Properties

Label 275.2.b.d.199.1
Level $275$
Weight $2$
Character 275.199
Analytic conductor $2.196$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 275.199
Dual form 275.2.b.d.199.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421i q^{2} -2.82843i q^{3} -3.82843 q^{4} -6.82843 q^{6} +2.00000i q^{7} +4.41421i q^{8} -5.00000 q^{9} +O(q^{10})\) \(q-2.41421i q^{2} -2.82843i q^{3} -3.82843 q^{4} -6.82843 q^{6} +2.00000i q^{7} +4.41421i q^{8} -5.00000 q^{9} +1.00000 q^{11} +10.8284i q^{12} -1.17157i q^{13} +4.82843 q^{14} +3.00000 q^{16} -6.82843i q^{17} +12.0711i q^{18} +5.65685 q^{21} -2.41421i q^{22} -2.82843i q^{23} +12.4853 q^{24} -2.82843 q^{26} +5.65685i q^{27} -7.65685i q^{28} +3.65685 q^{29} +1.58579i q^{32} -2.82843i q^{33} -16.4853 q^{34} +19.1421 q^{36} +7.65685i q^{37} -3.31371 q^{39} +6.00000 q^{41} -13.6569i q^{42} -6.00000i q^{43} -3.82843 q^{44} -6.82843 q^{46} -2.82843i q^{47} -8.48528i q^{48} +3.00000 q^{49} -19.3137 q^{51} +4.48528i q^{52} +11.6569i q^{53} +13.6569 q^{54} -8.82843 q^{56} -8.82843i q^{58} -1.65685 q^{59} -9.31371 q^{61} -10.0000i q^{63} +9.82843 q^{64} -6.82843 q^{66} -12.4853i q^{67} +26.1421i q^{68} -8.00000 q^{69} +11.3137 q^{71} -22.0711i q^{72} -1.17157i q^{73} +18.4853 q^{74} +2.00000i q^{77} +8.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} -14.4853i q^{82} -6.00000i q^{83} -21.6569 q^{84} -14.4853 q^{86} -10.3431i q^{87} +4.41421i q^{88} +13.3137 q^{89} +2.34315 q^{91} +10.8284i q^{92} -6.82843 q^{94} +4.48528 q^{96} -3.65685i q^{97} -7.24264i q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 16 q^{6} - 20 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{4} - 16 q^{6} - 20 q^{9} + 4 q^{11} + 8 q^{14} + 12 q^{16} + 16 q^{24} - 8 q^{29} - 32 q^{34} + 20 q^{36} + 32 q^{39} + 24 q^{41} - 4 q^{44} - 16 q^{46} + 12 q^{49} - 32 q^{51} + 32 q^{54} - 24 q^{56} + 16 q^{59} + 8 q^{61} + 28 q^{64} - 16 q^{66} - 32 q^{69} + 40 q^{74} - 16 q^{79} + 4 q^{81} - 64 q^{84} - 24 q^{86} + 8 q^{89} + 32 q^{91} - 16 q^{94} - 16 q^{96} - 20 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.41421i − 1.70711i −0.521005 0.853553i \(-0.674443\pi\)
0.521005 0.853553i \(-0.325557\pi\)
\(3\) − 2.82843i − 1.63299i −0.577350 0.816497i \(-0.695913\pi\)
0.577350 0.816497i \(-0.304087\pi\)
\(4\) −3.82843 −1.91421
\(5\) 0 0
\(6\) −6.82843 −2.78769
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 4.41421i 1.56066i
\(9\) −5.00000 −1.66667
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 10.8284i 3.12590i
\(13\) − 1.17157i − 0.324936i −0.986714 0.162468i \(-0.948055\pi\)
0.986714 0.162468i \(-0.0519454\pi\)
\(14\) 4.82843 1.29045
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) − 6.82843i − 1.65614i −0.560627 0.828068i \(-0.689440\pi\)
0.560627 0.828068i \(-0.310560\pi\)
\(18\) 12.0711i 2.84518i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 5.65685 1.23443
\(22\) − 2.41421i − 0.514712i
\(23\) − 2.82843i − 0.589768i −0.955533 0.294884i \(-0.904719\pi\)
0.955533 0.294884i \(-0.0952810\pi\)
\(24\) 12.4853 2.54855
\(25\) 0 0
\(26\) −2.82843 −0.554700
\(27\) 5.65685i 1.08866i
\(28\) − 7.65685i − 1.44701i
\(29\) 3.65685 0.679061 0.339530 0.940595i \(-0.389732\pi\)
0.339530 + 0.940595i \(0.389732\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.58579i 0.280330i
\(33\) − 2.82843i − 0.492366i
\(34\) −16.4853 −2.82720
\(35\) 0 0
\(36\) 19.1421 3.19036
\(37\) 7.65685i 1.25878i 0.777090 + 0.629390i \(0.216695\pi\)
−0.777090 + 0.629390i \(0.783305\pi\)
\(38\) 0 0
\(39\) −3.31371 −0.530618
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) − 13.6569i − 2.10730i
\(43\) − 6.00000i − 0.914991i −0.889212 0.457496i \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) −3.82843 −0.577157
\(45\) 0 0
\(46\) −6.82843 −1.00680
\(47\) − 2.82843i − 0.412568i −0.978492 0.206284i \(-0.933863\pi\)
0.978492 0.206284i \(-0.0661372\pi\)
\(48\) − 8.48528i − 1.22474i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −19.3137 −2.70446
\(52\) 4.48528i 0.621997i
\(53\) 11.6569i 1.60119i 0.599204 + 0.800596i \(0.295484\pi\)
−0.599204 + 0.800596i \(0.704516\pi\)
\(54\) 13.6569 1.85846
\(55\) 0 0
\(56\) −8.82843 −1.17975
\(57\) 0 0
\(58\) − 8.82843i − 1.15923i
\(59\) −1.65685 −0.215704 −0.107852 0.994167i \(-0.534397\pi\)
−0.107852 + 0.994167i \(0.534397\pi\)
\(60\) 0 0
\(61\) −9.31371 −1.19250 −0.596249 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) 0 0
\(63\) − 10.0000i − 1.25988i
\(64\) 9.82843 1.22855
\(65\) 0 0
\(66\) −6.82843 −0.840521
\(67\) − 12.4853i − 1.52532i −0.646800 0.762660i \(-0.723893\pi\)
0.646800 0.762660i \(-0.276107\pi\)
\(68\) 26.1421i 3.17020i
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 11.3137 1.34269 0.671345 0.741145i \(-0.265717\pi\)
0.671345 + 0.741145i \(0.265717\pi\)
\(72\) − 22.0711i − 2.60110i
\(73\) − 1.17157i − 0.137122i −0.997647 0.0685611i \(-0.978159\pi\)
0.997647 0.0685611i \(-0.0218408\pi\)
\(74\) 18.4853 2.14887
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000i 0.227921i
\(78\) 8.00000i 0.905822i
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 14.4853i − 1.59963i
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) −21.6569 −2.36296
\(85\) 0 0
\(86\) −14.4853 −1.56199
\(87\) − 10.3431i − 1.10890i
\(88\) 4.41421i 0.470557i
\(89\) 13.3137 1.41125 0.705625 0.708585i \(-0.250666\pi\)
0.705625 + 0.708585i \(0.250666\pi\)
\(90\) 0 0
\(91\) 2.34315 0.245628
\(92\) 10.8284i 1.12894i
\(93\) 0 0
\(94\) −6.82843 −0.704298
\(95\) 0 0
\(96\) 4.48528 0.457777
\(97\) − 3.65685i − 0.371297i −0.982616 0.185649i \(-0.940561\pi\)
0.982616 0.185649i \(-0.0594386\pi\)
\(98\) − 7.24264i − 0.731617i
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) 9.31371 0.926749 0.463374 0.886163i \(-0.346639\pi\)
0.463374 + 0.886163i \(0.346639\pi\)
\(102\) 46.6274i 4.61680i
\(103\) 6.82843i 0.672825i 0.941715 + 0.336412i \(0.109214\pi\)
−0.941715 + 0.336412i \(0.890786\pi\)
\(104\) 5.17157 0.507114
\(105\) 0 0
\(106\) 28.1421 2.73341
\(107\) − 7.65685i − 0.740216i −0.928989 0.370108i \(-0.879321\pi\)
0.928989 0.370108i \(-0.120679\pi\)
\(108\) − 21.6569i − 2.08393i
\(109\) 7.65685 0.733394 0.366697 0.930341i \(-0.380489\pi\)
0.366697 + 0.930341i \(0.380489\pi\)
\(110\) 0 0
\(111\) 21.6569 2.05558
\(112\) 6.00000i 0.566947i
\(113\) 19.6569i 1.84916i 0.380986 + 0.924581i \(0.375584\pi\)
−0.380986 + 0.924581i \(0.624416\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −14.0000 −1.29987
\(117\) 5.85786i 0.541560i
\(118\) 4.00000i 0.368230i
\(119\) 13.6569 1.25192
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 22.4853i 2.03572i
\(123\) − 16.9706i − 1.53018i
\(124\) 0 0
\(125\) 0 0
\(126\) −24.1421 −2.15075
\(127\) − 4.34315i − 0.385392i −0.981259 0.192696i \(-0.938277\pi\)
0.981259 0.192696i \(-0.0617231\pi\)
\(128\) − 20.5563i − 1.81694i
\(129\) −16.9706 −1.49417
\(130\) 0 0
\(131\) −11.3137 −0.988483 −0.494242 0.869325i \(-0.664554\pi\)
−0.494242 + 0.869325i \(0.664554\pi\)
\(132\) 10.8284i 0.942494i
\(133\) 0 0
\(134\) −30.1421 −2.60388
\(135\) 0 0
\(136\) 30.1421 2.58467
\(137\) 10.9706i 0.937278i 0.883390 + 0.468639i \(0.155256\pi\)
−0.883390 + 0.468639i \(0.844744\pi\)
\(138\) 19.3137i 1.64409i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) − 27.3137i − 2.29212i
\(143\) − 1.17157i − 0.0979718i
\(144\) −15.0000 −1.25000
\(145\) 0 0
\(146\) −2.82843 −0.234082
\(147\) − 8.48528i − 0.699854i
\(148\) − 29.3137i − 2.40957i
\(149\) −0.343146 −0.0281116 −0.0140558 0.999901i \(-0.504474\pi\)
−0.0140558 + 0.999901i \(0.504474\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 34.1421i 2.76023i
\(154\) 4.82843 0.389086
\(155\) 0 0
\(156\) 12.6863 1.01572
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) 9.65685i 0.768258i
\(159\) 32.9706 2.61474
\(160\) 0 0
\(161\) 5.65685 0.445823
\(162\) − 2.41421i − 0.189679i
\(163\) 16.4853i 1.29123i 0.763664 + 0.645613i \(0.223398\pi\)
−0.763664 + 0.645613i \(0.776602\pi\)
\(164\) −22.9706 −1.79370
\(165\) 0 0
\(166\) −14.4853 −1.12428
\(167\) 22.9706i 1.77752i 0.458377 + 0.888758i \(0.348431\pi\)
−0.458377 + 0.888758i \(0.651569\pi\)
\(168\) 24.9706i 1.92652i
\(169\) 11.6274 0.894417
\(170\) 0 0
\(171\) 0 0
\(172\) 22.9706i 1.75149i
\(173\) − 22.1421i − 1.68344i −0.539918 0.841718i \(-0.681545\pi\)
0.539918 0.841718i \(-0.318455\pi\)
\(174\) −24.9706 −1.89301
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 4.68629i 0.352243i
\(178\) − 32.1421i − 2.40915i
\(179\) −9.65685 −0.721787 −0.360894 0.932607i \(-0.617528\pi\)
−0.360894 + 0.932607i \(0.617528\pi\)
\(180\) 0 0
\(181\) 21.3137 1.58424 0.792118 0.610368i \(-0.208979\pi\)
0.792118 + 0.610368i \(0.208979\pi\)
\(182\) − 5.65685i − 0.419314i
\(183\) 26.3431i 1.94734i
\(184\) 12.4853 0.920427
\(185\) 0 0
\(186\) 0 0
\(187\) − 6.82843i − 0.499344i
\(188\) 10.8284i 0.789744i
\(189\) −11.3137 −0.822951
\(190\) 0 0
\(191\) 3.31371 0.239772 0.119886 0.992788i \(-0.461747\pi\)
0.119886 + 0.992788i \(0.461747\pi\)
\(192\) − 27.7990i − 2.00622i
\(193\) − 1.17157i − 0.0843317i −0.999111 0.0421658i \(-0.986574\pi\)
0.999111 0.0421658i \(-0.0134258\pi\)
\(194\) −8.82843 −0.633844
\(195\) 0 0
\(196\) −11.4853 −0.820377
\(197\) 10.8284i 0.771493i 0.922605 + 0.385747i \(0.126056\pi\)
−0.922605 + 0.385747i \(0.873944\pi\)
\(198\) 12.0711i 0.857853i
\(199\) −10.3431 −0.733206 −0.366603 0.930377i \(-0.619479\pi\)
−0.366603 + 0.930377i \(0.619479\pi\)
\(200\) 0 0
\(201\) −35.3137 −2.49084
\(202\) − 22.4853i − 1.58206i
\(203\) 7.31371i 0.513322i
\(204\) 73.9411 5.17691
\(205\) 0 0
\(206\) 16.4853 1.14858
\(207\) 14.1421i 0.982946i
\(208\) − 3.51472i − 0.243702i
\(209\) 0 0
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) − 44.6274i − 3.06502i
\(213\) − 32.0000i − 2.19260i
\(214\) −18.4853 −1.26363
\(215\) 0 0
\(216\) −24.9706 −1.69903
\(217\) 0 0
\(218\) − 18.4853i − 1.25198i
\(219\) −3.31371 −0.223920
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) − 52.2843i − 3.50909i
\(223\) − 10.8284i − 0.725125i −0.931959 0.362563i \(-0.881902\pi\)
0.931959 0.362563i \(-0.118098\pi\)
\(224\) −3.17157 −0.211910
\(225\) 0 0
\(226\) 47.4558 3.15672
\(227\) − 25.3137i − 1.68013i −0.542486 0.840065i \(-0.682517\pi\)
0.542486 0.840065i \(-0.317483\pi\)
\(228\) 0 0
\(229\) −1.31371 −0.0868123 −0.0434062 0.999058i \(-0.513821\pi\)
−0.0434062 + 0.999058i \(0.513821\pi\)
\(230\) 0 0
\(231\) 5.65685 0.372194
\(232\) 16.1421i 1.05978i
\(233\) − 6.14214i − 0.402385i −0.979552 0.201192i \(-0.935518\pi\)
0.979552 0.201192i \(-0.0644816\pi\)
\(234\) 14.1421 0.924500
\(235\) 0 0
\(236\) 6.34315 0.412904
\(237\) 11.3137i 0.734904i
\(238\) − 32.9706i − 2.13716i
\(239\) 23.3137 1.50804 0.754019 0.656852i \(-0.228113\pi\)
0.754019 + 0.656852i \(0.228113\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) − 2.41421i − 0.155192i
\(243\) 14.1421i 0.907218i
\(244\) 35.6569 2.28270
\(245\) 0 0
\(246\) −40.9706 −2.61219
\(247\) 0 0
\(248\) 0 0
\(249\) −16.9706 −1.07547
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 38.2843i 2.41168i
\(253\) − 2.82843i − 0.177822i
\(254\) −10.4853 −0.657905
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) 9.31371i 0.580973i 0.956879 + 0.290487i \(0.0938172\pi\)
−0.956879 + 0.290487i \(0.906183\pi\)
\(258\) 40.9706i 2.55072i
\(259\) −15.3137 −0.951548
\(260\) 0 0
\(261\) −18.2843 −1.13177
\(262\) 27.3137i 1.68745i
\(263\) − 10.9706i − 0.676474i −0.941061 0.338237i \(-0.890169\pi\)
0.941061 0.338237i \(-0.109831\pi\)
\(264\) 12.4853 0.768416
\(265\) 0 0
\(266\) 0 0
\(267\) − 37.6569i − 2.30456i
\(268\) 47.7990i 2.91979i
\(269\) −17.3137 −1.05564 −0.527818 0.849358i \(-0.676990\pi\)
−0.527818 + 0.849358i \(0.676990\pi\)
\(270\) 0 0
\(271\) 7.31371 0.444276 0.222138 0.975015i \(-0.428696\pi\)
0.222138 + 0.975015i \(0.428696\pi\)
\(272\) − 20.4853i − 1.24210i
\(273\) − 6.62742i − 0.401110i
\(274\) 26.4853 1.60003
\(275\) 0 0
\(276\) 30.6274 1.84355
\(277\) − 6.82843i − 0.410280i −0.978733 0.205140i \(-0.934235\pi\)
0.978733 0.205140i \(-0.0657650\pi\)
\(278\) − 9.65685i − 0.579180i
\(279\) 0 0
\(280\) 0 0
\(281\) 17.3137 1.03285 0.516425 0.856333i \(-0.327263\pi\)
0.516425 + 0.856333i \(0.327263\pi\)
\(282\) 19.3137i 1.15011i
\(283\) 32.6274i 1.93950i 0.244103 + 0.969749i \(0.421507\pi\)
−0.244103 + 0.969749i \(0.578493\pi\)
\(284\) −43.3137 −2.57020
\(285\) 0 0
\(286\) −2.82843 −0.167248
\(287\) 12.0000i 0.708338i
\(288\) − 7.92893i − 0.467217i
\(289\) −29.6274 −1.74279
\(290\) 0 0
\(291\) −10.3431 −0.606326
\(292\) 4.48528i 0.262481i
\(293\) − 9.17157i − 0.535809i −0.963445 0.267905i \(-0.913669\pi\)
0.963445 0.267905i \(-0.0863312\pi\)
\(294\) −20.4853 −1.19473
\(295\) 0 0
\(296\) −33.7990 −1.96453
\(297\) 5.65685i 0.328244i
\(298\) 0.828427i 0.0479895i
\(299\) −3.31371 −0.191637
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 28.9706i 1.66707i
\(303\) − 26.3431i − 1.51337i
\(304\) 0 0
\(305\) 0 0
\(306\) 82.4264 4.71200
\(307\) 16.3431i 0.932753i 0.884586 + 0.466376i \(0.154441\pi\)
−0.884586 + 0.466376i \(0.845559\pi\)
\(308\) − 7.65685i − 0.436290i
\(309\) 19.3137 1.09872
\(310\) 0 0
\(311\) 4.68629 0.265735 0.132868 0.991134i \(-0.457581\pi\)
0.132868 + 0.991134i \(0.457581\pi\)
\(312\) − 14.6274i − 0.828114i
\(313\) − 1.31371i − 0.0742552i −0.999311 0.0371276i \(-0.988179\pi\)
0.999311 0.0371276i \(-0.0118208\pi\)
\(314\) 33.7990 1.90739
\(315\) 0 0
\(316\) 15.3137 0.861463
\(317\) 1.31371i 0.0737852i 0.999319 + 0.0368926i \(0.0117459\pi\)
−0.999319 + 0.0368926i \(0.988254\pi\)
\(318\) − 79.5980i − 4.46363i
\(319\) 3.65685 0.204745
\(320\) 0 0
\(321\) −21.6569 −1.20877
\(322\) − 13.6569i − 0.761067i
\(323\) 0 0
\(324\) −3.82843 −0.212690
\(325\) 0 0
\(326\) 39.7990 2.20426
\(327\) − 21.6569i − 1.19763i
\(328\) 26.4853i 1.46241i
\(329\) 5.65685 0.311872
\(330\) 0 0
\(331\) −7.31371 −0.401998 −0.200999 0.979591i \(-0.564419\pi\)
−0.200999 + 0.979591i \(0.564419\pi\)
\(332\) 22.9706i 1.26067i
\(333\) − 38.2843i − 2.09797i
\(334\) 55.4558 3.03441
\(335\) 0 0
\(336\) 16.9706 0.925820
\(337\) 20.4853i 1.11590i 0.829873 + 0.557952i \(0.188413\pi\)
−0.829873 + 0.557952i \(0.811587\pi\)
\(338\) − 28.0711i − 1.52686i
\(339\) 55.5980 3.01967
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 26.4853 1.42799
\(345\) 0 0
\(346\) −53.4558 −2.87380
\(347\) − 10.9706i − 0.588931i −0.955662 0.294465i \(-0.904858\pi\)
0.955662 0.294465i \(-0.0951416\pi\)
\(348\) 39.5980i 2.12267i
\(349\) −26.9706 −1.44370 −0.721851 0.692049i \(-0.756708\pi\)
−0.721851 + 0.692049i \(0.756708\pi\)
\(350\) 0 0
\(351\) 6.62742 0.353745
\(352\) 1.58579i 0.0845227i
\(353\) 21.3137i 1.13441i 0.823575 + 0.567207i \(0.191976\pi\)
−0.823575 + 0.567207i \(0.808024\pi\)
\(354\) 11.3137 0.601317
\(355\) 0 0
\(356\) −50.9706 −2.70143
\(357\) − 38.6274i − 2.04438i
\(358\) 23.3137i 1.23217i
\(359\) −0.686292 −0.0362211 −0.0181105 0.999836i \(-0.505765\pi\)
−0.0181105 + 0.999836i \(0.505765\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) − 51.4558i − 2.70446i
\(363\) − 2.82843i − 0.148454i
\(364\) −8.97056 −0.470185
\(365\) 0 0
\(366\) 63.5980 3.32432
\(367\) 8.48528i 0.442928i 0.975169 + 0.221464i \(0.0710835\pi\)
−0.975169 + 0.221464i \(0.928916\pi\)
\(368\) − 8.48528i − 0.442326i
\(369\) −30.0000 −1.56174
\(370\) 0 0
\(371\) −23.3137 −1.21039
\(372\) 0 0
\(373\) 35.7990i 1.85360i 0.375554 + 0.926801i \(0.377453\pi\)
−0.375554 + 0.926801i \(0.622547\pi\)
\(374\) −16.4853 −0.852434
\(375\) 0 0
\(376\) 12.4853 0.643879
\(377\) − 4.28427i − 0.220651i
\(378\) 27.3137i 1.40487i
\(379\) −33.6569 −1.72884 −0.864418 0.502773i \(-0.832313\pi\)
−0.864418 + 0.502773i \(0.832313\pi\)
\(380\) 0 0
\(381\) −12.2843 −0.629342
\(382\) − 8.00000i − 0.409316i
\(383\) − 5.85786i − 0.299323i −0.988737 0.149661i \(-0.952182\pi\)
0.988737 0.149661i \(-0.0478184\pi\)
\(384\) −58.1421 −2.96705
\(385\) 0 0
\(386\) −2.82843 −0.143963
\(387\) 30.0000i 1.52499i
\(388\) 14.0000i 0.710742i
\(389\) −20.6274 −1.04585 −0.522926 0.852378i \(-0.675160\pi\)
−0.522926 + 0.852378i \(0.675160\pi\)
\(390\) 0 0
\(391\) −19.3137 −0.976736
\(392\) 13.2426i 0.668854i
\(393\) 32.0000i 1.61419i
\(394\) 26.1421 1.31702
\(395\) 0 0
\(396\) 19.1421 0.961929
\(397\) 9.31371i 0.467442i 0.972304 + 0.233721i \(0.0750902\pi\)
−0.972304 + 0.233721i \(0.924910\pi\)
\(398\) 24.9706i 1.25166i
\(399\) 0 0
\(400\) 0 0
\(401\) −5.31371 −0.265354 −0.132677 0.991159i \(-0.542357\pi\)
−0.132677 + 0.991159i \(0.542357\pi\)
\(402\) 85.2548i 4.25212i
\(403\) 0 0
\(404\) −35.6569 −1.77399
\(405\) 0 0
\(406\) 17.6569 0.876295
\(407\) 7.65685i 0.379536i
\(408\) − 85.2548i − 4.22074i
\(409\) −1.02944 −0.0509024 −0.0254512 0.999676i \(-0.508102\pi\)
−0.0254512 + 0.999676i \(0.508102\pi\)
\(410\) 0 0
\(411\) 31.0294 1.53057
\(412\) − 26.1421i − 1.28793i
\(413\) − 3.31371i − 0.163057i
\(414\) 34.1421 1.67799
\(415\) 0 0
\(416\) 1.85786 0.0910893
\(417\) − 11.3137i − 0.554035i
\(418\) 0 0
\(419\) 25.6569 1.25342 0.626710 0.779253i \(-0.284401\pi\)
0.626710 + 0.779253i \(0.284401\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 38.6274i 1.88035i
\(423\) 14.1421i 0.687614i
\(424\) −51.4558 −2.49892
\(425\) 0 0
\(426\) −77.2548 −3.74301
\(427\) − 18.6274i − 0.901444i
\(428\) 29.3137i 1.41693i
\(429\) −3.31371 −0.159987
\(430\) 0 0
\(431\) −11.3137 −0.544962 −0.272481 0.962161i \(-0.587844\pi\)
−0.272481 + 0.962161i \(0.587844\pi\)
\(432\) 16.9706i 0.816497i
\(433\) − 7.65685i − 0.367965i −0.982930 0.183982i \(-0.941101\pi\)
0.982930 0.183982i \(-0.0588990\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −29.3137 −1.40387
\(437\) 0 0
\(438\) 8.00000i 0.382255i
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) 19.3137i 0.918659i
\(443\) − 26.8284i − 1.27466i −0.770592 0.637329i \(-0.780039\pi\)
0.770592 0.637329i \(-0.219961\pi\)
\(444\) −82.9117 −3.93481
\(445\) 0 0
\(446\) −26.1421 −1.23787
\(447\) 0.970563i 0.0459060i
\(448\) 19.6569i 0.928699i
\(449\) −28.6274 −1.35101 −0.675506 0.737355i \(-0.736075\pi\)
−0.675506 + 0.737355i \(0.736075\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) − 75.2548i − 3.53969i
\(453\) 33.9411i 1.59469i
\(454\) −61.1127 −2.86816
\(455\) 0 0
\(456\) 0 0
\(457\) − 0.485281i − 0.0227005i −0.999936 0.0113503i \(-0.996387\pi\)
0.999936 0.0113503i \(-0.00361298\pi\)
\(458\) 3.17157i 0.148198i
\(459\) 38.6274 1.80297
\(460\) 0 0
\(461\) 12.6274 0.588117 0.294059 0.955787i \(-0.404994\pi\)
0.294059 + 0.955787i \(0.404994\pi\)
\(462\) − 13.6569i − 0.635374i
\(463\) − 6.14214i − 0.285449i −0.989762 0.142725i \(-0.954414\pi\)
0.989762 0.142725i \(-0.0455863\pi\)
\(464\) 10.9706 0.509296
\(465\) 0 0
\(466\) −14.8284 −0.686914
\(467\) 14.8284i 0.686178i 0.939303 + 0.343089i \(0.111473\pi\)
−0.939303 + 0.343089i \(0.888527\pi\)
\(468\) − 22.4264i − 1.03666i
\(469\) 24.9706 1.15303
\(470\) 0 0
\(471\) 39.5980 1.82458
\(472\) − 7.31371i − 0.336641i
\(473\) − 6.00000i − 0.275880i
\(474\) 27.3137 1.25456
\(475\) 0 0
\(476\) −52.2843 −2.39645
\(477\) − 58.2843i − 2.66865i
\(478\) − 56.2843i − 2.57438i
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) 8.97056 0.409022
\(482\) − 14.4853i − 0.659786i
\(483\) − 16.0000i − 0.728025i
\(484\) −3.82843 −0.174019
\(485\) 0 0
\(486\) 34.1421 1.54872
\(487\) 24.4853i 1.10953i 0.832006 + 0.554767i \(0.187193\pi\)
−0.832006 + 0.554767i \(0.812807\pi\)
\(488\) − 41.1127i − 1.86108i
\(489\) 46.6274 2.10856
\(490\) 0 0
\(491\) −0.686292 −0.0309719 −0.0154860 0.999880i \(-0.504930\pi\)
−0.0154860 + 0.999880i \(0.504930\pi\)
\(492\) 64.9706i 2.92910i
\(493\) − 24.9706i − 1.12462i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 22.6274i 1.01498i
\(498\) 40.9706i 1.83593i
\(499\) −9.65685 −0.432300 −0.216150 0.976360i \(-0.569350\pi\)
−0.216150 + 0.976360i \(0.569350\pi\)
\(500\) 0 0
\(501\) 64.9706 2.90267
\(502\) − 28.9706i − 1.29302i
\(503\) 16.6274i 0.741380i 0.928757 + 0.370690i \(0.120879\pi\)
−0.928757 + 0.370690i \(0.879121\pi\)
\(504\) 44.1421 1.96625
\(505\) 0 0
\(506\) −6.82843 −0.303561
\(507\) − 32.8873i − 1.46058i
\(508\) 16.6274i 0.737722i
\(509\) 13.3137 0.590120 0.295060 0.955479i \(-0.404660\pi\)
0.295060 + 0.955479i \(0.404660\pi\)
\(510\) 0 0
\(511\) 2.34315 0.103655
\(512\) 31.2426i 1.38074i
\(513\) 0 0
\(514\) 22.4853 0.991783
\(515\) 0 0
\(516\) 64.9706 2.86017
\(517\) − 2.82843i − 0.124394i
\(518\) 36.9706i 1.62439i
\(519\) −62.6274 −2.74904
\(520\) 0 0
\(521\) 25.3137 1.10901 0.554507 0.832179i \(-0.312907\pi\)
0.554507 + 0.832179i \(0.312907\pi\)
\(522\) 44.1421i 1.93205i
\(523\) − 41.5980i − 1.81895i −0.415756 0.909476i \(-0.636483\pi\)
0.415756 0.909476i \(-0.363517\pi\)
\(524\) 43.3137 1.89217
\(525\) 0 0
\(526\) −26.4853 −1.15481
\(527\) 0 0
\(528\) − 8.48528i − 0.369274i
\(529\) 15.0000 0.652174
\(530\) 0 0
\(531\) 8.28427 0.359507
\(532\) 0 0
\(533\) − 7.02944i − 0.304479i
\(534\) −90.9117 −3.93413
\(535\) 0 0
\(536\) 55.1127 2.38051
\(537\) 27.3137i 1.17867i
\(538\) 41.7990i 1.80208i
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) − 17.6569i − 0.758427i
\(543\) − 60.2843i − 2.58705i
\(544\) 10.8284 0.464265
\(545\) 0 0
\(546\) −16.0000 −0.684737
\(547\) 34.0000i 1.45374i 0.686778 + 0.726868i \(0.259025\pi\)
−0.686778 + 0.726868i \(0.740975\pi\)
\(548\) − 42.0000i − 1.79415i
\(549\) 46.5685 1.98750
\(550\) 0 0
\(551\) 0 0
\(552\) − 35.3137i − 1.50305i
\(553\) − 8.00000i − 0.340195i
\(554\) −16.4853 −0.700392
\(555\) 0 0
\(556\) −15.3137 −0.649446
\(557\) − 9.85786i − 0.417691i −0.977949 0.208846i \(-0.933029\pi\)
0.977949 0.208846i \(-0.0669706\pi\)
\(558\) 0 0
\(559\) −7.02944 −0.297314
\(560\) 0 0
\(561\) −19.3137 −0.815425
\(562\) − 41.7990i − 1.76318i
\(563\) 0.343146i 0.0144619i 0.999974 + 0.00723093i \(0.00230170\pi\)
−0.999974 + 0.00723093i \(0.997698\pi\)
\(564\) 30.6274 1.28965
\(565\) 0 0
\(566\) 78.7696 3.31093
\(567\) 2.00000i 0.0839921i
\(568\) 49.9411i 2.09548i
\(569\) −31.6569 −1.32712 −0.663562 0.748121i \(-0.730956\pi\)
−0.663562 + 0.748121i \(0.730956\pi\)
\(570\) 0 0
\(571\) −21.9411 −0.918208 −0.459104 0.888383i \(-0.651829\pi\)
−0.459104 + 0.888383i \(0.651829\pi\)
\(572\) 4.48528i 0.187539i
\(573\) − 9.37258i − 0.391545i
\(574\) 28.9706 1.20921
\(575\) 0 0
\(576\) −49.1421 −2.04759
\(577\) 26.9706i 1.12280i 0.827545 + 0.561400i \(0.189737\pi\)
−0.827545 + 0.561400i \(0.810263\pi\)
\(578\) 71.5269i 2.97513i
\(579\) −3.31371 −0.137713
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 24.9706i 1.03506i
\(583\) 11.6569i 0.482778i
\(584\) 5.17157 0.214001
\(585\) 0 0
\(586\) −22.1421 −0.914683
\(587\) 2.14214i 0.0884154i 0.999022 + 0.0442077i \(0.0140763\pi\)
−0.999022 + 0.0442077i \(0.985924\pi\)
\(588\) 32.4853i 1.33967i
\(589\) 0 0
\(590\) 0 0
\(591\) 30.6274 1.25984
\(592\) 22.9706i 0.944084i
\(593\) 3.51472i 0.144332i 0.997393 + 0.0721661i \(0.0229912\pi\)
−0.997393 + 0.0721661i \(0.977009\pi\)
\(594\) 13.6569 0.560348
\(595\) 0 0
\(596\) 1.31371 0.0538116
\(597\) 29.2548i 1.19732i
\(598\) 8.00000i 0.327144i
\(599\) 5.65685 0.231133 0.115566 0.993300i \(-0.463132\pi\)
0.115566 + 0.993300i \(0.463132\pi\)
\(600\) 0 0
\(601\) 23.9411 0.976579 0.488289 0.872682i \(-0.337621\pi\)
0.488289 + 0.872682i \(0.337621\pi\)
\(602\) − 28.9706i − 1.18075i
\(603\) 62.4264i 2.54220i
\(604\) 45.9411 1.86932
\(605\) 0 0
\(606\) −63.5980 −2.58349
\(607\) − 38.2843i − 1.55391i −0.629556 0.776955i \(-0.716763\pi\)
0.629556 0.776955i \(-0.283237\pi\)
\(608\) 0 0
\(609\) 20.6863 0.838251
\(610\) 0 0
\(611\) −3.31371 −0.134058
\(612\) − 130.711i − 5.28367i
\(613\) − 25.4558i − 1.02815i −0.857745 0.514076i \(-0.828135\pi\)
0.857745 0.514076i \(-0.171865\pi\)
\(614\) 39.4558 1.59231
\(615\) 0 0
\(616\) −8.82843 −0.355707
\(617\) − 0.343146i − 0.0138145i −0.999976 0.00690726i \(-0.997801\pi\)
0.999976 0.00690726i \(-0.00219867\pi\)
\(618\) − 46.6274i − 1.87563i
\(619\) 14.3431 0.576500 0.288250 0.957555i \(-0.406927\pi\)
0.288250 + 0.957555i \(0.406927\pi\)
\(620\) 0 0
\(621\) 16.0000 0.642058
\(622\) − 11.3137i − 0.453638i
\(623\) 26.6274i 1.06680i
\(624\) −9.94113 −0.397964
\(625\) 0 0
\(626\) −3.17157 −0.126762
\(627\) 0 0
\(628\) − 53.5980i − 2.13879i
\(629\) 52.2843 2.08471
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) − 17.6569i − 0.702352i
\(633\) 45.2548i 1.79872i
\(634\) 3.17157 0.125959
\(635\) 0 0
\(636\) −126.225 −5.00516
\(637\) − 3.51472i − 0.139258i
\(638\) − 8.82843i − 0.349521i
\(639\) −56.5685 −2.23782
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 52.2843i 2.06350i
\(643\) − 1.45584i − 0.0574129i −0.999588 0.0287064i \(-0.990861\pi\)
0.999588 0.0287064i \(-0.00913880\pi\)
\(644\) −21.6569 −0.853400
\(645\) 0 0
\(646\) 0 0
\(647\) − 27.1127i − 1.06591i −0.846144 0.532955i \(-0.821081\pi\)
0.846144 0.532955i \(-0.178919\pi\)
\(648\) 4.41421i 0.173407i
\(649\) −1.65685 −0.0650372
\(650\) 0 0
\(651\) 0 0
\(652\) − 63.1127i − 2.47168i
\(653\) 11.6569i 0.456168i 0.973641 + 0.228084i \(0.0732461\pi\)
−0.973641 + 0.228084i \(0.926754\pi\)
\(654\) −52.2843 −2.04448
\(655\) 0 0
\(656\) 18.0000 0.702782
\(657\) 5.85786i 0.228537i
\(658\) − 13.6569i − 0.532400i
\(659\) −45.9411 −1.78961 −0.894806 0.446455i \(-0.852686\pi\)
−0.894806 + 0.446455i \(0.852686\pi\)
\(660\) 0 0
\(661\) 44.6274 1.73581 0.867903 0.496734i \(-0.165468\pi\)
0.867903 + 0.496734i \(0.165468\pi\)
\(662\) 17.6569i 0.686253i
\(663\) 22.6274i 0.878776i
\(664\) 26.4853 1.02783
\(665\) 0 0
\(666\) −92.4264 −3.58145
\(667\) − 10.3431i − 0.400488i
\(668\) − 87.9411i − 3.40254i
\(669\) −30.6274 −1.18412
\(670\) 0 0
\(671\) −9.31371 −0.359552
\(672\) 8.97056i 0.346047i
\(673\) − 12.4853i − 0.481272i −0.970615 0.240636i \(-0.922644\pi\)
0.970615 0.240636i \(-0.0773560\pi\)
\(674\) 49.4558 1.90497
\(675\) 0 0
\(676\) −44.5147 −1.71210
\(677\) − 22.8284i − 0.877368i −0.898641 0.438684i \(-0.855445\pi\)
0.898641 0.438684i \(-0.144555\pi\)
\(678\) − 134.225i − 5.15490i
\(679\) 7.31371 0.280674
\(680\) 0 0
\(681\) −71.5980 −2.74364
\(682\) 0 0
\(683\) − 7.79899i − 0.298420i −0.988806 0.149210i \(-0.952327\pi\)
0.988806 0.149210i \(-0.0476731\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 48.2843 1.84350
\(687\) 3.71573i 0.141764i
\(688\) − 18.0000i − 0.686244i
\(689\) 13.6569 0.520285
\(690\) 0 0
\(691\) −39.3137 −1.49556 −0.747782 0.663944i \(-0.768881\pi\)
−0.747782 + 0.663944i \(0.768881\pi\)
\(692\) 84.7696i 3.22245i
\(693\) − 10.0000i − 0.379869i
\(694\) −26.4853 −1.00537
\(695\) 0 0
\(696\) 45.6569 1.73062
\(697\) − 40.9706i − 1.55187i
\(698\) 65.1127i 2.46455i
\(699\) −17.3726 −0.657091
\(700\) 0 0
\(701\) −12.6274 −0.476931 −0.238465 0.971151i \(-0.576644\pi\)
−0.238465 + 0.971151i \(0.576644\pi\)
\(702\) − 16.0000i − 0.603881i
\(703\) 0 0
\(704\) 9.82843 0.370423
\(705\) 0 0
\(706\) 51.4558 1.93657
\(707\) 18.6274i 0.700556i
\(708\) − 17.9411i − 0.674269i
\(709\) −24.6274 −0.924902 −0.462451 0.886645i \(-0.653030\pi\)
−0.462451 + 0.886645i \(0.653030\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 58.7696i 2.20248i
\(713\) 0 0
\(714\) −93.2548 −3.48997
\(715\) 0 0
\(716\) 36.9706 1.38165
\(717\) − 65.9411i − 2.46262i
\(718\) 1.65685i 0.0618333i
\(719\) −18.3431 −0.684084 −0.342042 0.939685i \(-0.611118\pi\)
−0.342042 + 0.939685i \(0.611118\pi\)
\(720\) 0 0
\(721\) −13.6569 −0.508608
\(722\) 45.8701i 1.70711i
\(723\) − 16.9706i − 0.631142i
\(724\) −81.5980 −3.03257
\(725\) 0 0
\(726\) −6.82843 −0.253427
\(727\) 19.5147i 0.723761i 0.932225 + 0.361880i \(0.117865\pi\)
−0.932225 + 0.361880i \(0.882135\pi\)
\(728\) 10.3431i 0.383342i
\(729\) 43.0000 1.59259
\(730\) 0 0
\(731\) −40.9706 −1.51535
\(732\) − 100.853i − 3.72763i
\(733\) − 17.4558i − 0.644746i −0.946613 0.322373i \(-0.895519\pi\)
0.946613 0.322373i \(-0.104481\pi\)
\(734\) 20.4853 0.756126
\(735\) 0 0
\(736\) 4.48528 0.165330
\(737\) − 12.4853i − 0.459901i
\(738\) 72.4264i 2.66605i
\(739\) −29.9411 −1.10140 −0.550701 0.834703i \(-0.685640\pi\)
−0.550701 + 0.834703i \(0.685640\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 56.2843i 2.06626i
\(743\) − 49.5980i − 1.81957i −0.415076 0.909787i \(-0.636245\pi\)
0.415076 0.909787i \(-0.363755\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 86.4264 3.16430
\(747\) 30.0000i 1.09764i
\(748\) 26.1421i 0.955851i
\(749\) 15.3137 0.559551
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) − 8.48528i − 0.309426i
\(753\) − 33.9411i − 1.23688i
\(754\) −10.3431 −0.376675
\(755\) 0 0
\(756\) 43.3137 1.57530
\(757\) − 13.3137i − 0.483895i −0.970289 0.241947i \(-0.922214\pi\)
0.970289 0.241947i \(-0.0777862\pi\)
\(758\) 81.2548i 2.95131i
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 29.6569i 1.07435i
\(763\) 15.3137i 0.554393i
\(764\) −12.6863 −0.458974
\(765\) 0 0
\(766\) −14.1421 −0.510976
\(767\) 1.94113i 0.0700900i
\(768\) 84.7696i 3.05886i
\(769\) 18.9706 0.684096 0.342048 0.939682i \(-0.388879\pi\)
0.342048 + 0.939682i \(0.388879\pi\)
\(770\) 0 0
\(771\) 26.3431 0.948725
\(772\) 4.48528i 0.161429i
\(773\) 26.2843i 0.945380i 0.881229 + 0.472690i \(0.156717\pi\)
−0.881229 + 0.472690i \(0.843283\pi\)
\(774\) 72.4264 2.60331
\(775\) 0 0
\(776\) 16.1421 0.579469
\(777\) 43.3137i 1.55387i
\(778\) 49.7990i 1.78538i
\(779\) 0 0
\(780\) 0 0
\(781\) 11.3137 0.404836
\(782\) 46.6274i 1.66739i
\(783\) 20.6863i 0.739268i
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 77.2548 2.75559
\(787\) 14.9706i 0.533643i 0.963746 + 0.266821i \(0.0859734\pi\)
−0.963746 + 0.266821i \(0.914027\pi\)
\(788\) − 41.4558i − 1.47680i
\(789\) −31.0294 −1.10468
\(790\) 0 0
\(791\) −39.3137 −1.39783
\(792\) − 22.0711i − 0.784261i
\(793\) 10.9117i 0.387485i
\(794\) 22.4853 0.797973
\(795\) 0 0
\(796\) 39.5980 1.40351
\(797\) − 32.6274i − 1.15572i −0.816135 0.577861i \(-0.803887\pi\)
0.816135 0.577861i \(-0.196113\pi\)
\(798\) 0 0
\(799\) −19.3137 −0.683270
\(800\) 0 0
\(801\) −66.5685 −2.35208
\(802\) 12.8284i 0.452988i
\(803\) − 1.17157i − 0.0413439i
\(804\) 135.196 4.76799
\(805\) 0 0
\(806\) 0 0
\(807\) 48.9706i 1.72385i
\(808\) 41.1127i 1.44634i
\(809\) 10.9706 0.385704 0.192852 0.981228i \(-0.438226\pi\)
0.192852 + 0.981228i \(0.438226\pi\)
\(810\) 0 0
\(811\) 53.9411 1.89413 0.947065 0.321043i \(-0.104033\pi\)
0.947065 + 0.321043i \(0.104033\pi\)
\(812\) − 28.0000i − 0.982607i
\(813\) − 20.6863i − 0.725500i
\(814\) 18.4853 0.647909
\(815\) 0 0
\(816\) −57.9411 −2.02835
\(817\) 0 0
\(818\) 2.48528i 0.0868958i
\(819\) −11.7157 −0.409381
\(820\) 0 0
\(821\) −41.3137 −1.44186 −0.720929 0.693009i \(-0.756285\pi\)
−0.720929 + 0.693009i \(0.756285\pi\)
\(822\) − 74.9117i − 2.61285i
\(823\) 19.5147i 0.680240i 0.940382 + 0.340120i \(0.110468\pi\)
−0.940382 + 0.340120i \(0.889532\pi\)
\(824\) −30.1421 −1.05005
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) − 22.2843i − 0.774900i −0.921891 0.387450i \(-0.873356\pi\)
0.921891 0.387450i \(-0.126644\pi\)
\(828\) − 54.1421i − 1.88157i
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 0 0
\(831\) −19.3137 −0.669985
\(832\) − 11.5147i − 0.399201i
\(833\) − 20.4853i − 0.709773i
\(834\) −27.3137 −0.945796
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) − 61.9411i − 2.13972i
\(839\) −26.3431 −0.909466 −0.454733 0.890628i \(-0.650265\pi\)
−0.454733 + 0.890628i \(0.650265\pi\)
\(840\) 0 0
\(841\) −15.6274 −0.538876
\(842\) 14.4853i 0.499196i
\(843\) − 48.9706i − 1.68664i
\(844\) 61.2548 2.10848
\(845\) 0 0
\(846\) 34.1421 1.17383
\(847\) 2.00000i 0.0687208i
\(848\) 34.9706i 1.20089i
\(849\) 92.2843 3.16719
\(850\) 0 0
\(851\) 21.6569 0.742387
\(852\) 122.510i 4.19711i
\(853\) − 15.5147i − 0.531214i −0.964081 0.265607i \(-0.914428\pi\)
0.964081 0.265607i \(-0.0855723\pi\)
\(854\) −44.9706 −1.53886
\(855\) 0 0
\(856\) 33.7990 1.15523
\(857\) − 24.7696i − 0.846112i −0.906104 0.423056i \(-0.860957\pi\)
0.906104 0.423056i \(-0.139043\pi\)
\(858\) 8.00000i 0.273115i
\(859\) −24.2843 −0.828569 −0.414284 0.910148i \(-0.635968\pi\)
−0.414284 + 0.910148i \(0.635968\pi\)
\(860\) 0 0
\(861\) 33.9411 1.15671
\(862\) 27.3137i 0.930309i
\(863\) − 9.17157i − 0.312204i −0.987741 0.156102i \(-0.950107\pi\)
0.987741 0.156102i \(-0.0498929\pi\)
\(864\) −8.97056 −0.305185
\(865\) 0 0
\(866\) −18.4853 −0.628155
\(867\) 83.7990i 2.84596i
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) −14.6274 −0.495631
\(872\) 33.7990i 1.14458i
\(873\) 18.2843i 0.618829i
\(874\) 0 0
\(875\) 0 0
\(876\) 12.6863 0.428630
\(877\) 49.4558i 1.67001i 0.550246 + 0.835003i \(0.314534\pi\)
−0.550246 + 0.835003i \(0.685466\pi\)
\(878\) − 38.6274i − 1.30361i
\(879\) −25.9411 −0.874972
\(880\) 0 0
\(881\) −7.37258 −0.248389 −0.124194 0.992258i \(-0.539635\pi\)
−0.124194 + 0.992258i \(0.539635\pi\)
\(882\) 36.2132i 1.21936i
\(883\) 37.1716i 1.25092i 0.780255 + 0.625462i \(0.215089\pi\)
−0.780255 + 0.625462i \(0.784911\pi\)
\(884\) 30.6274 1.03011
\(885\) 0 0
\(886\) −64.7696 −2.17598
\(887\) − 38.2843i − 1.28546i −0.766093 0.642730i \(-0.777802\pi\)
0.766093 0.642730i \(-0.222198\pi\)
\(888\) 95.5980i 3.20806i
\(889\) 8.68629 0.291329
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 41.4558i 1.38804i
\(893\) 0 0
\(894\) 2.34315 0.0783665
\(895\) 0 0
\(896\) 41.1127 1.37348
\(897\) 9.37258i 0.312941i
\(898\) 69.1127i 2.30632i
\(899\) 0 0
\(900\) 0 0
\(901\) 79.5980 2.65179
\(902\) − 14.4853i − 0.482307i
\(903\) − 33.9411i − 1.12949i
\(904\) −86.7696 −2.88591
\(905\) 0 0
\(906\) 81.9411 2.72231
\(907\) 27.5147i 0.913611i 0.889567 + 0.456806i \(0.151007\pi\)
−0.889567 + 0.456806i \(0.848993\pi\)
\(908\) 96.9117i 3.21613i
\(909\) −46.5685 −1.54458
\(910\) 0 0
\(911\) −9.94113 −0.329364 −0.164682 0.986347i \(-0.552660\pi\)
−0.164682 + 0.986347i \(0.552660\pi\)
\(912\) 0 0
\(913\) − 6.00000i − 0.198571i
\(914\) −1.17157 −0.0387522
\(915\) 0 0
\(916\) 5.02944 0.166177
\(917\) − 22.6274i − 0.747223i
\(918\) − 93.2548i − 3.07787i
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 46.2254 1.52318
\(922\) − 30.4853i − 1.00398i
\(923\) − 13.2548i − 0.436288i
\(924\) −21.6569 −0.712458
\(925\) 0 0
\(926\) −14.8284 −0.487292
\(927\) − 34.1421i − 1.12137i
\(928\) 5.79899i 0.190361i
\(929\) −5.31371 −0.174337 −0.0871686 0.996194i \(-0.527782\pi\)
−0.0871686 + 0.996194i \(0.527782\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 23.5147i 0.770250i
\(933\) − 13.2548i − 0.433944i
\(934\) 35.7990 1.17138
\(935\) 0 0
\(936\) −25.8579 −0.845191
\(937\) 1.45584i 0.0475604i 0.999717 + 0.0237802i \(0.00757018\pi\)
−0.999717 + 0.0237802i \(0.992430\pi\)
\(938\) − 60.2843i − 1.96835i
\(939\) −3.71573 −0.121258
\(940\) 0 0
\(941\) −6.68629 −0.217967 −0.108983 0.994044i \(-0.534760\pi\)
−0.108983 + 0.994044i \(0.534760\pi\)
\(942\) − 95.5980i − 3.11475i
\(943\) − 16.9706i − 0.552638i
\(944\) −4.97056 −0.161778
\(945\) 0 0
\(946\) −14.4853 −0.470957
\(947\) − 41.1716i − 1.33790i −0.743309 0.668948i \(-0.766745\pi\)
0.743309 0.668948i \(-0.233255\pi\)
\(948\) − 43.3137i − 1.40676i
\(949\) −1.37258 −0.0445559
\(950\) 0 0
\(951\) 3.71573 0.120491
\(952\) 60.2843i 1.95382i
\(953\) 53.1716i 1.72240i 0.508269 + 0.861198i \(0.330285\pi\)
−0.508269 + 0.861198i \(0.669715\pi\)
\(954\) −140.711 −4.55568
\(955\) 0 0
\(956\) −89.2548 −2.88671
\(957\) − 10.3431i − 0.334346i
\(958\) − 86.9117i − 2.80799i
\(959\) −21.9411 −0.708516
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) − 21.6569i − 0.698245i
\(963\) 38.2843i 1.23369i
\(964\) −22.9706 −0.739832
\(965\) 0 0
\(966\) −38.6274 −1.24282
\(967\) 14.9706i 0.481421i 0.970597 + 0.240710i \(0.0773804\pi\)
−0.970597 + 0.240710i \(0.922620\pi\)
\(968\) 4.41421i 0.141878i
\(969\) 0 0
\(970\) 0 0
\(971\) −8.68629 −0.278756 −0.139378 0.990239i \(-0.544510\pi\)
−0.139378 + 0.990239i \(0.544510\pi\)
\(972\) − 54.1421i − 1.73661i
\(973\) 8.00000i 0.256468i
\(974\) 59.1127 1.89409
\(975\) 0 0
\(976\) −27.9411 −0.894374
\(977\) − 32.3431i − 1.03475i −0.855759 0.517374i \(-0.826909\pi\)
0.855759 0.517374i \(-0.173091\pi\)
\(978\) − 112.569i − 3.59955i
\(979\) 13.3137 0.425508
\(980\) 0 0
\(981\) −38.2843 −1.22232
\(982\) 1.65685i 0.0528723i
\(983\) − 21.8579i − 0.697158i −0.937279 0.348579i \(-0.886664\pi\)
0.937279 0.348579i \(-0.113336\pi\)
\(984\) 74.9117 2.38810
\(985\) 0 0
\(986\) −60.2843 −1.91984
\(987\) − 16.0000i − 0.509286i
\(988\) 0 0
\(989\) −16.9706 −0.539633
\(990\) 0 0
\(991\) −57.9411 −1.84056 −0.920280 0.391260i \(-0.872039\pi\)
−0.920280 + 0.391260i \(0.872039\pi\)
\(992\) 0 0
\(993\) 20.6863i 0.656460i
\(994\) 54.6274 1.73268
\(995\) 0 0
\(996\) 64.9706 2.05867
\(997\) 41.4558i 1.31292i 0.754361 + 0.656460i \(0.227947\pi\)
−0.754361 + 0.656460i \(0.772053\pi\)
\(998\) 23.3137i 0.737983i
\(999\) −43.3137 −1.37039
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 275.2.b.d.199.1 4
3.2 odd 2 2475.2.c.l.199.4 4
4.3 odd 2 4400.2.b.q.4049.3 4
5.2 odd 4 55.2.a.b.1.2 2
5.3 odd 4 275.2.a.c.1.1 2
5.4 even 2 inner 275.2.b.d.199.4 4
15.2 even 4 495.2.a.b.1.1 2
15.8 even 4 2475.2.a.x.1.2 2
15.14 odd 2 2475.2.c.l.199.1 4
20.3 even 4 4400.2.a.bn.1.1 2
20.7 even 4 880.2.a.m.1.2 2
20.19 odd 2 4400.2.b.q.4049.2 4
35.27 even 4 2695.2.a.f.1.2 2
40.27 even 4 3520.2.a.bo.1.1 2
40.37 odd 4 3520.2.a.bn.1.2 2
55.2 even 20 605.2.g.l.81.2 8
55.7 even 20 605.2.g.l.511.1 8
55.17 even 20 605.2.g.l.366.2 8
55.27 odd 20 605.2.g.f.366.1 8
55.32 even 4 605.2.a.d.1.1 2
55.37 odd 20 605.2.g.f.511.2 8
55.42 odd 20 605.2.g.f.81.1 8
55.43 even 4 3025.2.a.o.1.2 2
55.47 odd 20 605.2.g.f.251.2 8
55.52 even 20 605.2.g.l.251.1 8
60.47 odd 4 7920.2.a.ch.1.2 2
65.12 odd 4 9295.2.a.g.1.1 2
165.32 odd 4 5445.2.a.y.1.2 2
220.87 odd 4 9680.2.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.2 2 5.2 odd 4
275.2.a.c.1.1 2 5.3 odd 4
275.2.b.d.199.1 4 1.1 even 1 trivial
275.2.b.d.199.4 4 5.4 even 2 inner
495.2.a.b.1.1 2 15.2 even 4
605.2.a.d.1.1 2 55.32 even 4
605.2.g.f.81.1 8 55.42 odd 20
605.2.g.f.251.2 8 55.47 odd 20
605.2.g.f.366.1 8 55.27 odd 20
605.2.g.f.511.2 8 55.37 odd 20
605.2.g.l.81.2 8 55.2 even 20
605.2.g.l.251.1 8 55.52 even 20
605.2.g.l.366.2 8 55.17 even 20
605.2.g.l.511.1 8 55.7 even 20
880.2.a.m.1.2 2 20.7 even 4
2475.2.a.x.1.2 2 15.8 even 4
2475.2.c.l.199.1 4 15.14 odd 2
2475.2.c.l.199.4 4 3.2 odd 2
2695.2.a.f.1.2 2 35.27 even 4
3025.2.a.o.1.2 2 55.43 even 4
3520.2.a.bn.1.2 2 40.37 odd 4
3520.2.a.bo.1.1 2 40.27 even 4
4400.2.a.bn.1.1 2 20.3 even 4
4400.2.b.q.4049.2 4 20.19 odd 2
4400.2.b.q.4049.3 4 4.3 odd 2
5445.2.a.y.1.2 2 165.32 odd 4
7920.2.a.ch.1.2 2 60.47 odd 4
9295.2.a.g.1.1 2 65.12 odd 4
9680.2.a.bn.1.2 2 220.87 odd 4