Properties

Label 1200.3.e.n.751.1
Level $1200$
Weight $3$
Character 1200.751
Analytic conductor $32.698$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,3,Mod(751,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.751");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.e (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 751.1
Root \(-0.309017 - 0.535233i\) of defining polynomial
Character \(\chi\) \(=\) 1200.751
Dual form 1200.3.e.n.751.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205i q^{3} -4.28187i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{3} -4.28187i q^{7} -3.00000 q^{9} +11.2101i q^{11} -5.41641 q^{13} +1.41641 q^{17} +8.56373i q^{19} -7.41641 q^{21} -24.0557i q^{23} +5.19615i q^{27} -25.4164 q^{29} +50.1329i q^{31} +19.4164 q^{33} -60.2492 q^{37} +9.38149i q^{39} -30.0000 q^{41} +37.9121i q^{43} +39.9337i q^{47} +30.6656 q^{49} -2.45329i q^{51} +16.2492 q^{53} +14.8328 q^{57} -81.7415i q^{59} +94.4984 q^{61} +12.8456i q^{63} +129.614i q^{67} -41.6656 q^{69} +102.288i q^{71} -24.8328 q^{73} +48.0000 q^{77} -91.7022i q^{79} +9.00000 q^{81} +88.0450i q^{83} +44.0225i q^{87} -20.8328 q^{89} +23.1923i q^{91} +86.8328 q^{93} +14.0000 q^{97} -33.6302i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{9} + 32 q^{13} - 48 q^{17} + 24 q^{21} - 48 q^{29} + 24 q^{33} - 80 q^{37} - 120 q^{41} - 92 q^{49} - 96 q^{53} - 48 q^{57} + 56 q^{61} + 48 q^{69} + 8 q^{73} + 192 q^{77} + 36 q^{81} + 24 q^{89} + 240 q^{93} + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.73205i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.28187i − 0.611695i −0.952081 0.305848i \(-0.901060\pi\)
0.952081 0.305848i \(-0.0989398\pi\)
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) 11.2101i 1.01910i 0.860442 + 0.509549i \(0.170188\pi\)
−0.860442 + 0.509549i \(0.829812\pi\)
\(12\) 0 0
\(13\) −5.41641 −0.416647 −0.208323 0.978060i \(-0.566801\pi\)
−0.208323 + 0.978060i \(0.566801\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.41641 0.0833181 0.0416591 0.999132i \(-0.486736\pi\)
0.0416591 + 0.999132i \(0.486736\pi\)
\(18\) 0 0
\(19\) 8.56373i 0.450723i 0.974275 + 0.225361i \(0.0723563\pi\)
−0.974275 + 0.225361i \(0.927644\pi\)
\(20\) 0 0
\(21\) −7.41641 −0.353162
\(22\) 0 0
\(23\) − 24.0557i − 1.04590i −0.852364 0.522949i \(-0.824832\pi\)
0.852364 0.522949i \(-0.175168\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) −25.4164 −0.876428 −0.438214 0.898871i \(-0.644389\pi\)
−0.438214 + 0.898871i \(0.644389\pi\)
\(30\) 0 0
\(31\) 50.1329i 1.61719i 0.588364 + 0.808596i \(0.299772\pi\)
−0.588364 + 0.808596i \(0.700228\pi\)
\(32\) 0 0
\(33\) 19.4164 0.588376
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −60.2492 −1.62836 −0.814179 0.580614i \(-0.802812\pi\)
−0.814179 + 0.580614i \(0.802812\pi\)
\(38\) 0 0
\(39\) 9.38149i 0.240551i
\(40\) 0 0
\(41\) −30.0000 −0.731707 −0.365854 0.930672i \(-0.619223\pi\)
−0.365854 + 0.930672i \(0.619223\pi\)
\(42\) 0 0
\(43\) 37.9121i 0.881676i 0.897587 + 0.440838i \(0.145319\pi\)
−0.897587 + 0.440838i \(0.854681\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 39.9337i 0.849653i 0.905275 + 0.424827i \(0.139665\pi\)
−0.905275 + 0.424827i \(0.860335\pi\)
\(48\) 0 0
\(49\) 30.6656 0.625829
\(50\) 0 0
\(51\) − 2.45329i − 0.0481037i
\(52\) 0 0
\(53\) 16.2492 0.306589 0.153295 0.988181i \(-0.451012\pi\)
0.153295 + 0.988181i \(0.451012\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 14.8328 0.260225
\(58\) 0 0
\(59\) − 81.7415i − 1.38545i −0.721202 0.692725i \(-0.756410\pi\)
0.721202 0.692725i \(-0.243590\pi\)
\(60\) 0 0
\(61\) 94.4984 1.54915 0.774577 0.632479i \(-0.217963\pi\)
0.774577 + 0.632479i \(0.217963\pi\)
\(62\) 0 0
\(63\) 12.8456i 0.203898i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 129.614i 1.93454i 0.253748 + 0.967270i \(0.418337\pi\)
−0.253748 + 0.967270i \(0.581663\pi\)
\(68\) 0 0
\(69\) −41.6656 −0.603850
\(70\) 0 0
\(71\) 102.288i 1.44067i 0.693627 + 0.720335i \(0.256012\pi\)
−0.693627 + 0.720335i \(0.743988\pi\)
\(72\) 0 0
\(73\) −24.8328 −0.340176 −0.170088 0.985429i \(-0.554405\pi\)
−0.170088 + 0.985429i \(0.554405\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 48.0000 0.623377
\(78\) 0 0
\(79\) − 91.7022i − 1.16079i −0.814336 0.580393i \(-0.802899\pi\)
0.814336 0.580393i \(-0.197101\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 88.0450i 1.06078i 0.847753 + 0.530392i \(0.177955\pi\)
−0.847753 + 0.530392i \(0.822045\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 44.0225i 0.506006i
\(88\) 0 0
\(89\) −20.8328 −0.234077 −0.117038 0.993127i \(-0.537340\pi\)
−0.117038 + 0.993127i \(0.537340\pi\)
\(90\) 0 0
\(91\) 23.1923i 0.254861i
\(92\) 0 0
\(93\) 86.8328 0.933686
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000 0.144330 0.0721649 0.997393i \(-0.477009\pi\)
0.0721649 + 0.997393i \(0.477009\pi\)
\(98\) 0 0
\(99\) − 33.6302i − 0.339699i
\(100\) 0 0
\(101\) −132.748 −1.31433 −0.657167 0.753745i \(-0.728245\pi\)
−0.657167 + 0.753745i \(0.728245\pi\)
\(102\) 0 0
\(103\) − 62.9785i − 0.611442i −0.952121 0.305721i \(-0.901102\pi\)
0.952121 0.305721i \(-0.0988975\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 174.455i 1.63042i 0.579168 + 0.815208i \(0.303377\pi\)
−0.579168 + 0.815208i \(0.696623\pi\)
\(108\) 0 0
\(109\) 96.8328 0.888374 0.444187 0.895934i \(-0.353493\pi\)
0.444187 + 0.895934i \(0.353493\pi\)
\(110\) 0 0
\(111\) 104.355i 0.940133i
\(112\) 0 0
\(113\) −211.082 −1.86798 −0.933991 0.357296i \(-0.883699\pi\)
−0.933991 + 0.357296i \(0.883699\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 16.2492 0.138882
\(118\) 0 0
\(119\) − 6.06487i − 0.0509653i
\(120\) 0 0
\(121\) −4.66563 −0.0385589
\(122\) 0 0
\(123\) 51.9615i 0.422451i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 44.6017i 0.351194i 0.984462 + 0.175597i \(0.0561856\pi\)
−0.984462 + 0.175597i \(0.943814\pi\)
\(128\) 0 0
\(129\) 65.6656 0.509036
\(130\) 0 0
\(131\) 161.609i 1.23366i 0.787098 + 0.616828i \(0.211583\pi\)
−0.787098 + 0.616828i \(0.788417\pi\)
\(132\) 0 0
\(133\) 36.6687 0.275705
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 106.584 0.777982 0.388991 0.921241i \(-0.372824\pi\)
0.388991 + 0.921241i \(0.372824\pi\)
\(138\) 0 0
\(139\) 24.4418i 0.175840i 0.996128 + 0.0879200i \(0.0280220\pi\)
−0.996128 + 0.0879200i \(0.971978\pi\)
\(140\) 0 0
\(141\) 69.1672 0.490547
\(142\) 0 0
\(143\) − 60.7183i − 0.424604i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 53.1144i − 0.361323i
\(148\) 0 0
\(149\) 91.0820 0.611289 0.305644 0.952146i \(-0.401128\pi\)
0.305644 + 0.952146i \(0.401128\pi\)
\(150\) 0 0
\(151\) 251.914i 1.66831i 0.551533 + 0.834153i \(0.314043\pi\)
−0.551533 + 0.834153i \(0.685957\pi\)
\(152\) 0 0
\(153\) −4.24922 −0.0277727
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −21.4164 −0.136410 −0.0682051 0.997671i \(-0.521727\pi\)
−0.0682051 + 0.997671i \(0.521727\pi\)
\(158\) 0 0
\(159\) − 28.1445i − 0.177009i
\(160\) 0 0
\(161\) −103.003 −0.639771
\(162\) 0 0
\(163\) − 14.7197i − 0.0903052i −0.998980 0.0451526i \(-0.985623\pi\)
0.998980 0.0451526i \(-0.0143774\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 78.7091i − 0.471312i −0.971837 0.235656i \(-0.924276\pi\)
0.971837 0.235656i \(-0.0757239\pi\)
\(168\) 0 0
\(169\) −139.663 −0.826405
\(170\) 0 0
\(171\) − 25.6912i − 0.150241i
\(172\) 0 0
\(173\) −146.085 −0.844423 −0.422211 0.906497i \(-0.638746\pi\)
−0.422211 + 0.906497i \(0.638746\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −141.580 −0.799890
\(178\) 0 0
\(179\) 238.682i 1.33342i 0.745316 + 0.666711i \(0.232298\pi\)
−0.745316 + 0.666711i \(0.767702\pi\)
\(180\) 0 0
\(181\) 115.167 0.636283 0.318141 0.948043i \(-0.396941\pi\)
0.318141 + 0.948043i \(0.396941\pi\)
\(182\) 0 0
\(183\) − 163.676i − 0.894405i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 15.8780i 0.0849092i
\(188\) 0 0
\(189\) 22.2492 0.117721
\(190\) 0 0
\(191\) 6.06487i 0.0317532i 0.999874 + 0.0158766i \(0.00505390\pi\)
−0.999874 + 0.0158766i \(0.994946\pi\)
\(192\) 0 0
\(193\) 358.827 1.85921 0.929603 0.368563i \(-0.120150\pi\)
0.929603 + 0.368563i \(0.120150\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 141.915 0.720380 0.360190 0.932879i \(-0.382712\pi\)
0.360190 + 0.932879i \(0.382712\pi\)
\(198\) 0 0
\(199\) − 124.708i − 0.626672i −0.949642 0.313336i \(-0.898553\pi\)
0.949642 0.313336i \(-0.101447\pi\)
\(200\) 0 0
\(201\) 224.498 1.11691
\(202\) 0 0
\(203\) 108.830i 0.536107i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 72.1670i 0.348633i
\(208\) 0 0
\(209\) −96.0000 −0.459330
\(210\) 0 0
\(211\) 102.765i 0.487037i 0.969896 + 0.243518i \(0.0783016\pi\)
−0.969896 + 0.243518i \(0.921698\pi\)
\(212\) 0 0
\(213\) 177.167 0.831771
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 214.663 0.989228
\(218\) 0 0
\(219\) 43.0117i 0.196400i
\(220\) 0 0
\(221\) −7.67184 −0.0347142
\(222\) 0 0
\(223\) 39.7862i 0.178414i 0.996013 + 0.0892068i \(0.0284332\pi\)
−0.996013 + 0.0892068i \(0.971567\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 14.2425i − 0.0627423i −0.999508 0.0313711i \(-0.990013\pi\)
0.999508 0.0313711i \(-0.00998739\pi\)
\(228\) 0 0
\(229\) −45.0031 −0.196520 −0.0982601 0.995161i \(-0.531328\pi\)
−0.0982601 + 0.995161i \(0.531328\pi\)
\(230\) 0 0
\(231\) − 83.1384i − 0.359907i
\(232\) 0 0
\(233\) −393.915 −1.69062 −0.845311 0.534275i \(-0.820585\pi\)
−0.845311 + 0.534275i \(0.820585\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −158.833 −0.670181
\(238\) 0 0
\(239\) − 329.760i − 1.37975i −0.723929 0.689874i \(-0.757666\pi\)
0.723929 0.689874i \(-0.242334\pi\)
\(240\) 0 0
\(241\) −210.328 −0.872731 −0.436365 0.899770i \(-0.643735\pi\)
−0.436365 + 0.899770i \(0.643735\pi\)
\(242\) 0 0
\(243\) − 15.5885i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 46.3847i − 0.187792i
\(248\) 0 0
\(249\) 152.498 0.612444
\(250\) 0 0
\(251\) − 254.083i − 1.01228i −0.862450 0.506142i \(-0.831071\pi\)
0.862450 0.506142i \(-0.168929\pi\)
\(252\) 0 0
\(253\) 269.666 1.06587
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −407.410 −1.58525 −0.792627 0.609707i \(-0.791287\pi\)
−0.792627 + 0.609707i \(0.791287\pi\)
\(258\) 0 0
\(259\) 257.979i 0.996058i
\(260\) 0 0
\(261\) 76.2492 0.292143
\(262\) 0 0
\(263\) − 138.950i − 0.528328i −0.964478 0.264164i \(-0.914904\pi\)
0.964478 0.264164i \(-0.0850960\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 36.0835i 0.135144i
\(268\) 0 0
\(269\) −55.0820 −0.204766 −0.102383 0.994745i \(-0.532647\pi\)
−0.102383 + 0.994745i \(0.532647\pi\)
\(270\) 0 0
\(271\) − 406.061i − 1.49838i −0.662354 0.749191i \(-0.730443\pi\)
0.662354 0.749191i \(-0.269557\pi\)
\(272\) 0 0
\(273\) 40.1703 0.147144
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −350.584 −1.26564 −0.632822 0.774297i \(-0.718104\pi\)
−0.632822 + 0.774297i \(0.718104\pi\)
\(278\) 0 0
\(279\) − 150.399i − 0.539064i
\(280\) 0 0
\(281\) −457.161 −1.62691 −0.813454 0.581630i \(-0.802415\pi\)
−0.813454 + 0.581630i \(0.802415\pi\)
\(282\) 0 0
\(283\) 339.959i 1.20127i 0.799524 + 0.600635i \(0.205085\pi\)
−0.799524 + 0.600635i \(0.794915\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 128.456i 0.447582i
\(288\) 0 0
\(289\) −286.994 −0.993058
\(290\) 0 0
\(291\) − 24.2487i − 0.0833289i
\(292\) 0 0
\(293\) −575.410 −1.96386 −0.981929 0.189251i \(-0.939394\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −58.2492 −0.196125
\(298\) 0 0
\(299\) 130.295i 0.435770i
\(300\) 0 0
\(301\) 162.334 0.539317
\(302\) 0 0
\(303\) 229.926i 0.758831i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 358.336i − 1.16722i −0.812035 0.583609i \(-0.801640\pi\)
0.812035 0.583609i \(-0.198360\pi\)
\(308\) 0 0
\(309\) −109.082 −0.353016
\(310\) 0 0
\(311\) − 364.787i − 1.17295i −0.809968 0.586474i \(-0.800515\pi\)
0.809968 0.586474i \(-0.199485\pi\)
\(312\) 0 0
\(313\) 265.502 0.848248 0.424124 0.905604i \(-0.360582\pi\)
0.424124 + 0.905604i \(0.360582\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 459.580 1.44978 0.724890 0.688864i \(-0.241890\pi\)
0.724890 + 0.688864i \(0.241890\pi\)
\(318\) 0 0
\(319\) − 284.920i − 0.893165i
\(320\) 0 0
\(321\) 302.164 0.941321
\(322\) 0 0
\(323\) 12.1297i 0.0375534i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 167.719i − 0.512903i
\(328\) 0 0
\(329\) 170.991 0.519729
\(330\) 0 0
\(331\) − 583.219i − 1.76199i −0.473126 0.880995i \(-0.656875\pi\)
0.473126 0.880995i \(-0.343125\pi\)
\(332\) 0 0
\(333\) 180.748 0.542786
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 123.495 0.366455 0.183228 0.983071i \(-0.441346\pi\)
0.183228 + 0.983071i \(0.441346\pi\)
\(338\) 0 0
\(339\) 365.605i 1.07848i
\(340\) 0 0
\(341\) −561.994 −1.64808
\(342\) 0 0
\(343\) − 341.117i − 0.994512i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 487.859i − 1.40593i −0.711222 0.702967i \(-0.751858\pi\)
0.711222 0.702967i \(-0.248142\pi\)
\(348\) 0 0
\(349\) −12.8328 −0.0367702 −0.0183851 0.999831i \(-0.505852\pi\)
−0.0183851 + 0.999831i \(0.505852\pi\)
\(350\) 0 0
\(351\) − 28.1445i − 0.0801837i
\(352\) 0 0
\(353\) 5.74457 0.0162736 0.00813678 0.999967i \(-0.497410\pi\)
0.00813678 + 0.999967i \(0.497410\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −10.5047 −0.0294248
\(358\) 0 0
\(359\) − 44.8403i − 0.124903i −0.998048 0.0624516i \(-0.980108\pi\)
0.998048 0.0624516i \(-0.0198919\pi\)
\(360\) 0 0
\(361\) 287.663 0.796849
\(362\) 0 0
\(363\) 8.08111i 0.0222620i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 83.8543i 0.228486i 0.993453 + 0.114243i \(0.0364442\pi\)
−0.993453 + 0.114243i \(0.963556\pi\)
\(368\) 0 0
\(369\) 90.0000 0.243902
\(370\) 0 0
\(371\) − 69.5770i − 0.187539i
\(372\) 0 0
\(373\) −110.413 −0.296014 −0.148007 0.988986i \(-0.547286\pi\)
−0.148007 + 0.988986i \(0.547286\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 137.666 0.365161
\(378\) 0 0
\(379\) 662.791i 1.74879i 0.485216 + 0.874394i \(0.338741\pi\)
−0.485216 + 0.874394i \(0.661259\pi\)
\(380\) 0 0
\(381\) 77.2523 0.202762
\(382\) 0 0
\(383\) − 84.7740i − 0.221342i −0.993857 0.110671i \(-0.964700\pi\)
0.993857 0.110671i \(-0.0353000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 113.736i − 0.293892i
\(388\) 0 0
\(389\) −379.240 −0.974910 −0.487455 0.873148i \(-0.662075\pi\)
−0.487455 + 0.873148i \(0.662075\pi\)
\(390\) 0 0
\(391\) − 34.0726i − 0.0871423i
\(392\) 0 0
\(393\) 279.915 0.712252
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −94.2430 −0.237388 −0.118694 0.992931i \(-0.537871\pi\)
−0.118694 + 0.992931i \(0.537871\pi\)
\(398\) 0 0
\(399\) − 63.5121i − 0.159178i
\(400\) 0 0
\(401\) 246.158 0.613860 0.306930 0.951732i \(-0.400698\pi\)
0.306930 + 0.951732i \(0.400698\pi\)
\(402\) 0 0
\(403\) − 271.541i − 0.673798i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 675.398i − 1.65945i
\(408\) 0 0
\(409\) 584.663 1.42949 0.714746 0.699384i \(-0.246542\pi\)
0.714746 + 0.699384i \(0.246542\pi\)
\(410\) 0 0
\(411\) − 184.608i − 0.449168i
\(412\) 0 0
\(413\) −350.006 −0.847473
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 42.3344 0.101521
\(418\) 0 0
\(419\) 645.039i 1.53947i 0.638362 + 0.769736i \(0.279612\pi\)
−0.638362 + 0.769736i \(0.720388\pi\)
\(420\) 0 0
\(421\) 468.991 1.11399 0.556996 0.830515i \(-0.311954\pi\)
0.556996 + 0.830515i \(0.311954\pi\)
\(422\) 0 0
\(423\) − 119.801i − 0.283218i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 404.630i − 0.947610i
\(428\) 0 0
\(429\) −105.167 −0.245145
\(430\) 0 0
\(431\) 150.876i 0.350061i 0.984563 + 0.175030i \(0.0560023\pi\)
−0.984563 + 0.175030i \(0.943998\pi\)
\(432\) 0 0
\(433\) 25.5016 0.0588950 0.0294475 0.999566i \(-0.490625\pi\)
0.0294475 + 0.999566i \(0.490625\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 206.006 0.471410
\(438\) 0 0
\(439\) 44.0681i 0.100383i 0.998740 + 0.0501914i \(0.0159831\pi\)
−0.998740 + 0.0501914i \(0.984017\pi\)
\(440\) 0 0
\(441\) −91.9969 −0.208610
\(442\) 0 0
\(443\) − 95.0644i − 0.214592i −0.994227 0.107296i \(-0.965781\pi\)
0.994227 0.107296i \(-0.0342193\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 157.759i − 0.352928i
\(448\) 0 0
\(449\) −194.498 −0.433181 −0.216591 0.976263i \(-0.569494\pi\)
−0.216591 + 0.976263i \(0.569494\pi\)
\(450\) 0 0
\(451\) − 336.302i − 0.745681i
\(452\) 0 0
\(453\) 436.328 0.963197
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −289.659 −0.633828 −0.316914 0.948454i \(-0.602647\pi\)
−0.316914 + 0.948454i \(0.602647\pi\)
\(458\) 0 0
\(459\) 7.35987i 0.0160346i
\(460\) 0 0
\(461\) −640.249 −1.38883 −0.694413 0.719576i \(-0.744336\pi\)
−0.694413 + 0.719576i \(0.744336\pi\)
\(462\) 0 0
\(463\) − 514.175i − 1.11053i −0.831674 0.555265i \(-0.812617\pi\)
0.831674 0.555265i \(-0.187383\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 69.3732i − 0.148551i −0.997238 0.0742754i \(-0.976336\pi\)
0.997238 0.0742754i \(-0.0236644\pi\)
\(468\) 0 0
\(469\) 554.991 1.18335
\(470\) 0 0
\(471\) 37.0943i 0.0787565i
\(472\) 0 0
\(473\) −424.997 −0.898514
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −48.7477 −0.102196
\(478\) 0 0
\(479\) − 502.579i − 1.04923i −0.851341 0.524613i \(-0.824210\pi\)
0.851341 0.524613i \(-0.175790\pi\)
\(480\) 0 0
\(481\) 326.334 0.678450
\(482\) 0 0
\(483\) 178.407i 0.369372i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 580.368i − 1.19172i −0.803088 0.595861i \(-0.796811\pi\)
0.803088 0.595861i \(-0.203189\pi\)
\(488\) 0 0
\(489\) −25.4953 −0.0521377
\(490\) 0 0
\(491\) − 619.348i − 1.26140i −0.776027 0.630700i \(-0.782768\pi\)
0.776027 0.630700i \(-0.217232\pi\)
\(492\) 0 0
\(493\) −36.0000 −0.0730223
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 437.981 0.881250
\(498\) 0 0
\(499\) 86.7044i 0.173756i 0.996219 + 0.0868782i \(0.0276891\pi\)
−0.996219 + 0.0868782i \(0.972311\pi\)
\(500\) 0 0
\(501\) −136.328 −0.272112
\(502\) 0 0
\(503\) − 442.542i − 0.879805i −0.898046 0.439902i \(-0.855013\pi\)
0.898046 0.439902i \(-0.144987\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 241.903i 0.477125i
\(508\) 0 0
\(509\) 861.246 1.69204 0.846018 0.533155i \(-0.178994\pi\)
0.846018 + 0.533155i \(0.178994\pi\)
\(510\) 0 0
\(511\) 106.331i 0.208084i
\(512\) 0 0
\(513\) −44.4984 −0.0867416
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −447.659 −0.865879
\(518\) 0 0
\(519\) 253.027i 0.487528i
\(520\) 0 0
\(521\) −331.495 −0.636267 −0.318134 0.948046i \(-0.603056\pi\)
−0.318134 + 0.948046i \(0.603056\pi\)
\(522\) 0 0
\(523\) 892.671i 1.70683i 0.521234 + 0.853414i \(0.325472\pi\)
−0.521234 + 0.853414i \(0.674528\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 71.0087i 0.134741i
\(528\) 0 0
\(529\) −49.6749 −0.0939035
\(530\) 0 0
\(531\) 245.225i 0.461817i
\(532\) 0 0
\(533\) 162.492 0.304863
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 413.410 0.769851
\(538\) 0 0
\(539\) 343.764i 0.637781i
\(540\) 0 0
\(541\) −243.495 −0.450084 −0.225042 0.974349i \(-0.572252\pi\)
−0.225042 + 0.974349i \(0.572252\pi\)
\(542\) 0 0
\(543\) − 199.475i − 0.367358i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 130.864i 0.239239i 0.992820 + 0.119619i \(0.0381674\pi\)
−0.992820 + 0.119619i \(0.961833\pi\)
\(548\) 0 0
\(549\) −283.495 −0.516385
\(550\) 0 0
\(551\) − 217.659i − 0.395026i
\(552\) 0 0
\(553\) −392.656 −0.710048
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 717.915 1.28890 0.644448 0.764648i \(-0.277087\pi\)
0.644448 + 0.764648i \(0.277087\pi\)
\(558\) 0 0
\(559\) − 205.347i − 0.367347i
\(560\) 0 0
\(561\) 27.5016 0.0490224
\(562\) 0 0
\(563\) 777.482i 1.38096i 0.723351 + 0.690481i \(0.242601\pi\)
−0.723351 + 0.690481i \(0.757399\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 38.5368i − 0.0679661i
\(568\) 0 0
\(569\) −964.152 −1.69447 −0.847233 0.531221i \(-0.821733\pi\)
−0.847233 + 0.531221i \(0.821733\pi\)
\(570\) 0 0
\(571\) − 45.1352i − 0.0790459i −0.999219 0.0395230i \(-0.987416\pi\)
0.999219 0.0395230i \(-0.0125838\pi\)
\(572\) 0 0
\(573\) 10.5047 0.0183327
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 286.000 0.495667 0.247834 0.968803i \(-0.420281\pi\)
0.247834 + 0.968803i \(0.420281\pi\)
\(578\) 0 0
\(579\) − 621.506i − 1.07341i
\(580\) 0 0
\(581\) 376.997 0.648876
\(582\) 0 0
\(583\) 182.155i 0.312444i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 719.080i 1.22501i 0.790467 + 0.612504i \(0.209838\pi\)
−0.790467 + 0.612504i \(0.790162\pi\)
\(588\) 0 0
\(589\) −429.325 −0.728905
\(590\) 0 0
\(591\) − 245.804i − 0.415912i
\(592\) 0 0
\(593\) 360.590 0.608077 0.304039 0.952660i \(-0.401665\pi\)
0.304039 + 0.952660i \(0.401665\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −216.000 −0.361809
\(598\) 0 0
\(599\) 58.8790i 0.0982954i 0.998792 + 0.0491477i \(0.0156505\pi\)
−0.998792 + 0.0491477i \(0.984349\pi\)
\(600\) 0 0
\(601\) 570.997 0.950078 0.475039 0.879965i \(-0.342434\pi\)
0.475039 + 0.879965i \(0.342434\pi\)
\(602\) 0 0
\(603\) − 388.843i − 0.644847i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 667.073i 1.09897i 0.835505 + 0.549483i \(0.185175\pi\)
−0.835505 + 0.549483i \(0.814825\pi\)
\(608\) 0 0
\(609\) 188.498 0.309521
\(610\) 0 0
\(611\) − 216.297i − 0.354005i
\(612\) 0 0
\(613\) −293.246 −0.478379 −0.239189 0.970973i \(-0.576882\pi\)
−0.239189 + 0.970973i \(0.576882\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −900.906 −1.46014 −0.730069 0.683373i \(-0.760512\pi\)
−0.730069 + 0.683373i \(0.760512\pi\)
\(618\) 0 0
\(619\) − 604.277i − 0.976214i −0.872784 0.488107i \(-0.837688\pi\)
0.872784 0.488107i \(-0.162312\pi\)
\(620\) 0 0
\(621\) 124.997 0.201283
\(622\) 0 0
\(623\) 89.2033i 0.143183i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 166.277i 0.265194i
\(628\) 0 0
\(629\) −85.3375 −0.135672
\(630\) 0 0
\(631\) 162.529i 0.257573i 0.991672 + 0.128787i \(0.0411082\pi\)
−0.991672 + 0.128787i \(0.958892\pi\)
\(632\) 0 0
\(633\) 177.994 0.281191
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −166.098 −0.260750
\(638\) 0 0
\(639\) − 306.863i − 0.480223i
\(640\) 0 0
\(641\) −507.325 −0.791459 −0.395729 0.918367i \(-0.629508\pi\)
−0.395729 + 0.918367i \(0.629508\pi\)
\(642\) 0 0
\(643\) 357.087i 0.555345i 0.960676 + 0.277672i \(0.0895630\pi\)
−0.960676 + 0.277672i \(0.910437\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 130.569i 0.201806i 0.994896 + 0.100903i \(0.0321732\pi\)
−0.994896 + 0.100903i \(0.967827\pi\)
\(648\) 0 0
\(649\) 916.328 1.41191
\(650\) 0 0
\(651\) − 371.806i − 0.571131i
\(652\) 0 0
\(653\) −1052.74 −1.61215 −0.806076 0.591812i \(-0.798413\pi\)
−0.806076 + 0.591812i \(0.798413\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 74.4984 0.113392
\(658\) 0 0
\(659\) − 0.442426i 0 0.000671360i −1.00000 0.000335680i \(-0.999893\pi\)
1.00000 0.000335680i \(-0.000106850\pi\)
\(660\) 0 0
\(661\) −255.508 −0.386547 −0.193274 0.981145i \(-0.561911\pi\)
−0.193274 + 0.981145i \(0.561911\pi\)
\(662\) 0 0
\(663\) 13.2880i 0.0200423i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 611.409i 0.916655i
\(668\) 0 0
\(669\) 68.9117 0.103007
\(670\) 0 0
\(671\) 1059.33i 1.57874i
\(672\) 0 0
\(673\) 532.492 0.791222 0.395611 0.918418i \(-0.370533\pi\)
0.395611 + 0.918418i \(0.370533\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 201.246 0.297262 0.148631 0.988893i \(-0.452513\pi\)
0.148631 + 0.988893i \(0.452513\pi\)
\(678\) 0 0
\(679\) − 59.9461i − 0.0882859i
\(680\) 0 0
\(681\) −24.6687 −0.0362243
\(682\) 0 0
\(683\) 619.586i 0.907154i 0.891217 + 0.453577i \(0.149852\pi\)
−0.891217 + 0.453577i \(0.850148\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 77.9477i 0.113461i
\(688\) 0 0
\(689\) −88.0124 −0.127739
\(690\) 0 0
\(691\) 835.133i 1.20859i 0.796762 + 0.604293i \(0.206544\pi\)
−0.796762 + 0.604293i \(0.793456\pi\)
\(692\) 0 0
\(693\) −144.000 −0.207792
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −42.4922 −0.0609645
\(698\) 0 0
\(699\) 682.281i 0.976081i
\(700\) 0 0
\(701\) 383.921 0.547676 0.273838 0.961776i \(-0.411707\pi\)
0.273838 + 0.961776i \(0.411707\pi\)
\(702\) 0 0
\(703\) − 515.958i − 0.733938i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 568.408i 0.803971i
\(708\) 0 0
\(709\) −1127.99 −1.59096 −0.795482 0.605977i \(-0.792782\pi\)
−0.795482 + 0.605977i \(0.792782\pi\)
\(710\) 0 0
\(711\) 275.107i 0.386929i
\(712\) 0 0
\(713\) 1205.98 1.69142
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −571.161 −0.796598
\(718\) 0 0
\(719\) − 480.159i − 0.667815i −0.942606 0.333907i \(-0.891633\pi\)
0.942606 0.333907i \(-0.108367\pi\)
\(720\) 0 0
\(721\) −269.666 −0.374016
\(722\) 0 0
\(723\) 364.299i 0.503871i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 626.753i − 0.862109i −0.902326 0.431054i \(-0.858142\pi\)
0.902326 0.431054i \(-0.141858\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) 53.6990i 0.0734596i
\(732\) 0 0
\(733\) 721.076 0.983732 0.491866 0.870671i \(-0.336315\pi\)
0.491866 + 0.870671i \(0.336315\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1452.98 −1.97149
\(738\) 0 0
\(739\) − 358.245i − 0.484770i −0.970180 0.242385i \(-0.922070\pi\)
0.970180 0.242385i \(-0.0779297\pi\)
\(740\) 0 0
\(741\) −80.3406 −0.108422
\(742\) 0 0
\(743\) − 847.059i − 1.14005i −0.821627 0.570026i \(-0.806933\pi\)
0.821627 0.570026i \(-0.193067\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 264.135i − 0.353594i
\(748\) 0 0
\(749\) 746.991 0.997317
\(750\) 0 0
\(751\) 1184.81i 1.57765i 0.614619 + 0.788824i \(0.289310\pi\)
−0.614619 + 0.788824i \(0.710690\pi\)
\(752\) 0 0
\(753\) −440.085 −0.584442
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 462.741 0.611283 0.305642 0.952147i \(-0.401129\pi\)
0.305642 + 0.952147i \(0.401129\pi\)
\(758\) 0 0
\(759\) − 467.075i − 0.615382i
\(760\) 0 0
\(761\) 390.158 0.512691 0.256346 0.966585i \(-0.417482\pi\)
0.256346 + 0.966585i \(0.417482\pi\)
\(762\) 0 0
\(763\) − 414.625i − 0.543414i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 442.745i 0.577243i
\(768\) 0 0
\(769\) −579.981 −0.754202 −0.377101 0.926172i \(-0.623079\pi\)
−0.377101 + 0.926172i \(0.623079\pi\)
\(770\) 0 0
\(771\) 705.655i 0.915247i
\(772\) 0 0
\(773\) 744.906 0.963655 0.481828 0.876266i \(-0.339973\pi\)
0.481828 + 0.876266i \(0.339973\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 446.833 0.575074
\(778\) 0 0
\(779\) − 256.912i − 0.329797i
\(780\) 0 0
\(781\) −1146.65 −1.46818
\(782\) 0 0
\(783\) − 132.068i − 0.168669i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1393.82i 1.77105i 0.464589 + 0.885526i \(0.346202\pi\)
−0.464589 + 0.885526i \(0.653798\pi\)
\(788\) 0 0
\(789\) −240.669 −0.305030
\(790\) 0 0
\(791\) 903.825i 1.14264i
\(792\) 0 0
\(793\) −511.842 −0.645450
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −406.073 −0.509502 −0.254751 0.967007i \(-0.581993\pi\)
−0.254751 + 0.967007i \(0.581993\pi\)
\(798\) 0 0
\(799\) 56.5624i 0.0707915i
\(800\) 0 0
\(801\) 62.4984 0.0780255
\(802\) 0 0
\(803\) − 278.378i − 0.346672i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 95.4049i 0.118222i
\(808\) 0 0
\(809\) −123.325 −0.152441 −0.0762207 0.997091i \(-0.524285\pi\)
−0.0762207 + 0.997091i \(0.524285\pi\)
\(810\) 0 0
\(811\) 183.404i 0.226146i 0.993587 + 0.113073i \(0.0360694\pi\)
−0.993587 + 0.113073i \(0.963931\pi\)
\(812\) 0 0
\(813\) −703.319 −0.865091
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −324.669 −0.397391
\(818\) 0 0
\(819\) − 69.5770i − 0.0849536i
\(820\) 0 0
\(821\) 746.912 0.909759 0.454879 0.890553i \(-0.349682\pi\)
0.454879 + 0.890553i \(0.349682\pi\)
\(822\) 0 0
\(823\) − 52.0982i − 0.0633028i −0.999499 0.0316514i \(-0.989923\pi\)
0.999499 0.0316514i \(-0.0100766\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1327.69i − 1.60543i −0.596360 0.802717i \(-0.703387\pi\)
0.596360 0.802717i \(-0.296613\pi\)
\(828\) 0 0
\(829\) −798.486 −0.963192 −0.481596 0.876393i \(-0.659943\pi\)
−0.481596 + 0.876393i \(0.659943\pi\)
\(830\) 0 0
\(831\) 607.229i 0.730720i
\(832\) 0 0
\(833\) 43.4350 0.0521429
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −260.498 −0.311229
\(838\) 0 0
\(839\) 608.138i 0.724836i 0.932016 + 0.362418i \(0.118049\pi\)
−0.932016 + 0.362418i \(0.881951\pi\)
\(840\) 0 0
\(841\) −195.006 −0.231874
\(842\) 0 0
\(843\) 791.826i 0.939295i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 19.9776i 0.0235863i
\(848\) 0 0
\(849\) 588.827 0.693553
\(850\) 0 0
\(851\) 1449.34i 1.70310i
\(852\) 0 0
\(853\) 661.404 0.775386 0.387693 0.921789i \(-0.373272\pi\)
0.387693 + 0.921789i \(0.373272\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 291.070 0.339638 0.169819 0.985475i \(-0.445682\pi\)
0.169819 + 0.985475i \(0.445682\pi\)
\(858\) 0 0
\(859\) 1024.97i 1.19321i 0.802535 + 0.596604i \(0.203484\pi\)
−0.802535 + 0.596604i \(0.796516\pi\)
\(860\) 0 0
\(861\) 222.492 0.258411
\(862\) 0 0
\(863\) 301.956i 0.349891i 0.984578 + 0.174946i \(0.0559749\pi\)
−0.984578 + 0.174946i \(0.944025\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 497.088i 0.573342i
\(868\) 0 0
\(869\) 1027.99 1.18295
\(870\) 0 0
\(871\) − 702.044i − 0.806020i
\(872\) 0 0
\(873\) −42.0000 −0.0481100
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1358.23 1.54872 0.774362 0.632743i \(-0.218071\pi\)
0.774362 + 0.632743i \(0.218071\pi\)
\(878\) 0 0
\(879\) 996.640i 1.13383i
\(880\) 0 0
\(881\) 1176.49 1.33541 0.667703 0.744428i \(-0.267278\pi\)
0.667703 + 0.744428i \(0.267278\pi\)
\(882\) 0 0
\(883\) − 506.418i − 0.573520i −0.958002 0.286760i \(-0.907422\pi\)
0.958002 0.286760i \(-0.0925783\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1497.24i 1.68798i 0.536355 + 0.843992i \(0.319801\pi\)
−0.536355 + 0.843992i \(0.680199\pi\)
\(888\) 0 0
\(889\) 190.978 0.214824
\(890\) 0 0
\(891\) 100.891i 0.113233i
\(892\) 0 0
\(893\) −341.981 −0.382958
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 225.678 0.251592
\(898\) 0 0
\(899\) − 1274.20i − 1.41735i
\(900\) 0 0
\(901\) 23.0155 0.0255444
\(902\) 0 0
\(903\) − 281.171i − 0.311375i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 199.374i 0.219816i 0.993942 + 0.109908i \(0.0350557\pi\)
−0.993942 + 0.109908i \(0.964944\pi\)
\(908\) 0 0
\(909\) 398.243 0.438111
\(910\) 0 0
\(911\) 1385.82i 1.52121i 0.649215 + 0.760605i \(0.275098\pi\)
−0.649215 + 0.760605i \(0.724902\pi\)
\(912\) 0 0
\(913\) −986.991 −1.08104
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 691.988 0.754621
\(918\) 0 0
\(919\) − 483.135i − 0.525718i −0.964834 0.262859i \(-0.915335\pi\)
0.964834 0.262859i \(-0.0846654\pi\)
\(920\) 0 0
\(921\) −620.656 −0.673894
\(922\) 0 0
\(923\) − 554.031i − 0.600250i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 188.936i 0.203814i
\(928\) 0 0
\(929\) −1149.15 −1.23697 −0.618487 0.785795i \(-0.712254\pi\)
−0.618487 + 0.785795i \(0.712254\pi\)
\(930\) 0 0
\(931\) 262.612i 0.282075i
\(932\) 0 0
\(933\) −631.830 −0.677202
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1349.00 1.43970 0.719852 0.694127i \(-0.244210\pi\)
0.719852 + 0.694127i \(0.244210\pi\)
\(938\) 0 0
\(939\) − 459.862i − 0.489736i
\(940\) 0 0
\(941\) 1160.07 1.23280 0.616401 0.787432i \(-0.288590\pi\)
0.616401 + 0.787432i \(0.288590\pi\)
\(942\) 0 0
\(943\) 721.670i 0.765292i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 620.063i 0.654766i 0.944892 + 0.327383i \(0.106167\pi\)
−0.944892 + 0.327383i \(0.893833\pi\)
\(948\) 0 0
\(949\) 134.505 0.141733
\(950\) 0 0
\(951\) − 796.017i − 0.837031i
\(952\) 0 0
\(953\) 545.587 0.572494 0.286247 0.958156i \(-0.407592\pi\)
0.286247 + 0.958156i \(0.407592\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −493.495 −0.515669
\(958\) 0 0
\(959\) − 456.377i − 0.475888i
\(960\) 0 0
\(961\) −1552.31 −1.61531
\(962\) 0 0
\(963\) − 523.364i − 0.543472i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 181.621i 0.187819i 0.995581 + 0.0939097i \(0.0299365\pi\)
−0.995581 + 0.0939097i \(0.970064\pi\)
\(968\) 0 0
\(969\) 21.0093 0.0216814
\(970\) 0 0
\(971\) − 337.222i − 0.347293i −0.984808 0.173647i \(-0.944445\pi\)
0.984808 0.173647i \(-0.0555550\pi\)
\(972\) 0 0
\(973\) 104.656 0.107560
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −940.249 −0.962384 −0.481192 0.876615i \(-0.659796\pi\)
−0.481192 + 0.876615i \(0.659796\pi\)
\(978\) 0 0
\(979\) − 233.537i − 0.238547i
\(980\) 0 0
\(981\) −290.498 −0.296125
\(982\) 0 0
\(983\) 1651.87i 1.68043i 0.542250 + 0.840217i \(0.317573\pi\)
−0.542250 + 0.840217i \(0.682427\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 296.165i − 0.300065i
\(988\) 0 0
\(989\) 912.000 0.922144
\(990\) 0 0
\(991\) 970.903i 0.979720i 0.871801 + 0.489860i \(0.162952\pi\)
−0.871801 + 0.489860i \(0.837048\pi\)
\(992\) 0 0
\(993\) −1010.16 −1.01729
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1285.08 −1.28894 −0.644471 0.764628i \(-0.722922\pi\)
−0.644471 + 0.764628i \(0.722922\pi\)
\(998\) 0 0
\(999\) − 313.064i − 0.313378i
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 1200.3.e.n.751.1 4
3.2 odd 2 3600.3.e.bh.3151.2 4
4.3 odd 2 inner 1200.3.e.n.751.4 4
5.2 odd 4 1200.3.j.f.799.4 8
5.3 odd 4 1200.3.j.f.799.6 8
5.4 even 2 240.3.e.a.31.4 yes 4
12.11 even 2 3600.3.e.bh.3151.3 4
15.2 even 4 3600.3.j.l.1999.5 8
15.8 even 4 3600.3.j.l.1999.3 8
15.14 odd 2 720.3.e.a.271.2 4
20.3 even 4 1200.3.j.f.799.3 8
20.7 even 4 1200.3.j.f.799.5 8
20.19 odd 2 240.3.e.a.31.2 4
40.19 odd 2 960.3.e.b.511.3 4
40.29 even 2 960.3.e.b.511.1 4
60.23 odd 4 3600.3.j.l.1999.6 8
60.47 odd 4 3600.3.j.l.1999.4 8
60.59 even 2 720.3.e.a.271.1 4
120.29 odd 2 2880.3.e.f.2431.4 4
120.59 even 2 2880.3.e.f.2431.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
240.3.e.a.31.2 4 20.19 odd 2
240.3.e.a.31.4 yes 4 5.4 even 2
720.3.e.a.271.1 4 60.59 even 2
720.3.e.a.271.2 4 15.14 odd 2
960.3.e.b.511.1 4 40.29 even 2
960.3.e.b.511.3 4 40.19 odd 2
1200.3.e.n.751.1 4 1.1 even 1 trivial
1200.3.e.n.751.4 4 4.3 odd 2 inner
1200.3.j.f.799.3 8 20.3 even 4
1200.3.j.f.799.4 8 5.2 odd 4
1200.3.j.f.799.5 8 20.7 even 4
1200.3.j.f.799.6 8 5.3 odd 4
2880.3.e.f.2431.3 4 120.59 even 2
2880.3.e.f.2431.4 4 120.29 odd 2
3600.3.e.bh.3151.2 4 3.2 odd 2
3600.3.e.bh.3151.3 4 12.11 even 2
3600.3.j.l.1999.3 8 15.8 even 4
3600.3.j.l.1999.4 8 60.47 odd 4
3600.3.j.l.1999.5 8 15.2 even 4
3600.3.j.l.1999.6 8 60.23 odd 4