Properties

Label 950.2.b.g.799.3
Level $950$
Weight $2$
Character 950.799
Analytic conductor $7.586$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.4227136.2
Defining polynomial: \(x^{6} + 9 x^{4} + 22 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.3
Root \(-0.713538i\) of defining polynomial
Character \(\chi\) \(=\) 950.799
Dual form 950.2.b.g.799.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +2.77733i q^{3} -1.00000 q^{4} +2.77733 q^{6} +4.69527i q^{7} +1.00000i q^{8} -4.71354 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +2.77733i q^{3} -1.00000 q^{4} +2.77733 q^{6} +4.69527i q^{7} +1.00000i q^{8} -4.71354 q^{9} +6.40880 q^{11} -2.77733i q^{12} +1.06379i q^{13} +4.69527 q^{14} +1.00000 q^{16} -1.91794i q^{17} +4.71354i q^{18} +1.00000 q^{19} -13.0403 q^{21} -6.40880i q^{22} +1.79560i q^{23} -2.77733 q^{24} +1.06379 q^{26} -4.75905i q^{27} -4.69527i q^{28} -2.93621 q^{29} -5.55465 q^{31} -1.00000i q^{32} +17.7993i q^{33} -1.91794 q^{34} +4.71354 q^{36} +11.4088i q^{37} -1.00000i q^{38} -2.95449 q^{39} -1.14585 q^{41} +13.0403i q^{42} -3.55465i q^{43} -6.40880 q^{44} +1.79560 q^{46} -10.8359i q^{47} +2.77733i q^{48} -15.0455 q^{49} +5.32674 q^{51} -1.06379i q^{52} -8.69527i q^{53} -4.75905 q^{54} -4.69527 q^{56} +2.77733i q^{57} +2.93621i q^{58} +5.63148 q^{59} -3.39053 q^{61} +5.55465i q^{62} -22.1313i q^{63} -1.00000 q^{64} +17.7993 q^{66} +8.82284i q^{67} +1.91794i q^{68} -4.98696 q^{69} -1.42708 q^{71} -4.71354i q^{72} -12.6132i q^{73} +11.4088 q^{74} -1.00000 q^{76} +30.0910i q^{77} +2.95449i q^{78} +1.96345 q^{79} -0.923174 q^{81} +1.14585i q^{82} +16.2447i q^{83} +13.0403 q^{84} -3.55465 q^{86} -8.15482i q^{87} +6.40880i q^{88} +10.0000 q^{89} -4.99477 q^{91} -1.79560i q^{92} -15.4271i q^{93} -10.8359 q^{94} +2.77733 q^{96} +14.9452i q^{97} +15.0455i q^{98} -30.2081 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{4} + 4q^{6} - 26q^{9} + O(q^{10}) \) \( 6q - 6q^{4} + 4q^{6} - 26q^{9} + 4q^{11} - 4q^{14} + 6q^{16} + 6q^{19} - 22q^{21} - 4q^{24} - 4q^{26} - 28q^{29} - 8q^{31} + 8q^{34} + 26q^{36} - 58q^{39} - 16q^{41} - 4q^{44} + 28q^{46} - 50q^{49} - 22q^{51} + 14q^{54} + 4q^{56} + 12q^{59} + 44q^{61} - 6q^{64} + 8q^{66} - 16q^{69} - 4q^{71} + 34q^{74} - 6q^{76} - 48q^{79} - 2q^{81} + 22q^{84} + 4q^{86} + 60q^{89} - 14q^{91} - 26q^{94} + 4q^{96} - 48q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 2.77733i 1.60349i 0.597666 + 0.801745i \(0.296095\pi\)
−0.597666 + 0.801745i \(0.703905\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.77733 1.13384
\(7\) 4.69527i 1.77464i 0.461151 + 0.887322i \(0.347437\pi\)
−0.461151 + 0.887322i \(0.652563\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −4.71354 −1.57118
\(10\) 0 0
\(11\) 6.40880 1.93233 0.966163 0.257931i \(-0.0830406\pi\)
0.966163 + 0.257931i \(0.0830406\pi\)
\(12\) − 2.77733i − 0.801745i
\(13\) 1.06379i 0.295042i 0.989059 + 0.147521i \(0.0471293\pi\)
−0.989059 + 0.147521i \(0.952871\pi\)
\(14\) 4.69527 1.25486
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 1.91794i − 0.465169i −0.972576 0.232584i \(-0.925282\pi\)
0.972576 0.232584i \(-0.0747182\pi\)
\(18\) 4.71354i 1.11099i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −13.0403 −2.84562
\(22\) − 6.40880i − 1.36636i
\(23\) 1.79560i 0.374408i 0.982321 + 0.187204i \(0.0599426\pi\)
−0.982321 + 0.187204i \(0.940057\pi\)
\(24\) −2.77733 −0.566919
\(25\) 0 0
\(26\) 1.06379 0.208626
\(27\) − 4.75905i − 0.915880i
\(28\) − 4.69527i − 0.887322i
\(29\) −2.93621 −0.545241 −0.272620 0.962122i \(-0.587890\pi\)
−0.272620 + 0.962122i \(0.587890\pi\)
\(30\) 0 0
\(31\) −5.55465 −0.997645 −0.498822 0.866704i \(-0.666234\pi\)
−0.498822 + 0.866704i \(0.666234\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 17.7993i 3.09847i
\(34\) −1.91794 −0.328924
\(35\) 0 0
\(36\) 4.71354 0.785590
\(37\) 11.4088i 1.87560i 0.347182 + 0.937798i \(0.387139\pi\)
−0.347182 + 0.937798i \(0.612861\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) −2.95449 −0.473096
\(40\) 0 0
\(41\) −1.14585 −0.178951 −0.0894757 0.995989i \(-0.528519\pi\)
−0.0894757 + 0.995989i \(0.528519\pi\)
\(42\) 13.0403i 2.01216i
\(43\) − 3.55465i − 0.542079i −0.962568 0.271040i \(-0.912633\pi\)
0.962568 0.271040i \(-0.0873674\pi\)
\(44\) −6.40880 −0.966163
\(45\) 0 0
\(46\) 1.79560 0.264747
\(47\) − 10.8359i − 1.58058i −0.612736 0.790288i \(-0.709931\pi\)
0.612736 0.790288i \(-0.290069\pi\)
\(48\) 2.77733i 0.400872i
\(49\) −15.0455 −2.14936
\(50\) 0 0
\(51\) 5.32674 0.745893
\(52\) − 1.06379i − 0.147521i
\(53\) − 8.69527i − 1.19439i −0.802097 0.597193i \(-0.796283\pi\)
0.802097 0.597193i \(-0.203717\pi\)
\(54\) −4.75905 −0.647625
\(55\) 0 0
\(56\) −4.69527 −0.627431
\(57\) 2.77733i 0.367866i
\(58\) 2.93621i 0.385544i
\(59\) 5.63148 0.733156 0.366578 0.930387i \(-0.380529\pi\)
0.366578 + 0.930387i \(0.380529\pi\)
\(60\) 0 0
\(61\) −3.39053 −0.434113 −0.217056 0.976159i \(-0.569646\pi\)
−0.217056 + 0.976159i \(0.569646\pi\)
\(62\) 5.55465i 0.705441i
\(63\) − 22.1313i − 2.78828i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 17.7993 2.19095
\(67\) 8.82284i 1.07788i 0.842344 + 0.538941i \(0.181175\pi\)
−0.842344 + 0.538941i \(0.818825\pi\)
\(68\) 1.91794i 0.232584i
\(69\) −4.98696 −0.600360
\(70\) 0 0
\(71\) −1.42708 −0.169363 −0.0846814 0.996408i \(-0.526987\pi\)
−0.0846814 + 0.996408i \(0.526987\pi\)
\(72\) − 4.71354i − 0.555496i
\(73\) − 12.6132i − 1.47626i −0.674656 0.738132i \(-0.735708\pi\)
0.674656 0.738132i \(-0.264292\pi\)
\(74\) 11.4088 1.32625
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 30.0910i 3.42919i
\(78\) 2.95449i 0.334530i
\(79\) 1.96345 0.220906 0.110453 0.993881i \(-0.464770\pi\)
0.110453 + 0.993881i \(0.464770\pi\)
\(80\) 0 0
\(81\) −0.923174 −0.102575
\(82\) 1.14585i 0.126538i
\(83\) 16.2447i 1.78309i 0.452937 + 0.891543i \(0.350376\pi\)
−0.452937 + 0.891543i \(0.649624\pi\)
\(84\) 13.0403 1.42281
\(85\) 0 0
\(86\) −3.55465 −0.383308
\(87\) − 8.15482i − 0.874288i
\(88\) 6.40880i 0.683181i
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −4.99477 −0.523594
\(92\) − 1.79560i − 0.187204i
\(93\) − 15.4271i − 1.59971i
\(94\) −10.8359 −1.11764
\(95\) 0 0
\(96\) 2.77733 0.283460
\(97\) 14.9452i 1.51745i 0.651409 + 0.758727i \(0.274178\pi\)
−0.651409 + 0.758727i \(0.725822\pi\)
\(98\) 15.0455i 1.51983i
\(99\) −30.2081 −3.03603
\(100\) 0 0
\(101\) 10.5364 1.04841 0.524204 0.851592i \(-0.324363\pi\)
0.524204 + 0.851592i \(0.324363\pi\)
\(102\) − 5.32674i − 0.527426i
\(103\) − 16.9817i − 1.67326i −0.547769 0.836630i \(-0.684523\pi\)
0.547769 0.836630i \(-0.315477\pi\)
\(104\) −1.06379 −0.104313
\(105\) 0 0
\(106\) −8.69527 −0.844559
\(107\) 1.79036i 0.173081i 0.996248 + 0.0865405i \(0.0275812\pi\)
−0.996248 + 0.0865405i \(0.972419\pi\)
\(108\) 4.75905i 0.457940i
\(109\) −2.41404 −0.231223 −0.115611 0.993295i \(-0.536883\pi\)
−0.115611 + 0.993295i \(0.536883\pi\)
\(110\) 0 0
\(111\) −31.6860 −3.00750
\(112\) 4.69527i 0.443661i
\(113\) 7.14585i 0.672225i 0.941822 + 0.336112i \(0.109112\pi\)
−0.941822 + 0.336112i \(0.890888\pi\)
\(114\) 2.77733 0.260120
\(115\) 0 0
\(116\) 2.93621 0.272620
\(117\) − 5.01420i − 0.463563i
\(118\) − 5.63148i − 0.518420i
\(119\) 9.00523 0.825508
\(120\) 0 0
\(121\) 30.0728 2.73389
\(122\) 3.39053i 0.306964i
\(123\) − 3.18239i − 0.286947i
\(124\) 5.55465 0.498822
\(125\) 0 0
\(126\) −22.1313 −1.97161
\(127\) − 9.26295i − 0.821954i −0.911646 0.410977i \(-0.865188\pi\)
0.911646 0.410977i \(-0.134812\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 9.87242 0.869219
\(130\) 0 0
\(131\) 11.5181 1.00634 0.503171 0.864187i \(-0.332167\pi\)
0.503171 + 0.864187i \(0.332167\pi\)
\(132\) − 17.7993i − 1.54923i
\(133\) 4.69527i 0.407131i
\(134\) 8.82284 0.762177
\(135\) 0 0
\(136\) 1.91794 0.164462
\(137\) − 15.1041i − 1.29043i −0.764002 0.645214i \(-0.776768\pi\)
0.764002 0.645214i \(-0.223232\pi\)
\(138\) 4.98696i 0.424518i
\(139\) 0.700500 0.0594156 0.0297078 0.999559i \(-0.490542\pi\)
0.0297078 + 0.999559i \(0.490542\pi\)
\(140\) 0 0
\(141\) 30.0948 2.53444
\(142\) 1.42708i 0.119758i
\(143\) 6.81761i 0.570117i
\(144\) −4.71354 −0.392795
\(145\) 0 0
\(146\) −12.6132 −1.04388
\(147\) − 41.7863i − 3.44648i
\(148\) − 11.4088i − 0.937798i
\(149\) −12.9452 −1.06051 −0.530255 0.847838i \(-0.677904\pi\)
−0.530255 + 0.847838i \(0.677904\pi\)
\(150\) 0 0
\(151\) 5.70830 0.464535 0.232268 0.972652i \(-0.425385\pi\)
0.232268 + 0.972652i \(0.425385\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) 9.04028i 0.730863i
\(154\) 30.0910 2.42480
\(155\) 0 0
\(156\) 2.95449 0.236548
\(157\) − 9.26295i − 0.739264i −0.929178 0.369632i \(-0.879484\pi\)
0.929178 0.369632i \(-0.120516\pi\)
\(158\) − 1.96345i − 0.156204i
\(159\) 24.1496 1.91519
\(160\) 0 0
\(161\) −8.43081 −0.664441
\(162\) 0.923174i 0.0725314i
\(163\) 4.28123i 0.335332i 0.985844 + 0.167666i \(0.0536230\pi\)
−0.985844 + 0.167666i \(0.946377\pi\)
\(164\) 1.14585 0.0894757
\(165\) 0 0
\(166\) 16.2447 1.26083
\(167\) 15.2264i 1.17825i 0.808040 + 0.589127i \(0.200528\pi\)
−0.808040 + 0.589127i \(0.799472\pi\)
\(168\) − 13.0403i − 1.00608i
\(169\) 11.8684 0.912950
\(170\) 0 0
\(171\) −4.71354 −0.360453
\(172\) 3.55465i 0.271040i
\(173\) − 12.2630i − 0.932335i −0.884696 0.466168i \(-0.845634\pi\)
0.884696 0.466168i \(-0.154366\pi\)
\(174\) −8.15482 −0.618215
\(175\) 0 0
\(176\) 6.40880 0.483082
\(177\) 15.6404i 1.17561i
\(178\) − 10.0000i − 0.749532i
\(179\) 1.57292 0.117566 0.0587829 0.998271i \(-0.481278\pi\)
0.0587829 + 0.998271i \(0.481278\pi\)
\(180\) 0 0
\(181\) 11.4088 0.848010 0.424005 0.905660i \(-0.360624\pi\)
0.424005 + 0.905660i \(0.360624\pi\)
\(182\) 4.99477i 0.370237i
\(183\) − 9.41661i − 0.696096i
\(184\) −1.79560 −0.132373
\(185\) 0 0
\(186\) −15.4271 −1.13117
\(187\) − 12.2917i − 0.898858i
\(188\) 10.8359i 0.790288i
\(189\) 22.3450 1.62536
\(190\) 0 0
\(191\) −6.35805 −0.460053 −0.230026 0.973184i \(-0.573881\pi\)
−0.230026 + 0.973184i \(0.573881\pi\)
\(192\) − 2.77733i − 0.200436i
\(193\) 14.4998i 1.04372i 0.853031 + 0.521860i \(0.174762\pi\)
−0.853031 + 0.521860i \(0.825238\pi\)
\(194\) 14.9452 1.07300
\(195\) 0 0
\(196\) 15.0455 1.07468
\(197\) 5.14585i 0.366627i 0.983055 + 0.183313i \(0.0586823\pi\)
−0.983055 + 0.183313i \(0.941318\pi\)
\(198\) 30.2081i 2.14680i
\(199\) −3.87766 −0.274880 −0.137440 0.990510i \(-0.543887\pi\)
−0.137440 + 0.990510i \(0.543887\pi\)
\(200\) 0 0
\(201\) −24.5039 −1.72837
\(202\) − 10.5364i − 0.741337i
\(203\) − 13.7863i − 0.967608i
\(204\) −5.32674 −0.372947
\(205\) 0 0
\(206\) −16.9817 −1.18317
\(207\) − 8.46362i − 0.588262i
\(208\) 1.06379i 0.0737604i
\(209\) 6.40880 0.443306
\(210\) 0 0
\(211\) 17.1496 1.18063 0.590313 0.807174i \(-0.299004\pi\)
0.590313 + 0.807174i \(0.299004\pi\)
\(212\) 8.69527i 0.597193i
\(213\) − 3.96345i − 0.271571i
\(214\) 1.79036 0.122387
\(215\) 0 0
\(216\) 4.75905 0.323813
\(217\) − 26.0806i − 1.77046i
\(218\) 2.41404i 0.163499i
\(219\) 35.0310 2.36717
\(220\) 0 0
\(221\) 2.04028 0.137244
\(222\) 31.6860i 2.12662i
\(223\) 7.84635i 0.525430i 0.964873 + 0.262715i \(0.0846180\pi\)
−0.964873 + 0.262715i \(0.915382\pi\)
\(224\) 4.69527 0.313716
\(225\) 0 0
\(226\) 7.14585 0.475335
\(227\) − 6.28646i − 0.417247i −0.977996 0.208624i \(-0.933102\pi\)
0.977996 0.208624i \(-0.0668983\pi\)
\(228\) − 2.77733i − 0.183933i
\(229\) 3.14585 0.207884 0.103942 0.994583i \(-0.466854\pi\)
0.103942 + 0.994583i \(0.466854\pi\)
\(230\) 0 0
\(231\) −83.5726 −5.49867
\(232\) − 2.93621i − 0.192772i
\(233\) 0.182394i 0.0119490i 0.999982 + 0.00597451i \(0.00190176\pi\)
−0.999982 + 0.00597451i \(0.998098\pi\)
\(234\) −5.01420 −0.327789
\(235\) 0 0
\(236\) −5.63148 −0.366578
\(237\) 5.45315i 0.354220i
\(238\) − 9.00523i − 0.583723i
\(239\) 11.5039 0.744126 0.372063 0.928208i \(-0.378651\pi\)
0.372063 + 0.928208i \(0.378651\pi\)
\(240\) 0 0
\(241\) −0.445349 −0.0286874 −0.0143437 0.999897i \(-0.504566\pi\)
−0.0143437 + 0.999897i \(0.504566\pi\)
\(242\) − 30.0728i − 1.93315i
\(243\) − 16.8411i − 1.08036i
\(244\) 3.39053 0.217056
\(245\) 0 0
\(246\) −3.18239 −0.202902
\(247\) 1.06379i 0.0676872i
\(248\) − 5.55465i − 0.352721i
\(249\) −45.1168 −2.85916
\(250\) 0 0
\(251\) 13.2630 0.837150 0.418575 0.908182i \(-0.362530\pi\)
0.418575 + 0.908182i \(0.362530\pi\)
\(252\) 22.1313i 1.39414i
\(253\) 11.5076i 0.723479i
\(254\) −9.26295 −0.581209
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 23.4816i 1.46474i 0.680907 + 0.732370i \(0.261586\pi\)
−0.680907 + 0.732370i \(0.738414\pi\)
\(258\) − 9.87242i − 0.614630i
\(259\) −53.5674 −3.32851
\(260\) 0 0
\(261\) 13.8399 0.856671
\(262\) − 11.5181i − 0.711591i
\(263\) 27.9269i 1.72205i 0.508565 + 0.861023i \(0.330176\pi\)
−0.508565 + 0.861023i \(0.669824\pi\)
\(264\) −17.7993 −1.09547
\(265\) 0 0
\(266\) 4.69527 0.287885
\(267\) 27.7733i 1.69970i
\(268\) − 8.82284i − 0.538941i
\(269\) −16.4816 −1.00490 −0.502449 0.864607i \(-0.667568\pi\)
−0.502449 + 0.864607i \(0.667568\pi\)
\(270\) 0 0
\(271\) −1.46736 −0.0891356 −0.0445678 0.999006i \(-0.514191\pi\)
−0.0445678 + 0.999006i \(0.514191\pi\)
\(272\) − 1.91794i − 0.116292i
\(273\) − 13.8721i − 0.839577i
\(274\) −15.1041 −0.912470
\(275\) 0 0
\(276\) 4.98696 0.300180
\(277\) − 15.8359i − 0.951486i −0.879584 0.475743i \(-0.842179\pi\)
0.879584 0.475743i \(-0.157821\pi\)
\(278\) − 0.700500i − 0.0420132i
\(279\) 26.1821 1.56748
\(280\) 0 0
\(281\) −6.25515 −0.373151 −0.186576 0.982441i \(-0.559739\pi\)
−0.186576 + 0.982441i \(0.559739\pi\)
\(282\) − 30.0948i − 1.79212i
\(283\) − 16.7005i − 0.992742i −0.868111 0.496371i \(-0.834666\pi\)
0.868111 0.496371i \(-0.165334\pi\)
\(284\) 1.42708 0.0846814
\(285\) 0 0
\(286\) 6.81761 0.403134
\(287\) − 5.38006i − 0.317575i
\(288\) 4.71354i 0.277748i
\(289\) 13.3215 0.783618
\(290\) 0 0
\(291\) −41.5076 −2.43322
\(292\) 12.6132i 0.738132i
\(293\) 0.644516i 0.0376530i 0.999823 + 0.0188265i \(0.00599302\pi\)
−0.999823 + 0.0188265i \(0.994007\pi\)
\(294\) −41.7863 −2.43703
\(295\) 0 0
\(296\) −11.4088 −0.663123
\(297\) − 30.4998i − 1.76978i
\(298\) 12.9452i 0.749894i
\(299\) −1.91014 −0.110466
\(300\) 0 0
\(301\) 16.6900 0.961997
\(302\) − 5.70830i − 0.328476i
\(303\) 29.2630i 1.68111i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 9.04028 0.516798
\(307\) − 26.6457i − 1.52075i −0.649485 0.760375i \(-0.725015\pi\)
0.649485 0.760375i \(-0.274985\pi\)
\(308\) − 30.0910i − 1.71460i
\(309\) 47.1638 2.68305
\(310\) 0 0
\(311\) 11.5494 0.654907 0.327454 0.944867i \(-0.393809\pi\)
0.327454 + 0.944867i \(0.393809\pi\)
\(312\) − 2.95449i − 0.167265i
\(313\) − 23.0638i − 1.30364i −0.758373 0.651821i \(-0.774005\pi\)
0.758373 0.651821i \(-0.225995\pi\)
\(314\) −9.26295 −0.522739
\(315\) 0 0
\(316\) −1.96345 −0.110453
\(317\) 13.9232i 0.782003i 0.920390 + 0.391002i \(0.127871\pi\)
−0.920390 + 0.391002i \(0.872129\pi\)
\(318\) − 24.1496i − 1.35424i
\(319\) −18.8176 −1.05358
\(320\) 0 0
\(321\) −4.97242 −0.277534
\(322\) 8.43081i 0.469831i
\(323\) − 1.91794i − 0.106717i
\(324\) 0.923174 0.0512874
\(325\) 0 0
\(326\) 4.28123 0.237115
\(327\) − 6.70457i − 0.370763i
\(328\) − 1.14585i − 0.0632689i
\(329\) 50.8773 2.80496
\(330\) 0 0
\(331\) −0.735546 −0.0404292 −0.0202146 0.999796i \(-0.506435\pi\)
−0.0202146 + 0.999796i \(0.506435\pi\)
\(332\) − 16.2447i − 0.891543i
\(333\) − 53.7758i − 2.94690i
\(334\) 15.2264 0.833152
\(335\) 0 0
\(336\) −13.0403 −0.711406
\(337\) 2.28123i 0.124266i 0.998068 + 0.0621332i \(0.0197904\pi\)
−0.998068 + 0.0621332i \(0.980210\pi\)
\(338\) − 11.8684i − 0.645553i
\(339\) −19.8463 −1.07791
\(340\) 0 0
\(341\) −35.5987 −1.92778
\(342\) 4.71354i 0.254879i
\(343\) − 37.7758i − 2.03970i
\(344\) 3.55465 0.191654
\(345\) 0 0
\(346\) −12.2630 −0.659261
\(347\) 2.56246i 0.137560i 0.997632 + 0.0687799i \(0.0219106\pi\)
−0.997632 + 0.0687799i \(0.978089\pi\)
\(348\) 8.15482i 0.437144i
\(349\) 5.67176 0.303602 0.151801 0.988411i \(-0.451493\pi\)
0.151801 + 0.988411i \(0.451493\pi\)
\(350\) 0 0
\(351\) 5.06262 0.270223
\(352\) − 6.40880i − 0.341590i
\(353\) − 23.6665i − 1.25964i −0.776740 0.629821i \(-0.783128\pi\)
0.776740 0.629821i \(-0.216872\pi\)
\(354\) 15.6404 0.831280
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 25.0105i 1.32369i
\(358\) − 1.57292i − 0.0831316i
\(359\) −31.9724 −1.68744 −0.843720 0.536784i \(-0.819639\pi\)
−0.843720 + 0.536784i \(0.819639\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 11.4088i − 0.599633i
\(363\) 83.5218i 4.38376i
\(364\) 4.99477 0.261797
\(365\) 0 0
\(366\) −9.41661 −0.492214
\(367\) − 1.14585i − 0.0598128i −0.999553 0.0299064i \(-0.990479\pi\)
0.999553 0.0299064i \(-0.00952092\pi\)
\(368\) 1.79560i 0.0936020i
\(369\) 5.40100 0.281165
\(370\) 0 0
\(371\) 40.8266 2.11961
\(372\) 15.4271i 0.799857i
\(373\) 32.8579i 1.70132i 0.525719 + 0.850658i \(0.323796\pi\)
−0.525719 + 0.850658i \(0.676204\pi\)
\(374\) −12.2917 −0.635588
\(375\) 0 0
\(376\) 10.8359 0.558818
\(377\) − 3.12351i − 0.160869i
\(378\) − 22.3450i − 1.14930i
\(379\) 18.2865 0.939312 0.469656 0.882849i \(-0.344378\pi\)
0.469656 + 0.882849i \(0.344378\pi\)
\(380\) 0 0
\(381\) 25.7262 1.31800
\(382\) 6.35805i 0.325306i
\(383\) 20.8542i 1.06560i 0.846242 + 0.532799i \(0.178860\pi\)
−0.846242 + 0.532799i \(0.821140\pi\)
\(384\) −2.77733 −0.141730
\(385\) 0 0
\(386\) 14.4998 0.738022
\(387\) 16.7550i 0.851704i
\(388\) − 14.9452i − 0.758727i
\(389\) 24.9086 1.26292 0.631459 0.775409i \(-0.282456\pi\)
0.631459 + 0.775409i \(0.282456\pi\)
\(390\) 0 0
\(391\) 3.44385 0.174163
\(392\) − 15.0455i − 0.759913i
\(393\) 31.9895i 1.61366i
\(394\) 5.14585 0.259244
\(395\) 0 0
\(396\) 30.2081 1.51802
\(397\) 24.7445i 1.24189i 0.783853 + 0.620946i \(0.213251\pi\)
−0.783853 + 0.620946i \(0.786749\pi\)
\(398\) 3.87766i 0.194369i
\(399\) −13.0403 −0.652831
\(400\) 0 0
\(401\) 15.6457 0.781308 0.390654 0.920538i \(-0.372249\pi\)
0.390654 + 0.920538i \(0.372249\pi\)
\(402\) 24.5039i 1.22214i
\(403\) − 5.90897i − 0.294347i
\(404\) −10.5364 −0.524204
\(405\) 0 0
\(406\) −13.7863 −0.684202
\(407\) 73.1168i 3.62426i
\(408\) 5.32674i 0.263713i
\(409\) −30.5804 −1.51210 −0.756052 0.654512i \(-0.772874\pi\)
−0.756052 + 0.654512i \(0.772874\pi\)
\(410\) 0 0
\(411\) 41.9489 2.06919
\(412\) 16.9817i 0.836630i
\(413\) 26.4413i 1.30109i
\(414\) −8.46362 −0.415964
\(415\) 0 0
\(416\) 1.06379 0.0521565
\(417\) 1.94552i 0.0952723i
\(418\) − 6.40880i − 0.313465i
\(419\) −1.96345 −0.0959210 −0.0479605 0.998849i \(-0.515272\pi\)
−0.0479605 + 0.998849i \(0.515272\pi\)
\(420\) 0 0
\(421\) −25.5949 −1.24742 −0.623710 0.781656i \(-0.714376\pi\)
−0.623710 + 0.781656i \(0.714376\pi\)
\(422\) − 17.1496i − 0.834829i
\(423\) 51.0753i 2.48337i
\(424\) 8.69527 0.422279
\(425\) 0 0
\(426\) −3.96345 −0.192030
\(427\) − 15.9194i − 0.770396i
\(428\) − 1.79036i − 0.0865405i
\(429\) −18.9347 −0.914177
\(430\) 0 0
\(431\) −15.3540 −0.739575 −0.369788 0.929116i \(-0.620570\pi\)
−0.369788 + 0.929116i \(0.620570\pi\)
\(432\) − 4.75905i − 0.228970i
\(433\) 16.6640i 0.800819i 0.916336 + 0.400409i \(0.131132\pi\)
−0.916336 + 0.400409i \(0.868868\pi\)
\(434\) −26.0806 −1.25191
\(435\) 0 0
\(436\) 2.41404 0.115611
\(437\) 1.79560i 0.0858951i
\(438\) − 35.0310i − 1.67384i
\(439\) 19.5987 0.935393 0.467697 0.883889i \(-0.345084\pi\)
0.467697 + 0.883889i \(0.345084\pi\)
\(440\) 0 0
\(441\) 70.9176 3.37703
\(442\) − 2.04028i − 0.0970462i
\(443\) 13.0183i 0.618517i 0.950978 + 0.309258i \(0.100081\pi\)
−0.950978 + 0.309258i \(0.899919\pi\)
\(444\) 31.6860 1.50375
\(445\) 0 0
\(446\) 7.84635 0.371535
\(447\) − 35.9530i − 1.70052i
\(448\) − 4.69527i − 0.221830i
\(449\) −6.77359 −0.319666 −0.159833 0.987144i \(-0.551095\pi\)
−0.159833 + 0.987144i \(0.551095\pi\)
\(450\) 0 0
\(451\) −7.34352 −0.345793
\(452\) − 7.14585i − 0.336112i
\(453\) 15.8538i 0.744877i
\(454\) −6.28646 −0.295038
\(455\) 0 0
\(456\) −2.77733 −0.130060
\(457\) 27.3189i 1.27793i 0.769237 + 0.638963i \(0.220636\pi\)
−0.769237 + 0.638963i \(0.779364\pi\)
\(458\) − 3.14585i − 0.146996i
\(459\) −9.12758 −0.426039
\(460\) 0 0
\(461\) −27.7993 −1.29474 −0.647372 0.762174i \(-0.724132\pi\)
−0.647372 + 0.762174i \(0.724132\pi\)
\(462\) 83.5726i 3.88815i
\(463\) 38.2369i 1.77702i 0.458859 + 0.888509i \(0.348258\pi\)
−0.458859 + 0.888509i \(0.651742\pi\)
\(464\) −2.93621 −0.136310
\(465\) 0 0
\(466\) 0.182394 0.00844923
\(467\) − 23.4711i − 1.08611i −0.839696 0.543056i \(-0.817267\pi\)
0.839696 0.543056i \(-0.182733\pi\)
\(468\) 5.01420i 0.231782i
\(469\) −41.4256 −1.91286
\(470\) 0 0
\(471\) 25.7262 1.18540
\(472\) 5.63148i 0.259210i
\(473\) − 22.7811i − 1.04747i
\(474\) 5.45315 0.250472
\(475\) 0 0
\(476\) −9.00523 −0.412754
\(477\) 40.9855i 1.87660i
\(478\) − 11.5039i − 0.526176i
\(479\) 19.6900 0.899660 0.449830 0.893114i \(-0.351484\pi\)
0.449830 + 0.893114i \(0.351484\pi\)
\(480\) 0 0
\(481\) −12.1365 −0.553379
\(482\) 0.445349i 0.0202851i
\(483\) − 23.4151i − 1.06542i
\(484\) −30.0728 −1.36694
\(485\) 0 0
\(486\) −16.8411 −0.763928
\(487\) − 16.3357i − 0.740242i −0.928984 0.370121i \(-0.879316\pi\)
0.928984 0.370121i \(-0.120684\pi\)
\(488\) − 3.39053i − 0.153482i
\(489\) −11.8904 −0.537701
\(490\) 0 0
\(491\) 27.1093 1.22343 0.611713 0.791080i \(-0.290481\pi\)
0.611713 + 0.791080i \(0.290481\pi\)
\(492\) 3.18239i 0.143473i
\(493\) 5.63148i 0.253629i
\(494\) 1.06379 0.0478621
\(495\) 0 0
\(496\) −5.55465 −0.249411
\(497\) − 6.70050i − 0.300558i
\(498\) 45.1168i 2.02173i
\(499\) 4.69003 0.209955 0.104977 0.994475i \(-0.466523\pi\)
0.104977 + 0.994475i \(0.466523\pi\)
\(500\) 0 0
\(501\) −42.2887 −1.88932
\(502\) − 13.2630i − 0.591955i
\(503\) 5.19136i 0.231471i 0.993280 + 0.115736i \(0.0369226\pi\)
−0.993280 + 0.115736i \(0.963077\pi\)
\(504\) 22.1313 0.985807
\(505\) 0 0
\(506\) 11.5076 0.511577
\(507\) 32.9623i 1.46391i
\(508\) 9.26295i 0.410977i
\(509\) 4.51811 0.200262 0.100131 0.994974i \(-0.468074\pi\)
0.100131 + 0.994974i \(0.468074\pi\)
\(510\) 0 0
\(511\) 59.2223 2.61984
\(512\) − 1.00000i − 0.0441942i
\(513\) − 4.75905i − 0.210117i
\(514\) 23.4816 1.03573
\(515\) 0 0
\(516\) −9.87242 −0.434609
\(517\) − 69.4450i − 3.05419i
\(518\) 53.5674i 2.35361i
\(519\) 34.0582 1.49499
\(520\) 0 0
\(521\) 14.4453 0.632862 0.316431 0.948615i \(-0.397515\pi\)
0.316431 + 0.948615i \(0.397515\pi\)
\(522\) − 13.8399i − 0.605758i
\(523\) − 18.2134i − 0.796415i −0.917295 0.398208i \(-0.869632\pi\)
0.917295 0.398208i \(-0.130368\pi\)
\(524\) −11.5181 −0.503171
\(525\) 0 0
\(526\) 27.9269 1.21767
\(527\) 10.6535i 0.464073i
\(528\) 17.7993i 0.774617i
\(529\) 19.7758 0.859819
\(530\) 0 0
\(531\) −26.5442 −1.15192
\(532\) − 4.69527i − 0.203566i
\(533\) − 1.21894i − 0.0527981i
\(534\) 27.7733 1.20187
\(535\) 0 0
\(536\) −8.82284 −0.381089
\(537\) 4.36852i 0.188516i
\(538\) 16.4816i 0.710571i
\(539\) −96.4237 −4.15326
\(540\) 0 0
\(541\) 37.3905 1.60754 0.803772 0.594937i \(-0.202823\pi\)
0.803772 + 0.594937i \(0.202823\pi\)
\(542\) 1.46736i 0.0630284i
\(543\) 31.6860i 1.35977i
\(544\) −1.91794 −0.0822310
\(545\) 0 0
\(546\) −13.8721 −0.593671
\(547\) − 29.5621i − 1.26399i −0.774974 0.631993i \(-0.782237\pi\)
0.774974 0.631993i \(-0.217763\pi\)
\(548\) 15.1041i 0.645214i
\(549\) 15.9814 0.682069
\(550\) 0 0
\(551\) −2.93621 −0.125087
\(552\) − 4.98696i − 0.212259i
\(553\) 9.21894i 0.392029i
\(554\) −15.8359 −0.672802
\(555\) 0 0
\(556\) −0.700500 −0.0297078
\(557\) − 14.3723i − 0.608972i −0.952517 0.304486i \(-0.901515\pi\)
0.952517 0.304486i \(-0.0984847\pi\)
\(558\) − 26.1821i − 1.10837i
\(559\) 3.78139 0.159936
\(560\) 0 0
\(561\) 34.1380 1.44131
\(562\) 6.25515i 0.263858i
\(563\) − 6.39053i − 0.269329i −0.990891 0.134664i \(-0.957004\pi\)
0.990891 0.134664i \(-0.0429956\pi\)
\(564\) −30.0948 −1.26722
\(565\) 0 0
\(566\) −16.7005 −0.701974
\(567\) − 4.33455i − 0.182034i
\(568\) − 1.42708i − 0.0598788i
\(569\) 44.9086 1.88267 0.941334 0.337476i \(-0.109573\pi\)
0.941334 + 0.337476i \(0.109573\pi\)
\(570\) 0 0
\(571\) 12.4193 0.519730 0.259865 0.965645i \(-0.416322\pi\)
0.259865 + 0.965645i \(0.416322\pi\)
\(572\) − 6.81761i − 0.285058i
\(573\) − 17.6584i − 0.737690i
\(574\) −5.38006 −0.224559
\(575\) 0 0
\(576\) 4.71354 0.196397
\(577\) 4.20440i 0.175032i 0.996163 + 0.0875158i \(0.0278928\pi\)
−0.996163 + 0.0875158i \(0.972107\pi\)
\(578\) − 13.3215i − 0.554102i
\(579\) −40.2708 −1.67360
\(580\) 0 0
\(581\) −76.2731 −3.16434
\(582\) 41.5076i 1.72055i
\(583\) − 55.7262i − 2.30795i
\(584\) 12.6132 0.521938
\(585\) 0 0
\(586\) 0.644516 0.0266247
\(587\) − 24.7915i − 1.02326i −0.859207 0.511628i \(-0.829043\pi\)
0.859207 0.511628i \(-0.170957\pi\)
\(588\) 41.7863i 1.72324i
\(589\) −5.55465 −0.228875
\(590\) 0 0
\(591\) −14.2917 −0.587882
\(592\) 11.4088i 0.468899i
\(593\) 4.67176i 0.191846i 0.995389 + 0.0959231i \(0.0305803\pi\)
−0.995389 + 0.0959231i \(0.969420\pi\)
\(594\) −30.4998 −1.25142
\(595\) 0 0
\(596\) 12.9452 0.530255
\(597\) − 10.7695i − 0.440767i
\(598\) 1.91014i 0.0781113i
\(599\) −39.2369 −1.60318 −0.801588 0.597877i \(-0.796011\pi\)
−0.801588 + 0.597877i \(0.796011\pi\)
\(600\) 0 0
\(601\) −42.4267 −1.73062 −0.865311 0.501235i \(-0.832879\pi\)
−0.865311 + 0.501235i \(0.832879\pi\)
\(602\) − 16.6900i − 0.680235i
\(603\) − 41.5868i − 1.69355i
\(604\) −5.70830 −0.232268
\(605\) 0 0
\(606\) 29.2630 1.18873
\(607\) 2.98173i 0.121025i 0.998167 + 0.0605123i \(0.0192734\pi\)
−0.998167 + 0.0605123i \(0.980727\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) 38.2890 1.55155
\(610\) 0 0
\(611\) 11.5271 0.466336
\(612\) − 9.04028i − 0.365432i
\(613\) − 28.9452i − 1.16908i −0.811363 0.584542i \(-0.801274\pi\)
0.811363 0.584542i \(-0.198726\pi\)
\(614\) −26.6457 −1.07533
\(615\) 0 0
\(616\) −30.0910 −1.21240
\(617\) 15.0365i 0.605349i 0.953094 + 0.302674i \(0.0978794\pi\)
−0.953094 + 0.302674i \(0.902121\pi\)
\(618\) − 47.1638i − 1.89721i
\(619\) −8.25515 −0.331803 −0.165901 0.986142i \(-0.553053\pi\)
−0.165901 + 0.986142i \(0.553053\pi\)
\(620\) 0 0
\(621\) 8.54535 0.342913
\(622\) − 11.5494i − 0.463089i
\(623\) 46.9527i 1.88112i
\(624\) −2.95449 −0.118274
\(625\) 0 0
\(626\) −23.0638 −0.921814
\(627\) 17.7993i 0.710837i
\(628\) 9.26295i 0.369632i
\(629\) 21.8814 0.872468
\(630\) 0 0
\(631\) −9.09103 −0.361908 −0.180954 0.983492i \(-0.557919\pi\)
−0.180954 + 0.983492i \(0.557919\pi\)
\(632\) 1.96345i 0.0781020i
\(633\) 47.6300i 1.89312i
\(634\) 13.9232 0.552960
\(635\) 0 0
\(636\) −24.1496 −0.957593
\(637\) − 16.0052i − 0.634150i
\(638\) 18.8176i 0.744996i
\(639\) 6.72658 0.266099
\(640\) 0 0
\(641\) 30.1171 1.18955 0.594777 0.803891i \(-0.297240\pi\)
0.594777 + 0.803891i \(0.297240\pi\)
\(642\) 4.97242i 0.196246i
\(643\) − 35.3174i − 1.39278i −0.717662 0.696392i \(-0.754788\pi\)
0.717662 0.696392i \(-0.245212\pi\)
\(644\) 8.43081 0.332220
\(645\) 0 0
\(646\) −1.91794 −0.0754603
\(647\) − 4.12234i − 0.162066i −0.996711 0.0810330i \(-0.974178\pi\)
0.996711 0.0810330i \(-0.0258219\pi\)
\(648\) − 0.923174i − 0.0362657i
\(649\) 36.0910 1.41670
\(650\) 0 0
\(651\) 72.4342 2.83892
\(652\) − 4.28123i − 0.167666i
\(653\) 8.62741i 0.337617i 0.985649 + 0.168808i \(0.0539919\pi\)
−0.985649 + 0.168808i \(0.946008\pi\)
\(654\) −6.70457 −0.262169
\(655\) 0 0
\(656\) −1.14585 −0.0447379
\(657\) 59.4528i 2.31948i
\(658\) − 50.8773i − 1.98340i
\(659\) 37.3853 1.45632 0.728162 0.685405i \(-0.240375\pi\)
0.728162 + 0.685405i \(0.240375\pi\)
\(660\) 0 0
\(661\) −12.8997 −0.501739 −0.250869 0.968021i \(-0.580716\pi\)
−0.250869 + 0.968021i \(0.580716\pi\)
\(662\) 0.735546i 0.0285878i
\(663\) 5.66652i 0.220070i
\(664\) −16.2447 −0.630416
\(665\) 0 0
\(666\) −53.7758 −2.08377
\(667\) − 5.27226i − 0.204143i
\(668\) − 15.2264i − 0.589127i
\(669\) −21.7919 −0.842522
\(670\) 0 0
\(671\) −21.7292 −0.838848
\(672\) 13.0403i 0.503040i
\(673\) 10.4349i 0.402235i 0.979567 + 0.201118i \(0.0644573\pi\)
−0.979567 + 0.201118i \(0.935543\pi\)
\(674\) 2.28123 0.0878696
\(675\) 0 0
\(676\) −11.8684 −0.456475
\(677\) − 12.4401i − 0.478112i −0.971006 0.239056i \(-0.923162\pi\)
0.971006 0.239056i \(-0.0768380\pi\)
\(678\) 19.8463i 0.762194i
\(679\) −70.1716 −2.69294
\(680\) 0 0
\(681\) 17.4596 0.669052
\(682\) 35.5987i 1.36314i
\(683\) 17.5364i 0.671011i 0.942038 + 0.335505i \(0.108907\pi\)
−0.942038 + 0.335505i \(0.891093\pi\)
\(684\) 4.71354 0.180227
\(685\) 0 0
\(686\) −37.7758 −1.44229
\(687\) 8.73705i 0.333339i
\(688\) − 3.55465i − 0.135520i
\(689\) 9.24992 0.352394
\(690\) 0 0
\(691\) −25.2734 −0.961446 −0.480723 0.876872i \(-0.659626\pi\)
−0.480723 + 0.876872i \(0.659626\pi\)
\(692\) 12.2630i 0.466168i
\(693\) − 141.835i − 5.38787i
\(694\) 2.56246 0.0972695
\(695\) 0 0
\(696\) 8.15482 0.309108
\(697\) 2.19767i 0.0832426i
\(698\) − 5.67176i − 0.214679i
\(699\) −0.506567 −0.0191601
\(700\) 0 0
\(701\) −16.0545 −0.606370 −0.303185 0.952932i \(-0.598050\pi\)
−0.303185 + 0.952932i \(0.598050\pi\)
\(702\) − 5.06262i − 0.191076i
\(703\) 11.4088i 0.430291i
\(704\) −6.40880 −0.241541
\(705\) 0 0
\(706\) −23.6665 −0.890701
\(707\) 49.4711i 1.86055i
\(708\) − 15.6404i − 0.587804i
\(709\) −13.5076 −0.507290 −0.253645 0.967297i \(-0.581629\pi\)
−0.253645 + 0.967297i \(0.581629\pi\)
\(710\) 0 0
\(711\) −9.25482 −0.347083
\(712\) 10.0000i 0.374766i
\(713\) − 9.97392i − 0.373526i
\(714\) 25.0105 0.935993
\(715\) 0 0
\(716\) −1.57292 −0.0587829
\(717\) 31.9501i 1.19320i
\(718\) 31.9724i 1.19320i
\(719\) −0.803402 −0.0299619 −0.0149809 0.999888i \(-0.504769\pi\)
−0.0149809 + 0.999888i \(0.504769\pi\)
\(720\) 0 0
\(721\) 79.7337 2.96944
\(722\) − 1.00000i − 0.0372161i
\(723\) − 1.23688i − 0.0460000i
\(724\) −11.4088 −0.424005
\(725\) 0 0
\(726\) 83.5218 3.09979
\(727\) − 51.6860i − 1.91693i −0.285217 0.958463i \(-0.592066\pi\)
0.285217 0.958463i \(-0.407934\pi\)
\(728\) − 4.99477i − 0.185118i
\(729\) 44.0037 1.62977
\(730\) 0 0
\(731\) −6.81761 −0.252158
\(732\) 9.41661i 0.348048i
\(733\) − 22.3723i − 0.826338i −0.910654 0.413169i \(-0.864422\pi\)
0.910654 0.413169i \(-0.135578\pi\)
\(734\) −1.14585 −0.0422940
\(735\) 0 0
\(736\) 1.79560 0.0661866
\(737\) 56.5438i 2.08282i
\(738\) − 5.40100i − 0.198814i
\(739\) −33.4271 −1.22963 −0.614817 0.788669i \(-0.710770\pi\)
−0.614817 + 0.788669i \(0.710770\pi\)
\(740\) 0 0
\(741\) −2.95449 −0.108536
\(742\) − 40.8266i − 1.49879i
\(743\) 14.4998i 0.531947i 0.963980 + 0.265974i \(0.0856934\pi\)
−0.963980 + 0.265974i \(0.914307\pi\)
\(744\) 15.4271 0.565584
\(745\) 0 0
\(746\) 32.8579 1.20301
\(747\) − 76.5699i − 2.80155i
\(748\) 12.2917i 0.449429i
\(749\) −8.40623 −0.307157
\(750\) 0 0
\(751\) −26.6169 −0.971266 −0.485633 0.874163i \(-0.661411\pi\)
−0.485633 + 0.874163i \(0.661411\pi\)
\(752\) − 10.8359i − 0.395144i
\(753\) 36.8355i 1.34236i
\(754\) −3.12351 −0.113751
\(755\) 0 0
\(756\) −22.3450 −0.812680
\(757\) 31.0362i 1.12803i 0.825764 + 0.564015i \(0.190744\pi\)
−0.825764 + 0.564015i \(0.809256\pi\)
\(758\) − 18.2865i − 0.664194i
\(759\) −31.9605 −1.16009
\(760\) 0 0
\(761\) −27.8083 −1.00805 −0.504025 0.863689i \(-0.668148\pi\)
−0.504025 + 0.863689i \(0.668148\pi\)
\(762\) − 25.7262i − 0.931963i
\(763\) − 11.3345i − 0.410338i
\(764\) 6.35805 0.230026
\(765\) 0 0
\(766\) 20.8542 0.753491
\(767\) 5.99070i 0.216312i
\(768\) 2.77733i 0.100218i
\(769\) 19.2279 0.693376 0.346688 0.937980i \(-0.387306\pi\)
0.346688 + 0.937980i \(0.387306\pi\)
\(770\) 0 0
\(771\) −65.2159 −2.34869
\(772\) − 14.4998i − 0.521860i
\(773\) 6.00373i 0.215939i 0.994154 + 0.107970i \(0.0344349\pi\)
−0.994154 + 0.107970i \(0.965565\pi\)
\(774\) 16.7550 0.602245
\(775\) 0 0
\(776\) −14.9452 −0.536501
\(777\) − 148.774i − 5.33724i
\(778\) − 24.9086i − 0.893018i
\(779\) −1.14585 −0.0410543
\(780\) 0 0
\(781\) −9.14585 −0.327264
\(782\) − 3.44385i − 0.123152i
\(783\) 13.9736i 0.499375i
\(784\) −15.0455 −0.537340
\(785\) 0 0
\(786\) 31.9895 1.14103
\(787\) − 22.2797i − 0.794187i −0.917778 0.397093i \(-0.870019\pi\)
0.917778 0.397093i \(-0.129981\pi\)
\(788\) − 5.14585i − 0.183313i
\(789\) −77.5621 −2.76128
\(790\) 0 0
\(791\) −33.5517 −1.19296
\(792\) − 30.2081i − 1.07340i
\(793\) − 3.60680i − 0.128081i
\(794\) 24.7445 0.878150
\(795\) 0 0
\(796\) 3.87766 0.137440
\(797\) − 26.6259i − 0.943138i −0.881829 0.471569i \(-0.843688\pi\)
0.881829 0.471569i \(-0.156312\pi\)
\(798\) 13.0403i 0.461621i
\(799\) −20.7826 −0.735234
\(800\) 0 0
\(801\) −47.1354 −1.66545
\(802\) − 15.6457i − 0.552468i
\(803\) − 80.8355i − 2.85262i
\(804\) 24.5039 0.864186
\(805\) 0 0
\(806\) −5.90897 −0.208135
\(807\) − 45.7747i − 1.61134i
\(808\) 10.5364i 0.370669i
\(809\) −0.651250 −0.0228967 −0.0114484 0.999934i \(-0.503644\pi\)
−0.0114484 + 0.999934i \(0.503644\pi\)
\(810\) 0 0
\(811\) −13.0780 −0.459230 −0.229615 0.973281i \(-0.573747\pi\)
−0.229615 + 0.973281i \(0.573747\pi\)
\(812\) 13.7863i 0.483804i
\(813\) − 4.07533i − 0.142928i
\(814\) 73.1168 2.56274
\(815\) 0 0
\(816\) 5.32674 0.186473
\(817\) − 3.55465i − 0.124362i
\(818\) 30.5804i 1.06922i
\(819\) 23.5430 0.822660
\(820\) 0 0
\(821\) 14.6640 0.511776 0.255888 0.966706i \(-0.417632\pi\)
0.255888 + 0.966706i \(0.417632\pi\)
\(822\) − 41.9489i − 1.46314i
\(823\) − 18.2850i − 0.637374i −0.947860 0.318687i \(-0.896758\pi\)
0.947860 0.318687i \(-0.103242\pi\)
\(824\) 16.9817 0.591587
\(825\) 0 0
\(826\) 26.4413 0.920010
\(827\) − 40.1910i − 1.39758i −0.715327 0.698790i \(-0.753722\pi\)
0.715327 0.698790i \(-0.246278\pi\)
\(828\) 8.46362i 0.294131i
\(829\) −28.3760 −0.985539 −0.492769 0.870160i \(-0.664015\pi\)
−0.492769 + 0.870160i \(0.664015\pi\)
\(830\) 0 0
\(831\) 43.9814 1.52570
\(832\) − 1.06379i − 0.0368802i
\(833\) 28.8564i 0.999815i
\(834\) 1.94552 0.0673677
\(835\) 0 0
\(836\) −6.40880 −0.221653
\(837\) 26.4349i 0.913723i
\(838\) 1.96345i 0.0678264i
\(839\) −6.93471 −0.239413 −0.119706 0.992809i \(-0.538195\pi\)
−0.119706 + 0.992809i \(0.538195\pi\)
\(840\) 0 0
\(841\) −20.3787 −0.702712
\(842\) 25.5949i 0.882060i
\(843\) − 17.3726i − 0.598344i
\(844\) −17.1496 −0.590313
\(845\) 0 0
\(846\) 51.0753 1.75601
\(847\) 141.200i 4.85167i
\(848\) − 8.69527i − 0.298597i
\(849\) 46.3827 1.59185
\(850\) 0 0
\(851\) −20.4856 −0.702238
\(852\) 3.96345i 0.135786i
\(853\) 21.9739i 0.752373i 0.926544 + 0.376186i \(0.122765\pi\)
−0.926544 + 0.376186i \(0.877235\pi\)
\(854\) −15.9194 −0.544752
\(855\) 0 0
\(856\) −1.79036 −0.0611934
\(857\) − 13.6091i − 0.464879i −0.972611 0.232440i \(-0.925329\pi\)
0.972611 0.232440i \(-0.0746708\pi\)
\(858\) 18.9347i 0.646420i
\(859\) −5.17192 −0.176464 −0.0882319 0.996100i \(-0.528122\pi\)
−0.0882319 + 0.996100i \(0.528122\pi\)
\(860\) 0 0
\(861\) 14.9422 0.509228
\(862\) 15.3540i 0.522959i
\(863\) 15.9635i 0.543402i 0.962382 + 0.271701i \(0.0875862\pi\)
−0.962382 + 0.271701i \(0.912414\pi\)
\(864\) −4.75905 −0.161906
\(865\) 0 0
\(866\) 16.6640 0.566264
\(867\) 36.9982i 1.25652i
\(868\) 26.0806i 0.885232i
\(869\) 12.5834 0.426862
\(870\) 0 0
\(871\) −9.38563 −0.318020
\(872\) − 2.41404i − 0.0817496i
\(873\) − 70.4447i − 2.38419i
\(874\) 1.79560 0.0607370
\(875\) 0 0
\(876\) −35.0310 −1.18359
\(877\) 15.8866i 0.536453i 0.963356 + 0.268227i \(0.0864376\pi\)
−0.963356 + 0.268227i \(0.913562\pi\)
\(878\) − 19.5987i − 0.661423i
\(879\) −1.79003 −0.0603762
\(880\) 0 0
\(881\) −16.1458 −0.543967 −0.271984 0.962302i \(-0.587680\pi\)
−0.271984 + 0.962302i \(0.587680\pi\)
\(882\) − 70.9176i − 2.38792i
\(883\) 38.2887i 1.28852i 0.764808 + 0.644259i \(0.222834\pi\)
−0.764808 + 0.644259i \(0.777166\pi\)
\(884\) −2.04028 −0.0686221
\(885\) 0 0
\(886\) 13.0183 0.437357
\(887\) − 19.2809i − 0.647389i −0.946162 0.323695i \(-0.895075\pi\)
0.946162 0.323695i \(-0.104925\pi\)
\(888\) − 31.6860i − 1.06331i
\(889\) 43.4920 1.45868
\(890\) 0 0
\(891\) −5.91644 −0.198208
\(892\) − 7.84635i − 0.262715i
\(893\) − 10.8359i − 0.362609i
\(894\) −35.9530 −1.20245
\(895\) 0 0
\(896\) −4.69527 −0.156858
\(897\) − 5.30507i − 0.177131i
\(898\) 6.77359i 0.226038i
\(899\) 16.3096 0.543957
\(900\) 0 0
\(901\) −16.6770 −0.555591
\(902\) 7.34352i 0.244512i
\(903\) 46.3537i 1.54255i
\(904\) −7.14585 −0.237667
\(905\) 0 0
\(906\) 15.8538 0.526708
\(907\) − 43.0205i − 1.42847i −0.699905 0.714236i \(-0.746774\pi\)
0.699905 0.714236i \(-0.253226\pi\)
\(908\) 6.28646i 0.208624i
\(909\) −49.6636 −1.64724
\(910\) 0 0
\(911\) −25.7733 −0.853906 −0.426953 0.904274i \(-0.640413\pi\)
−0.426953 + 0.904274i \(0.640413\pi\)
\(912\) 2.77733i 0.0919664i
\(913\) 104.109i 3.44550i
\(914\) 27.3189 0.903630
\(915\) 0 0
\(916\) −3.14585 −0.103942
\(917\) 54.0806i 1.78590i
\(918\) 9.12758i 0.301255i
\(919\) 29.5897 0.976074 0.488037 0.872823i \(-0.337713\pi\)
0.488037 + 0.872823i \(0.337713\pi\)
\(920\) 0 0
\(921\) 74.0037 2.43851
\(922\) 27.7993i 0.915522i
\(923\) − 1.51811i − 0.0499691i
\(924\) 83.5726 2.74934
\(925\) 0 0
\(926\) 38.2369 1.25654
\(927\) 80.0440i 2.62899i
\(928\) 2.93621i 0.0963859i
\(929\) 1.42334 0.0466983 0.0233491 0.999727i \(-0.492567\pi\)
0.0233491 + 0.999727i \(0.492567\pi\)
\(930\) 0 0
\(931\) −15.0455 −0.493097
\(932\) − 0.182394i − 0.00597451i
\(933\) 32.0765i 1.05014i
\(934\) −23.4711 −0.767998
\(935\) 0 0
\(936\) 5.01420 0.163894
\(937\) − 28.2276i − 0.922155i −0.887360 0.461077i \(-0.847463\pi\)
0.887360 0.461077i \(-0.152537\pi\)
\(938\) 41.4256i 1.35259i
\(939\) 64.0556 2.09038
\(940\) 0 0
\(941\) −15.8672 −0.517256 −0.258628 0.965977i \(-0.583270\pi\)
−0.258628 + 0.965977i \(0.583270\pi\)
\(942\) − 25.7262i − 0.838206i
\(943\) − 2.05748i − 0.0670009i
\(944\) 5.63148 0.183289
\(945\) 0 0
\(946\) −22.7811 −0.740676
\(947\) − 43.6718i − 1.41914i −0.704634 0.709571i \(-0.748889\pi\)
0.704634 0.709571i \(-0.251111\pi\)
\(948\) − 5.45315i − 0.177110i
\(949\) 13.4178 0.435559
\(950\) 0 0
\(951\) −38.6692 −1.25393
\(952\) 9.00523i 0.291861i
\(953\) 16.0261i 0.519136i 0.965725 + 0.259568i \(0.0835801\pi\)
−0.965725 + 0.259568i \(0.916420\pi\)
\(954\) 40.9855 1.32695
\(955\) 0 0
\(956\) −11.5039 −0.372063
\(957\) − 52.2626i − 1.68941i
\(958\) − 19.6900i − 0.636156i
\(959\) 70.9176 2.29005
\(960\) 0 0
\(961\) −0.145848 −0.00470478
\(962\) 12.1365i 0.391298i
\(963\) − 8.43895i − 0.271941i
\(964\) 0.445349 0.0143437
\(965\) 0 0
\(966\) −23.4151 −0.753369
\(967\) 11.5987i 0.372988i 0.982456 + 0.186494i \(0.0597125\pi\)
−0.982456 + 0.186494i \(0.940288\pi\)
\(968\) 30.0728i 0.966575i
\(969\) 5.32674 0.171120
\(970\) 0 0
\(971\) −27.0362 −0.867633 −0.433817 0.901001i \(-0.642833\pi\)
−0.433817 + 0.901001i \(0.642833\pi\)
\(972\) 16.8411i 0.540179i
\(973\) 3.28903i 0.105442i
\(974\) −16.3357 −0.523430
\(975\) 0 0
\(976\) −3.39053 −0.108528
\(977\) − 14.1537i − 0.452815i −0.974033 0.226408i \(-0.927302\pi\)
0.974033 0.226408i \(-0.0726982\pi\)
\(978\) 11.8904i 0.380212i
\(979\) 64.0880 2.04826
\(980\) 0 0
\(981\) 11.3787 0.363293
\(982\) − 27.1093i − 0.865093i
\(983\) − 32.8542i − 1.04788i −0.851754 0.523942i \(-0.824461\pi\)
0.851754 0.523942i \(-0.175539\pi\)
\(984\) 3.18239 0.101451
\(985\) 0 0
\(986\) 5.63148 0.179343
\(987\) 141.303i 4.49772i
\(988\) − 1.06379i − 0.0338436i
\(989\) 6.38273 0.202959
\(990\) 0 0
\(991\) 47.9709 1.52385 0.761923 0.647667i \(-0.224255\pi\)
0.761923 + 0.647667i \(0.224255\pi\)
\(992\) 5.55465i 0.176360i
\(993\) − 2.04285i − 0.0648279i
\(994\) −6.70050 −0.212527
\(995\) 0 0
\(996\) 45.1168 1.42958
\(997\) − 44.1716i − 1.39893i −0.714668 0.699464i \(-0.753422\pi\)
0.714668 0.699464i \(-0.246578\pi\)
\(998\) − 4.69003i − 0.148460i
\(999\) 54.2951 1.71782
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 950.2.b.g.799.3 6
5.2 odd 4 950.2.a.m.1.3 yes 3
5.3 odd 4 950.2.a.k.1.1 3
5.4 even 2 inner 950.2.b.g.799.4 6
15.2 even 4 8550.2.a.cj.1.1 3
15.8 even 4 8550.2.a.co.1.3 3
20.3 even 4 7600.2.a.cb.1.3 3
20.7 even 4 7600.2.a.bm.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.1 3 5.3 odd 4
950.2.a.m.1.3 yes 3 5.2 odd 4
950.2.b.g.799.3 6 1.1 even 1 trivial
950.2.b.g.799.4 6 5.4 even 2 inner
7600.2.a.bm.1.1 3 20.7 even 4
7600.2.a.cb.1.3 3 20.3 even 4
8550.2.a.cj.1.1 3 15.2 even 4
8550.2.a.co.1.3 3 15.8 even 4