Properties

Label 1014.4.b.d.337.2
Level $1014$
Weight $4$
Character 1014.337
Analytic conductor $59.828$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1014.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(59.8279367458\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1014.337
Dual form 1014.4.b.d.337.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} -6.00000i q^{5} -6.00000i q^{6} -16.0000i q^{7} -8.00000i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} -6.00000i q^{5} -6.00000i q^{6} -16.0000i q^{7} -8.00000i q^{8} +9.00000 q^{9} +12.0000 q^{10} +12.0000i q^{11} +12.0000 q^{12} +32.0000 q^{14} +18.0000i q^{15} +16.0000 q^{16} +126.000 q^{17} +18.0000i q^{18} -20.0000i q^{19} +24.0000i q^{20} +48.0000i q^{21} -24.0000 q^{22} -168.000 q^{23} +24.0000i q^{24} +89.0000 q^{25} -27.0000 q^{27} +64.0000i q^{28} +30.0000 q^{29} -36.0000 q^{30} +88.0000i q^{31} +32.0000i q^{32} -36.0000i q^{33} +252.000i q^{34} -96.0000 q^{35} -36.0000 q^{36} +254.000i q^{37} +40.0000 q^{38} -48.0000 q^{40} -42.0000i q^{41} -96.0000 q^{42} +52.0000 q^{43} -48.0000i q^{44} -54.0000i q^{45} -336.000i q^{46} -96.0000i q^{47} -48.0000 q^{48} +87.0000 q^{49} +178.000i q^{50} -378.000 q^{51} +198.000 q^{53} -54.0000i q^{54} +72.0000 q^{55} -128.000 q^{56} +60.0000i q^{57} +60.0000i q^{58} -660.000i q^{59} -72.0000i q^{60} -538.000 q^{61} -176.000 q^{62} -144.000i q^{63} -64.0000 q^{64} +72.0000 q^{66} -884.000i q^{67} -504.000 q^{68} +504.000 q^{69} -192.000i q^{70} -792.000i q^{71} -72.0000i q^{72} +218.000i q^{73} -508.000 q^{74} -267.000 q^{75} +80.0000i q^{76} +192.000 q^{77} -520.000 q^{79} -96.0000i q^{80} +81.0000 q^{81} +84.0000 q^{82} +492.000i q^{83} -192.000i q^{84} -756.000i q^{85} +104.000i q^{86} -90.0000 q^{87} +96.0000 q^{88} +810.000i q^{89} +108.000 q^{90} +672.000 q^{92} -264.000i q^{93} +192.000 q^{94} -120.000 q^{95} -96.0000i q^{96} -1154.00i q^{97} +174.000i q^{98} +108.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} - 8q^{4} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} - 8q^{4} + 18q^{9} + 24q^{10} + 24q^{12} + 64q^{14} + 32q^{16} + 252q^{17} - 48q^{22} - 336q^{23} + 178q^{25} - 54q^{27} + 60q^{29} - 72q^{30} - 192q^{35} - 72q^{36} + 80q^{38} - 96q^{40} - 192q^{42} + 104q^{43} - 96q^{48} + 174q^{49} - 756q^{51} + 396q^{53} + 144q^{55} - 256q^{56} - 1076q^{61} - 352q^{62} - 128q^{64} + 144q^{66} - 1008q^{68} + 1008q^{69} - 1016q^{74} - 534q^{75} + 384q^{77} - 1040q^{79} + 162q^{81} + 168q^{82} - 180q^{87} + 192q^{88} + 216q^{90} + 1344q^{92} + 384q^{94} - 240q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) −3.00000 −0.577350
\(4\) −4.00000 −0.500000
\(5\) − 6.00000i − 0.536656i −0.963328 0.268328i \(-0.913529\pi\)
0.963328 0.268328i \(-0.0864711\pi\)
\(6\) − 6.00000i − 0.408248i
\(7\) − 16.0000i − 0.863919i −0.901893 0.431959i \(-0.857822\pi\)
0.901893 0.431959i \(-0.142178\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) 9.00000 0.333333
\(10\) 12.0000 0.379473
\(11\) 12.0000i 0.328921i 0.986384 + 0.164461i \(0.0525884\pi\)
−0.986384 + 0.164461i \(0.947412\pi\)
\(12\) 12.0000 0.288675
\(13\) 0 0
\(14\) 32.0000 0.610883
\(15\) 18.0000i 0.309839i
\(16\) 16.0000 0.250000
\(17\) 126.000 1.79762 0.898808 0.438342i \(-0.144434\pi\)
0.898808 + 0.438342i \(0.144434\pi\)
\(18\) 18.0000i 0.235702i
\(19\) − 20.0000i − 0.241490i −0.992684 0.120745i \(-0.961472\pi\)
0.992684 0.120745i \(-0.0385284\pi\)
\(20\) 24.0000i 0.268328i
\(21\) 48.0000i 0.498784i
\(22\) −24.0000 −0.232583
\(23\) −168.000 −1.52306 −0.761531 0.648129i \(-0.775552\pi\)
−0.761531 + 0.648129i \(0.775552\pi\)
\(24\) 24.0000i 0.204124i
\(25\) 89.0000 0.712000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 64.0000i 0.431959i
\(29\) 30.0000 0.192099 0.0960493 0.995377i \(-0.469379\pi\)
0.0960493 + 0.995377i \(0.469379\pi\)
\(30\) −36.0000 −0.219089
\(31\) 88.0000i 0.509847i 0.966961 + 0.254924i \(0.0820503\pi\)
−0.966961 + 0.254924i \(0.917950\pi\)
\(32\) 32.0000i 0.176777i
\(33\) − 36.0000i − 0.189903i
\(34\) 252.000i 1.27111i
\(35\) −96.0000 −0.463627
\(36\) −36.0000 −0.166667
\(37\) 254.000i 1.12858i 0.825578 + 0.564288i \(0.190849\pi\)
−0.825578 + 0.564288i \(0.809151\pi\)
\(38\) 40.0000 0.170759
\(39\) 0 0
\(40\) −48.0000 −0.189737
\(41\) − 42.0000i − 0.159983i −0.996796 0.0799914i \(-0.974511\pi\)
0.996796 0.0799914i \(-0.0254893\pi\)
\(42\) −96.0000 −0.352693
\(43\) 52.0000 0.184417 0.0922084 0.995740i \(-0.470607\pi\)
0.0922084 + 0.995740i \(0.470607\pi\)
\(44\) − 48.0000i − 0.164461i
\(45\) − 54.0000i − 0.178885i
\(46\) − 336.000i − 1.07697i
\(47\) − 96.0000i − 0.297937i −0.988842 0.148969i \(-0.952405\pi\)
0.988842 0.148969i \(-0.0475953\pi\)
\(48\) −48.0000 −0.144338
\(49\) 87.0000 0.253644
\(50\) 178.000i 0.503460i
\(51\) −378.000 −1.03785
\(52\) 0 0
\(53\) 198.000 0.513158 0.256579 0.966523i \(-0.417405\pi\)
0.256579 + 0.966523i \(0.417405\pi\)
\(54\) − 54.0000i − 0.136083i
\(55\) 72.0000 0.176518
\(56\) −128.000 −0.305441
\(57\) 60.0000i 0.139424i
\(58\) 60.0000i 0.135834i
\(59\) − 660.000i − 1.45635i −0.685391 0.728175i \(-0.740369\pi\)
0.685391 0.728175i \(-0.259631\pi\)
\(60\) − 72.0000i − 0.154919i
\(61\) −538.000 −1.12924 −0.564622 0.825350i \(-0.690978\pi\)
−0.564622 + 0.825350i \(0.690978\pi\)
\(62\) −176.000 −0.360516
\(63\) − 144.000i − 0.287973i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 72.0000 0.134282
\(67\) − 884.000i − 1.61191i −0.591979 0.805954i \(-0.701653\pi\)
0.591979 0.805954i \(-0.298347\pi\)
\(68\) −504.000 −0.898808
\(69\) 504.000 0.879340
\(70\) − 192.000i − 0.327834i
\(71\) − 792.000i − 1.32385i −0.749572 0.661923i \(-0.769740\pi\)
0.749572 0.661923i \(-0.230260\pi\)
\(72\) − 72.0000i − 0.117851i
\(73\) 218.000i 0.349520i 0.984611 + 0.174760i \(0.0559150\pi\)
−0.984611 + 0.174760i \(0.944085\pi\)
\(74\) −508.000 −0.798024
\(75\) −267.000 −0.411073
\(76\) 80.0000i 0.120745i
\(77\) 192.000 0.284161
\(78\) 0 0
\(79\) −520.000 −0.740564 −0.370282 0.928919i \(-0.620739\pi\)
−0.370282 + 0.928919i \(0.620739\pi\)
\(80\) − 96.0000i − 0.134164i
\(81\) 81.0000 0.111111
\(82\) 84.0000 0.113125
\(83\) 492.000i 0.650651i 0.945602 + 0.325325i \(0.105474\pi\)
−0.945602 + 0.325325i \(0.894526\pi\)
\(84\) − 192.000i − 0.249392i
\(85\) − 756.000i − 0.964703i
\(86\) 104.000i 0.130402i
\(87\) −90.0000 −0.110908
\(88\) 96.0000 0.116291
\(89\) 810.000i 0.964717i 0.875974 + 0.482359i \(0.160220\pi\)
−0.875974 + 0.482359i \(0.839780\pi\)
\(90\) 108.000 0.126491
\(91\) 0 0
\(92\) 672.000 0.761531
\(93\) − 264.000i − 0.294360i
\(94\) 192.000 0.210673
\(95\) −120.000 −0.129597
\(96\) − 96.0000i − 0.102062i
\(97\) − 1154.00i − 1.20795i −0.797004 0.603974i \(-0.793583\pi\)
0.797004 0.603974i \(-0.206417\pi\)
\(98\) 174.000i 0.179354i
\(99\) 108.000i 0.109640i
\(100\) −356.000 −0.356000
\(101\) 618.000 0.608845 0.304422 0.952537i \(-0.401537\pi\)
0.304422 + 0.952537i \(0.401537\pi\)
\(102\) − 756.000i − 0.733874i
\(103\) −128.000 −0.122449 −0.0612243 0.998124i \(-0.519501\pi\)
−0.0612243 + 0.998124i \(0.519501\pi\)
\(104\) 0 0
\(105\) 288.000 0.267675
\(106\) 396.000i 0.362858i
\(107\) −1476.00 −1.33355 −0.666777 0.745257i \(-0.732327\pi\)
−0.666777 + 0.745257i \(0.732327\pi\)
\(108\) 108.000 0.0962250
\(109\) − 1190.00i − 1.04570i −0.852425 0.522850i \(-0.824869\pi\)
0.852425 0.522850i \(-0.175131\pi\)
\(110\) 144.000i 0.124817i
\(111\) − 762.000i − 0.651584i
\(112\) − 256.000i − 0.215980i
\(113\) −462.000 −0.384613 −0.192307 0.981335i \(-0.561597\pi\)
−0.192307 + 0.981335i \(0.561597\pi\)
\(114\) −120.000 −0.0985880
\(115\) 1008.00i 0.817361i
\(116\) −120.000 −0.0960493
\(117\) 0 0
\(118\) 1320.00 1.02980
\(119\) − 2016.00i − 1.55300i
\(120\) 144.000 0.109545
\(121\) 1187.00 0.891811
\(122\) − 1076.00i − 0.798496i
\(123\) 126.000i 0.0923662i
\(124\) − 352.000i − 0.254924i
\(125\) − 1284.00i − 0.918756i
\(126\) 288.000 0.203628
\(127\) 2536.00 1.77192 0.885959 0.463763i \(-0.153501\pi\)
0.885959 + 0.463763i \(0.153501\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) −156.000 −0.106473
\(130\) 0 0
\(131\) 2292.00 1.52865 0.764324 0.644832i \(-0.223073\pi\)
0.764324 + 0.644832i \(0.223073\pi\)
\(132\) 144.000i 0.0949514i
\(133\) −320.000 −0.208628
\(134\) 1768.00 1.13979
\(135\) 162.000i 0.103280i
\(136\) − 1008.00i − 0.635554i
\(137\) − 726.000i − 0.452747i −0.974041 0.226374i \(-0.927313\pi\)
0.974041 0.226374i \(-0.0726870\pi\)
\(138\) 1008.00i 0.621787i
\(139\) 380.000 0.231879 0.115939 0.993256i \(-0.463012\pi\)
0.115939 + 0.993256i \(0.463012\pi\)
\(140\) 384.000 0.231814
\(141\) 288.000i 0.172014i
\(142\) 1584.00 0.936101
\(143\) 0 0
\(144\) 144.000 0.0833333
\(145\) − 180.000i − 0.103091i
\(146\) −436.000 −0.247148
\(147\) −261.000 −0.146442
\(148\) − 1016.00i − 0.564288i
\(149\) − 1590.00i − 0.874214i −0.899410 0.437107i \(-0.856003\pi\)
0.899410 0.437107i \(-0.143997\pi\)
\(150\) − 534.000i − 0.290673i
\(151\) 2432.00i 1.31068i 0.755332 + 0.655342i \(0.227476\pi\)
−0.755332 + 0.655342i \(0.772524\pi\)
\(152\) −160.000 −0.0853797
\(153\) 1134.00 0.599206
\(154\) 384.000i 0.200932i
\(155\) 528.000 0.273613
\(156\) 0 0
\(157\) 614.000 0.312118 0.156059 0.987748i \(-0.450121\pi\)
0.156059 + 0.987748i \(0.450121\pi\)
\(158\) − 1040.00i − 0.523658i
\(159\) −594.000 −0.296272
\(160\) 192.000 0.0948683
\(161\) 2688.00i 1.31580i
\(162\) 162.000i 0.0785674i
\(163\) − 1852.00i − 0.889938i −0.895546 0.444969i \(-0.853215\pi\)
0.895546 0.444969i \(-0.146785\pi\)
\(164\) 168.000i 0.0799914i
\(165\) −216.000 −0.101913
\(166\) −984.000 −0.460080
\(167\) − 2136.00i − 0.989752i −0.868964 0.494876i \(-0.835213\pi\)
0.868964 0.494876i \(-0.164787\pi\)
\(168\) 384.000 0.176347
\(169\) 0 0
\(170\) 1512.00 0.682148
\(171\) − 180.000i − 0.0804967i
\(172\) −208.000 −0.0922084
\(173\) −1758.00 −0.772591 −0.386296 0.922375i \(-0.626246\pi\)
−0.386296 + 0.922375i \(0.626246\pi\)
\(174\) − 180.000i − 0.0784239i
\(175\) − 1424.00i − 0.615110i
\(176\) 192.000i 0.0822304i
\(177\) 1980.00i 0.840824i
\(178\) −1620.00 −0.682158
\(179\) 540.000 0.225483 0.112742 0.993624i \(-0.464037\pi\)
0.112742 + 0.993624i \(0.464037\pi\)
\(180\) 216.000i 0.0894427i
\(181\) −1982.00 −0.813928 −0.406964 0.913444i \(-0.633412\pi\)
−0.406964 + 0.913444i \(0.633412\pi\)
\(182\) 0 0
\(183\) 1614.00 0.651969
\(184\) 1344.00i 0.538484i
\(185\) 1524.00 0.605658
\(186\) 528.000 0.208144
\(187\) 1512.00i 0.591275i
\(188\) 384.000i 0.148969i
\(189\) 432.000i 0.166261i
\(190\) − 240.000i − 0.0916391i
\(191\) −2688.00 −1.01831 −0.509154 0.860675i \(-0.670042\pi\)
−0.509154 + 0.860675i \(0.670042\pi\)
\(192\) 192.000 0.0721688
\(193\) − 2302.00i − 0.858557i −0.903172 0.429279i \(-0.858768\pi\)
0.903172 0.429279i \(-0.141232\pi\)
\(194\) 2308.00 0.854148
\(195\) 0 0
\(196\) −348.000 −0.126822
\(197\) − 4374.00i − 1.58190i −0.611880 0.790951i \(-0.709586\pi\)
0.611880 0.790951i \(-0.290414\pi\)
\(198\) −216.000 −0.0775275
\(199\) 1600.00 0.569955 0.284977 0.958534i \(-0.408014\pi\)
0.284977 + 0.958534i \(0.408014\pi\)
\(200\) − 712.000i − 0.251730i
\(201\) 2652.00i 0.930635i
\(202\) 1236.00i 0.430518i
\(203\) − 480.000i − 0.165958i
\(204\) 1512.00 0.518927
\(205\) −252.000 −0.0858558
\(206\) − 256.000i − 0.0865843i
\(207\) −1512.00 −0.507687
\(208\) 0 0
\(209\) 240.000 0.0794313
\(210\) 576.000i 0.189275i
\(211\) 3332.00 1.08713 0.543565 0.839367i \(-0.317074\pi\)
0.543565 + 0.839367i \(0.317074\pi\)
\(212\) −792.000 −0.256579
\(213\) 2376.00i 0.764323i
\(214\) − 2952.00i − 0.942965i
\(215\) − 312.000i − 0.0989685i
\(216\) 216.000i 0.0680414i
\(217\) 1408.00 0.440467
\(218\) 2380.00 0.739422
\(219\) − 654.000i − 0.201796i
\(220\) −288.000 −0.0882589
\(221\) 0 0
\(222\) 1524.00 0.460740
\(223\) − 2648.00i − 0.795171i −0.917565 0.397586i \(-0.869848\pi\)
0.917565 0.397586i \(-0.130152\pi\)
\(224\) 512.000 0.152721
\(225\) 801.000 0.237333
\(226\) − 924.000i − 0.271963i
\(227\) − 2244.00i − 0.656121i −0.944657 0.328061i \(-0.893605\pi\)
0.944657 0.328061i \(-0.106395\pi\)
\(228\) − 240.000i − 0.0697122i
\(229\) − 5650.00i − 1.63040i −0.579177 0.815202i \(-0.696626\pi\)
0.579177 0.815202i \(-0.303374\pi\)
\(230\) −2016.00 −0.577961
\(231\) −576.000 −0.164061
\(232\) − 240.000i − 0.0679171i
\(233\) −4698.00 −1.32093 −0.660464 0.750858i \(-0.729640\pi\)
−0.660464 + 0.750858i \(0.729640\pi\)
\(234\) 0 0
\(235\) −576.000 −0.159890
\(236\) 2640.00i 0.728175i
\(237\) 1560.00 0.427565
\(238\) 4032.00 1.09813
\(239\) 1200.00i 0.324776i 0.986727 + 0.162388i \(0.0519197\pi\)
−0.986727 + 0.162388i \(0.948080\pi\)
\(240\) 288.000i 0.0774597i
\(241\) − 718.000i − 0.191911i −0.995386 0.0959553i \(-0.969409\pi\)
0.995386 0.0959553i \(-0.0305906\pi\)
\(242\) 2374.00i 0.630605i
\(243\) −243.000 −0.0641500
\(244\) 2152.00 0.564622
\(245\) − 522.000i − 0.136120i
\(246\) −252.000 −0.0653127
\(247\) 0 0
\(248\) 704.000 0.180258
\(249\) − 1476.00i − 0.375653i
\(250\) 2568.00 0.649658
\(251\) −6012.00 −1.51185 −0.755924 0.654659i \(-0.772812\pi\)
−0.755924 + 0.654659i \(0.772812\pi\)
\(252\) 576.000i 0.143986i
\(253\) − 2016.00i − 0.500968i
\(254\) 5072.00i 1.25294i
\(255\) 2268.00i 0.556971i
\(256\) 256.000 0.0625000
\(257\) 2046.00 0.496599 0.248300 0.968683i \(-0.420128\pi\)
0.248300 + 0.968683i \(0.420128\pi\)
\(258\) − 312.000i − 0.0752879i
\(259\) 4064.00 0.974999
\(260\) 0 0
\(261\) 270.000 0.0640329
\(262\) 4584.00i 1.08092i
\(263\) −6072.00 −1.42363 −0.711817 0.702365i \(-0.752127\pi\)
−0.711817 + 0.702365i \(0.752127\pi\)
\(264\) −288.000 −0.0671408
\(265\) − 1188.00i − 0.275390i
\(266\) − 640.000i − 0.147522i
\(267\) − 2430.00i − 0.556980i
\(268\) 3536.00i 0.805954i
\(269\) −6930.00 −1.57074 −0.785371 0.619025i \(-0.787528\pi\)
−0.785371 + 0.619025i \(0.787528\pi\)
\(270\) −324.000 −0.0730297
\(271\) 1352.00i 0.303056i 0.988453 + 0.151528i \(0.0484194\pi\)
−0.988453 + 0.151528i \(0.951581\pi\)
\(272\) 2016.00 0.449404
\(273\) 0 0
\(274\) 1452.00 0.320141
\(275\) 1068.00i 0.234192i
\(276\) −2016.00 −0.439670
\(277\) 1186.00 0.257256 0.128628 0.991693i \(-0.458943\pi\)
0.128628 + 0.991693i \(0.458943\pi\)
\(278\) 760.000i 0.163963i
\(279\) 792.000i 0.169949i
\(280\) 768.000i 0.163917i
\(281\) 2442.00i 0.518425i 0.965820 + 0.259213i \(0.0834630\pi\)
−0.965820 + 0.259213i \(0.916537\pi\)
\(282\) −576.000 −0.121632
\(283\) −2828.00 −0.594018 −0.297009 0.954875i \(-0.595989\pi\)
−0.297009 + 0.954875i \(0.595989\pi\)
\(284\) 3168.00i 0.661923i
\(285\) 360.000 0.0748230
\(286\) 0 0
\(287\) −672.000 −0.138212
\(288\) 288.000i 0.0589256i
\(289\) 10963.0 2.23143
\(290\) 360.000 0.0728963
\(291\) 3462.00i 0.697409i
\(292\) − 872.000i − 0.174760i
\(293\) 4758.00i 0.948687i 0.880340 + 0.474344i \(0.157315\pi\)
−0.880340 + 0.474344i \(0.842685\pi\)
\(294\) − 522.000i − 0.103550i
\(295\) −3960.00 −0.781560
\(296\) 2032.00 0.399012
\(297\) − 324.000i − 0.0633010i
\(298\) 3180.00 0.618163
\(299\) 0 0
\(300\) 1068.00 0.205537
\(301\) − 832.000i − 0.159321i
\(302\) −4864.00 −0.926794
\(303\) −1854.00 −0.351517
\(304\) − 320.000i − 0.0603726i
\(305\) 3228.00i 0.606016i
\(306\) 2268.00i 0.423702i
\(307\) − 8476.00i − 1.57574i −0.615844 0.787868i \(-0.711185\pi\)
0.615844 0.787868i \(-0.288815\pi\)
\(308\) −768.000 −0.142081
\(309\) 384.000 0.0706958
\(310\) 1056.00i 0.193473i
\(311\) −4632.00 −0.844555 −0.422278 0.906467i \(-0.638769\pi\)
−0.422278 + 0.906467i \(0.638769\pi\)
\(312\) 0 0
\(313\) −4822.00 −0.870785 −0.435392 0.900241i \(-0.643390\pi\)
−0.435392 + 0.900241i \(0.643390\pi\)
\(314\) 1228.00i 0.220701i
\(315\) −864.000 −0.154542
\(316\) 2080.00 0.370282
\(317\) 3426.00i 0.607014i 0.952829 + 0.303507i \(0.0981575\pi\)
−0.952829 + 0.303507i \(0.901842\pi\)
\(318\) − 1188.00i − 0.209496i
\(319\) 360.000i 0.0631854i
\(320\) 384.000i 0.0670820i
\(321\) 4428.00 0.769928
\(322\) −5376.00 −0.930412
\(323\) − 2520.00i − 0.434107i
\(324\) −324.000 −0.0555556
\(325\) 0 0
\(326\) 3704.00 0.629281
\(327\) 3570.00i 0.603735i
\(328\) −336.000 −0.0565625
\(329\) −1536.00 −0.257393
\(330\) − 432.000i − 0.0720631i
\(331\) 2788.00i 0.462968i 0.972839 + 0.231484i \(0.0743581\pi\)
−0.972839 + 0.231484i \(0.925642\pi\)
\(332\) − 1968.00i − 0.325325i
\(333\) 2286.00i 0.376192i
\(334\) 4272.00 0.699861
\(335\) −5304.00 −0.865040
\(336\) 768.000i 0.124696i
\(337\) −434.000 −0.0701528 −0.0350764 0.999385i \(-0.511167\pi\)
−0.0350764 + 0.999385i \(0.511167\pi\)
\(338\) 0 0
\(339\) 1386.00 0.222057
\(340\) 3024.00i 0.482351i
\(341\) −1056.00 −0.167700
\(342\) 360.000 0.0569198
\(343\) − 6880.00i − 1.08305i
\(344\) − 416.000i − 0.0652012i
\(345\) − 3024.00i − 0.471903i
\(346\) − 3516.00i − 0.546304i
\(347\) 6684.00 1.03405 0.517026 0.855970i \(-0.327039\pi\)
0.517026 + 0.855970i \(0.327039\pi\)
\(348\) 360.000 0.0554541
\(349\) 2630.00i 0.403383i 0.979449 + 0.201692i \(0.0646438\pi\)
−0.979449 + 0.201692i \(0.935356\pi\)
\(350\) 2848.00 0.434949
\(351\) 0 0
\(352\) −384.000 −0.0581456
\(353\) 7422.00i 1.11907i 0.828805 + 0.559537i \(0.189021\pi\)
−0.828805 + 0.559537i \(0.810979\pi\)
\(354\) −3960.00 −0.594553
\(355\) −4752.00 −0.710451
\(356\) − 3240.00i − 0.482359i
\(357\) 6048.00i 0.896622i
\(358\) 1080.00i 0.159441i
\(359\) − 10440.0i − 1.53482i −0.641154 0.767412i \(-0.721544\pi\)
0.641154 0.767412i \(-0.278456\pi\)
\(360\) −432.000 −0.0632456
\(361\) 6459.00 0.941682
\(362\) − 3964.00i − 0.575534i
\(363\) −3561.00 −0.514887
\(364\) 0 0
\(365\) 1308.00 0.187572
\(366\) 3228.00i 0.461012i
\(367\) 10424.0 1.48264 0.741319 0.671153i \(-0.234200\pi\)
0.741319 + 0.671153i \(0.234200\pi\)
\(368\) −2688.00 −0.380765
\(369\) − 378.000i − 0.0533276i
\(370\) 3048.00i 0.428265i
\(371\) − 3168.00i − 0.443327i
\(372\) 1056.00i 0.147180i
\(373\) 3278.00 0.455036 0.227518 0.973774i \(-0.426939\pi\)
0.227518 + 0.973774i \(0.426939\pi\)
\(374\) −3024.00 −0.418094
\(375\) 3852.00i 0.530444i
\(376\) −768.000 −0.105337
\(377\) 0 0
\(378\) −864.000 −0.117564
\(379\) − 6140.00i − 0.832165i −0.909327 0.416083i \(-0.863403\pi\)
0.909327 0.416083i \(-0.136597\pi\)
\(380\) 480.000 0.0647986
\(381\) −7608.00 −1.02302
\(382\) − 5376.00i − 0.720053i
\(383\) 3072.00i 0.409848i 0.978778 + 0.204924i \(0.0656948\pi\)
−0.978778 + 0.204924i \(0.934305\pi\)
\(384\) 384.000i 0.0510310i
\(385\) − 1152.00i − 0.152497i
\(386\) 4604.00 0.607092
\(387\) 468.000 0.0614723
\(388\) 4616.00i 0.603974i
\(389\) −6150.00 −0.801587 −0.400794 0.916168i \(-0.631266\pi\)
−0.400794 + 0.916168i \(0.631266\pi\)
\(390\) 0 0
\(391\) −21168.0 −2.73788
\(392\) − 696.000i − 0.0896768i
\(393\) −6876.00 −0.882566
\(394\) 8748.00 1.11857
\(395\) 3120.00i 0.397428i
\(396\) − 432.000i − 0.0548202i
\(397\) − 106.000i − 0.0134005i −0.999978 0.00670024i \(-0.997867\pi\)
0.999978 0.00670024i \(-0.00213277\pi\)
\(398\) 3200.00i 0.403019i
\(399\) 960.000 0.120451
\(400\) 1424.00 0.178000
\(401\) − 1758.00i − 0.218929i −0.993991 0.109464i \(-0.965086\pi\)
0.993991 0.109464i \(-0.0349135\pi\)
\(402\) −5304.00 −0.658058
\(403\) 0 0
\(404\) −2472.00 −0.304422
\(405\) − 486.000i − 0.0596285i
\(406\) 960.000 0.117350
\(407\) −3048.00 −0.371213
\(408\) 3024.00i 0.366937i
\(409\) 3670.00i 0.443691i 0.975082 + 0.221846i \(0.0712081\pi\)
−0.975082 + 0.221846i \(0.928792\pi\)
\(410\) − 504.000i − 0.0607092i
\(411\) 2178.00i 0.261394i
\(412\) 512.000 0.0612243
\(413\) −10560.0 −1.25817
\(414\) − 3024.00i − 0.358989i
\(415\) 2952.00 0.349176
\(416\) 0 0
\(417\) −1140.00 −0.133875
\(418\) 480.000i 0.0561664i
\(419\) −9660.00 −1.12631 −0.563153 0.826353i \(-0.690412\pi\)
−0.563153 + 0.826353i \(0.690412\pi\)
\(420\) −1152.00 −0.133838
\(421\) − 8462.00i − 0.979602i −0.871834 0.489801i \(-0.837069\pi\)
0.871834 0.489801i \(-0.162931\pi\)
\(422\) 6664.00i 0.768717i
\(423\) − 864.000i − 0.0993123i
\(424\) − 1584.00i − 0.181429i
\(425\) 11214.0 1.27990
\(426\) −4752.00 −0.540458
\(427\) 8608.00i 0.975575i
\(428\) 5904.00 0.666777
\(429\) 0 0
\(430\) 624.000 0.0699813
\(431\) − 9792.00i − 1.09435i −0.837019 0.547174i \(-0.815704\pi\)
0.837019 0.547174i \(-0.184296\pi\)
\(432\) −432.000 −0.0481125
\(433\) 7342.00 0.814859 0.407430 0.913237i \(-0.366425\pi\)
0.407430 + 0.913237i \(0.366425\pi\)
\(434\) 2816.00i 0.311457i
\(435\) 540.000i 0.0595196i
\(436\) 4760.00i 0.522850i
\(437\) 3360.00i 0.367805i
\(438\) 1308.00 0.142691
\(439\) −10640.0 −1.15676 −0.578382 0.815766i \(-0.696316\pi\)
−0.578382 + 0.815766i \(0.696316\pi\)
\(440\) − 576.000i − 0.0624085i
\(441\) 783.000 0.0845481
\(442\) 0 0
\(443\) −17412.0 −1.86742 −0.933712 0.358024i \(-0.883451\pi\)
−0.933712 + 0.358024i \(0.883451\pi\)
\(444\) 3048.00i 0.325792i
\(445\) 4860.00 0.517722
\(446\) 5296.00 0.562271
\(447\) 4770.00i 0.504728i
\(448\) 1024.00i 0.107990i
\(449\) − 1710.00i − 0.179732i −0.995954 0.0898662i \(-0.971356\pi\)
0.995954 0.0898662i \(-0.0286440\pi\)
\(450\) 1602.00i 0.167820i
\(451\) 504.000 0.0526218
\(452\) 1848.00 0.192307
\(453\) − 7296.00i − 0.756724i
\(454\) 4488.00 0.463948
\(455\) 0 0
\(456\) 480.000 0.0492940
\(457\) 646.000i 0.0661239i 0.999453 + 0.0330619i \(0.0105259\pi\)
−0.999453 + 0.0330619i \(0.989474\pi\)
\(458\) 11300.0 1.15287
\(459\) −3402.00 −0.345952
\(460\) − 4032.00i − 0.408680i
\(461\) 6018.00i 0.607996i 0.952673 + 0.303998i \(0.0983216\pi\)
−0.952673 + 0.303998i \(0.901678\pi\)
\(462\) − 1152.00i − 0.116008i
\(463\) − 6712.00i − 0.673722i −0.941554 0.336861i \(-0.890635\pi\)
0.941554 0.336861i \(-0.109365\pi\)
\(464\) 480.000 0.0480247
\(465\) −1584.00 −0.157970
\(466\) − 9396.00i − 0.934037i
\(467\) −5364.00 −0.531512 −0.265756 0.964040i \(-0.585622\pi\)
−0.265756 + 0.964040i \(0.585622\pi\)
\(468\) 0 0
\(469\) −14144.0 −1.39256
\(470\) − 1152.00i − 0.113059i
\(471\) −1842.00 −0.180201
\(472\) −5280.00 −0.514898
\(473\) 624.000i 0.0606587i
\(474\) 3120.00i 0.302334i
\(475\) − 1780.00i − 0.171941i
\(476\) 8064.00i 0.776498i
\(477\) 1782.00 0.171053
\(478\) −2400.00 −0.229652
\(479\) 9840.00i 0.938624i 0.883032 + 0.469312i \(0.155498\pi\)
−0.883032 + 0.469312i \(0.844502\pi\)
\(480\) −576.000 −0.0547723
\(481\) 0 0
\(482\) 1436.00 0.135701
\(483\) − 8064.00i − 0.759678i
\(484\) −4748.00 −0.445905
\(485\) −6924.00 −0.648253
\(486\) − 486.000i − 0.0453609i
\(487\) − 1424.00i − 0.132500i −0.997803 0.0662501i \(-0.978896\pi\)
0.997803 0.0662501i \(-0.0211035\pi\)
\(488\) 4304.00i 0.399248i
\(489\) 5556.00i 0.513806i
\(490\) 1044.00 0.0962513
\(491\) 4548.00 0.418021 0.209011 0.977913i \(-0.432976\pi\)
0.209011 + 0.977913i \(0.432976\pi\)
\(492\) − 504.000i − 0.0461831i
\(493\) 3780.00 0.345320
\(494\) 0 0
\(495\) 648.000 0.0588393
\(496\) 1408.00i 0.127462i
\(497\) −12672.0 −1.14370
\(498\) 2952.00 0.265627
\(499\) − 6500.00i − 0.583126i −0.956552 0.291563i \(-0.905825\pi\)
0.956552 0.291563i \(-0.0941753\pi\)
\(500\) 5136.00i 0.459378i
\(501\) 6408.00i 0.571434i
\(502\) − 12024.0i − 1.06904i
\(503\) 12168.0 1.07862 0.539308 0.842108i \(-0.318686\pi\)
0.539308 + 0.842108i \(0.318686\pi\)
\(504\) −1152.00 −0.101814
\(505\) − 3708.00i − 0.326740i
\(506\) 4032.00 0.354238
\(507\) 0 0
\(508\) −10144.0 −0.885959
\(509\) 21090.0i 1.83654i 0.395957 + 0.918269i \(0.370413\pi\)
−0.395957 + 0.918269i \(0.629587\pi\)
\(510\) −4536.00 −0.393838
\(511\) 3488.00 0.301957
\(512\) 512.000i 0.0441942i
\(513\) 540.000i 0.0464748i
\(514\) 4092.00i 0.351149i
\(515\) 768.000i 0.0657129i
\(516\) 624.000 0.0532366
\(517\) 1152.00 0.0979979
\(518\) 8128.00i 0.689428i
\(519\) 5274.00 0.446056
\(520\) 0 0
\(521\) −5238.00 −0.440462 −0.220231 0.975448i \(-0.570681\pi\)
−0.220231 + 0.975448i \(0.570681\pi\)
\(522\) 540.000i 0.0452781i
\(523\) 8588.00 0.718025 0.359012 0.933333i \(-0.383114\pi\)
0.359012 + 0.933333i \(0.383114\pi\)
\(524\) −9168.00 −0.764324
\(525\) 4272.00i 0.355134i
\(526\) − 12144.0i − 1.00666i
\(527\) 11088.0i 0.916510i
\(528\) − 576.000i − 0.0474757i
\(529\) 16057.0 1.31972
\(530\) 2376.00 0.194730
\(531\) − 5940.00i − 0.485450i
\(532\) 1280.00 0.104314
\(533\) 0 0
\(534\) 4860.00 0.393844
\(535\) 8856.00i 0.715660i
\(536\) −7072.00 −0.569895
\(537\) −1620.00 −0.130183
\(538\) − 13860.0i − 1.11068i
\(539\) 1044.00i 0.0834291i
\(540\) − 648.000i − 0.0516398i
\(541\) 3062.00i 0.243338i 0.992571 + 0.121669i \(0.0388246\pi\)
−0.992571 + 0.121669i \(0.961175\pi\)
\(542\) −2704.00 −0.214293
\(543\) 5946.00 0.469921
\(544\) 4032.00i 0.317777i
\(545\) −7140.00 −0.561182
\(546\) 0 0
\(547\) −8476.00 −0.662537 −0.331268 0.943537i \(-0.607477\pi\)
−0.331268 + 0.943537i \(0.607477\pi\)
\(548\) 2904.00i 0.226374i
\(549\) −4842.00 −0.376414
\(550\) −2136.00 −0.165599
\(551\) − 600.000i − 0.0463899i
\(552\) − 4032.00i − 0.310894i
\(553\) 8320.00i 0.639787i
\(554\) 2372.00i 0.181907i
\(555\) −4572.00 −0.349677
\(556\) −1520.00 −0.115939
\(557\) − 12546.0i − 0.954383i −0.878799 0.477191i \(-0.841655\pi\)
0.878799 0.477191i \(-0.158345\pi\)
\(558\) −1584.00 −0.120172
\(559\) 0 0
\(560\) −1536.00 −0.115907
\(561\) − 4536.00i − 0.341373i
\(562\) −4884.00 −0.366582
\(563\) 12.0000 0.000898294 0 0.000449147 1.00000i \(-0.499857\pi\)
0.000449147 1.00000i \(0.499857\pi\)
\(564\) − 1152.00i − 0.0860070i
\(565\) 2772.00i 0.206405i
\(566\) − 5656.00i − 0.420034i
\(567\) − 1296.00i − 0.0959910i
\(568\) −6336.00 −0.468050
\(569\) −19290.0 −1.42123 −0.710614 0.703582i \(-0.751583\pi\)
−0.710614 + 0.703582i \(0.751583\pi\)
\(570\) 720.000i 0.0529079i
\(571\) 12148.0 0.890329 0.445165 0.895449i \(-0.353145\pi\)
0.445165 + 0.895449i \(0.353145\pi\)
\(572\) 0 0
\(573\) 8064.00 0.587920
\(574\) − 1344.00i − 0.0977308i
\(575\) −14952.0 −1.08442
\(576\) −576.000 −0.0416667
\(577\) 10366.0i 0.747907i 0.927447 + 0.373953i \(0.121998\pi\)
−0.927447 + 0.373953i \(0.878002\pi\)
\(578\) 21926.0i 1.57786i
\(579\) 6906.00i 0.495688i
\(580\) 720.000i 0.0515455i
\(581\) 7872.00 0.562109
\(582\) −6924.00 −0.493143
\(583\) 2376.00i 0.168789i
\(584\) 1744.00 0.123574
\(585\) 0 0
\(586\) −9516.00 −0.670823
\(587\) − 7644.00i − 0.537482i −0.963213 0.268741i \(-0.913393\pi\)
0.963213 0.268741i \(-0.0866075\pi\)
\(588\) 1044.00 0.0732208
\(589\) 1760.00 0.123123
\(590\) − 7920.00i − 0.552646i
\(591\) 13122.0i 0.913311i
\(592\) 4064.00i 0.282144i
\(593\) 8658.00i 0.599564i 0.954008 + 0.299782i \(0.0969139\pi\)
−0.954008 + 0.299782i \(0.903086\pi\)
\(594\) 648.000 0.0447605
\(595\) −12096.0 −0.833425
\(596\) 6360.00i 0.437107i
\(597\) −4800.00 −0.329064
\(598\) 0 0
\(599\) 25800.0 1.75987 0.879933 0.475098i \(-0.157587\pi\)
0.879933 + 0.475098i \(0.157587\pi\)
\(600\) 2136.00i 0.145336i
\(601\) 16202.0 1.09966 0.549828 0.835278i \(-0.314693\pi\)
0.549828 + 0.835278i \(0.314693\pi\)
\(602\) 1664.00 0.112657
\(603\) − 7956.00i − 0.537302i
\(604\) − 9728.00i − 0.655342i
\(605\) − 7122.00i − 0.478596i
\(606\) − 3708.00i − 0.248560i
\(607\) −24136.0 −1.61392 −0.806960 0.590605i \(-0.798889\pi\)
−0.806960 + 0.590605i \(0.798889\pi\)
\(608\) 640.000 0.0426898
\(609\) 1440.00i 0.0958157i
\(610\) −6456.00 −0.428518
\(611\) 0 0
\(612\) −4536.00 −0.299603
\(613\) 4642.00i 0.305854i 0.988237 + 0.152927i \(0.0488700\pi\)
−0.988237 + 0.152927i \(0.951130\pi\)
\(614\) 16952.0 1.11421
\(615\) 756.000 0.0495689
\(616\) − 1536.00i − 0.100466i
\(617\) 6726.00i 0.438863i 0.975628 + 0.219432i \(0.0704203\pi\)
−0.975628 + 0.219432i \(0.929580\pi\)
\(618\) 768.000i 0.0499895i
\(619\) − 21220.0i − 1.37787i −0.724821 0.688937i \(-0.758078\pi\)
0.724821 0.688937i \(-0.241922\pi\)
\(620\) −2112.00 −0.136806
\(621\) 4536.00 0.293113
\(622\) − 9264.00i − 0.597191i
\(623\) 12960.0 0.833437
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) − 9644.00i − 0.615738i
\(627\) −720.000 −0.0458597
\(628\) −2456.00 −0.156059
\(629\) 32004.0i 2.02875i
\(630\) − 1728.00i − 0.109278i
\(631\) 29792.0i 1.87956i 0.341783 + 0.939779i \(0.388969\pi\)
−0.341783 + 0.939779i \(0.611031\pi\)
\(632\) 4160.00i 0.261829i
\(633\) −9996.00 −0.627655
\(634\) −6852.00 −0.429223
\(635\) − 15216.0i − 0.950911i
\(636\) 2376.00 0.148136
\(637\) 0 0
\(638\) −720.000 −0.0446788
\(639\) − 7128.00i − 0.441282i
\(640\) −768.000 −0.0474342
\(641\) 10158.0 0.625923 0.312962 0.949766i \(-0.398679\pi\)
0.312962 + 0.949766i \(0.398679\pi\)
\(642\) 8856.00i 0.544421i
\(643\) − 29828.0i − 1.82940i −0.404138 0.914698i \(-0.632429\pi\)
0.404138 0.914698i \(-0.367571\pi\)
\(644\) − 10752.0i − 0.657901i
\(645\) 936.000i 0.0571395i
\(646\) 5040.00 0.306960
\(647\) −1944.00 −0.118124 −0.0590622 0.998254i \(-0.518811\pi\)
−0.0590622 + 0.998254i \(0.518811\pi\)
\(648\) − 648.000i − 0.0392837i
\(649\) 7920.00 0.479025
\(650\) 0 0
\(651\) −4224.00 −0.254304
\(652\) 7408.00i 0.444969i
\(653\) 26718.0 1.60116 0.800579 0.599227i \(-0.204525\pi\)
0.800579 + 0.599227i \(0.204525\pi\)
\(654\) −7140.00 −0.426905
\(655\) − 13752.0i − 0.820359i
\(656\) − 672.000i − 0.0399957i
\(657\) 1962.00i 0.116507i
\(658\) − 3072.00i − 0.182005i
\(659\) 4260.00 0.251815 0.125907 0.992042i \(-0.459816\pi\)
0.125907 + 0.992042i \(0.459816\pi\)
\(660\) 864.000 0.0509563
\(661\) 22862.0i 1.34528i 0.739971 + 0.672639i \(0.234839\pi\)
−0.739971 + 0.672639i \(0.765161\pi\)
\(662\) −5576.00 −0.327368
\(663\) 0 0
\(664\) 3936.00 0.230040
\(665\) 1920.00i 0.111962i
\(666\) −4572.00 −0.266008
\(667\) −5040.00 −0.292578
\(668\) 8544.00i 0.494876i
\(669\) 7944.00i 0.459092i
\(670\) − 10608.0i − 0.611676i
\(671\) − 6456.00i − 0.371432i
\(672\) −1536.00 −0.0881733
\(673\) 32542.0 1.86390 0.931948 0.362592i \(-0.118108\pi\)
0.931948 + 0.362592i \(0.118108\pi\)
\(674\) − 868.000i − 0.0496055i
\(675\) −2403.00 −0.137024
\(676\) 0 0
\(677\) 14214.0 0.806925 0.403463 0.914996i \(-0.367807\pi\)
0.403463 + 0.914996i \(0.367807\pi\)
\(678\) 2772.00i 0.157018i
\(679\) −18464.0 −1.04357
\(680\) −6048.00 −0.341074
\(681\) 6732.00i 0.378812i
\(682\) − 2112.00i − 0.118582i
\(683\) − 7092.00i − 0.397317i −0.980069 0.198659i \(-0.936341\pi\)
0.980069 0.198659i \(-0.0636585\pi\)
\(684\) 720.000i 0.0402484i
\(685\) −4356.00 −0.242970
\(686\) 13760.0 0.765830
\(687\) 16950.0i 0.941314i
\(688\) 832.000 0.0461042
\(689\) 0 0
\(690\) 6048.00 0.333686
\(691\) 13228.0i 0.728244i 0.931351 + 0.364122i \(0.118631\pi\)
−0.931351 + 0.364122i \(0.881369\pi\)
\(692\) 7032.00 0.386296
\(693\) 1728.00 0.0947205
\(694\) 13368.0i 0.731185i
\(695\) − 2280.00i − 0.124439i
\(696\) 720.000i 0.0392120i
\(697\) − 5292.00i − 0.287588i
\(698\) −5260.00 −0.285235
\(699\) 14094.0 0.762638
\(700\) 5696.00i 0.307555i
\(701\) −28062.0 −1.51196 −0.755982 0.654592i \(-0.772840\pi\)
−0.755982 + 0.654592i \(0.772840\pi\)
\(702\) 0 0
\(703\) 5080.00 0.272540
\(704\) − 768.000i − 0.0411152i
\(705\) 1728.00 0.0923124
\(706\) −14844.0 −0.791305
\(707\) − 9888.00i − 0.525992i
\(708\) − 7920.00i − 0.420412i
\(709\) − 27250.0i − 1.44343i −0.692188 0.721717i \(-0.743353\pi\)
0.692188 0.721717i \(-0.256647\pi\)
\(710\) − 9504.00i − 0.502364i
\(711\) −4680.00 −0.246855
\(712\) 6480.00 0.341079
\(713\) − 14784.0i − 0.776529i
\(714\) −12096.0 −0.634008
\(715\) 0 0
\(716\) −2160.00 −0.112742
\(717\) − 3600.00i − 0.187510i
\(718\) 20880.0 1.08529
\(719\) 14400.0 0.746912 0.373456 0.927648i \(-0.378173\pi\)
0.373456 + 0.927648i \(0.378173\pi\)
\(720\) − 864.000i − 0.0447214i
\(721\) 2048.00i 0.105786i
\(722\) 12918.0i 0.665870i
\(723\) 2154.00i 0.110800i
\(724\) 7928.00 0.406964
\(725\) 2670.00 0.136774
\(726\) − 7122.00i − 0.364080i
\(727\) −17984.0 −0.917455 −0.458727 0.888577i \(-0.651695\pi\)
−0.458727 + 0.888577i \(0.651695\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 2616.00i 0.132634i
\(731\) 6552.00 0.331511
\(732\) −6456.00 −0.325984
\(733\) − 16598.0i − 0.836373i −0.908361 0.418186i \(-0.862666\pi\)
0.908361 0.418186i \(-0.137334\pi\)
\(734\) 20848.0i 1.04838i
\(735\) 1566.00i 0.0785888i
\(736\) − 5376.00i − 0.269242i
\(737\) 10608.0 0.530191
\(738\) 756.000 0.0377083
\(739\) 1460.00i 0.0726752i 0.999340 + 0.0363376i \(0.0115692\pi\)
−0.999340 + 0.0363376i \(0.988431\pi\)
\(740\) −6096.00 −0.302829
\(741\) 0 0
\(742\) 6336.00 0.313480
\(743\) 30072.0i 1.48484i 0.669936 + 0.742419i \(0.266322\pi\)
−0.669936 + 0.742419i \(0.733678\pi\)
\(744\) −2112.00 −0.104072
\(745\) −9540.00 −0.469152
\(746\) 6556.00i 0.321759i
\(747\) 4428.00i 0.216884i
\(748\) − 6048.00i − 0.295637i
\(749\) 23616.0i 1.15208i
\(750\) −7704.00 −0.375080
\(751\) 18088.0 0.878882 0.439441 0.898271i \(-0.355177\pi\)
0.439441 + 0.898271i \(0.355177\pi\)
\(752\) − 1536.00i − 0.0744843i
\(753\) 18036.0 0.872866
\(754\) 0 0
\(755\) 14592.0 0.703387
\(756\) − 1728.00i − 0.0831306i
\(757\) 24734.0 1.18755 0.593773 0.804633i \(-0.297638\pi\)
0.593773 + 0.804633i \(0.297638\pi\)
\(758\) 12280.0 0.588430
\(759\) 6048.00i 0.289234i
\(760\) 960.000i 0.0458196i
\(761\) − 22278.0i − 1.06120i −0.847621 0.530602i \(-0.821966\pi\)
0.847621 0.530602i \(-0.178034\pi\)
\(762\) − 15216.0i − 0.723383i
\(763\) −19040.0 −0.903400
\(764\) 10752.0 0.509154
\(765\) − 6804.00i − 0.321568i
\(766\) −6144.00 −0.289806
\(767\) 0 0
\(768\) −768.000 −0.0360844
\(769\) − 16130.0i − 0.756388i −0.925726 0.378194i \(-0.876545\pi\)
0.925726 0.378194i \(-0.123455\pi\)
\(770\) 2304.00 0.107832
\(771\) −6138.00 −0.286712
\(772\) 9208.00i 0.429279i
\(773\) − 29718.0i − 1.38277i −0.722486 0.691386i \(-0.757001\pi\)
0.722486 0.691386i \(-0.242999\pi\)
\(774\) 936.000i 0.0434675i
\(775\) 7832.00i 0.363011i
\(776\) −9232.00 −0.427074
\(777\) −12192.0 −0.562916
\(778\) − 12300.0i − 0.566808i
\(779\) −840.000 −0.0386343
\(780\) 0 0
\(781\) 9504.00 0.435442
\(782\) − 42336.0i − 1.93597i
\(783\) −810.000 −0.0369694
\(784\) 1392.00 0.0634111
\(785\) − 3684.00i − 0.167500i
\(786\) − 13752.0i − 0.624068i
\(787\) 9524.00i 0.431377i 0.976462 + 0.215689i \(0.0691996\pi\)
−0.976462 + 0.215689i \(0.930800\pi\)
\(788\) 17496.0i 0.790951i
\(789\) 18216.0 0.821935
\(790\) −6240.00 −0.281024
\(791\) 7392.00i 0.332275i
\(792\) 864.000 0.0387638
\(793\) 0 0
\(794\) 212.000 0.00947556
\(795\) 3564.00i 0.158996i
\(796\) −6400.00 −0.284977
\(797\) 33906.0 1.50692 0.753458 0.657496i \(-0.228384\pi\)
0.753458 + 0.657496i \(0.228384\pi\)
\(798\) 1920.00i 0.0851720i
\(799\) − 12096.0i − 0.535577i
\(800\) 2848.00i 0.125865i
\(801\) 7290.00i 0.321572i
\(802\) 3516.00 0.154806
\(803\) −2616.00 −0.114965
\(804\) − 10608.0i − 0.465318i
\(805\) 16128.0 0.706133
\(806\) 0 0
\(807\) 20790.0 0.906868
\(808\) − 4944.00i − 0.215259i
\(809\) −630.000 −0.0273790 −0.0136895 0.999906i \(-0.504358\pi\)
−0.0136895 + 0.999906i \(0.504358\pi\)
\(810\) 972.000 0.0421637
\(811\) 20788.0i 0.900081i 0.893008 + 0.450040i \(0.148590\pi\)
−0.893008 + 0.450040i \(0.851410\pi\)
\(812\) 1920.00i 0.0829788i
\(813\) − 4056.00i − 0.174969i
\(814\) − 6096.00i − 0.262487i
\(815\) −11112.0 −0.477591
\(816\) −6048.00 −0.259464
\(817\) − 1040.00i − 0.0445349i
\(818\) −7340.00 −0.313737
\(819\) 0 0
\(820\) 1008.00 0.0429279
\(821\) 43098.0i 1.83207i 0.401097 + 0.916036i \(0.368629\pi\)
−0.401097 + 0.916036i \(0.631371\pi\)
\(822\) −4356.00 −0.184833
\(823\) 14272.0 0.604484 0.302242 0.953231i \(-0.402265\pi\)
0.302242 + 0.953231i \(0.402265\pi\)
\(824\) 1024.00i 0.0432921i
\(825\) − 3204.00i − 0.135211i
\(826\) − 21120.0i − 0.889660i
\(827\) 13644.0i 0.573698i 0.957976 + 0.286849i \(0.0926078\pi\)
−0.957976 + 0.286849i \(0.907392\pi\)
\(828\) 6048.00 0.253844
\(829\) 2410.00 0.100968 0.0504842 0.998725i \(-0.483924\pi\)
0.0504842 + 0.998725i \(0.483924\pi\)
\(830\) 5904.00i 0.246905i
\(831\) −3558.00 −0.148527
\(832\) 0 0
\(833\) 10962.0 0.455955
\(834\) − 2280.00i − 0.0946642i
\(835\) −12816.0 −0.531157
\(836\) −960.000 −0.0397157
\(837\) − 2376.00i − 0.0981202i
\(838\) − 19320.0i − 0.796418i
\(839\) 23160.0i 0.953006i 0.879173 + 0.476503i \(0.158096\pi\)
−0.879173 + 0.476503i \(0.841904\pi\)
\(840\) − 2304.00i − 0.0946376i
\(841\) −23489.0 −0.963098
\(842\) 16924.0 0.692684
\(843\) − 7326.00i − 0.299313i
\(844\) −13328.0 −0.543565
\(845\) 0 0
\(846\) 1728.00 0.0702244
\(847\) − 18992.0i − 0.770452i
\(848\) 3168.00 0.128290
\(849\) 8484.00 0.342957
\(850\) 22428.0i 0.905028i
\(851\) − 42672.0i − 1.71889i
\(852\) − 9504.00i − 0.382162i
\(853\) 32078.0i 1.28761i 0.765190 + 0.643804i \(0.222645\pi\)
−0.765190 + 0.643804i \(0.777355\pi\)
\(854\) −17216.0 −0.689835
\(855\) −1080.00 −0.0431991
\(856\) 11808.0i 0.471483i
\(857\) 14406.0 0.574212 0.287106 0.957899i \(-0.407307\pi\)
0.287106 + 0.957899i \(0.407307\pi\)
\(858\) 0 0
\(859\) 30620.0 1.21623 0.608115 0.793849i \(-0.291926\pi\)
0.608115 + 0.793849i \(0.291926\pi\)
\(860\) 1248.00i 0.0494842i
\(861\) 2016.00 0.0797969
\(862\) 19584.0 0.773821
\(863\) − 17568.0i − 0.692957i −0.938058 0.346478i \(-0.887377\pi\)
0.938058 0.346478i \(-0.112623\pi\)
\(864\) − 864.000i − 0.0340207i
\(865\) 10548.0i 0.414616i
\(866\) 14684.0i 0.576192i
\(867\) −32889.0 −1.28831
\(868\) −5632.00 −0.220233
\(869\) − 6240.00i − 0.243587i
\(870\) −1080.00 −0.0420867
\(871\) 0 0
\(872\) −9520.00 −0.369711
\(873\) − 10386.0i − 0.402649i
\(874\) −6720.00 −0.260077
\(875\) −20544.0 −0.793730
\(876\) 2616.00i 0.100898i
\(877\) 21706.0i 0.835758i 0.908503 + 0.417879i \(0.137226\pi\)
−0.908503 + 0.417879i \(0.862774\pi\)
\(878\) − 21280.0i − 0.817956i
\(879\) − 14274.0i − 0.547725i
\(880\) 1152.00 0.0441294
\(881\) 14958.0 0.572018 0.286009 0.958227i \(-0.407671\pi\)
0.286009 + 0.958227i \(0.407671\pi\)
\(882\) 1566.00i 0.0597845i
\(883\) 32812.0 1.25052 0.625261 0.780415i \(-0.284992\pi\)
0.625261 + 0.780415i \(0.284992\pi\)
\(884\) 0 0
\(885\) 11880.0 0.451234
\(886\) − 34824.0i − 1.32047i
\(887\) −38856.0 −1.47086 −0.735432 0.677598i \(-0.763021\pi\)
−0.735432 + 0.677598i \(0.763021\pi\)
\(888\) −6096.00 −0.230370
\(889\) − 40576.0i − 1.53079i
\(890\) 9720.00i 0.366084i
\(891\) 972.000i 0.0365468i
\(892\) 10592.0i 0.397586i
\(893\) −1920.00 −0.0719489
\(894\) −9540.00 −0.356896
\(895\) − 3240.00i − 0.121007i
\(896\) −2048.00 −0.0763604
\(897\) 0 0
\(898\) 3420.00 0.127090
\(899\) 2640.00i 0.0979410i
\(900\) −3204.00 −0.118667
\(901\) 24948.0 0.922462
\(902\) 1008.00i 0.0372092i
\(903\) 2496.00i 0.0919841i
\(904\) 3696.00i 0.135981i
\(905\) 11892.0i 0.436799i
\(906\) 14592.0 0.535085
\(907\) 28276.0 1.03516 0.517579 0.855635i \(-0.326833\pi\)
0.517579 + 0.855635i \(0.326833\pi\)
\(908\) 8976.00i 0.328061i
\(909\) 5562.00 0.202948
\(910\) 0 0
\(911\) 8112.00 0.295019 0.147510 0.989061i \(-0.452874\pi\)
0.147510 + 0.989061i \(0.452874\pi\)
\(912\) 960.000i 0.0348561i
\(913\) −5904.00 −0.214013
\(914\) −1292.00 −0.0467566
\(915\) − 9684.00i − 0.349883i
\(916\) 22600.0i 0.815202i
\(917\) − 36672.0i − 1.32063i
\(918\) − 6804.00i − 0.244625i
\(919\) −26080.0 −0.936126 −0.468063 0.883695i \(-0.655048\pi\)
−0.468063 + 0.883695i \(0.655048\pi\)
\(920\) 8064.00 0.288981
\(921\) 25428.0i 0.909751i
\(922\) −12036.0 −0.429918
\(923\) 0 0
\(924\) 2304.00 0.0820303
\(925\) 22606.0i 0.803547i
\(926\) 13424.0 0.476393
\(927\) −1152.00 −0.0408162
\(928\) 960.000i 0.0339586i
\(929\) − 49170.0i − 1.73651i −0.496120 0.868254i \(-0.665243\pi\)
0.496120 0.868254i \(-0.334757\pi\)
\(930\) − 3168.00i − 0.111702i
\(931\) − 1740.00i − 0.0612526i
\(932\) 18792.0 0.660464
\(933\) 13896.0 0.487604
\(934\) − 10728.0i − 0.375836i
\(935\) 9072.00 0.317311
\(936\) 0 0
\(937\) 48314.0 1.68447 0.842236 0.539110i \(-0.181239\pi\)
0.842236 + 0.539110i \(0.181239\pi\)
\(938\) − 28288.0i − 0.984687i
\(939\) 14466.0 0.502748
\(940\) 2304.00 0.0799449
\(941\) − 34782.0i − 1.20495i −0.798137 0.602477i \(-0.794181\pi\)
0.798137 0.602477i \(-0.205819\pi\)
\(942\) − 3684.00i − 0.127422i
\(943\) 7056.00i 0.243664i
\(944\) − 10560.0i − 0.364088i
\(945\) 2592.00 0.0892251
\(946\) −1248.00 −0.0428922
\(947\) − 25116.0i − 0.861838i −0.902391 0.430919i \(-0.858190\pi\)
0.902391 0.430919i \(-0.141810\pi\)
\(948\) −6240.00 −0.213782
\(949\) 0 0
\(950\) 3560.00 0.121581
\(951\) − 10278.0i − 0.350460i
\(952\) −16128.0 −0.549067
\(953\) 15462.0 0.525565 0.262782 0.964855i \(-0.415360\pi\)
0.262782 + 0.964855i \(0.415360\pi\)
\(954\) 3564.00i 0.120953i
\(955\) 16128.0i 0.546481i
\(956\) − 4800.00i − 0.162388i
\(957\) − 1080.00i − 0.0364801i
\(958\) −19680.0 −0.663708
\(959\) −11616.0 −0.391137
\(960\) − 1152.00i − 0.0387298i
\(961\) 22047.0 0.740056
\(962\) 0 0
\(963\) −13284.0 −0.444518
\(964\) 2872.00i 0.0959553i
\(965\) −13812.0 −0.460750
\(966\) 16128.0 0.537174
\(967\) 736.000i 0.0244759i 0.999925 + 0.0122379i \(0.00389555\pi\)
−0.999925 + 0.0122379i \(0.996104\pi\)
\(968\) − 9496.00i − 0.315303i
\(969\) 7560.00i 0.250632i
\(970\) − 13848.0i − 0.458384i
\(971\) −29268.0 −0.967307 −0.483653 0.875260i \(-0.660690\pi\)
−0.483653 + 0.875260i \(0.660690\pi\)
\(972\) 972.000 0.0320750
\(973\) − 6080.00i − 0.200325i
\(974\) 2848.00 0.0936918
\(975\) 0 0
\(976\) −8608.00 −0.282311
\(977\) − 16674.0i − 0.546007i −0.962013 0.273003i \(-0.911983\pi\)
0.962013 0.273003i \(-0.0880170\pi\)
\(978\) −11112.0 −0.363316
\(979\) −9720.00 −0.317316
\(980\) 2088.00i 0.0680599i
\(981\) − 10710.0i − 0.348567i
\(982\) 9096.00i 0.295586i
\(983\) − 31272.0i − 1.01467i −0.861749 0.507336i \(-0.830630\pi\)
0.861749 0.507336i \(-0.169370\pi\)
\(984\) 1008.00 0.0326564
\(985\) −26244.0 −0.848937
\(986\) 7560.00i 0.244178i
\(987\) 4608.00 0.148606
\(988\) 0 0
\(989\) −8736.00 −0.280878
\(990\) 1296.00i 0.0416056i
\(991\) −15928.0 −0.510565 −0.255282 0.966867i \(-0.582168\pi\)
−0.255282 + 0.966867i \(0.582168\pi\)
\(992\) −2816.00 −0.0901291
\(993\) − 8364.00i − 0.267295i
\(994\) − 25344.0i − 0.808715i
\(995\) − 9600.00i − 0.305870i
\(996\) 5904.00i 0.187827i
\(997\) 42014.0 1.33460 0.667300 0.744789i \(-0.267450\pi\)
0.667300 + 0.744789i \(0.267450\pi\)
\(998\) 13000.0 0.412332
\(999\) − 6858.00i − 0.217195i
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 1014.4.b.d.337.2 2
13.5 odd 4 1014.4.a.g.1.1 1
13.8 odd 4 6.4.a.a.1.1 1
13.12 even 2 inner 1014.4.b.d.337.1 2
39.8 even 4 18.4.a.a.1.1 1
52.47 even 4 48.4.a.c.1.1 1
65.8 even 4 150.4.c.d.49.2 2
65.34 odd 4 150.4.a.i.1.1 1
65.47 even 4 150.4.c.d.49.1 2
91.34 even 4 294.4.a.e.1.1 1
91.47 even 12 294.4.e.g.67.1 2
91.60 odd 12 294.4.e.h.79.1 2
91.73 even 12 294.4.e.g.79.1 2
91.86 odd 12 294.4.e.h.67.1 2
104.21 odd 4 192.4.a.i.1.1 1
104.99 even 4 192.4.a.c.1.1 1
117.34 odd 12 162.4.c.f.109.1 2
117.47 even 12 162.4.c.c.109.1 2
117.86 even 12 162.4.c.c.55.1 2
117.112 odd 12 162.4.c.f.55.1 2
143.21 even 4 726.4.a.f.1.1 1
156.47 odd 4 144.4.a.c.1.1 1
195.8 odd 4 450.4.c.e.199.1 2
195.47 odd 4 450.4.c.e.199.2 2
195.164 even 4 450.4.a.h.1.1 1
208.21 odd 4 768.4.d.n.385.2 2
208.99 even 4 768.4.d.c.385.2 2
208.125 odd 4 768.4.d.n.385.1 2
208.203 even 4 768.4.d.c.385.1 2
221.203 odd 4 1734.4.a.d.1.1 1
247.151 even 4 2166.4.a.i.1.1 1
260.47 odd 4 1200.4.f.j.49.1 2
260.99 even 4 1200.4.a.b.1.1 1
260.203 odd 4 1200.4.f.j.49.2 2
273.47 odd 12 882.4.g.f.361.1 2
273.86 even 12 882.4.g.i.361.1 2
273.125 odd 4 882.4.a.n.1.1 1
273.164 odd 12 882.4.g.f.667.1 2
273.242 even 12 882.4.g.i.667.1 2
312.125 even 4 576.4.a.q.1.1 1
312.203 odd 4 576.4.a.r.1.1 1
364.307 odd 4 2352.4.a.e.1.1 1
429.164 odd 4 2178.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.4.a.a.1.1 1 13.8 odd 4
18.4.a.a.1.1 1 39.8 even 4
48.4.a.c.1.1 1 52.47 even 4
144.4.a.c.1.1 1 156.47 odd 4
150.4.a.i.1.1 1 65.34 odd 4
150.4.c.d.49.1 2 65.47 even 4
150.4.c.d.49.2 2 65.8 even 4
162.4.c.c.55.1 2 117.86 even 12
162.4.c.c.109.1 2 117.47 even 12
162.4.c.f.55.1 2 117.112 odd 12
162.4.c.f.109.1 2 117.34 odd 12
192.4.a.c.1.1 1 104.99 even 4
192.4.a.i.1.1 1 104.21 odd 4
294.4.a.e.1.1 1 91.34 even 4
294.4.e.g.67.1 2 91.47 even 12
294.4.e.g.79.1 2 91.73 even 12
294.4.e.h.67.1 2 91.86 odd 12
294.4.e.h.79.1 2 91.60 odd 12
450.4.a.h.1.1 1 195.164 even 4
450.4.c.e.199.1 2 195.8 odd 4
450.4.c.e.199.2 2 195.47 odd 4
576.4.a.q.1.1 1 312.125 even 4
576.4.a.r.1.1 1 312.203 odd 4
726.4.a.f.1.1 1 143.21 even 4
768.4.d.c.385.1 2 208.203 even 4
768.4.d.c.385.2 2 208.99 even 4
768.4.d.n.385.1 2 208.125 odd 4
768.4.d.n.385.2 2 208.21 odd 4
882.4.a.n.1.1 1 273.125 odd 4
882.4.g.f.361.1 2 273.47 odd 12
882.4.g.f.667.1 2 273.164 odd 12
882.4.g.i.361.1 2 273.86 even 12
882.4.g.i.667.1 2 273.242 even 12
1014.4.a.g.1.1 1 13.5 odd 4
1014.4.b.d.337.1 2 13.12 even 2 inner
1014.4.b.d.337.2 2 1.1 even 1 trivial
1200.4.a.b.1.1 1 260.99 even 4
1200.4.f.j.49.1 2 260.47 odd 4
1200.4.f.j.49.2 2 260.203 odd 4
1734.4.a.d.1.1 1 221.203 odd 4
2166.4.a.i.1.1 1 247.151 even 4
2178.4.a.e.1.1 1 429.164 odd 4
2352.4.a.e.1.1 1 364.307 odd 4