Properties

Label 567.2.a.d.1.2
Level $567$
Weight $2$
Character 567.1
Self dual yes
Analytic conductor $4.528$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.52751779461\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.239123\) of defining polynomial
Character \(\chi\) \(=\) 567.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.239123 q^{2} -1.94282 q^{4} +1.18194 q^{5} -1.00000 q^{7} +0.942820 q^{8} +O(q^{10})\) \(q-0.239123 q^{2} -1.94282 q^{4} +1.18194 q^{5} -1.00000 q^{7} +0.942820 q^{8} -0.282630 q^{10} -3.70370 q^{11} +1.00000 q^{13} +0.239123 q^{14} +3.66019 q^{16} -6.94282 q^{17} +1.94282 q^{19} -2.29630 q^{20} +0.885640 q^{22} -5.60301 q^{23} -3.60301 q^{25} -0.239123 q^{26} +1.94282 q^{28} +0.239123 q^{29} +1.66019 q^{31} -2.76088 q^{32} +1.66019 q^{34} -1.18194 q^{35} -9.54583 q^{37} -0.464574 q^{38} +1.11436 q^{40} -10.1819 q^{41} +2.22545 q^{43} +7.19562 q^{44} +1.33981 q^{46} +5.82846 q^{47} +1.00000 q^{49} +0.861564 q^{50} -1.94282 q^{52} -11.6030 q^{53} -4.37756 q^{55} -0.942820 q^{56} -0.0571799 q^{58} +2.60301 q^{59} -7.60301 q^{61} -0.396990 q^{62} -6.66019 q^{64} +1.18194 q^{65} +3.50808 q^{67} +13.4887 q^{68} +0.282630 q^{70} +8.60301 q^{71} +15.1488 q^{73} +2.28263 q^{74} -3.77455 q^{76} +3.70370 q^{77} +7.37756 q^{79} +4.32614 q^{80} +2.43474 q^{82} -6.94282 q^{83} -8.20602 q^{85} -0.532157 q^{86} -3.49192 q^{88} +2.74720 q^{89} -1.00000 q^{91} +10.8856 q^{92} -1.39372 q^{94} +2.29630 q^{95} +7.16827 q^{97} -0.239123 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 3q^{4} - 5q^{5} - 3q^{7} - 6q^{8} + O(q^{10}) \) \( 3q - q^{2} + 3q^{4} - 5q^{5} - 3q^{7} - 6q^{8} - 2q^{11} + 3q^{13} + q^{14} + 3q^{16} - 12q^{17} - 3q^{19} - 16q^{20} - 15q^{22} + 6q^{25} - q^{26} - 3q^{28} + q^{29} - 3q^{31} - 8q^{32} - 3q^{34} + 5q^{35} - 3q^{37} + 8q^{38} + 21q^{40} - 22q^{41} - 3q^{43} + 23q^{44} + 12q^{46} - 9q^{47} + 3q^{49} + 10q^{50} + 3q^{52} - 18q^{53} - 6q^{55} + 6q^{56} - 9q^{58} - 9q^{59} - 6q^{61} - 18q^{62} - 12q^{64} - 5q^{65} + 6q^{68} + 9q^{71} + 3q^{73} + 6q^{74} - 21q^{76} + 2q^{77} + 15q^{79} + 11q^{80} + 9q^{82} - 12q^{83} + 9q^{85} + 34q^{86} - 21q^{88} - 2q^{89} - 3q^{91} + 15q^{92} + 24q^{94} + 16q^{95} + 3q^{97} - q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.239123 −0.169086 −0.0845428 0.996420i \(-0.526943\pi\)
−0.0845428 + 0.996420i \(0.526943\pi\)
\(3\) 0 0
\(4\) −1.94282 −0.971410
\(5\) 1.18194 0.528581 0.264291 0.964443i \(-0.414862\pi\)
0.264291 + 0.964443i \(0.414862\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0.942820 0.333337
\(9\) 0 0
\(10\) −0.282630 −0.0893755
\(11\) −3.70370 −1.11671 −0.558353 0.829603i \(-0.688567\pi\)
−0.558353 + 0.829603i \(0.688567\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0.239123 0.0639084
\(15\) 0 0
\(16\) 3.66019 0.915047
\(17\) −6.94282 −1.68388 −0.841941 0.539570i \(-0.818587\pi\)
−0.841941 + 0.539570i \(0.818587\pi\)
\(18\) 0 0
\(19\) 1.94282 0.445713 0.222857 0.974851i \(-0.428462\pi\)
0.222857 + 0.974851i \(0.428462\pi\)
\(20\) −2.29630 −0.513469
\(21\) 0 0
\(22\) 0.885640 0.188819
\(23\) −5.60301 −1.16831 −0.584154 0.811643i \(-0.698574\pi\)
−0.584154 + 0.811643i \(0.698574\pi\)
\(24\) 0 0
\(25\) −3.60301 −0.720602
\(26\) −0.239123 −0.0468959
\(27\) 0 0
\(28\) 1.94282 0.367158
\(29\) 0.239123 0.0444041 0.0222020 0.999754i \(-0.492932\pi\)
0.0222020 + 0.999754i \(0.492932\pi\)
\(30\) 0 0
\(31\) 1.66019 0.298179 0.149089 0.988824i \(-0.452366\pi\)
0.149089 + 0.988824i \(0.452366\pi\)
\(32\) −2.76088 −0.488059
\(33\) 0 0
\(34\) 1.66019 0.284720
\(35\) −1.18194 −0.199785
\(36\) 0 0
\(37\) −9.54583 −1.56932 −0.784662 0.619923i \(-0.787164\pi\)
−0.784662 + 0.619923i \(0.787164\pi\)
\(38\) −0.464574 −0.0753638
\(39\) 0 0
\(40\) 1.11436 0.176196
\(41\) −10.1819 −1.59015 −0.795076 0.606510i \(-0.792569\pi\)
−0.795076 + 0.606510i \(0.792569\pi\)
\(42\) 0 0
\(43\) 2.22545 0.339378 0.169689 0.985498i \(-0.445724\pi\)
0.169689 + 0.985498i \(0.445724\pi\)
\(44\) 7.19562 1.08478
\(45\) 0 0
\(46\) 1.33981 0.197544
\(47\) 5.82846 0.850168 0.425084 0.905154i \(-0.360245\pi\)
0.425084 + 0.905154i \(0.360245\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0.861564 0.121843
\(51\) 0 0
\(52\) −1.94282 −0.269421
\(53\) −11.6030 −1.59380 −0.796898 0.604114i \(-0.793527\pi\)
−0.796898 + 0.604114i \(0.793527\pi\)
\(54\) 0 0
\(55\) −4.37756 −0.590270
\(56\) −0.942820 −0.125990
\(57\) 0 0
\(58\) −0.0571799 −0.00750809
\(59\) 2.60301 0.338883 0.169442 0.985540i \(-0.445804\pi\)
0.169442 + 0.985540i \(0.445804\pi\)
\(60\) 0 0
\(61\) −7.60301 −0.973466 −0.486733 0.873551i \(-0.661811\pi\)
−0.486733 + 0.873551i \(0.661811\pi\)
\(62\) −0.396990 −0.0504178
\(63\) 0 0
\(64\) −6.66019 −0.832524
\(65\) 1.18194 0.146602
\(66\) 0 0
\(67\) 3.50808 0.428580 0.214290 0.976770i \(-0.431256\pi\)
0.214290 + 0.976770i \(0.431256\pi\)
\(68\) 13.4887 1.63574
\(69\) 0 0
\(70\) 0.282630 0.0337808
\(71\) 8.60301 1.02099 0.510495 0.859881i \(-0.329462\pi\)
0.510495 + 0.859881i \(0.329462\pi\)
\(72\) 0 0
\(73\) 15.1488 1.77304 0.886519 0.462693i \(-0.153117\pi\)
0.886519 + 0.462693i \(0.153117\pi\)
\(74\) 2.28263 0.265350
\(75\) 0 0
\(76\) −3.77455 −0.432971
\(77\) 3.70370 0.422075
\(78\) 0 0
\(79\) 7.37756 0.830040 0.415020 0.909812i \(-0.363775\pi\)
0.415020 + 0.909812i \(0.363775\pi\)
\(80\) 4.32614 0.483677
\(81\) 0 0
\(82\) 2.43474 0.268872
\(83\) −6.94282 −0.762074 −0.381037 0.924560i \(-0.624433\pi\)
−0.381037 + 0.924560i \(0.624433\pi\)
\(84\) 0 0
\(85\) −8.20602 −0.890068
\(86\) −0.532157 −0.0573840
\(87\) 0 0
\(88\) −3.49192 −0.372240
\(89\) 2.74720 0.291203 0.145602 0.989343i \(-0.453488\pi\)
0.145602 + 0.989343i \(0.453488\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 10.8856 1.13491
\(93\) 0 0
\(94\) −1.39372 −0.143751
\(95\) 2.29630 0.235596
\(96\) 0 0
\(97\) 7.16827 0.727828 0.363914 0.931433i \(-0.381440\pi\)
0.363914 + 0.931433i \(0.381440\pi\)
\(98\) −0.239123 −0.0241551
\(99\) 0 0
\(100\) 7.00000 0.700000
\(101\) −12.7850 −1.27215 −0.636075 0.771627i \(-0.719443\pi\)
−0.636075 + 0.771627i \(0.719443\pi\)
\(102\) 0 0
\(103\) −4.39699 −0.433248 −0.216624 0.976255i \(-0.569505\pi\)
−0.216624 + 0.976255i \(0.569505\pi\)
\(104\) 0.942820 0.0924511
\(105\) 0 0
\(106\) 2.77455 0.269488
\(107\) 13.7278 1.32711 0.663557 0.748126i \(-0.269046\pi\)
0.663557 + 0.748126i \(0.269046\pi\)
\(108\) 0 0
\(109\) 1.26320 0.120993 0.0604963 0.998168i \(-0.480732\pi\)
0.0604963 + 0.998168i \(0.480732\pi\)
\(110\) 1.04678 0.0998062
\(111\) 0 0
\(112\) −3.66019 −0.345855
\(113\) 12.1625 1.14415 0.572076 0.820200i \(-0.306138\pi\)
0.572076 + 0.820200i \(0.306138\pi\)
\(114\) 0 0
\(115\) −6.62244 −0.617546
\(116\) −0.464574 −0.0431346
\(117\) 0 0
\(118\) −0.622440 −0.0573003
\(119\) 6.94282 0.636447
\(120\) 0 0
\(121\) 2.71737 0.247034
\(122\) 1.81806 0.164599
\(123\) 0 0
\(124\) −3.22545 −0.289654
\(125\) −10.1683 −0.909478
\(126\) 0 0
\(127\) 1.33981 0.118889 0.0594445 0.998232i \(-0.481067\pi\)
0.0594445 + 0.998232i \(0.481067\pi\)
\(128\) 7.11436 0.628827
\(129\) 0 0
\(130\) −0.282630 −0.0247883
\(131\) −4.96690 −0.433960 −0.216980 0.976176i \(-0.569621\pi\)
−0.216980 + 0.976176i \(0.569621\pi\)
\(132\) 0 0
\(133\) −1.94282 −0.168464
\(134\) −0.838864 −0.0724668
\(135\) 0 0
\(136\) −6.54583 −0.561300
\(137\) −4.33981 −0.370775 −0.185387 0.982665i \(-0.559354\pi\)
−0.185387 + 0.982665i \(0.559354\pi\)
\(138\) 0 0
\(139\) 3.94282 0.334426 0.167213 0.985921i \(-0.446523\pi\)
0.167213 + 0.985921i \(0.446523\pi\)
\(140\) 2.29630 0.194073
\(141\) 0 0
\(142\) −2.05718 −0.172635
\(143\) −3.70370 −0.309719
\(144\) 0 0
\(145\) 0.282630 0.0234712
\(146\) −3.62244 −0.299795
\(147\) 0 0
\(148\) 18.5458 1.52446
\(149\) 11.1111 0.910256 0.455128 0.890426i \(-0.349594\pi\)
0.455128 + 0.890426i \(0.349594\pi\)
\(150\) 0 0
\(151\) 13.9234 1.13307 0.566535 0.824038i \(-0.308284\pi\)
0.566535 + 0.824038i \(0.308284\pi\)
\(152\) 1.83173 0.148573
\(153\) 0 0
\(154\) −0.885640 −0.0713669
\(155\) 1.96225 0.157612
\(156\) 0 0
\(157\) 0.0571799 0.00456346 0.00228173 0.999997i \(-0.499274\pi\)
0.00228173 + 0.999997i \(0.499274\pi\)
\(158\) −1.76415 −0.140348
\(159\) 0 0
\(160\) −3.26320 −0.257979
\(161\) 5.60301 0.441579
\(162\) 0 0
\(163\) −1.50808 −0.118122 −0.0590610 0.998254i \(-0.518811\pi\)
−0.0590610 + 0.998254i \(0.518811\pi\)
\(164\) 19.7817 1.54469
\(165\) 0 0
\(166\) 1.66019 0.128856
\(167\) −14.6843 −1.13630 −0.568151 0.822924i \(-0.692341\pi\)
−0.568151 + 0.822924i \(0.692341\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 1.96225 0.150498
\(171\) 0 0
\(172\) −4.32365 −0.329675
\(173\) 0.252796 0.0192197 0.00960987 0.999954i \(-0.496941\pi\)
0.00960987 + 0.999954i \(0.496941\pi\)
\(174\) 0 0
\(175\) 3.60301 0.272362
\(176\) −13.5562 −1.02184
\(177\) 0 0
\(178\) −0.656920 −0.0492383
\(179\) 14.1923 1.06079 0.530393 0.847752i \(-0.322044\pi\)
0.530393 + 0.847752i \(0.322044\pi\)
\(180\) 0 0
\(181\) −1.43147 −0.106400 −0.0532002 0.998584i \(-0.516942\pi\)
−0.0532002 + 0.998584i \(0.516942\pi\)
\(182\) 0.239123 0.0177250
\(183\) 0 0
\(184\) −5.28263 −0.389441
\(185\) −11.2826 −0.829515
\(186\) 0 0
\(187\) 25.7141 1.88040
\(188\) −11.3236 −0.825862
\(189\) 0 0
\(190\) −0.549100 −0.0398359
\(191\) 15.0676 1.09025 0.545126 0.838354i \(-0.316482\pi\)
0.545126 + 0.838354i \(0.316482\pi\)
\(192\) 0 0
\(193\) −7.84789 −0.564904 −0.282452 0.959282i \(-0.591148\pi\)
−0.282452 + 0.959282i \(0.591148\pi\)
\(194\) −1.71410 −0.123065
\(195\) 0 0
\(196\) −1.94282 −0.138773
\(197\) 6.69002 0.476644 0.238322 0.971186i \(-0.423403\pi\)
0.238322 + 0.971186i \(0.423403\pi\)
\(198\) 0 0
\(199\) −19.9396 −1.41348 −0.706739 0.707475i \(-0.749834\pi\)
−0.706739 + 0.707475i \(0.749834\pi\)
\(200\) −3.39699 −0.240203
\(201\) 0 0
\(202\) 3.05718 0.215102
\(203\) −0.239123 −0.0167832
\(204\) 0 0
\(205\) −12.0345 −0.840525
\(206\) 1.05142 0.0732561
\(207\) 0 0
\(208\) 3.66019 0.253789
\(209\) −7.19562 −0.497731
\(210\) 0 0
\(211\) −18.0917 −1.24548 −0.622741 0.782428i \(-0.713981\pi\)
−0.622741 + 0.782428i \(0.713981\pi\)
\(212\) 22.5426 1.54823
\(213\) 0 0
\(214\) −3.28263 −0.224396
\(215\) 2.63036 0.179389
\(216\) 0 0
\(217\) −1.66019 −0.112701
\(218\) −0.302060 −0.0204581
\(219\) 0 0
\(220\) 8.50481 0.573394
\(221\) −6.94282 −0.467025
\(222\) 0 0
\(223\) −22.6569 −1.51722 −0.758610 0.651545i \(-0.774121\pi\)
−0.758610 + 0.651545i \(0.774121\pi\)
\(224\) 2.76088 0.184469
\(225\) 0 0
\(226\) −2.90834 −0.193460
\(227\) −5.28263 −0.350620 −0.175310 0.984513i \(-0.556093\pi\)
−0.175310 + 0.984513i \(0.556093\pi\)
\(228\) 0 0
\(229\) 19.3365 1.27779 0.638897 0.769292i \(-0.279391\pi\)
0.638897 + 0.769292i \(0.279391\pi\)
\(230\) 1.58358 0.104418
\(231\) 0 0
\(232\) 0.225450 0.0148015
\(233\) −16.9806 −1.11243 −0.556217 0.831037i \(-0.687748\pi\)
−0.556217 + 0.831037i \(0.687748\pi\)
\(234\) 0 0
\(235\) 6.88891 0.449383
\(236\) −5.05718 −0.329194
\(237\) 0 0
\(238\) −1.66019 −0.107614
\(239\) 16.8856 1.09224 0.546121 0.837707i \(-0.316104\pi\)
0.546121 + 0.837707i \(0.316104\pi\)
\(240\) 0 0
\(241\) 27.1456 1.74860 0.874300 0.485386i \(-0.161321\pi\)
0.874300 + 0.485386i \(0.161321\pi\)
\(242\) −0.649786 −0.0417699
\(243\) 0 0
\(244\) 14.7713 0.945634
\(245\) 1.18194 0.0755116
\(246\) 0 0
\(247\) 1.94282 0.123619
\(248\) 1.56526 0.0993941
\(249\) 0 0
\(250\) 2.43147 0.153780
\(251\) −19.0780 −1.20419 −0.602096 0.798424i \(-0.705668\pi\)
−0.602096 + 0.798424i \(0.705668\pi\)
\(252\) 0 0
\(253\) 20.7518 1.30466
\(254\) −0.320380 −0.0201024
\(255\) 0 0
\(256\) 11.6192 0.726198
\(257\) −14.8421 −0.925827 −0.462913 0.886404i \(-0.653196\pi\)
−0.462913 + 0.886404i \(0.653196\pi\)
\(258\) 0 0
\(259\) 9.54583 0.593149
\(260\) −2.29630 −0.142411
\(261\) 0 0
\(262\) 1.18770 0.0733764
\(263\) −7.74145 −0.477358 −0.238679 0.971099i \(-0.576714\pi\)
−0.238679 + 0.971099i \(0.576714\pi\)
\(264\) 0 0
\(265\) −13.7141 −0.842450
\(266\) 0.464574 0.0284848
\(267\) 0 0
\(268\) −6.81557 −0.416327
\(269\) −1.51135 −0.0921486 −0.0460743 0.998938i \(-0.514671\pi\)
−0.0460743 + 0.998938i \(0.514671\pi\)
\(270\) 0 0
\(271\) −21.9806 −1.33522 −0.667612 0.744509i \(-0.732684\pi\)
−0.667612 + 0.744509i \(0.732684\pi\)
\(272\) −25.4120 −1.54083
\(273\) 0 0
\(274\) 1.03775 0.0626927
\(275\) 13.3445 0.804701
\(276\) 0 0
\(277\) −10.8285 −0.650619 −0.325310 0.945608i \(-0.605469\pi\)
−0.325310 + 0.945608i \(0.605469\pi\)
\(278\) −0.942820 −0.0565466
\(279\) 0 0
\(280\) −1.11436 −0.0665957
\(281\) −16.8766 −1.00677 −0.503387 0.864061i \(-0.667913\pi\)
−0.503387 + 0.864061i \(0.667913\pi\)
\(282\) 0 0
\(283\) −15.3171 −0.910508 −0.455254 0.890362i \(-0.650451\pi\)
−0.455254 + 0.890362i \(0.650451\pi\)
\(284\) −16.7141 −0.991799
\(285\) 0 0
\(286\) 0.885640 0.0523690
\(287\) 10.1819 0.601021
\(288\) 0 0
\(289\) 31.2028 1.83546
\(290\) −0.0675835 −0.00396864
\(291\) 0 0
\(292\) −29.4315 −1.72235
\(293\) −9.36964 −0.547380 −0.273690 0.961818i \(-0.588244\pi\)
−0.273690 + 0.961818i \(0.588244\pi\)
\(294\) 0 0
\(295\) 3.07661 0.179127
\(296\) −9.00000 −0.523114
\(297\) 0 0
\(298\) −2.65692 −0.153911
\(299\) −5.60301 −0.324030
\(300\) 0 0
\(301\) −2.22545 −0.128273
\(302\) −3.32941 −0.191586
\(303\) 0 0
\(304\) 7.11109 0.407849
\(305\) −8.98633 −0.514556
\(306\) 0 0
\(307\) 2.71410 0.154902 0.0774509 0.996996i \(-0.475322\pi\)
0.0774509 + 0.996996i \(0.475322\pi\)
\(308\) −7.19562 −0.410008
\(309\) 0 0
\(310\) −0.469220 −0.0266499
\(311\) −13.9806 −0.792765 −0.396383 0.918085i \(-0.629735\pi\)
−0.396383 + 0.918085i \(0.629735\pi\)
\(312\) 0 0
\(313\) −19.0539 −1.07699 −0.538495 0.842628i \(-0.681007\pi\)
−0.538495 + 0.842628i \(0.681007\pi\)
\(314\) −0.0136731 −0.000771615 0
\(315\) 0 0
\(316\) −14.3333 −0.806309
\(317\) −4.01943 −0.225754 −0.112877 0.993609i \(-0.536007\pi\)
−0.112877 + 0.993609i \(0.536007\pi\)
\(318\) 0 0
\(319\) −0.885640 −0.0495863
\(320\) −7.87197 −0.440056
\(321\) 0 0
\(322\) −1.33981 −0.0746647
\(323\) −13.4887 −0.750529
\(324\) 0 0
\(325\) −3.60301 −0.199859
\(326\) 0.360617 0.0199727
\(327\) 0 0
\(328\) −9.59974 −0.530057
\(329\) −5.82846 −0.321333
\(330\) 0 0
\(331\) −12.3776 −0.680332 −0.340166 0.940365i \(-0.610483\pi\)
−0.340166 + 0.940365i \(0.610483\pi\)
\(332\) 13.4887 0.740286
\(333\) 0 0
\(334\) 3.51135 0.192133
\(335\) 4.14635 0.226539
\(336\) 0 0
\(337\) 12.2599 0.667841 0.333920 0.942601i \(-0.391628\pi\)
0.333920 + 0.942601i \(0.391628\pi\)
\(338\) 2.86948 0.156079
\(339\) 0 0
\(340\) 15.9428 0.864621
\(341\) −6.14884 −0.332978
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 2.09820 0.113127
\(345\) 0 0
\(346\) −0.0604495 −0.00324978
\(347\) 6.64979 0.356979 0.178490 0.983942i \(-0.442879\pi\)
0.178490 + 0.983942i \(0.442879\pi\)
\(348\) 0 0
\(349\) 11.4347 0.612088 0.306044 0.952017i \(-0.400995\pi\)
0.306044 + 0.952017i \(0.400995\pi\)
\(350\) −0.861564 −0.0460525
\(351\) 0 0
\(352\) 10.2255 0.545018
\(353\) −22.1956 −1.18135 −0.590677 0.806908i \(-0.701139\pi\)
−0.590677 + 0.806908i \(0.701139\pi\)
\(354\) 0 0
\(355\) 10.1683 0.539676
\(356\) −5.33732 −0.282878
\(357\) 0 0
\(358\) −3.39372 −0.179364
\(359\) −7.55623 −0.398803 −0.199401 0.979918i \(-0.563900\pi\)
−0.199401 + 0.979918i \(0.563900\pi\)
\(360\) 0 0
\(361\) −15.2255 −0.801339
\(362\) 0.342298 0.0179908
\(363\) 0 0
\(364\) 1.94282 0.101831
\(365\) 17.9051 0.937194
\(366\) 0 0
\(367\) −18.5231 −0.966900 −0.483450 0.875372i \(-0.660616\pi\)
−0.483450 + 0.875372i \(0.660616\pi\)
\(368\) −20.5081 −1.06906
\(369\) 0 0
\(370\) 2.69794 0.140259
\(371\) 11.6030 0.602398
\(372\) 0 0
\(373\) 15.6602 0.810854 0.405427 0.914127i \(-0.367123\pi\)
0.405427 + 0.914127i \(0.367123\pi\)
\(374\) −6.14884 −0.317949
\(375\) 0 0
\(376\) 5.49519 0.283393
\(377\) 0.239123 0.0123155
\(378\) 0 0
\(379\) 4.03775 0.207405 0.103703 0.994608i \(-0.466931\pi\)
0.103703 + 0.994608i \(0.466931\pi\)
\(380\) −4.46130 −0.228860
\(381\) 0 0
\(382\) −3.60301 −0.184346
\(383\) 0.225450 0.0115200 0.00575998 0.999983i \(-0.498167\pi\)
0.00575998 + 0.999983i \(0.498167\pi\)
\(384\) 0 0
\(385\) 4.37756 0.223101
\(386\) 1.87661 0.0955171
\(387\) 0 0
\(388\) −13.9267 −0.707019
\(389\) 25.2632 1.28090 0.640448 0.768002i \(-0.278749\pi\)
0.640448 + 0.768002i \(0.278749\pi\)
\(390\) 0 0
\(391\) 38.9007 1.96729
\(392\) 0.942820 0.0476196
\(393\) 0 0
\(394\) −1.59974 −0.0805938
\(395\) 8.71986 0.438744
\(396\) 0 0
\(397\) −20.3009 −1.01888 −0.509438 0.860508i \(-0.670147\pi\)
−0.509438 + 0.860508i \(0.670147\pi\)
\(398\) 4.76801 0.238999
\(399\) 0 0
\(400\) −13.1877 −0.659385
\(401\) −15.2255 −0.760323 −0.380161 0.924920i \(-0.624132\pi\)
−0.380161 + 0.924920i \(0.624132\pi\)
\(402\) 0 0
\(403\) 1.66019 0.0826999
\(404\) 24.8389 1.23578
\(405\) 0 0
\(406\) 0.0571799 0.00283779
\(407\) 35.3549 1.75248
\(408\) 0 0
\(409\) 1.65692 0.0819294 0.0409647 0.999161i \(-0.486957\pi\)
0.0409647 + 0.999161i \(0.486957\pi\)
\(410\) 2.87772 0.142121
\(411\) 0 0
\(412\) 8.54256 0.420862
\(413\) −2.60301 −0.128086
\(414\) 0 0
\(415\) −8.20602 −0.402818
\(416\) −2.76088 −0.135363
\(417\) 0 0
\(418\) 1.72064 0.0841592
\(419\) 33.3743 1.63044 0.815220 0.579151i \(-0.196616\pi\)
0.815220 + 0.579151i \(0.196616\pi\)
\(420\) 0 0
\(421\) 18.2405 0.888988 0.444494 0.895782i \(-0.353384\pi\)
0.444494 + 0.895782i \(0.353384\pi\)
\(422\) 4.32614 0.210593
\(423\) 0 0
\(424\) −10.9396 −0.531272
\(425\) 25.0150 1.21341
\(426\) 0 0
\(427\) 7.60301 0.367935
\(428\) −26.6706 −1.28917
\(429\) 0 0
\(430\) −0.628979 −0.0303321
\(431\) 29.2826 1.41049 0.705247 0.708961i \(-0.250836\pi\)
0.705247 + 0.708961i \(0.250836\pi\)
\(432\) 0 0
\(433\) −12.2449 −0.588451 −0.294226 0.955736i \(-0.595062\pi\)
−0.294226 + 0.955736i \(0.595062\pi\)
\(434\) 0.396990 0.0190561
\(435\) 0 0
\(436\) −2.45417 −0.117533
\(437\) −10.8856 −0.520731
\(438\) 0 0
\(439\) −4.83173 −0.230606 −0.115303 0.993330i \(-0.536784\pi\)
−0.115303 + 0.993330i \(0.536784\pi\)
\(440\) −4.12725 −0.196759
\(441\) 0 0
\(442\) 1.66019 0.0789672
\(443\) 1.24488 0.0591461 0.0295730 0.999563i \(-0.490585\pi\)
0.0295730 + 0.999563i \(0.490585\pi\)
\(444\) 0 0
\(445\) 3.24704 0.153924
\(446\) 5.41780 0.256540
\(447\) 0 0
\(448\) 6.66019 0.314664
\(449\) 8.82846 0.416641 0.208320 0.978061i \(-0.433200\pi\)
0.208320 + 0.978061i \(0.433200\pi\)
\(450\) 0 0
\(451\) 37.7108 1.77573
\(452\) −23.6296 −1.11144
\(453\) 0 0
\(454\) 1.26320 0.0592849
\(455\) −1.18194 −0.0554104
\(456\) 0 0
\(457\) −10.5081 −0.491547 −0.245774 0.969327i \(-0.579042\pi\)
−0.245774 + 0.969327i \(0.579042\pi\)
\(458\) −4.62382 −0.216057
\(459\) 0 0
\(460\) 12.8662 0.599890
\(461\) 22.5516 1.05033 0.525166 0.851000i \(-0.324003\pi\)
0.525166 + 0.851000i \(0.324003\pi\)
\(462\) 0 0
\(463\) 10.3970 0.483189 0.241595 0.970377i \(-0.422330\pi\)
0.241595 + 0.970377i \(0.422330\pi\)
\(464\) 0.875237 0.0406318
\(465\) 0 0
\(466\) 4.06045 0.188097
\(467\) 13.3171 0.616242 0.308121 0.951347i \(-0.400300\pi\)
0.308121 + 0.951347i \(0.400300\pi\)
\(468\) 0 0
\(469\) −3.50808 −0.161988
\(470\) −1.64730 −0.0759842
\(471\) 0 0
\(472\) 2.45417 0.112962
\(473\) −8.24239 −0.378986
\(474\) 0 0
\(475\) −7.00000 −0.321182
\(476\) −13.4887 −0.618251
\(477\) 0 0
\(478\) −4.03775 −0.184682
\(479\) 14.5354 0.664141 0.332070 0.943255i \(-0.392253\pi\)
0.332070 + 0.943255i \(0.392253\pi\)
\(480\) 0 0
\(481\) −9.54583 −0.435252
\(482\) −6.49114 −0.295663
\(483\) 0 0
\(484\) −5.27936 −0.239971
\(485\) 8.47249 0.384716
\(486\) 0 0
\(487\) 13.0539 0.591529 0.295765 0.955261i \(-0.404426\pi\)
0.295765 + 0.955261i \(0.404426\pi\)
\(488\) −7.16827 −0.324492
\(489\) 0 0
\(490\) −0.282630 −0.0127679
\(491\) 19.3445 0.873003 0.436502 0.899704i \(-0.356217\pi\)
0.436502 + 0.899704i \(0.356217\pi\)
\(492\) 0 0
\(493\) −1.66019 −0.0747712
\(494\) −0.464574 −0.0209022
\(495\) 0 0
\(496\) 6.07661 0.272848
\(497\) −8.60301 −0.385898
\(498\) 0 0
\(499\) −36.2222 −1.62153 −0.810764 0.585374i \(-0.800948\pi\)
−0.810764 + 0.585374i \(0.800948\pi\)
\(500\) 19.7551 0.883476
\(501\) 0 0
\(502\) 4.56199 0.203612
\(503\) 15.6764 0.698974 0.349487 0.936941i \(-0.386356\pi\)
0.349487 + 0.936941i \(0.386356\pi\)
\(504\) 0 0
\(505\) −15.1111 −0.672435
\(506\) −4.96225 −0.220599
\(507\) 0 0
\(508\) −2.60301 −0.115490
\(509\) 34.3034 1.52047 0.760237 0.649646i \(-0.225083\pi\)
0.760237 + 0.649646i \(0.225083\pi\)
\(510\) 0 0
\(511\) −15.1488 −0.670145
\(512\) −17.0071 −0.751616
\(513\) 0 0
\(514\) 3.54910 0.156544
\(515\) −5.19699 −0.229007
\(516\) 0 0
\(517\) −21.5868 −0.949389
\(518\) −2.28263 −0.100293
\(519\) 0 0
\(520\) 1.11436 0.0488679
\(521\) −10.2449 −0.448836 −0.224418 0.974493i \(-0.572048\pi\)
−0.224418 + 0.974493i \(0.572048\pi\)
\(522\) 0 0
\(523\) 30.6030 1.33818 0.669088 0.743183i \(-0.266685\pi\)
0.669088 + 0.743183i \(0.266685\pi\)
\(524\) 9.64979 0.421553
\(525\) 0 0
\(526\) 1.85116 0.0807144
\(527\) −11.5264 −0.502098
\(528\) 0 0
\(529\) 8.39372 0.364944
\(530\) 3.27936 0.142446
\(531\) 0 0
\(532\) 3.77455 0.163647
\(533\) −10.1819 −0.441029
\(534\) 0 0
\(535\) 16.2255 0.701487
\(536\) 3.30749 0.142862
\(537\) 0 0
\(538\) 0.361399 0.0155810
\(539\) −3.70370 −0.159530
\(540\) 0 0
\(541\) −26.0917 −1.12177 −0.560884 0.827894i \(-0.689539\pi\)
−0.560884 + 0.827894i \(0.689539\pi\)
\(542\) 5.25607 0.225767
\(543\) 0 0
\(544\) 19.1683 0.821833
\(545\) 1.49303 0.0639544
\(546\) 0 0
\(547\) −10.9234 −0.467050 −0.233525 0.972351i \(-0.575026\pi\)
−0.233525 + 0.972351i \(0.575026\pi\)
\(548\) 8.43147 0.360175
\(549\) 0 0
\(550\) −3.19097 −0.136063
\(551\) 0.464574 0.0197915
\(552\) 0 0
\(553\) −7.37756 −0.313726
\(554\) 2.58934 0.110010
\(555\) 0 0
\(556\) −7.66019 −0.324864
\(557\) −13.9442 −0.590835 −0.295417 0.955368i \(-0.595459\pi\)
−0.295417 + 0.955368i \(0.595459\pi\)
\(558\) 0 0
\(559\) 2.22545 0.0941265
\(560\) −4.32614 −0.182813
\(561\) 0 0
\(562\) 4.03559 0.170231
\(563\) 30.2574 1.27520 0.637600 0.770368i \(-0.279927\pi\)
0.637600 + 0.770368i \(0.279927\pi\)
\(564\) 0 0
\(565\) 14.3754 0.604778
\(566\) 3.66268 0.153954
\(567\) 0 0
\(568\) 8.11109 0.340334
\(569\) −21.1352 −0.886032 −0.443016 0.896514i \(-0.646092\pi\)
−0.443016 + 0.896514i \(0.646092\pi\)
\(570\) 0 0
\(571\) −32.7863 −1.37207 −0.686033 0.727571i \(-0.740649\pi\)
−0.686033 + 0.727571i \(0.740649\pi\)
\(572\) 7.19562 0.300864
\(573\) 0 0
\(574\) −2.43474 −0.101624
\(575\) 20.1877 0.841885
\(576\) 0 0
\(577\) −17.3743 −0.723301 −0.361651 0.932314i \(-0.617787\pi\)
−0.361651 + 0.932314i \(0.617787\pi\)
\(578\) −7.46130 −0.310349
\(579\) 0 0
\(580\) −0.549100 −0.0228001
\(581\) 6.94282 0.288037
\(582\) 0 0
\(583\) 42.9740 1.77980
\(584\) 14.2826 0.591019
\(585\) 0 0
\(586\) 2.24050 0.0925542
\(587\) 16.9759 0.700671 0.350336 0.936624i \(-0.386068\pi\)
0.350336 + 0.936624i \(0.386068\pi\)
\(588\) 0 0
\(589\) 3.22545 0.132902
\(590\) −0.735689 −0.0302878
\(591\) 0 0
\(592\) −34.9396 −1.43601
\(593\) −13.0733 −0.536858 −0.268429 0.963300i \(-0.586504\pi\)
−0.268429 + 0.963300i \(0.586504\pi\)
\(594\) 0 0
\(595\) 8.20602 0.336414
\(596\) −21.5868 −0.884232
\(597\) 0 0
\(598\) 1.33981 0.0547889
\(599\) 29.2060 1.19333 0.596663 0.802492i \(-0.296493\pi\)
0.596663 + 0.802492i \(0.296493\pi\)
\(600\) 0 0
\(601\) 7.79071 0.317790 0.158895 0.987296i \(-0.449207\pi\)
0.158895 + 0.987296i \(0.449207\pi\)
\(602\) 0.532157 0.0216891
\(603\) 0 0
\(604\) −27.0506 −1.10067
\(605\) 3.21178 0.130577
\(606\) 0 0
\(607\) −19.6408 −0.797194 −0.398597 0.917126i \(-0.630503\pi\)
−0.398597 + 0.917126i \(0.630503\pi\)
\(608\) −5.36389 −0.217534
\(609\) 0 0
\(610\) 2.14884 0.0870040
\(611\) 5.82846 0.235794
\(612\) 0 0
\(613\) 23.5653 0.951792 0.475896 0.879502i \(-0.342124\pi\)
0.475896 + 0.879502i \(0.342124\pi\)
\(614\) −0.649005 −0.0261917
\(615\) 0 0
\(616\) 3.49192 0.140693
\(617\) −10.6602 −0.429163 −0.214582 0.976706i \(-0.568839\pi\)
−0.214582 + 0.976706i \(0.568839\pi\)
\(618\) 0 0
\(619\) −18.0150 −0.724086 −0.362043 0.932161i \(-0.617921\pi\)
−0.362043 + 0.932161i \(0.617921\pi\)
\(620\) −3.81230 −0.153106
\(621\) 0 0
\(622\) 3.34308 0.134045
\(623\) −2.74720 −0.110064
\(624\) 0 0
\(625\) 5.99673 0.239869
\(626\) 4.55623 0.182104
\(627\) 0 0
\(628\) −0.111090 −0.00443299
\(629\) 66.2750 2.64256
\(630\) 0 0
\(631\) 12.4703 0.496436 0.248218 0.968704i \(-0.420155\pi\)
0.248218 + 0.968704i \(0.420155\pi\)
\(632\) 6.95571 0.276683
\(633\) 0 0
\(634\) 0.961139 0.0381717
\(635\) 1.58358 0.0628424
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0.211777 0.00838434
\(639\) 0 0
\(640\) 8.40877 0.332386
\(641\) 19.1456 0.756205 0.378102 0.925764i \(-0.376577\pi\)
0.378102 + 0.925764i \(0.376577\pi\)
\(642\) 0 0
\(643\) 6.48865 0.255887 0.127944 0.991781i \(-0.459162\pi\)
0.127944 + 0.991781i \(0.459162\pi\)
\(644\) −10.8856 −0.428954
\(645\) 0 0
\(646\) 3.22545 0.126904
\(647\) −48.0988 −1.89096 −0.945479 0.325682i \(-0.894406\pi\)
−0.945479 + 0.325682i \(0.894406\pi\)
\(648\) 0 0
\(649\) −9.64076 −0.378433
\(650\) 0.861564 0.0337933
\(651\) 0 0
\(652\) 2.92993 0.114745
\(653\) −43.2405 −1.69213 −0.846066 0.533079i \(-0.821035\pi\)
−0.846066 + 0.533079i \(0.821035\pi\)
\(654\) 0 0
\(655\) −5.87059 −0.229383
\(656\) −37.2678 −1.45506
\(657\) 0 0
\(658\) 1.39372 0.0543329
\(659\) −2.50808 −0.0977009 −0.0488505 0.998806i \(-0.515556\pi\)
−0.0488505 + 0.998806i \(0.515556\pi\)
\(660\) 0 0
\(661\) −42.3354 −1.64666 −0.823329 0.567565i \(-0.807886\pi\)
−0.823329 + 0.567565i \(0.807886\pi\)
\(662\) 2.95976 0.115034
\(663\) 0 0
\(664\) −6.54583 −0.254027
\(665\) −2.29630 −0.0890468
\(666\) 0 0
\(667\) −1.33981 −0.0518777
\(668\) 28.5289 1.10382
\(669\) 0 0
\(670\) −0.991489 −0.0383046
\(671\) 28.1592 1.08708
\(672\) 0 0
\(673\) 13.4153 0.517122 0.258561 0.965995i \(-0.416752\pi\)
0.258561 + 0.965995i \(0.416752\pi\)
\(674\) −2.93163 −0.112922
\(675\) 0 0
\(676\) 23.3138 0.896686
\(677\) 1.96225 0.0754154 0.0377077 0.999289i \(-0.487994\pi\)
0.0377077 + 0.999289i \(0.487994\pi\)
\(678\) 0 0
\(679\) −7.16827 −0.275093
\(680\) −7.73680 −0.296693
\(681\) 0 0
\(682\) 1.47033 0.0563019
\(683\) 27.1672 1.03952 0.519761 0.854312i \(-0.326021\pi\)
0.519761 + 0.854312i \(0.326021\pi\)
\(684\) 0 0
\(685\) −5.12941 −0.195985
\(686\) 0.239123 0.00912977
\(687\) 0 0
\(688\) 8.14557 0.310547
\(689\) −11.6030 −0.442039
\(690\) 0 0
\(691\) −50.3171 −1.91415 −0.957077 0.289835i \(-0.906399\pi\)
−0.957077 + 0.289835i \(0.906399\pi\)
\(692\) −0.491138 −0.0186703
\(693\) 0 0
\(694\) −1.59012 −0.0603601
\(695\) 4.66019 0.176771
\(696\) 0 0
\(697\) 70.6914 2.67763
\(698\) −2.73431 −0.103495
\(699\) 0 0
\(700\) −7.00000 −0.264575
\(701\) 45.1672 1.70594 0.852970 0.521960i \(-0.174799\pi\)
0.852970 + 0.521960i \(0.174799\pi\)
\(702\) 0 0
\(703\) −18.5458 −0.699469
\(704\) 24.6673 0.929685
\(705\) 0 0
\(706\) 5.30749 0.199750
\(707\) 12.7850 0.480828
\(708\) 0 0
\(709\) 39.6181 1.48789 0.743944 0.668242i \(-0.232953\pi\)
0.743944 + 0.668242i \(0.232953\pi\)
\(710\) −2.43147 −0.0912514
\(711\) 0 0
\(712\) 2.59012 0.0970688
\(713\) −9.30206 −0.348365
\(714\) 0 0
\(715\) −4.37756 −0.163711
\(716\) −27.5732 −1.03046
\(717\) 0 0
\(718\) 1.80687 0.0674318
\(719\) −22.0377 −0.821869 −0.410935 0.911665i \(-0.634798\pi\)
−0.410935 + 0.911665i \(0.634798\pi\)
\(720\) 0 0
\(721\) 4.39699 0.163752
\(722\) 3.64076 0.135495
\(723\) 0 0
\(724\) 2.78109 0.103358
\(725\) −0.861564 −0.0319977
\(726\) 0 0
\(727\) 28.1111 1.04258 0.521291 0.853379i \(-0.325450\pi\)
0.521291 + 0.853379i \(0.325450\pi\)
\(728\) −0.942820 −0.0349432
\(729\) 0 0
\(730\) −4.28152 −0.158466
\(731\) −15.4509 −0.571472
\(732\) 0 0
\(733\) 11.8695 0.438409 0.219205 0.975679i \(-0.429654\pi\)
0.219205 + 0.975679i \(0.429654\pi\)
\(734\) 4.42931 0.163489
\(735\) 0 0
\(736\) 15.4692 0.570203
\(737\) −12.9929 −0.478598
\(738\) 0 0
\(739\) −12.1844 −0.448212 −0.224106 0.974565i \(-0.571946\pi\)
−0.224106 + 0.974565i \(0.571946\pi\)
\(740\) 21.9201 0.805800
\(741\) 0 0
\(742\) −2.77455 −0.101857
\(743\) −44.4854 −1.63201 −0.816005 0.578045i \(-0.803816\pi\)
−0.816005 + 0.578045i \(0.803816\pi\)
\(744\) 0 0
\(745\) 13.1327 0.481144
\(746\) −3.74472 −0.137104
\(747\) 0 0
\(748\) −49.9579 −1.82664
\(749\) −13.7278 −0.501602
\(750\) 0 0
\(751\) 42.8058 1.56200 0.781002 0.624528i \(-0.214709\pi\)
0.781002 + 0.624528i \(0.214709\pi\)
\(752\) 21.3333 0.777944
\(753\) 0 0
\(754\) −0.0571799 −0.00208237
\(755\) 16.4567 0.598919
\(756\) 0 0
\(757\) −22.4919 −0.817483 −0.408741 0.912650i \(-0.634032\pi\)
−0.408741 + 0.912650i \(0.634032\pi\)
\(758\) −0.965520 −0.0350693
\(759\) 0 0
\(760\) 2.16500 0.0785328
\(761\) −14.3365 −0.519699 −0.259850 0.965649i \(-0.583673\pi\)
−0.259850 + 0.965649i \(0.583673\pi\)
\(762\) 0 0
\(763\) −1.26320 −0.0457309
\(764\) −29.2736 −1.05908
\(765\) 0 0
\(766\) −0.0539104 −0.00194786
\(767\) 2.60301 0.0939892
\(768\) 0 0
\(769\) −31.2211 −1.12586 −0.562930 0.826504i \(-0.690326\pi\)
−0.562930 + 0.826504i \(0.690326\pi\)
\(770\) −1.04678 −0.0377232
\(771\) 0 0
\(772\) 15.2470 0.548753
\(773\) 4.38005 0.157539 0.0787697 0.996893i \(-0.474901\pi\)
0.0787697 + 0.996893i \(0.474901\pi\)
\(774\) 0 0
\(775\) −5.98168 −0.214868
\(776\) 6.75839 0.242612
\(777\) 0 0
\(778\) −6.04102 −0.216581
\(779\) −19.7817 −0.708752
\(780\) 0 0
\(781\) −31.8629 −1.14015
\(782\) −9.30206 −0.332641
\(783\) 0 0
\(784\) 3.66019 0.130721
\(785\) 0.0675835 0.00241216
\(786\) 0 0
\(787\) 27.6213 0.984594 0.492297 0.870427i \(-0.336157\pi\)
0.492297 + 0.870427i \(0.336157\pi\)
\(788\) −12.9975 −0.463017
\(789\) 0 0
\(790\) −2.08512 −0.0741853
\(791\) −12.1625 −0.432449
\(792\) 0 0
\(793\) −7.60301 −0.269991
\(794\) 4.85443 0.172277
\(795\) 0 0
\(796\) 38.7390 1.37307
\(797\) −2.96363 −0.104977 −0.0524885 0.998622i \(-0.516715\pi\)
−0.0524885 + 0.998622i \(0.516715\pi\)
\(798\) 0 0
\(799\) −40.4660 −1.43158
\(800\) 9.94747 0.351696
\(801\) 0 0
\(802\) 3.64076 0.128560
\(803\) −56.1067 −1.97996
\(804\) 0 0
\(805\) 6.62244 0.233410
\(806\) −0.396990 −0.0139834
\(807\) 0 0
\(808\) −12.0539 −0.424055
\(809\) −24.7896 −0.871556 −0.435778 0.900054i \(-0.643527\pi\)
−0.435778 + 0.900054i \(0.643527\pi\)
\(810\) 0 0
\(811\) 8.24377 0.289478 0.144739 0.989470i \(-0.453766\pi\)
0.144739 + 0.989470i \(0.453766\pi\)
\(812\) 0.464574 0.0163033
\(813\) 0 0
\(814\) −8.45417 −0.296319
\(815\) −1.78247 −0.0624370
\(816\) 0 0
\(817\) 4.32365 0.151265
\(818\) −0.396208 −0.0138531
\(819\) 0 0
\(820\) 23.3808 0.816494
\(821\) −28.8993 −1.00859 −0.504296 0.863531i \(-0.668248\pi\)
−0.504296 + 0.863531i \(0.668248\pi\)
\(822\) 0 0
\(823\) −36.0000 −1.25488 −0.627441 0.778664i \(-0.715897\pi\)
−0.627441 + 0.778664i \(0.715897\pi\)
\(824\) −4.14557 −0.144418
\(825\) 0 0
\(826\) 0.622440 0.0216575
\(827\) −50.7108 −1.76339 −0.881694 0.471821i \(-0.843597\pi\)
−0.881694 + 0.471821i \(0.843597\pi\)
\(828\) 0 0
\(829\) 14.8123 0.514452 0.257226 0.966351i \(-0.417191\pi\)
0.257226 + 0.966351i \(0.417191\pi\)
\(830\) 1.96225 0.0681107
\(831\) 0 0
\(832\) −6.66019 −0.230901
\(833\) −6.94282 −0.240554
\(834\) 0 0
\(835\) −17.3560 −0.600628
\(836\) 13.9798 0.483501
\(837\) 0 0
\(838\) −7.98057 −0.275684
\(839\) 33.7212 1.16419 0.582093 0.813122i \(-0.302234\pi\)
0.582093 + 0.813122i \(0.302234\pi\)
\(840\) 0 0
\(841\) −28.9428 −0.998028
\(842\) −4.36173 −0.150315
\(843\) 0 0
\(844\) 35.1488 1.20987
\(845\) −14.1833 −0.487921
\(846\) 0 0
\(847\) −2.71737 −0.0933699
\(848\) −42.4692 −1.45840
\(849\) 0 0
\(850\) −5.98168 −0.205170
\(851\) 53.4854 1.83346
\(852\) 0 0
\(853\) 11.7896 0.403668 0.201834 0.979420i \(-0.435310\pi\)
0.201834 + 0.979420i \(0.435310\pi\)
\(854\) −1.81806 −0.0622126
\(855\) 0 0
\(856\) 12.9428 0.442376
\(857\) −31.3261 −1.07008 −0.535040 0.844827i \(-0.679704\pi\)
−0.535040 + 0.844827i \(0.679704\pi\)
\(858\) 0 0
\(859\) −50.3893 −1.71926 −0.859631 0.510915i \(-0.829307\pi\)
−0.859631 + 0.510915i \(0.829307\pi\)
\(860\) −5.11031 −0.174260
\(861\) 0 0
\(862\) −7.00216 −0.238494
\(863\) 1.13268 0.0385568 0.0192784 0.999814i \(-0.493863\pi\)
0.0192784 + 0.999814i \(0.493863\pi\)
\(864\) 0 0
\(865\) 0.298791 0.0101592
\(866\) 2.92804 0.0994987
\(867\) 0 0
\(868\) 3.22545 0.109479
\(869\) −27.3242 −0.926911
\(870\) 0 0
\(871\) 3.50808 0.118867
\(872\) 1.19097 0.0403313
\(873\) 0 0
\(874\) 2.60301 0.0880481
\(875\) 10.1683 0.343750
\(876\) 0 0
\(877\) −27.3937 −0.925020 −0.462510 0.886614i \(-0.653051\pi\)
−0.462510 + 0.886614i \(0.653051\pi\)
\(878\) 1.15538 0.0389922
\(879\) 0 0
\(880\) −16.0227 −0.540125
\(881\) 1.20929 0.0407420 0.0203710 0.999792i \(-0.493515\pi\)
0.0203710 + 0.999792i \(0.493515\pi\)
\(882\) 0 0
\(883\) −51.0884 −1.71926 −0.859631 0.510916i \(-0.829306\pi\)
−0.859631 + 0.510916i \(0.829306\pi\)
\(884\) 13.4887 0.453672
\(885\) 0 0
\(886\) −0.297680 −0.0100008
\(887\) −41.5757 −1.39597 −0.697987 0.716110i \(-0.745921\pi\)
−0.697987 + 0.716110i \(0.745921\pi\)
\(888\) 0 0
\(889\) −1.33981 −0.0449358
\(890\) −0.776443 −0.0260264
\(891\) 0 0
\(892\) 44.0183 1.47384
\(893\) 11.3236 0.378931
\(894\) 0 0
\(895\) 16.7745 0.560711
\(896\) −7.11436 −0.237674
\(897\) 0 0
\(898\) −2.11109 −0.0704480
\(899\) 0.396990 0.0132404
\(900\) 0 0
\(901\) 80.5576 2.68376
\(902\) −9.01754 −0.300251
\(903\) 0 0
\(904\) 11.4671 0.381389
\(905\) −1.69192 −0.0562412
\(906\) 0 0
\(907\) 35.4509 1.17713 0.588564 0.808451i \(-0.299694\pi\)
0.588564 + 0.808451i \(0.299694\pi\)
\(908\) 10.2632 0.340596
\(909\) 0 0
\(910\) 0.282630 0.00936910
\(911\) −20.7108 −0.686180 −0.343090 0.939302i \(-0.611474\pi\)
−0.343090 + 0.939302i \(0.611474\pi\)
\(912\) 0 0
\(913\) 25.7141 0.851013
\(914\) 2.51273 0.0831136
\(915\) 0 0
\(916\) −37.5674 −1.24126
\(917\) 4.96690 0.164021
\(918\) 0 0
\(919\) 14.3926 0.474768 0.237384 0.971416i \(-0.423710\pi\)
0.237384 + 0.971416i \(0.423710\pi\)
\(920\) −6.24377 −0.205851
\(921\) 0 0
\(922\) −5.39261 −0.177596
\(923\) 8.60301 0.283172
\(924\) 0 0
\(925\) 34.3937 1.13086
\(926\) −2.48616 −0.0817004
\(927\) 0 0
\(928\) −0.660190 −0.0216718
\(929\) −41.7428 −1.36954 −0.684769 0.728760i \(-0.740097\pi\)
−0.684769 + 0.728760i \(0.740097\pi\)
\(930\) 0 0
\(931\) 1.94282 0.0636734
\(932\) 32.9902 1.08063
\(933\) 0 0
\(934\) −3.18443 −0.104198
\(935\) 30.3926 0.993945
\(936\) 0 0
\(937\) 3.17154 0.103610 0.0518048 0.998657i \(-0.483503\pi\)
0.0518048 + 0.998657i \(0.483503\pi\)
\(938\) 0.838864 0.0273899
\(939\) 0 0
\(940\) −13.3839 −0.436535
\(941\) −3.22080 −0.104995 −0.0524976 0.998621i \(-0.516718\pi\)
−0.0524976 + 0.998621i \(0.516718\pi\)
\(942\) 0 0
\(943\) 57.0495 1.85779
\(944\) 9.52751 0.310094
\(945\) 0 0
\(946\) 1.97095 0.0640810
\(947\) −45.3469 −1.47358 −0.736789 0.676123i \(-0.763659\pi\)
−0.736789 + 0.676123i \(0.763659\pi\)
\(948\) 0 0
\(949\) 15.1488 0.491752
\(950\) 1.67386 0.0543073
\(951\) 0 0
\(952\) 6.54583 0.212152
\(953\) −54.2703 −1.75799 −0.878994 0.476832i \(-0.841785\pi\)
−0.878994 + 0.476832i \(0.841785\pi\)
\(954\) 0 0
\(955\) 17.8090 0.576287
\(956\) −32.8058 −1.06101
\(957\) 0 0
\(958\) −3.47576 −0.112297
\(959\) 4.33981 0.140140
\(960\) 0 0
\(961\) −28.2438 −0.911089
\(962\) 2.28263 0.0735950
\(963\) 0 0
\(964\) −52.7390 −1.69861
\(965\) −9.27576 −0.298597
\(966\) 0 0
\(967\) 25.6591 0.825140 0.412570 0.910926i \(-0.364631\pi\)
0.412570 + 0.910926i \(0.364631\pi\)
\(968\) 2.56199 0.0823455
\(969\) 0 0
\(970\) −2.02597 −0.0650500
\(971\) −21.0183 −0.674510 −0.337255 0.941413i \(-0.609498\pi\)
−0.337255 + 0.941413i \(0.609498\pi\)
\(972\) 0 0
\(973\) −3.94282 −0.126401
\(974\) −3.12149 −0.100019
\(975\) 0 0
\(976\) −27.8285 −0.890767
\(977\) −2.09820 −0.0671273 −0.0335637 0.999437i \(-0.510686\pi\)
−0.0335637 + 0.999437i \(0.510686\pi\)
\(978\) 0 0
\(979\) −10.1748 −0.325188
\(980\) −2.29630 −0.0733527
\(981\) 0 0
\(982\) −4.62571 −0.147612
\(983\) 42.9923 1.37124 0.685622 0.727958i \(-0.259531\pi\)
0.685622 + 0.727958i \(0.259531\pi\)
\(984\) 0 0
\(985\) 7.90723 0.251945
\(986\) 0.396990 0.0126427
\(987\) 0 0
\(988\) −3.77455 −0.120084
\(989\) −12.4692 −0.396498
\(990\) 0 0
\(991\) −17.2632 −0.548384 −0.274192 0.961675i \(-0.588410\pi\)
−0.274192 + 0.961675i \(0.588410\pi\)
\(992\) −4.58358 −0.145529
\(993\) 0 0
\(994\) 2.05718 0.0652498
\(995\) −23.5674 −0.747137
\(996\) 0 0
\(997\) −38.9018 −1.23203 −0.616016 0.787733i \(-0.711254\pi\)
−0.616016 + 0.787733i \(0.711254\pi\)
\(998\) 8.66157 0.274177
\(999\) 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 567.2.a.d.1.2 3
3.2 odd 2 567.2.a.g.1.2 3
4.3 odd 2 9072.2.a.bq.1.3 3
7.6 odd 2 3969.2.a.m.1.2 3
9.2 odd 6 189.2.f.a.64.2 6
9.4 even 3 63.2.f.b.43.2 yes 6
9.5 odd 6 189.2.f.a.127.2 6
9.7 even 3 63.2.f.b.22.2 6
12.11 even 2 9072.2.a.cd.1.1 3
21.20 even 2 3969.2.a.p.1.2 3
36.7 odd 6 1008.2.r.k.337.2 6
36.11 even 6 3024.2.r.g.1009.3 6
36.23 even 6 3024.2.r.g.2017.3 6
36.31 odd 6 1008.2.r.k.673.2 6
63.2 odd 6 1323.2.g.c.361.2 6
63.4 even 3 441.2.g.e.79.2 6
63.5 even 6 1323.2.h.e.802.2 6
63.11 odd 6 1323.2.h.d.226.2 6
63.13 odd 6 441.2.f.d.295.2 6
63.16 even 3 441.2.g.e.67.2 6
63.20 even 6 1323.2.f.c.442.2 6
63.23 odd 6 1323.2.h.d.802.2 6
63.25 even 3 441.2.h.c.373.2 6
63.31 odd 6 441.2.g.d.79.2 6
63.32 odd 6 1323.2.g.c.667.2 6
63.34 odd 6 441.2.f.d.148.2 6
63.38 even 6 1323.2.h.e.226.2 6
63.40 odd 6 441.2.h.b.214.2 6
63.41 even 6 1323.2.f.c.883.2 6
63.47 even 6 1323.2.g.b.361.2 6
63.52 odd 6 441.2.h.b.373.2 6
63.58 even 3 441.2.h.c.214.2 6
63.59 even 6 1323.2.g.b.667.2 6
63.61 odd 6 441.2.g.d.67.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.2 6 9.7 even 3
63.2.f.b.43.2 yes 6 9.4 even 3
189.2.f.a.64.2 6 9.2 odd 6
189.2.f.a.127.2 6 9.5 odd 6
441.2.f.d.148.2 6 63.34 odd 6
441.2.f.d.295.2 6 63.13 odd 6
441.2.g.d.67.2 6 63.61 odd 6
441.2.g.d.79.2 6 63.31 odd 6
441.2.g.e.67.2 6 63.16 even 3
441.2.g.e.79.2 6 63.4 even 3
441.2.h.b.214.2 6 63.40 odd 6
441.2.h.b.373.2 6 63.52 odd 6
441.2.h.c.214.2 6 63.58 even 3
441.2.h.c.373.2 6 63.25 even 3
567.2.a.d.1.2 3 1.1 even 1 trivial
567.2.a.g.1.2 3 3.2 odd 2
1008.2.r.k.337.2 6 36.7 odd 6
1008.2.r.k.673.2 6 36.31 odd 6
1323.2.f.c.442.2 6 63.20 even 6
1323.2.f.c.883.2 6 63.41 even 6
1323.2.g.b.361.2 6 63.47 even 6
1323.2.g.b.667.2 6 63.59 even 6
1323.2.g.c.361.2 6 63.2 odd 6
1323.2.g.c.667.2 6 63.32 odd 6
1323.2.h.d.226.2 6 63.11 odd 6
1323.2.h.d.802.2 6 63.23 odd 6
1323.2.h.e.226.2 6 63.38 even 6
1323.2.h.e.802.2 6 63.5 even 6
3024.2.r.g.1009.3 6 36.11 even 6
3024.2.r.g.2017.3 6 36.23 even 6
3969.2.a.m.1.2 3 7.6 odd 2
3969.2.a.p.1.2 3 21.20 even 2
9072.2.a.bq.1.3 3 4.3 odd 2
9072.2.a.cd.1.1 3 12.11 even 2