Properties

Label 405.4.a.n.1.4
Level $405$
Weight $4$
Character 405.1
Self dual yes
Analytic conductor $23.896$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.8957735523\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - 2x^{6} - 44x^{5} + 74x^{4} + 479x^{3} - 460x^{2} - 1200x + 288 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.225250\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.225250 q^{2} -7.94926 q^{4} +5.00000 q^{5} -31.1940 q^{7} -3.59257 q^{8} +O(q^{10})\) \(q+0.225250 q^{2} -7.94926 q^{4} +5.00000 q^{5} -31.1940 q^{7} -3.59257 q^{8} +1.12625 q^{10} -18.1285 q^{11} -50.1562 q^{13} -7.02645 q^{14} +62.7849 q^{16} +131.631 q^{17} +23.2428 q^{19} -39.7463 q^{20} -4.08344 q^{22} +32.9856 q^{23} +25.0000 q^{25} -11.2977 q^{26} +247.969 q^{28} -125.817 q^{29} +125.113 q^{31} +42.8828 q^{32} +29.6499 q^{34} -155.970 q^{35} +99.9894 q^{37} +5.23544 q^{38} -17.9628 q^{40} -245.326 q^{41} +139.176 q^{43} +144.108 q^{44} +7.43001 q^{46} +472.961 q^{47} +630.067 q^{49} +5.63125 q^{50} +398.705 q^{52} -421.529 q^{53} -90.6424 q^{55} +112.067 q^{56} -28.3404 q^{58} +742.413 q^{59} +8.97736 q^{61} +28.1816 q^{62} -492.620 q^{64} -250.781 q^{65} +588.906 q^{67} -1046.37 q^{68} -35.1322 q^{70} +48.5526 q^{71} +409.800 q^{73} +22.5226 q^{74} -184.763 q^{76} +565.500 q^{77} +530.527 q^{79} +313.924 q^{80} -55.2595 q^{82} +294.590 q^{83} +658.155 q^{85} +31.3495 q^{86} +65.1278 q^{88} -852.817 q^{89} +1564.57 q^{91} -262.211 q^{92} +106.534 q^{94} +116.214 q^{95} -388.091 q^{97} +141.922 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8} + 10 q^{10} + 23 q^{11} + 96 q^{13} - 21 q^{14} + 324 q^{16} + 161 q^{17} + 279 q^{19} + 180 q^{20} + 311 q^{22} + 96 q^{23} + 175 q^{25} - 358 q^{26} + 337 q^{28} - 296 q^{29} + 244 q^{31} - 314 q^{32} + 125 q^{34} + 110 q^{35} + 404 q^{37} + 305 q^{38} + 90 q^{40} - 47 q^{41} + 525 q^{43} + 55 q^{44} + 717 q^{46} + 164 q^{47} + 1225 q^{49} + 50 q^{50} + 1682 q^{52} + 506 q^{53} + 115 q^{55} - 981 q^{56} + 1183 q^{58} - 85 q^{59} + 828 q^{61} - 786 q^{62} + 2236 q^{64} + 480 q^{65} + 1093 q^{67} + 2473 q^{68} - 105 q^{70} + 328 q^{71} + 2085 q^{73} - 1316 q^{74} + 2789 q^{76} + 24 q^{77} + 2110 q^{79} + 1620 q^{80} - 62 q^{82} + 1290 q^{83} + 805 q^{85} - 2569 q^{86} + 2271 q^{88} - 3048 q^{89} + 3338 q^{91} + 2763 q^{92} - 517 q^{94} + 1395 q^{95} + 1787 q^{97} + 1279 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.225250 0.0796378 0.0398189 0.999207i \(-0.487322\pi\)
0.0398189 + 0.999207i \(0.487322\pi\)
\(3\) 0 0
\(4\) −7.94926 −0.993658
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −31.1940 −1.68432 −0.842159 0.539229i \(-0.818716\pi\)
−0.842159 + 0.539229i \(0.818716\pi\)
\(8\) −3.59257 −0.158771
\(9\) 0 0
\(10\) 1.12625 0.0356151
\(11\) −18.1285 −0.496904 −0.248452 0.968644i \(-0.579922\pi\)
−0.248452 + 0.968644i \(0.579922\pi\)
\(12\) 0 0
\(13\) −50.1562 −1.07006 −0.535032 0.844832i \(-0.679700\pi\)
−0.535032 + 0.844832i \(0.679700\pi\)
\(14\) −7.02645 −0.134136
\(15\) 0 0
\(16\) 62.7849 0.981014
\(17\) 131.631 1.87795 0.938977 0.343981i \(-0.111776\pi\)
0.938977 + 0.343981i \(0.111776\pi\)
\(18\) 0 0
\(19\) 23.2428 0.280646 0.140323 0.990106i \(-0.455186\pi\)
0.140323 + 0.990106i \(0.455186\pi\)
\(20\) −39.7463 −0.444377
\(21\) 0 0
\(22\) −4.08344 −0.0395723
\(23\) 32.9856 0.299043 0.149521 0.988759i \(-0.452227\pi\)
0.149521 + 0.988759i \(0.452227\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −11.2977 −0.0852176
\(27\) 0 0
\(28\) 247.969 1.67364
\(29\) −125.817 −0.805645 −0.402823 0.915278i \(-0.631971\pi\)
−0.402823 + 0.915278i \(0.631971\pi\)
\(30\) 0 0
\(31\) 125.113 0.724868 0.362434 0.932009i \(-0.381946\pi\)
0.362434 + 0.932009i \(0.381946\pi\)
\(32\) 42.8828 0.236896
\(33\) 0 0
\(34\) 29.6499 0.149556
\(35\) −155.970 −0.753250
\(36\) 0 0
\(37\) 99.9894 0.444274 0.222137 0.975015i \(-0.428697\pi\)
0.222137 + 0.975015i \(0.428697\pi\)
\(38\) 5.23544 0.0223500
\(39\) 0 0
\(40\) −17.9628 −0.0710044
\(41\) −245.326 −0.934473 −0.467237 0.884132i \(-0.654750\pi\)
−0.467237 + 0.884132i \(0.654750\pi\)
\(42\) 0 0
\(43\) 139.176 0.493586 0.246793 0.969068i \(-0.420623\pi\)
0.246793 + 0.969068i \(0.420623\pi\)
\(44\) 144.108 0.493752
\(45\) 0 0
\(46\) 7.43001 0.0238151
\(47\) 472.961 1.46784 0.733919 0.679237i \(-0.237689\pi\)
0.733919 + 0.679237i \(0.237689\pi\)
\(48\) 0 0
\(49\) 630.067 1.83693
\(50\) 5.63125 0.0159276
\(51\) 0 0
\(52\) 398.705 1.06328
\(53\) −421.529 −1.09248 −0.546240 0.837628i \(-0.683941\pi\)
−0.546240 + 0.837628i \(0.683941\pi\)
\(54\) 0 0
\(55\) −90.6424 −0.222222
\(56\) 112.067 0.267420
\(57\) 0 0
\(58\) −28.3404 −0.0641598
\(59\) 742.413 1.63820 0.819101 0.573649i \(-0.194473\pi\)
0.819101 + 0.573649i \(0.194473\pi\)
\(60\) 0 0
\(61\) 8.97736 0.0188432 0.00942158 0.999956i \(-0.497001\pi\)
0.00942158 + 0.999956i \(0.497001\pi\)
\(62\) 28.1816 0.0577269
\(63\) 0 0
\(64\) −492.620 −0.962148
\(65\) −250.781 −0.478547
\(66\) 0 0
\(67\) 588.906 1.07383 0.536913 0.843638i \(-0.319590\pi\)
0.536913 + 0.843638i \(0.319590\pi\)
\(68\) −1046.37 −1.86604
\(69\) 0 0
\(70\) −35.1322 −0.0599872
\(71\) 48.5526 0.0811568 0.0405784 0.999176i \(-0.487080\pi\)
0.0405784 + 0.999176i \(0.487080\pi\)
\(72\) 0 0
\(73\) 409.800 0.657034 0.328517 0.944498i \(-0.393451\pi\)
0.328517 + 0.944498i \(0.393451\pi\)
\(74\) 22.5226 0.0353811
\(75\) 0 0
\(76\) −184.763 −0.278866
\(77\) 565.500 0.836944
\(78\) 0 0
\(79\) 530.527 0.755556 0.377778 0.925896i \(-0.376688\pi\)
0.377778 + 0.925896i \(0.376688\pi\)
\(80\) 313.924 0.438723
\(81\) 0 0
\(82\) −55.2595 −0.0744195
\(83\) 294.590 0.389584 0.194792 0.980845i \(-0.437597\pi\)
0.194792 + 0.980845i \(0.437597\pi\)
\(84\) 0 0
\(85\) 658.155 0.839846
\(86\) 31.3495 0.0393081
\(87\) 0 0
\(88\) 65.1278 0.0788937
\(89\) −852.817 −1.01571 −0.507856 0.861442i \(-0.669562\pi\)
−0.507856 + 0.861442i \(0.669562\pi\)
\(90\) 0 0
\(91\) 1564.57 1.80233
\(92\) −262.211 −0.297146
\(93\) 0 0
\(94\) 106.534 0.116896
\(95\) 116.214 0.125509
\(96\) 0 0
\(97\) −388.091 −0.406233 −0.203117 0.979155i \(-0.565107\pi\)
−0.203117 + 0.979155i \(0.565107\pi\)
\(98\) 141.922 0.146289
\(99\) 0 0
\(100\) −198.732 −0.198732
\(101\) 1079.99 1.06399 0.531997 0.846746i \(-0.321442\pi\)
0.531997 + 0.846746i \(0.321442\pi\)
\(102\) 0 0
\(103\) −594.997 −0.569192 −0.284596 0.958648i \(-0.591859\pi\)
−0.284596 + 0.958648i \(0.591859\pi\)
\(104\) 180.190 0.169895
\(105\) 0 0
\(106\) −94.9494 −0.0870028
\(107\) 498.693 0.450565 0.225282 0.974293i \(-0.427670\pi\)
0.225282 + 0.974293i \(0.427670\pi\)
\(108\) 0 0
\(109\) −959.301 −0.842976 −0.421488 0.906834i \(-0.638492\pi\)
−0.421488 + 0.906834i \(0.638492\pi\)
\(110\) −20.4172 −0.0176973
\(111\) 0 0
\(112\) −1958.51 −1.65234
\(113\) −1728.79 −1.43921 −0.719607 0.694382i \(-0.755678\pi\)
−0.719607 + 0.694382i \(0.755678\pi\)
\(114\) 0 0
\(115\) 164.928 0.133736
\(116\) 1000.16 0.800536
\(117\) 0 0
\(118\) 167.228 0.130463
\(119\) −4106.10 −3.16307
\(120\) 0 0
\(121\) −1002.36 −0.753087
\(122\) 2.02215 0.00150063
\(123\) 0 0
\(124\) −994.554 −0.720271
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 1019.95 0.712647 0.356324 0.934363i \(-0.384030\pi\)
0.356324 + 0.934363i \(0.384030\pi\)
\(128\) −454.025 −0.313520
\(129\) 0 0
\(130\) −56.4884 −0.0381105
\(131\) 1702.39 1.13541 0.567705 0.823232i \(-0.307831\pi\)
0.567705 + 0.823232i \(0.307831\pi\)
\(132\) 0 0
\(133\) −725.037 −0.472697
\(134\) 132.651 0.0855172
\(135\) 0 0
\(136\) −472.893 −0.298164
\(137\) 1049.68 0.654599 0.327300 0.944921i \(-0.393861\pi\)
0.327300 + 0.944921i \(0.393861\pi\)
\(138\) 0 0
\(139\) 1434.82 0.875540 0.437770 0.899087i \(-0.355768\pi\)
0.437770 + 0.899087i \(0.355768\pi\)
\(140\) 1239.85 0.748473
\(141\) 0 0
\(142\) 10.9365 0.00646316
\(143\) 909.256 0.531719
\(144\) 0 0
\(145\) −629.087 −0.360295
\(146\) 92.3075 0.0523248
\(147\) 0 0
\(148\) −794.842 −0.441457
\(149\) −1218.46 −0.669935 −0.334968 0.942230i \(-0.608725\pi\)
−0.334968 + 0.942230i \(0.608725\pi\)
\(150\) 0 0
\(151\) 1609.50 0.867414 0.433707 0.901054i \(-0.357205\pi\)
0.433707 + 0.901054i \(0.357205\pi\)
\(152\) −83.5014 −0.0445583
\(153\) 0 0
\(154\) 127.379 0.0666524
\(155\) 625.563 0.324171
\(156\) 0 0
\(157\) 1286.34 0.653890 0.326945 0.945043i \(-0.393981\pi\)
0.326945 + 0.945043i \(0.393981\pi\)
\(158\) 119.501 0.0601708
\(159\) 0 0
\(160\) 214.414 0.105943
\(161\) −1028.95 −0.503683
\(162\) 0 0
\(163\) 1416.84 0.680830 0.340415 0.940275i \(-0.389432\pi\)
0.340415 + 0.940275i \(0.389432\pi\)
\(164\) 1950.16 0.928547
\(165\) 0 0
\(166\) 66.3563 0.0310256
\(167\) −894.975 −0.414702 −0.207351 0.978267i \(-0.566484\pi\)
−0.207351 + 0.978267i \(0.566484\pi\)
\(168\) 0 0
\(169\) 318.645 0.145037
\(170\) 148.249 0.0668835
\(171\) 0 0
\(172\) −1106.35 −0.490456
\(173\) 1735.10 0.762526 0.381263 0.924467i \(-0.375489\pi\)
0.381263 + 0.924467i \(0.375489\pi\)
\(174\) 0 0
\(175\) −779.850 −0.336864
\(176\) −1138.19 −0.487469
\(177\) 0 0
\(178\) −192.097 −0.0808892
\(179\) −2133.37 −0.890815 −0.445408 0.895328i \(-0.646941\pi\)
−0.445408 + 0.895328i \(0.646941\pi\)
\(180\) 0 0
\(181\) −3611.98 −1.48330 −0.741648 0.670789i \(-0.765955\pi\)
−0.741648 + 0.670789i \(0.765955\pi\)
\(182\) 352.420 0.143534
\(183\) 0 0
\(184\) −118.503 −0.0474792
\(185\) 499.947 0.198686
\(186\) 0 0
\(187\) −2386.27 −0.933162
\(188\) −3759.69 −1.45853
\(189\) 0 0
\(190\) 26.1772 0.00999523
\(191\) 596.990 0.226160 0.113080 0.993586i \(-0.463928\pi\)
0.113080 + 0.993586i \(0.463928\pi\)
\(192\) 0 0
\(193\) −1207.74 −0.450442 −0.225221 0.974308i \(-0.572310\pi\)
−0.225221 + 0.974308i \(0.572310\pi\)
\(194\) −87.4173 −0.0323515
\(195\) 0 0
\(196\) −5008.57 −1.82528
\(197\) 3268.56 1.18211 0.591054 0.806632i \(-0.298712\pi\)
0.591054 + 0.806632i \(0.298712\pi\)
\(198\) 0 0
\(199\) −2109.88 −0.751585 −0.375793 0.926704i \(-0.622629\pi\)
−0.375793 + 0.926704i \(0.622629\pi\)
\(200\) −89.8142 −0.0317541
\(201\) 0 0
\(202\) 243.268 0.0847342
\(203\) 3924.75 1.35696
\(204\) 0 0
\(205\) −1226.63 −0.417909
\(206\) −134.023 −0.0453292
\(207\) 0 0
\(208\) −3149.05 −1.04975
\(209\) −421.357 −0.139454
\(210\) 0 0
\(211\) 658.642 0.214895 0.107447 0.994211i \(-0.465732\pi\)
0.107447 + 0.994211i \(0.465732\pi\)
\(212\) 3350.85 1.08555
\(213\) 0 0
\(214\) 112.330 0.0358820
\(215\) 695.882 0.220738
\(216\) 0 0
\(217\) −3902.77 −1.22091
\(218\) −216.082 −0.0671328
\(219\) 0 0
\(220\) 720.540 0.220813
\(221\) −6602.11 −2.00953
\(222\) 0 0
\(223\) 1218.79 0.365993 0.182997 0.983114i \(-0.441420\pi\)
0.182997 + 0.983114i \(0.441420\pi\)
\(224\) −1337.69 −0.399009
\(225\) 0 0
\(226\) −389.410 −0.114616
\(227\) 1525.57 0.446060 0.223030 0.974812i \(-0.428405\pi\)
0.223030 + 0.974812i \(0.428405\pi\)
\(228\) 0 0
\(229\) −6182.65 −1.78411 −0.892055 0.451927i \(-0.850737\pi\)
−0.892055 + 0.451927i \(0.850737\pi\)
\(230\) 37.1500 0.0106504
\(231\) 0 0
\(232\) 452.008 0.127913
\(233\) 2013.82 0.566222 0.283111 0.959087i \(-0.408633\pi\)
0.283111 + 0.959087i \(0.408633\pi\)
\(234\) 0 0
\(235\) 2364.80 0.656438
\(236\) −5901.64 −1.62781
\(237\) 0 0
\(238\) −924.898 −0.251900
\(239\) −5086.31 −1.37659 −0.688297 0.725429i \(-0.741641\pi\)
−0.688297 + 0.725429i \(0.741641\pi\)
\(240\) 0 0
\(241\) 3286.83 0.878519 0.439259 0.898360i \(-0.355241\pi\)
0.439259 + 0.898360i \(0.355241\pi\)
\(242\) −225.781 −0.0599742
\(243\) 0 0
\(244\) −71.3634 −0.0187237
\(245\) 3150.33 0.821500
\(246\) 0 0
\(247\) −1165.77 −0.300309
\(248\) −449.476 −0.115088
\(249\) 0 0
\(250\) 28.1562 0.00712303
\(251\) −3480.55 −0.875260 −0.437630 0.899155i \(-0.644182\pi\)
−0.437630 + 0.899155i \(0.644182\pi\)
\(252\) 0 0
\(253\) −597.979 −0.148595
\(254\) 229.744 0.0567537
\(255\) 0 0
\(256\) 3838.69 0.937180
\(257\) 6794.98 1.64926 0.824628 0.565675i \(-0.191384\pi\)
0.824628 + 0.565675i \(0.191384\pi\)
\(258\) 0 0
\(259\) −3119.07 −0.748300
\(260\) 1993.52 0.475512
\(261\) 0 0
\(262\) 383.464 0.0904216
\(263\) −5364.24 −1.25769 −0.628846 0.777530i \(-0.716472\pi\)
−0.628846 + 0.777530i \(0.716472\pi\)
\(264\) 0 0
\(265\) −2107.65 −0.488572
\(266\) −163.314 −0.0376445
\(267\) 0 0
\(268\) −4681.37 −1.06702
\(269\) −48.4985 −0.0109926 −0.00549629 0.999985i \(-0.501750\pi\)
−0.00549629 + 0.999985i \(0.501750\pi\)
\(270\) 0 0
\(271\) 7643.16 1.71324 0.856622 0.515945i \(-0.172559\pi\)
0.856622 + 0.515945i \(0.172559\pi\)
\(272\) 8264.43 1.84230
\(273\) 0 0
\(274\) 236.440 0.0521309
\(275\) −453.212 −0.0993807
\(276\) 0 0
\(277\) 5146.02 1.11623 0.558113 0.829765i \(-0.311526\pi\)
0.558113 + 0.829765i \(0.311526\pi\)
\(278\) 323.193 0.0697261
\(279\) 0 0
\(280\) 560.333 0.119594
\(281\) −5854.74 −1.24293 −0.621467 0.783440i \(-0.713463\pi\)
−0.621467 + 0.783440i \(0.713463\pi\)
\(282\) 0 0
\(283\) 9393.84 1.97317 0.986583 0.163260i \(-0.0522009\pi\)
0.986583 + 0.163260i \(0.0522009\pi\)
\(284\) −385.957 −0.0806421
\(285\) 0 0
\(286\) 204.810 0.0423449
\(287\) 7652.69 1.57395
\(288\) 0 0
\(289\) 12413.7 2.52671
\(290\) −141.702 −0.0286932
\(291\) 0 0
\(292\) −3257.61 −0.652867
\(293\) 1780.58 0.355026 0.177513 0.984118i \(-0.443195\pi\)
0.177513 + 0.984118i \(0.443195\pi\)
\(294\) 0 0
\(295\) 3712.07 0.732627
\(296\) −359.219 −0.0705377
\(297\) 0 0
\(298\) −274.458 −0.0533522
\(299\) −1654.43 −0.319995
\(300\) 0 0
\(301\) −4341.47 −0.831356
\(302\) 362.540 0.0690790
\(303\) 0 0
\(304\) 1459.30 0.275317
\(305\) 44.8868 0.00842692
\(306\) 0 0
\(307\) 8480.18 1.57651 0.788256 0.615347i \(-0.210984\pi\)
0.788256 + 0.615347i \(0.210984\pi\)
\(308\) −4495.31 −0.831636
\(309\) 0 0
\(310\) 140.908 0.0258163
\(311\) 6810.79 1.24182 0.620908 0.783884i \(-0.286764\pi\)
0.620908 + 0.783884i \(0.286764\pi\)
\(312\) 0 0
\(313\) 8046.81 1.45314 0.726570 0.687093i \(-0.241113\pi\)
0.726570 + 0.687093i \(0.241113\pi\)
\(314\) 289.747 0.0520744
\(315\) 0 0
\(316\) −4217.30 −0.750764
\(317\) −1566.22 −0.277501 −0.138751 0.990327i \(-0.544309\pi\)
−0.138751 + 0.990327i \(0.544309\pi\)
\(318\) 0 0
\(319\) 2280.88 0.400328
\(320\) −2463.10 −0.430286
\(321\) 0 0
\(322\) −231.772 −0.0401122
\(323\) 3059.47 0.527039
\(324\) 0 0
\(325\) −1253.91 −0.214013
\(326\) 319.142 0.0542198
\(327\) 0 0
\(328\) 881.349 0.148367
\(329\) −14753.5 −2.47231
\(330\) 0 0
\(331\) 2442.56 0.405605 0.202802 0.979220i \(-0.434995\pi\)
0.202802 + 0.979220i \(0.434995\pi\)
\(332\) −2341.77 −0.387113
\(333\) 0 0
\(334\) −201.593 −0.0330260
\(335\) 2944.53 0.480229
\(336\) 0 0
\(337\) 9472.98 1.53123 0.765617 0.643296i \(-0.222434\pi\)
0.765617 + 0.643296i \(0.222434\pi\)
\(338\) 71.7748 0.0115504
\(339\) 0 0
\(340\) −5231.85 −0.834520
\(341\) −2268.10 −0.360190
\(342\) 0 0
\(343\) −8954.76 −1.40966
\(344\) −500.001 −0.0783670
\(345\) 0 0
\(346\) 390.830 0.0607259
\(347\) 3312.47 0.512457 0.256229 0.966616i \(-0.417520\pi\)
0.256229 + 0.966616i \(0.417520\pi\)
\(348\) 0 0
\(349\) 5668.95 0.869490 0.434745 0.900554i \(-0.356838\pi\)
0.434745 + 0.900554i \(0.356838\pi\)
\(350\) −175.661 −0.0268271
\(351\) 0 0
\(352\) −777.400 −0.117715
\(353\) 1739.08 0.262215 0.131108 0.991368i \(-0.458147\pi\)
0.131108 + 0.991368i \(0.458147\pi\)
\(354\) 0 0
\(355\) 242.763 0.0362944
\(356\) 6779.27 1.00927
\(357\) 0 0
\(358\) −480.542 −0.0709426
\(359\) −8624.58 −1.26793 −0.633966 0.773361i \(-0.718574\pi\)
−0.633966 + 0.773361i \(0.718574\pi\)
\(360\) 0 0
\(361\) −6318.77 −0.921238
\(362\) −813.598 −0.118126
\(363\) 0 0
\(364\) −12437.2 −1.79090
\(365\) 2049.00 0.293835
\(366\) 0 0
\(367\) −6109.17 −0.868927 −0.434463 0.900689i \(-0.643062\pi\)
−0.434463 + 0.900689i \(0.643062\pi\)
\(368\) 2071.00 0.293365
\(369\) 0 0
\(370\) 112.613 0.0158229
\(371\) 13149.2 1.84009
\(372\) 0 0
\(373\) 6555.33 0.909979 0.454990 0.890497i \(-0.349643\pi\)
0.454990 + 0.890497i \(0.349643\pi\)
\(374\) −537.507 −0.0743150
\(375\) 0 0
\(376\) −1699.14 −0.233050
\(377\) 6310.52 0.862092
\(378\) 0 0
\(379\) 5032.40 0.682050 0.341025 0.940054i \(-0.389226\pi\)
0.341025 + 0.940054i \(0.389226\pi\)
\(380\) −923.816 −0.124713
\(381\) 0 0
\(382\) 134.472 0.0180109
\(383\) −3662.99 −0.488695 −0.244347 0.969688i \(-0.578574\pi\)
−0.244347 + 0.969688i \(0.578574\pi\)
\(384\) 0 0
\(385\) 2827.50 0.374293
\(386\) −272.044 −0.0358722
\(387\) 0 0
\(388\) 3085.03 0.403657
\(389\) 4870.96 0.634878 0.317439 0.948279i \(-0.397177\pi\)
0.317439 + 0.948279i \(0.397177\pi\)
\(390\) 0 0
\(391\) 4341.93 0.561588
\(392\) −2263.56 −0.291650
\(393\) 0 0
\(394\) 736.243 0.0941406
\(395\) 2652.63 0.337895
\(396\) 0 0
\(397\) −3744.62 −0.473393 −0.236696 0.971584i \(-0.576065\pi\)
−0.236696 + 0.971584i \(0.576065\pi\)
\(398\) −475.250 −0.0598546
\(399\) 0 0
\(400\) 1569.62 0.196203
\(401\) 1800.87 0.224267 0.112134 0.993693i \(-0.464232\pi\)
0.112134 + 0.993693i \(0.464232\pi\)
\(402\) 0 0
\(403\) −6275.18 −0.775655
\(404\) −8585.16 −1.05725
\(405\) 0 0
\(406\) 884.049 0.108066
\(407\) −1812.66 −0.220762
\(408\) 0 0
\(409\) 15932.0 1.92613 0.963064 0.269274i \(-0.0867837\pi\)
0.963064 + 0.269274i \(0.0867837\pi\)
\(410\) −276.298 −0.0332814
\(411\) 0 0
\(412\) 4729.79 0.565582
\(413\) −23158.8 −2.75926
\(414\) 0 0
\(415\) 1472.95 0.174227
\(416\) −2150.84 −0.253494
\(417\) 0 0
\(418\) −94.9106 −0.0111058
\(419\) 3447.75 0.401989 0.200995 0.979592i \(-0.435583\pi\)
0.200995 + 0.979592i \(0.435583\pi\)
\(420\) 0 0
\(421\) 5615.98 0.650133 0.325066 0.945691i \(-0.394613\pi\)
0.325066 + 0.945691i \(0.394613\pi\)
\(422\) 148.359 0.0171137
\(423\) 0 0
\(424\) 1514.37 0.173454
\(425\) 3290.77 0.375591
\(426\) 0 0
\(427\) −280.040 −0.0317379
\(428\) −3964.24 −0.447707
\(429\) 0 0
\(430\) 156.747 0.0175791
\(431\) 3534.04 0.394962 0.197481 0.980307i \(-0.436724\pi\)
0.197481 + 0.980307i \(0.436724\pi\)
\(432\) 0 0
\(433\) 6674.80 0.740809 0.370405 0.928871i \(-0.379219\pi\)
0.370405 + 0.928871i \(0.379219\pi\)
\(434\) −879.098 −0.0972305
\(435\) 0 0
\(436\) 7625.74 0.837630
\(437\) 766.679 0.0839250
\(438\) 0 0
\(439\) −6543.52 −0.711401 −0.355700 0.934600i \(-0.615758\pi\)
−0.355700 + 0.934600i \(0.615758\pi\)
\(440\) 325.639 0.0352823
\(441\) 0 0
\(442\) −1487.12 −0.160035
\(443\) −14309.7 −1.53471 −0.767353 0.641225i \(-0.778427\pi\)
−0.767353 + 0.641225i \(0.778427\pi\)
\(444\) 0 0
\(445\) −4264.09 −0.454241
\(446\) 274.533 0.0291469
\(447\) 0 0
\(448\) 15366.8 1.62056
\(449\) 14587.4 1.53324 0.766618 0.642104i \(-0.221938\pi\)
0.766618 + 0.642104i \(0.221938\pi\)
\(450\) 0 0
\(451\) 4447.38 0.464343
\(452\) 13742.6 1.43009
\(453\) 0 0
\(454\) 343.634 0.0355232
\(455\) 7822.87 0.806026
\(456\) 0 0
\(457\) 2025.77 0.207356 0.103678 0.994611i \(-0.466939\pi\)
0.103678 + 0.994611i \(0.466939\pi\)
\(458\) −1392.64 −0.142083
\(459\) 0 0
\(460\) −1311.06 −0.132888
\(461\) 3556.19 0.359280 0.179640 0.983732i \(-0.442507\pi\)
0.179640 + 0.983732i \(0.442507\pi\)
\(462\) 0 0
\(463\) −10282.7 −1.03213 −0.516066 0.856549i \(-0.672604\pi\)
−0.516066 + 0.856549i \(0.672604\pi\)
\(464\) −7899.43 −0.790349
\(465\) 0 0
\(466\) 453.613 0.0450927
\(467\) −8217.44 −0.814256 −0.407128 0.913371i \(-0.633470\pi\)
−0.407128 + 0.913371i \(0.633470\pi\)
\(468\) 0 0
\(469\) −18370.3 −1.80866
\(470\) 532.672 0.0522773
\(471\) 0 0
\(472\) −2667.17 −0.260098
\(473\) −2523.06 −0.245265
\(474\) 0 0
\(475\) 581.070 0.0561291
\(476\) 32640.5 3.14301
\(477\) 0 0
\(478\) −1145.69 −0.109629
\(479\) −13685.2 −1.30541 −0.652705 0.757612i \(-0.726366\pi\)
−0.652705 + 0.757612i \(0.726366\pi\)
\(480\) 0 0
\(481\) −5015.09 −0.475402
\(482\) 740.357 0.0699633
\(483\) 0 0
\(484\) 7968.01 0.748310
\(485\) −1940.45 −0.181673
\(486\) 0 0
\(487\) −3239.59 −0.301437 −0.150718 0.988577i \(-0.548159\pi\)
−0.150718 + 0.988577i \(0.548159\pi\)
\(488\) −32.2518 −0.00299174
\(489\) 0 0
\(490\) 709.612 0.0654225
\(491\) −11128.1 −1.02282 −0.511409 0.859338i \(-0.670876\pi\)
−0.511409 + 0.859338i \(0.670876\pi\)
\(492\) 0 0
\(493\) −16561.5 −1.51296
\(494\) −262.590 −0.0239159
\(495\) 0 0
\(496\) 7855.18 0.711105
\(497\) −1514.55 −0.136694
\(498\) 0 0
\(499\) −5427.22 −0.486885 −0.243443 0.969915i \(-0.578277\pi\)
−0.243443 + 0.969915i \(0.578277\pi\)
\(500\) −993.658 −0.0888755
\(501\) 0 0
\(502\) −783.993 −0.0697038
\(503\) 9600.22 0.850999 0.425500 0.904959i \(-0.360098\pi\)
0.425500 + 0.904959i \(0.360098\pi\)
\(504\) 0 0
\(505\) 5399.97 0.475833
\(506\) −134.695 −0.0118338
\(507\) 0 0
\(508\) −8107.87 −0.708127
\(509\) 18939.9 1.64930 0.824652 0.565641i \(-0.191371\pi\)
0.824652 + 0.565641i \(0.191371\pi\)
\(510\) 0 0
\(511\) −12783.3 −1.10666
\(512\) 4496.87 0.388155
\(513\) 0 0
\(514\) 1530.57 0.131343
\(515\) −2974.98 −0.254550
\(516\) 0 0
\(517\) −8574.06 −0.729375
\(518\) −702.570 −0.0595930
\(519\) 0 0
\(520\) 900.948 0.0759792
\(521\) −19292.6 −1.62231 −0.811155 0.584831i \(-0.801161\pi\)
−0.811155 + 0.584831i \(0.801161\pi\)
\(522\) 0 0
\(523\) 17967.5 1.50223 0.751114 0.660172i \(-0.229517\pi\)
0.751114 + 0.660172i \(0.229517\pi\)
\(524\) −13532.8 −1.12821
\(525\) 0 0
\(526\) −1208.29 −0.100160
\(527\) 16468.7 1.36127
\(528\) 0 0
\(529\) −11078.9 −0.910574
\(530\) −474.747 −0.0389088
\(531\) 0 0
\(532\) 5763.51 0.469699
\(533\) 12304.6 0.999946
\(534\) 0 0
\(535\) 2493.46 0.201499
\(536\) −2115.69 −0.170492
\(537\) 0 0
\(538\) −10.9243 −0.000875426 0
\(539\) −11422.1 −0.912777
\(540\) 0 0
\(541\) −8299.36 −0.659552 −0.329776 0.944059i \(-0.606973\pi\)
−0.329776 + 0.944059i \(0.606973\pi\)
\(542\) 1721.62 0.136439
\(543\) 0 0
\(544\) 5644.71 0.444880
\(545\) −4796.50 −0.376990
\(546\) 0 0
\(547\) 84.7210 0.00662232 0.00331116 0.999995i \(-0.498946\pi\)
0.00331116 + 0.999995i \(0.498946\pi\)
\(548\) −8344.17 −0.650448
\(549\) 0 0
\(550\) −102.086 −0.00791447
\(551\) −2924.35 −0.226101
\(552\) 0 0
\(553\) −16549.3 −1.27260
\(554\) 1159.14 0.0888938
\(555\) 0 0
\(556\) −11405.8 −0.869987
\(557\) 20914.0 1.59094 0.795469 0.605994i \(-0.207224\pi\)
0.795469 + 0.605994i \(0.207224\pi\)
\(558\) 0 0
\(559\) −6980.56 −0.528169
\(560\) −9792.56 −0.738949
\(561\) 0 0
\(562\) −1318.78 −0.0989846
\(563\) −11041.0 −0.826503 −0.413252 0.910617i \(-0.635607\pi\)
−0.413252 + 0.910617i \(0.635607\pi\)
\(564\) 0 0
\(565\) −8643.96 −0.643636
\(566\) 2115.96 0.157139
\(567\) 0 0
\(568\) −174.429 −0.0128853
\(569\) 6014.62 0.443139 0.221569 0.975145i \(-0.428882\pi\)
0.221569 + 0.975145i \(0.428882\pi\)
\(570\) 0 0
\(571\) −21937.3 −1.60779 −0.803896 0.594770i \(-0.797243\pi\)
−0.803896 + 0.594770i \(0.797243\pi\)
\(572\) −7227.91 −0.528346
\(573\) 0 0
\(574\) 1723.77 0.125346
\(575\) 824.641 0.0598085
\(576\) 0 0
\(577\) 473.507 0.0341635 0.0170818 0.999854i \(-0.494562\pi\)
0.0170818 + 0.999854i \(0.494562\pi\)
\(578\) 2796.19 0.201222
\(579\) 0 0
\(580\) 5000.78 0.358010
\(581\) −9189.44 −0.656183
\(582\) 0 0
\(583\) 7641.68 0.542858
\(584\) −1472.24 −0.104318
\(585\) 0 0
\(586\) 401.076 0.0282735
\(587\) −13020.3 −0.915514 −0.457757 0.889077i \(-0.651347\pi\)
−0.457757 + 0.889077i \(0.651347\pi\)
\(588\) 0 0
\(589\) 2907.97 0.203431
\(590\) 836.142 0.0583448
\(591\) 0 0
\(592\) 6277.82 0.435839
\(593\) 12887.3 0.892441 0.446220 0.894923i \(-0.352770\pi\)
0.446220 + 0.894923i \(0.352770\pi\)
\(594\) 0 0
\(595\) −20530.5 −1.41457
\(596\) 9685.88 0.665686
\(597\) 0 0
\(598\) −372.661 −0.0254837
\(599\) 8674.48 0.591702 0.295851 0.955234i \(-0.404397\pi\)
0.295851 + 0.955234i \(0.404397\pi\)
\(600\) 0 0
\(601\) 10935.7 0.742225 0.371112 0.928588i \(-0.378976\pi\)
0.371112 + 0.928588i \(0.378976\pi\)
\(602\) −977.916 −0.0662074
\(603\) 0 0
\(604\) −12794.4 −0.861913
\(605\) −5011.79 −0.336791
\(606\) 0 0
\(607\) 21670.4 1.44905 0.724527 0.689246i \(-0.242058\pi\)
0.724527 + 0.689246i \(0.242058\pi\)
\(608\) 996.718 0.0664840
\(609\) 0 0
\(610\) 10.1107 0.000671102 0
\(611\) −23721.9 −1.57068
\(612\) 0 0
\(613\) −15571.2 −1.02596 −0.512982 0.858399i \(-0.671459\pi\)
−0.512982 + 0.858399i \(0.671459\pi\)
\(614\) 1910.16 0.125550
\(615\) 0 0
\(616\) −2031.60 −0.132882
\(617\) −9726.54 −0.634645 −0.317322 0.948318i \(-0.602784\pi\)
−0.317322 + 0.948318i \(0.602784\pi\)
\(618\) 0 0
\(619\) 6297.59 0.408920 0.204460 0.978875i \(-0.434456\pi\)
0.204460 + 0.978875i \(0.434456\pi\)
\(620\) −4972.77 −0.322115
\(621\) 0 0
\(622\) 1534.13 0.0988955
\(623\) 26602.8 1.71078
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 1812.54 0.115725
\(627\) 0 0
\(628\) −10225.4 −0.649743
\(629\) 13161.7 0.834327
\(630\) 0 0
\(631\) −5670.98 −0.357778 −0.178889 0.983869i \(-0.557250\pi\)
−0.178889 + 0.983869i \(0.557250\pi\)
\(632\) −1905.95 −0.119960
\(633\) 0 0
\(634\) −352.792 −0.0220996
\(635\) 5099.76 0.318705
\(636\) 0 0
\(637\) −31601.8 −1.96563
\(638\) 513.767 0.0318813
\(639\) 0 0
\(640\) −2270.13 −0.140210
\(641\) −21664.5 −1.33494 −0.667470 0.744637i \(-0.732623\pi\)
−0.667470 + 0.744637i \(0.732623\pi\)
\(642\) 0 0
\(643\) 6223.72 0.381710 0.190855 0.981618i \(-0.438874\pi\)
0.190855 + 0.981618i \(0.438874\pi\)
\(644\) 8179.43 0.500488
\(645\) 0 0
\(646\) 689.146 0.0419723
\(647\) −20451.9 −1.24273 −0.621365 0.783521i \(-0.713422\pi\)
−0.621365 + 0.783521i \(0.713422\pi\)
\(648\) 0 0
\(649\) −13458.8 −0.814029
\(650\) −282.442 −0.0170435
\(651\) 0 0
\(652\) −11262.8 −0.676512
\(653\) −7448.44 −0.446371 −0.223185 0.974776i \(-0.571646\pi\)
−0.223185 + 0.974776i \(0.571646\pi\)
\(654\) 0 0
\(655\) 8511.96 0.507771
\(656\) −15402.7 −0.916731
\(657\) 0 0
\(658\) −3323.23 −0.196889
\(659\) −826.370 −0.0488480 −0.0244240 0.999702i \(-0.507775\pi\)
−0.0244240 + 0.999702i \(0.507775\pi\)
\(660\) 0 0
\(661\) −6258.22 −0.368255 −0.184127 0.982902i \(-0.558946\pi\)
−0.184127 + 0.982902i \(0.558946\pi\)
\(662\) 550.186 0.0323015
\(663\) 0 0
\(664\) −1058.33 −0.0618544
\(665\) −3625.18 −0.211396
\(666\) 0 0
\(667\) −4150.17 −0.240922
\(668\) 7114.39 0.412072
\(669\) 0 0
\(670\) 663.255 0.0382444
\(671\) −162.746 −0.00936324
\(672\) 0 0
\(673\) −10079.8 −0.577337 −0.288668 0.957429i \(-0.593213\pi\)
−0.288668 + 0.957429i \(0.593213\pi\)
\(674\) 2133.79 0.121944
\(675\) 0 0
\(676\) −2532.99 −0.144117
\(677\) 25710.4 1.45958 0.729788 0.683674i \(-0.239619\pi\)
0.729788 + 0.683674i \(0.239619\pi\)
\(678\) 0 0
\(679\) 12106.1 0.684226
\(680\) −2364.47 −0.133343
\(681\) 0 0
\(682\) −510.890 −0.0286847
\(683\) 32091.0 1.79784 0.898922 0.438109i \(-0.144352\pi\)
0.898922 + 0.438109i \(0.144352\pi\)
\(684\) 0 0
\(685\) 5248.39 0.292746
\(686\) −2017.06 −0.112262
\(687\) 0 0
\(688\) 8738.17 0.484215
\(689\) 21142.3 1.16902
\(690\) 0 0
\(691\) 10565.7 0.581679 0.290839 0.956772i \(-0.406066\pi\)
0.290839 + 0.956772i \(0.406066\pi\)
\(692\) −13792.7 −0.757690
\(693\) 0 0
\(694\) 746.133 0.0408110
\(695\) 7174.11 0.391553
\(696\) 0 0
\(697\) −32292.4 −1.75490
\(698\) 1276.93 0.0692443
\(699\) 0 0
\(700\) 6199.24 0.334727
\(701\) −13081.8 −0.704837 −0.352419 0.935842i \(-0.614641\pi\)
−0.352419 + 0.935842i \(0.614641\pi\)
\(702\) 0 0
\(703\) 2324.04 0.124684
\(704\) 8930.44 0.478095
\(705\) 0 0
\(706\) 391.728 0.0208823
\(707\) −33689.4 −1.79211
\(708\) 0 0
\(709\) −28221.2 −1.49488 −0.747440 0.664329i \(-0.768717\pi\)
−0.747440 + 0.664329i \(0.768717\pi\)
\(710\) 54.6823 0.00289041
\(711\) 0 0
\(712\) 3063.80 0.161265
\(713\) 4126.92 0.216766
\(714\) 0 0
\(715\) 4546.28 0.237792
\(716\) 16958.8 0.885165
\(717\) 0 0
\(718\) −1942.68 −0.100975
\(719\) 11471.4 0.595010 0.297505 0.954720i \(-0.403846\pi\)
0.297505 + 0.954720i \(0.403846\pi\)
\(720\) 0 0
\(721\) 18560.3 0.958701
\(722\) −1423.30 −0.0733654
\(723\) 0 0
\(724\) 28712.6 1.47389
\(725\) −3145.44 −0.161129
\(726\) 0 0
\(727\) 21659.3 1.10495 0.552475 0.833530i \(-0.313684\pi\)
0.552475 + 0.833530i \(0.313684\pi\)
\(728\) −5620.84 −0.286157
\(729\) 0 0
\(730\) 461.537 0.0234004
\(731\) 18319.9 0.926932
\(732\) 0 0
\(733\) 5681.72 0.286302 0.143151 0.989701i \(-0.454277\pi\)
0.143151 + 0.989701i \(0.454277\pi\)
\(734\) −1376.09 −0.0691995
\(735\) 0 0
\(736\) 1414.52 0.0708421
\(737\) −10676.0 −0.533588
\(738\) 0 0
\(739\) 261.324 0.0130080 0.00650402 0.999979i \(-0.497930\pi\)
0.00650402 + 0.999979i \(0.497930\pi\)
\(740\) −3974.21 −0.197425
\(741\) 0 0
\(742\) 2961.85 0.146540
\(743\) 16405.9 0.810061 0.405031 0.914303i \(-0.367261\pi\)
0.405031 + 0.914303i \(0.367261\pi\)
\(744\) 0 0
\(745\) −6092.31 −0.299604
\(746\) 1476.59 0.0724688
\(747\) 0 0
\(748\) 18969.1 0.927244
\(749\) −15556.2 −0.758895
\(750\) 0 0
\(751\) 21474.6 1.04343 0.521716 0.853119i \(-0.325292\pi\)
0.521716 + 0.853119i \(0.325292\pi\)
\(752\) 29694.8 1.43997
\(753\) 0 0
\(754\) 1421.44 0.0686551
\(755\) 8047.51 0.387919
\(756\) 0 0
\(757\) −13643.2 −0.655046 −0.327523 0.944843i \(-0.606214\pi\)
−0.327523 + 0.944843i \(0.606214\pi\)
\(758\) 1133.55 0.0543170
\(759\) 0 0
\(760\) −417.507 −0.0199271
\(761\) −26937.9 −1.28318 −0.641589 0.767048i \(-0.721725\pi\)
−0.641589 + 0.767048i \(0.721725\pi\)
\(762\) 0 0
\(763\) 29924.4 1.41984
\(764\) −4745.63 −0.224726
\(765\) 0 0
\(766\) −825.088 −0.0389186
\(767\) −37236.6 −1.75298
\(768\) 0 0
\(769\) −28885.4 −1.35453 −0.677265 0.735739i \(-0.736835\pi\)
−0.677265 + 0.735739i \(0.736835\pi\)
\(770\) 636.894 0.0298079
\(771\) 0 0
\(772\) 9600.67 0.447585
\(773\) −3031.34 −0.141048 −0.0705238 0.997510i \(-0.522467\pi\)
−0.0705238 + 0.997510i \(0.522467\pi\)
\(774\) 0 0
\(775\) 3127.82 0.144974
\(776\) 1394.24 0.0644979
\(777\) 0 0
\(778\) 1097.18 0.0505603
\(779\) −5702.06 −0.262256
\(780\) 0 0
\(781\) −880.185 −0.0403271
\(782\) 978.019 0.0447236
\(783\) 0 0
\(784\) 39558.7 1.80205
\(785\) 6431.68 0.292429
\(786\) 0 0
\(787\) 15154.8 0.686416 0.343208 0.939259i \(-0.388486\pi\)
0.343208 + 0.939259i \(0.388486\pi\)
\(788\) −25982.7 −1.17461
\(789\) 0 0
\(790\) 597.505 0.0269092
\(791\) 53928.0 2.42409
\(792\) 0 0
\(793\) −450.270 −0.0201634
\(794\) −843.474 −0.0377000
\(795\) 0 0
\(796\) 16772.0 0.746819
\(797\) −28380.6 −1.26135 −0.630673 0.776048i \(-0.717221\pi\)
−0.630673 + 0.776048i \(0.717221\pi\)
\(798\) 0 0
\(799\) 62256.3 2.75653
\(800\) 1072.07 0.0473793
\(801\) 0 0
\(802\) 405.646 0.0178602
\(803\) −7429.06 −0.326483
\(804\) 0 0
\(805\) −5144.77 −0.225254
\(806\) −1413.48 −0.0617715
\(807\) 0 0
\(808\) −3879.95 −0.168931
\(809\) 14569.6 0.633175 0.316588 0.948563i \(-0.397463\pi\)
0.316588 + 0.948563i \(0.397463\pi\)
\(810\) 0 0
\(811\) 27927.7 1.20921 0.604607 0.796524i \(-0.293330\pi\)
0.604607 + 0.796524i \(0.293330\pi\)
\(812\) −31198.9 −1.34836
\(813\) 0 0
\(814\) −408.300 −0.0175810
\(815\) 7084.19 0.304476
\(816\) 0 0
\(817\) 3234.85 0.138523
\(818\) 3588.68 0.153393
\(819\) 0 0
\(820\) 9750.79 0.415259
\(821\) −18044.1 −0.767043 −0.383521 0.923532i \(-0.625289\pi\)
−0.383521 + 0.923532i \(0.625289\pi\)
\(822\) 0 0
\(823\) −32252.8 −1.36606 −0.683028 0.730393i \(-0.739337\pi\)
−0.683028 + 0.730393i \(0.739337\pi\)
\(824\) 2137.57 0.0903710
\(825\) 0 0
\(826\) −5216.53 −0.219741
\(827\) −13569.5 −0.570566 −0.285283 0.958443i \(-0.592088\pi\)
−0.285283 + 0.958443i \(0.592088\pi\)
\(828\) 0 0
\(829\) 742.559 0.0311099 0.0155550 0.999879i \(-0.495049\pi\)
0.0155550 + 0.999879i \(0.495049\pi\)
\(830\) 331.782 0.0138751
\(831\) 0 0
\(832\) 24707.9 1.02956
\(833\) 82936.3 3.44967
\(834\) 0 0
\(835\) −4474.87 −0.185460
\(836\) 3349.48 0.138569
\(837\) 0 0
\(838\) 776.605 0.0320136
\(839\) 35504.7 1.46097 0.730487 0.682927i \(-0.239293\pi\)
0.730487 + 0.682927i \(0.239293\pi\)
\(840\) 0 0
\(841\) −8558.98 −0.350936
\(842\) 1265.00 0.0517752
\(843\) 0 0
\(844\) −5235.72 −0.213532
\(845\) 1593.23 0.0648623
\(846\) 0 0
\(847\) 31267.6 1.26844
\(848\) −26465.7 −1.07174
\(849\) 0 0
\(850\) 741.246 0.0299112
\(851\) 3298.21 0.132857
\(852\) 0 0
\(853\) 375.048 0.0150544 0.00752719 0.999972i \(-0.497604\pi\)
0.00752719 + 0.999972i \(0.497604\pi\)
\(854\) −63.0790 −0.00252754
\(855\) 0 0
\(856\) −1791.59 −0.0715365
\(857\) −32194.8 −1.28326 −0.641629 0.767015i \(-0.721741\pi\)
−0.641629 + 0.767015i \(0.721741\pi\)
\(858\) 0 0
\(859\) −28137.5 −1.11763 −0.558813 0.829294i \(-0.688743\pi\)
−0.558813 + 0.829294i \(0.688743\pi\)
\(860\) −5531.75 −0.219338
\(861\) 0 0
\(862\) 796.042 0.0314539
\(863\) −15724.6 −0.620244 −0.310122 0.950697i \(-0.600370\pi\)
−0.310122 + 0.950697i \(0.600370\pi\)
\(864\) 0 0
\(865\) 8675.48 0.341012
\(866\) 1503.50 0.0589964
\(867\) 0 0
\(868\) 31024.1 1.21317
\(869\) −9617.64 −0.375438
\(870\) 0 0
\(871\) −29537.3 −1.14906
\(872\) 3446.35 0.133840
\(873\) 0 0
\(874\) 172.694 0.00668361
\(875\) −3899.25 −0.150650
\(876\) 0 0
\(877\) 15280.5 0.588352 0.294176 0.955751i \(-0.404955\pi\)
0.294176 + 0.955751i \(0.404955\pi\)
\(878\) −1473.93 −0.0566544
\(879\) 0 0
\(880\) −5690.97 −0.218003
\(881\) 24687.6 0.944092 0.472046 0.881574i \(-0.343516\pi\)
0.472046 + 0.881574i \(0.343516\pi\)
\(882\) 0 0
\(883\) 6562.59 0.250112 0.125056 0.992150i \(-0.460089\pi\)
0.125056 + 0.992150i \(0.460089\pi\)
\(884\) 52481.9 1.99678
\(885\) 0 0
\(886\) −3223.26 −0.122221
\(887\) −50511.1 −1.91206 −0.956030 0.293270i \(-0.905256\pi\)
−0.956030 + 0.293270i \(0.905256\pi\)
\(888\) 0 0
\(889\) −31816.4 −1.20032
\(890\) −960.485 −0.0361747
\(891\) 0 0
\(892\) −9688.51 −0.363672
\(893\) 10992.9 0.411943
\(894\) 0 0
\(895\) −10666.9 −0.398385
\(896\) 14162.9 0.528067
\(897\) 0 0
\(898\) 3285.81 0.122104
\(899\) −15741.4 −0.583986
\(900\) 0 0
\(901\) −55486.3 −2.05163
\(902\) 1001.77 0.0369793
\(903\) 0 0
\(904\) 6210.81 0.228505
\(905\) −18059.9 −0.663350
\(906\) 0 0
\(907\) −7017.44 −0.256902 −0.128451 0.991716i \(-0.541001\pi\)
−0.128451 + 0.991716i \(0.541001\pi\)
\(908\) −12127.2 −0.443231
\(909\) 0 0
\(910\) 1762.10 0.0641902
\(911\) 6124.59 0.222741 0.111370 0.993779i \(-0.464476\pi\)
0.111370 + 0.993779i \(0.464476\pi\)
\(912\) 0 0
\(913\) −5340.47 −0.193586
\(914\) 456.305 0.0165134
\(915\) 0 0
\(916\) 49147.5 1.77279
\(917\) −53104.4 −1.91239
\(918\) 0 0
\(919\) 23780.7 0.853594 0.426797 0.904347i \(-0.359642\pi\)
0.426797 + 0.904347i \(0.359642\pi\)
\(920\) −592.516 −0.0212333
\(921\) 0 0
\(922\) 801.031 0.0286123
\(923\) −2435.22 −0.0868430
\(924\) 0 0
\(925\) 2499.74 0.0888549
\(926\) −2316.17 −0.0821967
\(927\) 0 0
\(928\) −5395.41 −0.190854
\(929\) −10189.1 −0.359842 −0.179921 0.983681i \(-0.557584\pi\)
−0.179921 + 0.983681i \(0.557584\pi\)
\(930\) 0 0
\(931\) 14644.5 0.515526
\(932\) −16008.4 −0.562631
\(933\) 0 0
\(934\) −1850.98 −0.0648456
\(935\) −11931.3 −0.417323
\(936\) 0 0
\(937\) −3880.22 −0.135284 −0.0676421 0.997710i \(-0.521548\pi\)
−0.0676421 + 0.997710i \(0.521548\pi\)
\(938\) −4137.92 −0.144038
\(939\) 0 0
\(940\) −18798.4 −0.652274
\(941\) 38673.9 1.33978 0.669890 0.742461i \(-0.266341\pi\)
0.669890 + 0.742461i \(0.266341\pi\)
\(942\) 0 0
\(943\) −8092.22 −0.279447
\(944\) 46612.3 1.60710
\(945\) 0 0
\(946\) −568.318 −0.0195324
\(947\) 7754.88 0.266103 0.133052 0.991109i \(-0.457522\pi\)
0.133052 + 0.991109i \(0.457522\pi\)
\(948\) 0 0
\(949\) −20554.0 −0.703069
\(950\) 130.886 0.00447000
\(951\) 0 0
\(952\) 14751.4 0.502203
\(953\) 22527.9 0.765739 0.382870 0.923802i \(-0.374936\pi\)
0.382870 + 0.923802i \(0.374936\pi\)
\(954\) 0 0
\(955\) 2984.95 0.101142
\(956\) 40432.4 1.36786
\(957\) 0 0
\(958\) −3082.58 −0.103960
\(959\) −32743.7 −1.10255
\(960\) 0 0
\(961\) −14137.8 −0.474567
\(962\) −1129.65 −0.0378600
\(963\) 0 0
\(964\) −26127.8 −0.872947
\(965\) −6038.72 −0.201444
\(966\) 0 0
\(967\) −5652.41 −0.187972 −0.0939861 0.995574i \(-0.529961\pi\)
−0.0939861 + 0.995574i \(0.529961\pi\)
\(968\) 3601.04 0.119568
\(969\) 0 0
\(970\) −437.087 −0.0144680
\(971\) −19612.4 −0.648188 −0.324094 0.946025i \(-0.605059\pi\)
−0.324094 + 0.946025i \(0.605059\pi\)
\(972\) 0 0
\(973\) −44757.9 −1.47469
\(974\) −729.716 −0.0240058
\(975\) 0 0
\(976\) 563.642 0.0184854
\(977\) 27683.6 0.906528 0.453264 0.891376i \(-0.350259\pi\)
0.453264 + 0.891376i \(0.350259\pi\)
\(978\) 0 0
\(979\) 15460.3 0.504712
\(980\) −25042.8 −0.816290
\(981\) 0 0
\(982\) −2506.60 −0.0814549
\(983\) 30616.4 0.993400 0.496700 0.867922i \(-0.334545\pi\)
0.496700 + 0.867922i \(0.334545\pi\)
\(984\) 0 0
\(985\) 16342.8 0.528655
\(986\) −3730.47 −0.120489
\(987\) 0 0
\(988\) 9267.02 0.298404
\(989\) 4590.82 0.147603
\(990\) 0 0
\(991\) −14462.4 −0.463587 −0.231793 0.972765i \(-0.574459\pi\)
−0.231793 + 0.972765i \(0.574459\pi\)
\(992\) 5365.19 0.171719
\(993\) 0 0
\(994\) −341.152 −0.0108860
\(995\) −10549.4 −0.336119
\(996\) 0 0
\(997\) −6897.26 −0.219096 −0.109548 0.993982i \(-0.534940\pi\)
−0.109548 + 0.993982i \(0.534940\pi\)
\(998\) −1222.48 −0.0387745
\(999\) 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 405.4.a.n.1.4 7
3.2 odd 2 405.4.a.m.1.4 7
5.4 even 2 2025.4.a.ba.1.4 7
9.2 odd 6 45.4.e.c.31.4 yes 14
9.4 even 3 135.4.e.c.46.4 14
9.5 odd 6 45.4.e.c.16.4 14
9.7 even 3 135.4.e.c.91.4 14
15.14 odd 2 2025.4.a.bb.1.4 7
45.2 even 12 225.4.k.d.49.7 28
45.14 odd 6 225.4.e.d.151.4 14
45.23 even 12 225.4.k.d.124.7 28
45.29 odd 6 225.4.e.d.76.4 14
45.32 even 12 225.4.k.d.124.8 28
45.38 even 12 225.4.k.d.49.8 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.4.e.c.16.4 14 9.5 odd 6
45.4.e.c.31.4 yes 14 9.2 odd 6
135.4.e.c.46.4 14 9.4 even 3
135.4.e.c.91.4 14 9.7 even 3
225.4.e.d.76.4 14 45.29 odd 6
225.4.e.d.151.4 14 45.14 odd 6
225.4.k.d.49.7 28 45.2 even 12
225.4.k.d.49.8 28 45.38 even 12
225.4.k.d.124.7 28 45.23 even 12
225.4.k.d.124.8 28 45.32 even 12
405.4.a.m.1.4 7 3.2 odd 2
405.4.a.n.1.4 7 1.1 even 1 trivial
2025.4.a.ba.1.4 7 5.4 even 2
2025.4.a.bb.1.4 7 15.14 odd 2