Properties

Label 825.6.a.m.1.7
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - 2x^{6} - 152x^{5} + 358x^{4} + 5771x^{3} - 13444x^{2} - 51316x + 92576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-10.2074\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.20737 q^{2} +9.00000 q^{3} +52.7757 q^{4} +82.8663 q^{6} -97.7871 q^{7} +191.289 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+9.20737 q^{2} +9.00000 q^{3} +52.7757 q^{4} +82.8663 q^{6} -97.7871 q^{7} +191.289 q^{8} +81.0000 q^{9} +121.000 q^{11} +474.981 q^{12} -490.118 q^{13} -900.362 q^{14} +72.4504 q^{16} -881.457 q^{17} +745.797 q^{18} +34.4568 q^{19} -880.084 q^{21} +1114.09 q^{22} -2905.50 q^{23} +1721.60 q^{24} -4512.70 q^{26} +729.000 q^{27} -5160.78 q^{28} -1410.45 q^{29} -2535.92 q^{31} -5454.18 q^{32} +1089.00 q^{33} -8115.90 q^{34} +4274.83 q^{36} +6262.31 q^{37} +317.257 q^{38} -4411.06 q^{39} -18279.4 q^{41} -8103.26 q^{42} +14176.4 q^{43} +6385.86 q^{44} -26752.0 q^{46} -7815.36 q^{47} +652.054 q^{48} -7244.68 q^{49} -7933.11 q^{51} -25866.3 q^{52} -4614.88 q^{53} +6712.17 q^{54} -18705.6 q^{56} +310.111 q^{57} -12986.6 q^{58} -1123.78 q^{59} -672.594 q^{61} -23349.2 q^{62} -7920.76 q^{63} -52537.1 q^{64} +10026.8 q^{66} +29266.6 q^{67} -46519.5 q^{68} -26149.5 q^{69} -9699.67 q^{71} +15494.4 q^{72} +40302.7 q^{73} +57659.5 q^{74} +1818.48 q^{76} -11832.2 q^{77} -40614.3 q^{78} -72842.2 q^{79} +6561.00 q^{81} -168305. q^{82} +100283. q^{83} -46447.0 q^{84} +130527. q^{86} -12694.1 q^{87} +23146.0 q^{88} -30687.7 q^{89} +47927.2 q^{91} -153340. q^{92} -22823.3 q^{93} -71958.9 q^{94} -49087.6 q^{96} -34455.3 q^{97} -66704.5 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9} + 847 q^{11} + 855 q^{12} + 113 q^{13} - 1212 q^{14} + 499 q^{16} - 1030 q^{17} - 729 q^{18} - 3803 q^{19} - 585 q^{21} - 1089 q^{22} - 514 q^{23} - 1863 q^{24} - 12111 q^{26} + 5103 q^{27} + 342 q^{28} - 2698 q^{29} - 17233 q^{31} - 9943 q^{32} + 7623 q^{33} + 4090 q^{34} + 7695 q^{36} - 23182 q^{37} + 11943 q^{38} + 1017 q^{39} - 16158 q^{41} - 10908 q^{42} + 4249 q^{43} + 11495 q^{44} - 28769 q^{46} - 7580 q^{47} + 4491 q^{48} + 37140 q^{49} - 9270 q^{51} + 23887 q^{52} - 20574 q^{53} - 6561 q^{54} - 73276 q^{56} - 34227 q^{57} - 8733 q^{58} - 364 q^{59} - 28127 q^{61} + 71917 q^{62} - 5265 q^{63} + 43379 q^{64} - 9801 q^{66} + 21493 q^{67} - 160660 q^{68} - 4626 q^{69} - 177084 q^{71} - 16767 q^{72} - 78670 q^{73} + 196750 q^{74} - 32701 q^{76} - 7865 q^{77} - 108999 q^{78} - 187432 q^{79} + 45927 q^{81} - 179552 q^{82} + 44592 q^{83} + 3078 q^{84} - 110433 q^{86} - 24282 q^{87} - 25047 q^{88} - 151168 q^{89} - 230153 q^{91} + 44767 q^{92} - 155097 q^{93} + 54040 q^{94} - 89487 q^{96} + 55589 q^{97} - 478341 q^{98} + 68607 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.20737 1.62765 0.813824 0.581111i \(-0.197382\pi\)
0.813824 + 0.581111i \(0.197382\pi\)
\(3\) 9.00000 0.577350
\(4\) 52.7757 1.64924
\(5\) 0 0
\(6\) 82.8663 0.939723
\(7\) −97.7871 −0.754287 −0.377143 0.926155i \(-0.623094\pi\)
−0.377143 + 0.926155i \(0.623094\pi\)
\(8\) 191.289 1.05673
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 474.981 0.952189
\(13\) −490.118 −0.804345 −0.402172 0.915564i \(-0.631745\pi\)
−0.402172 + 0.915564i \(0.631745\pi\)
\(14\) −900.362 −1.22771
\(15\) 0 0
\(16\) 72.4504 0.0707524
\(17\) −881.457 −0.739739 −0.369870 0.929084i \(-0.620598\pi\)
−0.369870 + 0.929084i \(0.620598\pi\)
\(18\) 745.797 0.542550
\(19\) 34.4568 0.0218973 0.0109487 0.999940i \(-0.496515\pi\)
0.0109487 + 0.999940i \(0.496515\pi\)
\(20\) 0 0
\(21\) −880.084 −0.435488
\(22\) 1114.09 0.490755
\(23\) −2905.50 −1.14525 −0.572627 0.819816i \(-0.694076\pi\)
−0.572627 + 0.819816i \(0.694076\pi\)
\(24\) 1721.60 0.610106
\(25\) 0 0
\(26\) −4512.70 −1.30919
\(27\) 729.000 0.192450
\(28\) −5160.78 −1.24400
\(29\) −1410.45 −0.311432 −0.155716 0.987802i \(-0.549768\pi\)
−0.155716 + 0.987802i \(0.549768\pi\)
\(30\) 0 0
\(31\) −2535.92 −0.473949 −0.236975 0.971516i \(-0.576156\pi\)
−0.236975 + 0.971516i \(0.576156\pi\)
\(32\) −5454.18 −0.941574
\(33\) 1089.00 0.174078
\(34\) −8115.90 −1.20404
\(35\) 0 0
\(36\) 4274.83 0.549747
\(37\) 6262.31 0.752022 0.376011 0.926615i \(-0.377295\pi\)
0.376011 + 0.926615i \(0.377295\pi\)
\(38\) 317.257 0.0356412
\(39\) −4411.06 −0.464389
\(40\) 0 0
\(41\) −18279.4 −1.69825 −0.849126 0.528190i \(-0.822871\pi\)
−0.849126 + 0.528190i \(0.822871\pi\)
\(42\) −8103.26 −0.708821
\(43\) 14176.4 1.16922 0.584608 0.811316i \(-0.301248\pi\)
0.584608 + 0.811316i \(0.301248\pi\)
\(44\) 6385.86 0.497265
\(45\) 0 0
\(46\) −26752.0 −1.86407
\(47\) −7815.36 −0.516065 −0.258032 0.966136i \(-0.583074\pi\)
−0.258032 + 0.966136i \(0.583074\pi\)
\(48\) 652.054 0.0408489
\(49\) −7244.68 −0.431051
\(50\) 0 0
\(51\) −7933.11 −0.427089
\(52\) −25866.3 −1.32656
\(53\) −4614.88 −0.225668 −0.112834 0.993614i \(-0.535993\pi\)
−0.112834 + 0.993614i \(0.535993\pi\)
\(54\) 6712.17 0.313241
\(55\) 0 0
\(56\) −18705.6 −0.797081
\(57\) 310.111 0.0126424
\(58\) −12986.6 −0.506902
\(59\) −1123.78 −0.0420293 −0.0210147 0.999779i \(-0.506690\pi\)
−0.0210147 + 0.999779i \(0.506690\pi\)
\(60\) 0 0
\(61\) −672.594 −0.0231435 −0.0115717 0.999933i \(-0.503683\pi\)
−0.0115717 + 0.999933i \(0.503683\pi\)
\(62\) −23349.2 −0.771423
\(63\) −7920.76 −0.251429
\(64\) −52537.1 −1.60330
\(65\) 0 0
\(66\) 10026.8 0.283337
\(67\) 29266.6 0.796499 0.398249 0.917277i \(-0.369618\pi\)
0.398249 + 0.917277i \(0.369618\pi\)
\(68\) −46519.5 −1.22001
\(69\) −26149.5 −0.661213
\(70\) 0 0
\(71\) −9699.67 −0.228355 −0.114178 0.993460i \(-0.536423\pi\)
−0.114178 + 0.993460i \(0.536423\pi\)
\(72\) 15494.4 0.352245
\(73\) 40302.7 0.885170 0.442585 0.896726i \(-0.354061\pi\)
0.442585 + 0.896726i \(0.354061\pi\)
\(74\) 57659.5 1.22403
\(75\) 0 0
\(76\) 1818.48 0.0361140
\(77\) −11832.2 −0.227426
\(78\) −40614.3 −0.755861
\(79\) −72842.2 −1.31315 −0.656576 0.754260i \(-0.727996\pi\)
−0.656576 + 0.754260i \(0.727996\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −168305. −2.76416
\(83\) 100283. 1.59784 0.798918 0.601440i \(-0.205406\pi\)
0.798918 + 0.601440i \(0.205406\pi\)
\(84\) −46447.0 −0.718224
\(85\) 0 0
\(86\) 130527. 1.90307
\(87\) −12694.1 −0.179805
\(88\) 23146.0 0.318617
\(89\) −30687.7 −0.410667 −0.205333 0.978692i \(-0.565828\pi\)
−0.205333 + 0.978692i \(0.565828\pi\)
\(90\) 0 0
\(91\) 47927.2 0.606707
\(92\) −153340. −1.88880
\(93\) −22823.3 −0.273635
\(94\) −71958.9 −0.839972
\(95\) 0 0
\(96\) −49087.6 −0.543618
\(97\) −34455.3 −0.371815 −0.185907 0.982567i \(-0.559522\pi\)
−0.185907 + 0.982567i \(0.559522\pi\)
\(98\) −66704.5 −0.701600
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 77516.8 0.756123 0.378062 0.925780i \(-0.376591\pi\)
0.378062 + 0.925780i \(0.376591\pi\)
\(102\) −73043.1 −0.695150
\(103\) −96973.6 −0.900659 −0.450330 0.892862i \(-0.648694\pi\)
−0.450330 + 0.892862i \(0.648694\pi\)
\(104\) −93754.3 −0.849979
\(105\) 0 0
\(106\) −42490.9 −0.367309
\(107\) −175905. −1.48531 −0.742656 0.669673i \(-0.766434\pi\)
−0.742656 + 0.669673i \(0.766434\pi\)
\(108\) 38473.5 0.317396
\(109\) −124853. −1.00654 −0.503271 0.864128i \(-0.667870\pi\)
−0.503271 + 0.864128i \(0.667870\pi\)
\(110\) 0 0
\(111\) 56360.8 0.434180
\(112\) −7084.72 −0.0533676
\(113\) −145912. −1.07496 −0.537481 0.843276i \(-0.680624\pi\)
−0.537481 + 0.843276i \(0.680624\pi\)
\(114\) 2855.31 0.0205774
\(115\) 0 0
\(116\) −74437.5 −0.513626
\(117\) −39699.5 −0.268115
\(118\) −10347.1 −0.0684089
\(119\) 86195.1 0.557976
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −6192.83 −0.0376695
\(123\) −164515. −0.980487
\(124\) −133835. −0.781656
\(125\) 0 0
\(126\) −72929.3 −0.409238
\(127\) −38194.3 −0.210131 −0.105065 0.994465i \(-0.533505\pi\)
−0.105065 + 0.994465i \(0.533505\pi\)
\(128\) −309195. −1.66804
\(129\) 127588. 0.675047
\(130\) 0 0
\(131\) 171732. 0.874327 0.437163 0.899382i \(-0.355983\pi\)
0.437163 + 0.899382i \(0.355983\pi\)
\(132\) 57472.7 0.287096
\(133\) −3369.43 −0.0165169
\(134\) 269468. 1.29642
\(135\) 0 0
\(136\) −168613. −0.781708
\(137\) 246624. 1.12262 0.561312 0.827604i \(-0.310297\pi\)
0.561312 + 0.827604i \(0.310297\pi\)
\(138\) −240768. −1.07622
\(139\) −13849.3 −0.0607982 −0.0303991 0.999538i \(-0.509678\pi\)
−0.0303991 + 0.999538i \(0.509678\pi\)
\(140\) 0 0
\(141\) −70338.2 −0.297950
\(142\) −89308.5 −0.371682
\(143\) −59304.3 −0.242519
\(144\) 5868.48 0.0235841
\(145\) 0 0
\(146\) 371082. 1.44075
\(147\) −65202.1 −0.248868
\(148\) 330498. 1.24026
\(149\) −195654. −0.721975 −0.360987 0.932571i \(-0.617560\pi\)
−0.360987 + 0.932571i \(0.617560\pi\)
\(150\) 0 0
\(151\) 205268. 0.732622 0.366311 0.930493i \(-0.380621\pi\)
0.366311 + 0.930493i \(0.380621\pi\)
\(152\) 6591.23 0.0231397
\(153\) −71398.0 −0.246580
\(154\) −108944. −0.370170
\(155\) 0 0
\(156\) −232797. −0.765888
\(157\) −89862.7 −0.290958 −0.145479 0.989361i \(-0.546472\pi\)
−0.145479 + 0.989361i \(0.546472\pi\)
\(158\) −670685. −2.13735
\(159\) −41533.9 −0.130290
\(160\) 0 0
\(161\) 284121. 0.863850
\(162\) 60409.6 0.180850
\(163\) 27924.3 0.0823214 0.0411607 0.999153i \(-0.486894\pi\)
0.0411607 + 0.999153i \(0.486894\pi\)
\(164\) −964708. −2.80083
\(165\) 0 0
\(166\) 923343. 2.60071
\(167\) 254410. 0.705901 0.352950 0.935642i \(-0.385178\pi\)
0.352950 + 0.935642i \(0.385178\pi\)
\(168\) −168351. −0.460195
\(169\) −131077. −0.353030
\(170\) 0 0
\(171\) 2791.00 0.00729911
\(172\) 748169. 1.92832
\(173\) −150277. −0.381748 −0.190874 0.981615i \(-0.561132\pi\)
−0.190874 + 0.981615i \(0.561132\pi\)
\(174\) −116879. −0.292660
\(175\) 0 0
\(176\) 8766.50 0.0213326
\(177\) −10114.0 −0.0242656
\(178\) −282553. −0.668421
\(179\) −50602.0 −0.118042 −0.0590208 0.998257i \(-0.518798\pi\)
−0.0590208 + 0.998257i \(0.518798\pi\)
\(180\) 0 0
\(181\) 569456. 1.29200 0.646002 0.763336i \(-0.276440\pi\)
0.646002 + 0.763336i \(0.276440\pi\)
\(182\) 441284. 0.987505
\(183\) −6053.35 −0.0133619
\(184\) −555792. −1.21023
\(185\) 0 0
\(186\) −210143. −0.445381
\(187\) −106656. −0.223040
\(188\) −412461. −0.851115
\(189\) −71286.8 −0.145163
\(190\) 0 0
\(191\) 379883. 0.753471 0.376735 0.926321i \(-0.377047\pi\)
0.376735 + 0.926321i \(0.377047\pi\)
\(192\) −472834. −0.925668
\(193\) 637445. 1.23183 0.615913 0.787814i \(-0.288787\pi\)
0.615913 + 0.787814i \(0.288787\pi\)
\(194\) −317243. −0.605184
\(195\) 0 0
\(196\) −382343. −0.710907
\(197\) −358863. −0.658814 −0.329407 0.944188i \(-0.606849\pi\)
−0.329407 + 0.944188i \(0.606849\pi\)
\(198\) 90241.4 0.163585
\(199\) −287892. −0.515344 −0.257672 0.966232i \(-0.582955\pi\)
−0.257672 + 0.966232i \(0.582955\pi\)
\(200\) 0 0
\(201\) 263399. 0.459859
\(202\) 713726. 1.23070
\(203\) 137924. 0.234909
\(204\) −418675. −0.704372
\(205\) 0 0
\(206\) −892872. −1.46596
\(207\) −235346. −0.381751
\(208\) −35509.2 −0.0569093
\(209\) 4169.28 0.00660230
\(210\) 0 0
\(211\) 44672.7 0.0690774 0.0345387 0.999403i \(-0.489004\pi\)
0.0345387 + 0.999403i \(0.489004\pi\)
\(212\) −243553. −0.372181
\(213\) −87297.0 −0.131841
\(214\) −1.61962e6 −2.41757
\(215\) 0 0
\(216\) 139450. 0.203369
\(217\) 247981. 0.357494
\(218\) −1.14957e6 −1.63830
\(219\) 362724. 0.511053
\(220\) 0 0
\(221\) 432018. 0.595005
\(222\) 518935. 0.706693
\(223\) 909913. 1.22529 0.612643 0.790359i \(-0.290106\pi\)
0.612643 + 0.790359i \(0.290106\pi\)
\(224\) 533349. 0.710217
\(225\) 0 0
\(226\) −1.34346e6 −1.74966
\(227\) 1.15501e6 1.48772 0.743859 0.668336i \(-0.232993\pi\)
0.743859 + 0.668336i \(0.232993\pi\)
\(228\) 16366.3 0.0208504
\(229\) −1.14608e6 −1.44420 −0.722099 0.691789i \(-0.756823\pi\)
−0.722099 + 0.691789i \(0.756823\pi\)
\(230\) 0 0
\(231\) −106490. −0.131304
\(232\) −269804. −0.329101
\(233\) 237894. 0.287074 0.143537 0.989645i \(-0.454152\pi\)
0.143537 + 0.989645i \(0.454152\pi\)
\(234\) −365528. −0.436397
\(235\) 0 0
\(236\) −59308.4 −0.0693164
\(237\) −655579. −0.758149
\(238\) 793630. 0.908188
\(239\) 38858.4 0.0440038 0.0220019 0.999758i \(-0.492996\pi\)
0.0220019 + 0.999758i \(0.492996\pi\)
\(240\) 0 0
\(241\) 73826.4 0.0818784 0.0409392 0.999162i \(-0.486965\pi\)
0.0409392 + 0.999162i \(0.486965\pi\)
\(242\) 134805. 0.147968
\(243\) 59049.0 0.0641500
\(244\) −35496.6 −0.0381692
\(245\) 0 0
\(246\) −1.51475e6 −1.59589
\(247\) −16887.9 −0.0176130
\(248\) −485095. −0.500839
\(249\) 902547. 0.922511
\(250\) 0 0
\(251\) −1.67060e6 −1.67374 −0.836870 0.547402i \(-0.815617\pi\)
−0.836870 + 0.547402i \(0.815617\pi\)
\(252\) −418023. −0.414667
\(253\) −351566. −0.345307
\(254\) −351669. −0.342019
\(255\) 0 0
\(256\) −1.16568e6 −1.11168
\(257\) 597714. 0.564496 0.282248 0.959342i \(-0.408920\pi\)
0.282248 + 0.959342i \(0.408920\pi\)
\(258\) 1.17475e6 1.09874
\(259\) −612374. −0.567240
\(260\) 0 0
\(261\) −114247. −0.103811
\(262\) 1.58120e6 1.42310
\(263\) −1.37719e6 −1.22773 −0.613867 0.789410i \(-0.710387\pi\)
−0.613867 + 0.789410i \(0.710387\pi\)
\(264\) 208314. 0.183954
\(265\) 0 0
\(266\) −31023.6 −0.0268837
\(267\) −276190. −0.237099
\(268\) 1.54456e6 1.31362
\(269\) −161719. −0.136264 −0.0681320 0.997676i \(-0.521704\pi\)
−0.0681320 + 0.997676i \(0.521704\pi\)
\(270\) 0 0
\(271\) 241445. 0.199708 0.0998538 0.995002i \(-0.468162\pi\)
0.0998538 + 0.995002i \(0.468162\pi\)
\(272\) −63861.9 −0.0523383
\(273\) 431345. 0.350282
\(274\) 2.27076e6 1.82724
\(275\) 0 0
\(276\) −1.38006e6 −1.09050
\(277\) 513505. 0.402110 0.201055 0.979580i \(-0.435563\pi\)
0.201055 + 0.979580i \(0.435563\pi\)
\(278\) −127516. −0.0989581
\(279\) −205410. −0.157983
\(280\) 0 0
\(281\) −39961.2 −0.0301907 −0.0150953 0.999886i \(-0.504805\pi\)
−0.0150953 + 0.999886i \(0.504805\pi\)
\(282\) −647630. −0.484958
\(283\) 1.66384e6 1.23494 0.617470 0.786594i \(-0.288158\pi\)
0.617470 + 0.786594i \(0.288158\pi\)
\(284\) −511907. −0.376613
\(285\) 0 0
\(286\) −546036. −0.394736
\(287\) 1.78749e6 1.28097
\(288\) −441789. −0.313858
\(289\) −642891. −0.452786
\(290\) 0 0
\(291\) −310098. −0.214667
\(292\) 2.12700e6 1.45986
\(293\) −1.08636e6 −0.739276 −0.369638 0.929176i \(-0.620518\pi\)
−0.369638 + 0.929176i \(0.620518\pi\)
\(294\) −600340. −0.405069
\(295\) 0 0
\(296\) 1.19791e6 0.794688
\(297\) 88209.0 0.0580259
\(298\) −1.80145e6 −1.17512
\(299\) 1.42404e6 0.921179
\(300\) 0 0
\(301\) −1.38627e6 −0.881924
\(302\) 1.88998e6 1.19245
\(303\) 697652. 0.436548
\(304\) 2496.41 0.00154929
\(305\) 0 0
\(306\) −657388. −0.401345
\(307\) −2.25020e6 −1.36262 −0.681310 0.731995i \(-0.738589\pi\)
−0.681310 + 0.731995i \(0.738589\pi\)
\(308\) −624454. −0.375080
\(309\) −872762. −0.519996
\(310\) 0 0
\(311\) 1.57755e6 0.924871 0.462435 0.886653i \(-0.346976\pi\)
0.462435 + 0.886653i \(0.346976\pi\)
\(312\) −843789. −0.490735
\(313\) 2.21406e6 1.27741 0.638703 0.769453i \(-0.279471\pi\)
0.638703 + 0.769453i \(0.279471\pi\)
\(314\) −827400. −0.473578
\(315\) 0 0
\(316\) −3.84429e6 −2.16570
\(317\) −2.33756e6 −1.30652 −0.653258 0.757136i \(-0.726598\pi\)
−0.653258 + 0.757136i \(0.726598\pi\)
\(318\) −382418. −0.212066
\(319\) −170665. −0.0939003
\(320\) 0 0
\(321\) −1.58314e6 −0.857545
\(322\) 2.61601e6 1.40604
\(323\) −30372.2 −0.0161983
\(324\) 346261. 0.183249
\(325\) 0 0
\(326\) 257109. 0.133990
\(327\) −1.12368e6 −0.581128
\(328\) −3.49665e6 −1.79460
\(329\) 764241. 0.389261
\(330\) 0 0
\(331\) −1.41311e6 −0.708934 −0.354467 0.935069i \(-0.615338\pi\)
−0.354467 + 0.935069i \(0.615338\pi\)
\(332\) 5.29250e6 2.63521
\(333\) 507247. 0.250674
\(334\) 2.34245e6 1.14896
\(335\) 0 0
\(336\) −63762.5 −0.0308118
\(337\) 1.96985e6 0.944841 0.472420 0.881373i \(-0.343380\pi\)
0.472420 + 0.881373i \(0.343380\pi\)
\(338\) −1.20688e6 −0.574608
\(339\) −1.31320e6 −0.620630
\(340\) 0 0
\(341\) −306847. −0.142901
\(342\) 25697.8 0.0118804
\(343\) 2.35194e6 1.07942
\(344\) 2.71179e6 1.23555
\(345\) 0 0
\(346\) −1.38366e6 −0.621352
\(347\) 2.86899e6 1.27910 0.639551 0.768749i \(-0.279120\pi\)
0.639551 + 0.768749i \(0.279120\pi\)
\(348\) −669938. −0.296542
\(349\) 730428. 0.321006 0.160503 0.987035i \(-0.448688\pi\)
0.160503 + 0.987035i \(0.448688\pi\)
\(350\) 0 0
\(351\) −357296. −0.154796
\(352\) −659956. −0.283895
\(353\) 641407. 0.273966 0.136983 0.990573i \(-0.456259\pi\)
0.136983 + 0.990573i \(0.456259\pi\)
\(354\) −93123.7 −0.0394959
\(355\) 0 0
\(356\) −1.61957e6 −0.677288
\(357\) 775756. 0.322147
\(358\) −465911. −0.192130
\(359\) −3.57087e6 −1.46230 −0.731152 0.682214i \(-0.761017\pi\)
−0.731152 + 0.682214i \(0.761017\pi\)
\(360\) 0 0
\(361\) −2.47491e6 −0.999521
\(362\) 5.24319e6 2.10293
\(363\) 131769. 0.0524864
\(364\) 2.52939e6 1.00060
\(365\) 0 0
\(366\) −55735.4 −0.0217485
\(367\) −1.70607e6 −0.661198 −0.330599 0.943771i \(-0.607251\pi\)
−0.330599 + 0.943771i \(0.607251\pi\)
\(368\) −210505. −0.0810294
\(369\) −1.48063e6 −0.566084
\(370\) 0 0
\(371\) 451276. 0.170219
\(372\) −1.20452e6 −0.451289
\(373\) −4.87852e6 −1.81558 −0.907791 0.419424i \(-0.862232\pi\)
−0.907791 + 0.419424i \(0.862232\pi\)
\(374\) −982024. −0.363030
\(375\) 0 0
\(376\) −1.49499e6 −0.545343
\(377\) 691288. 0.250499
\(378\) −656364. −0.236274
\(379\) −2.02923e6 −0.725659 −0.362829 0.931856i \(-0.618189\pi\)
−0.362829 + 0.931856i \(0.618189\pi\)
\(380\) 0 0
\(381\) −343749. −0.121319
\(382\) 3.49772e6 1.22639
\(383\) 3.39494e6 1.18259 0.591296 0.806455i \(-0.298617\pi\)
0.591296 + 0.806455i \(0.298617\pi\)
\(384\) −2.78275e6 −0.963045
\(385\) 0 0
\(386\) 5.86919e6 2.00498
\(387\) 1.14829e6 0.389739
\(388\) −1.81840e6 −0.613211
\(389\) 2.90688e6 0.973986 0.486993 0.873406i \(-0.338094\pi\)
0.486993 + 0.873406i \(0.338094\pi\)
\(390\) 0 0
\(391\) 2.56108e6 0.847190
\(392\) −1.38583e6 −0.455507
\(393\) 1.54559e6 0.504793
\(394\) −3.30418e6 −1.07232
\(395\) 0 0
\(396\) 517254. 0.165755
\(397\) 2.71276e6 0.863845 0.431923 0.901911i \(-0.357835\pi\)
0.431923 + 0.901911i \(0.357835\pi\)
\(398\) −2.65073e6 −0.838798
\(399\) −30324.9 −0.00953602
\(400\) 0 0
\(401\) 4.93699e6 1.53321 0.766605 0.642119i \(-0.221944\pi\)
0.766605 + 0.642119i \(0.221944\pi\)
\(402\) 2.42522e6 0.748488
\(403\) 1.24290e6 0.381219
\(404\) 4.09100e6 1.24703
\(405\) 0 0
\(406\) 1.26992e6 0.382349
\(407\) 757740. 0.226743
\(408\) −1.51752e6 −0.451319
\(409\) 411523. 0.121643 0.0608213 0.998149i \(-0.480628\pi\)
0.0608213 + 0.998149i \(0.480628\pi\)
\(410\) 0 0
\(411\) 2.21962e6 0.648147
\(412\) −5.11785e6 −1.48540
\(413\) 109891. 0.0317022
\(414\) −2.16692e6 −0.621357
\(415\) 0 0
\(416\) 2.67319e6 0.757350
\(417\) −124644. −0.0351018
\(418\) 38388.1 0.0107462
\(419\) −1.06540e6 −0.296469 −0.148235 0.988952i \(-0.547359\pi\)
−0.148235 + 0.988952i \(0.547359\pi\)
\(420\) 0 0
\(421\) −5.18377e6 −1.42541 −0.712706 0.701462i \(-0.752531\pi\)
−0.712706 + 0.701462i \(0.752531\pi\)
\(422\) 411318. 0.112434
\(423\) −633044. −0.172022
\(424\) −882778. −0.238472
\(425\) 0 0
\(426\) −803776. −0.214591
\(427\) 65771.1 0.0174568
\(428\) −9.28348e6 −2.44964
\(429\) −533738. −0.140018
\(430\) 0 0
\(431\) 7.59325e6 1.96895 0.984474 0.175530i \(-0.0561639\pi\)
0.984474 + 0.175530i \(0.0561639\pi\)
\(432\) 52816.4 0.0136163
\(433\) −6.48244e6 −1.66157 −0.830786 0.556592i \(-0.812109\pi\)
−0.830786 + 0.556592i \(0.812109\pi\)
\(434\) 2.28325e6 0.581874
\(435\) 0 0
\(436\) −6.58919e6 −1.66003
\(437\) −100114. −0.0250780
\(438\) 3.33974e6 0.831815
\(439\) 2.75964e6 0.683425 0.341713 0.939804i \(-0.388993\pi\)
0.341713 + 0.939804i \(0.388993\pi\)
\(440\) 0 0
\(441\) −586819. −0.143684
\(442\) 3.97775e6 0.968460
\(443\) 5.64065e6 1.36559 0.682795 0.730610i \(-0.260764\pi\)
0.682795 + 0.730610i \(0.260764\pi\)
\(444\) 2.97448e6 0.716067
\(445\) 0 0
\(446\) 8.37791e6 1.99434
\(447\) −1.76088e6 −0.416832
\(448\) 5.13745e6 1.20935
\(449\) 1.73569e6 0.406309 0.203154 0.979147i \(-0.434881\pi\)
0.203154 + 0.979147i \(0.434881\pi\)
\(450\) 0 0
\(451\) −2.21181e6 −0.512042
\(452\) −7.70058e6 −1.77287
\(453\) 1.84742e6 0.422979
\(454\) 1.06346e7 2.42148
\(455\) 0 0
\(456\) 59321.0 0.0133597
\(457\) 2.30513e6 0.516303 0.258151 0.966105i \(-0.416887\pi\)
0.258151 + 0.966105i \(0.416887\pi\)
\(458\) −1.05524e7 −2.35065
\(459\) −642582. −0.142363
\(460\) 0 0
\(461\) 5.17550e6 1.13423 0.567113 0.823640i \(-0.308060\pi\)
0.567113 + 0.823640i \(0.308060\pi\)
\(462\) −980494. −0.213718
\(463\) 4.51839e6 0.979560 0.489780 0.871846i \(-0.337077\pi\)
0.489780 + 0.871846i \(0.337077\pi\)
\(464\) −102188. −0.0220345
\(465\) 0 0
\(466\) 2.19038e6 0.467256
\(467\) −8.84437e6 −1.87661 −0.938306 0.345806i \(-0.887606\pi\)
−0.938306 + 0.345806i \(0.887606\pi\)
\(468\) −2.09517e6 −0.442186
\(469\) −2.86190e6 −0.600788
\(470\) 0 0
\(471\) −808765. −0.167985
\(472\) −214968. −0.0444138
\(473\) 1.71534e6 0.352532
\(474\) −6.03616e6 −1.23400
\(475\) 0 0
\(476\) 4.54901e6 0.920236
\(477\) −373805. −0.0752228
\(478\) 357784. 0.0716227
\(479\) −7.60699e6 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(480\) 0 0
\(481\) −3.06927e6 −0.604885
\(482\) 679747. 0.133269
\(483\) 2.55709e6 0.498744
\(484\) 772689. 0.149931
\(485\) 0 0
\(486\) 543686. 0.104414
\(487\) −4.53589e6 −0.866642 −0.433321 0.901240i \(-0.642658\pi\)
−0.433321 + 0.901240i \(0.642658\pi\)
\(488\) −128660. −0.0244565
\(489\) 251318. 0.0475283
\(490\) 0 0
\(491\) 2.60325e6 0.487317 0.243658 0.969861i \(-0.421652\pi\)
0.243658 + 0.969861i \(0.421652\pi\)
\(492\) −8.68237e6 −1.61706
\(493\) 1.24325e6 0.230379
\(494\) −155493. −0.0286678
\(495\) 0 0
\(496\) −183729. −0.0335330
\(497\) 948503. 0.172245
\(498\) 8.31008e6 1.50152
\(499\) −5.53534e6 −0.995160 −0.497580 0.867418i \(-0.665778\pi\)
−0.497580 + 0.867418i \(0.665778\pi\)
\(500\) 0 0
\(501\) 2.28969e6 0.407552
\(502\) −1.53818e7 −2.72426
\(503\) −1.85710e6 −0.327277 −0.163638 0.986520i \(-0.552323\pi\)
−0.163638 + 0.986520i \(0.552323\pi\)
\(504\) −1.51516e6 −0.265694
\(505\) 0 0
\(506\) −3.23700e6 −0.562039
\(507\) −1.17970e6 −0.203822
\(508\) −2.01573e6 −0.346556
\(509\) −3.23529e6 −0.553502 −0.276751 0.960942i \(-0.589258\pi\)
−0.276751 + 0.960942i \(0.589258\pi\)
\(510\) 0 0
\(511\) −3.94108e6 −0.667672
\(512\) −838646. −0.141385
\(513\) 25119.0 0.00421414
\(514\) 5.50337e6 0.918801
\(515\) 0 0
\(516\) 6.73352e6 1.11331
\(517\) −945658. −0.155599
\(518\) −5.63835e6 −0.923268
\(519\) −1.35249e6 −0.220402
\(520\) 0 0
\(521\) −1.26541e6 −0.204237 −0.102119 0.994772i \(-0.532562\pi\)
−0.102119 + 0.994772i \(0.532562\pi\)
\(522\) −1.05191e6 −0.168967
\(523\) −2.77343e6 −0.443366 −0.221683 0.975119i \(-0.571155\pi\)
−0.221683 + 0.975119i \(0.571155\pi\)
\(524\) 9.06329e6 1.44197
\(525\) 0 0
\(526\) −1.26803e7 −1.99832
\(527\) 2.23531e6 0.350599
\(528\) 78898.5 0.0123164
\(529\) 2.00561e6 0.311607
\(530\) 0 0
\(531\) −91026.4 −0.0140098
\(532\) −177824. −0.0272403
\(533\) 8.95906e6 1.36598
\(534\) −2.54298e6 −0.385913
\(535\) 0 0
\(536\) 5.59839e6 0.841688
\(537\) −455418. −0.0681513
\(538\) −1.48901e6 −0.221790
\(539\) −876606. −0.129967
\(540\) 0 0
\(541\) 1.16086e7 1.70524 0.852619 0.522533i \(-0.175013\pi\)
0.852619 + 0.522533i \(0.175013\pi\)
\(542\) 2.22307e6 0.325054
\(543\) 5.12510e6 0.745938
\(544\) 4.80763e6 0.696520
\(545\) 0 0
\(546\) 3.97155e6 0.570136
\(547\) −1.21550e7 −1.73695 −0.868475 0.495733i \(-0.834899\pi\)
−0.868475 + 0.495733i \(0.834899\pi\)
\(548\) 1.30158e7 1.85148
\(549\) −54480.2 −0.00771449
\(550\) 0 0
\(551\) −48599.7 −0.00681953
\(552\) −5.00213e6 −0.698726
\(553\) 7.12302e6 0.990493
\(554\) 4.72803e6 0.654494
\(555\) 0 0
\(556\) −730906. −0.100271
\(557\) −1.17133e7 −1.59971 −0.799856 0.600191i \(-0.795091\pi\)
−0.799856 + 0.600191i \(0.795091\pi\)
\(558\) −1.89128e6 −0.257141
\(559\) −6.94810e6 −0.940452
\(560\) 0 0
\(561\) −959907. −0.128772
\(562\) −367938. −0.0491398
\(563\) −2.89057e6 −0.384337 −0.192168 0.981362i \(-0.561552\pi\)
−0.192168 + 0.981362i \(0.561552\pi\)
\(564\) −3.71215e6 −0.491391
\(565\) 0 0
\(566\) 1.53196e7 2.01005
\(567\) −641581. −0.0838096
\(568\) −1.85544e6 −0.241311
\(569\) −8.57114e6 −1.10983 −0.554917 0.831906i \(-0.687250\pi\)
−0.554917 + 0.831906i \(0.687250\pi\)
\(570\) 0 0
\(571\) 2.49555e6 0.320315 0.160157 0.987092i \(-0.448800\pi\)
0.160157 + 0.987092i \(0.448800\pi\)
\(572\) −3.12982e6 −0.399972
\(573\) 3.41895e6 0.435016
\(574\) 1.64581e7 2.08497
\(575\) 0 0
\(576\) −4.25550e6 −0.534435
\(577\) −4.74905e6 −0.593837 −0.296918 0.954903i \(-0.595959\pi\)
−0.296918 + 0.954903i \(0.595959\pi\)
\(578\) −5.91933e6 −0.736976
\(579\) 5.73701e6 0.711195
\(580\) 0 0
\(581\) −9.80638e6 −1.20523
\(582\) −2.85518e6 −0.349403
\(583\) −558401. −0.0680416
\(584\) 7.70948e6 0.935390
\(585\) 0 0
\(586\) −1.00026e7 −1.20328
\(587\) 3.99522e6 0.478570 0.239285 0.970949i \(-0.423087\pi\)
0.239285 + 0.970949i \(0.423087\pi\)
\(588\) −3.44109e6 −0.410442
\(589\) −87379.9 −0.0103782
\(590\) 0 0
\(591\) −3.22977e6 −0.380367
\(592\) 453707. 0.0532073
\(593\) 8.72437e6 1.01882 0.509410 0.860524i \(-0.329864\pi\)
0.509410 + 0.860524i \(0.329864\pi\)
\(594\) 812173. 0.0944457
\(595\) 0 0
\(596\) −1.03257e7 −1.19071
\(597\) −2.59103e6 −0.297534
\(598\) 1.31117e7 1.49936
\(599\) −9.05896e6 −1.03160 −0.515800 0.856709i \(-0.672505\pi\)
−0.515800 + 0.856709i \(0.672505\pi\)
\(600\) 0 0
\(601\) 947925. 0.107050 0.0535251 0.998567i \(-0.482954\pi\)
0.0535251 + 0.998567i \(0.482954\pi\)
\(602\) −1.27639e7 −1.43546
\(603\) 2.37059e6 0.265500
\(604\) 1.08332e7 1.20827
\(605\) 0 0
\(606\) 6.42354e6 0.710547
\(607\) 9.91130e6 1.09184 0.545920 0.837837i \(-0.316180\pi\)
0.545920 + 0.837837i \(0.316180\pi\)
\(608\) −187934. −0.0206180
\(609\) 1.24132e6 0.135625
\(610\) 0 0
\(611\) 3.83045e6 0.415094
\(612\) −3.76808e6 −0.406669
\(613\) 1.74657e7 1.87730 0.938652 0.344865i \(-0.112075\pi\)
0.938652 + 0.344865i \(0.112075\pi\)
\(614\) −2.07184e7 −2.21787
\(615\) 0 0
\(616\) −2.26338e6 −0.240329
\(617\) 8.70104e6 0.920149 0.460075 0.887880i \(-0.347823\pi\)
0.460075 + 0.887880i \(0.347823\pi\)
\(618\) −8.03585e6 −0.846370
\(619\) −5.31547e6 −0.557590 −0.278795 0.960351i \(-0.589935\pi\)
−0.278795 + 0.960351i \(0.589935\pi\)
\(620\) 0 0
\(621\) −2.11811e6 −0.220404
\(622\) 1.45250e7 1.50536
\(623\) 3.00086e6 0.309761
\(624\) −319583. −0.0328566
\(625\) 0 0
\(626\) 2.03857e7 2.07917
\(627\) 37523.5 0.00381184
\(628\) −4.74257e6 −0.479860
\(629\) −5.51996e6 −0.556300
\(630\) 0 0
\(631\) 7.86646e6 0.786513 0.393257 0.919429i \(-0.371348\pi\)
0.393257 + 0.919429i \(0.371348\pi\)
\(632\) −1.39339e7 −1.38765
\(633\) 402054. 0.0398818
\(634\) −2.15228e7 −2.12655
\(635\) 0 0
\(636\) −2.19198e6 −0.214879
\(637\) 3.55075e6 0.346714
\(638\) −1.57137e6 −0.152837
\(639\) −785673. −0.0761184
\(640\) 0 0
\(641\) −4.86198e6 −0.467377 −0.233689 0.972311i \(-0.575080\pi\)
−0.233689 + 0.972311i \(0.575080\pi\)
\(642\) −1.45766e7 −1.39578
\(643\) 2.34243e6 0.223429 0.111714 0.993740i \(-0.464366\pi\)
0.111714 + 0.993740i \(0.464366\pi\)
\(644\) 1.49947e7 1.42470
\(645\) 0 0
\(646\) −279648. −0.0263652
\(647\) 512871. 0.0481668 0.0240834 0.999710i \(-0.492333\pi\)
0.0240834 + 0.999710i \(0.492333\pi\)
\(648\) 1.25505e6 0.117415
\(649\) −135978. −0.0126723
\(650\) 0 0
\(651\) 2.23183e6 0.206399
\(652\) 1.47372e6 0.135768
\(653\) −4.47741e6 −0.410907 −0.205454 0.978667i \(-0.565867\pi\)
−0.205454 + 0.978667i \(0.565867\pi\)
\(654\) −1.03461e7 −0.945872
\(655\) 0 0
\(656\) −1.32435e6 −0.120155
\(657\) 3.26452e6 0.295057
\(658\) 7.03665e6 0.633580
\(659\) −8.24188e6 −0.739286 −0.369643 0.929174i \(-0.620520\pi\)
−0.369643 + 0.929174i \(0.620520\pi\)
\(660\) 0 0
\(661\) 5.80782e6 0.517023 0.258511 0.966008i \(-0.416768\pi\)
0.258511 + 0.966008i \(0.416768\pi\)
\(662\) −1.30110e7 −1.15389
\(663\) 3.88816e6 0.343527
\(664\) 1.91831e7 1.68849
\(665\) 0 0
\(666\) 4.67042e6 0.408009
\(667\) 4.09807e6 0.356669
\(668\) 1.34267e7 1.16420
\(669\) 8.18922e6 0.707420
\(670\) 0 0
\(671\) −81383.9 −0.00697802
\(672\) 4.80014e6 0.410044
\(673\) 1.00424e7 0.854671 0.427336 0.904093i \(-0.359452\pi\)
0.427336 + 0.904093i \(0.359452\pi\)
\(674\) 1.81371e7 1.53787
\(675\) 0 0
\(676\) −6.91770e6 −0.582231
\(677\) 6.91467e6 0.579829 0.289914 0.957053i \(-0.406373\pi\)
0.289914 + 0.957053i \(0.406373\pi\)
\(678\) −1.20912e7 −1.01017
\(679\) 3.36928e6 0.280455
\(680\) 0 0
\(681\) 1.03951e7 0.858935
\(682\) −2.82525e6 −0.232593
\(683\) −1.21705e7 −0.998290 −0.499145 0.866518i \(-0.666353\pi\)
−0.499145 + 0.866518i \(0.666353\pi\)
\(684\) 147297. 0.0120380
\(685\) 0 0
\(686\) 2.16552e7 1.75692
\(687\) −1.03147e7 −0.833808
\(688\) 1.02709e6 0.0827248
\(689\) 2.26184e6 0.181515
\(690\) 0 0
\(691\) −1.55796e7 −1.24126 −0.620629 0.784104i \(-0.713123\pi\)
−0.620629 + 0.784104i \(0.713123\pi\)
\(692\) −7.93097e6 −0.629595
\(693\) −958411. −0.0758087
\(694\) 2.64158e7 2.08193
\(695\) 0 0
\(696\) −2.42824e6 −0.190006
\(697\) 1.61125e7 1.25626
\(698\) 6.72532e6 0.522485
\(699\) 2.14105e6 0.165742
\(700\) 0 0
\(701\) −1.83816e7 −1.41282 −0.706411 0.707802i \(-0.749687\pi\)
−0.706411 + 0.707802i \(0.749687\pi\)
\(702\) −3.28976e6 −0.251954
\(703\) 215780. 0.0164673
\(704\) −6.35699e6 −0.483415
\(705\) 0 0
\(706\) 5.90567e6 0.445921
\(707\) −7.58015e6 −0.570334
\(708\) −533775. −0.0400198
\(709\) −1.54867e7 −1.15702 −0.578512 0.815674i \(-0.696366\pi\)
−0.578512 + 0.815674i \(0.696366\pi\)
\(710\) 0 0
\(711\) −5.90021e6 −0.437717
\(712\) −5.87024e6 −0.433966
\(713\) 7.36814e6 0.542792
\(714\) 7.14267e6 0.524343
\(715\) 0 0
\(716\) −2.67055e6 −0.194679
\(717\) 349725. 0.0254056
\(718\) −3.28783e7 −2.38012
\(719\) −1.29862e7 −0.936829 −0.468415 0.883509i \(-0.655175\pi\)
−0.468415 + 0.883509i \(0.655175\pi\)
\(720\) 0 0
\(721\) 9.48277e6 0.679355
\(722\) −2.27874e7 −1.62687
\(723\) 664438. 0.0472725
\(724\) 3.00534e7 2.13082
\(725\) 0 0
\(726\) 1.21325e6 0.0854294
\(727\) 5.51579e6 0.387054 0.193527 0.981095i \(-0.438007\pi\)
0.193527 + 0.981095i \(0.438007\pi\)
\(728\) 9.16797e6 0.641128
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.24959e7 −0.864915
\(732\) −319470. −0.0220370
\(733\) −563402. −0.0387310 −0.0193655 0.999812i \(-0.506165\pi\)
−0.0193655 + 0.999812i \(0.506165\pi\)
\(734\) −1.57084e7 −1.07620
\(735\) 0 0
\(736\) 1.58471e7 1.07834
\(737\) 3.54126e6 0.240153
\(738\) −1.36327e7 −0.921386
\(739\) 2.10879e7 1.42044 0.710220 0.703980i \(-0.248595\pi\)
0.710220 + 0.703980i \(0.248595\pi\)
\(740\) 0 0
\(741\) −151991. −0.0101689
\(742\) 4.15506e6 0.277056
\(743\) −1.32554e7 −0.880891 −0.440445 0.897779i \(-0.645179\pi\)
−0.440445 + 0.897779i \(0.645179\pi\)
\(744\) −4.36586e6 −0.289159
\(745\) 0 0
\(746\) −4.49183e7 −2.95513
\(747\) 8.12292e6 0.532612
\(748\) −5.62886e6 −0.367846
\(749\) 1.72012e7 1.12035
\(750\) 0 0
\(751\) 8.54268e6 0.552706 0.276353 0.961056i \(-0.410874\pi\)
0.276353 + 0.961056i \(0.410874\pi\)
\(752\) −566226. −0.0365128
\(753\) −1.50354e7 −0.966334
\(754\) 6.36494e6 0.407724
\(755\) 0 0
\(756\) −3.76221e6 −0.239408
\(757\) 1.20390e7 0.763570 0.381785 0.924251i \(-0.375309\pi\)
0.381785 + 0.924251i \(0.375309\pi\)
\(758\) −1.86838e7 −1.18112
\(759\) −3.16409e6 −0.199363
\(760\) 0 0
\(761\) −2.02480e6 −0.126742 −0.0633708 0.997990i \(-0.520185\pi\)
−0.0633708 + 0.997990i \(0.520185\pi\)
\(762\) −3.16502e6 −0.197465
\(763\) 1.22090e7 0.759222
\(764\) 2.00486e7 1.24265
\(765\) 0 0
\(766\) 3.12584e7 1.92484
\(767\) 550786. 0.0338060
\(768\) −1.04911e7 −0.641830
\(769\) 1.97142e7 1.20216 0.601082 0.799187i \(-0.294737\pi\)
0.601082 + 0.799187i \(0.294737\pi\)
\(770\) 0 0
\(771\) 5.37943e6 0.325912
\(772\) 3.36416e7 2.03158
\(773\) −5.83885e6 −0.351462 −0.175731 0.984438i \(-0.556229\pi\)
−0.175731 + 0.984438i \(0.556229\pi\)
\(774\) 1.05727e7 0.634357
\(775\) 0 0
\(776\) −6.59093e6 −0.392909
\(777\) −5.51136e6 −0.327496
\(778\) 2.67647e7 1.58531
\(779\) −629850. −0.0371872
\(780\) 0 0
\(781\) −1.17366e6 −0.0688517
\(782\) 2.35808e7 1.37893
\(783\) −1.02822e6 −0.0599351
\(784\) −524880. −0.0304979
\(785\) 0 0
\(786\) 1.42308e7 0.821625
\(787\) 1.03958e7 0.598301 0.299150 0.954206i \(-0.403297\pi\)
0.299150 + 0.954206i \(0.403297\pi\)
\(788\) −1.89392e7 −1.08654
\(789\) −1.23947e7 −0.708832
\(790\) 0 0
\(791\) 1.42683e7 0.810830
\(792\) 1.87483e6 0.106206
\(793\) 329651. 0.0186153
\(794\) 2.49774e7 1.40604
\(795\) 0 0
\(796\) −1.51937e7 −0.849925
\(797\) −1.11544e7 −0.622017 −0.311009 0.950407i \(-0.600667\pi\)
−0.311009 + 0.950407i \(0.600667\pi\)
\(798\) −279213. −0.0155213
\(799\) 6.88890e6 0.381753
\(800\) 0 0
\(801\) −2.48571e6 −0.136889
\(802\) 4.54567e7 2.49553
\(803\) 4.87663e6 0.266889
\(804\) 1.39011e7 0.758417
\(805\) 0 0
\(806\) 1.14439e7 0.620490
\(807\) −1.45547e6 −0.0786721
\(808\) 1.48281e7 0.799022
\(809\) 2.19701e7 1.18022 0.590108 0.807325i \(-0.299085\pi\)
0.590108 + 0.807325i \(0.299085\pi\)
\(810\) 0 0
\(811\) 1.53658e7 0.820358 0.410179 0.912005i \(-0.365466\pi\)
0.410179 + 0.912005i \(0.365466\pi\)
\(812\) 7.27903e6 0.387421
\(813\) 2.17300e6 0.115301
\(814\) 6.97679e6 0.369058
\(815\) 0 0
\(816\) −574757. −0.0302175
\(817\) 488474. 0.0256027
\(818\) 3.78905e6 0.197992
\(819\) 3.88210e6 0.202236
\(820\) 0 0
\(821\) 4.41770e6 0.228738 0.114369 0.993438i \(-0.463515\pi\)
0.114369 + 0.993438i \(0.463515\pi\)
\(822\) 2.04368e7 1.05496
\(823\) 3.83104e7 1.97159 0.985796 0.167947i \(-0.0537137\pi\)
0.985796 + 0.167947i \(0.0537137\pi\)
\(824\) −1.85500e7 −0.951758
\(825\) 0 0
\(826\) 1.01181e6 0.0516000
\(827\) 2.51868e7 1.28059 0.640293 0.768131i \(-0.278813\pi\)
0.640293 + 0.768131i \(0.278813\pi\)
\(828\) −1.24205e7 −0.629600
\(829\) −3.85882e6 −0.195015 −0.0975077 0.995235i \(-0.531087\pi\)
−0.0975077 + 0.995235i \(0.531087\pi\)
\(830\) 0 0
\(831\) 4.62154e6 0.232158
\(832\) 2.57494e7 1.28961
\(833\) 6.38587e6 0.318866
\(834\) −1.14764e6 −0.0571335
\(835\) 0 0
\(836\) 220036. 0.0108888
\(837\) −1.84869e6 −0.0912116
\(838\) −9.80957e6 −0.482547
\(839\) 2.92410e7 1.43413 0.717063 0.697008i \(-0.245486\pi\)
0.717063 + 0.697008i \(0.245486\pi\)
\(840\) 0 0
\(841\) −1.85218e7 −0.903010
\(842\) −4.77289e7 −2.32007
\(843\) −359651. −0.0174306
\(844\) 2.35763e6 0.113925
\(845\) 0 0
\(846\) −5.82867e6 −0.279991
\(847\) −1.43170e6 −0.0685715
\(848\) −334350. −0.0159666
\(849\) 1.49746e7 0.712993
\(850\) 0 0
\(851\) −1.81952e7 −0.861256
\(852\) −4.60716e6 −0.217437
\(853\) 7.44565e6 0.350373 0.175186 0.984535i \(-0.443947\pi\)
0.175186 + 0.984535i \(0.443947\pi\)
\(854\) 605579. 0.0284136
\(855\) 0 0
\(856\) −3.36487e7 −1.56958
\(857\) 2.31717e7 1.07772 0.538859 0.842396i \(-0.318855\pi\)
0.538859 + 0.842396i \(0.318855\pi\)
\(858\) −4.91433e6 −0.227901
\(859\) 2.29217e7 1.05990 0.529949 0.848030i \(-0.322211\pi\)
0.529949 + 0.848030i \(0.322211\pi\)
\(860\) 0 0
\(861\) 1.60874e7 0.739568
\(862\) 6.99138e7 3.20476
\(863\) −3.28455e6 −0.150124 −0.0750619 0.997179i \(-0.523915\pi\)
−0.0750619 + 0.997179i \(0.523915\pi\)
\(864\) −3.97610e6 −0.181206
\(865\) 0 0
\(866\) −5.96863e7 −2.70445
\(867\) −5.78602e6 −0.261416
\(868\) 1.30873e7 0.589593
\(869\) −8.81390e6 −0.395930
\(870\) 0 0
\(871\) −1.43441e7 −0.640660
\(872\) −2.38830e7 −1.06365
\(873\) −2.79088e6 −0.123938
\(874\) −921791. −0.0408182
\(875\) 0 0
\(876\) 1.91430e7 0.842850
\(877\) −6.15691e6 −0.270311 −0.135156 0.990824i \(-0.543153\pi\)
−0.135156 + 0.990824i \(0.543153\pi\)
\(878\) 2.54090e7 1.11238
\(879\) −9.77728e6 −0.426821
\(880\) 0 0
\(881\) −2.47226e7 −1.07314 −0.536568 0.843857i \(-0.680280\pi\)
−0.536568 + 0.843857i \(0.680280\pi\)
\(882\) −5.40306e6 −0.233867
\(883\) −1.67969e7 −0.724980 −0.362490 0.931988i \(-0.618073\pi\)
−0.362490 + 0.931988i \(0.618073\pi\)
\(884\) 2.28000e7 0.981307
\(885\) 0 0
\(886\) 5.19356e7 2.22270
\(887\) −2.91188e7 −1.24270 −0.621348 0.783535i \(-0.713415\pi\)
−0.621348 + 0.783535i \(0.713415\pi\)
\(888\) 1.07812e7 0.458813
\(889\) 3.73491e6 0.158499
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 4.80213e7 2.02079
\(893\) −269292. −0.0113004
\(894\) −1.62131e7 −0.678456
\(895\) 0 0
\(896\) 3.02352e7 1.25818
\(897\) 1.28164e7 0.531843
\(898\) 1.59811e7 0.661328
\(899\) 3.57680e6 0.147603
\(900\) 0 0
\(901\) 4.06782e6 0.166936
\(902\) −2.03649e7 −0.833425
\(903\) −1.24764e7 −0.509179
\(904\) −2.79113e7 −1.13595
\(905\) 0 0
\(906\) 1.70098e7 0.688462
\(907\) −2.39464e7 −0.966544 −0.483272 0.875470i \(-0.660552\pi\)
−0.483272 + 0.875470i \(0.660552\pi\)
\(908\) 6.09564e7 2.45360
\(909\) 6.27886e6 0.252041
\(910\) 0 0
\(911\) −4.74117e7 −1.89273 −0.946366 0.323096i \(-0.895276\pi\)
−0.946366 + 0.323096i \(0.895276\pi\)
\(912\) 22467.7 0.000894482 0
\(913\) 1.21342e7 0.481765
\(914\) 2.12242e7 0.840359
\(915\) 0 0
\(916\) −6.04852e7 −2.38183
\(917\) −1.67932e7 −0.659493
\(918\) −5.91649e6 −0.231717
\(919\) −4.59032e7 −1.79289 −0.896446 0.443152i \(-0.853860\pi\)
−0.896446 + 0.443152i \(0.853860\pi\)
\(920\) 0 0
\(921\) −2.02518e7 −0.786709
\(922\) 4.76527e7 1.84612
\(923\) 4.75398e6 0.183676
\(924\) −5.62009e6 −0.216553
\(925\) 0 0
\(926\) 4.16025e7 1.59438
\(927\) −7.85486e6 −0.300220
\(928\) 7.69286e6 0.293236
\(929\) −3.87522e7 −1.47319 −0.736593 0.676336i \(-0.763567\pi\)
−0.736593 + 0.676336i \(0.763567\pi\)
\(930\) 0 0
\(931\) −249629. −0.00943888
\(932\) 1.25550e7 0.473454
\(933\) 1.41979e7 0.533974
\(934\) −8.14334e7 −3.05447
\(935\) 0 0
\(936\) −7.59410e6 −0.283326
\(937\) −329903. −0.0122755 −0.00613773 0.999981i \(-0.501954\pi\)
−0.00613773 + 0.999981i \(0.501954\pi\)
\(938\) −2.63505e7 −0.977873
\(939\) 1.99266e7 0.737511
\(940\) 0 0
\(941\) −4.87766e7 −1.79572 −0.897858 0.440284i \(-0.854878\pi\)
−0.897858 + 0.440284i \(0.854878\pi\)
\(942\) −7.44660e6 −0.273420
\(943\) 5.31109e7 1.94493
\(944\) −81418.5 −0.00297367
\(945\) 0 0
\(946\) 1.57938e7 0.573798
\(947\) 2.02123e7 0.732387 0.366194 0.930539i \(-0.380661\pi\)
0.366194 + 0.930539i \(0.380661\pi\)
\(948\) −3.45986e7 −1.25037
\(949\) −1.97531e7 −0.711982
\(950\) 0 0
\(951\) −2.10380e7 −0.754317
\(952\) 1.64882e7 0.589632
\(953\) 3.92102e7 1.39852 0.699258 0.714870i \(-0.253514\pi\)
0.699258 + 0.714870i \(0.253514\pi\)
\(954\) −3.44176e6 −0.122436
\(955\) 0 0
\(956\) 2.05078e6 0.0725728
\(957\) −1.53598e6 −0.0542133
\(958\) −7.00404e7 −2.46567
\(959\) −2.41167e7 −0.846780
\(960\) 0 0
\(961\) −2.21982e7 −0.775372
\(962\) −2.82599e7 −0.984540
\(963\) −1.42483e7 −0.495104
\(964\) 3.89624e6 0.135037
\(965\) 0 0
\(966\) 2.35441e7 0.811780
\(967\) −4.99839e7 −1.71895 −0.859476 0.511175i \(-0.829210\pi\)
−0.859476 + 0.511175i \(0.829210\pi\)
\(968\) 2.80067e6 0.0960668
\(969\) −273350. −0.00935211
\(970\) 0 0
\(971\) −1.81402e7 −0.617441 −0.308720 0.951153i \(-0.599901\pi\)
−0.308720 + 0.951153i \(0.599901\pi\)
\(972\) 3.11635e6 0.105799
\(973\) 1.35428e6 0.0458593
\(974\) −4.17636e7 −1.41059
\(975\) 0 0
\(976\) −48729.8 −0.00163746
\(977\) −3.35402e7 −1.12416 −0.562082 0.827082i \(-0.689999\pi\)
−0.562082 + 0.827082i \(0.689999\pi\)
\(978\) 2.31398e6 0.0773593
\(979\) −3.71322e6 −0.123821
\(980\) 0 0
\(981\) −1.01131e7 −0.335514
\(982\) 2.39691e7 0.793181
\(983\) 1.34660e7 0.444482 0.222241 0.974992i \(-0.428663\pi\)
0.222241 + 0.974992i \(0.428663\pi\)
\(984\) −3.14699e7 −1.03611
\(985\) 0 0
\(986\) 1.14471e7 0.374975
\(987\) 6.87817e6 0.224740
\(988\) −891271. −0.0290481
\(989\) −4.11896e7 −1.33905
\(990\) 0 0
\(991\) −1.89814e7 −0.613965 −0.306982 0.951715i \(-0.599319\pi\)
−0.306982 + 0.951715i \(0.599319\pi\)
\(992\) 1.38314e7 0.446259
\(993\) −1.27180e7 −0.409303
\(994\) 8.73322e6 0.280355
\(995\) 0 0
\(996\) 4.76325e7 1.52144
\(997\) −2.73471e7 −0.871311 −0.435656 0.900113i \(-0.643483\pi\)
−0.435656 + 0.900113i \(0.643483\pi\)
\(998\) −5.09659e7 −1.61977
\(999\) 4.56523e6 0.144727
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.m.1.7 7
5.4 even 2 825.6.a.o.1.1 yes 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.m.1.7 7 1.1 even 1 trivial
825.6.a.o.1.1 yes 7 5.4 even 2