625\) −16304.5 −1.04349
\(626\) −17046.1 −1.08834
\(627\) 12011.1 0.765038
\(628\) −30834.2 −1.95926
\(629\) 2357.67 0.149454
\(630\) 0 0
\(631\) −22410.9 −1.41389 −0.706945 0.707269i \(-0.749927\pi\)
−0.706945 + 0.707269i \(0.749927\pi\)
\(632\) −2392.46 −0.150580
\(633\) −3694.77 −0.231997
\(634\) 13696.1 0.857950
\(635\) −3555.62 −0.222206
\(636\) −2331.40 −0.145356
\(637\) 0 0
\(638\) −2227.73 −0.138239
\(639\) 2232.34 0.138200
\(640\) −10326.6 −0.637804
\(641\) 6827.81 0.420721 0.210361 0.977624i \(-0.432536\pi\)
0.210361 + 0.977624i \(0.432536\pi\)
\(642\) −7679.77 −0.472113
\(643\) 23264.3 1.42684 0.713418 0.700738i \(-0.247146\pi\)
0.713418 + 0.700738i \(0.247146\pi\)
\(644\) 0 0
\(645\) −5192.76 −0.316999
\(646\) −13237.1 −0.806204
\(647\) −14745.9 −0.896014 −0.448007 0.894030i \(-0.647866\pi\)
−0.448007 + 0.894030i \(0.647866\pi\)
\(648\) 585.075 0.0354690
\(649\) 22610.3 1.36754
\(650\) −311.628 −0.0188047
\(651\) 0 0
\(652\) −22489.3 −1.35084
\(653\) 10909.0 0.653755 0.326878 0.945067i \(-0.394003\pi\)
0.326878 + 0.945067i \(0.394003\pi\)
\(654\) 16415.8 0.981509
\(655\) −22875.7 −1.36462
\(656\) 18513.2 1.10186
\(657\) −7674.89 −0.455747
\(658\) 0 0
\(659\) −4182.99 −0.247263 −0.123631 0.992328i \(-0.539454\pi\)
−0.123631 + 0.992328i \(0.539454\pi\)
\(660\) −8623.47 −0.508588
\(661\) −2224.23 −0.130881 −0.0654406 0.997856i \(-0.520845\pi\)
−0.0654406 + 0.997856i \(0.520845\pi\)
\(662\) −14405.2 −0.845734
\(663\) 792.776 0.0464387
\(664\) 3145.19 0.183821
\(665\) 0 0
\(666\) 4393.63 0.255630
\(667\) 3689.95 0.214206
\(668\) 25899.7 1.50013
\(669\) −6561.72 −0.379209
\(670\) −29314.1 −1.69030
\(671\) 4841.04 0.278519
\(672\) 0 0
\(673\) −24152.5 −1.38337 −0.691687 0.722197i \(-0.743132\pi\)
−0.691687 + 0.722197i \(0.743132\pi\)
\(674\) −39151.2 −2.23746
\(675\) −153.770 −0.00876833
\(676\) 1642.02 0.0934240
\(677\) 15310.7 0.869187 0.434593 0.900627i \(-0.356892\pi\)
0.434593 + 0.900627i \(0.356892\pi\)
\(678\) −532.806 −0.0301804
\(679\) 0 0
\(680\) 1678.58 0.0946628
\(681\) 12416.0 0.698652
\(682\) −29020.0 −1.62937
\(683\) 11399.6 0.638646 0.319323 0.947646i \(-0.396545\pi\)
0.319323 + 0.947646i \(0.396545\pi\)
\(684\) −13528.8 −0.756265
\(685\) 11872.6 0.662233
\(686\) 0 0
\(687\) −2506.06 −0.139174
\(688\) −7165.54 −0.397069
\(689\) −1039.79 −0.0574935
\(690\) 26044.5 1.43695
\(691\) 3323.23 0.182955 0.0914773 0.995807i \(-0.470841\pi\)
0.0914773 + 0.995807i \(0.470841\pi\)
\(692\) 1605.52 0.0881976
\(693\) 0 0
\(694\) 912.936 0.0499345
\(695\) 32678.5 1.78355
\(696\) 443.188 0.0241365
\(697\) −7951.82 −0.432133
\(698\) 20244.8 1.09782
\(699\) −11056.5 −0.598279
\(700\) 0 0
\(701\) −12670.4 −0.682673 −0.341336 0.939941i \(-0.610880\pi\)
−0.341336 + 0.939941i \(0.610880\pi\)
\(702\) 1477.38 0.0794302
\(703\) −17944.0 −0.962692
\(704\) −18193.8 −0.974012
\(705\) −16029.0 −0.856295
\(706\) 12036.4 0.641635
\(707\) 0 0
\(708\) −25467.2 −1.35186
\(709\) 13075.2 0.692594 0.346297 0.938125i \(-0.387439\pi\)
0.346297 + 0.938125i \(0.387439\pi\)
\(710\) 11935.3 0.630877
\(711\) −2980.99 −0.157237
\(712\) −1872.48 −0.0985592
\(713\) 48067.8 2.52476
\(714\) 0 0
\(715\) −3846.03 −0.201165
\(716\) 6921.16 0.361251
\(717\) −9078.62 −0.472869
\(718\) 15514.7 0.806412
\(719\) 2988.41 0.155005 0.0775026 0.996992i \(-0.475305\pi\)
0.0775026 + 0.996992i \(0.475305\pi\)
\(720\) −4869.38 −0.252043
\(721\) 0 0
\(722\) 71877.0 3.70496
\(723\) 9796.73 0.503934
\(724\) −21438.9 −1.10051
\(725\) −116.479 −0.00596680
\(726\) 8350.37 0.426875
\(727\) 5507.46 0.280963 0.140482 0.990083i \(-0.455135\pi\)
0.140482 + 0.990083i \(0.455135\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −41033.9 −2.08046
\(731\) 3077.75 0.155725
\(732\) −5452.71 −0.275325
\(733\) 36585.2 1.84353 0.921764 0.387751i \(-0.126748\pi\)
0.921764 + 0.387751i \(0.126748\pi\)
\(734\) 14607.9 0.734587
\(735\) 0 0
\(736\) 46364.6 2.32204
\(737\) −15765.3 −0.787953
\(738\) −14818.6 −0.739133
\(739\) 6425.89 0.319865 0.159933 0.987128i \(-0.448872\pi\)
0.159933 + 0.987128i \(0.448872\pi\)
\(740\) 12883.0 0.639986
\(741\) −6033.76 −0.299131
\(742\) 0 0
\(743\) 20411.0 1.00782 0.503908 0.863757i \(-0.331895\pi\)
0.503908 + 0.863757i \(0.331895\pi\)
\(744\) 5773.27 0.284487
\(745\) 8499.60 0.417988
\(746\) −50354.6 −2.47133
\(747\) 3918.89 0.191947
\(748\) 5111.13 0.249842
\(749\) 0 0
\(750\) 17222.4 0.838496
\(751\) −24259.5 −1.17875 −0.589375 0.807860i \(-0.700626\pi\)
−0.589375 + 0.807860i \(0.700626\pi\)
\(752\) −22118.6 −1.07258
\(753\) −19089.5 −0.923850
\(754\) 1119.10 0.0540518
\(755\) 26033.9 1.25493
\(756\) 0 0
\(757\) 9295.39 0.446297 0.223148 0.974785i \(-0.428367\pi\)
0.223148 + 0.974785i \(0.428367\pi\)
\(758\) 1454.66 0.0697042
\(759\) 14006.9 0.669851
\(760\) −12775.6 −0.609761
\(761\) 21974.7 1.04676 0.523378 0.852101i \(-0.324672\pi\)
0.523378 + 0.852101i \(0.324672\pi\)
\(762\) 3927.27 0.186706
\(763\) 0 0
\(764\) 14288.9 0.676641
\(765\) 2091.50 0.0988476
\(766\) 14253.5 0.672324
\(767\) −11358.2 −0.534709
\(768\) −5467.13 −0.256873
\(769\) −22987.4 −1.07795 −0.538977 0.842320i \(-0.681189\pi\)
−0.538977 + 0.842320i \(0.681189\pi\)
\(770\) 0 0
\(771\) −18256.1 −0.852759
\(772\) 3590.68 0.167398
\(773\) −31970.9 −1.48760 −0.743799 0.668404i \(-0.766978\pi\)
−0.743799 + 0.668404i \(0.766978\pi\)
\(774\) 5735.53 0.266356
\(775\) −1517.34 −0.0703283
\(776\) −8849.59 −0.409384
\(777\) 0 0
\(778\) −6860.25 −0.316133
\(779\) 60520.8 2.78354
\(780\) 4331.98 0.198859
\(781\) 6418.85 0.294090
\(782\) −15436.6 −0.705896
\(783\) 552.209 0.0252035
\(784\) 0 0
\(785\) −36280.2 −1.64955
\(786\) 25266.7 1.14661
\(787\) −6087.26 −0.275715 −0.137857 0.990452i \(-0.544022\pi\)
−0.137857 + 0.990452i \(0.544022\pi\)
\(788\) −41520.9 −1.87706
\(789\) −369.680 −0.0166805
\(790\) −15937.9 −0.717779
\(791\) 0 0
\(792\) 1682.32 0.0754780
\(793\) −2431.88 −0.108901
\(794\) −33415.4 −1.49354
\(795\) −2743.19 −0.122378
\(796\) −40363.6 −1.79730
\(797\) −23080.0 −1.02577 −0.512883 0.858458i \(-0.671423\pi\)
−0.512883 + 0.858458i \(0.671423\pi\)
\(798\) 0 0
\(799\) 9500.41 0.420651
\(800\) −1463.57 −0.0646813
\(801\) −2333.10 −0.102916
\(802\) 902.429 0.0397330
\(803\) −22068.3 −0.969829
\(804\) 17757.3 0.778918
\(805\) 0 0
\(806\) 14578.1 0.637087
\(807\) −5807.37 −0.253320
\(808\) −4661.94 −0.202978
\(809\) −32377.8 −1.40710 −0.703550 0.710646i \(-0.748403\pi\)
−0.703550 + 0.710646i \(0.748403\pi\)
\(810\) 3897.61 0.169072
\(811\) −26352.8 −1.14103 −0.570513 0.821288i \(-0.693256\pi\)
−0.570513 + 0.821288i \(0.693256\pi\)
\(812\) 0 0
\(813\) −13838.1 −0.596952
\(814\) 12633.4 0.543980
\(815\) −26461.5 −1.13731
\(816\) 2886.08 0.123815
\(817\) −23424.5 −1.00308
\(818\) 20135.0 0.860638
\(819\) 0 0
\(820\) −43451.3 −1.85047
\(821\) 35355.3 1.50294 0.751468 0.659770i \(-0.229346\pi\)
0.751468 + 0.659770i \(0.229346\pi\)
\(822\) −13113.6 −0.556435
\(823\) −12663.3 −0.536347 −0.268173 0.963371i \(-0.586420\pi\)
−0.268173 + 0.963371i \(0.586420\pi\)
\(824\) 3692.02 0.156089
\(825\) −442.149 −0.0186590
\(826\) 0 0
\(827\) −16295.2 −0.685176 −0.342588 0.939486i \(-0.611303\pi\)
−0.342588 + 0.939486i \(0.611303\pi\)
\(828\) −15776.7 −0.662170
\(829\) −13638.9 −0.571411 −0.285705 0.958318i \(-0.592228\pi\)
−0.285705 + 0.958318i \(0.592228\pi\)
\(830\) 20952.4 0.876229
\(831\) 17502.9 0.730649
\(832\) 9139.61 0.380840
\(833\) 0 0
\(834\) −36094.2 −1.49861
\(835\) 30474.2 1.26300
\(836\) −38900.5 −1.60933
\(837\) 7193.46 0.297064
\(838\) 41685.0 1.71836
\(839\) 1890.31 0.0777838 0.0388919 0.999243i \(-0.487617\pi\)
0.0388919 + 0.999243i \(0.487617\pi\)
\(840\) 0 0
\(841\) −23970.7 −0.982849
\(842\) −51016.4 −2.08805
\(843\) −14075.7 −0.575082
\(844\) 11966.3 0.488028
\(845\) 1932.04 0.0786559
\(846\) 17704.5 0.719494
\(847\) 0 0
\(848\) −3785.35 −0.153289
\(849\) 10396.8 0.420279
\(850\) 487.280 0.0196630
\(851\) −20925.6 −0.842914
\(852\) −7229.89 −0.290718
\(853\) −1620.21 −0.0650351 −0.0325175 0.999471i \(-0.510352\pi\)
−0.0325175 + 0.999471i \(0.510352\pi\)
\(854\) 0 0
\(855\) −15918.3 −0.636718
\(856\) 4393.08 0.175412
\(857\) 14508.4 0.578292 0.289146 0.957285i \(-0.406629\pi\)
0.289146 + 0.957285i \(0.406629\pi\)
\(858\) 4248.03 0.169027
\(859\) −29639.8 −1.17730 −0.588648 0.808389i \(-0.700340\pi\)
−0.588648 + 0.808389i \(0.700340\pi\)
\(860\) 16817.8 0.666839
\(861\) 0 0
\(862\) −57548.8 −2.27392
\(863\) 21528.8 0.849186 0.424593 0.905384i \(-0.360417\pi\)
0.424593 + 0.905384i \(0.360417\pi\)
\(864\) 6938.56 0.273211
\(865\) 1889.10 0.0742557
\(866\) −29939.4 −1.17481
\(867\) 13499.4 0.528792
\(868\) 0 0
\(869\) −8571.50 −0.334601
\(870\) 2952.40 0.115053
\(871\) 7919.65 0.308091
\(872\) −9390.36 −0.364676
\(873\) −11026.5 −0.427482
\(874\) 117487. 4.54696
\(875\) 0 0
\(876\) 24856.7 0.958708
\(877\) 14865.3 0.572366 0.286183 0.958175i \(-0.407613\pi\)
0.286183 + 0.958175i \(0.407613\pi\)
\(878\) 25347.1 0.974285
\(879\) 8031.92 0.308202
\(880\) −14001.4 −0.536348
\(881\) 21336.0 0.815921 0.407961 0.913000i \(-0.366240\pi\)
0.407961 + 0.913000i \(0.366240\pi\)
\(882\) 0 0
\(883\) 37538.2 1.43065 0.715323 0.698794i \(-0.246280\pi\)
0.715323 + 0.698794i \(0.246280\pi\)
\(884\) −2567.57 −0.0976884
\(885\) −29965.3 −1.13816
\(886\) −54694.1 −2.07391
\(887\) −34575.0 −1.30881 −0.654406 0.756144i \(-0.727081\pi\)
−0.654406 + 0.756144i \(0.727081\pi\)
\(888\) −2513.30 −0.0949784
\(889\) 0 0
\(890\) −12474.0 −0.469807
\(891\) 2096.16 0.0788148
\(892\) 21251.4 0.797703
\(893\) −72306.9 −2.70958
\(894\) −9388.02 −0.351211
\(895\) 8143.61 0.304146
\(896\) 0 0
\(897\) −7036.32 −0.261913
\(898\) 46233.3 1.71807
\(899\) 5448.96 0.202150
\(900\) 498.016 0.0184450
\(901\) 1625.89 0.0601178
\(902\) −42609.2 −1.57287
\(903\) 0 0
\(904\) 304.783 0.0112134
\(905\) −25225.5 −0.926545
\(906\) −28755.0 −1.05444
\(907\) −10424.8 −0.381641 −0.190820 0.981625i \(-0.561115\pi\)
−0.190820 + 0.981625i \(0.561115\pi\)
\(908\) −40211.7 −1.46968
\(909\) −5808.75 −0.211952
\(910\) 0 0
\(911\) 10961.8 0.398661 0.199331 0.979932i \(-0.436123\pi\)
0.199331 + 0.979932i \(0.436123\pi\)
\(912\) −21965.8 −0.797543
\(913\) 11268.3 0.408464
\(914\) 41392.2 1.49796
\(915\) −6415.80 −0.231803
\(916\) 8116.38 0.292765
\(917\) 0 0
\(918\) −2310.12 −0.0830558
\(919\) −10779.2 −0.386914 −0.193457 0.981109i \(-0.561970\pi\)
−0.193457 + 0.981109i \(0.561970\pi\)
\(920\) −14898.3 −0.533895
\(921\) 1415.74 0.0506519
\(922\) −14316.7 −0.511384
\(923\) −3224.49 −0.114990
\(924\) 0 0
\(925\) 660.549 0.0234797
\(926\) 7321.32 0.259820
\(927\) 4600.23 0.162990
\(928\) 5255.88 0.185919
\(929\) −5429.07 −0.191735 −0.0958675 0.995394i \(-0.530563\pi\)
−0.0958675 + 0.995394i \(0.530563\pi\)
\(930\) 38460.0 1.35608
\(931\) 0 0
\(932\) 35808.8 1.25854
\(933\) −4555.55 −0.159852
\(934\) 33499.1 1.17358
\(935\) 6013.88 0.210348
\(936\) −845.109 −0.0295120
\(937\) 21300.1 0.742631 0.371315 0.928507i \(-0.378907\pi\)
0.371315 + 0.928507i \(0.378907\pi\)
\(938\) 0 0
\(939\) 12149.6 0.422244
\(940\) 51913.2 1.80130
\(941\) −26851.2 −0.930207 −0.465103 0.885256i \(-0.653983\pi\)
−0.465103 + 0.885256i \(0.653983\pi\)
\(942\) 40072.4 1.38602
\(943\) 70576.7 2.43722
\(944\) −41349.4 −1.42564
\(945\) 0 0
\(946\) 16491.9 0.566805
\(947\) −8021.68 −0.275258 −0.137629 0.990484i \(-0.543948\pi\)
−0.137629 + 0.990484i \(0.543948\pi\)
\(948\) 9654.52 0.330764
\(949\) 11085.9 0.379204
\(950\) −3708.65 −0.126658
\(951\) −9761.88 −0.332861
\(952\) 0 0
\(953\) 35715.0 1.21398 0.606990 0.794709i \(-0.292377\pi\)
0.606990 + 0.794709i \(0.292377\pi\)
\(954\) 3029.92 0.102827
\(955\) 16812.6 0.569680
\(956\) 29402.9 0.994726
\(957\) 1587.82 0.0536330
\(958\) −35490.6 −1.19692
\(959\) 0 0
\(960\) 24112.1 0.810641
\(961\) 41190.9 1.38266
\(962\) −6346.35 −0.212697
\(963\) 5473.76 0.183166
\(964\) −31728.7 −1.06007
\(965\) 4224.88 0.140936
\(966\) 0 0
\(967\) 53338.8 1.77380 0.886898 0.461965i \(-0.152855\pi\)
0.886898 + 0.461965i \(0.152855\pi\)
\(968\) −4776.69 −0.158604
\(969\) 9434.77 0.312785
\(970\) −58953.6 −1.95143
\(971\) −23112.9 −0.763882 −0.381941 0.924187i \(-0.624744\pi\)
−0.381941 + 0.924187i \(0.624744\pi\)
\(972\) −2361.01 −0.0779110
\(973\) 0 0
\(974\) −49181.3 −1.61794
\(975\) 222.113 0.00729569
\(976\) −8853.22 −0.290353
\(977\) −52874.6 −1.73143 −0.865715 0.500538i \(-0.833136\pi\)
−0.865715 + 0.500538i \(0.833136\pi\)
\(978\) 29227.4 0.955612
\(979\) −6708.57 −0.219006
\(980\) 0 0
\(981\) −11700.3 −0.380798
\(982\) −16645.5 −0.540917
\(983\) −45173.1 −1.46572 −0.732858 0.680381i \(-0.761814\pi\)
−0.732858 + 0.680381i \(0.761814\pi\)
\(984\) 8476.73 0.274622
\(985\) −48854.5 −1.58034
\(986\) −1749.89 −0.0565190
\(987\) 0 0
\(988\) 19541.6 0.629251
\(989\) −27316.7 −0.878281
\(990\) 11207.2 0.359785
\(991\) 60485.6 1.93884 0.969418 0.245414i \(-0.0789239\pi\)
0.969418 + 0.245414i \(0.0789239\pi\)
\(992\) 68466.7 2.19135
\(993\) 10267.3 0.328121
\(994\) 0 0
\(995\) −47492.8 −1.51319
\(996\) −12692.1 −0.403780
\(997\) 18108.1 0.575214 0.287607 0.957749i \(-0.407140\pi\)
0.287607 + 0.957749i \(0.407140\pi\)
\(998\) −23951.1 −0.759677
\(999\) −3131.56 −0.0991773
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 1911.4.a.k.1.3 3
7.6 odd 2 39.4.a.c.1.3 3
21.20 even 2 117.4.a.f.1.1 3
28.27 even 2 624.4.a.t.1.1 3
35.34 odd 2 975.4.a.l.1.1 3
56.13 odd 2 2496.4.a.bl.1.3 3
56.27 even 2 2496.4.a.bp.1.3 3
84.83 odd 2 1872.4.a.bk.1.3 3
91.34 even 4 507.4.b.g.337.6 6
91.83 even 4 507.4.b.g.337.1 6
91.90 odd 2 507.4.a.h.1.1 3
273.272 even 2 1521.4.a.u.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.3 3 7.6 odd 2
117.4.a.f.1.1 3 21.20 even 2
507.4.a.h.1.1 3 91.90 odd 2
507.4.b.g.337.1 6 91.83 even 4
507.4.b.g.337.6 6 91.34 even 4
624.4.a.t.1.1 3 28.27 even 2
975.4.a.l.1.1 3 35.34 odd 2
1521.4.a.u.1.3 3 273.272 even 2
1872.4.a.bk.1.3 3 84.83 odd 2
1911.4.a.k.1.3 3 1.1 even 1 trivial
2496.4.a.bl.1.3 3 56.13 odd 2
2496.4.a.bp.1.3 3 56.27 even 2