Properties

Label 162.7.b.a.161.3
Level $162$
Weight $7$
Character 162.161
Analytic conductor $37.269$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,7,Mod(161,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.161");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 162.b (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: \(37.2687615464\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.3
Root \(1.93185i\) of defining polynomial
Character \(\chi\) \(=\) 162.161
Dual form 162.7.b.a.161.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.65685i q^{2} -32.0000 q^{4} -208.528i q^{5} +4.19615 q^{7} -181.019i q^{8} +O(q^{10})\) \(q+5.65685i q^{2} -32.0000 q^{4} -208.528i q^{5} +4.19615 q^{7} -181.019i q^{8} +1179.62 q^{10} +2261.95i q^{11} +2840.01 q^{13} +23.7370i q^{14} +1024.00 q^{16} -1965.86i q^{17} -281.295 q^{19} +6672.91i q^{20} -12795.5 q^{22} -16744.3i q^{23} -27859.1 q^{25} +16065.5i q^{26} -134.277 q^{28} -37114.1i q^{29} -24708.3 q^{31} +5792.62i q^{32} +11120.6 q^{34} -875.017i q^{35} -17016.7 q^{37} -1591.25i q^{38} -37747.7 q^{40} -116236. i q^{41} -30662.9 q^{43} -72382.5i q^{44} +94720.0 q^{46} +77666.6i q^{47} -117631. q^{49} -157595. i q^{50} -90880.2 q^{52} -138657. i q^{53} +471682. q^{55} -759.585i q^{56} +209949. q^{58} +152958. i q^{59} -16138.1 q^{61} -139771. i q^{62} -32768.0 q^{64} -592222. i q^{65} -474667. q^{67} +62907.4i q^{68} +4949.85 q^{70} +150338. i q^{71} +331690. q^{73} -96261.2i q^{74} +9001.45 q^{76} +9491.50i q^{77} -896112. q^{79} -213533. i q^{80} +657529. q^{82} -945090. i q^{83} -409937. q^{85} -173455. i q^{86} +409457. q^{88} -790302. i q^{89} +11917.1 q^{91} +535817. i q^{92} -439349. q^{94} +58658.1i q^{95} +1.39268e6 q^{97} -665424. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 128 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 128 q^{4} - 4 q^{7} + 2640 q^{10} + 3836 q^{13} + 4096 q^{16} + 10244 q^{19} - 27072 q^{22} - 25700 q^{25} + 128 q^{28} - 40096 q^{31} + 60528 q^{34} - 20200 q^{37} - 84480 q^{40} + 184940 q^{43} + 162720 q^{46} - 470484 q^{49} - 122752 q^{52} + 949860 q^{55} + 24624 q^{58} + 609056 q^{61} - 131072 q^{64} - 2008972 q^{67} + 8160 q^{70} + 525824 q^{73} - 327808 q^{76} - 848716 q^{79} + 705792 q^{82} - 987840 q^{85} + 866304 q^{88} + 35260 q^{91} - 1965408 q^{94} + 3621728 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.65685i 0.707107i
\(3\) 0 0
\(4\) −32.0000 −0.500000
\(5\) − 208.528i − 1.66823i −0.551592 0.834114i \(-0.685980\pi\)
0.551592 0.834114i \(-0.314020\pi\)
\(6\) 0 0
\(7\) 4.19615 0.0122337 0.00611684 0.999981i \(-0.498053\pi\)
0.00611684 + 0.999981i \(0.498053\pi\)
\(8\) − 181.019i − 0.353553i
\(9\) 0 0
\(10\) 1179.62 1.17962
\(11\) 2261.95i 1.69944i 0.527235 + 0.849719i \(0.323229\pi\)
−0.527235 + 0.849719i \(0.676771\pi\)
\(12\) 0 0
\(13\) 2840.01 1.29268 0.646338 0.763052i \(-0.276300\pi\)
0.646338 + 0.763052i \(0.276300\pi\)
\(14\) 23.7370i 0.00865052i
\(15\) 0 0
\(16\) 1024.00 0.250000
\(17\) − 1965.86i − 0.400134i −0.979782 0.200067i \(-0.935884\pi\)
0.979782 0.200067i \(-0.0641160\pi\)
\(18\) 0 0
\(19\) −281.295 −0.0410111 −0.0205056 0.999790i \(-0.506528\pi\)
−0.0205056 + 0.999790i \(0.506528\pi\)
\(20\) 6672.91i 0.834114i
\(21\) 0 0
\(22\) −12795.5 −1.20168
\(23\) − 16744.3i − 1.37620i −0.725614 0.688102i \(-0.758444\pi\)
0.725614 0.688102i \(-0.241556\pi\)
\(24\) 0 0
\(25\) −27859.1 −1.78298
\(26\) 16065.5i 0.914059i
\(27\) 0 0
\(28\) −134.277 −0.00611684
\(29\) − 37114.1i − 1.52176i −0.648895 0.760878i \(-0.724769\pi\)
0.648895 0.760878i \(-0.275231\pi\)
\(30\) 0 0
\(31\) −24708.3 −0.829389 −0.414694 0.909961i \(-0.636112\pi\)
−0.414694 + 0.909961i \(0.636112\pi\)
\(32\) 5792.62i 0.176777i
\(33\) 0 0
\(34\) 11120.6 0.282937
\(35\) − 875.017i − 0.0204086i
\(36\) 0 0
\(37\) −17016.7 −0.335947 −0.167974 0.985791i \(-0.553722\pi\)
−0.167974 + 0.985791i \(0.553722\pi\)
\(38\) − 1591.25i − 0.0289993i
\(39\) 0 0
\(40\) −37747.7 −0.589808
\(41\) − 116236.i − 1.68651i −0.537516 0.843253i \(-0.680637\pi\)
0.537516 0.843253i \(-0.319363\pi\)
\(42\) 0 0
\(43\) −30662.9 −0.385662 −0.192831 0.981232i \(-0.561767\pi\)
−0.192831 + 0.981232i \(0.561767\pi\)
\(44\) − 72382.5i − 0.849719i
\(45\) 0 0
\(46\) 94720.0 0.973124
\(47\) 77666.6i 0.748068i 0.927415 + 0.374034i \(0.122026\pi\)
−0.927415 + 0.374034i \(0.877974\pi\)
\(48\) 0 0
\(49\) −117631. −0.999850
\(50\) − 157595.i − 1.26076i
\(51\) 0 0
\(52\) −90880.2 −0.646338
\(53\) − 138657.i − 0.931354i −0.884955 0.465677i \(-0.845811\pi\)
0.884955 0.465677i \(-0.154189\pi\)
\(54\) 0 0
\(55\) 471682. 2.83505
\(56\) − 759.585i − 0.00432526i
\(57\) 0 0
\(58\) 209949. 1.07604
\(59\) 152958.i 0.744758i 0.928081 + 0.372379i \(0.121458\pi\)
−0.928081 + 0.372379i \(0.878542\pi\)
\(60\) 0 0
\(61\) −16138.1 −0.0710989 −0.0355495 0.999368i \(-0.511318\pi\)
−0.0355495 + 0.999368i \(0.511318\pi\)
\(62\) − 139771.i − 0.586467i
\(63\) 0 0
\(64\) −32768.0 −0.125000
\(65\) − 592222.i − 2.15648i
\(66\) 0 0
\(67\) −474667. −1.57821 −0.789105 0.614259i \(-0.789455\pi\)
−0.789105 + 0.614259i \(0.789455\pi\)
\(68\) 62907.4i 0.200067i
\(69\) 0 0
\(70\) 4949.85 0.0144310
\(71\) 150338.i 0.420043i 0.977697 + 0.210021i \(0.0673534\pi\)
−0.977697 + 0.210021i \(0.932647\pi\)
\(72\) 0 0
\(73\) 331690. 0.852636 0.426318 0.904573i \(-0.359811\pi\)
0.426318 + 0.904573i \(0.359811\pi\)
\(74\) − 96261.2i − 0.237551i
\(75\) 0 0
\(76\) 9001.45 0.0205056
\(77\) 9491.50i 0.0207904i
\(78\) 0 0
\(79\) −896112. −1.81753 −0.908764 0.417310i \(-0.862973\pi\)
−0.908764 + 0.417310i \(0.862973\pi\)
\(80\) − 213533.i − 0.417057i
\(81\) 0 0
\(82\) 657529. 1.19254
\(83\) − 945090.i − 1.65287i −0.563032 0.826435i \(-0.690365\pi\)
0.563032 0.826435i \(-0.309635\pi\)
\(84\) 0 0
\(85\) −409937. −0.667514
\(86\) − 173455.i − 0.272704i
\(87\) 0 0
\(88\) 409457. 0.600842
\(89\) − 790302.i − 1.12105i −0.828139 0.560523i \(-0.810600\pi\)
0.828139 0.560523i \(-0.189400\pi\)
\(90\) 0 0
\(91\) 11917.1 0.0158142
\(92\) 535817.i 0.688102i
\(93\) 0 0
\(94\) −439349. −0.528964
\(95\) 58658.1i 0.0684159i
\(96\) 0 0
\(97\) 1.39268e6 1.52593 0.762965 0.646440i \(-0.223743\pi\)
0.762965 + 0.646440i \(0.223743\pi\)
\(98\) − 665424.i − 0.707001i
\(99\) 0 0
\(100\) 891492. 0.891492
\(101\) − 131552.i − 0.127683i −0.997960 0.0638414i \(-0.979665\pi\)
0.997960 0.0638414i \(-0.0203352\pi\)
\(102\) 0 0
\(103\) 1.93461e6 1.77044 0.885221 0.465171i \(-0.154007\pi\)
0.885221 + 0.465171i \(0.154007\pi\)
\(104\) − 514096.i − 0.457030i
\(105\) 0 0
\(106\) 784364. 0.658567
\(107\) 621916.i 0.507669i 0.967248 + 0.253834i \(0.0816918\pi\)
−0.967248 + 0.253834i \(0.918308\pi\)
\(108\) 0 0
\(109\) −1.34578e6 −1.03919 −0.519594 0.854413i \(-0.673917\pi\)
−0.519594 + 0.854413i \(0.673917\pi\)
\(110\) 2.66823e6i 2.00468i
\(111\) 0 0
\(112\) 4296.86 0.00305842
\(113\) − 1.23932e6i − 0.858914i −0.903087 0.429457i \(-0.858705\pi\)
0.903087 0.429457i \(-0.141295\pi\)
\(114\) 0 0
\(115\) −3.49166e6 −2.29582
\(116\) 1.18765e6i 0.760878i
\(117\) 0 0
\(118\) −865259. −0.526623
\(119\) − 8249.04i − 0.00489511i
\(120\) 0 0
\(121\) −3.34487e6 −1.88809
\(122\) − 91290.9i − 0.0502745i
\(123\) 0 0
\(124\) 790666. 0.414694
\(125\) 2.55116e6i 1.30620i
\(126\) 0 0
\(127\) −1.65297e6 −0.806962 −0.403481 0.914988i \(-0.632200\pi\)
−0.403481 + 0.914988i \(0.632200\pi\)
\(128\) − 185364.i − 0.0883883i
\(129\) 0 0
\(130\) 3.35012e6 1.52486
\(131\) 2.66126e6i 1.18379i 0.806017 + 0.591893i \(0.201619\pi\)
−0.806017 + 0.591893i \(0.798381\pi\)
\(132\) 0 0
\(133\) −1180.36 −0.000501717 0
\(134\) − 2.68512e6i − 1.11596i
\(135\) 0 0
\(136\) −355858. −0.141469
\(137\) − 2.61015e6i − 1.01509i −0.861626 0.507544i \(-0.830553\pi\)
0.861626 0.507544i \(-0.169447\pi\)
\(138\) 0 0
\(139\) −701321. −0.261139 −0.130570 0.991439i \(-0.541681\pi\)
−0.130570 + 0.991439i \(0.541681\pi\)
\(140\) 28000.6i 0.0102043i
\(141\) 0 0
\(142\) −850440. −0.297015
\(143\) 6.42396e6i 2.19682i
\(144\) 0 0
\(145\) −7.73935e6 −2.53864
\(146\) 1.87632e6i 0.602904i
\(147\) 0 0
\(148\) 544536. 0.167974
\(149\) 4.73344e6i 1.43093i 0.698650 + 0.715464i \(0.253785\pi\)
−0.698650 + 0.715464i \(0.746215\pi\)
\(150\) 0 0
\(151\) 3.73729e6 1.08549 0.542746 0.839897i \(-0.317385\pi\)
0.542746 + 0.839897i \(0.317385\pi\)
\(152\) 50919.9i 0.0144996i
\(153\) 0 0
\(154\) −53692.0 −0.0147010
\(155\) 5.15239e6i 1.38361i
\(156\) 0 0
\(157\) 3.20662e6 0.828607 0.414303 0.910139i \(-0.364025\pi\)
0.414303 + 0.910139i \(0.364025\pi\)
\(158\) − 5.06918e6i − 1.28519i
\(159\) 0 0
\(160\) 1.20793e6 0.294904
\(161\) − 70261.6i − 0.0168361i
\(162\) 0 0
\(163\) −964418. −0.222691 −0.111345 0.993782i \(-0.535516\pi\)
−0.111345 + 0.993782i \(0.535516\pi\)
\(164\) 3.71954e6i 0.843253i
\(165\) 0 0
\(166\) 5.34624e6 1.16876
\(167\) 1.03499e6i 0.222222i 0.993808 + 0.111111i \(0.0354410\pi\)
−0.993808 + 0.111111i \(0.964559\pi\)
\(168\) 0 0
\(169\) 3.23883e6 0.671009
\(170\) − 2.31896e6i − 0.472004i
\(171\) 0 0
\(172\) 981212. 0.192831
\(173\) − 648596.i − 0.125267i −0.998037 0.0626334i \(-0.980050\pi\)
0.998037 0.0626334i \(-0.0199499\pi\)
\(174\) 0 0
\(175\) −116901. −0.0218125
\(176\) 2.31624e6i 0.424860i
\(177\) 0 0
\(178\) 4.47063e6 0.792699
\(179\) − 4.65189e6i − 0.811092i −0.914075 0.405546i \(-0.867081\pi\)
0.914075 0.405546i \(-0.132919\pi\)
\(180\) 0 0
\(181\) 3.43720e6 0.579654 0.289827 0.957079i \(-0.406402\pi\)
0.289827 + 0.957079i \(0.406402\pi\)
\(182\) 67413.3i 0.0111823i
\(183\) 0 0
\(184\) −3.03104e6 −0.486562
\(185\) 3.54847e6i 0.560437i
\(186\) 0 0
\(187\) 4.44668e6 0.680003
\(188\) − 2.48533e6i − 0.374034i
\(189\) 0 0
\(190\) −331820. −0.0483774
\(191\) − 1.17509e7i − 1.68644i −0.537565 0.843222i \(-0.680656\pi\)
0.537565 0.843222i \(-0.319344\pi\)
\(192\) 0 0
\(193\) 6.95599e6 0.967581 0.483790 0.875184i \(-0.339260\pi\)
0.483790 + 0.875184i \(0.339260\pi\)
\(194\) 7.87816e6i 1.07900i
\(195\) 0 0
\(196\) 3.76420e6 0.499925
\(197\) − 409645.i − 0.0535807i −0.999641 0.0267904i \(-0.991471\pi\)
0.999641 0.0267904i \(-0.00852866\pi\)
\(198\) 0 0
\(199\) −3.24847e6 −0.412212 −0.206106 0.978530i \(-0.566079\pi\)
−0.206106 + 0.978530i \(0.566079\pi\)
\(200\) 5.04304e6i 0.630380i
\(201\) 0 0
\(202\) 744169. 0.0902854
\(203\) − 155736.i − 0.0186167i
\(204\) 0 0
\(205\) −2.42385e7 −2.81348
\(206\) 1.09438e7i 1.25189i
\(207\) 0 0
\(208\) 2.90817e6 0.323169
\(209\) − 636277.i − 0.0696959i
\(210\) 0 0
\(211\) −4.99487e6 −0.531712 −0.265856 0.964013i \(-0.585655\pi\)
−0.265856 + 0.964013i \(0.585655\pi\)
\(212\) 4.43703e6i 0.465677i
\(213\) 0 0
\(214\) −3.51809e6 −0.358976
\(215\) 6.39408e6i 0.643373i
\(216\) 0 0
\(217\) −103680. −0.0101465
\(218\) − 7.61287e6i − 0.734817i
\(219\) 0 0
\(220\) −1.50938e7 −1.41753
\(221\) − 5.58305e6i − 0.517243i
\(222\) 0 0
\(223\) 5.68924e6 0.513027 0.256513 0.966541i \(-0.417426\pi\)
0.256513 + 0.966541i \(0.417426\pi\)
\(224\) 24306.7i 0.00216263i
\(225\) 0 0
\(226\) 7.01068e6 0.607344
\(227\) − 1.16064e7i − 0.992246i −0.868252 0.496123i \(-0.834757\pi\)
0.868252 0.496123i \(-0.165243\pi\)
\(228\) 0 0
\(229\) −3.45690e6 −0.287859 −0.143930 0.989588i \(-0.545974\pi\)
−0.143930 + 0.989588i \(0.545974\pi\)
\(230\) − 1.97518e7i − 1.62339i
\(231\) 0 0
\(232\) −6.71837e6 −0.538022
\(233\) − 1.27739e7i − 1.00985i −0.863164 0.504924i \(-0.831520\pi\)
0.863164 0.504924i \(-0.168480\pi\)
\(234\) 0 0
\(235\) 1.61957e7 1.24795
\(236\) − 4.89464e6i − 0.372379i
\(237\) 0 0
\(238\) 46663.6 0.00346137
\(239\) − 1.29231e7i − 0.946617i −0.880897 0.473309i \(-0.843060\pi\)
0.880897 0.473309i \(-0.156940\pi\)
\(240\) 0 0
\(241\) 7.28032e6 0.520115 0.260057 0.965593i \(-0.416259\pi\)
0.260057 + 0.965593i \(0.416259\pi\)
\(242\) − 1.89214e7i − 1.33508i
\(243\) 0 0
\(244\) 516419. 0.0355495
\(245\) 2.45295e7i 1.66798i
\(246\) 0 0
\(247\) −798881. −0.0530141
\(248\) 4.47268e6i 0.293233i
\(249\) 0 0
\(250\) −1.44316e7 −0.923620
\(251\) − 1.71927e7i − 1.08723i −0.839334 0.543616i \(-0.817055\pi\)
0.839334 0.543616i \(-0.182945\pi\)
\(252\) 0 0
\(253\) 3.78748e7 2.33878
\(254\) − 9.35059e6i − 0.570608i
\(255\) 0 0
\(256\) 1.04858e6 0.0625000
\(257\) − 8.01111e6i − 0.471947i −0.971759 0.235974i \(-0.924172\pi\)
0.971759 0.235974i \(-0.0758279\pi\)
\(258\) 0 0
\(259\) −71404.8 −0.00410987
\(260\) 1.89511e7i 1.07824i
\(261\) 0 0
\(262\) −1.50543e7 −0.837062
\(263\) 1.68209e7i 0.924661i 0.886708 + 0.462331i \(0.152987\pi\)
−0.886708 + 0.462331i \(0.847013\pi\)
\(264\) 0 0
\(265\) −2.89140e7 −1.55371
\(266\) − 6677.11i 0 0.000354768i
\(267\) 0 0
\(268\) 1.51893e7 0.789105
\(269\) − 3.33514e6i − 0.171339i −0.996324 0.0856697i \(-0.972697\pi\)
0.996324 0.0856697i \(-0.0273030\pi\)
\(270\) 0 0
\(271\) −1.47048e7 −0.738840 −0.369420 0.929263i \(-0.620444\pi\)
−0.369420 + 0.929263i \(0.620444\pi\)
\(272\) − 2.01304e6i − 0.100033i
\(273\) 0 0
\(274\) 1.47652e7 0.717776
\(275\) − 6.30160e7i − 3.03007i
\(276\) 0 0
\(277\) −1.09201e7 −0.513790 −0.256895 0.966439i \(-0.582699\pi\)
−0.256895 + 0.966439i \(0.582699\pi\)
\(278\) − 3.96727e6i − 0.184653i
\(279\) 0 0
\(280\) −158395. −0.00721552
\(281\) 1.86747e7i 0.841654i 0.907141 + 0.420827i \(0.138260\pi\)
−0.907141 + 0.420827i \(0.861740\pi\)
\(282\) 0 0
\(283\) −1.46255e7 −0.645284 −0.322642 0.946521i \(-0.604571\pi\)
−0.322642 + 0.946521i \(0.604571\pi\)
\(284\) − 4.81081e6i − 0.210021i
\(285\) 0 0
\(286\) −3.63394e7 −1.55339
\(287\) − 487743.i − 0.0206322i
\(288\) 0 0
\(289\) 2.02730e7 0.839893
\(290\) − 4.37804e7i − 1.79509i
\(291\) 0 0
\(292\) −1.06141e7 −0.426318
\(293\) 4.78429e7i 1.90202i 0.309162 + 0.951010i \(0.399952\pi\)
−0.309162 + 0.951010i \(0.600048\pi\)
\(294\) 0 0
\(295\) 3.18960e7 1.24243
\(296\) 3.08036e6i 0.118775i
\(297\) 0 0
\(298\) −2.67764e7 −1.01182
\(299\) − 4.75539e7i − 1.77899i
\(300\) 0 0
\(301\) −128666. −0.00471807
\(302\) 2.11413e7i 0.767558i
\(303\) 0 0
\(304\) −288046. −0.0102528
\(305\) 3.36525e6i 0.118609i
\(306\) 0 0
\(307\) 4.24782e7 1.46808 0.734042 0.679104i \(-0.237631\pi\)
0.734042 + 0.679104i \(0.237631\pi\)
\(308\) − 303728.i − 0.0103952i
\(309\) 0 0
\(310\) −2.91463e7 −0.978360
\(311\) 3.09996e7i 1.03056i 0.857021 + 0.515282i \(0.172313\pi\)
−0.857021 + 0.515282i \(0.827687\pi\)
\(312\) 0 0
\(313\) −2.97828e7 −0.971254 −0.485627 0.874166i \(-0.661409\pi\)
−0.485627 + 0.874166i \(0.661409\pi\)
\(314\) 1.81394e7i 0.585913i
\(315\) 0 0
\(316\) 2.86756e7 0.908764
\(317\) − 2.17911e6i − 0.0684072i −0.999415 0.0342036i \(-0.989111\pi\)
0.999415 0.0342036i \(-0.0108895\pi\)
\(318\) 0 0
\(319\) 8.39504e7 2.58613
\(320\) 6.83306e6i 0.208528i
\(321\) 0 0
\(322\) 397459. 0.0119049
\(323\) 552987.i 0.0164099i
\(324\) 0 0
\(325\) −7.91201e7 −2.30482
\(326\) − 5.45557e6i − 0.157466i
\(327\) 0 0
\(328\) −2.10409e7 −0.596270
\(329\) 325901.i 0.00915162i
\(330\) 0 0
\(331\) 1.33671e7 0.368598 0.184299 0.982870i \(-0.440999\pi\)
0.184299 + 0.982870i \(0.440999\pi\)
\(332\) 3.02429e7i 0.826435i
\(333\) 0 0
\(334\) −5.85480e6 −0.157135
\(335\) 9.89816e7i 2.63281i
\(336\) 0 0
\(337\) 4.45488e7 1.16398 0.581991 0.813195i \(-0.302274\pi\)
0.581991 + 0.813195i \(0.302274\pi\)
\(338\) 1.83216e7i 0.474475i
\(339\) 0 0
\(340\) 1.31180e7 0.333757
\(341\) − 5.58891e7i − 1.40950i
\(342\) 0 0
\(343\) −987272. −0.0244655
\(344\) 5.55057e6i 0.136352i
\(345\) 0 0
\(346\) 3.66902e6 0.0885771
\(347\) 2.08986e7i 0.500182i 0.968222 + 0.250091i \(0.0804605\pi\)
−0.968222 + 0.250091i \(0.919540\pi\)
\(348\) 0 0
\(349\) 6.45154e7 1.51770 0.758852 0.651263i \(-0.225761\pi\)
0.758852 + 0.651263i \(0.225761\pi\)
\(350\) − 661293.i − 0.0154237i
\(351\) 0 0
\(352\) −1.31026e7 −0.300421
\(353\) 4.20921e6i 0.0956923i 0.998855 + 0.0478461i \(0.0152357\pi\)
−0.998855 + 0.0478461i \(0.984764\pi\)
\(354\) 0 0
\(355\) 3.13497e7 0.700727
\(356\) 2.52897e7i 0.560523i
\(357\) 0 0
\(358\) 2.63151e7 0.573529
\(359\) 5.61320e7i 1.21319i 0.795012 + 0.606593i \(0.207464\pi\)
−0.795012 + 0.606593i \(0.792536\pi\)
\(360\) 0 0
\(361\) −4.69668e7 −0.998318
\(362\) 1.94437e7i 0.409877i
\(363\) 0 0
\(364\) −381347. −0.00790709
\(365\) − 6.91668e7i − 1.42239i
\(366\) 0 0
\(367\) 3.35103e7 0.677922 0.338961 0.940800i \(-0.389925\pi\)
0.338961 + 0.940800i \(0.389925\pi\)
\(368\) − 1.71461e7i − 0.344051i
\(369\) 0 0
\(370\) −2.00732e7 −0.396289
\(371\) − 581827.i − 0.0113939i
\(372\) 0 0
\(373\) 2.22777e7 0.429283 0.214642 0.976693i \(-0.431142\pi\)
0.214642 + 0.976693i \(0.431142\pi\)
\(374\) 2.51542e7i 0.480835i
\(375\) 0 0
\(376\) 1.40592e7 0.264482
\(377\) − 1.05404e8i − 1.96714i
\(378\) 0 0
\(379\) 7.72205e7 1.41845 0.709226 0.704981i \(-0.249044\pi\)
0.709226 + 0.704981i \(0.249044\pi\)
\(380\) − 1.87706e6i − 0.0342080i
\(381\) 0 0
\(382\) 6.64733e7 1.19250
\(383\) 3.32594e7i 0.591996i 0.955189 + 0.295998i \(0.0956521\pi\)
−0.955189 + 0.295998i \(0.904348\pi\)
\(384\) 0 0
\(385\) 1.97925e6 0.0346831
\(386\) 3.93490e7i 0.684183i
\(387\) 0 0
\(388\) −4.45656e7 −0.762965
\(389\) 1.50959e7i 0.256454i 0.991745 + 0.128227i \(0.0409286\pi\)
−0.991745 + 0.128227i \(0.959071\pi\)
\(390\) 0 0
\(391\) −3.29169e7 −0.550666
\(392\) 2.12936e7i 0.353500i
\(393\) 0 0
\(394\) 2.31730e6 0.0378873
\(395\) 1.86865e8i 3.03205i
\(396\) 0 0
\(397\) −8.91350e7 −1.42455 −0.712273 0.701902i \(-0.752334\pi\)
−0.712273 + 0.701902i \(0.752334\pi\)
\(398\) − 1.83761e7i − 0.291478i
\(399\) 0 0
\(400\) −2.85277e7 −0.445746
\(401\) 1.12043e8i 1.73761i 0.495158 + 0.868803i \(0.335110\pi\)
−0.495158 + 0.868803i \(0.664890\pi\)
\(402\) 0 0
\(403\) −7.01718e7 −1.07213
\(404\) 4.20966e6i 0.0638414i
\(405\) 0 0
\(406\) 880978. 0.0131640
\(407\) − 3.84911e7i − 0.570922i
\(408\) 0 0
\(409\) −6.67941e7 −0.976266 −0.488133 0.872769i \(-0.662322\pi\)
−0.488133 + 0.872769i \(0.662322\pi\)
\(410\) − 1.37113e8i − 1.98943i
\(411\) 0 0
\(412\) −6.19075e7 −0.885221
\(413\) 641833.i 0.00911113i
\(414\) 0 0
\(415\) −1.97078e8 −2.75736
\(416\) 1.64511e7i 0.228515i
\(417\) 0 0
\(418\) 3.59933e6 0.0492824
\(419\) − 4.73571e7i − 0.643789i −0.946776 0.321894i \(-0.895680\pi\)
0.946776 0.321894i \(-0.104320\pi\)
\(420\) 0 0
\(421\) 6.61569e7 0.886602 0.443301 0.896373i \(-0.353807\pi\)
0.443301 + 0.896373i \(0.353807\pi\)
\(422\) − 2.82552e7i − 0.375977i
\(423\) 0 0
\(424\) −2.50996e7 −0.329283
\(425\) 5.47671e7i 0.713432i
\(426\) 0 0
\(427\) −67717.9 −0.000869801 0
\(428\) − 1.99013e7i − 0.253834i
\(429\) 0 0
\(430\) −3.61704e7 −0.454933
\(431\) 9.06666e7i 1.13244i 0.824254 + 0.566221i \(0.191595\pi\)
−0.824254 + 0.566221i \(0.808405\pi\)
\(432\) 0 0
\(433\) −699414. −0.00861530 −0.00430765 0.999991i \(-0.501371\pi\)
−0.00430765 + 0.999991i \(0.501371\pi\)
\(434\) − 586502.i − 0.00717464i
\(435\) 0 0
\(436\) 4.30649e7 0.519594
\(437\) 4.71009e6i 0.0564397i
\(438\) 0 0
\(439\) −2.57296e7 −0.304116 −0.152058 0.988372i \(-0.548590\pi\)
−0.152058 + 0.988372i \(0.548590\pi\)
\(440\) − 8.53835e7i − 1.00234i
\(441\) 0 0
\(442\) 3.15825e7 0.365746
\(443\) − 165578.i − 0.00190455i −1.00000 0.000952273i \(-0.999697\pi\)
1.00000 0.000952273i \(-0.000303118\pi\)
\(444\) 0 0
\(445\) −1.64801e8 −1.87016
\(446\) 3.21832e7i 0.362765i
\(447\) 0 0
\(448\) −137500. −0.00152921
\(449\) 4.30456e7i 0.475542i 0.971321 + 0.237771i \(0.0764169\pi\)
−0.971321 + 0.237771i \(0.923583\pi\)
\(450\) 0 0
\(451\) 2.62920e8 2.86611
\(452\) 3.96584e7i 0.429457i
\(453\) 0 0
\(454\) 6.56556e7 0.701624
\(455\) − 2.48506e6i − 0.0263816i
\(456\) 0 0
\(457\) −1.35491e8 −1.41959 −0.709793 0.704411i \(-0.751211\pi\)
−0.709793 + 0.704411i \(0.751211\pi\)
\(458\) − 1.95552e7i − 0.203547i
\(459\) 0 0
\(460\) 1.11733e8 1.14791
\(461\) − 3.20349e7i − 0.326980i −0.986545 0.163490i \(-0.947725\pi\)
0.986545 0.163490i \(-0.0522751\pi\)
\(462\) 0 0
\(463\) 1.29333e8 1.30306 0.651532 0.758621i \(-0.274127\pi\)
0.651532 + 0.758621i \(0.274127\pi\)
\(464\) − 3.80048e7i − 0.380439i
\(465\) 0 0
\(466\) 7.22602e7 0.714071
\(467\) − 8.49294e7i − 0.833887i −0.908932 0.416944i \(-0.863101\pi\)
0.908932 0.416944i \(-0.136899\pi\)
\(468\) 0 0
\(469\) −1.99178e6 −0.0193073
\(470\) 9.16168e7i 0.882432i
\(471\) 0 0
\(472\) 2.76883e7 0.263312
\(473\) − 6.93579e7i − 0.655410i
\(474\) 0 0
\(475\) 7.83664e6 0.0731222
\(476\) 263969.i 0.00244755i
\(477\) 0 0
\(478\) 7.31043e7 0.669360
\(479\) − 1.15719e8i − 1.05292i −0.850199 0.526461i \(-0.823518\pi\)
0.850199 0.526461i \(-0.176482\pi\)
\(480\) 0 0
\(481\) −4.83277e7 −0.434271
\(482\) 4.11837e7i 0.367777i
\(483\) 0 0
\(484\) 1.07036e8 0.944046
\(485\) − 2.90412e8i − 2.54560i
\(486\) 0 0
\(487\) 4.52839e7 0.392064 0.196032 0.980597i \(-0.437194\pi\)
0.196032 + 0.980597i \(0.437194\pi\)
\(488\) 2.92131e6i 0.0251373i
\(489\) 0 0
\(490\) −1.38760e8 −1.17944
\(491\) − 2.04312e8i − 1.72603i −0.505175 0.863017i \(-0.668572\pi\)
0.505175 0.863017i \(-0.331428\pi\)
\(492\) 0 0
\(493\) −7.29611e7 −0.608906
\(494\) − 4.51915e6i − 0.0374866i
\(495\) 0 0
\(496\) −2.53013e7 −0.207347
\(497\) 630841.i 0.00513867i
\(498\) 0 0
\(499\) −2.92990e7 −0.235804 −0.117902 0.993025i \(-0.537617\pi\)
−0.117902 + 0.993025i \(0.537617\pi\)
\(500\) − 8.16373e7i − 0.653098i
\(501\) 0 0
\(502\) 9.72564e7 0.768789
\(503\) 1.64834e8i 1.29522i 0.761973 + 0.647609i \(0.224231\pi\)
−0.761973 + 0.647609i \(0.775769\pi\)
\(504\) 0 0
\(505\) −2.74323e7 −0.213004
\(506\) 2.14252e8i 1.65376i
\(507\) 0 0
\(508\) 5.28949e7 0.403481
\(509\) 1.67339e8i 1.26895i 0.772944 + 0.634474i \(0.218783\pi\)
−0.772944 + 0.634474i \(0.781217\pi\)
\(510\) 0 0
\(511\) 1.39182e6 0.0104309
\(512\) 5.93164e6i 0.0441942i
\(513\) 0 0
\(514\) 4.53177e7 0.333717
\(515\) − 4.03421e8i − 2.95350i
\(516\) 0 0
\(517\) −1.75678e8 −1.27130
\(518\) − 403927.i − 0.00290612i
\(519\) 0 0
\(520\) −1.07204e8 −0.762430
\(521\) 6.01419e7i 0.425270i 0.977132 + 0.212635i \(0.0682045\pi\)
−0.977132 + 0.212635i \(0.931796\pi\)
\(522\) 0 0
\(523\) 1.09251e8 0.763698 0.381849 0.924225i \(-0.375287\pi\)
0.381849 + 0.924225i \(0.375287\pi\)
\(524\) − 8.51602e7i − 0.591893i
\(525\) 0 0
\(526\) −9.51535e7 −0.653834
\(527\) 4.85731e7i 0.331867i
\(528\) 0 0
\(529\) −1.32335e8 −0.893940
\(530\) − 1.63562e8i − 1.09864i
\(531\) 0 0
\(532\) 37771.5 0.000250859 0
\(533\) − 3.30110e8i − 2.18011i
\(534\) 0 0
\(535\) 1.29687e8 0.846907
\(536\) 8.59239e7i 0.557981i
\(537\) 0 0
\(538\) 1.88664e7 0.121155
\(539\) − 2.66077e8i − 1.69918i
\(540\) 0 0
\(541\) 4.52159e7 0.285561 0.142781 0.989754i \(-0.454396\pi\)
0.142781 + 0.989754i \(0.454396\pi\)
\(542\) − 8.31827e7i − 0.522439i
\(543\) 0 0
\(544\) 1.13875e7 0.0707343
\(545\) 2.80633e8i 1.73360i
\(546\) 0 0
\(547\) 5.11451e7 0.312494 0.156247 0.987718i \(-0.450060\pi\)
0.156247 + 0.987718i \(0.450060\pi\)
\(548\) 8.35248e7i 0.507544i
\(549\) 0 0
\(550\) 3.56473e8 2.14258
\(551\) 1.04400e7i 0.0624089i
\(552\) 0 0
\(553\) −3.76022e6 −0.0222351
\(554\) − 6.17732e7i − 0.363304i
\(555\) 0 0
\(556\) 2.24423e7 0.130570
\(557\) 2.43306e7i 0.140795i 0.997519 + 0.0703974i \(0.0224267\pi\)
−0.997519 + 0.0703974i \(0.977573\pi\)
\(558\) 0 0
\(559\) −8.70827e7 −0.498536
\(560\) − 896018.i − 0.00510214i
\(561\) 0 0
\(562\) −1.05640e8 −0.595139
\(563\) − 1.44398e8i − 0.809166i −0.914501 0.404583i \(-0.867417\pi\)
0.914501 0.404583i \(-0.132583\pi\)
\(564\) 0 0
\(565\) −2.58434e8 −1.43286
\(566\) − 8.27342e7i − 0.456284i
\(567\) 0 0
\(568\) 2.72141e7 0.148508
\(569\) 8.31553e7i 0.451391i 0.974198 + 0.225696i \(0.0724655\pi\)
−0.974198 + 0.225696i \(0.927534\pi\)
\(570\) 0 0
\(571\) −2.26080e8 −1.21438 −0.607188 0.794558i \(-0.707703\pi\)
−0.607188 + 0.794558i \(0.707703\pi\)
\(572\) − 2.05567e8i − 1.09841i
\(573\) 0 0
\(574\) 2.75909e6 0.0145892
\(575\) 4.66481e8i 2.45375i
\(576\) 0 0
\(577\) 4.11006e7 0.213954 0.106977 0.994261i \(-0.465883\pi\)
0.106977 + 0.994261i \(0.465883\pi\)
\(578\) 1.14681e8i 0.593894i
\(579\) 0 0
\(580\) 2.47659e8 1.26932
\(581\) − 3.96574e6i − 0.0202207i
\(582\) 0 0
\(583\) 3.13636e8 1.58278
\(584\) − 6.00423e7i − 0.301452i
\(585\) 0 0
\(586\) −2.70640e8 −1.34493
\(587\) 1.14693e8i 0.567052i 0.958965 + 0.283526i \(0.0915042\pi\)
−0.958965 + 0.283526i \(0.908496\pi\)
\(588\) 0 0
\(589\) 6.95034e6 0.0340142
\(590\) 1.80431e8i 0.878528i
\(591\) 0 0
\(592\) −1.74251e7 −0.0839868
\(593\) − 2.90487e8i − 1.39304i −0.717539 0.696518i \(-0.754731\pi\)
0.717539 0.696518i \(-0.245269\pi\)
\(594\) 0 0
\(595\) −1.72016e6 −0.00816616
\(596\) − 1.51470e8i − 0.715464i
\(597\) 0 0
\(598\) 2.69005e8 1.25793
\(599\) − 3.57152e8i − 1.66178i −0.556439 0.830888i \(-0.687833\pi\)
0.556439 0.830888i \(-0.312167\pi\)
\(600\) 0 0
\(601\) −8.40094e7 −0.386994 −0.193497 0.981101i \(-0.561983\pi\)
−0.193497 + 0.981101i \(0.561983\pi\)
\(602\) − 727845.i − 0.00333618i
\(603\) 0 0
\(604\) −1.19593e8 −0.542746
\(605\) 6.97500e8i 3.14977i
\(606\) 0 0
\(607\) 5.16287e7 0.230847 0.115424 0.993316i \(-0.463177\pi\)
0.115424 + 0.993316i \(0.463177\pi\)
\(608\) − 1.62944e6i − 0.00724981i
\(609\) 0 0
\(610\) −1.90368e7 −0.0838694
\(611\) 2.20574e8i 0.967009i
\(612\) 0 0
\(613\) 4.44326e7 0.192895 0.0964473 0.995338i \(-0.469252\pi\)
0.0964473 + 0.995338i \(0.469252\pi\)
\(614\) 2.40293e8i 1.03809i
\(615\) 0 0
\(616\) 1.71814e6 0.00735051
\(617\) 1.65408e8i 0.704208i 0.935961 + 0.352104i \(0.114534\pi\)
−0.935961 + 0.352104i \(0.885466\pi\)
\(618\) 0 0
\(619\) 2.68162e8 1.13064 0.565320 0.824871i \(-0.308753\pi\)
0.565320 + 0.824871i \(0.308753\pi\)
\(620\) − 1.64876e8i − 0.691805i
\(621\) 0 0
\(622\) −1.75360e8 −0.728718
\(623\) − 3.31623e6i − 0.0137145i
\(624\) 0 0
\(625\) 9.66915e7 0.396049
\(626\) − 1.68477e8i − 0.686780i
\(627\) 0 0
\(628\) −1.02612e8 −0.414303
\(629\) 3.34525e7i 0.134424i
\(630\) 0 0
\(631\) 1.13998e8 0.453741 0.226871 0.973925i \(-0.427151\pi\)
0.226871 + 0.973925i \(0.427151\pi\)
\(632\) 1.62214e8i 0.642593i
\(633\) 0 0
\(634\) 1.23269e7 0.0483712
\(635\) 3.44691e8i 1.34620i
\(636\) 0 0
\(637\) −3.34074e8 −1.29248
\(638\) 4.74895e8i 1.82867i
\(639\) 0 0
\(640\) −3.86536e7 −0.147452
\(641\) − 3.65639e7i − 0.138828i −0.997588 0.0694142i \(-0.977887\pi\)
0.997588 0.0694142i \(-0.0221130\pi\)
\(642\) 0 0
\(643\) −2.82268e8 −1.06177 −0.530883 0.847445i \(-0.678140\pi\)
−0.530883 + 0.847445i \(0.678140\pi\)
\(644\) 2.24837e6i 0.00841803i
\(645\) 0 0
\(646\) −3.12817e6 −0.0116036
\(647\) − 1.45187e8i − 0.536062i −0.963410 0.268031i \(-0.913627\pi\)
0.963410 0.268031i \(-0.0863729\pi\)
\(648\) 0 0
\(649\) −3.45983e8 −1.26567
\(650\) − 4.47571e8i − 1.62975i
\(651\) 0 0
\(652\) 3.08614e7 0.111345
\(653\) − 9.91905e7i − 0.356230i −0.984010 0.178115i \(-0.943000\pi\)
0.984010 0.178115i \(-0.0569999\pi\)
\(654\) 0 0
\(655\) 5.54948e8 1.97482
\(656\) − 1.19025e8i − 0.421627i
\(657\) 0 0
\(658\) −1.84357e6 −0.00647117
\(659\) − 1.69578e8i − 0.592535i −0.955105 0.296268i \(-0.904258\pi\)
0.955105 0.296268i \(-0.0957420\pi\)
\(660\) 0 0
\(661\) 5.38050e8 1.86302 0.931511 0.363713i \(-0.118491\pi\)
0.931511 + 0.363713i \(0.118491\pi\)
\(662\) 7.56157e7i 0.260638i
\(663\) 0 0
\(664\) −1.71080e8 −0.584378
\(665\) 246138.i 0 0.000836978i
\(666\) 0 0
\(667\) −6.21449e8 −2.09425
\(668\) − 3.31198e7i − 0.111111i
\(669\) 0 0
\(670\) −5.59924e8 −1.86168
\(671\) − 3.65036e7i − 0.120828i
\(672\) 0 0
\(673\) −3.85365e7 −0.126423 −0.0632116 0.998000i \(-0.520134\pi\)
−0.0632116 + 0.998000i \(0.520134\pi\)
\(674\) 2.52006e8i 0.823060i
\(675\) 0 0
\(676\) −1.03643e8 −0.335504
\(677\) 1.90802e8i 0.614916i 0.951562 + 0.307458i \(0.0994784\pi\)
−0.951562 + 0.307458i \(0.900522\pi\)
\(678\) 0 0
\(679\) 5.84388e6 0.0186677
\(680\) 7.42066e7i 0.236002i
\(681\) 0 0
\(682\) 3.16156e8 0.996664
\(683\) 5.45896e8i 1.71336i 0.515850 + 0.856679i \(0.327476\pi\)
−0.515850 + 0.856679i \(0.672524\pi\)
\(684\) 0 0
\(685\) −5.44291e8 −1.69340
\(686\) − 5.58486e6i − 0.0172997i
\(687\) 0 0
\(688\) −3.13988e7 −0.0964156
\(689\) − 3.93788e8i − 1.20394i
\(690\) 0 0
\(691\) −1.70447e8 −0.516601 −0.258301 0.966065i \(-0.583162\pi\)
−0.258301 + 0.966065i \(0.583162\pi\)
\(692\) 2.07551e7i 0.0626334i
\(693\) 0 0
\(694\) −1.18220e8 −0.353682
\(695\) 1.46245e8i 0.435640i
\(696\) 0 0
\(697\) −2.28503e8 −0.674828
\(698\) 3.64954e8i 1.07318i
\(699\) 0 0
\(700\) 3.74084e6 0.0109062
\(701\) 8.02305e7i 0.232909i 0.993196 + 0.116454i \(0.0371529\pi\)
−0.993196 + 0.116454i \(0.962847\pi\)
\(702\) 0 0
\(703\) 4.78673e6 0.0137776
\(704\) − 7.41197e7i − 0.212430i
\(705\) 0 0
\(706\) −2.38109e7 −0.0676647
\(707\) − 552011.i − 0.00156203i
\(708\) 0 0
\(709\) 4.01132e8 1.12551 0.562754 0.826625i \(-0.309742\pi\)
0.562754 + 0.826625i \(0.309742\pi\)
\(710\) 1.77341e8i 0.495489i
\(711\) 0 0
\(712\) −1.43060e8 −0.396349
\(713\) 4.13723e8i 1.14141i
\(714\) 0 0
\(715\) 1.33958e9 3.66480
\(716\) 1.48860e8i 0.405546i
\(717\) 0 0
\(718\) −3.17531e8 −0.857852
\(719\) 1.02302e8i 0.275232i 0.990486 + 0.137616i \(0.0439439\pi\)
−0.990486 + 0.137616i \(0.956056\pi\)
\(720\) 0 0
\(721\) 8.11792e6 0.0216590
\(722\) − 2.65684e8i − 0.705917i
\(723\) 0 0
\(724\) −1.09990e8 −0.289827
\(725\) 1.03397e9i 2.71327i
\(726\) 0 0
\(727\) −5.87675e8 −1.52944 −0.764722 0.644360i \(-0.777124\pi\)
−0.764722 + 0.644360i \(0.777124\pi\)
\(728\) − 2.15723e6i − 0.00559115i
\(729\) 0 0
\(730\) 3.91266e8 1.00578
\(731\) 6.02788e7i 0.154317i
\(732\) 0 0
\(733\) −1.12015e8 −0.284424 −0.142212 0.989836i \(-0.545421\pi\)
−0.142212 + 0.989836i \(0.545421\pi\)
\(734\) 1.89563e8i 0.479363i
\(735\) 0 0
\(736\) 9.69933e7 0.243281
\(737\) − 1.07367e9i − 2.68207i
\(738\) 0 0
\(739\) 6.86968e8 1.70217 0.851086 0.525027i \(-0.175945\pi\)
0.851086 + 0.525027i \(0.175945\pi\)
\(740\) − 1.13551e8i − 0.280218i
\(741\) 0 0
\(742\) 3.29131e6 0.00805670
\(743\) − 5.57615e8i − 1.35946i −0.733460 0.679732i \(-0.762096\pi\)
0.733460 0.679732i \(-0.237904\pi\)
\(744\) 0 0
\(745\) 9.87057e8 2.38711
\(746\) 1.26022e8i 0.303549i
\(747\) 0 0
\(748\) −1.42294e8 −0.340001
\(749\) 2.60966e6i 0.00621066i
\(750\) 0 0
\(751\) 4.04996e8 0.956160 0.478080 0.878316i \(-0.341333\pi\)
0.478080 + 0.878316i \(0.341333\pi\)
\(752\) 7.95306e7i 0.187017i
\(753\) 0 0
\(754\) 5.96257e8 1.39098
\(755\) − 7.79332e8i − 1.81085i
\(756\) 0 0
\(757\) 6.51769e8 1.50247 0.751235 0.660035i \(-0.229458\pi\)
0.751235 + 0.660035i \(0.229458\pi\)
\(758\) 4.36825e8i 1.00300i
\(759\) 0 0
\(760\) 1.06182e7 0.0241887
\(761\) − 3.11208e8i − 0.706149i −0.935595 0.353074i \(-0.885136\pi\)
0.935595 0.353074i \(-0.114864\pi\)
\(762\) 0 0
\(763\) −5.64709e6 −0.0127131
\(764\) 3.76030e8i 0.843222i
\(765\) 0 0
\(766\) −1.88144e8 −0.418604
\(767\) 4.34401e8i 0.962730i
\(768\) 0 0
\(769\) −4.37211e8 −0.961417 −0.480708 0.876880i \(-0.659620\pi\)
−0.480708 + 0.876880i \(0.659620\pi\)
\(770\) 1.11963e7i 0.0245247i
\(771\) 0 0
\(772\) −2.22592e8 −0.483790
\(773\) 5.41906e7i 0.117324i 0.998278 + 0.0586619i \(0.0186834\pi\)
−0.998278 + 0.0586619i \(0.981317\pi\)
\(774\) 0 0
\(775\) 6.88352e8 1.47879
\(776\) − 2.52101e8i − 0.539498i
\(777\) 0 0
\(778\) −8.53951e7 −0.181340
\(779\) 3.26966e7i 0.0691656i
\(780\) 0 0
\(781\) −3.40057e8 −0.713837
\(782\) − 1.86206e8i − 0.389380i
\(783\) 0 0
\(784\) −1.20455e8 −0.249963
\(785\) − 6.68671e8i − 1.38230i
\(786\) 0 0
\(787\) −4.02320e7 −0.0825367 −0.0412684 0.999148i \(-0.513140\pi\)
−0.0412684 + 0.999148i \(0.513140\pi\)
\(788\) 1.31086e7i 0.0267904i
\(789\) 0 0
\(790\) −1.05707e9 −2.14398
\(791\) − 5.20039e6i − 0.0105077i
\(792\) 0 0
\(793\) −4.58323e7 −0.0919078
\(794\) − 5.04223e8i − 1.00731i
\(795\) 0 0
\(796\) 1.03951e8 0.206106
\(797\) − 8.94029e7i − 0.176594i −0.996094 0.0882972i \(-0.971857\pi\)
0.996094 0.0882972i \(-0.0281425\pi\)
\(798\) 0 0
\(799\) 1.52682e8 0.299327
\(800\) − 1.61377e8i − 0.315190i
\(801\) 0 0
\(802\) −6.33810e8 −1.22867
\(803\) 7.50266e8i 1.44900i
\(804\) 0 0
\(805\) −1.46515e7 −0.0280864
\(806\) − 3.96952e8i − 0.758111i
\(807\) 0 0
\(808\) −2.38134e7 −0.0451427
\(809\) 7.82985e8i 1.47879i 0.673270 + 0.739397i \(0.264889\pi\)
−0.673270 + 0.739397i \(0.735111\pi\)
\(810\) 0 0
\(811\) −6.76238e6 −0.0126776 −0.00633880 0.999980i \(-0.502018\pi\)
−0.00633880 + 0.999980i \(0.502018\pi\)
\(812\) 4.98357e6i 0.00930834i
\(813\) 0 0
\(814\) 2.17738e8 0.403703
\(815\) 2.01109e8i 0.371499i
\(816\) 0 0
\(817\) 8.62532e6 0.0158165
\(818\) − 3.77844e8i − 0.690324i
\(819\) 0 0
\(820\) 7.75631e8 1.40674
\(821\) − 2.93295e7i − 0.0530000i −0.999649 0.0265000i \(-0.991564\pi\)
0.999649 0.0265000i \(-0.00843619\pi\)
\(822\) 0 0
\(823\) −8.90952e8 −1.59829 −0.799144 0.601140i \(-0.794713\pi\)
−0.799144 + 0.601140i \(0.794713\pi\)
\(824\) − 3.50202e8i − 0.625946i
\(825\) 0 0
\(826\) −3.63076e6 −0.00644254
\(827\) 4.36886e8i 0.772417i 0.922411 + 0.386209i \(0.126216\pi\)
−0.922411 + 0.386209i \(0.873784\pi\)
\(828\) 0 0
\(829\) 2.08489e8 0.365948 0.182974 0.983118i \(-0.441428\pi\)
0.182974 + 0.983118i \(0.441428\pi\)
\(830\) − 1.11484e9i − 1.94975i
\(831\) 0 0
\(832\) −9.30614e7 −0.161584
\(833\) 2.31247e8i 0.400074i
\(834\) 0 0
\(835\) 2.15825e8 0.370718
\(836\) 2.03609e7i 0.0348480i
\(837\) 0 0
\(838\) 2.67892e8 0.455227
\(839\) − 1.11428e9i − 1.88672i −0.331767 0.943361i \(-0.607645\pi\)
0.331767 0.943361i \(-0.392355\pi\)
\(840\) 0 0
\(841\) −7.82634e8 −1.31574
\(842\) 3.74240e8i 0.626922i
\(843\) 0 0
\(844\) 1.59836e8 0.265856
\(845\) − 6.75389e8i − 1.11940i
\(846\) 0 0
\(847\) −1.40356e7 −0.0230983
\(848\) − 1.41985e8i − 0.232839i
\(849\) 0 0
\(850\) −3.09809e8 −0.504473
\(851\) 2.84933e8i 0.462332i
\(852\) 0 0
\(853\) −7.03808e7 −0.113399 −0.0566993 0.998391i \(-0.518058\pi\)
−0.0566993 + 0.998391i \(0.518058\pi\)
\(854\) − 383071.i 0 0.000615043i
\(855\) 0 0
\(856\) 1.12579e8 0.179488
\(857\) − 1.58916e8i − 0.252479i −0.992000 0.126240i \(-0.959709\pi\)
0.992000 0.126240i \(-0.0402908\pi\)
\(858\) 0 0
\(859\) 3.14241e8 0.495774 0.247887 0.968789i \(-0.420264\pi\)
0.247887 + 0.968789i \(0.420264\pi\)
\(860\) − 2.04611e8i − 0.321686i
\(861\) 0 0
\(862\) −5.12888e8 −0.800757
\(863\) − 3.90351e8i − 0.607328i −0.952779 0.303664i \(-0.901790\pi\)
0.952779 0.303664i \(-0.0982101\pi\)
\(864\) 0 0
\(865\) −1.35251e8 −0.208974
\(866\) − 3.95648e6i − 0.00609194i
\(867\) 0 0
\(868\) 3.31776e6 0.00507324
\(869\) − 2.02696e9i − 3.08878i
\(870\) 0 0
\(871\) −1.34806e9 −2.04011
\(872\) 2.43612e8i 0.367408i
\(873\) 0 0
\(874\) −2.66443e7 −0.0399089
\(875\) 1.07051e7i 0.0159796i
\(876\) 0 0
\(877\) −1.15579e9 −1.71349 −0.856745 0.515741i \(-0.827517\pi\)
−0.856745 + 0.515741i \(0.827517\pi\)
\(878\) − 1.45549e8i − 0.215043i
\(879\) 0 0
\(880\) 4.83002e8 0.708763
\(881\) − 7.82859e8i − 1.14487i −0.819951 0.572434i \(-0.805999\pi\)
0.819951 0.572434i \(-0.194001\pi\)
\(882\) 0 0
\(883\) 1.01047e9 1.46772 0.733860 0.679300i \(-0.237717\pi\)
0.733860 + 0.679300i \(0.237717\pi\)
\(884\) 1.78658e8i 0.258622i
\(885\) 0 0
\(886\) 936650. 0.00134672
\(887\) − 3.28343e8i − 0.470498i −0.971935 0.235249i \(-0.924410\pi\)
0.971935 0.235249i \(-0.0755905\pi\)
\(888\) 0 0
\(889\) −6.93610e6 −0.00987211
\(890\) − 9.32253e8i − 1.32240i
\(891\) 0 0
\(892\) −1.82056e8 −0.256513
\(893\) − 2.18473e7i − 0.0306791i
\(894\) 0 0
\(895\) −9.70051e8 −1.35309
\(896\) − 777815.i − 0.00108131i
\(897\) 0 0
\(898\) −2.43502e8 −0.336259
\(899\) 9.17028e8i 1.26213i
\(900\) 0 0
\(901\) −2.72580e8 −0.372666
\(902\) 1.48730e9i 2.02665i
\(903\) 0 0
\(904\) −2.24342e8 −0.303672
\(905\) − 7.16753e8i − 0.966994i
\(906\) 0 0
\(907\) 2.15438e8 0.288736 0.144368 0.989524i \(-0.453885\pi\)
0.144368 + 0.989524i \(0.453885\pi\)
\(908\) 3.71404e8i 0.496123i
\(909\) 0 0
\(910\) 1.40576e7 0.0186546
\(911\) 7.52234e8i 0.994942i 0.867481 + 0.497471i \(0.165738\pi\)
−0.867481 + 0.497471i \(0.834262\pi\)
\(912\) 0 0
\(913\) 2.13775e9 2.80895
\(914\) − 7.66452e8i − 1.00380i
\(915\) 0 0
\(916\) 1.10621e8 0.143930
\(917\) 1.11670e7i 0.0144820i
\(918\) 0 0
\(919\) 6.81968e8 0.878653 0.439327 0.898327i \(-0.355217\pi\)
0.439327 + 0.898327i \(0.355217\pi\)
\(920\) 6.32058e8i 0.811696i
\(921\) 0 0
\(922\) 1.81217e8 0.231210
\(923\) 4.26961e8i 0.542979i
\(924\) 0 0
\(925\) 4.74072e8 0.598989
\(926\) 7.31617e8i 0.921406i
\(927\) 0 0
\(928\) 2.14988e8 0.269011
\(929\) 1.11639e8i 0.139242i 0.997574 + 0.0696208i \(0.0221789\pi\)
−0.997574 + 0.0696208i \(0.977821\pi\)
\(930\) 0 0
\(931\) 3.30892e7 0.0410050
\(932\) 4.08765e8i 0.504924i
\(933\) 0 0
\(934\) 4.80433e8 0.589647
\(935\) − 9.27259e8i − 1.13440i
\(936\) 0 0
\(937\) 7.66443e8 0.931668 0.465834 0.884872i \(-0.345754\pi\)
0.465834 + 0.884872i \(0.345754\pi\)
\(938\) − 1.12672e7i − 0.0136523i
\(939\) 0 0
\(940\) −5.18263e8 −0.623974
\(941\) 5.32996e8i 0.639669i 0.947473 + 0.319834i \(0.103627\pi\)
−0.947473 + 0.319834i \(0.896373\pi\)
\(942\) 0 0
\(943\) −1.94628e9 −2.32098
\(944\) 1.56629e8i 0.186189i
\(945\) 0 0
\(946\) 3.92348e8 0.463445
\(947\) 3.13628e7i 0.0369288i 0.999830 + 0.0184644i \(0.00587773\pi\)
−0.999830 + 0.0184644i \(0.994122\pi\)
\(948\) 0 0
\(949\) 9.42001e8 1.10218
\(950\) 4.43308e7i 0.0517052i
\(951\) 0 0
\(952\) −1.49324e6 −0.00173068
\(953\) 6.21244e8i 0.717767i 0.933382 + 0.358883i \(0.116842\pi\)
−0.933382 + 0.358883i \(0.883158\pi\)
\(954\) 0 0
\(955\) −2.45040e9 −2.81337
\(956\) 4.13541e8i 0.473309i
\(957\) 0 0
\(958\) 6.54603e8 0.744529
\(959\) − 1.09526e7i − 0.0124183i
\(960\) 0 0
\(961\) −2.77002e8 −0.312114
\(962\) − 2.73383e8i − 0.307076i
\(963\) 0 0
\(964\) −2.32970e8 −0.260057
\(965\) − 1.45052e9i − 1.61415i
\(966\) 0 0
\(967\) −6.29984e8 −0.696707 −0.348354 0.937363i \(-0.613259\pi\)
−0.348354 + 0.937363i \(0.613259\pi\)
\(968\) 6.05486e8i 0.667541i
\(969\) 0 0
\(970\) 1.64282e9 1.80001
\(971\) 1.38333e8i 0.151101i 0.997142 + 0.0755507i \(0.0240715\pi\)
−0.997142 + 0.0755507i \(0.975929\pi\)
\(972\) 0 0
\(973\) −2.94285e6 −0.00319470
\(974\) 2.56165e8i 0.277231i
\(975\) 0 0
\(976\) −1.65254e7 −0.0177747
\(977\) 8.23975e8i 0.883549i 0.897126 + 0.441774i \(0.145651\pi\)
−0.897126 + 0.441774i \(0.854349\pi\)
\(978\) 0 0
\(979\) 1.78763e9 1.90515
\(980\) − 7.84944e8i − 0.833989i
\(981\) 0 0
\(982\) 1.15576e9 1.22049
\(983\) − 1.28601e9i − 1.35389i −0.736032 0.676947i \(-0.763303\pi\)
0.736032 0.676947i \(-0.236697\pi\)
\(984\) 0 0
\(985\) −8.54226e7 −0.0893849
\(986\) − 4.12730e8i − 0.430562i
\(987\) 0 0
\(988\) 2.55642e7 0.0265070
\(989\) 5.13428e8i 0.530751i
\(990\) 0 0
\(991\) −1.70244e8 −0.174924 −0.0874621 0.996168i \(-0.527876\pi\)
−0.0874621 + 0.996168i \(0.527876\pi\)
\(992\) − 1.43126e8i − 0.146617i
\(993\) 0 0
\(994\) −3.56858e6 −0.00363359
\(995\) 6.77399e8i 0.687663i
\(996\) 0 0
\(997\) −7.01078e8 −0.707426 −0.353713 0.935354i \(-0.615081\pi\)
−0.353713 + 0.935354i \(0.615081\pi\)
\(998\) − 1.65740e8i − 0.166738i
\(999\) 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 162.7.b.a.161.3 yes 4
3.2 odd 2 inner 162.7.b.a.161.2 4
9.2 odd 6 162.7.d.f.53.1 8
9.4 even 3 162.7.d.f.107.1 8
9.5 odd 6 162.7.d.f.107.4 8
9.7 even 3 162.7.d.f.53.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.7.b.a.161.2 4 3.2 odd 2 inner
162.7.b.a.161.3 yes 4 1.1 even 1 trivial
162.7.d.f.53.1 8 9.2 odd 6
162.7.d.f.53.4 8 9.7 even 3
162.7.d.f.107.1 8 9.4 even 3
162.7.d.f.107.4 8 9.5 odd 6