Properties

Label 80.10.a.g.1.1
Level $80$
Weight $10$
Character 80.1
Self dual yes
Analytic conductor $41.203$
Analytic rank $1$
Dimension $2$
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] = [80,10,Mod(1,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 80.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: \(41.2028668931\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{46}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 46 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.78233\) of defining polynomial
Character \(\chi\) \(=\) 80.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-216.776 q^{3} -625.000 q^{5} -1662.09 q^{7} +27308.8 q^{9} +O(q^{10})\) \(q-216.776 q^{3} -625.000 q^{5} -1662.09 q^{7} +27308.8 q^{9} -34372.0 q^{11} +162675. q^{13} +135485. q^{15} +91796.1 q^{17} +536227. q^{19} +360300. q^{21} -234899. q^{23} +390625. q^{25} -1.65309e6 q^{27} +7.25706e6 q^{29} -8.23051e6 q^{31} +7.45102e6 q^{33} +1.03880e6 q^{35} -1.56850e7 q^{37} -3.52641e7 q^{39} -1.67318e7 q^{41} -9.36208e6 q^{43} -1.70680e7 q^{45} +2.69410e7 q^{47} -3.75911e7 q^{49} -1.98992e7 q^{51} +3.41105e7 q^{53} +2.14825e7 q^{55} -1.16241e8 q^{57} -1.12155e8 q^{59} +5.49531e6 q^{61} -4.53896e7 q^{63} -1.01672e8 q^{65} +2.40686e8 q^{67} +5.09204e7 q^{69} +2.58502e8 q^{71} -3.53583e8 q^{73} -8.46781e7 q^{75} +5.71292e7 q^{77} -3.66190e8 q^{79} -1.79169e8 q^{81} +9.74953e6 q^{83} -5.73726e7 q^{85} -1.57316e9 q^{87} -1.14201e9 q^{89} -2.70380e8 q^{91} +1.78418e9 q^{93} -3.35142e8 q^{95} +1.25555e9 q^{97} -9.38657e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 108 q^{3} - 1250 q^{5} + 908 q^{7} + 19458 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 108 q^{3} - 1250 q^{5} + 908 q^{7} + 19458 q^{9} - 25120 q^{11} + 146948 q^{13} + 67500 q^{15} + 169268 q^{17} + 25480 q^{19} + 639864 q^{21} - 1782748 q^{23} + 781250 q^{25} - 4648104 q^{27} + 7323340 q^{29} - 10677272 q^{31} + 8457408 q^{33} - 567500 q^{35} - 5750460 q^{37} - 36974808 q^{39} - 7795764 q^{41} - 16770524 q^{43} - 12161250 q^{45} - 15393892 q^{47} - 71339334 q^{49} - 11472120 q^{51} - 58529292 q^{53} + 15700000 q^{55} - 171798192 q^{57} - 59618264 q^{59} - 188163772 q^{61} - 65566836 q^{63} - 91842500 q^{65} + 105998252 q^{67} - 117448344 q^{69} + 168665592 q^{71} - 390212412 q^{73} - 42187500 q^{75} + 80907584 q^{77} - 466946256 q^{79} - 350427222 q^{81} - 329535164 q^{83} - 105792500 q^{85} - 1565947656 q^{87} - 108334860 q^{89} - 310800616 q^{91} + 1518028560 q^{93} - 15925000 q^{95} + 2043058628 q^{97} - 1011292704 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −216.776 −1.54513 −0.772566 0.634935i \(-0.781027\pi\)
−0.772566 + 0.634935i \(0.781027\pi\)
\(4\) 0 0
\(5\) −625.000 −0.447214
\(6\) 0 0
\(7\) −1662.09 −0.261645 −0.130823 0.991406i \(-0.541762\pi\)
−0.130823 + 0.991406i \(0.541762\pi\)
\(8\) 0 0
\(9\) 27308.8 1.38743
\(10\) 0 0
\(11\) −34372.0 −0.707844 −0.353922 0.935275i \(-0.615152\pi\)
−0.353922 + 0.935275i \(0.615152\pi\)
\(12\) 0 0
\(13\) 162675. 1.57971 0.789853 0.613296i \(-0.210157\pi\)
0.789853 + 0.613296i \(0.210157\pi\)
\(14\) 0 0
\(15\) 135485. 0.691004
\(16\) 0 0
\(17\) 91796.1 0.266566 0.133283 0.991078i \(-0.457448\pi\)
0.133283 + 0.991078i \(0.457448\pi\)
\(18\) 0 0
\(19\) 536227. 0.943969 0.471985 0.881607i \(-0.343538\pi\)
0.471985 + 0.881607i \(0.343538\pi\)
\(20\) 0 0
\(21\) 360300. 0.404276
\(22\) 0 0
\(23\) −234899. −0.175027 −0.0875136 0.996163i \(-0.527892\pi\)
−0.0875136 + 0.996163i \(0.527892\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 0 0
\(27\) −1.65309e6 −0.598631
\(28\) 0 0
\(29\) 7.25706e6 1.90533 0.952665 0.304023i \(-0.0983301\pi\)
0.952665 + 0.304023i \(0.0983301\pi\)
\(30\) 0 0
\(31\) −8.23051e6 −1.60066 −0.800330 0.599559i \(-0.795343\pi\)
−0.800330 + 0.599559i \(0.795343\pi\)
\(32\) 0 0
\(33\) 7.45102e6 1.09371
\(34\) 0 0
\(35\) 1.03880e6 0.117011
\(36\) 0 0
\(37\) −1.56850e7 −1.37587 −0.687936 0.725771i \(-0.741483\pi\)
−0.687936 + 0.725771i \(0.741483\pi\)
\(38\) 0 0
\(39\) −3.52641e7 −2.44085
\(40\) 0 0
\(41\) −1.67318e7 −0.924730 −0.462365 0.886690i \(-0.652999\pi\)
−0.462365 + 0.886690i \(0.652999\pi\)
\(42\) 0 0
\(43\) −9.36208e6 −0.417604 −0.208802 0.977958i \(-0.566956\pi\)
−0.208802 + 0.977958i \(0.566956\pi\)
\(44\) 0 0
\(45\) −1.70680e7 −0.620478
\(46\) 0 0
\(47\) 2.69410e7 0.805328 0.402664 0.915348i \(-0.368084\pi\)
0.402664 + 0.915348i \(0.368084\pi\)
\(48\) 0 0
\(49\) −3.75911e7 −0.931542
\(50\) 0 0
\(51\) −1.98992e7 −0.411879
\(52\) 0 0
\(53\) 3.41105e7 0.593809 0.296905 0.954907i \(-0.404046\pi\)
0.296905 + 0.954907i \(0.404046\pi\)
\(54\) 0 0
\(55\) 2.14825e7 0.316557
\(56\) 0 0
\(57\) −1.16241e8 −1.45856
\(58\) 0 0
\(59\) −1.12155e8 −1.20499 −0.602496 0.798122i \(-0.705827\pi\)
−0.602496 + 0.798122i \(0.705827\pi\)
\(60\) 0 0
\(61\) 5.49531e6 0.0508168 0.0254084 0.999677i \(-0.491911\pi\)
0.0254084 + 0.999677i \(0.491911\pi\)
\(62\) 0 0
\(63\) −4.53896e7 −0.363014
\(64\) 0 0
\(65\) −1.01672e8 −0.706466
\(66\) 0 0
\(67\) 2.40686e8 1.45920 0.729598 0.683876i \(-0.239707\pi\)
0.729598 + 0.683876i \(0.239707\pi\)
\(68\) 0 0
\(69\) 5.09204e7 0.270440
\(70\) 0 0
\(71\) 2.58502e8 1.20726 0.603631 0.797264i \(-0.293720\pi\)
0.603631 + 0.797264i \(0.293720\pi\)
\(72\) 0 0
\(73\) −3.53583e8 −1.45726 −0.728632 0.684905i \(-0.759844\pi\)
−0.728632 + 0.684905i \(0.759844\pi\)
\(74\) 0 0
\(75\) −8.46781e7 −0.309026
\(76\) 0 0
\(77\) 5.71292e7 0.185204
\(78\) 0 0
\(79\) −3.66190e8 −1.05775 −0.528876 0.848699i \(-0.677386\pi\)
−0.528876 + 0.848699i \(0.677386\pi\)
\(80\) 0 0
\(81\) −1.79169e8 −0.462467
\(82\) 0 0
\(83\) 9.74953e6 0.0225493 0.0112746 0.999936i \(-0.496411\pi\)
0.0112746 + 0.999936i \(0.496411\pi\)
\(84\) 0 0
\(85\) −5.73726e7 −0.119212
\(86\) 0 0
\(87\) −1.57316e9 −2.94398
\(88\) 0 0
\(89\) −1.14201e9 −1.92937 −0.964687 0.263397i \(-0.915157\pi\)
−0.964687 + 0.263397i \(0.915157\pi\)
\(90\) 0 0
\(91\) −2.70380e8 −0.413322
\(92\) 0 0
\(93\) 1.78418e9 2.47323
\(94\) 0 0
\(95\) −3.35142e8 −0.422156
\(96\) 0 0
\(97\) 1.25555e9 1.44000 0.719999 0.693975i \(-0.244142\pi\)
0.719999 + 0.693975i \(0.244142\pi\)
\(98\) 0 0
\(99\) −9.38657e8 −0.982084
\(100\) 0 0
\(101\) 2.63563e8 0.252022 0.126011 0.992029i \(-0.459783\pi\)
0.126011 + 0.992029i \(0.459783\pi\)
\(102\) 0 0
\(103\) 2.09532e9 1.83435 0.917176 0.398482i \(-0.130463\pi\)
0.917176 + 0.398482i \(0.130463\pi\)
\(104\) 0 0
\(105\) −2.25188e8 −0.180798
\(106\) 0 0
\(107\) 3.30509e8 0.243757 0.121878 0.992545i \(-0.461108\pi\)
0.121878 + 0.992545i \(0.461108\pi\)
\(108\) 0 0
\(109\) −1.65293e9 −1.12160 −0.560798 0.827953i \(-0.689505\pi\)
−0.560798 + 0.827953i \(0.689505\pi\)
\(110\) 0 0
\(111\) 3.40014e9 2.12590
\(112\) 0 0
\(113\) 1.15695e9 0.667513 0.333757 0.942659i \(-0.391684\pi\)
0.333757 + 0.942659i \(0.391684\pi\)
\(114\) 0 0
\(115\) 1.46812e8 0.0782745
\(116\) 0 0
\(117\) 4.44246e9 2.19173
\(118\) 0 0
\(119\) −1.52573e8 −0.0697456
\(120\) 0 0
\(121\) −1.17652e9 −0.498957
\(122\) 0 0
\(123\) 3.62705e9 1.42883
\(124\) 0 0
\(125\) −2.44141e8 −0.0894427
\(126\) 0 0
\(127\) −1.56129e9 −0.532559 −0.266280 0.963896i \(-0.585794\pi\)
−0.266280 + 0.963896i \(0.585794\pi\)
\(128\) 0 0
\(129\) 2.02947e9 0.645253
\(130\) 0 0
\(131\) 4.75445e9 1.41052 0.705261 0.708948i \(-0.250830\pi\)
0.705261 + 0.708948i \(0.250830\pi\)
\(132\) 0 0
\(133\) −8.91256e8 −0.246985
\(134\) 0 0
\(135\) 1.03318e9 0.267716
\(136\) 0 0
\(137\) 1.57706e9 0.382478 0.191239 0.981543i \(-0.438749\pi\)
0.191239 + 0.981543i \(0.438749\pi\)
\(138\) 0 0
\(139\) −3.40723e9 −0.774167 −0.387084 0.922045i \(-0.626518\pi\)
−0.387084 + 0.922045i \(0.626518\pi\)
\(140\) 0 0
\(141\) −5.84015e9 −1.24434
\(142\) 0 0
\(143\) −5.59147e9 −1.11818
\(144\) 0 0
\(145\) −4.53567e9 −0.852089
\(146\) 0 0
\(147\) 8.14884e9 1.43935
\(148\) 0 0
\(149\) −4.20713e8 −0.0699274 −0.0349637 0.999389i \(-0.511132\pi\)
−0.0349637 + 0.999389i \(0.511132\pi\)
\(150\) 0 0
\(151\) 1.74377e9 0.272957 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(152\) 0 0
\(153\) 2.50684e9 0.369841
\(154\) 0 0
\(155\) 5.14407e9 0.715837
\(156\) 0 0
\(157\) −6.13142e9 −0.805402 −0.402701 0.915332i \(-0.631929\pi\)
−0.402701 + 0.915332i \(0.631929\pi\)
\(158\) 0 0
\(159\) −7.39434e9 −0.917513
\(160\) 0 0
\(161\) 3.90422e8 0.0457950
\(162\) 0 0
\(163\) −1.07953e10 −1.19782 −0.598910 0.800816i \(-0.704399\pi\)
−0.598910 + 0.800816i \(0.704399\pi\)
\(164\) 0 0
\(165\) −4.65689e9 −0.489123
\(166\) 0 0
\(167\) −1.55417e10 −1.54623 −0.773115 0.634266i \(-0.781302\pi\)
−0.773115 + 0.634266i \(0.781302\pi\)
\(168\) 0 0
\(169\) 1.58587e10 1.49547
\(170\) 0 0
\(171\) 1.46437e10 1.30969
\(172\) 0 0
\(173\) 3.21498e9 0.272880 0.136440 0.990648i \(-0.456434\pi\)
0.136440 + 0.990648i \(0.456434\pi\)
\(174\) 0 0
\(175\) −6.49253e8 −0.0523290
\(176\) 0 0
\(177\) 2.43125e10 1.86187
\(178\) 0 0
\(179\) −2.10234e10 −1.53061 −0.765305 0.643667i \(-0.777412\pi\)
−0.765305 + 0.643667i \(0.777412\pi\)
\(180\) 0 0
\(181\) −1.54514e10 −1.07008 −0.535038 0.844828i \(-0.679703\pi\)
−0.535038 + 0.844828i \(0.679703\pi\)
\(182\) 0 0
\(183\) −1.19125e9 −0.0785187
\(184\) 0 0
\(185\) 9.80315e9 0.615309
\(186\) 0 0
\(187\) −3.15521e9 −0.188687
\(188\) 0 0
\(189\) 2.74758e9 0.156629
\(190\) 0 0
\(191\) −2.53988e10 −1.38090 −0.690451 0.723380i \(-0.742588\pi\)
−0.690451 + 0.723380i \(0.742588\pi\)
\(192\) 0 0
\(193\) −1.70167e10 −0.882809 −0.441404 0.897308i \(-0.645520\pi\)
−0.441404 + 0.897308i \(0.645520\pi\)
\(194\) 0 0
\(195\) 2.20400e10 1.09158
\(196\) 0 0
\(197\) 4.38218e9 0.207297 0.103648 0.994614i \(-0.466948\pi\)
0.103648 + 0.994614i \(0.466948\pi\)
\(198\) 0 0
\(199\) 1.37237e10 0.620343 0.310171 0.950681i \(-0.399614\pi\)
0.310171 + 0.950681i \(0.399614\pi\)
\(200\) 0 0
\(201\) −5.21749e10 −2.25465
\(202\) 0 0
\(203\) −1.20619e10 −0.498520
\(204\) 0 0
\(205\) 1.04574e10 0.413552
\(206\) 0 0
\(207\) −6.41480e9 −0.242838
\(208\) 0 0
\(209\) −1.84312e10 −0.668183
\(210\) 0 0
\(211\) 1.21366e10 0.421526 0.210763 0.977537i \(-0.432405\pi\)
0.210763 + 0.977537i \(0.432405\pi\)
\(212\) 0 0
\(213\) −5.60370e10 −1.86538
\(214\) 0 0
\(215\) 5.85130e9 0.186758
\(216\) 0 0
\(217\) 1.36798e10 0.418805
\(218\) 0 0
\(219\) 7.66483e10 2.25166
\(220\) 0 0
\(221\) 1.49330e10 0.421095
\(222\) 0 0
\(223\) −6.20297e10 −1.67968 −0.839842 0.542830i \(-0.817353\pi\)
−0.839842 + 0.542830i \(0.817353\pi\)
\(224\) 0 0
\(225\) 1.06675e10 0.277486
\(226\) 0 0
\(227\) −3.51959e10 −0.879783 −0.439891 0.898051i \(-0.644983\pi\)
−0.439891 + 0.898051i \(0.644983\pi\)
\(228\) 0 0
\(229\) −3.83973e10 −0.922659 −0.461329 0.887229i \(-0.652627\pi\)
−0.461329 + 0.887229i \(0.652627\pi\)
\(230\) 0 0
\(231\) −1.23842e10 −0.286164
\(232\) 0 0
\(233\) −2.59156e10 −0.576049 −0.288024 0.957623i \(-0.592998\pi\)
−0.288024 + 0.957623i \(0.592998\pi\)
\(234\) 0 0
\(235\) −1.68381e10 −0.360153
\(236\) 0 0
\(237\) 7.93811e10 1.63437
\(238\) 0 0
\(239\) −5.76903e10 −1.14370 −0.571850 0.820358i \(-0.693774\pi\)
−0.571850 + 0.820358i \(0.693774\pi\)
\(240\) 0 0
\(241\) −4.59383e10 −0.877200 −0.438600 0.898682i \(-0.644525\pi\)
−0.438600 + 0.898682i \(0.644525\pi\)
\(242\) 0 0
\(243\) 7.13773e10 1.31320
\(244\) 0 0
\(245\) 2.34944e10 0.416598
\(246\) 0 0
\(247\) 8.72309e10 1.49119
\(248\) 0 0
\(249\) −2.11346e9 −0.0348416
\(250\) 0 0
\(251\) 6.33143e10 1.00686 0.503431 0.864035i \(-0.332071\pi\)
0.503431 + 0.864035i \(0.332071\pi\)
\(252\) 0 0
\(253\) 8.07393e9 0.123892
\(254\) 0 0
\(255\) 1.24370e10 0.184198
\(256\) 0 0
\(257\) 4.21910e10 0.603283 0.301642 0.953421i \(-0.402465\pi\)
0.301642 + 0.953421i \(0.402465\pi\)
\(258\) 0 0
\(259\) 2.60699e10 0.359990
\(260\) 0 0
\(261\) 1.98182e11 2.64351
\(262\) 0 0
\(263\) −5.01270e10 −0.646057 −0.323028 0.946389i \(-0.604701\pi\)
−0.323028 + 0.946389i \(0.604701\pi\)
\(264\) 0 0
\(265\) −2.13191e10 −0.265560
\(266\) 0 0
\(267\) 2.47561e11 2.98114
\(268\) 0 0
\(269\) 7.21552e10 0.840199 0.420099 0.907478i \(-0.361995\pi\)
0.420099 + 0.907478i \(0.361995\pi\)
\(270\) 0 0
\(271\) −5.23276e10 −0.589344 −0.294672 0.955598i \(-0.595210\pi\)
−0.294672 + 0.955598i \(0.595210\pi\)
\(272\) 0 0
\(273\) 5.86119e10 0.638637
\(274\) 0 0
\(275\) −1.34266e10 −0.141569
\(276\) 0 0
\(277\) −5.78954e10 −0.590860 −0.295430 0.955364i \(-0.595463\pi\)
−0.295430 + 0.955364i \(0.595463\pi\)
\(278\) 0 0
\(279\) −2.24765e11 −2.22081
\(280\) 0 0
\(281\) 7.31211e10 0.699623 0.349812 0.936820i \(-0.386246\pi\)
0.349812 + 0.936820i \(0.386246\pi\)
\(282\) 0 0
\(283\) 5.12368e10 0.474835 0.237418 0.971408i \(-0.423699\pi\)
0.237418 + 0.971408i \(0.423699\pi\)
\(284\) 0 0
\(285\) 7.26507e10 0.652286
\(286\) 0 0
\(287\) 2.78097e10 0.241951
\(288\) 0 0
\(289\) −1.10161e11 −0.928943
\(290\) 0 0
\(291\) −2.72173e11 −2.22499
\(292\) 0 0
\(293\) 1.95934e11 1.55312 0.776561 0.630043i \(-0.216963\pi\)
0.776561 + 0.630043i \(0.216963\pi\)
\(294\) 0 0
\(295\) 7.00968e10 0.538889
\(296\) 0 0
\(297\) 5.68200e10 0.423737
\(298\) 0 0
\(299\) −3.82122e10 −0.276491
\(300\) 0 0
\(301\) 1.55606e10 0.109264
\(302\) 0 0
\(303\) −5.71341e10 −0.389407
\(304\) 0 0
\(305\) −3.43457e9 −0.0227260
\(306\) 0 0
\(307\) 2.29564e11 1.47496 0.737481 0.675368i \(-0.236015\pi\)
0.737481 + 0.675368i \(0.236015\pi\)
\(308\) 0 0
\(309\) −4.54215e11 −2.83431
\(310\) 0 0
\(311\) −4.99151e10 −0.302559 −0.151280 0.988491i \(-0.548339\pi\)
−0.151280 + 0.988491i \(0.548339\pi\)
\(312\) 0 0
\(313\) −1.19272e11 −0.702408 −0.351204 0.936299i \(-0.614228\pi\)
−0.351204 + 0.936299i \(0.614228\pi\)
\(314\) 0 0
\(315\) 2.83685e10 0.162345
\(316\) 0 0
\(317\) −2.85879e11 −1.59007 −0.795034 0.606565i \(-0.792547\pi\)
−0.795034 + 0.606565i \(0.792547\pi\)
\(318\) 0 0
\(319\) −2.49440e11 −1.34868
\(320\) 0 0
\(321\) −7.16464e10 −0.376636
\(322\) 0 0
\(323\) 4.92236e10 0.251630
\(324\) 0 0
\(325\) 6.35450e10 0.315941
\(326\) 0 0
\(327\) 3.58316e11 1.73301
\(328\) 0 0
\(329\) −4.47782e10 −0.210710
\(330\) 0 0
\(331\) 2.12047e11 0.970973 0.485486 0.874244i \(-0.338643\pi\)
0.485486 + 0.874244i \(0.338643\pi\)
\(332\) 0 0
\(333\) −4.28340e11 −1.90893
\(334\) 0 0
\(335\) −1.50429e11 −0.652573
\(336\) 0 0
\(337\) 2.00430e11 0.846502 0.423251 0.906012i \(-0.360889\pi\)
0.423251 + 0.906012i \(0.360889\pi\)
\(338\) 0 0
\(339\) −2.50798e11 −1.03140
\(340\) 0 0
\(341\) 2.82899e11 1.13302
\(342\) 0 0
\(343\) 1.29551e11 0.505378
\(344\) 0 0
\(345\) −3.18252e10 −0.120944
\(346\) 0 0
\(347\) 9.86210e10 0.365163 0.182582 0.983191i \(-0.441555\pi\)
0.182582 + 0.983191i \(0.441555\pi\)
\(348\) 0 0
\(349\) −4.79143e11 −1.72882 −0.864411 0.502785i \(-0.832309\pi\)
−0.864411 + 0.502785i \(0.832309\pi\)
\(350\) 0 0
\(351\) −2.68917e11 −0.945662
\(352\) 0 0
\(353\) 4.56755e11 1.56566 0.782830 0.622235i \(-0.213775\pi\)
0.782830 + 0.622235i \(0.213775\pi\)
\(354\) 0 0
\(355\) −1.61564e11 −0.539904
\(356\) 0 0
\(357\) 3.30742e10 0.107766
\(358\) 0 0
\(359\) 1.54059e11 0.489509 0.244755 0.969585i \(-0.421293\pi\)
0.244755 + 0.969585i \(0.421293\pi\)
\(360\) 0 0
\(361\) −3.51479e10 −0.108922
\(362\) 0 0
\(363\) 2.55040e11 0.770954
\(364\) 0 0
\(365\) 2.20989e11 0.651708
\(366\) 0 0
\(367\) −3.42604e11 −0.985815 −0.492908 0.870082i \(-0.664066\pi\)
−0.492908 + 0.870082i \(0.664066\pi\)
\(368\) 0 0
\(369\) −4.56925e11 −1.28300
\(370\) 0 0
\(371\) −5.66947e10 −0.155367
\(372\) 0 0
\(373\) −1.68130e11 −0.449734 −0.224867 0.974389i \(-0.572195\pi\)
−0.224867 + 0.974389i \(0.572195\pi\)
\(374\) 0 0
\(375\) 5.29238e10 0.138201
\(376\) 0 0
\(377\) 1.18054e12 3.00986
\(378\) 0 0
\(379\) 1.00434e11 0.250038 0.125019 0.992154i \(-0.460101\pi\)
0.125019 + 0.992154i \(0.460101\pi\)
\(380\) 0 0
\(381\) 3.38451e11 0.822874
\(382\) 0 0
\(383\) −5.07533e11 −1.20523 −0.602615 0.798032i \(-0.705874\pi\)
−0.602615 + 0.798032i \(0.705874\pi\)
\(384\) 0 0
\(385\) −3.57058e10 −0.0828257
\(386\) 0 0
\(387\) −2.55667e11 −0.579396
\(388\) 0 0
\(389\) −1.26566e11 −0.280249 −0.140125 0.990134i \(-0.544750\pi\)
−0.140125 + 0.990134i \(0.544750\pi\)
\(390\) 0 0
\(391\) −2.15628e10 −0.0466562
\(392\) 0 0
\(393\) −1.03065e12 −2.17944
\(394\) 0 0
\(395\) 2.28868e11 0.473041
\(396\) 0 0
\(397\) −4.70667e11 −0.950948 −0.475474 0.879730i \(-0.657723\pi\)
−0.475474 + 0.879730i \(0.657723\pi\)
\(398\) 0 0
\(399\) 1.93203e11 0.381624
\(400\) 0 0
\(401\) −9.20810e11 −1.77836 −0.889182 0.457554i \(-0.848726\pi\)
−0.889182 + 0.457554i \(0.848726\pi\)
\(402\) 0 0
\(403\) −1.33890e12 −2.52857
\(404\) 0 0
\(405\) 1.11981e11 0.206821
\(406\) 0 0
\(407\) 5.39126e11 0.973902
\(408\) 0 0
\(409\) 8.23952e11 1.45595 0.727976 0.685603i \(-0.240461\pi\)
0.727976 + 0.685603i \(0.240461\pi\)
\(410\) 0 0
\(411\) −3.41870e11 −0.590979
\(412\) 0 0
\(413\) 1.86411e11 0.315280
\(414\) 0 0
\(415\) −6.09345e9 −0.0100843
\(416\) 0 0
\(417\) 7.38605e11 1.19619
\(418\) 0 0
\(419\) −7.01758e11 −1.11231 −0.556153 0.831080i \(-0.687723\pi\)
−0.556153 + 0.831080i \(0.687723\pi\)
\(420\) 0 0
\(421\) 1.22669e11 0.190311 0.0951555 0.995462i \(-0.469665\pi\)
0.0951555 + 0.995462i \(0.469665\pi\)
\(422\) 0 0
\(423\) 7.35725e11 1.11734
\(424\) 0 0
\(425\) 3.58579e10 0.0533131
\(426\) 0 0
\(427\) −9.13368e9 −0.0132960
\(428\) 0 0
\(429\) 1.21210e12 1.72774
\(430\) 0 0
\(431\) 4.82937e11 0.674129 0.337064 0.941482i \(-0.390566\pi\)
0.337064 + 0.941482i \(0.390566\pi\)
\(432\) 0 0
\(433\) 7.71175e11 1.05428 0.527142 0.849777i \(-0.323264\pi\)
0.527142 + 0.849777i \(0.323264\pi\)
\(434\) 0 0
\(435\) 9.83223e11 1.31659
\(436\) 0 0
\(437\) −1.25959e11 −0.165220
\(438\) 0 0
\(439\) 6.62301e11 0.851070 0.425535 0.904942i \(-0.360086\pi\)
0.425535 + 0.904942i \(0.360086\pi\)
\(440\) 0 0
\(441\) −1.02657e12 −1.29245
\(442\) 0 0
\(443\) 3.76774e11 0.464798 0.232399 0.972621i \(-0.425343\pi\)
0.232399 + 0.972621i \(0.425343\pi\)
\(444\) 0 0
\(445\) 7.13759e11 0.862843
\(446\) 0 0
\(447\) 9.12004e10 0.108047
\(448\) 0 0
\(449\) 5.77774e11 0.670888 0.335444 0.942060i \(-0.391114\pi\)
0.335444 + 0.942060i \(0.391114\pi\)
\(450\) 0 0
\(451\) 5.75105e11 0.654564
\(452\) 0 0
\(453\) −3.78008e11 −0.421754
\(454\) 0 0
\(455\) 1.68988e11 0.184843
\(456\) 0 0
\(457\) −8.28175e11 −0.888176 −0.444088 0.895983i \(-0.646472\pi\)
−0.444088 + 0.895983i \(0.646472\pi\)
\(458\) 0 0
\(459\) −1.51747e11 −0.159575
\(460\) 0 0
\(461\) 2.77554e11 0.286216 0.143108 0.989707i \(-0.454290\pi\)
0.143108 + 0.989707i \(0.454290\pi\)
\(462\) 0 0
\(463\) 1.01088e11 0.102232 0.0511159 0.998693i \(-0.483722\pi\)
0.0511159 + 0.998693i \(0.483722\pi\)
\(464\) 0 0
\(465\) −1.11511e12 −1.10606
\(466\) 0 0
\(467\) 1.30742e12 1.27201 0.636004 0.771686i \(-0.280586\pi\)
0.636004 + 0.771686i \(0.280586\pi\)
\(468\) 0 0
\(469\) −4.00041e11 −0.381792
\(470\) 0 0
\(471\) 1.32914e12 1.24445
\(472\) 0 0
\(473\) 3.21793e11 0.295598
\(474\) 0 0
\(475\) 2.09464e11 0.188794
\(476\) 0 0
\(477\) 9.31518e11 0.823869
\(478\) 0 0
\(479\) 1.38500e12 1.20210 0.601049 0.799212i \(-0.294750\pi\)
0.601049 + 0.799212i \(0.294750\pi\)
\(480\) 0 0
\(481\) −2.55157e12 −2.17347
\(482\) 0 0
\(483\) −8.46341e10 −0.0707593
\(484\) 0 0
\(485\) −7.84720e11 −0.643987
\(486\) 0 0
\(487\) −9.54305e11 −0.768789 −0.384394 0.923169i \(-0.625590\pi\)
−0.384394 + 0.923169i \(0.625590\pi\)
\(488\) 0 0
\(489\) 2.34017e12 1.85079
\(490\) 0 0
\(491\) 7.41251e11 0.575571 0.287785 0.957695i \(-0.407081\pi\)
0.287785 + 0.957695i \(0.407081\pi\)
\(492\) 0 0
\(493\) 6.66171e11 0.507895
\(494\) 0 0
\(495\) 5.86661e11 0.439201
\(496\) 0 0
\(497\) −4.29653e11 −0.315874
\(498\) 0 0
\(499\) −1.51393e12 −1.09308 −0.546542 0.837432i \(-0.684056\pi\)
−0.546542 + 0.837432i \(0.684056\pi\)
\(500\) 0 0
\(501\) 3.36906e12 2.38913
\(502\) 0 0
\(503\) −1.58191e12 −1.10186 −0.550929 0.834552i \(-0.685726\pi\)
−0.550929 + 0.834552i \(0.685726\pi\)
\(504\) 0 0
\(505\) −1.64727e11 −0.112708
\(506\) 0 0
\(507\) −3.43779e12 −2.31070
\(508\) 0 0
\(509\) −2.29553e11 −0.151584 −0.0757918 0.997124i \(-0.524148\pi\)
−0.0757918 + 0.997124i \(0.524148\pi\)
\(510\) 0 0
\(511\) 5.87686e11 0.381286
\(512\) 0 0
\(513\) −8.86432e11 −0.565089
\(514\) 0 0
\(515\) −1.30957e12 −0.820347
\(516\) 0 0
\(517\) −9.26014e11 −0.570046
\(518\) 0 0
\(519\) −6.96931e11 −0.421635
\(520\) 0 0
\(521\) −1.73274e12 −1.03030 −0.515149 0.857101i \(-0.672263\pi\)
−0.515149 + 0.857101i \(0.672263\pi\)
\(522\) 0 0
\(523\) −2.45691e12 −1.43592 −0.717961 0.696083i \(-0.754925\pi\)
−0.717961 + 0.696083i \(0.754925\pi\)
\(524\) 0 0
\(525\) 1.40742e11 0.0808552
\(526\) 0 0
\(527\) −7.55529e11 −0.426681
\(528\) 0 0
\(529\) −1.74598e12 −0.969366
\(530\) 0 0
\(531\) −3.06281e12 −1.67184
\(532\) 0 0
\(533\) −2.72185e12 −1.46080
\(534\) 0 0
\(535\) −2.06568e11 −0.109011
\(536\) 0 0
\(537\) 4.55737e12 2.36499
\(538\) 0 0
\(539\) 1.29208e12 0.659386
\(540\) 0 0
\(541\) −2.10080e12 −1.05438 −0.527190 0.849748i \(-0.676754\pi\)
−0.527190 + 0.849748i \(0.676754\pi\)
\(542\) 0 0
\(543\) 3.34949e12 1.65341
\(544\) 0 0
\(545\) 1.03308e12 0.501593
\(546\) 0 0
\(547\) −3.58621e12 −1.71275 −0.856373 0.516357i \(-0.827288\pi\)
−0.856373 + 0.516357i \(0.827288\pi\)
\(548\) 0 0
\(549\) 1.50070e11 0.0705048
\(550\) 0 0
\(551\) 3.89144e12 1.79857
\(552\) 0 0
\(553\) 6.08639e11 0.276756
\(554\) 0 0
\(555\) −2.12509e12 −0.950733
\(556\) 0 0
\(557\) −3.15166e12 −1.38737 −0.693683 0.720280i \(-0.744013\pi\)
−0.693683 + 0.720280i \(0.744013\pi\)
\(558\) 0 0
\(559\) −1.52298e12 −0.659691
\(560\) 0 0
\(561\) 6.83975e11 0.291546
\(562\) 0 0
\(563\) 3.70986e12 1.55621 0.778107 0.628131i \(-0.216180\pi\)
0.778107 + 0.628131i \(0.216180\pi\)
\(564\) 0 0
\(565\) −7.23091e11 −0.298521
\(566\) 0 0
\(567\) 2.97795e11 0.121002
\(568\) 0 0
\(569\) −1.52350e12 −0.609308 −0.304654 0.952463i \(-0.598541\pi\)
−0.304654 + 0.952463i \(0.598541\pi\)
\(570\) 0 0
\(571\) −2.81914e12 −1.10982 −0.554911 0.831909i \(-0.687248\pi\)
−0.554911 + 0.831909i \(0.687248\pi\)
\(572\) 0 0
\(573\) 5.50584e12 2.13367
\(574\) 0 0
\(575\) −9.17573e10 −0.0350054
\(576\) 0 0
\(577\) −2.80815e11 −0.105470 −0.0527350 0.998609i \(-0.516794\pi\)
−0.0527350 + 0.998609i \(0.516794\pi\)
\(578\) 0 0
\(579\) 3.68880e12 1.36406
\(580\) 0 0
\(581\) −1.62046e10 −0.00589990
\(582\) 0 0
\(583\) −1.17245e12 −0.420324
\(584\) 0 0
\(585\) −2.77654e12 −0.980173
\(586\) 0 0
\(587\) −1.71929e12 −0.597691 −0.298845 0.954302i \(-0.596601\pi\)
−0.298845 + 0.954302i \(0.596601\pi\)
\(588\) 0 0
\(589\) −4.41343e12 −1.51097
\(590\) 0 0
\(591\) −9.49952e11 −0.320301
\(592\) 0 0
\(593\) 4.82940e12 1.60379 0.801895 0.597465i \(-0.203825\pi\)
0.801895 + 0.597465i \(0.203825\pi\)
\(594\) 0 0
\(595\) 9.53582e10 0.0311912
\(596\) 0 0
\(597\) −2.97496e12 −0.958511
\(598\) 0 0
\(599\) −1.72589e12 −0.547763 −0.273881 0.961763i \(-0.588308\pi\)
−0.273881 + 0.961763i \(0.588308\pi\)
\(600\) 0 0
\(601\) 5.06692e12 1.58420 0.792098 0.610394i \(-0.208989\pi\)
0.792098 + 0.610394i \(0.208989\pi\)
\(602\) 0 0
\(603\) 6.57284e12 2.02453
\(604\) 0 0
\(605\) 7.35322e11 0.223140
\(606\) 0 0
\(607\) 1.57706e12 0.471520 0.235760 0.971811i \(-0.424242\pi\)
0.235760 + 0.971811i \(0.424242\pi\)
\(608\) 0 0
\(609\) 2.61472e12 0.770279
\(610\) 0 0
\(611\) 4.38263e12 1.27218
\(612\) 0 0
\(613\) −1.35557e12 −0.387747 −0.193874 0.981026i \(-0.562105\pi\)
−0.193874 + 0.981026i \(0.562105\pi\)
\(614\) 0 0
\(615\) −2.26691e12 −0.638992
\(616\) 0 0
\(617\) −4.35935e12 −1.21098 −0.605492 0.795852i \(-0.707023\pi\)
−0.605492 + 0.795852i \(0.707023\pi\)
\(618\) 0 0
\(619\) −1.67780e12 −0.459338 −0.229669 0.973269i \(-0.573764\pi\)
−0.229669 + 0.973269i \(0.573764\pi\)
\(620\) 0 0
\(621\) 3.88309e11 0.104777
\(622\) 0 0
\(623\) 1.89813e12 0.504811
\(624\) 0 0
\(625\) 1.52588e11 0.0400000
\(626\) 0 0
\(627\) 3.99544e12 1.03243
\(628\) 0 0
\(629\) −1.43983e12 −0.366760
\(630\) 0 0
\(631\) −4.12439e12 −1.03568 −0.517842 0.855476i \(-0.673264\pi\)
−0.517842 + 0.855476i \(0.673264\pi\)
\(632\) 0 0
\(633\) −2.63091e12 −0.651313
\(634\) 0 0
\(635\) 9.75808e11 0.238168
\(636\) 0 0
\(637\) −6.11514e12 −1.47156
\(638\) 0 0
\(639\) 7.05938e12 1.67499
\(640\) 0 0
\(641\) 2.54925e12 0.596420 0.298210 0.954500i \(-0.403610\pi\)
0.298210 + 0.954500i \(0.403610\pi\)
\(642\) 0 0
\(643\) 1.49409e11 0.0344688 0.0172344 0.999851i \(-0.494514\pi\)
0.0172344 + 0.999851i \(0.494514\pi\)
\(644\) 0 0
\(645\) −1.26842e12 −0.288566
\(646\) 0 0
\(647\) 7.88552e11 0.176913 0.0884567 0.996080i \(-0.471806\pi\)
0.0884567 + 0.996080i \(0.471806\pi\)
\(648\) 0 0
\(649\) 3.85498e12 0.852946
\(650\) 0 0
\(651\) −2.96546e12 −0.647109
\(652\) 0 0
\(653\) 1.87557e12 0.403668 0.201834 0.979420i \(-0.435310\pi\)
0.201834 + 0.979420i \(0.435310\pi\)
\(654\) 0 0
\(655\) −2.97153e12 −0.630805
\(656\) 0 0
\(657\) −9.65592e12 −2.02185
\(658\) 0 0
\(659\) 2.02453e11 0.0418158 0.0209079 0.999781i \(-0.493344\pi\)
0.0209079 + 0.999781i \(0.493344\pi\)
\(660\) 0 0
\(661\) 2.41063e12 0.491160 0.245580 0.969376i \(-0.421022\pi\)
0.245580 + 0.969376i \(0.421022\pi\)
\(662\) 0 0
\(663\) −3.23711e12 −0.650648
\(664\) 0 0
\(665\) 5.57035e11 0.110455
\(666\) 0 0
\(667\) −1.70468e12 −0.333484
\(668\) 0 0
\(669\) 1.34465e13 2.59533
\(670\) 0 0
\(671\) −1.88885e11 −0.0359704
\(672\) 0 0
\(673\) −5.63582e11 −0.105898 −0.0529491 0.998597i \(-0.516862\pi\)
−0.0529491 + 0.998597i \(0.516862\pi\)
\(674\) 0 0
\(675\) −6.45738e11 −0.119726
\(676\) 0 0
\(677\) 8.43518e11 0.154328 0.0771641 0.997018i \(-0.475413\pi\)
0.0771641 + 0.997018i \(0.475413\pi\)
\(678\) 0 0
\(679\) −2.08684e12 −0.376768
\(680\) 0 0
\(681\) 7.62962e12 1.35938
\(682\) 0 0
\(683\) −5.43927e12 −0.956418 −0.478209 0.878246i \(-0.658714\pi\)
−0.478209 + 0.878246i \(0.658714\pi\)
\(684\) 0 0
\(685\) −9.85665e11 −0.171050
\(686\) 0 0
\(687\) 8.32361e12 1.42563
\(688\) 0 0
\(689\) 5.54894e12 0.938044
\(690\) 0 0
\(691\) 1.41900e12 0.236773 0.118386 0.992968i \(-0.462228\pi\)
0.118386 + 0.992968i \(0.462228\pi\)
\(692\) 0 0
\(693\) 1.56013e12 0.256957
\(694\) 0 0
\(695\) 2.12952e12 0.346218
\(696\) 0 0
\(697\) −1.53591e12 −0.246501
\(698\) 0 0
\(699\) 5.61787e12 0.890071
\(700\) 0 0
\(701\) 9.10116e12 1.42353 0.711763 0.702419i \(-0.247897\pi\)
0.711763 + 0.702419i \(0.247897\pi\)
\(702\) 0 0
\(703\) −8.41075e12 −1.29878
\(704\) 0 0
\(705\) 3.65009e12 0.556484
\(706\) 0 0
\(707\) −4.38064e11 −0.0659402
\(708\) 0 0
\(709\) 3.00283e12 0.446296 0.223148 0.974785i \(-0.428367\pi\)
0.223148 + 0.974785i \(0.428367\pi\)
\(710\) 0 0
\(711\) −1.00002e13 −1.46756
\(712\) 0 0
\(713\) 1.93334e12 0.280159
\(714\) 0 0
\(715\) 3.49467e12 0.500068
\(716\) 0 0
\(717\) 1.25059e13 1.76717
\(718\) 0 0
\(719\) 8.40313e12 1.17263 0.586315 0.810083i \(-0.300578\pi\)
0.586315 + 0.810083i \(0.300578\pi\)
\(720\) 0 0
\(721\) −3.48260e12 −0.479949
\(722\) 0 0
\(723\) 9.95833e12 1.35539
\(724\) 0 0
\(725\) 2.83479e12 0.381066
\(726\) 0 0
\(727\) −6.70907e12 −0.890753 −0.445377 0.895343i \(-0.646930\pi\)
−0.445377 + 0.895343i \(0.646930\pi\)
\(728\) 0 0
\(729\) −1.19463e13 −1.56660
\(730\) 0 0
\(731\) −8.59403e11 −0.111319
\(732\) 0 0
\(733\) 1.50813e12 0.192962 0.0964810 0.995335i \(-0.469241\pi\)
0.0964810 + 0.995335i \(0.469241\pi\)
\(734\) 0 0
\(735\) −5.09302e12 −0.643699
\(736\) 0 0
\(737\) −8.27285e12 −1.03288
\(738\) 0 0
\(739\) 1.48289e13 1.82898 0.914492 0.404603i \(-0.132590\pi\)
0.914492 + 0.404603i \(0.132590\pi\)
\(740\) 0 0
\(741\) −1.89096e13 −2.30409
\(742\) 0 0
\(743\) 1.91312e12 0.230299 0.115150 0.993348i \(-0.463265\pi\)
0.115150 + 0.993348i \(0.463265\pi\)
\(744\) 0 0
\(745\) 2.62945e11 0.0312725
\(746\) 0 0
\(747\) 2.66248e11 0.0312855
\(748\) 0 0
\(749\) −5.49335e11 −0.0637777
\(750\) 0 0
\(751\) 6.50309e12 0.746002 0.373001 0.927831i \(-0.378329\pi\)
0.373001 + 0.927831i \(0.378329\pi\)
\(752\) 0 0
\(753\) −1.37250e13 −1.55573
\(754\) 0 0
\(755\) −1.08986e12 −0.122070
\(756\) 0 0
\(757\) −1.11069e11 −0.0122931 −0.00614653 0.999981i \(-0.501957\pi\)
−0.00614653 + 0.999981i \(0.501957\pi\)
\(758\) 0 0
\(759\) −1.75023e12 −0.191429
\(760\) 0 0
\(761\) 1.21294e13 1.31102 0.655510 0.755186i \(-0.272454\pi\)
0.655510 + 0.755186i \(0.272454\pi\)
\(762\) 0 0
\(763\) 2.74732e12 0.293460
\(764\) 0 0
\(765\) −1.56678e12 −0.165398
\(766\) 0 0
\(767\) −1.82448e13 −1.90353
\(768\) 0 0
\(769\) −1.19313e13 −1.23033 −0.615164 0.788399i \(-0.710910\pi\)
−0.615164 + 0.788399i \(0.710910\pi\)
\(770\) 0 0
\(771\) −9.14600e12 −0.932152
\(772\) 0 0
\(773\) −1.71471e13 −1.72736 −0.863681 0.504038i \(-0.831847\pi\)
−0.863681 + 0.504038i \(0.831847\pi\)
\(774\) 0 0
\(775\) −3.21504e12 −0.320132
\(776\) 0 0
\(777\) −5.65133e12 −0.556232
\(778\) 0 0
\(779\) −8.97204e12 −0.872917
\(780\) 0 0
\(781\) −8.88523e12 −0.854553
\(782\) 0 0
\(783\) −1.19966e13 −1.14059
\(784\) 0 0
\(785\) 3.83214e12 0.360187
\(786\) 0 0
\(787\) 2.12129e11 0.0197112 0.00985561 0.999951i \(-0.496863\pi\)
0.00985561 + 0.999951i \(0.496863\pi\)
\(788\) 0 0
\(789\) 1.08663e13 0.998242
\(790\) 0 0
\(791\) −1.92294e12 −0.174652
\(792\) 0 0
\(793\) 8.93950e11 0.0802757
\(794\) 0 0
\(795\) 4.62146e12 0.410325
\(796\) 0 0
\(797\) 9.75044e12 0.855976 0.427988 0.903784i \(-0.359223\pi\)
0.427988 + 0.903784i \(0.359223\pi\)
\(798\) 0 0
\(799\) 2.47308e12 0.214673
\(800\) 0 0
\(801\) −3.11870e13 −2.67687
\(802\) 0 0
\(803\) 1.21533e13 1.03152
\(804\) 0 0
\(805\) −2.44014e11 −0.0204801
\(806\) 0 0
\(807\) −1.56415e13 −1.29822
\(808\) 0 0
\(809\) 1.00162e13 0.822120 0.411060 0.911608i \(-0.365159\pi\)
0.411060 + 0.911608i \(0.365159\pi\)
\(810\) 0 0
\(811\) −1.09433e11 −0.00888287 −0.00444143 0.999990i \(-0.501414\pi\)
−0.00444143 + 0.999990i \(0.501414\pi\)
\(812\) 0 0
\(813\) 1.13434e13 0.910614
\(814\) 0 0
\(815\) 6.74708e12 0.535681
\(816\) 0 0
\(817\) −5.02020e12 −0.394205
\(818\) 0 0
\(819\) −7.38376e12 −0.573456
\(820\) 0 0
\(821\) −2.10914e13 −1.62017 −0.810086 0.586312i \(-0.800579\pi\)
−0.810086 + 0.586312i \(0.800579\pi\)
\(822\) 0 0
\(823\) −6.07029e12 −0.461222 −0.230611 0.973046i \(-0.574073\pi\)
−0.230611 + 0.973046i \(0.574073\pi\)
\(824\) 0 0
\(825\) 2.91055e12 0.218742
\(826\) 0 0
\(827\) 1.09879e13 0.816846 0.408423 0.912793i \(-0.366079\pi\)
0.408423 + 0.912793i \(0.366079\pi\)
\(828\) 0 0
\(829\) −1.34601e13 −0.989811 −0.494906 0.868947i \(-0.664797\pi\)
−0.494906 + 0.868947i \(0.664797\pi\)
\(830\) 0 0
\(831\) 1.25503e13 0.912957
\(832\) 0 0
\(833\) −3.45072e12 −0.248317
\(834\) 0 0
\(835\) 9.71355e12 0.691495
\(836\) 0 0
\(837\) 1.36058e13 0.958206
\(838\) 0 0
\(839\) 4.34425e12 0.302681 0.151341 0.988482i \(-0.451641\pi\)
0.151341 + 0.988482i \(0.451641\pi\)
\(840\) 0 0
\(841\) 3.81578e13 2.63028
\(842\) 0 0
\(843\) −1.58509e13 −1.08101
\(844\) 0 0
\(845\) −9.91170e12 −0.668795
\(846\) 0 0
\(847\) 1.95547e12 0.130550
\(848\) 0 0
\(849\) −1.11069e13 −0.733683
\(850\) 0 0
\(851\) 3.68440e12 0.240815
\(852\) 0 0
\(853\) −1.33638e13 −0.864290 −0.432145 0.901804i \(-0.642243\pi\)
−0.432145 + 0.901804i \(0.642243\pi\)
\(854\) 0 0
\(855\) −9.15233e12 −0.585712
\(856\) 0 0
\(857\) 2.60497e13 1.64964 0.824819 0.565397i \(-0.191277\pi\)
0.824819 + 0.565397i \(0.191277\pi\)
\(858\) 0 0
\(859\) −2.01187e12 −0.126076 −0.0630379 0.998011i \(-0.520079\pi\)
−0.0630379 + 0.998011i \(0.520079\pi\)
\(860\) 0 0
\(861\) −6.02847e12 −0.373846
\(862\) 0 0
\(863\) −2.36870e13 −1.45366 −0.726828 0.686819i \(-0.759006\pi\)
−0.726828 + 0.686819i \(0.759006\pi\)
\(864\) 0 0
\(865\) −2.00936e12 −0.122036
\(866\) 0 0
\(867\) 2.38803e13 1.43534
\(868\) 0 0
\(869\) 1.25867e13 0.748723
\(870\) 0 0
\(871\) 3.91536e13 2.30510
\(872\) 0 0
\(873\) 3.42876e13 1.99790
\(874\) 0 0
\(875\) 4.05783e11 0.0234022
\(876\) 0 0
\(877\) 1.83122e13 1.04531 0.522653 0.852545i \(-0.324942\pi\)
0.522653 + 0.852545i \(0.324942\pi\)
\(878\) 0 0
\(879\) −4.24738e13 −2.39978
\(880\) 0 0
\(881\) 3.34824e13 1.87251 0.936256 0.351320i \(-0.114267\pi\)
0.936256 + 0.351320i \(0.114267\pi\)
\(882\) 0 0
\(883\) 1.73556e13 0.960763 0.480382 0.877060i \(-0.340498\pi\)
0.480382 + 0.877060i \(0.340498\pi\)
\(884\) 0 0
\(885\) −1.51953e13 −0.832654
\(886\) 0 0
\(887\) 1.34047e13 0.727109 0.363555 0.931573i \(-0.381563\pi\)
0.363555 + 0.931573i \(0.381563\pi\)
\(888\) 0 0
\(889\) 2.59501e12 0.139341
\(890\) 0 0
\(891\) 6.15839e12 0.327354
\(892\) 0 0
\(893\) 1.44465e13 0.760204
\(894\) 0 0
\(895\) 1.31396e13 0.684510
\(896\) 0 0
\(897\) 8.28348e12 0.427216
\(898\) 0 0
\(899\) −5.97294e13 −3.04979
\(900\) 0 0
\(901\) 3.13122e12 0.158289
\(902\) 0 0
\(903\) −3.37316e12 −0.168827
\(904\) 0 0
\(905\) 9.65713e12 0.478552
\(906\) 0 0
\(907\) −9.01114e12 −0.442127 −0.221064 0.975259i \(-0.570953\pi\)
−0.221064 + 0.975259i \(0.570953\pi\)
\(908\) 0 0
\(909\) 7.19758e12 0.349663
\(910\) 0 0
\(911\) −7.64836e12 −0.367905 −0.183952 0.982935i \(-0.558889\pi\)
−0.183952 + 0.982935i \(0.558889\pi\)
\(912\) 0 0
\(913\) −3.35110e11 −0.0159613
\(914\) 0 0
\(915\) 7.44531e11 0.0351146
\(916\) 0 0
\(917\) −7.90232e12 −0.369056
\(918\) 0 0
\(919\) −3.19451e13 −1.47735 −0.738676 0.674060i \(-0.764549\pi\)
−0.738676 + 0.674060i \(0.764549\pi\)
\(920\) 0 0
\(921\) −4.97639e13 −2.27901
\(922\) 0 0
\(923\) 4.20519e13 1.90712
\(924\) 0 0
\(925\) −6.12697e12 −0.275174
\(926\) 0 0
\(927\) 5.72207e13 2.54504
\(928\) 0 0
\(929\) −1.36029e13 −0.599186 −0.299593 0.954067i \(-0.596851\pi\)
−0.299593 + 0.954067i \(0.596851\pi\)
\(930\) 0 0
\(931\) −2.01574e13 −0.879347
\(932\) 0 0
\(933\) 1.08204e13 0.467494
\(934\) 0 0
\(935\) 1.97201e12 0.0843833
\(936\) 0 0
\(937\) −1.10891e13 −0.469966 −0.234983 0.971999i \(-0.575503\pi\)
−0.234983 + 0.971999i \(0.575503\pi\)
\(938\) 0 0
\(939\) 2.58553e13 1.08531
\(940\) 0 0
\(941\) −2.46259e13 −1.02385 −0.511927 0.859029i \(-0.671068\pi\)
−0.511927 + 0.859029i \(0.671068\pi\)
\(942\) 0 0
\(943\) 3.93028e12 0.161853
\(944\) 0 0
\(945\) −1.71724e12 −0.0700466
\(946\) 0 0
\(947\) −4.82162e13 −1.94813 −0.974065 0.226267i \(-0.927348\pi\)
−0.974065 + 0.226267i \(0.927348\pi\)
\(948\) 0 0
\(949\) −5.75192e13 −2.30205
\(950\) 0 0
\(951\) 6.19717e13 2.45686
\(952\) 0 0
\(953\) −3.00142e12 −0.117872 −0.0589358 0.998262i \(-0.518771\pi\)
−0.0589358 + 0.998262i \(0.518771\pi\)
\(954\) 0 0
\(955\) 1.58742e13 0.617558
\(956\) 0 0
\(957\) 5.40725e13 2.08388
\(958\) 0 0
\(959\) −2.62122e12 −0.100074
\(960\) 0 0
\(961\) 4.13017e13 1.56211
\(962\) 0 0
\(963\) 9.02581e12 0.338195
\(964\) 0 0
\(965\) 1.06354e13 0.394804
\(966\) 0 0
\(967\) 1.36956e13 0.503690 0.251845 0.967768i \(-0.418963\pi\)
0.251845 + 0.967768i \(0.418963\pi\)
\(968\) 0 0
\(969\) −1.06705e13 −0.388801
\(970\) 0 0
\(971\) −2.83646e12 −0.102398 −0.0511988 0.998688i \(-0.516304\pi\)
−0.0511988 + 0.998688i \(0.516304\pi\)
\(972\) 0 0
\(973\) 5.66311e12 0.202557
\(974\) 0 0
\(975\) −1.37750e13 −0.488171
\(976\) 0 0
\(977\) −9.66226e12 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(978\) 0 0
\(979\) 3.92533e13 1.36570
\(980\) 0 0
\(981\) −4.51396e13 −1.55614
\(982\) 0 0
\(983\) 1.05400e13 0.360039 0.180019 0.983663i \(-0.442384\pi\)
0.180019 + 0.983663i \(0.442384\pi\)
\(984\) 0 0
\(985\) −2.73886e12 −0.0927059
\(986\) 0 0
\(987\) 9.70684e12 0.325575
\(988\) 0 0
\(989\) 2.19914e12 0.0730920
\(990\) 0 0
\(991\) −3.01597e13 −0.993335 −0.496668 0.867941i \(-0.665443\pi\)
−0.496668 + 0.867941i \(0.665443\pi\)
\(992\) 0 0
\(993\) −4.59668e13 −1.50028
\(994\) 0 0
\(995\) −8.57730e12 −0.277426
\(996\) 0 0
\(997\) 1.98318e13 0.635675 0.317837 0.948145i \(-0.397043\pi\)
0.317837 + 0.948145i \(0.397043\pi\)
\(998\) 0 0
\(999\) 2.59288e13 0.823640
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 80.10.a.g.1.1 2
4.3 odd 2 40.10.a.c.1.2 2
5.2 odd 4 400.10.c.m.49.4 4
5.3 odd 4 400.10.c.m.49.1 4
5.4 even 2 400.10.a.r.1.2 2
8.3 odd 2 320.10.a.n.1.1 2
8.5 even 2 320.10.a.q.1.2 2
12.11 even 2 360.10.a.g.1.2 2
20.3 even 4 200.10.c.d.49.4 4
20.7 even 4 200.10.c.d.49.1 4
20.19 odd 2 200.10.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.10.a.c.1.2 2 4.3 odd 2
80.10.a.g.1.1 2 1.1 even 1 trivial
200.10.a.c.1.1 2 20.19 odd 2
200.10.c.d.49.1 4 20.7 even 4
200.10.c.d.49.4 4 20.3 even 4
320.10.a.n.1.1 2 8.3 odd 2
320.10.a.q.1.2 2 8.5 even 2
360.10.a.g.1.2 2 12.11 even 2
400.10.a.r.1.2 2 5.4 even 2
400.10.c.m.49.1 4 5.3 odd 4
400.10.c.m.49.4 4 5.2 odd 4