Properties

Label 207.8.a.a.1.2
Level $207$
Weight $8$
Character 207.1
Self dual yes
Analytic conductor $64.664$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [207,8,Mod(1,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 207.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(64.6637002752\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 455x^{3} - 474x^{2} + 42284x + 127016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(12.4672\) of defining polynomial
Character \(\chi\) \(=\) 207.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-12.4672 q^{2} +27.4323 q^{4} +40.8788 q^{5} +1126.70 q^{7} +1253.80 q^{8} +O(q^{10})\) \(q-12.4672 q^{2} +27.4323 q^{4} +40.8788 q^{5} +1126.70 q^{7} +1253.80 q^{8} -509.647 q^{10} -500.265 q^{11} -12895.9 q^{13} -14046.8 q^{14} -19142.8 q^{16} +23798.2 q^{17} -32879.5 q^{19} +1121.40 q^{20} +6236.93 q^{22} -12167.0 q^{23} -76453.9 q^{25} +160777. q^{26} +30907.8 q^{28} +152213. q^{29} +252935. q^{31} +78171.4 q^{32} -296698. q^{34} +46058.0 q^{35} +89429.5 q^{37} +409916. q^{38} +51254.0 q^{40} -549274. q^{41} -330324. q^{43} -13723.4 q^{44} +151689. q^{46} -834394. q^{47} +445899. q^{49} +953170. q^{50} -353765. q^{52} +1.73470e6 q^{53} -20450.3 q^{55} +1.41265e6 q^{56} -1.89768e6 q^{58} +2.58024e6 q^{59} -2.40055e6 q^{61} -3.15340e6 q^{62} +1.47570e6 q^{64} -527171. q^{65} +3.57070e6 q^{67} +652839. q^{68} -574216. q^{70} -709607. q^{71} +797398. q^{73} -1.11494e6 q^{74} -901959. q^{76} -563647. q^{77} -1.98775e6 q^{79} -782535. q^{80} +6.84793e6 q^{82} -8.20364e6 q^{83} +972842. q^{85} +4.11823e6 q^{86} -627234. q^{88} -8.23613e6 q^{89} -1.45298e7 q^{91} -333769. q^{92} +1.04026e7 q^{94} -1.34407e6 q^{95} +1.54790e6 q^{97} -5.55914e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 270 q^{4} + 266 q^{5} - 496 q^{7} - 1422 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 270 q^{4} + 266 q^{5} - 496 q^{7} - 1422 q^{8} + 1452 q^{10} + 1148 q^{11} - 642 q^{13} - 5756 q^{14} - 22606 q^{16} + 5798 q^{17} - 6036 q^{19} + 27376 q^{20} - 97896 q^{22} - 60835 q^{23} - 262477 q^{25} + 355992 q^{26} - 507124 q^{28} + 169162 q^{29} - 199640 q^{31} + 284794 q^{32} - 1027740 q^{34} + 137680 q^{35} - 202002 q^{37} + 554924 q^{38} - 340904 q^{40} - 541282 q^{41} - 909596 q^{43} + 1236032 q^{44} - 80208 q^{47} + 850589 q^{49} + 941416 q^{50} + 146940 q^{52} + 278138 q^{53} - 933560 q^{55} + 539932 q^{56} - 3522712 q^{58} + 3177380 q^{59} + 147782 q^{61} - 4606456 q^{62} - 4142622 q^{64} - 3877332 q^{65} - 464916 q^{67} - 7513072 q^{68} + 2093200 q^{70} - 1576792 q^{71} - 38190 q^{73} - 12164864 q^{74} + 6889436 q^{76} - 10332384 q^{77} - 3913336 q^{79} - 6334776 q^{80} + 6799360 q^{82} - 15774716 q^{83} - 8520740 q^{85} - 24874084 q^{86} + 53216 q^{88} - 1116482 q^{89} - 27369552 q^{91} - 3285090 q^{92} - 7153744 q^{94} + 6067832 q^{95} - 15738566 q^{97} - 11730488 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −12.4672 −1.10196 −0.550980 0.834519i \(-0.685746\pi\)
−0.550980 + 0.834519i \(0.685746\pi\)
\(3\) 0 0
\(4\) 27.4323 0.214315
\(5\) 40.8788 0.146253 0.0731263 0.997323i \(-0.476702\pi\)
0.0731263 + 0.997323i \(0.476702\pi\)
\(6\) 0 0
\(7\) 1126.70 1.24155 0.620774 0.783990i \(-0.286819\pi\)
0.620774 + 0.783990i \(0.286819\pi\)
\(8\) 1253.80 0.865793
\(9\) 0 0
\(10\) −509.647 −0.161164
\(11\) −500.265 −0.113325 −0.0566626 0.998393i \(-0.518046\pi\)
−0.0566626 + 0.998393i \(0.518046\pi\)
\(12\) 0 0
\(13\) −12895.9 −1.62799 −0.813994 0.580873i \(-0.802711\pi\)
−0.813994 + 0.580873i \(0.802711\pi\)
\(14\) −14046.8 −1.36814
\(15\) 0 0
\(16\) −19142.8 −1.16838
\(17\) 23798.2 1.17482 0.587412 0.809288i \(-0.300147\pi\)
0.587412 + 0.809288i \(0.300147\pi\)
\(18\) 0 0
\(19\) −32879.5 −1.09973 −0.549866 0.835253i \(-0.685321\pi\)
−0.549866 + 0.835253i \(0.685321\pi\)
\(20\) 1121.40 0.0313441
\(21\) 0 0
\(22\) 6236.93 0.124880
\(23\) −12167.0 −0.208514
\(24\) 0 0
\(25\) −76453.9 −0.978610
\(26\) 160777. 1.79398
\(27\) 0 0
\(28\) 30907.8 0.266082
\(29\) 152213. 1.15894 0.579468 0.814995i \(-0.303260\pi\)
0.579468 + 0.814995i \(0.303260\pi\)
\(30\) 0 0
\(31\) 252935. 1.52490 0.762452 0.647045i \(-0.223995\pi\)
0.762452 + 0.647045i \(0.223995\pi\)
\(32\) 78171.4 0.421719
\(33\) 0 0
\(34\) −296698. −1.29461
\(35\) 46058.0 0.181580
\(36\) 0 0
\(37\) 89429.5 0.290252 0.145126 0.989413i \(-0.453641\pi\)
0.145126 + 0.989413i \(0.453641\pi\)
\(38\) 409916. 1.21186
\(39\) 0 0
\(40\) 51254.0 0.126624
\(41\) −549274. −1.24464 −0.622322 0.782761i \(-0.713811\pi\)
−0.622322 + 0.782761i \(0.713811\pi\)
\(42\) 0 0
\(43\) −330324. −0.633578 −0.316789 0.948496i \(-0.602605\pi\)
−0.316789 + 0.948496i \(0.602605\pi\)
\(44\) −13723.4 −0.0242873
\(45\) 0 0
\(46\) 151689. 0.229774
\(47\) −834394. −1.17227 −0.586136 0.810213i \(-0.699352\pi\)
−0.586136 + 0.810213i \(0.699352\pi\)
\(48\) 0 0
\(49\) 445899. 0.541440
\(50\) 953170. 1.07839
\(51\) 0 0
\(52\) −353765. −0.348902
\(53\) 1.73470e6 1.60051 0.800257 0.599658i \(-0.204697\pi\)
0.800257 + 0.599658i \(0.204697\pi\)
\(54\) 0 0
\(55\) −20450.3 −0.0165741
\(56\) 1.41265e6 1.07492
\(57\) 0 0
\(58\) −1.89768e6 −1.27710
\(59\) 2.58024e6 1.63560 0.817801 0.575502i \(-0.195193\pi\)
0.817801 + 0.575502i \(0.195193\pi\)
\(60\) 0 0
\(61\) −2.40055e6 −1.35412 −0.677060 0.735928i \(-0.736746\pi\)
−0.677060 + 0.735928i \(0.736746\pi\)
\(62\) −3.15340e6 −1.68038
\(63\) 0 0
\(64\) 1.47570e6 0.703667
\(65\) −527171. −0.238097
\(66\) 0 0
\(67\) 3.57070e6 1.45041 0.725206 0.688532i \(-0.241744\pi\)
0.725206 + 0.688532i \(0.241744\pi\)
\(68\) 652839. 0.251782
\(69\) 0 0
\(70\) −574216. −0.200093
\(71\) −709607. −0.235296 −0.117648 0.993055i \(-0.537535\pi\)
−0.117648 + 0.993055i \(0.537535\pi\)
\(72\) 0 0
\(73\) 797398. 0.239908 0.119954 0.992779i \(-0.461725\pi\)
0.119954 + 0.992779i \(0.461725\pi\)
\(74\) −1.11494e6 −0.319846
\(75\) 0 0
\(76\) −901959. −0.235689
\(77\) −563647. −0.140699
\(78\) 0 0
\(79\) −1.98775e6 −0.453594 −0.226797 0.973942i \(-0.572825\pi\)
−0.226797 + 0.973942i \(0.572825\pi\)
\(80\) −782535. −0.170879
\(81\) 0 0
\(82\) 6.84793e6 1.37155
\(83\) −8.20364e6 −1.57483 −0.787414 0.616425i \(-0.788581\pi\)
−0.787414 + 0.616425i \(0.788581\pi\)
\(84\) 0 0
\(85\) 972842. 0.171821
\(86\) 4.11823e6 0.698177
\(87\) 0 0
\(88\) −627234. −0.0981161
\(89\) −8.23613e6 −1.23839 −0.619196 0.785236i \(-0.712541\pi\)
−0.619196 + 0.785236i \(0.712541\pi\)
\(90\) 0 0
\(91\) −1.45298e7 −2.02123
\(92\) −333769. −0.0446877
\(93\) 0 0
\(94\) 1.04026e7 1.29180
\(95\) −1.34407e6 −0.160839
\(96\) 0 0
\(97\) 1.54790e6 0.172203 0.0861015 0.996286i \(-0.472559\pi\)
0.0861015 + 0.996286i \(0.472559\pi\)
\(98\) −5.55914e6 −0.596645
\(99\) 0 0
\(100\) −2.09731e6 −0.209731
\(101\) −1.26141e7 −1.21824 −0.609119 0.793079i \(-0.708477\pi\)
−0.609119 + 0.793079i \(0.708477\pi\)
\(102\) 0 0
\(103\) −1.53226e7 −1.38166 −0.690830 0.723017i \(-0.742755\pi\)
−0.690830 + 0.723017i \(0.742755\pi\)
\(104\) −1.61690e7 −1.40950
\(105\) 0 0
\(106\) −2.16270e7 −1.76370
\(107\) −3.58516e6 −0.282921 −0.141461 0.989944i \(-0.545180\pi\)
−0.141461 + 0.989944i \(0.545180\pi\)
\(108\) 0 0
\(109\) −1.14000e7 −0.843163 −0.421582 0.906790i \(-0.638525\pi\)
−0.421582 + 0.906790i \(0.638525\pi\)
\(110\) 254959. 0.0182640
\(111\) 0 0
\(112\) −2.15681e7 −1.45060
\(113\) 1.16966e7 0.762579 0.381290 0.924456i \(-0.375480\pi\)
0.381290 + 0.924456i \(0.375480\pi\)
\(114\) 0 0
\(115\) −497373. −0.0304958
\(116\) 4.17556e6 0.248377
\(117\) 0 0
\(118\) −3.21685e7 −1.80237
\(119\) 2.68133e7 1.45860
\(120\) 0 0
\(121\) −1.92369e7 −0.987157
\(122\) 2.99283e7 1.49219
\(123\) 0 0
\(124\) 6.93858e6 0.326810
\(125\) −6.31901e6 −0.289377
\(126\) 0 0
\(127\) 4.09153e7 1.77244 0.886222 0.463262i \(-0.153321\pi\)
0.886222 + 0.463262i \(0.153321\pi\)
\(128\) −2.84038e7 −1.19713
\(129\) 0 0
\(130\) 6.57237e6 0.262374
\(131\) −9.54150e6 −0.370823 −0.185412 0.982661i \(-0.559362\pi\)
−0.185412 + 0.982661i \(0.559362\pi\)
\(132\) 0 0
\(133\) −3.70451e7 −1.36537
\(134\) −4.45168e7 −1.59830
\(135\) 0 0
\(136\) 2.98382e7 1.01715
\(137\) 3.72077e7 1.23626 0.618132 0.786075i \(-0.287890\pi\)
0.618132 + 0.786075i \(0.287890\pi\)
\(138\) 0 0
\(139\) −4.43290e7 −1.40003 −0.700013 0.714130i \(-0.746823\pi\)
−0.700013 + 0.714130i \(0.746823\pi\)
\(140\) 1.26348e6 0.0389152
\(141\) 0 0
\(142\) 8.84685e6 0.259286
\(143\) 6.45139e6 0.184492
\(144\) 0 0
\(145\) 6.22230e6 0.169497
\(146\) −9.94137e6 −0.264369
\(147\) 0 0
\(148\) 2.45326e6 0.0622053
\(149\) −2.13993e7 −0.529965 −0.264983 0.964253i \(-0.585366\pi\)
−0.264983 + 0.964253i \(0.585366\pi\)
\(150\) 0 0
\(151\) −5.39107e7 −1.27425 −0.637126 0.770759i \(-0.719877\pi\)
−0.637126 + 0.770759i \(0.719877\pi\)
\(152\) −4.12243e7 −0.952141
\(153\) 0 0
\(154\) 7.02712e6 0.155044
\(155\) 1.03397e7 0.223021
\(156\) 0 0
\(157\) −4.92467e6 −0.101561 −0.0507807 0.998710i \(-0.516171\pi\)
−0.0507807 + 0.998710i \(0.516171\pi\)
\(158\) 2.47818e7 0.499842
\(159\) 0 0
\(160\) 3.19555e6 0.0616774
\(161\) −1.37085e7 −0.258881
\(162\) 0 0
\(163\) −5.24341e7 −0.948326 −0.474163 0.880437i \(-0.657249\pi\)
−0.474163 + 0.880437i \(0.657249\pi\)
\(164\) −1.50678e7 −0.266746
\(165\) 0 0
\(166\) 1.02277e8 1.73540
\(167\) −5.66790e7 −0.941704 −0.470852 0.882212i \(-0.656053\pi\)
−0.470852 + 0.882212i \(0.656053\pi\)
\(168\) 0 0
\(169\) 1.03557e8 1.65035
\(170\) −1.21287e7 −0.189340
\(171\) 0 0
\(172\) −9.06154e6 −0.135785
\(173\) 4.71933e7 0.692977 0.346489 0.938054i \(-0.387374\pi\)
0.346489 + 0.938054i \(0.387374\pi\)
\(174\) 0 0
\(175\) −8.61403e7 −1.21499
\(176\) 9.57648e6 0.132407
\(177\) 0 0
\(178\) 1.02682e8 1.36466
\(179\) −9.63068e6 −0.125508 −0.0627540 0.998029i \(-0.519988\pi\)
−0.0627540 + 0.998029i \(0.519988\pi\)
\(180\) 0 0
\(181\) 1.36975e8 1.71699 0.858494 0.512823i \(-0.171400\pi\)
0.858494 + 0.512823i \(0.171400\pi\)
\(182\) 1.81147e8 2.22731
\(183\) 0 0
\(184\) −1.52550e7 −0.180530
\(185\) 3.65577e6 0.0424501
\(186\) 0 0
\(187\) −1.19054e7 −0.133137
\(188\) −2.28893e7 −0.251235
\(189\) 0 0
\(190\) 1.67569e7 0.177238
\(191\) −5.87953e7 −0.610556 −0.305278 0.952263i \(-0.598749\pi\)
−0.305278 + 0.952263i \(0.598749\pi\)
\(192\) 0 0
\(193\) 1.44972e6 0.0145155 0.00725776 0.999974i \(-0.497690\pi\)
0.00725776 + 0.999974i \(0.497690\pi\)
\(194\) −1.92980e7 −0.189761
\(195\) 0 0
\(196\) 1.22320e7 0.116039
\(197\) 6.62738e7 0.617604 0.308802 0.951126i \(-0.400072\pi\)
0.308802 + 0.951126i \(0.400072\pi\)
\(198\) 0 0
\(199\) −1.78732e8 −1.60774 −0.803871 0.594803i \(-0.797230\pi\)
−0.803871 + 0.594803i \(0.797230\pi\)
\(200\) −9.58581e7 −0.847274
\(201\) 0 0
\(202\) 1.57263e8 1.34245
\(203\) 1.71498e8 1.43888
\(204\) 0 0
\(205\) −2.24537e7 −0.182032
\(206\) 1.91030e8 1.52253
\(207\) 0 0
\(208\) 2.46864e8 1.90212
\(209\) 1.64485e7 0.124627
\(210\) 0 0
\(211\) −1.07572e7 −0.0788335 −0.0394167 0.999223i \(-0.512550\pi\)
−0.0394167 + 0.999223i \(0.512550\pi\)
\(212\) 4.75869e7 0.343014
\(213\) 0 0
\(214\) 4.46971e7 0.311768
\(215\) −1.35032e7 −0.0926624
\(216\) 0 0
\(217\) 2.84980e8 1.89324
\(218\) 1.42126e8 0.929132
\(219\) 0 0
\(220\) −560998. −0.00355207
\(221\) −3.06900e8 −1.91260
\(222\) 0 0
\(223\) −2.14285e8 −1.29397 −0.646986 0.762502i \(-0.723971\pi\)
−0.646986 + 0.762502i \(0.723971\pi\)
\(224\) 8.80753e7 0.523584
\(225\) 0 0
\(226\) −1.45824e8 −0.840331
\(227\) 7.33851e7 0.416406 0.208203 0.978086i \(-0.433238\pi\)
0.208203 + 0.978086i \(0.433238\pi\)
\(228\) 0 0
\(229\) −2.83323e7 −0.155904 −0.0779519 0.996957i \(-0.524838\pi\)
−0.0779519 + 0.996957i \(0.524838\pi\)
\(230\) 6.20087e6 0.0336051
\(231\) 0 0
\(232\) 1.90845e8 1.00340
\(233\) −1.15251e8 −0.596896 −0.298448 0.954426i \(-0.596469\pi\)
−0.298448 + 0.954426i \(0.596469\pi\)
\(234\) 0 0
\(235\) −3.41090e7 −0.171448
\(236\) 7.07818e7 0.350534
\(237\) 0 0
\(238\) −3.34288e8 −1.60732
\(239\) −5.89008e7 −0.279080 −0.139540 0.990216i \(-0.544562\pi\)
−0.139540 + 0.990216i \(0.544562\pi\)
\(240\) 0 0
\(241\) −5.12886e7 −0.236027 −0.118013 0.993012i \(-0.537653\pi\)
−0.118013 + 0.993012i \(0.537653\pi\)
\(242\) 2.39831e8 1.08781
\(243\) 0 0
\(244\) −6.58527e7 −0.290208
\(245\) 1.82278e7 0.0791870
\(246\) 0 0
\(247\) 4.24012e8 1.79035
\(248\) 3.17130e8 1.32025
\(249\) 0 0
\(250\) 7.87806e7 0.318882
\(251\) 3.11308e7 0.124260 0.0621300 0.998068i \(-0.480211\pi\)
0.0621300 + 0.998068i \(0.480211\pi\)
\(252\) 0 0
\(253\) 6.08673e6 0.0236299
\(254\) −5.10101e8 −1.95316
\(255\) 0 0
\(256\) 1.65228e8 0.615523
\(257\) −2.38296e8 −0.875690 −0.437845 0.899051i \(-0.644258\pi\)
−0.437845 + 0.899051i \(0.644258\pi\)
\(258\) 0 0
\(259\) 1.00760e8 0.360361
\(260\) −1.44615e7 −0.0510278
\(261\) 0 0
\(262\) 1.18956e8 0.408632
\(263\) −2.84255e8 −0.963525 −0.481762 0.876302i \(-0.660003\pi\)
−0.481762 + 0.876302i \(0.660003\pi\)
\(264\) 0 0
\(265\) 7.09126e7 0.234079
\(266\) 4.61851e8 1.50458
\(267\) 0 0
\(268\) 9.79525e7 0.310845
\(269\) 1.94289e8 0.608576 0.304288 0.952580i \(-0.401581\pi\)
0.304288 + 0.952580i \(0.401581\pi\)
\(270\) 0 0
\(271\) −3.69086e8 −1.12651 −0.563254 0.826284i \(-0.690451\pi\)
−0.563254 + 0.826284i \(0.690451\pi\)
\(272\) −4.55564e8 −1.37264
\(273\) 0 0
\(274\) −4.63878e8 −1.36231
\(275\) 3.82473e7 0.110901
\(276\) 0 0
\(277\) −5.28946e8 −1.49531 −0.747657 0.664086i \(-0.768821\pi\)
−0.747657 + 0.664086i \(0.768821\pi\)
\(278\) 5.52661e8 1.54277
\(279\) 0 0
\(280\) 5.77476e7 0.157210
\(281\) −4.54274e7 −0.122137 −0.0610683 0.998134i \(-0.519451\pi\)
−0.0610683 + 0.998134i \(0.519451\pi\)
\(282\) 0 0
\(283\) 2.28233e8 0.598585 0.299293 0.954161i \(-0.403249\pi\)
0.299293 + 0.954161i \(0.403249\pi\)
\(284\) −1.94662e7 −0.0504274
\(285\) 0 0
\(286\) −8.04311e7 −0.203303
\(287\) −6.18864e8 −1.54528
\(288\) 0 0
\(289\) 1.56015e8 0.380210
\(290\) −7.75750e7 −0.186779
\(291\) 0 0
\(292\) 2.18745e7 0.0514159
\(293\) −4.49349e8 −1.04363 −0.521815 0.853058i \(-0.674745\pi\)
−0.521815 + 0.853058i \(0.674745\pi\)
\(294\) 0 0
\(295\) 1.05477e8 0.239211
\(296\) 1.12127e8 0.251298
\(297\) 0 0
\(298\) 2.66790e8 0.584000
\(299\) 1.56905e8 0.339459
\(300\) 0 0
\(301\) −3.72174e8 −0.786617
\(302\) 6.72118e8 1.40418
\(303\) 0 0
\(304\) 6.29405e8 1.28491
\(305\) −9.81319e7 −0.198044
\(306\) 0 0
\(307\) −3.56204e8 −0.702609 −0.351305 0.936261i \(-0.614262\pi\)
−0.351305 + 0.936261i \(0.614262\pi\)
\(308\) −1.54621e7 −0.0301538
\(309\) 0 0
\(310\) −1.28907e8 −0.245760
\(311\) −7.14279e8 −1.34650 −0.673250 0.739415i \(-0.735102\pi\)
−0.673250 + 0.739415i \(0.735102\pi\)
\(312\) 0 0
\(313\) 9.57174e7 0.176435 0.0882177 0.996101i \(-0.471883\pi\)
0.0882177 + 0.996101i \(0.471883\pi\)
\(314\) 6.13971e7 0.111916
\(315\) 0 0
\(316\) −5.45286e7 −0.0972120
\(317\) 6.88984e8 1.21479 0.607396 0.794399i \(-0.292214\pi\)
0.607396 + 0.794399i \(0.292214\pi\)
\(318\) 0 0
\(319\) −7.61471e7 −0.131337
\(320\) 6.03248e7 0.102913
\(321\) 0 0
\(322\) 1.70907e8 0.285276
\(323\) −7.82471e8 −1.29199
\(324\) 0 0
\(325\) 9.85945e8 1.59317
\(326\) 6.53709e8 1.04502
\(327\) 0 0
\(328\) −6.88681e8 −1.07760
\(329\) −9.40108e8 −1.45543
\(330\) 0 0
\(331\) 8.37330e8 1.26911 0.634554 0.772878i \(-0.281184\pi\)
0.634554 + 0.772878i \(0.281184\pi\)
\(332\) −2.25045e8 −0.337509
\(333\) 0 0
\(334\) 7.06631e8 1.03772
\(335\) 1.45966e8 0.212127
\(336\) 0 0
\(337\) −1.24811e9 −1.77643 −0.888213 0.459431i \(-0.848053\pi\)
−0.888213 + 0.459431i \(0.848053\pi\)
\(338\) −1.29107e9 −1.81862
\(339\) 0 0
\(340\) 2.66873e7 0.0368238
\(341\) −1.26534e8 −0.172810
\(342\) 0 0
\(343\) −4.25489e8 −0.569324
\(344\) −4.14161e8 −0.548547
\(345\) 0 0
\(346\) −5.88371e8 −0.763633
\(347\) 2.11977e8 0.272356 0.136178 0.990684i \(-0.456518\pi\)
0.136178 + 0.990684i \(0.456518\pi\)
\(348\) 0 0
\(349\) −1.03849e8 −0.130772 −0.0653858 0.997860i \(-0.520828\pi\)
−0.0653858 + 0.997860i \(0.520828\pi\)
\(350\) 1.07393e9 1.33887
\(351\) 0 0
\(352\) −3.91064e7 −0.0477913
\(353\) −2.33920e8 −0.283045 −0.141523 0.989935i \(-0.545200\pi\)
−0.141523 + 0.989935i \(0.545200\pi\)
\(354\) 0 0
\(355\) −2.90079e7 −0.0344126
\(356\) −2.25936e8 −0.265406
\(357\) 0 0
\(358\) 1.20068e8 0.138305
\(359\) 1.66286e9 1.89682 0.948408 0.317054i \(-0.102694\pi\)
0.948408 + 0.317054i \(0.102694\pi\)
\(360\) 0 0
\(361\) 1.87187e8 0.209412
\(362\) −1.70771e9 −1.89205
\(363\) 0 0
\(364\) −3.98586e8 −0.433179
\(365\) 3.25967e7 0.0350872
\(366\) 0 0
\(367\) −5.37695e8 −0.567812 −0.283906 0.958852i \(-0.591630\pi\)
−0.283906 + 0.958852i \(0.591630\pi\)
\(368\) 2.32910e8 0.243625
\(369\) 0 0
\(370\) −4.55774e7 −0.0467782
\(371\) 1.95448e9 1.98711
\(372\) 0 0
\(373\) −2.19239e8 −0.218744 −0.109372 0.994001i \(-0.534884\pi\)
−0.109372 + 0.994001i \(0.534884\pi\)
\(374\) 1.48428e8 0.146712
\(375\) 0 0
\(376\) −1.04617e9 −1.01495
\(377\) −1.96293e9 −1.88674
\(378\) 0 0
\(379\) 1.09529e9 1.03346 0.516730 0.856148i \(-0.327149\pi\)
0.516730 + 0.856148i \(0.327149\pi\)
\(380\) −3.68710e7 −0.0344701
\(381\) 0 0
\(382\) 7.33015e8 0.672808
\(383\) 1.49495e9 1.35966 0.679831 0.733368i \(-0.262053\pi\)
0.679831 + 0.733368i \(0.262053\pi\)
\(384\) 0 0
\(385\) −2.30412e7 −0.0205775
\(386\) −1.80740e7 −0.0159955
\(387\) 0 0
\(388\) 4.24623e7 0.0369057
\(389\) 1.06612e9 0.918296 0.459148 0.888360i \(-0.348155\pi\)
0.459148 + 0.888360i \(0.348155\pi\)
\(390\) 0 0
\(391\) −2.89552e8 −0.244968
\(392\) 5.59070e8 0.468775
\(393\) 0 0
\(394\) −8.26252e8 −0.680575
\(395\) −8.12570e7 −0.0663393
\(396\) 0 0
\(397\) 1.02859e9 0.825044 0.412522 0.910948i \(-0.364648\pi\)
0.412522 + 0.910948i \(0.364648\pi\)
\(398\) 2.22830e9 1.77167
\(399\) 0 0
\(400\) 1.46354e9 1.14339
\(401\) −2.90455e8 −0.224943 −0.112472 0.993655i \(-0.535877\pi\)
−0.112472 + 0.993655i \(0.535877\pi\)
\(402\) 0 0
\(403\) −3.26183e9 −2.48253
\(404\) −3.46034e8 −0.261087
\(405\) 0 0
\(406\) −2.13811e9 −1.58558
\(407\) −4.47385e7 −0.0328928
\(408\) 0 0
\(409\) −6.42289e8 −0.464193 −0.232097 0.972693i \(-0.574559\pi\)
−0.232097 + 0.972693i \(0.574559\pi\)
\(410\) 2.79935e8 0.200592
\(411\) 0 0
\(412\) −4.20333e8 −0.296110
\(413\) 2.90714e9 2.03068
\(414\) 0 0
\(415\) −3.35355e8 −0.230323
\(416\) −1.00809e9 −0.686553
\(417\) 0 0
\(418\) −2.05067e8 −0.137334
\(419\) −1.95033e9 −1.29526 −0.647632 0.761953i \(-0.724241\pi\)
−0.647632 + 0.761953i \(0.724241\pi\)
\(420\) 0 0
\(421\) −2.66647e9 −1.74160 −0.870801 0.491635i \(-0.836399\pi\)
−0.870801 + 0.491635i \(0.836399\pi\)
\(422\) 1.34113e8 0.0868713
\(423\) 0 0
\(424\) 2.17497e9 1.38571
\(425\) −1.81946e9 −1.14969
\(426\) 0 0
\(427\) −2.70469e9 −1.68120
\(428\) −9.83492e7 −0.0606342
\(429\) 0 0
\(430\) 1.68348e8 0.102110
\(431\) 1.45000e9 0.872366 0.436183 0.899858i \(-0.356330\pi\)
0.436183 + 0.899858i \(0.356330\pi\)
\(432\) 0 0
\(433\) 8.59337e8 0.508693 0.254347 0.967113i \(-0.418140\pi\)
0.254347 + 0.967113i \(0.418140\pi\)
\(434\) −3.55292e9 −2.08627
\(435\) 0 0
\(436\) −3.12728e8 −0.180702
\(437\) 4.00044e8 0.229310
\(438\) 0 0
\(439\) −2.27752e9 −1.28480 −0.642402 0.766368i \(-0.722062\pi\)
−0.642402 + 0.766368i \(0.722062\pi\)
\(440\) −2.56406e7 −0.0143497
\(441\) 0 0
\(442\) 3.82620e9 2.10761
\(443\) −1.83034e8 −0.100027 −0.0500136 0.998749i \(-0.515926\pi\)
−0.0500136 + 0.998749i \(0.515926\pi\)
\(444\) 0 0
\(445\) −3.36683e8 −0.181118
\(446\) 2.67154e9 1.42590
\(447\) 0 0
\(448\) 1.66266e9 0.873636
\(449\) −1.10637e9 −0.576819 −0.288409 0.957507i \(-0.593126\pi\)
−0.288409 + 0.957507i \(0.593126\pi\)
\(450\) 0 0
\(451\) 2.74783e8 0.141049
\(452\) 3.20865e8 0.163432
\(453\) 0 0
\(454\) −9.14910e8 −0.458863
\(455\) −5.93961e8 −0.295609
\(456\) 0 0
\(457\) −1.67986e8 −0.0823314 −0.0411657 0.999152i \(-0.513107\pi\)
−0.0411657 + 0.999152i \(0.513107\pi\)
\(458\) 3.53225e8 0.171800
\(459\) 0 0
\(460\) −1.36441e7 −0.00653570
\(461\) −2.56850e9 −1.22103 −0.610516 0.792004i \(-0.709038\pi\)
−0.610516 + 0.792004i \(0.709038\pi\)
\(462\) 0 0
\(463\) −1.10660e9 −0.518150 −0.259075 0.965857i \(-0.583418\pi\)
−0.259075 + 0.965857i \(0.583418\pi\)
\(464\) −2.91379e9 −1.35408
\(465\) 0 0
\(466\) 1.43686e9 0.657756
\(467\) −2.79649e9 −1.27059 −0.635294 0.772271i \(-0.719121\pi\)
−0.635294 + 0.772271i \(0.719121\pi\)
\(468\) 0 0
\(469\) 4.02309e9 1.80076
\(470\) 4.25246e8 0.188929
\(471\) 0 0
\(472\) 3.23511e9 1.41609
\(473\) 1.65249e8 0.0718003
\(474\) 0 0
\(475\) 2.51376e9 1.07621
\(476\) 7.35551e8 0.312599
\(477\) 0 0
\(478\) 7.34331e8 0.307535
\(479\) −2.19334e9 −0.911866 −0.455933 0.890014i \(-0.650694\pi\)
−0.455933 + 0.890014i \(0.650694\pi\)
\(480\) 0 0
\(481\) −1.15328e9 −0.472526
\(482\) 6.39428e8 0.260092
\(483\) 0 0
\(484\) −5.27713e8 −0.211563
\(485\) 6.32762e7 0.0251851
\(486\) 0 0
\(487\) −1.48441e9 −0.582373 −0.291187 0.956666i \(-0.594050\pi\)
−0.291187 + 0.956666i \(0.594050\pi\)
\(488\) −3.00982e9 −1.17239
\(489\) 0 0
\(490\) −2.27251e8 −0.0872609
\(491\) 3.58617e9 1.36724 0.683621 0.729837i \(-0.260404\pi\)
0.683621 + 0.729837i \(0.260404\pi\)
\(492\) 0 0
\(493\) 3.62240e9 1.36155
\(494\) −5.28626e9 −1.97290
\(495\) 0 0
\(496\) −4.84188e9 −1.78167
\(497\) −7.99511e8 −0.292131
\(498\) 0 0
\(499\) −2.64545e9 −0.953122 −0.476561 0.879141i \(-0.658117\pi\)
−0.476561 + 0.879141i \(0.658117\pi\)
\(500\) −1.73345e8 −0.0620177
\(501\) 0 0
\(502\) −3.88115e8 −0.136930
\(503\) −4.47733e8 −0.156867 −0.0784334 0.996919i \(-0.524992\pi\)
−0.0784334 + 0.996919i \(0.524992\pi\)
\(504\) 0 0
\(505\) −5.15651e8 −0.178170
\(506\) −7.58848e7 −0.0260392
\(507\) 0 0
\(508\) 1.12240e9 0.379861
\(509\) 1.50184e9 0.504792 0.252396 0.967624i \(-0.418782\pi\)
0.252396 + 0.967624i \(0.418782\pi\)
\(510\) 0 0
\(511\) 8.98425e8 0.297858
\(512\) 1.57575e9 0.518850
\(513\) 0 0
\(514\) 2.97089e9 0.964975
\(515\) −6.26368e8 −0.202071
\(516\) 0 0
\(517\) 4.17418e8 0.132848
\(518\) −1.25620e9 −0.397104
\(519\) 0 0
\(520\) −6.60968e8 −0.206143
\(521\) −1.31386e9 −0.407020 −0.203510 0.979073i \(-0.565235\pi\)
−0.203510 + 0.979073i \(0.565235\pi\)
\(522\) 0 0
\(523\) 4.83261e9 1.47715 0.738577 0.674169i \(-0.235498\pi\)
0.738577 + 0.674169i \(0.235498\pi\)
\(524\) −2.61745e8 −0.0794729
\(525\) 0 0
\(526\) 3.54388e9 1.06177
\(527\) 6.01938e9 1.79149
\(528\) 0 0
\(529\) 1.48036e8 0.0434783
\(530\) −8.84085e8 −0.257946
\(531\) 0 0
\(532\) −1.01623e9 −0.292619
\(533\) 7.08340e9 2.02627
\(534\) 0 0
\(535\) −1.46557e8 −0.0413779
\(536\) 4.47695e9 1.25576
\(537\) 0 0
\(538\) −2.42225e9 −0.670626
\(539\) −2.23068e8 −0.0613588
\(540\) 0 0
\(541\) 8.09486e8 0.219796 0.109898 0.993943i \(-0.464948\pi\)
0.109898 + 0.993943i \(0.464948\pi\)
\(542\) 4.60148e9 1.24137
\(543\) 0 0
\(544\) 1.86034e9 0.495445
\(545\) −4.66018e8 −0.123315
\(546\) 0 0
\(547\) 2.70806e9 0.707460 0.353730 0.935348i \(-0.384913\pi\)
0.353730 + 0.935348i \(0.384913\pi\)
\(548\) 1.02069e9 0.264950
\(549\) 0 0
\(550\) −4.76838e8 −0.122209
\(551\) −5.00469e9 −1.27452
\(552\) 0 0
\(553\) −2.23959e9 −0.563159
\(554\) 6.59450e9 1.64777
\(555\) 0 0
\(556\) −1.21605e9 −0.300047
\(557\) 2.74885e9 0.673997 0.336999 0.941505i \(-0.390588\pi\)
0.336999 + 0.941505i \(0.390588\pi\)
\(558\) 0 0
\(559\) 4.25983e9 1.03146
\(560\) −8.81679e8 −0.212155
\(561\) 0 0
\(562\) 5.66355e8 0.134590
\(563\) 4.75796e9 1.12368 0.561838 0.827247i \(-0.310094\pi\)
0.561838 + 0.827247i \(0.310094\pi\)
\(564\) 0 0
\(565\) 4.78143e8 0.111529
\(566\) −2.84544e9 −0.659617
\(567\) 0 0
\(568\) −8.89708e8 −0.203717
\(569\) −3.85644e9 −0.877595 −0.438797 0.898586i \(-0.644595\pi\)
−0.438797 + 0.898586i \(0.644595\pi\)
\(570\) 0 0
\(571\) −1.05862e9 −0.237966 −0.118983 0.992896i \(-0.537963\pi\)
−0.118983 + 0.992896i \(0.537963\pi\)
\(572\) 1.76977e8 0.0395394
\(573\) 0 0
\(574\) 7.71553e9 1.70284
\(575\) 9.30215e8 0.204054
\(576\) 0 0
\(577\) −5.02825e9 −1.08969 −0.544843 0.838538i \(-0.683411\pi\)
−0.544843 + 0.838538i \(0.683411\pi\)
\(578\) −1.94507e9 −0.418976
\(579\) 0 0
\(580\) 1.70692e8 0.0363258
\(581\) −9.24300e9 −1.95522
\(582\) 0 0
\(583\) −8.67811e8 −0.181378
\(584\) 9.99780e8 0.207711
\(585\) 0 0
\(586\) 5.60214e9 1.15004
\(587\) 8.27222e9 1.68806 0.844032 0.536293i \(-0.180176\pi\)
0.844032 + 0.536293i \(0.180176\pi\)
\(588\) 0 0
\(589\) −8.31636e9 −1.67699
\(590\) −1.31501e9 −0.263601
\(591\) 0 0
\(592\) −1.71193e9 −0.339125
\(593\) 2.13907e9 0.421245 0.210622 0.977568i \(-0.432451\pi\)
0.210622 + 0.977568i \(0.432451\pi\)
\(594\) 0 0
\(595\) 1.09610e9 0.213324
\(596\) −5.87032e8 −0.113579
\(597\) 0 0
\(598\) −1.95617e9 −0.374070
\(599\) 6.33073e9 1.20354 0.601769 0.798670i \(-0.294463\pi\)
0.601769 + 0.798670i \(0.294463\pi\)
\(600\) 0 0
\(601\) −4.50671e8 −0.0846835 −0.0423418 0.999103i \(-0.513482\pi\)
−0.0423418 + 0.999103i \(0.513482\pi\)
\(602\) 4.63999e9 0.866820
\(603\) 0 0
\(604\) −1.47889e9 −0.273091
\(605\) −7.86382e8 −0.144374
\(606\) 0 0
\(607\) −6.88005e9 −1.24862 −0.624311 0.781176i \(-0.714620\pi\)
−0.624311 + 0.781176i \(0.714620\pi\)
\(608\) −2.57023e9 −0.463778
\(609\) 0 0
\(610\) 1.22343e9 0.218236
\(611\) 1.07603e10 1.90845
\(612\) 0 0
\(613\) 8.99737e9 1.57763 0.788813 0.614633i \(-0.210696\pi\)
0.788813 + 0.614633i \(0.210696\pi\)
\(614\) 4.44088e9 0.774247
\(615\) 0 0
\(616\) −7.06702e8 −0.121816
\(617\) 2.14587e9 0.367794 0.183897 0.982946i \(-0.441129\pi\)
0.183897 + 0.982946i \(0.441129\pi\)
\(618\) 0 0
\(619\) 3.60649e9 0.611177 0.305589 0.952164i \(-0.401147\pi\)
0.305589 + 0.952164i \(0.401147\pi\)
\(620\) 2.83641e8 0.0477967
\(621\) 0 0
\(622\) 8.90509e9 1.48379
\(623\) −9.27961e9 −1.53752
\(624\) 0 0
\(625\) 5.71465e9 0.936288
\(626\) −1.19333e9 −0.194425
\(627\) 0 0
\(628\) −1.35095e8 −0.0217661
\(629\) 2.12826e9 0.340994
\(630\) 0 0
\(631\) −1.96983e9 −0.312123 −0.156062 0.987747i \(-0.549880\pi\)
−0.156062 + 0.987747i \(0.549880\pi\)
\(632\) −2.49225e9 −0.392719
\(633\) 0 0
\(634\) −8.58973e9 −1.33865
\(635\) 1.67257e9 0.259224
\(636\) 0 0
\(637\) −5.75029e9 −0.881459
\(638\) 9.49345e8 0.144728
\(639\) 0 0
\(640\) −1.16111e9 −0.175084
\(641\) −1.07839e9 −0.161724 −0.0808620 0.996725i \(-0.525767\pi\)
−0.0808620 + 0.996725i \(0.525767\pi\)
\(642\) 0 0
\(643\) −9.43215e9 −1.39918 −0.699588 0.714547i \(-0.746633\pi\)
−0.699588 + 0.714547i \(0.746633\pi\)
\(644\) −3.76056e8 −0.0554820
\(645\) 0 0
\(646\) 9.75526e9 1.42372
\(647\) 1.91605e9 0.278126 0.139063 0.990284i \(-0.455591\pi\)
0.139063 + 0.990284i \(0.455591\pi\)
\(648\) 0 0
\(649\) −1.29080e9 −0.185355
\(650\) −1.22920e10 −1.75560
\(651\) 0 0
\(652\) −1.43839e9 −0.203240
\(653\) −1.28782e10 −1.80991 −0.904956 0.425505i \(-0.860096\pi\)
−0.904956 + 0.425505i \(0.860096\pi\)
\(654\) 0 0
\(655\) −3.90045e8 −0.0542339
\(656\) 1.05146e10 1.45422
\(657\) 0 0
\(658\) 1.17206e10 1.60383
\(659\) 1.04515e10 1.42259 0.711294 0.702895i \(-0.248110\pi\)
0.711294 + 0.702895i \(0.248110\pi\)
\(660\) 0 0
\(661\) 9.36029e9 1.26062 0.630310 0.776344i \(-0.282928\pi\)
0.630310 + 0.776344i \(0.282928\pi\)
\(662\) −1.04392e10 −1.39851
\(663\) 0 0
\(664\) −1.02857e10 −1.36348
\(665\) −1.51436e9 −0.199689
\(666\) 0 0
\(667\) −1.85198e9 −0.241655
\(668\) −1.55484e9 −0.201821
\(669\) 0 0
\(670\) −1.81979e9 −0.233755
\(671\) 1.20091e9 0.153456
\(672\) 0 0
\(673\) −2.34925e9 −0.297083 −0.148541 0.988906i \(-0.547458\pi\)
−0.148541 + 0.988906i \(0.547458\pi\)
\(674\) 1.55605e10 1.95755
\(675\) 0 0
\(676\) 2.84080e9 0.353694
\(677\) −1.18065e10 −1.46238 −0.731188 0.682176i \(-0.761034\pi\)
−0.731188 + 0.682176i \(0.761034\pi\)
\(678\) 0 0
\(679\) 1.74401e9 0.213798
\(680\) 1.21975e9 0.148761
\(681\) 0 0
\(682\) 1.57754e9 0.190429
\(683\) −5.88145e9 −0.706337 −0.353169 0.935560i \(-0.614896\pi\)
−0.353169 + 0.935560i \(0.614896\pi\)
\(684\) 0 0
\(685\) 1.52101e9 0.180807
\(686\) 5.30468e9 0.627372
\(687\) 0 0
\(688\) 6.32332e9 0.740262
\(689\) −2.23706e10 −2.60562
\(690\) 0 0
\(691\) −4.16582e9 −0.480316 −0.240158 0.970734i \(-0.577199\pi\)
−0.240158 + 0.970734i \(0.577199\pi\)
\(692\) 1.29462e9 0.148515
\(693\) 0 0
\(694\) −2.64277e9 −0.300125
\(695\) −1.81212e9 −0.204757
\(696\) 0 0
\(697\) −1.30717e10 −1.46224
\(698\) 1.29471e9 0.144105
\(699\) 0 0
\(700\) −2.36303e9 −0.260391
\(701\) 1.38729e10 1.52109 0.760546 0.649284i \(-0.224931\pi\)
0.760546 + 0.649284i \(0.224931\pi\)
\(702\) 0 0
\(703\) −2.94039e9 −0.319199
\(704\) −7.38240e8 −0.0797431
\(705\) 0 0
\(706\) 2.91634e9 0.311905
\(707\) −1.42123e10 −1.51250
\(708\) 0 0
\(709\) 3.27840e9 0.345462 0.172731 0.984969i \(-0.444741\pi\)
0.172731 + 0.984969i \(0.444741\pi\)
\(710\) 3.61649e8 0.0379213
\(711\) 0 0
\(712\) −1.03265e10 −1.07219
\(713\) −3.07746e9 −0.317964
\(714\) 0 0
\(715\) 2.63725e8 0.0269824
\(716\) −2.64192e8 −0.0268982
\(717\) 0 0
\(718\) −2.07313e10 −2.09021
\(719\) 6.37084e9 0.639213 0.319607 0.947550i \(-0.396449\pi\)
0.319607 + 0.947550i \(0.396449\pi\)
\(720\) 0 0
\(721\) −1.72639e10 −1.71540
\(722\) −2.33371e9 −0.230763
\(723\) 0 0
\(724\) 3.75755e9 0.367976
\(725\) −1.16373e10 −1.13415
\(726\) 0 0
\(727\) 1.86871e10 1.80373 0.901867 0.432013i \(-0.142197\pi\)
0.901867 + 0.432013i \(0.142197\pi\)
\(728\) −1.82175e10 −1.74996
\(729\) 0 0
\(730\) −4.06391e8 −0.0386647
\(731\) −7.86110e9 −0.744342
\(732\) 0 0
\(733\) 1.28678e10 1.20682 0.603409 0.797432i \(-0.293809\pi\)
0.603409 + 0.797432i \(0.293809\pi\)
\(734\) 6.70358e9 0.625706
\(735\) 0 0
\(736\) −9.51111e8 −0.0879344
\(737\) −1.78630e9 −0.164368
\(738\) 0 0
\(739\) −5.37369e9 −0.489798 −0.244899 0.969549i \(-0.578755\pi\)
−0.244899 + 0.969549i \(0.578755\pi\)
\(740\) 1.00286e8 0.00909768
\(741\) 0 0
\(742\) −2.43670e10 −2.18972
\(743\) 5.20876e9 0.465879 0.232940 0.972491i \(-0.425166\pi\)
0.232940 + 0.972491i \(0.425166\pi\)
\(744\) 0 0
\(745\) −8.74778e8 −0.0775088
\(746\) 2.73331e9 0.241047
\(747\) 0 0
\(748\) −3.26593e8 −0.0285332
\(749\) −4.03938e9 −0.351260
\(750\) 0 0
\(751\) 7.73437e9 0.666324 0.333162 0.942870i \(-0.391884\pi\)
0.333162 + 0.942870i \(0.391884\pi\)
\(752\) 1.59726e10 1.36966
\(753\) 0 0
\(754\) 2.44724e10 2.07911
\(755\) −2.20381e9 −0.186363
\(756\) 0 0
\(757\) −1.32525e9 −0.111036 −0.0555178 0.998458i \(-0.517681\pi\)
−0.0555178 + 0.998458i \(0.517681\pi\)
\(758\) −1.36553e10 −1.13883
\(759\) 0 0
\(760\) −1.68520e9 −0.139253
\(761\) −1.43521e10 −1.18050 −0.590252 0.807219i \(-0.700972\pi\)
−0.590252 + 0.807219i \(0.700972\pi\)
\(762\) 0 0
\(763\) −1.28443e10 −1.04683
\(764\) −1.61289e9 −0.130851
\(765\) 0 0
\(766\) −1.86379e10 −1.49829
\(767\) −3.32746e10 −2.66274
\(768\) 0 0
\(769\) 1.52167e10 1.20664 0.603320 0.797499i \(-0.293844\pi\)
0.603320 + 0.797499i \(0.293844\pi\)
\(770\) 2.87261e8 0.0226756
\(771\) 0 0
\(772\) 3.97691e7 0.00311089
\(773\) −1.74767e10 −1.36092 −0.680459 0.732786i \(-0.738219\pi\)
−0.680459 + 0.732786i \(0.738219\pi\)
\(774\) 0 0
\(775\) −1.93378e10 −1.49229
\(776\) 1.94076e9 0.149092
\(777\) 0 0
\(778\) −1.32916e10 −1.01193
\(779\) 1.80598e10 1.36878
\(780\) 0 0
\(781\) 3.54992e8 0.0266649
\(782\) 3.60992e9 0.269944
\(783\) 0 0
\(784\) −8.53576e9 −0.632610
\(785\) −2.01315e8 −0.0148536
\(786\) 0 0
\(787\) −7.75898e8 −0.0567405 −0.0283702 0.999597i \(-0.509032\pi\)
−0.0283702 + 0.999597i \(0.509032\pi\)
\(788\) 1.81804e9 0.132362
\(789\) 0 0
\(790\) 1.01305e9 0.0731032
\(791\) 1.31785e10 0.946778
\(792\) 0 0
\(793\) 3.09574e10 2.20449
\(794\) −1.28237e10 −0.909165
\(795\) 0 0
\(796\) −4.90303e9 −0.344563
\(797\) 3.17179e9 0.221922 0.110961 0.993825i \(-0.464607\pi\)
0.110961 + 0.993825i \(0.464607\pi\)
\(798\) 0 0
\(799\) −1.98571e10 −1.37721
\(800\) −5.97651e9 −0.412698
\(801\) 0 0
\(802\) 3.62117e9 0.247879
\(803\) −3.98911e8 −0.0271876
\(804\) 0 0
\(805\) −5.60388e8 −0.0378619
\(806\) 4.06661e10 2.73564
\(807\) 0 0
\(808\) −1.58156e10 −1.05474
\(809\) 2.96650e10 1.96981 0.984905 0.173096i \(-0.0553772\pi\)
0.984905 + 0.173096i \(0.0553772\pi\)
\(810\) 0 0
\(811\) 2.28837e10 1.50645 0.753223 0.657765i \(-0.228498\pi\)
0.753223 + 0.657765i \(0.228498\pi\)
\(812\) 4.70459e9 0.308372
\(813\) 0 0
\(814\) 5.57766e8 0.0362465
\(815\) −2.14345e9 −0.138695
\(816\) 0 0
\(817\) 1.08609e10 0.696766
\(818\) 8.00757e9 0.511522
\(819\) 0 0
\(820\) −6.15956e8 −0.0390122
\(821\) 9.31507e9 0.587469 0.293734 0.955887i \(-0.405102\pi\)
0.293734 + 0.955887i \(0.405102\pi\)
\(822\) 0 0
\(823\) 2.28072e10 1.42618 0.713088 0.701074i \(-0.247296\pi\)
0.713088 + 0.701074i \(0.247296\pi\)
\(824\) −1.92115e10 −1.19623
\(825\) 0 0
\(826\) −3.62440e10 −2.23772
\(827\) −1.04877e10 −0.644781 −0.322391 0.946607i \(-0.604486\pi\)
−0.322391 + 0.946607i \(0.604486\pi\)
\(828\) 0 0
\(829\) −1.95304e10 −1.19061 −0.595307 0.803498i \(-0.702970\pi\)
−0.595307 + 0.803498i \(0.702970\pi\)
\(830\) 4.18096e9 0.253806
\(831\) 0 0
\(832\) −1.90305e10 −1.14556
\(833\) 1.06116e10 0.636097
\(834\) 0 0
\(835\) −2.31697e9 −0.137727
\(836\) 4.51219e8 0.0267095
\(837\) 0 0
\(838\) 2.43152e10 1.42733
\(839\) 1.46981e10 0.859200 0.429600 0.903019i \(-0.358655\pi\)
0.429600 + 0.903019i \(0.358655\pi\)
\(840\) 0 0
\(841\) 5.91903e9 0.343134
\(842\) 3.32435e10 1.91918
\(843\) 0 0
\(844\) −2.95095e8 −0.0168952
\(845\) 4.23328e9 0.241367
\(846\) 0 0
\(847\) −2.16741e10 −1.22560
\(848\) −3.32071e10 −1.87001
\(849\) 0 0
\(850\) 2.26837e10 1.26692
\(851\) −1.08809e9 −0.0605217
\(852\) 0 0
\(853\) 1.80045e10 0.993253 0.496626 0.867964i \(-0.334572\pi\)
0.496626 + 0.867964i \(0.334572\pi\)
\(854\) 3.37201e10 1.85262
\(855\) 0 0
\(856\) −4.49508e9 −0.244951
\(857\) 1.06997e10 0.580681 0.290340 0.956923i \(-0.406232\pi\)
0.290340 + 0.956923i \(0.406232\pi\)
\(858\) 0 0
\(859\) 2.30270e10 1.23954 0.619771 0.784783i \(-0.287225\pi\)
0.619771 + 0.784783i \(0.287225\pi\)
\(860\) −3.70425e8 −0.0198589
\(861\) 0 0
\(862\) −1.80776e10 −0.961312
\(863\) −1.75568e10 −0.929837 −0.464919 0.885353i \(-0.653916\pi\)
−0.464919 + 0.885353i \(0.653916\pi\)
\(864\) 0 0
\(865\) 1.92921e9 0.101350
\(866\) −1.07136e10 −0.560560
\(867\) 0 0
\(868\) 7.81767e9 0.405750
\(869\) 9.94404e8 0.0514036
\(870\) 0 0
\(871\) −4.60475e10 −2.36125
\(872\) −1.42933e10 −0.730005
\(873\) 0 0
\(874\) −4.98745e9 −0.252690
\(875\) −7.11959e9 −0.359275
\(876\) 0 0
\(877\) −3.55406e10 −1.77920 −0.889602 0.456737i \(-0.849018\pi\)
−0.889602 + 0.456737i \(0.849018\pi\)
\(878\) 2.83944e10 1.41580
\(879\) 0 0
\(880\) 3.91475e8 0.0193649
\(881\) 1.43619e10 0.707616 0.353808 0.935318i \(-0.384887\pi\)
0.353808 + 0.935318i \(0.384887\pi\)
\(882\) 0 0
\(883\) 7.01212e9 0.342757 0.171379 0.985205i \(-0.445178\pi\)
0.171379 + 0.985205i \(0.445178\pi\)
\(884\) −8.41897e9 −0.409898
\(885\) 0 0
\(886\) 2.28193e9 0.110226
\(887\) −1.65555e10 −0.796546 −0.398273 0.917267i \(-0.630390\pi\)
−0.398273 + 0.917267i \(0.630390\pi\)
\(888\) 0 0
\(889\) 4.60990e10 2.20057
\(890\) 4.19752e9 0.199585
\(891\) 0 0
\(892\) −5.87833e9 −0.277317
\(893\) 2.74344e10 1.28919
\(894\) 0 0
\(895\) −3.93691e8 −0.0183559
\(896\) −3.20024e10 −1.48630
\(897\) 0 0
\(898\) 1.37934e10 0.635631
\(899\) 3.85000e10 1.76727
\(900\) 0 0
\(901\) 4.12827e10 1.88032
\(902\) −3.42578e9 −0.155431
\(903\) 0 0
\(904\) 1.46652e10 0.660236
\(905\) 5.59939e9 0.251114
\(906\) 0 0
\(907\) 3.79998e10 1.69105 0.845525 0.533936i \(-0.179288\pi\)
0.845525 + 0.533936i \(0.179288\pi\)
\(908\) 2.01312e9 0.0892421
\(909\) 0 0
\(910\) 7.40506e9 0.325750
\(911\) −9.57936e9 −0.419781 −0.209890 0.977725i \(-0.567311\pi\)
−0.209890 + 0.977725i \(0.567311\pi\)
\(912\) 0 0
\(913\) 4.10400e9 0.178468
\(914\) 2.09432e9 0.0907258
\(915\) 0 0
\(916\) −7.77219e8 −0.0334125
\(917\) −1.07504e10 −0.460395
\(918\) 0 0
\(919\) 2.53060e10 1.07552 0.537760 0.843098i \(-0.319271\pi\)
0.537760 + 0.843098i \(0.319271\pi\)
\(920\) −6.23607e8 −0.0264030
\(921\) 0 0
\(922\) 3.20222e10 1.34553
\(923\) 9.15106e9 0.383059
\(924\) 0 0
\(925\) −6.83724e9 −0.284043
\(926\) 1.37962e10 0.570980
\(927\) 0 0
\(928\) 1.18987e10 0.488745
\(929\) 8.03531e9 0.328812 0.164406 0.986393i \(-0.447429\pi\)
0.164406 + 0.986393i \(0.447429\pi\)
\(930\) 0 0
\(931\) −1.46609e10 −0.595440
\(932\) −3.16160e9 −0.127924
\(933\) 0 0
\(934\) 3.48646e10 1.40014
\(935\) −4.86679e8 −0.0194716
\(936\) 0 0
\(937\) −1.25462e10 −0.498223 −0.249112 0.968475i \(-0.580139\pi\)
−0.249112 + 0.968475i \(0.580139\pi\)
\(938\) −5.01569e10 −1.98436
\(939\) 0 0
\(940\) −9.35690e8 −0.0367438
\(941\) 1.99670e10 0.781176 0.390588 0.920566i \(-0.372272\pi\)
0.390588 + 0.920566i \(0.372272\pi\)
\(942\) 0 0
\(943\) 6.68301e9 0.259526
\(944\) −4.93930e10 −1.91101
\(945\) 0 0
\(946\) −2.06021e9 −0.0791210
\(947\) −3.32659e10 −1.27284 −0.636422 0.771341i \(-0.719586\pi\)
−0.636422 + 0.771341i \(0.719586\pi\)
\(948\) 0 0
\(949\) −1.02832e10 −0.390568
\(950\) −3.13397e10 −1.18594
\(951\) 0 0
\(952\) 3.36186e10 1.26285
\(953\) −4.41016e8 −0.0165055 −0.00825277 0.999966i \(-0.502627\pi\)
−0.00825277 + 0.999966i \(0.502627\pi\)
\(954\) 0 0
\(955\) −2.40348e9 −0.0892954
\(956\) −1.61578e9 −0.0598109
\(957\) 0 0
\(958\) 2.73449e10 1.00484
\(959\) 4.19218e10 1.53488
\(960\) 0 0
\(961\) 3.64633e10 1.32533
\(962\) 1.43782e10 0.520705
\(963\) 0 0
\(964\) −1.40696e9 −0.0505840
\(965\) 5.92628e7 0.00212293
\(966\) 0 0
\(967\) −4.20440e9 −0.149524 −0.0747620 0.997201i \(-0.523820\pi\)
−0.0747620 + 0.997201i \(0.523820\pi\)
\(968\) −2.41193e10 −0.854674
\(969\) 0 0
\(970\) −7.88880e8 −0.0277530
\(971\) 3.02108e10 1.05900 0.529499 0.848311i \(-0.322380\pi\)
0.529499 + 0.848311i \(0.322380\pi\)
\(972\) 0 0
\(973\) −4.99453e10 −1.73820
\(974\) 1.85065e10 0.641752
\(975\) 0 0
\(976\) 4.59533e10 1.58213
\(977\) 3.60793e10 1.23773 0.618866 0.785496i \(-0.287592\pi\)
0.618866 + 0.785496i \(0.287592\pi\)
\(978\) 0 0
\(979\) 4.12025e9 0.140341
\(980\) 5.00032e8 0.0169710
\(981\) 0 0
\(982\) −4.47097e10 −1.50665
\(983\) 3.38986e9 0.113827 0.0569133 0.998379i \(-0.481874\pi\)
0.0569133 + 0.998379i \(0.481874\pi\)
\(984\) 0 0
\(985\) 2.70920e9 0.0903262
\(986\) −4.51614e10 −1.50037
\(987\) 0 0
\(988\) 1.16316e10 0.383699
\(989\) 4.01905e9 0.132110
\(990\) 0 0
\(991\) 8.93953e9 0.291781 0.145890 0.989301i \(-0.453395\pi\)
0.145890 + 0.989301i \(0.453395\pi\)
\(992\) 1.97723e10 0.643080
\(993\) 0 0
\(994\) 9.96771e9 0.321916
\(995\) −7.30635e9 −0.235136
\(996\) 0 0
\(997\) −4.01968e10 −1.28457 −0.642286 0.766465i \(-0.722014\pi\)
−0.642286 + 0.766465i \(0.722014\pi\)
\(998\) 3.29815e10 1.05030
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 207.8.a.a.1.2 5
3.2 odd 2 69.8.a.a.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.8.a.a.1.4 5 3.2 odd 2
207.8.a.a.1.2 5 1.1 even 1 trivial