Properties

Label 33.6.a.c.1.2
Level $33$
Weight $6$
Character 33.1
Self dual yes
Analytic conductor $5.293$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.29266605383\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
Defining polynomial: \(x^{2} - x - 44\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-6.15207\) of defining polynomial
Character \(\chi\) \(=\) 33.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.15207 q^{2} -9.00000 q^{3} -14.7603 q^{4} -37.5207 q^{5} -37.3686 q^{6} -76.4793 q^{7} -194.152 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+4.15207 q^{2} -9.00000 q^{3} -14.7603 q^{4} -37.5207 q^{5} -37.3686 q^{6} -76.4793 q^{7} -194.152 q^{8} +81.0000 q^{9} -155.788 q^{10} -121.000 q^{11} +132.843 q^{12} +169.779 q^{13} -317.547 q^{14} +337.686 q^{15} -333.802 q^{16} -0.875959 q^{17} +336.317 q^{18} -817.825 q^{19} +553.818 q^{20} +688.314 q^{21} -502.400 q^{22} +749.657 q^{23} +1747.37 q^{24} -1717.20 q^{25} +704.932 q^{26} -729.000 q^{27} +1128.86 q^{28} +6040.65 q^{29} +1402.10 q^{30} -1475.69 q^{31} +4826.90 q^{32} +1089.00 q^{33} -3.63704 q^{34} +2869.56 q^{35} -1195.59 q^{36} -15862.7 q^{37} -3395.66 q^{38} -1528.01 q^{39} +7284.72 q^{40} -7626.27 q^{41} +2857.93 q^{42} -18279.5 q^{43} +1786.00 q^{44} -3039.17 q^{45} +3112.62 q^{46} -12878.1 q^{47} +3004.22 q^{48} -10957.9 q^{49} -7129.93 q^{50} +7.88363 q^{51} -2505.99 q^{52} +21763.5 q^{53} -3026.86 q^{54} +4540.00 q^{55} +14848.6 q^{56} +7360.42 q^{57} +25081.2 q^{58} -1168.45 q^{59} -4984.36 q^{60} +14026.3 q^{61} -6127.16 q^{62} -6194.83 q^{63} +30723.3 q^{64} -6370.21 q^{65} +4521.60 q^{66} +36954.6 q^{67} +12.9295 q^{68} -6746.91 q^{69} +11914.6 q^{70} -37513.1 q^{71} -15726.3 q^{72} +80452.0 q^{73} -65862.9 q^{74} +15454.8 q^{75} +12071.4 q^{76} +9254.00 q^{77} -6344.39 q^{78} -62139.9 q^{79} +12524.5 q^{80} +6561.00 q^{81} -31664.8 q^{82} -11905.8 q^{83} -10159.7 q^{84} +32.8666 q^{85} -75897.7 q^{86} -54365.9 q^{87} +23492.4 q^{88} +146348. q^{89} -12618.9 q^{90} -12984.5 q^{91} -11065.2 q^{92} +13281.2 q^{93} -53470.8 q^{94} +30685.3 q^{95} -43442.1 q^{96} -138129. q^{97} -45498.0 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 5q^{2} - 18q^{3} + 37q^{4} + 58q^{5} + 45q^{6} - 286q^{7} - 375q^{8} + 162q^{9} + O(q^{10}) \) \( 2q - 5q^{2} - 18q^{3} + 37q^{4} + 58q^{5} + 45q^{6} - 286q^{7} - 375q^{8} + 162q^{9} - 1030q^{10} - 242q^{11} - 333q^{12} - 166q^{13} + 1600q^{14} - 522q^{15} - 335q^{16} - 800q^{17} - 405q^{18} - 1476q^{19} + 5498q^{20} + 2574q^{21} + 605q^{22} - 3370q^{23} + 3375q^{24} + 4282q^{25} + 3778q^{26} - 1458q^{27} - 9716q^{28} + 6600q^{29} + 9270q^{30} - 7528q^{31} + 10625q^{32} + 2178q^{33} + 7310q^{34} - 17144q^{35} + 2997q^{36} - 29916q^{37} + 2628q^{38} + 1494q^{39} - 9990q^{40} - 5780q^{41} - 14400q^{42} - 16656q^{43} - 4477q^{44} + 4698q^{45} + 40816q^{46} + 7850q^{47} + 3015q^{48} + 16134q^{49} - 62035q^{50} + 7200q^{51} - 19886q^{52} + 14178q^{53} + 3645q^{54} - 7018q^{55} + 52740q^{56} + 13284q^{57} + 19962q^{58} + 17300q^{59} - 49482q^{60} - 2946q^{61} + 49264q^{62} - 23166q^{63} - 22303q^{64} - 38444q^{65} - 5445q^{66} + 31336q^{67} - 41350q^{68} + 30330q^{69} + 195080q^{70} - 33810q^{71} - 30375q^{72} + 60644q^{73} + 62754q^{74} - 38538q^{75} - 21996q^{76} + 34606q^{77} - 34002q^{78} + 1870q^{79} + 12410q^{80} + 13122q^{81} - 48562q^{82} - 58296q^{83} + 87444q^{84} - 76300q^{85} - 90756q^{86} - 59400q^{87} + 45375q^{88} + 92388q^{89} - 83430q^{90} + 57368q^{91} - 224300q^{92} + 67752q^{93} - 243176q^{94} - 32184q^{95} - 95625q^{96} + 7120q^{97} - 293445q^{98} - 19602q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.15207 0.733989 0.366994 0.930223i \(-0.380387\pi\)
0.366994 + 0.930223i \(0.380387\pi\)
\(3\) −9.00000 −0.577350
\(4\) −14.7603 −0.461261
\(5\) −37.5207 −0.671190 −0.335595 0.942006i \(-0.608937\pi\)
−0.335595 + 0.942006i \(0.608937\pi\)
\(6\) −37.3686 −0.423769
\(7\) −76.4793 −0.589928 −0.294964 0.955508i \(-0.595308\pi\)
−0.294964 + 0.955508i \(0.595308\pi\)
\(8\) −194.152 −1.07255
\(9\) 81.0000 0.333333
\(10\) −155.788 −0.492646
\(11\) −121.000 −0.301511
\(12\) 132.843 0.266309
\(13\) 169.779 0.278628 0.139314 0.990248i \(-0.455510\pi\)
0.139314 + 0.990248i \(0.455510\pi\)
\(14\) −317.547 −0.433000
\(15\) 337.686 0.387512
\(16\) −333.802 −0.325978
\(17\) −0.875959 −0.000735126 0 −0.000367563 1.00000i \(-0.500117\pi\)
−0.000367563 1.00000i \(0.500117\pi\)
\(18\) 336.317 0.244663
\(19\) −817.825 −0.519728 −0.259864 0.965645i \(-0.583678\pi\)
−0.259864 + 0.965645i \(0.583678\pi\)
\(20\) 553.818 0.309594
\(21\) 688.314 0.340595
\(22\) −502.400 −0.221306
\(23\) 749.657 0.295490 0.147745 0.989025i \(-0.452799\pi\)
0.147745 + 0.989025i \(0.452799\pi\)
\(24\) 1747.37 0.619236
\(25\) −1717.20 −0.549504
\(26\) 704.932 0.204510
\(27\) −729.000 −0.192450
\(28\) 1128.86 0.272110
\(29\) 6040.65 1.33379 0.666897 0.745150i \(-0.267622\pi\)
0.666897 + 0.745150i \(0.267622\pi\)
\(30\) 1402.10 0.284429
\(31\) −1475.69 −0.275798 −0.137899 0.990446i \(-0.544035\pi\)
−0.137899 + 0.990446i \(0.544035\pi\)
\(32\) 4826.90 0.833284
\(33\) 1089.00 0.174078
\(34\) −3.63704 −0.000539574 0
\(35\) 2869.56 0.395954
\(36\) −1195.59 −0.153754
\(37\) −15862.7 −1.90490 −0.952450 0.304694i \(-0.901446\pi\)
−0.952450 + 0.304694i \(0.901446\pi\)
\(38\) −3395.66 −0.381475
\(39\) −1528.01 −0.160866
\(40\) 7284.72 0.719884
\(41\) −7626.27 −0.708521 −0.354260 0.935147i \(-0.615267\pi\)
−0.354260 + 0.935147i \(0.615267\pi\)
\(42\) 2857.93 0.249993
\(43\) −18279.5 −1.50762 −0.753812 0.657090i \(-0.771787\pi\)
−0.753812 + 0.657090i \(0.771787\pi\)
\(44\) 1786.00 0.139075
\(45\) −3039.17 −0.223730
\(46\) 3112.62 0.216886
\(47\) −12878.1 −0.850370 −0.425185 0.905106i \(-0.639791\pi\)
−0.425185 + 0.905106i \(0.639791\pi\)
\(48\) 3004.22 0.188204
\(49\) −10957.9 −0.651985
\(50\) −7129.93 −0.403330
\(51\) 7.88363 0.000424425 0
\(52\) −2505.99 −0.128520
\(53\) 21763.5 1.06424 0.532118 0.846670i \(-0.321396\pi\)
0.532118 + 0.846670i \(0.321396\pi\)
\(54\) −3026.86 −0.141256
\(55\) 4540.00 0.202371
\(56\) 14848.6 0.632726
\(57\) 7360.42 0.300065
\(58\) 25081.2 0.978990
\(59\) −1168.45 −0.0436999 −0.0218500 0.999761i \(-0.506956\pi\)
−0.0218500 + 0.999761i \(0.506956\pi\)
\(60\) −4984.36 −0.178744
\(61\) 14026.3 0.482635 0.241318 0.970446i \(-0.422420\pi\)
0.241318 + 0.970446i \(0.422420\pi\)
\(62\) −6127.16 −0.202432
\(63\) −6194.83 −0.196643
\(64\) 30723.3 0.937600
\(65\) −6370.21 −0.187012
\(66\) 4521.60 0.127771
\(67\) 36954.6 1.00573 0.502865 0.864365i \(-0.332279\pi\)
0.502865 + 0.864365i \(0.332279\pi\)
\(68\) 12.9295 0.000339084 0
\(69\) −6746.91 −0.170601
\(70\) 11914.6 0.290626
\(71\) −37513.1 −0.883155 −0.441578 0.897223i \(-0.645581\pi\)
−0.441578 + 0.897223i \(0.645581\pi\)
\(72\) −15726.3 −0.357516
\(73\) 80452.0 1.76697 0.883486 0.468458i \(-0.155190\pi\)
0.883486 + 0.468458i \(0.155190\pi\)
\(74\) −65862.9 −1.39818
\(75\) 15454.8 0.317256
\(76\) 12071.4 0.239730
\(77\) 9254.00 0.177870
\(78\) −6344.39 −0.118074
\(79\) −62139.9 −1.12022 −0.560109 0.828419i \(-0.689241\pi\)
−0.560109 + 0.828419i \(0.689241\pi\)
\(80\) 12524.5 0.218793
\(81\) 6561.00 0.111111
\(82\) −31664.8 −0.520046
\(83\) −11905.8 −0.189699 −0.0948495 0.995492i \(-0.530237\pi\)
−0.0948495 + 0.995492i \(0.530237\pi\)
\(84\) −10159.7 −0.157103
\(85\) 32.8666 0.000493409 0
\(86\) −75897.7 −1.10658
\(87\) −54365.9 −0.770066
\(88\) 23492.4 0.323386
\(89\) 146348. 1.95844 0.979220 0.202800i \(-0.0650041\pi\)
0.979220 + 0.202800i \(0.0650041\pi\)
\(90\) −12618.9 −0.164215
\(91\) −12984.5 −0.164370
\(92\) −11065.2 −0.136298
\(93\) 13281.2 0.159232
\(94\) −53470.8 −0.624162
\(95\) 30685.3 0.348836
\(96\) −43442.1 −0.481097
\(97\) −138129. −1.49058 −0.745291 0.666740i \(-0.767689\pi\)
−0.745291 + 0.666740i \(0.767689\pi\)
\(98\) −45498.0 −0.478550
\(99\) −9801.00 −0.100504
\(100\) 25346.4 0.253464
\(101\) 36615.1 0.357155 0.178577 0.983926i \(-0.442850\pi\)
0.178577 + 0.983926i \(0.442850\pi\)
\(102\) 32.7334 0.000311523 0
\(103\) 45198.5 0.419789 0.209895 0.977724i \(-0.432688\pi\)
0.209895 + 0.977724i \(0.432688\pi\)
\(104\) −32962.9 −0.298842
\(105\) −25826.0 −0.228604
\(106\) 90363.4 0.781138
\(107\) −28086.8 −0.237161 −0.118580 0.992944i \(-0.537834\pi\)
−0.118580 + 0.992944i \(0.537834\pi\)
\(108\) 10760.3 0.0887696
\(109\) −33852.9 −0.272916 −0.136458 0.990646i \(-0.543572\pi\)
−0.136458 + 0.990646i \(0.543572\pi\)
\(110\) 18850.4 0.148538
\(111\) 142764. 1.09979
\(112\) 25528.9 0.192304
\(113\) 168559. 1.24181 0.620904 0.783886i \(-0.286766\pi\)
0.620904 + 0.783886i \(0.286766\pi\)
\(114\) 30561.0 0.220244
\(115\) −28127.6 −0.198330
\(116\) −89162.1 −0.615227
\(117\) 13752.1 0.0928759
\(118\) −4851.49 −0.0320752
\(119\) 66.9928 0.000433671 0
\(120\) −65562.4 −0.415625
\(121\) 14641.0 0.0909091
\(122\) 58238.2 0.354249
\(123\) 68636.4 0.409065
\(124\) 21781.7 0.127215
\(125\) 181683. 1.04001
\(126\) −25721.3 −0.144333
\(127\) −336859. −1.85327 −0.926636 0.375960i \(-0.877313\pi\)
−0.926636 + 0.375960i \(0.877313\pi\)
\(128\) −26895.7 −0.145097
\(129\) 164515. 0.870427
\(130\) −26449.5 −0.137265
\(131\) −83896.5 −0.427135 −0.213568 0.976928i \(-0.568508\pi\)
−0.213568 + 0.976928i \(0.568508\pi\)
\(132\) −16074.0 −0.0802952
\(133\) 62546.7 0.306602
\(134\) 153438. 0.738195
\(135\) 27352.6 0.129171
\(136\) 170.069 0.000788458 0
\(137\) −196507. −0.894492 −0.447246 0.894411i \(-0.647595\pi\)
−0.447246 + 0.894411i \(0.647595\pi\)
\(138\) −28013.6 −0.125219
\(139\) 263232. 1.15559 0.577793 0.816183i \(-0.303914\pi\)
0.577793 + 0.816183i \(0.303914\pi\)
\(140\) −42355.6 −0.182638
\(141\) 115903. 0.490961
\(142\) −155757. −0.648226
\(143\) −20543.2 −0.0840094
\(144\) −27037.9 −0.108659
\(145\) −226649. −0.895230
\(146\) 334042. 1.29694
\(147\) 98621.2 0.376424
\(148\) 234139. 0.878655
\(149\) −337412. −1.24507 −0.622536 0.782591i \(-0.713898\pi\)
−0.622536 + 0.782591i \(0.713898\pi\)
\(150\) 64169.3 0.232862
\(151\) −157583. −0.562428 −0.281214 0.959645i \(-0.590737\pi\)
−0.281214 + 0.959645i \(0.590737\pi\)
\(152\) 158782. 0.557434
\(153\) −70.9527 −0.000245042 0
\(154\) 38423.2 0.130555
\(155\) 55368.8 0.185113
\(156\) 22553.9 0.0742011
\(157\) 5369.59 0.0173857 0.00869285 0.999962i \(-0.497233\pi\)
0.00869285 + 0.999962i \(0.497233\pi\)
\(158\) −258009. −0.822228
\(159\) −195871. −0.614437
\(160\) −181109. −0.559292
\(161\) −57333.2 −0.174318
\(162\) 27241.7 0.0815543
\(163\) −503102. −1.48316 −0.741579 0.670866i \(-0.765923\pi\)
−0.741579 + 0.670866i \(0.765923\pi\)
\(164\) 112566. 0.326813
\(165\) −40860.0 −0.116839
\(166\) −49433.9 −0.139237
\(167\) −584523. −1.62185 −0.810924 0.585151i \(-0.801035\pi\)
−0.810924 + 0.585151i \(0.801035\pi\)
\(168\) −133638. −0.365305
\(169\) −342468. −0.922367
\(170\) 136.464 0.000362157 0
\(171\) −66243.8 −0.173243
\(172\) 269811. 0.695407
\(173\) −55544.0 −0.141098 −0.0705491 0.997508i \(-0.522475\pi\)
−0.0705491 + 0.997508i \(0.522475\pi\)
\(174\) −225731. −0.565220
\(175\) 131330. 0.324168
\(176\) 40390.0 0.0982861
\(177\) 10516.1 0.0252302
\(178\) 607645. 1.43747
\(179\) 215136. 0.501858 0.250929 0.968006i \(-0.419264\pi\)
0.250929 + 0.968006i \(0.419264\pi\)
\(180\) 44859.2 0.103198
\(181\) 682113. 1.54761 0.773803 0.633427i \(-0.218352\pi\)
0.773803 + 0.633427i \(0.218352\pi\)
\(182\) −53912.7 −0.120646
\(183\) −126237. −0.278650
\(184\) −145547. −0.316927
\(185\) 595178. 1.27855
\(186\) 55144.4 0.116874
\(187\) 105.991 0.000221649 0
\(188\) 190085. 0.392242
\(189\) 55753.4 0.113532
\(190\) 127408. 0.256042
\(191\) −89697.4 −0.177908 −0.0889542 0.996036i \(-0.528352\pi\)
−0.0889542 + 0.996036i \(0.528352\pi\)
\(192\) −276509. −0.541323
\(193\) 657062. 1.26973 0.634867 0.772621i \(-0.281055\pi\)
0.634867 + 0.772621i \(0.281055\pi\)
\(194\) −573521. −1.09407
\(195\) 57331.9 0.107972
\(196\) 161742. 0.300735
\(197\) 720419. 1.32257 0.661286 0.750134i \(-0.270011\pi\)
0.661286 + 0.750134i \(0.270011\pi\)
\(198\) −40694.4 −0.0737686
\(199\) 140848. 0.252127 0.126063 0.992022i \(-0.459766\pi\)
0.126063 + 0.992022i \(0.459766\pi\)
\(200\) 333398. 0.589370
\(201\) −332592. −0.580659
\(202\) 152028. 0.262148
\(203\) −461985. −0.786842
\(204\) −116.365 −0.000195770 0
\(205\) 286143. 0.475552
\(206\) 187667. 0.308121
\(207\) 60722.2 0.0984967
\(208\) −56672.4 −0.0908266
\(209\) 98956.8 0.156704
\(210\) −107231. −0.167793
\(211\) −111929. −0.173075 −0.0865377 0.996249i \(-0.527580\pi\)
−0.0865377 + 0.996249i \(0.527580\pi\)
\(212\) −321236. −0.490890
\(213\) 337618. 0.509890
\(214\) −116618. −0.174073
\(215\) 685859. 1.01190
\(216\) 141537. 0.206412
\(217\) 112860. 0.162701
\(218\) −140559. −0.200317
\(219\) −724068. −1.02016
\(220\) −67012.0 −0.0933460
\(221\) −148.719 −0.000204826 0
\(222\) 592766. 0.807237
\(223\) −906826. −1.22113 −0.610565 0.791966i \(-0.709058\pi\)
−0.610565 + 0.791966i \(0.709058\pi\)
\(224\) −369158. −0.491578
\(225\) −139093. −0.183168
\(226\) 699867. 0.911474
\(227\) −1.35715e6 −1.74809 −0.874046 0.485844i \(-0.838512\pi\)
−0.874046 + 0.485844i \(0.838512\pi\)
\(228\) −108642. −0.138408
\(229\) −344956. −0.434685 −0.217342 0.976095i \(-0.569739\pi\)
−0.217342 + 0.976095i \(0.569739\pi\)
\(230\) −116788. −0.145572
\(231\) −83286.0 −0.102693
\(232\) −1.17281e6 −1.43056
\(233\) −1.18163e6 −1.42591 −0.712955 0.701210i \(-0.752643\pi\)
−0.712955 + 0.701210i \(0.752643\pi\)
\(234\) 57099.5 0.0681699
\(235\) 483196. 0.570760
\(236\) 17246.7 0.0201570
\(237\) 559259. 0.646759
\(238\) 278.158 0.000318310 0
\(239\) −723612. −0.819429 −0.409714 0.912214i \(-0.634372\pi\)
−0.409714 + 0.912214i \(0.634372\pi\)
\(240\) −112720. −0.126320
\(241\) 394322. 0.437329 0.218664 0.975800i \(-0.429830\pi\)
0.218664 + 0.975800i \(0.429830\pi\)
\(242\) 60790.4 0.0667262
\(243\) −59049.0 −0.0641500
\(244\) −207033. −0.222621
\(245\) 411148. 0.437606
\(246\) 284983. 0.300249
\(247\) −138849. −0.144811
\(248\) 286508. 0.295806
\(249\) 107153. 0.109523
\(250\) 754358. 0.763357
\(251\) 1.49884e6 1.50165 0.750827 0.660499i \(-0.229655\pi\)
0.750827 + 0.660499i \(0.229655\pi\)
\(252\) 91437.7 0.0907035
\(253\) −90708.5 −0.0890936
\(254\) −1.39866e6 −1.36028
\(255\) −295.799 −0.000284870 0
\(256\) −1.09482e6 −1.04410
\(257\) −512138. −0.483676 −0.241838 0.970317i \(-0.577750\pi\)
−0.241838 + 0.970317i \(0.577750\pi\)
\(258\) 683079. 0.638884
\(259\) 1.21317e6 1.12375
\(260\) 94026.4 0.0862614
\(261\) 489293. 0.444598
\(262\) −348344. −0.313513
\(263\) 154350. 0.137600 0.0687998 0.997630i \(-0.478083\pi\)
0.0687998 + 0.997630i \(0.478083\pi\)
\(264\) −211432. −0.186707
\(265\) −816580. −0.714305
\(266\) 259698. 0.225043
\(267\) −1.31713e6 −1.13071
\(268\) −545463. −0.463904
\(269\) 2.27104e6 1.91357 0.956783 0.290802i \(-0.0939221\pi\)
0.956783 + 0.290802i \(0.0939221\pi\)
\(270\) 113570. 0.0948098
\(271\) 304916. 0.252207 0.126103 0.992017i \(-0.459753\pi\)
0.126103 + 0.992017i \(0.459753\pi\)
\(272\) 292.397 0.000239635 0
\(273\) 116861. 0.0948993
\(274\) −815910. −0.656547
\(275\) 207781. 0.165682
\(276\) 99586.7 0.0786916
\(277\) −884418. −0.692561 −0.346280 0.938131i \(-0.612555\pi\)
−0.346280 + 0.938131i \(0.612555\pi\)
\(278\) 1.09296e6 0.848187
\(279\) −119531. −0.0919325
\(280\) −557130. −0.424680
\(281\) 2.03032e6 1.53391 0.766955 0.641701i \(-0.221771\pi\)
0.766955 + 0.641701i \(0.221771\pi\)
\(282\) 481237. 0.360360
\(283\) 1.32361e6 0.982415 0.491207 0.871043i \(-0.336556\pi\)
0.491207 + 0.871043i \(0.336556\pi\)
\(284\) 553706. 0.407365
\(285\) −276168. −0.201401
\(286\) −85296.8 −0.0616620
\(287\) 583252. 0.417976
\(288\) 390979. 0.277761
\(289\) −1.41986e6 −0.999999
\(290\) −941063. −0.657088
\(291\) 1.24316e6 0.860587
\(292\) −1.18750e6 −0.815034
\(293\) −2.50642e6 −1.70563 −0.852816 0.522211i \(-0.825107\pi\)
−0.852816 + 0.522211i \(0.825107\pi\)
\(294\) 409482. 0.276291
\(295\) 43841.1 0.0293310
\(296\) 3.07977e6 2.04310
\(297\) 88209.0 0.0580259
\(298\) −1.40096e6 −0.913869
\(299\) 127276. 0.0823317
\(300\) −228118. −0.146338
\(301\) 1.39800e6 0.889389
\(302\) −654295. −0.412816
\(303\) −329536. −0.206203
\(304\) 272991. 0.169420
\(305\) −526277. −0.323940
\(306\) −294.600 −0.000179858 0
\(307\) 871198. 0.527559 0.263779 0.964583i \(-0.415031\pi\)
0.263779 + 0.964583i \(0.415031\pi\)
\(308\) −136592. −0.0820444
\(309\) −406787. −0.242365
\(310\) 229895. 0.135871
\(311\) 1.26077e6 0.739156 0.369578 0.929200i \(-0.379502\pi\)
0.369578 + 0.929200i \(0.379502\pi\)
\(312\) 296666. 0.172536
\(313\) 326948. 0.188633 0.0943164 0.995542i \(-0.469933\pi\)
0.0943164 + 0.995542i \(0.469933\pi\)
\(314\) 22294.9 0.0127609
\(315\) 232434. 0.131985
\(316\) 917206. 0.516713
\(317\) −969525. −0.541890 −0.270945 0.962595i \(-0.587336\pi\)
−0.270945 + 0.962595i \(0.587336\pi\)
\(318\) −813270. −0.450990
\(319\) −730919. −0.402154
\(320\) −1.15276e6 −0.629308
\(321\) 252781. 0.136925
\(322\) −238051. −0.127947
\(323\) 716.381 0.000382065 0
\(324\) −96842.6 −0.0512512
\(325\) −291544. −0.153107
\(326\) −2.08891e6 −1.08862
\(327\) 304676. 0.157568
\(328\) 1.48066e6 0.759923
\(329\) 984910. 0.501657
\(330\) −169654. −0.0857587
\(331\) 2.06283e6 1.03489 0.517444 0.855717i \(-0.326884\pi\)
0.517444 + 0.855717i \(0.326884\pi\)
\(332\) 175734. 0.0875006
\(333\) −1.28488e6 −0.634967
\(334\) −2.42698e6 −1.19042
\(335\) −1.38656e6 −0.675037
\(336\) −229760. −0.111027
\(337\) −2.86366e6 −1.37356 −0.686778 0.726867i \(-0.740976\pi\)
−0.686778 + 0.726867i \(0.740976\pi\)
\(338\) −1.42195e6 −0.677007
\(339\) −1.51703e6 −0.716959
\(340\) −485.122 −0.000227590 0
\(341\) 178558. 0.0831561
\(342\) −275049. −0.127158
\(343\) 2.12344e6 0.974552
\(344\) 3.54900e6 1.61700
\(345\) 253149. 0.114506
\(346\) −230622. −0.103565
\(347\) −2.33015e6 −1.03887 −0.519434 0.854511i \(-0.673857\pi\)
−0.519434 + 0.854511i \(0.673857\pi\)
\(348\) 802458. 0.355201
\(349\) 949067. 0.417094 0.208547 0.978012i \(-0.433127\pi\)
0.208547 + 0.978012i \(0.433127\pi\)
\(350\) 545292. 0.237935
\(351\) −123769. −0.0536219
\(352\) −584055. −0.251245
\(353\) −2.05584e6 −0.878119 −0.439059 0.898458i \(-0.644688\pi\)
−0.439059 + 0.898458i \(0.644688\pi\)
\(354\) 43663.4 0.0185187
\(355\) 1.40752e6 0.592765
\(356\) −2.16014e6 −0.903351
\(357\) −602.935 −0.000250380 0
\(358\) 893259. 0.368358
\(359\) −1.11901e6 −0.458244 −0.229122 0.973398i \(-0.573585\pi\)
−0.229122 + 0.973398i \(0.573585\pi\)
\(360\) 590062. 0.239961
\(361\) −1.80726e6 −0.729883
\(362\) 2.83218e6 1.13592
\(363\) −131769. −0.0524864
\(364\) 191656. 0.0758175
\(365\) −3.01861e6 −1.18597
\(366\) −524144. −0.204526
\(367\) 4.10134e6 1.58950 0.794749 0.606938i \(-0.207602\pi\)
0.794749 + 0.606938i \(0.207602\pi\)
\(368\) −250237. −0.0963233
\(369\) −617728. −0.236174
\(370\) 2.47122e6 0.938442
\(371\) −1.66445e6 −0.627823
\(372\) −196035. −0.0734474
\(373\) −3.54538e6 −1.31945 −0.659723 0.751509i \(-0.729326\pi\)
−0.659723 + 0.751509i \(0.729326\pi\)
\(374\) 440.082 0.000162688 0
\(375\) −1.63514e6 −0.600451
\(376\) 2.50031e6 0.912063
\(377\) 1.02557e6 0.371632
\(378\) 231492. 0.0833310
\(379\) 3.23295e6 1.15612 0.578058 0.815996i \(-0.303811\pi\)
0.578058 + 0.815996i \(0.303811\pi\)
\(380\) −452926. −0.160904
\(381\) 3.03173e6 1.06999
\(382\) −372430. −0.130583
\(383\) −3.72375e6 −1.29713 −0.648565 0.761159i \(-0.724631\pi\)
−0.648565 + 0.761159i \(0.724631\pi\)
\(384\) 242061. 0.0837717
\(385\) −347216. −0.119385
\(386\) 2.72817e6 0.931971
\(387\) −1.48064e6 −0.502541
\(388\) 2.03883e6 0.687546
\(389\) −1.00909e6 −0.338108 −0.169054 0.985607i \(-0.554071\pi\)
−0.169054 + 0.985607i \(0.554071\pi\)
\(390\) 238046. 0.0792499
\(391\) −656.669 −0.000217222 0
\(392\) 2.12750e6 0.699286
\(393\) 755068. 0.246607
\(394\) 2.99123e6 0.970753
\(395\) 2.33153e6 0.751880
\(396\) 144666. 0.0463584
\(397\) −366919. −0.116841 −0.0584203 0.998292i \(-0.518606\pi\)
−0.0584203 + 0.998292i \(0.518606\pi\)
\(398\) 584811. 0.185058
\(399\) −562920. −0.177017
\(400\) 573204. 0.179126
\(401\) 1.21086e6 0.376038 0.188019 0.982165i \(-0.439793\pi\)
0.188019 + 0.982165i \(0.439793\pi\)
\(402\) −1.38094e6 −0.426197
\(403\) −250540. −0.0768449
\(404\) −540451. −0.164741
\(405\) −246173. −0.0745767
\(406\) −1.91819e6 −0.577533
\(407\) 1.91938e6 0.574349
\(408\) −1530.62 −0.000455216 0
\(409\) 314741. 0.0930346 0.0465173 0.998917i \(-0.485188\pi\)
0.0465173 + 0.998917i \(0.485188\pi\)
\(410\) 1.18808e6 0.349050
\(411\) 1.76856e6 0.516435
\(412\) −667145. −0.193632
\(413\) 89362.4 0.0257798
\(414\) 252123. 0.0722954
\(415\) 446715. 0.127324
\(416\) 819504. 0.232176
\(417\) −2.36909e6 −0.667178
\(418\) 410875. 0.115019
\(419\) 5.98987e6 1.66679 0.833397 0.552674i \(-0.186393\pi\)
0.833397 + 0.552674i \(0.186393\pi\)
\(420\) 381200. 0.105446
\(421\) −3.19815e6 −0.879414 −0.439707 0.898141i \(-0.644918\pi\)
−0.439707 + 0.898141i \(0.644918\pi\)
\(422\) −464735. −0.127035
\(423\) −1.04313e6 −0.283457
\(424\) −4.22542e6 −1.14145
\(425\) 1504.20 0.000403954 0
\(426\) 1.40181e6 0.374254
\(427\) −1.07272e6 −0.284720
\(428\) 414571. 0.109393
\(429\) 184889. 0.0485029
\(430\) 2.84773e6 0.742725
\(431\) −5.24498e6 −1.36004 −0.680019 0.733195i \(-0.738028\pi\)
−0.680019 + 0.733195i \(0.738028\pi\)
\(432\) 243341. 0.0627345
\(433\) 5.44629e6 1.39599 0.697993 0.716105i \(-0.254077\pi\)
0.697993 + 0.716105i \(0.254077\pi\)
\(434\) 468601. 0.119421
\(435\) 2.03984e6 0.516861
\(436\) 499680. 0.125885
\(437\) −613088. −0.153574
\(438\) −3.00638e6 −0.748787
\(439\) −6.38391e6 −1.58098 −0.790488 0.612477i \(-0.790173\pi\)
−0.790488 + 0.612477i \(0.790173\pi\)
\(440\) −881451. −0.217053
\(441\) −887591. −0.217328
\(442\) −617.492 −0.000150340 0
\(443\) 2.26340e6 0.547964 0.273982 0.961735i \(-0.411659\pi\)
0.273982 + 0.961735i \(0.411659\pi\)
\(444\) −2.10725e6 −0.507292
\(445\) −5.49106e6 −1.31449
\(446\) −3.76520e6 −0.896296
\(447\) 3.03671e6 0.718843
\(448\) −2.34969e6 −0.553116
\(449\) 4.05102e6 0.948306 0.474153 0.880442i \(-0.342754\pi\)
0.474153 + 0.880442i \(0.342754\pi\)
\(450\) −577524. −0.134443
\(451\) 922779. 0.213627
\(452\) −2.48798e6 −0.572797
\(453\) 1.41825e6 0.324718
\(454\) −5.63499e6 −1.28308
\(455\) 487189. 0.110324
\(456\) −1.42904e6 −0.321835
\(457\) −5.07731e6 −1.13722 −0.568608 0.822609i \(-0.692518\pi\)
−0.568608 + 0.822609i \(0.692518\pi\)
\(458\) −1.43228e6 −0.319054
\(459\) 638.574 0.000141475 0
\(460\) 415173. 0.0914818
\(461\) −3.81197e6 −0.835405 −0.417702 0.908584i \(-0.637165\pi\)
−0.417702 + 0.908584i \(0.637165\pi\)
\(462\) −345809. −0.0753757
\(463\) 6.00414e6 1.30166 0.650832 0.759222i \(-0.274420\pi\)
0.650832 + 0.759222i \(0.274420\pi\)
\(464\) −2.01638e6 −0.434788
\(465\) −498320. −0.106875
\(466\) −4.90621e6 −1.04660
\(467\) −3.29877e6 −0.699939 −0.349969 0.936761i \(-0.613808\pi\)
−0.349969 + 0.936761i \(0.613808\pi\)
\(468\) −202985. −0.0428400
\(469\) −2.82626e6 −0.593309
\(470\) 2.00626e6 0.418931
\(471\) −48326.3 −0.0100376
\(472\) 226857. 0.0468703
\(473\) 2.21182e6 0.454566
\(474\) 2.32208e6 0.474714
\(475\) 1.40437e6 0.285593
\(476\) −988.836 −0.000200035 0
\(477\) 1.76284e6 0.354746
\(478\) −3.00449e6 −0.601452
\(479\) 5.86911e6 1.16878 0.584391 0.811472i \(-0.301333\pi\)
0.584391 + 0.811472i \(0.301333\pi\)
\(480\) 1.62998e6 0.322908
\(481\) −2.69314e6 −0.530758
\(482\) 1.63725e6 0.320995
\(483\) 515999. 0.100642
\(484\) −216106. −0.0419328
\(485\) 5.18269e6 1.00046
\(486\) −245175. −0.0470854
\(487\) 981038. 0.187441 0.0937203 0.995599i \(-0.470124\pi\)
0.0937203 + 0.995599i \(0.470124\pi\)
\(488\) −2.72324e6 −0.517650
\(489\) 4.52792e6 0.856301
\(490\) 1.70712e6 0.321198
\(491\) 5.94395e6 1.11268 0.556342 0.830954i \(-0.312205\pi\)
0.556342 + 0.830954i \(0.312205\pi\)
\(492\) −1.01310e6 −0.188685
\(493\) −5291.36 −0.000980506 0
\(494\) −576511. −0.106289
\(495\) 367740. 0.0674572
\(496\) 492587. 0.0899040
\(497\) 2.86898e6 0.520998
\(498\) 444905. 0.0803884
\(499\) 6.59714e6 1.18605 0.593027 0.805183i \(-0.297933\pi\)
0.593027 + 0.805183i \(0.297933\pi\)
\(500\) −2.68170e6 −0.479716
\(501\) 5.26070e6 0.936374
\(502\) 6.22327e6 1.10220
\(503\) −7.55854e6 −1.33204 −0.666021 0.745933i \(-0.732004\pi\)
−0.666021 + 0.745933i \(0.732004\pi\)
\(504\) 1.20274e6 0.210909
\(505\) −1.37382e6 −0.239719
\(506\) −376628. −0.0653937
\(507\) 3.08221e6 0.532529
\(508\) 4.97216e6 0.854841
\(509\) 4.67934e6 0.800553 0.400276 0.916394i \(-0.368914\pi\)
0.400276 + 0.916394i \(0.368914\pi\)
\(510\) −1228.18 −0.000209091 0
\(511\) −6.15291e6 −1.04239
\(512\) −3.68509e6 −0.621260
\(513\) 596194. 0.100022
\(514\) −2.12643e6 −0.355013
\(515\) −1.69588e6 −0.281758
\(516\) −2.42830e6 −0.401494
\(517\) 1.55825e6 0.256396
\(518\) 5.03715e6 0.824823
\(519\) 499896. 0.0814631
\(520\) 1.23679e6 0.200580
\(521\) 557794. 0.0900284 0.0450142 0.998986i \(-0.485667\pi\)
0.0450142 + 0.998986i \(0.485667\pi\)
\(522\) 2.03158e6 0.326330
\(523\) 3.87401e6 0.619308 0.309654 0.950849i \(-0.399787\pi\)
0.309654 + 0.950849i \(0.399787\pi\)
\(524\) 1.23834e6 0.197021
\(525\) −1.18197e6 −0.187158
\(526\) 640871. 0.100997
\(527\) 1292.64 0.000202746 0
\(528\) −363510. −0.0567455
\(529\) −5.87436e6 −0.912686
\(530\) −3.39049e6 −0.524292
\(531\) −94644.6 −0.0145666
\(532\) −923210. −0.141423
\(533\) −1.29478e6 −0.197414
\(534\) −5.46880e6 −0.829926
\(535\) 1.05384e6 0.159180
\(536\) −7.17481e6 −1.07870
\(537\) −1.93622e6 −0.289748
\(538\) 9.42950e6 1.40454
\(539\) 1.32591e6 0.196581
\(540\) −403733. −0.0595813
\(541\) −5.42985e6 −0.797618 −0.398809 0.917034i \(-0.630576\pi\)
−0.398809 + 0.917034i \(0.630576\pi\)
\(542\) 1.26603e6 0.185117
\(543\) −6.13902e6 −0.893510
\(544\) −4228.17 −0.000612569 0
\(545\) 1.27018e6 0.183179
\(546\) 485215. 0.0696550
\(547\) −853295. −0.121936 −0.0609679 0.998140i \(-0.519419\pi\)
−0.0609679 + 0.998140i \(0.519419\pi\)
\(548\) 2.90051e6 0.412594
\(549\) 1.13613e6 0.160878
\(550\) 862721. 0.121608
\(551\) −4.94019e6 −0.693210
\(552\) 1.30993e6 0.182978
\(553\) 4.75242e6 0.660848
\(554\) −3.67216e6 −0.508332
\(555\) −5.35661e6 −0.738171
\(556\) −3.88540e6 −0.533026
\(557\) 1.08605e6 0.148324 0.0741618 0.997246i \(-0.476372\pi\)
0.0741618 + 0.997246i \(0.476372\pi\)
\(558\) −496300. −0.0674775
\(559\) −3.10347e6 −0.420066
\(560\) −957863. −0.129072
\(561\) −953.919 −0.000127969 0
\(562\) 8.43005e6 1.12587
\(563\) 6.05610e6 0.805235 0.402617 0.915368i \(-0.368101\pi\)
0.402617 + 0.915368i \(0.368101\pi\)
\(564\) −1.71077e6 −0.226461
\(565\) −6.32443e6 −0.833490
\(566\) 5.49573e6 0.721081
\(567\) −501781. −0.0655475
\(568\) 7.28325e6 0.947227
\(569\) 4.03513e6 0.522489 0.261245 0.965273i \(-0.415867\pi\)
0.261245 + 0.965273i \(0.415867\pi\)
\(570\) −1.14667e6 −0.147826
\(571\) 597325. 0.0766691 0.0383345 0.999265i \(-0.487795\pi\)
0.0383345 + 0.999265i \(0.487795\pi\)
\(572\) 303225. 0.0387502
\(573\) 807277. 0.102715
\(574\) 2.42170e6 0.306790
\(575\) −1.28731e6 −0.162373
\(576\) 2.48858e6 0.312533
\(577\) 4.08910e6 0.511315 0.255657 0.966767i \(-0.417708\pi\)
0.255657 + 0.966767i \(0.417708\pi\)
\(578\) −5.89534e6 −0.733988
\(579\) −5.91356e6 −0.733082
\(580\) 3.34542e6 0.412934
\(581\) 910551. 0.111909
\(582\) 5.16169e6 0.631662
\(583\) −2.63338e6 −0.320879
\(584\) −1.56199e7 −1.89516
\(585\) −515987. −0.0623374
\(586\) −1.04068e7 −1.25191
\(587\) 1.56181e6 0.187082 0.0935409 0.995615i \(-0.470181\pi\)
0.0935409 + 0.995615i \(0.470181\pi\)
\(588\) −1.45568e6 −0.173629
\(589\) 1.20685e6 0.143340
\(590\) 182031. 0.0215286
\(591\) −6.48377e6 −0.763587
\(592\) 5.29499e6 0.620956
\(593\) −8.87674e6 −1.03661 −0.518307 0.855195i \(-0.673437\pi\)
−0.518307 + 0.855195i \(0.673437\pi\)
\(594\) 366250. 0.0425903
\(595\) −2513.61 −0.000291076 0
\(596\) 4.98031e6 0.574303
\(597\) −1.26763e6 −0.145565
\(598\) 528457. 0.0604306
\(599\) −7.17157e6 −0.816671 −0.408336 0.912832i \(-0.633891\pi\)
−0.408336 + 0.912832i \(0.633891\pi\)
\(600\) −3.00058e6 −0.340273
\(601\) −4.79599e6 −0.541617 −0.270809 0.962633i \(-0.587291\pi\)
−0.270809 + 0.962633i \(0.587291\pi\)
\(602\) 5.80460e6 0.652802
\(603\) 2.99332e6 0.335244
\(604\) 2.32598e6 0.259426
\(605\) −549340. −0.0610173
\(606\) −1.36825e6 −0.151351
\(607\) −8.49023e6 −0.935293 −0.467646 0.883916i \(-0.654898\pi\)
−0.467646 + 0.883916i \(0.654898\pi\)
\(608\) −3.94756e6 −0.433081
\(609\) 4.15786e6 0.454284
\(610\) −2.18514e6 −0.237768
\(611\) −2.18643e6 −0.236937
\(612\) 1047.29 0.000113028 0
\(613\) 6.52329e6 0.701158 0.350579 0.936533i \(-0.385985\pi\)
0.350579 + 0.936533i \(0.385985\pi\)
\(614\) 3.61727e6 0.387222
\(615\) −2.57529e6 −0.274560
\(616\) −1.79668e6 −0.190774
\(617\) 1.20215e7 1.27129 0.635647 0.771980i \(-0.280733\pi\)
0.635647 + 0.771980i \(0.280733\pi\)
\(618\) −1.68901e6 −0.177893
\(619\) 1.34686e7 1.41285 0.706424 0.707789i \(-0.250307\pi\)
0.706424 + 0.707789i \(0.250307\pi\)
\(620\) −817263. −0.0853852
\(621\) −546500. −0.0568671
\(622\) 5.23482e6 0.542532
\(623\) −1.11926e7 −1.15534
\(624\) 510051. 0.0524388
\(625\) −1.45061e6 −0.148542
\(626\) 1.35751e6 0.138454
\(627\) −890611. −0.0904731
\(628\) −79257.0 −0.00801934
\(629\) 13895.1 0.00140034
\(630\) 965082. 0.0968752
\(631\) −1.08509e7 −1.08490 −0.542452 0.840087i \(-0.682504\pi\)
−0.542452 + 0.840087i \(0.682504\pi\)
\(632\) 1.20646e7 1.20149
\(633\) 1.00736e6 0.0999251
\(634\) −4.02554e6 −0.397741
\(635\) 1.26392e7 1.24390
\(636\) 2.89112e6 0.283416
\(637\) −1.86042e6 −0.181661
\(638\) −3.03482e6 −0.295177
\(639\) −3.03856e6 −0.294385
\(640\) 1.00915e6 0.0973876
\(641\) −9.31338e6 −0.895287 −0.447643 0.894212i \(-0.647737\pi\)
−0.447643 + 0.894212i \(0.647737\pi\)
\(642\) 1.04956e6 0.100501
\(643\) 1.35760e7 1.29493 0.647464 0.762096i \(-0.275830\pi\)
0.647464 + 0.762096i \(0.275830\pi\)
\(644\) 846258. 0.0804059
\(645\) −6.17273e6 −0.584222
\(646\) 2974.46 0.000280432 0
\(647\) −1.18628e7 −1.11411 −0.557053 0.830477i \(-0.688068\pi\)
−0.557053 + 0.830477i \(0.688068\pi\)
\(648\) −1.27383e6 −0.119172
\(649\) 141383. 0.0131760
\(650\) −1.21051e6 −0.112379
\(651\) −1.01574e6 −0.0939353
\(652\) 7.42596e6 0.684122
\(653\) 1.22670e7 1.12579 0.562893 0.826530i \(-0.309688\pi\)
0.562893 + 0.826530i \(0.309688\pi\)
\(654\) 1.26503e6 0.115653
\(655\) 3.14785e6 0.286689
\(656\) 2.54566e6 0.230962
\(657\) 6.51661e6 0.588991
\(658\) 4.08941e6 0.368211
\(659\) 4.60998e6 0.413509 0.206755 0.978393i \(-0.433710\pi\)
0.206755 + 0.978393i \(0.433710\pi\)
\(660\) 603108. 0.0538933
\(661\) −1.04708e7 −0.932130 −0.466065 0.884750i \(-0.654329\pi\)
−0.466065 + 0.884750i \(0.654329\pi\)
\(662\) 8.56500e6 0.759596
\(663\) 1338.47 0.000118257 0
\(664\) 2.31154e6 0.203461
\(665\) −2.34679e6 −0.205788
\(666\) −5.33490e6 −0.466058
\(667\) 4.52841e6 0.394123
\(668\) 8.62775e6 0.748094
\(669\) 8.16143e6 0.705020
\(670\) −5.75710e6 −0.495469
\(671\) −1.69718e6 −0.145520
\(672\) 3.32242e6 0.283813
\(673\) −5.32923e6 −0.453552 −0.226776 0.973947i \(-0.572819\pi\)
−0.226776 + 0.973947i \(0.572819\pi\)
\(674\) −1.18901e7 −1.00818
\(675\) 1.25184e6 0.105752
\(676\) 5.05495e6 0.425451
\(677\) 1.75693e6 0.147327 0.0736634 0.997283i \(-0.476531\pi\)
0.0736634 + 0.997283i \(0.476531\pi\)
\(678\) −6.29880e6 −0.526240
\(679\) 1.05640e7 0.879335
\(680\) −6381.11 −0.000529205 0
\(681\) 1.22144e7 1.00926
\(682\) 741386. 0.0610357
\(683\) −1.42170e7 −1.16615 −0.583077 0.812417i \(-0.698152\pi\)
−0.583077 + 0.812417i \(0.698152\pi\)
\(684\) 977781. 0.0799100
\(685\) 7.37307e6 0.600374
\(686\) 8.81667e6 0.715310
\(687\) 3.10460e6 0.250965
\(688\) 6.10173e6 0.491453
\(689\) 3.69497e6 0.296526
\(690\) 1.05109e6 0.0840460
\(691\) −2.40296e7 −1.91448 −0.957240 0.289294i \(-0.906579\pi\)
−0.957240 + 0.289294i \(0.906579\pi\)
\(692\) 819848. 0.0650831
\(693\) 749574. 0.0592900
\(694\) −9.67494e6 −0.762517
\(695\) −9.87665e6 −0.775618
\(696\) 1.05552e7 0.825934
\(697\) 6680.30 0.000520852 0
\(698\) 3.94059e6 0.306142
\(699\) 1.06347e7 0.823249
\(700\) −1.93848e6 −0.149526
\(701\) −5.04194e6 −0.387528 −0.193764 0.981048i \(-0.562070\pi\)
−0.193764 + 0.981048i \(0.562070\pi\)
\(702\) −513895. −0.0393579
\(703\) 1.29729e7 0.990030
\(704\) −3.71751e6 −0.282697
\(705\) −4.34876e6 −0.329528
\(706\) −8.53600e6 −0.644529
\(707\) −2.80030e6 −0.210696
\(708\) −155221. −0.0116377
\(709\) 1.87351e7 1.39972 0.699860 0.714280i \(-0.253246\pi\)
0.699860 + 0.714280i \(0.253246\pi\)
\(710\) 5.84411e6 0.435083
\(711\) −5.03333e6 −0.373406
\(712\) −2.84137e7 −2.10052
\(713\) −1.10626e6 −0.0814955
\(714\) −2503.43 −0.000183776 0
\(715\) 770795. 0.0563863
\(716\) −3.17548e6 −0.231487
\(717\) 6.51251e6 0.473097
\(718\) −4.64619e6 −0.336346
\(719\) 9.84863e6 0.710483 0.355242 0.934775i \(-0.384399\pi\)
0.355242 + 0.934775i \(0.384399\pi\)
\(720\) 1.01448e6 0.0729311
\(721\) −3.45675e6 −0.247645
\(722\) −7.50387e6 −0.535726
\(723\) −3.54890e6 −0.252492
\(724\) −1.00682e7 −0.713849
\(725\) −1.03730e7 −0.732925
\(726\) −547114. −0.0385244
\(727\) −1.76700e7 −1.23994 −0.619968 0.784627i \(-0.712854\pi\)
−0.619968 + 0.784627i \(0.712854\pi\)
\(728\) 2.52098e6 0.176295
\(729\) 531441. 0.0370370
\(730\) −1.25335e7 −0.870492
\(731\) 16012.1 0.00110829
\(732\) 1.86330e6 0.128530
\(733\) −1.72881e7 −1.18847 −0.594235 0.804291i \(-0.702545\pi\)
−0.594235 + 0.804291i \(0.702545\pi\)
\(734\) 1.70290e7 1.16667
\(735\) −3.70033e6 −0.252652
\(736\) 3.61852e6 0.246227
\(737\) −4.47151e6 −0.303239
\(738\) −2.56485e6 −0.173349
\(739\) −5.26063e6 −0.354345 −0.177172 0.984180i \(-0.556695\pi\)
−0.177172 + 0.984180i \(0.556695\pi\)
\(740\) −8.78503e6 −0.589745
\(741\) 1.24964e6 0.0836065
\(742\) −6.91093e6 −0.460815
\(743\) 5.40803e6 0.359391 0.179695 0.983722i \(-0.442489\pi\)
0.179695 + 0.983722i \(0.442489\pi\)
\(744\) −2.57857e6 −0.170784
\(745\) 1.26599e7 0.835681
\(746\) −1.47207e7 −0.968458
\(747\) −964373. −0.0632330
\(748\) −1564.46 −0.000102238 0
\(749\) 2.14806e6 0.139908
\(750\) −6.78922e6 −0.440724
\(751\) −1.50076e7 −0.970980 −0.485490 0.874242i \(-0.661359\pi\)
−0.485490 + 0.874242i \(0.661359\pi\)
\(752\) 4.29874e6 0.277202
\(753\) −1.34895e7 −0.866980
\(754\) 4.25825e6 0.272774
\(755\) 5.91262e6 0.377496
\(756\) −822939. −0.0523677
\(757\) 5.65482e6 0.358657 0.179328 0.983789i \(-0.442608\pi\)
0.179328 + 0.983789i \(0.442608\pi\)
\(758\) 1.34234e7 0.848576
\(759\) 816376. 0.0514382
\(760\) −5.95762e6 −0.374144
\(761\) 1.90760e7 1.19406 0.597030 0.802219i \(-0.296347\pi\)
0.597030 + 0.802219i \(0.296347\pi\)
\(762\) 1.25880e7 0.785358
\(763\) 2.58904e6 0.161001
\(764\) 1.32396e6 0.0820621
\(765\) 2662.19 0.000164470 0
\(766\) −1.54613e7 −0.952079
\(767\) −198378. −0.0121760
\(768\) 9.85336e6 0.602811
\(769\) 2.03557e7 1.24128 0.620639 0.784096i \(-0.286873\pi\)
0.620639 + 0.784096i \(0.286873\pi\)
\(770\) −1.44167e6 −0.0876269
\(771\) 4.60925e6 0.279250
\(772\) −9.69845e6 −0.585678
\(773\) −9.29224e6 −0.559334 −0.279667 0.960097i \(-0.590224\pi\)
−0.279667 + 0.960097i \(0.590224\pi\)
\(774\) −6.14771e6 −0.368860
\(775\) 2.53405e6 0.151552
\(776\) 2.68180e7 1.59872
\(777\) −1.09185e7 −0.648800
\(778\) −4.18981e6 −0.248168
\(779\) 6.23695e6 0.368238
\(780\) −846237. −0.0498030
\(781\) 4.53909e6 0.266281
\(782\) −2726.53 −0.000159439 0
\(783\) −4.40364e6 −0.256689
\(784\) 3.65777e6 0.212533
\(785\) −201471. −0.0116691
\(786\) 3.13509e6 0.181007
\(787\) 1.28703e7 0.740716 0.370358 0.928889i \(-0.379235\pi\)
0.370358 + 0.928889i \(0.379235\pi\)
\(788\) −1.06336e7 −0.610050
\(789\) −1.38915e6 −0.0794431
\(790\) 9.68067e6 0.551871
\(791\) −1.28912e7 −0.732578
\(792\) 1.90288e6 0.107795
\(793\) 2.38137e6 0.134476
\(794\) −1.52347e6 −0.0857597
\(795\) 7.34922e6 0.412404
\(796\) −2.07897e6 −0.116296
\(797\) 8.82772e6 0.492269 0.246135 0.969236i \(-0.420839\pi\)
0.246135 + 0.969236i \(0.420839\pi\)
\(798\) −2.33728e6 −0.129928
\(799\) 11280.7 0.000625129 0
\(800\) −8.28875e6 −0.457893
\(801\) 1.18541e7 0.652813
\(802\) 5.02756e6 0.276008
\(803\) −9.73469e6 −0.532762
\(804\) 4.90916e6 0.267835
\(805\) 2.15118e6 0.117000
\(806\) −1.04026e6 −0.0564033
\(807\) −2.04393e7 −1.10480
\(808\) −7.10890e6 −0.383066
\(809\) 640426. 0.0344031 0.0172016 0.999852i \(-0.494524\pi\)
0.0172016 + 0.999852i \(0.494524\pi\)
\(810\) −1.02213e6 −0.0547385
\(811\) 1.30571e7 0.697099 0.348550 0.937290i \(-0.386674\pi\)
0.348550 + 0.937290i \(0.386674\pi\)
\(812\) 6.81905e6 0.362939
\(813\) −2.74424e6 −0.145612
\(814\) 7.96941e6 0.421566
\(815\) 1.88767e7 0.995481
\(816\) −2631.57 −0.000138353 0
\(817\) 1.49494e7 0.783555
\(818\) 1.30682e6 0.0682864
\(819\) −1.05175e6 −0.0547901
\(820\) −4.22356e6 −0.219353
\(821\) 3.01122e7 1.55914 0.779570 0.626315i \(-0.215438\pi\)
0.779570 + 0.626315i \(0.215438\pi\)
\(822\) 7.34319e6 0.379057
\(823\) 1.95327e7 1.00523 0.502613 0.864512i \(-0.332372\pi\)
0.502613 + 0.864512i \(0.332372\pi\)
\(824\) −8.77539e6 −0.450244
\(825\) −1.87003e6 −0.0956563
\(826\) 371039. 0.0189221
\(827\) 6.83832e6 0.347685 0.173842 0.984773i \(-0.444382\pi\)
0.173842 + 0.984773i \(0.444382\pi\)
\(828\) −896280. −0.0454326
\(829\) 7.34516e6 0.371206 0.185603 0.982625i \(-0.440576\pi\)
0.185603 + 0.982625i \(0.440576\pi\)
\(830\) 1.85479e6 0.0934544
\(831\) 7.95976e6 0.399850
\(832\) 5.21615e6 0.261241
\(833\) 9598.68 0.000479291 0
\(834\) −9.83662e6 −0.489701
\(835\) 2.19317e7 1.08857
\(836\) −1.46064e6 −0.0722813
\(837\) 1.07578e6 0.0530773
\(838\) 2.48703e7 1.22341
\(839\) −3.88085e7 −1.90336 −0.951682 0.307084i \(-0.900647\pi\)
−0.951682 + 0.307084i \(0.900647\pi\)
\(840\) 5.01417e6 0.245189
\(841\) 1.59783e7 0.779007
\(842\) −1.32789e7 −0.645480
\(843\) −1.82729e7 −0.885603
\(844\) 1.65211e6 0.0798328
\(845\) 1.28496e7 0.619083
\(846\) −4.33114e6 −0.208054
\(847\) −1.11973e6 −0.0536298
\(848\) −7.26468e6 −0.346918
\(849\) −1.19125e7 −0.567197
\(850\) 6245.52 0.000296498 0
\(851\) −1.18916e7 −0.562879
\(852\) −4.98335e6 −0.235192
\(853\) −1.11803e7 −0.526115 −0.263058 0.964780i \(-0.584731\pi\)
−0.263058 + 0.964780i \(0.584731\pi\)
\(854\) −4.45402e6 −0.208981
\(855\) 2.48551e6 0.116279
\(856\) 5.45311e6 0.254367
\(857\) −2.30926e7 −1.07404 −0.537020 0.843569i \(-0.680450\pi\)
−0.537020 + 0.843569i \(0.680450\pi\)
\(858\) 767671. 0.0356006
\(859\) −1.28419e7 −0.593809 −0.296904 0.954907i \(-0.595954\pi\)
−0.296904 + 0.954907i \(0.595954\pi\)
\(860\) −1.01235e7 −0.466751
\(861\) −5.24927e6 −0.241319
\(862\) −2.17775e7 −0.998252
\(863\) −2.98309e7 −1.36345 −0.681725 0.731608i \(-0.738770\pi\)
−0.681725 + 0.731608i \(0.738770\pi\)
\(864\) −3.51881e6 −0.160366
\(865\) 2.08405e6 0.0947038
\(866\) 2.26134e7 1.02464
\(867\) 1.27787e7 0.577350
\(868\) −1.66585e6 −0.0750474
\(869\) 7.51893e6 0.337759
\(870\) 8.46957e6 0.379370
\(871\) 6.27410e6 0.280225
\(872\) 6.57260e6 0.292716
\(873\) −1.11885e7 −0.496860
\(874\) −2.54558e6 −0.112722
\(875\) −1.38950e7 −0.613532
\(876\) 1.06875e7 0.470560
\(877\) −3.85241e7 −1.69135 −0.845674 0.533700i \(-0.820801\pi\)
−0.845674 + 0.533700i \(0.820801\pi\)
\(878\) −2.65064e7 −1.16042
\(879\) 2.25578e7 0.984747
\(880\) −1.51546e6 −0.0659687
\(881\) −3.12023e7 −1.35440 −0.677200 0.735799i \(-0.736807\pi\)
−0.677200 + 0.735799i \(0.736807\pi\)
\(882\) −3.68534e6 −0.159517
\(883\) −1.08832e7 −0.469738 −0.234869 0.972027i \(-0.575466\pi\)
−0.234869 + 0.972027i \(0.575466\pi\)
\(884\) 2195.14 9.44783e−5 0
\(885\) −394570. −0.0169342
\(886\) 9.39779e6 0.402199
\(887\) 4.34842e7 1.85576 0.927882 0.372875i \(-0.121628\pi\)
0.927882 + 0.372875i \(0.121628\pi\)
\(888\) −2.77180e7 −1.17958
\(889\) 2.57628e7 1.09330
\(890\) −2.27992e7 −0.964818
\(891\) −793881. −0.0335013
\(892\) 1.33851e7 0.563259
\(893\) 1.05320e7 0.441961
\(894\) 1.26086e7 0.527623
\(895\) −8.07205e6 −0.336842
\(896\) 2.05697e6 0.0855967
\(897\) −1.14548e6 −0.0475342
\(898\) 1.68201e7 0.696046
\(899\) −8.91412e6 −0.367857
\(900\) 2.05306e6 0.0844881
\(901\) −19063.9 −0.000782348 0
\(902\) 3.83144e6 0.156800
\(903\) −1.25820e7 −0.513489
\(904\) −3.27260e7 −1.33190
\(905\) −2.55934e7 −1.03874
\(906\) 5.88866e6 0.238339
\(907\) −9.01861e6 −0.364017 −0.182008 0.983297i \(-0.558260\pi\)
−0.182008 + 0.983297i \(0.558260\pi\)
\(908\) 2.00320e7 0.806326
\(909\) 2.96582e6 0.119052
\(910\) 2.02284e6 0.0809764
\(911\) −6.69826e6 −0.267403 −0.133701 0.991022i \(-0.542686\pi\)
−0.133701 + 0.991022i \(0.542686\pi\)
\(912\) −2.45692e6 −0.0978147
\(913\) 1.44061e6 0.0571964
\(914\) −2.10813e7 −0.834704
\(915\) 4.73649e6 0.187027
\(916\) 5.09166e6 0.200503
\(917\) 6.41635e6 0.251979
\(918\) 2651.40 0.000103841 0
\(919\) −1.01836e7 −0.397753 −0.198877 0.980025i \(-0.563729\pi\)
−0.198877 + 0.980025i \(0.563729\pi\)
\(920\) 5.46104e6 0.212719
\(921\) −7.84078e6 −0.304586
\(922\) −1.58275e7 −0.613178
\(923\) −6.36892e6 −0.246072
\(924\) 1.22933e6 0.0473684
\(925\) 2.72394e7 1.04675
\(926\) 2.49296e7 0.955406
\(927\) 3.66108e6 0.139930
\(928\) 2.91576e7 1.11143
\(929\) 7.05588e6 0.268233 0.134116 0.990966i \(-0.457180\pi\)
0.134116 + 0.990966i \(0.457180\pi\)
\(930\) −2.06906e6 −0.0784449
\(931\) 8.96165e6 0.338855
\(932\) 1.74413e7 0.657716
\(933\) −1.13470e7 −0.426752
\(934\) −1.36967e7 −0.513747
\(935\) −3976.86 −0.000148768 0
\(936\) −2.66999e6 −0.0996140
\(937\) 1.62634e6 0.0605150 0.0302575 0.999542i \(-0.490367\pi\)
0.0302575 + 0.999542i \(0.490367\pi\)
\(938\) −1.17348e7 −0.435482
\(939\) −2.94253e6 −0.108907
\(940\) −7.13213e6 −0.263269
\(941\) −2.73619e7 −1.00733 −0.503665 0.863899i \(-0.668015\pi\)
−0.503665 + 0.863899i \(0.668015\pi\)
\(942\) −200654. −0.00736752
\(943\) −5.71709e6 −0.209361
\(944\) 390031. 0.0142452
\(945\) −2.09191e6 −0.0762014
\(946\) 9.18362e6 0.333646
\(947\) 4.28329e7 1.55204 0.776019 0.630709i \(-0.217236\pi\)
0.776019 + 0.630709i \(0.217236\pi\)
\(948\) −8.25485e6 −0.298324
\(949\) 1.36590e7 0.492327
\(950\) 5.83103e6 0.209622
\(951\) 8.72573e6 0.312860
\(952\) −13006.8 −0.000465133 0
\(953\) 2.80611e7 1.00086 0.500429 0.865777i \(-0.333176\pi\)
0.500429 + 0.865777i \(0.333176\pi\)
\(954\) 7.31943e6 0.260379
\(955\) 3.36551e6 0.119410
\(956\) 1.06808e7 0.377970
\(957\) 6.57827e6 0.232184
\(958\) 2.43690e7 0.857873
\(959\) 1.50287e7 0.527686
\(960\) 1.03748e7 0.363331
\(961\) −2.64515e7 −0.923936
\(962\) −1.11821e7 −0.389571
\(963\) −2.27503e6 −0.0790536
\(964\) −5.82032e6 −0.201723
\(965\) −2.46534e7 −0.852233
\(966\) 2.14246e6 0.0738704
\(967\) −3.20738e7 −1.10302 −0.551512 0.834167i \(-0.685949\pi\)
−0.551512 + 0.834167i \(0.685949\pi\)
\(968\) −2.84258e6 −0.0975044
\(969\) −6447.43 −0.000220586 0
\(970\) 2.15189e7 0.734329
\(971\) 4.92583e7 1.67661 0.838304 0.545203i \(-0.183548\pi\)
0.838304 + 0.545203i \(0.183548\pi\)
\(972\) 871583. 0.0295899
\(973\) −2.01318e7 −0.681712
\(974\) 4.07334e6 0.137579
\(975\) 2.62389e6 0.0883964
\(976\) −4.68201e6 −0.157329
\(977\) −1.26363e7 −0.423530 −0.211765 0.977321i \(-0.567921\pi\)
−0.211765 + 0.977321i \(0.567921\pi\)
\(978\) 1.88002e7 0.628516
\(979\) −1.77081e7 −0.590492
\(980\) −6.06869e6 −0.201850
\(981\) −2.74208e6 −0.0909720
\(982\) 2.46797e7 0.816697
\(983\) 6.52333e6 0.215321 0.107660 0.994188i \(-0.465664\pi\)
0.107660 + 0.994188i \(0.465664\pi\)
\(984\) −1.33259e7 −0.438742
\(985\) −2.70306e7 −0.887697
\(986\) −21970.1 −0.000719681 0
\(987\) −8.86419e6 −0.289632
\(988\) 2.04946e6 0.0667955
\(989\) −1.37033e7 −0.445488
\(990\) 1.52688e6 0.0495128
\(991\) 2.28897e7 0.740382 0.370191 0.928956i \(-0.379292\pi\)
0.370191 + 0.928956i \(0.379292\pi\)
\(992\) −7.12300e6 −0.229818
\(993\) −1.85655e7 −0.597492
\(994\) 1.19122e7 0.382407
\(995\) −5.28472e6 −0.169225
\(996\) −1.58161e6 −0.0505185
\(997\) −1.82397e7 −0.581138 −0.290569 0.956854i \(-0.593845\pi\)
−0.290569 + 0.956854i \(0.593845\pi\)
\(998\) 2.73918e7 0.870550
\(999\) 1.15639e7 0.366598
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 33.6.a.c.1.2 2
3.2 odd 2 99.6.a.f.1.1 2
4.3 odd 2 528.6.a.s.1.1 2
5.4 even 2 825.6.a.e.1.1 2
11.10 odd 2 363.6.a.j.1.1 2
33.32 even 2 1089.6.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.6.a.c.1.2 2 1.1 even 1 trivial
99.6.a.f.1.1 2 3.2 odd 2
363.6.a.j.1.1 2 11.10 odd 2
528.6.a.s.1.1 2 4.3 odd 2
825.6.a.e.1.1 2 5.4 even 2
1089.6.a.j.1.2 2 33.32 even 2