Properties

Label 950.2.a.k.1.2
Level $950$
Weight $2$
Character 950.1
Self dual yes
Analytic conductor $7.586$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.91223\) of defining polynomial
Character \(\chi\) \(=\) 950.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.25561 q^{3} +1.00000 q^{4} +2.25561 q^{6} -4.22547 q^{7} -1.00000 q^{8} +2.08777 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.25561 q^{3} +1.00000 q^{4} +2.25561 q^{6} -4.22547 q^{7} -1.00000 q^{8} +2.08777 q^{9} -5.13770 q^{11} -2.25561 q^{12} -3.16784 q^{13} +4.22547 q^{14} +1.00000 q^{16} +6.48108 q^{17} -2.08777 q^{18} -1.00000 q^{19} +9.53101 q^{21} +5.13770 q^{22} -7.56885 q^{23} +2.25561 q^{24} +3.16784 q^{26} +2.05763 q^{27} -4.22547 q^{28} +0.832162 q^{29} -4.51122 q^{31} -1.00000 q^{32} +11.5886 q^{33} -6.48108 q^{34} +2.08777 q^{36} -0.137699 q^{37} +1.00000 q^{38} +7.14540 q^{39} -11.6489 q^{41} -9.53101 q^{42} +2.51122 q^{43} -5.13770 q^{44} +7.56885 q^{46} +5.96216 q^{47} -2.25561 q^{48} +10.8546 q^{49} -14.6188 q^{51} -3.16784 q^{52} -0.225470 q^{53} -2.05763 q^{54} +4.22547 q^{56} +2.25561 q^{57} -0.832162 q^{58} +5.39331 q^{59} +14.4509 q^{61} +4.51122 q^{62} -8.82181 q^{63} +1.00000 q^{64} -11.5886 q^{66} +4.11021 q^{67} +6.48108 q^{68} +17.0724 q^{69} +3.82446 q^{71} -2.08777 q^{72} -4.70655 q^{73} +0.137699 q^{74} -1.00000 q^{76} +21.7092 q^{77} -7.14540 q^{78} +10.6265 q^{79} -10.9045 q^{81} +11.6489 q^{82} +12.0999 q^{83} +9.53101 q^{84} -2.51122 q^{86} -1.87703 q^{87} +5.13770 q^{88} -10.0000 q^{89} +13.3856 q^{91} -7.56885 q^{92} +10.1755 q^{93} -5.96216 q^{94} +2.25561 q^{96} -3.93972 q^{97} -10.8546 q^{98} -10.7263 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} - 2q^{3} + 3q^{4} + 2q^{6} - 2q^{7} - 3q^{8} + 13q^{9} + O(q^{10}) \) \( 3q - 3q^{2} - 2q^{3} + 3q^{4} + 2q^{6} - 2q^{7} - 3q^{8} + 13q^{9} + 2q^{11} - 2q^{12} + 2q^{13} + 2q^{14} + 3q^{16} + 4q^{17} - 13q^{18} - 3q^{19} - 11q^{21} - 2q^{22} - 14q^{23} + 2q^{24} - 2q^{26} + 7q^{27} - 2q^{28} + 14q^{29} - 4q^{31} - 3q^{32} - 4q^{33} - 4q^{34} + 13q^{36} + 17q^{37} + 3q^{38} + 29q^{39} - 8q^{41} + 11q^{42} - 2q^{43} + 2q^{44} + 14q^{46} - 13q^{47} - 2q^{48} + 25q^{49} - 11q^{51} + 2q^{52} + 10q^{53} - 7q^{54} + 2q^{56} + 2q^{57} - 14q^{58} - 6q^{59} + 22q^{61} + 4q^{62} - 2q^{63} + 3q^{64} + 4q^{66} + 4q^{68} + 8q^{69} - 2q^{71} - 13q^{72} + 12q^{73} - 17q^{74} - 3q^{76} + 50q^{77} - 29q^{78} + 24q^{79} - q^{81} + 8q^{82} - 12q^{83} - 11q^{84} + 2q^{86} + 21q^{87} - 2q^{88} - 30q^{89} - 7q^{91} - 14q^{92} + 44q^{93} + 13q^{94} + 2q^{96} - 25q^{98} + 24q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.25561 −1.30228 −0.651138 0.758959i \(-0.725708\pi\)
−0.651138 + 0.758959i \(0.725708\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.25561 0.920848
\(7\) −4.22547 −1.59708 −0.798539 0.601943i \(-0.794393\pi\)
−0.798539 + 0.601943i \(0.794393\pi\)
\(8\) −1.00000 −0.353553
\(9\) 2.08777 0.695924
\(10\) 0 0
\(11\) −5.13770 −1.54907 −0.774537 0.632528i \(-0.782017\pi\)
−0.774537 + 0.632528i \(0.782017\pi\)
\(12\) −2.25561 −0.651138
\(13\) −3.16784 −0.878600 −0.439300 0.898340i \(-0.644774\pi\)
−0.439300 + 0.898340i \(0.644774\pi\)
\(14\) 4.22547 1.12930
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.48108 1.57189 0.785946 0.618295i \(-0.212176\pi\)
0.785946 + 0.618295i \(0.212176\pi\)
\(18\) −2.08777 −0.492092
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 9.53101 2.07984
\(22\) 5.13770 1.09536
\(23\) −7.56885 −1.57821 −0.789107 0.614256i \(-0.789456\pi\)
−0.789107 + 0.614256i \(0.789456\pi\)
\(24\) 2.25561 0.460424
\(25\) 0 0
\(26\) 3.16784 0.621264
\(27\) 2.05763 0.395991
\(28\) −4.22547 −0.798539
\(29\) 0.832162 0.154529 0.0772643 0.997011i \(-0.475381\pi\)
0.0772643 + 0.997011i \(0.475381\pi\)
\(30\) 0 0
\(31\) −4.51122 −0.810239 −0.405119 0.914264i \(-0.632770\pi\)
−0.405119 + 0.914264i \(0.632770\pi\)
\(32\) −1.00000 −0.176777
\(33\) 11.5886 2.01732
\(34\) −6.48108 −1.11150
\(35\) 0 0
\(36\) 2.08777 0.347962
\(37\) −0.137699 −0.0226376 −0.0113188 0.999936i \(-0.503603\pi\)
−0.0113188 + 0.999936i \(0.503603\pi\)
\(38\) 1.00000 0.162221
\(39\) 7.14540 1.14418
\(40\) 0 0
\(41\) −11.6489 −1.81926 −0.909628 0.415425i \(-0.863633\pi\)
−0.909628 + 0.415425i \(0.863633\pi\)
\(42\) −9.53101 −1.47067
\(43\) 2.51122 0.382957 0.191479 0.981497i \(-0.438672\pi\)
0.191479 + 0.981497i \(0.438672\pi\)
\(44\) −5.13770 −0.774537
\(45\) 0 0
\(46\) 7.56885 1.11597
\(47\) 5.96216 0.869670 0.434835 0.900510i \(-0.356807\pi\)
0.434835 + 0.900510i \(0.356807\pi\)
\(48\) −2.25561 −0.325569
\(49\) 10.8546 1.55066
\(50\) 0 0
\(51\) −14.6188 −2.04704
\(52\) −3.16784 −0.439300
\(53\) −0.225470 −0.0309707 −0.0154853 0.999880i \(-0.504929\pi\)
−0.0154853 + 0.999880i \(0.504929\pi\)
\(54\) −2.05763 −0.280008
\(55\) 0 0
\(56\) 4.22547 0.564652
\(57\) 2.25561 0.298763
\(58\) −0.832162 −0.109268
\(59\) 5.39331 0.702149 0.351074 0.936348i \(-0.385816\pi\)
0.351074 + 0.936348i \(0.385816\pi\)
\(60\) 0 0
\(61\) 14.4509 1.85025 0.925127 0.379659i \(-0.123959\pi\)
0.925127 + 0.379659i \(0.123959\pi\)
\(62\) 4.51122 0.572925
\(63\) −8.82181 −1.11144
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −11.5886 −1.42646
\(67\) 4.11021 0.502142 0.251071 0.967969i \(-0.419217\pi\)
0.251071 + 0.967969i \(0.419217\pi\)
\(68\) 6.48108 0.785946
\(69\) 17.0724 2.05527
\(70\) 0 0
\(71\) 3.82446 0.453880 0.226940 0.973909i \(-0.427128\pi\)
0.226940 + 0.973909i \(0.427128\pi\)
\(72\) −2.08777 −0.246046
\(73\) −4.70655 −0.550860 −0.275430 0.961321i \(-0.588820\pi\)
−0.275430 + 0.961321i \(0.588820\pi\)
\(74\) 0.137699 0.0160072
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 21.7092 2.47399
\(78\) −7.14540 −0.809058
\(79\) 10.6265 1.19557 0.597786 0.801655i \(-0.296047\pi\)
0.597786 + 0.801655i \(0.296047\pi\)
\(80\) 0 0
\(81\) −10.9045 −1.21161
\(82\) 11.6489 1.28641
\(83\) 12.0999 1.32813 0.664066 0.747674i \(-0.268829\pi\)
0.664066 + 0.747674i \(0.268829\pi\)
\(84\) 9.53101 1.03992
\(85\) 0 0
\(86\) −2.51122 −0.270792
\(87\) −1.87703 −0.201239
\(88\) 5.13770 0.547681
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 13.3856 1.40319
\(92\) −7.56885 −0.789107
\(93\) 10.1755 1.05515
\(94\) −5.96216 −0.614950
\(95\) 0 0
\(96\) 2.25561 0.230212
\(97\) −3.93972 −0.400018 −0.200009 0.979794i \(-0.564097\pi\)
−0.200009 + 0.979794i \(0.564097\pi\)
\(98\) −10.8546 −1.09648
\(99\) −10.7263 −1.07804
\(100\) 0 0
\(101\) 3.19798 0.318211 0.159105 0.987262i \(-0.449139\pi\)
0.159105 + 0.987262i \(0.449139\pi\)
\(102\) 14.6188 1.44747
\(103\) 10.6868 1.05300 0.526499 0.850176i \(-0.323504\pi\)
0.526499 + 0.850176i \(0.323504\pi\)
\(104\) 3.16784 0.310632
\(105\) 0 0
\(106\) 0.225470 0.0218996
\(107\) −10.8168 −1.04570 −0.522848 0.852426i \(-0.675130\pi\)
−0.522848 + 0.852426i \(0.675130\pi\)
\(108\) 2.05763 0.197996
\(109\) 9.24791 0.885789 0.442894 0.896574i \(-0.353952\pi\)
0.442894 + 0.896574i \(0.353952\pi\)
\(110\) 0 0
\(111\) 0.310596 0.0294804
\(112\) −4.22547 −0.399269
\(113\) −17.6489 −1.66027 −0.830135 0.557562i \(-0.811737\pi\)
−0.830135 + 0.557562i \(0.811737\pi\)
\(114\) −2.25561 −0.211257
\(115\) 0 0
\(116\) 0.832162 0.0772643
\(117\) −6.61372 −0.611439
\(118\) −5.39331 −0.496494
\(119\) −27.3856 −2.51043
\(120\) 0 0
\(121\) 15.3960 1.39963
\(122\) −14.4509 −1.30833
\(123\) 26.2754 2.36917
\(124\) −4.51122 −0.405119
\(125\) 0 0
\(126\) 8.82181 0.785910
\(127\) 12.7866 1.13463 0.567314 0.823501i \(-0.307982\pi\)
0.567314 + 0.823501i \(0.307982\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.66432 −0.498716
\(130\) 0 0
\(131\) −2.11526 −0.184811 −0.0924057 0.995721i \(-0.529456\pi\)
−0.0924057 + 0.995721i \(0.529456\pi\)
\(132\) 11.5886 1.00866
\(133\) 4.22547 0.366395
\(134\) −4.11021 −0.355068
\(135\) 0 0
\(136\) −6.48108 −0.555748
\(137\) 5.36317 0.458206 0.229103 0.973402i \(-0.426421\pi\)
0.229103 + 0.973402i \(0.426421\pi\)
\(138\) −17.0724 −1.45330
\(139\) −10.1601 −0.861771 −0.430886 0.902407i \(-0.641799\pi\)
−0.430886 + 0.902407i \(0.641799\pi\)
\(140\) 0 0
\(141\) −13.4483 −1.13255
\(142\) −3.82446 −0.320941
\(143\) 16.2754 1.36102
\(144\) 2.08777 0.173981
\(145\) 0 0
\(146\) 4.70655 0.389517
\(147\) −24.4837 −2.01938
\(148\) −0.137699 −0.0113188
\(149\) −5.93972 −0.486601 −0.243301 0.969951i \(-0.578230\pi\)
−0.243301 + 0.969951i \(0.578230\pi\)
\(150\) 0 0
\(151\) −15.2978 −1.24492 −0.622460 0.782652i \(-0.713867\pi\)
−0.622460 + 0.782652i \(0.713867\pi\)
\(152\) 1.00000 0.0811107
\(153\) 13.5310 1.09392
\(154\) −21.7092 −1.74938
\(155\) 0 0
\(156\) 7.14540 0.572090
\(157\) 12.7866 1.02048 0.510242 0.860031i \(-0.329556\pi\)
0.510242 + 0.860031i \(0.329556\pi\)
\(158\) −10.6265 −0.845397
\(159\) 0.508572 0.0403324
\(160\) 0 0
\(161\) 31.9819 2.52053
\(162\) 10.9045 0.856740
\(163\) 11.4734 0.898664 0.449332 0.893365i \(-0.351662\pi\)
0.449332 + 0.893365i \(0.351662\pi\)
\(164\) −11.6489 −0.909628
\(165\) 0 0
\(166\) −12.0999 −0.939131
\(167\) −19.4131 −1.50223 −0.751115 0.660171i \(-0.770484\pi\)
−0.751115 + 0.660171i \(0.770484\pi\)
\(168\) −9.53101 −0.735333
\(169\) −2.96480 −0.228062
\(170\) 0 0
\(171\) −2.08777 −0.159656
\(172\) 2.51122 0.191479
\(173\) −9.78662 −0.744063 −0.372031 0.928220i \(-0.621339\pi\)
−0.372031 + 0.928220i \(0.621339\pi\)
\(174\) 1.87703 0.142297
\(175\) 0 0
\(176\) −5.13770 −0.387269
\(177\) −12.1652 −0.914392
\(178\) 10.0000 0.749532
\(179\) −6.82446 −0.510084 −0.255042 0.966930i \(-0.582089\pi\)
−0.255042 + 0.966930i \(0.582089\pi\)
\(180\) 0 0
\(181\) −0.137699 −0.0102351 −0.00511755 0.999987i \(-0.501629\pi\)
−0.00511755 + 0.999987i \(0.501629\pi\)
\(182\) −13.3856 −0.992207
\(183\) −32.5957 −2.40954
\(184\) 7.56885 0.557983
\(185\) 0 0
\(186\) −10.1755 −0.746107
\(187\) −33.2978 −2.43498
\(188\) 5.96216 0.434835
\(189\) −8.69446 −0.632429
\(190\) 0 0
\(191\) 19.3779 1.40214 0.701068 0.713095i \(-0.252707\pi\)
0.701068 + 0.713095i \(0.252707\pi\)
\(192\) −2.25561 −0.162785
\(193\) 5.42851 0.390752 0.195376 0.980728i \(-0.437407\pi\)
0.195376 + 0.980728i \(0.437407\pi\)
\(194\) 3.93972 0.282856
\(195\) 0 0
\(196\) 10.8546 0.775328
\(197\) 15.6489 1.11494 0.557470 0.830197i \(-0.311772\pi\)
0.557470 + 0.830197i \(0.311772\pi\)
\(198\) 10.7263 0.762288
\(199\) 18.0499 1.27953 0.639763 0.768572i \(-0.279033\pi\)
0.639763 + 0.768572i \(0.279033\pi\)
\(200\) 0 0
\(201\) −9.27102 −0.653927
\(202\) −3.19798 −0.225009
\(203\) −3.51628 −0.246794
\(204\) −14.6188 −1.02352
\(205\) 0 0
\(206\) −10.6868 −0.744582
\(207\) −15.8020 −1.09832
\(208\) −3.16784 −0.219650
\(209\) 5.13770 0.355382
\(210\) 0 0
\(211\) −7.50857 −0.516911 −0.258456 0.966023i \(-0.583214\pi\)
−0.258456 + 0.966023i \(0.583214\pi\)
\(212\) −0.225470 −0.0154853
\(213\) −8.62648 −0.591077
\(214\) 10.8168 0.739418
\(215\) 0 0
\(216\) −2.05763 −0.140004
\(217\) 19.0620 1.29401
\(218\) −9.24791 −0.626347
\(219\) 10.6161 0.717372
\(220\) 0 0
\(221\) −20.5310 −1.38106
\(222\) −0.310596 −0.0208458
\(223\) −27.8091 −1.86223 −0.931116 0.364723i \(-0.881164\pi\)
−0.931116 + 0.364723i \(0.881164\pi\)
\(224\) 4.22547 0.282326
\(225\) 0 0
\(226\) 17.6489 1.17399
\(227\) −8.91223 −0.591525 −0.295763 0.955261i \(-0.595574\pi\)
−0.295763 + 0.955261i \(0.595574\pi\)
\(228\) 2.25561 0.149381
\(229\) −13.6489 −0.901946 −0.450973 0.892538i \(-0.648923\pi\)
−0.450973 + 0.892538i \(0.648923\pi\)
\(230\) 0 0
\(231\) −48.9674 −3.22182
\(232\) −0.832162 −0.0546341
\(233\) −23.2754 −1.52482 −0.762411 0.647093i \(-0.775985\pi\)
−0.762411 + 0.647093i \(0.775985\pi\)
\(234\) 6.61372 0.432352
\(235\) 0 0
\(236\) 5.39331 0.351074
\(237\) −23.9692 −1.55697
\(238\) 27.3856 1.77515
\(239\) 3.72898 0.241208 0.120604 0.992701i \(-0.461517\pi\)
0.120604 + 0.992701i \(0.461517\pi\)
\(240\) 0 0
\(241\) −1.48878 −0.0959009 −0.0479505 0.998850i \(-0.515269\pi\)
−0.0479505 + 0.998850i \(0.515269\pi\)
\(242\) −15.3960 −0.989689
\(243\) 18.4234 1.18186
\(244\) 14.4509 0.925127
\(245\) 0 0
\(246\) −26.2754 −1.67526
\(247\) 3.16784 0.201565
\(248\) 4.51122 0.286463
\(249\) −27.2925 −1.72959
\(250\) 0 0
\(251\) −8.78662 −0.554606 −0.277303 0.960782i \(-0.589441\pi\)
−0.277303 + 0.960782i \(0.589441\pi\)
\(252\) −8.82181 −0.555722
\(253\) 38.8865 2.44477
\(254\) −12.7866 −0.802304
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.74175 −0.171025 −0.0855127 0.996337i \(-0.527253\pi\)
−0.0855127 + 0.996337i \(0.527253\pi\)
\(258\) 5.66432 0.352645
\(259\) 0.581844 0.0361540
\(260\) 0 0
\(261\) 1.73736 0.107540
\(262\) 2.11526 0.130681
\(263\) −2.74704 −0.169390 −0.0846948 0.996407i \(-0.526992\pi\)
−0.0846948 + 0.996407i \(0.526992\pi\)
\(264\) −11.5886 −0.713231
\(265\) 0 0
\(266\) −4.22547 −0.259080
\(267\) 22.5561 1.38041
\(268\) 4.11021 0.251071
\(269\) −9.74175 −0.593965 −0.296982 0.954883i \(-0.595980\pi\)
−0.296982 + 0.954883i \(0.595980\pi\)
\(270\) 0 0
\(271\) 26.3555 1.60098 0.800490 0.599346i \(-0.204573\pi\)
0.800490 + 0.599346i \(0.204573\pi\)
\(272\) 6.48108 0.392973
\(273\) −30.1927 −1.82734
\(274\) −5.36317 −0.324001
\(275\) 0 0
\(276\) 17.0724 1.02764
\(277\) 0.962158 0.0578104 0.0289052 0.999582i \(-0.490798\pi\)
0.0289052 + 0.999582i \(0.490798\pi\)
\(278\) 10.1601 0.609364
\(279\) −9.41839 −0.563864
\(280\) 0 0
\(281\) −14.6714 −0.875219 −0.437610 0.899165i \(-0.644175\pi\)
−0.437610 + 0.899165i \(0.644175\pi\)
\(282\) 13.4483 0.800834
\(283\) 26.1601 1.55506 0.777529 0.628847i \(-0.216473\pi\)
0.777529 + 0.628847i \(0.216473\pi\)
\(284\) 3.82446 0.226940
\(285\) 0 0
\(286\) −16.2754 −0.962384
\(287\) 49.2221 2.90549
\(288\) −2.08777 −0.123023
\(289\) 25.0044 1.47085
\(290\) 0 0
\(291\) 8.88647 0.520934
\(292\) −4.70655 −0.275430
\(293\) 22.4657 1.31246 0.656229 0.754562i \(-0.272150\pi\)
0.656229 + 0.754562i \(0.272150\pi\)
\(294\) 24.4837 1.42792
\(295\) 0 0
\(296\) 0.137699 0.00800360
\(297\) −10.5715 −0.613420
\(298\) 5.93972 0.344079
\(299\) 23.9769 1.38662
\(300\) 0 0
\(301\) −10.6111 −0.611612
\(302\) 15.2978 0.880291
\(303\) −7.21338 −0.414398
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −13.5310 −0.773516
\(307\) −17.2204 −0.982821 −0.491410 0.870928i \(-0.663518\pi\)
−0.491410 + 0.870928i \(0.663518\pi\)
\(308\) 21.7092 1.23700
\(309\) −24.1051 −1.37129
\(310\) 0 0
\(311\) −7.87439 −0.446516 −0.223258 0.974759i \(-0.571669\pi\)
−0.223258 + 0.974759i \(0.571669\pi\)
\(312\) −7.14540 −0.404529
\(313\) 25.1678 1.42257 0.711285 0.702904i \(-0.248113\pi\)
0.711285 + 0.702904i \(0.248113\pi\)
\(314\) −12.7866 −0.721590
\(315\) 0 0
\(316\) 10.6265 0.597786
\(317\) 23.9045 1.34261 0.671306 0.741180i \(-0.265734\pi\)
0.671306 + 0.741180i \(0.265734\pi\)
\(318\) −0.508572 −0.0285193
\(319\) −4.27540 −0.239376
\(320\) 0 0
\(321\) 24.3984 1.36178
\(322\) −31.9819 −1.78228
\(323\) −6.48108 −0.360617
\(324\) −10.9045 −0.605807
\(325\) 0 0
\(326\) −11.4734 −0.635451
\(327\) −20.8597 −1.15354
\(328\) 11.6489 0.643204
\(329\) −25.1929 −1.38893
\(330\) 0 0
\(331\) 30.7565 1.69053 0.845264 0.534348i \(-0.179443\pi\)
0.845264 + 0.534348i \(0.179443\pi\)
\(332\) 12.0999 0.664066
\(333\) −0.287484 −0.0157540
\(334\) 19.4131 1.06224
\(335\) 0 0
\(336\) 9.53101 0.519959
\(337\) −13.4734 −0.733942 −0.366971 0.930232i \(-0.619605\pi\)
−0.366971 + 0.930232i \(0.619605\pi\)
\(338\) 2.96480 0.161264
\(339\) 39.8091 2.16213
\(340\) 0 0
\(341\) 23.1773 1.25512
\(342\) 2.08777 0.112894
\(343\) −16.2875 −0.879441
\(344\) −2.51122 −0.135396
\(345\) 0 0
\(346\) 9.78662 0.526132
\(347\) −28.9468 −1.55394 −0.776971 0.629536i \(-0.783245\pi\)
−0.776971 + 0.629536i \(0.783245\pi\)
\(348\) −1.87703 −0.100619
\(349\) 27.9243 1.49475 0.747377 0.664400i \(-0.231313\pi\)
0.747377 + 0.664400i \(0.231313\pi\)
\(350\) 0 0
\(351\) −6.51825 −0.347918
\(352\) 5.13770 0.273840
\(353\) −28.3099 −1.50678 −0.753392 0.657571i \(-0.771584\pi\)
−0.753392 + 0.657571i \(0.771584\pi\)
\(354\) 12.1652 0.646573
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 61.7712 3.26928
\(358\) 6.82446 0.360684
\(359\) 2.60163 0.137309 0.0686545 0.997640i \(-0.478129\pi\)
0.0686545 + 0.997640i \(0.478129\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0.137699 0.00723731
\(363\) −34.7272 −1.82271
\(364\) 13.3856 0.701596
\(365\) 0 0
\(366\) 32.5957 1.70380
\(367\) −11.6489 −0.608069 −0.304034 0.952661i \(-0.598334\pi\)
−0.304034 + 0.952661i \(0.598334\pi\)
\(368\) −7.56885 −0.394554
\(369\) −24.3203 −1.26606
\(370\) 0 0
\(371\) 0.952717 0.0494626
\(372\) 10.1755 0.527577
\(373\) 12.8064 0.663091 0.331545 0.943439i \(-0.392430\pi\)
0.331545 + 0.943439i \(0.392430\pi\)
\(374\) 33.2978 1.72179
\(375\) 0 0
\(376\) −5.96216 −0.307475
\(377\) −2.63615 −0.135769
\(378\) 8.69446 0.447195
\(379\) −20.9122 −1.07419 −0.537095 0.843522i \(-0.680478\pi\)
−0.537095 + 0.843522i \(0.680478\pi\)
\(380\) 0 0
\(381\) −28.8416 −1.47760
\(382\) −19.3779 −0.991460
\(383\) −10.3511 −0.528916 −0.264458 0.964397i \(-0.585193\pi\)
−0.264458 + 0.964397i \(0.585193\pi\)
\(384\) 2.25561 0.115106
\(385\) 0 0
\(386\) −5.42851 −0.276304
\(387\) 5.24285 0.266509
\(388\) −3.93972 −0.200009
\(389\) 6.56620 0.332920 0.166460 0.986048i \(-0.446766\pi\)
0.166460 + 0.986048i \(0.446766\pi\)
\(390\) 0 0
\(391\) −49.0543 −2.48078
\(392\) −10.8546 −0.548240
\(393\) 4.77121 0.240676
\(394\) −15.6489 −0.788381
\(395\) 0 0
\(396\) −10.7263 −0.539019
\(397\) −23.5284 −1.18085 −0.590427 0.807091i \(-0.701041\pi\)
−0.590427 + 0.807091i \(0.701041\pi\)
\(398\) −18.0499 −0.904761
\(399\) −9.53101 −0.477147
\(400\) 0 0
\(401\) 6.22041 0.310633 0.155316 0.987865i \(-0.450360\pi\)
0.155316 + 0.987865i \(0.450360\pi\)
\(402\) 9.27102 0.462396
\(403\) 14.2908 0.711876
\(404\) 3.19798 0.159105
\(405\) 0 0
\(406\) 3.51628 0.174510
\(407\) 0.707457 0.0350674
\(408\) 14.6188 0.723737
\(409\) −34.4905 −1.70545 −0.852723 0.522363i \(-0.825051\pi\)
−0.852723 + 0.522363i \(0.825051\pi\)
\(410\) 0 0
\(411\) −12.0972 −0.596711
\(412\) 10.6868 0.526499
\(413\) −22.7893 −1.12139
\(414\) 15.8020 0.776627
\(415\) 0 0
\(416\) 3.16784 0.155316
\(417\) 22.9173 1.12226
\(418\) −5.13770 −0.251293
\(419\) −10.6265 −0.519138 −0.259569 0.965725i \(-0.583580\pi\)
−0.259569 + 0.965725i \(0.583580\pi\)
\(420\) 0 0
\(421\) −1.98021 −0.0965095 −0.0482548 0.998835i \(-0.515366\pi\)
−0.0482548 + 0.998835i \(0.515366\pi\)
\(422\) 7.50857 0.365512
\(423\) 12.4476 0.605224
\(424\) 0.225470 0.0109498
\(425\) 0 0
\(426\) 8.62648 0.417954
\(427\) −61.0620 −2.95500
\(428\) −10.8168 −0.522848
\(429\) −36.7109 −1.77242
\(430\) 0 0
\(431\) 15.0774 0.726254 0.363127 0.931740i \(-0.381709\pi\)
0.363127 + 0.931740i \(0.381709\pi\)
\(432\) 2.05763 0.0989979
\(433\) −13.5337 −0.650386 −0.325193 0.945648i \(-0.605429\pi\)
−0.325193 + 0.945648i \(0.605429\pi\)
\(434\) −19.0620 −0.915006
\(435\) 0 0
\(436\) 9.24791 0.442894
\(437\) 7.56885 0.362067
\(438\) −10.6161 −0.507258
\(439\) 39.1773 1.86983 0.934915 0.354872i \(-0.115476\pi\)
0.934915 + 0.354872i \(0.115476\pi\)
\(440\) 0 0
\(441\) 22.6619 1.07914
\(442\) 20.5310 0.976560
\(443\) −19.3132 −0.917600 −0.458800 0.888540i \(-0.651721\pi\)
−0.458800 + 0.888540i \(0.651721\pi\)
\(444\) 0.310596 0.0147402
\(445\) 0 0
\(446\) 27.8091 1.31680
\(447\) 13.3977 0.633689
\(448\) −4.22547 −0.199635
\(449\) 41.4131 1.95440 0.977202 0.212310i \(-0.0680985\pi\)
0.977202 + 0.212310i \(0.0680985\pi\)
\(450\) 0 0
\(451\) 59.8486 2.81816
\(452\) −17.6489 −0.830135
\(453\) 34.5059 1.62123
\(454\) 8.91223 0.418272
\(455\) 0 0
\(456\) −2.25561 −0.105629
\(457\) 37.8392 1.77004 0.885021 0.465551i \(-0.154144\pi\)
0.885021 + 0.465551i \(0.154144\pi\)
\(458\) 13.6489 0.637772
\(459\) 13.3357 0.622456
\(460\) 0 0
\(461\) 1.58864 0.0739903 0.0369952 0.999315i \(-0.488221\pi\)
0.0369952 + 0.999315i \(0.488221\pi\)
\(462\) 48.9674 2.27817
\(463\) −40.3581 −1.87560 −0.937800 0.347175i \(-0.887141\pi\)
−0.937800 + 0.347175i \(0.887141\pi\)
\(464\) 0.832162 0.0386322
\(465\) 0 0
\(466\) 23.2754 1.07821
\(467\) 39.5130 1.82844 0.914221 0.405217i \(-0.132804\pi\)
0.914221 + 0.405217i \(0.132804\pi\)
\(468\) −6.61372 −0.305719
\(469\) −17.3676 −0.801959
\(470\) 0 0
\(471\) −28.8416 −1.32895
\(472\) −5.39331 −0.248247
\(473\) −12.9019 −0.593229
\(474\) 23.9692 1.10094
\(475\) 0 0
\(476\) −27.3856 −1.25522
\(477\) −0.470730 −0.0215532
\(478\) −3.72898 −0.170560
\(479\) 7.61107 0.347759 0.173879 0.984767i \(-0.444370\pi\)
0.173879 + 0.984767i \(0.444370\pi\)
\(480\) 0 0
\(481\) 0.436209 0.0198894
\(482\) 1.48878 0.0678122
\(483\) −72.1388 −3.28243
\(484\) 15.3960 0.699816
\(485\) 0 0
\(486\) −18.4234 −0.835705
\(487\) 20.3907 0.923989 0.461995 0.886883i \(-0.347134\pi\)
0.461995 + 0.886883i \(0.347134\pi\)
\(488\) −14.4509 −0.654163
\(489\) −25.8794 −1.17031
\(490\) 0 0
\(491\) 25.0224 1.12925 0.564623 0.825349i \(-0.309021\pi\)
0.564623 + 0.825349i \(0.309021\pi\)
\(492\) 26.2754 1.18459
\(493\) 5.39331 0.242902
\(494\) −3.16784 −0.142528
\(495\) 0 0
\(496\) −4.51122 −0.202560
\(497\) −16.1601 −0.724881
\(498\) 27.2925 1.22301
\(499\) 22.6111 1.01221 0.506105 0.862472i \(-0.331085\pi\)
0.506105 + 0.862472i \(0.331085\pi\)
\(500\) 0 0
\(501\) 43.7884 1.95632
\(502\) 8.78662 0.392166
\(503\) −11.5035 −0.512916 −0.256458 0.966555i \(-0.582556\pi\)
−0.256458 + 0.966555i \(0.582556\pi\)
\(504\) 8.82181 0.392955
\(505\) 0 0
\(506\) −38.8865 −1.72871
\(507\) 6.68744 0.296999
\(508\) 12.7866 0.567314
\(509\) 9.11526 0.404027 0.202013 0.979383i \(-0.435252\pi\)
0.202013 + 0.979383i \(0.435252\pi\)
\(510\) 0 0
\(511\) 19.8874 0.879766
\(512\) −1.00000 −0.0441942
\(513\) −2.05763 −0.0908467
\(514\) 2.74175 0.120933
\(515\) 0 0
\(516\) −5.66432 −0.249358
\(517\) −30.6318 −1.34718
\(518\) −0.581844 −0.0255648
\(519\) 22.0748 0.968975
\(520\) 0 0
\(521\) 15.4888 0.678576 0.339288 0.940683i \(-0.389814\pi\)
0.339288 + 0.940683i \(0.389814\pi\)
\(522\) −1.73736 −0.0760423
\(523\) −4.34073 −0.189807 −0.0949035 0.995486i \(-0.530254\pi\)
−0.0949035 + 0.995486i \(0.530254\pi\)
\(524\) −2.11526 −0.0924057
\(525\) 0 0
\(526\) 2.74704 0.119776
\(527\) −29.2376 −1.27361
\(528\) 11.5886 0.504331
\(529\) 34.2875 1.49076
\(530\) 0 0
\(531\) 11.2600 0.488642
\(532\) 4.22547 0.183197
\(533\) 36.9019 1.59840
\(534\) −22.5561 −0.976097
\(535\) 0 0
\(536\) −4.11021 −0.177534
\(537\) 15.3933 0.664270
\(538\) 9.74175 0.419996
\(539\) −55.7677 −2.40208
\(540\) 0 0
\(541\) 19.5491 0.840480 0.420240 0.907413i \(-0.361946\pi\)
0.420240 + 0.907413i \(0.361946\pi\)
\(542\) −26.3555 −1.13206
\(543\) 0.310596 0.0133289
\(544\) −6.48108 −0.277874
\(545\) 0 0
\(546\) 30.1927 1.29213
\(547\) 41.8038 1.78740 0.893700 0.448665i \(-0.148100\pi\)
0.893700 + 0.448665i \(0.148100\pi\)
\(548\) 5.36317 0.229103
\(549\) 30.1703 1.28763
\(550\) 0 0
\(551\) −0.832162 −0.0354513
\(552\) −17.0724 −0.726648
\(553\) −44.9019 −1.90942
\(554\) −0.962158 −0.0408782
\(555\) 0 0
\(556\) −10.1601 −0.430886
\(557\) 9.76418 0.413722 0.206861 0.978370i \(-0.433675\pi\)
0.206861 + 0.978370i \(0.433675\pi\)
\(558\) 9.41839 0.398712
\(559\) −7.95513 −0.336466
\(560\) 0 0
\(561\) 75.1069 3.17102
\(562\) 14.6714 0.618874
\(563\) −11.4509 −0.482600 −0.241300 0.970451i \(-0.577574\pi\)
−0.241300 + 0.970451i \(0.577574\pi\)
\(564\) −13.4483 −0.566275
\(565\) 0 0
\(566\) −26.1601 −1.09959
\(567\) 46.0767 1.93504
\(568\) −3.82446 −0.160471
\(569\) −13.4338 −0.563174 −0.281587 0.959536i \(-0.590861\pi\)
−0.281587 + 0.959536i \(0.590861\pi\)
\(570\) 0 0
\(571\) 37.6335 1.57491 0.787457 0.616370i \(-0.211397\pi\)
0.787457 + 0.616370i \(0.211397\pi\)
\(572\) 16.2754 0.680509
\(573\) −43.7090 −1.82597
\(574\) −49.2221 −2.05449
\(575\) 0 0
\(576\) 2.08777 0.0869905
\(577\) −1.56885 −0.0653121 −0.0326560 0.999467i \(-0.510397\pi\)
−0.0326560 + 0.999467i \(0.510397\pi\)
\(578\) −25.0044 −1.04005
\(579\) −12.2446 −0.508868
\(580\) 0 0
\(581\) −51.1276 −2.12113
\(582\) −8.88647 −0.368356
\(583\) 1.15840 0.0479759
\(584\) 4.70655 0.194758
\(585\) 0 0
\(586\) −22.4657 −0.928048
\(587\) −25.8693 −1.06774 −0.533871 0.845566i \(-0.679263\pi\)
−0.533871 + 0.845566i \(0.679263\pi\)
\(588\) −24.4837 −1.00969
\(589\) 4.51122 0.185881
\(590\) 0 0
\(591\) −35.2978 −1.45196
\(592\) −0.137699 −0.00565940
\(593\) 28.9243 1.18778 0.593890 0.804547i \(-0.297592\pi\)
0.593890 + 0.804547i \(0.297592\pi\)
\(594\) 10.5715 0.433754
\(595\) 0 0
\(596\) −5.93972 −0.243301
\(597\) −40.7136 −1.66630
\(598\) −23.9769 −0.980488
\(599\) 41.3581 1.68985 0.844923 0.534887i \(-0.179646\pi\)
0.844923 + 0.534887i \(0.179646\pi\)
\(600\) 0 0
\(601\) 2.68147 0.109379 0.0546897 0.998503i \(-0.482583\pi\)
0.0546897 + 0.998503i \(0.482583\pi\)
\(602\) 10.6111 0.432475
\(603\) 8.58117 0.349452
\(604\) −15.2978 −0.622460
\(605\) 0 0
\(606\) 7.21338 0.293024
\(607\) −3.31324 −0.134480 −0.0672401 0.997737i \(-0.521419\pi\)
−0.0672401 + 0.997737i \(0.521419\pi\)
\(608\) 1.00000 0.0405554
\(609\) 7.93134 0.321394
\(610\) 0 0
\(611\) −18.8871 −0.764092
\(612\) 13.5310 0.546959
\(613\) 10.0603 0.406331 0.203165 0.979144i \(-0.434877\pi\)
0.203165 + 0.979144i \(0.434877\pi\)
\(614\) 17.2204 0.694959
\(615\) 0 0
\(616\) −21.7092 −0.874688
\(617\) 27.6265 1.11220 0.556100 0.831115i \(-0.312297\pi\)
0.556100 + 0.831115i \(0.312297\pi\)
\(618\) 24.1051 0.969651
\(619\) 16.6714 0.670078 0.335039 0.942204i \(-0.391250\pi\)
0.335039 + 0.942204i \(0.391250\pi\)
\(620\) 0 0
\(621\) −15.5739 −0.624959
\(622\) 7.87439 0.315734
\(623\) 42.2547 1.69290
\(624\) 7.14540 0.286045
\(625\) 0 0
\(626\) −25.1678 −1.00591
\(627\) −11.5886 −0.462806
\(628\) 12.7866 0.510242
\(629\) −0.892439 −0.0355839
\(630\) 0 0
\(631\) −0.709194 −0.0282326 −0.0141163 0.999900i \(-0.504494\pi\)
−0.0141163 + 0.999900i \(0.504494\pi\)
\(632\) −10.6265 −0.422699
\(633\) 16.9364 0.673162
\(634\) −23.9045 −0.949370
\(635\) 0 0
\(636\) 0.508572 0.0201662
\(637\) −34.3856 −1.36241
\(638\) 4.27540 0.169265
\(639\) 7.98459 0.315866
\(640\) 0 0
\(641\) −2.43553 −0.0961978 −0.0480989 0.998843i \(-0.515316\pi\)
−0.0480989 + 0.998843i \(0.515316\pi\)
\(642\) −24.3984 −0.962927
\(643\) −7.70390 −0.303812 −0.151906 0.988395i \(-0.548541\pi\)
−0.151906 + 0.988395i \(0.548541\pi\)
\(644\) 31.9819 1.26027
\(645\) 0 0
\(646\) 6.48108 0.254995
\(647\) 10.0499 0.395103 0.197552 0.980292i \(-0.436701\pi\)
0.197552 + 0.980292i \(0.436701\pi\)
\(648\) 10.9045 0.428370
\(649\) −27.7092 −1.08768
\(650\) 0 0
\(651\) −42.9964 −1.68516
\(652\) 11.4734 0.449332
\(653\) 7.09283 0.277564 0.138782 0.990323i \(-0.455681\pi\)
0.138782 + 0.990323i \(0.455681\pi\)
\(654\) 20.8597 0.815677
\(655\) 0 0
\(656\) −11.6489 −0.454814
\(657\) −9.82620 −0.383356
\(658\) 25.1929 0.982122
\(659\) −1.16346 −0.0453218 −0.0226609 0.999743i \(-0.507214\pi\)
−0.0226609 + 0.999743i \(0.507214\pi\)
\(660\) 0 0
\(661\) 1.79432 0.0697909 0.0348955 0.999391i \(-0.488890\pi\)
0.0348955 + 0.999391i \(0.488890\pi\)
\(662\) −30.7565 −1.19538
\(663\) 46.3099 1.79853
\(664\) −12.0999 −0.469566
\(665\) 0 0
\(666\) 0.287484 0.0111398
\(667\) −6.29851 −0.243879
\(668\) −19.4131 −0.751115
\(669\) 62.7263 2.42514
\(670\) 0 0
\(671\) −74.2446 −2.86618
\(672\) −9.53101 −0.367667
\(673\) 25.2824 0.974566 0.487283 0.873244i \(-0.337988\pi\)
0.487283 + 0.873244i \(0.337988\pi\)
\(674\) 13.4734 0.518975
\(675\) 0 0
\(676\) −2.96480 −0.114031
\(677\) 4.89682 0.188200 0.0941001 0.995563i \(-0.470003\pi\)
0.0941001 + 0.995563i \(0.470003\pi\)
\(678\) −39.8091 −1.52886
\(679\) 16.6472 0.638860
\(680\) 0 0
\(681\) 20.1025 0.770330
\(682\) −23.1773 −0.887504
\(683\) −10.1980 −0.390215 −0.195107 0.980782i \(-0.562506\pi\)
−0.195107 + 0.980782i \(0.562506\pi\)
\(684\) −2.08777 −0.0798279
\(685\) 0 0
\(686\) 16.2875 0.621859
\(687\) 30.7866 1.17458
\(688\) 2.51122 0.0957393
\(689\) 0.714253 0.0272109
\(690\) 0 0
\(691\) −39.9846 −1.52109 −0.760543 0.649288i \(-0.775067\pi\)
−0.760543 + 0.649288i \(0.775067\pi\)
\(692\) −9.78662 −0.372031
\(693\) 45.3238 1.72171
\(694\) 28.9468 1.09880
\(695\) 0 0
\(696\) 1.87703 0.0711487
\(697\) −75.4975 −2.85967
\(698\) −27.9243 −1.05695
\(699\) 52.5002 1.98574
\(700\) 0 0
\(701\) 4.91729 0.185723 0.0928617 0.995679i \(-0.470399\pi\)
0.0928617 + 0.995679i \(0.470399\pi\)
\(702\) 6.51825 0.246015
\(703\) 0.137699 0.00519342
\(704\) −5.13770 −0.193634
\(705\) 0 0
\(706\) 28.3099 1.06546
\(707\) −13.5130 −0.508207
\(708\) −12.1652 −0.457196
\(709\) −36.8865 −1.38530 −0.692650 0.721274i \(-0.743557\pi\)
−0.692650 + 0.721274i \(0.743557\pi\)
\(710\) 0 0
\(711\) 22.1857 0.832027
\(712\) 10.0000 0.374766
\(713\) 34.1447 1.27873
\(714\) −61.7712 −2.31173
\(715\) 0 0
\(716\) −6.82446 −0.255042
\(717\) −8.41113 −0.314119
\(718\) −2.60163 −0.0970921
\(719\) −23.8891 −0.890914 −0.445457 0.895303i \(-0.646959\pi\)
−0.445457 + 0.895303i \(0.646959\pi\)
\(720\) 0 0
\(721\) −45.1566 −1.68172
\(722\) −1.00000 −0.0372161
\(723\) 3.35811 0.124889
\(724\) −0.137699 −0.00511755
\(725\) 0 0
\(726\) 34.7272 1.28885
\(727\) −19.6894 −0.730240 −0.365120 0.930961i \(-0.618972\pi\)
−0.365120 + 0.930961i \(0.618972\pi\)
\(728\) −13.3856 −0.496104
\(729\) −8.84251 −0.327500
\(730\) 0 0
\(731\) 16.2754 0.601967
\(732\) −32.5957 −1.20477
\(733\) −1.76418 −0.0651615 −0.0325808 0.999469i \(-0.510373\pi\)
−0.0325808 + 0.999469i \(0.510373\pi\)
\(734\) 11.6489 0.429969
\(735\) 0 0
\(736\) 7.56885 0.278991
\(737\) −21.1170 −0.777855
\(738\) 24.3203 0.895241
\(739\) 28.1755 1.03645 0.518227 0.855243i \(-0.326592\pi\)
0.518227 + 0.855243i \(0.326592\pi\)
\(740\) 0 0
\(741\) −7.14540 −0.262493
\(742\) −0.952717 −0.0349753
\(743\) 5.42851 0.199153 0.0995763 0.995030i \(-0.468251\pi\)
0.0995763 + 0.995030i \(0.468251\pi\)
\(744\) −10.1755 −0.373053
\(745\) 0 0
\(746\) −12.8064 −0.468876
\(747\) 25.2617 0.924278
\(748\) −33.2978 −1.21749
\(749\) 45.7059 1.67006
\(750\) 0 0
\(751\) 25.8640 0.943792 0.471896 0.881654i \(-0.343570\pi\)
0.471896 + 0.881654i \(0.343570\pi\)
\(752\) 5.96216 0.217418
\(753\) 19.8192 0.722251
\(754\) 2.63615 0.0960031
\(755\) 0 0
\(756\) −8.69446 −0.316215
\(757\) 3.76947 0.137004 0.0685019 0.997651i \(-0.478178\pi\)
0.0685019 + 0.997651i \(0.478178\pi\)
\(758\) 20.9122 0.759566
\(759\) −87.7127 −3.18377
\(760\) 0 0
\(761\) 18.3605 0.665568 0.332784 0.943003i \(-0.392012\pi\)
0.332784 + 0.943003i \(0.392012\pi\)
\(762\) 28.8416 1.04482
\(763\) −39.0767 −1.41467
\(764\) 19.3779 0.701068
\(765\) 0 0
\(766\) 10.3511 0.374000
\(767\) −17.0851 −0.616908
\(768\) −2.25561 −0.0813923
\(769\) −38.1300 −1.37500 −0.687501 0.726183i \(-0.741292\pi\)
−0.687501 + 0.726183i \(0.741292\pi\)
\(770\) 0 0
\(771\) 6.18431 0.222722
\(772\) 5.42851 0.195376
\(773\) 29.1575 1.04872 0.524361 0.851496i \(-0.324304\pi\)
0.524361 + 0.851496i \(0.324304\pi\)
\(774\) −5.24285 −0.188450
\(775\) 0 0
\(776\) 3.93972 0.141428
\(777\) −1.31241 −0.0470825
\(778\) −6.56620 −0.235410
\(779\) 11.6489 0.417366
\(780\) 0 0
\(781\) −19.6489 −0.703094
\(782\) 49.0543 1.75418
\(783\) 1.71228 0.0611920
\(784\) 10.8546 0.387664
\(785\) 0 0
\(786\) −4.77121 −0.170183
\(787\) 47.0165 1.67596 0.837978 0.545704i \(-0.183738\pi\)
0.837978 + 0.545704i \(0.183738\pi\)
\(788\) 15.6489 0.557470
\(789\) 6.19624 0.220592
\(790\) 0 0
\(791\) 74.5750 2.65158
\(792\) 10.7263 0.381144
\(793\) −45.7782 −1.62563
\(794\) 23.5284 0.834990
\(795\) 0 0
\(796\) 18.0499 0.639763
\(797\) 42.6359 1.51024 0.755121 0.655586i \(-0.227578\pi\)
0.755121 + 0.655586i \(0.227578\pi\)
\(798\) 9.53101 0.337394
\(799\) 38.6412 1.36703
\(800\) 0 0
\(801\) −20.8777 −0.737678
\(802\) −6.22041 −0.219650
\(803\) 24.1808 0.853323
\(804\) −9.27102 −0.326964
\(805\) 0 0
\(806\) −14.2908 −0.503372
\(807\) 21.9736 0.773506
\(808\) −3.19798 −0.112504
\(809\) 49.4630 1.73903 0.869514 0.493909i \(-0.164432\pi\)
0.869514 + 0.493909i \(0.164432\pi\)
\(810\) 0 0
\(811\) −16.7816 −0.589280 −0.294640 0.955608i \(-0.595200\pi\)
−0.294640 + 0.955608i \(0.595200\pi\)
\(812\) −3.51628 −0.123397
\(813\) −59.4476 −2.08492
\(814\) −0.707457 −0.0247964
\(815\) 0 0
\(816\) −14.6188 −0.511760
\(817\) −2.51122 −0.0878564
\(818\) 34.4905 1.20593
\(819\) 27.9461 0.976515
\(820\) 0 0
\(821\) 11.5337 0.402527 0.201264 0.979537i \(-0.435495\pi\)
0.201264 + 0.979537i \(0.435495\pi\)
\(822\) 12.0972 0.421939
\(823\) −32.6309 −1.13744 −0.568720 0.822531i \(-0.692561\pi\)
−0.568720 + 0.822531i \(0.692561\pi\)
\(824\) −10.6868 −0.372291
\(825\) 0 0
\(826\) 22.7893 0.792940
\(827\) −6.64650 −0.231122 −0.115561 0.993300i \(-0.536867\pi\)
−0.115561 + 0.993300i \(0.536867\pi\)
\(828\) −15.8020 −0.549158
\(829\) −30.9217 −1.07395 −0.536977 0.843597i \(-0.680434\pi\)
−0.536977 + 0.843597i \(0.680434\pi\)
\(830\) 0 0
\(831\) −2.17025 −0.0752852
\(832\) −3.16784 −0.109825
\(833\) 70.3495 2.43747
\(834\) −22.9173 −0.793561
\(835\) 0 0
\(836\) 5.13770 0.177691
\(837\) −9.28243 −0.320848
\(838\) 10.6265 0.367086
\(839\) −48.7109 −1.68169 −0.840844 0.541277i \(-0.817941\pi\)
−0.840844 + 0.541277i \(0.817941\pi\)
\(840\) 0 0
\(841\) −28.3075 −0.976121
\(842\) 1.98021 0.0682426
\(843\) 33.0928 1.13978
\(844\) −7.50857 −0.258456
\(845\) 0 0
\(846\) −12.4476 −0.427958
\(847\) −65.0551 −2.23532
\(848\) −0.225470 −0.00774267
\(849\) −59.0070 −2.02512
\(850\) 0 0
\(851\) 1.04222 0.0357270
\(852\) −8.62648 −0.295538
\(853\) −46.1447 −1.57997 −0.789983 0.613129i \(-0.789911\pi\)
−0.789983 + 0.613129i \(0.789911\pi\)
\(854\) 61.0620 2.08950
\(855\) 0 0
\(856\) 10.8168 0.369709
\(857\) 8.40607 0.287146 0.143573 0.989640i \(-0.454141\pi\)
0.143573 + 0.989640i \(0.454141\pi\)
\(858\) 36.7109 1.25329
\(859\) −8.49581 −0.289873 −0.144937 0.989441i \(-0.546298\pi\)
−0.144937 + 0.989441i \(0.546298\pi\)
\(860\) 0 0
\(861\) −111.026 −3.78375
\(862\) −15.0774 −0.513539
\(863\) −3.37352 −0.114836 −0.0574179 0.998350i \(-0.518287\pi\)
−0.0574179 + 0.998350i \(0.518287\pi\)
\(864\) −2.05763 −0.0700021
\(865\) 0 0
\(866\) 13.5337 0.459892
\(867\) −56.4001 −1.91545
\(868\) 19.0620 0.647007
\(869\) −54.5957 −1.85203
\(870\) 0 0
\(871\) −13.0205 −0.441182
\(872\) −9.24791 −0.313174
\(873\) −8.22524 −0.278382
\(874\) −7.56885 −0.256020
\(875\) 0 0
\(876\) 10.6161 0.358686
\(877\) 13.2780 0.448368 0.224184 0.974547i \(-0.428028\pi\)
0.224184 + 0.974547i \(0.428028\pi\)
\(878\) −39.1773 −1.32217
\(879\) −50.6738 −1.70918
\(880\) 0 0
\(881\) −26.6489 −0.897825 −0.448912 0.893576i \(-0.648188\pi\)
−0.448912 + 0.893576i \(0.648188\pi\)
\(882\) −22.6619 −0.763066
\(883\) 47.7884 1.60821 0.804103 0.594490i \(-0.202646\pi\)
0.804103 + 0.594490i \(0.202646\pi\)
\(884\) −20.5310 −0.690532
\(885\) 0 0
\(886\) 19.3132 0.648841
\(887\) 36.3304 1.21985 0.609927 0.792457i \(-0.291199\pi\)
0.609927 + 0.792457i \(0.291199\pi\)
\(888\) −0.310596 −0.0104229
\(889\) −54.0295 −1.81209
\(890\) 0 0
\(891\) 56.0242 1.87688
\(892\) −27.8091 −0.931116
\(893\) −5.96216 −0.199516
\(894\) −13.3977 −0.448086
\(895\) 0 0
\(896\) 4.22547 0.141163
\(897\) −54.0825 −1.80576
\(898\) −41.4131 −1.38197
\(899\) −3.75406 −0.125205
\(900\) 0 0
\(901\) −1.46129 −0.0486826
\(902\) −59.8486 −1.99274
\(903\) 23.9344 0.796488
\(904\) 17.6489 0.586994
\(905\) 0 0
\(906\) −34.5059 −1.14638
\(907\) 39.3873 1.30784 0.653918 0.756566i \(-0.273124\pi\)
0.653918 + 0.756566i \(0.273124\pi\)
\(908\) −8.91223 −0.295763
\(909\) 6.67664 0.221450
\(910\) 0 0
\(911\) −20.5561 −0.681054 −0.340527 0.940235i \(-0.610605\pi\)
−0.340527 + 0.940235i \(0.610605\pi\)
\(912\) 2.25561 0.0746907
\(913\) −62.1654 −2.05738
\(914\) −37.8392 −1.25161
\(915\) 0 0
\(916\) −13.6489 −0.450973
\(917\) 8.93799 0.295158
\(918\) −13.3357 −0.440143
\(919\) 12.4054 0.409216 0.204608 0.978844i \(-0.434408\pi\)
0.204608 + 0.978844i \(0.434408\pi\)
\(920\) 0 0
\(921\) 38.8425 1.27990
\(922\) −1.58864 −0.0523190
\(923\) −12.1153 −0.398779
\(924\) −48.9674 −1.61091
\(925\) 0 0
\(926\) 40.3581 1.32625
\(927\) 22.3115 0.732806
\(928\) −0.832162 −0.0273171
\(929\) −31.3330 −1.02800 −0.514002 0.857789i \(-0.671838\pi\)
−0.514002 + 0.857789i \(0.671838\pi\)
\(930\) 0 0
\(931\) −10.8546 −0.355745
\(932\) −23.2754 −0.762411
\(933\) 17.7615 0.581487
\(934\) −39.5130 −1.29290
\(935\) 0 0
\(936\) 6.61372 0.216176
\(937\) −7.27299 −0.237598 −0.118799 0.992918i \(-0.537904\pi\)
−0.118799 + 0.992918i \(0.537904\pi\)
\(938\) 17.3676 0.567071
\(939\) −56.7688 −1.85258
\(940\) 0 0
\(941\) 6.72128 0.219107 0.109554 0.993981i \(-0.465058\pi\)
0.109554 + 0.993981i \(0.465058\pi\)
\(942\) 28.8416 0.939710
\(943\) 88.1689 2.87117
\(944\) 5.39331 0.175537
\(945\) 0 0
\(946\) 12.9019 0.419476
\(947\) −10.0757 −0.327416 −0.163708 0.986509i \(-0.552345\pi\)
−0.163708 + 0.986509i \(0.552345\pi\)
\(948\) −23.9692 −0.778483
\(949\) 14.9096 0.483986
\(950\) 0 0
\(951\) −53.9193 −1.74845
\(952\) 27.3856 0.887573
\(953\) 8.14473 0.263834 0.131917 0.991261i \(-0.457887\pi\)
0.131917 + 0.991261i \(0.457887\pi\)
\(954\) 0.470730 0.0152404
\(955\) 0 0
\(956\) 3.72898 0.120604
\(957\) 9.64363 0.311734
\(958\) −7.61107 −0.245903
\(959\) −22.6619 −0.731791
\(960\) 0 0
\(961\) −10.6489 −0.343513
\(962\) −0.436209 −0.0140639
\(963\) −22.5829 −0.727724
\(964\) −1.48878 −0.0479505
\(965\) 0 0
\(966\) 72.1388 2.32103
\(967\) −47.1773 −1.51712 −0.758560 0.651604i \(-0.774097\pi\)
−0.758560 + 0.651604i \(0.774097\pi\)
\(968\) −15.3960 −0.494845
\(969\) 14.6188 0.469623
\(970\) 0 0
\(971\) 0.230528 0.00739801 0.00369901 0.999993i \(-0.498823\pi\)
0.00369901 + 0.999993i \(0.498823\pi\)
\(972\) 18.4234 0.590932
\(973\) 42.9313 1.37632
\(974\) −20.3907 −0.653359
\(975\) 0 0
\(976\) 14.4509 0.462563
\(977\) 5.80905 0.185848 0.0929240 0.995673i \(-0.470379\pi\)
0.0929240 + 0.995673i \(0.470379\pi\)
\(978\) 25.8794 0.827533
\(979\) 51.3770 1.64202
\(980\) 0 0
\(981\) 19.3075 0.616441
\(982\) −25.0224 −0.798498
\(983\) 22.3511 0.712889 0.356444 0.934317i \(-0.383989\pi\)
0.356444 + 0.934317i \(0.383989\pi\)
\(984\) −26.2754 −0.837629
\(985\) 0 0
\(986\) −5.39331 −0.171758
\(987\) 56.8254 1.80877
\(988\) 3.16784 0.100782
\(989\) −19.0070 −0.604388
\(990\) 0 0
\(991\) −34.9415 −1.10995 −0.554976 0.831866i \(-0.687273\pi\)
−0.554976 + 0.831866i \(0.687273\pi\)
\(992\) 4.51122 0.143231
\(993\) −69.3746 −2.20154
\(994\) 16.1601 0.512568
\(995\) 0 0
\(996\) −27.2925 −0.864797
\(997\) 9.35282 0.296207 0.148103 0.988972i \(-0.452683\pi\)
0.148103 + 0.988972i \(0.452683\pi\)
\(998\) −22.6111 −0.715741
\(999\) −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.a.k.1.2 3
3.2 odd 2 8550.2.a.co.1.1 3
4.3 odd 2 7600.2.a.cb.1.2 3
5.2 odd 4 950.2.b.g.799.2 6
5.3 odd 4 950.2.b.g.799.5 6
5.4 even 2 950.2.a.m.1.2 yes 3
15.14 odd 2 8550.2.a.cj.1.3 3
20.19 odd 2 7600.2.a.bm.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.2 3 1.1 even 1 trivial
950.2.a.m.1.2 yes 3 5.4 even 2
950.2.b.g.799.2 6 5.2 odd 4
950.2.b.g.799.5 6 5.3 odd 4
7600.2.a.bm.1.2 3 20.19 odd 2
7600.2.a.cb.1.2 3 4.3 odd 2
8550.2.a.cj.1.3 3 15.14 odd 2
8550.2.a.co.1.1 3 3.2 odd 2