Properties

Label 495.6.a.b.1.1
Level $495$
Weight $6$
Character 495.1
Self dual yes
Analytic conductor $79.390$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [495,6,Mod(1,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 495.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(79.3899908074\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.18257.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 26x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.78415\) of defining polynomial
Character \(\chi\) \(=\) 495.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.78415 q^{2} +1.45634 q^{4} -25.0000 q^{5} +17.6498 q^{7} +176.669 q^{8} +O(q^{10})\) \(q-5.78415 q^{2} +1.45634 q^{4} -25.0000 q^{5} +17.6498 q^{7} +176.669 q^{8} +144.604 q^{10} -121.000 q^{11} +674.396 q^{13} -102.089 q^{14} -1068.48 q^{16} +2117.62 q^{17} -2307.79 q^{19} -36.4084 q^{20} +699.882 q^{22} +3072.47 q^{23} +625.000 q^{25} -3900.80 q^{26} +25.7040 q^{28} +1437.44 q^{29} -5157.66 q^{31} +526.846 q^{32} -12248.6 q^{34} -441.244 q^{35} +6928.88 q^{37} +13348.6 q^{38} -4416.72 q^{40} -2844.78 q^{41} -11665.3 q^{43} -176.217 q^{44} -17771.6 q^{46} -3451.40 q^{47} -16495.5 q^{49} -3615.09 q^{50} +982.146 q^{52} +18167.6 q^{53} +3025.00 q^{55} +3118.17 q^{56} -8314.36 q^{58} +8976.88 q^{59} -378.820 q^{61} +29832.7 q^{62} +31144.1 q^{64} -16859.9 q^{65} -12233.7 q^{67} +3083.97 q^{68} +2552.22 q^{70} -55867.0 q^{71} +19739.6 q^{73} -40077.6 q^{74} -3360.92 q^{76} -2135.62 q^{77} -13495.8 q^{79} +26712.0 q^{80} +16454.6 q^{82} -42078.3 q^{83} -52940.5 q^{85} +67474.0 q^{86} -21376.9 q^{88} +81223.2 q^{89} +11902.9 q^{91} +4474.55 q^{92} +19963.4 q^{94} +57694.8 q^{95} +152101. q^{97} +95412.3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 42 q^{4} - 75 q^{5} - 68 q^{7} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} - 42 q^{4} - 75 q^{5} - 68 q^{7} + 24 q^{8} + 50 q^{10} - 363 q^{11} + 290 q^{13} - 916 q^{14} - 590 q^{16} - 434 q^{17} - 2856 q^{19} + 1050 q^{20} + 242 q^{22} + 640 q^{23} + 1875 q^{25} - 2132 q^{26} - 580 q^{28} + 4538 q^{29} - 14968 q^{31} + 2496 q^{32} - 13704 q^{34} + 1700 q^{35} - 6190 q^{37} + 11668 q^{38} - 600 q^{40} + 8926 q^{41} - 33592 q^{43} + 5082 q^{44} - 35680 q^{46} + 24640 q^{47} - 14693 q^{49} - 1250 q^{50} + 18780 q^{52} + 22934 q^{53} + 9075 q^{55} + 40012 q^{56} - 32304 q^{58} + 13756 q^{59} + 24602 q^{61} + 7704 q^{62} + 35474 q^{64} - 7250 q^{65} + 16868 q^{67} + 71288 q^{68} + 22900 q^{70} - 4856 q^{71} + 1910 q^{73} - 29404 q^{74} + 6116 q^{76} + 8228 q^{77} - 36844 q^{79} + 14750 q^{80} + 84000 q^{82} + 48796 q^{83} + 10850 q^{85} + 83492 q^{86} - 2904 q^{88} + 188978 q^{89} - 93208 q^{91} + 6976 q^{92} + 70472 q^{94} + 71400 q^{95} + 247526 q^{97} + 154654 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.78415 −1.02250 −0.511251 0.859431i \(-0.670818\pi\)
−0.511251 + 0.859431i \(0.670818\pi\)
\(3\) 0 0
\(4\) 1.45634 0.0455105
\(5\) −25.0000 −0.447214
\(6\) 0 0
\(7\) 17.6498 0.136143 0.0680713 0.997680i \(-0.478315\pi\)
0.0680713 + 0.997680i \(0.478315\pi\)
\(8\) 176.669 0.975968
\(9\) 0 0
\(10\) 144.604 0.457277
\(11\) −121.000 −0.301511
\(12\) 0 0
\(13\) 674.396 1.10677 0.553384 0.832926i \(-0.313336\pi\)
0.553384 + 0.832926i \(0.313336\pi\)
\(14\) −102.089 −0.139206
\(15\) 0 0
\(16\) −1068.48 −1.04344
\(17\) 2117.62 1.77716 0.888579 0.458724i \(-0.151693\pi\)
0.888579 + 0.458724i \(0.151693\pi\)
\(18\) 0 0
\(19\) −2307.79 −1.46660 −0.733302 0.679903i \(-0.762022\pi\)
−0.733302 + 0.679903i \(0.762022\pi\)
\(20\) −36.4084 −0.0203529
\(21\) 0 0
\(22\) 699.882 0.308296
\(23\) 3072.47 1.21107 0.605534 0.795820i \(-0.292960\pi\)
0.605534 + 0.795820i \(0.292960\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) −3900.80 −1.13167
\(27\) 0 0
\(28\) 25.7040 0.00619591
\(29\) 1437.44 0.317391 0.158696 0.987328i \(-0.449271\pi\)
0.158696 + 0.987328i \(0.449271\pi\)
\(30\) 0 0
\(31\) −5157.66 −0.963937 −0.481969 0.876188i \(-0.660078\pi\)
−0.481969 + 0.876188i \(0.660078\pi\)
\(32\) 526.846 0.0909513
\(33\) 0 0
\(34\) −12248.6 −1.81715
\(35\) −441.244 −0.0608848
\(36\) 0 0
\(37\) 6928.88 0.832067 0.416034 0.909349i \(-0.363420\pi\)
0.416034 + 0.909349i \(0.363420\pi\)
\(38\) 13348.6 1.49961
\(39\) 0 0
\(40\) −4416.72 −0.436466
\(41\) −2844.78 −0.264295 −0.132147 0.991230i \(-0.542187\pi\)
−0.132147 + 0.991230i \(0.542187\pi\)
\(42\) 0 0
\(43\) −11665.3 −0.962113 −0.481057 0.876689i \(-0.659747\pi\)
−0.481057 + 0.876689i \(0.659747\pi\)
\(44\) −176.217 −0.0137219
\(45\) 0 0
\(46\) −17771.6 −1.23832
\(47\) −3451.40 −0.227903 −0.113952 0.993486i \(-0.536351\pi\)
−0.113952 + 0.993486i \(0.536351\pi\)
\(48\) 0 0
\(49\) −16495.5 −0.981465
\(50\) −3615.09 −0.204500
\(51\) 0 0
\(52\) 982.146 0.0503695
\(53\) 18167.6 0.888401 0.444200 0.895927i \(-0.353488\pi\)
0.444200 + 0.895927i \(0.353488\pi\)
\(54\) 0 0
\(55\) 3025.00 0.134840
\(56\) 3118.17 0.132871
\(57\) 0 0
\(58\) −8314.36 −0.324533
\(59\) 8976.88 0.335734 0.167867 0.985810i \(-0.446312\pi\)
0.167867 + 0.985810i \(0.446312\pi\)
\(60\) 0 0
\(61\) −378.820 −0.0130349 −0.00651746 0.999979i \(-0.502075\pi\)
−0.00651746 + 0.999979i \(0.502075\pi\)
\(62\) 29832.7 0.985628
\(63\) 0 0
\(64\) 31144.1 0.950441
\(65\) −16859.9 −0.494962
\(66\) 0 0
\(67\) −12233.7 −0.332945 −0.166473 0.986046i \(-0.553238\pi\)
−0.166473 + 0.986046i \(0.553238\pi\)
\(68\) 3083.97 0.0808793
\(69\) 0 0
\(70\) 2552.22 0.0622548
\(71\) −55867.0 −1.31525 −0.657627 0.753344i \(-0.728440\pi\)
−0.657627 + 0.753344i \(0.728440\pi\)
\(72\) 0 0
\(73\) 19739.6 0.433541 0.216771 0.976223i \(-0.430448\pi\)
0.216771 + 0.976223i \(0.430448\pi\)
\(74\) −40077.6 −0.850791
\(75\) 0 0
\(76\) −3360.92 −0.0667459
\(77\) −2135.62 −0.0410485
\(78\) 0 0
\(79\) −13495.8 −0.243293 −0.121647 0.992573i \(-0.538817\pi\)
−0.121647 + 0.992573i \(0.538817\pi\)
\(80\) 26712.0 0.466640
\(81\) 0 0
\(82\) 16454.6 0.270242
\(83\) −42078.3 −0.670444 −0.335222 0.942139i \(-0.608811\pi\)
−0.335222 + 0.942139i \(0.608811\pi\)
\(84\) 0 0
\(85\) −52940.5 −0.794769
\(86\) 67474.0 0.983763
\(87\) 0 0
\(88\) −21376.9 −0.294265
\(89\) 81223.2 1.08694 0.543469 0.839429i \(-0.317110\pi\)
0.543469 + 0.839429i \(0.317110\pi\)
\(90\) 0 0
\(91\) 11902.9 0.150678
\(92\) 4474.55 0.0551163
\(93\) 0 0
\(94\) 19963.4 0.233031
\(95\) 57694.8 0.655885
\(96\) 0 0
\(97\) 152101. 1.64136 0.820680 0.571388i \(-0.193595\pi\)
0.820680 + 0.571388i \(0.193595\pi\)
\(98\) 95412.3 1.00355
\(99\) 0 0
\(100\) 910.210 0.00910210
\(101\) −95207.6 −0.928684 −0.464342 0.885656i \(-0.653709\pi\)
−0.464342 + 0.885656i \(0.653709\pi\)
\(102\) 0 0
\(103\) −150824. −1.40081 −0.700403 0.713748i \(-0.746996\pi\)
−0.700403 + 0.713748i \(0.746996\pi\)
\(104\) 119145. 1.08017
\(105\) 0 0
\(106\) −105084. −0.908392
\(107\) −192242. −1.62327 −0.811633 0.584168i \(-0.801421\pi\)
−0.811633 + 0.584168i \(0.801421\pi\)
\(108\) 0 0
\(109\) 96302.8 0.776377 0.388188 0.921580i \(-0.373101\pi\)
0.388188 + 0.921580i \(0.373101\pi\)
\(110\) −17497.0 −0.137874
\(111\) 0 0
\(112\) −18858.5 −0.142056
\(113\) 210371. 1.54985 0.774926 0.632052i \(-0.217787\pi\)
0.774926 + 0.632052i \(0.217787\pi\)
\(114\) 0 0
\(115\) −76811.8 −0.541606
\(116\) 2093.39 0.0144446
\(117\) 0 0
\(118\) −51923.6 −0.343289
\(119\) 37375.5 0.241947
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 2191.15 0.0133282
\(123\) 0 0
\(124\) −7511.29 −0.0438692
\(125\) −15625.0 −0.0894427
\(126\) 0 0
\(127\) −63529.8 −0.349517 −0.174758 0.984611i \(-0.555914\pi\)
−0.174758 + 0.984611i \(0.555914\pi\)
\(128\) −197001. −1.06278
\(129\) 0 0
\(130\) 97520.1 0.506099
\(131\) 88396.0 0.450043 0.225022 0.974354i \(-0.427755\pi\)
0.225022 + 0.974354i \(0.427755\pi\)
\(132\) 0 0
\(133\) −40732.0 −0.199667
\(134\) 70761.8 0.340437
\(135\) 0 0
\(136\) 374118. 1.73445
\(137\) 42642.4 0.194106 0.0970532 0.995279i \(-0.469058\pi\)
0.0970532 + 0.995279i \(0.469058\pi\)
\(138\) 0 0
\(139\) 169503. 0.744114 0.372057 0.928210i \(-0.378653\pi\)
0.372057 + 0.928210i \(0.378653\pi\)
\(140\) −642.599 −0.00277090
\(141\) 0 0
\(142\) 323143. 1.34485
\(143\) −81601.9 −0.333703
\(144\) 0 0
\(145\) −35936.0 −0.141942
\(146\) −114176. −0.443297
\(147\) 0 0
\(148\) 10090.8 0.0378678
\(149\) 72462.8 0.267393 0.133696 0.991022i \(-0.457315\pi\)
0.133696 + 0.991022i \(0.457315\pi\)
\(150\) 0 0
\(151\) −61316.2 −0.218843 −0.109422 0.993995i \(-0.534900\pi\)
−0.109422 + 0.993995i \(0.534900\pi\)
\(152\) −407716. −1.43136
\(153\) 0 0
\(154\) 12352.7 0.0419722
\(155\) 128942. 0.431086
\(156\) 0 0
\(157\) 505789. 1.63765 0.818824 0.574045i \(-0.194627\pi\)
0.818824 + 0.574045i \(0.194627\pi\)
\(158\) 78061.5 0.248768
\(159\) 0 0
\(160\) −13171.2 −0.0406747
\(161\) 54228.4 0.164878
\(162\) 0 0
\(163\) 387702. 1.14295 0.571477 0.820618i \(-0.306371\pi\)
0.571477 + 0.820618i \(0.306371\pi\)
\(164\) −4142.95 −0.0120282
\(165\) 0 0
\(166\) 243387. 0.685530
\(167\) 635422. 1.76308 0.881538 0.472113i \(-0.156509\pi\)
0.881538 + 0.472113i \(0.156509\pi\)
\(168\) 0 0
\(169\) 83516.8 0.224935
\(170\) 306216. 0.812653
\(171\) 0 0
\(172\) −16988.6 −0.0437862
\(173\) 679820. 1.72695 0.863473 0.504395i \(-0.168284\pi\)
0.863473 + 0.504395i \(0.168284\pi\)
\(174\) 0 0
\(175\) 11031.1 0.0272285
\(176\) 129286. 0.314609
\(177\) 0 0
\(178\) −469807. −1.11140
\(179\) 63083.3 0.147157 0.0735787 0.997289i \(-0.476558\pi\)
0.0735787 + 0.997289i \(0.476558\pi\)
\(180\) 0 0
\(181\) 523614. 1.18800 0.593998 0.804466i \(-0.297549\pi\)
0.593998 + 0.804466i \(0.297549\pi\)
\(182\) −68848.3 −0.154069
\(183\) 0 0
\(184\) 542810. 1.18196
\(185\) −173222. −0.372112
\(186\) 0 0
\(187\) −256232. −0.535833
\(188\) −5026.39 −0.0103720
\(189\) 0 0
\(190\) −333715. −0.670644
\(191\) 684311. 1.35728 0.678641 0.734470i \(-0.262569\pi\)
0.678641 + 0.734470i \(0.262569\pi\)
\(192\) 0 0
\(193\) 527417. 1.01920 0.509601 0.860411i \(-0.329793\pi\)
0.509601 + 0.860411i \(0.329793\pi\)
\(194\) −879777. −1.67829
\(195\) 0 0
\(196\) −24023.0 −0.0446669
\(197\) 467139. 0.857591 0.428796 0.903402i \(-0.358938\pi\)
0.428796 + 0.903402i \(0.358938\pi\)
\(198\) 0 0
\(199\) −436613. −0.781563 −0.390782 0.920483i \(-0.627795\pi\)
−0.390782 + 0.920483i \(0.627795\pi\)
\(200\) 110418. 0.195194
\(201\) 0 0
\(202\) 550694. 0.949581
\(203\) 25370.5 0.0432104
\(204\) 0 0
\(205\) 71119.4 0.118196
\(206\) 872389. 1.43233
\(207\) 0 0
\(208\) −720580. −1.15484
\(209\) 279243. 0.442198
\(210\) 0 0
\(211\) −747312. −1.15557 −0.577785 0.816189i \(-0.696083\pi\)
−0.577785 + 0.816189i \(0.696083\pi\)
\(212\) 26458.2 0.0404315
\(213\) 0 0
\(214\) 1.11196e6 1.65979
\(215\) 291634. 0.430270
\(216\) 0 0
\(217\) −91031.6 −0.131233
\(218\) −557029. −0.793847
\(219\) 0 0
\(220\) 4405.41 0.00613663
\(221\) 1.42812e6 1.96690
\(222\) 0 0
\(223\) 870654. 1.17242 0.586211 0.810159i \(-0.300619\pi\)
0.586211 + 0.810159i \(0.300619\pi\)
\(224\) 9298.71 0.0123823
\(225\) 0 0
\(226\) −1.21682e6 −1.58473
\(227\) −82323.8 −0.106038 −0.0530189 0.998594i \(-0.516884\pi\)
−0.0530189 + 0.998594i \(0.516884\pi\)
\(228\) 0 0
\(229\) −340134. −0.428609 −0.214305 0.976767i \(-0.568748\pi\)
−0.214305 + 0.976767i \(0.568748\pi\)
\(230\) 444291. 0.553793
\(231\) 0 0
\(232\) 253951. 0.309763
\(233\) 248384. 0.299732 0.149866 0.988706i \(-0.452116\pi\)
0.149866 + 0.988706i \(0.452116\pi\)
\(234\) 0 0
\(235\) 86284.9 0.101921
\(236\) 13073.4 0.0152794
\(237\) 0 0
\(238\) −216185. −0.247391
\(239\) 1.37978e6 1.56248 0.781242 0.624229i \(-0.214587\pi\)
0.781242 + 0.624229i \(0.214587\pi\)
\(240\) 0 0
\(241\) −1.53382e6 −1.70110 −0.850552 0.525891i \(-0.823732\pi\)
−0.850552 + 0.525891i \(0.823732\pi\)
\(242\) −84685.7 −0.0929547
\(243\) 0 0
\(244\) −551.689 −0.000593225 0
\(245\) 412387. 0.438925
\(246\) 0 0
\(247\) −1.55637e6 −1.62319
\(248\) −911199. −0.940772
\(249\) 0 0
\(250\) 90377.3 0.0914554
\(251\) −338591. −0.339227 −0.169614 0.985511i \(-0.554252\pi\)
−0.169614 + 0.985511i \(0.554252\pi\)
\(252\) 0 0
\(253\) −371769. −0.365151
\(254\) 367466. 0.357382
\(255\) 0 0
\(256\) 142872. 0.136253
\(257\) −1.60985e6 −1.52038 −0.760189 0.649702i \(-0.774894\pi\)
−0.760189 + 0.649702i \(0.774894\pi\)
\(258\) 0 0
\(259\) 122293. 0.113280
\(260\) −24553.7 −0.0225259
\(261\) 0 0
\(262\) −511295. −0.460170
\(263\) −458118. −0.408402 −0.204201 0.978929i \(-0.565460\pi\)
−0.204201 + 0.978929i \(0.565460\pi\)
\(264\) 0 0
\(265\) −454191. −0.397305
\(266\) 235600. 0.204160
\(267\) 0 0
\(268\) −17816.4 −0.0151525
\(269\) 692380. 0.583397 0.291698 0.956510i \(-0.405780\pi\)
0.291698 + 0.956510i \(0.405780\pi\)
\(270\) 0 0
\(271\) −486690. −0.402558 −0.201279 0.979534i \(-0.564510\pi\)
−0.201279 + 0.979534i \(0.564510\pi\)
\(272\) −2.26264e6 −1.85436
\(273\) 0 0
\(274\) −246650. −0.198474
\(275\) −75625.0 −0.0603023
\(276\) 0 0
\(277\) 2.38271e6 1.86583 0.932915 0.360098i \(-0.117257\pi\)
0.932915 + 0.360098i \(0.117257\pi\)
\(278\) −980428. −0.760858
\(279\) 0 0
\(280\) −77954.1 −0.0594216
\(281\) −1.12154e6 −0.847320 −0.423660 0.905821i \(-0.639255\pi\)
−0.423660 + 0.905821i \(0.639255\pi\)
\(282\) 0 0
\(283\) 965956. 0.716954 0.358477 0.933539i \(-0.383296\pi\)
0.358477 + 0.933539i \(0.383296\pi\)
\(284\) −81361.1 −0.0598578
\(285\) 0 0
\(286\) 471997. 0.341212
\(287\) −50209.6 −0.0359817
\(288\) 0 0
\(289\) 3.06446e6 2.15829
\(290\) 207859. 0.145136
\(291\) 0 0
\(292\) 28747.4 0.0197307
\(293\) −2.20819e6 −1.50269 −0.751343 0.659912i \(-0.770594\pi\)
−0.751343 + 0.659912i \(0.770594\pi\)
\(294\) 0 0
\(295\) −224422. −0.150145
\(296\) 1.22412e6 0.812071
\(297\) 0 0
\(298\) −419135. −0.273410
\(299\) 2.07206e6 1.34037
\(300\) 0 0
\(301\) −205891. −0.130985
\(302\) 354662. 0.223768
\(303\) 0 0
\(304\) 2.46584e6 1.53031
\(305\) 9470.49 0.00582939
\(306\) 0 0
\(307\) 805480. 0.487763 0.243881 0.969805i \(-0.421579\pi\)
0.243881 + 0.969805i \(0.421579\pi\)
\(308\) −3110.18 −0.00186814
\(309\) 0 0
\(310\) −745817. −0.440786
\(311\) 870766. 0.510505 0.255253 0.966874i \(-0.417841\pi\)
0.255253 + 0.966874i \(0.417841\pi\)
\(312\) 0 0
\(313\) −3.16630e6 −1.82680 −0.913400 0.407063i \(-0.866553\pi\)
−0.913400 + 0.407063i \(0.866553\pi\)
\(314\) −2.92556e6 −1.67450
\(315\) 0 0
\(316\) −19654.4 −0.0110724
\(317\) −1.72288e6 −0.962955 −0.481478 0.876458i \(-0.659900\pi\)
−0.481478 + 0.876458i \(0.659900\pi\)
\(318\) 0 0
\(319\) −173930. −0.0956970
\(320\) −778602. −0.425050
\(321\) 0 0
\(322\) −313665. −0.168588
\(323\) −4.88703e6 −2.60639
\(324\) 0 0
\(325\) 421497. 0.221354
\(326\) −2.24252e6 −1.16867
\(327\) 0 0
\(328\) −502584. −0.257943
\(329\) −60916.3 −0.0310273
\(330\) 0 0
\(331\) −3.20493e6 −1.60786 −0.803932 0.594722i \(-0.797262\pi\)
−0.803932 + 0.594722i \(0.797262\pi\)
\(332\) −61280.1 −0.0305122
\(333\) 0 0
\(334\) −3.67537e6 −1.80275
\(335\) 305844. 0.148898
\(336\) 0 0
\(337\) 2.44638e6 1.17341 0.586703 0.809802i \(-0.300425\pi\)
0.586703 + 0.809802i \(0.300425\pi\)
\(338\) −483073. −0.229996
\(339\) 0 0
\(340\) −77099.2 −0.0361703
\(341\) 624077. 0.290638
\(342\) 0 0
\(343\) −587781. −0.269762
\(344\) −2.06090e6 −0.938992
\(345\) 0 0
\(346\) −3.93218e6 −1.76581
\(347\) 3.05055e6 1.36005 0.680025 0.733189i \(-0.261969\pi\)
0.680025 + 0.733189i \(0.261969\pi\)
\(348\) 0 0
\(349\) −3.08755e6 −1.35691 −0.678454 0.734643i \(-0.737350\pi\)
−0.678454 + 0.734643i \(0.737350\pi\)
\(350\) −63805.5 −0.0278412
\(351\) 0 0
\(352\) −63748.4 −0.0274228
\(353\) −2.98932e6 −1.27684 −0.638418 0.769690i \(-0.720411\pi\)
−0.638418 + 0.769690i \(0.720411\pi\)
\(354\) 0 0
\(355\) 1.39668e6 0.588200
\(356\) 118288. 0.0494671
\(357\) 0 0
\(358\) −364883. −0.150469
\(359\) 2.29991e6 0.941836 0.470918 0.882177i \(-0.343923\pi\)
0.470918 + 0.882177i \(0.343923\pi\)
\(360\) 0 0
\(361\) 2.84981e6 1.15093
\(362\) −3.02866e6 −1.21473
\(363\) 0 0
\(364\) 17334.7 0.00685743
\(365\) −493489. −0.193885
\(366\) 0 0
\(367\) 3.58207e6 1.38825 0.694127 0.719852i \(-0.255790\pi\)
0.694127 + 0.719852i \(0.255790\pi\)
\(368\) −3.28288e6 −1.26368
\(369\) 0 0
\(370\) 1.00194e6 0.380485
\(371\) 320655. 0.120949
\(372\) 0 0
\(373\) 511254. 0.190268 0.0951338 0.995464i \(-0.469672\pi\)
0.0951338 + 0.995464i \(0.469672\pi\)
\(374\) 1.48208e6 0.547891
\(375\) 0 0
\(376\) −609755. −0.222426
\(377\) 969404. 0.351278
\(378\) 0 0
\(379\) 2.45209e6 0.876877 0.438438 0.898761i \(-0.355532\pi\)
0.438438 + 0.898761i \(0.355532\pi\)
\(380\) 84023.0 0.0298497
\(381\) 0 0
\(382\) −3.95815e6 −1.38782
\(383\) 1.46315e6 0.509674 0.254837 0.966984i \(-0.417978\pi\)
0.254837 + 0.966984i \(0.417978\pi\)
\(384\) 0 0
\(385\) 53390.5 0.0183575
\(386\) −3.05065e6 −1.04214
\(387\) 0 0
\(388\) 221511. 0.0746991
\(389\) 803978. 0.269383 0.134691 0.990888i \(-0.456996\pi\)
0.134691 + 0.990888i \(0.456996\pi\)
\(390\) 0 0
\(391\) 6.50633e6 2.15226
\(392\) −2.91424e6 −0.957878
\(393\) 0 0
\(394\) −2.70200e6 −0.876889
\(395\) 337394. 0.108804
\(396\) 0 0
\(397\) −2.87304e6 −0.914883 −0.457441 0.889240i \(-0.651234\pi\)
−0.457441 + 0.889240i \(0.651234\pi\)
\(398\) 2.52543e6 0.799150
\(399\) 0 0
\(400\) −667801. −0.208688
\(401\) 3.45072e6 1.07164 0.535820 0.844332i \(-0.320002\pi\)
0.535820 + 0.844332i \(0.320002\pi\)
\(402\) 0 0
\(403\) −3.47831e6 −1.06685
\(404\) −138654. −0.0422649
\(405\) 0 0
\(406\) −146747. −0.0441827
\(407\) −838394. −0.250878
\(408\) 0 0
\(409\) −1.17776e6 −0.348136 −0.174068 0.984734i \(-0.555691\pi\)
−0.174068 + 0.984734i \(0.555691\pi\)
\(410\) −411365. −0.120856
\(411\) 0 0
\(412\) −219651. −0.0637513
\(413\) 158440. 0.0457077
\(414\) 0 0
\(415\) 1.05196e6 0.299832
\(416\) 355303. 0.100662
\(417\) 0 0
\(418\) −1.61518e6 −0.452148
\(419\) 443011. 0.123276 0.0616381 0.998099i \(-0.480368\pi\)
0.0616381 + 0.998099i \(0.480368\pi\)
\(420\) 0 0
\(421\) −3.41894e6 −0.940127 −0.470063 0.882633i \(-0.655769\pi\)
−0.470063 + 0.882633i \(0.655769\pi\)
\(422\) 4.32256e6 1.18157
\(423\) 0 0
\(424\) 3.20966e6 0.867050
\(425\) 1.32351e6 0.355432
\(426\) 0 0
\(427\) −6686.08 −0.00177461
\(428\) −279969. −0.0738756
\(429\) 0 0
\(430\) −1.68685e6 −0.439952
\(431\) 4.24640e6 1.10110 0.550551 0.834801i \(-0.314417\pi\)
0.550551 + 0.834801i \(0.314417\pi\)
\(432\) 0 0
\(433\) −1.03407e6 −0.265050 −0.132525 0.991180i \(-0.542309\pi\)
−0.132525 + 0.991180i \(0.542309\pi\)
\(434\) 526540. 0.134186
\(435\) 0 0
\(436\) 140249. 0.0353333
\(437\) −7.09063e6 −1.77616
\(438\) 0 0
\(439\) 3.55155e6 0.879542 0.439771 0.898110i \(-0.355060\pi\)
0.439771 + 0.898110i \(0.355060\pi\)
\(440\) 534424. 0.131599
\(441\) 0 0
\(442\) −8.26043e6 −2.01116
\(443\) −2.35135e6 −0.569257 −0.284629 0.958638i \(-0.591870\pi\)
−0.284629 + 0.958638i \(0.591870\pi\)
\(444\) 0 0
\(445\) −2.03058e6 −0.486094
\(446\) −5.03599e6 −1.19880
\(447\) 0 0
\(448\) 549685. 0.129395
\(449\) −3.01947e6 −0.706831 −0.353415 0.935467i \(-0.614980\pi\)
−0.353415 + 0.935467i \(0.614980\pi\)
\(450\) 0 0
\(451\) 344218. 0.0796878
\(452\) 306371. 0.0705345
\(453\) 0 0
\(454\) 476173. 0.108424
\(455\) −297573. −0.0673853
\(456\) 0 0
\(457\) 3.80671e6 0.852626 0.426313 0.904576i \(-0.359812\pi\)
0.426313 + 0.904576i \(0.359812\pi\)
\(458\) 1.96738e6 0.438254
\(459\) 0 0
\(460\) −111864. −0.0246487
\(461\) −7.43481e6 −1.62936 −0.814680 0.579910i \(-0.803088\pi\)
−0.814680 + 0.579910i \(0.803088\pi\)
\(462\) 0 0
\(463\) −108747. −0.0235757 −0.0117879 0.999931i \(-0.503752\pi\)
−0.0117879 + 0.999931i \(0.503752\pi\)
\(464\) −1.53588e6 −0.331178
\(465\) 0 0
\(466\) −1.43669e6 −0.306477
\(467\) 6.85473e6 1.45445 0.727224 0.686400i \(-0.240810\pi\)
0.727224 + 0.686400i \(0.240810\pi\)
\(468\) 0 0
\(469\) −215923. −0.0453280
\(470\) −499085. −0.104215
\(471\) 0 0
\(472\) 1.58594e6 0.327666
\(473\) 1.41151e6 0.290088
\(474\) 0 0
\(475\) −1.44237e6 −0.293321
\(476\) 54431.3 0.0110111
\(477\) 0 0
\(478\) −7.98085e6 −1.59764
\(479\) 9.48659e6 1.88917 0.944585 0.328266i \(-0.106464\pi\)
0.944585 + 0.328266i \(0.106464\pi\)
\(480\) 0 0
\(481\) 4.67281e6 0.920905
\(482\) 8.87182e6 1.73938
\(483\) 0 0
\(484\) 21322.2 0.00413732
\(485\) −3.80253e6 −0.734038
\(486\) 0 0
\(487\) 9.79771e6 1.87198 0.935992 0.352021i \(-0.114506\pi\)
0.935992 + 0.352021i \(0.114506\pi\)
\(488\) −66925.7 −0.0127216
\(489\) 0 0
\(490\) −2.38531e6 −0.448801
\(491\) −9.17094e6 −1.71676 −0.858381 0.513012i \(-0.828530\pi\)
−0.858381 + 0.513012i \(0.828530\pi\)
\(492\) 0 0
\(493\) 3.04395e6 0.564054
\(494\) 9.00225e6 1.65972
\(495\) 0 0
\(496\) 5.51087e6 1.00581
\(497\) −986040. −0.179062
\(498\) 0 0
\(499\) −1.91031e6 −0.343441 −0.171720 0.985146i \(-0.554933\pi\)
−0.171720 + 0.985146i \(0.554933\pi\)
\(500\) −22755.2 −0.00407058
\(501\) 0 0
\(502\) 1.95846e6 0.346861
\(503\) −7.08413e6 −1.24844 −0.624219 0.781250i \(-0.714583\pi\)
−0.624219 + 0.781250i \(0.714583\pi\)
\(504\) 0 0
\(505\) 2.38019e6 0.415320
\(506\) 2.15037e6 0.373367
\(507\) 0 0
\(508\) −92520.7 −0.0159067
\(509\) 7.51321e6 1.28538 0.642689 0.766127i \(-0.277819\pi\)
0.642689 + 0.766127i \(0.277819\pi\)
\(510\) 0 0
\(511\) 348398. 0.0590234
\(512\) 5.47764e6 0.923461
\(513\) 0 0
\(514\) 9.31158e6 1.55459
\(515\) 3.77060e6 0.626459
\(516\) 0 0
\(517\) 417619. 0.0687154
\(518\) −707361. −0.115829
\(519\) 0 0
\(520\) −2.97862e6 −0.483066
\(521\) 1.03994e7 1.67847 0.839235 0.543770i \(-0.183003\pi\)
0.839235 + 0.543770i \(0.183003\pi\)
\(522\) 0 0
\(523\) −9.14037e6 −1.46120 −0.730600 0.682806i \(-0.760759\pi\)
−0.730600 + 0.682806i \(0.760759\pi\)
\(524\) 128734. 0.0204817
\(525\) 0 0
\(526\) 2.64982e6 0.417592
\(527\) −1.09220e7 −1.71307
\(528\) 0 0
\(529\) 3.00374e6 0.466684
\(530\) 2.62711e6 0.406245
\(531\) 0 0
\(532\) −59319.5 −0.00908695
\(533\) −1.91850e6 −0.292513
\(534\) 0 0
\(535\) 4.80606e6 0.725947
\(536\) −2.16132e6 −0.324944
\(537\) 0 0
\(538\) −4.00483e6 −0.596524
\(539\) 1.99595e6 0.295923
\(540\) 0 0
\(541\) 6.47380e6 0.950969 0.475484 0.879724i \(-0.342273\pi\)
0.475484 + 0.879724i \(0.342273\pi\)
\(542\) 2.81508e6 0.411617
\(543\) 0 0
\(544\) 1.11566e6 0.161635
\(545\) −2.40757e6 −0.347206
\(546\) 0 0
\(547\) −226463. −0.0323616 −0.0161808 0.999869i \(-0.505151\pi\)
−0.0161808 + 0.999869i \(0.505151\pi\)
\(548\) 62101.6 0.00883388
\(549\) 0 0
\(550\) 437426. 0.0616592
\(551\) −3.31731e6 −0.465487
\(552\) 0 0
\(553\) −238197. −0.0331226
\(554\) −1.37819e7 −1.90781
\(555\) 0 0
\(556\) 246853. 0.0338650
\(557\) 5.45555e6 0.745076 0.372538 0.928017i \(-0.378488\pi\)
0.372538 + 0.928017i \(0.378488\pi\)
\(558\) 0 0
\(559\) −7.86706e6 −1.06484
\(560\) 471461. 0.0635296
\(561\) 0 0
\(562\) 6.48712e6 0.866386
\(563\) 1.30266e7 1.73205 0.866024 0.500002i \(-0.166667\pi\)
0.866024 + 0.500002i \(0.166667\pi\)
\(564\) 0 0
\(565\) −5.25928e6 −0.693115
\(566\) −5.58723e6 −0.733087
\(567\) 0 0
\(568\) −9.86997e6 −1.28365
\(569\) −3.02736e6 −0.391998 −0.195999 0.980604i \(-0.562795\pi\)
−0.195999 + 0.980604i \(0.562795\pi\)
\(570\) 0 0
\(571\) 9.79560e6 1.25731 0.628653 0.777686i \(-0.283607\pi\)
0.628653 + 0.777686i \(0.283607\pi\)
\(572\) −118840. −0.0151870
\(573\) 0 0
\(574\) 290420. 0.0367914
\(575\) 1.92029e6 0.242213
\(576\) 0 0
\(577\) 56282.7 0.00703778 0.00351889 0.999994i \(-0.498880\pi\)
0.00351889 + 0.999994i \(0.498880\pi\)
\(578\) −1.77253e7 −2.20686
\(579\) 0 0
\(580\) −52334.9 −0.00645983
\(581\) −742671. −0.0912759
\(582\) 0 0
\(583\) −2.19829e6 −0.267863
\(584\) 3.48737e6 0.423122
\(585\) 0 0
\(586\) 1.27725e7 1.53650
\(587\) 7.92842e6 0.949711 0.474855 0.880064i \(-0.342500\pi\)
0.474855 + 0.880064i \(0.342500\pi\)
\(588\) 0 0
\(589\) 1.19028e7 1.41371
\(590\) 1.29809e6 0.153523
\(591\) 0 0
\(592\) −7.40338e6 −0.868212
\(593\) 5.11910e6 0.597801 0.298900 0.954284i \(-0.403380\pi\)
0.298900 + 0.954284i \(0.403380\pi\)
\(594\) 0 0
\(595\) −934388. −0.108202
\(596\) 105530. 0.0121692
\(597\) 0 0
\(598\) −1.19851e7 −1.37053
\(599\) 1.00831e7 1.14822 0.574110 0.818778i \(-0.305348\pi\)
0.574110 + 0.818778i \(0.305348\pi\)
\(600\) 0 0
\(601\) 1.32233e7 1.49332 0.746659 0.665207i \(-0.231657\pi\)
0.746659 + 0.665207i \(0.231657\pi\)
\(602\) 1.19090e6 0.133932
\(603\) 0 0
\(604\) −89296.9 −0.00995965
\(605\) −366025. −0.0406558
\(606\) 0 0
\(607\) 937160. 0.103239 0.0516193 0.998667i \(-0.483562\pi\)
0.0516193 + 0.998667i \(0.483562\pi\)
\(608\) −1.21585e6 −0.133390
\(609\) 0 0
\(610\) −54778.7 −0.00596056
\(611\) −2.32761e6 −0.252236
\(612\) 0 0
\(613\) 3.05919e6 0.328818 0.164409 0.986392i \(-0.447428\pi\)
0.164409 + 0.986392i \(0.447428\pi\)
\(614\) −4.65901e6 −0.498738
\(615\) 0 0
\(616\) −377298. −0.0400620
\(617\) −8.74474e6 −0.924771 −0.462385 0.886679i \(-0.653006\pi\)
−0.462385 + 0.886679i \(0.653006\pi\)
\(618\) 0 0
\(619\) −4.14486e6 −0.434793 −0.217397 0.976083i \(-0.569757\pi\)
−0.217397 + 0.976083i \(0.569757\pi\)
\(620\) 187782. 0.0196189
\(621\) 0 0
\(622\) −5.03664e6 −0.521993
\(623\) 1.43357e6 0.147979
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 1.83143e7 1.86791
\(627\) 0 0
\(628\) 736598. 0.0745301
\(629\) 1.46727e7 1.47872
\(630\) 0 0
\(631\) −4.44508e6 −0.444433 −0.222216 0.974997i \(-0.571329\pi\)
−0.222216 + 0.974997i \(0.571329\pi\)
\(632\) −2.38429e6 −0.237446
\(633\) 0 0
\(634\) 9.96537e6 0.984624
\(635\) 1.58825e6 0.156309
\(636\) 0 0
\(637\) −1.11245e7 −1.08625
\(638\) 1.00604e6 0.0978504
\(639\) 0 0
\(640\) 4.92502e6 0.475289
\(641\) −1.08278e7 −1.04086 −0.520432 0.853903i \(-0.674229\pi\)
−0.520432 + 0.853903i \(0.674229\pi\)
\(642\) 0 0
\(643\) −1.23064e7 −1.17383 −0.586915 0.809649i \(-0.699658\pi\)
−0.586915 + 0.809649i \(0.699658\pi\)
\(644\) 78974.7 0.00750367
\(645\) 0 0
\(646\) 2.82673e7 2.66504
\(647\) 1.26477e7 1.18782 0.593912 0.804530i \(-0.297583\pi\)
0.593912 + 0.804530i \(0.297583\pi\)
\(648\) 0 0
\(649\) −1.08620e6 −0.101228
\(650\) −2.43800e6 −0.226334
\(651\) 0 0
\(652\) 564624. 0.0520164
\(653\) 578855. 0.0531235 0.0265618 0.999647i \(-0.491544\pi\)
0.0265618 + 0.999647i \(0.491544\pi\)
\(654\) 0 0
\(655\) −2.20990e6 −0.201265
\(656\) 3.03959e6 0.275775
\(657\) 0 0
\(658\) 352349. 0.0317255
\(659\) −1.71481e7 −1.53816 −0.769081 0.639151i \(-0.779286\pi\)
−0.769081 + 0.639151i \(0.779286\pi\)
\(660\) 0 0
\(661\) 6.99329e6 0.622555 0.311278 0.950319i \(-0.399243\pi\)
0.311278 + 0.950319i \(0.399243\pi\)
\(662\) 1.85378e7 1.64404
\(663\) 0 0
\(664\) −7.43392e6 −0.654332
\(665\) 1.01830e6 0.0892939
\(666\) 0 0
\(667\) 4.41649e6 0.384382
\(668\) 925387. 0.0802384
\(669\) 0 0
\(670\) −1.76904e6 −0.152248
\(671\) 45837.2 0.00393017
\(672\) 0 0
\(673\) 1.98109e7 1.68604 0.843018 0.537885i \(-0.180777\pi\)
0.843018 + 0.537885i \(0.180777\pi\)
\(674\) −1.41502e7 −1.19981
\(675\) 0 0
\(676\) 121628. 0.0102369
\(677\) 2.85012e6 0.238996 0.119498 0.992834i \(-0.461871\pi\)
0.119498 + 0.992834i \(0.461871\pi\)
\(678\) 0 0
\(679\) 2.68455e6 0.223459
\(680\) −9.35295e6 −0.775669
\(681\) 0 0
\(682\) −3.60975e6 −0.297178
\(683\) −4.58865e6 −0.376386 −0.188193 0.982132i \(-0.560263\pi\)
−0.188193 + 0.982132i \(0.560263\pi\)
\(684\) 0 0
\(685\) −1.06606e6 −0.0868070
\(686\) 3.39981e6 0.275832
\(687\) 0 0
\(688\) 1.24642e7 1.00391
\(689\) 1.22522e7 0.983254
\(690\) 0 0
\(691\) −2.65609e6 −0.211616 −0.105808 0.994387i \(-0.533743\pi\)
−0.105808 + 0.994387i \(0.533743\pi\)
\(692\) 990046. 0.0785942
\(693\) 0 0
\(694\) −1.76448e7 −1.39065
\(695\) −4.23757e6 −0.332778
\(696\) 0 0
\(697\) −6.02416e6 −0.469693
\(698\) 1.78588e7 1.38744
\(699\) 0 0
\(700\) 16065.0 0.00123918
\(701\) 7.90907e6 0.607898 0.303949 0.952688i \(-0.401695\pi\)
0.303949 + 0.952688i \(0.401695\pi\)
\(702\) 0 0
\(703\) −1.59904e7 −1.22031
\(704\) −3.76843e6 −0.286569
\(705\) 0 0
\(706\) 1.72906e7 1.30557
\(707\) −1.68039e6 −0.126433
\(708\) 0 0
\(709\) −2.38057e7 −1.77855 −0.889273 0.457378i \(-0.848789\pi\)
−0.889273 + 0.457378i \(0.848789\pi\)
\(710\) −8.07857e6 −0.601435
\(711\) 0 0
\(712\) 1.43496e7 1.06082
\(713\) −1.58468e7 −1.16739
\(714\) 0 0
\(715\) 2.04005e6 0.149237
\(716\) 91870.5 0.00669720
\(717\) 0 0
\(718\) −1.33030e7 −0.963029
\(719\) 2.00866e7 1.44905 0.724527 0.689246i \(-0.242058\pi\)
0.724527 + 0.689246i \(0.242058\pi\)
\(720\) 0 0
\(721\) −2.66201e6 −0.190709
\(722\) −1.64837e7 −1.17683
\(723\) 0 0
\(724\) 762558. 0.0540663
\(725\) 898400. 0.0634782
\(726\) 0 0
\(727\) −1.00897e7 −0.708012 −0.354006 0.935243i \(-0.615181\pi\)
−0.354006 + 0.935243i \(0.615181\pi\)
\(728\) 2.10288e6 0.147057
\(729\) 0 0
\(730\) 2.85441e6 0.198248
\(731\) −2.47028e7 −1.70983
\(732\) 0 0
\(733\) −1.48230e7 −1.01901 −0.509504 0.860468i \(-0.670171\pi\)
−0.509504 + 0.860468i \(0.670171\pi\)
\(734\) −2.07192e7 −1.41949
\(735\) 0 0
\(736\) 1.61872e6 0.110148
\(737\) 1.48028e6 0.100387
\(738\) 0 0
\(739\) −1.85393e7 −1.24877 −0.624383 0.781118i \(-0.714650\pi\)
−0.624383 + 0.781118i \(0.714650\pi\)
\(740\) −252269. −0.0169350
\(741\) 0 0
\(742\) −1.85471e6 −0.123671
\(743\) 8.68829e6 0.577381 0.288690 0.957423i \(-0.406780\pi\)
0.288690 + 0.957423i \(0.406780\pi\)
\(744\) 0 0
\(745\) −1.81157e6 −0.119582
\(746\) −2.95717e6 −0.194549
\(747\) 0 0
\(748\) −373160. −0.0243860
\(749\) −3.39303e6 −0.220995
\(750\) 0 0
\(751\) −1.37377e7 −0.888821 −0.444411 0.895823i \(-0.646587\pi\)
−0.444411 + 0.895823i \(0.646587\pi\)
\(752\) 3.68776e6 0.237803
\(753\) 0 0
\(754\) −5.60717e6 −0.359183
\(755\) 1.53291e6 0.0978696
\(756\) 0 0
\(757\) 5.59097e6 0.354607 0.177304 0.984156i \(-0.443263\pi\)
0.177304 + 0.984156i \(0.443263\pi\)
\(758\) −1.41832e7 −0.896608
\(759\) 0 0
\(760\) 1.01929e7 0.640123
\(761\) 2.44147e7 1.52823 0.764117 0.645078i \(-0.223175\pi\)
0.764117 + 0.645078i \(0.223175\pi\)
\(762\) 0 0
\(763\) 1.69972e6 0.105698
\(764\) 996586. 0.0617706
\(765\) 0 0
\(766\) −8.46309e6 −0.521143
\(767\) 6.05397e6 0.371580
\(768\) 0 0
\(769\) 1.99742e7 1.21802 0.609008 0.793164i \(-0.291568\pi\)
0.609008 + 0.793164i \(0.291568\pi\)
\(770\) −308819. −0.0187705
\(771\) 0 0
\(772\) 768096. 0.0463844
\(773\) 2.64430e7 1.59170 0.795852 0.605492i \(-0.207024\pi\)
0.795852 + 0.605492i \(0.207024\pi\)
\(774\) 0 0
\(775\) −3.22354e6 −0.192787
\(776\) 2.68716e7 1.60191
\(777\) 0 0
\(778\) −4.65032e6 −0.275445
\(779\) 6.56515e6 0.387616
\(780\) 0 0
\(781\) 6.75991e6 0.396564
\(782\) −3.76336e7 −2.20069
\(783\) 0 0
\(784\) 1.76251e7 1.02410
\(785\) −1.26447e7 −0.732378
\(786\) 0 0
\(787\) −2.87892e6 −0.165689 −0.0828443 0.996562i \(-0.526400\pi\)
−0.0828443 + 0.996562i \(0.526400\pi\)
\(788\) 680310. 0.0390294
\(789\) 0 0
\(790\) −1.95154e6 −0.111252
\(791\) 3.71300e6 0.211001
\(792\) 0 0
\(793\) −255474. −0.0144266
\(794\) 1.66181e7 0.935469
\(795\) 0 0
\(796\) −635855. −0.0355693
\(797\) 2.55634e7 1.42552 0.712759 0.701409i \(-0.247446\pi\)
0.712759 + 0.701409i \(0.247446\pi\)
\(798\) 0 0
\(799\) −7.30875e6 −0.405020
\(800\) 329279. 0.0181903
\(801\) 0 0
\(802\) −1.99595e7 −1.09575
\(803\) −2.38849e6 −0.130718
\(804\) 0 0
\(805\) −1.35571e6 −0.0737356
\(806\) 2.01190e7 1.09086
\(807\) 0 0
\(808\) −1.68202e7 −0.906365
\(809\) 2.32458e6 0.124874 0.0624371 0.998049i \(-0.480113\pi\)
0.0624371 + 0.998049i \(0.480113\pi\)
\(810\) 0 0
\(811\) 1.44240e7 0.770076 0.385038 0.922901i \(-0.374188\pi\)
0.385038 + 0.922901i \(0.374188\pi\)
\(812\) 36947.9 0.00196653
\(813\) 0 0
\(814\) 4.84939e6 0.256523
\(815\) −9.69255e6 −0.511145
\(816\) 0 0
\(817\) 2.69212e7 1.41104
\(818\) 6.81233e6 0.355969
\(819\) 0 0
\(820\) 103574. 0.00537916
\(821\) −9.95289e6 −0.515337 −0.257668 0.966233i \(-0.582954\pi\)
−0.257668 + 0.966233i \(0.582954\pi\)
\(822\) 0 0
\(823\) 2.46452e7 1.26833 0.634165 0.773198i \(-0.281344\pi\)
0.634165 + 0.773198i \(0.281344\pi\)
\(824\) −2.66460e7 −1.36714
\(825\) 0 0
\(826\) −916439. −0.0467362
\(827\) 2.09014e7 1.06270 0.531351 0.847152i \(-0.321685\pi\)
0.531351 + 0.847152i \(0.321685\pi\)
\(828\) 0 0
\(829\) −8.31387e6 −0.420162 −0.210081 0.977684i \(-0.567373\pi\)
−0.210081 + 0.977684i \(0.567373\pi\)
\(830\) −6.08467e6 −0.306579
\(831\) 0 0
\(832\) 2.10034e7 1.05192
\(833\) −3.49312e7 −1.74422
\(834\) 0 0
\(835\) −1.58855e7 −0.788472
\(836\) 406671. 0.0201246
\(837\) 0 0
\(838\) −2.56244e6 −0.126050
\(839\) 4.03176e7 1.97738 0.988690 0.149977i \(-0.0479198\pi\)
0.988690 + 0.149977i \(0.0479198\pi\)
\(840\) 0 0
\(841\) −1.84449e7 −0.899263
\(842\) 1.97757e7 0.961282
\(843\) 0 0
\(844\) −1.08834e6 −0.0525905
\(845\) −2.08792e6 −0.100594
\(846\) 0 0
\(847\) 258410. 0.0123766
\(848\) −1.94118e7 −0.926992
\(849\) 0 0
\(850\) −7.65539e6 −0.363430
\(851\) 2.12888e7 1.00769
\(852\) 0 0
\(853\) −2.11646e7 −0.995951 −0.497976 0.867191i \(-0.665923\pi\)
−0.497976 + 0.867191i \(0.665923\pi\)
\(854\) 38673.3 0.00181454
\(855\) 0 0
\(856\) −3.39633e7 −1.58425
\(857\) 1.03622e7 0.481946 0.240973 0.970532i \(-0.422533\pi\)
0.240973 + 0.970532i \(0.422533\pi\)
\(858\) 0 0
\(859\) −1.77409e7 −0.820336 −0.410168 0.912010i \(-0.634530\pi\)
−0.410168 + 0.912010i \(0.634530\pi\)
\(860\) 424716. 0.0195818
\(861\) 0 0
\(862\) −2.45618e7 −1.12588
\(863\) 2.62577e6 0.120014 0.0600068 0.998198i \(-0.480888\pi\)
0.0600068 + 0.998198i \(0.480888\pi\)
\(864\) 0 0
\(865\) −1.69955e7 −0.772314
\(866\) 5.98118e6 0.271014
\(867\) 0 0
\(868\) −132572. −0.00597247
\(869\) 1.63299e6 0.0733557
\(870\) 0 0
\(871\) −8.25039e6 −0.368493
\(872\) 1.70137e7 0.757718
\(873\) 0 0
\(874\) 4.10132e7 1.81612
\(875\) −275778. −0.0121770
\(876\) 0 0
\(877\) 7.30171e6 0.320572 0.160286 0.987071i \(-0.448758\pi\)
0.160286 + 0.987071i \(0.448758\pi\)
\(878\) −2.05427e7 −0.899334
\(879\) 0 0
\(880\) −3.23216e6 −0.140697
\(881\) −6.08524e6 −0.264142 −0.132071 0.991240i \(-0.542163\pi\)
−0.132071 + 0.991240i \(0.542163\pi\)
\(882\) 0 0
\(883\) 1.66311e7 0.717828 0.358914 0.933371i \(-0.383147\pi\)
0.358914 + 0.933371i \(0.383147\pi\)
\(884\) 2.07981e6 0.0895146
\(885\) 0 0
\(886\) 1.36006e7 0.582067
\(887\) 1.47741e7 0.630511 0.315256 0.949007i \(-0.397910\pi\)
0.315256 + 0.949007i \(0.397910\pi\)
\(888\) 0 0
\(889\) −1.12129e6 −0.0475841
\(890\) 1.17452e7 0.497032
\(891\) 0 0
\(892\) 1.26796e6 0.0533574
\(893\) 7.96511e6 0.334244
\(894\) 0 0
\(895\) −1.57708e6 −0.0658108
\(896\) −3.47702e6 −0.144689
\(897\) 0 0
\(898\) 1.74651e7 0.722736
\(899\) −7.41383e6 −0.305945
\(900\) 0 0
\(901\) 3.84722e7 1.57883
\(902\) −1.99101e6 −0.0814810
\(903\) 0 0
\(904\) 3.71661e7 1.51261
\(905\) −1.30904e7 −0.531288
\(906\) 0 0
\(907\) −9.23123e6 −0.372599 −0.186299 0.982493i \(-0.559649\pi\)
−0.186299 + 0.982493i \(0.559649\pi\)
\(908\) −119891. −0.00482583
\(909\) 0 0
\(910\) 1.72121e6 0.0689016
\(911\) 1.55931e7 0.622495 0.311247 0.950329i \(-0.399253\pi\)
0.311247 + 0.950329i \(0.399253\pi\)
\(912\) 0 0
\(913\) 5.09147e6 0.202146
\(914\) −2.20185e7 −0.871812
\(915\) 0 0
\(916\) −495349. −0.0195062
\(917\) 1.56017e6 0.0612700
\(918\) 0 0
\(919\) −4.21091e6 −0.164470 −0.0822351 0.996613i \(-0.526206\pi\)
−0.0822351 + 0.996613i \(0.526206\pi\)
\(920\) −1.35703e7 −0.528590
\(921\) 0 0
\(922\) 4.30040e7 1.66602
\(923\) −3.76765e7 −1.45568
\(924\) 0 0
\(925\) 4.33055e6 0.166413
\(926\) 629009. 0.0241062
\(927\) 0 0
\(928\) 757310. 0.0288671
\(929\) −4.10363e7 −1.56002 −0.780008 0.625770i \(-0.784785\pi\)
−0.780008 + 0.625770i \(0.784785\pi\)
\(930\) 0 0
\(931\) 3.80682e7 1.43942
\(932\) 361730. 0.0136410
\(933\) 0 0
\(934\) −3.96488e7 −1.48718
\(935\) 6.40581e6 0.239632
\(936\) 0 0
\(937\) 2.41565e7 0.898844 0.449422 0.893320i \(-0.351630\pi\)
0.449422 + 0.893320i \(0.351630\pi\)
\(938\) 1.24893e6 0.0463479
\(939\) 0 0
\(940\) 125660. 0.00463849
\(941\) −3.13080e7 −1.15261 −0.576304 0.817236i \(-0.695505\pi\)
−0.576304 + 0.817236i \(0.695505\pi\)
\(942\) 0 0
\(943\) −8.74049e6 −0.320079
\(944\) −9.59164e6 −0.350318
\(945\) 0 0
\(946\) −8.16436e6 −0.296616
\(947\) −5699.21 −0.000206509 0 −0.000103255 1.00000i \(-0.500033\pi\)
−0.000103255 1.00000i \(0.500033\pi\)
\(948\) 0 0
\(949\) 1.33123e7 0.479829
\(950\) 8.34288e6 0.299921
\(951\) 0 0
\(952\) 6.60310e6 0.236132
\(953\) 4.24159e7 1.51285 0.756427 0.654078i \(-0.226943\pi\)
0.756427 + 0.654078i \(0.226943\pi\)
\(954\) 0 0
\(955\) −1.71078e7 −0.606995
\(956\) 2.00942e6 0.0711094
\(957\) 0 0
\(958\) −5.48718e7 −1.93168
\(959\) 752628. 0.0264261
\(960\) 0 0
\(961\) −2.02765e6 −0.0708247
\(962\) −2.70282e7 −0.941628
\(963\) 0 0
\(964\) −2.23375e6 −0.0774180
\(965\) −1.31854e7 −0.455801
\(966\) 0 0
\(967\) 1.43147e7 0.492284 0.246142 0.969234i \(-0.420837\pi\)
0.246142 + 0.969234i \(0.420837\pi\)
\(968\) 2.58661e6 0.0887243
\(969\) 0 0
\(970\) 2.19944e7 0.750556
\(971\) 1.06931e7 0.363960 0.181980 0.983302i \(-0.441749\pi\)
0.181980 + 0.983302i \(0.441749\pi\)
\(972\) 0 0
\(973\) 2.99168e6 0.101306
\(974\) −5.66714e7 −1.91411
\(975\) 0 0
\(976\) 404762. 0.0136011
\(977\) −5.44429e7 −1.82476 −0.912378 0.409349i \(-0.865756\pi\)
−0.912378 + 0.409349i \(0.865756\pi\)
\(978\) 0 0
\(979\) −9.82800e6 −0.327724
\(980\) 600574. 0.0199757
\(981\) 0 0
\(982\) 5.30461e7 1.75539
\(983\) −4.16457e7 −1.37463 −0.687316 0.726359i \(-0.741211\pi\)
−0.687316 + 0.726359i \(0.741211\pi\)
\(984\) 0 0
\(985\) −1.16785e7 −0.383526
\(986\) −1.76067e7 −0.576746
\(987\) 0 0
\(988\) −2.26659e6 −0.0738722
\(989\) −3.58414e7 −1.16518
\(990\) 0 0
\(991\) −9.62055e6 −0.311183 −0.155591 0.987821i \(-0.549728\pi\)
−0.155591 + 0.987821i \(0.549728\pi\)
\(992\) −2.71730e6 −0.0876714
\(993\) 0 0
\(994\) 5.70340e6 0.183091
\(995\) 1.09153e7 0.349526
\(996\) 0 0
\(997\) 3.05865e7 0.974523 0.487262 0.873256i \(-0.337996\pi\)
0.487262 + 0.873256i \(0.337996\pi\)
\(998\) 1.10495e7 0.351169
\(999\) 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 495.6.a.b.1.1 3
3.2 odd 2 165.6.a.c.1.3 3
15.14 odd 2 825.6.a.g.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.c.1.3 3 3.2 odd 2
495.6.a.b.1.1 3 1.1 even 1 trivial
825.6.a.g.1.1 3 15.14 odd 2