Properties

Label 384.5.b.c.319.6
Level $384$
Weight $5$
Character 384.319
Analytic conductor $39.694$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.6940658242\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.49787136.1
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.6
Root \(-1.09445 - 0.895644i\) of defining polynomial
Character \(\chi\) \(=\) 384.319
Dual form 384.5.b.c.319.7

$q$-expansion

\(f(q)\) \(=\) \(q+5.19615 q^{3} -23.7490i q^{5} -9.38251i q^{7} +27.0000 q^{9} +O(q^{10})\) \(q+5.19615 q^{3} -23.7490i q^{5} -9.38251i q^{7} +27.0000 q^{9} +112.001 q^{11} -56.5020i q^{13} -123.404i q^{15} -79.9921 q^{17} +211.297 q^{19} -48.7530i q^{21} +217.355i q^{23} +60.9843 q^{25} +140.296 q^{27} -616.753i q^{29} -1111.81i q^{31} +581.976 q^{33} -222.825 q^{35} -802.980i q^{37} -293.593i q^{39} -2411.93 q^{41} -2130.38 q^{43} -641.223i q^{45} -3596.58i q^{47} +2312.97 q^{49} -415.651 q^{51} +833.725i q^{53} -2659.92i q^{55} +1097.93 q^{57} -1309.43 q^{59} +4785.91i q^{61} -253.328i q^{63} -1341.87 q^{65} +4025.23 q^{67} +1129.41i q^{69} -9487.13i q^{71} -266.031 q^{73} +316.883 q^{75} -1050.85i q^{77} -5756.11i q^{79} +729.000 q^{81} +7287.16 q^{83} +1899.73i q^{85} -3204.74i q^{87} -1414.08 q^{89} -530.130 q^{91} -5777.13i q^{93} -5018.09i q^{95} -3110.08 q^{97} +3024.04 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 216 q^{9} + 1392 q^{17} - 3576 q^{25} - 1440 q^{33} - 1008 q^{41} + 10376 q^{49} - 9504 q^{57} + 23808 q^{65} - 10256 q^{73} + 5832 q^{81} - 31632 q^{89} - 45200 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 5.19615 0.577350
\(4\) 0 0
\(5\) − 23.7490i − 0.949961i −0.879996 0.474980i \(-0.842455\pi\)
0.879996 0.474980i \(-0.157545\pi\)
\(6\) 0 0
\(7\) − 9.38251i − 0.191480i −0.995406 0.0957399i \(-0.969478\pi\)
0.995406 0.0957399i \(-0.0305217\pi\)
\(8\) 0 0
\(9\) 27.0000 0.333333
\(10\) 0 0
\(11\) 112.001 0.925631 0.462816 0.886455i \(-0.346839\pi\)
0.462816 + 0.886455i \(0.346839\pi\)
\(12\) 0 0
\(13\) − 56.5020i − 0.334331i −0.985929 0.167166i \(-0.946539\pi\)
0.985929 0.167166i \(-0.0534615\pi\)
\(14\) 0 0
\(15\) − 123.404i − 0.548460i
\(16\) 0 0
\(17\) −79.9921 −0.276789 −0.138395 0.990377i \(-0.544194\pi\)
−0.138395 + 0.990377i \(0.544194\pi\)
\(18\) 0 0
\(19\) 211.297 0.585309 0.292655 0.956218i \(-0.405461\pi\)
0.292655 + 0.956218i \(0.405461\pi\)
\(20\) 0 0
\(21\) − 48.7530i − 0.110551i
\(22\) 0 0
\(23\) 217.355i 0.410880i 0.978670 + 0.205440i \(0.0658625\pi\)
−0.978670 + 0.205440i \(0.934138\pi\)
\(24\) 0 0
\(25\) 60.9843 0.0975748
\(26\) 0 0
\(27\) 140.296 0.192450
\(28\) 0 0
\(29\) − 616.753i − 0.733357i −0.930348 0.366678i \(-0.880495\pi\)
0.930348 0.366678i \(-0.119505\pi\)
\(30\) 0 0
\(31\) − 1111.81i − 1.15693i −0.815707 0.578465i \(-0.803652\pi\)
0.815707 0.578465i \(-0.196348\pi\)
\(32\) 0 0
\(33\) 581.976 0.534414
\(34\) 0 0
\(35\) −222.825 −0.181898
\(36\) 0 0
\(37\) − 802.980i − 0.586545i −0.956029 0.293273i \(-0.905256\pi\)
0.956029 0.293273i \(-0.0947443\pi\)
\(38\) 0 0
\(39\) − 293.593i − 0.193026i
\(40\) 0 0
\(41\) −2411.93 −1.43482 −0.717409 0.696652i \(-0.754672\pi\)
−0.717409 + 0.696652i \(0.754672\pi\)
\(42\) 0 0
\(43\) −2130.38 −1.15218 −0.576090 0.817386i \(-0.695422\pi\)
−0.576090 + 0.817386i \(0.695422\pi\)
\(44\) 0 0
\(45\) − 641.223i − 0.316654i
\(46\) 0 0
\(47\) − 3596.58i − 1.62815i −0.580761 0.814074i \(-0.697245\pi\)
0.580761 0.814074i \(-0.302755\pi\)
\(48\) 0 0
\(49\) 2312.97 0.963335
\(50\) 0 0
\(51\) −415.651 −0.159804
\(52\) 0 0
\(53\) 833.725i 0.296805i 0.988927 + 0.148403i \(0.0474131\pi\)
−0.988927 + 0.148403i \(0.952587\pi\)
\(54\) 0 0
\(55\) − 2659.92i − 0.879313i
\(56\) 0 0
\(57\) 1097.93 0.337928
\(58\) 0 0
\(59\) −1309.43 −0.376165 −0.188083 0.982153i \(-0.560227\pi\)
−0.188083 + 0.982153i \(0.560227\pi\)
\(60\) 0 0
\(61\) 4785.91i 1.28619i 0.765786 + 0.643095i \(0.222350\pi\)
−0.765786 + 0.643095i \(0.777650\pi\)
\(62\) 0 0
\(63\) − 253.328i − 0.0638266i
\(64\) 0 0
\(65\) −1341.87 −0.317601
\(66\) 0 0
\(67\) 4025.23 0.896688 0.448344 0.893861i \(-0.352014\pi\)
0.448344 + 0.893861i \(0.352014\pi\)
\(68\) 0 0
\(69\) 1129.41i 0.237221i
\(70\) 0 0
\(71\) − 9487.13i − 1.88199i −0.338416 0.940997i \(-0.609891\pi\)
0.338416 0.940997i \(-0.390109\pi\)
\(72\) 0 0
\(73\) −266.031 −0.0499215 −0.0249607 0.999688i \(-0.507946\pi\)
−0.0249607 + 0.999688i \(0.507946\pi\)
\(74\) 0 0
\(75\) 316.883 0.0563348
\(76\) 0 0
\(77\) − 1050.85i − 0.177240i
\(78\) 0 0
\(79\) − 5756.11i − 0.922306i −0.887321 0.461153i \(-0.847436\pi\)
0.887321 0.461153i \(-0.152564\pi\)
\(80\) 0 0
\(81\) 729.000 0.111111
\(82\) 0 0
\(83\) 7287.16 1.05780 0.528898 0.848685i \(-0.322605\pi\)
0.528898 + 0.848685i \(0.322605\pi\)
\(84\) 0 0
\(85\) 1899.73i 0.262939i
\(86\) 0 0
\(87\) − 3204.74i − 0.423404i
\(88\) 0 0
\(89\) −1414.08 −0.178523 −0.0892614 0.996008i \(-0.528451\pi\)
−0.0892614 + 0.996008i \(0.528451\pi\)
\(90\) 0 0
\(91\) −530.130 −0.0640177
\(92\) 0 0
\(93\) − 5777.13i − 0.667953i
\(94\) 0 0
\(95\) − 5018.09i − 0.556021i
\(96\) 0 0
\(97\) −3110.08 −0.330543 −0.165271 0.986248i \(-0.552850\pi\)
−0.165271 + 0.986248i \(0.552850\pi\)
\(98\) 0 0
\(99\) 3024.04 0.308544
\(100\) 0 0
\(101\) 488.501i 0.0478876i 0.999713 + 0.0239438i \(0.00762227\pi\)
−0.999713 + 0.0239438i \(0.992378\pi\)
\(102\) 0 0
\(103\) − 20097.0i − 1.89434i −0.320739 0.947168i \(-0.603931\pi\)
0.320739 0.947168i \(-0.396069\pi\)
\(104\) 0 0
\(105\) −1157.83 −0.105019
\(106\) 0 0
\(107\) −12383.0 −1.08158 −0.540790 0.841158i \(-0.681874\pi\)
−0.540790 + 0.841158i \(0.681874\pi\)
\(108\) 0 0
\(109\) 15542.1i 1.30814i 0.756433 + 0.654072i \(0.226941\pi\)
−0.756433 + 0.654072i \(0.773059\pi\)
\(110\) 0 0
\(111\) − 4172.41i − 0.338642i
\(112\) 0 0
\(113\) 11653.6 0.912651 0.456325 0.889813i \(-0.349165\pi\)
0.456325 + 0.889813i \(0.349165\pi\)
\(114\) 0 0
\(115\) 5161.98 0.390319
\(116\) 0 0
\(117\) − 1525.55i − 0.111444i
\(118\) 0 0
\(119\) 750.527i 0.0529996i
\(120\) 0 0
\(121\) −2096.69 −0.143206
\(122\) 0 0
\(123\) −12532.8 −0.828393
\(124\) 0 0
\(125\) − 16291.5i − 1.04265i
\(126\) 0 0
\(127\) − 4724.04i − 0.292891i −0.989219 0.146445i \(-0.953217\pi\)
0.989219 0.146445i \(-0.0467833\pi\)
\(128\) 0 0
\(129\) −11069.8 −0.665212
\(130\) 0 0
\(131\) 9632.86 0.561323 0.280661 0.959807i \(-0.409446\pi\)
0.280661 + 0.959807i \(0.409446\pi\)
\(132\) 0 0
\(133\) − 1982.49i − 0.112075i
\(134\) 0 0
\(135\) − 3331.89i − 0.182820i
\(136\) 0 0
\(137\) 16407.6 0.874185 0.437093 0.899417i \(-0.356008\pi\)
0.437093 + 0.899417i \(0.356008\pi\)
\(138\) 0 0
\(139\) −25544.1 −1.32209 −0.661046 0.750346i \(-0.729887\pi\)
−0.661046 + 0.750346i \(0.729887\pi\)
\(140\) 0 0
\(141\) − 18688.4i − 0.940012i
\(142\) 0 0
\(143\) − 6328.30i − 0.309467i
\(144\) 0 0
\(145\) −14647.3 −0.696660
\(146\) 0 0
\(147\) 12018.5 0.556182
\(148\) 0 0
\(149\) − 18716.6i − 0.843054i −0.906816 0.421527i \(-0.861494\pi\)
0.906816 0.421527i \(-0.138506\pi\)
\(150\) 0 0
\(151\) 28860.0i 1.26574i 0.774260 + 0.632868i \(0.218122\pi\)
−0.774260 + 0.632868i \(0.781878\pi\)
\(152\) 0 0
\(153\) −2159.79 −0.0922631
\(154\) 0 0
\(155\) −26404.4 −1.09904
\(156\) 0 0
\(157\) 23963.1i 0.972172i 0.873911 + 0.486086i \(0.161576\pi\)
−0.873911 + 0.486086i \(0.838424\pi\)
\(158\) 0 0
\(159\) 4332.16i 0.171360i
\(160\) 0 0
\(161\) 2039.34 0.0786752
\(162\) 0 0
\(163\) 24424.8 0.919297 0.459648 0.888101i \(-0.347975\pi\)
0.459648 + 0.888101i \(0.347975\pi\)
\(164\) 0 0
\(165\) − 13821.4i − 0.507672i
\(166\) 0 0
\(167\) 3088.40i 0.110739i 0.998466 + 0.0553696i \(0.0176337\pi\)
−0.998466 + 0.0553696i \(0.982366\pi\)
\(168\) 0 0
\(169\) 25368.5 0.888223
\(170\) 0 0
\(171\) 5705.01 0.195103
\(172\) 0 0
\(173\) − 18621.0i − 0.622171i −0.950382 0.311085i \(-0.899307\pi\)
0.950382 0.311085i \(-0.100693\pi\)
\(174\) 0 0
\(175\) − 572.185i − 0.0186836i
\(176\) 0 0
\(177\) −6804.00 −0.217179
\(178\) 0 0
\(179\) 50739.3 1.58357 0.791787 0.610797i \(-0.209151\pi\)
0.791787 + 0.610797i \(0.209151\pi\)
\(180\) 0 0
\(181\) 58050.5i 1.77194i 0.463742 + 0.885970i \(0.346506\pi\)
−0.463742 + 0.885970i \(0.653494\pi\)
\(182\) 0 0
\(183\) 24868.3i 0.742582i
\(184\) 0 0
\(185\) −19070.0 −0.557195
\(186\) 0 0
\(187\) −8959.23 −0.256205
\(188\) 0 0
\(189\) − 1316.33i − 0.0368503i
\(190\) 0 0
\(191\) 21285.6i 0.583471i 0.956499 + 0.291736i \(0.0942327\pi\)
−0.956499 + 0.291736i \(0.905767\pi\)
\(192\) 0 0
\(193\) −26074.3 −0.700001 −0.350001 0.936749i \(-0.613819\pi\)
−0.350001 + 0.936749i \(0.613819\pi\)
\(194\) 0 0
\(195\) −6972.54 −0.183367
\(196\) 0 0
\(197\) − 4731.26i − 0.121911i −0.998140 0.0609557i \(-0.980585\pi\)
0.998140 0.0609557i \(-0.0194149\pi\)
\(198\) 0 0
\(199\) 38561.7i 0.973756i 0.873470 + 0.486878i \(0.161864\pi\)
−0.873470 + 0.486878i \(0.838136\pi\)
\(200\) 0 0
\(201\) 20915.7 0.517703
\(202\) 0 0
\(203\) −5786.69 −0.140423
\(204\) 0 0
\(205\) 57280.9i 1.36302i
\(206\) 0 0
\(207\) 5868.59i 0.136960i
\(208\) 0 0
\(209\) 23665.5 0.541780
\(210\) 0 0
\(211\) 48656.4 1.09289 0.546444 0.837496i \(-0.315981\pi\)
0.546444 + 0.837496i \(0.315981\pi\)
\(212\) 0 0
\(213\) − 49296.6i − 1.08657i
\(214\) 0 0
\(215\) 50594.5i 1.09453i
\(216\) 0 0
\(217\) −10431.6 −0.221529
\(218\) 0 0
\(219\) −1382.34 −0.0288222
\(220\) 0 0
\(221\) 4519.71i 0.0925393i
\(222\) 0 0
\(223\) 73299.7i 1.47398i 0.675902 + 0.736992i \(0.263754\pi\)
−0.675902 + 0.736992i \(0.736246\pi\)
\(224\) 0 0
\(225\) 1646.57 0.0325249
\(226\) 0 0
\(227\) −14996.0 −0.291020 −0.145510 0.989357i \(-0.546482\pi\)
−0.145510 + 0.989357i \(0.546482\pi\)
\(228\) 0 0
\(229\) 74694.9i 1.42436i 0.701996 + 0.712181i \(0.252292\pi\)
−0.701996 + 0.712181i \(0.747708\pi\)
\(230\) 0 0
\(231\) − 5460.40i − 0.102329i
\(232\) 0 0
\(233\) −69476.6 −1.27975 −0.639877 0.768477i \(-0.721015\pi\)
−0.639877 + 0.768477i \(0.721015\pi\)
\(234\) 0 0
\(235\) −85415.2 −1.54668
\(236\) 0 0
\(237\) − 29909.6i − 0.532494i
\(238\) 0 0
\(239\) 87478.2i 1.53145i 0.643166 + 0.765727i \(0.277620\pi\)
−0.643166 + 0.765727i \(0.722380\pi\)
\(240\) 0 0
\(241\) 74951.4 1.29046 0.645232 0.763987i \(-0.276761\pi\)
0.645232 + 0.763987i \(0.276761\pi\)
\(242\) 0 0
\(243\) 3788.00 0.0641500
\(244\) 0 0
\(245\) − 54930.7i − 0.915131i
\(246\) 0 0
\(247\) − 11938.7i − 0.195687i
\(248\) 0 0
\(249\) 37865.2 0.610719
\(250\) 0 0
\(251\) −43677.8 −0.693287 −0.346643 0.937997i \(-0.612679\pi\)
−0.346643 + 0.937997i \(0.612679\pi\)
\(252\) 0 0
\(253\) 24344.1i 0.380323i
\(254\) 0 0
\(255\) 9871.31i 0.151808i
\(256\) 0 0
\(257\) −1164.55 −0.0176316 −0.00881581 0.999961i \(-0.502806\pi\)
−0.00881581 + 0.999961i \(0.502806\pi\)
\(258\) 0 0
\(259\) −7533.97 −0.112312
\(260\) 0 0
\(261\) − 16652.3i − 0.244452i
\(262\) 0 0
\(263\) 35001.0i 0.506021i 0.967464 + 0.253010i \(0.0814207\pi\)
−0.967464 + 0.253010i \(0.918579\pi\)
\(264\) 0 0
\(265\) 19800.2 0.281953
\(266\) 0 0
\(267\) −7347.77 −0.103070
\(268\) 0 0
\(269\) − 41077.9i − 0.567681i −0.958872 0.283840i \(-0.908391\pi\)
0.958872 0.283840i \(-0.0916085\pi\)
\(270\) 0 0
\(271\) 46419.1i 0.632060i 0.948749 + 0.316030i \(0.102350\pi\)
−0.948749 + 0.316030i \(0.897650\pi\)
\(272\) 0 0
\(273\) −2754.64 −0.0369606
\(274\) 0 0
\(275\) 6830.32 0.0903183
\(276\) 0 0
\(277\) 150297.i 1.95881i 0.201907 + 0.979405i \(0.435286\pi\)
−0.201907 + 0.979405i \(0.564714\pi\)
\(278\) 0 0
\(279\) − 30018.8i − 0.385643i
\(280\) 0 0
\(281\) −117793. −1.49179 −0.745895 0.666063i \(-0.767978\pi\)
−0.745895 + 0.666063i \(0.767978\pi\)
\(282\) 0 0
\(283\) 41243.6 0.514973 0.257486 0.966282i \(-0.417106\pi\)
0.257486 + 0.966282i \(0.417106\pi\)
\(284\) 0 0
\(285\) − 26074.7i − 0.321019i
\(286\) 0 0
\(287\) 22629.9i 0.274739i
\(288\) 0 0
\(289\) −77122.3 −0.923388
\(290\) 0 0
\(291\) −16160.4 −0.190839
\(292\) 0 0
\(293\) − 71937.8i − 0.837957i −0.907996 0.418979i \(-0.862388\pi\)
0.907996 0.418979i \(-0.137612\pi\)
\(294\) 0 0
\(295\) 31097.7i 0.357342i
\(296\) 0 0
\(297\) 15713.4 0.178138
\(298\) 0 0
\(299\) 12281.0 0.137370
\(300\) 0 0
\(301\) 19988.3i 0.220619i
\(302\) 0 0
\(303\) 2538.33i 0.0276479i
\(304\) 0 0
\(305\) 113661. 1.22183
\(306\) 0 0
\(307\) 54572.7 0.579027 0.289514 0.957174i \(-0.406506\pi\)
0.289514 + 0.957174i \(0.406506\pi\)
\(308\) 0 0
\(309\) − 104427.i − 1.09370i
\(310\) 0 0
\(311\) − 60945.8i − 0.630119i −0.949072 0.315060i \(-0.897975\pi\)
0.949072 0.315060i \(-0.102025\pi\)
\(312\) 0 0
\(313\) 102774. 1.04904 0.524521 0.851397i \(-0.324244\pi\)
0.524521 + 0.851397i \(0.324244\pi\)
\(314\) 0 0
\(315\) −6016.29 −0.0606328
\(316\) 0 0
\(317\) − 39779.2i − 0.395857i −0.980217 0.197928i \(-0.936579\pi\)
0.980217 0.197928i \(-0.0634213\pi\)
\(318\) 0 0
\(319\) − 69077.2i − 0.678818i
\(320\) 0 0
\(321\) −64344.0 −0.624450
\(322\) 0 0
\(323\) −16902.1 −0.162007
\(324\) 0 0
\(325\) − 3445.73i − 0.0326223i
\(326\) 0 0
\(327\) 80758.9i 0.755257i
\(328\) 0 0
\(329\) −33744.9 −0.311758
\(330\) 0 0
\(331\) 87804.6 0.801422 0.400711 0.916204i \(-0.368763\pi\)
0.400711 + 0.916204i \(0.368763\pi\)
\(332\) 0 0
\(333\) − 21680.5i − 0.195515i
\(334\) 0 0
\(335\) − 95595.3i − 0.851818i
\(336\) 0 0
\(337\) −57972.0 −0.510456 −0.255228 0.966881i \(-0.582151\pi\)
−0.255228 + 0.966881i \(0.582151\pi\)
\(338\) 0 0
\(339\) 60554.1 0.526919
\(340\) 0 0
\(341\) − 124524.i − 1.07089i
\(342\) 0 0
\(343\) − 44228.9i − 0.375939i
\(344\) 0 0
\(345\) 26822.4 0.225351
\(346\) 0 0
\(347\) 159188. 1.32206 0.661031 0.750359i \(-0.270119\pi\)
0.661031 + 0.750359i \(0.270119\pi\)
\(348\) 0 0
\(349\) 17396.8i 0.142830i 0.997447 + 0.0714149i \(0.0227514\pi\)
−0.997447 + 0.0714149i \(0.977249\pi\)
\(350\) 0 0
\(351\) − 7927.01i − 0.0643421i
\(352\) 0 0
\(353\) 150532. 1.20803 0.604017 0.796972i \(-0.293566\pi\)
0.604017 + 0.796972i \(0.293566\pi\)
\(354\) 0 0
\(355\) −225310. −1.78782
\(356\) 0 0
\(357\) 3899.85i 0.0305993i
\(358\) 0 0
\(359\) − 24818.5i − 0.192569i −0.995354 0.0962846i \(-0.969304\pi\)
0.995354 0.0962846i \(-0.0306959\pi\)
\(360\) 0 0
\(361\) −85674.8 −0.657413
\(362\) 0 0
\(363\) −10894.7 −0.0826803
\(364\) 0 0
\(365\) 6317.99i 0.0474234i
\(366\) 0 0
\(367\) − 89484.6i − 0.664379i −0.943213 0.332190i \(-0.892213\pi\)
0.943213 0.332190i \(-0.107787\pi\)
\(368\) 0 0
\(369\) −65122.1 −0.478273
\(370\) 0 0
\(371\) 7822.44 0.0568322
\(372\) 0 0
\(373\) 168536.i 1.21137i 0.795705 + 0.605684i \(0.207100\pi\)
−0.795705 + 0.605684i \(0.792900\pi\)
\(374\) 0 0
\(375\) − 84652.9i − 0.601976i
\(376\) 0 0
\(377\) −34847.8 −0.245184
\(378\) 0 0
\(379\) 126865. 0.883208 0.441604 0.897210i \(-0.354410\pi\)
0.441604 + 0.897210i \(0.354410\pi\)
\(380\) 0 0
\(381\) − 24546.8i − 0.169101i
\(382\) 0 0
\(383\) − 257277.i − 1.75389i −0.480587 0.876947i \(-0.659576\pi\)
0.480587 0.876947i \(-0.340424\pi\)
\(384\) 0 0
\(385\) −24956.8 −0.168371
\(386\) 0 0
\(387\) −57520.3 −0.384060
\(388\) 0 0
\(389\) 150562.i 0.994983i 0.867469 + 0.497492i \(0.165745\pi\)
−0.867469 + 0.497492i \(0.834255\pi\)
\(390\) 0 0
\(391\) − 17386.7i − 0.113727i
\(392\) 0 0
\(393\) 50053.8 0.324080
\(394\) 0 0
\(395\) −136702. −0.876155
\(396\) 0 0
\(397\) 185402.i 1.17634i 0.808736 + 0.588171i \(0.200152\pi\)
−0.808736 + 0.588171i \(0.799848\pi\)
\(398\) 0 0
\(399\) − 10301.3i − 0.0647064i
\(400\) 0 0
\(401\) 167681. 1.04279 0.521393 0.853317i \(-0.325413\pi\)
0.521393 + 0.853317i \(0.325413\pi\)
\(402\) 0 0
\(403\) −62819.4 −0.386798
\(404\) 0 0
\(405\) − 17313.0i − 0.105551i
\(406\) 0 0
\(407\) − 89934.9i − 0.542925i
\(408\) 0 0
\(409\) 60544.6 0.361934 0.180967 0.983489i \(-0.442077\pi\)
0.180967 + 0.983489i \(0.442077\pi\)
\(410\) 0 0
\(411\) 85256.3 0.504711
\(412\) 0 0
\(413\) 12285.7i 0.0720280i
\(414\) 0 0
\(415\) − 173063.i − 1.00486i
\(416\) 0 0
\(417\) −132731. −0.763310
\(418\) 0 0
\(419\) −307306. −1.75042 −0.875211 0.483741i \(-0.839277\pi\)
−0.875211 + 0.483741i \(0.839277\pi\)
\(420\) 0 0
\(421\) − 172189.i − 0.971496i −0.874099 0.485748i \(-0.838547\pi\)
0.874099 0.485748i \(-0.161453\pi\)
\(422\) 0 0
\(423\) − 97107.7i − 0.542716i
\(424\) 0 0
\(425\) −4878.26 −0.0270077
\(426\) 0 0
\(427\) 44903.9 0.246279
\(428\) 0 0
\(429\) − 32882.8i − 0.178671i
\(430\) 0 0
\(431\) − 28153.8i − 0.151559i −0.997125 0.0757797i \(-0.975855\pi\)
0.997125 0.0757797i \(-0.0241446\pi\)
\(432\) 0 0
\(433\) −322037. −1.71763 −0.858815 0.512285i \(-0.828799\pi\)
−0.858815 + 0.512285i \(0.828799\pi\)
\(434\) 0 0
\(435\) −76109.5 −0.402217
\(436\) 0 0
\(437\) 45926.4i 0.240492i
\(438\) 0 0
\(439\) − 292438.i − 1.51742i −0.651430 0.758709i \(-0.725830\pi\)
0.651430 0.758709i \(-0.274170\pi\)
\(440\) 0 0
\(441\) 62450.1 0.321112
\(442\) 0 0
\(443\) 227681. 1.16016 0.580082 0.814558i \(-0.303020\pi\)
0.580082 + 0.814558i \(0.303020\pi\)
\(444\) 0 0
\(445\) 33583.0i 0.169590i
\(446\) 0 0
\(447\) − 97254.5i − 0.486737i
\(448\) 0 0
\(449\) 159097. 0.789169 0.394585 0.918860i \(-0.370889\pi\)
0.394585 + 0.918860i \(0.370889\pi\)
\(450\) 0 0
\(451\) −270139. −1.32811
\(452\) 0 0
\(453\) 149961.i 0.730773i
\(454\) 0 0
\(455\) 12590.1i 0.0608143i
\(456\) 0 0
\(457\) 340590. 1.63079 0.815397 0.578902i \(-0.196519\pi\)
0.815397 + 0.578902i \(0.196519\pi\)
\(458\) 0 0
\(459\) −11222.6 −0.0532681
\(460\) 0 0
\(461\) − 178561.i − 0.840204i −0.907477 0.420102i \(-0.861994\pi\)
0.907477 0.420102i \(-0.138006\pi\)
\(462\) 0 0
\(463\) 4179.11i 0.0194950i 0.999952 + 0.00974748i \(0.00310277\pi\)
−0.999952 + 0.00974748i \(0.996897\pi\)
\(464\) 0 0
\(465\) −137201. −0.634529
\(466\) 0 0
\(467\) 160583. 0.736318 0.368159 0.929763i \(-0.379988\pi\)
0.368159 + 0.929763i \(0.379988\pi\)
\(468\) 0 0
\(469\) − 37766.8i − 0.171698i
\(470\) 0 0
\(471\) 124516.i 0.561284i
\(472\) 0 0
\(473\) −238606. −1.06649
\(474\) 0 0
\(475\) 12885.8 0.0571114
\(476\) 0 0
\(477\) 22510.6i 0.0989350i
\(478\) 0 0
\(479\) 153260.i 0.667970i 0.942578 + 0.333985i \(0.108393\pi\)
−0.942578 + 0.333985i \(0.891607\pi\)
\(480\) 0 0
\(481\) −45370.0 −0.196100
\(482\) 0 0
\(483\) 10596.7 0.0454231
\(484\) 0 0
\(485\) 73861.3i 0.314003i
\(486\) 0 0
\(487\) − 113347.i − 0.477917i −0.971030 0.238958i \(-0.923194\pi\)
0.971030 0.238958i \(-0.0768059\pi\)
\(488\) 0 0
\(489\) 126915. 0.530756
\(490\) 0 0
\(491\) 248336. 1.03009 0.515046 0.857162i \(-0.327775\pi\)
0.515046 + 0.857162i \(0.327775\pi\)
\(492\) 0 0
\(493\) 49335.4i 0.202985i
\(494\) 0 0
\(495\) − 71817.9i − 0.293104i
\(496\) 0 0
\(497\) −89013.1 −0.360364
\(498\) 0 0
\(499\) 378612. 1.52052 0.760262 0.649617i \(-0.225071\pi\)
0.760262 + 0.649617i \(0.225071\pi\)
\(500\) 0 0
\(501\) 16047.8i 0.0639353i
\(502\) 0 0
\(503\) − 255639.i − 1.01040i −0.863004 0.505198i \(-0.831420\pi\)
0.863004 0.505198i \(-0.168580\pi\)
\(504\) 0 0
\(505\) 11601.4 0.0454913
\(506\) 0 0
\(507\) 131819. 0.512816
\(508\) 0 0
\(509\) 204106.i 0.787809i 0.919151 + 0.393904i \(0.128876\pi\)
−0.919151 + 0.393904i \(0.871124\pi\)
\(510\) 0 0
\(511\) 2496.04i 0.00955895i
\(512\) 0 0
\(513\) 29644.1 0.112643
\(514\) 0 0
\(515\) −477284. −1.79954
\(516\) 0 0
\(517\) − 402822.i − 1.50707i
\(518\) 0 0
\(519\) − 96757.3i − 0.359211i
\(520\) 0 0
\(521\) −295131. −1.08727 −0.543637 0.839320i \(-0.682953\pi\)
−0.543637 + 0.839320i \(0.682953\pi\)
\(522\) 0 0
\(523\) 212950. 0.778527 0.389263 0.921127i \(-0.372730\pi\)
0.389263 + 0.921127i \(0.372730\pi\)
\(524\) 0 0
\(525\) − 2973.16i − 0.0107870i
\(526\) 0 0
\(527\) 88936.0i 0.320226i
\(528\) 0 0
\(529\) 232598. 0.831178
\(530\) 0 0
\(531\) −35354.6 −0.125388
\(532\) 0 0
\(533\) 136279.i 0.479704i
\(534\) 0 0
\(535\) 294084.i 1.02746i
\(536\) 0 0
\(537\) 263649. 0.914277
\(538\) 0 0
\(539\) 259056. 0.891694
\(540\) 0 0
\(541\) − 387756.i − 1.32484i −0.749132 0.662421i \(-0.769529\pi\)
0.749132 0.662421i \(-0.230471\pi\)
\(542\) 0 0
\(543\) 301639.i 1.02303i
\(544\) 0 0
\(545\) 369108. 1.24268
\(546\) 0 0
\(547\) −241227. −0.806215 −0.403107 0.915153i \(-0.632070\pi\)
−0.403107 + 0.915153i \(0.632070\pi\)
\(548\) 0 0
\(549\) 129220.i 0.428730i
\(550\) 0 0
\(551\) − 130318.i − 0.429240i
\(552\) 0 0
\(553\) −54006.8 −0.176603
\(554\) 0 0
\(555\) −99090.6 −0.321697
\(556\) 0 0
\(557\) − 86454.9i − 0.278663i −0.990246 0.139331i \(-0.955505\pi\)
0.990246 0.139331i \(-0.0444954\pi\)
\(558\) 0 0
\(559\) 120371.i 0.385210i
\(560\) 0 0
\(561\) −46553.5 −0.147920
\(562\) 0 0
\(563\) 99585.2 0.314179 0.157090 0.987584i \(-0.449789\pi\)
0.157090 + 0.987584i \(0.449789\pi\)
\(564\) 0 0
\(565\) − 276762.i − 0.866982i
\(566\) 0 0
\(567\) − 6839.85i − 0.0212755i
\(568\) 0 0
\(569\) −524085. −1.61874 −0.809371 0.587298i \(-0.800192\pi\)
−0.809371 + 0.587298i \(0.800192\pi\)
\(570\) 0 0
\(571\) 32749.2 0.100445 0.0502225 0.998738i \(-0.484007\pi\)
0.0502225 + 0.998738i \(0.484007\pi\)
\(572\) 0 0
\(573\) 110603.i 0.336867i
\(574\) 0 0
\(575\) 13255.3i 0.0400915i
\(576\) 0 0
\(577\) −140276. −0.421338 −0.210669 0.977557i \(-0.567564\pi\)
−0.210669 + 0.977557i \(0.567564\pi\)
\(578\) 0 0
\(579\) −135486. −0.404146
\(580\) 0 0
\(581\) − 68371.8i − 0.202547i
\(582\) 0 0
\(583\) 93378.4i 0.274732i
\(584\) 0 0
\(585\) −36230.4 −0.105867
\(586\) 0 0
\(587\) −303094. −0.879634 −0.439817 0.898087i \(-0.644957\pi\)
−0.439817 + 0.898087i \(0.644957\pi\)
\(588\) 0 0
\(589\) − 234921.i − 0.677161i
\(590\) 0 0
\(591\) − 24584.4i − 0.0703856i
\(592\) 0 0
\(593\) 117438. 0.333963 0.166981 0.985960i \(-0.446598\pi\)
0.166981 + 0.985960i \(0.446598\pi\)
\(594\) 0 0
\(595\) 17824.3 0.0503475
\(596\) 0 0
\(597\) 200373.i 0.562198i
\(598\) 0 0
\(599\) 537187.i 1.49717i 0.663037 + 0.748587i \(0.269267\pi\)
−0.663037 + 0.748587i \(0.730733\pi\)
\(600\) 0 0
\(601\) 382705. 1.05953 0.529767 0.848143i \(-0.322279\pi\)
0.529767 + 0.848143i \(0.322279\pi\)
\(602\) 0 0
\(603\) 108681. 0.298896
\(604\) 0 0
\(605\) 49794.2i 0.136040i
\(606\) 0 0
\(607\) 578934.i 1.57127i 0.618688 + 0.785637i \(0.287664\pi\)
−0.618688 + 0.785637i \(0.712336\pi\)
\(608\) 0 0
\(609\) −30068.5 −0.0810732
\(610\) 0 0
\(611\) −203214. −0.544341
\(612\) 0 0
\(613\) − 287542.i − 0.765209i −0.923912 0.382605i \(-0.875027\pi\)
0.923912 0.382605i \(-0.124973\pi\)
\(614\) 0 0
\(615\) 297641.i 0.786940i
\(616\) 0 0
\(617\) −400806. −1.05284 −0.526422 0.850223i \(-0.676467\pi\)
−0.526422 + 0.850223i \(0.676467\pi\)
\(618\) 0 0
\(619\) −233469. −0.609323 −0.304661 0.952461i \(-0.598543\pi\)
−0.304661 + 0.952461i \(0.598543\pi\)
\(620\) 0 0
\(621\) 30494.1i 0.0790738i
\(622\) 0 0
\(623\) 13267.6i 0.0341835i
\(624\) 0 0
\(625\) −348791. −0.892904
\(626\) 0 0
\(627\) 122970. 0.312797
\(628\) 0 0
\(629\) 64232.1i 0.162349i
\(630\) 0 0
\(631\) 499538.i 1.25461i 0.778773 + 0.627306i \(0.215843\pi\)
−0.778773 + 0.627306i \(0.784157\pi\)
\(632\) 0 0
\(633\) 252826. 0.630979
\(634\) 0 0
\(635\) −112191. −0.278235
\(636\) 0 0
\(637\) − 130687.i − 0.322073i
\(638\) 0 0
\(639\) − 256152.i − 0.627331i
\(640\) 0 0
\(641\) 405774. 0.987571 0.493786 0.869584i \(-0.335613\pi\)
0.493786 + 0.869584i \(0.335613\pi\)
\(642\) 0 0
\(643\) −550571. −1.33165 −0.665827 0.746106i \(-0.731921\pi\)
−0.665827 + 0.746106i \(0.731921\pi\)
\(644\) 0 0
\(645\) 262897.i 0.631925i
\(646\) 0 0
\(647\) 428220.i 1.02296i 0.859295 + 0.511480i \(0.170903\pi\)
−0.859295 + 0.511480i \(0.829097\pi\)
\(648\) 0 0
\(649\) −146658. −0.348190
\(650\) 0 0
\(651\) −54204.0 −0.127900
\(652\) 0 0
\(653\) 802147.i 1.88117i 0.339561 + 0.940584i \(0.389721\pi\)
−0.339561 + 0.940584i \(0.610279\pi\)
\(654\) 0 0
\(655\) − 228771.i − 0.533234i
\(656\) 0 0
\(657\) −7182.85 −0.0166405
\(658\) 0 0
\(659\) 662958. 1.52656 0.763282 0.646065i \(-0.223586\pi\)
0.763282 + 0.646065i \(0.223586\pi\)
\(660\) 0 0
\(661\) − 231396.i − 0.529607i −0.964302 0.264804i \(-0.914693\pi\)
0.964302 0.264804i \(-0.0853071\pi\)
\(662\) 0 0
\(663\) 23485.1i 0.0534276i
\(664\) 0 0
\(665\) −47082.2 −0.106467
\(666\) 0 0
\(667\) 134055. 0.301321
\(668\) 0 0
\(669\) 380877.i 0.851005i
\(670\) 0 0
\(671\) 536029.i 1.19054i
\(672\) 0 0
\(673\) 311867. 0.688557 0.344278 0.938868i \(-0.388124\pi\)
0.344278 + 0.938868i \(0.388124\pi\)
\(674\) 0 0
\(675\) 8555.85 0.0187783
\(676\) 0 0
\(677\) 655173.i 1.42948i 0.699389 + 0.714741i \(0.253455\pi\)
−0.699389 + 0.714741i \(0.746545\pi\)
\(678\) 0 0
\(679\) 29180.3i 0.0632923i
\(680\) 0 0
\(681\) −77921.4 −0.168021
\(682\) 0 0
\(683\) −358002. −0.767439 −0.383719 0.923450i \(-0.625357\pi\)
−0.383719 + 0.923450i \(0.625357\pi\)
\(684\) 0 0
\(685\) − 389664.i − 0.830442i
\(686\) 0 0
\(687\) 388126.i 0.822355i
\(688\) 0 0
\(689\) 47107.1 0.0992312
\(690\) 0 0
\(691\) 826006. 1.72992 0.864962 0.501838i \(-0.167343\pi\)
0.864962 + 0.501838i \(0.167343\pi\)
\(692\) 0 0
\(693\) − 28373.1i − 0.0590799i
\(694\) 0 0
\(695\) 606648.i 1.25593i
\(696\) 0 0
\(697\) 192935. 0.397142
\(698\) 0 0
\(699\) −361011. −0.738866
\(700\) 0 0
\(701\) − 229057.i − 0.466130i −0.972461 0.233065i \(-0.925124\pi\)
0.972461 0.233065i \(-0.0748756\pi\)
\(702\) 0 0
\(703\) − 169667.i − 0.343310i
\(704\) 0 0
\(705\) −443831. −0.892974
\(706\) 0 0
\(707\) 4583.37 0.00916950
\(708\) 0 0
\(709\) − 6312.88i − 0.0125584i −0.999980 0.00627921i \(-0.998001\pi\)
0.999980 0.00627921i \(-0.00199875\pi\)
\(710\) 0 0
\(711\) − 155415.i − 0.307435i
\(712\) 0 0
\(713\) 241658. 0.475359
\(714\) 0 0
\(715\) −150291. −0.293982
\(716\) 0 0
\(717\) 454550.i 0.884185i
\(718\) 0 0
\(719\) 179483.i 0.347189i 0.984817 + 0.173594i \(0.0555382\pi\)
−0.984817 + 0.173594i \(0.944462\pi\)
\(720\) 0 0
\(721\) −188560. −0.362727
\(722\) 0 0
\(723\) 389459. 0.745049
\(724\) 0 0
\(725\) − 37612.2i − 0.0715571i
\(726\) 0 0
\(727\) 310931.i 0.588295i 0.955760 + 0.294147i \(0.0950357\pi\)
−0.955760 + 0.294147i \(0.904964\pi\)
\(728\) 0 0
\(729\) 19683.0 0.0370370
\(730\) 0 0
\(731\) 170414. 0.318911
\(732\) 0 0
\(733\) − 602271.i − 1.12094i −0.828173 0.560472i \(-0.810620\pi\)
0.828173 0.560472i \(-0.189380\pi\)
\(734\) 0 0
\(735\) − 285428.i − 0.528351i
\(736\) 0 0
\(737\) 450832. 0.830002
\(738\) 0 0
\(739\) −64180.3 −0.117520 −0.0587602 0.998272i \(-0.518715\pi\)
−0.0587602 + 0.998272i \(0.518715\pi\)
\(740\) 0 0
\(741\) − 62035.2i − 0.112980i
\(742\) 0 0
\(743\) − 525793.i − 0.952440i −0.879326 0.476220i \(-0.842007\pi\)
0.879326 0.476220i \(-0.157993\pi\)
\(744\) 0 0
\(745\) −444502. −0.800868
\(746\) 0 0
\(747\) 196753. 0.352599
\(748\) 0 0
\(749\) 116184.i 0.207101i
\(750\) 0 0
\(751\) 405381.i 0.718759i 0.933191 + 0.359380i \(0.117012\pi\)
−0.933191 + 0.359380i \(0.882988\pi\)
\(752\) 0 0
\(753\) −226956. −0.400269
\(754\) 0 0
\(755\) 685398. 1.20240
\(756\) 0 0
\(757\) − 5071.58i − 0.00885016i −0.999990 0.00442508i \(-0.998591\pi\)
0.999990 0.00442508i \(-0.00140855\pi\)
\(758\) 0 0
\(759\) 126496.i 0.219580i
\(760\) 0 0
\(761\) −29104.4 −0.0502561 −0.0251280 0.999684i \(-0.507999\pi\)
−0.0251280 + 0.999684i \(0.507999\pi\)
\(762\) 0 0
\(763\) 145823. 0.250483
\(764\) 0 0
\(765\) 51292.8i 0.0876463i
\(766\) 0 0
\(767\) 73985.4i 0.125764i
\(768\) 0 0
\(769\) −235095. −0.397548 −0.198774 0.980045i \(-0.563696\pi\)
−0.198774 + 0.980045i \(0.563696\pi\)
\(770\) 0 0
\(771\) −6051.19 −0.0101796
\(772\) 0 0
\(773\) − 89776.8i − 0.150247i −0.997174 0.0751234i \(-0.976065\pi\)
0.997174 0.0751234i \(-0.0239351\pi\)
\(774\) 0 0
\(775\) − 67802.8i − 0.112887i
\(776\) 0 0
\(777\) −39147.7 −0.0648431
\(778\) 0 0
\(779\) −509632. −0.839812
\(780\) 0 0
\(781\) − 1.06257e6i − 1.74203i
\(782\) 0 0
\(783\) − 86528.0i − 0.141135i
\(784\) 0 0
\(785\) 569099. 0.923525
\(786\) 0 0
\(787\) 406715. 0.656660 0.328330 0.944563i \(-0.393514\pi\)
0.328330 + 0.944563i \(0.393514\pi\)
\(788\) 0 0
\(789\) 181870.i 0.292151i
\(790\) 0 0
\(791\) − 109340.i − 0.174754i
\(792\) 0 0
\(793\) 270414. 0.430013
\(794\) 0 0
\(795\) 102885. 0.162786
\(796\) 0 0
\(797\) 1.02634e6i 1.61576i 0.589350 + 0.807878i \(0.299384\pi\)
−0.589350 + 0.807878i \(0.700616\pi\)
\(798\) 0 0
\(799\) 287698.i 0.450654i
\(800\) 0 0
\(801\) −38180.1 −0.0595076
\(802\) 0 0
\(803\) −29795.9 −0.0462089
\(804\) 0 0
\(805\) − 48432.3i − 0.0747383i
\(806\) 0 0
\(807\) − 213447.i − 0.327751i
\(808\) 0 0
\(809\) 245306. 0.374810 0.187405 0.982283i \(-0.439992\pi\)
0.187405 + 0.982283i \(0.439992\pi\)
\(810\) 0 0
\(811\) −1.26760e6 −1.92726 −0.963628 0.267246i \(-0.913886\pi\)
−0.963628 + 0.267246i \(0.913886\pi\)
\(812\) 0 0
\(813\) 241201.i 0.364920i
\(814\) 0 0
\(815\) − 580065.i − 0.873296i
\(816\) 0 0
\(817\) −450142. −0.674382
\(818\) 0 0
\(819\) −14313.5 −0.0213392
\(820\) 0 0
\(821\) 163484.i 0.242542i 0.992619 + 0.121271i \(0.0386971\pi\)
−0.992619 + 0.121271i \(0.961303\pi\)
\(822\) 0 0
\(823\) − 913000.i − 1.34794i −0.738758 0.673971i \(-0.764587\pi\)
0.738758 0.673971i \(-0.235413\pi\)
\(824\) 0 0
\(825\) 35491.4 0.0521453
\(826\) 0 0
\(827\) −378235. −0.553032 −0.276516 0.961009i \(-0.589180\pi\)
−0.276516 + 0.961009i \(0.589180\pi\)
\(828\) 0 0
\(829\) 70195.3i 0.102141i 0.998695 + 0.0510704i \(0.0162633\pi\)
−0.998695 + 0.0510704i \(0.983737\pi\)
\(830\) 0 0
\(831\) 780969.i 1.13092i
\(832\) 0 0
\(833\) −185019. −0.266641
\(834\) 0 0
\(835\) 73346.5 0.105198
\(836\) 0 0
\(837\) − 155982.i − 0.222651i
\(838\) 0 0
\(839\) − 410513.i − 0.583180i −0.956543 0.291590i \(-0.905816\pi\)
0.956543 0.291590i \(-0.0941843\pi\)
\(840\) 0 0
\(841\) 326897. 0.462188
\(842\) 0 0
\(843\) −612072. −0.861286
\(844\) 0 0
\(845\) − 602478.i − 0.843777i
\(846\) 0 0
\(847\) 19672.2i 0.0274211i
\(848\) 0 0
\(849\) 214308. 0.297320
\(850\) 0 0
\(851\) 174532. 0.240999
\(852\) 0 0
\(853\) − 577725.i − 0.794005i −0.917817 0.397003i \(-0.870050\pi\)
0.917817 0.397003i \(-0.129950\pi\)
\(854\) 0 0
\(855\) − 135488.i − 0.185340i
\(856\) 0 0
\(857\) −388294. −0.528687 −0.264344 0.964429i \(-0.585155\pi\)
−0.264344 + 0.964429i \(0.585155\pi\)
\(858\) 0 0
\(859\) −124443. −0.168650 −0.0843248 0.996438i \(-0.526873\pi\)
−0.0843248 + 0.996438i \(0.526873\pi\)
\(860\) 0 0
\(861\) 117589.i 0.158620i
\(862\) 0 0
\(863\) − 652166.i − 0.875662i −0.899057 0.437831i \(-0.855747\pi\)
0.899057 0.437831i \(-0.144253\pi\)
\(864\) 0 0
\(865\) −442229. −0.591038
\(866\) 0 0
\(867\) −400739. −0.533118
\(868\) 0 0
\(869\) − 644693.i − 0.853716i
\(870\) 0 0
\(871\) − 227434.i − 0.299791i
\(872\) 0 0
\(873\) −83972.1 −0.110181
\(874\) 0 0
\(875\) −152855. −0.199647
\(876\) 0 0
\(877\) − 75235.9i − 0.0978196i −0.998803 0.0489098i \(-0.984425\pi\)
0.998803 0.0489098i \(-0.0155747\pi\)
\(878\) 0 0
\(879\) − 373800.i − 0.483795i
\(880\) 0 0
\(881\) −916209. −1.18044 −0.590218 0.807244i \(-0.700958\pi\)
−0.590218 + 0.807244i \(0.700958\pi\)
\(882\) 0 0
\(883\) 1.14981e6 1.47471 0.737354 0.675506i \(-0.236075\pi\)
0.737354 + 0.675506i \(0.236075\pi\)
\(884\) 0 0
\(885\) 161588.i 0.206311i
\(886\) 0 0
\(887\) − 1.01630e6i − 1.29174i −0.763448 0.645870i \(-0.776495\pi\)
0.763448 0.645870i \(-0.223505\pi\)
\(888\) 0 0
\(889\) −44323.3 −0.0560827
\(890\) 0 0
\(891\) 81649.0 0.102848
\(892\) 0 0
\(893\) − 759945.i − 0.952970i
\(894\) 0 0
\(895\) − 1.20501e6i − 1.50433i
\(896\) 0 0
\(897\) 63814.0 0.0793105
\(898\) 0 0
\(899\) −685711. −0.848442
\(900\) 0 0
\(901\) − 66691.5i − 0.0821525i
\(902\) 0 0
\(903\) 103862.i 0.127375i
\(904\) 0 0
\(905\) 1.37864e6 1.68327
\(906\) 0 0
\(907\) −1.00415e6 −1.22063 −0.610315 0.792159i \(-0.708957\pi\)
−0.610315 + 0.792159i \(0.708957\pi\)
\(908\) 0 0
\(909\) 13189.5i 0.0159625i
\(910\) 0 0
\(911\) − 231087.i − 0.278444i −0.990261 0.139222i \(-0.955540\pi\)
0.990261 0.139222i \(-0.0444602\pi\)
\(912\) 0 0
\(913\) 816172. 0.979129
\(914\) 0 0
\(915\) 590598. 0.705424
\(916\) 0 0
\(917\) − 90380.4i − 0.107482i
\(918\) 0 0
\(919\) − 545658.i − 0.646084i −0.946385 0.323042i \(-0.895294\pi\)
0.946385 0.323042i \(-0.104706\pi\)
\(920\) 0 0
\(921\) 283568. 0.334302
\(922\) 0 0
\(923\) −536041. −0.629209
\(924\) 0 0
\(925\) − 48969.2i − 0.0572320i
\(926\) 0 0
\(927\) − 542619.i − 0.631445i
\(928\) 0 0
\(929\) −243550. −0.282200 −0.141100 0.989995i \(-0.545064\pi\)
−0.141100 + 0.989995i \(0.545064\pi\)
\(930\) 0 0
\(931\) 488722. 0.563849
\(932\) 0 0
\(933\) − 316684.i − 0.363800i
\(934\) 0 0
\(935\) 212773.i 0.243385i
\(936\) 0 0
\(937\) 479076. 0.545664 0.272832 0.962062i \(-0.412040\pi\)
0.272832 + 0.962062i \(0.412040\pi\)
\(938\) 0 0
\(939\) 534027. 0.605665
\(940\) 0 0
\(941\) 1.25655e6i 1.41906i 0.704675 + 0.709530i \(0.251093\pi\)
−0.704675 + 0.709530i \(0.748907\pi\)
\(942\) 0 0
\(943\) − 524246.i − 0.589538i
\(944\) 0 0
\(945\) −31261.5 −0.0350063
\(946\) 0 0
\(947\) 1.22304e6 1.36377 0.681886 0.731459i \(-0.261160\pi\)
0.681886 + 0.731459i \(0.261160\pi\)
\(948\) 0 0
\(949\) 15031.3i 0.0166903i
\(950\) 0 0
\(951\) − 206699.i − 0.228548i
\(952\) 0 0
\(953\) −1.12329e6 −1.23682 −0.618409 0.785856i \(-0.712223\pi\)
−0.618409 + 0.785856i \(0.712223\pi\)
\(954\) 0 0
\(955\) 505512. 0.554275
\(956\) 0 0
\(957\) − 358936.i − 0.391916i
\(958\) 0 0
\(959\) − 153944.i − 0.167389i
\(960\) 0 0
\(961\) −312598. −0.338485
\(962\) 0 0
\(963\) −334341. −0.360526
\(964\) 0 0
\(965\) 619240.i 0.664974i
\(966\) 0 0
\(967\) − 310108.i − 0.331635i −0.986156 0.165818i \(-0.946974\pi\)
0.986156 0.165818i \(-0.0530263\pi\)
\(968\) 0 0
\(969\) −87825.7 −0.0935350
\(970\) 0 0
\(971\) 459519. 0.487376 0.243688 0.969854i \(-0.421643\pi\)
0.243688 + 0.969854i \(0.421643\pi\)
\(972\) 0 0
\(973\) 239668.i 0.253154i
\(974\) 0 0
\(975\) − 17904.5i − 0.0188345i
\(976\) 0 0
\(977\) −645770. −0.676532 −0.338266 0.941050i \(-0.609840\pi\)
−0.338266 + 0.941050i \(0.609840\pi\)
\(978\) 0 0
\(979\) −158379. −0.165246
\(980\) 0 0
\(981\) 419635.i 0.436048i
\(982\) 0 0
\(983\) − 828941.i − 0.857860i −0.903338 0.428930i \(-0.858891\pi\)
0.903338 0.428930i \(-0.141109\pi\)
\(984\) 0 0
\(985\) −112363. −0.115811
\(986\) 0 0
\(987\) −175344. −0.179993
\(988\) 0 0
\(989\) − 463050.i − 0.473407i
\(990\) 0 0
\(991\) − 460245.i − 0.468642i −0.972159 0.234321i \(-0.924713\pi\)
0.972159 0.234321i \(-0.0752867\pi\)
\(992\) 0 0
\(993\) 456246. 0.462701
\(994\) 0 0
\(995\) 915803. 0.925030
\(996\) 0 0
\(997\) 1.82730e6i 1.83831i 0.393894 + 0.919156i \(0.371128\pi\)
−0.393894 + 0.919156i \(0.628872\pi\)
\(998\) 0 0
\(999\) − 112655.i − 0.112881i
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 384.5.b.c.319.6 yes 8
3.2 odd 2 1152.5.b.k.703.5 8
4.3 odd 2 inner 384.5.b.c.319.2 8
8.3 odd 2 inner 384.5.b.c.319.7 yes 8
8.5 even 2 inner 384.5.b.c.319.3 yes 8
12.11 even 2 1152.5.b.k.703.6 8
16.3 odd 4 768.5.g.g.511.1 4
16.5 even 4 768.5.g.c.511.2 4
16.11 odd 4 768.5.g.c.511.4 4
16.13 even 4 768.5.g.g.511.3 4
24.5 odd 2 1152.5.b.k.703.3 8
24.11 even 2 1152.5.b.k.703.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.5.b.c.319.2 8 4.3 odd 2 inner
384.5.b.c.319.3 yes 8 8.5 even 2 inner
384.5.b.c.319.6 yes 8 1.1 even 1 trivial
384.5.b.c.319.7 yes 8 8.3 odd 2 inner
768.5.g.c.511.2 4 16.5 even 4
768.5.g.c.511.4 4 16.11 odd 4
768.5.g.g.511.1 4 16.3 odd 4
768.5.g.g.511.3 4 16.13 even 4
1152.5.b.k.703.3 8 24.5 odd 2
1152.5.b.k.703.4 8 24.11 even 2
1152.5.b.k.703.5 8 3.2 odd 2
1152.5.b.k.703.6 8 12.11 even 2