Properties

Label 1521.4.a.o.1.1
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{2} -5.00000 q^{4} -1.73205 q^{5} -13.8564 q^{7} +22.5167 q^{8} +O(q^{10})\) \(q-1.73205 q^{2} -5.00000 q^{4} -1.73205 q^{5} -13.8564 q^{7} +22.5167 q^{8} +3.00000 q^{10} +13.8564 q^{11} +24.0000 q^{14} +1.00000 q^{16} +117.000 q^{17} +114.315 q^{19} +8.66025 q^{20} -24.0000 q^{22} -78.0000 q^{23} -122.000 q^{25} +69.2820 q^{28} +141.000 q^{29} -155.885 q^{31} -181.865 q^{32} -202.650 q^{34} +24.0000 q^{35} +143.760 q^{37} -198.000 q^{38} -39.0000 q^{40} -271.932 q^{41} -104.000 q^{43} -69.2820 q^{44} +135.100 q^{46} -301.377 q^{47} -151.000 q^{49} +211.310 q^{50} -93.0000 q^{53} -24.0000 q^{55} -312.000 q^{56} -244.219 q^{58} +284.056 q^{59} +145.000 q^{61} +270.000 q^{62} +307.000 q^{64} -786.351 q^{67} -585.000 q^{68} -41.5692 q^{70} +1056.55 q^{71} +458.993 q^{73} -249.000 q^{74} -571.577 q^{76} -192.000 q^{77} +1276.00 q^{79} -1.73205 q^{80} +471.000 q^{82} -789.815 q^{83} -202.650 q^{85} +180.133 q^{86} +312.000 q^{88} -976.877 q^{89} +390.000 q^{92} +522.000 q^{94} -198.000 q^{95} -200.918 q^{97} +261.540 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{4} + 6 q^{10} + 48 q^{14} + 2 q^{16} + 234 q^{17} - 48 q^{22} - 156 q^{23} - 244 q^{25} + 282 q^{29} + 48 q^{35} - 396 q^{38} - 78 q^{40} - 208 q^{43} - 302 q^{49} - 186 q^{53} - 48 q^{55} - 624 q^{56} + 290 q^{61} + 540 q^{62} + 614 q^{64} - 1170 q^{68} - 498 q^{74} - 384 q^{77} + 2552 q^{79} + 942 q^{82} + 624 q^{88} + 780 q^{92} + 1044 q^{94} - 396 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −0.612372 −0.306186 0.951972i \(-0.599053\pi\)
−0.306186 + 0.951972i \(0.599053\pi\)
\(3\) 0 0
\(4\) −5.00000 −0.625000
\(5\) −1.73205 −0.154919 −0.0774597 0.996995i \(-0.524681\pi\)
−0.0774597 + 0.996995i \(0.524681\pi\)
\(6\) 0 0
\(7\) −13.8564 −0.748176 −0.374088 0.927393i \(-0.622044\pi\)
−0.374088 + 0.927393i \(0.622044\pi\)
\(8\) 22.5167 0.995105
\(9\) 0 0
\(10\) 3.00000 0.0948683
\(11\) 13.8564 0.379806 0.189903 0.981803i \(-0.439183\pi\)
0.189903 + 0.981803i \(0.439183\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 24.0000 0.458162
\(15\) 0 0
\(16\) 1.00000 0.0156250
\(17\) 117.000 1.66922 0.834608 0.550845i \(-0.185694\pi\)
0.834608 + 0.550845i \(0.185694\pi\)
\(18\) 0 0
\(19\) 114.315 1.38030 0.690151 0.723665i \(-0.257544\pi\)
0.690151 + 0.723665i \(0.257544\pi\)
\(20\) 8.66025 0.0968246
\(21\) 0 0
\(22\) −24.0000 −0.232583
\(23\) −78.0000 −0.707136 −0.353568 0.935409i \(-0.615032\pi\)
−0.353568 + 0.935409i \(0.615032\pi\)
\(24\) 0 0
\(25\) −122.000 −0.976000
\(26\) 0 0
\(27\) 0 0
\(28\) 69.2820 0.467610
\(29\) 141.000 0.902864 0.451432 0.892306i \(-0.350913\pi\)
0.451432 + 0.892306i \(0.350913\pi\)
\(30\) 0 0
\(31\) −155.885 −0.903151 −0.451576 0.892233i \(-0.649138\pi\)
−0.451576 + 0.892233i \(0.649138\pi\)
\(32\) −181.865 −1.00467
\(33\) 0 0
\(34\) −202.650 −1.02218
\(35\) 24.0000 0.115907
\(36\) 0 0
\(37\) 143.760 0.638758 0.319379 0.947627i \(-0.396526\pi\)
0.319379 + 0.947627i \(0.396526\pi\)
\(38\) −198.000 −0.845259
\(39\) 0 0
\(40\) −39.0000 −0.154161
\(41\) −271.932 −1.03582 −0.517910 0.855435i \(-0.673290\pi\)
−0.517910 + 0.855435i \(0.673290\pi\)
\(42\) 0 0
\(43\) −104.000 −0.368834 −0.184417 0.982848i \(-0.559040\pi\)
−0.184417 + 0.982848i \(0.559040\pi\)
\(44\) −69.2820 −0.237379
\(45\) 0 0
\(46\) 135.100 0.433030
\(47\) −301.377 −0.935326 −0.467663 0.883907i \(-0.654904\pi\)
−0.467663 + 0.883907i \(0.654904\pi\)
\(48\) 0 0
\(49\) −151.000 −0.440233
\(50\) 211.310 0.597675
\(51\) 0 0
\(52\) 0 0
\(53\) −93.0000 −0.241029 −0.120514 0.992712i \(-0.538454\pi\)
−0.120514 + 0.992712i \(0.538454\pi\)
\(54\) 0 0
\(55\) −24.0000 −0.0588393
\(56\) −312.000 −0.744513
\(57\) 0 0
\(58\) −244.219 −0.552889
\(59\) 284.056 0.626796 0.313398 0.949622i \(-0.398533\pi\)
0.313398 + 0.949622i \(0.398533\pi\)
\(60\) 0 0
\(61\) 145.000 0.304350 0.152175 0.988354i \(-0.451372\pi\)
0.152175 + 0.988354i \(0.451372\pi\)
\(62\) 270.000 0.553065
\(63\) 0 0
\(64\) 307.000 0.599609
\(65\) 0 0
\(66\) 0 0
\(67\) −786.351 −1.43385 −0.716926 0.697149i \(-0.754451\pi\)
−0.716926 + 0.697149i \(0.754451\pi\)
\(68\) −585.000 −1.04326
\(69\) 0 0
\(70\) −41.5692 −0.0709782
\(71\) 1056.55 1.76605 0.883025 0.469326i \(-0.155503\pi\)
0.883025 + 0.469326i \(0.155503\pi\)
\(72\) 0 0
\(73\) 458.993 0.735906 0.367953 0.929844i \(-0.380059\pi\)
0.367953 + 0.929844i \(0.380059\pi\)
\(74\) −249.000 −0.391158
\(75\) 0 0
\(76\) −571.577 −0.862689
\(77\) −192.000 −0.284161
\(78\) 0 0
\(79\) 1276.00 1.81723 0.908615 0.417634i \(-0.137141\pi\)
0.908615 + 0.417634i \(0.137141\pi\)
\(80\) −1.73205 −0.00242061
\(81\) 0 0
\(82\) 471.000 0.634308
\(83\) −789.815 −1.04450 −0.522250 0.852793i \(-0.674907\pi\)
−0.522250 + 0.852793i \(0.674907\pi\)
\(84\) 0 0
\(85\) −202.650 −0.258594
\(86\) 180.133 0.225864
\(87\) 0 0
\(88\) 312.000 0.377947
\(89\) −976.877 −1.16347 −0.581734 0.813379i \(-0.697626\pi\)
−0.581734 + 0.813379i \(0.697626\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 390.000 0.441960
\(93\) 0 0
\(94\) 522.000 0.572768
\(95\) −198.000 −0.213835
\(96\) 0 0
\(97\) −200.918 −0.210311 −0.105155 0.994456i \(-0.533534\pi\)
−0.105155 + 0.994456i \(0.533534\pi\)
\(98\) 261.540 0.269587
\(99\) 0 0
\(100\) 610.000 0.610000
\(101\) 429.000 0.422645 0.211322 0.977416i \(-0.432223\pi\)
0.211322 + 0.977416i \(0.432223\pi\)
\(102\) 0 0
\(103\) −182.000 −0.174107 −0.0870534 0.996204i \(-0.527745\pi\)
−0.0870534 + 0.996204i \(0.527745\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 161.081 0.147599
\(107\) 1506.00 1.36066 0.680330 0.732906i \(-0.261837\pi\)
0.680330 + 0.732906i \(0.261837\pi\)
\(108\) 0 0
\(109\) −1551.92 −1.36373 −0.681866 0.731477i \(-0.738831\pi\)
−0.681866 + 0.731477i \(0.738831\pi\)
\(110\) 41.5692 0.0360315
\(111\) 0 0
\(112\) −13.8564 −0.0116902
\(113\) 687.000 0.571925 0.285962 0.958241i \(-0.407687\pi\)
0.285962 + 0.958241i \(0.407687\pi\)
\(114\) 0 0
\(115\) 135.100 0.109549
\(116\) −705.000 −0.564290
\(117\) 0 0
\(118\) −492.000 −0.383833
\(119\) −1621.20 −1.24887
\(120\) 0 0
\(121\) −1139.00 −0.855748
\(122\) −251.147 −0.186376
\(123\) 0 0
\(124\) 779.423 0.564470
\(125\) 427.817 0.306121
\(126\) 0 0
\(127\) −286.000 −0.199830 −0.0999149 0.994996i \(-0.531857\pi\)
−0.0999149 + 0.994996i \(0.531857\pi\)
\(128\) 923.183 0.637489
\(129\) 0 0
\(130\) 0 0
\(131\) 1974.00 1.31656 0.658279 0.752774i \(-0.271285\pi\)
0.658279 + 0.752774i \(0.271285\pi\)
\(132\) 0 0
\(133\) −1584.00 −1.03271
\(134\) 1362.00 0.878051
\(135\) 0 0
\(136\) 2634.45 1.66105
\(137\) 846.973 0.528188 0.264094 0.964497i \(-0.414927\pi\)
0.264094 + 0.964497i \(0.414927\pi\)
\(138\) 0 0
\(139\) 236.000 0.144009 0.0720045 0.997404i \(-0.477060\pi\)
0.0720045 + 0.997404i \(0.477060\pi\)
\(140\) −120.000 −0.0724418
\(141\) 0 0
\(142\) −1830.00 −1.08148
\(143\) 0 0
\(144\) 0 0
\(145\) −244.219 −0.139871
\(146\) −795.000 −0.450648
\(147\) 0 0
\(148\) −718.801 −0.399224
\(149\) −46.7654 −0.0257125 −0.0128563 0.999917i \(-0.504092\pi\)
−0.0128563 + 0.999917i \(0.504092\pi\)
\(150\) 0 0
\(151\) −1770.16 −0.953995 −0.476998 0.878905i \(-0.658275\pi\)
−0.476998 + 0.878905i \(0.658275\pi\)
\(152\) 2574.00 1.37355
\(153\) 0 0
\(154\) 332.554 0.174013
\(155\) 270.000 0.139916
\(156\) 0 0
\(157\) 1211.00 0.615594 0.307797 0.951452i \(-0.400408\pi\)
0.307797 + 0.951452i \(0.400408\pi\)
\(158\) −2210.10 −1.11282
\(159\) 0 0
\(160\) 315.000 0.155643
\(161\) 1080.80 0.529062
\(162\) 0 0
\(163\) −1004.59 −0.482733 −0.241367 0.970434i \(-0.577596\pi\)
−0.241367 + 0.970434i \(0.577596\pi\)
\(164\) 1359.66 0.647388
\(165\) 0 0
\(166\) 1368.00 0.639623
\(167\) 914.523 0.423760 0.211880 0.977296i \(-0.432041\pi\)
0.211880 + 0.977296i \(0.432041\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 351.000 0.158356
\(171\) 0 0
\(172\) 520.000 0.230521
\(173\) 2574.00 1.13120 0.565600 0.824680i \(-0.308645\pi\)
0.565600 + 0.824680i \(0.308645\pi\)
\(174\) 0 0
\(175\) 1690.48 0.730219
\(176\) 13.8564 0.00593447
\(177\) 0 0
\(178\) 1692.00 0.712476
\(179\) 3744.00 1.56335 0.781675 0.623686i \(-0.214366\pi\)
0.781675 + 0.623686i \(0.214366\pi\)
\(180\) 0 0
\(181\) 637.000 0.261590 0.130795 0.991409i \(-0.458247\pi\)
0.130795 + 0.991409i \(0.458247\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1756.30 −0.703675
\(185\) −249.000 −0.0989559
\(186\) 0 0
\(187\) 1621.20 0.633978
\(188\) 1506.88 0.584579
\(189\) 0 0
\(190\) 342.946 0.130947
\(191\) 2598.00 0.984213 0.492106 0.870535i \(-0.336227\pi\)
0.492106 + 0.870535i \(0.336227\pi\)
\(192\) 0 0
\(193\) −1117.17 −0.416662 −0.208331 0.978058i \(-0.566803\pi\)
−0.208331 + 0.978058i \(0.566803\pi\)
\(194\) 348.000 0.128788
\(195\) 0 0
\(196\) 755.000 0.275146
\(197\) −2050.75 −0.741674 −0.370837 0.928698i \(-0.620929\pi\)
−0.370837 + 0.928698i \(0.620929\pi\)
\(198\) 0 0
\(199\) 2522.00 0.898391 0.449196 0.893433i \(-0.351711\pi\)
0.449196 + 0.893433i \(0.351711\pi\)
\(200\) −2747.03 −0.971223
\(201\) 0 0
\(202\) −743.050 −0.258816
\(203\) −1953.75 −0.675500
\(204\) 0 0
\(205\) 471.000 0.160469
\(206\) 315.233 0.106618
\(207\) 0 0
\(208\) 0 0
\(209\) 1584.00 0.524247
\(210\) 0 0
\(211\) 1042.00 0.339973 0.169986 0.985446i \(-0.445628\pi\)
0.169986 + 0.985446i \(0.445628\pi\)
\(212\) 465.000 0.150643
\(213\) 0 0
\(214\) −2608.47 −0.833230
\(215\) 180.133 0.0571395
\(216\) 0 0
\(217\) 2160.00 0.675716
\(218\) 2688.00 0.835112
\(219\) 0 0
\(220\) 120.000 0.0367745
\(221\) 0 0
\(222\) 0 0
\(223\) −2407.55 −0.722966 −0.361483 0.932379i \(-0.617730\pi\)
−0.361483 + 0.932379i \(0.617730\pi\)
\(224\) 2520.00 0.751672
\(225\) 0 0
\(226\) −1189.92 −0.350231
\(227\) 2407.55 0.703942 0.351971 0.936011i \(-0.385512\pi\)
0.351971 + 0.936011i \(0.385512\pi\)
\(228\) 0 0
\(229\) 2508.01 0.723729 0.361864 0.932231i \(-0.382140\pi\)
0.361864 + 0.932231i \(0.382140\pi\)
\(230\) −234.000 −0.0670848
\(231\) 0 0
\(232\) 3174.85 0.898444
\(233\) −5850.00 −1.64483 −0.822417 0.568885i \(-0.807375\pi\)
−0.822417 + 0.568885i \(0.807375\pi\)
\(234\) 0 0
\(235\) 522.000 0.144900
\(236\) −1420.28 −0.391748
\(237\) 0 0
\(238\) 2808.00 0.764771
\(239\) −5383.21 −1.45695 −0.728475 0.685072i \(-0.759771\pi\)
−0.728475 + 0.685072i \(0.759771\pi\)
\(240\) 0 0
\(241\) 4917.29 1.31432 0.657159 0.753752i \(-0.271758\pi\)
0.657159 + 0.753752i \(0.271758\pi\)
\(242\) 1972.81 0.524036
\(243\) 0 0
\(244\) −725.000 −0.190219
\(245\) 261.540 0.0682006
\(246\) 0 0
\(247\) 0 0
\(248\) −3510.00 −0.898731
\(249\) 0 0
\(250\) −741.000 −0.187460
\(251\) −3978.00 −1.00036 −0.500178 0.865923i \(-0.666732\pi\)
−0.500178 + 0.865923i \(0.666732\pi\)
\(252\) 0 0
\(253\) −1080.80 −0.268574
\(254\) 495.367 0.122370
\(255\) 0 0
\(256\) −4055.00 −0.989990
\(257\) 2067.00 0.501696 0.250848 0.968026i \(-0.419291\pi\)
0.250848 + 0.968026i \(0.419291\pi\)
\(258\) 0 0
\(259\) −1992.00 −0.477903
\(260\) 0 0
\(261\) 0 0
\(262\) −3419.07 −0.806224
\(263\) 2052.00 0.481109 0.240555 0.970636i \(-0.422671\pi\)
0.240555 + 0.970636i \(0.422671\pi\)
\(264\) 0 0
\(265\) 161.081 0.0373400
\(266\) 2743.57 0.632402
\(267\) 0 0
\(268\) 3931.76 0.896157
\(269\) −3330.00 −0.754772 −0.377386 0.926056i \(-0.623177\pi\)
−0.377386 + 0.926056i \(0.623177\pi\)
\(270\) 0 0
\(271\) −2805.92 −0.628958 −0.314479 0.949264i \(-0.601830\pi\)
−0.314479 + 0.949264i \(0.601830\pi\)
\(272\) 117.000 0.0260815
\(273\) 0 0
\(274\) −1467.00 −0.323448
\(275\) −1690.48 −0.370690
\(276\) 0 0
\(277\) −377.000 −0.0817752 −0.0408876 0.999164i \(-0.513019\pi\)
−0.0408876 + 0.999164i \(0.513019\pi\)
\(278\) −408.764 −0.0881872
\(279\) 0 0
\(280\) 540.400 0.115340
\(281\) 36.3731 0.00772183 0.00386092 0.999993i \(-0.498771\pi\)
0.00386092 + 0.999993i \(0.498771\pi\)
\(282\) 0 0
\(283\) 7124.00 1.49639 0.748194 0.663480i \(-0.230921\pi\)
0.748194 + 0.663480i \(0.230921\pi\)
\(284\) −5282.75 −1.10378
\(285\) 0 0
\(286\) 0 0
\(287\) 3768.00 0.774976
\(288\) 0 0
\(289\) 8776.00 1.78628
\(290\) 423.000 0.0856532
\(291\) 0 0
\(292\) −2294.97 −0.459941
\(293\) 8322.50 1.65941 0.829703 0.558205i \(-0.188510\pi\)
0.829703 + 0.558205i \(0.188510\pi\)
\(294\) 0 0
\(295\) −492.000 −0.0971029
\(296\) 3237.00 0.635631
\(297\) 0 0
\(298\) 81.0000 0.0157457
\(299\) 0 0
\(300\) 0 0
\(301\) 1441.07 0.275952
\(302\) 3066.00 0.584200
\(303\) 0 0
\(304\) 114.315 0.0215672
\(305\) −251.147 −0.0471497
\(306\) 0 0
\(307\) −2220.49 −0.412801 −0.206401 0.978468i \(-0.566175\pi\)
−0.206401 + 0.978468i \(0.566175\pi\)
\(308\) 960.000 0.177601
\(309\) 0 0
\(310\) −467.654 −0.0856805
\(311\) 4914.00 0.895972 0.447986 0.894041i \(-0.352141\pi\)
0.447986 + 0.894041i \(0.352141\pi\)
\(312\) 0 0
\(313\) −518.000 −0.0935434 −0.0467717 0.998906i \(-0.514893\pi\)
−0.0467717 + 0.998906i \(0.514893\pi\)
\(314\) −2097.51 −0.376973
\(315\) 0 0
\(316\) −6380.00 −1.13577
\(317\) 3916.17 0.693861 0.346930 0.937891i \(-0.387224\pi\)
0.346930 + 0.937891i \(0.387224\pi\)
\(318\) 0 0
\(319\) 1953.75 0.342913
\(320\) −531.740 −0.0928911
\(321\) 0 0
\(322\) −1872.00 −0.323983
\(323\) 13374.9 2.30402
\(324\) 0 0
\(325\) 0 0
\(326\) 1740.00 0.295613
\(327\) 0 0
\(328\) −6123.00 −1.03075
\(329\) 4176.00 0.699788
\(330\) 0 0
\(331\) 7454.75 1.23792 0.618958 0.785424i \(-0.287555\pi\)
0.618958 + 0.785424i \(0.287555\pi\)
\(332\) 3949.08 0.652812
\(333\) 0 0
\(334\) −1584.00 −0.259499
\(335\) 1362.00 0.222131
\(336\) 0 0
\(337\) 3575.00 0.577871 0.288936 0.957349i \(-0.406699\pi\)
0.288936 + 0.957349i \(0.406699\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 1013.25 0.161621
\(341\) −2160.00 −0.343022
\(342\) 0 0
\(343\) 6845.06 1.07755
\(344\) −2341.73 −0.367028
\(345\) 0 0
\(346\) −4458.30 −0.692716
\(347\) 6966.00 1.07768 0.538839 0.842409i \(-0.318863\pi\)
0.538839 + 0.842409i \(0.318863\pi\)
\(348\) 0 0
\(349\) 6651.08 1.02013 0.510063 0.860137i \(-0.329622\pi\)
0.510063 + 0.860137i \(0.329622\pi\)
\(350\) −2928.00 −0.447166
\(351\) 0 0
\(352\) −2520.00 −0.381581
\(353\) −5630.90 −0.849015 −0.424508 0.905424i \(-0.639553\pi\)
−0.424508 + 0.905424i \(0.639553\pi\)
\(354\) 0 0
\(355\) −1830.00 −0.273595
\(356\) 4884.38 0.727168
\(357\) 0 0
\(358\) −6484.80 −0.957353
\(359\) 7129.12 1.04808 0.524040 0.851694i \(-0.324424\pi\)
0.524040 + 0.851694i \(0.324424\pi\)
\(360\) 0 0
\(361\) 6209.00 0.905234
\(362\) −1103.32 −0.160191
\(363\) 0 0
\(364\) 0 0
\(365\) −795.000 −0.114006
\(366\) 0 0
\(367\) 2.00000 0.000284466 0 0.000142233 1.00000i \(-0.499955\pi\)
0.000142233 1.00000i \(0.499955\pi\)
\(368\) −78.0000 −0.0110490
\(369\) 0 0
\(370\) 431.281 0.0605979
\(371\) 1288.65 0.180332
\(372\) 0 0
\(373\) 3499.00 0.485714 0.242857 0.970062i \(-0.421915\pi\)
0.242857 + 0.970062i \(0.421915\pi\)
\(374\) −2808.00 −0.388231
\(375\) 0 0
\(376\) −6786.00 −0.930748
\(377\) 0 0
\(378\) 0 0
\(379\) 5518.31 0.747907 0.373953 0.927447i \(-0.378002\pi\)
0.373953 + 0.927447i \(0.378002\pi\)
\(380\) 990.000 0.133647
\(381\) 0 0
\(382\) −4499.87 −0.602705
\(383\) −7364.68 −0.982552 −0.491276 0.871004i \(-0.663469\pi\)
−0.491276 + 0.871004i \(0.663469\pi\)
\(384\) 0 0
\(385\) 332.554 0.0440221
\(386\) 1935.00 0.255153
\(387\) 0 0
\(388\) 1004.59 0.131444
\(389\) −1209.00 −0.157580 −0.0787901 0.996891i \(-0.525106\pi\)
−0.0787901 + 0.996891i \(0.525106\pi\)
\(390\) 0 0
\(391\) −9126.00 −1.18036
\(392\) −3400.02 −0.438078
\(393\) 0 0
\(394\) 3552.00 0.454181
\(395\) −2210.10 −0.281524
\(396\) 0 0
\(397\) 11694.8 1.47845 0.739226 0.673457i \(-0.235191\pi\)
0.739226 + 0.673457i \(0.235191\pi\)
\(398\) −4368.23 −0.550150
\(399\) 0 0
\(400\) −122.000 −0.0152500
\(401\) −2980.86 −0.371215 −0.185607 0.982624i \(-0.559425\pi\)
−0.185607 + 0.982624i \(0.559425\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −2145.00 −0.264153
\(405\) 0 0
\(406\) 3384.00 0.413658
\(407\) 1992.00 0.242604
\(408\) 0 0
\(409\) −43.3013 −0.00523499 −0.00261749 0.999997i \(-0.500833\pi\)
−0.00261749 + 0.999997i \(0.500833\pi\)
\(410\) −815.796 −0.0982666
\(411\) 0 0
\(412\) 910.000 0.108817
\(413\) −3936.00 −0.468954
\(414\) 0 0
\(415\) 1368.00 0.161813
\(416\) 0 0
\(417\) 0 0
\(418\) −2743.57 −0.321034
\(419\) 9462.00 1.10322 0.551610 0.834102i \(-0.314014\pi\)
0.551610 + 0.834102i \(0.314014\pi\)
\(420\) 0 0
\(421\) −7068.50 −0.818284 −0.409142 0.912471i \(-0.634172\pi\)
−0.409142 + 0.912471i \(0.634172\pi\)
\(422\) −1804.80 −0.208190
\(423\) 0 0
\(424\) −2094.05 −0.239849
\(425\) −14274.0 −1.62915
\(426\) 0 0
\(427\) −2009.18 −0.227707
\(428\) −7530.00 −0.850412
\(429\) 0 0
\(430\) −312.000 −0.0349906
\(431\) 9928.12 1.10956 0.554780 0.831997i \(-0.312802\pi\)
0.554780 + 0.831997i \(0.312802\pi\)
\(432\) 0 0
\(433\) 6617.00 0.734394 0.367197 0.930143i \(-0.380317\pi\)
0.367197 + 0.930143i \(0.380317\pi\)
\(434\) −3741.23 −0.413790
\(435\) 0 0
\(436\) 7759.59 0.852332
\(437\) −8916.60 −0.976061
\(438\) 0 0
\(439\) −13988.0 −1.52075 −0.760377 0.649482i \(-0.774986\pi\)
−0.760377 + 0.649482i \(0.774986\pi\)
\(440\) −540.400 −0.0585513
\(441\) 0 0
\(442\) 0 0
\(443\) −2004.00 −0.214928 −0.107464 0.994209i \(-0.534273\pi\)
−0.107464 + 0.994209i \(0.534273\pi\)
\(444\) 0 0
\(445\) 1692.00 0.180244
\(446\) 4170.00 0.442725
\(447\) 0 0
\(448\) −4253.92 −0.448613
\(449\) −9082.87 −0.954671 −0.477336 0.878721i \(-0.658397\pi\)
−0.477336 + 0.878721i \(0.658397\pi\)
\(450\) 0 0
\(451\) −3768.00 −0.393411
\(452\) −3435.00 −0.357453
\(453\) 0 0
\(454\) −4170.00 −0.431074
\(455\) 0 0
\(456\) 0 0
\(457\) 2523.60 0.258313 0.129156 0.991624i \(-0.458773\pi\)
0.129156 + 0.991624i \(0.458773\pi\)
\(458\) −4344.00 −0.443192
\(459\) 0 0
\(460\) −675.500 −0.0684681
\(461\) 19587.8 1.97894 0.989472 0.144725i \(-0.0462299\pi\)
0.989472 + 0.144725i \(0.0462299\pi\)
\(462\) 0 0
\(463\) 8632.54 0.866497 0.433249 0.901274i \(-0.357367\pi\)
0.433249 + 0.901274i \(0.357367\pi\)
\(464\) 141.000 0.0141072
\(465\) 0 0
\(466\) 10132.5 1.00725
\(467\) 5460.00 0.541025 0.270512 0.962716i \(-0.412807\pi\)
0.270512 + 0.962716i \(0.412807\pi\)
\(468\) 0 0
\(469\) 10896.0 1.07277
\(470\) −904.131 −0.0887328
\(471\) 0 0
\(472\) 6396.00 0.623728
\(473\) −1441.07 −0.140085
\(474\) 0 0
\(475\) −13946.5 −1.34717
\(476\) 8106.00 0.780542
\(477\) 0 0
\(478\) 9324.00 0.892196
\(479\) −2553.04 −0.243531 −0.121766 0.992559i \(-0.538856\pi\)
−0.121766 + 0.992559i \(0.538856\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −8517.00 −0.804852
\(483\) 0 0
\(484\) 5695.00 0.534842
\(485\) 348.000 0.0325812
\(486\) 0 0
\(487\) −10828.8 −1.00760 −0.503798 0.863822i \(-0.668064\pi\)
−0.503798 + 0.863822i \(0.668064\pi\)
\(488\) 3264.92 0.302860
\(489\) 0 0
\(490\) −453.000 −0.0417642
\(491\) 11388.0 1.04671 0.523354 0.852116i \(-0.324681\pi\)
0.523354 + 0.852116i \(0.324681\pi\)
\(492\) 0 0
\(493\) 16497.0 1.50707
\(494\) 0 0
\(495\) 0 0
\(496\) −155.885 −0.0141117
\(497\) −14640.0 −1.32132
\(498\) 0 0
\(499\) 17677.3 1.58586 0.792931 0.609311i \(-0.208554\pi\)
0.792931 + 0.609311i \(0.208554\pi\)
\(500\) −2139.08 −0.191325
\(501\) 0 0
\(502\) 6890.10 0.612590
\(503\) −3876.00 −0.343583 −0.171792 0.985133i \(-0.554956\pi\)
−0.171792 + 0.985133i \(0.554956\pi\)
\(504\) 0 0
\(505\) −743.050 −0.0654758
\(506\) 1872.00 0.164467
\(507\) 0 0
\(508\) 1430.00 0.124894
\(509\) 17065.9 1.48612 0.743058 0.669228i \(-0.233375\pi\)
0.743058 + 0.669228i \(0.233375\pi\)
\(510\) 0 0
\(511\) −6360.00 −0.550587
\(512\) −361.999 −0.0312465
\(513\) 0 0
\(514\) −3580.15 −0.307225
\(515\) 315.233 0.0269725
\(516\) 0 0
\(517\) −4176.00 −0.355242
\(518\) 3450.25 0.292655
\(519\) 0 0
\(520\) 0 0
\(521\) −2121.00 −0.178355 −0.0891773 0.996016i \(-0.528424\pi\)
−0.0891773 + 0.996016i \(0.528424\pi\)
\(522\) 0 0
\(523\) −11464.0 −0.958481 −0.479241 0.877684i \(-0.659088\pi\)
−0.479241 + 0.877684i \(0.659088\pi\)
\(524\) −9870.00 −0.822849
\(525\) 0 0
\(526\) −3554.17 −0.294618
\(527\) −18238.5 −1.50755
\(528\) 0 0
\(529\) −6083.00 −0.499959
\(530\) −279.000 −0.0228660
\(531\) 0 0
\(532\) 7920.00 0.645443
\(533\) 0 0
\(534\) 0 0
\(535\) −2608.47 −0.210792
\(536\) −17706.0 −1.42683
\(537\) 0 0
\(538\) 5767.73 0.462202
\(539\) −2092.32 −0.167203
\(540\) 0 0
\(541\) −4764.87 −0.378665 −0.189333 0.981913i \(-0.560632\pi\)
−0.189333 + 0.981913i \(0.560632\pi\)
\(542\) 4860.00 0.385157
\(543\) 0 0
\(544\) −21278.2 −1.67702
\(545\) 2688.00 0.211268
\(546\) 0 0
\(547\) 6554.00 0.512301 0.256151 0.966637i \(-0.417546\pi\)
0.256151 + 0.966637i \(0.417546\pi\)
\(548\) −4234.86 −0.330118
\(549\) 0 0
\(550\) 2928.00 0.227001
\(551\) 16118.5 1.24622
\(552\) 0 0
\(553\) −17680.8 −1.35961
\(554\) 652.983 0.0500769
\(555\) 0 0
\(556\) −1180.00 −0.0900057
\(557\) −18112.1 −1.37780 −0.688898 0.724858i \(-0.741905\pi\)
−0.688898 + 0.724858i \(0.741905\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 24.0000 0.00181104
\(561\) 0 0
\(562\) −63.0000 −0.00472864
\(563\) −12168.0 −0.910870 −0.455435 0.890269i \(-0.650516\pi\)
−0.455435 + 0.890269i \(0.650516\pi\)
\(564\) 0 0
\(565\) −1189.92 −0.0886022
\(566\) −12339.1 −0.916347
\(567\) 0 0
\(568\) 23790.0 1.75741
\(569\) −7722.00 −0.568933 −0.284467 0.958686i \(-0.591817\pi\)
−0.284467 + 0.958686i \(0.591817\pi\)
\(570\) 0 0
\(571\) −11440.0 −0.838440 −0.419220 0.907885i \(-0.637696\pi\)
−0.419220 + 0.907885i \(0.637696\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6526.37 −0.474574
\(575\) 9516.00 0.690165
\(576\) 0 0
\(577\) −15444.7 −1.11433 −0.557167 0.830400i \(-0.688112\pi\)
−0.557167 + 0.830400i \(0.688112\pi\)
\(578\) −15200.5 −1.09387
\(579\) 0 0
\(580\) 1221.10 0.0874194
\(581\) 10944.0 0.781469
\(582\) 0 0
\(583\) −1288.65 −0.0915442
\(584\) 10335.0 0.732304
\(585\) 0 0
\(586\) −14415.0 −1.01617
\(587\) 14071.2 0.989403 0.494702 0.869063i \(-0.335277\pi\)
0.494702 + 0.869063i \(0.335277\pi\)
\(588\) 0 0
\(589\) −17820.0 −1.24662
\(590\) 852.169 0.0594631
\(591\) 0 0
\(592\) 143.760 0.00998059
\(593\) −26938.6 −1.86549 −0.932745 0.360538i \(-0.882593\pi\)
−0.932745 + 0.360538i \(0.882593\pi\)
\(594\) 0 0
\(595\) 2808.00 0.193474
\(596\) 233.827 0.0160703
\(597\) 0 0
\(598\) 0 0
\(599\) 10554.0 0.719908 0.359954 0.932970i \(-0.382792\pi\)
0.359954 + 0.932970i \(0.382792\pi\)
\(600\) 0 0
\(601\) −14831.0 −1.00660 −0.503302 0.864111i \(-0.667882\pi\)
−0.503302 + 0.864111i \(0.667882\pi\)
\(602\) −2496.00 −0.168986
\(603\) 0 0
\(604\) 8850.78 0.596247
\(605\) 1972.81 0.132572
\(606\) 0 0
\(607\) −7954.00 −0.531866 −0.265933 0.963991i \(-0.585680\pi\)
−0.265933 + 0.963991i \(0.585680\pi\)
\(608\) −20790.0 −1.38675
\(609\) 0 0
\(610\) 435.000 0.0288732
\(611\) 0 0
\(612\) 0 0
\(613\) 25220.4 1.66173 0.830866 0.556472i \(-0.187845\pi\)
0.830866 + 0.556472i \(0.187845\pi\)
\(614\) 3846.00 0.252788
\(615\) 0 0
\(616\) −4323.20 −0.282771
\(617\) −17384.6 −1.13432 −0.567162 0.823607i \(-0.691959\pi\)
−0.567162 + 0.823607i \(0.691959\pi\)
\(618\) 0 0
\(619\) −8209.92 −0.533093 −0.266547 0.963822i \(-0.585883\pi\)
−0.266547 + 0.963822i \(0.585883\pi\)
\(620\) −1350.00 −0.0874473
\(621\) 0 0
\(622\) −8511.30 −0.548669
\(623\) 13536.0 0.870479
\(624\) 0 0
\(625\) 14509.0 0.928576
\(626\) 897.202 0.0572834
\(627\) 0 0
\(628\) −6055.00 −0.384747
\(629\) 16819.9 1.06622
\(630\) 0 0
\(631\) 12865.7 0.811687 0.405843 0.913943i \(-0.366978\pi\)
0.405843 + 0.913943i \(0.366978\pi\)
\(632\) 28731.3 1.80834
\(633\) 0 0
\(634\) −6783.00 −0.424901
\(635\) 495.367 0.0309575
\(636\) 0 0
\(637\) 0 0
\(638\) −3384.00 −0.209990
\(639\) 0 0
\(640\) −1599.00 −0.0987594
\(641\) 6201.00 0.382098 0.191049 0.981581i \(-0.438811\pi\)
0.191049 + 0.981581i \(0.438811\pi\)
\(642\) 0 0
\(643\) 16821.7 1.03170 0.515849 0.856679i \(-0.327476\pi\)
0.515849 + 0.856679i \(0.327476\pi\)
\(644\) −5404.00 −0.330664
\(645\) 0 0
\(646\) −23166.0 −1.41092
\(647\) −13494.0 −0.819944 −0.409972 0.912098i \(-0.634462\pi\)
−0.409972 + 0.912098i \(0.634462\pi\)
\(648\) 0 0
\(649\) 3936.00 0.238061
\(650\) 0 0
\(651\) 0 0
\(652\) 5022.95 0.301708
\(653\) 11334.0 0.679225 0.339612 0.940566i \(-0.389704\pi\)
0.339612 + 0.940566i \(0.389704\pi\)
\(654\) 0 0
\(655\) −3419.07 −0.203960
\(656\) −271.932 −0.0161847
\(657\) 0 0
\(658\) −7233.04 −0.428531
\(659\) −13236.0 −0.782400 −0.391200 0.920306i \(-0.627940\pi\)
−0.391200 + 0.920306i \(0.627940\pi\)
\(660\) 0 0
\(661\) 11852.4 0.697437 0.348718 0.937228i \(-0.386617\pi\)
0.348718 + 0.937228i \(0.386617\pi\)
\(662\) −12912.0 −0.758065
\(663\) 0 0
\(664\) −17784.0 −1.03939
\(665\) 2743.57 0.159986
\(666\) 0 0
\(667\) −10998.0 −0.638447
\(668\) −4572.61 −0.264850
\(669\) 0 0
\(670\) −2359.05 −0.136027
\(671\) 2009.18 0.115594
\(672\) 0 0
\(673\) −8021.00 −0.459416 −0.229708 0.973260i \(-0.573777\pi\)
−0.229708 + 0.973260i \(0.573777\pi\)
\(674\) −6192.08 −0.353873
\(675\) 0 0
\(676\) 0 0
\(677\) 21630.0 1.22793 0.613965 0.789333i \(-0.289574\pi\)
0.613965 + 0.789333i \(0.289574\pi\)
\(678\) 0 0
\(679\) 2784.00 0.157349
\(680\) −4563.00 −0.257328
\(681\) 0 0
\(682\) 3741.23 0.210057
\(683\) 26538.5 1.48677 0.743387 0.668861i \(-0.233218\pi\)
0.743387 + 0.668861i \(0.233218\pi\)
\(684\) 0 0
\(685\) −1467.00 −0.0818266
\(686\) −11856.0 −0.659860
\(687\) 0 0
\(688\) −104.000 −0.00576303
\(689\) 0 0
\(690\) 0 0
\(691\) −831.384 −0.0457704 −0.0228852 0.999738i \(-0.507285\pi\)
−0.0228852 + 0.999738i \(0.507285\pi\)
\(692\) −12870.0 −0.707000
\(693\) 0 0
\(694\) −12065.5 −0.659941
\(695\) −408.764 −0.0223098
\(696\) 0 0
\(697\) −31816.0 −1.72901
\(698\) −11520.0 −0.624697
\(699\) 0 0
\(700\) −8452.41 −0.456387
\(701\) 30186.0 1.62640 0.813202 0.581981i \(-0.197722\pi\)
0.813202 + 0.581981i \(0.197722\pi\)
\(702\) 0 0
\(703\) 16434.0 0.881679
\(704\) 4253.92 0.227735
\(705\) 0 0
\(706\) 9753.00 0.519914
\(707\) −5944.40 −0.316212
\(708\) 0 0
\(709\) −11880.1 −0.629292 −0.314646 0.949209i \(-0.601886\pi\)
−0.314646 + 0.949209i \(0.601886\pi\)
\(710\) 3169.65 0.167542
\(711\) 0 0
\(712\) −21996.0 −1.15777
\(713\) 12159.0 0.638651
\(714\) 0 0
\(715\) 0 0
\(716\) −18720.0 −0.977094
\(717\) 0 0
\(718\) −12348.0 −0.641815
\(719\) 18408.0 0.954802 0.477401 0.878686i \(-0.341579\pi\)
0.477401 + 0.878686i \(0.341579\pi\)
\(720\) 0 0
\(721\) 2521.87 0.130262
\(722\) −10754.3 −0.554340
\(723\) 0 0
\(724\) −3185.00 −0.163494
\(725\) −17202.0 −0.881195
\(726\) 0 0
\(727\) −21112.0 −1.07703 −0.538515 0.842616i \(-0.681014\pi\)
−0.538515 + 0.842616i \(0.681014\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1376.98 0.0698142
\(731\) −12168.0 −0.615663
\(732\) 0 0
\(733\) 23959.5 1.20732 0.603658 0.797243i \(-0.293709\pi\)
0.603658 + 0.797243i \(0.293709\pi\)
\(734\) −3.46410 −0.000174199 0
\(735\) 0 0
\(736\) 14185.5 0.710441
\(737\) −10896.0 −0.544585
\(738\) 0 0
\(739\) −3166.19 −0.157605 −0.0788025 0.996890i \(-0.525110\pi\)
−0.0788025 + 0.996890i \(0.525110\pi\)
\(740\) 1245.00 0.0618474
\(741\) 0 0
\(742\) −2232.00 −0.110430
\(743\) 30103.0 1.48637 0.743185 0.669086i \(-0.233314\pi\)
0.743185 + 0.669086i \(0.233314\pi\)
\(744\) 0 0
\(745\) 81.0000 0.00398337
\(746\) −6060.45 −0.297438
\(747\) 0 0
\(748\) −8106.00 −0.396236
\(749\) −20867.7 −1.01801
\(750\) 0 0
\(751\) −28496.0 −1.38460 −0.692299 0.721610i \(-0.743402\pi\)
−0.692299 + 0.721610i \(0.743402\pi\)
\(752\) −301.377 −0.0146145
\(753\) 0 0
\(754\) 0 0
\(755\) 3066.00 0.147792
\(756\) 0 0
\(757\) 17422.0 0.836477 0.418239 0.908337i \(-0.362648\pi\)
0.418239 + 0.908337i \(0.362648\pi\)
\(758\) −9558.00 −0.457998
\(759\) 0 0
\(760\) −4458.30 −0.212789
\(761\) −41326.7 −1.96858 −0.984292 0.176547i \(-0.943507\pi\)
−0.984292 + 0.176547i \(0.943507\pi\)
\(762\) 0 0
\(763\) 21504.0 1.02031
\(764\) −12990.0 −0.615133
\(765\) 0 0
\(766\) 12756.0 0.601688
\(767\) 0 0
\(768\) 0 0
\(769\) −14071.2 −0.659844 −0.329922 0.944008i \(-0.607022\pi\)
−0.329922 + 0.944008i \(0.607022\pi\)
\(770\) −576.000 −0.0269579
\(771\) 0 0
\(772\) 5585.86 0.260414
\(773\) 200.918 0.00934866 0.00467433 0.999989i \(-0.498512\pi\)
0.00467433 + 0.999989i \(0.498512\pi\)
\(774\) 0 0
\(775\) 19017.9 0.881476
\(776\) −4524.00 −0.209281
\(777\) 0 0
\(778\) 2094.05 0.0964978
\(779\) −31086.0 −1.42975
\(780\) 0 0
\(781\) 14640.0 0.670756
\(782\) 15806.7 0.722821
\(783\) 0 0
\(784\) −151.000 −0.00687864
\(785\) −2097.51 −0.0953675
\(786\) 0 0
\(787\) −6903.95 −0.312706 −0.156353 0.987701i \(-0.549974\pi\)
−0.156353 + 0.987701i \(0.549974\pi\)
\(788\) 10253.7 0.463546
\(789\) 0 0
\(790\) 3828.00 0.172398
\(791\) −9519.35 −0.427900
\(792\) 0 0
\(793\) 0 0
\(794\) −20256.0 −0.905363
\(795\) 0 0
\(796\) −12610.0 −0.561494
\(797\) −31278.0 −1.39012 −0.695059 0.718953i \(-0.744622\pi\)
−0.695059 + 0.718953i \(0.744622\pi\)
\(798\) 0 0
\(799\) −35261.1 −1.56126
\(800\) 22187.6 0.980561
\(801\) 0 0
\(802\) 5163.00 0.227322
\(803\) 6360.00 0.279501
\(804\) 0 0
\(805\) −1872.00 −0.0819619
\(806\) 0 0
\(807\) 0 0
\(808\) 9659.65 0.420576
\(809\) −8049.00 −0.349799 −0.174900 0.984586i \(-0.555960\pi\)
−0.174900 + 0.984586i \(0.555960\pi\)
\(810\) 0 0
\(811\) −14026.1 −0.607305 −0.303653 0.952783i \(-0.598206\pi\)
−0.303653 + 0.952783i \(0.598206\pi\)
\(812\) 9768.77 0.422188
\(813\) 0 0
\(814\) −3450.25 −0.148564
\(815\) 1740.00 0.0747847
\(816\) 0 0
\(817\) −11888.8 −0.509102
\(818\) 75.0000 0.00320576
\(819\) 0 0
\(820\) −2355.00 −0.100293
\(821\) 8036.72 0.341636 0.170818 0.985303i \(-0.445359\pi\)
0.170818 + 0.985303i \(0.445359\pi\)
\(822\) 0 0
\(823\) 40300.0 1.70689 0.853445 0.521184i \(-0.174509\pi\)
0.853445 + 0.521184i \(0.174509\pi\)
\(824\) −4098.03 −0.173255
\(825\) 0 0
\(826\) 6817.35 0.287174
\(827\) −39525.4 −1.66195 −0.830975 0.556310i \(-0.812217\pi\)
−0.830975 + 0.556310i \(0.812217\pi\)
\(828\) 0 0
\(829\) 12311.0 0.515776 0.257888 0.966175i \(-0.416973\pi\)
0.257888 + 0.966175i \(0.416973\pi\)
\(830\) −2369.45 −0.0990899
\(831\) 0 0
\(832\) 0 0
\(833\) −17667.0 −0.734844
\(834\) 0 0
\(835\) −1584.00 −0.0656486
\(836\) −7920.00 −0.327654
\(837\) 0 0
\(838\) −16388.7 −0.675581
\(839\) −21467.0 −0.883343 −0.441671 0.897177i \(-0.645614\pi\)
−0.441671 + 0.897177i \(0.645614\pi\)
\(840\) 0 0
\(841\) −4508.00 −0.184837
\(842\) 12243.0 0.501095
\(843\) 0 0
\(844\) −5210.00 −0.212483
\(845\) 0 0
\(846\) 0 0
\(847\) 15782.4 0.640249
\(848\) −93.0000 −0.00376608
\(849\) 0 0
\(850\) 24723.3 0.997649
\(851\) −11213.3 −0.451688
\(852\) 0 0
\(853\) 774.227 0.0310774 0.0155387 0.999879i \(-0.495054\pi\)
0.0155387 + 0.999879i \(0.495054\pi\)
\(854\) 3480.00 0.139442
\(855\) 0 0
\(856\) 33910.1 1.35400
\(857\) 13923.0 0.554960 0.277480 0.960731i \(-0.410501\pi\)
0.277480 + 0.960731i \(0.410501\pi\)
\(858\) 0 0
\(859\) −22358.0 −0.888062 −0.444031 0.896011i \(-0.646452\pi\)
−0.444031 + 0.896011i \(0.646452\pi\)
\(860\) −900.666 −0.0357122
\(861\) 0 0
\(862\) −17196.0 −0.679464
\(863\) −2230.88 −0.0879955 −0.0439977 0.999032i \(-0.514009\pi\)
−0.0439977 + 0.999032i \(0.514009\pi\)
\(864\) 0 0
\(865\) −4458.30 −0.175245
\(866\) −11461.0 −0.449723
\(867\) 0 0
\(868\) −10800.0 −0.422322
\(869\) 17680.8 0.690195
\(870\) 0 0
\(871\) 0 0
\(872\) −34944.0 −1.35706
\(873\) 0 0
\(874\) 15444.0 0.597713
\(875\) −5928.00 −0.229032
\(876\) 0 0
\(877\) 16754.1 0.645093 0.322547 0.946554i \(-0.395461\pi\)
0.322547 + 0.946554i \(0.395461\pi\)
\(878\) 24227.9 0.931268
\(879\) 0 0
\(880\) −24.0000 −0.000919363 0
\(881\) 17355.0 0.663683 0.331842 0.943335i \(-0.392330\pi\)
0.331842 + 0.943335i \(0.392330\pi\)
\(882\) 0 0
\(883\) 46982.0 1.79057 0.895283 0.445497i \(-0.146973\pi\)
0.895283 + 0.445497i \(0.146973\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 3471.03 0.131616
\(887\) 8916.00 0.337508 0.168754 0.985658i \(-0.446026\pi\)
0.168754 + 0.985658i \(0.446026\pi\)
\(888\) 0 0
\(889\) 3962.93 0.149508
\(890\) −2930.63 −0.110376
\(891\) 0 0
\(892\) 12037.8 0.451854
\(893\) −34452.0 −1.29103
\(894\) 0 0
\(895\) −6484.80 −0.242193
\(896\) −12792.0 −0.476954
\(897\) 0 0
\(898\) 15732.0 0.584614
\(899\) −21979.7 −0.815423
\(900\) 0 0
\(901\) −10881.0 −0.402329
\(902\) 6526.37 0.240914
\(903\) 0 0
\(904\) 15468.9 0.569126
\(905\) −1103.32 −0.0405254
\(906\) 0 0
\(907\) 30836.0 1.12888 0.564439 0.825475i \(-0.309092\pi\)
0.564439 + 0.825475i \(0.309092\pi\)
\(908\) −12037.8 −0.439964
\(909\) 0 0
\(910\) 0 0
\(911\) 27480.0 0.999400 0.499700 0.866199i \(-0.333444\pi\)
0.499700 + 0.866199i \(0.333444\pi\)
\(912\) 0 0
\(913\) −10944.0 −0.396707
\(914\) −4371.00 −0.158184
\(915\) 0 0
\(916\) −12540.0 −0.452331
\(917\) −27352.5 −0.985017
\(918\) 0 0
\(919\) −28442.0 −1.02091 −0.510454 0.859905i \(-0.670523\pi\)
−0.510454 + 0.859905i \(0.670523\pi\)
\(920\) 3042.00 0.109013
\(921\) 0 0
\(922\) −33927.0 −1.21185
\(923\) 0 0
\(924\) 0 0
\(925\) −17538.7 −0.623427
\(926\) −14952.0 −0.530619
\(927\) 0 0
\(928\) −25643.0 −0.907083
\(929\) 6978.43 0.246453 0.123227 0.992379i \(-0.460676\pi\)
0.123227 + 0.992379i \(0.460676\pi\)
\(930\) 0 0
\(931\) −17261.6 −0.607655
\(932\) 29250.0 1.02802
\(933\) 0 0
\(934\) −9457.00 −0.331309
\(935\) −2808.00 −0.0982154
\(936\) 0 0
\(937\) −38465.0 −1.34109 −0.670543 0.741871i \(-0.733939\pi\)
−0.670543 + 0.741871i \(0.733939\pi\)
\(938\) −18872.4 −0.656937
\(939\) 0 0
\(940\) −2610.00 −0.0905626
\(941\) 4884.38 0.169210 0.0846049 0.996415i \(-0.473037\pi\)
0.0846049 + 0.996415i \(0.473037\pi\)
\(942\) 0 0
\(943\) 21210.7 0.732466
\(944\) 284.056 0.00979369
\(945\) 0 0
\(946\) 2496.00 0.0857843
\(947\) 21765.0 0.746849 0.373424 0.927661i \(-0.378184\pi\)
0.373424 + 0.927661i \(0.378184\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 24156.0 0.824973
\(951\) 0 0
\(952\) −36504.0 −1.24275
\(953\) 6474.00 0.220056 0.110028 0.993928i \(-0.464906\pi\)
0.110028 + 0.993928i \(0.464906\pi\)
\(954\) 0 0
\(955\) −4499.87 −0.152474
\(956\) 26916.1 0.910594
\(957\) 0 0
\(958\) 4422.00 0.149132
\(959\) −11736.0 −0.395177
\(960\) 0 0
\(961\) −5491.00 −0.184317
\(962\) 0 0
\(963\) 0 0
\(964\) −24586.5 −0.821449
\(965\) 1935.00 0.0645491
\(966\) 0 0
\(967\) −7541.35 −0.250789 −0.125395 0.992107i \(-0.540020\pi\)
−0.125395 + 0.992107i \(0.540020\pi\)
\(968\) −25646.5 −0.851559
\(969\) 0 0
\(970\) −602.754 −0.0199518
\(971\) −34998.0 −1.15668 −0.578342 0.815795i \(-0.696300\pi\)
−0.578342 + 0.815795i \(0.696300\pi\)
\(972\) 0 0
\(973\) −3270.11 −0.107744
\(974\) 18756.0 0.617024
\(975\) 0 0
\(976\) 145.000 0.00475547
\(977\) 25216.9 0.825753 0.412877 0.910787i \(-0.364524\pi\)
0.412877 + 0.910787i \(0.364524\pi\)
\(978\) 0 0
\(979\) −13536.0 −0.441892
\(980\) −1307.70 −0.0426254
\(981\) 0 0
\(982\) −19724.6 −0.640975
\(983\) 56440.6 1.83131 0.915654 0.401967i \(-0.131673\pi\)
0.915654 + 0.401967i \(0.131673\pi\)
\(984\) 0 0
\(985\) 3552.00 0.114900
\(986\) −28573.6 −0.922891
\(987\) 0 0
\(988\) 0 0
\(989\) 8112.00 0.260816
\(990\) 0 0
\(991\) 59282.0 1.90026 0.950129 0.311859i \(-0.100951\pi\)
0.950129 + 0.311859i \(0.100951\pi\)
\(992\) 28350.0 0.907372
\(993\) 0 0
\(994\) 25357.2 0.809137
\(995\) −4368.23 −0.139178
\(996\) 0 0
\(997\) −37711.0 −1.19791 −0.598957 0.800782i \(-0.704418\pi\)
−0.598957 + 0.800782i \(0.704418\pi\)
\(998\) −30618.0 −0.971138
\(999\) 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 1521.4.a.o.1.1 2
3.2 odd 2 169.4.a.i.1.2 2
13.2 odd 12 117.4.q.a.82.1 2
13.7 odd 12 117.4.q.a.10.1 2
13.12 even 2 inner 1521.4.a.o.1.2 2
39.2 even 12 13.4.e.b.4.1 2
39.5 even 4 169.4.b.d.168.1 2
39.8 even 4 169.4.b.d.168.2 2
39.11 even 12 169.4.e.a.147.1 2
39.17 odd 6 169.4.c.h.146.2 4
39.20 even 12 13.4.e.b.10.1 yes 2
39.23 odd 6 169.4.c.h.22.2 4
39.29 odd 6 169.4.c.h.22.1 4
39.32 even 12 169.4.e.a.23.1 2
39.35 odd 6 169.4.c.h.146.1 4
39.38 odd 2 169.4.a.i.1.1 2
156.59 odd 12 208.4.w.b.49.1 2
156.119 odd 12 208.4.w.b.17.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.e.b.4.1 2 39.2 even 12
13.4.e.b.10.1 yes 2 39.20 even 12
117.4.q.a.10.1 2 13.7 odd 12
117.4.q.a.82.1 2 13.2 odd 12
169.4.a.i.1.1 2 39.38 odd 2
169.4.a.i.1.2 2 3.2 odd 2
169.4.b.d.168.1 2 39.5 even 4
169.4.b.d.168.2 2 39.8 even 4
169.4.c.h.22.1 4 39.29 odd 6
169.4.c.h.22.2 4 39.23 odd 6
169.4.c.h.146.1 4 39.35 odd 6
169.4.c.h.146.2 4 39.17 odd 6
169.4.e.a.23.1 2 39.32 even 12
169.4.e.a.147.1 2 39.11 even 12
208.4.w.b.17.1 2 156.119 odd 12
208.4.w.b.49.1 2 156.59 odd 12
1521.4.a.o.1.1 2 1.1 even 1 trivial
1521.4.a.o.1.2 2 13.12 even 2 inner