Properties

Label 117.4.a.f.1.1
Level $117$
Weight $4$
Character 117.1
Self dual yes
Analytic conductor $6.903$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.90322347067\)
Analytic rank: \(0\)
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 1.1
Root \(-3.20905\) of defining polynomial
Character \(\chi\) \(=\) 117.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.20905 q^{2} +9.71610 q^{4} +11.4322 q^{5} -11.2543 q^{7} -7.22315 q^{8} +O(q^{10})\) \(q-4.20905 q^{2} +9.71610 q^{4} +11.4322 q^{5} -11.2543 q^{7} -7.22315 q^{8} -48.1187 q^{10} -25.8785 q^{11} +13.0000 q^{13} +47.3699 q^{14} -47.3262 q^{16} +20.3276 q^{17} +154.712 q^{19} +111.076 q^{20} +108.924 q^{22} +180.418 q^{23} +5.69520 q^{25} -54.7176 q^{26} -109.348 q^{28} +20.4522 q^{29} +266.424 q^{31} +256.984 q^{32} -85.5599 q^{34} -128.661 q^{35} +115.984 q^{37} -651.190 q^{38} -82.5765 q^{40} -391.184 q^{41} +151.407 q^{43} -251.438 q^{44} -759.390 q^{46} +467.365 q^{47} -216.341 q^{49} -23.9714 q^{50} +126.309 q^{52} -79.9842 q^{53} -295.848 q^{55} +81.2915 q^{56} -86.0843 q^{58} +873.710 q^{59} -187.068 q^{61} -1121.39 q^{62} -703.047 q^{64} +148.619 q^{65} -609.204 q^{67} +197.505 q^{68} +541.542 q^{70} -248.038 q^{71} +852.765 q^{73} -488.181 q^{74} +1503.20 q^{76} +291.244 q^{77} -331.221 q^{79} -541.043 q^{80} +1646.51 q^{82} +435.432 q^{83} +232.389 q^{85} -637.281 q^{86} +186.924 q^{88} -259.233 q^{89} -146.306 q^{91} +1752.96 q^{92} -1967.16 q^{94} +1768.70 q^{95} +1225.17 q^{97} +910.589 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 30 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 30 q^{7} + 6 q^{8} - 4 q^{10} + 16 q^{11} + 39 q^{13} + 176 q^{14} - 110 q^{16} + 146 q^{17} + 94 q^{19} + 244 q^{20} - 56 q^{22} + 48 q^{23} + 145 q^{25} - 26 q^{26} + 80 q^{28} + 2 q^{29} + 302 q^{31} - 154 q^{32} + 164 q^{34} - 80 q^{35} + 374 q^{37} - 312 q^{38} - 516 q^{40} - 480 q^{41} - 260 q^{43} - 712 q^{44} - 1104 q^{46} + 24 q^{47} + 447 q^{49} - 814 q^{50} + 130 q^{52} + 678 q^{53} - 1552 q^{55} - 96 q^{56} - 628 q^{58} + 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 750 q^{64} - 52 q^{65} + 74 q^{67} + 460 q^{68} + 1216 q^{70} + 948 q^{71} - 222 q^{73} - 1724 q^{74} + 2392 q^{76} - 112 q^{77} - 24 q^{79} - 1100 q^{80} + 564 q^{82} + 796 q^{83} - 248 q^{85} - 1800 q^{86} + 1608 q^{88} - 1436 q^{89} + 390 q^{91} + 1296 q^{92} - 1920 q^{94} + 4032 q^{95} + 3242 q^{97} + 5070 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.20905 −1.48812 −0.744062 0.668111i \(-0.767103\pi\)
−0.744062 + 0.668111i \(0.767103\pi\)
\(3\) 0 0
\(4\) 9.71610 1.21451
\(5\) 11.4322 1.02253 0.511264 0.859424i \(-0.329178\pi\)
0.511264 + 0.859424i \(0.329178\pi\)
\(6\) 0 0
\(7\) −11.2543 −0.607675 −0.303838 0.952724i \(-0.598268\pi\)
−0.303838 + 0.952724i \(0.598268\pi\)
\(8\) −7.22315 −0.319221
\(9\) 0 0
\(10\) −48.1187 −1.52165
\(11\) −25.8785 −0.709333 −0.354666 0.934993i \(-0.615406\pi\)
−0.354666 + 0.934993i \(0.615406\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 47.3699 0.904296
\(15\) 0 0
\(16\) −47.3262 −0.739472
\(17\) 20.3276 0.290010 0.145005 0.989431i \(-0.453680\pi\)
0.145005 + 0.989431i \(0.453680\pi\)
\(18\) 0 0
\(19\) 154.712 1.86807 0.934035 0.357181i \(-0.116262\pi\)
0.934035 + 0.357181i \(0.116262\pi\)
\(20\) 111.076 1.24187
\(21\) 0 0
\(22\) 108.924 1.05558
\(23\) 180.418 1.63565 0.817823 0.575471i \(-0.195181\pi\)
0.817823 + 0.575471i \(0.195181\pi\)
\(24\) 0 0
\(25\) 5.69520 0.0455616
\(26\) −54.7176 −0.412731
\(27\) 0 0
\(28\) −109.348 −0.738029
\(29\) 20.4522 0.130961 0.0654806 0.997854i \(-0.479142\pi\)
0.0654806 + 0.997854i \(0.479142\pi\)
\(30\) 0 0
\(31\) 266.424 1.54359 0.771794 0.635873i \(-0.219360\pi\)
0.771794 + 0.635873i \(0.219360\pi\)
\(32\) 256.984 1.41965
\(33\) 0 0
\(34\) −85.5599 −0.431571
\(35\) −128.661 −0.621364
\(36\) 0 0
\(37\) 115.984 0.515340 0.257670 0.966233i \(-0.417045\pi\)
0.257670 + 0.966233i \(0.417045\pi\)
\(38\) −651.190 −2.77992
\(39\) 0 0
\(40\) −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
\(45\) 0 0
\(46\) −759.390 −2.43404
\(47\) 467.365 1.45047 0.725236 0.688500i \(-0.241731\pi\)
0.725236 + 0.688500i \(0.241731\pi\)
\(48\) 0 0
\(49\) −216.341 −0.630731
\(50\) −23.9714 −0.0678012
\(51\) 0 0
\(52\) 126.309 0.336845
\(53\) −79.9842 −0.207296 −0.103648 0.994614i \(-0.533051\pi\)
−0.103648 + 0.994614i \(0.533051\pi\)
\(54\) 0 0
\(55\) −295.848 −0.725312
\(56\) 81.2915 0.193983
\(57\) 0 0
\(58\) −86.0843 −0.194887
\(59\) 873.710 1.92792 0.963960 0.266045i \(-0.0857171\pi\)
0.963960 + 0.266045i \(0.0857171\pi\)
\(60\) 0 0
\(61\) −187.068 −0.392649 −0.196325 0.980539i \(-0.562901\pi\)
−0.196325 + 0.980539i \(0.562901\pi\)
\(62\) −1121.39 −2.29705
\(63\) 0 0
\(64\) −703.047 −1.37314
\(65\) 148.619 0.283598
\(66\) 0 0
\(67\) −609.204 −1.11084 −0.555418 0.831571i \(-0.687442\pi\)
−0.555418 + 0.831571i \(0.687442\pi\)
\(68\) 197.505 0.352221
\(69\) 0 0
\(70\) 541.542 0.924667
\(71\) −248.038 −0.414601 −0.207301 0.978277i \(-0.566468\pi\)
−0.207301 + 0.978277i \(0.566468\pi\)
\(72\) 0 0
\(73\) 852.765 1.36724 0.683621 0.729838i \(-0.260404\pi\)
0.683621 + 0.729838i \(0.260404\pi\)
\(74\) −488.181 −0.766890
\(75\) 0 0
\(76\) 1503.20 2.26880
\(77\) 291.244 0.431044
\(78\) 0 0
\(79\) −331.221 −0.471712 −0.235856 0.971788i \(-0.575789\pi\)
−0.235856 + 0.971788i \(0.575789\pi\)
\(80\) −541.043 −0.756130
\(81\) 0 0
\(82\) 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
\(87\) 0 0
\(88\) 186.924 0.226434
\(89\) −259.233 −0.308749 −0.154375 0.988012i \(-0.549336\pi\)
−0.154375 + 0.988012i \(0.549336\pi\)
\(90\) 0 0
\(91\) −146.306 −0.168539
\(92\) 1752.96 1.98651
\(93\) 0 0
\(94\) −1967.16 −2.15848
\(95\) 1768.70 1.91015
\(96\) 0 0
\(97\) 1225.17 1.28245 0.641223 0.767355i \(-0.278428\pi\)
0.641223 + 0.767355i \(0.278428\pi\)
\(98\) 910.589 0.938606
\(99\) 0 0
\(100\) 55.3351 0.0553351
\(101\) −645.416 −0.635855 −0.317927 0.948115i \(-0.602987\pi\)
−0.317927 + 0.948115i \(0.602987\pi\)
\(102\) 0 0
\(103\) −511.137 −0.488969 −0.244484 0.969653i \(-0.578619\pi\)
−0.244484 + 0.969653i \(0.578619\pi\)
\(104\) −93.9010 −0.0885360
\(105\) 0 0
\(106\) 336.657 0.308482
\(107\) −608.195 −0.549499 −0.274750 0.961516i \(-0.588595\pi\)
−0.274750 + 0.961516i \(0.588595\pi\)
\(108\) 0 0
\(109\) −1300.04 −1.14239 −0.571197 0.820813i \(-0.693521\pi\)
−0.571197 + 0.820813i \(0.693521\pi\)
\(110\) 1245.24 1.07935
\(111\) 0 0
\(112\) 532.623 0.449359
\(113\) −42.1953 −0.0351274 −0.0175637 0.999846i \(-0.505591\pi\)
−0.0175637 + 0.999846i \(0.505591\pi\)
\(114\) 0 0
\(115\) 2062.58 1.67249
\(116\) 198.716 0.159054
\(117\) 0 0
\(118\) −3677.49 −2.86899
\(119\) −228.773 −0.176232
\(120\) 0 0
\(121\) −661.303 −0.496847
\(122\) 787.378 0.584311
\(123\) 0 0
\(124\) 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
\(129\) 0 0
\(130\) −625.543 −0.422029
\(131\) −2000.98 −1.33456 −0.667278 0.744809i \(-0.732541\pi\)
−0.667278 + 0.744809i \(0.732541\pi\)
\(132\) 0 0
\(133\) −1741.17 −1.13518
\(134\) 2564.17 1.65306
\(135\) 0 0
\(136\) −146.829 −0.0925773
\(137\) −1038.53 −0.647644 −0.323822 0.946118i \(-0.604968\pi\)
−0.323822 + 0.946118i \(0.604968\pi\)
\(138\) 0 0
\(139\) −2858.46 −1.74426 −0.872128 0.489277i \(-0.837261\pi\)
−0.872128 + 0.489277i \(0.837261\pi\)
\(140\) −1250.09 −0.754655
\(141\) 0 0
\(142\) 1044.00 0.616978
\(143\) −336.421 −0.196734
\(144\) 0 0
\(145\) 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 \(-0.565521\pi\)
−0.204390 + 0.978890i \(0.565521\pi\)
\(150\) 0 0
\(151\) 2277.24 1.22728 0.613640 0.789586i \(-0.289705\pi\)
0.613640 + 0.789586i \(0.289705\pi\)
\(152\) −1117.51 −0.596328
\(153\) 0 0
\(154\) −1225.86 −0.641447
\(155\) 3045.82 1.57836
\(156\) 0 0
\(157\) 3173.51 1.61321 0.806605 0.591091i \(-0.201303\pi\)
0.806605 + 0.591091i \(0.201303\pi\)
\(158\) 1394.12 0.701966
\(159\) 0 0
\(160\) 2937.89 1.45163
\(161\) −2030.48 −0.993941
\(162\) 0 0
\(163\) −2314.65 −1.11225 −0.556126 0.831098i \(-0.687713\pi\)
−0.556126 + 0.831098i \(0.687713\pi\)
\(164\) −3800.78 −1.80970
\(165\) 0 0
\(166\) −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
\(171\) 0 0
\(172\) 1471.09 0.652148
\(173\) 165.243 0.0726198 0.0363099 0.999341i \(-0.488440\pi\)
0.0363099 + 0.999341i \(0.488440\pi\)
\(174\) 0 0
\(175\) −64.0954 −0.0276866
\(176\) 1224.73 0.524532
\(177\) 0 0
\(178\) 1091.13 0.459457
\(179\) −712.339 −0.297446 −0.148723 0.988879i \(-0.547516\pi\)
−0.148723 + 0.988879i \(0.547516\pi\)
\(180\) 0 0
\(181\) 2206.53 0.906133 0.453066 0.891477i \(-0.350330\pi\)
0.453066 + 0.891477i \(0.350330\pi\)
\(182\) 615.809 0.250806
\(183\) 0 0
\(184\) −1303.19 −0.522132
\(185\) 1325.95 0.526949
\(186\) 0 0
\(187\) −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 \(-0.589859\pi\)
−0.278565 + 0.960417i \(0.589859\pi\)
\(192\) 0 0
\(193\) 369.560 0.137832 0.0689158 0.997622i \(-0.478046\pi\)
0.0689158 + 0.997622i \(0.478046\pi\)
\(194\) −5156.80 −1.90844
\(195\) 0 0
\(196\) −2101.99 −0.766031
\(197\) 4273.41 1.54552 0.772761 0.634697i \(-0.218875\pi\)
0.772761 + 0.634697i \(0.218875\pi\)
\(198\) 0 0
\(199\) 4154.31 1.47985 0.739927 0.672687i \(-0.234860\pi\)
0.739927 + 0.672687i \(0.234860\pi\)
\(200\) −41.1373 −0.0145442
\(201\) 0 0
\(202\) 2716.59 0.946230
\(203\) −230.175 −0.0795819
\(204\) 0 0
\(205\) −4472.09 −1.52363
\(206\) 2151.40 0.727646
\(207\) 0 0
\(208\) −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
\(213\) 0 0
\(214\) 2559.92 0.817723
\(215\) 1730.92 0.549059
\(216\) 0 0
\(217\) −2998.42 −0.938000
\(218\) 5471.92 1.70002
\(219\) 0 0
\(220\) −2874.49 −0.880901
\(221\) 264.259 0.0804343
\(222\) 0 0
\(223\) −2187.24 −0.656809 −0.328404 0.944537i \(-0.606511\pi\)
−0.328404 + 0.944537i \(0.606511\pi\)
\(224\) −2892.17 −0.862684
\(225\) 0 0
\(226\) 177.602 0.0522739
\(227\) −4138.67 −1.21010 −0.605051 0.796187i \(-0.706847\pi\)
−0.605051 + 0.796187i \(0.706847\pi\)
\(228\) 0 0
\(229\) −835.354 −0.241056 −0.120528 0.992710i \(-0.538459\pi\)
−0.120528 + 0.992710i \(0.538459\pi\)
\(230\) −8681.50 −2.48887
\(231\) 0 0
\(232\) −147.729 −0.0418056
\(233\) −3685.51 −1.03625 −0.518124 0.855305i \(-0.673370\pi\)
−0.518124 + 0.855305i \(0.673370\pi\)
\(234\) 0 0
\(235\) 5343.01 1.48315
\(236\) 8489.05 2.34148
\(237\) 0 0
\(238\) 962.917 0.262255
\(239\) −3026.21 −0.819034 −0.409517 0.912303i \(-0.634303\pi\)
−0.409517 + 0.912303i \(0.634303\pi\)
\(240\) 0 0
\(241\) 3265.58 0.872839 0.436420 0.899743i \(-0.356246\pi\)
0.436420 + 0.899743i \(0.356246\pi\)
\(242\) 2783.46 0.739370
\(243\) 0 0
\(244\) −1817.57 −0.476877
\(245\) −2473.25 −0.644940
\(246\) 0 0
\(247\) 2011.25 0.518110
\(248\) −1924.42 −0.492746
\(249\) 0 0
\(250\) 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
\(255\) 0 0
\(256\) 1822.38 0.444917
\(257\) 6085.36 1.47702 0.738511 0.674242i \(-0.235529\pi\)
0.738511 + 0.674242i \(0.235529\pi\)
\(258\) 0 0
\(259\) −1305.31 −0.313159
\(260\) 1443.99 0.344433
\(261\) 0 0
\(262\) 8422.24 1.98598
\(263\) −123.227 −0.0288916 −0.0144458 0.999896i \(-0.504598\pi\)
−0.0144458 + 0.999896i \(0.504598\pi\)
\(264\) 0 0
\(265\) −914.395 −0.211965
\(266\) 7328.69 1.68929
\(267\) 0 0
\(268\) −5919.08 −1.34913
\(269\) 1935.79 0.438763 0.219381 0.975639i \(-0.429596\pi\)
0.219381 + 0.975639i \(0.429596\pi\)
\(270\) 0 0
\(271\) −4612.69 −1.03395 −0.516976 0.856000i \(-0.672942\pi\)
−0.516976 + 0.856000i \(0.672942\pi\)
\(272\) −962.028 −0.214454
\(273\) 0 0
\(274\) 4371.20 0.963774
\(275\) −147.383 −0.0323183
\(276\) 0 0
\(277\) −5834.30 −1.26552 −0.632761 0.774347i \(-0.718078\pi\)
−0.632761 + 0.774347i \(0.718078\pi\)
\(278\) 12031.4 2.59567
\(279\) 0 0
\(280\) 929.341 0.198353
\(281\) −4691.91 −0.996071 −0.498036 0.867157i \(-0.665945\pi\)
−0.498036 + 0.867157i \(0.665945\pi\)
\(282\) 0 0
\(283\) 3465.60 0.727945 0.363973 0.931410i \(-0.381420\pi\)
0.363973 + 0.931410i \(0.381420\pi\)
\(284\) −2409.96 −0.503539
\(285\) 0 0
\(286\) 1416.01 0.292764
\(287\) 4402.50 0.905475
\(288\) 0 0
\(289\) −4499.79 −0.915894
\(290\) −984.133 −0.199277
\(291\) 0 0
\(292\) 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
\(297\) 0 0
\(298\) 3129.34 0.608315
\(299\) 2345.44 0.453646
\(300\) 0 0
\(301\) −1703.98 −0.326299
\(302\) −9585.02 −1.82634
\(303\) 0 0
\(304\) −7321.93 −1.38139
\(305\) −2138.60 −0.401494
\(306\) 0 0
\(307\) 471.915 0.0877316 0.0438658 0.999037i \(-0.486033\pi\)
0.0438658 + 0.999037i \(0.486033\pi\)
\(308\) 2829.76 0.523508
\(309\) 0 0
\(310\) −12820.0 −2.34880
\(311\) 1518.52 0.276872 0.138436 0.990371i \(-0.455793\pi\)
0.138436 + 0.990371i \(0.455793\pi\)
\(312\) 0 0
\(313\) 4049.86 0.731348 0.365674 0.930743i \(-0.380839\pi\)
0.365674 + 0.930743i \(0.380839\pi\)
\(314\) −13357.5 −2.40066
\(315\) 0 0
\(316\) −3218.17 −0.572900
\(317\) −3253.96 −0.576532 −0.288266 0.957550i \(-0.593079\pi\)
−0.288266 + 0.957550i \(0.593079\pi\)
\(318\) 0 0
\(319\) −529.272 −0.0928951
\(320\) −8037.37 −1.40407
\(321\) 0 0
\(322\) 8546.40 1.47911
\(323\) 3144.92 0.541759
\(324\) 0 0
\(325\) 74.0375 0.0126365
\(326\) 9742.46 1.65517
\(327\) 0 0
\(328\) 2825.58 0.475660
\(329\) −5259.86 −0.881415
\(330\) 0 0
\(331\) −3422.45 −0.568322 −0.284161 0.958777i \(-0.591715\pi\)
−0.284161 + 0.958777i \(0.591715\pi\)
\(332\) 4230.71 0.699368
\(333\) 0 0
\(334\) −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
\(339\) 0 0
\(340\) 2257.92 0.360155
\(341\) −6894.66 −1.09492
\(342\) 0 0
\(343\) 6294.99 0.990955
\(344\) −1093.64 −0.171410
\(345\) 0 0
\(346\) −695.518 −0.108067
\(347\) −216.898 −0.0335554 −0.0167777 0.999859i \(-0.505341\pi\)
−0.0167777 + 0.999859i \(0.505341\pi\)
\(348\) 0 0
\(349\) −4809.84 −0.737721 −0.368861 0.929485i \(-0.620252\pi\)
−0.368861 + 0.929485i \(0.620252\pi\)
\(350\) 269.781 0.0412011
\(351\) 0 0
\(352\) −6650.35 −1.00700
\(353\) 2859.64 0.431170 0.215585 0.976485i \(-0.430834\pi\)
0.215585 + 0.976485i \(0.430834\pi\)
\(354\) 0 0
\(355\) −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 \(-0.587338\pi\)
−0.270949 + 0.962594i \(0.587338\pi\)
\(360\) 0 0
\(361\) 17076.8 2.48969
\(362\) −9287.39 −1.34844
\(363\) 0 0
\(364\) −1421.52 −0.204692
\(365\) 9748.98 1.39804
\(366\) 0 0
\(367\) −3470.59 −0.493633 −0.246816 0.969062i \(-0.579384\pi\)
−0.246816 + 0.969062i \(0.579384\pi\)
\(368\) −8538.52 −1.20951
\(369\) 0 0
\(370\) −5580.98 −0.784166
\(371\) 900.166 0.125968
\(372\) 0 0
\(373\) −11963.4 −1.66070 −0.830352 0.557240i \(-0.811860\pi\)
−0.830352 + 0.557240i \(0.811860\pi\)
\(374\) 2214.16 0.306127
\(375\) 0 0
\(376\) −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
\(381\) 0 0
\(382\) 6189.99 0.829078
\(383\) 3386.40 0.451793 0.225897 0.974151i \(-0.427469\pi\)
0.225897 + 0.974151i \(0.427469\pi\)
\(384\) 0 0
\(385\) 3329.56 0.440754
\(386\) −1555.49 −0.205110
\(387\) 0 0
\(388\) 11903.9 1.55755
\(389\) 1629.88 0.212438 0.106219 0.994343i \(-0.466126\pi\)
0.106219 + 0.994343i \(0.466126\pi\)
\(390\) 0 0
\(391\) 3667.47 0.474353
\(392\) 1562.66 0.201343
\(393\) 0 0
\(394\) −17987.0 −2.29993
\(395\) −3786.58 −0.482338
\(396\) 0 0
\(397\) 7938.94 1.00364 0.501819 0.864973i \(-0.332664\pi\)
0.501819 + 0.864973i \(0.332664\pi\)
\(398\) −17485.7 −2.20221
\(399\) 0 0
\(400\) −269.532 −0.0336915
\(401\) −214.402 −0.0267001 −0.0133500 0.999911i \(-0.504250\pi\)
−0.0133500 + 0.999911i \(0.504250\pi\)
\(402\) 0 0
\(403\) 3463.52 0.428114
\(404\) −6270.93 −0.772253
\(405\) 0 0
\(406\) 968.819 0.118428
\(407\) −3001.48 −0.365548
\(408\) 0 0
\(409\) −4783.73 −0.578338 −0.289169 0.957278i \(-0.593379\pi\)
−0.289169 + 0.957278i \(0.593379\pi\)
\(410\) 18823.2 2.26735
\(411\) 0 0
\(412\) −4966.25 −0.593859
\(413\) −9832.99 −1.17155
\(414\) 0 0
\(415\) 4977.95 0.588815
\(416\) 3340.79 0.393739
\(417\) 0 0
\(418\) 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
\(423\) 0 0
\(424\) 577.738 0.0661732
\(425\) 115.770 0.0132133
\(426\) 0 0
\(427\) 2105.32 0.238603
\(428\) −5909.28 −0.667374
\(429\) 0 0
\(430\) −7285.53 −0.817068
\(431\) 13672.6 1.52805 0.764023 0.645189i \(-0.223221\pi\)
0.764023 + 0.645189i \(0.223221\pi\)
\(432\) 0 0
\(433\) 7113.10 0.789455 0.394727 0.918798i \(-0.370839\pi\)
0.394727 + 0.918798i \(0.370839\pi\)
\(434\) 12620.5 1.39586
\(435\) 0 0
\(436\) −12631.3 −1.38745
\(437\) 27912.9 3.05550
\(438\) 0 0
\(439\) −6022.04 −0.654707 −0.327353 0.944902i \(-0.606157\pi\)
−0.327353 + 0.944902i \(0.606157\pi\)
\(440\) 2136.96 0.231535
\(441\) 0 0
\(442\) −1112.28 −0.119696
\(443\) 12994.4 1.39364 0.696821 0.717245i \(-0.254597\pi\)
0.696821 + 0.717245i \(0.254597\pi\)
\(444\) 0 0
\(445\) −2963.61 −0.315704
\(446\) 9206.20 0.977413
\(447\) 0 0
\(448\) 7912.30 0.834422
\(449\) −10984.3 −1.15452 −0.577260 0.816560i \(-0.695878\pi\)
−0.577260 + 0.816560i \(0.695878\pi\)
\(450\) 0 0
\(451\) 10123.2 1.05695
\(452\) −409.973 −0.0426627
\(453\) 0 0
\(454\) 17419.9 1.80078
\(455\) −1672.60 −0.172335
\(456\) 0 0
\(457\) 9834.10 1.00661 0.503304 0.864109i \(-0.332118\pi\)
0.503304 + 0.864109i \(0.332118\pi\)
\(458\) 3516.05 0.358721
\(459\) 0 0
\(460\) 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
\(465\) 0 0
\(466\) 15512.5 1.54207
\(467\) 7958.82 0.788630 0.394315 0.918975i \(-0.370982\pi\)
0.394315 + 0.918975i \(0.370982\pi\)
\(468\) 0 0
\(469\) 6856.16 0.675028
\(470\) −22489.0 −2.20711
\(471\) 0 0
\(472\) −6310.94 −0.615433
\(473\) −3918.20 −0.380886
\(474\) 0 0
\(475\) 881.114 0.0851122
\(476\) −2222.78 −0.214036
\(477\) 0 0
\(478\) 12737.5 1.21882
\(479\) −8431.98 −0.804315 −0.402158 0.915570i \(-0.631740\pi\)
−0.402158 + 0.915570i \(0.631740\pi\)
\(480\) 0 0
\(481\) 1507.79 0.142930
\(482\) −13745.0 −1.29889
\(483\) 0 0
\(484\) −6425.29 −0.603427
\(485\) 14006.4 1.31133
\(486\) 0 0
\(487\) −11684.7 −1.08723 −0.543617 0.839334i \(-0.682945\pi\)
−0.543617 + 0.839334i \(0.682945\pi\)
\(488\) 1351.22 0.125342
\(489\) 0 0
\(490\) 10410.0 0.959750
\(491\) 3954.70 0.363489 0.181745 0.983346i \(-0.441826\pi\)
0.181745 + 0.983346i \(0.441826\pi\)
\(492\) 0 0
\(493\) 415.744 0.0379801
\(494\) −8465.47 −0.771011
\(495\) 0 0
\(496\) −12608.8 −1.14144
\(497\) 2791.49 0.251943
\(498\) 0 0
\(499\) −5690.37 −0.510493 −0.255246 0.966876i \(-0.582157\pi\)
−0.255246 + 0.966876i \(0.582157\pi\)
\(500\) −13251.9 −1.18529
\(501\) 0 0
\(502\) −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
\(507\) 0 0
\(508\) −3021.88 −0.263926
\(509\) 18558.6 1.61610 0.808049 0.589115i \(-0.200524\pi\)
0.808049 + 0.589115i \(0.200524\pi\)
\(510\) 0 0
\(511\) −9597.27 −0.830838
\(512\) −14896.8 −1.28584
\(513\) 0 0
\(514\) −25613.6 −2.19799
\(515\) −5843.42 −0.499984
\(516\) 0 0
\(517\) −12094.7 −1.02887
\(518\) 5494.13 0.466020
\(519\) 0 0
\(520\) −1073.49 −0.0905305
\(521\) −17297.5 −1.45454 −0.727271 0.686350i \(-0.759212\pi\)
−0.727271 + 0.686350i \(0.759212\pi\)
\(522\) 0 0
\(523\) −5016.11 −0.419386 −0.209693 0.977767i \(-0.567247\pi\)
−0.209693 + 0.977767i \(0.567247\pi\)
\(524\) −19441.8 −1.62084
\(525\) 0 0
\(526\) 518.667 0.0429942
\(527\) 5415.77 0.447656
\(528\) 0 0
\(529\) 20383.8 1.67533
\(530\) 3848.73 0.315431
\(531\) 0 0
\(532\) −16917.4 −1.37869
\(533\) −5085.39 −0.413269
\(534\) 0 0
\(535\) −6953.01 −0.561878
\(536\) 4400.37 0.354603
\(537\) 0 0
\(538\) −8147.83 −0.652933
\(539\) 5598.57 0.447398
\(540\) 0 0
\(541\) 17642.3 1.40204 0.701018 0.713144i \(-0.252729\pi\)
0.701018 + 0.713144i \(0.252729\pi\)
\(542\) 19415.0 1.53865
\(543\) 0 0
\(544\) 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
\(549\) 0 0
\(550\) 620.343 0.0480937
\(551\) 3164.20 0.244645
\(552\) 0 0
\(553\) 3727.66 0.286648
\(554\) 24556.9 1.88325
\(555\) 0 0
\(556\) −27773.1 −2.11842
\(557\) −8179.15 −0.622193 −0.311096 0.950378i \(-0.600696\pi\)
−0.311096 + 0.950378i \(0.600696\pi\)
\(558\) 0 0
\(559\) 1968.30 0.148927
\(560\) 6089.05 0.459481
\(561\) 0 0
\(562\) 19748.5 1.48228
\(563\) 1880.07 0.140738 0.0703690 0.997521i \(-0.477582\pi\)
0.0703690 + 0.997521i \(0.477582\pi\)
\(564\) 0 0
\(565\) −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 \(-0.621583\pi\)
−0.372743 + 0.927935i \(0.621583\pi\)
\(570\) 0 0
\(571\) 23428.9 1.71711 0.858555 0.512721i \(-0.171362\pi\)
0.858555 + 0.512721i \(0.171362\pi\)
\(572\) −3268.70 −0.238935
\(573\) 0 0
\(574\) −18530.3 −1.34746
\(575\) 1027.52 0.0745225
\(576\) 0 0
\(577\) 20508.1 1.47966 0.739831 0.672793i \(-0.234906\pi\)
0.739831 + 0.672793i \(0.234906\pi\)
\(578\) 18939.8 1.36296
\(579\) 0 0
\(580\) 2271.76 0.162637
\(581\) −4900.49 −0.349925
\(582\) 0 0
\(583\) 2069.87 0.147042
\(584\) −6159.65 −0.436452
\(585\) 0 0
\(586\) 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
\(591\) 0 0
\(592\) −5489.06 −0.381080
\(593\) 14659.5 1.01517 0.507584 0.861602i \(-0.330539\pi\)
0.507584 + 0.861602i \(0.330539\pi\)
\(594\) 0 0
\(595\) −2615.38 −0.180202
\(596\) −7223.72 −0.496468
\(597\) 0 0
\(598\) −9872.07 −0.675082
\(599\) −23635.9 −1.61225 −0.806125 0.591746i \(-0.798439\pi\)
−0.806125 + 0.591746i \(0.798439\pi\)
\(600\) 0 0
\(601\) −11527.0 −0.782356 −0.391178 0.920315i \(-0.627932\pi\)
−0.391178 + 0.920315i \(0.627932\pi\)
\(602\) 7172.15 0.485573
\(603\) 0 0
\(604\) 22125.9 1.49055
\(605\) −7560.15 −0.508039
\(606\) 0 0
\(607\) −5098.56 −0.340930 −0.170465 0.985364i \(-0.554527\pi\)
−0.170465 + 0.985364i \(0.554527\pi\)
\(608\) 39758.4 2.65200
\(609\) 0 0
\(610\) 9001.47 0.597473
\(611\) 6075.74 0.402288
\(612\) 0 0
\(613\) 1516.39 0.0999128 0.0499564 0.998751i \(-0.484092\pi\)
0.0499564 + 0.998751i \(0.484092\pi\)
\(614\) −1986.31 −0.130556
\(615\) 0 0
\(616\) −2103.70 −0.137598
\(617\) −18539.3 −1.20966 −0.604832 0.796353i \(-0.706760\pi\)
−0.604832 + 0.796353i \(0.706760\pi\)
\(618\) 0 0
\(619\) 25684.9 1.66779 0.833897 0.551920i \(-0.186105\pi\)
0.833897 + 0.551920i \(0.186105\pi\)
\(620\) 29593.5 1.91694
\(621\) 0 0
\(622\) −6391.51 −0.412020
\(623\) 2917.49 0.187619
\(624\) 0 0
\(625\) −16304.5 −1.04349
\(626\) −17046.1 −1.08834
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 13696.1 0.857950
\(635\) −3555.62 −0.222206
\(636\) 0 0
\(637\) −2812.43 −0.174933
\(638\) 2227.73 0.138239
\(639\) 0 0
\(640\) 10326.6 0.637804
\(641\) −6827.81 −0.420721 −0.210361 0.977624i \(-0.567464\pi\)
−0.210361 + 0.977624i \(0.567464\pi\)
\(642\) 0 0
\(643\) −23264.3 −1.42684 −0.713418 0.700738i \(-0.752854\pi\)
−0.713418 + 0.700738i \(0.752854\pi\)
\(644\) −19728.4 −1.20715
\(645\) 0 0
\(646\) −13237.1 −0.806204
\(647\) −14745.9 −0.896014 −0.448007 0.894030i \(-0.647866\pi\)
−0.448007 + 0.894030i \(0.647866\pi\)
\(648\) 0 0
\(649\) −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 \(-0.605997\pi\)
−0.326878 + 0.945067i \(0.605997\pi\)
\(654\) 0 0
\(655\) −22875.7 −1.36462
\(656\) 18513.2 1.10186
\(657\) 0 0
\(658\) 22139.0 1.31166
\(659\) 4182.99 0.247263 0.123631 0.992328i \(-0.460546\pi\)
0.123631 + 0.992328i \(0.460546\pi\)
\(660\) 0 0
\(661\) 2224.23 0.130881 0.0654406 0.997856i \(-0.479155\pi\)
0.0654406 + 0.997856i \(0.479155\pi\)
\(662\) 14405.2 0.845734
\(663\) 0 0
\(664\) −3145.19 −0.183821
\(665\) −19905.4 −1.16075
\(666\) 0 0
\(667\) 3689.95 0.214206
\(668\) 25899.7 1.50013
\(669\) 0 0
\(670\) 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
\(675\) 0 0
\(676\) 1642.02 0.0934240
\(677\) 15310.7 0.869187 0.434593 0.900627i \(-0.356892\pi\)
0.434593 + 0.900627i \(0.356892\pi\)
\(678\) 0 0
\(679\) −13788.4 −0.779310
\(680\) −1678.58 −0.0946628
\(681\) 0 0
\(682\) 29020.0 1.62937
\(683\) −11399.6 −0.638646 −0.319323 0.947646i \(-0.603455\pi\)
−0.319323 + 0.947646i \(0.603455\pi\)
\(684\) 0 0
\(685\) −11872.6 −0.662233
\(686\) −26495.9 −1.47466
\(687\) 0 0
\(688\) −7165.54 −0.397069
\(689\) −1039.79 −0.0574935
\(690\) 0 0
\(691\) −3323.23 −0.182955 −0.0914773 0.995807i \(-0.529159\pi\)
−0.0914773 + 0.995807i \(0.529159\pi\)
\(692\) 1605.52 0.0881976
\(693\) 0 0
\(694\) 912.936 0.0499345
\(695\) −32678.5 −1.78355
\(696\) 0 0
\(697\) −7951.82 −0.432133
\(698\) 20244.8 1.09782
\(699\) 0 0
\(700\) −622.758 −0.0336257
\(701\) 12670.4 0.682673 0.341336 0.939941i \(-0.389120\pi\)
0.341336 + 0.939941i \(0.389120\pi\)
\(702\) 0 0
\(703\) 17944.0 0.962692
\(704\) 18193.8 0.974012
\(705\) 0 0
\(706\) −12036.4 −0.641635
\(707\) 7263.71 0.386393
\(708\) 0 0
\(709\) 13075.2 0.692594 0.346297 0.938125i \(-0.387439\pi\)
0.346297 + 0.938125i \(0.387439\pi\)
\(710\) 11935.3 0.630877
\(711\) 0 0
\(712\) 1872.48 0.0985592
\(713\) 48067.8 2.52476
\(714\) 0 0
\(715\) −3846.03 −0.201165
\(716\) −6921.16 −0.361251
\(717\) 0 0
\(718\) 15514.7 0.806412
\(719\) 2988.41 0.155005 0.0775026 0.996992i \(-0.475305\pi\)
0.0775026 + 0.996992i \(0.475305\pi\)
\(720\) 0 0
\(721\) 5752.48 0.297134
\(722\) −71877.0 −3.70496
\(723\) 0 0
\(724\) 21438.9 1.10051
\(725\) 116.479 0.00596680
\(726\) 0 0
\(727\) −5507.46 −0.280963 −0.140482 0.990083i \(-0.544865\pi\)
−0.140482 + 0.990083i \(0.544865\pi\)
\(728\) 1056.79 0.0538011
\(729\) 0 0
\(730\) −41033.9 −2.08046
\(731\) 3077.75 0.155725
\(732\) 0 0
\(733\) −36585.2 −1.84353 −0.921764 0.387751i \(-0.873252\pi\)
−0.921764 + 0.387751i \(0.873252\pi\)
\(734\) 14607.9 0.734587
\(735\) 0 0
\(736\) 46364.6 2.32204
\(737\) 15765.3 0.787953
\(738\) 0 0
\(739\) 6425.89 0.319865 0.159933 0.987128i \(-0.448872\pi\)
0.159933 + 0.987128i \(0.448872\pi\)
\(740\) 12883.0 0.639986
\(741\) 0 0
\(742\) −3788.84 −0.187457
\(743\) −20411.0 −1.00782 −0.503908 0.863757i \(-0.668105\pi\)
−0.503908 + 0.863757i \(0.668105\pi\)
\(744\) 0 0
\(745\) −8499.60 −0.417988
\(746\) 50354.6 2.47133
\(747\) 0 0
\(748\) −5111.13 −0.249842
\(749\) 6844.81 0.333917
\(750\) 0 0
\(751\) −24259.5 −1.17875 −0.589375 0.807860i \(-0.700626\pi\)
−0.589375 + 0.807860i \(0.700626\pi\)
\(752\) −22118.6 −1.07258
\(753\) 0 0
\(754\) −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
\(759\) 0 0
\(760\) −12775.6 −0.609761
\(761\) 21974.7 1.04676 0.523378 0.852101i \(-0.324672\pi\)
0.523378 + 0.852101i \(0.324672\pi\)
\(762\) 0 0
\(763\) 14631.0 0.694204
\(764\) −14288.9 −0.676641
\(765\) 0 0
\(766\) −14253.5 −0.672324
\(767\) 11358.2 0.534709
\(768\) 0 0
\(769\) 22987.4 1.07795 0.538977 0.842320i \(-0.318811\pi\)
0.538977 + 0.842320i \(0.318811\pi\)
\(770\) −14014.3 −0.655897
\(771\) 0 0
\(772\) 3590.68 0.167398
\(773\) −31970.9 −1.48760 −0.743799 0.668404i \(-0.766978\pi\)
−0.743799 + 0.668404i \(0.766978\pi\)
\(774\) 0 0
\(775\) 1517.34 0.0703283
\(776\) −8849.59 −0.409384
\(777\) 0 0
\(778\) −6860.25 −0.316133
\(779\) −60520.8 −2.78354
\(780\) 0 0
\(781\) 6418.85 0.294090
\(782\) −15436.6 −0.705896
\(783\) 0 0
\(784\) 10238.6 0.466408
\(785\) 36280.2 1.64955
\(786\) 0 0
\(787\) 6087.26 0.275715 0.137857 0.990452i \(-0.455978\pi\)
0.137857 + 0.990452i \(0.455978\pi\)
\(788\) 41520.9 1.87706
\(789\) 0 0
\(790\) 15937.9 0.717779
\(791\) 474.878 0.0213460
\(792\) 0 0
\(793\) −2431.88 −0.108901
\(794\) −33415.4 −1.49354
\(795\) 0 0
\(796\) 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
\(801\) 0 0
\(802\) 902.429 0.0397330
\(803\) −22068.3 −0.969829
\(804\) 0 0
\(805\) −23212.9 −1.01633
\(806\) −14578.1 −0.637087
\(807\) 0 0
\(808\) 4661.94 0.202978
\(809\) 32377.8 1.40710 0.703550 0.710646i \(-0.251597\pi\)
0.703550 + 0.710646i \(0.251597\pi\)
\(810\) 0 0
\(811\) 26352.8 1.14103 0.570513 0.821288i \(-0.306744\pi\)
0.570513 + 0.821288i \(0.306744\pi\)
\(812\) −2236.40 −0.0966532
\(813\) 0 0
\(814\) 12633.4 0.543980
\(815\) −26461.5 −1.13731
\(816\) 0 0
\(817\) 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 \(-0.770654\pi\)
−0.751468 + 0.659770i \(0.770654\pi\)
\(822\) 0 0
\(823\) −12663.3 −0.536347 −0.268173 0.963371i \(-0.586420\pi\)
−0.268173 + 0.963371i \(0.586420\pi\)
\(824\) 3692.02 0.156089
\(825\) 0 0
\(826\) 41387.6 1.74341
\(827\) 16295.2 0.685176 0.342588 0.939486i \(-0.388697\pi\)
0.342588 + 0.939486i \(0.388697\pi\)
\(828\) 0 0
\(829\) 13638.9 0.571411 0.285705 0.958318i \(-0.407772\pi\)
0.285705 + 0.958318i \(0.407772\pi\)
\(830\) −20952.4 −0.876229
\(831\) 0 0
\(832\) −9139.61 −0.380840
\(833\) −4397.69 −0.182918
\(834\) 0 0
\(835\) 30474.2 1.26300
\(836\) −38900.5 −1.60933
\(837\) 0 0
\(838\) −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
\(843\) 0 0
\(844\) 11966.3 0.488028
\(845\) 1932.04 0.0786559
\(846\) 0 0
\(847\) 7442.50 0.301921
\(848\) 3785.35 0.153289
\(849\) 0 0
\(850\) −487.280 −0.0196630
\(851\) 20925.6 0.842914
\(852\) 0 0
\(853\) 1620.21 0.0650351 0.0325175 0.999471i \(-0.489648\pi\)
0.0325175 + 0.999471i \(0.489648\pi\)
\(854\) −8861.39 −0.355071
\(855\) 0 0
\(856\) 4393.08 0.175412
\(857\) 14508.4 0.578292 0.289146 0.957285i \(-0.406629\pi\)
0.289146 + 0.957285i \(0.406629\pi\)
\(858\) 0 0
\(859\) 29639.8 1.17730 0.588648 0.808389i \(-0.299660\pi\)
0.588648 + 0.808389i \(0.299660\pi\)
\(860\) 16817.8 0.666839
\(861\) 0 0
\(862\) −57548.8 −2.27392
\(863\) −21528.8 −0.849186 −0.424593 0.905384i \(-0.639583\pi\)
−0.424593 + 0.905384i \(0.639583\pi\)
\(864\) 0 0
\(865\) 1889.10 0.0742557
\(866\) −29939.4 −1.17481
\(867\) 0 0
\(868\) −29132.9 −1.13921
\(869\) 8571.50 0.334601
\(870\) 0 0
\(871\) −7919.65 −0.308091
\(872\) 9390.36 0.364676
\(873\) 0 0
\(874\) −117487. −4.54696
\(875\) 15349.9 0.593054
\(876\) 0 0
\(877\) 14865.3 0.572366 0.286183 0.958175i \(-0.407613\pi\)
0.286183 + 0.958175i \(0.407613\pi\)
\(878\) 25347.1 0.974285
\(879\) 0 0
\(880\) 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
\(885\) 0 0
\(886\) −54694.1 −2.07391
\(887\) −34575.0 −1.30881 −0.654406 0.756144i \(-0.727081\pi\)
−0.654406 + 0.756144i \(0.727081\pi\)
\(888\) 0 0
\(889\) 3500.29 0.132054
\(890\) 12474.0 0.469807
\(891\) 0 0
\(892\) −21251.4 −0.797703
\(893\) 72306.9 2.70958
\(894\) 0 0
\(895\) −8143.61 −0.304146
\(896\) −10165.9 −0.379039
\(897\) 0 0
\(898\) 46233.3 1.71807
\(899\) 5448.96 0.202150
\(900\) 0 0
\(901\) −1625.89 −0.0601178
\(902\) −42609.2 −1.57287
\(903\) 0 0
\(904\) 304.783 0.0112134
\(905\) 25225.5 0.926545
\(906\) 0 0
\(907\) −10424.8 −0.381641 −0.190820 0.981625i \(-0.561115\pi\)
−0.190820 + 0.981625i \(0.561115\pi\)
\(908\) −40211.7 −1.46968
\(909\) 0 0
\(910\) 7040.05 0.256456
\(911\) −10961.8 −0.398661 −0.199331 0.979932i \(-0.563877\pi\)
−0.199331 + 0.979932i \(0.563877\pi\)
\(912\) 0 0
\(913\) −11268.3 −0.408464
\(914\) −41392.2 −1.49796
\(915\) 0 0
\(916\) −8116.38 −0.292765
\(917\) 22519.7 0.810976
\(918\) 0 0
\(919\) −10779.2 −0.386914 −0.193457 0.981109i \(-0.561970\pi\)
−0.193457 + 0.981109i \(0.561970\pi\)
\(920\) −14898.3 −0.533895
\(921\) 0 0
\(922\) 14316.7 0.511384
\(923\) −3224.49 −0.114990
\(924\) 0 0
\(925\) 660.549 0.0234797
\(926\) −7321.32 −0.259820
\(927\) 0 0
\(928\) 5255.88 0.185919
\(929\) −5429.07 −0.191735 −0.0958675 0.995394i \(-0.530563\pi\)
−0.0958675 + 0.995394i \(0.530563\pi\)
\(930\) 0 0
\(931\) −33470.5 −1.17825
\(932\) −35808.8 −1.25854
\(933\) 0 0
\(934\) −33499.1 −1.17358
\(935\) −6013.88 −0.210348
\(936\) 0 0
\(937\) −21300.1 −0.742631 −0.371315 0.928507i \(-0.621093\pi\)
−0.371315 + 0.928507i \(0.621093\pi\)
\(938\) −28857.9 −1.00453
\(939\) 0 0
\(940\) 51913.2 1.80130
\(941\) −26851.2 −0.930207 −0.465103 0.885256i \(-0.653983\pi\)
−0.465103 + 0.885256i \(0.653983\pi\)
\(942\) 0 0
\(943\) −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 \(-0.456052\pi\)
0.137629 + 0.990484i \(0.456052\pi\)
\(948\) 0 0
\(949\) 11085.9 0.379204
\(950\) −3708.65 −0.126658
\(951\) 0 0
\(952\) 1652.46 0.0562569
\(953\) −35715.0 −1.21398 −0.606990 0.794709i \(-0.707623\pi\)
−0.606990 + 0.794709i \(0.707623\pi\)
\(954\) 0 0
\(955\) −16812.6 −0.569680
\(956\) −29402.9 −0.994726
\(957\) 0 0
\(958\) 35490.6 1.19692
\(959\) 11687.9 0.393557
\(960\) 0 0
\(961\) 41190.9 1.38266
\(962\) −6346.35 −0.212697
\(963\) 0 0
\(964\) 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
\(969\) 0 0
\(970\) −58953.6 −1.95143
\(971\) −23112.9 −0.763882 −0.381941 0.924187i \(-0.624744\pi\)
−0.381941 + 0.924187i \(0.624744\pi\)
\(972\) 0 0
\(973\) 32170.0 1.05994
\(974\) 49181.3 1.61794
\(975\) 0 0
\(976\) 8853.22 0.290353
\(977\) 52874.6 1.73143 0.865715 0.500538i \(-0.166864\pi\)
0.865715 + 0.500538i \(0.166864\pi\)
\(978\) 0 0
\(979\) 6708.57 0.219006
\(980\) −24030.4 −0.783287
\(981\) 0 0
\(982\) −16645.5 −0.540917
\(983\) −45173.1 −1.46572 −0.732858 0.680381i \(-0.761814\pi\)
−0.732858 + 0.680381i \(0.761814\pi\)
\(984\) 0 0
\(985\) 48854.5 1.58034
\(986\) −1749.89 −0.0565190
\(987\) 0 0
\(988\) 19541.6 0.629251
\(989\) 27316.7 0.878281
\(990\) 0 0
\(991\) 60485.6 1.93884 0.969418 0.245414i \(-0.0789239\pi\)
0.969418 + 0.245414i \(0.0789239\pi\)
\(992\) 68466.7 2.19135
\(993\) 0 0
\(994\) −11749.5 −0.374922
\(995\) 47492.8 1.51319
\(996\) 0 0
\(997\) −18108.1 −0.575214 −0.287607 0.957749i \(-0.592860\pi\)
−0.287607 + 0.957749i \(0.592860\pi\)
\(998\) 23951.1 0.759677
\(999\) 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 117.4.a.f.1.1 3
3.2 odd 2 39.4.a.c.1.3 3
4.3 odd 2 1872.4.a.bk.1.3 3
12.11 even 2 624.4.a.t.1.1 3
13.12 even 2 1521.4.a.u.1.3 3
15.14 odd 2 975.4.a.l.1.1 3
21.20 even 2 1911.4.a.k.1.3 3
24.5 odd 2 2496.4.a.bl.1.3 3
24.11 even 2 2496.4.a.bp.1.3 3
39.5 even 4 507.4.b.g.337.1 6
39.8 even 4 507.4.b.g.337.6 6
39.38 odd 2 507.4.a.h.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.3 3 3.2 odd 2
117.4.a.f.1.1 3 1.1 even 1 trivial
507.4.a.h.1.1 3 39.38 odd 2
507.4.b.g.337.1 6 39.5 even 4
507.4.b.g.337.6 6 39.8 even 4
624.4.a.t.1.1 3 12.11 even 2
975.4.a.l.1.1 3 15.14 odd 2
1521.4.a.u.1.3 3 13.12 even 2
1872.4.a.bk.1.3 3 4.3 odd 2
1911.4.a.k.1.3 3 21.20 even 2
2496.4.a.bl.1.3 3 24.5 odd 2
2496.4.a.bp.1.3 3 24.11 even 2