Properties

Label 3675.2.a.ba.1.1
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 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 3675.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +2.23607 q^{8} +1.00000 q^{9} +2.23607 q^{11} +1.61803 q^{12} -2.47214 q^{13} +1.85410 q^{16} +6.47214 q^{17} -0.618034 q^{18} +6.47214 q^{19} -1.38197 q^{22} -0.236068 q^{23} -2.23607 q^{24} +1.52786 q^{26} -1.00000 q^{27} -3.00000 q^{29} +4.00000 q^{31} -5.61803 q^{32} -2.23607 q^{33} -4.00000 q^{34} -1.61803 q^{36} +5.47214 q^{37} -4.00000 q^{38} +2.47214 q^{39} -8.94427 q^{41} -8.70820 q^{43} -3.61803 q^{44} +0.145898 q^{46} +1.52786 q^{47} -1.85410 q^{48} -6.47214 q^{51} +4.00000 q^{52} -6.00000 q^{53} +0.618034 q^{54} -6.47214 q^{57} +1.85410 q^{58} -2.47214 q^{59} +12.0000 q^{61} -2.47214 q^{62} -0.236068 q^{64} +1.38197 q^{66} +0.708204 q^{67} -10.4721 q^{68} +0.236068 q^{69} +3.76393 q^{71} +2.23607 q^{72} -5.52786 q^{73} -3.38197 q^{74} -10.4721 q^{76} -1.52786 q^{78} +10.7082 q^{79} +1.00000 q^{81} +5.52786 q^{82} -0.944272 q^{83} +5.38197 q^{86} +3.00000 q^{87} +5.00000 q^{88} +10.4721 q^{89} +0.381966 q^{92} -4.00000 q^{93} -0.944272 q^{94} +5.61803 q^{96} -4.00000 q^{97} +2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - 2q^{3} - q^{4} - q^{6} + 2q^{9} + O(q^{10}) \) \( 2q + q^{2} - 2q^{3} - q^{4} - q^{6} + 2q^{9} + q^{12} + 4q^{13} - 3q^{16} + 4q^{17} + q^{18} + 4q^{19} - 5q^{22} + 4q^{23} + 12q^{26} - 2q^{27} - 6q^{29} + 8q^{31} - 9q^{32} - 8q^{34} - q^{36} + 2q^{37} - 8q^{38} - 4q^{39} - 4q^{43} - 5q^{44} + 7q^{46} + 12q^{47} + 3q^{48} - 4q^{51} + 8q^{52} - 12q^{53} - q^{54} - 4q^{57} - 3q^{58} + 4q^{59} + 24q^{61} + 4q^{62} + 4q^{64} + 5q^{66} - 12q^{67} - 12q^{68} - 4q^{69} + 12q^{71} - 20q^{73} - 9q^{74} - 12q^{76} - 12q^{78} + 8q^{79} + 2q^{81} + 20q^{82} + 16q^{83} + 13q^{86} + 6q^{87} + 10q^{88} + 12q^{89} + 3q^{92} - 8q^{93} + 16q^{94} + 9q^{96} - 8q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) 0.618034 0.252311
\(7\) 0 0
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.23607 0.674200 0.337100 0.941469i \(-0.390554\pi\)
0.337100 + 0.941469i \(0.390554\pi\)
\(12\) 1.61803 0.467086
\(13\) −2.47214 −0.685647 −0.342824 0.939400i \(-0.611383\pi\)
−0.342824 + 0.939400i \(0.611383\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 6.47214 1.56972 0.784862 0.619671i \(-0.212734\pi\)
0.784862 + 0.619671i \(0.212734\pi\)
\(18\) −0.618034 −0.145672
\(19\) 6.47214 1.48481 0.742405 0.669951i \(-0.233685\pi\)
0.742405 + 0.669951i \(0.233685\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.38197 −0.294636
\(23\) −0.236068 −0.0492236 −0.0246118 0.999697i \(-0.507835\pi\)
−0.0246118 + 0.999697i \(0.507835\pi\)
\(24\) −2.23607 −0.456435
\(25\) 0 0
\(26\) 1.52786 0.299639
\(27\) −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 \(0.383046\pi\)
\(32\) −5.61803 −0.993137
\(33\) −2.23607 −0.389249
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) −1.61803 −0.269672
\(37\) 5.47214 0.899614 0.449807 0.893126i \(-0.351493\pi\)
0.449807 + 0.893126i \(0.351493\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.47214 0.395859
\(40\) 0 0
\(41\) −8.94427 −1.39686 −0.698430 0.715678i \(-0.746118\pi\)
−0.698430 + 0.715678i \(0.746118\pi\)
\(42\) 0 0
\(43\) −8.70820 −1.32799 −0.663994 0.747738i \(-0.731140\pi\)
−0.663994 + 0.747738i \(0.731140\pi\)
\(44\) −3.61803 −0.545439
\(45\) 0 0
\(46\) 0.145898 0.0215115
\(47\) 1.52786 0.222862 0.111431 0.993772i \(-0.464457\pi\)
0.111431 + 0.993772i \(0.464457\pi\)
\(48\) −1.85410 −0.267617
\(49\) 0 0
\(50\) 0 0
\(51\) −6.47214 −0.906280
\(52\) 4.00000 0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0.618034 0.0841038
\(55\) 0 0
\(56\) 0 0
\(57\) −6.47214 −0.857255
\(58\) 1.85410 0.243456
\(59\) −2.47214 −0.321845 −0.160922 0.986967i \(-0.551447\pi\)
−0.160922 + 0.986967i \(0.551447\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) −2.47214 −0.313962
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) 0 0
\(66\) 1.38197 0.170108
\(67\) 0.708204 0.0865209 0.0432604 0.999064i \(-0.486225\pi\)
0.0432604 + 0.999064i \(0.486225\pi\)
\(68\) −10.4721 −1.26993
\(69\) 0.236068 0.0284192
\(70\) 0 0
\(71\) 3.76393 0.446697 0.223348 0.974739i \(-0.428301\pi\)
0.223348 + 0.974739i \(0.428301\pi\)
\(72\) 2.23607 0.263523
\(73\) −5.52786 −0.646988 −0.323494 0.946230i \(-0.604857\pi\)
−0.323494 + 0.946230i \(0.604857\pi\)
\(74\) −3.38197 −0.393146
\(75\) 0 0
\(76\) −10.4721 −1.20124
\(77\) 0 0
\(78\) −1.52786 −0.172997
\(79\) 10.7082 1.20477 0.602384 0.798207i \(-0.294218\pi\)
0.602384 + 0.798207i \(0.294218\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 5.52786 0.610450
\(83\) −0.944272 −0.103647 −0.0518237 0.998656i \(-0.516503\pi\)
−0.0518237 + 0.998656i \(0.516503\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 5.38197 0.580352
\(87\) 3.00000 0.321634
\(88\) 5.00000 0.533002
\(89\) 10.4721 1.11004 0.555022 0.831836i \(-0.312710\pi\)
0.555022 + 0.831836i \(0.312710\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.381966 0.0398227
\(93\) −4.00000 −0.414781
\(94\) −0.944272 −0.0973942
\(95\) 0 0
\(96\) 5.61803 0.573388
\(97\) −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
\(101\) 18.4721 1.83805 0.919023 0.394204i \(-0.128980\pi\)
0.919023 + 0.394204i \(0.128980\pi\)
\(102\) 4.00000 0.396059
\(103\) −19.4164 −1.91316 −0.956578 0.291477i \(-0.905853\pi\)
−0.956578 + 0.291477i \(0.905853\pi\)
\(104\) −5.52786 −0.542052
\(105\) 0 0
\(106\) 3.70820 0.360173
\(107\) −8.94427 −0.864675 −0.432338 0.901712i \(-0.642311\pi\)
−0.432338 + 0.901712i \(0.642311\pi\)
\(108\) 1.61803 0.155695
\(109\) −10.4164 −0.997711 −0.498855 0.866685i \(-0.666246\pi\)
−0.498855 + 0.866685i \(0.666246\pi\)
\(110\) 0 0
\(111\) −5.47214 −0.519392
\(112\) 0 0
\(113\) 19.4721 1.83178 0.915892 0.401424i \(-0.131485\pi\)
0.915892 + 0.401424i \(0.131485\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 4.85410 0.450692
\(117\) −2.47214 −0.228549
\(118\) 1.52786 0.140651
\(119\) 0 0
\(120\) 0 0
\(121\) −6.00000 −0.545455
\(122\) −7.41641 −0.671450
\(123\) 8.94427 0.806478
\(124\) −6.47214 −0.581215
\(125\) 0 0
\(126\) 0 0
\(127\) 3.18034 0.282210 0.141105 0.989995i \(-0.454935\pi\)
0.141105 + 0.989995i \(0.454935\pi\)
\(128\) 11.3820 1.00603
\(129\) 8.70820 0.766715
\(130\) 0 0
\(131\) 22.4721 1.96340 0.981700 0.190435i \(-0.0609898\pi\)
0.981700 + 0.190435i \(0.0609898\pi\)
\(132\) 3.61803 0.314909
\(133\) 0 0
\(134\) −0.437694 −0.0378110
\(135\) 0 0
\(136\) 14.4721 1.24098
\(137\) −15.8885 −1.35745 −0.678725 0.734393i \(-0.737467\pi\)
−0.678725 + 0.734393i \(0.737467\pi\)
\(138\) −0.145898 −0.0124197
\(139\) 15.4164 1.30760 0.653801 0.756666i \(-0.273173\pi\)
0.653801 + 0.756666i \(0.273173\pi\)
\(140\) 0 0
\(141\) −1.52786 −0.128669
\(142\) −2.32624 −0.195214
\(143\) −5.52786 −0.462263
\(144\) 1.85410 0.154508
\(145\) 0 0
\(146\) 3.41641 0.282744
\(147\) 0 0
\(148\) −8.85410 −0.727803
\(149\) −22.8885 −1.87510 −0.937551 0.347847i \(-0.886913\pi\)
−0.937551 + 0.347847i \(0.886913\pi\)
\(150\) 0 0
\(151\) −6.70820 −0.545906 −0.272953 0.962027i \(-0.588000\pi\)
−0.272953 + 0.962027i \(0.588000\pi\)
\(152\) 14.4721 1.17385
\(153\) 6.47214 0.523241
\(154\) 0 0
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 16.9443 1.35230 0.676150 0.736764i \(-0.263647\pi\)
0.676150 + 0.736764i \(0.263647\pi\)
\(158\) −6.61803 −0.526503
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) −0.618034 −0.0485573
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 14.4721 1.13008
\(165\) 0 0
\(166\) 0.583592 0.0452955
\(167\) −13.5279 −1.04682 −0.523409 0.852082i \(-0.675340\pi\)
−0.523409 + 0.852082i \(0.675340\pi\)
\(168\) 0 0
\(169\) −6.88854 −0.529888
\(170\) 0 0
\(171\) 6.47214 0.494937
\(172\) 14.0902 1.07437
\(173\) −6.47214 −0.492067 −0.246034 0.969261i \(-0.579127\pi\)
−0.246034 + 0.969261i \(0.579127\pi\)
\(174\) −1.85410 −0.140559
\(175\) 0 0
\(176\) 4.14590 0.312509
\(177\) 2.47214 0.185817
\(178\) −6.47214 −0.485107
\(179\) 16.9443 1.26647 0.633237 0.773958i \(-0.281726\pi\)
0.633237 + 0.773958i \(0.281726\pi\)
\(180\) 0 0
\(181\) −2.47214 −0.183752 −0.0918762 0.995770i \(-0.529286\pi\)
−0.0918762 + 0.995770i \(0.529286\pi\)
\(182\) 0 0
\(183\) −12.0000 −0.887066
\(184\) −0.527864 −0.0389147
\(185\) 0 0
\(186\) 2.47214 0.181266
\(187\) 14.4721 1.05831
\(188\) −2.47214 −0.180299
\(189\) 0 0
\(190\) 0 0
\(191\) 20.9443 1.51547 0.757737 0.652560i \(-0.226305\pi\)
0.757737 + 0.652560i \(0.226305\pi\)
\(192\) 0.236068 0.0170367
\(193\) −12.8885 −0.927738 −0.463869 0.885904i \(-0.653539\pi\)
−0.463869 + 0.885904i \(0.653539\pi\)
\(194\) 2.47214 0.177489
\(195\) 0 0
\(196\) 0 0
\(197\) 21.0000 1.49619 0.748094 0.663593i \(-0.230969\pi\)
0.748094 + 0.663593i \(0.230969\pi\)
\(198\) −1.38197 −0.0982120
\(199\) 13.8885 0.984533 0.492266 0.870445i \(-0.336169\pi\)
0.492266 + 0.870445i \(0.336169\pi\)
\(200\) 0 0
\(201\) −0.708204 −0.0499529
\(202\) −11.4164 −0.803256
\(203\) 0 0
\(204\) 10.4721 0.733196
\(205\) 0 0
\(206\) 12.0000 0.836080
\(207\) −0.236068 −0.0164079
\(208\) −4.58359 −0.317815
\(209\) 14.4721 1.00106
\(210\) 0 0
\(211\) −21.8885 −1.50687 −0.753435 0.657523i \(-0.771604\pi\)
−0.753435 + 0.657523i \(0.771604\pi\)
\(212\) 9.70820 0.666762
\(213\) −3.76393 −0.257900
\(214\) 5.52786 0.377877
\(215\) 0 0
\(216\) −2.23607 −0.152145
\(217\) 0 0
\(218\) 6.43769 0.436016
\(219\) 5.52786 0.373538
\(220\) 0 0
\(221\) −16.0000 −1.07628
\(222\) 3.38197 0.226983
\(223\) 23.4164 1.56808 0.784039 0.620711i \(-0.213156\pi\)
0.784039 + 0.620711i \(0.213156\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −12.0344 −0.800519
\(227\) 19.4164 1.28871 0.644356 0.764726i \(-0.277125\pi\)
0.644356 + 0.764726i \(0.277125\pi\)
\(228\) 10.4721 0.693534
\(229\) 11.0557 0.730583 0.365292 0.930893i \(-0.380969\pi\)
0.365292 + 0.930893i \(0.380969\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.70820 −0.440415
\(233\) 13.4721 0.882589 0.441294 0.897362i \(-0.354519\pi\)
0.441294 + 0.897362i \(0.354519\pi\)
\(234\) 1.52786 0.0998796
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) −10.7082 −0.695573
\(238\) 0 0
\(239\) −3.05573 −0.197659 −0.0988293 0.995104i \(-0.531510\pi\)
−0.0988293 + 0.995104i \(0.531510\pi\)
\(240\) 0 0
\(241\) 9.52786 0.613744 0.306872 0.951751i \(-0.400718\pi\)
0.306872 + 0.951751i \(0.400718\pi\)
\(242\) 3.70820 0.238372
\(243\) −1.00000 −0.0641500
\(244\) −19.4164 −1.24301
\(245\) 0 0
\(246\) −5.52786 −0.352444
\(247\) −16.0000 −1.01806
\(248\) 8.94427 0.567962
\(249\) 0.944272 0.0598408
\(250\) 0 0
\(251\) −1.52786 −0.0964379 −0.0482190 0.998837i \(-0.515355\pi\)
−0.0482190 + 0.998837i \(0.515355\pi\)
\(252\) 0 0
\(253\) −0.527864 −0.0331865
\(254\) −1.96556 −0.123330
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) 8.94427 0.557928 0.278964 0.960302i \(-0.410009\pi\)
0.278964 + 0.960302i \(0.410009\pi\)
\(258\) −5.38197 −0.335067
\(259\) 0 0
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) −13.8885 −0.858037
\(263\) 1.18034 0.0727829 0.0363914 0.999338i \(-0.488414\pi\)
0.0363914 + 0.999338i \(0.488414\pi\)
\(264\) −5.00000 −0.307729
\(265\) 0 0
\(266\) 0 0
\(267\) −10.4721 −0.640884
\(268\) −1.14590 −0.0699969
\(269\) 8.94427 0.545342 0.272671 0.962107i \(-0.412093\pi\)
0.272671 + 0.962107i \(0.412093\pi\)
\(270\) 0 0
\(271\) 23.4164 1.42245 0.711223 0.702967i \(-0.248142\pi\)
0.711223 + 0.702967i \(0.248142\pi\)
\(272\) 12.0000 0.727607
\(273\) 0 0
\(274\) 9.81966 0.593227
\(275\) 0 0
\(276\) −0.381966 −0.0229917
\(277\) 19.8885 1.19499 0.597493 0.801874i \(-0.296163\pi\)
0.597493 + 0.801874i \(0.296163\pi\)
\(278\) −9.52786 −0.571443
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 25.3607 1.51289 0.756446 0.654057i \(-0.226934\pi\)
0.756446 + 0.654057i \(0.226934\pi\)
\(282\) 0.944272 0.0562306
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −6.09017 −0.361385
\(285\) 0 0
\(286\) 3.41641 0.202016
\(287\) 0 0
\(288\) −5.61803 −0.331046
\(289\) 24.8885 1.46403
\(290\) 0 0
\(291\) 4.00000 0.234484
\(292\) 8.94427 0.523424
\(293\) 28.3607 1.65685 0.828424 0.560101i \(-0.189238\pi\)
0.828424 + 0.560101i \(0.189238\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 12.2361 0.711207
\(297\) −2.23607 −0.129750
\(298\) 14.1459 0.819450
\(299\) 0.583592 0.0337500
\(300\) 0 0
\(301\) 0 0
\(302\) 4.14590 0.238570
\(303\) −18.4721 −1.06120
\(304\) 12.0000 0.688247
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) −4.58359 −0.261599 −0.130800 0.991409i \(-0.541754\pi\)
−0.130800 + 0.991409i \(0.541754\pi\)
\(308\) 0 0
\(309\) 19.4164 1.10456
\(310\) 0 0
\(311\) −24.3607 −1.38137 −0.690684 0.723157i \(-0.742690\pi\)
−0.690684 + 0.723157i \(0.742690\pi\)
\(312\) 5.52786 0.312954
\(313\) 19.4164 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(314\) −10.4721 −0.590977
\(315\) 0 0
\(316\) −17.3262 −0.974677
\(317\) 9.94427 0.558526 0.279263 0.960215i \(-0.409910\pi\)
0.279263 + 0.960215i \(0.409910\pi\)
\(318\) −3.70820 −0.207946
\(319\) −6.70820 −0.375587
\(320\) 0 0
\(321\) 8.94427 0.499221
\(322\) 0 0
\(323\) 41.8885 2.33074
\(324\) −1.61803 −0.0898908
\(325\) 0 0
\(326\) 7.41641 0.410757
\(327\) 10.4164 0.576029
\(328\) −20.0000 −1.10432
\(329\) 0 0
\(330\) 0 0
\(331\) −10.8197 −0.594702 −0.297351 0.954768i \(-0.596103\pi\)
−0.297351 + 0.954768i \(0.596103\pi\)
\(332\) 1.52786 0.0838524
\(333\) 5.47214 0.299871
\(334\) 8.36068 0.457476
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 4.25735 0.231570
\(339\) −19.4721 −1.05758
\(340\) 0 0
\(341\) 8.94427 0.484359
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) −19.4721 −1.04987
\(345\) 0 0
\(346\) 4.00000 0.215041
\(347\) 18.7082 1.00431 0.502155 0.864778i \(-0.332541\pi\)
0.502155 + 0.864778i \(0.332541\pi\)
\(348\) −4.85410 −0.260207
\(349\) −3.41641 −0.182876 −0.0914381 0.995811i \(-0.529146\pi\)
−0.0914381 + 0.995811i \(0.529146\pi\)
\(350\) 0 0
\(351\) 2.47214 0.131953
\(352\) −12.5623 −0.669573
\(353\) −21.8885 −1.16501 −0.582505 0.812827i \(-0.697927\pi\)
−0.582505 + 0.812827i \(0.697927\pi\)
\(354\) −1.52786 −0.0812051
\(355\) 0 0
\(356\) −16.9443 −0.898045
\(357\) 0 0
\(358\) −10.4721 −0.553470
\(359\) −16.2361 −0.856907 −0.428453 0.903564i \(-0.640941\pi\)
−0.428453 + 0.903564i \(0.640941\pi\)
\(360\) 0 0
\(361\) 22.8885 1.20466
\(362\) 1.52786 0.0803028
\(363\) 6.00000 0.314918
\(364\) 0 0
\(365\) 0 0
\(366\) 7.41641 0.387662
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) −0.437694 −0.0228164
\(369\) −8.94427 −0.465620
\(370\) 0 0
\(371\) 0 0
\(372\) 6.47214 0.335565
\(373\) −8.52786 −0.441556 −0.220778 0.975324i \(-0.570860\pi\)
−0.220778 + 0.975324i \(0.570860\pi\)
\(374\) −8.94427 −0.462497
\(375\) 0 0
\(376\) 3.41641 0.176188
\(377\) 7.41641 0.381964
\(378\) 0 0
\(379\) −4.70820 −0.241844 −0.120922 0.992662i \(-0.538585\pi\)
−0.120922 + 0.992662i \(0.538585\pi\)
\(380\) 0 0
\(381\) −3.18034 −0.162934
\(382\) −12.9443 −0.662287
\(383\) 7.41641 0.378961 0.189480 0.981885i \(-0.439320\pi\)
0.189480 + 0.981885i \(0.439320\pi\)
\(384\) −11.3820 −0.580834
\(385\) 0 0
\(386\) 7.96556 0.405436
\(387\) −8.70820 −0.442663
\(388\) 6.47214 0.328573
\(389\) 29.9443 1.51823 0.759117 0.650954i \(-0.225631\pi\)
0.759117 + 0.650954i \(0.225631\pi\)
\(390\) 0 0
\(391\) −1.52786 −0.0772674
\(392\) 0 0
\(393\) −22.4721 −1.13357
\(394\) −12.9787 −0.653858
\(395\) 0 0
\(396\) −3.61803 −0.181813
\(397\) −25.8885 −1.29931 −0.649654 0.760230i \(-0.725086\pi\)
−0.649654 + 0.760230i \(0.725086\pi\)
\(398\) −8.58359 −0.430257
\(399\) 0 0
\(400\) 0 0
\(401\) 1.58359 0.0790808 0.0395404 0.999218i \(-0.487411\pi\)
0.0395404 + 0.999218i \(0.487411\pi\)
\(402\) 0.437694 0.0218302
\(403\) −9.88854 −0.492583
\(404\) −29.8885 −1.48701
\(405\) 0 0
\(406\) 0 0
\(407\) 12.2361 0.606519
\(408\) −14.4721 −0.716477
\(409\) −19.4164 −0.960080 −0.480040 0.877247i \(-0.659378\pi\)
−0.480040 + 0.877247i \(0.659378\pi\)
\(410\) 0 0
\(411\) 15.8885 0.783724
\(412\) 31.4164 1.54778
\(413\) 0 0
\(414\) 0.145898 0.00717050
\(415\) 0 0
\(416\) 13.8885 0.680942
\(417\) −15.4164 −0.754945
\(418\) −8.94427 −0.437479
\(419\) −5.88854 −0.287674 −0.143837 0.989601i \(-0.545944\pi\)
−0.143837 + 0.989601i \(0.545944\pi\)
\(420\) 0 0
\(421\) −16.4164 −0.800087 −0.400043 0.916496i \(-0.631005\pi\)
−0.400043 + 0.916496i \(0.631005\pi\)
\(422\) 13.5279 0.658526
\(423\) 1.52786 0.0742873
\(424\) −13.4164 −0.651558
\(425\) 0 0
\(426\) 2.32624 0.112707
\(427\) 0 0
\(428\) 14.4721 0.699537
\(429\) 5.52786 0.266888
\(430\) 0 0
\(431\) −28.9443 −1.39420 −0.697098 0.716976i \(-0.745526\pi\)
−0.697098 + 0.716976i \(0.745526\pi\)
\(432\) −1.85410 −0.0892055
\(433\) 4.94427 0.237607 0.118803 0.992918i \(-0.462094\pi\)
0.118803 + 0.992918i \(0.462094\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 16.8541 0.807165
\(437\) −1.52786 −0.0730876
\(438\) −3.41641 −0.163242
\(439\) 0.583592 0.0278533 0.0139267 0.999903i \(-0.495567\pi\)
0.0139267 + 0.999903i \(0.495567\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 9.88854 0.470350
\(443\) 32.9443 1.56523 0.782615 0.622506i \(-0.213885\pi\)
0.782615 + 0.622506i \(0.213885\pi\)
\(444\) 8.85410 0.420197
\(445\) 0 0
\(446\) −14.4721 −0.685275
\(447\) 22.8885 1.08259
\(448\) 0 0
\(449\) −15.4721 −0.730175 −0.365088 0.930973i \(-0.618961\pi\)
−0.365088 + 0.930973i \(0.618961\pi\)
\(450\) 0 0
\(451\) −20.0000 −0.941763
\(452\) −31.5066 −1.48194
\(453\) 6.70820 0.315179
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −14.4721 −0.677720
\(457\) 8.05573 0.376831 0.188416 0.982089i \(-0.439665\pi\)
0.188416 + 0.982089i \(0.439665\pi\)
\(458\) −6.83282 −0.319277
\(459\) −6.47214 −0.302093
\(460\) 0 0
\(461\) 21.5279 1.00265 0.501326 0.865258i \(-0.332846\pi\)
0.501326 + 0.865258i \(0.332846\pi\)
\(462\) 0 0
\(463\) −33.8885 −1.57493 −0.787467 0.616357i \(-0.788608\pi\)
−0.787467 + 0.616357i \(0.788608\pi\)
\(464\) −5.56231 −0.258224
\(465\) 0 0
\(466\) −8.32624 −0.385706
\(467\) −33.8885 −1.56817 −0.784087 0.620650i \(-0.786869\pi\)
−0.784087 + 0.620650i \(0.786869\pi\)
\(468\) 4.00000 0.184900
\(469\) 0 0
\(470\) 0 0
\(471\) −16.9443 −0.780751
\(472\) −5.52786 −0.254441
\(473\) −19.4721 −0.895330
\(474\) 6.61803 0.303976
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 1.88854 0.0863800
\(479\) −21.5279 −0.983633 −0.491817 0.870699i \(-0.663667\pi\)
−0.491817 + 0.870699i \(0.663667\pi\)
\(480\) 0 0
\(481\) −13.5279 −0.616818
\(482\) −5.88854 −0.268216
\(483\) 0 0
\(484\) 9.70820 0.441282
\(485\) 0 0
\(486\) 0.618034 0.0280346
\(487\) −1.29180 −0.0585369 −0.0292684 0.999572i \(-0.509318\pi\)
−0.0292684 + 0.999572i \(0.509318\pi\)
\(488\) 26.8328 1.21466
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 27.1803 1.22663 0.613316 0.789838i \(-0.289835\pi\)
0.613316 + 0.789838i \(0.289835\pi\)
\(492\) −14.4721 −0.652454
\(493\) −19.4164 −0.874471
\(494\) 9.88854 0.444907
\(495\) 0 0
\(496\) 7.41641 0.333007
\(497\) 0 0
\(498\) −0.583592 −0.0261514
\(499\) −12.0000 −0.537194 −0.268597 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) 13.5279 0.604380
\(502\) 0.944272 0.0421449
\(503\) 4.94427 0.220454 0.110227 0.993906i \(-0.464842\pi\)
0.110227 + 0.993906i \(0.464842\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0.326238 0.0145030
\(507\) 6.88854 0.305931
\(508\) −5.14590 −0.228312
\(509\) 18.4721 0.818763 0.409382 0.912363i \(-0.365745\pi\)
0.409382 + 0.912363i \(0.365745\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −18.7082 −0.826794
\(513\) −6.47214 −0.285752
\(514\) −5.52786 −0.243824
\(515\) 0 0
\(516\) −14.0902 −0.620285
\(517\) 3.41641 0.150253
\(518\) 0 0
\(519\) 6.47214 0.284095
\(520\) 0 0
\(521\) −3.05573 −0.133874 −0.0669369 0.997757i \(-0.521323\pi\)
−0.0669369 + 0.997757i \(0.521323\pi\)
\(522\) 1.85410 0.0811518
\(523\) 33.8885 1.48184 0.740921 0.671592i \(-0.234389\pi\)
0.740921 + 0.671592i \(0.234389\pi\)
\(524\) −36.3607 −1.58842
\(525\) 0 0
\(526\) −0.729490 −0.0318073
\(527\) 25.8885 1.12772
\(528\) −4.14590 −0.180427
\(529\) −22.9443 −0.997577
\(530\) 0 0
\(531\) −2.47214 −0.107282
\(532\) 0 0
\(533\) 22.1115 0.957753
\(534\) 6.47214 0.280077
\(535\) 0 0
\(536\) 1.58359 0.0684008
\(537\) −16.9443 −0.731199
\(538\) −5.52786 −0.238323
\(539\) 0 0
\(540\) 0 0
\(541\) −1.58359 −0.0680839 −0.0340420 0.999420i \(-0.510838\pi\)
−0.0340420 + 0.999420i \(0.510838\pi\)
\(542\) −14.4721 −0.621631
\(543\) 2.47214 0.106090
\(544\) −36.3607 −1.55895
\(545\) 0 0
\(546\) 0 0
\(547\) −37.1803 −1.58972 −0.794858 0.606795i \(-0.792455\pi\)
−0.794858 + 0.606795i \(0.792455\pi\)
\(548\) 25.7082 1.09820
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) −19.4164 −0.827167
\(552\) 0.527864 0.0224674
\(553\) 0 0
\(554\) −12.2918 −0.522228
\(555\) 0 0
\(556\) −24.9443 −1.05787
\(557\) 14.8885 0.630848 0.315424 0.948951i \(-0.397853\pi\)
0.315424 + 0.948951i \(0.397853\pi\)
\(558\) −2.47214 −0.104654
\(559\) 21.5279 0.910532
\(560\) 0 0
\(561\) −14.4721 −0.611014
\(562\) −15.6738 −0.661158
\(563\) 12.9443 0.545536 0.272768 0.962080i \(-0.412061\pi\)
0.272768 + 0.962080i \(0.412061\pi\)
\(564\) 2.47214 0.104096
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 8.41641 0.353145
\(569\) 26.3050 1.10276 0.551380 0.834254i \(-0.314101\pi\)
0.551380 + 0.834254i \(0.314101\pi\)
\(570\) 0 0
\(571\) 16.2361 0.679458 0.339729 0.940523i \(-0.389665\pi\)
0.339729 + 0.940523i \(0.389665\pi\)
\(572\) 8.94427 0.373979
\(573\) −20.9443 −0.874960
\(574\) 0 0
\(575\) 0 0
\(576\) −0.236068 −0.00983617
\(577\) 27.4164 1.14136 0.570680 0.821173i \(-0.306680\pi\)
0.570680 + 0.821173i \(0.306680\pi\)
\(578\) −15.3820 −0.639805
\(579\) 12.8885 0.535630
\(580\) 0 0
\(581\) 0 0
\(582\) −2.47214 −0.102473
\(583\) −13.4164 −0.555651
\(584\) −12.3607 −0.511489
\(585\) 0 0
\(586\) −17.5279 −0.724069
\(587\) −5.52786 −0.228159 −0.114080 0.993472i \(-0.536392\pi\)
−0.114080 + 0.993472i \(0.536392\pi\)
\(588\) 0 0
\(589\) 25.8885 1.06672
\(590\) 0 0
\(591\) −21.0000 −0.863825
\(592\) 10.1459 0.416994
\(593\) 27.0557 1.11105 0.555523 0.831501i \(-0.312518\pi\)
0.555523 + 0.831501i \(0.312518\pi\)
\(594\) 1.38197 0.0567028
\(595\) 0 0
\(596\) 37.0344 1.51699
\(597\) −13.8885 −0.568420
\(598\) −0.360680 −0.0147493
\(599\) −0.708204 −0.0289364 −0.0144682 0.999895i \(-0.504606\pi\)
−0.0144682 + 0.999895i \(0.504606\pi\)
\(600\) 0 0
\(601\) 19.0557 0.777299 0.388650 0.921386i \(-0.372942\pi\)
0.388650 + 0.921386i \(0.372942\pi\)
\(602\) 0 0
\(603\) 0.708204 0.0288403
\(604\) 10.8541 0.441647
\(605\) 0 0
\(606\) 11.4164 0.463760
\(607\) −8.58359 −0.348397 −0.174199 0.984711i \(-0.555733\pi\)
−0.174199 + 0.984711i \(0.555733\pi\)
\(608\) −36.3607 −1.47462
\(609\) 0 0
\(610\) 0 0
\(611\) −3.77709 −0.152805
\(612\) −10.4721 −0.423311
\(613\) 45.2492 1.82760 0.913799 0.406166i \(-0.133134\pi\)
0.913799 + 0.406166i \(0.133134\pi\)
\(614\) 2.83282 0.114323
\(615\) 0 0
\(616\) 0 0
\(617\) 7.58359 0.305304 0.152652 0.988280i \(-0.451219\pi\)
0.152652 + 0.988280i \(0.451219\pi\)
\(618\) −12.0000 −0.482711
\(619\) −26.4721 −1.06400 −0.532002 0.846743i \(-0.678560\pi\)
−0.532002 + 0.846743i \(0.678560\pi\)
\(620\) 0 0
\(621\) 0.236068 0.00947308
\(622\) 15.0557 0.603680
\(623\) 0 0
\(624\) 4.58359 0.183491
\(625\) 0 0
\(626\) −12.0000 −0.479616
\(627\) −14.4721 −0.577961
\(628\) −27.4164 −1.09403
\(629\) 35.4164 1.41214
\(630\) 0 0
\(631\) 31.6525 1.26007 0.630033 0.776569i \(-0.283042\pi\)
0.630033 + 0.776569i \(0.283042\pi\)
\(632\) 23.9443 0.952452
\(633\) 21.8885 0.869992
\(634\) −6.14590 −0.244085
\(635\) 0 0
\(636\) −9.70820 −0.384955
\(637\) 0 0
\(638\) 4.14590 0.164138
\(639\) 3.76393 0.148899
\(640\) 0 0
\(641\) −19.3607 −0.764701 −0.382350 0.924017i \(-0.624885\pi\)
−0.382350 + 0.924017i \(0.624885\pi\)
\(642\) −5.52786 −0.218167
\(643\) 24.0000 0.946468 0.473234 0.880937i \(-0.343087\pi\)
0.473234 + 0.880937i \(0.343087\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −25.8885 −1.01857
\(647\) −33.8885 −1.33230 −0.666148 0.745820i \(-0.732058\pi\)
−0.666148 + 0.745820i \(0.732058\pi\)
\(648\) 2.23607 0.0878410
\(649\) −5.52786 −0.216988
\(650\) 0 0
\(651\) 0 0
\(652\) 19.4164 0.760405
\(653\) 23.8885 0.934831 0.467415 0.884038i \(-0.345185\pi\)
0.467415 + 0.884038i \(0.345185\pi\)
\(654\) −6.43769 −0.251734
\(655\) 0 0
\(656\) −16.5836 −0.647480
\(657\) −5.52786 −0.215663
\(658\) 0 0
\(659\) −32.9443 −1.28333 −0.641663 0.766986i \(-0.721755\pi\)
−0.641663 + 0.766986i \(0.721755\pi\)
\(660\) 0 0
\(661\) −24.3607 −0.947521 −0.473760 0.880654i \(-0.657104\pi\)
−0.473760 + 0.880654i \(0.657104\pi\)
\(662\) 6.68692 0.259894
\(663\) 16.0000 0.621389
\(664\) −2.11146 −0.0819404
\(665\) 0 0
\(666\) −3.38197 −0.131049
\(667\) 0.708204 0.0274218
\(668\) 21.8885 0.846893
\(669\) −23.4164 −0.905331
\(670\) 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 \(0.696248\pi\)
\(674\) −1.23607 −0.0476116
\(675\) 0 0
\(676\) 11.1459 0.428688
\(677\) −14.1115 −0.542347 −0.271174 0.962530i \(-0.587412\pi\)
−0.271174 + 0.962530i \(0.587412\pi\)
\(678\) 12.0344 0.462180
\(679\) 0 0
\(680\) 0 0
\(681\) −19.4164 −0.744038
\(682\) −5.52786 −0.211673
\(683\) 4.81966 0.184419 0.0922096 0.995740i \(-0.470607\pi\)
0.0922096 + 0.995740i \(0.470607\pi\)
\(684\) −10.4721 −0.400412
\(685\) 0 0
\(686\) 0 0
\(687\) −11.0557 −0.421802
\(688\) −16.1459 −0.615557
\(689\) 14.8328 0.565085
\(690\) 0 0
\(691\) −42.8328 −1.62944 −0.814719 0.579857i \(-0.803109\pi\)
−0.814719 + 0.579857i \(0.803109\pi\)
\(692\) 10.4721 0.398091
\(693\) 0 0
\(694\) −11.5623 −0.438899
\(695\) 0 0
\(696\) 6.70820 0.254274
\(697\) −57.8885 −2.19268
\(698\) 2.11146 0.0799198
\(699\) −13.4721 −0.509563
\(700\) 0 0
\(701\) −33.7771 −1.27574 −0.637871 0.770143i \(-0.720185\pi\)
−0.637871 + 0.770143i \(0.720185\pi\)
\(702\) −1.52786 −0.0576655
\(703\) 35.4164 1.33576
\(704\) −0.527864 −0.0198946
\(705\) 0 0
\(706\) 13.5279 0.509128
\(707\) 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
\(711\) 10.7082 0.401589
\(712\) 23.4164 0.877567
\(713\) −0.944272 −0.0353633
\(714\) 0 0
\(715\) 0 0
\(716\) −27.4164 −1.02460
\(717\) 3.05573 0.114118
\(718\) 10.0344 0.374482
\(719\) 16.9443 0.631915 0.315957 0.948773i \(-0.397674\pi\)
0.315957 + 0.948773i \(0.397674\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −14.1459 −0.526456
\(723\) −9.52786 −0.354345
\(724\) 4.00000 0.148659
\(725\) 0 0
\(726\) −3.70820 −0.137624
\(727\) −45.3050 −1.68027 −0.840134 0.542379i \(-0.817524\pi\)
−0.840134 + 0.542379i \(0.817524\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −56.3607 −2.08458
\(732\) 19.4164 0.717651
\(733\) 32.0000 1.18195 0.590973 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) 2.47214 0.0912482
\(735\) 0 0
\(736\) 1.32624 0.0488858
\(737\) 1.58359 0.0583324
\(738\) 5.52786 0.203483
\(739\) −7.29180 −0.268233 −0.134117 0.990966i \(-0.542820\pi\)
−0.134117 + 0.990966i \(0.542820\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) 0 0
\(743\) 20.9443 0.768371 0.384185 0.923256i \(-0.374482\pi\)
0.384185 + 0.923256i \(0.374482\pi\)
\(744\) −8.94427 −0.327913
\(745\) 0 0
\(746\) 5.27051 0.192967
\(747\) −0.944272 −0.0345491
\(748\) −23.4164 −0.856189
\(749\) 0 0
\(750\) 0 0
\(751\) 51.7771 1.88937 0.944686 0.327975i \(-0.106366\pi\)
0.944686 + 0.327975i \(0.106366\pi\)
\(752\) 2.83282 0.103302
\(753\) 1.52786 0.0556785
\(754\) −4.58359 −0.166925
\(755\) 0 0
\(756\) 0 0
\(757\) −3.47214 −0.126197 −0.0630985 0.998007i \(-0.520098\pi\)
−0.0630985 + 0.998007i \(0.520098\pi\)
\(758\) 2.90983 0.105690
\(759\) 0.527864 0.0191603
\(760\) 0 0
\(761\) −35.0557 −1.27077 −0.635385 0.772196i \(-0.719158\pi\)
−0.635385 + 0.772196i \(0.719158\pi\)
\(762\) 1.96556 0.0712047
\(763\) 0 0
\(764\) −33.8885 −1.22604
\(765\) 0 0
\(766\) −4.58359 −0.165612
\(767\) 6.11146 0.220672
\(768\) 6.56231 0.236797
\(769\) 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
\(772\) 20.8541 0.750556
\(773\) 31.7771 1.14294 0.571471 0.820622i \(-0.306373\pi\)
0.571471 + 0.820622i \(0.306373\pi\)
\(774\) 5.38197 0.193451
\(775\) 0 0
\(776\) −8.94427 −0.321081
\(777\) 0 0
\(778\) −18.5066 −0.663493
\(779\) −57.8885 −2.07407
\(780\) 0 0
\(781\) 8.41641 0.301163
\(782\) 0.944272 0.0337671
\(783\) 3.00000 0.107211
\(784\) 0 0
\(785\) 0 0
\(786\) 13.8885 0.495388
\(787\) −42.4721 −1.51397 −0.756984 0.653433i \(-0.773328\pi\)
−0.756984 + 0.653433i \(0.773328\pi\)
\(788\) −33.9787 −1.21044
\(789\) −1.18034 −0.0420212
\(790\) 0 0
\(791\) 0 0
\(792\) 5.00000 0.177667
\(793\) −29.6656 −1.05346
\(794\) 16.0000 0.567819
\(795\) 0 0
\(796\) −22.4721 −0.796504
\(797\) 25.5279 0.904243 0.452122 0.891956i \(-0.350667\pi\)
0.452122 + 0.891956i \(0.350667\pi\)
\(798\) 0 0
\(799\) 9.88854 0.349832
\(800\) 0 0
\(801\) 10.4721 0.370015
\(802\) −0.978714 −0.0345596
\(803\) −12.3607 −0.436199
\(804\) 1.14590 0.0404127
\(805\) 0 0
\(806\) 6.11146 0.215267
\(807\) −8.94427 −0.314853
\(808\) 41.3050 1.45310
\(809\) 17.4721 0.614288 0.307144 0.951663i \(-0.400627\pi\)
0.307144 + 0.951663i \(0.400627\pi\)
\(810\) 0 0
\(811\) −14.8328 −0.520851 −0.260425 0.965494i \(-0.583863\pi\)
−0.260425 + 0.965494i \(0.583863\pi\)
\(812\) 0 0
\(813\) −23.4164 −0.821249
\(814\) −7.56231 −0.265059
\(815\) 0 0
\(816\) −12.0000 −0.420084
\(817\) −56.3607 −1.97181
\(818\) 12.0000 0.419570
\(819\) 0 0
\(820\) 0 0
\(821\) 25.7771 0.899627 0.449813 0.893123i \(-0.351491\pi\)
0.449813 + 0.893123i \(0.351491\pi\)
\(822\) −9.81966 −0.342500
\(823\) −49.5410 −1.72689 −0.863446 0.504442i \(-0.831698\pi\)
−0.863446 + 0.504442i \(0.831698\pi\)
\(824\) −43.4164 −1.51248
\(825\) 0 0
\(826\) 0 0
\(827\) 9.76393 0.339525 0.169763 0.985485i \(-0.445700\pi\)
0.169763 + 0.985485i \(0.445700\pi\)
\(828\) 0.381966 0.0132742
\(829\) 6.47214 0.224787 0.112393 0.993664i \(-0.464148\pi\)
0.112393 + 0.993664i \(0.464148\pi\)
\(830\) 0 0
\(831\) −19.8885 −0.689926
\(832\) 0.583592 0.0202324
\(833\) 0 0
\(834\) 9.52786 0.329923
\(835\) 0 0
\(836\) −23.4164 −0.809873
\(837\) −4.00000 −0.138260
\(838\) 3.63932 0.125718
\(839\) 43.1935 1.49121 0.745603 0.666391i \(-0.232162\pi\)
0.745603 + 0.666391i \(0.232162\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 10.1459 0.349651
\(843\) −25.3607 −0.873468
\(844\) 35.4164 1.21908
\(845\) 0 0
\(846\) −0.944272 −0.0324647
\(847\) 0 0
\(848\) −11.1246 −0.382021
\(849\) 0 0
\(850\) 0 0
\(851\) −1.29180 −0.0442822
\(852\) 6.09017 0.208646
\(853\) −46.8328 −1.60353 −0.801763 0.597643i \(-0.796104\pi\)
−0.801763 + 0.597643i \(0.796104\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −20.0000 −0.683586
\(857\) 2.11146 0.0721260 0.0360630 0.999350i \(-0.488518\pi\)
0.0360630 + 0.999350i \(0.488518\pi\)
\(858\) −3.41641 −0.116634
\(859\) 9.88854 0.337393 0.168696 0.985668i \(-0.446044\pi\)
0.168696 + 0.985668i \(0.446044\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 17.8885 0.609286
\(863\) 38.1246 1.29778 0.648888 0.760884i \(-0.275234\pi\)
0.648888 + 0.760884i \(0.275234\pi\)
\(864\) 5.61803 0.191129
\(865\) 0 0
\(866\) −3.05573 −0.103838
\(867\) −24.8885 −0.845259
\(868\) 0 0
\(869\) 23.9443 0.812254
\(870\) 0 0
\(871\) −1.75078 −0.0593228
\(872\) −23.2918 −0.788760
\(873\) −4.00000 −0.135379
\(874\) 0.944272 0.0319405
\(875\) 0 0
\(876\) −8.94427 −0.302199
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) −0.360680 −0.0121724
\(879\) −28.3607 −0.956582
\(880\) 0 0
\(881\) 23.0557 0.776767 0.388384 0.921498i \(-0.373034\pi\)
0.388384 + 0.921498i \(0.373034\pi\)
\(882\) 0 0
\(883\) 23.5410 0.792218 0.396109 0.918203i \(-0.370360\pi\)
0.396109 + 0.918203i \(0.370360\pi\)
\(884\) 25.8885 0.870726
\(885\) 0 0
\(886\) −20.3607 −0.684030
\(887\) −26.8328 −0.900958 −0.450479 0.892787i \(-0.648747\pi\)
−0.450479 + 0.892787i \(0.648747\pi\)
\(888\) −12.2361 −0.410616
\(889\) 0 0
\(890\) 0 0
\(891\) 2.23607 0.0749111
\(892\) −37.8885 −1.26860
\(893\) 9.88854 0.330908
\(894\) −14.1459 −0.473110
\(895\) 0 0
\(896\) 0 0
\(897\) −0.583592 −0.0194856
\(898\) 9.56231 0.319098
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) −38.8328 −1.29371
\(902\) 12.3607 0.411566
\(903\) 0 0
\(904\) 43.5410 1.44815
\(905\) 0 0
\(906\) −4.14590 −0.137738
\(907\) 10.1115 0.335745 0.167873 0.985809i \(-0.446310\pi\)
0.167873 + 0.985809i \(0.446310\pi\)
\(908\) −31.4164 −1.04259
\(909\) 18.4721 0.612682
\(910\) 0 0
\(911\) 20.0132 0.663065 0.331533 0.943444i \(-0.392434\pi\)
0.331533 + 0.943444i \(0.392434\pi\)
\(912\) −12.0000 −0.397360
\(913\) −2.11146 −0.0698790
\(914\) −4.97871 −0.164681
\(915\) 0 0
\(916\) −17.8885 −0.591054
\(917\) 0 0
\(918\) 4.00000 0.132020
\(919\) −7.18034 −0.236858 −0.118429 0.992963i \(-0.537786\pi\)
−0.118429 + 0.992963i \(0.537786\pi\)
\(920\) 0 0
\(921\) 4.58359 0.151034
\(922\) −13.3050 −0.438175
\(923\) −9.30495 −0.306276
\(924\) 0 0
\(925\) 0 0
\(926\) 20.9443 0.688271
\(927\) −19.4164 −0.637719
\(928\) 16.8541 0.553263
\(929\) −31.4164 −1.03074 −0.515369 0.856968i \(-0.672345\pi\)
−0.515369 + 0.856968i \(0.672345\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −21.7984 −0.714029
\(933\) 24.3607 0.797533
\(934\) 20.9443 0.685318
\(935\) 0 0
\(936\) −5.52786 −0.180684
\(937\) 28.0000 0.914720 0.457360 0.889282i \(-0.348795\pi\)
0.457360 + 0.889282i \(0.348795\pi\)
\(938\) 0 0
\(939\) −19.4164 −0.633631
\(940\) 0 0
\(941\) 39.7771 1.29670 0.648348 0.761344i \(-0.275460\pi\)
0.648348 + 0.761344i \(0.275460\pi\)
\(942\) 10.4721 0.341201
\(943\) 2.11146 0.0687585
\(944\) −4.58359 −0.149183
\(945\) 0 0
\(946\) 12.0344 0.391273
\(947\) −15.0557 −0.489245 −0.244623 0.969618i \(-0.578664\pi\)
−0.244623 + 0.969618i \(0.578664\pi\)
\(948\) 17.3262 0.562730
\(949\) 13.6656 0.443605
\(950\) 0 0
\(951\) −9.94427 −0.322465
\(952\) 0 0
\(953\) 33.3607 1.08066 0.540329 0.841454i \(-0.318300\pi\)
0.540329 + 0.841454i \(0.318300\pi\)
\(954\) 3.70820 0.120058
\(955\) 0 0
\(956\) 4.94427 0.159909
\(957\) 6.70820 0.216845
\(958\) 13.3050 0.429863
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 8.36068 0.269559
\(963\) −8.94427 −0.288225
\(964\) −15.4164 −0.496529
\(965\) 0 0
\(966\) 0 0
\(967\) −27.7771 −0.893251 −0.446625 0.894721i \(-0.647374\pi\)
−0.446625 + 0.894721i \(0.647374\pi\)
\(968\) −13.4164 −0.431220
\(969\) −41.8885 −1.34565
\(970\) 0 0
\(971\) −21.3050 −0.683708 −0.341854 0.939753i \(-0.611055\pi\)
−0.341854 + 0.939753i \(0.611055\pi\)
\(972\) 1.61803 0.0518985
\(973\) 0 0
\(974\) 0.798374 0.0255815
\(975\) 0 0
\(976\) 22.2492 0.712180
\(977\) 34.4164 1.10108 0.550539 0.834809i \(-0.314422\pi\)
0.550539 + 0.834809i \(0.314422\pi\)
\(978\) −7.41641 −0.237151
\(979\) 23.4164 0.748392
\(980\) 0 0
\(981\) −10.4164 −0.332570
\(982\) −16.7984 −0.536058
\(983\) 40.9443 1.30592 0.652960 0.757393i \(-0.273527\pi\)
0.652960 + 0.757393i \(0.273527\pi\)
\(984\) 20.0000 0.637577
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 25.8885 0.823624
\(989\) 2.05573 0.0653684
\(990\) 0 0
\(991\) 9.54102 0.303080 0.151540 0.988451i \(-0.451577\pi\)
0.151540 + 0.988451i \(0.451577\pi\)
\(992\) −22.4721 −0.713491
\(993\) 10.8197 0.343352
\(994\) 0 0
\(995\) 0 0
\(996\) −1.52786 −0.0484122
\(997\) −19.4164 −0.614924 −0.307462 0.951560i \(-0.599480\pi\)
−0.307462 + 0.951560i \(0.599480\pi\)
\(998\) 7.41641 0.234762
\(999\) −5.47214 −0.173131
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.1.1 yes 2
5.4 even 2 3675.2.a.v.1.2 yes 2
7.6 odd 2 3675.2.a.bc.1.1 yes 2
35.34 odd 2 3675.2.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3675.2.a.u.1.2 2 35.34 odd 2
3675.2.a.v.1.2 yes 2 5.4 even 2
3675.2.a.ba.1.1 yes 2 1.1 even 1 trivial
3675.2.a.bc.1.1 yes 2 7.6 odd 2