Properties

Label 147.4.a.j.1.1
Level $147$
Weight $4$
Character 147.1
Self dual yes
Analytic conductor $8.673$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
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 \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 147.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.414214 q^{2} -3.00000 q^{3} -7.82843 q^{4} +0.100505 q^{5} +1.24264 q^{6} +6.55635 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-0.414214 q^{2} -3.00000 q^{3} -7.82843 q^{4} +0.100505 q^{5} +1.24264 q^{6} +6.55635 q^{8} +9.00000 q^{9} -0.0416306 q^{10} -43.9411 q^{11} +23.4853 q^{12} +16.6447 q^{13} -0.301515 q^{15} +59.9117 q^{16} +121.640 q^{17} -3.72792 q^{18} +127.113 q^{19} -0.786797 q^{20} +18.2010 q^{22} +53.5980 q^{23} -19.6690 q^{24} -124.990 q^{25} -6.89444 q^{26} -27.0000 q^{27} +235.681 q^{29} +0.124892 q^{30} +18.7107 q^{31} -77.2670 q^{32} +131.823 q^{33} -50.3848 q^{34} -70.4558 q^{36} -191.882 q^{37} -52.6518 q^{38} -49.9340 q^{39} +0.658946 q^{40} +319.713 q^{41} -218.579 q^{43} +343.990 q^{44} +0.904546 q^{45} -22.2010 q^{46} -401.553 q^{47} -179.735 q^{48} +51.7725 q^{50} -364.919 q^{51} -130.302 q^{52} +643.117 q^{53} +11.1838 q^{54} -4.41631 q^{55} -381.338 q^{57} -97.6224 q^{58} +11.6123 q^{59} +2.36039 q^{60} +12.2426 q^{61} -7.75022 q^{62} -447.288 q^{64} +1.67287 q^{65} -54.6030 q^{66} +669.048 q^{67} -952.247 q^{68} -160.794 q^{69} +822.098 q^{71} +59.0071 q^{72} -515.100 q^{73} +79.4802 q^{74} +374.970 q^{75} -995.092 q^{76} +20.6833 q^{78} -805.754 q^{79} +6.02143 q^{80} +81.0000 q^{81} -132.429 q^{82} +394.863 q^{83} +12.2254 q^{85} +90.5382 q^{86} -707.044 q^{87} -288.093 q^{88} -673.418 q^{89} -0.374675 q^{90} -419.588 q^{92} -56.1320 q^{93} +166.329 q^{94} +12.7755 q^{95} +231.801 q^{96} +1091.11 q^{97} -395.470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 6q^{3} - 10q^{4} + 20q^{5} - 6q^{6} - 18q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 6q^{3} - 10q^{4} + 20q^{5} - 6q^{6} - 18q^{8} + 18q^{9} + 48q^{10} - 20q^{11} + 30q^{12} + 104q^{13} - 60q^{15} + 18q^{16} + 116q^{17} + 18q^{18} + 192q^{19} - 44q^{20} + 76q^{22} + 28q^{23} + 54q^{24} + 146q^{25} + 204q^{26} - 54q^{27} + 296q^{29} - 144q^{30} - 104q^{31} + 18q^{32} + 60q^{33} - 64q^{34} - 90q^{36} - 248q^{37} + 104q^{38} - 312q^{39} - 488q^{40} + 20q^{41} - 720q^{43} + 292q^{44} + 180q^{45} - 84q^{46} - 96q^{47} - 54q^{48} + 706q^{50} - 348q^{51} - 320q^{52} + 268q^{53} - 54q^{54} + 472q^{55} - 576q^{57} + 48q^{58} - 616q^{59} + 132q^{60} + 16q^{61} - 304q^{62} + 118q^{64} + 1740q^{65} - 228q^{66} - 144q^{67} - 940q^{68} - 84q^{69} + 988q^{71} - 162q^{72} + 104q^{73} - 56q^{74} - 438q^{75} - 1136q^{76} - 612q^{78} - 944q^{79} - 828q^{80} + 162q^{81} - 856q^{82} + 1016q^{83} - 100q^{85} - 1120q^{86} - 888q^{87} - 876q^{88} - 388q^{89} + 432q^{90} - 364q^{92} + 312q^{93} + 904q^{94} + 1304q^{95} - 54q^{96} + 488q^{97} - 180q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.414214 −0.146447 −0.0732233 0.997316i \(-0.523329\pi\)
−0.0732233 + 0.997316i \(0.523329\pi\)
\(3\) −3.00000 −0.577350
\(4\) −7.82843 −0.978553
\(5\) 0.100505 0.00898945 0.00449472 0.999990i \(-0.498569\pi\)
0.00449472 + 0.999990i \(0.498569\pi\)
\(6\) 1.24264 0.0845510
\(7\) 0 0
\(8\) 6.55635 0.289752
\(9\) 9.00000 0.333333
\(10\) −0.0416306 −0.00131647
\(11\) −43.9411 −1.20443 −0.602216 0.798333i \(-0.705715\pi\)
−0.602216 + 0.798333i \(0.705715\pi\)
\(12\) 23.4853 0.564968
\(13\) 16.6447 0.355108 0.177554 0.984111i \(-0.443182\pi\)
0.177554 + 0.984111i \(0.443182\pi\)
\(14\) 0 0
\(15\) −0.301515 −0.00519006
\(16\) 59.9117 0.936120
\(17\) 121.640 1.73541 0.867704 0.497081i \(-0.165595\pi\)
0.867704 + 0.497081i \(0.165595\pi\)
\(18\) −3.72792 −0.0488155
\(19\) 127.113 1.53482 0.767412 0.641154i \(-0.221544\pi\)
0.767412 + 0.641154i \(0.221544\pi\)
\(20\) −0.786797 −0.00879665
\(21\) 0 0
\(22\) 18.2010 0.176385
\(23\) 53.5980 0.485911 0.242955 0.970037i \(-0.421883\pi\)
0.242955 + 0.970037i \(0.421883\pi\)
\(24\) −19.6690 −0.167289
\(25\) −124.990 −0.999919
\(26\) −6.89444 −0.0520043
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 235.681 1.50913 0.754567 0.656223i \(-0.227847\pi\)
0.754567 + 0.656223i \(0.227847\pi\)
\(30\) 0.124892 0.000760067 0
\(31\) 18.7107 0.108404 0.0542022 0.998530i \(-0.482738\pi\)
0.0542022 + 0.998530i \(0.482738\pi\)
\(32\) −77.2670 −0.426844
\(33\) 131.823 0.695379
\(34\) −50.3848 −0.254145
\(35\) 0 0
\(36\) −70.4558 −0.326184
\(37\) −191.882 −0.852574 −0.426287 0.904588i \(-0.640179\pi\)
−0.426287 + 0.904588i \(0.640179\pi\)
\(38\) −52.6518 −0.224770
\(39\) −49.9340 −0.205021
\(40\) 0.658946 0.00260471
\(41\) 319.713 1.21782 0.608912 0.793238i \(-0.291606\pi\)
0.608912 + 0.793238i \(0.291606\pi\)
\(42\) 0 0
\(43\) −218.579 −0.775184 −0.387592 0.921831i \(-0.626693\pi\)
−0.387592 + 0.921831i \(0.626693\pi\)
\(44\) 343.990 1.17860
\(45\) 0.904546 0.00299648
\(46\) −22.2010 −0.0711600
\(47\) −401.553 −1.24623 −0.623113 0.782132i \(-0.714132\pi\)
−0.623113 + 0.782132i \(0.714132\pi\)
\(48\) −179.735 −0.540469
\(49\) 0 0
\(50\) 51.7725 0.146435
\(51\) −364.919 −1.00194
\(52\) −130.302 −0.347492
\(53\) 643.117 1.66677 0.833386 0.552692i \(-0.186399\pi\)
0.833386 + 0.552692i \(0.186399\pi\)
\(54\) 11.1838 0.0281837
\(55\) −4.41631 −0.0108272
\(56\) 0 0
\(57\) −381.338 −0.886131
\(58\) −97.6224 −0.221008
\(59\) 11.6123 0.0256235 0.0128118 0.999918i \(-0.495922\pi\)
0.0128118 + 0.999918i \(0.495922\pi\)
\(60\) 2.36039 0.00507875
\(61\) 12.2426 0.0256969 0.0128484 0.999917i \(-0.495910\pi\)
0.0128484 + 0.999917i \(0.495910\pi\)
\(62\) −7.75022 −0.0158755
\(63\) 0 0
\(64\) −447.288 −0.873610
\(65\) 1.67287 0.00319222
\(66\) −54.6030 −0.101836
\(67\) 669.048 1.21996 0.609979 0.792417i \(-0.291178\pi\)
0.609979 + 0.792417i \(0.291178\pi\)
\(68\) −952.247 −1.69819
\(69\) −160.794 −0.280541
\(70\) 0 0
\(71\) 822.098 1.37416 0.687078 0.726584i \(-0.258893\pi\)
0.687078 + 0.726584i \(0.258893\pi\)
\(72\) 59.0071 0.0965841
\(73\) −515.100 −0.825861 −0.412930 0.910763i \(-0.635495\pi\)
−0.412930 + 0.910763i \(0.635495\pi\)
\(74\) 79.4802 0.124857
\(75\) 374.970 0.577304
\(76\) −995.092 −1.50191
\(77\) 0 0
\(78\) 20.6833 0.0300247
\(79\) −805.754 −1.14752 −0.573762 0.819022i \(-0.694517\pi\)
−0.573762 + 0.819022i \(0.694517\pi\)
\(80\) 6.02143 0.00841520
\(81\) 81.0000 0.111111
\(82\) −132.429 −0.178346
\(83\) 394.863 0.522191 0.261095 0.965313i \(-0.415916\pi\)
0.261095 + 0.965313i \(0.415916\pi\)
\(84\) 0 0
\(85\) 12.2254 0.0156004
\(86\) 90.5382 0.113523
\(87\) −707.044 −0.871299
\(88\) −288.093 −0.348987
\(89\) −673.418 −0.802047 −0.401024 0.916068i \(-0.631345\pi\)
−0.401024 + 0.916068i \(0.631345\pi\)
\(90\) −0.374675 −0.000438825 0
\(91\) 0 0
\(92\) −419.588 −0.475490
\(93\) −56.1320 −0.0625873
\(94\) 166.329 0.182505
\(95\) 12.7755 0.0137972
\(96\) 231.801 0.246439
\(97\) 1091.11 1.14212 0.571061 0.820908i \(-0.306532\pi\)
0.571061 + 0.820908i \(0.306532\pi\)
\(98\) 0 0
\(99\) −395.470 −0.401477
\(100\) 978.474 0.978474
\(101\) 1370.79 1.35048 0.675242 0.737597i \(-0.264039\pi\)
0.675242 + 0.737597i \(0.264039\pi\)
\(102\) 151.154 0.146730
\(103\) 1413.96 1.35263 0.676316 0.736611i \(-0.263575\pi\)
0.676316 + 0.736611i \(0.263575\pi\)
\(104\) 109.128 0.102893
\(105\) 0 0
\(106\) −266.388 −0.244093
\(107\) −343.539 −0.310385 −0.155192 0.987884i \(-0.549600\pi\)
−0.155192 + 0.987884i \(0.549600\pi\)
\(108\) 211.368 0.188323
\(109\) −317.657 −0.279138 −0.139569 0.990212i \(-0.544572\pi\)
−0.139569 + 0.990212i \(0.544572\pi\)
\(110\) 1.82929 0.00158560
\(111\) 575.647 0.492234
\(112\) 0 0
\(113\) 798.373 0.664643 0.332321 0.943166i \(-0.392168\pi\)
0.332321 + 0.943166i \(0.392168\pi\)
\(114\) 157.955 0.129771
\(115\) 5.38687 0.00436807
\(116\) −1845.01 −1.47677
\(117\) 149.802 0.118369
\(118\) −4.80996 −0.00375248
\(119\) 0 0
\(120\) −1.97684 −0.00150383
\(121\) 599.823 0.450656
\(122\) −5.07107 −0.00376322
\(123\) −959.138 −0.703110
\(124\) −146.475 −0.106080
\(125\) −25.1253 −0.0179782
\(126\) 0 0
\(127\) 1071.40 0.748593 0.374297 0.927309i \(-0.377884\pi\)
0.374297 + 0.927309i \(0.377884\pi\)
\(128\) 803.409 0.554781
\(129\) 655.736 0.447553
\(130\) −0.692927 −0.000467490 0
\(131\) 2515.02 1.67739 0.838695 0.544601i \(-0.183319\pi\)
0.838695 + 0.544601i \(0.183319\pi\)
\(132\) −1031.97 −0.680465
\(133\) 0 0
\(134\) −277.129 −0.178659
\(135\) −2.71364 −0.00173002
\(136\) 797.512 0.502839
\(137\) 251.064 0.156568 0.0782841 0.996931i \(-0.475056\pi\)
0.0782841 + 0.996931i \(0.475056\pi\)
\(138\) 66.6030 0.0410842
\(139\) −886.067 −0.540685 −0.270343 0.962764i \(-0.587137\pi\)
−0.270343 + 0.962764i \(0.587137\pi\)
\(140\) 0 0
\(141\) 1204.66 0.719508
\(142\) −340.524 −0.201240
\(143\) −731.385 −0.427703
\(144\) 539.205 0.312040
\(145\) 23.6872 0.0135663
\(146\) 213.361 0.120945
\(147\) 0 0
\(148\) 1502.14 0.834289
\(149\) 582.626 0.320339 0.160170 0.987090i \(-0.448796\pi\)
0.160170 + 0.987090i \(0.448796\pi\)
\(150\) −155.318 −0.0845442
\(151\) −2811.76 −1.51535 −0.757676 0.652631i \(-0.773665\pi\)
−0.757676 + 0.652631i \(0.773665\pi\)
\(152\) 833.395 0.444719
\(153\) 1094.76 0.578469
\(154\) 0 0
\(155\) 1.88052 0.000974496 0
\(156\) 390.905 0.200624
\(157\) 1691.34 0.859770 0.429885 0.902884i \(-0.358554\pi\)
0.429885 + 0.902884i \(0.358554\pi\)
\(158\) 333.754 0.168051
\(159\) −1929.35 −0.962311
\(160\) −7.76573 −0.00383709
\(161\) 0 0
\(162\) −33.5513 −0.0162718
\(163\) −40.7232 −0.0195686 −0.00978432 0.999952i \(-0.503114\pi\)
−0.00978432 + 0.999952i \(0.503114\pi\)
\(164\) −2502.85 −1.19170
\(165\) 13.2489 0.00625107
\(166\) −163.558 −0.0764731
\(167\) −2900.47 −1.34398 −0.671990 0.740560i \(-0.734560\pi\)
−0.671990 + 0.740560i \(0.734560\pi\)
\(168\) 0 0
\(169\) −1919.96 −0.873899
\(170\) −5.06393 −0.00228462
\(171\) 1144.01 0.511608
\(172\) 1711.13 0.758559
\(173\) 2146.15 0.943171 0.471585 0.881820i \(-0.343682\pi\)
0.471585 + 0.881820i \(0.343682\pi\)
\(174\) 292.867 0.127599
\(175\) 0 0
\(176\) −2632.59 −1.12749
\(177\) −34.8368 −0.0147938
\(178\) 278.939 0.117457
\(179\) 1203.54 0.502552 0.251276 0.967916i \(-0.419150\pi\)
0.251276 + 0.967916i \(0.419150\pi\)
\(180\) −7.08117 −0.00293222
\(181\) 2990.47 1.22807 0.614033 0.789280i \(-0.289546\pi\)
0.614033 + 0.789280i \(0.289546\pi\)
\(182\) 0 0
\(183\) −36.7279 −0.0148361
\(184\) 351.407 0.140794
\(185\) −19.2851 −0.00766417
\(186\) 23.2506 0.00916570
\(187\) −5344.98 −2.09018
\(188\) 3143.53 1.21950
\(189\) 0 0
\(190\) −5.29177 −0.00202056
\(191\) −2807.41 −1.06354 −0.531772 0.846887i \(-0.678474\pi\)
−0.531772 + 0.846887i \(0.678474\pi\)
\(192\) 1341.87 0.504379
\(193\) 3336.37 1.24434 0.622169 0.782883i \(-0.286252\pi\)
0.622169 + 0.782883i \(0.286252\pi\)
\(194\) −451.954 −0.167260
\(195\) −5.01862 −0.00184303
\(196\) 0 0
\(197\) −4226.65 −1.52861 −0.764305 0.644855i \(-0.776918\pi\)
−0.764305 + 0.644855i \(0.776918\pi\)
\(198\) 163.809 0.0587950
\(199\) −4385.69 −1.56228 −0.781140 0.624356i \(-0.785361\pi\)
−0.781140 + 0.624356i \(0.785361\pi\)
\(200\) −819.477 −0.289729
\(201\) −2007.14 −0.704343
\(202\) −567.800 −0.197774
\(203\) 0 0
\(204\) 2856.74 0.980450
\(205\) 32.1328 0.0109476
\(206\) −585.680 −0.198088
\(207\) 482.382 0.161970
\(208\) 997.210 0.332423
\(209\) −5585.48 −1.84859
\(210\) 0 0
\(211\) 2291.56 0.747665 0.373833 0.927496i \(-0.378043\pi\)
0.373833 + 0.927496i \(0.378043\pi\)
\(212\) −5034.59 −1.63103
\(213\) −2466.29 −0.793369
\(214\) 142.299 0.0454548
\(215\) −21.9683 −0.00696848
\(216\) −177.021 −0.0557629
\(217\) 0 0
\(218\) 131.578 0.0408788
\(219\) 1545.30 0.476811
\(220\) 34.5727 0.0105950
\(221\) 2024.65 0.616257
\(222\) −238.441 −0.0720860
\(223\) 217.970 0.0654544 0.0327272 0.999464i \(-0.489581\pi\)
0.0327272 + 0.999464i \(0.489581\pi\)
\(224\) 0 0
\(225\) −1124.91 −0.333306
\(226\) −330.697 −0.0973347
\(227\) −1835.36 −0.536639 −0.268320 0.963330i \(-0.586468\pi\)
−0.268320 + 0.963330i \(0.586468\pi\)
\(228\) 2985.28 0.867126
\(229\) −2774.01 −0.800488 −0.400244 0.916409i \(-0.631075\pi\)
−0.400244 + 0.916409i \(0.631075\pi\)
\(230\) −2.23131 −0.000639689 0
\(231\) 0 0
\(232\) 1545.21 0.437275
\(233\) −988.712 −0.277994 −0.138997 0.990293i \(-0.544388\pi\)
−0.138997 + 0.990293i \(0.544388\pi\)
\(234\) −62.0500 −0.0173348
\(235\) −40.3581 −0.0112029
\(236\) −90.9058 −0.0250740
\(237\) 2417.26 0.662524
\(238\) 0 0
\(239\) −837.928 −0.226783 −0.113391 0.993550i \(-0.536171\pi\)
−0.113391 + 0.993550i \(0.536171\pi\)
\(240\) −18.0643 −0.00485852
\(241\) 3454.99 0.923466 0.461733 0.887019i \(-0.347228\pi\)
0.461733 + 0.887019i \(0.347228\pi\)
\(242\) −248.455 −0.0659970
\(243\) −243.000 −0.0641500
\(244\) −95.8406 −0.0251458
\(245\) 0 0
\(246\) 397.288 0.102968
\(247\) 2115.75 0.545028
\(248\) 122.674 0.0314104
\(249\) −1184.59 −0.301487
\(250\) 10.4072 0.00263284
\(251\) −5635.01 −1.41705 −0.708523 0.705688i \(-0.750638\pi\)
−0.708523 + 0.705688i \(0.750638\pi\)
\(252\) 0 0
\(253\) −2355.16 −0.585246
\(254\) −443.788 −0.109629
\(255\) −36.6762 −0.00900687
\(256\) 3245.52 0.792364
\(257\) 2271.16 0.551248 0.275624 0.961265i \(-0.411115\pi\)
0.275624 + 0.961265i \(0.411115\pi\)
\(258\) −271.615 −0.0655426
\(259\) 0 0
\(260\) −13.0960 −0.00312376
\(261\) 2121.13 0.503045
\(262\) −1041.75 −0.245648
\(263\) 163.867 0.0384201 0.0192101 0.999815i \(-0.493885\pi\)
0.0192101 + 0.999815i \(0.493885\pi\)
\(264\) 864.280 0.201488
\(265\) 64.6365 0.0149834
\(266\) 0 0
\(267\) 2020.26 0.463062
\(268\) −5237.59 −1.19379
\(269\) 5167.10 1.17116 0.585582 0.810613i \(-0.300866\pi\)
0.585582 + 0.810613i \(0.300866\pi\)
\(270\) 1.12403 0.000253356 0
\(271\) 1622.27 0.363638 0.181819 0.983332i \(-0.441801\pi\)
0.181819 + 0.983332i \(0.441801\pi\)
\(272\) 7287.63 1.62455
\(273\) 0 0
\(274\) −103.994 −0.0229289
\(275\) 5492.20 1.20433
\(276\) 1258.76 0.274524
\(277\) −4612.37 −1.00047 −0.500235 0.865890i \(-0.666753\pi\)
−0.500235 + 0.865890i \(0.666753\pi\)
\(278\) 367.021 0.0791815
\(279\) 168.396 0.0361348
\(280\) 0 0
\(281\) −2125.22 −0.451174 −0.225587 0.974223i \(-0.572430\pi\)
−0.225587 + 0.974223i \(0.572430\pi\)
\(282\) −498.987 −0.105370
\(283\) −2571.54 −0.540149 −0.270075 0.962839i \(-0.587048\pi\)
−0.270075 + 0.962839i \(0.587048\pi\)
\(284\) −6435.73 −1.34468
\(285\) −38.3264 −0.00796583
\(286\) 302.950 0.0626356
\(287\) 0 0
\(288\) −695.403 −0.142281
\(289\) 9883.19 2.01164
\(290\) −9.81154 −0.00198674
\(291\) −3273.34 −0.659404
\(292\) 4032.42 0.808149
\(293\) 3324.96 0.662957 0.331478 0.943463i \(-0.392453\pi\)
0.331478 + 0.943463i \(0.392453\pi\)
\(294\) 0 0
\(295\) 1.16709 0.000230341 0
\(296\) −1258.05 −0.247035
\(297\) 1186.41 0.231793
\(298\) −241.331 −0.0469126
\(299\) 892.120 0.172551
\(300\) −2935.42 −0.564922
\(301\) 0 0
\(302\) 1164.67 0.221918
\(303\) −4112.37 −0.779702
\(304\) 7615.54 1.43678
\(305\) 1.23045 0.000231001 0
\(306\) −453.463 −0.0847149
\(307\) −887.096 −0.164916 −0.0824580 0.996595i \(-0.526277\pi\)
−0.0824580 + 0.996595i \(0.526277\pi\)
\(308\) 0 0
\(309\) −4241.87 −0.780943
\(310\) −0.778936 −0.000142712 0
\(311\) −4510.82 −0.822460 −0.411230 0.911532i \(-0.634901\pi\)
−0.411230 + 0.911532i \(0.634901\pi\)
\(312\) −327.385 −0.0594055
\(313\) 3715.78 0.671018 0.335509 0.942037i \(-0.391092\pi\)
0.335509 + 0.942037i \(0.391092\pi\)
\(314\) −700.577 −0.125910
\(315\) 0 0
\(316\) 6307.79 1.12291
\(317\) 6954.52 1.23219 0.616096 0.787671i \(-0.288713\pi\)
0.616096 + 0.787671i \(0.288713\pi\)
\(318\) 799.163 0.140927
\(319\) −10356.1 −1.81765
\(320\) −44.9548 −0.00785327
\(321\) 1030.62 0.179201
\(322\) 0 0
\(323\) 15461.9 2.66355
\(324\) −634.103 −0.108728
\(325\) −2080.41 −0.355079
\(326\) 16.8681 0.00286576
\(327\) 952.971 0.161160
\(328\) 2096.15 0.352867
\(329\) 0 0
\(330\) −5.48788 −0.000915448 0
\(331\) −9863.18 −1.63785 −0.818926 0.573899i \(-0.805430\pi\)
−0.818926 + 0.573899i \(0.805430\pi\)
\(332\) −3091.16 −0.510992
\(333\) −1726.94 −0.284191
\(334\) 1201.41 0.196821
\(335\) 67.2427 0.0109668
\(336\) 0 0
\(337\) −5945.06 −0.960974 −0.480487 0.877002i \(-0.659540\pi\)
−0.480487 + 0.877002i \(0.659540\pi\)
\(338\) 795.272 0.127979
\(339\) −2395.12 −0.383732
\(340\) −95.7056 −0.0152658
\(341\) −822.168 −0.130566
\(342\) −473.866 −0.0749232
\(343\) 0 0
\(344\) −1433.08 −0.224612
\(345\) −16.1606 −0.00252191
\(346\) −888.963 −0.138124
\(347\) 1169.57 0.180939 0.0904697 0.995899i \(-0.471163\pi\)
0.0904697 + 0.995899i \(0.471163\pi\)
\(348\) 5535.04 0.852613
\(349\) −9176.66 −1.40749 −0.703747 0.710451i \(-0.748491\pi\)
−0.703747 + 0.710451i \(0.748491\pi\)
\(350\) 0 0
\(351\) −449.406 −0.0683405
\(352\) 3395.20 0.514104
\(353\) 10587.2 1.59632 0.798158 0.602448i \(-0.205808\pi\)
0.798158 + 0.602448i \(0.205808\pi\)
\(354\) 14.4299 0.00216649
\(355\) 82.6250 0.0123529
\(356\) 5271.81 0.784846
\(357\) 0 0
\(358\) −498.522 −0.0735970
\(359\) 8615.21 1.26656 0.633278 0.773924i \(-0.281709\pi\)
0.633278 + 0.773924i \(0.281709\pi\)
\(360\) 5.93052 0.000868238 0
\(361\) 9298.64 1.35568
\(362\) −1238.69 −0.179846
\(363\) −1799.47 −0.260186
\(364\) 0 0
\(365\) −51.7701 −0.00742403
\(366\) 15.2132 0.00217270
\(367\) 8297.79 1.18022 0.590110 0.807323i \(-0.299084\pi\)
0.590110 + 0.807323i \(0.299084\pi\)
\(368\) 3211.15 0.454871
\(369\) 2877.41 0.405941
\(370\) 7.98817 0.00112239
\(371\) 0 0
\(372\) 439.426 0.0612450
\(373\) −5123.86 −0.711269 −0.355634 0.934625i \(-0.615735\pi\)
−0.355634 + 0.934625i \(0.615735\pi\)
\(374\) 2213.96 0.306100
\(375\) 75.3758 0.0103797
\(376\) −2632.72 −0.361097
\(377\) 3922.83 0.535905
\(378\) 0 0
\(379\) −1502.49 −0.203635 −0.101817 0.994803i \(-0.532466\pi\)
−0.101817 + 0.994803i \(0.532466\pi\)
\(380\) −100.012 −0.0135013
\(381\) −3214.20 −0.432200
\(382\) 1162.87 0.155753
\(383\) −10872.9 −1.45060 −0.725301 0.688431i \(-0.758300\pi\)
−0.725301 + 0.688431i \(0.758300\pi\)
\(384\) −2410.23 −0.320303
\(385\) 0 0
\(386\) −1381.97 −0.182229
\(387\) −1967.21 −0.258395
\(388\) −8541.71 −1.11763
\(389\) 4618.99 0.602036 0.301018 0.953618i \(-0.402674\pi\)
0.301018 + 0.953618i \(0.402674\pi\)
\(390\) 2.07878 0.000269905 0
\(391\) 6519.64 0.843254
\(392\) 0 0
\(393\) −7545.05 −0.968442
\(394\) 1750.73 0.223860
\(395\) −80.9824 −0.0103156
\(396\) 3095.91 0.392867
\(397\) 9606.95 1.21451 0.607253 0.794508i \(-0.292271\pi\)
0.607253 + 0.794508i \(0.292271\pi\)
\(398\) 1816.61 0.228791
\(399\) 0 0
\(400\) −7488.36 −0.936044
\(401\) −10501.0 −1.30772 −0.653862 0.756614i \(-0.726852\pi\)
−0.653862 + 0.756614i \(0.726852\pi\)
\(402\) 831.386 0.103149
\(403\) 311.433 0.0384952
\(404\) −10731.1 −1.32152
\(405\) 8.14091 0.000998827 0
\(406\) 0 0
\(407\) 8431.52 1.02687
\(408\) −2392.54 −0.290314
\(409\) −12066.9 −1.45885 −0.729427 0.684059i \(-0.760213\pi\)
−0.729427 + 0.684059i \(0.760213\pi\)
\(410\) −13.3098 −0.00160323
\(411\) −753.192 −0.0903947
\(412\) −11069.0 −1.32362
\(413\) 0 0
\(414\) −199.809 −0.0237200
\(415\) 39.6857 0.00469421
\(416\) −1286.08 −0.151576
\(417\) 2658.20 0.312165
\(418\) 2313.58 0.270720
\(419\) −6366.31 −0.742278 −0.371139 0.928577i \(-0.621033\pi\)
−0.371139 + 0.928577i \(0.621033\pi\)
\(420\) 0 0
\(421\) −4731.84 −0.547781 −0.273890 0.961761i \(-0.588311\pi\)
−0.273890 + 0.961761i \(0.588311\pi\)
\(422\) −949.194 −0.109493
\(423\) −3613.98 −0.415408
\(424\) 4216.50 0.482951
\(425\) −15203.7 −1.73527
\(426\) 1021.57 0.116186
\(427\) 0 0
\(428\) 2689.37 0.303728
\(429\) 2194.16 0.246934
\(430\) 9.09955 0.00102051
\(431\) 3752.78 0.419409 0.209704 0.977765i \(-0.432750\pi\)
0.209704 + 0.977765i \(0.432750\pi\)
\(432\) −1617.62 −0.180156
\(433\) 11709.2 1.29956 0.649780 0.760122i \(-0.274861\pi\)
0.649780 + 0.760122i \(0.274861\pi\)
\(434\) 0 0
\(435\) −71.0615 −0.00783250
\(436\) 2486.75 0.273151
\(437\) 6812.98 0.745788
\(438\) −640.084 −0.0698274
\(439\) −14924.7 −1.62259 −0.811296 0.584635i \(-0.801238\pi\)
−0.811296 + 0.584635i \(0.801238\pi\)
\(440\) −28.9548 −0.00313720
\(441\) 0 0
\(442\) −838.638 −0.0902487
\(443\) −8517.66 −0.913513 −0.456756 0.889592i \(-0.650989\pi\)
−0.456756 + 0.889592i \(0.650989\pi\)
\(444\) −4506.41 −0.481677
\(445\) −67.6820 −0.00720996
\(446\) −90.2860 −0.00958557
\(447\) −1747.88 −0.184948
\(448\) 0 0
\(449\) −5965.73 −0.627038 −0.313519 0.949582i \(-0.601508\pi\)
−0.313519 + 0.949582i \(0.601508\pi\)
\(450\) 465.953 0.0488116
\(451\) −14048.5 −1.46678
\(452\) −6250.01 −0.650389
\(453\) 8435.29 0.874889
\(454\) 760.231 0.0785890
\(455\) 0 0
\(456\) −2500.19 −0.256759
\(457\) −13860.6 −1.41875 −0.709376 0.704830i \(-0.751023\pi\)
−0.709376 + 0.704830i \(0.751023\pi\)
\(458\) 1149.03 0.117229
\(459\) −3284.27 −0.333979
\(460\) −42.1707 −0.00427439
\(461\) −149.312 −0.0150850 −0.00754249 0.999972i \(-0.502401\pi\)
−0.00754249 + 0.999972i \(0.502401\pi\)
\(462\) 0 0
\(463\) 5403.95 0.542425 0.271213 0.962519i \(-0.412575\pi\)
0.271213 + 0.962519i \(0.412575\pi\)
\(464\) 14120.1 1.41273
\(465\) −5.64155 −0.000562625 0
\(466\) 409.538 0.0407113
\(467\) 3704.66 0.367090 0.183545 0.983011i \(-0.441243\pi\)
0.183545 + 0.983011i \(0.441243\pi\)
\(468\) −1172.71 −0.115831
\(469\) 0 0
\(470\) 16.7169 0.00164062
\(471\) −5074.03 −0.496389
\(472\) 76.1341 0.00742448
\(473\) 9604.59 0.933657
\(474\) −1001.26 −0.0970244
\(475\) −15887.8 −1.53470
\(476\) 0 0
\(477\) 5788.05 0.555591
\(478\) 347.081 0.0332115
\(479\) −10671.8 −1.01796 −0.508982 0.860777i \(-0.669978\pi\)
−0.508982 + 0.860777i \(0.669978\pi\)
\(480\) 23.2972 0.00221535
\(481\) −3193.82 −0.302756
\(482\) −1431.10 −0.135238
\(483\) 0 0
\(484\) −4695.67 −0.440990
\(485\) 109.662 0.0102670
\(486\) 100.654 0.00939455
\(487\) 5853.92 0.544695 0.272348 0.962199i \(-0.412200\pi\)
0.272348 + 0.962199i \(0.412200\pi\)
\(488\) 80.2670 0.00744573
\(489\) 122.170 0.0112980
\(490\) 0 0
\(491\) 4065.31 0.373656 0.186828 0.982393i \(-0.440179\pi\)
0.186828 + 0.982393i \(0.440179\pi\)
\(492\) 7508.54 0.688031
\(493\) 28668.2 2.61896
\(494\) −876.371 −0.0798174
\(495\) −39.7468 −0.00360906
\(496\) 1120.99 0.101480
\(497\) 0 0
\(498\) 490.673 0.0441517
\(499\) 4811.19 0.431620 0.215810 0.976435i \(-0.430761\pi\)
0.215810 + 0.976435i \(0.430761\pi\)
\(500\) 196.691 0.0175926
\(501\) 8701.40 0.775947
\(502\) 2334.10 0.207522
\(503\) 17001.2 1.50705 0.753526 0.657418i \(-0.228351\pi\)
0.753526 + 0.657418i \(0.228351\pi\)
\(504\) 0 0
\(505\) 137.771 0.0121401
\(506\) 975.537 0.0857074
\(507\) 5759.87 0.504546
\(508\) −8387.37 −0.732538
\(509\) 13797.2 1.20148 0.600738 0.799446i \(-0.294874\pi\)
0.600738 + 0.799446i \(0.294874\pi\)
\(510\) 15.1918 0.00131903
\(511\) 0 0
\(512\) −7771.61 −0.670820
\(513\) −3432.04 −0.295377
\(514\) −940.744 −0.0807284
\(515\) 142.110 0.0121594
\(516\) −5133.38 −0.437954
\(517\) 17644.7 1.50099
\(518\) 0 0
\(519\) −6438.44 −0.544540
\(520\) 10.9679 0.000924954 0
\(521\) 3936.61 0.331029 0.165515 0.986207i \(-0.447072\pi\)
0.165515 + 0.986207i \(0.447072\pi\)
\(522\) −878.601 −0.0736692
\(523\) 17459.3 1.45973 0.729866 0.683590i \(-0.239582\pi\)
0.729866 + 0.683590i \(0.239582\pi\)
\(524\) −19688.6 −1.64142
\(525\) 0 0
\(526\) −67.8760 −0.00562649
\(527\) 2275.96 0.188126
\(528\) 7897.76 0.650958
\(529\) −9294.26 −0.763891
\(530\) −26.7733 −0.00219426
\(531\) 104.510 0.00854118
\(532\) 0 0
\(533\) 5321.51 0.432458
\(534\) −836.817 −0.0678139
\(535\) −34.5274 −0.00279019
\(536\) 4386.51 0.353486
\(537\) −3610.62 −0.290148
\(538\) −2140.28 −0.171513
\(539\) 0 0
\(540\) 21.2435 0.00169292
\(541\) 19101.3 1.51798 0.758992 0.651100i \(-0.225692\pi\)
0.758992 + 0.651100i \(0.225692\pi\)
\(542\) −671.967 −0.0532536
\(543\) −8971.41 −0.709024
\(544\) −9398.73 −0.740749
\(545\) −31.9261 −0.00250929
\(546\) 0 0
\(547\) 15413.5 1.20481 0.602407 0.798189i \(-0.294208\pi\)
0.602407 + 0.798189i \(0.294208\pi\)
\(548\) −1965.44 −0.153210
\(549\) 110.184 0.00856563
\(550\) −2274.94 −0.176371
\(551\) 29958.1 2.31626
\(552\) −1054.22 −0.0812874
\(553\) 0 0
\(554\) 1910.51 0.146516
\(555\) 57.8554 0.00442491
\(556\) 6936.51 0.529089
\(557\) −20492.9 −1.55891 −0.779453 0.626460i \(-0.784503\pi\)
−0.779453 + 0.626460i \(0.784503\pi\)
\(558\) −69.7519 −0.00529182
\(559\) −3638.17 −0.275274
\(560\) 0 0
\(561\) 16034.9 1.20677
\(562\) 880.294 0.0660729
\(563\) −7142.49 −0.534671 −0.267336 0.963603i \(-0.586143\pi\)
−0.267336 + 0.963603i \(0.586143\pi\)
\(564\) −9430.59 −0.704077
\(565\) 80.2406 0.00597477
\(566\) 1065.17 0.0791030
\(567\) 0 0
\(568\) 5389.96 0.398165
\(569\) 4097.91 0.301922 0.150961 0.988540i \(-0.451763\pi\)
0.150961 + 0.988540i \(0.451763\pi\)
\(570\) 15.8753 0.00116657
\(571\) −2838.48 −0.208033 −0.104016 0.994576i \(-0.533169\pi\)
−0.104016 + 0.994576i \(0.533169\pi\)
\(572\) 5725.60 0.418530
\(573\) 8422.23 0.614038
\(574\) 0 0
\(575\) −6699.21 −0.485872
\(576\) −4025.60 −0.291203
\(577\) −15464.5 −1.11576 −0.557881 0.829921i \(-0.688385\pi\)
−0.557881 + 0.829921i \(0.688385\pi\)
\(578\) −4093.75 −0.294598
\(579\) −10009.1 −0.718418
\(580\) −185.433 −0.0132753
\(581\) 0 0
\(582\) 1355.86 0.0965675
\(583\) −28259.3 −2.00751
\(584\) −3377.17 −0.239295
\(585\) 15.0559 0.00106407
\(586\) −1377.24 −0.0970878
\(587\) 14003.6 0.984652 0.492326 0.870411i \(-0.336147\pi\)
0.492326 + 0.870411i \(0.336147\pi\)
\(588\) 0 0
\(589\) 2378.36 0.166382
\(590\) −0.483425 −3.37327e−5 0
\(591\) 12679.9 0.882543
\(592\) −11496.0 −0.798112
\(593\) 6504.50 0.450435 0.225217 0.974309i \(-0.427691\pi\)
0.225217 + 0.974309i \(0.427691\pi\)
\(594\) −491.427 −0.0339453
\(595\) 0 0
\(596\) −4561.04 −0.313469
\(597\) 13157.1 0.901982
\(598\) −369.528 −0.0252695
\(599\) −12616.1 −0.860567 −0.430284 0.902694i \(-0.641586\pi\)
−0.430284 + 0.902694i \(0.641586\pi\)
\(600\) 2458.43 0.167275
\(601\) 8270.87 0.561358 0.280679 0.959802i \(-0.409440\pi\)
0.280679 + 0.959802i \(0.409440\pi\)
\(602\) 0 0
\(603\) 6021.43 0.406653
\(604\) 22011.7 1.48285
\(605\) 60.2852 0.00405114
\(606\) 1703.40 0.114185
\(607\) 3811.84 0.254889 0.127445 0.991846i \(-0.459323\pi\)
0.127445 + 0.991846i \(0.459323\pi\)
\(608\) −9821.62 −0.655130
\(609\) 0 0
\(610\) −0.509668 −3.38293e−5 0
\(611\) −6683.72 −0.442544
\(612\) −8570.22 −0.566063
\(613\) 11359.3 0.748445 0.374222 0.927339i \(-0.377910\pi\)
0.374222 + 0.927339i \(0.377910\pi\)
\(614\) 367.447 0.0241514
\(615\) −96.3983 −0.00632057
\(616\) 0 0
\(617\) −18272.2 −1.19224 −0.596118 0.802896i \(-0.703291\pi\)
−0.596118 + 0.802896i \(0.703291\pi\)
\(618\) 1757.04 0.114366
\(619\) 29600.2 1.92203 0.961013 0.276503i \(-0.0891756\pi\)
0.961013 + 0.276503i \(0.0891756\pi\)
\(620\) −14.7215 −0.000953596 0
\(621\) −1447.15 −0.0935136
\(622\) 1868.44 0.120447
\(623\) 0 0
\(624\) −2991.63 −0.191925
\(625\) 15621.2 0.999758
\(626\) −1539.13 −0.0982682
\(627\) 16756.4 1.06728
\(628\) −13240.6 −0.841331
\(629\) −23340.5 −1.47956
\(630\) 0 0
\(631\) 7185.41 0.453322 0.226661 0.973974i \(-0.427219\pi\)
0.226661 + 0.973974i \(0.427219\pi\)
\(632\) −5282.81 −0.332498
\(633\) −6874.67 −0.431665
\(634\) −2880.66 −0.180450
\(635\) 107.681 0.00672944
\(636\) 15103.8 0.941673
\(637\) 0 0
\(638\) 4289.64 0.266189
\(639\) 7398.88 0.458052
\(640\) 80.7467 0.00498718
\(641\) −232.982 −0.0143561 −0.00717803 0.999974i \(-0.502285\pi\)
−0.00717803 + 0.999974i \(0.502285\pi\)
\(642\) −426.896 −0.0262433
\(643\) −1837.96 −0.112725 −0.0563624 0.998410i \(-0.517950\pi\)
−0.0563624 + 0.998410i \(0.517950\pi\)
\(644\) 0 0
\(645\) 65.9048 0.00402325
\(646\) −6404.54 −0.390067
\(647\) −18594.7 −1.12988 −0.564941 0.825131i \(-0.691101\pi\)
−0.564941 + 0.825131i \(0.691101\pi\)
\(648\) 531.064 0.0321947
\(649\) −510.256 −0.0308618
\(650\) 861.736 0.0520001
\(651\) 0 0
\(652\) 318.799 0.0191490
\(653\) −28864.7 −1.72980 −0.864902 0.501940i \(-0.832620\pi\)
−0.864902 + 0.501940i \(0.832620\pi\)
\(654\) −394.733 −0.0236014
\(655\) 252.772 0.0150788
\(656\) 19154.5 1.14003
\(657\) −4635.90 −0.275287
\(658\) 0 0
\(659\) −29066.3 −1.71815 −0.859076 0.511847i \(-0.828961\pi\)
−0.859076 + 0.511847i \(0.828961\pi\)
\(660\) −103.718 −0.00611701
\(661\) −3979.51 −0.234168 −0.117084 0.993122i \(-0.537355\pi\)
−0.117084 + 0.993122i \(0.537355\pi\)
\(662\) 4085.46 0.239858
\(663\) −6073.95 −0.355796
\(664\) 2588.86 0.151306
\(665\) 0 0
\(666\) 715.322 0.0416189
\(667\) 12632.0 0.733305
\(668\) 22706.1 1.31516
\(669\) −653.909 −0.0377901
\(670\) −27.8528 −0.00160604
\(671\) −537.955 −0.0309501
\(672\) 0 0
\(673\) −184.229 −0.0105520 −0.00527601 0.999986i \(-0.501679\pi\)
−0.00527601 + 0.999986i \(0.501679\pi\)
\(674\) 2462.52 0.140731
\(675\) 3374.73 0.192435
\(676\) 15030.2 0.855156
\(677\) −16683.5 −0.947116 −0.473558 0.880763i \(-0.657031\pi\)
−0.473558 + 0.880763i \(0.657031\pi\)
\(678\) 992.091 0.0561962
\(679\) 0 0
\(680\) 80.1540 0.00452024
\(681\) 5506.08 0.309829
\(682\) 340.553 0.0191209
\(683\) 17808.2 0.997676 0.498838 0.866695i \(-0.333760\pi\)
0.498838 + 0.866695i \(0.333760\pi\)
\(684\) −8955.83 −0.500636
\(685\) 25.2332 0.00140746
\(686\) 0 0
\(687\) 8322.03 0.462162
\(688\) −13095.4 −0.725666
\(689\) 10704.5 0.591883
\(690\) 6.69394 0.000369325 0
\(691\) −20145.7 −1.10908 −0.554542 0.832156i \(-0.687106\pi\)
−0.554542 + 0.832156i \(0.687106\pi\)
\(692\) −16801.0 −0.922943
\(693\) 0 0
\(694\) −484.453 −0.0264980
\(695\) −89.0542 −0.00486046
\(696\) −4635.63 −0.252461
\(697\) 38889.7 2.11342
\(698\) 3801.10 0.206123
\(699\) 2966.14 0.160500
\(700\) 0 0
\(701\) −2719.67 −0.146534 −0.0732672 0.997312i \(-0.523343\pi\)
−0.0732672 + 0.997312i \(0.523343\pi\)
\(702\) 186.150 0.0100082
\(703\) −24390.7 −1.30855
\(704\) 19654.4 1.05220
\(705\) 121.074 0.00646798
\(706\) −4385.36 −0.233775
\(707\) 0 0
\(708\) 272.717 0.0144765
\(709\) −625.708 −0.0331438 −0.0165719 0.999863i \(-0.505275\pi\)
−0.0165719 + 0.999863i \(0.505275\pi\)
\(710\) −34.2244 −0.00180904
\(711\) −7251.79 −0.382508
\(712\) −4415.17 −0.232395
\(713\) 1002.85 0.0526749
\(714\) 0 0
\(715\) −73.5079 −0.00384481
\(716\) −9421.82 −0.491774
\(717\) 2513.78 0.130933
\(718\) −3568.54 −0.185483
\(719\) −9577.54 −0.496776 −0.248388 0.968661i \(-0.579901\pi\)
−0.248388 + 0.968661i \(0.579901\pi\)
\(720\) 54.1929 0.00280507
\(721\) 0 0
\(722\) −3851.62 −0.198535
\(723\) −10365.0 −0.533163
\(724\) −23410.7 −1.20173
\(725\) −29457.8 −1.50901
\(726\) 745.364 0.0381034
\(727\) −16741.2 −0.854053 −0.427027 0.904239i \(-0.640439\pi\)
−0.427027 + 0.904239i \(0.640439\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 21.4439 0.00108722
\(731\) −26587.8 −1.34526
\(732\) 287.522 0.0145179
\(733\) −6496.04 −0.327335 −0.163668 0.986516i \(-0.552332\pi\)
−0.163668 + 0.986516i \(0.552332\pi\)
\(734\) −3437.06 −0.172839
\(735\) 0 0
\(736\) −4141.36 −0.207408
\(737\) −29398.7 −1.46936
\(738\) −1191.86 −0.0594487
\(739\) 499.280 0.0248529 0.0124265 0.999923i \(-0.496044\pi\)
0.0124265 + 0.999923i \(0.496044\pi\)
\(740\) 150.972 0.00749980
\(741\) −6347.24 −0.314672
\(742\) 0 0
\(743\) −6367.30 −0.314393 −0.157196 0.987567i \(-0.550246\pi\)
−0.157196 + 0.987567i \(0.550246\pi\)
\(744\) −368.021 −0.0181348
\(745\) 58.5568 0.00287967
\(746\) 2122.37 0.104163
\(747\) 3553.77 0.174064
\(748\) 41842.8 2.04535
\(749\) 0 0
\(750\) −31.2217 −0.00152007
\(751\) 496.098 0.0241050 0.0120525 0.999927i \(-0.496163\pi\)
0.0120525 + 0.999927i \(0.496163\pi\)
\(752\) −24057.7 −1.16662
\(753\) 16905.0 0.818132
\(754\) −1624.89 −0.0784815
\(755\) −282.597 −0.0136222
\(756\) 0 0
\(757\) 13025.9 0.625408 0.312704 0.949851i \(-0.398765\pi\)
0.312704 + 0.949851i \(0.398765\pi\)
\(758\) 622.350 0.0298216
\(759\) 7065.47 0.337892
\(760\) 83.7604 0.00399778
\(761\) −25474.6 −1.21347 −0.606736 0.794904i \(-0.707521\pi\)
−0.606736 + 0.794904i \(0.707521\pi\)
\(762\) 1331.36 0.0632943
\(763\) 0 0
\(764\) 21977.6 1.04074
\(765\) 110.029 0.00520012
\(766\) 4503.72 0.212436
\(767\) 193.282 0.00909911
\(768\) −9736.57 −0.457472
\(769\) −29054.0 −1.36244 −0.681218 0.732080i \(-0.738550\pi\)
−0.681218 + 0.732080i \(0.738550\pi\)
\(770\) 0 0
\(771\) −6813.47 −0.318263
\(772\) −26118.5 −1.21765
\(773\) 1897.35 0.0882834 0.0441417 0.999025i \(-0.485945\pi\)
0.0441417 + 0.999025i \(0.485945\pi\)
\(774\) 814.844 0.0378410
\(775\) −2338.65 −0.108396
\(776\) 7153.72 0.330933
\(777\) 0 0
\(778\) −1913.25 −0.0881662
\(779\) 40639.6 1.86914
\(780\) 39.2879 0.00180350
\(781\) −36123.9 −1.65508
\(782\) −2700.52 −0.123492
\(783\) −6363.39 −0.290433
\(784\) 0 0
\(785\) 169.989 0.00772886
\(786\) 3125.26 0.141825
\(787\) 42650.3 1.93179 0.965895 0.258936i \(-0.0833718\pi\)
0.965895 + 0.258936i \(0.0833718\pi\)
\(788\) 33088.0 1.49583
\(789\) −491.602 −0.0221819
\(790\) 33.5440 0.00151069
\(791\) 0 0
\(792\) −2592.84 −0.116329
\(793\) 203.775 0.00912516
\(794\) −3979.33 −0.177860
\(795\) −193.910 −0.00865064
\(796\) 34333.1 1.52877
\(797\) 36822.8 1.63655 0.818275 0.574828i \(-0.194931\pi\)
0.818275 + 0.574828i \(0.194931\pi\)
\(798\) 0 0
\(799\) −48844.8 −2.16271
\(800\) 9657.60 0.426810
\(801\) −6060.77 −0.267349
\(802\) 4349.68 0.191512
\(803\) 22634.1 0.994693
\(804\) 15712.8 0.689238
\(805\) 0 0
\(806\) −129.000 −0.00563750
\(807\) −15501.3 −0.676172
\(808\) 8987.38 0.391306
\(809\) 5081.94 0.220855 0.110427 0.993884i \(-0.464778\pi\)
0.110427 + 0.993884i \(0.464778\pi\)
\(810\) −3.37208 −0.000146275 0
\(811\) 11873.4 0.514097 0.257048 0.966399i \(-0.417250\pi\)
0.257048 + 0.966399i \(0.417250\pi\)
\(812\) 0 0
\(813\) −4866.82 −0.209947
\(814\) −3492.45 −0.150381
\(815\) −4.09289 −0.000175911 0
\(816\) −21862.9 −0.937935
\(817\) −27784.1 −1.18977
\(818\) 4998.29 0.213644
\(819\) 0 0
\(820\) −251.549 −0.0107128
\(821\) 16969.4 0.721361 0.360681 0.932689i \(-0.382544\pi\)
0.360681 + 0.932689i \(0.382544\pi\)
\(822\) 311.982 0.0132380
\(823\) 3995.94 0.169246 0.0846231 0.996413i \(-0.473031\pi\)
0.0846231 + 0.996413i \(0.473031\pi\)
\(824\) 9270.39 0.391929
\(825\) −16476.6 −0.695323
\(826\) 0 0
\(827\) −13589.7 −0.571417 −0.285708 0.958317i \(-0.592229\pi\)
−0.285708 + 0.958317i \(0.592229\pi\)
\(828\) −3776.29 −0.158497
\(829\) −30646.0 −1.28393 −0.641966 0.766733i \(-0.721881\pi\)
−0.641966 + 0.766733i \(0.721881\pi\)
\(830\) −16.4384 −0.000687451 0
\(831\) 13837.1 0.577622
\(832\) −7444.96 −0.310226
\(833\) 0 0
\(834\) −1101.06 −0.0457155
\(835\) −291.511 −0.0120816
\(836\) 43725.5 1.80894
\(837\) −505.188 −0.0208624
\(838\) 2637.01 0.108704
\(839\) −7497.57 −0.308516 −0.154258 0.988031i \(-0.549299\pi\)
−0.154258 + 0.988031i \(0.549299\pi\)
\(840\) 0 0
\(841\) 31156.6 1.27749
\(842\) 1959.99 0.0802207
\(843\) 6375.65 0.260485
\(844\) −17939.3 −0.731630
\(845\) −192.965 −0.00785586
\(846\) 1496.96 0.0608351
\(847\) 0 0
\(848\) 38530.2 1.56030
\(849\) 7714.62 0.311855
\(850\) 6297.59 0.254124
\(851\) −10284.5 −0.414275
\(852\) 19307.2 0.776354
\(853\) −10347.6 −0.415352 −0.207676 0.978198i \(-0.566590\pi\)
−0.207676 + 0.978198i \(0.566590\pi\)
\(854\) 0 0
\(855\) 114.979 0.00459907
\(856\) −2252.36 −0.0899348
\(857\) 1550.00 0.0617817 0.0308909 0.999523i \(-0.490166\pi\)
0.0308909 + 0.999523i \(0.490166\pi\)
\(858\) −908.849 −0.0361627
\(859\) 15187.5 0.603250 0.301625 0.953427i \(-0.402471\pi\)
0.301625 + 0.953427i \(0.402471\pi\)
\(860\) 171.977 0.00681903
\(861\) 0 0
\(862\) −1554.45 −0.0614210
\(863\) 41550.2 1.63892 0.819459 0.573138i \(-0.194274\pi\)
0.819459 + 0.573138i \(0.194274\pi\)
\(864\) 2086.21 0.0821462
\(865\) 215.699 0.00847858
\(866\) −4850.12 −0.190316
\(867\) −29649.6 −1.16142
\(868\) 0 0
\(869\) 35405.8 1.38212
\(870\) 29.4346 0.00114704
\(871\) 11136.1 0.433216
\(872\) −2082.67 −0.0808808
\(873\) 9820.03 0.380707
\(874\) −2822.03 −0.109218
\(875\) 0 0
\(876\) −12097.3 −0.466585
\(877\) 27837.4 1.07184 0.535919 0.844269i \(-0.319965\pi\)
0.535919 + 0.844269i \(0.319965\pi\)
\(878\) 6182.02 0.237623
\(879\) −9974.89 −0.382758
\(880\) −264.588 −0.0101355
\(881\) −2587.85 −0.0989635 −0.0494817 0.998775i \(-0.515757\pi\)
−0.0494817 + 0.998775i \(0.515757\pi\)
\(882\) 0 0
\(883\) −16382.0 −0.624346 −0.312173 0.950025i \(-0.601057\pi\)
−0.312173 + 0.950025i \(0.601057\pi\)
\(884\) −15849.8 −0.603040
\(885\) −3.50127 −0.000132988 0
\(886\) 3528.13 0.133781
\(887\) −22980.2 −0.869896 −0.434948 0.900456i \(-0.643233\pi\)
−0.434948 + 0.900456i \(0.643233\pi\)
\(888\) 3774.14 0.142626
\(889\) 0 0
\(890\) 28.0348 0.00105587
\(891\) −3559.23 −0.133826
\(892\) −1706.36 −0.0640506
\(893\) −51042.5 −1.91274
\(894\) 723.994 0.0270850
\(895\) 120.962 0.00451766
\(896\) 0 0
\(897\) −2676.36 −0.0996222
\(898\) 2471.09 0.0918276
\(899\) 4409.76 0.163597
\(900\) 8806.27 0.326158
\(901\) 78228.5 2.89253
\(902\) 5819.10 0.214806
\(903\) 0 0
\(904\) 5234.42 0.192582
\(905\) 300.557 0.0110396
\(906\) −3494.01 −0.128125
\(907\) −22715.4 −0.831590 −0.415795 0.909458i \(-0.636497\pi\)
−0.415795 + 0.909458i \(0.636497\pi\)
\(908\) 14368.0 0.525130
\(909\) 12337.1 0.450161
\(910\) 0 0
\(911\) 34922.3 1.27006 0.635031 0.772487i \(-0.280987\pi\)
0.635031 + 0.772487i \(0.280987\pi\)
\(912\) −22846.6 −0.829525
\(913\) −17350.7 −0.628943
\(914\) 5741.24 0.207772
\(915\) −3.69134 −0.000133368 0
\(916\) 21716.1 0.783320
\(917\) 0 0
\(918\) 1360.39 0.0489102
\(919\) 11702.6 0.420059 0.210030 0.977695i \(-0.432644\pi\)
0.210030 + 0.977695i \(0.432644\pi\)
\(920\) 35.3182 0.00126566
\(921\) 2661.29 0.0952144
\(922\) 61.8473 0.00220914
\(923\) 13683.5 0.487973
\(924\) 0 0
\(925\) 23983.3 0.852505
\(926\) −2238.39 −0.0794364
\(927\) 12725.6 0.450878
\(928\) −18210.4 −0.644165
\(929\) −53096.5 −1.87518 −0.937588 0.347747i \(-0.886947\pi\)
−0.937588 + 0.347747i \(0.886947\pi\)
\(930\) 2.33681 8.23946e−5 0
\(931\) 0 0
\(932\) 7740.06 0.272032
\(933\) 13532.5 0.474848
\(934\) −1534.52 −0.0537591
\(935\) −537.198 −0.0187896
\(936\) 982.154 0.0342978
\(937\) −39020.6 −1.36046 −0.680229 0.733000i \(-0.738120\pi\)
−0.680229 + 0.733000i \(0.738120\pi\)
\(938\) 0 0
\(939\) −11147.4 −0.387412
\(940\) 315.941 0.0109626
\(941\) 30743.8 1.06506 0.532528 0.846412i \(-0.321242\pi\)
0.532528 + 0.846412i \(0.321242\pi\)
\(942\) 2101.73 0.0726944
\(943\) 17136.0 0.591754
\(944\) 695.710 0.0239867
\(945\) 0 0
\(946\) −3978.35 −0.136731
\(947\) 16300.3 0.559332 0.279666 0.960097i \(-0.409776\pi\)
0.279666 + 0.960097i \(0.409776\pi\)
\(948\) −18923.4 −0.648315
\(949\) −8573.66 −0.293269
\(950\) 6580.94 0.224752
\(951\) −20863.6 −0.711406
\(952\) 0 0
\(953\) −11512.7 −0.391325 −0.195663 0.980671i \(-0.562686\pi\)
−0.195663 + 0.980671i \(0.562686\pi\)
\(954\) −2397.49 −0.0813644
\(955\) −282.159 −0.00956068
\(956\) 6559.65 0.221919
\(957\) 31068.3 1.04942
\(958\) 4420.39 0.149078
\(959\) 0 0
\(960\) 134.864 0.00453409
\(961\) −29440.9 −0.988248
\(962\) 1322.92 0.0443375
\(963\) −3091.85 −0.103462
\(964\) −27047.1 −0.903661
\(965\) 335.322 0.0111859
\(966\) 0 0
\(967\) −18178.4 −0.604528 −0.302264 0.953224i \(-0.597742\pi\)
−0.302264 + 0.953224i \(0.597742\pi\)
\(968\) 3932.65 0.130579
\(969\) −46385.8 −1.53780
\(970\) −45.4237 −0.00150357
\(971\) −28276.7 −0.934543 −0.467271 0.884114i \(-0.654763\pi\)
−0.467271 + 0.884114i \(0.654763\pi\)
\(972\) 1902.31 0.0627742
\(973\) 0 0
\(974\) −2424.77 −0.0797688
\(975\) 6241.24 0.205005
\(976\) 733.477 0.0240554
\(977\) −13947.1 −0.456710 −0.228355 0.973578i \(-0.573335\pi\)
−0.228355 + 0.973578i \(0.573335\pi\)
\(978\) −50.6043 −0.00165455
\(979\) 29590.8 0.966011
\(980\) 0 0
\(981\) −2858.91 −0.0930459
\(982\) −1683.91 −0.0547206
\(983\) 26576.8 0.862327 0.431164 0.902274i \(-0.358103\pi\)
0.431164 + 0.902274i \(0.358103\pi\)
\(984\) −6288.45 −0.203728
\(985\) −424.799 −0.0137414
\(986\) −11874.7 −0.383539
\(987\) 0 0
\(988\) −16563.0 −0.533339
\(989\) −11715.4 −0.376671
\(990\) 16.4636 0.000528534 0
\(991\) 16249.9 0.520884 0.260442 0.965490i \(-0.416132\pi\)
0.260442 + 0.965490i \(0.416132\pi\)
\(992\) −1445.72 −0.0462718
\(993\) 29589.5 0.945615
\(994\) 0 0
\(995\) −440.784 −0.0140440
\(996\) 9273.47 0.295021
\(997\) −18814.1 −0.597642 −0.298821 0.954309i \(-0.596593\pi\)
−0.298821 + 0.954309i \(0.596593\pi\)
\(998\) −1992.86 −0.0632092
\(999\) 5180.82 0.164078
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 147.4.a.j.1.1 2
3.2 odd 2 441.4.a.n.1.2 2
4.3 odd 2 2352.4.a.cf.1.1 2
7.2 even 3 147.4.e.k.67.2 4
7.3 odd 6 147.4.e.j.79.2 4
7.4 even 3 147.4.e.k.79.2 4
7.5 odd 6 147.4.e.j.67.2 4
7.6 odd 2 147.4.a.k.1.1 yes 2
21.2 odd 6 441.4.e.v.361.1 4
21.5 even 6 441.4.e.u.361.1 4
21.11 odd 6 441.4.e.v.226.1 4
21.17 even 6 441.4.e.u.226.1 4
21.20 even 2 441.4.a.o.1.2 2
28.27 even 2 2352.4.a.bl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.j.1.1 2 1.1 even 1 trivial
147.4.a.k.1.1 yes 2 7.6 odd 2
147.4.e.j.67.2 4 7.5 odd 6
147.4.e.j.79.2 4 7.3 odd 6
147.4.e.k.67.2 4 7.2 even 3
147.4.e.k.79.2 4 7.4 even 3
441.4.a.n.1.2 2 3.2 odd 2
441.4.a.o.1.2 2 21.20 even 2
441.4.e.u.226.1 4 21.17 even 6
441.4.e.u.361.1 4 21.5 even 6
441.4.e.v.226.1 4 21.11 odd 6
441.4.e.v.361.1 4 21.2 odd 6
2352.4.a.bl.1.2 2 28.27 even 2
2352.4.a.cf.1.1 2 4.3 odd 2