Properties

Label 32.8.b.a.17.4
Level $32$
Weight $8$
Character 32.17
Analytic conductor $9.996$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 32.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.99632081549\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{30} \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 17.4
Root \(0.776001 + 5.60338i\) of defining polynomial
Character \(\chi\) \(=\) 32.17
Dual form 32.8.b.a.17.3

$q$-expansion

\(f(q)\) \(=\) \(q+21.9408i q^{3} -184.916i q^{5} -1051.96 q^{7} +1705.60 q^{9} +O(q^{10})\) \(q+21.9408i q^{3} -184.916i q^{5} -1051.96 q^{7} +1705.60 q^{9} -4324.35i q^{11} -11253.2i q^{13} +4057.20 q^{15} -21746.4 q^{17} -45466.5i q^{19} -23080.8i q^{21} -4414.37 q^{23} +43931.2 q^{25} +85406.8i q^{27} +23687.1i q^{29} -72941.9 q^{31} +94879.8 q^{33} +194523. i q^{35} +483338. i q^{37} +246905. q^{39} -411040. q^{41} +96165.3i q^{43} -315392. i q^{45} +156171. q^{47} +283070. q^{49} -477134. i q^{51} -686962. i q^{53} -799641. q^{55} +997572. q^{57} -1.79961e6i q^{59} -1.36394e6i q^{61} -1.79422e6 q^{63} -2.08090e6 q^{65} +1.08853e6i q^{67} -96854.8i q^{69} +5.60830e6 q^{71} +21698.7 q^{73} +963886. i q^{75} +4.54903e6i q^{77} -2.34010e6 q^{79} +1.85626e6 q^{81} +882169. i q^{83} +4.02125e6i q^{85} -519714. q^{87} -1.34738e6 q^{89} +1.18379e7i q^{91} -1.60040e6i q^{93} -8.40746e6 q^{95} +7.32798e6 q^{97} -7.37562e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 688 q^{7} - 2918 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 688 q^{7} - 2918 q^{9} - 17872 q^{15} + 1452 q^{17} + 1296 q^{23} - 39314 q^{25} + 89280 q^{31} + 53880 q^{33} + 328208 q^{39} + 521244 q^{41} - 1566432 q^{47} - 511050 q^{49} + 3270256 q^{55} - 1889896 q^{57} - 5776816 q^{63} + 1416480 q^{65} + 7597104 q^{71} + 2089564 q^{73} - 16015904 q^{79} - 723058 q^{81} + 37453776 q^{87} + 2169084 q^{89} - 48537936 q^{95} - 1088308 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 21.9408i 0.469168i 0.972096 + 0.234584i \(0.0753728\pi\)
−0.972096 + 0.234584i \(0.924627\pi\)
\(4\) 0 0
\(5\) − 184.916i − 0.661574i −0.943705 0.330787i \(-0.892686\pi\)
0.943705 0.330787i \(-0.107314\pi\)
\(6\) 0 0
\(7\) −1051.96 −1.15919 −0.579595 0.814905i \(-0.696789\pi\)
−0.579595 + 0.814905i \(0.696789\pi\)
\(8\) 0 0
\(9\) 1705.60 0.779882
\(10\) 0 0
\(11\) − 4324.35i − 0.979596i −0.871836 0.489798i \(-0.837071\pi\)
0.871836 0.489798i \(-0.162929\pi\)
\(12\) 0 0
\(13\) − 11253.2i − 1.42061i −0.703892 0.710307i \(-0.748556\pi\)
0.703892 0.710307i \(-0.251444\pi\)
\(14\) 0 0
\(15\) 4057.20 0.310389
\(16\) 0 0
\(17\) −21746.4 −1.07354 −0.536768 0.843730i \(-0.680355\pi\)
−0.536768 + 0.843730i \(0.680355\pi\)
\(18\) 0 0
\(19\) − 45466.5i − 1.52074i −0.649492 0.760368i \(-0.725019\pi\)
0.649492 0.760368i \(-0.274981\pi\)
\(20\) 0 0
\(21\) − 23080.8i − 0.543855i
\(22\) 0 0
\(23\) −4414.37 −0.0756522 −0.0378261 0.999284i \(-0.512043\pi\)
−0.0378261 + 0.999284i \(0.512043\pi\)
\(24\) 0 0
\(25\) 43931.2 0.562320
\(26\) 0 0
\(27\) 85406.8i 0.835063i
\(28\) 0 0
\(29\) 23687.1i 0.180351i 0.995926 + 0.0901756i \(0.0287428\pi\)
−0.995926 + 0.0901756i \(0.971257\pi\)
\(30\) 0 0
\(31\) −72941.9 −0.439755 −0.219878 0.975527i \(-0.570566\pi\)
−0.219878 + 0.975527i \(0.570566\pi\)
\(32\) 0 0
\(33\) 94879.8 0.459595
\(34\) 0 0
\(35\) 194523.i 0.766890i
\(36\) 0 0
\(37\) 483338.i 1.56872i 0.620307 + 0.784359i \(0.287008\pi\)
−0.620307 + 0.784359i \(0.712992\pi\)
\(38\) 0 0
\(39\) 246905. 0.666506
\(40\) 0 0
\(41\) −411040. −0.931410 −0.465705 0.884940i \(-0.654199\pi\)
−0.465705 + 0.884940i \(0.654199\pi\)
\(42\) 0 0
\(43\) 96165.3i 0.184450i 0.995738 + 0.0922250i \(0.0293979\pi\)
−0.995738 + 0.0922250i \(0.970602\pi\)
\(44\) 0 0
\(45\) − 315392.i − 0.515949i
\(46\) 0 0
\(47\) 156171. 0.219411 0.109705 0.993964i \(-0.465009\pi\)
0.109705 + 0.993964i \(0.465009\pi\)
\(48\) 0 0
\(49\) 283070. 0.343722
\(50\) 0 0
\(51\) − 477134.i − 0.503668i
\(52\) 0 0
\(53\) − 686962.i − 0.633822i −0.948455 0.316911i \(-0.897354\pi\)
0.948455 0.316911i \(-0.102646\pi\)
\(54\) 0 0
\(55\) −799641. −0.648075
\(56\) 0 0
\(57\) 997572. 0.713480
\(58\) 0 0
\(59\) − 1.79961e6i − 1.14077i −0.821378 0.570384i \(-0.806794\pi\)
0.821378 0.570384i \(-0.193206\pi\)
\(60\) 0 0
\(61\) − 1.36394e6i − 0.769379i −0.923046 0.384690i \(-0.874308\pi\)
0.923046 0.384690i \(-0.125692\pi\)
\(62\) 0 0
\(63\) −1.79422e6 −0.904031
\(64\) 0 0
\(65\) −2.08090e6 −0.939841
\(66\) 0 0
\(67\) 1.08853e6i 0.442160i 0.975256 + 0.221080i \(0.0709582\pi\)
−0.975256 + 0.221080i \(0.929042\pi\)
\(68\) 0 0
\(69\) − 96854.8i − 0.0354936i
\(70\) 0 0
\(71\) 5.60830e6 1.85963 0.929816 0.368025i \(-0.119966\pi\)
0.929816 + 0.368025i \(0.119966\pi\)
\(72\) 0 0
\(73\) 21698.7 0.00652837 0.00326418 0.999995i \(-0.498961\pi\)
0.00326418 + 0.999995i \(0.498961\pi\)
\(74\) 0 0
\(75\) 963886.i 0.263822i
\(76\) 0 0
\(77\) 4.54903e6i 1.13554i
\(78\) 0 0
\(79\) −2.34010e6 −0.533998 −0.266999 0.963697i \(-0.586032\pi\)
−0.266999 + 0.963697i \(0.586032\pi\)
\(80\) 0 0
\(81\) 1.85626e6 0.388097
\(82\) 0 0
\(83\) 882169.i 0.169347i 0.996409 + 0.0846736i \(0.0269848\pi\)
−0.996409 + 0.0846736i \(0.973015\pi\)
\(84\) 0 0
\(85\) 4.02125e6i 0.710223i
\(86\) 0 0
\(87\) −519714. −0.0846150
\(88\) 0 0
\(89\) −1.34738e6 −0.202594 −0.101297 0.994856i \(-0.532299\pi\)
−0.101297 + 0.994856i \(0.532299\pi\)
\(90\) 0 0
\(91\) 1.18379e7i 1.64676i
\(92\) 0 0
\(93\) − 1.60040e6i − 0.206319i
\(94\) 0 0
\(95\) −8.40746e6 −1.00608
\(96\) 0 0
\(97\) 7.32798e6 0.815236 0.407618 0.913153i \(-0.366360\pi\)
0.407618 + 0.913153i \(0.366360\pi\)
\(98\) 0 0
\(99\) − 7.37562e6i − 0.763969i
\(100\) 0 0
\(101\) − 9.66027e6i − 0.932963i −0.884531 0.466482i \(-0.845521\pi\)
0.884531 0.466482i \(-0.154479\pi\)
\(102\) 0 0
\(103\) 1.38659e7 1.25031 0.625154 0.780501i \(-0.285036\pi\)
0.625154 + 0.780501i \(0.285036\pi\)
\(104\) 0 0
\(105\) −4.26800e6 −0.359800
\(106\) 0 0
\(107\) − 1.88354e7i − 1.48638i −0.669078 0.743192i \(-0.733311\pi\)
0.669078 0.743192i \(-0.266689\pi\)
\(108\) 0 0
\(109\) 8.60619e6i 0.636529i 0.948002 + 0.318265i \(0.103100\pi\)
−0.948002 + 0.318265i \(0.896900\pi\)
\(110\) 0 0
\(111\) −1.06048e7 −0.735992
\(112\) 0 0
\(113\) −1.31340e7 −0.856292 −0.428146 0.903709i \(-0.640833\pi\)
−0.428146 + 0.903709i \(0.640833\pi\)
\(114\) 0 0
\(115\) 816286.i 0.0500495i
\(116\) 0 0
\(117\) − 1.91935e7i − 1.10791i
\(118\) 0 0
\(119\) 2.28763e7 1.24443
\(120\) 0 0
\(121\) 787128. 0.0403921
\(122\) 0 0
\(123\) − 9.01856e6i − 0.436988i
\(124\) 0 0
\(125\) − 2.25701e7i − 1.03359i
\(126\) 0 0
\(127\) −3.18768e7 −1.38090 −0.690449 0.723381i \(-0.742587\pi\)
−0.690449 + 0.723381i \(0.742587\pi\)
\(128\) 0 0
\(129\) −2.10994e6 −0.0865380
\(130\) 0 0
\(131\) 1.17050e7i 0.454905i 0.973789 + 0.227453i \(0.0730397\pi\)
−0.973789 + 0.227453i \(0.926960\pi\)
\(132\) 0 0
\(133\) 4.78288e7i 1.76282i
\(134\) 0 0
\(135\) 1.57931e7 0.552456
\(136\) 0 0
\(137\) 3.79174e7 1.25984 0.629921 0.776659i \(-0.283087\pi\)
0.629921 + 0.776659i \(0.283087\pi\)
\(138\) 0 0
\(139\) 2.80300e7i 0.885260i 0.896704 + 0.442630i \(0.145955\pi\)
−0.896704 + 0.442630i \(0.854045\pi\)
\(140\) 0 0
\(141\) 3.42652e6i 0.102941i
\(142\) 0 0
\(143\) −4.86630e7 −1.39163
\(144\) 0 0
\(145\) 4.38012e6 0.119316
\(146\) 0 0
\(147\) 6.21078e6i 0.161263i
\(148\) 0 0
\(149\) 6.82083e6i 0.168922i 0.996427 + 0.0844608i \(0.0269168\pi\)
−0.996427 + 0.0844608i \(0.973083\pi\)
\(150\) 0 0
\(151\) 7.62068e7 1.80125 0.900626 0.434595i \(-0.143109\pi\)
0.900626 + 0.434595i \(0.143109\pi\)
\(152\) 0 0
\(153\) −3.70907e7 −0.837230
\(154\) 0 0
\(155\) 1.34881e7i 0.290931i
\(156\) 0 0
\(157\) 4.28737e7i 0.884183i 0.896970 + 0.442092i \(0.145763\pi\)
−0.896970 + 0.442092i \(0.854237\pi\)
\(158\) 0 0
\(159\) 1.50725e7 0.297369
\(160\) 0 0
\(161\) 4.64373e6 0.0876952
\(162\) 0 0
\(163\) 4.12127e7i 0.745374i 0.927957 + 0.372687i \(0.121563\pi\)
−0.927957 + 0.372687i \(0.878437\pi\)
\(164\) 0 0
\(165\) − 1.75448e7i − 0.304056i
\(166\) 0 0
\(167\) −2.41703e7 −0.401582 −0.200791 0.979634i \(-0.564351\pi\)
−0.200791 + 0.979634i \(0.564351\pi\)
\(168\) 0 0
\(169\) −6.38869e7 −1.01814
\(170\) 0 0
\(171\) − 7.75477e7i − 1.18599i
\(172\) 0 0
\(173\) − 5.90454e7i − 0.867011i −0.901151 0.433505i \(-0.857277\pi\)
0.901151 0.433505i \(-0.142723\pi\)
\(174\) 0 0
\(175\) −4.62137e7 −0.651835
\(176\) 0 0
\(177\) 3.94850e7 0.535212
\(178\) 0 0
\(179\) − 1.20311e8i − 1.56791i −0.620819 0.783954i \(-0.713200\pi\)
0.620819 0.783954i \(-0.286800\pi\)
\(180\) 0 0
\(181\) 3.21539e6i 0.0403049i 0.999797 + 0.0201525i \(0.00641517\pi\)
−0.999797 + 0.0201525i \(0.993585\pi\)
\(182\) 0 0
\(183\) 2.99259e7 0.360968
\(184\) 0 0
\(185\) 8.93768e7 1.03782
\(186\) 0 0
\(187\) 9.40392e7i 1.05163i
\(188\) 0 0
\(189\) − 8.98443e7i − 0.967997i
\(190\) 0 0
\(191\) −7.04122e7 −0.731191 −0.365596 0.930774i \(-0.619135\pi\)
−0.365596 + 0.930774i \(0.619135\pi\)
\(192\) 0 0
\(193\) 1.17861e7 0.118010 0.0590050 0.998258i \(-0.481207\pi\)
0.0590050 + 0.998258i \(0.481207\pi\)
\(194\) 0 0
\(195\) − 4.56566e7i − 0.440943i
\(196\) 0 0
\(197\) − 8.97008e7i − 0.835920i −0.908465 0.417960i \(-0.862745\pi\)
0.908465 0.417960i \(-0.137255\pi\)
\(198\) 0 0
\(199\) 3.98955e7 0.358871 0.179436 0.983770i \(-0.442573\pi\)
0.179436 + 0.983770i \(0.442573\pi\)
\(200\) 0 0
\(201\) −2.38833e7 −0.207447
\(202\) 0 0
\(203\) − 2.49178e7i − 0.209061i
\(204\) 0 0
\(205\) 7.60078e7i 0.616197i
\(206\) 0 0
\(207\) −7.52915e6 −0.0589997
\(208\) 0 0
\(209\) −1.96613e8 −1.48971
\(210\) 0 0
\(211\) 6.63604e7i 0.486318i 0.969986 + 0.243159i \(0.0781837\pi\)
−0.969986 + 0.243159i \(0.921816\pi\)
\(212\) 0 0
\(213\) 1.23051e8i 0.872479i
\(214\) 0 0
\(215\) 1.77825e7 0.122027
\(216\) 0 0
\(217\) 7.67317e7 0.509760
\(218\) 0 0
\(219\) 476088.i 0.00306290i
\(220\) 0 0
\(221\) 2.44718e8i 1.52508i
\(222\) 0 0
\(223\) 1.57269e8 0.949675 0.474837 0.880074i \(-0.342507\pi\)
0.474837 + 0.880074i \(0.342507\pi\)
\(224\) 0 0
\(225\) 7.49291e7 0.438543
\(226\) 0 0
\(227\) 1.95129e8i 1.10721i 0.832778 + 0.553607i \(0.186749\pi\)
−0.832778 + 0.553607i \(0.813251\pi\)
\(228\) 0 0
\(229\) − 6.95136e7i − 0.382512i −0.981540 0.191256i \(-0.938744\pi\)
0.981540 0.191256i \(-0.0612561\pi\)
\(230\) 0 0
\(231\) −9.98095e7 −0.532758
\(232\) 0 0
\(233\) −3.81872e7 −0.197776 −0.0988878 0.995099i \(-0.531528\pi\)
−0.0988878 + 0.995099i \(0.531528\pi\)
\(234\) 0 0
\(235\) − 2.88785e7i − 0.145157i
\(236\) 0 0
\(237\) − 5.13437e7i − 0.250535i
\(238\) 0 0
\(239\) 2.63549e8 1.24873 0.624366 0.781132i \(-0.285357\pi\)
0.624366 + 0.781132i \(0.285357\pi\)
\(240\) 0 0
\(241\) 1.21337e8 0.558386 0.279193 0.960235i \(-0.409933\pi\)
0.279193 + 0.960235i \(0.409933\pi\)
\(242\) 0 0
\(243\) 2.27512e8i 1.01715i
\(244\) 0 0
\(245\) − 5.23441e7i − 0.227398i
\(246\) 0 0
\(247\) −5.11645e8 −2.16038
\(248\) 0 0
\(249\) −1.93555e7 −0.0794523
\(250\) 0 0
\(251\) − 3.62042e7i − 0.144511i −0.997386 0.0722555i \(-0.976980\pi\)
0.997386 0.0722555i \(-0.0230197\pi\)
\(252\) 0 0
\(253\) 1.90893e7i 0.0741085i
\(254\) 0 0
\(255\) −8.82295e7 −0.333214
\(256\) 0 0
\(257\) 1.75550e8 0.645111 0.322555 0.946551i \(-0.395458\pi\)
0.322555 + 0.946551i \(0.395458\pi\)
\(258\) 0 0
\(259\) − 5.08451e8i − 1.81844i
\(260\) 0 0
\(261\) 4.04008e7i 0.140653i
\(262\) 0 0
\(263\) 1.10733e8 0.375348 0.187674 0.982231i \(-0.439905\pi\)
0.187674 + 0.982231i \(0.439905\pi\)
\(264\) 0 0
\(265\) −1.27030e8 −0.419320
\(266\) 0 0
\(267\) − 2.95627e7i − 0.0950505i
\(268\) 0 0
\(269\) − 3.07561e8i − 0.963382i −0.876341 0.481691i \(-0.840023\pi\)
0.876341 0.481691i \(-0.159977\pi\)
\(270\) 0 0
\(271\) 2.86036e8 0.873027 0.436513 0.899698i \(-0.356213\pi\)
0.436513 + 0.899698i \(0.356213\pi\)
\(272\) 0 0
\(273\) −2.59734e8 −0.772607
\(274\) 0 0
\(275\) − 1.89974e8i − 0.550846i
\(276\) 0 0
\(277\) − 3.15642e8i − 0.892309i −0.894956 0.446155i \(-0.852793\pi\)
0.894956 0.446155i \(-0.147207\pi\)
\(278\) 0 0
\(279\) −1.24410e8 −0.342957
\(280\) 0 0
\(281\) −2.50451e8 −0.673366 −0.336683 0.941618i \(-0.609305\pi\)
−0.336683 + 0.941618i \(0.609305\pi\)
\(282\) 0 0
\(283\) − 3.47387e7i − 0.0911088i −0.998962 0.0455544i \(-0.985495\pi\)
0.998962 0.0455544i \(-0.0145054\pi\)
\(284\) 0 0
\(285\) − 1.84467e8i − 0.472020i
\(286\) 0 0
\(287\) 4.32397e8 1.07968
\(288\) 0 0
\(289\) 6.25676e7 0.152478
\(290\) 0 0
\(291\) 1.60782e8i 0.382482i
\(292\) 0 0
\(293\) 5.34336e8i 1.24102i 0.784200 + 0.620508i \(0.213074\pi\)
−0.784200 + 0.620508i \(0.786926\pi\)
\(294\) 0 0
\(295\) −3.32777e8 −0.754703
\(296\) 0 0
\(297\) 3.69329e8 0.818024
\(298\) 0 0
\(299\) 4.96760e7i 0.107472i
\(300\) 0 0
\(301\) − 1.01162e8i − 0.213813i
\(302\) 0 0
\(303\) 2.11954e8 0.437716
\(304\) 0 0
\(305\) −2.52214e8 −0.509002
\(306\) 0 0
\(307\) − 1.68851e8i − 0.333057i −0.986037 0.166529i \(-0.946744\pi\)
0.986037 0.166529i \(-0.0532558\pi\)
\(308\) 0 0
\(309\) 3.04229e8i 0.586605i
\(310\) 0 0
\(311\) −6.99677e8 −1.31897 −0.659487 0.751716i \(-0.729226\pi\)
−0.659487 + 0.751716i \(0.729226\pi\)
\(312\) 0 0
\(313\) 5.06345e8 0.933343 0.466671 0.884431i \(-0.345453\pi\)
0.466671 + 0.884431i \(0.345453\pi\)
\(314\) 0 0
\(315\) 3.31779e8i 0.598084i
\(316\) 0 0
\(317\) 4.78996e8i 0.844548i 0.906468 + 0.422274i \(0.138768\pi\)
−0.906468 + 0.422274i \(0.861232\pi\)
\(318\) 0 0
\(319\) 1.02431e8 0.176671
\(320\) 0 0
\(321\) 4.13263e8 0.697364
\(322\) 0 0
\(323\) 9.88733e8i 1.63256i
\(324\) 0 0
\(325\) − 4.94369e8i − 0.798839i
\(326\) 0 0
\(327\) −1.88827e8 −0.298639
\(328\) 0 0
\(329\) −1.64285e8 −0.254339
\(330\) 0 0
\(331\) 4.20825e8i 0.637828i 0.947784 + 0.318914i \(0.103318\pi\)
−0.947784 + 0.318914i \(0.896682\pi\)
\(332\) 0 0
\(333\) 8.24382e8i 1.22341i
\(334\) 0 0
\(335\) 2.01287e8 0.292522
\(336\) 0 0
\(337\) −1.02987e9 −1.46580 −0.732902 0.680334i \(-0.761835\pi\)
−0.732902 + 0.680334i \(0.761835\pi\)
\(338\) 0 0
\(339\) − 2.88170e8i − 0.401745i
\(340\) 0 0
\(341\) 3.15427e8i 0.430782i
\(342\) 0 0
\(343\) 5.68554e8 0.760751
\(344\) 0 0
\(345\) −1.79100e7 −0.0234816
\(346\) 0 0
\(347\) − 5.31496e8i − 0.682883i −0.939903 0.341442i \(-0.889085\pi\)
0.939903 0.341442i \(-0.110915\pi\)
\(348\) 0 0
\(349\) − 1.31436e9i − 1.65511i −0.561385 0.827555i \(-0.689731\pi\)
0.561385 0.827555i \(-0.310269\pi\)
\(350\) 0 0
\(351\) 9.61103e8 1.18630
\(352\) 0 0
\(353\) 1.42031e9 1.71858 0.859291 0.511486i \(-0.170905\pi\)
0.859291 + 0.511486i \(0.170905\pi\)
\(354\) 0 0
\(355\) − 1.03706e9i − 1.23028i
\(356\) 0 0
\(357\) 5.01924e8i 0.583847i
\(358\) 0 0
\(359\) −7.59524e8 −0.866385 −0.433193 0.901301i \(-0.642613\pi\)
−0.433193 + 0.901301i \(0.642613\pi\)
\(360\) 0 0
\(361\) −1.17333e9 −1.31264
\(362\) 0 0
\(363\) 1.72702e7i 0.0189507i
\(364\) 0 0
\(365\) − 4.01244e6i − 0.00431900i
\(366\) 0 0
\(367\) −3.76159e8 −0.397228 −0.198614 0.980078i \(-0.563644\pi\)
−0.198614 + 0.980078i \(0.563644\pi\)
\(368\) 0 0
\(369\) −7.01071e8 −0.726390
\(370\) 0 0
\(371\) 7.22655e8i 0.734720i
\(372\) 0 0
\(373\) − 4.57130e8i − 0.456099i −0.973650 0.228049i \(-0.926765\pi\)
0.973650 0.228049i \(-0.0732347\pi\)
\(374\) 0 0
\(375\) 4.95206e8 0.484927
\(376\) 0 0
\(377\) 2.66557e8 0.256209
\(378\) 0 0
\(379\) − 1.91565e9i − 1.80750i −0.428056 0.903752i \(-0.640801\pi\)
0.428056 0.903752i \(-0.359199\pi\)
\(380\) 0 0
\(381\) − 6.99402e8i − 0.647873i
\(382\) 0 0
\(383\) −1.57853e9 −1.43568 −0.717839 0.696209i \(-0.754869\pi\)
−0.717839 + 0.696209i \(0.754869\pi\)
\(384\) 0 0
\(385\) 8.41187e8 0.751243
\(386\) 0 0
\(387\) 1.64020e8i 0.143849i
\(388\) 0 0
\(389\) − 2.21214e9i − 1.90541i −0.303890 0.952707i \(-0.598285\pi\)
0.303890 0.952707i \(-0.401715\pi\)
\(390\) 0 0
\(391\) 9.59967e7 0.0812153
\(392\) 0 0
\(393\) −2.56817e8 −0.213427
\(394\) 0 0
\(395\) 4.32721e8i 0.353279i
\(396\) 0 0
\(397\) 1.30879e9i 1.04979i 0.851167 + 0.524896i \(0.175896\pi\)
−0.851167 + 0.524896i \(0.824104\pi\)
\(398\) 0 0
\(399\) −1.04940e9 −0.827060
\(400\) 0 0
\(401\) 1.04514e8 0.0809413 0.0404706 0.999181i \(-0.487114\pi\)
0.0404706 + 0.999181i \(0.487114\pi\)
\(402\) 0 0
\(403\) 8.20832e8i 0.624722i
\(404\) 0 0
\(405\) − 3.43251e8i − 0.256755i
\(406\) 0 0
\(407\) 2.09013e9 1.53671
\(408\) 0 0
\(409\) 1.94107e9 1.40284 0.701421 0.712747i \(-0.252549\pi\)
0.701421 + 0.712747i \(0.252549\pi\)
\(410\) 0 0
\(411\) 8.31937e8i 0.591077i
\(412\) 0 0
\(413\) 1.89312e9i 1.32237i
\(414\) 0 0
\(415\) 1.63127e8 0.112036
\(416\) 0 0
\(417\) −6.15000e8 −0.415336
\(418\) 0 0
\(419\) − 1.96429e9i − 1.30454i −0.757989 0.652268i \(-0.773818\pi\)
0.757989 0.652268i \(-0.226182\pi\)
\(420\) 0 0
\(421\) 2.33176e9i 1.52299i 0.648174 + 0.761493i \(0.275533\pi\)
−0.648174 + 0.761493i \(0.724467\pi\)
\(422\) 0 0
\(423\) 2.66366e8 0.171115
\(424\) 0 0
\(425\) −9.55346e8 −0.603670
\(426\) 0 0
\(427\) 1.43480e9i 0.891857i
\(428\) 0 0
\(429\) − 1.06771e9i − 0.652907i
\(430\) 0 0
\(431\) −1.72745e8 −0.103929 −0.0519643 0.998649i \(-0.516548\pi\)
−0.0519643 + 0.998649i \(0.516548\pi\)
\(432\) 0 0
\(433\) −2.71381e9 −1.60647 −0.803233 0.595665i \(-0.796889\pi\)
−0.803233 + 0.595665i \(0.796889\pi\)
\(434\) 0 0
\(435\) 9.61033e7i 0.0559791i
\(436\) 0 0
\(437\) 2.00706e8i 0.115047i
\(438\) 0 0
\(439\) 2.73808e8 0.154461 0.0772307 0.997013i \(-0.475392\pi\)
0.0772307 + 0.997013i \(0.475392\pi\)
\(440\) 0 0
\(441\) 4.82804e8 0.268063
\(442\) 0 0
\(443\) 2.72048e9i 1.48673i 0.668886 + 0.743365i \(0.266771\pi\)
−0.668886 + 0.743365i \(0.733229\pi\)
\(444\) 0 0
\(445\) 2.49152e8i 0.134031i
\(446\) 0 0
\(447\) −1.49654e8 −0.0792526
\(448\) 0 0
\(449\) 1.69111e9 0.881675 0.440837 0.897587i \(-0.354682\pi\)
0.440837 + 0.897587i \(0.354682\pi\)
\(450\) 0 0
\(451\) 1.77748e9i 0.912406i
\(452\) 0 0
\(453\) 1.67204e9i 0.845089i
\(454\) 0 0
\(455\) 2.18902e9 1.08945
\(456\) 0 0
\(457\) −1.15773e8 −0.0567413 −0.0283706 0.999597i \(-0.509032\pi\)
−0.0283706 + 0.999597i \(0.509032\pi\)
\(458\) 0 0
\(459\) − 1.85729e9i − 0.896470i
\(460\) 0 0
\(461\) − 9.34713e8i − 0.444350i −0.975007 0.222175i \(-0.928684\pi\)
0.975007 0.222175i \(-0.0713156\pi\)
\(462\) 0 0
\(463\) −1.91738e9 −0.897790 −0.448895 0.893584i \(-0.648182\pi\)
−0.448895 + 0.893584i \(0.648182\pi\)
\(464\) 0 0
\(465\) −2.95940e8 −0.136495
\(466\) 0 0
\(467\) − 1.65838e9i − 0.753485i −0.926318 0.376742i \(-0.877044\pi\)
0.926318 0.376742i \(-0.122956\pi\)
\(468\) 0 0
\(469\) − 1.14509e9i − 0.512548i
\(470\) 0 0
\(471\) −9.40684e8 −0.414830
\(472\) 0 0
\(473\) 4.15853e8 0.180686
\(474\) 0 0
\(475\) − 1.99740e9i − 0.855140i
\(476\) 0 0
\(477\) − 1.17168e9i − 0.494306i
\(478\) 0 0
\(479\) −4.09755e9 −1.70353 −0.851766 0.523923i \(-0.824468\pi\)
−0.851766 + 0.523923i \(0.824468\pi\)
\(480\) 0 0
\(481\) 5.43912e9 2.22854
\(482\) 0 0
\(483\) 1.01887e8i 0.0411438i
\(484\) 0 0
\(485\) − 1.35506e9i − 0.539339i
\(486\) 0 0
\(487\) 3.24167e9 1.27180 0.635898 0.771773i \(-0.280630\pi\)
0.635898 + 0.771773i \(0.280630\pi\)
\(488\) 0 0
\(489\) −9.04239e8 −0.349705
\(490\) 0 0
\(491\) − 2.16351e9i − 0.824847i −0.910992 0.412423i \(-0.864682\pi\)
0.910992 0.412423i \(-0.135318\pi\)
\(492\) 0 0
\(493\) − 5.15110e8i − 0.193613i
\(494\) 0 0
\(495\) −1.36387e9 −0.505422
\(496\) 0 0
\(497\) −5.89969e9 −2.15567
\(498\) 0 0
\(499\) 2.16713e9i 0.780787i 0.920648 + 0.390394i \(0.127661\pi\)
−0.920648 + 0.390394i \(0.872339\pi\)
\(500\) 0 0
\(501\) − 5.30316e8i − 0.188409i
\(502\) 0 0
\(503\) 2.58480e9 0.905606 0.452803 0.891611i \(-0.350424\pi\)
0.452803 + 0.891611i \(0.350424\pi\)
\(504\) 0 0
\(505\) −1.78633e9 −0.617224
\(506\) 0 0
\(507\) − 1.40173e9i − 0.477680i
\(508\) 0 0
\(509\) − 2.11747e9i − 0.711712i −0.934541 0.355856i \(-0.884189\pi\)
0.934541 0.355856i \(-0.115811\pi\)
\(510\) 0 0
\(511\) −2.28261e7 −0.00756762
\(512\) 0 0
\(513\) 3.88315e9 1.26991
\(514\) 0 0
\(515\) − 2.56402e9i − 0.827172i
\(516\) 0 0
\(517\) − 6.75339e8i − 0.214934i
\(518\) 0 0
\(519\) 1.29550e9 0.406773
\(520\) 0 0
\(521\) −2.34354e9 −0.726006 −0.363003 0.931788i \(-0.618249\pi\)
−0.363003 + 0.931788i \(0.618249\pi\)
\(522\) 0 0
\(523\) 3.26155e9i 0.996939i 0.866907 + 0.498470i \(0.166104\pi\)
−0.866907 + 0.498470i \(0.833896\pi\)
\(524\) 0 0
\(525\) − 1.01397e9i − 0.305820i
\(526\) 0 0
\(527\) 1.58622e9 0.472093
\(528\) 0 0
\(529\) −3.38534e9 −0.994277
\(530\) 0 0
\(531\) − 3.06942e9i − 0.889664i
\(532\) 0 0
\(533\) 4.62554e9i 1.32317i
\(534\) 0 0
\(535\) −3.48296e9 −0.983354
\(536\) 0 0
\(537\) 2.63973e9 0.735612
\(538\) 0 0
\(539\) − 1.22409e9i − 0.336709i
\(540\) 0 0
\(541\) 7.24638e9i 1.96757i 0.179344 + 0.983787i \(0.442603\pi\)
−0.179344 + 0.983787i \(0.557397\pi\)
\(542\) 0 0
\(543\) −7.05482e7 −0.0189098
\(544\) 0 0
\(545\) 1.59142e9 0.421111
\(546\) 0 0
\(547\) 6.41460e9i 1.67577i 0.545848 + 0.837884i \(0.316207\pi\)
−0.545848 + 0.837884i \(0.683793\pi\)
\(548\) 0 0
\(549\) − 2.32634e9i − 0.600025i
\(550\) 0 0
\(551\) 1.07697e9 0.274267
\(552\) 0 0
\(553\) 2.46168e9 0.619005
\(554\) 0 0
\(555\) 1.96100e9i 0.486914i
\(556\) 0 0
\(557\) − 1.76223e9i − 0.432085i −0.976384 0.216042i \(-0.930685\pi\)
0.976384 0.216042i \(-0.0693149\pi\)
\(558\) 0 0
\(559\) 1.08217e9 0.262032
\(560\) 0 0
\(561\) −2.06330e9 −0.493391
\(562\) 0 0
\(563\) 2.33270e9i 0.550909i 0.961314 + 0.275454i \(0.0888283\pi\)
−0.961314 + 0.275454i \(0.911172\pi\)
\(564\) 0 0
\(565\) 2.42868e9i 0.566501i
\(566\) 0 0
\(567\) −1.95270e9 −0.449878
\(568\) 0 0
\(569\) 5.53591e9 1.25978 0.629892 0.776683i \(-0.283099\pi\)
0.629892 + 0.776683i \(0.283099\pi\)
\(570\) 0 0
\(571\) 3.70890e9i 0.833717i 0.908971 + 0.416858i \(0.136869\pi\)
−0.908971 + 0.416858i \(0.863131\pi\)
\(572\) 0 0
\(573\) − 1.54490e9i − 0.343051i
\(574\) 0 0
\(575\) −1.93929e8 −0.0425407
\(576\) 0 0
\(577\) 3.63713e9 0.788212 0.394106 0.919065i \(-0.371054\pi\)
0.394106 + 0.919065i \(0.371054\pi\)
\(578\) 0 0
\(579\) 2.58596e8i 0.0553665i
\(580\) 0 0
\(581\) − 9.28003e8i − 0.196306i
\(582\) 0 0
\(583\) −2.97067e9 −0.620889
\(584\) 0 0
\(585\) −3.54919e9 −0.732965
\(586\) 0 0
\(587\) − 2.04311e9i − 0.416926i −0.978030 0.208463i \(-0.933154\pi\)
0.978030 0.208463i \(-0.0668461\pi\)
\(588\) 0 0
\(589\) 3.31641e9i 0.668752i
\(590\) 0 0
\(591\) 1.96811e9 0.392187
\(592\) 0 0
\(593\) −5.74764e9 −1.13187 −0.565937 0.824448i \(-0.691486\pi\)
−0.565937 + 0.824448i \(0.691486\pi\)
\(594\) 0 0
\(595\) − 4.23018e9i − 0.823284i
\(596\) 0 0
\(597\) 8.75341e8i 0.168371i
\(598\) 0 0
\(599\) −3.14564e8 −0.0598020 −0.0299010 0.999553i \(-0.509519\pi\)
−0.0299010 + 0.999553i \(0.509519\pi\)
\(600\) 0 0
\(601\) −2.49586e9 −0.468985 −0.234493 0.972118i \(-0.575343\pi\)
−0.234493 + 0.972118i \(0.575343\pi\)
\(602\) 0 0
\(603\) 1.85660e9i 0.344833i
\(604\) 0 0
\(605\) − 1.45552e8i − 0.0267224i
\(606\) 0 0
\(607\) 2.81126e9 0.510200 0.255100 0.966915i \(-0.417892\pi\)
0.255100 + 0.966915i \(0.417892\pi\)
\(608\) 0 0
\(609\) 5.46717e8 0.0980849
\(610\) 0 0
\(611\) − 1.75743e9i − 0.311698i
\(612\) 0 0
\(613\) − 2.34343e9i − 0.410904i −0.978667 0.205452i \(-0.934134\pi\)
0.978667 0.205452i \(-0.0658664\pi\)
\(614\) 0 0
\(615\) −1.66767e9 −0.289100
\(616\) 0 0
\(617\) −6.26737e8 −0.107420 −0.0537102 0.998557i \(-0.517105\pi\)
−0.0537102 + 0.998557i \(0.517105\pi\)
\(618\) 0 0
\(619\) − 8.90879e9i − 1.50974i −0.655876 0.754869i \(-0.727700\pi\)
0.655876 0.754869i \(-0.272300\pi\)
\(620\) 0 0
\(621\) − 3.77017e8i − 0.0631743i
\(622\) 0 0
\(623\) 1.41739e9 0.234845
\(624\) 0 0
\(625\) −7.41437e8 −0.121477
\(626\) 0 0
\(627\) − 4.31385e9i − 0.698922i
\(628\) 0 0
\(629\) − 1.05109e10i − 1.68407i
\(630\) 0 0
\(631\) 2.21942e9 0.351671 0.175835 0.984420i \(-0.443737\pi\)
0.175835 + 0.984420i \(0.443737\pi\)
\(632\) 0 0
\(633\) −1.45600e9 −0.228165
\(634\) 0 0
\(635\) 5.89451e9i 0.913566i
\(636\) 0 0
\(637\) − 3.18545e9i − 0.488296i
\(638\) 0 0
\(639\) 9.56552e9 1.45029
\(640\) 0 0
\(641\) 8.58104e9 1.28688 0.643438 0.765498i \(-0.277507\pi\)
0.643438 + 0.765498i \(0.277507\pi\)
\(642\) 0 0
\(643\) − 7.71279e9i − 1.14412i −0.820210 0.572062i \(-0.806144\pi\)
0.820210 0.572062i \(-0.193856\pi\)
\(644\) 0 0
\(645\) 3.90161e8i 0.0572513i
\(646\) 0 0
\(647\) −1.18898e10 −1.72587 −0.862935 0.505316i \(-0.831376\pi\)
−0.862935 + 0.505316i \(0.831376\pi\)
\(648\) 0 0
\(649\) −7.78217e9 −1.11749
\(650\) 0 0
\(651\) 1.68356e9i 0.239163i
\(652\) 0 0
\(653\) 2.00345e9i 0.281567i 0.990040 + 0.140784i \(0.0449622\pi\)
−0.990040 + 0.140784i \(0.955038\pi\)
\(654\) 0 0
\(655\) 2.16443e9 0.300954
\(656\) 0 0
\(657\) 3.70094e7 0.00509135
\(658\) 0 0
\(659\) 7.66944e9i 1.04391i 0.852972 + 0.521957i \(0.174798\pi\)
−0.852972 + 0.521957i \(0.825202\pi\)
\(660\) 0 0
\(661\) − 4.04041e9i − 0.544152i −0.962276 0.272076i \(-0.912290\pi\)
0.962276 0.272076i \(-0.0877103\pi\)
\(662\) 0 0
\(663\) −5.36930e9 −0.715518
\(664\) 0 0
\(665\) 8.84429e9 1.16624
\(666\) 0 0
\(667\) − 1.04564e8i − 0.0136440i
\(668\) 0 0
\(669\) 3.45060e9i 0.445557i
\(670\) 0 0
\(671\) −5.89816e9 −0.753681
\(672\) 0 0
\(673\) 4.11138e9 0.519918 0.259959 0.965620i \(-0.416291\pi\)
0.259959 + 0.965620i \(0.416291\pi\)
\(674\) 0 0
\(675\) 3.75203e9i 0.469572i
\(676\) 0 0
\(677\) 1.12671e10i 1.39557i 0.716305 + 0.697787i \(0.245832\pi\)
−0.716305 + 0.697787i \(0.754168\pi\)
\(678\) 0 0
\(679\) −7.70872e9 −0.945014
\(680\) 0 0
\(681\) −4.28129e9 −0.519469
\(682\) 0 0
\(683\) − 8.00228e9i − 0.961039i −0.876984 0.480520i \(-0.840448\pi\)
0.876984 0.480520i \(-0.159552\pi\)
\(684\) 0 0
\(685\) − 7.01151e9i − 0.833479i
\(686\) 0 0
\(687\) 1.52518e9 0.179462
\(688\) 0 0
\(689\) −7.73055e9 −0.900416
\(690\) 0 0
\(691\) 7.24716e9i 0.835593i 0.908541 + 0.417797i \(0.137197\pi\)
−0.908541 + 0.417797i \(0.862803\pi\)
\(692\) 0 0
\(693\) 7.75884e9i 0.885585i
\(694\) 0 0
\(695\) 5.18318e9 0.585665
\(696\) 0 0
\(697\) 8.93865e9 0.999902
\(698\) 0 0
\(699\) − 8.37859e8i − 0.0927899i
\(700\) 0 0
\(701\) 4.68293e8i 0.0513458i 0.999670 + 0.0256729i \(0.00817283\pi\)
−0.999670 + 0.0256729i \(0.991827\pi\)
\(702\) 0 0
\(703\) 2.19757e10 2.38561
\(704\) 0 0
\(705\) 6.33617e8 0.0681028
\(706\) 0 0
\(707\) 1.01622e10i 1.08148i
\(708\) 0 0
\(709\) 1.08926e10i 1.14781i 0.818923 + 0.573903i \(0.194571\pi\)
−0.818923 + 0.573903i \(0.805429\pi\)
\(710\) 0 0
\(711\) −3.99128e9 −0.416455
\(712\) 0 0
\(713\) 3.21992e8 0.0332684
\(714\) 0 0
\(715\) 8.99855e9i 0.920664i
\(716\) 0 0
\(717\) 5.78249e9i 0.585865i
\(718\) 0 0
\(719\) 7.40284e9 0.742758 0.371379 0.928481i \(-0.378885\pi\)
0.371379 + 0.928481i \(0.378885\pi\)
\(720\) 0 0
\(721\) −1.45863e10 −1.44935
\(722\) 0 0
\(723\) 2.66224e9i 0.261977i
\(724\) 0 0
\(725\) 1.04060e9i 0.101415i
\(726\) 0 0
\(727\) −1.04295e10 −1.00668 −0.503342 0.864087i \(-0.667897\pi\)
−0.503342 + 0.864087i \(0.667897\pi\)
\(728\) 0 0
\(729\) −9.32176e8 −0.0891152
\(730\) 0 0
\(731\) − 2.09125e9i − 0.198014i
\(732\) 0 0
\(733\) − 1.55538e10i − 1.45873i −0.684127 0.729363i \(-0.739816\pi\)
0.684127 0.729363i \(-0.260184\pi\)
\(734\) 0 0
\(735\) 1.14847e9 0.106688
\(736\) 0 0
\(737\) 4.70720e9 0.433138
\(738\) 0 0
\(739\) − 1.07140e10i − 0.976554i −0.872689 0.488277i \(-0.837626\pi\)
0.872689 0.488277i \(-0.162374\pi\)
\(740\) 0 0
\(741\) − 1.12259e10i − 1.01358i
\(742\) 0 0
\(743\) 5.46982e9 0.489229 0.244615 0.969620i \(-0.421339\pi\)
0.244615 + 0.969620i \(0.421339\pi\)
\(744\) 0 0
\(745\) 1.26128e9 0.111754
\(746\) 0 0
\(747\) 1.50463e9i 0.132071i
\(748\) 0 0
\(749\) 1.98140e10i 1.72300i
\(750\) 0 0
\(751\) −5.68185e9 −0.489497 −0.244749 0.969587i \(-0.578705\pi\)
−0.244749 + 0.969587i \(0.578705\pi\)
\(752\) 0 0
\(753\) 7.94349e8 0.0677999
\(754\) 0 0
\(755\) − 1.40918e10i − 1.19166i
\(756\) 0 0
\(757\) − 5.38304e9i − 0.451016i −0.974241 0.225508i \(-0.927596\pi\)
0.974241 0.225508i \(-0.0724043\pi\)
\(758\) 0 0
\(759\) −4.18835e8 −0.0347693
\(760\) 0 0
\(761\) −3.28593e9 −0.270279 −0.135139 0.990827i \(-0.543148\pi\)
−0.135139 + 0.990827i \(0.543148\pi\)
\(762\) 0 0
\(763\) − 9.05334e9i − 0.737858i
\(764\) 0 0
\(765\) 6.85865e9i 0.553890i
\(766\) 0 0
\(767\) −2.02515e10 −1.62059
\(768\) 0 0
\(769\) 7.10914e9 0.563735 0.281868 0.959453i \(-0.409046\pi\)
0.281868 + 0.959453i \(0.409046\pi\)
\(770\) 0 0
\(771\) 3.85170e9i 0.302665i
\(772\) 0 0
\(773\) − 1.08690e10i − 0.846370i −0.906043 0.423185i \(-0.860912\pi\)
0.906043 0.423185i \(-0.139088\pi\)
\(774\) 0 0
\(775\) −3.20443e9 −0.247283
\(776\) 0 0
\(777\) 1.11558e10 0.853155
\(778\) 0 0
\(779\) 1.86886e10i 1.41643i
\(780\) 0 0
\(781\) − 2.42523e10i − 1.82169i
\(782\) 0 0
\(783\) −2.02304e9 −0.150605
\(784\) 0 0
\(785\) 7.92802e9 0.584953
\(786\) 0 0
\(787\) − 1.94246e10i − 1.42050i −0.703951 0.710248i \(-0.748583\pi\)
0.703951 0.710248i \(-0.251417\pi\)
\(788\) 0 0
\(789\) 2.42958e9i 0.176101i
\(790\) 0 0
\(791\) 1.38164e10 0.992606
\(792\) 0 0
\(793\) −1.53487e10 −1.09299
\(794\) 0 0
\(795\) − 2.78714e9i − 0.196732i
\(796\) 0 0
\(797\) − 2.30309e10i − 1.61141i −0.592314 0.805707i \(-0.701786\pi\)
0.592314 0.805707i \(-0.298214\pi\)
\(798\) 0 0
\(799\) −3.39616e9 −0.235545
\(800\) 0 0
\(801\) −2.29810e9 −0.157999
\(802\) 0 0
\(803\) − 9.38331e7i − 0.00639516i
\(804\) 0 0
\(805\) − 8.58697e8i − 0.0580169i
\(806\) 0 0
\(807\) 6.74814e9 0.451988
\(808\) 0 0
\(809\) −1.50546e10 −0.999656 −0.499828 0.866125i \(-0.666603\pi\)
−0.499828 + 0.866125i \(0.666603\pi\)
\(810\) 0 0
\(811\) − 1.98821e10i − 1.30885i −0.756128 0.654424i \(-0.772911\pi\)
0.756128 0.654424i \(-0.227089\pi\)
\(812\) 0 0
\(813\) 6.27585e9i 0.409596i
\(814\) 0 0
\(815\) 7.62086e9 0.493120
\(816\) 0 0
\(817\) 4.37230e9 0.280500
\(818\) 0 0
\(819\) 2.01908e10i 1.28428i
\(820\) 0 0
\(821\) 2.29870e10i 1.44971i 0.688901 + 0.724855i \(0.258093\pi\)
−0.688901 + 0.724855i \(0.741907\pi\)
\(822\) 0 0
\(823\) −1.46696e10 −0.917316 −0.458658 0.888613i \(-0.651670\pi\)
−0.458658 + 0.888613i \(0.651670\pi\)
\(824\) 0 0
\(825\) 4.16819e9 0.258439
\(826\) 0 0
\(827\) 1.26591e9i 0.0778277i 0.999243 + 0.0389138i \(0.0123898\pi\)
−0.999243 + 0.0389138i \(0.987610\pi\)
\(828\) 0 0
\(829\) 2.49778e10i 1.52270i 0.648342 + 0.761349i \(0.275463\pi\)
−0.648342 + 0.761349i \(0.724537\pi\)
\(830\) 0 0
\(831\) 6.92544e9 0.418643
\(832\) 0 0
\(833\) −6.15576e9 −0.368998
\(834\) 0 0
\(835\) 4.46947e9i 0.265676i
\(836\) 0 0
\(837\) − 6.22973e9i − 0.367223i
\(838\) 0 0
\(839\) 3.09477e10 1.80909 0.904547 0.426373i \(-0.140209\pi\)
0.904547 + 0.426373i \(0.140209\pi\)
\(840\) 0 0
\(841\) 1.66888e10 0.967473
\(842\) 0 0
\(843\) − 5.49510e9i − 0.315921i
\(844\) 0 0
\(845\) 1.18137e10i 0.673577i
\(846\) 0 0
\(847\) −8.28024e8 −0.0468221
\(848\) 0 0
\(849\) 7.62194e8 0.0427453
\(850\) 0 0
\(851\) − 2.13363e9i − 0.118677i
\(852\) 0 0
\(853\) − 1.86385e10i − 1.02823i −0.857723 0.514113i \(-0.828121\pi\)
0.857723 0.514113i \(-0.171879\pi\)
\(854\) 0 0
\(855\) −1.43398e10 −0.784623
\(856\) 0 0
\(857\) −2.37366e10 −1.28821 −0.644103 0.764939i \(-0.722769\pi\)
−0.644103 + 0.764939i \(0.722769\pi\)
\(858\) 0 0
\(859\) 2.46268e10i 1.32566i 0.748770 + 0.662830i \(0.230645\pi\)
−0.748770 + 0.662830i \(0.769355\pi\)
\(860\) 0 0
\(861\) 9.48713e9i 0.506552i
\(862\) 0 0
\(863\) −2.50051e10 −1.32431 −0.662157 0.749365i \(-0.730359\pi\)
−0.662157 + 0.749365i \(0.730359\pi\)
\(864\) 0 0
\(865\) −1.09184e10 −0.573592
\(866\) 0 0
\(867\) 1.37278e9i 0.0715378i
\(868\) 0 0
\(869\) 1.01194e10i 0.523102i
\(870\) 0 0
\(871\) 1.22495e10 0.628139
\(872\) 0 0
\(873\) 1.24986e10 0.635788
\(874\) 0 0
\(875\) 2.37428e10i 1.19813i
\(876\) 0 0
\(877\) 1.07280e10i 0.537057i 0.963272 + 0.268528i \(0.0865373\pi\)
−0.963272 + 0.268528i \(0.913463\pi\)
\(878\) 0 0
\(879\) −1.17238e10 −0.582245
\(880\) 0 0
\(881\) 5.94499e9 0.292911 0.146455 0.989217i \(-0.453214\pi\)
0.146455 + 0.989217i \(0.453214\pi\)
\(882\) 0 0
\(883\) 1.60870e10i 0.786342i 0.919465 + 0.393171i \(0.128622\pi\)
−0.919465 + 0.393171i \(0.871378\pi\)
\(884\) 0 0
\(885\) − 7.30139e9i − 0.354082i
\(886\) 0 0
\(887\) 2.61884e10 1.26001 0.630007 0.776589i \(-0.283052\pi\)
0.630007 + 0.776589i \(0.283052\pi\)
\(888\) 0 0
\(889\) 3.35330e10 1.60072
\(890\) 0 0
\(891\) − 8.02711e9i − 0.380178i
\(892\) 0 0
\(893\) − 7.10056e9i − 0.333666i
\(894\) 0 0
\(895\) −2.22474e10 −1.03729
\(896\) 0 0
\(897\) −1.08993e9 −0.0504226
\(898\) 0 0
\(899\) − 1.72778e9i − 0.0793104i
\(900\) 0 0
\(901\) 1.49390e10i 0.680430i
\(902\) 0 0
\(903\) 2.21957e9 0.100314
\(904\) 0 0
\(905\) 5.94575e8 0.0266647
\(906\) 0 0
\(907\) − 1.79753e9i − 0.0799929i −0.999200 0.0399964i \(-0.987265\pi\)
0.999200 0.0399964i \(-0.0127347\pi\)
\(908\) 0 0
\(909\) − 1.64766e10i − 0.727601i
\(910\) 0 0
\(911\) −3.70021e10 −1.62148 −0.810740 0.585406i \(-0.800935\pi\)
−0.810740 + 0.585406i \(0.800935\pi\)
\(912\) 0 0
\(913\) 3.81481e9 0.165892
\(914\) 0 0
\(915\) − 5.53377e9i − 0.238807i
\(916\) 0 0
\(917\) − 1.23131e10i − 0.527322i
\(918\) 0 0
\(919\) 2.44812e10 1.04047 0.520234 0.854024i \(-0.325845\pi\)
0.520234 + 0.854024i \(0.325845\pi\)
\(920\) 0 0
\(921\) 3.70472e9 0.156260
\(922\) 0 0
\(923\) − 6.31115e10i − 2.64182i
\(924\) 0 0
\(925\) 2.12336e10i 0.882121i
\(926\) 0 0
\(927\) 2.36497e10 0.975093
\(928\) 0 0
\(929\) −6.19981e9 −0.253702 −0.126851 0.991922i \(-0.540487\pi\)
−0.126851 + 0.991922i \(0.540487\pi\)
\(930\) 0 0
\(931\) − 1.28702e10i − 0.522711i
\(932\) 0 0
\(933\) − 1.53515e10i − 0.618820i
\(934\) 0 0
\(935\) 1.73893e10 0.695732
\(936\) 0 0
\(937\) −2.78747e10 −1.10693 −0.553466 0.832871i \(-0.686695\pi\)
−0.553466 + 0.832871i \(0.686695\pi\)
\(938\) 0 0
\(939\) 1.11096e10i 0.437894i
\(940\) 0 0
\(941\) − 3.45921e10i − 1.35336i −0.736277 0.676680i \(-0.763418\pi\)
0.736277 0.676680i \(-0.236582\pi\)
\(942\) 0 0
\(943\) 1.81448e9 0.0704632
\(944\) 0 0
\(945\) −1.66136e10 −0.640402
\(946\) 0 0
\(947\) 2.20475e10i 0.843598i 0.906689 + 0.421799i \(0.138601\pi\)
−0.906689 + 0.421799i \(0.861399\pi\)
\(948\) 0 0
\(949\) − 2.44181e8i − 0.00927429i
\(950\) 0 0
\(951\) −1.05096e10 −0.396235
\(952\) 0 0
\(953\) −3.95876e9 −0.148161 −0.0740805 0.997252i \(-0.523602\pi\)
−0.0740805 + 0.997252i \(0.523602\pi\)
\(954\) 0 0
\(955\) 1.30203e10i 0.483737i
\(956\) 0 0
\(957\) 2.24743e9i 0.0828885i
\(958\) 0 0
\(959\) −3.98874e10 −1.46040
\(960\) 0 0
\(961\) −2.21921e10 −0.806615
\(962\) 0 0
\(963\) − 3.21256e10i − 1.15920i
\(964\) 0 0
\(965\) − 2.17943e9i − 0.0780724i
\(966\) 0 0
\(967\) 9.76598e9 0.347315 0.173657 0.984806i \(-0.444441\pi\)
0.173657 + 0.984806i \(0.444441\pi\)
\(968\) 0 0
\(969\) −2.16936e10 −0.765946
\(970\) 0 0
\(971\) − 2.15553e10i − 0.755589i −0.925889 0.377795i \(-0.876683\pi\)
0.925889 0.377795i \(-0.123317\pi\)
\(972\) 0 0
\(973\) − 2.94863e10i − 1.02619i
\(974\) 0 0
\(975\) 1.08468e10 0.374789
\(976\) 0 0
\(977\) 5.24340e10 1.79880 0.899398 0.437131i \(-0.144005\pi\)
0.899398 + 0.437131i \(0.144005\pi\)
\(978\) 0 0
\(979\) 5.82657e9i 0.198460i
\(980\) 0 0
\(981\) 1.46787e10i 0.496417i
\(982\) 0 0
\(983\) 2.85116e10 0.957380 0.478690 0.877984i \(-0.341112\pi\)
0.478690 + 0.877984i \(0.341112\pi\)
\(984\) 0 0
\(985\) −1.65871e10 −0.553023
\(986\) 0 0
\(987\) − 3.60455e9i − 0.119328i
\(988\) 0 0
\(989\) − 4.24509e8i − 0.0139540i
\(990\) 0 0
\(991\) 2.72290e10 0.888738 0.444369 0.895844i \(-0.353428\pi\)
0.444369 + 0.895844i \(0.353428\pi\)
\(992\) 0 0
\(993\) −9.23324e9 −0.299248
\(994\) 0 0
\(995\) − 7.37731e9i − 0.237420i
\(996\) 0 0
\(997\) 1.30057e10i 0.415624i 0.978169 + 0.207812i \(0.0666343\pi\)
−0.978169 + 0.207812i \(0.933366\pi\)
\(998\) 0 0
\(999\) −4.12804e10 −1.30998
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 32.8.b.a.17.4 6
3.2 odd 2 288.8.d.b.145.4 6
4.3 odd 2 8.8.b.a.5.3 6
8.3 odd 2 8.8.b.a.5.4 yes 6
8.5 even 2 inner 32.8.b.a.17.3 6
12.11 even 2 72.8.d.b.37.4 6
16.3 odd 4 256.8.a.r.1.4 6
16.5 even 4 256.8.a.q.1.4 6
16.11 odd 4 256.8.a.r.1.3 6
16.13 even 4 256.8.a.q.1.3 6
24.5 odd 2 288.8.d.b.145.3 6
24.11 even 2 72.8.d.b.37.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.b.a.5.3 6 4.3 odd 2
8.8.b.a.5.4 yes 6 8.3 odd 2
32.8.b.a.17.3 6 8.5 even 2 inner
32.8.b.a.17.4 6 1.1 even 1 trivial
72.8.d.b.37.3 6 24.11 even 2
72.8.d.b.37.4 6 12.11 even 2
256.8.a.q.1.3 6 16.13 even 4
256.8.a.q.1.4 6 16.5 even 4
256.8.a.r.1.3 6 16.11 odd 4
256.8.a.r.1.4 6 16.3 odd 4
288.8.d.b.145.3 6 24.5 odd 2
288.8.d.b.145.4 6 3.2 odd 2