Properties

Label 5225.2.a.o.1.6
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $8$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 28x^{4} - 17x^{3} - 28x^{2} + 6x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.865980\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.86598 q^{2} -2.01288 q^{3} +1.48188 q^{4} -3.75600 q^{6} +2.04136 q^{7} -0.966800 q^{8} +1.05170 q^{9} +O(q^{10})\) \(q+1.86598 q^{2} -2.01288 q^{3} +1.48188 q^{4} -3.75600 q^{6} +2.04136 q^{7} -0.966800 q^{8} +1.05170 q^{9} +1.00000 q^{11} -2.98285 q^{12} +4.91831 q^{13} +3.80913 q^{14} -4.76779 q^{16} +6.79891 q^{17} +1.96245 q^{18} -1.00000 q^{19} -4.10901 q^{21} +1.86598 q^{22} -1.43601 q^{23} +1.94606 q^{24} +9.17747 q^{26} +3.92170 q^{27} +3.02505 q^{28} -0.704749 q^{29} -4.21862 q^{31} -6.96300 q^{32} -2.01288 q^{33} +12.6866 q^{34} +1.55849 q^{36} +1.40495 q^{37} -1.86598 q^{38} -9.89999 q^{39} -9.38658 q^{41} -7.66733 q^{42} -3.80388 q^{43} +1.48188 q^{44} -2.67956 q^{46} +7.13622 q^{47} +9.59701 q^{48} -2.83286 q^{49} -13.6854 q^{51} +7.28835 q^{52} -2.65027 q^{53} +7.31782 q^{54} -1.97358 q^{56} +2.01288 q^{57} -1.31505 q^{58} +13.3502 q^{59} +7.58071 q^{61} -7.87185 q^{62} +2.14689 q^{63} -3.45724 q^{64} -3.75600 q^{66} -10.5714 q^{67} +10.0752 q^{68} +2.89051 q^{69} +5.75745 q^{71} -1.01678 q^{72} +5.49519 q^{73} +2.62161 q^{74} -1.48188 q^{76} +2.04136 q^{77} -18.4732 q^{78} +7.12218 q^{79} -11.0490 q^{81} -17.5152 q^{82} +10.8625 q^{83} -6.08907 q^{84} -7.09797 q^{86} +1.41858 q^{87} -0.966800 q^{88} -3.26592 q^{89} +10.0400 q^{91} -2.12799 q^{92} +8.49158 q^{93} +13.3160 q^{94} +14.0157 q^{96} +1.64413 q^{97} -5.28607 q^{98} +1.05170 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{2} + 7 q^{3} + 10 q^{4} + 11 q^{7} + 18 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{2} + 7 q^{3} + 10 q^{4} + 11 q^{7} + 18 q^{8} + 11 q^{9} + 8 q^{11} + 7 q^{12} + 17 q^{13} + 12 q^{14} + 18 q^{16} + 9 q^{17} + 2 q^{18} - 8 q^{19} + q^{21} + 6 q^{22} + 8 q^{23} + q^{24} + 10 q^{26} + 34 q^{27} + 22 q^{28} - 3 q^{29} - q^{31} + 37 q^{32} + 7 q^{33} - 8 q^{34} + 30 q^{36} + 17 q^{37} - 6 q^{38} + 14 q^{39} - 5 q^{41} - 15 q^{42} + 21 q^{43} + 10 q^{44} - 2 q^{46} + 8 q^{47} - 10 q^{48} + 19 q^{49} - 16 q^{51} - 9 q^{52} + 19 q^{53} - 3 q^{54} + 24 q^{56} - 7 q^{57} - 37 q^{58} - 33 q^{59} - q^{61} + 42 q^{62} + 20 q^{63} + 48 q^{64} + 18 q^{67} + 37 q^{68} + 16 q^{69} - 18 q^{71} - 13 q^{72} + 18 q^{73} + 15 q^{74} - 10 q^{76} + 11 q^{77} + 51 q^{78} - 5 q^{79} + 32 q^{81} - 12 q^{82} + 33 q^{83} - 51 q^{84} - 16 q^{86} + 26 q^{87} + 18 q^{88} - 20 q^{89} + 6 q^{91} + 3 q^{92} - 18 q^{93} + 30 q^{94} + 21 q^{96} + 69 q^{98} + 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.86598 1.31945 0.659723 0.751508i \(-0.270673\pi\)
0.659723 + 0.751508i \(0.270673\pi\)
\(3\) −2.01288 −1.16214 −0.581069 0.813854i \(-0.697365\pi\)
−0.581069 + 0.813854i \(0.697365\pi\)
\(4\) 1.48188 0.740940
\(5\) 0 0
\(6\) −3.75600 −1.53338
\(7\) 2.04136 0.771560 0.385780 0.922591i \(-0.373932\pi\)
0.385780 + 0.922591i \(0.373932\pi\)
\(8\) −0.966800 −0.341815
\(9\) 1.05170 0.350566
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −2.98285 −0.861075
\(13\) 4.91831 1.36409 0.682047 0.731308i \(-0.261090\pi\)
0.682047 + 0.731308i \(0.261090\pi\)
\(14\) 3.80913 1.01803
\(15\) 0 0
\(16\) −4.76779 −1.19195
\(17\) 6.79891 1.64898 0.824489 0.565878i \(-0.191463\pi\)
0.824489 + 0.565878i \(0.191463\pi\)
\(18\) 1.96245 0.462554
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −4.10901 −0.896660
\(22\) 1.86598 0.397828
\(23\) −1.43601 −0.299428 −0.149714 0.988729i \(-0.547835\pi\)
−0.149714 + 0.988729i \(0.547835\pi\)
\(24\) 1.94606 0.397237
\(25\) 0 0
\(26\) 9.17747 1.79985
\(27\) 3.92170 0.754732
\(28\) 3.02505 0.571680
\(29\) −0.704749 −0.130869 −0.0654343 0.997857i \(-0.520843\pi\)
−0.0654343 + 0.997857i \(0.520843\pi\)
\(30\) 0 0
\(31\) −4.21862 −0.757686 −0.378843 0.925461i \(-0.623678\pi\)
−0.378843 + 0.925461i \(0.623678\pi\)
\(32\) −6.96300 −1.23090
\(33\) −2.01288 −0.350398
\(34\) 12.6866 2.17574
\(35\) 0 0
\(36\) 1.55849 0.259749
\(37\) 1.40495 0.230973 0.115486 0.993309i \(-0.463157\pi\)
0.115486 + 0.993309i \(0.463157\pi\)
\(38\) −1.86598 −0.302702
\(39\) −9.89999 −1.58527
\(40\) 0 0
\(41\) −9.38658 −1.46594 −0.732969 0.680262i \(-0.761866\pi\)
−0.732969 + 0.680262i \(0.761866\pi\)
\(42\) −7.66733 −1.18310
\(43\) −3.80388 −0.580087 −0.290043 0.957014i \(-0.593670\pi\)
−0.290043 + 0.957014i \(0.593670\pi\)
\(44\) 1.48188 0.223402
\(45\) 0 0
\(46\) −2.67956 −0.395079
\(47\) 7.13622 1.04093 0.520463 0.853885i \(-0.325760\pi\)
0.520463 + 0.853885i \(0.325760\pi\)
\(48\) 9.59701 1.38521
\(49\) −2.83286 −0.404695
\(50\) 0 0
\(51\) −13.6854 −1.91634
\(52\) 7.28835 1.01071
\(53\) −2.65027 −0.364043 −0.182021 0.983295i \(-0.558264\pi\)
−0.182021 + 0.983295i \(0.558264\pi\)
\(54\) 7.31782 0.995829
\(55\) 0 0
\(56\) −1.97358 −0.263731
\(57\) 2.01288 0.266613
\(58\) −1.31505 −0.172674
\(59\) 13.3502 1.73805 0.869023 0.494772i \(-0.164748\pi\)
0.869023 + 0.494772i \(0.164748\pi\)
\(60\) 0 0
\(61\) 7.58071 0.970610 0.485305 0.874345i \(-0.338709\pi\)
0.485305 + 0.874345i \(0.338709\pi\)
\(62\) −7.87185 −0.999726
\(63\) 2.14689 0.270483
\(64\) −3.45724 −0.432155
\(65\) 0 0
\(66\) −3.75600 −0.462332
\(67\) −10.5714 −1.29150 −0.645751 0.763548i \(-0.723455\pi\)
−0.645751 + 0.763548i \(0.723455\pi\)
\(68\) 10.0752 1.22179
\(69\) 2.89051 0.347977
\(70\) 0 0
\(71\) 5.75745 0.683283 0.341642 0.939830i \(-0.389017\pi\)
0.341642 + 0.939830i \(0.389017\pi\)
\(72\) −1.01678 −0.119829
\(73\) 5.49519 0.643163 0.321582 0.946882i \(-0.395786\pi\)
0.321582 + 0.946882i \(0.395786\pi\)
\(74\) 2.62161 0.304756
\(75\) 0 0
\(76\) −1.48188 −0.169983
\(77\) 2.04136 0.232634
\(78\) −18.4732 −2.09168
\(79\) 7.12218 0.801307 0.400654 0.916230i \(-0.368783\pi\)
0.400654 + 0.916230i \(0.368783\pi\)
\(80\) 0 0
\(81\) −11.0490 −1.22767
\(82\) −17.5152 −1.93423
\(83\) 10.8625 1.19231 0.596157 0.802868i \(-0.296694\pi\)
0.596157 + 0.802868i \(0.296694\pi\)
\(84\) −6.08907 −0.664371
\(85\) 0 0
\(86\) −7.09797 −0.765393
\(87\) 1.41858 0.152087
\(88\) −0.966800 −0.103061
\(89\) −3.26592 −0.346187 −0.173094 0.984905i \(-0.555376\pi\)
−0.173094 + 0.984905i \(0.555376\pi\)
\(90\) 0 0
\(91\) 10.0400 1.05248
\(92\) −2.12799 −0.221858
\(93\) 8.49158 0.880536
\(94\) 13.3160 1.37345
\(95\) 0 0
\(96\) 14.0157 1.43047
\(97\) 1.64413 0.166936 0.0834680 0.996510i \(-0.473400\pi\)
0.0834680 + 0.996510i \(0.473400\pi\)
\(98\) −5.28607 −0.533974
\(99\) 1.05170 0.105700
\(100\) 0 0
\(101\) −2.89820 −0.288382 −0.144191 0.989550i \(-0.546058\pi\)
−0.144191 + 0.989550i \(0.546058\pi\)
\(102\) −25.5367 −2.52851
\(103\) 12.7793 1.25918 0.629590 0.776927i \(-0.283223\pi\)
0.629590 + 0.776927i \(0.283223\pi\)
\(104\) −4.75502 −0.466268
\(105\) 0 0
\(106\) −4.94535 −0.480335
\(107\) 17.2758 1.67012 0.835059 0.550160i \(-0.185433\pi\)
0.835059 + 0.550160i \(0.185433\pi\)
\(108\) 5.81150 0.559211
\(109\) 6.56114 0.628444 0.314222 0.949350i \(-0.398256\pi\)
0.314222 + 0.949350i \(0.398256\pi\)
\(110\) 0 0
\(111\) −2.82800 −0.268422
\(112\) −9.73276 −0.919659
\(113\) 1.66685 0.156804 0.0784018 0.996922i \(-0.475018\pi\)
0.0784018 + 0.996922i \(0.475018\pi\)
\(114\) 3.75600 0.351782
\(115\) 0 0
\(116\) −1.04435 −0.0969658
\(117\) 5.17258 0.478205
\(118\) 24.9112 2.29326
\(119\) 13.8790 1.27229
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 14.1455 1.28067
\(123\) 18.8941 1.70362
\(124\) −6.25149 −0.561400
\(125\) 0 0
\(126\) 4.00606 0.356888
\(127\) 5.24891 0.465766 0.232883 0.972505i \(-0.425184\pi\)
0.232883 + 0.972505i \(0.425184\pi\)
\(128\) 7.47487 0.660691
\(129\) 7.65677 0.674141
\(130\) 0 0
\(131\) −9.05742 −0.791350 −0.395675 0.918391i \(-0.629489\pi\)
−0.395675 + 0.918391i \(0.629489\pi\)
\(132\) −2.98285 −0.259624
\(133\) −2.04136 −0.177008
\(134\) −19.7260 −1.70407
\(135\) 0 0
\(136\) −6.57319 −0.563646
\(137\) 15.5957 1.33243 0.666217 0.745758i \(-0.267912\pi\)
0.666217 + 0.745758i \(0.267912\pi\)
\(138\) 5.39364 0.459137
\(139\) 9.26096 0.785505 0.392752 0.919644i \(-0.371523\pi\)
0.392752 + 0.919644i \(0.371523\pi\)
\(140\) 0 0
\(141\) −14.3644 −1.20970
\(142\) 10.7433 0.901556
\(143\) 4.91831 0.411290
\(144\) −5.01428 −0.417857
\(145\) 0 0
\(146\) 10.2539 0.848620
\(147\) 5.70223 0.470312
\(148\) 2.08197 0.171137
\(149\) 5.22633 0.428158 0.214079 0.976816i \(-0.431325\pi\)
0.214079 + 0.976816i \(0.431325\pi\)
\(150\) 0 0
\(151\) 20.7869 1.69161 0.845805 0.533492i \(-0.179120\pi\)
0.845805 + 0.533492i \(0.179120\pi\)
\(152\) 0.966800 0.0784178
\(153\) 7.15041 0.578076
\(154\) 3.80913 0.306948
\(155\) 0 0
\(156\) −14.6706 −1.17459
\(157\) −11.2308 −0.896317 −0.448158 0.893954i \(-0.647920\pi\)
−0.448158 + 0.893954i \(0.647920\pi\)
\(158\) 13.2898 1.05728
\(159\) 5.33469 0.423068
\(160\) 0 0
\(161\) −2.93140 −0.231027
\(162\) −20.6173 −1.61984
\(163\) −4.43995 −0.347763 −0.173882 0.984767i \(-0.555631\pi\)
−0.173882 + 0.984767i \(0.555631\pi\)
\(164\) −13.9098 −1.08617
\(165\) 0 0
\(166\) 20.2692 1.57319
\(167\) 14.1852 1.09768 0.548841 0.835927i \(-0.315069\pi\)
0.548841 + 0.835927i \(0.315069\pi\)
\(168\) 3.97259 0.306492
\(169\) 11.1898 0.860753
\(170\) 0 0
\(171\) −1.05170 −0.0804254
\(172\) −5.63690 −0.429810
\(173\) 5.44057 0.413639 0.206819 0.978379i \(-0.433689\pi\)
0.206819 + 0.978379i \(0.433689\pi\)
\(174\) 2.64704 0.200671
\(175\) 0 0
\(176\) −4.76779 −0.359386
\(177\) −26.8724 −2.01985
\(178\) −6.09415 −0.456776
\(179\) −8.45415 −0.631893 −0.315946 0.948777i \(-0.602322\pi\)
−0.315946 + 0.948777i \(0.602322\pi\)
\(180\) 0 0
\(181\) −13.6320 −1.01326 −0.506629 0.862164i \(-0.669109\pi\)
−0.506629 + 0.862164i \(0.669109\pi\)
\(182\) 18.7345 1.38869
\(183\) −15.2591 −1.12798
\(184\) 1.38833 0.102349
\(185\) 0 0
\(186\) 15.8451 1.16182
\(187\) 6.79891 0.497186
\(188\) 10.5750 0.771263
\(189\) 8.00559 0.582321
\(190\) 0 0
\(191\) −12.4905 −0.903778 −0.451889 0.892074i \(-0.649250\pi\)
−0.451889 + 0.892074i \(0.649250\pi\)
\(192\) 6.95902 0.502224
\(193\) 9.94947 0.716179 0.358089 0.933687i \(-0.383428\pi\)
0.358089 + 0.933687i \(0.383428\pi\)
\(194\) 3.06791 0.220263
\(195\) 0 0
\(196\) −4.19797 −0.299855
\(197\) −24.3654 −1.73596 −0.867982 0.496595i \(-0.834583\pi\)
−0.867982 + 0.496595i \(0.834583\pi\)
\(198\) 1.96245 0.139465
\(199\) −11.5042 −0.815509 −0.407754 0.913092i \(-0.633688\pi\)
−0.407754 + 0.913092i \(0.633688\pi\)
\(200\) 0 0
\(201\) 21.2790 1.50090
\(202\) −5.40798 −0.380504
\(203\) −1.43864 −0.100973
\(204\) −20.2802 −1.41989
\(205\) 0 0
\(206\) 23.8459 1.66142
\(207\) −1.51025 −0.104969
\(208\) −23.4495 −1.62593
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 10.8040 0.743778 0.371889 0.928277i \(-0.378710\pi\)
0.371889 + 0.928277i \(0.378710\pi\)
\(212\) −3.92739 −0.269734
\(213\) −11.5891 −0.794070
\(214\) 32.2364 2.20363
\(215\) 0 0
\(216\) −3.79150 −0.257979
\(217\) −8.61170 −0.584600
\(218\) 12.2430 0.829198
\(219\) −11.0612 −0.747445
\(220\) 0 0
\(221\) 33.4392 2.24936
\(222\) −5.27700 −0.354169
\(223\) 4.86983 0.326108 0.163054 0.986617i \(-0.447865\pi\)
0.163054 + 0.986617i \(0.447865\pi\)
\(224\) −14.2140 −0.949711
\(225\) 0 0
\(226\) 3.11030 0.206894
\(227\) 27.6885 1.83775 0.918876 0.394547i \(-0.129098\pi\)
0.918876 + 0.394547i \(0.129098\pi\)
\(228\) 2.98285 0.197544
\(229\) 5.51690 0.364567 0.182284 0.983246i \(-0.441651\pi\)
0.182284 + 0.983246i \(0.441651\pi\)
\(230\) 0 0
\(231\) −4.10901 −0.270353
\(232\) 0.681351 0.0447329
\(233\) −24.3012 −1.59202 −0.796012 0.605281i \(-0.793061\pi\)
−0.796012 + 0.605281i \(0.793061\pi\)
\(234\) 9.65193 0.630967
\(235\) 0 0
\(236\) 19.7834 1.28779
\(237\) −14.3361 −0.931230
\(238\) 25.8979 1.67871
\(239\) −10.6660 −0.689928 −0.344964 0.938616i \(-0.612109\pi\)
−0.344964 + 0.938616i \(0.612109\pi\)
\(240\) 0 0
\(241\) −4.78267 −0.308079 −0.154039 0.988065i \(-0.549228\pi\)
−0.154039 + 0.988065i \(0.549228\pi\)
\(242\) 1.86598 0.119950
\(243\) 10.4753 0.671990
\(244\) 11.2337 0.719164
\(245\) 0 0
\(246\) 35.2560 2.24784
\(247\) −4.91831 −0.312945
\(248\) 4.07856 0.258989
\(249\) −21.8649 −1.38563
\(250\) 0 0
\(251\) 29.0361 1.83274 0.916370 0.400333i \(-0.131105\pi\)
0.916370 + 0.400333i \(0.131105\pi\)
\(252\) 3.18144 0.200412
\(253\) −1.43601 −0.0902809
\(254\) 9.79437 0.614553
\(255\) 0 0
\(256\) 20.8624 1.30390
\(257\) 2.04952 0.127845 0.0639227 0.997955i \(-0.479639\pi\)
0.0639227 + 0.997955i \(0.479639\pi\)
\(258\) 14.2874 0.889493
\(259\) 2.86801 0.178209
\(260\) 0 0
\(261\) −0.741183 −0.0458781
\(262\) −16.9010 −1.04414
\(263\) −27.0100 −1.66551 −0.832755 0.553642i \(-0.813238\pi\)
−0.832755 + 0.553642i \(0.813238\pi\)
\(264\) 1.94606 0.119771
\(265\) 0 0
\(266\) −3.80913 −0.233553
\(267\) 6.57392 0.402317
\(268\) −15.6655 −0.956925
\(269\) 12.3963 0.755818 0.377909 0.925843i \(-0.376643\pi\)
0.377909 + 0.925843i \(0.376643\pi\)
\(270\) 0 0
\(271\) 32.8661 1.99647 0.998236 0.0593651i \(-0.0189076\pi\)
0.998236 + 0.0593651i \(0.0189076\pi\)
\(272\) −32.4158 −1.96550
\(273\) −20.2094 −1.22313
\(274\) 29.1013 1.75808
\(275\) 0 0
\(276\) 4.28340 0.257830
\(277\) −14.8509 −0.892304 −0.446152 0.894957i \(-0.647206\pi\)
−0.446152 + 0.894957i \(0.647206\pi\)
\(278\) 17.2808 1.03643
\(279\) −4.43671 −0.265619
\(280\) 0 0
\(281\) 7.48442 0.446483 0.223242 0.974763i \(-0.428336\pi\)
0.223242 + 0.974763i \(0.428336\pi\)
\(282\) −26.8036 −1.59613
\(283\) 10.6009 0.630155 0.315078 0.949066i \(-0.397969\pi\)
0.315078 + 0.949066i \(0.397969\pi\)
\(284\) 8.53185 0.506272
\(285\) 0 0
\(286\) 9.17747 0.542675
\(287\) −19.1614 −1.13106
\(288\) −7.32298 −0.431511
\(289\) 29.2252 1.71913
\(290\) 0 0
\(291\) −3.30944 −0.194003
\(292\) 8.14321 0.476546
\(293\) 22.4854 1.31361 0.656805 0.754060i \(-0.271907\pi\)
0.656805 + 0.754060i \(0.271907\pi\)
\(294\) 10.6402 0.620551
\(295\) 0 0
\(296\) −1.35831 −0.0789500
\(297\) 3.92170 0.227560
\(298\) 9.75223 0.564932
\(299\) −7.06273 −0.408448
\(300\) 0 0
\(301\) −7.76508 −0.447572
\(302\) 38.7879 2.23199
\(303\) 5.83374 0.335140
\(304\) 4.76779 0.273452
\(305\) 0 0
\(306\) 13.3425 0.762741
\(307\) −13.3031 −0.759250 −0.379625 0.925140i \(-0.623947\pi\)
−0.379625 + 0.925140i \(0.623947\pi\)
\(308\) 3.02505 0.172368
\(309\) −25.7232 −1.46334
\(310\) 0 0
\(311\) −8.03547 −0.455650 −0.227825 0.973702i \(-0.573161\pi\)
−0.227825 + 0.973702i \(0.573161\pi\)
\(312\) 9.57131 0.541869
\(313\) −17.1926 −0.971785 −0.485892 0.874019i \(-0.661505\pi\)
−0.485892 + 0.874019i \(0.661505\pi\)
\(314\) −20.9565 −1.18264
\(315\) 0 0
\(316\) 10.5542 0.593721
\(317\) −3.68497 −0.206969 −0.103484 0.994631i \(-0.532999\pi\)
−0.103484 + 0.994631i \(0.532999\pi\)
\(318\) 9.95442 0.558216
\(319\) −0.704749 −0.0394584
\(320\) 0 0
\(321\) −34.7742 −1.94091
\(322\) −5.46993 −0.304828
\(323\) −6.79891 −0.378302
\(324\) −16.3733 −0.909630
\(325\) 0 0
\(326\) −8.28485 −0.458855
\(327\) −13.2068 −0.730339
\(328\) 9.07495 0.501080
\(329\) 14.5676 0.803136
\(330\) 0 0
\(331\) −16.1229 −0.886192 −0.443096 0.896474i \(-0.646120\pi\)
−0.443096 + 0.896474i \(0.646120\pi\)
\(332\) 16.0969 0.883433
\(333\) 1.47759 0.0809712
\(334\) 26.4693 1.44833
\(335\) 0 0
\(336\) 19.5909 1.06877
\(337\) 9.25224 0.504001 0.252001 0.967727i \(-0.418911\pi\)
0.252001 + 0.967727i \(0.418911\pi\)
\(338\) 20.8799 1.13572
\(339\) −3.35516 −0.182228
\(340\) 0 0
\(341\) −4.21862 −0.228451
\(342\) −1.96245 −0.106117
\(343\) −20.0724 −1.08381
\(344\) 3.67759 0.198283
\(345\) 0 0
\(346\) 10.1520 0.545775
\(347\) −1.38539 −0.0743715 −0.0371858 0.999308i \(-0.511839\pi\)
−0.0371858 + 0.999308i \(0.511839\pi\)
\(348\) 2.10216 0.112688
\(349\) −31.0884 −1.66412 −0.832062 0.554683i \(-0.812840\pi\)
−0.832062 + 0.554683i \(0.812840\pi\)
\(350\) 0 0
\(351\) 19.2882 1.02953
\(352\) −6.96300 −0.371129
\(353\) 29.4747 1.56878 0.784389 0.620269i \(-0.212977\pi\)
0.784389 + 0.620269i \(0.212977\pi\)
\(354\) −50.1433 −2.66509
\(355\) 0 0
\(356\) −4.83971 −0.256504
\(357\) −27.9368 −1.47857
\(358\) −15.7753 −0.833749
\(359\) −9.74843 −0.514502 −0.257251 0.966345i \(-0.582817\pi\)
−0.257251 + 0.966345i \(0.582817\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −25.4370 −1.33694
\(363\) −2.01288 −0.105649
\(364\) 14.8781 0.779826
\(365\) 0 0
\(366\) −28.4731 −1.48831
\(367\) 23.1973 1.21089 0.605444 0.795888i \(-0.292996\pi\)
0.605444 + 0.795888i \(0.292996\pi\)
\(368\) 6.84658 0.356903
\(369\) −9.87186 −0.513908
\(370\) 0 0
\(371\) −5.41015 −0.280881
\(372\) 12.5835 0.652425
\(373\) −32.5197 −1.68381 −0.841903 0.539629i \(-0.818565\pi\)
−0.841903 + 0.539629i \(0.818565\pi\)
\(374\) 12.6866 0.656010
\(375\) 0 0
\(376\) −6.89930 −0.355804
\(377\) −3.46617 −0.178517
\(378\) 14.9383 0.768342
\(379\) 18.8601 0.968780 0.484390 0.874852i \(-0.339042\pi\)
0.484390 + 0.874852i \(0.339042\pi\)
\(380\) 0 0
\(381\) −10.5654 −0.541284
\(382\) −23.3069 −1.19249
\(383\) 8.07858 0.412796 0.206398 0.978468i \(-0.433826\pi\)
0.206398 + 0.978468i \(0.433826\pi\)
\(384\) −15.0460 −0.767815
\(385\) 0 0
\(386\) 18.5655 0.944960
\(387\) −4.00054 −0.203359
\(388\) 2.43640 0.123690
\(389\) 32.3244 1.63891 0.819456 0.573143i \(-0.194276\pi\)
0.819456 + 0.573143i \(0.194276\pi\)
\(390\) 0 0
\(391\) −9.76328 −0.493750
\(392\) 2.73881 0.138331
\(393\) 18.2315 0.919659
\(394\) −45.4654 −2.29051
\(395\) 0 0
\(396\) 1.55849 0.0783172
\(397\) −2.99126 −0.150127 −0.0750634 0.997179i \(-0.523916\pi\)
−0.0750634 + 0.997179i \(0.523916\pi\)
\(398\) −21.4665 −1.07602
\(399\) 4.10901 0.205708
\(400\) 0 0
\(401\) 3.52554 0.176057 0.0880286 0.996118i \(-0.471943\pi\)
0.0880286 + 0.996118i \(0.471943\pi\)
\(402\) 39.7061 1.98036
\(403\) −20.7485 −1.03356
\(404\) −4.29479 −0.213674
\(405\) 0 0
\(406\) −2.68448 −0.133228
\(407\) 1.40495 0.0696409
\(408\) 13.2311 0.655035
\(409\) −23.0387 −1.13919 −0.569595 0.821926i \(-0.692900\pi\)
−0.569595 + 0.821926i \(0.692900\pi\)
\(410\) 0 0
\(411\) −31.3924 −1.54847
\(412\) 18.9374 0.932978
\(413\) 27.2525 1.34101
\(414\) −2.81809 −0.138501
\(415\) 0 0
\(416\) −34.2462 −1.67906
\(417\) −18.6412 −0.912865
\(418\) −1.86598 −0.0912681
\(419\) −29.8644 −1.45897 −0.729486 0.683996i \(-0.760241\pi\)
−0.729486 + 0.683996i \(0.760241\pi\)
\(420\) 0 0
\(421\) 21.8866 1.06669 0.533345 0.845898i \(-0.320935\pi\)
0.533345 + 0.845898i \(0.320935\pi\)
\(422\) 20.1600 0.981375
\(423\) 7.50516 0.364913
\(424\) 2.56228 0.124435
\(425\) 0 0
\(426\) −21.6250 −1.04773
\(427\) 15.4749 0.748884
\(428\) 25.6007 1.23746
\(429\) −9.89999 −0.477976
\(430\) 0 0
\(431\) −12.7937 −0.616250 −0.308125 0.951346i \(-0.599701\pi\)
−0.308125 + 0.951346i \(0.599701\pi\)
\(432\) −18.6979 −0.899601
\(433\) −8.06755 −0.387702 −0.193851 0.981031i \(-0.562098\pi\)
−0.193851 + 0.981031i \(0.562098\pi\)
\(434\) −16.0693 −0.771349
\(435\) 0 0
\(436\) 9.72283 0.465639
\(437\) 1.43601 0.0686935
\(438\) −20.6399 −0.986214
\(439\) 1.25574 0.0599330 0.0299665 0.999551i \(-0.490460\pi\)
0.0299665 + 0.999551i \(0.490460\pi\)
\(440\) 0 0
\(441\) −2.97932 −0.141872
\(442\) 62.3968 2.96791
\(443\) −30.4664 −1.44750 −0.723752 0.690060i \(-0.757584\pi\)
−0.723752 + 0.690060i \(0.757584\pi\)
\(444\) −4.19076 −0.198885
\(445\) 0 0
\(446\) 9.08701 0.430282
\(447\) −10.5200 −0.497579
\(448\) −7.05746 −0.333433
\(449\) 2.35876 0.111317 0.0556584 0.998450i \(-0.482274\pi\)
0.0556584 + 0.998450i \(0.482274\pi\)
\(450\) 0 0
\(451\) −9.38658 −0.441997
\(452\) 2.47007 0.116182
\(453\) −41.8415 −1.96589
\(454\) 51.6662 2.42482
\(455\) 0 0
\(456\) −1.94606 −0.0911324
\(457\) 37.1810 1.73926 0.869628 0.493708i \(-0.164359\pi\)
0.869628 + 0.493708i \(0.164359\pi\)
\(458\) 10.2944 0.481027
\(459\) 26.6633 1.24454
\(460\) 0 0
\(461\) 10.4784 0.488030 0.244015 0.969771i \(-0.421535\pi\)
0.244015 + 0.969771i \(0.421535\pi\)
\(462\) −7.66733 −0.356717
\(463\) 24.2378 1.12642 0.563212 0.826312i \(-0.309565\pi\)
0.563212 + 0.826312i \(0.309565\pi\)
\(464\) 3.36010 0.155989
\(465\) 0 0
\(466\) −45.3455 −2.10059
\(467\) −21.5240 −0.996012 −0.498006 0.867174i \(-0.665934\pi\)
−0.498006 + 0.867174i \(0.665934\pi\)
\(468\) 7.66515 0.354322
\(469\) −21.5800 −0.996471
\(470\) 0 0
\(471\) 22.6063 1.04164
\(472\) −12.9070 −0.594091
\(473\) −3.80388 −0.174903
\(474\) −26.7509 −1.22871
\(475\) 0 0
\(476\) 20.5670 0.942688
\(477\) −2.78729 −0.127621
\(478\) −19.9026 −0.910323
\(479\) −7.29664 −0.333392 −0.166696 0.986008i \(-0.553310\pi\)
−0.166696 + 0.986008i \(0.553310\pi\)
\(480\) 0 0
\(481\) 6.90999 0.315068
\(482\) −8.92436 −0.406493
\(483\) 5.90057 0.268485
\(484\) 1.48188 0.0673582
\(485\) 0 0
\(486\) 19.5467 0.886655
\(487\) 30.8245 1.39679 0.698396 0.715712i \(-0.253898\pi\)
0.698396 + 0.715712i \(0.253898\pi\)
\(488\) −7.32903 −0.331770
\(489\) 8.93709 0.404149
\(490\) 0 0
\(491\) −30.9271 −1.39572 −0.697860 0.716234i \(-0.745864\pi\)
−0.697860 + 0.716234i \(0.745864\pi\)
\(492\) 27.9988 1.26228
\(493\) −4.79153 −0.215799
\(494\) −9.17747 −0.412914
\(495\) 0 0
\(496\) 20.1135 0.903122
\(497\) 11.7530 0.527194
\(498\) −40.7995 −1.82827
\(499\) 38.2965 1.71439 0.857193 0.514995i \(-0.172206\pi\)
0.857193 + 0.514995i \(0.172206\pi\)
\(500\) 0 0
\(501\) −28.5531 −1.27566
\(502\) 54.1807 2.41820
\(503\) −38.2324 −1.70470 −0.852349 0.522973i \(-0.824823\pi\)
−0.852349 + 0.522973i \(0.824823\pi\)
\(504\) −2.07561 −0.0924552
\(505\) 0 0
\(506\) −2.67956 −0.119121
\(507\) −22.5237 −1.00031
\(508\) 7.77826 0.345105
\(509\) 13.8877 0.615561 0.307780 0.951457i \(-0.400414\pi\)
0.307780 + 0.951457i \(0.400414\pi\)
\(510\) 0 0
\(511\) 11.2176 0.496239
\(512\) 23.9791 1.05974
\(513\) −3.92170 −0.173147
\(514\) 3.82436 0.168685
\(515\) 0 0
\(516\) 11.3464 0.499498
\(517\) 7.13622 0.313851
\(518\) 5.35164 0.235138
\(519\) −10.9512 −0.480706
\(520\) 0 0
\(521\) −30.9995 −1.35811 −0.679057 0.734085i \(-0.737611\pi\)
−0.679057 + 0.734085i \(0.737611\pi\)
\(522\) −1.38303 −0.0605337
\(523\) 14.4767 0.633020 0.316510 0.948589i \(-0.397489\pi\)
0.316510 + 0.948589i \(0.397489\pi\)
\(524\) −13.4220 −0.586343
\(525\) 0 0
\(526\) −50.4002 −2.19755
\(527\) −28.6820 −1.24941
\(528\) 9.59701 0.417656
\(529\) −20.9379 −0.910343
\(530\) 0 0
\(531\) 14.0404 0.609300
\(532\) −3.02505 −0.131152
\(533\) −46.1661 −1.99968
\(534\) 12.2668 0.530837
\(535\) 0 0
\(536\) 10.2204 0.441455
\(537\) 17.0172 0.734347
\(538\) 23.1313 0.997261
\(539\) −2.83286 −0.122020
\(540\) 0 0
\(541\) 11.0122 0.473453 0.236726 0.971576i \(-0.423926\pi\)
0.236726 + 0.971576i \(0.423926\pi\)
\(542\) 61.3275 2.63424
\(543\) 27.4396 1.17755
\(544\) −47.3408 −2.02972
\(545\) 0 0
\(546\) −37.7103 −1.61385
\(547\) −2.09356 −0.0895142 −0.0447571 0.998998i \(-0.514251\pi\)
−0.0447571 + 0.998998i \(0.514251\pi\)
\(548\) 23.1110 0.987255
\(549\) 7.97262 0.340263
\(550\) 0 0
\(551\) 0.704749 0.0300233
\(552\) −2.79455 −0.118944
\(553\) 14.5389 0.618257
\(554\) −27.7115 −1.17735
\(555\) 0 0
\(556\) 13.7236 0.582012
\(557\) −27.0490 −1.14610 −0.573052 0.819519i \(-0.694241\pi\)
−0.573052 + 0.819519i \(0.694241\pi\)
\(558\) −8.27882 −0.350470
\(559\) −18.7087 −0.791293
\(560\) 0 0
\(561\) −13.6854 −0.577799
\(562\) 13.9658 0.589111
\(563\) −11.5944 −0.488646 −0.244323 0.969694i \(-0.578566\pi\)
−0.244323 + 0.969694i \(0.578566\pi\)
\(564\) −21.2863 −0.896315
\(565\) 0 0
\(566\) 19.7810 0.831456
\(567\) −22.5550 −0.947221
\(568\) −5.56630 −0.233557
\(569\) −26.0673 −1.09280 −0.546399 0.837525i \(-0.684002\pi\)
−0.546399 + 0.837525i \(0.684002\pi\)
\(570\) 0 0
\(571\) 43.1051 1.80389 0.901946 0.431849i \(-0.142139\pi\)
0.901946 + 0.431849i \(0.142139\pi\)
\(572\) 7.28835 0.304741
\(573\) 25.1418 1.05032
\(574\) −35.7547 −1.49237
\(575\) 0 0
\(576\) −3.63597 −0.151499
\(577\) 7.45466 0.310341 0.155171 0.987888i \(-0.450407\pi\)
0.155171 + 0.987888i \(0.450407\pi\)
\(578\) 54.5337 2.26830
\(579\) −20.0271 −0.832299
\(580\) 0 0
\(581\) 22.1742 0.919941
\(582\) −6.17535 −0.255976
\(583\) −2.65027 −0.109763
\(584\) −5.31275 −0.219843
\(585\) 0 0
\(586\) 41.9573 1.73324
\(587\) 27.8753 1.15054 0.575269 0.817964i \(-0.304897\pi\)
0.575269 + 0.817964i \(0.304897\pi\)
\(588\) 8.45002 0.348473
\(589\) 4.21862 0.173825
\(590\) 0 0
\(591\) 49.0447 2.01743
\(592\) −6.69852 −0.275307
\(593\) −29.0611 −1.19339 −0.596697 0.802466i \(-0.703521\pi\)
−0.596697 + 0.802466i \(0.703521\pi\)
\(594\) 7.31782 0.300254
\(595\) 0 0
\(596\) 7.74480 0.317240
\(597\) 23.1565 0.947734
\(598\) −13.1789 −0.538926
\(599\) −39.5579 −1.61629 −0.808146 0.588982i \(-0.799529\pi\)
−0.808146 + 0.588982i \(0.799529\pi\)
\(600\) 0 0
\(601\) −4.44168 −0.181180 −0.0905900 0.995888i \(-0.528875\pi\)
−0.0905900 + 0.995888i \(0.528875\pi\)
\(602\) −14.4895 −0.590547
\(603\) −11.1179 −0.452757
\(604\) 30.8036 1.25338
\(605\) 0 0
\(606\) 10.8856 0.442199
\(607\) 15.6237 0.634145 0.317072 0.948401i \(-0.397300\pi\)
0.317072 + 0.948401i \(0.397300\pi\)
\(608\) 6.96300 0.282387
\(609\) 2.89582 0.117345
\(610\) 0 0
\(611\) 35.0982 1.41992
\(612\) 10.5961 0.428320
\(613\) 29.5049 1.19169 0.595845 0.803099i \(-0.296817\pi\)
0.595845 + 0.803099i \(0.296817\pi\)
\(614\) −24.8234 −1.00179
\(615\) 0 0
\(616\) −1.97358 −0.0795179
\(617\) 27.5007 1.10714 0.553569 0.832804i \(-0.313266\pi\)
0.553569 + 0.832804i \(0.313266\pi\)
\(618\) −47.9990 −1.93080
\(619\) −24.6833 −0.992107 −0.496054 0.868292i \(-0.665218\pi\)
−0.496054 + 0.868292i \(0.665218\pi\)
\(620\) 0 0
\(621\) −5.63159 −0.225988
\(622\) −14.9940 −0.601206
\(623\) −6.66691 −0.267104
\(624\) 47.2011 1.88956
\(625\) 0 0
\(626\) −32.0811 −1.28222
\(627\) 2.01288 0.0803868
\(628\) −16.6427 −0.664117
\(629\) 9.55214 0.380869
\(630\) 0 0
\(631\) −7.77127 −0.309370 −0.154685 0.987964i \(-0.549436\pi\)
−0.154685 + 0.987964i \(0.549436\pi\)
\(632\) −6.88572 −0.273899
\(633\) −21.7472 −0.864373
\(634\) −6.87608 −0.273084
\(635\) 0 0
\(636\) 7.90537 0.313468
\(637\) −13.9329 −0.552042
\(638\) −1.31505 −0.0520632
\(639\) 6.05510 0.239536
\(640\) 0 0
\(641\) −36.9773 −1.46051 −0.730257 0.683172i \(-0.760600\pi\)
−0.730257 + 0.683172i \(0.760600\pi\)
\(642\) −64.8880 −2.56093
\(643\) 1.65306 0.0651902 0.0325951 0.999469i \(-0.489623\pi\)
0.0325951 + 0.999469i \(0.489623\pi\)
\(644\) −4.34399 −0.171177
\(645\) 0 0
\(646\) −12.6866 −0.499149
\(647\) 5.12322 0.201415 0.100707 0.994916i \(-0.467889\pi\)
0.100707 + 0.994916i \(0.467889\pi\)
\(648\) 10.6822 0.419636
\(649\) 13.3502 0.524041
\(650\) 0 0
\(651\) 17.3343 0.679387
\(652\) −6.57947 −0.257672
\(653\) −41.0422 −1.60611 −0.803053 0.595908i \(-0.796792\pi\)
−0.803053 + 0.595908i \(0.796792\pi\)
\(654\) −24.6437 −0.963643
\(655\) 0 0
\(656\) 44.7533 1.74732
\(657\) 5.77928 0.225471
\(658\) 27.1828 1.05970
\(659\) −39.9155 −1.55489 −0.777443 0.628953i \(-0.783484\pi\)
−0.777443 + 0.628953i \(0.783484\pi\)
\(660\) 0 0
\(661\) −35.6144 −1.38524 −0.692619 0.721303i \(-0.743543\pi\)
−0.692619 + 0.721303i \(0.743543\pi\)
\(662\) −30.0849 −1.16928
\(663\) −67.3092 −2.61407
\(664\) −10.5019 −0.407551
\(665\) 0 0
\(666\) 2.75714 0.106837
\(667\) 1.01202 0.0391857
\(668\) 21.0208 0.813317
\(669\) −9.80241 −0.378983
\(670\) 0 0
\(671\) 7.58071 0.292650
\(672\) 28.6111 1.10370
\(673\) 20.6495 0.795982 0.397991 0.917389i \(-0.369708\pi\)
0.397991 + 0.917389i \(0.369708\pi\)
\(674\) 17.2645 0.665003
\(675\) 0 0
\(676\) 16.5819 0.637767
\(677\) −8.49349 −0.326431 −0.163216 0.986590i \(-0.552187\pi\)
−0.163216 + 0.986590i \(0.552187\pi\)
\(678\) −6.26067 −0.240440
\(679\) 3.35625 0.128801
\(680\) 0 0
\(681\) −55.7338 −2.13572
\(682\) −7.87185 −0.301429
\(683\) 34.3263 1.31346 0.656729 0.754126i \(-0.271939\pi\)
0.656729 + 0.754126i \(0.271939\pi\)
\(684\) −1.55849 −0.0595904
\(685\) 0 0
\(686\) −37.4547 −1.43003
\(687\) −11.1049 −0.423677
\(688\) 18.1361 0.691433
\(689\) −13.0349 −0.496589
\(690\) 0 0
\(691\) −20.0621 −0.763198 −0.381599 0.924328i \(-0.624626\pi\)
−0.381599 + 0.924328i \(0.624626\pi\)
\(692\) 8.06228 0.306482
\(693\) 2.14689 0.0815537
\(694\) −2.58511 −0.0981293
\(695\) 0 0
\(696\) −1.37148 −0.0519858
\(697\) −63.8186 −2.41730
\(698\) −58.0103 −2.19572
\(699\) 48.9155 1.85015
\(700\) 0 0
\(701\) −0.877252 −0.0331334 −0.0165667 0.999863i \(-0.505274\pi\)
−0.0165667 + 0.999863i \(0.505274\pi\)
\(702\) 35.9913 1.35840
\(703\) −1.40495 −0.0529888
\(704\) −3.45724 −0.130300
\(705\) 0 0
\(706\) 54.9991 2.06992
\(707\) −5.91626 −0.222504
\(708\) −39.8216 −1.49659
\(709\) 31.6482 1.18857 0.594286 0.804254i \(-0.297435\pi\)
0.594286 + 0.804254i \(0.297435\pi\)
\(710\) 0 0
\(711\) 7.49038 0.280911
\(712\) 3.15749 0.118332
\(713\) 6.05796 0.226872
\(714\) −52.1295 −1.95090
\(715\) 0 0
\(716\) −12.5280 −0.468195
\(717\) 21.4695 0.801791
\(718\) −18.1904 −0.678859
\(719\) −26.0542 −0.971656 −0.485828 0.874054i \(-0.661482\pi\)
−0.485828 + 0.874054i \(0.661482\pi\)
\(720\) 0 0
\(721\) 26.0871 0.971533
\(722\) 1.86598 0.0694446
\(723\) 9.62695 0.358030
\(724\) −20.2010 −0.750763
\(725\) 0 0
\(726\) −3.75600 −0.139398
\(727\) 36.4040 1.35015 0.675075 0.737749i \(-0.264111\pi\)
0.675075 + 0.737749i \(0.264111\pi\)
\(728\) −9.70670 −0.359754
\(729\) 12.0615 0.446724
\(730\) 0 0
\(731\) −25.8623 −0.956550
\(732\) −22.6121 −0.835769
\(733\) −28.3926 −1.04870 −0.524352 0.851502i \(-0.675692\pi\)
−0.524352 + 0.851502i \(0.675692\pi\)
\(734\) 43.2856 1.59770
\(735\) 0 0
\(736\) 9.99891 0.368565
\(737\) −10.5714 −0.389402
\(738\) −18.4207 −0.678075
\(739\) 22.7331 0.836251 0.418126 0.908389i \(-0.362687\pi\)
0.418126 + 0.908389i \(0.362687\pi\)
\(740\) 0 0
\(741\) 9.89999 0.363685
\(742\) −10.0952 −0.370607
\(743\) 11.9651 0.438958 0.219479 0.975617i \(-0.429564\pi\)
0.219479 + 0.975617i \(0.429564\pi\)
\(744\) −8.20966 −0.300981
\(745\) 0 0
\(746\) −60.6811 −2.22169
\(747\) 11.4241 0.417985
\(748\) 10.0752 0.368385
\(749\) 35.2661 1.28860
\(750\) 0 0
\(751\) 33.1233 1.20869 0.604343 0.796724i \(-0.293436\pi\)
0.604343 + 0.796724i \(0.293436\pi\)
\(752\) −34.0240 −1.24073
\(753\) −58.4462 −2.12990
\(754\) −6.46781 −0.235544
\(755\) 0 0
\(756\) 11.8633 0.431465
\(757\) −31.0693 −1.12923 −0.564617 0.825353i \(-0.690976\pi\)
−0.564617 + 0.825353i \(0.690976\pi\)
\(758\) 35.1926 1.27825
\(759\) 2.89051 0.104919
\(760\) 0 0
\(761\) 9.25808 0.335605 0.167803 0.985821i \(-0.446333\pi\)
0.167803 + 0.985821i \(0.446333\pi\)
\(762\) −19.7149 −0.714196
\(763\) 13.3936 0.484882
\(764\) −18.5094 −0.669646
\(765\) 0 0
\(766\) 15.0745 0.544662
\(767\) 65.6604 2.37086
\(768\) −41.9936 −1.51531
\(769\) −18.6181 −0.671388 −0.335694 0.941971i \(-0.608971\pi\)
−0.335694 + 0.941971i \(0.608971\pi\)
\(770\) 0 0
\(771\) −4.12544 −0.148574
\(772\) 14.7439 0.530646
\(773\) 24.0735 0.865864 0.432932 0.901427i \(-0.357479\pi\)
0.432932 + 0.901427i \(0.357479\pi\)
\(774\) −7.46492 −0.268321
\(775\) 0 0
\(776\) −1.58954 −0.0570613
\(777\) −5.77296 −0.207104
\(778\) 60.3166 2.16246
\(779\) 9.38658 0.336309
\(780\) 0 0
\(781\) 5.75745 0.206018
\(782\) −18.2181 −0.651477
\(783\) −2.76382 −0.0987707
\(784\) 13.5065 0.482375
\(785\) 0 0
\(786\) 34.0197 1.21344
\(787\) −14.1014 −0.502660 −0.251330 0.967901i \(-0.580868\pi\)
−0.251330 + 0.967901i \(0.580868\pi\)
\(788\) −36.1066 −1.28625
\(789\) 54.3680 1.93555
\(790\) 0 0
\(791\) 3.40262 0.120983
\(792\) −1.01678 −0.0361298
\(793\) 37.2843 1.32400
\(794\) −5.58163 −0.198084
\(795\) 0 0
\(796\) −17.0478 −0.604243
\(797\) 42.7967 1.51594 0.757969 0.652290i \(-0.226192\pi\)
0.757969 + 0.652290i \(0.226192\pi\)
\(798\) 7.66733 0.271421
\(799\) 48.5186 1.71646
\(800\) 0 0
\(801\) −3.43477 −0.121362
\(802\) 6.57859 0.232298
\(803\) 5.49519 0.193921
\(804\) 31.5329 1.11208
\(805\) 0 0
\(806\) −38.7162 −1.36372
\(807\) −24.9524 −0.878365
\(808\) 2.80198 0.0985733
\(809\) 28.5997 1.00551 0.502756 0.864428i \(-0.332319\pi\)
0.502756 + 0.864428i \(0.332319\pi\)
\(810\) 0 0
\(811\) −35.3795 −1.24234 −0.621171 0.783675i \(-0.713343\pi\)
−0.621171 + 0.783675i \(0.713343\pi\)
\(812\) −2.13190 −0.0748150
\(813\) −66.1556 −2.32018
\(814\) 2.62161 0.0918874
\(815\) 0 0
\(816\) 65.2492 2.28418
\(817\) 3.80388 0.133081
\(818\) −42.9897 −1.50310
\(819\) 10.5591 0.368964
\(820\) 0 0
\(821\) 24.0141 0.838097 0.419048 0.907964i \(-0.362364\pi\)
0.419048 + 0.907964i \(0.362364\pi\)
\(822\) −58.5776 −2.04313
\(823\) 20.7259 0.722458 0.361229 0.932477i \(-0.382357\pi\)
0.361229 + 0.932477i \(0.382357\pi\)
\(824\) −12.3550 −0.430407
\(825\) 0 0
\(826\) 50.8526 1.76939
\(827\) −28.6655 −0.996797 −0.498398 0.866948i \(-0.666078\pi\)
−0.498398 + 0.866948i \(0.666078\pi\)
\(828\) −2.23800 −0.0777760
\(829\) 21.3397 0.741158 0.370579 0.928801i \(-0.379159\pi\)
0.370579 + 0.928801i \(0.379159\pi\)
\(830\) 0 0
\(831\) 29.8931 1.03698
\(832\) −17.0038 −0.589500
\(833\) −19.2604 −0.667333
\(834\) −34.7842 −1.20448
\(835\) 0 0
\(836\) −1.48188 −0.0512519
\(837\) −16.5442 −0.571850
\(838\) −55.7264 −1.92504
\(839\) −40.3234 −1.39212 −0.696060 0.717984i \(-0.745065\pi\)
−0.696060 + 0.717984i \(0.745065\pi\)
\(840\) 0 0
\(841\) −28.5033 −0.982873
\(842\) 40.8400 1.40744
\(843\) −15.0653 −0.518875
\(844\) 16.0102 0.551095
\(845\) 0 0
\(846\) 14.0045 0.481484
\(847\) 2.04136 0.0701418
\(848\) 12.6359 0.433920
\(849\) −21.3383 −0.732328
\(850\) 0 0
\(851\) −2.01752 −0.0691597
\(852\) −17.1736 −0.588359
\(853\) −46.6593 −1.59758 −0.798792 0.601607i \(-0.794527\pi\)
−0.798792 + 0.601607i \(0.794527\pi\)
\(854\) 28.8759 0.988113
\(855\) 0 0
\(856\) −16.7023 −0.570872
\(857\) −52.7689 −1.80255 −0.901275 0.433248i \(-0.857368\pi\)
−0.901275 + 0.433248i \(0.857368\pi\)
\(858\) −18.4732 −0.630664
\(859\) −3.98543 −0.135981 −0.0679905 0.997686i \(-0.521659\pi\)
−0.0679905 + 0.997686i \(0.521659\pi\)
\(860\) 0 0
\(861\) 38.5696 1.31445
\(862\) −23.8728 −0.813109
\(863\) −14.6185 −0.497621 −0.248811 0.968552i \(-0.580040\pi\)
−0.248811 + 0.968552i \(0.580040\pi\)
\(864\) −27.3068 −0.928997
\(865\) 0 0
\(866\) −15.0539 −0.511552
\(867\) −58.8269 −1.99787
\(868\) −12.7615 −0.433154
\(869\) 7.12218 0.241603
\(870\) 0 0
\(871\) −51.9934 −1.76173
\(872\) −6.34331 −0.214812
\(873\) 1.72913 0.0585221
\(874\) 2.67956 0.0906374
\(875\) 0 0
\(876\) −16.3913 −0.553812
\(877\) 24.7365 0.835294 0.417647 0.908609i \(-0.362855\pi\)
0.417647 + 0.908609i \(0.362855\pi\)
\(878\) 2.34318 0.0790784
\(879\) −45.2605 −1.52660
\(880\) 0 0
\(881\) −30.9602 −1.04308 −0.521538 0.853228i \(-0.674642\pi\)
−0.521538 + 0.853228i \(0.674642\pi\)
\(882\) −5.55935 −0.187193
\(883\) −17.8695 −0.601358 −0.300679 0.953725i \(-0.597213\pi\)
−0.300679 + 0.953725i \(0.597213\pi\)
\(884\) 49.5529 1.66664
\(885\) 0 0
\(886\) −56.8498 −1.90991
\(887\) −28.5051 −0.957108 −0.478554 0.878058i \(-0.658839\pi\)
−0.478554 + 0.878058i \(0.658839\pi\)
\(888\) 2.73411 0.0917509
\(889\) 10.7149 0.359366
\(890\) 0 0
\(891\) −11.0490 −0.370156
\(892\) 7.21651 0.241627
\(893\) −7.13622 −0.238805
\(894\) −19.6301 −0.656529
\(895\) 0 0
\(896\) 15.2589 0.509763
\(897\) 14.2164 0.474673
\(898\) 4.40140 0.146877
\(899\) 2.97307 0.0991573
\(900\) 0 0
\(901\) −18.0190 −0.600299
\(902\) −17.5152 −0.583192
\(903\) 15.6302 0.520140
\(904\) −1.61151 −0.0535979
\(905\) 0 0
\(906\) −78.0754 −2.59388
\(907\) 26.4797 0.879245 0.439622 0.898183i \(-0.355112\pi\)
0.439622 + 0.898183i \(0.355112\pi\)
\(908\) 41.0311 1.36166
\(909\) −3.04803 −0.101097
\(910\) 0 0
\(911\) 14.7377 0.488281 0.244140 0.969740i \(-0.421494\pi\)
0.244140 + 0.969740i \(0.421494\pi\)
\(912\) −9.59701 −0.317789
\(913\) 10.8625 0.359496
\(914\) 69.3791 2.29486
\(915\) 0 0
\(916\) 8.17539 0.270122
\(917\) −18.4894 −0.610574
\(918\) 49.7532 1.64210
\(919\) 27.1844 0.896729 0.448365 0.893851i \(-0.352007\pi\)
0.448365 + 0.893851i \(0.352007\pi\)
\(920\) 0 0
\(921\) 26.7777 0.882354
\(922\) 19.5526 0.643929
\(923\) 28.3169 0.932063
\(924\) −6.08907 −0.200316
\(925\) 0 0
\(926\) 45.2272 1.48626
\(927\) 13.4400 0.441426
\(928\) 4.90717 0.161086
\(929\) 28.3245 0.929295 0.464648 0.885496i \(-0.346181\pi\)
0.464648 + 0.885496i \(0.346181\pi\)
\(930\) 0 0
\(931\) 2.83286 0.0928434
\(932\) −36.0115 −1.17959
\(933\) 16.1745 0.529528
\(934\) −40.1634 −1.31419
\(935\) 0 0
\(936\) −5.00085 −0.163458
\(937\) −37.8767 −1.23738 −0.618689 0.785636i \(-0.712336\pi\)
−0.618689 + 0.785636i \(0.712336\pi\)
\(938\) −40.2678 −1.31479
\(939\) 34.6067 1.12935
\(940\) 0 0
\(941\) −49.2677 −1.60608 −0.803040 0.595925i \(-0.796786\pi\)
−0.803040 + 0.595925i \(0.796786\pi\)
\(942\) 42.1829 1.37439
\(943\) 13.4792 0.438943
\(944\) −63.6509 −2.07166
\(945\) 0 0
\(946\) −7.09797 −0.230775
\(947\) 31.2750 1.01630 0.508151 0.861268i \(-0.330329\pi\)
0.508151 + 0.861268i \(0.330329\pi\)
\(948\) −21.2444 −0.689986
\(949\) 27.0271 0.877335
\(950\) 0 0
\(951\) 7.41741 0.240526
\(952\) −13.4182 −0.434887
\(953\) −17.0312 −0.551694 −0.275847 0.961202i \(-0.588958\pi\)
−0.275847 + 0.961202i \(0.588958\pi\)
\(954\) −5.20102 −0.168389
\(955\) 0 0
\(956\) −15.8058 −0.511195
\(957\) 1.41858 0.0458561
\(958\) −13.6154 −0.439893
\(959\) 31.8365 1.02805
\(960\) 0 0
\(961\) −13.2033 −0.425912
\(962\) 12.8939 0.415716
\(963\) 18.1690 0.585487
\(964\) −7.08734 −0.228268
\(965\) 0 0
\(966\) 11.0103 0.354252
\(967\) 46.9228 1.50894 0.754468 0.656337i \(-0.227895\pi\)
0.754468 + 0.656337i \(0.227895\pi\)
\(968\) −0.966800 −0.0310741
\(969\) 13.6854 0.439639
\(970\) 0 0
\(971\) −38.3346 −1.23022 −0.615108 0.788443i \(-0.710888\pi\)
−0.615108 + 0.788443i \(0.710888\pi\)
\(972\) 15.5231 0.497905
\(973\) 18.9049 0.606064
\(974\) 57.5179 1.84299
\(975\) 0 0
\(976\) −36.1432 −1.15692
\(977\) 45.8913 1.46819 0.734096 0.679045i \(-0.237606\pi\)
0.734096 + 0.679045i \(0.237606\pi\)
\(978\) 16.6764 0.533254
\(979\) −3.26592 −0.104379
\(980\) 0 0
\(981\) 6.90035 0.220311
\(982\) −57.7093 −1.84158
\(983\) 29.4166 0.938244 0.469122 0.883133i \(-0.344570\pi\)
0.469122 + 0.883133i \(0.344570\pi\)
\(984\) −18.2668 −0.582325
\(985\) 0 0
\(986\) −8.94089 −0.284736
\(987\) −29.3228 −0.933356
\(988\) −7.28835 −0.231873
\(989\) 5.46240 0.173694
\(990\) 0 0
\(991\) 1.56657 0.0497637 0.0248819 0.999690i \(-0.492079\pi\)
0.0248819 + 0.999690i \(0.492079\pi\)
\(992\) 29.3742 0.932633
\(993\) 32.4534 1.02988
\(994\) 21.9309 0.695605
\(995\) 0 0
\(996\) −32.4012 −1.02667
\(997\) −54.1570 −1.71517 −0.857585 0.514343i \(-0.828036\pi\)
−0.857585 + 0.514343i \(0.828036\pi\)
\(998\) 71.4605 2.26204
\(999\) 5.50980 0.174322
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 5225.2.a.o.1.6 8
5.4 even 2 1045.2.a.i.1.3 8
15.14 odd 2 9405.2.a.bf.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.i.1.3 8 5.4 even 2
5225.2.a.o.1.6 8 1.1 even 1 trivial
9405.2.a.bf.1.6 8 15.14 odd 2