Properties

Label 825.6.a.s.1.1
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 1.1
Root \(-11.1064\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-11.1064 q^{2} -9.00000 q^{3} +91.3531 q^{4} +99.9580 q^{6} -76.1461 q^{7} -659.202 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-11.1064 q^{2} -9.00000 q^{3} +91.3531 q^{4} +99.9580 q^{6} -76.1461 q^{7} -659.202 q^{8} +81.0000 q^{9} -121.000 q^{11} -822.178 q^{12} +470.783 q^{13} +845.713 q^{14} +4398.09 q^{16} +494.756 q^{17} -899.622 q^{18} -44.8076 q^{19} +685.315 q^{21} +1343.88 q^{22} -4242.60 q^{23} +5932.82 q^{24} -5228.73 q^{26} -729.000 q^{27} -6956.19 q^{28} -786.920 q^{29} +5594.34 q^{31} -27752.7 q^{32} +1089.00 q^{33} -5494.98 q^{34} +7399.60 q^{36} -6453.89 q^{37} +497.653 q^{38} -4237.05 q^{39} +7703.22 q^{41} -7611.42 q^{42} +2425.31 q^{43} -11053.7 q^{44} +47120.2 q^{46} -18816.2 q^{47} -39582.8 q^{48} -11008.8 q^{49} -4452.80 q^{51} +43007.5 q^{52} +34459.8 q^{53} +8096.60 q^{54} +50195.7 q^{56} +403.268 q^{57} +8739.88 q^{58} +27937.1 q^{59} +14661.0 q^{61} -62133.2 q^{62} -6167.84 q^{63} +167495. q^{64} -12094.9 q^{66} +47205.0 q^{67} +45197.5 q^{68} +38183.4 q^{69} -39501.5 q^{71} -53395.4 q^{72} -26993.8 q^{73} +71679.8 q^{74} -4093.31 q^{76} +9213.68 q^{77} +47058.6 q^{78} -12293.2 q^{79} +6561.00 q^{81} -85555.4 q^{82} +70308.6 q^{83} +62605.7 q^{84} -26936.6 q^{86} +7082.28 q^{87} +79763.4 q^{88} -143142. q^{89} -35848.3 q^{91} -387574. q^{92} -50349.1 q^{93} +208981. q^{94} +249774. q^{96} -91322.4 q^{97} +122268. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −11.1064 −1.96336 −0.981680 0.190536i \(-0.938978\pi\)
−0.981680 + 0.190536i \(0.938978\pi\)
\(3\) −9.00000 −0.577350
\(4\) 91.3531 2.85478
\(5\) 0 0
\(6\) 99.9580 1.13355
\(7\) −76.1461 −0.587358 −0.293679 0.955904i \(-0.594880\pi\)
−0.293679 + 0.955904i \(0.594880\pi\)
\(8\) −659.202 −3.64161
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −822.178 −1.64821
\(13\) 470.783 0.772614 0.386307 0.922370i \(-0.373750\pi\)
0.386307 + 0.922370i \(0.373750\pi\)
\(14\) 845.713 1.15320
\(15\) 0 0
\(16\) 4398.09 4.29501
\(17\) 494.756 0.415211 0.207605 0.978213i \(-0.433433\pi\)
0.207605 + 0.978213i \(0.433433\pi\)
\(18\) −899.622 −0.654454
\(19\) −44.8076 −0.0284752 −0.0142376 0.999899i \(-0.504532\pi\)
−0.0142376 + 0.999899i \(0.504532\pi\)
\(20\) 0 0
\(21\) 685.315 0.339111
\(22\) 1343.88 0.591975
\(23\) −4242.60 −1.67229 −0.836146 0.548507i \(-0.815196\pi\)
−0.836146 + 0.548507i \(0.815196\pi\)
\(24\) 5932.82 2.10249
\(25\) 0 0
\(26\) −5228.73 −1.51692
\(27\) −729.000 −0.192450
\(28\) −6956.19 −1.67678
\(29\) −786.920 −0.173754 −0.0868772 0.996219i \(-0.527689\pi\)
−0.0868772 + 0.996219i \(0.527689\pi\)
\(30\) 0 0
\(31\) 5594.34 1.04555 0.522775 0.852471i \(-0.324897\pi\)
0.522775 + 0.852471i \(0.324897\pi\)
\(32\) −27752.7 −4.79104
\(33\) 1089.00 0.174078
\(34\) −5494.98 −0.815209
\(35\) 0 0
\(36\) 7399.60 0.951595
\(37\) −6453.89 −0.775028 −0.387514 0.921864i \(-0.626666\pi\)
−0.387514 + 0.921864i \(0.626666\pi\)
\(38\) 497.653 0.0559072
\(39\) −4237.05 −0.446069
\(40\) 0 0
\(41\) 7703.22 0.715670 0.357835 0.933785i \(-0.383515\pi\)
0.357835 + 0.933785i \(0.383515\pi\)
\(42\) −7611.42 −0.665798
\(43\) 2425.31 0.200031 0.100015 0.994986i \(-0.468111\pi\)
0.100015 + 0.994986i \(0.468111\pi\)
\(44\) −11053.7 −0.860750
\(45\) 0 0
\(46\) 47120.2 3.28331
\(47\) −18816.2 −1.24247 −0.621237 0.783622i \(-0.713370\pi\)
−0.621237 + 0.783622i \(0.713370\pi\)
\(48\) −39582.8 −2.47973
\(49\) −11008.8 −0.655011
\(50\) 0 0
\(51\) −4452.80 −0.239722
\(52\) 43007.5 2.20565
\(53\) 34459.8 1.68509 0.842545 0.538626i \(-0.181056\pi\)
0.842545 + 0.538626i \(0.181056\pi\)
\(54\) 8096.60 0.377849
\(55\) 0 0
\(56\) 50195.7 2.13893
\(57\) 403.268 0.0164402
\(58\) 8739.88 0.341142
\(59\) 27937.1 1.04485 0.522423 0.852687i \(-0.325028\pi\)
0.522423 + 0.852687i \(0.325028\pi\)
\(60\) 0 0
\(61\) 14661.0 0.504475 0.252237 0.967665i \(-0.418834\pi\)
0.252237 + 0.967665i \(0.418834\pi\)
\(62\) −62133.2 −2.05279
\(63\) −6167.84 −0.195786
\(64\) 167495. 5.11154
\(65\) 0 0
\(66\) −12094.9 −0.341777
\(67\) 47205.0 1.28470 0.642349 0.766412i \(-0.277960\pi\)
0.642349 + 0.766412i \(0.277960\pi\)
\(68\) 45197.5 1.18534
\(69\) 38183.4 0.965498
\(70\) 0 0
\(71\) −39501.5 −0.929967 −0.464983 0.885319i \(-0.653940\pi\)
−0.464983 + 0.885319i \(0.653940\pi\)
\(72\) −53395.4 −1.21387
\(73\) −26993.8 −0.592867 −0.296433 0.955054i \(-0.595797\pi\)
−0.296433 + 0.955054i \(0.595797\pi\)
\(74\) 71679.8 1.52166
\(75\) 0 0
\(76\) −4093.31 −0.0812907
\(77\) 9213.68 0.177095
\(78\) 47058.6 0.875795
\(79\) −12293.2 −0.221614 −0.110807 0.993842i \(-0.535344\pi\)
−0.110807 + 0.993842i \(0.535344\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −85555.4 −1.40512
\(83\) 70308.6 1.12025 0.560123 0.828409i \(-0.310754\pi\)
0.560123 + 0.828409i \(0.310754\pi\)
\(84\) 62605.7 0.968090
\(85\) 0 0
\(86\) −26936.6 −0.392733
\(87\) 7082.28 0.100317
\(88\) 79763.4 1.09799
\(89\) −143142. −1.91554 −0.957769 0.287539i \(-0.907163\pi\)
−0.957769 + 0.287539i \(0.907163\pi\)
\(90\) 0 0
\(91\) −35848.3 −0.453801
\(92\) −387574. −4.77403
\(93\) −50349.1 −0.603648
\(94\) 208981. 2.43943
\(95\) 0 0
\(96\) 249774. 2.76611
\(97\) −91322.4 −0.985480 −0.492740 0.870177i \(-0.664005\pi\)
−0.492740 + 0.870177i \(0.664005\pi\)
\(98\) 122268. 1.28602
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) −141045. −1.37580 −0.687899 0.725806i \(-0.741467\pi\)
−0.687899 + 0.725806i \(0.741467\pi\)
\(102\) 49454.8 0.470661
\(103\) 139369. 1.29442 0.647208 0.762314i \(-0.275937\pi\)
0.647208 + 0.762314i \(0.275937\pi\)
\(104\) −310341. −2.81356
\(105\) 0 0
\(106\) −382726. −3.30844
\(107\) 189660. 1.60146 0.800729 0.599027i \(-0.204446\pi\)
0.800729 + 0.599027i \(0.204446\pi\)
\(108\) −66596.4 −0.549404
\(109\) −61287.5 −0.494089 −0.247045 0.969004i \(-0.579459\pi\)
−0.247045 + 0.969004i \(0.579459\pi\)
\(110\) 0 0
\(111\) 58085.0 0.447462
\(112\) −334898. −2.52271
\(113\) −243824. −1.79631 −0.898153 0.439683i \(-0.855091\pi\)
−0.898153 + 0.439683i \(0.855091\pi\)
\(114\) −4478.88 −0.0322780
\(115\) 0 0
\(116\) −71887.6 −0.496031
\(117\) 38133.5 0.257538
\(118\) −310282. −2.05141
\(119\) −37673.8 −0.243877
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −162832. −0.990466
\(123\) −69329.0 −0.413192
\(124\) 511060. 2.98482
\(125\) 0 0
\(126\) 68502.7 0.384398
\(127\) −101881. −0.560512 −0.280256 0.959925i \(-0.590419\pi\)
−0.280256 + 0.959925i \(0.590419\pi\)
\(128\) −972186. −5.24474
\(129\) −21827.8 −0.115488
\(130\) 0 0
\(131\) −235642. −1.19971 −0.599853 0.800110i \(-0.704774\pi\)
−0.599853 + 0.800110i \(0.704774\pi\)
\(132\) 99483.5 0.496954
\(133\) 3411.92 0.0167252
\(134\) −524280. −2.52233
\(135\) 0 0
\(136\) −326144. −1.51204
\(137\) −180612. −0.822139 −0.411070 0.911604i \(-0.634845\pi\)
−0.411070 + 0.911604i \(0.634845\pi\)
\(138\) −424081. −1.89562
\(139\) 307919. 1.35176 0.675879 0.737013i \(-0.263764\pi\)
0.675879 + 0.737013i \(0.263764\pi\)
\(140\) 0 0
\(141\) 169346. 0.717343
\(142\) 438721. 1.82586
\(143\) −56964.8 −0.232952
\(144\) 356245. 1.43167
\(145\) 0 0
\(146\) 299805. 1.16401
\(147\) 99078.9 0.378171
\(148\) −589583. −2.21254
\(149\) 325175. 1.19992 0.599959 0.800030i \(-0.295183\pi\)
0.599959 + 0.800030i \(0.295183\pi\)
\(150\) 0 0
\(151\) 44535.7 0.158952 0.0794761 0.996837i \(-0.474675\pi\)
0.0794761 + 0.996837i \(0.474675\pi\)
\(152\) 29537.2 0.103696
\(153\) 40075.2 0.138404
\(154\) −102331. −0.347701
\(155\) 0 0
\(156\) −387068. −1.27343
\(157\) 354519. 1.14786 0.573932 0.818903i \(-0.305417\pi\)
0.573932 + 0.818903i \(0.305417\pi\)
\(158\) 136534. 0.435108
\(159\) −310138. −0.972887
\(160\) 0 0
\(161\) 323057. 0.982234
\(162\) −72869.4 −0.218151
\(163\) 97011.8 0.285993 0.142996 0.989723i \(-0.454326\pi\)
0.142996 + 0.989723i \(0.454326\pi\)
\(164\) 703713. 2.04308
\(165\) 0 0
\(166\) −780879. −2.19945
\(167\) −385119. −1.06857 −0.534287 0.845303i \(-0.679420\pi\)
−0.534287 + 0.845303i \(0.679420\pi\)
\(168\) −451761. −1.23491
\(169\) −149656. −0.403067
\(170\) 0 0
\(171\) −3629.41 −0.00949175
\(172\) 221560. 0.571045
\(173\) −544249. −1.38255 −0.691277 0.722590i \(-0.742952\pi\)
−0.691277 + 0.722590i \(0.742952\pi\)
\(174\) −78659.0 −0.196959
\(175\) 0 0
\(176\) −532169. −1.29499
\(177\) −251434. −0.603242
\(178\) 1.58979e6 3.76089
\(179\) −265639. −0.619668 −0.309834 0.950791i \(-0.600273\pi\)
−0.309834 + 0.950791i \(0.600273\pi\)
\(180\) 0 0
\(181\) −748969. −1.69929 −0.849645 0.527355i \(-0.823184\pi\)
−0.849645 + 0.527355i \(0.823184\pi\)
\(182\) 398148. 0.890975
\(183\) −131949. −0.291259
\(184\) 2.79673e6 6.08984
\(185\) 0 0
\(186\) 559199. 1.18518
\(187\) −59865.5 −0.125191
\(188\) −1.71892e6 −3.54700
\(189\) 55510.5 0.113037
\(190\) 0 0
\(191\) −964220. −1.91246 −0.956231 0.292613i \(-0.905475\pi\)
−0.956231 + 0.292613i \(0.905475\pi\)
\(192\) −1.50745e6 −2.95115
\(193\) 167314. 0.323325 0.161662 0.986846i \(-0.448314\pi\)
0.161662 + 0.986846i \(0.448314\pi\)
\(194\) 1.01427e6 1.93485
\(195\) 0 0
\(196\) −1.00568e6 −1.86991
\(197\) 99359.5 0.182408 0.0912040 0.995832i \(-0.470928\pi\)
0.0912040 + 0.995832i \(0.470928\pi\)
\(198\) 108854. 0.197325
\(199\) −56809.9 −0.101693 −0.0508466 0.998706i \(-0.516192\pi\)
−0.0508466 + 0.998706i \(0.516192\pi\)
\(200\) 0 0
\(201\) −424845. −0.741721
\(202\) 1.56651e6 2.70119
\(203\) 59920.9 0.102056
\(204\) −406778. −0.684355
\(205\) 0 0
\(206\) −1.54790e6 −2.54140
\(207\) −343650. −0.557431
\(208\) 2.07055e6 3.31839
\(209\) 5421.72 0.00858561
\(210\) 0 0
\(211\) 529807. 0.819241 0.409620 0.912256i \(-0.365661\pi\)
0.409620 + 0.912256i \(0.365661\pi\)
\(212\) 3.14801e6 4.81057
\(213\) 355513. 0.536916
\(214\) −2.10645e6 −3.14424
\(215\) 0 0
\(216\) 480558. 0.700828
\(217\) −425987. −0.614112
\(218\) 680686. 0.970076
\(219\) 242944. 0.342292
\(220\) 0 0
\(221\) 232923. 0.320798
\(222\) −645118. −0.878530
\(223\) 757914. 1.02061 0.510303 0.859995i \(-0.329533\pi\)
0.510303 + 0.859995i \(0.329533\pi\)
\(224\) 2.11326e6 2.81406
\(225\) 0 0
\(226\) 2.70802e6 3.52680
\(227\) 794647. 1.02355 0.511776 0.859119i \(-0.328988\pi\)
0.511776 + 0.859119i \(0.328988\pi\)
\(228\) 36839.8 0.0469332
\(229\) −191327. −0.241095 −0.120548 0.992708i \(-0.538465\pi\)
−0.120548 + 0.992708i \(0.538465\pi\)
\(230\) 0 0
\(231\) −82923.2 −0.102246
\(232\) 518739. 0.632746
\(233\) −845860. −1.02072 −0.510362 0.859960i \(-0.670489\pi\)
−0.510362 + 0.859960i \(0.670489\pi\)
\(234\) −423527. −0.505640
\(235\) 0 0
\(236\) 2.55214e6 2.98281
\(237\) 110639. 0.127949
\(238\) 418422. 0.478819
\(239\) 1.41892e6 1.60681 0.803403 0.595435i \(-0.203020\pi\)
0.803403 + 0.595435i \(0.203020\pi\)
\(240\) 0 0
\(241\) −902862. −1.00133 −0.500667 0.865640i \(-0.666912\pi\)
−0.500667 + 0.865640i \(0.666912\pi\)
\(242\) −162609. −0.178487
\(243\) −59049.0 −0.0641500
\(244\) 1.33933e6 1.44017
\(245\) 0 0
\(246\) 769999. 0.811245
\(247\) −21094.7 −0.0220004
\(248\) −3.68780e6 −3.80748
\(249\) −632778. −0.646774
\(250\) 0 0
\(251\) −58009.8 −0.0581188 −0.0290594 0.999578i \(-0.509251\pi\)
−0.0290594 + 0.999578i \(0.509251\pi\)
\(252\) −563451. −0.558927
\(253\) 513354. 0.504215
\(254\) 1.13154e6 1.10049
\(255\) 0 0
\(256\) 5.43769e6 5.18579
\(257\) −661274. −0.624523 −0.312262 0.949996i \(-0.601087\pi\)
−0.312262 + 0.949996i \(0.601087\pi\)
\(258\) 242430. 0.226744
\(259\) 491439. 0.455219
\(260\) 0 0
\(261\) −63740.5 −0.0579181
\(262\) 2.61715e6 2.35546
\(263\) 1.09488e6 0.976058 0.488029 0.872827i \(-0.337716\pi\)
0.488029 + 0.872827i \(0.337716\pi\)
\(264\) −717871. −0.633923
\(265\) 0 0
\(266\) −37894.3 −0.0328375
\(267\) 1.28827e6 1.10594
\(268\) 4.31232e6 3.66754
\(269\) −206823. −0.174268 −0.0871341 0.996197i \(-0.527771\pi\)
−0.0871341 + 0.996197i \(0.527771\pi\)
\(270\) 0 0
\(271\) −298237. −0.246683 −0.123341 0.992364i \(-0.539361\pi\)
−0.123341 + 0.992364i \(0.539361\pi\)
\(272\) 2.17598e6 1.78334
\(273\) 322635. 0.262002
\(274\) 2.00596e6 1.61416
\(275\) 0 0
\(276\) 3.48817e6 2.75629
\(277\) −1.71987e6 −1.34678 −0.673390 0.739287i \(-0.735163\pi\)
−0.673390 + 0.739287i \(0.735163\pi\)
\(278\) −3.41988e6 −2.65399
\(279\) 453141. 0.348516
\(280\) 0 0
\(281\) 2.04137e6 1.54225 0.771126 0.636683i \(-0.219694\pi\)
0.771126 + 0.636683i \(0.219694\pi\)
\(282\) −1.88083e6 −1.40840
\(283\) 1.62392e6 1.20531 0.602656 0.798001i \(-0.294109\pi\)
0.602656 + 0.798001i \(0.294109\pi\)
\(284\) −3.60858e6 −2.65485
\(285\) 0 0
\(286\) 632676. 0.457369
\(287\) −586571. −0.420354
\(288\) −2.24797e6 −1.59701
\(289\) −1.17507e6 −0.827600
\(290\) 0 0
\(291\) 821901. 0.568967
\(292\) −2.46597e6 −1.69251
\(293\) 2.37278e6 1.61469 0.807344 0.590082i \(-0.200904\pi\)
0.807344 + 0.590082i \(0.200904\pi\)
\(294\) −1.10041e6 −0.742485
\(295\) 0 0
\(296\) 4.25442e6 2.82235
\(297\) 88209.0 0.0580259
\(298\) −3.61154e6 −2.35587
\(299\) −1.99734e6 −1.29204
\(300\) 0 0
\(301\) −184678. −0.117490
\(302\) −494634. −0.312080
\(303\) 1.26941e6 0.794318
\(304\) −197068. −0.122301
\(305\) 0 0
\(306\) −445093. −0.271736
\(307\) 1.29238e6 0.782608 0.391304 0.920261i \(-0.372024\pi\)
0.391304 + 0.920261i \(0.372024\pi\)
\(308\) 841699. 0.505568
\(309\) −1.25432e6 −0.747331
\(310\) 0 0
\(311\) −1.55279e6 −0.910357 −0.455179 0.890400i \(-0.650425\pi\)
−0.455179 + 0.890400i \(0.650425\pi\)
\(312\) 2.79307e6 1.62441
\(313\) −2.14996e6 −1.24042 −0.620210 0.784436i \(-0.712953\pi\)
−0.620210 + 0.784436i \(0.712953\pi\)
\(314\) −3.93745e6 −2.25367
\(315\) 0 0
\(316\) −1.12302e6 −0.632660
\(317\) 1.34306e6 0.750669 0.375335 0.926889i \(-0.377528\pi\)
0.375335 + 0.926889i \(0.377528\pi\)
\(318\) 3.44453e6 1.91013
\(319\) 95217.3 0.0523889
\(320\) 0 0
\(321\) −1.70694e6 −0.924602
\(322\) −3.58802e6 −1.92848
\(323\) −22168.8 −0.0118232
\(324\) 599368. 0.317198
\(325\) 0 0
\(326\) −1.07746e6 −0.561507
\(327\) 551587. 0.285263
\(328\) −5.07798e6 −2.60619
\(329\) 1.43278e6 0.729777
\(330\) 0 0
\(331\) 1.24921e6 0.626706 0.313353 0.949637i \(-0.398548\pi\)
0.313353 + 0.949637i \(0.398548\pi\)
\(332\) 6.42291e6 3.19806
\(333\) −522765. −0.258343
\(334\) 4.27731e6 2.09799
\(335\) 0 0
\(336\) 3.01408e6 1.45649
\(337\) −3.36237e6 −1.61277 −0.806383 0.591394i \(-0.798578\pi\)
−0.806383 + 0.591394i \(0.798578\pi\)
\(338\) 1.66215e6 0.791366
\(339\) 2.19442e6 1.03710
\(340\) 0 0
\(341\) −676915. −0.315245
\(342\) 40309.9 0.0186357
\(343\) 2.11806e6 0.972084
\(344\) −1.59877e6 −0.728435
\(345\) 0 0
\(346\) 6.04467e6 2.71445
\(347\) −3.03255e6 −1.35202 −0.676012 0.736890i \(-0.736293\pi\)
−0.676012 + 0.736890i \(0.736293\pi\)
\(348\) 646988. 0.286384
\(349\) 690808. 0.303595 0.151797 0.988412i \(-0.451494\pi\)
0.151797 + 0.988412i \(0.451494\pi\)
\(350\) 0 0
\(351\) −343201. −0.148690
\(352\) 3.35808e6 1.44455
\(353\) 3.16374e6 1.35134 0.675669 0.737205i \(-0.263855\pi\)
0.675669 + 0.737205i \(0.263855\pi\)
\(354\) 2.79254e6 1.18438
\(355\) 0 0
\(356\) −1.30764e7 −5.46845
\(357\) 339064. 0.140803
\(358\) 2.95030e6 1.21663
\(359\) −392579. −0.160765 −0.0803825 0.996764i \(-0.525614\pi\)
−0.0803825 + 0.996764i \(0.525614\pi\)
\(360\) 0 0
\(361\) −2.47409e6 −0.999189
\(362\) 8.31839e6 3.33632
\(363\) −131769. −0.0524864
\(364\) −3.27486e6 −1.29550
\(365\) 0 0
\(366\) 1.46549e6 0.571846
\(367\) 2.92576e6 1.13390 0.566949 0.823753i \(-0.308124\pi\)
0.566949 + 0.823753i \(0.308124\pi\)
\(368\) −1.86593e7 −7.18251
\(369\) 623961. 0.238557
\(370\) 0 0
\(371\) −2.62398e6 −0.989751
\(372\) −4.59954e6 −1.72329
\(373\) 3.29066e6 1.22465 0.612323 0.790607i \(-0.290235\pi\)
0.612323 + 0.790607i \(0.290235\pi\)
\(374\) 664893. 0.245795
\(375\) 0 0
\(376\) 1.24037e7 4.52461
\(377\) −370469. −0.134245
\(378\) −616525. −0.221933
\(379\) 2.02091e6 0.722683 0.361342 0.932434i \(-0.382319\pi\)
0.361342 + 0.932434i \(0.382319\pi\)
\(380\) 0 0
\(381\) 916932. 0.323612
\(382\) 1.07091e7 3.75485
\(383\) 1.49093e6 0.519349 0.259674 0.965696i \(-0.416385\pi\)
0.259674 + 0.965696i \(0.416385\pi\)
\(384\) 8.74967e6 3.02805
\(385\) 0 0
\(386\) −1.85826e6 −0.634803
\(387\) 196450. 0.0666769
\(388\) −8.34258e6 −2.81333
\(389\) 4.98939e6 1.67176 0.835878 0.548915i \(-0.184959\pi\)
0.835878 + 0.548915i \(0.184959\pi\)
\(390\) 0 0
\(391\) −2.09905e6 −0.694354
\(392\) 7.25700e6 2.38529
\(393\) 2.12078e6 0.692651
\(394\) −1.10353e6 −0.358133
\(395\) 0 0
\(396\) −895352. −0.286917
\(397\) 2.73054e6 0.869504 0.434752 0.900550i \(-0.356836\pi\)
0.434752 + 0.900550i \(0.356836\pi\)
\(398\) 630956. 0.199660
\(399\) −30707.3 −0.00965628
\(400\) 0 0
\(401\) −2.44533e6 −0.759409 −0.379705 0.925108i \(-0.623974\pi\)
−0.379705 + 0.925108i \(0.623974\pi\)
\(402\) 4.71852e6 1.45627
\(403\) 2.63372e6 0.807807
\(404\) −1.28849e7 −3.92761
\(405\) 0 0
\(406\) −665509. −0.200373
\(407\) 780921. 0.233680
\(408\) 2.93530e6 0.872975
\(409\) 5.04690e6 1.49182 0.745909 0.666047i \(-0.232015\pi\)
0.745909 + 0.666047i \(0.232015\pi\)
\(410\) 0 0
\(411\) 1.62551e6 0.474662
\(412\) 1.27318e7 3.69528
\(413\) −2.12731e6 −0.613698
\(414\) 3.81673e6 1.09444
\(415\) 0 0
\(416\) −1.30655e7 −3.70163
\(417\) −2.77127e6 −0.780438
\(418\) −60216.0 −0.0168566
\(419\) −30652.8 −0.00852971 −0.00426486 0.999991i \(-0.501358\pi\)
−0.00426486 + 0.999991i \(0.501358\pi\)
\(420\) 0 0
\(421\) 2.31067e6 0.635379 0.317689 0.948195i \(-0.397093\pi\)
0.317689 + 0.948195i \(0.397093\pi\)
\(422\) −5.88427e6 −1.60847
\(423\) −1.52411e6 −0.414158
\(424\) −2.27160e7 −6.13644
\(425\) 0 0
\(426\) −3.94849e6 −1.05416
\(427\) −1.11638e6 −0.296307
\(428\) 1.73260e7 4.57182
\(429\) 512683. 0.134495
\(430\) 0 0
\(431\) 7.38790e6 1.91570 0.957851 0.287266i \(-0.0927464\pi\)
0.957851 + 0.287266i \(0.0927464\pi\)
\(432\) −3.20621e6 −0.826575
\(433\) 3.90981e6 1.00216 0.501079 0.865402i \(-0.332937\pi\)
0.501079 + 0.865402i \(0.332937\pi\)
\(434\) 4.73120e6 1.20572
\(435\) 0 0
\(436\) −5.59880e6 −1.41052
\(437\) 190100. 0.0476189
\(438\) −2.69825e6 −0.672042
\(439\) 2.57832e6 0.638522 0.319261 0.947667i \(-0.396565\pi\)
0.319261 + 0.947667i \(0.396565\pi\)
\(440\) 0 0
\(441\) −891710. −0.218337
\(442\) −2.58695e6 −0.629842
\(443\) 300253. 0.0726905 0.0363452 0.999339i \(-0.488428\pi\)
0.0363452 + 0.999339i \(0.488428\pi\)
\(444\) 5.30625e6 1.27741
\(445\) 0 0
\(446\) −8.41774e6 −2.00382
\(447\) −2.92658e6 −0.692773
\(448\) −1.27541e7 −3.00230
\(449\) −3.40803e6 −0.797788 −0.398894 0.916997i \(-0.630606\pi\)
−0.398894 + 0.916997i \(0.630606\pi\)
\(450\) 0 0
\(451\) −932090. −0.215783
\(452\) −2.22741e7 −5.12807
\(453\) −400822. −0.0917710
\(454\) −8.82571e6 −2.00960
\(455\) 0 0
\(456\) −265835. −0.0598688
\(457\) −3.76619e6 −0.843552 −0.421776 0.906700i \(-0.638593\pi\)
−0.421776 + 0.906700i \(0.638593\pi\)
\(458\) 2.12497e6 0.473356
\(459\) −360677. −0.0799074
\(460\) 0 0
\(461\) −8.18562e6 −1.79390 −0.896952 0.442128i \(-0.854224\pi\)
−0.896952 + 0.442128i \(0.854224\pi\)
\(462\) 920981. 0.200746
\(463\) −950861. −0.206141 −0.103071 0.994674i \(-0.532867\pi\)
−0.103071 + 0.994674i \(0.532867\pi\)
\(464\) −3.46095e6 −0.746277
\(465\) 0 0
\(466\) 9.39449e6 2.00405
\(467\) 3.61259e6 0.766526 0.383263 0.923639i \(-0.374800\pi\)
0.383263 + 0.923639i \(0.374800\pi\)
\(468\) 3.48361e6 0.735216
\(469\) −3.59448e6 −0.754577
\(470\) 0 0
\(471\) −3.19067e6 −0.662720
\(472\) −1.84162e7 −3.80492
\(473\) −293463. −0.0603116
\(474\) −1.22880e6 −0.251210
\(475\) 0 0
\(476\) −3.44162e6 −0.696218
\(477\) 2.79124e6 0.561697
\(478\) −1.57592e7 −3.15474
\(479\) 8.09916e6 1.61288 0.806439 0.591318i \(-0.201392\pi\)
0.806439 + 0.591318i \(0.201392\pi\)
\(480\) 0 0
\(481\) −3.03838e6 −0.598797
\(482\) 1.00276e7 1.96598
\(483\) −2.90752e6 −0.567093
\(484\) 1.33750e6 0.259526
\(485\) 0 0
\(486\) 655824. 0.125950
\(487\) −2.22319e6 −0.424771 −0.212386 0.977186i \(-0.568123\pi\)
−0.212386 + 0.977186i \(0.568123\pi\)
\(488\) −9.66457e6 −1.83710
\(489\) −873106. −0.165118
\(490\) 0 0
\(491\) 1.80556e6 0.337993 0.168996 0.985617i \(-0.445947\pi\)
0.168996 + 0.985617i \(0.445947\pi\)
\(492\) −6.33342e6 −1.17958
\(493\) −389333. −0.0721447
\(494\) 234287. 0.0431947
\(495\) 0 0
\(496\) 2.46044e7 4.49065
\(497\) 3.00788e6 0.546223
\(498\) 7.02791e6 1.26985
\(499\) −3.16604e6 −0.569201 −0.284600 0.958646i \(-0.591861\pi\)
−0.284600 + 0.958646i \(0.591861\pi\)
\(500\) 0 0
\(501\) 3.46607e6 0.616941
\(502\) 644282. 0.114108
\(503\) 4.86714e6 0.857737 0.428869 0.903367i \(-0.358912\pi\)
0.428869 + 0.903367i \(0.358912\pi\)
\(504\) 4.06585e6 0.712976
\(505\) 0 0
\(506\) −5.70154e6 −0.989956
\(507\) 1.34690e6 0.232711
\(508\) −9.30717e6 −1.60014
\(509\) 4.22678e6 0.723128 0.361564 0.932347i \(-0.382243\pi\)
0.361564 + 0.932347i \(0.382243\pi\)
\(510\) 0 0
\(511\) 2.05547e6 0.348225
\(512\) −2.92835e7 −4.93683
\(513\) 32664.7 0.00548006
\(514\) 7.34441e6 1.22616
\(515\) 0 0
\(516\) −1.99404e6 −0.329693
\(517\) 2.27676e6 0.374620
\(518\) −5.45814e6 −0.893758
\(519\) 4.89824e6 0.798218
\(520\) 0 0
\(521\) 1.11140e7 1.79380 0.896901 0.442232i \(-0.145813\pi\)
0.896901 + 0.442232i \(0.145813\pi\)
\(522\) 707931. 0.113714
\(523\) −532067. −0.0850574 −0.0425287 0.999095i \(-0.513541\pi\)
−0.0425287 + 0.999095i \(0.513541\pi\)
\(524\) −2.15267e7 −3.42490
\(525\) 0 0
\(526\) −1.21602e7 −1.91635
\(527\) 2.76783e6 0.434124
\(528\) 4.78952e6 0.747665
\(529\) 1.15633e7 1.79656
\(530\) 0 0
\(531\) 2.26291e6 0.348282
\(532\) 311690. 0.0477467
\(533\) 3.62655e6 0.552937
\(534\) −1.43081e7 −2.17135
\(535\) 0 0
\(536\) −3.11176e7 −4.67837
\(537\) 2.39075e6 0.357765
\(538\) 2.29707e6 0.342151
\(539\) 1.33206e6 0.197493
\(540\) 0 0
\(541\) −9.77951e6 −1.43656 −0.718280 0.695754i \(-0.755070\pi\)
−0.718280 + 0.695754i \(0.755070\pi\)
\(542\) 3.31235e6 0.484327
\(543\) 6.74072e6 0.981086
\(544\) −1.37308e7 −1.98929
\(545\) 0 0
\(546\) −3.58333e6 −0.514405
\(547\) −8.61940e6 −1.23171 −0.615855 0.787859i \(-0.711189\pi\)
−0.615855 + 0.787859i \(0.711189\pi\)
\(548\) −1.64995e7 −2.34703
\(549\) 1.18754e6 0.168158
\(550\) 0 0
\(551\) 35260.0 0.00494770
\(552\) −2.51706e7 −3.51597
\(553\) 936079. 0.130167
\(554\) 1.91017e7 2.64422
\(555\) 0 0
\(556\) 2.81293e7 3.85898
\(557\) 1.37932e7 1.88376 0.941881 0.335946i \(-0.109056\pi\)
0.941881 + 0.335946i \(0.109056\pi\)
\(558\) −5.03279e6 −0.684263
\(559\) 1.14180e6 0.154547
\(560\) 0 0
\(561\) 538789. 0.0722789
\(562\) −2.26723e7 −3.02800
\(563\) 8.21998e6 1.09295 0.546474 0.837476i \(-0.315970\pi\)
0.546474 + 0.837476i \(0.315970\pi\)
\(564\) 1.54703e7 2.04786
\(565\) 0 0
\(566\) −1.80360e7 −2.36646
\(567\) −499595. −0.0652620
\(568\) 2.60394e7 3.38658
\(569\) 8.25501e6 1.06890 0.534450 0.845200i \(-0.320519\pi\)
0.534450 + 0.845200i \(0.320519\pi\)
\(570\) 0 0
\(571\) 9.47816e6 1.21656 0.608280 0.793722i \(-0.291860\pi\)
0.608280 + 0.793722i \(0.291860\pi\)
\(572\) −5.20391e6 −0.665028
\(573\) 8.67798e6 1.10416
\(574\) 6.51472e6 0.825307
\(575\) 0 0
\(576\) 1.35671e7 1.70385
\(577\) −5.03255e6 −0.629287 −0.314644 0.949210i \(-0.601885\pi\)
−0.314644 + 0.949210i \(0.601885\pi\)
\(578\) 1.30509e7 1.62488
\(579\) −1.50583e6 −0.186672
\(580\) 0 0
\(581\) −5.35373e6 −0.657985
\(582\) −9.12840e6 −1.11709
\(583\) −4.16964e6 −0.508074
\(584\) 1.77944e7 2.15899
\(585\) 0 0
\(586\) −2.63531e7 −3.17021
\(587\) −2.93358e6 −0.351401 −0.175700 0.984444i \(-0.556219\pi\)
−0.175700 + 0.984444i \(0.556219\pi\)
\(588\) 9.05116e6 1.07960
\(589\) −250669. −0.0297723
\(590\) 0 0
\(591\) −894236. −0.105313
\(592\) −2.83848e7 −3.32875
\(593\) 1.08060e7 1.26191 0.630953 0.775821i \(-0.282664\pi\)
0.630953 + 0.775821i \(0.282664\pi\)
\(594\) −979688. −0.113926
\(595\) 0 0
\(596\) 2.97058e7 3.42551
\(597\) 511289. 0.0587125
\(598\) 2.21834e7 2.53673
\(599\) 3.28617e6 0.374217 0.187108 0.982339i \(-0.440088\pi\)
0.187108 + 0.982339i \(0.440088\pi\)
\(600\) 0 0
\(601\) 6.11530e6 0.690608 0.345304 0.938491i \(-0.387776\pi\)
0.345304 + 0.938491i \(0.387776\pi\)
\(602\) 2.05112e6 0.230675
\(603\) 3.82360e6 0.428233
\(604\) 4.06848e6 0.453774
\(605\) 0 0
\(606\) −1.40986e7 −1.55953
\(607\) −7.27336e6 −0.801241 −0.400621 0.916244i \(-0.631205\pi\)
−0.400621 + 0.916244i \(0.631205\pi\)
\(608\) 1.24353e6 0.136426
\(609\) −539288. −0.0589220
\(610\) 0 0
\(611\) −8.85836e6 −0.959954
\(612\) 3.66100e6 0.395113
\(613\) −7.10635e6 −0.763827 −0.381914 0.924198i \(-0.624735\pi\)
−0.381914 + 0.924198i \(0.624735\pi\)
\(614\) −1.43538e7 −1.53654
\(615\) 0 0
\(616\) −6.07368e6 −0.644911
\(617\) −5.11598e6 −0.541023 −0.270512 0.962717i \(-0.587193\pi\)
−0.270512 + 0.962717i \(0.587193\pi\)
\(618\) 1.39311e7 1.46728
\(619\) −2.88443e6 −0.302576 −0.151288 0.988490i \(-0.548342\pi\)
−0.151288 + 0.988490i \(0.548342\pi\)
\(620\) 0 0
\(621\) 3.09285e6 0.321833
\(622\) 1.72460e7 1.78736
\(623\) 1.08997e7 1.12511
\(624\) −1.86349e7 −1.91587
\(625\) 0 0
\(626\) 2.38784e7 2.43539
\(627\) −48795.5 −0.00495690
\(628\) 3.23864e7 3.27690
\(629\) −3.19310e6 −0.321800
\(630\) 0 0
\(631\) −8.37374e6 −0.837233 −0.418616 0.908163i \(-0.637485\pi\)
−0.418616 + 0.908163i \(0.637485\pi\)
\(632\) 8.10370e6 0.807031
\(633\) −4.76826e6 −0.472989
\(634\) −1.49167e7 −1.47383
\(635\) 0 0
\(636\) −2.83321e7 −2.77738
\(637\) −5.18274e6 −0.506071
\(638\) −1.05753e6 −0.102858
\(639\) −3.19962e6 −0.309989
\(640\) 0 0
\(641\) 1.14615e7 1.10179 0.550893 0.834576i \(-0.314287\pi\)
0.550893 + 0.834576i \(0.314287\pi\)
\(642\) 1.89580e7 1.81533
\(643\) 1.69107e7 1.61300 0.806498 0.591237i \(-0.201360\pi\)
0.806498 + 0.591237i \(0.201360\pi\)
\(644\) 2.95123e7 2.80407
\(645\) 0 0
\(646\) 246217. 0.0232133
\(647\) 1.21273e7 1.13895 0.569473 0.822010i \(-0.307147\pi\)
0.569473 + 0.822010i \(0.307147\pi\)
\(648\) −4.32502e6 −0.404623
\(649\) −3.38039e6 −0.315033
\(650\) 0 0
\(651\) 3.83389e6 0.354558
\(652\) 8.86233e6 0.816448
\(653\) −1.09620e7 −1.00602 −0.503010 0.864281i \(-0.667774\pi\)
−0.503010 + 0.864281i \(0.667774\pi\)
\(654\) −6.12617e6 −0.560073
\(655\) 0 0
\(656\) 3.38795e7 3.07381
\(657\) −2.18650e6 −0.197622
\(658\) −1.59131e7 −1.43282
\(659\) 2.65796e6 0.238416 0.119208 0.992869i \(-0.461964\pi\)
0.119208 + 0.992869i \(0.461964\pi\)
\(660\) 0 0
\(661\) 8.86632e6 0.789295 0.394648 0.918833i \(-0.370867\pi\)
0.394648 + 0.918833i \(0.370867\pi\)
\(662\) −1.38742e7 −1.23045
\(663\) −2.09631e6 −0.185213
\(664\) −4.63476e7 −4.07950
\(665\) 0 0
\(666\) 5.80606e6 0.507220
\(667\) 3.33858e6 0.290568
\(668\) −3.51819e7 −3.05055
\(669\) −6.82123e6 −0.589247
\(670\) 0 0
\(671\) −1.77398e6 −0.152105
\(672\) −1.90193e7 −1.62470
\(673\) 1.56559e7 1.33242 0.666209 0.745765i \(-0.267916\pi\)
0.666209 + 0.745765i \(0.267916\pi\)
\(674\) 3.73440e7 3.16644
\(675\) 0 0
\(676\) −1.36715e7 −1.15067
\(677\) −2.51293e6 −0.210721 −0.105361 0.994434i \(-0.533600\pi\)
−0.105361 + 0.994434i \(0.533600\pi\)
\(678\) −2.43722e7 −2.03620
\(679\) 6.95385e6 0.578830
\(680\) 0 0
\(681\) −7.15183e6 −0.590948
\(682\) 7.51812e6 0.618940
\(683\) −1.40231e7 −1.15025 −0.575125 0.818066i \(-0.695047\pi\)
−0.575125 + 0.818066i \(0.695047\pi\)
\(684\) −331558. −0.0270969
\(685\) 0 0
\(686\) −2.35242e7 −1.90855
\(687\) 1.72195e6 0.139196
\(688\) 1.06668e7 0.859135
\(689\) 1.62231e7 1.30192
\(690\) 0 0
\(691\) −2.03284e7 −1.61960 −0.809799 0.586707i \(-0.800424\pi\)
−0.809799 + 0.586707i \(0.800424\pi\)
\(692\) −4.97188e7 −3.94690
\(693\) 746308. 0.0590317
\(694\) 3.36809e7 2.65451
\(695\) 0 0
\(696\) −4.66865e6 −0.365316
\(697\) 3.81122e6 0.297154
\(698\) −7.67243e6 −0.596066
\(699\) 7.61274e6 0.589315
\(700\) 0 0
\(701\) 8.73851e6 0.671649 0.335825 0.941925i \(-0.390985\pi\)
0.335825 + 0.941925i \(0.390985\pi\)
\(702\) 3.81174e6 0.291932
\(703\) 289183. 0.0220691
\(704\) −2.02669e7 −1.54119
\(705\) 0 0
\(706\) −3.51379e7 −2.65316
\(707\) 1.07401e7 0.808086
\(708\) −2.29693e7 −1.72212
\(709\) 6.30404e6 0.470981 0.235491 0.971877i \(-0.424330\pi\)
0.235491 + 0.971877i \(0.424330\pi\)
\(710\) 0 0
\(711\) −995748. −0.0738713
\(712\) 9.43592e7 6.97564
\(713\) −2.37345e7 −1.74846
\(714\) −3.76579e6 −0.276446
\(715\) 0 0
\(716\) −2.42669e7 −1.76902
\(717\) −1.27703e7 −0.927690
\(718\) 4.36016e6 0.315640
\(719\) 6.08432e6 0.438924 0.219462 0.975621i \(-0.429570\pi\)
0.219462 + 0.975621i \(0.429570\pi\)
\(720\) 0 0
\(721\) −1.06124e7 −0.760285
\(722\) 2.74784e7 1.96177
\(723\) 8.12575e6 0.578120
\(724\) −6.84207e7 −4.85111
\(725\) 0 0
\(726\) 1.46349e6 0.103050
\(727\) 2.10256e7 1.47541 0.737704 0.675125i \(-0.235910\pi\)
0.737704 + 0.675125i \(0.235910\pi\)
\(728\) 2.36313e7 1.65257
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 1.19994e6 0.0830550
\(732\) −1.20540e7 −0.831480
\(733\) −9.82965e6 −0.675738 −0.337869 0.941193i \(-0.609706\pi\)
−0.337869 + 0.941193i \(0.609706\pi\)
\(734\) −3.24948e7 −2.22625
\(735\) 0 0
\(736\) 1.17743e8 8.01202
\(737\) −5.71180e6 −0.387351
\(738\) −6.92999e6 −0.468373
\(739\) −1.51937e7 −1.02342 −0.511709 0.859159i \(-0.670987\pi\)
−0.511709 + 0.859159i \(0.670987\pi\)
\(740\) 0 0
\(741\) 189852. 0.0127019
\(742\) 2.91431e7 1.94324
\(743\) −4.92706e6 −0.327428 −0.163714 0.986508i \(-0.552347\pi\)
−0.163714 + 0.986508i \(0.552347\pi\)
\(744\) 3.31902e7 2.19825
\(745\) 0 0
\(746\) −3.65475e7 −2.40442
\(747\) 5.69500e6 0.373415
\(748\) −5.46890e6 −0.357393
\(749\) −1.44419e7 −0.940629
\(750\) 0 0
\(751\) −1.07515e6 −0.0695617 −0.0347809 0.999395i \(-0.511073\pi\)
−0.0347809 + 0.999395i \(0.511073\pi\)
\(752\) −8.27554e7 −5.33644
\(753\) 522088. 0.0335549
\(754\) 4.11459e6 0.263572
\(755\) 0 0
\(756\) 5.07106e6 0.322697
\(757\) −1.80869e7 −1.14716 −0.573580 0.819150i \(-0.694446\pi\)
−0.573580 + 0.819150i \(0.694446\pi\)
\(758\) −2.24451e7 −1.41889
\(759\) −4.62019e6 −0.291109
\(760\) 0 0
\(761\) 540730. 0.0338469 0.0169234 0.999857i \(-0.494613\pi\)
0.0169234 + 0.999857i \(0.494613\pi\)
\(762\) −1.01838e7 −0.635367
\(763\) 4.66681e6 0.290207
\(764\) −8.80845e7 −5.45967
\(765\) 0 0
\(766\) −1.65589e7 −1.01967
\(767\) 1.31523e7 0.807262
\(768\) −4.89392e7 −2.99402
\(769\) 8.60213e6 0.524554 0.262277 0.964993i \(-0.415527\pi\)
0.262277 + 0.964993i \(0.415527\pi\)
\(770\) 0 0
\(771\) 5.95147e6 0.360569
\(772\) 1.52847e7 0.923023
\(773\) 1.45721e7 0.877150 0.438575 0.898695i \(-0.355483\pi\)
0.438575 + 0.898695i \(0.355483\pi\)
\(774\) −2.18187e6 −0.130911
\(775\) 0 0
\(776\) 6.01999e7 3.58874
\(777\) −4.42295e6 −0.262821
\(778\) −5.54143e7 −3.28226
\(779\) −345163. −0.0203789
\(780\) 0 0
\(781\) 4.77968e6 0.280395
\(782\) 2.33130e7 1.36327
\(783\) 573665. 0.0334390
\(784\) −4.84176e7 −2.81328
\(785\) 0 0
\(786\) −2.35543e7 −1.35992
\(787\) 9.86427e6 0.567712 0.283856 0.958867i \(-0.408386\pi\)
0.283856 + 0.958867i \(0.408386\pi\)
\(788\) 9.07680e6 0.520736
\(789\) −9.85388e6 −0.563527
\(790\) 0 0
\(791\) 1.85663e7 1.05507
\(792\) 6.46084e6 0.365996
\(793\) 6.90216e6 0.389764
\(794\) −3.03266e7 −1.70715
\(795\) 0 0
\(796\) −5.18976e6 −0.290312
\(797\) 1.18967e7 0.663410 0.331705 0.943383i \(-0.392376\pi\)
0.331705 + 0.943383i \(0.392376\pi\)
\(798\) 341049. 0.0189588
\(799\) −9.30943e6 −0.515889
\(800\) 0 0
\(801\) −1.15945e7 −0.638513
\(802\) 2.71589e7 1.49099
\(803\) 3.26625e6 0.178756
\(804\) −3.88109e7 −2.11745
\(805\) 0 0
\(806\) −2.92513e7 −1.58602
\(807\) 1.86141e6 0.100614
\(808\) 9.29773e7 5.01012
\(809\) 1.09831e7 0.590005 0.295002 0.955497i \(-0.404680\pi\)
0.295002 + 0.955497i \(0.404680\pi\)
\(810\) 0 0
\(811\) −3.01879e6 −0.161169 −0.0805845 0.996748i \(-0.525679\pi\)
−0.0805845 + 0.996748i \(0.525679\pi\)
\(812\) 5.47396e6 0.291348
\(813\) 2.68413e6 0.142422
\(814\) −8.67325e6 −0.458797
\(815\) 0 0
\(816\) −1.95838e7 −1.02961
\(817\) −108672. −0.00569593
\(818\) −5.60531e7 −2.92898
\(819\) −2.90372e6 −0.151267
\(820\) 0 0
\(821\) −5.39706e6 −0.279447 −0.139723 0.990191i \(-0.544621\pi\)
−0.139723 + 0.990191i \(0.544621\pi\)
\(822\) −1.80536e7 −0.931933
\(823\) 2.28710e7 1.17702 0.588512 0.808488i \(-0.299714\pi\)
0.588512 + 0.808488i \(0.299714\pi\)
\(824\) −9.18725e7 −4.71376
\(825\) 0 0
\(826\) 2.36268e7 1.20491
\(827\) −7.63569e6 −0.388226 −0.194113 0.980979i \(-0.562183\pi\)
−0.194113 + 0.980979i \(0.562183\pi\)
\(828\) −3.13935e7 −1.59134
\(829\) −7.03185e6 −0.355372 −0.177686 0.984087i \(-0.556861\pi\)
−0.177686 + 0.984087i \(0.556861\pi\)
\(830\) 0 0
\(831\) 1.54789e7 0.777564
\(832\) 7.88538e7 3.94925
\(833\) −5.44665e6 −0.271968
\(834\) 3.07789e7 1.53228
\(835\) 0 0
\(836\) 495291. 0.0245101
\(837\) −4.07827e6 −0.201216
\(838\) 340443. 0.0167469
\(839\) 6.66791e6 0.327028 0.163514 0.986541i \(-0.447717\pi\)
0.163514 + 0.986541i \(0.447717\pi\)
\(840\) 0 0
\(841\) −1.98919e7 −0.969809
\(842\) −2.56633e7 −1.24748
\(843\) −1.83723e7 −0.890420
\(844\) 4.83995e7 2.33876
\(845\) 0 0
\(846\) 1.69275e7 0.813142
\(847\) −1.11486e6 −0.0533962
\(848\) 1.51557e8 7.23748
\(849\) −1.46153e7 −0.695888
\(850\) 0 0
\(851\) 2.73812e7 1.29607
\(852\) 3.24772e7 1.53278
\(853\) 3.51845e7 1.65569 0.827845 0.560956i \(-0.189566\pi\)
0.827845 + 0.560956i \(0.189566\pi\)
\(854\) 1.23990e7 0.581758
\(855\) 0 0
\(856\) −1.25024e8 −5.83189
\(857\) −1.17588e7 −0.546905 −0.273452 0.961886i \(-0.588166\pi\)
−0.273452 + 0.961886i \(0.588166\pi\)
\(858\) −5.69409e6 −0.264062
\(859\) 3.68112e7 1.70215 0.851074 0.525045i \(-0.175952\pi\)
0.851074 + 0.525045i \(0.175952\pi\)
\(860\) 0 0
\(861\) 5.27914e6 0.242692
\(862\) −8.20533e7 −3.76121
\(863\) 3.57608e7 1.63448 0.817242 0.576295i \(-0.195502\pi\)
0.817242 + 0.576295i \(0.195502\pi\)
\(864\) 2.02317e7 0.922037
\(865\) 0 0
\(866\) −4.34241e7 −1.96760
\(867\) 1.05757e7 0.477815
\(868\) −3.89153e7 −1.75316
\(869\) 1.48748e6 0.0668191
\(870\) 0 0
\(871\) 2.22233e7 0.992576
\(872\) 4.04008e7 1.79928
\(873\) −7.39711e6 −0.328493
\(874\) −2.11134e6 −0.0934931
\(875\) 0 0
\(876\) 2.21937e7 0.977169
\(877\) 4.23836e7 1.86080 0.930398 0.366550i \(-0.119461\pi\)
0.930398 + 0.366550i \(0.119461\pi\)
\(878\) −2.86360e7 −1.25365
\(879\) −2.13550e7 −0.932240
\(880\) 0 0
\(881\) 1.28486e7 0.557721 0.278861 0.960332i \(-0.410043\pi\)
0.278861 + 0.960332i \(0.410043\pi\)
\(882\) 9.90373e6 0.428674
\(883\) 1.50474e7 0.649469 0.324735 0.945805i \(-0.394725\pi\)
0.324735 + 0.945805i \(0.394725\pi\)
\(884\) 2.12782e7 0.915809
\(885\) 0 0
\(886\) −3.33474e6 −0.142718
\(887\) −1.40467e7 −0.599466 −0.299733 0.954023i \(-0.596898\pi\)
−0.299733 + 0.954023i \(0.596898\pi\)
\(888\) −3.82897e7 −1.62948
\(889\) 7.75787e6 0.329221
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) 6.92378e7 2.91361
\(893\) 843109. 0.0353798
\(894\) 3.25039e7 1.36016
\(895\) 0 0
\(896\) 7.40282e7 3.08054
\(897\) 1.79761e7 0.745958
\(898\) 3.78511e7 1.56634
\(899\) −4.40230e6 −0.181669
\(900\) 0 0
\(901\) 1.70492e7 0.699668
\(902\) 1.03522e7 0.423659
\(903\) 1.66211e6 0.0678327
\(904\) 1.60729e8 6.54145
\(905\) 0 0
\(906\) 4.45170e6 0.180180
\(907\) −8.25845e6 −0.333335 −0.166667 0.986013i \(-0.553301\pi\)
−0.166667 + 0.986013i \(0.553301\pi\)
\(908\) 7.25935e7 2.92202
\(909\) −1.14247e7 −0.458600
\(910\) 0 0
\(911\) 2.01230e7 0.803335 0.401667 0.915786i \(-0.368431\pi\)
0.401667 + 0.915786i \(0.368431\pi\)
\(912\) 1.77361e6 0.0706108
\(913\) −8.50735e6 −0.337767
\(914\) 4.18290e7 1.65620
\(915\) 0 0
\(916\) −1.74783e7 −0.688274
\(917\) 1.79433e7 0.704657
\(918\) 4.00584e6 0.156887
\(919\) 1.70445e7 0.665725 0.332862 0.942975i \(-0.391986\pi\)
0.332862 + 0.942975i \(0.391986\pi\)
\(920\) 0 0
\(921\) −1.16314e7 −0.451839
\(922\) 9.09131e7 3.52208
\(923\) −1.85966e7 −0.718506
\(924\) −7.57529e6 −0.291890
\(925\) 0 0
\(926\) 1.05607e7 0.404729
\(927\) 1.12889e7 0.431472
\(928\) 2.18392e7 0.832465
\(929\) 1.53002e7 0.581644 0.290822 0.956777i \(-0.406071\pi\)
0.290822 + 0.956777i \(0.406071\pi\)
\(930\) 0 0
\(931\) 493276. 0.0186516
\(932\) −7.72719e7 −2.91395
\(933\) 1.39751e7 0.525595
\(934\) −4.01231e7 −1.50497
\(935\) 0 0
\(936\) −2.51377e7 −0.937854
\(937\) −2.49555e7 −0.928574 −0.464287 0.885685i \(-0.653689\pi\)
−0.464287 + 0.885685i \(0.653689\pi\)
\(938\) 3.99219e7 1.48151
\(939\) 1.93496e7 0.716157
\(940\) 0 0
\(941\) 2.67710e7 0.985576 0.492788 0.870149i \(-0.335978\pi\)
0.492788 + 0.870149i \(0.335978\pi\)
\(942\) 3.54370e7 1.30116
\(943\) −3.26817e7 −1.19681
\(944\) 1.22870e8 4.48762
\(945\) 0 0
\(946\) 3.25933e6 0.118413
\(947\) −3.32255e7 −1.20392 −0.601958 0.798528i \(-0.705612\pi\)
−0.601958 + 0.798528i \(0.705612\pi\)
\(948\) 1.01072e7 0.365266
\(949\) −1.27082e7 −0.458057
\(950\) 0 0
\(951\) −1.20876e7 −0.433399
\(952\) 2.48346e7 0.888107
\(953\) 2.27164e7 0.810229 0.405114 0.914266i \(-0.367232\pi\)
0.405114 + 0.914266i \(0.367232\pi\)
\(954\) −3.10008e7 −1.10281
\(955\) 0 0
\(956\) 1.29623e8 4.58709
\(957\) −856956. −0.0302467
\(958\) −8.99529e7 −3.16666
\(959\) 1.37529e7 0.482890
\(960\) 0 0
\(961\) 2.66748e6 0.0931736
\(962\) 3.37456e7 1.17566
\(963\) 1.53624e7 0.533819
\(964\) −8.24792e7 −2.85859
\(965\) 0 0
\(966\) 3.22922e7 1.11341
\(967\) −4.69627e7 −1.61505 −0.807526 0.589832i \(-0.799194\pi\)
−0.807526 + 0.589832i \(0.799194\pi\)
\(968\) −9.65138e6 −0.331056
\(969\) 199519. 0.00682615
\(970\) 0 0
\(971\) −261687. −0.00890706 −0.00445353 0.999990i \(-0.501418\pi\)
−0.00445353 + 0.999990i \(0.501418\pi\)
\(972\) −5.39431e6 −0.183135
\(973\) −2.34468e7 −0.793966
\(974\) 2.46918e7 0.833979
\(975\) 0 0
\(976\) 6.44805e7 2.16672
\(977\) −3.02629e7 −1.01432 −0.507159 0.861853i \(-0.669304\pi\)
−0.507159 + 0.861853i \(0.669304\pi\)
\(978\) 9.69710e6 0.324186
\(979\) 1.73201e7 0.577556
\(980\) 0 0
\(981\) −4.96429e6 −0.164696
\(982\) −2.00533e7 −0.663601
\(983\) −2.18315e7 −0.720610 −0.360305 0.932835i \(-0.617327\pi\)
−0.360305 + 0.932835i \(0.617327\pi\)
\(984\) 4.57018e7 1.50469
\(985\) 0 0
\(986\) 4.32411e6 0.141646
\(987\) −1.28950e7 −0.421337
\(988\) −1.92706e6 −0.0628064
\(989\) −1.02896e7 −0.334510
\(990\) 0 0
\(991\) −5.46573e7 −1.76792 −0.883962 0.467558i \(-0.845134\pi\)
−0.883962 + 0.467558i \(0.845134\pi\)
\(992\) −1.55258e8 −5.00927
\(993\) −1.12428e7 −0.361829
\(994\) −3.34069e7 −1.07243
\(995\) 0 0
\(996\) −5.78062e7 −1.84640
\(997\) −1.96397e7 −0.625745 −0.312872 0.949795i \(-0.601291\pi\)
−0.312872 + 0.949795i \(0.601291\pi\)
\(998\) 3.51635e7 1.11755
\(999\) 4.70488e6 0.149154
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.1.1 yes 9
5.4 even 2 825.6.a.r.1.9 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.r.1.9 9 5.4 even 2
825.6.a.s.1.1 yes 9 1.1 even 1 trivial