Properties

Label 825.6.a.s.1.3
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - x^{8} - 229 x^{7} + 267 x^{6} + 16434 x^{5} - 16568 x^{4} - 405504 x^{3} + 202288 x^{2} + 2184608 x + 1190400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-5.77674\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.77674 q^{2} -9.00000 q^{3} +1.37069 q^{4} +51.9906 q^{6} -150.201 q^{7} +176.937 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-5.77674 q^{2} -9.00000 q^{3} +1.37069 q^{4} +51.9906 q^{6} -150.201 q^{7} +176.937 q^{8} +81.0000 q^{9} -121.000 q^{11} -12.3362 q^{12} +1146.84 q^{13} +867.675 q^{14} -1065.98 q^{16} -1570.59 q^{17} -467.916 q^{18} +731.596 q^{19} +1351.81 q^{21} +698.985 q^{22} +2182.68 q^{23} -1592.44 q^{24} -6624.97 q^{26} -729.000 q^{27} -205.880 q^{28} +8603.71 q^{29} +6700.21 q^{31} +495.906 q^{32} +1089.00 q^{33} +9072.88 q^{34} +111.026 q^{36} -3056.37 q^{37} -4226.24 q^{38} -10321.5 q^{39} -440.525 q^{41} -7809.07 q^{42} -13592.0 q^{43} -165.854 q^{44} -12608.8 q^{46} +15365.3 q^{47} +9593.85 q^{48} +5753.49 q^{49} +14135.3 q^{51} +1571.96 q^{52} -25436.6 q^{53} +4211.24 q^{54} -26576.3 q^{56} -6584.36 q^{57} -49701.4 q^{58} -8795.36 q^{59} +15256.9 q^{61} -38705.3 q^{62} -12166.3 q^{63} +31246.7 q^{64} -6290.87 q^{66} -46228.1 q^{67} -2152.79 q^{68} -19644.2 q^{69} +68040.5 q^{71} +14331.9 q^{72} -12850.7 q^{73} +17655.8 q^{74} +1002.79 q^{76} +18174.4 q^{77} +59624.8 q^{78} +10936.2 q^{79} +6561.00 q^{81} +2544.80 q^{82} +66842.2 q^{83} +1852.92 q^{84} +78517.1 q^{86} -77433.4 q^{87} -21409.4 q^{88} -66587.8 q^{89} -172257. q^{91} +2991.78 q^{92} -60301.9 q^{93} -88761.4 q^{94} -4463.16 q^{96} +145903. q^{97} -33236.4 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9} - 1089 q^{11} - 1539 q^{12} + 723 q^{13} + 720 q^{14} + 4147 q^{16} - 2804 q^{17} + 81 q^{18} - 1601 q^{19} + 513 q^{21} - 121 q^{22} - 2392 q^{23} + 1593 q^{24} - 1987 q^{26} - 6561 q^{27} + 4436 q^{28} + 5966 q^{29} + 21575 q^{31} - 25493 q^{32} + 9801 q^{33} - 3098 q^{34} + 13851 q^{36} - 10228 q^{37} + 12765 q^{38} - 6507 q^{39} + 13304 q^{41} - 6480 q^{42} - 13829 q^{43} - 20691 q^{44} + 81283 q^{46} - 13998 q^{47} - 37323 q^{48} - 19468 q^{49} + 25236 q^{51} + 37131 q^{52} - 44166 q^{53} - 729 q^{54} + 79160 q^{56} + 14409 q^{57} - 7635 q^{58} + 11626 q^{59} + 49481 q^{61} - 94479 q^{62} - 4617 q^{63} + 109367 q^{64} + 1089 q^{66} + 26567 q^{67} - 119506 q^{68} + 21528 q^{69} + 78454 q^{71} - 14337 q^{72} - 100086 q^{73} + 162360 q^{74} + 194115 q^{76} + 6897 q^{77} + 17883 q^{78} - 8478 q^{79} + 59049 q^{81} - 52700 q^{82} - 157476 q^{83} - 39924 q^{84} + 251663 q^{86} - 53694 q^{87} + 21417 q^{88} - 65548 q^{89} - 106849 q^{91} - 350115 q^{92} - 194175 q^{93} - 35742 q^{94} + 229437 q^{96} + 116757 q^{97} - 14949 q^{98} - 88209 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.77674 −1.02119 −0.510596 0.859821i \(-0.670575\pi\)
−0.510596 + 0.859821i \(0.670575\pi\)
\(3\) −9.00000 −0.577350
\(4\) 1.37069 0.0428341
\(5\) 0 0
\(6\) 51.9906 0.589586
\(7\) −150.201 −1.15859 −0.579294 0.815119i \(-0.696672\pi\)
−0.579294 + 0.815119i \(0.696672\pi\)
\(8\) 176.937 0.977451
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −12.3362 −0.0247303
\(13\) 1146.84 1.88210 0.941051 0.338265i \(-0.109840\pi\)
0.941051 + 0.338265i \(0.109840\pi\)
\(14\) 867.675 1.18314
\(15\) 0 0
\(16\) −1065.98 −1.04100
\(17\) −1570.59 −1.31808 −0.659038 0.752110i \(-0.729036\pi\)
−0.659038 + 0.752110i \(0.729036\pi\)
\(18\) −467.916 −0.340397
\(19\) 731.596 0.464930 0.232465 0.972605i \(-0.425321\pi\)
0.232465 + 0.972605i \(0.425321\pi\)
\(20\) 0 0
\(21\) 1351.81 0.668911
\(22\) 698.985 0.307901
\(23\) 2182.68 0.860342 0.430171 0.902747i \(-0.358453\pi\)
0.430171 + 0.902747i \(0.358453\pi\)
\(24\) −1592.44 −0.564331
\(25\) 0 0
\(26\) −6624.97 −1.92199
\(27\) −729.000 −0.192450
\(28\) −205.880 −0.0496271
\(29\) 8603.71 1.89973 0.949863 0.312667i \(-0.101222\pi\)
0.949863 + 0.312667i \(0.101222\pi\)
\(30\) 0 0
\(31\) 6700.21 1.25223 0.626115 0.779731i \(-0.284644\pi\)
0.626115 + 0.779731i \(0.284644\pi\)
\(32\) 495.906 0.0856100
\(33\) 1089.00 0.174078
\(34\) 9072.88 1.34601
\(35\) 0 0
\(36\) 111.026 0.0142780
\(37\) −3056.37 −0.367029 −0.183515 0.983017i \(-0.558748\pi\)
−0.183515 + 0.983017i \(0.558748\pi\)
\(38\) −4226.24 −0.474783
\(39\) −10321.5 −1.08663
\(40\) 0 0
\(41\) −440.525 −0.0409271 −0.0204636 0.999791i \(-0.506514\pi\)
−0.0204636 + 0.999791i \(0.506514\pi\)
\(42\) −7809.07 −0.683087
\(43\) −13592.0 −1.12101 −0.560507 0.828150i \(-0.689394\pi\)
−0.560507 + 0.828150i \(0.689394\pi\)
\(44\) −165.854 −0.0129150
\(45\) 0 0
\(46\) −12608.8 −0.878575
\(47\) 15365.3 1.01460 0.507302 0.861768i \(-0.330643\pi\)
0.507302 + 0.861768i \(0.330643\pi\)
\(48\) 9593.85 0.601021
\(49\) 5753.49 0.342327
\(50\) 0 0
\(51\) 14135.3 0.760991
\(52\) 1571.96 0.0806181
\(53\) −25436.6 −1.24385 −0.621926 0.783076i \(-0.713650\pi\)
−0.621926 + 0.783076i \(0.713650\pi\)
\(54\) 4211.24 0.196529
\(55\) 0 0
\(56\) −26576.3 −1.13246
\(57\) −6584.36 −0.268427
\(58\) −49701.4 −1.93999
\(59\) −8795.36 −0.328945 −0.164473 0.986382i \(-0.552592\pi\)
−0.164473 + 0.986382i \(0.552592\pi\)
\(60\) 0 0
\(61\) 15256.9 0.524977 0.262489 0.964935i \(-0.415457\pi\)
0.262489 + 0.964935i \(0.415457\pi\)
\(62\) −38705.3 −1.27877
\(63\) −12166.3 −0.386196
\(64\) 31246.7 0.953575
\(65\) 0 0
\(66\) −6290.87 −0.177767
\(67\) −46228.1 −1.25811 −0.629056 0.777360i \(-0.716558\pi\)
−0.629056 + 0.777360i \(0.716558\pi\)
\(68\) −2152.79 −0.0564585
\(69\) −19644.2 −0.496719
\(70\) 0 0
\(71\) 68040.5 1.60185 0.800924 0.598766i \(-0.204342\pi\)
0.800924 + 0.598766i \(0.204342\pi\)
\(72\) 14331.9 0.325817
\(73\) −12850.7 −0.282240 −0.141120 0.989993i \(-0.545070\pi\)
−0.141120 + 0.989993i \(0.545070\pi\)
\(74\) 17655.8 0.374808
\(75\) 0 0
\(76\) 1002.79 0.0199148
\(77\) 18174.4 0.349328
\(78\) 59624.8 1.10966
\(79\) 10936.2 0.197151 0.0985756 0.995130i \(-0.468571\pi\)
0.0985756 + 0.995130i \(0.468571\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 2544.80 0.0417944
\(83\) 66842.2 1.06501 0.532507 0.846426i \(-0.321250\pi\)
0.532507 + 0.846426i \(0.321250\pi\)
\(84\) 1852.92 0.0286522
\(85\) 0 0
\(86\) 78517.1 1.14477
\(87\) −77433.4 −1.09681
\(88\) −21409.4 −0.294712
\(89\) −66587.8 −0.891085 −0.445543 0.895261i \(-0.646989\pi\)
−0.445543 + 0.895261i \(0.646989\pi\)
\(90\) 0 0
\(91\) −172257. −2.18058
\(92\) 2991.78 0.0368520
\(93\) −60301.9 −0.722975
\(94\) −88761.4 −1.03611
\(95\) 0 0
\(96\) −4463.16 −0.0494270
\(97\) 145903. 1.57447 0.787236 0.616652i \(-0.211511\pi\)
0.787236 + 0.616652i \(0.211511\pi\)
\(98\) −33236.4 −0.349582
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) −125698. −1.22610 −0.613050 0.790044i \(-0.710057\pi\)
−0.613050 + 0.790044i \(0.710057\pi\)
\(102\) −81655.9 −0.777118
\(103\) −63242.3 −0.587374 −0.293687 0.955902i \(-0.594882\pi\)
−0.293687 + 0.955902i \(0.594882\pi\)
\(104\) 202918. 1.83966
\(105\) 0 0
\(106\) 146940. 1.27021
\(107\) −103931. −0.877582 −0.438791 0.898589i \(-0.644593\pi\)
−0.438791 + 0.898589i \(0.644593\pi\)
\(108\) −999.233 −0.00824342
\(109\) −115234. −0.928995 −0.464498 0.885574i \(-0.653765\pi\)
−0.464498 + 0.885574i \(0.653765\pi\)
\(110\) 0 0
\(111\) 27507.3 0.211905
\(112\) 160112. 1.20609
\(113\) −200077. −1.47401 −0.737005 0.675887i \(-0.763761\pi\)
−0.737005 + 0.675887i \(0.763761\pi\)
\(114\) 38036.1 0.274116
\(115\) 0 0
\(116\) 11793.0 0.0813730
\(117\) 92893.7 0.627367
\(118\) 50808.5 0.335917
\(119\) 235905. 1.52711
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −88134.9 −0.536103
\(123\) 3964.73 0.0236293
\(124\) 9183.91 0.0536381
\(125\) 0 0
\(126\) 70281.6 0.394381
\(127\) −304565. −1.67560 −0.837802 0.545974i \(-0.816160\pi\)
−0.837802 + 0.545974i \(0.816160\pi\)
\(128\) −196373. −1.05939
\(129\) 122328. 0.647217
\(130\) 0 0
\(131\) 108252. 0.551135 0.275567 0.961282i \(-0.411134\pi\)
0.275567 + 0.961282i \(0.411134\pi\)
\(132\) 1492.68 0.00745646
\(133\) −109887. −0.538662
\(134\) 267048. 1.28477
\(135\) 0 0
\(136\) −277896. −1.28835
\(137\) −329333. −1.49911 −0.749557 0.661940i \(-0.769733\pi\)
−0.749557 + 0.661940i \(0.769733\pi\)
\(138\) 113479. 0.507245
\(139\) 40752.3 0.178902 0.0894510 0.995991i \(-0.471489\pi\)
0.0894510 + 0.995991i \(0.471489\pi\)
\(140\) 0 0
\(141\) −138288. −0.585782
\(142\) −393052. −1.63580
\(143\) −138767. −0.567475
\(144\) −86344.6 −0.347000
\(145\) 0 0
\(146\) 74234.9 0.288221
\(147\) −51781.4 −0.197643
\(148\) −4189.33 −0.0157214
\(149\) 9127.01 0.0336793 0.0168397 0.999858i \(-0.494640\pi\)
0.0168397 + 0.999858i \(0.494640\pi\)
\(150\) 0 0
\(151\) −273166. −0.974955 −0.487478 0.873135i \(-0.662083\pi\)
−0.487478 + 0.873135i \(0.662083\pi\)
\(152\) 129447. 0.454446
\(153\) −127218. −0.439358
\(154\) −104989. −0.356731
\(155\) 0 0
\(156\) −14147.6 −0.0465449
\(157\) 59757.7 0.193484 0.0967419 0.995309i \(-0.469158\pi\)
0.0967419 + 0.995309i \(0.469158\pi\)
\(158\) −63175.7 −0.201329
\(159\) 228929. 0.718138
\(160\) 0 0
\(161\) −327842. −0.996782
\(162\) −37901.2 −0.113466
\(163\) 101366. 0.298828 0.149414 0.988775i \(-0.452261\pi\)
0.149414 + 0.988775i \(0.452261\pi\)
\(164\) −603.823 −0.00175307
\(165\) 0 0
\(166\) −386130. −1.08758
\(167\) 332221. 0.921799 0.460900 0.887452i \(-0.347527\pi\)
0.460900 + 0.887452i \(0.347527\pi\)
\(168\) 239186. 0.653828
\(169\) 943941. 2.54231
\(170\) 0 0
\(171\) 59259.3 0.154977
\(172\) −18630.4 −0.0480176
\(173\) −156146. −0.396658 −0.198329 0.980135i \(-0.563551\pi\)
−0.198329 + 0.980135i \(0.563551\pi\)
\(174\) 447313. 1.12005
\(175\) 0 0
\(176\) 128984. 0.313873
\(177\) 79158.3 0.189917
\(178\) 384660. 0.909970
\(179\) 798293. 1.86221 0.931107 0.364746i \(-0.118844\pi\)
0.931107 + 0.364746i \(0.118844\pi\)
\(180\) 0 0
\(181\) 534674. 1.21309 0.606544 0.795050i \(-0.292555\pi\)
0.606544 + 0.795050i \(0.292555\pi\)
\(182\) 995081. 2.22679
\(183\) −137312. −0.303096
\(184\) 386198. 0.840942
\(185\) 0 0
\(186\) 348348. 0.738297
\(187\) 190041. 0.397415
\(188\) 21061.1 0.0434596
\(189\) 109497. 0.222970
\(190\) 0 0
\(191\) −226409. −0.449066 −0.224533 0.974466i \(-0.572086\pi\)
−0.224533 + 0.974466i \(0.572086\pi\)
\(192\) −281221. −0.550547
\(193\) −78423.8 −0.151550 −0.0757748 0.997125i \(-0.524143\pi\)
−0.0757748 + 0.997125i \(0.524143\pi\)
\(194\) −842843. −1.60784
\(195\) 0 0
\(196\) 7886.25 0.0146633
\(197\) 79280.1 0.145545 0.0727727 0.997349i \(-0.476815\pi\)
0.0727727 + 0.997349i \(0.476815\pi\)
\(198\) 56617.8 0.102634
\(199\) 750688. 1.34378 0.671888 0.740653i \(-0.265484\pi\)
0.671888 + 0.740653i \(0.265484\pi\)
\(200\) 0 0
\(201\) 416053. 0.726372
\(202\) 726126. 1.25208
\(203\) −1.29229e6 −2.20100
\(204\) 19375.1 0.0325963
\(205\) 0 0
\(206\) 365334. 0.599822
\(207\) 176797. 0.286781
\(208\) −1.22251e6 −1.95927
\(209\) −88523.1 −0.140182
\(210\) 0 0
\(211\) 648787. 1.00322 0.501610 0.865094i \(-0.332741\pi\)
0.501610 + 0.865094i \(0.332741\pi\)
\(212\) −34865.7 −0.0532793
\(213\) −612364. −0.924828
\(214\) 600385. 0.896180
\(215\) 0 0
\(216\) −128987. −0.188110
\(217\) −1.00638e6 −1.45082
\(218\) 665675. 0.948683
\(219\) 115656. 0.162951
\(220\) 0 0
\(221\) −1.80121e6 −2.48075
\(222\) −158902. −0.216395
\(223\) 880655. 1.18589 0.592944 0.805244i \(-0.297966\pi\)
0.592944 + 0.805244i \(0.297966\pi\)
\(224\) −74485.9 −0.0991868
\(225\) 0 0
\(226\) 1.15579e6 1.50525
\(227\) −22215.5 −0.0286148 −0.0143074 0.999898i \(-0.504554\pi\)
−0.0143074 + 0.999898i \(0.504554\pi\)
\(228\) −9025.12 −0.0114978
\(229\) −753330. −0.949285 −0.474642 0.880179i \(-0.657423\pi\)
−0.474642 + 0.880179i \(0.657423\pi\)
\(230\) 0 0
\(231\) −163569. −0.201684
\(232\) 1.52232e6 1.85689
\(233\) 727015. 0.877311 0.438655 0.898655i \(-0.355455\pi\)
0.438655 + 0.898655i \(0.355455\pi\)
\(234\) −536623. −0.640663
\(235\) 0 0
\(236\) −12055.7 −0.0140901
\(237\) −98426.0 −0.113825
\(238\) −1.36276e6 −1.55947
\(239\) −1.37068e6 −1.55217 −0.776086 0.630627i \(-0.782798\pi\)
−0.776086 + 0.630627i \(0.782798\pi\)
\(240\) 0 0
\(241\) 1.26915e6 1.40758 0.703788 0.710410i \(-0.251491\pi\)
0.703788 + 0.710410i \(0.251491\pi\)
\(242\) −84577.2 −0.0928357
\(243\) −59049.0 −0.0641500
\(244\) 20912.4 0.0224869
\(245\) 0 0
\(246\) −22903.2 −0.0241300
\(247\) 839021. 0.875045
\(248\) 1.18552e6 1.22399
\(249\) −601580. −0.614886
\(250\) 0 0
\(251\) −1.22546e6 −1.22777 −0.613884 0.789396i \(-0.710394\pi\)
−0.613884 + 0.789396i \(0.710394\pi\)
\(252\) −16676.3 −0.0165424
\(253\) −264105. −0.259403
\(254\) 1.75939e6 1.71111
\(255\) 0 0
\(256\) 134501. 0.128270
\(257\) 704646. 0.665485 0.332742 0.943018i \(-0.392026\pi\)
0.332742 + 0.943018i \(0.392026\pi\)
\(258\) −706654. −0.660933
\(259\) 459071. 0.425236
\(260\) 0 0
\(261\) 696901. 0.633242
\(262\) −625343. −0.562815
\(263\) −225428. −0.200964 −0.100482 0.994939i \(-0.532039\pi\)
−0.100482 + 0.994939i \(0.532039\pi\)
\(264\) 192685. 0.170152
\(265\) 0 0
\(266\) 634787. 0.550078
\(267\) 599290. 0.514468
\(268\) −63364.5 −0.0538901
\(269\) 177633. 0.149673 0.0748365 0.997196i \(-0.476157\pi\)
0.0748365 + 0.997196i \(0.476157\pi\)
\(270\) 0 0
\(271\) 1.72248e6 1.42473 0.712363 0.701812i \(-0.247625\pi\)
0.712363 + 0.701812i \(0.247625\pi\)
\(272\) 1.67422e6 1.37212
\(273\) 1.55031e6 1.25896
\(274\) 1.90247e6 1.53088
\(275\) 0 0
\(276\) −26926.0 −0.0212765
\(277\) 2.32275e6 1.81888 0.909438 0.415839i \(-0.136512\pi\)
0.909438 + 0.415839i \(0.136512\pi\)
\(278\) −235415. −0.182693
\(279\) 542717. 0.417410
\(280\) 0 0
\(281\) 993777. 0.750798 0.375399 0.926863i \(-0.377506\pi\)
0.375399 + 0.926863i \(0.377506\pi\)
\(282\) 798852. 0.598196
\(283\) 1.99675e6 1.48203 0.741017 0.671486i \(-0.234344\pi\)
0.741017 + 0.671486i \(0.234344\pi\)
\(284\) 93262.4 0.0686137
\(285\) 0 0
\(286\) 801622. 0.579501
\(287\) 66167.5 0.0474177
\(288\) 40168.4 0.0285367
\(289\) 1.04689e6 0.737322
\(290\) 0 0
\(291\) −1.31313e6 −0.909022
\(292\) −17614.3 −0.0120895
\(293\) −2.10760e6 −1.43423 −0.717116 0.696954i \(-0.754538\pi\)
−0.717116 + 0.696954i \(0.754538\pi\)
\(294\) 299127. 0.201831
\(295\) 0 0
\(296\) −540786. −0.358753
\(297\) 88209.0 0.0580259
\(298\) −52724.4 −0.0343930
\(299\) 2.50318e6 1.61925
\(300\) 0 0
\(301\) 2.04153e6 1.29879
\(302\) 1.57801e6 0.995617
\(303\) 1.13128e6 0.707889
\(304\) −779869. −0.483991
\(305\) 0 0
\(306\) 734903. 0.448670
\(307\) 831340. 0.503423 0.251711 0.967802i \(-0.419007\pi\)
0.251711 + 0.967802i \(0.419007\pi\)
\(308\) 24911.4 0.0149631
\(309\) 569181. 0.339120
\(310\) 0 0
\(311\) −985788. −0.577940 −0.288970 0.957338i \(-0.593313\pi\)
−0.288970 + 0.957338i \(0.593313\pi\)
\(312\) −1.82626e6 −1.06213
\(313\) 3.23805e6 1.86820 0.934098 0.357018i \(-0.116207\pi\)
0.934098 + 0.357018i \(0.116207\pi\)
\(314\) −345205. −0.197584
\(315\) 0 0
\(316\) 14990.2 0.00844479
\(317\) −1.92482e6 −1.07582 −0.537912 0.843001i \(-0.680787\pi\)
−0.537912 + 0.843001i \(0.680787\pi\)
\(318\) −1.32246e6 −0.733357
\(319\) −1.04105e6 −0.572789
\(320\) 0 0
\(321\) 935383. 0.506672
\(322\) 1.89386e6 1.01791
\(323\) −1.14904e6 −0.612812
\(324\) 8993.10 0.00475934
\(325\) 0 0
\(326\) −585563. −0.305161
\(327\) 1.03710e6 0.536356
\(328\) −77945.4 −0.0400042
\(329\) −2.30789e6 −1.17551
\(330\) 0 0
\(331\) −481029. −0.241324 −0.120662 0.992694i \(-0.538502\pi\)
−0.120662 + 0.992694i \(0.538502\pi\)
\(332\) 91619.9 0.0456189
\(333\) −247566. −0.122343
\(334\) −1.91916e6 −0.941334
\(335\) 0 0
\(336\) −1.44101e6 −0.696336
\(337\) −2.39601e6 −1.14925 −0.574623 0.818418i \(-0.694851\pi\)
−0.574623 + 0.818418i \(0.694851\pi\)
\(338\) −5.45290e6 −2.59618
\(339\) 1.80069e6 0.851020
\(340\) 0 0
\(341\) −810725. −0.377562
\(342\) −342325. −0.158261
\(343\) 1.66025e6 0.761972
\(344\) −2.40493e6 −1.09574
\(345\) 0 0
\(346\) 902016. 0.405064
\(347\) 2.16490e6 0.965192 0.482596 0.875843i \(-0.339694\pi\)
0.482596 + 0.875843i \(0.339694\pi\)
\(348\) −106137. −0.0469807
\(349\) 2.95271e6 1.29765 0.648824 0.760938i \(-0.275261\pi\)
0.648824 + 0.760938i \(0.275261\pi\)
\(350\) 0 0
\(351\) −836044. −0.362211
\(352\) −60004.7 −0.0258124
\(353\) −537762. −0.229696 −0.114848 0.993383i \(-0.536638\pi\)
−0.114848 + 0.993383i \(0.536638\pi\)
\(354\) −457277. −0.193941
\(355\) 0 0
\(356\) −91271.2 −0.0381688
\(357\) −2.12314e6 −0.881675
\(358\) −4.61153e6 −1.90168
\(359\) 3.19303e6 1.30757 0.653787 0.756678i \(-0.273179\pi\)
0.653787 + 0.756678i \(0.273179\pi\)
\(360\) 0 0
\(361\) −1.94087e6 −0.783840
\(362\) −3.08867e6 −1.23880
\(363\) −131769. −0.0524864
\(364\) −236110. −0.0934032
\(365\) 0 0
\(366\) 793214. 0.309519
\(367\) 3.24705e6 1.25842 0.629208 0.777237i \(-0.283379\pi\)
0.629208 + 0.777237i \(0.283379\pi\)
\(368\) −2.32670e6 −0.895616
\(369\) −35682.5 −0.0136424
\(370\) 0 0
\(371\) 3.82061e6 1.44111
\(372\) −82655.2 −0.0309680
\(373\) −2.23197e6 −0.830647 −0.415323 0.909674i \(-0.636332\pi\)
−0.415323 + 0.909674i \(0.636332\pi\)
\(374\) −1.09782e6 −0.405837
\(375\) 0 0
\(376\) 2.71870e6 0.991726
\(377\) 9.86705e6 3.57548
\(378\) −632535. −0.227696
\(379\) −1.67352e6 −0.598456 −0.299228 0.954182i \(-0.596729\pi\)
−0.299228 + 0.954182i \(0.596729\pi\)
\(380\) 0 0
\(381\) 2.74109e6 0.967410
\(382\) 1.30791e6 0.458583
\(383\) −1.51540e6 −0.527874 −0.263937 0.964540i \(-0.585021\pi\)
−0.263937 + 0.964540i \(0.585021\pi\)
\(384\) 1.76736e6 0.611641
\(385\) 0 0
\(386\) 453034. 0.154761
\(387\) −1.10095e6 −0.373671
\(388\) 199988. 0.0674410
\(389\) −3.15291e6 −1.05642 −0.528211 0.849113i \(-0.677137\pi\)
−0.528211 + 0.849113i \(0.677137\pi\)
\(390\) 0 0
\(391\) −3.42810e6 −1.13400
\(392\) 1.01801e6 0.334608
\(393\) −974268. −0.318198
\(394\) −457980. −0.148630
\(395\) 0 0
\(396\) −13434.1 −0.00430499
\(397\) −5.36348e6 −1.70793 −0.853966 0.520330i \(-0.825809\pi\)
−0.853966 + 0.520330i \(0.825809\pi\)
\(398\) −4.33653e6 −1.37225
\(399\) 988981. 0.310997
\(400\) 0 0
\(401\) −2.02367e6 −0.628462 −0.314231 0.949347i \(-0.601747\pi\)
−0.314231 + 0.949347i \(0.601747\pi\)
\(402\) −2.40343e6 −0.741765
\(403\) 7.68404e6 2.35682
\(404\) −172293. −0.0525188
\(405\) 0 0
\(406\) 7.46522e6 2.24764
\(407\) 369820. 0.110664
\(408\) 2.50106e6 0.743831
\(409\) −6.34651e6 −1.87597 −0.937986 0.346672i \(-0.887312\pi\)
−0.937986 + 0.346672i \(0.887312\pi\)
\(410\) 0 0
\(411\) 2.96400e6 0.865513
\(412\) −86685.6 −0.0251596
\(413\) 1.32108e6 0.381112
\(414\) −1.02131e6 −0.292858
\(415\) 0 0
\(416\) 568723. 0.161127
\(417\) −366771. −0.103289
\(418\) 511375. 0.143152
\(419\) 5.32755e6 1.48249 0.741246 0.671234i \(-0.234235\pi\)
0.741246 + 0.671234i \(0.234235\pi\)
\(420\) 0 0
\(421\) 4.62270e6 1.27113 0.635566 0.772046i \(-0.280767\pi\)
0.635566 + 0.772046i \(0.280767\pi\)
\(422\) −3.74787e6 −1.02448
\(423\) 1.24459e6 0.338201
\(424\) −4.50068e6 −1.21580
\(425\) 0 0
\(426\) 3.53747e6 0.944427
\(427\) −2.29160e6 −0.608233
\(428\) −142458. −0.0375904
\(429\) 1.24890e6 0.327632
\(430\) 0 0
\(431\) −1.98012e6 −0.513450 −0.256725 0.966484i \(-0.582644\pi\)
−0.256725 + 0.966484i \(0.582644\pi\)
\(432\) 777102. 0.200340
\(433\) −1.49070e6 −0.382094 −0.191047 0.981581i \(-0.561188\pi\)
−0.191047 + 0.981581i \(0.561188\pi\)
\(434\) 5.81360e6 1.48157
\(435\) 0 0
\(436\) −157950. −0.0397927
\(437\) 1.59684e6 0.399999
\(438\) −668114. −0.166405
\(439\) −2.31922e6 −0.574355 −0.287178 0.957877i \(-0.592717\pi\)
−0.287178 + 0.957877i \(0.592717\pi\)
\(440\) 0 0
\(441\) 466033. 0.114109
\(442\) 1.04051e7 2.53332
\(443\) 1.40179e6 0.339371 0.169686 0.985498i \(-0.445725\pi\)
0.169686 + 0.985498i \(0.445725\pi\)
\(444\) 37704.0 0.00907674
\(445\) 0 0
\(446\) −5.08731e6 −1.21102
\(447\) −82143.1 −0.0194448
\(448\) −4.69331e6 −1.10480
\(449\) 6.34855e6 1.48614 0.743069 0.669215i \(-0.233370\pi\)
0.743069 + 0.669215i \(0.233370\pi\)
\(450\) 0 0
\(451\) 53303.5 0.0123400
\(452\) −274243. −0.0631378
\(453\) 2.45850e6 0.562891
\(454\) 128333. 0.0292212
\(455\) 0 0
\(456\) −1.16502e6 −0.262374
\(457\) −3.44389e6 −0.771364 −0.385682 0.922632i \(-0.626034\pi\)
−0.385682 + 0.922632i \(0.626034\pi\)
\(458\) 4.35179e6 0.969402
\(459\) 1.14496e6 0.253664
\(460\) 0 0
\(461\) −7.33956e6 −1.60849 −0.804244 0.594299i \(-0.797429\pi\)
−0.804244 + 0.594299i \(0.797429\pi\)
\(462\) 944898. 0.205959
\(463\) 4.47030e6 0.969135 0.484567 0.874754i \(-0.338977\pi\)
0.484567 + 0.874754i \(0.338977\pi\)
\(464\) −9.17141e6 −1.97761
\(465\) 0 0
\(466\) −4.19977e6 −0.895903
\(467\) 6.94168e6 1.47290 0.736449 0.676493i \(-0.236501\pi\)
0.736449 + 0.676493i \(0.236501\pi\)
\(468\) 127329. 0.0268727
\(469\) 6.94354e6 1.45763
\(470\) 0 0
\(471\) −537819. −0.111708
\(472\) −1.55623e6 −0.321528
\(473\) 1.64463e6 0.337998
\(474\) 568581. 0.116238
\(475\) 0 0
\(476\) 323352. 0.0654122
\(477\) −2.06036e6 −0.414617
\(478\) 7.91803e6 1.58507
\(479\) −2.57692e6 −0.513170 −0.256585 0.966522i \(-0.582597\pi\)
−0.256585 + 0.966522i \(0.582597\pi\)
\(480\) 0 0
\(481\) −3.50515e6 −0.690787
\(482\) −7.33157e6 −1.43741
\(483\) 2.95058e6 0.575493
\(484\) 20068.3 0.00389401
\(485\) 0 0
\(486\) 341111. 0.0655095
\(487\) −267618. −0.0511321 −0.0255660 0.999673i \(-0.508139\pi\)
−0.0255660 + 0.999673i \(0.508139\pi\)
\(488\) 2.69951e6 0.513140
\(489\) −912291. −0.172529
\(490\) 0 0
\(491\) −5.20972e6 −0.975239 −0.487619 0.873056i \(-0.662135\pi\)
−0.487619 + 0.873056i \(0.662135\pi\)
\(492\) 5434.41 0.00101214
\(493\) −1.35129e7 −2.50398
\(494\) −4.84680e6 −0.893589
\(495\) 0 0
\(496\) −7.14231e6 −1.30357
\(497\) −1.02198e7 −1.85588
\(498\) 3.47517e6 0.627917
\(499\) 7.98482e6 1.43553 0.717767 0.696283i \(-0.245164\pi\)
0.717767 + 0.696283i \(0.245164\pi\)
\(500\) 0 0
\(501\) −2.98999e6 −0.532201
\(502\) 7.07919e6 1.25379
\(503\) 2.12921e6 0.375231 0.187615 0.982243i \(-0.439924\pi\)
0.187615 + 0.982243i \(0.439924\pi\)
\(504\) −2.15268e6 −0.377488
\(505\) 0 0
\(506\) 1.52566e6 0.264900
\(507\) −8.49547e6 −1.46780
\(508\) −417465. −0.0717729
\(509\) 8.45839e6 1.44708 0.723541 0.690281i \(-0.242513\pi\)
0.723541 + 0.690281i \(0.242513\pi\)
\(510\) 0 0
\(511\) 1.93019e6 0.327000
\(512\) 5.50697e6 0.928405
\(513\) −533333. −0.0894758
\(514\) −4.07055e6 −0.679588
\(515\) 0 0
\(516\) 167673. 0.0277230
\(517\) −1.85920e6 −0.305915
\(518\) −2.65193e6 −0.434248
\(519\) 1.40532e6 0.229011
\(520\) 0 0
\(521\) −473943. −0.0764948 −0.0382474 0.999268i \(-0.512177\pi\)
−0.0382474 + 0.999268i \(0.512177\pi\)
\(522\) −4.02581e6 −0.646662
\(523\) 9.60139e6 1.53490 0.767449 0.641110i \(-0.221526\pi\)
0.767449 + 0.641110i \(0.221526\pi\)
\(524\) 148380. 0.0236073
\(525\) 0 0
\(526\) 1.30224e6 0.205223
\(527\) −1.05233e7 −1.65053
\(528\) −1.16086e6 −0.181215
\(529\) −1.67224e6 −0.259811
\(530\) 0 0
\(531\) −712424. −0.109648
\(532\) −150621. −0.0230731
\(533\) −505210. −0.0770290
\(534\) −3.46194e6 −0.525371
\(535\) 0 0
\(536\) −8.17949e6 −1.22974
\(537\) −7.18464e6 −1.07515
\(538\) −1.02614e6 −0.152845
\(539\) −696172. −0.103215
\(540\) 0 0
\(541\) −5.96439e6 −0.876138 −0.438069 0.898941i \(-0.644338\pi\)
−0.438069 + 0.898941i \(0.644338\pi\)
\(542\) −9.95032e6 −1.45492
\(543\) −4.81206e6 −0.700377
\(544\) −778865. −0.112840
\(545\) 0 0
\(546\) −8.95573e6 −1.28564
\(547\) 5.67751e6 0.811314 0.405657 0.914025i \(-0.367043\pi\)
0.405657 + 0.914025i \(0.367043\pi\)
\(548\) −451414. −0.0642131
\(549\) 1.23581e6 0.174992
\(550\) 0 0
\(551\) 6.29444e6 0.883239
\(552\) −3.47579e6 −0.485518
\(553\) −1.64264e6 −0.228417
\(554\) −1.34179e7 −1.85742
\(555\) 0 0
\(556\) 55858.8 0.00766310
\(557\) −8.02484e6 −1.09597 −0.547985 0.836488i \(-0.684605\pi\)
−0.547985 + 0.836488i \(0.684605\pi\)
\(558\) −3.13513e6 −0.426256
\(559\) −1.55877e7 −2.10986
\(560\) 0 0
\(561\) −1.71037e6 −0.229447
\(562\) −5.74079e6 −0.766709
\(563\) 5.79184e6 0.770098 0.385049 0.922896i \(-0.374185\pi\)
0.385049 + 0.922896i \(0.374185\pi\)
\(564\) −189550. −0.0250914
\(565\) 0 0
\(566\) −1.15347e7 −1.51344
\(567\) −985472. −0.128732
\(568\) 1.20389e7 1.56573
\(569\) 668243. 0.0865274 0.0432637 0.999064i \(-0.486224\pi\)
0.0432637 + 0.999064i \(0.486224\pi\)
\(570\) 0 0
\(571\) 1.78527e6 0.229147 0.114573 0.993415i \(-0.463450\pi\)
0.114573 + 0.993415i \(0.463450\pi\)
\(572\) −190207. −0.0243073
\(573\) 2.03768e6 0.259268
\(574\) −382232. −0.0484226
\(575\) 0 0
\(576\) 2.53099e6 0.317858
\(577\) 621090. 0.0776631 0.0388316 0.999246i \(-0.487636\pi\)
0.0388316 + 0.999246i \(0.487636\pi\)
\(578\) −6.04762e6 −0.752948
\(579\) 705815. 0.0874972
\(580\) 0 0
\(581\) −1.00398e7 −1.23391
\(582\) 7.58559e6 0.928286
\(583\) 3.07782e6 0.375036
\(584\) −2.27376e6 −0.275875
\(585\) 0 0
\(586\) 1.21751e7 1.46463
\(587\) 3.14798e6 0.377083 0.188542 0.982065i \(-0.439624\pi\)
0.188542 + 0.982065i \(0.439624\pi\)
\(588\) −70976.3 −0.00846583
\(589\) 4.90184e6 0.582199
\(590\) 0 0
\(591\) −713521. −0.0840307
\(592\) 3.25803e6 0.382077
\(593\) −1.45313e6 −0.169695 −0.0848474 0.996394i \(-0.527040\pi\)
−0.0848474 + 0.996394i \(0.527040\pi\)
\(594\) −509560. −0.0592556
\(595\) 0 0
\(596\) 12510.3 0.00144262
\(597\) −6.75619e6 −0.775829
\(598\) −1.44602e7 −1.65357
\(599\) 7.62041e6 0.867784 0.433892 0.900965i \(-0.357140\pi\)
0.433892 + 0.900965i \(0.357140\pi\)
\(600\) 0 0
\(601\) 5.30235e6 0.598801 0.299400 0.954128i \(-0.403213\pi\)
0.299400 + 0.954128i \(0.403213\pi\)
\(602\) −1.17934e7 −1.32632
\(603\) −3.74448e6 −0.419371
\(604\) −374426. −0.0417613
\(605\) 0 0
\(606\) −6.53513e6 −0.722891
\(607\) −1.09581e7 −1.20716 −0.603580 0.797302i \(-0.706260\pi\)
−0.603580 + 0.797302i \(0.706260\pi\)
\(608\) 362803. 0.0398026
\(609\) 1.16306e7 1.27075
\(610\) 0 0
\(611\) 1.76215e7 1.90959
\(612\) −174376. −0.0188195
\(613\) 7.21856e6 0.775889 0.387944 0.921683i \(-0.373185\pi\)
0.387944 + 0.921683i \(0.373185\pi\)
\(614\) −4.80244e6 −0.514092
\(615\) 0 0
\(616\) 3.21573e6 0.341450
\(617\) −2.11311e6 −0.223465 −0.111733 0.993738i \(-0.535640\pi\)
−0.111733 + 0.993738i \(0.535640\pi\)
\(618\) −3.28801e6 −0.346307
\(619\) −1.80296e7 −1.89130 −0.945648 0.325191i \(-0.894571\pi\)
−0.945648 + 0.325191i \(0.894571\pi\)
\(620\) 0 0
\(621\) −1.59118e6 −0.165573
\(622\) 5.69464e6 0.590188
\(623\) 1.00016e7 1.03240
\(624\) 1.10026e7 1.13118
\(625\) 0 0
\(626\) −1.87053e7 −1.90779
\(627\) 796708. 0.0809339
\(628\) 81909.3 0.00828770
\(629\) 4.80029e6 0.483772
\(630\) 0 0
\(631\) 5.34708e6 0.534618 0.267309 0.963611i \(-0.413866\pi\)
0.267309 + 0.963611i \(0.413866\pi\)
\(632\) 1.93503e6 0.192706
\(633\) −5.83908e6 −0.579209
\(634\) 1.11192e7 1.09862
\(635\) 0 0
\(636\) 313791. 0.0307608
\(637\) 6.59831e6 0.644294
\(638\) 6.01387e6 0.584928
\(639\) 5.51128e6 0.533949
\(640\) 0 0
\(641\) −6.87676e6 −0.661057 −0.330529 0.943796i \(-0.607227\pi\)
−0.330529 + 0.943796i \(0.607227\pi\)
\(642\) −5.40346e6 −0.517410
\(643\) −537253. −0.0512450 −0.0256225 0.999672i \(-0.508157\pi\)
−0.0256225 + 0.999672i \(0.508157\pi\)
\(644\) −449370. −0.0426962
\(645\) 0 0
\(646\) 6.63768e6 0.625799
\(647\) −1.77812e7 −1.66994 −0.834970 0.550295i \(-0.814515\pi\)
−0.834970 + 0.550295i \(0.814515\pi\)
\(648\) 1.16089e6 0.108606
\(649\) 1.06424e6 0.0991808
\(650\) 0 0
\(651\) 9.05743e6 0.837631
\(652\) 138941. 0.0128000
\(653\) 850319. 0.0780367 0.0390184 0.999238i \(-0.487577\pi\)
0.0390184 + 0.999238i \(0.487577\pi\)
\(654\) −5.99108e6 −0.547722
\(655\) 0 0
\(656\) 469592. 0.0426051
\(657\) −1.04090e6 −0.0940799
\(658\) 1.33321e7 1.20042
\(659\) −9.40035e6 −0.843199 −0.421600 0.906782i \(-0.638531\pi\)
−0.421600 + 0.906782i \(0.638531\pi\)
\(660\) 0 0
\(661\) −3.01606e6 −0.268495 −0.134248 0.990948i \(-0.542862\pi\)
−0.134248 + 0.990948i \(0.542862\pi\)
\(662\) 2.77878e6 0.246439
\(663\) 1.62109e7 1.43226
\(664\) 1.18269e7 1.04100
\(665\) 0 0
\(666\) 1.43012e6 0.124936
\(667\) 1.87792e7 1.63441
\(668\) 455373. 0.0394844
\(669\) −7.92590e6 −0.684673
\(670\) 0 0
\(671\) −1.84608e6 −0.158287
\(672\) 670373. 0.0572655
\(673\) 1.95542e7 1.66419 0.832095 0.554634i \(-0.187142\pi\)
0.832095 + 0.554634i \(0.187142\pi\)
\(674\) 1.38411e7 1.17360
\(675\) 0 0
\(676\) 1.29385e6 0.108897
\(677\) −1.81798e7 −1.52446 −0.762231 0.647305i \(-0.775896\pi\)
−0.762231 + 0.647305i \(0.775896\pi\)
\(678\) −1.04021e7 −0.869055
\(679\) −2.19148e7 −1.82416
\(680\) 0 0
\(681\) 199939. 0.0165208
\(682\) 4.68335e6 0.385563
\(683\) −9.35931e6 −0.767701 −0.383850 0.923395i \(-0.625402\pi\)
−0.383850 + 0.923395i \(0.625402\pi\)
\(684\) 81226.1 0.00663828
\(685\) 0 0
\(686\) −9.59085e6 −0.778120
\(687\) 6.77997e6 0.548070
\(688\) 1.44888e7 1.16697
\(689\) −2.91716e7 −2.34106
\(690\) 0 0
\(691\) −6.10756e6 −0.486601 −0.243300 0.969951i \(-0.578230\pi\)
−0.243300 + 0.969951i \(0.578230\pi\)
\(692\) −214028. −0.0169905
\(693\) 1.47212e6 0.116443
\(694\) −1.25060e7 −0.985647
\(695\) 0 0
\(696\) −1.37009e7 −1.07207
\(697\) 691884. 0.0539450
\(698\) −1.70570e7 −1.32515
\(699\) −6.54313e6 −0.506516
\(700\) 0 0
\(701\) −1.64071e7 −1.26106 −0.630531 0.776164i \(-0.717163\pi\)
−0.630531 + 0.776164i \(0.717163\pi\)
\(702\) 4.82960e6 0.369887
\(703\) −2.23602e6 −0.170643
\(704\) −3.78086e6 −0.287514
\(705\) 0 0
\(706\) 3.10651e6 0.234564
\(707\) 1.88801e7 1.42054
\(708\) 108501. 0.00813491
\(709\) −1.77082e7 −1.32300 −0.661499 0.749946i \(-0.730079\pi\)
−0.661499 + 0.749946i \(0.730079\pi\)
\(710\) 0 0
\(711\) 885834. 0.0657171
\(712\) −1.17819e7 −0.870992
\(713\) 1.46244e7 1.07735
\(714\) 1.22648e7 0.900360
\(715\) 0 0
\(716\) 1.09421e6 0.0797662
\(717\) 1.23361e7 0.896147
\(718\) −1.84453e7 −1.33529
\(719\) −2.63661e6 −0.190206 −0.0951029 0.995467i \(-0.530318\pi\)
−0.0951029 + 0.995467i \(0.530318\pi\)
\(720\) 0 0
\(721\) 9.49909e6 0.680524
\(722\) 1.12119e7 0.800452
\(723\) −1.14224e7 −0.812664
\(724\) 732872. 0.0519615
\(725\) 0 0
\(726\) 761195. 0.0535987
\(727\) 8.88827e6 0.623708 0.311854 0.950130i \(-0.399050\pi\)
0.311854 + 0.950130i \(0.399050\pi\)
\(728\) −3.04786e7 −2.13141
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 2.13474e7 1.47758
\(732\) −188212. −0.0129828
\(733\) 2.21059e7 1.51967 0.759834 0.650118i \(-0.225280\pi\)
0.759834 + 0.650118i \(0.225280\pi\)
\(734\) −1.87574e7 −1.28509
\(735\) 0 0
\(736\) 1.08241e6 0.0736539
\(737\) 5.59361e6 0.379335
\(738\) 206129. 0.0139315
\(739\) 1.93561e7 1.30379 0.651893 0.758311i \(-0.273975\pi\)
0.651893 + 0.758311i \(0.273975\pi\)
\(740\) 0 0
\(741\) −7.55119e6 −0.505207
\(742\) −2.20707e7 −1.47165
\(743\) −8.35581e6 −0.555286 −0.277643 0.960684i \(-0.589553\pi\)
−0.277643 + 0.960684i \(0.589553\pi\)
\(744\) −1.06697e7 −0.706673
\(745\) 0 0
\(746\) 1.28935e7 0.848250
\(747\) 5.41422e6 0.355005
\(748\) 260488. 0.0170229
\(749\) 1.56107e7 1.01676
\(750\) 0 0
\(751\) −2092.53 −0.000135385 0 −6.76926e−5 1.00000i \(-0.500022\pi\)
−6.76926e−5 1.00000i \(0.500022\pi\)
\(752\) −1.63792e7 −1.05620
\(753\) 1.10292e7 0.708852
\(754\) −5.69994e7 −3.65125
\(755\) 0 0
\(756\) 150086. 0.00955073
\(757\) 1.32562e7 0.840773 0.420386 0.907345i \(-0.361895\pi\)
0.420386 + 0.907345i \(0.361895\pi\)
\(758\) 9.66746e6 0.611138
\(759\) 2.37694e6 0.149766
\(760\) 0 0
\(761\) 1.61517e7 1.01101 0.505507 0.862822i \(-0.331305\pi\)
0.505507 + 0.862822i \(0.331305\pi\)
\(762\) −1.58346e7 −0.987912
\(763\) 1.73083e7 1.07632
\(764\) −310337. −0.0192353
\(765\) 0 0
\(766\) 8.75407e6 0.539061
\(767\) −1.00868e7 −0.619109
\(768\) −1.21051e6 −0.0740566
\(769\) −7.83699e6 −0.477896 −0.238948 0.971032i \(-0.576803\pi\)
−0.238948 + 0.971032i \(0.576803\pi\)
\(770\) 0 0
\(771\) −6.34181e6 −0.384218
\(772\) −107495. −0.00649149
\(773\) 1.22821e7 0.739306 0.369653 0.929170i \(-0.379477\pi\)
0.369653 + 0.929170i \(0.379477\pi\)
\(774\) 6.35989e6 0.381590
\(775\) 0 0
\(776\) 2.58157e7 1.53897
\(777\) −4.13164e6 −0.245510
\(778\) 1.82135e7 1.07881
\(779\) −322286. −0.0190282
\(780\) 0 0
\(781\) −8.23290e6 −0.482975
\(782\) 1.98032e7 1.15803
\(783\) −6.27211e6 −0.365602
\(784\) −6.13312e6 −0.356362
\(785\) 0 0
\(786\) 5.62809e6 0.324941
\(787\) 1.33913e7 0.770703 0.385351 0.922770i \(-0.374080\pi\)
0.385351 + 0.922770i \(0.374080\pi\)
\(788\) 108668. 0.00623431
\(789\) 2.02885e6 0.116027
\(790\) 0 0
\(791\) 3.00518e7 1.70777
\(792\) −1.73416e6 −0.0982375
\(793\) 1.74971e7 0.988061
\(794\) 3.09834e7 1.74413
\(795\) 0 0
\(796\) 1.02896e6 0.0575594
\(797\) 1.58405e7 0.883332 0.441666 0.897180i \(-0.354388\pi\)
0.441666 + 0.897180i \(0.354388\pi\)
\(798\) −5.71308e6 −0.317587
\(799\) −2.41326e7 −1.33732
\(800\) 0 0
\(801\) −5.39361e6 −0.297028
\(802\) 1.16902e7 0.641781
\(803\) 1.55493e6 0.0850985
\(804\) 570280. 0.0311135
\(805\) 0 0
\(806\) −4.43887e7 −2.40677
\(807\) −1.59870e6 −0.0864137
\(808\) −2.22407e7 −1.19845
\(809\) 2.23526e7 1.20076 0.600380 0.799715i \(-0.295016\pi\)
0.600380 + 0.799715i \(0.295016\pi\)
\(810\) 0 0
\(811\) 3.12932e7 1.67070 0.835348 0.549722i \(-0.185266\pi\)
0.835348 + 0.549722i \(0.185266\pi\)
\(812\) −1.77133e6 −0.0942778
\(813\) −1.55023e7 −0.822566
\(814\) −2.13635e6 −0.113009
\(815\) 0 0
\(816\) −1.50680e7 −0.792191
\(817\) −9.94382e6 −0.521192
\(818\) 3.66621e7 1.91573
\(819\) −1.39528e7 −0.726860
\(820\) 0 0
\(821\) −2.77034e6 −0.143442 −0.0717208 0.997425i \(-0.522849\pi\)
−0.0717208 + 0.997425i \(0.522849\pi\)
\(822\) −1.71223e7 −0.883856
\(823\) −1.42994e7 −0.735899 −0.367949 0.929846i \(-0.619940\pi\)
−0.367949 + 0.929846i \(0.619940\pi\)
\(824\) −1.11899e7 −0.574129
\(825\) 0 0
\(826\) −7.63151e6 −0.389189
\(827\) −7.81938e6 −0.397565 −0.198783 0.980044i \(-0.563699\pi\)
−0.198783 + 0.980044i \(0.563699\pi\)
\(828\) 242334. 0.0122840
\(829\) 2.08778e7 1.05511 0.527557 0.849520i \(-0.323108\pi\)
0.527557 + 0.849520i \(0.323108\pi\)
\(830\) 0 0
\(831\) −2.09048e7 −1.05013
\(832\) 3.58349e7 1.79473
\(833\) −9.03636e6 −0.451213
\(834\) 2.11874e6 0.105478
\(835\) 0 0
\(836\) −121338. −0.00600455
\(837\) −4.88445e6 −0.240992
\(838\) −3.07758e7 −1.51391
\(839\) −1.70951e7 −0.838429 −0.419215 0.907887i \(-0.637694\pi\)
−0.419215 + 0.907887i \(0.637694\pi\)
\(840\) 0 0
\(841\) 5.35127e7 2.60896
\(842\) −2.67041e7 −1.29807
\(843\) −8.94399e6 −0.433474
\(844\) 889286. 0.0429720
\(845\) 0 0
\(846\) −7.18967e6 −0.345369
\(847\) −2.19910e6 −0.105326
\(848\) 2.71150e7 1.29485
\(849\) −1.79708e7 −0.855653
\(850\) 0 0
\(851\) −6.67108e6 −0.315771
\(852\) −839362. −0.0396141
\(853\) −1.04193e7 −0.490306 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(854\) 1.32380e7 0.621123
\(855\) 0 0
\(856\) −1.83894e7 −0.857793
\(857\) 867466. 0.0403460 0.0201730 0.999797i \(-0.493578\pi\)
0.0201730 + 0.999797i \(0.493578\pi\)
\(858\) −7.21459e6 −0.334575
\(859\) −1.41813e7 −0.655741 −0.327871 0.944723i \(-0.606331\pi\)
−0.327871 + 0.944723i \(0.606331\pi\)
\(860\) 0 0
\(861\) −595508. −0.0273766
\(862\) 1.14386e7 0.524332
\(863\) −1.99795e7 −0.913185 −0.456592 0.889676i \(-0.650930\pi\)
−0.456592 + 0.889676i \(0.650930\pi\)
\(864\) −361516. −0.0164757
\(865\) 0 0
\(866\) 8.61138e6 0.390192
\(867\) −9.42203e6 −0.425693
\(868\) −1.37944e6 −0.0621445
\(869\) −1.32328e6 −0.0594433
\(870\) 0 0
\(871\) −5.30161e7 −2.36790
\(872\) −2.03892e7 −0.908047
\(873\) 1.18181e7 0.524824
\(874\) −9.22454e6 −0.408476
\(875\) 0 0
\(876\) 158528. 0.00697986
\(877\) 3.18448e7 1.39810 0.699052 0.715071i \(-0.253606\pi\)
0.699052 + 0.715071i \(0.253606\pi\)
\(878\) 1.33975e7 0.586527
\(879\) 1.89684e7 0.828054
\(880\) 0 0
\(881\) 1.62021e6 0.0703284 0.0351642 0.999382i \(-0.488805\pi\)
0.0351642 + 0.999382i \(0.488805\pi\)
\(882\) −2.69215e6 −0.116527
\(883\) 1.23112e7 0.531371 0.265685 0.964060i \(-0.414402\pi\)
0.265685 + 0.964060i \(0.414402\pi\)
\(884\) −2.46890e6 −0.106261
\(885\) 0 0
\(886\) −8.09780e6 −0.346563
\(887\) 9.83752e6 0.419833 0.209916 0.977719i \(-0.432681\pi\)
0.209916 + 0.977719i \(0.432681\pi\)
\(888\) 4.86707e6 0.207126
\(889\) 4.57462e7 1.94134
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) 1.20711e6 0.0507964
\(893\) 1.12412e7 0.471720
\(894\) 474519. 0.0198568
\(895\) 0 0
\(896\) 2.94956e7 1.22740
\(897\) −2.25286e7 −0.934875
\(898\) −3.66739e7 −1.51763
\(899\) 5.76467e7 2.37889
\(900\) 0 0
\(901\) 3.99504e7 1.63949
\(902\) −307920. −0.0126015
\(903\) −1.83738e7 −0.749858
\(904\) −3.54011e7 −1.44077
\(905\) 0 0
\(906\) −1.42021e7 −0.574820
\(907\) 3.26593e7 1.31822 0.659110 0.752046i \(-0.270933\pi\)
0.659110 + 0.752046i \(0.270933\pi\)
\(908\) −30450.5 −0.00122569
\(909\) −1.01816e7 −0.408700
\(910\) 0 0
\(911\) 7.81356e6 0.311927 0.155963 0.987763i \(-0.450152\pi\)
0.155963 + 0.987763i \(0.450152\pi\)
\(912\) 7.01882e6 0.279433
\(913\) −8.08790e6 −0.321114
\(914\) 1.98945e7 0.787711
\(915\) 0 0
\(916\) −1.03258e6 −0.0406617
\(917\) −1.62596e7 −0.638538
\(918\) −6.61413e6 −0.259039
\(919\) 2.97666e7 1.16263 0.581313 0.813680i \(-0.302539\pi\)
0.581313 + 0.813680i \(0.302539\pi\)
\(920\) 0 0
\(921\) −7.48206e6 −0.290651
\(922\) 4.23987e7 1.64258
\(923\) 7.80313e7 3.01484
\(924\) −224203. −0.00863896
\(925\) 0 0
\(926\) −2.58237e7 −0.989673
\(927\) −5.12263e6 −0.195791
\(928\) 4.26664e6 0.162636
\(929\) −3.15595e7 −1.19975 −0.599875 0.800094i \(-0.704783\pi\)
−0.599875 + 0.800094i \(0.704783\pi\)
\(930\) 0 0
\(931\) 4.20923e6 0.159158
\(932\) 996512. 0.0375788
\(933\) 8.87209e6 0.333674
\(934\) −4.01003e7 −1.50411
\(935\) 0 0
\(936\) 1.64364e7 0.613221
\(937\) −2.29765e7 −0.854940 −0.427470 0.904030i \(-0.640595\pi\)
−0.427470 + 0.904030i \(0.640595\pi\)
\(938\) −4.01110e7 −1.48853
\(939\) −2.91424e7 −1.07860
\(940\) 0 0
\(941\) 3.71098e7 1.36620 0.683100 0.730325i \(-0.260631\pi\)
0.683100 + 0.730325i \(0.260631\pi\)
\(942\) 3.10684e6 0.114075
\(943\) −961527. −0.0352113
\(944\) 9.37571e6 0.342432
\(945\) 0 0
\(946\) −9.50057e6 −0.345161
\(947\) −9.02929e6 −0.327174 −0.163587 0.986529i \(-0.552306\pi\)
−0.163587 + 0.986529i \(0.552306\pi\)
\(948\) −134912. −0.00487560
\(949\) −1.47376e7 −0.531204
\(950\) 0 0
\(951\) 1.73234e7 0.621127
\(952\) 4.17404e7 1.49267
\(953\) −3.24538e7 −1.15753 −0.578766 0.815494i \(-0.696466\pi\)
−0.578766 + 0.815494i \(0.696466\pi\)
\(954\) 1.19022e7 0.423404
\(955\) 0 0
\(956\) −1.87877e6 −0.0664859
\(957\) 9.36944e6 0.330700
\(958\) 1.48862e7 0.524046
\(959\) 4.94664e7 1.73686
\(960\) 0 0
\(961\) 1.62636e7 0.568080
\(962\) 2.02483e7 0.705426
\(963\) −8.41845e6 −0.292527
\(964\) 1.73962e6 0.0602922
\(965\) 0 0
\(966\) −1.70447e7 −0.587689
\(967\) 1.38234e7 0.475387 0.237693 0.971340i \(-0.423609\pi\)
0.237693 + 0.971340i \(0.423609\pi\)
\(968\) 2.59054e6 0.0888591
\(969\) 1.03413e7 0.353807
\(970\) 0 0
\(971\) 2.63597e7 0.897208 0.448604 0.893731i \(-0.351921\pi\)
0.448604 + 0.893731i \(0.351921\pi\)
\(972\) −80937.9 −0.00274781
\(973\) −6.12106e6 −0.207274
\(974\) 1.54596e6 0.0522157
\(975\) 0 0
\(976\) −1.62636e7 −0.546501
\(977\) −4.42890e7 −1.48443 −0.742214 0.670163i \(-0.766224\pi\)
−0.742214 + 0.670163i \(0.766224\pi\)
\(978\) 5.27006e6 0.176185
\(979\) 8.05712e6 0.268672
\(980\) 0 0
\(981\) −9.33393e6 −0.309665
\(982\) 3.00952e7 0.995906
\(983\) −3.40185e7 −1.12288 −0.561438 0.827519i \(-0.689752\pi\)
−0.561438 + 0.827519i \(0.689752\pi\)
\(984\) 701509. 0.0230964
\(985\) 0 0
\(986\) 7.80604e7 2.55705
\(987\) 2.07710e7 0.678680
\(988\) 1.15004e6 0.0374817
\(989\) −2.96669e7 −0.964455
\(990\) 0 0
\(991\) −1.32232e7 −0.427712 −0.213856 0.976865i \(-0.568602\pi\)
−0.213856 + 0.976865i \(0.568602\pi\)
\(992\) 3.32268e6 0.107203
\(993\) 4.32926e6 0.139329
\(994\) 5.90370e7 1.89521
\(995\) 0 0
\(996\) −824579. −0.0263381
\(997\) 5.33000e7 1.69820 0.849101 0.528231i \(-0.177145\pi\)
0.849101 + 0.528231i \(0.177145\pi\)
\(998\) −4.61262e7 −1.46596
\(999\) 2.22809e6 0.0706349
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 825.6.a.s.1.3 yes 9
5.4 even 2 825.6.a.r.1.7 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.r.1.7 9 5.4 even 2
825.6.a.s.1.3 yes 9 1.1 even 1 trivial