Properties

Label 76.8.a.a.1.2
Level $76$
Weight $8$
Character 76.1
Self dual yes
Analytic conductor $23.741$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 76.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.7412619368\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 5014 x^{3} + 113222 x^{2} - 625803 x + 567036\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-79.8251\) of defining polynomial
Character \(\chi\) \(=\) 76.1

$q$-expansion

\(f(q)\) \(=\) \(q-36.5155 q^{3} -266.507 q^{5} +1627.87 q^{7} -853.617 q^{9} +O(q^{10})\) \(q-36.5155 q^{3} -266.507 q^{5} +1627.87 q^{7} -853.617 q^{9} +3312.50 q^{11} +6655.56 q^{13} +9731.63 q^{15} -30542.3 q^{17} -6859.00 q^{19} -59442.4 q^{21} -29491.6 q^{23} -7099.17 q^{25} +111030. q^{27} -119966. q^{29} +6726.33 q^{31} -120958. q^{33} -433838. q^{35} -297861. q^{37} -243031. q^{39} -111994. q^{41} -451608. q^{43} +227495. q^{45} -1.04264e6 q^{47} +1.82641e6 q^{49} +1.11527e6 q^{51} +24808.4 q^{53} -882803. q^{55} +250460. q^{57} -1.72955e6 q^{59} -1.66100e6 q^{61} -1.38958e6 q^{63} -1.77375e6 q^{65} +2.99362e6 q^{67} +1.07690e6 q^{69} -2.90204e6 q^{71} +4.36673e6 q^{73} +259230. q^{75} +5.39231e6 q^{77} +3.76244e6 q^{79} -2.18745e6 q^{81} -1.62199e6 q^{83} +8.13972e6 q^{85} +4.38064e6 q^{87} +2.25656e6 q^{89} +1.08344e7 q^{91} -245615. q^{93} +1.82797e6 q^{95} -1.60257e7 q^{97} -2.82760e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 14q^{3} - 280q^{5} + 414q^{7} + 3779q^{9} + O(q^{10}) \) \( 5q - 14q^{3} - 280q^{5} + 414q^{7} + 3779q^{9} - 2662q^{11} - 602q^{13} - 20800q^{15} - 27366q^{17} - 34295q^{19} - 59964q^{21} - 67096q^{23} - 109115q^{25} - 178778q^{27} - 372398q^{29} - 271372q^{31} - 792700q^{33} - 608250q^{35} - 562630q^{37} - 963904q^{39} - 956714q^{41} - 827362q^{43} - 1165100q^{45} - 1812982q^{47} - 862031q^{49} - 2458254q^{51} + 486998q^{53} + 467930q^{55} + 96026q^{57} - 367182q^{59} + 1879732q^{61} - 1007274q^{63} + 1790920q^{65} - 1046394q^{67} + 7261712q^{69} - 4664572q^{71} + 4224942q^{73} + 8194850q^{75} + 8611110q^{77} + 9574024q^{79} + 11351813q^{81} + 11754804q^{83} + 18711750q^{85} + 3801472q^{87} + 2782542q^{89} + 7385214q^{91} + 29535004q^{93} + 1920520q^{95} + 1291574q^{97} - 9760310q^{99} + O(q^{100}) \)

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
\(3\) −36.5155 −0.780824 −0.390412 0.920640i \(-0.627667\pi\)
−0.390412 + 0.920640i \(0.627667\pi\)
\(4\) 0 0
\(5\) −266.507 −0.953483 −0.476742 0.879043i \(-0.658182\pi\)
−0.476742 + 0.879043i \(0.658182\pi\)
\(6\) 0 0
\(7\) 1627.87 1.79381 0.896904 0.442225i \(-0.145811\pi\)
0.896904 + 0.442225i \(0.145811\pi\)
\(8\) 0 0
\(9\) −853.617 −0.390314
\(10\) 0 0
\(11\) 3312.50 0.750380 0.375190 0.926948i \(-0.377578\pi\)
0.375190 + 0.926948i \(0.377578\pi\)
\(12\) 0 0
\(13\) 6655.56 0.840201 0.420100 0.907478i \(-0.361995\pi\)
0.420100 + 0.907478i \(0.361995\pi\)
\(14\) 0 0
\(15\) 9731.63 0.744503
\(16\) 0 0
\(17\) −30542.3 −1.50775 −0.753876 0.657016i \(-0.771818\pi\)
−0.753876 + 0.657016i \(0.771818\pi\)
\(18\) 0 0
\(19\) −6859.00 −0.229416
\(20\) 0 0
\(21\) −59442.4 −1.40065
\(22\) 0 0
\(23\) −29491.6 −0.505417 −0.252709 0.967542i \(-0.581321\pi\)
−0.252709 + 0.967542i \(0.581321\pi\)
\(24\) 0 0
\(25\) −7099.17 −0.0908693
\(26\) 0 0
\(27\) 111030. 1.08559
\(28\) 0 0
\(29\) −119966. −0.913412 −0.456706 0.889618i \(-0.650971\pi\)
−0.456706 + 0.889618i \(0.650971\pi\)
\(30\) 0 0
\(31\) 6726.33 0.0405520 0.0202760 0.999794i \(-0.493546\pi\)
0.0202760 + 0.999794i \(0.493546\pi\)
\(32\) 0 0
\(33\) −120958. −0.585914
\(34\) 0 0
\(35\) −433838. −1.71037
\(36\) 0 0
\(37\) −297861. −0.966735 −0.483367 0.875418i \(-0.660586\pi\)
−0.483367 + 0.875418i \(0.660586\pi\)
\(38\) 0 0
\(39\) −243031. −0.656049
\(40\) 0 0
\(41\) −111994. −0.253777 −0.126889 0.991917i \(-0.540499\pi\)
−0.126889 + 0.991917i \(0.540499\pi\)
\(42\) 0 0
\(43\) −451608. −0.866209 −0.433104 0.901344i \(-0.642582\pi\)
−0.433104 + 0.901344i \(0.642582\pi\)
\(44\) 0 0
\(45\) 227495. 0.372158
\(46\) 0 0
\(47\) −1.04264e6 −1.46485 −0.732426 0.680847i \(-0.761612\pi\)
−0.732426 + 0.680847i \(0.761612\pi\)
\(48\) 0 0
\(49\) 1.82641e6 2.21775
\(50\) 0 0
\(51\) 1.11527e6 1.17729
\(52\) 0 0
\(53\) 24808.4 0.0228894 0.0114447 0.999935i \(-0.496357\pi\)
0.0114447 + 0.999935i \(0.496357\pi\)
\(54\) 0 0
\(55\) −882803. −0.715474
\(56\) 0 0
\(57\) 250460. 0.179133
\(58\) 0 0
\(59\) −1.72955e6 −1.09635 −0.548177 0.836362i \(-0.684678\pi\)
−0.548177 + 0.836362i \(0.684678\pi\)
\(60\) 0 0
\(61\) −1.66100e6 −0.936945 −0.468472 0.883478i \(-0.655195\pi\)
−0.468472 + 0.883478i \(0.655195\pi\)
\(62\) 0 0
\(63\) −1.38958e6 −0.700149
\(64\) 0 0
\(65\) −1.77375e6 −0.801118
\(66\) 0 0
\(67\) 2.99362e6 1.21600 0.608001 0.793936i \(-0.291972\pi\)
0.608001 + 0.793936i \(0.291972\pi\)
\(68\) 0 0
\(69\) 1.07690e6 0.394642
\(70\) 0 0
\(71\) −2.90204e6 −0.962274 −0.481137 0.876646i \(-0.659776\pi\)
−0.481137 + 0.876646i \(0.659776\pi\)
\(72\) 0 0
\(73\) 4.36673e6 1.31379 0.656895 0.753982i \(-0.271870\pi\)
0.656895 + 0.753982i \(0.271870\pi\)
\(74\) 0 0
\(75\) 259230. 0.0709529
\(76\) 0 0
\(77\) 5.39231e6 1.34604
\(78\) 0 0
\(79\) 3.76244e6 0.858568 0.429284 0.903170i \(-0.358766\pi\)
0.429284 + 0.903170i \(0.358766\pi\)
\(80\) 0 0
\(81\) −2.18745e6 −0.457341
\(82\) 0 0
\(83\) −1.62199e6 −0.311369 −0.155685 0.987807i \(-0.549758\pi\)
−0.155685 + 0.987807i \(0.549758\pi\)
\(84\) 0 0
\(85\) 8.13972e6 1.43762
\(86\) 0 0
\(87\) 4.38064e6 0.713214
\(88\) 0 0
\(89\) 2.25656e6 0.339298 0.169649 0.985505i \(-0.445737\pi\)
0.169649 + 0.985505i \(0.445737\pi\)
\(90\) 0 0
\(91\) 1.08344e7 1.50716
\(92\) 0 0
\(93\) −245615. −0.0316640
\(94\) 0 0
\(95\) 1.82797e6 0.218744
\(96\) 0 0
\(97\) −1.60257e7 −1.78285 −0.891426 0.453166i \(-0.850294\pi\)
−0.891426 + 0.453166i \(0.850294\pi\)
\(98\) 0 0
\(99\) −2.82760e6 −0.292884
\(100\) 0 0
\(101\) 1.91280e7 1.84733 0.923667 0.383195i \(-0.125176\pi\)
0.923667 + 0.383195i \(0.125176\pi\)
\(102\) 0 0
\(103\) −1.76135e7 −1.58824 −0.794118 0.607764i \(-0.792067\pi\)
−0.794118 + 0.607764i \(0.792067\pi\)
\(104\) 0 0
\(105\) 1.58418e7 1.33549
\(106\) 0 0
\(107\) 9.05349e6 0.714451 0.357226 0.934018i \(-0.383723\pi\)
0.357226 + 0.934018i \(0.383723\pi\)
\(108\) 0 0
\(109\) −1.44855e7 −1.07137 −0.535687 0.844417i \(-0.679947\pi\)
−0.535687 + 0.844417i \(0.679947\pi\)
\(110\) 0 0
\(111\) 1.08765e7 0.754849
\(112\) 0 0
\(113\) −1.94862e7 −1.27043 −0.635217 0.772333i \(-0.719089\pi\)
−0.635217 + 0.772333i \(0.719089\pi\)
\(114\) 0 0
\(115\) 7.85970e6 0.481907
\(116\) 0 0
\(117\) −5.68130e6 −0.327942
\(118\) 0 0
\(119\) −4.97188e7 −2.70462
\(120\) 0 0
\(121\) −8.51454e6 −0.436931
\(122\) 0 0
\(123\) 4.08953e6 0.198155
\(124\) 0 0
\(125\) 2.27128e7 1.04013
\(126\) 0 0
\(127\) −3.95790e7 −1.71456 −0.857278 0.514853i \(-0.827846\pi\)
−0.857278 + 0.514853i \(0.827846\pi\)
\(128\) 0 0
\(129\) 1.64907e7 0.676356
\(130\) 0 0
\(131\) 3.22041e7 1.25159 0.625794 0.779988i \(-0.284775\pi\)
0.625794 + 0.779988i \(0.284775\pi\)
\(132\) 0 0
\(133\) −1.11655e7 −0.411528
\(134\) 0 0
\(135\) −2.95902e7 −1.03509
\(136\) 0 0
\(137\) −2.71458e7 −0.901946 −0.450973 0.892538i \(-0.648923\pi\)
−0.450973 + 0.892538i \(0.648923\pi\)
\(138\) 0 0
\(139\) −3.12951e7 −0.988381 −0.494191 0.869354i \(-0.664536\pi\)
−0.494191 + 0.869354i \(0.664536\pi\)
\(140\) 0 0
\(141\) 3.80727e7 1.14379
\(142\) 0 0
\(143\) 2.20465e7 0.630470
\(144\) 0 0
\(145\) 3.19719e7 0.870924
\(146\) 0 0
\(147\) −6.66923e7 −1.73167
\(148\) 0 0
\(149\) −2.76956e7 −0.685896 −0.342948 0.939354i \(-0.611426\pi\)
−0.342948 + 0.939354i \(0.611426\pi\)
\(150\) 0 0
\(151\) 2.59448e7 0.613240 0.306620 0.951832i \(-0.400802\pi\)
0.306620 + 0.951832i \(0.400802\pi\)
\(152\) 0 0
\(153\) 2.60714e7 0.588497
\(154\) 0 0
\(155\) −1.79261e6 −0.0386656
\(156\) 0 0
\(157\) 6.65464e6 0.137238 0.0686192 0.997643i \(-0.478141\pi\)
0.0686192 + 0.997643i \(0.478141\pi\)
\(158\) 0 0
\(159\) −905893. −0.0178726
\(160\) 0 0
\(161\) −4.80084e7 −0.906622
\(162\) 0 0
\(163\) 9.46484e6 0.171181 0.0855907 0.996330i \(-0.472722\pi\)
0.0855907 + 0.996330i \(0.472722\pi\)
\(164\) 0 0
\(165\) 3.22360e7 0.558659
\(166\) 0 0
\(167\) −8.12016e7 −1.34914 −0.674570 0.738211i \(-0.735671\pi\)
−0.674570 + 0.738211i \(0.735671\pi\)
\(168\) 0 0
\(169\) −1.84520e7 −0.294063
\(170\) 0 0
\(171\) 5.85496e6 0.0895442
\(172\) 0 0
\(173\) −5.31944e7 −0.781097 −0.390548 0.920582i \(-0.627715\pi\)
−0.390548 + 0.920582i \(0.627715\pi\)
\(174\) 0 0
\(175\) −1.15565e7 −0.163002
\(176\) 0 0
\(177\) 6.31554e7 0.856060
\(178\) 0 0
\(179\) −5.79650e7 −0.755405 −0.377703 0.925927i \(-0.623286\pi\)
−0.377703 + 0.925927i \(0.623286\pi\)
\(180\) 0 0
\(181\) 1.25903e8 1.57820 0.789100 0.614265i \(-0.210547\pi\)
0.789100 + 0.614265i \(0.210547\pi\)
\(182\) 0 0
\(183\) 6.06521e7 0.731589
\(184\) 0 0
\(185\) 7.93819e7 0.921765
\(186\) 0 0
\(187\) −1.01171e8 −1.13139
\(188\) 0 0
\(189\) 1.80742e8 1.94734
\(190\) 0 0
\(191\) 1.60824e7 0.167007 0.0835033 0.996508i \(-0.473389\pi\)
0.0835033 + 0.996508i \(0.473389\pi\)
\(192\) 0 0
\(193\) 1.34454e8 1.34625 0.673123 0.739530i \(-0.264952\pi\)
0.673123 + 0.739530i \(0.264952\pi\)
\(194\) 0 0
\(195\) 6.47695e7 0.625532
\(196\) 0 0
\(197\) 9.49248e7 0.884602 0.442301 0.896867i \(-0.354162\pi\)
0.442301 + 0.896867i \(0.354162\pi\)
\(198\) 0 0
\(199\) −7.98652e7 −0.718409 −0.359205 0.933259i \(-0.616952\pi\)
−0.359205 + 0.933259i \(0.616952\pi\)
\(200\) 0 0
\(201\) −1.09313e8 −0.949483
\(202\) 0 0
\(203\) −1.95290e8 −1.63849
\(204\) 0 0
\(205\) 2.98472e7 0.241972
\(206\) 0 0
\(207\) 2.51745e7 0.197272
\(208\) 0 0
\(209\) −2.27204e7 −0.172149
\(210\) 0 0
\(211\) 9.73927e7 0.713736 0.356868 0.934155i \(-0.383845\pi\)
0.356868 + 0.934155i \(0.383845\pi\)
\(212\) 0 0
\(213\) 1.05969e8 0.751366
\(214\) 0 0
\(215\) 1.20357e8 0.825916
\(216\) 0 0
\(217\) 1.09496e7 0.0727425
\(218\) 0 0
\(219\) −1.59453e8 −1.02584
\(220\) 0 0
\(221\) −2.03276e8 −1.26682
\(222\) 0 0
\(223\) 2.21179e8 1.33560 0.667800 0.744341i \(-0.267236\pi\)
0.667800 + 0.744341i \(0.267236\pi\)
\(224\) 0 0
\(225\) 6.05997e6 0.0354676
\(226\) 0 0
\(227\) 3.17166e8 1.79968 0.899842 0.436217i \(-0.143682\pi\)
0.899842 + 0.436217i \(0.143682\pi\)
\(228\) 0 0
\(229\) 1.24652e8 0.685920 0.342960 0.939350i \(-0.388570\pi\)
0.342960 + 0.939350i \(0.388570\pi\)
\(230\) 0 0
\(231\) −1.96903e8 −1.05102
\(232\) 0 0
\(233\) 1.09237e8 0.565747 0.282874 0.959157i \(-0.408712\pi\)
0.282874 + 0.959157i \(0.408712\pi\)
\(234\) 0 0
\(235\) 2.77872e8 1.39671
\(236\) 0 0
\(237\) −1.37387e8 −0.670390
\(238\) 0 0
\(239\) 5.73871e7 0.271908 0.135954 0.990715i \(-0.456590\pi\)
0.135954 + 0.990715i \(0.456590\pi\)
\(240\) 0 0
\(241\) −1.03993e8 −0.478570 −0.239285 0.970949i \(-0.576913\pi\)
−0.239285 + 0.970949i \(0.576913\pi\)
\(242\) 0 0
\(243\) −1.62946e8 −0.728488
\(244\) 0 0
\(245\) −4.86751e8 −2.11459
\(246\) 0 0
\(247\) −4.56505e7 −0.192755
\(248\) 0 0
\(249\) 5.92279e7 0.243124
\(250\) 0 0
\(251\) 2.83460e8 1.13145 0.565724 0.824595i \(-0.308597\pi\)
0.565724 + 0.824595i \(0.308597\pi\)
\(252\) 0 0
\(253\) −9.76907e7 −0.379255
\(254\) 0 0
\(255\) −2.97226e8 −1.12253
\(256\) 0 0
\(257\) −4.16932e7 −0.153214 −0.0766072 0.997061i \(-0.524409\pi\)
−0.0766072 + 0.997061i \(0.524409\pi\)
\(258\) 0 0
\(259\) −4.84878e8 −1.73414
\(260\) 0 0
\(261\) 1.02405e8 0.356518
\(262\) 0 0
\(263\) −1.89687e8 −0.642972 −0.321486 0.946914i \(-0.604182\pi\)
−0.321486 + 0.946914i \(0.604182\pi\)
\(264\) 0 0
\(265\) −6.61162e6 −0.0218246
\(266\) 0 0
\(267\) −8.23994e7 −0.264932
\(268\) 0 0
\(269\) −5.69865e8 −1.78500 −0.892501 0.451045i \(-0.851051\pi\)
−0.892501 + 0.451045i \(0.851051\pi\)
\(270\) 0 0
\(271\) −4.73797e7 −0.144610 −0.0723052 0.997383i \(-0.523036\pi\)
−0.0723052 + 0.997383i \(0.523036\pi\)
\(272\) 0 0
\(273\) −3.95623e8 −1.17683
\(274\) 0 0
\(275\) −2.35160e7 −0.0681865
\(276\) 0 0
\(277\) −5.59400e7 −0.158140 −0.0790702 0.996869i \(-0.525195\pi\)
−0.0790702 + 0.996869i \(0.525195\pi\)
\(278\) 0 0
\(279\) −5.74171e6 −0.0158280
\(280\) 0 0
\(281\) −4.53360e8 −1.21891 −0.609454 0.792821i \(-0.708611\pi\)
−0.609454 + 0.792821i \(0.708611\pi\)
\(282\) 0 0
\(283\) 4.64833e8 1.21911 0.609557 0.792743i \(-0.291347\pi\)
0.609557 + 0.792743i \(0.291347\pi\)
\(284\) 0 0
\(285\) −6.67493e7 −0.170801
\(286\) 0 0
\(287\) −1.82312e8 −0.455227
\(288\) 0 0
\(289\) 5.22492e8 1.27332
\(290\) 0 0
\(291\) 5.85186e8 1.39209
\(292\) 0 0
\(293\) 6.77277e8 1.57300 0.786502 0.617588i \(-0.211890\pi\)
0.786502 + 0.617588i \(0.211890\pi\)
\(294\) 0 0
\(295\) 4.60937e8 1.04536
\(296\) 0 0
\(297\) 3.67785e8 0.814605
\(298\) 0 0
\(299\) −1.96283e8 −0.424652
\(300\) 0 0
\(301\) −7.35159e8 −1.55381
\(302\) 0 0
\(303\) −6.98470e8 −1.44244
\(304\) 0 0
\(305\) 4.42666e8 0.893361
\(306\) 0 0
\(307\) 8.20513e8 1.61846 0.809229 0.587494i \(-0.199885\pi\)
0.809229 + 0.587494i \(0.199885\pi\)
\(308\) 0 0
\(309\) 6.43165e8 1.24013
\(310\) 0 0
\(311\) −7.38919e8 −1.39295 −0.696475 0.717581i \(-0.745249\pi\)
−0.696475 + 0.717581i \(0.745249\pi\)
\(312\) 0 0
\(313\) 1.29886e7 0.0239419 0.0119709 0.999928i \(-0.496189\pi\)
0.0119709 + 0.999928i \(0.496189\pi\)
\(314\) 0 0
\(315\) 3.70331e8 0.667580
\(316\) 0 0
\(317\) −9.28698e7 −0.163745 −0.0818723 0.996643i \(-0.526090\pi\)
−0.0818723 + 0.996643i \(0.526090\pi\)
\(318\) 0 0
\(319\) −3.97388e8 −0.685406
\(320\) 0 0
\(321\) −3.30593e8 −0.557861
\(322\) 0 0
\(323\) 2.09489e8 0.345902
\(324\) 0 0
\(325\) −4.72490e7 −0.0763485
\(326\) 0 0
\(327\) 5.28946e8 0.836554
\(328\) 0 0
\(329\) −1.69729e9 −2.62766
\(330\) 0 0
\(331\) 1.25428e7 0.0190107 0.00950534 0.999955i \(-0.496974\pi\)
0.00950534 + 0.999955i \(0.496974\pi\)
\(332\) 0 0
\(333\) 2.54259e8 0.377330
\(334\) 0 0
\(335\) −7.97819e8 −1.15944
\(336\) 0 0
\(337\) −2.22454e8 −0.316619 −0.158309 0.987390i \(-0.550604\pi\)
−0.158309 + 0.987390i \(0.550604\pi\)
\(338\) 0 0
\(339\) 7.11548e8 0.991986
\(340\) 0 0
\(341\) 2.22809e7 0.0304294
\(342\) 0 0
\(343\) 1.63254e9 2.18441
\(344\) 0 0
\(345\) −2.87001e8 −0.376285
\(346\) 0 0
\(347\) −6.27729e8 −0.806527 −0.403264 0.915084i \(-0.632124\pi\)
−0.403264 + 0.915084i \(0.632124\pi\)
\(348\) 0 0
\(349\) −4.40544e7 −0.0554754 −0.0277377 0.999615i \(-0.508830\pi\)
−0.0277377 + 0.999615i \(0.508830\pi\)
\(350\) 0 0
\(351\) 7.38965e8 0.912114
\(352\) 0 0
\(353\) 8.73399e8 1.05682 0.528410 0.848989i \(-0.322788\pi\)
0.528410 + 0.848989i \(0.322788\pi\)
\(354\) 0 0
\(355\) 7.73412e8 0.917512
\(356\) 0 0
\(357\) 1.81551e9 2.11183
\(358\) 0 0
\(359\) 3.65022e8 0.416379 0.208189 0.978089i \(-0.433243\pi\)
0.208189 + 0.978089i \(0.433243\pi\)
\(360\) 0 0
\(361\) 4.70459e7 0.0526316
\(362\) 0 0
\(363\) 3.10913e8 0.341166
\(364\) 0 0
\(365\) −1.16376e9 −1.25268
\(366\) 0 0
\(367\) 1.74121e9 1.83873 0.919367 0.393401i \(-0.128701\pi\)
0.919367 + 0.393401i \(0.128701\pi\)
\(368\) 0 0
\(369\) 9.56002e7 0.0990528
\(370\) 0 0
\(371\) 4.03849e7 0.0410592
\(372\) 0 0
\(373\) −1.36952e9 −1.36643 −0.683214 0.730218i \(-0.739419\pi\)
−0.683214 + 0.730218i \(0.739419\pi\)
\(374\) 0 0
\(375\) −8.29370e8 −0.812155
\(376\) 0 0
\(377\) −7.98445e8 −0.767450
\(378\) 0 0
\(379\) −1.63943e9 −1.54688 −0.773439 0.633871i \(-0.781465\pi\)
−0.773439 + 0.633871i \(0.781465\pi\)
\(380\) 0 0
\(381\) 1.44525e9 1.33877
\(382\) 0 0
\(383\) 6.63452e8 0.603412 0.301706 0.953401i \(-0.402444\pi\)
0.301706 + 0.953401i \(0.402444\pi\)
\(384\) 0 0
\(385\) −1.43709e9 −1.28342
\(386\) 0 0
\(387\) 3.85501e8 0.338094
\(388\) 0 0
\(389\) 1.82458e9 1.57159 0.785797 0.618485i \(-0.212253\pi\)
0.785797 + 0.618485i \(0.212253\pi\)
\(390\) 0 0
\(391\) 9.00739e8 0.762045
\(392\) 0 0
\(393\) −1.17595e9 −0.977270
\(394\) 0 0
\(395\) −1.00271e9 −0.818630
\(396\) 0 0
\(397\) 1.47702e9 1.18473 0.592367 0.805668i \(-0.298194\pi\)
0.592367 + 0.805668i \(0.298194\pi\)
\(398\) 0 0
\(399\) 4.07716e8 0.321331
\(400\) 0 0
\(401\) 1.63247e8 0.126427 0.0632134 0.998000i \(-0.479865\pi\)
0.0632134 + 0.998000i \(0.479865\pi\)
\(402\) 0 0
\(403\) 4.47675e7 0.0340718
\(404\) 0 0
\(405\) 5.82969e8 0.436067
\(406\) 0 0
\(407\) −9.86663e8 −0.725418
\(408\) 0 0
\(409\) −1.34540e9 −0.972343 −0.486171 0.873863i \(-0.661607\pi\)
−0.486171 + 0.873863i \(0.661607\pi\)
\(410\) 0 0
\(411\) 9.91242e8 0.704261
\(412\) 0 0
\(413\) −2.81548e9 −1.96665
\(414\) 0 0
\(415\) 4.32272e8 0.296885
\(416\) 0 0
\(417\) 1.14276e9 0.771752
\(418\) 0 0
\(419\) −1.22787e9 −0.815464 −0.407732 0.913102i \(-0.633680\pi\)
−0.407732 + 0.913102i \(0.633680\pi\)
\(420\) 0 0
\(421\) −1.01531e9 −0.663149 −0.331574 0.943429i \(-0.607580\pi\)
−0.331574 + 0.943429i \(0.607580\pi\)
\(422\) 0 0
\(423\) 8.90019e8 0.571752
\(424\) 0 0
\(425\) 2.16825e8 0.137009
\(426\) 0 0
\(427\) −2.70388e9 −1.68070
\(428\) 0 0
\(429\) −8.05040e8 −0.492286
\(430\) 0 0
\(431\) −4.35620e8 −0.262082 −0.131041 0.991377i \(-0.541832\pi\)
−0.131041 + 0.991377i \(0.541832\pi\)
\(432\) 0 0
\(433\) 2.31031e9 1.36761 0.683807 0.729663i \(-0.260323\pi\)
0.683807 + 0.729663i \(0.260323\pi\)
\(434\) 0 0
\(435\) −1.16747e9 −0.680038
\(436\) 0 0
\(437\) 2.02283e8 0.115951
\(438\) 0 0
\(439\) −1.38581e9 −0.781765 −0.390883 0.920441i \(-0.627830\pi\)
−0.390883 + 0.920441i \(0.627830\pi\)
\(440\) 0 0
\(441\) −1.55906e9 −0.865618
\(442\) 0 0
\(443\) 9.50566e8 0.519481 0.259740 0.965679i \(-0.416363\pi\)
0.259740 + 0.965679i \(0.416363\pi\)
\(444\) 0 0
\(445\) −6.01388e8 −0.323515
\(446\) 0 0
\(447\) 1.01132e9 0.535564
\(448\) 0 0
\(449\) −2.53281e9 −1.32050 −0.660252 0.751044i \(-0.729551\pi\)
−0.660252 + 0.751044i \(0.729551\pi\)
\(450\) 0 0
\(451\) −3.70981e8 −0.190429
\(452\) 0 0
\(453\) −9.47387e8 −0.478833
\(454\) 0 0
\(455\) −2.88743e9 −1.43705
\(456\) 0 0
\(457\) −5.18819e8 −0.254278 −0.127139 0.991885i \(-0.540579\pi\)
−0.127139 + 0.991885i \(0.540579\pi\)
\(458\) 0 0
\(459\) −3.39110e9 −1.63680
\(460\) 0 0
\(461\) 2.74067e9 1.30288 0.651438 0.758702i \(-0.274166\pi\)
0.651438 + 0.758702i \(0.274166\pi\)
\(462\) 0 0
\(463\) 1.96766e9 0.921331 0.460665 0.887574i \(-0.347611\pi\)
0.460665 + 0.887574i \(0.347611\pi\)
\(464\) 0 0
\(465\) 6.54581e7 0.0301911
\(466\) 0 0
\(467\) −3.60124e9 −1.63622 −0.818112 0.575059i \(-0.804979\pi\)
−0.818112 + 0.575059i \(0.804979\pi\)
\(468\) 0 0
\(469\) 4.87321e9 2.18127
\(470\) 0 0
\(471\) −2.42997e8 −0.107159
\(472\) 0 0
\(473\) −1.49595e9 −0.649985
\(474\) 0 0
\(475\) 4.86932e7 0.0208469
\(476\) 0 0
\(477\) −2.11769e7 −0.00893405
\(478\) 0 0
\(479\) 1.75192e9 0.728350 0.364175 0.931330i \(-0.381351\pi\)
0.364175 + 0.931330i \(0.381351\pi\)
\(480\) 0 0
\(481\) −1.98243e9 −0.812251
\(482\) 0 0
\(483\) 1.75305e9 0.707912
\(484\) 0 0
\(485\) 4.27095e9 1.69992
\(486\) 0 0
\(487\) 3.79184e9 1.48764 0.743821 0.668379i \(-0.233011\pi\)
0.743821 + 0.668379i \(0.233011\pi\)
\(488\) 0 0
\(489\) −3.45614e8 −0.133663
\(490\) 0 0
\(491\) −1.11484e8 −0.0425039 −0.0212519 0.999774i \(-0.506765\pi\)
−0.0212519 + 0.999774i \(0.506765\pi\)
\(492\) 0 0
\(493\) 3.66405e9 1.37720
\(494\) 0 0
\(495\) 7.53575e8 0.279260
\(496\) 0 0
\(497\) −4.72413e9 −1.72613
\(498\) 0 0
\(499\) −1.31431e9 −0.473527 −0.236764 0.971567i \(-0.576087\pi\)
−0.236764 + 0.971567i \(0.576087\pi\)
\(500\) 0 0
\(501\) 2.96512e9 1.05344
\(502\) 0 0
\(503\) −4.40143e9 −1.54208 −0.771039 0.636788i \(-0.780263\pi\)
−0.771039 + 0.636788i \(0.780263\pi\)
\(504\) 0 0
\(505\) −5.09775e9 −1.76140
\(506\) 0 0
\(507\) 6.73784e8 0.229611
\(508\) 0 0
\(509\) −7.60902e8 −0.255751 −0.127875 0.991790i \(-0.540816\pi\)
−0.127875 + 0.991790i \(0.540816\pi\)
\(510\) 0 0
\(511\) 7.10845e9 2.35669
\(512\) 0 0
\(513\) −7.61553e8 −0.249052
\(514\) 0 0
\(515\) 4.69411e9 1.51436
\(516\) 0 0
\(517\) −3.45375e9 −1.09919
\(518\) 0 0
\(519\) 1.94242e9 0.609899
\(520\) 0 0
\(521\) 4.05257e9 1.25545 0.627724 0.778436i \(-0.283986\pi\)
0.627724 + 0.778436i \(0.283986\pi\)
\(522\) 0 0
\(523\) 1.45055e9 0.443379 0.221690 0.975117i \(-0.428843\pi\)
0.221690 + 0.975117i \(0.428843\pi\)
\(524\) 0 0
\(525\) 4.21992e8 0.127276
\(526\) 0 0
\(527\) −2.05437e8 −0.0611424
\(528\) 0 0
\(529\) −2.53507e9 −0.744553
\(530\) 0 0
\(531\) 1.47637e9 0.427923
\(532\) 0 0
\(533\) −7.45385e8 −0.213224
\(534\) 0 0
\(535\) −2.41282e9 −0.681218
\(536\) 0 0
\(537\) 2.11662e9 0.589838
\(538\) 0 0
\(539\) 6.04998e9 1.66415
\(540\) 0 0
\(541\) 9.90839e8 0.269037 0.134519 0.990911i \(-0.457051\pi\)
0.134519 + 0.990911i \(0.457051\pi\)
\(542\) 0 0
\(543\) −4.59742e9 −1.23230
\(544\) 0 0
\(545\) 3.86049e9 1.02154
\(546\) 0 0
\(547\) −1.99783e9 −0.521918 −0.260959 0.965350i \(-0.584039\pi\)
−0.260959 + 0.965350i \(0.584039\pi\)
\(548\) 0 0
\(549\) 1.41785e9 0.365703
\(550\) 0 0
\(551\) 8.22850e8 0.209551
\(552\) 0 0
\(553\) 6.12475e9 1.54011
\(554\) 0 0
\(555\) −2.89867e9 −0.719736
\(556\) 0 0
\(557\) 4.06763e9 0.997351 0.498676 0.866789i \(-0.333820\pi\)
0.498676 + 0.866789i \(0.333820\pi\)
\(558\) 0 0
\(559\) −3.00571e9 −0.727789
\(560\) 0 0
\(561\) 3.69432e9 0.883414
\(562\) 0 0
\(563\) −6.98740e9 −1.65020 −0.825100 0.564987i \(-0.808881\pi\)
−0.825100 + 0.564987i \(0.808881\pi\)
\(564\) 0 0
\(565\) 5.19320e9 1.21134
\(566\) 0 0
\(567\) −3.56087e9 −0.820381
\(568\) 0 0
\(569\) −4.99115e7 −0.0113581 −0.00567907 0.999984i \(-0.501808\pi\)
−0.00567907 + 0.999984i \(0.501808\pi\)
\(570\) 0 0
\(571\) −3.32850e9 −0.748207 −0.374103 0.927387i \(-0.622049\pi\)
−0.374103 + 0.927387i \(0.622049\pi\)
\(572\) 0 0
\(573\) −5.87256e8 −0.130403
\(574\) 0 0
\(575\) 2.09365e8 0.0459269
\(576\) 0 0
\(577\) 6.98807e9 1.51440 0.757202 0.653181i \(-0.226566\pi\)
0.757202 + 0.653181i \(0.226566\pi\)
\(578\) 0 0
\(579\) −4.90967e9 −1.05118
\(580\) 0 0
\(581\) −2.64039e9 −0.558536
\(582\) 0 0
\(583\) 8.21779e7 0.0171757
\(584\) 0 0
\(585\) 1.51411e9 0.312688
\(586\) 0 0
\(587\) 2.65909e8 0.0542624 0.0271312 0.999632i \(-0.491363\pi\)
0.0271312 + 0.999632i \(0.491363\pi\)
\(588\) 0 0
\(589\) −4.61359e7 −0.00930326
\(590\) 0 0
\(591\) −3.46623e9 −0.690718
\(592\) 0 0
\(593\) −6.30318e9 −1.24128 −0.620638 0.784098i \(-0.713126\pi\)
−0.620638 + 0.784098i \(0.713126\pi\)
\(594\) 0 0
\(595\) 1.32504e10 2.57881
\(596\) 0 0
\(597\) 2.91632e9 0.560951
\(598\) 0 0
\(599\) −9.29186e9 −1.76648 −0.883240 0.468921i \(-0.844643\pi\)
−0.883240 + 0.468921i \(0.844643\pi\)
\(600\) 0 0
\(601\) −6.64397e9 −1.24844 −0.624219 0.781249i \(-0.714583\pi\)
−0.624219 + 0.781249i \(0.714583\pi\)
\(602\) 0 0
\(603\) −2.55540e9 −0.474623
\(604\) 0 0
\(605\) 2.26918e9 0.416606
\(606\) 0 0
\(607\) −4.32378e9 −0.784700 −0.392350 0.919816i \(-0.628338\pi\)
−0.392350 + 0.919816i \(0.628338\pi\)
\(608\) 0 0
\(609\) 7.13110e9 1.27937
\(610\) 0 0
\(611\) −6.93938e9 −1.23077
\(612\) 0 0
\(613\) 1.22685e9 0.215119 0.107559 0.994199i \(-0.465696\pi\)
0.107559 + 0.994199i \(0.465696\pi\)
\(614\) 0 0
\(615\) −1.08989e9 −0.188938
\(616\) 0 0
\(617\) 7.41752e9 1.27134 0.635668 0.771963i \(-0.280725\pi\)
0.635668 + 0.771963i \(0.280725\pi\)
\(618\) 0 0
\(619\) −8.99268e9 −1.52395 −0.761977 0.647605i \(-0.775771\pi\)
−0.761977 + 0.647605i \(0.775771\pi\)
\(620\) 0 0
\(621\) −3.27444e9 −0.548676
\(622\) 0 0
\(623\) 3.67338e9 0.608636
\(624\) 0 0
\(625\) −5.49850e9 −0.900873
\(626\) 0 0
\(627\) 8.29648e8 0.134418
\(628\) 0 0
\(629\) 9.09735e9 1.45760
\(630\) 0 0
\(631\) −4.43901e9 −0.703370 −0.351685 0.936118i \(-0.614391\pi\)
−0.351685 + 0.936118i \(0.614391\pi\)
\(632\) 0 0
\(633\) −3.55634e9 −0.557302
\(634\) 0 0
\(635\) 1.05481e10 1.63480
\(636\) 0 0
\(637\) 1.21558e10 1.86335
\(638\) 0 0
\(639\) 2.47723e9 0.375589
\(640\) 0 0
\(641\) −9.77940e9 −1.46659 −0.733296 0.679910i \(-0.762019\pi\)
−0.733296 + 0.679910i \(0.762019\pi\)
\(642\) 0 0
\(643\) −4.55347e9 −0.675467 −0.337734 0.941242i \(-0.609660\pi\)
−0.337734 + 0.941242i \(0.609660\pi\)
\(644\) 0 0
\(645\) −4.39489e9 −0.644895
\(646\) 0 0
\(647\) 1.33205e10 1.93354 0.966772 0.255640i \(-0.0822861\pi\)
0.966772 + 0.255640i \(0.0822861\pi\)
\(648\) 0 0
\(649\) −5.72913e9 −0.822682
\(650\) 0 0
\(651\) −3.99829e8 −0.0567991
\(652\) 0 0
\(653\) −6.99125e9 −0.982559 −0.491279 0.871002i \(-0.663471\pi\)
−0.491279 + 0.871002i \(0.663471\pi\)
\(654\) 0 0
\(655\) −8.58261e9 −1.19337
\(656\) 0 0
\(657\) −3.72751e9 −0.512791
\(658\) 0 0
\(659\) 6.12019e9 0.833040 0.416520 0.909127i \(-0.363250\pi\)
0.416520 + 0.909127i \(0.363250\pi\)
\(660\) 0 0
\(661\) 3.45719e9 0.465606 0.232803 0.972524i \(-0.425210\pi\)
0.232803 + 0.972524i \(0.425210\pi\)
\(662\) 0 0
\(663\) 7.42273e9 0.989160
\(664\) 0 0
\(665\) 2.97569e9 0.392385
\(666\) 0 0
\(667\) 3.53800e9 0.461655
\(668\) 0 0
\(669\) −8.07646e9 −1.04287
\(670\) 0 0
\(671\) −5.50204e9 −0.703064
\(672\) 0 0
\(673\) −1.52296e10 −1.92591 −0.962953 0.269669i \(-0.913086\pi\)
−0.962953 + 0.269669i \(0.913086\pi\)
\(674\) 0 0
\(675\) −7.88218e8 −0.0986469
\(676\) 0 0
\(677\) −9.91870e9 −1.22855 −0.614277 0.789091i \(-0.710552\pi\)
−0.614277 + 0.789091i \(0.710552\pi\)
\(678\) 0 0
\(679\) −2.60877e10 −3.19809
\(680\) 0 0
\(681\) −1.15815e10 −1.40524
\(682\) 0 0
\(683\) 4.36189e9 0.523844 0.261922 0.965089i \(-0.415644\pi\)
0.261922 + 0.965089i \(0.415644\pi\)
\(684\) 0 0
\(685\) 7.23453e9 0.859990
\(686\) 0 0
\(687\) −4.55172e9 −0.535583
\(688\) 0 0
\(689\) 1.65114e8 0.0192317
\(690\) 0 0
\(691\) 1.11399e8 0.0128443 0.00642213 0.999979i \(-0.497956\pi\)
0.00642213 + 0.999979i \(0.497956\pi\)
\(692\) 0 0
\(693\) −4.60296e9 −0.525377
\(694\) 0 0
\(695\) 8.34036e9 0.942405
\(696\) 0 0
\(697\) 3.42056e9 0.382633
\(698\) 0 0
\(699\) −3.98883e9 −0.441749
\(700\) 0 0
\(701\) 4.01144e9 0.439833 0.219916 0.975519i \(-0.429422\pi\)
0.219916 + 0.975519i \(0.429422\pi\)
\(702\) 0 0
\(703\) 2.04303e9 0.221784
\(704\) 0 0
\(705\) −1.01466e10 −1.09059
\(706\) 0 0
\(707\) 3.11379e10 3.31376
\(708\) 0 0
\(709\) 5.93278e9 0.625168 0.312584 0.949890i \(-0.398805\pi\)
0.312584 + 0.949890i \(0.398805\pi\)
\(710\) 0 0
\(711\) −3.21168e9 −0.335111
\(712\) 0 0
\(713\) −1.98370e8 −0.0204957
\(714\) 0 0
\(715\) −5.87555e9 −0.601142
\(716\) 0 0
\(717\) −2.09552e9 −0.212312
\(718\) 0 0
\(719\) 4.51400e9 0.452908 0.226454 0.974022i \(-0.427287\pi\)
0.226454 + 0.974022i \(0.427287\pi\)
\(720\) 0 0
\(721\) −2.86724e10 −2.84899
\(722\) 0 0
\(723\) 3.79737e9 0.373679
\(724\) 0 0
\(725\) 8.51662e8 0.0830012
\(726\) 0 0
\(727\) 8.96487e9 0.865314 0.432657 0.901559i \(-0.357576\pi\)
0.432657 + 0.901559i \(0.357576\pi\)
\(728\) 0 0
\(729\) 1.07340e10 1.02616
\(730\) 0 0
\(731\) 1.37932e10 1.30603
\(732\) 0 0
\(733\) 2.02584e10 1.89994 0.949971 0.312338i \(-0.101112\pi\)
0.949971 + 0.312338i \(0.101112\pi\)
\(734\) 0 0
\(735\) 1.77740e10 1.65112
\(736\) 0 0
\(737\) 9.91634e9 0.912463
\(738\) 0 0
\(739\) 4.12238e9 0.375744 0.187872 0.982193i \(-0.439841\pi\)
0.187872 + 0.982193i \(0.439841\pi\)
\(740\) 0 0
\(741\) 1.66695e9 0.150508
\(742\) 0 0
\(743\) 2.33266e9 0.208636 0.104318 0.994544i \(-0.466734\pi\)
0.104318 + 0.994544i \(0.466734\pi\)
\(744\) 0 0
\(745\) 7.38106e9 0.653991
\(746\) 0 0
\(747\) 1.38456e9 0.121532
\(748\) 0 0
\(749\) 1.47379e10 1.28159
\(750\) 0 0
\(751\) 2.29418e10 1.97646 0.988230 0.152975i \(-0.0488855\pi\)
0.988230 + 0.152975i \(0.0488855\pi\)
\(752\) 0 0
\(753\) −1.03507e10 −0.883461
\(754\) 0 0
\(755\) −6.91446e9 −0.584715
\(756\) 0 0
\(757\) −8.83607e9 −0.740327 −0.370164 0.928967i \(-0.620698\pi\)
−0.370164 + 0.928967i \(0.620698\pi\)
\(758\) 0 0
\(759\) 3.56723e9 0.296131
\(760\) 0 0
\(761\) 1.93655e10 1.59287 0.796437 0.604721i \(-0.206715\pi\)
0.796437 + 0.604721i \(0.206715\pi\)
\(762\) 0 0
\(763\) −2.35805e10 −1.92184
\(764\) 0 0
\(765\) −6.94821e9 −0.561123
\(766\) 0 0
\(767\) −1.15111e10 −0.921158
\(768\) 0 0
\(769\) −3.79471e9 −0.300910 −0.150455 0.988617i \(-0.548074\pi\)
−0.150455 + 0.988617i \(0.548074\pi\)
\(770\) 0 0
\(771\) 1.52245e9 0.119633
\(772\) 0 0
\(773\) −2.83902e9 −0.221075 −0.110537 0.993872i \(-0.535257\pi\)
−0.110537 + 0.993872i \(0.535257\pi\)
\(774\) 0 0
\(775\) −4.77513e7 −0.00368493
\(776\) 0 0
\(777\) 1.77056e10 1.35405
\(778\) 0 0
\(779\) 7.68169e8 0.0582205
\(780\) 0 0
\(781\) −9.61298e9 −0.722070
\(782\) 0 0
\(783\) −1.33198e10 −0.991592
\(784\) 0 0
\(785\) −1.77351e9 −0.130855
\(786\) 0 0
\(787\) 1.30629e10 0.955274 0.477637 0.878557i \(-0.341493\pi\)
0.477637 + 0.878557i \(0.341493\pi\)
\(788\) 0 0
\(789\) 6.92651e9 0.502048
\(790\) 0 0
\(791\) −3.17209e10 −2.27892
\(792\) 0 0
\(793\) −1.10549e10 −0.787222
\(794\) 0 0
\(795\) 2.41427e8 0.0170412
\(796\) 0 0
\(797\) −1.85425e10 −1.29737 −0.648687 0.761055i \(-0.724682\pi\)
−0.648687 + 0.761055i \(0.724682\pi\)
\(798\) 0 0
\(799\) 3.18447e10 2.20863
\(800\) 0 0
\(801\) −1.92624e9 −0.132433
\(802\) 0 0
\(803\) 1.44648e10 0.985841
\(804\) 0 0
\(805\) 1.27945e10 0.864449
\(806\) 0 0
\(807\) 2.08089e10 1.39377
\(808\) 0 0
\(809\) −3.36442e9 −0.223404 −0.111702 0.993742i \(-0.535630\pi\)
−0.111702 + 0.993742i \(0.535630\pi\)
\(810\) 0 0
\(811\) −1.71841e10 −1.13124 −0.565618 0.824668i \(-0.691362\pi\)
−0.565618 + 0.824668i \(0.691362\pi\)
\(812\) 0 0
\(813\) 1.73009e9 0.112915
\(814\) 0 0
\(815\) −2.52244e9 −0.163219
\(816\) 0 0
\(817\) 3.09758e9 0.198722
\(818\) 0 0
\(819\) −9.24841e9 −0.588266
\(820\) 0 0
\(821\) 2.23026e10 1.40655 0.703274 0.710919i \(-0.251721\pi\)
0.703274 + 0.710919i \(0.251721\pi\)
\(822\) 0 0
\(823\) 2.62811e10 1.64340 0.821701 0.569918i \(-0.193025\pi\)
0.821701 + 0.569918i \(0.193025\pi\)
\(824\) 0 0
\(825\) 8.58697e8 0.0532416
\(826\) 0 0
\(827\) −1.55677e10 −0.957093 −0.478546 0.878062i \(-0.658836\pi\)
−0.478546 + 0.878062i \(0.658836\pi\)
\(828\) 0 0
\(829\) −8.16683e7 −0.00497866 −0.00248933 0.999997i \(-0.500792\pi\)
−0.00248933 + 0.999997i \(0.500792\pi\)
\(830\) 0 0
\(831\) 2.04268e9 0.123480
\(832\) 0 0
\(833\) −5.57827e10 −3.34382
\(834\) 0 0
\(835\) 2.16408e10 1.28638
\(836\) 0 0
\(837\) 7.46822e8 0.0440228
\(838\) 0 0
\(839\) −1.44911e10 −0.847101 −0.423551 0.905872i \(-0.639216\pi\)
−0.423551 + 0.905872i \(0.639216\pi\)
\(840\) 0 0
\(841\) −2.85792e9 −0.165678
\(842\) 0 0
\(843\) 1.65547e10 0.951753
\(844\) 0 0
\(845\) 4.91758e9 0.280384
\(846\) 0 0
\(847\) −1.38605e10 −0.783770
\(848\) 0 0
\(849\) −1.69736e10 −0.951913
\(850\) 0 0
\(851\) 8.78438e9 0.488605
\(852\) 0 0
\(853\) −2.37032e10 −1.30763 −0.653815 0.756654i \(-0.726833\pi\)
−0.653815 + 0.756654i \(0.726833\pi\)
\(854\) 0 0
\(855\) −1.56039e9 −0.0853789
\(856\) 0 0
\(857\) 1.75078e10 0.950163 0.475081 0.879942i \(-0.342419\pi\)
0.475081 + 0.879942i \(0.342419\pi\)
\(858\) 0 0
\(859\) −2.17897e10 −1.17294 −0.586469 0.809972i \(-0.699482\pi\)
−0.586469 + 0.809972i \(0.699482\pi\)
\(860\) 0 0
\(861\) 6.65721e9 0.355452
\(862\) 0 0
\(863\) −2.11606e10 −1.12070 −0.560351 0.828255i \(-0.689334\pi\)
−0.560351 + 0.828255i \(0.689334\pi\)
\(864\) 0 0
\(865\) 1.41767e10 0.744763
\(866\) 0 0
\(867\) −1.90791e10 −0.994238
\(868\) 0 0
\(869\) 1.24631e10 0.644252
\(870\) 0 0
\(871\) 1.99242e10 1.02169
\(872\) 0 0
\(873\) 1.36798e10 0.695872
\(874\) 0 0
\(875\) 3.69735e10 1.86579
\(876\) 0 0
\(877\) −1.68537e10 −0.843718 −0.421859 0.906661i \(-0.638622\pi\)
−0.421859 + 0.906661i \(0.638622\pi\)
\(878\) 0 0
\(879\) −2.47311e10 −1.22824
\(880\) 0 0
\(881\) 7.63499e9 0.376178 0.188089 0.982152i \(-0.439771\pi\)
0.188089 + 0.982152i \(0.439771\pi\)
\(882\) 0 0
\(883\) −3.53640e10 −1.72862 −0.864310 0.502960i \(-0.832244\pi\)
−0.864310 + 0.502960i \(0.832244\pi\)
\(884\) 0 0
\(885\) −1.68313e10 −0.816239
\(886\) 0 0
\(887\) 5.18891e9 0.249657 0.124829 0.992178i \(-0.460162\pi\)
0.124829 + 0.992178i \(0.460162\pi\)
\(888\) 0 0
\(889\) −6.44294e10 −3.07559
\(890\) 0 0
\(891\) −7.24591e9 −0.343179
\(892\) 0 0
\(893\) 7.15150e9 0.336060
\(894\) 0 0
\(895\) 1.54481e10 0.720266
\(896\) 0 0
\(897\) 7.16737e9 0.331579
\(898\) 0 0
\(899\) −8.06934e8 −0.0370407
\(900\) 0 0
\(901\) −7.57707e8 −0.0345115
\(902\) 0 0
\(903\) 2.68447e10 1.21325
\(904\) 0 0
\(905\) −3.35541e10 −1.50479
\(906\) 0 0
\(907\) 1.58922e10 0.707225 0.353612 0.935392i \(-0.384953\pi\)
0.353612 + 0.935392i \(0.384953\pi\)
\(908\) 0 0
\(909\) −1.63280e10 −0.721041
\(910\) 0 0
\(911\) −1.07719e10 −0.472041 −0.236020 0.971748i \(-0.575843\pi\)
−0.236020 + 0.971748i \(0.575843\pi\)
\(912\) 0 0
\(913\) −5.37284e9 −0.233645
\(914\) 0 0
\(915\) −1.61642e10 −0.697558
\(916\) 0 0
\(917\) 5.24240e10 2.24511
\(918\) 0 0
\(919\) −2.44674e10 −1.03988 −0.519941 0.854202i \(-0.674046\pi\)
−0.519941 + 0.854202i \(0.674046\pi\)
\(920\) 0 0
\(921\) −2.99615e10 −1.26373
\(922\) 0 0
\(923\) −1.93147e10 −0.808503
\(924\) 0 0
\(925\) 2.11456e9 0.0878465
\(926\) 0 0
\(927\) 1.50352e10 0.619911
\(928\) 0 0
\(929\) 3.69154e10 1.51061 0.755306 0.655372i \(-0.227488\pi\)
0.755306 + 0.655372i \(0.227488\pi\)
\(930\) 0 0
\(931\) −1.25274e10 −0.508786
\(932\) 0 0
\(933\) 2.69820e10 1.08765
\(934\) 0 0
\(935\) 2.69628e10 1.07876
\(936\) 0 0
\(937\) 3.71448e10 1.47506 0.737530 0.675314i \(-0.235992\pi\)
0.737530 + 0.675314i \(0.235992\pi\)
\(938\) 0 0
\(939\) −4.74286e8 −0.0186944
\(940\) 0 0
\(941\) 1.56830e10 0.613573 0.306787 0.951778i \(-0.400746\pi\)
0.306787 + 0.951778i \(0.400746\pi\)
\(942\) 0 0
\(943\) 3.30289e9 0.128263
\(944\) 0 0
\(945\) −4.81689e10 −1.85676
\(946\) 0 0
\(947\) −2.19871e10 −0.841284 −0.420642 0.907227i \(-0.638195\pi\)
−0.420642 + 0.907227i \(0.638195\pi\)
\(948\) 0 0
\(949\) 2.90630e10 1.10385
\(950\) 0 0
\(951\) 3.39119e9 0.127856
\(952\) 0 0
\(953\) 3.88326e10 1.45335 0.726676 0.686980i \(-0.241064\pi\)
0.726676 + 0.686980i \(0.241064\pi\)
\(954\) 0 0
\(955\) −4.28606e9 −0.159238
\(956\) 0 0
\(957\) 1.45108e10 0.535181
\(958\) 0 0
\(959\) −4.41897e10 −1.61792
\(960\) 0 0
\(961\) −2.74674e10 −0.998356
\(962\) 0 0
\(963\) −7.72821e9 −0.278861
\(964\) 0 0
\(965\) −3.58330e10 −1.28362
\(966\) 0 0
\(967\) −2.05650e10 −0.731367 −0.365683 0.930739i \(-0.619165\pi\)
−0.365683 + 0.930739i \(0.619165\pi\)
\(968\) 0 0
\(969\) −7.64962e9 −0.270089
\(970\) 0 0
\(971\) 2.17717e10 0.763178 0.381589 0.924332i \(-0.375377\pi\)
0.381589 + 0.924332i \(0.375377\pi\)
\(972\) 0 0
\(973\) −5.09443e10 −1.77297
\(974\) 0 0
\(975\) 1.72532e9 0.0596147
\(976\) 0 0
\(977\) 1.55819e10 0.534551 0.267276 0.963620i \(-0.413877\pi\)
0.267276 + 0.963620i \(0.413877\pi\)
\(978\) 0 0
\(979\) 7.47485e9 0.254603
\(980\) 0 0
\(981\) 1.23651e10 0.418172
\(982\) 0 0
\(983\) −2.07083e10 −0.695356 −0.347678 0.937614i \(-0.613030\pi\)
−0.347678 + 0.937614i \(0.613030\pi\)
\(984\) 0 0
\(985\) −2.52981e10 −0.843453
\(986\) 0 0
\(987\) 6.19773e10 2.05174
\(988\) 0 0
\(989\) 1.33186e10 0.437797
\(990\) 0 0
\(991\) −2.48904e10 −0.812407 −0.406204 0.913783i \(-0.633148\pi\)
−0.406204 + 0.913783i \(0.633148\pi\)
\(992\) 0 0
\(993\) −4.58008e8 −0.0148440
\(994\) 0 0
\(995\) 2.12846e10 0.684992
\(996\) 0 0
\(997\) −1.81625e10 −0.580421 −0.290211 0.956963i \(-0.593725\pi\)
−0.290211 + 0.956963i \(0.593725\pi\)
\(998\) 0 0
\(999\) −3.30714e10 −1.04948
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 76.8.a.a.1.2 5
4.3 odd 2 304.8.a.g.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.8.a.a.1.2 5 1.1 even 1 trivial
304.8.a.g.1.4 5 4.3 odd 2