Properties

Label 1521.4.a.bi.1.4
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.588238\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.213700 q^{2} -7.95433 q^{4} -15.3391 q^{5} -32.3928 q^{7} +3.40944 q^{8} +O(q^{10})\) \(q-0.213700 q^{2} -7.95433 q^{4} -15.3391 q^{5} -32.3928 q^{7} +3.40944 q^{8} +3.27797 q^{10} -29.5925 q^{11} +6.92233 q^{14} +62.9061 q^{16} +78.1958 q^{17} +10.6600 q^{19} +122.013 q^{20} +6.32392 q^{22} +26.8789 q^{23} +110.289 q^{25} +257.663 q^{28} +190.785 q^{29} -128.108 q^{31} -40.7185 q^{32} -16.7104 q^{34} +496.877 q^{35} -379.934 q^{37} -2.27805 q^{38} -52.2979 q^{40} +464.631 q^{41} +322.758 q^{43} +235.389 q^{44} -5.74401 q^{46} +248.529 q^{47} +706.292 q^{49} -23.5688 q^{50} -740.167 q^{53} +453.924 q^{55} -110.441 q^{56} -40.7706 q^{58} +340.673 q^{59} -590.834 q^{61} +27.3767 q^{62} -494.547 q^{64} -340.777 q^{67} -621.995 q^{68} -106.183 q^{70} +36.2243 q^{71} -164.572 q^{73} +81.1919 q^{74} -84.7935 q^{76} +958.584 q^{77} -327.242 q^{79} -964.925 q^{80} -99.2916 q^{82} +1404.48 q^{83} -1199.46 q^{85} -68.9734 q^{86} -100.894 q^{88} -736.986 q^{89} -213.803 q^{92} -53.1107 q^{94} -163.516 q^{95} +1494.87 q^{97} -150.935 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8} - 54 q^{10} + 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} - 352 q^{19} + 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} - 940 q^{28} + 97 q^{29} - 717 q^{31} + 707 q^{32} - 632 q^{34} + 418 q^{35} - 1108 q^{37} + 660 q^{38} - 1506 q^{40} + 334 q^{41} + 242 q^{43} - 307 q^{44} - 979 q^{46} - 184 q^{47} - 38 q^{49} - 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} - 1161 q^{58} + 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} - 2308 q^{67} - 2785 q^{68} + 1420 q^{70} + 96 q^{71} - 2505 q^{73} + 1191 q^{74} - 2409 q^{76} + 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 1517 q^{82} + 1539 q^{83} - 4296 q^{85} - 3763 q^{86} - 3716 q^{88} - 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} - 1445 q^{97} + 1457 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.213700 −0.0755543 −0.0377772 0.999286i \(-0.512028\pi\)
−0.0377772 + 0.999286i \(0.512028\pi\)
\(3\) 0 0
\(4\) −7.95433 −0.994292
\(5\) −15.3391 −1.37197 −0.685987 0.727614i \(-0.740629\pi\)
−0.685987 + 0.727614i \(0.740629\pi\)
\(6\) 0 0
\(7\) −32.3928 −1.74905 −0.874523 0.484984i \(-0.838825\pi\)
−0.874523 + 0.484984i \(0.838825\pi\)
\(8\) 3.40944 0.150677
\(9\) 0 0
\(10\) 3.27797 0.103659
\(11\) −29.5925 −0.811135 −0.405567 0.914065i \(-0.632926\pi\)
−0.405567 + 0.914065i \(0.632926\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 6.92233 0.132148
\(15\) 0 0
\(16\) 62.9061 0.982907
\(17\) 78.1958 1.11560 0.557802 0.829974i \(-0.311645\pi\)
0.557802 + 0.829974i \(0.311645\pi\)
\(18\) 0 0
\(19\) 10.6600 0.128715 0.0643574 0.997927i \(-0.479500\pi\)
0.0643574 + 0.997927i \(0.479500\pi\)
\(20\) 122.013 1.36414
\(21\) 0 0
\(22\) 6.32392 0.0612847
\(23\) 26.8789 0.243680 0.121840 0.992550i \(-0.461121\pi\)
0.121840 + 0.992550i \(0.461121\pi\)
\(24\) 0 0
\(25\) 110.289 0.882314
\(26\) 0 0
\(27\) 0 0
\(28\) 257.663 1.73906
\(29\) 190.785 1.22165 0.610824 0.791766i \(-0.290838\pi\)
0.610824 + 0.791766i \(0.290838\pi\)
\(30\) 0 0
\(31\) −128.108 −0.742222 −0.371111 0.928589i \(-0.621023\pi\)
−0.371111 + 0.928589i \(0.621023\pi\)
\(32\) −40.7185 −0.224940
\(33\) 0 0
\(34\) −16.7104 −0.0842887
\(35\) 496.877 2.39965
\(36\) 0 0
\(37\) −379.934 −1.68813 −0.844065 0.536241i \(-0.819844\pi\)
−0.844065 + 0.536241i \(0.819844\pi\)
\(38\) −2.27805 −0.00972496
\(39\) 0 0
\(40\) −52.2979 −0.206725
\(41\) 464.631 1.76983 0.884917 0.465749i \(-0.154215\pi\)
0.884917 + 0.465749i \(0.154215\pi\)
\(42\) 0 0
\(43\) 322.758 1.14466 0.572328 0.820025i \(-0.306041\pi\)
0.572328 + 0.820025i \(0.306041\pi\)
\(44\) 235.389 0.806504
\(45\) 0 0
\(46\) −5.74401 −0.0184110
\(47\) 248.529 0.771314 0.385657 0.922642i \(-0.373975\pi\)
0.385657 + 0.922642i \(0.373975\pi\)
\(48\) 0 0
\(49\) 706.292 2.05916
\(50\) −23.5688 −0.0666626
\(51\) 0 0
\(52\) 0 0
\(53\) −740.167 −1.91830 −0.959148 0.282904i \(-0.908702\pi\)
−0.959148 + 0.282904i \(0.908702\pi\)
\(54\) 0 0
\(55\) 453.924 1.11286
\(56\) −110.441 −0.263542
\(57\) 0 0
\(58\) −40.7706 −0.0923008
\(59\) 340.673 0.751727 0.375864 0.926675i \(-0.377346\pi\)
0.375864 + 0.926675i \(0.377346\pi\)
\(60\) 0 0
\(61\) −590.834 −1.24014 −0.620070 0.784547i \(-0.712896\pi\)
−0.620070 + 0.784547i \(0.712896\pi\)
\(62\) 27.3767 0.0560781
\(63\) 0 0
\(64\) −494.547 −0.965912
\(65\) 0 0
\(66\) 0 0
\(67\) −340.777 −0.621382 −0.310691 0.950511i \(-0.600560\pi\)
−0.310691 + 0.950511i \(0.600560\pi\)
\(68\) −621.995 −1.10924
\(69\) 0 0
\(70\) −106.183 −0.181304
\(71\) 36.2243 0.0605498 0.0302749 0.999542i \(-0.490362\pi\)
0.0302749 + 0.999542i \(0.490362\pi\)
\(72\) 0 0
\(73\) −164.572 −0.263859 −0.131929 0.991259i \(-0.542117\pi\)
−0.131929 + 0.991259i \(0.542117\pi\)
\(74\) 81.1919 0.127546
\(75\) 0 0
\(76\) −84.7935 −0.127980
\(77\) 958.584 1.41871
\(78\) 0 0
\(79\) −327.242 −0.466046 −0.233023 0.972471i \(-0.574862\pi\)
−0.233023 + 0.972471i \(0.574862\pi\)
\(80\) −964.925 −1.34852
\(81\) 0 0
\(82\) −99.2916 −0.133719
\(83\) 1404.48 1.85737 0.928684 0.370871i \(-0.120941\pi\)
0.928684 + 0.370871i \(0.120941\pi\)
\(84\) 0 0
\(85\) −1199.46 −1.53058
\(86\) −68.9734 −0.0864837
\(87\) 0 0
\(88\) −100.894 −0.122220
\(89\) −736.986 −0.877756 −0.438878 0.898547i \(-0.644624\pi\)
−0.438878 + 0.898547i \(0.644624\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −213.803 −0.242289
\(93\) 0 0
\(94\) −53.1107 −0.0582761
\(95\) −163.516 −0.176593
\(96\) 0 0
\(97\) 1494.87 1.56476 0.782378 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(98\) −150.935 −0.155578
\(99\) 0 0
\(100\) −877.277 −0.877277
\(101\) 484.010 0.476839 0.238420 0.971162i \(-0.423371\pi\)
0.238420 + 0.971162i \(0.423371\pi\)
\(102\) 0 0
\(103\) 214.049 0.204766 0.102383 0.994745i \(-0.467353\pi\)
0.102383 + 0.994745i \(0.467353\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 158.174 0.144936
\(107\) −148.288 −0.133977 −0.0669884 0.997754i \(-0.521339\pi\)
−0.0669884 + 0.997754i \(0.521339\pi\)
\(108\) 0 0
\(109\) −504.881 −0.443659 −0.221829 0.975086i \(-0.571203\pi\)
−0.221829 + 0.975086i \(0.571203\pi\)
\(110\) −97.0035 −0.0840811
\(111\) 0 0
\(112\) −2037.70 −1.71915
\(113\) 1136.49 0.946120 0.473060 0.881030i \(-0.343149\pi\)
0.473060 + 0.881030i \(0.343149\pi\)
\(114\) 0 0
\(115\) −412.299 −0.334322
\(116\) −1517.56 −1.21467
\(117\) 0 0
\(118\) −72.8019 −0.0567962
\(119\) −2532.98 −1.95124
\(120\) 0 0
\(121\) −455.282 −0.342060
\(122\) 126.261 0.0936979
\(123\) 0 0
\(124\) 1019.01 0.737985
\(125\) 225.651 0.161463
\(126\) 0 0
\(127\) −1286.84 −0.899122 −0.449561 0.893250i \(-0.648420\pi\)
−0.449561 + 0.893250i \(0.648420\pi\)
\(128\) 431.433 0.297919
\(129\) 0 0
\(130\) 0 0
\(131\) −2229.88 −1.48722 −0.743610 0.668614i \(-0.766888\pi\)
−0.743610 + 0.668614i \(0.766888\pi\)
\(132\) 0 0
\(133\) −345.308 −0.225128
\(134\) 72.8241 0.0469481
\(135\) 0 0
\(136\) 266.604 0.168096
\(137\) −474.745 −0.296060 −0.148030 0.988983i \(-0.547293\pi\)
−0.148030 + 0.988983i \(0.547293\pi\)
\(138\) 0 0
\(139\) 1927.02 1.17588 0.587941 0.808904i \(-0.299939\pi\)
0.587941 + 0.808904i \(0.299939\pi\)
\(140\) −3952.33 −2.38595
\(141\) 0 0
\(142\) −7.74113 −0.00457480
\(143\) 0 0
\(144\) 0 0
\(145\) −2926.47 −1.67607
\(146\) 35.1690 0.0199357
\(147\) 0 0
\(148\) 3022.12 1.67849
\(149\) 1065.73 0.585959 0.292979 0.956119i \(-0.405353\pi\)
0.292979 + 0.956119i \(0.405353\pi\)
\(150\) 0 0
\(151\) −877.888 −0.473123 −0.236561 0.971617i \(-0.576020\pi\)
−0.236561 + 0.971617i \(0.576020\pi\)
\(152\) 36.3448 0.0193944
\(153\) 0 0
\(154\) −204.849 −0.107190
\(155\) 1965.07 1.01831
\(156\) 0 0
\(157\) 2314.94 1.17676 0.588382 0.808583i \(-0.299765\pi\)
0.588382 + 0.808583i \(0.299765\pi\)
\(158\) 69.9317 0.0352118
\(159\) 0 0
\(160\) 624.587 0.308612
\(161\) −870.681 −0.426207
\(162\) 0 0
\(163\) −331.399 −0.159246 −0.0796232 0.996825i \(-0.525372\pi\)
−0.0796232 + 0.996825i \(0.525372\pi\)
\(164\) −3695.83 −1.75973
\(165\) 0 0
\(166\) −300.137 −0.140332
\(167\) −2171.62 −1.00626 −0.503129 0.864212i \(-0.667818\pi\)
−0.503129 + 0.864212i \(0.667818\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 256.324 0.115642
\(171\) 0 0
\(172\) −2567.33 −1.13812
\(173\) 2936.69 1.29059 0.645295 0.763933i \(-0.276734\pi\)
0.645295 + 0.763933i \(0.276734\pi\)
\(174\) 0 0
\(175\) −3572.57 −1.54321
\(176\) −1861.55 −0.797270
\(177\) 0 0
\(178\) 157.494 0.0663183
\(179\) −1615.57 −0.674599 −0.337300 0.941397i \(-0.609514\pi\)
−0.337300 + 0.941397i \(0.609514\pi\)
\(180\) 0 0
\(181\) 725.019 0.297736 0.148868 0.988857i \(-0.452437\pi\)
0.148868 + 0.988857i \(0.452437\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 91.6419 0.0367170
\(185\) 5827.87 2.31607
\(186\) 0 0
\(187\) −2314.01 −0.904905
\(188\) −1976.89 −0.766911
\(189\) 0 0
\(190\) 34.9433 0.0133424
\(191\) −1717.08 −0.650490 −0.325245 0.945630i \(-0.605447\pi\)
−0.325245 + 0.945630i \(0.605447\pi\)
\(192\) 0 0
\(193\) −435.830 −0.162548 −0.0812740 0.996692i \(-0.525899\pi\)
−0.0812740 + 0.996692i \(0.525899\pi\)
\(194\) −319.454 −0.118224
\(195\) 0 0
\(196\) −5618.08 −2.04741
\(197\) 694.556 0.251193 0.125597 0.992081i \(-0.459915\pi\)
0.125597 + 0.992081i \(0.459915\pi\)
\(198\) 0 0
\(199\) 2899.24 1.03277 0.516386 0.856356i \(-0.327277\pi\)
0.516386 + 0.856356i \(0.327277\pi\)
\(200\) 376.024 0.132945
\(201\) 0 0
\(202\) −103.433 −0.0360273
\(203\) −6180.04 −2.13672
\(204\) 0 0
\(205\) −7127.04 −2.42817
\(206\) −45.7423 −0.0154709
\(207\) 0 0
\(208\) 0 0
\(209\) −315.457 −0.104405
\(210\) 0 0
\(211\) 5250.30 1.71301 0.856507 0.516136i \(-0.172630\pi\)
0.856507 + 0.516136i \(0.172630\pi\)
\(212\) 5887.53 1.90735
\(213\) 0 0
\(214\) 31.6891 0.0101225
\(215\) −4950.84 −1.57044
\(216\) 0 0
\(217\) 4149.77 1.29818
\(218\) 107.893 0.0335203
\(219\) 0 0
\(220\) −3610.66 −1.10650
\(221\) 0 0
\(222\) 0 0
\(223\) 1382.46 0.415141 0.207570 0.978220i \(-0.433444\pi\)
0.207570 + 0.978220i \(0.433444\pi\)
\(224\) 1318.99 0.393431
\(225\) 0 0
\(226\) −242.867 −0.0714835
\(227\) 1223.57 0.357760 0.178880 0.983871i \(-0.442753\pi\)
0.178880 + 0.983871i \(0.442753\pi\)
\(228\) 0 0
\(229\) 4019.61 1.15993 0.579964 0.814642i \(-0.303067\pi\)
0.579964 + 0.814642i \(0.303067\pi\)
\(230\) 88.1082 0.0252595
\(231\) 0 0
\(232\) 650.468 0.184075
\(233\) −1658.40 −0.466290 −0.233145 0.972442i \(-0.574902\pi\)
−0.233145 + 0.972442i \(0.574902\pi\)
\(234\) 0 0
\(235\) −3812.23 −1.05822
\(236\) −2709.83 −0.747436
\(237\) 0 0
\(238\) 541.297 0.147425
\(239\) −618.554 −0.167410 −0.0837049 0.996491i \(-0.526675\pi\)
−0.0837049 + 0.996491i \(0.526675\pi\)
\(240\) 0 0
\(241\) 1135.25 0.303434 0.151717 0.988424i \(-0.451520\pi\)
0.151717 + 0.988424i \(0.451520\pi\)
\(242\) 97.2938 0.0258441
\(243\) 0 0
\(244\) 4699.69 1.23306
\(245\) −10833.9 −2.82512
\(246\) 0 0
\(247\) 0 0
\(248\) −436.777 −0.111836
\(249\) 0 0
\(250\) −48.2215 −0.0121992
\(251\) −2287.83 −0.575326 −0.287663 0.957732i \(-0.592878\pi\)
−0.287663 + 0.957732i \(0.592878\pi\)
\(252\) 0 0
\(253\) −795.414 −0.197657
\(254\) 274.997 0.0679326
\(255\) 0 0
\(256\) 3864.18 0.943403
\(257\) 5113.49 1.24113 0.620566 0.784154i \(-0.286903\pi\)
0.620566 + 0.784154i \(0.286903\pi\)
\(258\) 0 0
\(259\) 12307.1 2.95262
\(260\) 0 0
\(261\) 0 0
\(262\) 476.526 0.112366
\(263\) −2760.17 −0.647145 −0.323573 0.946203i \(-0.604884\pi\)
−0.323573 + 0.946203i \(0.604884\pi\)
\(264\) 0 0
\(265\) 11353.5 2.63185
\(266\) 73.7923 0.0170094
\(267\) 0 0
\(268\) 2710.66 0.617834
\(269\) 3310.32 0.750312 0.375156 0.926962i \(-0.377589\pi\)
0.375156 + 0.926962i \(0.377589\pi\)
\(270\) 0 0
\(271\) −2522.26 −0.565374 −0.282687 0.959212i \(-0.591226\pi\)
−0.282687 + 0.959212i \(0.591226\pi\)
\(272\) 4918.99 1.09654
\(273\) 0 0
\(274\) 101.453 0.0223686
\(275\) −3263.74 −0.715675
\(276\) 0 0
\(277\) −3455.48 −0.749530 −0.374765 0.927120i \(-0.622277\pi\)
−0.374765 + 0.927120i \(0.622277\pi\)
\(278\) −411.804 −0.0888430
\(279\) 0 0
\(280\) 1694.07 0.361572
\(281\) −981.649 −0.208400 −0.104200 0.994556i \(-0.533228\pi\)
−0.104200 + 0.994556i \(0.533228\pi\)
\(282\) 0 0
\(283\) 5331.10 1.11979 0.559896 0.828563i \(-0.310841\pi\)
0.559896 + 0.828563i \(0.310841\pi\)
\(284\) −288.140 −0.0602041
\(285\) 0 0
\(286\) 0 0
\(287\) −15050.7 −3.09552
\(288\) 0 0
\(289\) 1201.58 0.244572
\(290\) 625.387 0.126634
\(291\) 0 0
\(292\) 1309.06 0.262353
\(293\) −2420.39 −0.482597 −0.241299 0.970451i \(-0.577573\pi\)
−0.241299 + 0.970451i \(0.577573\pi\)
\(294\) 0 0
\(295\) −5225.64 −1.03135
\(296\) −1295.36 −0.254363
\(297\) 0 0
\(298\) −227.746 −0.0442717
\(299\) 0 0
\(300\) 0 0
\(301\) −10455.0 −2.00205
\(302\) 187.605 0.0357465
\(303\) 0 0
\(304\) 670.581 0.126515
\(305\) 9062.88 1.70144
\(306\) 0 0
\(307\) 875.509 0.162762 0.0813810 0.996683i \(-0.474067\pi\)
0.0813810 + 0.996683i \(0.474067\pi\)
\(308\) −7624.90 −1.41061
\(309\) 0 0
\(310\) −419.935 −0.0769377
\(311\) −6419.10 −1.17040 −0.585199 0.810890i \(-0.698984\pi\)
−0.585199 + 0.810890i \(0.698984\pi\)
\(312\) 0 0
\(313\) −5015.58 −0.905742 −0.452871 0.891576i \(-0.649600\pi\)
−0.452871 + 0.891576i \(0.649600\pi\)
\(314\) −494.701 −0.0889096
\(315\) 0 0
\(316\) 2602.99 0.463386
\(317\) 5228.81 0.926433 0.463217 0.886245i \(-0.346695\pi\)
0.463217 + 0.886245i \(0.346695\pi\)
\(318\) 0 0
\(319\) −5645.80 −0.990922
\(320\) 7585.92 1.32521
\(321\) 0 0
\(322\) 186.064 0.0322018
\(323\) 833.570 0.143595
\(324\) 0 0
\(325\) 0 0
\(326\) 70.8199 0.0120318
\(327\) 0 0
\(328\) 1584.13 0.266674
\(329\) −8050.56 −1.34906
\(330\) 0 0
\(331\) 3186.81 0.529193 0.264596 0.964359i \(-0.414761\pi\)
0.264596 + 0.964359i \(0.414761\pi\)
\(332\) −11171.7 −1.84677
\(333\) 0 0
\(334\) 464.075 0.0760271
\(335\) 5227.23 0.852520
\(336\) 0 0
\(337\) −5600.69 −0.905309 −0.452655 0.891686i \(-0.649523\pi\)
−0.452655 + 0.891686i \(0.649523\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 9540.87 1.52184
\(341\) 3791.04 0.602042
\(342\) 0 0
\(343\) −11768.0 −1.85252
\(344\) 1100.42 0.172474
\(345\) 0 0
\(346\) −627.570 −0.0975097
\(347\) 6982.33 1.08020 0.540102 0.841599i \(-0.318386\pi\)
0.540102 + 0.841599i \(0.318386\pi\)
\(348\) 0 0
\(349\) −2872.20 −0.440531 −0.220265 0.975440i \(-0.570692\pi\)
−0.220265 + 0.975440i \(0.570692\pi\)
\(350\) 763.459 0.116596
\(351\) 0 0
\(352\) 1204.96 0.182457
\(353\) −5104.55 −0.769655 −0.384827 0.922989i \(-0.625739\pi\)
−0.384827 + 0.922989i \(0.625739\pi\)
\(354\) 0 0
\(355\) −555.650 −0.0830728
\(356\) 5862.23 0.872746
\(357\) 0 0
\(358\) 345.247 0.0509689
\(359\) −10771.0 −1.58348 −0.791741 0.610857i \(-0.790825\pi\)
−0.791741 + 0.610857i \(0.790825\pi\)
\(360\) 0 0
\(361\) −6745.36 −0.983433
\(362\) −154.936 −0.0224952
\(363\) 0 0
\(364\) 0 0
\(365\) 2524.39 0.362008
\(366\) 0 0
\(367\) −11380.5 −1.61868 −0.809340 0.587340i \(-0.800175\pi\)
−0.809340 + 0.587340i \(0.800175\pi\)
\(368\) 1690.84 0.239514
\(369\) 0 0
\(370\) −1245.41 −0.174989
\(371\) 23976.1 3.35519
\(372\) 0 0
\(373\) 3196.75 0.443758 0.221879 0.975074i \(-0.428781\pi\)
0.221879 + 0.975074i \(0.428781\pi\)
\(374\) 494.504 0.0683695
\(375\) 0 0
\(376\) 847.346 0.116220
\(377\) 0 0
\(378\) 0 0
\(379\) −2050.75 −0.277942 −0.138971 0.990296i \(-0.544379\pi\)
−0.138971 + 0.990296i \(0.544379\pi\)
\(380\) 1300.66 0.175585
\(381\) 0 0
\(382\) 366.940 0.0491473
\(383\) −3507.54 −0.467955 −0.233978 0.972242i \(-0.575174\pi\)
−0.233978 + 0.972242i \(0.575174\pi\)
\(384\) 0 0
\(385\) −14703.9 −1.94644
\(386\) 93.1369 0.0122812
\(387\) 0 0
\(388\) −11890.7 −1.55582
\(389\) 6572.66 0.856676 0.428338 0.903618i \(-0.359099\pi\)
0.428338 + 0.903618i \(0.359099\pi\)
\(390\) 0 0
\(391\) 2101.81 0.271850
\(392\) 2408.06 0.310269
\(393\) 0 0
\(394\) −148.427 −0.0189787
\(395\) 5019.62 0.639403
\(396\) 0 0
\(397\) −5285.98 −0.668251 −0.334126 0.942529i \(-0.608441\pi\)
−0.334126 + 0.942529i \(0.608441\pi\)
\(398\) −619.567 −0.0780304
\(399\) 0 0
\(400\) 6937.86 0.867233
\(401\) 919.541 0.114513 0.0572565 0.998360i \(-0.481765\pi\)
0.0572565 + 0.998360i \(0.481765\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −3849.97 −0.474117
\(405\) 0 0
\(406\) 1320.67 0.161438
\(407\) 11243.2 1.36930
\(408\) 0 0
\(409\) −1539.17 −0.186081 −0.0930403 0.995662i \(-0.529659\pi\)
−0.0930403 + 0.995662i \(0.529659\pi\)
\(410\) 1523.05 0.183458
\(411\) 0 0
\(412\) −1702.62 −0.203597
\(413\) −11035.4 −1.31481
\(414\) 0 0
\(415\) −21543.5 −2.54826
\(416\) 0 0
\(417\) 0 0
\(418\) 67.4132 0.00788825
\(419\) −10579.9 −1.23357 −0.616783 0.787133i \(-0.711564\pi\)
−0.616783 + 0.787133i \(0.711564\pi\)
\(420\) 0 0
\(421\) 74.9351 0.00867485 0.00433743 0.999991i \(-0.498619\pi\)
0.00433743 + 0.999991i \(0.498619\pi\)
\(422\) −1121.99 −0.129426
\(423\) 0 0
\(424\) −2523.55 −0.289044
\(425\) 8624.15 0.984313
\(426\) 0 0
\(427\) 19138.8 2.16906
\(428\) 1179.53 0.133212
\(429\) 0 0
\(430\) 1057.99 0.118653
\(431\) 12165.6 1.35962 0.679810 0.733388i \(-0.262062\pi\)
0.679810 + 0.733388i \(0.262062\pi\)
\(432\) 0 0
\(433\) −2869.23 −0.318445 −0.159222 0.987243i \(-0.550899\pi\)
−0.159222 + 0.987243i \(0.550899\pi\)
\(434\) −886.806 −0.0980831
\(435\) 0 0
\(436\) 4015.99 0.441126
\(437\) 286.530 0.0313652
\(438\) 0 0
\(439\) −3845.12 −0.418035 −0.209018 0.977912i \(-0.567027\pi\)
−0.209018 + 0.977912i \(0.567027\pi\)
\(440\) 1547.63 0.167682
\(441\) 0 0
\(442\) 0 0
\(443\) 3858.30 0.413799 0.206900 0.978362i \(-0.433663\pi\)
0.206900 + 0.978362i \(0.433663\pi\)
\(444\) 0 0
\(445\) 11304.7 1.20426
\(446\) −295.432 −0.0313657
\(447\) 0 0
\(448\) 16019.7 1.68942
\(449\) 5550.51 0.583395 0.291698 0.956511i \(-0.405780\pi\)
0.291698 + 0.956511i \(0.405780\pi\)
\(450\) 0 0
\(451\) −13749.6 −1.43557
\(452\) −9039.99 −0.940719
\(453\) 0 0
\(454\) −261.478 −0.0270303
\(455\) 0 0
\(456\) 0 0
\(457\) 8102.42 0.829355 0.414677 0.909968i \(-0.363894\pi\)
0.414677 + 0.909968i \(0.363894\pi\)
\(458\) −858.991 −0.0876376
\(459\) 0 0
\(460\) 3279.56 0.332414
\(461\) −9230.86 −0.932590 −0.466295 0.884629i \(-0.654412\pi\)
−0.466295 + 0.884629i \(0.654412\pi\)
\(462\) 0 0
\(463\) −13934.7 −1.39870 −0.699352 0.714777i \(-0.746528\pi\)
−0.699352 + 0.714777i \(0.746528\pi\)
\(464\) 12001.5 1.20077
\(465\) 0 0
\(466\) 354.400 0.0352302
\(467\) −10918.0 −1.08185 −0.540926 0.841070i \(-0.681926\pi\)
−0.540926 + 0.841070i \(0.681926\pi\)
\(468\) 0 0
\(469\) 11038.7 1.08682
\(470\) 814.673 0.0799533
\(471\) 0 0
\(472\) 1161.51 0.113268
\(473\) −9551.23 −0.928470
\(474\) 0 0
\(475\) 1175.69 0.113567
\(476\) 20148.2 1.94010
\(477\) 0 0
\(478\) 132.185 0.0126485
\(479\) 11028.4 1.05198 0.525992 0.850490i \(-0.323694\pi\)
0.525992 + 0.850490i \(0.323694\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −242.602 −0.0229258
\(483\) 0 0
\(484\) 3621.47 0.340108
\(485\) −22930.0 −2.14680
\(486\) 0 0
\(487\) −1078.10 −0.100315 −0.0501576 0.998741i \(-0.515972\pi\)
−0.0501576 + 0.998741i \(0.515972\pi\)
\(488\) −2014.41 −0.186861
\(489\) 0 0
\(490\) 2315.21 0.213450
\(491\) −6572.88 −0.604134 −0.302067 0.953287i \(-0.597677\pi\)
−0.302067 + 0.953287i \(0.597677\pi\)
\(492\) 0 0
\(493\) 14918.6 1.36288
\(494\) 0 0
\(495\) 0 0
\(496\) −8058.77 −0.729535
\(497\) −1173.41 −0.105904
\(498\) 0 0
\(499\) −9956.79 −0.893241 −0.446620 0.894724i \(-0.647373\pi\)
−0.446620 + 0.894724i \(0.647373\pi\)
\(500\) −1794.90 −0.160541
\(501\) 0 0
\(502\) 488.910 0.0434683
\(503\) 11965.0 1.06062 0.530310 0.847804i \(-0.322076\pi\)
0.530310 + 0.847804i \(0.322076\pi\)
\(504\) 0 0
\(505\) −7424.29 −0.654211
\(506\) 169.980 0.0149338
\(507\) 0 0
\(508\) 10235.9 0.893990
\(509\) −7160.44 −0.623538 −0.311769 0.950158i \(-0.600922\pi\)
−0.311769 + 0.950158i \(0.600922\pi\)
\(510\) 0 0
\(511\) 5330.94 0.461501
\(512\) −4277.24 −0.369197
\(513\) 0 0
\(514\) −1092.75 −0.0937729
\(515\) −3283.33 −0.280933
\(516\) 0 0
\(517\) −7354.61 −0.625639
\(518\) −2630.03 −0.223083
\(519\) 0 0
\(520\) 0 0
\(521\) 17213.1 1.44745 0.723725 0.690089i \(-0.242428\pi\)
0.723725 + 0.690089i \(0.242428\pi\)
\(522\) 0 0
\(523\) −10110.1 −0.845281 −0.422640 0.906297i \(-0.638897\pi\)
−0.422640 + 0.906297i \(0.638897\pi\)
\(524\) 17737.2 1.47873
\(525\) 0 0
\(526\) 589.847 0.0488946
\(527\) −10017.5 −0.828026
\(528\) 0 0
\(529\) −11444.5 −0.940620
\(530\) −2426.25 −0.198848
\(531\) 0 0
\(532\) 2746.70 0.223843
\(533\) 0 0
\(534\) 0 0
\(535\) 2274.61 0.183813
\(536\) −1161.86 −0.0936281
\(537\) 0 0
\(538\) −707.416 −0.0566893
\(539\) −20901.0 −1.67026
\(540\) 0 0
\(541\) 5951.54 0.472970 0.236485 0.971635i \(-0.424005\pi\)
0.236485 + 0.971635i \(0.424005\pi\)
\(542\) 539.007 0.0427165
\(543\) 0 0
\(544\) −3184.02 −0.250944
\(545\) 7744.43 0.608688
\(546\) 0 0
\(547\) 5157.62 0.403152 0.201576 0.979473i \(-0.435394\pi\)
0.201576 + 0.979473i \(0.435394\pi\)
\(548\) 3776.28 0.294370
\(549\) 0 0
\(550\) 697.460 0.0540724
\(551\) 2033.77 0.157244
\(552\) 0 0
\(553\) 10600.3 0.815136
\(554\) 738.436 0.0566302
\(555\) 0 0
\(556\) −15328.2 −1.16917
\(557\) 23314.4 1.77354 0.886772 0.462206i \(-0.152942\pi\)
0.886772 + 0.462206i \(0.152942\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 31256.6 2.35863
\(561\) 0 0
\(562\) 209.778 0.0157455
\(563\) −2329.12 −0.174353 −0.0871764 0.996193i \(-0.527784\pi\)
−0.0871764 + 0.996193i \(0.527784\pi\)
\(564\) 0 0
\(565\) −17432.7 −1.29805
\(566\) −1139.26 −0.0846051
\(567\) 0 0
\(568\) 123.505 0.00912348
\(569\) 17446.6 1.28541 0.642706 0.766113i \(-0.277812\pi\)
0.642706 + 0.766113i \(0.277812\pi\)
\(570\) 0 0
\(571\) −413.526 −0.0303074 −0.0151537 0.999885i \(-0.504824\pi\)
−0.0151537 + 0.999885i \(0.504824\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 3216.33 0.233880
\(575\) 2964.45 0.215002
\(576\) 0 0
\(577\) −21258.8 −1.53382 −0.766911 0.641754i \(-0.778207\pi\)
−0.766911 + 0.641754i \(0.778207\pi\)
\(578\) −256.778 −0.0184785
\(579\) 0 0
\(580\) 23278.1 1.66650
\(581\) −45495.0 −3.24862
\(582\) 0 0
\(583\) 21903.4 1.55600
\(584\) −561.098 −0.0397575
\(585\) 0 0
\(586\) 517.238 0.0364623
\(587\) −2965.69 −0.208530 −0.104265 0.994550i \(-0.533249\pi\)
−0.104265 + 0.994550i \(0.533249\pi\)
\(588\) 0 0
\(589\) −1365.64 −0.0955349
\(590\) 1116.72 0.0779230
\(591\) 0 0
\(592\) −23900.2 −1.65928
\(593\) 17786.8 1.23173 0.615865 0.787852i \(-0.288807\pi\)
0.615865 + 0.787852i \(0.288807\pi\)
\(594\) 0 0
\(595\) 38853.7 2.67705
\(596\) −8477.16 −0.582614
\(597\) 0 0
\(598\) 0 0
\(599\) −21796.4 −1.48677 −0.743387 0.668861i \(-0.766782\pi\)
−0.743387 + 0.668861i \(0.766782\pi\)
\(600\) 0 0
\(601\) 20468.7 1.38924 0.694622 0.719375i \(-0.255572\pi\)
0.694622 + 0.719375i \(0.255572\pi\)
\(602\) 2234.24 0.151264
\(603\) 0 0
\(604\) 6983.01 0.470422
\(605\) 6983.64 0.469298
\(606\) 0 0
\(607\) 7852.53 0.525081 0.262541 0.964921i \(-0.415440\pi\)
0.262541 + 0.964921i \(0.415440\pi\)
\(608\) −434.061 −0.0289531
\(609\) 0 0
\(610\) −1936.74 −0.128551
\(611\) 0 0
\(612\) 0 0
\(613\) −2460.21 −0.162099 −0.0810497 0.996710i \(-0.525827\pi\)
−0.0810497 + 0.996710i \(0.525827\pi\)
\(614\) −187.096 −0.0122974
\(615\) 0 0
\(616\) 3268.23 0.213768
\(617\) 17829.3 1.16334 0.581670 0.813425i \(-0.302400\pi\)
0.581670 + 0.813425i \(0.302400\pi\)
\(618\) 0 0
\(619\) 16901.7 1.09748 0.548739 0.835994i \(-0.315108\pi\)
0.548739 + 0.835994i \(0.315108\pi\)
\(620\) −15630.8 −1.01250
\(621\) 0 0
\(622\) 1371.76 0.0884286
\(623\) 23873.0 1.53524
\(624\) 0 0
\(625\) −17247.4 −1.10384
\(626\) 1071.83 0.0684327
\(627\) 0 0
\(628\) −18413.8 −1.17005
\(629\) −29709.3 −1.88328
\(630\) 0 0
\(631\) −1833.82 −0.115695 −0.0578473 0.998325i \(-0.518424\pi\)
−0.0578473 + 0.998325i \(0.518424\pi\)
\(632\) −1115.71 −0.0702226
\(633\) 0 0
\(634\) −1117.40 −0.0699960
\(635\) 19739.0 1.23357
\(636\) 0 0
\(637\) 0 0
\(638\) 1206.51 0.0748684
\(639\) 0 0
\(640\) −6617.81 −0.408737
\(641\) −29172.1 −1.79755 −0.898773 0.438414i \(-0.855540\pi\)
−0.898773 + 0.438414i \(0.855540\pi\)
\(642\) 0 0
\(643\) −25103.8 −1.53965 −0.769826 0.638254i \(-0.779657\pi\)
−0.769826 + 0.638254i \(0.779657\pi\)
\(644\) 6925.69 0.423774
\(645\) 0 0
\(646\) −178.134 −0.0108492
\(647\) 24471.6 1.48698 0.743492 0.668745i \(-0.233168\pi\)
0.743492 + 0.668745i \(0.233168\pi\)
\(648\) 0 0
\(649\) −10081.4 −0.609752
\(650\) 0 0
\(651\) 0 0
\(652\) 2636.06 0.158337
\(653\) −6632.68 −0.397484 −0.198742 0.980052i \(-0.563686\pi\)
−0.198742 + 0.980052i \(0.563686\pi\)
\(654\) 0 0
\(655\) 34204.5 2.04043
\(656\) 29228.1 1.73958
\(657\) 0 0
\(658\) 1720.40 0.101928
\(659\) −26436.9 −1.56272 −0.781361 0.624080i \(-0.785474\pi\)
−0.781361 + 0.624080i \(0.785474\pi\)
\(660\) 0 0
\(661\) −27596.7 −1.62389 −0.811943 0.583737i \(-0.801590\pi\)
−0.811943 + 0.583737i \(0.801590\pi\)
\(662\) −681.020 −0.0399828
\(663\) 0 0
\(664\) 4788.49 0.279863
\(665\) 5296.73 0.308870
\(666\) 0 0
\(667\) 5128.07 0.297691
\(668\) 17273.8 1.00051
\(669\) 0 0
\(670\) −1117.06 −0.0644115
\(671\) 17484.3 1.00592
\(672\) 0 0
\(673\) 18860.1 1.08024 0.540122 0.841587i \(-0.318378\pi\)
0.540122 + 0.841587i \(0.318378\pi\)
\(674\) 1196.87 0.0684000
\(675\) 0 0
\(676\) 0 0
\(677\) 14705.5 0.834825 0.417412 0.908717i \(-0.362937\pi\)
0.417412 + 0.908717i \(0.362937\pi\)
\(678\) 0 0
\(679\) −48423.1 −2.73683
\(680\) −4089.47 −0.230624
\(681\) 0 0
\(682\) −810.145 −0.0454869
\(683\) −17950.2 −1.00563 −0.502814 0.864395i \(-0.667702\pi\)
−0.502814 + 0.864395i \(0.667702\pi\)
\(684\) 0 0
\(685\) 7282.18 0.406186
\(686\) 2514.83 0.139966
\(687\) 0 0
\(688\) 20303.5 1.12509
\(689\) 0 0
\(690\) 0 0
\(691\) 8692.38 0.478544 0.239272 0.970953i \(-0.423091\pi\)
0.239272 + 0.970953i \(0.423091\pi\)
\(692\) −23359.4 −1.28322
\(693\) 0 0
\(694\) −1492.12 −0.0816141
\(695\) −29558.8 −1.61328
\(696\) 0 0
\(697\) 36332.2 1.97443
\(698\) 613.789 0.0332840
\(699\) 0 0
\(700\) 28417.4 1.53440
\(701\) −23275.9 −1.25409 −0.627047 0.778982i \(-0.715736\pi\)
−0.627047 + 0.778982i \(0.715736\pi\)
\(702\) 0 0
\(703\) −4050.11 −0.217287
\(704\) 14634.9 0.783485
\(705\) 0 0
\(706\) 1090.84 0.0581507
\(707\) −15678.4 −0.834014
\(708\) 0 0
\(709\) 34164.7 1.80971 0.904854 0.425722i \(-0.139980\pi\)
0.904854 + 0.425722i \(0.139980\pi\)
\(710\) 118.742 0.00627651
\(711\) 0 0
\(712\) −2512.71 −0.132258
\(713\) −3443.40 −0.180864
\(714\) 0 0
\(715\) 0 0
\(716\) 12850.8 0.670748
\(717\) 0 0
\(718\) 2301.75 0.119639
\(719\) 8391.74 0.435270 0.217635 0.976030i \(-0.430166\pi\)
0.217635 + 0.976030i \(0.430166\pi\)
\(720\) 0 0
\(721\) −6933.64 −0.358145
\(722\) 1441.48 0.0743026
\(723\) 0 0
\(724\) −5767.04 −0.296036
\(725\) 21041.5 1.07788
\(726\) 0 0
\(727\) 23500.4 1.19887 0.599436 0.800423i \(-0.295392\pi\)
0.599436 + 0.800423i \(0.295392\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −539.462 −0.0273512
\(731\) 25238.3 1.27698
\(732\) 0 0
\(733\) −1289.07 −0.0649563 −0.0324781 0.999472i \(-0.510340\pi\)
−0.0324781 + 0.999472i \(0.510340\pi\)
\(734\) 2432.01 0.122298
\(735\) 0 0
\(736\) −1094.47 −0.0548133
\(737\) 10084.5 0.504024
\(738\) 0 0
\(739\) 18197.8 0.905844 0.452922 0.891550i \(-0.350382\pi\)
0.452922 + 0.891550i \(0.350382\pi\)
\(740\) −46356.8 −2.30285
\(741\) 0 0
\(742\) −5123.68 −0.253499
\(743\) 21277.2 1.05059 0.525294 0.850921i \(-0.323956\pi\)
0.525294 + 0.850921i \(0.323956\pi\)
\(744\) 0 0
\(745\) −16347.4 −0.803920
\(746\) −683.146 −0.0335278
\(747\) 0 0
\(748\) 18406.4 0.899739
\(749\) 4803.45 0.234331
\(750\) 0 0
\(751\) −25653.4 −1.24648 −0.623240 0.782030i \(-0.714184\pi\)
−0.623240 + 0.782030i \(0.714184\pi\)
\(752\) 15634.0 0.758130
\(753\) 0 0
\(754\) 0 0
\(755\) 13466.0 0.649112
\(756\) 0 0
\(757\) 26521.2 1.27336 0.636678 0.771130i \(-0.280308\pi\)
0.636678 + 0.771130i \(0.280308\pi\)
\(758\) 438.245 0.0209997
\(759\) 0 0
\(760\) −557.497 −0.0266086
\(761\) 21607.2 1.02925 0.514625 0.857415i \(-0.327931\pi\)
0.514625 + 0.857415i \(0.327931\pi\)
\(762\) 0 0
\(763\) 16354.5 0.775979
\(764\) 13658.2 0.646776
\(765\) 0 0
\(766\) 749.560 0.0353560
\(767\) 0 0
\(768\) 0 0
\(769\) 17179.2 0.805591 0.402795 0.915290i \(-0.368039\pi\)
0.402795 + 0.915290i \(0.368039\pi\)
\(770\) 3142.21 0.147062
\(771\) 0 0
\(772\) 3466.74 0.161620
\(773\) −12822.1 −0.596609 −0.298304 0.954471i \(-0.596421\pi\)
−0.298304 + 0.954471i \(0.596421\pi\)
\(774\) 0 0
\(775\) −14128.9 −0.654873
\(776\) 5096.67 0.235773
\(777\) 0 0
\(778\) −1404.58 −0.0647256
\(779\) 4952.98 0.227804
\(780\) 0 0
\(781\) −1071.97 −0.0491140
\(782\) −449.157 −0.0205394
\(783\) 0 0
\(784\) 44430.0 2.02396
\(785\) −35509.1 −1.61449
\(786\) 0 0
\(787\) 35213.4 1.59495 0.797474 0.603354i \(-0.206169\pi\)
0.797474 + 0.603354i \(0.206169\pi\)
\(788\) −5524.73 −0.249759
\(789\) 0 0
\(790\) −1072.69 −0.0483097
\(791\) −36813.9 −1.65481
\(792\) 0 0
\(793\) 0 0
\(794\) 1129.61 0.0504893
\(795\) 0 0
\(796\) −23061.5 −1.02688
\(797\) −7500.15 −0.333336 −0.166668 0.986013i \(-0.553301\pi\)
−0.166668 + 0.986013i \(0.553301\pi\)
\(798\) 0 0
\(799\) 19434.0 0.860481
\(800\) −4490.82 −0.198468
\(801\) 0 0
\(802\) −196.506 −0.00865195
\(803\) 4870.10 0.214025
\(804\) 0 0
\(805\) 13355.5 0.584745
\(806\) 0 0
\(807\) 0 0
\(808\) 1650.20 0.0718489
\(809\) 22020.5 0.956982 0.478491 0.878092i \(-0.341184\pi\)
0.478491 + 0.878092i \(0.341184\pi\)
\(810\) 0 0
\(811\) −20444.6 −0.885210 −0.442605 0.896717i \(-0.645946\pi\)
−0.442605 + 0.896717i \(0.645946\pi\)
\(812\) 49158.1 2.12452
\(813\) 0 0
\(814\) −2402.67 −0.103457
\(815\) 5083.37 0.218482
\(816\) 0 0
\(817\) 3440.62 0.147334
\(818\) 328.920 0.0140592
\(819\) 0 0
\(820\) 56690.8 2.41431
\(821\) 40353.0 1.71538 0.857692 0.514164i \(-0.171898\pi\)
0.857692 + 0.514164i \(0.171898\pi\)
\(822\) 0 0
\(823\) −33110.7 −1.40239 −0.701194 0.712971i \(-0.747349\pi\)
−0.701194 + 0.712971i \(0.747349\pi\)
\(824\) 729.787 0.0308536
\(825\) 0 0
\(826\) 2358.26 0.0993392
\(827\) −43410.4 −1.82531 −0.912653 0.408735i \(-0.865970\pi\)
−0.912653 + 0.408735i \(0.865970\pi\)
\(828\) 0 0
\(829\) −5502.87 −0.230546 −0.115273 0.993334i \(-0.536774\pi\)
−0.115273 + 0.993334i \(0.536774\pi\)
\(830\) 4603.84 0.192532
\(831\) 0 0
\(832\) 0 0
\(833\) 55229.1 2.29721
\(834\) 0 0
\(835\) 33310.8 1.38056
\(836\) 2509.25 0.103809
\(837\) 0 0
\(838\) 2260.93 0.0932013
\(839\) −1698.27 −0.0698819 −0.0349409 0.999389i \(-0.511124\pi\)
−0.0349409 + 0.999389i \(0.511124\pi\)
\(840\) 0 0
\(841\) 12009.8 0.492425
\(842\) −16.0136 −0.000655423 0
\(843\) 0 0
\(844\) −41762.7 −1.70323
\(845\) 0 0
\(846\) 0 0
\(847\) 14747.9 0.598279
\(848\) −46561.0 −1.88551
\(849\) 0 0
\(850\) −1842.98 −0.0743691
\(851\) −10212.2 −0.411363
\(852\) 0 0
\(853\) −18686.8 −0.750087 −0.375044 0.927007i \(-0.622372\pi\)
−0.375044 + 0.927007i \(0.622372\pi\)
\(854\) −4089.95 −0.163882
\(855\) 0 0
\(856\) −505.578 −0.0201873
\(857\) 9659.16 0.385007 0.192503 0.981296i \(-0.438339\pi\)
0.192503 + 0.981296i \(0.438339\pi\)
\(858\) 0 0
\(859\) −1559.90 −0.0619595 −0.0309798 0.999520i \(-0.509863\pi\)
−0.0309798 + 0.999520i \(0.509863\pi\)
\(860\) 39380.6 1.56147
\(861\) 0 0
\(862\) −2599.79 −0.102725
\(863\) −11358.7 −0.448034 −0.224017 0.974585i \(-0.571917\pi\)
−0.224017 + 0.974585i \(0.571917\pi\)
\(864\) 0 0
\(865\) −45046.3 −1.77066
\(866\) 613.155 0.0240599
\(867\) 0 0
\(868\) −33008.7 −1.29077
\(869\) 9683.93 0.378026
\(870\) 0 0
\(871\) 0 0
\(872\) −1721.36 −0.0668493
\(873\) 0 0
\(874\) −61.2314 −0.00236977
\(875\) −7309.45 −0.282405
\(876\) 0 0
\(877\) 33301.6 1.28223 0.641115 0.767444i \(-0.278472\pi\)
0.641115 + 0.767444i \(0.278472\pi\)
\(878\) 821.701 0.0315844
\(879\) 0 0
\(880\) 28554.6 1.09383
\(881\) −28912.3 −1.10565 −0.552826 0.833296i \(-0.686451\pi\)
−0.552826 + 0.833296i \(0.686451\pi\)
\(882\) 0 0
\(883\) −49100.9 −1.87132 −0.935661 0.352901i \(-0.885195\pi\)
−0.935661 + 0.352901i \(0.885195\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −824.517 −0.0312643
\(887\) −4566.49 −0.172861 −0.0864304 0.996258i \(-0.527546\pi\)
−0.0864304 + 0.996258i \(0.527546\pi\)
\(888\) 0 0
\(889\) 41684.3 1.57261
\(890\) −2415.82 −0.0909870
\(891\) 0 0
\(892\) −10996.5 −0.412771
\(893\) 2649.33 0.0992795
\(894\) 0 0
\(895\) 24781.4 0.925533
\(896\) −13975.3 −0.521074
\(897\) 0 0
\(898\) −1186.14 −0.0440780
\(899\) −24441.0 −0.906734
\(900\) 0 0
\(901\) −57877.9 −2.14006
\(902\) 2938.29 0.108464
\(903\) 0 0
\(904\) 3874.78 0.142559
\(905\) −11121.2 −0.408486
\(906\) 0 0
\(907\) 48873.3 1.78921 0.894604 0.446860i \(-0.147458\pi\)
0.894604 + 0.446860i \(0.147458\pi\)
\(908\) −9732.72 −0.355718
\(909\) 0 0
\(910\) 0 0
\(911\) −31550.1 −1.14742 −0.573710 0.819058i \(-0.694497\pi\)
−0.573710 + 0.819058i \(0.694497\pi\)
\(912\) 0 0
\(913\) −41562.1 −1.50658
\(914\) −1731.49 −0.0626613
\(915\) 0 0
\(916\) −31973.3 −1.15331
\(917\) 72232.1 2.60121
\(918\) 0 0
\(919\) −11720.3 −0.420694 −0.210347 0.977627i \(-0.567459\pi\)
−0.210347 + 0.977627i \(0.567459\pi\)
\(920\) −1405.71 −0.0503748
\(921\) 0 0
\(922\) 1972.63 0.0704612
\(923\) 0 0
\(924\) 0 0
\(925\) −41902.7 −1.48946
\(926\) 2977.84 0.105678
\(927\) 0 0
\(928\) −7768.47 −0.274798
\(929\) −6911.23 −0.244080 −0.122040 0.992525i \(-0.538944\pi\)
−0.122040 + 0.992525i \(0.538944\pi\)
\(930\) 0 0
\(931\) 7529.10 0.265044
\(932\) 13191.5 0.463628
\(933\) 0 0
\(934\) 2333.18 0.0817386
\(935\) 35494.9 1.24151
\(936\) 0 0
\(937\) −28673.9 −0.999717 −0.499858 0.866107i \(-0.666615\pi\)
−0.499858 + 0.866107i \(0.666615\pi\)
\(938\) −2358.97 −0.0821143
\(939\) 0 0
\(940\) 30323.7 1.05218
\(941\) 22463.6 0.778207 0.389103 0.921194i \(-0.372785\pi\)
0.389103 + 0.921194i \(0.372785\pi\)
\(942\) 0 0
\(943\) 12488.8 0.431272
\(944\) 21430.4 0.738878
\(945\) 0 0
\(946\) 2041.10 0.0701499
\(947\) 36806.6 1.26299 0.631497 0.775379i \(-0.282441\pi\)
0.631497 + 0.775379i \(0.282441\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −251.244 −0.00858046
\(951\) 0 0
\(952\) −8636.04 −0.294008
\(953\) 1781.07 0.0605400 0.0302700 0.999542i \(-0.490363\pi\)
0.0302700 + 0.999542i \(0.490363\pi\)
\(954\) 0 0
\(955\) 26338.5 0.892455
\(956\) 4920.19 0.166454
\(957\) 0 0
\(958\) −2356.77 −0.0794819
\(959\) 15378.3 0.517822
\(960\) 0 0
\(961\) −13379.3 −0.449107
\(962\) 0 0
\(963\) 0 0
\(964\) −9030.12 −0.301702
\(965\) 6685.26 0.223012
\(966\) 0 0
\(967\) −54777.2 −1.82163 −0.910815 0.412815i \(-0.864545\pi\)
−0.910815 + 0.412815i \(0.864545\pi\)
\(968\) −1552.26 −0.0515408
\(969\) 0 0
\(970\) 4900.15 0.162200
\(971\) 2391.54 0.0790403 0.0395201 0.999219i \(-0.487417\pi\)
0.0395201 + 0.999219i \(0.487417\pi\)
\(972\) 0 0
\(973\) −62421.5 −2.05667
\(974\) 230.391 0.00757925
\(975\) 0 0
\(976\) −37167.0 −1.21894
\(977\) −43738.5 −1.43226 −0.716130 0.697967i \(-0.754088\pi\)
−0.716130 + 0.697967i \(0.754088\pi\)
\(978\) 0 0
\(979\) 21809.3 0.711979
\(980\) 86176.5 2.80899
\(981\) 0 0
\(982\) 1404.62 0.0456449
\(983\) 27712.4 0.899174 0.449587 0.893236i \(-0.351571\pi\)
0.449587 + 0.893236i \(0.351571\pi\)
\(984\) 0 0
\(985\) −10653.9 −0.344631
\(986\) −3188.09 −0.102971
\(987\) 0 0
\(988\) 0 0
\(989\) 8675.38 0.278929
\(990\) 0 0
\(991\) 3550.23 0.113801 0.0569005 0.998380i \(-0.481878\pi\)
0.0569005 + 0.998380i \(0.481878\pi\)
\(992\) 5216.37 0.166956
\(993\) 0 0
\(994\) 250.757 0.00800153
\(995\) −44471.8 −1.41694
\(996\) 0 0
\(997\) −28338.4 −0.900188 −0.450094 0.892981i \(-0.648610\pi\)
−0.450094 + 0.892981i \(0.648610\pi\)
\(998\) 2127.76 0.0674882
\(999\) 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.bi.1.4 9
3.2 odd 2 507.4.a.o.1.6 9
13.12 even 2 1521.4.a.bf.1.6 9
39.5 even 4 507.4.b.k.337.9 18
39.8 even 4 507.4.b.k.337.10 18
39.38 odd 2 507.4.a.p.1.4 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.6 9 3.2 odd 2
507.4.a.p.1.4 yes 9 39.38 odd 2
507.4.b.k.337.9 18 39.5 even 4
507.4.b.k.337.10 18 39.8 even 4
1521.4.a.bf.1.6 9 13.12 even 2
1521.4.a.bi.1.4 9 1.1 even 1 trivial