Properties

Label 450.8.c.d.199.1
Level $450$
Weight $8$
Character 450.199
Analytic conductor $140.573$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,8,Mod(199,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.199");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(140.573261468\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 450.199
Dual form 450.8.c.d.199.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000i q^{2} -64.0000 q^{4} +1427.00i q^{7} +512.000i q^{8} +O(q^{10})\) \(q-8.00000i q^{2} -64.0000 q^{4} +1427.00i q^{7} +512.000i q^{8} -5070.00 q^{11} +11899.0i q^{13} +11416.0 q^{14} +4096.00 q^{16} -19242.0i q^{17} +43711.0 q^{19} +40560.0i q^{22} -3150.00i q^{23} +95192.0 q^{26} -91328.0i q^{28} +140106. q^{29} +147563. q^{31} -32768.0i q^{32} -153936. q^{34} +561674. i q^{37} -349688. i q^{38} +270336. q^{41} -180683. i q^{43} +324480. q^{44} -25200.0 q^{46} -97470.0i q^{47} -1.21279e6 q^{49} -761536. i q^{52} +2.13013e6i q^{53} -730624. q^{56} -1.12085e6i q^{58} -935070. q^{59} +135875. q^{61} -1.18050e6i q^{62} -262144. q^{64} -1.44343e6i q^{67} +1.23149e6i q^{68} +2.68584e6 q^{71} -3.28047e6i q^{73} +4.49339e6 q^{74} -2.79750e6 q^{76} -7.23489e6i q^{77} -5.67267e6 q^{79} -2.16269e6i q^{82} +4.22791e6i q^{83} -1.44546e6 q^{86} -2.59584e6i q^{88} -1.18908e7 q^{89} -1.69799e7 q^{91} +201600. i q^{92} -779760. q^{94} +3.80853e6i q^{97} +9.70229e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 128 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 128 q^{4} - 10140 q^{11} + 22832 q^{14} + 8192 q^{16} + 87422 q^{19} + 190384 q^{26} + 280212 q^{29} + 295126 q^{31} - 307872 q^{34} + 540672 q^{41} + 648960 q^{44} - 50400 q^{46} - 2425572 q^{49} - 1461248 q^{56} - 1870140 q^{59} + 271750 q^{61} - 524288 q^{64} + 5371680 q^{71} + 8986784 q^{74} - 5595008 q^{76} - 11345344 q^{79} - 2890928 q^{86} - 23781696 q^{89} - 33959746 q^{91} - 1559520 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 8.00000i − 0.707107i
\(3\) 0 0
\(4\) −64.0000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1427.00i 1.57246i 0.617931 + 0.786232i \(0.287971\pi\)
−0.617931 + 0.786232i \(0.712029\pi\)
\(8\) 512.000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −5070.00 −1.14851 −0.574253 0.818678i \(-0.694708\pi\)
−0.574253 + 0.818678i \(0.694708\pi\)
\(12\) 0 0
\(13\) 11899.0i 1.50213i 0.660226 + 0.751067i \(0.270461\pi\)
−0.660226 + 0.751067i \(0.729539\pi\)
\(14\) 11416.0 1.11190
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) − 19242.0i − 0.949902i −0.880012 0.474951i \(-0.842466\pi\)
0.880012 0.474951i \(-0.157534\pi\)
\(18\) 0 0
\(19\) 43711.0 1.46202 0.731010 0.682367i \(-0.239049\pi\)
0.731010 + 0.682367i \(0.239049\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 40560.0i 0.812117i
\(23\) − 3150.00i − 0.0539838i −0.999636 0.0269919i \(-0.991407\pi\)
0.999636 0.0269919i \(-0.00859283\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 95192.0 1.06217
\(27\) 0 0
\(28\) − 91328.0i − 0.786232i
\(29\) 140106. 1.06675 0.533376 0.845878i \(-0.320923\pi\)
0.533376 + 0.845878i \(0.320923\pi\)
\(30\) 0 0
\(31\) 147563. 0.889634 0.444817 0.895621i \(-0.353269\pi\)
0.444817 + 0.895621i \(0.353269\pi\)
\(32\) − 32768.0i − 0.176777i
\(33\) 0 0
\(34\) −153936. −0.671682
\(35\) 0 0
\(36\) 0 0
\(37\) 561674.i 1.82296i 0.411339 + 0.911482i \(0.365061\pi\)
−0.411339 + 0.911482i \(0.634939\pi\)
\(38\) − 349688.i − 1.03380i
\(39\) 0 0
\(40\) 0 0
\(41\) 270336. 0.612577 0.306288 0.951939i \(-0.400913\pi\)
0.306288 + 0.951939i \(0.400913\pi\)
\(42\) 0 0
\(43\) − 180683.i − 0.346559i −0.984873 0.173280i \(-0.944564\pi\)
0.984873 0.173280i \(-0.0554365\pi\)
\(44\) 324480. 0.574253
\(45\) 0 0
\(46\) −25200.0 −0.0381723
\(47\) − 97470.0i − 0.136939i −0.997653 0.0684697i \(-0.978188\pi\)
0.997653 0.0684697i \(-0.0218116\pi\)
\(48\) 0 0
\(49\) −1.21279e6 −1.47264
\(50\) 0 0
\(51\) 0 0
\(52\) − 761536.i − 0.751067i
\(53\) 2.13013e6i 1.96535i 0.185324 + 0.982677i \(0.440666\pi\)
−0.185324 + 0.982677i \(0.559334\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −730624. −0.555950
\(57\) 0 0
\(58\) − 1.12085e6i − 0.754308i
\(59\) −935070. −0.592737 −0.296369 0.955074i \(-0.595776\pi\)
−0.296369 + 0.955074i \(0.595776\pi\)
\(60\) 0 0
\(61\) 135875. 0.0766452 0.0383226 0.999265i \(-0.487799\pi\)
0.0383226 + 0.999265i \(0.487799\pi\)
\(62\) − 1.18050e6i − 0.629066i
\(63\) 0 0
\(64\) −262144. −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 1.44343e6i − 0.586320i −0.956063 0.293160i \(-0.905293\pi\)
0.956063 0.293160i \(-0.0947069\pi\)
\(68\) 1.23149e6i 0.474951i
\(69\) 0 0
\(70\) 0 0
\(71\) 2.68584e6 0.890586 0.445293 0.895385i \(-0.353100\pi\)
0.445293 + 0.895385i \(0.353100\pi\)
\(72\) 0 0
\(73\) − 3.28047e6i − 0.986974i −0.869753 0.493487i \(-0.835722\pi\)
0.869753 0.493487i \(-0.164278\pi\)
\(74\) 4.49339e6 1.28903
\(75\) 0 0
\(76\) −2.79750e6 −0.731010
\(77\) − 7.23489e6i − 1.80599i
\(78\) 0 0
\(79\) −5.67267e6 −1.29447 −0.647236 0.762289i \(-0.724075\pi\)
−0.647236 + 0.762289i \(0.724075\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 2.16269e6i − 0.433157i
\(83\) 4.22791e6i 0.811619i 0.913958 + 0.405809i \(0.133010\pi\)
−0.913958 + 0.405809i \(0.866990\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.44546e6 −0.245055
\(87\) 0 0
\(88\) − 2.59584e6i − 0.406058i
\(89\) −1.18908e7 −1.78792 −0.893959 0.448148i \(-0.852084\pi\)
−0.893959 + 0.448148i \(0.852084\pi\)
\(90\) 0 0
\(91\) −1.69799e7 −2.36205
\(92\) 201600.i 0.0269919i
\(93\) 0 0
\(94\) −779760. −0.0968308
\(95\) 0 0
\(96\) 0 0
\(97\) 3.80853e6i 0.423698i 0.977302 + 0.211849i \(0.0679485\pi\)
−0.977302 + 0.211849i \(0.932052\pi\)
\(98\) 9.70229e6i 1.04132i
\(99\) 0 0
\(100\) 0 0
\(101\) 1.20593e7 1.16465 0.582327 0.812955i \(-0.302142\pi\)
0.582327 + 0.812955i \(0.302142\pi\)
\(102\) 0 0
\(103\) − 3.61951e6i − 0.326377i −0.986595 0.163188i \(-0.947822\pi\)
0.986595 0.163188i \(-0.0521778\pi\)
\(104\) −6.09229e6 −0.531085
\(105\) 0 0
\(106\) 1.70411e7 1.38972
\(107\) 2.56928e6i 0.202754i 0.994848 + 0.101377i \(0.0323248\pi\)
−0.994848 + 0.101377i \(0.967675\pi\)
\(108\) 0 0
\(109\) −6.71514e6 −0.496664 −0.248332 0.968675i \(-0.579882\pi\)
−0.248332 + 0.968675i \(0.579882\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 5.84499e6i 0.393116i
\(113\) 1.37565e7i 0.896876i 0.893814 + 0.448438i \(0.148020\pi\)
−0.893814 + 0.448438i \(0.851980\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −8.96678e6 −0.533376
\(117\) 0 0
\(118\) 7.48056e6i 0.419128i
\(119\) 2.74583e7 1.49369
\(120\) 0 0
\(121\) 6.21773e6 0.319068
\(122\) − 1.08700e6i − 0.0541964i
\(123\) 0 0
\(124\) −9.44403e6 −0.444817
\(125\) 0 0
\(126\) 0 0
\(127\) 4.01769e7i 1.74046i 0.492647 + 0.870229i \(0.336029\pi\)
−0.492647 + 0.870229i \(0.663971\pi\)
\(128\) 2.09715e6i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −3.44456e7 −1.33870 −0.669351 0.742946i \(-0.733428\pi\)
−0.669351 + 0.742946i \(0.733428\pi\)
\(132\) 0 0
\(133\) 6.23756e7i 2.29897i
\(134\) −1.15475e7 −0.414591
\(135\) 0 0
\(136\) 9.85190e6 0.335841
\(137\) − 2.78217e7i − 0.924405i −0.886774 0.462203i \(-0.847059\pi\)
0.886774 0.462203i \(-0.152941\pi\)
\(138\) 0 0
\(139\) 8.75852e6 0.276617 0.138308 0.990389i \(-0.455833\pi\)
0.138308 + 0.990389i \(0.455833\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 2.14867e7i − 0.629739i
\(143\) − 6.03279e7i − 1.72521i
\(144\) 0 0
\(145\) 0 0
\(146\) −2.62437e7 −0.697896
\(147\) 0 0
\(148\) − 3.59471e7i − 0.911482i
\(149\) −7.68194e7 −1.90247 −0.951237 0.308460i \(-0.900186\pi\)
−0.951237 + 0.308460i \(0.900186\pi\)
\(150\) 0 0
\(151\) −1.42992e7 −0.337981 −0.168991 0.985618i \(-0.554051\pi\)
−0.168991 + 0.985618i \(0.554051\pi\)
\(152\) 2.23800e7i 0.516902i
\(153\) 0 0
\(154\) −5.78791e7 −1.27702
\(155\) 0 0
\(156\) 0 0
\(157\) − 6.38587e7i − 1.31696i −0.752600 0.658478i \(-0.771200\pi\)
0.752600 0.658478i \(-0.228800\pi\)
\(158\) 4.53814e7i 0.915330i
\(159\) 0 0
\(160\) 0 0
\(161\) 4.49505e6 0.0848875
\(162\) 0 0
\(163\) 8.81015e7i 1.59341i 0.604371 + 0.796703i \(0.293425\pi\)
−0.604371 + 0.796703i \(0.706575\pi\)
\(164\) −1.73015e7 −0.306288
\(165\) 0 0
\(166\) 3.38232e7 0.573901
\(167\) − 6.22383e7i − 1.03407i −0.855964 0.517035i \(-0.827036\pi\)
0.855964 0.517035i \(-0.172964\pi\)
\(168\) 0 0
\(169\) −7.88377e7 −1.25641
\(170\) 0 0
\(171\) 0 0
\(172\) 1.15637e7i 0.173280i
\(173\) − 8.69882e7i − 1.27732i −0.769490 0.638659i \(-0.779490\pi\)
0.769490 0.638659i \(-0.220510\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.07667e7 −0.287127
\(177\) 0 0
\(178\) 9.51268e7i 1.26425i
\(179\) −1.03821e8 −1.35300 −0.676500 0.736443i \(-0.736504\pi\)
−0.676500 + 0.736443i \(0.736504\pi\)
\(180\) 0 0
\(181\) −1.21159e7 −0.151873 −0.0759365 0.997113i \(-0.524195\pi\)
−0.0759365 + 0.997113i \(0.524195\pi\)
\(182\) 1.35839e8i 1.67022i
\(183\) 0 0
\(184\) 1.61280e6 0.0190861
\(185\) 0 0
\(186\) 0 0
\(187\) 9.75569e7i 1.09097i
\(188\) 6.23808e6i 0.0684697i
\(189\) 0 0
\(190\) 0 0
\(191\) 8.64043e7 0.897260 0.448630 0.893718i \(-0.351912\pi\)
0.448630 + 0.893718i \(0.351912\pi\)
\(192\) 0 0
\(193\) 8.89367e7i 0.890493i 0.895408 + 0.445247i \(0.146884\pi\)
−0.895408 + 0.445247i \(0.853116\pi\)
\(194\) 3.04682e7 0.299600
\(195\) 0 0
\(196\) 7.76183e7 0.736322
\(197\) 1.32518e8i 1.23493i 0.786599 + 0.617465i \(0.211840\pi\)
−0.786599 + 0.617465i \(0.788160\pi\)
\(198\) 0 0
\(199\) 3.94534e7 0.354894 0.177447 0.984130i \(-0.443216\pi\)
0.177447 + 0.984130i \(0.443216\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 9.64743e7i − 0.823535i
\(203\) 1.99931e8i 1.67743i
\(204\) 0 0
\(205\) 0 0
\(206\) −2.89561e7 −0.230783
\(207\) 0 0
\(208\) 4.87383e7i 0.375534i
\(209\) −2.21615e8 −1.67914
\(210\) 0 0
\(211\) 1.93707e8 1.41957 0.709786 0.704417i \(-0.248792\pi\)
0.709786 + 0.704417i \(0.248792\pi\)
\(212\) − 1.36328e8i − 0.982677i
\(213\) 0 0
\(214\) 2.05543e7 0.143369
\(215\) 0 0
\(216\) 0 0
\(217\) 2.10572e8i 1.39892i
\(218\) 5.37212e7i 0.351194i
\(219\) 0 0
\(220\) 0 0
\(221\) 2.28961e8 1.42688
\(222\) 0 0
\(223\) − 1.36236e8i − 0.822671i −0.911484 0.411335i \(-0.865062\pi\)
0.911484 0.411335i \(-0.134938\pi\)
\(224\) 4.67599e7 0.277975
\(225\) 0 0
\(226\) 1.10052e8 0.634187
\(227\) 2.86433e8i 1.62530i 0.582755 + 0.812648i \(0.301975\pi\)
−0.582755 + 0.812648i \(0.698025\pi\)
\(228\) 0 0
\(229\) −2.84427e8 −1.56512 −0.782558 0.622578i \(-0.786085\pi\)
−0.782558 + 0.622578i \(0.786085\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 7.17343e7i 0.377154i
\(233\) − 3.24032e8i − 1.67820i −0.543981 0.839098i \(-0.683083\pi\)
0.543981 0.839098i \(-0.316917\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 5.98445e7 0.296369
\(237\) 0 0
\(238\) − 2.19667e8i − 1.05620i
\(239\) 2.53375e8 1.20053 0.600263 0.799802i \(-0.295062\pi\)
0.600263 + 0.799802i \(0.295062\pi\)
\(240\) 0 0
\(241\) 1.07795e8 0.496067 0.248034 0.968751i \(-0.420216\pi\)
0.248034 + 0.968751i \(0.420216\pi\)
\(242\) − 4.97418e7i − 0.225615i
\(243\) 0 0
\(244\) −8.69600e6 −0.0383226
\(245\) 0 0
\(246\) 0 0
\(247\) 5.20117e8i 2.19615i
\(248\) 7.55523e7i 0.314533i
\(249\) 0 0
\(250\) 0 0
\(251\) −2.09909e8 −0.837863 −0.418932 0.908018i \(-0.637595\pi\)
−0.418932 + 0.908018i \(0.637595\pi\)
\(252\) 0 0
\(253\) 1.59705e7i 0.0620007i
\(254\) 3.21415e8 1.23069
\(255\) 0 0
\(256\) 1.67772e7 0.0625000
\(257\) 1.88661e8i 0.693293i 0.937996 + 0.346646i \(0.112680\pi\)
−0.937996 + 0.346646i \(0.887320\pi\)
\(258\) 0 0
\(259\) −8.01509e8 −2.86655
\(260\) 0 0
\(261\) 0 0
\(262\) 2.75565e8i 0.946605i
\(263\) 4.38730e8i 1.48714i 0.668657 + 0.743571i \(0.266869\pi\)
−0.668657 + 0.743571i \(0.733131\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.99005e8 1.62562
\(267\) 0 0
\(268\) 9.23797e7i 0.293160i
\(269\) −1.19910e8 −0.375598 −0.187799 0.982207i \(-0.560135\pi\)
−0.187799 + 0.982207i \(0.560135\pi\)
\(270\) 0 0
\(271\) −3.99299e8 −1.21872 −0.609362 0.792892i \(-0.708574\pi\)
−0.609362 + 0.792892i \(0.708574\pi\)
\(272\) − 7.88152e7i − 0.237476i
\(273\) 0 0
\(274\) −2.22574e8 −0.653653
\(275\) 0 0
\(276\) 0 0
\(277\) − 9.45520e7i − 0.267295i −0.991029 0.133648i \(-0.957331\pi\)
0.991029 0.133648i \(-0.0426691\pi\)
\(278\) − 7.00681e7i − 0.195598i
\(279\) 0 0
\(280\) 0 0
\(281\) 2.37829e8 0.639430 0.319715 0.947514i \(-0.396413\pi\)
0.319715 + 0.947514i \(0.396413\pi\)
\(282\) 0 0
\(283\) 5.34471e8i 1.40175i 0.713282 + 0.700877i \(0.247208\pi\)
−0.713282 + 0.700877i \(0.752792\pi\)
\(284\) −1.71894e8 −0.445293
\(285\) 0 0
\(286\) −4.82623e8 −1.21991
\(287\) 3.85769e8i 0.963255i
\(288\) 0 0
\(289\) 4.00841e7 0.0976854
\(290\) 0 0
\(291\) 0 0
\(292\) 2.09950e8i 0.493487i
\(293\) 6.18845e8i 1.43729i 0.695375 + 0.718647i \(0.255238\pi\)
−0.695375 + 0.718647i \(0.744762\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.87577e8 −0.644515
\(297\) 0 0
\(298\) 6.14555e8i 1.34525i
\(299\) 3.74818e7 0.0810909
\(300\) 0 0
\(301\) 2.57835e8 0.544952
\(302\) 1.14394e8i 0.238989i
\(303\) 0 0
\(304\) 1.79040e8 0.365505
\(305\) 0 0
\(306\) 0 0
\(307\) − 1.74921e8i − 0.345030i −0.985007 0.172515i \(-0.944811\pi\)
0.985007 0.172515i \(-0.0551893\pi\)
\(308\) 4.63033e8i 0.902993i
\(309\) 0 0
\(310\) 0 0
\(311\) 4.03443e7 0.0760537 0.0380269 0.999277i \(-0.487893\pi\)
0.0380269 + 0.999277i \(0.487893\pi\)
\(312\) 0 0
\(313\) − 8.33163e8i − 1.53577i −0.640590 0.767883i \(-0.721310\pi\)
0.640590 0.767883i \(-0.278690\pi\)
\(314\) −5.10870e8 −0.931229
\(315\) 0 0
\(316\) 3.63051e8 0.647236
\(317\) − 1.06497e9i − 1.87771i −0.344307 0.938857i \(-0.611886\pi\)
0.344307 0.938857i \(-0.388114\pi\)
\(318\) 0 0
\(319\) −7.10337e8 −1.22517
\(320\) 0 0
\(321\) 0 0
\(322\) − 3.59604e7i − 0.0600246i
\(323\) − 8.41087e8i − 1.38878i
\(324\) 0 0
\(325\) 0 0
\(326\) 7.04812e8 1.12671
\(327\) 0 0
\(328\) 1.38412e8i 0.216579i
\(329\) 1.39090e8 0.215332
\(330\) 0 0
\(331\) 1.92701e8 0.292070 0.146035 0.989279i \(-0.453349\pi\)
0.146035 + 0.989279i \(0.453349\pi\)
\(332\) − 2.70586e8i − 0.405809i
\(333\) 0 0
\(334\) −4.97906e8 −0.731198
\(335\) 0 0
\(336\) 0 0
\(337\) 2.05520e8i 0.292516i 0.989247 + 0.146258i \(0.0467229\pi\)
−0.989247 + 0.146258i \(0.953277\pi\)
\(338\) 6.30701e8i 0.888414i
\(339\) 0 0
\(340\) 0 0
\(341\) −7.48144e8 −1.02175
\(342\) 0 0
\(343\) − 5.55450e8i − 0.743217i
\(344\) 9.25097e7 0.122527
\(345\) 0 0
\(346\) −6.95905e8 −0.903200
\(347\) 1.24873e9i 1.60441i 0.597050 + 0.802204i \(0.296340\pi\)
−0.597050 + 0.802204i \(0.703660\pi\)
\(348\) 0 0
\(349\) 9.75467e8 1.22835 0.614177 0.789168i \(-0.289488\pi\)
0.614177 + 0.789168i \(0.289488\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.66134e8i 0.203029i
\(353\) − 8.12177e8i − 0.982742i −0.870950 0.491371i \(-0.836496\pi\)
0.870950 0.491371i \(-0.163504\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 7.61014e8 0.893959
\(357\) 0 0
\(358\) 8.30565e8i 0.956716i
\(359\) −8.19316e8 −0.934589 −0.467294 0.884102i \(-0.654771\pi\)
−0.467294 + 0.884102i \(0.654771\pi\)
\(360\) 0 0
\(361\) 1.01678e9 1.13750
\(362\) 9.69271e7i 0.107390i
\(363\) 0 0
\(364\) 1.08671e9 1.18103
\(365\) 0 0
\(366\) 0 0
\(367\) − 4.44464e8i − 0.469360i −0.972073 0.234680i \(-0.924596\pi\)
0.972073 0.234680i \(-0.0754042\pi\)
\(368\) − 1.29024e7i − 0.0134959i
\(369\) 0 0
\(370\) 0 0
\(371\) −3.03970e9 −3.09045
\(372\) 0 0
\(373\) − 5.23822e8i − 0.522640i −0.965252 0.261320i \(-0.915842\pi\)
0.965252 0.261320i \(-0.0841578\pi\)
\(374\) 7.80456e8 0.771432
\(375\) 0 0
\(376\) 4.99046e7 0.0484154
\(377\) 1.66712e9i 1.60241i
\(378\) 0 0
\(379\) 6.01805e8 0.567830 0.283915 0.958849i \(-0.408367\pi\)
0.283915 + 0.958849i \(0.408367\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 6.91234e8i − 0.634459i
\(383\) − 4.67737e8i − 0.425408i −0.977117 0.212704i \(-0.931773\pi\)
0.977117 0.212704i \(-0.0682270\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7.11494e8 0.629674
\(387\) 0 0
\(388\) − 2.43746e8i − 0.211849i
\(389\) −1.11784e9 −0.962841 −0.481421 0.876490i \(-0.659879\pi\)
−0.481421 + 0.876490i \(0.659879\pi\)
\(390\) 0 0
\(391\) −6.06123e7 −0.0512793
\(392\) − 6.20946e8i − 0.520658i
\(393\) 0 0
\(394\) 1.06014e9 0.873227
\(395\) 0 0
\(396\) 0 0
\(397\) − 1.01062e9i − 0.810626i −0.914178 0.405313i \(-0.867162\pi\)
0.914178 0.405313i \(-0.132838\pi\)
\(398\) − 3.15627e8i − 0.250948i
\(399\) 0 0
\(400\) 0 0
\(401\) 4.30852e8 0.333675 0.166837 0.985984i \(-0.446645\pi\)
0.166837 + 0.985984i \(0.446645\pi\)
\(402\) 0 0
\(403\) 1.75585e9i 1.33635i
\(404\) −7.71795e8 −0.582327
\(405\) 0 0
\(406\) 1.59945e9 1.18612
\(407\) − 2.84769e9i − 2.09369i
\(408\) 0 0
\(409\) −7.35181e8 −0.531328 −0.265664 0.964066i \(-0.585591\pi\)
−0.265664 + 0.964066i \(0.585591\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.31649e8i 0.163188i
\(413\) − 1.33434e9i − 0.932058i
\(414\) 0 0
\(415\) 0 0
\(416\) 3.89906e8 0.265542
\(417\) 0 0
\(418\) 1.77292e9i 1.18733i
\(419\) −7.97408e8 −0.529580 −0.264790 0.964306i \(-0.585303\pi\)
−0.264790 + 0.964306i \(0.585303\pi\)
\(420\) 0 0
\(421\) −2.16840e8 −0.141629 −0.0708144 0.997490i \(-0.522560\pi\)
−0.0708144 + 0.997490i \(0.522560\pi\)
\(422\) − 1.54966e9i − 1.00379i
\(423\) 0 0
\(424\) −1.09063e9 −0.694858
\(425\) 0 0
\(426\) 0 0
\(427\) 1.93894e8i 0.120522i
\(428\) − 1.64434e8i − 0.101377i
\(429\) 0 0
\(430\) 0 0
\(431\) −1.05196e9 −0.632890 −0.316445 0.948611i \(-0.602489\pi\)
−0.316445 + 0.948611i \(0.602489\pi\)
\(432\) 0 0
\(433\) − 8.42421e8i − 0.498680i −0.968416 0.249340i \(-0.919786\pi\)
0.968416 0.249340i \(-0.0802136\pi\)
\(434\) 1.68458e9 0.989185
\(435\) 0 0
\(436\) 4.29769e8 0.248332
\(437\) − 1.37690e8i − 0.0789253i
\(438\) 0 0
\(439\) −8.78820e8 −0.495763 −0.247882 0.968790i \(-0.579734\pi\)
−0.247882 + 0.968790i \(0.579734\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 1.83168e9i − 1.00896i
\(443\) − 2.09919e9i − 1.14720i −0.819136 0.573599i \(-0.805547\pi\)
0.819136 0.573599i \(-0.194453\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −1.08989e9 −0.581716
\(447\) 0 0
\(448\) − 3.74079e8i − 0.196558i
\(449\) −1.30937e9 −0.682656 −0.341328 0.939944i \(-0.610877\pi\)
−0.341328 + 0.939944i \(0.610877\pi\)
\(450\) 0 0
\(451\) −1.37060e9 −0.703548
\(452\) − 8.80414e8i − 0.448438i
\(453\) 0 0
\(454\) 2.29146e9 1.14926
\(455\) 0 0
\(456\) 0 0
\(457\) − 2.54163e9i − 1.24568i −0.782350 0.622839i \(-0.785979\pi\)
0.782350 0.622839i \(-0.214021\pi\)
\(458\) 2.27541e9i 1.10670i
\(459\) 0 0
\(460\) 0 0
\(461\) −5.86141e8 −0.278644 −0.139322 0.990247i \(-0.544492\pi\)
−0.139322 + 0.990247i \(0.544492\pi\)
\(462\) 0 0
\(463\) − 1.57279e9i − 0.736439i −0.929739 0.368220i \(-0.879967\pi\)
0.929739 0.368220i \(-0.120033\pi\)
\(464\) 5.73874e8 0.266688
\(465\) 0 0
\(466\) −2.59226e9 −1.18666
\(467\) − 1.58046e9i − 0.718083i −0.933322 0.359042i \(-0.883104\pi\)
0.933322 0.359042i \(-0.116896\pi\)
\(468\) 0 0
\(469\) 2.05978e9 0.921968
\(470\) 0 0
\(471\) 0 0
\(472\) − 4.78756e8i − 0.209564i
\(473\) 9.16063e8i 0.398026i
\(474\) 0 0
\(475\) 0 0
\(476\) −1.75733e9 −0.746844
\(477\) 0 0
\(478\) − 2.02700e9i − 0.848901i
\(479\) 3.70114e9 1.53873 0.769364 0.638811i \(-0.220573\pi\)
0.769364 + 0.638811i \(0.220573\pi\)
\(480\) 0 0
\(481\) −6.68336e9 −2.73834
\(482\) − 8.62363e8i − 0.350773i
\(483\) 0 0
\(484\) −3.97935e8 −0.159534
\(485\) 0 0
\(486\) 0 0
\(487\) − 9.78261e8i − 0.383798i −0.981415 0.191899i \(-0.938535\pi\)
0.981415 0.191899i \(-0.0614647\pi\)
\(488\) 6.95680e7i 0.0270982i
\(489\) 0 0
\(490\) 0 0
\(491\) 8.54921e8 0.325942 0.162971 0.986631i \(-0.447892\pi\)
0.162971 + 0.986631i \(0.447892\pi\)
\(492\) 0 0
\(493\) − 2.69592e9i − 1.01331i
\(494\) 4.16094e9 1.55291
\(495\) 0 0
\(496\) 6.04418e8 0.222409
\(497\) 3.83269e9i 1.40041i
\(498\) 0 0
\(499\) 2.00214e9 0.721345 0.360672 0.932693i \(-0.382547\pi\)
0.360672 + 0.932693i \(0.382547\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1.67927e9i 0.592459i
\(503\) − 3.51483e9i − 1.23145i −0.787962 0.615724i \(-0.788864\pi\)
0.787962 0.615724i \(-0.211136\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.27764e8 0.0438411
\(507\) 0 0
\(508\) − 2.57132e9i − 0.870229i
\(509\) −3.70703e9 −1.24599 −0.622995 0.782226i \(-0.714084\pi\)
−0.622995 + 0.782226i \(0.714084\pi\)
\(510\) 0 0
\(511\) 4.68122e9 1.55198
\(512\) − 1.34218e8i − 0.0441942i
\(513\) 0 0
\(514\) 1.50929e9 0.490232
\(515\) 0 0
\(516\) 0 0
\(517\) 4.94173e8i 0.157276i
\(518\) 6.41207e9i 2.02696i
\(519\) 0 0
\(520\) 0 0
\(521\) 1.31752e9 0.408156 0.204078 0.978955i \(-0.434580\pi\)
0.204078 + 0.978955i \(0.434580\pi\)
\(522\) 0 0
\(523\) 4.45839e8i 0.136277i 0.997676 + 0.0681385i \(0.0217060\pi\)
−0.997676 + 0.0681385i \(0.978294\pi\)
\(524\) 2.20452e9 0.669351
\(525\) 0 0
\(526\) 3.50984e9 1.05157
\(527\) − 2.83941e9i − 0.845066i
\(528\) 0 0
\(529\) 3.39490e9 0.997086
\(530\) 0 0
\(531\) 0 0
\(532\) − 3.99204e9i − 1.14949i
\(533\) 3.21673e9i 0.920172i
\(534\) 0 0
\(535\) 0 0
\(536\) 7.39038e8 0.207295
\(537\) 0 0
\(538\) 9.59282e8i 0.265588i
\(539\) 6.14883e9 1.69134
\(540\) 0 0
\(541\) −7.31894e9 −1.98727 −0.993637 0.112631i \(-0.964072\pi\)
−0.993637 + 0.112631i \(0.964072\pi\)
\(542\) 3.19439e9i 0.861768i
\(543\) 0 0
\(544\) −6.30522e8 −0.167921
\(545\) 0 0
\(546\) 0 0
\(547\) 5.21813e9i 1.36320i 0.731725 + 0.681600i \(0.238715\pi\)
−0.731725 + 0.681600i \(0.761285\pi\)
\(548\) 1.78059e9i 0.462203i
\(549\) 0 0
\(550\) 0 0
\(551\) 6.12417e9 1.55961
\(552\) 0 0
\(553\) − 8.09490e9i − 2.03551i
\(554\) −7.56416e8 −0.189006
\(555\) 0 0
\(556\) −5.60545e8 −0.138308
\(557\) − 5.22742e7i − 0.0128172i −0.999979 0.00640862i \(-0.997960\pi\)
0.999979 0.00640862i \(-0.00203994\pi\)
\(558\) 0 0
\(559\) 2.14995e9 0.520579
\(560\) 0 0
\(561\) 0 0
\(562\) − 1.90263e9i − 0.452145i
\(563\) − 7.63499e9i − 1.80314i −0.432636 0.901569i \(-0.642416\pi\)
0.432636 0.901569i \(-0.357584\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 4.27577e9 0.991190
\(567\) 0 0
\(568\) 1.37515e9i 0.314870i
\(569\) −2.52617e9 −0.574871 −0.287435 0.957800i \(-0.592803\pi\)
−0.287435 + 0.957800i \(0.592803\pi\)
\(570\) 0 0
\(571\) −4.75335e9 −1.06850 −0.534248 0.845328i \(-0.679405\pi\)
−0.534248 + 0.845328i \(0.679405\pi\)
\(572\) 3.86099e9i 0.862606i
\(573\) 0 0
\(574\) 3.08616e9 0.681124
\(575\) 0 0
\(576\) 0 0
\(577\) 7.07337e9i 1.53289i 0.642309 + 0.766445i \(0.277976\pi\)
−0.642309 + 0.766445i \(0.722024\pi\)
\(578\) − 3.20673e8i − 0.0690740i
\(579\) 0 0
\(580\) 0 0
\(581\) −6.03322e9 −1.27624
\(582\) 0 0
\(583\) − 1.07998e10i − 2.25722i
\(584\) 1.67960e9 0.348948
\(585\) 0 0
\(586\) 4.95076e9 1.01632
\(587\) 6.82995e8i 0.139375i 0.997569 + 0.0696874i \(0.0222002\pi\)
−0.997569 + 0.0696874i \(0.977800\pi\)
\(588\) 0 0
\(589\) 6.45013e9 1.30066
\(590\) 0 0
\(591\) 0 0
\(592\) 2.30062e9i 0.455741i
\(593\) 4.11466e9i 0.810294i 0.914251 + 0.405147i \(0.132780\pi\)
−0.914251 + 0.405147i \(0.867220\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.91644e9 0.951237
\(597\) 0 0
\(598\) − 2.99855e8i − 0.0573399i
\(599\) −2.71835e8 −0.0516787 −0.0258394 0.999666i \(-0.508226\pi\)
−0.0258394 + 0.999666i \(0.508226\pi\)
\(600\) 0 0
\(601\) −1.63646e9 −0.307500 −0.153750 0.988110i \(-0.549135\pi\)
−0.153750 + 0.988110i \(0.549135\pi\)
\(602\) − 2.06268e9i − 0.385340i
\(603\) 0 0
\(604\) 9.15149e8 0.168991
\(605\) 0 0
\(606\) 0 0
\(607\) − 7.25840e8i − 0.131729i −0.997829 0.0658644i \(-0.979020\pi\)
0.997829 0.0658644i \(-0.0209805\pi\)
\(608\) − 1.43232e9i − 0.258451i
\(609\) 0 0
\(610\) 0 0
\(611\) 1.15980e9 0.205701
\(612\) 0 0
\(613\) 9.79507e9i 1.71750i 0.512397 + 0.858749i \(0.328758\pi\)
−0.512397 + 0.858749i \(0.671242\pi\)
\(614\) −1.39937e9 −0.243973
\(615\) 0 0
\(616\) 3.70426e9 0.638512
\(617\) 7.54736e8i 0.129359i 0.997906 + 0.0646796i \(0.0206025\pi\)
−0.997906 + 0.0646796i \(0.979397\pi\)
\(618\) 0 0
\(619\) −9.48066e9 −1.60665 −0.803325 0.595541i \(-0.796938\pi\)
−0.803325 + 0.595541i \(0.796938\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 3.22754e8i − 0.0537781i
\(623\) − 1.69682e10i − 2.81144i
\(624\) 0 0
\(625\) 0 0
\(626\) −6.66531e9 −1.08595
\(627\) 0 0
\(628\) 4.08696e9i 0.658478i
\(629\) 1.08077e10 1.73164
\(630\) 0 0
\(631\) 2.11281e8 0.0334779 0.0167389 0.999860i \(-0.494672\pi\)
0.0167389 + 0.999860i \(0.494672\pi\)
\(632\) − 2.90441e9i − 0.457665i
\(633\) 0 0
\(634\) −8.51975e9 −1.32774
\(635\) 0 0
\(636\) 0 0
\(637\) − 1.44309e10i − 2.21211i
\(638\) 5.68270e9i 0.866328i
\(639\) 0 0
\(640\) 0 0
\(641\) −7.52664e9 −1.12875 −0.564375 0.825518i \(-0.690883\pi\)
−0.564375 + 0.825518i \(0.690883\pi\)
\(642\) 0 0
\(643\) 3.52375e9i 0.522717i 0.965242 + 0.261358i \(0.0841704\pi\)
−0.965242 + 0.261358i \(0.915830\pi\)
\(644\) −2.87683e8 −0.0424438
\(645\) 0 0
\(646\) −6.72870e9 −0.982013
\(647\) − 3.90185e9i − 0.566377i −0.959064 0.283189i \(-0.908608\pi\)
0.959064 0.283189i \(-0.0913922\pi\)
\(648\) 0 0
\(649\) 4.74080e9 0.680763
\(650\) 0 0
\(651\) 0 0
\(652\) − 5.63850e9i − 0.796703i
\(653\) − 8.20768e9i − 1.15352i −0.816915 0.576759i \(-0.804317\pi\)
0.816915 0.576759i \(-0.195683\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.10730e9 0.153144
\(657\) 0 0
\(658\) − 1.11272e9i − 0.152263i
\(659\) 4.49137e9 0.611336 0.305668 0.952138i \(-0.401120\pi\)
0.305668 + 0.952138i \(0.401120\pi\)
\(660\) 0 0
\(661\) 9.78484e9 1.31780 0.658898 0.752232i \(-0.271023\pi\)
0.658898 + 0.752232i \(0.271023\pi\)
\(662\) − 1.54161e9i − 0.206525i
\(663\) 0 0
\(664\) −2.16469e9 −0.286951
\(665\) 0 0
\(666\) 0 0
\(667\) − 4.41334e8i − 0.0575873i
\(668\) 3.98325e9i 0.517035i
\(669\) 0 0
\(670\) 0 0
\(671\) −6.88886e8 −0.0880276
\(672\) 0 0
\(673\) 1.99688e9i 0.252522i 0.991997 + 0.126261i \(0.0402978\pi\)
−0.991997 + 0.126261i \(0.959702\pi\)
\(674\) 1.64416e9 0.206840
\(675\) 0 0
\(676\) 5.04561e9 0.628204
\(677\) 1.76852e9i 0.219053i 0.993984 + 0.109526i \(0.0349334\pi\)
−0.993984 + 0.109526i \(0.965067\pi\)
\(678\) 0 0
\(679\) −5.43477e9 −0.666250
\(680\) 0 0
\(681\) 0 0
\(682\) 5.98516e9i 0.722487i
\(683\) 1.33982e10i 1.60906i 0.593910 + 0.804531i \(0.297583\pi\)
−0.593910 + 0.804531i \(0.702417\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −4.44360e9 −0.525533
\(687\) 0 0
\(688\) − 7.40078e8i − 0.0866399i
\(689\) −2.53464e10 −2.95223
\(690\) 0 0
\(691\) 1.36908e10 1.57854 0.789270 0.614046i \(-0.210459\pi\)
0.789270 + 0.614046i \(0.210459\pi\)
\(692\) 5.56724e9i 0.638659i
\(693\) 0 0
\(694\) 9.98983e9 1.13449
\(695\) 0 0
\(696\) 0 0
\(697\) − 5.20181e9i − 0.581888i
\(698\) − 7.80374e9i − 0.868578i
\(699\) 0 0
\(700\) 0 0
\(701\) 8.98742e9 0.985422 0.492711 0.870193i \(-0.336006\pi\)
0.492711 + 0.870193i \(0.336006\pi\)
\(702\) 0 0
\(703\) 2.45513e10i 2.66521i
\(704\) 1.32907e9 0.143563
\(705\) 0 0
\(706\) −6.49742e9 −0.694903
\(707\) 1.72086e10i 1.83138i
\(708\) 0 0
\(709\) 4.36499e9 0.459962 0.229981 0.973195i \(-0.426134\pi\)
0.229981 + 0.973195i \(0.426134\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 6.08811e9i − 0.632125i
\(713\) − 4.64823e8i − 0.0480258i
\(714\) 0 0
\(715\) 0 0
\(716\) 6.64452e9 0.676500
\(717\) 0 0
\(718\) 6.55452e9i 0.660854i
\(719\) −1.46045e10 −1.46534 −0.732668 0.680587i \(-0.761725\pi\)
−0.732668 + 0.680587i \(0.761725\pi\)
\(720\) 0 0
\(721\) 5.16504e9 0.513216
\(722\) − 8.13424e9i − 0.804335i
\(723\) 0 0
\(724\) 7.75417e8 0.0759365
\(725\) 0 0
\(726\) 0 0
\(727\) 1.42461e10i 1.37507i 0.726150 + 0.687536i \(0.241308\pi\)
−0.726150 + 0.687536i \(0.758692\pi\)
\(728\) − 8.69369e9i − 0.835112i
\(729\) 0 0
\(730\) 0 0
\(731\) −3.47670e9 −0.329198
\(732\) 0 0
\(733\) − 8.78749e9i − 0.824140i −0.911152 0.412070i \(-0.864806\pi\)
0.911152 0.412070i \(-0.135194\pi\)
\(734\) −3.55571e9 −0.331887
\(735\) 0 0
\(736\) −1.03219e8 −0.00954307
\(737\) 7.31821e9i 0.673393i
\(738\) 0 0
\(739\) −1.59660e10 −1.45526 −0.727632 0.685968i \(-0.759379\pi\)
−0.727632 + 0.685968i \(0.759379\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.43176e10i 2.18528i
\(743\) 4.01209e9i 0.358847i 0.983772 + 0.179424i \(0.0574232\pi\)
−0.983772 + 0.179424i \(0.942577\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −4.19057e9 −0.369562
\(747\) 0 0
\(748\) − 6.24364e9i − 0.545485i
\(749\) −3.66637e9 −0.318823
\(750\) 0 0
\(751\) 7.41663e9 0.638950 0.319475 0.947595i \(-0.396493\pi\)
0.319475 + 0.947595i \(0.396493\pi\)
\(752\) − 3.99237e8i − 0.0342349i
\(753\) 0 0
\(754\) 1.33370e10 1.13307
\(755\) 0 0
\(756\) 0 0
\(757\) 1.51143e8i 0.0126634i 0.999980 + 0.00633171i \(0.00201546\pi\)
−0.999980 + 0.00633171i \(0.997985\pi\)
\(758\) − 4.81444e9i − 0.401516i
\(759\) 0 0
\(760\) 0 0
\(761\) −3.85268e9 −0.316896 −0.158448 0.987367i \(-0.550649\pi\)
−0.158448 + 0.987367i \(0.550649\pi\)
\(762\) 0 0
\(763\) − 9.58251e9i − 0.780986i
\(764\) −5.52987e9 −0.448630
\(765\) 0 0
\(766\) −3.74189e9 −0.300809
\(767\) − 1.11264e10i − 0.890371i
\(768\) 0 0
\(769\) −1.56192e10 −1.23856 −0.619280 0.785170i \(-0.712576\pi\)
−0.619280 + 0.785170i \(0.712576\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 5.69195e9i − 0.445247i
\(773\) 1.44828e10i 1.12778i 0.825850 + 0.563889i \(0.190696\pi\)
−0.825850 + 0.563889i \(0.809304\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.94997e9 −0.149800
\(777\) 0 0
\(778\) 8.94269e9i 0.680831i
\(779\) 1.18167e10 0.895599
\(780\) 0 0
\(781\) −1.36172e10 −1.02284
\(782\) 4.84898e8i 0.0362599i
\(783\) 0 0
\(784\) −4.96757e9 −0.368161
\(785\) 0 0
\(786\) 0 0
\(787\) 1.37729e10i 1.00720i 0.863938 + 0.503599i \(0.167991\pi\)
−0.863938 + 0.503599i \(0.832009\pi\)
\(788\) − 8.48113e9i − 0.617465i
\(789\) 0 0
\(790\) 0 0
\(791\) −1.96305e10 −1.41031
\(792\) 0 0
\(793\) 1.61678e9i 0.115131i
\(794\) −8.08495e9 −0.573199
\(795\) 0 0
\(796\) −2.52502e9 −0.177447
\(797\) 6.88529e9i 0.481746i 0.970557 + 0.240873i \(0.0774338\pi\)
−0.970557 + 0.240873i \(0.922566\pi\)
\(798\) 0 0
\(799\) −1.87552e9 −0.130079
\(800\) 0 0
\(801\) 0 0
\(802\) − 3.44682e9i − 0.235944i
\(803\) 1.66320e10i 1.13355i
\(804\) 0 0
\(805\) 0 0
\(806\) 1.40468e10 0.944942
\(807\) 0 0
\(808\) 6.17436e9i 0.411767i
\(809\) −1.28256e10 −0.851644 −0.425822 0.904807i \(-0.640015\pi\)
−0.425822 + 0.904807i \(0.640015\pi\)
\(810\) 0 0
\(811\) −7.36094e9 −0.484574 −0.242287 0.970205i \(-0.577898\pi\)
−0.242287 + 0.970205i \(0.577898\pi\)
\(812\) − 1.27956e10i − 0.838715i
\(813\) 0 0
\(814\) −2.27815e10 −1.48046
\(815\) 0 0
\(816\) 0 0
\(817\) − 7.89783e9i − 0.506677i
\(818\) 5.88144e9i 0.375705i
\(819\) 0 0
\(820\) 0 0
\(821\) 1.09919e10 0.693219 0.346610 0.938009i \(-0.387333\pi\)
0.346610 + 0.938009i \(0.387333\pi\)
\(822\) 0 0
\(823\) 1.15788e10i 0.724041i 0.932170 + 0.362020i \(0.117913\pi\)
−0.932170 + 0.362020i \(0.882087\pi\)
\(824\) 1.85319e9 0.115392
\(825\) 0 0
\(826\) −1.06748e10 −0.659064
\(827\) − 4.97823e9i − 0.306059i −0.988222 0.153030i \(-0.951097\pi\)
0.988222 0.153030i \(-0.0489030\pi\)
\(828\) 0 0
\(829\) 1.15589e10 0.704655 0.352328 0.935877i \(-0.385390\pi\)
0.352328 + 0.935877i \(0.385390\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 3.11925e9i − 0.187767i
\(833\) 2.33364e10i 1.39887i
\(834\) 0 0
\(835\) 0 0
\(836\) 1.41833e10 0.839570
\(837\) 0 0
\(838\) 6.37926e9i 0.374469i
\(839\) 4.50868e9 0.263562 0.131781 0.991279i \(-0.457930\pi\)
0.131781 + 0.991279i \(0.457930\pi\)
\(840\) 0 0
\(841\) 2.37981e9 0.137961
\(842\) 1.73472e9i 0.100147i
\(843\) 0 0
\(844\) −1.23973e10 −0.709786
\(845\) 0 0
\(846\) 0 0
\(847\) 8.87270e9i 0.501723i
\(848\) 8.72502e9i 0.491339i
\(849\) 0 0
\(850\) 0 0
\(851\) 1.76927e9 0.0984105
\(852\) 0 0
\(853\) 9.35143e9i 0.515889i 0.966160 + 0.257945i \(0.0830452\pi\)
−0.966160 + 0.257945i \(0.916955\pi\)
\(854\) 1.55115e9 0.0852219
\(855\) 0 0
\(856\) −1.31547e9 −0.0716843
\(857\) − 1.85745e9i − 0.100806i −0.998729 0.0504029i \(-0.983949\pi\)
0.998729 0.0504029i \(-0.0160505\pi\)
\(858\) 0 0
\(859\) −3.34304e10 −1.79956 −0.899779 0.436346i \(-0.856272\pi\)
−0.899779 + 0.436346i \(0.856272\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8.41567e9i 0.447521i
\(863\) 1.34259e10i 0.711061i 0.934665 + 0.355531i \(0.115700\pi\)
−0.934665 + 0.355531i \(0.884300\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −6.73937e9 −0.352620
\(867\) 0 0
\(868\) − 1.34766e10i − 0.699459i
\(869\) 2.87604e10 1.48671
\(870\) 0 0
\(871\) 1.71754e10 0.880732
\(872\) − 3.43815e9i − 0.175597i
\(873\) 0 0
\(874\) −1.10152e9 −0.0558086
\(875\) 0 0
\(876\) 0 0
\(877\) 3.08992e9i 0.154685i 0.997005 + 0.0773426i \(0.0246435\pi\)
−0.997005 + 0.0773426i \(0.975356\pi\)
\(878\) 7.03056e9i 0.350558i
\(879\) 0 0
\(880\) 0 0
\(881\) −8.72396e9 −0.429831 −0.214916 0.976633i \(-0.568948\pi\)
−0.214916 + 0.976633i \(0.568948\pi\)
\(882\) 0 0
\(883\) 3.90723e9i 0.190988i 0.995430 + 0.0954942i \(0.0304431\pi\)
−0.995430 + 0.0954942i \(0.969557\pi\)
\(884\) −1.46535e10 −0.713440
\(885\) 0 0
\(886\) −1.67935e10 −0.811192
\(887\) − 1.53854e10i − 0.740246i −0.928983 0.370123i \(-0.879316\pi\)
0.928983 0.370123i \(-0.120684\pi\)
\(888\) 0 0
\(889\) −5.73325e10 −2.73681
\(890\) 0 0
\(891\) 0 0
\(892\) 8.71913e9i 0.411335i
\(893\) − 4.26051e9i − 0.200208i
\(894\) 0 0
\(895\) 0 0
\(896\) −2.99264e9 −0.138988
\(897\) 0 0
\(898\) 1.04750e10i 0.482711i
\(899\) 2.06745e10 0.949020
\(900\) 0 0
\(901\) 4.09880e10 1.86690
\(902\) 1.09648e10i 0.497484i
\(903\) 0 0
\(904\) −7.04332e9 −0.317094
\(905\) 0 0
\(906\) 0 0
\(907\) − 2.63986e10i − 1.17478i −0.809305 0.587388i \(-0.800156\pi\)
0.809305 0.587388i \(-0.199844\pi\)
\(908\) − 1.83317e10i − 0.812648i
\(909\) 0 0
\(910\) 0 0
\(911\) 2.12340e10 0.930503 0.465251 0.885179i \(-0.345964\pi\)
0.465251 + 0.885179i \(0.345964\pi\)
\(912\) 0 0
\(913\) − 2.14355e10i − 0.932149i
\(914\) −2.03331e10 −0.880828
\(915\) 0 0
\(916\) 1.82033e10 0.782558
\(917\) − 4.91538e10i − 2.10506i
\(918\) 0 0
\(919\) −9.50036e9 −0.403771 −0.201886 0.979409i \(-0.564707\pi\)
−0.201886 + 0.979409i \(0.564707\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 4.68913e9i 0.197031i
\(923\) 3.19588e10i 1.33778i
\(924\) 0 0
\(925\) 0 0
\(926\) −1.25823e10 −0.520741
\(927\) 0 0
\(928\) − 4.59099e9i − 0.188577i
\(929\) −4.29492e10 −1.75752 −0.878760 0.477264i \(-0.841629\pi\)
−0.878760 + 0.477264i \(0.841629\pi\)
\(930\) 0 0
\(931\) −5.30121e10 −2.15303
\(932\) 2.07381e10i 0.839098i
\(933\) 0 0
\(934\) −1.26437e10 −0.507761
\(935\) 0 0
\(936\) 0 0
\(937\) 1.52405e10i 0.605216i 0.953115 + 0.302608i \(0.0978573\pi\)
−0.953115 + 0.302608i \(0.902143\pi\)
\(938\) − 1.64782e10i − 0.651930i
\(939\) 0 0
\(940\) 0 0
\(941\) 2.27448e10 0.889853 0.444926 0.895567i \(-0.353230\pi\)
0.444926 + 0.895567i \(0.353230\pi\)
\(942\) 0 0
\(943\) − 8.51558e8i − 0.0330692i
\(944\) −3.83005e9 −0.148184
\(945\) 0 0
\(946\) 7.32850e9 0.281447
\(947\) 8.80667e9i 0.336967i 0.985705 + 0.168483i \(0.0538869\pi\)
−0.985705 + 0.168483i \(0.946113\pi\)
\(948\) 0 0
\(949\) 3.90343e10 1.48257
\(950\) 0 0
\(951\) 0 0
\(952\) 1.40587e10i 0.528098i
\(953\) − 1.37186e9i − 0.0513434i −0.999670 0.0256717i \(-0.991828\pi\)
0.999670 0.0256717i \(-0.00817246\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.62160e10 −0.600263
\(957\) 0 0
\(958\) − 2.96092e10i − 1.08804i
\(959\) 3.97016e10 1.45359
\(960\) 0 0
\(961\) −5.73778e9 −0.208551
\(962\) 5.34669e10i 1.93630i
\(963\) 0 0
\(964\) −6.89891e9 −0.248034
\(965\) 0 0
\(966\) 0 0
\(967\) 2.51907e10i 0.895874i 0.894065 + 0.447937i \(0.147841\pi\)
−0.894065 + 0.447937i \(0.852159\pi\)
\(968\) 3.18348e9i 0.112808i
\(969\) 0 0
\(970\) 0 0
\(971\) 5.11915e9 0.179445 0.0897223 0.995967i \(-0.471402\pi\)
0.0897223 + 0.995967i \(0.471402\pi\)
\(972\) 0 0
\(973\) 1.24984e10i 0.434970i
\(974\) −7.82608e9 −0.271386
\(975\) 0 0
\(976\) 5.56544e8 0.0191613
\(977\) − 5.06512e8i − 0.0173764i −0.999962 0.00868818i \(-0.997234\pi\)
0.999962 0.00868818i \(-0.00276557\pi\)
\(978\) 0 0
\(979\) 6.02866e10 2.05344
\(980\) 0 0
\(981\) 0 0
\(982\) − 6.83937e9i − 0.230476i
\(983\) − 1.00756e10i − 0.338323i −0.985588 0.169162i \(-0.945894\pi\)
0.985588 0.169162i \(-0.0541060\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.15674e10 −0.716519
\(987\) 0 0
\(988\) − 3.32875e10i − 1.09807i
\(989\) −5.69151e8 −0.0187086
\(990\) 0 0
\(991\) 4.56127e10 1.48877 0.744386 0.667749i \(-0.232742\pi\)
0.744386 + 0.667749i \(0.232742\pi\)
\(992\) − 4.83534e9i − 0.157267i
\(993\) 0 0
\(994\) 3.06615e10 0.990243
\(995\) 0 0
\(996\) 0 0
\(997\) 3.05555e10i 0.976463i 0.872714 + 0.488232i \(0.162358\pi\)
−0.872714 + 0.488232i \(0.837642\pi\)
\(998\) − 1.60171e10i − 0.510068i
\(999\) 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 450.8.c.d.199.1 2
3.2 odd 2 150.8.c.d.49.2 2
5.2 odd 4 450.8.a.o.1.1 1
5.3 odd 4 450.8.a.m.1.1 1
5.4 even 2 inner 450.8.c.d.199.2 2
15.2 even 4 150.8.a.f.1.1 1
15.8 even 4 150.8.a.l.1.1 yes 1
15.14 odd 2 150.8.c.d.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.8.a.f.1.1 1 15.2 even 4
150.8.a.l.1.1 yes 1 15.8 even 4
150.8.c.d.49.1 2 15.14 odd 2
150.8.c.d.49.2 2 3.2 odd 2
450.8.a.m.1.1 1 5.3 odd 4
450.8.a.o.1.1 1 5.2 odd 4
450.8.c.d.199.1 2 1.1 even 1 trivial
450.8.c.d.199.2 2 5.4 even 2 inner