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Properties

Label 12.8.a.a.1.1
Level $12$
Weight $8$
Character 12.1
Self dual yes
Analytic conductor $3.749$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [12,8,Mod(1,12)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 12.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-27] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.74862030581\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 12.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} -378.000 q^{5} -832.000 q^{7} +729.000 q^{9} -2484.00 q^{11} +14870.0 q^{13} +10206.0 q^{15} -22302.0 q^{17} -16300.0 q^{19} +22464.0 q^{21} -115128. q^{23} +64759.0 q^{25} -19683.0 q^{27} +157086. q^{29} -16456.0 q^{31} +67068.0 q^{33} +314496. q^{35} -149266. q^{37} -401490. q^{39} -241110. q^{41} -443188. q^{43} -275562. q^{45} +922752. q^{47} -131319. q^{49} +602154. q^{51} -697626. q^{53} +938952. q^{55} +440100. q^{57} +870156. q^{59} +2.06706e6 q^{61} -606528. q^{63} -5.62086e6 q^{65} -1.68075e6 q^{67} +3.10846e6 q^{69} -1.07028e6 q^{71} -2.40333e6 q^{73} -1.74849e6 q^{75} +2.06669e6 q^{77} +2.30151e6 q^{79} +531441. q^{81} +4.70869e6 q^{83} +8.43016e6 q^{85} -4.24132e6 q^{87} +4.14369e6 q^{89} -1.23718e7 q^{91} +444312. q^{93} +6.16140e6 q^{95} -1.62297e6 q^{97} -1.81084e6 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −27.0000 −0.577350
\(4\) 0 0
\(5\) −378.000 −1.35237 −0.676187 − 0.736730i \(-0.736369\pi\)
−0.676187 + 0.736730i \(0.736369\pi\)
\(6\) 0 0
\(7\) −832.000 −0.916812 −0.458406 − 0.888743i \(-0.651579\pi\)
−0.458406 + 0.888743i \(0.651579\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −2484.00 −0.562700 −0.281350 − 0.959605i \(-0.590782\pi\)
−0.281350 + 0.959605i \(0.590782\pi\)
\(12\) 0 0
\(13\) 14870.0 1.87719 0.938597 − 0.345015i \(-0.112126\pi\)
0.938597 + 0.345015i \(0.112126\pi\)
\(14\) 0 0
\(15\) 10206.0 0.780793
\(16\) 0 0
\(17\) −22302.0 −1.10096 −0.550481 − 0.834847i \(-0.685556\pi\)
−0.550481 + 0.834847i \(0.685556\pi\)
\(18\) 0 0
\(19\) −16300.0 −0.545193 −0.272596 − 0.962128i \(-0.587882\pi\)
−0.272596 + 0.962128i \(0.587882\pi\)
\(20\) 0 0
\(21\) 22464.0 0.529322
\(22\) 0 0
\(23\) −115128. −1.97303 −0.986515 − 0.163673i \(-0.947666\pi\)
−0.986515 + 0.163673i \(0.947666\pi\)
\(24\) 0 0
\(25\) 64759.0 0.828915
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) 157086. 1.19604 0.598018 − 0.801482i \(-0.295955\pi\)
0.598018 + 0.801482i \(0.295955\pi\)
\(30\) 0 0
\(31\) −16456.0 −0.0992107 −0.0496053 − 0.998769i \(-0.515796\pi\)
−0.0496053 + 0.998769i \(0.515796\pi\)
\(32\) 0 0
\(33\) 67068.0 0.324875
\(34\) 0 0
\(35\) 314496. 1.23987
\(36\) 0 0
\(37\) −149266. −0.484457 −0.242228 − 0.970219i \(-0.577878\pi\)
−0.242228 + 0.970219i \(0.577878\pi\)
\(38\) 0 0
\(39\) −401490. −1.08380
\(40\) 0 0
\(41\) −241110. −0.546351 −0.273175 − 0.961964i \(-0.588074\pi\)
−0.273175 + 0.961964i \(0.588074\pi\)
\(42\) 0 0
\(43\) −443188. −0.850058 −0.425029 − 0.905180i \(-0.639736\pi\)
−0.425029 + 0.905180i \(0.639736\pi\)
\(44\) 0 0
\(45\) −275562. −0.450791
\(46\) 0 0
\(47\) 922752. 1.29641 0.648205 − 0.761466i \(-0.275520\pi\)
0.648205 + 0.761466i \(0.275520\pi\)
\(48\) 0 0
\(49\) −131319. −0.159456
\(50\) 0 0
\(51\) 602154. 0.635641
\(52\) 0 0
\(53\) −697626. −0.643661 −0.321830 − 0.946797i \(-0.604298\pi\)
−0.321830 + 0.946797i \(0.604298\pi\)
\(54\) 0 0
\(55\) 938952. 0.760981
\(56\) 0 0
\(57\) 440100. 0.314767
\(58\) 0 0
\(59\) 870156. 0.551588 0.275794 − 0.961217i \(-0.411059\pi\)
0.275794 + 0.961217i \(0.411059\pi\)
\(60\) 0 0
\(61\) 2.06706e6 1.16600 0.583001 − 0.812472i \(-0.301878\pi\)
0.583001 + 0.812472i \(0.301878\pi\)
\(62\) 0 0
\(63\) −606528. −0.305604
\(64\) 0 0
\(65\) −5.62086e6 −2.53867
\(66\) 0 0
\(67\) −1.68075e6 −0.682717 −0.341359 − 0.939933i \(-0.610887\pi\)
−0.341359 + 0.939933i \(0.610887\pi\)
\(68\) 0 0
\(69\) 3.10846e6 1.13913
\(70\) 0 0
\(71\) −1.07028e6 −0.354890 −0.177445 − 0.984131i \(-0.556783\pi\)
−0.177445 + 0.984131i \(0.556783\pi\)
\(72\) 0 0
\(73\) −2.40333e6 −0.723076 −0.361538 − 0.932357i \(-0.617748\pi\)
−0.361538 + 0.932357i \(0.617748\pi\)
\(74\) 0 0
\(75\) −1.74849e6 −0.478574
\(76\) 0 0
\(77\) 2.06669e6 0.515890
\(78\) 0 0
\(79\) 2.30151e6 0.525192 0.262596 − 0.964906i \(-0.415421\pi\)
0.262596 + 0.964906i \(0.415421\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 4.70869e6 0.903914 0.451957 − 0.892040i \(-0.350726\pi\)
0.451957 + 0.892040i \(0.350726\pi\)
\(84\) 0 0
\(85\) 8.43016e6 1.48891
\(86\) 0 0
\(87\) −4.24132e6 −0.690532
\(88\) 0 0
\(89\) 4.14369e6 0.623049 0.311525 − 0.950238i \(-0.399160\pi\)
0.311525 + 0.950238i \(0.399160\pi\)
\(90\) 0 0
\(91\) −1.23718e7 −1.72103
\(92\) 0 0
\(93\) 444312. 0.0572793
\(94\) 0 0
\(95\) 6.16140e6 0.737304
\(96\) 0 0
\(97\) −1.62297e6 −0.180555 −0.0902777 − 0.995917i \(-0.528775\pi\)
−0.0902777 + 0.995917i \(0.528775\pi\)
\(98\) 0 0
\(99\) −1.81084e6 −0.187567
\(100\) 0 0
\(101\) 1.09388e6 0.105644 0.0528219 − 0.998604i \(-0.483178\pi\)
0.0528219 + 0.998604i \(0.483178\pi\)
\(102\) 0 0
\(103\) −1.56332e7 −1.40967 −0.704837 − 0.709370i \(-0.748980\pi\)
−0.704837 + 0.709370i \(0.748980\pi\)
\(104\) 0 0
\(105\) −8.49139e6 −0.715841
\(106\) 0 0
\(107\) −2.37028e6 −0.187049 −0.0935246 − 0.995617i \(-0.529813\pi\)
−0.0935246 + 0.995617i \(0.529813\pi\)
\(108\) 0 0
\(109\) 2.12706e6 0.157321 0.0786606 − 0.996901i \(-0.474936\pi\)
0.0786606 + 0.996901i \(0.474936\pi\)
\(110\) 0 0
\(111\) 4.03018e6 0.279701
\(112\) 0 0
\(113\) −2.17091e7 −1.41536 −0.707682 − 0.706531i \(-0.750259\pi\)
−0.707682 + 0.706531i \(0.750259\pi\)
\(114\) 0 0
\(115\) 4.35184e7 2.66827
\(116\) 0 0
\(117\) 1.08402e7 0.625731
\(118\) 0 0
\(119\) 1.85553e7 1.00938
\(120\) 0 0
\(121\) −1.33169e7 −0.683368
\(122\) 0 0
\(123\) 6.50997e6 0.315436
\(124\) 0 0
\(125\) 5.05235e6 0.231371
\(126\) 0 0
\(127\) −1.02998e7 −0.446187 −0.223094 − 0.974797i \(-0.571616\pi\)
−0.223094 + 0.974797i \(0.571616\pi\)
\(128\) 0 0
\(129\) 1.19661e7 0.490781
\(130\) 0 0
\(131\) −3.98823e7 −1.55000 −0.774999 − 0.631962i \(-0.782250\pi\)
−0.774999 + 0.631962i \(0.782250\pi\)
\(132\) 0 0
\(133\) 1.35616e7 0.499839
\(134\) 0 0
\(135\) 7.44017e6 0.260264
\(136\) 0 0
\(137\) −2.57412e7 −0.855277 −0.427639 − 0.903950i \(-0.640654\pi\)
−0.427639 + 0.903950i \(0.640654\pi\)
\(138\) 0 0
\(139\) 2.62409e7 0.828757 0.414379 − 0.910105i \(-0.363999\pi\)
0.414379 + 0.910105i \(0.363999\pi\)
\(140\) 0 0
\(141\) −2.49143e7 −0.748483
\(142\) 0 0
\(143\) −3.69371e7 −1.05630
\(144\) 0 0
\(145\) −5.93785e7 −1.61749
\(146\) 0 0
\(147\) 3.54561e6 0.0920621
\(148\) 0 0
\(149\) 6.98130e7 1.72896 0.864479 − 0.502670i \(-0.167649\pi\)
0.864479 + 0.502670i \(0.167649\pi\)
\(150\) 0 0
\(151\) 2.71335e7 0.641338 0.320669 − 0.947191i \(-0.396092\pi\)
0.320669 + 0.947191i \(0.396092\pi\)
\(152\) 0 0
\(153\) −1.62582e7 −0.366988
\(154\) 0 0
\(155\) 6.22037e6 0.134170
\(156\) 0 0
\(157\) −1.09857e7 −0.226557 −0.113279 − 0.993563i \(-0.536135\pi\)
−0.113279 + 0.993563i \(0.536135\pi\)
\(158\) 0 0
\(159\) 1.88359e7 0.371618
\(160\) 0 0
\(161\) 9.57865e7 1.80890
\(162\) 0 0
\(163\) −8.43924e7 −1.52632 −0.763162 − 0.646207i \(-0.776354\pi\)
−0.763162 + 0.646207i \(0.776354\pi\)
\(164\) 0 0
\(165\) −2.53517e7 −0.439353
\(166\) 0 0
\(167\) 5.95467e7 0.989350 0.494675 − 0.869078i \(-0.335287\pi\)
0.494675 + 0.869078i \(0.335287\pi\)
\(168\) 0 0
\(169\) 1.58368e8 2.52386
\(170\) 0 0
\(171\) −1.18827e7 −0.181731
\(172\) 0 0
\(173\) −2.12157e7 −0.311527 −0.155763 − 0.987794i \(-0.549784\pi\)
−0.155763 + 0.987794i \(0.549784\pi\)
\(174\) 0 0
\(175\) −5.38795e7 −0.759959
\(176\) 0 0
\(177\) −2.34942e7 −0.318460
\(178\) 0 0
\(179\) −9.50932e7 −1.23926 −0.619632 − 0.784892i \(-0.712718\pi\)
−0.619632 + 0.784892i \(0.712718\pi\)
\(180\) 0 0
\(181\) −1.31732e8 −1.65126 −0.825629 − 0.564214i \(-0.809179\pi\)
−0.825629 + 0.564214i \(0.809179\pi\)
\(182\) 0 0
\(183\) −5.58107e7 −0.673191
\(184\) 0 0
\(185\) 5.64225e7 0.655166
\(186\) 0 0
\(187\) 5.53982e7 0.619512
\(188\) 0 0
\(189\) 1.63763e7 0.176441
\(190\) 0 0
\(191\) −5.80470e7 −0.602786 −0.301393 − 0.953500i \(-0.597452\pi\)
−0.301393 + 0.953500i \(0.597452\pi\)
\(192\) 0 0
\(193\) −6.92087e7 −0.692963 −0.346482 − 0.938057i \(-0.612624\pi\)
−0.346482 + 0.938057i \(0.612624\pi\)
\(194\) 0 0
\(195\) 1.51763e8 1.46570
\(196\) 0 0
\(197\) −5.33848e7 −0.497491 −0.248746 − 0.968569i \(-0.580018\pi\)
−0.248746 + 0.968569i \(0.580018\pi\)
\(198\) 0 0
\(199\) 7.19134e7 0.646880 0.323440 − 0.946249i \(-0.395161\pi\)
0.323440 + 0.946249i \(0.395161\pi\)
\(200\) 0 0
\(201\) 4.53802e7 0.394167
\(202\) 0 0
\(203\) −1.30696e8 −1.09654
\(204\) 0 0
\(205\) 9.11396e7 0.738871
\(206\) 0 0
\(207\) −8.39283e7 −0.657676
\(208\) 0 0
\(209\) 4.04892e7 0.306780
\(210\) 0 0
\(211\) 2.05463e8 1.50572 0.752861 − 0.658180i \(-0.228673\pi\)
0.752861 + 0.658180i \(0.228673\pi\)
\(212\) 0 0
\(213\) 2.88976e7 0.204896
\(214\) 0 0
\(215\) 1.67525e8 1.14960
\(216\) 0 0
\(217\) 1.36914e7 0.0909575
\(218\) 0 0
\(219\) 6.48900e7 0.417468
\(220\) 0 0
\(221\) −3.31631e8 −2.06672
\(222\) 0 0
\(223\) 5.12508e7 0.309480 0.154740 − 0.987955i \(-0.450546\pi\)
0.154740 + 0.987955i \(0.450546\pi\)
\(224\) 0 0
\(225\) 4.72093e7 0.276305
\(226\) 0 0
\(227\) 1.17076e8 0.664319 0.332159 − 0.943223i \(-0.392223\pi\)
0.332159 + 0.943223i \(0.392223\pi\)
\(228\) 0 0
\(229\) −1.95800e8 −1.07743 −0.538715 − 0.842488i \(-0.681090\pi\)
−0.538715 + 0.842488i \(0.681090\pi\)
\(230\) 0 0
\(231\) −5.58006e7 −0.297849
\(232\) 0 0
\(233\) −1.30949e8 −0.678199 −0.339099 − 0.940751i \(-0.610122\pi\)
−0.339099 + 0.940751i \(0.610122\pi\)
\(234\) 0 0
\(235\) −3.48800e8 −1.75323
\(236\) 0 0
\(237\) −6.21408e7 −0.303220
\(238\) 0 0
\(239\) 1.72546e8 0.817544 0.408772 − 0.912637i \(-0.365957\pi\)
0.408772 + 0.912637i \(0.365957\pi\)
\(240\) 0 0
\(241\) 1.05073e8 0.483538 0.241769 − 0.970334i \(-0.422272\pi\)
0.241769 + 0.970334i \(0.422272\pi\)
\(242\) 0 0
\(243\) −1.43489e7 −0.0641500
\(244\) 0 0
\(245\) 4.96386e7 0.215644
\(246\) 0 0
\(247\) −2.42381e8 −1.02343
\(248\) 0 0
\(249\) −1.27135e8 −0.521875
\(250\) 0 0
\(251\) 1.33754e8 0.533885 0.266943 − 0.963712i \(-0.413987\pi\)
0.266943 + 0.963712i \(0.413987\pi\)
\(252\) 0 0
\(253\) 2.85978e8 1.11022
\(254\) 0 0
\(255\) −2.27614e8 −0.859624
\(256\) 0 0
\(257\) 2.97590e8 1.09359 0.546793 − 0.837268i \(-0.315849\pi\)
0.546793 + 0.837268i \(0.315849\pi\)
\(258\) 0 0
\(259\) 1.24189e8 0.444156
\(260\) 0 0
\(261\) 1.14516e8 0.398679
\(262\) 0 0
\(263\) 3.39541e8 1.15092 0.575462 − 0.817829i \(-0.304822\pi\)
0.575462 + 0.817829i \(0.304822\pi\)
\(264\) 0 0
\(265\) 2.63703e8 0.870470
\(266\) 0 0
\(267\) −1.11880e8 −0.359718
\(268\) 0 0
\(269\) −2.68088e7 −0.0839739 −0.0419869 − 0.999118i \(-0.513369\pi\)
−0.0419869 + 0.999118i \(0.513369\pi\)
\(270\) 0 0
\(271\) −3.49721e8 −1.06741 −0.533703 − 0.845672i \(-0.679200\pi\)
−0.533703 + 0.845672i \(0.679200\pi\)
\(272\) 0 0
\(273\) 3.34040e8 0.993639
\(274\) 0 0
\(275\) −1.60861e8 −0.466431
\(276\) 0 0
\(277\) −4.34818e8 −1.22922 −0.614608 − 0.788833i \(-0.710686\pi\)
−0.614608 + 0.788833i \(0.710686\pi\)
\(278\) 0 0
\(279\) −1.19964e7 −0.0330702
\(280\) 0 0
\(281\) −1.27669e8 −0.343252 −0.171626 − 0.985162i \(-0.554902\pi\)
−0.171626 + 0.985162i \(0.554902\pi\)
\(282\) 0 0
\(283\) 2.78115e8 0.729411 0.364705 − 0.931123i \(-0.381170\pi\)
0.364705 + 0.931123i \(0.381170\pi\)
\(284\) 0 0
\(285\) −1.66358e8 −0.425683
\(286\) 0 0
\(287\) 2.00604e8 0.500901
\(288\) 0 0
\(289\) 8.70405e7 0.212119
\(290\) 0 0
\(291\) 4.38203e7 0.104244
\(292\) 0 0
\(293\) −2.26385e7 −0.0525788 −0.0262894 − 0.999654i \(-0.508369\pi\)
−0.0262894 + 0.999654i \(0.508369\pi\)
\(294\) 0 0
\(295\) −3.28919e8 −0.745954
\(296\) 0 0
\(297\) 4.88926e7 0.108292
\(298\) 0 0
\(299\) −1.71195e9 −3.70376
\(300\) 0 0
\(301\) 3.68732e8 0.779343
\(302\) 0 0
\(303\) −2.95347e7 −0.0609935
\(304\) 0 0
\(305\) −7.81349e8 −1.57687
\(306\) 0 0
\(307\) 8.19497e8 1.61645 0.808226 − 0.588872i \(-0.200428\pi\)
0.808226 + 0.588872i \(0.200428\pi\)
\(308\) 0 0
\(309\) 4.22097e8 0.813875
\(310\) 0 0
\(311\) 9.24456e8 1.74271 0.871355 − 0.490654i \(-0.163242\pi\)
0.871355 + 0.490654i \(0.163242\pi\)
\(312\) 0 0
\(313\) −3.23653e8 −0.596588 −0.298294 − 0.954474i \(-0.596418\pi\)
−0.298294 + 0.954474i \(0.596418\pi\)
\(314\) 0 0
\(315\) 2.29268e8 0.413291
\(316\) 0 0
\(317\) 7.12783e8 1.25675 0.628376 − 0.777910i \(-0.283720\pi\)
0.628376 + 0.777910i \(0.283720\pi\)
\(318\) 0 0
\(319\) −3.90202e8 −0.673010
\(320\) 0 0
\(321\) 6.39975e7 0.107993
\(322\) 0 0
\(323\) 3.63523e8 0.600237
\(324\) 0 0
\(325\) 9.62966e8 1.55603
\(326\) 0 0
\(327\) −5.74307e7 −0.0908295
\(328\) 0 0
\(329\) −7.67730e8 −1.18856
\(330\) 0 0
\(331\) −1.65095e8 −0.250229 −0.125114 − 0.992142i \(-0.539930\pi\)
−0.125114 + 0.992142i \(0.539930\pi\)
\(332\) 0 0
\(333\) −1.08815e8 −0.161486
\(334\) 0 0
\(335\) 6.35323e8 0.923289
\(336\) 0 0
\(337\) −8.09761e8 −1.15253 −0.576265 − 0.817263i \(-0.695490\pi\)
−0.576265 + 0.817263i \(0.695490\pi\)
\(338\) 0 0
\(339\) 5.86147e8 0.817160
\(340\) 0 0
\(341\) 4.08767e7 0.0558259
\(342\) 0 0
\(343\) 7.94445e8 1.06300
\(344\) 0 0
\(345\) −1.17500e9 −1.54053
\(346\) 0 0
\(347\) −4.44349e8 −0.570915 −0.285458 − 0.958391i \(-0.592146\pi\)
−0.285458 + 0.958391i \(0.592146\pi\)
\(348\) 0 0
\(349\) −9.48806e8 −1.19478 −0.597390 − 0.801951i \(-0.703796\pi\)
−0.597390 + 0.801951i \(0.703796\pi\)
\(350\) 0 0
\(351\) −2.92686e8 −0.361266
\(352\) 0 0
\(353\) 4.38524e8 0.530618 0.265309 − 0.964163i \(-0.414526\pi\)
0.265309 + 0.964163i \(0.414526\pi\)
\(354\) 0 0
\(355\) 4.04566e8 0.479943
\(356\) 0 0
\(357\) −5.00992e8 −0.582763
\(358\) 0 0
\(359\) −5.00653e8 −0.571092 −0.285546 − 0.958365i \(-0.592175\pi\)
−0.285546 + 0.958365i \(0.592175\pi\)
\(360\) 0 0
\(361\) −6.28182e8 −0.702765
\(362\) 0 0
\(363\) 3.59557e8 0.394543
\(364\) 0 0
\(365\) 9.08460e8 0.977870
\(366\) 0 0
\(367\) −1.61229e9 −1.70259 −0.851297 − 0.524685i \(-0.824183\pi\)
−0.851297 + 0.524685i \(0.824183\pi\)
\(368\) 0 0
\(369\) −1.75769e8 −0.182117
\(370\) 0 0
\(371\) 5.80425e8 0.590116
\(372\) 0 0
\(373\) 9.55028e7 0.0952873 0.0476437 − 0.998864i \(-0.484829\pi\)
0.0476437 + 0.998864i \(0.484829\pi\)
\(374\) 0 0
\(375\) −1.36413e8 −0.133582
\(376\) 0 0
\(377\) 2.33587e9 2.24519
\(378\) 0 0
\(379\) 3.51478e8 0.331635 0.165818 − 0.986156i \(-0.446974\pi\)
0.165818 + 0.986156i \(0.446974\pi\)
\(380\) 0 0
\(381\) 2.78095e8 0.257606
\(382\) 0 0
\(383\) −1.21120e9 −1.10159 −0.550797 − 0.834639i \(-0.685676\pi\)
−0.550797 + 0.834639i \(0.685676\pi\)
\(384\) 0 0
\(385\) −7.81208e8 −0.697677
\(386\) 0 0
\(387\) −3.23084e8 −0.283353
\(388\) 0 0
\(389\) −1.43013e7 −0.0123183 −0.00615917 − 0.999981i \(-0.501961\pi\)
−0.00615917 + 0.999981i \(0.501961\pi\)
\(390\) 0 0
\(391\) 2.56758e9 2.17223
\(392\) 0 0
\(393\) 1.07682e9 0.894892
\(394\) 0 0
\(395\) −8.69972e8 −0.710256
\(396\) 0 0
\(397\) 1.83153e9 1.46908 0.734541 − 0.678564i \(-0.237397\pi\)
0.734541 + 0.678564i \(0.237397\pi\)
\(398\) 0 0
\(399\) −3.66163e8 −0.288582
\(400\) 0 0
\(401\) −3.14244e8 −0.243367 −0.121684 − 0.992569i \(-0.538829\pi\)
−0.121684 + 0.992569i \(0.538829\pi\)
\(402\) 0 0
\(403\) −2.44701e8 −0.186238
\(404\) 0 0
\(405\) −2.00885e8 −0.150264
\(406\) 0 0
\(407\) 3.70777e8 0.272604
\(408\) 0 0
\(409\) 1.39490e8 0.100812 0.0504059 − 0.998729i \(-0.483948\pi\)
0.0504059 + 0.998729i \(0.483948\pi\)
\(410\) 0 0
\(411\) 6.95013e8 0.493795
\(412\) 0 0
\(413\) −7.23970e8 −0.505703
\(414\) 0 0
\(415\) −1.77989e9 −1.22243
\(416\) 0 0
\(417\) −7.08505e8 −0.478483
\(418\) 0 0
\(419\) −2.31748e9 −1.53910 −0.769549 − 0.638588i \(-0.779519\pi\)
−0.769549 + 0.638588i \(0.779519\pi\)
\(420\) 0 0
\(421\) −1.05935e9 −0.691913 −0.345957 − 0.938250i \(-0.612446\pi\)
−0.345957 + 0.938250i \(0.612446\pi\)
\(422\) 0 0
\(423\) 6.72686e8 0.432137
\(424\) 0 0
\(425\) −1.44426e9 −0.912605
\(426\) 0 0
\(427\) −1.71980e9 −1.06900
\(428\) 0 0
\(429\) 9.97301e8 0.609854
\(430\) 0 0
\(431\) −1.70587e9 −1.02630 −0.513150 − 0.858299i \(-0.671522\pi\)
−0.513150 + 0.858299i \(0.671522\pi\)
\(432\) 0 0
\(433\) 2.71320e9 1.60611 0.803054 − 0.595906i \(-0.203207\pi\)
0.803054 + 0.595906i \(0.203207\pi\)
\(434\) 0 0
\(435\) 1.60322e9 0.933858
\(436\) 0 0
\(437\) 1.87659e9 1.07568
\(438\) 0 0
\(439\) 1.66778e9 0.940832 0.470416 − 0.882445i \(-0.344104\pi\)
0.470416 + 0.882445i \(0.344104\pi\)
\(440\) 0 0
\(441\) −9.57316e7 −0.0531521
\(442\) 0 0
\(443\) −7.96843e8 −0.435472 −0.217736 − 0.976008i \(-0.569867\pi\)
−0.217736 + 0.976008i \(0.569867\pi\)
\(444\) 0 0
\(445\) −1.56631e9 −0.842595
\(446\) 0 0
\(447\) −1.88495e9 −0.998214
\(448\) 0 0
\(449\) −9.88997e8 −0.515624 −0.257812 − 0.966195i \(-0.583001\pi\)
−0.257812 + 0.966195i \(0.583001\pi\)
\(450\) 0 0
\(451\) 5.98917e8 0.307432
\(452\) 0 0
\(453\) −7.32605e8 −0.370277
\(454\) 0 0
\(455\) 4.67656e9 2.32748
\(456\) 0 0
\(457\) 7.56748e8 0.370890 0.185445 − 0.982655i \(-0.440627\pi\)
0.185445 + 0.982655i \(0.440627\pi\)
\(458\) 0 0
\(459\) 4.38970e8 0.211880
\(460\) 0 0
\(461\) −2.54215e9 −1.20851 −0.604253 − 0.796793i \(-0.706528\pi\)
−0.604253 + 0.796793i \(0.706528\pi\)
\(462\) 0 0
\(463\) 1.35745e9 0.635611 0.317806 − 0.948156i \(-0.397054\pi\)
0.317806 + 0.948156i \(0.397054\pi\)
\(464\) 0 0
\(465\) −1.67950e8 −0.0774630
\(466\) 0 0
\(467\) 3.65551e9 1.66088 0.830441 − 0.557107i \(-0.188089\pi\)
0.830441 + 0.557107i \(0.188089\pi\)
\(468\) 0 0
\(469\) 1.39838e9 0.625923
\(470\) 0 0
\(471\) 2.96613e8 0.130803
\(472\) 0 0
\(473\) 1.10088e9 0.478328
\(474\) 0 0
\(475\) −1.05557e9 −0.451918
\(476\) 0 0
\(477\) −5.08569e8 −0.214554
\(478\) 0 0
\(479\) 1.22597e9 0.509687 0.254844 − 0.966982i \(-0.417976\pi\)
0.254844 + 0.966982i \(0.417976\pi\)
\(480\) 0 0
\(481\) −2.21959e9 −0.909419
\(482\) 0 0
\(483\) −2.58624e9 −1.04437
\(484\) 0 0
\(485\) 6.13484e8 0.244178
\(486\) 0 0
\(487\) −4.96327e9 −1.94723 −0.973613 − 0.228204i \(-0.926715\pi\)
−0.973613 + 0.228204i \(0.926715\pi\)
\(488\) 0 0
\(489\) 2.27859e9 0.881223
\(490\) 0 0
\(491\) 2.58972e8 0.0987344 0.0493672 − 0.998781i \(-0.484280\pi\)
0.0493672 + 0.998781i \(0.484280\pi\)
\(492\) 0 0
\(493\) −3.50333e9 −1.31679
\(494\) 0 0
\(495\) 6.84496e8 0.253660
\(496\) 0 0
\(497\) 8.90473e8 0.325367
\(498\) 0 0
\(499\) 3.60546e9 1.29900 0.649500 − 0.760362i \(-0.274978\pi\)
0.649500 + 0.760362i \(0.274978\pi\)
\(500\) 0 0
\(501\) −1.60776e9 −0.571201
\(502\) 0 0
\(503\) −2.25140e9 −0.788795 −0.394398 − 0.918940i \(-0.629047\pi\)
−0.394398 + 0.918940i \(0.629047\pi\)
\(504\) 0 0
\(505\) −4.13486e8 −0.142870
\(506\) 0 0
\(507\) −4.27595e9 −1.45715
\(508\) 0 0
\(509\) −5.03059e9 −1.69086 −0.845428 − 0.534090i \(-0.820654\pi\)
−0.845428 + 0.534090i \(0.820654\pi\)
\(510\) 0 0
\(511\) 1.99957e9 0.662925
\(512\) 0 0
\(513\) 3.20833e8 0.104922
\(514\) 0 0
\(515\) 5.90936e9 1.90641
\(516\) 0 0
\(517\) −2.29212e9 −0.729491
\(518\) 0 0
\(519\) 5.72823e8 0.179860
\(520\) 0 0
\(521\) 8.62267e8 0.267122 0.133561 − 0.991041i \(-0.457359\pi\)
0.133561 + 0.991041i \(0.457359\pi\)
\(522\) 0 0
\(523\) −2.60059e9 −0.794907 −0.397453 − 0.917622i \(-0.630106\pi\)
−0.397453 + 0.917622i \(0.630106\pi\)
\(524\) 0 0
\(525\) 1.45475e9 0.438763
\(526\) 0 0
\(527\) 3.67002e8 0.109227
\(528\) 0 0
\(529\) 9.84963e9 2.89284
\(530\) 0 0
\(531\) 6.34344e8 0.183863
\(532\) 0 0
\(533\) −3.58531e9 −1.02561
\(534\) 0 0
\(535\) 8.95964e8 0.252960
\(536\) 0 0
\(537\) 2.56752e9 0.715490
\(538\) 0 0
\(539\) 3.26196e8 0.0897260
\(540\) 0 0
\(541\) −3.78391e8 −0.102742 −0.0513712 − 0.998680i \(-0.516359\pi\)
−0.0513712 + 0.998680i \(0.516359\pi\)
\(542\) 0 0
\(543\) 3.55675e9 0.953354
\(544\) 0 0
\(545\) −8.04029e8 −0.212757
\(546\) 0 0
\(547\) 1.35980e9 0.355238 0.177619 − 0.984099i \(-0.443160\pi\)
0.177619 + 0.984099i \(0.443160\pi\)
\(548\) 0 0
\(549\) 1.50689e9 0.388667
\(550\) 0 0
\(551\) −2.56050e9 −0.652070
\(552\) 0 0
\(553\) −1.91486e9 −0.481503
\(554\) 0 0
\(555\) −1.52341e9 −0.378261
\(556\) 0 0
\(557\) 3.33022e9 0.816544 0.408272 − 0.912860i \(-0.366131\pi\)
0.408272 + 0.912860i \(0.366131\pi\)
\(558\) 0 0
\(559\) −6.59021e9 −1.59572
\(560\) 0 0
\(561\) −1.49575e9 −0.357675
\(562\) 0 0
\(563\) −6.37690e9 −1.50602 −0.753009 − 0.658010i \(-0.771398\pi\)
−0.753009 + 0.658010i \(0.771398\pi\)
\(564\) 0 0
\(565\) 8.20605e9 1.91410
\(566\) 0 0
\(567\) −4.42159e8 −0.101868
\(568\) 0 0
\(569\) 2.87492e9 0.654235 0.327117 − 0.944984i \(-0.393923\pi\)
0.327117 + 0.944984i \(0.393923\pi\)
\(570\) 0 0
\(571\) −2.26452e8 −0.0509037 −0.0254519 − 0.999676i \(-0.508102\pi\)
−0.0254519 + 0.999676i \(0.508102\pi\)
\(572\) 0 0
\(573\) 1.56727e9 0.348018
\(574\) 0 0
\(575\) −7.45557e9 −1.63547
\(576\) 0 0
\(577\) 6.03280e8 0.130738 0.0653692 − 0.997861i \(-0.479177\pi\)
0.0653692 + 0.997861i \(0.479177\pi\)
\(578\) 0 0
\(579\) 1.86863e9 0.400082
\(580\) 0 0
\(581\) −3.91763e9 −0.828719
\(582\) 0 0
\(583\) 1.73290e9 0.362188
\(584\) 0 0
\(585\) −4.09761e9 −0.846223
\(586\) 0 0
\(587\) −6.79902e7 −0.0138744 −0.00693718 − 0.999976i \(-0.502208\pi\)
−0.00693718 + 0.999976i \(0.502208\pi\)
\(588\) 0 0
\(589\) 2.68233e8 0.0540889
\(590\) 0 0
\(591\) 1.44139e9 0.287227
\(592\) 0 0
\(593\) −5.80175e9 −1.14253 −0.571265 − 0.820766i \(-0.693547\pi\)
−0.571265 + 0.820766i \(0.693547\pi\)
\(594\) 0 0
\(595\) −7.01389e9 −1.36505
\(596\) 0 0
\(597\) −1.94166e9 −0.373477
\(598\) 0 0
\(599\) 5.05519e9 0.961046 0.480523 − 0.876982i \(-0.340447\pi\)
0.480523 + 0.876982i \(0.340447\pi\)
\(600\) 0 0
\(601\) −5.97061e9 −1.12191 −0.560955 − 0.827846i \(-0.689566\pi\)
−0.560955 + 0.827846i \(0.689566\pi\)
\(602\) 0 0
\(603\) −1.22527e9 −0.227572
\(604\) 0 0
\(605\) 5.03379e9 0.924169
\(606\) 0 0
\(607\) 4.03712e9 0.732675 0.366337 − 0.930482i \(-0.380612\pi\)
0.366337 + 0.930482i \(0.380612\pi\)
\(608\) 0 0
\(609\) 3.52878e9 0.633088
\(610\) 0 0
\(611\) 1.37213e10 2.43361
\(612\) 0 0
\(613\) −8.33838e8 −0.146208 −0.0731038 − 0.997324i \(-0.523290\pi\)
−0.0731038 + 0.997324i \(0.523290\pi\)
\(614\) 0 0
\(615\) −2.46077e9 −0.426587
\(616\) 0 0
\(617\) 8.21270e9 1.40763 0.703814 − 0.710384i \(-0.251479\pi\)
0.703814 + 0.710384i \(0.251479\pi\)
\(618\) 0 0
\(619\) 5.43752e9 0.921475 0.460737 − 0.887537i \(-0.347585\pi\)
0.460737 + 0.887537i \(0.347585\pi\)
\(620\) 0 0
\(621\) 2.26606e9 0.379710
\(622\) 0 0
\(623\) −3.44755e9 −0.571219
\(624\) 0 0
\(625\) −6.96908e9 −1.14181
\(626\) 0 0
\(627\) −1.09321e9 −0.177120
\(628\) 0 0
\(629\) 3.32893e9 0.533369
\(630\) 0 0
\(631\) 6.20797e9 0.983664 0.491832 − 0.870690i \(-0.336327\pi\)
0.491832 + 0.870690i \(0.336327\pi\)
\(632\) 0 0
\(633\) −5.54750e9 −0.869329
\(634\) 0 0
\(635\) 3.89334e9 0.603412
\(636\) 0 0
\(637\) −1.95271e9 −0.299330
\(638\) 0 0
\(639\) −7.80234e8 −0.118297
\(640\) 0 0
\(641\) −3.31503e8 −0.0497147 −0.0248573 − 0.999691i \(-0.507913\pi\)
−0.0248573 + 0.999691i \(0.507913\pi\)
\(642\) 0 0
\(643\) −6.37327e8 −0.0945418 −0.0472709 − 0.998882i \(-0.515052\pi\)
−0.0472709 + 0.998882i \(0.515052\pi\)
\(644\) 0 0
\(645\) −4.52318e9 −0.663719
\(646\) 0 0
\(647\) −7.36821e9 −1.06954 −0.534770 − 0.844998i \(-0.679602\pi\)
−0.534770 + 0.844998i \(0.679602\pi\)
\(648\) 0 0
\(649\) −2.16147e9 −0.310379
\(650\) 0 0
\(651\) −3.69668e8 −0.0525143
\(652\) 0 0
\(653\) 9.87503e9 1.38785 0.693924 − 0.720048i \(-0.255880\pi\)
0.693924 + 0.720048i \(0.255880\pi\)
\(654\) 0 0
\(655\) 1.50755e10 2.09618
\(656\) 0 0
\(657\) −1.75203e9 −0.241025
\(658\) 0 0
\(659\) −1.21767e10 −1.65741 −0.828705 − 0.559686i \(-0.810922\pi\)
−0.828705 + 0.559686i \(0.810922\pi\)
\(660\) 0 0
\(661\) −1.11032e10 −1.49536 −0.747678 − 0.664062i \(-0.768831\pi\)
−0.747678 + 0.664062i \(0.768831\pi\)
\(662\) 0 0
\(663\) 8.95403e9 1.19322
\(664\) 0 0
\(665\) −5.12628e9 −0.675969
\(666\) 0 0
\(667\) −1.80850e10 −2.35982
\(668\) 0 0
\(669\) −1.38377e9 −0.178679
\(670\) 0 0
\(671\) −5.13458e9 −0.656109
\(672\) 0 0
\(673\) −3.49663e9 −0.442177 −0.221089 − 0.975254i \(-0.570961\pi\)
−0.221089 + 0.975254i \(0.570961\pi\)
\(674\) 0 0
\(675\) −1.27465e9 −0.159525
\(676\) 0 0
\(677\) 6.40744e9 0.793641 0.396820 − 0.917896i \(-0.370114\pi\)
0.396820 + 0.917896i \(0.370114\pi\)
\(678\) 0 0
\(679\) 1.35031e9 0.165535
\(680\) 0 0
\(681\) −3.16105e9 −0.383545
\(682\) 0 0
\(683\) 1.48217e10 1.78002 0.890011 − 0.455938i \(-0.150696\pi\)
0.890011 + 0.455938i \(0.150696\pi\)
\(684\) 0 0
\(685\) 9.73018e9 1.15665
\(686\) 0 0
\(687\) 5.28661e9 0.622055
\(688\) 0 0
\(689\) −1.03737e10 −1.20828
\(690\) 0 0
\(691\) 1.91226e9 0.220482 0.110241 − 0.993905i \(-0.464838\pi\)
0.110241 + 0.993905i \(0.464838\pi\)
\(692\) 0 0
\(693\) 1.50662e9 0.171963
\(694\) 0 0
\(695\) −9.91907e9 −1.12079
\(696\) 0 0
\(697\) 5.37724e9 0.601512
\(698\) 0 0
\(699\) 3.53563e9 0.391558
\(700\) 0 0
\(701\) −1.15650e10 −1.26804 −0.634020 − 0.773317i \(-0.718596\pi\)
−0.634020 + 0.773317i \(0.718596\pi\)
\(702\) 0 0
\(703\) 2.43304e9 0.264122
\(704\) 0 0
\(705\) 9.41761e9 1.01223
\(706\) 0 0
\(707\) −9.10106e8 −0.0968555
\(708\) 0 0
\(709\) 5.17466e9 0.545281 0.272641 − 0.962116i \(-0.412103\pi\)
0.272641 + 0.962116i \(0.412103\pi\)
\(710\) 0 0
\(711\) 1.67780e9 0.175064
\(712\) 0 0
\(713\) 1.89455e9 0.195746
\(714\) 0 0
\(715\) 1.39622e10 1.42851
\(716\) 0 0
\(717\) −4.65873e9 −0.472009
\(718\) 0 0
\(719\) −9.97915e9 −1.00125 −0.500625 − 0.865664i \(-0.666896\pi\)
−0.500625 + 0.865664i \(0.666896\pi\)
\(720\) 0 0
\(721\) 1.30068e10 1.29241
\(722\) 0 0
\(723\) −2.83697e9 −0.279171
\(724\) 0 0
\(725\) 1.01727e10 0.991413
\(726\) 0 0
\(727\) −5.25105e9 −0.506846 −0.253423 − 0.967356i \(-0.581556\pi\)
−0.253423 + 0.967356i \(0.581556\pi\)
\(728\) 0 0
\(729\) 3.87420e8 0.0370370
\(730\) 0 0
\(731\) 9.88398e9 0.935882
\(732\) 0 0
\(733\) 1.21211e10 1.13679 0.568393 − 0.822757i \(-0.307565\pi\)
0.568393 + 0.822757i \(0.307565\pi\)
\(734\) 0 0
\(735\) −1.34024e9 −0.124502
\(736\) 0 0
\(737\) 4.17498e9 0.384165
\(738\) 0 0
\(739\) −1.85860e10 −1.69406 −0.847032 − 0.531542i \(-0.821613\pi\)
−0.847032 + 0.531542i \(0.821613\pi\)
\(740\) 0 0
\(741\) 6.54429e9 0.590879
\(742\) 0 0
\(743\) −1.45147e10 −1.29822 −0.649109 − 0.760696i \(-0.724858\pi\)
−0.649109 + 0.760696i \(0.724858\pi\)
\(744\) 0 0
\(745\) −2.63893e10 −2.33820
\(746\) 0 0
\(747\) 3.43264e9 0.301305
\(748\) 0 0
\(749\) 1.97207e9 0.171489
\(750\) 0 0
\(751\) −4.06991e9 −0.350626 −0.175313 − 0.984513i \(-0.556094\pi\)
−0.175313 + 0.984513i \(0.556094\pi\)
\(752\) 0 0
\(753\) −3.61135e9 −0.308239
\(754\) 0 0
\(755\) −1.02565e10 −0.867328
\(756\) 0 0
\(757\) 5.68835e9 0.476596 0.238298 − 0.971192i \(-0.423410\pi\)
0.238298 + 0.971192i \(0.423410\pi\)
\(758\) 0 0
\(759\) −7.72140e9 −0.640988
\(760\) 0 0
\(761\) −1.26392e10 −1.03962 −0.519809 − 0.854282i \(-0.673997\pi\)
−0.519809 + 0.854282i \(0.673997\pi\)
\(762\) 0 0
\(763\) −1.76972e9 −0.144234
\(764\) 0 0
\(765\) 6.14558e9 0.496304
\(766\) 0 0
\(767\) 1.29392e10 1.03544
\(768\) 0 0
\(769\) −1.40530e10 −1.11436 −0.557182 − 0.830391i \(-0.688117\pi\)
−0.557182 + 0.830391i \(0.688117\pi\)
\(770\) 0 0
\(771\) −8.03494e9 −0.631382
\(772\) 0 0
\(773\) 1.74521e10 1.35900 0.679500 − 0.733675i \(-0.262197\pi\)
0.679500 + 0.733675i \(0.262197\pi\)
\(774\) 0 0
\(775\) −1.06567e9 −0.0822372
\(776\) 0 0
\(777\) −3.35311e9 −0.256433
\(778\) 0 0
\(779\) 3.93009e9 0.297867
\(780\) 0 0
\(781\) 2.65858e9 0.199696
\(782\) 0 0
\(783\) −3.09192e9 −0.230177
\(784\) 0 0
\(785\) 4.15258e9 0.306390
\(786\) 0 0
\(787\) 1.57180e10 1.14944 0.574720 − 0.818350i \(-0.305111\pi\)
0.574720 + 0.818350i \(0.305111\pi\)
\(788\) 0 0
\(789\) −9.16759e9 −0.664486
\(790\) 0 0
\(791\) 1.80620e10 1.29762
\(792\) 0 0
\(793\) 3.07372e10 2.18881
\(794\) 0 0
\(795\) −7.11997e9 −0.502566
\(796\) 0 0
\(797\) −1.80766e10 −1.26477 −0.632385 − 0.774654i \(-0.717924\pi\)
−0.632385 + 0.774654i \(0.717924\pi\)
\(798\) 0 0
\(799\) −2.05792e10 −1.42730
\(800\) 0 0
\(801\) 3.02075e9 0.207683
\(802\) 0 0
\(803\) 5.96988e9 0.406875
\(804\) 0 0
\(805\) −3.62073e10 −2.44630
\(806\) 0 0
\(807\) 7.23837e8 0.0484823
\(808\) 0 0
\(809\) 2.14745e10 1.42595 0.712973 − 0.701192i \(-0.247348\pi\)
0.712973 + 0.701192i \(0.247348\pi\)
\(810\) 0 0
\(811\) 2.05508e10 1.35287 0.676436 − 0.736502i \(-0.263524\pi\)
0.676436 + 0.736502i \(0.263524\pi\)
\(812\) 0 0
\(813\) 9.44248e9 0.616267
\(814\) 0 0
\(815\) 3.19003e10 2.06416
\(816\) 0 0
\(817\) 7.22396e9 0.463445
\(818\) 0 0
\(819\) −9.01907e9 −0.573678
\(820\) 0 0
\(821\) −5.11303e9 −0.322461 −0.161231 − 0.986917i \(-0.551546\pi\)
−0.161231 + 0.986917i \(0.551546\pi\)
\(822\) 0 0
\(823\) −1.96455e9 −0.122847 −0.0614233 − 0.998112i \(-0.519564\pi\)
−0.0614233 + 0.998112i \(0.519564\pi\)
\(824\) 0 0
\(825\) 4.34326e9 0.269294
\(826\) 0 0
\(827\) −8.45961e9 −0.520093 −0.260047 − 0.965596i \(-0.583738\pi\)
−0.260047 + 0.965596i \(0.583738\pi\)
\(828\) 0 0
\(829\) 3.79163e9 0.231145 0.115573 − 0.993299i \(-0.463130\pi\)
0.115573 + 0.993299i \(0.463130\pi\)
\(830\) 0 0
\(831\) 1.17401e10 0.709688
\(832\) 0 0
\(833\) 2.92868e9 0.175555
\(834\) 0 0
\(835\) −2.25086e10 −1.33797
\(836\) 0 0
\(837\) 3.23903e8 0.0190931
\(838\) 0 0
\(839\) −5.75205e9 −0.336245 −0.168122 − 0.985766i \(-0.553770\pi\)
−0.168122 + 0.985766i \(0.553770\pi\)
\(840\) 0 0
\(841\) 7.42614e9 0.430504
\(842\) 0 0
\(843\) 3.44706e9 0.198177
\(844\) 0 0
\(845\) −5.98632e10 −3.41320
\(846\) 0 0
\(847\) 1.10797e10 0.626520
\(848\) 0 0
\(849\) −7.50911e9 −0.421125
\(850\) 0 0
\(851\) 1.71847e10 0.955847
\(852\) 0 0
\(853\) −1.54604e10 −0.852904 −0.426452 − 0.904510i \(-0.640237\pi\)
−0.426452 + 0.904510i \(0.640237\pi\)
\(854\) 0 0
\(855\) 4.49166e9 0.245768
\(856\) 0 0
\(857\) −2.91908e10 −1.58421 −0.792105 − 0.610385i \(-0.791015\pi\)
−0.792105 + 0.610385i \(0.791015\pi\)
\(858\) 0 0
\(859\) 1.51850e10 0.817410 0.408705 − 0.912667i \(-0.365981\pi\)
0.408705 + 0.912667i \(0.365981\pi\)
\(860\) 0 0
\(861\) −5.41630e9 −0.289195
\(862\) 0 0
\(863\) 1.19335e10 0.632021 0.316011 − 0.948756i \(-0.397656\pi\)
0.316011 + 0.948756i \(0.397656\pi\)
\(864\) 0 0
\(865\) 8.01953e9 0.421301
\(866\) 0 0
\(867\) −2.35009e9 −0.122467
\(868\) 0 0
\(869\) −5.71696e9 −0.295526
\(870\) 0 0
\(871\) −2.49927e10 −1.28159
\(872\) 0 0
\(873\) −1.18315e9 −0.0601851
\(874\) 0 0
\(875\) −4.20355e9 −0.212123
\(876\) 0 0
\(877\) 2.25544e10 1.12910 0.564549 − 0.825399i \(-0.309050\pi\)
0.564549 + 0.825399i \(0.309050\pi\)
\(878\) 0 0
\(879\) 6.11239e8 0.0303564
\(880\) 0 0
\(881\) −6.33664e9 −0.312208 −0.156104 − 0.987741i \(-0.549893\pi\)
−0.156104 + 0.987741i \(0.549893\pi\)
\(882\) 0 0
\(883\) −9.72527e9 −0.475378 −0.237689 − 0.971341i \(-0.576390\pi\)
−0.237689 + 0.971341i \(0.576390\pi\)
\(884\) 0 0
\(885\) 8.88081e9 0.430677
\(886\) 0 0
\(887\) −3.36353e10 −1.61831 −0.809156 − 0.587593i \(-0.800075\pi\)
−0.809156 + 0.587593i \(0.800075\pi\)
\(888\) 0 0
\(889\) 8.56946e9 0.409070
\(890\) 0 0
\(891\) −1.32010e9 −0.0625223
\(892\) 0 0
\(893\) −1.50409e10 −0.706793
\(894\) 0 0
\(895\) 3.59452e10 1.67595
\(896\) 0 0
\(897\) 4.62227e10 2.13837
\(898\) 0 0
\(899\) −2.58501e9 −0.118660
\(900\) 0 0
\(901\) 1.55585e10 0.708647
\(902\) 0 0
\(903\) −9.95578e9 −0.449954
\(904\) 0 0
\(905\) 4.97945e10 2.23312
\(906\) 0 0
\(907\) −1.24681e10 −0.554851 −0.277425 − 0.960747i \(-0.589481\pi\)
−0.277425 + 0.960747i \(0.589481\pi\)
\(908\) 0 0
\(909\) 7.97437e8 0.0352146
\(910\) 0 0
\(911\) 1.89363e10 0.829814 0.414907 − 0.909864i \(-0.363814\pi\)
0.414907 + 0.909864i \(0.363814\pi\)
\(912\) 0 0
\(913\) −1.16964e10 −0.508633
\(914\) 0 0
\(915\) 2.10964e10 0.910406
\(916\) 0 0
\(917\) 3.31821e10 1.42106
\(918\) 0 0
\(919\) −1.24971e10 −0.531135 −0.265568 − 0.964092i \(-0.585559\pi\)
−0.265568 + 0.964092i \(0.585559\pi\)
\(920\) 0 0
\(921\) −2.21264e10 −0.933259
\(922\) 0 0
\(923\) −1.59151e10 −0.666197
\(924\) 0 0
\(925\) −9.66632e9 −0.401573
\(926\) 0 0
\(927\) −1.13966e10 −0.469891
\(928\) 0 0
\(929\) −7.35427e9 −0.300943 −0.150472 − 0.988614i \(-0.548079\pi\)
−0.150472 + 0.988614i \(0.548079\pi\)
\(930\) 0 0
\(931\) 2.14050e9 0.0869343
\(932\) 0 0
\(933\) −2.49603e10 −1.00615
\(934\) 0 0
\(935\) −2.09405e10 −0.837812
\(936\) 0 0
\(937\) 1.36199e10 0.540861 0.270430 − 0.962740i \(-0.412834\pi\)
0.270430 + 0.962740i \(0.412834\pi\)
\(938\) 0 0
\(939\) 8.73863e9 0.344440
\(940\) 0 0
\(941\) 1.02933e10 0.402707 0.201353 − 0.979519i \(-0.435466\pi\)
0.201353 + 0.979519i \(0.435466\pi\)
\(942\) 0 0
\(943\) 2.77585e10 1.07797
\(944\) 0 0
\(945\) −6.19022e9 −0.238614
\(946\) 0 0
\(947\) −2.39142e10 −0.915022 −0.457511 − 0.889204i \(-0.651259\pi\)
−0.457511 + 0.889204i \(0.651259\pi\)
\(948\) 0 0
\(949\) −3.57376e10 −1.35735
\(950\) 0 0
\(951\) −1.92451e10 −0.725586
\(952\) 0 0
\(953\) 2.83252e10 1.06010 0.530051 − 0.847966i \(-0.322173\pi\)
0.530051 + 0.847966i \(0.322173\pi\)
\(954\) 0 0
\(955\) 2.19418e10 0.815191
\(956\) 0 0
\(957\) 1.05354e10 0.388563
\(958\) 0 0
\(959\) 2.14167e10 0.784128
\(960\) 0 0
\(961\) −2.72418e10 −0.990157
\(962\) 0 0
\(963\) −1.72793e9 −0.0623497
\(964\) 0 0
\(965\) 2.61609e10 0.937145
\(966\) 0 0
\(967\) −3.64982e10 −1.29801 −0.649007 − 0.760783i \(-0.724815\pi\)
−0.649007 + 0.760783i \(0.724815\pi\)
\(968\) 0 0
\(969\) −9.81511e9 −0.346547
\(970\) 0 0
\(971\) −2.88087e10 −1.00985 −0.504924 − 0.863164i \(-0.668479\pi\)
−0.504924 + 0.863164i \(0.668479\pi\)
\(972\) 0 0
\(973\) −2.18324e10 −0.759814
\(974\) 0 0
\(975\) −2.60001e10 −0.898377
\(976\) 0 0
\(977\) −4.93806e10 −1.69405 −0.847023 − 0.531556i \(-0.821607\pi\)
−0.847023 + 0.531556i \(0.821607\pi\)
\(978\) 0 0
\(979\) −1.02929e10 −0.350590
\(980\) 0 0
\(981\) 1.55063e9 0.0524404
\(982\) 0 0
\(983\) 5.11399e10 1.71721 0.858603 − 0.512641i \(-0.171333\pi\)
0.858603 + 0.512641i \(0.171333\pi\)
\(984\) 0 0
\(985\) 2.01794e10 0.672795
\(986\) 0 0
\(987\) 2.07287e10 0.686218
\(988\) 0 0
\(989\) 5.10233e10 1.67719
\(990\) 0 0
\(991\) −5.74784e10 −1.87606 −0.938030 − 0.346553i \(-0.887352\pi\)
−0.938030 + 0.346553i \(0.887352\pi\)
\(992\) 0 0
\(993\) 4.45758e9 0.144470
\(994\) 0 0
\(995\) −2.71833e10 −0.874824
\(996\) 0 0
\(997\) 1.98127e10 0.633155 0.316577 − 0.948567i \(-0.397466\pi\)
0.316577 + 0.948567i \(0.397466\pi\)
\(998\) 0 0
\(999\) 2.93800e9 0.0932337
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 12.8.a.a.1.1 ✓ 1
3.2 odd 2 36.8.a.c.1.1 1
4.3 odd 2 48.8.a.e.1.1 1
5.2 odd 4 300.8.d.c.49.2 2
5.3 odd 4 300.8.d.c.49.1 2
5.4 even 2 300.8.a.g.1.1 1
7.2 even 3 588.8.i.h.361.1 2
7.3 odd 6 588.8.i.a.373.1 2
7.4 even 3 588.8.i.h.373.1 2
7.5 odd 6 588.8.i.a.361.1 2
7.6 odd 2 588.8.a.d.1.1 1
8.3 odd 2 192.8.a.g.1.1 1
8.5 even 2 192.8.a.o.1.1 1
9.2 odd 6 324.8.e.a.109.1 2
9.4 even 3 324.8.e.f.217.1 2
9.5 odd 6 324.8.e.a.217.1 2
9.7 even 3 324.8.e.f.109.1 2
12.11 even 2 144.8.a.j.1.1 1
24.5 odd 2 576.8.a.d.1.1 1
24.11 even 2 576.8.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.8.a.a.1.1 ✓ 1 1.1 even 1 trivial
36.8.a.c.1.1 1 3.2 odd 2
48.8.a.e.1.1 1 4.3 odd 2
144.8.a.j.1.1 1 12.11 even 2
192.8.a.g.1.1 1 8.3 odd 2
192.8.a.o.1.1 1 8.5 even 2
300.8.a.g.1.1 1 5.4 even 2
300.8.d.c.49.1 2 5.3 odd 4
300.8.d.c.49.2 2 5.2 odd 4
324.8.e.a.109.1 2 9.2 odd 6
324.8.e.a.217.1 2 9.5 odd 6
324.8.e.f.109.1 2 9.7 even 3
324.8.e.f.217.1 2 9.4 even 3
576.8.a.d.1.1 1 24.5 odd 2
576.8.a.e.1.1 1 24.11 even 2
588.8.a.d.1.1 1 7.6 odd 2
588.8.i.a.361.1 2 7.5 odd 6
588.8.i.a.373.1 2 7.3 odd 6
588.8.i.h.361.1 2 7.2 even 3
588.8.i.h.373.1 2 7.4 even 3