Properties

Label 588.6.a.n.1.2
Level $588$
Weight $6$
Character 588.1
Self dual yes
Analytic conductor $94.306$
Analytic rank $0$
Dimension $4$
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] = [588,6,Mod(1,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, 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("588.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 588.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: \(94.3056860500\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 699x^{2} - 686x + 59664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2}\cdot 7 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(26.2941\) of defining polynomial
Character \(\chi\) \(=\) 588.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.00000 q^{3} -31.9616 q^{5} +81.0000 q^{9} +O(q^{10})\) \(q-9.00000 q^{3} -31.9616 q^{5} +81.0000 q^{9} +260.884 q^{11} +769.735 q^{13} +287.654 q^{15} -1553.32 q^{17} +750.049 q^{19} +754.854 q^{23} -2103.46 q^{25} -729.000 q^{27} +6008.93 q^{29} -6420.04 q^{31} -2347.95 q^{33} -4775.73 q^{37} -6927.62 q^{39} -5423.27 q^{41} -11896.4 q^{43} -2588.89 q^{45} -17428.0 q^{47} +13979.9 q^{51} +37650.6 q^{53} -8338.26 q^{55} -6750.44 q^{57} +22078.1 q^{59} -8173.38 q^{61} -24602.0 q^{65} +13001.7 q^{67} -6793.69 q^{69} -12349.6 q^{71} +43600.3 q^{73} +18931.1 q^{75} +76749.5 q^{79} +6561.00 q^{81} +21893.6 q^{83} +49646.5 q^{85} -54080.4 q^{87} -136967. q^{89} +57780.4 q^{93} -23972.8 q^{95} -93050.1 q^{97} +21131.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{3} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 36 q^{3} + 324 q^{9} + 462 q^{11} - 602 q^{13} - 228 q^{17} - 358 q^{19} + 2148 q^{23} + 5454 q^{25} - 2916 q^{27} - 5532 q^{29} - 830 q^{31} - 4158 q^{33} + 3914 q^{37} + 5418 q^{39} - 8316 q^{41} - 14518 q^{43} - 41700 q^{47} + 2052 q^{51} - 22164 q^{53} + 3892 q^{55} + 3222 q^{57} - 32886 q^{59} - 83732 q^{61} + 93192 q^{65} + 80034 q^{67} - 19332 q^{69} + 44772 q^{71} + 22470 q^{73} - 49086 q^{75} + 75286 q^{79} + 26244 q^{81} - 17418 q^{83} + 139252 q^{85} + 49788 q^{87} - 28944 q^{89} + 7470 q^{93} + 144120 q^{95} - 216678 q^{97} + 37422 q^{99}+O(q^{100}) \) 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
\(3\) −9.00000 −0.577350
\(4\) 0 0
\(5\) −31.9616 −0.571746 −0.285873 0.958267i \(-0.592284\pi\)
−0.285873 + 0.958267i \(0.592284\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 260.884 0.650077 0.325039 0.945701i \(-0.394623\pi\)
0.325039 + 0.945701i \(0.394623\pi\)
\(12\) 0 0
\(13\) 769.735 1.26323 0.631616 0.775281i \(-0.282392\pi\)
0.631616 + 0.775281i \(0.282392\pi\)
\(14\) 0 0
\(15\) 287.654 0.330098
\(16\) 0 0
\(17\) −1553.32 −1.30358 −0.651791 0.758399i \(-0.725982\pi\)
−0.651791 + 0.758399i \(0.725982\pi\)
\(18\) 0 0
\(19\) 750.049 0.476656 0.238328 0.971185i \(-0.423401\pi\)
0.238328 + 0.971185i \(0.423401\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 754.854 0.297539 0.148769 0.988872i \(-0.452469\pi\)
0.148769 + 0.988872i \(0.452469\pi\)
\(24\) 0 0
\(25\) −2103.46 −0.673106
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) 6008.93 1.32679 0.663395 0.748269i \(-0.269115\pi\)
0.663395 + 0.748269i \(0.269115\pi\)
\(30\) 0 0
\(31\) −6420.04 −1.19987 −0.599934 0.800049i \(-0.704807\pi\)
−0.599934 + 0.800049i \(0.704807\pi\)
\(32\) 0 0
\(33\) −2347.95 −0.375322
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4775.73 −0.573502 −0.286751 0.958005i \(-0.592575\pi\)
−0.286751 + 0.958005i \(0.592575\pi\)
\(38\) 0 0
\(39\) −6927.62 −0.729327
\(40\) 0 0
\(41\) −5423.27 −0.503850 −0.251925 0.967747i \(-0.581064\pi\)
−0.251925 + 0.967747i \(0.581064\pi\)
\(42\) 0 0
\(43\) −11896.4 −0.981171 −0.490585 0.871393i \(-0.663217\pi\)
−0.490585 + 0.871393i \(0.663217\pi\)
\(44\) 0 0
\(45\) −2588.89 −0.190582
\(46\) 0 0
\(47\) −17428.0 −1.15081 −0.575404 0.817869i \(-0.695155\pi\)
−0.575404 + 0.817869i \(0.695155\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 13979.9 0.752623
\(52\) 0 0
\(53\) 37650.6 1.84112 0.920561 0.390599i \(-0.127732\pi\)
0.920561 + 0.390599i \(0.127732\pi\)
\(54\) 0 0
\(55\) −8338.26 −0.371679
\(56\) 0 0
\(57\) −6750.44 −0.275198
\(58\) 0 0
\(59\) 22078.1 0.825717 0.412859 0.910795i \(-0.364530\pi\)
0.412859 + 0.910795i \(0.364530\pi\)
\(60\) 0 0
\(61\) −8173.38 −0.281240 −0.140620 0.990064i \(-0.544910\pi\)
−0.140620 + 0.990064i \(0.544910\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −24602.0 −0.722248
\(66\) 0 0
\(67\) 13001.7 0.353846 0.176923 0.984225i \(-0.443386\pi\)
0.176923 + 0.984225i \(0.443386\pi\)
\(68\) 0 0
\(69\) −6793.69 −0.171784
\(70\) 0 0
\(71\) −12349.6 −0.290742 −0.145371 0.989377i \(-0.546438\pi\)
−0.145371 + 0.989377i \(0.546438\pi\)
\(72\) 0 0
\(73\) 43600.3 0.957596 0.478798 0.877925i \(-0.341073\pi\)
0.478798 + 0.877925i \(0.341073\pi\)
\(74\) 0 0
\(75\) 18931.1 0.388618
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 76749.5 1.38359 0.691796 0.722093i \(-0.256820\pi\)
0.691796 + 0.722093i \(0.256820\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) 21893.6 0.348836 0.174418 0.984672i \(-0.444196\pi\)
0.174418 + 0.984672i \(0.444196\pi\)
\(84\) 0 0
\(85\) 49646.5 0.745318
\(86\) 0 0
\(87\) −54080.4 −0.766023
\(88\) 0 0
\(89\) −136967. −1.83291 −0.916454 0.400139i \(-0.868962\pi\)
−0.916454 + 0.400139i \(0.868962\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 57780.4 0.692744
\(94\) 0 0
\(95\) −23972.8 −0.272527
\(96\) 0 0
\(97\) −93050.1 −1.00412 −0.502062 0.864832i \(-0.667425\pi\)
−0.502062 + 0.864832i \(0.667425\pi\)
\(98\) 0 0
\(99\) 21131.6 0.216692
\(100\) 0 0
\(101\) 122302. 1.19297 0.596484 0.802625i \(-0.296564\pi\)
0.596484 + 0.802625i \(0.296564\pi\)
\(102\) 0 0
\(103\) 74801.3 0.694730 0.347365 0.937730i \(-0.387076\pi\)
0.347365 + 0.937730i \(0.387076\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 70772.4 0.597592 0.298796 0.954317i \(-0.403415\pi\)
0.298796 + 0.954317i \(0.403415\pi\)
\(108\) 0 0
\(109\) −142262. −1.14689 −0.573446 0.819243i \(-0.694394\pi\)
−0.573446 + 0.819243i \(0.694394\pi\)
\(110\) 0 0
\(111\) 42981.5 0.331112
\(112\) 0 0
\(113\) 198899. 1.46533 0.732667 0.680588i \(-0.238275\pi\)
0.732667 + 0.680588i \(0.238275\pi\)
\(114\) 0 0
\(115\) −24126.3 −0.170117
\(116\) 0 0
\(117\) 62348.6 0.421077
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −92990.7 −0.577399
\(122\) 0 0
\(123\) 48809.4 0.290898
\(124\) 0 0
\(125\) 167110. 0.956592
\(126\) 0 0
\(127\) 219619. 1.20826 0.604131 0.796885i \(-0.293521\pi\)
0.604131 + 0.796885i \(0.293521\pi\)
\(128\) 0 0
\(129\) 107068. 0.566479
\(130\) 0 0
\(131\) 54622.1 0.278093 0.139047 0.990286i \(-0.455596\pi\)
0.139047 + 0.990286i \(0.455596\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 23300.0 0.110033
\(136\) 0 0
\(137\) 72942.8 0.332033 0.166017 0.986123i \(-0.446909\pi\)
0.166017 + 0.986123i \(0.446909\pi\)
\(138\) 0 0
\(139\) −110668. −0.485831 −0.242915 0.970047i \(-0.578104\pi\)
−0.242915 + 0.970047i \(0.578104\pi\)
\(140\) 0 0
\(141\) 156852. 0.664419
\(142\) 0 0
\(143\) 200811. 0.821199
\(144\) 0 0
\(145\) −192055. −0.758587
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 327025. 1.20675 0.603373 0.797459i \(-0.293823\pi\)
0.603373 + 0.797459i \(0.293823\pi\)
\(150\) 0 0
\(151\) 174490. 0.622771 0.311386 0.950284i \(-0.399207\pi\)
0.311386 + 0.950284i \(0.399207\pi\)
\(152\) 0 0
\(153\) −125819. −0.434527
\(154\) 0 0
\(155\) 205195. 0.686020
\(156\) 0 0
\(157\) 440292. 1.42558 0.712790 0.701377i \(-0.247431\pi\)
0.712790 + 0.701377i \(0.247431\pi\)
\(158\) 0 0
\(159\) −338856. −1.06297
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 56491.9 0.166540 0.0832698 0.996527i \(-0.473464\pi\)
0.0832698 + 0.996527i \(0.473464\pi\)
\(164\) 0 0
\(165\) 75044.3 0.214589
\(166\) 0 0
\(167\) −688014. −1.90900 −0.954501 0.298209i \(-0.903611\pi\)
−0.954501 + 0.298209i \(0.903611\pi\)
\(168\) 0 0
\(169\) 221200. 0.595755
\(170\) 0 0
\(171\) 60754.0 0.158885
\(172\) 0 0
\(173\) 282558. 0.717782 0.358891 0.933379i \(-0.383155\pi\)
0.358891 + 0.933379i \(0.383155\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −198703. −0.476728
\(178\) 0 0
\(179\) 151782. 0.354069 0.177034 0.984205i \(-0.443350\pi\)
0.177034 + 0.984205i \(0.443350\pi\)
\(180\) 0 0
\(181\) −322258. −0.731151 −0.365575 0.930782i \(-0.619128\pi\)
−0.365575 + 0.930782i \(0.619128\pi\)
\(182\) 0 0
\(183\) 73560.4 0.162374
\(184\) 0 0
\(185\) 152640. 0.327898
\(186\) 0 0
\(187\) −405235. −0.847429
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 538521. 1.06812 0.534059 0.845447i \(-0.320666\pi\)
0.534059 + 0.845447i \(0.320666\pi\)
\(192\) 0 0
\(193\) 303440. 0.586380 0.293190 0.956054i \(-0.405283\pi\)
0.293190 + 0.956054i \(0.405283\pi\)
\(194\) 0 0
\(195\) 221418. 0.416990
\(196\) 0 0
\(197\) 656475. 1.20518 0.602591 0.798050i \(-0.294135\pi\)
0.602591 + 0.798050i \(0.294135\pi\)
\(198\) 0 0
\(199\) 842204. 1.50759 0.753797 0.657107i \(-0.228220\pi\)
0.753797 + 0.657107i \(0.228220\pi\)
\(200\) 0 0
\(201\) −117016. −0.204293
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 173336. 0.288074
\(206\) 0 0
\(207\) 61143.2 0.0991796
\(208\) 0 0
\(209\) 195675. 0.309864
\(210\) 0 0
\(211\) 1.01133e6 1.56382 0.781908 0.623394i \(-0.214247\pi\)
0.781908 + 0.623394i \(0.214247\pi\)
\(212\) 0 0
\(213\) 111147. 0.167860
\(214\) 0 0
\(215\) 380228. 0.560981
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −392403. −0.552869
\(220\) 0 0
\(221\) −1.19564e6 −1.64673
\(222\) 0 0
\(223\) 957124. 1.28886 0.644431 0.764663i \(-0.277094\pi\)
0.644431 + 0.764663i \(0.277094\pi\)
\(224\) 0 0
\(225\) −170380. −0.224369
\(226\) 0 0
\(227\) 471480. 0.607293 0.303647 0.952785i \(-0.401796\pi\)
0.303647 + 0.952785i \(0.401796\pi\)
\(228\) 0 0
\(229\) −268196. −0.337959 −0.168980 0.985620i \(-0.554047\pi\)
−0.168980 + 0.985620i \(0.554047\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.54722e6 1.86708 0.933538 0.358478i \(-0.116704\pi\)
0.933538 + 0.358478i \(0.116704\pi\)
\(234\) 0 0
\(235\) 557027. 0.657970
\(236\) 0 0
\(237\) −690745. −0.798817
\(238\) 0 0
\(239\) −937542. −1.06169 −0.530843 0.847470i \(-0.678125\pi\)
−0.530843 + 0.847470i \(0.678125\pi\)
\(240\) 0 0
\(241\) −1.28486e6 −1.42499 −0.712497 0.701675i \(-0.752436\pi\)
−0.712497 + 0.701675i \(0.752436\pi\)
\(242\) 0 0
\(243\) −59049.0 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 577339. 0.602128
\(248\) 0 0
\(249\) −197042. −0.201401
\(250\) 0 0
\(251\) 709769. 0.711103 0.355552 0.934657i \(-0.384293\pi\)
0.355552 + 0.934657i \(0.384293\pi\)
\(252\) 0 0
\(253\) 196929. 0.193423
\(254\) 0 0
\(255\) −446819. −0.430309
\(256\) 0 0
\(257\) −1.82016e6 −1.71900 −0.859502 0.511132i \(-0.829226\pi\)
−0.859502 + 0.511132i \(0.829226\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 486723. 0.442263
\(262\) 0 0
\(263\) 359187. 0.320208 0.160104 0.987100i \(-0.448817\pi\)
0.160104 + 0.987100i \(0.448817\pi\)
\(264\) 0 0
\(265\) −1.20337e6 −1.05265
\(266\) 0 0
\(267\) 1.23270e6 1.05823
\(268\) 0 0
\(269\) 1.15767e6 0.975446 0.487723 0.872998i \(-0.337828\pi\)
0.487723 + 0.872998i \(0.337828\pi\)
\(270\) 0 0
\(271\) −780175. −0.645311 −0.322655 0.946516i \(-0.604576\pi\)
−0.322655 + 0.946516i \(0.604576\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −548757. −0.437571
\(276\) 0 0
\(277\) −326331. −0.255540 −0.127770 0.991804i \(-0.540782\pi\)
−0.127770 + 0.991804i \(0.540782\pi\)
\(278\) 0 0
\(279\) −520023. −0.399956
\(280\) 0 0
\(281\) 364094. 0.275073 0.137536 0.990497i \(-0.456082\pi\)
0.137536 + 0.990497i \(0.456082\pi\)
\(282\) 0 0
\(283\) −64019.2 −0.0475165 −0.0237582 0.999718i \(-0.507563\pi\)
−0.0237582 + 0.999718i \(0.507563\pi\)
\(284\) 0 0
\(285\) 215755. 0.157343
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 992939. 0.699324
\(290\) 0 0
\(291\) 837451. 0.579731
\(292\) 0 0
\(293\) 398020. 0.270855 0.135427 0.990787i \(-0.456759\pi\)
0.135427 + 0.990787i \(0.456759\pi\)
\(294\) 0 0
\(295\) −705651. −0.472101
\(296\) 0 0
\(297\) −190184. −0.125107
\(298\) 0 0
\(299\) 581038. 0.375860
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −1.10071e6 −0.688760
\(304\) 0 0
\(305\) 261234. 0.160798
\(306\) 0 0
\(307\) −638841. −0.386854 −0.193427 0.981115i \(-0.561960\pi\)
−0.193427 + 0.981115i \(0.561960\pi\)
\(308\) 0 0
\(309\) −673212. −0.401103
\(310\) 0 0
\(311\) 2.48188e6 1.45506 0.727529 0.686077i \(-0.240669\pi\)
0.727529 + 0.686077i \(0.240669\pi\)
\(312\) 0 0
\(313\) 1.47653e6 0.851889 0.425944 0.904749i \(-0.359942\pi\)
0.425944 + 0.904749i \(0.359942\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.23126e6 −1.24710 −0.623551 0.781783i \(-0.714310\pi\)
−0.623551 + 0.781783i \(0.714310\pi\)
\(318\) 0 0
\(319\) 1.56763e6 0.862516
\(320\) 0 0
\(321\) −636952. −0.345020
\(322\) 0 0
\(323\) −1.16506e6 −0.621360
\(324\) 0 0
\(325\) −1.61910e6 −0.850289
\(326\) 0 0
\(327\) 1.28036e6 0.662159
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.41122e6 −1.20967 −0.604836 0.796350i \(-0.706761\pi\)
−0.604836 + 0.796350i \(0.706761\pi\)
\(332\) 0 0
\(333\) −386834. −0.191167
\(334\) 0 0
\(335\) −415556. −0.202310
\(336\) 0 0
\(337\) −1.79586e6 −0.861388 −0.430694 0.902498i \(-0.641731\pi\)
−0.430694 + 0.902498i \(0.641731\pi\)
\(338\) 0 0
\(339\) −1.79009e6 −0.846011
\(340\) 0 0
\(341\) −1.67488e6 −0.780007
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 217137. 0.0982169
\(346\) 0 0
\(347\) 2.20731e6 0.984103 0.492051 0.870566i \(-0.336247\pi\)
0.492051 + 0.870566i \(0.336247\pi\)
\(348\) 0 0
\(349\) 148188. 0.0651252 0.0325626 0.999470i \(-0.489633\pi\)
0.0325626 + 0.999470i \(0.489633\pi\)
\(350\) 0 0
\(351\) −561137. −0.243109
\(352\) 0 0
\(353\) 1.97040e6 0.841623 0.420811 0.907148i \(-0.361745\pi\)
0.420811 + 0.907148i \(0.361745\pi\)
\(354\) 0 0
\(355\) 394714. 0.166231
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.43395e6 0.996723 0.498362 0.866969i \(-0.333935\pi\)
0.498362 + 0.866969i \(0.333935\pi\)
\(360\) 0 0
\(361\) −1.91353e6 −0.772799
\(362\) 0 0
\(363\) 836917. 0.333362
\(364\) 0 0
\(365\) −1.39354e6 −0.547502
\(366\) 0 0
\(367\) −4.30348e6 −1.66784 −0.833920 0.551886i \(-0.813908\pi\)
−0.833920 + 0.551886i \(0.813908\pi\)
\(368\) 0 0
\(369\) −439285. −0.167950
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.20305e6 −0.447726 −0.223863 0.974621i \(-0.571867\pi\)
−0.223863 + 0.974621i \(0.571867\pi\)
\(374\) 0 0
\(375\) −1.50399e6 −0.552289
\(376\) 0 0
\(377\) 4.62529e6 1.67604
\(378\) 0 0
\(379\) 5.11488e6 1.82910 0.914550 0.404473i \(-0.132545\pi\)
0.914550 + 0.404473i \(0.132545\pi\)
\(380\) 0 0
\(381\) −1.97657e6 −0.697590
\(382\) 0 0
\(383\) 5.01055e6 1.74537 0.872687 0.488280i \(-0.162375\pi\)
0.872687 + 0.488280i \(0.162375\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −963609. −0.327057
\(388\) 0 0
\(389\) −4.08326e6 −1.36815 −0.684073 0.729413i \(-0.739793\pi\)
−0.684073 + 0.729413i \(0.739793\pi\)
\(390\) 0 0
\(391\) −1.17253e6 −0.387866
\(392\) 0 0
\(393\) −491599. −0.160557
\(394\) 0 0
\(395\) −2.45304e6 −0.791063
\(396\) 0 0
\(397\) −1.07300e6 −0.341682 −0.170841 0.985299i \(-0.554648\pi\)
−0.170841 + 0.985299i \(0.554648\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.89948e6 −0.589894 −0.294947 0.955514i \(-0.595302\pi\)
−0.294947 + 0.955514i \(0.595302\pi\)
\(402\) 0 0
\(403\) −4.94173e6 −1.51571
\(404\) 0 0
\(405\) −209700. −0.0635274
\(406\) 0 0
\(407\) −1.24591e6 −0.372821
\(408\) 0 0
\(409\) 1.93577e6 0.572198 0.286099 0.958200i \(-0.407641\pi\)
0.286099 + 0.958200i \(0.407641\pi\)
\(410\) 0 0
\(411\) −656486. −0.191699
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −699754. −0.199446
\(416\) 0 0
\(417\) 996012. 0.280495
\(418\) 0 0
\(419\) 3.93448e6 1.09484 0.547422 0.836856i \(-0.315609\pi\)
0.547422 + 0.836856i \(0.315609\pi\)
\(420\) 0 0
\(421\) −5.16927e6 −1.42142 −0.710712 0.703483i \(-0.751627\pi\)
−0.710712 + 0.703483i \(0.751627\pi\)
\(422\) 0 0
\(423\) −1.41167e6 −0.383603
\(424\) 0 0
\(425\) 3.26734e6 0.877448
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.80730e6 −0.474119
\(430\) 0 0
\(431\) −2.12951e6 −0.552187 −0.276093 0.961131i \(-0.589040\pi\)
−0.276093 + 0.961131i \(0.589040\pi\)
\(432\) 0 0
\(433\) 3.20879e6 0.822473 0.411237 0.911529i \(-0.365097\pi\)
0.411237 + 0.911529i \(0.365097\pi\)
\(434\) 0 0
\(435\) 1.72849e6 0.437971
\(436\) 0 0
\(437\) 566177. 0.141824
\(438\) 0 0
\(439\) 6.99077e6 1.73126 0.865632 0.500680i \(-0.166917\pi\)
0.865632 + 0.500680i \(0.166917\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.46008e6 1.80607 0.903034 0.429568i \(-0.141334\pi\)
0.903034 + 0.429568i \(0.141334\pi\)
\(444\) 0 0
\(445\) 4.37768e6 1.04796
\(446\) 0 0
\(447\) −2.94323e6 −0.696715
\(448\) 0 0
\(449\) −5.61620e6 −1.31470 −0.657350 0.753586i \(-0.728323\pi\)
−0.657350 + 0.753586i \(0.728323\pi\)
\(450\) 0 0
\(451\) −1.41484e6 −0.327542
\(452\) 0 0
\(453\) −1.57041e6 −0.359557
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.81271e6 −1.97387 −0.986936 0.161111i \(-0.948492\pi\)
−0.986936 + 0.161111i \(0.948492\pi\)
\(458\) 0 0
\(459\) 1.13237e6 0.250874
\(460\) 0 0
\(461\) 3.24339e6 0.710798 0.355399 0.934715i \(-0.384345\pi\)
0.355399 + 0.934715i \(0.384345\pi\)
\(462\) 0 0
\(463\) −5.70039e6 −1.23581 −0.617906 0.786252i \(-0.712019\pi\)
−0.617906 + 0.786252i \(0.712019\pi\)
\(464\) 0 0
\(465\) −1.84675e6 −0.396074
\(466\) 0 0
\(467\) 2.18014e6 0.462585 0.231292 0.972884i \(-0.425705\pi\)
0.231292 + 0.972884i \(0.425705\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −3.96263e6 −0.823059
\(472\) 0 0
\(473\) −3.10358e6 −0.637837
\(474\) 0 0
\(475\) −1.57770e6 −0.320840
\(476\) 0 0
\(477\) 3.04970e6 0.613707
\(478\) 0 0
\(479\) 3.93616e6 0.783852 0.391926 0.919997i \(-0.371809\pi\)
0.391926 + 0.919997i \(0.371809\pi\)
\(480\) 0 0
\(481\) −3.67605e6 −0.724466
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.97403e6 0.574104
\(486\) 0 0
\(487\) 7.00176e6 1.33778 0.668890 0.743361i \(-0.266770\pi\)
0.668890 + 0.743361i \(0.266770\pi\)
\(488\) 0 0
\(489\) −508428. −0.0961517
\(490\) 0 0
\(491\) −9.37114e6 −1.75424 −0.877119 0.480273i \(-0.840537\pi\)
−0.877119 + 0.480273i \(0.840537\pi\)
\(492\) 0 0
\(493\) −9.33378e6 −1.72958
\(494\) 0 0
\(495\) −675399. −0.123893
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −9.75932e6 −1.75456 −0.877280 0.479978i \(-0.840645\pi\)
−0.877280 + 0.479978i \(0.840645\pi\)
\(500\) 0 0
\(501\) 6.19213e6 1.10216
\(502\) 0 0
\(503\) −1.69667e6 −0.299004 −0.149502 0.988761i \(-0.547767\pi\)
−0.149502 + 0.988761i \(0.547767\pi\)
\(504\) 0 0
\(505\) −3.90895e6 −0.682075
\(506\) 0 0
\(507\) −1.99080e6 −0.343959
\(508\) 0 0
\(509\) −3.73566e6 −0.639106 −0.319553 0.947568i \(-0.603533\pi\)
−0.319553 + 0.947568i \(0.603533\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −546786. −0.0917326
\(514\) 0 0
\(515\) −2.39077e6 −0.397209
\(516\) 0 0
\(517\) −4.54668e6 −0.748114
\(518\) 0 0
\(519\) −2.54302e6 −0.414412
\(520\) 0 0
\(521\) 7.32718e6 1.18261 0.591306 0.806447i \(-0.298612\pi\)
0.591306 + 0.806447i \(0.298612\pi\)
\(522\) 0 0
\(523\) −9591.43 −0.00153331 −0.000766653 1.00000i \(-0.500244\pi\)
−0.000766653 1.00000i \(0.500244\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.97236e6 1.56413
\(528\) 0 0
\(529\) −5.86654e6 −0.911471
\(530\) 0 0
\(531\) 1.78832e6 0.275239
\(532\) 0 0
\(533\) −4.17448e6 −0.636479
\(534\) 0 0
\(535\) −2.26200e6 −0.341671
\(536\) 0 0
\(537\) −1.36604e6 −0.204422
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −3.75692e6 −0.551872 −0.275936 0.961176i \(-0.588988\pi\)
−0.275936 + 0.961176i \(0.588988\pi\)
\(542\) 0 0
\(543\) 2.90032e6 0.422130
\(544\) 0 0
\(545\) 4.54692e6 0.655731
\(546\) 0 0
\(547\) 1.03391e7 1.47746 0.738729 0.674002i \(-0.235426\pi\)
0.738729 + 0.674002i \(0.235426\pi\)
\(548\) 0 0
\(549\) −662044. −0.0937467
\(550\) 0 0
\(551\) 4.50699e6 0.632423
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1.37376e6 −0.189312
\(556\) 0 0
\(557\) −1.00476e7 −1.37222 −0.686108 0.727500i \(-0.740682\pi\)
−0.686108 + 0.727500i \(0.740682\pi\)
\(558\) 0 0
\(559\) −9.15708e6 −1.23945
\(560\) 0 0
\(561\) 3.64712e6 0.489263
\(562\) 0 0
\(563\) −8.49380e6 −1.12936 −0.564678 0.825311i \(-0.691000\pi\)
−0.564678 + 0.825311i \(0.691000\pi\)
\(564\) 0 0
\(565\) −6.35713e6 −0.837799
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.81531e6 0.494025 0.247012 0.969012i \(-0.420551\pi\)
0.247012 + 0.969012i \(0.420551\pi\)
\(570\) 0 0
\(571\) 1.22133e7 1.56762 0.783812 0.620998i \(-0.213272\pi\)
0.783812 + 0.620998i \(0.213272\pi\)
\(572\) 0 0
\(573\) −4.84669e6 −0.616678
\(574\) 0 0
\(575\) −1.58780e6 −0.200275
\(576\) 0 0
\(577\) −2.29471e6 −0.286938 −0.143469 0.989655i \(-0.545826\pi\)
−0.143469 + 0.989655i \(0.545826\pi\)
\(578\) 0 0
\(579\) −2.73096e6 −0.338547
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 9.82243e6 1.19687
\(584\) 0 0
\(585\) −1.99276e6 −0.240749
\(586\) 0 0
\(587\) 7.53391e6 0.902454 0.451227 0.892409i \(-0.350986\pi\)
0.451227 + 0.892409i \(0.350986\pi\)
\(588\) 0 0
\(589\) −4.81534e6 −0.571925
\(590\) 0 0
\(591\) −5.90827e6 −0.695812
\(592\) 0 0
\(593\) 7.23521e6 0.844918 0.422459 0.906382i \(-0.361167\pi\)
0.422459 + 0.906382i \(0.361167\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −7.57984e6 −0.870410
\(598\) 0 0
\(599\) 3.33856e6 0.380182 0.190091 0.981766i \(-0.439122\pi\)
0.190091 + 0.981766i \(0.439122\pi\)
\(600\) 0 0
\(601\) 1.48297e7 1.67474 0.837370 0.546636i \(-0.184092\pi\)
0.837370 + 0.546636i \(0.184092\pi\)
\(602\) 0 0
\(603\) 1.05314e6 0.117949
\(604\) 0 0
\(605\) 2.97213e6 0.330126
\(606\) 0 0
\(607\) −6.37473e6 −0.702247 −0.351124 0.936329i \(-0.614200\pi\)
−0.351124 + 0.936329i \(0.614200\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.34149e7 −1.45374
\(612\) 0 0
\(613\) −1.54869e7 −1.66461 −0.832306 0.554317i \(-0.812980\pi\)
−0.832306 + 0.554317i \(0.812980\pi\)
\(614\) 0 0
\(615\) −1.56003e6 −0.166320
\(616\) 0 0
\(617\) −4.55324e6 −0.481513 −0.240756 0.970586i \(-0.577396\pi\)
−0.240756 + 0.970586i \(0.577396\pi\)
\(618\) 0 0
\(619\) −1.16422e7 −1.22126 −0.610628 0.791918i \(-0.709083\pi\)
−0.610628 + 0.791918i \(0.709083\pi\)
\(620\) 0 0
\(621\) −550289. −0.0572613
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.23221e6 0.126178
\(626\) 0 0
\(627\) −1.76108e6 −0.178900
\(628\) 0 0
\(629\) 7.41822e6 0.747607
\(630\) 0 0
\(631\) 1.07436e7 1.07417 0.537087 0.843527i \(-0.319525\pi\)
0.537087 + 0.843527i \(0.319525\pi\)
\(632\) 0 0
\(633\) −9.10195e6 −0.902870
\(634\) 0 0
\(635\) −7.01938e6 −0.690819
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −1.00032e6 −0.0969140
\(640\) 0 0
\(641\) 1.23138e7 1.18371 0.591857 0.806043i \(-0.298395\pi\)
0.591857 + 0.806043i \(0.298395\pi\)
\(642\) 0 0
\(643\) −1.67944e7 −1.60191 −0.800955 0.598724i \(-0.795675\pi\)
−0.800955 + 0.598724i \(0.795675\pi\)
\(644\) 0 0
\(645\) −3.42205e6 −0.323882
\(646\) 0 0
\(647\) 1.99108e7 1.86994 0.934970 0.354726i \(-0.115426\pi\)
0.934970 + 0.354726i \(0.115426\pi\)
\(648\) 0 0
\(649\) 5.75981e6 0.536780
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.10990e7 1.93633 0.968163 0.250319i \(-0.0805355\pi\)
0.968163 + 0.250319i \(0.0805355\pi\)
\(654\) 0 0
\(655\) −1.74581e6 −0.158999
\(656\) 0 0
\(657\) 3.53163e6 0.319199
\(658\) 0 0
\(659\) 1.75070e7 1.57035 0.785177 0.619271i \(-0.212572\pi\)
0.785177 + 0.619271i \(0.212572\pi\)
\(660\) 0 0
\(661\) 1.80169e7 1.60389 0.801947 0.597395i \(-0.203797\pi\)
0.801947 + 0.597395i \(0.203797\pi\)
\(662\) 0 0
\(663\) 1.07608e7 0.950737
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.53587e6 0.394771
\(668\) 0 0
\(669\) −8.61412e6 −0.744124
\(670\) 0 0
\(671\) −2.13230e6 −0.182828
\(672\) 0 0
\(673\) 1.48593e7 1.26462 0.632312 0.774714i \(-0.282106\pi\)
0.632312 + 0.774714i \(0.282106\pi\)
\(674\) 0 0
\(675\) 1.53342e6 0.129539
\(676\) 0 0
\(677\) 7.18320e6 0.602347 0.301173 0.953569i \(-0.402622\pi\)
0.301173 + 0.953569i \(0.402622\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −4.24332e6 −0.350621
\(682\) 0 0
\(683\) 1.65092e6 0.135417 0.0677087 0.997705i \(-0.478431\pi\)
0.0677087 + 0.997705i \(0.478431\pi\)
\(684\) 0 0
\(685\) −2.33137e6 −0.189839
\(686\) 0 0
\(687\) 2.41377e6 0.195121
\(688\) 0 0
\(689\) 2.89810e7 2.32576
\(690\) 0 0
\(691\) 777444. 0.0619404 0.0309702 0.999520i \(-0.490140\pi\)
0.0309702 + 0.999520i \(0.490140\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.53713e6 0.277772
\(696\) 0 0
\(697\) 8.42406e6 0.656809
\(698\) 0 0
\(699\) −1.39250e7 −1.07796
\(700\) 0 0
\(701\) 1.65671e6 0.127336 0.0636681 0.997971i \(-0.479720\pi\)
0.0636681 + 0.997971i \(0.479720\pi\)
\(702\) 0 0
\(703\) −3.58203e6 −0.273364
\(704\) 0 0
\(705\) −5.01324e6 −0.379879
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.99901e6 0.373481 0.186740 0.982409i \(-0.440208\pi\)
0.186740 + 0.982409i \(0.440208\pi\)
\(710\) 0 0
\(711\) 6.21671e6 0.461197
\(712\) 0 0
\(713\) −4.84619e6 −0.357007
\(714\) 0 0
\(715\) −6.41825e6 −0.469517
\(716\) 0 0
\(717\) 8.43788e6 0.612965
\(718\) 0 0
\(719\) −1.81384e7 −1.30851 −0.654253 0.756276i \(-0.727017\pi\)
−0.654253 + 0.756276i \(0.727017\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1.15637e7 0.822721
\(724\) 0 0
\(725\) −1.26395e7 −0.893070
\(726\) 0 0
\(727\) 1.17918e7 0.827454 0.413727 0.910401i \(-0.364227\pi\)
0.413727 + 0.910401i \(0.364227\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 1.84789e7 1.27904
\(732\) 0 0
\(733\) −4.96469e6 −0.341297 −0.170648 0.985332i \(-0.554586\pi\)
−0.170648 + 0.985332i \(0.554586\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.39194e6 0.230027
\(738\) 0 0
\(739\) 1.86292e7 1.25483 0.627413 0.778686i \(-0.284114\pi\)
0.627413 + 0.778686i \(0.284114\pi\)
\(740\) 0 0
\(741\) −5.19605e6 −0.347639
\(742\) 0 0
\(743\) 1.77727e7 1.18109 0.590544 0.807006i \(-0.298913\pi\)
0.590544 + 0.807006i \(0.298913\pi\)
\(744\) 0 0
\(745\) −1.04523e7 −0.689953
\(746\) 0 0
\(747\) 1.77338e6 0.116279
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4.66059e6 0.301537 0.150769 0.988569i \(-0.451825\pi\)
0.150769 + 0.988569i \(0.451825\pi\)
\(752\) 0 0
\(753\) −6.38792e6 −0.410556
\(754\) 0 0
\(755\) −5.57699e6 −0.356067
\(756\) 0 0
\(757\) −2.36520e6 −0.150013 −0.0750064 0.997183i \(-0.523898\pi\)
−0.0750064 + 0.997183i \(0.523898\pi\)
\(758\) 0 0
\(759\) −1.77236e6 −0.111673
\(760\) 0 0
\(761\) −1.76946e7 −1.10759 −0.553794 0.832654i \(-0.686820\pi\)
−0.553794 + 0.832654i \(0.686820\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 4.02137e6 0.248439
\(766\) 0 0
\(767\) 1.69943e7 1.04307
\(768\) 0 0
\(769\) −7.90324e6 −0.481936 −0.240968 0.970533i \(-0.577465\pi\)
−0.240968 + 0.970533i \(0.577465\pi\)
\(770\) 0 0
\(771\) 1.63814e7 0.992468
\(772\) 0 0
\(773\) 2.22460e7 1.33907 0.669534 0.742782i \(-0.266494\pi\)
0.669534 + 0.742782i \(0.266494\pi\)
\(774\) 0 0
\(775\) 1.35043e7 0.807639
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.06771e6 −0.240163
\(780\) 0 0
\(781\) −3.22181e6 −0.189005
\(782\) 0 0
\(783\) −4.38051e6 −0.255341
\(784\) 0 0
\(785\) −1.40724e7 −0.815070
\(786\) 0 0
\(787\) 2.49897e7 1.43822 0.719108 0.694898i \(-0.244551\pi\)
0.719108 + 0.694898i \(0.244551\pi\)
\(788\) 0 0
\(789\) −3.23269e6 −0.184872
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −6.29134e6 −0.355271
\(794\) 0 0
\(795\) 1.08304e7 0.607750
\(796\) 0 0
\(797\) 1.40897e7 0.785696 0.392848 0.919603i \(-0.371490\pi\)
0.392848 + 0.919603i \(0.371490\pi\)
\(798\) 0 0
\(799\) 2.70712e7 1.50017
\(800\) 0 0
\(801\) −1.10943e7 −0.610970
\(802\) 0 0
\(803\) 1.13746e7 0.622512
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1.04190e7 −0.563174
\(808\) 0 0
\(809\) −2.54305e7 −1.36611 −0.683053 0.730369i \(-0.739348\pi\)
−0.683053 + 0.730369i \(0.739348\pi\)
\(810\) 0 0
\(811\) 1.97155e6 0.105258 0.0526291 0.998614i \(-0.483240\pi\)
0.0526291 + 0.998614i \(0.483240\pi\)
\(812\) 0 0
\(813\) 7.02158e6 0.372570
\(814\) 0 0
\(815\) −1.80557e6 −0.0952184
\(816\) 0 0
\(817\) −8.92288e6 −0.467681
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.11451e6 −0.0577067 −0.0288534 0.999584i \(-0.509186\pi\)
−0.0288534 + 0.999584i \(0.509186\pi\)
\(822\) 0 0
\(823\) 2.79838e6 0.144015 0.0720073 0.997404i \(-0.477060\pi\)
0.0720073 + 0.997404i \(0.477060\pi\)
\(824\) 0 0
\(825\) 4.93882e6 0.252632
\(826\) 0 0
\(827\) 1.56547e6 0.0795940 0.0397970 0.999208i \(-0.487329\pi\)
0.0397970 + 0.999208i \(0.487329\pi\)
\(828\) 0 0
\(829\) −1.33721e7 −0.675791 −0.337895 0.941184i \(-0.609715\pi\)
−0.337895 + 0.941184i \(0.609715\pi\)
\(830\) 0 0
\(831\) 2.93698e6 0.147536
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 2.19900e7 1.09146
\(836\) 0 0
\(837\) 4.68021e6 0.230915
\(838\) 0 0
\(839\) 4.37187e6 0.214418 0.107209 0.994236i \(-0.465809\pi\)
0.107209 + 0.994236i \(0.465809\pi\)
\(840\) 0 0
\(841\) 1.55961e7 0.760372
\(842\) 0 0
\(843\) −3.27685e6 −0.158813
\(844\) 0 0
\(845\) −7.06989e6 −0.340621
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 576173. 0.0274336
\(850\) 0 0
\(851\) −3.60498e6 −0.170639
\(852\) 0 0
\(853\) −1.45910e7 −0.686612 −0.343306 0.939224i \(-0.611547\pi\)
−0.343306 + 0.939224i \(0.611547\pi\)
\(854\) 0 0
\(855\) −1.94179e6 −0.0908422
\(856\) 0 0
\(857\) −1.31276e7 −0.610565 −0.305283 0.952262i \(-0.598751\pi\)
−0.305283 + 0.952262i \(0.598751\pi\)
\(858\) 0 0
\(859\) 1.42131e7 0.657211 0.328605 0.944467i \(-0.393421\pi\)
0.328605 + 0.944467i \(0.393421\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.66170e7 1.21656 0.608278 0.793724i \(-0.291861\pi\)
0.608278 + 0.793724i \(0.291861\pi\)
\(864\) 0 0
\(865\) −9.03101e6 −0.410389
\(866\) 0 0
\(867\) −8.93646e6 −0.403755
\(868\) 0 0
\(869\) 2.00227e7 0.899441
\(870\) 0 0
\(871\) 1.00079e7 0.446990
\(872\) 0 0
\(873\) −7.53706e6 −0.334708
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.42383e7 −0.625114 −0.312557 0.949899i \(-0.601186\pi\)
−0.312557 + 0.949899i \(0.601186\pi\)
\(878\) 0 0
\(879\) −3.58218e6 −0.156378
\(880\) 0 0
\(881\) −1.51921e7 −0.659445 −0.329722 0.944078i \(-0.606955\pi\)
−0.329722 + 0.944078i \(0.606955\pi\)
\(882\) 0 0
\(883\) 1.03997e7 0.448868 0.224434 0.974489i \(-0.427947\pi\)
0.224434 + 0.974489i \(0.427947\pi\)
\(884\) 0 0
\(885\) 6.35086e6 0.272567
\(886\) 0 0
\(887\) 2.02129e7 0.862618 0.431309 0.902204i \(-0.358052\pi\)
0.431309 + 0.902204i \(0.358052\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.71166e6 0.0722308
\(892\) 0 0
\(893\) −1.30718e7 −0.548540
\(894\) 0 0
\(895\) −4.85119e6 −0.202437
\(896\) 0 0
\(897\) −5.22934e6 −0.217003
\(898\) 0 0
\(899\) −3.85776e7 −1.59197
\(900\) 0 0
\(901\) −5.84834e7 −2.40005
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.02999e7 0.418033
\(906\) 0 0
\(907\) −4.35643e6 −0.175838 −0.0879189 0.996128i \(-0.528022\pi\)
−0.0879189 + 0.996128i \(0.528022\pi\)
\(908\) 0 0
\(909\) 9.90643e6 0.397656
\(910\) 0 0
\(911\) 1.99500e7 0.796427 0.398214 0.917293i \(-0.369630\pi\)
0.398214 + 0.917293i \(0.369630\pi\)
\(912\) 0 0
\(913\) 5.71168e6 0.226771
\(914\) 0 0
\(915\) −2.35111e6 −0.0928368
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −9.55172e6 −0.373072 −0.186536 0.982448i \(-0.559726\pi\)
−0.186536 + 0.982448i \(0.559726\pi\)
\(920\) 0 0
\(921\) 5.74957e6 0.223350
\(922\) 0 0
\(923\) −9.50594e6 −0.367275
\(924\) 0 0
\(925\) 1.00455e7 0.386028
\(926\) 0 0
\(927\) 6.05891e6 0.231577
\(928\) 0 0
\(929\) −1.26025e7 −0.479090 −0.239545 0.970885i \(-0.576998\pi\)
−0.239545 + 0.970885i \(0.576998\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −2.23370e7 −0.840078
\(934\) 0 0
\(935\) 1.29520e7 0.484514
\(936\) 0 0
\(937\) 1.00216e7 0.372897 0.186448 0.982465i \(-0.440302\pi\)
0.186448 + 0.982465i \(0.440302\pi\)
\(938\) 0 0
\(939\) −1.32888e7 −0.491838
\(940\) 0 0
\(941\) −1.54113e6 −0.0567369 −0.0283684 0.999598i \(-0.509031\pi\)
−0.0283684 + 0.999598i \(0.509031\pi\)
\(942\) 0 0
\(943\) −4.09377e6 −0.149915
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.07046e7 1.11257 0.556287 0.830990i \(-0.312226\pi\)
0.556287 + 0.830990i \(0.312226\pi\)
\(948\) 0 0
\(949\) 3.35607e7 1.20967
\(950\) 0 0
\(951\) 2.00813e7 0.720014
\(952\) 0 0
\(953\) −4.39769e7 −1.56853 −0.784263 0.620428i \(-0.786959\pi\)
−0.784263 + 0.620428i \(0.786959\pi\)
\(954\) 0 0
\(955\) −1.72120e7 −0.610692
\(956\) 0 0
\(957\) −1.41087e7 −0.497974
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.25878e7 0.439684
\(962\) 0 0
\(963\) 5.73257e6 0.199197
\(964\) 0 0
\(965\) −9.69841e6 −0.335261
\(966\) 0 0
\(967\) −4.20732e7 −1.44690 −0.723452 0.690375i \(-0.757445\pi\)
−0.723452 + 0.690375i \(0.757445\pi\)
\(968\) 0 0
\(969\) 1.04856e7 0.358743
\(970\) 0 0
\(971\) 2.08006e7 0.707992 0.353996 0.935247i \(-0.384823\pi\)
0.353996 + 0.935247i \(0.384823\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 1.45719e7 0.490915
\(976\) 0 0
\(977\) 3.74338e7 1.25466 0.627332 0.778752i \(-0.284147\pi\)
0.627332 + 0.778752i \(0.284147\pi\)
\(978\) 0 0
\(979\) −3.57324e7 −1.19153
\(980\) 0 0
\(981\) −1.15232e7 −0.382297
\(982\) 0 0
\(983\) −2.41962e7 −0.798664 −0.399332 0.916806i \(-0.630758\pi\)
−0.399332 + 0.916806i \(0.630758\pi\)
\(984\) 0 0
\(985\) −2.09820e7 −0.689058
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −8.98005e6 −0.291936
\(990\) 0 0
\(991\) −2.52080e7 −0.815369 −0.407684 0.913123i \(-0.633664\pi\)
−0.407684 + 0.913123i \(0.633664\pi\)
\(992\) 0 0
\(993\) 2.17010e7 0.698404
\(994\) 0 0
\(995\) −2.69182e7 −0.861962
\(996\) 0 0
\(997\) 4.95511e7 1.57876 0.789379 0.613907i \(-0.210403\pi\)
0.789379 + 0.613907i \(0.210403\pi\)
\(998\) 0 0
\(999\) 3.48150e6 0.110371
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 588.6.a.n.1.2 4
7.2 even 3 84.6.i.c.25.3 8
7.3 odd 6 588.6.i.o.373.2 8
7.4 even 3 84.6.i.c.37.3 yes 8
7.5 odd 6 588.6.i.o.361.2 8
7.6 odd 2 588.6.a.p.1.3 4
21.2 odd 6 252.6.k.f.109.2 8
21.11 odd 6 252.6.k.f.37.2 8
28.11 odd 6 336.6.q.i.289.3 8
28.23 odd 6 336.6.q.i.193.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.i.c.25.3 8 7.2 even 3
84.6.i.c.37.3 yes 8 7.4 even 3
252.6.k.f.37.2 8 21.11 odd 6
252.6.k.f.109.2 8 21.2 odd 6
336.6.q.i.193.3 8 28.23 odd 6
336.6.q.i.289.3 8 28.11 odd 6
588.6.a.n.1.2 4 1.1 even 1 trivial
588.6.a.p.1.3 4 7.6 odd 2
588.6.i.o.361.2 8 7.5 odd 6
588.6.i.o.373.2 8 7.3 odd 6