Properties

Label 1521.2.b.m.1351.6
Level $1521$
Weight $2$
Character 1521.1351
Analytic conductor $12.145$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.6
Root \(1.80194i\) of defining polynomial
Character \(\chi\) \(=\) 1521.1351
Dual form 1521.2.b.m.1351.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.80194i q^{2} -1.24698 q^{4} +1.44504i q^{5} +3.44504i q^{7} +1.35690i q^{8} +O(q^{10})\) \(q+1.80194i q^{2} -1.24698 q^{4} +1.44504i q^{5} +3.44504i q^{7} +1.35690i q^{8} -2.60388 q^{10} +5.18598i q^{11} -6.20775 q^{14} -4.93900 q^{16} -0.753020 q^{17} -7.96077i q^{19} -1.80194i q^{20} -9.34481 q^{22} -2.82908 q^{23} +2.91185 q^{25} -4.29590i q^{28} +3.91185 q^{29} +4.89977i q^{31} -6.18598i q^{32} -1.35690i q^{34} -4.97823 q^{35} +6.24698i q^{37} +14.3448 q^{38} -1.96077 q^{40} +1.80194i q^{41} +7.09783 q^{43} -6.46681i q^{44} -5.09783i q^{46} -10.5526i q^{47} -4.86831 q^{49} +5.24698i q^{50} +3.08815 q^{53} -7.49396 q^{55} -4.67456 q^{56} +7.04892i q^{58} -1.87800i q^{59} +3.34481 q^{61} -8.82908 q^{62} +1.26875 q^{64} +4.54288i q^{67} +0.939001 q^{68} -8.97046i q^{70} -9.11960i q^{71} +2.95108i q^{73} -11.2567 q^{74} +9.92692i q^{76} -17.8659 q^{77} -9.43296 q^{79} -7.13706i q^{80} -3.24698 q^{82} -6.46681i q^{83} -1.08815i q^{85} +12.7899i q^{86} -7.03684 q^{88} -1.15883i q^{89} +3.52781 q^{92} +19.0151 q^{94} +11.5036 q^{95} -8.65817i q^{97} -8.77240i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{4} + O(q^{10}) \) \( 6q + 2q^{4} + 2q^{10} - 2q^{14} - 10q^{16} - 14q^{17} - 10q^{22} + 4q^{23} + 10q^{25} + 16q^{29} - 36q^{35} + 40q^{38} + 14q^{40} + 6q^{43} - 34q^{49} + 26q^{53} - 26q^{55} + 14q^{56} - 26q^{61} - 32q^{62} - 8q^{64} - 14q^{68} - 14q^{74} - 30q^{77} - 18q^{79} - 10q^{82} + 14q^{88} + 34q^{92} + 64q^{94} + 6q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.80194i 1.27416i 0.770797 + 0.637081i \(0.219858\pi\)
−0.770797 + 0.637081i \(0.780142\pi\)
\(3\) 0 0
\(4\) −1.24698 −0.623490
\(5\) 1.44504i 0.646242i 0.946358 + 0.323121i \(0.104732\pi\)
−0.946358 + 0.323121i \(0.895268\pi\)
\(6\) 0 0
\(7\) 3.44504i 1.30210i 0.759033 + 0.651052i \(0.225672\pi\)
−0.759033 + 0.651052i \(0.774328\pi\)
\(8\) 1.35690i 0.479735i
\(9\) 0 0
\(10\) −2.60388 −0.823418
\(11\) 5.18598i 1.56363i 0.623509 + 0.781816i \(0.285706\pi\)
−0.623509 + 0.781816i \(0.714294\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −6.20775 −1.65909
\(15\) 0 0
\(16\) −4.93900 −1.23475
\(17\) −0.753020 −0.182634 −0.0913171 0.995822i \(-0.529108\pi\)
−0.0913171 + 0.995822i \(0.529108\pi\)
\(18\) 0 0
\(19\) − 7.96077i − 1.82633i −0.407594 0.913163i \(-0.633632\pi\)
0.407594 0.913163i \(-0.366368\pi\)
\(20\) − 1.80194i − 0.402926i
\(21\) 0 0
\(22\) −9.34481 −1.99232
\(23\) −2.82908 −0.589905 −0.294952 0.955512i \(-0.595304\pi\)
−0.294952 + 0.955512i \(0.595304\pi\)
\(24\) 0 0
\(25\) 2.91185 0.582371
\(26\) 0 0
\(27\) 0 0
\(28\) − 4.29590i − 0.811848i
\(29\) 3.91185 0.726413 0.363207 0.931709i \(-0.381682\pi\)
0.363207 + 0.931709i \(0.381682\pi\)
\(30\) 0 0
\(31\) 4.89977i 0.880025i 0.897992 + 0.440013i \(0.145026\pi\)
−0.897992 + 0.440013i \(0.854974\pi\)
\(32\) − 6.18598i − 1.09354i
\(33\) 0 0
\(34\) − 1.35690i − 0.232706i
\(35\) −4.97823 −0.841474
\(36\) 0 0
\(37\) 6.24698i 1.02700i 0.858090 + 0.513499i \(0.171651\pi\)
−0.858090 + 0.513499i \(0.828349\pi\)
\(38\) 14.3448 2.32704
\(39\) 0 0
\(40\) −1.96077 −0.310025
\(41\) 1.80194i 0.281415i 0.990051 + 0.140708i \(0.0449378\pi\)
−0.990051 + 0.140708i \(0.955062\pi\)
\(42\) 0 0
\(43\) 7.09783 1.08241 0.541205 0.840891i \(-0.317968\pi\)
0.541205 + 0.840891i \(0.317968\pi\)
\(44\) − 6.46681i − 0.974909i
\(45\) 0 0
\(46\) − 5.09783i − 0.751635i
\(47\) − 10.5526i − 1.53925i −0.638496 0.769625i \(-0.720443\pi\)
0.638496 0.769625i \(-0.279557\pi\)
\(48\) 0 0
\(49\) −4.86831 −0.695473
\(50\) 5.24698i 0.742035i
\(51\) 0 0
\(52\) 0 0
\(53\) 3.08815 0.424189 0.212095 0.977249i \(-0.431971\pi\)
0.212095 + 0.977249i \(0.431971\pi\)
\(54\) 0 0
\(55\) −7.49396 −1.01049
\(56\) −4.67456 −0.624665
\(57\) 0 0
\(58\) 7.04892i 0.925568i
\(59\) − 1.87800i − 0.244495i −0.992500 0.122248i \(-0.960990\pi\)
0.992500 0.122248i \(-0.0390102\pi\)
\(60\) 0 0
\(61\) 3.34481 0.428260 0.214130 0.976805i \(-0.431308\pi\)
0.214130 + 0.976805i \(0.431308\pi\)
\(62\) −8.82908 −1.12129
\(63\) 0 0
\(64\) 1.26875 0.158594
\(65\) 0 0
\(66\) 0 0
\(67\) 4.54288i 0.555001i 0.960726 + 0.277500i \(0.0895060\pi\)
−0.960726 + 0.277500i \(0.910494\pi\)
\(68\) 0.939001 0.113871
\(69\) 0 0
\(70\) − 8.97046i − 1.07218i
\(71\) − 9.11960i − 1.08230i −0.840927 0.541149i \(-0.817989\pi\)
0.840927 0.541149i \(-0.182011\pi\)
\(72\) 0 0
\(73\) 2.95108i 0.345398i 0.984975 + 0.172699i \(0.0552488\pi\)
−0.984975 + 0.172699i \(0.944751\pi\)
\(74\) −11.2567 −1.30856
\(75\) 0 0
\(76\) 9.92692i 1.13870i
\(77\) −17.8659 −2.03601
\(78\) 0 0
\(79\) −9.43296 −1.06129 −0.530645 0.847594i \(-0.678050\pi\)
−0.530645 + 0.847594i \(0.678050\pi\)
\(80\) − 7.13706i − 0.797948i
\(81\) 0 0
\(82\) −3.24698 −0.358569
\(83\) − 6.46681i − 0.709825i −0.934900 0.354912i \(-0.884511\pi\)
0.934900 0.354912i \(-0.115489\pi\)
\(84\) 0 0
\(85\) − 1.08815i − 0.118026i
\(86\) 12.7899i 1.37917i
\(87\) 0 0
\(88\) −7.03684 −0.750129
\(89\) − 1.15883i − 0.122836i −0.998112 0.0614181i \(-0.980438\pi\)
0.998112 0.0614181i \(-0.0195623\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3.52781 0.367800
\(93\) 0 0
\(94\) 19.0151 1.96125
\(95\) 11.5036 1.18025
\(96\) 0 0
\(97\) − 8.65817i − 0.879104i −0.898217 0.439552i \(-0.855137\pi\)
0.898217 0.439552i \(-0.144863\pi\)
\(98\) − 8.77240i − 0.886146i
\(99\) 0 0
\(100\) −3.63102 −0.363102
\(101\) −8.47650 −0.843443 −0.421722 0.906725i \(-0.638574\pi\)
−0.421722 + 0.906725i \(0.638574\pi\)
\(102\) 0 0
\(103\) −5.64742 −0.556456 −0.278228 0.960515i \(-0.589747\pi\)
−0.278228 + 0.960515i \(0.589747\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 5.56465i 0.540486i
\(107\) 6.73556 0.651151 0.325576 0.945516i \(-0.394442\pi\)
0.325576 + 0.945516i \(0.394442\pi\)
\(108\) 0 0
\(109\) 2.07606i 0.198851i 0.995045 + 0.0994255i \(0.0317005\pi\)
−0.995045 + 0.0994255i \(0.968300\pi\)
\(110\) − 13.5036i − 1.28752i
\(111\) 0 0
\(112\) − 17.0151i − 1.60777i
\(113\) −6.16852 −0.580286 −0.290143 0.956983i \(-0.593703\pi\)
−0.290143 + 0.956983i \(0.593703\pi\)
\(114\) 0 0
\(115\) − 4.08815i − 0.381222i
\(116\) −4.87800 −0.452911
\(117\) 0 0
\(118\) 3.38404 0.311526
\(119\) − 2.59419i − 0.237809i
\(120\) 0 0
\(121\) −15.8944 −1.44495
\(122\) 6.02715i 0.545672i
\(123\) 0 0
\(124\) − 6.10992i − 0.548687i
\(125\) 11.4330i 1.02260i
\(126\) 0 0
\(127\) 14.2620 1.26555 0.632776 0.774335i \(-0.281915\pi\)
0.632776 + 0.774335i \(0.281915\pi\)
\(128\) − 10.0858i − 0.891463i
\(129\) 0 0
\(130\) 0 0
\(131\) −22.6015 −1.97470 −0.987350 0.158554i \(-0.949317\pi\)
−0.987350 + 0.158554i \(0.949317\pi\)
\(132\) 0 0
\(133\) 27.4252 2.37807
\(134\) −8.18598 −0.707161
\(135\) 0 0
\(136\) − 1.02177i − 0.0876161i
\(137\) 13.6353i 1.16495i 0.812850 + 0.582473i \(0.197915\pi\)
−0.812850 + 0.582473i \(0.802085\pi\)
\(138\) 0 0
\(139\) 17.6015 1.49294 0.746469 0.665420i \(-0.231748\pi\)
0.746469 + 0.665420i \(0.231748\pi\)
\(140\) 6.20775 0.524651
\(141\) 0 0
\(142\) 16.4330 1.37902
\(143\) 0 0
\(144\) 0 0
\(145\) 5.65279i 0.469439i
\(146\) −5.31767 −0.440093
\(147\) 0 0
\(148\) − 7.78986i − 0.640322i
\(149\) 12.7385i 1.04358i 0.853073 + 0.521791i \(0.174736\pi\)
−0.853073 + 0.521791i \(0.825264\pi\)
\(150\) 0 0
\(151\) 15.6407i 1.27282i 0.771350 + 0.636412i \(0.219582\pi\)
−0.771350 + 0.636412i \(0.780418\pi\)
\(152\) 10.8019 0.876153
\(153\) 0 0
\(154\) − 32.1933i − 2.59421i
\(155\) −7.08038 −0.568710
\(156\) 0 0
\(157\) −0.823708 −0.0657391 −0.0328695 0.999460i \(-0.510465\pi\)
−0.0328695 + 0.999460i \(0.510465\pi\)
\(158\) − 16.9976i − 1.35226i
\(159\) 0 0
\(160\) 8.93900 0.706690
\(161\) − 9.74632i − 0.768117i
\(162\) 0 0
\(163\) 6.26875i 0.491006i 0.969396 + 0.245503i \(0.0789532\pi\)
−0.969396 + 0.245503i \(0.921047\pi\)
\(164\) − 2.24698i − 0.175460i
\(165\) 0 0
\(166\) 11.6528 0.904432
\(167\) 7.45042i 0.576531i 0.957551 + 0.288265i \(0.0930785\pi\)
−0.957551 + 0.288265i \(0.906921\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 1.96077 0.150384
\(171\) 0 0
\(172\) −8.85086 −0.674871
\(173\) −2.00969 −0.152794 −0.0763969 0.997077i \(-0.524342\pi\)
−0.0763969 + 0.997077i \(0.524342\pi\)
\(174\) 0 0
\(175\) 10.0315i 0.758307i
\(176\) − 25.6136i − 1.93070i
\(177\) 0 0
\(178\) 2.08815 0.156513
\(179\) 20.0368 1.49762 0.748812 0.662783i \(-0.230625\pi\)
0.748812 + 0.662783i \(0.230625\pi\)
\(180\) 0 0
\(181\) −24.1226 −1.79302 −0.896509 0.443026i \(-0.853905\pi\)
−0.896509 + 0.443026i \(0.853905\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) − 3.83877i − 0.282998i
\(185\) −9.02715 −0.663689
\(186\) 0 0
\(187\) − 3.90515i − 0.285573i
\(188\) 13.1588i 0.959707i
\(189\) 0 0
\(190\) 20.7289i 1.50383i
\(191\) 7.08038 0.512318 0.256159 0.966635i \(-0.417543\pi\)
0.256159 + 0.966635i \(0.417543\pi\)
\(192\) 0 0
\(193\) 9.76809i 0.703122i 0.936165 + 0.351561i \(0.114349\pi\)
−0.936165 + 0.351561i \(0.885651\pi\)
\(194\) 15.6015 1.12012
\(195\) 0 0
\(196\) 6.07069 0.433621
\(197\) 23.4112i 1.66798i 0.551781 + 0.833989i \(0.313948\pi\)
−0.551781 + 0.833989i \(0.686052\pi\)
\(198\) 0 0
\(199\) −4.02475 −0.285307 −0.142654 0.989773i \(-0.545563\pi\)
−0.142654 + 0.989773i \(0.545563\pi\)
\(200\) 3.95108i 0.279384i
\(201\) 0 0
\(202\) − 15.2741i − 1.07468i
\(203\) 13.4765i 0.945865i
\(204\) 0 0
\(205\) −2.60388 −0.181863
\(206\) − 10.1763i − 0.709016i
\(207\) 0 0
\(208\) 0 0
\(209\) 41.2844 2.85570
\(210\) 0 0
\(211\) −3.91185 −0.269303 −0.134652 0.990893i \(-0.542992\pi\)
−0.134652 + 0.990893i \(0.542992\pi\)
\(212\) −3.85086 −0.264478
\(213\) 0 0
\(214\) 12.1371i 0.829673i
\(215\) 10.2567i 0.699499i
\(216\) 0 0
\(217\) −16.8799 −1.14588
\(218\) −3.74094 −0.253368
\(219\) 0 0
\(220\) 9.34481 0.630027
\(221\) 0 0
\(222\) 0 0
\(223\) − 7.44935i − 0.498846i −0.968395 0.249423i \(-0.919759\pi\)
0.968395 0.249423i \(-0.0802409\pi\)
\(224\) 21.3110 1.42390
\(225\) 0 0
\(226\) − 11.1153i − 0.739378i
\(227\) − 21.2500i − 1.41041i −0.709004 0.705205i \(-0.750855\pi\)
0.709004 0.705205i \(-0.249145\pi\)
\(228\) 0 0
\(229\) 9.29590i 0.614290i 0.951663 + 0.307145i \(0.0993737\pi\)
−0.951663 + 0.307145i \(0.900626\pi\)
\(230\) 7.36658 0.485738
\(231\) 0 0
\(232\) 5.30798i 0.348486i
\(233\) 16.2107 1.06200 0.531000 0.847372i \(-0.321816\pi\)
0.531000 + 0.847372i \(0.321816\pi\)
\(234\) 0 0
\(235\) 15.2489 0.994728
\(236\) 2.34183i 0.152440i
\(237\) 0 0
\(238\) 4.67456 0.303007
\(239\) − 13.5090i − 0.873826i −0.899504 0.436913i \(-0.856072\pi\)
0.899504 0.436913i \(-0.143928\pi\)
\(240\) 0 0
\(241\) 6.26875i 0.403806i 0.979406 + 0.201903i \(0.0647125\pi\)
−0.979406 + 0.201903i \(0.935287\pi\)
\(242\) − 28.6407i − 1.84109i
\(243\) 0 0
\(244\) −4.17092 −0.267015
\(245\) − 7.03492i − 0.449444i
\(246\) 0 0
\(247\) 0 0
\(248\) −6.64848 −0.422179
\(249\) 0 0
\(250\) −20.6015 −1.30295
\(251\) −0.753020 −0.0475302 −0.0237651 0.999718i \(-0.507565\pi\)
−0.0237651 + 0.999718i \(0.507565\pi\)
\(252\) 0 0
\(253\) − 14.6716i − 0.922394i
\(254\) 25.6993i 1.61252i
\(255\) 0 0
\(256\) 20.7114 1.29446
\(257\) 19.7265 1.23050 0.615252 0.788331i \(-0.289054\pi\)
0.615252 + 0.788331i \(0.289054\pi\)
\(258\) 0 0
\(259\) −21.5211 −1.33726
\(260\) 0 0
\(261\) 0 0
\(262\) − 40.7265i − 2.51609i
\(263\) 17.6093 1.08583 0.542917 0.839787i \(-0.317320\pi\)
0.542917 + 0.839787i \(0.317320\pi\)
\(264\) 0 0
\(265\) 4.46250i 0.274129i
\(266\) 49.4185i 3.03004i
\(267\) 0 0
\(268\) − 5.66487i − 0.346037i
\(269\) 16.3870 0.999135 0.499567 0.866275i \(-0.333492\pi\)
0.499567 + 0.866275i \(0.333492\pi\)
\(270\) 0 0
\(271\) − 0.795233i − 0.0483070i −0.999708 0.0241535i \(-0.992311\pi\)
0.999708 0.0241535i \(-0.00768904\pi\)
\(272\) 3.71917 0.225508
\(273\) 0 0
\(274\) −24.5700 −1.48433
\(275\) 15.1008i 0.910614i
\(276\) 0 0
\(277\) 4.83340 0.290411 0.145205 0.989402i \(-0.453616\pi\)
0.145205 + 0.989402i \(0.453616\pi\)
\(278\) 31.7168i 1.90225i
\(279\) 0 0
\(280\) − 6.75494i − 0.403685i
\(281\) − 18.7748i − 1.12001i −0.828489 0.560005i \(-0.810799\pi\)
0.828489 0.560005i \(-0.189201\pi\)
\(282\) 0 0
\(283\) 7.91723 0.470631 0.235315 0.971919i \(-0.424388\pi\)
0.235315 + 0.971919i \(0.424388\pi\)
\(284\) 11.3720i 0.674802i
\(285\) 0 0
\(286\) 0 0
\(287\) −6.20775 −0.366432
\(288\) 0 0
\(289\) −16.4330 −0.966645
\(290\) −10.1860 −0.598141
\(291\) 0 0
\(292\) − 3.67994i − 0.215352i
\(293\) − 6.57912i − 0.384356i −0.981360 0.192178i \(-0.938445\pi\)
0.981360 0.192178i \(-0.0615552\pi\)
\(294\) 0 0
\(295\) 2.71379 0.158003
\(296\) −8.47650 −0.492687
\(297\) 0 0
\(298\) −22.9541 −1.32969
\(299\) 0 0
\(300\) 0 0
\(301\) 24.4523i 1.40941i
\(302\) −28.1836 −1.62178
\(303\) 0 0
\(304\) 39.3183i 2.25506i
\(305\) 4.83340i 0.276759i
\(306\) 0 0
\(307\) − 24.8649i − 1.41911i −0.704649 0.709556i \(-0.748895\pi\)
0.704649 0.709556i \(-0.251105\pi\)
\(308\) 22.2784 1.26943
\(309\) 0 0
\(310\) − 12.7584i − 0.724628i
\(311\) −17.0804 −0.968539 −0.484270 0.874919i \(-0.660915\pi\)
−0.484270 + 0.874919i \(0.660915\pi\)
\(312\) 0 0
\(313\) 15.6974 0.887269 0.443635 0.896208i \(-0.353689\pi\)
0.443635 + 0.896208i \(0.353689\pi\)
\(314\) − 1.48427i − 0.0837622i
\(315\) 0 0
\(316\) 11.7627 0.661704
\(317\) 32.7821i 1.84123i 0.390477 + 0.920613i \(0.372310\pi\)
−0.390477 + 0.920613i \(0.627690\pi\)
\(318\) 0 0
\(319\) 20.2868i 1.13584i
\(320\) 1.83340i 0.102490i
\(321\) 0 0
\(322\) 17.5623 0.978706
\(323\) 5.99462i 0.333550i
\(324\) 0 0
\(325\) 0 0
\(326\) −11.2959 −0.625622
\(327\) 0 0
\(328\) −2.44504 −0.135005
\(329\) 36.3540 2.00426
\(330\) 0 0
\(331\) 29.1618i 1.60288i 0.598076 + 0.801439i \(0.295932\pi\)
−0.598076 + 0.801439i \(0.704068\pi\)
\(332\) 8.06398i 0.442569i
\(333\) 0 0
\(334\) −13.4252 −0.734594
\(335\) −6.56465 −0.358665
\(336\) 0 0
\(337\) 33.2911 1.81348 0.906741 0.421688i \(-0.138562\pi\)
0.906741 + 0.421688i \(0.138562\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 1.35690i 0.0735880i
\(341\) −25.4101 −1.37604
\(342\) 0 0
\(343\) 7.34375i 0.396525i
\(344\) 9.63102i 0.519270i
\(345\) 0 0
\(346\) − 3.62133i − 0.194684i
\(347\) 0.873690 0.0469022 0.0234511 0.999725i \(-0.492535\pi\)
0.0234511 + 0.999725i \(0.492535\pi\)
\(348\) 0 0
\(349\) − 3.23191i − 0.173000i −0.996252 0.0865002i \(-0.972432\pi\)
0.996252 0.0865002i \(-0.0275683\pi\)
\(350\) −18.0761 −0.966206
\(351\) 0 0
\(352\) 32.0804 1.70989
\(353\) 8.14675i 0.433608i 0.976215 + 0.216804i \(0.0695632\pi\)
−0.976215 + 0.216804i \(0.930437\pi\)
\(354\) 0 0
\(355\) 13.1782 0.699427
\(356\) 1.44504i 0.0765871i
\(357\) 0 0
\(358\) 36.1051i 1.90822i
\(359\) − 2.64071i − 0.139371i −0.997569 0.0696857i \(-0.977800\pi\)
0.997569 0.0696857i \(-0.0221996\pi\)
\(360\) 0 0
\(361\) −44.3739 −2.33547
\(362\) − 43.4674i − 2.28460i
\(363\) 0 0
\(364\) 0 0
\(365\) −4.26444 −0.223211
\(366\) 0 0
\(367\) −2.90408 −0.151592 −0.0757960 0.997123i \(-0.524150\pi\)
−0.0757960 + 0.997123i \(0.524150\pi\)
\(368\) 13.9729 0.728385
\(369\) 0 0
\(370\) − 16.2664i − 0.845648i
\(371\) 10.6388i 0.552339i
\(372\) 0 0
\(373\) −8.39852 −0.434859 −0.217429 0.976076i \(-0.569767\pi\)
−0.217429 + 0.976076i \(0.569767\pi\)
\(374\) 7.03684 0.363866
\(375\) 0 0
\(376\) 14.3187 0.738432
\(377\) 0 0
\(378\) 0 0
\(379\) − 15.7482i − 0.808932i −0.914553 0.404466i \(-0.867457\pi\)
0.914553 0.404466i \(-0.132543\pi\)
\(380\) −14.3448 −0.735873
\(381\) 0 0
\(382\) 12.7584i 0.652776i
\(383\) 12.7385i 0.650909i 0.945558 + 0.325455i \(0.105517\pi\)
−0.945558 + 0.325455i \(0.894483\pi\)
\(384\) 0 0
\(385\) − 25.8170i − 1.31576i
\(386\) −17.6015 −0.895892
\(387\) 0 0
\(388\) 10.7966i 0.548112i
\(389\) −0.310371 −0.0157365 −0.00786823 0.999969i \(-0.502505\pi\)
−0.00786823 + 0.999969i \(0.502505\pi\)
\(390\) 0 0
\(391\) 2.13036 0.107737
\(392\) − 6.60579i − 0.333643i
\(393\) 0 0
\(394\) −42.1855 −2.12528
\(395\) − 13.6310i − 0.685851i
\(396\) 0 0
\(397\) 1.49098i 0.0748299i 0.999300 + 0.0374150i \(0.0119123\pi\)
−0.999300 + 0.0374150i \(0.988088\pi\)
\(398\) − 7.25236i − 0.363528i
\(399\) 0 0
\(400\) −14.3817 −0.719083
\(401\) 23.8334i 1.19018i 0.803658 + 0.595092i \(0.202884\pi\)
−0.803658 + 0.595092i \(0.797116\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 10.5700 0.525878
\(405\) 0 0
\(406\) −24.2838 −1.20519
\(407\) −32.3967 −1.60585
\(408\) 0 0
\(409\) − 4.26742i − 0.211010i −0.994419 0.105505i \(-0.966354\pi\)
0.994419 0.105505i \(-0.0336460\pi\)
\(410\) − 4.69202i − 0.231722i
\(411\) 0 0
\(412\) 7.04221 0.346945
\(413\) 6.46980 0.318358
\(414\) 0 0
\(415\) 9.34481 0.458719
\(416\) 0 0
\(417\) 0 0
\(418\) 74.3919i 3.63863i
\(419\) −29.6896 −1.45043 −0.725217 0.688521i \(-0.758260\pi\)
−0.725217 + 0.688521i \(0.758260\pi\)
\(420\) 0 0
\(421\) − 29.3991i − 1.43282i −0.697677 0.716412i \(-0.745783\pi\)
0.697677 0.716412i \(-0.254217\pi\)
\(422\) − 7.04892i − 0.343136i
\(423\) 0 0
\(424\) 4.19029i 0.203499i
\(425\) −2.19269 −0.106361
\(426\) 0 0
\(427\) 11.5230i 0.557638i
\(428\) −8.39911 −0.405986
\(429\) 0 0
\(430\) −18.4819 −0.891275
\(431\) − 33.0562i − 1.59226i −0.605124 0.796131i \(-0.706877\pi\)
0.605124 0.796131i \(-0.293123\pi\)
\(432\) 0 0
\(433\) 29.2664 1.40645 0.703226 0.710967i \(-0.251742\pi\)
0.703226 + 0.710967i \(0.251742\pi\)
\(434\) − 30.4166i − 1.46004i
\(435\) 0 0
\(436\) − 2.58881i − 0.123982i
\(437\) 22.5217i 1.07736i
\(438\) 0 0
\(439\) 2.13169 0.101740 0.0508699 0.998705i \(-0.483801\pi\)
0.0508699 + 0.998705i \(0.483801\pi\)
\(440\) − 10.1685i − 0.484765i
\(441\) 0 0
\(442\) 0 0
\(443\) 22.9922 1.09239 0.546197 0.837657i \(-0.316075\pi\)
0.546197 + 0.837657i \(0.316075\pi\)
\(444\) 0 0
\(445\) 1.67456 0.0793819
\(446\) 13.4233 0.635610
\(447\) 0 0
\(448\) 4.37090i 0.206505i
\(449\) 12.9379i 0.610579i 0.952260 + 0.305289i \(0.0987532\pi\)
−0.952260 + 0.305289i \(0.901247\pi\)
\(450\) 0 0
\(451\) −9.34481 −0.440030
\(452\) 7.69202 0.361802
\(453\) 0 0
\(454\) 38.2911 1.79709
\(455\) 0 0
\(456\) 0 0
\(457\) 4.85325i 0.227025i 0.993537 + 0.113513i \(0.0362103\pi\)
−0.993537 + 0.113513i \(0.963790\pi\)
\(458\) −16.7506 −0.782705
\(459\) 0 0
\(460\) 5.09783i 0.237688i
\(461\) 18.8345i 0.877208i 0.898680 + 0.438604i \(0.144527\pi\)
−0.898680 + 0.438604i \(0.855473\pi\)
\(462\) 0 0
\(463\) − 22.8767i − 1.06317i −0.847005 0.531585i \(-0.821597\pi\)
0.847005 0.531585i \(-0.178403\pi\)
\(464\) −19.3207 −0.896939
\(465\) 0 0
\(466\) 29.2107i 1.35316i
\(467\) 13.0000 0.601568 0.300784 0.953692i \(-0.402752\pi\)
0.300784 + 0.953692i \(0.402752\pi\)
\(468\) 0 0
\(469\) −15.6504 −0.722668
\(470\) 27.4776i 1.26745i
\(471\) 0 0
\(472\) 2.54825 0.117293
\(473\) 36.8092i 1.69249i
\(474\) 0 0
\(475\) − 23.1806i − 1.06360i
\(476\) 3.23490i 0.148271i
\(477\) 0 0
\(478\) 24.3424 1.11340
\(479\) 38.0901i 1.74038i 0.492717 + 0.870190i \(0.336004\pi\)
−0.492717 + 0.870190i \(0.663996\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −11.2959 −0.514514
\(483\) 0 0
\(484\) 19.8200 0.900909
\(485\) 12.5114 0.568114
\(486\) 0 0
\(487\) 21.2500i 0.962928i 0.876466 + 0.481464i \(0.159895\pi\)
−0.876466 + 0.481464i \(0.840105\pi\)
\(488\) 4.53856i 0.205451i
\(489\) 0 0
\(490\) 12.6765 0.572665
\(491\) −6.35019 −0.286580 −0.143290 0.989681i \(-0.545768\pi\)
−0.143290 + 0.989681i \(0.545768\pi\)
\(492\) 0 0
\(493\) −2.94571 −0.132668
\(494\) 0 0
\(495\) 0 0
\(496\) − 24.2000i − 1.08661i
\(497\) 31.4174 1.40926
\(498\) 0 0
\(499\) 4.65087i 0.208202i 0.994567 + 0.104101i \(0.0331965\pi\)
−0.994567 + 0.104101i \(0.966804\pi\)
\(500\) − 14.2567i − 0.637578i
\(501\) 0 0
\(502\) − 1.35690i − 0.0605612i
\(503\) −15.4752 −0.690004 −0.345002 0.938602i \(-0.612122\pi\)
−0.345002 + 0.938602i \(0.612122\pi\)
\(504\) 0 0
\(505\) − 12.2489i − 0.545069i
\(506\) 26.4373 1.17528
\(507\) 0 0
\(508\) −17.7845 −0.789059
\(509\) − 20.5047i − 0.908855i −0.890784 0.454428i \(-0.849844\pi\)
0.890784 0.454428i \(-0.150156\pi\)
\(510\) 0 0
\(511\) −10.1666 −0.449744
\(512\) 17.1491i 0.757892i
\(513\) 0 0
\(514\) 35.5459i 1.56786i
\(515\) − 8.16075i − 0.359606i
\(516\) 0 0
\(517\) 54.7254 2.40682
\(518\) − 38.7797i − 1.70388i
\(519\) 0 0
\(520\) 0 0
\(521\) 42.0267 1.84122 0.920611 0.390481i \(-0.127691\pi\)
0.920611 + 0.390481i \(0.127691\pi\)
\(522\) 0 0
\(523\) −29.9885 −1.31131 −0.655653 0.755062i \(-0.727607\pi\)
−0.655653 + 0.755062i \(0.727607\pi\)
\(524\) 28.1836 1.23121
\(525\) 0 0
\(526\) 31.7308i 1.38353i
\(527\) − 3.68963i − 0.160723i
\(528\) 0 0
\(529\) −14.9963 −0.652012
\(530\) −8.04115 −0.349285
\(531\) 0 0
\(532\) −34.1987 −1.48270
\(533\) 0 0
\(534\) 0 0
\(535\) 9.73317i 0.420802i
\(536\) −6.16421 −0.266253
\(537\) 0 0
\(538\) 29.5284i 1.27306i
\(539\) − 25.2470i − 1.08746i
\(540\) 0 0
\(541\) 36.3803i 1.56411i 0.623208 + 0.782056i \(0.285829\pi\)
−0.623208 + 0.782056i \(0.714171\pi\)
\(542\) 1.43296 0.0615509
\(543\) 0 0
\(544\) 4.65817i 0.199717i
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) −25.8159 −1.10381 −0.551905 0.833907i \(-0.686099\pi\)
−0.551905 + 0.833907i \(0.686099\pi\)
\(548\) − 17.0030i − 0.726331i
\(549\) 0 0
\(550\) −27.2107 −1.16027
\(551\) − 31.1414i − 1.32667i
\(552\) 0 0
\(553\) − 32.4969i − 1.38191i
\(554\) 8.70948i 0.370030i
\(555\) 0 0
\(556\) −21.9487 −0.930832
\(557\) − 17.9903i − 0.762274i −0.924519 0.381137i \(-0.875533\pi\)
0.924519 0.381137i \(-0.124467\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 24.5875 1.03901
\(561\) 0 0
\(562\) 33.8310 1.42707
\(563\) −39.1323 −1.64923 −0.824614 0.565695i \(-0.808608\pi\)
−0.824614 + 0.565695i \(0.808608\pi\)
\(564\) 0 0
\(565\) − 8.91377i − 0.375005i
\(566\) 14.2664i 0.599660i
\(567\) 0 0
\(568\) 12.3744 0.519216
\(569\) 30.6002 1.28283 0.641413 0.767196i \(-0.278349\pi\)
0.641413 + 0.767196i \(0.278349\pi\)
\(570\) 0 0
\(571\) 2.96184 0.123949 0.0619745 0.998078i \(-0.480260\pi\)
0.0619745 + 0.998078i \(0.480260\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 11.1860i − 0.466894i
\(575\) −8.23788 −0.343543
\(576\) 0 0
\(577\) 0.819396i 0.0341119i 0.999855 + 0.0170560i \(0.00542934\pi\)
−0.999855 + 0.0170560i \(0.994571\pi\)
\(578\) − 29.6112i − 1.23166i
\(579\) 0 0
\(580\) − 7.04892i − 0.292690i
\(581\) 22.2784 0.924265
\(582\) 0 0
\(583\) 16.0151i 0.663276i
\(584\) −4.00431 −0.165700
\(585\) 0 0
\(586\) 11.8552 0.489732
\(587\) − 31.7995i − 1.31251i −0.754540 0.656254i \(-0.772140\pi\)
0.754540 0.656254i \(-0.227860\pi\)
\(588\) 0 0
\(589\) 39.0060 1.60721
\(590\) 4.89008i 0.201322i
\(591\) 0 0
\(592\) − 30.8538i − 1.26808i
\(593\) 4.26337i 0.175076i 0.996161 + 0.0875379i \(0.0278999\pi\)
−0.996161 + 0.0875379i \(0.972100\pi\)
\(594\) 0 0
\(595\) 3.74871 0.153682
\(596\) − 15.8847i − 0.650663i
\(597\) 0 0
\(598\) 0 0
\(599\) −24.7278 −1.01035 −0.505175 0.863017i \(-0.668572\pi\)
−0.505175 + 0.863017i \(0.668572\pi\)
\(600\) 0 0
\(601\) 6.82371 0.278345 0.139172 0.990268i \(-0.455556\pi\)
0.139172 + 0.990268i \(0.455556\pi\)
\(602\) −44.0616 −1.79582
\(603\) 0 0
\(604\) − 19.5036i − 0.793592i
\(605\) − 22.9681i − 0.933785i
\(606\) 0 0
\(607\) 31.9963 1.29869 0.649344 0.760494i \(-0.275043\pi\)
0.649344 + 0.760494i \(0.275043\pi\)
\(608\) −49.2452 −1.99716
\(609\) 0 0
\(610\) −8.70948 −0.352637
\(611\) 0 0
\(612\) 0 0
\(613\) − 33.5875i − 1.35659i −0.734792 0.678293i \(-0.762720\pi\)
0.734792 0.678293i \(-0.237280\pi\)
\(614\) 44.8049 1.80818
\(615\) 0 0
\(616\) − 24.2422i − 0.976746i
\(617\) − 26.5870i − 1.07035i −0.844740 0.535176i \(-0.820245\pi\)
0.844740 0.535176i \(-0.179755\pi\)
\(618\) 0 0
\(619\) − 9.17928i − 0.368946i −0.982838 0.184473i \(-0.940942\pi\)
0.982838 0.184473i \(-0.0590579\pi\)
\(620\) 8.82908 0.354585
\(621\) 0 0
\(622\) − 30.7778i − 1.23408i
\(623\) 3.99223 0.159945
\(624\) 0 0
\(625\) −1.96184 −0.0784735
\(626\) 28.2857i 1.13053i
\(627\) 0 0
\(628\) 1.02715 0.0409876
\(629\) − 4.70410i − 0.187565i
\(630\) 0 0
\(631\) 17.0043i 0.676931i 0.940979 + 0.338465i \(0.109908\pi\)
−0.940979 + 0.338465i \(0.890092\pi\)
\(632\) − 12.7995i − 0.509139i
\(633\) 0 0
\(634\) −59.0713 −2.34602
\(635\) 20.6093i 0.817853i
\(636\) 0 0
\(637\) 0 0
\(638\) −36.5555 −1.44725
\(639\) 0 0
\(640\) 14.5743 0.576101
\(641\) −21.6649 −0.855711 −0.427856 0.903847i \(-0.640731\pi\)
−0.427856 + 0.903847i \(0.640731\pi\)
\(642\) 0 0
\(643\) − 9.35557i − 0.368948i −0.982837 0.184474i \(-0.940942\pi\)
0.982837 0.184474i \(-0.0590581\pi\)
\(644\) 12.1535i 0.478913i
\(645\) 0 0
\(646\) −10.8019 −0.424997
\(647\) −0.702775 −0.0276289 −0.0138145 0.999905i \(-0.504397\pi\)
−0.0138145 + 0.999905i \(0.504397\pi\)
\(648\) 0 0
\(649\) 9.73928 0.382300
\(650\) 0 0
\(651\) 0 0
\(652\) − 7.81700i − 0.306137i
\(653\) 37.3411 1.46127 0.730635 0.682768i \(-0.239224\pi\)
0.730635 + 0.682768i \(0.239224\pi\)
\(654\) 0 0
\(655\) − 32.6601i − 1.27614i
\(656\) − 8.89977i − 0.347478i
\(657\) 0 0
\(658\) 65.5077i 2.55376i
\(659\) 0.735562 0.0286534 0.0143267 0.999897i \(-0.495440\pi\)
0.0143267 + 0.999897i \(0.495440\pi\)
\(660\) 0 0
\(661\) 13.8485i 0.538643i 0.963050 + 0.269321i \(0.0867994\pi\)
−0.963050 + 0.269321i \(0.913201\pi\)
\(662\) −52.5478 −2.04233
\(663\) 0 0
\(664\) 8.77479 0.340528
\(665\) 39.6305i 1.53681i
\(666\) 0 0
\(667\) −11.0670 −0.428515
\(668\) − 9.29052i − 0.359461i
\(669\) 0 0
\(670\) − 11.8291i − 0.456997i
\(671\) 17.3461i 0.669640i
\(672\) 0 0
\(673\) 6.35019 0.244782 0.122391 0.992482i \(-0.460944\pi\)
0.122391 + 0.992482i \(0.460944\pi\)
\(674\) 59.9885i 2.31067i
\(675\) 0 0
\(676\) 0 0
\(677\) −33.7241 −1.29612 −0.648061 0.761589i \(-0.724420\pi\)
−0.648061 + 0.761589i \(0.724420\pi\)
\(678\) 0 0
\(679\) 29.8278 1.14468
\(680\) 1.47650 0.0566212
\(681\) 0 0
\(682\) − 45.7875i − 1.75329i
\(683\) − 19.2687i − 0.737298i −0.929569 0.368649i \(-0.879820\pi\)
0.929569 0.368649i \(-0.120180\pi\)
\(684\) 0 0
\(685\) −19.7036 −0.752837
\(686\) −13.2330 −0.505237
\(687\) 0 0
\(688\) −35.0562 −1.33651
\(689\) 0 0
\(690\) 0 0
\(691\) 39.4010i 1.49889i 0.662069 + 0.749443i \(0.269679\pi\)
−0.662069 + 0.749443i \(0.730321\pi\)
\(692\) 2.50604 0.0952654
\(693\) 0 0
\(694\) 1.57434i 0.0597610i
\(695\) 25.4349i 0.964800i
\(696\) 0 0
\(697\) − 1.35690i − 0.0513961i
\(698\) 5.82371 0.220431
\(699\) 0 0
\(700\) − 12.5090i − 0.472797i
\(701\) 18.3985 0.694902 0.347451 0.937698i \(-0.387047\pi\)
0.347451 + 0.937698i \(0.387047\pi\)
\(702\) 0 0
\(703\) 49.7308 1.87563
\(704\) 6.57971i 0.247982i
\(705\) 0 0
\(706\) −14.6799 −0.552487
\(707\) − 29.2019i − 1.09825i
\(708\) 0 0
\(709\) − 38.4553i − 1.44422i −0.691778 0.722110i \(-0.743172\pi\)
0.691778 0.722110i \(-0.256828\pi\)
\(710\) 23.7463i 0.891183i
\(711\) 0 0
\(712\) 1.57242 0.0589288
\(713\) − 13.8619i − 0.519131i
\(714\) 0 0
\(715\) 0 0
\(716\) −24.9855 −0.933753
\(717\) 0 0
\(718\) 4.75840 0.177582
\(719\) 48.4999 1.80874 0.904371 0.426747i \(-0.140341\pi\)
0.904371 + 0.426747i \(0.140341\pi\)
\(720\) 0 0
\(721\) − 19.4556i − 0.724564i
\(722\) − 79.9590i − 2.97576i
\(723\) 0 0
\(724\) 30.0804 1.11793
\(725\) 11.3907 0.423042
\(726\) 0 0
\(727\) −19.0344 −0.705948 −0.352974 0.935633i \(-0.614830\pi\)
−0.352974 + 0.935633i \(0.614830\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) − 7.68425i − 0.284407i
\(731\) −5.34481 −0.197685
\(732\) 0 0
\(733\) − 24.6213i − 0.909410i −0.890642 0.454705i \(-0.849745\pi\)
0.890642 0.454705i \(-0.150255\pi\)
\(734\) − 5.23298i − 0.193153i
\(735\) 0 0
\(736\) 17.5007i 0.645083i
\(737\) −23.5593 −0.867817
\(738\) 0 0
\(739\) − 44.5115i − 1.63738i −0.574233 0.818692i \(-0.694700\pi\)
0.574233 0.818692i \(-0.305300\pi\)
\(740\) 11.2567 0.413803
\(741\) 0 0
\(742\) −19.1704 −0.703769
\(743\) 10.4112i 0.381950i 0.981595 + 0.190975i \(0.0611649\pi\)
−0.981595 + 0.190975i \(0.938835\pi\)
\(744\) 0 0
\(745\) −18.4077 −0.674407
\(746\) − 15.1336i − 0.554081i
\(747\) 0 0
\(748\) 4.86964i 0.178052i
\(749\) 23.2043i 0.847866i
\(750\) 0 0
\(751\) −1.69979 −0.0620263 −0.0310131 0.999519i \(-0.509873\pi\)
−0.0310131 + 0.999519i \(0.509873\pi\)
\(752\) 52.1191i 1.90059i
\(753\) 0 0
\(754\) 0 0
\(755\) −22.6015 −0.822552
\(756\) 0 0
\(757\) 27.4252 0.996786 0.498393 0.866951i \(-0.333924\pi\)
0.498393 + 0.866951i \(0.333924\pi\)
\(758\) 28.3773 1.03071
\(759\) 0 0
\(760\) 15.6093i 0.566207i
\(761\) 5.02608i 0.182195i 0.995842 + 0.0910977i \(0.0290375\pi\)
−0.995842 + 0.0910977i \(0.970962\pi\)
\(762\) 0 0
\(763\) −7.15213 −0.258924
\(764\) −8.82908 −0.319425
\(765\) 0 0
\(766\) −22.9541 −0.829364
\(767\) 0 0
\(768\) 0 0
\(769\) − 42.4456i − 1.53063i −0.643657 0.765314i \(-0.722584\pi\)
0.643657 0.765314i \(-0.277416\pi\)
\(770\) 46.5206 1.67649
\(771\) 0 0
\(772\) − 12.1806i − 0.438390i
\(773\) 26.3593i 0.948078i 0.880504 + 0.474039i \(0.157204\pi\)
−0.880504 + 0.474039i \(0.842796\pi\)
\(774\) 0 0
\(775\) 14.2674i 0.512501i
\(776\) 11.7482 0.421737
\(777\) 0 0
\(778\) − 0.559270i − 0.0200508i
\(779\) 14.3448 0.513956
\(780\) 0 0
\(781\) 47.2941 1.69232
\(782\) 3.83877i 0.137274i
\(783\) 0 0
\(784\) 24.0446 0.858736
\(785\) − 1.19029i − 0.0424834i
\(786\) 0 0
\(787\) 17.1424i 0.611062i 0.952182 + 0.305531i \(0.0988340\pi\)
−0.952182 + 0.305531i \(0.901166\pi\)
\(788\) − 29.1933i − 1.03997i
\(789\) 0 0
\(790\) 24.5623 0.873886
\(791\) − 21.2508i − 0.755592i
\(792\) 0 0
\(793\) 0 0
\(794\) −2.68664 −0.0953455
\(795\) 0 0
\(796\) 5.01879 0.177886
\(797\) 30.1629 1.06842 0.534212 0.845351i \(-0.320608\pi\)
0.534212 + 0.845351i \(0.320608\pi\)
\(798\) 0 0
\(799\) 7.94630i 0.281120i
\(800\) − 18.0127i − 0.636844i
\(801\) 0 0
\(802\) −42.9463 −1.51649
\(803\) −15.3043 −0.540076
\(804\) 0 0
\(805\) 14.0838 0.496390
\(806\) 0 0
\(807\) 0 0
\(808\) − 11.5017i − 0.404629i
\(809\) 29.8504 1.04948 0.524742 0.851261i \(-0.324162\pi\)
0.524742 + 0.851261i \(0.324162\pi\)
\(810\) 0 0
\(811\) − 47.7362i − 1.67624i −0.545484 0.838122i \(-0.683654\pi\)
0.545484 0.838122i \(-0.316346\pi\)
\(812\) − 16.8049i − 0.589737i
\(813\) 0 0
\(814\) − 58.3769i − 2.04611i
\(815\) −9.05861 −0.317309
\(816\) 0 0
\(817\) − 56.5042i − 1.97683i
\(818\) 7.68963 0.268862
\(819\) 0 0
\(820\) 3.24698 0.113389
\(821\) − 17.9299i − 0.625758i −0.949793 0.312879i \(-0.898707\pi\)
0.949793 0.312879i \(-0.101293\pi\)
\(822\) 0 0
\(823\) −54.3196 −1.89346 −0.946731 0.322026i \(-0.895636\pi\)
−0.946731 + 0.322026i \(0.895636\pi\)
\(824\) − 7.66296i − 0.266952i
\(825\) 0 0
\(826\) 11.6582i 0.405640i
\(827\) − 49.1041i − 1.70752i −0.520670 0.853758i \(-0.674318\pi\)
0.520670 0.853758i \(-0.325682\pi\)
\(828\) 0 0
\(829\) 7.35796 0.255553 0.127776 0.991803i \(-0.459216\pi\)
0.127776 + 0.991803i \(0.459216\pi\)
\(830\) 16.8388i 0.584482i
\(831\) 0 0
\(832\) 0 0
\(833\) 3.66594 0.127017
\(834\) 0 0
\(835\) −10.7662 −0.372579
\(836\) −51.4808 −1.78050
\(837\) 0 0
\(838\) − 53.4989i − 1.84809i
\(839\) 36.5013i 1.26016i 0.776529 + 0.630082i \(0.216979\pi\)
−0.776529 + 0.630082i \(0.783021\pi\)
\(840\) 0 0
\(841\) −13.6974 −0.472324
\(842\) 52.9754 1.82565
\(843\) 0 0
\(844\) 4.87800 0.167908
\(845\) 0 0
\(846\) 0 0
\(847\) − 54.7569i − 1.88147i
\(848\) −15.2524 −0.523768
\(849\) 0 0
\(850\) − 3.95108i − 0.135521i
\(851\) − 17.6732i − 0.605831i
\(852\) 0 0
\(853\) − 9.73855i − 0.333441i −0.986004 0.166721i \(-0.946682\pi\)
0.986004 0.166721i \(-0.0533178\pi\)
\(854\) −20.7638 −0.710522
\(855\) 0 0
\(856\) 9.13946i 0.312380i
\(857\) −15.2030 −0.519323 −0.259662 0.965700i \(-0.583611\pi\)
−0.259662 + 0.965700i \(0.583611\pi\)
\(858\) 0 0
\(859\) −31.9885 −1.09143 −0.545717 0.837970i \(-0.683743\pi\)
−0.545717 + 0.837970i \(0.683743\pi\)
\(860\) − 12.7899i − 0.436130i
\(861\) 0 0
\(862\) 59.5652 2.02880
\(863\) − 35.2905i − 1.20130i −0.799511 0.600652i \(-0.794908\pi\)
0.799511 0.600652i \(-0.205092\pi\)
\(864\) 0 0
\(865\) − 2.90408i − 0.0987418i
\(866\) 52.7362i 1.79205i
\(867\) 0 0
\(868\) 21.0489 0.714447
\(869\) − 48.9191i − 1.65947i
\(870\) 0 0
\(871\) 0 0
\(872\) −2.81700 −0.0953958
\(873\) 0 0
\(874\) −40.5827 −1.37273
\(875\) −39.3870 −1.33152
\(876\) 0 0
\(877\) 15.3263i 0.517532i 0.965940 + 0.258766i \(0.0833159\pi\)
−0.965940 + 0.258766i \(0.916684\pi\)
\(878\) 3.84117i 0.129633i
\(879\) 0 0
\(880\) 37.0127 1.24770
\(881\) −36.4306 −1.22738 −0.613689 0.789548i \(-0.710315\pi\)
−0.613689 + 0.789548i \(0.710315\pi\)
\(882\) 0 0
\(883\) 37.6819 1.26810 0.634048 0.773294i \(-0.281392\pi\)
0.634048 + 0.773294i \(0.281392\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 41.4306i 1.39189i
\(887\) −4.24890 −0.142664 −0.0713320 0.997453i \(-0.522725\pi\)
−0.0713320 + 0.997453i \(0.522725\pi\)
\(888\) 0 0
\(889\) 49.1333i 1.64788i
\(890\) 3.01746i 0.101145i
\(891\) 0 0
\(892\) 9.28919i 0.311025i
\(893\) −84.0066 −2.81117
\(894\) 0 0
\(895\) 28.9541i 0.967828i
\(896\) 34.7458 1.16078
\(897\) 0 0
\(898\) −23.3134 −0.777977
\(899\) 19.1672i 0.639262i
\(900\) 0 0
\(901\) −2.32544 −0.0774715
\(902\) − 16.8388i − 0.560670i
\(903\) 0 0
\(904\) − 8.37004i − 0.278383i
\(905\) − 34.8582i − 1.15872i
\(906\) 0 0
\(907\) 12.6183 0.418985 0.209493 0.977810i \(-0.432819\pi\)
0.209493 + 0.977810i \(0.432819\pi\)
\(908\) 26.4983i 0.879376i
\(909\) 0 0
\(910\) 0 0
\(911\) −6.77777 −0.224558 −0.112279 0.993677i \(-0.535815\pi\)
−0.112279 + 0.993677i \(0.535815\pi\)
\(912\) 0 0
\(913\) 33.5368 1.10990
\(914\) −8.74525 −0.289267
\(915\) 0 0
\(916\) − 11.5918i − 0.383004i
\(917\) − 77.8631i − 2.57126i
\(918\) 0 0
\(919\) −20.4674 −0.675157 −0.337579 0.941297i \(-0.609608\pi\)
−0.337579 + 0.941297i \(0.609608\pi\)
\(920\) 5.54719 0.182885
\(921\) 0 0
\(922\) −33.9385 −1.11771
\(923\) 0 0
\(924\) 0 0
\(925\) 18.1903i 0.598093i
\(926\) 41.2223 1.35465
\(927\) 0 0
\(928\) − 24.1987i − 0.794360i
\(929\) − 4.65220i − 0.152634i −0.997084 0.0763169i \(-0.975684\pi\)
0.997084 0.0763169i \(-0.0243161\pi\)
\(930\) 0 0
\(931\) 38.7555i 1.27016i
\(932\) −20.2145 −0.662147
\(933\) 0 0
\(934\) 23.4252i 0.766496i
\(935\) 5.64310 0.184549
\(936\) 0 0
\(937\) −41.8544 −1.36732 −0.683662 0.729798i \(-0.739614\pi\)
−0.683662 + 0.729798i \(0.739614\pi\)
\(938\) − 28.2010i − 0.920797i
\(939\) 0 0
\(940\) −19.0151 −0.620203
\(941\) 30.3454i 0.989232i 0.869112 + 0.494616i \(0.164691\pi\)
−0.869112 + 0.494616i \(0.835309\pi\)
\(942\) 0 0
\(943\) − 5.09783i − 0.166008i
\(944\) 9.27545i 0.301890i
\(945\) 0 0
\(946\) −66.3279 −2.15651
\(947\) − 12.0325i − 0.391004i −0.980703 0.195502i \(-0.937366\pi\)
0.980703 0.195502i \(-0.0626337\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 41.7700 1.35520
\(951\) 0 0
\(952\) 3.52004 0.114085
\(953\) 22.9825 0.744478 0.372239 0.928137i \(-0.378590\pi\)
0.372239 + 0.928137i \(0.378590\pi\)
\(954\) 0 0
\(955\) 10.2314i 0.331082i
\(956\) 16.8455i 0.544822i
\(957\) 0 0
\(958\) −68.6359 −2.21753
\(959\) −46.9743 −1.51688
\(960\) 0 0
\(961\) 6.99223 0.225556
\(962\) 0 0
\(963\) 0 0
\(964\) − 7.81700i − 0.251769i
\(965\) −14.1153 −0.454387
\(966\) 0 0
\(967\) 38.8883i 1.25056i 0.780399 + 0.625281i \(0.215016\pi\)
−0.780399 + 0.625281i \(0.784984\pi\)
\(968\) − 21.5670i − 0.693191i
\(969\) 0 0
\(970\) 22.5448i 0.723870i
\(971\) 57.5133 1.84569 0.922845 0.385171i \(-0.125857\pi\)
0.922845 + 0.385171i \(0.125857\pi\)
\(972\) 0 0
\(973\) 60.6378i 1.94396i
\(974\) −38.2911 −1.22693
\(975\) 0 0
\(976\) −16.5200 −0.528794
\(977\) 16.3690i 0.523690i 0.965110 + 0.261845i \(0.0843309\pi\)
−0.965110 + 0.261845i \(0.915669\pi\)
\(978\) 0 0
\(979\) 6.00969 0.192070
\(980\) 8.77240i 0.280224i
\(981\) 0 0
\(982\) − 11.4426i − 0.365150i
\(983\) 15.6963i 0.500635i 0.968164 + 0.250318i \(0.0805351\pi\)
−0.968164 + 0.250318i \(0.919465\pi\)
\(984\) 0 0
\(985\) −33.8301 −1.07792
\(986\) − 5.30798i − 0.169040i
\(987\) 0 0
\(988\) 0 0
\(989\) −20.0804 −0.638519
\(990\) 0 0
\(991\) −11.2644 −0.357827 −0.178913 0.983865i \(-0.557258\pi\)
−0.178913 + 0.983865i \(0.557258\pi\)
\(992\) 30.3099 0.962340
\(993\) 0 0
\(994\) 56.6122i 1.79563i
\(995\) − 5.81594i − 0.184378i
\(996\) 0 0
\(997\) −7.70112 −0.243897 −0.121948 0.992536i \(-0.538914\pi\)
−0.121948 + 0.992536i \(0.538914\pi\)
\(998\) −8.38059 −0.265283
\(999\) 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 1521.2.b.m.1351.6 6
3.2 odd 2 507.2.b.g.337.1 6
13.5 odd 4 1521.2.a.q.1.3 3
13.8 odd 4 1521.2.a.p.1.1 3
13.12 even 2 inner 1521.2.b.m.1351.1 6
39.2 even 12 507.2.e.k.22.3 6
39.5 even 4 507.2.a.j.1.1 3
39.8 even 4 507.2.a.k.1.3 yes 3
39.11 even 12 507.2.e.j.22.1 6
39.17 odd 6 507.2.j.h.361.6 12
39.20 even 12 507.2.e.j.484.1 6
39.23 odd 6 507.2.j.h.316.1 12
39.29 odd 6 507.2.j.h.316.6 12
39.32 even 12 507.2.e.k.484.3 6
39.35 odd 6 507.2.j.h.361.1 12
39.38 odd 2 507.2.b.g.337.6 6
156.47 odd 4 8112.2.a.cf.1.2 3
156.83 odd 4 8112.2.a.by.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.1 3 39.5 even 4
507.2.a.k.1.3 yes 3 39.8 even 4
507.2.b.g.337.1 6 3.2 odd 2
507.2.b.g.337.6 6 39.38 odd 2
507.2.e.j.22.1 6 39.11 even 12
507.2.e.j.484.1 6 39.20 even 12
507.2.e.k.22.3 6 39.2 even 12
507.2.e.k.484.3 6 39.32 even 12
507.2.j.h.316.1 12 39.23 odd 6
507.2.j.h.316.6 12 39.29 odd 6
507.2.j.h.361.1 12 39.35 odd 6
507.2.j.h.361.6 12 39.17 odd 6
1521.2.a.p.1.1 3 13.8 odd 4
1521.2.a.q.1.3 3 13.5 odd 4
1521.2.b.m.1351.1 6 13.12 even 2 inner
1521.2.b.m.1351.6 6 1.1 even 1 trivial
8112.2.a.by.1.2 3 156.83 odd 4
8112.2.a.cf.1.2 3 156.47 odd 4