Properties

Label 1859.4.a.a.1.1
Level $1859$
Weight $4$
Character 1859.1
Self dual yes
Analytic conductor $109.685$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1859.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.684550701\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1859.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.73205 q^{2} -7.92820 q^{3} -0.535898 q^{4} -14.8564 q^{5} +21.6603 q^{6} -3.07180 q^{7} +23.3205 q^{8} +35.8564 q^{9} +O(q^{10})\) \(q-2.73205 q^{2} -7.92820 q^{3} -0.535898 q^{4} -14.8564 q^{5} +21.6603 q^{6} -3.07180 q^{7} +23.3205 q^{8} +35.8564 q^{9} +40.5885 q^{10} +11.0000 q^{11} +4.24871 q^{12} +8.39230 q^{14} +117.785 q^{15} -59.4256 q^{16} -41.2154 q^{17} -97.9615 q^{18} -139.923 q^{19} +7.96152 q^{20} +24.3538 q^{21} -30.0526 q^{22} -111.354 q^{23} -184.890 q^{24} +95.7128 q^{25} -70.2154 q^{27} +1.64617 q^{28} -24.9948 q^{29} -321.794 q^{30} -31.4974 q^{31} -24.2102 q^{32} -87.2102 q^{33} +112.603 q^{34} +45.6359 q^{35} -19.2154 q^{36} -13.1436 q^{37} +382.277 q^{38} -346.459 q^{40} -261.072 q^{41} -66.5359 q^{42} -57.7128 q^{43} -5.89488 q^{44} -532.697 q^{45} +304.224 q^{46} +343.846 q^{47} +471.138 q^{48} -333.564 q^{49} -261.492 q^{50} +326.764 q^{51} -342.995 q^{53} +191.832 q^{54} -163.420 q^{55} -71.6359 q^{56} +1109.34 q^{57} +68.2872 q^{58} -88.3693 q^{59} -63.1206 q^{60} +738.697 q^{61} +86.0526 q^{62} -110.144 q^{63} +541.549 q^{64} +238.263 q^{66} -342.359 q^{67} +22.0873 q^{68} +882.836 q^{69} -124.679 q^{70} +207.364 q^{71} +836.190 q^{72} +1010.60 q^{73} +35.9090 q^{74} -758.831 q^{75} +74.9845 q^{76} -33.7898 q^{77} +1294.23 q^{79} +882.851 q^{80} -411.441 q^{81} +713.261 q^{82} -441.846 q^{83} -13.0512 q^{84} +612.313 q^{85} +157.674 q^{86} +198.164 q^{87} +256.526 q^{88} +1489.11 q^{89} +1455.36 q^{90} +59.6743 q^{92} +249.718 q^{93} -939.405 q^{94} +2078.75 q^{95} +191.944 q^{96} -1346.42 q^{97} +911.314 q^{98} +394.420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} - 8q^{4} - 2q^{5} + 26q^{6} - 20q^{7} + 12q^{8} + 44q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} - 8q^{4} - 2q^{5} + 26q^{6} - 20q^{7} + 12q^{8} + 44q^{9} + 50q^{10} + 22q^{11} - 40q^{12} - 4q^{14} + 194q^{15} - 8q^{16} - 124q^{17} - 92q^{18} - 72q^{19} - 88q^{20} - 76q^{21} - 22q^{22} - 98q^{23} - 252q^{24} + 136q^{25} - 182q^{27} + 128q^{28} + 144q^{29} - 266q^{30} + 34q^{31} + 104q^{32} - 22q^{33} + 52q^{34} - 172q^{35} - 80q^{36} - 54q^{37} + 432q^{38} - 492q^{40} - 536q^{41} - 140q^{42} - 60q^{43} - 88q^{44} - 428q^{45} + 314q^{46} + 272q^{47} + 776q^{48} - 390q^{49} - 232q^{50} - 164q^{51} - 492q^{53} + 110q^{54} - 22q^{55} + 120q^{56} + 1512q^{57} + 192q^{58} - 634q^{59} - 632q^{60} + 840q^{61} + 134q^{62} - 248q^{63} + 224q^{64} + 286q^{66} - 754q^{67} + 640q^{68} + 962q^{69} - 284q^{70} + 678q^{71} + 744q^{72} + 400q^{73} + 6q^{74} - 520q^{75} - 432q^{76} - 220q^{77} + 316q^{79} + 1544q^{80} - 1294q^{81} + 512q^{82} - 468q^{83} + 736q^{84} - 452q^{85} + 156q^{86} + 1200q^{87} + 132q^{88} + 1842q^{89} + 1532q^{90} - 40q^{92} + 638q^{93} - 992q^{94} + 2952q^{95} + 952q^{96} - 2194q^{97} + 870q^{98} + 484q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.73205 −0.965926 −0.482963 0.875641i \(-0.660439\pi\)
−0.482963 + 0.875641i \(0.660439\pi\)
\(3\) −7.92820 −1.52578 −0.762892 0.646526i \(-0.776221\pi\)
−0.762892 + 0.646526i \(0.776221\pi\)
\(4\) −0.535898 −0.0669873
\(5\) −14.8564 −1.32880 −0.664399 0.747378i \(-0.731312\pi\)
−0.664399 + 0.747378i \(0.731312\pi\)
\(6\) 21.6603 1.47379
\(7\) −3.07180 −0.165861 −0.0829307 0.996555i \(-0.526428\pi\)
−0.0829307 + 0.996555i \(0.526428\pi\)
\(8\) 23.3205 1.03063
\(9\) 35.8564 1.32802
\(10\) 40.5885 1.28352
\(11\) 11.0000 0.301511
\(12\) 4.24871 0.102208
\(13\) 0 0
\(14\) 8.39230 0.160210
\(15\) 117.785 2.02746
\(16\) −59.4256 −0.928525
\(17\) −41.2154 −0.588012 −0.294006 0.955804i \(-0.594989\pi\)
−0.294006 + 0.955804i \(0.594989\pi\)
\(18\) −97.9615 −1.28276
\(19\) −139.923 −1.68950 −0.844751 0.535159i \(-0.820252\pi\)
−0.844751 + 0.535159i \(0.820252\pi\)
\(20\) 7.96152 0.0890125
\(21\) 24.3538 0.253069
\(22\) −30.0526 −0.291238
\(23\) −111.354 −1.00952 −0.504758 0.863261i \(-0.668418\pi\)
−0.504758 + 0.863261i \(0.668418\pi\)
\(24\) −184.890 −1.57252
\(25\) 95.7128 0.765703
\(26\) 0 0
\(27\) −70.2154 −0.500480
\(28\) 1.64617 0.0111106
\(29\) −24.9948 −0.160049 −0.0800246 0.996793i \(-0.525500\pi\)
−0.0800246 + 0.996793i \(0.525500\pi\)
\(30\) −321.794 −1.95837
\(31\) −31.4974 −0.182487 −0.0912436 0.995829i \(-0.529084\pi\)
−0.0912436 + 0.995829i \(0.529084\pi\)
\(32\) −24.2102 −0.133744
\(33\) −87.2102 −0.460041
\(34\) 112.603 0.567976
\(35\) 45.6359 0.220396
\(36\) −19.2154 −0.0889601
\(37\) −13.1436 −0.0583998 −0.0291999 0.999574i \(-0.509296\pi\)
−0.0291999 + 0.999574i \(0.509296\pi\)
\(38\) 382.277 1.63193
\(39\) 0 0
\(40\) −346.459 −1.36950
\(41\) −261.072 −0.994453 −0.497226 0.867621i \(-0.665648\pi\)
−0.497226 + 0.867621i \(0.665648\pi\)
\(42\) −66.5359 −0.244446
\(43\) −57.7128 −0.204677 −0.102339 0.994750i \(-0.532633\pi\)
−0.102339 + 0.994750i \(0.532633\pi\)
\(44\) −5.89488 −0.0201974
\(45\) −532.697 −1.76466
\(46\) 304.224 0.975118
\(47\) 343.846 1.06713 0.533565 0.845759i \(-0.320852\pi\)
0.533565 + 0.845759i \(0.320852\pi\)
\(48\) 471.138 1.41673
\(49\) −333.564 −0.972490
\(50\) −261.492 −0.739612
\(51\) 326.764 0.897179
\(52\) 0 0
\(53\) −342.995 −0.888943 −0.444471 0.895793i \(-0.646608\pi\)
−0.444471 + 0.895793i \(0.646608\pi\)
\(54\) 191.832 0.483426
\(55\) −163.420 −0.400647
\(56\) −71.6359 −0.170942
\(57\) 1109.34 2.57782
\(58\) 68.2872 0.154596
\(59\) −88.3693 −0.194995 −0.0974975 0.995236i \(-0.531084\pi\)
−0.0974975 + 0.995236i \(0.531084\pi\)
\(60\) −63.1206 −0.135814
\(61\) 738.697 1.55050 0.775250 0.631654i \(-0.217624\pi\)
0.775250 + 0.631654i \(0.217624\pi\)
\(62\) 86.0526 0.176269
\(63\) −110.144 −0.220266
\(64\) 541.549 1.05771
\(65\) 0 0
\(66\) 238.263 0.444365
\(67\) −342.359 −0.624266 −0.312133 0.950038i \(-0.601043\pi\)
−0.312133 + 0.950038i \(0.601043\pi\)
\(68\) 22.0873 0.0393893
\(69\) 882.836 1.54030
\(70\) −124.679 −0.212886
\(71\) 207.364 0.346614 0.173307 0.984868i \(-0.444555\pi\)
0.173307 + 0.984868i \(0.444555\pi\)
\(72\) 836.190 1.36869
\(73\) 1010.60 1.62030 0.810149 0.586224i \(-0.199386\pi\)
0.810149 + 0.586224i \(0.199386\pi\)
\(74\) 35.9090 0.0564099
\(75\) −758.831 −1.16830
\(76\) 74.9845 0.113175
\(77\) −33.7898 −0.0500091
\(78\) 0 0
\(79\) 1294.23 1.84319 0.921593 0.388157i \(-0.126888\pi\)
0.921593 + 0.388157i \(0.126888\pi\)
\(80\) 882.851 1.23382
\(81\) −411.441 −0.564391
\(82\) 713.261 0.960568
\(83\) −441.846 −0.584324 −0.292162 0.956369i \(-0.594375\pi\)
−0.292162 + 0.956369i \(0.594375\pi\)
\(84\) −13.0512 −0.0169524
\(85\) 612.313 0.781349
\(86\) 157.674 0.197703
\(87\) 198.164 0.244200
\(88\) 256.526 0.310747
\(89\) 1489.11 1.77355 0.886773 0.462205i \(-0.152942\pi\)
0.886773 + 0.462205i \(0.152942\pi\)
\(90\) 1455.36 1.70453
\(91\) 0 0
\(92\) 59.6743 0.0676248
\(93\) 249.718 0.278436
\(94\) −939.405 −1.03077
\(95\) 2078.75 2.24501
\(96\) 191.944 0.204064
\(97\) −1346.42 −1.40936 −0.704679 0.709526i \(-0.748909\pi\)
−0.704679 + 0.709526i \(0.748909\pi\)
\(98\) 911.314 0.939353
\(99\) 394.420 0.400412
\(100\) −51.2923 −0.0512923
\(101\) −161.461 −0.159069 −0.0795347 0.996832i \(-0.525343\pi\)
−0.0795347 + 0.996832i \(0.525343\pi\)
\(102\) −892.736 −0.866608
\(103\) −34.7592 −0.0332517 −0.0166259 0.999862i \(-0.505292\pi\)
−0.0166259 + 0.999862i \(0.505292\pi\)
\(104\) 0 0
\(105\) −361.810 −0.336277
\(106\) 937.079 0.858653
\(107\) 832.179 0.751867 0.375934 0.926647i \(-0.377322\pi\)
0.375934 + 0.926647i \(0.377322\pi\)
\(108\) 37.6283 0.0335258
\(109\) −1044.26 −0.917629 −0.458815 0.888532i \(-0.651726\pi\)
−0.458815 + 0.888532i \(0.651726\pi\)
\(110\) 446.473 0.386996
\(111\) 104.205 0.0891055
\(112\) 182.543 0.154007
\(113\) 295.082 0.245654 0.122827 0.992428i \(-0.460804\pi\)
0.122827 + 0.992428i \(0.460804\pi\)
\(114\) −3030.77 −2.48998
\(115\) 1654.32 1.34144
\(116\) 13.3947 0.0107213
\(117\) 0 0
\(118\) 241.429 0.188351
\(119\) 126.605 0.0975285
\(120\) 2746.80 2.08956
\(121\) 121.000 0.0909091
\(122\) −2018.16 −1.49767
\(123\) 2069.83 1.51732
\(124\) 16.8794 0.0122243
\(125\) 435.102 0.311334
\(126\) 300.918 0.212761
\(127\) −1317.60 −0.920618 −0.460309 0.887759i \(-0.652261\pi\)
−0.460309 + 0.887759i \(0.652261\pi\)
\(128\) −1285.86 −0.887928
\(129\) 457.559 0.312293
\(130\) 0 0
\(131\) −1600.71 −1.06759 −0.533797 0.845612i \(-0.679235\pi\)
−0.533797 + 0.845612i \(0.679235\pi\)
\(132\) 46.7358 0.0308169
\(133\) 429.815 0.280223
\(134\) 935.342 0.602994
\(135\) 1043.15 0.665036
\(136\) −961.164 −0.606023
\(137\) −1611.68 −1.00507 −0.502536 0.864556i \(-0.667600\pi\)
−0.502536 + 0.864556i \(0.667600\pi\)
\(138\) −2411.95 −1.48782
\(139\) −31.8619 −0.0194424 −0.00972120 0.999953i \(-0.503094\pi\)
−0.00972120 + 0.999953i \(0.503094\pi\)
\(140\) −24.4562 −0.0147637
\(141\) −2726.08 −1.62821
\(142\) −566.529 −0.334803
\(143\) 0 0
\(144\) −2130.79 −1.23310
\(145\) 371.334 0.212673
\(146\) −2761.01 −1.56509
\(147\) 2644.56 1.48381
\(148\) 7.04363 0.00391205
\(149\) 2428.34 1.33515 0.667576 0.744542i \(-0.267332\pi\)
0.667576 + 0.744542i \(0.267332\pi\)
\(150\) 2073.16 1.12849
\(151\) 2576.68 1.38866 0.694328 0.719659i \(-0.255702\pi\)
0.694328 + 0.719659i \(0.255702\pi\)
\(152\) −3263.08 −1.74125
\(153\) −1477.84 −0.780889
\(154\) 92.3154 0.0483051
\(155\) 467.939 0.242489
\(156\) 0 0
\(157\) 2475.94 1.25861 0.629305 0.777158i \(-0.283340\pi\)
0.629305 + 0.777158i \(0.283340\pi\)
\(158\) −3535.89 −1.78038
\(159\) 2719.33 1.35633
\(160\) 359.677 0.177719
\(161\) 342.056 0.167440
\(162\) 1124.08 0.545160
\(163\) 2725.11 1.30949 0.654745 0.755850i \(-0.272776\pi\)
0.654745 + 0.755850i \(0.272776\pi\)
\(164\) 139.908 0.0666157
\(165\) 1295.63 0.611301
\(166\) 1207.15 0.564414
\(167\) −2737.30 −1.26837 −0.634187 0.773180i \(-0.718665\pi\)
−0.634187 + 0.773180i \(0.718665\pi\)
\(168\) 567.944 0.260820
\(169\) 0 0
\(170\) −1672.87 −0.754725
\(171\) −5017.14 −2.24368
\(172\) 30.9282 0.0137108
\(173\) 2307.42 1.01404 0.507022 0.861933i \(-0.330746\pi\)
0.507022 + 0.861933i \(0.330746\pi\)
\(174\) −541.395 −0.235879
\(175\) −294.010 −0.127001
\(176\) −653.682 −0.279961
\(177\) 700.610 0.297520
\(178\) −4068.33 −1.71311
\(179\) −1312.15 −0.547905 −0.273953 0.961743i \(-0.588331\pi\)
−0.273953 + 0.961743i \(0.588331\pi\)
\(180\) 285.472 0.118210
\(181\) −803.174 −0.329831 −0.164916 0.986308i \(-0.552735\pi\)
−0.164916 + 0.986308i \(0.552735\pi\)
\(182\) 0 0
\(183\) −5856.54 −2.36573
\(184\) −2596.83 −1.04044
\(185\) 195.267 0.0776015
\(186\) −682.242 −0.268949
\(187\) −453.369 −0.177292
\(188\) −184.267 −0.0714842
\(189\) 215.687 0.0830103
\(190\) −5679.26 −2.16851
\(191\) 1718.25 0.650932 0.325466 0.945554i \(-0.394479\pi\)
0.325466 + 0.945554i \(0.394479\pi\)
\(192\) −4293.51 −1.61384
\(193\) −1340.18 −0.499837 −0.249919 0.968267i \(-0.580404\pi\)
−0.249919 + 0.968267i \(0.580404\pi\)
\(194\) 3678.48 1.36134
\(195\) 0 0
\(196\) 178.756 0.0651445
\(197\) 3518.33 1.27244 0.636220 0.771508i \(-0.280497\pi\)
0.636220 + 0.771508i \(0.280497\pi\)
\(198\) −1077.58 −0.386768
\(199\) 823.692 0.293417 0.146709 0.989180i \(-0.453132\pi\)
0.146709 + 0.989180i \(0.453132\pi\)
\(200\) 2232.07 0.789156
\(201\) 2714.29 0.952494
\(202\) 441.121 0.153649
\(203\) 76.7791 0.0265460
\(204\) −175.112 −0.0600996
\(205\) 3878.59 1.32143
\(206\) 94.9639 0.0321187
\(207\) −3992.75 −1.34065
\(208\) 0 0
\(209\) −1539.15 −0.509404
\(210\) 988.484 0.324819
\(211\) −107.343 −0.0350228 −0.0175114 0.999847i \(-0.505574\pi\)
−0.0175114 + 0.999847i \(0.505574\pi\)
\(212\) 183.810 0.0595479
\(213\) −1644.03 −0.528858
\(214\) −2273.56 −0.726248
\(215\) 857.405 0.271975
\(216\) −1637.46 −0.515810
\(217\) 96.7537 0.0302676
\(218\) 2852.96 0.886362
\(219\) −8012.24 −2.47222
\(220\) 87.5768 0.0268383
\(221\) 0 0
\(222\) −284.694 −0.0860693
\(223\) 3933.68 1.18125 0.590625 0.806946i \(-0.298881\pi\)
0.590625 + 0.806946i \(0.298881\pi\)
\(224\) 74.3689 0.0221830
\(225\) 3431.92 1.01686
\(226\) −806.178 −0.237284
\(227\) 1771.90 0.518085 0.259042 0.965866i \(-0.416593\pi\)
0.259042 + 0.965866i \(0.416593\pi\)
\(228\) −594.493 −0.172681
\(229\) −1915.37 −0.552713 −0.276356 0.961055i \(-0.589127\pi\)
−0.276356 + 0.961055i \(0.589127\pi\)
\(230\) −4519.68 −1.29573
\(231\) 267.892 0.0763031
\(232\) −582.892 −0.164952
\(233\) 4396.32 1.23610 0.618052 0.786137i \(-0.287922\pi\)
0.618052 + 0.786137i \(0.287922\pi\)
\(234\) 0 0
\(235\) −5108.32 −1.41800
\(236\) 47.3570 0.0130622
\(237\) −10260.9 −2.81230
\(238\) −345.892 −0.0942053
\(239\) 4084.49 1.10546 0.552728 0.833362i \(-0.313587\pi\)
0.552728 + 0.833362i \(0.313587\pi\)
\(240\) −6999.42 −1.88255
\(241\) −3908.58 −1.04471 −0.522353 0.852730i \(-0.674946\pi\)
−0.522353 + 0.852730i \(0.674946\pi\)
\(242\) −330.578 −0.0878114
\(243\) 5157.80 1.36162
\(244\) −395.867 −0.103864
\(245\) 4955.56 1.29224
\(246\) −5654.88 −1.46562
\(247\) 0 0
\(248\) −734.536 −0.188077
\(249\) 3503.05 0.891552
\(250\) −1188.72 −0.300725
\(251\) 1094.89 0.275335 0.137667 0.990479i \(-0.456040\pi\)
0.137667 + 0.990479i \(0.456040\pi\)
\(252\) 59.0258 0.0147551
\(253\) −1224.89 −0.304381
\(254\) 3599.76 0.889249
\(255\) −4854.54 −1.19217
\(256\) −819.364 −0.200040
\(257\) 783.179 0.190091 0.0950454 0.995473i \(-0.469700\pi\)
0.0950454 + 0.995473i \(0.469700\pi\)
\(258\) −1250.07 −0.301652
\(259\) 40.3744 0.00968628
\(260\) 0 0
\(261\) −896.225 −0.212548
\(262\) 4373.23 1.03122
\(263\) 6180.06 1.44897 0.724484 0.689292i \(-0.242078\pi\)
0.724484 + 0.689292i \(0.242078\pi\)
\(264\) −2033.79 −0.474132
\(265\) 5095.67 1.18122
\(266\) −1174.28 −0.270675
\(267\) −11806.0 −2.70605
\(268\) 183.470 0.0418179
\(269\) 986.965 0.223704 0.111852 0.993725i \(-0.464322\pi\)
0.111852 + 0.993725i \(0.464322\pi\)
\(270\) −2849.93 −0.642376
\(271\) −4576.99 −1.02595 −0.512975 0.858404i \(-0.671457\pi\)
−0.512975 + 0.858404i \(0.671457\pi\)
\(272\) 2449.25 0.545984
\(273\) 0 0
\(274\) 4403.18 0.970825
\(275\) 1052.84 0.230868
\(276\) −473.110 −0.103181
\(277\) 567.836 0.123169 0.0615847 0.998102i \(-0.480385\pi\)
0.0615847 + 0.998102i \(0.480385\pi\)
\(278\) 87.0484 0.0187799
\(279\) −1129.38 −0.242346
\(280\) 1064.25 0.227147
\(281\) −5311.01 −1.12750 −0.563752 0.825944i \(-0.690643\pi\)
−0.563752 + 0.825944i \(0.690643\pi\)
\(282\) 7447.79 1.57273
\(283\) −4728.44 −0.993204 −0.496602 0.867978i \(-0.665419\pi\)
−0.496602 + 0.867978i \(0.665419\pi\)
\(284\) −111.126 −0.0232187
\(285\) −16480.8 −3.42539
\(286\) 0 0
\(287\) 801.960 0.164941
\(288\) −868.092 −0.177614
\(289\) −3214.29 −0.654242
\(290\) −1014.50 −0.205426
\(291\) 10674.7 2.15038
\(292\) −541.579 −0.108539
\(293\) −2328.92 −0.464358 −0.232179 0.972673i \(-0.574585\pi\)
−0.232179 + 0.972673i \(0.574585\pi\)
\(294\) −7225.08 −1.43325
\(295\) 1312.85 0.259109
\(296\) −306.515 −0.0601886
\(297\) −772.369 −0.150900
\(298\) −6634.36 −1.28966
\(299\) 0 0
\(300\) 406.656 0.0782610
\(301\) 177.282 0.0339481
\(302\) −7039.61 −1.34134
\(303\) 1280.10 0.242705
\(304\) 8315.01 1.56875
\(305\) −10974.4 −2.06030
\(306\) 4037.52 0.754280
\(307\) 1678.07 0.311962 0.155981 0.987760i \(-0.450146\pi\)
0.155981 + 0.987760i \(0.450146\pi\)
\(308\) 18.1079 0.00334997
\(309\) 275.578 0.0507349
\(310\) −1278.43 −0.234226
\(311\) 3572.71 0.651413 0.325707 0.945471i \(-0.394398\pi\)
0.325707 + 0.945471i \(0.394398\pi\)
\(312\) 0 0
\(313\) 7184.36 1.29739 0.648697 0.761047i \(-0.275314\pi\)
0.648697 + 0.761047i \(0.275314\pi\)
\(314\) −6764.40 −1.21572
\(315\) 1636.34 0.292690
\(316\) −693.573 −0.123470
\(317\) 15.7077 0.00278306 0.00139153 0.999999i \(-0.499557\pi\)
0.00139153 + 0.999999i \(0.499557\pi\)
\(318\) −7429.36 −1.31012
\(319\) −274.943 −0.0482566
\(320\) −8045.47 −1.40549
\(321\) −6597.69 −1.14719
\(322\) −934.515 −0.161734
\(323\) 5766.98 0.993447
\(324\) 220.491 0.0378070
\(325\) 0 0
\(326\) −7445.13 −1.26487
\(327\) 8279.08 1.40010
\(328\) −6088.33 −1.02491
\(329\) −1056.23 −0.176996
\(330\) −3539.73 −0.590472
\(331\) 1318.95 0.219022 0.109511 0.993986i \(-0.465072\pi\)
0.109511 + 0.993986i \(0.465072\pi\)
\(332\) 236.785 0.0391423
\(333\) −471.282 −0.0775558
\(334\) 7478.43 1.22515
\(335\) 5086.22 0.829523
\(336\) −1447.24 −0.234981
\(337\) −239.183 −0.0386621 −0.0193310 0.999813i \(-0.506154\pi\)
−0.0193310 + 0.999813i \(0.506154\pi\)
\(338\) 0 0
\(339\) −2339.47 −0.374816
\(340\) −328.137 −0.0523404
\(341\) −346.472 −0.0550220
\(342\) 13707.1 2.16723
\(343\) 2078.27 0.327160
\(344\) −1345.89 −0.210947
\(345\) −13115.8 −2.04675
\(346\) −6303.98 −0.979491
\(347\) −5862.79 −0.907006 −0.453503 0.891255i \(-0.649826\pi\)
−0.453503 + 0.891255i \(0.649826\pi\)
\(348\) −106.196 −0.0163583
\(349\) −3491.73 −0.535553 −0.267776 0.963481i \(-0.586289\pi\)
−0.267776 + 0.963481i \(0.586289\pi\)
\(350\) 803.251 0.122673
\(351\) 0 0
\(352\) −266.313 −0.0403253
\(353\) 10916.7 1.64600 0.822999 0.568043i \(-0.192299\pi\)
0.822999 + 0.568043i \(0.192299\pi\)
\(354\) −1914.10 −0.287382
\(355\) −3080.69 −0.460580
\(356\) −798.013 −0.118805
\(357\) −1003.75 −0.148807
\(358\) 3584.87 0.529236
\(359\) 11500.7 1.69077 0.845384 0.534160i \(-0.179372\pi\)
0.845384 + 0.534160i \(0.179372\pi\)
\(360\) −12422.8 −1.81872
\(361\) 12719.5 1.85442
\(362\) 2194.31 0.318592
\(363\) −959.313 −0.138708
\(364\) 0 0
\(365\) −15013.9 −2.15305
\(366\) 16000.4 2.28512
\(367\) 6767.01 0.962493 0.481246 0.876585i \(-0.340184\pi\)
0.481246 + 0.876585i \(0.340184\pi\)
\(368\) 6617.27 0.937362
\(369\) −9361.10 −1.32065
\(370\) −533.478 −0.0749573
\(371\) 1053.61 0.147441
\(372\) −133.823 −0.0186517
\(373\) −5310.22 −0.737139 −0.368569 0.929600i \(-0.620152\pi\)
−0.368569 + 0.929600i \(0.620152\pi\)
\(374\) 1238.63 0.171251
\(375\) −3449.58 −0.475028
\(376\) 8018.67 1.09982
\(377\) 0 0
\(378\) −589.269 −0.0801818
\(379\) 838.267 0.113612 0.0568059 0.998385i \(-0.481908\pi\)
0.0568059 + 0.998385i \(0.481908\pi\)
\(380\) −1114.00 −0.150387
\(381\) 10446.2 1.40466
\(382\) −4694.34 −0.628752
\(383\) 2832.16 0.377851 0.188925 0.981991i \(-0.439500\pi\)
0.188925 + 0.981991i \(0.439500\pi\)
\(384\) 10194.5 1.35479
\(385\) 501.994 0.0664520
\(386\) 3661.45 0.482806
\(387\) −2069.37 −0.271814
\(388\) 721.542 0.0944091
\(389\) 3111.25 0.405519 0.202759 0.979229i \(-0.435009\pi\)
0.202759 + 0.979229i \(0.435009\pi\)
\(390\) 0 0
\(391\) 4589.49 0.593608
\(392\) −7778.88 −1.00228
\(393\) 12690.8 1.62892
\(394\) −9612.25 −1.22908
\(395\) −19227.5 −2.44922
\(396\) −211.369 −0.0268225
\(397\) −14208.7 −1.79626 −0.898131 0.439728i \(-0.855075\pi\)
−0.898131 + 0.439728i \(0.855075\pi\)
\(398\) −2250.37 −0.283419
\(399\) −3407.66 −0.427560
\(400\) −5687.79 −0.710974
\(401\) 6261.68 0.779784 0.389892 0.920861i \(-0.372512\pi\)
0.389892 + 0.920861i \(0.372512\pi\)
\(402\) −7415.58 −0.920039
\(403\) 0 0
\(404\) 86.5269 0.0106556
\(405\) 6112.54 0.749961
\(406\) −209.764 −0.0256415
\(407\) −144.580 −0.0176082
\(408\) 7620.30 0.924660
\(409\) 4192.50 0.506860 0.253430 0.967354i \(-0.418441\pi\)
0.253430 + 0.967354i \(0.418441\pi\)
\(410\) −10596.5 −1.27640
\(411\) 12777.7 1.53352
\(412\) 18.6274 0.00222744
\(413\) 271.453 0.0323421
\(414\) 10908.4 1.29497
\(415\) 6564.25 0.776448
\(416\) 0 0
\(417\) 252.608 0.0296649
\(418\) 4205.05 0.492047
\(419\) −9287.15 −1.08283 −0.541416 0.840755i \(-0.682112\pi\)
−0.541416 + 0.840755i \(0.682112\pi\)
\(420\) 193.894 0.0225263
\(421\) −13146.0 −1.52185 −0.760923 0.648842i \(-0.775254\pi\)
−0.760923 + 0.648842i \(0.775254\pi\)
\(422\) 293.267 0.0338294
\(423\) 12329.1 1.41716
\(424\) −7998.81 −0.916172
\(425\) −3944.84 −0.450242
\(426\) 4491.56 0.510838
\(427\) −2269.13 −0.257168
\(428\) −445.964 −0.0503656
\(429\) 0 0
\(430\) −2342.47 −0.262707
\(431\) −4909.67 −0.548701 −0.274351 0.961630i \(-0.588463\pi\)
−0.274351 + 0.961630i \(0.588463\pi\)
\(432\) 4172.59 0.464708
\(433\) −11743.3 −1.30334 −0.651671 0.758502i \(-0.725932\pi\)
−0.651671 + 0.758502i \(0.725932\pi\)
\(434\) −264.336 −0.0292363
\(435\) −2944.01 −0.324493
\(436\) 559.615 0.0614695
\(437\) 15581.0 1.70558
\(438\) 21889.8 2.38798
\(439\) −11824.2 −1.28551 −0.642754 0.766073i \(-0.722208\pi\)
−0.642754 + 0.766073i \(0.722208\pi\)
\(440\) −3811.05 −0.412920
\(441\) −11960.4 −1.29148
\(442\) 0 0
\(443\) 10102.1 1.08344 0.541722 0.840558i \(-0.317772\pi\)
0.541722 + 0.840558i \(0.317772\pi\)
\(444\) −55.8433 −0.00596894
\(445\) −22122.9 −2.35668
\(446\) −10747.0 −1.14100
\(447\) −19252.4 −2.03715
\(448\) −1663.53 −0.175434
\(449\) 345.254 0.0362885 0.0181443 0.999835i \(-0.494224\pi\)
0.0181443 + 0.999835i \(0.494224\pi\)
\(450\) −9376.17 −0.982216
\(451\) −2871.79 −0.299839
\(452\) −158.134 −0.0164557
\(453\) −20428.4 −2.11879
\(454\) −4840.93 −0.500431
\(455\) 0 0
\(456\) 25870.3 2.65678
\(457\) 10567.1 1.08164 0.540821 0.841138i \(-0.318114\pi\)
0.540821 + 0.841138i \(0.318114\pi\)
\(458\) 5232.89 0.533879
\(459\) 2893.95 0.294288
\(460\) −886.546 −0.0898596
\(461\) −4733.96 −0.478270 −0.239135 0.970986i \(-0.576864\pi\)
−0.239135 + 0.970986i \(0.576864\pi\)
\(462\) −731.895 −0.0737031
\(463\) −3431.20 −0.344409 −0.172204 0.985061i \(-0.555089\pi\)
−0.172204 + 0.985061i \(0.555089\pi\)
\(464\) 1485.33 0.148610
\(465\) −3709.91 −0.369985
\(466\) −12011.0 −1.19399
\(467\) 5116.96 0.507034 0.253517 0.967331i \(-0.418413\pi\)
0.253517 + 0.967331i \(0.418413\pi\)
\(468\) 0 0
\(469\) 1051.66 0.103542
\(470\) 13956.2 1.36968
\(471\) −19629.8 −1.92037
\(472\) −2060.82 −0.200968
\(473\) −634.841 −0.0617125
\(474\) 28033.2 2.71648
\(475\) −13392.4 −1.29366
\(476\) −67.8476 −0.00653317
\(477\) −12298.6 −1.18053
\(478\) −11159.0 −1.06779
\(479\) −11566.9 −1.10335 −0.551675 0.834059i \(-0.686011\pi\)
−0.551675 + 0.834059i \(0.686011\pi\)
\(480\) −2851.59 −0.271160
\(481\) 0 0
\(482\) 10678.4 1.00911
\(483\) −2711.89 −0.255477
\(484\) −64.8437 −0.00608975
\(485\) 20002.9 1.87275
\(486\) −14091.4 −1.31522
\(487\) 18326.5 1.70525 0.852623 0.522527i \(-0.175010\pi\)
0.852623 + 0.522527i \(0.175010\pi\)
\(488\) 17226.8 1.59799
\(489\) −21605.2 −1.99800
\(490\) −13538.9 −1.24821
\(491\) −7617.58 −0.700156 −0.350078 0.936721i \(-0.613845\pi\)
−0.350078 + 0.936721i \(0.613845\pi\)
\(492\) −1109.22 −0.101641
\(493\) 1030.17 0.0941108
\(494\) 0 0
\(495\) −5859.67 −0.532066
\(496\) 1871.75 0.169444
\(497\) −636.980 −0.0574899
\(498\) −9570.50 −0.861173
\(499\) −12909.1 −1.15810 −0.579050 0.815292i \(-0.696576\pi\)
−0.579050 + 0.815292i \(0.696576\pi\)
\(500\) −233.171 −0.0208554
\(501\) 21701.8 1.93526
\(502\) −2991.30 −0.265953
\(503\) 10165.7 0.901121 0.450561 0.892746i \(-0.351224\pi\)
0.450561 + 0.892746i \(0.351224\pi\)
\(504\) −2568.60 −0.227013
\(505\) 2398.74 0.211371
\(506\) 3346.47 0.294009
\(507\) 0 0
\(508\) 706.102 0.0616697
\(509\) −6449.93 −0.561666 −0.280833 0.959757i \(-0.590611\pi\)
−0.280833 + 0.959757i \(0.590611\pi\)
\(510\) 13262.8 1.15155
\(511\) −3104.36 −0.268745
\(512\) 12525.4 1.08115
\(513\) 9824.75 0.845562
\(514\) −2139.68 −0.183614
\(515\) 516.397 0.0441848
\(516\) −245.205 −0.0209197
\(517\) 3782.31 0.321752
\(518\) −110.305 −0.00935623
\(519\) −18293.7 −1.54721
\(520\) 0 0
\(521\) −19327.4 −1.62524 −0.812620 0.582794i \(-0.801959\pi\)
−0.812620 + 0.582794i \(0.801959\pi\)
\(522\) 2448.53 0.205305
\(523\) 6259.09 0.523310 0.261655 0.965161i \(-0.415732\pi\)
0.261655 + 0.965161i \(0.415732\pi\)
\(524\) 857.819 0.0715153
\(525\) 2330.97 0.193775
\(526\) −16884.2 −1.39960
\(527\) 1298.18 0.107305
\(528\) 5182.52 0.427160
\(529\) 232.675 0.0191235
\(530\) −13921.6 −1.14098
\(531\) −3168.61 −0.258956
\(532\) −230.337 −0.0187714
\(533\) 0 0
\(534\) 32254.6 2.61384
\(535\) −12363.2 −0.999079
\(536\) −7983.99 −0.643387
\(537\) 10403.0 0.835985
\(538\) −2696.44 −0.216081
\(539\) −3669.20 −0.293217
\(540\) −559.022 −0.0445490
\(541\) 14008.2 1.11323 0.556616 0.830770i \(-0.312100\pi\)
0.556616 + 0.830770i \(0.312100\pi\)
\(542\) 12504.6 0.990991
\(543\) 6367.72 0.503251
\(544\) 997.834 0.0786430
\(545\) 15513.9 1.21934
\(546\) 0 0
\(547\) −4949.45 −0.386879 −0.193440 0.981112i \(-0.561964\pi\)
−0.193440 + 0.981112i \(0.561964\pi\)
\(548\) 863.695 0.0673270
\(549\) 26487.0 2.05909
\(550\) −2876.41 −0.223001
\(551\) 3497.35 0.270404
\(552\) 20588.2 1.58748
\(553\) −3975.60 −0.305714
\(554\) −1551.36 −0.118973
\(555\) −1548.11 −0.118403
\(556\) 17.0748 0.00130239
\(557\) 3801.58 0.289188 0.144594 0.989491i \(-0.453812\pi\)
0.144594 + 0.989491i \(0.453812\pi\)
\(558\) 3085.54 0.234088
\(559\) 0 0
\(560\) −2711.94 −0.204644
\(561\) 3594.40 0.270510
\(562\) 14510.0 1.08908
\(563\) −9900.11 −0.741101 −0.370551 0.928812i \(-0.620831\pi\)
−0.370551 + 0.928812i \(0.620831\pi\)
\(564\) 1460.90 0.109069
\(565\) −4383.85 −0.326425
\(566\) 12918.3 0.959361
\(567\) 1263.86 0.0936107
\(568\) 4835.84 0.357231
\(569\) 5329.16 0.392636 0.196318 0.980540i \(-0.437102\pi\)
0.196318 + 0.980540i \(0.437102\pi\)
\(570\) 45026.3 3.30868
\(571\) −16962.6 −1.24319 −0.621597 0.783337i \(-0.713516\pi\)
−0.621597 + 0.783337i \(0.713516\pi\)
\(572\) 0 0
\(573\) −13622.6 −0.993181
\(574\) −2190.99 −0.159321
\(575\) −10658.0 −0.772989
\(576\) 19418.0 1.40466
\(577\) 15487.0 1.11738 0.558692 0.829375i \(-0.311303\pi\)
0.558692 + 0.829375i \(0.311303\pi\)
\(578\) 8781.61 0.631949
\(579\) 10625.3 0.762643
\(580\) −198.997 −0.0142464
\(581\) 1357.26 0.0969169
\(582\) −29163.7 −2.07710
\(583\) −3772.94 −0.268026
\(584\) 23567.7 1.66993
\(585\) 0 0
\(586\) 6362.72 0.448535
\(587\) −11084.2 −0.779373 −0.389686 0.920948i \(-0.627417\pi\)
−0.389686 + 0.920948i \(0.627417\pi\)
\(588\) −1417.22 −0.0993964
\(589\) 4407.22 0.308313
\(590\) −3586.77 −0.250280
\(591\) −27894.0 −1.94147
\(592\) 781.066 0.0542257
\(593\) −4349.68 −0.301214 −0.150607 0.988594i \(-0.548123\pi\)
−0.150607 + 0.988594i \(0.548123\pi\)
\(594\) 2110.15 0.145759
\(595\) −1880.90 −0.129596
\(596\) −1301.34 −0.0894382
\(597\) −6530.40 −0.447691
\(598\) 0 0
\(599\) 13183.9 0.899299 0.449650 0.893205i \(-0.351549\pi\)
0.449650 + 0.893205i \(0.351549\pi\)
\(600\) −17696.3 −1.20408
\(601\) −18765.0 −1.27361 −0.636806 0.771024i \(-0.719745\pi\)
−0.636806 + 0.771024i \(0.719745\pi\)
\(602\) −484.344 −0.0327913
\(603\) −12275.8 −0.829034
\(604\) −1380.84 −0.0930223
\(605\) −1797.63 −0.120800
\(606\) −3497.29 −0.234435
\(607\) 21871.4 1.46249 0.731244 0.682116i \(-0.238940\pi\)
0.731244 + 0.682116i \(0.238940\pi\)
\(608\) 3387.57 0.225961
\(609\) −608.720 −0.0405034
\(610\) 29982.6 1.99010
\(611\) 0 0
\(612\) 791.970 0.0523096
\(613\) 3527.85 0.232445 0.116222 0.993223i \(-0.462921\pi\)
0.116222 + 0.993223i \(0.462921\pi\)
\(614\) −4584.56 −0.301332
\(615\) −30750.2 −2.01621
\(616\) −787.994 −0.0515409
\(617\) 22728.1 1.48298 0.741490 0.670963i \(-0.234119\pi\)
0.741490 + 0.670963i \(0.234119\pi\)
\(618\) −752.893 −0.0490062
\(619\) 21443.3 1.39237 0.696187 0.717861i \(-0.254879\pi\)
0.696187 + 0.717861i \(0.254879\pi\)
\(620\) −250.767 −0.0162437
\(621\) 7818.75 0.505243
\(622\) −9760.81 −0.629217
\(623\) −4574.25 −0.294163
\(624\) 0 0
\(625\) −18428.2 −1.17940
\(626\) −19628.0 −1.25319
\(627\) 12202.7 0.777240
\(628\) −1326.85 −0.0843109
\(629\) 541.718 0.0343398
\(630\) −4470.56 −0.282716
\(631\) −21532.0 −1.35844 −0.679219 0.733936i \(-0.737681\pi\)
−0.679219 + 0.733936i \(0.737681\pi\)
\(632\) 30182.0 1.89964
\(633\) 851.038 0.0534372
\(634\) −42.9141 −0.00268823
\(635\) 19574.9 1.22332
\(636\) −1457.29 −0.0908572
\(637\) 0 0
\(638\) 751.159 0.0466123
\(639\) 7435.33 0.460309
\(640\) 19103.2 1.17988
\(641\) 20148.3 1.24151 0.620756 0.784004i \(-0.286826\pi\)
0.620756 + 0.784004i \(0.286826\pi\)
\(642\) 18025.2 1.10810
\(643\) −28869.7 −1.77062 −0.885310 0.465000i \(-0.846054\pi\)
−0.885310 + 0.465000i \(0.846054\pi\)
\(644\) −183.307 −0.0112163
\(645\) −6797.68 −0.414974
\(646\) −15755.7 −0.959597
\(647\) −1590.02 −0.0966155 −0.0483077 0.998833i \(-0.515383\pi\)
−0.0483077 + 0.998833i \(0.515383\pi\)
\(648\) −9595.02 −0.581679
\(649\) −972.062 −0.0587932
\(650\) 0 0
\(651\) −767.083 −0.0461818
\(652\) −1460.38 −0.0877192
\(653\) 20028.1 1.20024 0.600122 0.799909i \(-0.295119\pi\)
0.600122 + 0.799909i \(0.295119\pi\)
\(654\) −22618.9 −1.35240
\(655\) 23780.8 1.41862
\(656\) 15514.4 0.923375
\(657\) 36236.5 2.15178
\(658\) 2885.66 0.170965
\(659\) −10520.7 −0.621897 −0.310948 0.950427i \(-0.600647\pi\)
−0.310948 + 0.950427i \(0.600647\pi\)
\(660\) −694.326 −0.0409494
\(661\) −3295.83 −0.193938 −0.0969690 0.995287i \(-0.530915\pi\)
−0.0969690 + 0.995287i \(0.530915\pi\)
\(662\) −3603.45 −0.211559
\(663\) 0 0
\(664\) −10304.1 −0.602222
\(665\) −6385.51 −0.372360
\(666\) 1287.57 0.0749132
\(667\) 2783.27 0.161572
\(668\) 1466.91 0.0849649
\(669\) −31187.0 −1.80233
\(670\) −13895.8 −0.801257
\(671\) 8125.67 0.467493
\(672\) −589.612 −0.0338464
\(673\) −1187.64 −0.0680239 −0.0340119 0.999421i \(-0.510828\pi\)
−0.0340119 + 0.999421i \(0.510828\pi\)
\(674\) 653.460 0.0373447
\(675\) −6720.51 −0.383219
\(676\) 0 0
\(677\) 13221.4 0.750574 0.375287 0.926909i \(-0.377544\pi\)
0.375287 + 0.926909i \(0.377544\pi\)
\(678\) 6391.55 0.362044
\(679\) 4135.91 0.233758
\(680\) 14279.4 0.805282
\(681\) −14048.0 −0.790485
\(682\) 946.578 0.0531471
\(683\) 13831.4 0.774882 0.387441 0.921894i \(-0.373359\pi\)
0.387441 + 0.921894i \(0.373359\pi\)
\(684\) 2688.68 0.150298
\(685\) 23943.7 1.33554
\(686\) −5677.93 −0.316012
\(687\) 15185.4 0.843320
\(688\) 3429.62 0.190048
\(689\) 0 0
\(690\) 35832.9 1.97701
\(691\) 9817.07 0.540462 0.270231 0.962796i \(-0.412900\pi\)
0.270231 + 0.962796i \(0.412900\pi\)
\(692\) −1236.54 −0.0679280
\(693\) −1211.58 −0.0664128
\(694\) 16017.4 0.876101
\(695\) 473.354 0.0258350
\(696\) 4621.29 0.251680
\(697\) 10760.2 0.584750
\(698\) 9539.58 0.517304
\(699\) −34854.9 −1.88603
\(700\) 157.560 0.00850742
\(701\) 29949.8 1.61368 0.806838 0.590773i \(-0.201177\pi\)
0.806838 + 0.590773i \(0.201177\pi\)
\(702\) 0 0
\(703\) 1839.09 0.0986667
\(704\) 5957.03 0.318912
\(705\) 40499.8 2.16356
\(706\) −29825.0 −1.58991
\(707\) 495.976 0.0263835
\(708\) −375.456 −0.0199301
\(709\) −11307.5 −0.598959 −0.299479 0.954103i \(-0.596813\pi\)
−0.299479 + 0.954103i \(0.596813\pi\)
\(710\) 8416.59 0.444886
\(711\) 46406.3 2.44778
\(712\) 34726.9 1.82787
\(713\) 3507.36 0.184224
\(714\) 2742.30 0.143737
\(715\) 0 0
\(716\) 703.181 0.0367027
\(717\) −32382.7 −1.68669
\(718\) −31420.6 −1.63316
\(719\) −32623.4 −1.69214 −0.846070 0.533071i \(-0.821038\pi\)
−0.846070 + 0.533071i \(0.821038\pi\)
\(720\) 31655.9 1.63853
\(721\) 106.773 0.00551518
\(722\) −34750.2 −1.79123
\(723\) 30988.0 1.59399
\(724\) 430.420 0.0220945
\(725\) −2392.33 −0.122550
\(726\) 2620.89 0.133981
\(727\) −502.545 −0.0256373 −0.0128187 0.999918i \(-0.504080\pi\)
−0.0128187 + 0.999918i \(0.504080\pi\)
\(728\) 0 0
\(729\) −29783.2 −1.51314
\(730\) 41018.7 2.07968
\(731\) 2378.66 0.120353
\(732\) 3138.51 0.158474
\(733\) −8631.37 −0.434935 −0.217467 0.976068i \(-0.569780\pi\)
−0.217467 + 0.976068i \(0.569780\pi\)
\(734\) −18487.8 −0.929697
\(735\) −39288.7 −1.97168
\(736\) 2695.90 0.135017
\(737\) −3765.95 −0.188223
\(738\) 25575.0 1.27565
\(739\) 18357.5 0.913792 0.456896 0.889520i \(-0.348961\pi\)
0.456896 + 0.889520i \(0.348961\pi\)
\(740\) −104.643 −0.00519832
\(741\) 0 0
\(742\) −2878.52 −0.142417
\(743\) −11182.6 −0.552155 −0.276078 0.961135i \(-0.589035\pi\)
−0.276078 + 0.961135i \(0.589035\pi\)
\(744\) 5823.55 0.286965
\(745\) −36076.4 −1.77415
\(746\) 14507.8 0.712021
\(747\) −15843.0 −0.775991
\(748\) 242.960 0.0118763
\(749\) −2556.29 −0.124706
\(750\) 9424.43 0.458842
\(751\) 16733.4 0.813063 0.406531 0.913637i \(-0.366738\pi\)
0.406531 + 0.913637i \(0.366738\pi\)
\(752\) −20433.3 −0.990857
\(753\) −8680.53 −0.420101
\(754\) 0 0
\(755\) −38280.2 −1.84524
\(756\) −115.587 −0.00556064
\(757\) −24402.4 −1.17163 −0.585813 0.810446i \(-0.699225\pi\)
−0.585813 + 0.810446i \(0.699225\pi\)
\(758\) −2290.19 −0.109741
\(759\) 9711.19 0.464419
\(760\) 48477.6 2.31377
\(761\) −8469.33 −0.403434 −0.201717 0.979444i \(-0.564652\pi\)
−0.201717 + 0.979444i \(0.564652\pi\)
\(762\) −28539.7 −1.35680
\(763\) 3207.74 0.152199
\(764\) −920.805 −0.0436042
\(765\) 21955.3 1.03764
\(766\) −7737.62 −0.364976
\(767\) 0 0
\(768\) 6496.08 0.305218
\(769\) −32834.7 −1.53973 −0.769864 0.638208i \(-0.779676\pi\)
−0.769864 + 0.638208i \(0.779676\pi\)
\(770\) −1371.47 −0.0641877
\(771\) −6209.20 −0.290038
\(772\) 718.202 0.0334827
\(773\) 35571.4 1.65513 0.827564 0.561371i \(-0.189726\pi\)
0.827564 + 0.561371i \(0.189726\pi\)
\(774\) 5653.64 0.262553
\(775\) −3014.71 −0.139731
\(776\) −31399.1 −1.45253
\(777\) −320.097 −0.0147792
\(778\) −8500.11 −0.391701
\(779\) 36530.0 1.68013
\(780\) 0 0
\(781\) 2281.01 0.104508
\(782\) −12538.7 −0.573381
\(783\) 1755.02 0.0801014
\(784\) 19822.3 0.902982
\(785\) −36783.6 −1.67244
\(786\) −34671.8 −1.57341
\(787\) −15729.6 −0.712452 −0.356226 0.934400i \(-0.615937\pi\)
−0.356226 + 0.934400i \(0.615937\pi\)
\(788\) −1885.47 −0.0852373
\(789\) −48996.7 −2.21081
\(790\) 52530.6 2.36577
\(791\) −906.431 −0.0407446
\(792\) 9198.09 0.412676
\(793\) 0 0
\(794\) 38819.0 1.73506
\(795\) −40399.5 −1.80229
\(796\) −441.415 −0.0196552
\(797\) 7888.07 0.350577 0.175288 0.984517i \(-0.443914\pi\)
0.175288 + 0.984517i \(0.443914\pi\)
\(798\) 9309.91 0.412991
\(799\) −14171.8 −0.627485
\(800\) −2317.23 −0.102408
\(801\) 53394.2 2.35530
\(802\) −17107.2 −0.753214
\(803\) 11116.6 0.488538
\(804\) −1454.58 −0.0638050
\(805\) −5081.73 −0.222494
\(806\) 0 0
\(807\) −7824.86 −0.341323
\(808\) −3765.36 −0.163942
\(809\) 5896.97 0.256275 0.128138 0.991756i \(-0.459100\pi\)
0.128138 + 0.991756i \(0.459100\pi\)
\(810\) −16699.8 −0.724407
\(811\) −14197.9 −0.614744 −0.307372 0.951589i \(-0.599450\pi\)
−0.307372 + 0.951589i \(0.599450\pi\)
\(812\) −41.1458 −0.00177824
\(813\) 36287.3 1.56538
\(814\) 394.999 0.0170082
\(815\) −40485.3 −1.74005
\(816\) −19418.2 −0.833053
\(817\) 8075.35 0.345803
\(818\) −11454.1 −0.489589
\(819\) 0 0
\(820\) −2078.53 −0.0885188
\(821\) 19841.7 0.843459 0.421729 0.906722i \(-0.361423\pi\)
0.421729 + 0.906722i \(0.361423\pi\)
\(822\) −34909.3 −1.48127
\(823\) −28202.2 −1.19449 −0.597246 0.802058i \(-0.703738\pi\)
−0.597246 + 0.802058i \(0.703738\pi\)
\(824\) −810.602 −0.0342702
\(825\) −8347.14 −0.352255
\(826\) −741.622 −0.0312401
\(827\) −34031.0 −1.43092 −0.715462 0.698651i \(-0.753784\pi\)
−0.715462 + 0.698651i \(0.753784\pi\)
\(828\) 2139.71 0.0898067
\(829\) 4931.55 0.206610 0.103305 0.994650i \(-0.467058\pi\)
0.103305 + 0.994650i \(0.467058\pi\)
\(830\) −17933.9 −0.749992
\(831\) −4501.92 −0.187930
\(832\) 0 0
\(833\) 13748.0 0.571836
\(834\) −690.138 −0.0286541
\(835\) 40666.4 1.68541
\(836\) 824.830 0.0341236
\(837\) 2211.60 0.0913312
\(838\) 25373.0 1.04594
\(839\) 38189.8 1.57146 0.785731 0.618568i \(-0.212287\pi\)
0.785731 + 0.618568i \(0.212287\pi\)
\(840\) −8437.60 −0.346577
\(841\) −23764.3 −0.974384
\(842\) 35915.6 1.46999
\(843\) 42106.8 1.72033
\(844\) 57.5250 0.00234608
\(845\) 0 0
\(846\) −33683.7 −1.36888
\(847\) −371.687 −0.0150783
\(848\) 20382.7 0.825406
\(849\) 37488.0 1.51541
\(850\) 10777.5 0.434900
\(851\) 1463.59 0.0589556
\(852\) 881.030 0.0354268
\(853\) −42966.8 −1.72469 −0.862343 0.506325i \(-0.831003\pi\)
−0.862343 + 0.506325i \(0.831003\pi\)
\(854\) 6199.37 0.248405
\(855\) 74536.6 2.98140
\(856\) 19406.8 0.774898
\(857\) −17281.5 −0.688828 −0.344414 0.938818i \(-0.611922\pi\)
−0.344414 + 0.938818i \(0.611922\pi\)
\(858\) 0 0
\(859\) 9316.75 0.370062 0.185031 0.982733i \(-0.440761\pi\)
0.185031 + 0.982733i \(0.440761\pi\)
\(860\) −459.482 −0.0182188
\(861\) −6358.10 −0.251665
\(862\) 13413.5 0.530005
\(863\) 9647.65 0.380544 0.190272 0.981731i \(-0.439063\pi\)
0.190272 + 0.981731i \(0.439063\pi\)
\(864\) 1699.93 0.0669361
\(865\) −34279.9 −1.34746
\(866\) 32083.3 1.25893
\(867\) 25483.6 0.998232
\(868\) −51.8501 −0.00202754
\(869\) 14236.5 0.555742
\(870\) 8043.18 0.313436
\(871\) 0 0
\(872\) −24352.6 −0.945737
\(873\) −48277.6 −1.87165
\(874\) −42568.0 −1.64746
\(875\) −1336.55 −0.0516383
\(876\) 4293.75 0.165608
\(877\) −19728.7 −0.759624 −0.379812 0.925064i \(-0.624011\pi\)
−0.379812 + 0.925064i \(0.624011\pi\)
\(878\) 32304.3 1.24171
\(879\) 18464.1 0.708509
\(880\) 9711.36 0.372011
\(881\) 19473.9 0.744712 0.372356 0.928090i \(-0.378550\pi\)
0.372356 + 0.928090i \(0.378550\pi\)
\(882\) 32676.4 1.24748
\(883\) 49092.4 1.87100 0.935499 0.353329i \(-0.114950\pi\)
0.935499 + 0.353329i \(0.114950\pi\)
\(884\) 0 0
\(885\) −10408.5 −0.395344
\(886\) −27599.5 −1.04653
\(887\) 9292.86 0.351774 0.175887 0.984410i \(-0.443721\pi\)
0.175887 + 0.984410i \(0.443721\pi\)
\(888\) 2430.12 0.0918348
\(889\) 4047.41 0.152695
\(890\) 60440.8 2.27638
\(891\) −4525.85 −0.170170
\(892\) −2108.05 −0.0791288
\(893\) −48112.0 −1.80292
\(894\) 52598.5 1.96774
\(895\) 19493.9 0.728055
\(896\) 3949.89 0.147273
\(897\) 0 0
\(898\) −943.252 −0.0350520
\(899\) 787.273 0.0292069
\(900\) −1839.16 −0.0681170
\(901\) 14136.7 0.522709
\(902\) 7845.88 0.289622
\(903\) −1405.53 −0.0517974
\(904\) 6881.46 0.253179
\(905\) 11932.3 0.438279
\(906\) 55811.5 2.04659
\(907\) 37688.7 1.37975 0.689875 0.723928i \(-0.257665\pi\)
0.689875 + 0.723928i \(0.257665\pi\)
\(908\) −949.559 −0.0347051
\(909\) −5789.42 −0.211246
\(910\) 0 0
\(911\) 33049.6 1.20196 0.600979 0.799265i \(-0.294778\pi\)
0.600979 + 0.799265i \(0.294778\pi\)
\(912\) −65923.1 −2.39357
\(913\) −4860.31 −0.176180
\(914\) −28870.0 −1.04479
\(915\) 87007.2 3.14357
\(916\) 1026.44 0.0370247
\(917\) 4917.06 0.177073
\(918\) −7906.43 −0.284260
\(919\) −23148.0 −0.830883 −0.415442 0.909620i \(-0.636373\pi\)
−0.415442 + 0.909620i \(0.636373\pi\)
\(920\) 38579.5 1.38253
\(921\) −13304.1 −0.475986
\(922\) 12933.4 0.461973
\(923\) 0 0
\(924\) −143.563 −0.00511134
\(925\) −1258.01 −0.0447169
\(926\) 9374.21 0.332673
\(927\) −1246.34 −0.0441588
\(928\) 605.131 0.0214056
\(929\) 23177.9 0.818561 0.409280 0.912409i \(-0.365780\pi\)
0.409280 + 0.912409i \(0.365780\pi\)
\(930\) 10135.7 0.357378
\(931\) 46673.3 1.64302
\(932\) −2355.98 −0.0828033
\(933\) −28325.1 −0.993916
\(934\) −13979.8 −0.489757
\(935\) 6735.44 0.235585
\(936\) 0 0
\(937\) −34574.7 −1.20545 −0.602724 0.797950i \(-0.705918\pi\)
−0.602724 + 0.797950i \(0.705918\pi\)
\(938\) −2873.18 −0.100014
\(939\) −56959.1 −1.97954
\(940\) 2737.54 0.0949880
\(941\) −41831.2 −1.44916 −0.724578 0.689192i \(-0.757966\pi\)
−0.724578 + 0.689192i \(0.757966\pi\)
\(942\) 53629.6 1.85493
\(943\) 29071.3 1.00392
\(944\) 5251.40 0.181058
\(945\) −3204.34 −0.110304
\(946\) 1734.42 0.0596097
\(947\) −27231.2 −0.934419 −0.467209 0.884147i \(-0.654741\pi\)
−0.467209 + 0.884147i \(0.654741\pi\)
\(948\) 5498.79 0.188389
\(949\) 0 0
\(950\) 36588.8 1.24958
\(951\) −124.534 −0.00424635
\(952\) 2952.50 0.100516
\(953\) 40939.4 1.39156 0.695781 0.718254i \(-0.255058\pi\)
0.695781 + 0.718254i \(0.255058\pi\)
\(954\) 33600.3 1.14030
\(955\) −25527.0 −0.864956
\(956\) −2188.87 −0.0740515
\(957\) 2179.81 0.0736292
\(958\) 31601.3 1.06575
\(959\) 4950.74 0.166703
\(960\) 63786.1 2.14447
\(961\) −28798.9 −0.966698
\(962\) 0 0
\(963\) 29839.0 0.998491
\(964\) 2094.60 0.0699820
\(965\) 19910.3 0.664182
\(966\) 7409.03 0.246772
\(967\) 46173.1 1.53550 0.767750 0.640750i \(-0.221376\pi\)
0.767750 + 0.640750i \(0.221376\pi\)
\(968\) 2821.78 0.0936937
\(969\) −45721.8 −1.51579
\(970\) −54648.9 −1.80894
\(971\) −5153.91 −0.170337 −0.0851683 0.996367i \(-0.527143\pi\)
−0.0851683 + 0.996367i \(0.527143\pi\)
\(972\) −2764.06 −0.0912111
\(973\) 97.8734 0.00322474
\(974\) −50069.0 −1.64714
\(975\) 0 0
\(976\) −43897.6 −1.43968
\(977\) −9692.13 −0.317378 −0.158689 0.987329i \(-0.550727\pi\)
−0.158689 + 0.987329i \(0.550727\pi\)
\(978\) 59026.5 1.92992
\(979\) 16380.2 0.534744
\(980\) −2655.68 −0.0865638
\(981\) −37443.3 −1.21863
\(982\) 20811.6 0.676299
\(983\) −32915.7 −1.06800 −0.534002 0.845483i \(-0.679313\pi\)
−0.534002 + 0.845483i \(0.679313\pi\)
\(984\) 48269.5 1.56380
\(985\) −52269.7 −1.69081
\(986\) −2814.48 −0.0909041
\(987\) 8373.97 0.270057
\(988\) 0 0
\(989\) 6426.54 0.206625
\(990\) 16008.9 0.513936
\(991\) 29477.9 0.944901 0.472451 0.881357i \(-0.343370\pi\)
0.472451 + 0.881357i \(0.343370\pi\)
\(992\) 762.560 0.0244066
\(993\) −10456.9 −0.334180
\(994\) 1740.26 0.0555310
\(995\) −12237.1 −0.389892
\(996\) −1877.28 −0.0597227
\(997\) −31944.4 −1.01473 −0.507366 0.861731i \(-0.669381\pi\)
−0.507366 + 0.861731i \(0.669381\pi\)
\(998\) 35268.4 1.11864
\(999\) 922.883 0.0292279
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 1859.4.a.a.1.1 2
13.12 even 2 11.4.a.a.1.2 2
39.38 odd 2 99.4.a.c.1.1 2
52.51 odd 2 176.4.a.i.1.2 2
65.12 odd 4 275.4.b.c.199.4 4
65.38 odd 4 275.4.b.c.199.1 4
65.64 even 2 275.4.a.b.1.1 2
91.90 odd 2 539.4.a.e.1.2 2
104.51 odd 2 704.4.a.n.1.1 2
104.77 even 2 704.4.a.p.1.2 2
143.25 even 10 121.4.c.c.9.2 8
143.38 even 10 121.4.c.c.3.1 8
143.51 odd 10 121.4.c.f.27.1 8
143.64 even 10 121.4.c.c.81.1 8
143.90 odd 10 121.4.c.f.81.2 8
143.103 even 10 121.4.c.c.27.2 8
143.116 odd 10 121.4.c.f.3.2 8
143.129 odd 10 121.4.c.f.9.1 8
143.142 odd 2 121.4.a.c.1.1 2
156.155 even 2 1584.4.a.bc.1.1 2
195.194 odd 2 2475.4.a.q.1.2 2
429.428 even 2 1089.4.a.v.1.2 2
572.571 even 2 1936.4.a.w.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.a.a.1.2 2 13.12 even 2
99.4.a.c.1.1 2 39.38 odd 2
121.4.a.c.1.1 2 143.142 odd 2
121.4.c.c.3.1 8 143.38 even 10
121.4.c.c.9.2 8 143.25 even 10
121.4.c.c.27.2 8 143.103 even 10
121.4.c.c.81.1 8 143.64 even 10
121.4.c.f.3.2 8 143.116 odd 10
121.4.c.f.9.1 8 143.129 odd 10
121.4.c.f.27.1 8 143.51 odd 10
121.4.c.f.81.2 8 143.90 odd 10
176.4.a.i.1.2 2 52.51 odd 2
275.4.a.b.1.1 2 65.64 even 2
275.4.b.c.199.1 4 65.38 odd 4
275.4.b.c.199.4 4 65.12 odd 4
539.4.a.e.1.2 2 91.90 odd 2
704.4.a.n.1.1 2 104.51 odd 2
704.4.a.p.1.2 2 104.77 even 2
1089.4.a.v.1.2 2 429.428 even 2
1584.4.a.bc.1.1 2 156.155 even 2
1859.4.a.a.1.1 2 1.1 even 1 trivial
1936.4.a.w.1.2 2 572.571 even 2
2475.4.a.q.1.2 2 195.194 odd 2