Properties

Label 2205.4.a.bz.1.6
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(130.099211563\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.1163891200.1
Defining polynomial: \( x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2\cdot 7 \)
Twist minimal: no (minimal twist has level 245)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(4.10376\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.51797 q^{2} +22.4480 q^{4} -5.00000 q^{5} +79.7239 q^{8} +O(q^{10})\) \(q+5.51797 q^{2} +22.4480 q^{4} -5.00000 q^{5} +79.7239 q^{8} -27.5899 q^{10} -34.5211 q^{11} -68.8935 q^{13} +260.330 q^{16} -91.4346 q^{17} -11.8278 q^{19} -112.240 q^{20} -190.486 q^{22} +0.104165 q^{23} +25.0000 q^{25} -380.152 q^{26} -190.863 q^{29} -159.802 q^{31} +798.703 q^{32} -504.534 q^{34} -177.908 q^{37} -65.2657 q^{38} -398.619 q^{40} +145.247 q^{41} +8.25729 q^{43} -774.930 q^{44} +0.574782 q^{46} +260.529 q^{47} +137.949 q^{50} -1546.52 q^{52} -353.107 q^{53} +172.605 q^{55} -1053.18 q^{58} +240.495 q^{59} -778.188 q^{61} -881.783 q^{62} +2324.58 q^{64} +344.467 q^{65} +151.945 q^{67} -2052.53 q^{68} +311.449 q^{71} -639.888 q^{73} -981.690 q^{74} -265.512 q^{76} +391.186 q^{79} -1301.65 q^{80} +801.472 q^{82} +493.205 q^{83} +457.173 q^{85} +45.5635 q^{86} -2752.15 q^{88} +473.850 q^{89} +2.33831 q^{92} +1437.59 q^{94} +59.1392 q^{95} -839.005 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 14 q^{4} - 30 q^{5} + 66 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} + 14 q^{4} - 30 q^{5} + 66 q^{8} - 10 q^{10} + 16 q^{11} - 168 q^{13} + 298 q^{16} - 4 q^{17} - 308 q^{19} - 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} + 56 q^{26} - 176 q^{29} - 392 q^{31} + 770 q^{32} - 812 q^{34} - 140 q^{37} + 20 q^{38} - 330 q^{40} + 656 q^{41} - 388 q^{43} + 160 q^{44} - 388 q^{46} + 628 q^{47} + 50 q^{50} - 1520 q^{52} + 676 q^{53} - 80 q^{55} - 2012 q^{58} + 996 q^{59} - 740 q^{61} - 364 q^{62} + 1426 q^{64} + 840 q^{65} + 1768 q^{67} - 2940 q^{68} + 224 q^{71} - 2640 q^{73} - 928 q^{74} + 1340 q^{76} + 1636 q^{79} - 1490 q^{80} + 1756 q^{82} + 140 q^{83} + 20 q^{85} - 1180 q^{86} - 5652 q^{88} - 1904 q^{89} + 1952 q^{92} + 3332 q^{94} + 1540 q^{95} - 516 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.51797 1.95090 0.975449 0.220225i \(-0.0706790\pi\)
0.975449 + 0.220225i \(0.0706790\pi\)
\(3\) 0 0
\(4\) 22.4480 2.80600
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 79.7239 3.52333
\(9\) 0 0
\(10\) −27.5899 −0.872468
\(11\) −34.5211 −0.946227 −0.473113 0.881002i \(-0.656870\pi\)
−0.473113 + 0.881002i \(0.656870\pi\)
\(12\) 0 0
\(13\) −68.8935 −1.46982 −0.734908 0.678167i \(-0.762775\pi\)
−0.734908 + 0.678167i \(0.762775\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 260.330 4.06766
\(17\) −91.4346 −1.30448 −0.652239 0.758013i \(-0.726170\pi\)
−0.652239 + 0.758013i \(0.726170\pi\)
\(18\) 0 0
\(19\) −11.8278 −0.142815 −0.0714077 0.997447i \(-0.522749\pi\)
−0.0714077 + 0.997447i \(0.522749\pi\)
\(20\) −112.240 −1.25488
\(21\) 0 0
\(22\) −190.486 −1.84599
\(23\) 0.104165 0.000944348 0 0.000472174 1.00000i \(-0.499850\pi\)
0.000472174 1.00000i \(0.499850\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −380.152 −2.86746
\(27\) 0 0
\(28\) 0 0
\(29\) −190.863 −1.22215 −0.611076 0.791572i \(-0.709263\pi\)
−0.611076 + 0.791572i \(0.709263\pi\)
\(30\) 0 0
\(31\) −159.802 −0.925847 −0.462924 0.886398i \(-0.653200\pi\)
−0.462924 + 0.886398i \(0.653200\pi\)
\(32\) 798.703 4.41225
\(33\) 0 0
\(34\) −504.534 −2.54491
\(35\) 0 0
\(36\) 0 0
\(37\) −177.908 −0.790482 −0.395241 0.918577i \(-0.629339\pi\)
−0.395241 + 0.918577i \(0.629339\pi\)
\(38\) −65.2657 −0.278618
\(39\) 0 0
\(40\) −398.619 −1.57568
\(41\) 145.247 0.553264 0.276632 0.960976i \(-0.410782\pi\)
0.276632 + 0.960976i \(0.410782\pi\)
\(42\) 0 0
\(43\) 8.25729 0.0292843 0.0146421 0.999893i \(-0.495339\pi\)
0.0146421 + 0.999893i \(0.495339\pi\)
\(44\) −774.930 −2.65512
\(45\) 0 0
\(46\) 0.574782 0.00184233
\(47\) 260.529 0.808553 0.404277 0.914637i \(-0.367523\pi\)
0.404277 + 0.914637i \(0.367523\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 137.949 0.390180
\(51\) 0 0
\(52\) −1546.52 −4.12431
\(53\) −353.107 −0.915151 −0.457576 0.889171i \(-0.651282\pi\)
−0.457576 + 0.889171i \(0.651282\pi\)
\(54\) 0 0
\(55\) 172.605 0.423165
\(56\) 0 0
\(57\) 0 0
\(58\) −1053.18 −2.38429
\(59\) 240.495 0.530673 0.265337 0.964156i \(-0.414517\pi\)
0.265337 + 0.964156i \(0.414517\pi\)
\(60\) 0 0
\(61\) −778.188 −1.63339 −0.816695 0.577069i \(-0.804196\pi\)
−0.816695 + 0.577069i \(0.804196\pi\)
\(62\) −881.783 −1.80623
\(63\) 0 0
\(64\) 2324.58 4.54020
\(65\) 344.467 0.657322
\(66\) 0 0
\(67\) 151.945 0.277059 0.138530 0.990358i \(-0.455762\pi\)
0.138530 + 0.990358i \(0.455762\pi\)
\(68\) −2052.53 −3.66037
\(69\) 0 0
\(70\) 0 0
\(71\) 311.449 0.520594 0.260297 0.965529i \(-0.416180\pi\)
0.260297 + 0.965529i \(0.416180\pi\)
\(72\) 0 0
\(73\) −639.888 −1.02593 −0.512967 0.858408i \(-0.671454\pi\)
−0.512967 + 0.858408i \(0.671454\pi\)
\(74\) −981.690 −1.54215
\(75\) 0 0
\(76\) −265.512 −0.400740
\(77\) 0 0
\(78\) 0 0
\(79\) 391.186 0.557112 0.278556 0.960420i \(-0.410144\pi\)
0.278556 + 0.960420i \(0.410144\pi\)
\(80\) −1301.65 −1.81911
\(81\) 0 0
\(82\) 801.472 1.07936
\(83\) 493.205 0.652245 0.326122 0.945328i \(-0.394258\pi\)
0.326122 + 0.945328i \(0.394258\pi\)
\(84\) 0 0
\(85\) 457.173 0.583381
\(86\) 45.5635 0.0571307
\(87\) 0 0
\(88\) −2752.15 −3.33387
\(89\) 473.850 0.564359 0.282180 0.959362i \(-0.408943\pi\)
0.282180 + 0.959362i \(0.408943\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.33831 0.00264984
\(93\) 0 0
\(94\) 1437.59 1.57741
\(95\) 59.1392 0.0638690
\(96\) 0 0
\(97\) −839.005 −0.878227 −0.439114 0.898431i \(-0.644707\pi\)
−0.439114 + 0.898431i \(0.644707\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 561.201 0.561201
\(101\) 887.440 0.874293 0.437146 0.899390i \(-0.355989\pi\)
0.437146 + 0.899390i \(0.355989\pi\)
\(102\) 0 0
\(103\) −619.087 −0.592238 −0.296119 0.955151i \(-0.595692\pi\)
−0.296119 + 0.955151i \(0.595692\pi\)
\(104\) −5492.46 −5.17865
\(105\) 0 0
\(106\) −1948.44 −1.78537
\(107\) 2151.30 1.94368 0.971841 0.235639i \(-0.0757183\pi\)
0.971841 + 0.235639i \(0.0757183\pi\)
\(108\) 0 0
\(109\) −407.076 −0.357714 −0.178857 0.983875i \(-0.557240\pi\)
−0.178857 + 0.983875i \(0.557240\pi\)
\(110\) 952.432 0.825553
\(111\) 0 0
\(112\) 0 0
\(113\) 349.581 0.291025 0.145513 0.989356i \(-0.453517\pi\)
0.145513 + 0.989356i \(0.453517\pi\)
\(114\) 0 0
\(115\) −0.520827 −0.000422325 0
\(116\) −4284.50 −3.42936
\(117\) 0 0
\(118\) 1327.04 1.03529
\(119\) 0 0
\(120\) 0 0
\(121\) −139.296 −0.104655
\(122\) −4294.02 −3.18658
\(123\) 0 0
\(124\) −3587.24 −2.59793
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1183.78 0.827114 0.413557 0.910478i \(-0.364286\pi\)
0.413557 + 0.910478i \(0.364286\pi\)
\(128\) 6437.36 4.44522
\(129\) 0 0
\(130\) 1900.76 1.28237
\(131\) 223.357 0.148968 0.0744840 0.997222i \(-0.476269\pi\)
0.0744840 + 0.997222i \(0.476269\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 838.426 0.540515
\(135\) 0 0
\(136\) −7289.52 −4.59611
\(137\) −2036.66 −1.27010 −0.635050 0.772471i \(-0.719021\pi\)
−0.635050 + 0.772471i \(0.719021\pi\)
\(138\) 0 0
\(139\) −2687.00 −1.63963 −0.819815 0.572629i \(-0.805924\pi\)
−0.819815 + 0.572629i \(0.805924\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1718.57 1.01563
\(143\) 2378.28 1.39078
\(144\) 0 0
\(145\) 954.316 0.546563
\(146\) −3530.88 −2.00149
\(147\) 0 0
\(148\) −3993.68 −2.21810
\(149\) −673.500 −0.370304 −0.185152 0.982710i \(-0.559278\pi\)
−0.185152 + 0.982710i \(0.559278\pi\)
\(150\) 0 0
\(151\) −2125.18 −1.14533 −0.572664 0.819790i \(-0.694090\pi\)
−0.572664 + 0.819790i \(0.694090\pi\)
\(152\) −942.961 −0.503186
\(153\) 0 0
\(154\) 0 0
\(155\) 799.010 0.414052
\(156\) 0 0
\(157\) 2813.03 1.42997 0.714983 0.699142i \(-0.246435\pi\)
0.714983 + 0.699142i \(0.246435\pi\)
\(158\) 2158.55 1.08687
\(159\) 0 0
\(160\) −3993.52 −1.97322
\(161\) 0 0
\(162\) 0 0
\(163\) −1344.42 −0.646032 −0.323016 0.946394i \(-0.604697\pi\)
−0.323016 + 0.946394i \(0.604697\pi\)
\(164\) 3260.52 1.55246
\(165\) 0 0
\(166\) 2721.49 1.27246
\(167\) 1451.24 0.672456 0.336228 0.941781i \(-0.390849\pi\)
0.336228 + 0.941781i \(0.390849\pi\)
\(168\) 0 0
\(169\) 2549.31 1.16036
\(170\) 2522.67 1.13812
\(171\) 0 0
\(172\) 185.360 0.0821718
\(173\) −1979.16 −0.869784 −0.434892 0.900483i \(-0.643213\pi\)
−0.434892 + 0.900483i \(0.643213\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −8986.87 −3.84893
\(177\) 0 0
\(178\) 2614.69 1.10101
\(179\) 4358.66 1.82001 0.910005 0.414598i \(-0.136078\pi\)
0.910005 + 0.414598i \(0.136078\pi\)
\(180\) 0 0
\(181\) 377.923 0.155198 0.0775988 0.996985i \(-0.475275\pi\)
0.0775988 + 0.996985i \(0.475275\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 8.30447 0.00332725
\(185\) 889.538 0.353514
\(186\) 0 0
\(187\) 3156.42 1.23433
\(188\) 5848.36 2.26880
\(189\) 0 0
\(190\) 326.328 0.124602
\(191\) 2425.95 0.919033 0.459517 0.888169i \(-0.348023\pi\)
0.459517 + 0.888169i \(0.348023\pi\)
\(192\) 0 0
\(193\) −622.923 −0.232326 −0.116163 0.993230i \(-0.537060\pi\)
−0.116163 + 0.993230i \(0.537060\pi\)
\(194\) −4629.61 −1.71333
\(195\) 0 0
\(196\) 0 0
\(197\) −2842.29 −1.02794 −0.513971 0.857807i \(-0.671826\pi\)
−0.513971 + 0.857807i \(0.671826\pi\)
\(198\) 0 0
\(199\) 867.364 0.308974 0.154487 0.987995i \(-0.450628\pi\)
0.154487 + 0.987995i \(0.450628\pi\)
\(200\) 1993.10 0.704666
\(201\) 0 0
\(202\) 4896.87 1.70566
\(203\) 0 0
\(204\) 0 0
\(205\) −726.237 −0.247427
\(206\) −3416.11 −1.15540
\(207\) 0 0
\(208\) −17935.0 −5.97871
\(209\) 408.309 0.135136
\(210\) 0 0
\(211\) −5975.92 −1.94976 −0.974880 0.222730i \(-0.928503\pi\)
−0.974880 + 0.222730i \(0.928503\pi\)
\(212\) −7926.57 −2.56792
\(213\) 0 0
\(214\) 11870.8 3.79192
\(215\) −41.2864 −0.0130963
\(216\) 0 0
\(217\) 0 0
\(218\) −2246.24 −0.697864
\(219\) 0 0
\(220\) 3874.65 1.18740
\(221\) 6299.25 1.91734
\(222\) 0 0
\(223\) 5181.58 1.55598 0.777992 0.628274i \(-0.216238\pi\)
0.777992 + 0.628274i \(0.216238\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1928.98 0.567761
\(227\) 3753.06 1.09735 0.548677 0.836035i \(-0.315132\pi\)
0.548677 + 0.836035i \(0.315132\pi\)
\(228\) 0 0
\(229\) 6258.14 1.80589 0.902947 0.429752i \(-0.141399\pi\)
0.902947 + 0.429752i \(0.141399\pi\)
\(230\) −2.87391 −0.000823913 0
\(231\) 0 0
\(232\) −15216.4 −4.30605
\(233\) −1779.96 −0.500469 −0.250234 0.968185i \(-0.580508\pi\)
−0.250234 + 0.968185i \(0.580508\pi\)
\(234\) 0 0
\(235\) −1302.64 −0.361596
\(236\) 5398.63 1.48907
\(237\) 0 0
\(238\) 0 0
\(239\) −3519.46 −0.952532 −0.476266 0.879301i \(-0.658010\pi\)
−0.476266 + 0.879301i \(0.658010\pi\)
\(240\) 0 0
\(241\) −362.930 −0.0970058 −0.0485029 0.998823i \(-0.515445\pi\)
−0.0485029 + 0.998823i \(0.515445\pi\)
\(242\) −768.629 −0.204171
\(243\) 0 0
\(244\) −17468.8 −4.58330
\(245\) 0 0
\(246\) 0 0
\(247\) 814.861 0.209912
\(248\) −12740.0 −3.26207
\(249\) 0 0
\(250\) −689.747 −0.174494
\(251\) −5333.85 −1.34131 −0.670656 0.741768i \(-0.733988\pi\)
−0.670656 + 0.741768i \(0.733988\pi\)
\(252\) 0 0
\(253\) −3.59590 −0.000893567 0
\(254\) 6532.06 1.61361
\(255\) 0 0
\(256\) 16924.5 4.13197
\(257\) −2438.78 −0.591933 −0.295966 0.955198i \(-0.595642\pi\)
−0.295966 + 0.955198i \(0.595642\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 7732.62 1.84445
\(261\) 0 0
\(262\) 1232.48 0.290621
\(263\) −1526.25 −0.357843 −0.178921 0.983863i \(-0.557261\pi\)
−0.178921 + 0.983863i \(0.557261\pi\)
\(264\) 0 0
\(265\) 1765.54 0.409268
\(266\) 0 0
\(267\) 0 0
\(268\) 3410.86 0.777430
\(269\) −7564.12 −1.71447 −0.857235 0.514925i \(-0.827820\pi\)
−0.857235 + 0.514925i \(0.827820\pi\)
\(270\) 0 0
\(271\) −4282.68 −0.959980 −0.479990 0.877274i \(-0.659360\pi\)
−0.479990 + 0.877274i \(0.659360\pi\)
\(272\) −23803.2 −5.30617
\(273\) 0 0
\(274\) −11238.2 −2.47784
\(275\) −863.027 −0.189245
\(276\) 0 0
\(277\) −4008.41 −0.869465 −0.434732 0.900560i \(-0.643157\pi\)
−0.434732 + 0.900560i \(0.643157\pi\)
\(278\) −14826.8 −3.19875
\(279\) 0 0
\(280\) 0 0
\(281\) −6935.44 −1.47236 −0.736181 0.676785i \(-0.763373\pi\)
−0.736181 + 0.676785i \(0.763373\pi\)
\(282\) 0 0
\(283\) 2666.64 0.560124 0.280062 0.959982i \(-0.409645\pi\)
0.280062 + 0.959982i \(0.409645\pi\)
\(284\) 6991.41 1.46079
\(285\) 0 0
\(286\) 13123.3 2.71327
\(287\) 0 0
\(288\) 0 0
\(289\) 3447.28 0.701665
\(290\) 5265.89 1.06629
\(291\) 0 0
\(292\) −14364.2 −2.87878
\(293\) 5939.10 1.18418 0.592092 0.805870i \(-0.298302\pi\)
0.592092 + 0.805870i \(0.298302\pi\)
\(294\) 0 0
\(295\) −1202.47 −0.237324
\(296\) −14183.5 −2.78513
\(297\) 0 0
\(298\) −3716.36 −0.722425
\(299\) −7.17632 −0.00138802
\(300\) 0 0
\(301\) 0 0
\(302\) −11726.7 −2.23442
\(303\) 0 0
\(304\) −3079.14 −0.580924
\(305\) 3890.94 0.730474
\(306\) 0 0
\(307\) −10381.5 −1.92998 −0.964992 0.262278i \(-0.915526\pi\)
−0.964992 + 0.262278i \(0.915526\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4408.91 0.807772
\(311\) 4240.54 0.773181 0.386590 0.922252i \(-0.373653\pi\)
0.386590 + 0.922252i \(0.373653\pi\)
\(312\) 0 0
\(313\) 283.903 0.0512688 0.0256344 0.999671i \(-0.491839\pi\)
0.0256344 + 0.999671i \(0.491839\pi\)
\(314\) 15522.2 2.78972
\(315\) 0 0
\(316\) 8781.36 1.56326
\(317\) −1739.49 −0.308201 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(318\) 0 0
\(319\) 6588.80 1.15643
\(320\) −11622.9 −2.03044
\(321\) 0 0
\(322\) 0 0
\(323\) 1081.47 0.186300
\(324\) 0 0
\(325\) −1722.34 −0.293963
\(326\) −7418.48 −1.26034
\(327\) 0 0
\(328\) 11579.7 1.94933
\(329\) 0 0
\(330\) 0 0
\(331\) 5606.96 0.931077 0.465538 0.885028i \(-0.345861\pi\)
0.465538 + 0.885028i \(0.345861\pi\)
\(332\) 11071.5 1.83020
\(333\) 0 0
\(334\) 8007.89 1.31189
\(335\) −759.723 −0.123905
\(336\) 0 0
\(337\) −9427.44 −1.52387 −0.761937 0.647652i \(-0.775751\pi\)
−0.761937 + 0.647652i \(0.775751\pi\)
\(338\) 14067.0 2.26375
\(339\) 0 0
\(340\) 10262.6 1.63697
\(341\) 5516.53 0.876062
\(342\) 0 0
\(343\) 0 0
\(344\) 658.303 0.103178
\(345\) 0 0
\(346\) −10920.9 −1.69686
\(347\) −11634.2 −1.79988 −0.899939 0.436017i \(-0.856389\pi\)
−0.899939 + 0.436017i \(0.856389\pi\)
\(348\) 0 0
\(349\) 1317.10 0.202013 0.101006 0.994886i \(-0.467794\pi\)
0.101006 + 0.994886i \(0.467794\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −27572.1 −4.17499
\(353\) 5848.82 0.881874 0.440937 0.897538i \(-0.354646\pi\)
0.440937 + 0.897538i \(0.354646\pi\)
\(354\) 0 0
\(355\) −1557.24 −0.232817
\(356\) 10637.0 1.58359
\(357\) 0 0
\(358\) 24051.0 3.55065
\(359\) 12422.0 1.82621 0.913104 0.407727i \(-0.133679\pi\)
0.913104 + 0.407727i \(0.133679\pi\)
\(360\) 0 0
\(361\) −6719.10 −0.979604
\(362\) 2085.37 0.302775
\(363\) 0 0
\(364\) 0 0
\(365\) 3199.44 0.458812
\(366\) 0 0
\(367\) 6590.78 0.937427 0.468713 0.883350i \(-0.344718\pi\)
0.468713 + 0.883350i \(0.344718\pi\)
\(368\) 27.1174 0.00384128
\(369\) 0 0
\(370\) 4908.45 0.689670
\(371\) 0 0
\(372\) 0 0
\(373\) 344.278 0.0477910 0.0238955 0.999714i \(-0.492393\pi\)
0.0238955 + 0.999714i \(0.492393\pi\)
\(374\) 17417.0 2.40806
\(375\) 0 0
\(376\) 20770.4 2.84880
\(377\) 13149.2 1.79634
\(378\) 0 0
\(379\) 5241.23 0.710353 0.355177 0.934799i \(-0.384421\pi\)
0.355177 + 0.934799i \(0.384421\pi\)
\(380\) 1327.56 0.179217
\(381\) 0 0
\(382\) 13386.3 1.79294
\(383\) 7597.37 1.01360 0.506798 0.862065i \(-0.330829\pi\)
0.506798 + 0.862065i \(0.330829\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3437.27 −0.453245
\(387\) 0 0
\(388\) −18834.0 −2.46431
\(389\) −3101.84 −0.404291 −0.202146 0.979355i \(-0.564791\pi\)
−0.202146 + 0.979355i \(0.564791\pi\)
\(390\) 0 0
\(391\) −9.52432 −0.00123188
\(392\) 0 0
\(393\) 0 0
\(394\) −15683.7 −2.00541
\(395\) −1955.93 −0.249148
\(396\) 0 0
\(397\) 2932.06 0.370669 0.185335 0.982675i \(-0.440663\pi\)
0.185335 + 0.982675i \(0.440663\pi\)
\(398\) 4786.09 0.602777
\(399\) 0 0
\(400\) 6508.25 0.813531
\(401\) −89.2375 −0.0111130 −0.00555649 0.999985i \(-0.501769\pi\)
−0.00555649 + 0.999985i \(0.501769\pi\)
\(402\) 0 0
\(403\) 11009.3 1.36083
\(404\) 19921.3 2.45327
\(405\) 0 0
\(406\) 0 0
\(407\) 6141.56 0.747975
\(408\) 0 0
\(409\) −147.388 −0.0178188 −0.00890940 0.999960i \(-0.502836\pi\)
−0.00890940 + 0.999960i \(0.502836\pi\)
\(410\) −4007.36 −0.482706
\(411\) 0 0
\(412\) −13897.3 −1.66182
\(413\) 0 0
\(414\) 0 0
\(415\) −2466.03 −0.291693
\(416\) −55025.4 −6.48520
\(417\) 0 0
\(418\) 2253.04 0.263636
\(419\) −3781.67 −0.440923 −0.220462 0.975396i \(-0.570756\pi\)
−0.220462 + 0.975396i \(0.570756\pi\)
\(420\) 0 0
\(421\) −10899.2 −1.26175 −0.630874 0.775885i \(-0.717304\pi\)
−0.630874 + 0.775885i \(0.717304\pi\)
\(422\) −32975.0 −3.80378
\(423\) 0 0
\(424\) −28151.1 −3.22438
\(425\) −2285.86 −0.260896
\(426\) 0 0
\(427\) 0 0
\(428\) 48292.4 5.45398
\(429\) 0 0
\(430\) −227.817 −0.0255496
\(431\) 11283.4 1.26102 0.630512 0.776180i \(-0.282845\pi\)
0.630512 + 0.776180i \(0.282845\pi\)
\(432\) 0 0
\(433\) 8906.19 0.988462 0.494231 0.869331i \(-0.335450\pi\)
0.494231 + 0.869331i \(0.335450\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −9138.07 −1.00375
\(437\) −1.23205 −0.000134867 0
\(438\) 0 0
\(439\) −8149.49 −0.886000 −0.443000 0.896522i \(-0.646086\pi\)
−0.443000 + 0.896522i \(0.646086\pi\)
\(440\) 13760.8 1.49095
\(441\) 0 0
\(442\) 34759.1 3.74054
\(443\) 11472.8 1.23045 0.615223 0.788353i \(-0.289066\pi\)
0.615223 + 0.788353i \(0.289066\pi\)
\(444\) 0 0
\(445\) −2369.25 −0.252389
\(446\) 28591.8 3.03557
\(447\) 0 0
\(448\) 0 0
\(449\) −1963.80 −0.206409 −0.103204 0.994660i \(-0.532910\pi\)
−0.103204 + 0.994660i \(0.532910\pi\)
\(450\) 0 0
\(451\) −5014.10 −0.523514
\(452\) 7847.42 0.816618
\(453\) 0 0
\(454\) 20709.3 2.14083
\(455\) 0 0
\(456\) 0 0
\(457\) 5589.85 0.572171 0.286086 0.958204i \(-0.407646\pi\)
0.286086 + 0.958204i \(0.407646\pi\)
\(458\) 34532.3 3.52311
\(459\) 0 0
\(460\) −11.6915 −0.00118505
\(461\) −18790.9 −1.89844 −0.949219 0.314618i \(-0.898124\pi\)
−0.949219 + 0.314618i \(0.898124\pi\)
\(462\) 0 0
\(463\) −7892.22 −0.792187 −0.396094 0.918210i \(-0.629634\pi\)
−0.396094 + 0.918210i \(0.629634\pi\)
\(464\) −49687.4 −4.97130
\(465\) 0 0
\(466\) −9821.78 −0.976363
\(467\) 385.511 0.0381998 0.0190999 0.999818i \(-0.493920\pi\)
0.0190999 + 0.999818i \(0.493920\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −7187.95 −0.705437
\(471\) 0 0
\(472\) 19173.2 1.86974
\(473\) −285.050 −0.0277096
\(474\) 0 0
\(475\) −295.696 −0.0285631
\(476\) 0 0
\(477\) 0 0
\(478\) −19420.3 −1.85829
\(479\) 1694.46 0.161632 0.0808160 0.996729i \(-0.474247\pi\)
0.0808160 + 0.996729i \(0.474247\pi\)
\(480\) 0 0
\(481\) 12256.7 1.16186
\(482\) −2002.64 −0.189248
\(483\) 0 0
\(484\) −3126.91 −0.293662
\(485\) 4195.03 0.392755
\(486\) 0 0
\(487\) 15711.2 1.46189 0.730945 0.682436i \(-0.239079\pi\)
0.730945 + 0.682436i \(0.239079\pi\)
\(488\) −62040.2 −5.75498
\(489\) 0 0
\(490\) 0 0
\(491\) 2716.34 0.249667 0.124834 0.992178i \(-0.460160\pi\)
0.124834 + 0.992178i \(0.460160\pi\)
\(492\) 0 0
\(493\) 17451.5 1.59427
\(494\) 4496.38 0.409518
\(495\) 0 0
\(496\) −41601.2 −3.76603
\(497\) 0 0
\(498\) 0 0
\(499\) 4295.34 0.385342 0.192671 0.981263i \(-0.438285\pi\)
0.192671 + 0.981263i \(0.438285\pi\)
\(500\) −2806.00 −0.250977
\(501\) 0 0
\(502\) −29432.0 −2.61677
\(503\) 6515.26 0.577537 0.288769 0.957399i \(-0.406754\pi\)
0.288769 + 0.957399i \(0.406754\pi\)
\(504\) 0 0
\(505\) −4437.20 −0.390996
\(506\) −19.8421 −0.00174326
\(507\) 0 0
\(508\) 26573.5 2.32088
\(509\) 14391.8 1.25325 0.626624 0.779321i \(-0.284436\pi\)
0.626624 + 0.779321i \(0.284436\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 41890.2 3.61583
\(513\) 0 0
\(514\) −13457.1 −1.15480
\(515\) 3095.44 0.264857
\(516\) 0 0
\(517\) −8993.73 −0.765075
\(518\) 0 0
\(519\) 0 0
\(520\) 27462.3 2.31596
\(521\) 2913.36 0.244984 0.122492 0.992469i \(-0.460911\pi\)
0.122492 + 0.992469i \(0.460911\pi\)
\(522\) 0 0
\(523\) −16870.2 −1.41048 −0.705242 0.708967i \(-0.749162\pi\)
−0.705242 + 0.708967i \(0.749162\pi\)
\(524\) 5013.93 0.418005
\(525\) 0 0
\(526\) −8421.82 −0.698115
\(527\) 14611.4 1.20775
\(528\) 0 0
\(529\) −12167.0 −0.999999
\(530\) 9742.18 0.798441
\(531\) 0 0
\(532\) 0 0
\(533\) −10006.6 −0.813197
\(534\) 0 0
\(535\) −10756.5 −0.869241
\(536\) 12113.6 0.976172
\(537\) 0 0
\(538\) −41738.6 −3.34476
\(539\) 0 0
\(540\) 0 0
\(541\) −2229.59 −0.177186 −0.0885929 0.996068i \(-0.528237\pi\)
−0.0885929 + 0.996068i \(0.528237\pi\)
\(542\) −23631.7 −1.87282
\(543\) 0 0
\(544\) −73029.1 −5.75569
\(545\) 2035.38 0.159975
\(546\) 0 0
\(547\) −1218.73 −0.0952639 −0.0476319 0.998865i \(-0.515167\pi\)
−0.0476319 + 0.998865i \(0.515167\pi\)
\(548\) −45719.1 −3.56391
\(549\) 0 0
\(550\) −4762.16 −0.369198
\(551\) 2257.50 0.174542
\(552\) 0 0
\(553\) 0 0
\(554\) −22118.3 −1.69624
\(555\) 0 0
\(556\) −60317.9 −4.60081
\(557\) 22734.5 1.72943 0.864714 0.502264i \(-0.167499\pi\)
0.864714 + 0.502264i \(0.167499\pi\)
\(558\) 0 0
\(559\) −568.873 −0.0430425
\(560\) 0 0
\(561\) 0 0
\(562\) −38269.6 −2.87243
\(563\) 4302.11 0.322047 0.161023 0.986951i \(-0.448521\pi\)
0.161023 + 0.986951i \(0.448521\pi\)
\(564\) 0 0
\(565\) −1747.91 −0.130150
\(566\) 14714.4 1.09275
\(567\) 0 0
\(568\) 24829.9 1.83422
\(569\) −14866.9 −1.09535 −0.547675 0.836691i \(-0.684487\pi\)
−0.547675 + 0.836691i \(0.684487\pi\)
\(570\) 0 0
\(571\) 16514.5 1.21035 0.605174 0.796093i \(-0.293103\pi\)
0.605174 + 0.796093i \(0.293103\pi\)
\(572\) 53387.6 3.90253
\(573\) 0 0
\(574\) 0 0
\(575\) 2.60414 0.000188870 0
\(576\) 0 0
\(577\) −1867.97 −0.134774 −0.0673870 0.997727i \(-0.521466\pi\)
−0.0673870 + 0.997727i \(0.521466\pi\)
\(578\) 19022.0 1.36888
\(579\) 0 0
\(580\) 21422.5 1.53366
\(581\) 0 0
\(582\) 0 0
\(583\) 12189.6 0.865941
\(584\) −51014.3 −3.61471
\(585\) 0 0
\(586\) 32771.8 2.31022
\(587\) 3926.15 0.276064 0.138032 0.990428i \(-0.455922\pi\)
0.138032 + 0.990428i \(0.455922\pi\)
\(588\) 0 0
\(589\) 1890.11 0.132225
\(590\) −6635.21 −0.462996
\(591\) 0 0
\(592\) −46314.7 −3.21541
\(593\) −7554.18 −0.523125 −0.261562 0.965187i \(-0.584238\pi\)
−0.261562 + 0.965187i \(0.584238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −15118.8 −1.03907
\(597\) 0 0
\(598\) −39.5988 −0.00270788
\(599\) −28210.9 −1.92432 −0.962160 0.272487i \(-0.912154\pi\)
−0.962160 + 0.272487i \(0.912154\pi\)
\(600\) 0 0
\(601\) 23181.4 1.57336 0.786679 0.617363i \(-0.211799\pi\)
0.786679 + 0.617363i \(0.211799\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −47706.1 −3.21380
\(605\) 696.478 0.0468031
\(606\) 0 0
\(607\) 24863.6 1.66257 0.831287 0.555844i \(-0.187605\pi\)
0.831287 + 0.555844i \(0.187605\pi\)
\(608\) −9446.93 −0.630137
\(609\) 0 0
\(610\) 21470.1 1.42508
\(611\) −17948.7 −1.18843
\(612\) 0 0
\(613\) −12792.7 −0.842893 −0.421446 0.906853i \(-0.638477\pi\)
−0.421446 + 0.906853i \(0.638477\pi\)
\(614\) −57285.0 −3.76520
\(615\) 0 0
\(616\) 0 0
\(617\) −19793.3 −1.29149 −0.645744 0.763554i \(-0.723453\pi\)
−0.645744 + 0.763554i \(0.723453\pi\)
\(618\) 0 0
\(619\) 20951.7 1.36045 0.680226 0.733002i \(-0.261882\pi\)
0.680226 + 0.733002i \(0.261882\pi\)
\(620\) 17936.2 1.16183
\(621\) 0 0
\(622\) 23399.2 1.50840
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 1566.57 0.100020
\(627\) 0 0
\(628\) 63147.1 4.01249
\(629\) 16266.9 1.03117
\(630\) 0 0
\(631\) 23273.5 1.46831 0.734156 0.678981i \(-0.237578\pi\)
0.734156 + 0.678981i \(0.237578\pi\)
\(632\) 31186.9 1.96289
\(633\) 0 0
\(634\) −9598.47 −0.601268
\(635\) −5918.90 −0.369896
\(636\) 0 0
\(637\) 0 0
\(638\) 36356.8 2.25608
\(639\) 0 0
\(640\) −32186.8 −1.98796
\(641\) 22117.7 1.36287 0.681434 0.731880i \(-0.261357\pi\)
0.681434 + 0.731880i \(0.261357\pi\)
\(642\) 0 0
\(643\) −20269.9 −1.24318 −0.621591 0.783342i \(-0.713513\pi\)
−0.621591 + 0.783342i \(0.713513\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 5967.54 0.363451
\(647\) −3145.25 −0.191117 −0.0955583 0.995424i \(-0.530464\pi\)
−0.0955583 + 0.995424i \(0.530464\pi\)
\(648\) 0 0
\(649\) −8302.13 −0.502137
\(650\) −9503.81 −0.573493
\(651\) 0 0
\(652\) −30179.6 −1.81277
\(653\) 5953.35 0.356773 0.178386 0.983961i \(-0.442912\pi\)
0.178386 + 0.983961i \(0.442912\pi\)
\(654\) 0 0
\(655\) −1116.79 −0.0666205
\(656\) 37812.3 2.25049
\(657\) 0 0
\(658\) 0 0
\(659\) −26277.5 −1.55330 −0.776652 0.629930i \(-0.783084\pi\)
−0.776652 + 0.629930i \(0.783084\pi\)
\(660\) 0 0
\(661\) 24004.3 1.41250 0.706248 0.707964i \(-0.250386\pi\)
0.706248 + 0.707964i \(0.250386\pi\)
\(662\) 30939.1 1.81644
\(663\) 0 0
\(664\) 39320.3 2.29808
\(665\) 0 0
\(666\) 0 0
\(667\) −19.8814 −0.00115414
\(668\) 32577.4 1.88691
\(669\) 0 0
\(670\) −4192.13 −0.241725
\(671\) 26863.9 1.54556
\(672\) 0 0
\(673\) 9205.36 0.527252 0.263626 0.964625i \(-0.415082\pi\)
0.263626 + 0.964625i \(0.415082\pi\)
\(674\) −52020.4 −2.97292
\(675\) 0 0
\(676\) 57227.1 3.25598
\(677\) −18773.1 −1.06575 −0.532873 0.846195i \(-0.678888\pi\)
−0.532873 + 0.846195i \(0.678888\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 36447.6 2.05544
\(681\) 0 0
\(682\) 30440.1 1.70911
\(683\) 10222.2 0.572684 0.286342 0.958127i \(-0.407561\pi\)
0.286342 + 0.958127i \(0.407561\pi\)
\(684\) 0 0
\(685\) 10183.3 0.568006
\(686\) 0 0
\(687\) 0 0
\(688\) 2149.62 0.119118
\(689\) 24326.8 1.34510
\(690\) 0 0
\(691\) −22355.1 −1.23072 −0.615361 0.788246i \(-0.710990\pi\)
−0.615361 + 0.788246i \(0.710990\pi\)
\(692\) −44428.2 −2.44062
\(693\) 0 0
\(694\) −64197.3 −3.51138
\(695\) 13435.0 0.733264
\(696\) 0 0
\(697\) −13280.6 −0.721722
\(698\) 7267.70 0.394107
\(699\) 0 0
\(700\) 0 0
\(701\) −16217.3 −0.873779 −0.436890 0.899515i \(-0.643920\pi\)
−0.436890 + 0.899515i \(0.643920\pi\)
\(702\) 0 0
\(703\) 2104.26 0.112893
\(704\) −80247.1 −4.29606
\(705\) 0 0
\(706\) 32273.7 1.72045
\(707\) 0 0
\(708\) 0 0
\(709\) 3922.92 0.207798 0.103899 0.994588i \(-0.466868\pi\)
0.103899 + 0.994588i \(0.466868\pi\)
\(710\) −8592.83 −0.454202
\(711\) 0 0
\(712\) 37777.1 1.98842
\(713\) −16.6458 −0.000874322 0
\(714\) 0 0
\(715\) −11891.4 −0.621976
\(716\) 97843.4 5.10695
\(717\) 0 0
\(718\) 68544.3 3.56275
\(719\) 23908.6 1.24011 0.620055 0.784558i \(-0.287110\pi\)
0.620055 + 0.784558i \(0.287110\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −37075.8 −1.91111
\(723\) 0 0
\(724\) 8483.62 0.435485
\(725\) −4771.58 −0.244430
\(726\) 0 0
\(727\) 26906.4 1.37263 0.686315 0.727305i \(-0.259227\pi\)
0.686315 + 0.727305i \(0.259227\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 17654.4 0.895095
\(731\) −755.001 −0.0382007
\(732\) 0 0
\(733\) −27637.5 −1.39265 −0.696325 0.717726i \(-0.745183\pi\)
−0.696325 + 0.717726i \(0.745183\pi\)
\(734\) 36367.7 1.82882
\(735\) 0 0
\(736\) 83.1973 0.00416670
\(737\) −5245.29 −0.262161
\(738\) 0 0
\(739\) −14038.9 −0.698821 −0.349410 0.936970i \(-0.613618\pi\)
−0.349410 + 0.936970i \(0.613618\pi\)
\(740\) 19968.4 0.991963
\(741\) 0 0
\(742\) 0 0
\(743\) −17698.7 −0.873893 −0.436946 0.899488i \(-0.643940\pi\)
−0.436946 + 0.899488i \(0.643940\pi\)
\(744\) 0 0
\(745\) 3367.50 0.165605
\(746\) 1899.72 0.0932354
\(747\) 0 0
\(748\) 70855.4 3.46354
\(749\) 0 0
\(750\) 0 0
\(751\) 17199.7 0.835719 0.417859 0.908512i \(-0.362780\pi\)
0.417859 + 0.908512i \(0.362780\pi\)
\(752\) 67823.4 3.28892
\(753\) 0 0
\(754\) 72557.1 3.50448
\(755\) 10625.9 0.512206
\(756\) 0 0
\(757\) −19200.7 −0.921879 −0.460939 0.887432i \(-0.652487\pi\)
−0.460939 + 0.887432i \(0.652487\pi\)
\(758\) 28921.0 1.38583
\(759\) 0 0
\(760\) 4714.80 0.225031
\(761\) −27918.7 −1.32990 −0.664949 0.746889i \(-0.731547\pi\)
−0.664949 + 0.746889i \(0.731547\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 54457.7 2.57881
\(765\) 0 0
\(766\) 41922.1 1.97742
\(767\) −16568.5 −0.779992
\(768\) 0 0
\(769\) 7436.29 0.348712 0.174356 0.984683i \(-0.444216\pi\)
0.174356 + 0.984683i \(0.444216\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −13983.4 −0.651908
\(773\) 4989.84 0.232176 0.116088 0.993239i \(-0.462965\pi\)
0.116088 + 0.993239i \(0.462965\pi\)
\(774\) 0 0
\(775\) −3995.05 −0.185169
\(776\) −66888.7 −3.09429
\(777\) 0 0
\(778\) −17115.9 −0.788731
\(779\) −1717.96 −0.0790146
\(780\) 0 0
\(781\) −10751.5 −0.492600
\(782\) −52.5550 −0.00240328
\(783\) 0 0
\(784\) 0 0
\(785\) −14065.2 −0.639500
\(786\) 0 0
\(787\) 2870.69 0.130024 0.0650122 0.997884i \(-0.479291\pi\)
0.0650122 + 0.997884i \(0.479291\pi\)
\(788\) −63803.8 −2.88441
\(789\) 0 0
\(790\) −10792.8 −0.486063
\(791\) 0 0
\(792\) 0 0
\(793\) 53612.1 2.40078
\(794\) 16179.0 0.723138
\(795\) 0 0
\(796\) 19470.6 0.866982
\(797\) 4676.61 0.207847 0.103923 0.994585i \(-0.466860\pi\)
0.103923 + 0.994585i \(0.466860\pi\)
\(798\) 0 0
\(799\) −23821.3 −1.05474
\(800\) 19967.6 0.882451
\(801\) 0 0
\(802\) −492.410 −0.0216803
\(803\) 22089.6 0.970766
\(804\) 0 0
\(805\) 0 0
\(806\) 60749.1 2.65483
\(807\) 0 0
\(808\) 70750.2 3.08042
\(809\) 15376.9 0.668261 0.334131 0.942527i \(-0.391557\pi\)
0.334131 + 0.942527i \(0.391557\pi\)
\(810\) 0 0
\(811\) −36422.9 −1.57704 −0.788522 0.615007i \(-0.789153\pi\)
−0.788522 + 0.615007i \(0.789153\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 33889.0 1.45922
\(815\) 6722.10 0.288914
\(816\) 0 0
\(817\) −97.6658 −0.00418225
\(818\) −813.285 −0.0347627
\(819\) 0 0
\(820\) −16302.6 −0.694282
\(821\) 25479.0 1.08310 0.541550 0.840669i \(-0.317838\pi\)
0.541550 + 0.840669i \(0.317838\pi\)
\(822\) 0 0
\(823\) −17933.3 −0.759557 −0.379779 0.925077i \(-0.624000\pi\)
−0.379779 + 0.925077i \(0.624000\pi\)
\(824\) −49356.1 −2.08665
\(825\) 0 0
\(826\) 0 0
\(827\) −13021.6 −0.547529 −0.273764 0.961797i \(-0.588269\pi\)
−0.273764 + 0.961797i \(0.588269\pi\)
\(828\) 0 0
\(829\) 11397.6 0.477509 0.238754 0.971080i \(-0.423261\pi\)
0.238754 + 0.971080i \(0.423261\pi\)
\(830\) −13607.5 −0.569063
\(831\) 0 0
\(832\) −160149. −6.67326
\(833\) 0 0
\(834\) 0 0
\(835\) −7256.19 −0.300731
\(836\) 9165.75 0.379191
\(837\) 0 0
\(838\) −20867.2 −0.860196
\(839\) −37681.2 −1.55053 −0.775267 0.631633i \(-0.782385\pi\)
−0.775267 + 0.631633i \(0.782385\pi\)
\(840\) 0 0
\(841\) 12039.8 0.493656
\(842\) −60141.7 −2.46154
\(843\) 0 0
\(844\) −134148. −5.47104
\(845\) −12746.6 −0.518929
\(846\) 0 0
\(847\) 0 0
\(848\) −91924.4 −3.72252
\(849\) 0 0
\(850\) −12613.3 −0.508981
\(851\) −18.5318 −0.000746490 0
\(852\) 0 0
\(853\) −21771.6 −0.873911 −0.436956 0.899483i \(-0.643943\pi\)
−0.436956 + 0.899483i \(0.643943\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 171510. 6.84823
\(857\) 29860.2 1.19021 0.595103 0.803649i \(-0.297111\pi\)
0.595103 + 0.803649i \(0.297111\pi\)
\(858\) 0 0
\(859\) −18530.6 −0.736039 −0.368019 0.929818i \(-0.619964\pi\)
−0.368019 + 0.929818i \(0.619964\pi\)
\(860\) −926.799 −0.0367484
\(861\) 0 0
\(862\) 62261.4 2.46013
\(863\) 21296.5 0.840024 0.420012 0.907519i \(-0.362026\pi\)
0.420012 + 0.907519i \(0.362026\pi\)
\(864\) 0 0
\(865\) 9895.79 0.388979
\(866\) 49144.1 1.92839
\(867\) 0 0
\(868\) 0 0
\(869\) −13504.2 −0.527155
\(870\) 0 0
\(871\) −10468.0 −0.407226
\(872\) −32453.7 −1.26035
\(873\) 0 0
\(874\) −6.79843 −0.000263112 0
\(875\) 0 0
\(876\) 0 0
\(877\) −18793.4 −0.723614 −0.361807 0.932253i \(-0.617840\pi\)
−0.361807 + 0.932253i \(0.617840\pi\)
\(878\) −44968.7 −1.72850
\(879\) 0 0
\(880\) 44934.4 1.72129
\(881\) 1638.62 0.0626634 0.0313317 0.999509i \(-0.490025\pi\)
0.0313317 + 0.999509i \(0.490025\pi\)
\(882\) 0 0
\(883\) −35424.1 −1.35008 −0.675038 0.737783i \(-0.735873\pi\)
−0.675038 + 0.737783i \(0.735873\pi\)
\(884\) 141406. 5.38008
\(885\) 0 0
\(886\) 63306.5 2.40048
\(887\) −5131.41 −0.194246 −0.0971229 0.995272i \(-0.530964\pi\)
−0.0971229 + 0.995272i \(0.530964\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −13073.5 −0.492386
\(891\) 0 0
\(892\) 116316. 4.36610
\(893\) −3081.49 −0.115474
\(894\) 0 0
\(895\) −21793.3 −0.813933
\(896\) 0 0
\(897\) 0 0
\(898\) −10836.2 −0.402682
\(899\) 30500.3 1.13153
\(900\) 0 0
\(901\) 32286.2 1.19380
\(902\) −27667.7 −1.02132
\(903\) 0 0
\(904\) 27870.0 1.02538
\(905\) −1889.61 −0.0694065
\(906\) 0 0
\(907\) −19934.6 −0.729787 −0.364893 0.931049i \(-0.618895\pi\)
−0.364893 + 0.931049i \(0.618895\pi\)
\(908\) 84248.8 3.07918
\(909\) 0 0
\(910\) 0 0
\(911\) −48387.4 −1.75976 −0.879882 0.475193i \(-0.842378\pi\)
−0.879882 + 0.475193i \(0.842378\pi\)
\(912\) 0 0
\(913\) −17026.0 −0.617172
\(914\) 30844.7 1.11625
\(915\) 0 0
\(916\) 140483. 5.06735
\(917\) 0 0
\(918\) 0 0
\(919\) −14431.7 −0.518019 −0.259009 0.965875i \(-0.583396\pi\)
−0.259009 + 0.965875i \(0.583396\pi\)
\(920\) −41.5224 −0.00148799
\(921\) 0 0
\(922\) −103688. −3.70366
\(923\) −21456.8 −0.765177
\(924\) 0 0
\(925\) −4447.69 −0.158096
\(926\) −43549.1 −1.54548
\(927\) 0 0
\(928\) −152443. −5.39244
\(929\) 5549.82 0.196000 0.0979998 0.995186i \(-0.468756\pi\)
0.0979998 + 0.995186i \(0.468756\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −39956.7 −1.40432
\(933\) 0 0
\(934\) 2127.24 0.0745239
\(935\) −15782.1 −0.552010
\(936\) 0 0
\(937\) −26231.9 −0.914576 −0.457288 0.889319i \(-0.651179\pi\)
−0.457288 + 0.889319i \(0.651179\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −29241.8 −1.01464
\(941\) 2836.80 0.0982753 0.0491376 0.998792i \(-0.484353\pi\)
0.0491376 + 0.998792i \(0.484353\pi\)
\(942\) 0 0
\(943\) 15.1298 0.000522474 0
\(944\) 62607.9 2.15860
\(945\) 0 0
\(946\) −1572.90 −0.0540586
\(947\) 40272.9 1.38193 0.690967 0.722886i \(-0.257185\pi\)
0.690967 + 0.722886i \(0.257185\pi\)
\(948\) 0 0
\(949\) 44084.1 1.50793
\(950\) −1631.64 −0.0557236
\(951\) 0 0
\(952\) 0 0
\(953\) −17770.1 −0.604019 −0.302010 0.953305i \(-0.597657\pi\)
−0.302010 + 0.953305i \(0.597657\pi\)
\(954\) 0 0
\(955\) −12129.7 −0.411004
\(956\) −79005.0 −2.67281
\(957\) 0 0
\(958\) 9349.97 0.315328
\(959\) 0 0
\(960\) 0 0
\(961\) −4254.35 −0.142807
\(962\) 67632.0 2.26668
\(963\) 0 0
\(964\) −8147.07 −0.272199
\(965\) 3114.61 0.103899
\(966\) 0 0
\(967\) −3530.14 −0.117396 −0.0586978 0.998276i \(-0.518695\pi\)
−0.0586978 + 0.998276i \(0.518695\pi\)
\(968\) −11105.2 −0.368734
\(969\) 0 0
\(970\) 23148.0 0.766226
\(971\) 17650.7 0.583356 0.291678 0.956517i \(-0.405786\pi\)
0.291678 + 0.956517i \(0.405786\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 86693.8 2.85200
\(975\) 0 0
\(976\) −202586. −6.64407
\(977\) 14477.6 0.474085 0.237042 0.971499i \(-0.423822\pi\)
0.237042 + 0.971499i \(0.423822\pi\)
\(978\) 0 0
\(979\) −16357.8 −0.534012
\(980\) 0 0
\(981\) 0 0
\(982\) 14988.7 0.487075
\(983\) −8764.09 −0.284365 −0.142183 0.989840i \(-0.545412\pi\)
−0.142183 + 0.989840i \(0.545412\pi\)
\(984\) 0 0
\(985\) 14211.4 0.459710
\(986\) 96296.9 3.11026
\(987\) 0 0
\(988\) 18292.0 0.589015
\(989\) 0.860124 2.76546e−5 0
\(990\) 0 0
\(991\) 33624.1 1.07781 0.538903 0.842368i \(-0.318839\pi\)
0.538903 + 0.842368i \(0.318839\pi\)
\(992\) −127634. −4.08507
\(993\) 0 0
\(994\) 0 0
\(995\) −4336.82 −0.138177
\(996\) 0 0
\(997\) −16631.3 −0.528302 −0.264151 0.964481i \(-0.585092\pi\)
−0.264151 + 0.964481i \(0.585092\pi\)
\(998\) 23701.6 0.751764
\(999\) 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 2205.4.a.bz.1.6 6
3.2 odd 2 245.4.a.o.1.1 6
7.6 odd 2 2205.4.a.ca.1.6 6
15.14 odd 2 1225.4.a.bj.1.6 6
21.2 odd 6 245.4.e.q.116.6 12
21.5 even 6 245.4.e.p.116.6 12
21.11 odd 6 245.4.e.q.226.6 12
21.17 even 6 245.4.e.p.226.6 12
21.20 even 2 245.4.a.p.1.1 yes 6
105.104 even 2 1225.4.a.bi.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.o.1.1 6 3.2 odd 2
245.4.a.p.1.1 yes 6 21.20 even 2
245.4.e.p.116.6 12 21.5 even 6
245.4.e.p.226.6 12 21.17 even 6
245.4.e.q.116.6 12 21.2 odd 6
245.4.e.q.226.6 12 21.11 odd 6
1225.4.a.bi.1.6 6 105.104 even 2
1225.4.a.bj.1.6 6 15.14 odd 2
2205.4.a.bz.1.6 6 1.1 even 1 trivial
2205.4.a.ca.1.6 6 7.6 odd 2