Properties

Label 7350.2.a.dg.1.2
Level 7350
Weight 2
Character 7350.1
Self dual yes
Analytic conductor 58.690
Analytic rank 0
Dimension 2
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(58.6900454856\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 7350.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.41421 q^{11} +1.00000 q^{12} +3.00000 q^{13} +1.00000 q^{16} +5.82843 q^{17} -1.00000 q^{18} +1.41421 q^{19} -3.41421 q^{22} -0.414214 q^{23} -1.00000 q^{24} -3.00000 q^{26} +1.00000 q^{27} +1.00000 q^{29} -2.41421 q^{31} -1.00000 q^{32} +3.41421 q^{33} -5.82843 q^{34} +1.00000 q^{36} +7.07107 q^{37} -1.41421 q^{38} +3.00000 q^{39} -5.82843 q^{41} +1.58579 q^{43} +3.41421 q^{44} +0.414214 q^{46} -1.65685 q^{47} +1.00000 q^{48} +5.82843 q^{51} +3.00000 q^{52} +12.6569 q^{53} -1.00000 q^{54} +1.41421 q^{57} -1.00000 q^{58} +7.24264 q^{59} -0.171573 q^{61} +2.41421 q^{62} +1.00000 q^{64} -3.41421 q^{66} -15.3137 q^{67} +5.82843 q^{68} -0.414214 q^{69} -9.31371 q^{71} -1.00000 q^{72} +15.0711 q^{73} -7.07107 q^{74} +1.41421 q^{76} -3.00000 q^{78} +1.07107 q^{79} +1.00000 q^{81} +5.82843 q^{82} -0.0710678 q^{83} -1.58579 q^{86} +1.00000 q^{87} -3.41421 q^{88} +4.00000 q^{89} -0.414214 q^{92} -2.41421 q^{93} +1.65685 q^{94} -1.00000 q^{96} -0.343146 q^{97} +3.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 2q^{8} + 2q^{9} + 4q^{11} + 2q^{12} + 6q^{13} + 2q^{16} + 6q^{17} - 2q^{18} - 4q^{22} + 2q^{23} - 2q^{24} - 6q^{26} + 2q^{27} + 2q^{29} - 2q^{31} - 2q^{32} + 4q^{33} - 6q^{34} + 2q^{36} + 6q^{39} - 6q^{41} + 6q^{43} + 4q^{44} - 2q^{46} + 8q^{47} + 2q^{48} + 6q^{51} + 6q^{52} + 14q^{53} - 2q^{54} - 2q^{58} + 6q^{59} - 6q^{61} + 2q^{62} + 2q^{64} - 4q^{66} - 8q^{67} + 6q^{68} + 2q^{69} + 4q^{71} - 2q^{72} + 16q^{73} - 6q^{78} - 12q^{79} + 2q^{81} + 6q^{82} + 14q^{83} - 6q^{86} + 2q^{87} - 4q^{88} + 8q^{89} + 2q^{92} - 2q^{93} - 8q^{94} - 2q^{96} - 12q^{97} + 4q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.41421 1.02942 0.514712 0.857363i \(-0.327899\pi\)
0.514712 + 0.857363i \(0.327899\pi\)
\(12\) 1.00000 0.288675
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.82843 1.41360 0.706801 0.707413i \(-0.250138\pi\)
0.706801 + 0.707413i \(0.250138\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.41421 0.324443 0.162221 0.986754i \(-0.448134\pi\)
0.162221 + 0.986754i \(0.448134\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.41421 −0.727913
\(23\) −0.414214 −0.0863695 −0.0431847 0.999067i \(-0.513750\pi\)
−0.0431847 + 0.999067i \(0.513750\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −3.00000 −0.588348
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) −2.41421 −0.433606 −0.216803 0.976215i \(-0.569563\pi\)
−0.216803 + 0.976215i \(0.569563\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.41421 0.594338
\(34\) −5.82843 −0.999567
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 7.07107 1.16248 0.581238 0.813733i \(-0.302568\pi\)
0.581238 + 0.813733i \(0.302568\pi\)
\(38\) −1.41421 −0.229416
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) −5.82843 −0.910247 −0.455124 0.890428i \(-0.650405\pi\)
−0.455124 + 0.890428i \(0.650405\pi\)
\(42\) 0 0
\(43\) 1.58579 0.241830 0.120915 0.992663i \(-0.461417\pi\)
0.120915 + 0.992663i \(0.461417\pi\)
\(44\) 3.41421 0.514712
\(45\) 0 0
\(46\) 0.414214 0.0610725
\(47\) −1.65685 −0.241677 −0.120839 0.992672i \(-0.538558\pi\)
−0.120839 + 0.992672i \(0.538558\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 0 0
\(51\) 5.82843 0.816143
\(52\) 3.00000 0.416025
\(53\) 12.6569 1.73855 0.869276 0.494326i \(-0.164585\pi\)
0.869276 + 0.494326i \(0.164585\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 1.41421 0.187317
\(58\) −1.00000 −0.131306
\(59\) 7.24264 0.942912 0.471456 0.881890i \(-0.343729\pi\)
0.471456 + 0.881890i \(0.343729\pi\)
\(60\) 0 0
\(61\) −0.171573 −0.0219677 −0.0109838 0.999940i \(-0.503496\pi\)
−0.0109838 + 0.999940i \(0.503496\pi\)
\(62\) 2.41421 0.306605
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.41421 −0.420261
\(67\) −15.3137 −1.87087 −0.935434 0.353502i \(-0.884991\pi\)
−0.935434 + 0.353502i \(0.884991\pi\)
\(68\) 5.82843 0.706801
\(69\) −0.414214 −0.0498655
\(70\) 0 0
\(71\) −9.31371 −1.10533 −0.552667 0.833402i \(-0.686390\pi\)
−0.552667 + 0.833402i \(0.686390\pi\)
\(72\) −1.00000 −0.117851
\(73\) 15.0711 1.76394 0.881968 0.471310i \(-0.156219\pi\)
0.881968 + 0.471310i \(0.156219\pi\)
\(74\) −7.07107 −0.821995
\(75\) 0 0
\(76\) 1.41421 0.162221
\(77\) 0 0
\(78\) −3.00000 −0.339683
\(79\) 1.07107 0.120505 0.0602523 0.998183i \(-0.480809\pi\)
0.0602523 + 0.998183i \(0.480809\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 5.82843 0.643642
\(83\) −0.0710678 −0.00780071 −0.00390035 0.999992i \(-0.501242\pi\)
−0.00390035 + 0.999992i \(0.501242\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.58579 −0.171000
\(87\) 1.00000 0.107211
\(88\) −3.41421 −0.363956
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.414214 −0.0431847
\(93\) −2.41421 −0.250342
\(94\) 1.65685 0.170891
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −0.343146 −0.0348412 −0.0174206 0.999848i \(-0.505545\pi\)
−0.0174206 + 0.999848i \(0.505545\pi\)
\(98\) 0 0
\(99\) 3.41421 0.343141
\(100\) 0 0
\(101\) 5.65685 0.562878 0.281439 0.959579i \(-0.409188\pi\)
0.281439 + 0.959579i \(0.409188\pi\)
\(102\) −5.82843 −0.577100
\(103\) −0.0710678 −0.00700252 −0.00350126 0.999994i \(-0.501114\pi\)
−0.00350126 + 0.999994i \(0.501114\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) −12.6569 −1.22934
\(107\) 6.48528 0.626956 0.313478 0.949595i \(-0.398506\pi\)
0.313478 + 0.949595i \(0.398506\pi\)
\(108\) 1.00000 0.0962250
\(109\) −11.3137 −1.08366 −0.541828 0.840489i \(-0.682268\pi\)
−0.541828 + 0.840489i \(0.682268\pi\)
\(110\) 0 0
\(111\) 7.07107 0.671156
\(112\) 0 0
\(113\) 6.58579 0.619539 0.309769 0.950812i \(-0.399748\pi\)
0.309769 + 0.950812i \(0.399748\pi\)
\(114\) −1.41421 −0.132453
\(115\) 0 0
\(116\) 1.00000 0.0928477
\(117\) 3.00000 0.277350
\(118\) −7.24264 −0.666739
\(119\) 0 0
\(120\) 0 0
\(121\) 0.656854 0.0597140
\(122\) 0.171573 0.0155335
\(123\) −5.82843 −0.525532
\(124\) −2.41421 −0.216803
\(125\) 0 0
\(126\) 0 0
\(127\) −3.17157 −0.281432 −0.140716 0.990050i \(-0.544940\pi\)
−0.140716 + 0.990050i \(0.544940\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.58579 0.139621
\(130\) 0 0
\(131\) −18.8284 −1.64505 −0.822524 0.568731i \(-0.807435\pi\)
−0.822524 + 0.568731i \(0.807435\pi\)
\(132\) 3.41421 0.297169
\(133\) 0 0
\(134\) 15.3137 1.32290
\(135\) 0 0
\(136\) −5.82843 −0.499784
\(137\) −23.2132 −1.98324 −0.991619 0.129197i \(-0.958760\pi\)
−0.991619 + 0.129197i \(0.958760\pi\)
\(138\) 0.414214 0.0352602
\(139\) −15.5563 −1.31947 −0.659736 0.751497i \(-0.729332\pi\)
−0.659736 + 0.751497i \(0.729332\pi\)
\(140\) 0 0
\(141\) −1.65685 −0.139532
\(142\) 9.31371 0.781589
\(143\) 10.2426 0.856533
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −15.0711 −1.24729
\(147\) 0 0
\(148\) 7.07107 0.581238
\(149\) 4.65685 0.381504 0.190752 0.981638i \(-0.438907\pi\)
0.190752 + 0.981638i \(0.438907\pi\)
\(150\) 0 0
\(151\) 7.89949 0.642852 0.321426 0.946935i \(-0.395838\pi\)
0.321426 + 0.946935i \(0.395838\pi\)
\(152\) −1.41421 −0.114708
\(153\) 5.82843 0.471200
\(154\) 0 0
\(155\) 0 0
\(156\) 3.00000 0.240192
\(157\) −19.7990 −1.58013 −0.790066 0.613022i \(-0.789954\pi\)
−0.790066 + 0.613022i \(0.789954\pi\)
\(158\) −1.07107 −0.0852096
\(159\) 12.6569 1.00375
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 5.24264 0.410635 0.205318 0.978695i \(-0.434177\pi\)
0.205318 + 0.978695i \(0.434177\pi\)
\(164\) −5.82843 −0.455124
\(165\) 0 0
\(166\) 0.0710678 0.00551593
\(167\) −8.82843 −0.683164 −0.341582 0.939852i \(-0.610963\pi\)
−0.341582 + 0.939852i \(0.610963\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 1.41421 0.108148
\(172\) 1.58579 0.120915
\(173\) 14.2426 1.08285 0.541424 0.840750i \(-0.317885\pi\)
0.541424 + 0.840750i \(0.317885\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 0 0
\(176\) 3.41421 0.257356
\(177\) 7.24264 0.544390
\(178\) −4.00000 −0.299813
\(179\) −5.07107 −0.379029 −0.189515 0.981878i \(-0.560691\pi\)
−0.189515 + 0.981878i \(0.560691\pi\)
\(180\) 0 0
\(181\) −23.3137 −1.73289 −0.866447 0.499269i \(-0.833602\pi\)
−0.866447 + 0.499269i \(0.833602\pi\)
\(182\) 0 0
\(183\) −0.171573 −0.0126830
\(184\) 0.414214 0.0305362
\(185\) 0 0
\(186\) 2.41421 0.177019
\(187\) 19.8995 1.45520
\(188\) −1.65685 −0.120839
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0711 1.16286 0.581431 0.813596i \(-0.302493\pi\)
0.581431 + 0.813596i \(0.302493\pi\)
\(192\) 1.00000 0.0721688
\(193\) −6.82843 −0.491521 −0.245760 0.969331i \(-0.579038\pi\)
−0.245760 + 0.969331i \(0.579038\pi\)
\(194\) 0.343146 0.0246364
\(195\) 0 0
\(196\) 0 0
\(197\) 23.1421 1.64881 0.824404 0.566001i \(-0.191510\pi\)
0.824404 + 0.566001i \(0.191510\pi\)
\(198\) −3.41421 −0.242638
\(199\) 19.6569 1.39344 0.696719 0.717344i \(-0.254643\pi\)
0.696719 + 0.717344i \(0.254643\pi\)
\(200\) 0 0
\(201\) −15.3137 −1.08015
\(202\) −5.65685 −0.398015
\(203\) 0 0
\(204\) 5.82843 0.408072
\(205\) 0 0
\(206\) 0.0710678 0.00495153
\(207\) −0.414214 −0.0287898
\(208\) 3.00000 0.208013
\(209\) 4.82843 0.333989
\(210\) 0 0
\(211\) −26.5563 −1.82821 −0.914107 0.405473i \(-0.867107\pi\)
−0.914107 + 0.405473i \(0.867107\pi\)
\(212\) 12.6569 0.869276
\(213\) −9.31371 −0.638165
\(214\) −6.48528 −0.443325
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 11.3137 0.766261
\(219\) 15.0711 1.01841
\(220\) 0 0
\(221\) 17.4853 1.17619
\(222\) −7.07107 −0.474579
\(223\) 24.5563 1.64441 0.822207 0.569188i \(-0.192742\pi\)
0.822207 + 0.569188i \(0.192742\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −6.58579 −0.438080
\(227\) 25.3848 1.68485 0.842423 0.538816i \(-0.181128\pi\)
0.842423 + 0.538816i \(0.181128\pi\)
\(228\) 1.41421 0.0936586
\(229\) 29.3137 1.93710 0.968552 0.248810i \(-0.0800396\pi\)
0.968552 + 0.248810i \(0.0800396\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) 14.2426 0.933066 0.466533 0.884504i \(-0.345503\pi\)
0.466533 + 0.884504i \(0.345503\pi\)
\(234\) −3.00000 −0.196116
\(235\) 0 0
\(236\) 7.24264 0.471456
\(237\) 1.07107 0.0695733
\(238\) 0 0
\(239\) −11.3137 −0.731823 −0.365911 0.930650i \(-0.619243\pi\)
−0.365911 + 0.930650i \(0.619243\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) −0.656854 −0.0422242
\(243\) 1.00000 0.0641500
\(244\) −0.171573 −0.0109838
\(245\) 0 0
\(246\) 5.82843 0.371607
\(247\) 4.24264 0.269953
\(248\) 2.41421 0.153303
\(249\) −0.0710678 −0.00450374
\(250\) 0 0
\(251\) −1.92893 −0.121753 −0.0608766 0.998145i \(-0.519390\pi\)
−0.0608766 + 0.998145i \(0.519390\pi\)
\(252\) 0 0
\(253\) −1.41421 −0.0889108
\(254\) 3.17157 0.199002
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 13.3431 0.832323 0.416161 0.909291i \(-0.363375\pi\)
0.416161 + 0.909291i \(0.363375\pi\)
\(258\) −1.58579 −0.0987268
\(259\) 0 0
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 18.8284 1.16322
\(263\) −10.5563 −0.650932 −0.325466 0.945554i \(-0.605521\pi\)
−0.325466 + 0.945554i \(0.605521\pi\)
\(264\) −3.41421 −0.210130
\(265\) 0 0
\(266\) 0 0
\(267\) 4.00000 0.244796
\(268\) −15.3137 −0.935434
\(269\) 0.443651 0.0270499 0.0135249 0.999909i \(-0.495695\pi\)
0.0135249 + 0.999909i \(0.495695\pi\)
\(270\) 0 0
\(271\) −11.5147 −0.699469 −0.349735 0.936849i \(-0.613728\pi\)
−0.349735 + 0.936849i \(0.613728\pi\)
\(272\) 5.82843 0.353400
\(273\) 0 0
\(274\) 23.2132 1.40236
\(275\) 0 0
\(276\) −0.414214 −0.0249327
\(277\) −24.0416 −1.44452 −0.722261 0.691621i \(-0.756897\pi\)
−0.722261 + 0.691621i \(0.756897\pi\)
\(278\) 15.5563 0.933008
\(279\) −2.41421 −0.144535
\(280\) 0 0
\(281\) 20.0000 1.19310 0.596550 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(282\) 1.65685 0.0986642
\(283\) −0.242641 −0.0144235 −0.00721175 0.999974i \(-0.502296\pi\)
−0.00721175 + 0.999974i \(0.502296\pi\)
\(284\) −9.31371 −0.552667
\(285\) 0 0
\(286\) −10.2426 −0.605660
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 16.9706 0.998268
\(290\) 0 0
\(291\) −0.343146 −0.0201156
\(292\) 15.0711 0.881968
\(293\) −11.3137 −0.660954 −0.330477 0.943814i \(-0.607210\pi\)
−0.330477 + 0.943814i \(0.607210\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −7.07107 −0.410997
\(297\) 3.41421 0.198113
\(298\) −4.65685 −0.269764
\(299\) −1.24264 −0.0718638
\(300\) 0 0
\(301\) 0 0
\(302\) −7.89949 −0.454565
\(303\) 5.65685 0.324978
\(304\) 1.41421 0.0811107
\(305\) 0 0
\(306\) −5.82843 −0.333189
\(307\) 8.58579 0.490017 0.245008 0.969521i \(-0.421209\pi\)
0.245008 + 0.969521i \(0.421209\pi\)
\(308\) 0 0
\(309\) −0.0710678 −0.00404291
\(310\) 0 0
\(311\) −11.4142 −0.647241 −0.323620 0.946187i \(-0.604900\pi\)
−0.323620 + 0.946187i \(0.604900\pi\)
\(312\) −3.00000 −0.169842
\(313\) −16.0416 −0.906727 −0.453363 0.891326i \(-0.649776\pi\)
−0.453363 + 0.891326i \(0.649776\pi\)
\(314\) 19.7990 1.11732
\(315\) 0 0
\(316\) 1.07107 0.0602523
\(317\) 19.9706 1.12166 0.560829 0.827931i \(-0.310482\pi\)
0.560829 + 0.827931i \(0.310482\pi\)
\(318\) −12.6569 −0.709761
\(319\) 3.41421 0.191159
\(320\) 0 0
\(321\) 6.48528 0.361973
\(322\) 0 0
\(323\) 8.24264 0.458633
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −5.24264 −0.290363
\(327\) −11.3137 −0.625650
\(328\) 5.82843 0.321821
\(329\) 0 0
\(330\) 0 0
\(331\) −24.0711 −1.32307 −0.661533 0.749916i \(-0.730094\pi\)
−0.661533 + 0.749916i \(0.730094\pi\)
\(332\) −0.0710678 −0.00390035
\(333\) 7.07107 0.387492
\(334\) 8.82843 0.483070
\(335\) 0 0
\(336\) 0 0
\(337\) 4.79899 0.261418 0.130709 0.991421i \(-0.458275\pi\)
0.130709 + 0.991421i \(0.458275\pi\)
\(338\) 4.00000 0.217571
\(339\) 6.58579 0.357691
\(340\) 0 0
\(341\) −8.24264 −0.446364
\(342\) −1.41421 −0.0764719
\(343\) 0 0
\(344\) −1.58579 −0.0854999
\(345\) 0 0
\(346\) −14.2426 −0.765689
\(347\) −9.79899 −0.526037 −0.263019 0.964791i \(-0.584718\pi\)
−0.263019 + 0.964791i \(0.584718\pi\)
\(348\) 1.00000 0.0536056
\(349\) 24.6569 1.31985 0.659926 0.751331i \(-0.270588\pi\)
0.659926 + 0.751331i \(0.270588\pi\)
\(350\) 0 0
\(351\) 3.00000 0.160128
\(352\) −3.41421 −0.181978
\(353\) 20.4853 1.09032 0.545161 0.838332i \(-0.316469\pi\)
0.545161 + 0.838332i \(0.316469\pi\)
\(354\) −7.24264 −0.384942
\(355\) 0 0
\(356\) 4.00000 0.212000
\(357\) 0 0
\(358\) 5.07107 0.268014
\(359\) 33.8701 1.78759 0.893797 0.448472i \(-0.148032\pi\)
0.893797 + 0.448472i \(0.148032\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 23.3137 1.22534
\(363\) 0.656854 0.0344759
\(364\) 0 0
\(365\) 0 0
\(366\) 0.171573 0.00896826
\(367\) 0.757359 0.0395338 0.0197669 0.999805i \(-0.493708\pi\)
0.0197669 + 0.999805i \(0.493708\pi\)
\(368\) −0.414214 −0.0215924
\(369\) −5.82843 −0.303416
\(370\) 0 0
\(371\) 0 0
\(372\) −2.41421 −0.125171
\(373\) 2.34315 0.121323 0.0606617 0.998158i \(-0.480679\pi\)
0.0606617 + 0.998158i \(0.480679\pi\)
\(374\) −19.8995 −1.02898
\(375\) 0 0
\(376\) 1.65685 0.0854457
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) −34.2132 −1.75741 −0.878707 0.477361i \(-0.841593\pi\)
−0.878707 + 0.477361i \(0.841593\pi\)
\(380\) 0 0
\(381\) −3.17157 −0.162485
\(382\) −16.0711 −0.822267
\(383\) 27.5563 1.40806 0.704032 0.710168i \(-0.251381\pi\)
0.704032 + 0.710168i \(0.251381\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 6.82843 0.347558
\(387\) 1.58579 0.0806101
\(388\) −0.343146 −0.0174206
\(389\) −10.3431 −0.524418 −0.262209 0.965011i \(-0.584451\pi\)
−0.262209 + 0.965011i \(0.584451\pi\)
\(390\) 0 0
\(391\) −2.41421 −0.122092
\(392\) 0 0
\(393\) −18.8284 −0.949769
\(394\) −23.1421 −1.16588
\(395\) 0 0
\(396\) 3.41421 0.171571
\(397\) 30.3137 1.52140 0.760701 0.649103i \(-0.224856\pi\)
0.760701 + 0.649103i \(0.224856\pi\)
\(398\) −19.6569 −0.985309
\(399\) 0 0
\(400\) 0 0
\(401\) 12.1005 0.604270 0.302135 0.953265i \(-0.402301\pi\)
0.302135 + 0.953265i \(0.402301\pi\)
\(402\) 15.3137 0.763778
\(403\) −7.24264 −0.360782
\(404\) 5.65685 0.281439
\(405\) 0 0
\(406\) 0 0
\(407\) 24.1421 1.19668
\(408\) −5.82843 −0.288550
\(409\) 22.5858 1.11680 0.558398 0.829573i \(-0.311416\pi\)
0.558398 + 0.829573i \(0.311416\pi\)
\(410\) 0 0
\(411\) −23.2132 −1.14502
\(412\) −0.0710678 −0.00350126
\(413\) 0 0
\(414\) 0.414214 0.0203575
\(415\) 0 0
\(416\) −3.00000 −0.147087
\(417\) −15.5563 −0.761798
\(418\) −4.82843 −0.236166
\(419\) 24.5563 1.19966 0.599828 0.800129i \(-0.295236\pi\)
0.599828 + 0.800129i \(0.295236\pi\)
\(420\) 0 0
\(421\) −35.6569 −1.73781 −0.868904 0.494980i \(-0.835175\pi\)
−0.868904 + 0.494980i \(0.835175\pi\)
\(422\) 26.5563 1.29274
\(423\) −1.65685 −0.0805590
\(424\) −12.6569 −0.614671
\(425\) 0 0
\(426\) 9.31371 0.451251
\(427\) 0 0
\(428\) 6.48528 0.313478
\(429\) 10.2426 0.494519
\(430\) 0 0
\(431\) 33.0416 1.59156 0.795780 0.605586i \(-0.207061\pi\)
0.795780 + 0.605586i \(0.207061\pi\)
\(432\) 1.00000 0.0481125
\(433\) 28.6274 1.37575 0.687873 0.725831i \(-0.258545\pi\)
0.687873 + 0.725831i \(0.258545\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −11.3137 −0.541828
\(437\) −0.585786 −0.0280220
\(438\) −15.0711 −0.720123
\(439\) 1.58579 0.0756855 0.0378427 0.999284i \(-0.487951\pi\)
0.0378427 + 0.999284i \(0.487951\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −17.4853 −0.831690
\(443\) −14.3431 −0.681463 −0.340732 0.940161i \(-0.610675\pi\)
−0.340732 + 0.940161i \(0.610675\pi\)
\(444\) 7.07107 0.335578
\(445\) 0 0
\(446\) −24.5563 −1.16278
\(447\) 4.65685 0.220262
\(448\) 0 0
\(449\) −25.0711 −1.18318 −0.591588 0.806240i \(-0.701499\pi\)
−0.591588 + 0.806240i \(0.701499\pi\)
\(450\) 0 0
\(451\) −19.8995 −0.937031
\(452\) 6.58579 0.309769
\(453\) 7.89949 0.371151
\(454\) −25.3848 −1.19137
\(455\) 0 0
\(456\) −1.41421 −0.0662266
\(457\) −3.82843 −0.179086 −0.0895431 0.995983i \(-0.528541\pi\)
−0.0895431 + 0.995983i \(0.528541\pi\)
\(458\) −29.3137 −1.36974
\(459\) 5.82843 0.272048
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 0 0
\(463\) 29.8995 1.38955 0.694774 0.719228i \(-0.255505\pi\)
0.694774 + 0.719228i \(0.255505\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) −14.2426 −0.659778
\(467\) −1.24264 −0.0575026 −0.0287513 0.999587i \(-0.509153\pi\)
−0.0287513 + 0.999587i \(0.509153\pi\)
\(468\) 3.00000 0.138675
\(469\) 0 0
\(470\) 0 0
\(471\) −19.7990 −0.912289
\(472\) −7.24264 −0.333370
\(473\) 5.41421 0.248946
\(474\) −1.07107 −0.0491958
\(475\) 0 0
\(476\) 0 0
\(477\) 12.6569 0.579518
\(478\) 11.3137 0.517477
\(479\) 20.2426 0.924910 0.462455 0.886643i \(-0.346969\pi\)
0.462455 + 0.886643i \(0.346969\pi\)
\(480\) 0 0
\(481\) 21.2132 0.967239
\(482\) 6.00000 0.273293
\(483\) 0 0
\(484\) 0.656854 0.0298570
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 5.51472 0.249896 0.124948 0.992163i \(-0.460124\pi\)
0.124948 + 0.992163i \(0.460124\pi\)
\(488\) 0.171573 0.00776674
\(489\) 5.24264 0.237080
\(490\) 0 0
\(491\) −0.142136 −0.00641449 −0.00320725 0.999995i \(-0.501021\pi\)
−0.00320725 + 0.999995i \(0.501021\pi\)
\(492\) −5.82843 −0.262766
\(493\) 5.82843 0.262499
\(494\) −4.24264 −0.190885
\(495\) 0 0
\(496\) −2.41421 −0.108401
\(497\) 0 0
\(498\) 0.0710678 0.00318462
\(499\) −26.0711 −1.16710 −0.583551 0.812077i \(-0.698337\pi\)
−0.583551 + 0.812077i \(0.698337\pi\)
\(500\) 0 0
\(501\) −8.82843 −0.394425
\(502\) 1.92893 0.0860925
\(503\) −24.8701 −1.10890 −0.554451 0.832217i \(-0.687072\pi\)
−0.554451 + 0.832217i \(0.687072\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.41421 0.0628695
\(507\) −4.00000 −0.177646
\(508\) −3.17157 −0.140716
\(509\) −6.04163 −0.267791 −0.133895 0.990995i \(-0.542749\pi\)
−0.133895 + 0.990995i \(0.542749\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 1.41421 0.0624391
\(514\) −13.3431 −0.588541
\(515\) 0 0
\(516\) 1.58579 0.0698104
\(517\) −5.65685 −0.248788
\(518\) 0 0
\(519\) 14.2426 0.625183
\(520\) 0 0
\(521\) −10.6569 −0.466885 −0.233443 0.972371i \(-0.574999\pi\)
−0.233443 + 0.972371i \(0.574999\pi\)
\(522\) −1.00000 −0.0437688
\(523\) −24.2843 −1.06188 −0.530939 0.847410i \(-0.678160\pi\)
−0.530939 + 0.847410i \(0.678160\pi\)
\(524\) −18.8284 −0.822524
\(525\) 0 0
\(526\) 10.5563 0.460279
\(527\) −14.0711 −0.612945
\(528\) 3.41421 0.148585
\(529\) −22.8284 −0.992540
\(530\) 0 0
\(531\) 7.24264 0.314304
\(532\) 0 0
\(533\) −17.4853 −0.757372
\(534\) −4.00000 −0.173097
\(535\) 0 0
\(536\) 15.3137 0.661451
\(537\) −5.07107 −0.218833
\(538\) −0.443651 −0.0191271
\(539\) 0 0
\(540\) 0 0
\(541\) −34.2426 −1.47221 −0.736103 0.676869i \(-0.763336\pi\)
−0.736103 + 0.676869i \(0.763336\pi\)
\(542\) 11.5147 0.494600
\(543\) −23.3137 −1.00049
\(544\) −5.82843 −0.249892
\(545\) 0 0
\(546\) 0 0
\(547\) 33.8701 1.44818 0.724090 0.689706i \(-0.242260\pi\)
0.724090 + 0.689706i \(0.242260\pi\)
\(548\) −23.2132 −0.991619
\(549\) −0.171573 −0.00732255
\(550\) 0 0
\(551\) 1.41421 0.0602475
\(552\) 0.414214 0.0176301
\(553\) 0 0
\(554\) 24.0416 1.02143
\(555\) 0 0
\(556\) −15.5563 −0.659736
\(557\) −19.3137 −0.818348 −0.409174 0.912456i \(-0.634183\pi\)
−0.409174 + 0.912456i \(0.634183\pi\)
\(558\) 2.41421 0.102202
\(559\) 4.75736 0.201215
\(560\) 0 0
\(561\) 19.8995 0.840157
\(562\) −20.0000 −0.843649
\(563\) 27.3848 1.15413 0.577065 0.816698i \(-0.304198\pi\)
0.577065 + 0.816698i \(0.304198\pi\)
\(564\) −1.65685 −0.0697661
\(565\) 0 0
\(566\) 0.242641 0.0101989
\(567\) 0 0
\(568\) 9.31371 0.390795
\(569\) −28.2426 −1.18399 −0.591997 0.805941i \(-0.701660\pi\)
−0.591997 + 0.805941i \(0.701660\pi\)
\(570\) 0 0
\(571\) 36.0122 1.50706 0.753532 0.657412i \(-0.228349\pi\)
0.753532 + 0.657412i \(0.228349\pi\)
\(572\) 10.2426 0.428266
\(573\) 16.0711 0.671378
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 19.6569 0.818326 0.409163 0.912461i \(-0.365821\pi\)
0.409163 + 0.912461i \(0.365821\pi\)
\(578\) −16.9706 −0.705882
\(579\) −6.82843 −0.283780
\(580\) 0 0
\(581\) 0 0
\(582\) 0.343146 0.0142238
\(583\) 43.2132 1.78971
\(584\) −15.0711 −0.623645
\(585\) 0 0
\(586\) 11.3137 0.467365
\(587\) −41.5269 −1.71400 −0.857000 0.515317i \(-0.827674\pi\)
−0.857000 + 0.515317i \(0.827674\pi\)
\(588\) 0 0
\(589\) −3.41421 −0.140680
\(590\) 0 0
\(591\) 23.1421 0.951940
\(592\) 7.07107 0.290619
\(593\) −26.0000 −1.06769 −0.533846 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) −3.41421 −0.140087
\(595\) 0 0
\(596\) 4.65685 0.190752
\(597\) 19.6569 0.804501
\(598\) 1.24264 0.0508154
\(599\) −30.4142 −1.24269 −0.621346 0.783537i \(-0.713414\pi\)
−0.621346 + 0.783537i \(0.713414\pi\)
\(600\) 0 0
\(601\) 1.27208 0.0518891 0.0259446 0.999663i \(-0.491741\pi\)
0.0259446 + 0.999663i \(0.491741\pi\)
\(602\) 0 0
\(603\) −15.3137 −0.623622
\(604\) 7.89949 0.321426
\(605\) 0 0
\(606\) −5.65685 −0.229794
\(607\) 39.9411 1.62116 0.810580 0.585628i \(-0.199152\pi\)
0.810580 + 0.585628i \(0.199152\pi\)
\(608\) −1.41421 −0.0573539
\(609\) 0 0
\(610\) 0 0
\(611\) −4.97056 −0.201087
\(612\) 5.82843 0.235600
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) −8.58579 −0.346494
\(615\) 0 0
\(616\) 0 0
\(617\) 22.3848 0.901177 0.450589 0.892732i \(-0.351214\pi\)
0.450589 + 0.892732i \(0.351214\pi\)
\(618\) 0.0710678 0.00285877
\(619\) −12.1421 −0.488034 −0.244017 0.969771i \(-0.578465\pi\)
−0.244017 + 0.969771i \(0.578465\pi\)
\(620\) 0 0
\(621\) −0.414214 −0.0166218
\(622\) 11.4142 0.457668
\(623\) 0 0
\(624\) 3.00000 0.120096
\(625\) 0 0
\(626\) 16.0416 0.641153
\(627\) 4.82843 0.192829
\(628\) −19.7990 −0.790066
\(629\) 41.2132 1.64328
\(630\) 0 0
\(631\) −40.8284 −1.62535 −0.812677 0.582714i \(-0.801991\pi\)
−0.812677 + 0.582714i \(0.801991\pi\)
\(632\) −1.07107 −0.0426048
\(633\) −26.5563 −1.05552
\(634\) −19.9706 −0.793132
\(635\) 0 0
\(636\) 12.6569 0.501877
\(637\) 0 0
\(638\) −3.41421 −0.135170
\(639\) −9.31371 −0.368445
\(640\) 0 0
\(641\) 31.7574 1.25434 0.627170 0.778882i \(-0.284213\pi\)
0.627170 + 0.778882i \(0.284213\pi\)
\(642\) −6.48528 −0.255954
\(643\) −13.7574 −0.542537 −0.271269 0.962504i \(-0.587443\pi\)
−0.271269 + 0.962504i \(0.587443\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.24264 −0.324302
\(647\) 4.87006 0.191462 0.0957309 0.995407i \(-0.469481\pi\)
0.0957309 + 0.995407i \(0.469481\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 24.7279 0.970656
\(650\) 0 0
\(651\) 0 0
\(652\) 5.24264 0.205318
\(653\) −12.6863 −0.496453 −0.248226 0.968702i \(-0.579848\pi\)
−0.248226 + 0.968702i \(0.579848\pi\)
\(654\) 11.3137 0.442401
\(655\) 0 0
\(656\) −5.82843 −0.227562
\(657\) 15.0711 0.587978
\(658\) 0 0
\(659\) 3.17157 0.123547 0.0617735 0.998090i \(-0.480324\pi\)
0.0617735 + 0.998090i \(0.480324\pi\)
\(660\) 0 0
\(661\) 34.3431 1.33579 0.667897 0.744254i \(-0.267195\pi\)
0.667897 + 0.744254i \(0.267195\pi\)
\(662\) 24.0711 0.935549
\(663\) 17.4853 0.679072
\(664\) 0.0710678 0.00275797
\(665\) 0 0
\(666\) −7.07107 −0.273998
\(667\) −0.414214 −0.0160384
\(668\) −8.82843 −0.341582
\(669\) 24.5563 0.949403
\(670\) 0 0
\(671\) −0.585786 −0.0226140
\(672\) 0 0
\(673\) −0.798990 −0.0307988 −0.0153994 0.999881i \(-0.504902\pi\)
−0.0153994 + 0.999881i \(0.504902\pi\)
\(674\) −4.79899 −0.184850
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) −4.38478 −0.168521 −0.0842603 0.996444i \(-0.526853\pi\)
−0.0842603 + 0.996444i \(0.526853\pi\)
\(678\) −6.58579 −0.252926
\(679\) 0 0
\(680\) 0 0
\(681\) 25.3848 0.972747
\(682\) 8.24264 0.315627
\(683\) 25.0711 0.959318 0.479659 0.877455i \(-0.340760\pi\)
0.479659 + 0.877455i \(0.340760\pi\)
\(684\) 1.41421 0.0540738
\(685\) 0 0
\(686\) 0 0
\(687\) 29.3137 1.11839
\(688\) 1.58579 0.0604575
\(689\) 37.9706 1.44656
\(690\) 0 0
\(691\) 33.1127 1.25967 0.629833 0.776730i \(-0.283123\pi\)
0.629833 + 0.776730i \(0.283123\pi\)
\(692\) 14.2426 0.541424
\(693\) 0 0
\(694\) 9.79899 0.371965
\(695\) 0 0
\(696\) −1.00000 −0.0379049
\(697\) −33.9706 −1.28673
\(698\) −24.6569 −0.933276
\(699\) 14.2426 0.538706
\(700\) 0 0
\(701\) 36.3137 1.37155 0.685775 0.727814i \(-0.259463\pi\)
0.685775 + 0.727814i \(0.259463\pi\)
\(702\) −3.00000 −0.113228
\(703\) 10.0000 0.377157
\(704\) 3.41421 0.128678
\(705\) 0 0
\(706\) −20.4853 −0.770974
\(707\) 0 0
\(708\) 7.24264 0.272195
\(709\) 19.8995 0.747341 0.373671 0.927561i \(-0.378099\pi\)
0.373671 + 0.927561i \(0.378099\pi\)
\(710\) 0 0
\(711\) 1.07107 0.0401682
\(712\) −4.00000 −0.149906
\(713\) 1.00000 0.0374503
\(714\) 0 0
\(715\) 0 0
\(716\) −5.07107 −0.189515
\(717\) −11.3137 −0.422518
\(718\) −33.8701 −1.26402
\(719\) 18.9289 0.705930 0.352965 0.935637i \(-0.385173\pi\)
0.352965 + 0.935637i \(0.385173\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 17.0000 0.632674
\(723\) −6.00000 −0.223142
\(724\) −23.3137 −0.866447
\(725\) 0 0
\(726\) −0.656854 −0.0243781
\(727\) −37.8701 −1.40452 −0.702261 0.711919i \(-0.747826\pi\)
−0.702261 + 0.711919i \(0.747826\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 9.24264 0.341851
\(732\) −0.171573 −0.00634152
\(733\) −0.514719 −0.0190116 −0.00950578 0.999955i \(-0.503026\pi\)
−0.00950578 + 0.999955i \(0.503026\pi\)
\(734\) −0.757359 −0.0279546
\(735\) 0 0
\(736\) 0.414214 0.0152681
\(737\) −52.2843 −1.92592
\(738\) 5.82843 0.214547
\(739\) 53.8701 1.98164 0.990821 0.135180i \(-0.0431613\pi\)
0.990821 + 0.135180i \(0.0431613\pi\)
\(740\) 0 0
\(741\) 4.24264 0.155857
\(742\) 0 0
\(743\) −36.5563 −1.34112 −0.670561 0.741854i \(-0.733947\pi\)
−0.670561 + 0.741854i \(0.733947\pi\)
\(744\) 2.41421 0.0885094
\(745\) 0 0
\(746\) −2.34315 −0.0857887
\(747\) −0.0710678 −0.00260024
\(748\) 19.8995 0.727598
\(749\) 0 0
\(750\) 0 0
\(751\) −12.9706 −0.473303 −0.236651 0.971595i \(-0.576050\pi\)
−0.236651 + 0.971595i \(0.576050\pi\)
\(752\) −1.65685 −0.0604193
\(753\) −1.92893 −0.0702942
\(754\) −3.00000 −0.109254
\(755\) 0 0
\(756\) 0 0
\(757\) −44.2426 −1.60803 −0.804013 0.594612i \(-0.797306\pi\)
−0.804013 + 0.594612i \(0.797306\pi\)
\(758\) 34.2132 1.24268
\(759\) −1.41421 −0.0513327
\(760\) 0 0
\(761\) 15.0294 0.544817 0.272408 0.962182i \(-0.412180\pi\)
0.272408 + 0.962182i \(0.412180\pi\)
\(762\) 3.17157 0.114894
\(763\) 0 0
\(764\) 16.0711 0.581431
\(765\) 0 0
\(766\) −27.5563 −0.995651
\(767\) 21.7279 0.784550
\(768\) 1.00000 0.0360844
\(769\) −15.3137 −0.552226 −0.276113 0.961125i \(-0.589046\pi\)
−0.276113 + 0.961125i \(0.589046\pi\)
\(770\) 0 0
\(771\) 13.3431 0.480542
\(772\) −6.82843 −0.245760
\(773\) −19.3137 −0.694666 −0.347333 0.937742i \(-0.612913\pi\)
−0.347333 + 0.937742i \(0.612913\pi\)
\(774\) −1.58579 −0.0569999
\(775\) 0 0
\(776\) 0.343146 0.0123182
\(777\) 0 0
\(778\) 10.3431 0.370820
\(779\) −8.24264 −0.295323
\(780\) 0 0
\(781\) −31.7990 −1.13786
\(782\) 2.41421 0.0863321
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) 0 0
\(786\) 18.8284 0.671588
\(787\) −43.4558 −1.54903 −0.774517 0.632553i \(-0.782007\pi\)
−0.774517 + 0.632553i \(0.782007\pi\)
\(788\) 23.1421 0.824404
\(789\) −10.5563 −0.375816
\(790\) 0 0
\(791\) 0 0
\(792\) −3.41421 −0.121319
\(793\) −0.514719 −0.0182782
\(794\) −30.3137 −1.07579
\(795\) 0 0
\(796\) 19.6569 0.696719
\(797\) 16.2426 0.575344 0.287672 0.957729i \(-0.407119\pi\)
0.287672 + 0.957729i \(0.407119\pi\)
\(798\) 0 0
\(799\) −9.65685 −0.341635
\(800\) 0 0
\(801\) 4.00000 0.141333
\(802\) −12.1005 −0.427284
\(803\) 51.4558 1.81584
\(804\) −15.3137 −0.540073
\(805\) 0 0
\(806\) 7.24264 0.255111
\(807\) 0.443651 0.0156172
\(808\) −5.65685 −0.199007
\(809\) −34.6274 −1.21744 −0.608718 0.793387i \(-0.708316\pi\)
−0.608718 + 0.793387i \(0.708316\pi\)
\(810\) 0 0
\(811\) 33.7574 1.18538 0.592691 0.805430i \(-0.298066\pi\)
0.592691 + 0.805430i \(0.298066\pi\)
\(812\) 0 0
\(813\) −11.5147 −0.403839
\(814\) −24.1421 −0.846181
\(815\) 0 0
\(816\) 5.82843 0.204036
\(817\) 2.24264 0.0784601
\(818\) −22.5858 −0.789694
\(819\) 0 0
\(820\) 0 0
\(821\) 13.1716 0.459691 0.229846 0.973227i \(-0.426178\pi\)
0.229846 + 0.973227i \(0.426178\pi\)
\(822\) 23.2132 0.809653
\(823\) 28.5858 0.996438 0.498219 0.867051i \(-0.333988\pi\)
0.498219 + 0.867051i \(0.333988\pi\)
\(824\) 0.0710678 0.00247576
\(825\) 0 0
\(826\) 0 0
\(827\) −11.8995 −0.413786 −0.206893 0.978364i \(-0.566335\pi\)
−0.206893 + 0.978364i \(0.566335\pi\)
\(828\) −0.414214 −0.0143949
\(829\) −0.798990 −0.0277501 −0.0138750 0.999904i \(-0.504417\pi\)
−0.0138750 + 0.999904i \(0.504417\pi\)
\(830\) 0 0
\(831\) −24.0416 −0.833995
\(832\) 3.00000 0.104006
\(833\) 0 0
\(834\) 15.5563 0.538672
\(835\) 0 0
\(836\) 4.82843 0.166995
\(837\) −2.41421 −0.0834474
\(838\) −24.5563 −0.848285
\(839\) 17.4142 0.601205 0.300603 0.953749i \(-0.402812\pi\)
0.300603 + 0.953749i \(0.402812\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 35.6569 1.22882
\(843\) 20.0000 0.688837
\(844\) −26.5563 −0.914107
\(845\) 0 0
\(846\) 1.65685 0.0569638
\(847\) 0 0
\(848\) 12.6569 0.434638
\(849\) −0.242641 −0.00832741
\(850\) 0 0
\(851\) −2.92893 −0.100403
\(852\) −9.31371 −0.319082
\(853\) 14.3137 0.490092 0.245046 0.969511i \(-0.421197\pi\)
0.245046 + 0.969511i \(0.421197\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −6.48528 −0.221662
\(857\) 29.4558 1.00619 0.503096 0.864230i \(-0.332194\pi\)
0.503096 + 0.864230i \(0.332194\pi\)
\(858\) −10.2426 −0.349678
\(859\) 21.9411 0.748622 0.374311 0.927303i \(-0.377879\pi\)
0.374311 + 0.927303i \(0.377879\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −33.0416 −1.12540
\(863\) 24.9706 0.850008 0.425004 0.905192i \(-0.360273\pi\)
0.425004 + 0.905192i \(0.360273\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −28.6274 −0.972799
\(867\) 16.9706 0.576351
\(868\) 0 0
\(869\) 3.65685 0.124050
\(870\) 0 0
\(871\) −45.9411 −1.55666
\(872\) 11.3137 0.383131
\(873\) −0.343146 −0.0116137
\(874\) 0.585786 0.0198145
\(875\) 0 0
\(876\) 15.0711 0.509204
\(877\) −26.3431 −0.889545 −0.444772 0.895644i \(-0.646715\pi\)
−0.444772 + 0.895644i \(0.646715\pi\)
\(878\) −1.58579 −0.0535177
\(879\) −11.3137 −0.381602
\(880\) 0 0
\(881\) 41.6274 1.40246 0.701232 0.712933i \(-0.252634\pi\)
0.701232 + 0.712933i \(0.252634\pi\)
\(882\) 0 0
\(883\) −31.5269 −1.06097 −0.530483 0.847696i \(-0.677989\pi\)
−0.530483 + 0.847696i \(0.677989\pi\)
\(884\) 17.4853 0.588094
\(885\) 0 0
\(886\) 14.3431 0.481867
\(887\) −25.7574 −0.864847 −0.432424 0.901671i \(-0.642342\pi\)
−0.432424 + 0.901671i \(0.642342\pi\)
\(888\) −7.07107 −0.237289
\(889\) 0 0
\(890\) 0 0
\(891\) 3.41421 0.114380
\(892\) 24.5563 0.822207
\(893\) −2.34315 −0.0784104
\(894\) −4.65685 −0.155749
\(895\) 0 0
\(896\) 0 0
\(897\) −1.24264 −0.0414906
\(898\) 25.0711 0.836632
\(899\) −2.41421 −0.0805185
\(900\) 0 0
\(901\) 73.7696 2.45762
\(902\) 19.8995 0.662581
\(903\) 0 0
\(904\) −6.58579 −0.219040
\(905\) 0 0
\(906\) −7.89949 −0.262443
\(907\) 40.6985 1.35137 0.675686 0.737190i \(-0.263848\pi\)
0.675686 + 0.737190i \(0.263848\pi\)
\(908\) 25.3848 0.842423
\(909\) 5.65685 0.187626
\(910\) 0 0
\(911\) 22.0711 0.731247 0.365624 0.930763i \(-0.380856\pi\)
0.365624 + 0.930763i \(0.380856\pi\)
\(912\) 1.41421 0.0468293
\(913\) −0.242641 −0.00803023
\(914\) 3.82843 0.126633
\(915\) 0 0
\(916\) 29.3137 0.968552
\(917\) 0 0
\(918\) −5.82843 −0.192367
\(919\) 4.44365 0.146583 0.0732913 0.997311i \(-0.476650\pi\)
0.0732913 + 0.997311i \(0.476650\pi\)
\(920\) 0 0
\(921\) 8.58579 0.282911
\(922\) −20.0000 −0.658665
\(923\) −27.9411 −0.919693
\(924\) 0 0
\(925\) 0 0
\(926\) −29.8995 −0.982558
\(927\) −0.0710678 −0.00233417
\(928\) −1.00000 −0.0328266
\(929\) −8.65685 −0.284022 −0.142011 0.989865i \(-0.545357\pi\)
−0.142011 + 0.989865i \(0.545357\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 14.2426 0.466533
\(933\) −11.4142 −0.373685
\(934\) 1.24264 0.0406604
\(935\) 0 0
\(936\) −3.00000 −0.0980581
\(937\) 0.242641 0.00792673 0.00396336 0.999992i \(-0.498738\pi\)
0.00396336 + 0.999992i \(0.498738\pi\)
\(938\) 0 0
\(939\) −16.0416 −0.523499
\(940\) 0 0
\(941\) 51.3137 1.67278 0.836390 0.548136i \(-0.184662\pi\)
0.836390 + 0.548136i \(0.184662\pi\)
\(942\) 19.7990 0.645086
\(943\) 2.41421 0.0786176
\(944\) 7.24264 0.235728
\(945\) 0 0
\(946\) −5.41421 −0.176031
\(947\) −25.2132 −0.819319 −0.409660 0.912239i \(-0.634352\pi\)
−0.409660 + 0.912239i \(0.634352\pi\)
\(948\) 1.07107 0.0347867
\(949\) 45.2132 1.46768
\(950\) 0 0
\(951\) 19.9706 0.647590
\(952\) 0 0
\(953\) −31.5980 −1.02356 −0.511779 0.859117i \(-0.671014\pi\)
−0.511779 + 0.859117i \(0.671014\pi\)
\(954\) −12.6569 −0.409781
\(955\) 0 0
\(956\) −11.3137 −0.365911
\(957\) 3.41421 0.110366
\(958\) −20.2426 −0.654010
\(959\) 0 0
\(960\) 0 0
\(961\) −25.1716 −0.811986
\(962\) −21.2132 −0.683941
\(963\) 6.48528 0.208985
\(964\) −6.00000 −0.193247
\(965\) 0 0
\(966\) 0 0
\(967\) −44.0416 −1.41628 −0.708142 0.706070i \(-0.750466\pi\)
−0.708142 + 0.706070i \(0.750466\pi\)
\(968\) −0.656854 −0.0211121
\(969\) 8.24264 0.264792
\(970\) 0 0
\(971\) −44.9706 −1.44317 −0.721587 0.692324i \(-0.756587\pi\)
−0.721587 + 0.692324i \(0.756587\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −5.51472 −0.176703
\(975\) 0 0
\(976\) −0.171573 −0.00549191
\(977\) −8.44365 −0.270136 −0.135068 0.990836i \(-0.543125\pi\)
−0.135068 + 0.990836i \(0.543125\pi\)
\(978\) −5.24264 −0.167641
\(979\) 13.6569 0.436475
\(980\) 0 0
\(981\) −11.3137 −0.361219
\(982\) 0.142136 0.00453573
\(983\) 28.6863 0.914951 0.457475 0.889222i \(-0.348754\pi\)
0.457475 + 0.889222i \(0.348754\pi\)
\(984\) 5.82843 0.185803
\(985\) 0 0
\(986\) −5.82843 −0.185615
\(987\) 0 0
\(988\) 4.24264 0.134976
\(989\) −0.656854 −0.0208867
\(990\) 0 0
\(991\) −18.3431 −0.582689 −0.291345 0.956618i \(-0.594103\pi\)
−0.291345 + 0.956618i \(0.594103\pi\)
\(992\) 2.41421 0.0766514
\(993\) −24.0711 −0.763872
\(994\) 0 0
\(995\) 0 0
\(996\) −0.0710678 −0.00225187
\(997\) −25.3137 −0.801693 −0.400847 0.916145i \(-0.631284\pi\)
−0.400847 + 0.916145i \(0.631284\pi\)
\(998\) 26.0711 0.825265
\(999\) 7.07107 0.223719
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 7350.2.a.dg.1.2 yes 2
5.4 even 2 7350.2.a.dj.1.2 yes 2
7.6 odd 2 7350.2.a.dc.1.2 2
35.34 odd 2 7350.2.a.dm.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7350.2.a.dc.1.2 2 7.6 odd 2
7350.2.a.dg.1.2 yes 2 1.1 even 1 trivial
7350.2.a.dj.1.2 yes 2 5.4 even 2
7350.2.a.dm.1.2 yes 2 35.34 odd 2