Properties

Label 450.6.a.bf.1.2
Level $450$
Weight $6$
Character 450.1
Self dual yes
Analytic conductor $72.173$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.1727189158\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{4081}) \)
Defining polynomial: \(x^{2} - x - 1020\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-31.4414\) of defining polynomial
Character \(\chi\) \(=\) 450.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} +16.0000 q^{4} +241.648 q^{7} +64.0000 q^{8} +O(q^{10})\) \(q+4.00000 q^{2} +16.0000 q^{4} +241.648 q^{7} +64.0000 q^{8} -653.296 q^{11} -828.296 q^{13} +966.592 q^{14} +256.000 q^{16} -2162.48 q^{17} +1254.24 q^{19} -2613.18 q^{22} -3746.48 q^{23} -3313.18 q^{26} +3866.37 q^{28} -2466.70 q^{29} -1895.76 q^{31} +1024.00 q^{32} -8649.92 q^{34} -1050.45 q^{37} +5016.96 q^{38} -1960.56 q^{41} -11017.6 q^{43} -10452.7 q^{44} -14985.9 q^{46} +23064.8 q^{47} +41586.8 q^{49} -13252.7 q^{52} +27329.4 q^{53} +15465.5 q^{56} -9866.82 q^{58} -35225.7 q^{59} +2685.33 q^{61} -7583.04 q^{62} +4096.00 q^{64} -48125.1 q^{67} -34599.7 q^{68} -27237.7 q^{71} +6547.77 q^{73} -4201.78 q^{74} +20067.8 q^{76} -157868. q^{77} -65037.3 q^{79} -7842.23 q^{82} +62567.2 q^{83} -44070.3 q^{86} -41811.0 q^{88} -17926.2 q^{89} -200156. q^{91} -59943.7 q^{92} +92259.2 q^{94} -95974.9 q^{97} +166347. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{2} + 32q^{4} + 100q^{7} + 128q^{8} + O(q^{10}) \) \( 2q + 8q^{2} + 32q^{4} + 100q^{7} + 128q^{8} - 540q^{11} - 890q^{13} + 400q^{14} + 512q^{16} - 492q^{17} + 592q^{19} - 2160q^{22} - 3660q^{23} - 3560q^{26} + 1600q^{28} - 5700q^{29} - 5708q^{31} + 2048q^{32} - 1968q^{34} - 11300q^{37} + 2368q^{38} - 15420q^{41} - 6320q^{43} - 8640q^{44} - 14640q^{46} + 7800q^{47} + 44844q^{49} - 14240q^{52} + 27828q^{53} + 6400q^{56} - 22800q^{58} - 50520q^{59} - 29126q^{61} - 22832q^{62} + 8192q^{64} - 97400q^{67} - 7872q^{68} - 6180q^{71} - 32900q^{73} - 45200q^{74} + 9472q^{76} - 173916q^{77} + 7912q^{79} - 61680q^{82} + 163464q^{83} - 25280q^{86} - 34560q^{88} - 164640q^{89} - 191416q^{91} - 58560q^{92} + 31200q^{94} - 52430q^{97} + 179376q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) 0 0
\(4\) 16.0000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 241.648 1.86397 0.931984 0.362500i \(-0.118077\pi\)
0.931984 + 0.362500i \(0.118077\pi\)
\(8\) 64.0000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −653.296 −1.62790 −0.813951 0.580933i \(-0.802688\pi\)
−0.813951 + 0.580933i \(0.802688\pi\)
\(12\) 0 0
\(13\) −828.296 −1.35934 −0.679669 0.733519i \(-0.737876\pi\)
−0.679669 + 0.733519i \(0.737876\pi\)
\(14\) 966.592 1.31802
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) −2162.48 −1.81481 −0.907403 0.420262i \(-0.861938\pi\)
−0.907403 + 0.420262i \(0.861938\pi\)
\(18\) 0 0
\(19\) 1254.24 0.797071 0.398535 0.917153i \(-0.369519\pi\)
0.398535 + 0.917153i \(0.369519\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2613.18 −1.15110
\(23\) −3746.48 −1.47674 −0.738370 0.674396i \(-0.764404\pi\)
−0.738370 + 0.674396i \(0.764404\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3313.18 −0.961197
\(27\) 0 0
\(28\) 3866.37 0.931984
\(29\) −2466.70 −0.544656 −0.272328 0.962205i \(-0.587794\pi\)
−0.272328 + 0.962205i \(0.587794\pi\)
\(30\) 0 0
\(31\) −1895.76 −0.354306 −0.177153 0.984183i \(-0.556689\pi\)
−0.177153 + 0.984183i \(0.556689\pi\)
\(32\) 1024.00 0.176777
\(33\) 0 0
\(34\) −8649.92 −1.28326
\(35\) 0 0
\(36\) 0 0
\(37\) −1050.45 −0.126145 −0.0630724 0.998009i \(-0.520090\pi\)
−0.0630724 + 0.998009i \(0.520090\pi\)
\(38\) 5016.96 0.563614
\(39\) 0 0
\(40\) 0 0
\(41\) −1960.56 −0.182146 −0.0910730 0.995844i \(-0.529030\pi\)
−0.0910730 + 0.995844i \(0.529030\pi\)
\(42\) 0 0
\(43\) −11017.6 −0.908688 −0.454344 0.890826i \(-0.650126\pi\)
−0.454344 + 0.890826i \(0.650126\pi\)
\(44\) −10452.7 −0.813951
\(45\) 0 0
\(46\) −14985.9 −1.04421
\(47\) 23064.8 1.52302 0.761509 0.648154i \(-0.224459\pi\)
0.761509 + 0.648154i \(0.224459\pi\)
\(48\) 0 0
\(49\) 41586.8 2.47437
\(50\) 0 0
\(51\) 0 0
\(52\) −13252.7 −0.679669
\(53\) 27329.4 1.33641 0.668205 0.743977i \(-0.267063\pi\)
0.668205 + 0.743977i \(0.267063\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 15465.5 0.659012
\(57\) 0 0
\(58\) −9866.82 −0.385130
\(59\) −35225.7 −1.31744 −0.658718 0.752390i \(-0.728901\pi\)
−0.658718 + 0.752390i \(0.728901\pi\)
\(60\) 0 0
\(61\) 2685.33 0.0924002 0.0462001 0.998932i \(-0.485289\pi\)
0.0462001 + 0.998932i \(0.485289\pi\)
\(62\) −7583.04 −0.250532
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −48125.1 −1.30974 −0.654869 0.755743i \(-0.727276\pi\)
−0.654869 + 0.755743i \(0.727276\pi\)
\(68\) −34599.7 −0.907403
\(69\) 0 0
\(70\) 0 0
\(71\) −27237.7 −0.641245 −0.320622 0.947207i \(-0.603892\pi\)
−0.320622 + 0.947207i \(0.603892\pi\)
\(72\) 0 0
\(73\) 6547.77 0.143809 0.0719046 0.997412i \(-0.477092\pi\)
0.0719046 + 0.997412i \(0.477092\pi\)
\(74\) −4201.78 −0.0891978
\(75\) 0 0
\(76\) 20067.8 0.398535
\(77\) −157868. −3.03436
\(78\) 0 0
\(79\) −65037.3 −1.17245 −0.586226 0.810148i \(-0.699387\pi\)
−0.586226 + 0.810148i \(0.699387\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −7842.23 −0.128797
\(83\) 62567.2 0.996900 0.498450 0.866919i \(-0.333903\pi\)
0.498450 + 0.866919i \(0.333903\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −44070.3 −0.642539
\(87\) 0 0
\(88\) −41811.0 −0.575551
\(89\) −17926.2 −0.239891 −0.119946 0.992780i \(-0.538272\pi\)
−0.119946 + 0.992780i \(0.538272\pi\)
\(90\) 0 0
\(91\) −200156. −2.53376
\(92\) −59943.7 −0.738370
\(93\) 0 0
\(94\) 92259.2 1.07694
\(95\) 0 0
\(96\) 0 0
\(97\) −95974.9 −1.03569 −0.517843 0.855475i \(-0.673265\pi\)
−0.517843 + 0.855475i \(0.673265\pi\)
\(98\) 166347. 1.74965
\(99\) 0 0
\(100\) 0 0
\(101\) −41533.7 −0.405133 −0.202567 0.979269i \(-0.564928\pi\)
−0.202567 + 0.979269i \(0.564928\pi\)
\(102\) 0 0
\(103\) 41380.4 0.384328 0.192164 0.981363i \(-0.438449\pi\)
0.192164 + 0.981363i \(0.438449\pi\)
\(104\) −53011.0 −0.480598
\(105\) 0 0
\(106\) 109317. 0.944985
\(107\) −11403.4 −0.0962885 −0.0481442 0.998840i \(-0.515331\pi\)
−0.0481442 + 0.998840i \(0.515331\pi\)
\(108\) 0 0
\(109\) 98362.4 0.792981 0.396490 0.918039i \(-0.370228\pi\)
0.396490 + 0.918039i \(0.370228\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 61861.9 0.465992
\(113\) 10076.8 0.0742377 0.0371189 0.999311i \(-0.488182\pi\)
0.0371189 + 0.999311i \(0.488182\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −39467.3 −0.272328
\(117\) 0 0
\(118\) −140903. −0.931568
\(119\) −522559. −3.38274
\(120\) 0 0
\(121\) 265745. 1.65007
\(122\) 10741.3 0.0653368
\(123\) 0 0
\(124\) −30332.2 −0.177153
\(125\) 0 0
\(126\) 0 0
\(127\) 255129. 1.40362 0.701812 0.712362i \(-0.252374\pi\)
0.701812 + 0.712362i \(0.252374\pi\)
\(128\) 16384.0 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 65433.3 0.333135 0.166568 0.986030i \(-0.446732\pi\)
0.166568 + 0.986030i \(0.446732\pi\)
\(132\) 0 0
\(133\) 303085. 1.48571
\(134\) −192500. −0.926124
\(135\) 0 0
\(136\) −138399. −0.641631
\(137\) −67354.0 −0.306593 −0.153296 0.988180i \(-0.548989\pi\)
−0.153296 + 0.988180i \(0.548989\pi\)
\(138\) 0 0
\(139\) −93837.8 −0.411946 −0.205973 0.978558i \(-0.566036\pi\)
−0.205973 + 0.978558i \(0.566036\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −108951. −0.453429
\(143\) 541123. 2.21287
\(144\) 0 0
\(145\) 0 0
\(146\) 26191.1 0.101688
\(147\) 0 0
\(148\) −16807.1 −0.0630724
\(149\) −432235. −1.59498 −0.797489 0.603334i \(-0.793839\pi\)
−0.797489 + 0.603334i \(0.793839\pi\)
\(150\) 0 0
\(151\) −255822. −0.913053 −0.456527 0.889710i \(-0.650907\pi\)
−0.456527 + 0.889710i \(0.650907\pi\)
\(152\) 80271.4 0.281807
\(153\) 0 0
\(154\) −631471. −2.14561
\(155\) 0 0
\(156\) 0 0
\(157\) 58837.3 0.190504 0.0952519 0.995453i \(-0.469634\pi\)
0.0952519 + 0.995453i \(0.469634\pi\)
\(158\) −260149. −0.829048
\(159\) 0 0
\(160\) 0 0
\(161\) −905330. −2.75259
\(162\) 0 0
\(163\) −155006. −0.456962 −0.228481 0.973548i \(-0.573376\pi\)
−0.228481 + 0.973548i \(0.573376\pi\)
\(164\) −31368.9 −0.0910730
\(165\) 0 0
\(166\) 250269. 0.704914
\(167\) −588656. −1.63332 −0.816658 0.577121i \(-0.804176\pi\)
−0.816658 + 0.577121i \(0.804176\pi\)
\(168\) 0 0
\(169\) 314782. 0.847798
\(170\) 0 0
\(171\) 0 0
\(172\) −176281. −0.454344
\(173\) 192049. 0.487862 0.243931 0.969793i \(-0.421563\pi\)
0.243931 + 0.969793i \(0.421563\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −167244. −0.406976
\(177\) 0 0
\(178\) −71704.9 −0.169629
\(179\) 350715. 0.818130 0.409065 0.912505i \(-0.365855\pi\)
0.409065 + 0.912505i \(0.365855\pi\)
\(180\) 0 0
\(181\) 524637. 1.19032 0.595158 0.803608i \(-0.297089\pi\)
0.595158 + 0.803608i \(0.297089\pi\)
\(182\) −800625. −1.79164
\(183\) 0 0
\(184\) −239775. −0.522106
\(185\) 0 0
\(186\) 0 0
\(187\) 1.41274e6 2.95433
\(188\) 369037. 0.761509
\(189\) 0 0
\(190\) 0 0
\(191\) −63662.7 −0.126270 −0.0631352 0.998005i \(-0.520110\pi\)
−0.0631352 + 0.998005i \(0.520110\pi\)
\(192\) 0 0
\(193\) 717477. 1.38648 0.693242 0.720705i \(-0.256182\pi\)
0.693242 + 0.720705i \(0.256182\pi\)
\(194\) −383900. −0.732341
\(195\) 0 0
\(196\) 665389. 1.23719
\(197\) −287465. −0.527740 −0.263870 0.964558i \(-0.584999\pi\)
−0.263870 + 0.964558i \(0.584999\pi\)
\(198\) 0 0
\(199\) 643619. 1.15212 0.576058 0.817409i \(-0.304590\pi\)
0.576058 + 0.817409i \(0.304590\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −166135. −0.286472
\(203\) −596074. −1.01522
\(204\) 0 0
\(205\) 0 0
\(206\) 165522. 0.271761
\(207\) 0 0
\(208\) −212044. −0.339834
\(209\) −819391. −1.29755
\(210\) 0 0
\(211\) −873827. −1.35120 −0.675600 0.737269i \(-0.736115\pi\)
−0.675600 + 0.737269i \(0.736115\pi\)
\(212\) 437270. 0.668205
\(213\) 0 0
\(214\) −45613.5 −0.0680862
\(215\) 0 0
\(216\) 0 0
\(217\) −458107. −0.660416
\(218\) 393449. 0.560722
\(219\) 0 0
\(220\) 0 0
\(221\) 1.79117e6 2.46693
\(222\) 0 0
\(223\) 288496. 0.388488 0.194244 0.980953i \(-0.437775\pi\)
0.194244 + 0.980953i \(0.437775\pi\)
\(224\) 247448. 0.329506
\(225\) 0 0
\(226\) 40307.0 0.0524940
\(227\) 125358. 0.161469 0.0807343 0.996736i \(-0.474273\pi\)
0.0807343 + 0.996736i \(0.474273\pi\)
\(228\) 0 0
\(229\) −698183. −0.879793 −0.439897 0.898048i \(-0.644985\pi\)
−0.439897 + 0.898048i \(0.644985\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −157869. −0.192565
\(233\) 984588. 1.18813 0.594066 0.804416i \(-0.297522\pi\)
0.594066 + 0.804416i \(0.297522\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −563611. −0.658718
\(237\) 0 0
\(238\) −2.09024e6 −2.39196
\(239\) −131049. −0.148402 −0.0742011 0.997243i \(-0.523641\pi\)
−0.0742011 + 0.997243i \(0.523641\pi\)
\(240\) 0 0
\(241\) −1.23632e6 −1.37116 −0.685579 0.727998i \(-0.740451\pi\)
−0.685579 + 0.727998i \(0.740451\pi\)
\(242\) 1.06298e6 1.16677
\(243\) 0 0
\(244\) 42965.3 0.0462001
\(245\) 0 0
\(246\) 0 0
\(247\) −1.03888e6 −1.08349
\(248\) −121329. −0.125266
\(249\) 0 0
\(250\) 0 0
\(251\) −611918. −0.613068 −0.306534 0.951860i \(-0.599169\pi\)
−0.306534 + 0.951860i \(0.599169\pi\)
\(252\) 0 0
\(253\) 2.44756e6 2.40399
\(254\) 1.02052e6 0.992513
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 1.64515e6 1.55372 0.776858 0.629675i \(-0.216812\pi\)
0.776858 + 0.629675i \(0.216812\pi\)
\(258\) 0 0
\(259\) −253838. −0.235130
\(260\) 0 0
\(261\) 0 0
\(262\) 261733. 0.235562
\(263\) −268273. −0.239159 −0.119580 0.992825i \(-0.538155\pi\)
−0.119580 + 0.992825i \(0.538155\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.21234e6 1.05056
\(267\) 0 0
\(268\) −770001. −0.654869
\(269\) 947177. 0.798087 0.399044 0.916932i \(-0.369342\pi\)
0.399044 + 0.916932i \(0.369342\pi\)
\(270\) 0 0
\(271\) 609159. 0.503857 0.251929 0.967746i \(-0.418935\pi\)
0.251929 + 0.967746i \(0.418935\pi\)
\(272\) −553595. −0.453701
\(273\) 0 0
\(274\) −269416. −0.216794
\(275\) 0 0
\(276\) 0 0
\(277\) −741976. −0.581019 −0.290510 0.956872i \(-0.593825\pi\)
−0.290510 + 0.956872i \(0.593825\pi\)
\(278\) −375351. −0.291290
\(279\) 0 0
\(280\) 0 0
\(281\) 423084. 0.319639 0.159820 0.987146i \(-0.448909\pi\)
0.159820 + 0.987146i \(0.448909\pi\)
\(282\) 0 0
\(283\) 445130. 0.330385 0.165192 0.986261i \(-0.447175\pi\)
0.165192 + 0.986261i \(0.447175\pi\)
\(284\) −435803. −0.320622
\(285\) 0 0
\(286\) 2.16449e6 1.56473
\(287\) −473765. −0.339514
\(288\) 0 0
\(289\) 3.25647e6 2.29352
\(290\) 0 0
\(291\) 0 0
\(292\) 104764. 0.0719046
\(293\) −2.45992e6 −1.67399 −0.836994 0.547211i \(-0.815689\pi\)
−0.836994 + 0.547211i \(0.815689\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −67228.5 −0.0445989
\(297\) 0 0
\(298\) −1.72894e6 −1.12782
\(299\) 3.10320e6 2.00739
\(300\) 0 0
\(301\) −2.66238e6 −1.69376
\(302\) −1.02329e6 −0.645626
\(303\) 0 0
\(304\) 321086. 0.199268
\(305\) 0 0
\(306\) 0 0
\(307\) −374253. −0.226631 −0.113315 0.993559i \(-0.536147\pi\)
−0.113315 + 0.993559i \(0.536147\pi\)
\(308\) −2.52588e6 −1.51718
\(309\) 0 0
\(310\) 0 0
\(311\) −2.22568e6 −1.30485 −0.652427 0.757851i \(-0.726249\pi\)
−0.652427 + 0.757851i \(0.726249\pi\)
\(312\) 0 0
\(313\) −3.05869e6 −1.76471 −0.882357 0.470580i \(-0.844045\pi\)
−0.882357 + 0.470580i \(0.844045\pi\)
\(314\) 235349. 0.134706
\(315\) 0 0
\(316\) −1.04060e6 −0.586226
\(317\) 520610. 0.290981 0.145490 0.989360i \(-0.453524\pi\)
0.145490 + 0.989360i \(0.453524\pi\)
\(318\) 0 0
\(319\) 1.61149e6 0.886646
\(320\) 0 0
\(321\) 0 0
\(322\) −3.62132e6 −1.94638
\(323\) −2.71227e6 −1.44653
\(324\) 0 0
\(325\) 0 0
\(326\) −620025. −0.323121
\(327\) 0 0
\(328\) −125476. −0.0643983
\(329\) 5.57357e6 2.83886
\(330\) 0 0
\(331\) −2.12755e6 −1.06736 −0.533679 0.845687i \(-0.679191\pi\)
−0.533679 + 0.845687i \(0.679191\pi\)
\(332\) 1.00108e6 0.498450
\(333\) 0 0
\(334\) −2.35462e6 −1.15493
\(335\) 0 0
\(336\) 0 0
\(337\) 3.08383e6 1.47916 0.739581 0.673067i \(-0.235024\pi\)
0.739581 + 0.673067i \(0.235024\pi\)
\(338\) 1.25913e6 0.599484
\(339\) 0 0
\(340\) 0 0
\(341\) 1.23849e6 0.576776
\(342\) 0 0
\(343\) 5.98799e6 2.74819
\(344\) −705125. −0.321270
\(345\) 0 0
\(346\) 768196. 0.344970
\(347\) 4.04955e6 1.80544 0.902719 0.430231i \(-0.141568\pi\)
0.902719 + 0.430231i \(0.141568\pi\)
\(348\) 0 0
\(349\) −493135. −0.216722 −0.108361 0.994112i \(-0.534560\pi\)
−0.108361 + 0.994112i \(0.534560\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −668975. −0.287775
\(353\) 1.34815e6 0.575841 0.287920 0.957654i \(-0.407036\pi\)
0.287920 + 0.957654i \(0.407036\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −286820. −0.119946
\(357\) 0 0
\(358\) 1.40286e6 0.578505
\(359\) −1.64749e6 −0.674664 −0.337332 0.941386i \(-0.609524\pi\)
−0.337332 + 0.941386i \(0.609524\pi\)
\(360\) 0 0
\(361\) −902980. −0.364678
\(362\) 2.09855e6 0.841681
\(363\) 0 0
\(364\) −3.20250e6 −1.26688
\(365\) 0 0
\(366\) 0 0
\(367\) 27918.4 0.0108199 0.00540997 0.999985i \(-0.498278\pi\)
0.00540997 + 0.999985i \(0.498278\pi\)
\(368\) −959099. −0.369185
\(369\) 0 0
\(370\) 0 0
\(371\) 6.60409e6 2.49103
\(372\) 0 0
\(373\) −2.80536e6 −1.04404 −0.522019 0.852934i \(-0.674821\pi\)
−0.522019 + 0.852934i \(0.674821\pi\)
\(374\) 5.65096e6 2.08902
\(375\) 0 0
\(376\) 1.47615e6 0.538468
\(377\) 2.04316e6 0.740371
\(378\) 0 0
\(379\) 1.70014e6 0.607976 0.303988 0.952676i \(-0.401682\pi\)
0.303988 + 0.952676i \(0.401682\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −254651. −0.0892867
\(383\) 627913. 0.218727 0.109363 0.994002i \(-0.465119\pi\)
0.109363 + 0.994002i \(0.465119\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.86991e6 0.980392
\(387\) 0 0
\(388\) −1.53560e6 −0.517843
\(389\) −3.14339e6 −1.05323 −0.526616 0.850103i \(-0.676540\pi\)
−0.526616 + 0.850103i \(0.676540\pi\)
\(390\) 0 0
\(391\) 8.10169e6 2.68000
\(392\) 2.66156e6 0.874823
\(393\) 0 0
\(394\) −1.14986e6 −0.373169
\(395\) 0 0
\(396\) 0 0
\(397\) 1.10773e6 0.352743 0.176371 0.984324i \(-0.443564\pi\)
0.176371 + 0.984324i \(0.443564\pi\)
\(398\) 2.57448e6 0.814669
\(399\) 0 0
\(400\) 0 0
\(401\) −2.41563e6 −0.750187 −0.375093 0.926987i \(-0.622389\pi\)
−0.375093 + 0.926987i \(0.622389\pi\)
\(402\) 0 0
\(403\) 1.57025e6 0.481622
\(404\) −664540. −0.202567
\(405\) 0 0
\(406\) −2.38430e6 −0.717869
\(407\) 686252. 0.205351
\(408\) 0 0
\(409\) 3.69830e6 1.09319 0.546593 0.837398i \(-0.315925\pi\)
0.546593 + 0.837398i \(0.315925\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 662087. 0.192164
\(413\) −8.51222e6 −2.45566
\(414\) 0 0
\(415\) 0 0
\(416\) −848175. −0.240299
\(417\) 0 0
\(418\) −3.27756e6 −0.917509
\(419\) 3.45478e6 0.961357 0.480679 0.876897i \(-0.340390\pi\)
0.480679 + 0.876897i \(0.340390\pi\)
\(420\) 0 0
\(421\) −4.83980e6 −1.33083 −0.665415 0.746474i \(-0.731745\pi\)
−0.665415 + 0.746474i \(0.731745\pi\)
\(422\) −3.49531e6 −0.955442
\(423\) 0 0
\(424\) 1.74908e6 0.472493
\(425\) 0 0
\(426\) 0 0
\(427\) 648905. 0.172231
\(428\) −182454. −0.0481442
\(429\) 0 0
\(430\) 0 0
\(431\) 5.54923e6 1.43893 0.719464 0.694529i \(-0.244387\pi\)
0.719464 + 0.694529i \(0.244387\pi\)
\(432\) 0 0
\(433\) 814302. 0.208721 0.104360 0.994540i \(-0.466720\pi\)
0.104360 + 0.994540i \(0.466720\pi\)
\(434\) −1.83243e6 −0.466984
\(435\) 0 0
\(436\) 1.57380e6 0.396490
\(437\) −4.69899e6 −1.17707
\(438\) 0 0
\(439\) −4.02462e6 −0.996697 −0.498349 0.866977i \(-0.666060\pi\)
−0.498349 + 0.866977i \(0.666060\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 7.16470e6 1.74438
\(443\) 4.46382e6 1.08068 0.540341 0.841446i \(-0.318295\pi\)
0.540341 + 0.841446i \(0.318295\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.15399e6 0.274703
\(447\) 0 0
\(448\) 989791. 0.232996
\(449\) 4.66889e6 1.09294 0.546472 0.837477i \(-0.315970\pi\)
0.546472 + 0.837477i \(0.315970\pi\)
\(450\) 0 0
\(451\) 1.28082e6 0.296516
\(452\) 161228. 0.0371189
\(453\) 0 0
\(454\) 501433. 0.114176
\(455\) 0 0
\(456\) 0 0
\(457\) 4.71340e6 1.05571 0.527854 0.849335i \(-0.322997\pi\)
0.527854 + 0.849335i \(0.322997\pi\)
\(458\) −2.79273e6 −0.622108
\(459\) 0 0
\(460\) 0 0
\(461\) −6.23975e6 −1.36746 −0.683731 0.729734i \(-0.739644\pi\)
−0.683731 + 0.729734i \(0.739644\pi\)
\(462\) 0 0
\(463\) 3.33992e6 0.724076 0.362038 0.932163i \(-0.382081\pi\)
0.362038 + 0.932163i \(0.382081\pi\)
\(464\) −631476. −0.136164
\(465\) 0 0
\(466\) 3.93835e6 0.840136
\(467\) −722497. −0.153301 −0.0766503 0.997058i \(-0.524422\pi\)
−0.0766503 + 0.997058i \(0.524422\pi\)
\(468\) 0 0
\(469\) −1.16293e7 −2.44131
\(470\) 0 0
\(471\) 0 0
\(472\) −2.25444e6 −0.465784
\(473\) 7.19774e6 1.47926
\(474\) 0 0
\(475\) 0 0
\(476\) −8.36095e6 −1.69137
\(477\) 0 0
\(478\) −524197. −0.104936
\(479\) 5.37714e6 1.07081 0.535405 0.844595i \(-0.320159\pi\)
0.535405 + 0.844595i \(0.320159\pi\)
\(480\) 0 0
\(481\) 870080. 0.171473
\(482\) −4.94527e6 −0.969556
\(483\) 0 0
\(484\) 4.25192e6 0.825034
\(485\) 0 0
\(486\) 0 0
\(487\) 1.72084e6 0.328790 0.164395 0.986395i \(-0.447433\pi\)
0.164395 + 0.986395i \(0.447433\pi\)
\(488\) 171861. 0.0326684
\(489\) 0 0
\(490\) 0 0
\(491\) −7.31449e6 −1.36924 −0.684621 0.728899i \(-0.740032\pi\)
−0.684621 + 0.728899i \(0.740032\pi\)
\(492\) 0 0
\(493\) 5.33420e6 0.988444
\(494\) −4.15553e6 −0.766142
\(495\) 0 0
\(496\) −485314. −0.0885766
\(497\) −6.58193e6 −1.19526
\(498\) 0 0
\(499\) −7.29878e6 −1.31220 −0.656098 0.754676i \(-0.727794\pi\)
−0.656098 + 0.754676i \(0.727794\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −2.44767e6 −0.433505
\(503\) −923591. −0.162764 −0.0813822 0.996683i \(-0.525933\pi\)
−0.0813822 + 0.996683i \(0.525933\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 9.79025e6 1.69988
\(507\) 0 0
\(508\) 4.08207e6 0.701812
\(509\) −8.81318e6 −1.50778 −0.753891 0.657000i \(-0.771825\pi\)
−0.753891 + 0.657000i \(0.771825\pi\)
\(510\) 0 0
\(511\) 1.58226e6 0.268056
\(512\) 262144. 0.0441942
\(513\) 0 0
\(514\) 6.58059e6 1.09864
\(515\) 0 0
\(516\) 0 0
\(517\) −1.50682e7 −2.47933
\(518\) −1.01535e6 −0.166262
\(519\) 0 0
\(520\) 0 0
\(521\) −1.66995e6 −0.269531 −0.134765 0.990878i \(-0.543028\pi\)
−0.134765 + 0.990878i \(0.543028\pi\)
\(522\) 0 0
\(523\) −2.85438e6 −0.456308 −0.228154 0.973625i \(-0.573269\pi\)
−0.228154 + 0.973625i \(0.573269\pi\)
\(524\) 1.04693e6 0.166568
\(525\) 0 0
\(526\) −1.07309e6 −0.169111
\(527\) 4.09954e6 0.642997
\(528\) 0 0
\(529\) 7.59978e6 1.18076
\(530\) 0 0
\(531\) 0 0
\(532\) 4.84936e6 0.742857
\(533\) 1.62392e6 0.247598
\(534\) 0 0
\(535\) 0 0
\(536\) −3.08000e6 −0.463062
\(537\) 0 0
\(538\) 3.78871e6 0.564333
\(539\) −2.71685e7 −4.02804
\(540\) 0 0
\(541\) −7.85655e6 −1.15409 −0.577044 0.816713i \(-0.695794\pi\)
−0.577044 + 0.816713i \(0.695794\pi\)
\(542\) 2.43664e6 0.356281
\(543\) 0 0
\(544\) −2.21438e6 −0.320815
\(545\) 0 0
\(546\) 0 0
\(547\) 1.07748e7 1.53972 0.769860 0.638213i \(-0.220326\pi\)
0.769860 + 0.638213i \(0.220326\pi\)
\(548\) −1.07766e6 −0.153296
\(549\) 0 0
\(550\) 0 0
\(551\) −3.09384e6 −0.434129
\(552\) 0 0
\(553\) −1.57161e7 −2.18541
\(554\) −2.96791e6 −0.410843
\(555\) 0 0
\(556\) −1.50140e6 −0.205973
\(557\) 7.62158e6 1.04090 0.520448 0.853893i \(-0.325765\pi\)
0.520448 + 0.853893i \(0.325765\pi\)
\(558\) 0 0
\(559\) 9.12581e6 1.23521
\(560\) 0 0
\(561\) 0 0
\(562\) 1.69233e6 0.226019
\(563\) 4.25551e6 0.565823 0.282911 0.959146i \(-0.408700\pi\)
0.282911 + 0.959146i \(0.408700\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.78052e6 0.233617
\(567\) 0 0
\(568\) −1.74321e6 −0.226714
\(569\) −5.99899e6 −0.776779 −0.388389 0.921495i \(-0.626968\pi\)
−0.388389 + 0.921495i \(0.626968\pi\)
\(570\) 0 0
\(571\) 9.20283e6 1.18122 0.590611 0.806957i \(-0.298887\pi\)
0.590611 + 0.806957i \(0.298887\pi\)
\(572\) 8.65796e6 1.10643
\(573\) 0 0
\(574\) −1.89506e6 −0.240073
\(575\) 0 0
\(576\) 0 0
\(577\) −4.03879e6 −0.505024 −0.252512 0.967594i \(-0.581257\pi\)
−0.252512 + 0.967594i \(0.581257\pi\)
\(578\) 1.30259e7 1.62176
\(579\) 0 0
\(580\) 0 0
\(581\) 1.51192e7 1.85819
\(582\) 0 0
\(583\) −1.78542e7 −2.17555
\(584\) 419058. 0.0508442
\(585\) 0 0
\(586\) −9.83969e6 −1.18369
\(587\) −1.65997e7 −1.98840 −0.994200 0.107544i \(-0.965701\pi\)
−0.994200 + 0.107544i \(0.965701\pi\)
\(588\) 0 0
\(589\) −2.37774e6 −0.282407
\(590\) 0 0
\(591\) 0 0
\(592\) −268914. −0.0315362
\(593\) 1.09900e7 1.28340 0.641698 0.766957i \(-0.278230\pi\)
0.641698 + 0.766957i \(0.278230\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.91577e6 −0.797489
\(597\) 0 0
\(598\) 1.24128e7 1.41944
\(599\) −4.47167e6 −0.509217 −0.254608 0.967044i \(-0.581947\pi\)
−0.254608 + 0.967044i \(0.581947\pi\)
\(600\) 0 0
\(601\) 1.60311e7 1.81041 0.905206 0.424974i \(-0.139717\pi\)
0.905206 + 0.424974i \(0.139717\pi\)
\(602\) −1.06495e7 −1.19767
\(603\) 0 0
\(604\) −4.09316e6 −0.456527
\(605\) 0 0
\(606\) 0 0
\(607\) 720250. 0.0793435 0.0396718 0.999213i \(-0.487369\pi\)
0.0396718 + 0.999213i \(0.487369\pi\)
\(608\) 1.28434e6 0.140904
\(609\) 0 0
\(610\) 0 0
\(611\) −1.91045e7 −2.07030
\(612\) 0 0
\(613\) 1.46218e7 1.57163 0.785813 0.618464i \(-0.212245\pi\)
0.785813 + 0.618464i \(0.212245\pi\)
\(614\) −1.49701e6 −0.160252
\(615\) 0 0
\(616\) −1.01035e7 −1.07281
\(617\) −4.57269e6 −0.483569 −0.241784 0.970330i \(-0.577733\pi\)
−0.241784 + 0.970330i \(0.577733\pi\)
\(618\) 0 0
\(619\) −4.62486e6 −0.485145 −0.242573 0.970133i \(-0.577991\pi\)
−0.242573 + 0.970133i \(0.577991\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −8.90273e6 −0.922671
\(623\) −4.33184e6 −0.447149
\(624\) 0 0
\(625\) 0 0
\(626\) −1.22348e7 −1.24784
\(627\) 0 0
\(628\) 941397. 0.0952519
\(629\) 2.27157e6 0.228928
\(630\) 0 0
\(631\) −1.03670e7 −1.03653 −0.518263 0.855221i \(-0.673421\pi\)
−0.518263 + 0.855221i \(0.673421\pi\)
\(632\) −4.16239e6 −0.414524
\(633\) 0 0
\(634\) 2.08244e6 0.205755
\(635\) 0 0
\(636\) 0 0
\(637\) −3.44462e7 −3.36351
\(638\) 6.44595e6 0.626954
\(639\) 0 0
\(640\) 0 0
\(641\) 4.62432e6 0.444532 0.222266 0.974986i \(-0.428655\pi\)
0.222266 + 0.974986i \(0.428655\pi\)
\(642\) 0 0
\(643\) −1.13460e7 −1.08222 −0.541111 0.840951i \(-0.681996\pi\)
−0.541111 + 0.840951i \(0.681996\pi\)
\(644\) −1.44853e7 −1.37630
\(645\) 0 0
\(646\) −1.08491e7 −1.02285
\(647\) −1.10482e7 −1.03760 −0.518800 0.854895i \(-0.673621\pi\)
−0.518800 + 0.854895i \(0.673621\pi\)
\(648\) 0 0
\(649\) 2.30128e7 2.14466
\(650\) 0 0
\(651\) 0 0
\(652\) −2.48010e6 −0.228481
\(653\) 1.37432e6 0.126126 0.0630631 0.998010i \(-0.479913\pi\)
0.0630631 + 0.998010i \(0.479913\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −501902. −0.0455365
\(657\) 0 0
\(658\) 2.22943e7 2.00738
\(659\) 1.28061e7 1.14869 0.574344 0.818614i \(-0.305257\pi\)
0.574344 + 0.818614i \(0.305257\pi\)
\(660\) 0 0
\(661\) −104794. −0.00932894 −0.00466447 0.999989i \(-0.501485\pi\)
−0.00466447 + 0.999989i \(0.501485\pi\)
\(662\) −8.51021e6 −0.754737
\(663\) 0 0
\(664\) 4.00430e6 0.352457
\(665\) 0 0
\(666\) 0 0
\(667\) 9.24146e6 0.804315
\(668\) −9.41850e6 −0.816658
\(669\) 0 0
\(670\) 0 0
\(671\) −1.75432e6 −0.150419
\(672\) 0 0
\(673\) −1.80490e7 −1.53608 −0.768042 0.640400i \(-0.778769\pi\)
−0.768042 + 0.640400i \(0.778769\pi\)
\(674\) 1.23353e7 1.04593
\(675\) 0 0
\(676\) 5.03651e6 0.423899
\(677\) −1.29914e7 −1.08939 −0.544697 0.838633i \(-0.683355\pi\)
−0.544697 + 0.838633i \(0.683355\pi\)
\(678\) 0 0
\(679\) −2.31922e7 −1.93049
\(680\) 0 0
\(681\) 0 0
\(682\) 4.95397e6 0.407842
\(683\) 3.91843e6 0.321411 0.160705 0.987002i \(-0.448623\pi\)
0.160705 + 0.987002i \(0.448623\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 2.39520e7 1.94326
\(687\) 0 0
\(688\) −2.82050e6 −0.227172
\(689\) −2.26368e7 −1.81663
\(690\) 0 0
\(691\) −3.11077e6 −0.247840 −0.123920 0.992292i \(-0.539547\pi\)
−0.123920 + 0.992292i \(0.539547\pi\)
\(692\) 3.07278e6 0.243931
\(693\) 0 0
\(694\) 1.61982e7 1.27664
\(695\) 0 0
\(696\) 0 0
\(697\) 4.23967e6 0.330560
\(698\) −1.97254e6 −0.153245
\(699\) 0 0
\(700\) 0 0
\(701\) −1.94916e7 −1.49814 −0.749070 0.662490i \(-0.769500\pi\)
−0.749070 + 0.662490i \(0.769500\pi\)
\(702\) 0 0
\(703\) −1.31751e6 −0.100546
\(704\) −2.67590e6 −0.203488
\(705\) 0 0
\(706\) 5.39261e6 0.407181
\(707\) −1.00366e7 −0.755155
\(708\) 0 0
\(709\) −8.13693e6 −0.607918 −0.303959 0.952685i \(-0.598309\pi\)
−0.303959 + 0.952685i \(0.598309\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.14728e6 −0.0848143
\(713\) 7.10243e6 0.523218
\(714\) 0 0
\(715\) 0 0
\(716\) 5.61144e6 0.409065
\(717\) 0 0
\(718\) −6.58997e6 −0.477059
\(719\) 1.14993e7 0.829566 0.414783 0.909920i \(-0.363857\pi\)
0.414783 + 0.909920i \(0.363857\pi\)
\(720\) 0 0
\(721\) 9.99951e6 0.716375
\(722\) −3.61192e6 −0.257867
\(723\) 0 0
\(724\) 8.39419e6 0.595158
\(725\) 0 0
\(726\) 0 0
\(727\) 2.10897e7 1.47991 0.739953 0.672658i \(-0.234848\pi\)
0.739953 + 0.672658i \(0.234848\pi\)
\(728\) −1.28100e7 −0.895820
\(729\) 0 0
\(730\) 0 0
\(731\) 2.38253e7 1.64909
\(732\) 0 0
\(733\) −2.20274e7 −1.51427 −0.757134 0.653259i \(-0.773401\pi\)
−0.757134 + 0.653259i \(0.773401\pi\)
\(734\) 111673. 0.00765085
\(735\) 0 0
\(736\) −3.83640e6 −0.261053
\(737\) 3.14399e7 2.13213
\(738\) 0 0
\(739\) −1.26088e7 −0.849305 −0.424653 0.905356i \(-0.639604\pi\)
−0.424653 + 0.905356i \(0.639604\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.64164e7 1.76142
\(743\) 5.90968e6 0.392728 0.196364 0.980531i \(-0.437087\pi\)
0.196364 + 0.980531i \(0.437087\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.12214e7 −0.738247
\(747\) 0 0
\(748\) 2.26039e7 1.47716
\(749\) −2.75561e6 −0.179479
\(750\) 0 0
\(751\) −1.15585e7 −0.747830 −0.373915 0.927463i \(-0.621985\pi\)
−0.373915 + 0.927463i \(0.621985\pi\)
\(752\) 5.90459e6 0.380755
\(753\) 0 0
\(754\) 8.17265e6 0.523521
\(755\) 0 0
\(756\) 0 0
\(757\) −2.64988e7 −1.68068 −0.840342 0.542057i \(-0.817646\pi\)
−0.840342 + 0.542057i \(0.817646\pi\)
\(758\) 6.80056e6 0.429904
\(759\) 0 0
\(760\) 0 0
\(761\) −1.54716e6 −0.0968442 −0.0484221 0.998827i \(-0.515419\pi\)
−0.0484221 + 0.998827i \(0.515419\pi\)
\(762\) 0 0
\(763\) 2.37691e7 1.47809
\(764\) −1.01860e6 −0.0631352
\(765\) 0 0
\(766\) 2.51165e6 0.154663
\(767\) 2.91773e7 1.79084
\(768\) 0 0
\(769\) 1.16191e7 0.708529 0.354265 0.935145i \(-0.384731\pi\)
0.354265 + 0.935145i \(0.384731\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.14796e7 0.693242
\(773\) 1.16233e7 0.699649 0.349824 0.936815i \(-0.386241\pi\)
0.349824 + 0.936815i \(0.386241\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −6.14239e6 −0.366171
\(777\) 0 0
\(778\) −1.25736e7 −0.744748
\(779\) −2.45901e6 −0.145183
\(780\) 0 0
\(781\) 1.77943e7 1.04388
\(782\) 3.24068e7 1.89504
\(783\) 0 0
\(784\) 1.06462e7 0.618594
\(785\) 0 0
\(786\) 0 0
\(787\) −2.03314e7 −1.17012 −0.585059 0.810991i \(-0.698929\pi\)
−0.585059 + 0.810991i \(0.698929\pi\)
\(788\) −4.59945e6 −0.263870
\(789\) 0 0
\(790\) 0 0
\(791\) 2.43503e6 0.138377
\(792\) 0 0
\(793\) −2.22425e6 −0.125603
\(794\) 4.43092e6 0.249427
\(795\) 0 0
\(796\) 1.02979e7 0.576058
\(797\) 6.32170e6 0.352523 0.176262 0.984343i \(-0.443599\pi\)
0.176262 + 0.984343i \(0.443599\pi\)
\(798\) 0 0
\(799\) −4.98772e7 −2.76398
\(800\) 0 0
\(801\) 0 0
\(802\) −9.66252e6 −0.530462
\(803\) −4.27764e6 −0.234107
\(804\) 0 0
\(805\) 0 0
\(806\) 6.28100e6 0.340558
\(807\) 0 0
\(808\) −2.65816e6 −0.143236
\(809\) 1.84549e7 0.991380 0.495690 0.868499i \(-0.334915\pi\)
0.495690 + 0.868499i \(0.334915\pi\)
\(810\) 0 0
\(811\) 3.00263e7 1.60306 0.801530 0.597954i \(-0.204020\pi\)
0.801530 + 0.597954i \(0.204020\pi\)
\(812\) −9.53719e6 −0.507610
\(813\) 0 0
\(814\) 2.74501e6 0.145205
\(815\) 0 0
\(816\) 0 0
\(817\) −1.38187e7 −0.724289
\(818\) 1.47932e7 0.772999
\(819\) 0 0
\(820\) 0 0
\(821\) 2.51609e7 1.30277 0.651387 0.758746i \(-0.274188\pi\)
0.651387 + 0.758746i \(0.274188\pi\)
\(822\) 0 0
\(823\) 7.97388e6 0.410365 0.205182 0.978724i \(-0.434221\pi\)
0.205182 + 0.978724i \(0.434221\pi\)
\(824\) 2.64835e6 0.135880
\(825\) 0 0
\(826\) −3.40489e7 −1.73641
\(827\) 1.37135e7 0.697243 0.348622 0.937264i \(-0.386650\pi\)
0.348622 + 0.937264i \(0.386650\pi\)
\(828\) 0 0
\(829\) −1.62566e7 −0.821569 −0.410785 0.911732i \(-0.634745\pi\)
−0.410785 + 0.911732i \(0.634745\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −3.39270e6 −0.169917
\(833\) −8.99307e7 −4.49051
\(834\) 0 0
\(835\) 0 0
\(836\) −1.31103e7 −0.648777
\(837\) 0 0
\(838\) 1.38191e7 0.679782
\(839\) 2.36734e7 1.16106 0.580532 0.814237i \(-0.302844\pi\)
0.580532 + 0.814237i \(0.302844\pi\)
\(840\) 0 0
\(841\) −1.44265e7 −0.703350
\(842\) −1.93592e7 −0.941039
\(843\) 0 0
\(844\) −1.39812e7 −0.675600
\(845\) 0 0
\(846\) 0 0
\(847\) 6.42168e7 3.07567
\(848\) 6.99632e6 0.334103
\(849\) 0 0
\(850\) 0 0
\(851\) 3.93547e6 0.186283
\(852\) 0 0
\(853\) 2.53714e7 1.19391 0.596956 0.802274i \(-0.296377\pi\)
0.596956 + 0.802274i \(0.296377\pi\)
\(854\) 2.59562e6 0.121786
\(855\) 0 0
\(856\) −729817. −0.0340431
\(857\) −2.37850e7 −1.10624 −0.553122 0.833100i \(-0.686564\pi\)
−0.553122 + 0.833100i \(0.686564\pi\)
\(858\) 0 0
\(859\) −8.55700e6 −0.395675 −0.197838 0.980235i \(-0.563392\pi\)
−0.197838 + 0.980235i \(0.563392\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 2.21969e7 1.01748
\(863\) −3.91623e7 −1.78995 −0.894977 0.446113i \(-0.852808\pi\)
−0.894977 + 0.446113i \(0.852808\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 3.25721e6 0.147588
\(867\) 0 0
\(868\) −7.32971e6 −0.330208
\(869\) 4.24886e7 1.90864
\(870\) 0 0
\(871\) 3.98618e7 1.78038
\(872\) 6.29519e6 0.280361
\(873\) 0 0
\(874\) −1.87960e7 −0.832311
\(875\) 0 0
\(876\) 0 0
\(877\) 2.35047e7 1.03194 0.515972 0.856605i \(-0.327431\pi\)
0.515972 + 0.856605i \(0.327431\pi\)
\(878\) −1.60985e7 −0.704771
\(879\) 0 0
\(880\) 0 0
\(881\) −4.50902e7 −1.95723 −0.978617 0.205691i \(-0.934056\pi\)
−0.978617 + 0.205691i \(0.934056\pi\)
\(882\) 0 0
\(883\) −1.95876e7 −0.845434 −0.422717 0.906262i \(-0.638924\pi\)
−0.422717 + 0.906262i \(0.638924\pi\)
\(884\) 2.86588e7 1.23347
\(885\) 0 0
\(886\) 1.78553e7 0.764157
\(887\) −9.54399e6 −0.407306 −0.203653 0.979043i \(-0.565281\pi\)
−0.203653 + 0.979043i \(0.565281\pi\)
\(888\) 0 0
\(889\) 6.16515e7 2.61631
\(890\) 0 0
\(891\) 0 0
\(892\) 4.61594e6 0.194244
\(893\) 2.89288e7 1.21395
\(894\) 0 0
\(895\) 0 0
\(896\) 3.95916e6 0.164753
\(897\) 0 0
\(898\) 1.86756e7 0.772828
\(899\) 4.67628e6 0.192975
\(900\) 0 0
\(901\) −5.90992e7 −2.42532
\(902\) 5.12330e6 0.209668
\(903\) 0 0
\(904\) 644912. 0.0262470
\(905\) 0 0
\(906\) 0 0
\(907\) 1.08034e7 0.436054 0.218027 0.975943i \(-0.430038\pi\)
0.218027 + 0.975943i \(0.430038\pi\)
\(908\) 2.00573e6 0.0807343
\(909\) 0 0
\(910\) 0 0
\(911\) 3.88914e7 1.55259 0.776296 0.630368i \(-0.217096\pi\)
0.776296 + 0.630368i \(0.217096\pi\)
\(912\) 0 0
\(913\) −4.08749e7 −1.62286
\(914\) 1.88536e7 0.746498
\(915\) 0 0
\(916\) −1.11709e7 −0.439897
\(917\) 1.58118e7 0.620953
\(918\) 0 0
\(919\) 2.89714e7 1.13157 0.565785 0.824553i \(-0.308573\pi\)
0.565785 + 0.824553i \(0.308573\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −2.49590e7 −0.966941
\(923\) 2.25609e7 0.871668
\(924\) 0 0
\(925\) 0 0
\(926\) 1.33597e7 0.511999
\(927\) 0 0
\(928\) −2.52590e6 −0.0962824
\(929\) 3.61023e7 1.37245 0.686224 0.727391i \(-0.259267\pi\)
0.686224 + 0.727391i \(0.259267\pi\)
\(930\) 0 0
\(931\) 5.21599e7 1.97225
\(932\) 1.57534e7 0.594066
\(933\) 0 0
\(934\) −2.88999e6 −0.108400
\(935\) 0 0
\(936\) 0 0
\(937\) −1.02966e7 −0.383129 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(938\) −4.65173e7 −1.72627
\(939\) 0 0
\(940\) 0 0
\(941\) −1.55696e7 −0.573195 −0.286597 0.958051i \(-0.592524\pi\)
−0.286597 + 0.958051i \(0.592524\pi\)
\(942\) 0 0
\(943\) 7.34519e6 0.268982
\(944\) −9.01778e6 −0.329359
\(945\) 0 0
\(946\) 2.87910e7 1.04599
\(947\) 3.54260e7 1.28365 0.641826 0.766850i \(-0.278177\pi\)
0.641826 + 0.766850i \(0.278177\pi\)
\(948\) 0 0
\(949\) −5.42350e6 −0.195485
\(950\) 0 0
\(951\) 0 0
\(952\) −3.34438e7 −1.19598
\(953\) −5.47490e6 −0.195274 −0.0976369 0.995222i \(-0.531128\pi\)
−0.0976369 + 0.995222i \(0.531128\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −2.09679e6 −0.0742011
\(957\) 0 0
\(958\) 2.15086e7 0.757177
\(959\) −1.62760e7 −0.571479
\(960\) 0 0
\(961\) −2.50352e7 −0.874467
\(962\) 3.48032e6 0.121250
\(963\) 0 0
\(964\) −1.97811e7 −0.685579
\(965\) 0 0
\(966\) 0 0
\(967\) −1.65533e7 −0.569269 −0.284635 0.958636i \(-0.591872\pi\)
−0.284635 + 0.958636i \(0.591872\pi\)
\(968\) 1.70077e7 0.583387
\(969\) 0 0
\(970\) 0 0
\(971\) −1.83751e6 −0.0625435 −0.0312718 0.999511i \(-0.509956\pi\)
−0.0312718 + 0.999511i \(0.509956\pi\)
\(972\) 0 0
\(973\) −2.26757e7 −0.767855
\(974\) 6.88338e6 0.232490
\(975\) 0 0
\(976\) 687445. 0.0231001
\(977\) −516874. −0.0173240 −0.00866200 0.999962i \(-0.502757\pi\)
−0.00866200 + 0.999962i \(0.502757\pi\)
\(978\) 0 0
\(979\) 1.17111e7 0.390519
\(980\) 0 0
\(981\) 0 0
\(982\) −2.92580e7 −0.968201
\(983\) 4.15798e7 1.37246 0.686229 0.727386i \(-0.259265\pi\)
0.686229 + 0.727386i \(0.259265\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 2.13368e7 0.698935
\(987\) 0 0
\(988\) −1.66221e7 −0.541744
\(989\) 4.12771e7 1.34190
\(990\) 0 0
\(991\) −1.61859e7 −0.523544 −0.261772 0.965130i \(-0.584307\pi\)
−0.261772 + 0.965130i \(0.584307\pi\)
\(992\) −1.94126e6 −0.0626331
\(993\) 0 0
\(994\) −2.63277e7 −0.845176
\(995\) 0 0
\(996\) 0 0
\(997\) −2.74426e7 −0.874355 −0.437178 0.899375i \(-0.644022\pi\)
−0.437178 + 0.899375i \(0.644022\pi\)
\(998\) −2.91951e7 −0.927863
\(999\) 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 450.6.a.bf.1.2 yes 2
3.2 odd 2 450.6.a.ba.1.2 yes 2
5.2 odd 4 450.6.c.p.199.4 4
5.3 odd 4 450.6.c.p.199.1 4
5.4 even 2 450.6.a.y.1.1 2
15.2 even 4 450.6.c.q.199.2 4
15.8 even 4 450.6.c.q.199.3 4
15.14 odd 2 450.6.a.bd.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.6.a.y.1.1 2 5.4 even 2
450.6.a.ba.1.2 yes 2 3.2 odd 2
450.6.a.bd.1.1 yes 2 15.14 odd 2
450.6.a.bf.1.2 yes 2 1.1 even 1 trivial
450.6.c.p.199.1 4 5.3 odd 4
450.6.c.p.199.4 4 5.2 odd 4
450.6.c.q.199.2 4 15.2 even 4
450.6.c.q.199.3 4 15.8 even 4