Properties

Label 576.7.e.b.449.1
Level $576$
Weight $7$
Character 576.449
Analytic conductor $132.511$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 576.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(132.511152165\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 576.449
Dual form 576.7.e.b.449.2

$q$-expansion

\(f(q)\) \(=\) \(q-173.948i q^{5} -484.000 q^{7} +O(q^{10})\) \(q-173.948i q^{5} -484.000 q^{7} -1340.67i q^{11} -3368.00 q^{13} +12.7279i q^{17} -5744.00 q^{19} -3377.14i q^{23} -14633.0 q^{25} -29354.8i q^{29} -39796.0 q^{31} +84191.0i q^{35} -52526.0 q^{37} -37042.5i q^{41} -3800.00 q^{43} +76791.8i q^{47} +116607. q^{49} -238738. i q^{53} -233208. q^{55} +249841. i q^{59} -13250.0 q^{61} +585858. i q^{65} -168968. q^{67} -531467. i q^{71} +236144. q^{73} +648886. i q^{77} -35116.0 q^{79} +10980.0i q^{83} +2214.00 q^{85} -129328. i q^{89} +1.63011e6 q^{91} +999159. i q^{95} -321424. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 968 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 968 q^{7} - 6736 q^{13} - 11488 q^{19} - 29266 q^{25} - 79592 q^{31} - 105052 q^{37} - 7600 q^{43} + 233214 q^{49} - 466416 q^{55} - 26500 q^{61} - 337936 q^{67} + 472288 q^{73} - 70232 q^{79} + 4428 q^{85} + 3260224 q^{91} - 642848 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 173.948i − 1.39159i −0.718242 0.695793i \(-0.755053\pi\)
0.718242 0.695793i \(-0.244947\pi\)
\(6\) 0 0
\(7\) −484.000 −1.41108 −0.705539 0.708671i \(-0.749295\pi\)
−0.705539 + 0.708671i \(0.749295\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1340.67i − 1.00727i −0.863917 0.503634i \(-0.831996\pi\)
0.863917 0.503634i \(-0.168004\pi\)
\(12\) 0 0
\(13\) −3368.00 −1.53300 −0.766500 0.642245i \(-0.778003\pi\)
−0.766500 + 0.642245i \(0.778003\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 12.7279i 0.00259066i 0.999999 + 0.00129533i \(0.000412317\pi\)
−0.999999 + 0.00129533i \(0.999588\pi\)
\(18\) 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
\(22\) 0 0
\(23\) − 3377.14i − 0.277566i −0.990323 0.138783i \(-0.955681\pi\)
0.990323 0.138783i \(-0.0443190\pi\)
\(24\) 0 0
\(25\) −14633.0 −0.936512
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 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 \(0.732815\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 84191.0i 1.96364i
\(36\) 0 0
\(37\) −52526.0 −1.03698 −0.518489 0.855085i \(-0.673505\pi\)
−0.518489 + 0.855085i \(0.673505\pi\)
\(38\) 0 0
\(39\) 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
\(43\) −3800.00 −0.0477945 −0.0238973 0.999714i \(-0.507607\pi\)
−0.0238973 + 0.999714i \(0.507607\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 76791.8i 0.739641i 0.929103 + 0.369821i \(0.120581\pi\)
−0.929103 + 0.369821i \(0.879419\pi\)
\(48\) 0 0
\(49\) 116607. 0.991143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 238738.i − 1.60359i −0.597599 0.801795i \(-0.703879\pi\)
0.597599 0.801795i \(-0.296121\pi\)
\(54\) 0 0
\(55\) −233208. −1.40170
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 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 \(-0.509292\pi\)
−0.0291875 + 0.999574i \(0.509292\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 585858.i 2.13330i
\(66\) 0 0
\(67\) −168968. −0.561798 −0.280899 0.959737i \(-0.590633\pi\)
−0.280899 + 0.959737i \(0.590633\pi\)
\(68\) 0 0
\(69\) 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. 0.607027 0.303514 0.952827i \(-0.401840\pi\)
0.303514 + 0.952827i \(0.401840\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 648886.i 1.42134i
\(78\) 0 0
\(79\) −35116.0 −0.0712236 −0.0356118 0.999366i \(-0.511338\pi\)
−0.0356118 + 0.999366i \(0.511338\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10980.0i 0.0192029i 0.999954 + 0.00960144i \(0.00305628\pi\)
−0.999954 + 0.00960144i \(0.996944\pi\)
\(84\) 0 0
\(85\) 2214.00 0.00360513
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 129328.i − 0.183453i −0.995784 0.0917263i \(-0.970762\pi\)
0.995784 0.0917263i \(-0.0292385\pi\)
\(90\) 0 0
\(91\) 1.63011e6 2.16318
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 999159.i 1.16537i
\(96\) 0 0
\(97\) −321424. −0.352179 −0.176089 0.984374i \(-0.556345\pi\)
−0.176089 + 0.984374i \(0.556345\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 668780.i − 0.649111i −0.945867 0.324556i \(-0.894785\pi\)
0.945867 0.324556i \(-0.105215\pi\)
\(102\) 0 0
\(103\) 1.99341e6 1.82425 0.912127 0.409907i \(-0.134439\pi\)
0.912127 + 0.409907i \(0.134439\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 260668.i 0.212783i 0.994324 + 0.106391i \(0.0339296\pi\)
−0.994324 + 0.106391i \(0.966070\pi\)
\(108\) 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
\(112\) 0 0
\(113\) − 821897.i − 0.569616i −0.958585 0.284808i \(-0.908070\pi\)
0.958585 0.284808i \(-0.0919298\pi\)
\(114\) 0 0
\(115\) −587448. −0.386257
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 6160.31i − 0.00365563i
\(120\) 0 0
\(121\) −25847.0 −0.0145900
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 172557.i − 0.0883490i
\(126\) 0 0
\(127\) 3.05721e6 1.49250 0.746250 0.665666i \(-0.231852\pi\)
0.746250 + 0.665666i \(0.231852\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 3.07388e6i − 1.36733i −0.729797 0.683664i \(-0.760385\pi\)
0.729797 0.683664i \(-0.239615\pi\)
\(132\) 0 0
\(133\) 2.78010e6 1.18169
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 4.48412e6i − 1.74388i −0.489617 0.871938i \(-0.662863\pi\)
0.489617 0.871938i \(-0.337137\pi\)
\(138\) 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
\(142\) 0 0
\(143\) 4.51539e6i 1.54414i
\(144\) 0 0
\(145\) −5.10622e6 −1.67493
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 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 \(0.701791\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.92245e6i 1.85894i
\(156\) 0 0
\(157\) −6.15568e6 −1.59066 −0.795329 0.606178i \(-0.792702\pi\)
−0.795329 + 0.606178i \(0.792702\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.63454e6i 0.391667i
\(162\) 0 0
\(163\) −800696. −0.184886 −0.0924432 0.995718i \(-0.529468\pi\)
−0.0924432 + 0.995718i \(0.529468\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.80467e6i 1.03161i 0.856707 + 0.515804i \(0.172507\pi\)
−0.856707 + 0.515804i \(0.827493\pi\)
\(168\) 0 0
\(169\) 6.51661e6 1.35009
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.56992e6i 0.689478i 0.938699 + 0.344739i \(0.112033\pi\)
−0.938699 + 0.344739i \(0.887967\pi\)
\(174\) 0 0
\(175\) 7.08237e6 1.32149
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 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 \(-0.160631\pi\)
0.875350 + 0.483491i \(0.160631\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 9.13681e6i 1.44304i
\(186\) 0 0
\(187\) 17064.0 0.00260949
\(188\) 0 0
\(189\) 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
\(193\) −3.93195e6 −0.546936 −0.273468 0.961881i \(-0.588171\pi\)
−0.273468 + 0.961881i \(0.588171\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 5.37967e6i − 0.703651i −0.936066 0.351825i \(-0.885561\pi\)
0.936066 0.351825i \(-0.114439\pi\)
\(198\) 0 0
\(199\) −565900. −0.0718093 −0.0359046 0.999355i \(-0.511431\pi\)
−0.0359046 + 0.999355i \(0.511431\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.42077e7i 1.69839i
\(204\) 0 0
\(205\) −6.44348e6 −0.747926
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 7.70083e6i 0.843527i
\(210\) 0 0
\(211\) 1.35165e7 1.43885 0.719427 0.694568i \(-0.244404\pi\)
0.719427 + 0.694568i \(0.244404\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 661003.i 0.0665102i
\(216\) 0 0
\(217\) 1.92613e7 1.88497
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 42867.6i − 0.00397148i
\(222\) 0 0
\(223\) −5.35484e6 −0.482872 −0.241436 0.970417i \(-0.577618\pi\)
−0.241436 + 0.970417i \(0.577618\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.36063e7i 1.16322i 0.813466 + 0.581612i \(0.197578\pi\)
−0.813466 + 0.581612i \(0.802422\pi\)
\(228\) 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
\(232\) 0 0
\(233\) − 2.02333e7i − 1.59956i −0.600297 0.799778i \(-0.704951\pi\)
0.600297 0.799778i \(-0.295049\pi\)
\(234\) 0 0
\(235\) 1.33578e7 1.02927
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 2.03947e7i − 1.49391i −0.664877 0.746953i \(-0.731516\pi\)
0.664877 0.746953i \(-0.268484\pi\)
\(240\) 0 0
\(241\) −3.12093e6 −0.222963 −0.111481 0.993767i \(-0.535560\pi\)
−0.111481 + 0.993767i \(0.535560\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 2.02836e7i − 1.37926i
\(246\) 0 0
\(247\) 1.93458e7 1.28379
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 5.09519e6i − 0.322210i −0.986937 0.161105i \(-0.948494\pi\)
0.986937 0.161105i \(-0.0515058\pi\)
\(252\) 0 0
\(253\) −4.52765e6 −0.279583
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.44374e7i 0.850529i 0.905069 + 0.425264i \(0.139819\pi\)
−0.905069 + 0.425264i \(0.860181\pi\)
\(258\) 0 0
\(259\) 2.54226e7 1.46326
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3.12567e7i 1.71821i 0.511801 + 0.859104i \(0.328978\pi\)
−0.511801 + 0.859104i \(0.671022\pi\)
\(264\) 0 0
\(265\) −4.15280e7 −2.23153
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 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 \(0.232627\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.96181e7i 0.943319i
\(276\) 0 0
\(277\) −1.32213e7 −0.622062 −0.311031 0.950400i \(-0.600674\pi\)
−0.311031 + 0.950400i \(0.600674\pi\)
\(278\) 0 0
\(279\) 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
\(283\) 6.74325e6 0.297516 0.148758 0.988874i \(-0.452473\pi\)
0.148758 + 0.988874i \(0.452473\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.79286e7i 0.758403i
\(288\) 0 0
\(289\) 2.41374e7 0.999993
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.00239e7i 0.398505i 0.979948 + 0.199253i \(0.0638514\pi\)
−0.979948 + 0.199253i \(0.936149\pi\)
\(294\) 0 0
\(295\) 4.34593e7 1.69284
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.13742e7i 0.425508i
\(300\) 0 0
\(301\) 1.83920e6 0.0674418
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.30481e6i 0.0812337i
\(306\) 0 0
\(307\) 5.23060e6 0.180774 0.0903871 0.995907i \(-0.471190\pi\)
0.0903871 + 0.995907i \(0.471190\pi\)
\(308\) 0 0
\(309\) 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
\(313\) 2.24778e7 0.733029 0.366515 0.930412i \(-0.380551\pi\)
0.366515 + 0.930412i \(0.380551\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.76211e7i 0.867088i 0.901132 + 0.433544i \(0.142737\pi\)
−0.901132 + 0.433544i \(0.857263\pi\)
\(318\) 0 0
\(319\) −3.93553e7 −1.21236
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 73109.2i − 0.00216952i
\(324\) 0 0
\(325\) 4.92839e7 1.43567
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 3.71672e7i − 1.04369i
\(330\) 0 0
\(331\) 5.76138e6 0.158870 0.0794352 0.996840i \(-0.474688\pi\)
0.0794352 + 0.996840i \(0.474688\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.93917e7i 0.781790i
\(336\) 0 0
\(337\) −4.01052e7 −1.04788 −0.523939 0.851756i \(-0.675538\pi\)
−0.523939 + 0.851756i \(0.675538\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.33535e7i 1.34555i
\(342\) 0 0
\(343\) 504328. 0.0124977
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 6.78127e7i − 1.62302i −0.584341 0.811508i \(-0.698647\pi\)
0.584341 0.811508i \(-0.301353\pi\)
\(348\) 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
\(352\) 0 0
\(353\) − 1.75976e7i − 0.400063i −0.979789 0.200032i \(-0.935896\pi\)
0.979789 0.200032i \(-0.0641045\pi\)
\(354\) 0 0
\(355\) −9.24478e7 −2.06639
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 1.39920e7i − 0.302410i −0.988502 0.151205i \(-0.951685\pi\)
0.988502 0.151205i \(-0.0483154\pi\)
\(360\) 0 0
\(361\) −1.40523e7 −0.298694
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 4.10768e7i − 0.844731i
\(366\) 0 0
\(367\) −2.65855e7 −0.537832 −0.268916 0.963164i \(-0.586665\pi\)
−0.268916 + 0.963164i \(0.586665\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.15549e8i 2.26279i
\(372\) 0 0
\(373\) −1.78829e7 −0.344598 −0.172299 0.985045i \(-0.555119\pi\)
−0.172299 + 0.985045i \(0.555119\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 9.88671e7i 1.84513i
\(378\) 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
\(382\) 0 0
\(383\) 8.68648e6i 0.154614i 0.997007 + 0.0773068i \(0.0246321\pi\)
−0.997007 + 0.0773068i \(0.975368\pi\)
\(384\) 0 0
\(385\) 1.12873e8 1.97791
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 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
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.10837e6i 0.0991137i
\(396\) 0 0
\(397\) −1.56911e7 −0.250774 −0.125387 0.992108i \(-0.540017\pi\)
−0.125387 + 0.992108i \(0.540017\pi\)
\(398\) 0 0
\(399\) 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
\(403\) 1.34033e8 2.04784
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.04203e7i 1.04451i
\(408\) 0 0
\(409\) −1.15512e8 −1.68832 −0.844162 0.536088i \(-0.819901\pi\)
−0.844162 + 0.536088i \(0.819901\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 1.20923e8i − 1.71656i
\(414\) 0 0
\(415\) 1.90994e6 0.0267225
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 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 \(-0.882824\pi\)
−0.933005 + 0.359863i \(0.882824\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 186248.i − 0.00242619i
\(426\) 0 0
\(427\) 6.41300e6 0.0823716
\(428\) 0 0
\(429\) 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
\(433\) −4.00631e7 −0.493493 −0.246747 0.969080i \(-0.579362\pi\)
−0.246747 + 0.969080i \(0.579362\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.93983e7i 0.232445i
\(438\) 0 0
\(439\) −1.38592e8 −1.63811 −0.819057 0.573712i \(-0.805503\pi\)
−0.819057 + 0.573712i \(0.805503\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.11443e8i 1.28186i 0.767600 + 0.640929i \(0.221451\pi\)
−0.767600 + 0.640929i \(0.778549\pi\)
\(444\) 0 0
\(445\) −2.24965e7 −0.255290
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 6.11166e7i − 0.675181i −0.941293 0.337591i \(-0.890388\pi\)
0.941293 0.337591i \(-0.109612\pi\)
\(450\) 0 0
\(451\) −4.96619e7 −0.541370
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 2.83555e8i − 3.01026i
\(456\) 0 0
\(457\) 3.56665e7 0.373690 0.186845 0.982389i \(-0.440174\pi\)
0.186845 + 0.982389i \(0.440174\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 1.51983e8i − 1.55128i −0.631173 0.775642i \(-0.717426\pi\)
0.631173 0.775642i \(-0.282574\pi\)
\(462\) 0 0
\(463\) 1.14978e8 1.15844 0.579218 0.815173i \(-0.303358\pi\)
0.579218 + 0.815173i \(0.303358\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 8.81705e7i − 0.865711i −0.901463 0.432855i \(-0.857506\pi\)
0.901463 0.432855i \(-0.142494\pi\)
\(468\) 0 0
\(469\) 8.17805e7 0.792741
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.09456e6i 0.0481419i
\(474\) 0 0
\(475\) 8.40520e7 0.784272
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 8.94388e7i − 0.813803i −0.913472 0.406902i \(-0.866609\pi\)
0.913472 0.406902i \(-0.133391\pi\)
\(480\) 0 0
\(481\) 1.76908e8 1.58969
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5.59111e7i 0.490087i
\(486\) 0 0
\(487\) −7.51688e7 −0.650805 −0.325403 0.945576i \(-0.605500\pi\)
−0.325403 + 0.945576i \(0.605500\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 4.50822e7i − 0.380856i −0.981701 0.190428i \(-0.939012\pi\)
0.981701 0.190428i \(-0.0609876\pi\)
\(492\) 0 0
\(493\) 373626. 0.00311815
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.57230e8i 2.09533i
\(498\) 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
\(502\) 0 0
\(503\) − 1.61043e8i − 1.26543i −0.774386 0.632713i \(-0.781941\pi\)
0.774386 0.632713i \(-0.218059\pi\)
\(504\) 0 0
\(505\) −1.16333e8 −0.903295
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 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
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 3.46751e8i − 2.53861i
\(516\) 0 0
\(517\) 1.02953e8 0.745018
\(518\) 0 0
\(519\) 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
\(523\) 3.77691e7 0.264016 0.132008 0.991249i \(-0.457857\pi\)
0.132008 + 0.991249i \(0.457857\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 506520.i − 0.00346071i
\(528\) 0 0
\(529\) 1.36631e8 0.922957
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.24759e8i 0.823931i
\(534\) 0 0
\(535\) 4.53427e7 0.296105
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 1.56332e8i − 0.998347i
\(540\) 0 0
\(541\) −2.54800e7 −0.160919 −0.0804595 0.996758i \(-0.525639\pi\)
−0.0804595 + 0.996758i \(0.525639\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.38253e7i 0.208955i
\(546\) 0 0
\(547\) −2.05216e8 −1.25386 −0.626930 0.779076i \(-0.715689\pi\)
−0.626930 + 0.779076i \(0.715689\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.68614e8i 1.00795i
\(552\) 0 0
\(553\) 1.69961e7 0.100502
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.41143e8i 1.39543i 0.716375 + 0.697715i \(0.245800\pi\)
−0.716375 + 0.697715i \(0.754200\pi\)
\(558\) 0 0
\(559\) 1.27984e7 0.0732690
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.68877e8i 0.946337i 0.880972 + 0.473168i \(0.156890\pi\)
−0.880972 + 0.473168i \(0.843110\pi\)
\(564\) 0 0
\(565\) −1.42968e8 −0.792670
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 2.43995e8i − 1.32448i −0.749293 0.662238i \(-0.769607\pi\)
0.749293 0.662238i \(-0.230393\pi\)
\(570\) 0 0
\(571\) −2.41502e8 −1.29722 −0.648608 0.761123i \(-0.724648\pi\)
−0.648608 + 0.761123i \(0.724648\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.94177e7i 0.259944i
\(576\) 0 0
\(577\) −4.93979e7 −0.257147 −0.128573 0.991700i \(-0.541040\pi\)
−0.128573 + 0.991700i \(0.541040\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 5.31430e6i − 0.0270968i
\(582\) 0 0
\(583\) −3.20069e8 −1.61525
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.72052e8i 0.850639i 0.905043 + 0.425320i \(0.139838\pi\)
−0.905043 + 0.425320i \(0.860162\pi\)
\(588\) 0 0
\(589\) 2.28588e8 1.11869
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.70643e8i 1.29788i 0.760841 + 0.648938i \(0.224787\pi\)
−0.760841 + 0.648938i \(0.775213\pi\)
\(594\) 0 0
\(595\) −1.07158e6 −0.00508712
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 1.73299e8i − 0.806337i −0.915126 0.403169i \(-0.867909\pi\)
0.915126 0.403169i \(-0.132091\pi\)
\(600\) 0 0
\(601\) −4.31090e8 −1.98584 −0.992921 0.118775i \(-0.962103\pi\)
−0.992921 + 0.118775i \(0.962103\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.49604e6i 0.0203032i
\(606\) 0 0
\(607\) 1.66991e7 0.0746665 0.0373332 0.999303i \(-0.488114\pi\)
0.0373332 + 0.999303i \(0.488114\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 2.58635e8i − 1.13387i
\(612\) 0 0
\(613\) 1.92321e8 0.834920 0.417460 0.908695i \(-0.362920\pi\)
0.417460 + 0.908695i \(0.362920\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 1.87023e8i − 0.796233i −0.917335 0.398117i \(-0.869664\pi\)
0.917335 0.398117i \(-0.130336\pi\)
\(618\) 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
\(622\) 0 0
\(623\) 6.25950e7i 0.258866i
\(624\) 0 0
\(625\) −2.58657e8 −1.05946
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 668547.i − 0.00268646i
\(630\) 0 0
\(631\) 9.23602e7 0.367618 0.183809 0.982962i \(-0.441157\pi\)
0.183809 + 0.982962i \(0.441157\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 5.31797e8i − 2.07694i
\(636\) 0 0
\(637\) −3.92732e8 −1.51942
\(638\) 0 0
\(639\) 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
\(643\) −3.75946e8 −1.41414 −0.707071 0.707143i \(-0.749984\pi\)
−0.707071 + 0.707143i \(0.749984\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.63747e7i 0.0973813i 0.998814 + 0.0486906i \(0.0155048\pi\)
−0.998814 + 0.0486906i \(0.984495\pi\)
\(648\) 0 0
\(649\) 3.34955e8 1.22533
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 2.58756e8i − 0.929291i −0.885497 0.464645i \(-0.846182\pi\)
0.885497 0.464645i \(-0.153818\pi\)
\(654\) 0 0
\(655\) −5.34696e8 −1.90275
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 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 \(-0.195025\pi\)
0.818104 + 0.575070i \(0.195025\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 4.83593e8i − 1.64443i
\(666\) 0 0
\(667\) −9.91354e7 −0.334081
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.77639e7i 0.0587992i
\(672\) 0 0
\(673\) 5.48833e8 1.80051 0.900254 0.435364i \(-0.143380\pi\)
0.900254 + 0.435364i \(0.143380\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.00760e8i 0.324731i 0.986731 + 0.162365i \(0.0519123\pi\)
−0.986731 + 0.162365i \(0.948088\pi\)
\(678\) 0 0
\(679\) 1.55569e8 0.496952
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 313056.i 0 0.000982562i 1.00000 0.000491281i \(0.000156380\pi\)
−1.00000 0.000491281i \(0.999844\pi\)
\(684\) 0 0
\(685\) −7.80005e8 −2.42675
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 8.04068e8i 2.45830i
\(690\) 0 0
\(691\) 3.72812e8 1.12994 0.564971 0.825111i \(-0.308887\pi\)
0.564971 + 0.825111i \(0.308887\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 1.90009e8i − 0.566003i
\(696\) 0 0
\(697\) 471474. 0.00139239
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 6.21170e8i − 1.80325i −0.432517 0.901626i \(-0.642374\pi\)
0.432517 0.901626i \(-0.357626\pi\)
\(702\) 0 0
\(703\) 3.01709e8 0.868406
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.23690e8i 0.915947i
\(708\) 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
\(712\) 0 0
\(713\) 1.34397e8i 0.370783i
\(714\) 0 0
\(715\) 7.85445e8 2.14881
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 9.60389e7i − 0.258381i −0.991620 0.129191i \(-0.958762\pi\)
0.991620 0.129191i \(-0.0412379\pi\)
\(720\) 0 0
\(721\) −9.64811e8 −2.57417
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.29549e8i 1.12719i
\(726\) 0 0
\(727\) 3.91371e8 1.01856 0.509278 0.860602i \(-0.329912\pi\)
0.509278 + 0.860602i \(0.329912\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 48366.1i 0 0.000123819i
\(732\) 0 0
\(733\) −3.49078e7 −0.0886361 −0.0443181 0.999017i \(-0.514111\pi\)
−0.0443181 + 0.999017i \(0.514111\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.26531e8i 0.565881i
\(738\) 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
\(742\) 0 0
\(743\) 2.45628e8i 0.598842i 0.954121 + 0.299421i \(0.0967935\pi\)
−0.954121 + 0.299421i \(0.903207\pi\)
\(744\) 0 0
\(745\) 3.86317e8 0.934276
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 1.26163e8i − 0.300253i
\(750\) 0 0
\(751\) −8.23270e7 −0.194367 −0.0971835 0.995266i \(-0.530983\pi\)
−0.0971835 + 0.995266i \(0.530983\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 7.09484e8i 1.64855i
\(756\) 0 0
\(757\) 6.03579e8 1.39138 0.695691 0.718341i \(-0.255098\pi\)
0.695691 + 0.718341i \(0.255098\pi\)
\(758\) 0 0
\(759\) 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
\(763\) 9.41167e7 0.211882
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 8.41463e8i − 1.86487i
\(768\) 0 0
\(769\) 8.15796e8 1.79392 0.896958 0.442115i \(-0.145772\pi\)
0.896958 + 0.442115i \(0.145772\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 3.66587e8i 0.793667i 0.917891 + 0.396833i \(0.129891\pi\)
−0.917891 + 0.396833i \(0.870109\pi\)
\(774\) 0 0
\(775\) 5.82335e8 1.25103
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.12772e8i 0.450093i
\(780\) 0 0
\(781\) −7.12524e8 −1.49571
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.07077e9i 2.21354i
\(786\) 0 0
\(787\) 4.02462e8 0.825659 0.412830 0.910808i \(-0.364540\pi\)
0.412830 + 0.910808i \(0.364540\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 3.97798e8i 0.803773i
\(792\) 0 0
\(793\) 4.46260e7 0.0894887
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5.18940e8i 1.02504i 0.858675 + 0.512521i \(0.171288\pi\)
−0.858675 + 0.512521i \(0.828712\pi\)
\(798\) 0 0
\(799\) −977400. −0.00191616
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 3.16592e8i − 0.611440i
\(804\) 0 0
\(805\) 2.84325e8 0.545038
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 3.04036e6i − 0.00574221i −0.999996 0.00287110i \(-0.999086\pi\)
0.999996 0.00287110i \(-0.000913902\pi\)
\(810\) 0 0
\(811\) −2.25521e8 −0.422790 −0.211395 0.977401i \(-0.567801\pi\)
−0.211395 + 0.977401i \(0.567801\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.39280e8i 0.257285i
\(816\) 0 0
\(817\) 2.18272e7 0.0400250
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 2.77035e8i − 0.500617i −0.968166 0.250309i \(-0.919468\pi\)
0.968166 0.250309i \(-0.0805321\pi\)
\(822\) 0 0
\(823\) −7.07336e8 −1.26890 −0.634448 0.772965i \(-0.718773\pi\)
−0.634448 + 0.772965i \(0.718773\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.66346e8i 0.470900i 0.971886 + 0.235450i \(0.0756564\pi\)
−0.971886 + 0.235450i \(0.924344\pi\)
\(828\) 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
\(832\) 0 0
\(833\) 1.48416e6i 0.00256772i
\(834\) 0 0
\(835\) 8.35764e8 1.43557
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7.63364e8i 1.29255i 0.763106 + 0.646273i \(0.223673\pi\)
−0.763106 + 0.646273i \(0.776327\pi\)
\(840\) 0 0
\(841\) −2.66883e8 −0.448676
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 1.13355e9i − 1.87876i
\(846\) 0 0
\(847\) 1.25099e7 0.0205876
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.77388e8i 0.287829i
\(852\) 0 0
\(853\) 1.87985e7 0.0302884 0.0151442 0.999885i \(-0.495179\pi\)
0.0151442 + 0.999885i \(0.495179\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 6.86427e8i − 1.09057i −0.838252 0.545283i \(-0.816422\pi\)
0.838252 0.545283i \(-0.183578\pi\)
\(858\) 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
\(862\) 0 0
\(863\) 3.65665e8i 0.568920i 0.958688 + 0.284460i \(0.0918142\pi\)
−0.958688 + 0.284460i \(0.908186\pi\)
\(864\) 0 0
\(865\) 6.20982e8 0.959468
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 4.70791e7i 0.0717413i
\(870\) 0 0
\(871\) 5.69084e8 0.861236
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 8.35174e7i 0.124667i
\(876\) 0 0
\(877\) 5.85387e8 0.867849 0.433925 0.900949i \(-0.357128\pi\)
0.433925 + 0.900949i \(0.357128\pi\)
\(878\) 0 0
\(879\) 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
\(883\) −2.20085e8 −0.319675 −0.159837 0.987143i \(-0.551097\pi\)
−0.159837 + 0.987143i \(0.551097\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.17196e9i 1.67936i 0.543084 + 0.839678i \(0.317256\pi\)
−0.543084 + 0.839678i \(0.682744\pi\)
\(888\) 0 0
\(889\) −1.47969e9 −2.10604
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 4.41092e8i − 0.619405i
\(894\) 0 0
\(895\) 1.29365e9 1.80446
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.16820e9i 1.60783i
\(900\) 0 0
\(901\) 3.03863e6 0.00415436
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 1.80579e9i − 2.43625i
\(906\) 0 0
\(907\) −7.31614e8 −0.980529 −0.490264 0.871574i \(-0.663100\pi\)
−0.490264 + 0.871574i \(0.663100\pi\)
\(908\) 0 0
\(909\) 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
\(913\) 1.47205e7 0.0193425
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.48776e9i 1.92941i
\(918\) 0 0
\(919\) −2.15987e8 −0.278279 −0.139139 0.990273i \(-0.544434\pi\)
−0.139139 + 0.990273i \(0.544434\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.78998e9i 2.27637i
\(924\) 0 0
\(925\) 7.68613e8 0.971141
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3.10124e8i 0.386802i 0.981120 + 0.193401i \(0.0619518\pi\)
−0.981120 + 0.193401i \(0.938048\pi\)
\(930\) 0 0
\(931\) −6.69791e8 −0.830023
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 2.96825e6i − 0.00363133i
\(936\) 0 0
\(937\) −7.42448e8 −0.902501 −0.451250 0.892397i \(-0.649022\pi\)
−0.451250 + 0.892397i \(0.649022\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 1.81766e8i − 0.218144i −0.994034 0.109072i \(-0.965212\pi\)
0.994034 0.109072i \(-0.0347879\pi\)
\(942\) 0 0
\(943\) −1.25098e8 −0.149181
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 8.59189e8i − 1.01167i −0.862630 0.505835i \(-0.831184\pi\)
0.862630 0.505835i \(-0.168816\pi\)
\(948\) 0 0
\(949\) −7.95333e8 −0.930573
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.86819e8i 0.793530i 0.917920 + 0.396765i \(0.129867\pi\)
−0.917920 + 0.396765i \(0.870133\pi\)
\(954\) 0 0
\(955\) 2.26029e9 2.59510
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.17031e9i 2.46075i
\(960\) 0 0
\(961\) 6.96218e8 0.784468
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 6.83957e8i 0.761109i
\(966\) 0 0
\(967\) −1.09411e9 −1.20999 −0.604995 0.796230i \(-0.706825\pi\)
−0.604995 + 0.796230i \(0.706825\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 4.43115e8i − 0.484014i −0.970274 0.242007i \(-0.922194\pi\)
0.970274 0.242007i \(-0.0778058\pi\)
\(972\) 0 0
\(973\) −5.28687e8 −0.573931
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.19004e9i 1.27608i 0.770001 + 0.638042i \(0.220256\pi\)
−0.770001 + 0.638042i \(0.779744\pi\)
\(978\) 0 0
\(979\) −1.73387e8 −0.184786
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1.18187e9i − 1.24425i −0.782918 0.622125i \(-0.786270\pi\)
0.782918 0.622125i \(-0.213730\pi\)
\(984\) 0 0
\(985\) −9.35785e8 −0.979191
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.28331e7i 0.0132661i
\(990\) 0 0
\(991\) 5.09602e8 0.523613 0.261806 0.965120i \(-0.415682\pi\)
0.261806 + 0.965120i \(0.415682\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 9.84373e7i 0.0999288i
\(996\) 0 0
\(997\) 9.90780e8 0.999751 0.499875 0.866097i \(-0.333379\pi\)
0.499875 + 0.866097i \(0.333379\pi\)
\(998\) 0 0
\(999\) 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 576.7.e.b.449.1 2
3.2 odd 2 inner 576.7.e.b.449.2 2
4.3 odd 2 576.7.e.k.449.1 2
8.3 odd 2 144.7.e.d.17.2 2
8.5 even 2 18.7.b.a.17.2 yes 2
12.11 even 2 576.7.e.k.449.2 2
24.5 odd 2 18.7.b.a.17.1 2
24.11 even 2 144.7.e.d.17.1 2
40.13 odd 4 450.7.b.a.449.4 4
40.29 even 2 450.7.d.a.251.1 2
40.37 odd 4 450.7.b.a.449.1 4
72.5 odd 6 162.7.d.d.107.2 4
72.13 even 6 162.7.d.d.107.1 4
72.29 odd 6 162.7.d.d.53.1 4
72.61 even 6 162.7.d.d.53.2 4
120.29 odd 2 450.7.d.a.251.2 2
120.53 even 4 450.7.b.a.449.2 4
120.77 even 4 450.7.b.a.449.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.b.a.17.1 2 24.5 odd 2
18.7.b.a.17.2 yes 2 8.5 even 2
144.7.e.d.17.1 2 24.11 even 2
144.7.e.d.17.2 2 8.3 odd 2
162.7.d.d.53.1 4 72.29 odd 6
162.7.d.d.53.2 4 72.61 even 6
162.7.d.d.107.1 4 72.13 even 6
162.7.d.d.107.2 4 72.5 odd 6
450.7.b.a.449.1 4 40.37 odd 4
450.7.b.a.449.2 4 120.53 even 4
450.7.b.a.449.3 4 120.77 even 4
450.7.b.a.449.4 4 40.13 odd 4
450.7.d.a.251.1 2 40.29 even 2
450.7.d.a.251.2 2 120.29 odd 2
576.7.e.b.449.1 2 1.1 even 1 trivial
576.7.e.b.449.2 2 3.2 odd 2 inner
576.7.e.k.449.1 2 4.3 odd 2
576.7.e.k.449.2 2 12.11 even 2