Properties

Label 112.10.a.c.1.2
Level $112$
Weight $10$
Character 112.1
Self dual yes
Analytic conductor $57.684$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 112.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.6840136504\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2305}) \)
Defining polynomial: \( x^{2} - x - 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(24.5052\) of defining polynomial
Character \(\chi\) \(=\) 112.1

$q$-expansion

\(f(q)\) \(=\) \(q+247.052 q^{3} -2373.22 q^{5} -2401.00 q^{7} +41351.7 q^{9} +O(q^{10})\) \(q+247.052 q^{3} -2373.22 q^{5} -2401.00 q^{7} +41351.7 q^{9} +27940.9 q^{11} +60943.3 q^{13} -586309. q^{15} -358867. q^{17} +391593. q^{19} -593172. q^{21} +302894. q^{23} +3.67904e6 q^{25} +5.35330e6 q^{27} +6.73993e6 q^{29} -2.98262e6 q^{31} +6.90287e6 q^{33} +5.69810e6 q^{35} -3.49582e6 q^{37} +1.50562e7 q^{39} +3.43724e7 q^{41} +1.45085e7 q^{43} -9.81367e7 q^{45} +2.76485e7 q^{47} +5.76480e6 q^{49} -8.86589e7 q^{51} -2.39217e7 q^{53} -6.63100e7 q^{55} +9.67439e7 q^{57} +1.20580e8 q^{59} +7.23140e7 q^{61} -9.92855e7 q^{63} -1.44632e8 q^{65} -8.70377e7 q^{67} +7.48307e7 q^{69} -2.19622e8 q^{71} +2.67792e8 q^{73} +9.08915e8 q^{75} -6.70862e7 q^{77} -2.85350e7 q^{79} +5.08619e8 q^{81} +3.83237e8 q^{83} +8.51671e8 q^{85} +1.66511e9 q^{87} +7.21581e8 q^{89} -1.46325e8 q^{91} -7.36861e8 q^{93} -9.29336e8 q^{95} -6.73736e8 q^{97} +1.15541e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{3} - 2730 q^{5} - 4802 q^{7} + 75982 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 14 q^{3} - 2730 q^{5} - 4802 q^{7} + 75982 q^{9} - 44940 q^{11} + 100282 q^{13} - 503160 q^{15} - 870408 q^{17} - 508774 q^{19} - 33614 q^{21} - 79800 q^{23} + 1853210 q^{25} + 1869812 q^{27} + 2006328 q^{29} - 2188732 q^{31} + 23887920 q^{33} + 6554730 q^{35} - 20723576 q^{37} + 5888224 q^{39} + 19016592 q^{41} - 4193716 q^{43} - 110492130 q^{45} + 74542524 q^{47} + 11529602 q^{49} + 30556644 q^{51} - 3239748 q^{53} - 40307400 q^{55} + 306576332 q^{57} + 133642362 q^{59} + 227801686 q^{61} - 182432782 q^{63} - 158667180 q^{65} - 332930272 q^{67} + 164018400 q^{69} + 167985720 q^{71} - 44684276 q^{73} + 1334428970 q^{75} + 107900940 q^{77} - 269642776 q^{79} + 638826478 q^{81} + 183105762 q^{83} + 1034179020 q^{85} + 2768288796 q^{87} + 791657748 q^{89} - 240777082 q^{91} - 921877624 q^{93} - 608102040 q^{95} - 4169480 q^{97} - 1368480540 q^{99}+O(q^{100}) \) 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
\(3\) 247.052 1.76093 0.880467 0.474108i \(-0.157229\pi\)
0.880467 + 0.474108i \(0.157229\pi\)
\(4\) 0 0
\(5\) −2373.22 −1.69814 −0.849069 0.528283i \(-0.822836\pi\)
−0.849069 + 0.528283i \(0.822836\pi\)
\(6\) 0 0
\(7\) −2401.00 −0.377964
\(8\) 0 0
\(9\) 41351.7 2.10089
\(10\) 0 0
\(11\) 27940.9 0.575405 0.287703 0.957720i \(-0.407109\pi\)
0.287703 + 0.957720i \(0.407109\pi\)
\(12\) 0 0
\(13\) 60943.3 0.591808 0.295904 0.955218i \(-0.404379\pi\)
0.295904 + 0.955218i \(0.404379\pi\)
\(14\) 0 0
\(15\) −586309. −2.99031
\(16\) 0 0
\(17\) −358867. −1.04211 −0.521055 0.853523i \(-0.674462\pi\)
−0.521055 + 0.853523i \(0.674462\pi\)
\(18\) 0 0
\(19\) 391593. 0.689356 0.344678 0.938721i \(-0.387988\pi\)
0.344678 + 0.938721i \(0.387988\pi\)
\(20\) 0 0
\(21\) −593172. −0.665570
\(22\) 0 0
\(23\) 302894. 0.225692 0.112846 0.993612i \(-0.464003\pi\)
0.112846 + 0.993612i \(0.464003\pi\)
\(24\) 0 0
\(25\) 3.67904e6 1.88367
\(26\) 0 0
\(27\) 5.35330e6 1.93859
\(28\) 0 0
\(29\) 6.73993e6 1.76956 0.884778 0.466013i \(-0.154310\pi\)
0.884778 + 0.466013i \(0.154310\pi\)
\(30\) 0 0
\(31\) −2.98262e6 −0.580056 −0.290028 0.957018i \(-0.593665\pi\)
−0.290028 + 0.957018i \(0.593665\pi\)
\(32\) 0 0
\(33\) 6.90287e6 1.01325
\(34\) 0 0
\(35\) 5.69810e6 0.641835
\(36\) 0 0
\(37\) −3.49582e6 −0.306649 −0.153324 0.988176i \(-0.548998\pi\)
−0.153324 + 0.988176i \(0.548998\pi\)
\(38\) 0 0
\(39\) 1.50562e7 1.04214
\(40\) 0 0
\(41\) 3.43724e7 1.89969 0.949845 0.312722i \(-0.101241\pi\)
0.949845 + 0.312722i \(0.101241\pi\)
\(42\) 0 0
\(43\) 1.45085e7 0.647164 0.323582 0.946200i \(-0.395113\pi\)
0.323582 + 0.946200i \(0.395113\pi\)
\(44\) 0 0
\(45\) −9.81367e7 −3.56759
\(46\) 0 0
\(47\) 2.76485e7 0.826479 0.413239 0.910622i \(-0.364397\pi\)
0.413239 + 0.910622i \(0.364397\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) −8.86589e7 −1.83509
\(52\) 0 0
\(53\) −2.39217e7 −0.416438 −0.208219 0.978082i \(-0.566767\pi\)
−0.208219 + 0.978082i \(0.566767\pi\)
\(54\) 0 0
\(55\) −6.63100e7 −0.977117
\(56\) 0 0
\(57\) 9.67439e7 1.21391
\(58\) 0 0
\(59\) 1.20580e8 1.29551 0.647756 0.761848i \(-0.275707\pi\)
0.647756 + 0.761848i \(0.275707\pi\)
\(60\) 0 0
\(61\) 7.23140e7 0.668710 0.334355 0.942447i \(-0.391482\pi\)
0.334355 + 0.942447i \(0.391482\pi\)
\(62\) 0 0
\(63\) −9.92855e7 −0.794060
\(64\) 0 0
\(65\) −1.44632e8 −1.00497
\(66\) 0 0
\(67\) −8.70377e7 −0.527680 −0.263840 0.964566i \(-0.584989\pi\)
−0.263840 + 0.964566i \(0.584989\pi\)
\(68\) 0 0
\(69\) 7.48307e7 0.397428
\(70\) 0 0
\(71\) −2.19622e8 −1.02568 −0.512842 0.858483i \(-0.671407\pi\)
−0.512842 + 0.858483i \(0.671407\pi\)
\(72\) 0 0
\(73\) 2.67792e8 1.10368 0.551841 0.833949i \(-0.313926\pi\)
0.551841 + 0.833949i \(0.313926\pi\)
\(74\) 0 0
\(75\) 9.08915e8 3.31702
\(76\) 0 0
\(77\) −6.70862e7 −0.217483
\(78\) 0 0
\(79\) −2.85350e7 −0.0824243 −0.0412122 0.999150i \(-0.513122\pi\)
−0.0412122 + 0.999150i \(0.513122\pi\)
\(80\) 0 0
\(81\) 5.08619e8 1.31283
\(82\) 0 0
\(83\) 3.83237e8 0.886372 0.443186 0.896430i \(-0.353848\pi\)
0.443186 + 0.896430i \(0.353848\pi\)
\(84\) 0 0
\(85\) 8.51671e8 1.76965
\(86\) 0 0
\(87\) 1.66511e9 3.11607
\(88\) 0 0
\(89\) 7.21581e8 1.21907 0.609537 0.792757i \(-0.291355\pi\)
0.609537 + 0.792757i \(0.291355\pi\)
\(90\) 0 0
\(91\) −1.46325e8 −0.223683
\(92\) 0 0
\(93\) −7.36861e8 −1.02144
\(94\) 0 0
\(95\) −9.29336e8 −1.17062
\(96\) 0 0
\(97\) −6.73736e8 −0.772710 −0.386355 0.922350i \(-0.626266\pi\)
−0.386355 + 0.922350i \(0.626266\pi\)
\(98\) 0 0
\(99\) 1.15541e9 1.20886
\(100\) 0 0
\(101\) −1.04805e7 −0.0100216 −0.00501081 0.999987i \(-0.501595\pi\)
−0.00501081 + 0.999987i \(0.501595\pi\)
\(102\) 0 0
\(103\) −1.39782e9 −1.22373 −0.611864 0.790963i \(-0.709580\pi\)
−0.611864 + 0.790963i \(0.709580\pi\)
\(104\) 0 0
\(105\) 1.40773e9 1.13023
\(106\) 0 0
\(107\) 4.84414e8 0.357264 0.178632 0.983916i \(-0.442833\pi\)
0.178632 + 0.983916i \(0.442833\pi\)
\(108\) 0 0
\(109\) −1.96298e8 −0.133198 −0.0665989 0.997780i \(-0.521215\pi\)
−0.0665989 + 0.997780i \(0.521215\pi\)
\(110\) 0 0
\(111\) −8.63649e8 −0.539988
\(112\) 0 0
\(113\) −5.80849e8 −0.335128 −0.167564 0.985861i \(-0.553590\pi\)
−0.167564 + 0.985861i \(0.553590\pi\)
\(114\) 0 0
\(115\) −7.18835e8 −0.383256
\(116\) 0 0
\(117\) 2.52011e9 1.24332
\(118\) 0 0
\(119\) 8.61641e8 0.393881
\(120\) 0 0
\(121\) −1.57725e9 −0.668909
\(122\) 0 0
\(123\) 8.49178e9 3.34523
\(124\) 0 0
\(125\) −4.09598e9 −1.50059
\(126\) 0 0
\(127\) −2.59088e9 −0.883751 −0.441876 0.897076i \(-0.645687\pi\)
−0.441876 + 0.897076i \(0.645687\pi\)
\(128\) 0 0
\(129\) 3.58436e9 1.13961
\(130\) 0 0
\(131\) −2.45598e9 −0.728626 −0.364313 0.931277i \(-0.618696\pi\)
−0.364313 + 0.931277i \(0.618696\pi\)
\(132\) 0 0
\(133\) −9.40215e8 −0.260552
\(134\) 0 0
\(135\) −1.27046e10 −3.29198
\(136\) 0 0
\(137\) 6.08364e9 1.47544 0.737719 0.675108i \(-0.235903\pi\)
0.737719 + 0.675108i \(0.235903\pi\)
\(138\) 0 0
\(139\) −2.25249e9 −0.511796 −0.255898 0.966704i \(-0.582371\pi\)
−0.255898 + 0.966704i \(0.582371\pi\)
\(140\) 0 0
\(141\) 6.83063e9 1.45537
\(142\) 0 0
\(143\) 1.70281e9 0.340530
\(144\) 0 0
\(145\) −1.59953e10 −3.00495
\(146\) 0 0
\(147\) 1.42421e9 0.251562
\(148\) 0 0
\(149\) −3.13517e9 −0.521103 −0.260551 0.965460i \(-0.583904\pi\)
−0.260551 + 0.965460i \(0.583904\pi\)
\(150\) 0 0
\(151\) 6.20938e9 0.971969 0.485984 0.873968i \(-0.338461\pi\)
0.485984 + 0.873968i \(0.338461\pi\)
\(152\) 0 0
\(153\) −1.48398e10 −2.18936
\(154\) 0 0
\(155\) 7.07840e9 0.985014
\(156\) 0 0
\(157\) 1.33378e10 1.75201 0.876006 0.482300i \(-0.160198\pi\)
0.876006 + 0.482300i \(0.160198\pi\)
\(158\) 0 0
\(159\) −5.90990e9 −0.733319
\(160\) 0 0
\(161\) −7.27249e8 −0.0853035
\(162\) 0 0
\(163\) −7.33621e9 −0.814006 −0.407003 0.913427i \(-0.633426\pi\)
−0.407003 + 0.913427i \(0.633426\pi\)
\(164\) 0 0
\(165\) −1.63820e10 −1.72064
\(166\) 0 0
\(167\) 6.42205e9 0.638925 0.319462 0.947599i \(-0.396498\pi\)
0.319462 + 0.947599i \(0.396498\pi\)
\(168\) 0 0
\(169\) −6.89041e9 −0.649763
\(170\) 0 0
\(171\) 1.61931e10 1.44826
\(172\) 0 0
\(173\) 1.91846e10 1.62834 0.814171 0.580626i \(-0.197192\pi\)
0.814171 + 0.580626i \(0.197192\pi\)
\(174\) 0 0
\(175\) −8.83338e9 −0.711960
\(176\) 0 0
\(177\) 2.97896e10 2.28131
\(178\) 0 0
\(179\) −1.53377e10 −1.11666 −0.558330 0.829619i \(-0.688558\pi\)
−0.558330 + 0.829619i \(0.688558\pi\)
\(180\) 0 0
\(181\) −1.73475e10 −1.20139 −0.600694 0.799479i \(-0.705109\pi\)
−0.600694 + 0.799479i \(0.705109\pi\)
\(182\) 0 0
\(183\) 1.78653e10 1.17755
\(184\) 0 0
\(185\) 8.29634e9 0.520732
\(186\) 0 0
\(187\) −1.00271e10 −0.599636
\(188\) 0 0
\(189\) −1.28533e10 −0.732717
\(190\) 0 0
\(191\) −2.70138e10 −1.46871 −0.734355 0.678765i \(-0.762515\pi\)
−0.734355 + 0.678765i \(0.762515\pi\)
\(192\) 0 0
\(193\) 2.70232e9 0.140194 0.0700970 0.997540i \(-0.477669\pi\)
0.0700970 + 0.997540i \(0.477669\pi\)
\(194\) 0 0
\(195\) −3.57316e10 −1.76969
\(196\) 0 0
\(197\) 2.39047e10 1.13080 0.565399 0.824818i \(-0.308722\pi\)
0.565399 + 0.824818i \(0.308722\pi\)
\(198\) 0 0
\(199\) 2.16111e9 0.0976872 0.0488436 0.998806i \(-0.484446\pi\)
0.0488436 + 0.998806i \(0.484446\pi\)
\(200\) 0 0
\(201\) −2.15028e10 −0.929209
\(202\) 0 0
\(203\) −1.61826e10 −0.668829
\(204\) 0 0
\(205\) −8.15732e10 −3.22593
\(206\) 0 0
\(207\) 1.25252e10 0.474153
\(208\) 0 0
\(209\) 1.09415e10 0.396659
\(210\) 0 0
\(211\) −3.61055e9 −0.125401 −0.0627007 0.998032i \(-0.519971\pi\)
−0.0627007 + 0.998032i \(0.519971\pi\)
\(212\) 0 0
\(213\) −5.42581e10 −1.80616
\(214\) 0 0
\(215\) −3.44319e10 −1.09897
\(216\) 0 0
\(217\) 7.16126e9 0.219240
\(218\) 0 0
\(219\) 6.61585e10 1.94351
\(220\) 0 0
\(221\) −2.18706e10 −0.616730
\(222\) 0 0
\(223\) 7.13001e10 1.93072 0.965358 0.260929i \(-0.0840287\pi\)
0.965358 + 0.260929i \(0.0840287\pi\)
\(224\) 0 0
\(225\) 1.52135e11 3.95737
\(226\) 0 0
\(227\) −7.15361e10 −1.78817 −0.894086 0.447896i \(-0.852173\pi\)
−0.894086 + 0.447896i \(0.852173\pi\)
\(228\) 0 0
\(229\) −3.56020e10 −0.855491 −0.427745 0.903899i \(-0.640692\pi\)
−0.427745 + 0.903899i \(0.640692\pi\)
\(230\) 0 0
\(231\) −1.65738e10 −0.382973
\(232\) 0 0
\(233\) −3.80069e10 −0.844814 −0.422407 0.906406i \(-0.638815\pi\)
−0.422407 + 0.906406i \(0.638815\pi\)
\(234\) 0 0
\(235\) −6.56160e10 −1.40347
\(236\) 0 0
\(237\) −7.04962e9 −0.145144
\(238\) 0 0
\(239\) 8.67126e10 1.71906 0.859531 0.511083i \(-0.170756\pi\)
0.859531 + 0.511083i \(0.170756\pi\)
\(240\) 0 0
\(241\) −8.18418e9 −0.156278 −0.0781391 0.996942i \(-0.524898\pi\)
−0.0781391 + 0.996942i \(0.524898\pi\)
\(242\) 0 0
\(243\) 2.02863e10 0.373228
\(244\) 0 0
\(245\) −1.36811e10 −0.242591
\(246\) 0 0
\(247\) 2.38650e10 0.407967
\(248\) 0 0
\(249\) 9.46795e10 1.56084
\(250\) 0 0
\(251\) −9.75467e10 −1.55125 −0.775624 0.631196i \(-0.782565\pi\)
−0.775624 + 0.631196i \(0.782565\pi\)
\(252\) 0 0
\(253\) 8.46315e9 0.129864
\(254\) 0 0
\(255\) 2.10407e11 3.11623
\(256\) 0 0
\(257\) −4.43042e9 −0.0633499 −0.0316750 0.999498i \(-0.510084\pi\)
−0.0316750 + 0.999498i \(0.510084\pi\)
\(258\) 0 0
\(259\) 8.39346e9 0.115902
\(260\) 0 0
\(261\) 2.78708e11 3.71763
\(262\) 0 0
\(263\) 1.20620e11 1.55460 0.777301 0.629129i \(-0.216588\pi\)
0.777301 + 0.629129i \(0.216588\pi\)
\(264\) 0 0
\(265\) 5.67714e10 0.707168
\(266\) 0 0
\(267\) 1.78268e11 2.14671
\(268\) 0 0
\(269\) 7.59025e10 0.883835 0.441917 0.897056i \(-0.354298\pi\)
0.441917 + 0.897056i \(0.354298\pi\)
\(270\) 0 0
\(271\) −7.11397e10 −0.801217 −0.400608 0.916249i \(-0.631201\pi\)
−0.400608 + 0.916249i \(0.631201\pi\)
\(272\) 0 0
\(273\) −3.61499e10 −0.393890
\(274\) 0 0
\(275\) 1.02796e11 1.08387
\(276\) 0 0
\(277\) 8.61542e10 0.879261 0.439630 0.898179i \(-0.355109\pi\)
0.439630 + 0.898179i \(0.355109\pi\)
\(278\) 0 0
\(279\) −1.23336e11 −1.21863
\(280\) 0 0
\(281\) −1.00179e11 −0.958511 −0.479256 0.877675i \(-0.659093\pi\)
−0.479256 + 0.877675i \(0.659093\pi\)
\(282\) 0 0
\(283\) 4.57444e10 0.423935 0.211967 0.977277i \(-0.432013\pi\)
0.211967 + 0.977277i \(0.432013\pi\)
\(284\) 0 0
\(285\) −2.29594e11 −2.06139
\(286\) 0 0
\(287\) −8.25282e10 −0.718015
\(288\) 0 0
\(289\) 1.01980e10 0.0859950
\(290\) 0 0
\(291\) −1.66448e11 −1.36069
\(292\) 0 0
\(293\) −1.01615e10 −0.0805476 −0.0402738 0.999189i \(-0.512823\pi\)
−0.0402738 + 0.999189i \(0.512823\pi\)
\(294\) 0 0
\(295\) −2.86163e11 −2.19996
\(296\) 0 0
\(297\) 1.49576e11 1.11547
\(298\) 0 0
\(299\) 1.84594e10 0.133566
\(300\) 0 0
\(301\) −3.48349e10 −0.244605
\(302\) 0 0
\(303\) −2.58924e9 −0.0176474
\(304\) 0 0
\(305\) −1.71617e11 −1.13556
\(306\) 0 0
\(307\) 3.68957e10 0.237057 0.118529 0.992951i \(-0.462182\pi\)
0.118529 + 0.992951i \(0.462182\pi\)
\(308\) 0 0
\(309\) −3.45335e11 −2.15490
\(310\) 0 0
\(311\) −1.88558e11 −1.14294 −0.571469 0.820624i \(-0.693626\pi\)
−0.571469 + 0.820624i \(0.693626\pi\)
\(312\) 0 0
\(313\) 1.31778e11 0.776058 0.388029 0.921647i \(-0.373156\pi\)
0.388029 + 0.921647i \(0.373156\pi\)
\(314\) 0 0
\(315\) 2.35626e11 1.34842
\(316\) 0 0
\(317\) −1.50686e11 −0.838118 −0.419059 0.907959i \(-0.637640\pi\)
−0.419059 + 0.907959i \(0.637640\pi\)
\(318\) 0 0
\(319\) 1.88320e11 1.01821
\(320\) 0 0
\(321\) 1.19675e11 0.629118
\(322\) 0 0
\(323\) −1.40530e11 −0.718386
\(324\) 0 0
\(325\) 2.24213e11 1.11477
\(326\) 0 0
\(327\) −4.84959e10 −0.234552
\(328\) 0 0
\(329\) −6.63841e10 −0.312380
\(330\) 0 0
\(331\) −3.38877e10 −0.155173 −0.0775865 0.996986i \(-0.524721\pi\)
−0.0775865 + 0.996986i \(0.524721\pi\)
\(332\) 0 0
\(333\) −1.44558e11 −0.644234
\(334\) 0 0
\(335\) 2.06559e11 0.896073
\(336\) 0 0
\(337\) 1.98312e11 0.837555 0.418778 0.908089i \(-0.362459\pi\)
0.418778 + 0.908089i \(0.362459\pi\)
\(338\) 0 0
\(339\) −1.43500e11 −0.590138
\(340\) 0 0
\(341\) −8.33371e10 −0.333767
\(342\) 0 0
\(343\) −1.38413e10 −0.0539949
\(344\) 0 0
\(345\) −1.77590e11 −0.674888
\(346\) 0 0
\(347\) 1.71762e11 0.635981 0.317990 0.948094i \(-0.396992\pi\)
0.317990 + 0.948094i \(0.396992\pi\)
\(348\) 0 0
\(349\) −1.88189e11 −0.679014 −0.339507 0.940603i \(-0.610260\pi\)
−0.339507 + 0.940603i \(0.610260\pi\)
\(350\) 0 0
\(351\) 3.26248e11 1.14727
\(352\) 0 0
\(353\) 1.97995e11 0.678686 0.339343 0.940663i \(-0.389795\pi\)
0.339343 + 0.940663i \(0.389795\pi\)
\(354\) 0 0
\(355\) 5.21211e11 1.74175
\(356\) 0 0
\(357\) 2.12870e11 0.693598
\(358\) 0 0
\(359\) −4.34669e11 −1.38113 −0.690563 0.723272i \(-0.742637\pi\)
−0.690563 + 0.723272i \(0.742637\pi\)
\(360\) 0 0
\(361\) −1.69343e11 −0.524788
\(362\) 0 0
\(363\) −3.89663e11 −1.17790
\(364\) 0 0
\(365\) −6.35528e11 −1.87420
\(366\) 0 0
\(367\) −6.02066e10 −0.173240 −0.0866198 0.996241i \(-0.527607\pi\)
−0.0866198 + 0.996241i \(0.527607\pi\)
\(368\) 0 0
\(369\) 1.42136e12 3.99103
\(370\) 0 0
\(371\) 5.74359e10 0.157399
\(372\) 0 0
\(373\) −6.85521e11 −1.83371 −0.916855 0.399219i \(-0.869281\pi\)
−0.916855 + 0.399219i \(0.869281\pi\)
\(374\) 0 0
\(375\) −1.01192e12 −2.64244
\(376\) 0 0
\(377\) 4.10754e11 1.04724
\(378\) 0 0
\(379\) −5.05522e11 −1.25853 −0.629266 0.777190i \(-0.716644\pi\)
−0.629266 + 0.777190i \(0.716644\pi\)
\(380\) 0 0
\(381\) −6.40082e11 −1.55623
\(382\) 0 0
\(383\) −2.02054e11 −0.479814 −0.239907 0.970796i \(-0.577117\pi\)
−0.239907 + 0.970796i \(0.577117\pi\)
\(384\) 0 0
\(385\) 1.59210e11 0.369316
\(386\) 0 0
\(387\) 5.99952e11 1.35962
\(388\) 0 0
\(389\) 5.51954e11 1.22216 0.611082 0.791567i \(-0.290735\pi\)
0.611082 + 0.791567i \(0.290735\pi\)
\(390\) 0 0
\(391\) −1.08699e11 −0.235196
\(392\) 0 0
\(393\) −6.06756e11 −1.28306
\(394\) 0 0
\(395\) 6.77197e10 0.139968
\(396\) 0 0
\(397\) 1.26816e11 0.256222 0.128111 0.991760i \(-0.459109\pi\)
0.128111 + 0.991760i \(0.459109\pi\)
\(398\) 0 0
\(399\) −2.32282e11 −0.458815
\(400\) 0 0
\(401\) 5.86957e11 1.13359 0.566795 0.823859i \(-0.308183\pi\)
0.566795 + 0.823859i \(0.308183\pi\)
\(402\) 0 0
\(403\) −1.81771e11 −0.343282
\(404\) 0 0
\(405\) −1.20706e12 −2.22937
\(406\) 0 0
\(407\) −9.76764e10 −0.176447
\(408\) 0 0
\(409\) −9.08262e10 −0.160493 −0.0802465 0.996775i \(-0.525571\pi\)
−0.0802465 + 0.996775i \(0.525571\pi\)
\(410\) 0 0
\(411\) 1.50298e12 2.59815
\(412\) 0 0
\(413\) −2.89513e11 −0.489658
\(414\) 0 0
\(415\) −9.09505e11 −1.50518
\(416\) 0 0
\(417\) −5.56483e11 −0.901238
\(418\) 0 0
\(419\) 8.34669e11 1.32297 0.661487 0.749957i \(-0.269926\pi\)
0.661487 + 0.749957i \(0.269926\pi\)
\(420\) 0 0
\(421\) −1.61360e10 −0.0250337 −0.0125169 0.999922i \(-0.503984\pi\)
−0.0125169 + 0.999922i \(0.503984\pi\)
\(422\) 0 0
\(423\) 1.14331e12 1.73634
\(424\) 0 0
\(425\) −1.32029e12 −1.96299
\(426\) 0 0
\(427\) −1.73626e11 −0.252749
\(428\) 0 0
\(429\) 4.20684e11 0.599650
\(430\) 0 0
\(431\) 4.81935e11 0.672730 0.336365 0.941732i \(-0.390802\pi\)
0.336365 + 0.941732i \(0.390802\pi\)
\(432\) 0 0
\(433\) 6.68314e11 0.913661 0.456830 0.889554i \(-0.348985\pi\)
0.456830 + 0.889554i \(0.348985\pi\)
\(434\) 0 0
\(435\) −3.95168e12 −5.29151
\(436\) 0 0
\(437\) 1.18611e11 0.155582
\(438\) 0 0
\(439\) 4.79819e11 0.616577 0.308288 0.951293i \(-0.400244\pi\)
0.308288 + 0.951293i \(0.400244\pi\)
\(440\) 0 0
\(441\) 2.38384e11 0.300126
\(442\) 0 0
\(443\) 6.53598e11 0.806295 0.403148 0.915135i \(-0.367916\pi\)
0.403148 + 0.915135i \(0.367916\pi\)
\(444\) 0 0
\(445\) −1.71247e12 −2.07016
\(446\) 0 0
\(447\) −7.74551e11 −0.917627
\(448\) 0 0
\(449\) −1.88660e11 −0.219064 −0.109532 0.993983i \(-0.534935\pi\)
−0.109532 + 0.993983i \(0.534935\pi\)
\(450\) 0 0
\(451\) 9.60397e11 1.09309
\(452\) 0 0
\(453\) 1.53404e12 1.71157
\(454\) 0 0
\(455\) 3.47261e11 0.379844
\(456\) 0 0
\(457\) 1.31205e12 1.40711 0.703556 0.710639i \(-0.251594\pi\)
0.703556 + 0.710639i \(0.251594\pi\)
\(458\) 0 0
\(459\) −1.92113e12 −2.02022
\(460\) 0 0
\(461\) 1.16594e12 1.20233 0.601165 0.799125i \(-0.294703\pi\)
0.601165 + 0.799125i \(0.294703\pi\)
\(462\) 0 0
\(463\) 3.87318e10 0.0391700 0.0195850 0.999808i \(-0.493766\pi\)
0.0195850 + 0.999808i \(0.493766\pi\)
\(464\) 0 0
\(465\) 1.74873e12 1.73454
\(466\) 0 0
\(467\) −1.17150e11 −0.113977 −0.0569883 0.998375i \(-0.518150\pi\)
−0.0569883 + 0.998375i \(0.518150\pi\)
\(468\) 0 0
\(469\) 2.08977e11 0.199444
\(470\) 0 0
\(471\) 3.29514e12 3.08518
\(472\) 0 0
\(473\) 4.05381e11 0.372382
\(474\) 0 0
\(475\) 1.44069e12 1.29852
\(476\) 0 0
\(477\) −9.89203e11 −0.874888
\(478\) 0 0
\(479\) 1.94734e12 1.69018 0.845089 0.534626i \(-0.179548\pi\)
0.845089 + 0.534626i \(0.179548\pi\)
\(480\) 0 0
\(481\) −2.13047e11 −0.181477
\(482\) 0 0
\(483\) −1.79668e11 −0.150214
\(484\) 0 0
\(485\) 1.59892e12 1.31217
\(486\) 0 0
\(487\) 1.95983e12 1.57884 0.789421 0.613852i \(-0.210381\pi\)
0.789421 + 0.613852i \(0.210381\pi\)
\(488\) 0 0
\(489\) −1.81243e12 −1.43341
\(490\) 0 0
\(491\) 2.93589e11 0.227967 0.113984 0.993483i \(-0.463639\pi\)
0.113984 + 0.993483i \(0.463639\pi\)
\(492\) 0 0
\(493\) −2.41874e12 −1.84407
\(494\) 0 0
\(495\) −2.74203e12 −2.05281
\(496\) 0 0
\(497\) 5.27313e11 0.387672
\(498\) 0 0
\(499\) −1.71390e12 −1.23746 −0.618732 0.785602i \(-0.712353\pi\)
−0.618732 + 0.785602i \(0.712353\pi\)
\(500\) 0 0
\(501\) 1.58658e12 1.12510
\(502\) 0 0
\(503\) −1.61385e12 −1.12411 −0.562053 0.827101i \(-0.689988\pi\)
−0.562053 + 0.827101i \(0.689988\pi\)
\(504\) 0 0
\(505\) 2.48726e10 0.0170181
\(506\) 0 0
\(507\) −1.70229e12 −1.14419
\(508\) 0 0
\(509\) 1.17088e12 0.773181 0.386590 0.922252i \(-0.373653\pi\)
0.386590 + 0.922252i \(0.373653\pi\)
\(510\) 0 0
\(511\) −6.42967e11 −0.417153
\(512\) 0 0
\(513\) 2.09632e12 1.33638
\(514\) 0 0
\(515\) 3.31734e12 2.07806
\(516\) 0 0
\(517\) 7.72526e11 0.475560
\(518\) 0 0
\(519\) 4.73960e12 2.86740
\(520\) 0 0
\(521\) 2.91339e12 1.73232 0.866161 0.499765i \(-0.166580\pi\)
0.866161 + 0.499765i \(0.166580\pi\)
\(522\) 0 0
\(523\) −2.53787e12 −1.48324 −0.741620 0.670820i \(-0.765942\pi\)
−0.741620 + 0.670820i \(0.765942\pi\)
\(524\) 0 0
\(525\) −2.18230e12 −1.25371
\(526\) 0 0
\(527\) 1.07036e12 0.604482
\(528\) 0 0
\(529\) −1.70941e12 −0.949063
\(530\) 0 0
\(531\) 4.98620e12 2.72172
\(532\) 0 0
\(533\) 2.09477e12 1.12425
\(534\) 0 0
\(535\) −1.14962e12 −0.606683
\(536\) 0 0
\(537\) −3.78920e12 −1.96636
\(538\) 0 0
\(539\) 1.61074e11 0.0822008
\(540\) 0 0
\(541\) 1.11804e12 0.561136 0.280568 0.959834i \(-0.409477\pi\)
0.280568 + 0.959834i \(0.409477\pi\)
\(542\) 0 0
\(543\) −4.28574e12 −2.11556
\(544\) 0 0
\(545\) 4.65859e11 0.226188
\(546\) 0 0
\(547\) −1.99438e12 −0.952500 −0.476250 0.879310i \(-0.658004\pi\)
−0.476250 + 0.879310i \(0.658004\pi\)
\(548\) 0 0
\(549\) 2.99031e12 1.40488
\(550\) 0 0
\(551\) 2.63931e12 1.21985
\(552\) 0 0
\(553\) 6.85125e10 0.0311535
\(554\) 0 0
\(555\) 2.04963e12 0.916973
\(556\) 0 0
\(557\) −4.69088e11 −0.206493 −0.103247 0.994656i \(-0.532923\pi\)
−0.103247 + 0.994656i \(0.532923\pi\)
\(558\) 0 0
\(559\) 8.84197e11 0.382997
\(560\) 0 0
\(561\) −2.47721e12 −1.05592
\(562\) 0 0
\(563\) −1.74518e12 −0.732069 −0.366034 0.930601i \(-0.619285\pi\)
−0.366034 + 0.930601i \(0.619285\pi\)
\(564\) 0 0
\(565\) 1.37848e12 0.569093
\(566\) 0 0
\(567\) −1.22119e12 −0.496205
\(568\) 0 0
\(569\) −4.13341e12 −1.65311 −0.826557 0.562853i \(-0.809704\pi\)
−0.826557 + 0.562853i \(0.809704\pi\)
\(570\) 0 0
\(571\) −4.14262e12 −1.63084 −0.815422 0.578867i \(-0.803495\pi\)
−0.815422 + 0.578867i \(0.803495\pi\)
\(572\) 0 0
\(573\) −6.67383e12 −2.58630
\(574\) 0 0
\(575\) 1.11436e12 0.425129
\(576\) 0 0
\(577\) 4.27130e12 1.60424 0.802120 0.597164i \(-0.203706\pi\)
0.802120 + 0.597164i \(0.203706\pi\)
\(578\) 0 0
\(579\) 6.67614e11 0.246872
\(580\) 0 0
\(581\) −9.20152e11 −0.335017
\(582\) 0 0
\(583\) −6.68394e11 −0.239621
\(584\) 0 0
\(585\) −5.98078e12 −2.11133
\(586\) 0 0
\(587\) −5.32282e12 −1.85042 −0.925210 0.379455i \(-0.876111\pi\)
−0.925210 + 0.379455i \(0.876111\pi\)
\(588\) 0 0
\(589\) −1.16797e12 −0.399865
\(590\) 0 0
\(591\) 5.90570e12 1.99126
\(592\) 0 0
\(593\) −2.69262e11 −0.0894190 −0.0447095 0.999000i \(-0.514236\pi\)
−0.0447095 + 0.999000i \(0.514236\pi\)
\(594\) 0 0
\(595\) −2.04486e12 −0.668864
\(596\) 0 0
\(597\) 5.33906e11 0.172021
\(598\) 0 0
\(599\) −3.51807e12 −1.11657 −0.558283 0.829651i \(-0.688540\pi\)
−0.558283 + 0.829651i \(0.688540\pi\)
\(600\) 0 0
\(601\) −1.97549e12 −0.617645 −0.308823 0.951120i \(-0.599935\pi\)
−0.308823 + 0.951120i \(0.599935\pi\)
\(602\) 0 0
\(603\) −3.59916e12 −1.10860
\(604\) 0 0
\(605\) 3.74316e12 1.13590
\(606\) 0 0
\(607\) 1.20562e12 0.360464 0.180232 0.983624i \(-0.442315\pi\)
0.180232 + 0.983624i \(0.442315\pi\)
\(608\) 0 0
\(609\) −3.99794e12 −1.17776
\(610\) 0 0
\(611\) 1.68499e12 0.489117
\(612\) 0 0
\(613\) −5.02026e11 −0.143600 −0.0717999 0.997419i \(-0.522874\pi\)
−0.0717999 + 0.997419i \(0.522874\pi\)
\(614\) 0 0
\(615\) −2.01528e13 −5.68065
\(616\) 0 0
\(617\) 5.17852e12 1.43854 0.719271 0.694730i \(-0.244476\pi\)
0.719271 + 0.694730i \(0.244476\pi\)
\(618\) 0 0
\(619\) 4.69963e12 1.28664 0.643318 0.765599i \(-0.277557\pi\)
0.643318 + 0.765599i \(0.277557\pi\)
\(620\) 0 0
\(621\) 1.62149e12 0.437523
\(622\) 0 0
\(623\) −1.73252e12 −0.460767
\(624\) 0 0
\(625\) 2.53502e12 0.664541
\(626\) 0 0
\(627\) 2.70312e12 0.698491
\(628\) 0 0
\(629\) 1.25454e12 0.319562
\(630\) 0 0
\(631\) 5.53678e12 1.39035 0.695177 0.718839i \(-0.255326\pi\)
0.695177 + 0.718839i \(0.255326\pi\)
\(632\) 0 0
\(633\) −8.91994e11 −0.220823
\(634\) 0 0
\(635\) 6.14872e12 1.50073
\(636\) 0 0
\(637\) 3.51326e11 0.0845441
\(638\) 0 0
\(639\) −9.08175e12 −2.15484
\(640\) 0 0
\(641\) −2.85506e12 −0.667966 −0.333983 0.942579i \(-0.608393\pi\)
−0.333983 + 0.942579i \(0.608393\pi\)
\(642\) 0 0
\(643\) 7.48209e12 1.72613 0.863065 0.505093i \(-0.168542\pi\)
0.863065 + 0.505093i \(0.168542\pi\)
\(644\) 0 0
\(645\) −8.50646e12 −1.93522
\(646\) 0 0
\(647\) 3.83872e12 0.861225 0.430613 0.902537i \(-0.358297\pi\)
0.430613 + 0.902537i \(0.358297\pi\)
\(648\) 0 0
\(649\) 3.36912e12 0.745445
\(650\) 0 0
\(651\) 1.76920e12 0.386068
\(652\) 0 0
\(653\) −1.09261e11 −0.0235157 −0.0117578 0.999931i \(-0.503743\pi\)
−0.0117578 + 0.999931i \(0.503743\pi\)
\(654\) 0 0
\(655\) 5.82859e12 1.23731
\(656\) 0 0
\(657\) 1.10736e13 2.31871
\(658\) 0 0
\(659\) 2.10319e12 0.434403 0.217202 0.976127i \(-0.430307\pi\)
0.217202 + 0.976127i \(0.430307\pi\)
\(660\) 0 0
\(661\) −1.76045e12 −0.358688 −0.179344 0.983786i \(-0.557397\pi\)
−0.179344 + 0.983786i \(0.557397\pi\)
\(662\) 0 0
\(663\) −5.40317e12 −1.08602
\(664\) 0 0
\(665\) 2.23134e12 0.442453
\(666\) 0 0
\(667\) 2.04149e12 0.399374
\(668\) 0 0
\(669\) 1.76148e13 3.39986
\(670\) 0 0
\(671\) 2.02052e12 0.384779
\(672\) 0 0
\(673\) 8.52389e12 1.60166 0.800830 0.598892i \(-0.204392\pi\)
0.800830 + 0.598892i \(0.204392\pi\)
\(674\) 0 0
\(675\) 1.96950e13 3.65165
\(676\) 0 0
\(677\) −2.88497e12 −0.527828 −0.263914 0.964546i \(-0.585014\pi\)
−0.263914 + 0.964546i \(0.585014\pi\)
\(678\) 0 0
\(679\) 1.61764e12 0.292057
\(680\) 0 0
\(681\) −1.76732e13 −3.14885
\(682\) 0 0
\(683\) −1.03086e13 −1.81262 −0.906309 0.422616i \(-0.861112\pi\)
−0.906309 + 0.422616i \(0.861112\pi\)
\(684\) 0 0
\(685\) −1.44378e13 −2.50550
\(686\) 0 0
\(687\) −8.79556e12 −1.50646
\(688\) 0 0
\(689\) −1.45787e12 −0.246451
\(690\) 0 0
\(691\) −7.85439e12 −1.31057 −0.655287 0.755380i \(-0.727452\pi\)
−0.655287 + 0.755380i \(0.727452\pi\)
\(692\) 0 0
\(693\) −2.77413e12 −0.456906
\(694\) 0 0
\(695\) 5.34566e12 0.869099
\(696\) 0 0
\(697\) −1.23351e13 −1.97969
\(698\) 0 0
\(699\) −9.38969e12 −1.48766
\(700\) 0 0
\(701\) 1.69579e12 0.265242 0.132621 0.991167i \(-0.457661\pi\)
0.132621 + 0.991167i \(0.457661\pi\)
\(702\) 0 0
\(703\) −1.36894e12 −0.211390
\(704\) 0 0
\(705\) −1.62106e13 −2.47142
\(706\) 0 0
\(707\) 2.51638e10 0.00378781
\(708\) 0 0
\(709\) −2.20703e12 −0.328020 −0.164010 0.986459i \(-0.552443\pi\)
−0.164010 + 0.986459i \(0.552443\pi\)
\(710\) 0 0
\(711\) −1.17997e12 −0.173164
\(712\) 0 0
\(713\) −9.03417e11 −0.130914
\(714\) 0 0
\(715\) −4.04115e12 −0.578266
\(716\) 0 0
\(717\) 2.14225e13 3.02715
\(718\) 0 0
\(719\) 1.46869e12 0.204952 0.102476 0.994735i \(-0.467324\pi\)
0.102476 + 0.994735i \(0.467324\pi\)
\(720\) 0 0
\(721\) 3.35617e12 0.462525
\(722\) 0 0
\(723\) −2.02192e12 −0.275195
\(724\) 0 0
\(725\) 2.47965e13 3.33326
\(726\) 0 0
\(727\) 4.97511e11 0.0660538 0.0330269 0.999454i \(-0.489485\pi\)
0.0330269 + 0.999454i \(0.489485\pi\)
\(728\) 0 0
\(729\) −4.99938e12 −0.655605
\(730\) 0 0
\(731\) −5.20663e12 −0.674417
\(732\) 0 0
\(733\) −5.22437e12 −0.668445 −0.334223 0.942494i \(-0.608474\pi\)
−0.334223 + 0.942494i \(0.608474\pi\)
\(734\) 0 0
\(735\) −3.37995e12 −0.427187
\(736\) 0 0
\(737\) −2.43191e12 −0.303630
\(738\) 0 0
\(739\) −7.35495e11 −0.0907152 −0.0453576 0.998971i \(-0.514443\pi\)
−0.0453576 + 0.998971i \(0.514443\pi\)
\(740\) 0 0
\(741\) 5.89590e12 0.718403
\(742\) 0 0
\(743\) 1.12076e12 0.134916 0.0674580 0.997722i \(-0.478511\pi\)
0.0674580 + 0.997722i \(0.478511\pi\)
\(744\) 0 0
\(745\) 7.44045e12 0.884904
\(746\) 0 0
\(747\) 1.58475e13 1.86217
\(748\) 0 0
\(749\) −1.16308e12 −0.135033
\(750\) 0 0
\(751\) 1.62183e12 0.186049 0.0930244 0.995664i \(-0.470347\pi\)
0.0930244 + 0.995664i \(0.470347\pi\)
\(752\) 0 0
\(753\) −2.40991e13 −2.73164
\(754\) 0 0
\(755\) −1.47362e13 −1.65054
\(756\) 0 0
\(757\) −1.55150e12 −0.171720 −0.0858600 0.996307i \(-0.527364\pi\)
−0.0858600 + 0.996307i \(0.527364\pi\)
\(758\) 0 0
\(759\) 2.09084e12 0.228682
\(760\) 0 0
\(761\) 1.62863e13 1.76032 0.880159 0.474679i \(-0.157436\pi\)
0.880159 + 0.474679i \(0.157436\pi\)
\(762\) 0 0
\(763\) 4.71312e11 0.0503440
\(764\) 0 0
\(765\) 3.52181e13 3.71783
\(766\) 0 0
\(767\) 7.34856e12 0.766695
\(768\) 0 0
\(769\) −1.32915e13 −1.37059 −0.685293 0.728267i \(-0.740326\pi\)
−0.685293 + 0.728267i \(0.740326\pi\)
\(770\) 0 0
\(771\) −1.09455e12 −0.111555
\(772\) 0 0
\(773\) −7.58593e12 −0.764190 −0.382095 0.924123i \(-0.624797\pi\)
−0.382095 + 0.924123i \(0.624797\pi\)
\(774\) 0 0
\(775\) −1.09732e13 −1.09263
\(776\) 0 0
\(777\) 2.07362e12 0.204096
\(778\) 0 0
\(779\) 1.34600e13 1.30956
\(780\) 0 0
\(781\) −6.13645e12 −0.590184
\(782\) 0 0
\(783\) 3.60809e13 3.43044
\(784\) 0 0
\(785\) −3.16536e13 −2.97516
\(786\) 0 0
\(787\) −3.30985e12 −0.307554 −0.153777 0.988106i \(-0.549144\pi\)
−0.153777 + 0.988106i \(0.549144\pi\)
\(788\) 0 0
\(789\) 2.97995e13 2.73755
\(790\) 0 0
\(791\) 1.39462e12 0.126666
\(792\) 0 0
\(793\) 4.40706e12 0.395748
\(794\) 0 0
\(795\) 1.40255e13 1.24528
\(796\) 0 0
\(797\) −3.26446e12 −0.286582 −0.143291 0.989681i \(-0.545768\pi\)
−0.143291 + 0.989681i \(0.545768\pi\)
\(798\) 0 0
\(799\) −9.92216e12 −0.861283
\(800\) 0 0
\(801\) 2.98386e13 2.56114
\(802\) 0 0
\(803\) 7.48235e12 0.635064
\(804\) 0 0
\(805\) 1.72592e12 0.144857
\(806\) 0 0
\(807\) 1.87519e13 1.55637
\(808\) 0 0
\(809\) 3.46768e12 0.284624 0.142312 0.989822i \(-0.454546\pi\)
0.142312 + 0.989822i \(0.454546\pi\)
\(810\) 0 0
\(811\) −1.02915e13 −0.835381 −0.417690 0.908589i \(-0.637160\pi\)
−0.417690 + 0.908589i \(0.637160\pi\)
\(812\) 0 0
\(813\) −1.75752e13 −1.41089
\(814\) 0 0
\(815\) 1.74104e13 1.38229
\(816\) 0 0
\(817\) 5.68143e12 0.446127
\(818\) 0 0
\(819\) −6.05079e12 −0.469931
\(820\) 0 0
\(821\) 6.23700e12 0.479106 0.239553 0.970883i \(-0.422999\pi\)
0.239553 + 0.970883i \(0.422999\pi\)
\(822\) 0 0
\(823\) −2.46689e13 −1.87435 −0.937175 0.348860i \(-0.886569\pi\)
−0.937175 + 0.348860i \(0.886569\pi\)
\(824\) 0 0
\(825\) 2.53959e13 1.90863
\(826\) 0 0
\(827\) 1.45027e13 1.07814 0.539068 0.842262i \(-0.318776\pi\)
0.539068 + 0.842262i \(0.318776\pi\)
\(828\) 0 0
\(829\) −1.07681e13 −0.791854 −0.395927 0.918282i \(-0.629577\pi\)
−0.395927 + 0.918282i \(0.629577\pi\)
\(830\) 0 0
\(831\) 2.12846e13 1.54832
\(832\) 0 0
\(833\) −2.06880e12 −0.148873
\(834\) 0 0
\(835\) −1.52409e13 −1.08498
\(836\) 0 0
\(837\) −1.59668e13 −1.12449
\(838\) 0 0
\(839\) 2.36129e13 1.64521 0.822603 0.568616i \(-0.192521\pi\)
0.822603 + 0.568616i \(0.192521\pi\)
\(840\) 0 0
\(841\) 3.09195e13 2.13133
\(842\) 0 0
\(843\) −2.47494e13 −1.68787
\(844\) 0 0
\(845\) 1.63524e13 1.10339
\(846\) 0 0
\(847\) 3.78698e12 0.252824
\(848\) 0 0
\(849\) 1.13013e13 0.746521
\(850\) 0 0
\(851\) −1.05886e12 −0.0692081
\(852\) 0 0
\(853\) −1.67503e13 −1.08331 −0.541655 0.840601i \(-0.682202\pi\)
−0.541655 + 0.840601i \(0.682202\pi\)
\(854\) 0 0
\(855\) −3.84297e13 −2.45934
\(856\) 0 0
\(857\) −1.16182e13 −0.735742 −0.367871 0.929877i \(-0.619913\pi\)
−0.367871 + 0.929877i \(0.619913\pi\)
\(858\) 0 0
\(859\) −1.58818e13 −0.995246 −0.497623 0.867393i \(-0.665794\pi\)
−0.497623 + 0.867393i \(0.665794\pi\)
\(860\) 0 0
\(861\) −2.03888e13 −1.26438
\(862\) 0 0
\(863\) −1.35971e13 −0.834446 −0.417223 0.908804i \(-0.636997\pi\)
−0.417223 + 0.908804i \(0.636997\pi\)
\(864\) 0 0
\(865\) −4.55293e13 −2.76515
\(866\) 0 0
\(867\) 2.51943e12 0.151431
\(868\) 0 0
\(869\) −7.97294e11 −0.0474274
\(870\) 0 0
\(871\) −5.30437e12 −0.312285
\(872\) 0 0
\(873\) −2.78601e13 −1.62338
\(874\) 0 0
\(875\) 9.83445e12 0.567170
\(876\) 0 0
\(877\) 3.40277e12 0.194238 0.0971189 0.995273i \(-0.469037\pi\)
0.0971189 + 0.995273i \(0.469037\pi\)
\(878\) 0 0
\(879\) −2.51041e12 −0.141839
\(880\) 0 0
\(881\) 9.30779e12 0.520541 0.260270 0.965536i \(-0.416188\pi\)
0.260270 + 0.965536i \(0.416188\pi\)
\(882\) 0 0
\(883\) −1.00154e13 −0.554430 −0.277215 0.960808i \(-0.589411\pi\)
−0.277215 + 0.960808i \(0.589411\pi\)
\(884\) 0 0
\(885\) −7.06972e13 −3.87398
\(886\) 0 0
\(887\) 2.58440e13 1.40186 0.700928 0.713232i \(-0.252769\pi\)
0.700928 + 0.713232i \(0.252769\pi\)
\(888\) 0 0
\(889\) 6.22070e12 0.334027
\(890\) 0 0
\(891\) 1.42113e13 0.755412
\(892\) 0 0
\(893\) 1.08270e13 0.569739
\(894\) 0 0
\(895\) 3.63997e13 1.89624
\(896\) 0 0
\(897\) 4.56043e12 0.235201
\(898\) 0 0
\(899\) −2.01026e13 −1.02644
\(900\) 0 0
\(901\) 8.58471e12 0.433974
\(902\) 0 0
\(903\) −8.60604e12 −0.430733
\(904\) 0 0
\(905\) 4.11694e13 2.04012
\(906\) 0 0
\(907\) −3.82049e13 −1.87451 −0.937253 0.348650i \(-0.886640\pi\)
−0.937253 + 0.348650i \(0.886640\pi\)
\(908\) 0 0
\(909\) −4.33389e11 −0.0210543
\(910\) 0 0
\(911\) −9.85478e12 −0.474039 −0.237020 0.971505i \(-0.576171\pi\)
−0.237020 + 0.971505i \(0.576171\pi\)
\(912\) 0 0
\(913\) 1.07080e13 0.510023
\(914\) 0 0
\(915\) −4.23983e13 −1.99965
\(916\) 0 0
\(917\) 5.89682e12 0.275395
\(918\) 0 0
\(919\) −2.60834e13 −1.20627 −0.603135 0.797639i \(-0.706082\pi\)
−0.603135 + 0.797639i \(0.706082\pi\)
\(920\) 0 0
\(921\) 9.11517e12 0.417442
\(922\) 0 0
\(923\) −1.33845e13 −0.607008
\(924\) 0 0
\(925\) −1.28613e13 −0.577625
\(926\) 0 0
\(927\) −5.78024e13 −2.57091
\(928\) 0 0
\(929\) −3.60592e13 −1.58835 −0.794174 0.607690i \(-0.792096\pi\)
−0.794174 + 0.607690i \(0.792096\pi\)
\(930\) 0 0
\(931\) 2.25746e12 0.0984795
\(932\) 0 0
\(933\) −4.65836e13 −2.01264
\(934\) 0 0
\(935\) 2.37965e13 1.01826
\(936\) 0 0
\(937\) 2.75358e13 1.16700 0.583499 0.812114i \(-0.301683\pi\)
0.583499 + 0.812114i \(0.301683\pi\)
\(938\) 0 0
\(939\) 3.25561e13 1.36659
\(940\) 0 0
\(941\) −7.60157e12 −0.316046 −0.158023 0.987435i \(-0.550512\pi\)
−0.158023 + 0.987435i \(0.550512\pi\)
\(942\) 0 0
\(943\) 1.04112e13 0.428745
\(944\) 0 0
\(945\) 3.05037e13 1.24425
\(946\) 0 0
\(947\) 2.35269e13 0.950582 0.475291 0.879829i \(-0.342343\pi\)
0.475291 + 0.879829i \(0.342343\pi\)
\(948\) 0 0
\(949\) 1.63201e13 0.653168
\(950\) 0 0
\(951\) −3.72272e13 −1.47587
\(952\) 0 0
\(953\) 3.76056e12 0.147684 0.0738422 0.997270i \(-0.476474\pi\)
0.0738422 + 0.997270i \(0.476474\pi\)
\(954\) 0 0
\(955\) 6.41098e13 2.49407
\(956\) 0 0
\(957\) 4.65248e13 1.79300
\(958\) 0 0
\(959\) −1.46068e13 −0.557663
\(960\) 0 0
\(961\) −1.75436e13 −0.663535
\(962\) 0 0
\(963\) 2.00313e13 0.750571
\(964\) 0 0
\(965\) −6.41320e12 −0.238068
\(966\) 0 0
\(967\) −3.59678e13 −1.32280 −0.661400 0.750033i \(-0.730037\pi\)
−0.661400 + 0.750033i \(0.730037\pi\)
\(968\) 0 0
\(969\) −3.47182e13 −1.26503
\(970\) 0 0
\(971\) 1.20049e13 0.433383 0.216692 0.976240i \(-0.430473\pi\)
0.216692 + 0.976240i \(0.430473\pi\)
\(972\) 0 0
\(973\) 5.40823e12 0.193441
\(974\) 0 0
\(975\) 5.53923e13 1.96304
\(976\) 0 0
\(977\) −4.59954e12 −0.161506 −0.0807530 0.996734i \(-0.525732\pi\)
−0.0807530 + 0.996734i \(0.525732\pi\)
\(978\) 0 0
\(979\) 2.01617e13 0.701462
\(980\) 0 0
\(981\) −8.11727e12 −0.279833
\(982\) 0 0
\(983\) 3.13830e12 0.107202 0.0536011 0.998562i \(-0.482930\pi\)
0.0536011 + 0.998562i \(0.482930\pi\)
\(984\) 0 0
\(985\) −5.67310e13 −1.92025
\(986\) 0 0
\(987\) −1.64003e13 −0.550080
\(988\) 0 0
\(989\) 4.39454e12 0.146060
\(990\) 0 0
\(991\) −1.38736e13 −0.456939 −0.228470 0.973551i \(-0.573372\pi\)
−0.228470 + 0.973551i \(0.573372\pi\)
\(992\) 0 0
\(993\) −8.37203e12 −0.273249
\(994\) 0 0
\(995\) −5.12878e12 −0.165886
\(996\) 0 0
\(997\) −1.65453e13 −0.530331 −0.265166 0.964203i \(-0.585427\pi\)
−0.265166 + 0.964203i \(0.585427\pi\)
\(998\) 0 0
\(999\) −1.87142e13 −0.594465
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 112.10.a.c.1.2 2
4.3 odd 2 14.10.a.c.1.1 2
12.11 even 2 126.10.a.o.1.2 2
20.3 even 4 350.10.c.j.99.3 4
20.7 even 4 350.10.c.j.99.2 4
20.19 odd 2 350.10.a.j.1.2 2
28.3 even 6 98.10.c.h.79.1 4
28.11 odd 6 98.10.c.j.79.2 4
28.19 even 6 98.10.c.h.67.1 4
28.23 odd 6 98.10.c.j.67.2 4
28.27 even 2 98.10.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.10.a.c.1.1 2 4.3 odd 2
98.10.a.e.1.2 2 28.27 even 2
98.10.c.h.67.1 4 28.19 even 6
98.10.c.h.79.1 4 28.3 even 6
98.10.c.j.67.2 4 28.23 odd 6
98.10.c.j.79.2 4 28.11 odd 6
112.10.a.c.1.2 2 1.1 even 1 trivial
126.10.a.o.1.2 2 12.11 even 2
350.10.a.j.1.2 2 20.19 odd 2
350.10.c.j.99.2 4 20.7 even 4
350.10.c.j.99.3 4 20.3 even 4