Properties

Label 1521.4.a.bi.1.8
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 1.8
Root \(-4.14324\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.69820 q^{2} +14.0731 q^{4} +4.47249 q^{5} -27.2096 q^{7} +28.5326 q^{8} +O(q^{10})\) \(q+4.69820 q^{2} +14.0731 q^{4} +4.47249 q^{5} -27.2096 q^{7} +28.5326 q^{8} +21.0127 q^{10} -5.99207 q^{11} -127.836 q^{14} +21.4672 q^{16} -105.037 q^{17} +156.462 q^{19} +62.9418 q^{20} -28.1519 q^{22} +175.423 q^{23} -104.997 q^{25} -382.923 q^{28} -204.886 q^{29} -31.9570 q^{31} -127.404 q^{32} -493.483 q^{34} -121.695 q^{35} -344.140 q^{37} +735.088 q^{38} +127.612 q^{40} +46.5921 q^{41} -173.286 q^{43} -84.3269 q^{44} +824.175 q^{46} -265.613 q^{47} +397.361 q^{49} -493.296 q^{50} -172.912 q^{53} -26.7995 q^{55} -776.360 q^{56} -962.596 q^{58} -137.566 q^{59} -58.9384 q^{61} -150.140 q^{62} -770.306 q^{64} -211.668 q^{67} -1478.19 q^{68} -571.746 q^{70} -436.317 q^{71} -1159.11 q^{73} -1616.84 q^{74} +2201.90 q^{76} +163.042 q^{77} -1017.51 q^{79} +96.0118 q^{80} +218.899 q^{82} -150.251 q^{83} -469.775 q^{85} -814.131 q^{86} -170.969 q^{88} +565.984 q^{89} +2468.75 q^{92} -1247.90 q^{94} +699.773 q^{95} +286.741 q^{97} +1866.88 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.69820 1.66106 0.830532 0.556970i \(-0.188036\pi\)
0.830532 + 0.556970i \(0.188036\pi\)
\(3\) 0 0
\(4\) 14.0731 1.75914
\(5\) 4.47249 0.400032 0.200016 0.979793i \(-0.435901\pi\)
0.200016 + 0.979793i \(0.435901\pi\)
\(6\) 0 0
\(7\) −27.2096 −1.46918 −0.734589 0.678512i \(-0.762625\pi\)
−0.734589 + 0.678512i \(0.762625\pi\)
\(8\) 28.5326 1.26098
\(9\) 0 0
\(10\) 21.0127 0.664479
\(11\) −5.99207 −0.164243 −0.0821216 0.996622i \(-0.526170\pi\)
−0.0821216 + 0.996622i \(0.526170\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −127.836 −2.44040
\(15\) 0 0
\(16\) 21.4672 0.335425
\(17\) −105.037 −1.49854 −0.749268 0.662267i \(-0.769594\pi\)
−0.749268 + 0.662267i \(0.769594\pi\)
\(18\) 0 0
\(19\) 156.462 1.88920 0.944599 0.328227i \(-0.106451\pi\)
0.944599 + 0.328227i \(0.106451\pi\)
\(20\) 62.9418 0.703711
\(21\) 0 0
\(22\) −28.1519 −0.272819
\(23\) 175.423 1.59036 0.795181 0.606372i \(-0.207376\pi\)
0.795181 + 0.606372i \(0.207376\pi\)
\(24\) 0 0
\(25\) −104.997 −0.839975
\(26\) 0 0
\(27\) 0 0
\(28\) −382.923 −2.58449
\(29\) −204.886 −1.31195 −0.655973 0.754785i \(-0.727741\pi\)
−0.655973 + 0.754785i \(0.727741\pi\)
\(30\) 0 0
\(31\) −31.9570 −0.185150 −0.0925748 0.995706i \(-0.529510\pi\)
−0.0925748 + 0.995706i \(0.529510\pi\)
\(32\) −127.404 −0.703813
\(33\) 0 0
\(34\) −493.483 −2.48916
\(35\) −121.695 −0.587718
\(36\) 0 0
\(37\) −344.140 −1.52909 −0.764545 0.644571i \(-0.777036\pi\)
−0.764545 + 0.644571i \(0.777036\pi\)
\(38\) 735.088 3.13808
\(39\) 0 0
\(40\) 127.612 0.504430
\(41\) 46.5921 0.177475 0.0887373 0.996055i \(-0.471717\pi\)
0.0887373 + 0.996055i \(0.471717\pi\)
\(42\) 0 0
\(43\) −173.286 −0.614554 −0.307277 0.951620i \(-0.599418\pi\)
−0.307277 + 0.951620i \(0.599418\pi\)
\(44\) −84.3269 −0.288926
\(45\) 0 0
\(46\) 824.175 2.64169
\(47\) −265.613 −0.824334 −0.412167 0.911108i \(-0.635228\pi\)
−0.412167 + 0.911108i \(0.635228\pi\)
\(48\) 0 0
\(49\) 397.361 1.15849
\(50\) −493.296 −1.39525
\(51\) 0 0
\(52\) 0 0
\(53\) −172.912 −0.448137 −0.224068 0.974573i \(-0.571934\pi\)
−0.224068 + 0.974573i \(0.571934\pi\)
\(54\) 0 0
\(55\) −26.7995 −0.0657025
\(56\) −776.360 −1.85260
\(57\) 0 0
\(58\) −962.596 −2.17923
\(59\) −137.566 −0.303551 −0.151775 0.988415i \(-0.548499\pi\)
−0.151775 + 0.988415i \(0.548499\pi\)
\(60\) 0 0
\(61\) −58.9384 −0.123710 −0.0618548 0.998085i \(-0.519702\pi\)
−0.0618548 + 0.998085i \(0.519702\pi\)
\(62\) −150.140 −0.307546
\(63\) 0 0
\(64\) −770.306 −1.50450
\(65\) 0 0
\(66\) 0 0
\(67\) −211.668 −0.385961 −0.192980 0.981203i \(-0.561815\pi\)
−0.192980 + 0.981203i \(0.561815\pi\)
\(68\) −1478.19 −2.63613
\(69\) 0 0
\(70\) −571.746 −0.976238
\(71\) −436.317 −0.729314 −0.364657 0.931142i \(-0.618814\pi\)
−0.364657 + 0.931142i \(0.618814\pi\)
\(72\) 0 0
\(73\) −1159.11 −1.85840 −0.929199 0.369579i \(-0.879502\pi\)
−0.929199 + 0.369579i \(0.879502\pi\)
\(74\) −1616.84 −2.53992
\(75\) 0 0
\(76\) 2201.90 3.32336
\(77\) 163.042 0.241303
\(78\) 0 0
\(79\) −1017.51 −1.44910 −0.724548 0.689224i \(-0.757952\pi\)
−0.724548 + 0.689224i \(0.757952\pi\)
\(80\) 96.0118 0.134181
\(81\) 0 0
\(82\) 218.899 0.294797
\(83\) −150.251 −0.198701 −0.0993504 0.995053i \(-0.531676\pi\)
−0.0993504 + 0.995053i \(0.531676\pi\)
\(84\) 0 0
\(85\) −469.775 −0.599462
\(86\) −814.131 −1.02081
\(87\) 0 0
\(88\) −170.969 −0.207107
\(89\) 565.984 0.674092 0.337046 0.941488i \(-0.390572\pi\)
0.337046 + 0.941488i \(0.390572\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2468.75 2.79766
\(93\) 0 0
\(94\) −1247.90 −1.36927
\(95\) 699.773 0.755739
\(96\) 0 0
\(97\) 286.741 0.300145 0.150073 0.988675i \(-0.452049\pi\)
0.150073 + 0.988675i \(0.452049\pi\)
\(98\) 1866.88 1.92432
\(99\) 0 0
\(100\) −1477.63 −1.47763
\(101\) 1218.57 1.20052 0.600258 0.799807i \(-0.295065\pi\)
0.600258 + 0.799807i \(0.295065\pi\)
\(102\) 0 0
\(103\) 74.2485 0.0710283 0.0355142 0.999369i \(-0.488693\pi\)
0.0355142 + 0.999369i \(0.488693\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −812.374 −0.744384
\(107\) 1253.46 1.13249 0.566246 0.824237i \(-0.308395\pi\)
0.566246 + 0.824237i \(0.308395\pi\)
\(108\) 0 0
\(109\) −722.643 −0.635015 −0.317507 0.948256i \(-0.602846\pi\)
−0.317507 + 0.948256i \(0.602846\pi\)
\(110\) −125.909 −0.109136
\(111\) 0 0
\(112\) −584.113 −0.492799
\(113\) 855.913 0.712544 0.356272 0.934382i \(-0.384048\pi\)
0.356272 + 0.934382i \(0.384048\pi\)
\(114\) 0 0
\(115\) 784.580 0.636195
\(116\) −2883.38 −2.30789
\(117\) 0 0
\(118\) −646.310 −0.504218
\(119\) 2858.00 2.20162
\(120\) 0 0
\(121\) −1295.10 −0.973024
\(122\) −276.905 −0.205490
\(123\) 0 0
\(124\) −449.733 −0.325703
\(125\) −1028.66 −0.736048
\(126\) 0 0
\(127\) 726.104 0.507333 0.253667 0.967292i \(-0.418363\pi\)
0.253667 + 0.967292i \(0.418363\pi\)
\(128\) −2599.82 −1.79526
\(129\) 0 0
\(130\) 0 0
\(131\) 1456.91 0.971685 0.485843 0.874046i \(-0.338513\pi\)
0.485843 + 0.874046i \(0.338513\pi\)
\(132\) 0 0
\(133\) −4257.25 −2.77557
\(134\) −994.460 −0.641106
\(135\) 0 0
\(136\) −2996.97 −1.88962
\(137\) 1806.80 1.12675 0.563377 0.826200i \(-0.309502\pi\)
0.563377 + 0.826200i \(0.309502\pi\)
\(138\) 0 0
\(139\) 1229.28 0.750117 0.375059 0.927001i \(-0.377623\pi\)
0.375059 + 0.927001i \(0.377623\pi\)
\(140\) −1712.62 −1.03388
\(141\) 0 0
\(142\) −2049.90 −1.21144
\(143\) 0 0
\(144\) 0 0
\(145\) −916.352 −0.524820
\(146\) −5445.71 −3.08692
\(147\) 0 0
\(148\) −4843.12 −2.68988
\(149\) 446.127 0.245290 0.122645 0.992451i \(-0.460862\pi\)
0.122645 + 0.992451i \(0.460862\pi\)
\(150\) 0 0
\(151\) −207.208 −0.111671 −0.0558355 0.998440i \(-0.517782\pi\)
−0.0558355 + 0.998440i \(0.517782\pi\)
\(152\) 4464.26 2.38223
\(153\) 0 0
\(154\) 766.002 0.400819
\(155\) −142.927 −0.0740658
\(156\) 0 0
\(157\) 1096.60 0.557442 0.278721 0.960372i \(-0.410089\pi\)
0.278721 + 0.960372i \(0.410089\pi\)
\(158\) −4780.46 −2.40704
\(159\) 0 0
\(160\) −569.812 −0.281547
\(161\) −4773.20 −2.33653
\(162\) 0 0
\(163\) 3154.25 1.51571 0.757853 0.652425i \(-0.226248\pi\)
0.757853 + 0.652425i \(0.226248\pi\)
\(164\) 655.694 0.312202
\(165\) 0 0
\(166\) −705.908 −0.330055
\(167\) 3679.65 1.70503 0.852514 0.522705i \(-0.175077\pi\)
0.852514 + 0.522705i \(0.175077\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −2207.10 −0.995745
\(171\) 0 0
\(172\) −2438.66 −1.08108
\(173\) 3666.99 1.61154 0.805768 0.592231i \(-0.201753\pi\)
0.805768 + 0.592231i \(0.201753\pi\)
\(174\) 0 0
\(175\) 2856.92 1.23407
\(176\) −128.633 −0.0550913
\(177\) 0 0
\(178\) 2659.11 1.11971
\(179\) −4173.50 −1.74269 −0.871347 0.490666i \(-0.836753\pi\)
−0.871347 + 0.490666i \(0.836753\pi\)
\(180\) 0 0
\(181\) −499.197 −0.205000 −0.102500 0.994733i \(-0.532684\pi\)
−0.102500 + 0.994733i \(0.532684\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 5005.29 2.00541
\(185\) −1539.16 −0.611685
\(186\) 0 0
\(187\) 629.386 0.246124
\(188\) −3738.00 −1.45012
\(189\) 0 0
\(190\) 3287.68 1.25533
\(191\) 3086.68 1.16934 0.584671 0.811271i \(-0.301224\pi\)
0.584671 + 0.811271i \(0.301224\pi\)
\(192\) 0 0
\(193\) −1644.14 −0.613201 −0.306601 0.951838i \(-0.599192\pi\)
−0.306601 + 0.951838i \(0.599192\pi\)
\(194\) 1347.17 0.498561
\(195\) 0 0
\(196\) 5592.10 2.03794
\(197\) −1371.21 −0.495911 −0.247956 0.968771i \(-0.579759\pi\)
−0.247956 + 0.968771i \(0.579759\pi\)
\(198\) 0 0
\(199\) −4627.33 −1.64836 −0.824178 0.566332i \(-0.808362\pi\)
−0.824178 + 0.566332i \(0.808362\pi\)
\(200\) −2995.83 −1.05919
\(201\) 0 0
\(202\) 5725.08 1.99413
\(203\) 5574.87 1.92748
\(204\) 0 0
\(205\) 208.383 0.0709955
\(206\) 348.834 0.117983
\(207\) 0 0
\(208\) 0 0
\(209\) −937.528 −0.310288
\(210\) 0 0
\(211\) 5088.31 1.66016 0.830080 0.557644i \(-0.188294\pi\)
0.830080 + 0.557644i \(0.188294\pi\)
\(212\) −2433.40 −0.788334
\(213\) 0 0
\(214\) 5889.01 1.88114
\(215\) −775.019 −0.245841
\(216\) 0 0
\(217\) 869.535 0.272018
\(218\) −3395.12 −1.05480
\(219\) 0 0
\(220\) −377.151 −0.115580
\(221\) 0 0
\(222\) 0 0
\(223\) −4744.82 −1.42483 −0.712415 0.701759i \(-0.752398\pi\)
−0.712415 + 0.701759i \(0.752398\pi\)
\(224\) 3466.60 1.03403
\(225\) 0 0
\(226\) 4021.25 1.18358
\(227\) 1145.52 0.334937 0.167469 0.985877i \(-0.446441\pi\)
0.167469 + 0.985877i \(0.446441\pi\)
\(228\) 0 0
\(229\) −1348.47 −0.389123 −0.194561 0.980890i \(-0.562328\pi\)
−0.194561 + 0.980890i \(0.562328\pi\)
\(230\) 3686.12 1.05676
\(231\) 0 0
\(232\) −5845.94 −1.65433
\(233\) 952.002 0.267673 0.133836 0.991003i \(-0.457270\pi\)
0.133836 + 0.991003i \(0.457270\pi\)
\(234\) 0 0
\(235\) −1187.95 −0.329760
\(236\) −1935.97 −0.533987
\(237\) 0 0
\(238\) 13427.5 3.65703
\(239\) −3069.12 −0.830647 −0.415323 0.909674i \(-0.636332\pi\)
−0.415323 + 0.909674i \(0.636332\pi\)
\(240\) 0 0
\(241\) −2508.17 −0.670396 −0.335198 0.942148i \(-0.608803\pi\)
−0.335198 + 0.942148i \(0.608803\pi\)
\(242\) −6084.62 −1.61626
\(243\) 0 0
\(244\) −829.446 −0.217622
\(245\) 1777.19 0.463432
\(246\) 0 0
\(247\) 0 0
\(248\) −911.815 −0.233469
\(249\) 0 0
\(250\) −4832.85 −1.22262
\(251\) −3405.91 −0.856490 −0.428245 0.903663i \(-0.640868\pi\)
−0.428245 + 0.903663i \(0.640868\pi\)
\(252\) 0 0
\(253\) −1051.15 −0.261206
\(254\) 3411.38 0.842714
\(255\) 0 0
\(256\) −6052.04 −1.47755
\(257\) 4733.95 1.14901 0.574506 0.818501i \(-0.305194\pi\)
0.574506 + 0.818501i \(0.305194\pi\)
\(258\) 0 0
\(259\) 9363.91 2.24651
\(260\) 0 0
\(261\) 0 0
\(262\) 6844.85 1.61403
\(263\) 1866.11 0.437526 0.218763 0.975778i \(-0.429798\pi\)
0.218763 + 0.975778i \(0.429798\pi\)
\(264\) 0 0
\(265\) −773.346 −0.179269
\(266\) −20001.4 −4.61040
\(267\) 0 0
\(268\) −2978.83 −0.678958
\(269\) −1580.19 −0.358164 −0.179082 0.983834i \(-0.557313\pi\)
−0.179082 + 0.983834i \(0.557313\pi\)
\(270\) 0 0
\(271\) −4922.09 −1.10330 −0.551652 0.834074i \(-0.686002\pi\)
−0.551652 + 0.834074i \(0.686002\pi\)
\(272\) −2254.84 −0.502646
\(273\) 0 0
\(274\) 8488.71 1.87161
\(275\) 629.148 0.137960
\(276\) 0 0
\(277\) 2687.08 0.582856 0.291428 0.956593i \(-0.405870\pi\)
0.291428 + 0.956593i \(0.405870\pi\)
\(278\) 5775.41 1.24599
\(279\) 0 0
\(280\) −3472.26 −0.741098
\(281\) 883.753 0.187617 0.0938083 0.995590i \(-0.470096\pi\)
0.0938083 + 0.995590i \(0.470096\pi\)
\(282\) 0 0
\(283\) −469.776 −0.0986759 −0.0493379 0.998782i \(-0.515711\pi\)
−0.0493379 + 0.998782i \(0.515711\pi\)
\(284\) −6140.33 −1.28296
\(285\) 0 0
\(286\) 0 0
\(287\) −1267.75 −0.260742
\(288\) 0 0
\(289\) 6119.68 1.24561
\(290\) −4305.20 −0.871760
\(291\) 0 0
\(292\) −16312.2 −3.26918
\(293\) −3403.76 −0.678668 −0.339334 0.940666i \(-0.610202\pi\)
−0.339334 + 0.940666i \(0.610202\pi\)
\(294\) 0 0
\(295\) −615.261 −0.121430
\(296\) −9819.22 −1.92814
\(297\) 0 0
\(298\) 2095.99 0.407442
\(299\) 0 0
\(300\) 0 0
\(301\) 4715.03 0.902889
\(302\) −973.504 −0.185493
\(303\) 0 0
\(304\) 3358.79 0.633684
\(305\) −263.602 −0.0494878
\(306\) 0 0
\(307\) 888.862 0.165244 0.0826222 0.996581i \(-0.473671\pi\)
0.0826222 + 0.996581i \(0.473671\pi\)
\(308\) 2294.50 0.424484
\(309\) 0 0
\(310\) −671.501 −0.123028
\(311\) 1218.36 0.222145 0.111072 0.993812i \(-0.464571\pi\)
0.111072 + 0.993812i \(0.464571\pi\)
\(312\) 0 0
\(313\) −4870.41 −0.879526 −0.439763 0.898114i \(-0.644938\pi\)
−0.439763 + 0.898114i \(0.644938\pi\)
\(314\) 5152.06 0.925948
\(315\) 0 0
\(316\) −14319.5 −2.54916
\(317\) 4340.93 0.769120 0.384560 0.923100i \(-0.374353\pi\)
0.384560 + 0.923100i \(0.374353\pi\)
\(318\) 0 0
\(319\) 1227.69 0.215478
\(320\) −3445.19 −0.601849
\(321\) 0 0
\(322\) −22425.4 −3.88112
\(323\) −16434.2 −2.83103
\(324\) 0 0
\(325\) 0 0
\(326\) 14819.3 2.51769
\(327\) 0 0
\(328\) 1329.39 0.223791
\(329\) 7227.23 1.21109
\(330\) 0 0
\(331\) −907.289 −0.150662 −0.0753310 0.997159i \(-0.524001\pi\)
−0.0753310 + 0.997159i \(0.524001\pi\)
\(332\) −2114.49 −0.349542
\(333\) 0 0
\(334\) 17287.7 2.83216
\(335\) −946.684 −0.154397
\(336\) 0 0
\(337\) 8660.88 1.39997 0.699983 0.714160i \(-0.253191\pi\)
0.699983 + 0.714160i \(0.253191\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −6611.19 −1.05454
\(341\) 191.488 0.0304096
\(342\) 0 0
\(343\) −1479.14 −0.232846
\(344\) −4944.29 −0.774937
\(345\) 0 0
\(346\) 17228.2 2.67687
\(347\) 347.605 0.0537763 0.0268882 0.999638i \(-0.491440\pi\)
0.0268882 + 0.999638i \(0.491440\pi\)
\(348\) 0 0
\(349\) −10970.2 −1.68258 −0.841292 0.540581i \(-0.818205\pi\)
−0.841292 + 0.540581i \(0.818205\pi\)
\(350\) 13422.4 2.04988
\(351\) 0 0
\(352\) 763.411 0.115596
\(353\) −10384.8 −1.56580 −0.782901 0.622146i \(-0.786261\pi\)
−0.782901 + 0.622146i \(0.786261\pi\)
\(354\) 0 0
\(355\) −1951.42 −0.291749
\(356\) 7965.15 1.18582
\(357\) 0 0
\(358\) −19608.0 −2.89473
\(359\) −8665.80 −1.27399 −0.636996 0.770867i \(-0.719823\pi\)
−0.636996 + 0.770867i \(0.719823\pi\)
\(360\) 0 0
\(361\) 17621.2 2.56907
\(362\) −2345.33 −0.340518
\(363\) 0 0
\(364\) 0 0
\(365\) −5184.09 −0.743419
\(366\) 0 0
\(367\) 1234.22 0.175547 0.0877734 0.996140i \(-0.472025\pi\)
0.0877734 + 0.996140i \(0.472025\pi\)
\(368\) 3765.85 0.533447
\(369\) 0 0
\(370\) −7231.31 −1.01605
\(371\) 4704.85 0.658393
\(372\) 0 0
\(373\) −427.483 −0.0593410 −0.0296705 0.999560i \(-0.509446\pi\)
−0.0296705 + 0.999560i \(0.509446\pi\)
\(374\) 2956.98 0.408829
\(375\) 0 0
\(376\) −7578.64 −1.03946
\(377\) 0 0
\(378\) 0 0
\(379\) −124.241 −0.0168386 −0.00841929 0.999965i \(-0.502680\pi\)
−0.00841929 + 0.999965i \(0.502680\pi\)
\(380\) 9847.98 1.32945
\(381\) 0 0
\(382\) 14501.8 1.94235
\(383\) 9341.21 1.24625 0.623125 0.782122i \(-0.285863\pi\)
0.623125 + 0.782122i \(0.285863\pi\)
\(384\) 0 0
\(385\) 729.202 0.0965288
\(386\) −7724.51 −1.01857
\(387\) 0 0
\(388\) 4035.33 0.527997
\(389\) −11368.9 −1.48182 −0.740908 0.671607i \(-0.765604\pi\)
−0.740908 + 0.671607i \(0.765604\pi\)
\(390\) 0 0
\(391\) −18425.9 −2.38321
\(392\) 11337.7 1.46082
\(393\) 0 0
\(394\) −6442.21 −0.823741
\(395\) −4550.80 −0.579685
\(396\) 0 0
\(397\) 12077.8 1.52687 0.763436 0.645883i \(-0.223511\pi\)
0.763436 + 0.645883i \(0.223511\pi\)
\(398\) −21740.1 −2.73802
\(399\) 0 0
\(400\) −2253.99 −0.281748
\(401\) −4856.74 −0.604823 −0.302411 0.953178i \(-0.597792\pi\)
−0.302411 + 0.953178i \(0.597792\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 17149.0 2.11187
\(405\) 0 0
\(406\) 26191.8 3.20167
\(407\) 2062.11 0.251143
\(408\) 0 0
\(409\) −2981.80 −0.360490 −0.180245 0.983622i \(-0.557689\pi\)
−0.180245 + 0.983622i \(0.557689\pi\)
\(410\) 979.023 0.117928
\(411\) 0 0
\(412\) 1044.91 0.124949
\(413\) 3743.10 0.445971
\(414\) 0 0
\(415\) −671.995 −0.0794866
\(416\) 0 0
\(417\) 0 0
\(418\) −4404.70 −0.515409
\(419\) −7774.01 −0.906408 −0.453204 0.891407i \(-0.649719\pi\)
−0.453204 + 0.891407i \(0.649719\pi\)
\(420\) 0 0
\(421\) −3959.71 −0.458396 −0.229198 0.973380i \(-0.573610\pi\)
−0.229198 + 0.973380i \(0.573610\pi\)
\(422\) 23905.9 2.75763
\(423\) 0 0
\(424\) −4933.62 −0.565089
\(425\) 11028.5 1.25873
\(426\) 0 0
\(427\) 1603.69 0.181752
\(428\) 17640.1 1.99221
\(429\) 0 0
\(430\) −3641.19 −0.408358
\(431\) 4375.12 0.488961 0.244480 0.969654i \(-0.421383\pi\)
0.244480 + 0.969654i \(0.421383\pi\)
\(432\) 0 0
\(433\) 8992.74 0.998068 0.499034 0.866582i \(-0.333688\pi\)
0.499034 + 0.866582i \(0.333688\pi\)
\(434\) 4085.25 0.451839
\(435\) 0 0
\(436\) −10169.8 −1.11708
\(437\) 27447.0 3.00451
\(438\) 0 0
\(439\) −195.465 −0.0212506 −0.0106253 0.999944i \(-0.503382\pi\)
−0.0106253 + 0.999944i \(0.503382\pi\)
\(440\) −764.659 −0.0828493
\(441\) 0 0
\(442\) 0 0
\(443\) −7369.97 −0.790424 −0.395212 0.918590i \(-0.629329\pi\)
−0.395212 + 0.918590i \(0.629329\pi\)
\(444\) 0 0
\(445\) 2531.36 0.269658
\(446\) −22292.1 −2.36673
\(447\) 0 0
\(448\) 20959.7 2.21038
\(449\) −164.281 −0.0172670 −0.00863351 0.999963i \(-0.502748\pi\)
−0.00863351 + 0.999963i \(0.502748\pi\)
\(450\) 0 0
\(451\) −279.183 −0.0291490
\(452\) 12045.3 1.25346
\(453\) 0 0
\(454\) 5381.88 0.556352
\(455\) 0 0
\(456\) 0 0
\(457\) −11687.0 −1.19627 −0.598133 0.801397i \(-0.704091\pi\)
−0.598133 + 0.801397i \(0.704091\pi\)
\(458\) −6335.36 −0.646358
\(459\) 0 0
\(460\) 11041.5 1.11915
\(461\) 1057.53 0.106842 0.0534211 0.998572i \(-0.482987\pi\)
0.0534211 + 0.998572i \(0.482987\pi\)
\(462\) 0 0
\(463\) −8554.74 −0.858688 −0.429344 0.903141i \(-0.641255\pi\)
−0.429344 + 0.903141i \(0.641255\pi\)
\(464\) −4398.33 −0.440059
\(465\) 0 0
\(466\) 4472.70 0.444622
\(467\) 7705.26 0.763506 0.381753 0.924264i \(-0.375321\pi\)
0.381753 + 0.924264i \(0.375321\pi\)
\(468\) 0 0
\(469\) 5759.40 0.567046
\(470\) −5581.24 −0.547752
\(471\) 0 0
\(472\) −3925.10 −0.382770
\(473\) 1038.34 0.100936
\(474\) 0 0
\(475\) −16428.0 −1.58688
\(476\) 40220.9 3.87295
\(477\) 0 0
\(478\) −14419.3 −1.37976
\(479\) −4508.59 −0.430069 −0.215034 0.976606i \(-0.568986\pi\)
−0.215034 + 0.976606i \(0.568986\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −11783.9 −1.11357
\(483\) 0 0
\(484\) −18226.0 −1.71168
\(485\) 1282.45 0.120068
\(486\) 0 0
\(487\) −6725.73 −0.625815 −0.312908 0.949784i \(-0.601303\pi\)
−0.312908 + 0.949784i \(0.601303\pi\)
\(488\) −1681.67 −0.155995
\(489\) 0 0
\(490\) 8349.61 0.769790
\(491\) −11517.5 −1.05861 −0.529303 0.848433i \(-0.677546\pi\)
−0.529303 + 0.848433i \(0.677546\pi\)
\(492\) 0 0
\(493\) 21520.5 1.96600
\(494\) 0 0
\(495\) 0 0
\(496\) −686.026 −0.0621038
\(497\) 11872.0 1.07149
\(498\) 0 0
\(499\) −19907.9 −1.78598 −0.892988 0.450080i \(-0.851395\pi\)
−0.892988 + 0.450080i \(0.851395\pi\)
\(500\) −14476.4 −1.29481
\(501\) 0 0
\(502\) −16001.6 −1.42269
\(503\) 5735.48 0.508415 0.254207 0.967150i \(-0.418185\pi\)
0.254207 + 0.967150i \(0.418185\pi\)
\(504\) 0 0
\(505\) 5450.04 0.480244
\(506\) −4938.51 −0.433881
\(507\) 0 0
\(508\) 10218.5 0.892469
\(509\) 9253.84 0.805834 0.402917 0.915237i \(-0.367996\pi\)
0.402917 + 0.915237i \(0.367996\pi\)
\(510\) 0 0
\(511\) 31538.8 2.73032
\(512\) −7635.12 −0.659039
\(513\) 0 0
\(514\) 22241.1 1.90858
\(515\) 332.076 0.0284136
\(516\) 0 0
\(517\) 1591.57 0.135391
\(518\) 43993.5 3.73159
\(519\) 0 0
\(520\) 0 0
\(521\) −3887.42 −0.326892 −0.163446 0.986552i \(-0.552261\pi\)
−0.163446 + 0.986552i \(0.552261\pi\)
\(522\) 0 0
\(523\) 4782.27 0.399836 0.199918 0.979813i \(-0.435932\pi\)
0.199918 + 0.979813i \(0.435932\pi\)
\(524\) 20503.2 1.70933
\(525\) 0 0
\(526\) 8767.37 0.726759
\(527\) 3356.65 0.277453
\(528\) 0 0
\(529\) 18606.4 1.52925
\(530\) −3633.34 −0.297777
\(531\) 0 0
\(532\) −59912.7 −4.88261
\(533\) 0 0
\(534\) 0 0
\(535\) 5606.09 0.453033
\(536\) −6039.44 −0.486687
\(537\) 0 0
\(538\) −7424.06 −0.594933
\(539\) −2381.01 −0.190274
\(540\) 0 0
\(541\) 14872.6 1.18192 0.590962 0.806699i \(-0.298748\pi\)
0.590962 + 0.806699i \(0.298748\pi\)
\(542\) −23125.0 −1.83266
\(543\) 0 0
\(544\) 13382.0 1.05469
\(545\) −3232.01 −0.254026
\(546\) 0 0
\(547\) 16965.3 1.32611 0.663055 0.748570i \(-0.269259\pi\)
0.663055 + 0.748570i \(0.269259\pi\)
\(548\) 25427.3 1.98212
\(549\) 0 0
\(550\) 2955.86 0.229161
\(551\) −32056.8 −2.47852
\(552\) 0 0
\(553\) 27686.0 2.12898
\(554\) 12624.5 0.968162
\(555\) 0 0
\(556\) 17299.8 1.31956
\(557\) 1934.05 0.147125 0.0735623 0.997291i \(-0.476563\pi\)
0.0735623 + 0.997291i \(0.476563\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2612.44 −0.197135
\(561\) 0 0
\(562\) 4152.05 0.311643
\(563\) −6592.13 −0.493473 −0.246736 0.969083i \(-0.579358\pi\)
−0.246736 + 0.969083i \(0.579358\pi\)
\(564\) 0 0
\(565\) 3828.06 0.285040
\(566\) −2207.10 −0.163907
\(567\) 0 0
\(568\) −12449.3 −0.919646
\(569\) 14477.7 1.06667 0.533336 0.845903i \(-0.320938\pi\)
0.533336 + 0.845903i \(0.320938\pi\)
\(570\) 0 0
\(571\) −4998.21 −0.366320 −0.183160 0.983083i \(-0.558633\pi\)
−0.183160 + 0.983083i \(0.558633\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −5956.15 −0.433109
\(575\) −18418.9 −1.33586
\(576\) 0 0
\(577\) −9291.73 −0.670398 −0.335199 0.942147i \(-0.608804\pi\)
−0.335199 + 0.942147i \(0.608804\pi\)
\(578\) 28751.5 2.06904
\(579\) 0 0
\(580\) −12895.9 −0.923230
\(581\) 4088.26 0.291927
\(582\) 0 0
\(583\) 1036.10 0.0736034
\(584\) −33072.3 −2.34339
\(585\) 0 0
\(586\) −15991.5 −1.12731
\(587\) −5602.64 −0.393945 −0.196972 0.980409i \(-0.563111\pi\)
−0.196972 + 0.980409i \(0.563111\pi\)
\(588\) 0 0
\(589\) −5000.04 −0.349784
\(590\) −2890.62 −0.201703
\(591\) 0 0
\(592\) −7387.73 −0.512895
\(593\) −10885.8 −0.753839 −0.376919 0.926246i \(-0.623017\pi\)
−0.376919 + 0.926246i \(0.623017\pi\)
\(594\) 0 0
\(595\) 12782.4 0.880717
\(596\) 6278.38 0.431498
\(597\) 0 0
\(598\) 0 0
\(599\) −20403.2 −1.39174 −0.695872 0.718166i \(-0.744982\pi\)
−0.695872 + 0.718166i \(0.744982\pi\)
\(600\) 0 0
\(601\) −6312.19 −0.428419 −0.214209 0.976788i \(-0.568717\pi\)
−0.214209 + 0.976788i \(0.568717\pi\)
\(602\) 22152.2 1.49976
\(603\) 0 0
\(604\) −2916.05 −0.196445
\(605\) −5792.30 −0.389241
\(606\) 0 0
\(607\) −21848.6 −1.46097 −0.730484 0.682930i \(-0.760705\pi\)
−0.730484 + 0.682930i \(0.760705\pi\)
\(608\) −19933.8 −1.32964
\(609\) 0 0
\(610\) −1238.45 −0.0822025
\(611\) 0 0
\(612\) 0 0
\(613\) 1335.14 0.0879704 0.0439852 0.999032i \(-0.485995\pi\)
0.0439852 + 0.999032i \(0.485995\pi\)
\(614\) 4176.05 0.274482
\(615\) 0 0
\(616\) 4652.00 0.304277
\(617\) 18908.3 1.23374 0.616871 0.787064i \(-0.288400\pi\)
0.616871 + 0.787064i \(0.288400\pi\)
\(618\) 0 0
\(619\) −6722.49 −0.436510 −0.218255 0.975892i \(-0.570036\pi\)
−0.218255 + 0.975892i \(0.570036\pi\)
\(620\) −2011.43 −0.130292
\(621\) 0 0
\(622\) 5724.11 0.368997
\(623\) −15400.2 −0.990362
\(624\) 0 0
\(625\) 8523.93 0.545532
\(626\) −22882.2 −1.46095
\(627\) 0 0
\(628\) 15432.6 0.980617
\(629\) 36147.3 2.29140
\(630\) 0 0
\(631\) 8098.44 0.510925 0.255463 0.966819i \(-0.417772\pi\)
0.255463 + 0.966819i \(0.417772\pi\)
\(632\) −29032.2 −1.82727
\(633\) 0 0
\(634\) 20394.6 1.27756
\(635\) 3247.50 0.202950
\(636\) 0 0
\(637\) 0 0
\(638\) 5767.94 0.357923
\(639\) 0 0
\(640\) −11627.7 −0.718163
\(641\) 10955.9 0.675090 0.337545 0.941309i \(-0.390403\pi\)
0.337545 + 0.941309i \(0.390403\pi\)
\(642\) 0 0
\(643\) 28125.0 1.72495 0.862473 0.506104i \(-0.168915\pi\)
0.862473 + 0.506104i \(0.168915\pi\)
\(644\) −67173.7 −4.11027
\(645\) 0 0
\(646\) −77211.1 −4.70253
\(647\) 29001.4 1.76223 0.881115 0.472901i \(-0.156793\pi\)
0.881115 + 0.472901i \(0.156793\pi\)
\(648\) 0 0
\(649\) 824.302 0.0498562
\(650\) 0 0
\(651\) 0 0
\(652\) 44390.1 2.66634
\(653\) −19506.3 −1.16898 −0.584488 0.811402i \(-0.698705\pi\)
−0.584488 + 0.811402i \(0.698705\pi\)
\(654\) 0 0
\(655\) 6516.01 0.388705
\(656\) 1000.20 0.0595294
\(657\) 0 0
\(658\) 33955.0 2.01171
\(659\) −5985.86 −0.353833 −0.176917 0.984226i \(-0.556612\pi\)
−0.176917 + 0.984226i \(0.556612\pi\)
\(660\) 0 0
\(661\) 280.836 0.0165254 0.00826268 0.999966i \(-0.497370\pi\)
0.00826268 + 0.999966i \(0.497370\pi\)
\(662\) −4262.63 −0.250259
\(663\) 0 0
\(664\) −4287.04 −0.250557
\(665\) −19040.5 −1.11032
\(666\) 0 0
\(667\) −35941.8 −2.08647
\(668\) 51784.0 2.99938
\(669\) 0 0
\(670\) −4447.71 −0.256463
\(671\) 353.163 0.0203185
\(672\) 0 0
\(673\) −14868.9 −0.851640 −0.425820 0.904808i \(-0.640014\pi\)
−0.425820 + 0.904808i \(0.640014\pi\)
\(674\) 40690.6 2.32543
\(675\) 0 0
\(676\) 0 0
\(677\) 20769.5 1.17908 0.589540 0.807739i \(-0.299309\pi\)
0.589540 + 0.807739i \(0.299309\pi\)
\(678\) 0 0
\(679\) −7802.09 −0.440967
\(680\) −13403.9 −0.755907
\(681\) 0 0
\(682\) 899.650 0.0505123
\(683\) 14980.8 0.839275 0.419638 0.907692i \(-0.362157\pi\)
0.419638 + 0.907692i \(0.362157\pi\)
\(684\) 0 0
\(685\) 8080.90 0.450738
\(686\) −6949.30 −0.386772
\(687\) 0 0
\(688\) −3719.96 −0.206137
\(689\) 0 0
\(690\) 0 0
\(691\) −9472.45 −0.521489 −0.260745 0.965408i \(-0.583968\pi\)
−0.260745 + 0.965408i \(0.583968\pi\)
\(692\) 51605.8 2.83491
\(693\) 0 0
\(694\) 1633.12 0.0893260
\(695\) 5497.95 0.300071
\(696\) 0 0
\(697\) −4893.87 −0.265952
\(698\) −51540.3 −2.79488
\(699\) 0 0
\(700\) 40205.7 2.17090
\(701\) −1035.34 −0.0557834 −0.0278917 0.999611i \(-0.508879\pi\)
−0.0278917 + 0.999611i \(0.508879\pi\)
\(702\) 0 0
\(703\) −53844.8 −2.88875
\(704\) 4615.72 0.247105
\(705\) 0 0
\(706\) −48790.0 −2.60090
\(707\) −33156.7 −1.76377
\(708\) 0 0
\(709\) −16800.4 −0.889917 −0.444958 0.895551i \(-0.646781\pi\)
−0.444958 + 0.895551i \(0.646781\pi\)
\(710\) −9168.18 −0.484614
\(711\) 0 0
\(712\) 16149.0 0.850013
\(713\) −5606.00 −0.294455
\(714\) 0 0
\(715\) 0 0
\(716\) −58734.1 −3.06564
\(717\) 0 0
\(718\) −40713.7 −2.11618
\(719\) −14873.1 −0.771449 −0.385724 0.922614i \(-0.626048\pi\)
−0.385724 + 0.922614i \(0.626048\pi\)
\(720\) 0 0
\(721\) −2020.27 −0.104353
\(722\) 82788.1 4.26739
\(723\) 0 0
\(724\) −7025.24 −0.360623
\(725\) 21512.4 1.10200
\(726\) 0 0
\(727\) −2318.92 −0.118300 −0.0591500 0.998249i \(-0.518839\pi\)
−0.0591500 + 0.998249i \(0.518839\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −24355.9 −1.23487
\(731\) 18201.3 0.920931
\(732\) 0 0
\(733\) 3350.95 0.168854 0.0844270 0.996430i \(-0.473094\pi\)
0.0844270 + 0.996430i \(0.473094\pi\)
\(734\) 5798.61 0.291595
\(735\) 0 0
\(736\) −22349.6 −1.11932
\(737\) 1268.33 0.0633915
\(738\) 0 0
\(739\) 29348.1 1.46088 0.730438 0.682979i \(-0.239316\pi\)
0.730438 + 0.682979i \(0.239316\pi\)
\(740\) −21660.8 −1.07604
\(741\) 0 0
\(742\) 22104.3 1.09363
\(743\) 23622.2 1.16637 0.583187 0.812338i \(-0.301805\pi\)
0.583187 + 0.812338i \(0.301805\pi\)
\(744\) 0 0
\(745\) 1995.30 0.0981236
\(746\) −2008.40 −0.0985693
\(747\) 0 0
\(748\) 8857.41 0.432966
\(749\) −34106.1 −1.66383
\(750\) 0 0
\(751\) 29751.4 1.44560 0.722798 0.691059i \(-0.242856\pi\)
0.722798 + 0.691059i \(0.242856\pi\)
\(752\) −5701.97 −0.276502
\(753\) 0 0
\(754\) 0 0
\(755\) −926.735 −0.0446720
\(756\) 0 0
\(757\) −36907.2 −1.77201 −0.886006 0.463673i \(-0.846531\pi\)
−0.886006 + 0.463673i \(0.846531\pi\)
\(758\) −583.709 −0.0279700
\(759\) 0 0
\(760\) 19966.4 0.952968
\(761\) −20208.2 −0.962613 −0.481306 0.876552i \(-0.659838\pi\)
−0.481306 + 0.876552i \(0.659838\pi\)
\(762\) 0 0
\(763\) 19662.8 0.932950
\(764\) 43439.1 2.05703
\(765\) 0 0
\(766\) 43886.9 2.07010
\(767\) 0 0
\(768\) 0 0
\(769\) −24751.0 −1.16066 −0.580329 0.814382i \(-0.697076\pi\)
−0.580329 + 0.814382i \(0.697076\pi\)
\(770\) 3425.94 0.160341
\(771\) 0 0
\(772\) −23138.2 −1.07871
\(773\) −30673.4 −1.42722 −0.713612 0.700541i \(-0.752942\pi\)
−0.713612 + 0.700541i \(0.752942\pi\)
\(774\) 0 0
\(775\) 3355.38 0.155521
\(776\) 8181.46 0.378476
\(777\) 0 0
\(778\) −53413.4 −2.46139
\(779\) 7289.87 0.335285
\(780\) 0 0
\(781\) 2614.44 0.119785
\(782\) −86568.5 −3.95867
\(783\) 0 0
\(784\) 8530.23 0.388585
\(785\) 4904.55 0.222995
\(786\) 0 0
\(787\) −29009.9 −1.31397 −0.656983 0.753905i \(-0.728168\pi\)
−0.656983 + 0.753905i \(0.728168\pi\)
\(788\) −19297.1 −0.872375
\(789\) 0 0
\(790\) −21380.6 −0.962894
\(791\) −23289.0 −1.04685
\(792\) 0 0
\(793\) 0 0
\(794\) 56744.0 2.53623
\(795\) 0 0
\(796\) −65120.8 −2.89968
\(797\) −6778.24 −0.301252 −0.150626 0.988591i \(-0.548129\pi\)
−0.150626 + 0.988591i \(0.548129\pi\)
\(798\) 0 0
\(799\) 27899.1 1.23529
\(800\) 13377.0 0.591185
\(801\) 0 0
\(802\) −22817.9 −1.00465
\(803\) 6945.44 0.305229
\(804\) 0 0
\(805\) −21348.1 −0.934685
\(806\) 0 0
\(807\) 0 0
\(808\) 34768.9 1.51382
\(809\) 34862.9 1.51510 0.757549 0.652778i \(-0.226397\pi\)
0.757549 + 0.652778i \(0.226397\pi\)
\(810\) 0 0
\(811\) −22665.4 −0.981370 −0.490685 0.871337i \(-0.663253\pi\)
−0.490685 + 0.871337i \(0.663253\pi\)
\(812\) 78455.6 3.39070
\(813\) 0 0
\(814\) 9688.21 0.417164
\(815\) 14107.4 0.606331
\(816\) 0 0
\(817\) −27112.6 −1.16101
\(818\) −14009.1 −0.598797
\(819\) 0 0
\(820\) 2932.59 0.124891
\(821\) 20748.7 0.882016 0.441008 0.897503i \(-0.354621\pi\)
0.441008 + 0.897503i \(0.354621\pi\)
\(822\) 0 0
\(823\) 6141.41 0.260117 0.130058 0.991506i \(-0.458484\pi\)
0.130058 + 0.991506i \(0.458484\pi\)
\(824\) 2118.50 0.0895650
\(825\) 0 0
\(826\) 17585.8 0.740786
\(827\) −28383.5 −1.19346 −0.596730 0.802442i \(-0.703533\pi\)
−0.596730 + 0.802442i \(0.703533\pi\)
\(828\) 0 0
\(829\) −908.734 −0.0380720 −0.0190360 0.999819i \(-0.506060\pi\)
−0.0190360 + 0.999819i \(0.506060\pi\)
\(830\) −3157.17 −0.132032
\(831\) 0 0
\(832\) 0 0
\(833\) −41737.4 −1.73603
\(834\) 0 0
\(835\) 16457.2 0.682065
\(836\) −13193.9 −0.545839
\(837\) 0 0
\(838\) −36523.8 −1.50560
\(839\) −27820.3 −1.14477 −0.572385 0.819985i \(-0.693982\pi\)
−0.572385 + 0.819985i \(0.693982\pi\)
\(840\) 0 0
\(841\) 17589.3 0.721200
\(842\) −18603.5 −0.761425
\(843\) 0 0
\(844\) 71608.3 2.92045
\(845\) 0 0
\(846\) 0 0
\(847\) 35239.0 1.42955
\(848\) −3711.93 −0.150316
\(849\) 0 0
\(850\) 51814.1 2.09084
\(851\) −60370.3 −2.43181
\(852\) 0 0
\(853\) −5802.11 −0.232896 −0.116448 0.993197i \(-0.537151\pi\)
−0.116448 + 0.993197i \(0.537151\pi\)
\(854\) 7534.45 0.301901
\(855\) 0 0
\(856\) 35764.5 1.42804
\(857\) 43311.1 1.72635 0.863173 0.504909i \(-0.168474\pi\)
0.863173 + 0.504909i \(0.168474\pi\)
\(858\) 0 0
\(859\) −16698.2 −0.663254 −0.331627 0.943411i \(-0.607598\pi\)
−0.331627 + 0.943411i \(0.607598\pi\)
\(860\) −10906.9 −0.432468
\(861\) 0 0
\(862\) 20555.2 0.812196
\(863\) 19429.2 0.766369 0.383185 0.923672i \(-0.374827\pi\)
0.383185 + 0.923672i \(0.374827\pi\)
\(864\) 0 0
\(865\) 16400.6 0.644666
\(866\) 42249.7 1.65786
\(867\) 0 0
\(868\) 12237.0 0.478517
\(869\) 6096.98 0.238004
\(870\) 0 0
\(871\) 0 0
\(872\) −20618.9 −0.800738
\(873\) 0 0
\(874\) 128952. 4.99068
\(875\) 27989.4 1.08139
\(876\) 0 0
\(877\) −11619.7 −0.447400 −0.223700 0.974658i \(-0.571814\pi\)
−0.223700 + 0.974658i \(0.571814\pi\)
\(878\) −918.332 −0.0352986
\(879\) 0 0
\(880\) −575.309 −0.0220383
\(881\) 51102.0 1.95422 0.977112 0.212726i \(-0.0682341\pi\)
0.977112 + 0.212726i \(0.0682341\pi\)
\(882\) 0 0
\(883\) −37838.5 −1.44209 −0.721046 0.692888i \(-0.756338\pi\)
−0.721046 + 0.692888i \(0.756338\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −34625.6 −1.31295
\(887\) 9626.92 0.364420 0.182210 0.983260i \(-0.441675\pi\)
0.182210 + 0.983260i \(0.441675\pi\)
\(888\) 0 0
\(889\) −19757.0 −0.745364
\(890\) 11892.8 0.447920
\(891\) 0 0
\(892\) −66774.3 −2.50647
\(893\) −41558.3 −1.55733
\(894\) 0 0
\(895\) −18666.0 −0.697133
\(896\) 70740.0 2.63757
\(897\) 0 0
\(898\) −771.825 −0.0286816
\(899\) 6547.54 0.242906
\(900\) 0 0
\(901\) 18162.0 0.671549
\(902\) −1311.66 −0.0484184
\(903\) 0 0
\(904\) 24421.4 0.898500
\(905\) −2232.65 −0.0820065
\(906\) 0 0
\(907\) 17066.0 0.624772 0.312386 0.949955i \(-0.398872\pi\)
0.312386 + 0.949955i \(0.398872\pi\)
\(908\) 16121.0 0.589200
\(909\) 0 0
\(910\) 0 0
\(911\) 37423.9 1.36104 0.680521 0.732729i \(-0.261754\pi\)
0.680521 + 0.732729i \(0.261754\pi\)
\(912\) 0 0
\(913\) 900.312 0.0326353
\(914\) −54907.7 −1.98708
\(915\) 0 0
\(916\) −18977.1 −0.684520
\(917\) −39641.9 −1.42758
\(918\) 0 0
\(919\) −2783.88 −0.0999256 −0.0499628 0.998751i \(-0.515910\pi\)
−0.0499628 + 0.998751i \(0.515910\pi\)
\(920\) 22386.1 0.802227
\(921\) 0 0
\(922\) 4968.50 0.177472
\(923\) 0 0
\(924\) 0 0
\(925\) 36133.6 1.28440
\(926\) −40191.9 −1.42634
\(927\) 0 0
\(928\) 26103.3 0.923363
\(929\) 26585.2 0.938894 0.469447 0.882961i \(-0.344453\pi\)
0.469447 + 0.882961i \(0.344453\pi\)
\(930\) 0 0
\(931\) 62171.8 2.18861
\(932\) 13397.6 0.470873
\(933\) 0 0
\(934\) 36200.9 1.26823
\(935\) 2814.92 0.0984576
\(936\) 0 0
\(937\) 34474.0 1.20194 0.600970 0.799272i \(-0.294781\pi\)
0.600970 + 0.799272i \(0.294781\pi\)
\(938\) 27058.8 0.941900
\(939\) 0 0
\(940\) −16718.2 −0.580092
\(941\) −41994.3 −1.45481 −0.727404 0.686209i \(-0.759274\pi\)
−0.727404 + 0.686209i \(0.759274\pi\)
\(942\) 0 0
\(943\) 8173.34 0.282249
\(944\) −2953.15 −0.101819
\(945\) 0 0
\(946\) 4878.32 0.167662
\(947\) −49352.0 −1.69348 −0.846739 0.532008i \(-0.821438\pi\)
−0.846739 + 0.532008i \(0.821438\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −77181.9 −2.63591
\(951\) 0 0
\(952\) 81546.2 2.77618
\(953\) 51144.5 1.73844 0.869220 0.494425i \(-0.164621\pi\)
0.869220 + 0.494425i \(0.164621\pi\)
\(954\) 0 0
\(955\) 13805.2 0.467774
\(956\) −43192.0 −1.46122
\(957\) 0 0
\(958\) −21182.3 −0.714372
\(959\) −49162.3 −1.65540
\(960\) 0 0
\(961\) −28769.8 −0.965720
\(962\) 0 0
\(963\) 0 0
\(964\) −35297.7 −1.17932
\(965\) −7353.41 −0.245300
\(966\) 0 0
\(967\) 24895.1 0.827892 0.413946 0.910301i \(-0.364150\pi\)
0.413946 + 0.910301i \(0.364150\pi\)
\(968\) −36952.4 −1.22696
\(969\) 0 0
\(970\) 6025.19 0.199440
\(971\) −42942.9 −1.41926 −0.709630 0.704574i \(-0.751138\pi\)
−0.709630 + 0.704574i \(0.751138\pi\)
\(972\) 0 0
\(973\) −33448.2 −1.10206
\(974\) −31598.8 −1.03952
\(975\) 0 0
\(976\) −1265.24 −0.0414953
\(977\) 42555.7 1.39353 0.696764 0.717301i \(-0.254623\pi\)
0.696764 + 0.717301i \(0.254623\pi\)
\(978\) 0 0
\(979\) −3391.41 −0.110715
\(980\) 25010.6 0.815240
\(981\) 0 0
\(982\) −54111.3 −1.75841
\(983\) −4345.38 −0.140993 −0.0704965 0.997512i \(-0.522458\pi\)
−0.0704965 + 0.997512i \(0.522458\pi\)
\(984\) 0 0
\(985\) −6132.71 −0.198380
\(986\) 101108. 3.26565
\(987\) 0 0
\(988\) 0 0
\(989\) −30398.4 −0.977363
\(990\) 0 0
\(991\) 7934.89 0.254349 0.127175 0.991880i \(-0.459409\pi\)
0.127175 + 0.991880i \(0.459409\pi\)
\(992\) 4071.43 0.130311
\(993\) 0 0
\(994\) 55777.0 1.77982
\(995\) −20695.7 −0.659395
\(996\) 0 0
\(997\) 2725.62 0.0865810 0.0432905 0.999063i \(-0.486216\pi\)
0.0432905 + 0.999063i \(0.486216\pi\)
\(998\) −93531.5 −2.96662
\(999\) 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.1.8 9
3.2 odd 2 507.4.a.o.1.2 9
13.12 even 2 1521.4.a.bf.1.2 9
39.5 even 4 507.4.b.k.337.16 18
39.8 even 4 507.4.b.k.337.3 18
39.38 odd 2 507.4.a.p.1.8 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.2 9 3.2 odd 2
507.4.a.p.1.8 yes 9 39.38 odd 2
507.4.b.k.337.3 18 39.8 even 4
507.4.b.k.337.16 18 39.5 even 4
1521.4.a.bf.1.2 9 13.12 even 2
1521.4.a.bi.1.8 9 1.1 even 1 trivial