Properties

Label 192.7.g.b.127.2
Level $192$
Weight $7$
Character 192.127
Analytic conductor $44.170$
Analytic rank $0$
Dimension $2$
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] = [192,7,Mod(127,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.127");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 192.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(44.1703840550\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 192.127
Dual form 192.7.g.b.127.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+15.5885i q^{3} -6.00000 q^{5} -200.918i q^{7} -243.000 q^{9} +O(q^{10})\) \(q+15.5885i q^{3} -6.00000 q^{5} -200.918i q^{7} -243.000 q^{9} +644.323i q^{11} +2654.00 q^{13} -93.5307i q^{15} -7206.00 q^{17} +3540.31i q^{19} +3132.00 q^{21} -17459.1i q^{23} -15589.0 q^{25} -3788.00i q^{27} -11550.0 q^{29} +8542.47i q^{31} -10044.0 q^{33} +1205.51i q^{35} -22346.0 q^{37} +41371.8i q^{39} +103626. q^{41} -127389. i q^{43} +1458.00 q^{45} -160831. i q^{47} +77281.0 q^{49} -112330. i q^{51} -168462. q^{53} -3865.94i q^{55} -55188.0 q^{57} +111593. i q^{59} +260470. q^{61} +48823.0i q^{63} -15924.0 q^{65} -317083. i q^{67} +272160. q^{69} -708423. i q^{71} -395918. q^{73} -243008. i q^{75} +129456. q^{77} -556743. i q^{79} +59049.0 q^{81} +89145.2i q^{83} +43236.0 q^{85} -180047. i q^{87} -251886. q^{89} -533236. i q^{91} -133164. q^{93} -21241.9i q^{95} +517474. q^{97} -156570. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{5} - 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{5} - 486 q^{9} + 5308 q^{13} - 14412 q^{17} + 6264 q^{21} - 31178 q^{25} - 23100 q^{29} - 20088 q^{33} - 44692 q^{37} + 207252 q^{41} + 2916 q^{45} + 154562 q^{49} - 336924 q^{53} - 110376 q^{57} + 520940 q^{61} - 31848 q^{65} + 544320 q^{69} - 791836 q^{73} + 258912 q^{77} + 118098 q^{81} + 86472 q^{85} - 503772 q^{89} - 266328 q^{93} + 1034948 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 15.5885i 0.577350i
\(4\) 0 0
\(5\) −6.00000 −0.0480000 −0.0240000 0.999712i \(-0.507640\pi\)
−0.0240000 + 0.999712i \(0.507640\pi\)
\(6\) 0 0
\(7\) − 200.918i − 0.585766i −0.956148 0.292883i \(-0.905385\pi\)
0.956148 0.292883i \(-0.0946147\pi\)
\(8\) 0 0
\(9\) −243.000 −0.333333
\(10\) 0 0
\(11\) 644.323i 0.484089i 0.970265 + 0.242045i \(0.0778181\pi\)
−0.970265 + 0.242045i \(0.922182\pi\)
\(12\) 0 0
\(13\) 2654.00 1.20801 0.604005 0.796980i \(-0.293571\pi\)
0.604005 + 0.796980i \(0.293571\pi\)
\(14\) 0 0
\(15\) − 93.5307i − 0.0277128i
\(16\) 0 0
\(17\) −7206.00 −1.46672 −0.733360 0.679840i \(-0.762049\pi\)
−0.733360 + 0.679840i \(0.762049\pi\)
\(18\) 0 0
\(19\) 3540.31i 0.516156i 0.966124 + 0.258078i \(0.0830891\pi\)
−0.966124 + 0.258078i \(0.916911\pi\)
\(20\) 0 0
\(21\) 3132.00 0.338192
\(22\) 0 0
\(23\) − 17459.1i − 1.43495i −0.696583 0.717476i \(-0.745297\pi\)
0.696583 0.717476i \(-0.254703\pi\)
\(24\) 0 0
\(25\) −15589.0 −0.997696
\(26\) 0 0
\(27\) − 3788.00i − 0.192450i
\(28\) 0 0
\(29\) −11550.0 −0.473574 −0.236787 0.971562i \(-0.576094\pi\)
−0.236787 + 0.971562i \(0.576094\pi\)
\(30\) 0 0
\(31\) 8542.47i 0.286747i 0.989669 + 0.143373i \(0.0457950\pi\)
−0.989669 + 0.143373i \(0.954205\pi\)
\(32\) 0 0
\(33\) −10044.0 −0.279489
\(34\) 0 0
\(35\) 1205.51i 0.0281168i
\(36\) 0 0
\(37\) −22346.0 −0.441158 −0.220579 0.975369i \(-0.570795\pi\)
−0.220579 + 0.975369i \(0.570795\pi\)
\(38\) 0 0
\(39\) 41371.8i 0.697445i
\(40\) 0 0
\(41\) 103626. 1.50355 0.751774 0.659421i \(-0.229199\pi\)
0.751774 + 0.659421i \(0.229199\pi\)
\(42\) 0 0
\(43\) − 127389.i − 1.60223i −0.598507 0.801117i \(-0.704239\pi\)
0.598507 0.801117i \(-0.295761\pi\)
\(44\) 0 0
\(45\) 1458.00 0.0160000
\(46\) 0 0
\(47\) − 160831.i − 1.54909i −0.632518 0.774546i \(-0.717979\pi\)
0.632518 0.774546i \(-0.282021\pi\)
\(48\) 0 0
\(49\) 77281.0 0.656878
\(50\) 0 0
\(51\) − 112330.i − 0.846812i
\(52\) 0 0
\(53\) −168462. −1.13155 −0.565776 0.824559i \(-0.691423\pi\)
−0.565776 + 0.824559i \(0.691423\pi\)
\(54\) 0 0
\(55\) − 3865.94i − 0.0232363i
\(56\) 0 0
\(57\) −55188.0 −0.298003
\(58\) 0 0
\(59\) 111593.i 0.543349i 0.962389 + 0.271675i \(0.0875775\pi\)
−0.962389 + 0.271675i \(0.912423\pi\)
\(60\) 0 0
\(61\) 260470. 1.14754 0.573770 0.819016i \(-0.305480\pi\)
0.573770 + 0.819016i \(0.305480\pi\)
\(62\) 0 0
\(63\) 48823.0i 0.195255i
\(64\) 0 0
\(65\) −15924.0 −0.0579845
\(66\) 0 0
\(67\) − 317083.i − 1.05426i −0.849784 0.527131i \(-0.823268\pi\)
0.849784 0.527131i \(-0.176732\pi\)
\(68\) 0 0
\(69\) 272160. 0.828470
\(70\) 0 0
\(71\) − 708423.i − 1.97933i −0.143412 0.989663i \(-0.545807\pi\)
0.143412 0.989663i \(-0.454193\pi\)
\(72\) 0 0
\(73\) −395918. −1.01774 −0.508870 0.860844i \(-0.669937\pi\)
−0.508870 + 0.860844i \(0.669937\pi\)
\(74\) 0 0
\(75\) − 243008.i − 0.576020i
\(76\) 0 0
\(77\) 129456. 0.283563
\(78\) 0 0
\(79\) − 556743.i − 1.12921i −0.825362 0.564604i \(-0.809029\pi\)
0.825362 0.564604i \(-0.190971\pi\)
\(80\) 0 0
\(81\) 59049.0 0.111111
\(82\) 0 0
\(83\) 89145.2i 0.155906i 0.996957 + 0.0779531i \(0.0248385\pi\)
−0.996957 + 0.0779531i \(0.975162\pi\)
\(84\) 0 0
\(85\) 43236.0 0.0704026
\(86\) 0 0
\(87\) − 180047.i − 0.273418i
\(88\) 0 0
\(89\) −251886. −0.357301 −0.178650 0.983913i \(-0.557173\pi\)
−0.178650 + 0.983913i \(0.557173\pi\)
\(90\) 0 0
\(91\) − 533236.i − 0.707612i
\(92\) 0 0
\(93\) −133164. −0.165553
\(94\) 0 0
\(95\) − 21241.9i − 0.0247755i
\(96\) 0 0
\(97\) 517474. 0.566987 0.283494 0.958974i \(-0.408507\pi\)
0.283494 + 0.958974i \(0.408507\pi\)
\(98\) 0 0
\(99\) − 156570.i − 0.161363i
\(100\) 0 0
\(101\) −1.02382e6 −0.993712 −0.496856 0.867833i \(-0.665512\pi\)
−0.496856 + 0.867833i \(0.665512\pi\)
\(102\) 0 0
\(103\) 90960.4i 0.0832416i 0.999133 + 0.0416208i \(0.0132521\pi\)
−0.999133 + 0.0416208i \(0.986748\pi\)
\(104\) 0 0
\(105\) −18792.0 −0.0162332
\(106\) 0 0
\(107\) − 2.20282e6i − 1.79815i −0.437791 0.899077i \(-0.644239\pi\)
0.437791 0.899077i \(-0.355761\pi\)
\(108\) 0 0
\(109\) −952594. −0.735577 −0.367789 0.929909i \(-0.619885\pi\)
−0.367789 + 0.929909i \(0.619885\pi\)
\(110\) 0 0
\(111\) − 348340.i − 0.254703i
\(112\) 0 0
\(113\) −121038. −0.0838854 −0.0419427 0.999120i \(-0.513355\pi\)
−0.0419427 + 0.999120i \(0.513355\pi\)
\(114\) 0 0
\(115\) 104754.i 0.0688777i
\(116\) 0 0
\(117\) −644922. −0.402670
\(118\) 0 0
\(119\) 1.44781e6i 0.859156i
\(120\) 0 0
\(121\) 1.35641e6 0.765658
\(122\) 0 0
\(123\) 1.61537e6i 0.868074i
\(124\) 0 0
\(125\) 187284. 0.0958894
\(126\) 0 0
\(127\) − 1.44941e6i − 0.707590i −0.935323 0.353795i \(-0.884891\pi\)
0.935323 0.353795i \(-0.115109\pi\)
\(128\) 0 0
\(129\) 1.98580e6 0.925051
\(130\) 0 0
\(131\) − 632787.i − 0.281478i −0.990047 0.140739i \(-0.955052\pi\)
0.990047 0.140739i \(-0.0449478\pi\)
\(132\) 0 0
\(133\) 711312. 0.302347
\(134\) 0 0
\(135\) 22728.0i 0.00923760i
\(136\) 0 0
\(137\) −4.47821e6 −1.74158 −0.870789 0.491656i \(-0.836392\pi\)
−0.870789 + 0.491656i \(0.836392\pi\)
\(138\) 0 0
\(139\) 2.24905e6i 0.837443i 0.908115 + 0.418722i \(0.137522\pi\)
−0.908115 + 0.418722i \(0.862478\pi\)
\(140\) 0 0
\(141\) 2.50711e6 0.894368
\(142\) 0 0
\(143\) 1.71003e6i 0.584785i
\(144\) 0 0
\(145\) 69300.0 0.0227316
\(146\) 0 0
\(147\) 1.20469e6i 0.379248i
\(148\) 0 0
\(149\) 1.84051e6 0.556389 0.278194 0.960525i \(-0.410264\pi\)
0.278194 + 0.960525i \(0.410264\pi\)
\(150\) 0 0
\(151\) 1.14464e6i 0.332460i 0.986087 + 0.166230i \(0.0531594\pi\)
−0.986087 + 0.166230i \(0.946841\pi\)
\(152\) 0 0
\(153\) 1.75106e6 0.488907
\(154\) 0 0
\(155\) − 51254.8i − 0.0137638i
\(156\) 0 0
\(157\) 5.28775e6 1.36638 0.683191 0.730240i \(-0.260592\pi\)
0.683191 + 0.730240i \(0.260592\pi\)
\(158\) 0 0
\(159\) − 2.62606e6i − 0.653302i
\(160\) 0 0
\(161\) −3.50784e6 −0.840547
\(162\) 0 0
\(163\) 6.45869e6i 1.49136i 0.666306 + 0.745678i \(0.267874\pi\)
−0.666306 + 0.745678i \(0.732126\pi\)
\(164\) 0 0
\(165\) 60264.0 0.0134155
\(166\) 0 0
\(167\) 3.97714e6i 0.853927i 0.904269 + 0.426964i \(0.140417\pi\)
−0.904269 + 0.426964i \(0.859583\pi\)
\(168\) 0 0
\(169\) 2.21691e6 0.459290
\(170\) 0 0
\(171\) − 860296.i − 0.172052i
\(172\) 0 0
\(173\) −5.73548e6 −1.10772 −0.553862 0.832609i \(-0.686846\pi\)
−0.553862 + 0.832609i \(0.686846\pi\)
\(174\) 0 0
\(175\) 3.13211e6i 0.584417i
\(176\) 0 0
\(177\) −1.73956e6 −0.313703
\(178\) 0 0
\(179\) 8.16883e6i 1.42430i 0.702029 + 0.712149i \(0.252278\pi\)
−0.702029 + 0.712149i \(0.747722\pi\)
\(180\) 0 0
\(181\) −3.24656e6 −0.547505 −0.273752 0.961800i \(-0.588265\pi\)
−0.273752 + 0.961800i \(0.588265\pi\)
\(182\) 0 0
\(183\) 4.06033e6i 0.662533i
\(184\) 0 0
\(185\) 134076. 0.0211756
\(186\) 0 0
\(187\) − 4.64299e6i − 0.710024i
\(188\) 0 0
\(189\) −761076. −0.112731
\(190\) 0 0
\(191\) 1.11705e6i 0.160314i 0.996782 + 0.0801571i \(0.0255422\pi\)
−0.996782 + 0.0801571i \(0.974458\pi\)
\(192\) 0 0
\(193\) 3.12375e6 0.434514 0.217257 0.976114i \(-0.430289\pi\)
0.217257 + 0.976114i \(0.430289\pi\)
\(194\) 0 0
\(195\) − 248231.i − 0.0334774i
\(196\) 0 0
\(197\) −1.08530e7 −1.41955 −0.709777 0.704426i \(-0.751204\pi\)
−0.709777 + 0.704426i \(0.751204\pi\)
\(198\) 0 0
\(199\) 8.99733e6i 1.14171i 0.821052 + 0.570853i \(0.193388\pi\)
−0.821052 + 0.570853i \(0.806612\pi\)
\(200\) 0 0
\(201\) 4.94284e6 0.608679
\(202\) 0 0
\(203\) 2.32060e6i 0.277404i
\(204\) 0 0
\(205\) −621756. −0.0721703
\(206\) 0 0
\(207\) 4.24255e6i 0.478318i
\(208\) 0 0
\(209\) −2.28110e6 −0.249865
\(210\) 0 0
\(211\) − 3.84129e6i − 0.408912i −0.978876 0.204456i \(-0.934457\pi\)
0.978876 0.204456i \(-0.0655426\pi\)
\(212\) 0 0
\(213\) 1.10432e7 1.14276
\(214\) 0 0
\(215\) 764333.i 0.0769073i
\(216\) 0 0
\(217\) 1.71634e6 0.167967
\(218\) 0 0
\(219\) − 6.17175e6i − 0.587592i
\(220\) 0 0
\(221\) −1.91247e7 −1.77181
\(222\) 0 0
\(223\) − 3.51732e6i − 0.317174i −0.987345 0.158587i \(-0.949306\pi\)
0.987345 0.158587i \(-0.0506938\pi\)
\(224\) 0 0
\(225\) 3.78813e6 0.332565
\(226\) 0 0
\(227\) 1.62115e7i 1.38595i 0.720964 + 0.692973i \(0.243699\pi\)
−0.720964 + 0.692973i \(0.756301\pi\)
\(228\) 0 0
\(229\) 1.46861e7 1.22293 0.611463 0.791273i \(-0.290582\pi\)
0.611463 + 0.791273i \(0.290582\pi\)
\(230\) 0 0
\(231\) 2.01802e6i 0.163715i
\(232\) 0 0
\(233\) −6.10117e6 −0.482331 −0.241165 0.970484i \(-0.577530\pi\)
−0.241165 + 0.970484i \(0.577530\pi\)
\(234\) 0 0
\(235\) 964988.i 0.0743564i
\(236\) 0 0
\(237\) 8.67877e6 0.651948
\(238\) 0 0
\(239\) − 2.63536e7i − 1.93039i −0.261528 0.965196i \(-0.584226\pi\)
0.261528 0.965196i \(-0.415774\pi\)
\(240\) 0 0
\(241\) −2.15214e6 −0.153752 −0.0768758 0.997041i \(-0.524495\pi\)
−0.0768758 + 0.997041i \(0.524495\pi\)
\(242\) 0 0
\(243\) 920483.i 0.0641500i
\(244\) 0 0
\(245\) −463686. −0.0315301
\(246\) 0 0
\(247\) 9.39599e6i 0.623522i
\(248\) 0 0
\(249\) −1.38964e6 −0.0900125
\(250\) 0 0
\(251\) − 7.99802e6i − 0.505780i −0.967495 0.252890i \(-0.918619\pi\)
0.967495 0.252890i \(-0.0813810\pi\)
\(252\) 0 0
\(253\) 1.12493e7 0.694645
\(254\) 0 0
\(255\) 673983.i 0.0406470i
\(256\) 0 0
\(257\) −2.86154e7 −1.68578 −0.842890 0.538086i \(-0.819147\pi\)
−0.842890 + 0.538086i \(0.819147\pi\)
\(258\) 0 0
\(259\) 4.48971e6i 0.258416i
\(260\) 0 0
\(261\) 2.80665e6 0.157858
\(262\) 0 0
\(263\) − 1.49487e6i − 0.0821744i −0.999156 0.0410872i \(-0.986918\pi\)
0.999156 0.0410872i \(-0.0130821\pi\)
\(264\) 0 0
\(265\) 1.01077e6 0.0543145
\(266\) 0 0
\(267\) − 3.92651e6i − 0.206288i
\(268\) 0 0
\(269\) 532290. 0.0273459 0.0136729 0.999907i \(-0.495648\pi\)
0.0136729 + 0.999907i \(0.495648\pi\)
\(270\) 0 0
\(271\) − 2.88563e7i − 1.44988i −0.688811 0.724941i \(-0.741867\pi\)
0.688811 0.724941i \(-0.258133\pi\)
\(272\) 0 0
\(273\) 8.31233e6 0.408540
\(274\) 0 0
\(275\) − 1.00443e7i − 0.482974i
\(276\) 0 0
\(277\) 1.82479e7 0.858567 0.429284 0.903170i \(-0.358766\pi\)
0.429284 + 0.903170i \(0.358766\pi\)
\(278\) 0 0
\(279\) − 2.07582e6i − 0.0955823i
\(280\) 0 0
\(281\) 9.52019e6 0.429069 0.214534 0.976716i \(-0.431177\pi\)
0.214534 + 0.976716i \(0.431177\pi\)
\(282\) 0 0
\(283\) − 3.81369e7i − 1.68262i −0.540552 0.841310i \(-0.681785\pi\)
0.540552 0.841310i \(-0.318215\pi\)
\(284\) 0 0
\(285\) 331128. 0.0143041
\(286\) 0 0
\(287\) − 2.08203e7i − 0.880728i
\(288\) 0 0
\(289\) 2.77889e7 1.15127
\(290\) 0 0
\(291\) 8.06662e6i 0.327350i
\(292\) 0 0
\(293\) −1.68345e7 −0.669266 −0.334633 0.942349i \(-0.608612\pi\)
−0.334633 + 0.942349i \(0.608612\pi\)
\(294\) 0 0
\(295\) − 669555.i − 0.0260808i
\(296\) 0 0
\(297\) 2.44069e6 0.0931630
\(298\) 0 0
\(299\) − 4.63364e7i − 1.73344i
\(300\) 0 0
\(301\) −2.55947e7 −0.938535
\(302\) 0 0
\(303\) − 1.59598e7i − 0.573720i
\(304\) 0 0
\(305\) −1.56282e6 −0.0550820
\(306\) 0 0
\(307\) 4.39154e7i 1.51775i 0.651234 + 0.758877i \(0.274252\pi\)
−0.651234 + 0.758877i \(0.725748\pi\)
\(308\) 0 0
\(309\) −1.41793e6 −0.0480596
\(310\) 0 0
\(311\) 5.08371e7i 1.69005i 0.534726 + 0.845025i \(0.320415\pi\)
−0.534726 + 0.845025i \(0.679585\pi\)
\(312\) 0 0
\(313\) 1.30721e7 0.426298 0.213149 0.977020i \(-0.431628\pi\)
0.213149 + 0.977020i \(0.431628\pi\)
\(314\) 0 0
\(315\) − 292938.i − 0.00937226i
\(316\) 0 0
\(317\) −5.07723e6 −0.159386 −0.0796928 0.996819i \(-0.525394\pi\)
−0.0796928 + 0.996819i \(0.525394\pi\)
\(318\) 0 0
\(319\) − 7.44193e6i − 0.229252i
\(320\) 0 0
\(321\) 3.43385e7 1.03816
\(322\) 0 0
\(323\) − 2.55115e7i − 0.757056i
\(324\) 0 0
\(325\) −4.13732e7 −1.20523
\(326\) 0 0
\(327\) − 1.48495e7i − 0.424686i
\(328\) 0 0
\(329\) −3.23139e7 −0.907406
\(330\) 0 0
\(331\) − 5.59418e7i − 1.54260i −0.636474 0.771298i \(-0.719608\pi\)
0.636474 0.771298i \(-0.280392\pi\)
\(332\) 0 0
\(333\) 5.43008e6 0.147053
\(334\) 0 0
\(335\) 1.90250e6i 0.0506046i
\(336\) 0 0
\(337\) −2.55737e7 −0.668197 −0.334099 0.942538i \(-0.608432\pi\)
−0.334099 + 0.942538i \(0.608432\pi\)
\(338\) 0 0
\(339\) − 1.88680e6i − 0.0484313i
\(340\) 0 0
\(341\) −5.50411e6 −0.138811
\(342\) 0 0
\(343\) − 3.91649e7i − 0.970543i
\(344\) 0 0
\(345\) −1.63296e6 −0.0397666
\(346\) 0 0
\(347\) − 4.96031e6i − 0.118719i −0.998237 0.0593595i \(-0.981094\pi\)
0.998237 0.0593595i \(-0.0189058\pi\)
\(348\) 0 0
\(349\) 3.06563e7 0.721179 0.360590 0.932725i \(-0.382576\pi\)
0.360590 + 0.932725i \(0.382576\pi\)
\(350\) 0 0
\(351\) − 1.00533e7i − 0.232482i
\(352\) 0 0
\(353\) −5.33353e7 −1.21252 −0.606262 0.795265i \(-0.707332\pi\)
−0.606262 + 0.795265i \(0.707332\pi\)
\(354\) 0 0
\(355\) 4.25054e6i 0.0950077i
\(356\) 0 0
\(357\) −2.25692e7 −0.496034
\(358\) 0 0
\(359\) 2.83885e7i 0.613564i 0.951780 + 0.306782i \(0.0992522\pi\)
−0.951780 + 0.306782i \(0.900748\pi\)
\(360\) 0 0
\(361\) 3.45121e7 0.733583
\(362\) 0 0
\(363\) 2.11443e7i 0.442053i
\(364\) 0 0
\(365\) 2.37551e6 0.0488515
\(366\) 0 0
\(367\) − 5.75671e7i − 1.16460i −0.812974 0.582299i \(-0.802153\pi\)
0.812974 0.582299i \(-0.197847\pi\)
\(368\) 0 0
\(369\) −2.51811e7 −0.501183
\(370\) 0 0
\(371\) 3.38470e7i 0.662825i
\(372\) 0 0
\(373\) 4.68328e7 0.902451 0.451226 0.892410i \(-0.350987\pi\)
0.451226 + 0.892410i \(0.350987\pi\)
\(374\) 0 0
\(375\) 2.91947e6i 0.0553618i
\(376\) 0 0
\(377\) −3.06537e7 −0.572083
\(378\) 0 0
\(379\) 7.92476e7i 1.45569i 0.685742 + 0.727845i \(0.259478\pi\)
−0.685742 + 0.727845i \(0.740522\pi\)
\(380\) 0 0
\(381\) 2.25941e7 0.408527
\(382\) 0 0
\(383\) − 2.00575e7i − 0.357010i −0.983939 0.178505i \(-0.942874\pi\)
0.983939 0.178505i \(-0.0571260\pi\)
\(384\) 0 0
\(385\) −776736. −0.0136110
\(386\) 0 0
\(387\) 3.09555e7i 0.534078i
\(388\) 0 0
\(389\) 4.79345e7 0.814328 0.407164 0.913355i \(-0.366518\pi\)
0.407164 + 0.913355i \(0.366518\pi\)
\(390\) 0 0
\(391\) 1.25810e8i 2.10468i
\(392\) 0 0
\(393\) 9.86418e6 0.162511
\(394\) 0 0
\(395\) 3.34046e6i 0.0542020i
\(396\) 0 0
\(397\) 3.43785e7 0.549434 0.274717 0.961525i \(-0.411416\pi\)
0.274717 + 0.961525i \(0.411416\pi\)
\(398\) 0 0
\(399\) 1.10883e7i 0.174560i
\(400\) 0 0
\(401\) 9.09234e7 1.41008 0.705038 0.709170i \(-0.250930\pi\)
0.705038 + 0.709170i \(0.250930\pi\)
\(402\) 0 0
\(403\) 2.26717e7i 0.346393i
\(404\) 0 0
\(405\) −354294. −0.00533333
\(406\) 0 0
\(407\) − 1.43980e7i − 0.213560i
\(408\) 0 0
\(409\) −1.15454e8 −1.68748 −0.843739 0.536753i \(-0.819651\pi\)
−0.843739 + 0.536753i \(0.819651\pi\)
\(410\) 0 0
\(411\) − 6.98084e7i − 1.00550i
\(412\) 0 0
\(413\) 2.24209e7 0.318276
\(414\) 0 0
\(415\) − 534871.i − 0.00748350i
\(416\) 0 0
\(417\) −3.50593e7 −0.483498
\(418\) 0 0
\(419\) 7.40305e7i 1.00640i 0.864171 + 0.503198i \(0.167843\pi\)
−0.864171 + 0.503198i \(0.832157\pi\)
\(420\) 0 0
\(421\) −1.14811e8 −1.53863 −0.769317 0.638867i \(-0.779404\pi\)
−0.769317 + 0.638867i \(0.779404\pi\)
\(422\) 0 0
\(423\) 3.90820e7i 0.516364i
\(424\) 0 0
\(425\) 1.12334e8 1.46334
\(426\) 0 0
\(427\) − 5.23331e7i − 0.672191i
\(428\) 0 0
\(429\) −2.66568e7 −0.337626
\(430\) 0 0
\(431\) − 3.42559e6i − 0.0427862i −0.999771 0.0213931i \(-0.993190\pi\)
0.999771 0.0213931i \(-0.00681016\pi\)
\(432\) 0 0
\(433\) 1.08786e8 1.34002 0.670008 0.742354i \(-0.266291\pi\)
0.670008 + 0.742354i \(0.266291\pi\)
\(434\) 0 0
\(435\) 1.08028e6i 0.0131241i
\(436\) 0 0
\(437\) 6.18106e7 0.740659
\(438\) 0 0
\(439\) 1.13400e8i 1.34036i 0.742199 + 0.670180i \(0.233783\pi\)
−0.742199 + 0.670180i \(0.766217\pi\)
\(440\) 0 0
\(441\) −1.87793e7 −0.218959
\(442\) 0 0
\(443\) 2.44463e7i 0.281191i 0.990067 + 0.140595i \(0.0449017\pi\)
−0.990067 + 0.140595i \(0.955098\pi\)
\(444\) 0 0
\(445\) 1.51132e6 0.0171504
\(446\) 0 0
\(447\) 2.86906e7i 0.321231i
\(448\) 0 0
\(449\) 9.03699e7 0.998354 0.499177 0.866500i \(-0.333636\pi\)
0.499177 + 0.866500i \(0.333636\pi\)
\(450\) 0 0
\(451\) 6.67686e7i 0.727851i
\(452\) 0 0
\(453\) −1.78432e7 −0.191946
\(454\) 0 0
\(455\) 3.19942e6i 0.0339654i
\(456\) 0 0
\(457\) −5.82048e7 −0.609832 −0.304916 0.952379i \(-0.598628\pi\)
−0.304916 + 0.952379i \(0.598628\pi\)
\(458\) 0 0
\(459\) 2.72963e7i 0.282271i
\(460\) 0 0
\(461\) −1.73271e8 −1.76858 −0.884288 0.466942i \(-0.845356\pi\)
−0.884288 + 0.466942i \(0.845356\pi\)
\(462\) 0 0
\(463\) 1.25400e8i 1.26344i 0.775197 + 0.631719i \(0.217650\pi\)
−0.775197 + 0.631719i \(0.782350\pi\)
\(464\) 0 0
\(465\) 798984. 0.00794656
\(466\) 0 0
\(467\) 1.45905e8i 1.43258i 0.697800 + 0.716292i \(0.254162\pi\)
−0.697800 + 0.716292i \(0.745838\pi\)
\(468\) 0 0
\(469\) −6.37077e7 −0.617551
\(470\) 0 0
\(471\) 8.24279e7i 0.788881i
\(472\) 0 0
\(473\) 8.20796e7 0.775625
\(474\) 0 0
\(475\) − 5.51899e7i − 0.514966i
\(476\) 0 0
\(477\) 4.09363e7 0.377184
\(478\) 0 0
\(479\) − 7.62417e6i − 0.0693723i −0.999398 0.0346861i \(-0.988957\pi\)
0.999398 0.0346861i \(-0.0110432\pi\)
\(480\) 0 0
\(481\) −5.93063e7 −0.532924
\(482\) 0 0
\(483\) − 5.46818e7i − 0.485290i
\(484\) 0 0
\(485\) −3.10484e6 −0.0272154
\(486\) 0 0
\(487\) − 1.71349e8i − 1.48352i −0.670665 0.741760i \(-0.733991\pi\)
0.670665 0.741760i \(-0.266009\pi\)
\(488\) 0 0
\(489\) −1.00681e8 −0.861035
\(490\) 0 0
\(491\) − 3.30473e6i − 0.0279185i −0.999903 0.0139592i \(-0.995556\pi\)
0.999903 0.0139592i \(-0.00444351\pi\)
\(492\) 0 0
\(493\) 8.32293e7 0.694601
\(494\) 0 0
\(495\) 939423.i 0.00774543i
\(496\) 0 0
\(497\) −1.42335e8 −1.15942
\(498\) 0 0
\(499\) − 1.26207e8i − 1.01574i −0.861435 0.507868i \(-0.830434\pi\)
0.861435 0.507868i \(-0.169566\pi\)
\(500\) 0 0
\(501\) −6.19974e7 −0.493015
\(502\) 0 0
\(503\) 212336.i 0.00166847i 1.00000 0.000834236i \(0.000265546\pi\)
−1.00000 0.000834236i \(0.999734\pi\)
\(504\) 0 0
\(505\) 6.14293e6 0.0476982
\(506\) 0 0
\(507\) 3.45582e7i 0.265171i
\(508\) 0 0
\(509\) 1.49792e8 1.13589 0.567944 0.823067i \(-0.307739\pi\)
0.567944 + 0.823067i \(0.307739\pi\)
\(510\) 0 0
\(511\) 7.95470e7i 0.596158i
\(512\) 0 0
\(513\) 1.34107e7 0.0993342
\(514\) 0 0
\(515\) − 545762.i − 0.00399560i
\(516\) 0 0
\(517\) 1.03627e8 0.749899
\(518\) 0 0
\(519\) − 8.94073e7i − 0.639544i
\(520\) 0 0
\(521\) 1.11422e8 0.787876 0.393938 0.919137i \(-0.371113\pi\)
0.393938 + 0.919137i \(0.371113\pi\)
\(522\) 0 0
\(523\) 2.69486e8i 1.88379i 0.335911 + 0.941894i \(0.390956\pi\)
−0.335911 + 0.941894i \(0.609044\pi\)
\(524\) 0 0
\(525\) −4.88247e7 −0.337413
\(526\) 0 0
\(527\) − 6.15571e7i − 0.420578i
\(528\) 0 0
\(529\) −1.56783e8 −1.05909
\(530\) 0 0
\(531\) − 2.71170e7i − 0.181116i
\(532\) 0 0
\(533\) 2.75023e8 1.81630
\(534\) 0 0
\(535\) 1.32169e7i 0.0863114i
\(536\) 0 0
\(537\) −1.27339e8 −0.822319
\(538\) 0 0
\(539\) 4.97939e7i 0.317987i
\(540\) 0 0
\(541\) −3.03425e8 −1.91628 −0.958142 0.286292i \(-0.907577\pi\)
−0.958142 + 0.286292i \(0.907577\pi\)
\(542\) 0 0
\(543\) − 5.06089e7i − 0.316102i
\(544\) 0 0
\(545\) 5.71556e6 0.0353077
\(546\) 0 0
\(547\) − 1.92937e7i − 0.117884i −0.998261 0.0589418i \(-0.981227\pi\)
0.998261 0.0589418i \(-0.0187726\pi\)
\(548\) 0 0
\(549\) −6.32942e7 −0.382514
\(550\) 0 0
\(551\) − 4.08906e7i − 0.244438i
\(552\) 0 0
\(553\) −1.11860e8 −0.661452
\(554\) 0 0
\(555\) 2.09004e6i 0.0122257i
\(556\) 0 0
\(557\) 1.71693e8 0.993544 0.496772 0.867881i \(-0.334518\pi\)
0.496772 + 0.867881i \(0.334518\pi\)
\(558\) 0 0
\(559\) − 3.38090e8i − 1.93552i
\(560\) 0 0
\(561\) 7.23771e7 0.409933
\(562\) 0 0
\(563\) − 2.21392e8i − 1.24062i −0.784358 0.620308i \(-0.787008\pi\)
0.784358 0.620308i \(-0.212992\pi\)
\(564\) 0 0
\(565\) 726228. 0.00402650
\(566\) 0 0
\(567\) − 1.18640e7i − 0.0650852i
\(568\) 0 0
\(569\) 2.56991e8 1.39502 0.697511 0.716574i \(-0.254291\pi\)
0.697511 + 0.716574i \(0.254291\pi\)
\(570\) 0 0
\(571\) − 2.71249e8i − 1.45700i −0.685045 0.728501i \(-0.740218\pi\)
0.685045 0.728501i \(-0.259782\pi\)
\(572\) 0 0
\(573\) −1.74131e7 −0.0925574
\(574\) 0 0
\(575\) 2.72169e8i 1.43165i
\(576\) 0 0
\(577\) −3.41573e6 −0.0177810 −0.00889049 0.999960i \(-0.502830\pi\)
−0.00889049 + 0.999960i \(0.502830\pi\)
\(578\) 0 0
\(579\) 4.86944e7i 0.250867i
\(580\) 0 0
\(581\) 1.79109e7 0.0913247
\(582\) 0 0
\(583\) − 1.08544e8i − 0.547772i
\(584\) 0 0
\(585\) 3.86953e6 0.0193282
\(586\) 0 0
\(587\) − 2.32848e8i − 1.15122i −0.817725 0.575609i \(-0.804765\pi\)
0.817725 0.575609i \(-0.195235\pi\)
\(588\) 0 0
\(589\) −3.02430e7 −0.148006
\(590\) 0 0
\(591\) − 1.69182e8i − 0.819580i
\(592\) 0 0
\(593\) −8.86693e7 −0.425216 −0.212608 0.977138i \(-0.568196\pi\)
−0.212608 + 0.977138i \(0.568196\pi\)
\(594\) 0 0
\(595\) − 8.68689e6i − 0.0412395i
\(596\) 0 0
\(597\) −1.40255e8 −0.659165
\(598\) 0 0
\(599\) − 2.37591e7i − 0.110547i −0.998471 0.0552737i \(-0.982397\pi\)
0.998471 0.0552737i \(-0.0176031\pi\)
\(600\) 0 0
\(601\) 2.56421e8 1.18122 0.590609 0.806958i \(-0.298888\pi\)
0.590609 + 0.806958i \(0.298888\pi\)
\(602\) 0 0
\(603\) 7.70512e7i 0.351421i
\(604\) 0 0
\(605\) −8.13845e6 −0.0367516
\(606\) 0 0
\(607\) 2.08193e8i 0.930895i 0.885076 + 0.465447i \(0.154107\pi\)
−0.885076 + 0.465447i \(0.845893\pi\)
\(608\) 0 0
\(609\) −3.61746e7 −0.160159
\(610\) 0 0
\(611\) − 4.26846e8i − 1.87132i
\(612\) 0 0
\(613\) −2.59662e8 −1.12727 −0.563634 0.826024i \(-0.690597\pi\)
−0.563634 + 0.826024i \(0.690597\pi\)
\(614\) 0 0
\(615\) − 9.69222e6i − 0.0416675i
\(616\) 0 0
\(617\) 1.40534e8 0.598309 0.299155 0.954205i \(-0.403295\pi\)
0.299155 + 0.954205i \(0.403295\pi\)
\(618\) 0 0
\(619\) 1.58033e8i 0.666307i 0.942873 + 0.333154i \(0.108113\pi\)
−0.942873 + 0.333154i \(0.891887\pi\)
\(620\) 0 0
\(621\) −6.61349e7 −0.276157
\(622\) 0 0
\(623\) 5.06084e7i 0.209295i
\(624\) 0 0
\(625\) 2.42454e8 0.993093
\(626\) 0 0
\(627\) − 3.55589e7i − 0.144260i
\(628\) 0 0
\(629\) 1.61025e8 0.647056
\(630\) 0 0
\(631\) − 2.49902e8i − 0.994677i −0.867556 0.497339i \(-0.834311\pi\)
0.867556 0.497339i \(-0.165689\pi\)
\(632\) 0 0
\(633\) 5.98798e7 0.236086
\(634\) 0 0
\(635\) 8.69649e6i 0.0339643i
\(636\) 0 0
\(637\) 2.05104e8 0.793515
\(638\) 0 0
\(639\) 1.72147e8i 0.659775i
\(640\) 0 0
\(641\) −2.94095e7 −0.111664 −0.0558321 0.998440i \(-0.517781\pi\)
−0.0558321 + 0.998440i \(0.517781\pi\)
\(642\) 0 0
\(643\) − 2.55925e8i − 0.962675i −0.876535 0.481337i \(-0.840151\pi\)
0.876535 0.481337i \(-0.159849\pi\)
\(644\) 0 0
\(645\) −1.19148e7 −0.0444024
\(646\) 0 0
\(647\) − 4.08342e8i − 1.50769i −0.657054 0.753844i \(-0.728197\pi\)
0.657054 0.753844i \(-0.271803\pi\)
\(648\) 0 0
\(649\) −7.19016e7 −0.263030
\(650\) 0 0
\(651\) 2.67550e7i 0.0969756i
\(652\) 0 0
\(653\) −2.06095e8 −0.740164 −0.370082 0.928999i \(-0.620670\pi\)
−0.370082 + 0.928999i \(0.620670\pi\)
\(654\) 0 0
\(655\) 3.79672e6i 0.0135109i
\(656\) 0 0
\(657\) 9.62081e7 0.339247
\(658\) 0 0
\(659\) − 7.85056e7i − 0.274312i −0.990549 0.137156i \(-0.956204\pi\)
0.990549 0.137156i \(-0.0437961\pi\)
\(660\) 0 0
\(661\) 5.50505e7 0.190615 0.0953074 0.995448i \(-0.469617\pi\)
0.0953074 + 0.995448i \(0.469617\pi\)
\(662\) 0 0
\(663\) − 2.98125e8i − 1.02296i
\(664\) 0 0
\(665\) −4.26787e6 −0.0145126
\(666\) 0 0
\(667\) 2.01652e8i 0.679557i
\(668\) 0 0
\(669\) 5.48295e7 0.183120
\(670\) 0 0
\(671\) 1.67827e8i 0.555512i
\(672\) 0 0
\(673\) −4.82818e7 −0.158394 −0.0791969 0.996859i \(-0.525236\pi\)
−0.0791969 + 0.996859i \(0.525236\pi\)
\(674\) 0 0
\(675\) 5.90511e7i 0.192007i
\(676\) 0 0
\(677\) 4.02892e8 1.29844 0.649221 0.760600i \(-0.275095\pi\)
0.649221 + 0.760600i \(0.275095\pi\)
\(678\) 0 0
\(679\) − 1.03970e8i − 0.332122i
\(680\) 0 0
\(681\) −2.52713e8 −0.800176
\(682\) 0 0
\(683\) 1.26946e8i 0.398434i 0.979955 + 0.199217i \(0.0638398\pi\)
−0.979955 + 0.199217i \(0.936160\pi\)
\(684\) 0 0
\(685\) 2.68693e7 0.0835958
\(686\) 0 0
\(687\) 2.28934e8i 0.706056i
\(688\) 0 0
\(689\) −4.47098e8 −1.36693
\(690\) 0 0
\(691\) − 1.39216e8i − 0.421944i −0.977492 0.210972i \(-0.932337\pi\)
0.977492 0.210972i \(-0.0676629\pi\)
\(692\) 0 0
\(693\) −3.14578e7 −0.0945211
\(694\) 0 0
\(695\) − 1.34943e7i − 0.0401973i
\(696\) 0 0
\(697\) −7.46729e8 −2.20528
\(698\) 0 0
\(699\) − 9.51078e7i − 0.278474i
\(700\) 0 0
\(701\) −4.45981e8 −1.29468 −0.647340 0.762201i \(-0.724119\pi\)
−0.647340 + 0.762201i \(0.724119\pi\)
\(702\) 0 0
\(703\) − 7.91118e7i − 0.227706i
\(704\) 0 0
\(705\) −1.50427e7 −0.0429297
\(706\) 0 0
\(707\) 2.05704e8i 0.582083i
\(708\) 0 0
\(709\) −1.59057e8 −0.446288 −0.223144 0.974786i \(-0.571632\pi\)
−0.223144 + 0.974786i \(0.571632\pi\)
\(710\) 0 0
\(711\) 1.35289e8i 0.376403i
\(712\) 0 0
\(713\) 1.49144e8 0.411468
\(714\) 0 0
\(715\) − 1.02602e7i − 0.0280697i
\(716\) 0 0
\(717\) 4.10811e8 1.11451
\(718\) 0 0
\(719\) − 3.34049e8i − 0.898718i −0.893351 0.449359i \(-0.851652\pi\)
0.893351 0.449359i \(-0.148348\pi\)
\(720\) 0 0
\(721\) 1.82756e7 0.0487602
\(722\) 0 0
\(723\) − 3.35486e7i − 0.0887686i
\(724\) 0 0
\(725\) 1.80053e8 0.472483
\(726\) 0 0
\(727\) 2.60960e8i 0.679157i 0.940578 + 0.339579i \(0.110284\pi\)
−0.940578 + 0.339579i \(0.889716\pi\)
\(728\) 0 0
\(729\) −1.43489e7 −0.0370370
\(730\) 0 0
\(731\) 9.17964e8i 2.35003i
\(732\) 0 0
\(733\) 3.56906e8 0.906238 0.453119 0.891450i \(-0.350311\pi\)
0.453119 + 0.891450i \(0.350311\pi\)
\(734\) 0 0
\(735\) − 7.22815e6i − 0.0182039i
\(736\) 0 0
\(737\) 2.04304e8 0.510357
\(738\) 0 0
\(739\) 8.61653e7i 0.213501i 0.994286 + 0.106750i \(0.0340446\pi\)
−0.994286 + 0.106750i \(0.965955\pi\)
\(740\) 0 0
\(741\) −1.46469e8 −0.359990
\(742\) 0 0
\(743\) − 2.61364e7i − 0.0637204i −0.999492 0.0318602i \(-0.989857\pi\)
0.999492 0.0318602i \(-0.0101431\pi\)
\(744\) 0 0
\(745\) −1.10430e7 −0.0267067
\(746\) 0 0
\(747\) − 2.16623e7i − 0.0519688i
\(748\) 0 0
\(749\) −4.42585e8 −1.05330
\(750\) 0 0
\(751\) 5.80052e8i 1.36945i 0.728800 + 0.684727i \(0.240078\pi\)
−0.728800 + 0.684727i \(0.759922\pi\)
\(752\) 0 0
\(753\) 1.24677e8 0.292012
\(754\) 0 0
\(755\) − 6.86786e6i − 0.0159581i
\(756\) 0 0
\(757\) −3.35602e8 −0.773636 −0.386818 0.922156i \(-0.626426\pi\)
−0.386818 + 0.922156i \(0.626426\pi\)
\(758\) 0 0
\(759\) 1.75359e8i 0.401054i
\(760\) 0 0
\(761\) 2.34765e8 0.532696 0.266348 0.963877i \(-0.414183\pi\)
0.266348 + 0.963877i \(0.414183\pi\)
\(762\) 0 0
\(763\) 1.91393e8i 0.430877i
\(764\) 0 0
\(765\) −1.05063e7 −0.0234675
\(766\) 0 0
\(767\) 2.96167e8i 0.656372i
\(768\) 0 0
\(769\) 3.78738e8 0.832837 0.416419 0.909173i \(-0.363285\pi\)
0.416419 + 0.909173i \(0.363285\pi\)
\(770\) 0 0
\(771\) − 4.46070e8i − 0.973285i
\(772\) 0 0
\(773\) 1.34904e8 0.292070 0.146035 0.989279i \(-0.453349\pi\)
0.146035 + 0.989279i \(0.453349\pi\)
\(774\) 0 0
\(775\) − 1.33169e8i − 0.286086i
\(776\) 0 0
\(777\) −6.99877e7 −0.149196
\(778\) 0 0
\(779\) 3.66868e8i 0.776065i
\(780\) 0 0
\(781\) 4.56453e8 0.958171
\(782\) 0 0
\(783\) 4.37513e7i 0.0911394i
\(784\) 0 0
\(785\) −3.17265e7 −0.0655863
\(786\) 0 0
\(787\) − 8.58679e7i − 0.176160i −0.996113 0.0880799i \(-0.971927\pi\)
0.996113 0.0880799i \(-0.0280731\pi\)
\(788\) 0 0
\(789\) 2.33027e7 0.0474434
\(790\) 0 0
\(791\) 2.43187e7i 0.0491373i
\(792\) 0 0
\(793\) 6.91287e8 1.38624
\(794\) 0 0
\(795\) 1.57564e7i 0.0313585i
\(796\) 0 0
\(797\) −8.82735e8 −1.74363 −0.871817 0.489832i \(-0.837058\pi\)
−0.871817 + 0.489832i \(0.837058\pi\)
\(798\) 0 0
\(799\) 1.15895e9i 2.27208i
\(800\) 0 0
\(801\) 6.12083e7 0.119100
\(802\) 0 0
\(803\) − 2.55099e8i − 0.492677i
\(804\) 0 0
\(805\) 2.10470e7 0.0403463
\(806\) 0 0
\(807\) 8.29758e6i 0.0157881i
\(808\) 0 0
\(809\) 3.19181e7 0.0602825 0.0301412 0.999546i \(-0.490404\pi\)
0.0301412 + 0.999546i \(0.490404\pi\)
\(810\) 0 0
\(811\) − 8.59397e7i − 0.161113i −0.996750 0.0805567i \(-0.974330\pi\)
0.996750 0.0805567i \(-0.0256698\pi\)
\(812\) 0 0
\(813\) 4.49825e8 0.837089
\(814\) 0 0
\(815\) − 3.87521e7i − 0.0715851i
\(816\) 0 0
\(817\) 4.50996e8 0.827003
\(818\) 0 0
\(819\) 1.29576e8i 0.235871i
\(820\) 0 0
\(821\) −7.71027e8 −1.39328 −0.696642 0.717419i \(-0.745324\pi\)
−0.696642 + 0.717419i \(0.745324\pi\)
\(822\) 0 0
\(823\) − 2.04669e8i − 0.367158i −0.983005 0.183579i \(-0.941232\pi\)
0.983005 0.183579i \(-0.0587683\pi\)
\(824\) 0 0
\(825\) 1.56576e8 0.278845
\(826\) 0 0
\(827\) 2.55414e8i 0.451574i 0.974177 + 0.225787i \(0.0724953\pi\)
−0.974177 + 0.225787i \(0.927505\pi\)
\(828\) 0 0
\(829\) 6.29211e8 1.10442 0.552208 0.833706i \(-0.313785\pi\)
0.552208 + 0.833706i \(0.313785\pi\)
\(830\) 0 0
\(831\) 2.84457e8i 0.495694i
\(832\) 0 0
\(833\) −5.56887e8 −0.963456
\(834\) 0 0
\(835\) − 2.38628e7i − 0.0409885i
\(836\) 0 0
\(837\) 3.23589e7 0.0551845
\(838\) 0 0
\(839\) 2.24118e8i 0.379481i 0.981834 + 0.189741i \(0.0607647\pi\)
−0.981834 + 0.189741i \(0.939235\pi\)
\(840\) 0 0
\(841\) −4.61421e8 −0.775728
\(842\) 0 0
\(843\) 1.48405e8i 0.247723i
\(844\) 0 0
\(845\) −1.33014e7 −0.0220459
\(846\) 0 0
\(847\) − 2.72527e8i − 0.448496i
\(848\) 0 0
\(849\) 5.94496e8 0.971461
\(850\) 0 0
\(851\) 3.90140e8i 0.633042i
\(852\) 0 0
\(853\) −2.09382e8 −0.337359 −0.168679 0.985671i \(-0.553950\pi\)
−0.168679 + 0.985671i \(0.553950\pi\)
\(854\) 0 0
\(855\) 5.16177e6i 0.00825849i
\(856\) 0 0
\(857\) −3.32297e8 −0.527940 −0.263970 0.964531i \(-0.585032\pi\)
−0.263970 + 0.964531i \(0.585032\pi\)
\(858\) 0 0
\(859\) 6.58920e8i 1.03957i 0.854297 + 0.519784i \(0.173988\pi\)
−0.854297 + 0.519784i \(0.826012\pi\)
\(860\) 0 0
\(861\) 3.24557e8 0.508488
\(862\) 0 0
\(863\) 1.00694e9i 1.56665i 0.621610 + 0.783327i \(0.286479\pi\)
−0.621610 + 0.783327i \(0.713521\pi\)
\(864\) 0 0
\(865\) 3.44129e7 0.0531707
\(866\) 0 0
\(867\) 4.33186e8i 0.664686i
\(868\) 0 0
\(869\) 3.58723e8 0.546637
\(870\) 0 0
\(871\) − 8.41538e8i − 1.27356i
\(872\) 0 0
\(873\) −1.25746e8 −0.188996
\(874\) 0 0
\(875\) − 3.76287e7i − 0.0561688i
\(876\) 0 0
\(877\) −7.31416e8 −1.08434 −0.542170 0.840269i \(-0.682397\pi\)
−0.542170 + 0.840269i \(0.682397\pi\)
\(878\) 0 0
\(879\) − 2.62425e8i − 0.386401i
\(880\) 0 0
\(881\) −1.16606e9 −1.70528 −0.852638 0.522502i \(-0.824999\pi\)
−0.852638 + 0.522502i \(0.824999\pi\)
\(882\) 0 0
\(883\) − 7.42439e7i − 0.107840i −0.998545 0.0539198i \(-0.982828\pi\)
0.998545 0.0539198i \(-0.0171715\pi\)
\(884\) 0 0
\(885\) 1.04373e7 0.0150577
\(886\) 0 0
\(887\) − 2.35897e8i − 0.338027i −0.985614 0.169014i \(-0.945942\pi\)
0.985614 0.169014i \(-0.0540582\pi\)
\(888\) 0 0
\(889\) −2.91213e8 −0.414482
\(890\) 0 0
\(891\) 3.80466e7i 0.0537877i
\(892\) 0 0
\(893\) 5.69393e8 0.799572
\(894\) 0 0
\(895\) − 4.90130e7i − 0.0683663i
\(896\) 0 0
\(897\) 7.22313e8 1.00080
\(898\) 0 0
\(899\) − 9.86656e7i − 0.135796i
\(900\) 0 0
\(901\) 1.21394e9 1.65967
\(902\) 0 0
\(903\) − 3.98982e8i − 0.541864i
\(904\) 0 0
\(905\) 1.94794e7 0.0262802
\(906\) 0 0
\(907\) 2.54036e8i 0.340466i 0.985404 + 0.170233i \(0.0544520\pi\)
−0.985404 + 0.170233i \(0.945548\pi\)
\(908\) 0 0
\(909\) 2.48789e8 0.331237
\(910\) 0 0
\(911\) − 1.03209e9i − 1.36509i −0.730844 0.682544i \(-0.760874\pi\)
0.730844 0.682544i \(-0.239126\pi\)
\(912\) 0 0
\(913\) −5.74383e7 −0.0754726
\(914\) 0 0
\(915\) − 2.43620e7i − 0.0318016i
\(916\) 0 0
\(917\) −1.27138e8 −0.164880
\(918\) 0 0
\(919\) 1.13004e9i 1.45595i 0.685602 + 0.727976i \(0.259539\pi\)
−0.685602 + 0.727976i \(0.740461\pi\)
\(920\) 0 0
\(921\) −6.84573e8 −0.876276
\(922\) 0 0
\(923\) − 1.88015e9i − 2.39105i
\(924\) 0 0
\(925\) 3.48352e8 0.440142
\(926\) 0 0
\(927\) − 2.21034e7i − 0.0277472i
\(928\) 0 0
\(929\) −1.40049e9 −1.74676 −0.873382 0.487036i \(-0.838078\pi\)
−0.873382 + 0.487036i \(0.838078\pi\)
\(930\) 0 0
\(931\) 2.73599e8i 0.339051i
\(932\) 0 0
\(933\) −7.92472e8 −0.975751
\(934\) 0 0
\(935\) 2.78579e7i 0.0340812i
\(936\) 0 0
\(937\) 3.55654e8 0.432324 0.216162 0.976357i \(-0.430646\pi\)
0.216162 + 0.976357i \(0.430646\pi\)
\(938\) 0 0
\(939\) 2.03774e8i 0.246123i
\(940\) 0 0
\(941\) −1.04392e8 −0.125285 −0.0626427 0.998036i \(-0.519953\pi\)
−0.0626427 + 0.998036i \(0.519953\pi\)
\(942\) 0 0
\(943\) − 1.80921e9i − 2.15752i
\(944\) 0 0
\(945\) 4.56646e6 0.00541108
\(946\) 0 0
\(947\) − 9.83576e8i − 1.15813i −0.815281 0.579066i \(-0.803417\pi\)
0.815281 0.579066i \(-0.196583\pi\)
\(948\) 0 0
\(949\) −1.05077e9 −1.22944
\(950\) 0 0
\(951\) − 7.91462e7i − 0.0920213i
\(952\) 0 0
\(953\) 3.13478e8 0.362184 0.181092 0.983466i \(-0.442037\pi\)
0.181092 + 0.983466i \(0.442037\pi\)
\(954\) 0 0
\(955\) − 6.70229e6i − 0.00769508i
\(956\) 0 0
\(957\) 1.16008e8 0.132359
\(958\) 0 0
\(959\) 8.99753e8i 1.02016i
\(960\) 0 0
\(961\) 8.14530e8 0.917776
\(962\) 0 0
\(963\) 5.35284e8i 0.599384i
\(964\) 0 0
\(965\) −1.87425e7 −0.0208567
\(966\) 0 0
\(967\) − 3.95187e8i − 0.437042i −0.975832 0.218521i \(-0.929877\pi\)
0.975832 0.218521i \(-0.0701232\pi\)
\(968\) 0 0
\(969\) 3.97685e8 0.437087
\(970\) 0 0
\(971\) 2.04334e8i 0.223194i 0.993754 + 0.111597i \(0.0355966\pi\)
−0.993754 + 0.111597i \(0.964403\pi\)
\(972\) 0 0
\(973\) 4.51875e8 0.490546
\(974\) 0 0
\(975\) − 6.44944e8i − 0.695839i
\(976\) 0 0
\(977\) −3.87988e7 −0.0416040 −0.0208020 0.999784i \(-0.506622\pi\)
−0.0208020 + 0.999784i \(0.506622\pi\)
\(978\) 0 0
\(979\) − 1.62296e8i − 0.172966i
\(980\) 0 0
\(981\) 2.31480e8 0.245192
\(982\) 0 0
\(983\) 7.59099e8i 0.799168i 0.916697 + 0.399584i \(0.130845\pi\)
−0.916697 + 0.399584i \(0.869155\pi\)
\(984\) 0 0
\(985\) 6.51181e7 0.0681386
\(986\) 0 0
\(987\) − 5.03724e8i − 0.523891i
\(988\) 0 0
\(989\) −2.22409e9 −2.29913
\(990\) 0 0
\(991\) 5.08413e8i 0.522391i 0.965286 + 0.261195i \(0.0841167\pi\)
−0.965286 + 0.261195i \(0.915883\pi\)
\(992\) 0 0
\(993\) 8.72046e8 0.890618
\(994\) 0 0
\(995\) − 5.39840e7i − 0.0548019i
\(996\) 0 0
\(997\) −1.50556e8 −0.151919 −0.0759594 0.997111i \(-0.524202\pi\)
−0.0759594 + 0.997111i \(0.524202\pi\)
\(998\) 0 0
\(999\) 8.46465e7i 0.0849010i
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 192.7.g.b.127.2 2
3.2 odd 2 576.7.g.g.127.1 2
4.3 odd 2 inner 192.7.g.b.127.1 2
8.3 odd 2 48.7.g.b.31.2 yes 2
8.5 even 2 48.7.g.b.31.1 2
12.11 even 2 576.7.g.g.127.2 2
16.3 odd 4 768.7.b.b.127.4 4
16.5 even 4 768.7.b.b.127.3 4
16.11 odd 4 768.7.b.b.127.1 4
16.13 even 4 768.7.b.b.127.2 4
24.5 odd 2 144.7.g.c.127.1 2
24.11 even 2 144.7.g.c.127.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.7.g.b.31.1 2 8.5 even 2
48.7.g.b.31.2 yes 2 8.3 odd 2
144.7.g.c.127.1 2 24.5 odd 2
144.7.g.c.127.2 2 24.11 even 2
192.7.g.b.127.1 2 4.3 odd 2 inner
192.7.g.b.127.2 2 1.1 even 1 trivial
576.7.g.g.127.1 2 3.2 odd 2
576.7.g.g.127.2 2 12.11 even 2
768.7.b.b.127.1 4 16.11 odd 4
768.7.b.b.127.2 4 16.13 even 4
768.7.b.b.127.3 4 16.5 even 4
768.7.b.b.127.4 4 16.3 odd 4