Properties

Label 75.22.b.a.49.1
Level $75$
Weight $22$
Character 75.49
Analytic conductor $209.608$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(209.608008215\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.22.b.a.49.2

$q$-expansion

\(f(q)\) \(=\) \(q-2844.00i q^{2} +59049.0i q^{3} -5.99118e6 q^{4} +1.67935e8 q^{6} +3.63304e8i q^{7} +1.10746e10i q^{8} -3.48678e9 q^{9} +O(q^{10})\) \(q-2844.00i q^{2} +59049.0i q^{3} -5.99118e6 q^{4} +1.67935e8 q^{6} +3.63304e8i q^{7} +1.10746e10i q^{8} -3.48678e9 q^{9} +1.45818e10 q^{11} -3.53773e11i q^{12} -1.13351e11i q^{13} +1.03324e12 q^{14} +1.89318e13 q^{16} -8.58939e12i q^{17} +9.91641e12i q^{18} +2.92029e13 q^{19} -2.14527e13 q^{21} -4.14707e13i q^{22} +1.55899e14i q^{23} -6.53946e14 q^{24} -3.22370e14 q^{26} -2.05891e14i q^{27} -2.17662e15i q^{28} -2.40079e15 q^{29} +2.23982e15 q^{31} -3.06169e16i q^{32} +8.61043e14i q^{33} -2.44282e16 q^{34} +2.08900e16 q^{36} -3.07851e16i q^{37} -8.30532e16i q^{38} +6.69325e15 q^{39} -1.03208e17 q^{41} +6.10116e16i q^{42} +1.65557e17i q^{43} -8.73624e16 q^{44} +4.43377e17 q^{46} -6.65872e16i q^{47} +1.11790e18i q^{48} +4.26556e17 q^{49} +5.07195e17 q^{51} +6.79105e17i q^{52} -4.35423e17i q^{53} -5.85554e17 q^{54} -4.02346e18 q^{56} +1.72440e18i q^{57} +6.82784e18i q^{58} -5.53437e18 q^{59} -7.17621e18 q^{61} -6.37005e18i q^{62} -1.26676e18i q^{63} -4.73716e19 q^{64} +2.44881e18 q^{66} -1.57554e19i q^{67} +5.14606e19i q^{68} -9.20569e18 q^{69} +2.64579e19 q^{71} -3.86148e19i q^{72} -1.34712e19i q^{73} -8.75527e19 q^{74} -1.74960e20 q^{76} +5.29764e18i q^{77} -1.90356e19i q^{78} +1.68861e19 q^{79} +1.21577e19 q^{81} +2.93522e20i q^{82} +1.70688e20i q^{83} +1.28527e20 q^{84} +4.70845e20 q^{86} -1.41764e20i q^{87} +1.61488e20i q^{88} +3.12592e20 q^{89} +4.11808e19 q^{91} -9.34021e20i q^{92} +1.32259e20i q^{93} -1.89374e20 q^{94} +1.80790e21 q^{96} +9.49015e20i q^{97} -1.21313e21i q^{98} -5.08437e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 11982368 q^{4} + 335870712 q^{6} - 6973568802 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 11982368 q^{4} + 335870712 q^{6} - 6973568802 q^{9} + 29163666312 q^{11} + 2066472696960 q^{14} + 37863631405568 q^{16} + 58405878547592 q^{19} - 42905466344160 q^{21} - 13\!\cdots\!84 q^{24}+ \cdots - 10\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2844.00i − 1.96388i −0.189196 0.981939i \(-0.560588\pi\)
0.189196 0.981939i \(-0.439412\pi\)
\(3\) 59049.0i 0.577350i
\(4\) −5.99118e6 −2.85682
\(5\) 0 0
\(6\) 1.67935e8 1.13385
\(7\) 3.63304e8i 0.486117i 0.970012 + 0.243058i \(0.0781507\pi\)
−0.970012 + 0.243058i \(0.921849\pi\)
\(8\) 1.10746e10i 3.64657i
\(9\) −3.48678e9 −0.333333
\(10\) 0 0
\(11\) 1.45818e10 0.169508 0.0847538 0.996402i \(-0.472990\pi\)
0.0847538 + 0.996402i \(0.472990\pi\)
\(12\) − 3.53773e11i − 1.64939i
\(13\) − 1.13351e11i − 0.228044i −0.993478 0.114022i \(-0.963627\pi\)
0.993478 0.114022i \(-0.0363735\pi\)
\(14\) 1.03324e12 0.954674
\(15\) 0 0
\(16\) 1.89318e13 4.30460
\(17\) − 8.58939e12i − 1.03335i −0.856181 0.516676i \(-0.827169\pi\)
0.856181 0.516676i \(-0.172831\pi\)
\(18\) 9.91641e12i 0.654626i
\(19\) 2.92029e13 1.09273 0.546366 0.837546i \(-0.316011\pi\)
0.546366 + 0.837546i \(0.316011\pi\)
\(20\) 0 0
\(21\) −2.14527e13 −0.280660
\(22\) − 4.14707e13i − 0.332892i
\(23\) 1.55899e14i 0.784696i 0.919817 + 0.392348i \(0.128337\pi\)
−0.919817 + 0.392348i \(0.871663\pi\)
\(24\) −6.53946e14 −2.10535
\(25\) 0 0
\(26\) −3.22370e14 −0.447851
\(27\) − 2.05891e14i − 0.192450i
\(28\) − 2.17662e15i − 1.38875i
\(29\) −2.40079e15 −1.05968 −0.529840 0.848097i \(-0.677748\pi\)
−0.529840 + 0.848097i \(0.677748\pi\)
\(30\) 0 0
\(31\) 2.23982e15 0.490812 0.245406 0.969420i \(-0.421079\pi\)
0.245406 + 0.969420i \(0.421079\pi\)
\(32\) − 3.06169e16i − 4.80714i
\(33\) 8.61043e14i 0.0978652i
\(34\) −2.44282e16 −2.02938
\(35\) 0 0
\(36\) 2.08900e16 0.952273
\(37\) − 3.07851e16i − 1.05250i −0.850330 0.526250i \(-0.823598\pi\)
0.850330 0.526250i \(-0.176402\pi\)
\(38\) − 8.30532e16i − 2.14599i
\(39\) 6.69325e15 0.131661
\(40\) 0 0
\(41\) −1.03208e17 −1.20083 −0.600414 0.799689i \(-0.704998\pi\)
−0.600414 + 0.799689i \(0.704998\pi\)
\(42\) 6.10116e16i 0.551182i
\(43\) 1.65557e17i 1.16823i 0.811670 + 0.584117i \(0.198559\pi\)
−0.811670 + 0.584117i \(0.801441\pi\)
\(44\) −8.73624e16 −0.484252
\(45\) 0 0
\(46\) 4.43377e17 1.54105
\(47\) − 6.65872e16i − 0.184656i −0.995729 0.0923280i \(-0.970569\pi\)
0.995729 0.0923280i \(-0.0294308\pi\)
\(48\) 1.11790e18i 2.48526i
\(49\) 4.26556e17 0.763690
\(50\) 0 0
\(51\) 5.07195e17 0.596607
\(52\) 6.79105e17i 0.651482i
\(53\) − 4.35423e17i − 0.341991i −0.985272 0.170995i \(-0.945302\pi\)
0.985272 0.170995i \(-0.0546983\pi\)
\(54\) −5.85554e17 −0.377949
\(55\) 0 0
\(56\) −4.02346e18 −1.77266
\(57\) 1.72440e18i 0.630889i
\(58\) 6.82784e18i 2.08108i
\(59\) −5.53437e18 −1.40968 −0.704842 0.709364i \(-0.748982\pi\)
−0.704842 + 0.709364i \(0.748982\pi\)
\(60\) 0 0
\(61\) −7.17621e18 −1.28805 −0.644023 0.765006i \(-0.722736\pi\)
−0.644023 + 0.765006i \(0.722736\pi\)
\(62\) − 6.37005e18i − 0.963896i
\(63\) − 1.26676e18i − 0.162039i
\(64\) −4.73716e19 −5.13604
\(65\) 0 0
\(66\) 2.44881e18 0.192195
\(67\) − 1.57554e19i − 1.05595i −0.849258 0.527977i \(-0.822951\pi\)
0.849258 0.527977i \(-0.177049\pi\)
\(68\) 5.14606e19i 2.95210i
\(69\) −9.20569e18 −0.453045
\(70\) 0 0
\(71\) 2.64579e19 0.964588 0.482294 0.876009i \(-0.339804\pi\)
0.482294 + 0.876009i \(0.339804\pi\)
\(72\) − 3.86148e19i − 1.21552i
\(73\) − 1.34712e19i − 0.366875i −0.983031 0.183437i \(-0.941278\pi\)
0.983031 0.183437i \(-0.0587225\pi\)
\(74\) −8.75527e19 −2.06698
\(75\) 0 0
\(76\) −1.74960e20 −3.12174
\(77\) 5.29764e18i 0.0824005i
\(78\) − 1.90356e19i − 0.258567i
\(79\) 1.68861e19 0.200653 0.100326 0.994955i \(-0.468011\pi\)
0.100326 + 0.994955i \(0.468011\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 2.93522e20i 2.35828i
\(83\) 1.70688e20i 1.20749i 0.797178 + 0.603744i \(0.206325\pi\)
−0.797178 + 0.603744i \(0.793675\pi\)
\(84\) 1.28527e20 0.801794
\(85\) 0 0
\(86\) 4.70845e20 2.29427
\(87\) − 1.41764e20i − 0.611807i
\(88\) 1.61488e20i 0.618121i
\(89\) 3.12592e20 1.06263 0.531317 0.847173i \(-0.321697\pi\)
0.531317 + 0.847173i \(0.321697\pi\)
\(90\) 0 0
\(91\) 4.11808e19 0.110856
\(92\) − 9.34021e20i − 2.24174i
\(93\) 1.32259e20i 0.283371i
\(94\) −1.89374e20 −0.362642
\(95\) 0 0
\(96\) 1.80790e21 2.77540
\(97\) 9.49015e20i 1.30668i 0.757064 + 0.653341i \(0.226633\pi\)
−0.757064 + 0.653341i \(0.773367\pi\)
\(98\) − 1.21313e21i − 1.49980i
\(99\) −5.08437e19 −0.0565025
\(100\) 0 0
\(101\) 1.44798e20 0.130433 0.0652166 0.997871i \(-0.479226\pi\)
0.0652166 + 0.997871i \(0.479226\pi\)
\(102\) − 1.44246e21i − 1.17166i
\(103\) − 2.19627e21i − 1.61025i −0.593102 0.805127i \(-0.702097\pi\)
0.593102 0.805127i \(-0.297903\pi\)
\(104\) 1.25532e21 0.831579
\(105\) 0 0
\(106\) −1.23834e21 −0.671629
\(107\) − 1.63087e20i − 0.0801473i −0.999197 0.0400737i \(-0.987241\pi\)
0.999197 0.0400737i \(-0.0127593\pi\)
\(108\) 1.23353e21i 0.549795i
\(109\) −2.24852e20 −0.0909743 −0.0454871 0.998965i \(-0.514484\pi\)
−0.0454871 + 0.998965i \(0.514484\pi\)
\(110\) 0 0
\(111\) 1.81783e21 0.607661
\(112\) 6.87800e21i 2.09254i
\(113\) 4.24118e21i 1.17534i 0.809101 + 0.587670i \(0.199955\pi\)
−0.809101 + 0.587670i \(0.800045\pi\)
\(114\) 4.90421e21 1.23899
\(115\) 0 0
\(116\) 1.43836e22 3.02732
\(117\) 3.95230e20i 0.0760148i
\(118\) 1.57397e22i 2.76845i
\(119\) 3.12056e21 0.502330
\(120\) 0 0
\(121\) −7.18762e21 −0.971267
\(122\) 2.04091e22i 2.52957i
\(123\) − 6.09430e21i − 0.693299i
\(124\) −1.34192e22 −1.40216
\(125\) 0 0
\(126\) −3.60267e21 −0.318225
\(127\) 1.66312e21i 0.135202i 0.997712 + 0.0676012i \(0.0215346\pi\)
−0.997712 + 0.0676012i \(0.978465\pi\)
\(128\) 7.05165e22i 5.27942i
\(129\) −9.77599e21 −0.674480
\(130\) 0 0
\(131\) 6.40663e21 0.376081 0.188040 0.982161i \(-0.439786\pi\)
0.188040 + 0.982161i \(0.439786\pi\)
\(132\) − 5.15867e21i − 0.279583i
\(133\) 1.06095e22i 0.531196i
\(134\) −4.48085e22 −2.07377
\(135\) 0 0
\(136\) 9.51243e22 3.76819
\(137\) − 1.98314e22i − 0.727423i −0.931512 0.363711i \(-0.881509\pi\)
0.931512 0.363711i \(-0.118491\pi\)
\(138\) 2.61810e22i 0.889725i
\(139\) 5.20143e21 0.163858 0.0819290 0.996638i \(-0.473892\pi\)
0.0819290 + 0.996638i \(0.473892\pi\)
\(140\) 0 0
\(141\) 3.93191e21 0.106611
\(142\) − 7.52461e22i − 1.89433i
\(143\) − 1.65286e21i − 0.0386552i
\(144\) −6.60112e22 −1.43487
\(145\) 0 0
\(146\) −3.83122e22 −0.720498
\(147\) 2.51877e22i 0.440917i
\(148\) 1.84439e23i 3.00680i
\(149\) 7.47631e22 1.13562 0.567808 0.823161i \(-0.307792\pi\)
0.567808 + 0.823161i \(0.307792\pi\)
\(150\) 0 0
\(151\) 1.11044e23 1.46635 0.733174 0.680042i \(-0.238038\pi\)
0.733174 + 0.680042i \(0.238038\pi\)
\(152\) 3.23412e23i 3.98472i
\(153\) 2.99493e22i 0.344451i
\(154\) 1.50665e22 0.161824
\(155\) 0 0
\(156\) −4.01005e22 −0.376133
\(157\) 4.36563e22i 0.382913i 0.981501 + 0.191457i \(0.0613211\pi\)
−0.981501 + 0.191457i \(0.938679\pi\)
\(158\) − 4.80241e22i − 0.394058i
\(159\) 2.57113e22 0.197449
\(160\) 0 0
\(161\) −5.66388e22 −0.381454
\(162\) − 3.45764e22i − 0.218209i
\(163\) − 2.85661e23i − 1.68998i −0.534783 0.844989i \(-0.679607\pi\)
0.534783 0.844989i \(-0.320393\pi\)
\(164\) 6.18336e23 3.43055
\(165\) 0 0
\(166\) 4.85437e23 2.37136
\(167\) 2.66950e23i 1.22435i 0.790720 + 0.612177i \(0.209706\pi\)
−0.790720 + 0.612177i \(0.790294\pi\)
\(168\) − 2.37581e23i − 1.02344i
\(169\) 2.34216e23 0.947996
\(170\) 0 0
\(171\) −1.01824e23 −0.364244
\(172\) − 9.91884e23i − 3.33743i
\(173\) 2.28496e23i 0.723425i 0.932290 + 0.361713i \(0.117808\pi\)
−0.932290 + 0.361713i \(0.882192\pi\)
\(174\) −4.03177e23 −1.20151
\(175\) 0 0
\(176\) 2.76061e23 0.729661
\(177\) − 3.26799e23i − 0.813881i
\(178\) − 8.89013e23i − 2.08688i
\(179\) 1.29151e22 0.0285852 0.0142926 0.999898i \(-0.495450\pi\)
0.0142926 + 0.999898i \(0.495450\pi\)
\(180\) 0 0
\(181\) −8.75338e23 −1.72405 −0.862026 0.506863i \(-0.830805\pi\)
−0.862026 + 0.506863i \(0.830805\pi\)
\(182\) − 1.17118e23i − 0.217708i
\(183\) − 4.23748e23i − 0.743654i
\(184\) −1.72653e24 −2.86145
\(185\) 0 0
\(186\) 3.76145e23 0.556506
\(187\) − 1.25249e23i − 0.175161i
\(188\) 3.98936e23i 0.527529i
\(189\) 7.48011e22 0.0935532
\(190\) 0 0
\(191\) 1.33961e24 1.50013 0.750066 0.661363i \(-0.230022\pi\)
0.750066 + 0.661363i \(0.230022\pi\)
\(192\) − 2.79725e24i − 2.96529i
\(193\) 2.13970e23i 0.214783i 0.994217 + 0.107392i \(0.0342499\pi\)
−0.994217 + 0.107392i \(0.965750\pi\)
\(194\) 2.69900e24 2.56616
\(195\) 0 0
\(196\) −2.55558e24 −2.18173
\(197\) 1.42895e24i 1.15644i 0.815881 + 0.578220i \(0.196252\pi\)
−0.815881 + 0.578220i \(0.803748\pi\)
\(198\) 1.44600e23i 0.110964i
\(199\) −7.45856e23 −0.542872 −0.271436 0.962456i \(-0.587499\pi\)
−0.271436 + 0.962456i \(0.587499\pi\)
\(200\) 0 0
\(201\) 9.30344e23 0.609656
\(202\) − 4.11805e23i − 0.256155i
\(203\) − 8.72216e23i − 0.515129i
\(204\) −3.03870e24 −1.70440
\(205\) 0 0
\(206\) −6.24619e24 −3.16234
\(207\) − 5.43587e23i − 0.261565i
\(208\) − 2.14594e24i − 0.981639i
\(209\) 4.25832e23 0.185226
\(210\) 0 0
\(211\) −2.65090e24 −1.04334 −0.521671 0.853147i \(-0.674691\pi\)
−0.521671 + 0.853147i \(0.674691\pi\)
\(212\) 2.60870e24i 0.977006i
\(213\) 1.56231e24i 0.556905i
\(214\) −4.63819e23 −0.157400
\(215\) 0 0
\(216\) 2.28017e24 0.701782
\(217\) 8.13736e23i 0.238592i
\(218\) 6.39479e23i 0.178662i
\(219\) 7.95464e23 0.211815
\(220\) 0 0
\(221\) −9.73614e23 −0.235650
\(222\) − 5.16990e24i − 1.19337i
\(223\) − 3.97174e24i − 0.874539i −0.899330 0.437270i \(-0.855946\pi\)
0.899330 0.437270i \(-0.144054\pi\)
\(224\) 1.11232e25 2.33683
\(225\) 0 0
\(226\) 1.20619e25 2.30823
\(227\) − 2.14690e24i − 0.392229i −0.980581 0.196114i \(-0.937168\pi\)
0.980581 0.196114i \(-0.0628324\pi\)
\(228\) − 1.03312e25i − 1.80234i
\(229\) −3.66024e24 −0.609869 −0.304934 0.952373i \(-0.598635\pi\)
−0.304934 + 0.952373i \(0.598635\pi\)
\(230\) 0 0
\(231\) −3.12820e23 −0.0475739
\(232\) − 2.65878e25i − 3.86420i
\(233\) 2.90921e24i 0.404145i 0.979371 + 0.202073i \(0.0647677\pi\)
−0.979371 + 0.202073i \(0.935232\pi\)
\(234\) 1.12403e24 0.149284
\(235\) 0 0
\(236\) 3.31574e25 4.02721
\(237\) 9.97109e23i 0.115847i
\(238\) − 8.87487e24i − 0.986516i
\(239\) 4.80224e24 0.510818 0.255409 0.966833i \(-0.417790\pi\)
0.255409 + 0.966833i \(0.417790\pi\)
\(240\) 0 0
\(241\) 7.86263e24 0.766282 0.383141 0.923690i \(-0.374842\pi\)
0.383141 + 0.923690i \(0.374842\pi\)
\(242\) 2.04416e25i 1.90745i
\(243\) 7.17898e23i 0.0641500i
\(244\) 4.29940e25 3.67972
\(245\) 0 0
\(246\) −1.73322e25 −1.36155
\(247\) − 3.31018e24i − 0.249191i
\(248\) 2.48052e25i 1.78978i
\(249\) −1.00790e25 −0.697144
\(250\) 0 0
\(251\) −1.99525e25 −1.26889 −0.634443 0.772969i \(-0.718771\pi\)
−0.634443 + 0.772969i \(0.718771\pi\)
\(252\) 7.58941e24i 0.462916i
\(253\) 2.27330e24i 0.133012i
\(254\) 4.72991e24 0.265521
\(255\) 0 0
\(256\) 1.01203e26 5.23210
\(257\) 1.33580e25i 0.662894i 0.943474 + 0.331447i \(0.107537\pi\)
−0.943474 + 0.331447i \(0.892463\pi\)
\(258\) 2.78029e25i 1.32460i
\(259\) 1.11843e25 0.511638
\(260\) 0 0
\(261\) 8.37103e24 0.353227
\(262\) − 1.82204e25i − 0.738576i
\(263\) 5.83637e24i 0.227304i 0.993521 + 0.113652i \(0.0362549\pi\)
−0.993521 + 0.113652i \(0.963745\pi\)
\(264\) −9.53573e24 −0.356872
\(265\) 0 0
\(266\) 3.01735e25 1.04320
\(267\) 1.84583e25i 0.613512i
\(268\) 9.43938e25i 3.01667i
\(269\) −5.35297e25 −1.64511 −0.822557 0.568683i \(-0.807453\pi\)
−0.822557 + 0.568683i \(0.807453\pi\)
\(270\) 0 0
\(271\) −1.04403e25 −0.296849 −0.148425 0.988924i \(-0.547420\pi\)
−0.148425 + 0.988924i \(0.547420\pi\)
\(272\) − 1.62613e26i − 4.44817i
\(273\) 2.43168e24i 0.0640029i
\(274\) −5.64004e25 −1.42857
\(275\) 0 0
\(276\) 5.51530e25 1.29427
\(277\) 3.14884e25i 0.711399i 0.934600 + 0.355699i \(0.115757\pi\)
−0.934600 + 0.355699i \(0.884243\pi\)
\(278\) − 1.47929e25i − 0.321797i
\(279\) −7.80977e24 −0.163604
\(280\) 0 0
\(281\) −1.15887e25 −0.225225 −0.112613 0.993639i \(-0.535922\pi\)
−0.112613 + 0.993639i \(0.535922\pi\)
\(282\) − 1.11823e25i − 0.209371i
\(283\) 4.80399e25i 0.866652i 0.901237 + 0.433326i \(0.142660\pi\)
−0.901237 + 0.433326i \(0.857340\pi\)
\(284\) −1.58514e26 −2.75565
\(285\) 0 0
\(286\) −4.70074e24 −0.0759142
\(287\) − 3.74957e25i − 0.583743i
\(288\) 1.06755e26i 1.60238i
\(289\) −4.68568e24 −0.0678180
\(290\) 0 0
\(291\) −5.60384e25 −0.754413
\(292\) 8.07087e25i 1.04809i
\(293\) 7.96714e25i 0.998142i 0.866561 + 0.499071i \(0.166325\pi\)
−0.866561 + 0.499071i \(0.833675\pi\)
\(294\) 7.16339e25 0.865907
\(295\) 0 0
\(296\) 3.40933e26 3.83801
\(297\) − 3.00227e24i − 0.0326217i
\(298\) − 2.12626e26i − 2.23021i
\(299\) 1.76713e25 0.178946
\(300\) 0 0
\(301\) −6.01476e25 −0.567898
\(302\) − 3.15809e26i − 2.87973i
\(303\) 8.55018e24i 0.0753056i
\(304\) 5.52865e26 4.70377
\(305\) 0 0
\(306\) 8.51760e25 0.676460
\(307\) 1.51498e26i 1.16266i 0.813666 + 0.581332i \(0.197468\pi\)
−0.813666 + 0.581332i \(0.802532\pi\)
\(308\) − 3.17391e25i − 0.235403i
\(309\) 1.29687e26 0.929681
\(310\) 0 0
\(311\) 1.41406e26 0.947292 0.473646 0.880715i \(-0.342938\pi\)
0.473646 + 0.880715i \(0.342938\pi\)
\(312\) 7.41253e25i 0.480112i
\(313\) − 4.99598e25i − 0.312899i −0.987686 0.156450i \(-0.949995\pi\)
0.987686 0.156450i \(-0.0500049\pi\)
\(314\) 1.24158e26 0.751995
\(315\) 0 0
\(316\) −1.01168e26 −0.573229
\(317\) 1.81535e26i 0.995033i 0.867454 + 0.497517i \(0.165755\pi\)
−0.867454 + 0.497517i \(0.834245\pi\)
\(318\) − 7.31229e25i − 0.387765i
\(319\) −3.50079e25 −0.179624
\(320\) 0 0
\(321\) 9.63011e24 0.0462731
\(322\) 1.61081e26i 0.749130i
\(323\) − 2.50835e26i − 1.12918i
\(324\) −7.28388e25 −0.317424
\(325\) 0 0
\(326\) −8.12420e26 −3.31891
\(327\) − 1.32773e25i − 0.0525240i
\(328\) − 1.14299e27i − 4.37890i
\(329\) 2.41914e25 0.0897644
\(330\) 0 0
\(331\) 1.44090e26 0.501695 0.250848 0.968027i \(-0.419291\pi\)
0.250848 + 0.968027i \(0.419291\pi\)
\(332\) − 1.02262e27i − 3.44958i
\(333\) 1.07341e26i 0.350833i
\(334\) 7.59206e26 2.40448
\(335\) 0 0
\(336\) −4.06139e26 −1.20813
\(337\) − 3.63051e26i − 1.04678i −0.852095 0.523388i \(-0.824668\pi\)
0.852095 0.523388i \(-0.175332\pi\)
\(338\) − 6.66111e26i − 1.86175i
\(339\) −2.50438e26 −0.678583
\(340\) 0 0
\(341\) 3.26607e25 0.0831964
\(342\) 2.89588e26i 0.715331i
\(343\) 3.57891e26i 0.857360i
\(344\) −1.83349e27 −4.26004
\(345\) 0 0
\(346\) 6.49841e26 1.42072
\(347\) − 7.09622e25i − 0.150511i −0.997164 0.0752554i \(-0.976023\pi\)
0.997164 0.0752554i \(-0.0239772\pi\)
\(348\) 8.49335e26i 1.74782i
\(349\) 7.03939e26 1.40562 0.702810 0.711378i \(-0.251928\pi\)
0.702810 + 0.711378i \(0.251928\pi\)
\(350\) 0 0
\(351\) −2.33379e25 −0.0438872
\(352\) − 4.46451e26i − 0.814846i
\(353\) 1.08085e26i 0.191483i 0.995406 + 0.0957414i \(0.0305222\pi\)
−0.995406 + 0.0957414i \(0.969478\pi\)
\(354\) −9.29416e26 −1.59836
\(355\) 0 0
\(356\) −1.87280e27 −3.03575
\(357\) 1.84266e26i 0.290020i
\(358\) − 3.67305e25i − 0.0561378i
\(359\) 1.67492e26 0.248601 0.124301 0.992245i \(-0.460331\pi\)
0.124301 + 0.992245i \(0.460331\pi\)
\(360\) 0 0
\(361\) 1.38602e26 0.194064
\(362\) 2.48946e27i 3.38583i
\(363\) − 4.24422e26i − 0.560761i
\(364\) −2.46722e26 −0.316696
\(365\) 0 0
\(366\) −1.20514e27 −1.46045
\(367\) 9.83667e26i 1.15839i 0.815190 + 0.579194i \(0.196633\pi\)
−0.815190 + 0.579194i \(0.803367\pi\)
\(368\) 2.95146e27i 3.37780i
\(369\) 3.59863e26 0.400276
\(370\) 0 0
\(371\) 1.58191e26 0.166248
\(372\) − 7.92389e26i − 0.809539i
\(373\) − 1.00058e26i − 0.0993824i −0.998765 0.0496912i \(-0.984176\pi\)
0.998765 0.0496912i \(-0.0158237\pi\)
\(374\) −3.56208e26 −0.343995
\(375\) 0 0
\(376\) 7.37429e26 0.673360
\(377\) 2.72131e26i 0.241654i
\(378\) − 2.12734e26i − 0.183727i
\(379\) −9.23905e25 −0.0776096 −0.0388048 0.999247i \(-0.512355\pi\)
−0.0388048 + 0.999247i \(0.512355\pi\)
\(380\) 0 0
\(381\) −9.82055e25 −0.0780591
\(382\) − 3.80986e27i − 2.94608i
\(383\) 2.13677e27i 1.60757i 0.594919 + 0.803786i \(0.297184\pi\)
−0.594919 + 0.803786i \(0.702816\pi\)
\(384\) −4.16393e27 −3.04807
\(385\) 0 0
\(386\) 6.08530e26 0.421809
\(387\) − 5.77263e26i − 0.389411i
\(388\) − 5.68572e27i − 3.73295i
\(389\) 1.25581e27 0.802516 0.401258 0.915965i \(-0.368573\pi\)
0.401258 + 0.915965i \(0.368573\pi\)
\(390\) 0 0
\(391\) 1.33908e27 0.810868
\(392\) 4.72395e27i 2.78485i
\(393\) 3.78305e26i 0.217130i
\(394\) 4.06394e27 2.27111
\(395\) 0 0
\(396\) 3.04614e26 0.161417
\(397\) 1.78165e27i 0.919439i 0.888064 + 0.459719i \(0.152050\pi\)
−0.888064 + 0.459719i \(0.847950\pi\)
\(398\) 2.12121e27i 1.06614i
\(399\) −6.26483e26 −0.306686
\(400\) 0 0
\(401\) 1.40181e27 0.651141 0.325570 0.945518i \(-0.394444\pi\)
0.325570 + 0.945518i \(0.394444\pi\)
\(402\) − 2.64590e27i − 1.19729i
\(403\) − 2.53885e26i − 0.111927i
\(404\) −8.67511e26 −0.372624
\(405\) 0 0
\(406\) −2.48058e27 −1.01165
\(407\) − 4.48903e26i − 0.178407i
\(408\) 5.61699e27i 2.17557i
\(409\) −2.17493e27 −0.821015 −0.410507 0.911857i \(-0.634648\pi\)
−0.410507 + 0.911857i \(0.634648\pi\)
\(410\) 0 0
\(411\) 1.17102e27 0.419978
\(412\) 1.31582e28i 4.60021i
\(413\) − 2.01066e27i − 0.685271i
\(414\) −1.54596e27 −0.513683
\(415\) 0 0
\(416\) −3.47045e27 −1.09624
\(417\) 3.07139e26i 0.0946034i
\(418\) − 1.21107e27i − 0.363762i
\(419\) 6.07636e27 1.77990 0.889952 0.456055i \(-0.150738\pi\)
0.889952 + 0.456055i \(0.150738\pi\)
\(420\) 0 0
\(421\) −1.89993e27 −0.529389 −0.264695 0.964332i \(-0.585271\pi\)
−0.264695 + 0.964332i \(0.585271\pi\)
\(422\) 7.53915e27i 2.04900i
\(423\) 2.32175e26i 0.0615520i
\(424\) 4.82214e27 1.24709
\(425\) 0 0
\(426\) 4.44321e27 1.09369
\(427\) − 2.60714e27i − 0.626141i
\(428\) 9.77083e26i 0.228966i
\(429\) 9.75999e25 0.0223176
\(430\) 0 0
\(431\) −8.08572e24 −0.00176079 −0.000880395 1.00000i \(-0.500280\pi\)
−0.000880395 1.00000i \(0.500280\pi\)
\(432\) − 3.89789e27i − 0.828420i
\(433\) 5.60439e27i 1.16253i 0.813713 + 0.581267i \(0.197443\pi\)
−0.813713 + 0.581267i \(0.802557\pi\)
\(434\) 2.31426e27 0.468566
\(435\) 0 0
\(436\) 1.34713e27 0.259897
\(437\) 4.55272e27i 0.857463i
\(438\) − 2.26230e27i − 0.415979i
\(439\) 8.51110e27 1.52795 0.763973 0.645248i \(-0.223246\pi\)
0.763973 + 0.645248i \(0.223246\pi\)
\(440\) 0 0
\(441\) −1.48731e27 −0.254563
\(442\) 2.76896e27i 0.462789i
\(443\) 6.63134e27i 1.08234i 0.840915 + 0.541168i \(0.182018\pi\)
−0.840915 + 0.541168i \(0.817982\pi\)
\(444\) −1.08909e28 −1.73598
\(445\) 0 0
\(446\) −1.12956e28 −1.71749
\(447\) 4.41468e27i 0.655648i
\(448\) − 1.72103e28i − 2.49671i
\(449\) −1.30394e28 −1.84787 −0.923933 0.382555i \(-0.875044\pi\)
−0.923933 + 0.382555i \(0.875044\pi\)
\(450\) 0 0
\(451\) −1.50496e27 −0.203549
\(452\) − 2.54097e28i − 3.35773i
\(453\) 6.55703e27i 0.846596i
\(454\) −6.10577e27 −0.770289
\(455\) 0 0
\(456\) −1.90971e28 −2.30058
\(457\) 4.72949e27i 0.556793i 0.960466 + 0.278397i \(0.0898030\pi\)
−0.960466 + 0.278397i \(0.910197\pi\)
\(458\) 1.04097e28i 1.19771i
\(459\) −1.76848e27 −0.198869
\(460\) 0 0
\(461\) 4.92722e27 0.529349 0.264675 0.964338i \(-0.414735\pi\)
0.264675 + 0.964338i \(0.414735\pi\)
\(462\) 8.89661e26i 0.0934294i
\(463\) − 1.20207e28i − 1.23404i −0.786947 0.617021i \(-0.788339\pi\)
0.786947 0.617021i \(-0.211661\pi\)
\(464\) −4.54513e28 −4.56150
\(465\) 0 0
\(466\) 8.27379e27 0.793693
\(467\) 1.09969e28i 1.03144i 0.856758 + 0.515719i \(0.172475\pi\)
−0.856758 + 0.515719i \(0.827525\pi\)
\(468\) − 2.36789e27i − 0.217161i
\(469\) 5.72402e27 0.513317
\(470\) 0 0
\(471\) −2.57786e27 −0.221075
\(472\) − 6.12910e28i − 5.14051i
\(473\) 2.41413e27i 0.198024i
\(474\) 2.83578e27 0.227510
\(475\) 0 0
\(476\) −1.86958e28 −1.43507
\(477\) 1.51823e27i 0.113997i
\(478\) − 1.36576e28i − 1.00318i
\(479\) 1.32717e28 0.953684 0.476842 0.878989i \(-0.341781\pi\)
0.476842 + 0.878989i \(0.341781\pi\)
\(480\) 0 0
\(481\) −3.48951e27 −0.240017
\(482\) − 2.23613e28i − 1.50489i
\(483\) − 3.34446e27i − 0.220233i
\(484\) 4.30624e28 2.77473
\(485\) 0 0
\(486\) 2.04170e27 0.125983
\(487\) 2.62576e28i 1.58563i 0.609466 + 0.792813i \(0.291384\pi\)
−0.609466 + 0.792813i \(0.708616\pi\)
\(488\) − 7.94738e28i − 4.69695i
\(489\) 1.68680e28 0.975710
\(490\) 0 0
\(491\) −2.54066e28 −1.40796 −0.703982 0.710218i \(-0.748596\pi\)
−0.703982 + 0.710218i \(0.748596\pi\)
\(492\) 3.65121e28i 1.98063i
\(493\) 2.06213e28i 1.09502i
\(494\) −9.41414e27 −0.489382
\(495\) 0 0
\(496\) 4.24039e28 2.11275
\(497\) 9.61224e27i 0.468903i
\(498\) 2.86645e28i 1.36911i
\(499\) 1.30048e28 0.608204 0.304102 0.952640i \(-0.401644\pi\)
0.304102 + 0.952640i \(0.401644\pi\)
\(500\) 0 0
\(501\) −1.57631e28 −0.706882
\(502\) 5.67449e28i 2.49194i
\(503\) − 1.34993e27i − 0.0580559i −0.999579 0.0290280i \(-0.990759\pi\)
0.999579 0.0290280i \(-0.00924119\pi\)
\(504\) 1.40289e28 0.590886
\(505\) 0 0
\(506\) 6.46525e27 0.261219
\(507\) 1.38302e28i 0.547326i
\(508\) − 9.96406e27i − 0.386249i
\(509\) 4.04902e28 1.53749 0.768746 0.639554i \(-0.220881\pi\)
0.768746 + 0.639554i \(0.220881\pi\)
\(510\) 0 0
\(511\) 4.89416e27 0.178344
\(512\) − 1.39939e29i − 4.99579i
\(513\) − 6.01263e27i − 0.210296i
\(514\) 3.79902e28 1.30184
\(515\) 0 0
\(516\) 5.85698e28 1.92687
\(517\) − 9.70964e26i − 0.0313006i
\(518\) − 3.18083e28i − 1.00479i
\(519\) −1.34924e28 −0.417670
\(520\) 0 0
\(521\) 5.40378e28 1.60658 0.803289 0.595590i \(-0.203082\pi\)
0.803289 + 0.595590i \(0.203082\pi\)
\(522\) − 2.38072e28i − 0.693695i
\(523\) − 1.54066e28i − 0.439988i −0.975501 0.219994i \(-0.929396\pi\)
0.975501 0.219994i \(-0.0706037\pi\)
\(524\) −3.83833e28 −1.07439
\(525\) 0 0
\(526\) 1.65986e28 0.446398
\(527\) − 1.92387e28i − 0.507182i
\(528\) 1.63011e28i 0.421270i
\(529\) 1.51670e28 0.384252
\(530\) 0 0
\(531\) 1.92971e28 0.469895
\(532\) − 6.35637e28i − 1.51753i
\(533\) 1.16987e28i 0.273842i
\(534\) 5.24953e28 1.20486
\(535\) 0 0
\(536\) 1.74486e29 3.85061
\(537\) 7.62623e26i 0.0165037i
\(538\) 1.52238e29i 3.23080i
\(539\) 6.21997e27 0.129451
\(540\) 0 0
\(541\) −7.54478e28 −1.51034 −0.755171 0.655528i \(-0.772446\pi\)
−0.755171 + 0.655528i \(0.772446\pi\)
\(542\) 2.96923e28i 0.582976i
\(543\) − 5.16878e28i − 0.995382i
\(544\) −2.62981e29 −4.96747
\(545\) 0 0
\(546\) 6.91571e27 0.125694
\(547\) 7.90524e28i 1.40944i 0.709483 + 0.704722i \(0.248928\pi\)
−0.709483 + 0.704722i \(0.751072\pi\)
\(548\) 1.18813e29i 2.07812i
\(549\) 2.50219e28 0.429349
\(550\) 0 0
\(551\) −7.01101e28 −1.15795
\(552\) − 1.01950e29i − 1.65206i
\(553\) 6.13480e27i 0.0975408i
\(554\) 8.95531e28 1.39710
\(555\) 0 0
\(556\) −3.11627e28 −0.468113
\(557\) 1.17729e29i 1.73541i 0.497075 + 0.867707i \(0.334407\pi\)
−0.497075 + 0.867707i \(0.665593\pi\)
\(558\) 2.22110e28i 0.321299i
\(559\) 1.87660e28 0.266409
\(560\) 0 0
\(561\) 7.39583e27 0.101129
\(562\) 3.29582e28i 0.442315i
\(563\) 9.20807e28i 1.21291i 0.795116 + 0.606457i \(0.207410\pi\)
−0.795116 + 0.606457i \(0.792590\pi\)
\(564\) −2.35568e28 −0.304569
\(565\) 0 0
\(566\) 1.36626e29 1.70200
\(567\) 4.41693e27i 0.0540130i
\(568\) 2.93011e29i 3.51744i
\(569\) 2.71795e27 0.0320304 0.0160152 0.999872i \(-0.494902\pi\)
0.0160152 + 0.999872i \(0.494902\pi\)
\(570\) 0 0
\(571\) 1.28086e28 0.145487 0.0727434 0.997351i \(-0.476825\pi\)
0.0727434 + 0.997351i \(0.476825\pi\)
\(572\) 9.90260e27i 0.110431i
\(573\) 7.91029e28i 0.866102i
\(574\) −1.06638e29 −1.14640
\(575\) 0 0
\(576\) 1.65175e29 1.71201
\(577\) 1.49329e29i 1.51984i 0.650015 + 0.759922i \(0.274763\pi\)
−0.650015 + 0.759922i \(0.725237\pi\)
\(578\) 1.33261e28i 0.133186i
\(579\) −1.26347e28 −0.124005
\(580\) 0 0
\(581\) −6.20116e28 −0.586980
\(582\) 1.59373e29i 1.48158i
\(583\) − 6.34926e27i − 0.0579700i
\(584\) 1.49189e29 1.33783
\(585\) 0 0
\(586\) 2.26586e29 1.96023
\(587\) 2.62975e28i 0.223468i 0.993738 + 0.111734i \(0.0356404\pi\)
−0.993738 + 0.111734i \(0.964360\pi\)
\(588\) − 1.50904e29i − 1.25962i
\(589\) 6.54093e28 0.536326
\(590\) 0 0
\(591\) −8.43783e28 −0.667671
\(592\) − 5.82817e29i − 4.53059i
\(593\) − 1.92294e29i − 1.46856i −0.678849 0.734278i \(-0.737521\pi\)
0.678849 0.734278i \(-0.262479\pi\)
\(594\) −8.53846e27 −0.0640651
\(595\) 0 0
\(596\) −4.47919e29 −3.24425
\(597\) − 4.40421e28i − 0.313428i
\(598\) − 5.02572e28i − 0.351427i
\(599\) 2.45874e29 1.68940 0.844699 0.535242i \(-0.179780\pi\)
0.844699 + 0.535242i \(0.179780\pi\)
\(600\) 0 0
\(601\) 7.47252e28 0.495776 0.247888 0.968789i \(-0.420264\pi\)
0.247888 + 0.968789i \(0.420264\pi\)
\(602\) 1.71060e29i 1.11528i
\(603\) 5.49359e28i 0.351985i
\(604\) −6.65285e29 −4.18909
\(605\) 0 0
\(606\) 2.43167e28 0.147891
\(607\) − 1.23466e29i − 0.738016i −0.929426 0.369008i \(-0.879698\pi\)
0.929426 0.369008i \(-0.120302\pi\)
\(608\) − 8.94104e29i − 5.25291i
\(609\) 5.15035e28 0.297410
\(610\) 0 0
\(611\) −7.54771e27 −0.0421098
\(612\) − 1.79432e29i − 0.984034i
\(613\) 1.59244e29i 0.858476i 0.903191 + 0.429238i \(0.141218\pi\)
−0.903191 + 0.429238i \(0.858782\pi\)
\(614\) 4.30861e29 2.28333
\(615\) 0 0
\(616\) −5.86694e28 −0.300479
\(617\) 1.23256e29i 0.620602i 0.950638 + 0.310301i \(0.100430\pi\)
−0.950638 + 0.310301i \(0.899570\pi\)
\(618\) − 3.68831e29i − 1.82578i
\(619\) 5.18990e28 0.252585 0.126292 0.991993i \(-0.459692\pi\)
0.126292 + 0.991993i \(0.459692\pi\)
\(620\) 0 0
\(621\) 3.20983e28 0.151015
\(622\) − 4.02159e29i − 1.86037i
\(623\) 1.13566e29i 0.516564i
\(624\) 1.26715e29 0.566750
\(625\) 0 0
\(626\) −1.42086e29 −0.614497
\(627\) 2.51450e28i 0.106940i
\(628\) − 2.61553e29i − 1.09391i
\(629\) −2.64425e29 −1.08760
\(630\) 0 0
\(631\) −2.05208e28 −0.0816366 −0.0408183 0.999167i \(-0.512996\pi\)
−0.0408183 + 0.999167i \(0.512996\pi\)
\(632\) 1.87008e29i 0.731694i
\(633\) − 1.56533e29i − 0.602374i
\(634\) 5.16285e29 1.95412
\(635\) 0 0
\(636\) −1.54041e29 −0.564075
\(637\) − 4.83505e28i − 0.174155i
\(638\) 9.95625e28i 0.352759i
\(639\) −9.22528e28 −0.321529
\(640\) 0 0
\(641\) 5.14342e29 1.73477 0.867386 0.497636i \(-0.165798\pi\)
0.867386 + 0.497636i \(0.165798\pi\)
\(642\) − 2.73880e28i − 0.0908747i
\(643\) 8.85766e28i 0.289137i 0.989495 + 0.144569i \(0.0461794\pi\)
−0.989495 + 0.144569i \(0.953821\pi\)
\(644\) 3.39333e29 1.08975
\(645\) 0 0
\(646\) −7.13376e29 −2.21757
\(647\) 5.46916e29i 1.67273i 0.548174 + 0.836364i \(0.315323\pi\)
−0.548174 + 0.836364i \(0.684677\pi\)
\(648\) 1.34642e29i 0.405174i
\(649\) −8.07012e28 −0.238952
\(650\) 0 0
\(651\) −4.80503e28 −0.137751
\(652\) 1.71145e30i 4.82796i
\(653\) 5.66153e29i 1.57161i 0.618473 + 0.785806i \(0.287752\pi\)
−0.618473 + 0.785806i \(0.712248\pi\)
\(654\) −3.77606e28 −0.103151
\(655\) 0 0
\(656\) −1.95391e30 −5.16908
\(657\) 4.69713e28i 0.122292i
\(658\) − 6.88003e28i − 0.176286i
\(659\) 1.48653e29 0.374867 0.187434 0.982277i \(-0.439983\pi\)
0.187434 + 0.982277i \(0.439983\pi\)
\(660\) 0 0
\(661\) −4.04669e29 −0.988517 −0.494259 0.869315i \(-0.664560\pi\)
−0.494259 + 0.869315i \(0.664560\pi\)
\(662\) − 4.09792e29i − 0.985268i
\(663\) − 5.74909e28i − 0.136053i
\(664\) −1.89031e30 −4.40319
\(665\) 0 0
\(666\) 3.05278e29 0.688994
\(667\) − 3.74281e29i − 0.831527i
\(668\) − 1.59935e30i − 3.49776i
\(669\) 2.34527e29 0.504916
\(670\) 0 0
\(671\) −1.04642e29 −0.218334
\(672\) 6.56816e29i 1.34917i
\(673\) 1.54590e29i 0.312625i 0.987708 + 0.156313i \(0.0499608\pi\)
−0.987708 + 0.156313i \(0.950039\pi\)
\(674\) −1.03252e30 −2.05574
\(675\) 0 0
\(676\) −1.40323e30 −2.70825
\(677\) − 9.88421e29i − 1.87828i −0.343530 0.939142i \(-0.611623\pi\)
0.343530 0.939142i \(-0.388377\pi\)
\(678\) 7.12245e29i 1.33265i
\(679\) −3.44781e29 −0.635200
\(680\) 0 0
\(681\) 1.26772e29 0.226453
\(682\) − 9.28870e28i − 0.163388i
\(683\) 1.41169e29i 0.244524i 0.992498 + 0.122262i \(0.0390147\pi\)
−0.992498 + 0.122262i \(0.960985\pi\)
\(684\) 6.10048e29 1.04058
\(685\) 0 0
\(686\) 1.01784e30 1.68375
\(687\) − 2.16133e29i − 0.352108i
\(688\) 3.13430e30i 5.02877i
\(689\) −4.93555e28 −0.0779891
\(690\) 0 0
\(691\) 7.11585e29 1.09070 0.545352 0.838207i \(-0.316396\pi\)
0.545352 + 0.838207i \(0.316396\pi\)
\(692\) − 1.36896e30i − 2.06669i
\(693\) − 1.84717e28i − 0.0274668i
\(694\) −2.01817e29 −0.295585
\(695\) 0 0
\(696\) 1.56999e30 2.23099
\(697\) 8.86490e29i 1.24088i
\(698\) − 2.00200e30i − 2.76047i
\(699\) −1.71786e29 −0.233334
\(700\) 0 0
\(701\) −8.80754e29 −1.16096 −0.580478 0.814276i \(-0.697134\pi\)
−0.580478 + 0.814276i \(0.697134\pi\)
\(702\) 6.63731e28i 0.0861891i
\(703\) − 8.99015e29i − 1.15010i
\(704\) −6.90765e29 −0.870597
\(705\) 0 0
\(706\) 3.07393e29 0.376049
\(707\) 5.26057e28i 0.0634058i
\(708\) 1.95791e30i 2.32511i
\(709\) 2.79000e29 0.326452 0.163226 0.986589i \(-0.447810\pi\)
0.163226 + 0.986589i \(0.447810\pi\)
\(710\) 0 0
\(711\) −5.88783e28 −0.0668843
\(712\) 3.46185e30i 3.87496i
\(713\) 3.49186e29i 0.385139i
\(714\) 5.24052e29 0.569565
\(715\) 0 0
\(716\) −7.73767e28 −0.0816626
\(717\) 2.83568e29i 0.294921i
\(718\) − 4.76349e29i − 0.488223i
\(719\) −1.21404e30 −1.22625 −0.613124 0.789987i \(-0.710087\pi\)
−0.613124 + 0.789987i \(0.710087\pi\)
\(720\) 0 0
\(721\) 7.97913e29 0.782772
\(722\) − 3.94185e29i − 0.381118i
\(723\) 4.64280e29i 0.442413i
\(724\) 5.24431e30 4.92531
\(725\) 0 0
\(726\) −1.20706e30 −1.10127
\(727\) − 6.54831e29i − 0.588868i −0.955672 0.294434i \(-0.904869\pi\)
0.955672 0.294434i \(-0.0951311\pi\)
\(728\) 4.56062e29i 0.404245i
\(729\) −4.23912e28 −0.0370370
\(730\) 0 0
\(731\) 1.42204e30 1.20720
\(732\) 2.53875e30i 2.12449i
\(733\) − 2.20665e29i − 0.182029i −0.995850 0.0910147i \(-0.970989\pi\)
0.995850 0.0910147i \(-0.0290110\pi\)
\(734\) 2.79755e30 2.27493
\(735\) 0 0
\(736\) 4.77315e30 3.77214
\(737\) − 2.29743e29i − 0.178992i
\(738\) − 1.02345e30i − 0.786094i
\(739\) 4.07297e29 0.308422 0.154211 0.988038i \(-0.450717\pi\)
0.154211 + 0.988038i \(0.450717\pi\)
\(740\) 0 0
\(741\) 1.95463e29 0.143871
\(742\) − 4.49895e29i − 0.326490i
\(743\) − 3.97218e29i − 0.284214i −0.989851 0.142107i \(-0.954612\pi\)
0.989851 0.142107i \(-0.0453878\pi\)
\(744\) −1.46472e30 −1.03333
\(745\) 0 0
\(746\) −2.84565e29 −0.195175
\(747\) − 5.95152e29i − 0.402496i
\(748\) 7.50390e29i 0.500403i
\(749\) 5.92500e28 0.0389610
\(750\) 0 0
\(751\) −1.86890e30 −1.19500 −0.597498 0.801871i \(-0.703838\pi\)
−0.597498 + 0.801871i \(0.703838\pi\)
\(752\) − 1.26062e30i − 0.794869i
\(753\) − 1.17817e30i − 0.732592i
\(754\) 7.73941e29 0.474579
\(755\) 0 0
\(756\) −4.48147e29 −0.267265
\(757\) − 2.99103e30i − 1.75919i −0.475720 0.879597i \(-0.657812\pi\)
0.475720 0.879597i \(-0.342188\pi\)
\(758\) 2.62759e29i 0.152416i
\(759\) −1.34236e29 −0.0767945
\(760\) 0 0
\(761\) 1.51341e30 0.842206 0.421103 0.907013i \(-0.361643\pi\)
0.421103 + 0.907013i \(0.361643\pi\)
\(762\) 2.79297e29i 0.153299i
\(763\) − 8.16896e28i − 0.0442241i
\(764\) −8.02588e30 −4.28561
\(765\) 0 0
\(766\) 6.07697e30 3.15707
\(767\) 6.27325e29i 0.321470i
\(768\) 5.97596e30i 3.02075i
\(769\) −2.53401e30 −1.26352 −0.631759 0.775165i \(-0.717667\pi\)
−0.631759 + 0.775165i \(0.717667\pi\)
\(770\) 0 0
\(771\) −7.88777e29 −0.382722
\(772\) − 1.28193e30i − 0.613598i
\(773\) 1.69545e30i 0.800571i 0.916390 + 0.400285i \(0.131089\pi\)
−0.916390 + 0.400285i \(0.868911\pi\)
\(774\) −1.64173e30 −0.764756
\(775\) 0 0
\(776\) −1.05100e31 −4.76490
\(777\) 6.60424e29i 0.295394i
\(778\) − 3.57153e30i − 1.57604i
\(779\) −3.01396e30 −1.31218
\(780\) 0 0
\(781\) 3.85804e29 0.163505
\(782\) − 3.80834e30i − 1.59245i
\(783\) 4.94301e29i 0.203936i
\(784\) 8.07548e30 3.28738
\(785\) 0 0
\(786\) 1.07590e30 0.426417
\(787\) − 1.87332e30i − 0.732616i −0.930494 0.366308i \(-0.880622\pi\)
0.930494 0.366308i \(-0.119378\pi\)
\(788\) − 8.56113e30i − 3.30374i
\(789\) −3.44632e29 −0.131234
\(790\) 0 0
\(791\) −1.54084e30 −0.571353
\(792\) − 5.63075e29i − 0.206040i
\(793\) 8.13429e29i 0.293732i
\(794\) 5.06702e30 1.80567
\(795\) 0 0
\(796\) 4.46856e30 1.55089
\(797\) − 7.79664e29i − 0.267051i −0.991045 0.133526i \(-0.957370\pi\)
0.991045 0.133526i \(-0.0426299\pi\)
\(798\) 1.78172e30i 0.602294i
\(799\) −5.71944e29 −0.190815
\(800\) 0 0
\(801\) −1.08994e30 −0.354211
\(802\) − 3.98676e30i − 1.27876i
\(803\) − 1.96436e29i − 0.0621880i
\(804\) −5.57386e30 −1.74168
\(805\) 0 0
\(806\) −7.22050e29 −0.219811
\(807\) − 3.16088e30i − 0.949807i
\(808\) 1.60358e30i 0.475633i
\(809\) −8.82262e29 −0.258308 −0.129154 0.991625i \(-0.541226\pi\)
−0.129154 + 0.991625i \(0.541226\pi\)
\(810\) 0 0
\(811\) −2.06044e30 −0.587815 −0.293907 0.955834i \(-0.594956\pi\)
−0.293907 + 0.955834i \(0.594956\pi\)
\(812\) 5.22561e30i 1.47163i
\(813\) − 6.16490e29i − 0.171386i
\(814\) −1.27668e30 −0.350369
\(815\) 0 0
\(816\) 9.60212e30 2.56815
\(817\) 4.83476e30i 1.27657i
\(818\) 6.18551e30i 1.61237i
\(819\) −1.43589e29 −0.0369521
\(820\) 0 0
\(821\) −1.83846e30 −0.461160 −0.230580 0.973053i \(-0.574062\pi\)
−0.230580 + 0.973053i \(0.574062\pi\)
\(822\) − 3.33039e30i − 0.824785i
\(823\) − 7.73766e29i − 0.189196i −0.995516 0.0945979i \(-0.969843\pi\)
0.995516 0.0945979i \(-0.0301565\pi\)
\(824\) 2.43229e31 5.87190
\(825\) 0 0
\(826\) −5.71831e30 −1.34579
\(827\) − 4.59989e30i − 1.06891i −0.845198 0.534453i \(-0.820518\pi\)
0.845198 0.534453i \(-0.179482\pi\)
\(828\) 3.25673e30i 0.747245i
\(829\) −7.93000e30 −1.79660 −0.898298 0.439386i \(-0.855196\pi\)
−0.898298 + 0.439386i \(0.855196\pi\)
\(830\) 0 0
\(831\) −1.85936e30 −0.410726
\(832\) 5.36961e30i 1.17124i
\(833\) − 3.66386e30i − 0.789162i
\(834\) 8.73504e29 0.185790
\(835\) 0 0
\(836\) −2.55124e30 −0.529158
\(837\) − 4.61159e29i − 0.0944569i
\(838\) − 1.72812e31i − 3.49551i
\(839\) −4.84033e30 −0.966884 −0.483442 0.875376i \(-0.660614\pi\)
−0.483442 + 0.875376i \(0.660614\pi\)
\(840\) 0 0
\(841\) 6.30944e29 0.122923
\(842\) 5.40340e30i 1.03966i
\(843\) − 6.84300e29i − 0.130034i
\(844\) 1.58820e31 2.98064
\(845\) 0 0
\(846\) 6.60306e29 0.120881
\(847\) − 2.61129e30i − 0.472149i
\(848\) − 8.24334e30i − 1.47213i
\(849\) −2.83671e30 −0.500362
\(850\) 0 0
\(851\) 4.79937e30 0.825893
\(852\) − 9.36009e30i − 1.59098i
\(853\) − 2.96903e30i − 0.498483i −0.968441 0.249242i \(-0.919819\pi\)
0.968441 0.249242i \(-0.0801813\pi\)
\(854\) −7.41472e30 −1.22967
\(855\) 0 0
\(856\) 1.80612e30 0.292263
\(857\) − 3.70009e30i − 0.591444i −0.955274 0.295722i \(-0.904440\pi\)
0.955274 0.295722i \(-0.0955602\pi\)
\(858\) − 2.77574e29i − 0.0438291i
\(859\) −7.75385e29 −0.120945 −0.0604726 0.998170i \(-0.519261\pi\)
−0.0604726 + 0.998170i \(0.519261\pi\)
\(860\) 0 0
\(861\) 2.21408e30 0.337024
\(862\) 2.29958e28i 0.00345798i
\(863\) 1.29544e31i 1.92443i 0.272283 + 0.962217i \(0.412221\pi\)
−0.272283 + 0.962217i \(0.587779\pi\)
\(864\) −6.30375e30 −0.925134
\(865\) 0 0
\(866\) 1.59389e31 2.28307
\(867\) − 2.76685e29i − 0.0391548i
\(868\) − 4.87524e30i − 0.681615i
\(869\) 2.46231e29 0.0340122
\(870\) 0 0
\(871\) −1.78589e30 −0.240805
\(872\) − 2.49015e30i − 0.331744i
\(873\) − 3.30901e30i − 0.435560i
\(874\) 1.29479e31 1.68395
\(875\) 0 0
\(876\) −4.76577e30 −0.605118
\(877\) − 1.57355e31i − 1.97417i −0.160207 0.987083i \(-0.551216\pi\)
0.160207 0.987083i \(-0.448784\pi\)
\(878\) − 2.42056e31i − 3.00070i
\(879\) −4.70452e30 −0.576278
\(880\) 0 0
\(881\) −1.47526e31 −1.76450 −0.882252 0.470778i \(-0.843973\pi\)
−0.882252 + 0.470778i \(0.843973\pi\)
\(882\) 4.22991e30i 0.499932i
\(883\) 5.64453e30i 0.659235i 0.944115 + 0.329617i \(0.106920\pi\)
−0.944115 + 0.329617i \(0.893080\pi\)
\(884\) 5.83310e30 0.673210
\(885\) 0 0
\(886\) 1.88595e31 2.12558
\(887\) 5.89300e30i 0.656354i 0.944616 + 0.328177i \(0.106434\pi\)
−0.944616 + 0.328177i \(0.893566\pi\)
\(888\) 2.01318e31i 2.21588i
\(889\) −6.04218e29 −0.0657242
\(890\) 0 0
\(891\) 1.77281e29 0.0188342
\(892\) 2.37954e31i 2.49840i
\(893\) − 1.94454e30i − 0.201780i
\(894\) 1.25554e31 1.28761
\(895\) 0 0
\(896\) −2.56189e31 −2.56641
\(897\) 1.04347e30i 0.103314i
\(898\) 3.70840e31i 3.62898i
\(899\) −5.37734e30 −0.520104
\(900\) 0 0
\(901\) −3.74002e30 −0.353397
\(902\) 4.28009e30i 0.399746i
\(903\) − 3.55166e30i − 0.327876i
\(904\) −4.69695e31 −4.28596
\(905\) 0 0
\(906\) 1.86482e31 1.66261
\(907\) 8.32782e30i 0.733931i 0.930235 + 0.366965i \(0.119603\pi\)
−0.930235 + 0.366965i \(0.880397\pi\)
\(908\) 1.28624e31i 1.12053i
\(909\) −5.04879e29 −0.0434777
\(910\) 0 0
\(911\) 1.08086e31 0.909548 0.454774 0.890607i \(-0.349720\pi\)
0.454774 + 0.890607i \(0.349720\pi\)
\(912\) 3.26461e31i 2.71572i
\(913\) 2.48894e30i 0.204678i
\(914\) 1.34507e31 1.09347
\(915\) 0 0
\(916\) 2.19291e31 1.74229
\(917\) 2.32755e30i 0.182819i
\(918\) 5.02955e30i 0.390554i
\(919\) −6.55205e30 −0.502996 −0.251498 0.967858i \(-0.580923\pi\)
−0.251498 + 0.967858i \(0.580923\pi\)
\(920\) 0 0
\(921\) −8.94582e30 −0.671265
\(922\) − 1.40130e31i − 1.03958i
\(923\) − 2.99902e30i − 0.219969i
\(924\) 1.87416e30 0.135910
\(925\) 0 0
\(926\) −3.41869e31 −2.42351
\(927\) 7.65791e30i 0.536752i
\(928\) 7.35047e31i 5.09403i
\(929\) −1.04036e31 −0.712883 −0.356442 0.934318i \(-0.616010\pi\)
−0.356442 + 0.934318i \(0.616010\pi\)
\(930\) 0 0
\(931\) 1.24567e31 0.834509
\(932\) − 1.74296e31i − 1.15457i
\(933\) 8.34989e30i 0.546919i
\(934\) 3.12752e31 2.02562
\(935\) 0 0
\(936\) −4.37702e30 −0.277193
\(937\) − 2.09831e31i − 1.31402i −0.753880 0.657012i \(-0.771820\pi\)
0.753880 0.657012i \(-0.228180\pi\)
\(938\) − 1.62791e31i − 1.00809i
\(939\) 2.95007e30 0.180653
\(940\) 0 0
\(941\) 2.20967e31 1.32323 0.661617 0.749842i \(-0.269870\pi\)
0.661617 + 0.749842i \(0.269870\pi\)
\(942\) 7.33143e30i 0.434165i
\(943\) − 1.60900e31i − 0.942286i
\(944\) −1.04776e32 −6.06812
\(945\) 0 0
\(946\) 6.86578e30 0.388896
\(947\) − 2.59604e31i − 1.45424i −0.686509 0.727121i \(-0.740858\pi\)
0.686509 0.727121i \(-0.259142\pi\)
\(948\) − 5.97386e30i − 0.330954i
\(949\) −1.52698e30 −0.0836637
\(950\) 0 0
\(951\) −1.07194e31 −0.574483
\(952\) 3.45590e31i 1.83178i
\(953\) 2.94729e31i 1.54507i 0.634973 + 0.772535i \(0.281011\pi\)
−0.634973 + 0.772535i \(0.718989\pi\)
\(954\) 4.31783e30 0.223876
\(955\) 0 0
\(956\) −2.87711e31 −1.45931
\(957\) − 2.06718e30i − 0.103706i
\(958\) − 3.77448e31i − 1.87292i
\(959\) 7.20482e30 0.353612
\(960\) 0 0
\(961\) −1.58087e31 −0.759103
\(962\) 9.92417e30i 0.471364i
\(963\) 5.68648e29i 0.0267158i
\(964\) −4.71064e31 −2.18913
\(965\) 0 0
\(966\) −9.51166e30 −0.432510
\(967\) 6.98077e30i 0.313997i 0.987599 + 0.156998i \(0.0501818\pi\)
−0.987599 + 0.156998i \(0.949818\pi\)
\(968\) − 7.96002e31i − 3.54179i
\(969\) 1.48116e31 0.651931
\(970\) 0 0
\(971\) −1.30234e30 −0.0560946 −0.0280473 0.999607i \(-0.508929\pi\)
−0.0280473 + 0.999607i \(0.508929\pi\)
\(972\) − 4.30106e30i − 0.183265i
\(973\) 1.88970e30i 0.0796541i
\(974\) 7.46765e31 3.11398
\(975\) 0 0
\(976\) −1.35859e32 −5.54452
\(977\) 3.06599e31i 1.23788i 0.785439 + 0.618939i \(0.212437\pi\)
−0.785439 + 0.618939i \(0.787563\pi\)
\(978\) − 4.79726e31i − 1.91618i
\(979\) 4.55817e30 0.180124
\(980\) 0 0
\(981\) 7.84011e29 0.0303248
\(982\) 7.22564e31i 2.76507i
\(983\) − 8.31325e30i − 0.314745i −0.987539 0.157373i \(-0.949698\pi\)
0.987539 0.157373i \(-0.0503023\pi\)
\(984\) 6.74921e31 2.52816
\(985\) 0 0
\(986\) 5.86470e31 2.15049
\(987\) 1.42848e30i 0.0518255i
\(988\) 1.98319e31i 0.711895i
\(989\) −2.58102e31 −0.916708
\(990\) 0 0
\(991\) −1.32568e31 −0.460962 −0.230481 0.973077i \(-0.574030\pi\)
−0.230481 + 0.973077i \(0.574030\pi\)
\(992\) − 6.85764e31i − 2.35940i
\(993\) 8.50837e30i 0.289654i
\(994\) 2.73372e31 0.920868
\(995\) 0 0
\(996\) 6.03849e31 1.99161
\(997\) 3.42048e31i 1.11632i 0.829734 + 0.558159i \(0.188492\pi\)
−0.829734 + 0.558159i \(0.811508\pi\)
\(998\) − 3.69857e31i − 1.19444i
\(999\) −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.b.a.49.1 2
5.2 odd 4 75.22.a.c.1.1 1
5.3 odd 4 3.22.a.a.1.1 1
5.4 even 2 inner 75.22.b.a.49.2 2
15.8 even 4 9.22.a.d.1.1 1
20.3 even 4 48.22.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.a.1.1 1 5.3 odd 4
9.22.a.d.1.1 1 15.8 even 4
48.22.a.e.1.1 1 20.3 even 4
75.22.a.c.1.1 1 5.2 odd 4
75.22.b.a.49.1 2 1.1 even 1 trivial
75.22.b.a.49.2 2 5.4 even 2 inner