Properties

Label 75.22.a.c.1.1
Level $75$
Weight $22$
Character 75.1
Self dual yes
Analytic conductor $209.608$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(209.608008215\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 75.1

$q$-expansion

\(f(q)\) \(=\) \(q+2844.00 q^{2} +59049.0 q^{3} +5.99118e6 q^{4} +1.67935e8 q^{6} -3.63304e8 q^{7} +1.10746e10 q^{8} +3.48678e9 q^{9} +O(q^{10})\) \(q+2844.00 q^{2} +59049.0 q^{3} +5.99118e6 q^{4} +1.67935e8 q^{6} -3.63304e8 q^{7} +1.10746e10 q^{8} +3.48678e9 q^{9} +1.45818e10 q^{11} +3.53773e11 q^{12} -1.13351e11 q^{13} -1.03324e12 q^{14} +1.89318e13 q^{16} +8.58939e12 q^{17} +9.91641e12 q^{18} -2.92029e13 q^{19} -2.14527e13 q^{21} +4.14707e13 q^{22} +1.55899e14 q^{23} +6.53946e14 q^{24} -3.22370e14 q^{26} +2.05891e14 q^{27} -2.17662e15 q^{28} +2.40079e15 q^{29} +2.23982e15 q^{31} +3.06169e16 q^{32} +8.61043e14 q^{33} +2.44282e16 q^{34} +2.08900e16 q^{36} +3.07851e16 q^{37} -8.30532e16 q^{38} -6.69325e15 q^{39} -1.03208e17 q^{41} -6.10116e16 q^{42} +1.65557e17 q^{43} +8.73624e16 q^{44} +4.43377e17 q^{46} +6.65872e16 q^{47} +1.11790e18 q^{48} -4.26556e17 q^{49} +5.07195e17 q^{51} -6.79105e17 q^{52} -4.35423e17 q^{53} +5.85554e17 q^{54} -4.02346e18 q^{56} -1.72440e18 q^{57} +6.82784e18 q^{58} +5.53437e18 q^{59} -7.17621e18 q^{61} +6.37005e18 q^{62} -1.26676e18 q^{63} +4.73716e19 q^{64} +2.44881e18 q^{66} +1.57554e19 q^{67} +5.14606e19 q^{68} +9.20569e18 q^{69} +2.64579e19 q^{71} +3.86148e19 q^{72} -1.34712e19 q^{73} +8.75527e19 q^{74} -1.74960e20 q^{76} -5.29764e18 q^{77} -1.90356e19 q^{78} -1.68861e19 q^{79} +1.21577e19 q^{81} -2.93522e20 q^{82} +1.70688e20 q^{83} -1.28527e20 q^{84} +4.70845e20 q^{86} +1.41764e20 q^{87} +1.61488e20 q^{88} -3.12592e20 q^{89} +4.11808e19 q^{91} +9.34021e20 q^{92} +1.32259e20 q^{93} +1.89374e20 q^{94} +1.80790e21 q^{96} -9.49015e20 q^{97} -1.21313e21 q^{98} +5.08437e19 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2844.00 1.96388 0.981939 0.189196i \(-0.0605882\pi\)
0.981939 + 0.189196i \(0.0605882\pi\)
\(3\) 59049.0 0.577350
\(4\) 5.99118e6 2.85682
\(5\) 0 0
\(6\) 1.67935e8 1.13385
\(7\) −3.63304e8 −0.486117 −0.243058 0.970012i \(-0.578151\pi\)
−0.243058 + 0.970012i \(0.578151\pi\)
\(8\) 1.10746e10 3.64657
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) 1.45818e10 0.169508 0.0847538 0.996402i \(-0.472990\pi\)
0.0847538 + 0.996402i \(0.472990\pi\)
\(12\) 3.53773e11 1.64939
\(13\) −1.13351e11 −0.228044 −0.114022 0.993478i \(-0.536373\pi\)
−0.114022 + 0.993478i \(0.536373\pi\)
\(14\) −1.03324e12 −0.954674
\(15\) 0 0
\(16\) 1.89318e13 4.30460
\(17\) 8.58939e12 1.03335 0.516676 0.856181i \(-0.327169\pi\)
0.516676 + 0.856181i \(0.327169\pi\)
\(18\) 9.91641e12 0.654626
\(19\) −2.92029e13 −1.09273 −0.546366 0.837546i \(-0.683989\pi\)
−0.546366 + 0.837546i \(0.683989\pi\)
\(20\) 0 0
\(21\) −2.14527e13 −0.280660
\(22\) 4.14707e13 0.332892
\(23\) 1.55899e14 0.784696 0.392348 0.919817i \(-0.371663\pi\)
0.392348 + 0.919817i \(0.371663\pi\)
\(24\) 6.53946e14 2.10535
\(25\) 0 0
\(26\) −3.22370e14 −0.447851
\(27\) 2.05891e14 0.192450
\(28\) −2.17662e15 −1.38875
\(29\) 2.40079e15 1.05968 0.529840 0.848097i \(-0.322252\pi\)
0.529840 + 0.848097i \(0.322252\pi\)
\(30\) 0 0
\(31\) 2.23982e15 0.490812 0.245406 0.969420i \(-0.421079\pi\)
0.245406 + 0.969420i \(0.421079\pi\)
\(32\) 3.06169e16 4.80714
\(33\) 8.61043e14 0.0978652
\(34\) 2.44282e16 2.02938
\(35\) 0 0
\(36\) 2.08900e16 0.952273
\(37\) 3.07851e16 1.05250 0.526250 0.850330i \(-0.323598\pi\)
0.526250 + 0.850330i \(0.323598\pi\)
\(38\) −8.30532e16 −2.14599
\(39\) −6.69325e15 −0.131661
\(40\) 0 0
\(41\) −1.03208e17 −1.20083 −0.600414 0.799689i \(-0.704998\pi\)
−0.600414 + 0.799689i \(0.704998\pi\)
\(42\) −6.10116e16 −0.551182
\(43\) 1.65557e17 1.16823 0.584117 0.811670i \(-0.301441\pi\)
0.584117 + 0.811670i \(0.301441\pi\)
\(44\) 8.73624e16 0.484252
\(45\) 0 0
\(46\) 4.43377e17 1.54105
\(47\) 6.65872e16 0.184656 0.0923280 0.995729i \(-0.470569\pi\)
0.0923280 + 0.995729i \(0.470569\pi\)
\(48\) 1.11790e18 2.48526
\(49\) −4.26556e17 −0.763690
\(50\) 0 0
\(51\) 5.07195e17 0.596607
\(52\) −6.79105e17 −0.651482
\(53\) −4.35423e17 −0.341991 −0.170995 0.985272i \(-0.554698\pi\)
−0.170995 + 0.985272i \(0.554698\pi\)
\(54\) 5.85554e17 0.377949
\(55\) 0 0
\(56\) −4.02346e18 −1.77266
\(57\) −1.72440e18 −0.630889
\(58\) 6.82784e18 2.08108
\(59\) 5.53437e18 1.40968 0.704842 0.709364i \(-0.251018\pi\)
0.704842 + 0.709364i \(0.251018\pi\)
\(60\) 0 0
\(61\) −7.17621e18 −1.28805 −0.644023 0.765006i \(-0.722736\pi\)
−0.644023 + 0.765006i \(0.722736\pi\)
\(62\) 6.37005e18 0.963896
\(63\) −1.26676e18 −0.162039
\(64\) 4.73716e19 5.13604
\(65\) 0 0
\(66\) 2.44881e18 0.192195
\(67\) 1.57554e19 1.05595 0.527977 0.849258i \(-0.322951\pi\)
0.527977 + 0.849258i \(0.322951\pi\)
\(68\) 5.14606e19 2.95210
\(69\) 9.20569e18 0.453045
\(70\) 0 0
\(71\) 2.64579e19 0.964588 0.482294 0.876009i \(-0.339804\pi\)
0.482294 + 0.876009i \(0.339804\pi\)
\(72\) 3.86148e19 1.21552
\(73\) −1.34712e19 −0.366875 −0.183437 0.983031i \(-0.558722\pi\)
−0.183437 + 0.983031i \(0.558722\pi\)
\(74\) 8.75527e19 2.06698
\(75\) 0 0
\(76\) −1.74960e20 −3.12174
\(77\) −5.29764e18 −0.0824005
\(78\) −1.90356e19 −0.258567
\(79\) −1.68861e19 −0.200653 −0.100326 0.994955i \(-0.531989\pi\)
−0.100326 + 0.994955i \(0.531989\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) −2.93522e20 −2.35828
\(83\) 1.70688e20 1.20749 0.603744 0.797178i \(-0.293675\pi\)
0.603744 + 0.797178i \(0.293675\pi\)
\(84\) −1.28527e20 −0.801794
\(85\) 0 0
\(86\) 4.70845e20 2.29427
\(87\) 1.41764e20 0.611807
\(88\) 1.61488e20 0.618121
\(89\) −3.12592e20 −1.06263 −0.531317 0.847173i \(-0.678303\pi\)
−0.531317 + 0.847173i \(0.678303\pi\)
\(90\) 0 0
\(91\) 4.11808e19 0.110856
\(92\) 9.34021e20 2.24174
\(93\) 1.32259e20 0.283371
\(94\) 1.89374e20 0.362642
\(95\) 0 0
\(96\) 1.80790e21 2.77540
\(97\) −9.49015e20 −1.30668 −0.653341 0.757064i \(-0.726633\pi\)
−0.653341 + 0.757064i \(0.726633\pi\)
\(98\) −1.21313e21 −1.49980
\(99\) 5.08437e19 0.0565025
\(100\) 0 0
\(101\) 1.44798e20 0.130433 0.0652166 0.997871i \(-0.479226\pi\)
0.0652166 + 0.997871i \(0.479226\pi\)
\(102\) 1.44246e21 1.17166
\(103\) −2.19627e21 −1.61025 −0.805127 0.593102i \(-0.797903\pi\)
−0.805127 + 0.593102i \(0.797903\pi\)
\(104\) −1.25532e21 −0.831579
\(105\) 0 0
\(106\) −1.23834e21 −0.671629
\(107\) 1.63087e20 0.0801473 0.0400737 0.999197i \(-0.487241\pi\)
0.0400737 + 0.999197i \(0.487241\pi\)
\(108\) 1.23353e21 0.549795
\(109\) 2.24852e20 0.0909743 0.0454871 0.998965i \(-0.485516\pi\)
0.0454871 + 0.998965i \(0.485516\pi\)
\(110\) 0 0
\(111\) 1.81783e21 0.607661
\(112\) −6.87800e21 −2.09254
\(113\) 4.24118e21 1.17534 0.587670 0.809101i \(-0.300045\pi\)
0.587670 + 0.809101i \(0.300045\pi\)
\(114\) −4.90421e21 −1.23899
\(115\) 0 0
\(116\) 1.43836e22 3.02732
\(117\) −3.95230e20 −0.0760148
\(118\) 1.57397e22 2.76845
\(119\) −3.12056e21 −0.502330
\(120\) 0 0
\(121\) −7.18762e21 −0.971267
\(122\) −2.04091e22 −2.52957
\(123\) −6.09430e21 −0.693299
\(124\) 1.34192e22 1.40216
\(125\) 0 0
\(126\) −3.60267e21 −0.318225
\(127\) −1.66312e21 −0.135202 −0.0676012 0.997712i \(-0.521535\pi\)
−0.0676012 + 0.997712i \(0.521535\pi\)
\(128\) 7.05165e22 5.27942
\(129\) 9.77599e21 0.674480
\(130\) 0 0
\(131\) 6.40663e21 0.376081 0.188040 0.982161i \(-0.439786\pi\)
0.188040 + 0.982161i \(0.439786\pi\)
\(132\) 5.15867e21 0.279583
\(133\) 1.06095e22 0.531196
\(134\) 4.48085e22 2.07377
\(135\) 0 0
\(136\) 9.51243e22 3.76819
\(137\) 1.98314e22 0.727423 0.363711 0.931512i \(-0.381509\pi\)
0.363711 + 0.931512i \(0.381509\pi\)
\(138\) 2.61810e22 0.889725
\(139\) −5.20143e21 −0.163858 −0.0819290 0.996638i \(-0.526108\pi\)
−0.0819290 + 0.996638i \(0.526108\pi\)
\(140\) 0 0
\(141\) 3.93191e21 0.106611
\(142\) 7.52461e22 1.89433
\(143\) −1.65286e21 −0.0386552
\(144\) 6.60112e22 1.43487
\(145\) 0 0
\(146\) −3.83122e22 −0.720498
\(147\) −2.51877e22 −0.440917
\(148\) 1.84439e23 3.00680
\(149\) −7.47631e22 −1.13562 −0.567808 0.823161i \(-0.692208\pi\)
−0.567808 + 0.823161i \(0.692208\pi\)
\(150\) 0 0
\(151\) 1.11044e23 1.46635 0.733174 0.680042i \(-0.238038\pi\)
0.733174 + 0.680042i \(0.238038\pi\)
\(152\) −3.23412e23 −3.98472
\(153\) 2.99493e22 0.344451
\(154\) −1.50665e22 −0.161824
\(155\) 0 0
\(156\) −4.01005e22 −0.376133
\(157\) −4.36563e22 −0.382913 −0.191457 0.981501i \(-0.561321\pi\)
−0.191457 + 0.981501i \(0.561321\pi\)
\(158\) −4.80241e22 −0.394058
\(159\) −2.57113e22 −0.197449
\(160\) 0 0
\(161\) −5.66388e22 −0.381454
\(162\) 3.45764e22 0.218209
\(163\) −2.85661e23 −1.68998 −0.844989 0.534783i \(-0.820393\pi\)
−0.844989 + 0.534783i \(0.820393\pi\)
\(164\) −6.18336e23 −3.43055
\(165\) 0 0
\(166\) 4.85437e23 2.37136
\(167\) −2.66950e23 −1.22435 −0.612177 0.790720i \(-0.709706\pi\)
−0.612177 + 0.790720i \(0.709706\pi\)
\(168\) −2.37581e23 −1.02344
\(169\) −2.34216e23 −0.947996
\(170\) 0 0
\(171\) −1.01824e23 −0.364244
\(172\) 9.91884e23 3.33743
\(173\) 2.28496e23 0.723425 0.361713 0.932290i \(-0.382192\pi\)
0.361713 + 0.932290i \(0.382192\pi\)
\(174\) 4.03177e23 1.20151
\(175\) 0 0
\(176\) 2.76061e23 0.729661
\(177\) 3.26799e23 0.813881
\(178\) −8.89013e23 −2.08688
\(179\) −1.29151e22 −0.0285852 −0.0142926 0.999898i \(-0.504550\pi\)
−0.0142926 + 0.999898i \(0.504550\pi\)
\(180\) 0 0
\(181\) −8.75338e23 −1.72405 −0.862026 0.506863i \(-0.830805\pi\)
−0.862026 + 0.506863i \(0.830805\pi\)
\(182\) 1.17118e23 0.217708
\(183\) −4.23748e23 −0.743654
\(184\) 1.72653e24 2.86145
\(185\) 0 0
\(186\) 3.76145e23 0.556506
\(187\) 1.25249e23 0.175161
\(188\) 3.98936e23 0.527529
\(189\) −7.48011e22 −0.0935532
\(190\) 0 0
\(191\) 1.33961e24 1.50013 0.750066 0.661363i \(-0.230022\pi\)
0.750066 + 0.661363i \(0.230022\pi\)
\(192\) 2.79725e24 2.96529
\(193\) 2.13970e23 0.214783 0.107392 0.994217i \(-0.465750\pi\)
0.107392 + 0.994217i \(0.465750\pi\)
\(194\) −2.69900e24 −2.56616
\(195\) 0 0
\(196\) −2.55558e24 −2.18173
\(197\) −1.42895e24 −1.15644 −0.578220 0.815881i \(-0.696252\pi\)
−0.578220 + 0.815881i \(0.696252\pi\)
\(198\) 1.44600e23 0.110964
\(199\) 7.45856e23 0.542872 0.271436 0.962456i \(-0.412501\pi\)
0.271436 + 0.962456i \(0.412501\pi\)
\(200\) 0 0
\(201\) 9.30344e23 0.609656
\(202\) 4.11805e23 0.256155
\(203\) −8.72216e23 −0.515129
\(204\) 3.03870e24 1.70440
\(205\) 0 0
\(206\) −6.24619e24 −3.16234
\(207\) 5.43587e23 0.261565
\(208\) −2.14594e24 −0.981639
\(209\) −4.25832e23 −0.185226
\(210\) 0 0
\(211\) −2.65090e24 −1.04334 −0.521671 0.853147i \(-0.674691\pi\)
−0.521671 + 0.853147i \(0.674691\pi\)
\(212\) −2.60870e24 −0.977006
\(213\) 1.56231e24 0.556905
\(214\) 4.63819e23 0.157400
\(215\) 0 0
\(216\) 2.28017e24 0.701782
\(217\) −8.13736e23 −0.238592
\(218\) 6.39479e23 0.178662
\(219\) −7.95464e23 −0.211815
\(220\) 0 0
\(221\) −9.73614e23 −0.235650
\(222\) 5.16990e24 1.19337
\(223\) −3.97174e24 −0.874539 −0.437270 0.899330i \(-0.644054\pi\)
−0.437270 + 0.899330i \(0.644054\pi\)
\(224\) −1.11232e25 −2.33683
\(225\) 0 0
\(226\) 1.20619e25 2.30823
\(227\) 2.14690e24 0.392229 0.196114 0.980581i \(-0.437168\pi\)
0.196114 + 0.980581i \(0.437168\pi\)
\(228\) −1.03312e25 −1.80234
\(229\) 3.66024e24 0.609869 0.304934 0.952373i \(-0.401365\pi\)
0.304934 + 0.952373i \(0.401365\pi\)
\(230\) 0 0
\(231\) −3.12820e23 −0.0475739
\(232\) 2.65878e25 3.86420
\(233\) 2.90921e24 0.404145 0.202073 0.979371i \(-0.435232\pi\)
0.202073 + 0.979371i \(0.435232\pi\)
\(234\) −1.12403e24 −0.149284
\(235\) 0 0
\(236\) 3.31574e25 4.02721
\(237\) −9.97109e23 −0.115847
\(238\) −8.87487e24 −0.986516
\(239\) −4.80224e24 −0.510818 −0.255409 0.966833i \(-0.582210\pi\)
−0.255409 + 0.966833i \(0.582210\pi\)
\(240\) 0 0
\(241\) 7.86263e24 0.766282 0.383141 0.923690i \(-0.374842\pi\)
0.383141 + 0.923690i \(0.374842\pi\)
\(242\) −2.04416e25 −1.90745
\(243\) 7.17898e23 0.0641500
\(244\) −4.29940e25 −3.67972
\(245\) 0 0
\(246\) −1.73322e25 −1.36155
\(247\) 3.31018e24 0.249191
\(248\) 2.48052e25 1.78978
\(249\) 1.00790e25 0.697144
\(250\) 0 0
\(251\) −1.99525e25 −1.26889 −0.634443 0.772969i \(-0.718771\pi\)
−0.634443 + 0.772969i \(0.718771\pi\)
\(252\) −7.58941e24 −0.462916
\(253\) 2.27330e24 0.133012
\(254\) −4.72991e24 −0.265521
\(255\) 0 0
\(256\) 1.01203e26 5.23210
\(257\) −1.33580e25 −0.662894 −0.331447 0.943474i \(-0.607537\pi\)
−0.331447 + 0.943474i \(0.607537\pi\)
\(258\) 2.78029e25 1.32460
\(259\) −1.11843e25 −0.511638
\(260\) 0 0
\(261\) 8.37103e24 0.353227
\(262\) 1.82204e25 0.738576
\(263\) 5.83637e24 0.227304 0.113652 0.993521i \(-0.463745\pi\)
0.113652 + 0.993521i \(0.463745\pi\)
\(264\) 9.53573e24 0.356872
\(265\) 0 0
\(266\) 3.01735e25 1.04320
\(267\) −1.84583e25 −0.613512
\(268\) 9.43938e25 3.01667
\(269\) 5.35297e25 1.64511 0.822557 0.568683i \(-0.192547\pi\)
0.822557 + 0.568683i \(0.192547\pi\)
\(270\) 0 0
\(271\) −1.04403e25 −0.296849 −0.148425 0.988924i \(-0.547420\pi\)
−0.148425 + 0.988924i \(0.547420\pi\)
\(272\) 1.62613e26 4.44817
\(273\) 2.43168e24 0.0640029
\(274\) 5.64004e25 1.42857
\(275\) 0 0
\(276\) 5.51530e25 1.29427
\(277\) −3.14884e25 −0.711399 −0.355699 0.934600i \(-0.615757\pi\)
−0.355699 + 0.934600i \(0.615757\pi\)
\(278\) −1.47929e25 −0.321797
\(279\) 7.80977e24 0.163604
\(280\) 0 0
\(281\) −1.15887e25 −0.225225 −0.112613 0.993639i \(-0.535922\pi\)
−0.112613 + 0.993639i \(0.535922\pi\)
\(282\) 1.11823e25 0.209371
\(283\) 4.80399e25 0.866652 0.433326 0.901237i \(-0.357340\pi\)
0.433326 + 0.901237i \(0.357340\pi\)
\(284\) 1.58514e26 2.75565
\(285\) 0 0
\(286\) −4.70074e24 −0.0759142
\(287\) 3.74957e25 0.583743
\(288\) 1.06755e26 1.60238
\(289\) 4.68568e24 0.0678180
\(290\) 0 0
\(291\) −5.60384e25 −0.754413
\(292\) −8.07087e25 −1.04809
\(293\) 7.96714e25 0.998142 0.499071 0.866561i \(-0.333675\pi\)
0.499071 + 0.866561i \(0.333675\pi\)
\(294\) −7.16339e25 −0.865907
\(295\) 0 0
\(296\) 3.40933e26 3.83801
\(297\) 3.00227e24 0.0326217
\(298\) −2.12626e26 −2.23021
\(299\) −1.76713e25 −0.178946
\(300\) 0 0
\(301\) −6.01476e25 −0.567898
\(302\) 3.15809e26 2.87973
\(303\) 8.55018e24 0.0753056
\(304\) −5.52865e26 −4.70377
\(305\) 0 0
\(306\) 8.51760e25 0.676460
\(307\) −1.51498e26 −1.16266 −0.581332 0.813666i \(-0.697468\pi\)
−0.581332 + 0.813666i \(0.697468\pi\)
\(308\) −3.17391e25 −0.235403
\(309\) −1.29687e26 −0.929681
\(310\) 0 0
\(311\) 1.41406e26 0.947292 0.473646 0.880715i \(-0.342938\pi\)
0.473646 + 0.880715i \(0.342938\pi\)
\(312\) −7.41253e25 −0.480112
\(313\) −4.99598e25 −0.312899 −0.156450 0.987686i \(-0.550005\pi\)
−0.156450 + 0.987686i \(0.550005\pi\)
\(314\) −1.24158e26 −0.751995
\(315\) 0 0
\(316\) −1.01168e26 −0.573229
\(317\) −1.81535e26 −0.995033 −0.497517 0.867454i \(-0.665755\pi\)
−0.497517 + 0.867454i \(0.665755\pi\)
\(318\) −7.31229e25 −0.387765
\(319\) 3.50079e25 0.179624
\(320\) 0 0
\(321\) 9.63011e24 0.0462731
\(322\) −1.61081e26 −0.749130
\(323\) −2.50835e26 −1.12918
\(324\) 7.28388e25 0.317424
\(325\) 0 0
\(326\) −8.12420e26 −3.31891
\(327\) 1.32773e25 0.0525240
\(328\) −1.14299e27 −4.37890
\(329\) −2.41914e25 −0.0897644
\(330\) 0 0
\(331\) 1.44090e26 0.501695 0.250848 0.968027i \(-0.419291\pi\)
0.250848 + 0.968027i \(0.419291\pi\)
\(332\) 1.02262e27 3.44958
\(333\) 1.07341e26 0.350833
\(334\) −7.59206e26 −2.40448
\(335\) 0 0
\(336\) −4.06139e26 −1.20813
\(337\) 3.63051e26 1.04678 0.523388 0.852095i \(-0.324668\pi\)
0.523388 + 0.852095i \(0.324668\pi\)
\(338\) −6.66111e26 −1.86175
\(339\) 2.50438e26 0.678583
\(340\) 0 0
\(341\) 3.26607e25 0.0831964
\(342\) −2.89588e26 −0.715331
\(343\) 3.57891e26 0.857360
\(344\) 1.83349e27 4.26004
\(345\) 0 0
\(346\) 6.49841e26 1.42072
\(347\) 7.09622e25 0.150511 0.0752554 0.997164i \(-0.476023\pi\)
0.0752554 + 0.997164i \(0.476023\pi\)
\(348\) 8.49335e26 1.74782
\(349\) −7.03939e26 −1.40562 −0.702810 0.711378i \(-0.748072\pi\)
−0.702810 + 0.711378i \(0.748072\pi\)
\(350\) 0 0
\(351\) −2.33379e25 −0.0438872
\(352\) 4.46451e26 0.814846
\(353\) 1.08085e26 0.191483 0.0957414 0.995406i \(-0.469478\pi\)
0.0957414 + 0.995406i \(0.469478\pi\)
\(354\) 9.29416e26 1.59836
\(355\) 0 0
\(356\) −1.87280e27 −3.03575
\(357\) −1.84266e26 −0.290020
\(358\) −3.67305e25 −0.0561378
\(359\) −1.67492e26 −0.248601 −0.124301 0.992245i \(-0.539669\pi\)
−0.124301 + 0.992245i \(0.539669\pi\)
\(360\) 0 0
\(361\) 1.38602e26 0.194064
\(362\) −2.48946e27 −3.38583
\(363\) −4.24422e26 −0.560761
\(364\) 2.46722e26 0.316696
\(365\) 0 0
\(366\) −1.20514e27 −1.46045
\(367\) −9.83667e26 −1.15839 −0.579194 0.815190i \(-0.696633\pi\)
−0.579194 + 0.815190i \(0.696633\pi\)
\(368\) 2.95146e27 3.37780
\(369\) −3.59863e26 −0.400276
\(370\) 0 0
\(371\) 1.58191e26 0.166248
\(372\) 7.92389e26 0.809539
\(373\) −1.00058e26 −0.0993824 −0.0496912 0.998765i \(-0.515824\pi\)
−0.0496912 + 0.998765i \(0.515824\pi\)
\(374\) 3.56208e26 0.343995
\(375\) 0 0
\(376\) 7.37429e26 0.673360
\(377\) −2.72131e26 −0.241654
\(378\) −2.12734e26 −0.183727
\(379\) 9.23905e25 0.0776096 0.0388048 0.999247i \(-0.487645\pi\)
0.0388048 + 0.999247i \(0.487645\pi\)
\(380\) 0 0
\(381\) −9.82055e25 −0.0780591
\(382\) 3.80986e27 2.94608
\(383\) 2.13677e27 1.60757 0.803786 0.594919i \(-0.202816\pi\)
0.803786 + 0.594919i \(0.202816\pi\)
\(384\) 4.16393e27 3.04807
\(385\) 0 0
\(386\) 6.08530e26 0.421809
\(387\) 5.77263e26 0.389411
\(388\) −5.68572e27 −3.73295
\(389\) −1.25581e27 −0.802516 −0.401258 0.915965i \(-0.631427\pi\)
−0.401258 + 0.915965i \(0.631427\pi\)
\(390\) 0 0
\(391\) 1.33908e27 0.810868
\(392\) −4.72395e27 −2.78485
\(393\) 3.78305e26 0.217130
\(394\) −4.06394e27 −2.27111
\(395\) 0 0
\(396\) 3.04614e26 0.161417
\(397\) −1.78165e27 −0.919439 −0.459719 0.888064i \(-0.652050\pi\)
−0.459719 + 0.888064i \(0.652050\pi\)
\(398\) 2.12121e27 1.06614
\(399\) 6.26483e26 0.306686
\(400\) 0 0
\(401\) 1.40181e27 0.651141 0.325570 0.945518i \(-0.394444\pi\)
0.325570 + 0.945518i \(0.394444\pi\)
\(402\) 2.64590e27 1.19729
\(403\) −2.53885e26 −0.111927
\(404\) 8.67511e26 0.372624
\(405\) 0 0
\(406\) −2.48058e27 −1.01165
\(407\) 4.48903e26 0.178407
\(408\) 5.61699e27 2.17557
\(409\) 2.17493e27 0.821015 0.410507 0.911857i \(-0.365352\pi\)
0.410507 + 0.911857i \(0.365352\pi\)
\(410\) 0 0
\(411\) 1.17102e27 0.419978
\(412\) −1.31582e28 −4.60021
\(413\) −2.01066e27 −0.685271
\(414\) 1.54596e27 0.513683
\(415\) 0 0
\(416\) −3.47045e27 −1.09624
\(417\) −3.07139e26 −0.0946034
\(418\) −1.21107e27 −0.363762
\(419\) −6.07636e27 −1.77990 −0.889952 0.456055i \(-0.849262\pi\)
−0.889952 + 0.456055i \(0.849262\pi\)
\(420\) 0 0
\(421\) −1.89993e27 −0.529389 −0.264695 0.964332i \(-0.585271\pi\)
−0.264695 + 0.964332i \(0.585271\pi\)
\(422\) −7.53915e27 −2.04900
\(423\) 2.32175e26 0.0615520
\(424\) −4.82214e27 −1.24709
\(425\) 0 0
\(426\) 4.44321e27 1.09369
\(427\) 2.60714e27 0.626141
\(428\) 9.77083e26 0.228966
\(429\) −9.75999e25 −0.0223176
\(430\) 0 0
\(431\) −8.08572e24 −0.00176079 −0.000880395 1.00000i \(-0.500280\pi\)
−0.000880395 1.00000i \(0.500280\pi\)
\(432\) 3.89789e27 0.828420
\(433\) 5.60439e27 1.16253 0.581267 0.813713i \(-0.302557\pi\)
0.581267 + 0.813713i \(0.302557\pi\)
\(434\) −2.31426e27 −0.468566
\(435\) 0 0
\(436\) 1.34713e27 0.259897
\(437\) −4.55272e27 −0.857463
\(438\) −2.26230e27 −0.415979
\(439\) −8.51110e27 −1.52795 −0.763973 0.645248i \(-0.776754\pi\)
−0.763973 + 0.645248i \(0.776754\pi\)
\(440\) 0 0
\(441\) −1.48731e27 −0.254563
\(442\) −2.76896e27 −0.462789
\(443\) 6.63134e27 1.08234 0.541168 0.840915i \(-0.317982\pi\)
0.541168 + 0.840915i \(0.317982\pi\)
\(444\) 1.08909e28 1.73598
\(445\) 0 0
\(446\) −1.12956e28 −1.71749
\(447\) −4.41468e27 −0.655648
\(448\) −1.72103e28 −2.49671
\(449\) 1.30394e28 1.84787 0.923933 0.382555i \(-0.124956\pi\)
0.923933 + 0.382555i \(0.124956\pi\)
\(450\) 0 0
\(451\) −1.50496e27 −0.203549
\(452\) 2.54097e28 3.35773
\(453\) 6.55703e27 0.846596
\(454\) 6.10577e27 0.770289
\(455\) 0 0
\(456\) −1.90971e28 −2.30058
\(457\) −4.72949e27 −0.556793 −0.278397 0.960466i \(-0.589803\pi\)
−0.278397 + 0.960466i \(0.589803\pi\)
\(458\) 1.04097e28 1.19771
\(459\) 1.76848e27 0.198869
\(460\) 0 0
\(461\) 4.92722e27 0.529349 0.264675 0.964338i \(-0.414735\pi\)
0.264675 + 0.964338i \(0.414735\pi\)
\(462\) −8.89661e26 −0.0934294
\(463\) −1.20207e28 −1.23404 −0.617021 0.786947i \(-0.711661\pi\)
−0.617021 + 0.786947i \(0.711661\pi\)
\(464\) 4.54513e28 4.56150
\(465\) 0 0
\(466\) 8.27379e27 0.793693
\(467\) −1.09969e28 −1.03144 −0.515719 0.856758i \(-0.672475\pi\)
−0.515719 + 0.856758i \(0.672475\pi\)
\(468\) −2.36789e27 −0.217161
\(469\) −5.72402e27 −0.513317
\(470\) 0 0
\(471\) −2.57786e27 −0.221075
\(472\) 6.12910e28 5.14051
\(473\) 2.41413e27 0.198024
\(474\) −2.83578e27 −0.227510
\(475\) 0 0
\(476\) −1.86958e28 −1.43507
\(477\) −1.51823e27 −0.113997
\(478\) −1.36576e28 −1.00318
\(479\) −1.32717e28 −0.953684 −0.476842 0.878989i \(-0.658219\pi\)
−0.476842 + 0.878989i \(0.658219\pi\)
\(480\) 0 0
\(481\) −3.48951e27 −0.240017
\(482\) 2.23613e28 1.50489
\(483\) −3.34446e27 −0.220233
\(484\) −4.30624e28 −2.77473
\(485\) 0 0
\(486\) 2.04170e27 0.125983
\(487\) −2.62576e28 −1.58563 −0.792813 0.609466i \(-0.791384\pi\)
−0.792813 + 0.609466i \(0.791384\pi\)
\(488\) −7.94738e28 −4.69695
\(489\) −1.68680e28 −0.975710
\(490\) 0 0
\(491\) −2.54066e28 −1.40796 −0.703982 0.710218i \(-0.748596\pi\)
−0.703982 + 0.710218i \(0.748596\pi\)
\(492\) −3.65121e28 −1.98063
\(493\) 2.06213e28 1.09502
\(494\) 9.41414e27 0.489382
\(495\) 0 0
\(496\) 4.24039e28 2.11275
\(497\) −9.61224e27 −0.468903
\(498\) 2.86645e28 1.36911
\(499\) −1.30048e28 −0.608204 −0.304102 0.952640i \(-0.598356\pi\)
−0.304102 + 0.952640i \(0.598356\pi\)
\(500\) 0 0
\(501\) −1.57631e28 −0.706882
\(502\) −5.67449e28 −2.49194
\(503\) −1.34993e27 −0.0580559 −0.0290280 0.999579i \(-0.509241\pi\)
−0.0290280 + 0.999579i \(0.509241\pi\)
\(504\) −1.40289e28 −0.590886
\(505\) 0 0
\(506\) 6.46525e27 0.261219
\(507\) −1.38302e28 −0.547326
\(508\) −9.96406e27 −0.386249
\(509\) −4.04902e28 −1.53749 −0.768746 0.639554i \(-0.779119\pi\)
−0.768746 + 0.639554i \(0.779119\pi\)
\(510\) 0 0
\(511\) 4.89416e27 0.178344
\(512\) 1.39939e29 4.99579
\(513\) −6.01263e27 −0.210296
\(514\) −3.79902e28 −1.30184
\(515\) 0 0
\(516\) 5.85698e28 1.92687
\(517\) 9.70964e26 0.0313006
\(518\) −3.18083e28 −1.00479
\(519\) 1.34924e28 0.417670
\(520\) 0 0
\(521\) 5.40378e28 1.60658 0.803289 0.595590i \(-0.203082\pi\)
0.803289 + 0.595590i \(0.203082\pi\)
\(522\) 2.38072e28 0.693695
\(523\) −1.54066e28 −0.439988 −0.219994 0.975501i \(-0.570604\pi\)
−0.219994 + 0.975501i \(0.570604\pi\)
\(524\) 3.83833e28 1.07439
\(525\) 0 0
\(526\) 1.65986e28 0.446398
\(527\) 1.92387e28 0.507182
\(528\) 1.63011e28 0.421270
\(529\) −1.51670e28 −0.384252
\(530\) 0 0
\(531\) 1.92971e28 0.469895
\(532\) 6.35637e28 1.51753
\(533\) 1.16987e28 0.273842
\(534\) −5.24953e28 −1.20486
\(535\) 0 0
\(536\) 1.74486e29 3.85061
\(537\) −7.62623e26 −0.0165037
\(538\) 1.52238e29 3.23080
\(539\) −6.21997e27 −0.129451
\(540\) 0 0
\(541\) −7.54478e28 −1.51034 −0.755171 0.655528i \(-0.772446\pi\)
−0.755171 + 0.655528i \(0.772446\pi\)
\(542\) −2.96923e28 −0.582976
\(543\) −5.16878e28 −0.995382
\(544\) 2.62981e29 4.96747
\(545\) 0 0
\(546\) 6.91571e27 0.125694
\(547\) −7.90524e28 −1.40944 −0.704722 0.709483i \(-0.748928\pi\)
−0.704722 + 0.709483i \(0.748928\pi\)
\(548\) 1.18813e29 2.07812
\(549\) −2.50219e28 −0.429349
\(550\) 0 0
\(551\) −7.01101e28 −1.15795
\(552\) 1.01950e29 1.65206
\(553\) 6.13480e27 0.0975408
\(554\) −8.95531e28 −1.39710
\(555\) 0 0
\(556\) −3.11627e28 −0.468113
\(557\) −1.17729e29 −1.73541 −0.867707 0.497075i \(-0.834407\pi\)
−0.867707 + 0.497075i \(0.834407\pi\)
\(558\) 2.22110e28 0.321299
\(559\) −1.87660e28 −0.266409
\(560\) 0 0
\(561\) 7.39583e27 0.101129
\(562\) −3.29582e28 −0.442315
\(563\) 9.20807e28 1.21291 0.606457 0.795116i \(-0.292590\pi\)
0.606457 + 0.795116i \(0.292590\pi\)
\(564\) 2.35568e28 0.304569
\(565\) 0 0
\(566\) 1.36626e29 1.70200
\(567\) −4.41693e27 −0.0540130
\(568\) 2.93011e29 3.51744
\(569\) −2.71795e27 −0.0320304 −0.0160152 0.999872i \(-0.505098\pi\)
−0.0160152 + 0.999872i \(0.505098\pi\)
\(570\) 0 0
\(571\) 1.28086e28 0.145487 0.0727434 0.997351i \(-0.476825\pi\)
0.0727434 + 0.997351i \(0.476825\pi\)
\(572\) −9.90260e27 −0.110431
\(573\) 7.91029e28 0.866102
\(574\) 1.06638e29 1.14640
\(575\) 0 0
\(576\) 1.65175e29 1.71201
\(577\) −1.49329e29 −1.51984 −0.759922 0.650015i \(-0.774763\pi\)
−0.759922 + 0.650015i \(0.774763\pi\)
\(578\) 1.33261e28 0.133186
\(579\) 1.26347e28 0.124005
\(580\) 0 0
\(581\) −6.20116e28 −0.586980
\(582\) −1.59373e29 −1.48158
\(583\) −6.34926e27 −0.0579700
\(584\) −1.49189e29 −1.33783
\(585\) 0 0
\(586\) 2.26586e29 1.96023
\(587\) −2.62975e28 −0.223468 −0.111734 0.993738i \(-0.535640\pi\)
−0.111734 + 0.993738i \(0.535640\pi\)
\(588\) −1.50904e29 −1.25962
\(589\) −6.54093e28 −0.536326
\(590\) 0 0
\(591\) −8.43783e28 −0.667671
\(592\) 5.82817e29 4.53059
\(593\) −1.92294e29 −1.46856 −0.734278 0.678849i \(-0.762479\pi\)
−0.734278 + 0.678849i \(0.762479\pi\)
\(594\) 8.53846e27 0.0640651
\(595\) 0 0
\(596\) −4.47919e29 −3.24425
\(597\) 4.40421e28 0.313428
\(598\) −5.02572e28 −0.351427
\(599\) −2.45874e29 −1.68940 −0.844699 0.535242i \(-0.820220\pi\)
−0.844699 + 0.535242i \(0.820220\pi\)
\(600\) 0 0
\(601\) 7.47252e28 0.495776 0.247888 0.968789i \(-0.420264\pi\)
0.247888 + 0.968789i \(0.420264\pi\)
\(602\) −1.71060e29 −1.11528
\(603\) 5.49359e28 0.351985
\(604\) 6.65285e29 4.18909
\(605\) 0 0
\(606\) 2.43167e28 0.147891
\(607\) 1.23466e29 0.738016 0.369008 0.929426i \(-0.379698\pi\)
0.369008 + 0.929426i \(0.379698\pi\)
\(608\) −8.94104e29 −5.25291
\(609\) −5.15035e28 −0.297410
\(610\) 0 0
\(611\) −7.54771e27 −0.0421098
\(612\) 1.79432e29 0.984034
\(613\) 1.59244e29 0.858476 0.429238 0.903191i \(-0.358782\pi\)
0.429238 + 0.903191i \(0.358782\pi\)
\(614\) −4.30861e29 −2.28333
\(615\) 0 0
\(616\) −5.86694e28 −0.300479
\(617\) −1.23256e29 −0.620602 −0.310301 0.950638i \(-0.600430\pi\)
−0.310301 + 0.950638i \(0.600430\pi\)
\(618\) −3.68831e29 −1.82578
\(619\) −5.18990e28 −0.252585 −0.126292 0.991993i \(-0.540308\pi\)
−0.126292 + 0.991993i \(0.540308\pi\)
\(620\) 0 0
\(621\) 3.20983e28 0.151015
\(622\) 4.02159e29 1.86037
\(623\) 1.13566e29 0.516564
\(624\) −1.26715e29 −0.566750
\(625\) 0 0
\(626\) −1.42086e29 −0.614497
\(627\) −2.51450e28 −0.106940
\(628\) −2.61553e29 −1.09391
\(629\) 2.64425e29 1.08760
\(630\) 0 0
\(631\) −2.05208e28 −0.0816366 −0.0408183 0.999167i \(-0.512996\pi\)
−0.0408183 + 0.999167i \(0.512996\pi\)
\(632\) −1.87008e29 −0.731694
\(633\) −1.56533e29 −0.602374
\(634\) −5.16285e29 −1.95412
\(635\) 0 0
\(636\) −1.54041e29 −0.564075
\(637\) 4.83505e28 0.174155
\(638\) 9.95625e28 0.352759
\(639\) 9.22528e28 0.321529
\(640\) 0 0
\(641\) 5.14342e29 1.73477 0.867386 0.497636i \(-0.165798\pi\)
0.867386 + 0.497636i \(0.165798\pi\)
\(642\) 2.73880e28 0.0908747
\(643\) 8.85766e28 0.289137 0.144569 0.989495i \(-0.453821\pi\)
0.144569 + 0.989495i \(0.453821\pi\)
\(644\) −3.39333e29 −1.08975
\(645\) 0 0
\(646\) −7.13376e29 −2.21757
\(647\) −5.46916e29 −1.67273 −0.836364 0.548174i \(-0.815323\pi\)
−0.836364 + 0.548174i \(0.815323\pi\)
\(648\) 1.34642e29 0.405174
\(649\) 8.07012e28 0.238952
\(650\) 0 0
\(651\) −4.80503e28 −0.137751
\(652\) −1.71145e30 −4.82796
\(653\) 5.66153e29 1.57161 0.785806 0.618473i \(-0.212248\pi\)
0.785806 + 0.618473i \(0.212248\pi\)
\(654\) 3.77606e28 0.103151
\(655\) 0 0
\(656\) −1.95391e30 −5.16908
\(657\) −4.69713e28 −0.122292
\(658\) −6.88003e28 −0.176286
\(659\) −1.48653e29 −0.374867 −0.187434 0.982277i \(-0.560017\pi\)
−0.187434 + 0.982277i \(0.560017\pi\)
\(660\) 0 0
\(661\) −4.04669e29 −0.988517 −0.494259 0.869315i \(-0.664560\pi\)
−0.494259 + 0.869315i \(0.664560\pi\)
\(662\) 4.09792e29 0.985268
\(663\) −5.74909e28 −0.136053
\(664\) 1.89031e30 4.40319
\(665\) 0 0
\(666\) 3.05278e29 0.688994
\(667\) 3.74281e29 0.831527
\(668\) −1.59935e30 −3.49776
\(669\) −2.34527e29 −0.504916
\(670\) 0 0
\(671\) −1.04642e29 −0.218334
\(672\) −6.56816e29 −1.34917
\(673\) 1.54590e29 0.312625 0.156313 0.987708i \(-0.450039\pi\)
0.156313 + 0.987708i \(0.450039\pi\)
\(674\) 1.03252e30 2.05574
\(675\) 0 0
\(676\) −1.40323e30 −2.70825
\(677\) 9.88421e29 1.87828 0.939142 0.343530i \(-0.111623\pi\)
0.939142 + 0.343530i \(0.111623\pi\)
\(678\) 7.12245e29 1.33265
\(679\) 3.44781e29 0.635200
\(680\) 0 0
\(681\) 1.26772e29 0.226453
\(682\) 9.28870e28 0.163388
\(683\) 1.41169e29 0.244524 0.122262 0.992498i \(-0.460985\pi\)
0.122262 + 0.992498i \(0.460985\pi\)
\(684\) −6.10048e29 −1.04058
\(685\) 0 0
\(686\) 1.01784e30 1.68375
\(687\) 2.16133e29 0.352108
\(688\) 3.13430e30 5.02877
\(689\) 4.93555e28 0.0779891
\(690\) 0 0
\(691\) 7.11585e29 1.09070 0.545352 0.838207i \(-0.316396\pi\)
0.545352 + 0.838207i \(0.316396\pi\)
\(692\) 1.36896e30 2.06669
\(693\) −1.84717e28 −0.0274668
\(694\) 2.01817e29 0.295585
\(695\) 0 0
\(696\) 1.56999e30 2.23099
\(697\) −8.86490e29 −1.24088
\(698\) −2.00200e30 −2.76047
\(699\) 1.71786e29 0.233334
\(700\) 0 0
\(701\) −8.80754e29 −1.16096 −0.580478 0.814276i \(-0.697134\pi\)
−0.580478 + 0.814276i \(0.697134\pi\)
\(702\) −6.63731e28 −0.0861891
\(703\) −8.99015e29 −1.15010
\(704\) 6.90765e29 0.870597
\(705\) 0 0
\(706\) 3.07393e29 0.376049
\(707\) −5.26057e28 −0.0634058
\(708\) 1.95791e30 2.32511
\(709\) −2.79000e29 −0.326452 −0.163226 0.986589i \(-0.552190\pi\)
−0.163226 + 0.986589i \(0.552190\pi\)
\(710\) 0 0
\(711\) −5.88783e28 −0.0668843
\(712\) −3.46185e30 −3.87496
\(713\) 3.49186e29 0.385139
\(714\) −5.24052e29 −0.569565
\(715\) 0 0
\(716\) −7.73767e28 −0.0816626
\(717\) −2.83568e29 −0.294921
\(718\) −4.76349e29 −0.488223
\(719\) 1.21404e30 1.22625 0.613124 0.789987i \(-0.289913\pi\)
0.613124 + 0.789987i \(0.289913\pi\)
\(720\) 0 0
\(721\) 7.97913e29 0.782772
\(722\) 3.94185e29 0.381118
\(723\) 4.64280e29 0.442413
\(724\) −5.24431e30 −4.92531
\(725\) 0 0
\(726\) −1.20706e30 −1.10127
\(727\) 6.54831e29 0.588868 0.294434 0.955672i \(-0.404869\pi\)
0.294434 + 0.955672i \(0.404869\pi\)
\(728\) 4.56062e29 0.404245
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) 1.42204e30 1.20720
\(732\) −2.53875e30 −2.12449
\(733\) −2.20665e29 −0.182029 −0.0910147 0.995850i \(-0.529011\pi\)
−0.0910147 + 0.995850i \(0.529011\pi\)
\(734\) −2.79755e30 −2.27493
\(735\) 0 0
\(736\) 4.77315e30 3.77214
\(737\) 2.29743e29 0.178992
\(738\) −1.02345e30 −0.786094
\(739\) −4.07297e29 −0.308422 −0.154211 0.988038i \(-0.549283\pi\)
−0.154211 + 0.988038i \(0.549283\pi\)
\(740\) 0 0
\(741\) 1.95463e29 0.143871
\(742\) 4.49895e29 0.326490
\(743\) −3.97218e29 −0.284214 −0.142107 0.989851i \(-0.545388\pi\)
−0.142107 + 0.989851i \(0.545388\pi\)
\(744\) 1.46472e30 1.03333
\(745\) 0 0
\(746\) −2.84565e29 −0.195175
\(747\) 5.95152e29 0.402496
\(748\) 7.50390e29 0.500403
\(749\) −5.92500e28 −0.0389610
\(750\) 0 0
\(751\) −1.86890e30 −1.19500 −0.597498 0.801871i \(-0.703838\pi\)
−0.597498 + 0.801871i \(0.703838\pi\)
\(752\) 1.26062e30 0.794869
\(753\) −1.17817e30 −0.732592
\(754\) −7.73941e29 −0.474579
\(755\) 0 0
\(756\) −4.48147e29 −0.267265
\(757\) 2.99103e30 1.75919 0.879597 0.475720i \(-0.157812\pi\)
0.879597 + 0.475720i \(0.157812\pi\)
\(758\) 2.62759e29 0.152416
\(759\) 1.34236e29 0.0767945
\(760\) 0 0
\(761\) 1.51341e30 0.842206 0.421103 0.907013i \(-0.361643\pi\)
0.421103 + 0.907013i \(0.361643\pi\)
\(762\) −2.79297e29 −0.153299
\(763\) −8.16896e28 −0.0442241
\(764\) 8.02588e30 4.28561
\(765\) 0 0
\(766\) 6.07697e30 3.15707
\(767\) −6.27325e29 −0.321470
\(768\) 5.97596e30 3.02075
\(769\) 2.53401e30 1.26352 0.631759 0.775165i \(-0.282333\pi\)
0.631759 + 0.775165i \(0.282333\pi\)
\(770\) 0 0
\(771\) −7.88777e29 −0.382722
\(772\) 1.28193e30 0.613598
\(773\) 1.69545e30 0.800571 0.400285 0.916390i \(-0.368911\pi\)
0.400285 + 0.916390i \(0.368911\pi\)
\(774\) 1.64173e30 0.764756
\(775\) 0 0
\(776\) −1.05100e31 −4.76490
\(777\) −6.60424e29 −0.295394
\(778\) −3.57153e30 −1.57604
\(779\) 3.01396e30 1.31218
\(780\) 0 0
\(781\) 3.85804e29 0.163505
\(782\) 3.80834e30 1.59245
\(783\) 4.94301e29 0.203936
\(784\) −8.07548e30 −3.28738
\(785\) 0 0
\(786\) 1.07590e30 0.426417
\(787\) 1.87332e30 0.732616 0.366308 0.930494i \(-0.380622\pi\)
0.366308 + 0.930494i \(0.380622\pi\)
\(788\) −8.56113e30 −3.30374
\(789\) 3.44632e29 0.131234
\(790\) 0 0
\(791\) −1.54084e30 −0.571353
\(792\) 5.63075e29 0.206040
\(793\) 8.13429e29 0.293732
\(794\) −5.06702e30 −1.80567
\(795\) 0 0
\(796\) 4.46856e30 1.55089
\(797\) 7.79664e29 0.267051 0.133526 0.991045i \(-0.457370\pi\)
0.133526 + 0.991045i \(0.457370\pi\)
\(798\) 1.78172e30 0.602294
\(799\) 5.71944e29 0.190815
\(800\) 0 0
\(801\) −1.08994e30 −0.354211
\(802\) 3.98676e30 1.27876
\(803\) −1.96436e29 −0.0621880
\(804\) 5.57386e30 1.74168
\(805\) 0 0
\(806\) −7.22050e29 −0.219811
\(807\) 3.16088e30 0.949807
\(808\) 1.60358e30 0.475633
\(809\) 8.82262e29 0.258308 0.129154 0.991625i \(-0.458774\pi\)
0.129154 + 0.991625i \(0.458774\pi\)
\(810\) 0 0
\(811\) −2.06044e30 −0.587815 −0.293907 0.955834i \(-0.594956\pi\)
−0.293907 + 0.955834i \(0.594956\pi\)
\(812\) −5.22561e30 −1.47163
\(813\) −6.16490e29 −0.171386
\(814\) 1.27668e30 0.350369
\(815\) 0 0
\(816\) 9.60212e30 2.56815
\(817\) −4.83476e30 −1.27657
\(818\) 6.18551e30 1.61237
\(819\) 1.43589e29 0.0369521
\(820\) 0 0
\(821\) −1.83846e30 −0.461160 −0.230580 0.973053i \(-0.574062\pi\)
−0.230580 + 0.973053i \(0.574062\pi\)
\(822\) 3.33039e30 0.824785
\(823\) −7.73766e29 −0.189196 −0.0945979 0.995516i \(-0.530157\pi\)
−0.0945979 + 0.995516i \(0.530157\pi\)
\(824\) −2.43229e31 −5.87190
\(825\) 0 0
\(826\) −5.71831e30 −1.34579
\(827\) 4.59989e30 1.06891 0.534453 0.845198i \(-0.320518\pi\)
0.534453 + 0.845198i \(0.320518\pi\)
\(828\) 3.25673e30 0.747245
\(829\) 7.93000e30 1.79660 0.898298 0.439386i \(-0.144804\pi\)
0.898298 + 0.439386i \(0.144804\pi\)
\(830\) 0 0
\(831\) −1.85936e30 −0.410726
\(832\) −5.36961e30 −1.17124
\(833\) −3.66386e30 −0.789162
\(834\) −8.73504e29 −0.185790
\(835\) 0 0
\(836\) −2.55124e30 −0.529158
\(837\) 4.61159e29 0.0944569
\(838\) −1.72812e31 −3.49551
\(839\) 4.84033e30 0.966884 0.483442 0.875376i \(-0.339386\pi\)
0.483442 + 0.875376i \(0.339386\pi\)
\(840\) 0 0
\(841\) 6.30944e29 0.122923
\(842\) −5.40340e30 −1.03966
\(843\) −6.84300e29 −0.130034
\(844\) −1.58820e31 −2.98064
\(845\) 0 0
\(846\) 6.60306e29 0.120881
\(847\) 2.61129e30 0.472149
\(848\) −8.24334e30 −1.47213
\(849\) 2.83671e30 0.500362
\(850\) 0 0
\(851\) 4.79937e30 0.825893
\(852\) 9.36009e30 1.59098
\(853\) −2.96903e30 −0.498483 −0.249242 0.968441i \(-0.580181\pi\)
−0.249242 + 0.968441i \(0.580181\pi\)
\(854\) 7.41472e30 1.22967
\(855\) 0 0
\(856\) 1.80612e30 0.292263
\(857\) 3.70009e30 0.591444 0.295722 0.955274i \(-0.404440\pi\)
0.295722 + 0.955274i \(0.404440\pi\)
\(858\) −2.77574e29 −0.0438291
\(859\) 7.75385e29 0.120945 0.0604726 0.998170i \(-0.480739\pi\)
0.0604726 + 0.998170i \(0.480739\pi\)
\(860\) 0 0
\(861\) 2.21408e30 0.337024
\(862\) −2.29958e28 −0.00345798
\(863\) 1.29544e31 1.92443 0.962217 0.272283i \(-0.0877786\pi\)
0.962217 + 0.272283i \(0.0877786\pi\)
\(864\) 6.30375e30 0.925134
\(865\) 0 0
\(866\) 1.59389e31 2.28307
\(867\) 2.76685e29 0.0391548
\(868\) −4.87524e30 −0.681615
\(869\) −2.46231e29 −0.0340122
\(870\) 0 0
\(871\) −1.78589e30 −0.240805
\(872\) 2.49015e30 0.331744
\(873\) −3.30901e30 −0.435560
\(874\) −1.29479e31 −1.68395
\(875\) 0 0
\(876\) −4.76577e30 −0.605118
\(877\) 1.57355e31 1.97417 0.987083 0.160207i \(-0.0512162\pi\)
0.987083 + 0.160207i \(0.0512162\pi\)
\(878\) −2.42056e31 −3.00070
\(879\) 4.70452e30 0.576278
\(880\) 0 0
\(881\) −1.47526e31 −1.76450 −0.882252 0.470778i \(-0.843973\pi\)
−0.882252 + 0.470778i \(0.843973\pi\)
\(882\) −4.22991e30 −0.499932
\(883\) 5.64453e30 0.659235 0.329617 0.944115i \(-0.393080\pi\)
0.329617 + 0.944115i \(0.393080\pi\)
\(884\) −5.83310e30 −0.673210
\(885\) 0 0
\(886\) 1.88595e31 2.12558
\(887\) −5.89300e30 −0.656354 −0.328177 0.944616i \(-0.606434\pi\)
−0.328177 + 0.944616i \(0.606434\pi\)
\(888\) 2.01318e31 2.21588
\(889\) 6.04218e29 0.0657242
\(890\) 0 0
\(891\) 1.77281e29 0.0188342
\(892\) −2.37954e31 −2.49840
\(893\) −1.94454e30 −0.201780
\(894\) −1.25554e31 −1.28761
\(895\) 0 0
\(896\) −2.56189e31 −2.56641
\(897\) −1.04347e30 −0.103314
\(898\) 3.70840e31 3.62898
\(899\) 5.37734e30 0.520104
\(900\) 0 0
\(901\) −3.74002e30 −0.353397
\(902\) −4.28009e30 −0.399746
\(903\) −3.55166e30 −0.327876
\(904\) 4.69695e31 4.28596
\(905\) 0 0
\(906\) 1.86482e31 1.66261
\(907\) −8.32782e30 −0.733931 −0.366965 0.930235i \(-0.619603\pi\)
−0.366965 + 0.930235i \(0.619603\pi\)
\(908\) 1.28624e31 1.12053
\(909\) 5.04879e29 0.0434777
\(910\) 0 0
\(911\) 1.08086e31 0.909548 0.454774 0.890607i \(-0.349720\pi\)
0.454774 + 0.890607i \(0.349720\pi\)
\(912\) −3.26461e31 −2.71572
\(913\) 2.48894e30 0.204678
\(914\) −1.34507e31 −1.09347
\(915\) 0 0
\(916\) 2.19291e31 1.74229
\(917\) −2.32755e30 −0.182819
\(918\) 5.02955e30 0.390554
\(919\) 6.55205e30 0.502996 0.251498 0.967858i \(-0.419077\pi\)
0.251498 + 0.967858i \(0.419077\pi\)
\(920\) 0 0
\(921\) −8.94582e30 −0.671265
\(922\) 1.40130e31 1.03958
\(923\) −2.99902e30 −0.219969
\(924\) −1.87416e30 −0.135910
\(925\) 0 0
\(926\) −3.41869e31 −2.42351
\(927\) −7.65791e30 −0.536752
\(928\) 7.35047e31 5.09403
\(929\) 1.04036e31 0.712883 0.356442 0.934318i \(-0.383990\pi\)
0.356442 + 0.934318i \(0.383990\pi\)
\(930\) 0 0
\(931\) 1.24567e31 0.834509
\(932\) 1.74296e31 1.15457
\(933\) 8.34989e30 0.546919
\(934\) −3.12752e31 −2.02562
\(935\) 0 0
\(936\) −4.37702e30 −0.277193
\(937\) 2.09831e31 1.31402 0.657012 0.753880i \(-0.271820\pi\)
0.657012 + 0.753880i \(0.271820\pi\)
\(938\) −1.62791e31 −1.00809
\(939\) −2.95007e30 −0.180653
\(940\) 0 0
\(941\) 2.20967e31 1.32323 0.661617 0.749842i \(-0.269870\pi\)
0.661617 + 0.749842i \(0.269870\pi\)
\(942\) −7.33143e30 −0.434165
\(943\) −1.60900e31 −0.942286
\(944\) 1.04776e32 6.06812
\(945\) 0 0
\(946\) 6.86578e30 0.388896
\(947\) 2.59604e31 1.45424 0.727121 0.686509i \(-0.240858\pi\)
0.727121 + 0.686509i \(0.240858\pi\)
\(948\) −5.97386e30 −0.330954
\(949\) 1.52698e30 0.0836637
\(950\) 0 0
\(951\) −1.07194e31 −0.574483
\(952\) −3.45590e31 −1.83178
\(953\) 2.94729e31 1.54507 0.772535 0.634973i \(-0.218989\pi\)
0.772535 + 0.634973i \(0.218989\pi\)
\(954\) −4.31783e30 −0.223876
\(955\) 0 0
\(956\) −2.87711e31 −1.45931
\(957\) 2.06718e30 0.103706
\(958\) −3.77448e31 −1.87292
\(959\) −7.20482e30 −0.353612
\(960\) 0 0
\(961\) −1.58087e31 −0.759103
\(962\) −9.92417e30 −0.471364
\(963\) 5.68648e29 0.0267158
\(964\) 4.71064e31 2.18913
\(965\) 0 0
\(966\) −9.51166e30 −0.432510
\(967\) −6.98077e30 −0.313997 −0.156998 0.987599i \(-0.550182\pi\)
−0.156998 + 0.987599i \(0.550182\pi\)
\(968\) −7.96002e31 −3.54179
\(969\) −1.48116e31 −0.651931
\(970\) 0 0
\(971\) −1.30234e30 −0.0560946 −0.0280473 0.999607i \(-0.508929\pi\)
−0.0280473 + 0.999607i \(0.508929\pi\)
\(972\) 4.30106e30 0.183265
\(973\) 1.88970e30 0.0796541
\(974\) −7.46765e31 −3.11398
\(975\) 0 0
\(976\) −1.35859e32 −5.54452
\(977\) −3.06599e31 −1.23788 −0.618939 0.785439i \(-0.712437\pi\)
−0.618939 + 0.785439i \(0.712437\pi\)
\(978\) −4.79726e31 −1.91618
\(979\) −4.55817e30 −0.180124
\(980\) 0 0
\(981\) 7.84011e29 0.0303248
\(982\) −7.22564e31 −2.76507
\(983\) −8.31325e30 −0.314745 −0.157373 0.987539i \(-0.550302\pi\)
−0.157373 + 0.987539i \(0.550302\pi\)
\(984\) −6.74921e31 −2.52816
\(985\) 0 0
\(986\) 5.86470e31 2.15049
\(987\) −1.42848e30 −0.0518255
\(988\) 1.98319e31 0.711895
\(989\) 2.58102e31 0.916708
\(990\) 0 0
\(991\) −1.32568e31 −0.460962 −0.230481 0.973077i \(-0.574030\pi\)
−0.230481 + 0.973077i \(0.574030\pi\)
\(992\) 6.85764e31 2.35940
\(993\) 8.50837e30 0.289654
\(994\) −2.73372e31 −0.920868
\(995\) 0 0
\(996\) 6.03849e31 1.99161
\(997\) −3.42048e31 −1.11632 −0.558159 0.829734i \(-0.688492\pi\)
−0.558159 + 0.829734i \(0.688492\pi\)
\(998\) −3.69857e31 −1.19444
\(999\) 6.33837e30 0.202554
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 75.22.a.c.1.1 1
5.2 odd 4 75.22.b.a.49.2 2
5.3 odd 4 75.22.b.a.49.1 2
5.4 even 2 3.22.a.a.1.1 1
15.14 odd 2 9.22.a.d.1.1 1
20.19 odd 2 48.22.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.a.1.1 1 5.4 even 2
9.22.a.d.1.1 1 15.14 odd 2
48.22.a.e.1.1 1 20.19 odd 2
75.22.a.c.1.1 1 1.1 even 1 trivial
75.22.b.a.49.1 2 5.3 odd 4
75.22.b.a.49.2 2 5.2 odd 4