Properties

Label 153.4.a.g.1.3
Level $153$
Weight $4$
Character 153.1
Self dual yes
Analytic conductor $9.027$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 153.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.02729223088\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2636.1
Defining polynomial: \( x^{3} - 14x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.58966\) of defining polynomial
Character \(\chi\) \(=\) 153.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.03251 q^{2} +17.3261 q^{4} -0.885690 q^{5} +3.81828 q^{7} +46.9339 q^{8} +O(q^{10})\) \(q+5.03251 q^{2} +17.3261 q^{4} -0.885690 q^{5} +3.81828 q^{7} +46.9339 q^{8} -4.45724 q^{10} +52.3720 q^{11} -8.06025 q^{13} +19.2156 q^{14} +97.5862 q^{16} +17.0000 q^{17} -66.5154 q^{19} -15.3456 q^{20} +263.563 q^{22} -180.226 q^{23} -124.216 q^{25} -40.5633 q^{26} +66.1562 q^{28} +41.2800 q^{29} -34.9114 q^{31} +115.632 q^{32} +85.5527 q^{34} -3.38182 q^{35} +130.368 q^{37} -334.739 q^{38} -41.5689 q^{40} +17.9081 q^{41} +277.620 q^{43} +907.405 q^{44} -906.987 q^{46} -463.789 q^{47} -328.421 q^{49} -625.116 q^{50} -139.653 q^{52} +329.944 q^{53} -46.3853 q^{55} +179.207 q^{56} +207.742 q^{58} -678.656 q^{59} +340.280 q^{61} -175.692 q^{62} -198.770 q^{64} +7.13888 q^{65} +15.3925 q^{67} +294.545 q^{68} -17.0190 q^{70} +670.203 q^{71} +193.480 q^{73} +656.080 q^{74} -1152.46 q^{76} +199.971 q^{77} +1080.15 q^{79} -86.4311 q^{80} +90.1229 q^{82} +865.668 q^{83} -15.0567 q^{85} +1397.13 q^{86} +2458.02 q^{88} -1129.46 q^{89} -30.7763 q^{91} -3122.61 q^{92} -2334.02 q^{94} +58.9120 q^{95} -379.412 q^{97} -1652.78 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8} - 56 q^{10} + 28 q^{11} + 30 q^{13} - 92 q^{14} + 137 q^{16} + 51 q^{17} + 80 q^{19} + 168 q^{20} + 286 q^{22} - 142 q^{23} - 223 q^{25} - 26 q^{26} + 476 q^{28} + 456 q^{29} + 230 q^{31} + 71 q^{32} - 17 q^{34} + 332 q^{35} + 356 q^{37} - 724 q^{38} - 424 q^{40} + 294 q^{41} + 556 q^{43} + 1122 q^{44} - 704 q^{46} - 640 q^{47} - 269 q^{49} - 547 q^{50} - 774 q^{52} - 302 q^{53} + 76 q^{55} - 684 q^{56} - 1304 q^{58} - 636 q^{59} - 84 q^{61} - 508 q^{62} - 919 q^{64} - 408 q^{65} + 1008 q^{67} + 425 q^{68} - 1504 q^{70} + 402 q^{71} + 838 q^{73} - 836 q^{74} - 908 q^{76} + 504 q^{77} - 594 q^{79} + 40 q^{80} + 358 q^{82} + 2396 q^{83} + 136 q^{85} + 1264 q^{86} + 1838 q^{88} + 170 q^{89} - 1016 q^{91} - 4896 q^{92} - 2016 q^{94} + 472 q^{95} - 270 q^{97} - 2857 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.03251 1.77926 0.889630 0.456681i \(-0.150962\pi\)
0.889630 + 0.456681i \(0.150962\pi\)
\(3\) 0 0
\(4\) 17.3261 2.16577
\(5\) −0.885690 −0.0792185 −0.0396092 0.999215i \(-0.512611\pi\)
−0.0396092 + 0.999215i \(0.512611\pi\)
\(6\) 0 0
\(7\) 3.81828 0.206168 0.103084 0.994673i \(-0.467129\pi\)
0.103084 + 0.994673i \(0.467129\pi\)
\(8\) 46.9339 2.07421
\(9\) 0 0
\(10\) −4.45724 −0.140950
\(11\) 52.3720 1.43552 0.717761 0.696289i \(-0.245167\pi\)
0.717761 + 0.696289i \(0.245167\pi\)
\(12\) 0 0
\(13\) −8.06025 −0.171962 −0.0859811 0.996297i \(-0.527402\pi\)
−0.0859811 + 0.996297i \(0.527402\pi\)
\(14\) 19.2156 0.366827
\(15\) 0 0
\(16\) 97.5862 1.52478
\(17\) 17.0000 0.242536
\(18\) 0 0
\(19\) −66.5154 −0.803141 −0.401570 0.915828i \(-0.631535\pi\)
−0.401570 + 0.915828i \(0.631535\pi\)
\(20\) −15.3456 −0.171569
\(21\) 0 0
\(22\) 263.563 2.55417
\(23\) −180.226 −1.63390 −0.816948 0.576711i \(-0.804336\pi\)
−0.816948 + 0.576711i \(0.804336\pi\)
\(24\) 0 0
\(25\) −124.216 −0.993724
\(26\) −40.5633 −0.305966
\(27\) 0 0
\(28\) 66.1562 0.446512
\(29\) 41.2800 0.264328 0.132164 0.991228i \(-0.457807\pi\)
0.132164 + 0.991228i \(0.457807\pi\)
\(30\) 0 0
\(31\) −34.9114 −0.202267 −0.101133 0.994873i \(-0.532247\pi\)
−0.101133 + 0.994873i \(0.532247\pi\)
\(32\) 115.632 0.638783
\(33\) 0 0
\(34\) 85.5527 0.431534
\(35\) −3.38182 −0.0163323
\(36\) 0 0
\(37\) 130.368 0.579255 0.289627 0.957139i \(-0.406469\pi\)
0.289627 + 0.957139i \(0.406469\pi\)
\(38\) −334.739 −1.42900
\(39\) 0 0
\(40\) −41.5689 −0.164315
\(41\) 17.9081 0.0682142 0.0341071 0.999418i \(-0.489141\pi\)
0.0341071 + 0.999418i \(0.489141\pi\)
\(42\) 0 0
\(43\) 277.620 0.984573 0.492287 0.870433i \(-0.336161\pi\)
0.492287 + 0.870433i \(0.336161\pi\)
\(44\) 907.405 3.10901
\(45\) 0 0
\(46\) −906.987 −2.90713
\(47\) −463.789 −1.43937 −0.719687 0.694299i \(-0.755715\pi\)
−0.719687 + 0.694299i \(0.755715\pi\)
\(48\) 0 0
\(49\) −328.421 −0.957495
\(50\) −625.116 −1.76809
\(51\) 0 0
\(52\) −139.653 −0.372431
\(53\) 329.944 0.855118 0.427559 0.903987i \(-0.359374\pi\)
0.427559 + 0.903987i \(0.359374\pi\)
\(54\) 0 0
\(55\) −46.3853 −0.113720
\(56\) 179.207 0.427635
\(57\) 0 0
\(58\) 207.742 0.470308
\(59\) −678.656 −1.49752 −0.748759 0.662843i \(-0.769350\pi\)
−0.748759 + 0.662843i \(0.769350\pi\)
\(60\) 0 0
\(61\) 340.280 0.714237 0.357118 0.934059i \(-0.383759\pi\)
0.357118 + 0.934059i \(0.383759\pi\)
\(62\) −175.692 −0.359885
\(63\) 0 0
\(64\) −198.770 −0.388223
\(65\) 7.13888 0.0136226
\(66\) 0 0
\(67\) 15.3925 0.0280671 0.0140336 0.999902i \(-0.495533\pi\)
0.0140336 + 0.999902i \(0.495533\pi\)
\(68\) 294.545 0.525276
\(69\) 0 0
\(70\) −17.0190 −0.0290595
\(71\) 670.203 1.12026 0.560130 0.828405i \(-0.310751\pi\)
0.560130 + 0.828405i \(0.310751\pi\)
\(72\) 0 0
\(73\) 193.480 0.310207 0.155103 0.987898i \(-0.450429\pi\)
0.155103 + 0.987898i \(0.450429\pi\)
\(74\) 656.080 1.03065
\(75\) 0 0
\(76\) −1152.46 −1.73942
\(77\) 199.971 0.295959
\(78\) 0 0
\(79\) 1080.15 1.53831 0.769156 0.639061i \(-0.220677\pi\)
0.769156 + 0.639061i \(0.220677\pi\)
\(80\) −86.4311 −0.120791
\(81\) 0 0
\(82\) 90.1229 0.121371
\(83\) 865.668 1.14481 0.572406 0.819970i \(-0.306010\pi\)
0.572406 + 0.819970i \(0.306010\pi\)
\(84\) 0 0
\(85\) −15.0567 −0.0192133
\(86\) 1397.13 1.75181
\(87\) 0 0
\(88\) 2458.02 2.97757
\(89\) −1129.46 −1.34520 −0.672599 0.740008i \(-0.734822\pi\)
−0.672599 + 0.740008i \(0.734822\pi\)
\(90\) 0 0
\(91\) −30.7763 −0.0354531
\(92\) −3122.61 −3.53864
\(93\) 0 0
\(94\) −2334.02 −2.56102
\(95\) 58.9120 0.0636236
\(96\) 0 0
\(97\) −379.412 −0.397149 −0.198574 0.980086i \(-0.563631\pi\)
−0.198574 + 0.980086i \(0.563631\pi\)
\(98\) −1652.78 −1.70363
\(99\) 0 0
\(100\) −2152.18 −2.15218
\(101\) −131.732 −0.129780 −0.0648902 0.997892i \(-0.520670\pi\)
−0.0648902 + 0.997892i \(0.520670\pi\)
\(102\) 0 0
\(103\) 195.988 0.187488 0.0937442 0.995596i \(-0.470116\pi\)
0.0937442 + 0.995596i \(0.470116\pi\)
\(104\) −378.299 −0.356685
\(105\) 0 0
\(106\) 1660.45 1.52148
\(107\) 485.147 0.438326 0.219163 0.975688i \(-0.429667\pi\)
0.219163 + 0.975688i \(0.429667\pi\)
\(108\) 0 0
\(109\) −1255.12 −1.10292 −0.551460 0.834201i \(-0.685929\pi\)
−0.551460 + 0.834201i \(0.685929\pi\)
\(110\) −233.435 −0.202337
\(111\) 0 0
\(112\) 372.612 0.314362
\(113\) 1013.35 0.843612 0.421806 0.906686i \(-0.361396\pi\)
0.421806 + 0.906686i \(0.361396\pi\)
\(114\) 0 0
\(115\) 159.624 0.129435
\(116\) 715.224 0.572473
\(117\) 0 0
\(118\) −3415.34 −2.66447
\(119\) 64.9108 0.0500031
\(120\) 0 0
\(121\) 1411.83 1.06073
\(122\) 1712.46 1.27081
\(123\) 0 0
\(124\) −604.880 −0.438063
\(125\) 220.728 0.157940
\(126\) 0 0
\(127\) 1927.72 1.34691 0.673456 0.739227i \(-0.264809\pi\)
0.673456 + 0.739227i \(0.264809\pi\)
\(128\) −1925.37 −1.32953
\(129\) 0 0
\(130\) 35.9265 0.0242381
\(131\) 406.738 0.271274 0.135637 0.990759i \(-0.456692\pi\)
0.135637 + 0.990759i \(0.456692\pi\)
\(132\) 0 0
\(133\) −253.975 −0.165582
\(134\) 77.4631 0.0499387
\(135\) 0 0
\(136\) 797.877 0.503069
\(137\) 130.552 0.0814149 0.0407074 0.999171i \(-0.487039\pi\)
0.0407074 + 0.999171i \(0.487039\pi\)
\(138\) 0 0
\(139\) 2073.54 1.26529 0.632644 0.774443i \(-0.281970\pi\)
0.632644 + 0.774443i \(0.281970\pi\)
\(140\) −58.5938 −0.0353720
\(141\) 0 0
\(142\) 3372.80 1.99323
\(143\) −422.131 −0.246856
\(144\) 0 0
\(145\) −36.5613 −0.0209397
\(146\) 973.689 0.551939
\(147\) 0 0
\(148\) 2258.78 1.25453
\(149\) 1852.73 1.01867 0.509334 0.860569i \(-0.329892\pi\)
0.509334 + 0.860569i \(0.329892\pi\)
\(150\) 0 0
\(151\) 2050.86 1.10527 0.552637 0.833422i \(-0.313622\pi\)
0.552637 + 0.833422i \(0.313622\pi\)
\(152\) −3121.83 −1.66588
\(153\) 0 0
\(154\) 1006.36 0.526588
\(155\) 30.9207 0.0160233
\(156\) 0 0
\(157\) −262.991 −0.133688 −0.0668438 0.997763i \(-0.521293\pi\)
−0.0668438 + 0.997763i \(0.521293\pi\)
\(158\) 5435.88 2.73706
\(159\) 0 0
\(160\) −102.414 −0.0506035
\(161\) −688.152 −0.336857
\(162\) 0 0
\(163\) −1444.98 −0.694354 −0.347177 0.937800i \(-0.612860\pi\)
−0.347177 + 0.937800i \(0.612860\pi\)
\(164\) 310.279 0.147736
\(165\) 0 0
\(166\) 4356.48 2.03692
\(167\) 501.565 0.232409 0.116204 0.993225i \(-0.462927\pi\)
0.116204 + 0.993225i \(0.462927\pi\)
\(168\) 0 0
\(169\) −2132.03 −0.970429
\(170\) −75.7731 −0.0341855
\(171\) 0 0
\(172\) 4810.08 2.13236
\(173\) 2590.14 1.13829 0.569146 0.822237i \(-0.307274\pi\)
0.569146 + 0.822237i \(0.307274\pi\)
\(174\) 0 0
\(175\) −474.290 −0.204874
\(176\) 5110.79 2.18886
\(177\) 0 0
\(178\) −5684.02 −2.39346
\(179\) −2165.65 −0.904294 −0.452147 0.891943i \(-0.649342\pi\)
−0.452147 + 0.891943i \(0.649342\pi\)
\(180\) 0 0
\(181\) −1925.56 −0.790750 −0.395375 0.918520i \(-0.629385\pi\)
−0.395375 + 0.918520i \(0.629385\pi\)
\(182\) −154.882 −0.0630803
\(183\) 0 0
\(184\) −8458.69 −3.38904
\(185\) −115.466 −0.0458877
\(186\) 0 0
\(187\) 890.324 0.348165
\(188\) −8035.68 −3.11735
\(189\) 0 0
\(190\) 296.475 0.113203
\(191\) 2783.52 1.05449 0.527247 0.849712i \(-0.323224\pi\)
0.527247 + 0.849712i \(0.323224\pi\)
\(192\) 0 0
\(193\) 2258.27 0.842246 0.421123 0.907004i \(-0.361636\pi\)
0.421123 + 0.907004i \(0.361636\pi\)
\(194\) −1909.39 −0.706631
\(195\) 0 0
\(196\) −5690.27 −2.07371
\(197\) 1270.70 0.459560 0.229780 0.973243i \(-0.426199\pi\)
0.229780 + 0.973243i \(0.426199\pi\)
\(198\) 0 0
\(199\) −4794.36 −1.70786 −0.853928 0.520392i \(-0.825786\pi\)
−0.853928 + 0.520392i \(0.825786\pi\)
\(200\) −5829.92 −2.06119
\(201\) 0 0
\(202\) −662.942 −0.230913
\(203\) 157.619 0.0544960
\(204\) 0 0
\(205\) −15.8611 −0.00540383
\(206\) 986.313 0.333591
\(207\) 0 0
\(208\) −786.569 −0.262205
\(209\) −3483.54 −1.15293
\(210\) 0 0
\(211\) −2807.00 −0.915837 −0.457918 0.888994i \(-0.651405\pi\)
−0.457918 + 0.888994i \(0.651405\pi\)
\(212\) 5716.66 1.85199
\(213\) 0 0
\(214\) 2441.50 0.779896
\(215\) −245.885 −0.0779964
\(216\) 0 0
\(217\) −133.302 −0.0417009
\(218\) −6316.38 −1.96238
\(219\) 0 0
\(220\) −803.679 −0.246291
\(221\) −137.024 −0.0417070
\(222\) 0 0
\(223\) 4684.30 1.40665 0.703327 0.710866i \(-0.251697\pi\)
0.703327 + 0.710866i \(0.251697\pi\)
\(224\) 441.516 0.131697
\(225\) 0 0
\(226\) 5099.70 1.50101
\(227\) 1395.72 0.408095 0.204047 0.978961i \(-0.434590\pi\)
0.204047 + 0.978961i \(0.434590\pi\)
\(228\) 0 0
\(229\) 894.638 0.258163 0.129082 0.991634i \(-0.458797\pi\)
0.129082 + 0.991634i \(0.458797\pi\)
\(230\) 803.309 0.230298
\(231\) 0 0
\(232\) 1937.43 0.548270
\(233\) −1196.13 −0.336313 −0.168156 0.985760i \(-0.553781\pi\)
−0.168156 + 0.985760i \(0.553781\pi\)
\(234\) 0 0
\(235\) 410.773 0.114025
\(236\) −11758.5 −3.24328
\(237\) 0 0
\(238\) 326.664 0.0889685
\(239\) −4948.82 −1.33938 −0.669691 0.742639i \(-0.733574\pi\)
−0.669691 + 0.742639i \(0.733574\pi\)
\(240\) 0 0
\(241\) −6702.73 −1.79154 −0.895770 0.444518i \(-0.853375\pi\)
−0.895770 + 0.444518i \(0.853375\pi\)
\(242\) 7105.03 1.88731
\(243\) 0 0
\(244\) 5895.75 1.54687
\(245\) 290.879 0.0758513
\(246\) 0 0
\(247\) 536.130 0.138110
\(248\) −1638.53 −0.419543
\(249\) 0 0
\(250\) 1110.81 0.281016
\(251\) 4756.08 1.19602 0.598010 0.801489i \(-0.295958\pi\)
0.598010 + 0.801489i \(0.295958\pi\)
\(252\) 0 0
\(253\) −9438.77 −2.34550
\(254\) 9701.29 2.39651
\(255\) 0 0
\(256\) −8099.28 −1.97736
\(257\) −2892.84 −0.702143 −0.351071 0.936349i \(-0.614183\pi\)
−0.351071 + 0.936349i \(0.614183\pi\)
\(258\) 0 0
\(259\) 497.784 0.119424
\(260\) 123.689 0.0295034
\(261\) 0 0
\(262\) 2046.92 0.482667
\(263\) −5415.48 −1.26971 −0.634853 0.772633i \(-0.718939\pi\)
−0.634853 + 0.772633i \(0.718939\pi\)
\(264\) 0 0
\(265\) −292.228 −0.0677412
\(266\) −1278.13 −0.294613
\(267\) 0 0
\(268\) 266.693 0.0607869
\(269\) −5787.00 −1.31167 −0.655835 0.754904i \(-0.727683\pi\)
−0.655835 + 0.754904i \(0.727683\pi\)
\(270\) 0 0
\(271\) 5465.13 1.22503 0.612515 0.790459i \(-0.290158\pi\)
0.612515 + 0.790459i \(0.290158\pi\)
\(272\) 1658.97 0.369815
\(273\) 0 0
\(274\) 657.006 0.144858
\(275\) −6505.42 −1.42651
\(276\) 0 0
\(277\) −1207.65 −0.261952 −0.130976 0.991386i \(-0.541811\pi\)
−0.130976 + 0.991386i \(0.541811\pi\)
\(278\) 10435.1 2.25128
\(279\) 0 0
\(280\) −158.722 −0.0338766
\(281\) 1197.18 0.254155 0.127077 0.991893i \(-0.459440\pi\)
0.127077 + 0.991893i \(0.459440\pi\)
\(282\) 0 0
\(283\) 3164.73 0.664748 0.332374 0.943148i \(-0.392150\pi\)
0.332374 + 0.943148i \(0.392150\pi\)
\(284\) 11612.0 2.42622
\(285\) 0 0
\(286\) −2124.38 −0.439221
\(287\) 68.3784 0.0140636
\(288\) 0 0
\(289\) 289.000 0.0588235
\(290\) −183.995 −0.0372571
\(291\) 0 0
\(292\) 3352.26 0.671836
\(293\) −7456.21 −1.48668 −0.743339 0.668915i \(-0.766759\pi\)
−0.743339 + 0.668915i \(0.766759\pi\)
\(294\) 0 0
\(295\) 601.079 0.118631
\(296\) 6118.70 1.20149
\(297\) 0 0
\(298\) 9323.89 1.81248
\(299\) 1452.66 0.280969
\(300\) 0 0
\(301\) 1060.03 0.202988
\(302\) 10321.0 1.96657
\(303\) 0 0
\(304\) −6490.98 −1.22462
\(305\) −301.383 −0.0565808
\(306\) 0 0
\(307\) −6535.48 −1.21498 −0.607491 0.794327i \(-0.707824\pi\)
−0.607491 + 0.794327i \(0.707824\pi\)
\(308\) 3464.73 0.640978
\(309\) 0 0
\(310\) 155.608 0.0285096
\(311\) 8935.89 1.62928 0.814642 0.579963i \(-0.196933\pi\)
0.814642 + 0.579963i \(0.196933\pi\)
\(312\) 0 0
\(313\) −2628.71 −0.474707 −0.237353 0.971423i \(-0.576280\pi\)
−0.237353 + 0.971423i \(0.576280\pi\)
\(314\) −1323.50 −0.237865
\(315\) 0 0
\(316\) 18714.9 3.33163
\(317\) −4268.54 −0.756293 −0.378147 0.925746i \(-0.623438\pi\)
−0.378147 + 0.925746i \(0.623438\pi\)
\(318\) 0 0
\(319\) 2161.92 0.379449
\(320\) 176.048 0.0307544
\(321\) 0 0
\(322\) −3463.13 −0.599357
\(323\) −1130.76 −0.194790
\(324\) 0 0
\(325\) 1001.21 0.170883
\(326\) −7271.89 −1.23544
\(327\) 0 0
\(328\) 840.500 0.141490
\(329\) −1770.88 −0.296753
\(330\) 0 0
\(331\) 992.298 0.164778 0.0823892 0.996600i \(-0.473745\pi\)
0.0823892 + 0.996600i \(0.473745\pi\)
\(332\) 14998.7 2.47940
\(333\) 0 0
\(334\) 2524.13 0.413516
\(335\) −13.6330 −0.00222344
\(336\) 0 0
\(337\) 8042.26 1.29997 0.649985 0.759947i \(-0.274775\pi\)
0.649985 + 0.759947i \(0.274775\pi\)
\(338\) −10729.5 −1.72665
\(339\) 0 0
\(340\) −260.875 −0.0416116
\(341\) −1828.38 −0.290359
\(342\) 0 0
\(343\) −2563.68 −0.403573
\(344\) 13029.8 2.04221
\(345\) 0 0
\(346\) 13034.9 2.02532
\(347\) 7414.16 1.14701 0.573506 0.819202i \(-0.305583\pi\)
0.573506 + 0.819202i \(0.305583\pi\)
\(348\) 0 0
\(349\) −859.194 −0.131781 −0.0658905 0.997827i \(-0.520989\pi\)
−0.0658905 + 0.997827i \(0.520989\pi\)
\(350\) −2386.87 −0.364525
\(351\) 0 0
\(352\) 6055.89 0.916988
\(353\) −569.084 −0.0858053 −0.0429027 0.999079i \(-0.513661\pi\)
−0.0429027 + 0.999079i \(0.513661\pi\)
\(354\) 0 0
\(355\) −593.592 −0.0887453
\(356\) −19569.2 −2.91339
\(357\) 0 0
\(358\) −10898.7 −1.60897
\(359\) 5005.21 0.735835 0.367918 0.929858i \(-0.380071\pi\)
0.367918 + 0.929858i \(0.380071\pi\)
\(360\) 0 0
\(361\) −2434.71 −0.354965
\(362\) −9690.40 −1.40695
\(363\) 0 0
\(364\) −533.235 −0.0767833
\(365\) −171.363 −0.0245741
\(366\) 0 0
\(367\) −10975.3 −1.56105 −0.780523 0.625127i \(-0.785047\pi\)
−0.780523 + 0.625127i \(0.785047\pi\)
\(368\) −17587.5 −2.49134
\(369\) 0 0
\(370\) −581.083 −0.0816462
\(371\) 1259.82 0.176298
\(372\) 0 0
\(373\) −3211.72 −0.445835 −0.222918 0.974837i \(-0.571558\pi\)
−0.222918 + 0.974837i \(0.571558\pi\)
\(374\) 4480.56 0.619477
\(375\) 0 0
\(376\) −21767.4 −2.98556
\(377\) −332.727 −0.0454544
\(378\) 0 0
\(379\) 8051.48 1.09123 0.545616 0.838035i \(-0.316296\pi\)
0.545616 + 0.838035i \(0.316296\pi\)
\(380\) 1020.72 0.137794
\(381\) 0 0
\(382\) 14008.1 1.87622
\(383\) 2584.16 0.344763 0.172382 0.985030i \(-0.444854\pi\)
0.172382 + 0.985030i \(0.444854\pi\)
\(384\) 0 0
\(385\) −177.112 −0.0234454
\(386\) 11364.7 1.49858
\(387\) 0 0
\(388\) −6573.74 −0.860132
\(389\) 5174.31 0.674417 0.337208 0.941430i \(-0.390517\pi\)
0.337208 + 0.941430i \(0.390517\pi\)
\(390\) 0 0
\(391\) −3063.83 −0.396278
\(392\) −15414.1 −1.98604
\(393\) 0 0
\(394\) 6394.79 0.817677
\(395\) −956.680 −0.121863
\(396\) 0 0
\(397\) −5149.36 −0.650980 −0.325490 0.945545i \(-0.605529\pi\)
−0.325490 + 0.945545i \(0.605529\pi\)
\(398\) −24127.7 −3.03872
\(399\) 0 0
\(400\) −12121.7 −1.51522
\(401\) −8700.49 −1.08350 −0.541748 0.840541i \(-0.682237\pi\)
−0.541748 + 0.840541i \(0.682237\pi\)
\(402\) 0 0
\(403\) 281.394 0.0347823
\(404\) −2282.41 −0.281074
\(405\) 0 0
\(406\) 793.219 0.0969625
\(407\) 6827.65 0.831533
\(408\) 0 0
\(409\) 12346.0 1.49260 0.746299 0.665611i \(-0.231829\pi\)
0.746299 + 0.665611i \(0.231829\pi\)
\(410\) −79.8209 −0.00961482
\(411\) 0 0
\(412\) 3395.72 0.406056
\(413\) −2591.30 −0.308740
\(414\) 0 0
\(415\) −766.713 −0.0906903
\(416\) −932.023 −0.109847
\(417\) 0 0
\(418\) −17531.0 −2.05136
\(419\) 5763.33 0.671974 0.335987 0.941867i \(-0.390930\pi\)
0.335987 + 0.941867i \(0.390930\pi\)
\(420\) 0 0
\(421\) −1876.12 −0.217188 −0.108594 0.994086i \(-0.534635\pi\)
−0.108594 + 0.994086i \(0.534635\pi\)
\(422\) −14126.2 −1.62951
\(423\) 0 0
\(424\) 15485.6 1.77369
\(425\) −2111.66 −0.241014
\(426\) 0 0
\(427\) 1299.29 0.147253
\(428\) 8405.72 0.949313
\(429\) 0 0
\(430\) −1237.42 −0.138776
\(431\) −83.9299 −0.00937996 −0.00468998 0.999989i \(-0.501493\pi\)
−0.00468998 + 0.999989i \(0.501493\pi\)
\(432\) 0 0
\(433\) −15345.0 −1.70308 −0.851539 0.524291i \(-0.824331\pi\)
−0.851539 + 0.524291i \(0.824331\pi\)
\(434\) −670.842 −0.0741968
\(435\) 0 0
\(436\) −21746.3 −2.38867
\(437\) 11987.8 1.31225
\(438\) 0 0
\(439\) 3064.74 0.333194 0.166597 0.986025i \(-0.446722\pi\)
0.166597 + 0.986025i \(0.446722\pi\)
\(440\) −2177.05 −0.235879
\(441\) 0 0
\(442\) −689.575 −0.0742076
\(443\) 1792.97 0.192295 0.0961474 0.995367i \(-0.469348\pi\)
0.0961474 + 0.995367i \(0.469348\pi\)
\(444\) 0 0
\(445\) 1000.35 0.106564
\(446\) 23573.8 2.50281
\(447\) 0 0
\(448\) −758.960 −0.0800391
\(449\) −2499.19 −0.262681 −0.131341 0.991337i \(-0.541928\pi\)
−0.131341 + 0.991337i \(0.541928\pi\)
\(450\) 0 0
\(451\) 937.885 0.0979231
\(452\) 17557.5 1.82707
\(453\) 0 0
\(454\) 7024.00 0.726107
\(455\) 27.2583 0.00280854
\(456\) 0 0
\(457\) 14784.4 1.51331 0.756656 0.653813i \(-0.226832\pi\)
0.756656 + 0.653813i \(0.226832\pi\)
\(458\) 4502.28 0.459340
\(459\) 0 0
\(460\) 2765.67 0.280326
\(461\) 17746.9 1.79297 0.896483 0.443078i \(-0.146113\pi\)
0.896483 + 0.443078i \(0.146113\pi\)
\(462\) 0 0
\(463\) 18486.4 1.85559 0.927793 0.373096i \(-0.121704\pi\)
0.927793 + 0.373096i \(0.121704\pi\)
\(464\) 4028.36 0.403043
\(465\) 0 0
\(466\) −6019.52 −0.598388
\(467\) −7406.57 −0.733908 −0.366954 0.930239i \(-0.619599\pi\)
−0.366954 + 0.930239i \(0.619599\pi\)
\(468\) 0 0
\(469\) 58.7731 0.00578655
\(470\) 2067.22 0.202880
\(471\) 0 0
\(472\) −31852.0 −3.10616
\(473\) 14539.5 1.41338
\(474\) 0 0
\(475\) 8262.24 0.798101
\(476\) 1124.65 0.108295
\(477\) 0 0
\(478\) −24905.0 −2.38311
\(479\) 18550.9 1.76955 0.884775 0.466019i \(-0.154312\pi\)
0.884775 + 0.466019i \(0.154312\pi\)
\(480\) 0 0
\(481\) −1050.80 −0.0996100
\(482\) −33731.6 −3.18762
\(483\) 0 0
\(484\) 24461.5 2.29729
\(485\) 336.041 0.0314615
\(486\) 0 0
\(487\) 10203.4 0.949406 0.474703 0.880146i \(-0.342556\pi\)
0.474703 + 0.880146i \(0.342556\pi\)
\(488\) 15970.7 1.48147
\(489\) 0 0
\(490\) 1463.85 0.134959
\(491\) 1247.46 0.114658 0.0573290 0.998355i \(-0.481742\pi\)
0.0573290 + 0.998355i \(0.481742\pi\)
\(492\) 0 0
\(493\) 701.760 0.0641089
\(494\) 2698.08 0.245734
\(495\) 0 0
\(496\) −3406.87 −0.308413
\(497\) 2559.03 0.230962
\(498\) 0 0
\(499\) 70.0303 0.00628254 0.00314127 0.999995i \(-0.499000\pi\)
0.00314127 + 0.999995i \(0.499000\pi\)
\(500\) 3824.36 0.342061
\(501\) 0 0
\(502\) 23935.0 2.12803
\(503\) −1444.29 −0.128028 −0.0640138 0.997949i \(-0.520390\pi\)
−0.0640138 + 0.997949i \(0.520390\pi\)
\(504\) 0 0
\(505\) 116.674 0.0102810
\(506\) −47500.7 −4.17325
\(507\) 0 0
\(508\) 33400.0 2.91710
\(509\) −14272.8 −1.24289 −0.621445 0.783458i \(-0.713454\pi\)
−0.621445 + 0.783458i \(0.713454\pi\)
\(510\) 0 0
\(511\) 738.761 0.0639547
\(512\) −25356.7 −2.18871
\(513\) 0 0
\(514\) −14558.3 −1.24929
\(515\) −173.585 −0.0148525
\(516\) 0 0
\(517\) −24289.6 −2.06625
\(518\) 2505.10 0.212486
\(519\) 0 0
\(520\) 335.055 0.0282561
\(521\) −14874.0 −1.25075 −0.625376 0.780324i \(-0.715054\pi\)
−0.625376 + 0.780324i \(0.715054\pi\)
\(522\) 0 0
\(523\) −8142.90 −0.680811 −0.340406 0.940279i \(-0.610564\pi\)
−0.340406 + 0.940279i \(0.610564\pi\)
\(524\) 7047.21 0.587517
\(525\) 0 0
\(526\) −27253.4 −2.25914
\(527\) −593.494 −0.0490569
\(528\) 0 0
\(529\) 20314.2 1.66962
\(530\) −1470.64 −0.120529
\(531\) 0 0
\(532\) −4400.40 −0.358612
\(533\) −144.344 −0.0117303
\(534\) 0 0
\(535\) −429.689 −0.0347235
\(536\) 722.432 0.0582170
\(537\) 0 0
\(538\) −29123.1 −2.33380
\(539\) −17200.0 −1.37451
\(540\) 0 0
\(541\) 3179.67 0.252689 0.126344 0.991986i \(-0.459676\pi\)
0.126344 + 0.991986i \(0.459676\pi\)
\(542\) 27503.3 2.17965
\(543\) 0 0
\(544\) 1965.75 0.154928
\(545\) 1111.64 0.0873716
\(546\) 0 0
\(547\) 2107.07 0.164702 0.0823509 0.996603i \(-0.473757\pi\)
0.0823509 + 0.996603i \(0.473757\pi\)
\(548\) 2261.97 0.176326
\(549\) 0 0
\(550\) −32738.6 −2.53814
\(551\) −2745.76 −0.212292
\(552\) 0 0
\(553\) 4124.33 0.317151
\(554\) −6077.51 −0.466081
\(555\) 0 0
\(556\) 35926.4 2.74032
\(557\) −467.382 −0.0355540 −0.0177770 0.999842i \(-0.505659\pi\)
−0.0177770 + 0.999842i \(0.505659\pi\)
\(558\) 0 0
\(559\) −2237.69 −0.169309
\(560\) −330.019 −0.0249033
\(561\) 0 0
\(562\) 6024.80 0.452208
\(563\) −14612.6 −1.09387 −0.546935 0.837175i \(-0.684206\pi\)
−0.546935 + 0.837175i \(0.684206\pi\)
\(564\) 0 0
\(565\) −897.515 −0.0668297
\(566\) 15926.5 1.18276
\(567\) 0 0
\(568\) 31455.3 2.32365
\(569\) −11602.3 −0.854821 −0.427410 0.904058i \(-0.640574\pi\)
−0.427410 + 0.904058i \(0.640574\pi\)
\(570\) 0 0
\(571\) −10534.9 −0.772104 −0.386052 0.922477i \(-0.626161\pi\)
−0.386052 + 0.922477i \(0.626161\pi\)
\(572\) −7313.91 −0.534633
\(573\) 0 0
\(574\) 344.115 0.0250228
\(575\) 22386.8 1.62364
\(576\) 0 0
\(577\) 14404.7 1.03930 0.519650 0.854379i \(-0.326062\pi\)
0.519650 + 0.854379i \(0.326062\pi\)
\(578\) 1454.40 0.104662
\(579\) 0 0
\(580\) −633.466 −0.0453504
\(581\) 3305.37 0.236024
\(582\) 0 0
\(583\) 17279.8 1.22754
\(584\) 9080.77 0.643433
\(585\) 0 0
\(586\) −37523.5 −2.64519
\(587\) 11004.9 0.773799 0.386900 0.922122i \(-0.373546\pi\)
0.386900 + 0.922122i \(0.373546\pi\)
\(588\) 0 0
\(589\) 2322.14 0.162449
\(590\) 3024.94 0.211076
\(591\) 0 0
\(592\) 12722.2 0.883239
\(593\) −1853.59 −0.128361 −0.0641804 0.997938i \(-0.520443\pi\)
−0.0641804 + 0.997938i \(0.520443\pi\)
\(594\) 0 0
\(595\) −57.4909 −0.00396117
\(596\) 32100.7 2.20620
\(597\) 0 0
\(598\) 7310.53 0.499916
\(599\) −19074.7 −1.30112 −0.650559 0.759456i \(-0.725465\pi\)
−0.650559 + 0.759456i \(0.725465\pi\)
\(600\) 0 0
\(601\) −27776.0 −1.88520 −0.942600 0.333923i \(-0.891627\pi\)
−0.942600 + 0.333923i \(0.891627\pi\)
\(602\) 5334.62 0.361168
\(603\) 0 0
\(604\) 35533.4 2.39377
\(605\) −1250.44 −0.0840291
\(606\) 0 0
\(607\) 18728.3 1.25232 0.626159 0.779695i \(-0.284626\pi\)
0.626159 + 0.779695i \(0.284626\pi\)
\(608\) −7691.32 −0.513033
\(609\) 0 0
\(610\) −1516.71 −0.100672
\(611\) 3738.25 0.247518
\(612\) 0 0
\(613\) −24405.3 −1.60802 −0.804012 0.594613i \(-0.797305\pi\)
−0.804012 + 0.594613i \(0.797305\pi\)
\(614\) −32889.8 −2.16177
\(615\) 0 0
\(616\) 9385.43 0.613880
\(617\) 22516.4 1.46917 0.734584 0.678518i \(-0.237377\pi\)
0.734584 + 0.678518i \(0.237377\pi\)
\(618\) 0 0
\(619\) −5146.53 −0.334179 −0.167089 0.985942i \(-0.553437\pi\)
−0.167089 + 0.985942i \(0.553437\pi\)
\(620\) 535.736 0.0347027
\(621\) 0 0
\(622\) 44969.9 2.89892
\(623\) −4312.60 −0.277337
\(624\) 0 0
\(625\) 15331.4 0.981213
\(626\) −13229.0 −0.844627
\(627\) 0 0
\(628\) −4556.62 −0.289536
\(629\) 2216.26 0.140490
\(630\) 0 0
\(631\) −3858.77 −0.243447 −0.121724 0.992564i \(-0.538842\pi\)
−0.121724 + 0.992564i \(0.538842\pi\)
\(632\) 50695.8 3.19078
\(633\) 0 0
\(634\) −21481.5 −1.34564
\(635\) −1707.36 −0.106700
\(636\) 0 0
\(637\) 2647.15 0.164653
\(638\) 10879.9 0.675138
\(639\) 0 0
\(640\) 1705.28 0.105324
\(641\) −18689.3 −1.15161 −0.575805 0.817587i \(-0.695311\pi\)
−0.575805 + 0.817587i \(0.695311\pi\)
\(642\) 0 0
\(643\) 26473.5 1.62366 0.811831 0.583893i \(-0.198471\pi\)
0.811831 + 0.583893i \(0.198471\pi\)
\(644\) −11923.0 −0.729555
\(645\) 0 0
\(646\) −5690.57 −0.346583
\(647\) −14397.7 −0.874855 −0.437427 0.899254i \(-0.644110\pi\)
−0.437427 + 0.899254i \(0.644110\pi\)
\(648\) 0 0
\(649\) −35542.6 −2.14972
\(650\) 5038.59 0.304046
\(651\) 0 0
\(652\) −25036.0 −1.50381
\(653\) −20939.5 −1.25486 −0.627431 0.778672i \(-0.715893\pi\)
−0.627431 + 0.778672i \(0.715893\pi\)
\(654\) 0 0
\(655\) −360.244 −0.0214899
\(656\) 1747.59 0.104012
\(657\) 0 0
\(658\) −8911.96 −0.528001
\(659\) −4031.76 −0.238323 −0.119162 0.992875i \(-0.538021\pi\)
−0.119162 + 0.992875i \(0.538021\pi\)
\(660\) 0 0
\(661\) 6691.52 0.393752 0.196876 0.980428i \(-0.436920\pi\)
0.196876 + 0.980428i \(0.436920\pi\)
\(662\) 4993.75 0.293184
\(663\) 0 0
\(664\) 40629.2 2.37458
\(665\) 224.943 0.0131171
\(666\) 0 0
\(667\) −7439.71 −0.431884
\(668\) 8690.19 0.503344
\(669\) 0 0
\(670\) −68.6083 −0.00395607
\(671\) 17821.2 1.02530
\(672\) 0 0
\(673\) 10319.2 0.591048 0.295524 0.955335i \(-0.404506\pi\)
0.295524 + 0.955335i \(0.404506\pi\)
\(674\) 40472.7 2.31298
\(675\) 0 0
\(676\) −36939.9 −2.10172
\(677\) 19813.3 1.12480 0.562398 0.826866i \(-0.309879\pi\)
0.562398 + 0.826866i \(0.309879\pi\)
\(678\) 0 0
\(679\) −1448.70 −0.0818793
\(680\) −706.671 −0.0398524
\(681\) 0 0
\(682\) −9201.33 −0.516624
\(683\) −5924.61 −0.331916 −0.165958 0.986133i \(-0.553072\pi\)
−0.165958 + 0.986133i \(0.553072\pi\)
\(684\) 0 0
\(685\) −115.629 −0.00644957
\(686\) −12901.7 −0.718061
\(687\) 0 0
\(688\) 27091.9 1.50126
\(689\) −2659.43 −0.147048
\(690\) 0 0
\(691\) 1973.16 0.108629 0.0543143 0.998524i \(-0.482703\pi\)
0.0543143 + 0.998524i \(0.482703\pi\)
\(692\) 44877.1 2.46528
\(693\) 0 0
\(694\) 37311.8 2.04083
\(695\) −1836.51 −0.100234
\(696\) 0 0
\(697\) 304.439 0.0165444
\(698\) −4323.90 −0.234473
\(699\) 0 0
\(700\) −8217.63 −0.443710
\(701\) 12840.1 0.691815 0.345907 0.938269i \(-0.387571\pi\)
0.345907 + 0.938269i \(0.387571\pi\)
\(702\) 0 0
\(703\) −8671.50 −0.465223
\(704\) −10410.0 −0.557302
\(705\) 0 0
\(706\) −2863.92 −0.152670
\(707\) −502.990 −0.0267566
\(708\) 0 0
\(709\) −27749.7 −1.46990 −0.734952 0.678119i \(-0.762796\pi\)
−0.734952 + 0.678119i \(0.762796\pi\)
\(710\) −2987.26 −0.157901
\(711\) 0 0
\(712\) −53010.0 −2.79022
\(713\) 6291.92 0.330483
\(714\) 0 0
\(715\) 373.877 0.0195555
\(716\) −37522.4 −1.95849
\(717\) 0 0
\(718\) 25188.8 1.30924
\(719\) −16888.3 −0.875979 −0.437989 0.898980i \(-0.644309\pi\)
−0.437989 + 0.898980i \(0.644309\pi\)
\(720\) 0 0
\(721\) 748.339 0.0386541
\(722\) −12252.7 −0.631575
\(723\) 0 0
\(724\) −33362.6 −1.71258
\(725\) −5127.62 −0.262669
\(726\) 0 0
\(727\) 2135.25 0.108930 0.0544649 0.998516i \(-0.482655\pi\)
0.0544649 + 0.998516i \(0.482655\pi\)
\(728\) −1444.45 −0.0735371
\(729\) 0 0
\(730\) −862.386 −0.0437238
\(731\) 4719.54 0.238794
\(732\) 0 0
\(733\) 4795.27 0.241633 0.120817 0.992675i \(-0.461449\pi\)
0.120817 + 0.992675i \(0.461449\pi\)
\(734\) −55233.1 −2.77751
\(735\) 0 0
\(736\) −20839.9 −1.04371
\(737\) 806.138 0.0402910
\(738\) 0 0
\(739\) −32747.6 −1.63010 −0.815048 0.579393i \(-0.803290\pi\)
−0.815048 + 0.579393i \(0.803290\pi\)
\(740\) −2000.58 −0.0993821
\(741\) 0 0
\(742\) 6340.05 0.313680
\(743\) −12299.4 −0.607298 −0.303649 0.952784i \(-0.598205\pi\)
−0.303649 + 0.952784i \(0.598205\pi\)
\(744\) 0 0
\(745\) −1640.94 −0.0806974
\(746\) −16163.0 −0.793257
\(747\) 0 0
\(748\) 15425.9 0.754046
\(749\) 1852.43 0.0903688
\(750\) 0 0
\(751\) 30102.6 1.46266 0.731332 0.682021i \(-0.238899\pi\)
0.731332 + 0.682021i \(0.238899\pi\)
\(752\) −45259.4 −2.19474
\(753\) 0 0
\(754\) −1674.45 −0.0808753
\(755\) −1816.42 −0.0875581
\(756\) 0 0
\(757\) 38826.3 1.86416 0.932078 0.362257i \(-0.117994\pi\)
0.932078 + 0.362257i \(0.117994\pi\)
\(758\) 40519.2 1.94159
\(759\) 0 0
\(760\) 2764.97 0.131968
\(761\) −19981.6 −0.951815 −0.475907 0.879495i \(-0.657880\pi\)
−0.475907 + 0.879495i \(0.657880\pi\)
\(762\) 0 0
\(763\) −4792.39 −0.227387
\(764\) 48227.7 2.28379
\(765\) 0 0
\(766\) 13004.8 0.613424
\(767\) 5470.14 0.257517
\(768\) 0 0
\(769\) −22407.7 −1.05077 −0.525384 0.850865i \(-0.676078\pi\)
−0.525384 + 0.850865i \(0.676078\pi\)
\(770\) −891.320 −0.0417155
\(771\) 0 0
\(772\) 39127.0 1.82411
\(773\) 6902.77 0.321184 0.160592 0.987021i \(-0.448660\pi\)
0.160592 + 0.987021i \(0.448660\pi\)
\(774\) 0 0
\(775\) 4336.54 0.200997
\(776\) −17807.3 −0.823768
\(777\) 0 0
\(778\) 26039.8 1.19996
\(779\) −1191.17 −0.0547856
\(780\) 0 0
\(781\) 35099.9 1.60816
\(782\) −15418.8 −0.705082
\(783\) 0 0
\(784\) −32049.3 −1.45997
\(785\) 232.928 0.0105905
\(786\) 0 0
\(787\) −22185.9 −1.00488 −0.502442 0.864611i \(-0.667565\pi\)
−0.502442 + 0.864611i \(0.667565\pi\)
\(788\) 22016.3 0.995301
\(789\) 0 0
\(790\) −4814.50 −0.216826
\(791\) 3869.27 0.173926
\(792\) 0 0
\(793\) −2742.74 −0.122822
\(794\) −25914.2 −1.15826
\(795\) 0 0
\(796\) −83067.8 −3.69882
\(797\) 16291.1 0.724040 0.362020 0.932170i \(-0.382087\pi\)
0.362020 + 0.932170i \(0.382087\pi\)
\(798\) 0 0
\(799\) −7884.41 −0.349100
\(800\) −14363.3 −0.634775
\(801\) 0 0
\(802\) −43785.3 −1.92782
\(803\) 10132.9 0.445309
\(804\) 0 0
\(805\) 609.489 0.0266853
\(806\) 1416.12 0.0618867
\(807\) 0 0
\(808\) −6182.70 −0.269191
\(809\) −17696.8 −0.769082 −0.384541 0.923108i \(-0.625640\pi\)
−0.384541 + 0.923108i \(0.625640\pi\)
\(810\) 0 0
\(811\) −3095.34 −0.134022 −0.0670111 0.997752i \(-0.521346\pi\)
−0.0670111 + 0.997752i \(0.521346\pi\)
\(812\) 2730.93 0.118026
\(813\) 0 0
\(814\) 34360.2 1.47951
\(815\) 1279.81 0.0550057
\(816\) 0 0
\(817\) −18466.0 −0.790751
\(818\) 62131.5 2.65572
\(819\) 0 0
\(820\) −274.811 −0.0117034
\(821\) −12323.5 −0.523864 −0.261932 0.965086i \(-0.584360\pi\)
−0.261932 + 0.965086i \(0.584360\pi\)
\(822\) 0 0
\(823\) −34436.5 −1.45854 −0.729271 0.684225i \(-0.760140\pi\)
−0.729271 + 0.684225i \(0.760140\pi\)
\(824\) 9198.50 0.388889
\(825\) 0 0
\(826\) −13040.8 −0.549329
\(827\) −18761.6 −0.788880 −0.394440 0.918922i \(-0.629061\pi\)
−0.394440 + 0.918922i \(0.629061\pi\)
\(828\) 0 0
\(829\) 22423.8 0.939457 0.469728 0.882811i \(-0.344352\pi\)
0.469728 + 0.882811i \(0.344352\pi\)
\(830\) −3858.49 −0.161362
\(831\) 0 0
\(832\) 1602.13 0.0667596
\(833\) −5583.15 −0.232227
\(834\) 0 0
\(835\) −444.231 −0.0184111
\(836\) −60356.4 −2.49697
\(837\) 0 0
\(838\) 29004.0 1.19562
\(839\) −9128.63 −0.375632 −0.187816 0.982204i \(-0.560141\pi\)
−0.187816 + 0.982204i \(0.560141\pi\)
\(840\) 0 0
\(841\) −22685.0 −0.930131
\(842\) −9441.58 −0.386435
\(843\) 0 0
\(844\) −48634.4 −1.98349
\(845\) 1888.32 0.0768759
\(846\) 0 0
\(847\) 5390.75 0.218688
\(848\) 32198.0 1.30387
\(849\) 0 0
\(850\) −10627.0 −0.428826
\(851\) −23495.7 −0.946442
\(852\) 0 0
\(853\) 27204.8 1.09200 0.545999 0.837786i \(-0.316150\pi\)
0.545999 + 0.837786i \(0.316150\pi\)
\(854\) 6538.68 0.262001
\(855\) 0 0
\(856\) 22769.8 0.909179
\(857\) 38060.0 1.51704 0.758520 0.651649i \(-0.225923\pi\)
0.758520 + 0.651649i \(0.225923\pi\)
\(858\) 0 0
\(859\) −33326.2 −1.32372 −0.661860 0.749627i \(-0.730233\pi\)
−0.661860 + 0.749627i \(0.730233\pi\)
\(860\) −4260.24 −0.168922
\(861\) 0 0
\(862\) −422.378 −0.0166894
\(863\) 41724.2 1.64578 0.822890 0.568201i \(-0.192360\pi\)
0.822890 + 0.568201i \(0.192360\pi\)
\(864\) 0 0
\(865\) −2294.06 −0.0901737
\(866\) −77223.8 −3.03022
\(867\) 0 0
\(868\) −2309.60 −0.0903146
\(869\) 56569.7 2.20828
\(870\) 0 0
\(871\) −124.068 −0.00482649
\(872\) −58907.5 −2.28768
\(873\) 0 0
\(874\) 60328.6 2.33483
\(875\) 842.801 0.0325621
\(876\) 0 0
\(877\) −49337.3 −1.89966 −0.949830 0.312767i \(-0.898744\pi\)
−0.949830 + 0.312767i \(0.898744\pi\)
\(878\) 15423.3 0.592838
\(879\) 0 0
\(880\) −4526.57 −0.173398
\(881\) −8845.46 −0.338265 −0.169132 0.985593i \(-0.554097\pi\)
−0.169132 + 0.985593i \(0.554097\pi\)
\(882\) 0 0
\(883\) 14724.2 0.561165 0.280582 0.959830i \(-0.409472\pi\)
0.280582 + 0.959830i \(0.409472\pi\)
\(884\) −2374.10 −0.0903277
\(885\) 0 0
\(886\) 9023.14 0.342143
\(887\) −3864.38 −0.146283 −0.0731415 0.997322i \(-0.523302\pi\)
−0.0731415 + 0.997322i \(0.523302\pi\)
\(888\) 0 0
\(889\) 7360.60 0.277690
\(890\) 5034.28 0.189606
\(891\) 0 0
\(892\) 81160.9 3.04649
\(893\) 30849.1 1.15602
\(894\) 0 0
\(895\) 1918.10 0.0716368
\(896\) −7351.61 −0.274107
\(897\) 0 0
\(898\) −12577.2 −0.467379
\(899\) −1441.14 −0.0534648
\(900\) 0 0
\(901\) 5609.04 0.207397
\(902\) 4719.92 0.174231
\(903\) 0 0
\(904\) 47560.6 1.74982
\(905\) 1705.45 0.0626421
\(906\) 0 0
\(907\) −743.409 −0.0272155 −0.0136078 0.999907i \(-0.504332\pi\)
−0.0136078 + 0.999907i \(0.504332\pi\)
\(908\) 24182.5 0.883839
\(909\) 0 0
\(910\) 137.177 0.00499713
\(911\) −16291.0 −0.592475 −0.296238 0.955114i \(-0.595732\pi\)
−0.296238 + 0.955114i \(0.595732\pi\)
\(912\) 0 0
\(913\) 45336.8 1.64340
\(914\) 74402.5 2.69258
\(915\) 0 0
\(916\) 15500.6 0.559122
\(917\) 1553.04 0.0559280
\(918\) 0 0
\(919\) −6188.99 −0.222150 −0.111075 0.993812i \(-0.535429\pi\)
−0.111075 + 0.993812i \(0.535429\pi\)
\(920\) 7491.78 0.268474
\(921\) 0 0
\(922\) 89311.6 3.19015
\(923\) −5402.00 −0.192643
\(924\) 0 0
\(925\) −16193.8 −0.575620
\(926\) 93033.0 3.30157
\(927\) 0 0
\(928\) 4773.30 0.168848
\(929\) 31661.7 1.11818 0.559089 0.829108i \(-0.311151\pi\)
0.559089 + 0.829108i \(0.311151\pi\)
\(930\) 0 0
\(931\) 21845.0 0.769003
\(932\) −20724.3 −0.728376
\(933\) 0 0
\(934\) −37273.6 −1.30581
\(935\) −788.551 −0.0275811
\(936\) 0 0
\(937\) 35010.5 1.22064 0.610322 0.792153i \(-0.291040\pi\)
0.610322 + 0.792153i \(0.291040\pi\)
\(938\) 295.776 0.0102958
\(939\) 0 0
\(940\) 7117.12 0.246952
\(941\) 45625.8 1.58061 0.790307 0.612711i \(-0.209921\pi\)
0.790307 + 0.612711i \(0.209921\pi\)
\(942\) 0 0
\(943\) −3227.51 −0.111455
\(944\) −66227.5 −2.28339
\(945\) 0 0
\(946\) 73170.2 2.51477
\(947\) 21508.4 0.738044 0.369022 0.929421i \(-0.379693\pi\)
0.369022 + 0.929421i \(0.379693\pi\)
\(948\) 0 0
\(949\) −1559.50 −0.0533439
\(950\) 41579.8 1.42003
\(951\) 0 0
\(952\) 3046.52 0.103717
\(953\) −35686.7 −1.21302 −0.606509 0.795076i \(-0.707431\pi\)
−0.606509 + 0.795076i \(0.707431\pi\)
\(954\) 0 0
\(955\) −2465.34 −0.0835355
\(956\) −85744.0 −2.90079
\(957\) 0 0
\(958\) 93357.8 3.14849
\(959\) 498.486 0.0167851
\(960\) 0 0
\(961\) −28572.2 −0.959088
\(962\) −5288.17 −0.177232
\(963\) 0 0
\(964\) −116133. −3.88006
\(965\) −2000.12 −0.0667215
\(966\) 0 0
\(967\) −3731.33 −0.124086 −0.0620432 0.998073i \(-0.519762\pi\)
−0.0620432 + 0.998073i \(0.519762\pi\)
\(968\) 66262.5 2.20016
\(969\) 0 0
\(970\) 1691.13 0.0559782
\(971\) −17645.1 −0.583171 −0.291585 0.956545i \(-0.594183\pi\)
−0.291585 + 0.956545i \(0.594183\pi\)
\(972\) 0 0
\(973\) 7917.35 0.260862
\(974\) 51348.7 1.68924
\(975\) 0 0
\(976\) 33206.7 1.08906
\(977\) −24941.2 −0.816723 −0.408362 0.912820i \(-0.633900\pi\)
−0.408362 + 0.912820i \(0.633900\pi\)
\(978\) 0 0
\(979\) −59152.1 −1.93106
\(980\) 5039.81 0.164276
\(981\) 0 0
\(982\) 6277.85 0.204006
\(983\) 22506.2 0.730252 0.365126 0.930958i \(-0.381026\pi\)
0.365126 + 0.930958i \(0.381026\pi\)
\(984\) 0 0
\(985\) −1125.44 −0.0364057
\(986\) 3531.62 0.114066
\(987\) 0 0
\(988\) 9289.07 0.299114
\(989\) −50034.2 −1.60869
\(990\) 0 0
\(991\) 32694.1 1.04799 0.523997 0.851720i \(-0.324440\pi\)
0.523997 + 0.851720i \(0.324440\pi\)
\(992\) −4036.88 −0.129205
\(993\) 0 0
\(994\) 12878.3 0.410941
\(995\) 4246.32 0.135294
\(996\) 0 0
\(997\) 18248.8 0.579686 0.289843 0.957074i \(-0.406397\pi\)
0.289843 + 0.957074i \(0.406397\pi\)
\(998\) 352.428 0.0111783
\(999\) 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 153.4.a.g.1.3 3
3.2 odd 2 17.4.a.b.1.1 3
4.3 odd 2 2448.4.a.bi.1.2 3
12.11 even 2 272.4.a.h.1.1 3
15.2 even 4 425.4.b.f.324.1 6
15.8 even 4 425.4.b.f.324.6 6
15.14 odd 2 425.4.a.g.1.3 3
21.20 even 2 833.4.a.d.1.1 3
24.5 odd 2 1088.4.a.v.1.1 3
24.11 even 2 1088.4.a.x.1.3 3
33.32 even 2 2057.4.a.e.1.3 3
51.38 odd 4 289.4.b.b.288.5 6
51.47 odd 4 289.4.b.b.288.6 6
51.50 odd 2 289.4.a.b.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.b.1.1 3 3.2 odd 2
153.4.a.g.1.3 3 1.1 even 1 trivial
272.4.a.h.1.1 3 12.11 even 2
289.4.a.b.1.1 3 51.50 odd 2
289.4.b.b.288.5 6 51.38 odd 4
289.4.b.b.288.6 6 51.47 odd 4
425.4.a.g.1.3 3 15.14 odd 2
425.4.b.f.324.1 6 15.2 even 4
425.4.b.f.324.6 6 15.8 even 4
833.4.a.d.1.1 3 21.20 even 2
1088.4.a.v.1.1 3 24.5 odd 2
1088.4.a.x.1.3 3 24.11 even 2
2057.4.a.e.1.3 3 33.32 even 2
2448.4.a.bi.1.2 3 4.3 odd 2