Properties

Label 784.6.a.bf.1.1
Level $784$
Weight $6$
Character 784.1
Self dual yes
Analytic conductor $125.741$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(125.740914733\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{113})\)
Defining polynomial: \( x^{4} - 2x^{3} - 59x^{2} + 60x + 674 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.40086\) of defining polynomial
Character \(\chi\) \(=\) 784.1

$q$-expansion

\(f(q)\) \(=\) \(q-23.5186 q^{3} -74.2753 q^{5} +310.123 q^{9} +O(q^{10})\) \(q-23.5186 q^{3} -74.2753 q^{5} +310.123 q^{9} +424.219 q^{11} -252.233 q^{13} +1746.85 q^{15} -1104.35 q^{17} +6.47100 q^{19} +3612.39 q^{23} +2391.82 q^{25} -1578.65 q^{27} -5005.02 q^{29} -2821.69 q^{31} -9977.03 q^{33} -2046.88 q^{37} +5932.17 q^{39} +9393.81 q^{41} -10320.8 q^{43} -23034.5 q^{45} -17035.6 q^{47} +25972.7 q^{51} -39506.7 q^{53} -31509.0 q^{55} -152.189 q^{57} -33949.8 q^{59} +28295.2 q^{61} +18734.7 q^{65} -56100.9 q^{67} -84958.2 q^{69} +15537.4 q^{71} -78219.5 q^{73} -56252.3 q^{75} +45335.5 q^{79} -38232.4 q^{81} -1381.82 q^{83} +82025.7 q^{85} +117711. q^{87} -68879.4 q^{89} +66362.1 q^{93} -480.635 q^{95} +108857. q^{97} +131560. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 220 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 220 q^{9} + 1952 q^{11} + 4096 q^{15} + 7136 q^{23} + 2764 q^{25} - 3352 q^{29} - 9208 q^{37} - 2464 q^{39} - 20448 q^{43} + 67408 q^{51} - 102920 q^{53} - 15576 q^{57} - 63168 q^{65} + 22896 q^{67} + 153824 q^{71} + 90688 q^{79} - 17204 q^{81} + 272656 q^{85} + 247760 q^{93} - 108224 q^{95} + 42272 q^{99}+O(q^{100}) \) 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 0
\(3\) −23.5186 −1.50872 −0.754359 0.656462i \(-0.772052\pi\)
−0.754359 + 0.656462i \(0.772052\pi\)
\(4\) 0 0
\(5\) −74.2753 −1.32868 −0.664339 0.747432i \(-0.731287\pi\)
−0.664339 + 0.747432i \(0.731287\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 310.123 1.27623
\(10\) 0 0
\(11\) 424.219 1.05708 0.528541 0.848908i \(-0.322739\pi\)
0.528541 + 0.848908i \(0.322739\pi\)
\(12\) 0 0
\(13\) −252.233 −0.413946 −0.206973 0.978347i \(-0.566361\pi\)
−0.206973 + 0.978347i \(0.566361\pi\)
\(14\) 0 0
\(15\) 1746.85 2.00460
\(16\) 0 0
\(17\) −1104.35 −0.926794 −0.463397 0.886151i \(-0.653370\pi\)
−0.463397 + 0.886151i \(0.653370\pi\)
\(18\) 0 0
\(19\) 6.47100 0.00411232 0.00205616 0.999998i \(-0.499346\pi\)
0.00205616 + 0.999998i \(0.499346\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3612.39 1.42388 0.711942 0.702239i \(-0.247816\pi\)
0.711942 + 0.702239i \(0.247816\pi\)
\(24\) 0 0
\(25\) 2391.82 0.765383
\(26\) 0 0
\(27\) −1578.65 −0.416751
\(28\) 0 0
\(29\) −5005.02 −1.10512 −0.552561 0.833472i \(-0.686350\pi\)
−0.552561 + 0.833472i \(0.686350\pi\)
\(30\) 0 0
\(31\) −2821.69 −0.527357 −0.263679 0.964611i \(-0.584936\pi\)
−0.263679 + 0.964611i \(0.584936\pi\)
\(32\) 0 0
\(33\) −9977.03 −1.59484
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2046.88 −0.245803 −0.122902 0.992419i \(-0.539220\pi\)
−0.122902 + 0.992419i \(0.539220\pi\)
\(38\) 0 0
\(39\) 5932.17 0.624528
\(40\) 0 0
\(41\) 9393.81 0.872734 0.436367 0.899769i \(-0.356265\pi\)
0.436367 + 0.899769i \(0.356265\pi\)
\(42\) 0 0
\(43\) −10320.8 −0.851218 −0.425609 0.904907i \(-0.639940\pi\)
−0.425609 + 0.904907i \(0.639940\pi\)
\(44\) 0 0
\(45\) −23034.5 −1.69570
\(46\) 0 0
\(47\) −17035.6 −1.12490 −0.562448 0.826833i \(-0.690140\pi\)
−0.562448 + 0.826833i \(0.690140\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 25972.7 1.39827
\(52\) 0 0
\(53\) −39506.7 −1.93188 −0.965941 0.258761i \(-0.916686\pi\)
−0.965941 + 0.258761i \(0.916686\pi\)
\(54\) 0 0
\(55\) −31509.0 −1.40452
\(56\) 0 0
\(57\) −152.189 −0.00620433
\(58\) 0 0
\(59\) −33949.8 −1.26972 −0.634859 0.772628i \(-0.718942\pi\)
−0.634859 + 0.772628i \(0.718942\pi\)
\(60\) 0 0
\(61\) 28295.2 0.973618 0.486809 0.873508i \(-0.338161\pi\)
0.486809 + 0.873508i \(0.338161\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 18734.7 0.550001
\(66\) 0 0
\(67\) −56100.9 −1.52680 −0.763402 0.645924i \(-0.776472\pi\)
−0.763402 + 0.645924i \(0.776472\pi\)
\(68\) 0 0
\(69\) −84958.2 −2.14824
\(70\) 0 0
\(71\) 15537.4 0.365791 0.182895 0.983132i \(-0.441453\pi\)
0.182895 + 0.983132i \(0.441453\pi\)
\(72\) 0 0
\(73\) −78219.5 −1.71794 −0.858970 0.512025i \(-0.828895\pi\)
−0.858970 + 0.512025i \(0.828895\pi\)
\(74\) 0 0
\(75\) −56252.3 −1.15475
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 45335.5 0.817279 0.408640 0.912696i \(-0.366003\pi\)
0.408640 + 0.912696i \(0.366003\pi\)
\(80\) 0 0
\(81\) −38232.4 −0.647469
\(82\) 0 0
\(83\) −1381.82 −0.0220169 −0.0110085 0.999939i \(-0.503504\pi\)
−0.0110085 + 0.999939i \(0.503504\pi\)
\(84\) 0 0
\(85\) 82025.7 1.23141
\(86\) 0 0
\(87\) 117711. 1.66732
\(88\) 0 0
\(89\) −68879.4 −0.921753 −0.460876 0.887464i \(-0.652465\pi\)
−0.460876 + 0.887464i \(0.652465\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 66362.1 0.795633
\(94\) 0 0
\(95\) −480.635 −0.00546395
\(96\) 0 0
\(97\) 108857. 1.17470 0.587351 0.809332i \(-0.300171\pi\)
0.587351 + 0.809332i \(0.300171\pi\)
\(98\) 0 0
\(99\) 131560. 1.34908
\(100\) 0 0
\(101\) 17972.3 0.175307 0.0876535 0.996151i \(-0.472063\pi\)
0.0876535 + 0.996151i \(0.472063\pi\)
\(102\) 0 0
\(103\) −31773.7 −0.295103 −0.147552 0.989054i \(-0.547139\pi\)
−0.147552 + 0.989054i \(0.547139\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8229.36 −0.0694875 −0.0347438 0.999396i \(-0.511062\pi\)
−0.0347438 + 0.999396i \(0.511062\pi\)
\(108\) 0 0
\(109\) −11068.7 −0.0892338 −0.0446169 0.999004i \(-0.514207\pi\)
−0.0446169 + 0.999004i \(0.514207\pi\)
\(110\) 0 0
\(111\) 48139.6 0.370847
\(112\) 0 0
\(113\) 65184.3 0.480228 0.240114 0.970745i \(-0.422815\pi\)
0.240114 + 0.970745i \(0.422815\pi\)
\(114\) 0 0
\(115\) −268311. −1.89188
\(116\) 0 0
\(117\) −78223.5 −0.528290
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 18910.9 0.117422
\(122\) 0 0
\(123\) −220929. −1.31671
\(124\) 0 0
\(125\) 54456.9 0.311730
\(126\) 0 0
\(127\) 194777. 1.07159 0.535796 0.844348i \(-0.320012\pi\)
0.535796 + 0.844348i \(0.320012\pi\)
\(128\) 0 0
\(129\) 242730. 1.28425
\(130\) 0 0
\(131\) 236503. 1.20409 0.602046 0.798462i \(-0.294353\pi\)
0.602046 + 0.798462i \(0.294353\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 117255. 0.553727
\(136\) 0 0
\(137\) −200903. −0.914503 −0.457252 0.889337i \(-0.651166\pi\)
−0.457252 + 0.889337i \(0.651166\pi\)
\(138\) 0 0
\(139\) −52985.2 −0.232604 −0.116302 0.993214i \(-0.537104\pi\)
−0.116302 + 0.993214i \(0.537104\pi\)
\(140\) 0 0
\(141\) 400653. 1.69715
\(142\) 0 0
\(143\) −107002. −0.437575
\(144\) 0 0
\(145\) 371749. 1.46835
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −100770. −0.371849 −0.185925 0.982564i \(-0.559528\pi\)
−0.185925 + 0.982564i \(0.559528\pi\)
\(150\) 0 0
\(151\) 457904. 1.63430 0.817150 0.576425i \(-0.195553\pi\)
0.817150 + 0.576425i \(0.195553\pi\)
\(152\) 0 0
\(153\) −342484. −1.18280
\(154\) 0 0
\(155\) 209582. 0.700688
\(156\) 0 0
\(157\) −179037. −0.579688 −0.289844 0.957074i \(-0.593603\pi\)
−0.289844 + 0.957074i \(0.593603\pi\)
\(158\) 0 0
\(159\) 929141. 2.91467
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −243610. −0.718168 −0.359084 0.933305i \(-0.616911\pi\)
−0.359084 + 0.933305i \(0.616911\pi\)
\(164\) 0 0
\(165\) 741047. 2.11902
\(166\) 0 0
\(167\) −117033. −0.324725 −0.162362 0.986731i \(-0.551911\pi\)
−0.162362 + 0.986731i \(0.551911\pi\)
\(168\) 0 0
\(169\) −307671. −0.828648
\(170\) 0 0
\(171\) 2006.81 0.00524826
\(172\) 0 0
\(173\) 269733. 0.685203 0.342602 0.939481i \(-0.388692\pi\)
0.342602 + 0.939481i \(0.388692\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 798451. 1.91564
\(178\) 0 0
\(179\) −376525. −0.878336 −0.439168 0.898405i \(-0.644727\pi\)
−0.439168 + 0.898405i \(0.644727\pi\)
\(180\) 0 0
\(181\) −434641. −0.986131 −0.493065 0.869992i \(-0.664124\pi\)
−0.493065 + 0.869992i \(0.664124\pi\)
\(182\) 0 0
\(183\) −665464. −1.46891
\(184\) 0 0
\(185\) 152032. 0.326593
\(186\) 0 0
\(187\) −468485. −0.979697
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −565940. −1.12250 −0.561250 0.827646i \(-0.689680\pi\)
−0.561250 + 0.827646i \(0.689680\pi\)
\(192\) 0 0
\(193\) 514461. 0.994167 0.497084 0.867703i \(-0.334404\pi\)
0.497084 + 0.867703i \(0.334404\pi\)
\(194\) 0 0
\(195\) −440614. −0.829796
\(196\) 0 0
\(197\) −298541. −0.548073 −0.274037 0.961719i \(-0.588359\pi\)
−0.274037 + 0.961719i \(0.588359\pi\)
\(198\) 0 0
\(199\) 591919. 1.05957 0.529785 0.848132i \(-0.322273\pi\)
0.529785 + 0.848132i \(0.322273\pi\)
\(200\) 0 0
\(201\) 1.31941e6 2.30351
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −697728. −1.15958
\(206\) 0 0
\(207\) 1.12029e6 1.81720
\(208\) 0 0
\(209\) 2745.12 0.00434706
\(210\) 0 0
\(211\) 140535. 0.217309 0.108655 0.994080i \(-0.465346\pi\)
0.108655 + 0.994080i \(0.465346\pi\)
\(212\) 0 0
\(213\) −365418. −0.551875
\(214\) 0 0
\(215\) 766579. 1.13099
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 1.83961e6 2.59189
\(220\) 0 0
\(221\) 278553. 0.383643
\(222\) 0 0
\(223\) −490.526 −0.000660541 0 −0.000330271 1.00000i \(-0.500105\pi\)
−0.000330271 1.00000i \(0.500105\pi\)
\(224\) 0 0
\(225\) 741761. 0.976804
\(226\) 0 0
\(227\) −593898. −0.764975 −0.382488 0.923961i \(-0.624933\pi\)
−0.382488 + 0.923961i \(0.624933\pi\)
\(228\) 0 0
\(229\) −35880.3 −0.0452135 −0.0226067 0.999744i \(-0.507197\pi\)
−0.0226067 + 0.999744i \(0.507197\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.24822e6 1.50626 0.753131 0.657871i \(-0.228543\pi\)
0.753131 + 0.657871i \(0.228543\pi\)
\(234\) 0 0
\(235\) 1.26532e6 1.49462
\(236\) 0 0
\(237\) −1.06623e6 −1.23304
\(238\) 0 0
\(239\) 576943. 0.653339 0.326669 0.945139i \(-0.394074\pi\)
0.326669 + 0.945139i \(0.394074\pi\)
\(240\) 0 0
\(241\) −1.38241e6 −1.53318 −0.766592 0.642135i \(-0.778049\pi\)
−0.766592 + 0.642135i \(0.778049\pi\)
\(242\) 0 0
\(243\) 1.28278e6 1.39360
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1632.20 −0.00170228
\(248\) 0 0
\(249\) 32498.5 0.0332173
\(250\) 0 0
\(251\) −323217. −0.323824 −0.161912 0.986805i \(-0.551766\pi\)
−0.161912 + 0.986805i \(0.551766\pi\)
\(252\) 0 0
\(253\) 1.53244e6 1.50516
\(254\) 0 0
\(255\) −1.92913e6 −1.85785
\(256\) 0 0
\(257\) 1.84601e6 1.74342 0.871711 0.490021i \(-0.163011\pi\)
0.871711 + 0.490021i \(0.163011\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.55217e6 −1.41039
\(262\) 0 0
\(263\) 458222. 0.408495 0.204248 0.978919i \(-0.434525\pi\)
0.204248 + 0.978919i \(0.434525\pi\)
\(264\) 0 0
\(265\) 2.93437e6 2.56685
\(266\) 0 0
\(267\) 1.61995e6 1.39066
\(268\) 0 0
\(269\) 416958. 0.351327 0.175663 0.984450i \(-0.443793\pi\)
0.175663 + 0.984450i \(0.443793\pi\)
\(270\) 0 0
\(271\) 900379. 0.744735 0.372368 0.928085i \(-0.378546\pi\)
0.372368 + 0.928085i \(0.378546\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.01466e6 0.809073
\(276\) 0 0
\(277\) −447641. −0.350535 −0.175267 0.984521i \(-0.556079\pi\)
−0.175267 + 0.984521i \(0.556079\pi\)
\(278\) 0 0
\(279\) −875072. −0.673029
\(280\) 0 0
\(281\) −768521. −0.580617 −0.290309 0.956933i \(-0.593758\pi\)
−0.290309 + 0.956933i \(0.593758\pi\)
\(282\) 0 0
\(283\) 2.13220e6 1.58256 0.791282 0.611452i \(-0.209414\pi\)
0.791282 + 0.611452i \(0.209414\pi\)
\(284\) 0 0
\(285\) 11303.9 0.00824356
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −200275. −0.141053
\(290\) 0 0
\(291\) −2.56017e6 −1.77229
\(292\) 0 0
\(293\) −2.42669e6 −1.65138 −0.825688 0.564128i \(-0.809213\pi\)
−0.825688 + 0.564128i \(0.809213\pi\)
\(294\) 0 0
\(295\) 2.52163e6 1.68704
\(296\) 0 0
\(297\) −669693. −0.440539
\(298\) 0 0
\(299\) −911164. −0.589411
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −422682. −0.264489
\(304\) 0 0
\(305\) −2.10164e6 −1.29362
\(306\) 0 0
\(307\) −2.44328e6 −1.47954 −0.739772 0.672857i \(-0.765067\pi\)
−0.739772 + 0.672857i \(0.765067\pi\)
\(308\) 0 0
\(309\) 747271. 0.445228
\(310\) 0 0
\(311\) 1.15465e6 0.676938 0.338469 0.940978i \(-0.390091\pi\)
0.338469 + 0.940978i \(0.390091\pi\)
\(312\) 0 0
\(313\) −1.65706e6 −0.956044 −0.478022 0.878348i \(-0.658646\pi\)
−0.478022 + 0.878348i \(0.658646\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 821361. 0.459077 0.229539 0.973300i \(-0.426278\pi\)
0.229539 + 0.973300i \(0.426278\pi\)
\(318\) 0 0
\(319\) −2.12322e6 −1.16821
\(320\) 0 0
\(321\) 193543. 0.104837
\(322\) 0 0
\(323\) −7146.23 −0.00381128
\(324\) 0 0
\(325\) −603298. −0.316828
\(326\) 0 0
\(327\) 260319. 0.134629
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 95670.7 0.0479964 0.0239982 0.999712i \(-0.492360\pi\)
0.0239982 + 0.999712i \(0.492360\pi\)
\(332\) 0 0
\(333\) −634785. −0.313701
\(334\) 0 0
\(335\) 4.16691e6 2.02863
\(336\) 0 0
\(337\) 2.37020e6 1.13687 0.568435 0.822728i \(-0.307549\pi\)
0.568435 + 0.822728i \(0.307549\pi\)
\(338\) 0 0
\(339\) −1.53304e6 −0.724528
\(340\) 0 0
\(341\) −1.19701e6 −0.557460
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 6.31029e6 2.85431
\(346\) 0 0
\(347\) −490571. −0.218715 −0.109358 0.994002i \(-0.534879\pi\)
−0.109358 + 0.994002i \(0.534879\pi\)
\(348\) 0 0
\(349\) −4.21208e6 −1.85111 −0.925557 0.378607i \(-0.876403\pi\)
−0.925557 + 0.378607i \(0.876403\pi\)
\(350\) 0 0
\(351\) 398188. 0.172512
\(352\) 0 0
\(353\) 3.17378e6 1.35563 0.677814 0.735234i \(-0.262928\pi\)
0.677814 + 0.735234i \(0.262928\pi\)
\(354\) 0 0
\(355\) −1.15405e6 −0.486018
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.40098e6 1.39273 0.696366 0.717687i \(-0.254799\pi\)
0.696366 + 0.717687i \(0.254799\pi\)
\(360\) 0 0
\(361\) −2.47606e6 −0.999983
\(362\) 0 0
\(363\) −444757. −0.177156
\(364\) 0 0
\(365\) 5.80978e6 2.28259
\(366\) 0 0
\(367\) −1.96872e6 −0.762988 −0.381494 0.924371i \(-0.624590\pi\)
−0.381494 + 0.924371i \(0.624590\pi\)
\(368\) 0 0
\(369\) 2.91324e6 1.11381
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −3.47889e6 −1.29470 −0.647349 0.762194i \(-0.724122\pi\)
−0.647349 + 0.762194i \(0.724122\pi\)
\(374\) 0 0
\(375\) −1.28075e6 −0.470312
\(376\) 0 0
\(377\) 1.26243e6 0.457462
\(378\) 0 0
\(379\) 421294. 0.150656 0.0753281 0.997159i \(-0.476000\pi\)
0.0753281 + 0.997159i \(0.476000\pi\)
\(380\) 0 0
\(381\) −4.58089e6 −1.61673
\(382\) 0 0
\(383\) −2.66910e6 −0.929754 −0.464877 0.885375i \(-0.653901\pi\)
−0.464877 + 0.885375i \(0.653901\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.20071e6 −1.08635
\(388\) 0 0
\(389\) 3.10178e6 1.03929 0.519645 0.854382i \(-0.326064\pi\)
0.519645 + 0.854382i \(0.326064\pi\)
\(390\) 0 0
\(391\) −3.98933e6 −1.31965
\(392\) 0 0
\(393\) −5.56223e6 −1.81663
\(394\) 0 0
\(395\) −3.36731e6 −1.08590
\(396\) 0 0
\(397\) 613257. 0.195284 0.0976419 0.995222i \(-0.468870\pi\)
0.0976419 + 0.995222i \(0.468870\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.82223e6 0.876459 0.438229 0.898863i \(-0.355606\pi\)
0.438229 + 0.898863i \(0.355606\pi\)
\(402\) 0 0
\(403\) 711724. 0.218298
\(404\) 0 0
\(405\) 2.83973e6 0.860278
\(406\) 0 0
\(407\) −868324. −0.259834
\(408\) 0 0
\(409\) 2.28350e6 0.674983 0.337492 0.941329i \(-0.390422\pi\)
0.337492 + 0.941329i \(0.390422\pi\)
\(410\) 0 0
\(411\) 4.72496e6 1.37973
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 102635. 0.0292534
\(416\) 0 0
\(417\) 1.24614e6 0.350934
\(418\) 0 0
\(419\) 2.65270e6 0.738163 0.369082 0.929397i \(-0.379672\pi\)
0.369082 + 0.929397i \(0.379672\pi\)
\(420\) 0 0
\(421\) 2.93674e6 0.807532 0.403766 0.914862i \(-0.367701\pi\)
0.403766 + 0.914862i \(0.367701\pi\)
\(422\) 0 0
\(423\) −5.28314e6 −1.43562
\(424\) 0 0
\(425\) −2.64140e6 −0.709353
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 2.51654e6 0.660177
\(430\) 0 0
\(431\) 2.44565e6 0.634164 0.317082 0.948398i \(-0.397297\pi\)
0.317082 + 0.948398i \(0.397297\pi\)
\(432\) 0 0
\(433\) −2.11718e6 −0.542673 −0.271336 0.962485i \(-0.587466\pi\)
−0.271336 + 0.962485i \(0.587466\pi\)
\(434\) 0 0
\(435\) −8.74301e6 −2.21533
\(436\) 0 0
\(437\) 23375.7 0.00585547
\(438\) 0 0
\(439\) −4.64764e6 −1.15099 −0.575495 0.817806i \(-0.695190\pi\)
−0.575495 + 0.817806i \(0.695190\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.42925e6 −1.07231 −0.536155 0.844119i \(-0.680124\pi\)
−0.536155 + 0.844119i \(0.680124\pi\)
\(444\) 0 0
\(445\) 5.11604e6 1.22471
\(446\) 0 0
\(447\) 2.36998e6 0.561016
\(448\) 0 0
\(449\) −6.70171e6 −1.56881 −0.784404 0.620250i \(-0.787031\pi\)
−0.784404 + 0.620250i \(0.787031\pi\)
\(450\) 0 0
\(451\) 3.98503e6 0.922551
\(452\) 0 0
\(453\) −1.07692e7 −2.46570
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −5.88344e6 −1.31777 −0.658887 0.752242i \(-0.728972\pi\)
−0.658887 + 0.752242i \(0.728972\pi\)
\(458\) 0 0
\(459\) 1.74338e6 0.386242
\(460\) 0 0
\(461\) 1.54764e6 0.339171 0.169585 0.985515i \(-0.445757\pi\)
0.169585 + 0.985515i \(0.445757\pi\)
\(462\) 0 0
\(463\) 3.93764e6 0.853656 0.426828 0.904333i \(-0.359631\pi\)
0.426828 + 0.904333i \(0.359631\pi\)
\(464\) 0 0
\(465\) −4.92907e6 −1.05714
\(466\) 0 0
\(467\) 6.81586e6 1.44620 0.723100 0.690743i \(-0.242716\pi\)
0.723100 + 0.690743i \(0.242716\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 4.21070e6 0.874585
\(472\) 0 0
\(473\) −4.37827e6 −0.899807
\(474\) 0 0
\(475\) 15477.5 0.00314750
\(476\) 0 0
\(477\) −1.22519e7 −2.46552
\(478\) 0 0
\(479\) 5.20406e6 1.03634 0.518171 0.855277i \(-0.326613\pi\)
0.518171 + 0.855277i \(0.326613\pi\)
\(480\) 0 0
\(481\) 516291. 0.101749
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.08540e6 −1.56080
\(486\) 0 0
\(487\) 154998. 0.0296145 0.0148073 0.999890i \(-0.495287\pi\)
0.0148073 + 0.999890i \(0.495287\pi\)
\(488\) 0 0
\(489\) 5.72936e6 1.08351
\(490\) 0 0
\(491\) 1.61951e6 0.303165 0.151583 0.988445i \(-0.451563\pi\)
0.151583 + 0.988445i \(0.451563\pi\)
\(492\) 0 0
\(493\) 5.52727e6 1.02422
\(494\) 0 0
\(495\) −9.77168e6 −1.79249
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −4.10674e6 −0.738322 −0.369161 0.929366i \(-0.620355\pi\)
−0.369161 + 0.929366i \(0.620355\pi\)
\(500\) 0 0
\(501\) 2.75244e6 0.489918
\(502\) 0 0
\(503\) 3.40748e6 0.600501 0.300250 0.953860i \(-0.402930\pi\)
0.300250 + 0.953860i \(0.402930\pi\)
\(504\) 0 0
\(505\) −1.33490e6 −0.232927
\(506\) 0 0
\(507\) 7.23599e6 1.25020
\(508\) 0 0
\(509\) 1.07091e7 1.83213 0.916067 0.401025i \(-0.131346\pi\)
0.916067 + 0.401025i \(0.131346\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −10215.4 −0.00171381
\(514\) 0 0
\(515\) 2.36000e6 0.392097
\(516\) 0 0
\(517\) −7.22682e6 −1.18911
\(518\) 0 0
\(519\) −6.34374e6 −1.03378
\(520\) 0 0
\(521\) 7.92001e6 1.27830 0.639148 0.769084i \(-0.279287\pi\)
0.639148 + 0.769084i \(0.279287\pi\)
\(522\) 0 0
\(523\) 8.32746e6 1.33125 0.665623 0.746288i \(-0.268166\pi\)
0.665623 + 0.746288i \(0.268166\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.11612e6 0.488752
\(528\) 0 0
\(529\) 6.61298e6 1.02744
\(530\) 0 0
\(531\) −1.05286e7 −1.62045
\(532\) 0 0
\(533\) −2.36943e6 −0.361265
\(534\) 0 0
\(535\) 611239. 0.0923265
\(536\) 0 0
\(537\) 8.85532e6 1.32516
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 623261. 0.0915539 0.0457770 0.998952i \(-0.485424\pi\)
0.0457770 + 0.998952i \(0.485424\pi\)
\(542\) 0 0
\(543\) 1.02221e7 1.48779
\(544\) 0 0
\(545\) 822129. 0.118563
\(546\) 0 0
\(547\) −1.05691e7 −1.51032 −0.755159 0.655541i \(-0.772441\pi\)
−0.755159 + 0.655541i \(0.772441\pi\)
\(548\) 0 0
\(549\) 8.77502e6 1.24256
\(550\) 0 0
\(551\) −32387.5 −0.00454462
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −3.57559e6 −0.492737
\(556\) 0 0
\(557\) −1.35398e7 −1.84916 −0.924579 0.380991i \(-0.875583\pi\)
−0.924579 + 0.380991i \(0.875583\pi\)
\(558\) 0 0
\(559\) 2.60324e6 0.352359
\(560\) 0 0
\(561\) 1.10181e7 1.47809
\(562\) 0 0
\(563\) 1.39757e7 1.85824 0.929122 0.369774i \(-0.120565\pi\)
0.929122 + 0.369774i \(0.120565\pi\)
\(564\) 0 0
\(565\) −4.84159e6 −0.638068
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5.61993e6 −0.727697 −0.363848 0.931458i \(-0.618537\pi\)
−0.363848 + 0.931458i \(0.618537\pi\)
\(570\) 0 0
\(571\) −8.70790e6 −1.11769 −0.558847 0.829271i \(-0.688756\pi\)
−0.558847 + 0.829271i \(0.688756\pi\)
\(572\) 0 0
\(573\) 1.33101e7 1.69354
\(574\) 0 0
\(575\) 8.64019e6 1.08982
\(576\) 0 0
\(577\) −6.63992e6 −0.830278 −0.415139 0.909758i \(-0.636267\pi\)
−0.415139 + 0.909758i \(0.636267\pi\)
\(578\) 0 0
\(579\) −1.20994e7 −1.49992
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.67595e7 −2.04216
\(584\) 0 0
\(585\) 5.81007e6 0.701927
\(586\) 0 0
\(587\) 1.36652e7 1.63689 0.818446 0.574583i \(-0.194836\pi\)
0.818446 + 0.574583i \(0.194836\pi\)
\(588\) 0 0
\(589\) −18259.2 −0.00216866
\(590\) 0 0
\(591\) 7.02126e6 0.826888
\(592\) 0 0
\(593\) −7.02589e6 −0.820474 −0.410237 0.911979i \(-0.634554\pi\)
−0.410237 + 0.911979i \(0.634554\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.39211e7 −1.59859
\(598\) 0 0
\(599\) 3.58663e6 0.408432 0.204216 0.978926i \(-0.434536\pi\)
0.204216 + 0.978926i \(0.434536\pi\)
\(600\) 0 0
\(601\) 1.58600e7 1.79108 0.895542 0.444977i \(-0.146788\pi\)
0.895542 + 0.444977i \(0.146788\pi\)
\(602\) 0 0
\(603\) −1.73982e7 −1.94855
\(604\) 0 0
\(605\) −1.40461e6 −0.156015
\(606\) 0 0
\(607\) −6.44170e6 −0.709625 −0.354812 0.934938i \(-0.615455\pi\)
−0.354812 + 0.934938i \(0.615455\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.29694e6 0.465647
\(612\) 0 0
\(613\) −4.58865e6 −0.493212 −0.246606 0.969116i \(-0.579315\pi\)
−0.246606 + 0.969116i \(0.579315\pi\)
\(614\) 0 0
\(615\) 1.64096e7 1.74948
\(616\) 0 0
\(617\) 1.47104e6 0.155565 0.0777825 0.996970i \(-0.475216\pi\)
0.0777825 + 0.996970i \(0.475216\pi\)
\(618\) 0 0
\(619\) 3.13569e6 0.328932 0.164466 0.986383i \(-0.447410\pi\)
0.164466 + 0.986383i \(0.447410\pi\)
\(620\) 0 0
\(621\) −5.70269e6 −0.593404
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.15193e7 −1.17957
\(626\) 0 0
\(627\) −64561.3 −0.00655849
\(628\) 0 0
\(629\) 2.26046e6 0.227809
\(630\) 0 0
\(631\) −484547. −0.0484465 −0.0242233 0.999707i \(-0.507711\pi\)
−0.0242233 + 0.999707i \(0.507711\pi\)
\(632\) 0 0
\(633\) −3.30518e6 −0.327858
\(634\) 0 0
\(635\) −1.44672e7 −1.42380
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 4.81851e6 0.466832
\(640\) 0 0
\(641\) 3.04085e6 0.292314 0.146157 0.989261i \(-0.453310\pi\)
0.146157 + 0.989261i \(0.453310\pi\)
\(642\) 0 0
\(643\) −5.25888e6 −0.501609 −0.250805 0.968038i \(-0.580695\pi\)
−0.250805 + 0.968038i \(0.580695\pi\)
\(644\) 0 0
\(645\) −1.80288e7 −1.70635
\(646\) 0 0
\(647\) −2.11970e7 −1.99074 −0.995368 0.0961386i \(-0.969351\pi\)
−0.995368 + 0.0961386i \(0.969351\pi\)
\(648\) 0 0
\(649\) −1.44021e7 −1.34219
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.30106e7 1.19403 0.597013 0.802232i \(-0.296354\pi\)
0.597013 + 0.802232i \(0.296354\pi\)
\(654\) 0 0
\(655\) −1.75664e7 −1.59985
\(656\) 0 0
\(657\) −2.42577e7 −2.19248
\(658\) 0 0
\(659\) 1.59874e7 1.43405 0.717024 0.697049i \(-0.245504\pi\)
0.717024 + 0.697049i \(0.245504\pi\)
\(660\) 0 0
\(661\) 4.03142e6 0.358884 0.179442 0.983769i \(-0.442571\pi\)
0.179442 + 0.983769i \(0.442571\pi\)
\(662\) 0 0
\(663\) −6.55117e6 −0.578809
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.80800e7 −1.57357
\(668\) 0 0
\(669\) 11536.5 0.000996570 0
\(670\) 0 0
\(671\) 1.20034e7 1.02919
\(672\) 0 0
\(673\) 2.98234e6 0.253816 0.126908 0.991914i \(-0.459495\pi\)
0.126908 + 0.991914i \(0.459495\pi\)
\(674\) 0 0
\(675\) −3.77585e6 −0.318974
\(676\) 0 0
\(677\) 1.94696e6 0.163262 0.0816309 0.996663i \(-0.473987\pi\)
0.0816309 + 0.996663i \(0.473987\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 1.39676e7 1.15413
\(682\) 0 0
\(683\) −1.19940e7 −0.983812 −0.491906 0.870648i \(-0.663700\pi\)
−0.491906 + 0.870648i \(0.663700\pi\)
\(684\) 0 0
\(685\) 1.49221e7 1.21508
\(686\) 0 0
\(687\) 843854. 0.0682143
\(688\) 0 0
\(689\) 9.96490e6 0.799696
\(690\) 0 0
\(691\) 8.66304e6 0.690201 0.345100 0.938566i \(-0.387845\pi\)
0.345100 + 0.938566i \(0.387845\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.93549e6 0.309056
\(696\) 0 0
\(697\) −1.03740e7 −0.808845
\(698\) 0 0
\(699\) −2.93563e7 −2.27252
\(700\) 0 0
\(701\) −8.13382e6 −0.625172 −0.312586 0.949889i \(-0.601195\pi\)
−0.312586 + 0.949889i \(0.601195\pi\)
\(702\) 0 0
\(703\) −13245.3 −0.00101082
\(704\) 0 0
\(705\) −2.97586e7 −2.25497
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.21326e7 1.65355 0.826773 0.562535i \(-0.190174\pi\)
0.826773 + 0.562535i \(0.190174\pi\)
\(710\) 0 0
\(711\) 1.40596e7 1.04303
\(712\) 0 0
\(713\) −1.01930e7 −0.750896
\(714\) 0 0
\(715\) 7.94762e6 0.581396
\(716\) 0 0
\(717\) −1.35689e7 −0.985704
\(718\) 0 0
\(719\) 7.23196e6 0.521716 0.260858 0.965377i \(-0.415995\pi\)
0.260858 + 0.965377i \(0.415995\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 3.25123e7 2.31314
\(724\) 0 0
\(725\) −1.19711e7 −0.845843
\(726\) 0 0
\(727\) 1.70200e7 1.19433 0.597163 0.802120i \(-0.296295\pi\)
0.597163 + 0.802120i \(0.296295\pi\)
\(728\) 0 0
\(729\) −2.08788e7 −1.45508
\(730\) 0 0
\(731\) 1.13977e7 0.788904
\(732\) 0 0
\(733\) 1.00011e6 0.0687525 0.0343763 0.999409i \(-0.489056\pi\)
0.0343763 + 0.999409i \(0.489056\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.37991e7 −1.61396
\(738\) 0 0
\(739\) −3.25979e6 −0.219572 −0.109786 0.993955i \(-0.535017\pi\)
−0.109786 + 0.993955i \(0.535017\pi\)
\(740\) 0 0
\(741\) 38387.1 0.00256826
\(742\) 0 0
\(743\) −1.36125e7 −0.904617 −0.452309 0.891861i \(-0.649399\pi\)
−0.452309 + 0.891861i \(0.649399\pi\)
\(744\) 0 0
\(745\) 7.48475e6 0.494068
\(746\) 0 0
\(747\) −428536. −0.0280986
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −6.56544e6 −0.424780 −0.212390 0.977185i \(-0.568125\pi\)
−0.212390 + 0.977185i \(0.568125\pi\)
\(752\) 0 0
\(753\) 7.60160e6 0.488560
\(754\) 0 0
\(755\) −3.40110e7 −2.17146
\(756\) 0 0
\(757\) 2.62531e7 1.66510 0.832551 0.553948i \(-0.186879\pi\)
0.832551 + 0.553948i \(0.186879\pi\)
\(758\) 0 0
\(759\) −3.60409e7 −2.27086
\(760\) 0 0
\(761\) −5.25111e6 −0.328692 −0.164346 0.986403i \(-0.552551\pi\)
−0.164346 + 0.986403i \(0.552551\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2.54381e7 1.57156
\(766\) 0 0
\(767\) 8.56327e6 0.525595
\(768\) 0 0
\(769\) −1.77307e7 −1.08121 −0.540605 0.841277i \(-0.681805\pi\)
−0.540605 + 0.841277i \(0.681805\pi\)
\(770\) 0 0
\(771\) −4.34156e7 −2.63033
\(772\) 0 0
\(773\) 3.82592e6 0.230296 0.115148 0.993348i \(-0.463266\pi\)
0.115148 + 0.993348i \(0.463266\pi\)
\(774\) 0 0
\(775\) −6.74898e6 −0.403631
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 60787.3 0.00358896
\(780\) 0 0
\(781\) 6.59126e6 0.386671
\(782\) 0 0
\(783\) 7.90117e6 0.460561
\(784\) 0 0
\(785\) 1.32980e7 0.770218
\(786\) 0 0
\(787\) −2.94263e7 −1.69355 −0.846777 0.531948i \(-0.821460\pi\)
−0.846777 + 0.531948i \(0.821460\pi\)
\(788\) 0 0
\(789\) −1.07767e7 −0.616304
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −7.13700e6 −0.403026
\(794\) 0 0
\(795\) −6.90122e7 −3.87265
\(796\) 0 0
\(797\) 1.35805e7 0.757305 0.378653 0.925539i \(-0.376387\pi\)
0.378653 + 0.925539i \(0.376387\pi\)
\(798\) 0 0
\(799\) 1.88132e7 1.04255
\(800\) 0 0
\(801\) −2.13611e7 −1.17637
\(802\) 0 0
\(803\) −3.31822e7 −1.81600
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −9.80625e6 −0.530053
\(808\) 0 0
\(809\) −1.15714e7 −0.621604 −0.310802 0.950475i \(-0.600598\pi\)
−0.310802 + 0.950475i \(0.600598\pi\)
\(810\) 0 0
\(811\) 3.52530e6 0.188210 0.0941052 0.995562i \(-0.470001\pi\)
0.0941052 + 0.995562i \(0.470001\pi\)
\(812\) 0 0
\(813\) −2.11756e7 −1.12360
\(814\) 0 0
\(815\) 1.80942e7 0.954214
\(816\) 0 0
\(817\) −66785.7 −0.00350049
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.78951e7 0.926567 0.463283 0.886210i \(-0.346671\pi\)
0.463283 + 0.886210i \(0.346671\pi\)
\(822\) 0 0
\(823\) 3.61421e7 1.86000 0.930001 0.367557i \(-0.119806\pi\)
0.930001 + 0.367557i \(0.119806\pi\)
\(824\) 0 0
\(825\) −2.38633e7 −1.22066
\(826\) 0 0
\(827\) 1.00605e7 0.511512 0.255756 0.966741i \(-0.417676\pi\)
0.255756 + 0.966741i \(0.417676\pi\)
\(828\) 0 0
\(829\) 2.03654e7 1.02921 0.514607 0.857426i \(-0.327938\pi\)
0.514607 + 0.857426i \(0.327938\pi\)
\(830\) 0 0
\(831\) 1.05279e7 0.528858
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 8.69263e6 0.431454
\(836\) 0 0
\(837\) 4.45446e6 0.219777
\(838\) 0 0
\(839\) 5.95014e6 0.291825 0.145912 0.989298i \(-0.453388\pi\)
0.145912 + 0.989298i \(0.453388\pi\)
\(840\) 0 0
\(841\) 4.53905e6 0.221297
\(842\) 0 0
\(843\) 1.80745e7 0.875987
\(844\) 0 0
\(845\) 2.28524e7 1.10101
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −5.01462e7 −2.38764
\(850\) 0 0
\(851\) −7.39411e6 −0.349995
\(852\) 0 0
\(853\) 1.59836e7 0.752146 0.376073 0.926590i \(-0.377274\pi\)
0.376073 + 0.926590i \(0.377274\pi\)
\(854\) 0 0
\(855\) −149056. −0.00697325
\(856\) 0 0
\(857\) −2.34591e6 −0.109109 −0.0545544 0.998511i \(-0.517374\pi\)
−0.0545544 + 0.998511i \(0.517374\pi\)
\(858\) 0 0
\(859\) 1.30223e7 0.602152 0.301076 0.953600i \(-0.402654\pi\)
0.301076 + 0.953600i \(0.402654\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.96688e7 −0.898983 −0.449492 0.893285i \(-0.648395\pi\)
−0.449492 + 0.893285i \(0.648395\pi\)
\(864\) 0 0
\(865\) −2.00345e7 −0.910414
\(866\) 0 0
\(867\) 4.71019e6 0.212809
\(868\) 0 0
\(869\) 1.92322e7 0.863931
\(870\) 0 0
\(871\) 1.41505e7 0.632015
\(872\) 0 0
\(873\) 3.37592e7 1.49919
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.34581e7 1.02990 0.514950 0.857221i \(-0.327811\pi\)
0.514950 + 0.857221i \(0.327811\pi\)
\(878\) 0 0
\(879\) 5.70724e7 2.49146
\(880\) 0 0
\(881\) 4.59257e6 0.199350 0.0996750 0.995020i \(-0.468220\pi\)
0.0996750 + 0.995020i \(0.468220\pi\)
\(882\) 0 0
\(883\) 1.23402e7 0.532622 0.266311 0.963887i \(-0.414195\pi\)
0.266311 + 0.963887i \(0.414195\pi\)
\(884\) 0 0
\(885\) −5.93052e7 −2.54527
\(886\) 0 0
\(887\) −1.36554e7 −0.582769 −0.291384 0.956606i \(-0.594116\pi\)
−0.291384 + 0.956606i \(0.594116\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.62189e7 −0.684428
\(892\) 0 0
\(893\) −110237. −0.00462594
\(894\) 0 0
\(895\) 2.79665e7 1.16703
\(896\) 0 0
\(897\) 2.14293e7 0.889255
\(898\) 0 0
\(899\) 1.41226e7 0.582795
\(900\) 0 0
\(901\) 4.36291e7 1.79046
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3.22831e7 1.31025
\(906\) 0 0
\(907\) −7.39599e6 −0.298523 −0.149262 0.988798i \(-0.547690\pi\)
−0.149262 + 0.988798i \(0.547690\pi\)
\(908\) 0 0
\(909\) 5.57362e6 0.223732
\(910\) 0 0
\(911\) 3.51041e7 1.40140 0.700699 0.713457i \(-0.252871\pi\)
0.700699 + 0.713457i \(0.252871\pi\)
\(912\) 0 0
\(913\) −586195. −0.0232737
\(914\) 0 0
\(915\) 4.94275e7 1.95171
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 4.81337e7 1.88001 0.940005 0.341159i \(-0.110820\pi\)
0.940005 + 0.341159i \(0.110820\pi\)
\(920\) 0 0
\(921\) 5.74626e7 2.23221
\(922\) 0 0
\(923\) −3.91905e6 −0.151418
\(924\) 0 0
\(925\) −4.89577e6 −0.188134
\(926\) 0 0
\(927\) −9.85376e6 −0.376619
\(928\) 0 0
\(929\) −1.83602e7 −0.697971 −0.348986 0.937128i \(-0.613474\pi\)
−0.348986 + 0.937128i \(0.613474\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −2.71557e7 −1.02131
\(934\) 0 0
\(935\) 3.47969e7 1.30170
\(936\) 0 0
\(937\) 2.27081e7 0.844951 0.422475 0.906374i \(-0.361161\pi\)
0.422475 + 0.906374i \(0.361161\pi\)
\(938\) 0 0
\(939\) 3.89718e7 1.44240
\(940\) 0 0
\(941\) 4.15801e7 1.53078 0.765389 0.643568i \(-0.222547\pi\)
0.765389 + 0.643568i \(0.222547\pi\)
\(942\) 0 0
\(943\) 3.39340e7 1.24267
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.55341e7 −0.562874 −0.281437 0.959580i \(-0.590811\pi\)
−0.281437 + 0.959580i \(0.590811\pi\)
\(948\) 0 0
\(949\) 1.97296e7 0.711135
\(950\) 0 0
\(951\) −1.93172e7 −0.692618
\(952\) 0 0
\(953\) 3.94908e7 1.40852 0.704262 0.709940i \(-0.251278\pi\)
0.704262 + 0.709940i \(0.251278\pi\)
\(954\) 0 0
\(955\) 4.20354e7 1.49144
\(956\) 0 0
\(957\) 4.99352e7 1.76249
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.06672e7 −0.721894
\(962\) 0 0
\(963\) −2.55212e6 −0.0886819
\(964\) 0 0
\(965\) −3.82118e7 −1.32093
\(966\) 0 0
\(967\) 1.87472e7 0.644718 0.322359 0.946617i \(-0.395524\pi\)
0.322359 + 0.946617i \(0.395524\pi\)
\(968\) 0 0
\(969\) 168069. 0.00575014
\(970\) 0 0
\(971\) 5.35423e7 1.82242 0.911211 0.411940i \(-0.135149\pi\)
0.911211 + 0.411940i \(0.135149\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 1.41887e7 0.478004
\(976\) 0 0
\(977\) 4.46104e7 1.49520 0.747600 0.664149i \(-0.231206\pi\)
0.747600 + 0.664149i \(0.231206\pi\)
\(978\) 0 0
\(979\) −2.92200e7 −0.974368
\(980\) 0 0
\(981\) −3.43265e6 −0.113883
\(982\) 0 0
\(983\) 2.00068e7 0.660380 0.330190 0.943914i \(-0.392887\pi\)
0.330190 + 0.943914i \(0.392887\pi\)
\(984\) 0 0
\(985\) 2.21742e7 0.728213
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −3.72826e7 −1.21204
\(990\) 0 0
\(991\) −3.24635e7 −1.05005 −0.525026 0.851086i \(-0.675944\pi\)
−0.525026 + 0.851086i \(0.675944\pi\)
\(992\) 0 0
\(993\) −2.25004e6 −0.0724130
\(994\) 0 0
\(995\) −4.39649e7 −1.40783
\(996\) 0 0
\(997\) 2.78940e7 0.888736 0.444368 0.895844i \(-0.353428\pi\)
0.444368 + 0.895844i \(0.353428\pi\)
\(998\) 0 0
\(999\) 3.23130e6 0.102439
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 784.6.a.bf.1.1 4
4.3 odd 2 49.6.a.g.1.2 yes 4
7.6 odd 2 inner 784.6.a.bf.1.4 4
12.11 even 2 441.6.a.z.1.4 4
28.3 even 6 49.6.c.h.30.4 8
28.11 odd 6 49.6.c.h.30.3 8
28.19 even 6 49.6.c.h.18.4 8
28.23 odd 6 49.6.c.h.18.3 8
28.27 even 2 49.6.a.g.1.1 4
84.83 odd 2 441.6.a.z.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.g.1.1 4 28.27 even 2
49.6.a.g.1.2 yes 4 4.3 odd 2
49.6.c.h.18.3 8 28.23 odd 6
49.6.c.h.18.4 8 28.19 even 6
49.6.c.h.30.3 8 28.11 odd 6
49.6.c.h.30.4 8 28.3 even 6
441.6.a.z.1.3 4 84.83 odd 2
441.6.a.z.1.4 4 12.11 even 2
784.6.a.bf.1.1 4 1.1 even 1 trivial
784.6.a.bf.1.4 4 7.6 odd 2 inner