Properties

Label 7623.2.a.ct.1.8
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-2.43045\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.43045 q^{2} +3.90710 q^{4} -1.25863 q^{5} +1.00000 q^{7} +4.63512 q^{8} +O(q^{10})\) \(q+2.43045 q^{2} +3.90710 q^{4} -1.25863 q^{5} +1.00000 q^{7} +4.63512 q^{8} -3.05904 q^{10} -3.17831 q^{13} +2.43045 q^{14} +3.45123 q^{16} -5.92418 q^{17} +2.86222 q^{19} -4.91760 q^{20} -6.76343 q^{23} -3.41585 q^{25} -7.72473 q^{26} +3.90710 q^{28} -4.49549 q^{29} +9.72751 q^{31} -0.882184 q^{32} -14.3984 q^{34} -1.25863 q^{35} -5.45495 q^{37} +6.95648 q^{38} -5.83390 q^{40} +0.314484 q^{41} -0.132562 q^{43} -16.4382 q^{46} -9.37505 q^{47} +1.00000 q^{49} -8.30206 q^{50} -12.4180 q^{52} -4.35192 q^{53} +4.63512 q^{56} -10.9261 q^{58} +6.94685 q^{59} -2.45215 q^{61} +23.6423 q^{62} -9.04656 q^{64} +4.00032 q^{65} -9.41987 q^{67} -23.1464 q^{68} -3.05904 q^{70} +0.116610 q^{71} +0.615032 q^{73} -13.2580 q^{74} +11.1830 q^{76} +8.52928 q^{79} -4.34382 q^{80} +0.764339 q^{82} -0.950532 q^{83} +7.45636 q^{85} -0.322186 q^{86} -10.0552 q^{89} -3.17831 q^{91} -26.4254 q^{92} -22.7856 q^{94} -3.60248 q^{95} +17.5770 q^{97} +2.43045 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - q^{2} + 7q^{4} - 10q^{5} + 8q^{7} + O(q^{10}) \) \( 8q - q^{2} + 7q^{4} - 10q^{5} + 8q^{7} + 6q^{10} - 6q^{13} - q^{14} + q^{16} + 5q^{17} - 13q^{19} - 23q^{20} - 16q^{23} + 16q^{25} + 6q^{26} + 7q^{28} - 9q^{29} + 9q^{31} - 16q^{32} - 12q^{34} - 10q^{35} + 7q^{37} + 10q^{38} + 5q^{40} + 10q^{41} - 4q^{43} + 4q^{46} - 16q^{47} + 8q^{49} - 6q^{50} - 41q^{52} - 37q^{53} - 15q^{58} - q^{59} + 19q^{61} + 18q^{62} - 4q^{64} + 4q^{65} - 19q^{67} - 9q^{68} + 6q^{70} - 13q^{71} - 25q^{73} - 33q^{74} + 26q^{76} - 4q^{80} - 13q^{82} + 25q^{83} + 3q^{85} - 4q^{86} - 37q^{89} - 6q^{91} - 35q^{92} - 42q^{94} - 21q^{95} + 15q^{97} - q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.43045 1.71859 0.859295 0.511481i \(-0.170903\pi\)
0.859295 + 0.511481i \(0.170903\pi\)
\(3\) 0 0
\(4\) 3.90710 1.95355
\(5\) −1.25863 −0.562877 −0.281438 0.959579i \(-0.590812\pi\)
−0.281438 + 0.959579i \(0.590812\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 4.63512 1.63876
\(9\) 0 0
\(10\) −3.05904 −0.967354
\(11\) 0 0
\(12\) 0 0
\(13\) −3.17831 −0.881504 −0.440752 0.897629i \(-0.645288\pi\)
−0.440752 + 0.897629i \(0.645288\pi\)
\(14\) 2.43045 0.649566
\(15\) 0 0
\(16\) 3.45123 0.862807
\(17\) −5.92418 −1.43682 −0.718412 0.695617i \(-0.755131\pi\)
−0.718412 + 0.695617i \(0.755131\pi\)
\(18\) 0 0
\(19\) 2.86222 0.656638 0.328319 0.944567i \(-0.393518\pi\)
0.328319 + 0.944567i \(0.393518\pi\)
\(20\) −4.91760 −1.09961
\(21\) 0 0
\(22\) 0 0
\(23\) −6.76343 −1.41027 −0.705136 0.709072i \(-0.749114\pi\)
−0.705136 + 0.709072i \(0.749114\pi\)
\(24\) 0 0
\(25\) −3.41585 −0.683170
\(26\) −7.72473 −1.51494
\(27\) 0 0
\(28\) 3.90710 0.738372
\(29\) −4.49549 −0.834792 −0.417396 0.908725i \(-0.637057\pi\)
−0.417396 + 0.908725i \(0.637057\pi\)
\(30\) 0 0
\(31\) 9.72751 1.74711 0.873556 0.486723i \(-0.161808\pi\)
0.873556 + 0.486723i \(0.161808\pi\)
\(32\) −0.882184 −0.155950
\(33\) 0 0
\(34\) −14.3984 −2.46931
\(35\) −1.25863 −0.212747
\(36\) 0 0
\(37\) −5.45495 −0.896789 −0.448394 0.893836i \(-0.648004\pi\)
−0.448394 + 0.893836i \(0.648004\pi\)
\(38\) 6.95648 1.12849
\(39\) 0 0
\(40\) −5.83390 −0.922421
\(41\) 0.314484 0.0491142 0.0245571 0.999698i \(-0.492182\pi\)
0.0245571 + 0.999698i \(0.492182\pi\)
\(42\) 0 0
\(43\) −0.132562 −0.0202155 −0.0101078 0.999949i \(-0.503217\pi\)
−0.0101078 + 0.999949i \(0.503217\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −16.4382 −2.42368
\(47\) −9.37505 −1.36749 −0.683746 0.729720i \(-0.739650\pi\)
−0.683746 + 0.729720i \(0.739650\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −8.30206 −1.17409
\(51\) 0 0
\(52\) −12.4180 −1.72206
\(53\) −4.35192 −0.597783 −0.298891 0.954287i \(-0.596617\pi\)
−0.298891 + 0.954287i \(0.596617\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.63512 0.619393
\(57\) 0 0
\(58\) −10.9261 −1.43466
\(59\) 6.94685 0.904403 0.452202 0.891916i \(-0.350639\pi\)
0.452202 + 0.891916i \(0.350639\pi\)
\(60\) 0 0
\(61\) −2.45215 −0.313966 −0.156983 0.987601i \(-0.550177\pi\)
−0.156983 + 0.987601i \(0.550177\pi\)
\(62\) 23.6423 3.00257
\(63\) 0 0
\(64\) −9.04656 −1.13082
\(65\) 4.00032 0.496179
\(66\) 0 0
\(67\) −9.41987 −1.15082 −0.575410 0.817865i \(-0.695158\pi\)
−0.575410 + 0.817865i \(0.695158\pi\)
\(68\) −23.1464 −2.80691
\(69\) 0 0
\(70\) −3.05904 −0.365626
\(71\) 0.116610 0.0138391 0.00691954 0.999976i \(-0.497797\pi\)
0.00691954 + 0.999976i \(0.497797\pi\)
\(72\) 0 0
\(73\) 0.615032 0.0719840 0.0359920 0.999352i \(-0.488541\pi\)
0.0359920 + 0.999352i \(0.488541\pi\)
\(74\) −13.2580 −1.54121
\(75\) 0 0
\(76\) 11.1830 1.28277
\(77\) 0 0
\(78\) 0 0
\(79\) 8.52928 0.959619 0.479810 0.877373i \(-0.340706\pi\)
0.479810 + 0.877373i \(0.340706\pi\)
\(80\) −4.34382 −0.485654
\(81\) 0 0
\(82\) 0.764339 0.0844072
\(83\) −0.950532 −0.104334 −0.0521672 0.998638i \(-0.516613\pi\)
−0.0521672 + 0.998638i \(0.516613\pi\)
\(84\) 0 0
\(85\) 7.45636 0.808756
\(86\) −0.322186 −0.0347422
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0552 −1.06585 −0.532923 0.846164i \(-0.678906\pi\)
−0.532923 + 0.846164i \(0.678906\pi\)
\(90\) 0 0
\(91\) −3.17831 −0.333177
\(92\) −26.4254 −2.75504
\(93\) 0 0
\(94\) −22.7856 −2.35016
\(95\) −3.60248 −0.369606
\(96\) 0 0
\(97\) 17.5770 1.78467 0.892337 0.451369i \(-0.149064\pi\)
0.892337 + 0.451369i \(0.149064\pi\)
\(98\) 2.43045 0.245513
\(99\) 0 0
\(100\) −13.3461 −1.33461
\(101\) 12.4761 1.24142 0.620711 0.784039i \(-0.286844\pi\)
0.620711 + 0.784039i \(0.286844\pi\)
\(102\) 0 0
\(103\) 8.21802 0.809746 0.404873 0.914373i \(-0.367316\pi\)
0.404873 + 0.914373i \(0.367316\pi\)
\(104\) −14.7318 −1.44457
\(105\) 0 0
\(106\) −10.5771 −1.02734
\(107\) −12.1885 −1.17831 −0.589154 0.808020i \(-0.700539\pi\)
−0.589154 + 0.808020i \(0.700539\pi\)
\(108\) 0 0
\(109\) 0.886088 0.0848718 0.0424359 0.999099i \(-0.486488\pi\)
0.0424359 + 0.999099i \(0.486488\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.45123 0.326110
\(113\) −4.54502 −0.427559 −0.213780 0.976882i \(-0.568577\pi\)
−0.213780 + 0.976882i \(0.568577\pi\)
\(114\) 0 0
\(115\) 8.51266 0.793810
\(116\) −17.5643 −1.63081
\(117\) 0 0
\(118\) 16.8840 1.55430
\(119\) −5.92418 −0.543069
\(120\) 0 0
\(121\) 0 0
\(122\) −5.95984 −0.539579
\(123\) 0 0
\(124\) 38.0064 3.41307
\(125\) 10.5924 0.947417
\(126\) 0 0
\(127\) −8.02779 −0.712351 −0.356175 0.934419i \(-0.615919\pi\)
−0.356175 + 0.934419i \(0.615919\pi\)
\(128\) −20.2229 −1.78747
\(129\) 0 0
\(130\) 9.72259 0.852727
\(131\) −0.101461 −0.00886466 −0.00443233 0.999990i \(-0.501411\pi\)
−0.00443233 + 0.999990i \(0.501411\pi\)
\(132\) 0 0
\(133\) 2.86222 0.248186
\(134\) −22.8945 −1.97779
\(135\) 0 0
\(136\) −27.4593 −2.35461
\(137\) 4.56409 0.389937 0.194968 0.980810i \(-0.437540\pi\)
0.194968 + 0.980810i \(0.437540\pi\)
\(138\) 0 0
\(139\) −3.82533 −0.324460 −0.162230 0.986753i \(-0.551869\pi\)
−0.162230 + 0.986753i \(0.551869\pi\)
\(140\) −4.91760 −0.415613
\(141\) 0 0
\(142\) 0.283415 0.0237837
\(143\) 0 0
\(144\) 0 0
\(145\) 5.65817 0.469885
\(146\) 1.49480 0.123711
\(147\) 0 0
\(148\) −21.3130 −1.75192
\(149\) 4.87223 0.399149 0.199574 0.979883i \(-0.436044\pi\)
0.199574 + 0.979883i \(0.436044\pi\)
\(150\) 0 0
\(151\) −16.9739 −1.38131 −0.690657 0.723183i \(-0.742678\pi\)
−0.690657 + 0.723183i \(0.742678\pi\)
\(152\) 13.2667 1.07607
\(153\) 0 0
\(154\) 0 0
\(155\) −12.2434 −0.983410
\(156\) 0 0
\(157\) 2.24483 0.179157 0.0895785 0.995980i \(-0.471448\pi\)
0.0895785 + 0.995980i \(0.471448\pi\)
\(158\) 20.7300 1.64919
\(159\) 0 0
\(160\) 1.11034 0.0877804
\(161\) −6.76343 −0.533033
\(162\) 0 0
\(163\) −15.8372 −1.24047 −0.620235 0.784416i \(-0.712963\pi\)
−0.620235 + 0.784416i \(0.712963\pi\)
\(164\) 1.22872 0.0959470
\(165\) 0 0
\(166\) −2.31022 −0.179308
\(167\) −20.7416 −1.60503 −0.802515 0.596632i \(-0.796505\pi\)
−0.802515 + 0.596632i \(0.796505\pi\)
\(168\) 0 0
\(169\) −2.89835 −0.222950
\(170\) 18.1223 1.38992
\(171\) 0 0
\(172\) −0.517933 −0.0394920
\(173\) 21.5554 1.63883 0.819414 0.573202i \(-0.194299\pi\)
0.819414 + 0.573202i \(0.194299\pi\)
\(174\) 0 0
\(175\) −3.41585 −0.258214
\(176\) 0 0
\(177\) 0 0
\(178\) −24.4386 −1.83175
\(179\) −4.78161 −0.357394 −0.178697 0.983904i \(-0.557188\pi\)
−0.178697 + 0.983904i \(0.557188\pi\)
\(180\) 0 0
\(181\) −7.85284 −0.583697 −0.291849 0.956465i \(-0.594270\pi\)
−0.291849 + 0.956465i \(0.594270\pi\)
\(182\) −7.72473 −0.572595
\(183\) 0 0
\(184\) −31.3493 −2.31110
\(185\) 6.86578 0.504782
\(186\) 0 0
\(187\) 0 0
\(188\) −36.6293 −2.67146
\(189\) 0 0
\(190\) −8.75565 −0.635201
\(191\) 8.83029 0.638937 0.319469 0.947597i \(-0.396496\pi\)
0.319469 + 0.947597i \(0.396496\pi\)
\(192\) 0 0
\(193\) 25.6023 1.84289 0.921446 0.388507i \(-0.127009\pi\)
0.921446 + 0.388507i \(0.127009\pi\)
\(194\) 42.7201 3.06712
\(195\) 0 0
\(196\) 3.90710 0.279079
\(197\) 11.1977 0.797802 0.398901 0.916994i \(-0.369392\pi\)
0.398901 + 0.916994i \(0.369392\pi\)
\(198\) 0 0
\(199\) −12.2503 −0.868400 −0.434200 0.900817i \(-0.642969\pi\)
−0.434200 + 0.900817i \(0.642969\pi\)
\(200\) −15.8328 −1.11955
\(201\) 0 0
\(202\) 30.3227 2.13350
\(203\) −4.49549 −0.315522
\(204\) 0 0
\(205\) −0.395820 −0.0276453
\(206\) 19.9735 1.39162
\(207\) 0 0
\(208\) −10.9691 −0.760568
\(209\) 0 0
\(210\) 0 0
\(211\) −14.2636 −0.981946 −0.490973 0.871175i \(-0.663359\pi\)
−0.490973 + 0.871175i \(0.663359\pi\)
\(212\) −17.0034 −1.16780
\(213\) 0 0
\(214\) −29.6236 −2.02503
\(215\) 0.166847 0.0113788
\(216\) 0 0
\(217\) 9.72751 0.660347
\(218\) 2.15359 0.145860
\(219\) 0 0
\(220\) 0 0
\(221\) 18.8289 1.26657
\(222\) 0 0
\(223\) −2.96400 −0.198484 −0.0992421 0.995063i \(-0.531642\pi\)
−0.0992421 + 0.995063i \(0.531642\pi\)
\(224\) −0.882184 −0.0589434
\(225\) 0 0
\(226\) −11.0465 −0.734799
\(227\) −8.25605 −0.547974 −0.273987 0.961733i \(-0.588342\pi\)
−0.273987 + 0.961733i \(0.588342\pi\)
\(228\) 0 0
\(229\) 13.1081 0.866208 0.433104 0.901344i \(-0.357418\pi\)
0.433104 + 0.901344i \(0.357418\pi\)
\(230\) 20.6896 1.36423
\(231\) 0 0
\(232\) −20.8371 −1.36802
\(233\) 12.8277 0.840369 0.420185 0.907439i \(-0.361965\pi\)
0.420185 + 0.907439i \(0.361965\pi\)
\(234\) 0 0
\(235\) 11.7997 0.769730
\(236\) 27.1420 1.76680
\(237\) 0 0
\(238\) −14.3984 −0.933312
\(239\) 4.98109 0.322200 0.161100 0.986938i \(-0.448496\pi\)
0.161100 + 0.986938i \(0.448496\pi\)
\(240\) 0 0
\(241\) −2.62686 −0.169211 −0.0846053 0.996415i \(-0.526963\pi\)
−0.0846053 + 0.996415i \(0.526963\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −9.58081 −0.613349
\(245\) −1.25863 −0.0804110
\(246\) 0 0
\(247\) −9.09701 −0.578829
\(248\) 45.0881 2.86310
\(249\) 0 0
\(250\) 25.7444 1.62822
\(251\) 26.2229 1.65518 0.827588 0.561337i \(-0.189713\pi\)
0.827588 + 0.561337i \(0.189713\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −19.5112 −1.22424
\(255\) 0 0
\(256\) −31.0576 −1.94110
\(257\) −29.9518 −1.86834 −0.934170 0.356827i \(-0.883859\pi\)
−0.934170 + 0.356827i \(0.883859\pi\)
\(258\) 0 0
\(259\) −5.45495 −0.338954
\(260\) 15.6296 0.969309
\(261\) 0 0
\(262\) −0.246595 −0.0152347
\(263\) −3.33709 −0.205774 −0.102887 0.994693i \(-0.532808\pi\)
−0.102887 + 0.994693i \(0.532808\pi\)
\(264\) 0 0
\(265\) 5.47747 0.336478
\(266\) 6.95648 0.426529
\(267\) 0 0
\(268\) −36.8044 −2.24818
\(269\) 1.73581 0.105834 0.0529172 0.998599i \(-0.483148\pi\)
0.0529172 + 0.998599i \(0.483148\pi\)
\(270\) 0 0
\(271\) −2.47596 −0.150404 −0.0752019 0.997168i \(-0.523960\pi\)
−0.0752019 + 0.997168i \(0.523960\pi\)
\(272\) −20.4457 −1.23970
\(273\) 0 0
\(274\) 11.0928 0.670141
\(275\) 0 0
\(276\) 0 0
\(277\) 5.15571 0.309776 0.154888 0.987932i \(-0.450498\pi\)
0.154888 + 0.987932i \(0.450498\pi\)
\(278\) −9.29728 −0.557614
\(279\) 0 0
\(280\) −5.83390 −0.348642
\(281\) 22.5705 1.34644 0.673222 0.739441i \(-0.264910\pi\)
0.673222 + 0.739441i \(0.264910\pi\)
\(282\) 0 0
\(283\) −8.48757 −0.504534 −0.252267 0.967658i \(-0.581176\pi\)
−0.252267 + 0.967658i \(0.581176\pi\)
\(284\) 0.455608 0.0270353
\(285\) 0 0
\(286\) 0 0
\(287\) 0.314484 0.0185634
\(288\) 0 0
\(289\) 18.0959 1.06447
\(290\) 13.7519 0.807540
\(291\) 0 0
\(292\) 2.40299 0.140624
\(293\) −24.6261 −1.43867 −0.719336 0.694662i \(-0.755554\pi\)
−0.719336 + 0.694662i \(0.755554\pi\)
\(294\) 0 0
\(295\) −8.74353 −0.509068
\(296\) −25.2843 −1.46962
\(297\) 0 0
\(298\) 11.8417 0.685973
\(299\) 21.4963 1.24316
\(300\) 0 0
\(301\) −0.132562 −0.00764075
\(302\) −41.2542 −2.37391
\(303\) 0 0
\(304\) 9.87817 0.566552
\(305\) 3.08636 0.176724
\(306\) 0 0
\(307\) 4.59391 0.262188 0.131094 0.991370i \(-0.458151\pi\)
0.131094 + 0.991370i \(0.458151\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −29.7569 −1.69008
\(311\) 2.21073 0.125359 0.0626796 0.998034i \(-0.480035\pi\)
0.0626796 + 0.998034i \(0.480035\pi\)
\(312\) 0 0
\(313\) −9.69928 −0.548236 −0.274118 0.961696i \(-0.588386\pi\)
−0.274118 + 0.961696i \(0.588386\pi\)
\(314\) 5.45595 0.307897
\(315\) 0 0
\(316\) 33.3248 1.87466
\(317\) −6.45539 −0.362571 −0.181286 0.983431i \(-0.558026\pi\)
−0.181286 + 0.983431i \(0.558026\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 11.3863 0.636513
\(321\) 0 0
\(322\) −16.4382 −0.916065
\(323\) −16.9563 −0.943474
\(324\) 0 0
\(325\) 10.8566 0.602217
\(326\) −38.4917 −2.13186
\(327\) 0 0
\(328\) 1.45767 0.0804864
\(329\) −9.37505 −0.516863
\(330\) 0 0
\(331\) 3.62076 0.199015 0.0995075 0.995037i \(-0.468273\pi\)
0.0995075 + 0.995037i \(0.468273\pi\)
\(332\) −3.71382 −0.203823
\(333\) 0 0
\(334\) −50.4114 −2.75839
\(335\) 11.8561 0.647770
\(336\) 0 0
\(337\) −6.40510 −0.348908 −0.174454 0.984665i \(-0.555816\pi\)
−0.174454 + 0.984665i \(0.555816\pi\)
\(338\) −7.04430 −0.383159
\(339\) 0 0
\(340\) 29.1327 1.57994
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −0.614440 −0.0331284
\(345\) 0 0
\(346\) 52.3894 2.81647
\(347\) 17.4600 0.937302 0.468651 0.883383i \(-0.344740\pi\)
0.468651 + 0.883383i \(0.344740\pi\)
\(348\) 0 0
\(349\) −21.6249 −1.15755 −0.578777 0.815486i \(-0.696470\pi\)
−0.578777 + 0.815486i \(0.696470\pi\)
\(350\) −8.30206 −0.443764
\(351\) 0 0
\(352\) 0 0
\(353\) −4.40150 −0.234268 −0.117134 0.993116i \(-0.537371\pi\)
−0.117134 + 0.993116i \(0.537371\pi\)
\(354\) 0 0
\(355\) −0.146769 −0.00778970
\(356\) −39.2865 −2.08218
\(357\) 0 0
\(358\) −11.6215 −0.614214
\(359\) 10.6817 0.563760 0.281880 0.959450i \(-0.409042\pi\)
0.281880 + 0.959450i \(0.409042\pi\)
\(360\) 0 0
\(361\) −10.8077 −0.568827
\(362\) −19.0860 −1.00314
\(363\) 0 0
\(364\) −12.4180 −0.650879
\(365\) −0.774098 −0.0405181
\(366\) 0 0
\(367\) 10.2167 0.533309 0.266655 0.963792i \(-0.414082\pi\)
0.266655 + 0.963792i \(0.414082\pi\)
\(368\) −23.3421 −1.21679
\(369\) 0 0
\(370\) 16.6869 0.867513
\(371\) −4.35192 −0.225941
\(372\) 0 0
\(373\) −27.8851 −1.44383 −0.721917 0.691980i \(-0.756739\pi\)
−0.721917 + 0.691980i \(0.756739\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −43.4544 −2.24099
\(377\) 14.2881 0.735873
\(378\) 0 0
\(379\) 1.14803 0.0589703 0.0294852 0.999565i \(-0.490613\pi\)
0.0294852 + 0.999565i \(0.490613\pi\)
\(380\) −14.0752 −0.722044
\(381\) 0 0
\(382\) 21.4616 1.09807
\(383\) 19.9516 1.01948 0.509740 0.860328i \(-0.329742\pi\)
0.509740 + 0.860328i \(0.329742\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 62.2251 3.16717
\(387\) 0 0
\(388\) 68.6751 3.48645
\(389\) −2.60000 −0.131825 −0.0659127 0.997825i \(-0.520996\pi\)
−0.0659127 + 0.997825i \(0.520996\pi\)
\(390\) 0 0
\(391\) 40.0678 2.02631
\(392\) 4.63512 0.234109
\(393\) 0 0
\(394\) 27.2154 1.37109
\(395\) −10.7352 −0.540148
\(396\) 0 0
\(397\) 8.77237 0.440272 0.220136 0.975469i \(-0.429350\pi\)
0.220136 + 0.975469i \(0.429350\pi\)
\(398\) −29.7737 −1.49242
\(399\) 0 0
\(400\) −11.7889 −0.589444
\(401\) −28.3535 −1.41591 −0.707953 0.706260i \(-0.750381\pi\)
−0.707953 + 0.706260i \(0.750381\pi\)
\(402\) 0 0
\(403\) −30.9170 −1.54009
\(404\) 48.7455 2.42518
\(405\) 0 0
\(406\) −10.9261 −0.542252
\(407\) 0 0
\(408\) 0 0
\(409\) 19.9055 0.984264 0.492132 0.870521i \(-0.336218\pi\)
0.492132 + 0.870521i \(0.336218\pi\)
\(410\) −0.962021 −0.0475108
\(411\) 0 0
\(412\) 32.1086 1.58188
\(413\) 6.94685 0.341832
\(414\) 0 0
\(415\) 1.19637 0.0587275
\(416\) 2.80385 0.137470
\(417\) 0 0
\(418\) 0 0
\(419\) 30.8957 1.50935 0.754676 0.656097i \(-0.227794\pi\)
0.754676 + 0.656097i \(0.227794\pi\)
\(420\) 0 0
\(421\) −24.1931 −1.17910 −0.589551 0.807732i \(-0.700695\pi\)
−0.589551 + 0.807732i \(0.700695\pi\)
\(422\) −34.6670 −1.68756
\(423\) 0 0
\(424\) −20.1717 −0.979623
\(425\) 20.2361 0.981595
\(426\) 0 0
\(427\) −2.45215 −0.118668
\(428\) −47.6218 −2.30188
\(429\) 0 0
\(430\) 0.405513 0.0195556
\(431\) −19.4001 −0.934468 −0.467234 0.884134i \(-0.654749\pi\)
−0.467234 + 0.884134i \(0.654749\pi\)
\(432\) 0 0
\(433\) −21.3877 −1.02783 −0.513914 0.857842i \(-0.671805\pi\)
−0.513914 + 0.857842i \(0.671805\pi\)
\(434\) 23.6423 1.13486
\(435\) 0 0
\(436\) 3.46203 0.165801
\(437\) −19.3584 −0.926038
\(438\) 0 0
\(439\) −31.7315 −1.51446 −0.757232 0.653146i \(-0.773449\pi\)
−0.757232 + 0.653146i \(0.773449\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 45.7627 2.17671
\(443\) 1.66610 0.0791590 0.0395795 0.999216i \(-0.487398\pi\)
0.0395795 + 0.999216i \(0.487398\pi\)
\(444\) 0 0
\(445\) 12.6557 0.599940
\(446\) −7.20386 −0.341113
\(447\) 0 0
\(448\) −9.04656 −0.427410
\(449\) 17.1858 0.811047 0.405524 0.914085i \(-0.367089\pi\)
0.405524 + 0.914085i \(0.367089\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −17.7578 −0.835258
\(453\) 0 0
\(454\) −20.0659 −0.941742
\(455\) 4.00032 0.187538
\(456\) 0 0
\(457\) 10.2206 0.478101 0.239051 0.971007i \(-0.423164\pi\)
0.239051 + 0.971007i \(0.423164\pi\)
\(458\) 31.8586 1.48866
\(459\) 0 0
\(460\) 33.2598 1.55075
\(461\) 22.1160 1.03004 0.515022 0.857177i \(-0.327784\pi\)
0.515022 + 0.857177i \(0.327784\pi\)
\(462\) 0 0
\(463\) −30.3717 −1.41149 −0.705747 0.708464i \(-0.749389\pi\)
−0.705747 + 0.708464i \(0.749389\pi\)
\(464\) −15.5150 −0.720265
\(465\) 0 0
\(466\) 31.1771 1.44425
\(467\) −25.1909 −1.16570 −0.582848 0.812581i \(-0.698062\pi\)
−0.582848 + 0.812581i \(0.698062\pi\)
\(468\) 0 0
\(469\) −9.41987 −0.434969
\(470\) 28.6787 1.32285
\(471\) 0 0
\(472\) 32.1995 1.48210
\(473\) 0 0
\(474\) 0 0
\(475\) −9.77690 −0.448595
\(476\) −23.1464 −1.06091
\(477\) 0 0
\(478\) 12.1063 0.553729
\(479\) 21.2040 0.968835 0.484417 0.874837i \(-0.339032\pi\)
0.484417 + 0.874837i \(0.339032\pi\)
\(480\) 0 0
\(481\) 17.3375 0.790523
\(482\) −6.38445 −0.290804
\(483\) 0 0
\(484\) 0 0
\(485\) −22.1230 −1.00455
\(486\) 0 0
\(487\) −16.8198 −0.762176 −0.381088 0.924539i \(-0.624451\pi\)
−0.381088 + 0.924539i \(0.624451\pi\)
\(488\) −11.3660 −0.514515
\(489\) 0 0
\(490\) −3.05904 −0.138193
\(491\) −4.83804 −0.218338 −0.109169 0.994023i \(-0.534819\pi\)
−0.109169 + 0.994023i \(0.534819\pi\)
\(492\) 0 0
\(493\) 26.6321 1.19945
\(494\) −22.1099 −0.994770
\(495\) 0 0
\(496\) 33.5719 1.50742
\(497\) 0.116610 0.00523068
\(498\) 0 0
\(499\) 30.5836 1.36911 0.684554 0.728962i \(-0.259997\pi\)
0.684554 + 0.728962i \(0.259997\pi\)
\(500\) 41.3858 1.85083
\(501\) 0 0
\(502\) 63.7335 2.84457
\(503\) 28.2407 1.25919 0.629594 0.776924i \(-0.283221\pi\)
0.629594 + 0.776924i \(0.283221\pi\)
\(504\) 0 0
\(505\) −15.7029 −0.698768
\(506\) 0 0
\(507\) 0 0
\(508\) −31.3654 −1.39161
\(509\) 4.20065 0.186190 0.0930952 0.995657i \(-0.470324\pi\)
0.0930952 + 0.995657i \(0.470324\pi\)
\(510\) 0 0
\(511\) 0.615032 0.0272074
\(512\) −35.0383 −1.54849
\(513\) 0 0
\(514\) −72.7964 −3.21091
\(515\) −10.3435 −0.455787
\(516\) 0 0
\(517\) 0 0
\(518\) −13.2580 −0.582523
\(519\) 0 0
\(520\) 18.5419 0.813118
\(521\) 20.6470 0.904560 0.452280 0.891876i \(-0.350611\pi\)
0.452280 + 0.891876i \(0.350611\pi\)
\(522\) 0 0
\(523\) −22.6341 −0.989719 −0.494859 0.868973i \(-0.664780\pi\)
−0.494859 + 0.868973i \(0.664780\pi\)
\(524\) −0.396417 −0.0173176
\(525\) 0 0
\(526\) −8.11063 −0.353640
\(527\) −57.6275 −2.51030
\(528\) 0 0
\(529\) 22.7440 0.988869
\(530\) 13.3127 0.578268
\(531\) 0 0
\(532\) 11.1830 0.484843
\(533\) −0.999529 −0.0432944
\(534\) 0 0
\(535\) 15.3409 0.663243
\(536\) −43.6622 −1.88592
\(537\) 0 0
\(538\) 4.21881 0.181886
\(539\) 0 0
\(540\) 0 0
\(541\) 42.7163 1.83652 0.918258 0.395982i \(-0.129596\pi\)
0.918258 + 0.395982i \(0.129596\pi\)
\(542\) −6.01770 −0.258482
\(543\) 0 0
\(544\) 5.22622 0.224072
\(545\) −1.11526 −0.0477724
\(546\) 0 0
\(547\) 44.4827 1.90194 0.950972 0.309278i \(-0.100087\pi\)
0.950972 + 0.309278i \(0.100087\pi\)
\(548\) 17.8324 0.761761
\(549\) 0 0
\(550\) 0 0
\(551\) −12.8671 −0.548156
\(552\) 0 0
\(553\) 8.52928 0.362702
\(554\) 12.5307 0.532379
\(555\) 0 0
\(556\) −14.9459 −0.633849
\(557\) 24.9911 1.05891 0.529454 0.848339i \(-0.322397\pi\)
0.529454 + 0.848339i \(0.322397\pi\)
\(558\) 0 0
\(559\) 0.421323 0.0178201
\(560\) −4.34382 −0.183560
\(561\) 0 0
\(562\) 54.8565 2.31398
\(563\) −9.01005 −0.379729 −0.189864 0.981810i \(-0.560805\pi\)
−0.189864 + 0.981810i \(0.560805\pi\)
\(564\) 0 0
\(565\) 5.72050 0.240663
\(566\) −20.6286 −0.867086
\(567\) 0 0
\(568\) 0.540502 0.0226789
\(569\) −29.1738 −1.22303 −0.611516 0.791232i \(-0.709440\pi\)
−0.611516 + 0.791232i \(0.709440\pi\)
\(570\) 0 0
\(571\) 1.78994 0.0749067 0.0374533 0.999298i \(-0.488075\pi\)
0.0374533 + 0.999298i \(0.488075\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0.764339 0.0319029
\(575\) 23.1028 0.963455
\(576\) 0 0
\(577\) −21.5850 −0.898594 −0.449297 0.893382i \(-0.648325\pi\)
−0.449297 + 0.893382i \(0.648325\pi\)
\(578\) 43.9813 1.82938
\(579\) 0 0
\(580\) 22.1070 0.917944
\(581\) −0.950532 −0.0394347
\(582\) 0 0
\(583\) 0 0
\(584\) 2.85074 0.117965
\(585\) 0 0
\(586\) −59.8526 −2.47249
\(587\) −5.52355 −0.227981 −0.113991 0.993482i \(-0.536363\pi\)
−0.113991 + 0.993482i \(0.536363\pi\)
\(588\) 0 0
\(589\) 27.8423 1.14722
\(590\) −21.2507 −0.874879
\(591\) 0 0
\(592\) −18.8263 −0.773756
\(593\) 40.7867 1.67491 0.837454 0.546508i \(-0.184043\pi\)
0.837454 + 0.546508i \(0.184043\pi\)
\(594\) 0 0
\(595\) 7.45636 0.305681
\(596\) 19.0363 0.779757
\(597\) 0 0
\(598\) 52.2457 2.13648
\(599\) 26.5149 1.08337 0.541685 0.840582i \(-0.317787\pi\)
0.541685 + 0.840582i \(0.317787\pi\)
\(600\) 0 0
\(601\) −37.7699 −1.54067 −0.770333 0.637641i \(-0.779910\pi\)
−0.770333 + 0.637641i \(0.779910\pi\)
\(602\) −0.322186 −0.0131313
\(603\) 0 0
\(604\) −66.3186 −2.69847
\(605\) 0 0
\(606\) 0 0
\(607\) −27.7244 −1.12530 −0.562649 0.826696i \(-0.690218\pi\)
−0.562649 + 0.826696i \(0.690218\pi\)
\(608\) −2.52500 −0.102402
\(609\) 0 0
\(610\) 7.50125 0.303717
\(611\) 29.7968 1.20545
\(612\) 0 0
\(613\) 36.2264 1.46317 0.731586 0.681749i \(-0.238780\pi\)
0.731586 + 0.681749i \(0.238780\pi\)
\(614\) 11.1653 0.450594
\(615\) 0 0
\(616\) 0 0
\(617\) −41.1920 −1.65833 −0.829163 0.559007i \(-0.811183\pi\)
−0.829163 + 0.559007i \(0.811183\pi\)
\(618\) 0 0
\(619\) −35.7395 −1.43649 −0.718247 0.695789i \(-0.755055\pi\)
−0.718247 + 0.695789i \(0.755055\pi\)
\(620\) −47.8360 −1.92114
\(621\) 0 0
\(622\) 5.37308 0.215441
\(623\) −10.0552 −0.402852
\(624\) 0 0
\(625\) 3.74725 0.149890
\(626\) −23.5736 −0.942192
\(627\) 0 0
\(628\) 8.77077 0.349992
\(629\) 32.3161 1.28853
\(630\) 0 0
\(631\) 2.76508 0.110076 0.0550380 0.998484i \(-0.482472\pi\)
0.0550380 + 0.998484i \(0.482472\pi\)
\(632\) 39.5342 1.57259
\(633\) 0 0
\(634\) −15.6895 −0.623111
\(635\) 10.1040 0.400966
\(636\) 0 0
\(637\) −3.17831 −0.125929
\(638\) 0 0
\(639\) 0 0
\(640\) 25.4531 1.00612
\(641\) −38.4864 −1.52012 −0.760061 0.649851i \(-0.774831\pi\)
−0.760061 + 0.649851i \(0.774831\pi\)
\(642\) 0 0
\(643\) −26.3781 −1.04025 −0.520126 0.854089i \(-0.674115\pi\)
−0.520126 + 0.854089i \(0.674115\pi\)
\(644\) −26.4254 −1.04131
\(645\) 0 0
\(646\) −41.2115 −1.62144
\(647\) 27.3211 1.07410 0.537051 0.843549i \(-0.319538\pi\)
0.537051 + 0.843549i \(0.319538\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 26.3865 1.03496
\(651\) 0 0
\(652\) −61.8777 −2.42332
\(653\) 8.46165 0.331130 0.165565 0.986199i \(-0.447055\pi\)
0.165565 + 0.986199i \(0.447055\pi\)
\(654\) 0 0
\(655\) 0.127702 0.00498971
\(656\) 1.08536 0.0423761
\(657\) 0 0
\(658\) −22.7856 −0.888276
\(659\) 5.29247 0.206165 0.103083 0.994673i \(-0.467129\pi\)
0.103083 + 0.994673i \(0.467129\pi\)
\(660\) 0 0
\(661\) −19.2700 −0.749517 −0.374759 0.927122i \(-0.622274\pi\)
−0.374759 + 0.927122i \(0.622274\pi\)
\(662\) 8.80008 0.342025
\(663\) 0 0
\(664\) −4.40583 −0.170979
\(665\) −3.60248 −0.139698
\(666\) 0 0
\(667\) 30.4050 1.17728
\(668\) −81.0393 −3.13551
\(669\) 0 0
\(670\) 28.8158 1.11325
\(671\) 0 0
\(672\) 0 0
\(673\) −18.9922 −0.732096 −0.366048 0.930596i \(-0.619289\pi\)
−0.366048 + 0.930596i \(0.619289\pi\)
\(674\) −15.5673 −0.599629
\(675\) 0 0
\(676\) −11.3241 −0.435544
\(677\) −32.4219 −1.24607 −0.623037 0.782192i \(-0.714101\pi\)
−0.623037 + 0.782192i \(0.714101\pi\)
\(678\) 0 0
\(679\) 17.5770 0.674544
\(680\) 34.5611 1.32536
\(681\) 0 0
\(682\) 0 0
\(683\) −15.1260 −0.578779 −0.289389 0.957211i \(-0.593452\pi\)
−0.289389 + 0.957211i \(0.593452\pi\)
\(684\) 0 0
\(685\) −5.74451 −0.219486
\(686\) 2.43045 0.0927951
\(687\) 0 0
\(688\) −0.457502 −0.0174421
\(689\) 13.8318 0.526948
\(690\) 0 0
\(691\) −31.4108 −1.19492 −0.597461 0.801898i \(-0.703824\pi\)
−0.597461 + 0.801898i \(0.703824\pi\)
\(692\) 84.2192 3.20153
\(693\) 0 0
\(694\) 42.4357 1.61084
\(695\) 4.81468 0.182631
\(696\) 0 0
\(697\) −1.86306 −0.0705685
\(698\) −52.5583 −1.98936
\(699\) 0 0
\(700\) −13.3461 −0.504434
\(701\) 16.0647 0.606755 0.303377 0.952870i \(-0.401886\pi\)
0.303377 + 0.952870i \(0.401886\pi\)
\(702\) 0 0
\(703\) −15.6133 −0.588865
\(704\) 0 0
\(705\) 0 0
\(706\) −10.6976 −0.402611
\(707\) 12.4761 0.469214
\(708\) 0 0
\(709\) 39.6978 1.49088 0.745441 0.666572i \(-0.232239\pi\)
0.745441 + 0.666572i \(0.232239\pi\)
\(710\) −0.356716 −0.0133873
\(711\) 0 0
\(712\) −46.6069 −1.74667
\(713\) −65.7913 −2.46391
\(714\) 0 0
\(715\) 0 0
\(716\) −18.6822 −0.698187
\(717\) 0 0
\(718\) 25.9614 0.968873
\(719\) 9.86241 0.367806 0.183903 0.982944i \(-0.441127\pi\)
0.183903 + 0.982944i \(0.441127\pi\)
\(720\) 0 0
\(721\) 8.21802 0.306055
\(722\) −26.2676 −0.977580
\(723\) 0 0
\(724\) −30.6818 −1.14028
\(725\) 15.3559 0.570305
\(726\) 0 0
\(727\) −31.5764 −1.17111 −0.585553 0.810634i \(-0.699122\pi\)
−0.585553 + 0.810634i \(0.699122\pi\)
\(728\) −14.7318 −0.545998
\(729\) 0 0
\(730\) −1.88141 −0.0696340
\(731\) 0.785321 0.0290462
\(732\) 0 0
\(733\) 41.2926 1.52518 0.762588 0.646885i \(-0.223928\pi\)
0.762588 + 0.646885i \(0.223928\pi\)
\(734\) 24.8313 0.916540
\(735\) 0 0
\(736\) 5.96659 0.219931
\(737\) 0 0
\(738\) 0 0
\(739\) 35.4095 1.30256 0.651281 0.758837i \(-0.274232\pi\)
0.651281 + 0.758837i \(0.274232\pi\)
\(740\) 26.8253 0.986116
\(741\) 0 0
\(742\) −10.5771 −0.388299
\(743\) 4.02408 0.147629 0.0738146 0.997272i \(-0.476483\pi\)
0.0738146 + 0.997272i \(0.476483\pi\)
\(744\) 0 0
\(745\) −6.13235 −0.224672
\(746\) −67.7733 −2.48136
\(747\) 0 0
\(748\) 0 0
\(749\) −12.1885 −0.445359
\(750\) 0 0
\(751\) 24.4190 0.891062 0.445531 0.895267i \(-0.353015\pi\)
0.445531 + 0.895267i \(0.353015\pi\)
\(752\) −32.3554 −1.17988
\(753\) 0 0
\(754\) 34.7265 1.26466
\(755\) 21.3638 0.777510
\(756\) 0 0
\(757\) −1.31723 −0.0478755 −0.0239377 0.999713i \(-0.507620\pi\)
−0.0239377 + 0.999713i \(0.507620\pi\)
\(758\) 2.79023 0.101346
\(759\) 0 0
\(760\) −16.6979 −0.605696
\(761\) 7.24404 0.262596 0.131298 0.991343i \(-0.458085\pi\)
0.131298 + 0.991343i \(0.458085\pi\)
\(762\) 0 0
\(763\) 0.886088 0.0320785
\(764\) 34.5008 1.24820
\(765\) 0 0
\(766\) 48.4915 1.75207
\(767\) −22.0792 −0.797235
\(768\) 0 0
\(769\) 44.3139 1.59800 0.798999 0.601332i \(-0.205363\pi\)
0.798999 + 0.601332i \(0.205363\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 100.031 3.60018
\(773\) −19.0559 −0.685395 −0.342697 0.939446i \(-0.611341\pi\)
−0.342697 + 0.939446i \(0.611341\pi\)
\(774\) 0 0
\(775\) −33.2277 −1.19357
\(776\) 81.4714 2.92465
\(777\) 0 0
\(778\) −6.31918 −0.226554
\(779\) 0.900123 0.0322502
\(780\) 0 0
\(781\) 0 0
\(782\) 97.3828 3.48240
\(783\) 0 0
\(784\) 3.45123 0.123258
\(785\) −2.82541 −0.100843
\(786\) 0 0
\(787\) 31.4600 1.12143 0.560714 0.828009i \(-0.310527\pi\)
0.560714 + 0.828009i \(0.310527\pi\)
\(788\) 43.7505 1.55855
\(789\) 0 0
\(790\) −26.0914 −0.928292
\(791\) −4.54502 −0.161602
\(792\) 0 0
\(793\) 7.79370 0.276763
\(794\) 21.3208 0.756648
\(795\) 0 0
\(796\) −47.8631 −1.69646
\(797\) 12.5453 0.444376 0.222188 0.975004i \(-0.428680\pi\)
0.222188 + 0.975004i \(0.428680\pi\)
\(798\) 0 0
\(799\) 55.5395 1.96485
\(800\) 3.01340 0.106540
\(801\) 0 0
\(802\) −68.9118 −2.43336
\(803\) 0 0
\(804\) 0 0
\(805\) 8.51266 0.300032
\(806\) −75.1424 −2.64678
\(807\) 0 0
\(808\) 57.8284 2.03440
\(809\) 5.38024 0.189159 0.0945796 0.995517i \(-0.469849\pi\)
0.0945796 + 0.995517i \(0.469849\pi\)
\(810\) 0 0
\(811\) 13.6999 0.481070 0.240535 0.970641i \(-0.422677\pi\)
0.240535 + 0.970641i \(0.422677\pi\)
\(812\) −17.5643 −0.616387
\(813\) 0 0
\(814\) 0 0
\(815\) 19.9333 0.698231
\(816\) 0 0
\(817\) −0.379421 −0.0132743
\(818\) 48.3794 1.69155
\(819\) 0 0
\(820\) −1.54651 −0.0540064
\(821\) 9.23116 0.322170 0.161085 0.986941i \(-0.448501\pi\)
0.161085 + 0.986941i \(0.448501\pi\)
\(822\) 0 0
\(823\) −12.4427 −0.433724 −0.216862 0.976202i \(-0.569582\pi\)
−0.216862 + 0.976202i \(0.569582\pi\)
\(824\) 38.0915 1.32698
\(825\) 0 0
\(826\) 16.8840 0.587469
\(827\) 4.48387 0.155919 0.0779597 0.996957i \(-0.475159\pi\)
0.0779597 + 0.996957i \(0.475159\pi\)
\(828\) 0 0
\(829\) −36.8776 −1.28081 −0.640405 0.768037i \(-0.721234\pi\)
−0.640405 + 0.768037i \(0.721234\pi\)
\(830\) 2.90772 0.100928
\(831\) 0 0
\(832\) 28.7528 0.996823
\(833\) −5.92418 −0.205261
\(834\) 0 0
\(835\) 26.1060 0.903435
\(836\) 0 0
\(837\) 0 0
\(838\) 75.0905 2.59396
\(839\) −9.16407 −0.316379 −0.158189 0.987409i \(-0.550566\pi\)
−0.158189 + 0.987409i \(0.550566\pi\)
\(840\) 0 0
\(841\) −8.79054 −0.303122
\(842\) −58.8003 −2.02639
\(843\) 0 0
\(844\) −55.7293 −1.91828
\(845\) 3.64795 0.125493
\(846\) 0 0
\(847\) 0 0
\(848\) −15.0195 −0.515771
\(849\) 0 0
\(850\) 49.1829 1.68696
\(851\) 36.8942 1.26472
\(852\) 0 0
\(853\) 6.52049 0.223257 0.111629 0.993750i \(-0.464393\pi\)
0.111629 + 0.993750i \(0.464393\pi\)
\(854\) −5.95984 −0.203942
\(855\) 0 0
\(856\) −56.4952 −1.93097
\(857\) −48.0736 −1.64216 −0.821082 0.570810i \(-0.806629\pi\)
−0.821082 + 0.570810i \(0.806629\pi\)
\(858\) 0 0
\(859\) −0.316298 −0.0107920 −0.00539598 0.999985i \(-0.501718\pi\)
−0.00539598 + 0.999985i \(0.501718\pi\)
\(860\) 0.651887 0.0222291
\(861\) 0 0
\(862\) −47.1509 −1.60597
\(863\) −3.62693 −0.123462 −0.0617311 0.998093i \(-0.519662\pi\)
−0.0617311 + 0.998093i \(0.519662\pi\)
\(864\) 0 0
\(865\) −27.1303 −0.922458
\(866\) −51.9818 −1.76641
\(867\) 0 0
\(868\) 38.0064 1.29002
\(869\) 0 0
\(870\) 0 0
\(871\) 29.9393 1.01445
\(872\) 4.10712 0.139085
\(873\) 0 0
\(874\) −47.0497 −1.59148
\(875\) 10.5924 0.358090
\(876\) 0 0
\(877\) 26.7076 0.901852 0.450926 0.892561i \(-0.351094\pi\)
0.450926 + 0.892561i \(0.351094\pi\)
\(878\) −77.1220 −2.60274
\(879\) 0 0
\(880\) 0 0
\(881\) −2.91937 −0.0983560 −0.0491780 0.998790i \(-0.515660\pi\)
−0.0491780 + 0.998790i \(0.515660\pi\)
\(882\) 0 0
\(883\) −45.1574 −1.51967 −0.759834 0.650117i \(-0.774720\pi\)
−0.759834 + 0.650117i \(0.774720\pi\)
\(884\) 73.5663 2.47430
\(885\) 0 0
\(886\) 4.04939 0.136042
\(887\) 24.0249 0.806676 0.403338 0.915051i \(-0.367850\pi\)
0.403338 + 0.915051i \(0.367850\pi\)
\(888\) 0 0
\(889\) −8.02779 −0.269243
\(890\) 30.7592 1.03105
\(891\) 0 0
\(892\) −11.5806 −0.387749
\(893\) −26.8334 −0.897947
\(894\) 0 0
\(895\) 6.01828 0.201169
\(896\) −20.2229 −0.675599
\(897\) 0 0
\(898\) 41.7692 1.39386
\(899\) −43.7300 −1.45848
\(900\) 0 0
\(901\) 25.7816 0.858909
\(902\) 0 0
\(903\) 0 0
\(904\) −21.0667 −0.700667
\(905\) 9.88384 0.328550
\(906\) 0 0
\(907\) −16.0270 −0.532166 −0.266083 0.963950i \(-0.585730\pi\)
−0.266083 + 0.963950i \(0.585730\pi\)
\(908\) −32.2572 −1.07049
\(909\) 0 0
\(910\) 9.72259 0.322301
\(911\) 16.4043 0.543500 0.271750 0.962368i \(-0.412398\pi\)
0.271750 + 0.962368i \(0.412398\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 24.8408 0.821660
\(915\) 0 0
\(916\) 51.2147 1.69218
\(917\) −0.101461 −0.00335053
\(918\) 0 0
\(919\) 32.3201 1.06614 0.533071 0.846070i \(-0.321038\pi\)
0.533071 + 0.846070i \(0.321038\pi\)
\(920\) 39.4572 1.30086
\(921\) 0 0
\(922\) 53.7518 1.77022
\(923\) −0.370623 −0.0121992
\(924\) 0 0
\(925\) 18.6333 0.612659
\(926\) −73.8171 −2.42578
\(927\) 0 0
\(928\) 3.96585 0.130185
\(929\) 8.19531 0.268879 0.134440 0.990922i \(-0.457077\pi\)
0.134440 + 0.990922i \(0.457077\pi\)
\(930\) 0 0
\(931\) 2.86222 0.0938054
\(932\) 50.1190 1.64170
\(933\) 0 0
\(934\) −61.2253 −2.00335
\(935\) 0 0
\(936\) 0 0
\(937\) 34.1374 1.11522 0.557610 0.830103i \(-0.311718\pi\)
0.557610 + 0.830103i \(0.311718\pi\)
\(938\) −22.8945 −0.747533
\(939\) 0 0
\(940\) 46.1027 1.50371
\(941\) −0.979371 −0.0319266 −0.0159633 0.999873i \(-0.505081\pi\)
−0.0159633 + 0.999873i \(0.505081\pi\)
\(942\) 0 0
\(943\) −2.12699 −0.0692644
\(944\) 23.9752 0.780326
\(945\) 0 0
\(946\) 0 0
\(947\) −0.935599 −0.0304029 −0.0152014 0.999884i \(-0.504839\pi\)
−0.0152014 + 0.999884i \(0.504839\pi\)
\(948\) 0 0
\(949\) −1.95476 −0.0634542
\(950\) −23.7623 −0.770950
\(951\) 0 0
\(952\) −27.4593 −0.889960
\(953\) −16.7321 −0.542004 −0.271002 0.962579i \(-0.587355\pi\)
−0.271002 + 0.962579i \(0.587355\pi\)
\(954\) 0 0
\(955\) −11.1141 −0.359643
\(956\) 19.4616 0.629434
\(957\) 0 0
\(958\) 51.5353 1.66503
\(959\) 4.56409 0.147382
\(960\) 0 0
\(961\) 63.6245 2.05240
\(962\) 42.1380 1.35859
\(963\) 0 0
\(964\) −10.2634 −0.330561
\(965\) −32.2238 −1.03732
\(966\) 0 0
\(967\) 36.4439 1.17196 0.585978 0.810327i \(-0.300710\pi\)
0.585978 + 0.810327i \(0.300710\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −53.7688 −1.72641
\(971\) 33.2417 1.06678 0.533389 0.845870i \(-0.320918\pi\)
0.533389 + 0.845870i \(0.320918\pi\)
\(972\) 0 0
\(973\) −3.82533 −0.122634
\(974\) −40.8796 −1.30987
\(975\) 0 0
\(976\) −8.46294 −0.270892
\(977\) −22.5666 −0.721970 −0.360985 0.932572i \(-0.617559\pi\)
−0.360985 + 0.932572i \(0.617559\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −4.91760 −0.157087
\(981\) 0 0
\(982\) −11.7586 −0.375233
\(983\) 15.1048 0.481767 0.240884 0.970554i \(-0.422563\pi\)
0.240884 + 0.970554i \(0.422563\pi\)
\(984\) 0 0
\(985\) −14.0938 −0.449064
\(986\) 64.7281 2.06136
\(987\) 0 0
\(988\) −35.5429 −1.13077
\(989\) 0.896574 0.0285094
\(990\) 0 0
\(991\) −55.1534 −1.75201 −0.876003 0.482305i \(-0.839800\pi\)
−0.876003 + 0.482305i \(0.839800\pi\)
\(992\) −8.58145 −0.272461
\(993\) 0 0
\(994\) 0.283415 0.00898939
\(995\) 15.4186 0.488802
\(996\) 0 0
\(997\) 49.7048 1.57417 0.787083 0.616847i \(-0.211590\pi\)
0.787083 + 0.616847i \(0.211590\pi\)
\(998\) 74.3319 2.35293
\(999\) 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 7623.2.a.ct.1.8 8
3.2 odd 2 847.2.a.p.1.1 8
11.3 even 5 693.2.m.i.64.4 16
11.4 even 5 693.2.m.i.379.4 16
11.10 odd 2 7623.2.a.cw.1.1 8
21.20 even 2 5929.2.a.bt.1.1 8
33.2 even 10 847.2.f.v.323.1 16
33.5 odd 10 847.2.f.w.729.4 16
33.8 even 10 847.2.f.x.372.4 16
33.14 odd 10 77.2.f.b.64.1 16
33.17 even 10 847.2.f.v.729.1 16
33.20 odd 10 847.2.f.w.323.4 16
33.26 odd 10 77.2.f.b.71.1 yes 16
33.29 even 10 847.2.f.x.148.4 16
33.32 even 2 847.2.a.o.1.8 8
231.26 even 30 539.2.q.f.214.1 32
231.47 even 30 539.2.q.f.361.4 32
231.59 even 30 539.2.q.f.324.4 32
231.80 even 30 539.2.q.f.471.1 32
231.125 even 10 539.2.f.e.148.1 16
231.146 even 10 539.2.f.e.295.1 16
231.158 odd 30 539.2.q.g.324.4 32
231.179 odd 30 539.2.q.g.471.1 32
231.191 odd 30 539.2.q.g.214.1 32
231.212 odd 30 539.2.q.g.361.4 32
231.230 odd 2 5929.2.a.bs.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.b.64.1 16 33.14 odd 10
77.2.f.b.71.1 yes 16 33.26 odd 10
539.2.f.e.148.1 16 231.125 even 10
539.2.f.e.295.1 16 231.146 even 10
539.2.q.f.214.1 32 231.26 even 30
539.2.q.f.324.4 32 231.59 even 30
539.2.q.f.361.4 32 231.47 even 30
539.2.q.f.471.1 32 231.80 even 30
539.2.q.g.214.1 32 231.191 odd 30
539.2.q.g.324.4 32 231.158 odd 30
539.2.q.g.361.4 32 231.212 odd 30
539.2.q.g.471.1 32 231.179 odd 30
693.2.m.i.64.4 16 11.3 even 5
693.2.m.i.379.4 16 11.4 even 5
847.2.a.o.1.8 8 33.32 even 2
847.2.a.p.1.1 8 3.2 odd 2
847.2.f.v.323.1 16 33.2 even 10
847.2.f.v.729.1 16 33.17 even 10
847.2.f.w.323.4 16 33.20 odd 10
847.2.f.w.729.4 16 33.5 odd 10
847.2.f.x.148.4 16 33.29 even 10
847.2.f.x.372.4 16 33.8 even 10
5929.2.a.bs.1.8 8 231.230 odd 2
5929.2.a.bt.1.1 8 21.20 even 2
7623.2.a.ct.1.8 8 1.1 even 1 trivial
7623.2.a.cw.1.1 8 11.10 odd 2