Properties

Label 950.2.b.g.799.5
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 799.5
Root \(-1.91223i\) of defining polynomial
Character \(\chi\) \(=\) 950.799
Dual form 950.2.b.g.799.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -2.25561i q^{3} -1.00000 q^{4} +2.25561 q^{6} +4.22547i q^{7} -1.00000i q^{8} -2.08777 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -2.25561i q^{3} -1.00000 q^{4} +2.25561 q^{6} +4.22547i q^{7} -1.00000i q^{8} -2.08777 q^{9} -5.13770 q^{11} +2.25561i q^{12} -3.16784i q^{13} -4.22547 q^{14} +1.00000 q^{16} -6.48108i q^{17} -2.08777i q^{18} +1.00000 q^{19} +9.53101 q^{21} -5.13770i q^{22} -7.56885i q^{23} -2.25561 q^{24} +3.16784 q^{26} -2.05763i q^{27} -4.22547i q^{28} -0.832162 q^{29} -4.51122 q^{31} +1.00000i q^{32} +11.5886i q^{33} +6.48108 q^{34} +2.08777 q^{36} +0.137699i q^{37} +1.00000i q^{38} -7.14540 q^{39} -11.6489 q^{41} +9.53101i q^{42} +2.51122i q^{43} +5.13770 q^{44} +7.56885 q^{46} -5.96216i q^{47} -2.25561i q^{48} -10.8546 q^{49} -14.6188 q^{51} +3.16784i q^{52} -0.225470i q^{53} +2.05763 q^{54} +4.22547 q^{56} -2.25561i q^{57} -0.832162i q^{58} -5.39331 q^{59} +14.4509 q^{61} -4.51122i q^{62} -8.82181i q^{63} -1.00000 q^{64} -11.5886 q^{66} -4.11021i q^{67} +6.48108i q^{68} -17.0724 q^{69} +3.82446 q^{71} +2.08777i q^{72} -4.70655i q^{73} -0.137699 q^{74} -1.00000 q^{76} -21.7092i q^{77} -7.14540i q^{78} -10.6265 q^{79} -10.9045 q^{81} -11.6489i q^{82} +12.0999i q^{83} -9.53101 q^{84} -2.51122 q^{86} +1.87703i q^{87} +5.13770i q^{88} +10.0000 q^{89} +13.3856 q^{91} +7.56885i q^{92} +10.1755i q^{93} +5.96216 q^{94} +2.25561 q^{96} +3.93972i q^{97} -10.8546i q^{98} +10.7263 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 4 q^{6} - 26 q^{9} + O(q^{10}) \) \( 6 q - 6 q^{4} + 4 q^{6} - 26 q^{9} + 4 q^{11} - 4 q^{14} + 6 q^{16} + 6 q^{19} - 22 q^{21} - 4 q^{24} - 4 q^{26} - 28 q^{29} - 8 q^{31} + 8 q^{34} + 26 q^{36} - 58 q^{39} - 16 q^{41} - 4 q^{44} + 28 q^{46} - 50 q^{49} - 22 q^{51} + 14 q^{54} + 4 q^{56} + 12 q^{59} + 44 q^{61} - 6 q^{64} + 8 q^{66} - 16 q^{69} - 4 q^{71} + 34 q^{74} - 6 q^{76} - 48 q^{79} - 2 q^{81} + 22 q^{84} + 4 q^{86} + 60 q^{89} - 14 q^{91} - 26 q^{94} + 4 q^{96} - 48 q^{99} + O(q^{100}) \)

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
\(3\) − 2.25561i − 1.30228i −0.758959 0.651138i \(-0.774292\pi\)
0.758959 0.651138i \(-0.225708\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.25561 0.920848
\(7\) 4.22547i 1.59708i 0.601943 + 0.798539i \(0.294393\pi\)
−0.601943 + 0.798539i \(0.705607\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −2.08777 −0.695924
\(10\) 0 0
\(11\) −5.13770 −1.54907 −0.774537 0.632528i \(-0.782017\pi\)
−0.774537 + 0.632528i \(0.782017\pi\)
\(12\) 2.25561i 0.651138i
\(13\) − 3.16784i − 0.878600i −0.898340 0.439300i \(-0.855226\pi\)
0.898340 0.439300i \(-0.144774\pi\)
\(14\) −4.22547 −1.12930
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.48108i − 1.57189i −0.618295 0.785946i \(-0.712176\pi\)
0.618295 0.785946i \(-0.287824\pi\)
\(18\) − 2.08777i − 0.492092i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 9.53101 2.07984
\(22\) − 5.13770i − 1.09536i
\(23\) − 7.56885i − 1.57821i −0.614256 0.789107i \(-0.710544\pi\)
0.614256 0.789107i \(-0.289456\pi\)
\(24\) −2.25561 −0.460424
\(25\) 0 0
\(26\) 3.16784 0.621264
\(27\) − 2.05763i − 0.395991i
\(28\) − 4.22547i − 0.798539i
\(29\) −0.832162 −0.154529 −0.0772643 0.997011i \(-0.524619\pi\)
−0.0772643 + 0.997011i \(0.524619\pi\)
\(30\) 0 0
\(31\) −4.51122 −0.810239 −0.405119 0.914264i \(-0.632770\pi\)
−0.405119 + 0.914264i \(0.632770\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 11.5886i 2.01732i
\(34\) 6.48108 1.11150
\(35\) 0 0
\(36\) 2.08777 0.347962
\(37\) 0.137699i 0.0226376i 0.999936 + 0.0113188i \(0.00360296\pi\)
−0.999936 + 0.0113188i \(0.996397\pi\)
\(38\) 1.00000i 0.162221i
\(39\) −7.14540 −1.14418
\(40\) 0 0
\(41\) −11.6489 −1.81926 −0.909628 0.415425i \(-0.863633\pi\)
−0.909628 + 0.415425i \(0.863633\pi\)
\(42\) 9.53101i 1.47067i
\(43\) 2.51122i 0.382957i 0.981497 + 0.191479i \(0.0613282\pi\)
−0.981497 + 0.191479i \(0.938672\pi\)
\(44\) 5.13770 0.774537
\(45\) 0 0
\(46\) 7.56885 1.11597
\(47\) − 5.96216i − 0.869670i −0.900510 0.434835i \(-0.856807\pi\)
0.900510 0.434835i \(-0.143193\pi\)
\(48\) − 2.25561i − 0.325569i
\(49\) −10.8546 −1.55066
\(50\) 0 0
\(51\) −14.6188 −2.04704
\(52\) 3.16784i 0.439300i
\(53\) − 0.225470i − 0.0309707i −0.999880 0.0154853i \(-0.995071\pi\)
0.999880 0.0154853i \(-0.00492934\pi\)
\(54\) 2.05763 0.280008
\(55\) 0 0
\(56\) 4.22547 0.564652
\(57\) − 2.25561i − 0.298763i
\(58\) − 0.832162i − 0.109268i
\(59\) −5.39331 −0.702149 −0.351074 0.936348i \(-0.614184\pi\)
−0.351074 + 0.936348i \(0.614184\pi\)
\(60\) 0 0
\(61\) 14.4509 1.85025 0.925127 0.379659i \(-0.123959\pi\)
0.925127 + 0.379659i \(0.123959\pi\)
\(62\) − 4.51122i − 0.572925i
\(63\) − 8.82181i − 1.11144i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −11.5886 −1.42646
\(67\) − 4.11021i − 0.502142i −0.967969 0.251071i \(-0.919217\pi\)
0.967969 0.251071i \(-0.0807827\pi\)
\(68\) 6.48108i 0.785946i
\(69\) −17.0724 −2.05527
\(70\) 0 0
\(71\) 3.82446 0.453880 0.226940 0.973909i \(-0.427128\pi\)
0.226940 + 0.973909i \(0.427128\pi\)
\(72\) 2.08777i 0.246046i
\(73\) − 4.70655i − 0.550860i −0.961321 0.275430i \(-0.911180\pi\)
0.961321 0.275430i \(-0.0888202\pi\)
\(74\) −0.137699 −0.0160072
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) − 21.7092i − 2.47399i
\(78\) − 7.14540i − 0.809058i
\(79\) −10.6265 −1.19557 −0.597786 0.801655i \(-0.703953\pi\)
−0.597786 + 0.801655i \(0.703953\pi\)
\(80\) 0 0
\(81\) −10.9045 −1.21161
\(82\) − 11.6489i − 1.28641i
\(83\) 12.0999i 1.32813i 0.747674 + 0.664066i \(0.231171\pi\)
−0.747674 + 0.664066i \(0.768829\pi\)
\(84\) −9.53101 −1.03992
\(85\) 0 0
\(86\) −2.51122 −0.270792
\(87\) 1.87703i 0.201239i
\(88\) 5.13770i 0.547681i
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 13.3856 1.40319
\(92\) 7.56885i 0.789107i
\(93\) 10.1755i 1.05515i
\(94\) 5.96216 0.614950
\(95\) 0 0
\(96\) 2.25561 0.230212
\(97\) 3.93972i 0.400018i 0.979794 + 0.200009i \(0.0640972\pi\)
−0.979794 + 0.200009i \(0.935903\pi\)
\(98\) − 10.8546i − 1.09648i
\(99\) 10.7263 1.07804
\(100\) 0 0
\(101\) 3.19798 0.318211 0.159105 0.987262i \(-0.449139\pi\)
0.159105 + 0.987262i \(0.449139\pi\)
\(102\) − 14.6188i − 1.44747i
\(103\) 10.6868i 1.05300i 0.850176 + 0.526499i \(0.176496\pi\)
−0.850176 + 0.526499i \(0.823504\pi\)
\(104\) −3.16784 −0.310632
\(105\) 0 0
\(106\) 0.225470 0.0218996
\(107\) 10.8168i 1.04570i 0.852426 + 0.522848i \(0.175130\pi\)
−0.852426 + 0.522848i \(0.824870\pi\)
\(108\) 2.05763i 0.197996i
\(109\) −9.24791 −0.885789 −0.442894 0.896574i \(-0.646048\pi\)
−0.442894 + 0.896574i \(0.646048\pi\)
\(110\) 0 0
\(111\) 0.310596 0.0294804
\(112\) 4.22547i 0.399269i
\(113\) − 17.6489i − 1.66027i −0.557562 0.830135i \(-0.688263\pi\)
0.557562 0.830135i \(-0.311737\pi\)
\(114\) 2.25561 0.211257
\(115\) 0 0
\(116\) 0.832162 0.0772643
\(117\) 6.61372i 0.611439i
\(118\) − 5.39331i − 0.496494i
\(119\) 27.3856 2.51043
\(120\) 0 0
\(121\) 15.3960 1.39963
\(122\) 14.4509i 1.30833i
\(123\) 26.2754i 2.36917i
\(124\) 4.51122 0.405119
\(125\) 0 0
\(126\) 8.82181 0.785910
\(127\) − 12.7866i − 1.13463i −0.823501 0.567314i \(-0.807982\pi\)
0.823501 0.567314i \(-0.192018\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 5.66432 0.498716
\(130\) 0 0
\(131\) −2.11526 −0.184811 −0.0924057 0.995721i \(-0.529456\pi\)
−0.0924057 + 0.995721i \(0.529456\pi\)
\(132\) − 11.5886i − 1.00866i
\(133\) 4.22547i 0.366395i
\(134\) 4.11021 0.355068
\(135\) 0 0
\(136\) −6.48108 −0.555748
\(137\) − 5.36317i − 0.458206i −0.973402 0.229103i \(-0.926421\pi\)
0.973402 0.229103i \(-0.0735793\pi\)
\(138\) − 17.0724i − 1.45330i
\(139\) 10.1601 0.861771 0.430886 0.902407i \(-0.358201\pi\)
0.430886 + 0.902407i \(0.358201\pi\)
\(140\) 0 0
\(141\) −13.4483 −1.13255
\(142\) 3.82446i 0.320941i
\(143\) 16.2754i 1.36102i
\(144\) −2.08777 −0.173981
\(145\) 0 0
\(146\) 4.70655 0.389517
\(147\) 24.4837i 2.01938i
\(148\) − 0.137699i − 0.0113188i
\(149\) 5.93972 0.486601 0.243301 0.969951i \(-0.421770\pi\)
0.243301 + 0.969951i \(0.421770\pi\)
\(150\) 0 0
\(151\) −15.2978 −1.24492 −0.622460 0.782652i \(-0.713867\pi\)
−0.622460 + 0.782652i \(0.713867\pi\)
\(152\) − 1.00000i − 0.0811107i
\(153\) 13.5310i 1.09392i
\(154\) 21.7092 1.74938
\(155\) 0 0
\(156\) 7.14540 0.572090
\(157\) − 12.7866i − 1.02048i −0.860031 0.510242i \(-0.829556\pi\)
0.860031 0.510242i \(-0.170444\pi\)
\(158\) − 10.6265i − 0.845397i
\(159\) −0.508572 −0.0403324
\(160\) 0 0
\(161\) 31.9819 2.52053
\(162\) − 10.9045i − 0.856740i
\(163\) 11.4734i 0.898664i 0.893365 + 0.449332i \(0.148338\pi\)
−0.893365 + 0.449332i \(0.851662\pi\)
\(164\) 11.6489 0.909628
\(165\) 0 0
\(166\) −12.0999 −0.939131
\(167\) 19.4131i 1.50223i 0.660171 + 0.751115i \(0.270484\pi\)
−0.660171 + 0.751115i \(0.729516\pi\)
\(168\) − 9.53101i − 0.735333i
\(169\) 2.96480 0.228062
\(170\) 0 0
\(171\) −2.08777 −0.159656
\(172\) − 2.51122i − 0.191479i
\(173\) − 9.78662i − 0.744063i −0.928220 0.372031i \(-0.878661\pi\)
0.928220 0.372031i \(-0.121339\pi\)
\(174\) −1.87703 −0.142297
\(175\) 0 0
\(176\) −5.13770 −0.387269
\(177\) 12.1652i 0.914392i
\(178\) 10.0000i 0.749532i
\(179\) 6.82446 0.510084 0.255042 0.966930i \(-0.417911\pi\)
0.255042 + 0.966930i \(0.417911\pi\)
\(180\) 0 0
\(181\) −0.137699 −0.0102351 −0.00511755 0.999987i \(-0.501629\pi\)
−0.00511755 + 0.999987i \(0.501629\pi\)
\(182\) 13.3856i 0.992207i
\(183\) − 32.5957i − 2.40954i
\(184\) −7.56885 −0.557983
\(185\) 0 0
\(186\) −10.1755 −0.746107
\(187\) 33.2978i 2.43498i
\(188\) 5.96216i 0.434835i
\(189\) 8.69446 0.632429
\(190\) 0 0
\(191\) 19.3779 1.40214 0.701068 0.713095i \(-0.252707\pi\)
0.701068 + 0.713095i \(0.252707\pi\)
\(192\) 2.25561i 0.162785i
\(193\) 5.42851i 0.390752i 0.980728 + 0.195376i \(0.0625928\pi\)
−0.980728 + 0.195376i \(0.937407\pi\)
\(194\) −3.93972 −0.282856
\(195\) 0 0
\(196\) 10.8546 0.775328
\(197\) − 15.6489i − 1.11494i −0.830197 0.557470i \(-0.811772\pi\)
0.830197 0.557470i \(-0.188228\pi\)
\(198\) 10.7263i 0.762288i
\(199\) −18.0499 −1.27953 −0.639763 0.768572i \(-0.720967\pi\)
−0.639763 + 0.768572i \(0.720967\pi\)
\(200\) 0 0
\(201\) −9.27102 −0.653927
\(202\) 3.19798i 0.225009i
\(203\) − 3.51628i − 0.246794i
\(204\) 14.6188 1.02352
\(205\) 0 0
\(206\) −10.6868 −0.744582
\(207\) 15.8020i 1.09832i
\(208\) − 3.16784i − 0.219650i
\(209\) −5.13770 −0.355382
\(210\) 0 0
\(211\) −7.50857 −0.516911 −0.258456 0.966023i \(-0.583214\pi\)
−0.258456 + 0.966023i \(0.583214\pi\)
\(212\) 0.225470i 0.0154853i
\(213\) − 8.62648i − 0.591077i
\(214\) −10.8168 −0.739418
\(215\) 0 0
\(216\) −2.05763 −0.140004
\(217\) − 19.0620i − 1.29401i
\(218\) − 9.24791i − 0.626347i
\(219\) −10.6161 −0.717372
\(220\) 0 0
\(221\) −20.5310 −1.38106
\(222\) 0.310596i 0.0208458i
\(223\) − 27.8091i − 1.86223i −0.364723 0.931116i \(-0.618836\pi\)
0.364723 0.931116i \(-0.381164\pi\)
\(224\) −4.22547 −0.282326
\(225\) 0 0
\(226\) 17.6489 1.17399
\(227\) 8.91223i 0.591525i 0.955261 + 0.295763i \(0.0955738\pi\)
−0.955261 + 0.295763i \(0.904426\pi\)
\(228\) 2.25561i 0.149381i
\(229\) 13.6489 0.901946 0.450973 0.892538i \(-0.351077\pi\)
0.450973 + 0.892538i \(0.351077\pi\)
\(230\) 0 0
\(231\) −48.9674 −3.22182
\(232\) 0.832162i 0.0546341i
\(233\) − 23.2754i − 1.52482i −0.647093 0.762411i \(-0.724015\pi\)
0.647093 0.762411i \(-0.275985\pi\)
\(234\) −6.61372 −0.432352
\(235\) 0 0
\(236\) 5.39331 0.351074
\(237\) 23.9692i 1.55697i
\(238\) 27.3856i 1.77515i
\(239\) −3.72898 −0.241208 −0.120604 0.992701i \(-0.538483\pi\)
−0.120604 + 0.992701i \(0.538483\pi\)
\(240\) 0 0
\(241\) −1.48878 −0.0959009 −0.0479505 0.998850i \(-0.515269\pi\)
−0.0479505 + 0.998850i \(0.515269\pi\)
\(242\) 15.3960i 0.989689i
\(243\) 18.4234i 1.18186i
\(244\) −14.4509 −0.925127
\(245\) 0 0
\(246\) −26.2754 −1.67526
\(247\) − 3.16784i − 0.201565i
\(248\) 4.51122i 0.286463i
\(249\) 27.2925 1.72959
\(250\) 0 0
\(251\) −8.78662 −0.554606 −0.277303 0.960782i \(-0.589441\pi\)
−0.277303 + 0.960782i \(0.589441\pi\)
\(252\) 8.82181i 0.555722i
\(253\) 38.8865i 2.44477i
\(254\) 12.7866 0.802304
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.74175i 0.171025i 0.996337 + 0.0855127i \(0.0272528\pi\)
−0.996337 + 0.0855127i \(0.972747\pi\)
\(258\) 5.66432i 0.352645i
\(259\) −0.581844 −0.0361540
\(260\) 0 0
\(261\) 1.73736 0.107540
\(262\) − 2.11526i − 0.130681i
\(263\) − 2.74704i − 0.169390i −0.996407 0.0846948i \(-0.973008\pi\)
0.996407 0.0846948i \(-0.0269915\pi\)
\(264\) 11.5886 0.713231
\(265\) 0 0
\(266\) −4.22547 −0.259080
\(267\) − 22.5561i − 1.38041i
\(268\) 4.11021i 0.251071i
\(269\) 9.74175 0.593965 0.296982 0.954883i \(-0.404020\pi\)
0.296982 + 0.954883i \(0.404020\pi\)
\(270\) 0 0
\(271\) 26.3555 1.60098 0.800490 0.599346i \(-0.204573\pi\)
0.800490 + 0.599346i \(0.204573\pi\)
\(272\) − 6.48108i − 0.392973i
\(273\) − 30.1927i − 1.82734i
\(274\) 5.36317 0.324001
\(275\) 0 0
\(276\) 17.0724 1.02764
\(277\) − 0.962158i − 0.0578104i −0.999582 0.0289052i \(-0.990798\pi\)
0.999582 0.0289052i \(-0.00920210\pi\)
\(278\) 10.1601i 0.609364i
\(279\) 9.41839 0.563864
\(280\) 0 0
\(281\) −14.6714 −0.875219 −0.437610 0.899165i \(-0.644175\pi\)
−0.437610 + 0.899165i \(0.644175\pi\)
\(282\) − 13.4483i − 0.800834i
\(283\) 26.1601i 1.55506i 0.628847 + 0.777529i \(0.283527\pi\)
−0.628847 + 0.777529i \(0.716473\pi\)
\(284\) −3.82446 −0.226940
\(285\) 0 0
\(286\) −16.2754 −0.962384
\(287\) − 49.2221i − 2.90549i
\(288\) − 2.08777i − 0.123023i
\(289\) −25.0044 −1.47085
\(290\) 0 0
\(291\) 8.88647 0.520934
\(292\) 4.70655i 0.275430i
\(293\) 22.4657i 1.31246i 0.754562 + 0.656229i \(0.227850\pi\)
−0.754562 + 0.656229i \(0.772150\pi\)
\(294\) −24.4837 −1.42792
\(295\) 0 0
\(296\) 0.137699 0.00800360
\(297\) 10.5715i 0.613420i
\(298\) 5.93972i 0.344079i
\(299\) −23.9769 −1.38662
\(300\) 0 0
\(301\) −10.6111 −0.611612
\(302\) − 15.2978i − 0.880291i
\(303\) − 7.21338i − 0.414398i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −13.5310 −0.773516
\(307\) 17.2204i 0.982821i 0.870928 + 0.491410i \(0.163518\pi\)
−0.870928 + 0.491410i \(0.836482\pi\)
\(308\) 21.7092i 1.23700i
\(309\) 24.1051 1.37129
\(310\) 0 0
\(311\) −7.87439 −0.446516 −0.223258 0.974759i \(-0.571669\pi\)
−0.223258 + 0.974759i \(0.571669\pi\)
\(312\) 7.14540i 0.404529i
\(313\) 25.1678i 1.42257i 0.702904 + 0.711285i \(0.251887\pi\)
−0.702904 + 0.711285i \(0.748113\pi\)
\(314\) 12.7866 0.721590
\(315\) 0 0
\(316\) 10.6265 0.597786
\(317\) − 23.9045i − 1.34261i −0.741180 0.671306i \(-0.765734\pi\)
0.741180 0.671306i \(-0.234266\pi\)
\(318\) − 0.508572i − 0.0285193i
\(319\) 4.27540 0.239376
\(320\) 0 0
\(321\) 24.3984 1.36178
\(322\) 31.9819i 1.78228i
\(323\) − 6.48108i − 0.360617i
\(324\) 10.9045 0.605807
\(325\) 0 0
\(326\) −11.4734 −0.635451
\(327\) 20.8597i 1.15354i
\(328\) 11.6489i 0.643204i
\(329\) 25.1929 1.38893
\(330\) 0 0
\(331\) 30.7565 1.69053 0.845264 0.534348i \(-0.179443\pi\)
0.845264 + 0.534348i \(0.179443\pi\)
\(332\) − 12.0999i − 0.664066i
\(333\) − 0.287484i − 0.0157540i
\(334\) −19.4131 −1.06224
\(335\) 0 0
\(336\) 9.53101 0.519959
\(337\) 13.4734i 0.733942i 0.930232 + 0.366971i \(0.119605\pi\)
−0.930232 + 0.366971i \(0.880395\pi\)
\(338\) 2.96480i 0.161264i
\(339\) −39.8091 −2.16213
\(340\) 0 0
\(341\) 23.1773 1.25512
\(342\) − 2.08777i − 0.112894i
\(343\) − 16.2875i − 0.879441i
\(344\) 2.51122 0.135396
\(345\) 0 0
\(346\) 9.78662 0.526132
\(347\) 28.9468i 1.55394i 0.629536 + 0.776971i \(0.283245\pi\)
−0.629536 + 0.776971i \(0.716755\pi\)
\(348\) − 1.87703i − 0.100619i
\(349\) −27.9243 −1.49475 −0.747377 0.664400i \(-0.768687\pi\)
−0.747377 + 0.664400i \(0.768687\pi\)
\(350\) 0 0
\(351\) −6.51825 −0.347918
\(352\) − 5.13770i − 0.273840i
\(353\) − 28.3099i − 1.50678i −0.657571 0.753392i \(-0.728416\pi\)
0.657571 0.753392i \(-0.271584\pi\)
\(354\) −12.1652 −0.646573
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) − 61.7712i − 3.26928i
\(358\) 6.82446i 0.360684i
\(359\) −2.60163 −0.137309 −0.0686545 0.997640i \(-0.521871\pi\)
−0.0686545 + 0.997640i \(0.521871\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 0.137699i − 0.00723731i
\(363\) − 34.7272i − 1.82271i
\(364\) −13.3856 −0.701596
\(365\) 0 0
\(366\) 32.5957 1.70380
\(367\) 11.6489i 0.608069i 0.952661 + 0.304034i \(0.0983337\pi\)
−0.952661 + 0.304034i \(0.901666\pi\)
\(368\) − 7.56885i − 0.394554i
\(369\) 24.3203 1.26606
\(370\) 0 0
\(371\) 0.952717 0.0494626
\(372\) − 10.1755i − 0.527577i
\(373\) 12.8064i 0.663091i 0.943439 + 0.331545i \(0.107570\pi\)
−0.943439 + 0.331545i \(0.892430\pi\)
\(374\) −33.2978 −1.72179
\(375\) 0 0
\(376\) −5.96216 −0.307475
\(377\) 2.63615i 0.135769i
\(378\) 8.69446i 0.447195i
\(379\) 20.9122 1.07419 0.537095 0.843522i \(-0.319522\pi\)
0.537095 + 0.843522i \(0.319522\pi\)
\(380\) 0 0
\(381\) −28.8416 −1.47760
\(382\) 19.3779i 0.991460i
\(383\) − 10.3511i − 0.528916i −0.964397 0.264458i \(-0.914807\pi\)
0.964397 0.264458i \(-0.0851930\pi\)
\(384\) −2.25561 −0.115106
\(385\) 0 0
\(386\) −5.42851 −0.276304
\(387\) − 5.24285i − 0.266509i
\(388\) − 3.93972i − 0.200009i
\(389\) −6.56620 −0.332920 −0.166460 0.986048i \(-0.553234\pi\)
−0.166460 + 0.986048i \(0.553234\pi\)
\(390\) 0 0
\(391\) −49.0543 −2.48078
\(392\) 10.8546i 0.548240i
\(393\) 4.77121i 0.240676i
\(394\) 15.6489 0.788381
\(395\) 0 0
\(396\) −10.7263 −0.539019
\(397\) 23.5284i 1.18085i 0.807091 + 0.590427i \(0.201041\pi\)
−0.807091 + 0.590427i \(0.798959\pi\)
\(398\) − 18.0499i − 0.904761i
\(399\) 9.53101 0.477147
\(400\) 0 0
\(401\) 6.22041 0.310633 0.155316 0.987865i \(-0.450360\pi\)
0.155316 + 0.987865i \(0.450360\pi\)
\(402\) − 9.27102i − 0.462396i
\(403\) 14.2908i 0.711876i
\(404\) −3.19798 −0.159105
\(405\) 0 0
\(406\) 3.51628 0.174510
\(407\) − 0.707457i − 0.0350674i
\(408\) 14.6188i 0.723737i
\(409\) 34.4905 1.70545 0.852723 0.522363i \(-0.174949\pi\)
0.852723 + 0.522363i \(0.174949\pi\)
\(410\) 0 0
\(411\) −12.0972 −0.596711
\(412\) − 10.6868i − 0.526499i
\(413\) − 22.7893i − 1.12139i
\(414\) −15.8020 −0.776627
\(415\) 0 0
\(416\) 3.16784 0.155316
\(417\) − 22.9173i − 1.12226i
\(418\) − 5.13770i − 0.251293i
\(419\) 10.6265 0.519138 0.259569 0.965725i \(-0.416420\pi\)
0.259569 + 0.965725i \(0.416420\pi\)
\(420\) 0 0
\(421\) −1.98021 −0.0965095 −0.0482548 0.998835i \(-0.515366\pi\)
−0.0482548 + 0.998835i \(0.515366\pi\)
\(422\) − 7.50857i − 0.365512i
\(423\) 12.4476i 0.605224i
\(424\) −0.225470 −0.0109498
\(425\) 0 0
\(426\) 8.62648 0.417954
\(427\) 61.0620i 2.95500i
\(428\) − 10.8168i − 0.522848i
\(429\) 36.7109 1.77242
\(430\) 0 0
\(431\) 15.0774 0.726254 0.363127 0.931740i \(-0.381709\pi\)
0.363127 + 0.931740i \(0.381709\pi\)
\(432\) − 2.05763i − 0.0989979i
\(433\) − 13.5337i − 0.650386i −0.945648 0.325193i \(-0.894571\pi\)
0.945648 0.325193i \(-0.105429\pi\)
\(434\) 19.0620 0.915006
\(435\) 0 0
\(436\) 9.24791 0.442894
\(437\) − 7.56885i − 0.362067i
\(438\) − 10.6161i − 0.507258i
\(439\) −39.1773 −1.86983 −0.934915 0.354872i \(-0.884524\pi\)
−0.934915 + 0.354872i \(0.884524\pi\)
\(440\) 0 0
\(441\) 22.6619 1.07914
\(442\) − 20.5310i − 0.976560i
\(443\) − 19.3132i − 0.917600i −0.888540 0.458800i \(-0.848279\pi\)
0.888540 0.458800i \(-0.151721\pi\)
\(444\) −0.310596 −0.0147402
\(445\) 0 0
\(446\) 27.8091 1.31680
\(447\) − 13.3977i − 0.633689i
\(448\) − 4.22547i − 0.199635i
\(449\) −41.4131 −1.95440 −0.977202 0.212310i \(-0.931901\pi\)
−0.977202 + 0.212310i \(0.931901\pi\)
\(450\) 0 0
\(451\) 59.8486 2.81816
\(452\) 17.6489i 0.830135i
\(453\) 34.5059i 1.62123i
\(454\) −8.91223 −0.418272
\(455\) 0 0
\(456\) −2.25561 −0.105629
\(457\) − 37.8392i − 1.77004i −0.465551 0.885021i \(-0.654144\pi\)
0.465551 0.885021i \(-0.345856\pi\)
\(458\) 13.6489i 0.637772i
\(459\) −13.3357 −0.622456
\(460\) 0 0
\(461\) 1.58864 0.0739903 0.0369952 0.999315i \(-0.488221\pi\)
0.0369952 + 0.999315i \(0.488221\pi\)
\(462\) − 48.9674i − 2.27817i
\(463\) − 40.3581i − 1.87560i −0.347175 0.937800i \(-0.612859\pi\)
0.347175 0.937800i \(-0.387141\pi\)
\(464\) −0.832162 −0.0386322
\(465\) 0 0
\(466\) 23.2754 1.07821
\(467\) − 39.5130i − 1.82844i −0.405217 0.914221i \(-0.632804\pi\)
0.405217 0.914221i \(-0.367196\pi\)
\(468\) − 6.61372i − 0.305719i
\(469\) 17.3676 0.801959
\(470\) 0 0
\(471\) −28.8416 −1.32895
\(472\) 5.39331i 0.248247i
\(473\) − 12.9019i − 0.593229i
\(474\) −23.9692 −1.10094
\(475\) 0 0
\(476\) −27.3856 −1.25522
\(477\) 0.470730i 0.0215532i
\(478\) − 3.72898i − 0.170560i
\(479\) −7.61107 −0.347759 −0.173879 0.984767i \(-0.555630\pi\)
−0.173879 + 0.984767i \(0.555630\pi\)
\(480\) 0 0
\(481\) 0.436209 0.0198894
\(482\) − 1.48878i − 0.0678122i
\(483\) − 72.1388i − 3.28243i
\(484\) −15.3960 −0.699816
\(485\) 0 0
\(486\) −18.4234 −0.835705
\(487\) − 20.3907i − 0.923989i −0.886883 0.461995i \(-0.847134\pi\)
0.886883 0.461995i \(-0.152866\pi\)
\(488\) − 14.4509i − 0.654163i
\(489\) 25.8794 1.17031
\(490\) 0 0
\(491\) 25.0224 1.12925 0.564623 0.825349i \(-0.309021\pi\)
0.564623 + 0.825349i \(0.309021\pi\)
\(492\) − 26.2754i − 1.18459i
\(493\) 5.39331i 0.242902i
\(494\) 3.16784 0.142528
\(495\) 0 0
\(496\) −4.51122 −0.202560
\(497\) 16.1601i 0.724881i
\(498\) 27.2925i 1.22301i
\(499\) −22.6111 −1.01221 −0.506105 0.862472i \(-0.668915\pi\)
−0.506105 + 0.862472i \(0.668915\pi\)
\(500\) 0 0
\(501\) 43.7884 1.95632
\(502\) − 8.78662i − 0.392166i
\(503\) − 11.5035i − 0.512916i −0.966555 0.256458i \(-0.917444\pi\)
0.966555 0.256458i \(-0.0825556\pi\)
\(504\) −8.82181 −0.392955
\(505\) 0 0
\(506\) −38.8865 −1.72871
\(507\) − 6.68744i − 0.296999i
\(508\) 12.7866i 0.567314i
\(509\) −9.11526 −0.404027 −0.202013 0.979383i \(-0.564748\pi\)
−0.202013 + 0.979383i \(0.564748\pi\)
\(510\) 0 0
\(511\) 19.8874 0.879766
\(512\) 1.00000i 0.0441942i
\(513\) − 2.05763i − 0.0908467i
\(514\) −2.74175 −0.120933
\(515\) 0 0
\(516\) −5.66432 −0.249358
\(517\) 30.6318i 1.34718i
\(518\) − 0.581844i − 0.0255648i
\(519\) −22.0748 −0.968975
\(520\) 0 0
\(521\) 15.4888 0.678576 0.339288 0.940683i \(-0.389814\pi\)
0.339288 + 0.940683i \(0.389814\pi\)
\(522\) 1.73736i 0.0760423i
\(523\) − 4.34073i − 0.189807i −0.995486 0.0949035i \(-0.969746\pi\)
0.995486 0.0949035i \(-0.0302543\pi\)
\(524\) 2.11526 0.0924057
\(525\) 0 0
\(526\) 2.74704 0.119776
\(527\) 29.2376i 1.27361i
\(528\) 11.5886i 0.504331i
\(529\) −34.2875 −1.49076
\(530\) 0 0
\(531\) 11.2600 0.488642
\(532\) − 4.22547i − 0.183197i
\(533\) 36.9019i 1.59840i
\(534\) 22.5561 0.976097
\(535\) 0 0
\(536\) −4.11021 −0.177534
\(537\) − 15.3933i − 0.664270i
\(538\) 9.74175i 0.419996i
\(539\) 55.7677 2.40208
\(540\) 0 0
\(541\) 19.5491 0.840480 0.420240 0.907413i \(-0.361946\pi\)
0.420240 + 0.907413i \(0.361946\pi\)
\(542\) 26.3555i 1.13206i
\(543\) 0.310596i 0.0133289i
\(544\) 6.48108 0.277874
\(545\) 0 0
\(546\) 30.1927 1.29213
\(547\) − 41.8038i − 1.78740i −0.448665 0.893700i \(-0.648100\pi\)
0.448665 0.893700i \(-0.351900\pi\)
\(548\) 5.36317i 0.229103i
\(549\) −30.1703 −1.28763
\(550\) 0 0
\(551\) −0.832162 −0.0354513
\(552\) 17.0724i 0.726648i
\(553\) − 44.9019i − 1.90942i
\(554\) 0.962158 0.0408782
\(555\) 0 0
\(556\) −10.1601 −0.430886
\(557\) − 9.76418i − 0.413722i −0.978370 0.206861i \(-0.933675\pi\)
0.978370 0.206861i \(-0.0663247\pi\)
\(558\) 9.41839i 0.398712i
\(559\) 7.95513 0.336466
\(560\) 0 0
\(561\) 75.1069 3.17102
\(562\) − 14.6714i − 0.618874i
\(563\) − 11.4509i − 0.482600i −0.970451 0.241300i \(-0.922426\pi\)
0.970451 0.241300i \(-0.0775737\pi\)
\(564\) 13.4483 0.566275
\(565\) 0 0
\(566\) −26.1601 −1.09959
\(567\) − 46.0767i − 1.93504i
\(568\) − 3.82446i − 0.160471i
\(569\) 13.4338 0.563174 0.281587 0.959536i \(-0.409139\pi\)
0.281587 + 0.959536i \(0.409139\pi\)
\(570\) 0 0
\(571\) 37.6335 1.57491 0.787457 0.616370i \(-0.211397\pi\)
0.787457 + 0.616370i \(0.211397\pi\)
\(572\) − 16.2754i − 0.680509i
\(573\) − 43.7090i − 1.82597i
\(574\) 49.2221 2.05449
\(575\) 0 0
\(576\) 2.08777 0.0869905
\(577\) 1.56885i 0.0653121i 0.999467 + 0.0326560i \(0.0103966\pi\)
−0.999467 + 0.0326560i \(0.989603\pi\)
\(578\) − 25.0044i − 1.04005i
\(579\) 12.2446 0.508868
\(580\) 0 0
\(581\) −51.1276 −2.12113
\(582\) 8.88647i 0.368356i
\(583\) 1.15840i 0.0479759i
\(584\) −4.70655 −0.194758
\(585\) 0 0
\(586\) −22.4657 −0.928048
\(587\) 25.8693i 1.06774i 0.845566 + 0.533871i \(0.179263\pi\)
−0.845566 + 0.533871i \(0.820737\pi\)
\(588\) − 24.4837i − 1.00969i
\(589\) −4.51122 −0.185881
\(590\) 0 0
\(591\) −35.2978 −1.45196
\(592\) 0.137699i 0.00565940i
\(593\) 28.9243i 1.18778i 0.804547 + 0.593890i \(0.202408\pi\)
−0.804547 + 0.593890i \(0.797592\pi\)
\(594\) −10.5715 −0.433754
\(595\) 0 0
\(596\) −5.93972 −0.243301
\(597\) 40.7136i 1.66630i
\(598\) − 23.9769i − 0.980488i
\(599\) −41.3581 −1.68985 −0.844923 0.534887i \(-0.820354\pi\)
−0.844923 + 0.534887i \(0.820354\pi\)
\(600\) 0 0
\(601\) 2.68147 0.109379 0.0546897 0.998503i \(-0.482583\pi\)
0.0546897 + 0.998503i \(0.482583\pi\)
\(602\) − 10.6111i − 0.432475i
\(603\) 8.58117i 0.349452i
\(604\) 15.2978 0.622460
\(605\) 0 0
\(606\) 7.21338 0.293024
\(607\) 3.31324i 0.134480i 0.997737 + 0.0672401i \(0.0214194\pi\)
−0.997737 + 0.0672401i \(0.978581\pi\)
\(608\) 1.00000i 0.0405554i
\(609\) −7.93134 −0.321394
\(610\) 0 0
\(611\) −18.8871 −0.764092
\(612\) − 13.5310i − 0.546959i
\(613\) 10.0603i 0.406331i 0.979144 + 0.203165i \(0.0651229\pi\)
−0.979144 + 0.203165i \(0.934877\pi\)
\(614\) −17.2204 −0.694959
\(615\) 0 0
\(616\) −21.7092 −0.874688
\(617\) − 27.6265i − 1.11220i −0.831115 0.556100i \(-0.812297\pi\)
0.831115 0.556100i \(-0.187703\pi\)
\(618\) 24.1051i 0.969651i
\(619\) −16.6714 −0.670078 −0.335039 0.942204i \(-0.608750\pi\)
−0.335039 + 0.942204i \(0.608750\pi\)
\(620\) 0 0
\(621\) −15.5739 −0.624959
\(622\) − 7.87439i − 0.315734i
\(623\) 42.2547i 1.69290i
\(624\) −7.14540 −0.286045
\(625\) 0 0
\(626\) −25.1678 −1.00591
\(627\) 11.5886i 0.462806i
\(628\) 12.7866i 0.510242i
\(629\) 0.892439 0.0355839
\(630\) 0 0
\(631\) −0.709194 −0.0282326 −0.0141163 0.999900i \(-0.504494\pi\)
−0.0141163 + 0.999900i \(0.504494\pi\)
\(632\) 10.6265i 0.422699i
\(633\) 16.9364i 0.673162i
\(634\) 23.9045 0.949370
\(635\) 0 0
\(636\) 0.508572 0.0201662
\(637\) 34.3856i 1.36241i
\(638\) 4.27540i 0.169265i
\(639\) −7.98459 −0.315866
\(640\) 0 0
\(641\) −2.43553 −0.0961978 −0.0480989 0.998843i \(-0.515316\pi\)
−0.0480989 + 0.998843i \(0.515316\pi\)
\(642\) 24.3984i 0.962927i
\(643\) − 7.70390i − 0.303812i −0.988395 0.151906i \(-0.951459\pi\)
0.988395 0.151906i \(-0.0485411\pi\)
\(644\) −31.9819 −1.26027
\(645\) 0 0
\(646\) 6.48108 0.254995
\(647\) − 10.0499i − 0.395103i −0.980292 0.197552i \(-0.936701\pi\)
0.980292 0.197552i \(-0.0632990\pi\)
\(648\) 10.9045i 0.428370i
\(649\) 27.7092 1.08768
\(650\) 0 0
\(651\) −42.9964 −1.68516
\(652\) − 11.4734i − 0.449332i
\(653\) 7.09283i 0.277564i 0.990323 + 0.138782i \(0.0443187\pi\)
−0.990323 + 0.138782i \(0.955681\pi\)
\(654\) −20.8597 −0.815677
\(655\) 0 0
\(656\) −11.6489 −0.454814
\(657\) 9.82620i 0.383356i
\(658\) 25.1929i 0.982122i
\(659\) 1.16346 0.0453218 0.0226609 0.999743i \(-0.492786\pi\)
0.0226609 + 0.999743i \(0.492786\pi\)
\(660\) 0 0
\(661\) 1.79432 0.0697909 0.0348955 0.999391i \(-0.488890\pi\)
0.0348955 + 0.999391i \(0.488890\pi\)
\(662\) 30.7565i 1.19538i
\(663\) 46.3099i 1.79853i
\(664\) 12.0999 0.469566
\(665\) 0 0
\(666\) 0.287484 0.0111398
\(667\) 6.29851i 0.243879i
\(668\) − 19.4131i − 0.751115i
\(669\) −62.7263 −2.42514
\(670\) 0 0
\(671\) −74.2446 −2.86618
\(672\) 9.53101i 0.367667i
\(673\) 25.2824i 0.974566i 0.873244 + 0.487283i \(0.162012\pi\)
−0.873244 + 0.487283i \(0.837988\pi\)
\(674\) −13.4734 −0.518975
\(675\) 0 0
\(676\) −2.96480 −0.114031
\(677\) − 4.89682i − 0.188200i −0.995563 0.0941001i \(-0.970003\pi\)
0.995563 0.0941001i \(-0.0299974\pi\)
\(678\) − 39.8091i − 1.52886i
\(679\) −16.6472 −0.638860
\(680\) 0 0
\(681\) 20.1025 0.770330
\(682\) 23.1773i 0.887504i
\(683\) − 10.1980i − 0.390215i −0.980782 0.195107i \(-0.937494\pi\)
0.980782 0.195107i \(-0.0625055\pi\)
\(684\) 2.08777 0.0798279
\(685\) 0 0
\(686\) 16.2875 0.621859
\(687\) − 30.7866i − 1.17458i
\(688\) 2.51122i 0.0957393i
\(689\) −0.714253 −0.0272109
\(690\) 0 0
\(691\) −39.9846 −1.52109 −0.760543 0.649288i \(-0.775067\pi\)
−0.760543 + 0.649288i \(0.775067\pi\)
\(692\) 9.78662i 0.372031i
\(693\) 45.3238i 1.72171i
\(694\) −28.9468 −1.09880
\(695\) 0 0
\(696\) 1.87703 0.0711487
\(697\) 75.4975i 2.85967i
\(698\) − 27.9243i − 1.05695i
\(699\) −52.5002 −1.98574
\(700\) 0 0
\(701\) 4.91729 0.185723 0.0928617 0.995679i \(-0.470399\pi\)
0.0928617 + 0.995679i \(0.470399\pi\)
\(702\) − 6.51825i − 0.246015i
\(703\) 0.137699i 0.00519342i
\(704\) 5.13770 0.193634
\(705\) 0 0
\(706\) 28.3099 1.06546
\(707\) 13.5130i 0.508207i
\(708\) − 12.1652i − 0.457196i
\(709\) 36.8865 1.38530 0.692650 0.721274i \(-0.256443\pi\)
0.692650 + 0.721274i \(0.256443\pi\)
\(710\) 0 0
\(711\) 22.1857 0.832027
\(712\) − 10.0000i − 0.374766i
\(713\) 34.1447i 1.27873i
\(714\) 61.7712 2.31173
\(715\) 0 0
\(716\) −6.82446 −0.255042
\(717\) 8.41113i 0.314119i
\(718\) − 2.60163i − 0.0970921i
\(719\) 23.8891 0.890914 0.445457 0.895303i \(-0.353041\pi\)
0.445457 + 0.895303i \(0.353041\pi\)
\(720\) 0 0
\(721\) −45.1566 −1.68172
\(722\) 1.00000i 0.0372161i
\(723\) 3.35811i 0.124889i
\(724\) 0.137699 0.00511755
\(725\) 0 0
\(726\) 34.7272 1.28885
\(727\) 19.6894i 0.730240i 0.930961 + 0.365120i \(0.118972\pi\)
−0.930961 + 0.365120i \(0.881028\pi\)
\(728\) − 13.3856i − 0.496104i
\(729\) 8.84251 0.327500
\(730\) 0 0
\(731\) 16.2754 0.601967
\(732\) 32.5957i 1.20477i
\(733\) − 1.76418i − 0.0651615i −0.999469 0.0325808i \(-0.989627\pi\)
0.999469 0.0325808i \(-0.0103726\pi\)
\(734\) −11.6489 −0.429969
\(735\) 0 0
\(736\) 7.56885 0.278991
\(737\) 21.1170i 0.777855i
\(738\) 24.3203i 0.895241i
\(739\) −28.1755 −1.03645 −0.518227 0.855243i \(-0.673408\pi\)
−0.518227 + 0.855243i \(0.673408\pi\)
\(740\) 0 0
\(741\) −7.14540 −0.262493
\(742\) 0.952717i 0.0349753i
\(743\) 5.42851i 0.199153i 0.995030 + 0.0995763i \(0.0317487\pi\)
−0.995030 + 0.0995763i \(0.968251\pi\)
\(744\) 10.1755 0.373053
\(745\) 0 0
\(746\) −12.8064 −0.468876
\(747\) − 25.2617i − 0.924278i
\(748\) − 33.2978i − 1.21749i
\(749\) −45.7059 −1.67006
\(750\) 0 0
\(751\) 25.8640 0.943792 0.471896 0.881654i \(-0.343570\pi\)
0.471896 + 0.881654i \(0.343570\pi\)
\(752\) − 5.96216i − 0.217418i
\(753\) 19.8192i 0.722251i
\(754\) −2.63615 −0.0960031
\(755\) 0 0
\(756\) −8.69446 −0.316215
\(757\) − 3.76947i − 0.137004i −0.997651 0.0685019i \(-0.978178\pi\)
0.997651 0.0685019i \(-0.0218219\pi\)
\(758\) 20.9122i 0.759566i
\(759\) 87.7127 3.18377
\(760\) 0 0
\(761\) 18.3605 0.665568 0.332784 0.943003i \(-0.392012\pi\)
0.332784 + 0.943003i \(0.392012\pi\)
\(762\) − 28.8416i − 1.04482i
\(763\) − 39.0767i − 1.41467i
\(764\) −19.3779 −0.701068
\(765\) 0 0
\(766\) 10.3511 0.374000
\(767\) 17.0851i 0.616908i
\(768\) − 2.25561i − 0.0813923i
\(769\) 38.1300 1.37500 0.687501 0.726183i \(-0.258708\pi\)
0.687501 + 0.726183i \(0.258708\pi\)
\(770\) 0 0
\(771\) 6.18431 0.222722
\(772\) − 5.42851i − 0.195376i
\(773\) 29.1575i 1.04872i 0.851496 + 0.524361i \(0.175696\pi\)
−0.851496 + 0.524361i \(0.824304\pi\)
\(774\) 5.24285 0.188450
\(775\) 0 0
\(776\) 3.93972 0.141428
\(777\) 1.31241i 0.0470825i
\(778\) − 6.56620i − 0.235410i
\(779\) −11.6489 −0.417366
\(780\) 0 0
\(781\) −19.6489 −0.703094
\(782\) − 49.0543i − 1.75418i
\(783\) 1.71228i 0.0611920i
\(784\) −10.8546 −0.387664
\(785\) 0 0
\(786\) −4.77121 −0.170183
\(787\) − 47.0165i − 1.67596i −0.545704 0.837978i \(-0.683738\pi\)
0.545704 0.837978i \(-0.316262\pi\)
\(788\) 15.6489i 0.557470i
\(789\) −6.19624 −0.220592
\(790\) 0 0
\(791\) 74.5750 2.65158
\(792\) − 10.7263i − 0.381144i
\(793\) − 45.7782i − 1.62563i
\(794\) −23.5284 −0.834990
\(795\) 0 0
\(796\) 18.0499 0.639763
\(797\) − 42.6359i − 1.51024i −0.655586 0.755121i \(-0.727578\pi\)
0.655586 0.755121i \(-0.272422\pi\)
\(798\) 9.53101i 0.337394i
\(799\) −38.6412 −1.36703
\(800\) 0 0
\(801\) −20.8777 −0.737678
\(802\) 6.22041i 0.219650i
\(803\) 24.1808i 0.853323i
\(804\) 9.27102 0.326964
\(805\) 0 0
\(806\) −14.2908 −0.503372
\(807\) − 21.9736i − 0.773506i
\(808\) − 3.19798i − 0.112504i
\(809\) −49.4630 −1.73903 −0.869514 0.493909i \(-0.835568\pi\)
−0.869514 + 0.493909i \(0.835568\pi\)
\(810\) 0 0
\(811\) −16.7816 −0.589280 −0.294640 0.955608i \(-0.595200\pi\)
−0.294640 + 0.955608i \(0.595200\pi\)
\(812\) 3.51628i 0.123397i
\(813\) − 59.4476i − 2.08492i
\(814\) 0.707457 0.0247964
\(815\) 0 0
\(816\) −14.6188 −0.511760
\(817\) 2.51122i 0.0878564i
\(818\) 34.4905i 1.20593i
\(819\) −27.9461 −0.976515
\(820\) 0 0
\(821\) 11.5337 0.402527 0.201264 0.979537i \(-0.435495\pi\)
0.201264 + 0.979537i \(0.435495\pi\)
\(822\) − 12.0972i − 0.421939i
\(823\) − 32.6309i − 1.13744i −0.822531 0.568720i \(-0.807439\pi\)
0.822531 0.568720i \(-0.192561\pi\)
\(824\) 10.6868 0.372291
\(825\) 0 0
\(826\) 22.7893 0.792940
\(827\) 6.64650i 0.231122i 0.993300 + 0.115561i \(0.0368665\pi\)
−0.993300 + 0.115561i \(0.963133\pi\)
\(828\) − 15.8020i − 0.549158i
\(829\) 30.9217 1.07395 0.536977 0.843597i \(-0.319566\pi\)
0.536977 + 0.843597i \(0.319566\pi\)
\(830\) 0 0
\(831\) −2.17025 −0.0752852
\(832\) 3.16784i 0.109825i
\(833\) 70.3495i 2.43747i
\(834\) 22.9173 0.793561
\(835\) 0 0
\(836\) 5.13770 0.177691
\(837\) 9.28243i 0.320848i
\(838\) 10.6265i 0.367086i
\(839\) 48.7109 1.68169 0.840844 0.541277i \(-0.182059\pi\)
0.840844 + 0.541277i \(0.182059\pi\)
\(840\) 0 0
\(841\) −28.3075 −0.976121
\(842\) − 1.98021i − 0.0682426i
\(843\) 33.0928i 1.13978i
\(844\) 7.50857 0.258456
\(845\) 0 0
\(846\) −12.4476 −0.427958
\(847\) 65.0551i 2.23532i
\(848\) − 0.225470i − 0.00774267i
\(849\) 59.0070 2.02512
\(850\) 0 0
\(851\) 1.04222 0.0357270
\(852\) 8.62648i 0.295538i
\(853\) − 46.1447i − 1.57997i −0.613129 0.789983i \(-0.710089\pi\)
0.613129 0.789983i \(-0.289911\pi\)
\(854\) −61.0620 −2.08950
\(855\) 0 0
\(856\) 10.8168 0.369709
\(857\) − 8.40607i − 0.287146i −0.989640 0.143573i \(-0.954141\pi\)
0.989640 0.143573i \(-0.0458592\pi\)
\(858\) 36.7109i 1.25329i
\(859\) 8.49581 0.289873 0.144937 0.989441i \(-0.453702\pi\)
0.144937 + 0.989441i \(0.453702\pi\)
\(860\) 0 0
\(861\) −111.026 −3.78375
\(862\) 15.0774i 0.513539i
\(863\) − 3.37352i − 0.114836i −0.998350 0.0574179i \(-0.981713\pi\)
0.998350 0.0574179i \(-0.0182868\pi\)
\(864\) 2.05763 0.0700021
\(865\) 0 0
\(866\) 13.5337 0.459892
\(867\) 56.4001i 1.91545i
\(868\) 19.0620i 0.647007i
\(869\) 54.5957 1.85203
\(870\) 0 0
\(871\) −13.0205 −0.441182
\(872\) 9.24791i 0.313174i
\(873\) − 8.22524i − 0.278382i
\(874\) 7.56885 0.256020
\(875\) 0 0
\(876\) 10.6161 0.358686
\(877\) − 13.2780i − 0.448368i −0.974547 0.224184i \(-0.928028\pi\)
0.974547 0.224184i \(-0.0719716\pi\)
\(878\) − 39.1773i − 1.32217i
\(879\) 50.6738 1.70918
\(880\) 0 0
\(881\) −26.6489 −0.897825 −0.448912 0.893576i \(-0.648188\pi\)
−0.448912 + 0.893576i \(0.648188\pi\)
\(882\) 22.6619i 0.763066i
\(883\) 47.7884i 1.60821i 0.594490 + 0.804103i \(0.297354\pi\)
−0.594490 + 0.804103i \(0.702646\pi\)
\(884\) 20.5310 0.690532
\(885\) 0 0
\(886\) 19.3132 0.648841
\(887\) − 36.3304i − 1.21985i −0.792457 0.609927i \(-0.791199\pi\)
0.792457 0.609927i \(-0.208801\pi\)
\(888\) − 0.310596i − 0.0104229i
\(889\) 54.0295 1.81209
\(890\) 0 0
\(891\) 56.0242 1.87688
\(892\) 27.8091i 0.931116i
\(893\) − 5.96216i − 0.199516i
\(894\) 13.3977 0.448086
\(895\) 0 0
\(896\) 4.22547 0.141163
\(897\) 54.0825i 1.80576i
\(898\) − 41.4131i − 1.38197i
\(899\) 3.75406 0.125205
\(900\) 0 0
\(901\) −1.46129 −0.0486826
\(902\) 59.8486i 1.99274i
\(903\) 23.9344i 0.796488i
\(904\) −17.6489 −0.586994
\(905\) 0 0
\(906\) −34.5059 −1.14638
\(907\) − 39.3873i − 1.30784i −0.756566 0.653918i \(-0.773124\pi\)
0.756566 0.653918i \(-0.226876\pi\)
\(908\) − 8.91223i − 0.295763i
\(909\) −6.67664 −0.221450
\(910\) 0 0
\(911\) −20.5561 −0.681054 −0.340527 0.940235i \(-0.610605\pi\)
−0.340527 + 0.940235i \(0.610605\pi\)
\(912\) − 2.25561i − 0.0746907i
\(913\) − 62.1654i − 2.05738i
\(914\) 37.8392 1.25161
\(915\) 0 0
\(916\) −13.6489 −0.450973
\(917\) − 8.93799i − 0.295158i
\(918\) − 13.3357i − 0.440143i
\(919\) −12.4054 −0.409216 −0.204608 0.978844i \(-0.565592\pi\)
−0.204608 + 0.978844i \(0.565592\pi\)
\(920\) 0 0
\(921\) 38.8425 1.27990
\(922\) 1.58864i 0.0523190i
\(923\) − 12.1153i − 0.398779i
\(924\) 48.9674 1.61091
\(925\) 0 0
\(926\) 40.3581 1.32625
\(927\) − 22.3115i − 0.732806i
\(928\) − 0.832162i − 0.0273171i
\(929\) 31.3330 1.02800 0.514002 0.857789i \(-0.328162\pi\)
0.514002 + 0.857789i \(0.328162\pi\)
\(930\) 0 0
\(931\) −10.8546 −0.355745
\(932\) 23.2754i 0.762411i
\(933\) 17.7615i 0.581487i
\(934\) 39.5130 1.29290
\(935\) 0 0
\(936\) 6.61372 0.216176
\(937\) 7.27299i 0.237598i 0.992918 + 0.118799i \(0.0379044\pi\)
−0.992918 + 0.118799i \(0.962096\pi\)
\(938\) 17.3676i 0.567071i
\(939\) 56.7688 1.85258
\(940\) 0 0
\(941\) 6.72128 0.219107 0.109554 0.993981i \(-0.465058\pi\)
0.109554 + 0.993981i \(0.465058\pi\)
\(942\) − 28.8416i − 0.939710i
\(943\) 88.1689i 2.87117i
\(944\) −5.39331 −0.175537
\(945\) 0 0
\(946\) 12.9019 0.419476
\(947\) 10.0757i 0.327416i 0.986509 + 0.163708i \(0.0523454\pi\)
−0.986509 + 0.163708i \(0.947655\pi\)
\(948\) − 23.9692i − 0.778483i
\(949\) −14.9096 −0.483986
\(950\) 0 0
\(951\) −53.9193 −1.74845
\(952\) − 27.3856i − 0.887573i
\(953\) 8.14473i 0.263834i 0.991261 + 0.131917i \(0.0421132\pi\)
−0.991261 + 0.131917i \(0.957887\pi\)
\(954\) −0.470730 −0.0152404
\(955\) 0 0
\(956\) 3.72898 0.120604
\(957\) − 9.64363i − 0.311734i
\(958\) − 7.61107i − 0.245903i
\(959\) 22.6619 0.731791
\(960\) 0 0
\(961\) −10.6489 −0.343513
\(962\) 0.436209i 0.0140639i
\(963\) − 22.5829i − 0.727724i
\(964\) 1.48878 0.0479505
\(965\) 0 0
\(966\) 72.1388 2.32103
\(967\) 47.1773i 1.51712i 0.651604 + 0.758560i \(0.274097\pi\)
−0.651604 + 0.758560i \(0.725903\pi\)
\(968\) − 15.3960i − 0.494845i
\(969\) −14.6188 −0.469623
\(970\) 0 0
\(971\) 0.230528 0.00739801 0.00369901 0.999993i \(-0.498823\pi\)
0.00369901 + 0.999993i \(0.498823\pi\)
\(972\) − 18.4234i − 0.590932i
\(973\) 42.9313i 1.37632i
\(974\) 20.3907 0.653359
\(975\) 0 0
\(976\) 14.4509 0.462563
\(977\) − 5.80905i − 0.185848i −0.995673 0.0929240i \(-0.970379\pi\)
0.995673 0.0929240i \(-0.0296214\pi\)
\(978\) 25.8794i 0.827533i
\(979\) −51.3770 −1.64202
\(980\) 0 0
\(981\) 19.3075 0.616441
\(982\) 25.0224i 0.798498i
\(983\) 22.3511i 0.712889i 0.934317 + 0.356444i \(0.116011\pi\)
−0.934317 + 0.356444i \(0.883989\pi\)
\(984\) 26.2754 0.837629
\(985\) 0 0
\(986\) −5.39331 −0.171758
\(987\) − 56.8254i − 1.80877i
\(988\) 3.16784i 0.100782i
\(989\) 19.0070 0.604388
\(990\) 0 0
\(991\) −34.9415 −1.10995 −0.554976 0.831866i \(-0.687273\pi\)
−0.554976 + 0.831866i \(0.687273\pi\)
\(992\) − 4.51122i − 0.143231i
\(993\) − 69.3746i − 2.20154i
\(994\) −16.1601 −0.512568
\(995\) 0 0
\(996\) −27.2925 −0.864797
\(997\) − 9.35282i − 0.296207i −0.988972 0.148103i \(-0.952683\pi\)
0.988972 0.148103i \(-0.0473168\pi\)
\(998\) − 22.6111i − 0.715741i
\(999\) 0.283334 0.00896430
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.799.5 6
5.2 odd 4 950.2.a.k.1.2 3
5.3 odd 4 950.2.a.m.1.2 yes 3
5.4 even 2 inner 950.2.b.g.799.2 6
15.2 even 4 8550.2.a.co.1.1 3
15.8 even 4 8550.2.a.cj.1.3 3
20.3 even 4 7600.2.a.bm.1.2 3
20.7 even 4 7600.2.a.cb.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.2 3 5.2 odd 4
950.2.a.m.1.2 yes 3 5.3 odd 4
950.2.b.g.799.2 6 5.4 even 2 inner
950.2.b.g.799.5 6 1.1 even 1 trivial
7600.2.a.bm.1.2 3 20.3 even 4
7600.2.a.cb.1.2 3 20.7 even 4
8550.2.a.cj.1.3 3 15.8 even 4
8550.2.a.co.1.1 3 15.2 even 4