Properties

Label 448.6.a.q.1.1
Level $448$
Weight $6$
Character 448.1
Self dual yes
Analytic conductor $71.852$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,6,Mod(1,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, 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("448.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 448.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: \(71.8519512762\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(7.15207\) of defining polynomial
Character \(\chi\) \(=\) 448.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-26.3041 q^{3} +97.5207 q^{5} -49.0000 q^{7} +448.908 q^{9} +O(q^{10})\) \(q-26.3041 q^{3} +97.5207 q^{5} -49.0000 q^{7} +448.908 q^{9} +406.175 q^{11} +905.594 q^{13} -2565.20 q^{15} +359.825 q^{17} +1813.25 q^{19} +1288.90 q^{21} +2163.43 q^{23} +6385.28 q^{25} -5416.22 q^{27} +4090.26 q^{29} -4451.54 q^{31} -10684.1 q^{33} -4778.51 q^{35} -8935.69 q^{37} -23820.9 q^{39} -3415.04 q^{41} -8694.80 q^{43} +43777.8 q^{45} -14204.5 q^{47} +2401.00 q^{49} -9464.88 q^{51} +14616.4 q^{53} +39610.5 q^{55} -47696.1 q^{57} +21396.8 q^{59} -54081.9 q^{61} -21996.5 q^{63} +88314.1 q^{65} +44973.1 q^{67} -56907.0 q^{69} -5700.89 q^{71} +47650.8 q^{73} -167959. q^{75} -19902.6 q^{77} -5809.20 q^{79} +33384.4 q^{81} +27058.2 q^{83} +35090.4 q^{85} -107591. q^{87} -96245.2 q^{89} -44374.1 q^{91} +117094. q^{93} +176830. q^{95} +101968. q^{97} +182335. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 26 q^{3} + 62 q^{5} - 98 q^{7} + 206 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 26 q^{3} + 62 q^{5} - 98 q^{7} + 206 q^{9} + 972 q^{11} - 78 q^{13} - 2576 q^{15} + 560 q^{17} + 2642 q^{19} + 1274 q^{21} - 2272 q^{23} + 4522 q^{25} - 5564 q^{27} + 7808 q^{29} - 5444 q^{31} - 10512 q^{33} - 3038 q^{35} - 576 q^{37} - 24120 q^{39} - 16888 q^{41} - 8396 q^{43} + 52406 q^{45} + 4532 q^{47} + 4802 q^{49} - 9404 q^{51} - 1420 q^{53} + 19512 q^{55} - 47444 q^{57} + 34146 q^{59} - 19106 q^{61} - 10094 q^{63} + 123252 q^{65} + 56952 q^{67} - 58256 q^{69} + 7224 q^{71} + 128828 q^{73} - 168526 q^{75} - 47628 q^{77} - 52808 q^{79} + 92366 q^{81} - 84486 q^{83} + 27980 q^{85} - 106460 q^{87} - 130972 q^{89} + 3822 q^{91} + 116792 q^{93} + 147392 q^{95} + 194624 q^{97} + 44892 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −26.3041 −1.68741 −0.843706 0.536806i \(-0.819631\pi\)
−0.843706 + 0.536806i \(0.819631\pi\)
\(4\) 0 0
\(5\) 97.5207 1.74450 0.872251 0.489058i \(-0.162659\pi\)
0.872251 + 0.489058i \(0.162659\pi\)
\(6\) 0 0
\(7\) −49.0000 −0.377964
\(8\) 0 0
\(9\) 448.908 1.84736
\(10\) 0 0
\(11\) 406.175 1.01212 0.506060 0.862498i \(-0.331102\pi\)
0.506060 + 0.862498i \(0.331102\pi\)
\(12\) 0 0
\(13\) 905.594 1.48619 0.743096 0.669185i \(-0.233357\pi\)
0.743096 + 0.669185i \(0.233357\pi\)
\(14\) 0 0
\(15\) −2565.20 −2.94369
\(16\) 0 0
\(17\) 359.825 0.301973 0.150987 0.988536i \(-0.451755\pi\)
0.150987 + 0.988536i \(0.451755\pi\)
\(18\) 0 0
\(19\) 1813.25 1.15232 0.576162 0.817336i \(-0.304550\pi\)
0.576162 + 0.817336i \(0.304550\pi\)
\(20\) 0 0
\(21\) 1288.90 0.637781
\(22\) 0 0
\(23\) 2163.43 0.852751 0.426376 0.904546i \(-0.359790\pi\)
0.426376 + 0.904546i \(0.359790\pi\)
\(24\) 0 0
\(25\) 6385.28 2.04329
\(26\) 0 0
\(27\) −5416.22 −1.42984
\(28\) 0 0
\(29\) 4090.26 0.903141 0.451571 0.892235i \(-0.350864\pi\)
0.451571 + 0.892235i \(0.350864\pi\)
\(30\) 0 0
\(31\) −4451.54 −0.831966 −0.415983 0.909372i \(-0.636562\pi\)
−0.415983 + 0.909372i \(0.636562\pi\)
\(32\) 0 0
\(33\) −10684.1 −1.70786
\(34\) 0 0
\(35\) −4778.51 −0.659360
\(36\) 0 0
\(37\) −8935.69 −1.07306 −0.536530 0.843882i \(-0.680265\pi\)
−0.536530 + 0.843882i \(0.680265\pi\)
\(38\) 0 0
\(39\) −23820.9 −2.50782
\(40\) 0 0
\(41\) −3415.04 −0.317275 −0.158637 0.987337i \(-0.550710\pi\)
−0.158637 + 0.987337i \(0.550710\pi\)
\(42\) 0 0
\(43\) −8694.80 −0.717114 −0.358557 0.933508i \(-0.616731\pi\)
−0.358557 + 0.933508i \(0.616731\pi\)
\(44\) 0 0
\(45\) 43777.8 3.22272
\(46\) 0 0
\(47\) −14204.5 −0.937955 −0.468977 0.883210i \(-0.655377\pi\)
−0.468977 + 0.883210i \(0.655377\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) −9464.88 −0.509553
\(52\) 0 0
\(53\) 14616.4 0.714743 0.357371 0.933962i \(-0.383673\pi\)
0.357371 + 0.933962i \(0.383673\pi\)
\(54\) 0 0
\(55\) 39610.5 1.76564
\(56\) 0 0
\(57\) −47696.1 −1.94444
\(58\) 0 0
\(59\) 21396.8 0.800239 0.400119 0.916463i \(-0.368969\pi\)
0.400119 + 0.916463i \(0.368969\pi\)
\(60\) 0 0
\(61\) −54081.9 −1.86092 −0.930460 0.366394i \(-0.880592\pi\)
−0.930460 + 0.366394i \(0.880592\pi\)
\(62\) 0 0
\(63\) −21996.5 −0.698235
\(64\) 0 0
\(65\) 88314.1 2.59267
\(66\) 0 0
\(67\) 44973.1 1.22396 0.611978 0.790874i \(-0.290374\pi\)
0.611978 + 0.790874i \(0.290374\pi\)
\(68\) 0 0
\(69\) −56907.0 −1.43894
\(70\) 0 0
\(71\) −5700.89 −0.134214 −0.0671069 0.997746i \(-0.521377\pi\)
−0.0671069 + 0.997746i \(0.521377\pi\)
\(72\) 0 0
\(73\) 47650.8 1.04656 0.523279 0.852162i \(-0.324709\pi\)
0.523279 + 0.852162i \(0.324709\pi\)
\(74\) 0 0
\(75\) −167959. −3.44787
\(76\) 0 0
\(77\) −19902.6 −0.382545
\(78\) 0 0
\(79\) −5809.20 −0.104725 −0.0523623 0.998628i \(-0.516675\pi\)
−0.0523623 + 0.998628i \(0.516675\pi\)
\(80\) 0 0
\(81\) 33384.4 0.565368
\(82\) 0 0
\(83\) 27058.2 0.431126 0.215563 0.976490i \(-0.430841\pi\)
0.215563 + 0.976490i \(0.430841\pi\)
\(84\) 0 0
\(85\) 35090.4 0.526794
\(86\) 0 0
\(87\) −107591. −1.52397
\(88\) 0 0
\(89\) −96245.2 −1.28796 −0.643982 0.765040i \(-0.722719\pi\)
−0.643982 + 0.765040i \(0.722719\pi\)
\(90\) 0 0
\(91\) −44374.1 −0.561728
\(92\) 0 0
\(93\) 117094. 1.40387
\(94\) 0 0
\(95\) 176830. 2.01023
\(96\) 0 0
\(97\) 101968. 1.10036 0.550182 0.835045i \(-0.314558\pi\)
0.550182 + 0.835045i \(0.314558\pi\)
\(98\) 0 0
\(99\) 182335. 1.86974
\(100\) 0 0
\(101\) 21856.8 0.213198 0.106599 0.994302i \(-0.466004\pi\)
0.106599 + 0.994302i \(0.466004\pi\)
\(102\) 0 0
\(103\) 136106. 1.26411 0.632056 0.774923i \(-0.282211\pi\)
0.632056 + 0.774923i \(0.282211\pi\)
\(104\) 0 0
\(105\) 125695. 1.11261
\(106\) 0 0
\(107\) 27394.7 0.231317 0.115658 0.993289i \(-0.463102\pi\)
0.115658 + 0.993289i \(0.463102\pi\)
\(108\) 0 0
\(109\) −88569.6 −0.714033 −0.357016 0.934098i \(-0.616206\pi\)
−0.357016 + 0.934098i \(0.616206\pi\)
\(110\) 0 0
\(111\) 235046. 1.81069
\(112\) 0 0
\(113\) 18947.6 0.139591 0.0697955 0.997561i \(-0.477765\pi\)
0.0697955 + 0.997561i \(0.477765\pi\)
\(114\) 0 0
\(115\) 210979. 1.48763
\(116\) 0 0
\(117\) 406528. 2.74553
\(118\) 0 0
\(119\) −17631.4 −0.114135
\(120\) 0 0
\(121\) 3927.29 0.0243854
\(122\) 0 0
\(123\) 89829.6 0.535373
\(124\) 0 0
\(125\) 317945. 1.82002
\(126\) 0 0
\(127\) 188535. 1.03725 0.518625 0.855002i \(-0.326444\pi\)
0.518625 + 0.855002i \(0.326444\pi\)
\(128\) 0 0
\(129\) 228709. 1.21007
\(130\) 0 0
\(131\) −214161. −1.09034 −0.545171 0.838325i \(-0.683535\pi\)
−0.545171 + 0.838325i \(0.683535\pi\)
\(132\) 0 0
\(133\) −88849.4 −0.435537
\(134\) 0 0
\(135\) −528193. −2.49436
\(136\) 0 0
\(137\) −44903.9 −0.204401 −0.102200 0.994764i \(-0.532588\pi\)
−0.102200 + 0.994764i \(0.532588\pi\)
\(138\) 0 0
\(139\) 262924. 1.15423 0.577116 0.816662i \(-0.304178\pi\)
0.577116 + 0.816662i \(0.304178\pi\)
\(140\) 0 0
\(141\) 373638. 1.58272
\(142\) 0 0
\(143\) 367830. 1.50420
\(144\) 0 0
\(145\) 398885. 1.57553
\(146\) 0 0
\(147\) −63156.2 −0.241059
\(148\) 0 0
\(149\) −209764. −0.774044 −0.387022 0.922070i \(-0.626496\pi\)
−0.387022 + 0.922070i \(0.626496\pi\)
\(150\) 0 0
\(151\) −330665. −1.18017 −0.590087 0.807340i \(-0.700907\pi\)
−0.590087 + 0.807340i \(0.700907\pi\)
\(152\) 0 0
\(153\) 161528. 0.557853
\(154\) 0 0
\(155\) −434117. −1.45137
\(156\) 0 0
\(157\) 510079. 1.65154 0.825769 0.564008i \(-0.190741\pi\)
0.825769 + 0.564008i \(0.190741\pi\)
\(158\) 0 0
\(159\) −384471. −1.20606
\(160\) 0 0
\(161\) −106008. −0.322310
\(162\) 0 0
\(163\) 138.797 0.000409176 0 0.000204588 1.00000i \(-0.499935\pi\)
0.000204588 1.00000i \(0.499935\pi\)
\(164\) 0 0
\(165\) −1.04192e6 −2.97937
\(166\) 0 0
\(167\) −527469. −1.46354 −0.731772 0.681549i \(-0.761307\pi\)
−0.731772 + 0.681549i \(0.761307\pi\)
\(168\) 0 0
\(169\) 448807. 1.20877
\(170\) 0 0
\(171\) 813983. 2.12875
\(172\) 0 0
\(173\) 345616. 0.877968 0.438984 0.898495i \(-0.355338\pi\)
0.438984 + 0.898495i \(0.355338\pi\)
\(174\) 0 0
\(175\) −312879. −0.772291
\(176\) 0 0
\(177\) −562825. −1.35033
\(178\) 0 0
\(179\) 349527. 0.815358 0.407679 0.913125i \(-0.366338\pi\)
0.407679 + 0.913125i \(0.366338\pi\)
\(180\) 0 0
\(181\) 469274. 1.06471 0.532353 0.846522i \(-0.321308\pi\)
0.532353 + 0.846522i \(0.321308\pi\)
\(182\) 0 0
\(183\) 1.42258e6 3.14014
\(184\) 0 0
\(185\) −871414. −1.87195
\(186\) 0 0
\(187\) 146152. 0.305633
\(188\) 0 0
\(189\) 265395. 0.540428
\(190\) 0 0
\(191\) −851324. −1.68854 −0.844270 0.535918i \(-0.819966\pi\)
−0.844270 + 0.535918i \(0.819966\pi\)
\(192\) 0 0
\(193\) 122886. 0.237470 0.118735 0.992926i \(-0.462116\pi\)
0.118735 + 0.992926i \(0.462116\pi\)
\(194\) 0 0
\(195\) −2.32303e6 −4.37489
\(196\) 0 0
\(197\) −744847. −1.36742 −0.683709 0.729755i \(-0.739634\pi\)
−0.683709 + 0.729755i \(0.739634\pi\)
\(198\) 0 0
\(199\) −326738. −0.584880 −0.292440 0.956284i \(-0.594467\pi\)
−0.292440 + 0.956284i \(0.594467\pi\)
\(200\) 0 0
\(201\) −1.18298e6 −2.06532
\(202\) 0 0
\(203\) −200423. −0.341355
\(204\) 0 0
\(205\) −333037. −0.553487
\(206\) 0 0
\(207\) 971178. 1.57534
\(208\) 0 0
\(209\) 736498. 1.16629
\(210\) 0 0
\(211\) 336421. 0.520209 0.260104 0.965581i \(-0.416243\pi\)
0.260104 + 0.965581i \(0.416243\pi\)
\(212\) 0 0
\(213\) 149957. 0.226474
\(214\) 0 0
\(215\) −847923. −1.25101
\(216\) 0 0
\(217\) 218125. 0.314454
\(218\) 0 0
\(219\) −1.25341e6 −1.76597
\(220\) 0 0
\(221\) 325855. 0.448791
\(222\) 0 0
\(223\) −633862. −0.853556 −0.426778 0.904356i \(-0.640351\pi\)
−0.426778 + 0.904356i \(0.640351\pi\)
\(224\) 0 0
\(225\) 2.86640e6 3.77468
\(226\) 0 0
\(227\) −297408. −0.383079 −0.191539 0.981485i \(-0.561348\pi\)
−0.191539 + 0.981485i \(0.561348\pi\)
\(228\) 0 0
\(229\) −594953. −0.749711 −0.374855 0.927083i \(-0.622308\pi\)
−0.374855 + 0.927083i \(0.622308\pi\)
\(230\) 0 0
\(231\) 523520. 0.645511
\(232\) 0 0
\(233\) 150894. 0.182088 0.0910442 0.995847i \(-0.470980\pi\)
0.0910442 + 0.995847i \(0.470980\pi\)
\(234\) 0 0
\(235\) −1.38523e6 −1.63626
\(236\) 0 0
\(237\) 152806. 0.176713
\(238\) 0 0
\(239\) 826500. 0.935941 0.467970 0.883744i \(-0.344985\pi\)
0.467970 + 0.883744i \(0.344985\pi\)
\(240\) 0 0
\(241\) 1.36773e6 1.51690 0.758450 0.651731i \(-0.225957\pi\)
0.758450 + 0.651731i \(0.225957\pi\)
\(242\) 0 0
\(243\) 437993. 0.475829
\(244\) 0 0
\(245\) 234147. 0.249215
\(246\) 0 0
\(247\) 1.64207e6 1.71257
\(248\) 0 0
\(249\) −711744. −0.727487
\(250\) 0 0
\(251\) 1.73585e6 1.73911 0.869557 0.493832i \(-0.164404\pi\)
0.869557 + 0.493832i \(0.164404\pi\)
\(252\) 0 0
\(253\) 878730. 0.863086
\(254\) 0 0
\(255\) −923021. −0.888917
\(256\) 0 0
\(257\) −2.04674e6 −1.93299 −0.966496 0.256680i \(-0.917371\pi\)
−0.966496 + 0.256680i \(0.917371\pi\)
\(258\) 0 0
\(259\) 437849. 0.405578
\(260\) 0 0
\(261\) 1.83615e6 1.66842
\(262\) 0 0
\(263\) 405737. 0.361705 0.180853 0.983510i \(-0.442114\pi\)
0.180853 + 0.983510i \(0.442114\pi\)
\(264\) 0 0
\(265\) 1.42540e6 1.24687
\(266\) 0 0
\(267\) 2.53165e6 2.17333
\(268\) 0 0
\(269\) 1.48037e6 1.24736 0.623678 0.781681i \(-0.285638\pi\)
0.623678 + 0.781681i \(0.285638\pi\)
\(270\) 0 0
\(271\) −457477. −0.378395 −0.189198 0.981939i \(-0.560589\pi\)
−0.189198 + 0.981939i \(0.560589\pi\)
\(272\) 0 0
\(273\) 1.16722e6 0.947866
\(274\) 0 0
\(275\) 2.59354e6 2.06805
\(276\) 0 0
\(277\) −1.45830e6 −1.14195 −0.570977 0.820966i \(-0.693435\pi\)
−0.570977 + 0.820966i \(0.693435\pi\)
\(278\) 0 0
\(279\) −1.99833e6 −1.53694
\(280\) 0 0
\(281\) −1.90172e6 −1.43675 −0.718373 0.695658i \(-0.755113\pi\)
−0.718373 + 0.695658i \(0.755113\pi\)
\(282\) 0 0
\(283\) 1.01187e6 0.751032 0.375516 0.926816i \(-0.377465\pi\)
0.375516 + 0.926816i \(0.377465\pi\)
\(284\) 0 0
\(285\) −4.65135e6 −3.39209
\(286\) 0 0
\(287\) 167337. 0.119919
\(288\) 0 0
\(289\) −1.29038e6 −0.908812
\(290\) 0 0
\(291\) −2.68219e6 −1.85677
\(292\) 0 0
\(293\) −1.30915e6 −0.890882 −0.445441 0.895311i \(-0.646953\pi\)
−0.445441 + 0.895311i \(0.646953\pi\)
\(294\) 0 0
\(295\) 2.08663e6 1.39602
\(296\) 0 0
\(297\) −2.19993e6 −1.44717
\(298\) 0 0
\(299\) 1.95918e6 1.26735
\(300\) 0 0
\(301\) 426045. 0.271044
\(302\) 0 0
\(303\) −574925. −0.359753
\(304\) 0 0
\(305\) −5.27411e6 −3.24638
\(306\) 0 0
\(307\) −1.83705e6 −1.11244 −0.556219 0.831036i \(-0.687748\pi\)
−0.556219 + 0.831036i \(0.687748\pi\)
\(308\) 0 0
\(309\) −3.58016e6 −2.13307
\(310\) 0 0
\(311\) −1.41953e6 −0.832233 −0.416116 0.909311i \(-0.636609\pi\)
−0.416116 + 0.909311i \(0.636609\pi\)
\(312\) 0 0
\(313\) 2.67261e6 1.54197 0.770983 0.636855i \(-0.219765\pi\)
0.770983 + 0.636855i \(0.219765\pi\)
\(314\) 0 0
\(315\) −2.14511e6 −1.21807
\(316\) 0 0
\(317\) −34066.4 −0.0190405 −0.00952024 0.999955i \(-0.503030\pi\)
−0.00952024 + 0.999955i \(0.503030\pi\)
\(318\) 0 0
\(319\) 1.66136e6 0.914087
\(320\) 0 0
\(321\) −720595. −0.390327
\(322\) 0 0
\(323\) 652453. 0.347971
\(324\) 0 0
\(325\) 5.78247e6 3.03672
\(326\) 0 0
\(327\) 2.32975e6 1.20487
\(328\) 0 0
\(329\) 696021. 0.354514
\(330\) 0 0
\(331\) 1.55050e6 0.777862 0.388931 0.921267i \(-0.372844\pi\)
0.388931 + 0.921267i \(0.372844\pi\)
\(332\) 0 0
\(333\) −4.01130e6 −1.98232
\(334\) 0 0
\(335\) 4.38581e6 2.13520
\(336\) 0 0
\(337\) −3.49728e6 −1.67747 −0.838736 0.544539i \(-0.816705\pi\)
−0.838736 + 0.544539i \(0.816705\pi\)
\(338\) 0 0
\(339\) −498399. −0.235547
\(340\) 0 0
\(341\) −1.80810e6 −0.842049
\(342\) 0 0
\(343\) −117649. −0.0539949
\(344\) 0 0
\(345\) −5.54961e6 −2.51024
\(346\) 0 0
\(347\) 3.25824e6 1.45265 0.726323 0.687354i \(-0.241228\pi\)
0.726323 + 0.687354i \(0.241228\pi\)
\(348\) 0 0
\(349\) 1.40547e6 0.617672 0.308836 0.951115i \(-0.400060\pi\)
0.308836 + 0.951115i \(0.400060\pi\)
\(350\) 0 0
\(351\) −4.90489e6 −2.12501
\(352\) 0 0
\(353\) 2.37361e6 1.01385 0.506924 0.861991i \(-0.330783\pi\)
0.506924 + 0.861991i \(0.330783\pi\)
\(354\) 0 0
\(355\) −555955. −0.234136
\(356\) 0 0
\(357\) 463779. 0.192593
\(358\) 0 0
\(359\) 785602. 0.321711 0.160856 0.986978i \(-0.448575\pi\)
0.160856 + 0.986978i \(0.448575\pi\)
\(360\) 0 0
\(361\) 811787. 0.327849
\(362\) 0 0
\(363\) −103304. −0.0411481
\(364\) 0 0
\(365\) 4.64694e6 1.82572
\(366\) 0 0
\(367\) 2.62687e6 1.01806 0.509031 0.860748i \(-0.330004\pi\)
0.509031 + 0.860748i \(0.330004\pi\)
\(368\) 0 0
\(369\) −1.53304e6 −0.586120
\(370\) 0 0
\(371\) −716202. −0.270147
\(372\) 0 0
\(373\) −2.43475e6 −0.906113 −0.453056 0.891482i \(-0.649666\pi\)
−0.453056 + 0.891482i \(0.649666\pi\)
\(374\) 0 0
\(375\) −8.36326e6 −3.07113
\(376\) 0 0
\(377\) 3.70411e6 1.34224
\(378\) 0 0
\(379\) 83774.3 0.0299580 0.0149790 0.999888i \(-0.495232\pi\)
0.0149790 + 0.999888i \(0.495232\pi\)
\(380\) 0 0
\(381\) −4.95926e6 −1.75027
\(382\) 0 0
\(383\) −1.75410e6 −0.611021 −0.305511 0.952189i \(-0.598827\pi\)
−0.305511 + 0.952189i \(0.598827\pi\)
\(384\) 0 0
\(385\) −1.94091e6 −0.667351
\(386\) 0 0
\(387\) −3.90316e6 −1.32477
\(388\) 0 0
\(389\) −3.75594e6 −1.25848 −0.629238 0.777213i \(-0.716633\pi\)
−0.629238 + 0.777213i \(0.716633\pi\)
\(390\) 0 0
\(391\) 778454. 0.257508
\(392\) 0 0
\(393\) 5.63333e6 1.83985
\(394\) 0 0
\(395\) −566517. −0.182692
\(396\) 0 0
\(397\) 1.00266e6 0.319286 0.159643 0.987175i \(-0.448966\pi\)
0.159643 + 0.987175i \(0.448966\pi\)
\(398\) 0 0
\(399\) 2.33711e6 0.734930
\(400\) 0 0
\(401\) 5.31342e6 1.65011 0.825056 0.565051i \(-0.191144\pi\)
0.825056 + 0.565051i \(0.191144\pi\)
\(402\) 0 0
\(403\) −4.03128e6 −1.23646
\(404\) 0 0
\(405\) 3.25567e6 0.986286
\(406\) 0 0
\(407\) −3.62945e6 −1.08606
\(408\) 0 0
\(409\) −4.58663e6 −1.35577 −0.677884 0.735169i \(-0.737103\pi\)
−0.677884 + 0.735169i \(0.737103\pi\)
\(410\) 0 0
\(411\) 1.18116e6 0.344908
\(412\) 0 0
\(413\) −1.04845e6 −0.302462
\(414\) 0 0
\(415\) 2.63874e6 0.752101
\(416\) 0 0
\(417\) −6.91599e6 −1.94767
\(418\) 0 0
\(419\) 2.29852e6 0.639607 0.319803 0.947484i \(-0.396383\pi\)
0.319803 + 0.947484i \(0.396383\pi\)
\(420\) 0 0
\(421\) 5554.63 0.00152739 0.000763695 1.00000i \(-0.499757\pi\)
0.000763695 1.00000i \(0.499757\pi\)
\(422\) 0 0
\(423\) −6.37652e6 −1.73274
\(424\) 0 0
\(425\) 2.29758e6 0.617019
\(426\) 0 0
\(427\) 2.65002e6 0.703362
\(428\) 0 0
\(429\) −9.67544e6 −2.53821
\(430\) 0 0
\(431\) 4.17192e6 1.08179 0.540895 0.841090i \(-0.318085\pi\)
0.540895 + 0.841090i \(0.318085\pi\)
\(432\) 0 0
\(433\) 1.68030e6 0.430693 0.215347 0.976538i \(-0.430912\pi\)
0.215347 + 0.976538i \(0.430912\pi\)
\(434\) 0 0
\(435\) −1.04923e7 −2.65857
\(436\) 0 0
\(437\) 3.92284e6 0.982645
\(438\) 0 0
\(439\) 2.12899e6 0.527245 0.263623 0.964626i \(-0.415083\pi\)
0.263623 + 0.964626i \(0.415083\pi\)
\(440\) 0 0
\(441\) 1.07783e6 0.263908
\(442\) 0 0
\(443\) −5.46150e6 −1.32222 −0.661108 0.750291i \(-0.729913\pi\)
−0.661108 + 0.750291i \(0.729913\pi\)
\(444\) 0 0
\(445\) −9.38589e6 −2.24686
\(446\) 0 0
\(447\) 5.51767e6 1.30613
\(448\) 0 0
\(449\) 1.17382e6 0.274779 0.137390 0.990517i \(-0.456129\pi\)
0.137390 + 0.990517i \(0.456129\pi\)
\(450\) 0 0
\(451\) −1.38710e6 −0.321120
\(452\) 0 0
\(453\) 8.69786e6 1.99144
\(454\) 0 0
\(455\) −4.32739e6 −0.979936
\(456\) 0 0
\(457\) 3.88231e6 0.869561 0.434780 0.900537i \(-0.356826\pi\)
0.434780 + 0.900537i \(0.356826\pi\)
\(458\) 0 0
\(459\) −1.94889e6 −0.431773
\(460\) 0 0
\(461\) 1.13112e6 0.247889 0.123945 0.992289i \(-0.460445\pi\)
0.123945 + 0.992289i \(0.460445\pi\)
\(462\) 0 0
\(463\) 1.74204e6 0.377664 0.188832 0.982009i \(-0.439530\pi\)
0.188832 + 0.982009i \(0.439530\pi\)
\(464\) 0 0
\(465\) 1.14191e7 2.44905
\(466\) 0 0
\(467\) 4.96811e6 1.05414 0.527071 0.849821i \(-0.323290\pi\)
0.527071 + 0.849821i \(0.323290\pi\)
\(468\) 0 0
\(469\) −2.20368e6 −0.462612
\(470\) 0 0
\(471\) −1.34172e7 −2.78682
\(472\) 0 0
\(473\) −3.53161e6 −0.725805
\(474\) 0 0
\(475\) 1.15781e7 2.35453
\(476\) 0 0
\(477\) 6.56140e6 1.32038
\(478\) 0 0
\(479\) −1.72949e6 −0.344412 −0.172206 0.985061i \(-0.555090\pi\)
−0.172206 + 0.985061i \(0.555090\pi\)
\(480\) 0 0
\(481\) −8.09210e6 −1.59477
\(482\) 0 0
\(483\) 2.78844e6 0.543869
\(484\) 0 0
\(485\) 9.94403e6 1.91959
\(486\) 0 0
\(487\) 3.90261e6 0.745647 0.372824 0.927902i \(-0.378390\pi\)
0.372824 + 0.927902i \(0.378390\pi\)
\(488\) 0 0
\(489\) −3650.93 −0.000690448 0
\(490\) 0 0
\(491\) −8.62067e6 −1.61375 −0.806877 0.590720i \(-0.798844\pi\)
−0.806877 + 0.590720i \(0.798844\pi\)
\(492\) 0 0
\(493\) 1.47178e6 0.272725
\(494\) 0 0
\(495\) 1.77814e7 3.26177
\(496\) 0 0
\(497\) 279344. 0.0507280
\(498\) 0 0
\(499\) −7.01787e6 −1.26169 −0.630846 0.775908i \(-0.717292\pi\)
−0.630846 + 0.775908i \(0.717292\pi\)
\(500\) 0 0
\(501\) 1.38746e7 2.46960
\(502\) 0 0
\(503\) 3.89602e6 0.686596 0.343298 0.939226i \(-0.388456\pi\)
0.343298 + 0.939226i \(0.388456\pi\)
\(504\) 0 0
\(505\) 2.13149e6 0.371925
\(506\) 0 0
\(507\) −1.18055e7 −2.03969
\(508\) 0 0
\(509\) −1.54805e6 −0.264843 −0.132422 0.991193i \(-0.542275\pi\)
−0.132422 + 0.991193i \(0.542275\pi\)
\(510\) 0 0
\(511\) −2.33489e6 −0.395561
\(512\) 0 0
\(513\) −9.82097e6 −1.64764
\(514\) 0 0
\(515\) 1.32732e7 2.20525
\(516\) 0 0
\(517\) −5.76952e6 −0.949322
\(518\) 0 0
\(519\) −9.09113e6 −1.48149
\(520\) 0 0
\(521\) 7.11069e6 1.14767 0.573836 0.818970i \(-0.305455\pi\)
0.573836 + 0.818970i \(0.305455\pi\)
\(522\) 0 0
\(523\) 5.37329e6 0.858986 0.429493 0.903070i \(-0.358692\pi\)
0.429493 + 0.903070i \(0.358692\pi\)
\(524\) 0 0
\(525\) 8.23001e6 1.30317
\(526\) 0 0
\(527\) −1.60177e6 −0.251232
\(528\) 0 0
\(529\) −1.75593e6 −0.272815
\(530\) 0 0
\(531\) 9.60520e6 1.47833
\(532\) 0 0
\(533\) −3.09264e6 −0.471532
\(534\) 0 0
\(535\) 2.67155e6 0.403533
\(536\) 0 0
\(537\) −9.19401e6 −1.37584
\(538\) 0 0
\(539\) 975227. 0.144588
\(540\) 0 0
\(541\) −5.26079e6 −0.772784 −0.386392 0.922335i \(-0.626279\pi\)
−0.386392 + 0.922335i \(0.626279\pi\)
\(542\) 0 0
\(543\) −1.23438e7 −1.79660
\(544\) 0 0
\(545\) −8.63736e6 −1.24563
\(546\) 0 0
\(547\) 4.26833e6 0.609943 0.304972 0.952361i \(-0.401353\pi\)
0.304972 + 0.952361i \(0.401353\pi\)
\(548\) 0 0
\(549\) −2.42778e7 −3.43778
\(550\) 0 0
\(551\) 7.41667e6 1.04071
\(552\) 0 0
\(553\) 284651. 0.0395822
\(554\) 0 0
\(555\) 2.29218e7 3.15876
\(556\) 0 0
\(557\) 5.28071e6 0.721198 0.360599 0.932721i \(-0.382572\pi\)
0.360599 + 0.932721i \(0.382572\pi\)
\(558\) 0 0
\(559\) −7.87395e6 −1.06577
\(560\) 0 0
\(561\) −3.84440e6 −0.515729
\(562\) 0 0
\(563\) 8.98229e6 1.19431 0.597154 0.802127i \(-0.296298\pi\)
0.597154 + 0.802127i \(0.296298\pi\)
\(564\) 0 0
\(565\) 1.84778e6 0.243517
\(566\) 0 0
\(567\) −1.63584e6 −0.213689
\(568\) 0 0
\(569\) 1.06571e7 1.37993 0.689966 0.723841i \(-0.257625\pi\)
0.689966 + 0.723841i \(0.257625\pi\)
\(570\) 0 0
\(571\) −3.52812e6 −0.452849 −0.226425 0.974029i \(-0.572704\pi\)
−0.226425 + 0.974029i \(0.572704\pi\)
\(572\) 0 0
\(573\) 2.23933e7 2.84926
\(574\) 0 0
\(575\) 1.38141e7 1.74242
\(576\) 0 0
\(577\) 2.63312e6 0.329254 0.164627 0.986356i \(-0.447358\pi\)
0.164627 + 0.986356i \(0.447358\pi\)
\(578\) 0 0
\(579\) −3.23241e6 −0.400709
\(580\) 0 0
\(581\) −1.32585e6 −0.162950
\(582\) 0 0
\(583\) 5.93680e6 0.723405
\(584\) 0 0
\(585\) 3.96449e7 4.78958
\(586\) 0 0
\(587\) −1.04984e7 −1.25755 −0.628776 0.777586i \(-0.716444\pi\)
−0.628776 + 0.777586i \(0.716444\pi\)
\(588\) 0 0
\(589\) −8.07176e6 −0.958694
\(590\) 0 0
\(591\) 1.95925e7 2.30740
\(592\) 0 0
\(593\) 4.46953e6 0.521946 0.260973 0.965346i \(-0.415957\pi\)
0.260973 + 0.965346i \(0.415957\pi\)
\(594\) 0 0
\(595\) −1.71943e6 −0.199109
\(596\) 0 0
\(597\) 8.59455e6 0.986933
\(598\) 0 0
\(599\) −8.44304e6 −0.961461 −0.480731 0.876868i \(-0.659629\pi\)
−0.480731 + 0.876868i \(0.659629\pi\)
\(600\) 0 0
\(601\) −1.38045e7 −1.55896 −0.779481 0.626426i \(-0.784517\pi\)
−0.779481 + 0.626426i \(0.784517\pi\)
\(602\) 0 0
\(603\) 2.01888e7 2.26108
\(604\) 0 0
\(605\) 382992. 0.0425403
\(606\) 0 0
\(607\) 6.43837e6 0.709258 0.354629 0.935007i \(-0.384607\pi\)
0.354629 + 0.935007i \(0.384607\pi\)
\(608\) 0 0
\(609\) 5.27194e6 0.576007
\(610\) 0 0
\(611\) −1.28635e7 −1.39398
\(612\) 0 0
\(613\) 1.29974e6 0.139703 0.0698513 0.997557i \(-0.477748\pi\)
0.0698513 + 0.997557i \(0.477748\pi\)
\(614\) 0 0
\(615\) 8.76024e6 0.933960
\(616\) 0 0
\(617\) −5.51795e6 −0.583532 −0.291766 0.956490i \(-0.594243\pi\)
−0.291766 + 0.956490i \(0.594243\pi\)
\(618\) 0 0
\(619\) −9.36773e6 −0.982671 −0.491335 0.870971i \(-0.663491\pi\)
−0.491335 + 0.870971i \(0.663491\pi\)
\(620\) 0 0
\(621\) −1.17176e7 −1.21930
\(622\) 0 0
\(623\) 4.71601e6 0.486805
\(624\) 0 0
\(625\) 1.10522e7 1.13174
\(626\) 0 0
\(627\) −1.93730e7 −1.96801
\(628\) 0 0
\(629\) −3.21528e6 −0.324035
\(630\) 0 0
\(631\) −7.50281e6 −0.750154 −0.375077 0.926994i \(-0.622384\pi\)
−0.375077 + 0.926994i \(0.622384\pi\)
\(632\) 0 0
\(633\) −8.84927e6 −0.877806
\(634\) 0 0
\(635\) 1.83861e7 1.80948
\(636\) 0 0
\(637\) 2.17433e6 0.212313
\(638\) 0 0
\(639\) −2.55917e6 −0.247941
\(640\) 0 0
\(641\) 112701. 0.0108338 0.00541690 0.999985i \(-0.498276\pi\)
0.00541690 + 0.999985i \(0.498276\pi\)
\(642\) 0 0
\(643\) 453508. 0.0432572 0.0216286 0.999766i \(-0.493115\pi\)
0.0216286 + 0.999766i \(0.493115\pi\)
\(644\) 0 0
\(645\) 2.23039e7 2.11096
\(646\) 0 0
\(647\) −7.71032e6 −0.724122 −0.362061 0.932154i \(-0.617927\pi\)
−0.362061 + 0.932154i \(0.617927\pi\)
\(648\) 0 0
\(649\) 8.69087e6 0.809937
\(650\) 0 0
\(651\) −5.73760e6 −0.530613
\(652\) 0 0
\(653\) 5.98665e6 0.549416 0.274708 0.961528i \(-0.411419\pi\)
0.274708 + 0.961528i \(0.411419\pi\)
\(654\) 0 0
\(655\) −2.08851e7 −1.90210
\(656\) 0 0
\(657\) 2.13908e7 1.93336
\(658\) 0 0
\(659\) −1.44274e7 −1.29412 −0.647058 0.762441i \(-0.724001\pi\)
−0.647058 + 0.762441i \(0.724001\pi\)
\(660\) 0 0
\(661\) 1.80369e7 1.60568 0.802838 0.596197i \(-0.203322\pi\)
0.802838 + 0.596197i \(0.203322\pi\)
\(662\) 0 0
\(663\) −8.57133e6 −0.757294
\(664\) 0 0
\(665\) −8.66465e6 −0.759796
\(666\) 0 0
\(667\) 8.84897e6 0.770155
\(668\) 0 0
\(669\) 1.66732e7 1.44030
\(670\) 0 0
\(671\) −2.19667e7 −1.88347
\(672\) 0 0
\(673\) −1.58113e7 −1.34564 −0.672820 0.739806i \(-0.734917\pi\)
−0.672820 + 0.739806i \(0.734917\pi\)
\(674\) 0 0
\(675\) −3.45841e7 −2.92157
\(676\) 0 0
\(677\) 1.27877e7 1.07231 0.536154 0.844120i \(-0.319876\pi\)
0.536154 + 0.844120i \(0.319876\pi\)
\(678\) 0 0
\(679\) −4.99645e6 −0.415899
\(680\) 0 0
\(681\) 7.82306e6 0.646411
\(682\) 0 0
\(683\) −4.17491e6 −0.342448 −0.171224 0.985232i \(-0.554772\pi\)
−0.171224 + 0.985232i \(0.554772\pi\)
\(684\) 0 0
\(685\) −4.37906e6 −0.356578
\(686\) 0 0
\(687\) 1.56497e7 1.26507
\(688\) 0 0
\(689\) 1.32365e7 1.06224
\(690\) 0 0
\(691\) −8.31609e6 −0.662558 −0.331279 0.943533i \(-0.607480\pi\)
−0.331279 + 0.943533i \(0.607480\pi\)
\(692\) 0 0
\(693\) −8.93442e6 −0.706697
\(694\) 0 0
\(695\) 2.56405e7 2.01356
\(696\) 0 0
\(697\) −1.22882e6 −0.0958086
\(698\) 0 0
\(699\) −3.96914e6 −0.307258
\(700\) 0 0
\(701\) 1.67718e6 0.128909 0.0644546 0.997921i \(-0.479469\pi\)
0.0644546 + 0.997921i \(0.479469\pi\)
\(702\) 0 0
\(703\) −1.62027e7 −1.23651
\(704\) 0 0
\(705\) 3.64374e7 2.76105
\(706\) 0 0
\(707\) −1.07099e6 −0.0805814
\(708\) 0 0
\(709\) −1.66093e7 −1.24090 −0.620450 0.784246i \(-0.713050\pi\)
−0.620450 + 0.784246i \(0.713050\pi\)
\(710\) 0 0
\(711\) −2.60779e6 −0.193464
\(712\) 0 0
\(713\) −9.63057e6 −0.709460
\(714\) 0 0
\(715\) 3.58710e7 2.62409
\(716\) 0 0
\(717\) −2.17404e7 −1.57932
\(718\) 0 0
\(719\) 2.32802e7 1.67944 0.839722 0.543017i \(-0.182718\pi\)
0.839722 + 0.543017i \(0.182718\pi\)
\(720\) 0 0
\(721\) −6.66921e6 −0.477789
\(722\) 0 0
\(723\) −3.59769e7 −2.55963
\(724\) 0 0
\(725\) 2.61174e7 1.84538
\(726\) 0 0
\(727\) 1.74527e7 1.22469 0.612345 0.790591i \(-0.290226\pi\)
0.612345 + 0.790591i \(0.290226\pi\)
\(728\) 0 0
\(729\) −1.96334e7 −1.36829
\(730\) 0 0
\(731\) −3.12860e6 −0.216549
\(732\) 0 0
\(733\) −940675. −0.0646666 −0.0323333 0.999477i \(-0.510294\pi\)
−0.0323333 + 0.999477i \(0.510294\pi\)
\(734\) 0 0
\(735\) −6.15904e6 −0.420528
\(736\) 0 0
\(737\) 1.82670e7 1.23879
\(738\) 0 0
\(739\) 1.13900e7 0.767204 0.383602 0.923498i \(-0.374683\pi\)
0.383602 + 0.923498i \(0.374683\pi\)
\(740\) 0 0
\(741\) −4.31932e7 −2.88982
\(742\) 0 0
\(743\) −262087. −0.0174170 −0.00870851 0.999962i \(-0.502772\pi\)
−0.00870851 + 0.999962i \(0.502772\pi\)
\(744\) 0 0
\(745\) −2.04563e7 −1.35032
\(746\) 0 0
\(747\) 1.21466e7 0.796443
\(748\) 0 0
\(749\) −1.34234e6 −0.0874296
\(750\) 0 0
\(751\) −1.10676e6 −0.0716069 −0.0358035 0.999359i \(-0.511399\pi\)
−0.0358035 + 0.999359i \(0.511399\pi\)
\(752\) 0 0
\(753\) −4.56601e7 −2.93460
\(754\) 0 0
\(755\) −3.22467e7 −2.05882
\(756\) 0 0
\(757\) −3.07218e7 −1.94853 −0.974264 0.225412i \(-0.927627\pi\)
−0.974264 + 0.225412i \(0.927627\pi\)
\(758\) 0 0
\(759\) −2.31142e7 −1.45638
\(760\) 0 0
\(761\) 1.18635e7 0.742593 0.371297 0.928514i \(-0.378913\pi\)
0.371297 + 0.928514i \(0.378913\pi\)
\(762\) 0 0
\(763\) 4.33991e6 0.269879
\(764\) 0 0
\(765\) 1.57523e7 0.973175
\(766\) 0 0
\(767\) 1.93768e7 1.18931
\(768\) 0 0
\(769\) 2.08006e7 1.26841 0.634204 0.773166i \(-0.281328\pi\)
0.634204 + 0.773166i \(0.281328\pi\)
\(770\) 0 0
\(771\) 5.38378e7 3.26175
\(772\) 0 0
\(773\) −1.12633e7 −0.677980 −0.338990 0.940790i \(-0.610085\pi\)
−0.338990 + 0.940790i \(0.610085\pi\)
\(774\) 0 0
\(775\) −2.84243e7 −1.69995
\(776\) 0 0
\(777\) −1.15172e7 −0.684377
\(778\) 0 0
\(779\) −6.19233e6 −0.365603
\(780\) 0 0
\(781\) −2.31556e6 −0.135840
\(782\) 0 0
\(783\) −2.21537e7 −1.29135
\(784\) 0 0
\(785\) 4.97433e7 2.88111
\(786\) 0 0
\(787\) 2.65941e7 1.53056 0.765278 0.643700i \(-0.222601\pi\)
0.765278 + 0.643700i \(0.222601\pi\)
\(788\) 0 0
\(789\) −1.06726e7 −0.610346
\(790\) 0 0
\(791\) −928431. −0.0527604
\(792\) 0 0
\(793\) −4.89763e7 −2.76568
\(794\) 0 0
\(795\) −3.74938e7 −2.10398
\(796\) 0 0
\(797\) −1.25159e7 −0.697938 −0.348969 0.937134i \(-0.613468\pi\)
−0.348969 + 0.937134i \(0.613468\pi\)
\(798\) 0 0
\(799\) −5.11114e6 −0.283237
\(800\) 0 0
\(801\) −4.32052e7 −2.37933
\(802\) 0 0
\(803\) 1.93546e7 1.05924
\(804\) 0 0
\(805\) −1.03380e7 −0.562270
\(806\) 0 0
\(807\) −3.89399e7 −2.10480
\(808\) 0 0
\(809\) 1.02591e7 0.551109 0.275555 0.961285i \(-0.411139\pi\)
0.275555 + 0.961285i \(0.411139\pi\)
\(810\) 0 0
\(811\) 1.32031e7 0.704892 0.352446 0.935832i \(-0.385350\pi\)
0.352446 + 0.935832i \(0.385350\pi\)
\(812\) 0 0
\(813\) 1.20335e7 0.638508
\(814\) 0 0
\(815\) 13535.5 0.000713809 0
\(816\) 0 0
\(817\) −1.57659e7 −0.826348
\(818\) 0 0
\(819\) −1.99199e7 −1.03771
\(820\) 0 0
\(821\) −2.88139e7 −1.49192 −0.745958 0.665993i \(-0.768008\pi\)
−0.745958 + 0.665993i \(0.768008\pi\)
\(822\) 0 0
\(823\) −2.72097e7 −1.40031 −0.700156 0.713990i \(-0.746886\pi\)
−0.700156 + 0.713990i \(0.746886\pi\)
\(824\) 0 0
\(825\) −6.82209e7 −3.48966
\(826\) 0 0
\(827\) 3.36307e7 1.70991 0.854954 0.518705i \(-0.173586\pi\)
0.854954 + 0.518705i \(0.173586\pi\)
\(828\) 0 0
\(829\) 3.30835e7 1.67196 0.835978 0.548763i \(-0.184901\pi\)
0.835978 + 0.548763i \(0.184901\pi\)
\(830\) 0 0
\(831\) 3.83594e7 1.92695
\(832\) 0 0
\(833\) 863939. 0.0431391
\(834\) 0 0
\(835\) −5.14392e7 −2.55316
\(836\) 0 0
\(837\) 2.41105e7 1.18958
\(838\) 0 0
\(839\) −1.85094e7 −0.907796 −0.453898 0.891054i \(-0.649967\pi\)
−0.453898 + 0.891054i \(0.649967\pi\)
\(840\) 0 0
\(841\) −3.78094e6 −0.184336
\(842\) 0 0
\(843\) 5.00230e7 2.42438
\(844\) 0 0
\(845\) 4.37679e7 2.10870
\(846\) 0 0
\(847\) −192437. −0.00921680
\(848\) 0 0
\(849\) −2.66164e7 −1.26730
\(850\) 0 0
\(851\) −1.93317e7 −0.915052
\(852\) 0 0
\(853\) −1.52810e7 −0.719086 −0.359543 0.933129i \(-0.617067\pi\)
−0.359543 + 0.933129i \(0.617067\pi\)
\(854\) 0 0
\(855\) 7.93802e7 3.71361
\(856\) 0 0
\(857\) 5.10629e6 0.237494 0.118747 0.992925i \(-0.462112\pi\)
0.118747 + 0.992925i \(0.462112\pi\)
\(858\) 0 0
\(859\) −2.32935e7 −1.07709 −0.538546 0.842596i \(-0.681026\pi\)
−0.538546 + 0.842596i \(0.681026\pi\)
\(860\) 0 0
\(861\) −4.40165e6 −0.202352
\(862\) 0 0
\(863\) −2.42566e7 −1.10867 −0.554337 0.832292i \(-0.687028\pi\)
−0.554337 + 0.832292i \(0.687028\pi\)
\(864\) 0 0
\(865\) 3.37047e7 1.53162
\(866\) 0 0
\(867\) 3.39424e7 1.53354
\(868\) 0 0
\(869\) −2.35955e6 −0.105994
\(870\) 0 0
\(871\) 4.07274e7 1.81903
\(872\) 0 0
\(873\) 4.57744e7 2.03276
\(874\) 0 0
\(875\) −1.55793e7 −0.687904
\(876\) 0 0
\(877\) 2.81339e7 1.23518 0.617592 0.786499i \(-0.288108\pi\)
0.617592 + 0.786499i \(0.288108\pi\)
\(878\) 0 0
\(879\) 3.44361e7 1.50328
\(880\) 0 0
\(881\) −8.51907e6 −0.369788 −0.184894 0.982759i \(-0.559194\pi\)
−0.184894 + 0.982759i \(0.559194\pi\)
\(882\) 0 0
\(883\) −1.36512e7 −0.589210 −0.294605 0.955619i \(-0.595188\pi\)
−0.294605 + 0.955619i \(0.595188\pi\)
\(884\) 0 0
\(885\) −5.48871e7 −2.35566
\(886\) 0 0
\(887\) −1.02750e7 −0.438502 −0.219251 0.975668i \(-0.570361\pi\)
−0.219251 + 0.975668i \(0.570361\pi\)
\(888\) 0 0
\(889\) −9.23823e6 −0.392043
\(890\) 0 0
\(891\) 1.35599e7 0.572220
\(892\) 0 0
\(893\) −2.57564e7 −1.08083
\(894\) 0 0
\(895\) 3.40861e7 1.42239
\(896\) 0 0
\(897\) −5.15346e7 −2.13854
\(898\) 0 0
\(899\) −1.82079e7 −0.751383
\(900\) 0 0
\(901\) 5.25933e6 0.215833
\(902\) 0 0
\(903\) −1.12067e7 −0.457362
\(904\) 0 0
\(905\) 4.57639e7 1.85738
\(906\) 0 0
\(907\) 3.08516e7 1.24526 0.622630 0.782516i \(-0.286064\pi\)
0.622630 + 0.782516i \(0.286064\pi\)
\(908\) 0 0
\(909\) 9.81170e6 0.393853
\(910\) 0 0
\(911\) −1.48424e7 −0.592525 −0.296263 0.955107i \(-0.595740\pi\)
−0.296263 + 0.955107i \(0.595740\pi\)
\(912\) 0 0
\(913\) 1.09904e7 0.436351
\(914\) 0 0
\(915\) 1.38731e8 5.47798
\(916\) 0 0
\(917\) 1.04939e7 0.412110
\(918\) 0 0
\(919\) 3.14065e7 1.22668 0.613340 0.789819i \(-0.289826\pi\)
0.613340 + 0.789819i \(0.289826\pi\)
\(920\) 0 0
\(921\) 4.83221e7 1.87714
\(922\) 0 0
\(923\) −5.16269e6 −0.199467
\(924\) 0 0
\(925\) −5.70569e7 −2.19257
\(926\) 0 0
\(927\) 6.10991e7 2.33526
\(928\) 0 0
\(929\) −7.75661e6 −0.294871 −0.147436 0.989072i \(-0.547102\pi\)
−0.147436 + 0.989072i \(0.547102\pi\)
\(930\) 0 0
\(931\) 4.35362e6 0.164618
\(932\) 0 0
\(933\) 3.73396e7 1.40432
\(934\) 0 0
\(935\) 1.42528e7 0.533178
\(936\) 0 0
\(937\) 2.51058e7 0.934167 0.467084 0.884213i \(-0.345305\pi\)
0.467084 + 0.884213i \(0.345305\pi\)
\(938\) 0 0
\(939\) −7.03007e7 −2.60193
\(940\) 0 0
\(941\) 4.07777e7 1.50123 0.750617 0.660738i \(-0.229756\pi\)
0.750617 + 0.660738i \(0.229756\pi\)
\(942\) 0 0
\(943\) −7.38818e6 −0.270557
\(944\) 0 0
\(945\) 2.58815e7 0.942778
\(946\) 0 0
\(947\) 3.11789e7 1.12976 0.564879 0.825174i \(-0.308923\pi\)
0.564879 + 0.825174i \(0.308923\pi\)
\(948\) 0 0
\(949\) 4.31522e7 1.55538
\(950\) 0 0
\(951\) 896086. 0.0321291
\(952\) 0 0
\(953\) −2.49371e7 −0.889433 −0.444716 0.895671i \(-0.646696\pi\)
−0.444716 + 0.895671i \(0.646696\pi\)
\(954\) 0 0
\(955\) −8.30217e7 −2.94566
\(956\) 0 0
\(957\) −4.37007e7 −1.54244
\(958\) 0 0
\(959\) 2.20029e6 0.0772562
\(960\) 0 0
\(961\) −8.81296e6 −0.307832
\(962\) 0 0
\(963\) 1.22977e7 0.427325
\(964\) 0 0
\(965\) 1.19839e7 0.414267
\(966\) 0 0
\(967\) 9.62902e6 0.331143 0.165572 0.986198i \(-0.447053\pi\)
0.165572 + 0.986198i \(0.447053\pi\)
\(968\) 0 0
\(969\) −1.71622e7 −0.587170
\(970\) 0 0
\(971\) −5.76987e7 −1.96389 −0.981946 0.189161i \(-0.939423\pi\)
−0.981946 + 0.189161i \(0.939423\pi\)
\(972\) 0 0
\(973\) −1.28833e7 −0.436259
\(974\) 0 0
\(975\) −1.52103e8 −5.12420
\(976\) 0 0
\(977\) 3.58334e7 1.20102 0.600512 0.799616i \(-0.294963\pi\)
0.600512 + 0.799616i \(0.294963\pi\)
\(978\) 0 0
\(979\) −3.90924e7 −1.30357
\(980\) 0 0
\(981\) −3.97595e7 −1.31907
\(982\) 0 0
\(983\) −5.26717e7 −1.73857 −0.869287 0.494307i \(-0.835422\pi\)
−0.869287 + 0.494307i \(0.835422\pi\)
\(984\) 0 0
\(985\) −7.26380e7 −2.38547
\(986\) 0 0
\(987\) −1.83082e7 −0.598210
\(988\) 0 0
\(989\) −1.88105e7 −0.611520
\(990\) 0 0
\(991\) −1.36827e7 −0.442577 −0.221288 0.975208i \(-0.571026\pi\)
−0.221288 + 0.975208i \(0.571026\pi\)
\(992\) 0 0
\(993\) −4.07846e7 −1.31257
\(994\) 0 0
\(995\) −3.18637e7 −1.02032
\(996\) 0 0
\(997\) −3.43844e7 −1.09553 −0.547763 0.836633i \(-0.684521\pi\)
−0.547763 + 0.836633i \(0.684521\pi\)
\(998\) 0 0
\(999\) 4.83976e7 1.53430
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 448.6.a.q.1.1 2
4.3 odd 2 448.6.a.z.1.2 2
8.3 odd 2 56.6.a.c.1.1 2
8.5 even 2 112.6.a.k.1.2 2
24.5 odd 2 1008.6.a.bt.1.2 2
24.11 even 2 504.6.a.s.1.2 2
56.3 even 6 392.6.i.g.177.1 4
56.11 odd 6 392.6.i.l.177.2 4
56.13 odd 2 784.6.a.p.1.1 2
56.19 even 6 392.6.i.g.361.1 4
56.27 even 2 392.6.a.f.1.2 2
56.51 odd 6 392.6.i.l.361.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.a.c.1.1 2 8.3 odd 2
112.6.a.k.1.2 2 8.5 even 2
392.6.a.f.1.2 2 56.27 even 2
392.6.i.g.177.1 4 56.3 even 6
392.6.i.g.361.1 4 56.19 even 6
392.6.i.l.177.2 4 56.11 odd 6
392.6.i.l.361.2 4 56.51 odd 6
448.6.a.q.1.1 2 1.1 even 1 trivial
448.6.a.z.1.2 2 4.3 odd 2
504.6.a.s.1.2 2 24.11 even 2
784.6.a.p.1.1 2 56.13 odd 2
1008.6.a.bt.1.2 2 24.5 odd 2