Properties

Label 966.4.a.e.1.1
Level $966$
Weight $4$
Character 966.1
Self dual yes
Analytic conductor $56.996$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [966,4,Mod(1,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(56.9958450655\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.65101.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 28x - 29 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.08080\) of defining polynomial
Character \(\chi\) \(=\) 966.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -13.6703 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -13.6703 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -27.3405 q^{10} -2.24894 q^{11} -12.0000 q^{12} +45.7445 q^{13} -14.0000 q^{14} +41.0108 q^{15} +16.0000 q^{16} +4.13496 q^{17} +18.0000 q^{18} +133.113 q^{19} -54.6810 q^{20} +21.0000 q^{21} -4.49789 q^{22} -23.0000 q^{23} -24.0000 q^{24} +61.8758 q^{25} +91.4891 q^{26} -27.0000 q^{27} -28.0000 q^{28} -197.341 q^{29} +82.0215 q^{30} +116.520 q^{31} +32.0000 q^{32} +6.74683 q^{33} +8.26993 q^{34} +95.6918 q^{35} +36.0000 q^{36} +54.4321 q^{37} +266.226 q^{38} -137.234 q^{39} -109.362 q^{40} -234.204 q^{41} +42.0000 q^{42} -266.216 q^{43} -8.99577 q^{44} -123.032 q^{45} -46.0000 q^{46} -341.707 q^{47} -48.0000 q^{48} +49.0000 q^{49} +123.752 q^{50} -12.4049 q^{51} +182.978 q^{52} +114.614 q^{53} -54.0000 q^{54} +30.7436 q^{55} -56.0000 q^{56} -399.339 q^{57} -394.682 q^{58} -292.868 q^{59} +164.043 q^{60} -648.641 q^{61} +233.040 q^{62} -63.0000 q^{63} +64.0000 q^{64} -625.339 q^{65} +13.4937 q^{66} -329.305 q^{67} +16.5399 q^{68} +69.0000 q^{69} +191.384 q^{70} -57.5035 q^{71} +72.0000 q^{72} -536.143 q^{73} +108.864 q^{74} -185.627 q^{75} +532.452 q^{76} +15.7426 q^{77} -274.467 q^{78} -245.033 q^{79} -218.724 q^{80} +81.0000 q^{81} -468.408 q^{82} +481.992 q^{83} +84.0000 q^{84} -56.5260 q^{85} -532.433 q^{86} +592.023 q^{87} -17.9915 q^{88} +750.007 q^{89} -246.065 q^{90} -320.212 q^{91} -92.0000 q^{92} -349.560 q^{93} -683.413 q^{94} -1819.69 q^{95} -96.0000 q^{96} +1258.45 q^{97} +98.0000 q^{98} -20.2405 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 5 q^{5} - 18 q^{6} - 21 q^{7} + 24 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 5 q^{5} - 18 q^{6} - 21 q^{7} + 24 q^{8} + 27 q^{9} + 10 q^{10} - 56 q^{11} - 36 q^{12} + 29 q^{13} - 42 q^{14} - 15 q^{15} + 48 q^{16} + 28 q^{17} + 54 q^{18} - 18 q^{19} + 20 q^{20} + 63 q^{21} - 112 q^{22} - 69 q^{23} - 72 q^{24} + 32 q^{25} + 58 q^{26} - 81 q^{27} - 84 q^{28} - 242 q^{29} - 30 q^{30} - 86 q^{31} + 96 q^{32} + 168 q^{33} + 56 q^{34} - 35 q^{35} + 108 q^{36} - 70 q^{37} - 36 q^{38} - 87 q^{39} + 40 q^{40} - 402 q^{41} + 126 q^{42} - 553 q^{43} - 224 q^{44} + 45 q^{45} - 138 q^{46} - 368 q^{47} - 144 q^{48} + 147 q^{49} + 64 q^{50} - 84 q^{51} + 116 q^{52} + 23 q^{53} - 162 q^{54} - 467 q^{55} - 168 q^{56} + 54 q^{57} - 484 q^{58} - 861 q^{59} - 60 q^{60} - 311 q^{61} - 172 q^{62} - 189 q^{63} + 192 q^{64} - 624 q^{65} + 336 q^{66} - 215 q^{67} + 112 q^{68} + 207 q^{69} - 70 q^{70} + 121 q^{71} + 216 q^{72} - 588 q^{73} - 140 q^{74} - 96 q^{75} - 72 q^{76} + 392 q^{77} - 174 q^{78} - 1418 q^{79} + 80 q^{80} + 243 q^{81} - 804 q^{82} - 352 q^{83} + 252 q^{84} - 827 q^{85} - 1106 q^{86} + 726 q^{87} - 448 q^{88} + 1275 q^{89} + 90 q^{90} - 203 q^{91} - 276 q^{92} + 258 q^{93} - 736 q^{94} - 3593 q^{95} - 288 q^{96} - 602 q^{97} + 294 q^{98} - 504 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) −13.6703 −1.22270 −0.611352 0.791359i \(-0.709374\pi\)
−0.611352 + 0.791359i \(0.709374\pi\)
\(6\) −6.00000 −0.408248
\(7\) −7.00000 −0.377964
\(8\) 8.00000 0.353553
\(9\) 9.00000 0.333333
\(10\) −27.3405 −0.864583
\(11\) −2.24894 −0.0616438 −0.0308219 0.999525i \(-0.509812\pi\)
−0.0308219 + 0.999525i \(0.509812\pi\)
\(12\) −12.0000 −0.288675
\(13\) 45.7445 0.975942 0.487971 0.872860i \(-0.337737\pi\)
0.487971 + 0.872860i \(0.337737\pi\)
\(14\) −14.0000 −0.267261
\(15\) 41.0108 0.705929
\(16\) 16.0000 0.250000
\(17\) 4.13496 0.0589927 0.0294964 0.999565i \(-0.490610\pi\)
0.0294964 + 0.999565i \(0.490610\pi\)
\(18\) 18.0000 0.235702
\(19\) 133.113 1.60728 0.803638 0.595119i \(-0.202895\pi\)
0.803638 + 0.595119i \(0.202895\pi\)
\(20\) −54.6810 −0.611352
\(21\) 21.0000 0.218218
\(22\) −4.49789 −0.0435888
\(23\) −23.0000 −0.208514
\(24\) −24.0000 −0.204124
\(25\) 61.8758 0.495006
\(26\) 91.4891 0.690095
\(27\) −27.0000 −0.192450
\(28\) −28.0000 −0.188982
\(29\) −197.341 −1.26363 −0.631816 0.775119i \(-0.717690\pi\)
−0.631816 + 0.775119i \(0.717690\pi\)
\(30\) 82.0215 0.499167
\(31\) 116.520 0.675084 0.337542 0.941311i \(-0.390405\pi\)
0.337542 + 0.941311i \(0.390405\pi\)
\(32\) 32.0000 0.176777
\(33\) 6.74683 0.0355901
\(34\) 8.26993 0.0417141
\(35\) 95.6918 0.462139
\(36\) 36.0000 0.166667
\(37\) 54.4321 0.241853 0.120927 0.992661i \(-0.461413\pi\)
0.120927 + 0.992661i \(0.461413\pi\)
\(38\) 266.226 1.13652
\(39\) −137.234 −0.563461
\(40\) −109.362 −0.432291
\(41\) −234.204 −0.892111 −0.446055 0.895005i \(-0.647172\pi\)
−0.446055 + 0.895005i \(0.647172\pi\)
\(42\) 42.0000 0.154303
\(43\) −266.216 −0.944130 −0.472065 0.881564i \(-0.656491\pi\)
−0.472065 + 0.881564i \(0.656491\pi\)
\(44\) −8.99577 −0.0308219
\(45\) −123.032 −0.407568
\(46\) −46.0000 −0.147442
\(47\) −341.707 −1.06049 −0.530245 0.847844i \(-0.677900\pi\)
−0.530245 + 0.847844i \(0.677900\pi\)
\(48\) −48.0000 −0.144338
\(49\) 49.0000 0.142857
\(50\) 123.752 0.350022
\(51\) −12.4049 −0.0340595
\(52\) 182.978 0.487971
\(53\) 114.614 0.297046 0.148523 0.988909i \(-0.452548\pi\)
0.148523 + 0.988909i \(0.452548\pi\)
\(54\) −54.0000 −0.136083
\(55\) 30.7436 0.0753722
\(56\) −56.0000 −0.133631
\(57\) −399.339 −0.927961
\(58\) −394.682 −0.893522
\(59\) −292.868 −0.646240 −0.323120 0.946358i \(-0.604732\pi\)
−0.323120 + 0.946358i \(0.604732\pi\)
\(60\) 164.043 0.352964
\(61\) −648.641 −1.36148 −0.680738 0.732527i \(-0.738341\pi\)
−0.680738 + 0.732527i \(0.738341\pi\)
\(62\) 233.040 0.477356
\(63\) −63.0000 −0.125988
\(64\) 64.0000 0.125000
\(65\) −625.339 −1.19329
\(66\) 13.4937 0.0251660
\(67\) −329.305 −0.600463 −0.300231 0.953866i \(-0.597064\pi\)
−0.300231 + 0.953866i \(0.597064\pi\)
\(68\) 16.5399 0.0294964
\(69\) 69.0000 0.120386
\(70\) 191.384 0.326782
\(71\) −57.5035 −0.0961185 −0.0480592 0.998844i \(-0.515304\pi\)
−0.0480592 + 0.998844i \(0.515304\pi\)
\(72\) 72.0000 0.117851
\(73\) −536.143 −0.859599 −0.429800 0.902924i \(-0.641416\pi\)
−0.429800 + 0.902924i \(0.641416\pi\)
\(74\) 108.864 0.171016
\(75\) −185.627 −0.285792
\(76\) 532.452 0.803638
\(77\) 15.7426 0.0232992
\(78\) −274.467 −0.398427
\(79\) −245.033 −0.348966 −0.174483 0.984660i \(-0.555825\pi\)
−0.174483 + 0.984660i \(0.555825\pi\)
\(80\) −218.724 −0.305676
\(81\) 81.0000 0.111111
\(82\) −468.408 −0.630817
\(83\) 481.992 0.637416 0.318708 0.947853i \(-0.396751\pi\)
0.318708 + 0.947853i \(0.396751\pi\)
\(84\) 84.0000 0.109109
\(85\) −56.5260 −0.0721307
\(86\) −532.433 −0.667601
\(87\) 592.023 0.729558
\(88\) −17.9915 −0.0217944
\(89\) 750.007 0.893265 0.446632 0.894718i \(-0.352623\pi\)
0.446632 + 0.894718i \(0.352623\pi\)
\(90\) −246.065 −0.288194
\(91\) −320.212 −0.368871
\(92\) −92.0000 −0.104257
\(93\) −349.560 −0.389760
\(94\) −683.413 −0.749880
\(95\) −1819.69 −1.96522
\(96\) −96.0000 −0.102062
\(97\) 1258.45 1.31729 0.658643 0.752456i \(-0.271131\pi\)
0.658643 + 0.752456i \(0.271131\pi\)
\(98\) 98.0000 0.101015
\(99\) −20.2405 −0.0205479
\(100\) 247.503 0.247503
\(101\) −187.049 −0.184278 −0.0921390 0.995746i \(-0.529370\pi\)
−0.0921390 + 0.995746i \(0.529370\pi\)
\(102\) −24.8098 −0.0240837
\(103\) −23.0657 −0.0220653 −0.0110327 0.999939i \(-0.503512\pi\)
−0.0110327 + 0.999939i \(0.503512\pi\)
\(104\) 365.956 0.345048
\(105\) −287.075 −0.266816
\(106\) 229.228 0.210043
\(107\) −2038.20 −1.84150 −0.920748 0.390158i \(-0.872420\pi\)
−0.920748 + 0.390158i \(0.872420\pi\)
\(108\) −108.000 −0.0962250
\(109\) −1090.83 −0.958559 −0.479280 0.877662i \(-0.659102\pi\)
−0.479280 + 0.877662i \(0.659102\pi\)
\(110\) 61.4872 0.0532962
\(111\) −163.296 −0.139634
\(112\) −112.000 −0.0944911
\(113\) −2043.44 −1.70116 −0.850580 0.525846i \(-0.823749\pi\)
−0.850580 + 0.525846i \(0.823749\pi\)
\(114\) −798.678 −0.656167
\(115\) 314.416 0.254952
\(116\) −789.364 −0.631816
\(117\) 411.701 0.325314
\(118\) −585.736 −0.456961
\(119\) −28.9447 −0.0222971
\(120\) 328.086 0.249584
\(121\) −1325.94 −0.996200
\(122\) −1297.28 −0.962709
\(123\) 702.612 0.515060
\(124\) 466.080 0.337542
\(125\) 862.924 0.617458
\(126\) −126.000 −0.0890871
\(127\) 2696.86 1.88431 0.942154 0.335179i \(-0.108797\pi\)
0.942154 + 0.335179i \(0.108797\pi\)
\(128\) 128.000 0.0883883
\(129\) 798.649 0.545094
\(130\) −1250.68 −0.843783
\(131\) −243.606 −0.162473 −0.0812366 0.996695i \(-0.525887\pi\)
−0.0812366 + 0.996695i \(0.525887\pi\)
\(132\) 26.9873 0.0177950
\(133\) −931.792 −0.607493
\(134\) −658.610 −0.424591
\(135\) 369.097 0.235310
\(136\) 33.0797 0.0208571
\(137\) 42.4481 0.0264714 0.0132357 0.999912i \(-0.495787\pi\)
0.0132357 + 0.999912i \(0.495787\pi\)
\(138\) 138.000 0.0851257
\(139\) −190.134 −0.116021 −0.0580106 0.998316i \(-0.518476\pi\)
−0.0580106 + 0.998316i \(0.518476\pi\)
\(140\) 382.767 0.231069
\(141\) 1025.12 0.612274
\(142\) −115.007 −0.0679660
\(143\) −102.877 −0.0601608
\(144\) 144.000 0.0833333
\(145\) 2697.70 1.54505
\(146\) −1072.29 −0.607829
\(147\) −147.000 −0.0824786
\(148\) 217.728 0.120927
\(149\) −743.917 −0.409020 −0.204510 0.978864i \(-0.565560\pi\)
−0.204510 + 0.978864i \(0.565560\pi\)
\(150\) −371.255 −0.202085
\(151\) −843.472 −0.454575 −0.227287 0.973828i \(-0.572986\pi\)
−0.227287 + 0.973828i \(0.572986\pi\)
\(152\) 1064.90 0.568258
\(153\) 37.2147 0.0196642
\(154\) 31.4852 0.0164750
\(155\) −1592.86 −0.825428
\(156\) −548.934 −0.281730
\(157\) 800.908 0.407130 0.203565 0.979061i \(-0.434747\pi\)
0.203565 + 0.979061i \(0.434747\pi\)
\(158\) −490.065 −0.246756
\(159\) −343.841 −0.171499
\(160\) −437.448 −0.216146
\(161\) 161.000 0.0788110
\(162\) 162.000 0.0785674
\(163\) −2882.00 −1.38488 −0.692442 0.721474i \(-0.743465\pi\)
−0.692442 + 0.721474i \(0.743465\pi\)
\(164\) −936.816 −0.446055
\(165\) −92.2309 −0.0435161
\(166\) 963.985 0.450721
\(167\) −1920.92 −0.890092 −0.445046 0.895508i \(-0.646813\pi\)
−0.445046 + 0.895508i \(0.646813\pi\)
\(168\) 168.000 0.0771517
\(169\) −104.438 −0.0475367
\(170\) −113.052 −0.0510041
\(171\) 1198.02 0.535758
\(172\) −1064.87 −0.472065
\(173\) 1054.30 0.463336 0.231668 0.972795i \(-0.425582\pi\)
0.231668 + 0.972795i \(0.425582\pi\)
\(174\) 1184.05 0.515875
\(175\) −433.131 −0.187095
\(176\) −35.9831 −0.0154110
\(177\) 878.604 0.373107
\(178\) 1500.01 0.631634
\(179\) 1791.12 0.747903 0.373952 0.927448i \(-0.378003\pi\)
0.373952 + 0.927448i \(0.378003\pi\)
\(180\) −492.129 −0.203784
\(181\) 2257.56 0.927088 0.463544 0.886074i \(-0.346578\pi\)
0.463544 + 0.886074i \(0.346578\pi\)
\(182\) −640.423 −0.260832
\(183\) 1945.92 0.786048
\(184\) −184.000 −0.0737210
\(185\) −744.100 −0.295715
\(186\) −699.120 −0.275602
\(187\) −9.29930 −0.00363654
\(188\) −1366.83 −0.530245
\(189\) 189.000 0.0727393
\(190\) −3639.38 −1.38962
\(191\) −1959.08 −0.742166 −0.371083 0.928600i \(-0.621014\pi\)
−0.371083 + 0.928600i \(0.621014\pi\)
\(192\) −192.000 −0.0721688
\(193\) 597.599 0.222882 0.111441 0.993771i \(-0.464453\pi\)
0.111441 + 0.993771i \(0.464453\pi\)
\(194\) 2516.91 0.931461
\(195\) 1876.02 0.688946
\(196\) 196.000 0.0714286
\(197\) 577.779 0.208960 0.104480 0.994527i \(-0.466682\pi\)
0.104480 + 0.994527i \(0.466682\pi\)
\(198\) −40.4810 −0.0145296
\(199\) 750.244 0.267253 0.133627 0.991032i \(-0.457338\pi\)
0.133627 + 0.991032i \(0.457338\pi\)
\(200\) 495.006 0.175011
\(201\) 987.915 0.346677
\(202\) −374.098 −0.130304
\(203\) 1381.39 0.477608
\(204\) −49.6196 −0.0170297
\(205\) 3201.63 1.09079
\(206\) −46.1313 −0.0156025
\(207\) −207.000 −0.0695048
\(208\) 731.912 0.243986
\(209\) −299.364 −0.0990786
\(210\) −574.151 −0.188667
\(211\) −2101.72 −0.685726 −0.342863 0.939385i \(-0.611397\pi\)
−0.342863 + 0.939385i \(0.611397\pi\)
\(212\) 458.455 0.148523
\(213\) 172.511 0.0554940
\(214\) −4076.40 −1.30213
\(215\) 3639.24 1.15439
\(216\) −216.000 −0.0680414
\(217\) −815.639 −0.255158
\(218\) −2181.67 −0.677804
\(219\) 1608.43 0.496290
\(220\) 122.974 0.0376861
\(221\) 189.152 0.0575735
\(222\) −326.592 −0.0987362
\(223\) 717.403 0.215430 0.107715 0.994182i \(-0.465647\pi\)
0.107715 + 0.994182i \(0.465647\pi\)
\(224\) −224.000 −0.0668153
\(225\) 556.882 0.165002
\(226\) −4086.89 −1.20290
\(227\) −2998.37 −0.876689 −0.438345 0.898807i \(-0.644435\pi\)
−0.438345 + 0.898807i \(0.644435\pi\)
\(228\) −1597.36 −0.463980
\(229\) −4217.18 −1.21694 −0.608470 0.793577i \(-0.708217\pi\)
−0.608470 + 0.793577i \(0.708217\pi\)
\(230\) 628.832 0.180278
\(231\) −47.2278 −0.0134518
\(232\) −1578.73 −0.446761
\(233\) −5781.45 −1.62556 −0.812779 0.582572i \(-0.802047\pi\)
−0.812779 + 0.582572i \(0.802047\pi\)
\(234\) 823.401 0.230032
\(235\) 4671.22 1.29667
\(236\) −1171.47 −0.323120
\(237\) 735.098 0.201476
\(238\) −57.8895 −0.0157665
\(239\) −1880.38 −0.508920 −0.254460 0.967083i \(-0.581898\pi\)
−0.254460 + 0.967083i \(0.581898\pi\)
\(240\) 656.172 0.176482
\(241\) −3640.87 −0.973149 −0.486575 0.873639i \(-0.661754\pi\)
−0.486575 + 0.873639i \(0.661754\pi\)
\(242\) −2651.88 −0.704420
\(243\) −243.000 −0.0641500
\(244\) −2594.57 −0.680738
\(245\) −669.842 −0.174672
\(246\) 1405.22 0.364203
\(247\) 6089.19 1.56861
\(248\) 932.159 0.238678
\(249\) −1445.98 −0.368012
\(250\) 1725.85 0.436609
\(251\) 3732.87 0.938711 0.469355 0.883009i \(-0.344486\pi\)
0.469355 + 0.883009i \(0.344486\pi\)
\(252\) −252.000 −0.0629941
\(253\) 51.7257 0.0128536
\(254\) 5393.71 1.33241
\(255\) 169.578 0.0416447
\(256\) 256.000 0.0625000
\(257\) 5136.35 1.24668 0.623339 0.781951i \(-0.285776\pi\)
0.623339 + 0.781951i \(0.285776\pi\)
\(258\) 1597.30 0.385440
\(259\) −381.024 −0.0914120
\(260\) −2501.36 −0.596644
\(261\) −1776.07 −0.421210
\(262\) −487.213 −0.114886
\(263\) 2813.09 0.659554 0.329777 0.944059i \(-0.393026\pi\)
0.329777 + 0.944059i \(0.393026\pi\)
\(264\) 53.9746 0.0125830
\(265\) −1566.80 −0.363199
\(266\) −1863.58 −0.429562
\(267\) −2250.02 −0.515727
\(268\) −1317.22 −0.300231
\(269\) 4514.17 1.02317 0.511587 0.859232i \(-0.329058\pi\)
0.511587 + 0.859232i \(0.329058\pi\)
\(270\) 738.194 0.166389
\(271\) −7839.12 −1.75717 −0.878584 0.477588i \(-0.841511\pi\)
−0.878584 + 0.477588i \(0.841511\pi\)
\(272\) 66.1594 0.0147482
\(273\) 960.635 0.212968
\(274\) 84.8961 0.0187181
\(275\) −139.155 −0.0305141
\(276\) 276.000 0.0601929
\(277\) −4158.00 −0.901912 −0.450956 0.892546i \(-0.648917\pi\)
−0.450956 + 0.892546i \(0.648917\pi\)
\(278\) −380.268 −0.0820394
\(279\) 1048.68 0.225028
\(280\) 765.534 0.163391
\(281\) −4081.76 −0.866539 −0.433270 0.901264i \(-0.642640\pi\)
−0.433270 + 0.901264i \(0.642640\pi\)
\(282\) 2050.24 0.432943
\(283\) 3452.60 0.725216 0.362608 0.931942i \(-0.381886\pi\)
0.362608 + 0.931942i \(0.381886\pi\)
\(284\) −230.014 −0.0480592
\(285\) 5459.07 1.13462
\(286\) −205.754 −0.0425401
\(287\) 1639.43 0.337186
\(288\) 288.000 0.0589256
\(289\) −4895.90 −0.996520
\(290\) 5395.40 1.09251
\(291\) −3775.36 −0.760535
\(292\) −2144.57 −0.429800
\(293\) −71.6634 −0.0142888 −0.00714440 0.999974i \(-0.502274\pi\)
−0.00714440 + 0.999974i \(0.502274\pi\)
\(294\) −294.000 −0.0583212
\(295\) 4003.58 0.790160
\(296\) 435.457 0.0855081
\(297\) 60.7215 0.0118634
\(298\) −1487.83 −0.289221
\(299\) −1052.12 −0.203498
\(300\) −742.510 −0.142896
\(301\) 1863.51 0.356848
\(302\) −1686.94 −0.321433
\(303\) 561.147 0.106393
\(304\) 2129.81 0.401819
\(305\) 8867.09 1.66468
\(306\) 74.4294 0.0139047
\(307\) −4057.18 −0.754252 −0.377126 0.926162i \(-0.623088\pi\)
−0.377126 + 0.926162i \(0.623088\pi\)
\(308\) 62.9704 0.0116496
\(309\) 69.1970 0.0127394
\(310\) −3185.71 −0.583666
\(311\) −2894.69 −0.527790 −0.263895 0.964551i \(-0.585007\pi\)
−0.263895 + 0.964551i \(0.585007\pi\)
\(312\) −1097.87 −0.199213
\(313\) 6860.31 1.23887 0.619437 0.785046i \(-0.287361\pi\)
0.619437 + 0.785046i \(0.287361\pi\)
\(314\) 1601.82 0.287884
\(315\) 861.226 0.154046
\(316\) −980.131 −0.174483
\(317\) −649.300 −0.115042 −0.0575210 0.998344i \(-0.518320\pi\)
−0.0575210 + 0.998344i \(0.518320\pi\)
\(318\) −687.683 −0.121268
\(319\) 443.809 0.0778951
\(320\) −874.896 −0.152838
\(321\) 6114.59 1.06319
\(322\) 322.000 0.0557278
\(323\) 550.418 0.0948175
\(324\) 324.000 0.0555556
\(325\) 2830.48 0.483098
\(326\) −5764.01 −0.979260
\(327\) 3272.50 0.553425
\(328\) −1873.63 −0.315409
\(329\) 2391.95 0.400828
\(330\) −184.462 −0.0307706
\(331\) 357.326 0.0593365 0.0296683 0.999560i \(-0.490555\pi\)
0.0296683 + 0.999560i \(0.490555\pi\)
\(332\) 1927.97 0.318708
\(333\) 489.889 0.0806178
\(334\) −3841.84 −0.629390
\(335\) 4501.68 0.734189
\(336\) 336.000 0.0545545
\(337\) 1082.26 0.174940 0.0874698 0.996167i \(-0.472122\pi\)
0.0874698 + 0.996167i \(0.472122\pi\)
\(338\) −208.876 −0.0336136
\(339\) 6130.33 0.982165
\(340\) −226.104 −0.0360653
\(341\) −262.047 −0.0416147
\(342\) 2396.04 0.378838
\(343\) −343.000 −0.0539949
\(344\) −2129.73 −0.333800
\(345\) −943.247 −0.147196
\(346\) 2108.60 0.327628
\(347\) −8970.38 −1.38777 −0.693883 0.720087i \(-0.744102\pi\)
−0.693883 + 0.720087i \(0.744102\pi\)
\(348\) 2368.09 0.364779
\(349\) 4741.49 0.727238 0.363619 0.931548i \(-0.381541\pi\)
0.363619 + 0.931548i \(0.381541\pi\)
\(350\) −866.261 −0.132296
\(351\) −1235.10 −0.187820
\(352\) −71.9662 −0.0108972
\(353\) −3746.71 −0.564921 −0.282460 0.959279i \(-0.591151\pi\)
−0.282460 + 0.959279i \(0.591151\pi\)
\(354\) 1757.21 0.263826
\(355\) 786.087 0.117524
\(356\) 3000.03 0.446632
\(357\) 86.8342 0.0128733
\(358\) 3582.24 0.528847
\(359\) 3071.00 0.451480 0.225740 0.974188i \(-0.427520\pi\)
0.225740 + 0.974188i \(0.427520\pi\)
\(360\) −984.258 −0.144097
\(361\) 10860.1 1.58333
\(362\) 4515.11 0.655550
\(363\) 3977.83 0.575156
\(364\) −1280.85 −0.184436
\(365\) 7329.21 1.05104
\(366\) 3891.85 0.555820
\(367\) 10815.2 1.53827 0.769137 0.639084i \(-0.220686\pi\)
0.769137 + 0.639084i \(0.220686\pi\)
\(368\) −368.000 −0.0521286
\(369\) −2107.84 −0.297370
\(370\) −1488.20 −0.209102
\(371\) −802.297 −0.112273
\(372\) −1398.24 −0.194880
\(373\) 6306.89 0.875491 0.437746 0.899099i \(-0.355777\pi\)
0.437746 + 0.899099i \(0.355777\pi\)
\(374\) −18.5986 −0.00257142
\(375\) −2588.77 −0.356490
\(376\) −2733.65 −0.374940
\(377\) −9027.27 −1.23323
\(378\) 378.000 0.0514344
\(379\) 10326.7 1.39960 0.699798 0.714341i \(-0.253273\pi\)
0.699798 + 0.714341i \(0.253273\pi\)
\(380\) −7278.76 −0.982611
\(381\) −8090.57 −1.08791
\(382\) −3918.15 −0.524791
\(383\) −2.17159 −0.000289721 0 −0.000144860 1.00000i \(-0.500046\pi\)
−0.000144860 1.00000i \(0.500046\pi\)
\(384\) −384.000 −0.0510310
\(385\) −215.205 −0.0284880
\(386\) 1195.20 0.157601
\(387\) −2395.95 −0.314710
\(388\) 5033.82 0.658643
\(389\) 5237.25 0.682620 0.341310 0.939951i \(-0.389129\pi\)
0.341310 + 0.939951i \(0.389129\pi\)
\(390\) 3752.04 0.487158
\(391\) −95.1042 −0.0123008
\(392\) 392.000 0.0505076
\(393\) 730.819 0.0938040
\(394\) 1155.56 0.147757
\(395\) 3349.66 0.426683
\(396\) −80.9620 −0.0102740
\(397\) −4123.42 −0.521281 −0.260641 0.965436i \(-0.583934\pi\)
−0.260641 + 0.965436i \(0.583934\pi\)
\(398\) 1500.49 0.188977
\(399\) 2795.37 0.350736
\(400\) 990.013 0.123752
\(401\) −3775.05 −0.470117 −0.235059 0.971981i \(-0.575528\pi\)
−0.235059 + 0.971981i \(0.575528\pi\)
\(402\) 1975.83 0.245138
\(403\) 5330.15 0.658843
\(404\) −748.196 −0.0921390
\(405\) −1107.29 −0.135856
\(406\) 2762.77 0.337720
\(407\) −122.415 −0.0149088
\(408\) −99.2391 −0.0120418
\(409\) −7150.35 −0.864455 −0.432227 0.901765i \(-0.642272\pi\)
−0.432227 + 0.901765i \(0.642272\pi\)
\(410\) 6403.26 0.771303
\(411\) −127.344 −0.0152833
\(412\) −92.2627 −0.0110327
\(413\) 2050.08 0.244256
\(414\) −414.000 −0.0491473
\(415\) −6588.96 −0.779372
\(416\) 1463.82 0.172524
\(417\) 570.402 0.0669849
\(418\) −598.728 −0.0700591
\(419\) 4943.81 0.576423 0.288211 0.957567i \(-0.406939\pi\)
0.288211 + 0.957567i \(0.406939\pi\)
\(420\) −1148.30 −0.133408
\(421\) 8461.47 0.979541 0.489771 0.871851i \(-0.337081\pi\)
0.489771 + 0.871851i \(0.337081\pi\)
\(422\) −4203.44 −0.484882
\(423\) −3075.36 −0.353497
\(424\) 916.910 0.105021
\(425\) 255.854 0.0292018
\(426\) 345.021 0.0392402
\(427\) 4540.49 0.514589
\(428\) −8152.79 −0.920748
\(429\) 308.631 0.0347339
\(430\) 7278.49 0.816279
\(431\) −4753.05 −0.531199 −0.265599 0.964084i \(-0.585570\pi\)
−0.265599 + 0.964084i \(0.585570\pi\)
\(432\) −432.000 −0.0481125
\(433\) 7712.94 0.856029 0.428014 0.903772i \(-0.359213\pi\)
0.428014 + 0.903772i \(0.359213\pi\)
\(434\) −1631.28 −0.180424
\(435\) −8093.10 −0.892034
\(436\) −4363.34 −0.479280
\(437\) −3061.60 −0.335140
\(438\) 3216.86 0.350930
\(439\) −7571.77 −0.823191 −0.411595 0.911367i \(-0.635028\pi\)
−0.411595 + 0.911367i \(0.635028\pi\)
\(440\) 245.949 0.0266481
\(441\) 441.000 0.0476190
\(442\) 378.304 0.0407106
\(443\) −12543.0 −1.34523 −0.672615 0.739992i \(-0.734829\pi\)
−0.672615 + 0.739992i \(0.734829\pi\)
\(444\) −653.185 −0.0698171
\(445\) −10252.8 −1.09220
\(446\) 1434.81 0.152332
\(447\) 2231.75 0.236148
\(448\) −448.000 −0.0472456
\(449\) 8755.46 0.920258 0.460129 0.887852i \(-0.347803\pi\)
0.460129 + 0.887852i \(0.347803\pi\)
\(450\) 1113.76 0.116674
\(451\) 526.712 0.0549931
\(452\) −8173.78 −0.850580
\(453\) 2530.42 0.262449
\(454\) −5996.73 −0.619913
\(455\) 4377.37 0.451021
\(456\) −3194.71 −0.328084
\(457\) −2061.64 −0.211027 −0.105514 0.994418i \(-0.533649\pi\)
−0.105514 + 0.994418i \(0.533649\pi\)
\(458\) −8434.37 −0.860507
\(459\) −111.644 −0.0113532
\(460\) 1257.66 0.127476
\(461\) −4924.04 −0.497473 −0.248737 0.968571i \(-0.580015\pi\)
−0.248737 + 0.968571i \(0.580015\pi\)
\(462\) −94.4556 −0.00951185
\(463\) 4482.52 0.449936 0.224968 0.974366i \(-0.427772\pi\)
0.224968 + 0.974366i \(0.427772\pi\)
\(464\) −3157.46 −0.315908
\(465\) 4778.57 0.476561
\(466\) −11562.9 −1.14944
\(467\) −9387.29 −0.930175 −0.465088 0.885265i \(-0.653977\pi\)
−0.465088 + 0.885265i \(0.653977\pi\)
\(468\) 1646.80 0.162657
\(469\) 2305.13 0.226954
\(470\) 9342.43 0.916882
\(471\) −2402.72 −0.235057
\(472\) −2342.94 −0.228480
\(473\) 598.705 0.0581998
\(474\) 1470.20 0.142465
\(475\) 8236.48 0.795612
\(476\) −115.779 −0.0111486
\(477\) 1031.52 0.0990152
\(478\) −3760.76 −0.359861
\(479\) −20437.4 −1.94949 −0.974747 0.223313i \(-0.928313\pi\)
−0.974747 + 0.223313i \(0.928313\pi\)
\(480\) 1312.34 0.124792
\(481\) 2489.97 0.236035
\(482\) −7281.74 −0.688120
\(483\) −483.000 −0.0455016
\(484\) −5303.77 −0.498100
\(485\) −17203.4 −1.61065
\(486\) −486.000 −0.0453609
\(487\) 5380.22 0.500618 0.250309 0.968166i \(-0.419468\pi\)
0.250309 + 0.968166i \(0.419468\pi\)
\(488\) −5189.13 −0.481354
\(489\) 8646.01 0.799563
\(490\) −1339.68 −0.123512
\(491\) −11763.2 −1.08120 −0.540598 0.841281i \(-0.681802\pi\)
−0.540598 + 0.841281i \(0.681802\pi\)
\(492\) 2810.45 0.257530
\(493\) −815.998 −0.0745450
\(494\) 12178.4 1.10917
\(495\) 276.693 0.0251241
\(496\) 1864.32 0.168771
\(497\) 402.525 0.0363294
\(498\) −2891.95 −0.260224
\(499\) 19677.6 1.76531 0.882655 0.470021i \(-0.155754\pi\)
0.882655 + 0.470021i \(0.155754\pi\)
\(500\) 3451.70 0.308729
\(501\) 5762.76 0.513895
\(502\) 7465.73 0.663769
\(503\) −1256.89 −0.111415 −0.0557075 0.998447i \(-0.517741\pi\)
−0.0557075 + 0.998447i \(0.517741\pi\)
\(504\) −504.000 −0.0445435
\(505\) 2557.01 0.225318
\(506\) 103.451 0.00908888
\(507\) 313.315 0.0274454
\(508\) 10787.4 0.942154
\(509\) 7115.85 0.619655 0.309828 0.950793i \(-0.399729\pi\)
0.309828 + 0.950793i \(0.399729\pi\)
\(510\) 339.156 0.0294472
\(511\) 3753.00 0.324898
\(512\) 512.000 0.0441942
\(513\) −3594.05 −0.309320
\(514\) 10272.7 0.881535
\(515\) 315.314 0.0269794
\(516\) 3194.60 0.272547
\(517\) 768.479 0.0653727
\(518\) −762.049 −0.0646380
\(519\) −3162.91 −0.267507
\(520\) −5002.71 −0.421891
\(521\) 2680.13 0.225372 0.112686 0.993631i \(-0.464055\pi\)
0.112686 + 0.993631i \(0.464055\pi\)
\(522\) −3552.14 −0.297841
\(523\) −6097.07 −0.509763 −0.254882 0.966972i \(-0.582037\pi\)
−0.254882 + 0.966972i \(0.582037\pi\)
\(524\) −974.426 −0.0812366
\(525\) 1299.39 0.108019
\(526\) 5626.18 0.466375
\(527\) 481.806 0.0398250
\(528\) 107.949 0.00889752
\(529\) 529.000 0.0434783
\(530\) −3133.60 −0.256820
\(531\) −2635.81 −0.215413
\(532\) −3727.17 −0.303747
\(533\) −10713.6 −0.870648
\(534\) −4500.04 −0.364674
\(535\) 27862.7 2.25160
\(536\) −2634.44 −0.212296
\(537\) −5373.36 −0.431802
\(538\) 9028.34 0.723493
\(539\) −110.198 −0.00880626
\(540\) 1476.39 0.117655
\(541\) −8998.60 −0.715120 −0.357560 0.933890i \(-0.616391\pi\)
−0.357560 + 0.933890i \(0.616391\pi\)
\(542\) −15678.2 −1.24251
\(543\) −6772.67 −0.535254
\(544\) 132.319 0.0104285
\(545\) 14912.0 1.17203
\(546\) 1921.27 0.150591
\(547\) 2587.96 0.202291 0.101145 0.994872i \(-0.467749\pi\)
0.101145 + 0.994872i \(0.467749\pi\)
\(548\) 169.792 0.0132357
\(549\) −5837.77 −0.453825
\(550\) −278.310 −0.0215767
\(551\) −26268.7 −2.03100
\(552\) 552.000 0.0425628
\(553\) 1715.23 0.131897
\(554\) −8315.99 −0.637748
\(555\) 2232.30 0.170731
\(556\) −760.536 −0.0580106
\(557\) 21691.8 1.65011 0.825054 0.565053i \(-0.191144\pi\)
0.825054 + 0.565053i \(0.191144\pi\)
\(558\) 2097.36 0.159119
\(559\) −12177.9 −0.921417
\(560\) 1531.07 0.115535
\(561\) 27.8979 0.00209955
\(562\) −8163.52 −0.612736
\(563\) 9737.57 0.728934 0.364467 0.931216i \(-0.381251\pi\)
0.364467 + 0.931216i \(0.381251\pi\)
\(564\) 4100.48 0.306137
\(565\) 27934.4 2.08002
\(566\) 6905.21 0.512805
\(567\) −567.000 −0.0419961
\(568\) −460.028 −0.0339830
\(569\) −7487.90 −0.551685 −0.275843 0.961203i \(-0.588957\pi\)
−0.275843 + 0.961203i \(0.588957\pi\)
\(570\) 10918.1 0.802299
\(571\) 21995.9 1.61208 0.806041 0.591859i \(-0.201606\pi\)
0.806041 + 0.591859i \(0.201606\pi\)
\(572\) −411.507 −0.0300804
\(573\) 5877.23 0.428490
\(574\) 3278.86 0.238427
\(575\) −1423.14 −0.103216
\(576\) 576.000 0.0416667
\(577\) −290.397 −0.0209521 −0.0104761 0.999945i \(-0.503335\pi\)
−0.0104761 + 0.999945i \(0.503335\pi\)
\(578\) −9791.80 −0.704646
\(579\) −1792.80 −0.128681
\(580\) 10790.8 0.772524
\(581\) −3373.95 −0.240921
\(582\) −7550.73 −0.537779
\(583\) −257.760 −0.0183110
\(584\) −4289.14 −0.303914
\(585\) −5628.05 −0.397763
\(586\) −143.327 −0.0101037
\(587\) −7867.97 −0.553230 −0.276615 0.960981i \(-0.589213\pi\)
−0.276615 + 0.960981i \(0.589213\pi\)
\(588\) −588.000 −0.0412393
\(589\) 15510.3 1.08505
\(590\) 8007.16 0.558728
\(591\) −1733.34 −0.120643
\(592\) 870.913 0.0604634
\(593\) −11265.5 −0.780132 −0.390066 0.920787i \(-0.627548\pi\)
−0.390066 + 0.920787i \(0.627548\pi\)
\(594\) 121.443 0.00838866
\(595\) 395.682 0.0272628
\(596\) −2975.67 −0.204510
\(597\) −2250.73 −0.154299
\(598\) −2104.25 −0.143895
\(599\) −19215.6 −1.31073 −0.655364 0.755313i \(-0.727485\pi\)
−0.655364 + 0.755313i \(0.727485\pi\)
\(600\) −1485.02 −0.101043
\(601\) 19551.2 1.32697 0.663486 0.748189i \(-0.269076\pi\)
0.663486 + 0.748189i \(0.269076\pi\)
\(602\) 3727.03 0.252329
\(603\) −2963.74 −0.200154
\(604\) −3373.89 −0.227287
\(605\) 18126.0 1.21806
\(606\) 1122.29 0.0752312
\(607\) −13803.0 −0.922979 −0.461490 0.887146i \(-0.652685\pi\)
−0.461490 + 0.887146i \(0.652685\pi\)
\(608\) 4259.62 0.284129
\(609\) −4144.16 −0.275747
\(610\) 17734.2 1.17711
\(611\) −15631.2 −1.03498
\(612\) 148.859 0.00983212
\(613\) −12878.1 −0.848519 −0.424260 0.905541i \(-0.639466\pi\)
−0.424260 + 0.905541i \(0.639466\pi\)
\(614\) −8114.35 −0.533336
\(615\) −9604.89 −0.629767
\(616\) 125.941 0.00823750
\(617\) 20412.1 1.33187 0.665933 0.746012i \(-0.268034\pi\)
0.665933 + 0.746012i \(0.268034\pi\)
\(618\) 138.394 0.00900813
\(619\) 5190.18 0.337013 0.168507 0.985701i \(-0.446106\pi\)
0.168507 + 0.985701i \(0.446106\pi\)
\(620\) −6371.43 −0.412714
\(621\) 621.000 0.0401286
\(622\) −5789.38 −0.373204
\(623\) −5250.05 −0.337622
\(624\) −2195.74 −0.140865
\(625\) −19530.9 −1.24998
\(626\) 13720.6 0.876016
\(627\) 898.091 0.0572030
\(628\) 3203.63 0.203565
\(629\) 225.075 0.0142676
\(630\) 1722.45 0.108927
\(631\) 10103.1 0.637401 0.318700 0.947856i \(-0.396754\pi\)
0.318700 + 0.947856i \(0.396754\pi\)
\(632\) −1960.26 −0.123378
\(633\) 6305.15 0.395904
\(634\) −1298.60 −0.0813470
\(635\) −36866.7 −2.30395
\(636\) −1375.37 −0.0857497
\(637\) 2241.48 0.139420
\(638\) 887.618 0.0550801
\(639\) −517.532 −0.0320395
\(640\) −1749.79 −0.108073
\(641\) 17138.8 1.05607 0.528037 0.849221i \(-0.322928\pi\)
0.528037 + 0.849221i \(0.322928\pi\)
\(642\) 12229.2 0.751787
\(643\) −16302.6 −0.999860 −0.499930 0.866066i \(-0.666641\pi\)
−0.499930 + 0.866066i \(0.666641\pi\)
\(644\) 644.000 0.0394055
\(645\) −10917.7 −0.666489
\(646\) 1100.84 0.0670461
\(647\) 26719.6 1.62358 0.811791 0.583949i \(-0.198493\pi\)
0.811791 + 0.583949i \(0.198493\pi\)
\(648\) 648.000 0.0392837
\(649\) 658.643 0.0398367
\(650\) 5660.96 0.341602
\(651\) 2446.92 0.147315
\(652\) −11528.0 −0.692442
\(653\) 27974.5 1.67646 0.838229 0.545319i \(-0.183591\pi\)
0.838229 + 0.545319i \(0.183591\pi\)
\(654\) 6545.01 0.391330
\(655\) 3330.16 0.198657
\(656\) −3747.27 −0.223028
\(657\) −4825.28 −0.286533
\(658\) 4783.89 0.283428
\(659\) −28367.4 −1.67684 −0.838421 0.545023i \(-0.816521\pi\)
−0.838421 + 0.545023i \(0.816521\pi\)
\(660\) −368.923 −0.0217581
\(661\) 22702.1 1.33587 0.667935 0.744219i \(-0.267178\pi\)
0.667935 + 0.744219i \(0.267178\pi\)
\(662\) 714.651 0.0419573
\(663\) −567.456 −0.0332401
\(664\) 3855.94 0.225361
\(665\) 12737.8 0.742784
\(666\) 979.777 0.0570054
\(667\) 4538.84 0.263485
\(668\) −7683.68 −0.445046
\(669\) −2152.21 −0.124378
\(670\) 9003.36 0.519150
\(671\) 1458.76 0.0839266
\(672\) 672.000 0.0385758
\(673\) 114.075 0.00653382 0.00326691 0.999995i \(-0.498960\pi\)
0.00326691 + 0.999995i \(0.498960\pi\)
\(674\) 2164.53 0.123701
\(675\) −1670.65 −0.0952640
\(676\) −417.753 −0.0237684
\(677\) −15152.5 −0.860201 −0.430101 0.902781i \(-0.641522\pi\)
−0.430101 + 0.902781i \(0.641522\pi\)
\(678\) 12260.7 0.694496
\(679\) −8809.18 −0.497887
\(680\) −452.208 −0.0255020
\(681\) 8995.10 0.506157
\(682\) −524.093 −0.0294261
\(683\) −15093.7 −0.845600 −0.422800 0.906223i \(-0.638953\pi\)
−0.422800 + 0.906223i \(0.638953\pi\)
\(684\) 4792.07 0.267879
\(685\) −580.276 −0.0323667
\(686\) −686.000 −0.0381802
\(687\) 12651.6 0.702601
\(688\) −4259.46 −0.236033
\(689\) 5242.95 0.289899
\(690\) −1886.49 −0.104084
\(691\) −21282.9 −1.17169 −0.585847 0.810422i \(-0.699238\pi\)
−0.585847 + 0.810422i \(0.699238\pi\)
\(692\) 4217.21 0.231668
\(693\) 141.683 0.00776639
\(694\) −17940.8 −0.981299
\(695\) 2599.18 0.141860
\(696\) 4736.18 0.257938
\(697\) −968.426 −0.0526280
\(698\) 9482.98 0.514235
\(699\) 17344.3 0.938517
\(700\) −1732.52 −0.0935474
\(701\) 34984.4 1.88494 0.942471 0.334288i \(-0.108496\pi\)
0.942471 + 0.334288i \(0.108496\pi\)
\(702\) −2470.20 −0.132809
\(703\) 7245.62 0.388725
\(704\) −143.932 −0.00770548
\(705\) −14013.7 −0.748631
\(706\) −7493.41 −0.399459
\(707\) 1309.34 0.0696505
\(708\) 3514.41 0.186553
\(709\) 28316.7 1.49994 0.749968 0.661474i \(-0.230069\pi\)
0.749968 + 0.661474i \(0.230069\pi\)
\(710\) 1572.17 0.0831024
\(711\) −2205.29 −0.116322
\(712\) 6000.06 0.315817
\(713\) −2679.96 −0.140765
\(714\) 173.668 0.00910277
\(715\) 1406.35 0.0735589
\(716\) 7164.48 0.373952
\(717\) 5641.15 0.293825
\(718\) 6142.00 0.319244
\(719\) −12607.4 −0.653930 −0.326965 0.945036i \(-0.606026\pi\)
−0.326965 + 0.945036i \(0.606026\pi\)
\(720\) −1968.52 −0.101892
\(721\) 161.460 0.00833991
\(722\) 21720.2 1.11959
\(723\) 10922.6 0.561848
\(724\) 9030.23 0.463544
\(725\) −12210.6 −0.625506
\(726\) 7955.65 0.406697
\(727\) −29247.4 −1.49206 −0.746029 0.665914i \(-0.768042\pi\)
−0.746029 + 0.665914i \(0.768042\pi\)
\(728\) −2561.69 −0.130416
\(729\) 729.000 0.0370370
\(730\) 14658.4 0.743195
\(731\) −1100.79 −0.0556968
\(732\) 7783.70 0.393024
\(733\) 22731.4 1.14543 0.572717 0.819753i \(-0.305890\pi\)
0.572717 + 0.819753i \(0.305890\pi\)
\(734\) 21630.3 1.08772
\(735\) 2009.53 0.100847
\(736\) −736.000 −0.0368605
\(737\) 740.588 0.0370148
\(738\) −4215.67 −0.210272
\(739\) −12055.5 −0.600095 −0.300047 0.953924i \(-0.597002\pi\)
−0.300047 + 0.953924i \(0.597002\pi\)
\(740\) −2976.40 −0.147858
\(741\) −18267.6 −0.905636
\(742\) −1604.59 −0.0793888
\(743\) −37806.7 −1.86675 −0.933375 0.358903i \(-0.883151\pi\)
−0.933375 + 0.358903i \(0.883151\pi\)
\(744\) −2796.48 −0.137801
\(745\) 10169.5 0.500111
\(746\) 12613.8 0.619066
\(747\) 4337.93 0.212472
\(748\) −37.1972 −0.00181827
\(749\) 14267.4 0.696020
\(750\) −5177.54 −0.252076
\(751\) −32936.0 −1.60033 −0.800167 0.599777i \(-0.795256\pi\)
−0.800167 + 0.599777i \(0.795256\pi\)
\(752\) −5467.31 −0.265123
\(753\) −11198.6 −0.541965
\(754\) −18054.5 −0.872026
\(755\) 11530.5 0.555811
\(756\) 756.000 0.0363696
\(757\) 9916.45 0.476116 0.238058 0.971251i \(-0.423489\pi\)
0.238058 + 0.971251i \(0.423489\pi\)
\(758\) 20653.4 0.989664
\(759\) −155.177 −0.00742104
\(760\) −14557.5 −0.694811
\(761\) −22886.3 −1.09018 −0.545092 0.838377i \(-0.683505\pi\)
−0.545092 + 0.838377i \(0.683505\pi\)
\(762\) −16181.1 −0.769266
\(763\) 7635.84 0.362301
\(764\) −7836.30 −0.371083
\(765\) −508.734 −0.0240436
\(766\) −4.34318 −0.000204864 0
\(767\) −13397.1 −0.630693
\(768\) −768.000 −0.0360844
\(769\) −26115.9 −1.22466 −0.612329 0.790603i \(-0.709767\pi\)
−0.612329 + 0.790603i \(0.709767\pi\)
\(770\) −430.411 −0.0201441
\(771\) −15409.0 −0.719770
\(772\) 2390.40 0.111441
\(773\) 4051.95 0.188536 0.0942681 0.995547i \(-0.469949\pi\)
0.0942681 + 0.995547i \(0.469949\pi\)
\(774\) −4791.89 −0.222534
\(775\) 7209.76 0.334171
\(776\) 10067.6 0.465731
\(777\) 1143.07 0.0527767
\(778\) 10474.5 0.482685
\(779\) −31175.6 −1.43387
\(780\) 7504.07 0.344473
\(781\) 129.322 0.00592511
\(782\) −190.208 −0.00869800
\(783\) 5328.21 0.243186
\(784\) 784.000 0.0357143
\(785\) −10948.6 −0.497800
\(786\) 1461.64 0.0663294
\(787\) 14327.0 0.648921 0.324461 0.945899i \(-0.394817\pi\)
0.324461 + 0.945899i \(0.394817\pi\)
\(788\) 2311.12 0.104480
\(789\) −8439.27 −0.380793
\(790\) 6699.32 0.301710
\(791\) 14304.1 0.642978
\(792\) −161.924 −0.00726479
\(793\) −29671.8 −1.32872
\(794\) −8246.85 −0.368602
\(795\) 4700.40 0.209693
\(796\) 3000.98 0.133627
\(797\) 6604.16 0.293515 0.146757 0.989173i \(-0.453116\pi\)
0.146757 + 0.989173i \(0.453116\pi\)
\(798\) 5590.75 0.248008
\(799\) −1412.95 −0.0625612
\(800\) 1980.03 0.0875056
\(801\) 6750.06 0.297755
\(802\) −7550.10 −0.332423
\(803\) 1205.75 0.0529890
\(804\) 3951.66 0.173339
\(805\) −2200.91 −0.0963626
\(806\) 10660.3 0.465872
\(807\) −13542.5 −0.590730
\(808\) −1496.39 −0.0651521
\(809\) −19434.3 −0.844592 −0.422296 0.906458i \(-0.638776\pi\)
−0.422296 + 0.906458i \(0.638776\pi\)
\(810\) −2214.58 −0.0960647
\(811\) −12521.9 −0.542174 −0.271087 0.962555i \(-0.587383\pi\)
−0.271087 + 0.962555i \(0.587383\pi\)
\(812\) 5525.55 0.238804
\(813\) 23517.4 1.01450
\(814\) −244.829 −0.0105421
\(815\) 39397.7 1.69330
\(816\) −198.478 −0.00851486
\(817\) −35436.9 −1.51748
\(818\) −14300.7 −0.611262
\(819\) −2881.91 −0.122957
\(820\) 12806.5 0.545394
\(821\) −22995.9 −0.977542 −0.488771 0.872412i \(-0.662555\pi\)
−0.488771 + 0.872412i \(0.662555\pi\)
\(822\) −254.688 −0.0108069
\(823\) 10120.8 0.428664 0.214332 0.976761i \(-0.431243\pi\)
0.214332 + 0.976761i \(0.431243\pi\)
\(824\) −184.525 −0.00780127
\(825\) 417.465 0.0176173
\(826\) 4100.15 0.172715
\(827\) 5060.60 0.212786 0.106393 0.994324i \(-0.466070\pi\)
0.106393 + 0.994324i \(0.466070\pi\)
\(828\) −828.000 −0.0347524
\(829\) 33161.5 1.38932 0.694661 0.719337i \(-0.255554\pi\)
0.694661 + 0.719337i \(0.255554\pi\)
\(830\) −13177.9 −0.551099
\(831\) 12474.0 0.520719
\(832\) 2927.65 0.121993
\(833\) 202.613 0.00842753
\(834\) 1140.80 0.0473655
\(835\) 26259.5 1.08832
\(836\) −1197.46 −0.0495393
\(837\) −3146.04 −0.129920
\(838\) 9887.63 0.407592
\(839\) −21458.6 −0.882997 −0.441498 0.897262i \(-0.645553\pi\)
−0.441498 + 0.897262i \(0.645553\pi\)
\(840\) −2296.60 −0.0943337
\(841\) 14554.5 0.596764
\(842\) 16922.9 0.692640
\(843\) 12245.3 0.500297
\(844\) −8406.87 −0.342863
\(845\) 1427.70 0.0581234
\(846\) −6150.72 −0.249960
\(847\) 9281.60 0.376528
\(848\) 1833.82 0.0742614
\(849\) −10357.8 −0.418703
\(850\) 511.708 0.0206488
\(851\) −1251.94 −0.0504299
\(852\) 690.042 0.0277470
\(853\) −3125.48 −0.125456 −0.0627282 0.998031i \(-0.519980\pi\)
−0.0627282 + 0.998031i \(0.519980\pi\)
\(854\) 9080.98 0.363870
\(855\) −16377.2 −0.655074
\(856\) −16305.6 −0.651067
\(857\) −7942.86 −0.316596 −0.158298 0.987391i \(-0.550601\pi\)
−0.158298 + 0.987391i \(0.550601\pi\)
\(858\) 617.261 0.0245605
\(859\) −46688.0 −1.85445 −0.927227 0.374500i \(-0.877814\pi\)
−0.927227 + 0.374500i \(0.877814\pi\)
\(860\) 14557.0 0.577196
\(861\) −4918.29 −0.194675
\(862\) −9506.11 −0.375614
\(863\) −17044.1 −0.672293 −0.336147 0.941810i \(-0.609124\pi\)
−0.336147 + 0.941810i \(0.609124\pi\)
\(864\) −864.000 −0.0340207
\(865\) −14412.6 −0.566523
\(866\) 15425.9 0.605304
\(867\) 14687.7 0.575341
\(868\) −3262.56 −0.127579
\(869\) 551.065 0.0215116
\(870\) −16186.2 −0.630763
\(871\) −15063.9 −0.586017
\(872\) −8726.67 −0.338902
\(873\) 11326.1 0.439095
\(874\) −6123.20 −0.236980
\(875\) −6040.47 −0.233377
\(876\) 6433.71 0.248145
\(877\) 47244.8 1.81909 0.909547 0.415602i \(-0.136429\pi\)
0.909547 + 0.415602i \(0.136429\pi\)
\(878\) −15143.5 −0.582084
\(879\) 214.990 0.00824964
\(880\) 491.898 0.0188430
\(881\) 17135.5 0.655290 0.327645 0.944801i \(-0.393745\pi\)
0.327645 + 0.944801i \(0.393745\pi\)
\(882\) 882.000 0.0336718
\(883\) 31343.5 1.19456 0.597278 0.802034i \(-0.296249\pi\)
0.597278 + 0.802034i \(0.296249\pi\)
\(884\) 756.608 0.0287867
\(885\) −12010.7 −0.456199
\(886\) −25086.0 −0.951221
\(887\) −19459.5 −0.736625 −0.368313 0.929702i \(-0.620064\pi\)
−0.368313 + 0.929702i \(0.620064\pi\)
\(888\) −1306.37 −0.0493681
\(889\) −18878.0 −0.712202
\(890\) −20505.6 −0.772301
\(891\) −182.164 −0.00684931
\(892\) 2869.61 0.107715
\(893\) −45485.6 −1.70450
\(894\) 4463.50 0.166982
\(895\) −24485.1 −0.914465
\(896\) −896.000 −0.0334077
\(897\) 3156.37 0.117490
\(898\) 17510.9 0.650721
\(899\) −22994.2 −0.853057
\(900\) 2227.53 0.0825011
\(901\) 473.924 0.0175235
\(902\) 1053.42 0.0388860
\(903\) −5590.54 −0.206026
\(904\) −16347.6 −0.601451
\(905\) −30861.4 −1.13355
\(906\) 5060.83 0.185579
\(907\) −9773.95 −0.357815 −0.178908 0.983866i \(-0.557256\pi\)
−0.178908 + 0.983866i \(0.557256\pi\)
\(908\) −11993.5 −0.438345
\(909\) −1683.44 −0.0614260
\(910\) 8754.75 0.318920
\(911\) −3629.19 −0.131987 −0.0659936 0.997820i \(-0.521022\pi\)
−0.0659936 + 0.997820i \(0.521022\pi\)
\(912\) −6389.43 −0.231990
\(913\) −1083.97 −0.0392928
\(914\) −4123.28 −0.149219
\(915\) −26601.3 −0.961105
\(916\) −16868.7 −0.608470
\(917\) 1705.24 0.0614091
\(918\) −223.288 −0.00802789
\(919\) −52390.8 −1.88054 −0.940268 0.340434i \(-0.889426\pi\)
−0.940268 + 0.340434i \(0.889426\pi\)
\(920\) 2515.33 0.0901390
\(921\) 12171.5 0.435467
\(922\) −9848.07 −0.351767
\(923\) −2630.47 −0.0938061
\(924\) −188.911 −0.00672589
\(925\) 3368.03 0.119719
\(926\) 8965.04 0.318153
\(927\) −207.591 −0.00735511
\(928\) −6314.91 −0.223381
\(929\) 1873.27 0.0661573 0.0330786 0.999453i \(-0.489469\pi\)
0.0330786 + 0.999453i \(0.489469\pi\)
\(930\) 9557.14 0.336980
\(931\) 6522.54 0.229611
\(932\) −23125.8 −0.812779
\(933\) 8684.06 0.304720
\(934\) −18774.6 −0.657733
\(935\) 127.124 0.00444641
\(936\) 3293.61 0.115016
\(937\) −44416.3 −1.54858 −0.774289 0.632832i \(-0.781892\pi\)
−0.774289 + 0.632832i \(0.781892\pi\)
\(938\) 4610.27 0.160480
\(939\) −20580.9 −0.715264
\(940\) 18684.9 0.648333
\(941\) −14295.2 −0.495227 −0.247614 0.968859i \(-0.579646\pi\)
−0.247614 + 0.968859i \(0.579646\pi\)
\(942\) −4805.45 −0.166210
\(943\) 5386.69 0.186018
\(944\) −4685.89 −0.161560
\(945\) −2583.68 −0.0889387
\(946\) 1197.41 0.0411535
\(947\) 29442.1 1.01029 0.505143 0.863036i \(-0.331440\pi\)
0.505143 + 0.863036i \(0.331440\pi\)
\(948\) 2940.39 0.100738
\(949\) −24525.6 −0.838919
\(950\) 16473.0 0.562582
\(951\) 1947.90 0.0664195
\(952\) −231.558 −0.00788323
\(953\) 12611.0 0.428658 0.214329 0.976762i \(-0.431244\pi\)
0.214329 + 0.976762i \(0.431244\pi\)
\(954\) 2063.05 0.0700143
\(955\) 26781.1 0.907450
\(956\) −7521.53 −0.254460
\(957\) −1331.43 −0.0449727
\(958\) −40874.7 −1.37850
\(959\) −297.136 −0.0100053
\(960\) 2624.69 0.0882411
\(961\) −16214.1 −0.544262
\(962\) 4979.94 0.166902
\(963\) −18343.8 −0.613832
\(964\) −14563.5 −0.486575
\(965\) −8169.33 −0.272518
\(966\) −966.000 −0.0321745
\(967\) 2874.35 0.0955871 0.0477936 0.998857i \(-0.484781\pi\)
0.0477936 + 0.998857i \(0.484781\pi\)
\(968\) −10607.5 −0.352210
\(969\) −1651.25 −0.0547429
\(970\) −34406.8 −1.13890
\(971\) 701.141 0.0231727 0.0115863 0.999933i \(-0.496312\pi\)
0.0115863 + 0.999933i \(0.496312\pi\)
\(972\) −972.000 −0.0320750
\(973\) 1330.94 0.0438519
\(974\) 10760.4 0.353991
\(975\) −8491.44 −0.278917
\(976\) −10378.3 −0.340369
\(977\) −27477.9 −0.899790 −0.449895 0.893081i \(-0.648539\pi\)
−0.449895 + 0.893081i \(0.648539\pi\)
\(978\) 17292.0 0.565376
\(979\) −1686.72 −0.0550643
\(980\) −2679.37 −0.0873360
\(981\) −9817.51 −0.319520
\(982\) −23526.4 −0.764521
\(983\) −30697.6 −0.996034 −0.498017 0.867167i \(-0.665938\pi\)
−0.498017 + 0.867167i \(0.665938\pi\)
\(984\) 5620.90 0.182101
\(985\) −7898.39 −0.255496
\(986\) −1632.00 −0.0527113
\(987\) −7175.84 −0.231418
\(988\) 24356.8 0.784304
\(989\) 6122.97 0.196865
\(990\) 553.385 0.0177654
\(991\) −34553.8 −1.10761 −0.553803 0.832648i \(-0.686824\pi\)
−0.553803 + 0.832648i \(0.686824\pi\)
\(992\) 3728.64 0.119339
\(993\) −1071.98 −0.0342580
\(994\) 805.049 0.0256887
\(995\) −10256.0 −0.326772
\(996\) −5783.91 −0.184006
\(997\) −58846.7 −1.86930 −0.934651 0.355566i \(-0.884288\pi\)
−0.934651 + 0.355566i \(0.884288\pi\)
\(998\) 39355.2 1.24826
\(999\) −1469.67 −0.0465447
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 966.4.a.e.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.4.a.e.1.1 3 1.1 even 1 trivial