Properties

Label 825.6.a.q.1.4
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $8$
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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - x^{7} - 176x^{6} + 272x^{5} + 9055x^{4} - 15851x^{3} - 118840x^{2} + 149572x - 33248 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.292443\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.29244 q^{2} -9.00000 q^{3} -30.3296 q^{4} -11.6320 q^{6} +91.7671 q^{7} -80.5574 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+1.29244 q^{2} -9.00000 q^{3} -30.3296 q^{4} -11.6320 q^{6} +91.7671 q^{7} -80.5574 q^{8} +81.0000 q^{9} -121.000 q^{11} +272.966 q^{12} +70.3337 q^{13} +118.604 q^{14} +866.431 q^{16} -1175.46 q^{17} +104.688 q^{18} -961.502 q^{19} -825.904 q^{21} -156.386 q^{22} +1303.06 q^{23} +725.017 q^{24} +90.9024 q^{26} -729.000 q^{27} -2783.26 q^{28} -3346.41 q^{29} +6451.89 q^{31} +3697.65 q^{32} +1089.00 q^{33} -1519.21 q^{34} -2456.70 q^{36} -13216.4 q^{37} -1242.69 q^{38} -633.004 q^{39} +18467.9 q^{41} -1067.43 q^{42} +19255.1 q^{43} +3669.88 q^{44} +1684.14 q^{46} +17064.0 q^{47} -7797.88 q^{48} -8385.80 q^{49} +10579.1 q^{51} -2133.19 q^{52} -31237.8 q^{53} -942.191 q^{54} -7392.52 q^{56} +8653.52 q^{57} -4325.05 q^{58} +43403.2 q^{59} -3568.78 q^{61} +8338.70 q^{62} +7433.14 q^{63} -22946.8 q^{64} +1407.47 q^{66} +70764.0 q^{67} +35651.1 q^{68} -11727.6 q^{69} +41227.7 q^{71} -6525.15 q^{72} -41255.5 q^{73} -17081.4 q^{74} +29162.0 q^{76} -11103.8 q^{77} -818.121 q^{78} -57458.9 q^{79} +6561.00 q^{81} +23868.7 q^{82} -17233.5 q^{83} +25049.3 q^{84} +24886.1 q^{86} +30117.7 q^{87} +9747.45 q^{88} -24409.7 q^{89} +6454.32 q^{91} -39521.4 q^{92} -58067.0 q^{93} +22054.3 q^{94} -33278.9 q^{96} -126547. q^{97} -10838.2 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9} - 968 q^{11} - 963 q^{12} + 382 q^{13} + 1048 q^{14} - 325 q^{16} + 288 q^{17} + 729 q^{18} - 988 q^{19} + 594 q^{21} - 1089 q^{22} + 5972 q^{23} - 1863 q^{24} - 14579 q^{26} - 5832 q^{27} + 5942 q^{28} + 1032 q^{29} - 4682 q^{31} + 3863 q^{32} + 8712 q^{33} + 2206 q^{34} + 8667 q^{36} - 17200 q^{37} + 11011 q^{38} - 3438 q^{39} - 13220 q^{41} - 9432 q^{42} + 22872 q^{43} - 12947 q^{44} + 9101 q^{46} + 6700 q^{47} + 2925 q^{48} - 43466 q^{49} - 2592 q^{51} + 5009 q^{52} + 6224 q^{53} - 6561 q^{54} + 20992 q^{56} + 8892 q^{57} + 33015 q^{58} - 77556 q^{59} + 11554 q^{61} + 12135 q^{62} - 5346 q^{63} - 149917 q^{64} + 9801 q^{66} + 20894 q^{67} + 91776 q^{68} - 53748 q^{69} - 21648 q^{71} + 16767 q^{72} + 64660 q^{73} - 179522 q^{74} + 24401 q^{76} + 7986 q^{77} + 131211 q^{78} - 22660 q^{79} + 52488 q^{81} - 56080 q^{82} + 100390 q^{83} - 53478 q^{84} + 47271 q^{86} - 9288 q^{87} - 25047 q^{88} - 25578 q^{89} + 73250 q^{91} + 95311 q^{92} + 42138 q^{93} - 170120 q^{94} - 34767 q^{96} + 142828 q^{97} + 303397 q^{98} - 78408 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.29244 0.228474 0.114237 0.993454i \(-0.463558\pi\)
0.114237 + 0.993454i \(0.463558\pi\)
\(3\) −9.00000 −0.577350
\(4\) −30.3296 −0.947800
\(5\) 0 0
\(6\) −11.6320 −0.131909
\(7\) 91.7671 0.707851 0.353926 0.935274i \(-0.384847\pi\)
0.353926 + 0.935274i \(0.384847\pi\)
\(8\) −80.5574 −0.445021
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 272.966 0.547212
\(13\) 70.3337 0.115426 0.0577132 0.998333i \(-0.481619\pi\)
0.0577132 + 0.998333i \(0.481619\pi\)
\(14\) 118.604 0.161725
\(15\) 0 0
\(16\) 866.431 0.846124
\(17\) −1175.46 −0.986471 −0.493235 0.869896i \(-0.664186\pi\)
−0.493235 + 0.869896i \(0.664186\pi\)
\(18\) 104.688 0.0761579
\(19\) −961.502 −0.611035 −0.305517 0.952186i \(-0.598829\pi\)
−0.305517 + 0.952186i \(0.598829\pi\)
\(20\) 0 0
\(21\) −825.904 −0.408678
\(22\) −156.386 −0.0688874
\(23\) 1303.06 0.513625 0.256813 0.966461i \(-0.417328\pi\)
0.256813 + 0.966461i \(0.417328\pi\)
\(24\) 725.017 0.256933
\(25\) 0 0
\(26\) 90.9024 0.0263719
\(27\) −729.000 −0.192450
\(28\) −2783.26 −0.670901
\(29\) −3346.41 −0.738898 −0.369449 0.929251i \(-0.620454\pi\)
−0.369449 + 0.929251i \(0.620454\pi\)
\(30\) 0 0
\(31\) 6451.89 1.20582 0.602911 0.797809i \(-0.294008\pi\)
0.602911 + 0.797809i \(0.294008\pi\)
\(32\) 3697.65 0.638338
\(33\) 1089.00 0.174078
\(34\) −1519.21 −0.225383
\(35\) 0 0
\(36\) −2456.70 −0.315933
\(37\) −13216.4 −1.58711 −0.793557 0.608496i \(-0.791773\pi\)
−0.793557 + 0.608496i \(0.791773\pi\)
\(38\) −1242.69 −0.139605
\(39\) −633.004 −0.0666415
\(40\) 0 0
\(41\) 18467.9 1.71577 0.857883 0.513844i \(-0.171779\pi\)
0.857883 + 0.513844i \(0.171779\pi\)
\(42\) −1067.43 −0.0933722
\(43\) 19255.1 1.58809 0.794044 0.607861i \(-0.207972\pi\)
0.794044 + 0.607861i \(0.207972\pi\)
\(44\) 3669.88 0.285772
\(45\) 0 0
\(46\) 1684.14 0.117350
\(47\) 17064.0 1.12677 0.563387 0.826193i \(-0.309498\pi\)
0.563387 + 0.826193i \(0.309498\pi\)
\(48\) −7797.88 −0.488510
\(49\) −8385.80 −0.498947
\(50\) 0 0
\(51\) 10579.1 0.569539
\(52\) −2133.19 −0.109401
\(53\) −31237.8 −1.52753 −0.763767 0.645491i \(-0.776653\pi\)
−0.763767 + 0.645491i \(0.776653\pi\)
\(54\) −942.191 −0.0439698
\(55\) 0 0
\(56\) −7392.52 −0.315009
\(57\) 8653.52 0.352781
\(58\) −4325.05 −0.168819
\(59\) 43403.2 1.62327 0.811637 0.584162i \(-0.198577\pi\)
0.811637 + 0.584162i \(0.198577\pi\)
\(60\) 0 0
\(61\) −3568.78 −0.122799 −0.0613996 0.998113i \(-0.519556\pi\)
−0.0613996 + 0.998113i \(0.519556\pi\)
\(62\) 8338.70 0.275499
\(63\) 7433.14 0.235950
\(64\) −22946.8 −0.700280
\(65\) 0 0
\(66\) 1407.47 0.0397722
\(67\) 70764.0 1.92586 0.962932 0.269745i \(-0.0869394\pi\)
0.962932 + 0.269745i \(0.0869394\pi\)
\(68\) 35651.1 0.934977
\(69\) −11727.6 −0.296542
\(70\) 0 0
\(71\) 41227.7 0.970607 0.485304 0.874346i \(-0.338709\pi\)
0.485304 + 0.874346i \(0.338709\pi\)
\(72\) −6525.15 −0.148340
\(73\) −41255.5 −0.906097 −0.453048 0.891486i \(-0.649664\pi\)
−0.453048 + 0.891486i \(0.649664\pi\)
\(74\) −17081.4 −0.362614
\(75\) 0 0
\(76\) 29162.0 0.579139
\(77\) −11103.8 −0.213425
\(78\) −818.121 −0.0152258
\(79\) −57458.9 −1.03583 −0.517916 0.855431i \(-0.673292\pi\)
−0.517916 + 0.855431i \(0.673292\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 23868.7 0.392008
\(83\) −17233.5 −0.274587 −0.137293 0.990530i \(-0.543840\pi\)
−0.137293 + 0.990530i \(0.543840\pi\)
\(84\) 25049.3 0.387345
\(85\) 0 0
\(86\) 24886.1 0.362836
\(87\) 30117.7 0.426603
\(88\) 9747.45 0.134179
\(89\) −24409.7 −0.326654 −0.163327 0.986572i \(-0.552223\pi\)
−0.163327 + 0.986572i \(0.552223\pi\)
\(90\) 0 0
\(91\) 6454.32 0.0817047
\(92\) −39521.4 −0.486814
\(93\) −58067.0 −0.696181
\(94\) 22054.3 0.257438
\(95\) 0 0
\(96\) −33278.9 −0.368545
\(97\) −126547. −1.36560 −0.682799 0.730606i \(-0.739238\pi\)
−0.682799 + 0.730606i \(0.739238\pi\)
\(98\) −10838.2 −0.113996
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) −89974.7 −0.877641 −0.438820 0.898575i \(-0.644604\pi\)
−0.438820 + 0.898575i \(0.644604\pi\)
\(102\) 13672.9 0.130125
\(103\) 106257. 0.986876 0.493438 0.869781i \(-0.335740\pi\)
0.493438 + 0.869781i \(0.335740\pi\)
\(104\) −5665.91 −0.0513672
\(105\) 0 0
\(106\) −40373.1 −0.349002
\(107\) 90929.4 0.767794 0.383897 0.923376i \(-0.374582\pi\)
0.383897 + 0.923376i \(0.374582\pi\)
\(108\) 22110.3 0.182404
\(109\) 46968.1 0.378649 0.189324 0.981915i \(-0.439370\pi\)
0.189324 + 0.981915i \(0.439370\pi\)
\(110\) 0 0
\(111\) 118947. 0.916320
\(112\) 79509.9 0.598930
\(113\) 137709. 1.01453 0.507267 0.861789i \(-0.330656\pi\)
0.507267 + 0.861789i \(0.330656\pi\)
\(114\) 11184.2 0.0806013
\(115\) 0 0
\(116\) 101495. 0.700327
\(117\) 5697.03 0.0384755
\(118\) 56096.2 0.370876
\(119\) −107868. −0.698274
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −4612.45 −0.0280564
\(123\) −166211. −0.990598
\(124\) −195683. −1.14288
\(125\) 0 0
\(126\) 9606.90 0.0539085
\(127\) −25160.6 −0.138424 −0.0692120 0.997602i \(-0.522048\pi\)
−0.0692120 + 0.997602i \(0.522048\pi\)
\(128\) −147982. −0.798334
\(129\) −173296. −0.916883
\(130\) 0 0
\(131\) −217042. −1.10501 −0.552505 0.833510i \(-0.686328\pi\)
−0.552505 + 0.833510i \(0.686328\pi\)
\(132\) −33028.9 −0.164991
\(133\) −88234.2 −0.432522
\(134\) 91458.5 0.440009
\(135\) 0 0
\(136\) 94691.8 0.439000
\(137\) −24093.9 −0.109675 −0.0548373 0.998495i \(-0.517464\pi\)
−0.0548373 + 0.998495i \(0.517464\pi\)
\(138\) −15157.2 −0.0677520
\(139\) −359804. −1.57953 −0.789766 0.613408i \(-0.789798\pi\)
−0.789766 + 0.613408i \(0.789798\pi\)
\(140\) 0 0
\(141\) −153576. −0.650543
\(142\) 53284.5 0.221758
\(143\) −8510.38 −0.0348024
\(144\) 70180.9 0.282041
\(145\) 0 0
\(146\) −53320.4 −0.207019
\(147\) 75472.2 0.288067
\(148\) 400847. 1.50427
\(149\) −307960. −1.13639 −0.568197 0.822892i \(-0.692359\pi\)
−0.568197 + 0.822892i \(0.692359\pi\)
\(150\) 0 0
\(151\) −531554. −1.89716 −0.948582 0.316531i \(-0.897482\pi\)
−0.948582 + 0.316531i \(0.897482\pi\)
\(152\) 77456.1 0.271924
\(153\) −95212.0 −0.328824
\(154\) −14351.1 −0.0487621
\(155\) 0 0
\(156\) 19198.7 0.0631628
\(157\) 428858. 1.38856 0.694280 0.719705i \(-0.255723\pi\)
0.694280 + 0.719705i \(0.255723\pi\)
\(158\) −74262.3 −0.236660
\(159\) 281140. 0.881923
\(160\) 0 0
\(161\) 119578. 0.363570
\(162\) 8479.72 0.0253860
\(163\) −20926.3 −0.0616912 −0.0308456 0.999524i \(-0.509820\pi\)
−0.0308456 + 0.999524i \(0.509820\pi\)
\(164\) −560124. −1.62620
\(165\) 0 0
\(166\) −22273.4 −0.0627358
\(167\) −550264. −1.52679 −0.763396 0.645931i \(-0.776469\pi\)
−0.763396 + 0.645931i \(0.776469\pi\)
\(168\) 66532.7 0.181870
\(169\) −366346. −0.986677
\(170\) 0 0
\(171\) −77881.6 −0.203678
\(172\) −583999. −1.50519
\(173\) −586340. −1.48948 −0.744740 0.667355i \(-0.767426\pi\)
−0.744740 + 0.667355i \(0.767426\pi\)
\(174\) 38925.4 0.0974676
\(175\) 0 0
\(176\) −104838. −0.255116
\(177\) −390629. −0.937198
\(178\) −31548.2 −0.0746318
\(179\) 556686. 1.29861 0.649303 0.760530i \(-0.275061\pi\)
0.649303 + 0.760530i \(0.275061\pi\)
\(180\) 0 0
\(181\) 399427. 0.906235 0.453117 0.891451i \(-0.350312\pi\)
0.453117 + 0.891451i \(0.350312\pi\)
\(182\) 8341.85 0.0186674
\(183\) 32119.0 0.0708981
\(184\) −104972. −0.228574
\(185\) 0 0
\(186\) −75048.3 −0.159059
\(187\) 142230. 0.297432
\(188\) −517545. −1.06796
\(189\) −66898.2 −0.136226
\(190\) 0 0
\(191\) −615748. −1.22129 −0.610646 0.791903i \(-0.709090\pi\)
−0.610646 + 0.791903i \(0.709090\pi\)
\(192\) 206521. 0.404307
\(193\) −422687. −0.816819 −0.408409 0.912799i \(-0.633916\pi\)
−0.408409 + 0.912799i \(0.633916\pi\)
\(194\) −163555. −0.312003
\(195\) 0 0
\(196\) 254338. 0.472902
\(197\) −116768. −0.214367 −0.107184 0.994239i \(-0.534183\pi\)
−0.107184 + 0.994239i \(0.534183\pi\)
\(198\) −12667.2 −0.0229625
\(199\) −674634. −1.20764 −0.603818 0.797123i \(-0.706354\pi\)
−0.603818 + 0.797123i \(0.706354\pi\)
\(200\) 0 0
\(201\) −636876. −1.11190
\(202\) −116287. −0.200518
\(203\) −307090. −0.523030
\(204\) −320860. −0.539809
\(205\) 0 0
\(206\) 137331. 0.225475
\(207\) 105548. 0.171208
\(208\) 60939.3 0.0976651
\(209\) 116342. 0.184234
\(210\) 0 0
\(211\) 347243. 0.536941 0.268471 0.963288i \(-0.413482\pi\)
0.268471 + 0.963288i \(0.413482\pi\)
\(212\) 947431. 1.44780
\(213\) −371050. −0.560380
\(214\) 117521. 0.175421
\(215\) 0 0
\(216\) 58726.4 0.0856444
\(217\) 592072. 0.853542
\(218\) 60703.6 0.0865113
\(219\) 371299. 0.523135
\(220\) 0 0
\(221\) −82674.2 −0.113865
\(222\) 153733. 0.209355
\(223\) −314163. −0.423051 −0.211525 0.977373i \(-0.567843\pi\)
−0.211525 + 0.977373i \(0.567843\pi\)
\(224\) 339323. 0.451849
\(225\) 0 0
\(226\) 177981. 0.231795
\(227\) 1.03995e6 1.33952 0.669759 0.742579i \(-0.266398\pi\)
0.669759 + 0.742579i \(0.266398\pi\)
\(228\) −262458. −0.334366
\(229\) −192732. −0.242866 −0.121433 0.992600i \(-0.538749\pi\)
−0.121433 + 0.992600i \(0.538749\pi\)
\(230\) 0 0
\(231\) 99934.4 0.123221
\(232\) 269578. 0.328825
\(233\) 644411. 0.777631 0.388815 0.921316i \(-0.372884\pi\)
0.388815 + 0.921316i \(0.372884\pi\)
\(234\) 7363.09 0.00879064
\(235\) 0 0
\(236\) −1.31640e6 −1.53854
\(237\) 517130. 0.598038
\(238\) −139414. −0.159537
\(239\) −605351. −0.685508 −0.342754 0.939425i \(-0.611360\pi\)
−0.342754 + 0.939425i \(0.611360\pi\)
\(240\) 0 0
\(241\) −432527. −0.479701 −0.239851 0.970810i \(-0.577098\pi\)
−0.239851 + 0.970810i \(0.577098\pi\)
\(242\) 18922.7 0.0207703
\(243\) −59049.0 −0.0641500
\(244\) 108240. 0.116389
\(245\) 0 0
\(246\) −214819. −0.226326
\(247\) −67626.0 −0.0705296
\(248\) −519748. −0.536616
\(249\) 155102. 0.158533
\(250\) 0 0
\(251\) −1.02207e6 −1.02400 −0.511998 0.858987i \(-0.671094\pi\)
−0.511998 + 0.858987i \(0.671094\pi\)
\(252\) −225444. −0.223634
\(253\) −157671. −0.154864
\(254\) −32518.6 −0.0316263
\(255\) 0 0
\(256\) 543039. 0.517882
\(257\) 909641. 0.859087 0.429544 0.903046i \(-0.358674\pi\)
0.429544 + 0.903046i \(0.358674\pi\)
\(258\) −223975. −0.209484
\(259\) −1.21283e6 −1.12344
\(260\) 0 0
\(261\) −271059. −0.246299
\(262\) −280515. −0.252466
\(263\) −999592. −0.891115 −0.445557 0.895253i \(-0.646994\pi\)
−0.445557 + 0.895253i \(0.646994\pi\)
\(264\) −87727.1 −0.0774682
\(265\) 0 0
\(266\) −114038. −0.0988199
\(267\) 219687. 0.188594
\(268\) −2.14624e6 −1.82533
\(269\) −2.14720e6 −1.80922 −0.904611 0.426238i \(-0.859839\pi\)
−0.904611 + 0.426238i \(0.859839\pi\)
\(270\) 0 0
\(271\) −743716. −0.615154 −0.307577 0.951523i \(-0.599518\pi\)
−0.307577 + 0.951523i \(0.599518\pi\)
\(272\) −1.01845e6 −0.834677
\(273\) −58088.9 −0.0471723
\(274\) −31140.0 −0.0250578
\(275\) 0 0
\(276\) 355693. 0.281062
\(277\) 2.30153e6 1.80226 0.901131 0.433547i \(-0.142738\pi\)
0.901131 + 0.433547i \(0.142738\pi\)
\(278\) −465026. −0.360882
\(279\) 522603. 0.401940
\(280\) 0 0
\(281\) 974755. 0.736427 0.368214 0.929741i \(-0.379970\pi\)
0.368214 + 0.929741i \(0.379970\pi\)
\(282\) −198488. −0.148632
\(283\) 1.23221e6 0.914577 0.457289 0.889318i \(-0.348821\pi\)
0.457289 + 0.889318i \(0.348821\pi\)
\(284\) −1.25042e6 −0.919941
\(285\) 0 0
\(286\) −10999.2 −0.00795143
\(287\) 1.69475e6 1.21451
\(288\) 299510. 0.212779
\(289\) −38159.6 −0.0268757
\(290\) 0 0
\(291\) 1.13892e6 0.788429
\(292\) 1.25126e6 0.858798
\(293\) −367972. −0.250407 −0.125203 0.992131i \(-0.539958\pi\)
−0.125203 + 0.992131i \(0.539958\pi\)
\(294\) 97543.5 0.0658158
\(295\) 0 0
\(296\) 1.06468e6 0.706299
\(297\) 88209.0 0.0580259
\(298\) −398021. −0.259636
\(299\) 91649.4 0.0592859
\(300\) 0 0
\(301\) 1.76698e6 1.12413
\(302\) −687003. −0.433452
\(303\) 809772. 0.506706
\(304\) −833075. −0.517011
\(305\) 0 0
\(306\) −123056. −0.0751276
\(307\) 760172. 0.460327 0.230163 0.973152i \(-0.426074\pi\)
0.230163 + 0.973152i \(0.426074\pi\)
\(308\) 336774. 0.202284
\(309\) −956309. −0.569773
\(310\) 0 0
\(311\) −2.20267e6 −1.29136 −0.645681 0.763607i \(-0.723426\pi\)
−0.645681 + 0.763607i \(0.723426\pi\)
\(312\) 50993.2 0.0296569
\(313\) 685921. 0.395743 0.197872 0.980228i \(-0.436597\pi\)
0.197872 + 0.980228i \(0.436597\pi\)
\(314\) 554275. 0.317250
\(315\) 0 0
\(316\) 1.74270e6 0.981761
\(317\) −1.87909e6 −1.05027 −0.525134 0.851019i \(-0.675985\pi\)
−0.525134 + 0.851019i \(0.675985\pi\)
\(318\) 363358. 0.201496
\(319\) 404916. 0.222786
\(320\) 0 0
\(321\) −818364. −0.443286
\(322\) 154548. 0.0830663
\(323\) 1.13020e6 0.602768
\(324\) −198992. −0.105311
\(325\) 0 0
\(326\) −27046.0 −0.0140948
\(327\) −422713. −0.218613
\(328\) −1.48773e6 −0.763553
\(329\) 1.56592e6 0.797588
\(330\) 0 0
\(331\) 1.94540e6 0.975974 0.487987 0.872851i \(-0.337731\pi\)
0.487987 + 0.872851i \(0.337731\pi\)
\(332\) 522686. 0.260253
\(333\) −1.07053e6 −0.529038
\(334\) −711185. −0.348832
\(335\) 0 0
\(336\) −715589. −0.345792
\(337\) −1.39092e6 −0.667154 −0.333577 0.942723i \(-0.608256\pi\)
−0.333577 + 0.942723i \(0.608256\pi\)
\(338\) −473482. −0.225430
\(339\) −1.23938e6 −0.585742
\(340\) 0 0
\(341\) −780679. −0.363569
\(342\) −100658. −0.0465352
\(343\) −2.31187e6 −1.06103
\(344\) −1.55114e6 −0.706733
\(345\) 0 0
\(346\) −757812. −0.340307
\(347\) 3.23565e6 1.44257 0.721286 0.692637i \(-0.243551\pi\)
0.721286 + 0.692637i \(0.243551\pi\)
\(348\) −913458. −0.404334
\(349\) −1.54020e6 −0.676883 −0.338442 0.940987i \(-0.609900\pi\)
−0.338442 + 0.940987i \(0.609900\pi\)
\(350\) 0 0
\(351\) −51273.3 −0.0222138
\(352\) −447416. −0.192466
\(353\) 696423. 0.297465 0.148733 0.988877i \(-0.452481\pi\)
0.148733 + 0.988877i \(0.452481\pi\)
\(354\) −504866. −0.214125
\(355\) 0 0
\(356\) 740337. 0.309602
\(357\) 970814. 0.403149
\(358\) 719485. 0.296698
\(359\) −2.60283e6 −1.06588 −0.532942 0.846152i \(-0.678914\pi\)
−0.532942 + 0.846152i \(0.678914\pi\)
\(360\) 0 0
\(361\) −1.55161e6 −0.626636
\(362\) 516236. 0.207051
\(363\) −131769. −0.0524864
\(364\) −195757. −0.0774397
\(365\) 0 0
\(366\) 41512.0 0.0161984
\(367\) −801086. −0.310466 −0.155233 0.987878i \(-0.549613\pi\)
−0.155233 + 0.987878i \(0.549613\pi\)
\(368\) 1.12902e6 0.434591
\(369\) 1.49590e6 0.571922
\(370\) 0 0
\(371\) −2.86660e6 −1.08127
\(372\) 1.76115e6 0.659840
\(373\) −1.94463e6 −0.723712 −0.361856 0.932234i \(-0.617857\pi\)
−0.361856 + 0.932234i \(0.617857\pi\)
\(374\) 183824. 0.0679554
\(375\) 0 0
\(376\) −1.37463e6 −0.501438
\(377\) −235366. −0.0852884
\(378\) −86462.1 −0.0311241
\(379\) −3.33496e6 −1.19260 −0.596298 0.802763i \(-0.703362\pi\)
−0.596298 + 0.802763i \(0.703362\pi\)
\(380\) 0 0
\(381\) 226445. 0.0799191
\(382\) −795820. −0.279033
\(383\) 2.16994e6 0.755875 0.377938 0.925831i \(-0.376633\pi\)
0.377938 + 0.925831i \(0.376633\pi\)
\(384\) 1.33184e6 0.460918
\(385\) 0 0
\(386\) −546299. −0.186622
\(387\) 1.55966e6 0.529362
\(388\) 3.83812e6 1.29431
\(389\) −561944. −0.188286 −0.0941432 0.995559i \(-0.530011\pi\)
−0.0941432 + 0.995559i \(0.530011\pi\)
\(390\) 0 0
\(391\) −1.53170e6 −0.506676
\(392\) 675539. 0.222042
\(393\) 1.95338e6 0.637978
\(394\) −150916. −0.0489773
\(395\) 0 0
\(396\) 297260. 0.0952575
\(397\) −3.98781e6 −1.26987 −0.634934 0.772566i \(-0.718973\pi\)
−0.634934 + 0.772566i \(0.718973\pi\)
\(398\) −871926. −0.275913
\(399\) 794108. 0.249717
\(400\) 0 0
\(401\) 3.07030e6 0.953499 0.476750 0.879039i \(-0.341815\pi\)
0.476750 + 0.879039i \(0.341815\pi\)
\(402\) −823126. −0.254040
\(403\) 453786. 0.139184
\(404\) 2.72890e6 0.831828
\(405\) 0 0
\(406\) −396897. −0.119499
\(407\) 1.59918e6 0.478533
\(408\) −852226. −0.253457
\(409\) −34399.2 −0.0101681 −0.00508405 0.999987i \(-0.501618\pi\)
−0.00508405 + 0.999987i \(0.501618\pi\)
\(410\) 0 0
\(411\) 216845. 0.0633206
\(412\) −3.22272e6 −0.935361
\(413\) 3.98299e6 1.14904
\(414\) 136415. 0.0391166
\(415\) 0 0
\(416\) 260070. 0.0736811
\(417\) 3.23823e6 0.911944
\(418\) 150365. 0.0420926
\(419\) 3.35733e6 0.934241 0.467121 0.884194i \(-0.345291\pi\)
0.467121 + 0.884194i \(0.345291\pi\)
\(420\) 0 0
\(421\) −693515. −0.190700 −0.0953499 0.995444i \(-0.530397\pi\)
−0.0953499 + 0.995444i \(0.530397\pi\)
\(422\) 448791. 0.122677
\(423\) 1.38219e6 0.375591
\(424\) 2.51644e6 0.679785
\(425\) 0 0
\(426\) −479560. −0.128032
\(427\) −327497. −0.0869235
\(428\) −2.75785e6 −0.727715
\(429\) 76593.5 0.0200932
\(430\) 0 0
\(431\) −3.60068e6 −0.933666 −0.466833 0.884346i \(-0.654605\pi\)
−0.466833 + 0.884346i \(0.654605\pi\)
\(432\) −631628. −0.162837
\(433\) −2.17347e6 −0.557100 −0.278550 0.960422i \(-0.589854\pi\)
−0.278550 + 0.960422i \(0.589854\pi\)
\(434\) 765219. 0.195012
\(435\) 0 0
\(436\) −1.42452e6 −0.358883
\(437\) −1.25290e6 −0.313843
\(438\) 479883. 0.119523
\(439\) −6.87194e6 −1.70184 −0.850919 0.525296i \(-0.823954\pi\)
−0.850919 + 0.525296i \(0.823954\pi\)
\(440\) 0 0
\(441\) −679250. −0.166316
\(442\) −106852. −0.0260151
\(443\) 4.97594e6 1.20466 0.602332 0.798246i \(-0.294238\pi\)
0.602332 + 0.798246i \(0.294238\pi\)
\(444\) −3.60762e6 −0.868488
\(445\) 0 0
\(446\) −406037. −0.0966560
\(447\) 2.77164e6 0.656098
\(448\) −2.10576e6 −0.495694
\(449\) −1.82131e6 −0.426352 −0.213176 0.977014i \(-0.568381\pi\)
−0.213176 + 0.977014i \(0.568381\pi\)
\(450\) 0 0
\(451\) −2.23462e6 −0.517323
\(452\) −4.17666e6 −0.961576
\(453\) 4.78398e6 1.09533
\(454\) 1.34408e6 0.306045
\(455\) 0 0
\(456\) −697105. −0.156995
\(457\) −1.52586e6 −0.341763 −0.170881 0.985292i \(-0.554661\pi\)
−0.170881 + 0.985292i \(0.554661\pi\)
\(458\) −249096. −0.0554885
\(459\) 856908. 0.189846
\(460\) 0 0
\(461\) 4.99516e6 1.09470 0.547352 0.836902i \(-0.315636\pi\)
0.547352 + 0.836902i \(0.315636\pi\)
\(462\) 129159. 0.0281528
\(463\) −5.20396e6 −1.12819 −0.564094 0.825711i \(-0.690774\pi\)
−0.564094 + 0.825711i \(0.690774\pi\)
\(464\) −2.89943e6 −0.625199
\(465\) 0 0
\(466\) 832865. 0.177668
\(467\) 2.28807e6 0.485487 0.242744 0.970090i \(-0.421953\pi\)
0.242744 + 0.970090i \(0.421953\pi\)
\(468\) −172789. −0.0364671
\(469\) 6.49381e6 1.36322
\(470\) 0 0
\(471\) −3.85973e6 −0.801686
\(472\) −3.49645e6 −0.722391
\(473\) −2.32987e6 −0.478826
\(474\) 668361. 0.136636
\(475\) 0 0
\(476\) 3.27160e6 0.661824
\(477\) −2.53026e6 −0.509178
\(478\) −782382. −0.156621
\(479\) 1.91319e6 0.380996 0.190498 0.981688i \(-0.438990\pi\)
0.190498 + 0.981688i \(0.438990\pi\)
\(480\) 0 0
\(481\) −929557. −0.183195
\(482\) −559017. −0.109599
\(483\) −1.07621e6 −0.209907
\(484\) −444056. −0.0861636
\(485\) 0 0
\(486\) −76317.5 −0.0146566
\(487\) −5.63300e6 −1.07626 −0.538130 0.842862i \(-0.680869\pi\)
−0.538130 + 0.842862i \(0.680869\pi\)
\(488\) 287492. 0.0546482
\(489\) 188336. 0.0356174
\(490\) 0 0
\(491\) −2.42388e6 −0.453741 −0.226871 0.973925i \(-0.572849\pi\)
−0.226871 + 0.973925i \(0.572849\pi\)
\(492\) 5.04112e6 0.938889
\(493\) 3.93356e6 0.728901
\(494\) −87402.8 −0.0161142
\(495\) 0 0
\(496\) 5.59012e6 1.02027
\(497\) 3.78335e6 0.687045
\(498\) 200460. 0.0362206
\(499\) 6.53921e6 1.17564 0.587819 0.808992i \(-0.299987\pi\)
0.587819 + 0.808992i \(0.299987\pi\)
\(500\) 0 0
\(501\) 4.95238e6 0.881494
\(502\) −1.32097e6 −0.233956
\(503\) 7.09673e6 1.25066 0.625329 0.780361i \(-0.284965\pi\)
0.625329 + 0.780361i \(0.284965\pi\)
\(504\) −598794. −0.105003
\(505\) 0 0
\(506\) −203781. −0.0353823
\(507\) 3.29712e6 0.569658
\(508\) 763110. 0.131198
\(509\) 1.40802e6 0.240887 0.120443 0.992720i \(-0.461568\pi\)
0.120443 + 0.992720i \(0.461568\pi\)
\(510\) 0 0
\(511\) −3.78590e6 −0.641382
\(512\) 5.43728e6 0.916657
\(513\) 700935. 0.117594
\(514\) 1.17566e6 0.196279
\(515\) 0 0
\(516\) 5.25599e6 0.869021
\(517\) −2.06475e6 −0.339735
\(518\) −1.56751e6 −0.256677
\(519\) 5.27706e6 0.859951
\(520\) 0 0
\(521\) −4.33371e6 −0.699465 −0.349732 0.936850i \(-0.613728\pi\)
−0.349732 + 0.936850i \(0.613728\pi\)
\(522\) −350329. −0.0562729
\(523\) −5.99430e6 −0.958262 −0.479131 0.877743i \(-0.659048\pi\)
−0.479131 + 0.877743i \(0.659048\pi\)
\(524\) 6.58280e6 1.04733
\(525\) 0 0
\(526\) −1.29192e6 −0.203596
\(527\) −7.58392e6 −1.18951
\(528\) 943543. 0.147291
\(529\) −4.73837e6 −0.736189
\(530\) 0 0
\(531\) 3.51566e6 0.541091
\(532\) 2.67611e6 0.409944
\(533\) 1.29892e6 0.198045
\(534\) 283933. 0.0430887
\(535\) 0 0
\(536\) −5.70057e6 −0.857050
\(537\) −5.01017e6 −0.749751
\(538\) −2.77513e6 −0.413360
\(539\) 1.01468e6 0.150438
\(540\) 0 0
\(541\) 9.71649e6 1.42730 0.713652 0.700501i \(-0.247040\pi\)
0.713652 + 0.700501i \(0.247040\pi\)
\(542\) −961211. −0.140547
\(543\) −3.59484e6 −0.523215
\(544\) −4.34643e6 −0.629702
\(545\) 0 0
\(546\) −75076.6 −0.0107776
\(547\) −1.18605e7 −1.69486 −0.847430 0.530908i \(-0.821851\pi\)
−0.847430 + 0.530908i \(0.821851\pi\)
\(548\) 730758. 0.103950
\(549\) −289071. −0.0409331
\(550\) 0 0
\(551\) 3.21758e6 0.451492
\(552\) 944744. 0.131967
\(553\) −5.27283e6 −0.733215
\(554\) 2.97460e6 0.411770
\(555\) 0 0
\(556\) 1.09127e7 1.49708
\(557\) −4.65955e6 −0.636364 −0.318182 0.948030i \(-0.603072\pi\)
−0.318182 + 0.948030i \(0.603072\pi\)
\(558\) 675435. 0.0918329
\(559\) 1.35428e6 0.183307
\(560\) 0 0
\(561\) −1.28007e6 −0.171722
\(562\) 1.25982e6 0.168254
\(563\) −7.30425e6 −0.971191 −0.485595 0.874184i \(-0.661397\pi\)
−0.485595 + 0.874184i \(0.661397\pi\)
\(564\) 4.65790e6 0.616584
\(565\) 0 0
\(566\) 1.59257e6 0.208957
\(567\) 602084. 0.0786501
\(568\) −3.32120e6 −0.431941
\(569\) −3.49414e6 −0.452439 −0.226219 0.974076i \(-0.572637\pi\)
−0.226219 + 0.974076i \(0.572637\pi\)
\(570\) 0 0
\(571\) 1.42919e7 1.83443 0.917213 0.398396i \(-0.130433\pi\)
0.917213 + 0.398396i \(0.130433\pi\)
\(572\) 258116. 0.0329857
\(573\) 5.54174e6 0.705114
\(574\) 2.19036e6 0.277483
\(575\) 0 0
\(576\) −1.85869e6 −0.233427
\(577\) −6.93782e6 −0.867528 −0.433764 0.901027i \(-0.642815\pi\)
−0.433764 + 0.901027i \(0.642815\pi\)
\(578\) −49319.1 −0.00614039
\(579\) 3.80418e6 0.471591
\(580\) 0 0
\(581\) −1.58147e6 −0.194366
\(582\) 1.47199e6 0.180135
\(583\) 3.77978e6 0.460569
\(584\) 3.32344e6 0.403232
\(585\) 0 0
\(586\) −475583. −0.0572114
\(587\) 2.56707e6 0.307498 0.153749 0.988110i \(-0.450865\pi\)
0.153749 + 0.988110i \(0.450865\pi\)
\(588\) −2.28904e6 −0.273030
\(589\) −6.20351e6 −0.736799
\(590\) 0 0
\(591\) 1.05091e6 0.123765
\(592\) −1.14511e7 −1.34289
\(593\) 8.76559e6 1.02363 0.511817 0.859095i \(-0.328973\pi\)
0.511817 + 0.859095i \(0.328973\pi\)
\(594\) 114005. 0.0132574
\(595\) 0 0
\(596\) 9.34031e6 1.07707
\(597\) 6.07171e6 0.697229
\(598\) 118452. 0.0135453
\(599\) 6.17238e6 0.702887 0.351444 0.936209i \(-0.385691\pi\)
0.351444 + 0.936209i \(0.385691\pi\)
\(600\) 0 0
\(601\) −1.19370e7 −1.34806 −0.674031 0.738703i \(-0.735439\pi\)
−0.674031 + 0.738703i \(0.735439\pi\)
\(602\) 2.28373e6 0.256834
\(603\) 5.73189e6 0.641955
\(604\) 1.61218e7 1.79813
\(605\) 0 0
\(606\) 1.04658e6 0.115769
\(607\) −2.00634e6 −0.221021 −0.110510 0.993875i \(-0.535249\pi\)
−0.110510 + 0.993875i \(0.535249\pi\)
\(608\) −3.55530e6 −0.390047
\(609\) 2.76381e6 0.301971
\(610\) 0 0
\(611\) 1.20018e6 0.130059
\(612\) 2.88774e6 0.311659
\(613\) −1.13241e6 −0.121718 −0.0608589 0.998146i \(-0.519384\pi\)
−0.0608589 + 0.998146i \(0.519384\pi\)
\(614\) 982479. 0.105173
\(615\) 0 0
\(616\) 894495. 0.0949787
\(617\) −1.20807e6 −0.127755 −0.0638775 0.997958i \(-0.520347\pi\)
−0.0638775 + 0.997958i \(0.520347\pi\)
\(618\) −1.23598e6 −0.130178
\(619\) 1.14027e7 1.19614 0.598071 0.801443i \(-0.295934\pi\)
0.598071 + 0.801443i \(0.295934\pi\)
\(620\) 0 0
\(621\) −949934. −0.0988472
\(622\) −2.84682e6 −0.295042
\(623\) −2.24001e6 −0.231222
\(624\) −548454. −0.0563870
\(625\) 0 0
\(626\) 886514. 0.0904170
\(627\) −1.04708e6 −0.106368
\(628\) −1.30071e7 −1.31608
\(629\) 1.55353e7 1.56564
\(630\) 0 0
\(631\) 1.32708e6 0.132685 0.0663426 0.997797i \(-0.478867\pi\)
0.0663426 + 0.997797i \(0.478867\pi\)
\(632\) 4.62874e6 0.460967
\(633\) −3.12518e6 −0.310003
\(634\) −2.42862e6 −0.239959
\(635\) 0 0
\(636\) −8.52688e6 −0.835886
\(637\) −589805. −0.0575917
\(638\) 523331. 0.0509008
\(639\) 3.33945e6 0.323536
\(640\) 0 0
\(641\) −9.61062e6 −0.923860 −0.461930 0.886916i \(-0.652843\pi\)
−0.461930 + 0.886916i \(0.652843\pi\)
\(642\) −1.05769e6 −0.101279
\(643\) 1.26822e7 1.20967 0.604835 0.796351i \(-0.293239\pi\)
0.604835 + 0.796351i \(0.293239\pi\)
\(644\) −3.62677e6 −0.344592
\(645\) 0 0
\(646\) 1.46072e6 0.137717
\(647\) −1.27269e7 −1.19526 −0.597629 0.801773i \(-0.703890\pi\)
−0.597629 + 0.801773i \(0.703890\pi\)
\(648\) −528537. −0.0494468
\(649\) −5.25179e6 −0.489435
\(650\) 0 0
\(651\) −5.32864e6 −0.492793
\(652\) 634685. 0.0584709
\(653\) 7.54035e6 0.692004 0.346002 0.938234i \(-0.387539\pi\)
0.346002 + 0.938234i \(0.387539\pi\)
\(654\) −546332. −0.0499473
\(655\) 0 0
\(656\) 1.60012e7 1.45175
\(657\) −3.34170e6 −0.302032
\(658\) 2.02386e6 0.182228
\(659\) 1.20388e7 1.07987 0.539933 0.841708i \(-0.318450\pi\)
0.539933 + 0.841708i \(0.318450\pi\)
\(660\) 0 0
\(661\) 6.36439e6 0.566569 0.283285 0.959036i \(-0.408576\pi\)
0.283285 + 0.959036i \(0.408576\pi\)
\(662\) 2.51432e6 0.222985
\(663\) 744068. 0.0657399
\(664\) 1.38829e6 0.122197
\(665\) 0 0
\(666\) −1.38359e6 −0.120871
\(667\) −4.36059e6 −0.379517
\(668\) 1.66893e7 1.44709
\(669\) 2.82746e6 0.244248
\(670\) 0 0
\(671\) 431823. 0.0370254
\(672\) −3.05390e6 −0.260875
\(673\) −741014. −0.0630650 −0.0315325 0.999503i \(-0.510039\pi\)
−0.0315325 + 0.999503i \(0.510039\pi\)
\(674\) −1.79768e6 −0.152427
\(675\) 0 0
\(676\) 1.11111e7 0.935172
\(677\) 1.43961e7 1.20718 0.603590 0.797295i \(-0.293736\pi\)
0.603590 + 0.797295i \(0.293736\pi\)
\(678\) −1.60183e6 −0.133827
\(679\) −1.16129e7 −0.966640
\(680\) 0 0
\(681\) −9.35956e6 −0.773371
\(682\) −1.00898e6 −0.0830659
\(683\) −7.32070e6 −0.600484 −0.300242 0.953863i \(-0.597067\pi\)
−0.300242 + 0.953863i \(0.597067\pi\)
\(684\) 2.36212e6 0.193046
\(685\) 0 0
\(686\) −2.98796e6 −0.242418
\(687\) 1.73459e6 0.140219
\(688\) 1.66832e7 1.34372
\(689\) −2.19707e6 −0.176318
\(690\) 0 0
\(691\) 1.51860e7 1.20989 0.604947 0.796266i \(-0.293194\pi\)
0.604947 + 0.796266i \(0.293194\pi\)
\(692\) 1.77835e7 1.41173
\(693\) −899409. −0.0711417
\(694\) 4.18189e6 0.329590
\(695\) 0 0
\(696\) −2.42621e6 −0.189847
\(697\) −2.17082e7 −1.69255
\(698\) −1.99062e6 −0.154650
\(699\) −5.79970e6 −0.448965
\(700\) 0 0
\(701\) −1.17236e7 −0.901089 −0.450544 0.892754i \(-0.648770\pi\)
−0.450544 + 0.892754i \(0.648770\pi\)
\(702\) −66267.8 −0.00507528
\(703\) 1.27076e7 0.969782
\(704\) 2.77656e6 0.211142
\(705\) 0 0
\(706\) 900087. 0.0679630
\(707\) −8.25672e6 −0.621239
\(708\) 1.18476e7 0.888276
\(709\) 6.09002e6 0.454991 0.227495 0.973779i \(-0.426946\pi\)
0.227495 + 0.973779i \(0.426946\pi\)
\(710\) 0 0
\(711\) −4.65417e6 −0.345277
\(712\) 1.96638e6 0.145368
\(713\) 8.40723e6 0.619340
\(714\) 1.25472e6 0.0921090
\(715\) 0 0
\(716\) −1.68841e7 −1.23082
\(717\) 5.44816e6 0.395778
\(718\) −3.36401e6 −0.243527
\(719\) 3.50864e6 0.253114 0.126557 0.991959i \(-0.459607\pi\)
0.126557 + 0.991959i \(0.459607\pi\)
\(720\) 0 0
\(721\) 9.75086e6 0.698562
\(722\) −2.00537e6 −0.143170
\(723\) 3.89274e6 0.276956
\(724\) −1.21145e7 −0.858929
\(725\) 0 0
\(726\) −170304. −0.0119918
\(727\) −222222. −0.0155937 −0.00779687 0.999970i \(-0.502482\pi\)
−0.00779687 + 0.999970i \(0.502482\pi\)
\(728\) −519944. −0.0363603
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −2.26335e7 −1.56660
\(732\) −974157. −0.0671972
\(733\) −1.95614e6 −0.134475 −0.0672373 0.997737i \(-0.521418\pi\)
−0.0672373 + 0.997737i \(0.521418\pi\)
\(734\) −1.03536e6 −0.0709333
\(735\) 0 0
\(736\) 4.81828e6 0.327867
\(737\) −8.56245e6 −0.580670
\(738\) 1.93337e6 0.130669
\(739\) −3.61998e6 −0.243834 −0.121917 0.992540i \(-0.538904\pi\)
−0.121917 + 0.992540i \(0.538904\pi\)
\(740\) 0 0
\(741\) 608634. 0.0407203
\(742\) −3.70492e6 −0.247041
\(743\) 4.37060e6 0.290449 0.145224 0.989399i \(-0.453610\pi\)
0.145224 + 0.989399i \(0.453610\pi\)
\(744\) 4.67773e6 0.309815
\(745\) 0 0
\(746\) −2.51333e6 −0.165349
\(747\) −1.39592e6 −0.0915289
\(748\) −4.31378e6 −0.281906
\(749\) 8.34432e6 0.543484
\(750\) 0 0
\(751\) 8.89727e6 0.575648 0.287824 0.957683i \(-0.407068\pi\)
0.287824 + 0.957683i \(0.407068\pi\)
\(752\) 1.47848e7 0.953390
\(753\) 9.19867e6 0.591204
\(754\) −304197. −0.0194862
\(755\) 0 0
\(756\) 2.02900e6 0.129115
\(757\) −1.36934e7 −0.868506 −0.434253 0.900791i \(-0.642988\pi\)
−0.434253 + 0.900791i \(0.642988\pi\)
\(758\) −4.31025e6 −0.272477
\(759\) 1.41904e6 0.0894107
\(760\) 0 0
\(761\) 1.75245e7 1.09694 0.548472 0.836169i \(-0.315210\pi\)
0.548472 + 0.836169i \(0.315210\pi\)
\(762\) 292668. 0.0182594
\(763\) 4.31012e6 0.268027
\(764\) 1.86754e7 1.15754
\(765\) 0 0
\(766\) 2.80452e6 0.172698
\(767\) 3.05271e6 0.187369
\(768\) −4.88735e6 −0.298999
\(769\) −1.68641e7 −1.02837 −0.514183 0.857681i \(-0.671905\pi\)
−0.514183 + 0.857681i \(0.671905\pi\)
\(770\) 0 0
\(771\) −8.18677e6 −0.495994
\(772\) 1.28199e7 0.774181
\(773\) 1.24437e7 0.749033 0.374516 0.927220i \(-0.377809\pi\)
0.374516 + 0.927220i \(0.377809\pi\)
\(774\) 2.01577e6 0.120945
\(775\) 0 0
\(776\) 1.01943e7 0.607720
\(777\) 1.09155e7 0.648618
\(778\) −726281. −0.0430185
\(779\) −1.77569e7 −1.04839
\(780\) 0 0
\(781\) −4.98856e6 −0.292649
\(782\) −1.97963e6 −0.115762
\(783\) 2.43953e6 0.142201
\(784\) −7.26572e6 −0.422171
\(785\) 0 0
\(786\) 2.52463e6 0.145761
\(787\) −2.62304e7 −1.50962 −0.754811 0.655943i \(-0.772271\pi\)
−0.754811 + 0.655943i \(0.772271\pi\)
\(788\) 3.54152e6 0.203177
\(789\) 8.99633e6 0.514485
\(790\) 0 0
\(791\) 1.26372e7 0.718139
\(792\) 789544. 0.0447263
\(793\) −251006. −0.0141743
\(794\) −5.15402e6 −0.290132
\(795\) 0 0
\(796\) 2.04614e7 1.14460
\(797\) −4.99841e6 −0.278731 −0.139366 0.990241i \(-0.544506\pi\)
−0.139366 + 0.990241i \(0.544506\pi\)
\(798\) 1.02634e6 0.0570537
\(799\) −2.00580e7 −1.11153
\(800\) 0 0
\(801\) −1.97719e6 −0.108885
\(802\) 3.96819e6 0.217850
\(803\) 4.99191e6 0.273199
\(804\) 1.93162e7 1.05386
\(805\) 0 0
\(806\) 586492. 0.0317998
\(807\) 1.93248e7 1.04456
\(808\) 7.24813e6 0.390569
\(809\) 2.76591e7 1.48582 0.742912 0.669389i \(-0.233444\pi\)
0.742912 + 0.669389i \(0.233444\pi\)
\(810\) 0 0
\(811\) −1.72731e7 −0.922184 −0.461092 0.887352i \(-0.652542\pi\)
−0.461092 + 0.887352i \(0.652542\pi\)
\(812\) 9.31393e6 0.495727
\(813\) 6.69344e6 0.355159
\(814\) 2.06685e6 0.109332
\(815\) 0 0
\(816\) 9.16607e6 0.481901
\(817\) −1.85138e7 −0.970377
\(818\) −44459.0 −0.00232315
\(819\) 522800. 0.0272349
\(820\) 0 0
\(821\) −1.70950e7 −0.885139 −0.442570 0.896734i \(-0.645933\pi\)
−0.442570 + 0.896734i \(0.645933\pi\)
\(822\) 280260. 0.0144671
\(823\) 1.39800e7 0.719464 0.359732 0.933056i \(-0.382868\pi\)
0.359732 + 0.933056i \(0.382868\pi\)
\(824\) −8.55976e6 −0.439181
\(825\) 0 0
\(826\) 5.14778e6 0.262525
\(827\) −2.33121e7 −1.18527 −0.592636 0.805471i \(-0.701913\pi\)
−0.592636 + 0.805471i \(0.701913\pi\)
\(828\) −3.20123e6 −0.162271
\(829\) −2.71774e7 −1.37348 −0.686738 0.726905i \(-0.740958\pi\)
−0.686738 + 0.726905i \(0.740958\pi\)
\(830\) 0 0
\(831\) −2.07138e7 −1.04054
\(832\) −1.61393e6 −0.0808309
\(833\) 9.85714e6 0.492196
\(834\) 4.18523e6 0.208355
\(835\) 0 0
\(836\) −3.52860e6 −0.174617
\(837\) −4.70343e6 −0.232060
\(838\) 4.33916e6 0.213450
\(839\) 6.44081e6 0.315890 0.157945 0.987448i \(-0.449513\pi\)
0.157945 + 0.987448i \(0.449513\pi\)
\(840\) 0 0
\(841\) −9.31268e6 −0.454030
\(842\) −896328. −0.0435699
\(843\) −8.77280e6 −0.425176
\(844\) −1.05317e7 −0.508913
\(845\) 0 0
\(846\) 1.78640e6 0.0858127
\(847\) 1.34356e6 0.0643501
\(848\) −2.70654e7 −1.29248
\(849\) −1.10899e7 −0.528031
\(850\) 0 0
\(851\) −1.72218e7 −0.815182
\(852\) 1.12538e7 0.531128
\(853\) 3.24050e6 0.152490 0.0762448 0.997089i \(-0.475707\pi\)
0.0762448 + 0.997089i \(0.475707\pi\)
\(854\) −423271. −0.0198598
\(855\) 0 0
\(856\) −7.32504e6 −0.341685
\(857\) 2.15579e7 1.00266 0.501331 0.865255i \(-0.332844\pi\)
0.501331 + 0.865255i \(0.332844\pi\)
\(858\) 98992.7 0.00459076
\(859\) 3.00808e7 1.39093 0.695467 0.718558i \(-0.255197\pi\)
0.695467 + 0.718558i \(0.255197\pi\)
\(860\) 0 0
\(861\) −1.52527e7 −0.701196
\(862\) −4.65367e6 −0.213318
\(863\) 3.68726e7 1.68530 0.842650 0.538462i \(-0.180995\pi\)
0.842650 + 0.538462i \(0.180995\pi\)
\(864\) −2.69559e6 −0.122848
\(865\) 0 0
\(866\) −2.80908e6 −0.127283
\(867\) 343436. 0.0155167
\(868\) −1.79573e7 −0.808987
\(869\) 6.95252e6 0.312315
\(870\) 0 0
\(871\) 4.97710e6 0.222296
\(872\) −3.78363e6 −0.168507
\(873\) −1.02503e7 −0.455200
\(874\) −1.61930e6 −0.0717049
\(875\) 0 0
\(876\) −1.12614e7 −0.495827
\(877\) 1.99151e7 0.874348 0.437174 0.899377i \(-0.355979\pi\)
0.437174 + 0.899377i \(0.355979\pi\)
\(878\) −8.88160e6 −0.388826
\(879\) 3.31175e6 0.144572
\(880\) 0 0
\(881\) 2.96980e7 1.28910 0.644551 0.764561i \(-0.277044\pi\)
0.644551 + 0.764561i \(0.277044\pi\)
\(882\) −877892. −0.0379988
\(883\) −1.61839e7 −0.698526 −0.349263 0.937025i \(-0.613568\pi\)
−0.349263 + 0.937025i \(0.613568\pi\)
\(884\) 2.50748e6 0.107921
\(885\) 0 0
\(886\) 6.43112e6 0.275234
\(887\) 3.22208e7 1.37508 0.687540 0.726147i \(-0.258691\pi\)
0.687540 + 0.726147i \(0.258691\pi\)
\(888\) −9.58209e6 −0.407782
\(889\) −2.30891e6 −0.0979836
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) 9.52842e6 0.400967
\(893\) −1.64071e7 −0.688498
\(894\) 3.58219e6 0.149901
\(895\) 0 0
\(896\) −1.35799e7 −0.565102
\(897\) −824845. −0.0342288
\(898\) −2.35394e6 −0.0974102
\(899\) −2.15907e7 −0.890979
\(900\) 0 0
\(901\) 3.67187e7 1.50687
\(902\) −2.88812e6 −0.118195
\(903\) −1.59029e7 −0.649016
\(904\) −1.10935e7 −0.451489
\(905\) 0 0
\(906\) 6.18303e6 0.250254
\(907\) −6.59225e6 −0.266082 −0.133041 0.991111i \(-0.542474\pi\)
−0.133041 + 0.991111i \(0.542474\pi\)
\(908\) −3.15413e7 −1.26959
\(909\) −7.28795e6 −0.292547
\(910\) 0 0
\(911\) −4.51432e7 −1.80217 −0.901086 0.433640i \(-0.857229\pi\)
−0.901086 + 0.433640i \(0.857229\pi\)
\(912\) 7.49767e6 0.298497
\(913\) 2.08526e6 0.0827910
\(914\) −1.97209e6 −0.0780838
\(915\) 0 0
\(916\) 5.84550e6 0.230188
\(917\) −1.99173e7 −0.782182
\(918\) 1.10750e6 0.0433749
\(919\) −4.40836e7 −1.72182 −0.860911 0.508756i \(-0.830106\pi\)
−0.860911 + 0.508756i \(0.830106\pi\)
\(920\) 0 0
\(921\) −6.84155e6 −0.265770
\(922\) 6.45596e6 0.250111
\(923\) 2.89970e6 0.112034
\(924\) −3.03097e6 −0.116789
\(925\) 0 0
\(926\) −6.72582e6 −0.257761
\(927\) 8.60678e6 0.328959
\(928\) −1.23739e7 −0.471667
\(929\) −3.78094e7 −1.43734 −0.718672 0.695349i \(-0.755250\pi\)
−0.718672 + 0.695349i \(0.755250\pi\)
\(930\) 0 0
\(931\) 8.06296e6 0.304874
\(932\) −1.95447e7 −0.737038
\(933\) 1.98240e7 0.745568
\(934\) 2.95721e6 0.110921
\(935\) 0 0
\(936\) −458938. −0.0171224
\(937\) −4.50928e7 −1.67787 −0.838935 0.544231i \(-0.816821\pi\)
−0.838935 + 0.544231i \(0.816821\pi\)
\(938\) 8.39288e6 0.311461
\(939\) −6.17329e6 −0.228482
\(940\) 0 0
\(941\) 4.32466e7 1.59213 0.796063 0.605213i \(-0.206912\pi\)
0.796063 + 0.605213i \(0.206912\pi\)
\(942\) −4.98847e6 −0.183164
\(943\) 2.40649e7 0.881261
\(944\) 3.76059e7 1.37349
\(945\) 0 0
\(946\) −3.01122e6 −0.109399
\(947\) 3.10917e7 1.12660 0.563300 0.826253i \(-0.309532\pi\)
0.563300 + 0.826253i \(0.309532\pi\)
\(948\) −1.56843e7 −0.566820
\(949\) −2.90165e6 −0.104588
\(950\) 0 0
\(951\) 1.69118e7 0.606373
\(952\) 8.68959e6 0.310747
\(953\) −4.53118e7 −1.61614 −0.808071 0.589085i \(-0.799488\pi\)
−0.808071 + 0.589085i \(0.799488\pi\)
\(954\) −3.27022e6 −0.116334
\(955\) 0 0
\(956\) 1.83601e7 0.649725
\(957\) −3.64424e6 −0.128626
\(958\) 2.47269e6 0.0870476
\(959\) −2.21103e6 −0.0776333
\(960\) 0 0
\(961\) 1.29978e7 0.454005
\(962\) −1.20140e6 −0.0418552
\(963\) 7.36528e6 0.255931
\(964\) 1.31184e7 0.454661
\(965\) 0 0
\(966\) −1.39094e6 −0.0479583
\(967\) 3.52998e7 1.21396 0.606982 0.794715i \(-0.292380\pi\)
0.606982 + 0.794715i \(0.292380\pi\)
\(968\) −1.17944e6 −0.0404565
\(969\) −1.01718e7 −0.348008
\(970\) 0 0
\(971\) −2.90825e7 −0.989883 −0.494941 0.868926i \(-0.664810\pi\)
−0.494941 + 0.868926i \(0.664810\pi\)
\(972\) 1.79093e6 0.0608014
\(973\) −3.30181e7 −1.11807
\(974\) −7.28033e6 −0.245897
\(975\) 0 0
\(976\) −3.09210e6 −0.103903
\(977\) 2.59538e7 0.869890 0.434945 0.900457i \(-0.356768\pi\)
0.434945 + 0.900457i \(0.356768\pi\)
\(978\) 243414. 0.00813764
\(979\) 2.95357e6 0.0984898
\(980\) 0 0
\(981\) 3.80441e6 0.126216
\(982\) −3.13273e6 −0.103668
\(983\) −2.55539e7 −0.843477 −0.421738 0.906718i \(-0.638580\pi\)
−0.421738 + 0.906718i \(0.638580\pi\)
\(984\) 1.33896e7 0.440837
\(985\) 0 0
\(986\) 5.08390e6 0.166535
\(987\) −1.40932e7 −0.460488
\(988\) 2.05107e6 0.0668479
\(989\) 2.50906e7 0.815682
\(990\) 0 0
\(991\) −5.48265e7 −1.77340 −0.886699 0.462346i \(-0.847008\pi\)
−0.886699 + 0.462346i \(0.847008\pi\)
\(992\) 2.38568e7 0.769722
\(993\) −1.75086e7 −0.563479
\(994\) 4.88976e6 0.156972
\(995\) 0 0
\(996\) −4.70418e6 −0.150257
\(997\) −2.30540e7 −0.734527 −0.367264 0.930117i \(-0.619705\pi\)
−0.367264 + 0.930117i \(0.619705\pi\)
\(998\) 8.45155e6 0.268603
\(999\) 9.63474e6 0.305440
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.q.1.4 yes 8
5.4 even 2 825.6.a.p.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.p.1.5 8 5.4 even 2
825.6.a.q.1.4 yes 8 1.1 even 1 trivial