Properties

Label 4009.2.a.d.1.7
Level $4009$
Weight $2$
Character 4009.1
Self dual yes
Analytic conductor $32.012$
Analytic rank $1$
Dimension $75$
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] = [4009,2,Mod(1,4009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4009, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4009 = 19 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4009.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: \(32.0120261703\)
Analytic rank: \(1\)
Dimension: \(75\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Character \(\chi\) \(=\) 4009.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.52406 q^{2} -2.15147 q^{3} +4.37089 q^{4} -1.88445 q^{5} +5.43045 q^{6} -0.0300064 q^{7} -5.98429 q^{8} +1.62883 q^{9} +O(q^{10})\) \(q-2.52406 q^{2} -2.15147 q^{3} +4.37089 q^{4} -1.88445 q^{5} +5.43045 q^{6} -0.0300064 q^{7} -5.98429 q^{8} +1.62883 q^{9} +4.75648 q^{10} -1.05669 q^{11} -9.40386 q^{12} -2.00226 q^{13} +0.0757381 q^{14} +4.05435 q^{15} +6.36293 q^{16} -1.58468 q^{17} -4.11127 q^{18} -1.00000 q^{19} -8.23675 q^{20} +0.0645579 q^{21} +2.66714 q^{22} +4.43409 q^{23} +12.8750 q^{24} -1.44883 q^{25} +5.05383 q^{26} +2.95003 q^{27} -0.131155 q^{28} -9.99262 q^{29} -10.2334 q^{30} +6.58144 q^{31} -4.09186 q^{32} +2.27343 q^{33} +3.99982 q^{34} +0.0565457 q^{35} +7.11945 q^{36} +9.77672 q^{37} +2.52406 q^{38} +4.30780 q^{39} +11.2771 q^{40} +0.483248 q^{41} -0.162948 q^{42} +2.46914 q^{43} -4.61867 q^{44} -3.06946 q^{45} -11.1919 q^{46} -11.1906 q^{47} -13.6897 q^{48} -6.99910 q^{49} +3.65694 q^{50} +3.40939 q^{51} -8.75166 q^{52} -1.30835 q^{53} -7.44606 q^{54} +1.99128 q^{55} +0.179567 q^{56} +2.15147 q^{57} +25.2220 q^{58} -4.32698 q^{59} +17.7211 q^{60} -1.78340 q^{61} -16.6120 q^{62} -0.0488754 q^{63} -2.39774 q^{64} +3.77317 q^{65} -5.73829 q^{66} +9.99586 q^{67} -6.92645 q^{68} -9.53982 q^{69} -0.142725 q^{70} +7.41827 q^{71} -9.74740 q^{72} +10.1761 q^{73} -24.6771 q^{74} +3.11712 q^{75} -4.37089 q^{76} +0.0317074 q^{77} -10.8732 q^{78} -8.83374 q^{79} -11.9906 q^{80} -11.2334 q^{81} -1.21975 q^{82} +5.61499 q^{83} +0.282176 q^{84} +2.98625 q^{85} -6.23226 q^{86} +21.4988 q^{87} +6.32352 q^{88} -1.84974 q^{89} +7.74751 q^{90} +0.0600806 q^{91} +19.3809 q^{92} -14.1598 q^{93} +28.2458 q^{94} +1.88445 q^{95} +8.80352 q^{96} +2.99708 q^{97} +17.6662 q^{98} -1.72117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.52406 −1.78478 −0.892391 0.451263i \(-0.850974\pi\)
−0.892391 + 0.451263i \(0.850974\pi\)
\(3\) −2.15147 −1.24215 −0.621076 0.783750i \(-0.713304\pi\)
−0.621076 + 0.783750i \(0.713304\pi\)
\(4\) 4.37089 2.18545
\(5\) −1.88445 −0.842754 −0.421377 0.906886i \(-0.638453\pi\)
−0.421377 + 0.906886i \(0.638453\pi\)
\(6\) 5.43045 2.21697
\(7\) −0.0300064 −0.0113414 −0.00567068 0.999984i \(-0.501805\pi\)
−0.00567068 + 0.999984i \(0.501805\pi\)
\(8\) −5.98429 −2.11576
\(9\) 1.62883 0.542944
\(10\) 4.75648 1.50413
\(11\) −1.05669 −0.318603 −0.159302 0.987230i \(-0.550924\pi\)
−0.159302 + 0.987230i \(0.550924\pi\)
\(12\) −9.40386 −2.71466
\(13\) −2.00226 −0.555327 −0.277663 0.960678i \(-0.589560\pi\)
−0.277663 + 0.960678i \(0.589560\pi\)
\(14\) 0.0757381 0.0202418
\(15\) 4.05435 1.04683
\(16\) 6.36293 1.59073
\(17\) −1.58468 −0.384340 −0.192170 0.981362i \(-0.561553\pi\)
−0.192170 + 0.981362i \(0.561553\pi\)
\(18\) −4.11127 −0.969037
\(19\) −1.00000 −0.229416
\(20\) −8.23675 −1.84179
\(21\) 0.0645579 0.0140877
\(22\) 2.66714 0.568637
\(23\) 4.43409 0.924572 0.462286 0.886731i \(-0.347029\pi\)
0.462286 + 0.886731i \(0.347029\pi\)
\(24\) 12.8750 2.62810
\(25\) −1.44883 −0.289766
\(26\) 5.05383 0.991137
\(27\) 2.95003 0.567733
\(28\) −0.131155 −0.0247859
\(29\) −9.99262 −1.85558 −0.927791 0.373100i \(-0.878295\pi\)
−0.927791 + 0.373100i \(0.878295\pi\)
\(30\) −10.2334 −1.86836
\(31\) 6.58144 1.18206 0.591030 0.806649i \(-0.298721\pi\)
0.591030 + 0.806649i \(0.298721\pi\)
\(32\) −4.09186 −0.723346
\(33\) 2.27343 0.395754
\(34\) 3.99982 0.685964
\(35\) 0.0565457 0.00955797
\(36\) 7.11945 1.18658
\(37\) 9.77672 1.60728 0.803641 0.595114i \(-0.202893\pi\)
0.803641 + 0.595114i \(0.202893\pi\)
\(38\) 2.52406 0.409457
\(39\) 4.30780 0.689801
\(40\) 11.2771 1.78307
\(41\) 0.483248 0.0754707 0.0377353 0.999288i \(-0.487986\pi\)
0.0377353 + 0.999288i \(0.487986\pi\)
\(42\) −0.162948 −0.0251435
\(43\) 2.46914 0.376540 0.188270 0.982117i \(-0.439712\pi\)
0.188270 + 0.982117i \(0.439712\pi\)
\(44\) −4.61867 −0.696290
\(45\) −3.06946 −0.457568
\(46\) −11.1919 −1.65016
\(47\) −11.1906 −1.63232 −0.816158 0.577829i \(-0.803900\pi\)
−0.816158 + 0.577829i \(0.803900\pi\)
\(48\) −13.6897 −1.97593
\(49\) −6.99910 −0.999871
\(50\) 3.65694 0.517170
\(51\) 3.40939 0.477409
\(52\) −8.75166 −1.21364
\(53\) −1.30835 −0.179716 −0.0898580 0.995955i \(-0.528641\pi\)
−0.0898580 + 0.995955i \(0.528641\pi\)
\(54\) −7.44606 −1.01328
\(55\) 1.99128 0.268504
\(56\) 0.179567 0.0239956
\(57\) 2.15147 0.284969
\(58\) 25.2220 3.31181
\(59\) −4.32698 −0.563324 −0.281662 0.959514i \(-0.590886\pi\)
−0.281662 + 0.959514i \(0.590886\pi\)
\(60\) 17.7211 2.28779
\(61\) −1.78340 −0.228340 −0.114170 0.993461i \(-0.536421\pi\)
−0.114170 + 0.993461i \(0.536421\pi\)
\(62\) −16.6120 −2.10972
\(63\) −0.0488754 −0.00615772
\(64\) −2.39774 −0.299718
\(65\) 3.77317 0.468004
\(66\) −5.73829 −0.706334
\(67\) 9.99586 1.22119 0.610594 0.791944i \(-0.290931\pi\)
0.610594 + 0.791944i \(0.290931\pi\)
\(68\) −6.92645 −0.839955
\(69\) −9.53982 −1.14846
\(70\) −0.142725 −0.0170589
\(71\) 7.41827 0.880386 0.440193 0.897903i \(-0.354910\pi\)
0.440193 + 0.897903i \(0.354910\pi\)
\(72\) −9.74740 −1.14874
\(73\) 10.1761 1.19102 0.595510 0.803348i \(-0.296950\pi\)
0.595510 + 0.803348i \(0.296950\pi\)
\(74\) −24.6771 −2.86865
\(75\) 3.11712 0.359934
\(76\) −4.37089 −0.501376
\(77\) 0.0317074 0.00361339
\(78\) −10.8732 −1.23114
\(79\) −8.83374 −0.993873 −0.496937 0.867787i \(-0.665542\pi\)
−0.496937 + 0.867787i \(0.665542\pi\)
\(80\) −11.9906 −1.34060
\(81\) −11.2334 −1.24816
\(82\) −1.21975 −0.134699
\(83\) 5.61499 0.616325 0.308163 0.951334i \(-0.400286\pi\)
0.308163 + 0.951334i \(0.400286\pi\)
\(84\) 0.282176 0.0307879
\(85\) 2.98625 0.323904
\(86\) −6.23226 −0.672042
\(87\) 21.4988 2.30492
\(88\) 6.32352 0.674089
\(89\) −1.84974 −0.196072 −0.0980362 0.995183i \(-0.531256\pi\)
−0.0980362 + 0.995183i \(0.531256\pi\)
\(90\) 7.74751 0.816659
\(91\) 0.0600806 0.00629816
\(92\) 19.3809 2.02060
\(93\) −14.1598 −1.46830
\(94\) 28.2458 2.91333
\(95\) 1.88445 0.193341
\(96\) 8.80352 0.898506
\(97\) 2.99708 0.304307 0.152154 0.988357i \(-0.451379\pi\)
0.152154 + 0.988357i \(0.451379\pi\)
\(98\) 17.6662 1.78455
\(99\) −1.72117 −0.172984
\(100\) −6.33269 −0.633269
\(101\) 1.50288 0.149542 0.0747712 0.997201i \(-0.476177\pi\)
0.0747712 + 0.997201i \(0.476177\pi\)
\(102\) −8.60550 −0.852072
\(103\) 15.1436 1.49214 0.746070 0.665868i \(-0.231938\pi\)
0.746070 + 0.665868i \(0.231938\pi\)
\(104\) 11.9821 1.17494
\(105\) −0.121656 −0.0118725
\(106\) 3.30236 0.320754
\(107\) 14.7355 1.42453 0.712267 0.701909i \(-0.247668\pi\)
0.712267 + 0.701909i \(0.247668\pi\)
\(108\) 12.8943 1.24075
\(109\) 0.211487 0.0202568 0.0101284 0.999949i \(-0.496776\pi\)
0.0101284 + 0.999949i \(0.496776\pi\)
\(110\) −5.02611 −0.479221
\(111\) −21.0343 −1.99649
\(112\) −0.190929 −0.0180411
\(113\) 13.9112 1.30866 0.654329 0.756210i \(-0.272951\pi\)
0.654329 + 0.756210i \(0.272951\pi\)
\(114\) −5.43045 −0.508608
\(115\) −8.35584 −0.779186
\(116\) −43.6767 −4.05528
\(117\) −3.26134 −0.301511
\(118\) 10.9216 1.00541
\(119\) 0.0475504 0.00435894
\(120\) −24.2624 −2.21484
\(121\) −9.88341 −0.898492
\(122\) 4.50140 0.407538
\(123\) −1.03969 −0.0937461
\(124\) 28.7668 2.58333
\(125\) 12.1525 1.08696
\(126\) 0.123365 0.0109902
\(127\) −21.5251 −1.91004 −0.955021 0.296538i \(-0.904168\pi\)
−0.955021 + 0.296538i \(0.904168\pi\)
\(128\) 14.2358 1.25828
\(129\) −5.31228 −0.467720
\(130\) −9.52371 −0.835284
\(131\) −3.62027 −0.316304 −0.158152 0.987415i \(-0.550554\pi\)
−0.158152 + 0.987415i \(0.550554\pi\)
\(132\) 9.93693 0.864899
\(133\) 0.0300064 0.00260189
\(134\) −25.2302 −2.17956
\(135\) −5.55920 −0.478459
\(136\) 9.48315 0.813174
\(137\) −21.7964 −1.86219 −0.931096 0.364774i \(-0.881146\pi\)
−0.931096 + 0.364774i \(0.881146\pi\)
\(138\) 24.0791 2.04975
\(139\) 10.8620 0.921306 0.460653 0.887580i \(-0.347615\pi\)
0.460653 + 0.887580i \(0.347615\pi\)
\(140\) 0.247155 0.0208884
\(141\) 24.0763 2.02759
\(142\) −18.7242 −1.57130
\(143\) 2.11576 0.176929
\(144\) 10.3641 0.863679
\(145\) 18.8306 1.56380
\(146\) −25.6851 −2.12571
\(147\) 15.0584 1.24199
\(148\) 42.7330 3.51263
\(149\) 0.201078 0.0164729 0.00823647 0.999966i \(-0.497378\pi\)
0.00823647 + 0.999966i \(0.497378\pi\)
\(150\) −7.86781 −0.642404
\(151\) 7.69857 0.626501 0.313250 0.949671i \(-0.398582\pi\)
0.313250 + 0.949671i \(0.398582\pi\)
\(152\) 5.98429 0.485390
\(153\) −2.58117 −0.208675
\(154\) −0.0800314 −0.00644911
\(155\) −12.4024 −0.996186
\(156\) 18.8290 1.50752
\(157\) 12.4128 0.990650 0.495325 0.868708i \(-0.335049\pi\)
0.495325 + 0.868708i \(0.335049\pi\)
\(158\) 22.2969 1.77385
\(159\) 2.81488 0.223235
\(160\) 7.71092 0.609602
\(161\) −0.133051 −0.0104859
\(162\) 28.3538 2.22769
\(163\) 14.1956 1.11189 0.555943 0.831220i \(-0.312357\pi\)
0.555943 + 0.831220i \(0.312357\pi\)
\(164\) 2.11223 0.164937
\(165\) −4.28418 −0.333523
\(166\) −14.1726 −1.10001
\(167\) −5.70430 −0.441412 −0.220706 0.975340i \(-0.570836\pi\)
−0.220706 + 0.975340i \(0.570836\pi\)
\(168\) −0.386333 −0.0298063
\(169\) −8.99096 −0.691612
\(170\) −7.53748 −0.578098
\(171\) −1.62883 −0.124560
\(172\) 10.7923 0.822908
\(173\) −3.06403 −0.232954 −0.116477 0.993193i \(-0.537160\pi\)
−0.116477 + 0.993193i \(0.537160\pi\)
\(174\) −54.2644 −4.11377
\(175\) 0.0434742 0.00328634
\(176\) −6.72362 −0.506812
\(177\) 9.30937 0.699735
\(178\) 4.66887 0.349946
\(179\) 11.6233 0.868764 0.434382 0.900729i \(-0.356967\pi\)
0.434382 + 0.900729i \(0.356967\pi\)
\(180\) −13.4163 −0.999991
\(181\) 25.4037 1.88824 0.944121 0.329598i \(-0.106913\pi\)
0.944121 + 0.329598i \(0.106913\pi\)
\(182\) −0.151647 −0.0112408
\(183\) 3.83693 0.283634
\(184\) −26.5349 −1.95618
\(185\) −18.4238 −1.35454
\(186\) 35.7402 2.62060
\(187\) 1.67451 0.122452
\(188\) −48.9129 −3.56734
\(189\) −0.0885198 −0.00643887
\(190\) −4.75648 −0.345071
\(191\) 16.1846 1.17108 0.585538 0.810645i \(-0.300883\pi\)
0.585538 + 0.810645i \(0.300883\pi\)
\(192\) 5.15868 0.372296
\(193\) 20.9037 1.50468 0.752340 0.658775i \(-0.228925\pi\)
0.752340 + 0.658775i \(0.228925\pi\)
\(194\) −7.56482 −0.543123
\(195\) −8.11786 −0.581332
\(196\) −30.5923 −2.18517
\(197\) 16.1742 1.15237 0.576183 0.817321i \(-0.304542\pi\)
0.576183 + 0.817321i \(0.304542\pi\)
\(198\) 4.34433 0.308738
\(199\) −1.43825 −0.101955 −0.0509775 0.998700i \(-0.516234\pi\)
−0.0509775 + 0.998700i \(0.516234\pi\)
\(200\) 8.67023 0.613078
\(201\) −21.5058 −1.51690
\(202\) −3.79337 −0.266900
\(203\) 0.299842 0.0210448
\(204\) 14.9021 1.04335
\(205\) −0.910659 −0.0636032
\(206\) −38.2233 −2.66314
\(207\) 7.22239 0.501991
\(208\) −12.7402 −0.883376
\(209\) 1.05669 0.0730925
\(210\) 0.307069 0.0211897
\(211\) −1.00000 −0.0688428
\(212\) −5.71867 −0.392760
\(213\) −15.9602 −1.09357
\(214\) −37.1933 −2.54248
\(215\) −4.65298 −0.317330
\(216\) −17.6538 −1.20119
\(217\) −0.197485 −0.0134062
\(218\) −0.533806 −0.0361539
\(219\) −21.8935 −1.47943
\(220\) 8.70366 0.586801
\(221\) 3.17293 0.213434
\(222\) 53.0920 3.56330
\(223\) −14.7601 −0.988406 −0.494203 0.869346i \(-0.664540\pi\)
−0.494203 + 0.869346i \(0.664540\pi\)
\(224\) 0.122782 0.00820372
\(225\) −2.35990 −0.157327
\(226\) −35.1128 −2.33567
\(227\) −7.39221 −0.490638 −0.245319 0.969442i \(-0.578893\pi\)
−0.245319 + 0.969442i \(0.578893\pi\)
\(228\) 9.40386 0.622786
\(229\) 17.7818 1.17505 0.587526 0.809206i \(-0.300102\pi\)
0.587526 + 0.809206i \(0.300102\pi\)
\(230\) 21.0907 1.39068
\(231\) −0.0682175 −0.00448838
\(232\) 59.7987 3.92598
\(233\) 28.3451 1.85695 0.928474 0.371399i \(-0.121122\pi\)
0.928474 + 0.371399i \(0.121122\pi\)
\(234\) 8.23184 0.538132
\(235\) 21.0882 1.37564
\(236\) −18.9128 −1.23112
\(237\) 19.0055 1.23454
\(238\) −0.120020 −0.00777976
\(239\) −15.2381 −0.985671 −0.492835 0.870123i \(-0.664039\pi\)
−0.492835 + 0.870123i \(0.664039\pi\)
\(240\) 25.7975 1.66522
\(241\) 15.5631 1.00251 0.501254 0.865300i \(-0.332872\pi\)
0.501254 + 0.865300i \(0.332872\pi\)
\(242\) 24.9464 1.60361
\(243\) 15.3183 0.982667
\(244\) −7.79503 −0.499026
\(245\) 13.1895 0.842645
\(246\) 2.62425 0.167316
\(247\) 2.00226 0.127401
\(248\) −39.3852 −2.50096
\(249\) −12.0805 −0.765570
\(250\) −30.6738 −1.93998
\(251\) −12.4932 −0.788566 −0.394283 0.918989i \(-0.629007\pi\)
−0.394283 + 0.918989i \(0.629007\pi\)
\(252\) −0.213629 −0.0134574
\(253\) −4.68544 −0.294571
\(254\) 54.3307 3.40901
\(255\) −6.42483 −0.402339
\(256\) −31.1365 −1.94603
\(257\) 15.0878 0.941149 0.470575 0.882360i \(-0.344047\pi\)
0.470575 + 0.882360i \(0.344047\pi\)
\(258\) 13.4085 0.834779
\(259\) −0.293364 −0.0182288
\(260\) 16.4921 1.02280
\(261\) −16.2763 −1.00748
\(262\) 9.13778 0.564534
\(263\) −24.2515 −1.49541 −0.747706 0.664030i \(-0.768845\pi\)
−0.747706 + 0.664030i \(0.768845\pi\)
\(264\) −13.6049 −0.837322
\(265\) 2.46553 0.151456
\(266\) −0.0757381 −0.00464380
\(267\) 3.97967 0.243552
\(268\) 43.6909 2.66884
\(269\) 17.3904 1.06031 0.530155 0.847901i \(-0.322134\pi\)
0.530155 + 0.847901i \(0.322134\pi\)
\(270\) 14.0318 0.853946
\(271\) −6.83729 −0.415336 −0.207668 0.978199i \(-0.566587\pi\)
−0.207668 + 0.978199i \(0.566587\pi\)
\(272\) −10.0832 −0.611383
\(273\) −0.129262 −0.00782328
\(274\) 55.0155 3.32361
\(275\) 1.53096 0.0923205
\(276\) −41.6976 −2.50990
\(277\) −31.0354 −1.86474 −0.932369 0.361509i \(-0.882262\pi\)
−0.932369 + 0.361509i \(0.882262\pi\)
\(278\) −27.4165 −1.64433
\(279\) 10.7201 0.641793
\(280\) −0.338386 −0.0202224
\(281\) −7.98321 −0.476239 −0.238119 0.971236i \(-0.576531\pi\)
−0.238119 + 0.971236i \(0.576531\pi\)
\(282\) −60.7700 −3.61880
\(283\) 23.3527 1.38817 0.694086 0.719892i \(-0.255809\pi\)
0.694086 + 0.719892i \(0.255809\pi\)
\(284\) 32.4245 1.92404
\(285\) −4.05435 −0.240159
\(286\) −5.34031 −0.315779
\(287\) −0.0145005 −0.000855940 0
\(288\) −6.66495 −0.392736
\(289\) −14.4888 −0.852283
\(290\) −47.5297 −2.79104
\(291\) −6.44814 −0.377996
\(292\) 44.4786 2.60291
\(293\) −21.7306 −1.26951 −0.634756 0.772712i \(-0.718899\pi\)
−0.634756 + 0.772712i \(0.718899\pi\)
\(294\) −38.0083 −2.21669
\(295\) 8.15399 0.474744
\(296\) −58.5067 −3.40063
\(297\) −3.11726 −0.180882
\(298\) −0.507533 −0.0294006
\(299\) −8.87820 −0.513439
\(300\) 13.6246 0.786617
\(301\) −0.0740899 −0.00427047
\(302\) −19.4317 −1.11817
\(303\) −3.23341 −0.185754
\(304\) −6.36293 −0.364939
\(305\) 3.36073 0.192435
\(306\) 6.51504 0.372440
\(307\) −22.0265 −1.25712 −0.628559 0.777762i \(-0.716355\pi\)
−0.628559 + 0.777762i \(0.716355\pi\)
\(308\) 0.138590 0.00789687
\(309\) −32.5810 −1.85347
\(310\) 31.3045 1.77798
\(311\) −9.59387 −0.544018 −0.272009 0.962295i \(-0.587688\pi\)
−0.272009 + 0.962295i \(0.587688\pi\)
\(312\) −25.7791 −1.45946
\(313\) −14.1696 −0.800915 −0.400457 0.916315i \(-0.631149\pi\)
−0.400457 + 0.916315i \(0.631149\pi\)
\(314\) −31.3307 −1.76809
\(315\) 0.0921034 0.00518944
\(316\) −38.6113 −2.17206
\(317\) −34.5057 −1.93803 −0.969017 0.246996i \(-0.920557\pi\)
−0.969017 + 0.246996i \(0.920557\pi\)
\(318\) −7.10494 −0.398426
\(319\) 10.5591 0.591194
\(320\) 4.51844 0.252588
\(321\) −31.7030 −1.76949
\(322\) 0.335829 0.0187150
\(323\) 1.58468 0.0881737
\(324\) −49.1000 −2.72778
\(325\) 2.90094 0.160915
\(326\) −35.8306 −1.98448
\(327\) −0.455008 −0.0251620
\(328\) −2.89189 −0.159678
\(329\) 0.335790 0.0185127
\(330\) 10.8135 0.595266
\(331\) 18.9592 1.04209 0.521045 0.853529i \(-0.325542\pi\)
0.521045 + 0.853529i \(0.325542\pi\)
\(332\) 24.5425 1.34695
\(333\) 15.9246 0.872664
\(334\) 14.3980 0.787824
\(335\) −18.8367 −1.02916
\(336\) 0.410778 0.0224098
\(337\) −13.1717 −0.717506 −0.358753 0.933433i \(-0.616798\pi\)
−0.358753 + 0.933433i \(0.616798\pi\)
\(338\) 22.6937 1.23438
\(339\) −29.9296 −1.62555
\(340\) 13.0526 0.707875
\(341\) −6.95452 −0.376608
\(342\) 4.11127 0.222312
\(343\) 0.420063 0.0226813
\(344\) −14.7760 −0.796670
\(345\) 17.9774 0.967868
\(346\) 7.73380 0.415772
\(347\) −29.8639 −1.60318 −0.801590 0.597874i \(-0.796012\pi\)
−0.801590 + 0.597874i \(0.796012\pi\)
\(348\) 93.9691 5.03727
\(349\) −3.38787 −0.181348 −0.0906742 0.995881i \(-0.528902\pi\)
−0.0906742 + 0.995881i \(0.528902\pi\)
\(350\) −0.109732 −0.00586541
\(351\) −5.90672 −0.315278
\(352\) 4.32381 0.230460
\(353\) −12.0195 −0.639736 −0.319868 0.947462i \(-0.603639\pi\)
−0.319868 + 0.947462i \(0.603639\pi\)
\(354\) −23.4974 −1.24887
\(355\) −13.9794 −0.741949
\(356\) −8.08503 −0.428506
\(357\) −0.102303 −0.00541447
\(358\) −29.3379 −1.55055
\(359\) −4.40718 −0.232602 −0.116301 0.993214i \(-0.537104\pi\)
−0.116301 + 0.993214i \(0.537104\pi\)
\(360\) 18.3685 0.968106
\(361\) 1.00000 0.0526316
\(362\) −64.1206 −3.37010
\(363\) 21.2639 1.11606
\(364\) 0.262606 0.0137643
\(365\) −19.1764 −1.00374
\(366\) −9.68464 −0.506224
\(367\) 8.79114 0.458894 0.229447 0.973321i \(-0.426308\pi\)
0.229447 + 0.973321i \(0.426308\pi\)
\(368\) 28.2138 1.47075
\(369\) 0.787130 0.0409763
\(370\) 46.5028 2.41756
\(371\) 0.0392590 0.00203822
\(372\) −61.8909 −3.20889
\(373\) −22.0627 −1.14236 −0.571182 0.820824i \(-0.693515\pi\)
−0.571182 + 0.820824i \(0.693515\pi\)
\(374\) −4.22656 −0.218550
\(375\) −26.1458 −1.35016
\(376\) 66.9677 3.45360
\(377\) 20.0078 1.03045
\(378\) 0.223430 0.0114920
\(379\) 37.6682 1.93488 0.967442 0.253094i \(-0.0814481\pi\)
0.967442 + 0.253094i \(0.0814481\pi\)
\(380\) 8.23675 0.422536
\(381\) 46.3106 2.37256
\(382\) −40.8510 −2.09012
\(383\) −6.76286 −0.345566 −0.172783 0.984960i \(-0.555276\pi\)
−0.172783 + 0.984960i \(0.555276\pi\)
\(384\) −30.6279 −1.56297
\(385\) −0.0597511 −0.00304520
\(386\) −52.7622 −2.68553
\(387\) 4.02181 0.204440
\(388\) 13.0999 0.665048
\(389\) 8.43093 0.427465 0.213733 0.976892i \(-0.431438\pi\)
0.213733 + 0.976892i \(0.431438\pi\)
\(390\) 20.4900 1.03755
\(391\) −7.02660 −0.355350
\(392\) 41.8846 2.11549
\(393\) 7.78890 0.392898
\(394\) −40.8247 −2.05672
\(395\) 16.6468 0.837590
\(396\) −7.52303 −0.378047
\(397\) 32.5114 1.63170 0.815851 0.578263i \(-0.196269\pi\)
0.815851 + 0.578263i \(0.196269\pi\)
\(398\) 3.63024 0.181968
\(399\) −0.0645579 −0.00323194
\(400\) −9.21882 −0.460941
\(401\) −18.0249 −0.900120 −0.450060 0.892998i \(-0.648597\pi\)
−0.450060 + 0.892998i \(0.648597\pi\)
\(402\) 54.2820 2.70734
\(403\) −13.1777 −0.656430
\(404\) 6.56894 0.326817
\(405\) 21.1688 1.05189
\(406\) −0.756821 −0.0375604
\(407\) −10.3309 −0.512085
\(408\) −20.4027 −1.01009
\(409\) −35.9550 −1.77786 −0.888931 0.458041i \(-0.848551\pi\)
−0.888931 + 0.458041i \(0.848551\pi\)
\(410\) 2.29856 0.113518
\(411\) 46.8943 2.31313
\(412\) 66.1909 3.26099
\(413\) 0.129837 0.00638886
\(414\) −18.2298 −0.895944
\(415\) −10.5812 −0.519410
\(416\) 8.19296 0.401693
\(417\) −23.3694 −1.14440
\(418\) −2.66714 −0.130454
\(419\) −32.6069 −1.59295 −0.796475 0.604672i \(-0.793304\pi\)
−0.796475 + 0.604672i \(0.793304\pi\)
\(420\) −0.531748 −0.0259466
\(421\) 28.4931 1.38867 0.694334 0.719653i \(-0.255699\pi\)
0.694334 + 0.719653i \(0.255699\pi\)
\(422\) 2.52406 0.122869
\(423\) −18.2276 −0.886256
\(424\) 7.82956 0.380237
\(425\) 2.29593 0.111369
\(426\) 40.2845 1.95179
\(427\) 0.0535133 0.00258969
\(428\) 64.4073 3.11324
\(429\) −4.55200 −0.219773
\(430\) 11.7444 0.566366
\(431\) −36.2652 −1.74683 −0.873417 0.486973i \(-0.838101\pi\)
−0.873417 + 0.486973i \(0.838101\pi\)
\(432\) 18.7708 0.903112
\(433\) −0.0466208 −0.00224045 −0.00112023 0.999999i \(-0.500357\pi\)
−0.00112023 + 0.999999i \(0.500357\pi\)
\(434\) 0.498465 0.0239271
\(435\) −40.5136 −1.94248
\(436\) 0.924387 0.0442701
\(437\) −4.43409 −0.212111
\(438\) 55.2607 2.64046
\(439\) −24.7229 −1.17996 −0.589980 0.807418i \(-0.700864\pi\)
−0.589980 + 0.807418i \(0.700864\pi\)
\(440\) −11.9164 −0.568091
\(441\) −11.4004 −0.542874
\(442\) −8.00868 −0.380934
\(443\) −0.893765 −0.0424640 −0.0212320 0.999775i \(-0.506759\pi\)
−0.0212320 + 0.999775i \(0.506759\pi\)
\(444\) −91.9389 −4.36323
\(445\) 3.48576 0.165241
\(446\) 37.2553 1.76409
\(447\) −0.432613 −0.0204619
\(448\) 0.0719477 0.00339921
\(449\) −30.2572 −1.42793 −0.713963 0.700184i \(-0.753101\pi\)
−0.713963 + 0.700184i \(0.753101\pi\)
\(450\) 5.95655 0.280794
\(451\) −0.510642 −0.0240452
\(452\) 60.8045 2.86000
\(453\) −16.5633 −0.778210
\(454\) 18.6584 0.875682
\(455\) −0.113219 −0.00530779
\(456\) −12.8750 −0.602928
\(457\) 16.9998 0.795215 0.397608 0.917556i \(-0.369841\pi\)
0.397608 + 0.917556i \(0.369841\pi\)
\(458\) −44.8823 −2.09721
\(459\) −4.67484 −0.218203
\(460\) −36.5225 −1.70287
\(461\) 32.9250 1.53347 0.766735 0.641964i \(-0.221880\pi\)
0.766735 + 0.641964i \(0.221880\pi\)
\(462\) 0.172185 0.00801079
\(463\) −39.7687 −1.84821 −0.924105 0.382139i \(-0.875188\pi\)
−0.924105 + 0.382139i \(0.875188\pi\)
\(464\) −63.5823 −2.95173
\(465\) 26.6835 1.23742
\(466\) −71.5447 −3.31425
\(467\) −27.3212 −1.26428 −0.632138 0.774856i \(-0.717823\pi\)
−0.632138 + 0.774856i \(0.717823\pi\)
\(468\) −14.2550 −0.658937
\(469\) −0.299940 −0.0138499
\(470\) −53.2279 −2.45522
\(471\) −26.7058 −1.23054
\(472\) 25.8939 1.19186
\(473\) −2.60910 −0.119967
\(474\) −47.9712 −2.20339
\(475\) 1.44883 0.0664770
\(476\) 0.207838 0.00952623
\(477\) −2.13109 −0.0975758
\(478\) 38.4619 1.75921
\(479\) 23.1598 1.05820 0.529100 0.848559i \(-0.322530\pi\)
0.529100 + 0.848559i \(0.322530\pi\)
\(480\) −16.5898 −0.757219
\(481\) −19.5755 −0.892567
\(482\) −39.2823 −1.78926
\(483\) 0.286256 0.0130251
\(484\) −43.1994 −1.96361
\(485\) −5.64786 −0.256456
\(486\) −38.6643 −1.75385
\(487\) −0.518795 −0.0235088 −0.0117544 0.999931i \(-0.503742\pi\)
−0.0117544 + 0.999931i \(0.503742\pi\)
\(488\) 10.6724 0.483115
\(489\) −30.5415 −1.38113
\(490\) −33.2911 −1.50394
\(491\) −10.1054 −0.456049 −0.228025 0.973655i \(-0.573227\pi\)
−0.228025 + 0.973655i \(0.573227\pi\)
\(492\) −4.54440 −0.204877
\(493\) 15.8351 0.713175
\(494\) −5.05383 −0.227382
\(495\) 3.24346 0.145783
\(496\) 41.8772 1.88034
\(497\) −0.222596 −0.00998477
\(498\) 30.4919 1.36638
\(499\) 0.989559 0.0442987 0.0221494 0.999755i \(-0.492949\pi\)
0.0221494 + 0.999755i \(0.492949\pi\)
\(500\) 53.1174 2.37548
\(501\) 12.2726 0.548301
\(502\) 31.5337 1.40742
\(503\) 18.7123 0.834339 0.417169 0.908829i \(-0.363022\pi\)
0.417169 + 0.908829i \(0.363022\pi\)
\(504\) 0.292484 0.0130283
\(505\) −2.83211 −0.126027
\(506\) 11.8264 0.525746
\(507\) 19.3438 0.859088
\(508\) −94.0839 −4.17430
\(509\) 28.4241 1.25988 0.629938 0.776646i \(-0.283080\pi\)
0.629938 + 0.776646i \(0.283080\pi\)
\(510\) 16.2167 0.718087
\(511\) −0.305348 −0.0135078
\(512\) 50.1190 2.21497
\(513\) −2.95003 −0.130247
\(514\) −38.0825 −1.67975
\(515\) −28.5374 −1.25751
\(516\) −23.2194 −1.02218
\(517\) 11.8250 0.520061
\(518\) 0.740470 0.0325344
\(519\) 6.59217 0.289364
\(520\) −22.5797 −0.990186
\(521\) −12.7716 −0.559534 −0.279767 0.960068i \(-0.590257\pi\)
−0.279767 + 0.960068i \(0.590257\pi\)
\(522\) 41.0824 1.79813
\(523\) 20.3194 0.888505 0.444252 0.895902i \(-0.353469\pi\)
0.444252 + 0.895902i \(0.353469\pi\)
\(524\) −15.8238 −0.691266
\(525\) −0.0935336 −0.00408214
\(526\) 61.2123 2.66898
\(527\) −10.4294 −0.454314
\(528\) 14.4657 0.629538
\(529\) −3.33884 −0.145167
\(530\) −6.22315 −0.270317
\(531\) −7.04792 −0.305854
\(532\) 0.131155 0.00568628
\(533\) −0.967588 −0.0419109
\(534\) −10.0449 −0.434687
\(535\) −27.7684 −1.20053
\(536\) −59.8181 −2.58375
\(537\) −25.0071 −1.07914
\(538\) −43.8944 −1.89242
\(539\) 7.39586 0.318562
\(540\) −24.2987 −1.04565
\(541\) −22.4990 −0.967307 −0.483654 0.875260i \(-0.660691\pi\)
−0.483654 + 0.875260i \(0.660691\pi\)
\(542\) 17.2577 0.741284
\(543\) −54.6554 −2.34549
\(544\) 6.48427 0.278011
\(545\) −0.398538 −0.0170715
\(546\) 0.326265 0.0139628
\(547\) 26.8522 1.14812 0.574059 0.818814i \(-0.305368\pi\)
0.574059 + 0.818814i \(0.305368\pi\)
\(548\) −95.2698 −4.06972
\(549\) −2.90485 −0.123976
\(550\) −3.86424 −0.164772
\(551\) 9.99262 0.425700
\(552\) 57.0890 2.42987
\(553\) 0.265069 0.0112719
\(554\) 78.3354 3.32815
\(555\) 39.6382 1.68255
\(556\) 47.4768 2.01347
\(557\) −28.6371 −1.21339 −0.606696 0.794934i \(-0.707505\pi\)
−0.606696 + 0.794934i \(0.707505\pi\)
\(558\) −27.0581 −1.14546
\(559\) −4.94385 −0.209103
\(560\) 0.359796 0.0152042
\(561\) −3.60265 −0.152104
\(562\) 20.1501 0.849982
\(563\) −16.5476 −0.697399 −0.348699 0.937235i \(-0.613377\pi\)
−0.348699 + 0.937235i \(0.613377\pi\)
\(564\) 105.235 4.43118
\(565\) −26.2151 −1.10288
\(566\) −58.9436 −2.47758
\(567\) 0.337074 0.0141558
\(568\) −44.3930 −1.86269
\(569\) 16.7226 0.701047 0.350523 0.936554i \(-0.386004\pi\)
0.350523 + 0.936554i \(0.386004\pi\)
\(570\) 10.2334 0.428632
\(571\) −34.9884 −1.46422 −0.732110 0.681187i \(-0.761464\pi\)
−0.732110 + 0.681187i \(0.761464\pi\)
\(572\) 9.24777 0.386669
\(573\) −34.8207 −1.45466
\(574\) 0.0366003 0.00152767
\(575\) −6.42425 −0.267910
\(576\) −3.90552 −0.162730
\(577\) −29.9566 −1.24711 −0.623555 0.781780i \(-0.714312\pi\)
−0.623555 + 0.781780i \(0.714312\pi\)
\(578\) 36.5707 1.52114
\(579\) −44.9737 −1.86904
\(580\) 82.3067 3.41760
\(581\) −0.168486 −0.00698996
\(582\) 16.2755 0.674641
\(583\) 1.38252 0.0572581
\(584\) −60.8966 −2.51992
\(585\) 6.14585 0.254100
\(586\) 54.8493 2.26580
\(587\) 18.0669 0.745700 0.372850 0.927892i \(-0.378381\pi\)
0.372850 + 0.927892i \(0.378381\pi\)
\(588\) 65.8185 2.71431
\(589\) −6.58144 −0.271183
\(590\) −20.5812 −0.847314
\(591\) −34.7984 −1.43141
\(592\) 62.2086 2.55676
\(593\) 18.4516 0.757715 0.378857 0.925455i \(-0.376317\pi\)
0.378857 + 0.925455i \(0.376317\pi\)
\(594\) 7.86815 0.322834
\(595\) −0.0896066 −0.00367351
\(596\) 0.878890 0.0360007
\(597\) 3.09436 0.126644
\(598\) 22.4091 0.916378
\(599\) 6.68751 0.273244 0.136622 0.990623i \(-0.456375\pi\)
0.136622 + 0.990623i \(0.456375\pi\)
\(600\) −18.6538 −0.761536
\(601\) −5.64057 −0.230084 −0.115042 0.993361i \(-0.536700\pi\)
−0.115042 + 0.993361i \(0.536700\pi\)
\(602\) 0.187008 0.00762186
\(603\) 16.2816 0.663037
\(604\) 33.6496 1.36918
\(605\) 18.6248 0.757207
\(606\) 8.16133 0.331531
\(607\) 18.8606 0.765527 0.382763 0.923846i \(-0.374972\pi\)
0.382763 + 0.923846i \(0.374972\pi\)
\(608\) 4.09186 0.165947
\(609\) −0.645103 −0.0261409
\(610\) −8.48269 −0.343454
\(611\) 22.4065 0.906469
\(612\) −11.2820 −0.456049
\(613\) −22.7479 −0.918777 −0.459389 0.888235i \(-0.651931\pi\)
−0.459389 + 0.888235i \(0.651931\pi\)
\(614\) 55.5963 2.24368
\(615\) 1.95926 0.0790049
\(616\) −0.189746 −0.00764508
\(617\) 6.49220 0.261366 0.130683 0.991424i \(-0.458283\pi\)
0.130683 + 0.991424i \(0.458283\pi\)
\(618\) 82.2364 3.30803
\(619\) −38.3622 −1.54191 −0.770953 0.636892i \(-0.780220\pi\)
−0.770953 + 0.636892i \(0.780220\pi\)
\(620\) −54.2097 −2.17711
\(621\) 13.0807 0.524910
\(622\) 24.2155 0.970954
\(623\) 0.0555041 0.00222373
\(624\) 27.4103 1.09729
\(625\) −15.6567 −0.626269
\(626\) 35.7650 1.42946
\(627\) −2.27343 −0.0907921
\(628\) 54.2551 2.16501
\(629\) −15.4929 −0.617743
\(630\) −0.232475 −0.00926202
\(631\) 31.3435 1.24776 0.623882 0.781519i \(-0.285555\pi\)
0.623882 + 0.781519i \(0.285555\pi\)
\(632\) 52.8636 2.10280
\(633\) 2.15147 0.0855133
\(634\) 87.0946 3.45897
\(635\) 40.5630 1.60969
\(636\) 12.3036 0.487868
\(637\) 14.0140 0.555255
\(638\) −26.6517 −1.05515
\(639\) 12.0831 0.478001
\(640\) −26.8267 −1.06042
\(641\) 3.78248 0.149399 0.0746995 0.997206i \(-0.476200\pi\)
0.0746995 + 0.997206i \(0.476200\pi\)
\(642\) 80.0204 3.15815
\(643\) −19.8824 −0.784086 −0.392043 0.919947i \(-0.628232\pi\)
−0.392043 + 0.919947i \(0.628232\pi\)
\(644\) −0.581552 −0.0229164
\(645\) 10.0107 0.394173
\(646\) −3.99982 −0.157371
\(647\) −38.0577 −1.49620 −0.748102 0.663584i \(-0.769035\pi\)
−0.748102 + 0.663584i \(0.769035\pi\)
\(648\) 67.2239 2.64080
\(649\) 4.57226 0.179477
\(650\) −7.32215 −0.287198
\(651\) 0.424884 0.0166525
\(652\) 62.0475 2.42997
\(653\) 9.68491 0.379000 0.189500 0.981881i \(-0.439313\pi\)
0.189500 + 0.981881i \(0.439313\pi\)
\(654\) 1.14847 0.0449087
\(655\) 6.82223 0.266566
\(656\) 3.07487 0.120054
\(657\) 16.5751 0.646657
\(658\) −0.847554 −0.0330411
\(659\) −10.9756 −0.427548 −0.213774 0.976883i \(-0.568576\pi\)
−0.213774 + 0.976883i \(0.568576\pi\)
\(660\) −18.7257 −0.728897
\(661\) 19.2154 0.747394 0.373697 0.927551i \(-0.378090\pi\)
0.373697 + 0.927551i \(0.378090\pi\)
\(662\) −47.8542 −1.85991
\(663\) −6.82647 −0.265118
\(664\) −33.6017 −1.30400
\(665\) −0.0565457 −0.00219275
\(666\) −40.1948 −1.55752
\(667\) −44.3082 −1.71562
\(668\) −24.9329 −0.964682
\(669\) 31.7558 1.22775
\(670\) 47.5451 1.83683
\(671\) 1.88449 0.0727499
\(672\) −0.264162 −0.0101903
\(673\) −13.1026 −0.505066 −0.252533 0.967588i \(-0.581264\pi\)
−0.252533 + 0.967588i \(0.581264\pi\)
\(674\) 33.2461 1.28059
\(675\) −4.27410 −0.164510
\(676\) −39.2985 −1.51148
\(677\) 33.3769 1.28278 0.641389 0.767216i \(-0.278359\pi\)
0.641389 + 0.767216i \(0.278359\pi\)
\(678\) 75.5443 2.90126
\(679\) −0.0899316 −0.00345126
\(680\) −17.8706 −0.685305
\(681\) 15.9041 0.609447
\(682\) 17.5536 0.672164
\(683\) 18.9880 0.726554 0.363277 0.931681i \(-0.381658\pi\)
0.363277 + 0.931681i \(0.381658\pi\)
\(684\) −7.11945 −0.272219
\(685\) 41.0743 1.56937
\(686\) −1.06026 −0.0404811
\(687\) −38.2569 −1.45959
\(688\) 15.7109 0.598974
\(689\) 2.61966 0.0998011
\(690\) −45.3760 −1.72743
\(691\) −36.1174 −1.37397 −0.686986 0.726671i \(-0.741067\pi\)
−0.686986 + 0.726671i \(0.741067\pi\)
\(692\) −13.3925 −0.509108
\(693\) 0.0516460 0.00196187
\(694\) 75.3785 2.86133
\(695\) −20.4690 −0.776434
\(696\) −128.655 −4.87666
\(697\) −0.765791 −0.0290064
\(698\) 8.55119 0.323667
\(699\) −60.9836 −2.30661
\(700\) 0.190021 0.00718213
\(701\) −48.4828 −1.83117 −0.915586 0.402123i \(-0.868273\pi\)
−0.915586 + 0.402123i \(0.868273\pi\)
\(702\) 14.9089 0.562702
\(703\) −9.77672 −0.368736
\(704\) 2.53366 0.0954911
\(705\) −45.3706 −1.70876
\(706\) 30.3381 1.14179
\(707\) −0.0450961 −0.00169601
\(708\) 40.6903 1.52923
\(709\) 8.53426 0.320511 0.160255 0.987076i \(-0.448768\pi\)
0.160255 + 0.987076i \(0.448768\pi\)
\(710\) 35.2849 1.32422
\(711\) −14.3887 −0.539617
\(712\) 11.0694 0.414843
\(713\) 29.1827 1.09290
\(714\) 0.258220 0.00966365
\(715\) −3.98705 −0.149107
\(716\) 50.8041 1.89864
\(717\) 32.7843 1.22435
\(718\) 11.1240 0.415144
\(719\) 40.1253 1.49642 0.748210 0.663462i \(-0.230914\pi\)
0.748210 + 0.663462i \(0.230914\pi\)
\(720\) −19.5308 −0.727868
\(721\) −0.454404 −0.0169229
\(722\) −2.52406 −0.0939359
\(723\) −33.4836 −1.24527
\(724\) 111.037 4.12665
\(725\) 14.4776 0.537685
\(726\) −53.6714 −1.99193
\(727\) −29.4684 −1.09292 −0.546461 0.837484i \(-0.684025\pi\)
−0.546461 + 0.837484i \(0.684025\pi\)
\(728\) −0.359540 −0.0133254
\(729\) 0.743392 0.0275330
\(730\) 48.4023 1.79145
\(731\) −3.91278 −0.144719
\(732\) 16.7708 0.619867
\(733\) −12.9069 −0.476728 −0.238364 0.971176i \(-0.576611\pi\)
−0.238364 + 0.971176i \(0.576611\pi\)
\(734\) −22.1894 −0.819025
\(735\) −28.3768 −1.04669
\(736\) −18.1437 −0.668785
\(737\) −10.5625 −0.389074
\(738\) −1.98677 −0.0731338
\(739\) 0.644649 0.0237138 0.0118569 0.999930i \(-0.496226\pi\)
0.0118569 + 0.999930i \(0.496226\pi\)
\(740\) −80.5284 −2.96028
\(741\) −4.30780 −0.158251
\(742\) −0.0990921 −0.00363778
\(743\) −4.80387 −0.176237 −0.0881184 0.996110i \(-0.528085\pi\)
−0.0881184 + 0.996110i \(0.528085\pi\)
\(744\) 84.7362 3.10658
\(745\) −0.378922 −0.0138826
\(746\) 55.6876 2.03887
\(747\) 9.14587 0.334630
\(748\) 7.31909 0.267612
\(749\) −0.442159 −0.0161561
\(750\) 65.9937 2.40975
\(751\) −15.5330 −0.566808 −0.283404 0.959001i \(-0.591464\pi\)
−0.283404 + 0.959001i \(0.591464\pi\)
\(752\) −71.2050 −2.59658
\(753\) 26.8789 0.979520
\(754\) −50.5010 −1.83914
\(755\) −14.5076 −0.527986
\(756\) −0.386911 −0.0140718
\(757\) −33.3109 −1.21071 −0.605353 0.795957i \(-0.706968\pi\)
−0.605353 + 0.795957i \(0.706968\pi\)
\(758\) −95.0768 −3.45335
\(759\) 10.0806 0.365903
\(760\) −11.2771 −0.409064
\(761\) −9.71815 −0.352283 −0.176141 0.984365i \(-0.556362\pi\)
−0.176141 + 0.984365i \(0.556362\pi\)
\(762\) −116.891 −4.23451
\(763\) −0.00634596 −0.000229739 0
\(764\) 70.7412 2.55933
\(765\) 4.86410 0.175862
\(766\) 17.0699 0.616760
\(767\) 8.66373 0.312829
\(768\) 66.9893 2.41727
\(769\) −21.1214 −0.761658 −0.380829 0.924646i \(-0.624361\pi\)
−0.380829 + 0.924646i \(0.624361\pi\)
\(770\) 0.150816 0.00543501
\(771\) −32.4609 −1.16905
\(772\) 91.3678 3.28840
\(773\) −11.3508 −0.408261 −0.204130 0.978944i \(-0.565437\pi\)
−0.204130 + 0.978944i \(0.565437\pi\)
\(774\) −10.1513 −0.364881
\(775\) −9.53540 −0.342522
\(776\) −17.9354 −0.643843
\(777\) 0.631165 0.0226429
\(778\) −21.2802 −0.762932
\(779\) −0.483248 −0.0173142
\(780\) −35.4823 −1.27047
\(781\) −7.83878 −0.280494
\(782\) 17.7356 0.634223
\(783\) −29.4785 −1.05348
\(784\) −44.5348 −1.59053
\(785\) −23.3914 −0.834874
\(786\) −19.6597 −0.701238
\(787\) −27.6980 −0.987327 −0.493664 0.869653i \(-0.664343\pi\)
−0.493664 + 0.869653i \(0.664343\pi\)
\(788\) 70.6958 2.51843
\(789\) 52.1764 1.85753
\(790\) −42.0175 −1.49492
\(791\) −0.417426 −0.0148420
\(792\) 10.2999 0.365993
\(793\) 3.57082 0.126804
\(794\) −82.0609 −2.91223
\(795\) −5.30452 −0.188132
\(796\) −6.28645 −0.222817
\(797\) 33.9368 1.20210 0.601052 0.799210i \(-0.294748\pi\)
0.601052 + 0.799210i \(0.294748\pi\)
\(798\) 0.162948 0.00576831
\(799\) 17.7335 0.627365
\(800\) 5.92842 0.209601
\(801\) −3.01292 −0.106456
\(802\) 45.4960 1.60652
\(803\) −10.7529 −0.379463
\(804\) −93.9996 −3.31511
\(805\) 0.250729 0.00883703
\(806\) 33.2615 1.17158
\(807\) −37.4149 −1.31707
\(808\) −8.99367 −0.316396
\(809\) 42.6190 1.49840 0.749201 0.662342i \(-0.230438\pi\)
0.749201 + 0.662342i \(0.230438\pi\)
\(810\) −53.4315 −1.87739
\(811\) −32.3710 −1.13670 −0.568350 0.822787i \(-0.692418\pi\)
−0.568350 + 0.822787i \(0.692418\pi\)
\(812\) 1.31058 0.0459923
\(813\) 14.7102 0.515910
\(814\) 26.0759 0.913960
\(815\) −26.7510 −0.937046
\(816\) 21.6937 0.759431
\(817\) −2.46914 −0.0863842
\(818\) 90.7528 3.17310
\(819\) 0.0978612 0.00341955
\(820\) −3.98039 −0.139001
\(821\) −5.95062 −0.207678 −0.103839 0.994594i \(-0.533113\pi\)
−0.103839 + 0.994594i \(0.533113\pi\)
\(822\) −118.364 −4.12843
\(823\) −1.00236 −0.0349399 −0.0174700 0.999847i \(-0.505561\pi\)
−0.0174700 + 0.999847i \(0.505561\pi\)
\(824\) −90.6234 −3.15702
\(825\) −3.29382 −0.114676
\(826\) −0.327717 −0.0114027
\(827\) 1.62336 0.0564496 0.0282248 0.999602i \(-0.491015\pi\)
0.0282248 + 0.999602i \(0.491015\pi\)
\(828\) 31.5683 1.09707
\(829\) 31.6385 1.09885 0.549425 0.835543i \(-0.314847\pi\)
0.549425 + 0.835543i \(0.314847\pi\)
\(830\) 26.7076 0.927034
\(831\) 66.7718 2.31629
\(832\) 4.80091 0.166441
\(833\) 11.0913 0.384291
\(834\) 58.9858 2.04251
\(835\) 10.7495 0.372001
\(836\) 4.61867 0.159740
\(837\) 19.4154 0.671096
\(838\) 82.3018 2.84307
\(839\) 30.1852 1.04211 0.521054 0.853524i \(-0.325539\pi\)
0.521054 + 0.853524i \(0.325539\pi\)
\(840\) 0.728027 0.0251193
\(841\) 70.8524 2.44318
\(842\) −71.9184 −2.47847
\(843\) 17.1757 0.591561
\(844\) −4.37089 −0.150452
\(845\) 16.9430 0.582859
\(846\) 46.0076 1.58177
\(847\) 0.296566 0.0101901
\(848\) −8.32495 −0.285880
\(849\) −50.2426 −1.72432
\(850\) −5.79507 −0.198769
\(851\) 43.3509 1.48605
\(852\) −69.7603 −2.38995
\(853\) 27.5434 0.943068 0.471534 0.881848i \(-0.343700\pi\)
0.471534 + 0.881848i \(0.343700\pi\)
\(854\) −0.135071 −0.00462203
\(855\) 3.06946 0.104973
\(856\) −88.1814 −3.01398
\(857\) 16.4943 0.563435 0.281718 0.959497i \(-0.409096\pi\)
0.281718 + 0.959497i \(0.409096\pi\)
\(858\) 11.4895 0.392246
\(859\) −38.1095 −1.30028 −0.650139 0.759815i \(-0.725289\pi\)
−0.650139 + 0.759815i \(0.725289\pi\)
\(860\) −20.3377 −0.693509
\(861\) 0.0311975 0.00106321
\(862\) 91.5357 3.11772
\(863\) −9.77818 −0.332853 −0.166427 0.986054i \(-0.553223\pi\)
−0.166427 + 0.986054i \(0.553223\pi\)
\(864\) −12.0711 −0.410667
\(865\) 5.77402 0.196323
\(866\) 0.117674 0.00399872
\(867\) 31.1723 1.05867
\(868\) −0.863187 −0.0292985
\(869\) 9.33449 0.316651
\(870\) 102.259 3.46690
\(871\) −20.0143 −0.678159
\(872\) −1.26560 −0.0428586
\(873\) 4.88174 0.165222
\(874\) 11.1919 0.378573
\(875\) −0.364654 −0.0123275
\(876\) −95.6944 −3.23321
\(877\) 28.6121 0.966163 0.483081 0.875575i \(-0.339517\pi\)
0.483081 + 0.875575i \(0.339517\pi\)
\(878\) 62.4022 2.10597
\(879\) 46.7527 1.57693
\(880\) 12.6704 0.427118
\(881\) −4.09192 −0.137860 −0.0689301 0.997621i \(-0.521959\pi\)
−0.0689301 + 0.997621i \(0.521959\pi\)
\(882\) 28.7752 0.968912
\(883\) 24.5185 0.825114 0.412557 0.910932i \(-0.364636\pi\)
0.412557 + 0.910932i \(0.364636\pi\)
\(884\) 13.8685 0.466450
\(885\) −17.5431 −0.589704
\(886\) 2.25592 0.0757891
\(887\) 31.9688 1.07341 0.536704 0.843771i \(-0.319669\pi\)
0.536704 + 0.843771i \(0.319669\pi\)
\(888\) 125.876 4.22411
\(889\) 0.645890 0.0216625
\(890\) −8.79827 −0.294919
\(891\) 11.8702 0.397666
\(892\) −64.5146 −2.16011
\(893\) 11.1906 0.374479
\(894\) 1.09194 0.0365201
\(895\) −21.9035 −0.732154
\(896\) −0.427165 −0.0142706
\(897\) 19.1012 0.637770
\(898\) 76.3711 2.54854
\(899\) −65.7658 −2.19341
\(900\) −10.3149 −0.343830
\(901\) 2.07331 0.0690721
\(902\) 1.28889 0.0429154
\(903\) 0.159402 0.00530458
\(904\) −83.2488 −2.76881
\(905\) −47.8721 −1.59132
\(906\) 41.8067 1.38894
\(907\) −37.6444 −1.24996 −0.624982 0.780640i \(-0.714894\pi\)
−0.624982 + 0.780640i \(0.714894\pi\)
\(908\) −32.3106 −1.07226
\(909\) 2.44794 0.0811931
\(910\) 0.285772 0.00947326
\(911\) −48.3751 −1.60274 −0.801369 0.598170i \(-0.795895\pi\)
−0.801369 + 0.598170i \(0.795895\pi\)
\(912\) 13.6897 0.453310
\(913\) −5.93328 −0.196363
\(914\) −42.9085 −1.41929
\(915\) −7.23051 −0.239033
\(916\) 77.7222 2.56801
\(917\) 0.108631 0.00358732
\(918\) 11.7996 0.389445
\(919\) 48.9429 1.61448 0.807239 0.590224i \(-0.200961\pi\)
0.807239 + 0.590224i \(0.200961\pi\)
\(920\) 50.0038 1.64857
\(921\) 47.3894 1.56153
\(922\) −83.1047 −2.73691
\(923\) −14.8533 −0.488902
\(924\) −0.298172 −0.00980912
\(925\) −14.1648 −0.465737
\(926\) 100.379 3.29865
\(927\) 24.6663 0.810148
\(928\) 40.8884 1.34223
\(929\) 42.0953 1.38110 0.690550 0.723284i \(-0.257368\pi\)
0.690550 + 0.723284i \(0.257368\pi\)
\(930\) −67.3507 −2.20852
\(931\) 6.99910 0.229386
\(932\) 123.893 4.05826
\(933\) 20.6409 0.675754
\(934\) 68.9606 2.25646
\(935\) −3.15553 −0.103197
\(936\) 19.5168 0.637927
\(937\) 12.4036 0.405207 0.202604 0.979261i \(-0.435060\pi\)
0.202604 + 0.979261i \(0.435060\pi\)
\(938\) 0.757067 0.0247191
\(939\) 30.4856 0.994859
\(940\) 92.1741 3.00639
\(941\) −40.4438 −1.31843 −0.659215 0.751955i \(-0.729111\pi\)
−0.659215 + 0.751955i \(0.729111\pi\)
\(942\) 67.4071 2.19624
\(943\) 2.14277 0.0697780
\(944\) −27.5323 −0.896098
\(945\) 0.166811 0.00542638
\(946\) 6.58554 0.214115
\(947\) 55.1836 1.79322 0.896612 0.442816i \(-0.146021\pi\)
0.896612 + 0.442816i \(0.146021\pi\)
\(948\) 83.0712 2.69803
\(949\) −20.3751 −0.661405
\(950\) −3.65694 −0.118647
\(951\) 74.2381 2.40733
\(952\) −0.284555 −0.00922249
\(953\) 30.2971 0.981420 0.490710 0.871323i \(-0.336737\pi\)
0.490710 + 0.871323i \(0.336737\pi\)
\(954\) 5.37900 0.174151
\(955\) −30.4991 −0.986929
\(956\) −66.6041 −2.15413
\(957\) −22.7175 −0.734353
\(958\) −58.4569 −1.88866
\(959\) 0.654031 0.0211198
\(960\) −9.72130 −0.313754
\(961\) 12.3153 0.397268
\(962\) 49.4099 1.59304
\(963\) 24.0016 0.773442
\(964\) 68.0247 2.19093
\(965\) −39.3920 −1.26807
\(966\) −0.722528 −0.0232469
\(967\) 4.26436 0.137133 0.0685663 0.997647i \(-0.478158\pi\)
0.0685663 + 0.997647i \(0.478158\pi\)
\(968\) 59.1452 1.90100
\(969\) −3.40939 −0.109525
\(970\) 14.2556 0.457718
\(971\) −9.23405 −0.296335 −0.148167 0.988962i \(-0.547337\pi\)
−0.148167 + 0.988962i \(0.547337\pi\)
\(972\) 66.9545 2.14757
\(973\) −0.325931 −0.0104489
\(974\) 1.30947 0.0419582
\(975\) −6.24129 −0.199881
\(976\) −11.3476 −0.363228
\(977\) 14.2786 0.456814 0.228407 0.973566i \(-0.426648\pi\)
0.228407 + 0.973566i \(0.426648\pi\)
\(978\) 77.0886 2.46502
\(979\) 1.95460 0.0624692
\(980\) 57.6498 1.84156
\(981\) 0.344477 0.0109983
\(982\) 25.5066 0.813948
\(983\) 41.8189 1.33381 0.666907 0.745141i \(-0.267618\pi\)
0.666907 + 0.745141i \(0.267618\pi\)
\(984\) 6.22183 0.198345
\(985\) −30.4796 −0.971160
\(986\) −39.9687 −1.27286
\(987\) −0.722442 −0.0229956
\(988\) 8.75166 0.278427
\(989\) 10.9484 0.348138
\(990\) −8.18669 −0.260190
\(991\) −55.0997 −1.75030 −0.875149 0.483853i \(-0.839237\pi\)
−0.875149 + 0.483853i \(0.839237\pi\)
\(992\) −26.9303 −0.855039
\(993\) −40.7902 −1.29444
\(994\) 0.561845 0.0178206
\(995\) 2.71032 0.0859230
\(996\) −52.8026 −1.67311
\(997\) 23.7035 0.750697 0.375349 0.926884i \(-0.377523\pi\)
0.375349 + 0.926884i \(0.377523\pi\)
\(998\) −2.49771 −0.0790636
\(999\) 28.8416 0.912508
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 4009.2.a.d.1.7 75
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4009.2.a.d.1.7 75 1.1 even 1 trivial