Properties

Label 825.6.a.s.1.5
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 1.5
Root \(-0.624305\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.624305 q^{2} -9.00000 q^{3} -31.6102 q^{4} +5.61875 q^{6} -106.290 q^{7} +39.7122 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-0.624305 q^{2} -9.00000 q^{3} -31.6102 q^{4} +5.61875 q^{6} -106.290 q^{7} +39.7122 q^{8} +81.0000 q^{9} -121.000 q^{11} +284.492 q^{12} -264.362 q^{13} +66.3575 q^{14} +986.735 q^{16} -509.491 q^{17} -50.5687 q^{18} +1662.79 q^{19} +956.611 q^{21} +75.5409 q^{22} -2212.06 q^{23} -357.410 q^{24} +165.043 q^{26} -729.000 q^{27} +3359.86 q^{28} -5424.15 q^{29} -8799.95 q^{31} -1886.81 q^{32} +1089.00 q^{33} +318.078 q^{34} -2560.43 q^{36} +2600.13 q^{37} -1038.09 q^{38} +2379.26 q^{39} -9444.87 q^{41} -597.217 q^{42} -9572.69 q^{43} +3824.84 q^{44} +1381.00 q^{46} -5521.13 q^{47} -8880.62 q^{48} -5509.41 q^{49} +4585.42 q^{51} +8356.55 q^{52} -24674.9 q^{53} +455.119 q^{54} -4221.01 q^{56} -14965.1 q^{57} +3386.32 q^{58} +31867.9 q^{59} +13479.2 q^{61} +5493.85 q^{62} -8609.50 q^{63} -30397.6 q^{64} -679.868 q^{66} +29599.5 q^{67} +16105.1 q^{68} +19908.6 q^{69} -20097.8 q^{71} +3216.69 q^{72} -63873.3 q^{73} -1623.27 q^{74} -52561.3 q^{76} +12861.1 q^{77} -1485.38 q^{78} -84623.1 q^{79} +6561.00 q^{81} +5896.48 q^{82} -81972.5 q^{83} -30238.7 q^{84} +5976.28 q^{86} +48817.3 q^{87} -4805.18 q^{88} +1898.12 q^{89} +28099.1 q^{91} +69923.8 q^{92} +79199.5 q^{93} +3446.87 q^{94} +16981.3 q^{96} -26096.5 q^{97} +3439.56 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.624305 −0.110363 −0.0551813 0.998476i \(-0.517574\pi\)
−0.0551813 + 0.998476i \(0.517574\pi\)
\(3\) −9.00000 −0.577350
\(4\) −31.6102 −0.987820
\(5\) 0 0
\(6\) 5.61875 0.0637179
\(7\) −106.290 −0.819875 −0.409938 0.912114i \(-0.634450\pi\)
−0.409938 + 0.912114i \(0.634450\pi\)
\(8\) 39.7122 0.219381
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 284.492 0.570318
\(13\) −264.362 −0.433851 −0.216926 0.976188i \(-0.569603\pi\)
−0.216926 + 0.976188i \(0.569603\pi\)
\(14\) 66.3575 0.0904836
\(15\) 0 0
\(16\) 986.735 0.963609
\(17\) −509.491 −0.427577 −0.213788 0.976880i \(-0.568580\pi\)
−0.213788 + 0.976880i \(0.568580\pi\)
\(18\) −50.5687 −0.0367875
\(19\) 1662.79 1.05671 0.528353 0.849025i \(-0.322810\pi\)
0.528353 + 0.849025i \(0.322810\pi\)
\(20\) 0 0
\(21\) 956.611 0.473355
\(22\) 75.5409 0.0332756
\(23\) −2212.06 −0.871922 −0.435961 0.899966i \(-0.643591\pi\)
−0.435961 + 0.899966i \(0.643591\pi\)
\(24\) −357.410 −0.126660
\(25\) 0 0
\(26\) 165.043 0.0478809
\(27\) −729.000 −0.192450
\(28\) 3359.86 0.809889
\(29\) −5424.15 −1.19767 −0.598834 0.800873i \(-0.704369\pi\)
−0.598834 + 0.800873i \(0.704369\pi\)
\(30\) 0 0
\(31\) −8799.95 −1.64466 −0.822329 0.569012i \(-0.807326\pi\)
−0.822329 + 0.569012i \(0.807326\pi\)
\(32\) −1886.81 −0.325727
\(33\) 1089.00 0.174078
\(34\) 318.078 0.0471885
\(35\) 0 0
\(36\) −2560.43 −0.329273
\(37\) 2600.13 0.312242 0.156121 0.987738i \(-0.450101\pi\)
0.156121 + 0.987738i \(0.450101\pi\)
\(38\) −1038.09 −0.116621
\(39\) 2379.26 0.250484
\(40\) 0 0
\(41\) −9444.87 −0.877478 −0.438739 0.898615i \(-0.644575\pi\)
−0.438739 + 0.898615i \(0.644575\pi\)
\(42\) −597.217 −0.0522407
\(43\) −9572.69 −0.789519 −0.394760 0.918784i \(-0.629172\pi\)
−0.394760 + 0.918784i \(0.629172\pi\)
\(44\) 3824.84 0.297839
\(45\) 0 0
\(46\) 1381.00 0.0962276
\(47\) −5521.13 −0.364572 −0.182286 0.983246i \(-0.558350\pi\)
−0.182286 + 0.983246i \(0.558350\pi\)
\(48\) −8880.62 −0.556340
\(49\) −5509.41 −0.327805
\(50\) 0 0
\(51\) 4585.42 0.246862
\(52\) 8356.55 0.428567
\(53\) −24674.9 −1.20661 −0.603303 0.797512i \(-0.706149\pi\)
−0.603303 + 0.797512i \(0.706149\pi\)
\(54\) 455.119 0.0212393
\(55\) 0 0
\(56\) −4221.01 −0.179865
\(57\) −14965.1 −0.610089
\(58\) 3386.32 0.132178
\(59\) 31867.9 1.19186 0.595928 0.803038i \(-0.296784\pi\)
0.595928 + 0.803038i \(0.296784\pi\)
\(60\) 0 0
\(61\) 13479.2 0.463810 0.231905 0.972738i \(-0.425504\pi\)
0.231905 + 0.972738i \(0.425504\pi\)
\(62\) 5493.85 0.181509
\(63\) −8609.50 −0.273292
\(64\) −30397.6 −0.927661
\(65\) 0 0
\(66\) −679.868 −0.0192117
\(67\) 29599.5 0.805560 0.402780 0.915297i \(-0.368044\pi\)
0.402780 + 0.915297i \(0.368044\pi\)
\(68\) 16105.1 0.422369
\(69\) 19908.6 0.503404
\(70\) 0 0
\(71\) −20097.8 −0.473155 −0.236578 0.971613i \(-0.576026\pi\)
−0.236578 + 0.971613i \(0.576026\pi\)
\(72\) 3216.69 0.0731270
\(73\) −63873.3 −1.40285 −0.701427 0.712742i \(-0.747453\pi\)
−0.701427 + 0.712742i \(0.747453\pi\)
\(74\) −1623.27 −0.0344598
\(75\) 0 0
\(76\) −52561.3 −1.04384
\(77\) 12861.1 0.247202
\(78\) −1485.38 −0.0276441
\(79\) −84623.1 −1.52553 −0.762766 0.646675i \(-0.776159\pi\)
−0.762766 + 0.646675i \(0.776159\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 5896.48 0.0968408
\(83\) −81972.5 −1.30609 −0.653045 0.757319i \(-0.726509\pi\)
−0.653045 + 0.757319i \(0.726509\pi\)
\(84\) −30238.7 −0.467590
\(85\) 0 0
\(86\) 5976.28 0.0871334
\(87\) 48817.3 0.691474
\(88\) −4805.18 −0.0661459
\(89\) 1898.12 0.0254009 0.0127004 0.999919i \(-0.495957\pi\)
0.0127004 + 0.999919i \(0.495957\pi\)
\(90\) 0 0
\(91\) 28099.1 0.355704
\(92\) 69923.8 0.861302
\(93\) 79199.5 0.949544
\(94\) 3446.87 0.0402352
\(95\) 0 0
\(96\) 16981.3 0.188059
\(97\) −26096.5 −0.281613 −0.140806 0.990037i \(-0.544970\pi\)
−0.140806 + 0.990037i \(0.544970\pi\)
\(98\) 3439.56 0.0361774
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) 114822. 1.12001 0.560003 0.828491i \(-0.310800\pi\)
0.560003 + 0.828491i \(0.310800\pi\)
\(102\) −2862.70 −0.0272443
\(103\) −194550. −1.80691 −0.903457 0.428679i \(-0.858979\pi\)
−0.903457 + 0.428679i \(0.858979\pi\)
\(104\) −10498.4 −0.0951787
\(105\) 0 0
\(106\) 15404.7 0.133164
\(107\) −6495.07 −0.0548434 −0.0274217 0.999624i \(-0.508730\pi\)
−0.0274217 + 0.999624i \(0.508730\pi\)
\(108\) 23043.9 0.190106
\(109\) −55792.2 −0.449787 −0.224894 0.974383i \(-0.572203\pi\)
−0.224894 + 0.974383i \(0.572203\pi\)
\(110\) 0 0
\(111\) −23401.2 −0.180273
\(112\) −104880. −0.790039
\(113\) 105074. 0.774107 0.387053 0.922057i \(-0.373493\pi\)
0.387053 + 0.922057i \(0.373493\pi\)
\(114\) 9342.81 0.0673311
\(115\) 0 0
\(116\) 171459. 1.18308
\(117\) −21413.3 −0.144617
\(118\) −19895.3 −0.131536
\(119\) 54153.9 0.350560
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −8415.15 −0.0511873
\(123\) 85003.8 0.506612
\(124\) 278168. 1.62463
\(125\) 0 0
\(126\) 5374.95 0.0301612
\(127\) 119862. 0.659437 0.329718 0.944079i \(-0.393046\pi\)
0.329718 + 0.944079i \(0.393046\pi\)
\(128\) 79355.4 0.428106
\(129\) 86154.2 0.455829
\(130\) 0 0
\(131\) −213844. −1.08873 −0.544364 0.838849i \(-0.683229\pi\)
−0.544364 + 0.838849i \(0.683229\pi\)
\(132\) −34423.6 −0.171957
\(133\) −176738. −0.866367
\(134\) −18479.1 −0.0889037
\(135\) 0 0
\(136\) −20233.0 −0.0938023
\(137\) −269753. −1.22791 −0.613953 0.789343i \(-0.710422\pi\)
−0.613953 + 0.789343i \(0.710422\pi\)
\(138\) −12429.0 −0.0555570
\(139\) 242482. 1.06449 0.532247 0.846589i \(-0.321348\pi\)
0.532247 + 0.846589i \(0.321348\pi\)
\(140\) 0 0
\(141\) 49690.2 0.210486
\(142\) 12547.2 0.0522186
\(143\) 31987.8 0.130811
\(144\) 79925.6 0.321203
\(145\) 0 0
\(146\) 39876.4 0.154823
\(147\) 49584.7 0.189258
\(148\) −82190.7 −0.308439
\(149\) 366868. 1.35377 0.676884 0.736090i \(-0.263330\pi\)
0.676884 + 0.736090i \(0.263330\pi\)
\(150\) 0 0
\(151\) 180382. 0.643800 0.321900 0.946774i \(-0.395678\pi\)
0.321900 + 0.946774i \(0.395678\pi\)
\(152\) 66033.2 0.231821
\(153\) −41268.8 −0.142526
\(154\) −8029.25 −0.0272818
\(155\) 0 0
\(156\) −75208.9 −0.247433
\(157\) −145886. −0.472349 −0.236175 0.971711i \(-0.575894\pi\)
−0.236175 + 0.971711i \(0.575894\pi\)
\(158\) 52830.6 0.168362
\(159\) 222074. 0.696634
\(160\) 0 0
\(161\) 235120. 0.714867
\(162\) −4096.07 −0.0122625
\(163\) −147546. −0.434969 −0.217484 0.976064i \(-0.569785\pi\)
−0.217484 + 0.976064i \(0.569785\pi\)
\(164\) 298555. 0.866790
\(165\) 0 0
\(166\) 51175.9 0.144143
\(167\) 366460. 1.01680 0.508400 0.861121i \(-0.330237\pi\)
0.508400 + 0.861121i \(0.330237\pi\)
\(168\) 37989.1 0.103845
\(169\) −301406. −0.811773
\(170\) 0 0
\(171\) 134686. 0.352235
\(172\) 302595. 0.779903
\(173\) −501960. −1.27513 −0.637564 0.770398i \(-0.720058\pi\)
−0.637564 + 0.770398i \(0.720058\pi\)
\(174\) −30476.9 −0.0763129
\(175\) 0 0
\(176\) −119395. −0.290539
\(177\) −286811. −0.688118
\(178\) −1185.01 −0.00280331
\(179\) 475285. 1.10872 0.554360 0.832277i \(-0.312963\pi\)
0.554360 + 0.832277i \(0.312963\pi\)
\(180\) 0 0
\(181\) 492648. 1.11774 0.558869 0.829256i \(-0.311235\pi\)
0.558869 + 0.829256i \(0.311235\pi\)
\(182\) −17542.4 −0.0392564
\(183\) −121313. −0.267781
\(184\) −87845.8 −0.191283
\(185\) 0 0
\(186\) −49444.7 −0.104794
\(187\) 61648.4 0.128919
\(188\) 174524. 0.360132
\(189\) 77485.5 0.157785
\(190\) 0 0
\(191\) −297473. −0.590016 −0.295008 0.955495i \(-0.595322\pi\)
−0.295008 + 0.955495i \(0.595322\pi\)
\(192\) 273578. 0.535585
\(193\) 397116. 0.767404 0.383702 0.923457i \(-0.374649\pi\)
0.383702 + 0.923457i \(0.374649\pi\)
\(194\) 16292.2 0.0310795
\(195\) 0 0
\(196\) 174154. 0.323812
\(197\) −283439. −0.520348 −0.260174 0.965562i \(-0.583780\pi\)
−0.260174 + 0.965562i \(0.583780\pi\)
\(198\) 6118.82 0.0110919
\(199\) 442088. 0.791364 0.395682 0.918388i \(-0.370508\pi\)
0.395682 + 0.918388i \(0.370508\pi\)
\(200\) 0 0
\(201\) −266396. −0.465090
\(202\) −71683.8 −0.123607
\(203\) 576533. 0.981938
\(204\) −144946. −0.243855
\(205\) 0 0
\(206\) 121458. 0.199416
\(207\) −179177. −0.290641
\(208\) −260855. −0.418063
\(209\) −201198. −0.318609
\(210\) 0 0
\(211\) 424099. 0.655785 0.327892 0.944715i \(-0.393662\pi\)
0.327892 + 0.944715i \(0.393662\pi\)
\(212\) 779979. 1.19191
\(213\) 180881. 0.273176
\(214\) 4054.91 0.00605266
\(215\) 0 0
\(216\) −28950.2 −0.0422199
\(217\) 935347. 1.34841
\(218\) 34831.4 0.0496397
\(219\) 574860. 0.809938
\(220\) 0 0
\(221\) 134690. 0.185505
\(222\) 14609.5 0.0198954
\(223\) −1.02158e6 −1.37566 −0.687830 0.725872i \(-0.741437\pi\)
−0.687830 + 0.725872i \(0.741437\pi\)
\(224\) 200550. 0.267056
\(225\) 0 0
\(226\) −65598.5 −0.0854324
\(227\) 1.15245e6 1.48442 0.742211 0.670166i \(-0.233777\pi\)
0.742211 + 0.670166i \(0.233777\pi\)
\(228\) 473052. 0.602659
\(229\) 1.17552e6 1.48130 0.740650 0.671891i \(-0.234518\pi\)
0.740650 + 0.671891i \(0.234518\pi\)
\(230\) 0 0
\(231\) −115750. −0.142722
\(232\) −215405. −0.262746
\(233\) −613235. −0.740009 −0.370004 0.929030i \(-0.620644\pi\)
−0.370004 + 0.929030i \(0.620644\pi\)
\(234\) 13368.5 0.0159603
\(235\) 0 0
\(236\) −1.00735e6 −1.17734
\(237\) 761608. 0.880766
\(238\) −33808.5 −0.0386887
\(239\) 1.11292e6 1.26029 0.630146 0.776477i \(-0.282995\pi\)
0.630146 + 0.776477i \(0.282995\pi\)
\(240\) 0 0
\(241\) 277378. 0.307630 0.153815 0.988100i \(-0.450844\pi\)
0.153815 + 0.988100i \(0.450844\pi\)
\(242\) −9140.45 −0.0100330
\(243\) −59049.0 −0.0641500
\(244\) −426082. −0.458161
\(245\) 0 0
\(246\) −53068.3 −0.0559110
\(247\) −439579. −0.458453
\(248\) −349465. −0.360807
\(249\) 737753. 0.754071
\(250\) 0 0
\(251\) 826834. 0.828388 0.414194 0.910189i \(-0.364063\pi\)
0.414194 + 0.910189i \(0.364063\pi\)
\(252\) 272148. 0.269963
\(253\) 267659. 0.262894
\(254\) −74830.6 −0.0727772
\(255\) 0 0
\(256\) 923181. 0.880414
\(257\) 591626. 0.558746 0.279373 0.960183i \(-0.409873\pi\)
0.279373 + 0.960183i \(0.409873\pi\)
\(258\) −53786.5 −0.0503065
\(259\) −276368. −0.255999
\(260\) 0 0
\(261\) −439356. −0.399223
\(262\) 133504. 0.120155
\(263\) −1.48336e6 −1.32238 −0.661192 0.750216i \(-0.729949\pi\)
−0.661192 + 0.750216i \(0.729949\pi\)
\(264\) 43246.6 0.0381893
\(265\) 0 0
\(266\) 110339. 0.0956145
\(267\) −17083.1 −0.0146652
\(268\) −935648. −0.795748
\(269\) 676733. 0.570213 0.285106 0.958496i \(-0.407971\pi\)
0.285106 + 0.958496i \(0.407971\pi\)
\(270\) 0 0
\(271\) 33731.6 0.0279006 0.0139503 0.999903i \(-0.495559\pi\)
0.0139503 + 0.999903i \(0.495559\pi\)
\(272\) −502733. −0.412017
\(273\) −252892. −0.205366
\(274\) 168408. 0.135515
\(275\) 0 0
\(276\) −629314. −0.497273
\(277\) 1.51556e6 1.18679 0.593394 0.804912i \(-0.297788\pi\)
0.593394 + 0.804912i \(0.297788\pi\)
\(278\) −151383. −0.117480
\(279\) −712796. −0.548220
\(280\) 0 0
\(281\) −1.97949e6 −1.49551 −0.747753 0.663978i \(-0.768867\pi\)
−0.747753 + 0.663978i \(0.768867\pi\)
\(282\) −31021.9 −0.0232298
\(283\) −691882. −0.513530 −0.256765 0.966474i \(-0.582657\pi\)
−0.256765 + 0.966474i \(0.582657\pi\)
\(284\) 635298. 0.467392
\(285\) 0 0
\(286\) −19970.2 −0.0144366
\(287\) 1.00390e6 0.719422
\(288\) −152832. −0.108576
\(289\) −1.16028e6 −0.817178
\(290\) 0 0
\(291\) 234868. 0.162589
\(292\) 2.01905e6 1.38577
\(293\) 1.72110e6 1.17122 0.585608 0.810594i \(-0.300856\pi\)
0.585608 + 0.810594i \(0.300856\pi\)
\(294\) −30956.0 −0.0208870
\(295\) 0 0
\(296\) 103257. 0.0684999
\(297\) 88209.0 0.0580259
\(298\) −229038. −0.149405
\(299\) 584785. 0.378284
\(300\) 0 0
\(301\) 1.01748e6 0.647307
\(302\) −112614. −0.0710515
\(303\) −1.03340e6 −0.646636
\(304\) 1.64074e6 1.01825
\(305\) 0 0
\(306\) 25764.3 0.0157295
\(307\) −929998. −0.563165 −0.281583 0.959537i \(-0.590859\pi\)
−0.281583 + 0.959537i \(0.590859\pi\)
\(308\) −406543. −0.244191
\(309\) 1.75095e6 1.04322
\(310\) 0 0
\(311\) −293753. −0.172219 −0.0861096 0.996286i \(-0.527444\pi\)
−0.0861096 + 0.996286i \(0.527444\pi\)
\(312\) 94485.6 0.0549515
\(313\) −665923. −0.384205 −0.192102 0.981375i \(-0.561531\pi\)
−0.192102 + 0.981375i \(0.561531\pi\)
\(314\) 91077.1 0.0521297
\(315\) 0 0
\(316\) 2.67496e6 1.50695
\(317\) −2.04535e6 −1.14320 −0.571598 0.820534i \(-0.693676\pi\)
−0.571598 + 0.820534i \(0.693676\pi\)
\(318\) −138642. −0.0768824
\(319\) 656322. 0.361110
\(320\) 0 0
\(321\) 58455.7 0.0316639
\(322\) −146787. −0.0788946
\(323\) −847178. −0.451823
\(324\) −207395. −0.109758
\(325\) 0 0
\(326\) 92113.6 0.0480043
\(327\) 502130. 0.259685
\(328\) −375076. −0.192502
\(329\) 586842. 0.298904
\(330\) 0 0
\(331\) −2.62633e6 −1.31759 −0.658793 0.752324i \(-0.728933\pi\)
−0.658793 + 0.752324i \(0.728933\pi\)
\(332\) 2.59117e6 1.29018
\(333\) 210611. 0.104081
\(334\) −228783. −0.112217
\(335\) 0 0
\(336\) 943922. 0.456129
\(337\) −966186. −0.463432 −0.231716 0.972783i \(-0.574434\pi\)
−0.231716 + 0.972783i \(0.574434\pi\)
\(338\) 188169. 0.0895894
\(339\) −945670. −0.446931
\(340\) 0 0
\(341\) 1.06479e6 0.495883
\(342\) −84085.3 −0.0388736
\(343\) 2.37201e6 1.08863
\(344\) −380152. −0.173206
\(345\) 0 0
\(346\) 313376. 0.140726
\(347\) −1.94573e6 −0.867479 −0.433740 0.901038i \(-0.642806\pi\)
−0.433740 + 0.901038i \(0.642806\pi\)
\(348\) −1.54313e6 −0.683052
\(349\) 1.24936e6 0.549065 0.274532 0.961578i \(-0.411477\pi\)
0.274532 + 0.961578i \(0.411477\pi\)
\(350\) 0 0
\(351\) 192720. 0.0834947
\(352\) 228305. 0.0982105
\(353\) −4.13513e6 −1.76625 −0.883126 0.469137i \(-0.844565\pi\)
−0.883126 + 0.469137i \(0.844565\pi\)
\(354\) 179058. 0.0759425
\(355\) 0 0
\(356\) −60000.1 −0.0250915
\(357\) −487385. −0.202396
\(358\) −296723. −0.122361
\(359\) −2.34776e6 −0.961431 −0.480716 0.876877i \(-0.659623\pi\)
−0.480716 + 0.876877i \(0.659623\pi\)
\(360\) 0 0
\(361\) 288781. 0.116627
\(362\) −307563. −0.123356
\(363\) −131769. −0.0524864
\(364\) −888218. −0.351371
\(365\) 0 0
\(366\) 75736.4 0.0295530
\(367\) 1.61235e6 0.624878 0.312439 0.949938i \(-0.398854\pi\)
0.312439 + 0.949938i \(0.398854\pi\)
\(368\) −2.18272e6 −0.840192
\(369\) −765034. −0.292493
\(370\) 0 0
\(371\) 2.62270e6 0.989266
\(372\) −2.50352e6 −0.937979
\(373\) −541048. −0.201355 −0.100678 0.994919i \(-0.532101\pi\)
−0.100678 + 0.994919i \(0.532101\pi\)
\(374\) −38487.4 −0.0142279
\(375\) 0 0
\(376\) −219256. −0.0799803
\(377\) 1.43394e6 0.519610
\(378\) −48374.6 −0.0174136
\(379\) 3.10147e6 1.10910 0.554549 0.832151i \(-0.312891\pi\)
0.554549 + 0.832151i \(0.312891\pi\)
\(380\) 0 0
\(381\) −1.07876e6 −0.380726
\(382\) 185714. 0.0651157
\(383\) 729397. 0.254078 0.127039 0.991898i \(-0.459453\pi\)
0.127039 + 0.991898i \(0.459453\pi\)
\(384\) −714199. −0.247167
\(385\) 0 0
\(386\) −247922. −0.0846927
\(387\) −775388. −0.263173
\(388\) 824916. 0.278183
\(389\) 5.35650e6 1.79476 0.897382 0.441255i \(-0.145467\pi\)
0.897382 + 0.441255i \(0.145467\pi\)
\(390\) 0 0
\(391\) 1.12703e6 0.372814
\(392\) −218791. −0.0719141
\(393\) 1.92460e6 0.628578
\(394\) 176952. 0.0574269
\(395\) 0 0
\(396\) 309812. 0.0992797
\(397\) 1.13420e6 0.361170 0.180585 0.983559i \(-0.442201\pi\)
0.180585 + 0.983559i \(0.442201\pi\)
\(398\) −275998. −0.0873370
\(399\) 1.59065e6 0.500197
\(400\) 0 0
\(401\) −626701. −0.194625 −0.0973127 0.995254i \(-0.531025\pi\)
−0.0973127 + 0.995254i \(0.531025\pi\)
\(402\) 166312. 0.0513286
\(403\) 2.32637e6 0.713537
\(404\) −3.62954e6 −1.10636
\(405\) 0 0
\(406\) −359933. −0.108369
\(407\) −314616. −0.0941444
\(408\) 182097. 0.0541568
\(409\) −471719. −0.139436 −0.0697180 0.997567i \(-0.522210\pi\)
−0.0697180 + 0.997567i \(0.522210\pi\)
\(410\) 0 0
\(411\) 2.42778e6 0.708932
\(412\) 6.14976e6 1.78491
\(413\) −3.38725e6 −0.977173
\(414\) 111861. 0.0320759
\(415\) 0 0
\(416\) 498802. 0.141317
\(417\) −2.18234e6 −0.614586
\(418\) 125609. 0.0351625
\(419\) 3.13616e6 0.872696 0.436348 0.899778i \(-0.356272\pi\)
0.436348 + 0.899778i \(0.356272\pi\)
\(420\) 0 0
\(421\) 2.61937e6 0.720265 0.360132 0.932901i \(-0.382732\pi\)
0.360132 + 0.932901i \(0.382732\pi\)
\(422\) −264767. −0.0723741
\(423\) −447212. −0.121524
\(424\) −979894. −0.264706
\(425\) 0 0
\(426\) −112925. −0.0301484
\(427\) −1.43271e6 −0.380267
\(428\) 205311. 0.0541754
\(429\) −287890. −0.0755238
\(430\) 0 0
\(431\) 6.05506e6 1.57009 0.785046 0.619438i \(-0.212639\pi\)
0.785046 + 0.619438i \(0.212639\pi\)
\(432\) −719330. −0.185447
\(433\) 1.93261e6 0.495365 0.247683 0.968841i \(-0.420331\pi\)
0.247683 + 0.968841i \(0.420331\pi\)
\(434\) −583942. −0.148815
\(435\) 0 0
\(436\) 1.76360e6 0.444309
\(437\) −3.67820e6 −0.921365
\(438\) −358888. −0.0893868
\(439\) −3.88023e6 −0.960939 −0.480469 0.877011i \(-0.659534\pi\)
−0.480469 + 0.877011i \(0.659534\pi\)
\(440\) 0 0
\(441\) −446263. −0.109268
\(442\) −84087.7 −0.0204728
\(443\) 6.92252e6 1.67593 0.837963 0.545727i \(-0.183746\pi\)
0.837963 + 0.545727i \(0.183746\pi\)
\(444\) 739717. 0.178077
\(445\) 0 0
\(446\) 637779. 0.151821
\(447\) −3.30181e6 −0.781598
\(448\) 3.23096e6 0.760566
\(449\) 3.31516e6 0.776048 0.388024 0.921649i \(-0.373158\pi\)
0.388024 + 0.921649i \(0.373158\pi\)
\(450\) 0 0
\(451\) 1.14283e6 0.264570
\(452\) −3.32143e6 −0.764678
\(453\) −1.62344e6 −0.371698
\(454\) −719480. −0.163825
\(455\) 0 0
\(456\) −594299. −0.133842
\(457\) 3.40904e6 0.763557 0.381778 0.924254i \(-0.375312\pi\)
0.381778 + 0.924254i \(0.375312\pi\)
\(458\) −733886. −0.163480
\(459\) 371419. 0.0822872
\(460\) 0 0
\(461\) −2.54110e6 −0.556891 −0.278445 0.960452i \(-0.589819\pi\)
−0.278445 + 0.960452i \(0.589819\pi\)
\(462\) 72263.3 0.0157512
\(463\) 5.00975e6 1.08608 0.543042 0.839705i \(-0.317272\pi\)
0.543042 + 0.839705i \(0.317272\pi\)
\(464\) −5.35220e6 −1.15408
\(465\) 0 0
\(466\) 382846. 0.0816693
\(467\) −4.45513e6 −0.945297 −0.472649 0.881251i \(-0.656702\pi\)
−0.472649 + 0.881251i \(0.656702\pi\)
\(468\) 676880. 0.142856
\(469\) −3.14614e6 −0.660459
\(470\) 0 0
\(471\) 1.31297e6 0.272711
\(472\) 1.26555e6 0.261471
\(473\) 1.15830e6 0.238049
\(474\) −475476. −0.0972036
\(475\) 0 0
\(476\) −1.71182e6 −0.346290
\(477\) −1.99867e6 −0.402202
\(478\) −694804. −0.139089
\(479\) −1.72313e6 −0.343147 −0.171573 0.985171i \(-0.554885\pi\)
−0.171573 + 0.985171i \(0.554885\pi\)
\(480\) 0 0
\(481\) −687376. −0.135466
\(482\) −173168. −0.0339509
\(483\) −2.11608e6 −0.412729
\(484\) −462806. −0.0898018
\(485\) 0 0
\(486\) 36864.6 0.00707976
\(487\) 7.34645e6 1.40364 0.701819 0.712356i \(-0.252372\pi\)
0.701819 + 0.712356i \(0.252372\pi\)
\(488\) 535290. 0.101751
\(489\) 1.32791e6 0.251129
\(490\) 0 0
\(491\) −1.57312e6 −0.294481 −0.147240 0.989101i \(-0.547039\pi\)
−0.147240 + 0.989101i \(0.547039\pi\)
\(492\) −2.68699e6 −0.500442
\(493\) 2.76355e6 0.512095
\(494\) 274432. 0.0505961
\(495\) 0 0
\(496\) −8.68322e6 −1.58481
\(497\) 2.13620e6 0.387928
\(498\) −460583. −0.0832213
\(499\) −5.80283e6 −1.04325 −0.521625 0.853175i \(-0.674674\pi\)
−0.521625 + 0.853175i \(0.674674\pi\)
\(500\) 0 0
\(501\) −3.29814e6 −0.587050
\(502\) −516197. −0.0914231
\(503\) −4.44523e6 −0.783384 −0.391692 0.920096i \(-0.628110\pi\)
−0.391692 + 0.920096i \(0.628110\pi\)
\(504\) −341902. −0.0599550
\(505\) 0 0
\(506\) −167101. −0.0290137
\(507\) 2.71265e6 0.468677
\(508\) −3.78888e6 −0.651405
\(509\) −5.34223e6 −0.913962 −0.456981 0.889476i \(-0.651069\pi\)
−0.456981 + 0.889476i \(0.651069\pi\)
\(510\) 0 0
\(511\) 6.78910e6 1.15016
\(512\) −3.11572e6 −0.525271
\(513\) −1.21218e6 −0.203363
\(514\) −369355. −0.0616647
\(515\) 0 0
\(516\) −2.72335e6 −0.450277
\(517\) 668057. 0.109923
\(518\) 172538. 0.0282527
\(519\) 4.51764e6 0.736195
\(520\) 0 0
\(521\) −1.08913e7 −1.75786 −0.878930 0.476951i \(-0.841742\pi\)
−0.878930 + 0.476951i \(0.841742\pi\)
\(522\) 274292. 0.0440592
\(523\) 2.27063e6 0.362988 0.181494 0.983392i \(-0.441907\pi\)
0.181494 + 0.983392i \(0.441907\pi\)
\(524\) 6.75967e6 1.07547
\(525\) 0 0
\(526\) 926071. 0.145942
\(527\) 4.48349e6 0.703218
\(528\) 1.07455e6 0.167743
\(529\) −1.54313e6 −0.239752
\(530\) 0 0
\(531\) 2.58130e6 0.397285
\(532\) 5.58674e6 0.855815
\(533\) 2.49686e6 0.380695
\(534\) 10665.1 0.00161849
\(535\) 0 0
\(536\) 1.17546e6 0.176725
\(537\) −4.27757e6 −0.640120
\(538\) −422488. −0.0629302
\(539\) 666639. 0.0988369
\(540\) 0 0
\(541\) −7.70067e6 −1.13119 −0.565595 0.824683i \(-0.691353\pi\)
−0.565595 + 0.824683i \(0.691353\pi\)
\(542\) −21058.8 −0.00307919
\(543\) −4.43383e6 −0.645326
\(544\) 961315. 0.139274
\(545\) 0 0
\(546\) 157882. 0.0226647
\(547\) −4.26005e6 −0.608760 −0.304380 0.952551i \(-0.598449\pi\)
−0.304380 + 0.952551i \(0.598449\pi\)
\(548\) 8.52696e6 1.21295
\(549\) 1.09182e6 0.154603
\(550\) 0 0
\(551\) −9.01923e6 −1.26558
\(552\) 790613. 0.110437
\(553\) 8.99460e6 1.25075
\(554\) −946170. −0.130977
\(555\) 0 0
\(556\) −7.66493e6 −1.05153
\(557\) −2.72296e6 −0.371881 −0.185940 0.982561i \(-0.559533\pi\)
−0.185940 + 0.982561i \(0.559533\pi\)
\(558\) 445002. 0.0605029
\(559\) 2.53066e6 0.342534
\(560\) 0 0
\(561\) −554836. −0.0744316
\(562\) 1.23581e6 0.165048
\(563\) 3.37881e6 0.449255 0.224627 0.974445i \(-0.427884\pi\)
0.224627 + 0.974445i \(0.427884\pi\)
\(564\) −1.57072e6 −0.207922
\(565\) 0 0
\(566\) 431946. 0.0566746
\(567\) −697369. −0.0910972
\(568\) −798130. −0.103801
\(569\) −1.98049e6 −0.256444 −0.128222 0.991746i \(-0.540927\pi\)
−0.128222 + 0.991746i \(0.540927\pi\)
\(570\) 0 0
\(571\) −1.02513e7 −1.31580 −0.657898 0.753107i \(-0.728554\pi\)
−0.657898 + 0.753107i \(0.728554\pi\)
\(572\) −1.01114e6 −0.129218
\(573\) 2.67725e6 0.340646
\(574\) −626737. −0.0793973
\(575\) 0 0
\(576\) −2.46220e6 −0.309220
\(577\) −1.56178e7 −1.95291 −0.976453 0.215729i \(-0.930787\pi\)
−0.976453 + 0.215729i \(0.930787\pi\)
\(578\) 724366. 0.0901859
\(579\) −3.57404e6 −0.443061
\(580\) 0 0
\(581\) 8.71287e6 1.07083
\(582\) −146630. −0.0179438
\(583\) 2.98566e6 0.363805
\(584\) −2.53655e6 −0.307759
\(585\) 0 0
\(586\) −1.07449e6 −0.129259
\(587\) −9.39734e6 −1.12567 −0.562833 0.826570i \(-0.690289\pi\)
−0.562833 + 0.826570i \(0.690289\pi\)
\(588\) −1.56739e6 −0.186953
\(589\) −1.46325e7 −1.73792
\(590\) 0 0
\(591\) 2.55095e6 0.300423
\(592\) 2.56564e6 0.300879
\(593\) −1.28737e7 −1.50337 −0.751684 0.659523i \(-0.770758\pi\)
−0.751684 + 0.659523i \(0.770758\pi\)
\(594\) −55069.3 −0.00640389
\(595\) 0 0
\(596\) −1.15968e7 −1.33728
\(597\) −3.97880e6 −0.456894
\(598\) −365084. −0.0417484
\(599\) −4.74896e6 −0.540793 −0.270397 0.962749i \(-0.587155\pi\)
−0.270397 + 0.962749i \(0.587155\pi\)
\(600\) 0 0
\(601\) 1.93066e6 0.218032 0.109016 0.994040i \(-0.465230\pi\)
0.109016 + 0.994040i \(0.465230\pi\)
\(602\) −635219. −0.0714385
\(603\) 2.39756e6 0.268520
\(604\) −5.70192e6 −0.635959
\(605\) 0 0
\(606\) 645154. 0.0713644
\(607\) 1.21875e7 1.34259 0.671293 0.741192i \(-0.265739\pi\)
0.671293 + 0.741192i \(0.265739\pi\)
\(608\) −3.13738e6 −0.344198
\(609\) −5.18880e6 −0.566922
\(610\) 0 0
\(611\) 1.45958e6 0.158170
\(612\) 1.30452e6 0.140790
\(613\) 1.32187e7 1.42082 0.710410 0.703788i \(-0.248510\pi\)
0.710410 + 0.703788i \(0.248510\pi\)
\(614\) 580602. 0.0621524
\(615\) 0 0
\(616\) 510743. 0.0542314
\(617\) −1.20057e7 −1.26962 −0.634811 0.772667i \(-0.718922\pi\)
−0.634811 + 0.772667i \(0.718922\pi\)
\(618\) −1.09313e6 −0.115133
\(619\) −39871.4 −0.00418249 −0.00209124 0.999998i \(-0.500666\pi\)
−0.00209124 + 0.999998i \(0.500666\pi\)
\(620\) 0 0
\(621\) 1.61259e6 0.167801
\(622\) 183392. 0.0190066
\(623\) −201752. −0.0208256
\(624\) 2.34770e6 0.241369
\(625\) 0 0
\(626\) 415739. 0.0424019
\(627\) 1.81078e6 0.183949
\(628\) 4.61148e6 0.466596
\(629\) −1.32474e6 −0.133507
\(630\) 0 0
\(631\) 4.92201e6 0.492118 0.246059 0.969255i \(-0.420864\pi\)
0.246059 + 0.969255i \(0.420864\pi\)
\(632\) −3.36057e6 −0.334673
\(633\) −3.81689e6 −0.378617
\(634\) 1.27693e6 0.126166
\(635\) 0 0
\(636\) −7.01981e6 −0.688149
\(637\) 1.45648e6 0.142218
\(638\) −409745. −0.0398531
\(639\) −1.62793e6 −0.157718
\(640\) 0 0
\(641\) −7.42514e6 −0.713772 −0.356886 0.934148i \(-0.616162\pi\)
−0.356886 + 0.934148i \(0.616162\pi\)
\(642\) −36494.2 −0.00349451
\(643\) 9.55282e6 0.911180 0.455590 0.890190i \(-0.349428\pi\)
0.455590 + 0.890190i \(0.349428\pi\)
\(644\) −7.43221e6 −0.706160
\(645\) 0 0
\(646\) 528898. 0.0498644
\(647\) −8.28391e6 −0.777992 −0.388996 0.921240i \(-0.627178\pi\)
−0.388996 + 0.921240i \(0.627178\pi\)
\(648\) 260552. 0.0243757
\(649\) −3.85602e6 −0.359358
\(650\) 0 0
\(651\) −8.41812e6 −0.778508
\(652\) 4.66396e6 0.429671
\(653\) −2.04353e6 −0.187542 −0.0937711 0.995594i \(-0.529892\pi\)
−0.0937711 + 0.995594i \(0.529892\pi\)
\(654\) −313482. −0.0286595
\(655\) 0 0
\(656\) −9.31958e6 −0.845545
\(657\) −5.17374e6 −0.467618
\(658\) −366369. −0.0329878
\(659\) −1.43938e7 −1.29110 −0.645552 0.763716i \(-0.723373\pi\)
−0.645552 + 0.763716i \(0.723373\pi\)
\(660\) 0 0
\(661\) 1.39764e7 1.24421 0.622103 0.782936i \(-0.286279\pi\)
0.622103 + 0.782936i \(0.286279\pi\)
\(662\) 1.63963e6 0.145412
\(663\) −1.21221e6 −0.107101
\(664\) −3.25531e6 −0.286531
\(665\) 0 0
\(666\) −131485. −0.0114866
\(667\) 1.19985e7 1.04427
\(668\) −1.15839e7 −1.00442
\(669\) 9.19424e6 0.794238
\(670\) 0 0
\(671\) −1.63099e6 −0.139844
\(672\) −1.80495e6 −0.154185
\(673\) −4.36237e6 −0.371266 −0.185633 0.982619i \(-0.559433\pi\)
−0.185633 + 0.982619i \(0.559433\pi\)
\(674\) 603195. 0.0511456
\(675\) 0 0
\(676\) 9.52751e6 0.801886
\(677\) −2.77129e6 −0.232386 −0.116193 0.993227i \(-0.537069\pi\)
−0.116193 + 0.993227i \(0.537069\pi\)
\(678\) 590386. 0.0493244
\(679\) 2.77380e6 0.230887
\(680\) 0 0
\(681\) −1.03720e7 −0.857031
\(682\) −664756. −0.0547270
\(683\) 1.27204e7 1.04340 0.521698 0.853130i \(-0.325299\pi\)
0.521698 + 0.853130i \(0.325299\pi\)
\(684\) −4.25746e6 −0.347945
\(685\) 0 0
\(686\) −1.48086e6 −0.120145
\(687\) −1.05797e7 −0.855229
\(688\) −9.44571e6 −0.760788
\(689\) 6.52310e6 0.523487
\(690\) 0 0
\(691\) 2.06818e7 1.64776 0.823879 0.566766i \(-0.191806\pi\)
0.823879 + 0.566766i \(0.191806\pi\)
\(692\) 1.58671e7 1.25960
\(693\) 1.04175e6 0.0824006
\(694\) 1.21473e6 0.0957373
\(695\) 0 0
\(696\) 1.93864e6 0.151696
\(697\) 4.81207e6 0.375189
\(698\) −779981. −0.0605962
\(699\) 5.51911e6 0.427244
\(700\) 0 0
\(701\) 1.40652e7 1.08106 0.540531 0.841324i \(-0.318224\pi\)
0.540531 + 0.841324i \(0.318224\pi\)
\(702\) −120316. −0.00921469
\(703\) 4.32348e6 0.329948
\(704\) 3.67811e6 0.279700
\(705\) 0 0
\(706\) 2.58158e6 0.194928
\(707\) −1.22044e7 −0.918265
\(708\) 9.06618e6 0.679737
\(709\) −2.36738e7 −1.76869 −0.884345 0.466833i \(-0.845395\pi\)
−0.884345 + 0.466833i \(0.845395\pi\)
\(710\) 0 0
\(711\) −6.85447e6 −0.508510
\(712\) 75378.6 0.00557247
\(713\) 1.94660e7 1.43401
\(714\) 304277. 0.0223369
\(715\) 0 0
\(716\) −1.50239e7 −1.09522
\(717\) −1.00163e7 −0.727629
\(718\) 1.46572e6 0.106106
\(719\) 1.03736e7 0.748354 0.374177 0.927357i \(-0.377925\pi\)
0.374177 + 0.927357i \(0.377925\pi\)
\(720\) 0 0
\(721\) 2.06787e7 1.48144
\(722\) −180287. −0.0128713
\(723\) −2.49640e6 −0.177610
\(724\) −1.55727e7 −1.10412
\(725\) 0 0
\(726\) 82264.1 0.00579253
\(727\) 2.09455e7 1.46979 0.734894 0.678182i \(-0.237232\pi\)
0.734894 + 0.678182i \(0.237232\pi\)
\(728\) 1.11588e6 0.0780347
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 4.87720e6 0.337580
\(732\) 3.83474e6 0.264520
\(733\) 1.15758e7 0.795780 0.397890 0.917433i \(-0.369743\pi\)
0.397890 + 0.917433i \(0.369743\pi\)
\(734\) −1.00660e6 −0.0689631
\(735\) 0 0
\(736\) 4.17375e6 0.284009
\(737\) −3.58154e6 −0.242885
\(738\) 477615. 0.0322803
\(739\) −1.84303e7 −1.24143 −0.620713 0.784038i \(-0.713157\pi\)
−0.620713 + 0.784038i \(0.713157\pi\)
\(740\) 0 0
\(741\) 3.95621e6 0.264688
\(742\) −1.63736e6 −0.109178
\(743\) 1.27772e7 0.849108 0.424554 0.905403i \(-0.360431\pi\)
0.424554 + 0.905403i \(0.360431\pi\)
\(744\) 3.14519e6 0.208312
\(745\) 0 0
\(746\) 337779. 0.0222221
\(747\) −6.63977e6 −0.435363
\(748\) −1.94872e6 −0.127349
\(749\) 690362. 0.0449648
\(750\) 0 0
\(751\) 1.85955e7 1.20311 0.601557 0.798830i \(-0.294547\pi\)
0.601557 + 0.798830i \(0.294547\pi\)
\(752\) −5.44790e6 −0.351305
\(753\) −7.44151e6 −0.478270
\(754\) −895215. −0.0573455
\(755\) 0 0
\(756\) −2.44933e6 −0.155863
\(757\) −1.07465e7 −0.681595 −0.340797 0.940137i \(-0.610697\pi\)
−0.340797 + 0.940137i \(0.610697\pi\)
\(758\) −1.93627e6 −0.122403
\(759\) −2.40893e6 −0.151782
\(760\) 0 0
\(761\) −1.04966e7 −0.657031 −0.328515 0.944499i \(-0.606548\pi\)
−0.328515 + 0.944499i \(0.606548\pi\)
\(762\) 673476. 0.0420179
\(763\) 5.93016e6 0.368769
\(764\) 9.40318e6 0.582829
\(765\) 0 0
\(766\) −455367. −0.0280407
\(767\) −8.42467e6 −0.517088
\(768\) −8.30862e6 −0.508307
\(769\) −1.94421e7 −1.18557 −0.592787 0.805360i \(-0.701972\pi\)
−0.592787 + 0.805360i \(0.701972\pi\)
\(770\) 0 0
\(771\) −5.32463e6 −0.322592
\(772\) −1.25529e7 −0.758057
\(773\) −1.32960e7 −0.800338 −0.400169 0.916441i \(-0.631049\pi\)
−0.400169 + 0.916441i \(0.631049\pi\)
\(774\) 484079. 0.0290445
\(775\) 0 0
\(776\) −1.03635e6 −0.0617805
\(777\) 2.48731e6 0.147801
\(778\) −3.34409e6 −0.198075
\(779\) −1.57049e7 −0.927236
\(780\) 0 0
\(781\) 2.43184e6 0.142662
\(782\) −703608. −0.0411447
\(783\) 3.95420e6 0.230491
\(784\) −5.43633e6 −0.315876
\(785\) 0 0
\(786\) −1.20154e6 −0.0693715
\(787\) 7.22388e6 0.415752 0.207876 0.978155i \(-0.433345\pi\)
0.207876 + 0.978155i \(0.433345\pi\)
\(788\) 8.95957e6 0.514010
\(789\) 1.33503e7 0.763479
\(790\) 0 0
\(791\) −1.11684e7 −0.634671
\(792\) −389219. −0.0220486
\(793\) −3.56340e6 −0.201225
\(794\) −708085. −0.0398597
\(795\) 0 0
\(796\) −1.39745e7 −0.781726
\(797\) 1.01501e7 0.566012 0.283006 0.959118i \(-0.408668\pi\)
0.283006 + 0.959118i \(0.408668\pi\)
\(798\) −993048. −0.0552031
\(799\) 2.81297e6 0.155883
\(800\) 0 0
\(801\) 153748. 0.00846696
\(802\) 391253. 0.0214794
\(803\) 7.72867e6 0.422976
\(804\) 8.42084e6 0.459425
\(805\) 0 0
\(806\) −1.45237e6 −0.0787478
\(807\) −6.09060e6 −0.329213
\(808\) 4.55982e6 0.245708
\(809\) 1.07717e7 0.578644 0.289322 0.957232i \(-0.406570\pi\)
0.289322 + 0.957232i \(0.406570\pi\)
\(810\) 0 0
\(811\) 2.92706e7 1.56271 0.781357 0.624084i \(-0.214528\pi\)
0.781357 + 0.624084i \(0.214528\pi\)
\(812\) −1.82243e7 −0.969978
\(813\) −303585. −0.0161084
\(814\) 196416. 0.0103900
\(815\) 0 0
\(816\) 4.52459e6 0.237878
\(817\) −1.59174e7 −0.834290
\(818\) 294496. 0.0153885
\(819\) 2.27602e6 0.118568
\(820\) 0 0
\(821\) 1.75303e7 0.907677 0.453838 0.891084i \(-0.350054\pi\)
0.453838 + 0.891084i \(0.350054\pi\)
\(822\) −1.51567e6 −0.0782396
\(823\) −3.94210e6 −0.202875 −0.101437 0.994842i \(-0.532344\pi\)
−0.101437 + 0.994842i \(0.532344\pi\)
\(824\) −7.72600e6 −0.396403
\(825\) 0 0
\(826\) 2.11467e6 0.107843
\(827\) −1.55928e7 −0.792794 −0.396397 0.918079i \(-0.629740\pi\)
−0.396397 + 0.918079i \(0.629740\pi\)
\(828\) 5.66383e6 0.287101
\(829\) 1.82374e7 0.921675 0.460837 0.887485i \(-0.347549\pi\)
0.460837 + 0.887485i \(0.347549\pi\)
\(830\) 0 0
\(831\) −1.36400e7 −0.685192
\(832\) 8.03597e6 0.402467
\(833\) 2.80700e6 0.140162
\(834\) 1.36245e6 0.0678273
\(835\) 0 0
\(836\) 6.35992e6 0.314728
\(837\) 6.41516e6 0.316515
\(838\) −1.95792e6 −0.0963130
\(839\) 8.42568e6 0.413238 0.206619 0.978421i \(-0.433754\pi\)
0.206619 + 0.978421i \(0.433754\pi\)
\(840\) 0 0
\(841\) 8.91021e6 0.434408
\(842\) −1.63529e6 −0.0794903
\(843\) 1.78154e7 0.863430
\(844\) −1.34059e7 −0.647797
\(845\) 0 0
\(846\) 279197. 0.0134117
\(847\) −1.55619e6 −0.0745341
\(848\) −2.43476e7 −1.16270
\(849\) 6.22694e6 0.296487
\(850\) 0 0
\(851\) −5.75165e6 −0.272250
\(852\) −5.71768e6 −0.269849
\(853\) −4.15991e7 −1.95755 −0.978773 0.204949i \(-0.934297\pi\)
−0.978773 + 0.204949i \(0.934297\pi\)
\(854\) 894448. 0.0419672
\(855\) 0 0
\(856\) −257934. −0.0120316
\(857\) −1.04420e7 −0.485659 −0.242829 0.970069i \(-0.578076\pi\)
−0.242829 + 0.970069i \(0.578076\pi\)
\(858\) 179731. 0.00833500
\(859\) −6.29331e6 −0.291002 −0.145501 0.989358i \(-0.546479\pi\)
−0.145501 + 0.989358i \(0.546479\pi\)
\(860\) 0 0
\(861\) −9.03506e6 −0.415359
\(862\) −3.78020e6 −0.173279
\(863\) 9.45460e6 0.432132 0.216066 0.976379i \(-0.430677\pi\)
0.216066 + 0.976379i \(0.430677\pi\)
\(864\) 1.37549e6 0.0626863
\(865\) 0 0
\(866\) −1.20654e6 −0.0546698
\(867\) 1.04425e7 0.471798
\(868\) −2.95666e7 −1.33199
\(869\) 1.02394e7 0.459965
\(870\) 0 0
\(871\) −7.82499e6 −0.349493
\(872\) −2.21563e6 −0.0986748
\(873\) −2.11381e6 −0.0938710
\(874\) 2.29632e6 0.101684
\(875\) 0 0
\(876\) −1.81715e7 −0.800073
\(877\) −432267. −0.0189781 −0.00948905 0.999955i \(-0.503021\pi\)
−0.00948905 + 0.999955i \(0.503021\pi\)
\(878\) 2.42245e6 0.106052
\(879\) −1.54899e7 −0.676202
\(880\) 0 0
\(881\) −1.03415e7 −0.448894 −0.224447 0.974486i \(-0.572058\pi\)
−0.224447 + 0.974486i \(0.572058\pi\)
\(882\) 278604. 0.0120591
\(883\) 1.89956e7 0.819881 0.409941 0.912112i \(-0.365549\pi\)
0.409941 + 0.912112i \(0.365549\pi\)
\(884\) −4.25759e6 −0.183245
\(885\) 0 0
\(886\) −4.32176e6 −0.184960
\(887\) 190605. 0.00813438 0.00406719 0.999992i \(-0.498705\pi\)
0.00406719 + 0.999992i \(0.498705\pi\)
\(888\) −929312. −0.0395484
\(889\) −1.27402e7 −0.540656
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) 3.22925e7 1.35890
\(893\) −9.18050e6 −0.385246
\(894\) 2.06134e6 0.0862592
\(895\) 0 0
\(896\) −8.43470e6 −0.350994
\(897\) −5.26307e6 −0.218403
\(898\) −2.06967e6 −0.0856467
\(899\) 4.77322e7 1.96975
\(900\) 0 0
\(901\) 1.25716e7 0.515917
\(902\) −713474. −0.0291986
\(903\) −9.15734e6 −0.373723
\(904\) 4.17274e6 0.169824
\(905\) 0 0
\(906\) 1.01352e6 0.0410216
\(907\) 2.99780e7 1.21000 0.604999 0.796226i \(-0.293173\pi\)
0.604999 + 0.796226i \(0.293173\pi\)
\(908\) −3.64292e7 −1.46634
\(909\) 9.30056e6 0.373335
\(910\) 0 0
\(911\) 4.55816e7 1.81967 0.909837 0.414966i \(-0.136206\pi\)
0.909837 + 0.414966i \(0.136206\pi\)
\(912\) −1.47666e7 −0.587887
\(913\) 9.91867e6 0.393801
\(914\) −2.12828e6 −0.0842681
\(915\) 0 0
\(916\) −3.71586e7 −1.46326
\(917\) 2.27295e7 0.892622
\(918\) −231879. −0.00908143
\(919\) −3.05489e6 −0.119318 −0.0596591 0.998219i \(-0.519001\pi\)
−0.0596591 + 0.998219i \(0.519001\pi\)
\(920\) 0 0
\(921\) 8.36998e6 0.325144
\(922\) 1.58642e6 0.0614599
\(923\) 5.31311e6 0.205279
\(924\) 3.65888e6 0.140984
\(925\) 0 0
\(926\) −3.12761e6 −0.119863
\(927\) −1.57585e7 −0.602305
\(928\) 1.02344e7 0.390113
\(929\) −1.75975e7 −0.668978 −0.334489 0.942400i \(-0.608564\pi\)
−0.334489 + 0.942400i \(0.608564\pi\)
\(930\) 0 0
\(931\) −9.16102e6 −0.346393
\(932\) 1.93845e7 0.730995
\(933\) 2.64378e6 0.0994309
\(934\) 2.78136e6 0.104326
\(935\) 0 0
\(936\) −850370. −0.0317262
\(937\) 2.79612e7 1.04042 0.520208 0.854040i \(-0.325854\pi\)
0.520208 + 0.854040i \(0.325854\pi\)
\(938\) 1.96415e6 0.0728899
\(939\) 5.99330e6 0.221821
\(940\) 0 0
\(941\) 3.56145e7 1.31115 0.655576 0.755129i \(-0.272426\pi\)
0.655576 + 0.755129i \(0.272426\pi\)
\(942\) −819694. −0.0300971
\(943\) 2.08926e7 0.765092
\(944\) 3.14452e7 1.14848
\(945\) 0 0
\(946\) −723130. −0.0262717
\(947\) 3.24704e7 1.17656 0.588278 0.808659i \(-0.299806\pi\)
0.588278 + 0.808659i \(0.299806\pi\)
\(948\) −2.40746e7 −0.870038
\(949\) 1.68857e7 0.608629
\(950\) 0 0
\(951\) 1.84082e7 0.660024
\(952\) 2.15057e6 0.0769061
\(953\) −4.13052e6 −0.147324 −0.0736619 0.997283i \(-0.523469\pi\)
−0.0736619 + 0.997283i \(0.523469\pi\)
\(954\) 1.24778e6 0.0443880
\(955\) 0 0
\(956\) −3.51798e7 −1.24494
\(957\) −5.90689e6 −0.208487
\(958\) 1.07576e6 0.0378705
\(959\) 2.86721e7 1.00673
\(960\) 0 0
\(961\) 4.88099e7 1.70490
\(962\) 429132. 0.0149504
\(963\) −526101. −0.0182811
\(964\) −8.76798e6 −0.303883
\(965\) 0 0
\(966\) 1.32108e6 0.0455498
\(967\) 4.10935e7 1.41321 0.706605 0.707608i \(-0.250226\pi\)
0.706605 + 0.707608i \(0.250226\pi\)
\(968\) 581426. 0.0199437
\(969\) 7.62460e6 0.260860
\(970\) 0 0
\(971\) −2.37238e7 −0.807487 −0.403744 0.914872i \(-0.632291\pi\)
−0.403744 + 0.914872i \(0.632291\pi\)
\(972\) 1.86655e6 0.0633687
\(973\) −2.57735e7 −0.872752
\(974\) −4.58642e6 −0.154909
\(975\) 0 0
\(976\) 1.33004e7 0.446932
\(977\) −1.39204e7 −0.466568 −0.233284 0.972409i \(-0.574947\pi\)
−0.233284 + 0.972409i \(0.574947\pi\)
\(978\) −829023. −0.0277153
\(979\) −229673. −0.00765866
\(980\) 0 0
\(981\) −4.51917e6 −0.149929
\(982\) 982104. 0.0324997
\(983\) −4.80216e6 −0.158509 −0.0792543 0.996854i \(-0.525254\pi\)
−0.0792543 + 0.996854i \(0.525254\pi\)
\(984\) 3.37569e6 0.111141
\(985\) 0 0
\(986\) −1.72530e6 −0.0565162
\(987\) −5.28158e6 −0.172572
\(988\) 1.38952e7 0.452869
\(989\) 2.11754e7 0.688399
\(990\) 0 0
\(991\) −4.05874e7 −1.31283 −0.656413 0.754401i \(-0.727927\pi\)
−0.656413 + 0.754401i \(0.727927\pi\)
\(992\) 1.66039e7 0.535710
\(993\) 2.36369e7 0.760708
\(994\) −1.33364e6 −0.0428128
\(995\) 0 0
\(996\) −2.33205e7 −0.744887
\(997\) 4.33820e7 1.38220 0.691101 0.722758i \(-0.257126\pi\)
0.691101 + 0.722758i \(0.257126\pi\)
\(998\) 3.62273e6 0.115136
\(999\) −1.89549e6 −0.0600909
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.1.5 yes 9
5.4 even 2 825.6.a.r.1.5 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.r.1.5 9 5.4 even 2
825.6.a.s.1.5 yes 9 1.1 even 1 trivial