Properties

Label 825.6.a.u.1.8
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - x^{9} - 271 x^{8} + 309 x^{7} + 24456 x^{6} - 33410 x^{5} - 822204 x^{4} + 1367872 x^{3} + 7443872 x^{2} - 12856224 x - 7036608 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(6.73090\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+6.73090 q^{2} -9.00000 q^{3} +13.3050 q^{4} -60.5781 q^{6} -2.41142 q^{7} -125.834 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+6.73090 q^{2} -9.00000 q^{3} +13.3050 q^{4} -60.5781 q^{6} -2.41142 q^{7} -125.834 q^{8} +81.0000 q^{9} +121.000 q^{11} -119.745 q^{12} -334.983 q^{13} -16.2310 q^{14} -1272.74 q^{16} +1211.39 q^{17} +545.203 q^{18} -746.587 q^{19} +21.7028 q^{21} +814.439 q^{22} -3387.32 q^{23} +1132.51 q^{24} -2254.73 q^{26} -729.000 q^{27} -32.0839 q^{28} -2012.09 q^{29} +8689.78 q^{31} -4539.97 q^{32} -1089.00 q^{33} +8153.73 q^{34} +1077.70 q^{36} -1392.25 q^{37} -5025.20 q^{38} +3014.84 q^{39} +2339.11 q^{41} +146.079 q^{42} -10534.9 q^{43} +1609.90 q^{44} -22799.7 q^{46} +22537.9 q^{47} +11454.6 q^{48} -16801.2 q^{49} -10902.5 q^{51} -4456.94 q^{52} -5835.91 q^{53} -4906.83 q^{54} +303.439 q^{56} +6719.28 q^{57} -13543.2 q^{58} +27573.2 q^{59} +13969.9 q^{61} +58490.0 q^{62} -195.325 q^{63} +10169.5 q^{64} -7329.95 q^{66} -18128.3 q^{67} +16117.5 q^{68} +30485.8 q^{69} +986.398 q^{71} -10192.6 q^{72} +11598.8 q^{73} -9371.12 q^{74} -9933.34 q^{76} -291.782 q^{77} +20292.6 q^{78} -48553.4 q^{79} +6561.00 q^{81} +15744.3 q^{82} +34168.5 q^{83} +288.755 q^{84} -70909.2 q^{86} +18108.8 q^{87} -15225.9 q^{88} +124210. q^{89} +807.783 q^{91} -45068.2 q^{92} -78208.0 q^{93} +151700. q^{94} +40859.7 q^{96} +138028. q^{97} -113087. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9} + 1210 q^{11} - 2007 q^{12} + 1102 q^{13} + 684 q^{14} + 7019 q^{16} - 1106 q^{17} + 81 q^{18} + 2586 q^{19} + 1692 q^{21} + 121 q^{22} - 2206 q^{23} + 1593 q^{24} + 12001 q^{26} - 7290 q^{27} - 14452 q^{28} + 5824 q^{29} - 4586 q^{31} + 10627 q^{32} - 10890 q^{33} + 7426 q^{34} + 18063 q^{36} + 18362 q^{37} - 44001 q^{38} - 9918 q^{39} - 5474 q^{41} - 6156 q^{42} - 20496 q^{43} + 26983 q^{44} + 19981 q^{46} + 14970 q^{47} - 63171 q^{48} + 68582 q^{49} + 9954 q^{51} + 58603 q^{52} - 61980 q^{53} - 729 q^{54} + 7132 q^{56} - 23274 q^{57} - 7161 q^{58} + 61190 q^{59} + 8230 q^{61} + 4509 q^{62} - 15228 q^{63} + 152223 q^{64} - 1089 q^{66} + 11930 q^{67} - 105598 q^{68} + 19854 q^{69} + 59822 q^{71} - 14337 q^{72} + 20680 q^{73} + 132564 q^{74} + 68165 q^{76} - 22748 q^{77} - 108009 q^{78} + 234494 q^{79} + 65610 q^{81} - 151948 q^{82} - 185478 q^{83} + 130068 q^{84} - 17825 q^{86} - 52416 q^{87} - 21417 q^{88} + 181834 q^{89} + 206274 q^{91} - 98373 q^{92} + 41274 q^{93} - 64998 q^{94} - 95643 q^{96} - 304358 q^{97} - 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.73090 1.18987 0.594933 0.803775i \(-0.297179\pi\)
0.594933 + 0.803775i \(0.297179\pi\)
\(3\) −9.00000 −0.577350
\(4\) 13.3050 0.415781
\(5\) 0 0
\(6\) −60.5781 −0.686969
\(7\) −2.41142 −0.0186006 −0.00930031 0.999957i \(-0.502960\pi\)
−0.00930031 + 0.999957i \(0.502960\pi\)
\(8\) −125.834 −0.695142
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −119.745 −0.240051
\(13\) −334.983 −0.549748 −0.274874 0.961480i \(-0.588636\pi\)
−0.274874 + 0.961480i \(0.588636\pi\)
\(14\) −16.2310 −0.0221322
\(15\) 0 0
\(16\) −1272.74 −1.24291
\(17\) 1211.39 1.01663 0.508313 0.861173i \(-0.330269\pi\)
0.508313 + 0.861173i \(0.330269\pi\)
\(18\) 545.203 0.396622
\(19\) −746.587 −0.474456 −0.237228 0.971454i \(-0.576239\pi\)
−0.237228 + 0.971454i \(0.576239\pi\)
\(20\) 0 0
\(21\) 21.7028 0.0107391
\(22\) 814.439 0.358758
\(23\) −3387.32 −1.33517 −0.667584 0.744534i \(-0.732672\pi\)
−0.667584 + 0.744534i \(0.732672\pi\)
\(24\) 1132.51 0.401341
\(25\) 0 0
\(26\) −2254.73 −0.654127
\(27\) −729.000 −0.192450
\(28\) −32.0839 −0.00773379
\(29\) −2012.09 −0.444275 −0.222137 0.975015i \(-0.571303\pi\)
−0.222137 + 0.975015i \(0.571303\pi\)
\(30\) 0 0
\(31\) 8689.78 1.62407 0.812035 0.583609i \(-0.198360\pi\)
0.812035 + 0.583609i \(0.198360\pi\)
\(32\) −4539.97 −0.783751
\(33\) −1089.00 −0.174078
\(34\) 8153.73 1.20965
\(35\) 0 0
\(36\) 1077.70 0.138594
\(37\) −1392.25 −0.167192 −0.0835958 0.996500i \(-0.526640\pi\)
−0.0835958 + 0.996500i \(0.526640\pi\)
\(38\) −5025.20 −0.564540
\(39\) 3014.84 0.317397
\(40\) 0 0
\(41\) 2339.11 0.217316 0.108658 0.994079i \(-0.465345\pi\)
0.108658 + 0.994079i \(0.465345\pi\)
\(42\) 146.079 0.0127781
\(43\) −10534.9 −0.868877 −0.434439 0.900701i \(-0.643053\pi\)
−0.434439 + 0.900701i \(0.643053\pi\)
\(44\) 1609.90 0.125363
\(45\) 0 0
\(46\) −22799.7 −1.58867
\(47\) 22537.9 1.48823 0.744113 0.668054i \(-0.232873\pi\)
0.744113 + 0.668054i \(0.232873\pi\)
\(48\) 11454.6 0.717593
\(49\) −16801.2 −0.999654
\(50\) 0 0
\(51\) −10902.5 −0.586949
\(52\) −4456.94 −0.228575
\(53\) −5835.91 −0.285377 −0.142688 0.989768i \(-0.545575\pi\)
−0.142688 + 0.989768i \(0.545575\pi\)
\(54\) −4906.83 −0.228990
\(55\) 0 0
\(56\) 303.439 0.0129301
\(57\) 6719.28 0.273928
\(58\) −13543.2 −0.528628
\(59\) 27573.2 1.03123 0.515617 0.856819i \(-0.327563\pi\)
0.515617 + 0.856819i \(0.327563\pi\)
\(60\) 0 0
\(61\) 13969.9 0.480693 0.240347 0.970687i \(-0.422739\pi\)
0.240347 + 0.970687i \(0.422739\pi\)
\(62\) 58490.0 1.93242
\(63\) −195.325 −0.00620021
\(64\) 10169.5 0.310349
\(65\) 0 0
\(66\) −7329.95 −0.207129
\(67\) −18128.3 −0.493366 −0.246683 0.969096i \(-0.579341\pi\)
−0.246683 + 0.969096i \(0.579341\pi\)
\(68\) 16117.5 0.422694
\(69\) 30485.8 0.770860
\(70\) 0 0
\(71\) 986.398 0.0232224 0.0116112 0.999933i \(-0.496304\pi\)
0.0116112 + 0.999933i \(0.496304\pi\)
\(72\) −10192.6 −0.231714
\(73\) 11598.8 0.254745 0.127372 0.991855i \(-0.459346\pi\)
0.127372 + 0.991855i \(0.459346\pi\)
\(74\) −9371.12 −0.198936
\(75\) 0 0
\(76\) −9933.34 −0.197270
\(77\) −291.782 −0.00560830
\(78\) 20292.6 0.377660
\(79\) −48553.4 −0.875289 −0.437645 0.899148i \(-0.644187\pi\)
−0.437645 + 0.899148i \(0.644187\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 15744.3 0.258577
\(83\) 34168.5 0.544416 0.272208 0.962239i \(-0.412246\pi\)
0.272208 + 0.962239i \(0.412246\pi\)
\(84\) 288.755 0.00446510
\(85\) 0 0
\(86\) −70909.2 −1.03385
\(87\) 18108.8 0.256502
\(88\) −15225.9 −0.209593
\(89\) 124210. 1.66219 0.831097 0.556128i \(-0.187713\pi\)
0.831097 + 0.556128i \(0.187713\pi\)
\(90\) 0 0
\(91\) 807.783 0.0102257
\(92\) −45068.2 −0.555138
\(93\) −78208.0 −0.937657
\(94\) 151700. 1.77079
\(95\) 0 0
\(96\) 40859.7 0.452499
\(97\) 138028. 1.48949 0.744744 0.667351i \(-0.232572\pi\)
0.744744 + 0.667351i \(0.232572\pi\)
\(98\) −113087. −1.18945
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −5951.66 −0.0580543 −0.0290272 0.999579i \(-0.509241\pi\)
−0.0290272 + 0.999579i \(0.509241\pi\)
\(102\) −73383.6 −0.698390
\(103\) 191198. 1.77578 0.887892 0.460052i \(-0.152169\pi\)
0.887892 + 0.460052i \(0.152169\pi\)
\(104\) 42152.3 0.382153
\(105\) 0 0
\(106\) −39280.9 −0.339560
\(107\) −88404.1 −0.746471 −0.373236 0.927737i \(-0.621752\pi\)
−0.373236 + 0.927737i \(0.621752\pi\)
\(108\) −9699.34 −0.0800171
\(109\) 223100. 1.79859 0.899296 0.437341i \(-0.144080\pi\)
0.899296 + 0.437341i \(0.144080\pi\)
\(110\) 0 0
\(111\) 12530.3 0.0965281
\(112\) 3069.10 0.0231188
\(113\) −12365.3 −0.0910978 −0.0455489 0.998962i \(-0.514504\pi\)
−0.0455489 + 0.998962i \(0.514504\pi\)
\(114\) 45226.8 0.325937
\(115\) 0 0
\(116\) −26770.8 −0.184721
\(117\) −27133.6 −0.183249
\(118\) 185592. 1.22703
\(119\) −2921.16 −0.0189099
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 94029.9 0.571961
\(123\) −21052.0 −0.125467
\(124\) 115618. 0.675257
\(125\) 0 0
\(126\) −1314.71 −0.00737741
\(127\) 207000. 1.13883 0.569417 0.822049i \(-0.307169\pi\)
0.569417 + 0.822049i \(0.307169\pi\)
\(128\) 213729. 1.15302
\(129\) 94813.9 0.501647
\(130\) 0 0
\(131\) −104541. −0.532239 −0.266119 0.963940i \(-0.585742\pi\)
−0.266119 + 0.963940i \(0.585742\pi\)
\(132\) −14489.1 −0.0723782
\(133\) 1800.33 0.00882518
\(134\) −122020. −0.587040
\(135\) 0 0
\(136\) −152434. −0.706699
\(137\) 159251. 0.724907 0.362453 0.932002i \(-0.381939\pi\)
0.362453 + 0.932002i \(0.381939\pi\)
\(138\) 205197. 0.917220
\(139\) 184879. 0.811614 0.405807 0.913959i \(-0.366990\pi\)
0.405807 + 0.913959i \(0.366990\pi\)
\(140\) 0 0
\(141\) −202841. −0.859227
\(142\) 6639.35 0.0276315
\(143\) −40532.9 −0.165755
\(144\) −103092. −0.414302
\(145\) 0 0
\(146\) 78070.3 0.303112
\(147\) 151211. 0.577151
\(148\) −18523.9 −0.0695151
\(149\) 270575. 0.998442 0.499221 0.866475i \(-0.333620\pi\)
0.499221 + 0.866475i \(0.333620\pi\)
\(150\) 0 0
\(151\) −287926. −1.02764 −0.513818 0.857899i \(-0.671769\pi\)
−0.513818 + 0.857899i \(0.671769\pi\)
\(152\) 93946.1 0.329815
\(153\) 98122.4 0.338875
\(154\) −1963.95 −0.00667312
\(155\) 0 0
\(156\) 40112.5 0.131968
\(157\) −400095. −1.29543 −0.647715 0.761883i \(-0.724275\pi\)
−0.647715 + 0.761883i \(0.724275\pi\)
\(158\) −326808. −1.04148
\(159\) 52523.1 0.164762
\(160\) 0 0
\(161\) 8168.23 0.0248350
\(162\) 44161.4 0.132207
\(163\) −230209. −0.678661 −0.339331 0.940667i \(-0.610201\pi\)
−0.339331 + 0.940667i \(0.610201\pi\)
\(164\) 31121.9 0.0903559
\(165\) 0 0
\(166\) 229985. 0.647782
\(167\) 505911. 1.40373 0.701864 0.712311i \(-0.252351\pi\)
0.701864 + 0.712311i \(0.252351\pi\)
\(168\) −2730.95 −0.00746518
\(169\) −259080. −0.697777
\(170\) 0 0
\(171\) −60473.5 −0.158152
\(172\) −140167. −0.361263
\(173\) 182355. 0.463237 0.231619 0.972807i \(-0.425598\pi\)
0.231619 + 0.972807i \(0.425598\pi\)
\(174\) 121888. 0.305203
\(175\) 0 0
\(176\) −154001. −0.374751
\(177\) −248159. −0.595383
\(178\) 836045. 1.97779
\(179\) 293039. 0.683586 0.341793 0.939775i \(-0.388966\pi\)
0.341793 + 0.939775i \(0.388966\pi\)
\(180\) 0 0
\(181\) 234051. 0.531024 0.265512 0.964108i \(-0.414459\pi\)
0.265512 + 0.964108i \(0.414459\pi\)
\(182\) 5437.11 0.0121672
\(183\) −125729. −0.277528
\(184\) 426240. 0.928132
\(185\) 0 0
\(186\) −526410. −1.11569
\(187\) 146578. 0.306524
\(188\) 299867. 0.618776
\(189\) 1757.92 0.00357969
\(190\) 0 0
\(191\) 560936. 1.11258 0.556288 0.830990i \(-0.312225\pi\)
0.556288 + 0.830990i \(0.312225\pi\)
\(192\) −91525.5 −0.179180
\(193\) 490651. 0.948155 0.474078 0.880483i \(-0.342782\pi\)
0.474078 + 0.880483i \(0.342782\pi\)
\(194\) 929050. 1.77229
\(195\) 0 0
\(196\) −223540. −0.415637
\(197\) −1.01719e6 −1.86740 −0.933698 0.358061i \(-0.883438\pi\)
−0.933698 + 0.358061i \(0.883438\pi\)
\(198\) 65969.5 0.119586
\(199\) −104543. −0.187138 −0.0935689 0.995613i \(-0.529828\pi\)
−0.0935689 + 0.995613i \(0.529828\pi\)
\(200\) 0 0
\(201\) 163154. 0.284845
\(202\) −40060.0 −0.0690769
\(203\) 4851.98 0.00826379
\(204\) −145058. −0.244042
\(205\) 0 0
\(206\) 1.28693e6 2.11295
\(207\) −274373. −0.445056
\(208\) 426345. 0.683286
\(209\) −90337.0 −0.143054
\(210\) 0 0
\(211\) 29310.5 0.0453228 0.0226614 0.999743i \(-0.492786\pi\)
0.0226614 + 0.999743i \(0.492786\pi\)
\(212\) −77646.7 −0.118654
\(213\) −8877.59 −0.0134074
\(214\) −595039. −0.888201
\(215\) 0 0
\(216\) 91733.1 0.133780
\(217\) −20954.7 −0.0302087
\(218\) 1.50166e6 2.14008
\(219\) −104389. −0.147077
\(220\) 0 0
\(221\) −405794. −0.558888
\(222\) 84340.1 0.114855
\(223\) 794924. 1.07044 0.535221 0.844712i \(-0.320228\pi\)
0.535221 + 0.844712i \(0.320228\pi\)
\(224\) 10947.8 0.0145782
\(225\) 0 0
\(226\) −83229.4 −0.108394
\(227\) 379790. 0.489191 0.244596 0.969625i \(-0.421345\pi\)
0.244596 + 0.969625i \(0.421345\pi\)
\(228\) 89400.0 0.113894
\(229\) 162001. 0.204140 0.102070 0.994777i \(-0.467453\pi\)
0.102070 + 0.994777i \(0.467453\pi\)
\(230\) 0 0
\(231\) 2626.03 0.00323795
\(232\) 253189. 0.308834
\(233\) −463134. −0.558878 −0.279439 0.960163i \(-0.590148\pi\)
−0.279439 + 0.960163i \(0.590148\pi\)
\(234\) −182634. −0.218042
\(235\) 0 0
\(236\) 366861. 0.428767
\(237\) 436980. 0.505348
\(238\) −19662.0 −0.0225002
\(239\) 932706. 1.05621 0.528105 0.849179i \(-0.322903\pi\)
0.528105 + 0.849179i \(0.322903\pi\)
\(240\) 0 0
\(241\) 414220. 0.459397 0.229699 0.973262i \(-0.426226\pi\)
0.229699 + 0.973262i \(0.426226\pi\)
\(242\) 98547.1 0.108170
\(243\) −59049.0 −0.0641500
\(244\) 185869. 0.199863
\(245\) 0 0
\(246\) −141699. −0.149289
\(247\) 250094. 0.260832
\(248\) −1.09347e6 −1.12896
\(249\) −307516. −0.314318
\(250\) 0 0
\(251\) 1.90536e6 1.90894 0.954472 0.298300i \(-0.0964195\pi\)
0.954472 + 0.298300i \(0.0964195\pi\)
\(252\) −2598.80 −0.00257793
\(253\) −409865. −0.402568
\(254\) 1.39329e6 1.35506
\(255\) 0 0
\(256\) 1.11316e6 1.06160
\(257\) −631671. −0.596565 −0.298283 0.954478i \(-0.596414\pi\)
−0.298283 + 0.954478i \(0.596414\pi\)
\(258\) 638183. 0.596892
\(259\) 3357.31 0.00310987
\(260\) 0 0
\(261\) −162979. −0.148092
\(262\) −703652. −0.633293
\(263\) −961547. −0.857198 −0.428599 0.903495i \(-0.640993\pi\)
−0.428599 + 0.903495i \(0.640993\pi\)
\(264\) 137033. 0.121009
\(265\) 0 0
\(266\) 12117.9 0.0105008
\(267\) −1.11789e6 −0.959668
\(268\) −241197. −0.205132
\(269\) 720581. 0.607158 0.303579 0.952806i \(-0.401818\pi\)
0.303579 + 0.952806i \(0.401818\pi\)
\(270\) 0 0
\(271\) −632223. −0.522934 −0.261467 0.965212i \(-0.584206\pi\)
−0.261467 + 0.965212i \(0.584206\pi\)
\(272\) −1.54178e6 −1.26357
\(273\) −7270.05 −0.00590379
\(274\) 1.07191e6 0.862542
\(275\) 0 0
\(276\) 405614. 0.320509
\(277\) −2.37840e6 −1.86246 −0.931228 0.364437i \(-0.881262\pi\)
−0.931228 + 0.364437i \(0.881262\pi\)
\(278\) 1.24440e6 0.965712
\(279\) 703872. 0.541356
\(280\) 0 0
\(281\) 2.11538e6 1.59817 0.799085 0.601218i \(-0.205317\pi\)
0.799085 + 0.601218i \(0.205317\pi\)
\(282\) −1.36530e6 −1.02237
\(283\) 40867.3 0.0303326 0.0151663 0.999885i \(-0.495172\pi\)
0.0151663 + 0.999885i \(0.495172\pi\)
\(284\) 13124.0 0.00965542
\(285\) 0 0
\(286\) −272823. −0.197227
\(287\) −5640.58 −0.00404221
\(288\) −367738. −0.261250
\(289\) 47603.2 0.0335267
\(290\) 0 0
\(291\) −1.24225e6 −0.859956
\(292\) 154322. 0.105918
\(293\) −837587. −0.569982 −0.284991 0.958530i \(-0.591991\pi\)
−0.284991 + 0.958530i \(0.591991\pi\)
\(294\) 1.01778e6 0.686732
\(295\) 0 0
\(296\) 175193. 0.116222
\(297\) −88209.0 −0.0580259
\(298\) 1.82122e6 1.18801
\(299\) 1.13469e6 0.734007
\(300\) 0 0
\(301\) 25404.0 0.0161617
\(302\) −1.93800e6 −1.22275
\(303\) 53564.9 0.0335177
\(304\) 950209. 0.589705
\(305\) 0 0
\(306\) 660452. 0.403216
\(307\) −632213. −0.382840 −0.191420 0.981508i \(-0.561309\pi\)
−0.191420 + 0.981508i \(0.561309\pi\)
\(308\) −3882.15 −0.00233182
\(309\) −1.72078e6 −1.02525
\(310\) 0 0
\(311\) 1.99159e6 1.16761 0.583806 0.811893i \(-0.301563\pi\)
0.583806 + 0.811893i \(0.301563\pi\)
\(312\) −379370. −0.220636
\(313\) −305651. −0.176346 −0.0881728 0.996105i \(-0.528103\pi\)
−0.0881728 + 0.996105i \(0.528103\pi\)
\(314\) −2.69300e6 −1.54139
\(315\) 0 0
\(316\) −646002. −0.363929
\(317\) 770054. 0.430401 0.215200 0.976570i \(-0.430960\pi\)
0.215200 + 0.976570i \(0.430960\pi\)
\(318\) 353528. 0.196045
\(319\) −243463. −0.133954
\(320\) 0 0
\(321\) 795637. 0.430975
\(322\) 54979.5 0.0295503
\(323\) −904406. −0.482344
\(324\) 87294.1 0.0461979
\(325\) 0 0
\(326\) −1.54951e6 −0.807516
\(327\) −2.00790e6 −1.03842
\(328\) −294340. −0.151065
\(329\) −54348.3 −0.0276819
\(330\) 0 0
\(331\) 965864. 0.484558 0.242279 0.970207i \(-0.422105\pi\)
0.242279 + 0.970207i \(0.422105\pi\)
\(332\) 454612. 0.226358
\(333\) −112773. −0.0557305
\(334\) 3.40524e6 1.67025
\(335\) 0 0
\(336\) −27621.9 −0.0133477
\(337\) −1.11434e6 −0.534494 −0.267247 0.963628i \(-0.586114\pi\)
−0.267247 + 0.963628i \(0.586114\pi\)
\(338\) −1.74384e6 −0.830261
\(339\) 111288. 0.0525953
\(340\) 0 0
\(341\) 1.05146e6 0.489675
\(342\) −407041. −0.188180
\(343\) 81043.4 0.0371948
\(344\) 1.32565e6 0.603993
\(345\) 0 0
\(346\) 1.22742e6 0.551190
\(347\) −2.70207e6 −1.20468 −0.602342 0.798238i \(-0.705766\pi\)
−0.602342 + 0.798238i \(0.705766\pi\)
\(348\) 240937. 0.106649
\(349\) −760601. −0.334267 −0.167133 0.985934i \(-0.553451\pi\)
−0.167133 + 0.985934i \(0.553451\pi\)
\(350\) 0 0
\(351\) 244202. 0.105799
\(352\) −549336. −0.236310
\(353\) −2.61113e6 −1.11530 −0.557650 0.830076i \(-0.688297\pi\)
−0.557650 + 0.830076i \(0.688297\pi\)
\(354\) −1.67033e6 −0.708426
\(355\) 0 0
\(356\) 1.65261e6 0.691109
\(357\) 26290.5 0.0109176
\(358\) 1.97242e6 0.813376
\(359\) −1.34676e6 −0.551512 −0.275756 0.961228i \(-0.588928\pi\)
−0.275756 + 0.961228i \(0.588928\pi\)
\(360\) 0 0
\(361\) −1.91871e6 −0.774891
\(362\) 1.57537e6 0.631847
\(363\) −131769. −0.0524864
\(364\) 10747.6 0.00425164
\(365\) 0 0
\(366\) −846269. −0.330222
\(367\) −1.80461e6 −0.699388 −0.349694 0.936864i \(-0.613714\pi\)
−0.349694 + 0.936864i \(0.613714\pi\)
\(368\) 4.31116e6 1.65949
\(369\) 189468. 0.0724386
\(370\) 0 0
\(371\) 14072.8 0.00530818
\(372\) −1.04056e6 −0.389860
\(373\) 715758. 0.266376 0.133188 0.991091i \(-0.457479\pi\)
0.133188 + 0.991091i \(0.457479\pi\)
\(374\) 986601. 0.364723
\(375\) 0 0
\(376\) −2.83604e6 −1.03453
\(377\) 674014. 0.244239
\(378\) 11832.4 0.00425935
\(379\) 4.22778e6 1.51187 0.755934 0.654648i \(-0.227183\pi\)
0.755934 + 0.654648i \(0.227183\pi\)
\(380\) 0 0
\(381\) −1.86300e6 −0.657506
\(382\) 3.77560e6 1.32382
\(383\) −4.99412e6 −1.73965 −0.869825 0.493361i \(-0.835768\pi\)
−0.869825 + 0.493361i \(0.835768\pi\)
\(384\) −1.92356e6 −0.665699
\(385\) 0 0
\(386\) 3.30252e6 1.12818
\(387\) −853325. −0.289626
\(388\) 1.83646e6 0.619301
\(389\) 1.96218e6 0.657455 0.328727 0.944425i \(-0.393380\pi\)
0.328727 + 0.944425i \(0.393380\pi\)
\(390\) 0 0
\(391\) −4.10335e6 −1.35737
\(392\) 2.11416e6 0.694902
\(393\) 940865. 0.307288
\(394\) −6.84660e6 −2.22195
\(395\) 0 0
\(396\) 130402. 0.0417876
\(397\) −4.49019e6 −1.42984 −0.714921 0.699205i \(-0.753537\pi\)
−0.714921 + 0.699205i \(0.753537\pi\)
\(398\) −703667. −0.222669
\(399\) −16203.0 −0.00509522
\(400\) 0 0
\(401\) 4.74027e6 1.47212 0.736059 0.676918i \(-0.236685\pi\)
0.736059 + 0.676918i \(0.236685\pi\)
\(402\) 1.09818e6 0.338928
\(403\) −2.91093e6 −0.892830
\(404\) −79186.8 −0.0241379
\(405\) 0 0
\(406\) 32658.2 0.00983280
\(407\) −168463. −0.0504101
\(408\) 1.37191e6 0.408013
\(409\) −1.77250e6 −0.523935 −0.261967 0.965077i \(-0.584371\pi\)
−0.261967 + 0.965077i \(0.584371\pi\)
\(410\) 0 0
\(411\) −1.43326e6 −0.418525
\(412\) 2.54389e6 0.738338
\(413\) −66490.5 −0.0191816
\(414\) −1.84677e6 −0.529557
\(415\) 0 0
\(416\) 1.52081e6 0.430866
\(417\) −1.66391e6 −0.468586
\(418\) −608049. −0.170215
\(419\) −2.99507e6 −0.833435 −0.416718 0.909036i \(-0.636820\pi\)
−0.416718 + 0.909036i \(0.636820\pi\)
\(420\) 0 0
\(421\) −5.49888e6 −1.51206 −0.756030 0.654537i \(-0.772864\pi\)
−0.756030 + 0.654537i \(0.772864\pi\)
\(422\) 197286. 0.0539281
\(423\) 1.82557e6 0.496075
\(424\) 734356. 0.198377
\(425\) 0 0
\(426\) −59754.1 −0.0159531
\(427\) −33687.2 −0.00894119
\(428\) −1.17622e6 −0.310369
\(429\) 364796. 0.0956989
\(430\) 0 0
\(431\) −597155. −0.154844 −0.0774219 0.996998i \(-0.524669\pi\)
−0.0774219 + 0.996998i \(0.524669\pi\)
\(432\) 927825. 0.239198
\(433\) 1.07785e6 0.276274 0.138137 0.990413i \(-0.455889\pi\)
0.138137 + 0.990413i \(0.455889\pi\)
\(434\) −141044. −0.0359443
\(435\) 0 0
\(436\) 2.96834e6 0.747820
\(437\) 2.52893e6 0.633479
\(438\) −702633. −0.175002
\(439\) 4.45787e6 1.10399 0.551996 0.833847i \(-0.313866\pi\)
0.551996 + 0.833847i \(0.313866\pi\)
\(440\) 0 0
\(441\) −1.36090e6 −0.333218
\(442\) −2.73136e6 −0.665002
\(443\) −2.35378e6 −0.569845 −0.284923 0.958551i \(-0.591968\pi\)
−0.284923 + 0.958551i \(0.591968\pi\)
\(444\) 166715. 0.0401346
\(445\) 0 0
\(446\) 5.35055e6 1.27368
\(447\) −2.43518e6 −0.576451
\(448\) −24522.9 −0.00577268
\(449\) 1.43645e6 0.336260 0.168130 0.985765i \(-0.446227\pi\)
0.168130 + 0.985765i \(0.446227\pi\)
\(450\) 0 0
\(451\) 283033. 0.0655232
\(452\) −164520. −0.0378767
\(453\) 2.59134e6 0.593305
\(454\) 2.55633e6 0.582072
\(455\) 0 0
\(456\) −845515. −0.190419
\(457\) 4.60085e6 1.03050 0.515250 0.857040i \(-0.327699\pi\)
0.515250 + 0.857040i \(0.327699\pi\)
\(458\) 1.09041e6 0.242899
\(459\) −883102. −0.195650
\(460\) 0 0
\(461\) −4.67111e6 −1.02369 −0.511844 0.859078i \(-0.671038\pi\)
−0.511844 + 0.859078i \(0.671038\pi\)
\(462\) 17675.6 0.00385273
\(463\) 326607. 0.0708065 0.0354032 0.999373i \(-0.488728\pi\)
0.0354032 + 0.999373i \(0.488728\pi\)
\(464\) 2.56086e6 0.552192
\(465\) 0 0
\(466\) −3.11731e6 −0.664990
\(467\) 1.63074e6 0.346014 0.173007 0.984921i \(-0.444652\pi\)
0.173007 + 0.984921i \(0.444652\pi\)
\(468\) −361012. −0.0761917
\(469\) 43714.8 0.00917692
\(470\) 0 0
\(471\) 3.60085e6 0.747916
\(472\) −3.46965e6 −0.716854
\(473\) −1.27472e6 −0.261976
\(474\) 2.94127e6 0.601297
\(475\) 0 0
\(476\) −38866.0 −0.00786236
\(477\) −472708. −0.0951256
\(478\) 6.27795e6 1.25675
\(479\) −7.31820e6 −1.45736 −0.728678 0.684856i \(-0.759865\pi\)
−0.728678 + 0.684856i \(0.759865\pi\)
\(480\) 0 0
\(481\) 466381. 0.0919133
\(482\) 2.78807e6 0.546621
\(483\) −73514.1 −0.0143385
\(484\) 194798. 0.0377983
\(485\) 0 0
\(486\) −397453. −0.0763299
\(487\) 2.80682e6 0.536281 0.268141 0.963380i \(-0.413591\pi\)
0.268141 + 0.963380i \(0.413591\pi\)
\(488\) −1.75789e6 −0.334150
\(489\) 2.07188e6 0.391825
\(490\) 0 0
\(491\) −2.39394e6 −0.448136 −0.224068 0.974574i \(-0.571934\pi\)
−0.224068 + 0.974574i \(0.571934\pi\)
\(492\) −280097. −0.0521670
\(493\) −2.43742e6 −0.451661
\(494\) 1.68336e6 0.310355
\(495\) 0 0
\(496\) −1.10598e7 −2.01857
\(497\) −2378.62 −0.000431950 0
\(498\) −2.06986e6 −0.373997
\(499\) 5.55266e6 0.998274 0.499137 0.866523i \(-0.333650\pi\)
0.499137 + 0.866523i \(0.333650\pi\)
\(500\) 0 0
\(501\) −4.55320e6 −0.810443
\(502\) 1.28248e7 2.27139
\(503\) −3.38576e6 −0.596673 −0.298336 0.954461i \(-0.596432\pi\)
−0.298336 + 0.954461i \(0.596432\pi\)
\(504\) 24578.5 0.00431002
\(505\) 0 0
\(506\) −2.75876e6 −0.479002
\(507\) 2.33172e6 0.402862
\(508\) 2.75413e6 0.473506
\(509\) 4.81925e6 0.824489 0.412244 0.911073i \(-0.364745\pi\)
0.412244 + 0.911073i \(0.364745\pi\)
\(510\) 0 0
\(511\) −27969.5 −0.00473841
\(512\) 653266. 0.110132
\(513\) 544262. 0.0913092
\(514\) −4.25171e6 −0.709833
\(515\) 0 0
\(516\) 1.26150e6 0.208575
\(517\) 2.72708e6 0.448717
\(518\) 22597.7 0.00370032
\(519\) −1.64120e6 −0.267450
\(520\) 0 0
\(521\) 6.32781e6 1.02131 0.510657 0.859785i \(-0.329402\pi\)
0.510657 + 0.859785i \(0.329402\pi\)
\(522\) −1.09700e6 −0.176209
\(523\) −3.74833e6 −0.599216 −0.299608 0.954062i \(-0.596856\pi\)
−0.299608 + 0.954062i \(0.596856\pi\)
\(524\) −1.39091e6 −0.221295
\(525\) 0 0
\(526\) −6.47208e6 −1.01995
\(527\) 1.05267e7 1.65107
\(528\) 1.38601e6 0.216362
\(529\) 5.03756e6 0.782674
\(530\) 0 0
\(531\) 2.23343e6 0.343744
\(532\) 23953.4 0.00366934
\(533\) −783562. −0.119469
\(534\) −7.52441e6 −1.14188
\(535\) 0 0
\(536\) 2.28116e6 0.342960
\(537\) −2.63735e6 −0.394669
\(538\) 4.85016e6 0.722437
\(539\) −2.03294e6 −0.301407
\(540\) 0 0
\(541\) 4.24091e6 0.622968 0.311484 0.950251i \(-0.399174\pi\)
0.311484 + 0.950251i \(0.399174\pi\)
\(542\) −4.25543e6 −0.622221
\(543\) −2.10646e6 −0.306587
\(544\) −5.49966e6 −0.796781
\(545\) 0 0
\(546\) −48934.0 −0.00702472
\(547\) 6.27240e6 0.896325 0.448162 0.893952i \(-0.352079\pi\)
0.448162 + 0.893952i \(0.352079\pi\)
\(548\) 2.11884e6 0.301402
\(549\) 1.13156e6 0.160231
\(550\) 0 0
\(551\) 1.50220e6 0.210789
\(552\) −3.83616e6 −0.535857
\(553\) 117082. 0.0162809
\(554\) −1.60088e7 −2.21607
\(555\) 0 0
\(556\) 2.45981e6 0.337454
\(557\) 7.08639e6 0.967803 0.483902 0.875122i \(-0.339219\pi\)
0.483902 + 0.875122i \(0.339219\pi\)
\(558\) 4.73769e6 0.644142
\(559\) 3.52900e6 0.477664
\(560\) 0 0
\(561\) −1.31920e6 −0.176972
\(562\) 1.42384e7 1.90161
\(563\) 3.35539e6 0.446140 0.223070 0.974802i \(-0.428392\pi\)
0.223070 + 0.974802i \(0.428392\pi\)
\(564\) −2.69880e6 −0.357250
\(565\) 0 0
\(566\) 275074. 0.0360918
\(567\) −15821.3 −0.00206674
\(568\) −124123. −0.0161428
\(569\) 7.59058e6 0.982865 0.491433 0.870916i \(-0.336473\pi\)
0.491433 + 0.870916i \(0.336473\pi\)
\(570\) 0 0
\(571\) −1.00981e7 −1.29613 −0.648067 0.761584i \(-0.724422\pi\)
−0.648067 + 0.761584i \(0.724422\pi\)
\(572\) −539290. −0.0689180
\(573\) −5.04842e6 −0.642346
\(574\) −37966.2 −0.00480969
\(575\) 0 0
\(576\) 823730. 0.103450
\(577\) 1.34784e7 1.68538 0.842692 0.538396i \(-0.180970\pi\)
0.842692 + 0.538396i \(0.180970\pi\)
\(578\) 320412. 0.0398923
\(579\) −4.41586e6 −0.547418
\(580\) 0 0
\(581\) −82394.5 −0.0101265
\(582\) −8.36145e6 −1.02323
\(583\) −706145. −0.0860443
\(584\) −1.45952e6 −0.177084
\(585\) 0 0
\(586\) −5.63771e6 −0.678202
\(587\) −3.66148e6 −0.438593 −0.219296 0.975658i \(-0.570376\pi\)
−0.219296 + 0.975658i \(0.570376\pi\)
\(588\) 2.01186e6 0.239968
\(589\) −6.48768e6 −0.770550
\(590\) 0 0
\(591\) 9.15471e6 1.07814
\(592\) 1.77197e6 0.207804
\(593\) 4.20157e6 0.490653 0.245327 0.969440i \(-0.421105\pi\)
0.245327 + 0.969440i \(0.421105\pi\)
\(594\) −593726. −0.0690430
\(595\) 0 0
\(596\) 3.60001e6 0.415133
\(597\) 940886. 0.108044
\(598\) 7.63750e6 0.873370
\(599\) 5.56521e6 0.633745 0.316873 0.948468i \(-0.397367\pi\)
0.316873 + 0.948468i \(0.397367\pi\)
\(600\) 0 0
\(601\) −455310. −0.0514187 −0.0257093 0.999669i \(-0.508184\pi\)
−0.0257093 + 0.999669i \(0.508184\pi\)
\(602\) 170992. 0.0192302
\(603\) −1.46839e6 −0.164455
\(604\) −3.83086e6 −0.427271
\(605\) 0 0
\(606\) 360540. 0.0398816
\(607\) 8.22381e6 0.905944 0.452972 0.891525i \(-0.350364\pi\)
0.452972 + 0.891525i \(0.350364\pi\)
\(608\) 3.38948e6 0.371856
\(609\) −43667.8 −0.00477110
\(610\) 0 0
\(611\) −7.54980e6 −0.818149
\(612\) 1.30552e6 0.140898
\(613\) −2.86125e6 −0.307543 −0.153771 0.988106i \(-0.549142\pi\)
−0.153771 + 0.988106i \(0.549142\pi\)
\(614\) −4.25536e6 −0.455528
\(615\) 0 0
\(616\) 36716.1 0.00389856
\(617\) −1.59938e7 −1.69137 −0.845685 0.533683i \(-0.820808\pi\)
−0.845685 + 0.533683i \(0.820808\pi\)
\(618\) −1.15824e7 −1.21991
\(619\) 6.22085e6 0.652564 0.326282 0.945273i \(-0.394204\pi\)
0.326282 + 0.945273i \(0.394204\pi\)
\(620\) 0 0
\(621\) 2.46935e6 0.256953
\(622\) 1.34052e7 1.38930
\(623\) −299522. −0.0309178
\(624\) −3.83710e6 −0.394496
\(625\) 0 0
\(626\) −2.05730e6 −0.209828
\(627\) 813033. 0.0825923
\(628\) −5.32326e6 −0.538615
\(629\) −1.68656e6 −0.169971
\(630\) 0 0
\(631\) −9.11719e6 −0.911565 −0.455783 0.890091i \(-0.650641\pi\)
−0.455783 + 0.890091i \(0.650641\pi\)
\(632\) 6.10967e6 0.608450
\(633\) −263795. −0.0261672
\(634\) 5.18316e6 0.512119
\(635\) 0 0
\(636\) 698820. 0.0685051
\(637\) 5.62811e6 0.549558
\(638\) −1.63872e6 −0.159387
\(639\) 79898.3 0.00774079
\(640\) 0 0
\(641\) 9.50107e6 0.913330 0.456665 0.889639i \(-0.349044\pi\)
0.456665 + 0.889639i \(0.349044\pi\)
\(642\) 5.35535e6 0.512803
\(643\) −5.32410e6 −0.507830 −0.253915 0.967227i \(-0.581718\pi\)
−0.253915 + 0.967227i \(0.581718\pi\)
\(644\) 108678. 0.0103259
\(645\) 0 0
\(646\) −6.08747e6 −0.573925
\(647\) 2.02469e6 0.190150 0.0950752 0.995470i \(-0.469691\pi\)
0.0950752 + 0.995470i \(0.469691\pi\)
\(648\) −825598. −0.0772380
\(649\) 3.33636e6 0.310929
\(650\) 0 0
\(651\) 188592. 0.0174410
\(652\) −3.06293e6 −0.282175
\(653\) −1.14621e7 −1.05192 −0.525959 0.850510i \(-0.676293\pi\)
−0.525959 + 0.850510i \(0.676293\pi\)
\(654\) −1.35149e7 −1.23558
\(655\) 0 0
\(656\) −2.97708e6 −0.270103
\(657\) 939502. 0.0849150
\(658\) −365813. −0.0329378
\(659\) −8.95330e6 −0.803100 −0.401550 0.915837i \(-0.631528\pi\)
−0.401550 + 0.915837i \(0.631528\pi\)
\(660\) 0 0
\(661\) −1.38575e7 −1.23362 −0.616812 0.787111i \(-0.711576\pi\)
−0.616812 + 0.787111i \(0.711576\pi\)
\(662\) 6.50113e6 0.576559
\(663\) 3.65215e6 0.322674
\(664\) −4.29956e6 −0.378446
\(665\) 0 0
\(666\) −759061. −0.0663118
\(667\) 6.81557e6 0.593182
\(668\) 6.73115e6 0.583644
\(669\) −7.15431e6 −0.618020
\(670\) 0 0
\(671\) 1.69036e6 0.144935
\(672\) −98529.9 −0.00841676
\(673\) −9.94769e6 −0.846612 −0.423306 0.905987i \(-0.639130\pi\)
−0.423306 + 0.905987i \(0.639130\pi\)
\(674\) −7.50051e6 −0.635976
\(675\) 0 0
\(676\) −3.44705e6 −0.290122
\(677\) 1.32600e7 1.11192 0.555960 0.831209i \(-0.312351\pi\)
0.555960 + 0.831209i \(0.312351\pi\)
\(678\) 749065. 0.0625814
\(679\) −332842. −0.0277054
\(680\) 0 0
\(681\) −3.41811e6 −0.282435
\(682\) 7.07729e6 0.582648
\(683\) −2.05349e7 −1.68438 −0.842192 0.539178i \(-0.818735\pi\)
−0.842192 + 0.539178i \(0.818735\pi\)
\(684\) −804600. −0.0657567
\(685\) 0 0
\(686\) 545495. 0.0442568
\(687\) −1.45801e6 −0.117860
\(688\) 1.34081e7 1.07993
\(689\) 1.95493e6 0.156885
\(690\) 0 0
\(691\) 1.74373e7 1.38926 0.694630 0.719367i \(-0.255568\pi\)
0.694630 + 0.719367i \(0.255568\pi\)
\(692\) 2.42624e6 0.192605
\(693\) −23634.3 −0.00186943
\(694\) −1.81874e7 −1.43341
\(695\) 0 0
\(696\) −2.27870e6 −0.178305
\(697\) 2.83357e6 0.220929
\(698\) −5.11953e6 −0.397733
\(699\) 4.16821e6 0.322668
\(700\) 0 0
\(701\) 1.83721e6 0.141209 0.0706046 0.997504i \(-0.477507\pi\)
0.0706046 + 0.997504i \(0.477507\pi\)
\(702\) 1.64370e6 0.125887
\(703\) 1.03944e6 0.0793251
\(704\) 1.23051e6 0.0935736
\(705\) 0 0
\(706\) −1.75752e7 −1.32706
\(707\) 14351.9 0.00107985
\(708\) −3.30175e6 −0.247549
\(709\) 4.37773e6 0.327065 0.163532 0.986538i \(-0.447711\pi\)
0.163532 + 0.986538i \(0.447711\pi\)
\(710\) 0 0
\(711\) −3.93282e6 −0.291763
\(712\) −1.56299e7 −1.15546
\(713\) −2.94350e7 −2.16841
\(714\) 176958. 0.0129905
\(715\) 0 0
\(716\) 3.89889e6 0.284222
\(717\) −8.39436e6 −0.609803
\(718\) −9.06492e6 −0.656225
\(719\) −1.65258e7 −1.19217 −0.596087 0.802920i \(-0.703279\pi\)
−0.596087 + 0.802920i \(0.703279\pi\)
\(720\) 0 0
\(721\) −461058. −0.0330307
\(722\) −1.29146e7 −0.922017
\(723\) −3.72798e6 −0.265233
\(724\) 3.11405e6 0.220790
\(725\) 0 0
\(726\) −886924. −0.0624518
\(727\) −1.37064e7 −0.961803 −0.480902 0.876774i \(-0.659691\pi\)
−0.480902 + 0.876774i \(0.659691\pi\)
\(728\) −101647. −0.00710829
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.27618e7 −0.883323
\(732\) −1.67282e6 −0.115391
\(733\) −2.48655e6 −0.170938 −0.0854688 0.996341i \(-0.527239\pi\)
−0.0854688 + 0.996341i \(0.527239\pi\)
\(734\) −1.21466e7 −0.832178
\(735\) 0 0
\(736\) 1.53783e7 1.04644
\(737\) −2.19352e6 −0.148756
\(738\) 1.27529e6 0.0861923
\(739\) −2.27900e7 −1.53509 −0.767544 0.640996i \(-0.778521\pi\)
−0.767544 + 0.640996i \(0.778521\pi\)
\(740\) 0 0
\(741\) −2.25084e6 −0.150591
\(742\) 94722.6 0.00631603
\(743\) 1.24390e7 0.826634 0.413317 0.910587i \(-0.364370\pi\)
0.413317 + 0.910587i \(0.364370\pi\)
\(744\) 9.84124e6 0.651805
\(745\) 0 0
\(746\) 4.81770e6 0.316951
\(747\) 2.76765e6 0.181472
\(748\) 1.95022e6 0.127447
\(749\) 213179. 0.0138848
\(750\) 0 0
\(751\) 2.85180e7 1.84510 0.922549 0.385880i \(-0.126102\pi\)
0.922549 + 0.385880i \(0.126102\pi\)
\(752\) −2.86848e7 −1.84973
\(753\) −1.71483e7 −1.10213
\(754\) 4.53672e6 0.290612
\(755\) 0 0
\(756\) 23389.2 0.00148837
\(757\) 1.53660e6 0.0974586 0.0487293 0.998812i \(-0.484483\pi\)
0.0487293 + 0.998812i \(0.484483\pi\)
\(758\) 2.84567e7 1.79892
\(759\) 3.68879e6 0.232423
\(760\) 0 0
\(761\) −3.69462e6 −0.231264 −0.115632 0.993292i \(-0.536889\pi\)
−0.115632 + 0.993292i \(0.536889\pi\)
\(762\) −1.25396e7 −0.782344
\(763\) −537986. −0.0334549
\(764\) 7.46325e6 0.462588
\(765\) 0 0
\(766\) −3.36149e7 −2.06995
\(767\) −9.23654e6 −0.566919
\(768\) −1.00185e7 −0.612913
\(769\) −1.49715e6 −0.0912956 −0.0456478 0.998958i \(-0.514535\pi\)
−0.0456478 + 0.998958i \(0.514535\pi\)
\(770\) 0 0
\(771\) 5.68504e6 0.344427
\(772\) 6.52811e6 0.394225
\(773\) −5.52587e6 −0.332623 −0.166311 0.986073i \(-0.553186\pi\)
−0.166311 + 0.986073i \(0.553186\pi\)
\(774\) −5.74365e6 −0.344616
\(775\) 0 0
\(776\) −1.73686e7 −1.03541
\(777\) −30215.8 −0.00179548
\(778\) 1.32073e7 0.782283
\(779\) −1.74635e6 −0.103107
\(780\) 0 0
\(781\) 119354. 0.00700181
\(782\) −2.76192e7 −1.61508
\(783\) 1.46681e6 0.0855007
\(784\) 2.13835e7 1.24248
\(785\) 0 0
\(786\) 6.33286e6 0.365632
\(787\) −6.04216e6 −0.347741 −0.173870 0.984769i \(-0.555627\pi\)
−0.173870 + 0.984769i \(0.555627\pi\)
\(788\) −1.35337e7 −0.776428
\(789\) 8.65393e6 0.494904
\(790\) 0 0
\(791\) 29817.9 0.00169448
\(792\) −1.23330e6 −0.0698644
\(793\) −4.67967e6 −0.264260
\(794\) −3.02230e7 −1.70132
\(795\) 0 0
\(796\) −1.39094e6 −0.0778084
\(797\) 3.45685e6 0.192768 0.0963840 0.995344i \(-0.469272\pi\)
0.0963840 + 0.995344i \(0.469272\pi\)
\(798\) −109061. −0.00606263
\(799\) 2.73021e7 1.51297
\(800\) 0 0
\(801\) 1.00610e7 0.554065
\(802\) 3.19063e7 1.75162
\(803\) 1.40345e6 0.0768085
\(804\) 2.17077e6 0.118433
\(805\) 0 0
\(806\) −1.95931e7 −1.06235
\(807\) −6.48523e6 −0.350543
\(808\) 748922. 0.0403560
\(809\) 7.36347e6 0.395559 0.197779 0.980247i \(-0.436627\pi\)
0.197779 + 0.980247i \(0.436627\pi\)
\(810\) 0 0
\(811\) 3.37010e7 1.79925 0.899623 0.436667i \(-0.143841\pi\)
0.899623 + 0.436667i \(0.143841\pi\)
\(812\) 64555.6 0.00343593
\(813\) 5.69000e6 0.301916
\(814\) −1.13391e6 −0.0599813
\(815\) 0 0
\(816\) 1.38760e7 0.729523
\(817\) 7.86520e6 0.412244
\(818\) −1.19305e7 −0.623412
\(819\) 65430.4 0.00340855
\(820\) 0 0
\(821\) −2.10298e7 −1.08888 −0.544438 0.838801i \(-0.683257\pi\)
−0.544438 + 0.838801i \(0.683257\pi\)
\(822\) −9.64715e6 −0.497989
\(823\) 1.99288e7 1.02561 0.512803 0.858506i \(-0.328607\pi\)
0.512803 + 0.858506i \(0.328607\pi\)
\(824\) −2.40592e7 −1.23442
\(825\) 0 0
\(826\) −447541. −0.0228235
\(827\) −1.87978e7 −0.955748 −0.477874 0.878428i \(-0.658593\pi\)
−0.477874 + 0.878428i \(0.658593\pi\)
\(828\) −3.65053e6 −0.185046
\(829\) 7.27718e6 0.367771 0.183885 0.982948i \(-0.441133\pi\)
0.183885 + 0.982948i \(0.441133\pi\)
\(830\) 0 0
\(831\) 2.14056e7 1.07529
\(832\) −3.40661e6 −0.170614
\(833\) −2.03527e7 −1.01627
\(834\) −1.11996e7 −0.557554
\(835\) 0 0
\(836\) −1.20193e6 −0.0594792
\(837\) −6.33485e6 −0.312552
\(838\) −2.01595e7 −0.991677
\(839\) 3.52339e6 0.172805 0.0864025 0.996260i \(-0.472463\pi\)
0.0864025 + 0.996260i \(0.472463\pi\)
\(840\) 0 0
\(841\) −1.64627e7 −0.802620
\(842\) −3.70124e7 −1.79915
\(843\) −1.90384e7 −0.922704
\(844\) 389976. 0.0188444
\(845\) 0 0
\(846\) 1.22877e7 0.590263
\(847\) −35305.6 −0.00169097
\(848\) 7.42757e6 0.354697
\(849\) −367806. −0.0175125
\(850\) 0 0
\(851\) 4.71600e6 0.223229
\(852\) −118116. −0.00557456
\(853\) 4.55380e6 0.214290 0.107145 0.994243i \(-0.465829\pi\)
0.107145 + 0.994243i \(0.465829\pi\)
\(854\) −226745. −0.0106388
\(855\) 0 0
\(856\) 1.11243e7 0.518904
\(857\) −2.88776e7 −1.34310 −0.671552 0.740958i \(-0.734372\pi\)
−0.671552 + 0.740958i \(0.734372\pi\)
\(858\) 2.45541e6 0.113869
\(859\) 1.25501e7 0.580316 0.290158 0.956979i \(-0.406292\pi\)
0.290158 + 0.956979i \(0.406292\pi\)
\(860\) 0 0
\(861\) 50765.2 0.00233377
\(862\) −4.01939e6 −0.184243
\(863\) 2.86046e7 1.30740 0.653701 0.756753i \(-0.273215\pi\)
0.653701 + 0.756753i \(0.273215\pi\)
\(864\) 3.30964e6 0.150833
\(865\) 0 0
\(866\) 7.25492e6 0.328729
\(867\) −428428. −0.0193567
\(868\) −278802. −0.0125602
\(869\) −5.87496e6 −0.263910
\(870\) 0 0
\(871\) 6.07266e6 0.271227
\(872\) −2.80736e7 −1.25028
\(873\) 1.11802e7 0.496496
\(874\) 1.70219e7 0.753755
\(875\) 0 0
\(876\) −1.38890e6 −0.0611519
\(877\) 3.42227e7 1.50250 0.751251 0.660017i \(-0.229451\pi\)
0.751251 + 0.660017i \(0.229451\pi\)
\(878\) 3.00055e7 1.31360
\(879\) 7.53828e6 0.329079
\(880\) 0 0
\(881\) −5.25799e6 −0.228234 −0.114117 0.993467i \(-0.536404\pi\)
−0.114117 + 0.993467i \(0.536404\pi\)
\(882\) −9.16005e6 −0.396485
\(883\) 8.94488e6 0.386076 0.193038 0.981191i \(-0.438166\pi\)
0.193038 + 0.981191i \(0.438166\pi\)
\(884\) −5.39909e6 −0.232375
\(885\) 0 0
\(886\) −1.58431e7 −0.678040
\(887\) 4.14864e7 1.77050 0.885252 0.465112i \(-0.153986\pi\)
0.885252 + 0.465112i \(0.153986\pi\)
\(888\) −1.57674e6 −0.0671007
\(889\) −499163. −0.0211830
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 1.05765e7 0.445070
\(893\) −1.68265e7 −0.706098
\(894\) −1.63909e7 −0.685899
\(895\) 0 0
\(896\) −515390. −0.0214470
\(897\) −1.02122e7 −0.423779
\(898\) 9.66861e6 0.400104
\(899\) −1.74846e7 −0.721533
\(900\) 0 0
\(901\) −7.06954e6 −0.290121
\(902\) 1.90506e6 0.0779638
\(903\) −228636. −0.00933094
\(904\) 1.55597e6 0.0633259
\(905\) 0 0
\(906\) 1.74420e7 0.705954
\(907\) 3.66068e7 1.47755 0.738777 0.673950i \(-0.235404\pi\)
0.738777 + 0.673950i \(0.235404\pi\)
\(908\) 5.05310e6 0.203396
\(909\) −482084. −0.0193514
\(910\) 0 0
\(911\) 2.81370e7 1.12326 0.561631 0.827388i \(-0.310174\pi\)
0.561631 + 0.827388i \(0.310174\pi\)
\(912\) −8.55188e6 −0.340467
\(913\) 4.13439e6 0.164147
\(914\) 3.09679e7 1.22616
\(915\) 0 0
\(916\) 2.15542e6 0.0848776
\(917\) 252091. 0.00989997
\(918\) −5.94407e6 −0.232797
\(919\) −9.39900e6 −0.367107 −0.183554 0.983010i \(-0.558760\pi\)
−0.183554 + 0.983010i \(0.558760\pi\)
\(920\) 0 0
\(921\) 5.68991e6 0.221033
\(922\) −3.14408e7 −1.21805
\(923\) −330426. −0.0127665
\(924\) 34939.4 0.00134628
\(925\) 0 0
\(926\) 2.19836e6 0.0842502
\(927\) 1.54870e7 0.591928
\(928\) 9.13481e6 0.348201
\(929\) −3.51723e7 −1.33709 −0.668546 0.743671i \(-0.733083\pi\)
−0.668546 + 0.743671i \(0.733083\pi\)
\(930\) 0 0
\(931\) 1.25435e7 0.474292
\(932\) −6.16200e6 −0.232371
\(933\) −1.79243e7 −0.674121
\(934\) 1.09764e7 0.411710
\(935\) 0 0
\(936\) 3.41433e6 0.127384
\(937\) −1.09773e7 −0.408457 −0.204229 0.978923i \(-0.565469\pi\)
−0.204229 + 0.978923i \(0.565469\pi\)
\(938\) 294240. 0.0109193
\(939\) 2.75086e6 0.101813
\(940\) 0 0
\(941\) 2.86362e7 1.05424 0.527122 0.849790i \(-0.323271\pi\)
0.527122 + 0.849790i \(0.323271\pi\)
\(942\) 2.42370e7 0.889920
\(943\) −7.92331e6 −0.290153
\(944\) −3.50934e7 −1.28173
\(945\) 0 0
\(946\) −8.58001e6 −0.311717
\(947\) −1.64152e7 −0.594800 −0.297400 0.954753i \(-0.596119\pi\)
−0.297400 + 0.954753i \(0.596119\pi\)
\(948\) 5.81402e6 0.210114
\(949\) −3.88539e6 −0.140046
\(950\) 0 0
\(951\) −6.93049e6 −0.248492
\(952\) 367582. 0.0131450
\(953\) −2.15286e7 −0.767862 −0.383931 0.923362i \(-0.625430\pi\)
−0.383931 + 0.923362i \(0.625430\pi\)
\(954\) −3.18175e6 −0.113187
\(955\) 0 0
\(956\) 1.24097e7 0.439152
\(957\) 2.19116e6 0.0773383
\(958\) −4.92581e7 −1.73406
\(959\) −384022. −0.0134837
\(960\) 0 0
\(961\) 4.68831e7 1.63760
\(962\) 3.13916e6 0.109365
\(963\) −7.16073e6 −0.248824
\(964\) 5.51119e6 0.191009
\(965\) 0 0
\(966\) −494816. −0.0170609
\(967\) 1.52551e7 0.524624 0.262312 0.964983i \(-0.415515\pi\)
0.262312 + 0.964983i \(0.415515\pi\)
\(968\) −1.84234e6 −0.0631947
\(969\) 8.13966e6 0.278482
\(970\) 0 0
\(971\) −1.95235e7 −0.664521 −0.332261 0.943188i \(-0.607811\pi\)
−0.332261 + 0.943188i \(0.607811\pi\)
\(972\) −785647. −0.0266724
\(973\) −445820. −0.0150965
\(974\) 1.88924e7 0.638103
\(975\) 0 0
\(976\) −1.77800e7 −0.597457
\(977\) −5.23977e7 −1.75621 −0.878104 0.478469i \(-0.841192\pi\)
−0.878104 + 0.478469i \(0.841192\pi\)
\(978\) 1.39456e7 0.466219
\(979\) 1.50294e7 0.501170
\(980\) 0 0
\(981\) 1.80711e7 0.599531
\(982\) −1.61134e7 −0.533222
\(983\) 3.04687e7 1.00570 0.502851 0.864373i \(-0.332284\pi\)
0.502851 + 0.864373i \(0.332284\pi\)
\(984\) 2.64906e6 0.0872177
\(985\) 0 0
\(986\) −1.64060e7 −0.537416
\(987\) 489134. 0.0159822
\(988\) 3.32750e6 0.108449
\(989\) 3.56850e7 1.16010
\(990\) 0 0
\(991\) 2.04432e6 0.0661250 0.0330625 0.999453i \(-0.489474\pi\)
0.0330625 + 0.999453i \(0.489474\pi\)
\(992\) −3.94513e7 −1.27287
\(993\) −8.69277e6 −0.279760
\(994\) −16010.2 −0.000513963 0
\(995\) 0 0
\(996\) −4.09150e6 −0.130688
\(997\) 4.98989e7 1.58984 0.794919 0.606715i \(-0.207513\pi\)
0.794919 + 0.606715i \(0.207513\pi\)
\(998\) 3.73744e7 1.18781
\(999\) 1.01495e6 0.0321760
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.u.1.8 yes 10
5.4 even 2 825.6.a.t.1.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.t.1.3 10 5.4 even 2
825.6.a.u.1.8 yes 10 1.1 even 1 trivial