Properties

Label 3675.2.a.ba.1.2
Level $3675$
Weight $2$
Character 3675.1
Self dual yes
Analytic conductor $29.345$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.3450227428\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 3675.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} -2.23607 q^{8} +1.00000 q^{9} -2.23607 q^{11} -0.618034 q^{12} +6.47214 q^{13} -4.85410 q^{16} -2.47214 q^{17} +1.61803 q^{18} -2.47214 q^{19} -3.61803 q^{22} +4.23607 q^{23} +2.23607 q^{24} +10.4721 q^{26} -1.00000 q^{27} -3.00000 q^{29} +4.00000 q^{31} -3.38197 q^{32} +2.23607 q^{33} -4.00000 q^{34} +0.618034 q^{36} -3.47214 q^{37} -4.00000 q^{38} -6.47214 q^{39} +8.94427 q^{41} +4.70820 q^{43} -1.38197 q^{44} +6.85410 q^{46} +10.4721 q^{47} +4.85410 q^{48} +2.47214 q^{51} +4.00000 q^{52} -6.00000 q^{53} -1.61803 q^{54} +2.47214 q^{57} -4.85410 q^{58} +6.47214 q^{59} +12.0000 q^{61} +6.47214 q^{62} +4.23607 q^{64} +3.61803 q^{66} -12.7082 q^{67} -1.52786 q^{68} -4.23607 q^{69} +8.23607 q^{71} -2.23607 q^{72} -14.4721 q^{73} -5.61803 q^{74} -1.52786 q^{76} -10.4721 q^{78} -2.70820 q^{79} +1.00000 q^{81} +14.4721 q^{82} +16.9443 q^{83} +7.61803 q^{86} +3.00000 q^{87} +5.00000 q^{88} +1.52786 q^{89} +2.61803 q^{92} -4.00000 q^{93} +16.9443 q^{94} +3.38197 q^{96} -4.00000 q^{97} -2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - 2q^{3} - q^{4} - q^{6} + 2q^{9} + O(q^{10}) \) \( 2q + q^{2} - 2q^{3} - q^{4} - q^{6} + 2q^{9} + q^{12} + 4q^{13} - 3q^{16} + 4q^{17} + q^{18} + 4q^{19} - 5q^{22} + 4q^{23} + 12q^{26} - 2q^{27} - 6q^{29} + 8q^{31} - 9q^{32} - 8q^{34} - q^{36} + 2q^{37} - 8q^{38} - 4q^{39} - 4q^{43} - 5q^{44} + 7q^{46} + 12q^{47} + 3q^{48} - 4q^{51} + 8q^{52} - 12q^{53} - q^{54} - 4q^{57} - 3q^{58} + 4q^{59} + 24q^{61} + 4q^{62} + 4q^{64} + 5q^{66} - 12q^{67} - 12q^{68} - 4q^{69} + 12q^{71} - 20q^{73} - 9q^{74} - 12q^{76} - 12q^{78} + 8q^{79} + 2q^{81} + 20q^{82} + 16q^{83} + 13q^{86} + 6q^{87} + 10q^{88} + 12q^{89} + 3q^{92} - 8q^{93} + 16q^{94} + 9q^{96} - 8q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.618034 0.309017
\(5\) 0 0
\(6\) −1.61803 −0.660560
\(7\) 0 0
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.23607 −0.674200 −0.337100 0.941469i \(-0.609446\pi\)
−0.337100 + 0.941469i \(0.609446\pi\)
\(12\) −0.618034 −0.178411
\(13\) 6.47214 1.79505 0.897524 0.440966i \(-0.145364\pi\)
0.897524 + 0.440966i \(0.145364\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) −2.47214 −0.599581 −0.299791 0.954005i \(-0.596917\pi\)
−0.299791 + 0.954005i \(0.596917\pi\)
\(18\) 1.61803 0.381374
\(19\) −2.47214 −0.567147 −0.283573 0.958951i \(-0.591520\pi\)
−0.283573 + 0.958951i \(0.591520\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.61803 −0.771367
\(23\) 4.23607 0.883281 0.441641 0.897192i \(-0.354397\pi\)
0.441641 + 0.897192i \(0.354397\pi\)
\(24\) 2.23607 0.456435
\(25\) 0 0
\(26\) 10.4721 2.05375
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −3.38197 −0.597853
\(33\) 2.23607 0.389249
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 0.618034 0.103006
\(37\) −3.47214 −0.570816 −0.285408 0.958406i \(-0.592129\pi\)
−0.285408 + 0.958406i \(0.592129\pi\)
\(38\) −4.00000 −0.648886
\(39\) −6.47214 −1.03637
\(40\) 0 0
\(41\) 8.94427 1.39686 0.698430 0.715678i \(-0.253882\pi\)
0.698430 + 0.715678i \(0.253882\pi\)
\(42\) 0 0
\(43\) 4.70820 0.717994 0.358997 0.933339i \(-0.383119\pi\)
0.358997 + 0.933339i \(0.383119\pi\)
\(44\) −1.38197 −0.208339
\(45\) 0 0
\(46\) 6.85410 1.01058
\(47\) 10.4721 1.52752 0.763759 0.645501i \(-0.223352\pi\)
0.763759 + 0.645501i \(0.223352\pi\)
\(48\) 4.85410 0.700629
\(49\) 0 0
\(50\) 0 0
\(51\) 2.47214 0.346168
\(52\) 4.00000 0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.61803 −0.220187
\(55\) 0 0
\(56\) 0 0
\(57\) 2.47214 0.327442
\(58\) −4.85410 −0.637375
\(59\) 6.47214 0.842600 0.421300 0.906921i \(-0.361574\pi\)
0.421300 + 0.906921i \(0.361574\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 6.47214 0.821962
\(63\) 0 0
\(64\) 4.23607 0.529508
\(65\) 0 0
\(66\) 3.61803 0.445349
\(67\) −12.7082 −1.55255 −0.776277 0.630392i \(-0.782894\pi\)
−0.776277 + 0.630392i \(0.782894\pi\)
\(68\) −1.52786 −0.185281
\(69\) −4.23607 −0.509963
\(70\) 0 0
\(71\) 8.23607 0.977441 0.488721 0.872440i \(-0.337464\pi\)
0.488721 + 0.872440i \(0.337464\pi\)
\(72\) −2.23607 −0.263523
\(73\) −14.4721 −1.69384 −0.846918 0.531724i \(-0.821544\pi\)
−0.846918 + 0.531724i \(0.821544\pi\)
\(74\) −5.61803 −0.653083
\(75\) 0 0
\(76\) −1.52786 −0.175258
\(77\) 0 0
\(78\) −10.4721 −1.18574
\(79\) −2.70820 −0.304697 −0.152348 0.988327i \(-0.548684\pi\)
−0.152348 + 0.988327i \(0.548684\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 14.4721 1.59818
\(83\) 16.9443 1.85988 0.929938 0.367717i \(-0.119860\pi\)
0.929938 + 0.367717i \(0.119860\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7.61803 0.821474
\(87\) 3.00000 0.321634
\(88\) 5.00000 0.533002
\(89\) 1.52786 0.161953 0.0809766 0.996716i \(-0.474196\pi\)
0.0809766 + 0.996716i \(0.474196\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.61803 0.272949
\(93\) −4.00000 −0.414781
\(94\) 16.9443 1.74767
\(95\) 0 0
\(96\) 3.38197 0.345170
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 0 0
\(99\) −2.23607 −0.224733
\(100\) 0 0
\(101\) 9.52786 0.948058 0.474029 0.880509i \(-0.342799\pi\)
0.474029 + 0.880509i \(0.342799\pi\)
\(102\) 4.00000 0.396059
\(103\) 7.41641 0.730760 0.365380 0.930858i \(-0.380939\pi\)
0.365380 + 0.930858i \(0.380939\pi\)
\(104\) −14.4721 −1.41911
\(105\) 0 0
\(106\) −9.70820 −0.942944
\(107\) 8.94427 0.864675 0.432338 0.901712i \(-0.357689\pi\)
0.432338 + 0.901712i \(0.357689\pi\)
\(108\) −0.618034 −0.0594703
\(109\) 16.4164 1.57241 0.786203 0.617968i \(-0.212044\pi\)
0.786203 + 0.617968i \(0.212044\pi\)
\(110\) 0 0
\(111\) 3.47214 0.329561
\(112\) 0 0
\(113\) 10.5279 0.990378 0.495189 0.868785i \(-0.335099\pi\)
0.495189 + 0.868785i \(0.335099\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −1.85410 −0.172149
\(117\) 6.47214 0.598349
\(118\) 10.4721 0.964038
\(119\) 0 0
\(120\) 0 0
\(121\) −6.00000 −0.545455
\(122\) 19.4164 1.75788
\(123\) −8.94427 −0.806478
\(124\) 2.47214 0.222004
\(125\) 0 0
\(126\) 0 0
\(127\) −19.1803 −1.70198 −0.850990 0.525182i \(-0.823997\pi\)
−0.850990 + 0.525182i \(0.823997\pi\)
\(128\) 13.6180 1.20368
\(129\) −4.70820 −0.414534
\(130\) 0 0
\(131\) 13.5279 1.18193 0.590967 0.806695i \(-0.298746\pi\)
0.590967 + 0.806695i \(0.298746\pi\)
\(132\) 1.38197 0.120285
\(133\) 0 0
\(134\) −20.5623 −1.77631
\(135\) 0 0
\(136\) 5.52786 0.474010
\(137\) 19.8885 1.69919 0.849596 0.527433i \(-0.176846\pi\)
0.849596 + 0.527433i \(0.176846\pi\)
\(138\) −6.85410 −0.583460
\(139\) −11.4164 −0.968327 −0.484164 0.874978i \(-0.660876\pi\)
−0.484164 + 0.874978i \(0.660876\pi\)
\(140\) 0 0
\(141\) −10.4721 −0.881913
\(142\) 13.3262 1.11831
\(143\) −14.4721 −1.21022
\(144\) −4.85410 −0.404508
\(145\) 0 0
\(146\) −23.4164 −1.93796
\(147\) 0 0
\(148\) −2.14590 −0.176392
\(149\) 12.8885 1.05587 0.527935 0.849285i \(-0.322966\pi\)
0.527935 + 0.849285i \(0.322966\pi\)
\(150\) 0 0
\(151\) 6.70820 0.545906 0.272953 0.962027i \(-0.412000\pi\)
0.272953 + 0.962027i \(0.412000\pi\)
\(152\) 5.52786 0.448369
\(153\) −2.47214 −0.199860
\(154\) 0 0
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) −0.944272 −0.0753611 −0.0376806 0.999290i \(-0.511997\pi\)
−0.0376806 + 0.999290i \(0.511997\pi\)
\(158\) −4.38197 −0.348610
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 1.61803 0.127125
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 5.52786 0.431654
\(165\) 0 0
\(166\) 27.4164 2.12793
\(167\) −22.4721 −1.73895 −0.869473 0.493980i \(-0.835541\pi\)
−0.869473 + 0.493980i \(0.835541\pi\)
\(168\) 0 0
\(169\) 28.8885 2.22220
\(170\) 0 0
\(171\) −2.47214 −0.189049
\(172\) 2.90983 0.221872
\(173\) 2.47214 0.187953 0.0939765 0.995574i \(-0.470042\pi\)
0.0939765 + 0.995574i \(0.470042\pi\)
\(174\) 4.85410 0.367989
\(175\) 0 0
\(176\) 10.8541 0.818159
\(177\) −6.47214 −0.486476
\(178\) 2.47214 0.185294
\(179\) −0.944272 −0.0705782 −0.0352891 0.999377i \(-0.511235\pi\)
−0.0352891 + 0.999377i \(0.511235\pi\)
\(180\) 0 0
\(181\) 6.47214 0.481070 0.240535 0.970640i \(-0.422677\pi\)
0.240535 + 0.970640i \(0.422677\pi\)
\(182\) 0 0
\(183\) −12.0000 −0.887066
\(184\) −9.47214 −0.698295
\(185\) 0 0
\(186\) −6.47214 −0.474560
\(187\) 5.52786 0.404237
\(188\) 6.47214 0.472029
\(189\) 0 0
\(190\) 0 0
\(191\) 3.05573 0.221105 0.110552 0.993870i \(-0.464738\pi\)
0.110552 + 0.993870i \(0.464738\pi\)
\(192\) −4.23607 −0.305712
\(193\) 22.8885 1.64755 0.823777 0.566914i \(-0.191863\pi\)
0.823777 + 0.566914i \(0.191863\pi\)
\(194\) −6.47214 −0.464672
\(195\) 0 0
\(196\) 0 0
\(197\) 21.0000 1.49619 0.748094 0.663593i \(-0.230969\pi\)
0.748094 + 0.663593i \(0.230969\pi\)
\(198\) −3.61803 −0.257122
\(199\) −21.8885 −1.55164 −0.775819 0.630956i \(-0.782663\pi\)
−0.775819 + 0.630956i \(0.782663\pi\)
\(200\) 0 0
\(201\) 12.7082 0.896368
\(202\) 15.4164 1.08469
\(203\) 0 0
\(204\) 1.52786 0.106972
\(205\) 0 0
\(206\) 12.0000 0.836080
\(207\) 4.23607 0.294427
\(208\) −31.4164 −2.17834
\(209\) 5.52786 0.382370
\(210\) 0 0
\(211\) 13.8885 0.956127 0.478063 0.878325i \(-0.341339\pi\)
0.478063 + 0.878325i \(0.341339\pi\)
\(212\) −3.70820 −0.254680
\(213\) −8.23607 −0.564326
\(214\) 14.4721 0.989295
\(215\) 0 0
\(216\) 2.23607 0.152145
\(217\) 0 0
\(218\) 26.5623 1.79903
\(219\) 14.4721 0.977936
\(220\) 0 0
\(221\) −16.0000 −1.07628
\(222\) 5.61803 0.377058
\(223\) −3.41641 −0.228780 −0.114390 0.993436i \(-0.536491\pi\)
−0.114390 + 0.993436i \(0.536491\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 17.0344 1.13311
\(227\) −7.41641 −0.492244 −0.246122 0.969239i \(-0.579156\pi\)
−0.246122 + 0.969239i \(0.579156\pi\)
\(228\) 1.52786 0.101185
\(229\) 28.9443 1.91269 0.956346 0.292238i \(-0.0943999\pi\)
0.956346 + 0.292238i \(0.0943999\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.70820 0.440415
\(233\) 4.52786 0.296630 0.148315 0.988940i \(-0.452615\pi\)
0.148315 + 0.988940i \(0.452615\pi\)
\(234\) 10.4721 0.684585
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 2.70820 0.175917
\(238\) 0 0
\(239\) −20.9443 −1.35477 −0.677386 0.735628i \(-0.736887\pi\)
−0.677386 + 0.735628i \(0.736887\pi\)
\(240\) 0 0
\(241\) 18.4721 1.18989 0.594947 0.803765i \(-0.297173\pi\)
0.594947 + 0.803765i \(0.297173\pi\)
\(242\) −9.70820 −0.624067
\(243\) −1.00000 −0.0641500
\(244\) 7.41641 0.474787
\(245\) 0 0
\(246\) −14.4721 −0.922710
\(247\) −16.0000 −1.01806
\(248\) −8.94427 −0.567962
\(249\) −16.9443 −1.07380
\(250\) 0 0
\(251\) −10.4721 −0.660995 −0.330498 0.943807i \(-0.607217\pi\)
−0.330498 + 0.943807i \(0.607217\pi\)
\(252\) 0 0
\(253\) −9.47214 −0.595508
\(254\) −31.0344 −1.94727
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) −8.94427 −0.557928 −0.278964 0.960302i \(-0.589991\pi\)
−0.278964 + 0.960302i \(0.589991\pi\)
\(258\) −7.61803 −0.474278
\(259\) 0 0
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) 21.8885 1.35228
\(263\) −21.1803 −1.30604 −0.653018 0.757343i \(-0.726497\pi\)
−0.653018 + 0.757343i \(0.726497\pi\)
\(264\) −5.00000 −0.307729
\(265\) 0 0
\(266\) 0 0
\(267\) −1.52786 −0.0935038
\(268\) −7.85410 −0.479766
\(269\) −8.94427 −0.545342 −0.272671 0.962107i \(-0.587907\pi\)
−0.272671 + 0.962107i \(0.587907\pi\)
\(270\) 0 0
\(271\) −3.41641 −0.207532 −0.103766 0.994602i \(-0.533089\pi\)
−0.103766 + 0.994602i \(0.533089\pi\)
\(272\) 12.0000 0.727607
\(273\) 0 0
\(274\) 32.1803 1.94409
\(275\) 0 0
\(276\) −2.61803 −0.157587
\(277\) −15.8885 −0.954650 −0.477325 0.878727i \(-0.658394\pi\)
−0.477325 + 0.878727i \(0.658394\pi\)
\(278\) −18.4721 −1.10789
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −19.3607 −1.15496 −0.577481 0.816404i \(-0.695964\pi\)
−0.577481 + 0.816404i \(0.695964\pi\)
\(282\) −16.9443 −1.00902
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 5.09017 0.302046
\(285\) 0 0
\(286\) −23.4164 −1.38464
\(287\) 0 0
\(288\) −3.38197 −0.199284
\(289\) −10.8885 −0.640503
\(290\) 0 0
\(291\) 4.00000 0.234484
\(292\) −8.94427 −0.523424
\(293\) −16.3607 −0.955801 −0.477901 0.878414i \(-0.658602\pi\)
−0.477901 + 0.878414i \(0.658602\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 7.76393 0.451269
\(297\) 2.23607 0.129750
\(298\) 20.8541 1.20805
\(299\) 27.4164 1.58553
\(300\) 0 0
\(301\) 0 0
\(302\) 10.8541 0.624583
\(303\) −9.52786 −0.547361
\(304\) 12.0000 0.688247
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) −31.4164 −1.79303 −0.896515 0.443014i \(-0.853909\pi\)
−0.896515 + 0.443014i \(0.853909\pi\)
\(308\) 0 0
\(309\) −7.41641 −0.421905
\(310\) 0 0
\(311\) 20.3607 1.15455 0.577274 0.816550i \(-0.304116\pi\)
0.577274 + 0.816550i \(0.304116\pi\)
\(312\) 14.4721 0.819323
\(313\) −7.41641 −0.419200 −0.209600 0.977787i \(-0.567216\pi\)
−0.209600 + 0.977787i \(0.567216\pi\)
\(314\) −1.52786 −0.0862224
\(315\) 0 0
\(316\) −1.67376 −0.0941565
\(317\) −7.94427 −0.446195 −0.223097 0.974796i \(-0.571617\pi\)
−0.223097 + 0.974796i \(0.571617\pi\)
\(318\) 9.70820 0.544409
\(319\) 6.70820 0.375587
\(320\) 0 0
\(321\) −8.94427 −0.499221
\(322\) 0 0
\(323\) 6.11146 0.340051
\(324\) 0.618034 0.0343352
\(325\) 0 0
\(326\) −19.4164 −1.07538
\(327\) −16.4164 −0.907829
\(328\) −20.0000 −1.10432
\(329\) 0 0
\(330\) 0 0
\(331\) −33.1803 −1.82376 −0.911878 0.410461i \(-0.865368\pi\)
−0.911878 + 0.410461i \(0.865368\pi\)
\(332\) 10.4721 0.574733
\(333\) −3.47214 −0.190272
\(334\) −36.3607 −1.98957
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 46.7426 2.54246
\(339\) −10.5279 −0.571795
\(340\) 0 0
\(341\) −8.94427 −0.484359
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) −10.5279 −0.567624
\(345\) 0 0
\(346\) 4.00000 0.215041
\(347\) 5.29180 0.284078 0.142039 0.989861i \(-0.454634\pi\)
0.142039 + 0.989861i \(0.454634\pi\)
\(348\) 1.85410 0.0993903
\(349\) 23.4164 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(350\) 0 0
\(351\) −6.47214 −0.345457
\(352\) 7.56231 0.403072
\(353\) 13.8885 0.739213 0.369606 0.929188i \(-0.379493\pi\)
0.369606 + 0.929188i \(0.379493\pi\)
\(354\) −10.4721 −0.556588
\(355\) 0 0
\(356\) 0.944272 0.0500463
\(357\) 0 0
\(358\) −1.52786 −0.0807501
\(359\) −11.7639 −0.620877 −0.310438 0.950594i \(-0.600476\pi\)
−0.310438 + 0.950594i \(0.600476\pi\)
\(360\) 0 0
\(361\) −12.8885 −0.678344
\(362\) 10.4721 0.550403
\(363\) 6.00000 0.314918
\(364\) 0 0
\(365\) 0 0
\(366\) −19.4164 −1.01491
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) −20.5623 −1.07188
\(369\) 8.94427 0.465620
\(370\) 0 0
\(371\) 0 0
\(372\) −2.47214 −0.128174
\(373\) −17.4721 −0.904673 −0.452336 0.891847i \(-0.649409\pi\)
−0.452336 + 0.891847i \(0.649409\pi\)
\(374\) 8.94427 0.462497
\(375\) 0 0
\(376\) −23.4164 −1.20761
\(377\) −19.4164 −0.999996
\(378\) 0 0
\(379\) 8.70820 0.447310 0.223655 0.974668i \(-0.428201\pi\)
0.223655 + 0.974668i \(0.428201\pi\)
\(380\) 0 0
\(381\) 19.1803 0.982639
\(382\) 4.94427 0.252971
\(383\) −19.4164 −0.992132 −0.496066 0.868285i \(-0.665223\pi\)
−0.496066 + 0.868285i \(0.665223\pi\)
\(384\) −13.6180 −0.694942
\(385\) 0 0
\(386\) 37.0344 1.88500
\(387\) 4.70820 0.239331
\(388\) −2.47214 −0.125504
\(389\) 12.0557 0.611250 0.305625 0.952152i \(-0.401135\pi\)
0.305625 + 0.952152i \(0.401135\pi\)
\(390\) 0 0
\(391\) −10.4721 −0.529599
\(392\) 0 0
\(393\) −13.5279 −0.682390
\(394\) 33.9787 1.71182
\(395\) 0 0
\(396\) −1.38197 −0.0694464
\(397\) 9.88854 0.496292 0.248146 0.968723i \(-0.420179\pi\)
0.248146 + 0.968723i \(0.420179\pi\)
\(398\) −35.4164 −1.77526
\(399\) 0 0
\(400\) 0 0
\(401\) 28.4164 1.41905 0.709524 0.704681i \(-0.248910\pi\)
0.709524 + 0.704681i \(0.248910\pi\)
\(402\) 20.5623 1.02555
\(403\) 25.8885 1.28960
\(404\) 5.88854 0.292966
\(405\) 0 0
\(406\) 0 0
\(407\) 7.76393 0.384844
\(408\) −5.52786 −0.273670
\(409\) 7.41641 0.366718 0.183359 0.983046i \(-0.441303\pi\)
0.183359 + 0.983046i \(0.441303\pi\)
\(410\) 0 0
\(411\) −19.8885 −0.981030
\(412\) 4.58359 0.225817
\(413\) 0 0
\(414\) 6.85410 0.336861
\(415\) 0 0
\(416\) −21.8885 −1.07317
\(417\) 11.4164 0.559064
\(418\) 8.94427 0.437479
\(419\) 29.8885 1.46015 0.730075 0.683367i \(-0.239485\pi\)
0.730075 + 0.683367i \(0.239485\pi\)
\(420\) 0 0
\(421\) 10.4164 0.507665 0.253832 0.967248i \(-0.418309\pi\)
0.253832 + 0.967248i \(0.418309\pi\)
\(422\) 22.4721 1.09393
\(423\) 10.4721 0.509173
\(424\) 13.4164 0.651558
\(425\) 0 0
\(426\) −13.3262 −0.645658
\(427\) 0 0
\(428\) 5.52786 0.267199
\(429\) 14.4721 0.698721
\(430\) 0 0
\(431\) −11.0557 −0.532536 −0.266268 0.963899i \(-0.585791\pi\)
−0.266268 + 0.963899i \(0.585791\pi\)
\(432\) 4.85410 0.233543
\(433\) −12.9443 −0.622062 −0.311031 0.950400i \(-0.600674\pi\)
−0.311031 + 0.950400i \(0.600674\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 10.1459 0.485900
\(437\) −10.4721 −0.500950
\(438\) 23.4164 1.11888
\(439\) 27.4164 1.30851 0.654257 0.756272i \(-0.272982\pi\)
0.654257 + 0.756272i \(0.272982\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −25.8885 −1.23139
\(443\) 15.0557 0.715319 0.357660 0.933852i \(-0.383575\pi\)
0.357660 + 0.933852i \(0.383575\pi\)
\(444\) 2.14590 0.101840
\(445\) 0 0
\(446\) −5.52786 −0.261752
\(447\) −12.8885 −0.609607
\(448\) 0 0
\(449\) −6.52786 −0.308069 −0.154034 0.988065i \(-0.549227\pi\)
−0.154034 + 0.988065i \(0.549227\pi\)
\(450\) 0 0
\(451\) −20.0000 −0.941763
\(452\) 6.50658 0.306044
\(453\) −6.70820 −0.315179
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −5.52786 −0.258866
\(457\) 25.9443 1.21362 0.606811 0.794846i \(-0.292449\pi\)
0.606811 + 0.794846i \(0.292449\pi\)
\(458\) 46.8328 2.18835
\(459\) 2.47214 0.115389
\(460\) 0 0
\(461\) 30.4721 1.41923 0.709614 0.704590i \(-0.248869\pi\)
0.709614 + 0.704590i \(0.248869\pi\)
\(462\) 0 0
\(463\) 1.88854 0.0877681 0.0438840 0.999037i \(-0.486027\pi\)
0.0438840 + 0.999037i \(0.486027\pi\)
\(464\) 14.5623 0.676038
\(465\) 0 0
\(466\) 7.32624 0.339381
\(467\) 1.88854 0.0873914 0.0436957 0.999045i \(-0.486087\pi\)
0.0436957 + 0.999045i \(0.486087\pi\)
\(468\) 4.00000 0.184900
\(469\) 0 0
\(470\) 0 0
\(471\) 0.944272 0.0435098
\(472\) −14.4721 −0.666134
\(473\) −10.5279 −0.484072
\(474\) 4.38197 0.201270
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −33.8885 −1.55003
\(479\) −30.4721 −1.39231 −0.696154 0.717893i \(-0.745107\pi\)
−0.696154 + 0.717893i \(0.745107\pi\)
\(480\) 0 0
\(481\) −22.4721 −1.02464
\(482\) 29.8885 1.36139
\(483\) 0 0
\(484\) −3.70820 −0.168555
\(485\) 0 0
\(486\) −1.61803 −0.0733955
\(487\) −14.7082 −0.666492 −0.333246 0.942840i \(-0.608144\pi\)
−0.333246 + 0.942840i \(0.608144\pi\)
\(488\) −26.8328 −1.21466
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 4.81966 0.217508 0.108754 0.994069i \(-0.465314\pi\)
0.108754 + 0.994069i \(0.465314\pi\)
\(492\) −5.52786 −0.249215
\(493\) 7.41641 0.334018
\(494\) −25.8885 −1.16478
\(495\) 0 0
\(496\) −19.4164 −0.871822
\(497\) 0 0
\(498\) −27.4164 −1.22856
\(499\) −12.0000 −0.537194 −0.268597 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) 22.4721 1.00398
\(502\) −16.9443 −0.756260
\(503\) −12.9443 −0.577157 −0.288578 0.957456i \(-0.593183\pi\)
−0.288578 + 0.957456i \(0.593183\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −15.3262 −0.681334
\(507\) −28.8885 −1.28299
\(508\) −11.8541 −0.525941
\(509\) 9.52786 0.422315 0.211158 0.977452i \(-0.432277\pi\)
0.211158 + 0.977452i \(0.432277\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −5.29180 −0.233867
\(513\) 2.47214 0.109147
\(514\) −14.4721 −0.638339
\(515\) 0 0
\(516\) −2.90983 −0.128098
\(517\) −23.4164 −1.02985
\(518\) 0 0
\(519\) −2.47214 −0.108515
\(520\) 0 0
\(521\) −20.9443 −0.917585 −0.458793 0.888543i \(-0.651718\pi\)
−0.458793 + 0.888543i \(0.651718\pi\)
\(522\) −4.85410 −0.212458
\(523\) −1.88854 −0.0825803 −0.0412901 0.999147i \(-0.513147\pi\)
−0.0412901 + 0.999147i \(0.513147\pi\)
\(524\) 8.36068 0.365238
\(525\) 0 0
\(526\) −34.2705 −1.49427
\(527\) −9.88854 −0.430752
\(528\) −10.8541 −0.472364
\(529\) −5.05573 −0.219814
\(530\) 0 0
\(531\) 6.47214 0.280867
\(532\) 0 0
\(533\) 57.8885 2.50743
\(534\) −2.47214 −0.106980
\(535\) 0 0
\(536\) 28.4164 1.22740
\(537\) 0.944272 0.0407483
\(538\) −14.4721 −0.623938
\(539\) 0 0
\(540\) 0 0
\(541\) −28.4164 −1.22172 −0.610858 0.791740i \(-0.709176\pi\)
−0.610858 + 0.791740i \(0.709176\pi\)
\(542\) −5.52786 −0.237442
\(543\) −6.47214 −0.277746
\(544\) 8.36068 0.358461
\(545\) 0 0
\(546\) 0 0
\(547\) −14.8197 −0.633643 −0.316821 0.948485i \(-0.602616\pi\)
−0.316821 + 0.948485i \(0.602616\pi\)
\(548\) 12.2918 0.525080
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) 7.41641 0.315950
\(552\) 9.47214 0.403161
\(553\) 0 0
\(554\) −25.7082 −1.09224
\(555\) 0 0
\(556\) −7.05573 −0.299230
\(557\) −20.8885 −0.885076 −0.442538 0.896750i \(-0.645922\pi\)
−0.442538 + 0.896750i \(0.645922\pi\)
\(558\) 6.47214 0.273987
\(559\) 30.4721 1.28883
\(560\) 0 0
\(561\) −5.52786 −0.233387
\(562\) −31.3262 −1.32142
\(563\) −4.94427 −0.208376 −0.104188 0.994558i \(-0.533224\pi\)
−0.104188 + 0.994558i \(0.533224\pi\)
\(564\) −6.47214 −0.272526
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −18.4164 −0.772735
\(569\) −36.3050 −1.52198 −0.760991 0.648762i \(-0.775287\pi\)
−0.760991 + 0.648762i \(0.775287\pi\)
\(570\) 0 0
\(571\) 11.7639 0.492305 0.246153 0.969231i \(-0.420834\pi\)
0.246153 + 0.969231i \(0.420834\pi\)
\(572\) −8.94427 −0.373979
\(573\) −3.05573 −0.127655
\(574\) 0 0
\(575\) 0 0
\(576\) 4.23607 0.176503
\(577\) 0.583592 0.0242953 0.0121476 0.999926i \(-0.496133\pi\)
0.0121476 + 0.999926i \(0.496133\pi\)
\(578\) −17.6180 −0.732814
\(579\) −22.8885 −0.951215
\(580\) 0 0
\(581\) 0 0
\(582\) 6.47214 0.268279
\(583\) 13.4164 0.555651
\(584\) 32.3607 1.33909
\(585\) 0 0
\(586\) −26.4721 −1.09355
\(587\) −14.4721 −0.597329 −0.298664 0.954358i \(-0.596541\pi\)
−0.298664 + 0.954358i \(0.596541\pi\)
\(588\) 0 0
\(589\) −9.88854 −0.407450
\(590\) 0 0
\(591\) −21.0000 −0.863825
\(592\) 16.8541 0.692699
\(593\) 44.9443 1.84564 0.922820 0.385231i \(-0.125878\pi\)
0.922820 + 0.385231i \(0.125878\pi\)
\(594\) 3.61803 0.148450
\(595\) 0 0
\(596\) 7.96556 0.326282
\(597\) 21.8885 0.895838
\(598\) 44.3607 1.81404
\(599\) 12.7082 0.519243 0.259622 0.965710i \(-0.416402\pi\)
0.259622 + 0.965710i \(0.416402\pi\)
\(600\) 0 0
\(601\) 36.9443 1.50699 0.753494 0.657455i \(-0.228367\pi\)
0.753494 + 0.657455i \(0.228367\pi\)
\(602\) 0 0
\(603\) −12.7082 −0.517518
\(604\) 4.14590 0.168694
\(605\) 0 0
\(606\) −15.4164 −0.626249
\(607\) −35.4164 −1.43751 −0.718754 0.695265i \(-0.755287\pi\)
−0.718754 + 0.695265i \(0.755287\pi\)
\(608\) 8.36068 0.339070
\(609\) 0 0
\(610\) 0 0
\(611\) 67.7771 2.74197
\(612\) −1.52786 −0.0617602
\(613\) −35.2492 −1.42370 −0.711851 0.702330i \(-0.752143\pi\)
−0.711851 + 0.702330i \(0.752143\pi\)
\(614\) −50.8328 −2.05145
\(615\) 0 0
\(616\) 0 0
\(617\) 34.4164 1.38555 0.692776 0.721153i \(-0.256387\pi\)
0.692776 + 0.721153i \(0.256387\pi\)
\(618\) −12.0000 −0.482711
\(619\) −17.5279 −0.704504 −0.352252 0.935905i \(-0.614584\pi\)
−0.352252 + 0.935905i \(0.614584\pi\)
\(620\) 0 0
\(621\) −4.23607 −0.169988
\(622\) 32.9443 1.32094
\(623\) 0 0
\(624\) 31.4164 1.25766
\(625\) 0 0
\(626\) −12.0000 −0.479616
\(627\) −5.52786 −0.220762
\(628\) −0.583592 −0.0232879
\(629\) 8.58359 0.342250
\(630\) 0 0
\(631\) 0.347524 0.0138347 0.00691736 0.999976i \(-0.497798\pi\)
0.00691736 + 0.999976i \(0.497798\pi\)
\(632\) 6.05573 0.240884
\(633\) −13.8885 −0.552020
\(634\) −12.8541 −0.510502
\(635\) 0 0
\(636\) 3.70820 0.147040
\(637\) 0 0
\(638\) 10.8541 0.429718
\(639\) 8.23607 0.325814
\(640\) 0 0
\(641\) 25.3607 1.00169 0.500843 0.865538i \(-0.333023\pi\)
0.500843 + 0.865538i \(0.333023\pi\)
\(642\) −14.4721 −0.571170
\(643\) 24.0000 0.946468 0.473234 0.880937i \(-0.343087\pi\)
0.473234 + 0.880937i \(0.343087\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 9.88854 0.389060
\(647\) 1.88854 0.0742463 0.0371232 0.999311i \(-0.488181\pi\)
0.0371232 + 0.999311i \(0.488181\pi\)
\(648\) −2.23607 −0.0878410
\(649\) −14.4721 −0.568081
\(650\) 0 0
\(651\) 0 0
\(652\) −7.41641 −0.290449
\(653\) −11.8885 −0.465235 −0.232617 0.972568i \(-0.574729\pi\)
−0.232617 + 0.972568i \(0.574729\pi\)
\(654\) −26.5623 −1.03867
\(655\) 0 0
\(656\) −43.4164 −1.69513
\(657\) −14.4721 −0.564612
\(658\) 0 0
\(659\) −15.0557 −0.586488 −0.293244 0.956038i \(-0.594735\pi\)
−0.293244 + 0.956038i \(0.594735\pi\)
\(660\) 0 0
\(661\) 20.3607 0.791939 0.395969 0.918264i \(-0.370409\pi\)
0.395969 + 0.918264i \(0.370409\pi\)
\(662\) −53.6869 −2.08660
\(663\) 16.0000 0.621389
\(664\) −37.8885 −1.47036
\(665\) 0 0
\(666\) −5.61803 −0.217694
\(667\) −12.7082 −0.492064
\(668\) −13.8885 −0.537364
\(669\) 3.41641 0.132086
\(670\) 0 0
\(671\) −26.8328 −1.03587
\(672\) 0 0
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 3.23607 0.124649
\(675\) 0 0
\(676\) 17.8541 0.686696
\(677\) −49.8885 −1.91737 −0.958686 0.284466i \(-0.908184\pi\)
−0.958686 + 0.284466i \(0.908184\pi\)
\(678\) −17.0344 −0.654204
\(679\) 0 0
\(680\) 0 0
\(681\) 7.41641 0.284197
\(682\) −14.4721 −0.554167
\(683\) 27.1803 1.04003 0.520013 0.854158i \(-0.325927\pi\)
0.520013 + 0.854158i \(0.325927\pi\)
\(684\) −1.52786 −0.0584193
\(685\) 0 0
\(686\) 0 0
\(687\) −28.9443 −1.10429
\(688\) −22.8541 −0.871304
\(689\) −38.8328 −1.47941
\(690\) 0 0
\(691\) 10.8328 0.412100 0.206050 0.978541i \(-0.433939\pi\)
0.206050 + 0.978541i \(0.433939\pi\)
\(692\) 1.52786 0.0580807
\(693\) 0 0
\(694\) 8.56231 0.325021
\(695\) 0 0
\(696\) −6.70820 −0.254274
\(697\) −22.1115 −0.837531
\(698\) 37.8885 1.43410
\(699\) −4.52786 −0.171260
\(700\) 0 0
\(701\) 37.7771 1.42682 0.713410 0.700746i \(-0.247150\pi\)
0.713410 + 0.700746i \(0.247150\pi\)
\(702\) −10.4721 −0.395245
\(703\) 8.58359 0.323736
\(704\) −9.47214 −0.356995
\(705\) 0 0
\(706\) 22.4721 0.845750
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) −2.70820 −0.101566
\(712\) −3.41641 −0.128035
\(713\) 16.9443 0.634568
\(714\) 0 0
\(715\) 0 0
\(716\) −0.583592 −0.0218099
\(717\) 20.9443 0.782178
\(718\) −19.0344 −0.710359
\(719\) −0.944272 −0.0352154 −0.0176077 0.999845i \(-0.505605\pi\)
−0.0176077 + 0.999845i \(0.505605\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −20.8541 −0.776109
\(723\) −18.4721 −0.686986
\(724\) 4.00000 0.148659
\(725\) 0 0
\(726\) 9.70820 0.360305
\(727\) 17.3050 0.641805 0.320903 0.947112i \(-0.396014\pi\)
0.320903 + 0.947112i \(0.396014\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −11.6393 −0.430496
\(732\) −7.41641 −0.274118
\(733\) 32.0000 1.18195 0.590973 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) −6.47214 −0.238891
\(735\) 0 0
\(736\) −14.3262 −0.528072
\(737\) 28.4164 1.04673
\(738\) 14.4721 0.532727
\(739\) −20.7082 −0.761764 −0.380882 0.924624i \(-0.624380\pi\)
−0.380882 + 0.924624i \(0.624380\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) 0 0
\(743\) 3.05573 0.112104 0.0560519 0.998428i \(-0.482149\pi\)
0.0560519 + 0.998428i \(0.482149\pi\)
\(744\) 8.94427 0.327913
\(745\) 0 0
\(746\) −28.2705 −1.03506
\(747\) 16.9443 0.619958
\(748\) 3.41641 0.124916
\(749\) 0 0
\(750\) 0 0
\(751\) −19.7771 −0.721676 −0.360838 0.932628i \(-0.617509\pi\)
−0.360838 + 0.932628i \(0.617509\pi\)
\(752\) −50.8328 −1.85368
\(753\) 10.4721 0.381626
\(754\) −31.4164 −1.14412
\(755\) 0 0
\(756\) 0 0
\(757\) 5.47214 0.198888 0.0994441 0.995043i \(-0.468294\pi\)
0.0994441 + 0.995043i \(0.468294\pi\)
\(758\) 14.0902 0.511778
\(759\) 9.47214 0.343817
\(760\) 0 0
\(761\) −52.9443 −1.91923 −0.959614 0.281319i \(-0.909228\pi\)
−0.959614 + 0.281319i \(0.909228\pi\)
\(762\) 31.0344 1.12426
\(763\) 0 0
\(764\) 1.88854 0.0683251
\(765\) 0 0
\(766\) −31.4164 −1.13512
\(767\) 41.8885 1.51251
\(768\) −13.5623 −0.489388
\(769\) 20.0000 0.721218 0.360609 0.932717i \(-0.382569\pi\)
0.360609 + 0.932717i \(0.382569\pi\)
\(770\) 0 0
\(771\) 8.94427 0.322120
\(772\) 14.1459 0.509122
\(773\) −39.7771 −1.43068 −0.715341 0.698775i \(-0.753729\pi\)
−0.715341 + 0.698775i \(0.753729\pi\)
\(774\) 7.61803 0.273825
\(775\) 0 0
\(776\) 8.94427 0.321081
\(777\) 0 0
\(778\) 19.5066 0.699345
\(779\) −22.1115 −0.792225
\(780\) 0 0
\(781\) −18.4164 −0.658991
\(782\) −16.9443 −0.605926
\(783\) 3.00000 0.107211
\(784\) 0 0
\(785\) 0 0
\(786\) −21.8885 −0.780739
\(787\) −33.5279 −1.19514 −0.597570 0.801817i \(-0.703867\pi\)
−0.597570 + 0.801817i \(0.703867\pi\)
\(788\) 12.9787 0.462348
\(789\) 21.1803 0.754040
\(790\) 0 0
\(791\) 0 0
\(792\) 5.00000 0.177667
\(793\) 77.6656 2.75799
\(794\) 16.0000 0.567819
\(795\) 0 0
\(796\) −13.5279 −0.479482
\(797\) 34.4721 1.22107 0.610533 0.791991i \(-0.290955\pi\)
0.610533 + 0.791991i \(0.290955\pi\)
\(798\) 0 0
\(799\) −25.8885 −0.915871
\(800\) 0 0
\(801\) 1.52786 0.0539844
\(802\) 45.9787 1.62356
\(803\) 32.3607 1.14198
\(804\) 7.85410 0.276993
\(805\) 0 0
\(806\) 41.8885 1.47546
\(807\) 8.94427 0.314853
\(808\) −21.3050 −0.749506
\(809\) 8.52786 0.299824 0.149912 0.988699i \(-0.452101\pi\)
0.149912 + 0.988699i \(0.452101\pi\)
\(810\) 0 0
\(811\) 38.8328 1.36360 0.681802 0.731536i \(-0.261196\pi\)
0.681802 + 0.731536i \(0.261196\pi\)
\(812\) 0 0
\(813\) 3.41641 0.119819
\(814\) 12.5623 0.440309
\(815\) 0 0
\(816\) −12.0000 −0.420084
\(817\) −11.6393 −0.407208
\(818\) 12.0000 0.419570
\(819\) 0 0
\(820\) 0 0
\(821\) −45.7771 −1.59763 −0.798816 0.601576i \(-0.794540\pi\)
−0.798816 + 0.601576i \(0.794540\pi\)
\(822\) −32.1803 −1.12242
\(823\) 17.5410 0.611442 0.305721 0.952121i \(-0.401103\pi\)
0.305721 + 0.952121i \(0.401103\pi\)
\(824\) −16.5836 −0.577717
\(825\) 0 0
\(826\) 0 0
\(827\) 14.2361 0.495037 0.247518 0.968883i \(-0.420385\pi\)
0.247518 + 0.968883i \(0.420385\pi\)
\(828\) 2.61803 0.0909830
\(829\) −2.47214 −0.0858608 −0.0429304 0.999078i \(-0.513669\pi\)
−0.0429304 + 0.999078i \(0.513669\pi\)
\(830\) 0 0
\(831\) 15.8885 0.551167
\(832\) 27.4164 0.950493
\(833\) 0 0
\(834\) 18.4721 0.639638
\(835\) 0 0
\(836\) 3.41641 0.118159
\(837\) −4.00000 −0.138260
\(838\) 48.3607 1.67059
\(839\) −55.1935 −1.90549 −0.952746 0.303770i \(-0.901755\pi\)
−0.952746 + 0.303770i \(0.901755\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 16.8541 0.580831
\(843\) 19.3607 0.666817
\(844\) 8.58359 0.295459
\(845\) 0 0
\(846\) 16.9443 0.582556
\(847\) 0 0
\(848\) 29.1246 1.00014
\(849\) 0 0
\(850\) 0 0
\(851\) −14.7082 −0.504191
\(852\) −5.09017 −0.174386
\(853\) 6.83282 0.233951 0.116976 0.993135i \(-0.462680\pi\)
0.116976 + 0.993135i \(0.462680\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −20.0000 −0.683586
\(857\) 37.8885 1.29425 0.647124 0.762385i \(-0.275972\pi\)
0.647124 + 0.762385i \(0.275972\pi\)
\(858\) 23.4164 0.799423
\(859\) −25.8885 −0.883306 −0.441653 0.897186i \(-0.645608\pi\)
−0.441653 + 0.897186i \(0.645608\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −17.8885 −0.609286
\(863\) −2.12461 −0.0723226 −0.0361613 0.999346i \(-0.511513\pi\)
−0.0361613 + 0.999346i \(0.511513\pi\)
\(864\) 3.38197 0.115057
\(865\) 0 0
\(866\) −20.9443 −0.711715
\(867\) 10.8885 0.369794
\(868\) 0 0
\(869\) 6.05573 0.205427
\(870\) 0 0
\(871\) −82.2492 −2.78691
\(872\) −36.7082 −1.24310
\(873\) −4.00000 −0.135379
\(874\) −16.9443 −0.573149
\(875\) 0 0
\(876\) 8.94427 0.302199
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 44.3607 1.49710
\(879\) 16.3607 0.551832
\(880\) 0 0
\(881\) 40.9443 1.37945 0.689724 0.724073i \(-0.257732\pi\)
0.689724 + 0.724073i \(0.257732\pi\)
\(882\) 0 0
\(883\) −43.5410 −1.46527 −0.732636 0.680621i \(-0.761710\pi\)
−0.732636 + 0.680621i \(0.761710\pi\)
\(884\) −9.88854 −0.332588
\(885\) 0 0
\(886\) 24.3607 0.818413
\(887\) 26.8328 0.900958 0.450479 0.892787i \(-0.351253\pi\)
0.450479 + 0.892787i \(0.351253\pi\)
\(888\) −7.76393 −0.260540
\(889\) 0 0
\(890\) 0 0
\(891\) −2.23607 −0.0749111
\(892\) −2.11146 −0.0706968
\(893\) −25.8885 −0.866327
\(894\) −20.8541 −0.697466
\(895\) 0 0
\(896\) 0 0
\(897\) −27.4164 −0.915407
\(898\) −10.5623 −0.352469
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) 14.8328 0.494153
\(902\) −32.3607 −1.07749
\(903\) 0 0
\(904\) −23.5410 −0.782963
\(905\) 0 0
\(906\) −10.8541 −0.360603
\(907\) 45.8885 1.52370 0.761852 0.647751i \(-0.224290\pi\)
0.761852 + 0.647751i \(0.224290\pi\)
\(908\) −4.58359 −0.152112
\(909\) 9.52786 0.316019
\(910\) 0 0
\(911\) −56.0132 −1.85580 −0.927899 0.372831i \(-0.878387\pi\)
−0.927899 + 0.372831i \(0.878387\pi\)
\(912\) −12.0000 −0.397360
\(913\) −37.8885 −1.25393
\(914\) 41.9787 1.38853
\(915\) 0 0
\(916\) 17.8885 0.591054
\(917\) 0 0
\(918\) 4.00000 0.132020
\(919\) 15.1803 0.500753 0.250377 0.968149i \(-0.419446\pi\)
0.250377 + 0.968149i \(0.419446\pi\)
\(920\) 0 0
\(921\) 31.4164 1.03521
\(922\) 49.3050 1.62377
\(923\) 53.3050 1.75455
\(924\) 0 0
\(925\) 0 0
\(926\) 3.05573 0.100417
\(927\) 7.41641 0.243587
\(928\) 10.1459 0.333055
\(929\) −4.58359 −0.150383 −0.0751914 0.997169i \(-0.523957\pi\)
−0.0751914 + 0.997169i \(0.523957\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2.79837 0.0916638
\(933\) −20.3607 −0.666579
\(934\) 3.05573 0.0999865
\(935\) 0 0
\(936\) −14.4721 −0.473037
\(937\) 28.0000 0.914720 0.457360 0.889282i \(-0.348795\pi\)
0.457360 + 0.889282i \(0.348795\pi\)
\(938\) 0 0
\(939\) 7.41641 0.242025
\(940\) 0 0
\(941\) −31.7771 −1.03590 −0.517952 0.855410i \(-0.673305\pi\)
−0.517952 + 0.855410i \(0.673305\pi\)
\(942\) 1.52786 0.0497805
\(943\) 37.8885 1.23382
\(944\) −31.4164 −1.02252
\(945\) 0 0
\(946\) −17.0344 −0.553837
\(947\) −32.9443 −1.07054 −0.535272 0.844679i \(-0.679791\pi\)
−0.535272 + 0.844679i \(0.679791\pi\)
\(948\) 1.67376 0.0543613
\(949\) −93.6656 −3.04052
\(950\) 0 0
\(951\) 7.94427 0.257611
\(952\) 0 0
\(953\) −11.3607 −0.368009 −0.184004 0.982925i \(-0.558906\pi\)
−0.184004 + 0.982925i \(0.558906\pi\)
\(954\) −9.70820 −0.314315
\(955\) 0 0
\(956\) −12.9443 −0.418648
\(957\) −6.70820 −0.216845
\(958\) −49.3050 −1.59297
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −36.3607 −1.17232
\(963\) 8.94427 0.288225
\(964\) 11.4164 0.367698
\(965\) 0 0
\(966\) 0 0
\(967\) 43.7771 1.40778 0.703888 0.710311i \(-0.251446\pi\)
0.703888 + 0.710311i \(0.251446\pi\)
\(968\) 13.4164 0.431220
\(969\) −6.11146 −0.196328
\(970\) 0 0
\(971\) 41.3050 1.32554 0.662769 0.748823i \(-0.269381\pi\)
0.662769 + 0.748823i \(0.269381\pi\)
\(972\) −0.618034 −0.0198234
\(973\) 0 0
\(974\) −23.7984 −0.762549
\(975\) 0 0
\(976\) −58.2492 −1.86451
\(977\) 7.58359 0.242621 0.121310 0.992615i \(-0.461290\pi\)
0.121310 + 0.992615i \(0.461290\pi\)
\(978\) 19.4164 0.620868
\(979\) −3.41641 −0.109189
\(980\) 0 0
\(981\) 16.4164 0.524136
\(982\) 7.79837 0.248856
\(983\) 23.0557 0.735364 0.367682 0.929952i \(-0.380152\pi\)
0.367682 + 0.929952i \(0.380152\pi\)
\(984\) 20.0000 0.637577
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) −9.88854 −0.314596
\(989\) 19.9443 0.634191
\(990\) 0 0
\(991\) −57.5410 −1.82785 −0.913925 0.405882i \(-0.866964\pi\)
−0.913925 + 0.405882i \(0.866964\pi\)
\(992\) −13.5279 −0.429510
\(993\) 33.1803 1.05295
\(994\) 0 0
\(995\) 0 0
\(996\) −10.4721 −0.331822
\(997\) 7.41641 0.234880 0.117440 0.993080i \(-0.462531\pi\)
0.117440 + 0.993080i \(0.462531\pi\)
\(998\) −19.4164 −0.614616
\(999\) 3.47214 0.109854
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 3675.2.a.ba.1.2 yes 2
5.4 even 2 3675.2.a.v.1.1 yes 2
7.6 odd 2 3675.2.a.bc.1.2 yes 2
35.34 odd 2 3675.2.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3675.2.a.u.1.1 2 35.34 odd 2
3675.2.a.v.1.1 yes 2 5.4 even 2
3675.2.a.ba.1.2 yes 2 1.1 even 1 trivial
3675.2.a.bc.1.2 yes 2 7.6 odd 2