Properties

Label 1008.6.a.bd.1.2
Level $1008$
Weight $6$
Character 1008.1
Self dual yes
Analytic conductor $161.667$
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] = [1008,6,Mod(1,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1008.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: \(161.666890371\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{345}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 86 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-8.78709\) of defining polynomial
Character \(\chi\) \(=\) 1008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+14.7225 q^{5} +49.0000 q^{7} +O(q^{10})\) \(q+14.7225 q^{5} +49.0000 q^{7} +58.5549 q^{11} +1179.39 q^{13} -1496.55 q^{17} -498.838 q^{19} -1889.34 q^{23} -2908.25 q^{25} -1914.54 q^{29} -794.577 q^{31} +721.404 q^{35} +2987.93 q^{37} -11941.3 q^{41} -9820.19 q^{43} -19636.0 q^{47} +2401.00 q^{49} +19875.0 q^{53} +862.077 q^{55} +35838.6 q^{59} +49975.9 q^{61} +17363.6 q^{65} -48176.2 q^{67} +77179.1 q^{71} -59667.3 q^{73} +2869.19 q^{77} -60743.1 q^{79} -46134.2 q^{83} -22033.1 q^{85} -78668.7 q^{89} +57790.2 q^{91} -7344.15 q^{95} -43573.3 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 82 q^{5} + 98 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 82 q^{5} + 98 q^{7} + 340 q^{11} + 910 q^{13} - 3216 q^{17} + 674 q^{19} - 1104 q^{23} + 3322 q^{25} - 8064 q^{29} + 6212 q^{31} - 4018 q^{35} - 8512 q^{37} + 1304 q^{41} + 10004 q^{43} - 12748 q^{47} + 4802 q^{49} + 11220 q^{53} - 26360 q^{55} - 12018 q^{59} + 102738 q^{61} + 43420 q^{65} - 24136 q^{67} + 89720 q^{71} - 55588 q^{73} + 16660 q^{77} - 48824 q^{79} + 35782 q^{83} + 144276 q^{85} + 18300 q^{89} + 44590 q^{91} - 120784 q^{95} - 69984 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 14.7225 0.263365 0.131682 0.991292i \(-0.457962\pi\)
0.131682 + 0.991292i \(0.457962\pi\)
\(6\) 0 0
\(7\) 49.0000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 58.5549 0.145909 0.0729545 0.997335i \(-0.476757\pi\)
0.0729545 + 0.997335i \(0.476757\pi\)
\(12\) 0 0
\(13\) 1179.39 1.93553 0.967765 0.251853i \(-0.0810400\pi\)
0.967765 + 0.251853i \(0.0810400\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1496.55 −1.25594 −0.627972 0.778236i \(-0.716115\pi\)
−0.627972 + 0.778236i \(0.716115\pi\)
\(18\) 0 0
\(19\) −498.838 −0.317012 −0.158506 0.987358i \(-0.550668\pi\)
−0.158506 + 0.987358i \(0.550668\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1889.34 −0.744716 −0.372358 0.928089i \(-0.621451\pi\)
−0.372358 + 0.928089i \(0.621451\pi\)
\(24\) 0 0
\(25\) −2908.25 −0.930639
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1914.54 −0.422737 −0.211369 0.977406i \(-0.567792\pi\)
−0.211369 + 0.977406i \(0.567792\pi\)
\(30\) 0 0
\(31\) −794.577 −0.148502 −0.0742509 0.997240i \(-0.523657\pi\)
−0.0742509 + 0.997240i \(0.523657\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 721.404 0.0995424
\(36\) 0 0
\(37\) 2987.93 0.358811 0.179406 0.983775i \(-0.442583\pi\)
0.179406 + 0.983775i \(0.442583\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11941.3 −1.10941 −0.554704 0.832047i \(-0.687169\pi\)
−0.554704 + 0.832047i \(0.687169\pi\)
\(42\) 0 0
\(43\) −9820.19 −0.809933 −0.404966 0.914332i \(-0.632717\pi\)
−0.404966 + 0.914332i \(0.632717\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −19636.0 −1.29660 −0.648302 0.761383i \(-0.724521\pi\)
−0.648302 + 0.761383i \(0.724521\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 19875.0 0.971889 0.485945 0.873990i \(-0.338476\pi\)
0.485945 + 0.873990i \(0.338476\pi\)
\(54\) 0 0
\(55\) 862.077 0.0384272
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 35838.6 1.34036 0.670180 0.742199i \(-0.266217\pi\)
0.670180 + 0.742199i \(0.266217\pi\)
\(60\) 0 0
\(61\) 49975.9 1.71964 0.859818 0.510601i \(-0.170577\pi\)
0.859818 + 0.510601i \(0.170577\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 17363.6 0.509750
\(66\) 0 0
\(67\) −48176.2 −1.31113 −0.655565 0.755139i \(-0.727569\pi\)
−0.655565 + 0.755139i \(0.727569\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 77179.1 1.81699 0.908497 0.417891i \(-0.137230\pi\)
0.908497 + 0.417891i \(0.137230\pi\)
\(72\) 0 0
\(73\) −59667.3 −1.31048 −0.655238 0.755422i \(-0.727432\pi\)
−0.655238 + 0.755422i \(0.727432\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2869.19 0.0551484
\(78\) 0 0
\(79\) −60743.1 −1.09504 −0.547519 0.836793i \(-0.684428\pi\)
−0.547519 + 0.836793i \(0.684428\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −46134.2 −0.735068 −0.367534 0.930010i \(-0.619798\pi\)
−0.367534 + 0.930010i \(0.619798\pi\)
\(84\) 0 0
\(85\) −22033.1 −0.330771
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −78668.7 −1.05275 −0.526377 0.850251i \(-0.676450\pi\)
−0.526377 + 0.850251i \(0.676450\pi\)
\(90\) 0 0
\(91\) 57790.2 0.731562
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7344.15 −0.0834897
\(96\) 0 0
\(97\) −43573.3 −0.470209 −0.235104 0.971970i \(-0.575543\pi\)
−0.235104 + 0.971970i \(0.575543\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 164188. 1.60154 0.800772 0.598970i \(-0.204423\pi\)
0.800772 + 0.598970i \(0.204423\pi\)
\(102\) 0 0
\(103\) −164547. −1.52825 −0.764127 0.645065i \(-0.776830\pi\)
−0.764127 + 0.645065i \(0.776830\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −185426. −1.56571 −0.782854 0.622206i \(-0.786237\pi\)
−0.782854 + 0.622206i \(0.786237\pi\)
\(108\) 0 0
\(109\) 190063. 1.53225 0.766127 0.642689i \(-0.222181\pi\)
0.766127 + 0.642689i \(0.222181\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 116023. 0.854769 0.427384 0.904070i \(-0.359435\pi\)
0.427384 + 0.904070i \(0.359435\pi\)
\(114\) 0 0
\(115\) −27815.9 −0.196132
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −73331.2 −0.474702
\(120\) 0 0
\(121\) −157622. −0.978711
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −88824.6 −0.508462
\(126\) 0 0
\(127\) −71785.5 −0.394936 −0.197468 0.980309i \(-0.563272\pi\)
−0.197468 + 0.980309i \(0.563272\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2016.22 −0.0102650 −0.00513250 0.999987i \(-0.501634\pi\)
−0.00513250 + 0.999987i \(0.501634\pi\)
\(132\) 0 0
\(133\) −24443.1 −0.119819
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 243904. 1.11024 0.555120 0.831770i \(-0.312672\pi\)
0.555120 + 0.831770i \(0.312672\pi\)
\(138\) 0 0
\(139\) −209413. −0.919319 −0.459660 0.888095i \(-0.652029\pi\)
−0.459660 + 0.888095i \(0.652029\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 69059.3 0.282411
\(144\) 0 0
\(145\) −28186.9 −0.111334
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −22954.3 −0.0847028 −0.0423514 0.999103i \(-0.513485\pi\)
−0.0423514 + 0.999103i \(0.513485\pi\)
\(150\) 0 0
\(151\) 276737. 0.987700 0.493850 0.869547i \(-0.335589\pi\)
0.493850 + 0.869547i \(0.335589\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −11698.2 −0.0391101
\(156\) 0 0
\(157\) 276692. 0.895874 0.447937 0.894065i \(-0.352159\pi\)
0.447937 + 0.894065i \(0.352159\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −92577.7 −0.281476
\(162\) 0 0
\(163\) −274258. −0.808518 −0.404259 0.914645i \(-0.632471\pi\)
−0.404259 + 0.914645i \(0.632471\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −215716. −0.598537 −0.299268 0.954169i \(-0.596743\pi\)
−0.299268 + 0.954169i \(0.596743\pi\)
\(168\) 0 0
\(169\) 1.01967e6 2.74628
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −86252.7 −0.219108 −0.109554 0.993981i \(-0.534942\pi\)
−0.109554 + 0.993981i \(0.534942\pi\)
\(174\) 0 0
\(175\) −142504. −0.351749
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −435822. −1.01666 −0.508331 0.861162i \(-0.669738\pi\)
−0.508331 + 0.861162i \(0.669738\pi\)
\(180\) 0 0
\(181\) 56972.2 0.129261 0.0646303 0.997909i \(-0.479413\pi\)
0.0646303 + 0.997909i \(0.479413\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 43989.9 0.0944981
\(186\) 0 0
\(187\) −87630.7 −0.183253
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −350077. −0.694353 −0.347177 0.937800i \(-0.612860\pi\)
−0.347177 + 0.937800i \(0.612860\pi\)
\(192\) 0 0
\(193\) −623227. −1.20435 −0.602175 0.798364i \(-0.705699\pi\)
−0.602175 + 0.798364i \(0.705699\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −223265. −0.409879 −0.204939 0.978775i \(-0.565700\pi\)
−0.204939 + 0.978775i \(0.565700\pi\)
\(198\) 0 0
\(199\) −756375. −1.35396 −0.676978 0.736004i \(-0.736711\pi\)
−0.676978 + 0.736004i \(0.736711\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −93812.7 −0.159780
\(204\) 0 0
\(205\) −175806. −0.292179
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −29209.4 −0.0462549
\(210\) 0 0
\(211\) 30644.3 0.0473853 0.0236927 0.999719i \(-0.492458\pi\)
0.0236927 + 0.999719i \(0.492458\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −144578. −0.213308
\(216\) 0 0
\(217\) −38934.3 −0.0561284
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.76503e6 −2.43092
\(222\) 0 0
\(223\) −461375. −0.621286 −0.310643 0.950527i \(-0.600544\pi\)
−0.310643 + 0.950527i \(0.600544\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 346439. 0.446234 0.223117 0.974792i \(-0.428377\pi\)
0.223117 + 0.974792i \(0.428377\pi\)
\(228\) 0 0
\(229\) −521650. −0.657341 −0.328671 0.944445i \(-0.606601\pi\)
−0.328671 + 0.944445i \(0.606601\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −154120. −0.185981 −0.0929907 0.995667i \(-0.529643\pi\)
−0.0929907 + 0.995667i \(0.529643\pi\)
\(234\) 0 0
\(235\) −289091. −0.341480
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −478940. −0.542358 −0.271179 0.962529i \(-0.587414\pi\)
−0.271179 + 0.962529i \(0.587414\pi\)
\(240\) 0 0
\(241\) 86551.8 0.0959917 0.0479958 0.998848i \(-0.484717\pi\)
0.0479958 + 0.998848i \(0.484717\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 35348.8 0.0376235
\(246\) 0 0
\(247\) −588326. −0.613586
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.90322e6 −1.90679 −0.953397 0.301719i \(-0.902439\pi\)
−0.953397 + 0.301719i \(0.902439\pi\)
\(252\) 0 0
\(253\) −110630. −0.108661
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 569922. 0.538248 0.269124 0.963105i \(-0.413266\pi\)
0.269124 + 0.963105i \(0.413266\pi\)
\(258\) 0 0
\(259\) 146408. 0.135618
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.29853e6 1.15761 0.578806 0.815465i \(-0.303519\pi\)
0.578806 + 0.815465i \(0.303519\pi\)
\(264\) 0 0
\(265\) 292610. 0.255961
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 53721.5 0.0452655 0.0226328 0.999744i \(-0.492795\pi\)
0.0226328 + 0.999744i \(0.492795\pi\)
\(270\) 0 0
\(271\) 758053. 0.627013 0.313506 0.949586i \(-0.398496\pi\)
0.313506 + 0.949586i \(0.398496\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −170292. −0.135789
\(276\) 0 0
\(277\) 2.07598e6 1.62564 0.812819 0.582516i \(-0.197932\pi\)
0.812819 + 0.582516i \(0.197932\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.61195e6 −1.97333 −0.986664 0.162770i \(-0.947957\pi\)
−0.986664 + 0.162770i \(0.947957\pi\)
\(282\) 0 0
\(283\) −992734. −0.736829 −0.368414 0.929662i \(-0.620099\pi\)
−0.368414 + 0.929662i \(0.620099\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −585123. −0.419317
\(288\) 0 0
\(289\) 819820. 0.577396
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −371728. −0.252962 −0.126481 0.991969i \(-0.540368\pi\)
−0.126481 + 0.991969i \(0.540368\pi\)
\(294\) 0 0
\(295\) 527635. 0.353003
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.22827e6 −1.44142
\(300\) 0 0
\(301\) −481189. −0.306126
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 735772. 0.452891
\(306\) 0 0
\(307\) 2.83906e6 1.71921 0.859606 0.510958i \(-0.170709\pi\)
0.859606 + 0.510958i \(0.170709\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 759362. 0.445193 0.222596 0.974911i \(-0.428547\pi\)
0.222596 + 0.974911i \(0.428547\pi\)
\(312\) 0 0
\(313\) −1.80191e6 −1.03961 −0.519806 0.854284i \(-0.673996\pi\)
−0.519806 + 0.854284i \(0.673996\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 441899. 0.246987 0.123494 0.992345i \(-0.460590\pi\)
0.123494 + 0.992345i \(0.460590\pi\)
\(318\) 0 0
\(319\) −112106. −0.0616811
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 746538. 0.398149
\(324\) 0 0
\(325\) −3.42997e6 −1.80128
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −962162. −0.490070
\(330\) 0 0
\(331\) 3.15779e6 1.58421 0.792107 0.610383i \(-0.208984\pi\)
0.792107 + 0.610383i \(0.208984\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −709275. −0.345305
\(336\) 0 0
\(337\) −1.66612e6 −0.799158 −0.399579 0.916699i \(-0.630844\pi\)
−0.399579 + 0.916699i \(0.630844\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −46526.4 −0.0216677
\(342\) 0 0
\(343\) 117649. 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.41953e6 −1.52455 −0.762277 0.647251i \(-0.775919\pi\)
−0.762277 + 0.647251i \(0.775919\pi\)
\(348\) 0 0
\(349\) −3.18396e6 −1.39928 −0.699638 0.714497i \(-0.746655\pi\)
−0.699638 + 0.714497i \(0.746655\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3.02506e6 −1.29210 −0.646051 0.763294i \(-0.723581\pi\)
−0.646051 + 0.763294i \(0.723581\pi\)
\(354\) 0 0
\(355\) 1.13627e6 0.478532
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.12353e6 −0.460098 −0.230049 0.973179i \(-0.573889\pi\)
−0.230049 + 0.973179i \(0.573889\pi\)
\(360\) 0 0
\(361\) −2.22726e6 −0.899504
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −878453. −0.345133
\(366\) 0 0
\(367\) −3.39115e6 −1.31426 −0.657131 0.753777i \(-0.728230\pi\)
−0.657131 + 0.753777i \(0.728230\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 973873. 0.367340
\(372\) 0 0
\(373\) 1.33823e6 0.498033 0.249017 0.968499i \(-0.419893\pi\)
0.249017 + 0.968499i \(0.419893\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.25800e6 −0.818221
\(378\) 0 0
\(379\) −3.11403e6 −1.11359 −0.556795 0.830650i \(-0.687969\pi\)
−0.556795 + 0.830650i \(0.687969\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −334533. −0.116531 −0.0582656 0.998301i \(-0.518557\pi\)
−0.0582656 + 0.998301i \(0.518557\pi\)
\(384\) 0 0
\(385\) 42241.8 0.0145241
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.59871e6 −0.535667 −0.267834 0.963465i \(-0.586308\pi\)
−0.267834 + 0.963465i \(0.586308\pi\)
\(390\) 0 0
\(391\) 2.82750e6 0.935322
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −894292. −0.288394
\(396\) 0 0
\(397\) −2.20789e6 −0.703074 −0.351537 0.936174i \(-0.614341\pi\)
−0.351537 + 0.936174i \(0.614341\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.89272e6 −0.587793 −0.293897 0.955837i \(-0.594952\pi\)
−0.293897 + 0.955837i \(0.594952\pi\)
\(402\) 0 0
\(403\) −937118. −0.287430
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 174958. 0.0523537
\(408\) 0 0
\(409\) 1.14418e6 0.338210 0.169105 0.985598i \(-0.445912\pi\)
0.169105 + 0.985598i \(0.445912\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.75609e6 0.506608
\(414\) 0 0
\(415\) −679212. −0.193591
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5.17298e6 1.43948 0.719740 0.694244i \(-0.244261\pi\)
0.719740 + 0.694244i \(0.244261\pi\)
\(420\) 0 0
\(421\) −1.40960e6 −0.387606 −0.193803 0.981040i \(-0.562082\pi\)
−0.193803 + 0.981040i \(0.562082\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.35235e6 1.16883
\(426\) 0 0
\(427\) 2.44882e6 0.649961
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.17076e6 1.60009 0.800047 0.599938i \(-0.204808\pi\)
0.800047 + 0.599938i \(0.204808\pi\)
\(432\) 0 0
\(433\) 387046. 0.0992071 0.0496036 0.998769i \(-0.484204\pi\)
0.0496036 + 0.998769i \(0.484204\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 942475. 0.236084
\(438\) 0 0
\(439\) −3.32974e6 −0.824611 −0.412305 0.911046i \(-0.635276\pi\)
−0.412305 + 0.911046i \(0.635276\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.44003e6 −0.348627 −0.174314 0.984690i \(-0.555771\pi\)
−0.174314 + 0.984690i \(0.555771\pi\)
\(444\) 0 0
\(445\) −1.15820e6 −0.277258
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4.28977e6 −1.00420 −0.502098 0.864811i \(-0.667438\pi\)
−0.502098 + 0.864811i \(0.667438\pi\)
\(450\) 0 0
\(451\) −699222. −0.161873
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 850819. 0.192667
\(456\) 0 0
\(457\) −5.27889e6 −1.18237 −0.591183 0.806537i \(-0.701339\pi\)
−0.591183 + 0.806537i \(0.701339\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.72322e6 0.815955 0.407977 0.912992i \(-0.366234\pi\)
0.407977 + 0.912992i \(0.366234\pi\)
\(462\) 0 0
\(463\) −1.04809e6 −0.227220 −0.113610 0.993525i \(-0.536241\pi\)
−0.113610 + 0.993525i \(0.536241\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.04244e6 1.28210 0.641048 0.767501i \(-0.278500\pi\)
0.641048 + 0.767501i \(0.278500\pi\)
\(468\) 0 0
\(469\) −2.36063e6 −0.495560
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −575021. −0.118176
\(474\) 0 0
\(475\) 1.45074e6 0.295024
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.06868e6 −0.411959 −0.205980 0.978556i \(-0.566038\pi\)
−0.205980 + 0.978556i \(0.566038\pi\)
\(480\) 0 0
\(481\) 3.52394e6 0.694490
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −641509. −0.123836
\(486\) 0 0
\(487\) 2.82966e6 0.540645 0.270323 0.962770i \(-0.412870\pi\)
0.270323 + 0.962770i \(0.412870\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −4.44554e6 −0.832186 −0.416093 0.909322i \(-0.636601\pi\)
−0.416093 + 0.909322i \(0.636601\pi\)
\(492\) 0 0
\(493\) 2.86522e6 0.530934
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.78177e6 0.686759
\(498\) 0 0
\(499\) 1.34623e6 0.242029 0.121015 0.992651i \(-0.461385\pi\)
0.121015 + 0.992651i \(0.461385\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4.88909e6 −0.861604 −0.430802 0.902446i \(-0.641769\pi\)
−0.430802 + 0.902446i \(0.641769\pi\)
\(504\) 0 0
\(505\) 2.41727e6 0.421790
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.55518e6 −1.12148 −0.560738 0.827993i \(-0.689483\pi\)
−0.560738 + 0.827993i \(0.689483\pi\)
\(510\) 0 0
\(511\) −2.92370e6 −0.495313
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.42254e6 −0.402488
\(516\) 0 0
\(517\) −1.14978e6 −0.189186
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.77290e6 0.770349 0.385175 0.922844i \(-0.374141\pi\)
0.385175 + 0.922844i \(0.374141\pi\)
\(522\) 0 0
\(523\) 828135. 0.132387 0.0661937 0.997807i \(-0.478914\pi\)
0.0661937 + 0.997807i \(0.478914\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.18913e6 0.186510
\(528\) 0 0
\(529\) −2.86673e6 −0.445398
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.40835e7 −2.14730
\(534\) 0 0
\(535\) −2.72994e6 −0.412352
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 140590. 0.0208441
\(540\) 0 0
\(541\) 6.93356e6 1.01851 0.509253 0.860617i \(-0.329922\pi\)
0.509253 + 0.860617i \(0.329922\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.79820e6 0.403541
\(546\) 0 0
\(547\) −559077. −0.0798920 −0.0399460 0.999202i \(-0.512719\pi\)
−0.0399460 + 0.999202i \(0.512719\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 955047. 0.134013
\(552\) 0 0
\(553\) −2.97641e6 −0.413885
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.90444e6 0.533237 0.266619 0.963802i \(-0.414094\pi\)
0.266619 + 0.963802i \(0.414094\pi\)
\(558\) 0 0
\(559\) −1.15819e7 −1.56765
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.02307e6 −0.534918 −0.267459 0.963569i \(-0.586184\pi\)
−0.267459 + 0.963569i \(0.586184\pi\)
\(564\) 0 0
\(565\) 1.70815e6 0.225116
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.35801e6 0.305327 0.152663 0.988278i \(-0.451215\pi\)
0.152663 + 0.988278i \(0.451215\pi\)
\(570\) 0 0
\(571\) 7.76733e6 0.996969 0.498484 0.866899i \(-0.333890\pi\)
0.498484 + 0.866899i \(0.333890\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 5.49467e6 0.693062
\(576\) 0 0
\(577\) 1.38363e7 1.73013 0.865066 0.501657i \(-0.167276\pi\)
0.865066 + 0.501657i \(0.167276\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.26057e6 −0.277830
\(582\) 0 0
\(583\) 1.16378e6 0.141807
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.13747e6 0.256038 0.128019 0.991772i \(-0.459138\pi\)
0.128019 + 0.991772i \(0.459138\pi\)
\(588\) 0 0
\(589\) 396365. 0.0470768
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.11381e7 1.30069 0.650345 0.759639i \(-0.274624\pi\)
0.650345 + 0.759639i \(0.274624\pi\)
\(594\) 0 0
\(595\) −1.07962e6 −0.125020
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.10044e7 1.25314 0.626572 0.779364i \(-0.284457\pi\)
0.626572 + 0.779364i \(0.284457\pi\)
\(600\) 0 0
\(601\) 1.28046e6 0.144604 0.0723019 0.997383i \(-0.476965\pi\)
0.0723019 + 0.997383i \(0.476965\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2.32060e6 −0.257758
\(606\) 0 0
\(607\) 2.08327e6 0.229496 0.114748 0.993395i \(-0.463394\pi\)
0.114748 + 0.993395i \(0.463394\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.31585e7 −2.50962
\(612\) 0 0
\(613\) −1.37116e7 −1.47380 −0.736899 0.676003i \(-0.763711\pi\)
−0.736899 + 0.676003i \(0.763711\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.65172e7 1.74672 0.873361 0.487073i \(-0.161936\pi\)
0.873361 + 0.487073i \(0.161936\pi\)
\(618\) 0 0
\(619\) −3.51321e6 −0.368534 −0.184267 0.982876i \(-0.558991\pi\)
−0.184267 + 0.982876i \(0.558991\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.85477e6 −0.397904
\(624\) 0 0
\(625\) 7.78055e6 0.796728
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −4.47160e6 −0.450647
\(630\) 0 0
\(631\) 8.54135e6 0.853991 0.426995 0.904254i \(-0.359572\pi\)
0.426995 + 0.904254i \(0.359572\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.05686e6 −0.104012
\(636\) 0 0
\(637\) 2.83172e6 0.276504
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.95519e6 0.956983 0.478492 0.878092i \(-0.341184\pi\)
0.478492 + 0.878092i \(0.341184\pi\)
\(642\) 0 0
\(643\) 3.98313e6 0.379924 0.189962 0.981791i \(-0.439163\pi\)
0.189962 + 0.981791i \(0.439163\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −3.84121e6 −0.360750 −0.180375 0.983598i \(-0.557731\pi\)
−0.180375 + 0.983598i \(0.557731\pi\)
\(648\) 0 0
\(649\) 2.09853e6 0.195570
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.84286e7 −1.69126 −0.845628 0.533772i \(-0.820774\pi\)
−0.845628 + 0.533772i \(0.820774\pi\)
\(654\) 0 0
\(655\) −29683.8 −0.00270344
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.15854e7 1.93619 0.968093 0.250590i \(-0.0806245\pi\)
0.968093 + 0.250590i \(0.0806245\pi\)
\(660\) 0 0
\(661\) 1.44072e7 1.28256 0.641278 0.767308i \(-0.278404\pi\)
0.641278 + 0.767308i \(0.278404\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −359864. −0.0315561
\(666\) 0 0
\(667\) 3.61723e6 0.314819
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.92634e6 0.250910
\(672\) 0 0
\(673\) −5.55230e6 −0.472536 −0.236268 0.971688i \(-0.575924\pi\)
−0.236268 + 0.971688i \(0.575924\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9.92009e6 0.831848 0.415924 0.909399i \(-0.363458\pi\)
0.415924 + 0.909399i \(0.363458\pi\)
\(678\) 0 0
\(679\) −2.13509e6 −0.177722
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.01291e7 −0.830846 −0.415423 0.909628i \(-0.636366\pi\)
−0.415423 + 0.909628i \(0.636366\pi\)
\(684\) 0 0
\(685\) 3.59088e6 0.292398
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.34404e7 1.88112
\(690\) 0 0
\(691\) −1.98747e7 −1.58345 −0.791727 0.610875i \(-0.790818\pi\)
−0.791727 + 0.610875i \(0.790818\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.08309e6 −0.242116
\(696\) 0 0
\(697\) 1.78708e7 1.39336
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.13147e7 1.63826 0.819132 0.573605i \(-0.194455\pi\)
0.819132 + 0.573605i \(0.194455\pi\)
\(702\) 0 0
\(703\) −1.49049e6 −0.113747
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.04523e6 0.605327
\(708\) 0 0
\(709\) 9.83200e6 0.734559 0.367279 0.930111i \(-0.380289\pi\)
0.367279 + 0.930111i \(0.380289\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.50123e6 0.110592
\(714\) 0 0
\(715\) 1.01673e6 0.0743771
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1.80674e7 −1.30338 −0.651692 0.758484i \(-0.725940\pi\)
−0.651692 + 0.758484i \(0.725940\pi\)
\(720\) 0 0
\(721\) −8.06278e6 −0.577626
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.56797e6 0.393416
\(726\) 0 0
\(727\) 1.82003e7 1.27715 0.638576 0.769559i \(-0.279524\pi\)
0.638576 + 0.769559i \(0.279524\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.46965e7 1.01723
\(732\) 0 0
\(733\) 6.70629e6 0.461023 0.230512 0.973070i \(-0.425960\pi\)
0.230512 + 0.973070i \(0.425960\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.82095e6 −0.191305
\(738\) 0 0
\(739\) −2.16502e7 −1.45831 −0.729156 0.684347i \(-0.760087\pi\)
−0.729156 + 0.684347i \(0.760087\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.40437e7 −0.933276 −0.466638 0.884448i \(-0.654535\pi\)
−0.466638 + 0.884448i \(0.654535\pi\)
\(744\) 0 0
\(745\) −337945. −0.0223077
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.08586e6 −0.591782
\(750\) 0 0
\(751\) −8.91381e6 −0.576718 −0.288359 0.957522i \(-0.593110\pi\)
−0.288359 + 0.957522i \(0.593110\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4.07427e6 0.260125
\(756\) 0 0
\(757\) −1.04419e7 −0.662276 −0.331138 0.943582i \(-0.607433\pi\)
−0.331138 + 0.943582i \(0.607433\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.96214e7 −1.85414 −0.927072 0.374883i \(-0.877683\pi\)
−0.927072 + 0.374883i \(0.877683\pi\)
\(762\) 0 0
\(763\) 9.31307e6 0.579138
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.22678e7 2.59431
\(768\) 0 0
\(769\) −4.24042e6 −0.258579 −0.129289 0.991607i \(-0.541270\pi\)
−0.129289 + 0.991607i \(0.541270\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.37072e7 0.825086 0.412543 0.910938i \(-0.364641\pi\)
0.412543 + 0.910938i \(0.364641\pi\)
\(774\) 0 0
\(775\) 2.31083e6 0.138202
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.95677e6 0.351696
\(780\) 0 0
\(781\) 4.51922e6 0.265116
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.07360e6 0.235942
\(786\) 0 0
\(787\) −1.02427e7 −0.589491 −0.294745 0.955576i \(-0.595235\pi\)
−0.294745 + 0.955576i \(0.595235\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.68513e6 0.323072
\(792\) 0 0
\(793\) 5.89413e7 3.32841
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.00087e7 −1.67340 −0.836702 0.547658i \(-0.815520\pi\)
−0.836702 + 0.547658i \(0.815520\pi\)
\(798\) 0 0
\(799\) 2.93863e7 1.62846
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.49381e6 −0.191210
\(804\) 0 0
\(805\) −1.36298e6 −0.0741309
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −922081. −0.0495333 −0.0247667 0.999693i \(-0.507884\pi\)
−0.0247667 + 0.999693i \(0.507884\pi\)
\(810\) 0 0
\(811\) 2.49323e7 1.33110 0.665549 0.746355i \(-0.268198\pi\)
0.665549 + 0.746355i \(0.268198\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −4.03776e6 −0.212935
\(816\) 0 0
\(817\) 4.89868e6 0.256758
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.06228e6 0.417446 0.208723 0.977975i \(-0.433069\pi\)
0.208723 + 0.977975i \(0.433069\pi\)
\(822\) 0 0
\(823\) 5.68168e6 0.292400 0.146200 0.989255i \(-0.453296\pi\)
0.146200 + 0.989255i \(0.453296\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −976611. −0.0496544 −0.0248272 0.999692i \(-0.507904\pi\)
−0.0248272 + 0.999692i \(0.507904\pi\)
\(828\) 0 0
\(829\) −4.94857e6 −0.250088 −0.125044 0.992151i \(-0.539907\pi\)
−0.125044 + 0.992151i \(0.539907\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.59323e6 −0.179421
\(834\) 0 0
\(835\) −3.17588e6 −0.157633
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −2.03716e7 −0.999125 −0.499562 0.866278i \(-0.666506\pi\)
−0.499562 + 0.866278i \(0.666506\pi\)
\(840\) 0 0
\(841\) −1.68457e7 −0.821293
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.50122e7 0.723273
\(846\) 0 0
\(847\) −7.72349e6 −0.369918
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −5.64521e6 −0.267212
\(852\) 0 0
\(853\) 1.41579e7 0.666234 0.333117 0.942885i \(-0.391900\pi\)
0.333117 + 0.942885i \(0.391900\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.24397e7 0.578575 0.289287 0.957242i \(-0.406582\pi\)
0.289287 + 0.957242i \(0.406582\pi\)
\(858\) 0 0
\(859\) −1.72105e7 −0.795810 −0.397905 0.917427i \(-0.630263\pi\)
−0.397905 + 0.917427i \(0.630263\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.04374e7 0.934113 0.467057 0.884227i \(-0.345314\pi\)
0.467057 + 0.884227i \(0.345314\pi\)
\(864\) 0 0
\(865\) −1.26986e6 −0.0577052
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −3.55681e6 −0.159776
\(870\) 0 0
\(871\) −5.68187e7 −2.53773
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.35241e6 −0.192181
\(876\) 0 0
\(877\) −1.72398e7 −0.756890 −0.378445 0.925624i \(-0.623541\pi\)
−0.378445 + 0.925624i \(0.623541\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 3.86589e7 1.67807 0.839034 0.544080i \(-0.183121\pi\)
0.839034 + 0.544080i \(0.183121\pi\)
\(882\) 0 0
\(883\) 3.29454e7 1.42198 0.710990 0.703203i \(-0.248247\pi\)
0.710990 + 0.703203i \(0.248247\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.61545e7 1.54295 0.771477 0.636258i \(-0.219518\pi\)
0.771477 + 0.636258i \(0.219518\pi\)
\(888\) 0 0
\(889\) −3.51749e6 −0.149272
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 9.79516e6 0.411039
\(894\) 0 0
\(895\) −6.41641e6 −0.267753
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.52125e6 0.0627772
\(900\) 0 0
\(901\) −2.97440e7 −1.22064
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 838774. 0.0340427
\(906\) 0 0
\(907\) 4.15620e7 1.67756 0.838781 0.544469i \(-0.183269\pi\)
0.838781 + 0.544469i \(0.183269\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −9.55917e6 −0.381614 −0.190807 0.981628i \(-0.561110\pi\)
−0.190807 + 0.981628i \(0.561110\pi\)
\(912\) 0 0
\(913\) −2.70138e6 −0.107253
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −98794.6 −0.00387980
\(918\) 0 0
\(919\) 1.20453e7 0.470469 0.235234 0.971939i \(-0.424414\pi\)
0.235234 + 0.971939i \(0.424414\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 9.10244e7 3.51685
\(924\) 0 0
\(925\) −8.68963e6 −0.333924
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.42707e7 0.922661 0.461331 0.887228i \(-0.347372\pi\)
0.461331 + 0.887228i \(0.347372\pi\)
\(930\) 0 0
\(931\) −1.19771e6 −0.0452874
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.29015e6 −0.0482625
\(936\) 0 0
\(937\) 2.02014e7 0.751679 0.375840 0.926685i \(-0.377354\pi\)
0.375840 + 0.926685i \(0.377354\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 6.68084e6 0.245956 0.122978 0.992409i \(-0.460756\pi\)
0.122978 + 0.992409i \(0.460756\pi\)
\(942\) 0 0
\(943\) 2.25612e7 0.826195
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −3.19373e7 −1.15724 −0.578619 0.815598i \(-0.696408\pi\)
−0.578619 + 0.815598i \(0.696408\pi\)
\(948\) 0 0
\(949\) −7.03712e7 −2.53647
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2.26242e7 −0.806938 −0.403469 0.914993i \(-0.632196\pi\)
−0.403469 + 0.914993i \(0.632196\pi\)
\(954\) 0 0
\(955\) −5.15402e6 −0.182868
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.19513e7 0.419631
\(960\) 0 0
\(961\) −2.79978e7 −0.977947
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −9.17547e6 −0.317183
\(966\) 0 0
\(967\) 2.38201e7 0.819178 0.409589 0.912270i \(-0.365672\pi\)
0.409589 + 0.912270i \(0.365672\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.87074e7 0.636746 0.318373 0.947965i \(-0.396864\pi\)
0.318373 + 0.947965i \(0.396864\pi\)
\(972\) 0 0
\(973\) −1.02612e7 −0.347470
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.50783e7 −0.505376 −0.252688 0.967548i \(-0.581315\pi\)
−0.252688 + 0.967548i \(0.581315\pi\)
\(978\) 0 0
\(979\) −4.60644e6 −0.153606
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.38312e6 0.0786614 0.0393307 0.999226i \(-0.487477\pi\)
0.0393307 + 0.999226i \(0.487477\pi\)
\(984\) 0 0
\(985\) −3.28703e6 −0.107948
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.85537e7 0.603170
\(990\) 0 0
\(991\) 4.86331e7 1.57307 0.786535 0.617546i \(-0.211873\pi\)
0.786535 + 0.617546i \(0.211873\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.11357e7 −0.356584
\(996\) 0 0
\(997\) 1.73445e7 0.552617 0.276309 0.961069i \(-0.410889\pi\)
0.276309 + 0.961069i \(0.410889\pi\)
\(998\) 0 0
\(999\) 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 1008.6.a.bd.1.2 2
3.2 odd 2 112.6.a.i.1.2 2
4.3 odd 2 504.6.a.i.1.2 2
12.11 even 2 56.6.a.e.1.1 2
21.20 even 2 784.6.a.u.1.1 2
24.5 odd 2 448.6.a.t.1.1 2
24.11 even 2 448.6.a.v.1.2 2
84.11 even 6 392.6.i.j.177.2 4
84.23 even 6 392.6.i.j.361.2 4
84.47 odd 6 392.6.i.i.361.1 4
84.59 odd 6 392.6.i.i.177.1 4
84.83 odd 2 392.6.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.a.e.1.1 2 12.11 even 2
112.6.a.i.1.2 2 3.2 odd 2
392.6.a.d.1.2 2 84.83 odd 2
392.6.i.i.177.1 4 84.59 odd 6
392.6.i.i.361.1 4 84.47 odd 6
392.6.i.j.177.2 4 84.11 even 6
392.6.i.j.361.2 4 84.23 even 6
448.6.a.t.1.1 2 24.5 odd 2
448.6.a.v.1.2 2 24.11 even 2
504.6.a.i.1.2 2 4.3 odd 2
784.6.a.u.1.1 2 21.20 even 2
1008.6.a.bd.1.2 2 1.1 even 1 trivial