Properties

Label 975.2.c.h.274.1
Level $975$
Weight $2$
Character 975.274
Analytic conductor $7.785$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 975.274
Dual form 975.2.c.h.274.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421i q^{2} -1.00000i q^{3} -3.82843 q^{4} -2.41421 q^{6} -2.82843i q^{7} +4.41421i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-2.41421i q^{2} -1.00000i q^{3} -3.82843 q^{4} -2.41421 q^{6} -2.82843i q^{7} +4.41421i q^{8} -1.00000 q^{9} -2.00000 q^{11} +3.82843i q^{12} +1.00000i q^{13} -6.82843 q^{14} +3.00000 q^{16} -3.65685i q^{17} +2.41421i q^{18} -2.82843 q^{19} -2.82843 q^{21} +4.82843i q^{22} +4.00000i q^{23} +4.41421 q^{24} +2.41421 q^{26} +1.00000i q^{27} +10.8284i q^{28} -2.00000 q^{29} -6.82843 q^{31} +1.58579i q^{32} +2.00000i q^{33} -8.82843 q^{34} +3.82843 q^{36} +3.65685i q^{37} +6.82843i q^{38} +1.00000 q^{39} +10.8284 q^{41} +6.82843i q^{42} -9.65685i q^{43} +7.65685 q^{44} +9.65685 q^{46} -0.343146i q^{47} -3.00000i q^{48} -1.00000 q^{49} -3.65685 q^{51} -3.82843i q^{52} +2.00000i q^{53} +2.41421 q^{54} +12.4853 q^{56} +2.82843i q^{57} +4.82843i q^{58} +3.65685 q^{59} -9.31371 q^{61} +16.4853i q^{62} +2.82843i q^{63} +9.82843 q^{64} +4.82843 q^{66} +1.17157i q^{67} +14.0000i q^{68} +4.00000 q^{69} +2.00000 q^{71} -4.41421i q^{72} -11.6569i q^{73} +8.82843 q^{74} +10.8284 q^{76} +5.65685i q^{77} -2.41421i q^{78} -11.3137 q^{79} +1.00000 q^{81} -26.1421i q^{82} +7.65685i q^{83} +10.8284 q^{84} -23.3137 q^{86} +2.00000i q^{87} -8.82843i q^{88} -9.17157 q^{89} +2.82843 q^{91} -15.3137i q^{92} +6.82843i q^{93} -0.828427 q^{94} +1.58579 q^{96} -7.65685i q^{97} +2.41421i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} - 8 q^{11} - 16 q^{14} + 12 q^{16} + 12 q^{24} + 4 q^{26} - 8 q^{29} - 16 q^{31} - 24 q^{34} + 4 q^{36} + 4 q^{39} + 32 q^{41} + 8 q^{44} + 16 q^{46} - 4 q^{49} + 8 q^{51} + 4 q^{54} + 16 q^{56} - 8 q^{59} + 8 q^{61} + 28 q^{64} + 8 q^{66} + 16 q^{69} + 8 q^{71} + 24 q^{74} + 32 q^{76} + 4 q^{81} + 32 q^{84} - 48 q^{86} - 48 q^{89} + 8 q^{94} + 12 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.41421i − 1.70711i −0.521005 0.853553i \(-0.674443\pi\)
0.521005 0.853553i \(-0.325557\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −3.82843 −1.91421
\(5\) 0 0
\(6\) −2.41421 −0.985599
\(7\) − 2.82843i − 1.06904i −0.845154 0.534522i \(-0.820491\pi\)
0.845154 0.534522i \(-0.179509\pi\)
\(8\) 4.41421i 1.56066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 3.82843i 1.10517i
\(13\) 1.00000i 0.277350i
\(14\) −6.82843 −1.82497
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) − 3.65685i − 0.886917i −0.896295 0.443459i \(-0.853751\pi\)
0.896295 0.443459i \(-0.146249\pi\)
\(18\) 2.41421i 0.569036i
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) −2.82843 −0.617213
\(22\) 4.82843i 1.02942i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 4.41421 0.901048
\(25\) 0 0
\(26\) 2.41421 0.473466
\(27\) 1.00000i 0.192450i
\(28\) 10.8284i 2.04638i
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −6.82843 −1.22642 −0.613211 0.789919i \(-0.710122\pi\)
−0.613211 + 0.789919i \(0.710122\pi\)
\(32\) 1.58579i 0.280330i
\(33\) 2.00000i 0.348155i
\(34\) −8.82843 −1.51406
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) 3.65685i 0.601183i 0.953753 + 0.300592i \(0.0971841\pi\)
−0.953753 + 0.300592i \(0.902816\pi\)
\(38\) 6.82843i 1.10772i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 10.8284 1.69112 0.845558 0.533883i \(-0.179268\pi\)
0.845558 + 0.533883i \(0.179268\pi\)
\(42\) 6.82843i 1.05365i
\(43\) − 9.65685i − 1.47266i −0.676625 0.736328i \(-0.736558\pi\)
0.676625 0.736328i \(-0.263442\pi\)
\(44\) 7.65685 1.15431
\(45\) 0 0
\(46\) 9.65685 1.42383
\(47\) − 0.343146i − 0.0500530i −0.999687 0.0250265i \(-0.992033\pi\)
0.999687 0.0250265i \(-0.00796701\pi\)
\(48\) − 3.00000i − 0.433013i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −3.65685 −0.512062
\(52\) − 3.82843i − 0.530907i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 2.41421 0.328533
\(55\) 0 0
\(56\) 12.4853 1.66842
\(57\) 2.82843i 0.374634i
\(58\) 4.82843i 0.634004i
\(59\) 3.65685 0.476082 0.238041 0.971255i \(-0.423495\pi\)
0.238041 + 0.971255i \(0.423495\pi\)
\(60\) 0 0
\(61\) −9.31371 −1.19250 −0.596249 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) 16.4853i 2.09363i
\(63\) 2.82843i 0.356348i
\(64\) 9.82843 1.22855
\(65\) 0 0
\(66\) 4.82843 0.594338
\(67\) 1.17157i 0.143130i 0.997436 + 0.0715652i \(0.0227994\pi\)
−0.997436 + 0.0715652i \(0.977201\pi\)
\(68\) 14.0000i 1.69775i
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) − 4.41421i − 0.520220i
\(73\) − 11.6569i − 1.36433i −0.731198 0.682166i \(-0.761038\pi\)
0.731198 0.682166i \(-0.238962\pi\)
\(74\) 8.82843 1.02628
\(75\) 0 0
\(76\) 10.8284 1.24211
\(77\) 5.65685i 0.644658i
\(78\) − 2.41421i − 0.273356i
\(79\) −11.3137 −1.27289 −0.636446 0.771321i \(-0.719596\pi\)
−0.636446 + 0.771321i \(0.719596\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 26.1421i − 2.88692i
\(83\) 7.65685i 0.840449i 0.907420 + 0.420224i \(0.138049\pi\)
−0.907420 + 0.420224i \(0.861951\pi\)
\(84\) 10.8284 1.18148
\(85\) 0 0
\(86\) −23.3137 −2.51398
\(87\) 2.00000i 0.214423i
\(88\) − 8.82843i − 0.941113i
\(89\) −9.17157 −0.972185 −0.486092 0.873907i \(-0.661578\pi\)
−0.486092 + 0.873907i \(0.661578\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) − 15.3137i − 1.59656i
\(93\) 6.82843i 0.708075i
\(94\) −0.828427 −0.0854457
\(95\) 0 0
\(96\) 1.58579 0.161849
\(97\) − 7.65685i − 0.777436i −0.921357 0.388718i \(-0.872918\pi\)
0.921357 0.388718i \(-0.127082\pi\)
\(98\) 2.41421i 0.243872i
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −3.65685 −0.363871 −0.181935 0.983311i \(-0.558236\pi\)
−0.181935 + 0.983311i \(0.558236\pi\)
\(102\) 8.82843i 0.874145i
\(103\) − 13.6569i − 1.34565i −0.739802 0.672825i \(-0.765081\pi\)
0.739802 0.672825i \(-0.234919\pi\)
\(104\) −4.41421 −0.432849
\(105\) 0 0
\(106\) 4.82843 0.468978
\(107\) 11.3137i 1.09374i 0.837218 + 0.546869i \(0.184180\pi\)
−0.837218 + 0.546869i \(0.815820\pi\)
\(108\) − 3.82843i − 0.368391i
\(109\) 17.3137 1.65835 0.829176 0.558987i \(-0.188810\pi\)
0.829176 + 0.558987i \(0.188810\pi\)
\(110\) 0 0
\(111\) 3.65685 0.347093
\(112\) − 8.48528i − 0.801784i
\(113\) − 17.3137i − 1.62874i −0.580348 0.814368i \(-0.697084\pi\)
0.580348 0.814368i \(-0.302916\pi\)
\(114\) 6.82843 0.639541
\(115\) 0 0
\(116\) 7.65685 0.710921
\(117\) − 1.00000i − 0.0924500i
\(118\) − 8.82843i − 0.812723i
\(119\) −10.3431 −0.948155
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 22.4853i 2.03572i
\(123\) − 10.8284i − 0.976366i
\(124\) 26.1421 2.34763
\(125\) 0 0
\(126\) 6.82843 0.608325
\(127\) − 5.65685i − 0.501965i −0.967992 0.250982i \(-0.919246\pi\)
0.967992 0.250982i \(-0.0807536\pi\)
\(128\) − 20.5563i − 1.81694i
\(129\) −9.65685 −0.850239
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) − 7.65685i − 0.666444i
\(133\) 8.00000i 0.693688i
\(134\) 2.82843 0.244339
\(135\) 0 0
\(136\) 16.1421 1.38418
\(137\) − 5.17157i − 0.441837i −0.975292 0.220919i \(-0.929094\pi\)
0.975292 0.220919i \(-0.0709055\pi\)
\(138\) − 9.65685i − 0.822046i
\(139\) −15.3137 −1.29889 −0.649446 0.760408i \(-0.724999\pi\)
−0.649446 + 0.760408i \(0.724999\pi\)
\(140\) 0 0
\(141\) −0.343146 −0.0288981
\(142\) − 4.82843i − 0.405193i
\(143\) − 2.00000i − 0.167248i
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) −28.1421 −2.32906
\(147\) 1.00000i 0.0824786i
\(148\) − 14.0000i − 1.15079i
\(149\) 14.8284 1.21479 0.607396 0.794399i \(-0.292214\pi\)
0.607396 + 0.794399i \(0.292214\pi\)
\(150\) 0 0
\(151\) −20.4853 −1.66707 −0.833534 0.552468i \(-0.813686\pi\)
−0.833534 + 0.552468i \(0.813686\pi\)
\(152\) − 12.4853i − 1.01269i
\(153\) 3.65685i 0.295639i
\(154\) 13.6569 1.10050
\(155\) 0 0
\(156\) −3.82843 −0.306519
\(157\) − 10.0000i − 0.798087i −0.916932 0.399043i \(-0.869342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(158\) 27.3137i 2.17296i
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 11.3137 0.891645
\(162\) − 2.41421i − 0.189679i
\(163\) − 13.1716i − 1.03168i −0.856686 0.515839i \(-0.827480\pi\)
0.856686 0.515839i \(-0.172520\pi\)
\(164\) −41.4558 −3.23716
\(165\) 0 0
\(166\) 18.4853 1.43474
\(167\) 7.65685i 0.592505i 0.955110 + 0.296253i \(0.0957370\pi\)
−0.955110 + 0.296253i \(0.904263\pi\)
\(168\) − 12.4853i − 0.963260i
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 2.82843 0.216295
\(172\) 36.9706i 2.81898i
\(173\) 0.343146i 0.0260889i 0.999915 + 0.0130444i \(0.00415229\pi\)
−0.999915 + 0.0130444i \(0.995848\pi\)
\(174\) 4.82843 0.366042
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) − 3.65685i − 0.274866i
\(178\) 22.1421i 1.65962i
\(179\) 0.686292 0.0512958 0.0256479 0.999671i \(-0.491835\pi\)
0.0256479 + 0.999671i \(0.491835\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) − 6.82843i − 0.506157i
\(183\) 9.31371i 0.688489i
\(184\) −17.6569 −1.30168
\(185\) 0 0
\(186\) 16.4853 1.20876
\(187\) 7.31371i 0.534831i
\(188\) 1.31371i 0.0958120i
\(189\) 2.82843 0.205738
\(190\) 0 0
\(191\) −19.3137 −1.39749 −0.698745 0.715370i \(-0.746258\pi\)
−0.698745 + 0.715370i \(0.746258\pi\)
\(192\) − 9.82843i − 0.709306i
\(193\) 17.3137i 1.24627i 0.782115 + 0.623134i \(0.214141\pi\)
−0.782115 + 0.623134i \(0.785859\pi\)
\(194\) −18.4853 −1.32717
\(195\) 0 0
\(196\) 3.82843 0.273459
\(197\) − 16.4853i − 1.17453i −0.809396 0.587264i \(-0.800205\pi\)
0.809396 0.587264i \(-0.199795\pi\)
\(198\) − 4.82843i − 0.343141i
\(199\) −10.3431 −0.733206 −0.366603 0.930377i \(-0.619479\pi\)
−0.366603 + 0.930377i \(0.619479\pi\)
\(200\) 0 0
\(201\) 1.17157 0.0826364
\(202\) 8.82843i 0.621166i
\(203\) 5.65685i 0.397033i
\(204\) 14.0000 0.980196
\(205\) 0 0
\(206\) −32.9706 −2.29717
\(207\) − 4.00000i − 0.278019i
\(208\) 3.00000i 0.208013i
\(209\) 5.65685 0.391293
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) − 7.65685i − 0.525875i
\(213\) − 2.00000i − 0.137038i
\(214\) 27.3137 1.86713
\(215\) 0 0
\(216\) −4.41421 −0.300349
\(217\) 19.3137i 1.31110i
\(218\) − 41.7990i − 2.83098i
\(219\) −11.6569 −0.787697
\(220\) 0 0
\(221\) 3.65685 0.245987
\(222\) − 8.82843i − 0.592525i
\(223\) − 4.48528i − 0.300357i −0.988659 0.150178i \(-0.952015\pi\)
0.988659 0.150178i \(-0.0479848\pi\)
\(224\) 4.48528 0.299685
\(225\) 0 0
\(226\) −41.7990 −2.78043
\(227\) 5.31371i 0.352683i 0.984329 + 0.176342i \(0.0564263\pi\)
−0.984329 + 0.176342i \(0.943574\pi\)
\(228\) − 10.8284i − 0.717130i
\(229\) −21.3137 −1.40845 −0.704225 0.709977i \(-0.748705\pi\)
−0.704225 + 0.709977i \(0.748705\pi\)
\(230\) 0 0
\(231\) 5.65685 0.372194
\(232\) − 8.82843i − 0.579615i
\(233\) 26.9706i 1.76690i 0.468525 + 0.883450i \(0.344786\pi\)
−0.468525 + 0.883450i \(0.655214\pi\)
\(234\) −2.41421 −0.157822
\(235\) 0 0
\(236\) −14.0000 −0.911322
\(237\) 11.3137i 0.734904i
\(238\) 24.9706i 1.61860i
\(239\) −2.00000 −0.129369 −0.0646846 0.997906i \(-0.520604\pi\)
−0.0646846 + 0.997906i \(0.520604\pi\)
\(240\) 0 0
\(241\) 11.6569 0.750884 0.375442 0.926846i \(-0.377491\pi\)
0.375442 + 0.926846i \(0.377491\pi\)
\(242\) 16.8995i 1.08634i
\(243\) − 1.00000i − 0.0641500i
\(244\) 35.6569 2.28270
\(245\) 0 0
\(246\) −26.1421 −1.66676
\(247\) − 2.82843i − 0.179969i
\(248\) − 30.1421i − 1.91403i
\(249\) 7.65685 0.485233
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) − 10.8284i − 0.682127i
\(253\) − 8.00000i − 0.502956i
\(254\) −13.6569 −0.856907
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) − 15.6569i − 0.976648i −0.872662 0.488324i \(-0.837608\pi\)
0.872662 0.488324i \(-0.162392\pi\)
\(258\) 23.3137i 1.45145i
\(259\) 10.3431 0.642692
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 19.3137i 1.19320i
\(263\) − 12.0000i − 0.739952i −0.929041 0.369976i \(-0.879366\pi\)
0.929041 0.369976i \(-0.120634\pi\)
\(264\) −8.82843 −0.543352
\(265\) 0 0
\(266\) 19.3137 1.18420
\(267\) 9.17157i 0.561291i
\(268\) − 4.48528i − 0.273982i
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 11.7990 0.716738 0.358369 0.933580i \(-0.383333\pi\)
0.358369 + 0.933580i \(0.383333\pi\)
\(272\) − 10.9706i − 0.665188i
\(273\) − 2.82843i − 0.171184i
\(274\) −12.4853 −0.754263
\(275\) 0 0
\(276\) −15.3137 −0.921777
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 36.9706i 2.21735i
\(279\) 6.82843 0.408807
\(280\) 0 0
\(281\) 26.8284 1.60045 0.800225 0.599700i \(-0.204713\pi\)
0.800225 + 0.599700i \(0.204713\pi\)
\(282\) 0.828427i 0.0493321i
\(283\) 4.97056i 0.295469i 0.989027 + 0.147735i \(0.0471982\pi\)
−0.989027 + 0.147735i \(0.952802\pi\)
\(284\) −7.65685 −0.454351
\(285\) 0 0
\(286\) −4.82843 −0.285511
\(287\) − 30.6274i − 1.80788i
\(288\) − 1.58579i − 0.0934434i
\(289\) 3.62742 0.213377
\(290\) 0 0
\(291\) −7.65685 −0.448853
\(292\) 44.6274i 2.61162i
\(293\) 26.1421i 1.52724i 0.645666 + 0.763620i \(0.276580\pi\)
−0.645666 + 0.763620i \(0.723420\pi\)
\(294\) 2.41421 0.140800
\(295\) 0 0
\(296\) −16.1421 −0.938243
\(297\) − 2.00000i − 0.116052i
\(298\) − 35.7990i − 2.07378i
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) −27.3137 −1.57434
\(302\) 49.4558i 2.84586i
\(303\) 3.65685i 0.210081i
\(304\) −8.48528 −0.486664
\(305\) 0 0
\(306\) 8.82843 0.504688
\(307\) − 17.1716i − 0.980033i −0.871713 0.490017i \(-0.836991\pi\)
0.871713 0.490017i \(-0.163009\pi\)
\(308\) − 21.6569i − 1.23401i
\(309\) −13.6569 −0.776911
\(310\) 0 0
\(311\) 34.6274 1.96354 0.981770 0.190071i \(-0.0608718\pi\)
0.981770 + 0.190071i \(0.0608718\pi\)
\(312\) 4.41421i 0.249906i
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) −24.1421 −1.36242
\(315\) 0 0
\(316\) 43.3137 2.43659
\(317\) − 8.48528i − 0.476581i −0.971194 0.238290i \(-0.923413\pi\)
0.971194 0.238290i \(-0.0765870\pi\)
\(318\) − 4.82843i − 0.270765i
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) 11.3137 0.631470
\(322\) − 27.3137i − 1.52213i
\(323\) 10.3431i 0.575508i
\(324\) −3.82843 −0.212690
\(325\) 0 0
\(326\) −31.7990 −1.76118
\(327\) − 17.3137i − 0.957450i
\(328\) 47.7990i 2.63926i
\(329\) −0.970563 −0.0535089
\(330\) 0 0
\(331\) −2.14214 −0.117742 −0.0588712 0.998266i \(-0.518750\pi\)
−0.0588712 + 0.998266i \(0.518750\pi\)
\(332\) − 29.3137i − 1.60880i
\(333\) − 3.65685i − 0.200394i
\(334\) 18.4853 1.01147
\(335\) 0 0
\(336\) −8.48528 −0.462910
\(337\) − 13.3137i − 0.725244i −0.931936 0.362622i \(-0.881882\pi\)
0.931936 0.362622i \(-0.118118\pi\)
\(338\) 2.41421i 0.131316i
\(339\) −17.3137 −0.940352
\(340\) 0 0
\(341\) 13.6569 0.739560
\(342\) − 6.82843i − 0.369239i
\(343\) − 16.9706i − 0.916324i
\(344\) 42.6274 2.29832
\(345\) 0 0
\(346\) 0.828427 0.0445365
\(347\) − 31.3137i − 1.68101i −0.541805 0.840504i \(-0.682259\pi\)
0.541805 0.840504i \(-0.317741\pi\)
\(348\) − 7.65685i − 0.410450i
\(349\) 7.65685 0.409862 0.204931 0.978776i \(-0.434303\pi\)
0.204931 + 0.978776i \(0.434303\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) − 3.17157i − 0.169045i
\(353\) − 17.4558i − 0.929081i −0.885552 0.464540i \(-0.846220\pi\)
0.885552 0.464540i \(-0.153780\pi\)
\(354\) −8.82843 −0.469226
\(355\) 0 0
\(356\) 35.1127 1.86097
\(357\) 10.3431i 0.547417i
\(358\) − 1.65685i − 0.0875675i
\(359\) −1.02944 −0.0543316 −0.0271658 0.999631i \(-0.508648\pi\)
−0.0271658 + 0.999631i \(0.508648\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) − 33.7990i − 1.77644i
\(363\) 7.00000i 0.367405i
\(364\) −10.8284 −0.567564
\(365\) 0 0
\(366\) 22.4853 1.17532
\(367\) − 24.0000i − 1.25279i −0.779506 0.626395i \(-0.784530\pi\)
0.779506 0.626395i \(-0.215470\pi\)
\(368\) 12.0000i 0.625543i
\(369\) −10.8284 −0.563705
\(370\) 0 0
\(371\) 5.65685 0.293689
\(372\) − 26.1421i − 1.35541i
\(373\) − 10.0000i − 0.517780i −0.965907 0.258890i \(-0.916643\pi\)
0.965907 0.258890i \(-0.0833568\pi\)
\(374\) 17.6569 0.913014
\(375\) 0 0
\(376\) 1.51472 0.0781156
\(377\) − 2.00000i − 0.103005i
\(378\) − 6.82843i − 0.351216i
\(379\) 16.4853 0.846792 0.423396 0.905945i \(-0.360838\pi\)
0.423396 + 0.905945i \(0.360838\pi\)
\(380\) 0 0
\(381\) −5.65685 −0.289809
\(382\) 46.6274i 2.38567i
\(383\) 2.97056i 0.151789i 0.997116 + 0.0758943i \(0.0241812\pi\)
−0.997116 + 0.0758943i \(0.975819\pi\)
\(384\) −20.5563 −1.04901
\(385\) 0 0
\(386\) 41.7990 2.12751
\(387\) 9.65685i 0.490885i
\(388\) 29.3137i 1.48818i
\(389\) −6.97056 −0.353422 −0.176711 0.984263i \(-0.556546\pi\)
−0.176711 + 0.984263i \(0.556546\pi\)
\(390\) 0 0
\(391\) 14.6274 0.739740
\(392\) − 4.41421i − 0.222951i
\(393\) 8.00000i 0.403547i
\(394\) −39.7990 −2.00504
\(395\) 0 0
\(396\) −7.65685 −0.384771
\(397\) − 2.97056i − 0.149088i −0.997218 0.0745441i \(-0.976250\pi\)
0.997218 0.0745441i \(-0.0237502\pi\)
\(398\) 24.9706i 1.25166i
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) −2.14214 −0.106973 −0.0534866 0.998569i \(-0.517033\pi\)
−0.0534866 + 0.998569i \(0.517033\pi\)
\(402\) − 2.82843i − 0.141069i
\(403\) − 6.82843i − 0.340148i
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) 13.6569 0.677778
\(407\) − 7.31371i − 0.362527i
\(408\) − 16.1421i − 0.799155i
\(409\) 1.02944 0.0509024 0.0254512 0.999676i \(-0.491898\pi\)
0.0254512 + 0.999676i \(0.491898\pi\)
\(410\) 0 0
\(411\) −5.17157 −0.255095
\(412\) 52.2843i 2.57586i
\(413\) − 10.3431i − 0.508953i
\(414\) −9.65685 −0.474608
\(415\) 0 0
\(416\) −1.58579 −0.0777496
\(417\) 15.3137i 0.749916i
\(418\) − 13.6569i − 0.667979i
\(419\) 30.6274 1.49625 0.748124 0.663559i \(-0.230955\pi\)
0.748124 + 0.663559i \(0.230955\pi\)
\(420\) 0 0
\(421\) 14.6863 0.715766 0.357883 0.933766i \(-0.383499\pi\)
0.357883 + 0.933766i \(0.383499\pi\)
\(422\) 28.9706i 1.41026i
\(423\) 0.343146i 0.0166843i
\(424\) −8.82843 −0.428746
\(425\) 0 0
\(426\) −4.82843 −0.233938
\(427\) 26.3431i 1.27483i
\(428\) − 43.3137i − 2.09365i
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) 19.6569 0.946837 0.473419 0.880838i \(-0.343020\pi\)
0.473419 + 0.880838i \(0.343020\pi\)
\(432\) 3.00000i 0.144338i
\(433\) − 1.31371i − 0.0631328i −0.999502 0.0315664i \(-0.989950\pi\)
0.999502 0.0315664i \(-0.0100496\pi\)
\(434\) 46.6274 2.23819
\(435\) 0 0
\(436\) −66.2843 −3.17444
\(437\) − 11.3137i − 0.541208i
\(438\) 28.1421i 1.34468i
\(439\) 16.9706 0.809961 0.404980 0.914325i \(-0.367278\pi\)
0.404980 + 0.914325i \(0.367278\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) − 8.82843i − 0.419925i
\(443\) − 41.9411i − 1.99268i −0.0854611 0.996342i \(-0.527236\pi\)
0.0854611 0.996342i \(-0.472764\pi\)
\(444\) −14.0000 −0.664411
\(445\) 0 0
\(446\) −10.8284 −0.512741
\(447\) − 14.8284i − 0.701361i
\(448\) − 27.7990i − 1.31338i
\(449\) −7.79899 −0.368057 −0.184029 0.982921i \(-0.558914\pi\)
−0.184029 + 0.982921i \(0.558914\pi\)
\(450\) 0 0
\(451\) −21.6569 −1.01978
\(452\) 66.2843i 3.11775i
\(453\) 20.4853i 0.962482i
\(454\) 12.8284 0.602068
\(455\) 0 0
\(456\) −12.4853 −0.584677
\(457\) 3.65685i 0.171060i 0.996336 + 0.0855302i \(0.0272584\pi\)
−0.996336 + 0.0855302i \(0.972742\pi\)
\(458\) 51.4558i 2.40437i
\(459\) 3.65685 0.170687
\(460\) 0 0
\(461\) 10.8284 0.504330 0.252165 0.967684i \(-0.418857\pi\)
0.252165 + 0.967684i \(0.418857\pi\)
\(462\) − 13.6569i − 0.635374i
\(463\) 7.51472i 0.349239i 0.984636 + 0.174619i \(0.0558695\pi\)
−0.984636 + 0.174619i \(0.944131\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 65.1127 3.01629
\(467\) − 8.00000i − 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) 3.82843i 0.176969i
\(469\) 3.31371 0.153013
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 16.1421i 0.743002i
\(473\) 19.3137i 0.888045i
\(474\) 27.3137 1.25456
\(475\) 0 0
\(476\) 39.5980 1.81497
\(477\) − 2.00000i − 0.0915737i
\(478\) 4.82843i 0.220847i
\(479\) −2.68629 −0.122740 −0.0613699 0.998115i \(-0.519547\pi\)
−0.0613699 + 0.998115i \(0.519547\pi\)
\(480\) 0 0
\(481\) −3.65685 −0.166738
\(482\) − 28.1421i − 1.28184i
\(483\) − 11.3137i − 0.514792i
\(484\) 26.7990 1.21814
\(485\) 0 0
\(486\) −2.41421 −0.109511
\(487\) 31.7990i 1.44095i 0.693481 + 0.720475i \(0.256076\pi\)
−0.693481 + 0.720475i \(0.743924\pi\)
\(488\) − 41.1127i − 1.86108i
\(489\) −13.1716 −0.595639
\(490\) 0 0
\(491\) −14.6274 −0.660126 −0.330063 0.943959i \(-0.607070\pi\)
−0.330063 + 0.943959i \(0.607070\pi\)
\(492\) 41.4558i 1.86897i
\(493\) 7.31371i 0.329393i
\(494\) −6.82843 −0.307225
\(495\) 0 0
\(496\) −20.4853 −0.919816
\(497\) − 5.65685i − 0.253745i
\(498\) − 18.4853i − 0.828345i
\(499\) 2.14214 0.0958952 0.0479476 0.998850i \(-0.484732\pi\)
0.0479476 + 0.998850i \(0.484732\pi\)
\(500\) 0 0
\(501\) 7.65685 0.342083
\(502\) 0 0
\(503\) − 15.3137i − 0.682805i −0.939917 0.341402i \(-0.889098\pi\)
0.939917 0.341402i \(-0.110902\pi\)
\(504\) −12.4853 −0.556139
\(505\) 0 0
\(506\) −19.3137 −0.858599
\(507\) 1.00000i 0.0444116i
\(508\) 21.6569i 0.960868i
\(509\) 27.7990 1.23217 0.616084 0.787680i \(-0.288718\pi\)
0.616084 + 0.787680i \(0.288718\pi\)
\(510\) 0 0
\(511\) −32.9706 −1.45853
\(512\) 31.2426i 1.38074i
\(513\) − 2.82843i − 0.124878i
\(514\) −37.7990 −1.66724
\(515\) 0 0
\(516\) 36.9706 1.62754
\(517\) 0.686292i 0.0301831i
\(518\) − 24.9706i − 1.09714i
\(519\) 0.343146 0.0150624
\(520\) 0 0
\(521\) 2.68629 0.117689 0.0588443 0.998267i \(-0.481258\pi\)
0.0588443 + 0.998267i \(0.481258\pi\)
\(522\) − 4.82843i − 0.211335i
\(523\) − 7.31371i − 0.319806i −0.987133 0.159903i \(-0.948882\pi\)
0.987133 0.159903i \(-0.0511182\pi\)
\(524\) 30.6274 1.33796
\(525\) 0 0
\(526\) −28.9706 −1.26318
\(527\) 24.9706i 1.08773i
\(528\) 6.00000i 0.261116i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −3.65685 −0.158694
\(532\) − 30.6274i − 1.32787i
\(533\) 10.8284i 0.469031i
\(534\) 22.1421 0.958184
\(535\) 0 0
\(536\) −5.17157 −0.223378
\(537\) − 0.686292i − 0.0296157i
\(538\) 43.4558i 1.87351i
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) − 28.4853i − 1.22355i
\(543\) − 14.0000i − 0.600798i
\(544\) 5.79899 0.248630
\(545\) 0 0
\(546\) −6.82843 −0.292230
\(547\) 0.686292i 0.0293437i 0.999892 + 0.0146719i \(0.00467036\pi\)
−0.999892 + 0.0146719i \(0.995330\pi\)
\(548\) 19.7990i 0.845771i
\(549\) 9.31371 0.397499
\(550\) 0 0
\(551\) 5.65685 0.240990
\(552\) 17.6569i 0.751526i
\(553\) 32.0000i 1.36078i
\(554\) −4.82843 −0.205140
\(555\) 0 0
\(556\) 58.6274 2.48636
\(557\) 31.7990i 1.34737i 0.739020 + 0.673683i \(0.235289\pi\)
−0.739020 + 0.673683i \(0.764711\pi\)
\(558\) − 16.4853i − 0.697878i
\(559\) 9.65685 0.408441
\(560\) 0 0
\(561\) 7.31371 0.308785
\(562\) − 64.7696i − 2.73214i
\(563\) − 4.00000i − 0.168580i −0.996441 0.0842900i \(-0.973138\pi\)
0.996441 0.0842900i \(-0.0268622\pi\)
\(564\) 1.31371 0.0553171
\(565\) 0 0
\(566\) 12.0000 0.504398
\(567\) − 2.82843i − 0.118783i
\(568\) 8.82843i 0.370433i
\(569\) 9.02944 0.378534 0.189267 0.981926i \(-0.439389\pi\)
0.189267 + 0.981926i \(0.439389\pi\)
\(570\) 0 0
\(571\) 20.9706 0.877591 0.438795 0.898587i \(-0.355405\pi\)
0.438795 + 0.898587i \(0.355405\pi\)
\(572\) 7.65685i 0.320149i
\(573\) 19.3137i 0.806842i
\(574\) −73.9411 −3.08624
\(575\) 0 0
\(576\) −9.82843 −0.409518
\(577\) 35.9411i 1.49625i 0.663559 + 0.748124i \(0.269045\pi\)
−0.663559 + 0.748124i \(0.730955\pi\)
\(578\) − 8.75736i − 0.364258i
\(579\) 17.3137 0.719533
\(580\) 0 0
\(581\) 21.6569 0.898478
\(582\) 18.4853i 0.766240i
\(583\) − 4.00000i − 0.165663i
\(584\) 51.4558 2.12926
\(585\) 0 0
\(586\) 63.1127 2.60716
\(587\) 22.9706i 0.948097i 0.880499 + 0.474048i \(0.157208\pi\)
−0.880499 + 0.474048i \(0.842792\pi\)
\(588\) − 3.82843i − 0.157882i
\(589\) 19.3137 0.795807
\(590\) 0 0
\(591\) −16.4853 −0.678114
\(592\) 10.9706i 0.450887i
\(593\) 3.51472i 0.144332i 0.997393 + 0.0721661i \(0.0229912\pi\)
−0.997393 + 0.0721661i \(0.977009\pi\)
\(594\) −4.82843 −0.198113
\(595\) 0 0
\(596\) −56.7696 −2.32537
\(597\) 10.3431i 0.423317i
\(598\) 9.65685i 0.394898i
\(599\) 0.686292 0.0280411 0.0140206 0.999902i \(-0.495537\pi\)
0.0140206 + 0.999902i \(0.495537\pi\)
\(600\) 0 0
\(601\) 44.6274 1.82039 0.910195 0.414180i \(-0.135931\pi\)
0.910195 + 0.414180i \(0.135931\pi\)
\(602\) 65.9411i 2.68756i
\(603\) − 1.17157i − 0.0477101i
\(604\) 78.4264 3.19113
\(605\) 0 0
\(606\) 8.82843 0.358630
\(607\) − 25.9411i − 1.05292i −0.850201 0.526459i \(-0.823519\pi\)
0.850201 0.526459i \(-0.176481\pi\)
\(608\) − 4.48528i − 0.181902i
\(609\) 5.65685 0.229227
\(610\) 0 0
\(611\) 0.343146 0.0138822
\(612\) − 14.0000i − 0.565916i
\(613\) 36.3431i 1.46789i 0.679211 + 0.733943i \(0.262322\pi\)
−0.679211 + 0.733943i \(0.737678\pi\)
\(614\) −41.4558 −1.67302
\(615\) 0 0
\(616\) −24.9706 −1.00609
\(617\) − 29.1716i − 1.17440i −0.809441 0.587202i \(-0.800230\pi\)
0.809441 0.587202i \(-0.199770\pi\)
\(618\) 32.9706i 1.32627i
\(619\) 15.7990 0.635015 0.317508 0.948256i \(-0.397154\pi\)
0.317508 + 0.948256i \(0.397154\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) − 83.5980i − 3.35197i
\(623\) 25.9411i 1.03931i
\(624\) 3.00000 0.120096
\(625\) 0 0
\(626\) −14.4853 −0.578948
\(627\) − 5.65685i − 0.225913i
\(628\) 38.2843i 1.52771i
\(629\) 13.3726 0.533200
\(630\) 0 0
\(631\) −19.1127 −0.760865 −0.380432 0.924809i \(-0.624225\pi\)
−0.380432 + 0.924809i \(0.624225\pi\)
\(632\) − 49.9411i − 1.98655i
\(633\) 12.0000i 0.476957i
\(634\) −20.4853 −0.813574
\(635\) 0 0
\(636\) −7.65685 −0.303614
\(637\) − 1.00000i − 0.0396214i
\(638\) − 9.65685i − 0.382319i
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) −26.2843 −1.03817 −0.519083 0.854724i \(-0.673727\pi\)
−0.519083 + 0.854724i \(0.673727\pi\)
\(642\) − 27.3137i − 1.07799i
\(643\) − 17.1716i − 0.677181i −0.940934 0.338590i \(-0.890050\pi\)
0.940934 0.338590i \(-0.109950\pi\)
\(644\) −43.3137 −1.70680
\(645\) 0 0
\(646\) 24.9706 0.982454
\(647\) − 11.3137i − 0.444788i −0.974957 0.222394i \(-0.928613\pi\)
0.974957 0.222394i \(-0.0713871\pi\)
\(648\) 4.41421i 0.173407i
\(649\) −7.31371 −0.287088
\(650\) 0 0
\(651\) 19.3137 0.756964
\(652\) 50.4264i 1.97485i
\(653\) 2.68629i 0.105123i 0.998618 + 0.0525614i \(0.0167385\pi\)
−0.998618 + 0.0525614i \(0.983261\pi\)
\(654\) −41.7990 −1.63447
\(655\) 0 0
\(656\) 32.4853 1.26834
\(657\) 11.6569i 0.454777i
\(658\) 2.34315i 0.0913453i
\(659\) 24.6863 0.961641 0.480821 0.876819i \(-0.340339\pi\)
0.480821 + 0.876819i \(0.340339\pi\)
\(660\) 0 0
\(661\) −1.02944 −0.0400405 −0.0200202 0.999800i \(-0.506373\pi\)
−0.0200202 + 0.999800i \(0.506373\pi\)
\(662\) 5.17157i 0.200999i
\(663\) − 3.65685i − 0.142020i
\(664\) −33.7990 −1.31166
\(665\) 0 0
\(666\) −8.82843 −0.342095
\(667\) − 8.00000i − 0.309761i
\(668\) − 29.3137i − 1.13418i
\(669\) −4.48528 −0.173411
\(670\) 0 0
\(671\) 18.6274 0.719103
\(672\) − 4.48528i − 0.173023i
\(673\) 28.6274i 1.10351i 0.834008 + 0.551753i \(0.186041\pi\)
−0.834008 + 0.551753i \(0.813959\pi\)
\(674\) −32.1421 −1.23807
\(675\) 0 0
\(676\) 3.82843 0.147247
\(677\) 49.3137i 1.89528i 0.319341 + 0.947640i \(0.396538\pi\)
−0.319341 + 0.947640i \(0.603462\pi\)
\(678\) 41.7990i 1.60528i
\(679\) −21.6569 −0.831114
\(680\) 0 0
\(681\) 5.31371 0.203622
\(682\) − 32.9706i − 1.26251i
\(683\) 19.9411i 0.763026i 0.924363 + 0.381513i \(0.124597\pi\)
−0.924363 + 0.381513i \(0.875403\pi\)
\(684\) −10.8284 −0.414035
\(685\) 0 0
\(686\) −40.9706 −1.56426
\(687\) 21.3137i 0.813169i
\(688\) − 28.9706i − 1.10449i
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) −34.1421 −1.29883 −0.649414 0.760435i \(-0.724986\pi\)
−0.649414 + 0.760435i \(0.724986\pi\)
\(692\) − 1.31371i − 0.0499397i
\(693\) − 5.65685i − 0.214886i
\(694\) −75.5980 −2.86966
\(695\) 0 0
\(696\) −8.82843 −0.334641
\(697\) − 39.5980i − 1.49988i
\(698\) − 18.4853i − 0.699678i
\(699\) 26.9706 1.02012
\(700\) 0 0
\(701\) 38.9706 1.47190 0.735949 0.677037i \(-0.236736\pi\)
0.735949 + 0.677037i \(0.236736\pi\)
\(702\) 2.41421i 0.0911186i
\(703\) − 10.3431i − 0.390099i
\(704\) −19.6569 −0.740846
\(705\) 0 0
\(706\) −42.1421 −1.58604
\(707\) 10.3431i 0.388994i
\(708\) 14.0000i 0.526152i
\(709\) −40.6274 −1.52579 −0.762897 0.646520i \(-0.776224\pi\)
−0.762897 + 0.646520i \(0.776224\pi\)
\(710\) 0 0
\(711\) 11.3137 0.424297
\(712\) − 40.4853i − 1.51725i
\(713\) − 27.3137i − 1.02291i
\(714\) 24.9706 0.934500
\(715\) 0 0
\(716\) −2.62742 −0.0981912
\(717\) 2.00000i 0.0746914i
\(718\) 2.48528i 0.0927499i
\(719\) −37.9411 −1.41497 −0.707483 0.706731i \(-0.750169\pi\)
−0.707483 + 0.706731i \(0.750169\pi\)
\(720\) 0 0
\(721\) −38.6274 −1.43856
\(722\) 26.5563i 0.988325i
\(723\) − 11.6569i − 0.433523i
\(724\) −53.5980 −1.99195
\(725\) 0 0
\(726\) 16.8995 0.627199
\(727\) − 21.6569i − 0.803208i −0.915813 0.401604i \(-0.868453\pi\)
0.915813 0.401604i \(-0.131547\pi\)
\(728\) 12.4853i 0.462735i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −35.3137 −1.30612
\(732\) − 35.6569i − 1.31792i
\(733\) − 8.62742i − 0.318661i −0.987225 0.159330i \(-0.949066\pi\)
0.987225 0.159330i \(-0.0509335\pi\)
\(734\) −57.9411 −2.13865
\(735\) 0 0
\(736\) −6.34315 −0.233811
\(737\) − 2.34315i − 0.0863109i
\(738\) 26.1421i 0.962305i
\(739\) −10.1421 −0.373084 −0.186542 0.982447i \(-0.559728\pi\)
−0.186542 + 0.982447i \(0.559728\pi\)
\(740\) 0 0
\(741\) −2.82843 −0.103905
\(742\) − 13.6569i − 0.501359i
\(743\) − 2.00000i − 0.0733729i −0.999327 0.0366864i \(-0.988320\pi\)
0.999327 0.0366864i \(-0.0116803\pi\)
\(744\) −30.1421 −1.10506
\(745\) 0 0
\(746\) −24.1421 −0.883906
\(747\) − 7.65685i − 0.280150i
\(748\) − 28.0000i − 1.02378i
\(749\) 32.0000 1.16925
\(750\) 0 0
\(751\) 32.9706 1.20311 0.601556 0.798830i \(-0.294547\pi\)
0.601556 + 0.798830i \(0.294547\pi\)
\(752\) − 1.02944i − 0.0375397i
\(753\) 0 0
\(754\) −4.82843 −0.175841
\(755\) 0 0
\(756\) −10.8284 −0.393826
\(757\) − 15.9411i − 0.579390i −0.957119 0.289695i \(-0.906446\pi\)
0.957119 0.289695i \(-0.0935539\pi\)
\(758\) − 39.7990i − 1.44556i
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 15.5147 0.562408 0.281204 0.959648i \(-0.409266\pi\)
0.281204 + 0.959648i \(0.409266\pi\)
\(762\) 13.6569i 0.494736i
\(763\) − 48.9706i − 1.77285i
\(764\) 73.9411 2.67510
\(765\) 0 0
\(766\) 7.17157 0.259119
\(767\) 3.65685i 0.132041i
\(768\) 29.9706i 1.08147i
\(769\) −42.0000 −1.51456 −0.757279 0.653091i \(-0.773472\pi\)
−0.757279 + 0.653091i \(0.773472\pi\)
\(770\) 0 0
\(771\) −15.6569 −0.563868
\(772\) − 66.2843i − 2.38562i
\(773\) − 5.85786i − 0.210693i −0.994436 0.105346i \(-0.966405\pi\)
0.994436 0.105346i \(-0.0335951\pi\)
\(774\) 23.3137 0.837994
\(775\) 0 0
\(776\) 33.7990 1.21331
\(777\) − 10.3431i − 0.371058i
\(778\) 16.8284i 0.603328i
\(779\) −30.6274 −1.09734
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) − 35.3137i − 1.26282i
\(783\) − 2.00000i − 0.0714742i
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 19.3137 0.688897
\(787\) 32.7696i 1.16811i 0.811715 + 0.584054i \(0.198534\pi\)
−0.811715 + 0.584054i \(0.801466\pi\)
\(788\) 63.1127i 2.24830i
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) −48.9706 −1.74119
\(792\) 8.82843i 0.313704i
\(793\) − 9.31371i − 0.330739i
\(794\) −7.17157 −0.254510
\(795\) 0 0
\(796\) 39.5980 1.40351
\(797\) 35.6569i 1.26303i 0.775363 + 0.631515i \(0.217567\pi\)
−0.775363 + 0.631515i \(0.782433\pi\)
\(798\) − 19.3137i − 0.683698i
\(799\) −1.25483 −0.0443928
\(800\) 0 0
\(801\) 9.17157 0.324062
\(802\) 5.17157i 0.182615i
\(803\) 23.3137i 0.822723i
\(804\) −4.48528 −0.158184
\(805\) 0 0
\(806\) −16.4853 −0.580669
\(807\) 18.0000i 0.633630i
\(808\) − 16.1421i − 0.567878i
\(809\) −41.3137 −1.45251 −0.726256 0.687424i \(-0.758741\pi\)
−0.726256 + 0.687424i \(0.758741\pi\)
\(810\) 0 0
\(811\) −1.85786 −0.0652384 −0.0326192 0.999468i \(-0.510385\pi\)
−0.0326192 + 0.999468i \(0.510385\pi\)
\(812\) − 21.6569i − 0.760007i
\(813\) − 11.7990i − 0.413809i
\(814\) −17.6569 −0.618872
\(815\) 0 0
\(816\) −10.9706 −0.384047
\(817\) 27.3137i 0.955586i
\(818\) − 2.48528i − 0.0868958i
\(819\) −2.82843 −0.0988332
\(820\) 0 0
\(821\) 15.7990 0.551389 0.275694 0.961245i \(-0.411092\pi\)
0.275694 + 0.961245i \(0.411092\pi\)
\(822\) 12.4853i 0.435474i
\(823\) − 48.9706i − 1.70701i −0.521088 0.853503i \(-0.674473\pi\)
0.521088 0.853503i \(-0.325527\pi\)
\(824\) 60.2843 2.10010
\(825\) 0 0
\(826\) −24.9706 −0.868837
\(827\) − 26.0000i − 0.904109i −0.891990 0.452054i \(-0.850691\pi\)
0.891990 0.452054i \(-0.149309\pi\)
\(828\) 15.3137i 0.532188i
\(829\) −5.31371 −0.184553 −0.0922764 0.995733i \(-0.529414\pi\)
−0.0922764 + 0.995733i \(0.529414\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 9.82843i 0.340739i
\(833\) 3.65685i 0.126702i
\(834\) 36.9706 1.28019
\(835\) 0 0
\(836\) −21.6569 −0.749018
\(837\) − 6.82843i − 0.236025i
\(838\) − 73.9411i − 2.55425i
\(839\) 47.2548 1.63142 0.815709 0.578462i \(-0.196347\pi\)
0.815709 + 0.578462i \(0.196347\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) − 35.4558i − 1.22189i
\(843\) − 26.8284i − 0.924020i
\(844\) 45.9411 1.58136
\(845\) 0 0
\(846\) 0.828427 0.0284819
\(847\) 19.7990i 0.680301i
\(848\) 6.00000i 0.206041i
\(849\) 4.97056 0.170589
\(850\) 0 0
\(851\) −14.6274 −0.501421
\(852\) 7.65685i 0.262320i
\(853\) 7.65685i 0.262166i 0.991371 + 0.131083i \(0.0418454\pi\)
−0.991371 + 0.131083i \(0.958155\pi\)
\(854\) 63.5980 2.17628
\(855\) 0 0
\(856\) −49.9411 −1.70695
\(857\) 29.5980i 1.01105i 0.862813 + 0.505524i \(0.168701\pi\)
−0.862813 + 0.505524i \(0.831299\pi\)
\(858\) 4.82843i 0.164840i
\(859\) 23.3137 0.795453 0.397727 0.917504i \(-0.369799\pi\)
0.397727 + 0.917504i \(0.369799\pi\)
\(860\) 0 0
\(861\) −30.6274 −1.04378
\(862\) − 47.4558i − 1.61635i
\(863\) 39.6569i 1.34994i 0.737847 + 0.674968i \(0.235842\pi\)
−0.737847 + 0.674968i \(0.764158\pi\)
\(864\) −1.58579 −0.0539496
\(865\) 0 0
\(866\) −3.17157 −0.107774
\(867\) − 3.62742i − 0.123194i
\(868\) − 73.9411i − 2.50973i
\(869\) 22.6274 0.767583
\(870\) 0 0
\(871\) −1.17157 −0.0396972
\(872\) 76.4264i 2.58812i
\(873\) 7.65685i 0.259145i
\(874\) −27.3137 −0.923900
\(875\) 0 0
\(876\) 44.6274 1.50782
\(877\) − 14.2843i − 0.482346i −0.970482 0.241173i \(-0.922468\pi\)
0.970482 0.241173i \(-0.0775321\pi\)
\(878\) − 40.9706i − 1.38269i
\(879\) 26.1421 0.881752
\(880\) 0 0
\(881\) 53.5980 1.80576 0.902881 0.429891i \(-0.141448\pi\)
0.902881 + 0.429891i \(0.141448\pi\)
\(882\) − 2.41421i − 0.0812908i
\(883\) 51.5980i 1.73641i 0.496205 + 0.868205i \(0.334726\pi\)
−0.496205 + 0.868205i \(0.665274\pi\)
\(884\) −14.0000 −0.470871
\(885\) 0 0
\(886\) −101.255 −3.40172
\(887\) − 8.00000i − 0.268614i −0.990940 0.134307i \(-0.957119\pi\)
0.990940 0.134307i \(-0.0428808\pi\)
\(888\) 16.1421i 0.541695i
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 17.1716i 0.574947i
\(893\) 0.970563i 0.0324786i
\(894\) −35.7990 −1.19730
\(895\) 0 0
\(896\) −58.1421 −1.94239
\(897\) 4.00000i 0.133556i
\(898\) 18.8284i 0.628313i
\(899\) 13.6569 0.455482
\(900\) 0 0
\(901\) 7.31371 0.243655
\(902\) 52.2843i 1.74088i
\(903\) 27.3137i 0.908943i
\(904\) 76.4264 2.54190
\(905\) 0 0
\(906\) 49.4558 1.64306
\(907\) 20.9706i 0.696316i 0.937436 + 0.348158i \(0.113193\pi\)
−0.937436 + 0.348158i \(0.886807\pi\)
\(908\) − 20.3431i − 0.675111i
\(909\) 3.65685 0.121290
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 8.48528i 0.280976i
\(913\) − 15.3137i − 0.506810i
\(914\) 8.82843 0.292018
\(915\) 0 0
\(916\) 81.5980 2.69607
\(917\) 22.6274i 0.747223i
\(918\) − 8.82843i − 0.291382i
\(919\) 19.3137 0.637100 0.318550 0.947906i \(-0.396804\pi\)
0.318550 + 0.947906i \(0.396804\pi\)
\(920\) 0 0
\(921\) −17.1716 −0.565823
\(922\) − 26.1421i − 0.860945i
\(923\) 2.00000i 0.0658308i
\(924\) −21.6569 −0.712458
\(925\) 0 0
\(926\) 18.1421 0.596188
\(927\) 13.6569i 0.448550i
\(928\) − 3.17157i − 0.104112i
\(929\) 27.7990 0.912055 0.456028 0.889966i \(-0.349272\pi\)
0.456028 + 0.889966i \(0.349272\pi\)
\(930\) 0 0
\(931\) 2.82843 0.0926980
\(932\) − 103.255i − 3.38222i
\(933\) − 34.6274i − 1.13365i
\(934\) −19.3137 −0.631964
\(935\) 0 0
\(936\) 4.41421 0.144283
\(937\) 1.31371i 0.0429170i 0.999770 + 0.0214585i \(0.00683097\pi\)
−0.999770 + 0.0214585i \(0.993169\pi\)
\(938\) − 8.00000i − 0.261209i
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) 5.85786 0.190961 0.0954805 0.995431i \(-0.469561\pi\)
0.0954805 + 0.995431i \(0.469561\pi\)
\(942\) 24.1421i 0.786593i
\(943\) 43.3137i 1.41049i
\(944\) 10.9706 0.357061
\(945\) 0 0
\(946\) 46.6274 1.51599
\(947\) − 54.9706i − 1.78630i −0.449756 0.893152i \(-0.648489\pi\)
0.449756 0.893152i \(-0.351511\pi\)
\(948\) − 43.3137i − 1.40676i
\(949\) 11.6569 0.378398
\(950\) 0 0
\(951\) −8.48528 −0.275154
\(952\) − 45.6569i − 1.47975i
\(953\) − 51.6569i − 1.67333i −0.547715 0.836665i \(-0.684502\pi\)
0.547715 0.836665i \(-0.315498\pi\)
\(954\) −4.82843 −0.156326
\(955\) 0 0
\(956\) 7.65685 0.247640
\(957\) − 4.00000i − 0.129302i
\(958\) 6.48528i 0.209530i
\(959\) −14.6274 −0.472344
\(960\) 0 0
\(961\) 15.6274 0.504110
\(962\) 8.82843i 0.284640i
\(963\) − 11.3137i − 0.364579i
\(964\) −44.6274 −1.43735
\(965\) 0 0
\(966\) −27.3137 −0.878804
\(967\) − 10.1421i − 0.326149i −0.986614 0.163075i \(-0.947859\pi\)
0.986614 0.163075i \(-0.0521411\pi\)
\(968\) − 30.8995i − 0.993147i
\(969\) 10.3431 0.332270
\(970\) 0 0
\(971\) 7.31371 0.234708 0.117354 0.993090i \(-0.462559\pi\)
0.117354 + 0.993090i \(0.462559\pi\)
\(972\) 3.82843i 0.122797i
\(973\) 43.3137i 1.38857i
\(974\) 76.7696 2.45986
\(975\) 0 0
\(976\) −27.9411 −0.894374
\(977\) 13.8579i 0.443352i 0.975120 + 0.221676i \(0.0711528\pi\)
−0.975120 + 0.221676i \(0.928847\pi\)
\(978\) 31.7990i 1.01682i
\(979\) 18.3431 0.586249
\(980\) 0 0
\(981\) −17.3137 −0.552784
\(982\) 35.3137i 1.12691i
\(983\) − 2.68629i − 0.0856794i −0.999082 0.0428397i \(-0.986360\pi\)
0.999082 0.0428397i \(-0.0136405\pi\)
\(984\) 47.7990 1.52378
\(985\) 0 0
\(986\) 17.6569 0.562309
\(987\) 0.970563i 0.0308934i
\(988\) 10.8284i 0.344498i
\(989\) 38.6274 1.22828
\(990\) 0 0
\(991\) 27.3137 0.867649 0.433824 0.900998i \(-0.357164\pi\)
0.433824 + 0.900998i \(0.357164\pi\)
\(992\) − 10.8284i − 0.343803i
\(993\) 2.14214i 0.0679786i
\(994\) −13.6569 −0.433169
\(995\) 0 0
\(996\) −29.3137 −0.928840
\(997\) 51.2548i 1.62326i 0.584174 + 0.811628i \(0.301419\pi\)
−0.584174 + 0.811628i \(0.698581\pi\)
\(998\) − 5.17157i − 0.163703i
\(999\) −3.65685 −0.115698
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 975.2.c.h.274.1 4
3.2 odd 2 2925.2.c.u.2224.4 4
5.2 odd 4 975.2.a.l.1.2 2
5.3 odd 4 39.2.a.b.1.1 2
5.4 even 2 inner 975.2.c.h.274.4 4
15.2 even 4 2925.2.a.v.1.1 2
15.8 even 4 117.2.a.c.1.2 2
15.14 odd 2 2925.2.c.u.2224.1 4
20.3 even 4 624.2.a.k.1.2 2
35.13 even 4 1911.2.a.h.1.1 2
40.3 even 4 2496.2.a.bi.1.1 2
40.13 odd 4 2496.2.a.bf.1.1 2
45.13 odd 12 1053.2.e.m.703.2 4
45.23 even 12 1053.2.e.e.703.1 4
45.38 even 12 1053.2.e.e.352.1 4
45.43 odd 12 1053.2.e.m.352.2 4
55.43 even 4 4719.2.a.p.1.2 2
60.23 odd 4 1872.2.a.w.1.1 2
65.3 odd 12 507.2.e.h.22.2 4
65.8 even 4 507.2.b.e.337.1 4
65.18 even 4 507.2.b.e.337.4 4
65.23 odd 12 507.2.e.d.22.1 4
65.28 even 12 507.2.j.f.316.1 8
65.33 even 12 507.2.j.f.361.1 8
65.38 odd 4 507.2.a.h.1.2 2
65.43 odd 12 507.2.e.d.484.1 4
65.48 odd 12 507.2.e.h.484.2 4
65.58 even 12 507.2.j.f.361.4 8
65.63 even 12 507.2.j.f.316.4 8
105.83 odd 4 5733.2.a.u.1.2 2
120.53 even 4 7488.2.a.cl.1.2 2
120.83 odd 4 7488.2.a.co.1.2 2
195.8 odd 4 1521.2.b.j.1351.4 4
195.38 even 4 1521.2.a.f.1.1 2
195.83 odd 4 1521.2.b.j.1351.1 4
260.103 even 4 8112.2.a.bm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.b.1.1 2 5.3 odd 4
117.2.a.c.1.2 2 15.8 even 4
507.2.a.h.1.2 2 65.38 odd 4
507.2.b.e.337.1 4 65.8 even 4
507.2.b.e.337.4 4 65.18 even 4
507.2.e.d.22.1 4 65.23 odd 12
507.2.e.d.484.1 4 65.43 odd 12
507.2.e.h.22.2 4 65.3 odd 12
507.2.e.h.484.2 4 65.48 odd 12
507.2.j.f.316.1 8 65.28 even 12
507.2.j.f.316.4 8 65.63 even 12
507.2.j.f.361.1 8 65.33 even 12
507.2.j.f.361.4 8 65.58 even 12
624.2.a.k.1.2 2 20.3 even 4
975.2.a.l.1.2 2 5.2 odd 4
975.2.c.h.274.1 4 1.1 even 1 trivial
975.2.c.h.274.4 4 5.4 even 2 inner
1053.2.e.e.352.1 4 45.38 even 12
1053.2.e.e.703.1 4 45.23 even 12
1053.2.e.m.352.2 4 45.43 odd 12
1053.2.e.m.703.2 4 45.13 odd 12
1521.2.a.f.1.1 2 195.38 even 4
1521.2.b.j.1351.1 4 195.83 odd 4
1521.2.b.j.1351.4 4 195.8 odd 4
1872.2.a.w.1.1 2 60.23 odd 4
1911.2.a.h.1.1 2 35.13 even 4
2496.2.a.bf.1.1 2 40.13 odd 4
2496.2.a.bi.1.1 2 40.3 even 4
2925.2.a.v.1.1 2 15.2 even 4
2925.2.c.u.2224.1 4 15.14 odd 2
2925.2.c.u.2224.4 4 3.2 odd 2
4719.2.a.p.1.2 2 55.43 even 4
5733.2.a.u.1.2 2 105.83 odd 4
7488.2.a.cl.1.2 2 120.53 even 4
7488.2.a.co.1.2 2 120.83 odd 4
8112.2.a.bm.1.1 2 260.103 even 4