Properties

Label 1681.2.a.m.1.3
Level $1681$
Weight $2$
Character 1681.1
Self dual yes
Analytic conductor $13.423$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1681,2,Mod(1,1681)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1681, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1681.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1681 = 41^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1681.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: \(13.4228525798\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 41)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Character \(\chi\) \(=\) 1681.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.71427 q^{2} -3.15154 q^{3} +0.938727 q^{4} -2.70948 q^{5} +5.40260 q^{6} -1.61292 q^{7} +1.81931 q^{8} +6.93221 q^{9} +O(q^{10})\) \(q-1.71427 q^{2} -3.15154 q^{3} +0.938727 q^{4} -2.70948 q^{5} +5.40260 q^{6} -1.61292 q^{7} +1.81931 q^{8} +6.93221 q^{9} +4.64479 q^{10} -0.692944 q^{11} -2.95844 q^{12} -1.52612 q^{13} +2.76498 q^{14} +8.53905 q^{15} -4.99625 q^{16} -1.05794 q^{17} -11.8837 q^{18} -0.128551 q^{19} -2.54347 q^{20} +5.08317 q^{21} +1.18789 q^{22} +1.88345 q^{23} -5.73363 q^{24} +2.34131 q^{25} +2.61618 q^{26} -12.3925 q^{27} -1.51409 q^{28} -8.29648 q^{29} -14.6383 q^{30} -1.20736 q^{31} +4.92630 q^{32} +2.18384 q^{33} +1.81360 q^{34} +4.37017 q^{35} +6.50745 q^{36} -6.04786 q^{37} +0.220371 q^{38} +4.80963 q^{39} -4.92939 q^{40} -8.71394 q^{42} -10.5792 q^{43} -0.650485 q^{44} -18.7827 q^{45} -3.22875 q^{46} -6.34015 q^{47} +15.7459 q^{48} -4.39850 q^{49} -4.01364 q^{50} +3.33414 q^{51} -1.43261 q^{52} -12.4903 q^{53} +21.2441 q^{54} +1.87752 q^{55} -2.93440 q^{56} +0.405133 q^{57} +14.2224 q^{58} +8.17953 q^{59} +8.01584 q^{60} +2.63308 q^{61} +2.06974 q^{62} -11.1811 q^{63} +1.54747 q^{64} +4.13500 q^{65} -3.74369 q^{66} -8.49003 q^{67} -0.993116 q^{68} -5.93577 q^{69} -7.49167 q^{70} +4.29060 q^{71} +12.6118 q^{72} -12.1957 q^{73} +10.3677 q^{74} -7.37872 q^{75} -0.120674 q^{76} +1.11766 q^{77} -8.24501 q^{78} -11.5144 q^{79} +13.5373 q^{80} +18.2589 q^{81} +6.49090 q^{83} +4.77171 q^{84} +2.86647 q^{85} +18.1356 q^{86} +26.1467 q^{87} -1.26068 q^{88} +2.88381 q^{89} +32.1987 q^{90} +2.46150 q^{91} +1.76805 q^{92} +3.80504 q^{93} +10.8687 q^{94} +0.348306 q^{95} -15.5254 q^{96} -1.20901 q^{97} +7.54022 q^{98} -4.80363 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 8 q^{2} + 20 q^{4} + 12 q^{5} + 24 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 8 q^{2} + 20 q^{4} + 12 q^{5} + 24 q^{8} + 24 q^{9} - 4 q^{10} + 20 q^{16} + 20 q^{18} + 40 q^{20} + 56 q^{21} + 48 q^{23} + 8 q^{25} + 32 q^{31} + 36 q^{32} + 24 q^{33} + 108 q^{36} + 60 q^{39} + 44 q^{40} - 40 q^{42} + 36 q^{43} - 36 q^{45} + 36 q^{46} + 32 q^{49} - 60 q^{50} + 36 q^{51} - 60 q^{57} + 116 q^{59} + 40 q^{61} + 48 q^{62} + 60 q^{64} + 16 q^{66} + 4 q^{72} + 16 q^{73} + 24 q^{74} - 16 q^{77} - 60 q^{78} + 84 q^{80} - 28 q^{81} + 80 q^{83} - 24 q^{84} - 24 q^{86} + 76 q^{87} + 56 q^{90} + 40 q^{91} - 40 q^{92} + 36 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.71427 −1.21217 −0.606087 0.795399i \(-0.707261\pi\)
−0.606087 + 0.795399i \(0.707261\pi\)
\(3\) −3.15154 −1.81954 −0.909771 0.415110i \(-0.863743\pi\)
−0.909771 + 0.415110i \(0.863743\pi\)
\(4\) 0.938727 0.469364
\(5\) −2.70948 −1.21172 −0.605859 0.795572i \(-0.707171\pi\)
−0.605859 + 0.795572i \(0.707171\pi\)
\(6\) 5.40260 2.20560
\(7\) −1.61292 −0.609625 −0.304813 0.952412i \(-0.598594\pi\)
−0.304813 + 0.952412i \(0.598594\pi\)
\(8\) 1.81931 0.643223
\(9\) 6.93221 2.31074
\(10\) 4.64479 1.46881
\(11\) −0.692944 −0.208930 −0.104465 0.994529i \(-0.533313\pi\)
−0.104465 + 0.994529i \(0.533313\pi\)
\(12\) −2.95844 −0.854027
\(13\) −1.52612 −0.423269 −0.211635 0.977349i \(-0.567879\pi\)
−0.211635 + 0.977349i \(0.567879\pi\)
\(14\) 2.76498 0.738972
\(15\) 8.53905 2.20477
\(16\) −4.99625 −1.24906
\(17\) −1.05794 −0.256588 −0.128294 0.991736i \(-0.540950\pi\)
−0.128294 + 0.991736i \(0.540950\pi\)
\(18\) −11.8837 −2.80101
\(19\) −0.128551 −0.0294916 −0.0147458 0.999891i \(-0.504694\pi\)
−0.0147458 + 0.999891i \(0.504694\pi\)
\(20\) −2.54347 −0.568736
\(21\) 5.08317 1.10924
\(22\) 1.18789 0.253260
\(23\) 1.88345 0.392727 0.196363 0.980531i \(-0.437087\pi\)
0.196363 + 0.980531i \(0.437087\pi\)
\(24\) −5.73363 −1.17037
\(25\) 2.34131 0.468261
\(26\) 2.61618 0.513076
\(27\) −12.3925 −2.38494
\(28\) −1.51409 −0.286136
\(29\) −8.29648 −1.54062 −0.770309 0.637671i \(-0.779898\pi\)
−0.770309 + 0.637671i \(0.779898\pi\)
\(30\) −14.6383 −2.67257
\(31\) −1.20736 −0.216848 −0.108424 0.994105i \(-0.534580\pi\)
−0.108424 + 0.994105i \(0.534580\pi\)
\(32\) 4.92630 0.870855
\(33\) 2.18384 0.380158
\(34\) 1.81360 0.311029
\(35\) 4.37017 0.738694
\(36\) 6.50745 1.08457
\(37\) −6.04786 −0.994262 −0.497131 0.867675i \(-0.665613\pi\)
−0.497131 + 0.867675i \(0.665613\pi\)
\(38\) 0.220371 0.0357489
\(39\) 4.80963 0.770157
\(40\) −4.92939 −0.779405
\(41\) 0 0
\(42\) −8.71394 −1.34459
\(43\) −10.5792 −1.61331 −0.806655 0.591022i \(-0.798724\pi\)
−0.806655 + 0.591022i \(0.798724\pi\)
\(44\) −0.650485 −0.0980643
\(45\) −18.7827 −2.79996
\(46\) −3.22875 −0.476053
\(47\) −6.34015 −0.924806 −0.462403 0.886670i \(-0.653013\pi\)
−0.462403 + 0.886670i \(0.653013\pi\)
\(48\) 15.7459 2.27272
\(49\) −4.39850 −0.628357
\(50\) −4.01364 −0.567614
\(51\) 3.33414 0.466873
\(52\) −1.43261 −0.198667
\(53\) −12.4903 −1.71567 −0.857835 0.513925i \(-0.828191\pi\)
−0.857835 + 0.513925i \(0.828191\pi\)
\(54\) 21.2441 2.89096
\(55\) 1.87752 0.253165
\(56\) −2.93440 −0.392125
\(57\) 0.405133 0.0536612
\(58\) 14.2224 1.86749
\(59\) 8.17953 1.06488 0.532442 0.846467i \(-0.321274\pi\)
0.532442 + 0.846467i \(0.321274\pi\)
\(60\) 8.01584 1.03484
\(61\) 2.63308 0.337131 0.168566 0.985690i \(-0.446087\pi\)
0.168566 + 0.985690i \(0.446087\pi\)
\(62\) 2.06974 0.262857
\(63\) −11.1811 −1.40868
\(64\) 1.54747 0.193434
\(65\) 4.13500 0.512883
\(66\) −3.74369 −0.460817
\(67\) −8.49003 −1.03722 −0.518611 0.855010i \(-0.673551\pi\)
−0.518611 + 0.855010i \(0.673551\pi\)
\(68\) −0.993116 −0.120433
\(69\) −5.93577 −0.714583
\(70\) −7.49167 −0.895425
\(71\) 4.29060 0.509201 0.254600 0.967046i \(-0.418056\pi\)
0.254600 + 0.967046i \(0.418056\pi\)
\(72\) 12.6118 1.48632
\(73\) −12.1957 −1.42739 −0.713697 0.700455i \(-0.752981\pi\)
−0.713697 + 0.700455i \(0.752981\pi\)
\(74\) 10.3677 1.20522
\(75\) −7.37872 −0.852021
\(76\) −0.120674 −0.0138423
\(77\) 1.11766 0.127369
\(78\) −8.24501 −0.933563
\(79\) −11.5144 −1.29547 −0.647736 0.761865i \(-0.724284\pi\)
−0.647736 + 0.761865i \(0.724284\pi\)
\(80\) 13.5373 1.51351
\(81\) 18.2589 2.02876
\(82\) 0 0
\(83\) 6.49090 0.712469 0.356235 0.934397i \(-0.384060\pi\)
0.356235 + 0.934397i \(0.384060\pi\)
\(84\) 4.77171 0.520637
\(85\) 2.86647 0.310912
\(86\) 18.1356 1.95561
\(87\) 26.1467 2.80322
\(88\) −1.26068 −0.134389
\(89\) 2.88381 0.305684 0.152842 0.988251i \(-0.451157\pi\)
0.152842 + 0.988251i \(0.451157\pi\)
\(90\) 32.1987 3.39404
\(91\) 2.46150 0.258036
\(92\) 1.76805 0.184332
\(93\) 3.80504 0.394564
\(94\) 10.8687 1.12103
\(95\) 0.348306 0.0357355
\(96\) −15.5254 −1.58456
\(97\) −1.20901 −0.122756 −0.0613781 0.998115i \(-0.519550\pi\)
−0.0613781 + 0.998115i \(0.519550\pi\)
\(98\) 7.54022 0.761677
\(99\) −4.80363 −0.482783
\(100\) 2.19785 0.219785
\(101\) −5.30173 −0.527541 −0.263771 0.964585i \(-0.584966\pi\)
−0.263771 + 0.964585i \(0.584966\pi\)
\(102\) −5.71562 −0.565931
\(103\) 13.8765 1.36730 0.683648 0.729812i \(-0.260392\pi\)
0.683648 + 0.729812i \(0.260392\pi\)
\(104\) −2.77648 −0.272257
\(105\) −13.7728 −1.34409
\(106\) 21.4117 2.07969
\(107\) −9.53341 −0.921630 −0.460815 0.887496i \(-0.652443\pi\)
−0.460815 + 0.887496i \(0.652443\pi\)
\(108\) −11.6332 −1.11940
\(109\) −2.35650 −0.225712 −0.112856 0.993611i \(-0.536000\pi\)
−0.112856 + 0.993611i \(0.536000\pi\)
\(110\) −3.21858 −0.306880
\(111\) 19.0601 1.80910
\(112\) 8.05853 0.761460
\(113\) 7.67722 0.722212 0.361106 0.932525i \(-0.382399\pi\)
0.361106 + 0.932525i \(0.382399\pi\)
\(114\) −0.694508 −0.0650466
\(115\) −5.10318 −0.475874
\(116\) −7.78813 −0.723110
\(117\) −10.5794 −0.978064
\(118\) −14.0219 −1.29082
\(119\) 1.70637 0.156423
\(120\) 15.5352 1.41816
\(121\) −10.5198 −0.956348
\(122\) −4.51381 −0.408661
\(123\) 0 0
\(124\) −1.13338 −0.101781
\(125\) 7.20369 0.644317
\(126\) 19.1674 1.70757
\(127\) 6.23197 0.552998 0.276499 0.961014i \(-0.410826\pi\)
0.276499 + 0.961014i \(0.410826\pi\)
\(128\) −12.5054 −1.10533
\(129\) 33.3407 2.93549
\(130\) −7.08851 −0.621703
\(131\) −0.725340 −0.0633732 −0.0316866 0.999498i \(-0.510088\pi\)
−0.0316866 + 0.999498i \(0.510088\pi\)
\(132\) 2.05003 0.178432
\(133\) 0.207342 0.0179788
\(134\) 14.5542 1.25729
\(135\) 33.5773 2.88987
\(136\) −1.92472 −0.165043
\(137\) −18.6603 −1.59426 −0.797131 0.603807i \(-0.793650\pi\)
−0.797131 + 0.603807i \(0.793650\pi\)
\(138\) 10.1755 0.866199
\(139\) 6.34313 0.538017 0.269009 0.963138i \(-0.413304\pi\)
0.269009 + 0.963138i \(0.413304\pi\)
\(140\) 4.10240 0.346716
\(141\) 19.9812 1.68272
\(142\) −7.35526 −0.617240
\(143\) 1.05751 0.0884338
\(144\) −34.6350 −2.88625
\(145\) 22.4792 1.86679
\(146\) 20.9067 1.73025
\(147\) 13.8620 1.14332
\(148\) −5.67729 −0.466670
\(149\) 12.9287 1.05916 0.529579 0.848261i \(-0.322350\pi\)
0.529579 + 0.848261i \(0.322350\pi\)
\(150\) 12.6491 1.03280
\(151\) 1.39618 0.113620 0.0568098 0.998385i \(-0.481907\pi\)
0.0568098 + 0.998385i \(0.481907\pi\)
\(152\) −0.233874 −0.0189697
\(153\) −7.33385 −0.592907
\(154\) −1.91597 −0.154394
\(155\) 3.27132 0.262759
\(156\) 4.51493 0.361483
\(157\) −17.5611 −1.40153 −0.700763 0.713394i \(-0.747157\pi\)
−0.700763 + 0.713394i \(0.747157\pi\)
\(158\) 19.7388 1.57034
\(159\) 39.3636 3.12174
\(160\) −13.3477 −1.05523
\(161\) −3.03785 −0.239416
\(162\) −31.3006 −2.45921
\(163\) −17.6081 −1.37918 −0.689588 0.724202i \(-0.742208\pi\)
−0.689588 + 0.724202i \(0.742208\pi\)
\(164\) 0 0
\(165\) −5.91708 −0.460644
\(166\) −11.1272 −0.863636
\(167\) 2.77606 0.214818 0.107409 0.994215i \(-0.465745\pi\)
0.107409 + 0.994215i \(0.465745\pi\)
\(168\) 9.24787 0.713489
\(169\) −10.6710 −0.820843
\(170\) −4.91391 −0.376880
\(171\) −0.891141 −0.0681472
\(172\) −9.93097 −0.757229
\(173\) −11.7096 −0.890265 −0.445132 0.895465i \(-0.646843\pi\)
−0.445132 + 0.895465i \(0.646843\pi\)
\(174\) −44.8225 −3.39799
\(175\) −3.77633 −0.285464
\(176\) 3.46212 0.260967
\(177\) −25.7781 −1.93760
\(178\) −4.94364 −0.370542
\(179\) −4.25363 −0.317932 −0.158966 0.987284i \(-0.550816\pi\)
−0.158966 + 0.987284i \(0.550816\pi\)
\(180\) −17.6318 −1.31420
\(181\) −4.63621 −0.344606 −0.172303 0.985044i \(-0.555121\pi\)
−0.172303 + 0.985044i \(0.555121\pi\)
\(182\) −4.21969 −0.312784
\(183\) −8.29825 −0.613425
\(184\) 3.42658 0.252611
\(185\) 16.3866 1.20477
\(186\) −6.52287 −0.478280
\(187\) 0.733093 0.0536090
\(188\) −5.95167 −0.434070
\(189\) 19.9881 1.45392
\(190\) −0.597092 −0.0433176
\(191\) 22.3688 1.61855 0.809275 0.587430i \(-0.199860\pi\)
0.809275 + 0.587430i \(0.199860\pi\)
\(192\) −4.87692 −0.351962
\(193\) −14.5740 −1.04906 −0.524530 0.851392i \(-0.675759\pi\)
−0.524530 + 0.851392i \(0.675759\pi\)
\(194\) 2.07257 0.148802
\(195\) −13.0316 −0.933213
\(196\) −4.12899 −0.294928
\(197\) 0.930554 0.0662992 0.0331496 0.999450i \(-0.489446\pi\)
0.0331496 + 0.999450i \(0.489446\pi\)
\(198\) 8.23472 0.585216
\(199\) −0.412246 −0.0292233 −0.0146117 0.999893i \(-0.504651\pi\)
−0.0146117 + 0.999893i \(0.504651\pi\)
\(200\) 4.25956 0.301197
\(201\) 26.7567 1.88727
\(202\) 9.08860 0.639472
\(203\) 13.3815 0.939199
\(204\) 3.12985 0.219133
\(205\) 0 0
\(206\) −23.7881 −1.65740
\(207\) 13.0565 0.907488
\(208\) 7.62487 0.528689
\(209\) 0.0890785 0.00616169
\(210\) 23.6103 1.62926
\(211\) −22.4233 −1.54369 −0.771843 0.635813i \(-0.780665\pi\)
−0.771843 + 0.635813i \(0.780665\pi\)
\(212\) −11.7250 −0.805273
\(213\) −13.5220 −0.926513
\(214\) 16.3429 1.11717
\(215\) 28.6641 1.95488
\(216\) −22.5458 −1.53405
\(217\) 1.94737 0.132196
\(218\) 4.03969 0.273602
\(219\) 38.4351 2.59720
\(220\) 1.76248 0.118826
\(221\) 1.61454 0.108606
\(222\) −32.6741 −2.19295
\(223\) −4.07615 −0.272959 −0.136480 0.990643i \(-0.543579\pi\)
−0.136480 + 0.990643i \(0.543579\pi\)
\(224\) −7.94572 −0.530896
\(225\) 16.2304 1.08203
\(226\) −13.1608 −0.875446
\(227\) 21.5929 1.43317 0.716584 0.697501i \(-0.245705\pi\)
0.716584 + 0.697501i \(0.245705\pi\)
\(228\) 0.380309 0.0251866
\(229\) 9.90746 0.654703 0.327352 0.944903i \(-0.393844\pi\)
0.327352 + 0.944903i \(0.393844\pi\)
\(230\) 8.74824 0.576842
\(231\) −3.52235 −0.231754
\(232\) −15.0939 −0.990961
\(233\) 1.28663 0.0842899 0.0421449 0.999112i \(-0.486581\pi\)
0.0421449 + 0.999112i \(0.486581\pi\)
\(234\) 18.1359 1.18558
\(235\) 17.1785 1.12060
\(236\) 7.67834 0.499818
\(237\) 36.2881 2.35717
\(238\) −2.92518 −0.189611
\(239\) 7.09666 0.459045 0.229522 0.973303i \(-0.426284\pi\)
0.229522 + 0.973303i \(0.426284\pi\)
\(240\) −42.6632 −2.75390
\(241\) −10.7877 −0.694899 −0.347449 0.937699i \(-0.612952\pi\)
−0.347449 + 0.937699i \(0.612952\pi\)
\(242\) 18.0338 1.15926
\(243\) −20.3660 −1.30648
\(244\) 2.47174 0.158237
\(245\) 11.9177 0.761391
\(246\) 0 0
\(247\) 0.196184 0.0124829
\(248\) −2.19656 −0.139482
\(249\) −20.4563 −1.29637
\(250\) −12.3491 −0.781024
\(251\) 5.62542 0.355073 0.177537 0.984114i \(-0.443187\pi\)
0.177537 + 0.984114i \(0.443187\pi\)
\(252\) −10.4960 −0.661184
\(253\) −1.30513 −0.0820526
\(254\) −10.6833 −0.670329
\(255\) −9.03380 −0.565718
\(256\) 18.3427 1.14642
\(257\) 21.9844 1.37135 0.685675 0.727908i \(-0.259507\pi\)
0.685675 + 0.727908i \(0.259507\pi\)
\(258\) −57.1551 −3.55832
\(259\) 9.75470 0.606128
\(260\) 3.88163 0.240729
\(261\) −57.5129 −3.55996
\(262\) 1.24343 0.0768193
\(263\) −5.86109 −0.361410 −0.180705 0.983537i \(-0.557838\pi\)
−0.180705 + 0.983537i \(0.557838\pi\)
\(264\) 3.97308 0.244526
\(265\) 33.8422 2.07891
\(266\) −0.355440 −0.0217934
\(267\) −9.08846 −0.556205
\(268\) −7.96982 −0.486834
\(269\) −13.7302 −0.837146 −0.418573 0.908183i \(-0.637470\pi\)
−0.418573 + 0.908183i \(0.637470\pi\)
\(270\) −57.5606 −3.50303
\(271\) 15.9129 0.966638 0.483319 0.875444i \(-0.339431\pi\)
0.483319 + 0.875444i \(0.339431\pi\)
\(272\) 5.28573 0.320494
\(273\) −7.75753 −0.469507
\(274\) 31.9889 1.93252
\(275\) −1.62239 −0.0978340
\(276\) −5.57207 −0.335399
\(277\) 19.1964 1.15340 0.576699 0.816957i \(-0.304341\pi\)
0.576699 + 0.816957i \(0.304341\pi\)
\(278\) −10.8738 −0.652170
\(279\) −8.36966 −0.501078
\(280\) 7.95070 0.475145
\(281\) 1.45103 0.0865611 0.0432806 0.999063i \(-0.486219\pi\)
0.0432806 + 0.999063i \(0.486219\pi\)
\(282\) −34.2533 −2.03975
\(283\) −9.42328 −0.560156 −0.280078 0.959977i \(-0.590360\pi\)
−0.280078 + 0.959977i \(0.590360\pi\)
\(284\) 4.02771 0.239000
\(285\) −1.09770 −0.0650222
\(286\) −1.81287 −0.107197
\(287\) 0 0
\(288\) 34.1501 2.01232
\(289\) −15.8808 −0.934163
\(290\) −38.5354 −2.26288
\(291\) 3.81024 0.223360
\(292\) −11.4484 −0.669967
\(293\) 1.08492 0.0633820 0.0316910 0.999498i \(-0.489911\pi\)
0.0316910 + 0.999498i \(0.489911\pi\)
\(294\) −23.7633 −1.38590
\(295\) −22.1623 −1.29034
\(296\) −11.0029 −0.639533
\(297\) 8.58731 0.498286
\(298\) −22.1633 −1.28388
\(299\) −2.87437 −0.166229
\(300\) −6.92661 −0.399908
\(301\) 17.0633 0.983515
\(302\) −2.39343 −0.137727
\(303\) 16.7086 0.959884
\(304\) 0.642271 0.0368368
\(305\) −7.13429 −0.408508
\(306\) 12.5722 0.718706
\(307\) −29.6465 −1.69202 −0.846008 0.533170i \(-0.821000\pi\)
−0.846008 + 0.533170i \(0.821000\pi\)
\(308\) 1.04918 0.0597825
\(309\) −43.7325 −2.48785
\(310\) −5.60793 −0.318509
\(311\) −17.2471 −0.977994 −0.488997 0.872286i \(-0.662637\pi\)
−0.488997 + 0.872286i \(0.662637\pi\)
\(312\) 8.75020 0.495383
\(313\) −1.62235 −0.0917008 −0.0458504 0.998948i \(-0.514600\pi\)
−0.0458504 + 0.998948i \(0.514600\pi\)
\(314\) 30.1045 1.69889
\(315\) 30.2949 1.70693
\(316\) −10.8089 −0.608047
\(317\) −16.2930 −0.915105 −0.457552 0.889183i \(-0.651274\pi\)
−0.457552 + 0.889183i \(0.651274\pi\)
\(318\) −67.4799 −3.78408
\(319\) 5.74899 0.321882
\(320\) −4.19285 −0.234388
\(321\) 30.0449 1.67694
\(322\) 5.20770 0.290214
\(323\) 0.135999 0.00756719
\(324\) 17.1401 0.952227
\(325\) −3.57311 −0.198201
\(326\) 30.1851 1.67180
\(327\) 7.42662 0.410693
\(328\) 0 0
\(329\) 10.2261 0.563786
\(330\) 10.1435 0.558380
\(331\) 25.3548 1.39362 0.696812 0.717254i \(-0.254601\pi\)
0.696812 + 0.717254i \(0.254601\pi\)
\(332\) 6.09318 0.334407
\(333\) −41.9250 −2.29748
\(334\) −4.75893 −0.260397
\(335\) 23.0036 1.25682
\(336\) −25.3968 −1.38551
\(337\) 26.2434 1.42957 0.714785 0.699344i \(-0.246525\pi\)
0.714785 + 0.699344i \(0.246525\pi\)
\(338\) 18.2929 0.995004
\(339\) −24.1951 −1.31410
\(340\) 2.69083 0.145931
\(341\) 0.836631 0.0453061
\(342\) 1.52766 0.0826062
\(343\) 18.3848 0.992688
\(344\) −19.2468 −1.03772
\(345\) 16.0829 0.865874
\(346\) 20.0734 1.07915
\(347\) −16.6674 −0.894751 −0.447375 0.894346i \(-0.647641\pi\)
−0.447375 + 0.894346i \(0.647641\pi\)
\(348\) 24.5446 1.31573
\(349\) 8.65063 0.463058 0.231529 0.972828i \(-0.425627\pi\)
0.231529 + 0.972828i \(0.425627\pi\)
\(350\) 6.47366 0.346032
\(351\) 18.9124 1.00947
\(352\) −3.41365 −0.181948
\(353\) −13.7928 −0.734115 −0.367058 0.930198i \(-0.619635\pi\)
−0.367058 + 0.930198i \(0.619635\pi\)
\(354\) 44.1907 2.34871
\(355\) −11.6253 −0.617008
\(356\) 2.70711 0.143477
\(357\) −5.37769 −0.284618
\(358\) 7.29188 0.385388
\(359\) 18.1150 0.956075 0.478038 0.878339i \(-0.341348\pi\)
0.478038 + 0.878339i \(0.341348\pi\)
\(360\) −34.1716 −1.80100
\(361\) −18.9835 −0.999130
\(362\) 7.94772 0.417723
\(363\) 33.1537 1.74012
\(364\) 2.31068 0.121113
\(365\) 33.0439 1.72960
\(366\) 14.2255 0.743577
\(367\) −23.0926 −1.20543 −0.602713 0.797958i \(-0.705913\pi\)
−0.602713 + 0.797958i \(0.705913\pi\)
\(368\) −9.41019 −0.490540
\(369\) 0 0
\(370\) −28.0911 −1.46038
\(371\) 20.1458 1.04592
\(372\) 3.57189 0.185194
\(373\) −14.1402 −0.732151 −0.366076 0.930585i \(-0.619299\pi\)
−0.366076 + 0.930585i \(0.619299\pi\)
\(374\) −1.25672 −0.0649834
\(375\) −22.7027 −1.17236
\(376\) −11.5347 −0.594857
\(377\) 12.6614 0.652096
\(378\) −34.2650 −1.76240
\(379\) −37.1747 −1.90953 −0.954767 0.297354i \(-0.903896\pi\)
−0.954767 + 0.297354i \(0.903896\pi\)
\(380\) 0.326965 0.0167729
\(381\) −19.6403 −1.00620
\(382\) −38.3462 −1.96196
\(383\) 6.92178 0.353686 0.176843 0.984239i \(-0.443411\pi\)
0.176843 + 0.984239i \(0.443411\pi\)
\(384\) 39.4112 2.01120
\(385\) −3.02828 −0.154336
\(386\) 24.9838 1.27164
\(387\) −73.3371 −3.72793
\(388\) −1.13493 −0.0576173
\(389\) −2.36723 −0.120023 −0.0600117 0.998198i \(-0.519114\pi\)
−0.0600117 + 0.998198i \(0.519114\pi\)
\(390\) 22.3397 1.13122
\(391\) −1.99258 −0.100769
\(392\) −8.00223 −0.404174
\(393\) 2.28594 0.115310
\(394\) −1.59522 −0.0803661
\(395\) 31.1981 1.56975
\(396\) −4.50930 −0.226601
\(397\) −16.1503 −0.810559 −0.405279 0.914193i \(-0.632826\pi\)
−0.405279 + 0.914193i \(0.632826\pi\)
\(398\) 0.706702 0.0354238
\(399\) −0.653446 −0.0327132
\(400\) −11.6977 −0.584887
\(401\) 4.77242 0.238323 0.119162 0.992875i \(-0.461979\pi\)
0.119162 + 0.992875i \(0.461979\pi\)
\(402\) −45.8682 −2.28770
\(403\) 1.84257 0.0917851
\(404\) −4.97687 −0.247609
\(405\) −49.4721 −2.45829
\(406\) −22.9396 −1.13847
\(407\) 4.19083 0.207732
\(408\) 6.06583 0.300303
\(409\) 9.57331 0.473370 0.236685 0.971586i \(-0.423939\pi\)
0.236685 + 0.971586i \(0.423939\pi\)
\(410\) 0 0
\(411\) 58.8088 2.90083
\(412\) 13.0263 0.641759
\(413\) −13.1929 −0.649180
\(414\) −22.3823 −1.10003
\(415\) −17.5870 −0.863312
\(416\) −7.51813 −0.368606
\(417\) −19.9906 −0.978945
\(418\) −0.152705 −0.00746903
\(419\) 8.06247 0.393878 0.196939 0.980416i \(-0.436900\pi\)
0.196939 + 0.980416i \(0.436900\pi\)
\(420\) −12.9289 −0.630865
\(421\) 17.6577 0.860585 0.430292 0.902690i \(-0.358410\pi\)
0.430292 + 0.902690i \(0.358410\pi\)
\(422\) 38.4397 1.87121
\(423\) −43.9512 −2.13698
\(424\) −22.7237 −1.10356
\(425\) −2.47696 −0.120150
\(426\) 23.1804 1.12309
\(427\) −4.24694 −0.205524
\(428\) −8.94927 −0.432579
\(429\) −3.33280 −0.160909
\(430\) −49.1381 −2.36965
\(431\) −10.1535 −0.489075 −0.244537 0.969640i \(-0.578636\pi\)
−0.244537 + 0.969640i \(0.578636\pi\)
\(432\) 61.9160 2.97893
\(433\) 33.2213 1.59651 0.798256 0.602318i \(-0.205756\pi\)
0.798256 + 0.602318i \(0.205756\pi\)
\(434\) −3.33832 −0.160244
\(435\) −70.8440 −3.39671
\(436\) −2.21211 −0.105941
\(437\) −0.242119 −0.0115821
\(438\) −65.8882 −3.14826
\(439\) 36.8515 1.75883 0.879413 0.476060i \(-0.157935\pi\)
0.879413 + 0.476060i \(0.157935\pi\)
\(440\) 3.41579 0.162841
\(441\) −30.4913 −1.45197
\(442\) −2.76776 −0.131649
\(443\) 30.8429 1.46539 0.732694 0.680558i \(-0.238262\pi\)
0.732694 + 0.680558i \(0.238262\pi\)
\(444\) 17.8922 0.849127
\(445\) −7.81365 −0.370403
\(446\) 6.98763 0.330874
\(447\) −40.7452 −1.92718
\(448\) −2.49595 −0.117922
\(449\) −29.6350 −1.39856 −0.699282 0.714846i \(-0.746497\pi\)
−0.699282 + 0.714846i \(0.746497\pi\)
\(450\) −27.8233 −1.31161
\(451\) 0 0
\(452\) 7.20681 0.338980
\(453\) −4.40012 −0.206736
\(454\) −37.0160 −1.73725
\(455\) −6.66941 −0.312667
\(456\) 0.737063 0.0345161
\(457\) −7.59375 −0.355221 −0.177610 0.984101i \(-0.556837\pi\)
−0.177610 + 0.984101i \(0.556837\pi\)
\(458\) −16.9841 −0.793614
\(459\) 13.1105 0.611947
\(460\) −4.79050 −0.223358
\(461\) −25.2841 −1.17760 −0.588800 0.808279i \(-0.700399\pi\)
−0.588800 + 0.808279i \(0.700399\pi\)
\(462\) 6.03827 0.280926
\(463\) 14.7447 0.685246 0.342623 0.939473i \(-0.388685\pi\)
0.342623 + 0.939473i \(0.388685\pi\)
\(464\) 41.4512 1.92433
\(465\) −10.3097 −0.478101
\(466\) −2.20563 −0.102174
\(467\) −22.1233 −1.02374 −0.511872 0.859062i \(-0.671048\pi\)
−0.511872 + 0.859062i \(0.671048\pi\)
\(468\) −9.93114 −0.459067
\(469\) 13.6937 0.632317
\(470\) −29.4487 −1.35837
\(471\) 55.3445 2.55014
\(472\) 14.8811 0.684958
\(473\) 7.33078 0.337070
\(474\) −62.2077 −2.85729
\(475\) −0.300977 −0.0138098
\(476\) 1.60181 0.0734191
\(477\) −86.5851 −3.96446
\(478\) −12.1656 −0.556441
\(479\) 15.8848 0.725795 0.362898 0.931829i \(-0.381787\pi\)
0.362898 + 0.931829i \(0.381787\pi\)
\(480\) 42.0659 1.92004
\(481\) 9.22976 0.420841
\(482\) 18.4931 0.842337
\(483\) 9.57391 0.435628
\(484\) −9.87525 −0.448875
\(485\) 3.27579 0.148746
\(486\) 34.9129 1.58368
\(487\) −21.0380 −0.953323 −0.476662 0.879087i \(-0.658153\pi\)
−0.476662 + 0.879087i \(0.658153\pi\)
\(488\) 4.79039 0.216851
\(489\) 55.4928 2.50947
\(490\) −20.4301 −0.922938
\(491\) 0.0541126 0.00244207 0.00122103 0.999999i \(-0.499611\pi\)
0.00122103 + 0.999999i \(0.499611\pi\)
\(492\) 0 0
\(493\) 8.77717 0.395304
\(494\) −0.336312 −0.0151314
\(495\) 13.0154 0.584997
\(496\) 6.03226 0.270856
\(497\) −6.92039 −0.310422
\(498\) 35.0677 1.57142
\(499\) 32.5037 1.45507 0.727533 0.686073i \(-0.240667\pi\)
0.727533 + 0.686073i \(0.240667\pi\)
\(500\) 6.76230 0.302419
\(501\) −8.74887 −0.390871
\(502\) −9.64350 −0.430410
\(503\) 32.2851 1.43952 0.719760 0.694223i \(-0.244252\pi\)
0.719760 + 0.694223i \(0.244252\pi\)
\(504\) −20.3418 −0.906098
\(505\) 14.3649 0.639232
\(506\) 2.23734 0.0994619
\(507\) 33.6300 1.49356
\(508\) 5.85011 0.259557
\(509\) 28.8657 1.27945 0.639724 0.768604i \(-0.279049\pi\)
0.639724 + 0.768604i \(0.279049\pi\)
\(510\) 15.4864 0.685749
\(511\) 19.6706 0.870176
\(512\) −6.43357 −0.284326
\(513\) 1.59307 0.0703356
\(514\) −37.6872 −1.66231
\(515\) −37.5983 −1.65678
\(516\) 31.2978 1.37781
\(517\) 4.39337 0.193220
\(518\) −16.7222 −0.734731
\(519\) 36.9033 1.61987
\(520\) 7.52284 0.329898
\(521\) −17.5099 −0.767121 −0.383560 0.923516i \(-0.625302\pi\)
−0.383560 + 0.923516i \(0.625302\pi\)
\(522\) 98.5927 4.31529
\(523\) −7.28359 −0.318489 −0.159245 0.987239i \(-0.550906\pi\)
−0.159245 + 0.987239i \(0.550906\pi\)
\(524\) −0.680896 −0.0297451
\(525\) 11.9013 0.519414
\(526\) 10.0475 0.438092
\(527\) 1.27731 0.0556406
\(528\) −10.9110 −0.474840
\(529\) −19.4526 −0.845766
\(530\) −58.0147 −2.52000
\(531\) 56.7022 2.46066
\(532\) 0.194637 0.00843860
\(533\) 0 0
\(534\) 15.5801 0.674216
\(535\) 25.8306 1.11676
\(536\) −15.4460 −0.667165
\(537\) 13.4055 0.578490
\(538\) 23.5373 1.01477
\(539\) 3.04791 0.131283
\(540\) 31.5199 1.35640
\(541\) 30.4703 1.31002 0.655011 0.755620i \(-0.272664\pi\)
0.655011 + 0.755620i \(0.272664\pi\)
\(542\) −27.2790 −1.17173
\(543\) 14.6112 0.627026
\(544\) −5.21173 −0.223451
\(545\) 6.38491 0.273500
\(546\) 13.2985 0.569124
\(547\) 32.4922 1.38927 0.694634 0.719364i \(-0.255566\pi\)
0.694634 + 0.719364i \(0.255566\pi\)
\(548\) −17.5170 −0.748288
\(549\) 18.2530 0.779021
\(550\) 2.78122 0.118592
\(551\) 1.06652 0.0454352
\(552\) −10.7990 −0.459637
\(553\) 18.5718 0.789753
\(554\) −32.9078 −1.39812
\(555\) −51.6430 −2.19212
\(556\) 5.95447 0.252526
\(557\) −24.8381 −1.05242 −0.526211 0.850354i \(-0.676388\pi\)
−0.526211 + 0.850354i \(0.676388\pi\)
\(558\) 14.3479 0.607394
\(559\) 16.1451 0.682865
\(560\) −21.8345 −0.922675
\(561\) −2.31037 −0.0975439
\(562\) −2.48746 −0.104927
\(563\) 24.9154 1.05006 0.525029 0.851085i \(-0.324055\pi\)
0.525029 + 0.851085i \(0.324055\pi\)
\(564\) 18.7569 0.789810
\(565\) −20.8013 −0.875118
\(566\) 16.1541 0.679006
\(567\) −29.4500 −1.23679
\(568\) 7.80594 0.327530
\(569\) 14.0530 0.589131 0.294565 0.955631i \(-0.404825\pi\)
0.294565 + 0.955631i \(0.404825\pi\)
\(570\) 1.88176 0.0788182
\(571\) −30.7133 −1.28531 −0.642657 0.766154i \(-0.722168\pi\)
−0.642657 + 0.766154i \(0.722168\pi\)
\(572\) 0.992718 0.0415076
\(573\) −70.4962 −2.94502
\(574\) 0 0
\(575\) 4.40974 0.183899
\(576\) 10.7274 0.446975
\(577\) −0.149336 −0.00621693 −0.00310846 0.999995i \(-0.500989\pi\)
−0.00310846 + 0.999995i \(0.500989\pi\)
\(578\) 27.2239 1.13237
\(579\) 45.9306 1.90881
\(580\) 21.1018 0.876205
\(581\) −10.4693 −0.434339
\(582\) −6.53178 −0.270751
\(583\) 8.65505 0.358456
\(584\) −22.1877 −0.918133
\(585\) 28.6647 1.18514
\(586\) −1.85985 −0.0768299
\(587\) 16.0049 0.660593 0.330296 0.943877i \(-0.392851\pi\)
0.330296 + 0.943877i \(0.392851\pi\)
\(588\) 13.0127 0.536634
\(589\) 0.155207 0.00639519
\(590\) 37.9922 1.56411
\(591\) −2.93268 −0.120634
\(592\) 30.2166 1.24189
\(593\) 1.96819 0.0808239 0.0404120 0.999183i \(-0.487133\pi\)
0.0404120 + 0.999183i \(0.487133\pi\)
\(594\) −14.7210 −0.604009
\(595\) −4.62338 −0.189540
\(596\) 12.1365 0.497130
\(597\) 1.29921 0.0531731
\(598\) 4.92746 0.201499
\(599\) 18.6180 0.760712 0.380356 0.924840i \(-0.375801\pi\)
0.380356 + 0.924840i \(0.375801\pi\)
\(600\) −13.4242 −0.548040
\(601\) 10.0007 0.407939 0.203970 0.978977i \(-0.434616\pi\)
0.203970 + 0.978977i \(0.434616\pi\)
\(602\) −29.2512 −1.19219
\(603\) −58.8546 −2.39675
\(604\) 1.31063 0.0533289
\(605\) 28.5033 1.15882
\(606\) −28.6431 −1.16355
\(607\) 37.9289 1.53948 0.769742 0.638355i \(-0.220385\pi\)
0.769742 + 0.638355i \(0.220385\pi\)
\(608\) −0.633280 −0.0256829
\(609\) −42.1724 −1.70891
\(610\) 12.2301 0.495182
\(611\) 9.67583 0.391442
\(612\) −6.88449 −0.278289
\(613\) 27.0143 1.09110 0.545549 0.838079i \(-0.316321\pi\)
0.545549 + 0.838079i \(0.316321\pi\)
\(614\) 50.8222 2.05102
\(615\) 0 0
\(616\) 2.03337 0.0819269
\(617\) −10.0246 −0.403574 −0.201787 0.979429i \(-0.564675\pi\)
−0.201787 + 0.979429i \(0.564675\pi\)
\(618\) 74.9693 3.01571
\(619\) −33.3272 −1.33953 −0.669766 0.742572i \(-0.733606\pi\)
−0.669766 + 0.742572i \(0.733606\pi\)
\(620\) 3.07088 0.123329
\(621\) −23.3407 −0.936629
\(622\) 29.5662 1.18550
\(623\) −4.65135 −0.186353
\(624\) −24.0301 −0.961973
\(625\) −31.2248 −1.24899
\(626\) 2.78115 0.111157
\(627\) −0.280734 −0.0112115
\(628\) −16.4851 −0.657826
\(629\) 6.39827 0.255116
\(630\) −51.9338 −2.06909
\(631\) −10.5324 −0.419289 −0.209644 0.977778i \(-0.567231\pi\)
−0.209644 + 0.977778i \(0.567231\pi\)
\(632\) −20.9483 −0.833278
\(633\) 70.6680 2.80880
\(634\) 27.9306 1.10927
\(635\) −16.8854 −0.670077
\(636\) 36.9517 1.46523
\(637\) 6.71263 0.265964
\(638\) −9.85533 −0.390176
\(639\) 29.7433 1.17663
\(640\) 33.8832 1.33935
\(641\) −4.64277 −0.183379 −0.0916893 0.995788i \(-0.529227\pi\)
−0.0916893 + 0.995788i \(0.529227\pi\)
\(642\) −51.5052 −2.03275
\(643\) 30.0438 1.18481 0.592407 0.805639i \(-0.298178\pi\)
0.592407 + 0.805639i \(0.298178\pi\)
\(644\) −2.85171 −0.112373
\(645\) −90.3362 −3.55698
\(646\) −0.233139 −0.00917274
\(647\) 16.4853 0.648104 0.324052 0.946039i \(-0.394955\pi\)
0.324052 + 0.946039i \(0.394955\pi\)
\(648\) 33.2185 1.30495
\(649\) −5.66795 −0.222487
\(650\) 6.12529 0.240254
\(651\) −6.13721 −0.240536
\(652\) −16.5292 −0.647335
\(653\) 33.3959 1.30688 0.653441 0.756977i \(-0.273325\pi\)
0.653441 + 0.756977i \(0.273325\pi\)
\(654\) −12.7312 −0.497831
\(655\) 1.96530 0.0767905
\(656\) 0 0
\(657\) −84.5428 −3.29833
\(658\) −17.5304 −0.683406
\(659\) 4.77185 0.185885 0.0929425 0.995671i \(-0.470373\pi\)
0.0929425 + 0.995671i \(0.470373\pi\)
\(660\) −5.55452 −0.216210
\(661\) 12.9311 0.502962 0.251481 0.967862i \(-0.419082\pi\)
0.251481 + 0.967862i \(0.419082\pi\)
\(662\) −43.4650 −1.68931
\(663\) −5.08829 −0.197613
\(664\) 11.8090 0.458277
\(665\) −0.561789 −0.0217853
\(666\) 71.8709 2.78494
\(667\) −15.6260 −0.605042
\(668\) 2.60597 0.100828
\(669\) 12.8462 0.496661
\(670\) −39.4344 −1.52348
\(671\) −1.82458 −0.0704369
\(672\) 25.0412 0.965987
\(673\) −6.35874 −0.245111 −0.122556 0.992462i \(-0.539109\pi\)
−0.122556 + 0.992462i \(0.539109\pi\)
\(674\) −44.9883 −1.73289
\(675\) −29.0147 −1.11677
\(676\) −10.0171 −0.385274
\(677\) −34.9105 −1.34172 −0.670859 0.741585i \(-0.734074\pi\)
−0.670859 + 0.741585i \(0.734074\pi\)
\(678\) 41.4769 1.59291
\(679\) 1.95003 0.0748353
\(680\) 5.21500 0.199986
\(681\) −68.0508 −2.60771
\(682\) −1.43421 −0.0549189
\(683\) 25.0272 0.957641 0.478820 0.877913i \(-0.341065\pi\)
0.478820 + 0.877913i \(0.341065\pi\)
\(684\) −0.836538 −0.0319858
\(685\) 50.5599 1.93180
\(686\) −31.5166 −1.20331
\(687\) −31.2238 −1.19126
\(688\) 52.8562 2.01512
\(689\) 19.0616 0.726191
\(690\) −27.5704 −1.04959
\(691\) −26.6246 −1.01285 −0.506424 0.862285i \(-0.669033\pi\)
−0.506424 + 0.862285i \(0.669033\pi\)
\(692\) −10.9921 −0.417858
\(693\) 7.74786 0.294317
\(694\) 28.5724 1.08459
\(695\) −17.1866 −0.651925
\(696\) 47.5689 1.80310
\(697\) 0 0
\(698\) −14.8295 −0.561306
\(699\) −4.05486 −0.153369
\(700\) −3.54495 −0.133986
\(701\) −20.7971 −0.785497 −0.392749 0.919646i \(-0.628476\pi\)
−0.392749 + 0.919646i \(0.628476\pi\)
\(702\) −32.4211 −1.22365
\(703\) 0.777457 0.0293224
\(704\) −1.07231 −0.0404143
\(705\) −54.1389 −2.03899
\(706\) 23.6446 0.889874
\(707\) 8.55125 0.321603
\(708\) −24.1986 −0.909439
\(709\) 8.79271 0.330217 0.165109 0.986275i \(-0.447203\pi\)
0.165109 + 0.986275i \(0.447203\pi\)
\(710\) 19.9290 0.747921
\(711\) −79.8202 −2.99349
\(712\) 5.24655 0.196623
\(713\) −2.27400 −0.0851620
\(714\) 9.21882 0.345006
\(715\) −2.86532 −0.107157
\(716\) −3.99300 −0.149225
\(717\) −22.3654 −0.835251
\(718\) −31.0541 −1.15893
\(719\) −13.0553 −0.486880 −0.243440 0.969916i \(-0.578276\pi\)
−0.243440 + 0.969916i \(0.578276\pi\)
\(720\) 93.8430 3.49732
\(721\) −22.3817 −0.833538
\(722\) 32.5428 1.21112
\(723\) 33.9980 1.26440
\(724\) −4.35213 −0.161746
\(725\) −19.4246 −0.721412
\(726\) −56.8344 −2.10932
\(727\) −27.6930 −1.02708 −0.513538 0.858067i \(-0.671665\pi\)
−0.513538 + 0.858067i \(0.671665\pi\)
\(728\) 4.47824 0.165975
\(729\) 9.40774 0.348435
\(730\) −56.6463 −2.09657
\(731\) 11.1921 0.413956
\(732\) −7.78979 −0.287919
\(733\) 24.4503 0.903091 0.451545 0.892248i \(-0.350873\pi\)
0.451545 + 0.892248i \(0.350873\pi\)
\(734\) 39.5870 1.46118
\(735\) −37.5590 −1.38538
\(736\) 9.27845 0.342008
\(737\) 5.88311 0.216707
\(738\) 0 0
\(739\) −35.1342 −1.29243 −0.646215 0.763155i \(-0.723649\pi\)
−0.646215 + 0.763155i \(0.723649\pi\)
\(740\) 15.3825 0.565473
\(741\) −0.618281 −0.0227131
\(742\) −34.5353 −1.26783
\(743\) 5.72093 0.209881 0.104940 0.994479i \(-0.466535\pi\)
0.104940 + 0.994479i \(0.466535\pi\)
\(744\) 6.92254 0.253793
\(745\) −35.0300 −1.28340
\(746\) 24.2401 0.887494
\(747\) 44.9963 1.64633
\(748\) 0.688174 0.0251621
\(749\) 15.3766 0.561849
\(750\) 38.9186 1.42111
\(751\) −11.6727 −0.425943 −0.212972 0.977058i \(-0.568314\pi\)
−0.212972 + 0.977058i \(0.568314\pi\)
\(752\) 31.6770 1.15514
\(753\) −17.7287 −0.646071
\(754\) −21.7051 −0.790453
\(755\) −3.78293 −0.137675
\(756\) 18.7634 0.682417
\(757\) 38.3826 1.39504 0.697519 0.716566i \(-0.254287\pi\)
0.697519 + 0.716566i \(0.254287\pi\)
\(758\) 63.7275 2.31469
\(759\) 4.11316 0.149298
\(760\) 0.633677 0.0229859
\(761\) 48.2836 1.75028 0.875140 0.483871i \(-0.160769\pi\)
0.875140 + 0.483871i \(0.160769\pi\)
\(762\) 33.6688 1.21969
\(763\) 3.80085 0.137600
\(764\) 20.9982 0.759689
\(765\) 19.8710 0.718436
\(766\) −11.8658 −0.428729
\(767\) −12.4829 −0.450733
\(768\) −57.8077 −2.08596
\(769\) −3.92461 −0.141525 −0.0707625 0.997493i \(-0.522543\pi\)
−0.0707625 + 0.997493i \(0.522543\pi\)
\(770\) 5.19130 0.187082
\(771\) −69.2847 −2.49523
\(772\) −13.6810 −0.492391
\(773\) 33.3650 1.20005 0.600027 0.799979i \(-0.295156\pi\)
0.600027 + 0.799979i \(0.295156\pi\)
\(774\) 125.720 4.51890
\(775\) −2.82680 −0.101542
\(776\) −2.19956 −0.0789596
\(777\) −30.7423 −1.10287
\(778\) 4.05808 0.145489
\(779\) 0 0
\(780\) −12.2331 −0.438016
\(781\) −2.97315 −0.106388
\(782\) 3.41582 0.122149
\(783\) 102.814 3.67428
\(784\) 21.9760 0.784856
\(785\) 47.5815 1.69826
\(786\) −3.91872 −0.139776
\(787\) 11.2780 0.402018 0.201009 0.979589i \(-0.435578\pi\)
0.201009 + 0.979589i \(0.435578\pi\)
\(788\) 0.873536 0.0311184
\(789\) 18.4715 0.657602
\(790\) −53.4820 −1.90280
\(791\) −12.3827 −0.440279
\(792\) −8.73929 −0.310537
\(793\) −4.01839 −0.142697
\(794\) 27.6859 0.982538
\(795\) −106.655 −3.78266
\(796\) −0.386986 −0.0137164
\(797\) −28.5583 −1.01159 −0.505793 0.862655i \(-0.668800\pi\)
−0.505793 + 0.862655i \(0.668800\pi\)
\(798\) 1.12018 0.0396541
\(799\) 6.70750 0.237294
\(800\) 11.5340 0.407788
\(801\) 19.9912 0.706354
\(802\) −8.18122 −0.288889
\(803\) 8.45090 0.298226
\(804\) 25.1172 0.885816
\(805\) 8.23101 0.290105
\(806\) −3.15867 −0.111259
\(807\) 43.2713 1.52322
\(808\) −9.64548 −0.339327
\(809\) 10.6581 0.374717 0.187359 0.982292i \(-0.440007\pi\)
0.187359 + 0.982292i \(0.440007\pi\)
\(810\) 84.8086 2.97987
\(811\) −33.8780 −1.18962 −0.594809 0.803867i \(-0.702772\pi\)
−0.594809 + 0.803867i \(0.702772\pi\)
\(812\) 12.5616 0.440826
\(813\) −50.1500 −1.75884
\(814\) −7.18422 −0.251807
\(815\) 47.7090 1.67117
\(816\) −16.6582 −0.583153
\(817\) 1.35996 0.0475791
\(818\) −16.4113 −0.573806
\(819\) 17.0637 0.596252
\(820\) 0 0
\(821\) 50.1853 1.75148 0.875739 0.482785i \(-0.160375\pi\)
0.875739 + 0.482785i \(0.160375\pi\)
\(822\) −100.814 −3.51630
\(823\) 45.0545 1.57050 0.785251 0.619178i \(-0.212534\pi\)
0.785251 + 0.619178i \(0.212534\pi\)
\(824\) 25.2457 0.879476
\(825\) 5.11304 0.178013
\(826\) 22.6162 0.786919
\(827\) −3.57914 −0.124459 −0.0622295 0.998062i \(-0.519821\pi\)
−0.0622295 + 0.998062i \(0.519821\pi\)
\(828\) 12.2565 0.425942
\(829\) 18.5904 0.645672 0.322836 0.946455i \(-0.395364\pi\)
0.322836 + 0.946455i \(0.395364\pi\)
\(830\) 30.1489 1.04648
\(831\) −60.4981 −2.09866
\(832\) −2.36163 −0.0818747
\(833\) 4.65334 0.161229
\(834\) 34.2694 1.18665
\(835\) −7.52170 −0.260299
\(836\) 0.0836204 0.00289207
\(837\) 14.9622 0.517169
\(838\) −13.8213 −0.477448
\(839\) −36.2824 −1.25261 −0.626304 0.779579i \(-0.715433\pi\)
−0.626304 + 0.779579i \(0.715433\pi\)
\(840\) −25.0570 −0.864547
\(841\) 39.8315 1.37350
\(842\) −30.2701 −1.04318
\(843\) −4.57297 −0.157502
\(844\) −21.0494 −0.724550
\(845\) 28.9128 0.994631
\(846\) 75.3444 2.59039
\(847\) 16.9676 0.583014
\(848\) 62.4045 2.14298
\(849\) 29.6979 1.01923
\(850\) 4.24618 0.145643
\(851\) −11.3909 −0.390473
\(852\) −12.6935 −0.434871
\(853\) −16.0614 −0.549933 −0.274966 0.961454i \(-0.588667\pi\)
−0.274966 + 0.961454i \(0.588667\pi\)
\(854\) 7.28041 0.249130
\(855\) 2.41453 0.0825752
\(856\) −17.3442 −0.592814
\(857\) −15.8027 −0.539809 −0.269904 0.962887i \(-0.586992\pi\)
−0.269904 + 0.962887i \(0.586992\pi\)
\(858\) 5.71333 0.195050
\(859\) 20.2212 0.689937 0.344969 0.938614i \(-0.387890\pi\)
0.344969 + 0.938614i \(0.387890\pi\)
\(860\) 26.9078 0.917548
\(861\) 0 0
\(862\) 17.4058 0.592843
\(863\) −10.7247 −0.365074 −0.182537 0.983199i \(-0.558431\pi\)
−0.182537 + 0.983199i \(0.558431\pi\)
\(864\) −61.0492 −2.07694
\(865\) 31.7270 1.07875
\(866\) −56.9503 −1.93525
\(867\) 50.0489 1.69975
\(868\) 1.82805 0.0620480
\(869\) 7.97883 0.270663
\(870\) 121.446 4.11740
\(871\) 12.9568 0.439024
\(872\) −4.28721 −0.145183
\(873\) −8.38109 −0.283657
\(874\) 0.415058 0.0140396
\(875\) −11.6190 −0.392792
\(876\) 36.0801 1.21903
\(877\) −12.0861 −0.408120 −0.204060 0.978958i \(-0.565414\pi\)
−0.204060 + 0.978958i \(0.565414\pi\)
\(878\) −63.1735 −2.13200
\(879\) −3.41918 −0.115326
\(880\) −9.38055 −0.316218
\(881\) −9.42951 −0.317688 −0.158844 0.987304i \(-0.550777\pi\)
−0.158844 + 0.987304i \(0.550777\pi\)
\(882\) 52.2704 1.76003
\(883\) 23.5030 0.790938 0.395469 0.918479i \(-0.370582\pi\)
0.395469 + 0.918479i \(0.370582\pi\)
\(884\) 1.51561 0.0509756
\(885\) 69.8454 2.34783
\(886\) −52.8730 −1.77630
\(887\) −48.5055 −1.62866 −0.814328 0.580405i \(-0.802894\pi\)
−0.814328 + 0.580405i \(0.802894\pi\)
\(888\) 34.6762 1.16366
\(889\) −10.0516 −0.337121
\(890\) 13.3947 0.448992
\(891\) −12.6524 −0.423870
\(892\) −3.82639 −0.128117
\(893\) 0.815032 0.0272740
\(894\) 69.8484 2.33608
\(895\) 11.5252 0.385243
\(896\) 20.1702 0.673838
\(897\) 9.05870 0.302461
\(898\) 50.8025 1.69530
\(899\) 10.0168 0.334080
\(900\) 15.2359 0.507864
\(901\) 13.2140 0.440221
\(902\) 0 0
\(903\) −53.7758 −1.78955
\(904\) 13.9672 0.464544
\(905\) 12.5617 0.417566
\(906\) 7.54300 0.250599
\(907\) −8.58540 −0.285074 −0.142537 0.989790i \(-0.545526\pi\)
−0.142537 + 0.989790i \(0.545526\pi\)
\(908\) 20.2698 0.672677
\(909\) −36.7527 −1.21901
\(910\) 11.4332 0.379006
\(911\) −21.0427 −0.697176 −0.348588 0.937276i \(-0.613339\pi\)
−0.348588 + 0.937276i \(0.613339\pi\)
\(912\) −2.02414 −0.0670261
\(913\) −4.49783 −0.148856
\(914\) 13.0178 0.430589
\(915\) 22.4840 0.743298
\(916\) 9.30040 0.307294
\(917\) 1.16991 0.0386339
\(918\) −22.4750 −0.741785
\(919\) −47.8628 −1.57885 −0.789424 0.613848i \(-0.789621\pi\)
−0.789424 + 0.613848i \(0.789621\pi\)
\(920\) −9.28427 −0.306093
\(921\) 93.4322 3.07869
\(922\) 43.3439 1.42746
\(923\) −6.54797 −0.215529
\(924\) −3.30653 −0.108777
\(925\) −14.1599 −0.465575
\(926\) −25.2765 −0.830637
\(927\) 96.1950 3.15946
\(928\) −40.8710 −1.34165
\(929\) −52.1656 −1.71150 −0.855748 0.517392i \(-0.826903\pi\)
−0.855748 + 0.517392i \(0.826903\pi\)
\(930\) 17.6736 0.579541
\(931\) 0.565430 0.0185312
\(932\) 1.20779 0.0395626
\(933\) 54.3549 1.77950
\(934\) 37.9253 1.24096
\(935\) −1.98630 −0.0649591
\(936\) −19.2472 −0.629113
\(937\) −15.2606 −0.498543 −0.249271 0.968434i \(-0.580191\pi\)
−0.249271 + 0.968434i \(0.580191\pi\)
\(938\) −23.4747 −0.766478
\(939\) 5.11291 0.166854
\(940\) 16.1260 0.525971
\(941\) −55.3336 −1.80382 −0.901912 0.431919i \(-0.857837\pi\)
−0.901912 + 0.431919i \(0.857837\pi\)
\(942\) −94.8754 −3.09121
\(943\) 0 0
\(944\) −40.8669 −1.33011
\(945\) −54.1574 −1.76174
\(946\) −12.5669 −0.408587
\(947\) 1.57167 0.0510723 0.0255362 0.999674i \(-0.491871\pi\)
0.0255362 + 0.999674i \(0.491871\pi\)
\(948\) 34.0646 1.10637
\(949\) 18.6120 0.604172
\(950\) 0.515956 0.0167398
\(951\) 51.3480 1.66507
\(952\) 3.10441 0.100615
\(953\) 42.6660 1.38209 0.691044 0.722812i \(-0.257151\pi\)
0.691044 + 0.722812i \(0.257151\pi\)
\(954\) 148.430 4.80561
\(955\) −60.6080 −1.96123
\(956\) 6.66182 0.215459
\(957\) −18.1182 −0.585678
\(958\) −27.2309 −0.879789
\(959\) 30.0976 0.971902
\(960\) 13.2139 0.426478
\(961\) −29.5423 −0.952977
\(962\) −15.8223 −0.510132
\(963\) −66.0876 −2.12964
\(964\) −10.1267 −0.326160
\(965\) 39.4881 1.27117
\(966\) −16.4123 −0.528057
\(967\) 4.13912 0.133105 0.0665525 0.997783i \(-0.478800\pi\)
0.0665525 + 0.997783i \(0.478800\pi\)
\(968\) −19.1388 −0.615145
\(969\) −0.428606 −0.0137688
\(970\) −5.61559 −0.180306
\(971\) −16.0357 −0.514611 −0.257306 0.966330i \(-0.582835\pi\)
−0.257306 + 0.966330i \(0.582835\pi\)
\(972\) −19.1181 −0.613214
\(973\) −10.2309 −0.327989
\(974\) 36.0649 1.15559
\(975\) 11.2608 0.360635
\(976\) −13.1555 −0.421098
\(977\) −42.7852 −1.36882 −0.684410 0.729098i \(-0.739940\pi\)
−0.684410 + 0.729098i \(0.739940\pi\)
\(978\) −95.1297 −3.04191
\(979\) −1.99832 −0.0638666
\(980\) 11.1874 0.357369
\(981\) −16.3358 −0.521561
\(982\) −0.0927636 −0.00296021
\(983\) 31.6046 1.00803 0.504015 0.863695i \(-0.331856\pi\)
0.504015 + 0.863695i \(0.331856\pi\)
\(984\) 0 0
\(985\) −2.52132 −0.0803360
\(986\) −15.0465 −0.479177
\(987\) −32.2281 −1.02583
\(988\) 0.184163 0.00585901
\(989\) −19.9254 −0.633590
\(990\) −22.3119 −0.709117
\(991\) 27.4148 0.870858 0.435429 0.900223i \(-0.356597\pi\)
0.435429 + 0.900223i \(0.356597\pi\)
\(992\) −5.94781 −0.188843
\(993\) −79.9066 −2.53576
\(994\) 11.8634 0.376285
\(995\) 1.11697 0.0354105
\(996\) −19.2029 −0.608468
\(997\) 24.8267 0.786270 0.393135 0.919481i \(-0.371391\pi\)
0.393135 + 0.919481i \(0.371391\pi\)
\(998\) −55.7202 −1.76379
\(999\) 74.9481 2.37125
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 1681.2.a.m.1.3 24
41.13 odd 40 41.2.g.a.5.1 24
41.19 odd 40 41.2.g.a.33.1 yes 24
41.40 even 2 inner 1681.2.a.m.1.4 24
123.95 even 40 369.2.u.a.46.3 24
123.101 even 40 369.2.u.a.361.3 24
164.19 even 40 656.2.bs.d.33.1 24
164.95 even 40 656.2.bs.d.497.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
41.2.g.a.5.1 24 41.13 odd 40
41.2.g.a.33.1 yes 24 41.19 odd 40
369.2.u.a.46.3 24 123.95 even 40
369.2.u.a.361.3 24 123.101 even 40
656.2.bs.d.33.1 24 164.19 even 40
656.2.bs.d.497.1 24 164.95 even 40
1681.2.a.m.1.3 24 1.1 even 1 trivial
1681.2.a.m.1.4 24 41.40 even 2 inner