Properties

Label 950.2.a.i.1.2
Level $950$
Weight $2$
Character 950.1
Self dual yes
Analytic conductor $7.586$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.363328\) of defining polynomial
Character \(\chi\) \(=\) 950.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.36333 q^{3} +1.00000 q^{4} +1.36333 q^{6} -0.636672 q^{7} -1.00000 q^{8} -1.14134 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.36333 q^{3} +1.00000 q^{4} +1.36333 q^{6} -0.636672 q^{7} -1.00000 q^{8} -1.14134 q^{9} +3.50466 q^{11} -1.36333 q^{12} +0.141336 q^{13} +0.636672 q^{14} +1.00000 q^{16} -2.14134 q^{17} +1.14134 q^{18} +1.00000 q^{19} +0.867993 q^{21} -3.50466 q^{22} -4.91934 q^{23} +1.36333 q^{24} -0.141336 q^{26} +5.64600 q^{27} -0.636672 q^{28} +7.15066 q^{29} -7.78734 q^{31} -1.00000 q^{32} -4.77801 q^{33} +2.14134 q^{34} -1.14134 q^{36} -3.27334 q^{37} -1.00000 q^{38} -0.192688 q^{39} -4.23132 q^{41} -0.867993 q^{42} -2.49534 q^{43} +3.50466 q^{44} +4.91934 q^{46} +10.2827 q^{47} -1.36333 q^{48} -6.59465 q^{49} +2.91934 q^{51} +0.141336 q^{52} -8.14134 q^{53} -5.64600 q^{54} +0.636672 q^{56} -1.36333 q^{57} -7.15066 q^{58} -5.64600 q^{59} -6.49534 q^{61} +7.78734 q^{62} +0.726656 q^{63} +1.00000 q^{64} +4.77801 q^{66} -8.37266 q^{67} -2.14134 q^{68} +6.70668 q^{69} -8.95798 q^{71} +1.14134 q^{72} +3.69735 q^{73} +3.27334 q^{74} +1.00000 q^{76} -2.23132 q^{77} +0.192688 q^{78} -4.17997 q^{79} -4.27334 q^{81} +4.23132 q^{82} +9.00933 q^{83} +0.867993 q^{84} +2.49534 q^{86} -9.74870 q^{87} -3.50466 q^{88} -6.77801 q^{89} -0.0899847 q^{91} -4.91934 q^{92} +10.6167 q^{93} -10.2827 q^{94} +1.36333 q^{96} -14.5653 q^{97} +6.59465 q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} - 2q^{3} + 3q^{4} + 2q^{6} - 4q^{7} - 3q^{8} + 5q^{9} + O(q^{10}) \) \( 3q - 3q^{2} - 2q^{3} + 3q^{4} + 2q^{6} - 4q^{7} - 3q^{8} + 5q^{9} - 2q^{12} - 8q^{13} + 4q^{14} + 3q^{16} + 2q^{17} - 5q^{18} + 3q^{19} - 10q^{21} + 2q^{24} + 8q^{26} - 2q^{27} - 4q^{28} - 8q^{29} + 4q^{31} - 3q^{32} - 8q^{33} - 2q^{34} + 5q^{36} - 14q^{37} - 3q^{38} + 10q^{39} + 2q^{41} + 10q^{42} - 18q^{43} + 14q^{47} - 2q^{48} - 3q^{49} - 6q^{51} - 8q^{52} - 16q^{53} + 2q^{54} + 4q^{56} - 2q^{57} + 8q^{58} + 2q^{59} - 30q^{61} - 4q^{62} - 2q^{63} + 3q^{64} + 8q^{66} - 2q^{67} + 2q^{68} - 22q^{69} - 8q^{71} - 5q^{72} - 10q^{73} + 14q^{74} + 3q^{76} + 8q^{77} - 10q^{78} - 17q^{81} - 2q^{82} + 6q^{83} - 10q^{84} + 18q^{86} - 6q^{87} - 14q^{89} + 6q^{91} - 4q^{93} - 14q^{94} + 2q^{96} - 10q^{97} + 3q^{98} - 12q^{99} + O(q^{100}) \)

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
\(3\) −1.36333 −0.787118 −0.393559 0.919299i \(-0.628756\pi\)
−0.393559 + 0.919299i \(0.628756\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.36333 0.556576
\(7\) −0.636672 −0.240639 −0.120320 0.992735i \(-0.538392\pi\)
−0.120320 + 0.992735i \(0.538392\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.14134 −0.380445
\(10\) 0 0
\(11\) 3.50466 1.05670 0.528348 0.849028i \(-0.322812\pi\)
0.528348 + 0.849028i \(0.322812\pi\)
\(12\) −1.36333 −0.393559
\(13\) 0.141336 0.0391996 0.0195998 0.999808i \(-0.493761\pi\)
0.0195998 + 0.999808i \(0.493761\pi\)
\(14\) 0.636672 0.170158
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.14134 −0.519350 −0.259675 0.965696i \(-0.583615\pi\)
−0.259675 + 0.965696i \(0.583615\pi\)
\(18\) 1.14134 0.269016
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.867993 0.189412
\(22\) −3.50466 −0.747197
\(23\) −4.91934 −1.02575 −0.512877 0.858462i \(-0.671420\pi\)
−0.512877 + 0.858462i \(0.671420\pi\)
\(24\) 1.36333 0.278288
\(25\) 0 0
\(26\) −0.141336 −0.0277183
\(27\) 5.64600 1.08657
\(28\) −0.636672 −0.120320
\(29\) 7.15066 1.32785 0.663923 0.747801i \(-0.268890\pi\)
0.663923 + 0.747801i \(0.268890\pi\)
\(30\) 0 0
\(31\) −7.78734 −1.39865 −0.699323 0.714805i \(-0.746515\pi\)
−0.699323 + 0.714805i \(0.746515\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.77801 −0.831744
\(34\) 2.14134 0.367236
\(35\) 0 0
\(36\) −1.14134 −0.190223
\(37\) −3.27334 −0.538134 −0.269067 0.963121i \(-0.586715\pi\)
−0.269067 + 0.963121i \(0.586715\pi\)
\(38\) −1.00000 −0.162221
\(39\) −0.192688 −0.0308547
\(40\) 0 0
\(41\) −4.23132 −0.660821 −0.330411 0.943837i \(-0.607187\pi\)
−0.330411 + 0.943837i \(0.607187\pi\)
\(42\) −0.867993 −0.133934
\(43\) −2.49534 −0.380535 −0.190268 0.981732i \(-0.560936\pi\)
−0.190268 + 0.981732i \(0.560936\pi\)
\(44\) 3.50466 0.528348
\(45\) 0 0
\(46\) 4.91934 0.725318
\(47\) 10.2827 1.49988 0.749941 0.661505i \(-0.230082\pi\)
0.749941 + 0.661505i \(0.230082\pi\)
\(48\) −1.36333 −0.196779
\(49\) −6.59465 −0.942093
\(50\) 0 0
\(51\) 2.91934 0.408790
\(52\) 0.141336 0.0195998
\(53\) −8.14134 −1.11830 −0.559149 0.829067i \(-0.688872\pi\)
−0.559149 + 0.829067i \(0.688872\pi\)
\(54\) −5.64600 −0.768323
\(55\) 0 0
\(56\) 0.636672 0.0850788
\(57\) −1.36333 −0.180577
\(58\) −7.15066 −0.938928
\(59\) −5.64600 −0.735047 −0.367523 0.930014i \(-0.619794\pi\)
−0.367523 + 0.930014i \(0.619794\pi\)
\(60\) 0 0
\(61\) −6.49534 −0.831643 −0.415821 0.909446i \(-0.636506\pi\)
−0.415821 + 0.909446i \(0.636506\pi\)
\(62\) 7.78734 0.988993
\(63\) 0.726656 0.0915501
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.77801 0.588132
\(67\) −8.37266 −1.02288 −0.511441 0.859318i \(-0.670888\pi\)
−0.511441 + 0.859318i \(0.670888\pi\)
\(68\) −2.14134 −0.259675
\(69\) 6.70668 0.807389
\(70\) 0 0
\(71\) −8.95798 −1.06312 −0.531558 0.847022i \(-0.678393\pi\)
−0.531558 + 0.847022i \(0.678393\pi\)
\(72\) 1.14134 0.134508
\(73\) 3.69735 0.432742 0.216371 0.976311i \(-0.430578\pi\)
0.216371 + 0.976311i \(0.430578\pi\)
\(74\) 3.27334 0.380518
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −2.23132 −0.254283
\(78\) 0.192688 0.0218176
\(79\) −4.17997 −0.470283 −0.235142 0.971961i \(-0.575555\pi\)
−0.235142 + 0.971961i \(0.575555\pi\)
\(80\) 0 0
\(81\) −4.27334 −0.474816
\(82\) 4.23132 0.467271
\(83\) 9.00933 0.988902 0.494451 0.869205i \(-0.335369\pi\)
0.494451 + 0.869205i \(0.335369\pi\)
\(84\) 0.867993 0.0947058
\(85\) 0 0
\(86\) 2.49534 0.269079
\(87\) −9.74870 −1.04517
\(88\) −3.50466 −0.373598
\(89\) −6.77801 −0.718467 −0.359234 0.933248i \(-0.616962\pi\)
−0.359234 + 0.933248i \(0.616962\pi\)
\(90\) 0 0
\(91\) −0.0899847 −0.00943296
\(92\) −4.91934 −0.512877
\(93\) 10.6167 1.10090
\(94\) −10.2827 −1.06058
\(95\) 0 0
\(96\) 1.36333 0.139144
\(97\) −14.5653 −1.47889 −0.739443 0.673219i \(-0.764911\pi\)
−0.739443 + 0.673219i \(0.764911\pi\)
\(98\) 6.59465 0.666160
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −16.6167 −1.65342 −0.826712 0.562626i \(-0.809791\pi\)
−0.826712 + 0.562626i \(0.809791\pi\)
\(102\) −2.91934 −0.289058
\(103\) −9.06068 −0.892775 −0.446388 0.894840i \(-0.647290\pi\)
−0.446388 + 0.894840i \(0.647290\pi\)
\(104\) −0.141336 −0.0138591
\(105\) 0 0
\(106\) 8.14134 0.790756
\(107\) −0.0899847 −0.00869915 −0.00434958 0.999991i \(-0.501385\pi\)
−0.00434958 + 0.999991i \(0.501385\pi\)
\(108\) 5.64600 0.543287
\(109\) −13.5946 −1.30213 −0.651066 0.759021i \(-0.725678\pi\)
−0.651066 + 0.759021i \(0.725678\pi\)
\(110\) 0 0
\(111\) 4.46264 0.423575
\(112\) −0.636672 −0.0601598
\(113\) −11.5233 −1.08402 −0.542011 0.840371i \(-0.682337\pi\)
−0.542011 + 0.840371i \(0.682337\pi\)
\(114\) 1.36333 0.127687
\(115\) 0 0
\(116\) 7.15066 0.663923
\(117\) −0.161312 −0.0149133
\(118\) 5.64600 0.519756
\(119\) 1.36333 0.124976
\(120\) 0 0
\(121\) 1.28267 0.116607
\(122\) 6.49534 0.588060
\(123\) 5.76868 0.520144
\(124\) −7.78734 −0.699323
\(125\) 0 0
\(126\) −0.726656 −0.0647357
\(127\) 3.29200 0.292118 0.146059 0.989276i \(-0.453341\pi\)
0.146059 + 0.989276i \(0.453341\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 3.40196 0.299526
\(130\) 0 0
\(131\) 18.0187 1.57430 0.787149 0.616763i \(-0.211556\pi\)
0.787149 + 0.616763i \(0.211556\pi\)
\(132\) −4.77801 −0.415872
\(133\) −0.636672 −0.0552064
\(134\) 8.37266 0.723287
\(135\) 0 0
\(136\) 2.14134 0.183618
\(137\) 14.4240 1.23233 0.616163 0.787619i \(-0.288686\pi\)
0.616163 + 0.787619i \(0.288686\pi\)
\(138\) −6.70668 −0.570911
\(139\) 15.4720 1.31232 0.656158 0.754624i \(-0.272181\pi\)
0.656158 + 0.754624i \(0.272181\pi\)
\(140\) 0 0
\(141\) −14.0187 −1.18058
\(142\) 8.95798 0.751737
\(143\) 0.495336 0.0414220
\(144\) −1.14134 −0.0951113
\(145\) 0 0
\(146\) −3.69735 −0.305995
\(147\) 8.99067 0.741538
\(148\) −3.27334 −0.269067
\(149\) −17.1893 −1.40820 −0.704101 0.710100i \(-0.748650\pi\)
−0.704101 + 0.710100i \(0.748650\pi\)
\(150\) 0 0
\(151\) 3.29200 0.267899 0.133950 0.990988i \(-0.457234\pi\)
0.133950 + 0.990988i \(0.457234\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 2.44398 0.197584
\(154\) 2.23132 0.179805
\(155\) 0 0
\(156\) −0.192688 −0.0154273
\(157\) −15.1893 −1.21224 −0.606119 0.795374i \(-0.707274\pi\)
−0.606119 + 0.795374i \(0.707274\pi\)
\(158\) 4.17997 0.332541
\(159\) 11.0993 0.880233
\(160\) 0 0
\(161\) 3.13201 0.246837
\(162\) 4.27334 0.335746
\(163\) 14.0700 1.10205 0.551024 0.834489i \(-0.314237\pi\)
0.551024 + 0.834489i \(0.314237\pi\)
\(164\) −4.23132 −0.330411
\(165\) 0 0
\(166\) −9.00933 −0.699260
\(167\) 14.7967 1.14500 0.572500 0.819905i \(-0.305974\pi\)
0.572500 + 0.819905i \(0.305974\pi\)
\(168\) −0.867993 −0.0669671
\(169\) −12.9800 −0.998463
\(170\) 0 0
\(171\) −1.14134 −0.0872802
\(172\) −2.49534 −0.190268
\(173\) 17.2920 1.31469 0.657343 0.753591i \(-0.271680\pi\)
0.657343 + 0.753591i \(0.271680\pi\)
\(174\) 9.74870 0.739047
\(175\) 0 0
\(176\) 3.50466 0.264174
\(177\) 7.69735 0.578568
\(178\) 6.77801 0.508033
\(179\) 17.7360 1.32565 0.662825 0.748774i \(-0.269357\pi\)
0.662825 + 0.748774i \(0.269357\pi\)
\(180\) 0 0
\(181\) 6.17997 0.459354 0.229677 0.973267i \(-0.426233\pi\)
0.229677 + 0.973267i \(0.426233\pi\)
\(182\) 0.0899847 0.00667011
\(183\) 8.85527 0.654601
\(184\) 4.91934 0.362659
\(185\) 0 0
\(186\) −10.6167 −0.778454
\(187\) −7.50466 −0.548795
\(188\) 10.2827 0.749941
\(189\) −3.59465 −0.261472
\(190\) 0 0
\(191\) 14.6367 1.05907 0.529536 0.848287i \(-0.322366\pi\)
0.529536 + 0.848287i \(0.322366\pi\)
\(192\) −1.36333 −0.0983897
\(193\) −20.0187 −1.44097 −0.720487 0.693468i \(-0.756082\pi\)
−0.720487 + 0.693468i \(0.756082\pi\)
\(194\) 14.5653 1.04573
\(195\) 0 0
\(196\) −6.59465 −0.471046
\(197\) 9.94865 0.708812 0.354406 0.935092i \(-0.384683\pi\)
0.354406 + 0.935092i \(0.384683\pi\)
\(198\) 4.00000 0.284268
\(199\) 9.74870 0.691067 0.345534 0.938406i \(-0.387698\pi\)
0.345534 + 0.938406i \(0.387698\pi\)
\(200\) 0 0
\(201\) 11.4147 0.805129
\(202\) 16.6167 1.16915
\(203\) −4.55263 −0.319532
\(204\) 2.91934 0.204395
\(205\) 0 0
\(206\) 9.06068 0.631287
\(207\) 5.61462 0.390243
\(208\) 0.141336 0.00979990
\(209\) 3.50466 0.242423
\(210\) 0 0
\(211\) −20.7580 −1.42904 −0.714521 0.699614i \(-0.753355\pi\)
−0.714521 + 0.699614i \(0.753355\pi\)
\(212\) −8.14134 −0.559149
\(213\) 12.2127 0.836798
\(214\) 0.0899847 0.00615123
\(215\) 0 0
\(216\) −5.64600 −0.384162
\(217\) 4.95798 0.336569
\(218\) 13.5946 0.920746
\(219\) −5.04070 −0.340619
\(220\) 0 0
\(221\) −0.302648 −0.0203583
\(222\) −4.46264 −0.299513
\(223\) 10.7267 0.718310 0.359155 0.933278i \(-0.383065\pi\)
0.359155 + 0.933278i \(0.383065\pi\)
\(224\) 0.636672 0.0425394
\(225\) 0 0
\(226\) 11.5233 0.766520
\(227\) −12.5526 −0.833147 −0.416574 0.909102i \(-0.636769\pi\)
−0.416574 + 0.909102i \(0.636769\pi\)
\(228\) −1.36333 −0.0902886
\(229\) 25.4720 1.68324 0.841618 0.540074i \(-0.181604\pi\)
0.841618 + 0.540074i \(0.181604\pi\)
\(230\) 0 0
\(231\) 3.04202 0.200150
\(232\) −7.15066 −0.469464
\(233\) 3.11203 0.203876 0.101938 0.994791i \(-0.467496\pi\)
0.101938 + 0.994791i \(0.467496\pi\)
\(234\) 0.161312 0.0105453
\(235\) 0 0
\(236\) −5.64600 −0.367523
\(237\) 5.69867 0.370168
\(238\) −1.36333 −0.0883714
\(239\) −1.54330 −0.0998276 −0.0499138 0.998754i \(-0.515895\pi\)
−0.0499138 + 0.998754i \(0.515895\pi\)
\(240\) 0 0
\(241\) 10.2827 0.662365 0.331183 0.943567i \(-0.392552\pi\)
0.331183 + 0.943567i \(0.392552\pi\)
\(242\) −1.28267 −0.0824533
\(243\) −11.1120 −0.712837
\(244\) −6.49534 −0.415821
\(245\) 0 0
\(246\) −5.76868 −0.367798
\(247\) 0.141336 0.00899300
\(248\) 7.78734 0.494496
\(249\) −12.2827 −0.778383
\(250\) 0 0
\(251\) −2.51399 −0.158682 −0.0793409 0.996848i \(-0.525282\pi\)
−0.0793409 + 0.996848i \(0.525282\pi\)
\(252\) 0.726656 0.0457751
\(253\) −17.2406 −1.08391
\(254\) −3.29200 −0.206559
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 24.7967 1.54677 0.773387 0.633934i \(-0.218561\pi\)
0.773387 + 0.633934i \(0.218561\pi\)
\(258\) −3.40196 −0.211797
\(259\) 2.08405 0.129496
\(260\) 0 0
\(261\) −8.16131 −0.505173
\(262\) −18.0187 −1.11320
\(263\) 22.5653 1.39144 0.695719 0.718314i \(-0.255086\pi\)
0.695719 + 0.718314i \(0.255086\pi\)
\(264\) 4.77801 0.294066
\(265\) 0 0
\(266\) 0.636672 0.0390369
\(267\) 9.24065 0.565519
\(268\) −8.37266 −0.511441
\(269\) −26.5653 −1.61972 −0.809859 0.586625i \(-0.800456\pi\)
−0.809859 + 0.586625i \(0.800456\pi\)
\(270\) 0 0
\(271\) −24.9380 −1.51488 −0.757438 0.652907i \(-0.773549\pi\)
−0.757438 + 0.652907i \(0.773549\pi\)
\(272\) −2.14134 −0.129838
\(273\) 0.122679 0.00742485
\(274\) −14.4240 −0.871386
\(275\) 0 0
\(276\) 6.70668 0.403695
\(277\) −18.5467 −1.11436 −0.557181 0.830391i \(-0.688117\pi\)
−0.557181 + 0.830391i \(0.688117\pi\)
\(278\) −15.4720 −0.927947
\(279\) 8.88797 0.532109
\(280\) 0 0
\(281\) 24.7967 1.47925 0.739623 0.673022i \(-0.235004\pi\)
0.739623 + 0.673022i \(0.235004\pi\)
\(282\) 14.0187 0.834799
\(283\) −13.5747 −0.806931 −0.403465 0.914995i \(-0.632194\pi\)
−0.403465 + 0.914995i \(0.632194\pi\)
\(284\) −8.95798 −0.531558
\(285\) 0 0
\(286\) −0.495336 −0.0292898
\(287\) 2.69396 0.159020
\(288\) 1.14134 0.0672539
\(289\) −12.4147 −0.730275
\(290\) 0 0
\(291\) 19.8573 1.16406
\(292\) 3.69735 0.216371
\(293\) 15.6133 0.912139 0.456070 0.889944i \(-0.349257\pi\)
0.456070 + 0.889944i \(0.349257\pi\)
\(294\) −8.99067 −0.524347
\(295\) 0 0
\(296\) 3.27334 0.190259
\(297\) 19.7873 1.14818
\(298\) 17.1893 0.995749
\(299\) −0.695281 −0.0402091
\(300\) 0 0
\(301\) 1.58871 0.0915717
\(302\) −3.29200 −0.189433
\(303\) 22.6540 1.30144
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −2.44398 −0.139713
\(307\) −34.0187 −1.94155 −0.970774 0.239997i \(-0.922854\pi\)
−0.970774 + 0.239997i \(0.922854\pi\)
\(308\) −2.23132 −0.127141
\(309\) 12.3527 0.702719
\(310\) 0 0
\(311\) 8.93800 0.506828 0.253414 0.967358i \(-0.418447\pi\)
0.253414 + 0.967358i \(0.418447\pi\)
\(312\) 0.192688 0.0109088
\(313\) 18.4240 1.04139 0.520693 0.853744i \(-0.325674\pi\)
0.520693 + 0.853744i \(0.325674\pi\)
\(314\) 15.1893 0.857182
\(315\) 0 0
\(316\) −4.17997 −0.235142
\(317\) −33.5547 −1.88462 −0.942310 0.334742i \(-0.891351\pi\)
−0.942310 + 0.334742i \(0.891351\pi\)
\(318\) −11.0993 −0.622419
\(319\) 25.0607 1.40313
\(320\) 0 0
\(321\) 0.122679 0.00684726
\(322\) −3.13201 −0.174540
\(323\) −2.14134 −0.119147
\(324\) −4.27334 −0.237408
\(325\) 0 0
\(326\) −14.0700 −0.779266
\(327\) 18.5340 1.02493
\(328\) 4.23132 0.233636
\(329\) −6.54669 −0.360931
\(330\) 0 0
\(331\) 2.25130 0.123742 0.0618712 0.998084i \(-0.480293\pi\)
0.0618712 + 0.998084i \(0.480293\pi\)
\(332\) 9.00933 0.494451
\(333\) 3.73599 0.204731
\(334\) −14.7967 −0.809638
\(335\) 0 0
\(336\) 0.867993 0.0473529
\(337\) 21.3620 1.16366 0.581831 0.813309i \(-0.302336\pi\)
0.581831 + 0.813309i \(0.302336\pi\)
\(338\) 12.9800 0.706020
\(339\) 15.7101 0.853254
\(340\) 0 0
\(341\) −27.2920 −1.47794
\(342\) 1.14134 0.0617164
\(343\) 8.65533 0.467344
\(344\) 2.49534 0.134539
\(345\) 0 0
\(346\) −17.2920 −0.929624
\(347\) −11.5560 −0.620359 −0.310180 0.950678i \(-0.600389\pi\)
−0.310180 + 0.950678i \(0.600389\pi\)
\(348\) −9.74870 −0.522585
\(349\) −17.1120 −0.915986 −0.457993 0.888956i \(-0.651432\pi\)
−0.457993 + 0.888956i \(0.651432\pi\)
\(350\) 0 0
\(351\) 0.797984 0.0425932
\(352\) −3.50466 −0.186799
\(353\) −11.6974 −0.622587 −0.311294 0.950314i \(-0.600762\pi\)
−0.311294 + 0.950314i \(0.600762\pi\)
\(354\) −7.69735 −0.409110
\(355\) 0 0
\(356\) −6.77801 −0.359234
\(357\) −1.85866 −0.0983709
\(358\) −17.7360 −0.937376
\(359\) −4.47536 −0.236200 −0.118100 0.993002i \(-0.537680\pi\)
−0.118100 + 0.993002i \(0.537680\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −6.17997 −0.324812
\(363\) −1.74870 −0.0917831
\(364\) −0.0899847 −0.00471648
\(365\) 0 0
\(366\) −8.85527 −0.462873
\(367\) −18.7453 −0.978497 −0.489249 0.872144i \(-0.662729\pi\)
−0.489249 + 0.872144i \(0.662729\pi\)
\(368\) −4.91934 −0.256439
\(369\) 4.82936 0.251406
\(370\) 0 0
\(371\) 5.18336 0.269107
\(372\) 10.6167 0.550450
\(373\) −1.69735 −0.0878855 −0.0439428 0.999034i \(-0.513992\pi\)
−0.0439428 + 0.999034i \(0.513992\pi\)
\(374\) 7.50466 0.388057
\(375\) 0 0
\(376\) −10.2827 −0.530288
\(377\) 1.01065 0.0520510
\(378\) 3.59465 0.184889
\(379\) 2.63667 0.135437 0.0677184 0.997704i \(-0.478428\pi\)
0.0677184 + 0.997704i \(0.478428\pi\)
\(380\) 0 0
\(381\) −4.48808 −0.229931
\(382\) −14.6367 −0.748877
\(383\) −12.4953 −0.638482 −0.319241 0.947674i \(-0.603428\pi\)
−0.319241 + 0.947674i \(0.603428\pi\)
\(384\) 1.36333 0.0695721
\(385\) 0 0
\(386\) 20.0187 1.01892
\(387\) 2.84802 0.144773
\(388\) −14.5653 −0.739443
\(389\) −4.51399 −0.228869 −0.114434 0.993431i \(-0.536506\pi\)
−0.114434 + 0.993431i \(0.536506\pi\)
\(390\) 0 0
\(391\) 10.5340 0.532726
\(392\) 6.59465 0.333080
\(393\) −24.5653 −1.23916
\(394\) −9.94865 −0.501206
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 35.6774 1.79060 0.895298 0.445468i \(-0.146963\pi\)
0.895298 + 0.445468i \(0.146963\pi\)
\(398\) −9.74870 −0.488658
\(399\) 0.867993 0.0434540
\(400\) 0 0
\(401\) 15.3434 0.766210 0.383105 0.923705i \(-0.374855\pi\)
0.383105 + 0.923705i \(0.374855\pi\)
\(402\) −11.4147 −0.569312
\(403\) −1.10063 −0.0548264
\(404\) −16.6167 −0.826712
\(405\) 0 0
\(406\) 4.55263 0.225943
\(407\) −11.4720 −0.568644
\(408\) −2.91934 −0.144529
\(409\) 29.3620 1.45186 0.725929 0.687770i \(-0.241410\pi\)
0.725929 + 0.687770i \(0.241410\pi\)
\(410\) 0 0
\(411\) −19.6647 −0.969986
\(412\) −9.06068 −0.446388
\(413\) 3.59465 0.176881
\(414\) −5.61462 −0.275944
\(415\) 0 0
\(416\) −0.141336 −0.00692957
\(417\) −21.0934 −1.03295
\(418\) −3.50466 −0.171419
\(419\) 25.1379 1.22807 0.614035 0.789279i \(-0.289546\pi\)
0.614035 + 0.789279i \(0.289546\pi\)
\(420\) 0 0
\(421\) 14.5454 0.708898 0.354449 0.935075i \(-0.384668\pi\)
0.354449 + 0.935075i \(0.384668\pi\)
\(422\) 20.7580 1.01049
\(423\) −11.7360 −0.570623
\(424\) 8.14134 0.395378
\(425\) 0 0
\(426\) −12.2127 −0.591705
\(427\) 4.13540 0.200126
\(428\) −0.0899847 −0.00434958
\(429\) −0.675305 −0.0326040
\(430\) 0 0
\(431\) 19.4020 0.934560 0.467280 0.884109i \(-0.345234\pi\)
0.467280 + 0.884109i \(0.345234\pi\)
\(432\) 5.64600 0.271643
\(433\) −5.50466 −0.264537 −0.132269 0.991214i \(-0.542226\pi\)
−0.132269 + 0.991214i \(0.542226\pi\)
\(434\) −4.95798 −0.237991
\(435\) 0 0
\(436\) −13.5946 −0.651066
\(437\) −4.91934 −0.235324
\(438\) 5.04070 0.240854
\(439\) 12.2500 0.584660 0.292330 0.956318i \(-0.405570\pi\)
0.292330 + 0.956318i \(0.405570\pi\)
\(440\) 0 0
\(441\) 7.52671 0.358415
\(442\) 0.302648 0.0143955
\(443\) −31.6006 −1.50139 −0.750695 0.660649i \(-0.770281\pi\)
−0.750695 + 0.660649i \(0.770281\pi\)
\(444\) 4.46264 0.211788
\(445\) 0 0
\(446\) −10.7267 −0.507922
\(447\) 23.4347 1.10842
\(448\) −0.636672 −0.0300799
\(449\) −36.0187 −1.69983 −0.849913 0.526923i \(-0.823345\pi\)
−0.849913 + 0.526923i \(0.823345\pi\)
\(450\) 0 0
\(451\) −14.8294 −0.698287
\(452\) −11.5233 −0.542011
\(453\) −4.48808 −0.210868
\(454\) 12.5526 0.589124
\(455\) 0 0
\(456\) 1.36333 0.0638437
\(457\) −22.1413 −1.03573 −0.517864 0.855463i \(-0.673273\pi\)
−0.517864 + 0.855463i \(0.673273\pi\)
\(458\) −25.4720 −1.19023
\(459\) −12.0900 −0.564312
\(460\) 0 0
\(461\) −2.31537 −0.107837 −0.0539187 0.998545i \(-0.517171\pi\)
−0.0539187 + 0.998545i \(0.517171\pi\)
\(462\) −3.04202 −0.141528
\(463\) −15.8387 −0.736086 −0.368043 0.929809i \(-0.619972\pi\)
−0.368043 + 0.929809i \(0.619972\pi\)
\(464\) 7.15066 0.331961
\(465\) 0 0
\(466\) −3.11203 −0.144162
\(467\) −23.1379 −1.07070 −0.535348 0.844631i \(-0.679820\pi\)
−0.535348 + 0.844631i \(0.679820\pi\)
\(468\) −0.161312 −0.00745665
\(469\) 5.33063 0.246146
\(470\) 0 0
\(471\) 20.7080 0.954174
\(472\) 5.64600 0.259878
\(473\) −8.74531 −0.402110
\(474\) −5.69867 −0.261749
\(475\) 0 0
\(476\) 1.36333 0.0624880
\(477\) 9.29200 0.425451
\(478\) 1.54330 0.0705888
\(479\) −10.1214 −0.462457 −0.231228 0.972900i \(-0.574274\pi\)
−0.231228 + 0.972900i \(0.574274\pi\)
\(480\) 0 0
\(481\) −0.462642 −0.0210946
\(482\) −10.2827 −0.468363
\(483\) −4.26995 −0.194290
\(484\) 1.28267 0.0583033
\(485\) 0 0
\(486\) 11.1120 0.504052
\(487\) −20.3854 −0.923750 −0.461875 0.886945i \(-0.652823\pi\)
−0.461875 + 0.886945i \(0.652823\pi\)
\(488\) 6.49534 0.294030
\(489\) −19.1820 −0.867442
\(490\) 0 0
\(491\) −7.78734 −0.351438 −0.175719 0.984440i \(-0.556225\pi\)
−0.175719 + 0.984440i \(0.556225\pi\)
\(492\) 5.76868 0.260072
\(493\) −15.3120 −0.689617
\(494\) −0.141336 −0.00635901
\(495\) 0 0
\(496\) −7.78734 −0.349662
\(497\) 5.70329 0.255828
\(498\) 12.2827 0.550400
\(499\) 4.31537 0.193182 0.0965912 0.995324i \(-0.469206\pi\)
0.0965912 + 0.995324i \(0.469206\pi\)
\(500\) 0 0
\(501\) −20.1727 −0.901250
\(502\) 2.51399 0.112205
\(503\) −18.5526 −0.827221 −0.413610 0.910454i \(-0.635732\pi\)
−0.413610 + 0.910454i \(0.635732\pi\)
\(504\) −0.726656 −0.0323679
\(505\) 0 0
\(506\) 17.2406 0.766440
\(507\) 17.6960 0.785908
\(508\) 3.29200 0.146059
\(509\) −7.73599 −0.342892 −0.171446 0.985194i \(-0.554844\pi\)
−0.171446 + 0.985194i \(0.554844\pi\)
\(510\) 0 0
\(511\) −2.35400 −0.104135
\(512\) −1.00000 −0.0441942
\(513\) 5.64600 0.249277
\(514\) −24.7967 −1.09373
\(515\) 0 0
\(516\) 3.40196 0.149763
\(517\) 36.0373 1.58492
\(518\) −2.08405 −0.0915677
\(519\) −23.5747 −1.03481
\(520\) 0 0
\(521\) 15.2080 0.666273 0.333136 0.942879i \(-0.391893\pi\)
0.333136 + 0.942879i \(0.391893\pi\)
\(522\) 8.16131 0.357211
\(523\) 18.2113 0.796327 0.398163 0.917315i \(-0.369648\pi\)
0.398163 + 0.917315i \(0.369648\pi\)
\(524\) 18.0187 0.787149
\(525\) 0 0
\(526\) −22.5653 −0.983896
\(527\) 16.6753 0.726388
\(528\) −4.77801 −0.207936
\(529\) 1.19995 0.0521715
\(530\) 0 0
\(531\) 6.44398 0.279645
\(532\) −0.636672 −0.0276032
\(533\) −0.598038 −0.0259039
\(534\) −9.24065 −0.399882
\(535\) 0 0
\(536\) 8.37266 0.361644
\(537\) −24.1800 −1.04344
\(538\) 26.5653 1.14531
\(539\) −23.1120 −0.995506
\(540\) 0 0
\(541\) 16.5140 0.709992 0.354996 0.934868i \(-0.384482\pi\)
0.354996 + 0.934868i \(0.384482\pi\)
\(542\) 24.9380 1.07118
\(543\) −8.42533 −0.361565
\(544\) 2.14134 0.0918090
\(545\) 0 0
\(546\) −0.122679 −0.00525016
\(547\) −16.2827 −0.696197 −0.348098 0.937458i \(-0.613172\pi\)
−0.348098 + 0.937458i \(0.613172\pi\)
\(548\) 14.4240 0.616163
\(549\) 7.41336 0.316395
\(550\) 0 0
\(551\) 7.15066 0.304629
\(552\) −6.70668 −0.285455
\(553\) 2.66127 0.113169
\(554\) 18.5467 0.787973
\(555\) 0 0
\(556\) 15.4720 0.656158
\(557\) −37.4533 −1.58695 −0.793474 0.608604i \(-0.791730\pi\)
−0.793474 + 0.608604i \(0.791730\pi\)
\(558\) −8.88797 −0.376258
\(559\) −0.352681 −0.0149168
\(560\) 0 0
\(561\) 10.2313 0.431967
\(562\) −24.7967 −1.04598
\(563\) 29.1307 1.22771 0.613856 0.789418i \(-0.289618\pi\)
0.613856 + 0.789418i \(0.289618\pi\)
\(564\) −14.0187 −0.590292
\(565\) 0 0
\(566\) 13.5747 0.570586
\(567\) 2.72072 0.114259
\(568\) 8.95798 0.375868
\(569\) 14.8480 0.622461 0.311231 0.950334i \(-0.399259\pi\)
0.311231 + 0.950334i \(0.399259\pi\)
\(570\) 0 0
\(571\) 41.9087 1.75382 0.876912 0.480651i \(-0.159599\pi\)
0.876912 + 0.480651i \(0.159599\pi\)
\(572\) 0.495336 0.0207110
\(573\) −19.9546 −0.833615
\(574\) −2.69396 −0.112444
\(575\) 0 0
\(576\) −1.14134 −0.0475557
\(577\) −16.4427 −0.684517 −0.342259 0.939606i \(-0.611192\pi\)
−0.342259 + 0.939606i \(0.611192\pi\)
\(578\) 12.4147 0.516383
\(579\) 27.2920 1.13422
\(580\) 0 0
\(581\) −5.73599 −0.237969
\(582\) −19.8573 −0.823113
\(583\) −28.5327 −1.18170
\(584\) −3.69735 −0.152998
\(585\) 0 0
\(586\) −15.6133 −0.644980
\(587\) 42.5327 1.75551 0.877755 0.479109i \(-0.159040\pi\)
0.877755 + 0.479109i \(0.159040\pi\)
\(588\) 8.99067 0.370769
\(589\) −7.78734 −0.320872
\(590\) 0 0
\(591\) −13.5633 −0.557919
\(592\) −3.27334 −0.134534
\(593\) −3.92273 −0.161087 −0.0805437 0.996751i \(-0.525666\pi\)
−0.0805437 + 0.996751i \(0.525666\pi\)
\(594\) −19.7873 −0.811884
\(595\) 0 0
\(596\) −17.1893 −0.704101
\(597\) −13.2907 −0.543951
\(598\) 0.695281 0.0284322
\(599\) −10.7594 −0.439615 −0.219808 0.975543i \(-0.570543\pi\)
−0.219808 + 0.975543i \(0.570543\pi\)
\(600\) 0 0
\(601\) 25.2220 1.02883 0.514413 0.857542i \(-0.328010\pi\)
0.514413 + 0.857542i \(0.328010\pi\)
\(602\) −1.58871 −0.0647510
\(603\) 9.55602 0.389151
\(604\) 3.29200 0.133950
\(605\) 0 0
\(606\) −22.6540 −0.920256
\(607\) −39.2920 −1.59481 −0.797407 0.603442i \(-0.793795\pi\)
−0.797407 + 0.603442i \(0.793795\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 6.20672 0.251509
\(610\) 0 0
\(611\) 1.45331 0.0587947
\(612\) 2.44398 0.0987922
\(613\) −9.80599 −0.396060 −0.198030 0.980196i \(-0.563454\pi\)
−0.198030 + 0.980196i \(0.563454\pi\)
\(614\) 34.0187 1.37288
\(615\) 0 0
\(616\) 2.23132 0.0899025
\(617\) 35.0093 1.40942 0.704711 0.709494i \(-0.251077\pi\)
0.704711 + 0.709494i \(0.251077\pi\)
\(618\) −12.3527 −0.496898
\(619\) 13.4206 0.539420 0.269710 0.962942i \(-0.413072\pi\)
0.269710 + 0.962942i \(0.413072\pi\)
\(620\) 0 0
\(621\) −27.7746 −1.11456
\(622\) −8.93800 −0.358381
\(623\) 4.31537 0.172891
\(624\) −0.192688 −0.00771367
\(625\) 0 0
\(626\) −18.4240 −0.736371
\(627\) −4.77801 −0.190815
\(628\) −15.1893 −0.606119
\(629\) 7.00933 0.279480
\(630\) 0 0
\(631\) 40.5254 1.61329 0.806645 0.591036i \(-0.201281\pi\)
0.806645 + 0.591036i \(0.201281\pi\)
\(632\) 4.17997 0.166270
\(633\) 28.3000 1.12482
\(634\) 33.5547 1.33263
\(635\) 0 0
\(636\) 11.0993 0.440116
\(637\) −0.932062 −0.0369296
\(638\) −25.0607 −0.992162
\(639\) 10.2241 0.404458
\(640\) 0 0
\(641\) −26.0700 −1.02970 −0.514852 0.857279i \(-0.672153\pi\)
−0.514852 + 0.857279i \(0.672153\pi\)
\(642\) −0.122679 −0.00484174
\(643\) −30.1400 −1.18861 −0.594303 0.804241i \(-0.702572\pi\)
−0.594303 + 0.804241i \(0.702572\pi\)
\(644\) 3.13201 0.123418
\(645\) 0 0
\(646\) 2.14134 0.0842497
\(647\) 20.1086 0.790552 0.395276 0.918562i \(-0.370649\pi\)
0.395276 + 0.918562i \(0.370649\pi\)
\(648\) 4.27334 0.167873
\(649\) −19.7873 −0.776721
\(650\) 0 0
\(651\) −6.75935 −0.264920
\(652\) 14.0700 0.551024
\(653\) 28.0373 1.09718 0.548592 0.836090i \(-0.315164\pi\)
0.548592 + 0.836090i \(0.315164\pi\)
\(654\) −18.5340 −0.724736
\(655\) 0 0
\(656\) −4.23132 −0.165205
\(657\) −4.21992 −0.164635
\(658\) 6.54669 0.255216
\(659\) 4.90069 0.190904 0.0954518 0.995434i \(-0.469570\pi\)
0.0954518 + 0.995434i \(0.469570\pi\)
\(660\) 0 0
\(661\) −8.03863 −0.312667 −0.156333 0.987704i \(-0.549967\pi\)
−0.156333 + 0.987704i \(0.549967\pi\)
\(662\) −2.25130 −0.0874991
\(663\) 0.412609 0.0160244
\(664\) −9.00933 −0.349630
\(665\) 0 0
\(666\) −3.73599 −0.144766
\(667\) −35.1766 −1.36204
\(668\) 14.7967 0.572500
\(669\) −14.6240 −0.565395
\(670\) 0 0
\(671\) −22.7640 −0.878793
\(672\) −0.867993 −0.0334835
\(673\) 4.82936 0.186158 0.0930791 0.995659i \(-0.470329\pi\)
0.0930791 + 0.995659i \(0.470329\pi\)
\(674\) −21.3620 −0.822834
\(675\) 0 0
\(676\) −12.9800 −0.499232
\(677\) −12.8094 −0.492305 −0.246152 0.969231i \(-0.579166\pi\)
−0.246152 + 0.969231i \(0.579166\pi\)
\(678\) −15.7101 −0.603342
\(679\) 9.27334 0.355878
\(680\) 0 0
\(681\) 17.1133 0.655785
\(682\) 27.2920 1.04506
\(683\) −37.1307 −1.42077 −0.710383 0.703815i \(-0.751478\pi\)
−0.710383 + 0.703815i \(0.751478\pi\)
\(684\) −1.14134 −0.0436401
\(685\) 0 0
\(686\) −8.65533 −0.330462
\(687\) −34.7267 −1.32490
\(688\) −2.49534 −0.0951338
\(689\) −1.15066 −0.0438368
\(690\) 0 0
\(691\) −18.1986 −0.692308 −0.346154 0.938178i \(-0.612513\pi\)
−0.346154 + 0.938178i \(0.612513\pi\)
\(692\) 17.2920 0.657343
\(693\) 2.54669 0.0967406
\(694\) 11.5560 0.438660
\(695\) 0 0
\(696\) 9.74870 0.369524
\(697\) 9.06068 0.343198
\(698\) 17.1120 0.647700
\(699\) −4.24272 −0.160474
\(700\) 0 0
\(701\) −26.2827 −0.992683 −0.496341 0.868127i \(-0.665324\pi\)
−0.496341 + 0.868127i \(0.665324\pi\)
\(702\) −0.797984 −0.0301180
\(703\) −3.27334 −0.123456
\(704\) 3.50466 0.132087
\(705\) 0 0
\(706\) 11.6974 0.440236
\(707\) 10.5794 0.397879
\(708\) 7.69735 0.289284
\(709\) −14.9253 −0.560531 −0.280265 0.959923i \(-0.590422\pi\)
−0.280265 + 0.959923i \(0.590422\pi\)
\(710\) 0 0
\(711\) 4.77075 0.178917
\(712\) 6.77801 0.254017
\(713\) 38.3086 1.43467
\(714\) 1.85866 0.0695587
\(715\) 0 0
\(716\) 17.7360 0.662825
\(717\) 2.10402 0.0785761
\(718\) 4.47536 0.167019
\(719\) 32.3327 1.20581 0.602903 0.797814i \(-0.294011\pi\)
0.602903 + 0.797814i \(0.294011\pi\)
\(720\) 0 0
\(721\) 5.76868 0.214837
\(722\) −1.00000 −0.0372161
\(723\) −14.0187 −0.521359
\(724\) 6.17997 0.229677
\(725\) 0 0
\(726\) 1.74870 0.0649005
\(727\) 42.0246 1.55861 0.779303 0.626647i \(-0.215573\pi\)
0.779303 + 0.626647i \(0.215573\pi\)
\(728\) 0.0899847 0.00333506
\(729\) 27.9694 1.03590
\(730\) 0 0
\(731\) 5.34335 0.197631
\(732\) 8.85527 0.327300
\(733\) −26.5840 −0.981903 −0.490951 0.871187i \(-0.663351\pi\)
−0.490951 + 0.871187i \(0.663351\pi\)
\(734\) 18.7453 0.691902
\(735\) 0 0
\(736\) 4.91934 0.181329
\(737\) −29.3434 −1.08088
\(738\) −4.82936 −0.177771
\(739\) 8.14728 0.299702 0.149851 0.988709i \(-0.452121\pi\)
0.149851 + 0.988709i \(0.452121\pi\)
\(740\) 0 0
\(741\) −0.192688 −0.00707855
\(742\) −5.18336 −0.190287
\(743\) −35.8247 −1.31428 −0.657139 0.753769i \(-0.728234\pi\)
−0.657139 + 0.753769i \(0.728234\pi\)
\(744\) −10.6167 −0.389227
\(745\) 0 0
\(746\) 1.69735 0.0621445
\(747\) −10.2827 −0.376223
\(748\) −7.50466 −0.274398
\(749\) 0.0572907 0.00209336
\(750\) 0 0
\(751\) −14.8994 −0.543686 −0.271843 0.962342i \(-0.587633\pi\)
−0.271843 + 0.962342i \(0.587633\pi\)
\(752\) 10.2827 0.374970
\(753\) 3.42740 0.124901
\(754\) −1.01065 −0.0368056
\(755\) 0 0
\(756\) −3.59465 −0.130736
\(757\) 47.7920 1.73703 0.868514 0.495664i \(-0.165075\pi\)
0.868514 + 0.495664i \(0.165075\pi\)
\(758\) −2.63667 −0.0957682
\(759\) 23.5047 0.853165
\(760\) 0 0
\(761\) −38.9053 −1.41032 −0.705158 0.709050i \(-0.749124\pi\)
−0.705158 + 0.709050i \(0.749124\pi\)
\(762\) 4.48808 0.162586
\(763\) 8.65533 0.313344
\(764\) 14.6367 0.529536
\(765\) 0 0
\(766\) 12.4953 0.451475
\(767\) −0.797984 −0.0288135
\(768\) −1.36333 −0.0491949
\(769\) 8.74663 0.315412 0.157706 0.987486i \(-0.449590\pi\)
0.157706 + 0.987486i \(0.449590\pi\)
\(770\) 0 0
\(771\) −33.8060 −1.21749
\(772\) −20.0187 −0.720487
\(773\) 23.4707 0.844181 0.422090 0.906554i \(-0.361297\pi\)
0.422090 + 0.906554i \(0.361297\pi\)
\(774\) −2.84802 −0.102370
\(775\) 0 0
\(776\) 14.5653 0.522865
\(777\) −2.84124 −0.101929
\(778\) 4.51399 0.161834
\(779\) −4.23132 −0.151603
\(780\) 0 0
\(781\) −31.3947 −1.12339
\(782\) −10.5340 −0.376694
\(783\) 40.3727 1.44280
\(784\) −6.59465 −0.235523
\(785\) 0 0
\(786\) 24.5653 0.876217
\(787\) −1.26063 −0.0449364 −0.0224682 0.999748i \(-0.507152\pi\)
−0.0224682 + 0.999748i \(0.507152\pi\)
\(788\) 9.94865 0.354406
\(789\) −30.7640 −1.09523
\(790\) 0 0
\(791\) 7.33657 0.260859
\(792\) 4.00000 0.142134
\(793\) −0.918026 −0.0326000
\(794\) −35.6774 −1.26614
\(795\) 0 0
\(796\) 9.74870 0.345534
\(797\) 38.6481 1.36898 0.684492 0.729020i \(-0.260024\pi\)
0.684492 + 0.729020i \(0.260024\pi\)
\(798\) −0.867993 −0.0307266
\(799\) −22.0187 −0.778964
\(800\) 0 0
\(801\) 7.73599 0.273338
\(802\) −15.3434 −0.541793
\(803\) 12.9580 0.457277
\(804\) 11.4147 0.402565
\(805\) 0 0
\(806\) 1.10063 0.0387681
\(807\) 36.2173 1.27491
\(808\) 16.6167 0.584573
\(809\) 51.6506 1.81594 0.907970 0.419036i \(-0.137632\pi\)
0.907970 + 0.419036i \(0.137632\pi\)
\(810\) 0 0
\(811\) 8.19269 0.287684 0.143842 0.989601i \(-0.454054\pi\)
0.143842 + 0.989601i \(0.454054\pi\)
\(812\) −4.55263 −0.159766
\(813\) 33.9987 1.19239
\(814\) 11.4720 0.402092
\(815\) 0 0
\(816\) 2.91934 0.102197
\(817\) −2.49534 −0.0873007
\(818\) −29.3620 −1.02662
\(819\) 0.102703 0.00358873
\(820\) 0 0
\(821\) −49.9600 −1.74362 −0.871809 0.489846i \(-0.837053\pi\)
−0.871809 + 0.489846i \(0.837053\pi\)
\(822\) 19.6647 0.685884
\(823\) 16.8421 0.587078 0.293539 0.955947i \(-0.405167\pi\)
0.293539 + 0.955947i \(0.405167\pi\)
\(824\) 9.06068 0.315644
\(825\) 0 0
\(826\) −3.59465 −0.125074
\(827\) 41.7487 1.45174 0.725872 0.687829i \(-0.241436\pi\)
0.725872 + 0.687829i \(0.241436\pi\)
\(828\) 5.61462 0.195122
\(829\) −5.98002 −0.207695 −0.103847 0.994593i \(-0.533115\pi\)
−0.103847 + 0.994593i \(0.533115\pi\)
\(830\) 0 0
\(831\) 25.2852 0.877135
\(832\) 0.141336 0.00489995
\(833\) 14.1214 0.489276
\(834\) 21.0934 0.730404
\(835\) 0 0
\(836\) 3.50466 0.121211
\(837\) −43.9673 −1.51973
\(838\) −25.1379 −0.868376
\(839\) 32.0373 1.10605 0.553025 0.833164i \(-0.313473\pi\)
0.553025 + 0.833164i \(0.313473\pi\)
\(840\) 0 0
\(841\) 22.1320 0.763173
\(842\) −14.5454 −0.501267
\(843\) −33.8060 −1.16434
\(844\) −20.7580 −0.714521
\(845\) 0 0
\(846\) 11.7360 0.403491
\(847\) −0.816641 −0.0280601
\(848\) −8.14134 −0.279575
\(849\) 18.5067 0.635150
\(850\) 0 0
\(851\) 16.1027 0.551994
\(852\) 12.2127 0.418399
\(853\) 40.8480 1.39861 0.699305 0.714824i \(-0.253493\pi\)
0.699305 + 0.714824i \(0.253493\pi\)
\(854\) −4.13540 −0.141510
\(855\) 0 0
\(856\) 0.0899847 0.00307561
\(857\) −28.8667 −0.986067 −0.493033 0.870010i \(-0.664112\pi\)
−0.493033 + 0.870010i \(0.664112\pi\)
\(858\) 0.675305 0.0230545
\(859\) −11.5047 −0.392534 −0.196267 0.980550i \(-0.562882\pi\)
−0.196267 + 0.980550i \(0.562882\pi\)
\(860\) 0 0
\(861\) −3.67276 −0.125167
\(862\) −19.4020 −0.660833
\(863\) 31.6074 1.07593 0.537964 0.842968i \(-0.319194\pi\)
0.537964 + 0.842968i \(0.319194\pi\)
\(864\) −5.64600 −0.192081
\(865\) 0 0
\(866\) 5.50466 0.187056
\(867\) 16.9253 0.574813
\(868\) 4.95798 0.168285
\(869\) −14.6494 −0.496947
\(870\) 0 0
\(871\) −1.18336 −0.0400966
\(872\) 13.5946 0.460373
\(873\) 16.6240 0.562636
\(874\) 4.91934 0.166399
\(875\) 0 0
\(876\) −5.04070 −0.170310
\(877\) −12.0641 −0.407375 −0.203687 0.979036i \(-0.565293\pi\)
−0.203687 + 0.979036i \(0.565293\pi\)
\(878\) −12.2500 −0.413417
\(879\) −21.2861 −0.717961
\(880\) 0 0
\(881\) −22.8480 −0.769769 −0.384885 0.922965i \(-0.625759\pi\)
−0.384885 + 0.922965i \(0.625759\pi\)
\(882\) −7.52671 −0.253438
\(883\) 34.1773 1.15016 0.575079 0.818098i \(-0.304971\pi\)
0.575079 + 0.818098i \(0.304971\pi\)
\(884\) −0.302648 −0.0101792
\(885\) 0 0
\(886\) 31.6006 1.06164
\(887\) −26.6167 −0.893701 −0.446851 0.894609i \(-0.647454\pi\)
−0.446851 + 0.894609i \(0.647454\pi\)
\(888\) −4.46264 −0.149756
\(889\) −2.09592 −0.0702950
\(890\) 0 0
\(891\) −14.9766 −0.501736
\(892\) 10.7267 0.359155
\(893\) 10.2827 0.344097
\(894\) −23.4347 −0.783772
\(895\) 0 0
\(896\) 0.636672 0.0212697
\(897\) 0.947896 0.0316493
\(898\) 36.0187 1.20196
\(899\) −55.6846 −1.85719
\(900\) 0 0
\(901\) 17.4333 0.580789
\(902\) 14.8294 0.493764
\(903\) −2.16593 −0.0720777
\(904\) 11.5233 0.383260
\(905\) 0 0
\(906\) 4.48808 0.149106
\(907\) −13.1434 −0.436420 −0.218210 0.975902i \(-0.570022\pi\)
−0.218210 + 0.975902i \(0.570022\pi\)
\(908\) −12.5526 −0.416574
\(909\) 18.9652 0.629037
\(910\) 0 0
\(911\) −32.9253 −1.09086 −0.545432 0.838155i \(-0.683634\pi\)
−0.545432 + 0.838155i \(0.683634\pi\)
\(912\) −1.36333 −0.0451443
\(913\) 31.5747 1.04497
\(914\) 22.1413 0.732370
\(915\) 0 0
\(916\) 25.4720 0.841618
\(917\) −11.4720 −0.378838
\(918\) 12.0900 0.399029
\(919\) −24.1273 −0.795886 −0.397943 0.917410i \(-0.630276\pi\)
−0.397943 + 0.917410i \(0.630276\pi\)
\(920\) 0 0
\(921\) 46.3786 1.52823
\(922\) 2.31537 0.0762525
\(923\) −1.26609 −0.0416737
\(924\) 3.04202 0.100075
\(925\) 0 0
\(926\) 15.8387 0.520492
\(927\) 10.3413 0.339652
\(928\) −7.15066 −0.234732
\(929\) −15.9359 −0.522841 −0.261420 0.965225i \(-0.584191\pi\)
−0.261420 + 0.965225i \(0.584191\pi\)
\(930\) 0 0
\(931\) −6.59465 −0.216131
\(932\) 3.11203 0.101938
\(933\) −12.1854 −0.398933
\(934\) 23.1379 0.757097
\(935\) 0 0
\(936\) 0.161312 0.00527265
\(937\) −23.5547 −0.769498 −0.384749 0.923021i \(-0.625712\pi\)
−0.384749 + 0.923021i \(0.625712\pi\)
\(938\) −5.33063 −0.174051
\(939\) −25.1180 −0.819694
\(940\) 0 0
\(941\) −25.6974 −0.837710 −0.418855 0.908053i \(-0.637568\pi\)
−0.418855 + 0.908053i \(0.637568\pi\)
\(942\) −20.7080 −0.674703
\(943\) 20.8153 0.677840
\(944\) −5.64600 −0.183762
\(945\) 0 0
\(946\) 8.74531 0.284335
\(947\) 16.3013 0.529722 0.264861 0.964287i \(-0.414674\pi\)
0.264861 + 0.964287i \(0.414674\pi\)
\(948\) 5.69867 0.185084
\(949\) 0.522569 0.0169633
\(950\) 0 0
\(951\) 45.7461 1.48342
\(952\) −1.36333 −0.0441857
\(953\) −10.2754 −0.332853 −0.166427 0.986054i \(-0.553223\pi\)
−0.166427 + 0.986054i \(0.553223\pi\)
\(954\) −9.29200 −0.300840
\(955\) 0 0
\(956\) −1.54330 −0.0499138
\(957\) −34.1659 −1.10443
\(958\) 10.1214 0.327006
\(959\) −9.18336 −0.296546
\(960\) 0 0
\(961\) 29.6426 0.956213
\(962\) 0.462642 0.0149162
\(963\) 0.102703 0.00330955
\(964\) 10.2827 0.331183
\(965\) 0 0
\(966\) 4.26995 0.137384
\(967\) 53.2920 1.71376 0.856878 0.515520i \(-0.172401\pi\)
0.856878 + 0.515520i \(0.172401\pi\)
\(968\) −1.28267 −0.0412266
\(969\) 2.91934 0.0937828
\(970\) 0 0
\(971\) 2.54669 0.0817271 0.0408635 0.999165i \(-0.486989\pi\)
0.0408635 + 0.999165i \(0.486989\pi\)
\(972\) −11.1120 −0.356419
\(973\) −9.85057 −0.315795
\(974\) 20.3854 0.653190
\(975\) 0 0
\(976\) −6.49534 −0.207911
\(977\) 49.2966 1.57714 0.788569 0.614946i \(-0.210822\pi\)
0.788569 + 0.614946i \(0.210822\pi\)
\(978\) 19.1820 0.613374
\(979\) −23.7546 −0.759202
\(980\) 0 0
\(981\) 15.5161 0.495390
\(982\) 7.78734 0.248504
\(983\) 36.7453 1.17199 0.585997 0.810313i \(-0.300703\pi\)
0.585997 + 0.810313i \(0.300703\pi\)
\(984\) −5.76868 −0.183899
\(985\) 0 0
\(986\) 15.3120 0.487633
\(987\) 8.92528 0.284095
\(988\) 0.141336 0.00449650
\(989\) 12.2754 0.390335
\(990\) 0 0
\(991\) −35.5674 −1.12984 −0.564918 0.825147i \(-0.691092\pi\)
−0.564918 + 0.825147i \(0.691092\pi\)
\(992\) 7.78734 0.247248
\(993\) −3.06926 −0.0973999
\(994\) −5.70329 −0.180897
\(995\) 0 0
\(996\) −12.2827 −0.389191
\(997\) 48.4299 1.53379 0.766896 0.641771i \(-0.221800\pi\)
0.766896 + 0.641771i \(0.221800\pi\)
\(998\) −4.31537 −0.136601
\(999\) −18.4813 −0.584722
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.i.1.2 3
3.2 odd 2 8550.2.a.cl.1.2 3
4.3 odd 2 7600.2.a.cd.1.2 3
5.2 odd 4 190.2.b.b.39.2 6
5.3 odd 4 190.2.b.b.39.5 yes 6
5.4 even 2 950.2.a.n.1.2 3
15.2 even 4 1710.2.d.d.1369.5 6
15.8 even 4 1710.2.d.d.1369.2 6
15.14 odd 2 8550.2.a.ck.1.2 3
20.3 even 4 1520.2.d.j.609.4 6
20.7 even 4 1520.2.d.j.609.3 6
20.19 odd 2 7600.2.a.bi.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.b.39.2 6 5.2 odd 4
190.2.b.b.39.5 yes 6 5.3 odd 4
950.2.a.i.1.2 3 1.1 even 1 trivial
950.2.a.n.1.2 3 5.4 even 2
1520.2.d.j.609.3 6 20.7 even 4
1520.2.d.j.609.4 6 20.3 even 4
1710.2.d.d.1369.2 6 15.8 even 4
1710.2.d.d.1369.5 6 15.2 even 4
7600.2.a.bi.1.2 3 20.19 odd 2
7600.2.a.cd.1.2 3 4.3 odd 2
8550.2.a.ck.1.2 3 15.14 odd 2
8550.2.a.cl.1.2 3 3.2 odd 2