Properties

Label 43.6.a.a.1.5
Level $43$
Weight $6$
Character 43.1
Self dual yes
Analytic conductor $6.897$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.89650425196\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} - 173 x^{6} + 462 x^{5} + 9118 x^{4} - 14192 x^{3} - 167688 x^{2} + 106368 x + 681984\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.58275\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.582753 q^{2} +3.05838 q^{3} -31.6604 q^{4} +27.7074 q^{5} +1.78228 q^{6} -103.690 q^{7} -37.0983 q^{8} -233.646 q^{9} +O(q^{10})\) \(q+0.582753 q^{2} +3.05838 q^{3} -31.6604 q^{4} +27.7074 q^{5} +1.78228 q^{6} -103.690 q^{7} -37.0983 q^{8} -233.646 q^{9} +16.1466 q^{10} -158.323 q^{11} -96.8296 q^{12} -578.882 q^{13} -60.4256 q^{14} +84.7398 q^{15} +991.514 q^{16} +253.871 q^{17} -136.158 q^{18} -3092.51 q^{19} -877.227 q^{20} -317.124 q^{21} -92.2633 q^{22} +4163.45 q^{23} -113.461 q^{24} -2357.30 q^{25} -337.345 q^{26} -1457.77 q^{27} +3282.87 q^{28} +6771.63 q^{29} +49.3823 q^{30} +6264.06 q^{31} +1764.95 q^{32} -484.213 q^{33} +147.944 q^{34} -2872.98 q^{35} +7397.34 q^{36} -3294.82 q^{37} -1802.17 q^{38} -1770.44 q^{39} -1027.90 q^{40} -6150.84 q^{41} -184.805 q^{42} -1849.00 q^{43} +5012.58 q^{44} -6473.73 q^{45} +2426.26 q^{46} +8157.17 q^{47} +3032.43 q^{48} -6055.38 q^{49} -1373.72 q^{50} +776.436 q^{51} +18327.6 q^{52} -30457.8 q^{53} -849.517 q^{54} -4386.72 q^{55} +3846.72 q^{56} -9458.07 q^{57} +3946.19 q^{58} -45236.1 q^{59} -2682.90 q^{60} -7251.18 q^{61} +3650.40 q^{62} +24226.8 q^{63} -30699.9 q^{64} -16039.3 q^{65} -282.176 q^{66} +19685.7 q^{67} -8037.67 q^{68} +12733.4 q^{69} -1674.24 q^{70} -48132.2 q^{71} +8667.87 q^{72} +42502.1 q^{73} -1920.07 q^{74} -7209.53 q^{75} +97910.0 q^{76} +16416.5 q^{77} -1031.73 q^{78} -65151.9 q^{79} +27472.3 q^{80} +52317.6 q^{81} -3584.42 q^{82} +70403.6 q^{83} +10040.3 q^{84} +7034.12 q^{85} -1077.51 q^{86} +20710.2 q^{87} +5873.52 q^{88} +29645.1 q^{89} -3772.58 q^{90} +60024.2 q^{91} -131816. q^{92} +19157.9 q^{93} +4753.61 q^{94} -85685.3 q^{95} +5397.90 q^{96} -89440.4 q^{97} -3528.79 q^{98} +36991.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{2} - 26 q^{3} + 122 q^{4} - 212 q^{5} - 69 q^{6} - 136 q^{7} - 666 q^{8} + 546 q^{9} + O(q^{10}) \) \( 8 q - 12 q^{2} - 26 q^{3} + 122 q^{4} - 212 q^{5} - 69 q^{6} - 136 q^{7} - 666 q^{8} + 546 q^{9} - 617 q^{10} - 532 q^{11} - 4195 q^{12} - 2492 q^{13} - 4240 q^{14} - 1780 q^{15} + 1882 q^{16} - 2534 q^{17} - 3711 q^{18} - 1678 q^{19} - 2607 q^{20} - 2256 q^{21} + 11502 q^{22} - 2488 q^{23} + 19953 q^{24} + 4378 q^{25} + 4586 q^{26} - 8960 q^{27} + 18640 q^{28} - 4360 q^{29} + 25092 q^{30} + 5704 q^{31} - 18294 q^{32} - 12852 q^{33} + 30007 q^{34} + 5640 q^{35} + 67969 q^{36} - 3772 q^{37} - 6559 q^{38} + 11120 q^{39} + 14869 q^{40} - 10698 q^{41} + 78698 q^{42} - 14792 q^{43} - 356 q^{44} - 44912 q^{45} - 19389 q^{46} - 77864 q^{47} - 118727 q^{48} + 7188 q^{49} + 26877 q^{50} - 80246 q^{51} - 60736 q^{52} - 62352 q^{53} + 61026 q^{54} - 49552 q^{55} - 144528 q^{56} - 808 q^{57} + 52951 q^{58} - 26224 q^{59} + 101500 q^{60} - 82540 q^{61} - 9023 q^{62} - 61768 q^{63} + 153858 q^{64} - 5000 q^{65} - 48516 q^{66} + 27784 q^{67} + 40507 q^{68} - 93776 q^{69} + 185910 q^{70} - 9504 q^{71} - 186687 q^{72} + 14260 q^{73} - 15239 q^{74} + 167420 q^{75} + 1279 q^{76} - 218140 q^{77} + 264170 q^{78} + 160248 q^{79} - 1291 q^{80} + 161076 q^{81} - 47781 q^{82} - 77176 q^{83} + 16382 q^{84} + 141096 q^{85} + 22188 q^{86} + 268136 q^{87} + 129544 q^{88} - 265692 q^{89} + 48990 q^{90} + 401148 q^{91} + 190391 q^{92} - 123860 q^{93} + 248737 q^{94} + 135884 q^{95} + 950817 q^{96} + 144742 q^{97} - 292244 q^{98} + 239516 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.582753 0.103017 0.0515085 0.998673i \(-0.483597\pi\)
0.0515085 + 0.998673i \(0.483597\pi\)
\(3\) 3.05838 0.196195 0.0980976 0.995177i \(-0.468724\pi\)
0.0980976 + 0.995177i \(0.468724\pi\)
\(4\) −31.6604 −0.989387
\(5\) 27.7074 0.495645 0.247822 0.968805i \(-0.420285\pi\)
0.247822 + 0.968805i \(0.420285\pi\)
\(6\) 1.78228 0.0202115
\(7\) −103.690 −0.799819 −0.399910 0.916555i \(-0.630958\pi\)
−0.399910 + 0.916555i \(0.630958\pi\)
\(8\) −37.0983 −0.204941
\(9\) −233.646 −0.961507
\(10\) 16.1466 0.0510599
\(11\) −158.323 −0.394515 −0.197257 0.980352i \(-0.563203\pi\)
−0.197257 + 0.980352i \(0.563203\pi\)
\(12\) −96.8296 −0.194113
\(13\) −578.882 −0.950017 −0.475009 0.879981i \(-0.657555\pi\)
−0.475009 + 0.879981i \(0.657555\pi\)
\(14\) −60.4256 −0.0823950
\(15\) 84.7398 0.0972432
\(16\) 991.514 0.968275
\(17\) 253.871 0.213055 0.106527 0.994310i \(-0.466027\pi\)
0.106527 + 0.994310i \(0.466027\pi\)
\(18\) −136.158 −0.0990517
\(19\) −3092.51 −1.96529 −0.982645 0.185494i \(-0.940612\pi\)
−0.982645 + 0.185494i \(0.940612\pi\)
\(20\) −877.227 −0.490385
\(21\) −317.124 −0.156921
\(22\) −92.2633 −0.0406417
\(23\) 4163.45 1.64109 0.820547 0.571579i \(-0.193669\pi\)
0.820547 + 0.571579i \(0.193669\pi\)
\(24\) −113.461 −0.0402084
\(25\) −2357.30 −0.754336
\(26\) −337.345 −0.0978680
\(27\) −1457.77 −0.384839
\(28\) 3282.87 0.791331
\(29\) 6771.63 1.49520 0.747598 0.664151i \(-0.231207\pi\)
0.747598 + 0.664151i \(0.231207\pi\)
\(30\) 49.3823 0.0100177
\(31\) 6264.06 1.17072 0.585358 0.810775i \(-0.300954\pi\)
0.585358 + 0.810775i \(0.300954\pi\)
\(32\) 1764.95 0.304690
\(33\) −484.213 −0.0774019
\(34\) 147.944 0.0219483
\(35\) −2872.98 −0.396426
\(36\) 7397.34 0.951303
\(37\) −3294.82 −0.395665 −0.197833 0.980236i \(-0.563390\pi\)
−0.197833 + 0.980236i \(0.563390\pi\)
\(38\) −1802.17 −0.202459
\(39\) −1770.44 −0.186389
\(40\) −1027.90 −0.101578
\(41\) −6150.84 −0.571445 −0.285723 0.958312i \(-0.592234\pi\)
−0.285723 + 0.958312i \(0.592234\pi\)
\(42\) −184.805 −0.0161655
\(43\) −1849.00 −0.152499
\(44\) 5012.58 0.390328
\(45\) −6473.73 −0.476566
\(46\) 2426.26 0.169061
\(47\) 8157.17 0.538635 0.269318 0.963051i \(-0.413202\pi\)
0.269318 + 0.963051i \(0.413202\pi\)
\(48\) 3032.43 0.189971
\(49\) −6055.38 −0.360289
\(50\) −1373.72 −0.0777095
\(51\) 776.436 0.0418004
\(52\) 18327.6 0.939935
\(53\) −30457.8 −1.48939 −0.744697 0.667403i \(-0.767406\pi\)
−0.744697 + 0.667403i \(0.767406\pi\)
\(54\) −849.517 −0.0396449
\(55\) −4386.72 −0.195539
\(56\) 3846.72 0.163916
\(57\) −9458.07 −0.385581
\(58\) 3946.19 0.154031
\(59\) −45236.1 −1.69182 −0.845912 0.533322i \(-0.820943\pi\)
−0.845912 + 0.533322i \(0.820943\pi\)
\(60\) −2682.90 −0.0962112
\(61\) −7251.18 −0.249508 −0.124754 0.992188i \(-0.539814\pi\)
−0.124754 + 0.992188i \(0.539814\pi\)
\(62\) 3650.40 0.120604
\(63\) 24226.8 0.769032
\(64\) −30699.9 −0.936887
\(65\) −16039.3 −0.470871
\(66\) −282.176 −0.00797372
\(67\) 19685.7 0.535751 0.267876 0.963454i \(-0.413678\pi\)
0.267876 + 0.963454i \(0.413678\pi\)
\(68\) −8037.67 −0.210794
\(69\) 12733.4 0.321975
\(70\) −1674.24 −0.0408387
\(71\) −48132.2 −1.13316 −0.566579 0.824008i \(-0.691733\pi\)
−0.566579 + 0.824008i \(0.691733\pi\)
\(72\) 8667.87 0.197052
\(73\) 42502.1 0.933475 0.466738 0.884396i \(-0.345429\pi\)
0.466738 + 0.884396i \(0.345429\pi\)
\(74\) −1920.07 −0.0407603
\(75\) −7209.53 −0.147997
\(76\) 97910.0 1.94443
\(77\) 16416.5 0.315540
\(78\) −1031.73 −0.0192012
\(79\) −65151.9 −1.17452 −0.587258 0.809400i \(-0.699793\pi\)
−0.587258 + 0.809400i \(0.699793\pi\)
\(80\) 27472.3 0.479921
\(81\) 52317.6 0.886004
\(82\) −3584.42 −0.0588686
\(83\) 70403.6 1.12176 0.560880 0.827897i \(-0.310463\pi\)
0.560880 + 0.827897i \(0.310463\pi\)
\(84\) 10040.3 0.155255
\(85\) 7034.12 0.105600
\(86\) −1077.51 −0.0157100
\(87\) 20710.2 0.293350
\(88\) 5873.52 0.0808522
\(89\) 29645.1 0.396714 0.198357 0.980130i \(-0.436439\pi\)
0.198357 + 0.980130i \(0.436439\pi\)
\(90\) −3772.58 −0.0490945
\(91\) 60024.2 0.759842
\(92\) −131816. −1.62368
\(93\) 19157.9 0.229689
\(94\) 4753.61 0.0554887
\(95\) −85685.3 −0.974086
\(96\) 5397.90 0.0597787
\(97\) −89440.4 −0.965171 −0.482585 0.875849i \(-0.660302\pi\)
−0.482585 + 0.875849i \(0.660302\pi\)
\(98\) −3528.79 −0.0371160
\(99\) 36991.6 0.379329
\(100\) 74633.1 0.746331
\(101\) −28633.0 −0.279295 −0.139647 0.990201i \(-0.544597\pi\)
−0.139647 + 0.990201i \(0.544597\pi\)
\(102\) 452.470 0.00430615
\(103\) 30124.9 0.279790 0.139895 0.990166i \(-0.455324\pi\)
0.139895 + 0.990166i \(0.455324\pi\)
\(104\) 21475.5 0.194697
\(105\) −8786.67 −0.0777770
\(106\) −17749.4 −0.153433
\(107\) 83073.3 0.701459 0.350729 0.936477i \(-0.385934\pi\)
0.350729 + 0.936477i \(0.385934\pi\)
\(108\) 46153.5 0.380754
\(109\) 58783.5 0.473903 0.236951 0.971522i \(-0.423852\pi\)
0.236951 + 0.971522i \(0.423852\pi\)
\(110\) −2556.38 −0.0201439
\(111\) −10076.8 −0.0776276
\(112\) −102810. −0.774445
\(113\) −232647. −1.71396 −0.856981 0.515347i \(-0.827663\pi\)
−0.856981 + 0.515347i \(0.827663\pi\)
\(114\) −5511.72 −0.0397214
\(115\) 115358. 0.813400
\(116\) −214392. −1.47933
\(117\) 135254. 0.913449
\(118\) −26361.5 −0.174287
\(119\) −26323.9 −0.170405
\(120\) −3143.70 −0.0199291
\(121\) −135985. −0.844358
\(122\) −4225.64 −0.0257036
\(123\) −18811.6 −0.112115
\(124\) −198323. −1.15829
\(125\) −151900. −0.869528
\(126\) 14118.2 0.0792234
\(127\) 109116. 0.600313 0.300156 0.953890i \(-0.402961\pi\)
0.300156 + 0.953890i \(0.402961\pi\)
\(128\) −74368.9 −0.401205
\(129\) −5654.95 −0.0299195
\(130\) −9346.94 −0.0485078
\(131\) 139424. 0.709839 0.354920 0.934897i \(-0.384508\pi\)
0.354920 + 0.934897i \(0.384508\pi\)
\(132\) 15330.4 0.0765805
\(133\) 320662. 1.57188
\(134\) 11471.9 0.0551915
\(135\) −40390.9 −0.190743
\(136\) −9418.19 −0.0436637
\(137\) −64194.6 −0.292211 −0.146106 0.989269i \(-0.546674\pi\)
−0.146106 + 0.989269i \(0.546674\pi\)
\(138\) 7420.43 0.0331689
\(139\) −281632. −1.23636 −0.618181 0.786036i \(-0.712130\pi\)
−0.618181 + 0.786036i \(0.712130\pi\)
\(140\) 90959.7 0.392219
\(141\) 24947.7 0.105678
\(142\) −28049.2 −0.116735
\(143\) 91650.4 0.374796
\(144\) −231664. −0.931004
\(145\) 187624. 0.741086
\(146\) 24768.2 0.0961639
\(147\) −18519.7 −0.0706871
\(148\) 104315. 0.391466
\(149\) 225391. 0.831710 0.415855 0.909431i \(-0.363482\pi\)
0.415855 + 0.909431i \(0.363482\pi\)
\(150\) −4201.37 −0.0152462
\(151\) 388390. 1.38620 0.693099 0.720842i \(-0.256245\pi\)
0.693099 + 0.720842i \(0.256245\pi\)
\(152\) 114727. 0.402768
\(153\) −59316.1 −0.204854
\(154\) 9566.78 0.0325060
\(155\) 173561. 0.580259
\(156\) 56052.9 0.184411
\(157\) −369612. −1.19673 −0.598366 0.801223i \(-0.704183\pi\)
−0.598366 + 0.801223i \(0.704183\pi\)
\(158\) −37967.4 −0.120995
\(159\) −93151.7 −0.292212
\(160\) 48902.2 0.151018
\(161\) −431708. −1.31258
\(162\) 30488.2 0.0912735
\(163\) 522204. 1.53947 0.769735 0.638363i \(-0.220388\pi\)
0.769735 + 0.638363i \(0.220388\pi\)
\(164\) 194738. 0.565381
\(165\) −13416.3 −0.0383638
\(166\) 41027.9 0.115560
\(167\) −17232.5 −0.0478141 −0.0239071 0.999714i \(-0.507611\pi\)
−0.0239071 + 0.999714i \(0.507611\pi\)
\(168\) 11764.7 0.0321595
\(169\) −36189.0 −0.0974674
\(170\) 4099.15 0.0108786
\(171\) 722553. 1.88964
\(172\) 58540.1 0.150880
\(173\) −571643. −1.45214 −0.726072 0.687618i \(-0.758656\pi\)
−0.726072 + 0.687618i \(0.758656\pi\)
\(174\) 12068.9 0.0302201
\(175\) 244429. 0.603333
\(176\) −156980. −0.381999
\(177\) −138349. −0.331928
\(178\) 17275.7 0.0408683
\(179\) −191963. −0.447802 −0.223901 0.974612i \(-0.571879\pi\)
−0.223901 + 0.974612i \(0.571879\pi\)
\(180\) 204961. 0.471509
\(181\) 110804. 0.251396 0.125698 0.992069i \(-0.459883\pi\)
0.125698 + 0.992069i \(0.459883\pi\)
\(182\) 34979.3 0.0782767
\(183\) −22176.9 −0.0489522
\(184\) −154457. −0.336327
\(185\) −91290.9 −0.196109
\(186\) 11164.3 0.0236619
\(187\) −40193.8 −0.0840533
\(188\) −258259. −0.532919
\(189\) 151156. 0.307801
\(190\) −49933.3 −0.100348
\(191\) 295668. 0.586437 0.293218 0.956046i \(-0.405274\pi\)
0.293218 + 0.956046i \(0.405274\pi\)
\(192\) −93892.0 −0.183813
\(193\) −375944. −0.726491 −0.363246 0.931693i \(-0.618331\pi\)
−0.363246 + 0.931693i \(0.618331\pi\)
\(194\) −52121.6 −0.0994291
\(195\) −49054.3 −0.0923827
\(196\) 191716. 0.356466
\(197\) −732825. −1.34535 −0.672674 0.739939i \(-0.734854\pi\)
−0.672674 + 0.739939i \(0.734854\pi\)
\(198\) 21557.0 0.0390773
\(199\) 589939. 1.05603 0.528013 0.849236i \(-0.322937\pi\)
0.528013 + 0.849236i \(0.322937\pi\)
\(200\) 87451.8 0.154594
\(201\) 60206.3 0.105112
\(202\) −16685.9 −0.0287722
\(203\) −702150. −1.19589
\(204\) −24582.3 −0.0413568
\(205\) −170424. −0.283234
\(206\) 17555.3 0.0288231
\(207\) −972774. −1.57792
\(208\) −573969. −0.919878
\(209\) 489616. 0.775336
\(210\) −5120.45 −0.00801236
\(211\) −371450. −0.574374 −0.287187 0.957875i \(-0.592720\pi\)
−0.287187 + 0.957875i \(0.592720\pi\)
\(212\) 964307. 1.47359
\(213\) −147207. −0.222320
\(214\) 48411.2 0.0722622
\(215\) −51231.0 −0.0755851
\(216\) 54080.6 0.0788692
\(217\) −649520. −0.936361
\(218\) 34256.2 0.0488201
\(219\) 129988. 0.183143
\(220\) 138885. 0.193464
\(221\) −146962. −0.202406
\(222\) −5872.30 −0.00799697
\(223\) −835552. −1.12515 −0.562576 0.826746i \(-0.690190\pi\)
−0.562576 + 0.826746i \(0.690190\pi\)
\(224\) −183008. −0.243697
\(225\) 550775. 0.725300
\(226\) −135576. −0.176567
\(227\) −363342. −0.468006 −0.234003 0.972236i \(-0.575182\pi\)
−0.234003 + 0.972236i \(0.575182\pi\)
\(228\) 299446. 0.381489
\(229\) −106091. −0.133688 −0.0668438 0.997763i \(-0.521293\pi\)
−0.0668438 + 0.997763i \(0.521293\pi\)
\(230\) 67225.3 0.0837941
\(231\) 50208.0 0.0619075
\(232\) −251216. −0.306427
\(233\) 713760. 0.861315 0.430658 0.902515i \(-0.358282\pi\)
0.430658 + 0.902515i \(0.358282\pi\)
\(234\) 78819.4 0.0941008
\(235\) 226014. 0.266972
\(236\) 1.43219e6 1.67387
\(237\) −199259. −0.230435
\(238\) −15340.3 −0.0175547
\(239\) −70400.0 −0.0797220 −0.0398610 0.999205i \(-0.512692\pi\)
−0.0398610 + 0.999205i \(0.512692\pi\)
\(240\) 84020.6 0.0941581
\(241\) 44600.6 0.0494650 0.0247325 0.999694i \(-0.492127\pi\)
0.0247325 + 0.999694i \(0.492127\pi\)
\(242\) −79245.5 −0.0869833
\(243\) 514245. 0.558668
\(244\) 229575. 0.246860
\(245\) −167779. −0.178576
\(246\) −10962.5 −0.0115497
\(247\) 1.79020e6 1.86706
\(248\) −232386. −0.239928
\(249\) 215321. 0.220084
\(250\) −88520.3 −0.0895762
\(251\) 1.05989e6 1.06188 0.530942 0.847408i \(-0.321838\pi\)
0.530942 + 0.847408i \(0.321838\pi\)
\(252\) −767030. −0.760871
\(253\) −659170. −0.647435
\(254\) 63587.4 0.0618425
\(255\) 21513.0 0.0207181
\(256\) 939058. 0.895556
\(257\) −966091. −0.912400 −0.456200 0.889877i \(-0.650790\pi\)
−0.456200 + 0.889877i \(0.650790\pi\)
\(258\) −3295.44 −0.00308222
\(259\) 341640. 0.316460
\(260\) 507811. 0.465874
\(261\) −1.58217e6 −1.43764
\(262\) 81249.8 0.0731255
\(263\) 855270. 0.762454 0.381227 0.924481i \(-0.375502\pi\)
0.381227 + 0.924481i \(0.375502\pi\)
\(264\) 17963.5 0.0158628
\(265\) −843907. −0.738210
\(266\) 186867. 0.161930
\(267\) 90665.9 0.0778334
\(268\) −623256. −0.530065
\(269\) 1.55224e6 1.30791 0.653955 0.756533i \(-0.273108\pi\)
0.653955 + 0.756533i \(0.273108\pi\)
\(270\) −23537.9 −0.0196498
\(271\) −4177.31 −0.00345520 −0.00172760 0.999999i \(-0.500550\pi\)
−0.00172760 + 0.999999i \(0.500550\pi\)
\(272\) 251717. 0.206296
\(273\) 183577. 0.149077
\(274\) −37409.5 −0.0301027
\(275\) 373216. 0.297597
\(276\) −403145. −0.318558
\(277\) −1.59749e6 −1.25095 −0.625475 0.780244i \(-0.715095\pi\)
−0.625475 + 0.780244i \(0.715095\pi\)
\(278\) −164122. −0.127366
\(279\) −1.46357e6 −1.12565
\(280\) 106583. 0.0812439
\(281\) 1.60304e6 1.21110 0.605549 0.795808i \(-0.292954\pi\)
0.605549 + 0.795808i \(0.292954\pi\)
\(282\) 14538.4 0.0108866
\(283\) 2.04664e6 1.51906 0.759530 0.650472i \(-0.225429\pi\)
0.759530 + 0.650472i \(0.225429\pi\)
\(284\) 1.52389e6 1.12113
\(285\) −262058. −0.191111
\(286\) 53409.5 0.0386103
\(287\) 637780. 0.457053
\(288\) −412374. −0.292961
\(289\) −1.35541e6 −0.954608
\(290\) 109338. 0.0763446
\(291\) −273543. −0.189362
\(292\) −1.34563e6 −0.923569
\(293\) 1.35028e6 0.918869 0.459434 0.888212i \(-0.348052\pi\)
0.459434 + 0.888212i \(0.348052\pi\)
\(294\) −10792.4 −0.00728198
\(295\) −1.25337e6 −0.838544
\(296\) 122232. 0.0810880
\(297\) 230798. 0.151824
\(298\) 131347. 0.0856803
\(299\) −2.41014e6 −1.55907
\(300\) 228256. 0.146427
\(301\) 191723. 0.121971
\(302\) 226335. 0.142802
\(303\) −87570.6 −0.0547964
\(304\) −3.06626e6 −1.90294
\(305\) −200911. −0.123667
\(306\) −34566.6 −0.0211035
\(307\) −361103. −0.218668 −0.109334 0.994005i \(-0.534872\pi\)
−0.109334 + 0.994005i \(0.534872\pi\)
\(308\) −519754. −0.312192
\(309\) 92133.4 0.0548935
\(310\) 101143. 0.0597766
\(311\) 2.12425e6 1.24539 0.622694 0.782466i \(-0.286038\pi\)
0.622694 + 0.782466i \(0.286038\pi\)
\(312\) 65680.3 0.0381987
\(313\) −541717. −0.312545 −0.156272 0.987714i \(-0.549948\pi\)
−0.156272 + 0.987714i \(0.549948\pi\)
\(314\) −215392. −0.123284
\(315\) 671261. 0.381167
\(316\) 2.06273e6 1.16205
\(317\) −3.06921e6 −1.71545 −0.857725 0.514110i \(-0.828122\pi\)
−0.857725 + 0.514110i \(0.828122\pi\)
\(318\) −54284.4 −0.0301028
\(319\) −1.07211e6 −0.589877
\(320\) −850614. −0.464363
\(321\) 254070. 0.137623
\(322\) −251579. −0.135218
\(323\) −785100. −0.418715
\(324\) −1.65640e6 −0.876601
\(325\) 1.36460e6 0.716632
\(326\) 304316. 0.158592
\(327\) 179782. 0.0929775
\(328\) 228185. 0.117112
\(329\) −845817. −0.430811
\(330\) −7818.37 −0.00395213
\(331\) −282184. −0.141567 −0.0707836 0.997492i \(-0.522550\pi\)
−0.0707836 + 0.997492i \(0.522550\pi\)
\(332\) −2.22901e6 −1.10986
\(333\) 769823. 0.380435
\(334\) −10042.3 −0.00492567
\(335\) 545438. 0.265542
\(336\) −314432. −0.151942
\(337\) 1.04584e6 0.501640 0.250820 0.968034i \(-0.419300\pi\)
0.250820 + 0.968034i \(0.419300\pi\)
\(338\) −21089.2 −0.0100408
\(339\) −711523. −0.336271
\(340\) −222703. −0.104479
\(341\) −991746. −0.461864
\(342\) 421070. 0.194665
\(343\) 2.37060e6 1.08799
\(344\) 68594.7 0.0312532
\(345\) 352810. 0.159585
\(346\) −333127. −0.149596
\(347\) 1.90831e6 0.850796 0.425398 0.905006i \(-0.360134\pi\)
0.425398 + 0.905006i \(0.360134\pi\)
\(348\) −655694. −0.290237
\(349\) 1.46350e6 0.643175 0.321588 0.946880i \(-0.395784\pi\)
0.321588 + 0.946880i \(0.395784\pi\)
\(350\) 142441. 0.0621536
\(351\) 843874. 0.365603
\(352\) −279433. −0.120205
\(353\) 867374. 0.370484 0.185242 0.982693i \(-0.440693\pi\)
0.185242 + 0.982693i \(0.440693\pi\)
\(354\) −80623.4 −0.0341942
\(355\) −1.33362e6 −0.561643
\(356\) −938575. −0.392504
\(357\) −80508.6 −0.0334327
\(358\) −111867. −0.0461313
\(359\) 3.45466e6 1.41472 0.707359 0.706855i \(-0.249887\pi\)
0.707359 + 0.706855i \(0.249887\pi\)
\(360\) 240164. 0.0976679
\(361\) 7.08751e6 2.86237
\(362\) 64571.2 0.0258981
\(363\) −415893. −0.165659
\(364\) −1.90039e6 −0.751778
\(365\) 1.17762e6 0.462672
\(366\) −12923.6 −0.00504292
\(367\) −2.78797e6 −1.08050 −0.540248 0.841506i \(-0.681670\pi\)
−0.540248 + 0.841506i \(0.681670\pi\)
\(368\) 4.12811e6 1.58903
\(369\) 1.43712e6 0.549449
\(370\) −53200.0 −0.0202026
\(371\) 3.15817e6 1.19125
\(372\) −606546. −0.227251
\(373\) −1.65578e6 −0.616212 −0.308106 0.951352i \(-0.599695\pi\)
−0.308106 + 0.951352i \(0.599695\pi\)
\(374\) −23423.0 −0.00865892
\(375\) −464569. −0.170597
\(376\) −302617. −0.110388
\(377\) −3.91997e6 −1.42046
\(378\) 88086.4 0.0317088
\(379\) −2.35538e6 −0.842291 −0.421146 0.906993i \(-0.638372\pi\)
−0.421146 + 0.906993i \(0.638372\pi\)
\(380\) 2.71283e6 0.963749
\(381\) 333717. 0.117779
\(382\) 172301. 0.0604130
\(383\) −4.13864e6 −1.44165 −0.720827 0.693115i \(-0.756238\pi\)
−0.720827 + 0.693115i \(0.756238\pi\)
\(384\) −227449. −0.0787146
\(385\) 454859. 0.156396
\(386\) −219083. −0.0748410
\(387\) 432012. 0.146629
\(388\) 2.83172e6 0.954928
\(389\) −5.27781e6 −1.76840 −0.884198 0.467112i \(-0.845294\pi\)
−0.884198 + 0.467112i \(0.845294\pi\)
\(390\) −28586.5 −0.00951700
\(391\) 1.05698e6 0.349643
\(392\) 224644. 0.0738380
\(393\) 426412. 0.139267
\(394\) −427056. −0.138594
\(395\) −1.80519e6 −0.582143
\(396\) −1.17117e6 −0.375303
\(397\) 3.71439e6 1.18280 0.591401 0.806378i \(-0.298575\pi\)
0.591401 + 0.806378i \(0.298575\pi\)
\(398\) 343789. 0.108789
\(399\) 980707. 0.308395
\(400\) −2.33730e6 −0.730405
\(401\) 3.60034e6 1.11810 0.559052 0.829132i \(-0.311165\pi\)
0.559052 + 0.829132i \(0.311165\pi\)
\(402\) 35085.4 0.0108283
\(403\) −3.62615e6 −1.11220
\(404\) 906532. 0.276331
\(405\) 1.44959e6 0.439143
\(406\) −409180. −0.123197
\(407\) 521647. 0.156096
\(408\) −28804.4 −0.00856661
\(409\) 1.65305e6 0.488627 0.244314 0.969696i \(-0.421437\pi\)
0.244314 + 0.969696i \(0.421437\pi\)
\(410\) −99314.8 −0.0291779
\(411\) −196331. −0.0573304
\(412\) −953765. −0.276821
\(413\) 4.69053e6 1.35315
\(414\) −566887. −0.162553
\(415\) 1.95070e6 0.555994
\(416\) −1.02170e6 −0.289461
\(417\) −861339. −0.242568
\(418\) 285325. 0.0798728
\(419\) 1.25123e6 0.348180 0.174090 0.984730i \(-0.444302\pi\)
0.174090 + 0.984730i \(0.444302\pi\)
\(420\) 278189. 0.0769515
\(421\) 3.12108e6 0.858221 0.429111 0.903252i \(-0.358827\pi\)
0.429111 + 0.903252i \(0.358827\pi\)
\(422\) −216464. −0.0591703
\(423\) −1.90589e6 −0.517902
\(424\) 1.12993e6 0.305238
\(425\) −598451. −0.160715
\(426\) −85785.1 −0.0229028
\(427\) 751875. 0.199561
\(428\) −2.63013e6 −0.694015
\(429\) 280302. 0.0735331
\(430\) −29855.0 −0.00778656
\(431\) 246469. 0.0639102 0.0319551 0.999489i \(-0.489827\pi\)
0.0319551 + 0.999489i \(0.489827\pi\)
\(432\) −1.44540e6 −0.372630
\(433\) −7.56031e6 −1.93785 −0.968924 0.247357i \(-0.920438\pi\)
−0.968924 + 0.247357i \(0.920438\pi\)
\(434\) −378510. −0.0964612
\(435\) 573826. 0.145398
\(436\) −1.86111e6 −0.468873
\(437\) −1.28755e7 −3.22523
\(438\) 75750.6 0.0188669
\(439\) −3.76145e6 −0.931525 −0.465763 0.884910i \(-0.654220\pi\)
−0.465763 + 0.884910i \(0.654220\pi\)
\(440\) 162740. 0.0400740
\(441\) 1.41482e6 0.346421
\(442\) −85642.2 −0.0208513
\(443\) −1.02986e6 −0.249327 −0.124664 0.992199i \(-0.539785\pi\)
−0.124664 + 0.992199i \(0.539785\pi\)
\(444\) 319036. 0.0768038
\(445\) 821387. 0.196629
\(446\) −486920. −0.115910
\(447\) 689333. 0.163177
\(448\) 3.18327e6 0.749340
\(449\) −4.29961e6 −1.00650 −0.503249 0.864141i \(-0.667862\pi\)
−0.503249 + 0.864141i \(0.667862\pi\)
\(450\) 320965. 0.0747183
\(451\) 973820. 0.225443
\(452\) 7.36570e6 1.69577
\(453\) 1.18784e6 0.271966
\(454\) −211739. −0.0482126
\(455\) 1.66312e6 0.376612
\(456\) 350878. 0.0790213
\(457\) −1.42831e6 −0.319913 −0.159957 0.987124i \(-0.551135\pi\)
−0.159957 + 0.987124i \(0.551135\pi\)
\(458\) −61825.0 −0.0137721
\(459\) −370085. −0.0819917
\(460\) −3.65229e6 −0.804768
\(461\) −3.06394e6 −0.671472 −0.335736 0.941956i \(-0.608985\pi\)
−0.335736 + 0.941956i \(0.608985\pi\)
\(462\) 29258.9 0.00637753
\(463\) −3.51890e6 −0.762877 −0.381438 0.924394i \(-0.624571\pi\)
−0.381438 + 0.924394i \(0.624571\pi\)
\(464\) 6.71416e6 1.44776
\(465\) 530815. 0.113844
\(466\) 415945. 0.0887302
\(467\) −6.90153e6 −1.46438 −0.732188 0.681102i \(-0.761501\pi\)
−0.732188 + 0.681102i \(0.761501\pi\)
\(468\) −4.28218e6 −0.903755
\(469\) −2.04121e6 −0.428504
\(470\) 131710. 0.0275027
\(471\) −1.13041e6 −0.234793
\(472\) 1.67818e6 0.346724
\(473\) 292740. 0.0601629
\(474\) −116119. −0.0237387
\(475\) 7.28997e6 1.48249
\(476\) 833426. 0.168597
\(477\) 7.11636e6 1.43206
\(478\) −41025.8 −0.00821272
\(479\) −4.37684e6 −0.871609 −0.435805 0.900041i \(-0.643536\pi\)
−0.435805 + 0.900041i \(0.643536\pi\)
\(480\) 149562. 0.0296290
\(481\) 1.90731e6 0.375889
\(482\) 25991.1 0.00509574
\(483\) −1.32033e6 −0.257522
\(484\) 4.30533e6 0.835398
\(485\) −2.47816e6 −0.478382
\(486\) 299677. 0.0575524
\(487\) 4.08443e6 0.780385 0.390192 0.920733i \(-0.372409\pi\)
0.390192 + 0.920733i \(0.372409\pi\)
\(488\) 269006. 0.0511343
\(489\) 1.59710e6 0.302037
\(490\) −97773.6 −0.0183963
\(491\) −4.67807e6 −0.875716 −0.437858 0.899044i \(-0.644263\pi\)
−0.437858 + 0.899044i \(0.644263\pi\)
\(492\) 595583. 0.110925
\(493\) 1.71912e6 0.318559
\(494\) 1.04324e6 0.192339
\(495\) 1.02494e6 0.188012
\(496\) 6.21090e6 1.13357
\(497\) 4.99083e6 0.906321
\(498\) 125479. 0.0226724
\(499\) 9.66466e6 1.73754 0.868771 0.495214i \(-0.164910\pi\)
0.868771 + 0.495214i \(0.164910\pi\)
\(500\) 4.80922e6 0.860300
\(501\) −52703.5 −0.00938091
\(502\) 617655. 0.109392
\(503\) −1.03319e7 −1.82079 −0.910396 0.413738i \(-0.864223\pi\)
−0.910396 + 0.413738i \(0.864223\pi\)
\(504\) −898772. −0.157606
\(505\) −793345. −0.138431
\(506\) −384133. −0.0666969
\(507\) −110680. −0.0191227
\(508\) −3.45464e6 −0.593942
\(509\) −4.20006e6 −0.718557 −0.359278 0.933230i \(-0.616977\pi\)
−0.359278 + 0.933230i \(0.616977\pi\)
\(510\) 12536.8 0.00213432
\(511\) −4.40704e6 −0.746611
\(512\) 2.92704e6 0.493463
\(513\) 4.50815e6 0.756320
\(514\) −562992. −0.0939928
\(515\) 834681. 0.138676
\(516\) 179038. 0.0296020
\(517\) −1.29147e6 −0.212500
\(518\) 199092. 0.0326008
\(519\) −1.74830e6 −0.284904
\(520\) 595030. 0.0965007
\(521\) 6.46853e6 1.04403 0.522013 0.852938i \(-0.325181\pi\)
0.522013 + 0.852938i \(0.325181\pi\)
\(522\) −922012. −0.148102
\(523\) −3.00248e6 −0.479983 −0.239991 0.970775i \(-0.577145\pi\)
−0.239991 + 0.970775i \(0.577145\pi\)
\(524\) −4.41423e6 −0.702306
\(525\) 747556. 0.118371
\(526\) 498411. 0.0785458
\(527\) 1.59027e6 0.249427
\(528\) −480104. −0.0749463
\(529\) 1.08979e7 1.69319
\(530\) −491789. −0.0760482
\(531\) 1.05692e7 1.62670
\(532\) −1.01523e7 −1.55520
\(533\) 3.56061e6 0.542883
\(534\) 52835.8 0.00801817
\(535\) 2.30175e6 0.347674
\(536\) −730304. −0.109797
\(537\) −587098. −0.0878566
\(538\) 904572. 0.134737
\(539\) 958708. 0.142139
\(540\) 1.27879e6 0.188719
\(541\) −5.21767e6 −0.766449 −0.383224 0.923655i \(-0.625186\pi\)
−0.383224 + 0.923655i \(0.625186\pi\)
\(542\) −2434.34 −0.000355945 0
\(543\) 338880. 0.0493227
\(544\) 448071. 0.0649157
\(545\) 1.62874e6 0.234887
\(546\) 106980. 0.0153575
\(547\) −7.86187e6 −1.12346 −0.561730 0.827321i \(-0.689864\pi\)
−0.561730 + 0.827321i \(0.689864\pi\)
\(548\) 2.03243e6 0.289110
\(549\) 1.69421e6 0.239903
\(550\) 217492. 0.0306575
\(551\) −2.09413e7 −2.93850
\(552\) −472387. −0.0659858
\(553\) 6.75560e6 0.939401
\(554\) −930944. −0.128869
\(555\) −279202. −0.0384757
\(556\) 8.91659e6 1.22324
\(557\) 1.21531e7 1.65978 0.829889 0.557929i \(-0.188404\pi\)
0.829889 + 0.557929i \(0.188404\pi\)
\(558\) −852902. −0.115961
\(559\) 1.07035e6 0.144876
\(560\) −2.84860e6 −0.383850
\(561\) −122928. −0.0164909
\(562\) 934177. 0.124764
\(563\) 5.15518e6 0.685446 0.342723 0.939437i \(-0.388651\pi\)
0.342723 + 0.939437i \(0.388651\pi\)
\(564\) −789856. −0.104556
\(565\) −6.44604e6 −0.849517
\(566\) 1.19268e6 0.156489
\(567\) −5.42482e6 −0.708643
\(568\) 1.78562e6 0.232230
\(569\) −1.42079e6 −0.183972 −0.0919858 0.995760i \(-0.529321\pi\)
−0.0919858 + 0.995760i \(0.529321\pi\)
\(570\) −152715. −0.0196877
\(571\) −1.14786e7 −1.47332 −0.736661 0.676262i \(-0.763599\pi\)
−0.736661 + 0.676262i \(0.763599\pi\)
\(572\) −2.90169e6 −0.370818
\(573\) 904266. 0.115056
\(574\) 371668. 0.0470842
\(575\) −9.81450e6 −1.23794
\(576\) 7.17292e6 0.900824
\(577\) 1.00683e7 1.25897 0.629484 0.777014i \(-0.283266\pi\)
0.629484 + 0.777014i \(0.283266\pi\)
\(578\) −789867. −0.0983409
\(579\) −1.14978e6 −0.142534
\(580\) −5.94026e6 −0.733221
\(581\) −7.30015e6 −0.897205
\(582\) −159408. −0.0195075
\(583\) 4.82218e6 0.587587
\(584\) −1.57675e6 −0.191307
\(585\) 3.74752e6 0.452746
\(586\) 786877. 0.0946592
\(587\) −1.33560e7 −1.59985 −0.799927 0.600097i \(-0.795129\pi\)
−0.799927 + 0.600097i \(0.795129\pi\)
\(588\) 586340. 0.0699369
\(589\) −1.93716e7 −2.30080
\(590\) −730407. −0.0863843
\(591\) −2.24126e6 −0.263951
\(592\) −3.26686e6 −0.383113
\(593\) 1.20725e7 1.40981 0.704906 0.709301i \(-0.250989\pi\)
0.704906 + 0.709301i \(0.250989\pi\)
\(594\) 134498. 0.0156405
\(595\) −729367. −0.0844606
\(596\) −7.13598e6 −0.822883
\(597\) 1.80426e6 0.207187
\(598\) −1.40452e6 −0.160611
\(599\) 1.59735e7 1.81900 0.909502 0.415700i \(-0.136463\pi\)
0.909502 + 0.415700i \(0.136463\pi\)
\(600\) 267461. 0.0303307
\(601\) 7.82358e6 0.883526 0.441763 0.897132i \(-0.354353\pi\)
0.441763 + 0.897132i \(0.354353\pi\)
\(602\) 111727. 0.0125651
\(603\) −4.59948e6 −0.515129
\(604\) −1.22966e7 −1.37149
\(605\) −3.76778e6 −0.418502
\(606\) −51032.0 −0.00564496
\(607\) 3.67967e6 0.405356 0.202678 0.979245i \(-0.435036\pi\)
0.202678 + 0.979245i \(0.435036\pi\)
\(608\) −5.45813e6 −0.598804
\(609\) −2.14744e6 −0.234627
\(610\) −117082. −0.0127398
\(611\) −4.72204e6 −0.511713
\(612\) 1.87797e6 0.202680
\(613\) 4.33193e6 0.465619 0.232810 0.972522i \(-0.425208\pi\)
0.232810 + 0.972522i \(0.425208\pi\)
\(614\) −210434. −0.0225265
\(615\) −521220. −0.0555691
\(616\) −609025. −0.0646671
\(617\) −6.45241e6 −0.682353 −0.341176 0.939999i \(-0.610825\pi\)
−0.341176 + 0.939999i \(0.610825\pi\)
\(618\) 53691.0 0.00565497
\(619\) 7.05387e6 0.739947 0.369974 0.929042i \(-0.379367\pi\)
0.369974 + 0.929042i \(0.379367\pi\)
\(620\) −5.49500e6 −0.574101
\(621\) −6.06933e6 −0.631556
\(622\) 1.23791e6 0.128296
\(623\) −3.07390e6 −0.317299
\(624\) −1.75542e6 −0.180476
\(625\) 3.15781e6 0.323359
\(626\) −315687. −0.0321974
\(627\) 1.49743e6 0.152117
\(628\) 1.17021e7 1.18403
\(629\) −836461. −0.0842984
\(630\) 391179. 0.0392667
\(631\) 1.46818e7 1.46793 0.733966 0.679186i \(-0.237667\pi\)
0.733966 + 0.679186i \(0.237667\pi\)
\(632\) 2.41702e6 0.240706
\(633\) −1.13604e6 −0.112689
\(634\) −1.78859e6 −0.176721
\(635\) 3.02331e6 0.297542
\(636\) 2.94922e6 0.289111
\(637\) 3.50535e6 0.342281
\(638\) −624773. −0.0607674
\(639\) 1.12459e7 1.08954
\(640\) −2.06057e6 −0.198855
\(641\) 3.24675e6 0.312107 0.156054 0.987749i \(-0.450123\pi\)
0.156054 + 0.987749i \(0.450123\pi\)
\(642\) 148060. 0.0141775
\(643\) −7.57964e6 −0.722971 −0.361486 0.932378i \(-0.617730\pi\)
−0.361486 + 0.932378i \(0.617730\pi\)
\(644\) 1.36680e7 1.29865
\(645\) −156684. −0.0148294
\(646\) −457519. −0.0431348
\(647\) 6.13290e6 0.575977 0.287988 0.957634i \(-0.407014\pi\)
0.287988 + 0.957634i \(0.407014\pi\)
\(648\) −1.94089e6 −0.181578
\(649\) 7.16193e6 0.667449
\(650\) 795223. 0.0738254
\(651\) −1.98648e6 −0.183710
\(652\) −1.65332e7 −1.52313
\(653\) −1.13174e6 −0.103864 −0.0519318 0.998651i \(-0.516538\pi\)
−0.0519318 + 0.998651i \(0.516538\pi\)
\(654\) 104769. 0.00957827
\(655\) 3.86308e6 0.351828
\(656\) −6.09864e6 −0.553316
\(657\) −9.93045e6 −0.897544
\(658\) −492902. −0.0443809
\(659\) 1.03411e7 0.927586 0.463793 0.885944i \(-0.346488\pi\)
0.463793 + 0.885944i \(0.346488\pi\)
\(660\) 424765. 0.0379567
\(661\) −8.30708e6 −0.739511 −0.369756 0.929129i \(-0.620559\pi\)
−0.369756 + 0.929129i \(0.620559\pi\)
\(662\) −164444. −0.0145838
\(663\) −449465. −0.0397111
\(664\) −2.61185e6 −0.229894
\(665\) 8.88471e6 0.779093
\(666\) 448616. 0.0391913
\(667\) 2.81933e7 2.45376
\(668\) 545587. 0.0473067
\(669\) −2.55544e6 −0.220750
\(670\) 317856. 0.0273554
\(671\) 1.14803e6 0.0984344
\(672\) −559708. −0.0478121
\(673\) −1.13225e7 −0.963617 −0.481809 0.876276i \(-0.660020\pi\)
−0.481809 + 0.876276i \(0.660020\pi\)
\(674\) 609468. 0.0516775
\(675\) 3.43639e6 0.290298
\(676\) 1.14576e6 0.0964331
\(677\) 2.12894e7 1.78522 0.892609 0.450831i \(-0.148872\pi\)
0.892609 + 0.450831i \(0.148872\pi\)
\(678\) −414642. −0.0346417
\(679\) 9.27407e6 0.771962
\(680\) −260954. −0.0216417
\(681\) −1.11124e6 −0.0918205
\(682\) −577943. −0.0475799
\(683\) −1.18724e7 −0.973841 −0.486920 0.873446i \(-0.661880\pi\)
−0.486920 + 0.873446i \(0.661880\pi\)
\(684\) −2.28763e7 −1.86959
\(685\) −1.77866e6 −0.144833
\(686\) 1.38147e6 0.112081
\(687\) −324468. −0.0262289
\(688\) −1.83331e6 −0.147661
\(689\) 1.76315e7 1.41495
\(690\) 205601. 0.0164400
\(691\) −1.96927e7 −1.56896 −0.784478 0.620157i \(-0.787069\pi\)
−0.784478 + 0.620157i \(0.787069\pi\)
\(692\) 1.80985e7 1.43673
\(693\) −3.83566e6 −0.303394
\(694\) 1.11207e6 0.0876465
\(695\) −7.80329e6 −0.612796
\(696\) −768314. −0.0601195
\(697\) −1.56152e6 −0.121749
\(698\) 852859. 0.0662581
\(699\) 2.18295e6 0.168986
\(700\) −7.73870e6 −0.596930
\(701\) −1.36097e7 −1.04605 −0.523027 0.852316i \(-0.675197\pi\)
−0.523027 + 0.852316i \(0.675197\pi\)
\(702\) 491770. 0.0376634
\(703\) 1.01893e7 0.777597
\(704\) 4.86051e6 0.369615
\(705\) 691237. 0.0523786
\(706\) 505464. 0.0381662
\(707\) 2.96895e6 0.223385
\(708\) 4.38019e6 0.328405
\(709\) 1.30274e7 0.973290 0.486645 0.873600i \(-0.338220\pi\)
0.486645 + 0.873600i \(0.338220\pi\)
\(710\) −777170. −0.0578589
\(711\) 1.52225e7 1.12931
\(712\) −1.09978e6 −0.0813029
\(713\) 2.60801e7 1.92125
\(714\) −46916.6 −0.00344414
\(715\) 2.53939e6 0.185765
\(716\) 6.07764e6 0.443050
\(717\) −215310. −0.0156411
\(718\) 2.01321e6 0.145740
\(719\) −9.11723e6 −0.657719 −0.328860 0.944379i \(-0.606664\pi\)
−0.328860 + 0.944379i \(0.606664\pi\)
\(720\) −6.41879e6 −0.461447
\(721\) −3.12365e6 −0.223781
\(722\) 4.13026e6 0.294873
\(723\) 136406. 0.00970480
\(724\) −3.50809e6 −0.248728
\(725\) −1.59628e7 −1.12788
\(726\) −242363. −0.0170657
\(727\) −1.52424e7 −1.06959 −0.534797 0.844981i \(-0.679612\pi\)
−0.534797 + 0.844981i \(0.679612\pi\)
\(728\) −2.22680e6 −0.155723
\(729\) −1.11404e7 −0.776396
\(730\) 686262. 0.0476631
\(731\) −469408. −0.0324906
\(732\) 702128. 0.0484327
\(733\) −1.61394e7 −1.10950 −0.554751 0.832016i \(-0.687187\pi\)
−0.554751 + 0.832016i \(0.687187\pi\)
\(734\) −1.62470e6 −0.111310
\(735\) −513132. −0.0350357
\(736\) 7.34828e6 0.500025
\(737\) −3.11670e6 −0.211362
\(738\) 837486. 0.0566026
\(739\) −9.24471e6 −0.622704 −0.311352 0.950295i \(-0.600782\pi\)
−0.311352 + 0.950295i \(0.600782\pi\)
\(740\) 2.89031e6 0.194028
\(741\) 5.47510e6 0.366308
\(742\) 1.84043e6 0.122719
\(743\) −3.89612e6 −0.258917 −0.129459 0.991585i \(-0.541324\pi\)
−0.129459 + 0.991585i \(0.541324\pi\)
\(744\) −710724. −0.0470727
\(745\) 6.24501e6 0.412233
\(746\) −964909. −0.0634804
\(747\) −1.64495e7 −1.07858
\(748\) 1.27255e6 0.0831613
\(749\) −8.61387e6 −0.561040
\(750\) −270729. −0.0175744
\(751\) −2.36303e7 −1.52887 −0.764433 0.644704i \(-0.776981\pi\)
−0.764433 + 0.644704i \(0.776981\pi\)
\(752\) 8.08795e6 0.521547
\(753\) 3.24155e6 0.208337
\(754\) −2.28437e6 −0.146332
\(755\) 1.07613e7 0.687062
\(756\) −4.78565e6 −0.304535
\(757\) 2.86009e6 0.181401 0.0907005 0.995878i \(-0.471089\pi\)
0.0907005 + 0.995878i \(0.471089\pi\)
\(758\) −1.37260e6 −0.0867704
\(759\) −2.01599e6 −0.127024
\(760\) 3.17878e6 0.199630
\(761\) −2.90585e7 −1.81891 −0.909456 0.415800i \(-0.863502\pi\)
−0.909456 + 0.415800i \(0.863502\pi\)
\(762\) 194475. 0.0121332
\(763\) −6.09526e6 −0.379036
\(764\) −9.36097e6 −0.580213
\(765\) −1.64350e6 −0.101535
\(766\) −2.41180e6 −0.148515
\(767\) 2.61864e7 1.60726
\(768\) 2.87200e6 0.175704
\(769\) −1.85731e7 −1.13258 −0.566290 0.824206i \(-0.691622\pi\)
−0.566290 + 0.824206i \(0.691622\pi\)
\(770\) 265071. 0.0161115
\(771\) −2.95468e6 −0.179009
\(772\) 1.19026e7 0.718781
\(773\) 1.38698e7 0.834872 0.417436 0.908706i \(-0.362929\pi\)
0.417436 + 0.908706i \(0.362929\pi\)
\(774\) 251756. 0.0151052
\(775\) −1.47663e7 −0.883113
\(776\) 3.31808e6 0.197803
\(777\) 1.04487e6 0.0620881
\(778\) −3.07566e6 −0.182175
\(779\) 1.90215e7 1.12306
\(780\) 1.55308e6 0.0914023
\(781\) 7.62045e6 0.447047
\(782\) 615958. 0.0360192
\(783\) −9.87145e6 −0.575409
\(784\) −6.00400e6 −0.348859
\(785\) −1.02410e7 −0.593154
\(786\) 248493. 0.0143469
\(787\) −1.95803e7 −1.12689 −0.563445 0.826154i \(-0.690524\pi\)
−0.563445 + 0.826154i \(0.690524\pi\)
\(788\) 2.32015e7 1.33107
\(789\) 2.61574e6 0.149590
\(790\) −1.05198e6 −0.0599707
\(791\) 2.41232e7 1.37086
\(792\) −1.37233e6 −0.0777400
\(793\) 4.19757e6 0.237037
\(794\) 2.16457e6 0.121849
\(795\) −2.58099e6 −0.144833
\(796\) −1.86777e7 −1.04482
\(797\) −2.66479e7 −1.48600 −0.742998 0.669293i \(-0.766597\pi\)
−0.742998 + 0.669293i \(0.766597\pi\)
\(798\) 571510. 0.0317699
\(799\) 2.07087e6 0.114759
\(800\) −4.16052e6 −0.229839
\(801\) −6.92646e6 −0.381443
\(802\) 2.09811e6 0.115184
\(803\) −6.72907e6 −0.368270
\(804\) −1.90615e6 −0.103996
\(805\) −1.19615e7 −0.650573
\(806\) −2.11315e6 −0.114576
\(807\) 4.74734e6 0.256606
\(808\) 1.06223e6 0.0572390
\(809\) 4.31569e6 0.231835 0.115917 0.993259i \(-0.463019\pi\)
0.115917 + 0.993259i \(0.463019\pi\)
\(810\) 844750. 0.0452393
\(811\) 1.07890e7 0.576007 0.288003 0.957629i \(-0.407009\pi\)
0.288003 + 0.957629i \(0.407009\pi\)
\(812\) 2.22304e7 1.18320
\(813\) −12775.8 −0.000677894 0
\(814\) 303991. 0.0160805
\(815\) 1.44689e7 0.763031
\(816\) 769847. 0.0404743
\(817\) 5.71805e6 0.299704
\(818\) 963319. 0.0503369
\(819\) −1.40244e7 −0.730594
\(820\) 5.39568e6 0.280228
\(821\) −2.12759e7 −1.10162 −0.550808 0.834632i \(-0.685680\pi\)
−0.550808 + 0.834632i \(0.685680\pi\)
\(822\) −114413. −0.00590602
\(823\) 7.04803e6 0.362717 0.181359 0.983417i \(-0.441950\pi\)
0.181359 + 0.983417i \(0.441950\pi\)
\(824\) −1.11758e6 −0.0573404
\(825\) 1.14144e6 0.0583870
\(826\) 2.73342e6 0.139398
\(827\) 2.00838e7 1.02113 0.510566 0.859839i \(-0.329436\pi\)
0.510566 + 0.859839i \(0.329436\pi\)
\(828\) 3.07984e7 1.56118
\(829\) 2.04193e7 1.03194 0.515970 0.856606i \(-0.327431\pi\)
0.515970 + 0.856606i \(0.327431\pi\)
\(830\) 1.13678e6 0.0572769
\(831\) −4.88575e6 −0.245431
\(832\) 1.77716e7 0.890059
\(833\) −1.53729e6 −0.0767614
\(834\) −501948. −0.0249887
\(835\) −477467. −0.0236988
\(836\) −1.55014e7 −0.767108
\(837\) −9.13153e6 −0.450537
\(838\) 729160. 0.0358685
\(839\) −4.92445e6 −0.241520 −0.120760 0.992682i \(-0.538533\pi\)
−0.120760 + 0.992682i \(0.538533\pi\)
\(840\) 325970. 0.0159397
\(841\) 2.53438e7 1.23561
\(842\) 1.81882e6 0.0884114
\(843\) 4.90271e6 0.237612
\(844\) 1.17603e7 0.568278
\(845\) −1.00270e6 −0.0483092
\(846\) −1.11066e6 −0.0533528
\(847\) 1.41003e7 0.675334
\(848\) −3.01994e7 −1.44214
\(849\) 6.25940e6 0.298032
\(850\) −348749. −0.0165564
\(851\) −1.37178e7 −0.649324
\(852\) 4.66062e6 0.219961
\(853\) −1.25088e7 −0.588632 −0.294316 0.955708i \(-0.595092\pi\)
−0.294316 + 0.955708i \(0.595092\pi\)
\(854\) 438157. 0.0205582
\(855\) 2.00201e7 0.936591
\(856\) −3.08188e6 −0.143758
\(857\) 2.21879e7 1.03196 0.515982 0.856600i \(-0.327427\pi\)
0.515982 + 0.856600i \(0.327427\pi\)
\(858\) 163347. 0.00757517
\(859\) −1.40835e7 −0.651218 −0.325609 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(860\) 1.62199e6 0.0747830
\(861\) 1.95058e6 0.0896716
\(862\) 143631. 0.00658384
\(863\) −7.94491e6 −0.363130 −0.181565 0.983379i \(-0.558116\pi\)
−0.181565 + 0.983379i \(0.558116\pi\)
\(864\) −2.57289e6 −0.117256
\(865\) −1.58387e7 −0.719748
\(866\) −4.40579e6 −0.199632
\(867\) −4.14535e6 −0.187290
\(868\) 2.05641e7 0.926424
\(869\) 1.03151e7 0.463364
\(870\) 334399. 0.0149784
\(871\) −1.13957e7 −0.508973
\(872\) −2.18077e6 −0.0971221
\(873\) 2.08974e7 0.928019
\(874\) −7.50323e6 −0.332253
\(875\) 1.57505e7 0.695465
\(876\) −4.11546e6 −0.181200
\(877\) −4.03970e6 −0.177358 −0.0886788 0.996060i \(-0.528264\pi\)
−0.0886788 + 0.996060i \(0.528264\pi\)
\(878\) −2.19200e6 −0.0959630
\(879\) 4.12966e6 0.180278
\(880\) −4.34950e6 −0.189336
\(881\) −1.99840e7 −0.867447 −0.433724 0.901046i \(-0.642801\pi\)
−0.433724 + 0.901046i \(0.642801\pi\)
\(882\) 824489. 0.0356873
\(883\) 1.66868e7 0.720230 0.360115 0.932908i \(-0.382737\pi\)
0.360115 + 0.932908i \(0.382737\pi\)
\(884\) 4.65286e6 0.200258
\(885\) −3.83330e6 −0.164518
\(886\) −600155. −0.0256850
\(887\) −1.79964e6 −0.0768027 −0.0384014 0.999262i \(-0.512227\pi\)
−0.0384014 + 0.999262i \(0.512227\pi\)
\(888\) 373833. 0.0159091
\(889\) −1.13142e7 −0.480141
\(890\) 478666. 0.0202562
\(891\) −8.28310e6 −0.349541
\(892\) 2.64539e7 1.11321
\(893\) −2.52261e7 −1.05858
\(894\) 401711. 0.0168101
\(895\) −5.31881e6 −0.221951
\(896\) 7.71131e6 0.320892
\(897\) −7.37114e6 −0.305882
\(898\) −2.50561e6 −0.103687
\(899\) 4.24179e7 1.75045
\(900\) −1.74377e7 −0.717603
\(901\) −7.73238e6 −0.317323
\(902\) 567496. 0.0232245
\(903\) 586362. 0.0239302
\(904\) 8.63080e6 0.351261
\(905\) 3.07008e6 0.124603
\(906\) 692219. 0.0280171
\(907\) −3.64748e6 −0.147223 −0.0736113 0.997287i \(-0.523452\pi\)
−0.0736113 + 0.997287i \(0.523452\pi\)
\(908\) 1.15036e7 0.463039
\(909\) 6.68999e6 0.268544
\(910\) 969185. 0.0387974
\(911\) 4.92223e6 0.196502 0.0982508 0.995162i \(-0.468675\pi\)
0.0982508 + 0.995162i \(0.468675\pi\)
\(912\) −9.37781e6 −0.373348
\(913\) −1.11465e7 −0.442551
\(914\) −832352. −0.0329565
\(915\) −614463. −0.0242629
\(916\) 3.35890e6 0.132269
\(917\) −1.44569e7 −0.567743
\(918\) −215668. −0.00844655
\(919\) 1.52960e7 0.597432 0.298716 0.954342i \(-0.403442\pi\)
0.298716 + 0.954342i \(0.403442\pi\)
\(920\) −4.27959e6 −0.166699
\(921\) −1.10439e6 −0.0429016
\(922\) −1.78552e6 −0.0691731
\(923\) 2.78629e7 1.07652
\(924\) −1.58961e6 −0.0612505
\(925\) 7.76689e6 0.298464
\(926\) −2.05065e6 −0.0785893
\(927\) −7.03856e6 −0.269020
\(928\) 1.19516e7 0.455571
\(929\) 3.10755e7 1.18135 0.590674 0.806910i \(-0.298862\pi\)
0.590674 + 0.806910i \(0.298862\pi\)
\(930\) 309334. 0.0117279
\(931\) 1.87263e7 0.708073
\(932\) −2.25979e7 −0.852175
\(933\) 6.49676e6 0.244339
\(934\) −4.02188e6 −0.150856
\(935\) −1.11366e6 −0.0416606
\(936\) −5.01767e6 −0.187203
\(937\) −3.19233e7 −1.18784 −0.593922 0.804523i \(-0.702421\pi\)
−0.593922 + 0.804523i \(0.702421\pi\)
\(938\) −1.18952e6 −0.0441432
\(939\) −1.65678e6 −0.0613198
\(940\) −7.15569e6 −0.264139
\(941\) −4.89415e7 −1.80179 −0.900894 0.434040i \(-0.857088\pi\)
−0.900894 + 0.434040i \(0.857088\pi\)
\(942\) −658752. −0.0241877
\(943\) −2.56087e7 −0.937795
\(944\) −4.48522e7 −1.63815
\(945\) 4.18813e6 0.152560
\(946\) 170595. 0.00619781
\(947\) −1.45580e7 −0.527505 −0.263753 0.964590i \(-0.584960\pi\)
−0.263753 + 0.964590i \(0.584960\pi\)
\(948\) 6.30863e6 0.227989
\(949\) −2.46037e7 −0.886818
\(950\) 4.24825e6 0.152722
\(951\) −9.38680e6 −0.336563
\(952\) 976572. 0.0349230
\(953\) −2.71191e7 −0.967261 −0.483631 0.875272i \(-0.660682\pi\)
−0.483631 + 0.875272i \(0.660682\pi\)
\(954\) 4.14708e6 0.147527
\(955\) 8.19219e6 0.290664
\(956\) 2.22889e6 0.0788759
\(957\) −3.27891e6 −0.115731
\(958\) −2.55062e6 −0.0897907
\(959\) 6.65633e6 0.233716
\(960\) −2.60150e6 −0.0911059
\(961\) 1.06093e7 0.370576
\(962\) 1.11149e6 0.0387229
\(963\) −1.94098e7 −0.674458
\(964\) −1.41207e6 −0.0489401
\(965\) −1.04164e7 −0.360082
\(966\) −769424. −0.0265291
\(967\) 1.16883e7 0.401963 0.200982 0.979595i \(-0.435587\pi\)
0.200982 + 0.979595i \(0.435587\pi\)
\(968\) 5.04480e6 0.173044
\(969\) −2.40113e6 −0.0821499
\(970\) −1.44415e6 −0.0492815
\(971\) 3.07622e6 0.104706 0.0523528 0.998629i \(-0.483328\pi\)
0.0523528 + 0.998629i \(0.483328\pi\)
\(972\) −1.62812e7 −0.552739
\(973\) 2.92024e7 0.988865
\(974\) 2.38021e6 0.0803930
\(975\) 4.17346e6 0.140600
\(976\) −7.18964e6 −0.241592
\(977\) 3.81186e7 1.27762 0.638809 0.769365i \(-0.279427\pi\)
0.638809 + 0.769365i \(0.279427\pi\)
\(978\) 930714. 0.0311150
\(979\) −4.69350e6 −0.156509
\(980\) 5.31195e6 0.176680
\(981\) −1.37345e7 −0.455661
\(982\) −2.72616e6 −0.0902137
\(983\) −3.14497e7 −1.03809 −0.519043 0.854748i \(-0.673711\pi\)
−0.519043 + 0.854748i \(0.673711\pi\)
\(984\) 697878. 0.0229769
\(985\) −2.03047e7 −0.666815
\(986\) 1.00182e6 0.0328170
\(987\) −2.58683e6 −0.0845231
\(988\) −5.66783e7 −1.84725
\(989\) −7.69821e6 −0.250264
\(990\) 597288. 0.0193685
\(991\) 1.08378e7 0.350555 0.175277 0.984519i \(-0.443918\pi\)
0.175277 + 0.984519i \(0.443918\pi\)
\(992\) 1.10558e7 0.356705
\(993\) −863027. −0.0277748
\(994\) 2.90842e6 0.0933665
\(995\) 1.63457e7 0.523414
\(996\) −6.81715e6 −0.217748
\(997\) −1.37214e7 −0.437179 −0.218590 0.975817i \(-0.570146\pi\)
−0.218590 + 0.975817i \(0.570146\pi\)
\(998\) 5.63211e6 0.178997
\(999\) 4.80308e6 0.152267
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 43.6.a.a.1.5 8
3.2 odd 2 387.6.a.c.1.4 8
4.3 odd 2 688.6.a.e.1.4 8
5.4 even 2 1075.6.a.a.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.a.1.5 8 1.1 even 1 trivial
387.6.a.c.1.4 8 3.2 odd 2
688.6.a.e.1.4 8 4.3 odd 2
1075.6.a.a.1.4 8 5.4 even 2