Properties

Label 9.24.a.d.1.3
Level $9$
Weight $24$
Character 9.1
Self dual yes
Analytic conductor $30.168$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.1683633611\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 29258x^{2} + 97377280 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{10}\cdot 5^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(61.8825\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

\(f(q)\) \(=\) \(q+1856.48 q^{2} -4.94211e6 q^{4} -1.25135e8 q^{5} -2.21402e9 q^{7} -2.47481e10 q^{8} +O(q^{10})\) \(q+1856.48 q^{2} -4.94211e6 q^{4} -1.25135e8 q^{5} -2.21402e9 q^{7} -2.47481e10 q^{8} -2.32310e11 q^{10} +9.54062e11 q^{11} +3.54107e12 q^{13} -4.11027e12 q^{14} -4.48694e12 q^{16} +1.97591e14 q^{17} -1.41772e14 q^{19} +6.18429e14 q^{20} +1.77119e15 q^{22} -1.35827e15 q^{23} +3.73779e15 q^{25} +6.57390e15 q^{26} +1.09419e16 q^{28} -1.17210e17 q^{29} +2.61332e17 q^{31} +1.99273e17 q^{32} +3.66823e17 q^{34} +2.77051e17 q^{35} +1.63064e18 q^{37} -2.63196e17 q^{38} +3.09685e18 q^{40} +1.39986e18 q^{41} +7.83002e18 q^{43} -4.71508e18 q^{44} -2.52160e18 q^{46} -1.10807e19 q^{47} -2.24669e19 q^{49} +6.93911e18 q^{50} -1.75003e19 q^{52} +6.84789e19 q^{53} -1.19386e20 q^{55} +5.47929e19 q^{56} -2.17598e20 q^{58} -3.80399e20 q^{59} -2.54300e20 q^{61} +4.85157e20 q^{62} +4.07584e20 q^{64} -4.43111e20 q^{65} +7.58191e20 q^{67} -9.76516e20 q^{68} +5.14338e20 q^{70} +1.65470e21 q^{71} +3.73535e21 q^{73} +3.02724e21 q^{74} +7.00652e20 q^{76} -2.11231e21 q^{77} +3.30977e21 q^{79} +5.61472e20 q^{80} +2.59880e21 q^{82} -1.73160e22 q^{83} -2.47255e22 q^{85} +1.45362e22 q^{86} -2.36113e22 q^{88} -7.98037e20 q^{89} -7.83999e21 q^{91} +6.71272e21 q^{92} -2.05711e22 q^{94} +1.77406e22 q^{95} +9.51761e20 q^{97} -4.17092e22 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 19109968 q^{4} + 8561438480 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 19109968 q^{4} + 8561438480 q^{7} + 898978809600 q^{10} - 13534569998680 q^{13} + 27399081725056 q^{16} + 823113907401824 q^{19} - 27\!\cdots\!00 q^{22}+ \cdots + 61\!\cdots\!80 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1856.48 0.640980 0.320490 0.947252i \(-0.396152\pi\)
0.320490 + 0.947252i \(0.396152\pi\)
\(3\) 0 0
\(4\) −4.94211e6 −0.589145
\(5\) −1.25135e8 −1.14610 −0.573051 0.819520i \(-0.694240\pi\)
−0.573051 + 0.819520i \(0.694240\pi\)
\(6\) 0 0
\(7\) −2.21402e9 −0.423208 −0.211604 0.977355i \(-0.567869\pi\)
−0.211604 + 0.977355i \(0.567869\pi\)
\(8\) −2.47481e10 −1.01861
\(9\) 0 0
\(10\) −2.32310e11 −0.734628
\(11\) 9.54062e11 1.00823 0.504116 0.863636i \(-0.331818\pi\)
0.504116 + 0.863636i \(0.331818\pi\)
\(12\) 0 0
\(13\) 3.54107e12 0.548006 0.274003 0.961729i \(-0.411652\pi\)
0.274003 + 0.961729i \(0.411652\pi\)
\(14\) −4.11027e12 −0.271268
\(15\) 0 0
\(16\) −4.48694e12 −0.0637632
\(17\) 1.97591e14 1.39831 0.699157 0.714968i \(-0.253559\pi\)
0.699157 + 0.714968i \(0.253559\pi\)
\(18\) 0 0
\(19\) −1.41772e14 −0.279205 −0.139603 0.990208i \(-0.544583\pi\)
−0.139603 + 0.990208i \(0.544583\pi\)
\(20\) 6.18429e14 0.675220
\(21\) 0 0
\(22\) 1.77119e15 0.646257
\(23\) −1.35827e15 −0.297246 −0.148623 0.988894i \(-0.547484\pi\)
−0.148623 + 0.988894i \(0.547484\pi\)
\(24\) 0 0
\(25\) 3.73779e15 0.313548
\(26\) 6.57390e15 0.351261
\(27\) 0 0
\(28\) 1.09419e16 0.249331
\(29\) −1.17210e17 −1.78397 −0.891986 0.452063i \(-0.850688\pi\)
−0.891986 + 0.452063i \(0.850688\pi\)
\(30\) 0 0
\(31\) 2.61332e17 1.84728 0.923642 0.383255i \(-0.125197\pi\)
0.923642 + 0.383255i \(0.125197\pi\)
\(32\) 1.99273e17 0.977739
\(33\) 0 0
\(34\) 3.66823e17 0.896291
\(35\) 2.77051e17 0.485040
\(36\) 0 0
\(37\) 1.63064e18 1.50674 0.753370 0.657597i \(-0.228427\pi\)
0.753370 + 0.657597i \(0.228427\pi\)
\(38\) −2.63196e17 −0.178965
\(39\) 0 0
\(40\) 3.09685e18 1.16743
\(41\) 1.39986e18 0.397255 0.198628 0.980075i \(-0.436352\pi\)
0.198628 + 0.980075i \(0.436352\pi\)
\(42\) 0 0
\(43\) 7.83002e18 1.28492 0.642459 0.766320i \(-0.277914\pi\)
0.642459 + 0.766320i \(0.277914\pi\)
\(44\) −4.71508e18 −0.593995
\(45\) 0 0
\(46\) −2.52160e18 −0.190529
\(47\) −1.10807e19 −0.653796 −0.326898 0.945060i \(-0.606003\pi\)
−0.326898 + 0.945060i \(0.606003\pi\)
\(48\) 0 0
\(49\) −2.24669e19 −0.820895
\(50\) 6.93911e18 0.200978
\(51\) 0 0
\(52\) −1.75003e19 −0.322855
\(53\) 6.84789e19 1.01481 0.507404 0.861708i \(-0.330605\pi\)
0.507404 + 0.861708i \(0.330605\pi\)
\(54\) 0 0
\(55\) −1.19386e20 −1.15554
\(56\) 5.47929e19 0.431084
\(57\) 0 0
\(58\) −2.17598e20 −1.14349
\(59\) −3.80399e20 −1.64226 −0.821128 0.570744i \(-0.806655\pi\)
−0.821128 + 0.570744i \(0.806655\pi\)
\(60\) 0 0
\(61\) −2.54300e20 −0.748260 −0.374130 0.927376i \(-0.622059\pi\)
−0.374130 + 0.927376i \(0.622059\pi\)
\(62\) 4.85157e20 1.18407
\(63\) 0 0
\(64\) 4.07584e20 0.690474
\(65\) −4.43111e20 −0.628071
\(66\) 0 0
\(67\) 7.58191e20 0.758435 0.379218 0.925308i \(-0.376193\pi\)
0.379218 + 0.925308i \(0.376193\pi\)
\(68\) −9.76516e20 −0.823810
\(69\) 0 0
\(70\) 5.14338e20 0.310901
\(71\) 1.65470e21 0.849668 0.424834 0.905271i \(-0.360333\pi\)
0.424834 + 0.905271i \(0.360333\pi\)
\(72\) 0 0
\(73\) 3.73535e21 1.39354 0.696768 0.717297i \(-0.254621\pi\)
0.696768 + 0.717297i \(0.254621\pi\)
\(74\) 3.02724e21 0.965790
\(75\) 0 0
\(76\) 7.00652e20 0.164493
\(77\) −2.11231e21 −0.426692
\(78\) 0 0
\(79\) 3.30977e21 0.497836 0.248918 0.968525i \(-0.419925\pi\)
0.248918 + 0.968525i \(0.419925\pi\)
\(80\) 5.61472e20 0.0730791
\(81\) 0 0
\(82\) 2.59880e21 0.254632
\(83\) −1.73160e22 −1.47588 −0.737939 0.674868i \(-0.764201\pi\)
−0.737939 + 0.674868i \(0.764201\pi\)
\(84\) 0 0
\(85\) −2.47255e22 −1.60261
\(86\) 1.45362e22 0.823607
\(87\) 0 0
\(88\) −2.36113e22 −1.02700
\(89\) −7.98037e20 −0.0304816 −0.0152408 0.999884i \(-0.504851\pi\)
−0.0152408 + 0.999884i \(0.504851\pi\)
\(90\) 0 0
\(91\) −7.83999e21 −0.231921
\(92\) 6.71272e21 0.175121
\(93\) 0 0
\(94\) −2.05711e22 −0.419070
\(95\) 1.77406e22 0.319998
\(96\) 0 0
\(97\) 9.51761e20 0.0135099 0.00675496 0.999977i \(-0.497850\pi\)
0.00675496 + 0.999977i \(0.497850\pi\)
\(98\) −4.17092e22 −0.526177
\(99\) 0 0
\(100\) −1.84725e22 −0.184725
\(101\) 1.42407e23 1.27009 0.635045 0.772476i \(-0.280982\pi\)
0.635045 + 0.772476i \(0.280982\pi\)
\(102\) 0 0
\(103\) 1.25632e23 0.894275 0.447137 0.894465i \(-0.352444\pi\)
0.447137 + 0.894465i \(0.352444\pi\)
\(104\) −8.76348e22 −0.558204
\(105\) 0 0
\(106\) 1.27129e23 0.650472
\(107\) −2.55948e23 −1.17554 −0.587772 0.809027i \(-0.699995\pi\)
−0.587772 + 0.809027i \(0.699995\pi\)
\(108\) 0 0
\(109\) 3.89375e23 1.44532 0.722658 0.691205i \(-0.242920\pi\)
0.722658 + 0.691205i \(0.242920\pi\)
\(110\) −2.21638e23 −0.740675
\(111\) 0 0
\(112\) 9.93417e21 0.0269851
\(113\) 5.25664e23 1.28916 0.644579 0.764538i \(-0.277033\pi\)
0.644579 + 0.764538i \(0.277033\pi\)
\(114\) 0 0
\(115\) 1.69967e23 0.340674
\(116\) 5.79265e23 1.05102
\(117\) 0 0
\(118\) −7.06201e23 −1.05265
\(119\) −4.37471e23 −0.591778
\(120\) 0 0
\(121\) 1.48038e22 0.0165326
\(122\) −4.72101e23 −0.479620
\(123\) 0 0
\(124\) −1.29153e24 −1.08832
\(125\) 1.02400e24 0.786743
\(126\) 0 0
\(127\) −7.27123e23 −0.465441 −0.232721 0.972544i \(-0.574763\pi\)
−0.232721 + 0.972544i \(0.574763\pi\)
\(128\) −9.14950e23 −0.535159
\(129\) 0 0
\(130\) −8.22624e23 −0.402581
\(131\) −1.29816e24 −0.581711 −0.290856 0.956767i \(-0.593940\pi\)
−0.290856 + 0.956767i \(0.593940\pi\)
\(132\) 0 0
\(133\) 3.13886e23 0.118162
\(134\) 1.40756e24 0.486142
\(135\) 0 0
\(136\) −4.89001e24 −1.42434
\(137\) −1.56938e24 −0.420187 −0.210093 0.977681i \(-0.567377\pi\)
−0.210093 + 0.977681i \(0.567377\pi\)
\(138\) 0 0
\(139\) 5.97766e24 1.35476 0.677378 0.735635i \(-0.263116\pi\)
0.677378 + 0.735635i \(0.263116\pi\)
\(140\) −1.36922e24 −0.285759
\(141\) 0 0
\(142\) 3.07192e24 0.544620
\(143\) 3.37840e24 0.552518
\(144\) 0 0
\(145\) 1.46671e25 2.04461
\(146\) 6.93458e24 0.893228
\(147\) 0 0
\(148\) −8.05880e24 −0.887688
\(149\) 1.50771e24 0.153700 0.0768502 0.997043i \(-0.475514\pi\)
0.0768502 + 0.997043i \(0.475514\pi\)
\(150\) 0 0
\(151\) 1.13154e25 0.989548 0.494774 0.869022i \(-0.335251\pi\)
0.494774 + 0.869022i \(0.335251\pi\)
\(152\) 3.50859e24 0.284401
\(153\) 0 0
\(154\) −3.92146e24 −0.273501
\(155\) −3.27018e25 −2.11718
\(156\) 0 0
\(157\) 4.43402e24 0.247715 0.123857 0.992300i \(-0.460473\pi\)
0.123857 + 0.992300i \(0.460473\pi\)
\(158\) 6.14450e24 0.319103
\(159\) 0 0
\(160\) −2.49359e25 −1.12059
\(161\) 3.00724e24 0.125797
\(162\) 0 0
\(163\) 7.82788e24 0.284110 0.142055 0.989859i \(-0.454629\pi\)
0.142055 + 0.989859i \(0.454629\pi\)
\(164\) −6.91824e24 −0.234041
\(165\) 0 0
\(166\) −3.21468e25 −0.946008
\(167\) 6.36666e25 1.74853 0.874264 0.485452i \(-0.161345\pi\)
0.874264 + 0.485452i \(0.161345\pi\)
\(168\) 0 0
\(169\) −2.92148e25 −0.699689
\(170\) −4.59023e25 −1.02724
\(171\) 0 0
\(172\) −3.86968e25 −0.757004
\(173\) −1.87367e25 −0.342896 −0.171448 0.985193i \(-0.554844\pi\)
−0.171448 + 0.985193i \(0.554844\pi\)
\(174\) 0 0
\(175\) −8.27554e24 −0.132696
\(176\) −4.28081e24 −0.0642881
\(177\) 0 0
\(178\) −1.48154e24 −0.0195381
\(179\) 1.38944e26 1.71802 0.859012 0.511955i \(-0.171079\pi\)
0.859012 + 0.511955i \(0.171079\pi\)
\(180\) 0 0
\(181\) 8.32272e25 0.905652 0.452826 0.891599i \(-0.350416\pi\)
0.452826 + 0.891599i \(0.350416\pi\)
\(182\) −1.45548e25 −0.148657
\(183\) 0 0
\(184\) 3.36147e25 0.302778
\(185\) −2.04050e26 −1.72688
\(186\) 0 0
\(187\) 1.88514e26 1.40983
\(188\) 5.47621e25 0.385181
\(189\) 0 0
\(190\) 3.29350e25 0.205112
\(191\) −1.26673e26 −0.742679 −0.371340 0.928497i \(-0.621101\pi\)
−0.371340 + 0.928497i \(0.621101\pi\)
\(192\) 0 0
\(193\) 3.03237e25 0.157715 0.0788577 0.996886i \(-0.474873\pi\)
0.0788577 + 0.996886i \(0.474873\pi\)
\(194\) 1.76692e24 0.00865958
\(195\) 0 0
\(196\) 1.11034e26 0.483626
\(197\) 2.46843e26 1.01405 0.507025 0.861932i \(-0.330745\pi\)
0.507025 + 0.861932i \(0.330745\pi\)
\(198\) 0 0
\(199\) −2.44174e26 −0.893078 −0.446539 0.894764i \(-0.647344\pi\)
−0.446539 + 0.894764i \(0.647344\pi\)
\(200\) −9.25033e25 −0.319383
\(201\) 0 0
\(202\) 2.64374e26 0.814101
\(203\) 2.59506e26 0.754992
\(204\) 0 0
\(205\) −1.75171e26 −0.455295
\(206\) 2.33232e26 0.573212
\(207\) 0 0
\(208\) −1.58885e25 −0.0349426
\(209\) −1.35259e26 −0.281504
\(210\) 0 0
\(211\) −8.76884e26 −1.63566 −0.817832 0.575457i \(-0.804824\pi\)
−0.817832 + 0.575457i \(0.804824\pi\)
\(212\) −3.38430e26 −0.597869
\(213\) 0 0
\(214\) −4.75162e26 −0.753500
\(215\) −9.79808e26 −1.47265
\(216\) 0 0
\(217\) −5.78595e26 −0.781786
\(218\) 7.22865e26 0.926419
\(219\) 0 0
\(220\) 5.90020e26 0.680779
\(221\) 6.99683e26 0.766285
\(222\) 0 0
\(223\) −2.83665e26 −0.280091 −0.140046 0.990145i \(-0.544725\pi\)
−0.140046 + 0.990145i \(0.544725\pi\)
\(224\) −4.41194e26 −0.413787
\(225\) 0 0
\(226\) 9.75882e26 0.826324
\(227\) −7.65440e26 −0.616047 −0.308024 0.951379i \(-0.599668\pi\)
−0.308024 + 0.951379i \(0.599668\pi\)
\(228\) 0 0
\(229\) −9.18669e26 −0.668422 −0.334211 0.942498i \(-0.608470\pi\)
−0.334211 + 0.942498i \(0.608470\pi\)
\(230\) 3.15539e26 0.218365
\(231\) 0 0
\(232\) 2.90073e27 1.81717
\(233\) −6.69203e26 −0.398993 −0.199496 0.979899i \(-0.563931\pi\)
−0.199496 + 0.979899i \(0.563931\pi\)
\(234\) 0 0
\(235\) 1.38658e27 0.749317
\(236\) 1.87997e27 0.967527
\(237\) 0 0
\(238\) −8.12154e26 −0.379318
\(239\) 1.32134e27 0.588081 0.294041 0.955793i \(-0.405000\pi\)
0.294041 + 0.955793i \(0.405000\pi\)
\(240\) 0 0
\(241\) 9.67571e26 0.391279 0.195640 0.980676i \(-0.437322\pi\)
0.195640 + 0.980676i \(0.437322\pi\)
\(242\) 2.74828e25 0.0105970
\(243\) 0 0
\(244\) 1.25678e27 0.440834
\(245\) 2.81139e27 0.940829
\(246\) 0 0
\(247\) −5.02024e26 −0.153006
\(248\) −6.46749e27 −1.88166
\(249\) 0 0
\(250\) 1.90102e27 0.504286
\(251\) −2.70487e27 −0.685328 −0.342664 0.939458i \(-0.611329\pi\)
−0.342664 + 0.939458i \(0.611329\pi\)
\(252\) 0 0
\(253\) −1.29587e27 −0.299693
\(254\) −1.34989e27 −0.298339
\(255\) 0 0
\(256\) −5.11764e27 −1.03350
\(257\) 1.75032e27 0.337977 0.168988 0.985618i \(-0.445950\pi\)
0.168988 + 0.985618i \(0.445950\pi\)
\(258\) 0 0
\(259\) −3.61027e27 −0.637665
\(260\) 2.18990e27 0.370025
\(261\) 0 0
\(262\) −2.41000e27 −0.372865
\(263\) 5.05191e27 0.748108 0.374054 0.927407i \(-0.377967\pi\)
0.374054 + 0.927407i \(0.377967\pi\)
\(264\) 0 0
\(265\) −8.56909e27 −1.16307
\(266\) 5.82722e26 0.0757395
\(267\) 0 0
\(268\) −3.74706e27 −0.446828
\(269\) −2.79663e27 −0.319509 −0.159754 0.987157i \(-0.551070\pi\)
−0.159754 + 0.987157i \(0.551070\pi\)
\(270\) 0 0
\(271\) 3.12408e27 0.327775 0.163887 0.986479i \(-0.447597\pi\)
0.163887 + 0.986479i \(0.447597\pi\)
\(272\) −8.86579e26 −0.0891610
\(273\) 0 0
\(274\) −2.91352e27 −0.269331
\(275\) 3.56608e27 0.316130
\(276\) 0 0
\(277\) 2.33237e28 1.90230 0.951152 0.308723i \(-0.0999016\pi\)
0.951152 + 0.308723i \(0.0999016\pi\)
\(278\) 1.10974e28 0.868371
\(279\) 0 0
\(280\) −6.85650e27 −0.494066
\(281\) −1.54956e28 −1.07173 −0.535866 0.844303i \(-0.680015\pi\)
−0.535866 + 0.844303i \(0.680015\pi\)
\(282\) 0 0
\(283\) 5.17195e27 0.329694 0.164847 0.986319i \(-0.447287\pi\)
0.164847 + 0.986319i \(0.447287\pi\)
\(284\) −8.17772e27 −0.500578
\(285\) 0 0
\(286\) 6.27191e27 0.354153
\(287\) −3.09931e27 −0.168122
\(288\) 0 0
\(289\) 1.90747e28 0.955283
\(290\) 2.72290e28 1.31056
\(291\) 0 0
\(292\) −1.84605e28 −0.820995
\(293\) −9.81588e27 −0.419712 −0.209856 0.977732i \(-0.567300\pi\)
−0.209856 + 0.977732i \(0.567300\pi\)
\(294\) 0 0
\(295\) 4.76011e28 1.88219
\(296\) −4.03553e28 −1.53478
\(297\) 0 0
\(298\) 2.79902e27 0.0985189
\(299\) −4.80972e27 −0.162893
\(300\) 0 0
\(301\) −1.73358e28 −0.543788
\(302\) 2.10069e28 0.634280
\(303\) 0 0
\(304\) 6.36122e26 0.0178030
\(305\) 3.18217e28 0.857582
\(306\) 0 0
\(307\) −1.51274e28 −0.378158 −0.189079 0.981962i \(-0.560550\pi\)
−0.189079 + 0.981962i \(0.560550\pi\)
\(308\) 1.04393e28 0.251384
\(309\) 0 0
\(310\) −6.07101e28 −1.35707
\(311\) 5.90519e28 1.27201 0.636003 0.771686i \(-0.280586\pi\)
0.636003 + 0.771686i \(0.280586\pi\)
\(312\) 0 0
\(313\) 7.74763e27 0.155028 0.0775138 0.996991i \(-0.475302\pi\)
0.0775138 + 0.996991i \(0.475302\pi\)
\(314\) 8.23165e27 0.158780
\(315\) 0 0
\(316\) −1.63572e28 −0.293297
\(317\) −8.76539e28 −1.51562 −0.757809 0.652476i \(-0.773730\pi\)
−0.757809 + 0.652476i \(0.773730\pi\)
\(318\) 0 0
\(319\) −1.11826e29 −1.79866
\(320\) −5.10029e28 −0.791353
\(321\) 0 0
\(322\) 5.58286e27 0.0806333
\(323\) −2.80129e28 −0.390417
\(324\) 0 0
\(325\) 1.32358e28 0.171826
\(326\) 1.45323e28 0.182109
\(327\) 0 0
\(328\) −3.46439e28 −0.404648
\(329\) 2.45329e28 0.276692
\(330\) 0 0
\(331\) 1.28558e29 1.35231 0.676155 0.736759i \(-0.263645\pi\)
0.676155 + 0.736759i \(0.263645\pi\)
\(332\) 8.55777e28 0.869506
\(333\) 0 0
\(334\) 1.18196e29 1.12077
\(335\) −9.48761e28 −0.869244
\(336\) 0 0
\(337\) −1.77046e29 −1.51475 −0.757377 0.652978i \(-0.773520\pi\)
−0.757377 + 0.652978i \(0.773520\pi\)
\(338\) −5.42365e28 −0.448487
\(339\) 0 0
\(340\) 1.22196e29 0.944170
\(341\) 2.49327e29 1.86249
\(342\) 0 0
\(343\) 1.10337e29 0.770618
\(344\) −1.93779e29 −1.30883
\(345\) 0 0
\(346\) −3.47841e28 −0.219789
\(347\) −1.60648e27 −0.00981942 −0.00490971 0.999988i \(-0.501563\pi\)
−0.00490971 + 0.999988i \(0.501563\pi\)
\(348\) 0 0
\(349\) 1.80178e29 1.03088 0.515442 0.856925i \(-0.327628\pi\)
0.515442 + 0.856925i \(0.327628\pi\)
\(350\) −1.53633e28 −0.0850556
\(351\) 0 0
\(352\) 1.90118e29 0.985788
\(353\) 3.12751e29 1.56960 0.784801 0.619748i \(-0.212765\pi\)
0.784801 + 0.619748i \(0.212765\pi\)
\(354\) 0 0
\(355\) −2.07061e29 −0.973805
\(356\) 3.94398e27 0.0179581
\(357\) 0 0
\(358\) 2.57946e29 1.10122
\(359\) −1.81842e29 −0.751811 −0.375906 0.926658i \(-0.622668\pi\)
−0.375906 + 0.926658i \(0.622668\pi\)
\(360\) 0 0
\(361\) −2.37730e29 −0.922044
\(362\) 1.54509e29 0.580505
\(363\) 0 0
\(364\) 3.87461e28 0.136635
\(365\) −4.67422e29 −1.59713
\(366\) 0 0
\(367\) −2.10392e29 −0.675103 −0.337552 0.941307i \(-0.609599\pi\)
−0.337552 + 0.941307i \(0.609599\pi\)
\(368\) 6.09447e27 0.0189534
\(369\) 0 0
\(370\) −3.78813e29 −1.10689
\(371\) −1.51614e29 −0.429475
\(372\) 0 0
\(373\) −6.19125e29 −1.64864 −0.824322 0.566122i \(-0.808443\pi\)
−0.824322 + 0.566122i \(0.808443\pi\)
\(374\) 3.49972e29 0.903670
\(375\) 0 0
\(376\) 2.74227e29 0.665963
\(377\) −4.15049e29 −0.977628
\(378\) 0 0
\(379\) −8.16748e28 −0.181024 −0.0905122 0.995895i \(-0.528850\pi\)
−0.0905122 + 0.995895i \(0.528850\pi\)
\(380\) −8.76760e28 −0.188525
\(381\) 0 0
\(382\) −2.35166e29 −0.476042
\(383\) 7.92901e29 1.55752 0.778759 0.627323i \(-0.215849\pi\)
0.778759 + 0.627323i \(0.215849\pi\)
\(384\) 0 0
\(385\) 2.64324e29 0.489033
\(386\) 5.62953e28 0.101092
\(387\) 0 0
\(388\) −4.70370e27 −0.00795930
\(389\) 2.41701e29 0.397062 0.198531 0.980095i \(-0.436383\pi\)
0.198531 + 0.980095i \(0.436383\pi\)
\(390\) 0 0
\(391\) −2.68382e29 −0.415643
\(392\) 5.56013e29 0.836171
\(393\) 0 0
\(394\) 4.58258e29 0.649985
\(395\) −4.14167e29 −0.570570
\(396\) 0 0
\(397\) 4.74027e29 0.616187 0.308094 0.951356i \(-0.400309\pi\)
0.308094 + 0.951356i \(0.400309\pi\)
\(398\) −4.53303e29 −0.572445
\(399\) 0 0
\(400\) −1.67712e28 −0.0199928
\(401\) −9.97411e29 −1.15535 −0.577676 0.816266i \(-0.696040\pi\)
−0.577676 + 0.816266i \(0.696040\pi\)
\(402\) 0 0
\(403\) 9.25395e29 1.01232
\(404\) −7.03788e29 −0.748267
\(405\) 0 0
\(406\) 4.81766e29 0.483934
\(407\) 1.55573e30 1.51914
\(408\) 0 0
\(409\) 1.82542e29 0.168478 0.0842391 0.996446i \(-0.473154\pi\)
0.0842391 + 0.996446i \(0.473154\pi\)
\(410\) −3.25200e29 −0.291835
\(411\) 0 0
\(412\) −6.20885e29 −0.526857
\(413\) 8.42211e29 0.695016
\(414\) 0 0
\(415\) 2.16684e30 1.69151
\(416\) 7.05637e29 0.535807
\(417\) 0 0
\(418\) −2.51105e29 −0.180438
\(419\) −2.31769e30 −1.62029 −0.810146 0.586228i \(-0.800612\pi\)
−0.810146 + 0.586228i \(0.800612\pi\)
\(420\) 0 0
\(421\) 2.84805e29 0.188497 0.0942483 0.995549i \(-0.469955\pi\)
0.0942483 + 0.995549i \(0.469955\pi\)
\(422\) −1.62791e30 −1.04843
\(423\) 0 0
\(424\) −1.69473e30 −1.03369
\(425\) 7.38554e29 0.438439
\(426\) 0 0
\(427\) 5.63025e29 0.316670
\(428\) 1.26492e30 0.692566
\(429\) 0 0
\(430\) −1.81899e30 −0.943937
\(431\) 2.68991e30 1.35910 0.679548 0.733632i \(-0.262176\pi\)
0.679548 + 0.733632i \(0.262176\pi\)
\(432\) 0 0
\(433\) −2.83636e30 −1.35879 −0.679393 0.733774i \(-0.737757\pi\)
−0.679393 + 0.733774i \(0.737757\pi\)
\(434\) −1.07415e30 −0.501109
\(435\) 0 0
\(436\) −1.92433e30 −0.851501
\(437\) 1.92565e29 0.0829927
\(438\) 0 0
\(439\) −6.34373e29 −0.259419 −0.129710 0.991552i \(-0.541405\pi\)
−0.129710 + 0.991552i \(0.541405\pi\)
\(440\) 2.95459e30 1.17704
\(441\) 0 0
\(442\) 1.29894e30 0.491173
\(443\) −1.41158e30 −0.520072 −0.260036 0.965599i \(-0.583734\pi\)
−0.260036 + 0.965599i \(0.583734\pi\)
\(444\) 0 0
\(445\) 9.98622e28 0.0349350
\(446\) −5.26617e29 −0.179533
\(447\) 0 0
\(448\) −9.02399e29 −0.292214
\(449\) −3.28307e30 −1.03621 −0.518104 0.855318i \(-0.673362\pi\)
−0.518104 + 0.855318i \(0.673362\pi\)
\(450\) 0 0
\(451\) 1.33555e30 0.400525
\(452\) −2.59789e30 −0.759501
\(453\) 0 0
\(454\) −1.42102e30 −0.394874
\(455\) 9.81056e29 0.265805
\(456\) 0 0
\(457\) 3.77120e29 0.0971501 0.0485750 0.998820i \(-0.484532\pi\)
0.0485750 + 0.998820i \(0.484532\pi\)
\(458\) −1.70549e30 −0.428445
\(459\) 0 0
\(460\) −8.39994e29 −0.200706
\(461\) 1.82043e30 0.424243 0.212121 0.977243i \(-0.431963\pi\)
0.212121 + 0.977243i \(0.431963\pi\)
\(462\) 0 0
\(463\) 8.46419e30 1.87674 0.938370 0.345633i \(-0.112336\pi\)
0.938370 + 0.345633i \(0.112336\pi\)
\(464\) 5.25914e29 0.113752
\(465\) 0 0
\(466\) −1.24236e30 −0.255746
\(467\) −1.55072e30 −0.311451 −0.155725 0.987800i \(-0.549771\pi\)
−0.155725 + 0.987800i \(0.549771\pi\)
\(468\) 0 0
\(469\) −1.67865e30 −0.320976
\(470\) 2.57416e30 0.480297
\(471\) 0 0
\(472\) 9.41416e30 1.67282
\(473\) 7.47033e30 1.29550
\(474\) 0 0
\(475\) −5.29914e29 −0.0875444
\(476\) 2.16203e30 0.348643
\(477\) 0 0
\(478\) 2.45303e30 0.376948
\(479\) −9.58203e30 −1.43747 −0.718736 0.695284i \(-0.755279\pi\)
−0.718736 + 0.695284i \(0.755279\pi\)
\(480\) 0 0
\(481\) 5.77420e30 0.825703
\(482\) 1.79627e30 0.250802
\(483\) 0 0
\(484\) −7.31618e28 −0.00974008
\(485\) −1.19098e29 −0.0154837
\(486\) 0 0
\(487\) −4.05601e30 −0.502940 −0.251470 0.967865i \(-0.580914\pi\)
−0.251470 + 0.967865i \(0.580914\pi\)
\(488\) 6.29344e30 0.762185
\(489\) 0 0
\(490\) 5.21927e30 0.603052
\(491\) −5.18224e30 −0.584898 −0.292449 0.956281i \(-0.594470\pi\)
−0.292449 + 0.956281i \(0.594470\pi\)
\(492\) 0 0
\(493\) −2.31597e31 −2.49455
\(494\) −9.31995e29 −0.0980740
\(495\) 0 0
\(496\) −1.17258e30 −0.117789
\(497\) −3.66355e30 −0.359586
\(498\) 0 0
\(499\) 4.83971e30 0.453590 0.226795 0.973943i \(-0.427175\pi\)
0.226795 + 0.973943i \(0.427175\pi\)
\(500\) −5.06070e30 −0.463506
\(501\) 0 0
\(502\) −5.02152e30 −0.439281
\(503\) 1.61218e31 1.37842 0.689211 0.724561i \(-0.257957\pi\)
0.689211 + 0.724561i \(0.257957\pi\)
\(504\) 0 0
\(505\) −1.78200e31 −1.45565
\(506\) −2.40576e30 −0.192097
\(507\) 0 0
\(508\) 3.59352e30 0.274212
\(509\) −1.33185e31 −0.993577 −0.496789 0.867872i \(-0.665488\pi\)
−0.496789 + 0.867872i \(0.665488\pi\)
\(510\) 0 0
\(511\) −8.27014e30 −0.589756
\(512\) −1.82562e30 −0.127294
\(513\) 0 0
\(514\) 3.24943e30 0.216636
\(515\) −1.57209e31 −1.02493
\(516\) 0 0
\(517\) −1.05717e31 −0.659179
\(518\) −6.70238e30 −0.408730
\(519\) 0 0
\(520\) 1.09662e31 0.639759
\(521\) 2.34032e31 1.33549 0.667745 0.744390i \(-0.267260\pi\)
0.667745 + 0.744390i \(0.267260\pi\)
\(522\) 0 0
\(523\) 1.49020e31 0.813722 0.406861 0.913490i \(-0.366623\pi\)
0.406861 + 0.913490i \(0.366623\pi\)
\(524\) 6.41563e30 0.342712
\(525\) 0 0
\(526\) 9.37874e30 0.479522
\(527\) 5.16370e31 2.58308
\(528\) 0 0
\(529\) −1.90356e31 −0.911645
\(530\) −1.59083e31 −0.745507
\(531\) 0 0
\(532\) −1.55126e30 −0.0696146
\(533\) 4.95699e30 0.217698
\(534\) 0 0
\(535\) 3.20281e31 1.34729
\(536\) −1.87638e31 −0.772549
\(537\) 0 0
\(538\) −5.19187e30 −0.204799
\(539\) −2.14348e31 −0.827653
\(540\) 0 0
\(541\) 8.98916e30 0.332622 0.166311 0.986073i \(-0.446814\pi\)
0.166311 + 0.986073i \(0.446814\pi\)
\(542\) 5.79978e30 0.210097
\(543\) 0 0
\(544\) 3.93745e31 1.36719
\(545\) −4.87243e31 −1.65648
\(546\) 0 0
\(547\) 2.92608e31 0.953743 0.476871 0.878973i \(-0.341771\pi\)
0.476871 + 0.878973i \(0.341771\pi\)
\(548\) 7.75605e30 0.247551
\(549\) 0 0
\(550\) 6.62034e30 0.202633
\(551\) 1.66171e31 0.498095
\(552\) 0 0
\(553\) −7.32790e30 −0.210688
\(554\) 4.32999e31 1.21934
\(555\) 0 0
\(556\) −2.95422e31 −0.798148
\(557\) −3.88195e31 −1.02734 −0.513671 0.857987i \(-0.671715\pi\)
−0.513671 + 0.857987i \(0.671715\pi\)
\(558\) 0 0
\(559\) 2.77266e31 0.704144
\(560\) −1.24311e30 −0.0309277
\(561\) 0 0
\(562\) −2.87673e31 −0.686959
\(563\) −1.23043e31 −0.287878 −0.143939 0.989587i \(-0.545977\pi\)
−0.143939 + 0.989587i \(0.545977\pi\)
\(564\) 0 0
\(565\) −6.57788e31 −1.47751
\(566\) 9.60160e30 0.211327
\(567\) 0 0
\(568\) −4.09508e31 −0.865480
\(569\) 3.60714e31 0.747088 0.373544 0.927612i \(-0.378142\pi\)
0.373544 + 0.927612i \(0.378142\pi\)
\(570\) 0 0
\(571\) 4.10890e31 0.817354 0.408677 0.912679i \(-0.365990\pi\)
0.408677 + 0.912679i \(0.365990\pi\)
\(572\) −1.66964e31 −0.325513
\(573\) 0 0
\(574\) −5.75380e30 −0.107763
\(575\) −5.07693e30 −0.0932010
\(576\) 0 0
\(577\) −5.71556e31 −1.00818 −0.504089 0.863652i \(-0.668172\pi\)
−0.504089 + 0.863652i \(0.668172\pi\)
\(578\) 3.54117e31 0.612317
\(579\) 0 0
\(580\) −7.24862e31 −1.20457
\(581\) 3.83381e31 0.624604
\(582\) 0 0
\(583\) 6.53331e31 1.02316
\(584\) −9.24429e31 −1.41947
\(585\) 0 0
\(586\) −1.82229e31 −0.269027
\(587\) 2.06703e30 0.0299233 0.0149616 0.999888i \(-0.495237\pi\)
0.0149616 + 0.999888i \(0.495237\pi\)
\(588\) 0 0
\(589\) −3.70496e31 −0.515772
\(590\) 8.83703e31 1.20645
\(591\) 0 0
\(592\) −7.31658e30 −0.0960745
\(593\) 3.79598e31 0.488872 0.244436 0.969665i \(-0.421397\pi\)
0.244436 + 0.969665i \(0.421397\pi\)
\(594\) 0 0
\(595\) 5.47428e31 0.678238
\(596\) −7.45125e30 −0.0905519
\(597\) 0 0
\(598\) −8.92914e30 −0.104411
\(599\) −1.17051e32 −1.34266 −0.671331 0.741158i \(-0.734277\pi\)
−0.671331 + 0.741158i \(0.734277\pi\)
\(600\) 0 0
\(601\) 6.55159e31 0.723254 0.361627 0.932323i \(-0.382221\pi\)
0.361627 + 0.932323i \(0.382221\pi\)
\(602\) −3.21835e31 −0.348557
\(603\) 0 0
\(604\) −5.59222e31 −0.582987
\(605\) −1.85247e30 −0.0189480
\(606\) 0 0
\(607\) 7.95239e31 0.783119 0.391560 0.920153i \(-0.371936\pi\)
0.391560 + 0.920153i \(0.371936\pi\)
\(608\) −2.82513e31 −0.272990
\(609\) 0 0
\(610\) 5.90763e31 0.549693
\(611\) −3.92375e31 −0.358284
\(612\) 0 0
\(613\) 3.67217e30 0.0322944 0.0161472 0.999870i \(-0.494860\pi\)
0.0161472 + 0.999870i \(0.494860\pi\)
\(614\) −2.80837e31 −0.242391
\(615\) 0 0
\(616\) 5.22758e31 0.434633
\(617\) −3.41957e31 −0.279057 −0.139528 0.990218i \(-0.544559\pi\)
−0.139528 + 0.990218i \(0.544559\pi\)
\(618\) 0 0
\(619\) 2.21499e32 1.74152 0.870762 0.491704i \(-0.163626\pi\)
0.870762 + 0.491704i \(0.163626\pi\)
\(620\) 1.61616e32 1.24732
\(621\) 0 0
\(622\) 1.09628e32 0.815330
\(623\) 1.76687e30 0.0129001
\(624\) 0 0
\(625\) −1.72695e32 −1.21524
\(626\) 1.43833e31 0.0993695
\(627\) 0 0
\(628\) −2.19134e31 −0.145940
\(629\) 3.22200e32 2.10690
\(630\) 0 0
\(631\) 1.58296e32 0.998002 0.499001 0.866601i \(-0.333700\pi\)
0.499001 + 0.866601i \(0.333700\pi\)
\(632\) −8.19106e31 −0.507100
\(633\) 0 0
\(634\) −1.62727e32 −0.971481
\(635\) 9.09884e31 0.533443
\(636\) 0 0
\(637\) −7.95566e31 −0.449855
\(638\) −2.07602e32 −1.15290
\(639\) 0 0
\(640\) 1.14492e32 0.613346
\(641\) −1.97318e32 −1.03824 −0.519121 0.854700i \(-0.673741\pi\)
−0.519121 + 0.854700i \(0.673741\pi\)
\(642\) 0 0
\(643\) −2.91777e32 −1.48124 −0.740618 0.671926i \(-0.765467\pi\)
−0.740618 + 0.671926i \(0.765467\pi\)
\(644\) −1.48621e31 −0.0741127
\(645\) 0 0
\(646\) −5.20052e31 −0.250249
\(647\) 6.05387e31 0.286176 0.143088 0.989710i \(-0.454297\pi\)
0.143088 + 0.989710i \(0.454297\pi\)
\(648\) 0 0
\(649\) −3.62924e32 −1.65578
\(650\) 2.45719e31 0.110137
\(651\) 0 0
\(652\) −3.86862e31 −0.167382
\(653\) −2.70051e32 −1.14801 −0.574003 0.818853i \(-0.694610\pi\)
−0.574003 + 0.818853i \(0.694610\pi\)
\(654\) 0 0
\(655\) 1.62445e32 0.666700
\(656\) −6.28107e30 −0.0253302
\(657\) 0 0
\(658\) 4.55448e31 0.177354
\(659\) 3.41256e32 1.30587 0.652933 0.757416i \(-0.273538\pi\)
0.652933 + 0.757416i \(0.273538\pi\)
\(660\) 0 0
\(661\) −4.65199e32 −1.71918 −0.859592 0.510982i \(-0.829282\pi\)
−0.859592 + 0.510982i \(0.829282\pi\)
\(662\) 2.38664e32 0.866804
\(663\) 0 0
\(664\) 4.28540e32 1.50334
\(665\) −3.92781e31 −0.135426
\(666\) 0 0
\(667\) 1.59203e32 0.530279
\(668\) −3.14647e32 −1.03014
\(669\) 0 0
\(670\) −1.76135e32 −0.557168
\(671\) −2.42618e32 −0.754420
\(672\) 0 0
\(673\) −4.76445e32 −1.43166 −0.715829 0.698276i \(-0.753951\pi\)
−0.715829 + 0.698276i \(0.753951\pi\)
\(674\) −3.28682e32 −0.970927
\(675\) 0 0
\(676\) 1.44382e32 0.412218
\(677\) −4.41103e31 −0.123814 −0.0619071 0.998082i \(-0.519718\pi\)
−0.0619071 + 0.998082i \(0.519718\pi\)
\(678\) 0 0
\(679\) −2.10722e30 −0.00571751
\(680\) 6.11911e32 1.63243
\(681\) 0 0
\(682\) 4.62870e32 1.19382
\(683\) 2.77520e32 0.703812 0.351906 0.936035i \(-0.385534\pi\)
0.351906 + 0.936035i \(0.385534\pi\)
\(684\) 0 0
\(685\) 1.96384e32 0.481576
\(686\) 2.04838e32 0.493950
\(687\) 0 0
\(688\) −3.51328e31 −0.0819305
\(689\) 2.42488e32 0.556121
\(690\) 0 0
\(691\) 8.25247e32 1.83057 0.915286 0.402806i \(-0.131965\pi\)
0.915286 + 0.402806i \(0.131965\pi\)
\(692\) 9.25986e31 0.202015
\(693\) 0 0
\(694\) −2.98239e30 −0.00629405
\(695\) −7.48014e32 −1.55269
\(696\) 0 0
\(697\) 2.76599e32 0.555487
\(698\) 3.34496e32 0.660775
\(699\) 0 0
\(700\) 4.08986e31 0.0781773
\(701\) −1.99680e32 −0.375472 −0.187736 0.982220i \(-0.560115\pi\)
−0.187736 + 0.982220i \(0.560115\pi\)
\(702\) 0 0
\(703\) −2.31179e32 −0.420690
\(704\) 3.88860e32 0.696158
\(705\) 0 0
\(706\) 5.80614e32 1.00608
\(707\) −3.15291e32 −0.537512
\(708\) 0 0
\(709\) −1.31374e32 −0.216809 −0.108404 0.994107i \(-0.534574\pi\)
−0.108404 + 0.994107i \(0.534574\pi\)
\(710\) −3.84404e32 −0.624190
\(711\) 0 0
\(712\) 1.97499e31 0.0310489
\(713\) −3.54960e32 −0.549098
\(714\) 0 0
\(715\) −4.22755e32 −0.633241
\(716\) −6.86675e32 −1.01217
\(717\) 0 0
\(718\) −3.37586e32 −0.481896
\(719\) 8.81786e32 1.23874 0.619371 0.785098i \(-0.287388\pi\)
0.619371 + 0.785098i \(0.287388\pi\)
\(720\) 0 0
\(721\) −2.78151e32 −0.378464
\(722\) −4.41341e32 −0.591012
\(723\) 0 0
\(724\) −4.11317e32 −0.533560
\(725\) −4.38106e32 −0.559362
\(726\) 0 0
\(727\) −1.53127e32 −0.189412 −0.0947059 0.995505i \(-0.530191\pi\)
−0.0947059 + 0.995505i \(0.530191\pi\)
\(728\) 1.94025e32 0.236237
\(729\) 0 0
\(730\) −8.67757e32 −1.02373
\(731\) 1.54714e33 1.79672
\(732\) 0 0
\(733\) −4.26963e32 −0.480502 −0.240251 0.970711i \(-0.577230\pi\)
−0.240251 + 0.970711i \(0.577230\pi\)
\(734\) −3.90588e32 −0.432728
\(735\) 0 0
\(736\) −2.70666e32 −0.290629
\(737\) 7.23361e32 0.764679
\(738\) 0 0
\(739\) 6.20770e32 0.636091 0.318046 0.948075i \(-0.396973\pi\)
0.318046 + 0.948075i \(0.396973\pi\)
\(740\) 1.00844e33 1.01738
\(741\) 0 0
\(742\) −2.81467e32 −0.275285
\(743\) −8.09984e32 −0.780019 −0.390010 0.920811i \(-0.627528\pi\)
−0.390010 + 0.920811i \(0.627528\pi\)
\(744\) 0 0
\(745\) −1.88667e32 −0.176156
\(746\) −1.14939e33 −1.05675
\(747\) 0 0
\(748\) −9.31657e32 −0.830592
\(749\) 5.66675e32 0.497500
\(750\) 0 0
\(751\) 1.33627e33 1.13772 0.568859 0.822435i \(-0.307385\pi\)
0.568859 + 0.822435i \(0.307385\pi\)
\(752\) 4.97185e31 0.0416881
\(753\) 0 0
\(754\) −7.70528e32 −0.626640
\(755\) −1.41596e33 −1.13412
\(756\) 0 0
\(757\) 6.60404e32 0.513106 0.256553 0.966530i \(-0.417413\pi\)
0.256553 + 0.966530i \(0.417413\pi\)
\(758\) −1.51627e32 −0.116033
\(759\) 0 0
\(760\) −4.39047e32 −0.325953
\(761\) 1.77866e33 1.30068 0.650338 0.759645i \(-0.274627\pi\)
0.650338 + 0.759645i \(0.274627\pi\)
\(762\) 0 0
\(763\) −8.62084e32 −0.611670
\(764\) 6.26033e32 0.437546
\(765\) 0 0
\(766\) 1.47200e33 0.998338
\(767\) −1.34702e33 −0.899967
\(768\) 0 0
\(769\) 2.31911e33 1.50372 0.751861 0.659322i \(-0.229156\pi\)
0.751861 + 0.659322i \(0.229156\pi\)
\(770\) 4.90711e32 0.313460
\(771\) 0 0
\(772\) −1.49863e32 −0.0929172
\(773\) −7.88012e32 −0.481359 −0.240679 0.970605i \(-0.577370\pi\)
−0.240679 + 0.970605i \(0.577370\pi\)
\(774\) 0 0
\(775\) 9.76805e32 0.579213
\(776\) −2.35543e31 −0.0137613
\(777\) 0 0
\(778\) 4.48712e32 0.254508
\(779\) −1.98461e32 −0.110916
\(780\) 0 0
\(781\) 1.57869e33 0.856663
\(782\) −4.98245e32 −0.266419
\(783\) 0 0
\(784\) 1.00807e32 0.0523429
\(785\) −5.54850e32 −0.283906
\(786\) 0 0
\(787\) 2.06876e33 1.02802 0.514010 0.857784i \(-0.328160\pi\)
0.514010 + 0.857784i \(0.328160\pi\)
\(788\) −1.21992e33 −0.597422
\(789\) 0 0
\(790\) −7.68891e32 −0.365724
\(791\) −1.16383e33 −0.545582
\(792\) 0 0
\(793\) −9.00492e32 −0.410051
\(794\) 8.80020e32 0.394964
\(795\) 0 0
\(796\) 1.20673e33 0.526152
\(797\) 3.07381e33 1.32101 0.660506 0.750821i \(-0.270342\pi\)
0.660506 + 0.750821i \(0.270342\pi\)
\(798\) 0 0
\(799\) −2.18945e33 −0.914213
\(800\) 7.44839e32 0.306568
\(801\) 0 0
\(802\) −1.85167e33 −0.740557
\(803\) 3.56375e33 1.40501
\(804\) 0 0
\(805\) −3.76310e32 −0.144176
\(806\) 1.71797e33 0.648879
\(807\) 0 0
\(808\) −3.52430e33 −1.29373
\(809\) −1.24720e33 −0.451367 −0.225683 0.974201i \(-0.572461\pi\)
−0.225683 + 0.974201i \(0.572461\pi\)
\(810\) 0 0
\(811\) −1.90918e33 −0.671593 −0.335797 0.941934i \(-0.609006\pi\)
−0.335797 + 0.941934i \(0.609006\pi\)
\(812\) −1.28250e33 −0.444800
\(813\) 0 0
\(814\) 2.88818e33 0.973740
\(815\) −9.79540e32 −0.325619
\(816\) 0 0
\(817\) −1.11008e33 −0.358756
\(818\) 3.38884e32 0.107991
\(819\) 0 0
\(820\) 8.65713e32 0.268234
\(821\) −5.44054e33 −1.66225 −0.831123 0.556088i \(-0.812302\pi\)
−0.831123 + 0.556088i \(0.812302\pi\)
\(822\) 0 0
\(823\) −4.68984e33 −1.39335 −0.696675 0.717387i \(-0.745338\pi\)
−0.696675 + 0.717387i \(0.745338\pi\)
\(824\) −3.10915e33 −0.910917
\(825\) 0 0
\(826\) 1.56354e33 0.445491
\(827\) 2.85324e33 0.801723 0.400861 0.916139i \(-0.368711\pi\)
0.400861 + 0.916139i \(0.368711\pi\)
\(828\) 0 0
\(829\) −6.68450e32 −0.182680 −0.0913402 0.995820i \(-0.529115\pi\)
−0.0913402 + 0.995820i \(0.529115\pi\)
\(830\) 4.02268e33 1.08422
\(831\) 0 0
\(832\) 1.44328e33 0.378384
\(833\) −4.43925e33 −1.14787
\(834\) 0 0
\(835\) −7.96691e33 −2.00399
\(836\) 6.68465e32 0.165847
\(837\) 0 0
\(838\) −4.30273e33 −1.03857
\(839\) −9.91584e32 −0.236084 −0.118042 0.993009i \(-0.537662\pi\)
−0.118042 + 0.993009i \(0.537662\pi\)
\(840\) 0 0
\(841\) 9.42149e33 2.18256
\(842\) 5.28734e32 0.120823
\(843\) 0 0
\(844\) 4.33365e33 0.963643
\(845\) 3.65578e33 0.801915
\(846\) 0 0
\(847\) −3.27758e31 −0.00699672
\(848\) −3.07260e32 −0.0647074
\(849\) 0 0
\(850\) 1.37111e33 0.281031
\(851\) −2.21485e33 −0.447872
\(852\) 0 0
\(853\) −5.46591e33 −1.07584 −0.537921 0.842995i \(-0.680790\pi\)
−0.537921 + 0.842995i \(0.680790\pi\)
\(854\) 1.04524e33 0.202979
\(855\) 0 0
\(856\) 6.33425e33 1.19742
\(857\) −4.01521e33 −0.748909 −0.374454 0.927245i \(-0.622170\pi\)
−0.374454 + 0.927245i \(0.622170\pi\)
\(858\) 0 0
\(859\) 1.42976e33 0.259622 0.129811 0.991539i \(-0.458563\pi\)
0.129811 + 0.991539i \(0.458563\pi\)
\(860\) 4.84232e33 0.867603
\(861\) 0 0
\(862\) 4.99376e33 0.871152
\(863\) 1.03231e34 1.77699 0.888493 0.458890i \(-0.151753\pi\)
0.888493 + 0.458890i \(0.151753\pi\)
\(864\) 0 0
\(865\) 2.34461e33 0.392993
\(866\) −5.26564e33 −0.870955
\(867\) 0 0
\(868\) 2.85948e33 0.460585
\(869\) 3.15772e33 0.501934
\(870\) 0 0
\(871\) 2.68480e33 0.415627
\(872\) −9.63630e33 −1.47221
\(873\) 0 0
\(874\) 3.57492e32 0.0531967
\(875\) −2.26715e33 −0.332956
\(876\) 0 0
\(877\) 1.03125e34 1.47526 0.737629 0.675207i \(-0.235946\pi\)
0.737629 + 0.675207i \(0.235946\pi\)
\(878\) −1.17770e33 −0.166283
\(879\) 0 0
\(880\) 5.35679e32 0.0736807
\(881\) 1.04287e34 1.41582 0.707908 0.706305i \(-0.249639\pi\)
0.707908 + 0.706305i \(0.249639\pi\)
\(882\) 0 0
\(883\) −1.35131e34 −1.78734 −0.893669 0.448727i \(-0.851878\pi\)
−0.893669 + 0.448727i \(0.851878\pi\)
\(884\) −3.45791e33 −0.451453
\(885\) 0 0
\(886\) −2.62057e33 −0.333356
\(887\) 6.46528e33 0.811831 0.405916 0.913911i \(-0.366953\pi\)
0.405916 + 0.913911i \(0.366953\pi\)
\(888\) 0 0
\(889\) 1.60986e33 0.196979
\(890\) 1.85392e32 0.0223926
\(891\) 0 0
\(892\) 1.40190e33 0.165014
\(893\) 1.57093e33 0.182543
\(894\) 0 0
\(895\) −1.73867e34 −1.96903
\(896\) 2.02572e33 0.226484
\(897\) 0 0
\(898\) −6.09493e33 −0.664188
\(899\) −3.06308e34 −3.29550
\(900\) 0 0
\(901\) 1.35308e34 1.41902
\(902\) 2.47942e33 0.256729
\(903\) 0 0
\(904\) −1.30092e34 −1.31315
\(905\) −1.04146e34 −1.03797
\(906\) 0 0
\(907\) 1.63933e33 0.159288 0.0796439 0.996823i \(-0.474622\pi\)
0.0796439 + 0.996823i \(0.474622\pi\)
\(908\) 3.78289e33 0.362941
\(909\) 0 0
\(910\) 1.82131e33 0.170375
\(911\) 1.72363e34 1.59214 0.796072 0.605202i \(-0.206908\pi\)
0.796072 + 0.605202i \(0.206908\pi\)
\(912\) 0 0
\(913\) −1.65206e34 −1.48803
\(914\) 7.00114e32 0.0622712
\(915\) 0 0
\(916\) 4.54016e33 0.393798
\(917\) 2.87415e33 0.246185
\(918\) 0 0
\(919\) −1.99436e33 −0.166600 −0.0833000 0.996525i \(-0.526546\pi\)
−0.0833000 + 0.996525i \(0.526546\pi\)
\(920\) −4.20636e33 −0.347014
\(921\) 0 0
\(922\) 3.37959e33 0.271931
\(923\) 5.85941e33 0.465623
\(924\) 0 0
\(925\) 6.09499e33 0.472436
\(926\) 1.57136e34 1.20295
\(927\) 0 0
\(928\) −2.33568e34 −1.74426
\(929\) 1.66371e34 1.22715 0.613573 0.789638i \(-0.289732\pi\)
0.613573 + 0.789638i \(0.289732\pi\)
\(930\) 0 0
\(931\) 3.18517e33 0.229198
\(932\) 3.30727e33 0.235064
\(933\) 0 0
\(934\) −2.87887e33 −0.199634
\(935\) −2.35897e34 −1.61580
\(936\) 0 0
\(937\) 2.29325e34 1.53266 0.766331 0.642446i \(-0.222080\pi\)
0.766331 + 0.642446i \(0.222080\pi\)
\(938\) −3.11637e33 −0.205739
\(939\) 0 0
\(940\) −6.85264e33 −0.441456
\(941\) −9.97814e33 −0.634993 −0.317496 0.948259i \(-0.602842\pi\)
−0.317496 + 0.948259i \(0.602842\pi\)
\(942\) 0 0
\(943\) −1.90138e33 −0.118082
\(944\) 1.70682e33 0.104715
\(945\) 0 0
\(946\) 1.38685e34 0.830387
\(947\) −9.45460e33 −0.559266 −0.279633 0.960107i \(-0.590213\pi\)
−0.279633 + 0.960107i \(0.590213\pi\)
\(948\) 0 0
\(949\) 1.32271e34 0.763666
\(950\) −9.83772e32 −0.0561142
\(951\) 0 0
\(952\) 1.08266e34 0.602791
\(953\) 2.62732e33 0.144525 0.0722627 0.997386i \(-0.476978\pi\)
0.0722627 + 0.997386i \(0.476978\pi\)
\(954\) 0 0
\(955\) 1.58512e34 0.851186
\(956\) −6.53018e33 −0.346465
\(957\) 0 0
\(958\) −1.77888e34 −0.921390
\(959\) 3.47464e33 0.177826
\(960\) 0 0
\(961\) 4.82813e34 2.41246
\(962\) 1.07197e34 0.529259
\(963\) 0 0
\(964\) −4.78184e33 −0.230520
\(965\) −3.79456e33 −0.180758
\(966\) 0 0
\(967\) −1.74056e33 −0.0809625 −0.0404813 0.999180i \(-0.512889\pi\)
−0.0404813 + 0.999180i \(0.512889\pi\)
\(968\) −3.66366e32 −0.0168402
\(969\) 0 0
\(970\) −2.21103e32 −0.00992476
\(971\) 7.22515e33 0.320498 0.160249 0.987077i \(-0.448770\pi\)
0.160249 + 0.987077i \(0.448770\pi\)
\(972\) 0 0
\(973\) −1.32347e34 −0.573344
\(974\) −7.52989e33 −0.322374
\(975\) 0 0
\(976\) 1.14103e33 0.0477115
\(977\) −4.44312e33 −0.183612 −0.0918059 0.995777i \(-0.529264\pi\)
−0.0918059 + 0.995777i \(0.529264\pi\)
\(978\) 0 0
\(979\) −7.61377e32 −0.0307326
\(980\) −1.38942e34 −0.554284
\(981\) 0 0
\(982\) −9.62070e33 −0.374908
\(983\) 2.62175e34 1.00978 0.504890 0.863184i \(-0.331533\pi\)
0.504890 + 0.863184i \(0.331533\pi\)
\(984\) 0 0
\(985\) −3.08886e34 −1.16220
\(986\) −4.29954e34 −1.59896
\(987\) 0 0
\(988\) 2.48106e33 0.0901429
\(989\) −1.06353e34 −0.381937
\(990\) 0 0
\(991\) 2.84277e33 0.0997458 0.0498729 0.998756i \(-0.484118\pi\)
0.0498729 + 0.998756i \(0.484118\pi\)
\(992\) 5.20764e34 1.80616
\(993\) 0 0
\(994\) −6.80129e33 −0.230488
\(995\) 3.05547e34 1.02356
\(996\) 0 0
\(997\) 8.04071e33 0.263209 0.131604 0.991302i \(-0.457987\pi\)
0.131604 + 0.991302i \(0.457987\pi\)
\(998\) 8.98481e33 0.290742
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9.24.a.d.1.3 yes 4
3.2 odd 2 inner 9.24.a.d.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.24.a.d.1.2 4 3.2 odd 2 inner
9.24.a.d.1.3 yes 4 1.1 even 1 trivial