Properties

Label 825.6.a.s.1.6
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 1.6
Root \(3.27244\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.27244 q^{2} -9.00000 q^{3} -21.2911 q^{4} -29.4519 q^{6} +115.628 q^{7} -174.392 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+3.27244 q^{2} -9.00000 q^{3} -21.2911 q^{4} -29.4519 q^{6} +115.628 q^{7} -174.392 q^{8} +81.0000 q^{9} -121.000 q^{11} +191.620 q^{12} +883.035 q^{13} +378.385 q^{14} +110.630 q^{16} -519.889 q^{17} +265.068 q^{18} -1590.33 q^{19} -1040.65 q^{21} -395.965 q^{22} +1462.43 q^{23} +1569.53 q^{24} +2889.68 q^{26} -729.000 q^{27} -2461.85 q^{28} -7184.81 q^{29} +7836.58 q^{31} +5942.57 q^{32} +1089.00 q^{33} -1701.31 q^{34} -1724.58 q^{36} +10331.6 q^{37} -5204.27 q^{38} -7947.32 q^{39} -3161.57 q^{41} -3405.46 q^{42} -15968.3 q^{43} +2576.23 q^{44} +4785.73 q^{46} -10774.0 q^{47} -995.666 q^{48} -3437.22 q^{49} +4679.00 q^{51} -18800.8 q^{52} +2690.05 q^{53} -2385.61 q^{54} -20164.6 q^{56} +14313.0 q^{57} -23511.9 q^{58} -46938.2 q^{59} +2111.96 q^{61} +25644.7 q^{62} +9365.85 q^{63} +15906.6 q^{64} +3563.69 q^{66} +58553.4 q^{67} +11069.0 q^{68} -13161.9 q^{69} -57097.1 q^{71} -14125.8 q^{72} -29208.2 q^{73} +33809.6 q^{74} +33860.0 q^{76} -13991.0 q^{77} -26007.1 q^{78} +73679.8 q^{79} +6561.00 q^{81} -10346.0 q^{82} +39084.3 q^{83} +22156.6 q^{84} -52255.3 q^{86} +64663.3 q^{87} +21101.4 q^{88} +12826.5 q^{89} +102103. q^{91} -31136.9 q^{92} -70529.2 q^{93} -35257.2 q^{94} -53483.2 q^{96} +23693.2 q^{97} -11248.1 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.27244 0.578491 0.289245 0.957255i \(-0.406596\pi\)
0.289245 + 0.957255i \(0.406596\pi\)
\(3\) −9.00000 −0.577350
\(4\) −21.2911 −0.665348
\(5\) 0 0
\(6\) −29.4519 −0.333992
\(7\) 115.628 0.891902 0.445951 0.895057i \(-0.352866\pi\)
0.445951 + 0.895057i \(0.352866\pi\)
\(8\) −174.392 −0.963389
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 191.620 0.384139
\(13\) 883.035 1.44917 0.724585 0.689185i \(-0.242031\pi\)
0.724585 + 0.689185i \(0.242031\pi\)
\(14\) 378.385 0.515957
\(15\) 0 0
\(16\) 110.630 0.108037
\(17\) −519.889 −0.436303 −0.218152 0.975915i \(-0.570003\pi\)
−0.218152 + 0.975915i \(0.570003\pi\)
\(18\) 265.068 0.192830
\(19\) −1590.33 −1.01066 −0.505329 0.862927i \(-0.668629\pi\)
−0.505329 + 0.862927i \(0.668629\pi\)
\(20\) 0 0
\(21\) −1040.65 −0.514940
\(22\) −395.965 −0.174422
\(23\) 1462.43 0.576444 0.288222 0.957564i \(-0.406936\pi\)
0.288222 + 0.957564i \(0.406936\pi\)
\(24\) 1569.53 0.556213
\(25\) 0 0
\(26\) 2889.68 0.838332
\(27\) −729.000 −0.192450
\(28\) −2461.85 −0.593425
\(29\) −7184.81 −1.58643 −0.793214 0.608943i \(-0.791594\pi\)
−0.793214 + 0.608943i \(0.791594\pi\)
\(30\) 0 0
\(31\) 7836.58 1.46461 0.732306 0.680976i \(-0.238444\pi\)
0.732306 + 0.680976i \(0.238444\pi\)
\(32\) 5942.57 1.02589
\(33\) 1089.00 0.174078
\(34\) −1701.31 −0.252398
\(35\) 0 0
\(36\) −1724.58 −0.221783
\(37\) 10331.6 1.24069 0.620346 0.784328i \(-0.286992\pi\)
0.620346 + 0.784328i \(0.286992\pi\)
\(38\) −5204.27 −0.584656
\(39\) −7947.32 −0.836679
\(40\) 0 0
\(41\) −3161.57 −0.293726 −0.146863 0.989157i \(-0.546918\pi\)
−0.146863 + 0.989157i \(0.546918\pi\)
\(42\) −3405.46 −0.297888
\(43\) −15968.3 −1.31701 −0.658503 0.752578i \(-0.728810\pi\)
−0.658503 + 0.752578i \(0.728810\pi\)
\(44\) 2576.23 0.200610
\(45\) 0 0
\(46\) 4785.73 0.333467
\(47\) −10774.0 −0.711429 −0.355715 0.934595i \(-0.615762\pi\)
−0.355715 + 0.934595i \(0.615762\pi\)
\(48\) −995.666 −0.0623750
\(49\) −3437.22 −0.204511
\(50\) 0 0
\(51\) 4679.00 0.251900
\(52\) −18800.8 −0.964203
\(53\) 2690.05 0.131544 0.0657719 0.997835i \(-0.479049\pi\)
0.0657719 + 0.997835i \(0.479049\pi\)
\(54\) −2385.61 −0.111331
\(55\) 0 0
\(56\) −20164.6 −0.859248
\(57\) 14313.0 0.583504
\(58\) −23511.9 −0.917734
\(59\) −46938.2 −1.75548 −0.877742 0.479134i \(-0.840951\pi\)
−0.877742 + 0.479134i \(0.840951\pi\)
\(60\) 0 0
\(61\) 2111.96 0.0726710 0.0363355 0.999340i \(-0.488432\pi\)
0.0363355 + 0.999340i \(0.488432\pi\)
\(62\) 25644.7 0.847264
\(63\) 9365.85 0.297301
\(64\) 15906.6 0.485430
\(65\) 0 0
\(66\) 3563.69 0.100702
\(67\) 58553.4 1.59355 0.796774 0.604277i \(-0.206538\pi\)
0.796774 + 0.604277i \(0.206538\pi\)
\(68\) 11069.0 0.290294
\(69\) −13161.9 −0.332810
\(70\) 0 0
\(71\) −57097.1 −1.34421 −0.672107 0.740454i \(-0.734611\pi\)
−0.672107 + 0.740454i \(0.734611\pi\)
\(72\) −14125.8 −0.321130
\(73\) −29208.2 −0.641502 −0.320751 0.947164i \(-0.603935\pi\)
−0.320751 + 0.947164i \(0.603935\pi\)
\(74\) 33809.6 0.717729
\(75\) 0 0
\(76\) 33860.0 0.672439
\(77\) −13991.0 −0.268919
\(78\) −26007.1 −0.484011
\(79\) 73679.8 1.32825 0.664126 0.747620i \(-0.268804\pi\)
0.664126 + 0.747620i \(0.268804\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −10346.0 −0.169918
\(83\) 39084.3 0.622741 0.311370 0.950289i \(-0.399212\pi\)
0.311370 + 0.950289i \(0.399212\pi\)
\(84\) 22156.6 0.342614
\(85\) 0 0
\(86\) −52255.3 −0.761876
\(87\) 64663.3 0.915925
\(88\) 21101.4 0.290473
\(89\) 12826.5 0.171646 0.0858231 0.996310i \(-0.472648\pi\)
0.0858231 + 0.996310i \(0.472648\pi\)
\(90\) 0 0
\(91\) 102103. 1.29252
\(92\) −31136.9 −0.383536
\(93\) −70529.2 −0.845594
\(94\) −35257.2 −0.411555
\(95\) 0 0
\(96\) −53483.2 −0.592296
\(97\) 23693.2 0.255678 0.127839 0.991795i \(-0.459196\pi\)
0.127839 + 0.991795i \(0.459196\pi\)
\(98\) −11248.1 −0.118308
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) 114273. 1.11465 0.557325 0.830294i \(-0.311828\pi\)
0.557325 + 0.830294i \(0.311828\pi\)
\(102\) 15311.8 0.145722
\(103\) 134822. 1.25218 0.626092 0.779749i \(-0.284653\pi\)
0.626092 + 0.779749i \(0.284653\pi\)
\(104\) −153994. −1.39612
\(105\) 0 0
\(106\) 8803.02 0.0760969
\(107\) −29366.7 −0.247968 −0.123984 0.992284i \(-0.539567\pi\)
−0.123984 + 0.992284i \(0.539567\pi\)
\(108\) 15521.2 0.128046
\(109\) 102755. 0.828392 0.414196 0.910188i \(-0.364063\pi\)
0.414196 + 0.910188i \(0.364063\pi\)
\(110\) 0 0
\(111\) −92984.5 −0.716314
\(112\) 12791.8 0.0963581
\(113\) −134021. −0.987362 −0.493681 0.869643i \(-0.664349\pi\)
−0.493681 + 0.869643i \(0.664349\pi\)
\(114\) 46838.4 0.337551
\(115\) 0 0
\(116\) 152973. 1.05553
\(117\) 71525.9 0.483057
\(118\) −153603. −1.01553
\(119\) −60113.6 −0.389140
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 6911.26 0.0420395
\(123\) 28454.1 0.169583
\(124\) −166850. −0.974476
\(125\) 0 0
\(126\) 30649.2 0.171986
\(127\) 138288. 0.760809 0.380405 0.924820i \(-0.375785\pi\)
0.380405 + 0.924820i \(0.375785\pi\)
\(128\) −138109. −0.745070
\(129\) 143715. 0.760374
\(130\) 0 0
\(131\) 241088. 1.22743 0.613716 0.789527i \(-0.289674\pi\)
0.613716 + 0.789527i \(0.289674\pi\)
\(132\) −23186.1 −0.115822
\(133\) −183887. −0.901407
\(134\) 191613. 0.921853
\(135\) 0 0
\(136\) 90664.5 0.420330
\(137\) 249641. 1.13636 0.568179 0.822905i \(-0.307648\pi\)
0.568179 + 0.822905i \(0.307648\pi\)
\(138\) −43071.6 −0.192528
\(139\) −167292. −0.734410 −0.367205 0.930140i \(-0.619685\pi\)
−0.367205 + 0.930140i \(0.619685\pi\)
\(140\) 0 0
\(141\) 96965.8 0.410744
\(142\) −186847. −0.777616
\(143\) −106847. −0.436941
\(144\) 8960.99 0.0360122
\(145\) 0 0
\(146\) −95582.1 −0.371103
\(147\) 30935.0 0.118075
\(148\) −219972. −0.825492
\(149\) 141666. 0.522758 0.261379 0.965236i \(-0.415823\pi\)
0.261379 + 0.965236i \(0.415823\pi\)
\(150\) 0 0
\(151\) 365960. 1.30614 0.653072 0.757296i \(-0.273480\pi\)
0.653072 + 0.757296i \(0.273480\pi\)
\(152\) 277341. 0.973656
\(153\) −42111.0 −0.145434
\(154\) −45784.6 −0.155567
\(155\) 0 0
\(156\) 169207. 0.556683
\(157\) −283151. −0.916789 −0.458395 0.888749i \(-0.651575\pi\)
−0.458395 + 0.888749i \(0.651575\pi\)
\(158\) 241113. 0.768382
\(159\) −24210.4 −0.0759468
\(160\) 0 0
\(161\) 169098. 0.514131
\(162\) 21470.5 0.0642768
\(163\) 601651. 1.77368 0.886840 0.462076i \(-0.152895\pi\)
0.886840 + 0.462076i \(0.152895\pi\)
\(164\) 67313.4 0.195430
\(165\) 0 0
\(166\) 127901. 0.360250
\(167\) 178294. 0.494703 0.247352 0.968926i \(-0.420440\pi\)
0.247352 + 0.968926i \(0.420440\pi\)
\(168\) 181481. 0.496087
\(169\) 408458. 1.10010
\(170\) 0 0
\(171\) −128817. −0.336886
\(172\) 339984. 0.876268
\(173\) 760854. 1.93280 0.966398 0.257051i \(-0.0827509\pi\)
0.966398 + 0.257051i \(0.0827509\pi\)
\(174\) 211607. 0.529854
\(175\) 0 0
\(176\) −13386.2 −0.0325743
\(177\) 422444. 1.01353
\(178\) 41974.0 0.0992957
\(179\) 707719. 1.65093 0.825464 0.564454i \(-0.190913\pi\)
0.825464 + 0.564454i \(0.190913\pi\)
\(180\) 0 0
\(181\) 531826. 1.20663 0.603313 0.797504i \(-0.293847\pi\)
0.603313 + 0.797504i \(0.293847\pi\)
\(182\) 334127. 0.747710
\(183\) −19007.6 −0.0419566
\(184\) −255037. −0.555339
\(185\) 0 0
\(186\) −230803. −0.489168
\(187\) 62906.6 0.131550
\(188\) 229390. 0.473348
\(189\) −84292.6 −0.171647
\(190\) 0 0
\(191\) −519646. −1.03068 −0.515340 0.856986i \(-0.672334\pi\)
−0.515340 + 0.856986i \(0.672334\pi\)
\(192\) −143159. −0.280263
\(193\) −624883. −1.20755 −0.603775 0.797155i \(-0.706337\pi\)
−0.603775 + 0.797155i \(0.706337\pi\)
\(194\) 77534.5 0.147908
\(195\) 0 0
\(196\) 73182.4 0.136071
\(197\) 34880.0 0.0640340 0.0320170 0.999487i \(-0.489807\pi\)
0.0320170 + 0.999487i \(0.489807\pi\)
\(198\) −32073.2 −0.0581405
\(199\) 230348. 0.412336 0.206168 0.978517i \(-0.433901\pi\)
0.206168 + 0.978517i \(0.433901\pi\)
\(200\) 0 0
\(201\) −526981. −0.920036
\(202\) 373950. 0.644815
\(203\) −830764. −1.41494
\(204\) −99621.3 −0.167601
\(205\) 0 0
\(206\) 441197. 0.724377
\(207\) 118457. 0.192148
\(208\) 97689.8 0.156564
\(209\) 192430. 0.304725
\(210\) 0 0
\(211\) −1.12880e6 −1.74546 −0.872730 0.488204i \(-0.837652\pi\)
−0.872730 + 0.488204i \(0.837652\pi\)
\(212\) −57274.2 −0.0875224
\(213\) 513874. 0.776082
\(214\) −96100.6 −0.143447
\(215\) 0 0
\(216\) 127132. 0.185404
\(217\) 906126. 1.30629
\(218\) 336259. 0.479217
\(219\) 262874. 0.370371
\(220\) 0 0
\(221\) −459081. −0.632278
\(222\) −304286. −0.414381
\(223\) −1.21861e6 −1.64097 −0.820487 0.571665i \(-0.806298\pi\)
−0.820487 + 0.571665i \(0.806298\pi\)
\(224\) 687126. 0.914990
\(225\) 0 0
\(226\) −438575. −0.571180
\(227\) −834702. −1.07514 −0.537572 0.843218i \(-0.680658\pi\)
−0.537572 + 0.843218i \(0.680658\pi\)
\(228\) −304740. −0.388233
\(229\) 781702. 0.985037 0.492519 0.870302i \(-0.336076\pi\)
0.492519 + 0.870302i \(0.336076\pi\)
\(230\) 0 0
\(231\) 125919. 0.155260
\(232\) 1.25297e6 1.52835
\(233\) −207066. −0.249873 −0.124936 0.992165i \(-0.539873\pi\)
−0.124936 + 0.992165i \(0.539873\pi\)
\(234\) 234064. 0.279444
\(235\) 0 0
\(236\) 999369. 1.16801
\(237\) −663118. −0.766867
\(238\) −196718. −0.225114
\(239\) 1.57438e6 1.78285 0.891424 0.453171i \(-0.149707\pi\)
0.891424 + 0.453171i \(0.149707\pi\)
\(240\) 0 0
\(241\) 1.02355e6 1.13519 0.567595 0.823308i \(-0.307874\pi\)
0.567595 + 0.823308i \(0.307874\pi\)
\(242\) 47911.8 0.0525901
\(243\) −59049.0 −0.0641500
\(244\) −44966.0 −0.0483515
\(245\) 0 0
\(246\) 93114.3 0.0981021
\(247\) −1.40432e6 −1.46462
\(248\) −1.36664e6 −1.41099
\(249\) −351759. −0.359539
\(250\) 0 0
\(251\) 572916. 0.573993 0.286996 0.957932i \(-0.407343\pi\)
0.286996 + 0.957932i \(0.407343\pi\)
\(252\) −199410. −0.197808
\(253\) −176955. −0.173804
\(254\) 452540. 0.440121
\(255\) 0 0
\(256\) −960963. −0.916446
\(257\) 607677. 0.573905 0.286953 0.957945i \(-0.407358\pi\)
0.286953 + 0.957945i \(0.407358\pi\)
\(258\) 470298. 0.439869
\(259\) 1.19462e6 1.10657
\(260\) 0 0
\(261\) −581970. −0.528809
\(262\) 788946. 0.710059
\(263\) 1.39142e6 1.24042 0.620211 0.784435i \(-0.287047\pi\)
0.620211 + 0.784435i \(0.287047\pi\)
\(264\) −189913. −0.167704
\(265\) 0 0
\(266\) −601758. −0.521456
\(267\) −115439. −0.0991000
\(268\) −1.24667e6 −1.06026
\(269\) 221782. 0.186872 0.0934362 0.995625i \(-0.470215\pi\)
0.0934362 + 0.995625i \(0.470215\pi\)
\(270\) 0 0
\(271\) 965446. 0.798554 0.399277 0.916830i \(-0.369261\pi\)
0.399277 + 0.916830i \(0.369261\pi\)
\(272\) −57515.1 −0.0471368
\(273\) −918930. −0.746236
\(274\) 816936. 0.657373
\(275\) 0 0
\(276\) 280232. 0.221435
\(277\) 157129. 0.123043 0.0615217 0.998106i \(-0.480405\pi\)
0.0615217 + 0.998106i \(0.480405\pi\)
\(278\) −547453. −0.424850
\(279\) 634763. 0.488204
\(280\) 0 0
\(281\) 2.07218e6 1.56553 0.782765 0.622318i \(-0.213809\pi\)
0.782765 + 0.622318i \(0.213809\pi\)
\(282\) 317315. 0.237612
\(283\) −586163. −0.435063 −0.217532 0.976053i \(-0.569801\pi\)
−0.217532 + 0.976053i \(0.569801\pi\)
\(284\) 1.21566e6 0.894371
\(285\) 0 0
\(286\) −349651. −0.252767
\(287\) −365565. −0.261975
\(288\) 481348. 0.341962
\(289\) −1.14957e6 −0.809639
\(290\) 0 0
\(291\) −213239. −0.147616
\(292\) 621877. 0.426822
\(293\) −2.54116e6 −1.72927 −0.864635 0.502400i \(-0.832450\pi\)
−0.864635 + 0.502400i \(0.832450\pi\)
\(294\) 101233. 0.0683051
\(295\) 0 0
\(296\) −1.80175e6 −1.19527
\(297\) 88209.0 0.0580259
\(298\) 463594. 0.302411
\(299\) 1.29138e6 0.835366
\(300\) 0 0
\(301\) −1.84638e6 −1.17464
\(302\) 1.19758e6 0.755593
\(303\) −1.02845e6 −0.643543
\(304\) −175938. −0.109188
\(305\) 0 0
\(306\) −137806. −0.0841325
\(307\) 1.45371e6 0.880302 0.440151 0.897924i \(-0.354925\pi\)
0.440151 + 0.897924i \(0.354925\pi\)
\(308\) 297884. 0.178924
\(309\) −1.21340e6 −0.722949
\(310\) 0 0
\(311\) 1.19477e6 0.700458 0.350229 0.936664i \(-0.386104\pi\)
0.350229 + 0.936664i \(0.386104\pi\)
\(312\) 1.38595e6 0.806047
\(313\) −2.43275e6 −1.40358 −0.701789 0.712385i \(-0.747615\pi\)
−0.701789 + 0.712385i \(0.747615\pi\)
\(314\) −926596. −0.530354
\(315\) 0 0
\(316\) −1.56873e6 −0.883751
\(317\) −931723. −0.520761 −0.260381 0.965506i \(-0.583848\pi\)
−0.260381 + 0.965506i \(0.583848\pi\)
\(318\) −79227.2 −0.0439346
\(319\) 869362. 0.478326
\(320\) 0 0
\(321\) 264300. 0.143164
\(322\) 553363. 0.297420
\(323\) 826797. 0.440953
\(324\) −139691. −0.0739276
\(325\) 0 0
\(326\) 1.96886e6 1.02606
\(327\) −924794. −0.478272
\(328\) 551352. 0.282972
\(329\) −1.24577e6 −0.634525
\(330\) 0 0
\(331\) 2.23321e6 1.12036 0.560181 0.828370i \(-0.310731\pi\)
0.560181 + 0.828370i \(0.310731\pi\)
\(332\) −832150. −0.414339
\(333\) 836861. 0.413564
\(334\) 583455. 0.286181
\(335\) 0 0
\(336\) −115127. −0.0556324
\(337\) 1.37340e6 0.658751 0.329376 0.944199i \(-0.393162\pi\)
0.329376 + 0.944199i \(0.393162\pi\)
\(338\) 1.33665e6 0.636396
\(339\) 1.20619e6 0.570054
\(340\) 0 0
\(341\) −948226. −0.441597
\(342\) −421546. −0.194885
\(343\) −2.34079e6 −1.07431
\(344\) 2.78475e6 1.26879
\(345\) 0 0
\(346\) 2.48985e6 1.11810
\(347\) −1.48507e6 −0.662100 −0.331050 0.943613i \(-0.607403\pi\)
−0.331050 + 0.943613i \(0.607403\pi\)
\(348\) −1.37676e6 −0.609409
\(349\) −296745. −0.130413 −0.0652063 0.997872i \(-0.520771\pi\)
−0.0652063 + 0.997872i \(0.520771\pi\)
\(350\) 0 0
\(351\) −643733. −0.278893
\(352\) −719051. −0.309317
\(353\) −1.02630e6 −0.438368 −0.219184 0.975684i \(-0.570339\pi\)
−0.219184 + 0.975684i \(0.570339\pi\)
\(354\) 1.38242e6 0.586317
\(355\) 0 0
\(356\) −273092. −0.114204
\(357\) 541023. 0.224670
\(358\) 2.31597e6 0.955047
\(359\) 3.46040e6 1.41707 0.708534 0.705677i \(-0.249357\pi\)
0.708534 + 0.705677i \(0.249357\pi\)
\(360\) 0 0
\(361\) 53060.4 0.0214290
\(362\) 1.74037e6 0.698022
\(363\) −131769. −0.0524864
\(364\) −2.17390e6 −0.859975
\(365\) 0 0
\(366\) −62201.3 −0.0242715
\(367\) 2.54792e6 0.987461 0.493731 0.869615i \(-0.335633\pi\)
0.493731 + 0.869615i \(0.335633\pi\)
\(368\) 161789. 0.0622771
\(369\) −256087. −0.0979087
\(370\) 0 0
\(371\) 311044. 0.117324
\(372\) 1.50165e6 0.562614
\(373\) −1.62873e6 −0.606145 −0.303073 0.952967i \(-0.598012\pi\)
−0.303073 + 0.952967i \(0.598012\pi\)
\(374\) 205858. 0.0761007
\(375\) 0 0
\(376\) 1.87890e6 0.685383
\(377\) −6.34444e6 −2.29901
\(378\) −275842. −0.0992960
\(379\) −3.96243e6 −1.41698 −0.708490 0.705721i \(-0.750623\pi\)
−0.708490 + 0.705721i \(0.750623\pi\)
\(380\) 0 0
\(381\) −1.24459e6 −0.439253
\(382\) −1.70051e6 −0.596239
\(383\) −552568. −0.192481 −0.0962407 0.995358i \(-0.530682\pi\)
−0.0962407 + 0.995358i \(0.530682\pi\)
\(384\) 1.24298e6 0.430167
\(385\) 0 0
\(386\) −2.04489e6 −0.698557
\(387\) −1.29343e6 −0.439002
\(388\) −504455. −0.170115
\(389\) −4.13340e6 −1.38495 −0.692473 0.721443i \(-0.743479\pi\)
−0.692473 + 0.721443i \(0.743479\pi\)
\(390\) 0 0
\(391\) −760304. −0.251504
\(392\) 599424. 0.197024
\(393\) −2.16979e6 −0.708659
\(394\) 114142. 0.0370431
\(395\) 0 0
\(396\) 208675. 0.0668700
\(397\) 2.28281e6 0.726933 0.363466 0.931607i \(-0.381593\pi\)
0.363466 + 0.931607i \(0.381593\pi\)
\(398\) 753799. 0.238533
\(399\) 1.65498e6 0.520428
\(400\) 0 0
\(401\) 3.83742e6 1.19173 0.595866 0.803084i \(-0.296809\pi\)
0.595866 + 0.803084i \(0.296809\pi\)
\(402\) −1.72451e6 −0.532232
\(403\) 6.91998e6 2.12247
\(404\) −2.43299e6 −0.741630
\(405\) 0 0
\(406\) −2.71862e6 −0.818529
\(407\) −1.25013e6 −0.374083
\(408\) −815981. −0.242678
\(409\) 4.06933e6 1.20286 0.601430 0.798926i \(-0.294598\pi\)
0.601430 + 0.798926i \(0.294598\pi\)
\(410\) 0 0
\(411\) −2.24677e6 −0.656077
\(412\) −2.87052e6 −0.833139
\(413\) −5.42736e6 −1.56572
\(414\) 387644. 0.111156
\(415\) 0 0
\(416\) 5.24750e6 1.48669
\(417\) 1.50563e6 0.424012
\(418\) 629716. 0.176280
\(419\) 956542. 0.266176 0.133088 0.991104i \(-0.457511\pi\)
0.133088 + 0.991104i \(0.457511\pi\)
\(420\) 0 0
\(421\) 3.40755e6 0.936994 0.468497 0.883465i \(-0.344796\pi\)
0.468497 + 0.883465i \(0.344796\pi\)
\(422\) −3.69392e6 −1.00973
\(423\) −872693. −0.237143
\(424\) −469123. −0.126728
\(425\) 0 0
\(426\) 1.68162e6 0.448957
\(427\) 244201. 0.0648154
\(428\) 625250. 0.164985
\(429\) 961625. 0.252268
\(430\) 0 0
\(431\) −1548.19 −0.000401450 0 −0.000200725 1.00000i \(-0.500064\pi\)
−0.000200725 1.00000i \(0.500064\pi\)
\(432\) −80649.0 −0.0207917
\(433\) 4.72970e6 1.21231 0.606155 0.795347i \(-0.292711\pi\)
0.606155 + 0.795347i \(0.292711\pi\)
\(434\) 2.96524e6 0.755676
\(435\) 0 0
\(436\) −2.18777e6 −0.551169
\(437\) −2.32576e6 −0.582587
\(438\) 860239. 0.214256
\(439\) 725064. 0.179562 0.0897811 0.995962i \(-0.471383\pi\)
0.0897811 + 0.995962i \(0.471383\pi\)
\(440\) 0 0
\(441\) −278415. −0.0681704
\(442\) −1.50231e6 −0.365767
\(443\) 2.29072e6 0.554578 0.277289 0.960787i \(-0.410564\pi\)
0.277289 + 0.960787i \(0.410564\pi\)
\(444\) 1.97975e6 0.476598
\(445\) 0 0
\(446\) −3.98782e6 −0.949288
\(447\) −1.27500e6 −0.301815
\(448\) 1.83924e6 0.432956
\(449\) −3.83348e6 −0.897382 −0.448691 0.893687i \(-0.648110\pi\)
−0.448691 + 0.893687i \(0.648110\pi\)
\(450\) 0 0
\(451\) 382550. 0.0885618
\(452\) 2.85346e6 0.656940
\(453\) −3.29364e6 −0.754103
\(454\) −2.73151e6 −0.621961
\(455\) 0 0
\(456\) −2.49607e6 −0.562141
\(457\) 675446. 0.151286 0.0756432 0.997135i \(-0.475899\pi\)
0.0756432 + 0.997135i \(0.475899\pi\)
\(458\) 2.55807e6 0.569835
\(459\) 378999. 0.0839666
\(460\) 0 0
\(461\) −987250. −0.216359 −0.108179 0.994131i \(-0.534502\pi\)
−0.108179 + 0.994131i \(0.534502\pi\)
\(462\) 412061. 0.0898166
\(463\) −3.27705e6 −0.710445 −0.355222 0.934782i \(-0.615595\pi\)
−0.355222 + 0.934782i \(0.615595\pi\)
\(464\) −794853. −0.171392
\(465\) 0 0
\(466\) −677610. −0.144549
\(467\) 1.97021e6 0.418042 0.209021 0.977911i \(-0.432972\pi\)
0.209021 + 0.977911i \(0.432972\pi\)
\(468\) −1.52287e6 −0.321401
\(469\) 6.77040e6 1.42129
\(470\) 0 0
\(471\) 2.54836e6 0.529309
\(472\) 8.18565e6 1.69121
\(473\) 1.93217e6 0.397092
\(474\) −2.17001e6 −0.443626
\(475\) 0 0
\(476\) 1.27989e6 0.258913
\(477\) 217894. 0.0438479
\(478\) 5.15205e6 1.03136
\(479\) 6.21301e6 1.23727 0.618633 0.785680i \(-0.287687\pi\)
0.618633 + 0.785680i \(0.287687\pi\)
\(480\) 0 0
\(481\) 9.12318e6 1.79797
\(482\) 3.34952e6 0.656697
\(483\) −1.52188e6 −0.296834
\(484\) −311724. −0.0604862
\(485\) 0 0
\(486\) −193234. −0.0371102
\(487\) −1.04014e6 −0.198732 −0.0993660 0.995051i \(-0.531681\pi\)
−0.0993660 + 0.995051i \(0.531681\pi\)
\(488\) −368309. −0.0700104
\(489\) −5.41486e6 −1.02403
\(490\) 0 0
\(491\) −7.20841e6 −1.34939 −0.674693 0.738099i \(-0.735724\pi\)
−0.674693 + 0.738099i \(0.735724\pi\)
\(492\) −605820. −0.112832
\(493\) 3.73531e6 0.692164
\(494\) −4.59555e6 −0.847267
\(495\) 0 0
\(496\) 866957. 0.158232
\(497\) −6.60201e6 −1.19891
\(498\) −1.15111e6 −0.207990
\(499\) −2.91483e6 −0.524037 −0.262019 0.965063i \(-0.584388\pi\)
−0.262019 + 0.965063i \(0.584388\pi\)
\(500\) 0 0
\(501\) −1.60464e6 −0.285617
\(502\) 1.87483e6 0.332050
\(503\) 5.60301e6 0.987419 0.493710 0.869627i \(-0.335641\pi\)
0.493710 + 0.869627i \(0.335641\pi\)
\(504\) −1.63333e6 −0.286416
\(505\) 0 0
\(506\) −579073. −0.100544
\(507\) −3.67612e6 −0.635141
\(508\) −2.94431e6 −0.506203
\(509\) −5.16044e6 −0.882861 −0.441430 0.897295i \(-0.645529\pi\)
−0.441430 + 0.897295i \(0.645529\pi\)
\(510\) 0 0
\(511\) −3.37728e6 −0.572157
\(512\) 1.27480e6 0.214915
\(513\) 1.15935e6 0.194501
\(514\) 1.98859e6 0.331999
\(515\) 0 0
\(516\) −3.05985e6 −0.505914
\(517\) 1.30365e6 0.214504
\(518\) 3.90932e6 0.640143
\(519\) −6.84768e6 −1.11590
\(520\) 0 0
\(521\) 6.65504e6 1.07413 0.537064 0.843541i \(-0.319533\pi\)
0.537064 + 0.843541i \(0.319533\pi\)
\(522\) −1.90446e6 −0.305911
\(523\) −6.22833e6 −0.995674 −0.497837 0.867271i \(-0.665872\pi\)
−0.497837 + 0.867271i \(0.665872\pi\)
\(524\) −5.13304e6 −0.816670
\(525\) 0 0
\(526\) 4.55334e6 0.717573
\(527\) −4.07415e6 −0.639015
\(528\) 120476. 0.0188068
\(529\) −4.29763e6 −0.667713
\(530\) 0 0
\(531\) −3.80200e6 −0.585161
\(532\) 3.91516e6 0.599750
\(533\) −2.79177e6 −0.425659
\(534\) −377766. −0.0573284
\(535\) 0 0
\(536\) −1.02112e7 −1.53521
\(537\) −6.36947e6 −0.953164
\(538\) 725767. 0.108104
\(539\) 415904. 0.0616625
\(540\) 0 0
\(541\) −5.01609e6 −0.736838 −0.368419 0.929660i \(-0.620101\pi\)
−0.368419 + 0.929660i \(0.620101\pi\)
\(542\) 3.15936e6 0.461956
\(543\) −4.78643e6 −0.696646
\(544\) −3.08948e6 −0.447598
\(545\) 0 0
\(546\) −3.00714e6 −0.431691
\(547\) 7.65398e6 1.09375 0.546876 0.837213i \(-0.315817\pi\)
0.546876 + 0.837213i \(0.315817\pi\)
\(548\) −5.31515e6 −0.756074
\(549\) 171069. 0.0242237
\(550\) 0 0
\(551\) 1.14262e7 1.60334
\(552\) 2.29533e6 0.320625
\(553\) 8.51943e6 1.18467
\(554\) 514197. 0.0711795
\(555\) 0 0
\(556\) 3.56184e6 0.488639
\(557\) −1.43563e6 −0.196067 −0.0980335 0.995183i \(-0.531255\pi\)
−0.0980335 + 0.995183i \(0.531255\pi\)
\(558\) 2.07722e6 0.282421
\(559\) −1.41006e7 −1.90857
\(560\) 0 0
\(561\) −566159. −0.0759507
\(562\) 6.78107e6 0.905644
\(563\) 2.98059e6 0.396307 0.198153 0.980171i \(-0.436506\pi\)
0.198153 + 0.980171i \(0.436506\pi\)
\(564\) −2.06451e6 −0.273288
\(565\) 0 0
\(566\) −1.91818e6 −0.251680
\(567\) 758634. 0.0991002
\(568\) 9.95728e6 1.29500
\(569\) −4.06173e6 −0.525934 −0.262967 0.964805i \(-0.584701\pi\)
−0.262967 + 0.964805i \(0.584701\pi\)
\(570\) 0 0
\(571\) 2.56327e6 0.329007 0.164503 0.986377i \(-0.447398\pi\)
0.164503 + 0.986377i \(0.447398\pi\)
\(572\) 2.27490e6 0.290718
\(573\) 4.67681e6 0.595064
\(574\) −1.19629e6 −0.151550
\(575\) 0 0
\(576\) 1.28843e6 0.161810
\(577\) 1.06121e7 1.32698 0.663489 0.748186i \(-0.269075\pi\)
0.663489 + 0.748186i \(0.269075\pi\)
\(578\) −3.76190e6 −0.468369
\(579\) 5.62394e6 0.697179
\(580\) 0 0
\(581\) 4.51923e6 0.555423
\(582\) −697810. −0.0853945
\(583\) −325496. −0.0396619
\(584\) 5.09368e6 0.618016
\(585\) 0 0
\(586\) −8.31579e6 −1.00037
\(587\) −1.12956e7 −1.35305 −0.676526 0.736418i \(-0.736516\pi\)
−0.676526 + 0.736418i \(0.736516\pi\)
\(588\) −658641. −0.0785607
\(589\) −1.24628e7 −1.48022
\(590\) 0 0
\(591\) −313920. −0.0369700
\(592\) 1.14298e6 0.134040
\(593\) 1.45361e7 1.69751 0.848753 0.528790i \(-0.177354\pi\)
0.848753 + 0.528790i \(0.177354\pi\)
\(594\) 288659. 0.0335674
\(595\) 0 0
\(596\) −3.01624e6 −0.347816
\(597\) −2.07313e6 −0.238062
\(598\) 4.22597e6 0.483251
\(599\) 9.35623e6 1.06545 0.532726 0.846288i \(-0.321168\pi\)
0.532726 + 0.846288i \(0.321168\pi\)
\(600\) 0 0
\(601\) −1.40395e6 −0.158549 −0.0792747 0.996853i \(-0.525260\pi\)
−0.0792747 + 0.996853i \(0.525260\pi\)
\(602\) −6.04217e6 −0.679519
\(603\) 4.74283e6 0.531183
\(604\) −7.79171e6 −0.869041
\(605\) 0 0
\(606\) −3.36555e6 −0.372284
\(607\) 9.82683e6 1.08253 0.541267 0.840851i \(-0.317945\pi\)
0.541267 + 0.840851i \(0.317945\pi\)
\(608\) −9.45067e6 −1.03682
\(609\) 7.47687e6 0.816915
\(610\) 0 0
\(611\) −9.51381e6 −1.03098
\(612\) 896592. 0.0967646
\(613\) 1.00133e6 0.107629 0.0538143 0.998551i \(-0.482862\pi\)
0.0538143 + 0.998551i \(0.482862\pi\)
\(614\) 4.75718e6 0.509247
\(615\) 0 0
\(616\) 2.43991e6 0.259073
\(617\) 1.27829e7 1.35181 0.675905 0.736988i \(-0.263753\pi\)
0.675905 + 0.736988i \(0.263753\pi\)
\(618\) −3.97078e6 −0.418219
\(619\) −1.03109e7 −1.08161 −0.540803 0.841149i \(-0.681880\pi\)
−0.540803 + 0.841149i \(0.681880\pi\)
\(620\) 0 0
\(621\) −1.06612e6 −0.110937
\(622\) 3.90980e6 0.405209
\(623\) 1.48310e6 0.153092
\(624\) −879208. −0.0903921
\(625\) 0 0
\(626\) −7.96103e6 −0.811957
\(627\) −1.73187e6 −0.175933
\(628\) 6.02862e6 0.609984
\(629\) −5.37129e6 −0.541318
\(630\) 0 0
\(631\) −104040. −0.0104023 −0.00520114 0.999986i \(-0.501656\pi\)
−0.00520114 + 0.999986i \(0.501656\pi\)
\(632\) −1.28492e7 −1.27962
\(633\) 1.01592e7 1.00774
\(634\) −3.04901e6 −0.301256
\(635\) 0 0
\(636\) 515468. 0.0505311
\(637\) −3.03519e6 −0.296372
\(638\) 2.84493e6 0.276707
\(639\) −4.62487e6 −0.448071
\(640\) 0 0
\(641\) −7.20754e6 −0.692855 −0.346427 0.938077i \(-0.612605\pi\)
−0.346427 + 0.938077i \(0.612605\pi\)
\(642\) 864906. 0.0828192
\(643\) 1.77389e7 1.69199 0.845997 0.533188i \(-0.179006\pi\)
0.845997 + 0.533188i \(0.179006\pi\)
\(644\) −3.60029e6 −0.342076
\(645\) 0 0
\(646\) 2.70564e6 0.255087
\(647\) −1.23330e7 −1.15826 −0.579130 0.815235i \(-0.696608\pi\)
−0.579130 + 0.815235i \(0.696608\pi\)
\(648\) −1.14419e6 −0.107043
\(649\) 5.67953e6 0.529298
\(650\) 0 0
\(651\) −8.15514e6 −0.754186
\(652\) −1.28098e7 −1.18012
\(653\) −1.12222e6 −0.102990 −0.0514950 0.998673i \(-0.516399\pi\)
−0.0514950 + 0.998673i \(0.516399\pi\)
\(654\) −3.02633e6 −0.276676
\(655\) 0 0
\(656\) −349763. −0.0317332
\(657\) −2.36587e6 −0.213834
\(658\) −4.07671e6 −0.367067
\(659\) −9.47379e6 −0.849787 −0.424894 0.905243i \(-0.639688\pi\)
−0.424894 + 0.905243i \(0.639688\pi\)
\(660\) 0 0
\(661\) −1.25237e7 −1.11488 −0.557441 0.830216i \(-0.688217\pi\)
−0.557441 + 0.830216i \(0.688217\pi\)
\(662\) 7.30803e6 0.648120
\(663\) 4.13172e6 0.365046
\(664\) −6.81599e6 −0.599941
\(665\) 0 0
\(666\) 2.73857e6 0.239243
\(667\) −1.05073e7 −0.914487
\(668\) −3.79608e6 −0.329150
\(669\) 1.09675e7 0.947417
\(670\) 0 0
\(671\) −255547. −0.0219111
\(672\) −6.18414e6 −0.528270
\(673\) 5.53368e6 0.470952 0.235476 0.971880i \(-0.424335\pi\)
0.235476 + 0.971880i \(0.424335\pi\)
\(674\) 4.49436e6 0.381082
\(675\) 0 0
\(676\) −8.69654e6 −0.731947
\(677\) 4.42320e6 0.370907 0.185453 0.982653i \(-0.440625\pi\)
0.185453 + 0.982653i \(0.440625\pi\)
\(678\) 3.94718e6 0.329771
\(679\) 2.73959e6 0.228040
\(680\) 0 0
\(681\) 7.51232e6 0.620735
\(682\) −3.10301e6 −0.255460
\(683\) 9.10450e6 0.746800 0.373400 0.927670i \(-0.378192\pi\)
0.373400 + 0.927670i \(0.378192\pi\)
\(684\) 2.74266e6 0.224146
\(685\) 0 0
\(686\) −7.66010e6 −0.621476
\(687\) −7.03532e6 −0.568712
\(688\) −1.76657e6 −0.142285
\(689\) 2.37541e6 0.190629
\(690\) 0 0
\(691\) 2.00055e7 1.59388 0.796939 0.604060i \(-0.206451\pi\)
0.796939 + 0.604060i \(0.206451\pi\)
\(692\) −1.61994e7 −1.28598
\(693\) −1.13327e6 −0.0896395
\(694\) −4.85981e6 −0.383019
\(695\) 0 0
\(696\) −1.12768e7 −0.882392
\(697\) 1.64366e6 0.128154
\(698\) −971080. −0.0754425
\(699\) 1.86359e6 0.144264
\(700\) 0 0
\(701\) 2.43992e7 1.87534 0.937672 0.347522i \(-0.112977\pi\)
0.937672 + 0.347522i \(0.112977\pi\)
\(702\) −2.10658e6 −0.161337
\(703\) −1.64307e7 −1.25391
\(704\) −1.92469e6 −0.146363
\(705\) 0 0
\(706\) −3.35851e6 −0.253592
\(707\) 1.32131e7 0.994158
\(708\) −8.99432e6 −0.674350
\(709\) −2.48282e6 −0.185494 −0.0927470 0.995690i \(-0.529565\pi\)
−0.0927470 + 0.995690i \(0.529565\pi\)
\(710\) 0 0
\(711\) 5.96806e6 0.442751
\(712\) −2.23684e6 −0.165362
\(713\) 1.14605e7 0.844266
\(714\) 1.77046e6 0.129970
\(715\) 0 0
\(716\) −1.50682e7 −1.09844
\(717\) −1.41694e7 −1.02933
\(718\) 1.13240e7 0.819761
\(719\) −1.24262e7 −0.896430 −0.448215 0.893926i \(-0.647940\pi\)
−0.448215 + 0.893926i \(0.647940\pi\)
\(720\) 0 0
\(721\) 1.55892e7 1.11683
\(722\) 173637. 0.0123965
\(723\) −9.21199e6 −0.655402
\(724\) −1.13232e7 −0.802827
\(725\) 0 0
\(726\) −431206. −0.0303629
\(727\) −789687. −0.0554139 −0.0277070 0.999616i \(-0.508821\pi\)
−0.0277070 + 0.999616i \(0.508821\pi\)
\(728\) −1.78060e7 −1.24520
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 8.30176e6 0.574614
\(732\) 404694. 0.0279158
\(733\) −1.22120e7 −0.839512 −0.419756 0.907637i \(-0.637884\pi\)
−0.419756 + 0.907637i \(0.637884\pi\)
\(734\) 8.33790e6 0.571237
\(735\) 0 0
\(736\) 8.69063e6 0.591366
\(737\) −7.08497e6 −0.480473
\(738\) −838029. −0.0566393
\(739\) −2.79083e7 −1.87985 −0.939923 0.341385i \(-0.889104\pi\)
−0.939923 + 0.341385i \(0.889104\pi\)
\(740\) 0 0
\(741\) 1.26389e7 0.845596
\(742\) 1.01787e6 0.0678709
\(743\) −2.25788e7 −1.50047 −0.750236 0.661170i \(-0.770060\pi\)
−0.750236 + 0.661170i \(0.770060\pi\)
\(744\) 1.22997e7 0.814635
\(745\) 0 0
\(746\) −5.32992e6 −0.350650
\(747\) 3.16583e6 0.207580
\(748\) −1.33935e6 −0.0875268
\(749\) −3.39560e6 −0.221163
\(750\) 0 0
\(751\) 3.67339e6 0.237666 0.118833 0.992914i \(-0.462085\pi\)
0.118833 + 0.992914i \(0.462085\pi\)
\(752\) −1.19192e6 −0.0768604
\(753\) −5.15624e6 −0.331395
\(754\) −2.07618e7 −1.32995
\(755\) 0 0
\(756\) 1.79469e6 0.114205
\(757\) 5.72422e6 0.363058 0.181529 0.983386i \(-0.441895\pi\)
0.181529 + 0.983386i \(0.441895\pi\)
\(758\) −1.29668e7 −0.819710
\(759\) 1.59259e6 0.100346
\(760\) 0 0
\(761\) −1.00462e7 −0.628842 −0.314421 0.949284i \(-0.601810\pi\)
−0.314421 + 0.949284i \(0.601810\pi\)
\(762\) −4.07286e6 −0.254104
\(763\) 1.18813e7 0.738845
\(764\) 1.10639e7 0.685762
\(765\) 0 0
\(766\) −1.80825e6 −0.111349
\(767\) −4.14481e7 −2.54400
\(768\) 8.64867e6 0.529110
\(769\) −1.09550e7 −0.668033 −0.334017 0.942567i \(-0.608404\pi\)
−0.334017 + 0.942567i \(0.608404\pi\)
\(770\) 0 0
\(771\) −5.46910e6 −0.331344
\(772\) 1.33045e7 0.803441
\(773\) 1.11558e7 0.671507 0.335753 0.941950i \(-0.391009\pi\)
0.335753 + 0.941950i \(0.391009\pi\)
\(774\) −4.23268e6 −0.253959
\(775\) 0 0
\(776\) −4.13190e6 −0.246318
\(777\) −1.07516e7 −0.638881
\(778\) −1.35263e7 −0.801179
\(779\) 5.02794e6 0.296857
\(780\) 0 0
\(781\) 6.90875e6 0.405296
\(782\) −2.48805e6 −0.145493
\(783\) 5.23773e6 0.305308
\(784\) −380258. −0.0220947
\(785\) 0 0
\(786\) −7.10052e6 −0.409953
\(787\) −2.62258e7 −1.50936 −0.754679 0.656094i \(-0.772207\pi\)
−0.754679 + 0.656094i \(0.772207\pi\)
\(788\) −742634. −0.0426049
\(789\) −1.25228e7 −0.716158
\(790\) 0 0
\(791\) −1.54965e7 −0.880630
\(792\) 1.70922e6 0.0968242
\(793\) 1.86494e6 0.105313
\(794\) 7.47037e6 0.420524
\(795\) 0 0
\(796\) −4.90437e6 −0.274347
\(797\) −2.91979e7 −1.62819 −0.814097 0.580729i \(-0.802768\pi\)
−0.814097 + 0.580729i \(0.802768\pi\)
\(798\) 5.41582e6 0.301063
\(799\) 5.60128e6 0.310399
\(800\) 0 0
\(801\) 1.03895e6 0.0572154
\(802\) 1.25577e7 0.689406
\(803\) 3.53420e6 0.193420
\(804\) 1.12200e7 0.612144
\(805\) 0 0
\(806\) 2.26452e7 1.22783
\(807\) −1.99604e6 −0.107891
\(808\) −1.99282e7 −1.07384
\(809\) 6.94941e6 0.373316 0.186658 0.982425i \(-0.440234\pi\)
0.186658 + 0.982425i \(0.440234\pi\)
\(810\) 0 0
\(811\) 6.55556e6 0.349992 0.174996 0.984569i \(-0.444009\pi\)
0.174996 + 0.984569i \(0.444009\pi\)
\(812\) 1.76879e7 0.941427
\(813\) −8.68901e6 −0.461046
\(814\) −4.09096e6 −0.216403
\(815\) 0 0
\(816\) 517636. 0.0272144
\(817\) 2.53949e7 1.33104
\(818\) 1.33166e7 0.695843
\(819\) 8.27037e6 0.430839
\(820\) 0 0
\(821\) −1.86984e7 −0.968160 −0.484080 0.875024i \(-0.660846\pi\)
−0.484080 + 0.875024i \(0.660846\pi\)
\(822\) −7.35242e6 −0.379534
\(823\) 7.32584e6 0.377014 0.188507 0.982072i \(-0.439635\pi\)
0.188507 + 0.982072i \(0.439635\pi\)
\(824\) −2.35119e7 −1.20634
\(825\) 0 0
\(826\) −1.77607e7 −0.905754
\(827\) 1.76737e7 0.898596 0.449298 0.893382i \(-0.351674\pi\)
0.449298 + 0.893382i \(0.351674\pi\)
\(828\) −2.52209e6 −0.127845
\(829\) −8.48151e6 −0.428634 −0.214317 0.976764i \(-0.568753\pi\)
−0.214317 + 0.976764i \(0.568753\pi\)
\(830\) 0 0
\(831\) −1.41417e6 −0.0710391
\(832\) 1.40460e7 0.703470
\(833\) 1.78697e6 0.0892289
\(834\) 4.92708e6 0.245287
\(835\) 0 0
\(836\) −4.09706e6 −0.202748
\(837\) −5.71287e6 −0.281865
\(838\) 3.13022e6 0.153980
\(839\) 3.35504e7 1.64548 0.822741 0.568417i \(-0.192444\pi\)
0.822741 + 0.568417i \(0.192444\pi\)
\(840\) 0 0
\(841\) 3.11104e7 1.51675
\(842\) 1.11510e7 0.542043
\(843\) −1.86496e7 −0.903859
\(844\) 2.40334e7 1.16134
\(845\) 0 0
\(846\) −2.85583e6 −0.137185
\(847\) 1.69291e6 0.0810820
\(848\) 297599. 0.0142116
\(849\) 5.27547e6 0.251184
\(850\) 0 0
\(851\) 1.51093e7 0.715189
\(852\) −1.09410e7 −0.516365
\(853\) 2.06919e7 0.973704 0.486852 0.873484i \(-0.338145\pi\)
0.486852 + 0.873484i \(0.338145\pi\)
\(854\) 799133. 0.0374951
\(855\) 0 0
\(856\) 5.12131e6 0.238889
\(857\) −2.99048e7 −1.39088 −0.695440 0.718584i \(-0.744790\pi\)
−0.695440 + 0.718584i \(0.744790\pi\)
\(858\) 3.14686e6 0.145935
\(859\) 2.64304e7 1.22214 0.611070 0.791577i \(-0.290739\pi\)
0.611070 + 0.791577i \(0.290739\pi\)
\(860\) 0 0
\(861\) 3.29008e6 0.151251
\(862\) −5066.36 −0.000232235 0
\(863\) −1.40568e6 −0.0642481 −0.0321241 0.999484i \(-0.510227\pi\)
−0.0321241 + 0.999484i \(0.510227\pi\)
\(864\) −4.33214e6 −0.197432
\(865\) 0 0
\(866\) 1.54776e7 0.701310
\(867\) 1.03461e7 0.467446
\(868\) −1.92925e7 −0.869137
\(869\) −8.91525e6 −0.400483
\(870\) 0 0
\(871\) 5.17047e7 2.30932
\(872\) −1.79196e7 −0.798064
\(873\) 1.91915e6 0.0852261
\(874\) −7.61090e6 −0.337021
\(875\) 0 0
\(876\) −5.59689e6 −0.246426
\(877\) 3.65903e7 1.60645 0.803225 0.595675i \(-0.203115\pi\)
0.803225 + 0.595675i \(0.203115\pi\)
\(878\) 2.37273e6 0.103875
\(879\) 2.28704e7 0.998395
\(880\) 0 0
\(881\) −4.07911e7 −1.77062 −0.885311 0.464999i \(-0.846055\pi\)
−0.885311 + 0.464999i \(0.846055\pi\)
\(882\) −911095. −0.0394360
\(883\) 3.47123e7 1.49824 0.749120 0.662434i \(-0.230476\pi\)
0.749120 + 0.662434i \(0.230476\pi\)
\(884\) 9.77435e6 0.420685
\(885\) 0 0
\(886\) 7.49623e6 0.320818
\(887\) 1.02664e7 0.438135 0.219068 0.975710i \(-0.429698\pi\)
0.219068 + 0.975710i \(0.429698\pi\)
\(888\) 1.62158e7 0.690088
\(889\) 1.59900e7 0.678567
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) 2.59455e7 1.09182
\(893\) 1.71342e7 0.719011
\(894\) −4.17235e6 −0.174597
\(895\) 0 0
\(896\) −1.59692e7 −0.664530
\(897\) −1.16224e7 −0.482299
\(898\) −1.25448e7 −0.519127
\(899\) −5.63043e7 −2.32350
\(900\) 0 0
\(901\) −1.39853e6 −0.0573930
\(902\) 1.25187e6 0.0512322
\(903\) 1.66174e7 0.678179
\(904\) 2.33722e7 0.951213
\(905\) 0 0
\(906\) −1.07782e7 −0.436242
\(907\) −2.61646e7 −1.05608 −0.528039 0.849220i \(-0.677072\pi\)
−0.528039 + 0.849220i \(0.677072\pi\)
\(908\) 1.77718e7 0.715345
\(909\) 9.25608e6 0.371550
\(910\) 0 0
\(911\) −3.76509e7 −1.50307 −0.751536 0.659692i \(-0.770687\pi\)
−0.751536 + 0.659692i \(0.770687\pi\)
\(912\) 1.58344e6 0.0630398
\(913\) −4.72920e6 −0.187763
\(914\) 2.21035e6 0.0875178
\(915\) 0 0
\(916\) −1.66433e7 −0.655393
\(917\) 2.78765e7 1.09475
\(918\) 1.24025e6 0.0485739
\(919\) 2.89048e7 1.12897 0.564484 0.825444i \(-0.309075\pi\)
0.564484 + 0.825444i \(0.309075\pi\)
\(920\) 0 0
\(921\) −1.30834e7 −0.508243
\(922\) −3.23072e6 −0.125162
\(923\) −5.04188e7 −1.94800
\(924\) −2.68095e6 −0.103302
\(925\) 0 0
\(926\) −1.07239e7 −0.410986
\(927\) 1.09206e7 0.417395
\(928\) −4.26963e7 −1.62750
\(929\) 1.70718e7 0.648993 0.324496 0.945887i \(-0.394805\pi\)
0.324496 + 0.945887i \(0.394805\pi\)
\(930\) 0 0
\(931\) 5.46633e6 0.206691
\(932\) 4.40867e6 0.166252
\(933\) −1.07529e7 −0.404410
\(934\) 6.44739e6 0.241834
\(935\) 0 0
\(936\) −1.24735e7 −0.465372
\(937\) 2.85410e7 1.06199 0.530994 0.847375i \(-0.321819\pi\)
0.530994 + 0.847375i \(0.321819\pi\)
\(938\) 2.21557e7 0.822203
\(939\) 2.18948e7 0.810356
\(940\) 0 0
\(941\) −1.24597e7 −0.458703 −0.229352 0.973344i \(-0.573661\pi\)
−0.229352 + 0.973344i \(0.573661\pi\)
\(942\) 8.33936e6 0.306200
\(943\) −4.62358e6 −0.169317
\(944\) −5.19276e6 −0.189657
\(945\) 0 0
\(946\) 6.32289e6 0.229714
\(947\) −1.05265e7 −0.381424 −0.190712 0.981646i \(-0.561080\pi\)
−0.190712 + 0.981646i \(0.561080\pi\)
\(948\) 1.41185e7 0.510234
\(949\) −2.57919e7 −0.929646
\(950\) 0 0
\(951\) 8.38551e6 0.300662
\(952\) 1.04833e7 0.374893
\(953\) −3.58903e7 −1.28010 −0.640051 0.768333i \(-0.721087\pi\)
−0.640051 + 0.768333i \(0.721087\pi\)
\(954\) 713044. 0.0253656
\(955\) 0 0
\(956\) −3.35203e7 −1.18621
\(957\) −7.82426e6 −0.276162
\(958\) 2.03317e7 0.715747
\(959\) 2.88655e7 1.01352
\(960\) 0 0
\(961\) 3.27828e7 1.14509
\(962\) 2.98550e7 1.04011
\(963\) −2.37870e6 −0.0826559
\(964\) −2.17926e7 −0.755296
\(965\) 0 0
\(966\) −4.98027e6 −0.171716
\(967\) −2.57707e7 −0.886259 −0.443129 0.896458i \(-0.646132\pi\)
−0.443129 + 0.896458i \(0.646132\pi\)
\(968\) −2.55327e6 −0.0875808
\(969\) −7.44117e6 −0.254585
\(970\) 0 0
\(971\) 1.07132e7 0.364645 0.182322 0.983239i \(-0.441639\pi\)
0.182322 + 0.983239i \(0.441639\pi\)
\(972\) 1.25722e6 0.0426821
\(973\) −1.93436e7 −0.655022
\(974\) −3.40378e6 −0.114965
\(975\) 0 0
\(976\) 233645. 0.00785113
\(977\) 6.88736e6 0.230843 0.115421 0.993317i \(-0.463178\pi\)
0.115421 + 0.993317i \(0.463178\pi\)
\(978\) −1.77198e7 −0.592395
\(979\) −1.55201e6 −0.0517533
\(980\) 0 0
\(981\) 8.32314e6 0.276131
\(982\) −2.35891e7 −0.780607
\(983\) −1.49277e7 −0.492730 −0.246365 0.969177i \(-0.579236\pi\)
−0.246365 + 0.969177i \(0.579236\pi\)
\(984\) −4.96217e6 −0.163374
\(985\) 0 0
\(986\) 1.22236e7 0.400410
\(987\) 1.12119e7 0.366343
\(988\) 2.98996e7 0.974480
\(989\) −2.33526e7 −0.759180
\(990\) 0 0
\(991\) 3.68190e7 1.19093 0.595467 0.803380i \(-0.296967\pi\)
0.595467 + 0.803380i \(0.296967\pi\)
\(992\) 4.65694e7 1.50253
\(993\) −2.00988e7 −0.646842
\(994\) −2.16047e7 −0.693557
\(995\) 0 0
\(996\) 7.48935e6 0.239219
\(997\) −2.71677e7 −0.865595 −0.432798 0.901491i \(-0.642474\pi\)
−0.432798 + 0.901491i \(0.642474\pi\)
\(998\) −9.53861e6 −0.303151
\(999\) −7.53175e6 −0.238771
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.1.6 yes 9
5.4 even 2 825.6.a.r.1.4 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.r.1.4 9 5.4 even 2
825.6.a.s.1.6 yes 9 1.1 even 1 trivial