Properties

Label 450.7.b.a.449.1
Level $450$
Weight $7$
Character 450.449
Analytic conductor $103.524$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 450.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(103.524337629\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 450.449
Dual form 450.7.b.a.449.2

$q$-expansion

\(f(q)\) \(=\) \(q-5.65685 q^{2} +32.0000 q^{4} -484.000i q^{7} -181.019 q^{8} +O(q^{10})\) \(q-5.65685 q^{2} +32.0000 q^{4} -484.000i q^{7} -181.019 q^{8} +1340.67i q^{11} -3368.00i q^{13} +2737.92i q^{14} +1024.00 q^{16} -12.7279 q^{17} -5744.00 q^{19} -7584.00i q^{22} -3377.14 q^{23} +19052.3i q^{26} -15488.0i q^{28} -29354.8i q^{29} -39796.0 q^{31} -5792.62 q^{32} +72.0000 q^{34} +52526.0i q^{37} +32493.0 q^{38} -37042.5i q^{41} -3800.00i q^{43} +42901.6i q^{44} +19104.0 q^{46} -76791.8 q^{47} -116607. q^{49} -107776. i q^{52} +238738. q^{53} +87613.4i q^{56} +166056. i q^{58} +249841. i q^{59} +13250.0 q^{61} +225120. q^{62} +32768.0 q^{64} +168968. i q^{67} -407.294 q^{68} -531467. i q^{71} -236144. i q^{73} -297132. i q^{74} -183808. q^{76} +648886. q^{77} +35116.0 q^{79} +209544. i q^{82} -10980.0 q^{83} +21496.0i q^{86} -242688. i q^{88} +129328. i q^{89} -1.63011e6 q^{91} -108069. q^{92} +434400. q^{94} -321424. i q^{97} +659629. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 128 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 128 q^{4} + 4096 q^{16} - 22976 q^{19} - 159184 q^{31} + 288 q^{34} + 76416 q^{46} - 466428 q^{49} + 53000 q^{61} + 131072 q^{64} - 735232 q^{76} + 140464 q^{79} - 6520448 q^{91} + 1737600 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.65685 −0.707107
\(3\) 0 0
\(4\) 32.0000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 484.000i − 1.41108i −0.708671 0.705539i \(-0.750705\pi\)
0.708671 0.705539i \(-0.249295\pi\)
\(8\) −181.019 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 1340.67i 1.00727i 0.863917 + 0.503634i \(0.168004\pi\)
−0.863917 + 0.503634i \(0.831996\pi\)
\(12\) 0 0
\(13\) − 3368.00i − 1.53300i −0.642245 0.766500i \(-0.721997\pi\)
0.642245 0.766500i \(-0.278003\pi\)
\(14\) 2737.92i 0.997783i
\(15\) 0 0
\(16\) 1024.00 0.250000
\(17\) −12.7279 −0.00259066 −0.00129533 0.999999i \(-0.500412\pi\)
−0.00129533 + 0.999999i \(0.500412\pi\)
\(18\) 0 0
\(19\) −5744.00 −0.837440 −0.418720 0.908115i \(-0.637521\pi\)
−0.418720 + 0.908115i \(0.637521\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 7584.00i − 0.712246i
\(23\) −3377.14 −0.277566 −0.138783 0.990323i \(-0.544319\pi\)
−0.138783 + 0.990323i \(0.544319\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 19052.3i 1.08399i
\(27\) 0 0
\(28\) − 15488.0i − 0.705539i
\(29\) − 29354.8i − 1.20361i −0.798643 0.601805i \(-0.794449\pi\)
0.798643 0.601805i \(-0.205551\pi\)
\(30\) 0 0
\(31\) −39796.0 −1.33584 −0.667920 0.744233i \(-0.732815\pi\)
−0.667920 + 0.744233i \(0.732815\pi\)
\(32\) −5792.62 −0.176777
\(33\) 0 0
\(34\) 72.0000 0.00183187
\(35\) 0 0
\(36\) 0 0
\(37\) 52526.0i 1.03698i 0.855085 + 0.518489i \(0.173505\pi\)
−0.855085 + 0.518489i \(0.826495\pi\)
\(38\) 32493.0 0.592159
\(39\) 0 0
\(40\) 0 0
\(41\) − 37042.5i − 0.537463i −0.963215 0.268732i \(-0.913396\pi\)
0.963215 0.268732i \(-0.0866045\pi\)
\(42\) 0 0
\(43\) − 3800.00i − 0.0477945i −0.999714 0.0238973i \(-0.992393\pi\)
0.999714 0.0238973i \(-0.00760746\pi\)
\(44\) 42901.6i 0.503634i
\(45\) 0 0
\(46\) 19104.0 0.196269
\(47\) −76791.8 −0.739641 −0.369821 0.929103i \(-0.620581\pi\)
−0.369821 + 0.929103i \(0.620581\pi\)
\(48\) 0 0
\(49\) −116607. −0.991143
\(50\) 0 0
\(51\) 0 0
\(52\) − 107776.i − 0.766500i
\(53\) 238738. 1.60359 0.801795 0.597599i \(-0.203879\pi\)
0.801795 + 0.597599i \(0.203879\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 87613.4i 0.498892i
\(57\) 0 0
\(58\) 166056.i 0.851080i
\(59\) 249841.i 1.21649i 0.793751 + 0.608243i \(0.208125\pi\)
−0.793751 + 0.608243i \(0.791875\pi\)
\(60\) 0 0
\(61\) 13250.0 0.0583749 0.0291875 0.999574i \(-0.490708\pi\)
0.0291875 + 0.999574i \(0.490708\pi\)
\(62\) 225120. 0.944581
\(63\) 0 0
\(64\) 32768.0 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 168968.i 0.561798i 0.959737 + 0.280899i \(0.0906326\pi\)
−0.959737 + 0.280899i \(0.909367\pi\)
\(68\) −407.294 −0.00129533
\(69\) 0 0
\(70\) 0 0
\(71\) − 531467.i − 1.48491i −0.669894 0.742457i \(-0.733660\pi\)
0.669894 0.742457i \(-0.266340\pi\)
\(72\) 0 0
\(73\) − 236144.i − 0.607027i −0.952827 0.303514i \(-0.901840\pi\)
0.952827 0.303514i \(-0.0981598\pi\)
\(74\) − 297132.i − 0.733254i
\(75\) 0 0
\(76\) −183808. −0.418720
\(77\) 648886. 1.42134
\(78\) 0 0
\(79\) 35116.0 0.0712236 0.0356118 0.999366i \(-0.488662\pi\)
0.0356118 + 0.999366i \(0.488662\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 209544.i 0.380044i
\(83\) −10980.0 −0.0192029 −0.00960144 0.999954i \(-0.503056\pi\)
−0.00960144 + 0.999954i \(0.503056\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 21496.0i 0.0337958i
\(87\) 0 0
\(88\) − 242688.i − 0.356123i
\(89\) 129328.i 0.183453i 0.995784 + 0.0917263i \(0.0292385\pi\)
−0.995784 + 0.0917263i \(0.970762\pi\)
\(90\) 0 0
\(91\) −1.63011e6 −2.16318
\(92\) −108069. −0.138783
\(93\) 0 0
\(94\) 434400. 0.523005
\(95\) 0 0
\(96\) 0 0
\(97\) − 321424.i − 0.352179i −0.984374 0.176089i \(-0.943655\pi\)
0.984374 0.176089i \(-0.0563448\pi\)
\(98\) 659629. 0.700844
\(99\) 0 0
\(100\) 0 0
\(101\) 668780.i 0.649111i 0.945867 + 0.324556i \(0.105215\pi\)
−0.945867 + 0.324556i \(0.894785\pi\)
\(102\) 0 0
\(103\) − 1.99341e6i − 1.82425i −0.409907 0.912127i \(-0.634439\pi\)
0.409907 0.912127i \(-0.365561\pi\)
\(104\) 609673.i 0.541997i
\(105\) 0 0
\(106\) −1.35050e6 −1.13391
\(107\) 260668. 0.212783 0.106391 0.994324i \(-0.466070\pi\)
0.106391 + 0.994324i \(0.466070\pi\)
\(108\) 0 0
\(109\) −194456. −0.150156 −0.0750779 0.997178i \(-0.523921\pi\)
−0.0750779 + 0.997178i \(0.523921\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 495616.i − 0.352770i
\(113\) −821897. −0.569616 −0.284808 0.958585i \(-0.591930\pi\)
−0.284808 + 0.958585i \(0.591930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) − 939355.i − 0.601805i
\(117\) 0 0
\(118\) − 1.41331e6i − 0.860185i
\(119\) 6160.31i 0.00365563i
\(120\) 0 0
\(121\) −25847.0 −0.0145900
\(122\) −74953.3 −0.0412773
\(123\) 0 0
\(124\) −1.27347e6 −0.667920
\(125\) 0 0
\(126\) 0 0
\(127\) 3.05721e6i 1.49250i 0.665666 + 0.746250i \(0.268148\pi\)
−0.665666 + 0.746250i \(0.731852\pi\)
\(128\) −185364. −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 3.07388e6i 1.36733i 0.729797 + 0.683664i \(0.239615\pi\)
−0.729797 + 0.683664i \(0.760385\pi\)
\(132\) 0 0
\(133\) 2.78010e6i 1.18169i
\(134\) − 955827.i − 0.397251i
\(135\) 0 0
\(136\) 2304.00 0.000915937 0
\(137\) 4.48412e6 1.74388 0.871938 0.489617i \(-0.162863\pi\)
0.871938 + 0.489617i \(0.162863\pi\)
\(138\) 0 0
\(139\) 1.09233e6 0.406732 0.203366 0.979103i \(-0.434812\pi\)
0.203366 + 0.979103i \(0.434812\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 3.00643e6i 1.04999i
\(143\) 4.51539e6 1.54414
\(144\) 0 0
\(145\) 0 0
\(146\) 1.33583e6i 0.429233i
\(147\) 0 0
\(148\) 1.68083e6i 0.518489i
\(149\) 2.22087e6i 0.671375i 0.941973 + 0.335687i \(0.108969\pi\)
−0.941973 + 0.335687i \(0.891031\pi\)
\(150\) 0 0
\(151\) −4.07871e6 −1.18465 −0.592327 0.805697i \(-0.701791\pi\)
−0.592327 + 0.805697i \(0.701791\pi\)
\(152\) 1.03978e6 0.296080
\(153\) 0 0
\(154\) −3.67066e6 −1.00504
\(155\) 0 0
\(156\) 0 0
\(157\) 6.15568e6i 1.59066i 0.606178 + 0.795329i \(0.292702\pi\)
−0.606178 + 0.795329i \(0.707298\pi\)
\(158\) −198646. −0.0503627
\(159\) 0 0
\(160\) 0 0
\(161\) 1.63454e6i 0.391667i
\(162\) 0 0
\(163\) − 800696.i − 0.184886i −0.995718 0.0924432i \(-0.970532\pi\)
0.995718 0.0924432i \(-0.0294676\pi\)
\(164\) − 1.18536e6i − 0.268732i
\(165\) 0 0
\(166\) 62112.0 0.0135785
\(167\) −4.80467e6 −1.03161 −0.515804 0.856707i \(-0.672507\pi\)
−0.515804 + 0.856707i \(0.672507\pi\)
\(168\) 0 0
\(169\) −6.51661e6 −1.35009
\(170\) 0 0
\(171\) 0 0
\(172\) − 121600.i − 0.0238973i
\(173\) −3.56992e6 −0.689478 −0.344739 0.938699i \(-0.612033\pi\)
−0.344739 + 0.938699i \(0.612033\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.37285e6i 0.251817i
\(177\) 0 0
\(178\) − 731592.i − 0.129721i
\(179\) 7.43698e6i 1.29669i 0.761345 + 0.648347i \(0.224539\pi\)
−0.761345 + 0.648347i \(0.775461\pi\)
\(180\) 0 0
\(181\) −1.03812e7 −1.75070 −0.875350 0.483491i \(-0.839369\pi\)
−0.875350 + 0.483491i \(0.839369\pi\)
\(182\) 9.22131e6 1.52960
\(183\) 0 0
\(184\) 611328. 0.0981343
\(185\) 0 0
\(186\) 0 0
\(187\) − 17064.0i − 0.00260949i
\(188\) −2.45734e6 −0.369821
\(189\) 0 0
\(190\) 0 0
\(191\) 1.29941e7i 1.86485i 0.361360 + 0.932426i \(0.382313\pi\)
−0.361360 + 0.932426i \(0.617687\pi\)
\(192\) 0 0
\(193\) 3.93195e6i 0.546936i 0.961881 + 0.273468i \(0.0881708\pi\)
−0.961881 + 0.273468i \(0.911829\pi\)
\(194\) 1.81825e6i 0.249028i
\(195\) 0 0
\(196\) −3.73142e6 −0.495572
\(197\) −5.37967e6 −0.703651 −0.351825 0.936066i \(-0.614439\pi\)
−0.351825 + 0.936066i \(0.614439\pi\)
\(198\) 0 0
\(199\) 565900. 0.0718093 0.0359046 0.999355i \(-0.488569\pi\)
0.0359046 + 0.999355i \(0.488569\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 3.78319e6i − 0.458991i
\(203\) −1.42077e7 −1.69839
\(204\) 0 0
\(205\) 0 0
\(206\) 1.12764e7i 1.28994i
\(207\) 0 0
\(208\) − 3.44883e6i − 0.383250i
\(209\) − 7.70083e6i − 0.843527i
\(210\) 0 0
\(211\) −1.35165e7 −1.43885 −0.719427 0.694568i \(-0.755596\pi\)
−0.719427 + 0.694568i \(0.755596\pi\)
\(212\) 7.63960e6 0.801795
\(213\) 0 0
\(214\) −1.47456e6 −0.150460
\(215\) 0 0
\(216\) 0 0
\(217\) 1.92613e7i 1.88497i
\(218\) 1.10001e6 0.106176
\(219\) 0 0
\(220\) 0 0
\(221\) 42867.6i 0.00397148i
\(222\) 0 0
\(223\) 5.35484e6i 0.482872i 0.970417 + 0.241436i \(0.0776183\pi\)
−0.970417 + 0.241436i \(0.922382\pi\)
\(224\) 2.80363e6i 0.249446i
\(225\) 0 0
\(226\) 4.64935e6 0.402779
\(227\) 1.36063e7 1.16322 0.581612 0.813466i \(-0.302422\pi\)
0.581612 + 0.813466i \(0.302422\pi\)
\(228\) 0 0
\(229\) −4.34641e6 −0.361930 −0.180965 0.983490i \(-0.557922\pi\)
−0.180965 + 0.983490i \(0.557922\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5.31379e6i 0.425540i
\(233\) −2.02333e7 −1.59956 −0.799778 0.600297i \(-0.795049\pi\)
−0.799778 + 0.600297i \(0.795049\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 7.99490e6i 0.608243i
\(237\) 0 0
\(238\) − 34848.0i − 0.00258492i
\(239\) 2.03947e7i 1.49391i 0.664877 + 0.746953i \(0.268484\pi\)
−0.664877 + 0.746953i \(0.731516\pi\)
\(240\) 0 0
\(241\) −3.12093e6 −0.222963 −0.111481 0.993767i \(-0.535560\pi\)
−0.111481 + 0.993767i \(0.535560\pi\)
\(242\) 146213. 0.0103167
\(243\) 0 0
\(244\) 424000. 0.0291875
\(245\) 0 0
\(246\) 0 0
\(247\) 1.93458e7i 1.28379i
\(248\) 7.20385e6 0.472291
\(249\) 0 0
\(250\) 0 0
\(251\) 5.09519e6i 0.322210i 0.986937 + 0.161105i \(0.0515058\pi\)
−0.986937 + 0.161105i \(0.948494\pi\)
\(252\) 0 0
\(253\) − 4.52765e6i − 0.279583i
\(254\) − 1.72942e7i − 1.05536i
\(255\) 0 0
\(256\) 1.04858e6 0.0625000
\(257\) −1.44374e7 −0.850529 −0.425264 0.905069i \(-0.639819\pi\)
−0.425264 + 0.905069i \(0.639819\pi\)
\(258\) 0 0
\(259\) 2.54226e7 1.46326
\(260\) 0 0
\(261\) 0 0
\(262\) − 1.73885e7i − 0.966847i
\(263\) 3.12567e7 1.71821 0.859104 0.511801i \(-0.171022\pi\)
0.859104 + 0.511801i \(0.171022\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) − 1.57266e7i − 0.835584i
\(267\) 0 0
\(268\) 5.40698e6i 0.280899i
\(269\) − 251338.i − 0.0129122i −0.999979 0.00645612i \(-0.997945\pi\)
0.999979 0.00645612i \(-0.00205506\pi\)
\(270\) 0 0
\(271\) 2.96399e7 1.48925 0.744627 0.667481i \(-0.232627\pi\)
0.744627 + 0.667481i \(0.232627\pi\)
\(272\) −13033.4 −0.000647665 0
\(273\) 0 0
\(274\) −2.53660e7 −1.23311
\(275\) 0 0
\(276\) 0 0
\(277\) 1.32213e7i 0.622062i 0.950400 + 0.311031i \(0.100674\pi\)
−0.950400 + 0.311031i \(0.899326\pi\)
\(278\) −6.17914e6 −0.287603
\(279\) 0 0
\(280\) 0 0
\(281\) − 6.12360e6i − 0.275987i −0.990433 0.137993i \(-0.955935\pi\)
0.990433 0.137993i \(-0.0440652\pi\)
\(282\) 0 0
\(283\) 6.74325e6i 0.297516i 0.988874 + 0.148758i \(0.0475275\pi\)
−0.988874 + 0.148758i \(0.952473\pi\)
\(284\) − 1.70069e7i − 0.742457i
\(285\) 0 0
\(286\) −2.55429e7 −1.09187
\(287\) −1.79286e7 −0.758403
\(288\) 0 0
\(289\) −2.41374e7 −0.999993
\(290\) 0 0
\(291\) 0 0
\(292\) − 7.55661e6i − 0.303514i
\(293\) −1.00239e7 −0.398505 −0.199253 0.979948i \(-0.563851\pi\)
−0.199253 + 0.979948i \(0.563851\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) − 9.50822e6i − 0.366627i
\(297\) 0 0
\(298\) − 1.25632e7i − 0.474734i
\(299\) 1.13742e7i 0.425508i
\(300\) 0 0
\(301\) −1.83920e6 −0.0674418
\(302\) 2.30727e7 0.837677
\(303\) 0 0
\(304\) −5.88186e6 −0.209360
\(305\) 0 0
\(306\) 0 0
\(307\) − 5.23060e6i − 0.180774i −0.995907 0.0903871i \(-0.971190\pi\)
0.995907 0.0903871i \(-0.0288104\pi\)
\(308\) 2.07644e7 0.710668
\(309\) 0 0
\(310\) 0 0
\(311\) 3.12221e7i 1.03796i 0.854786 + 0.518981i \(0.173688\pi\)
−0.854786 + 0.518981i \(0.826312\pi\)
\(312\) 0 0
\(313\) − 2.24778e7i − 0.733029i −0.930412 0.366515i \(-0.880551\pi\)
0.930412 0.366515i \(-0.119449\pi\)
\(314\) − 3.48218e7i − 1.12477i
\(315\) 0 0
\(316\) 1.12371e6 0.0356118
\(317\) 2.76211e7 0.867088 0.433544 0.901132i \(-0.357263\pi\)
0.433544 + 0.901132i \(0.357263\pi\)
\(318\) 0 0
\(319\) 3.93553e7 1.21236
\(320\) 0 0
\(321\) 0 0
\(322\) − 9.24634e6i − 0.276950i
\(323\) 73109.2 0.00216952
\(324\) 0 0
\(325\) 0 0
\(326\) 4.52942e6i 0.130734i
\(327\) 0 0
\(328\) 6.70541e6i 0.190022i
\(329\) 3.71672e7i 1.04369i
\(330\) 0 0
\(331\) −5.76138e6 −0.158870 −0.0794352 0.996840i \(-0.525312\pi\)
−0.0794352 + 0.996840i \(0.525312\pi\)
\(332\) −351359. −0.00960144
\(333\) 0 0
\(334\) 2.71793e7 0.729456
\(335\) 0 0
\(336\) 0 0
\(337\) − 4.01052e7i − 1.04788i −0.851756 0.523939i \(-0.824462\pi\)
0.851756 0.523939i \(-0.175538\pi\)
\(338\) 3.68635e7 0.954656
\(339\) 0 0
\(340\) 0 0
\(341\) − 5.33535e7i − 1.34555i
\(342\) 0 0
\(343\) − 504328.i − 0.0124977i
\(344\) 687873.i 0.0168979i
\(345\) 0 0
\(346\) 2.01945e7 0.487535
\(347\) −6.78127e7 −1.62302 −0.811508 0.584341i \(-0.801353\pi\)
−0.811508 + 0.584341i \(0.801353\pi\)
\(348\) 0 0
\(349\) 4.20638e7 0.989538 0.494769 0.869024i \(-0.335253\pi\)
0.494769 + 0.869024i \(0.335253\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 7.76602e6i − 0.178062i
\(353\) −1.75976e7 −0.400063 −0.200032 0.979789i \(-0.564104\pi\)
−0.200032 + 0.979789i \(0.564104\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 4.13851e6i 0.0917263i
\(357\) 0 0
\(358\) − 4.20699e7i − 0.916901i
\(359\) 1.39920e7i 0.302410i 0.988502 + 0.151205i \(0.0483154\pi\)
−0.988502 + 0.151205i \(0.951685\pi\)
\(360\) 0 0
\(361\) −1.40523e7 −0.298694
\(362\) 5.87249e7 1.23793
\(363\) 0 0
\(364\) −5.21636e7 −1.08159
\(365\) 0 0
\(366\) 0 0
\(367\) − 2.65855e7i − 0.537832i −0.963164 0.268916i \(-0.913335\pi\)
0.963164 0.268916i \(-0.0866653\pi\)
\(368\) −3.45819e6 −0.0693914
\(369\) 0 0
\(370\) 0 0
\(371\) − 1.15549e8i − 2.26279i
\(372\) 0 0
\(373\) − 1.78829e7i − 0.344598i −0.985045 0.172299i \(-0.944881\pi\)
0.985045 0.172299i \(-0.0551195\pi\)
\(374\) 96528.6i 0.00184519i
\(375\) 0 0
\(376\) 1.39008e7 0.261503
\(377\) −9.88671e7 −1.84513
\(378\) 0 0
\(379\) −7.20978e7 −1.32435 −0.662177 0.749347i \(-0.730367\pi\)
−0.662177 + 0.749347i \(0.730367\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 7.35055e7i − 1.31865i
\(383\) 8.68648e6 0.154614 0.0773068 0.997007i \(-0.475368\pi\)
0.0773068 + 0.997007i \(0.475368\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 2.22425e7i − 0.386742i
\(387\) 0 0
\(388\) − 1.02856e7i − 0.176089i
\(389\) 4.94411e7i 0.839923i 0.907542 + 0.419962i \(0.137956\pi\)
−0.907542 + 0.419962i \(0.862044\pi\)
\(390\) 0 0
\(391\) 42984.0 0.000719079 0
\(392\) 2.11081e7 0.350422
\(393\) 0 0
\(394\) 3.04320e7 0.497556
\(395\) 0 0
\(396\) 0 0
\(397\) 1.56911e7i 0.250774i 0.992108 + 0.125387i \(0.0400172\pi\)
−0.992108 + 0.125387i \(0.959983\pi\)
\(398\) −3.20121e6 −0.0507768
\(399\) 0 0
\(400\) 0 0
\(401\) − 4.74514e7i − 0.735895i −0.929847 0.367947i \(-0.880061\pi\)
0.929847 0.367947i \(-0.119939\pi\)
\(402\) 0 0
\(403\) 1.34033e8i 2.04784i
\(404\) 2.14010e7i 0.324556i
\(405\) 0 0
\(406\) 8.03711e7 1.20094
\(407\) −7.04203e7 −1.04451
\(408\) 0 0
\(409\) 1.15512e8 1.68832 0.844162 0.536088i \(-0.180099\pi\)
0.844162 + 0.536088i \(0.180099\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 6.37892e7i − 0.912127i
\(413\) 1.20923e8 1.71656
\(414\) 0 0
\(415\) 0 0
\(416\) 1.95095e7i 0.270999i
\(417\) 0 0
\(418\) 4.35625e7i 0.596464i
\(419\) − 1.46693e8i − 1.99420i −0.0761306 0.997098i \(-0.524257\pi\)
0.0761306 0.997098i \(-0.475743\pi\)
\(420\) 0 0
\(421\) 1.39239e8 1.86601 0.933005 0.359863i \(-0.117176\pi\)
0.933005 + 0.359863i \(0.117176\pi\)
\(422\) 7.64609e7 1.01742
\(423\) 0 0
\(424\) −4.32161e7 −0.566955
\(425\) 0 0
\(426\) 0 0
\(427\) − 6.41300e6i − 0.0823716i
\(428\) 8.34137e6 0.106391
\(429\) 0 0
\(430\) 0 0
\(431\) − 1.00392e8i − 1.25391i −0.779056 0.626954i \(-0.784301\pi\)
0.779056 0.626954i \(-0.215699\pi\)
\(432\) 0 0
\(433\) 4.00631e7i 0.493493i 0.969080 + 0.246747i \(0.0793616\pi\)
−0.969080 + 0.246747i \(0.920638\pi\)
\(434\) − 1.08958e8i − 1.33288i
\(435\) 0 0
\(436\) −6.22259e6 −0.0750779
\(437\) 1.93983e7 0.232445
\(438\) 0 0
\(439\) 1.38592e8 1.63811 0.819057 0.573712i \(-0.194497\pi\)
0.819057 + 0.573712i \(0.194497\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 242496.i − 0.00280826i
\(443\) −1.11443e8 −1.28186 −0.640929 0.767600i \(-0.721451\pi\)
−0.640929 + 0.767600i \(0.721451\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) − 3.02915e7i − 0.341442i
\(447\) 0 0
\(448\) − 1.58597e7i − 0.176385i
\(449\) 6.11166e7i 0.675181i 0.941293 + 0.337591i \(0.109612\pi\)
−0.941293 + 0.337591i \(0.890388\pi\)
\(450\) 0 0
\(451\) 4.96619e7 0.541370
\(452\) −2.63007e7 −0.284808
\(453\) 0 0
\(454\) −7.69691e7 −0.822524
\(455\) 0 0
\(456\) 0 0
\(457\) 3.56665e7i 0.373690i 0.982389 + 0.186845i \(0.0598262\pi\)
−0.982389 + 0.186845i \(0.940174\pi\)
\(458\) 2.45870e7 0.255923
\(459\) 0 0
\(460\) 0 0
\(461\) 1.51983e8i 1.55128i 0.631173 + 0.775642i \(0.282574\pi\)
−0.631173 + 0.775642i \(0.717426\pi\)
\(462\) 0 0
\(463\) − 1.14978e8i − 1.15844i −0.815173 0.579218i \(-0.803358\pi\)
0.815173 0.579218i \(-0.196642\pi\)
\(464\) − 3.00593e7i − 0.300902i
\(465\) 0 0
\(466\) 1.14457e8 1.13106
\(467\) −8.81705e7 −0.865711 −0.432855 0.901463i \(-0.642494\pi\)
−0.432855 + 0.901463i \(0.642494\pi\)
\(468\) 0 0
\(469\) 8.17805e7 0.792741
\(470\) 0 0
\(471\) 0 0
\(472\) − 4.52260e7i − 0.430093i
\(473\) 5.09456e6 0.0481419
\(474\) 0 0
\(475\) 0 0
\(476\) 197130.i 0.00182781i
\(477\) 0 0
\(478\) − 1.15370e8i − 1.05635i
\(479\) 8.94388e7i 0.813803i 0.913472 + 0.406902i \(0.133391\pi\)
−0.913472 + 0.406902i \(0.866609\pi\)
\(480\) 0 0
\(481\) 1.76908e8 1.58969
\(482\) 1.76546e7 0.157659
\(483\) 0 0
\(484\) −827104. −0.00729498
\(485\) 0 0
\(486\) 0 0
\(487\) − 7.51688e7i − 0.650805i −0.945576 0.325403i \(-0.894500\pi\)
0.945576 0.325403i \(-0.105500\pi\)
\(488\) −2.39851e6 −0.0206387
\(489\) 0 0
\(490\) 0 0
\(491\) 4.50822e7i 0.380856i 0.981701 + 0.190428i \(0.0609876\pi\)
−0.981701 + 0.190428i \(0.939012\pi\)
\(492\) 0 0
\(493\) 373626.i 0.00311815i
\(494\) − 1.09436e8i − 0.907780i
\(495\) 0 0
\(496\) −4.07511e7 −0.333960
\(497\) −2.57230e8 −2.09533
\(498\) 0 0
\(499\) −9.15458e7 −0.736778 −0.368389 0.929672i \(-0.620091\pi\)
−0.368389 + 0.929672i \(0.620091\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 2.88228e7i − 0.227837i
\(503\) −1.61043e8 −1.26543 −0.632713 0.774386i \(-0.718059\pi\)
−0.632713 + 0.774386i \(0.718059\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.56122e7i 0.197695i
\(507\) 0 0
\(508\) 9.78308e7i 0.746250i
\(509\) − 2.39995e7i − 0.181990i −0.995851 0.0909951i \(-0.970995\pi\)
0.995851 0.0909951i \(-0.0290048\pi\)
\(510\) 0 0
\(511\) −1.14294e8 −0.856564
\(512\) −5.93164e6 −0.0441942
\(513\) 0 0
\(514\) 8.16702e7 0.601415
\(515\) 0 0
\(516\) 0 0
\(517\) − 1.02953e8i − 0.745018i
\(518\) −1.43812e8 −1.03468
\(519\) 0 0
\(520\) 0 0
\(521\) − 9.00897e7i − 0.637033i −0.947917 0.318517i \(-0.896815\pi\)
0.947917 0.318517i \(-0.103185\pi\)
\(522\) 0 0
\(523\) 3.77691e7i 0.264016i 0.991249 + 0.132008i \(0.0421425\pi\)
−0.991249 + 0.132008i \(0.957857\pi\)
\(524\) 9.83641e7i 0.683664i
\(525\) 0 0
\(526\) −1.76815e8 −1.21496
\(527\) 506520. 0.00346071
\(528\) 0 0
\(529\) −1.36631e8 −0.922957
\(530\) 0 0
\(531\) 0 0
\(532\) 8.89631e7i 0.590847i
\(533\) −1.24759e8 −0.823931
\(534\) 0 0
\(535\) 0 0
\(536\) − 3.05865e7i − 0.198626i
\(537\) 0 0
\(538\) 1.42178e6i 0.00913034i
\(539\) − 1.56332e8i − 0.998347i
\(540\) 0 0
\(541\) 2.54800e7 0.160919 0.0804595 0.996758i \(-0.474361\pi\)
0.0804595 + 0.996758i \(0.474361\pi\)
\(542\) −1.67669e8 −1.05306
\(543\) 0 0
\(544\) 73728.0 0.000457969 0
\(545\) 0 0
\(546\) 0 0
\(547\) 2.05216e8i 1.25386i 0.779076 + 0.626930i \(0.215689\pi\)
−0.779076 + 0.626930i \(0.784311\pi\)
\(548\) 1.43492e8 0.871938
\(549\) 0 0
\(550\) 0 0
\(551\) 1.68614e8i 1.00795i
\(552\) 0 0
\(553\) − 1.69961e7i − 0.100502i
\(554\) − 7.47908e7i − 0.439865i
\(555\) 0 0
\(556\) 3.49545e7 0.203366
\(557\) 2.41143e8 1.39543 0.697715 0.716375i \(-0.254200\pi\)
0.697715 + 0.716375i \(0.254200\pi\)
\(558\) 0 0
\(559\) −1.27984e7 −0.0732690
\(560\) 0 0
\(561\) 0 0
\(562\) 3.46403e7i 0.195152i
\(563\) −1.68877e8 −0.946337 −0.473168 0.880972i \(-0.656890\pi\)
−0.473168 + 0.880972i \(0.656890\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) − 3.81456e7i − 0.210375i
\(567\) 0 0
\(568\) 9.62058e7i 0.524996i
\(569\) 2.43995e8i 1.32448i 0.749293 + 0.662238i \(0.230393\pi\)
−0.749293 + 0.662238i \(0.769607\pi\)
\(570\) 0 0
\(571\) 2.41502e8 1.29722 0.648608 0.761123i \(-0.275352\pi\)
0.648608 + 0.761123i \(0.275352\pi\)
\(572\) 1.44493e8 0.772071
\(573\) 0 0
\(574\) 1.01419e8 0.536272
\(575\) 0 0
\(576\) 0 0
\(577\) − 4.93979e7i − 0.257147i −0.991700 0.128573i \(-0.958960\pi\)
0.991700 0.128573i \(-0.0410398\pi\)
\(578\) 1.36542e8 0.707102
\(579\) 0 0
\(580\) 0 0
\(581\) 5.31430e6i 0.0270968i
\(582\) 0 0
\(583\) 3.20069e8i 1.61525i
\(584\) 4.27466e7i 0.214617i
\(585\) 0 0
\(586\) 5.67038e7 0.281786
\(587\) 1.72052e8 0.850639 0.425320 0.905043i \(-0.360162\pi\)
0.425320 + 0.905043i \(0.360162\pi\)
\(588\) 0 0
\(589\) 2.28588e8 1.11869
\(590\) 0 0
\(591\) 0 0
\(592\) 5.37866e7i 0.259244i
\(593\) 2.70643e8 1.29788 0.648938 0.760841i \(-0.275213\pi\)
0.648938 + 0.760841i \(0.275213\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.10680e7i 0.335687i
\(597\) 0 0
\(598\) − 6.43423e7i − 0.300880i
\(599\) 1.73299e8i 0.806337i 0.915126 + 0.403169i \(0.132091\pi\)
−0.915126 + 0.403169i \(0.867909\pi\)
\(600\) 0 0
\(601\) −4.31090e8 −1.98584 −0.992921 0.118775i \(-0.962103\pi\)
−0.992921 + 0.118775i \(0.962103\pi\)
\(602\) 1.04041e7 0.0476886
\(603\) 0 0
\(604\) −1.30519e8 −0.592327
\(605\) 0 0
\(606\) 0 0
\(607\) 1.66991e7i 0.0746665i 0.999303 + 0.0373332i \(0.0118863\pi\)
−0.999303 + 0.0373332i \(0.988114\pi\)
\(608\) 3.32728e7 0.148040
\(609\) 0 0
\(610\) 0 0
\(611\) 2.58635e8i 1.13387i
\(612\) 0 0
\(613\) 1.92321e8i 0.834920i 0.908695 + 0.417460i \(0.137080\pi\)
−0.908695 + 0.417460i \(0.862920\pi\)
\(614\) 2.95887e7i 0.127827i
\(615\) 0 0
\(616\) −1.17461e8 −0.502518
\(617\) 1.87023e8 0.796233 0.398117 0.917335i \(-0.369664\pi\)
0.398117 + 0.917335i \(0.369664\pi\)
\(618\) 0 0
\(619\) −2.54873e8 −1.07461 −0.537307 0.843387i \(-0.680558\pi\)
−0.537307 + 0.843387i \(0.680558\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 1.76619e8i − 0.733950i
\(623\) 6.25950e7 0.258866
\(624\) 0 0
\(625\) 0 0
\(626\) 1.27154e8i 0.518330i
\(627\) 0 0
\(628\) 1.96982e8i 0.795329i
\(629\) − 668547.i − 0.00268646i
\(630\) 0 0
\(631\) 9.23602e7 0.367618 0.183809 0.982962i \(-0.441157\pi\)
0.183809 + 0.982962i \(0.441157\pi\)
\(632\) −6.35668e6 −0.0251813
\(633\) 0 0
\(634\) −1.56249e8 −0.613124
\(635\) 0 0
\(636\) 0 0
\(637\) 3.92732e8i 1.51942i
\(638\) −2.22627e8 −0.857267
\(639\) 0 0
\(640\) 0 0
\(641\) − 4.24666e8i − 1.61240i −0.591643 0.806200i \(-0.701520\pi\)
0.591643 0.806200i \(-0.298480\pi\)
\(642\) 0 0
\(643\) − 3.75946e8i − 1.41414i −0.707143 0.707071i \(-0.750016\pi\)
0.707143 0.707071i \(-0.249984\pi\)
\(644\) 5.23052e7i 0.195834i
\(645\) 0 0
\(646\) −413568. −0.00153408
\(647\) −2.63747e7 −0.0973813 −0.0486906 0.998814i \(-0.515505\pi\)
−0.0486906 + 0.998814i \(0.515505\pi\)
\(648\) 0 0
\(649\) −3.34955e8 −1.22533
\(650\) 0 0
\(651\) 0 0
\(652\) − 2.56223e7i − 0.0924432i
\(653\) 2.58756e8 0.929291 0.464645 0.885497i \(-0.346182\pi\)
0.464645 + 0.885497i \(0.346182\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) − 3.79315e7i − 0.134366i
\(657\) 0 0
\(658\) − 2.10250e8i − 0.738002i
\(659\) 1.39345e8i 0.486895i 0.969914 + 0.243447i \(0.0782783\pi\)
−0.969914 + 0.243447i \(0.921722\pi\)
\(660\) 0 0
\(661\) −4.72545e8 −1.63621 −0.818104 0.575070i \(-0.804975\pi\)
−0.818104 + 0.575070i \(0.804975\pi\)
\(662\) 3.25913e7 0.112338
\(663\) 0 0
\(664\) 1.98758e6 0.00678924
\(665\) 0 0
\(666\) 0 0
\(667\) 9.91354e7i 0.334081i
\(668\) −1.53749e8 −0.515804
\(669\) 0 0
\(670\) 0 0
\(671\) 1.77639e7i 0.0587992i
\(672\) 0 0
\(673\) − 5.48833e8i − 1.80051i −0.435364 0.900254i \(-0.643380\pi\)
0.435364 0.900254i \(-0.356620\pi\)
\(674\) 2.26869e8i 0.740961i
\(675\) 0 0
\(676\) −2.08532e8 −0.675044
\(677\) 1.00760e8 0.324731 0.162365 0.986731i \(-0.448088\pi\)
0.162365 + 0.986731i \(0.448088\pi\)
\(678\) 0 0
\(679\) −1.55569e8 −0.496952
\(680\) 0 0
\(681\) 0 0
\(682\) 3.01813e8i 0.951447i
\(683\) −313056. −0.000982562 0 −0.000491281 1.00000i \(-0.500156\pi\)
−0.000491281 1.00000i \(0.500156\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 2.85291e6i 0.00883722i
\(687\) 0 0
\(688\) − 3.89120e6i − 0.0119486i
\(689\) − 8.04068e8i − 2.45830i
\(690\) 0 0
\(691\) −3.72812e8 −1.12994 −0.564971 0.825111i \(-0.691113\pi\)
−0.564971 + 0.825111i \(0.691113\pi\)
\(692\) −1.14238e8 −0.344739
\(693\) 0 0
\(694\) 3.83607e8 1.14765
\(695\) 0 0
\(696\) 0 0
\(697\) 471474.i 0.00139239i
\(698\) −2.37949e8 −0.699709
\(699\) 0 0
\(700\) 0 0
\(701\) 6.21170e8i 1.80325i 0.432517 + 0.901626i \(0.357626\pi\)
−0.432517 + 0.901626i \(0.642374\pi\)
\(702\) 0 0
\(703\) − 3.01709e8i − 0.868406i
\(704\) 4.39312e7i 0.125909i
\(705\) 0 0
\(706\) 9.95469e7 0.282887
\(707\) 3.23690e8 0.915947
\(708\) 0 0
\(709\) 2.46510e8 0.691666 0.345833 0.938296i \(-0.387596\pi\)
0.345833 + 0.938296i \(0.387596\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 2.34109e7i − 0.0648603i
\(713\) 1.34397e8 0.370783
\(714\) 0 0
\(715\) 0 0
\(716\) 2.37983e8i 0.648347i
\(717\) 0 0
\(718\) − 7.91508e7i − 0.213836i
\(719\) 9.60389e7i 0.258381i 0.991620 + 0.129191i \(0.0412379\pi\)
−0.991620 + 0.129191i \(0.958762\pi\)
\(720\) 0 0
\(721\) −9.64811e8 −2.57417
\(722\) 7.94921e7 0.211209
\(723\) 0 0
\(724\) −3.32198e8 −0.875350
\(725\) 0 0
\(726\) 0 0
\(727\) 3.91371e8i 1.01856i 0.860602 + 0.509278i \(0.170088\pi\)
−0.860602 + 0.509278i \(0.829912\pi\)
\(728\) 2.95082e8 0.764801
\(729\) 0 0
\(730\) 0 0
\(731\) 48366.1i 0 0.000123819i
\(732\) 0 0
\(733\) − 3.49078e7i − 0.0886361i −0.999017 0.0443181i \(-0.985889\pi\)
0.999017 0.0443181i \(-0.0141115\pi\)
\(734\) 1.50390e8i 0.380304i
\(735\) 0 0
\(736\) 1.95625e7 0.0490671
\(737\) −2.26531e8 −0.565881
\(738\) 0 0
\(739\) 3.02999e8 0.750773 0.375386 0.926868i \(-0.377510\pi\)
0.375386 + 0.926868i \(0.377510\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 6.53644e8i 1.60004i
\(743\) 2.45628e8 0.598842 0.299421 0.954121i \(-0.403207\pi\)
0.299421 + 0.954121i \(0.403207\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1.01161e8i 0.243667i
\(747\) 0 0
\(748\) − 546048.i − 0.00130475i
\(749\) − 1.26163e8i − 0.300253i
\(750\) 0 0
\(751\) −8.23270e7 −0.194367 −0.0971835 0.995266i \(-0.530983\pi\)
−0.0971835 + 0.995266i \(0.530983\pi\)
\(752\) −7.86348e7 −0.184910
\(753\) 0 0
\(754\) 5.59277e8 1.30471
\(755\) 0 0
\(756\) 0 0
\(757\) − 6.03579e8i − 1.39138i −0.718341 0.695691i \(-0.755098\pi\)
0.718341 0.695691i \(-0.244902\pi\)
\(758\) 4.07847e8 0.936460
\(759\) 0 0
\(760\) 0 0
\(761\) − 2.32982e8i − 0.528651i −0.964434 0.264325i \(-0.914851\pi\)
0.964434 0.264325i \(-0.0851493\pi\)
\(762\) 0 0
\(763\) 9.41167e7i 0.211882i
\(764\) 4.15810e8i 0.932426i
\(765\) 0 0
\(766\) −4.91382e7 −0.109328
\(767\) 8.41463e8 1.86487
\(768\) 0 0
\(769\) −8.15796e8 −1.79392 −0.896958 0.442115i \(-0.854228\pi\)
−0.896958 + 0.442115i \(0.854228\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.25823e8i 0.273468i
\(773\) −3.66587e8 −0.793667 −0.396833 0.917891i \(-0.629891\pi\)
−0.396833 + 0.917891i \(0.629891\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 5.81840e7i 0.124514i
\(777\) 0 0
\(778\) − 2.79681e8i − 0.593915i
\(779\) 2.12772e8i 0.450093i
\(780\) 0 0
\(781\) 7.12524e8 1.49571
\(782\) −243154. −0.000508466 0
\(783\) 0 0
\(784\) −1.19406e8 −0.247786
\(785\) 0 0
\(786\) 0 0
\(787\) − 4.02462e8i − 0.825659i −0.910808 0.412830i \(-0.864540\pi\)
0.910808 0.412830i \(-0.135460\pi\)
\(788\) −1.72150e8 −0.351825
\(789\) 0 0
\(790\) 0 0
\(791\) 3.97798e8i 0.803773i
\(792\) 0 0
\(793\) − 4.46260e7i − 0.0894887i
\(794\) − 8.87623e7i − 0.177324i
\(795\) 0 0
\(796\) 1.81088e7 0.0359046
\(797\) 5.18940e8 1.02504 0.512521 0.858675i \(-0.328712\pi\)
0.512521 + 0.858675i \(0.328712\pi\)
\(798\) 0 0
\(799\) 977400. 0.00191616
\(800\) 0 0
\(801\) 0 0
\(802\) 2.68426e8i 0.520356i
\(803\) 3.16592e8 0.611440
\(804\) 0 0
\(805\) 0 0
\(806\) − 7.58205e8i − 1.44804i
\(807\) 0 0
\(808\) − 1.21062e8i − 0.229496i
\(809\) 3.04036e6i 0.00574221i 0.999996 + 0.00287110i \(0.000913902\pi\)
−0.999996 + 0.00287110i \(0.999086\pi\)
\(810\) 0 0
\(811\) 2.25521e8 0.422790 0.211395 0.977401i \(-0.432199\pi\)
0.211395 + 0.977401i \(0.432199\pi\)
\(812\) −4.54648e8 −0.849194
\(813\) 0 0
\(814\) 3.98357e8 0.738583
\(815\) 0 0
\(816\) 0 0
\(817\) 2.18272e7i 0.0400250i
\(818\) −6.53432e8 −1.19383
\(819\) 0 0
\(820\) 0 0
\(821\) 2.77035e8i 0.500617i 0.968166 + 0.250309i \(0.0805321\pi\)
−0.968166 + 0.250309i \(0.919468\pi\)
\(822\) 0 0
\(823\) 7.07336e8i 1.26890i 0.772965 + 0.634448i \(0.218773\pi\)
−0.772965 + 0.634448i \(0.781227\pi\)
\(824\) 3.60846e8i 0.644971i
\(825\) 0 0
\(826\) −6.84043e8 −1.21379
\(827\) 2.66346e8 0.470900 0.235450 0.971886i \(-0.424344\pi\)
0.235450 + 0.971886i \(0.424344\pi\)
\(828\) 0 0
\(829\) −5.03826e8 −0.884336 −0.442168 0.896932i \(-0.645791\pi\)
−0.442168 + 0.896932i \(0.645791\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 1.10363e8i − 0.191625i
\(833\) 1.48416e6 0.00256772
\(834\) 0 0
\(835\) 0 0
\(836\) − 2.46427e8i − 0.421763i
\(837\) 0 0
\(838\) 8.29822e8i 1.41011i
\(839\) − 7.63364e8i − 1.29255i −0.763106 0.646273i \(-0.776327\pi\)
0.763106 0.646273i \(-0.223673\pi\)
\(840\) 0 0
\(841\) −2.66883e8 −0.448676
\(842\) −7.87654e8 −1.31947
\(843\) 0 0
\(844\) −4.32528e8 −0.719427
\(845\) 0 0
\(846\) 0 0
\(847\) 1.25099e7i 0.0205876i
\(848\) 2.44467e8 0.400897
\(849\) 0 0
\(850\) 0 0
\(851\) − 1.77388e8i − 0.287829i
\(852\) 0 0
\(853\) 1.87985e7i 0.0302884i 0.999885 + 0.0151442i \(0.00482073\pi\)
−0.999885 + 0.0151442i \(0.995179\pi\)
\(854\) 3.62774e7i 0.0582455i
\(855\) 0 0
\(856\) −4.71859e7 −0.0752300
\(857\) 6.86427e8 1.09057 0.545283 0.838252i \(-0.316422\pi\)
0.545283 + 0.838252i \(0.316422\pi\)
\(858\) 0 0
\(859\) −5.51932e8 −0.870775 −0.435387 0.900243i \(-0.643389\pi\)
−0.435387 + 0.900243i \(0.643389\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 5.67901e8i 0.886647i
\(863\) 3.65665e8 0.568920 0.284460 0.958688i \(-0.408186\pi\)
0.284460 + 0.958688i \(0.408186\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) − 2.26631e8i − 0.348952i
\(867\) 0 0
\(868\) 6.16360e8i 0.942487i
\(869\) 4.70791e7i 0.0717413i
\(870\) 0 0
\(871\) 5.69084e8 0.861236
\(872\) 3.52003e7 0.0530881
\(873\) 0 0
\(874\) −1.09733e8 −0.164363
\(875\) 0 0
\(876\) 0 0
\(877\) − 5.85387e8i − 0.867849i −0.900949 0.433925i \(-0.857128\pi\)
0.900949 0.433925i \(-0.142872\pi\)
\(878\) −7.83994e8 −1.15832
\(879\) 0 0
\(880\) 0 0
\(881\) − 4.29761e8i − 0.628491i −0.949342 0.314246i \(-0.898248\pi\)
0.949342 0.314246i \(-0.101752\pi\)
\(882\) 0 0
\(883\) − 2.20085e8i − 0.319675i −0.987143 0.159837i \(-0.948903\pi\)
0.987143 0.159837i \(-0.0510970\pi\)
\(884\) 1.37176e6i 0.00198574i
\(885\) 0 0
\(886\) 6.30415e8 0.906411
\(887\) −1.17196e9 −1.67936 −0.839678 0.543084i \(-0.817256\pi\)
−0.839678 + 0.543084i \(0.817256\pi\)
\(888\) 0 0
\(889\) 1.47969e9 2.10604
\(890\) 0 0
\(891\) 0 0
\(892\) 1.71355e8i 0.241436i
\(893\) 4.41092e8 0.619405
\(894\) 0 0
\(895\) 0 0
\(896\) 8.97161e7i 0.124723i
\(897\) 0 0
\(898\) − 3.45728e8i − 0.477425i
\(899\) 1.16820e9i 1.60783i
\(900\) 0 0
\(901\) −3.03863e6 −0.00415436
\(902\) −2.80930e8 −0.382806
\(903\) 0 0
\(904\) 1.48779e8 0.201390
\(905\) 0 0
\(906\) 0 0
\(907\) 7.31614e8i 0.980529i 0.871574 + 0.490264i \(0.163100\pi\)
−0.871574 + 0.490264i \(0.836900\pi\)
\(908\) 4.35403e8 0.581612
\(909\) 0 0
\(910\) 0 0
\(911\) 9.18595e8i 1.21498i 0.794327 + 0.607490i \(0.207823\pi\)
−0.794327 + 0.607490i \(0.792177\pi\)
\(912\) 0 0
\(913\) − 1.47205e7i − 0.0193425i
\(914\) − 2.01760e8i − 0.264239i
\(915\) 0 0
\(916\) −1.39085e8 −0.180965
\(917\) 1.48776e9 1.92941
\(918\) 0 0
\(919\) 2.15987e8 0.278279 0.139139 0.990273i \(-0.455566\pi\)
0.139139 + 0.990273i \(0.455566\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 8.59744e8i − 1.09692i
\(923\) −1.78998e9 −2.27637
\(924\) 0 0
\(925\) 0 0
\(926\) 6.50414e8i 0.819138i
\(927\) 0 0
\(928\) 1.70041e8i 0.212770i
\(929\) − 3.10124e8i − 0.386802i −0.981120 0.193401i \(-0.938048\pi\)
0.981120 0.193401i \(-0.0619518\pi\)
\(930\) 0 0
\(931\) 6.69791e8 0.830023
\(932\) −6.47466e8 −0.799778
\(933\) 0 0
\(934\) 4.98768e8 0.612150
\(935\) 0 0
\(936\) 0 0
\(937\) − 7.42448e8i − 0.902501i −0.892397 0.451250i \(-0.850978\pi\)
0.892397 0.451250i \(-0.149022\pi\)
\(938\) −4.62620e8 −0.560553
\(939\) 0 0
\(940\) 0 0
\(941\) 1.81766e8i 0.218144i 0.994034 + 0.109072i \(0.0347879\pi\)
−0.994034 + 0.109072i \(0.965212\pi\)
\(942\) 0 0
\(943\) 1.25098e8i 0.149181i
\(944\) 2.55837e8i 0.304121i
\(945\) 0 0
\(946\) −2.88192e7 −0.0340415
\(947\) −8.59189e8 −1.01167 −0.505835 0.862630i \(-0.668816\pi\)
−0.505835 + 0.862630i \(0.668816\pi\)
\(948\) 0 0
\(949\) −7.95333e8 −0.930573
\(950\) 0 0
\(951\) 0 0
\(952\) − 1.11514e6i − 0.00129246i
\(953\) 6.86819e8 0.793530 0.396765 0.917920i \(-0.370133\pi\)
0.396765 + 0.917920i \(0.370133\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 6.52630e8i 0.746953i
\(957\) 0 0
\(958\) − 5.05942e8i − 0.575446i
\(959\) − 2.17031e9i − 2.46075i
\(960\) 0 0
\(961\) 6.96218e8 0.784468
\(962\) −1.00074e9 −1.12408
\(963\) 0 0
\(964\) −9.98697e7 −0.111481
\(965\) 0 0
\(966\) 0 0
\(967\) − 1.09411e9i − 1.20999i −0.796230 0.604995i \(-0.793175\pi\)
0.796230 0.604995i \(-0.206825\pi\)
\(968\) 4.67881e6 0.00515833
\(969\) 0 0
\(970\) 0 0
\(971\) 4.43115e8i 0.484014i 0.970274 + 0.242007i \(0.0778058\pi\)
−0.970274 + 0.242007i \(0.922194\pi\)
\(972\) 0 0
\(973\) − 5.28687e8i − 0.573931i
\(974\) 4.25219e8i 0.460189i
\(975\) 0 0
\(976\) 1.35680e7 0.0145937
\(977\) −1.19004e9 −1.27608 −0.638042 0.770001i \(-0.720256\pi\)
−0.638042 + 0.770001i \(0.720256\pi\)
\(978\) 0 0
\(979\) −1.73387e8 −0.184786
\(980\) 0 0
\(981\) 0 0
\(982\) − 2.55024e8i − 0.269306i
\(983\) −1.18187e9 −1.24425 −0.622125 0.782918i \(-0.713730\pi\)
−0.622125 + 0.782918i \(0.713730\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) − 2.11355e6i − 0.00220486i
\(987\) 0 0
\(988\) 6.19065e8i 0.641897i
\(989\) 1.28331e7i 0.0132661i
\(990\) 0 0
\(991\) 5.09602e8 0.523613 0.261806 0.965120i \(-0.415682\pi\)
0.261806 + 0.965120i \(0.415682\pi\)
\(992\) 2.30523e8 0.236145
\(993\) 0 0
\(994\) 1.45511e9 1.48162
\(995\) 0 0
\(996\) 0 0
\(997\) − 9.90780e8i − 0.999751i −0.866097 0.499875i \(-0.833379\pi\)
0.866097 0.499875i \(-0.166621\pi\)
\(998\) 5.17861e8 0.520981
\(999\) 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 450.7.b.a.449.1 4
3.2 odd 2 inner 450.7.b.a.449.3 4
5.2 odd 4 450.7.d.a.251.1 2
5.3 odd 4 18.7.b.a.17.2 yes 2
5.4 even 2 inner 450.7.b.a.449.4 4
15.2 even 4 450.7.d.a.251.2 2
15.8 even 4 18.7.b.a.17.1 2
15.14 odd 2 inner 450.7.b.a.449.2 4
20.3 even 4 144.7.e.d.17.2 2
40.3 even 4 576.7.e.k.449.1 2
40.13 odd 4 576.7.e.b.449.1 2
45.13 odd 12 162.7.d.d.107.1 4
45.23 even 12 162.7.d.d.107.2 4
45.38 even 12 162.7.d.d.53.1 4
45.43 odd 12 162.7.d.d.53.2 4
60.23 odd 4 144.7.e.d.17.1 2
120.53 even 4 576.7.e.b.449.2 2
120.83 odd 4 576.7.e.k.449.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.b.a.17.1 2 15.8 even 4
18.7.b.a.17.2 yes 2 5.3 odd 4
144.7.e.d.17.1 2 60.23 odd 4
144.7.e.d.17.2 2 20.3 even 4
162.7.d.d.53.1 4 45.38 even 12
162.7.d.d.53.2 4 45.43 odd 12
162.7.d.d.107.1 4 45.13 odd 12
162.7.d.d.107.2 4 45.23 even 12
450.7.b.a.449.1 4 1.1 even 1 trivial
450.7.b.a.449.2 4 15.14 odd 2 inner
450.7.b.a.449.3 4 3.2 odd 2 inner
450.7.b.a.449.4 4 5.4 even 2 inner
450.7.d.a.251.1 2 5.2 odd 4
450.7.d.a.251.2 2 15.2 even 4
576.7.e.b.449.1 2 40.13 odd 4
576.7.e.b.449.2 2 120.53 even 4
576.7.e.k.449.1 2 40.3 even 4
576.7.e.k.449.2 2 120.83 odd 4