Properties

Label 3969.2.a.be.1.3
Level $3969$
Weight $2$
Character 3969.1
Self dual yes
Analytic conductor $31.693$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.6926245622\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.59351616.1
Defining polynomial: \(x^{6} - 12 x^{4} + 21 x^{2} - 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 441)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.15750\) of defining polynomial
Character \(\chi\) \(=\) 3969.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.239123 q^{2} -1.94282 q^{4} -2.59179 q^{5} -0.942820 q^{8} +O(q^{10})\) \(q+0.239123 q^{2} -1.94282 q^{4} -2.59179 q^{5} -0.942820 q^{8} -0.619757 q^{10} +4.18194 q^{11} +3.68310 q^{13} +3.66019 q^{16} +1.71107 q^{17} -7.15561 q^{19} +5.03538 q^{20} +1.00000 q^{22} -5.12476 q^{23} +1.71737 q^{25} +0.880716 q^{26} -2.12476 q^{29} -6.53585 q^{31} +2.76088 q^{32} +0.409157 q^{34} +1.66019 q^{37} -1.71107 q^{38} +2.44359 q^{40} -10.2190 q^{41} -1.66019 q^{43} -8.12476 q^{44} -1.22545 q^{46} +9.33824 q^{47} +0.410663 q^{50} -7.15561 q^{52} +10.6465 q^{53} -10.8387 q^{55} -0.508080 q^{58} +6.06429 q^{59} -7.98597 q^{61} -1.56287 q^{62} -6.66019 q^{64} -9.54583 q^{65} +8.26320 q^{67} -3.32431 q^{68} +6.23912 q^{71} +7.15561 q^{73} +0.396990 q^{74} +13.9021 q^{76} -9.82846 q^{79} -9.48644 q^{80} -2.44359 q^{82} +6.89465 q^{83} -4.43474 q^{85} -0.396990 q^{86} -3.94282 q^{88} -5.03538 q^{89} +9.95649 q^{92} +2.23299 q^{94} +18.5458 q^{95} +3.06335 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{2} + 6q^{4} + 12q^{8} + O(q^{10}) \) \( 6q + 2q^{2} + 6q^{4} + 12q^{8} + 8q^{11} + 6q^{16} + 6q^{22} + 4q^{23} + 12q^{25} + 22q^{29} + 16q^{32} - 6q^{37} + 6q^{43} - 14q^{44} + 12q^{46} + 56q^{50} + 28q^{53} + 18q^{58} - 24q^{64} - 6q^{65} + 38q^{71} + 36q^{74} - 6q^{79} - 30q^{85} - 36q^{86} - 6q^{88} + 62q^{92} + 60q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.239123 0.169086 0.0845428 0.996420i \(-0.473057\pi\)
0.0845428 + 0.996420i \(0.473057\pi\)
\(3\) 0 0
\(4\) −1.94282 −0.971410
\(5\) −2.59179 −1.15908 −0.579542 0.814943i \(-0.696768\pi\)
−0.579542 + 0.814943i \(0.696768\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −0.942820 −0.333337
\(9\) 0 0
\(10\) −0.619757 −0.195984
\(11\) 4.18194 1.26090 0.630452 0.776228i \(-0.282870\pi\)
0.630452 + 0.776228i \(0.282870\pi\)
\(12\) 0 0
\(13\) 3.68310 1.02151 0.510755 0.859726i \(-0.329366\pi\)
0.510755 + 0.859726i \(0.329366\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.66019 0.915047
\(17\) 1.71107 0.414996 0.207498 0.978235i \(-0.433468\pi\)
0.207498 + 0.978235i \(0.433468\pi\)
\(18\) 0 0
\(19\) −7.15561 −1.64161 −0.820805 0.571209i \(-0.806475\pi\)
−0.820805 + 0.571209i \(0.806475\pi\)
\(20\) 5.03538 1.12595
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −5.12476 −1.06859 −0.534294 0.845299i \(-0.679422\pi\)
−0.534294 + 0.845299i \(0.679422\pi\)
\(24\) 0 0
\(25\) 1.71737 0.343474
\(26\) 0.880716 0.172723
\(27\) 0 0
\(28\) 0 0
\(29\) −2.12476 −0.394559 −0.197279 0.980347i \(-0.563211\pi\)
−0.197279 + 0.980347i \(0.563211\pi\)
\(30\) 0 0
\(31\) −6.53585 −1.17387 −0.586937 0.809633i \(-0.699666\pi\)
−0.586937 + 0.809633i \(0.699666\pi\)
\(32\) 2.76088 0.488059
\(33\) 0 0
\(34\) 0.409157 0.0701699
\(35\) 0 0
\(36\) 0 0
\(37\) 1.66019 0.272934 0.136467 0.990645i \(-0.456425\pi\)
0.136467 + 0.990645i \(0.456425\pi\)
\(38\) −1.71107 −0.277573
\(39\) 0 0
\(40\) 2.44359 0.386366
\(41\) −10.2190 −1.59593 −0.797967 0.602702i \(-0.794091\pi\)
−0.797967 + 0.602702i \(0.794091\pi\)
\(42\) 0 0
\(43\) −1.66019 −0.253177 −0.126588 0.991955i \(-0.540403\pi\)
−0.126588 + 0.991955i \(0.540403\pi\)
\(44\) −8.12476 −1.22485
\(45\) 0 0
\(46\) −1.22545 −0.180683
\(47\) 9.33824 1.36212 0.681061 0.732226i \(-0.261519\pi\)
0.681061 + 0.732226i \(0.261519\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0.410663 0.0580765
\(51\) 0 0
\(52\) −7.15561 −0.992305
\(53\) 10.6465 1.46241 0.731206 0.682157i \(-0.238958\pi\)
0.731206 + 0.682157i \(0.238958\pi\)
\(54\) 0 0
\(55\) −10.8387 −1.46149
\(56\) 0 0
\(57\) 0 0
\(58\) −0.508080 −0.0667142
\(59\) 6.06429 0.789504 0.394752 0.918788i \(-0.370831\pi\)
0.394752 + 0.918788i \(0.370831\pi\)
\(60\) 0 0
\(61\) −7.98597 −1.02250 −0.511249 0.859433i \(-0.670817\pi\)
−0.511249 + 0.859433i \(0.670817\pi\)
\(62\) −1.56287 −0.198485
\(63\) 0 0
\(64\) −6.66019 −0.832524
\(65\) −9.54583 −1.18401
\(66\) 0 0
\(67\) 8.26320 1.00951 0.504755 0.863262i \(-0.331583\pi\)
0.504755 + 0.863262i \(0.331583\pi\)
\(68\) −3.32431 −0.403131
\(69\) 0 0
\(70\) 0 0
\(71\) 6.23912 0.740448 0.370224 0.928943i \(-0.379281\pi\)
0.370224 + 0.928943i \(0.379281\pi\)
\(72\) 0 0
\(73\) 7.15561 0.837501 0.418750 0.908101i \(-0.362468\pi\)
0.418750 + 0.908101i \(0.362468\pi\)
\(74\) 0.396990 0.0461492
\(75\) 0 0
\(76\) 13.9021 1.59468
\(77\) 0 0
\(78\) 0 0
\(79\) −9.82846 −1.10579 −0.552894 0.833252i \(-0.686477\pi\)
−0.552894 + 0.833252i \(0.686477\pi\)
\(80\) −9.48644 −1.06062
\(81\) 0 0
\(82\) −2.44359 −0.269849
\(83\) 6.89465 0.756786 0.378393 0.925645i \(-0.376477\pi\)
0.378393 + 0.925645i \(0.376477\pi\)
\(84\) 0 0
\(85\) −4.43474 −0.481015
\(86\) −0.396990 −0.0428085
\(87\) 0 0
\(88\) −3.94282 −0.420306
\(89\) −5.03538 −0.533749 −0.266875 0.963731i \(-0.585991\pi\)
−0.266875 + 0.963731i \(0.585991\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 9.95649 1.03804
\(93\) 0 0
\(94\) 2.23299 0.230315
\(95\) 18.5458 1.90276
\(96\) 0 0
\(97\) 3.06335 0.311036 0.155518 0.987833i \(-0.450295\pi\)
0.155518 + 0.987833i \(0.450295\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −3.33654 −0.333654
\(101\) −11.0997 −1.10446 −0.552229 0.833692i \(-0.686223\pi\)
−0.552229 + 0.833692i \(0.686223\pi\)
\(102\) 0 0
\(103\) 7.98597 0.786881 0.393440 0.919350i \(-0.371285\pi\)
0.393440 + 0.919350i \(0.371285\pi\)
\(104\) −3.47250 −0.340507
\(105\) 0 0
\(106\) 2.54583 0.247273
\(107\) 3.95649 0.382489 0.191244 0.981542i \(-0.438748\pi\)
0.191244 + 0.981542i \(0.438748\pi\)
\(108\) 0 0
\(109\) 7.26320 0.695688 0.347844 0.937552i \(-0.386914\pi\)
0.347844 + 0.937552i \(0.386914\pi\)
\(110\) −2.59179 −0.247117
\(111\) 0 0
\(112\) 0 0
\(113\) 6.92915 0.651839 0.325920 0.945397i \(-0.394326\pi\)
0.325920 + 0.945397i \(0.394326\pi\)
\(114\) 0 0
\(115\) 13.2823 1.23858
\(116\) 4.12803 0.383278
\(117\) 0 0
\(118\) 1.45011 0.133494
\(119\) 0 0
\(120\) 0 0
\(121\) 6.48865 0.589877
\(122\) −1.90963 −0.172890
\(123\) 0 0
\(124\) 12.6980 1.14031
\(125\) 8.50788 0.760968
\(126\) 0 0
\(127\) 9.11109 0.808479 0.404239 0.914653i \(-0.367536\pi\)
0.404239 + 0.914653i \(0.367536\pi\)
\(128\) −7.11436 −0.628827
\(129\) 0 0
\(130\) −2.28263 −0.200200
\(131\) −4.30286 −0.375943 −0.187971 0.982175i \(-0.560191\pi\)
−0.187971 + 0.982175i \(0.560191\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.97592 0.170694
\(135\) 0 0
\(136\) −1.61323 −0.138334
\(137\) 20.5893 1.75907 0.879533 0.475838i \(-0.157855\pi\)
0.879533 + 0.475838i \(0.157855\pi\)
\(138\) 0 0
\(139\) −15.7613 −1.33686 −0.668429 0.743776i \(-0.733033\pi\)
−0.668429 + 0.743776i \(0.733033\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.49192 0.125199
\(143\) 15.4025 1.28802
\(144\) 0 0
\(145\) 5.50694 0.457326
\(146\) 1.71107 0.141609
\(147\) 0 0
\(148\) −3.22545 −0.265130
\(149\) 6.06758 0.497076 0.248538 0.968622i \(-0.420050\pi\)
0.248538 + 0.968622i \(0.420050\pi\)
\(150\) 0 0
\(151\) 4.48865 0.365281 0.182641 0.983180i \(-0.441536\pi\)
0.182641 + 0.983180i \(0.441536\pi\)
\(152\) 6.74645 0.547210
\(153\) 0 0
\(154\) 0 0
\(155\) 16.9396 1.36062
\(156\) 0 0
\(157\) 1.02891 0.0821163 0.0410582 0.999157i \(-0.486927\pi\)
0.0410582 + 0.999157i \(0.486927\pi\)
\(158\) −2.35021 −0.186973
\(159\) 0 0
\(160\) −7.15561 −0.565701
\(161\) 0 0
\(162\) 0 0
\(163\) 6.82846 0.534846 0.267423 0.963579i \(-0.413828\pi\)
0.267423 + 0.963579i \(0.413828\pi\)
\(164\) 19.8536 1.55031
\(165\) 0 0
\(166\) 1.64867 0.127962
\(167\) 17.9943 1.39244 0.696221 0.717827i \(-0.254863\pi\)
0.696221 + 0.717827i \(0.254863\pi\)
\(168\) 0 0
\(169\) 0.565260 0.0434816
\(170\) −1.06045 −0.0813328
\(171\) 0 0
\(172\) 3.22545 0.245938
\(173\) −0.830357 −0.0631309 −0.0315654 0.999502i \(-0.510049\pi\)
−0.0315654 + 0.999502i \(0.510049\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 15.3067 1.15379
\(177\) 0 0
\(178\) −1.20408 −0.0902493
\(179\) 7.57893 0.566476 0.283238 0.959050i \(-0.408591\pi\)
0.283238 + 0.959050i \(0.408591\pi\)
\(180\) 0 0
\(181\) 0.409157 0.0304124 0.0152062 0.999884i \(-0.495160\pi\)
0.0152062 + 0.999884i \(0.495160\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 4.83173 0.356200
\(185\) −4.30286 −0.316353
\(186\) 0 0
\(187\) 7.15561 0.523270
\(188\) −18.1425 −1.32318
\(189\) 0 0
\(190\) 4.43474 0.321730
\(191\) 16.0241 1.15946 0.579731 0.814808i \(-0.303158\pi\)
0.579731 + 0.814808i \(0.303158\pi\)
\(192\) 0 0
\(193\) −12.3743 −0.890721 −0.445360 0.895351i \(-0.646924\pi\)
−0.445360 + 0.895351i \(0.646924\pi\)
\(194\) 0.732518 0.0525917
\(195\) 0 0
\(196\) 0 0
\(197\) 23.1021 1.64595 0.822977 0.568075i \(-0.192312\pi\)
0.822977 + 0.568075i \(0.192312\pi\)
\(198\) 0 0
\(199\) −6.74645 −0.478243 −0.239122 0.970990i \(-0.576859\pi\)
−0.239122 + 0.970990i \(0.576859\pi\)
\(200\) −1.61917 −0.114493
\(201\) 0 0
\(202\) −2.65419 −0.186748
\(203\) 0 0
\(204\) 0 0
\(205\) 26.4854 1.84982
\(206\) 1.90963 0.133050
\(207\) 0 0
\(208\) 13.4809 0.934730
\(209\) −29.9244 −2.06991
\(210\) 0 0
\(211\) 16.8856 1.16246 0.581228 0.813741i \(-0.302573\pi\)
0.581228 + 0.813741i \(0.302573\pi\)
\(212\) −20.6843 −1.42060
\(213\) 0 0
\(214\) 0.946090 0.0646734
\(215\) 4.30286 0.293453
\(216\) 0 0
\(217\) 0 0
\(218\) 1.73680 0.117631
\(219\) 0 0
\(220\) 21.0577 1.41971
\(221\) 6.30206 0.423922
\(222\) 0 0
\(223\) −4.50142 −0.301437 −0.150719 0.988577i \(-0.548159\pi\)
−0.150719 + 0.988577i \(0.548159\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1.65692 0.110217
\(227\) −6.06429 −0.402501 −0.201251 0.979540i \(-0.564501\pi\)
−0.201251 + 0.979540i \(0.564501\pi\)
\(228\) 0 0
\(229\) −11.0493 −0.730159 −0.365080 0.930976i \(-0.618958\pi\)
−0.365080 + 0.930976i \(0.618958\pi\)
\(230\) 3.17611 0.209426
\(231\) 0 0
\(232\) 2.00327 0.131521
\(233\) −8.13844 −0.533167 −0.266583 0.963812i \(-0.585895\pi\)
−0.266583 + 0.963812i \(0.585895\pi\)
\(234\) 0 0
\(235\) −24.2028 −1.57881
\(236\) −11.7818 −0.766932
\(237\) 0 0
\(238\) 0 0
\(239\) 21.1625 1.36889 0.684445 0.729065i \(-0.260045\pi\)
0.684445 + 0.729065i \(0.260045\pi\)
\(240\) 0 0
\(241\) 13.6915 0.881945 0.440972 0.897521i \(-0.354634\pi\)
0.440972 + 0.897521i \(0.354634\pi\)
\(242\) 1.55159 0.0997398
\(243\) 0 0
\(244\) 15.5153 0.993265
\(245\) 0 0
\(246\) 0 0
\(247\) −26.3549 −1.67692
\(248\) 6.16213 0.391296
\(249\) 0 0
\(250\) 2.03443 0.128669
\(251\) 15.2040 0.959667 0.479833 0.877360i \(-0.340697\pi\)
0.479833 + 0.877360i \(0.340697\pi\)
\(252\) 0 0
\(253\) −21.4315 −1.34738
\(254\) 2.17867 0.136702
\(255\) 0 0
\(256\) 11.6192 0.726198
\(257\) 25.6215 1.59822 0.799112 0.601182i \(-0.205303\pi\)
0.799112 + 0.601182i \(0.205303\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 18.5458 1.15016
\(261\) 0 0
\(262\) −1.02891 −0.0635665
\(263\) 7.10069 0.437847 0.218924 0.975742i \(-0.429745\pi\)
0.218924 + 0.975742i \(0.429745\pi\)
\(264\) 0 0
\(265\) −27.5935 −1.69506
\(266\) 0 0
\(267\) 0 0
\(268\) −16.0539 −0.980649
\(269\) −16.4314 −1.00184 −0.500922 0.865493i \(-0.667006\pi\)
−0.500922 + 0.865493i \(0.667006\pi\)
\(270\) 0 0
\(271\) 12.6980 0.771348 0.385674 0.922635i \(-0.373969\pi\)
0.385674 + 0.922635i \(0.373969\pi\)
\(272\) 6.26285 0.379741
\(273\) 0 0
\(274\) 4.92339 0.297433
\(275\) 7.18194 0.433087
\(276\) 0 0
\(277\) −0.828460 −0.0497773 −0.0248887 0.999690i \(-0.507923\pi\)
−0.0248887 + 0.999690i \(0.507923\pi\)
\(278\) −3.76890 −0.226044
\(279\) 0 0
\(280\) 0 0
\(281\) 5.21969 0.311381 0.155690 0.987806i \(-0.450240\pi\)
0.155690 + 0.987806i \(0.450240\pi\)
\(282\) 0 0
\(283\) −7.35417 −0.437160 −0.218580 0.975819i \(-0.570142\pi\)
−0.218580 + 0.975819i \(0.570142\pi\)
\(284\) −12.1215 −0.719278
\(285\) 0 0
\(286\) 3.68310 0.217787
\(287\) 0 0
\(288\) 0 0
\(289\) −14.0722 −0.827778
\(290\) 1.31684 0.0773273
\(291\) 0 0
\(292\) −13.9021 −0.813557
\(293\) 7.82573 0.457184 0.228592 0.973522i \(-0.426588\pi\)
0.228592 + 0.973522i \(0.426588\pi\)
\(294\) 0 0
\(295\) −15.7174 −0.915101
\(296\) −1.56526 −0.0909789
\(297\) 0 0
\(298\) 1.45090 0.0840484
\(299\) −18.8750 −1.09157
\(300\) 0 0
\(301\) 0 0
\(302\) 1.07334 0.0617638
\(303\) 0 0
\(304\) −26.1909 −1.50215
\(305\) 20.6979 1.18516
\(306\) 0 0
\(307\) 22.6709 1.29390 0.646948 0.762534i \(-0.276045\pi\)
0.646948 + 0.762534i \(0.276045\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4.05064 0.230061
\(311\) −32.3176 −1.83256 −0.916281 0.400536i \(-0.868824\pi\)
−0.916281 + 0.400536i \(0.868824\pi\)
\(312\) 0 0
\(313\) −24.3196 −1.37462 −0.687312 0.726362i \(-0.741209\pi\)
−0.687312 + 0.726362i \(0.741209\pi\)
\(314\) 0.246037 0.0138847
\(315\) 0 0
\(316\) 19.0949 1.07417
\(317\) 5.13844 0.288603 0.144302 0.989534i \(-0.453906\pi\)
0.144302 + 0.989534i \(0.453906\pi\)
\(318\) 0 0
\(319\) −8.88564 −0.497500
\(320\) 17.2618 0.964964
\(321\) 0 0
\(322\) 0 0
\(323\) −12.2438 −0.681262
\(324\) 0 0
\(325\) 6.32525 0.350862
\(326\) 1.63284 0.0904349
\(327\) 0 0
\(328\) 9.63464 0.531984
\(329\) 0 0
\(330\) 0 0
\(331\) −11.6979 −0.642977 −0.321488 0.946913i \(-0.604183\pi\)
−0.321488 + 0.946913i \(0.604183\pi\)
\(332\) −13.3951 −0.735150
\(333\) 0 0
\(334\) 4.30286 0.235442
\(335\) −21.4165 −1.17011
\(336\) 0 0
\(337\) −33.6947 −1.83547 −0.917733 0.397198i \(-0.869983\pi\)
−0.917733 + 0.397198i \(0.869983\pi\)
\(338\) 0.135167 0.00735211
\(339\) 0 0
\(340\) 8.61590 0.467263
\(341\) −27.3326 −1.48014
\(342\) 0 0
\(343\) 0 0
\(344\) 1.56526 0.0843932
\(345\) 0 0
\(346\) −0.198558 −0.0106745
\(347\) 27.3114 1.46615 0.733075 0.680148i \(-0.238084\pi\)
0.733075 + 0.680148i \(0.238084\pi\)
\(348\) 0 0
\(349\) 22.9169 1.22672 0.613358 0.789805i \(-0.289818\pi\)
0.613358 + 0.789805i \(0.289818\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 11.5458 0.615395
\(353\) 10.2693 0.546581 0.273290 0.961932i \(-0.411888\pi\)
0.273290 + 0.961932i \(0.411888\pi\)
\(354\) 0 0
\(355\) −16.1705 −0.858241
\(356\) 9.78284 0.518489
\(357\) 0 0
\(358\) 1.81230 0.0957830
\(359\) 10.1007 0.533094 0.266547 0.963822i \(-0.414117\pi\)
0.266547 + 0.963822i \(0.414117\pi\)
\(360\) 0 0
\(361\) 32.2028 1.69488
\(362\) 0.0978390 0.00514231
\(363\) 0 0
\(364\) 0 0
\(365\) −18.5458 −0.970733
\(366\) 0 0
\(367\) −7.77537 −0.405871 −0.202935 0.979192i \(-0.565048\pi\)
−0.202935 + 0.979192i \(0.565048\pi\)
\(368\) −18.7576 −0.977808
\(369\) 0 0
\(370\) −1.02891 −0.0534907
\(371\) 0 0
\(372\) 0 0
\(373\) 24.1111 1.24842 0.624212 0.781255i \(-0.285420\pi\)
0.624212 + 0.781255i \(0.285420\pi\)
\(374\) 1.71107 0.0884775
\(375\) 0 0
\(376\) −8.80428 −0.454046
\(377\) −7.82573 −0.403045
\(378\) 0 0
\(379\) −13.3581 −0.686161 −0.343081 0.939306i \(-0.611470\pi\)
−0.343081 + 0.939306i \(0.611470\pi\)
\(380\) −36.0312 −1.84836
\(381\) 0 0
\(382\) 3.83173 0.196048
\(383\) −9.24040 −0.472162 −0.236081 0.971733i \(-0.575863\pi\)
−0.236081 + 0.971733i \(0.575863\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.95898 −0.150608
\(387\) 0 0
\(388\) −5.95153 −0.302143
\(389\) 10.4484 0.529756 0.264878 0.964282i \(-0.414668\pi\)
0.264878 + 0.964282i \(0.414668\pi\)
\(390\) 0 0
\(391\) −8.76884 −0.443459
\(392\) 0 0
\(393\) 0 0
\(394\) 5.52424 0.278307
\(395\) 25.4733 1.28170
\(396\) 0 0
\(397\) −0.409157 −0.0205350 −0.0102675 0.999947i \(-0.503268\pi\)
−0.0102675 + 0.999947i \(0.503268\pi\)
\(398\) −1.61323 −0.0808641
\(399\) 0 0
\(400\) 6.28590 0.314295
\(401\) 15.2528 0.761688 0.380844 0.924639i \(-0.375633\pi\)
0.380844 + 0.924639i \(0.375633\pi\)
\(402\) 0 0
\(403\) −24.0722 −1.19912
\(404\) 21.5647 1.07288
\(405\) 0 0
\(406\) 0 0
\(407\) 6.94282 0.344143
\(408\) 0 0
\(409\) −6.12670 −0.302946 −0.151473 0.988461i \(-0.548402\pi\)
−0.151473 + 0.988461i \(0.548402\pi\)
\(410\) 6.33327 0.312778
\(411\) 0 0
\(412\) −15.5153 −0.764384
\(413\) 0 0
\(414\) 0 0
\(415\) −17.8695 −0.877178
\(416\) 10.1686 0.498557
\(417\) 0 0
\(418\) −7.15561 −0.349992
\(419\) 1.56287 0.0763514 0.0381757 0.999271i \(-0.487845\pi\)
0.0381757 + 0.999271i \(0.487845\pi\)
\(420\) 0 0
\(421\) 23.2632 1.13378 0.566889 0.823794i \(-0.308147\pi\)
0.566889 + 0.823794i \(0.308147\pi\)
\(422\) 4.03775 0.196555
\(423\) 0 0
\(424\) −10.0377 −0.487476
\(425\) 2.93854 0.142540
\(426\) 0 0
\(427\) 0 0
\(428\) −7.68675 −0.371553
\(429\) 0 0
\(430\) 1.02891 0.0496187
\(431\) 1.00576 0.0484456 0.0242228 0.999707i \(-0.492289\pi\)
0.0242228 + 0.999707i \(0.492289\pi\)
\(432\) 0 0
\(433\) 13.1071 0.629889 0.314945 0.949110i \(-0.398014\pi\)
0.314945 + 0.949110i \(0.398014\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.1111 −0.675799
\(437\) 36.6708 1.75420
\(438\) 0 0
\(439\) 18.6141 0.888402 0.444201 0.895927i \(-0.353488\pi\)
0.444201 + 0.895927i \(0.353488\pi\)
\(440\) 10.2190 0.487170
\(441\) 0 0
\(442\) 1.50697 0.0716792
\(443\) −1.11901 −0.0531656 −0.0265828 0.999647i \(-0.508463\pi\)
−0.0265828 + 0.999647i \(0.508463\pi\)
\(444\) 0 0
\(445\) 13.0506 0.618660
\(446\) −1.07639 −0.0509687
\(447\) 0 0
\(448\) 0 0
\(449\) −39.4419 −1.86138 −0.930689 0.365813i \(-0.880791\pi\)
−0.930689 + 0.365813i \(0.880791\pi\)
\(450\) 0 0
\(451\) −42.7351 −2.01232
\(452\) −13.4621 −0.633203
\(453\) 0 0
\(454\) −1.45011 −0.0680572
\(455\) 0 0
\(456\) 0 0
\(457\) −34.2405 −1.60170 −0.800852 0.598863i \(-0.795619\pi\)
−0.800852 + 0.598863i \(0.795619\pi\)
\(458\) −2.64215 −0.123459
\(459\) 0 0
\(460\) −25.8051 −1.20317
\(461\) 20.3876 0.949543 0.474772 0.880109i \(-0.342531\pi\)
0.474772 + 0.880109i \(0.342531\pi\)
\(462\) 0 0
\(463\) 6.80903 0.316442 0.158221 0.987404i \(-0.449424\pi\)
0.158221 + 0.987404i \(0.449424\pi\)
\(464\) −7.77704 −0.361040
\(465\) 0 0
\(466\) −1.94609 −0.0901509
\(467\) 24.7911 1.14720 0.573598 0.819137i \(-0.305547\pi\)
0.573598 + 0.819137i \(0.305547\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −5.78744 −0.266955
\(471\) 0 0
\(472\) −5.71754 −0.263171
\(473\) −6.94282 −0.319231
\(474\) 0 0
\(475\) −12.2888 −0.563850
\(476\) 0 0
\(477\) 0 0
\(478\) 5.06045 0.231460
\(479\) −11.0997 −0.507157 −0.253579 0.967315i \(-0.581608\pi\)
−0.253579 + 0.967315i \(0.581608\pi\)
\(480\) 0 0
\(481\) 6.11465 0.278804
\(482\) 3.27395 0.149124
\(483\) 0 0
\(484\) −12.6063 −0.573013
\(485\) −7.93955 −0.360516
\(486\) 0 0
\(487\) −10.0377 −0.454854 −0.227427 0.973795i \(-0.573031\pi\)
−0.227427 + 0.973795i \(0.573031\pi\)
\(488\) 7.52933 0.340837
\(489\) 0 0
\(490\) 0 0
\(491\) −12.3880 −0.559061 −0.279530 0.960137i \(-0.590179\pi\)
−0.279530 + 0.960137i \(0.590179\pi\)
\(492\) 0 0
\(493\) −3.63562 −0.163740
\(494\) −6.30206 −0.283543
\(495\) 0 0
\(496\) −23.9225 −1.07415
\(497\) 0 0
\(498\) 0 0
\(499\) 10.2222 0.457608 0.228804 0.973473i \(-0.426519\pi\)
0.228804 + 0.973473i \(0.426519\pi\)
\(500\) −16.5293 −0.739212
\(501\) 0 0
\(502\) 3.63562 0.162266
\(503\) −8.45753 −0.377102 −0.188551 0.982063i \(-0.560379\pi\)
−0.188551 + 0.982063i \(0.560379\pi\)
\(504\) 0 0
\(505\) 28.7680 1.28016
\(506\) −5.12476 −0.227824
\(507\) 0 0
\(508\) −17.7012 −0.785364
\(509\) −10.5657 −0.468317 −0.234159 0.972198i \(-0.575233\pi\)
−0.234159 + 0.972198i \(0.575233\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 17.0071 0.751616
\(513\) 0 0
\(514\) 6.12670 0.270237
\(515\) −20.6979 −0.912060
\(516\) 0 0
\(517\) 39.0520 1.71750
\(518\) 0 0
\(519\) 0 0
\(520\) 9.00000 0.394676
\(521\) 19.7558 0.865515 0.432758 0.901510i \(-0.357541\pi\)
0.432758 + 0.901510i \(0.357541\pi\)
\(522\) 0 0
\(523\) −32.5282 −1.42236 −0.711179 0.703011i \(-0.751839\pi\)
−0.711179 + 0.703011i \(0.751839\pi\)
\(524\) 8.35969 0.365195
\(525\) 0 0
\(526\) 1.69794 0.0740337
\(527\) −11.1833 −0.487153
\(528\) 0 0
\(529\) 3.26320 0.141878
\(530\) −6.59825 −0.286610
\(531\) 0 0
\(532\) 0 0
\(533\) −37.6375 −1.63026
\(534\) 0 0
\(535\) −10.2544 −0.443336
\(536\) −7.79071 −0.336507
\(537\) 0 0
\(538\) −3.92914 −0.169397
\(539\) 0 0
\(540\) 0 0
\(541\) 15.2222 0.654453 0.327226 0.944946i \(-0.393886\pi\)
0.327226 + 0.944946i \(0.393886\pi\)
\(542\) 3.03638 0.130424
\(543\) 0 0
\(544\) 4.72406 0.202542
\(545\) −18.8247 −0.806361
\(546\) 0 0
\(547\) 23.3743 0.999412 0.499706 0.866195i \(-0.333441\pi\)
0.499706 + 0.866195i \(0.333441\pi\)
\(548\) −40.0014 −1.70877
\(549\) 0 0
\(550\) 1.71737 0.0732289
\(551\) 15.2040 0.647711
\(552\) 0 0
\(553\) 0 0
\(554\) −0.198104 −0.00841664
\(555\) 0 0
\(556\) 30.6214 1.29864
\(557\) 27.6673 1.17230 0.586151 0.810202i \(-0.300643\pi\)
0.586151 + 0.810202i \(0.300643\pi\)
\(558\) 0 0
\(559\) −6.11465 −0.258622
\(560\) 0 0
\(561\) 0 0
\(562\) 1.24815 0.0526500
\(563\) −8.55824 −0.360687 −0.180343 0.983604i \(-0.557721\pi\)
−0.180343 + 0.983604i \(0.557721\pi\)
\(564\) 0 0
\(565\) −17.9589 −0.755536
\(566\) −1.75855 −0.0739175
\(567\) 0 0
\(568\) −5.88237 −0.246819
\(569\) 13.7278 0.575498 0.287749 0.957706i \(-0.407093\pi\)
0.287749 + 0.957706i \(0.407093\pi\)
\(570\) 0 0
\(571\) 10.7174 0.448508 0.224254 0.974531i \(-0.428005\pi\)
0.224254 + 0.974531i \(0.428005\pi\)
\(572\) −29.9244 −1.25120
\(573\) 0 0
\(574\) 0 0
\(575\) −8.80111 −0.367032
\(576\) 0 0
\(577\) 45.6353 1.89982 0.949912 0.312518i \(-0.101173\pi\)
0.949912 + 0.312518i \(0.101173\pi\)
\(578\) −3.36500 −0.139965
\(579\) 0 0
\(580\) −10.6990 −0.444251
\(581\) 0 0
\(582\) 0 0
\(583\) 44.5231 1.84396
\(584\) −6.74645 −0.279170
\(585\) 0 0
\(586\) 1.87131 0.0773032
\(587\) 10.2190 0.421782 0.210891 0.977510i \(-0.432364\pi\)
0.210891 + 0.977510i \(0.432364\pi\)
\(588\) 0 0
\(589\) 46.7680 1.92704
\(590\) −3.75839 −0.154730
\(591\) 0 0
\(592\) 6.07661 0.249747
\(593\) −11.3961 −0.467981 −0.233990 0.972239i \(-0.575178\pi\)
−0.233990 + 0.972239i \(0.575178\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −11.7882 −0.482864
\(597\) 0 0
\(598\) −4.51346 −0.184569
\(599\) −34.5746 −1.41268 −0.706339 0.707874i \(-0.749655\pi\)
−0.706339 + 0.707874i \(0.749655\pi\)
\(600\) 0 0
\(601\) 38.8414 1.58437 0.792187 0.610279i \(-0.208943\pi\)
0.792187 + 0.610279i \(0.208943\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −8.72064 −0.354838
\(605\) −16.8172 −0.683717
\(606\) 0 0
\(607\) 41.3325 1.67763 0.838817 0.544414i \(-0.183248\pi\)
0.838817 + 0.544414i \(0.183248\pi\)
\(608\) −19.7558 −0.801202
\(609\) 0 0
\(610\) 4.94936 0.200394
\(611\) 34.3937 1.39142
\(612\) 0 0
\(613\) −28.6569 −1.15744 −0.578721 0.815526i \(-0.696448\pi\)
−0.578721 + 0.815526i \(0.696448\pi\)
\(614\) 5.42114 0.218779
\(615\) 0 0
\(616\) 0 0
\(617\) −33.7037 −1.35686 −0.678430 0.734665i \(-0.737339\pi\)
−0.678430 + 0.734665i \(0.737339\pi\)
\(618\) 0 0
\(619\) −1.43807 −0.0578010 −0.0289005 0.999582i \(-0.509201\pi\)
−0.0289005 + 0.999582i \(0.509201\pi\)
\(620\) −32.9105 −1.32172
\(621\) 0 0
\(622\) −7.72789 −0.309860
\(623\) 0 0
\(624\) 0 0
\(625\) −30.6375 −1.22550
\(626\) −5.81538 −0.232429
\(627\) 0 0
\(628\) −1.99900 −0.0797686
\(629\) 2.84071 0.113266
\(630\) 0 0
\(631\) −30.7680 −1.22486 −0.612428 0.790527i \(-0.709807\pi\)
−0.612428 + 0.790527i \(0.709807\pi\)
\(632\) 9.26647 0.368600
\(633\) 0 0
\(634\) 1.22872 0.0487987
\(635\) −23.6140 −0.937094
\(636\) 0 0
\(637\) 0 0
\(638\) −2.12476 −0.0841202
\(639\) 0 0
\(640\) 18.4389 0.728862
\(641\) 9.23912 0.364923 0.182462 0.983213i \(-0.441593\pi\)
0.182462 + 0.983213i \(0.441593\pi\)
\(642\) 0 0
\(643\) 25.5591 1.00795 0.503976 0.863718i \(-0.331870\pi\)
0.503976 + 0.863718i \(0.331870\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −2.92777 −0.115192
\(647\) −28.3111 −1.11302 −0.556512 0.830839i \(-0.687861\pi\)
−0.556512 + 0.830839i \(0.687861\pi\)
\(648\) 0 0
\(649\) 25.3605 0.995488
\(650\) 1.51252 0.0593257
\(651\) 0 0
\(652\) −13.2665 −0.519555
\(653\) −8.35021 −0.326769 −0.163385 0.986562i \(-0.552241\pi\)
−0.163385 + 0.986562i \(0.552241\pi\)
\(654\) 0 0
\(655\) 11.1521 0.435749
\(656\) −37.4033 −1.46035
\(657\) 0 0
\(658\) 0 0
\(659\) −33.5724 −1.30779 −0.653897 0.756583i \(-0.726867\pi\)
−0.653897 + 0.756583i \(0.726867\pi\)
\(660\) 0 0
\(661\) 16.9534 0.659410 0.329705 0.944084i \(-0.393051\pi\)
0.329705 + 0.944084i \(0.393051\pi\)
\(662\) −2.79725 −0.108718
\(663\) 0 0
\(664\) −6.50041 −0.252265
\(665\) 0 0
\(666\) 0 0
\(667\) 10.8889 0.421620
\(668\) −34.9597 −1.35263
\(669\) 0 0
\(670\) −5.12118 −0.197848
\(671\) −33.3969 −1.28927
\(672\) 0 0
\(673\) −44.4315 −1.71271 −0.856354 0.516390i \(-0.827276\pi\)
−0.856354 + 0.516390i \(0.827276\pi\)
\(674\) −8.05718 −0.310351
\(675\) 0 0
\(676\) −1.09820 −0.0422384
\(677\) 14.3736 0.552423 0.276212 0.961097i \(-0.410921\pi\)
0.276212 + 0.961097i \(0.410921\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 4.18116 0.160340
\(681\) 0 0
\(682\) −6.53585 −0.250271
\(683\) −32.3092 −1.23628 −0.618138 0.786069i \(-0.712113\pi\)
−0.618138 + 0.786069i \(0.712113\pi\)
\(684\) 0 0
\(685\) −53.3632 −2.03890
\(686\) 0 0
\(687\) 0 0
\(688\) −6.07661 −0.231669
\(689\) 39.2122 1.49387
\(690\) 0 0
\(691\) 28.9962 1.10307 0.551533 0.834153i \(-0.314043\pi\)
0.551533 + 0.834153i \(0.314043\pi\)
\(692\) 1.61323 0.0613259
\(693\) 0 0
\(694\) 6.53078 0.247905
\(695\) 40.8500 1.54953
\(696\) 0 0
\(697\) −17.4854 −0.662306
\(698\) 5.47997 0.207420
\(699\) 0 0
\(700\) 0 0
\(701\) −26.3912 −0.996783 −0.498392 0.866952i \(-0.666076\pi\)
−0.498392 + 0.866952i \(0.666076\pi\)
\(702\) 0 0
\(703\) −11.8797 −0.448050
\(704\) −27.8525 −1.04973
\(705\) 0 0
\(706\) 2.45563 0.0924190
\(707\) 0 0
\(708\) 0 0
\(709\) −7.88564 −0.296151 −0.148076 0.988976i \(-0.547308\pi\)
−0.148076 + 0.988976i \(0.547308\pi\)
\(710\) −3.86674 −0.145116
\(711\) 0 0
\(712\) 4.74746 0.177918
\(713\) 33.4947 1.25439
\(714\) 0 0
\(715\) −39.9201 −1.49293
\(716\) −14.7245 −0.550281
\(717\) 0 0
\(718\) 2.41531 0.0901385
\(719\) 33.1508 1.23632 0.618159 0.786053i \(-0.287879\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 7.70043 0.286580
\(723\) 0 0
\(724\) −0.794919 −0.0295429
\(725\) −3.64900 −0.135521
\(726\) 0 0
\(727\) −33.1005 −1.22763 −0.613814 0.789451i \(-0.710366\pi\)
−0.613814 + 0.789451i \(0.710366\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −4.43474 −0.164137
\(731\) −2.84071 −0.105067
\(732\) 0 0
\(733\) 44.5589 1.64582 0.822911 0.568170i \(-0.192349\pi\)
0.822911 + 0.568170i \(0.192349\pi\)
\(734\) −1.85927 −0.0686270
\(735\) 0 0
\(736\) −14.1488 −0.521533
\(737\) 34.5562 1.27290
\(738\) 0 0
\(739\) 39.8090 1.46440 0.732199 0.681090i \(-0.238494\pi\)
0.732199 + 0.681090i \(0.238494\pi\)
\(740\) 8.35969 0.307308
\(741\) 0 0
\(742\) 0 0
\(743\) 10.7414 0.394065 0.197033 0.980397i \(-0.436869\pi\)
0.197033 + 0.980397i \(0.436869\pi\)
\(744\) 0 0
\(745\) −15.7259 −0.576152
\(746\) 5.76552 0.211091
\(747\) 0 0
\(748\) −13.9021 −0.508310
\(749\) 0 0
\(750\) 0 0
\(751\) 19.7141 0.719378 0.359689 0.933072i \(-0.382883\pi\)
0.359689 + 0.933072i \(0.382883\pi\)
\(752\) 34.1797 1.24641
\(753\) 0 0
\(754\) −1.87131 −0.0681492
\(755\) −11.6336 −0.423391
\(756\) 0 0
\(757\) 35.3549 1.28499 0.642497 0.766288i \(-0.277898\pi\)
0.642497 + 0.766288i \(0.277898\pi\)
\(758\) −3.19424 −0.116020
\(759\) 0 0
\(760\) −17.4854 −0.634261
\(761\) 39.1144 1.41790 0.708948 0.705261i \(-0.249170\pi\)
0.708948 + 0.705261i \(0.249170\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −31.1319 −1.12631
\(765\) 0 0
\(766\) −2.20960 −0.0798359
\(767\) 22.3354 0.806486
\(768\) 0 0
\(769\) 37.8479 1.36483 0.682415 0.730965i \(-0.260930\pi\)
0.682415 + 0.730965i \(0.260930\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 24.0410 0.865255
\(773\) −29.8265 −1.07279 −0.536393 0.843969i \(-0.680213\pi\)
−0.536393 + 0.843969i \(0.680213\pi\)
\(774\) 0 0
\(775\) −11.2245 −0.403195
\(776\) −2.88819 −0.103680
\(777\) 0 0
\(778\) 2.49846 0.0895741
\(779\) 73.1229 2.61990
\(780\) 0 0
\(781\) 26.0917 0.933633
\(782\) −2.09683 −0.0749827
\(783\) 0 0
\(784\) 0 0
\(785\) −2.66673 −0.0951796
\(786\) 0 0
\(787\) −17.6206 −0.628107 −0.314053 0.949405i \(-0.601687\pi\)
−0.314053 + 0.949405i \(0.601687\pi\)
\(788\) −44.8832 −1.59890
\(789\) 0 0
\(790\) 6.09126 0.216717
\(791\) 0 0
\(792\) 0 0
\(793\) −29.4132 −1.04449
\(794\) −0.0978390 −0.00347218
\(795\) 0 0
\(796\) 13.1071 0.464570
\(797\) −10.1211 −0.358508 −0.179254 0.983803i \(-0.557368\pi\)
−0.179254 + 0.983803i \(0.557368\pi\)
\(798\) 0 0
\(799\) 15.9784 0.565276
\(800\) 4.74145 0.167635
\(801\) 0 0
\(802\) 3.64730 0.128791
\(803\) 29.9244 1.05601
\(804\) 0 0
\(805\) 0 0
\(806\) −5.75623 −0.202755
\(807\) 0 0
\(808\) 10.4650 0.368157
\(809\) −47.1469 −1.65760 −0.828799 0.559546i \(-0.810975\pi\)
−0.828799 + 0.559546i \(0.810975\pi\)
\(810\) 0 0
\(811\) −21.0577 −0.739435 −0.369717 0.929144i \(-0.620546\pi\)
−0.369717 + 0.929144i \(0.620546\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 1.66019 0.0581896
\(815\) −17.6979 −0.619931
\(816\) 0 0
\(817\) 11.8797 0.415617
\(818\) −1.46504 −0.0512238
\(819\) 0 0
\(820\) −51.4563 −1.79693
\(821\) 11.1604 0.389499 0.194750 0.980853i \(-0.437611\pi\)
0.194750 + 0.980853i \(0.437611\pi\)
\(822\) 0 0
\(823\) 9.43474 0.328874 0.164437 0.986388i \(-0.447419\pi\)
0.164437 + 0.986388i \(0.447419\pi\)
\(824\) −7.52933 −0.262297
\(825\) 0 0
\(826\) 0 0
\(827\) 17.2646 0.600348 0.300174 0.953884i \(-0.402955\pi\)
0.300174 + 0.953884i \(0.402955\pi\)
\(828\) 0 0
\(829\) 48.4526 1.68283 0.841415 0.540390i \(-0.181723\pi\)
0.841415 + 0.540390i \(0.181723\pi\)
\(830\) −4.27301 −0.148318
\(831\) 0 0
\(832\) −24.5302 −0.850431
\(833\) 0 0
\(834\) 0 0
\(835\) −46.6375 −1.61396
\(836\) 58.1376 2.01073
\(837\) 0 0
\(838\) 0.373720 0.0129099
\(839\) 14.8686 0.513320 0.256660 0.966502i \(-0.417378\pi\)
0.256660 + 0.966502i \(0.417378\pi\)
\(840\) 0 0
\(841\) −24.4854 −0.844323
\(842\) 5.56277 0.191706
\(843\) 0 0
\(844\) −32.8058 −1.12922
\(845\) −1.46504 −0.0503988
\(846\) 0 0
\(847\) 0 0
\(848\) 38.9683 1.33818
\(849\) 0 0
\(850\) 0.702674 0.0241015
\(851\) −8.50808 −0.291653
\(852\) 0 0
\(853\) −7.99801 −0.273847 −0.136923 0.990582i \(-0.543721\pi\)
−0.136923 + 0.990582i \(0.543721\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −3.73026 −0.127498
\(857\) 43.1322 1.47337 0.736684 0.676237i \(-0.236390\pi\)
0.736684 + 0.676237i \(0.236390\pi\)
\(858\) 0 0
\(859\) 2.44359 0.0833742 0.0416871 0.999131i \(-0.486727\pi\)
0.0416871 + 0.999131i \(0.486727\pi\)
\(860\) −8.35969 −0.285063
\(861\) 0 0
\(862\) 0.240500 0.00819146
\(863\) 25.7187 0.875476 0.437738 0.899103i \(-0.355780\pi\)
0.437738 + 0.899103i \(0.355780\pi\)
\(864\) 0 0
\(865\) 2.15211 0.0731739
\(866\) 3.13422 0.106505
\(867\) 0 0
\(868\) 0 0
\(869\) −41.1021 −1.39429
\(870\) 0 0
\(871\) 30.4342 1.03122
\(872\) −6.84789 −0.231899
\(873\) 0 0
\(874\) 8.76884 0.296611
\(875\) 0 0
\(876\) 0 0
\(877\) −21.9590 −0.741502 −0.370751 0.928732i \(-0.620900\pi\)
−0.370751 + 0.928732i \(0.620900\pi\)
\(878\) 4.45106 0.150216
\(879\) 0 0
\(880\) −39.6718 −1.33733
\(881\) 35.0576 1.18112 0.590560 0.806994i \(-0.298907\pi\)
0.590560 + 0.806994i \(0.298907\pi\)
\(882\) 0 0
\(883\) 26.3009 0.885097 0.442549 0.896744i \(-0.354074\pi\)
0.442549 + 0.896744i \(0.354074\pi\)
\(884\) −12.2438 −0.411803
\(885\) 0 0
\(886\) −0.267580 −0.00898954
\(887\) −47.8180 −1.60557 −0.802785 0.596269i \(-0.796649\pi\)
−0.802785 + 0.596269i \(0.796649\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 3.12071 0.104607
\(891\) 0 0
\(892\) 8.74545 0.292819
\(893\) −66.8208 −2.23607
\(894\) 0 0
\(895\) −19.6430 −0.656593
\(896\) 0 0
\(897\) 0 0
\(898\) −9.43147 −0.314732
\(899\) 13.8871 0.463162
\(900\) 0 0
\(901\) 18.2170 0.606895
\(902\) −10.2190 −0.340254
\(903\) 0 0
\(904\) −6.53294 −0.217282
\(905\) −1.06045 −0.0352505
\(906\) 0 0
\(907\) −19.1144 −0.634682 −0.317341 0.948312i \(-0.602790\pi\)
−0.317341 + 0.948312i \(0.602790\pi\)
\(908\) 11.7818 0.390994
\(909\) 0 0
\(910\) 0 0
\(911\) −18.0586 −0.598307 −0.299153 0.954205i \(-0.596704\pi\)
−0.299153 + 0.954205i \(0.596704\pi\)
\(912\) 0 0
\(913\) 28.8330 0.954234
\(914\) −8.18770 −0.270825
\(915\) 0 0
\(916\) 21.4668 0.709284
\(917\) 0 0
\(918\) 0 0
\(919\) 16.2093 0.534695 0.267348 0.963600i \(-0.413853\pi\)
0.267348 + 0.963600i \(0.413853\pi\)
\(920\) −12.5228 −0.412865
\(921\) 0 0
\(922\) 4.87514 0.160554
\(923\) 22.9793 0.756374
\(924\) 0 0
\(925\) 2.85116 0.0937456
\(926\) 1.62820 0.0535059
\(927\) 0 0
\(928\) −5.86621 −0.192568
\(929\) −22.6829 −0.744203 −0.372102 0.928192i \(-0.621363\pi\)
−0.372102 + 0.928192i \(0.621363\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 15.8115 0.517923
\(933\) 0 0
\(934\) 5.92814 0.193975
\(935\) −18.5458 −0.606513
\(936\) 0 0
\(937\) 51.2933 1.67568 0.837840 0.545915i \(-0.183818\pi\)
0.837840 + 0.545915i \(0.183818\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 47.0216 1.53368
\(941\) −31.9318 −1.04095 −0.520474 0.853878i \(-0.674245\pi\)
−0.520474 + 0.853878i \(0.674245\pi\)
\(942\) 0 0
\(943\) 52.3697 1.70539
\(944\) 22.1965 0.722433
\(945\) 0 0
\(946\) −1.66019 −0.0539774
\(947\) −4.49330 −0.146013 −0.0730063 0.997331i \(-0.523259\pi\)
−0.0730063 + 0.997331i \(0.523259\pi\)
\(948\) 0 0
\(949\) 26.3549 0.855515
\(950\) −2.93854 −0.0953390
\(951\) 0 0
\(952\) 0 0
\(953\) 1.14635 0.0371340 0.0185670 0.999828i \(-0.494090\pi\)
0.0185670 + 0.999828i \(0.494090\pi\)
\(954\) 0 0
\(955\) −41.5310 −1.34391
\(956\) −41.1150 −1.32975
\(957\) 0 0
\(958\) −2.65419 −0.0857530
\(959\) 0 0
\(960\) 0 0
\(961\) 11.7174 0.377980
\(962\) 1.46216 0.0471418
\(963\) 0 0
\(964\) −26.6000 −0.856730
\(965\) 32.0715 1.03242
\(966\) 0 0
\(967\) 49.6159 1.59554 0.797770 0.602962i \(-0.206013\pi\)
0.797770 + 0.602962i \(0.206013\pi\)
\(968\) −6.11763 −0.196628
\(969\) 0 0
\(970\) −1.89853 −0.0609582
\(971\) −5.13322 −0.164733 −0.0823664 0.996602i \(-0.526248\pi\)
−0.0823664 + 0.996602i \(0.526248\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −2.40026 −0.0769093
\(975\) 0 0
\(976\) −29.2302 −0.935634
\(977\) 31.1948 0.998011 0.499006 0.866599i \(-0.333699\pi\)
0.499006 + 0.866599i \(0.333699\pi\)
\(978\) 0 0
\(979\) −21.0577 −0.673006
\(980\) 0 0
\(981\) 0 0
\(982\) −2.96225 −0.0945292
\(983\) 20.3401 0.648748 0.324374 0.945929i \(-0.394846\pi\)
0.324374 + 0.945929i \(0.394846\pi\)
\(984\) 0 0
\(985\) −59.8757 −1.90780
\(986\) −0.869363 −0.0276861
\(987\) 0 0
\(988\) 51.2028 1.62898
\(989\) 8.50808 0.270541
\(990\) 0 0
\(991\) −12.9655 −0.411863 −0.205932 0.978566i \(-0.566022\pi\)
−0.205932 + 0.978566i \(0.566022\pi\)
\(992\) −18.0447 −0.572919
\(993\) 0 0
\(994\) 0 0
\(995\) 17.4854 0.554324
\(996\) 0 0
\(997\) −49.4816 −1.56710 −0.783548 0.621331i \(-0.786592\pi\)
−0.783548 + 0.621331i \(0.786592\pi\)
\(998\) 2.44436 0.0773749
\(999\) 0 0
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 3969.2.a.be.1.3 6
3.2 odd 2 3969.2.a.bd.1.4 6
7.6 odd 2 inner 3969.2.a.be.1.4 6
9.2 odd 6 1323.2.f.g.442.3 12
9.4 even 3 441.2.f.g.295.3 yes 12
9.5 odd 6 1323.2.f.g.883.3 12
9.7 even 3 441.2.f.g.148.3 12
21.20 even 2 3969.2.a.bd.1.3 6
63.2 odd 6 1323.2.g.g.361.4 12
63.4 even 3 441.2.g.g.79.3 12
63.5 even 6 1323.2.h.g.802.4 12
63.11 odd 6 1323.2.h.g.226.3 12
63.13 odd 6 441.2.f.g.295.4 yes 12
63.16 even 3 441.2.g.g.67.3 12
63.20 even 6 1323.2.f.g.442.4 12
63.23 odd 6 1323.2.h.g.802.3 12
63.25 even 3 441.2.h.g.373.4 12
63.31 odd 6 441.2.g.g.79.4 12
63.32 odd 6 1323.2.g.g.667.4 12
63.34 odd 6 441.2.f.g.148.4 yes 12
63.38 even 6 1323.2.h.g.226.4 12
63.40 odd 6 441.2.h.g.214.3 12
63.41 even 6 1323.2.f.g.883.4 12
63.47 even 6 1323.2.g.g.361.3 12
63.52 odd 6 441.2.h.g.373.3 12
63.58 even 3 441.2.h.g.214.4 12
63.59 even 6 1323.2.g.g.667.3 12
63.61 odd 6 441.2.g.g.67.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.f.g.148.3 12 9.7 even 3
441.2.f.g.148.4 yes 12 63.34 odd 6
441.2.f.g.295.3 yes 12 9.4 even 3
441.2.f.g.295.4 yes 12 63.13 odd 6
441.2.g.g.67.3 12 63.16 even 3
441.2.g.g.67.4 12 63.61 odd 6
441.2.g.g.79.3 12 63.4 even 3
441.2.g.g.79.4 12 63.31 odd 6
441.2.h.g.214.3 12 63.40 odd 6
441.2.h.g.214.4 12 63.58 even 3
441.2.h.g.373.3 12 63.52 odd 6
441.2.h.g.373.4 12 63.25 even 3
1323.2.f.g.442.3 12 9.2 odd 6
1323.2.f.g.442.4 12 63.20 even 6
1323.2.f.g.883.3 12 9.5 odd 6
1323.2.f.g.883.4 12 63.41 even 6
1323.2.g.g.361.3 12 63.47 even 6
1323.2.g.g.361.4 12 63.2 odd 6
1323.2.g.g.667.3 12 63.59 even 6
1323.2.g.g.667.4 12 63.32 odd 6
1323.2.h.g.226.3 12 63.11 odd 6
1323.2.h.g.226.4 12 63.38 even 6
1323.2.h.g.802.3 12 63.23 odd 6
1323.2.h.g.802.4 12 63.5 even 6
3969.2.a.bd.1.3 6 21.20 even 2
3969.2.a.bd.1.4 6 3.2 odd 2
3969.2.a.be.1.3 6 1.1 even 1 trivial
3969.2.a.be.1.4 6 7.6 odd 2 inner