Properties

Label 1334.2.a.d.1.4
Level $1334$
Weight $2$
Character 1334.1
Self dual yes
Analytic conductor $10.652$
Analytic rank $1$
Dimension $4$
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] = [1334,2,Mod(1,1334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1334, 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("1334.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1334 = 2 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1334.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: \(10.6520436296\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.37108.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 7x^{2} + 5x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.836038\) of defining polynomial
Character \(\chi\) \(=\) 1334.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.30104 q^{3} +1.00000 q^{4} -3.30104 q^{5} -3.30104 q^{6} -4.13708 q^{7} -1.00000 q^{8} +7.89687 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.30104 q^{3} +1.00000 q^{4} -3.30104 q^{5} -3.30104 q^{6} -4.13708 q^{7} -1.00000 q^{8} +7.89687 q^{9} +3.30104 q^{10} +0.836038 q^{11} +3.30104 q^{12} -1.16396 q^{13} +4.13708 q^{14} -10.8969 q^{15} +1.00000 q^{16} -1.13083 q^{17} -7.89687 q^{18} -3.59583 q^{19} -3.30104 q^{20} -13.6567 q^{21} -0.836038 q^{22} -1.00000 q^{23} -3.30104 q^{24} +5.89687 q^{25} +1.16396 q^{26} +16.1648 q^{27} -4.13708 q^{28} -1.00000 q^{29} +10.8969 q^{30} -7.16396 q^{31} -1.00000 q^{32} +2.75979 q^{33} +1.13083 q^{34} +13.6567 q^{35} +7.89687 q^{36} -2.79293 q^{37} +3.59583 q^{38} -3.84229 q^{39} +3.30104 q^{40} -7.80915 q^{41} +13.6567 q^{42} -6.35563 q^{43} +0.836038 q^{44} -26.0679 q^{45} +1.00000 q^{46} -0.246456 q^{47} +3.30104 q^{48} +10.1154 q^{49} -5.89687 q^{50} -3.73291 q^{51} -1.16396 q^{52} -7.14855 q^{53} -16.1648 q^{54} -2.75979 q^{55} +4.13708 q^{56} -11.8700 q^{57} +1.00000 q^{58} -1.67208 q^{59} -10.8969 q^{60} -10.1371 q^{61} +7.16396 q^{62} -32.6700 q^{63} +1.00000 q^{64} +3.84229 q^{65} -2.75979 q^{66} -0.869172 q^{67} -1.13083 q^{68} -3.30104 q^{69} -13.6567 q^{70} +13.7937 q^{71} -7.89687 q^{72} -11.8092 q^{73} +2.79293 q^{74} +19.4658 q^{75} -3.59583 q^{76} -3.45875 q^{77} +3.84229 q^{78} -16.4927 q^{79} -3.30104 q^{80} +29.6700 q^{81} +7.80915 q^{82} +9.19166 q^{83} -13.6567 q^{84} +3.73291 q^{85} +6.35563 q^{86} -3.30104 q^{87} -0.836038 q^{88} -12.3225 q^{89} +26.0679 q^{90} +4.81540 q^{91} -1.00000 q^{92} -23.6485 q^{93} +0.246456 q^{94} +11.8700 q^{95} -3.30104 q^{96} +13.2042 q^{97} -10.1154 q^{98} +6.60208 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + q^{3} + 4 q^{4} - q^{5} - q^{6} - 4 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + q^{3} + 4 q^{4} - q^{5} - q^{6} - 4 q^{8} + 7 q^{9} + q^{10} - q^{11} + q^{12} - 9 q^{13} - 19 q^{15} + 4 q^{16} - 7 q^{18} - 2 q^{19} - q^{20} - 22 q^{21} + q^{22} - 4 q^{23} - q^{24} - q^{25} + 9 q^{26} + 19 q^{27} - 4 q^{29} + 19 q^{30} - 33 q^{31} - 4 q^{32} + 3 q^{33} + 22 q^{35} + 7 q^{36} - 12 q^{37} + 2 q^{38} + q^{39} + q^{40} - 6 q^{41} + 22 q^{42} - 5 q^{43} - q^{44} - 22 q^{45} + 4 q^{46} + 3 q^{47} + q^{48} + 12 q^{49} + q^{50} + 14 q^{51} - 9 q^{52} - 9 q^{53} - 19 q^{54} - 3 q^{55} - 2 q^{57} + 4 q^{58} + 2 q^{59} - 19 q^{60} - 24 q^{61} + 33 q^{62} - 24 q^{63} + 4 q^{64} - q^{65} - 3 q^{66} - 8 q^{67} - q^{69} - 22 q^{70} + 6 q^{71} - 7 q^{72} - 22 q^{73} + 12 q^{74} + 20 q^{75} - 2 q^{76} - 18 q^{77} - q^{78} - 29 q^{79} - q^{80} + 12 q^{81} + 6 q^{82} + 12 q^{83} - 22 q^{84} - 14 q^{85} + 5 q^{86} - q^{87} + q^{88} - 20 q^{89} + 22 q^{90} - 18 q^{91} - 4 q^{92} - 5 q^{93} - 3 q^{94} + 2 q^{95} - q^{96} + 4 q^{97} - 12 q^{98} + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 3.30104 1.90586 0.952929 0.303195i \(-0.0980534\pi\)
0.952929 + 0.303195i \(0.0980534\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.30104 −1.47627 −0.738135 0.674653i \(-0.764293\pi\)
−0.738135 + 0.674653i \(0.764293\pi\)
\(6\) −3.30104 −1.34764
\(7\) −4.13708 −1.56367 −0.781834 0.623486i \(-0.785716\pi\)
−0.781834 + 0.623486i \(0.785716\pi\)
\(8\) −1.00000 −0.353553
\(9\) 7.89687 2.63229
\(10\) 3.30104 1.04388
\(11\) 0.836038 0.252075 0.126037 0.992025i \(-0.459774\pi\)
0.126037 + 0.992025i \(0.459774\pi\)
\(12\) 3.30104 0.952929
\(13\) −1.16396 −0.322825 −0.161413 0.986887i \(-0.551605\pi\)
−0.161413 + 0.986887i \(0.551605\pi\)
\(14\) 4.13708 1.10568
\(15\) −10.8969 −2.81356
\(16\) 1.00000 0.250000
\(17\) −1.13083 −0.274266 −0.137133 0.990553i \(-0.543789\pi\)
−0.137133 + 0.990553i \(0.543789\pi\)
\(18\) −7.89687 −1.86131
\(19\) −3.59583 −0.824940 −0.412470 0.910971i \(-0.635334\pi\)
−0.412470 + 0.910971i \(0.635334\pi\)
\(20\) −3.30104 −0.738135
\(21\) −13.6567 −2.98013
\(22\) −0.836038 −0.178244
\(23\) −1.00000 −0.208514
\(24\) −3.30104 −0.673822
\(25\) 5.89687 1.17937
\(26\) 1.16396 0.228272
\(27\) 16.1648 3.11091
\(28\) −4.13708 −0.781834
\(29\) −1.00000 −0.185695
\(30\) 10.8969 1.98949
\(31\) −7.16396 −1.28669 −0.643343 0.765578i \(-0.722453\pi\)
−0.643343 + 0.765578i \(0.722453\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.75979 0.480419
\(34\) 1.13083 0.193935
\(35\) 13.6567 2.30840
\(36\) 7.89687 1.31615
\(37\) −2.79293 −0.459155 −0.229577 0.973290i \(-0.573734\pi\)
−0.229577 + 0.973290i \(0.573734\pi\)
\(38\) 3.59583 0.583321
\(39\) −3.84229 −0.615258
\(40\) 3.30104 0.521940
\(41\) −7.80915 −1.21958 −0.609792 0.792561i \(-0.708747\pi\)
−0.609792 + 0.792561i \(0.708747\pi\)
\(42\) 13.6567 2.10727
\(43\) −6.35563 −0.969224 −0.484612 0.874729i \(-0.661039\pi\)
−0.484612 + 0.874729i \(0.661039\pi\)
\(44\) 0.836038 0.126037
\(45\) −26.0679 −3.88597
\(46\) 1.00000 0.147442
\(47\) −0.246456 −0.0359493 −0.0179747 0.999838i \(-0.505722\pi\)
−0.0179747 + 0.999838i \(0.505722\pi\)
\(48\) 3.30104 0.476464
\(49\) 10.1154 1.44506
\(50\) −5.89687 −0.833944
\(51\) −3.73291 −0.522712
\(52\) −1.16396 −0.161413
\(53\) −7.14855 −0.981930 −0.490965 0.871179i \(-0.663356\pi\)
−0.490965 + 0.871179i \(0.663356\pi\)
\(54\) −16.1648 −2.19975
\(55\) −2.75979 −0.372131
\(56\) 4.13708 0.552840
\(57\) −11.8700 −1.57222
\(58\) 1.00000 0.131306
\(59\) −1.67208 −0.217686 −0.108843 0.994059i \(-0.534715\pi\)
−0.108843 + 0.994059i \(0.534715\pi\)
\(60\) −10.8969 −1.40678
\(61\) −10.1371 −1.29792 −0.648960 0.760823i \(-0.724796\pi\)
−0.648960 + 0.760823i \(0.724796\pi\)
\(62\) 7.16396 0.909824
\(63\) −32.6700 −4.11603
\(64\) 1.00000 0.125000
\(65\) 3.84229 0.476577
\(66\) −2.75979 −0.339707
\(67\) −0.869172 −0.106186 −0.0530931 0.998590i \(-0.516908\pi\)
−0.0530931 + 0.998590i \(0.516908\pi\)
\(68\) −1.13083 −0.137133
\(69\) −3.30104 −0.397399
\(70\) −13.6567 −1.63228
\(71\) 13.7937 1.63702 0.818508 0.574495i \(-0.194801\pi\)
0.818508 + 0.574495i \(0.194801\pi\)
\(72\) −7.89687 −0.930655
\(73\) −11.8092 −1.38216 −0.691078 0.722780i \(-0.742864\pi\)
−0.691078 + 0.722780i \(0.742864\pi\)
\(74\) 2.79293 0.324671
\(75\) 19.4658 2.24772
\(76\) −3.59583 −0.412470
\(77\) −3.45875 −0.394162
\(78\) 3.84229 0.435053
\(79\) −16.4927 −1.85557 −0.927787 0.373110i \(-0.878291\pi\)
−0.927787 + 0.373110i \(0.878291\pi\)
\(80\) −3.30104 −0.369068
\(81\) 29.6700 3.29667
\(82\) 7.80915 0.862377
\(83\) 9.19166 1.00892 0.504458 0.863436i \(-0.331692\pi\)
0.504458 + 0.863436i \(0.331692\pi\)
\(84\) −13.6567 −1.49006
\(85\) 3.73291 0.404891
\(86\) 6.35563 0.685345
\(87\) −3.30104 −0.353909
\(88\) −0.836038 −0.0891219
\(89\) −12.3225 −1.30618 −0.653091 0.757280i \(-0.726528\pi\)
−0.653091 + 0.757280i \(0.726528\pi\)
\(90\) 26.0679 2.74780
\(91\) 4.81540 0.504792
\(92\) −1.00000 −0.104257
\(93\) −23.6485 −2.45224
\(94\) 0.246456 0.0254200
\(95\) 11.8700 1.21784
\(96\) −3.30104 −0.336911
\(97\) 13.2042 1.34068 0.670340 0.742054i \(-0.266148\pi\)
0.670340 + 0.742054i \(0.266148\pi\)
\(98\) −10.1154 −1.02181
\(99\) 6.60208 0.663534
\(100\) 5.89687 0.589687
\(101\) 8.40126 0.835957 0.417978 0.908457i \(-0.362739\pi\)
0.417978 + 0.908457i \(0.362739\pi\)
\(102\) 3.73291 0.369613
\(103\) 14.8100 1.45927 0.729635 0.683837i \(-0.239690\pi\)
0.729635 + 0.683837i \(0.239690\pi\)
\(104\) 1.16396 0.114136
\(105\) 45.0812 4.39948
\(106\) 7.14855 0.694329
\(107\) 13.1917 1.27529 0.637643 0.770332i \(-0.279909\pi\)
0.637643 + 0.770332i \(0.279909\pi\)
\(108\) 16.1648 1.55546
\(109\) 16.1010 1.54220 0.771100 0.636714i \(-0.219707\pi\)
0.771100 + 0.636714i \(0.219707\pi\)
\(110\) 2.75979 0.263136
\(111\) −9.21957 −0.875083
\(112\) −4.13708 −0.390917
\(113\) 2.32792 0.218993 0.109496 0.993987i \(-0.465076\pi\)
0.109496 + 0.993987i \(0.465076\pi\)
\(114\) 11.8700 1.11173
\(115\) 3.30104 0.307824
\(116\) −1.00000 −0.0928477
\(117\) −9.19166 −0.849770
\(118\) 1.67208 0.153927
\(119\) 4.67833 0.428861
\(120\) 10.8969 0.994744
\(121\) −10.3010 −0.936458
\(122\) 10.1371 0.917768
\(123\) −25.7783 −2.32435
\(124\) −7.16396 −0.643343
\(125\) −2.96062 −0.264805
\(126\) 32.6700 2.91047
\(127\) 17.7075 1.57129 0.785645 0.618678i \(-0.212332\pi\)
0.785645 + 0.618678i \(0.212332\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −20.9802 −1.84720
\(130\) −3.84229 −0.336991
\(131\) −14.1979 −1.24048 −0.620239 0.784413i \(-0.712964\pi\)
−0.620239 + 0.784413i \(0.712964\pi\)
\(132\) 2.75979 0.240209
\(133\) 14.8762 1.28993
\(134\) 0.869172 0.0750850
\(135\) −53.3606 −4.59255
\(136\) 1.13083 0.0969677
\(137\) 19.8546 1.69629 0.848146 0.529763i \(-0.177719\pi\)
0.848146 + 0.529763i \(0.177719\pi\)
\(138\) 3.30104 0.281003
\(139\) −9.21957 −0.781994 −0.390997 0.920392i \(-0.627870\pi\)
−0.390997 + 0.920392i \(0.627870\pi\)
\(140\) 13.6567 1.15420
\(141\) −0.813562 −0.0685143
\(142\) −13.7937 −1.15755
\(143\) −0.973116 −0.0813761
\(144\) 7.89687 0.658073
\(145\) 3.30104 0.274137
\(146\) 11.8092 0.977333
\(147\) 33.3914 2.75408
\(148\) −2.79293 −0.229577
\(149\) 6.49271 0.531903 0.265952 0.963986i \(-0.414314\pi\)
0.265952 + 0.963986i \(0.414314\pi\)
\(150\) −19.4658 −1.58938
\(151\) −14.3833 −1.17050 −0.585249 0.810853i \(-0.699003\pi\)
−0.585249 + 0.810853i \(0.699003\pi\)
\(152\) 3.59583 0.291660
\(153\) −8.93001 −0.721948
\(154\) 3.45875 0.278714
\(155\) 23.6485 1.89950
\(156\) −3.84229 −0.307629
\(157\) −9.79665 −0.781858 −0.390929 0.920421i \(-0.627846\pi\)
−0.390929 + 0.920421i \(0.627846\pi\)
\(158\) 16.4927 1.31209
\(159\) −23.5977 −1.87142
\(160\) 3.30104 0.260970
\(161\) 4.13708 0.326047
\(162\) −29.6700 −2.33109
\(163\) −19.8494 −1.55472 −0.777361 0.629055i \(-0.783442\pi\)
−0.777361 + 0.629055i \(0.783442\pi\)
\(164\) −7.80915 −0.609792
\(165\) −9.11020 −0.709228
\(166\) −9.19166 −0.713411
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 13.6567 1.05363
\(169\) −11.6452 −0.895784
\(170\) −3.73291 −0.286301
\(171\) −28.3958 −2.17148
\(172\) −6.35563 −0.484612
\(173\) 18.5483 1.41020 0.705101 0.709107i \(-0.250902\pi\)
0.705101 + 0.709107i \(0.250902\pi\)
\(174\) 3.30104 0.250251
\(175\) −24.3958 −1.84415
\(176\) 0.836038 0.0630187
\(177\) −5.51959 −0.414878
\(178\) 12.3225 0.923610
\(179\) −3.39501 −0.253755 −0.126878 0.991918i \(-0.540496\pi\)
−0.126878 + 0.991918i \(0.540496\pi\)
\(180\) −26.0679 −1.94299
\(181\) 22.9523 1.70603 0.853015 0.521887i \(-0.174772\pi\)
0.853015 + 0.521887i \(0.174772\pi\)
\(182\) −4.81540 −0.356942
\(183\) −33.4629 −2.47365
\(184\) 1.00000 0.0737210
\(185\) 9.21957 0.677836
\(186\) 23.6485 1.73399
\(187\) −0.945415 −0.0691356
\(188\) −0.246456 −0.0179747
\(189\) −66.8750 −4.86444
\(190\) −11.8700 −0.861140
\(191\) 3.67208 0.265702 0.132851 0.991136i \(-0.457587\pi\)
0.132851 + 0.991136i \(0.457587\pi\)
\(192\) 3.30104 0.238232
\(193\) 18.7113 1.34687 0.673433 0.739249i \(-0.264819\pi\)
0.673433 + 0.739249i \(0.264819\pi\)
\(194\) −13.2042 −0.948004
\(195\) 12.6836 0.908288
\(196\) 10.1154 0.722530
\(197\) 15.2042 1.08325 0.541626 0.840620i \(-0.317809\pi\)
0.541626 + 0.840620i \(0.317809\pi\)
\(198\) −6.60208 −0.469190
\(199\) 6.32792 0.448575 0.224287 0.974523i \(-0.427995\pi\)
0.224287 + 0.974523i \(0.427995\pi\)
\(200\) −5.89687 −0.416972
\(201\) −2.86917 −0.202376
\(202\) −8.40126 −0.591111
\(203\) 4.13708 0.290366
\(204\) −3.73291 −0.261356
\(205\) 25.7783 1.80044
\(206\) −14.8100 −1.03186
\(207\) −7.89687 −0.548871
\(208\) −1.16396 −0.0807063
\(209\) −3.00625 −0.207947
\(210\) −45.0812 −3.11090
\(211\) −13.5627 −0.933695 −0.466847 0.884338i \(-0.654610\pi\)
−0.466847 + 0.884338i \(0.654610\pi\)
\(212\) −7.14855 −0.490965
\(213\) 45.5337 3.11992
\(214\) −13.1917 −0.901764
\(215\) 20.9802 1.43084
\(216\) −16.1648 −1.09987
\(217\) 29.6379 2.01195
\(218\) −16.1010 −1.09050
\(219\) −38.9825 −2.63419
\(220\) −2.75979 −0.186065
\(221\) 1.31624 0.0885400
\(222\) 9.21957 0.618777
\(223\) −5.20416 −0.348497 −0.174248 0.984702i \(-0.555750\pi\)
−0.174248 + 0.984702i \(0.555750\pi\)
\(224\) 4.13708 0.276420
\(225\) 46.5669 3.10446
\(226\) −2.32792 −0.154851
\(227\) 2.62746 0.174391 0.0871955 0.996191i \(-0.472210\pi\)
0.0871955 + 0.996191i \(0.472210\pi\)
\(228\) −11.8700 −0.786109
\(229\) −8.80543 −0.581879 −0.290940 0.956741i \(-0.593968\pi\)
−0.290940 + 0.956741i \(0.593968\pi\)
\(230\) −3.30104 −0.217664
\(231\) −11.4175 −0.751215
\(232\) 1.00000 0.0656532
\(233\) −21.3590 −1.39927 −0.699636 0.714499i \(-0.746655\pi\)
−0.699636 + 0.714499i \(0.746655\pi\)
\(234\) 9.19166 0.600878
\(235\) 0.813562 0.0530709
\(236\) −1.67208 −0.108843
\(237\) −54.4431 −3.53646
\(238\) −4.67833 −0.303251
\(239\) 15.6666 1.01339 0.506695 0.862125i \(-0.330867\pi\)
0.506695 + 0.862125i \(0.330867\pi\)
\(240\) −10.8969 −0.703390
\(241\) −20.1369 −1.29713 −0.648565 0.761159i \(-0.724631\pi\)
−0.648565 + 0.761159i \(0.724631\pi\)
\(242\) 10.3010 0.662176
\(243\) 49.4475 3.17206
\(244\) −10.1371 −0.648960
\(245\) −33.3914 −2.13330
\(246\) 25.7783 1.64357
\(247\) 4.18541 0.266311
\(248\) 7.16396 0.454912
\(249\) 30.3421 1.92285
\(250\) 2.96062 0.187246
\(251\) −10.5436 −0.665504 −0.332752 0.943014i \(-0.607977\pi\)
−0.332752 + 0.943014i \(0.607977\pi\)
\(252\) −32.6700 −2.05802
\(253\) −0.836038 −0.0525612
\(254\) −17.7075 −1.11107
\(255\) 12.3225 0.771664
\(256\) 1.00000 0.0625000
\(257\) 3.54275 0.220991 0.110495 0.993877i \(-0.464756\pi\)
0.110495 + 0.993877i \(0.464756\pi\)
\(258\) 20.9802 1.30617
\(259\) 11.5546 0.717966
\(260\) 3.84229 0.238289
\(261\) −7.89687 −0.488804
\(262\) 14.1979 0.877150
\(263\) 0.777728 0.0479568 0.0239784 0.999712i \(-0.492367\pi\)
0.0239784 + 0.999712i \(0.492367\pi\)
\(264\) −2.75979 −0.169854
\(265\) 23.5977 1.44959
\(266\) −14.8762 −0.912121
\(267\) −40.6771 −2.48940
\(268\) −0.869172 −0.0530931
\(269\) −25.6358 −1.56304 −0.781522 0.623878i \(-0.785556\pi\)
−0.781522 + 0.623878i \(0.785556\pi\)
\(270\) 53.3606 3.24742
\(271\) 10.2590 0.623187 0.311594 0.950215i \(-0.399137\pi\)
0.311594 + 0.950215i \(0.399137\pi\)
\(272\) −1.13083 −0.0685665
\(273\) 15.8958 0.962061
\(274\) −19.8546 −1.19946
\(275\) 4.93001 0.297291
\(276\) −3.30104 −0.198699
\(277\) −7.88249 −0.473613 −0.236806 0.971557i \(-0.576101\pi\)
−0.236806 + 0.971557i \(0.576101\pi\)
\(278\) 9.21957 0.552953
\(279\) −56.5729 −3.38693
\(280\) −13.6567 −0.816142
\(281\) 11.1102 0.662779 0.331389 0.943494i \(-0.392483\pi\)
0.331389 + 0.943494i \(0.392483\pi\)
\(282\) 0.813562 0.0484469
\(283\) 11.7762 0.700024 0.350012 0.936745i \(-0.386178\pi\)
0.350012 + 0.936745i \(0.386178\pi\)
\(284\) 13.7937 0.818508
\(285\) 39.1833 2.32102
\(286\) 0.973116 0.0575416
\(287\) 32.3071 1.90703
\(288\) −7.89687 −0.465328
\(289\) −15.7212 −0.924778
\(290\) −3.30104 −0.193844
\(291\) 43.5875 2.55514
\(292\) −11.8092 −0.691078
\(293\) −2.20335 −0.128721 −0.0643605 0.997927i \(-0.520501\pi\)
−0.0643605 + 0.997927i \(0.520501\pi\)
\(294\) −33.3914 −1.94743
\(295\) 5.51959 0.321363
\(296\) 2.79293 0.162336
\(297\) 13.5144 0.784183
\(298\) −6.49271 −0.376112
\(299\) 1.16396 0.0673137
\(300\) 19.4658 1.12386
\(301\) 26.2937 1.51555
\(302\) 14.3833 0.827668
\(303\) 27.7329 1.59321
\(304\) −3.59583 −0.206235
\(305\) 33.4629 1.91608
\(306\) 8.93001 0.510495
\(307\) −21.3335 −1.21757 −0.608783 0.793337i \(-0.708342\pi\)
−0.608783 + 0.793337i \(0.708342\pi\)
\(308\) −3.45875 −0.197081
\(309\) 48.8883 2.78116
\(310\) −23.6485 −1.34315
\(311\) −10.6575 −0.604331 −0.302165 0.953256i \(-0.597709\pi\)
−0.302165 + 0.953256i \(0.597709\pi\)
\(312\) 3.84229 0.217527
\(313\) −13.7227 −0.775654 −0.387827 0.921732i \(-0.626774\pi\)
−0.387827 + 0.921732i \(0.626774\pi\)
\(314\) 9.79665 0.552857
\(315\) 107.845 6.07638
\(316\) −16.4927 −0.927787
\(317\) 3.99457 0.224357 0.112179 0.993688i \(-0.464217\pi\)
0.112179 + 0.993688i \(0.464217\pi\)
\(318\) 23.5977 1.32329
\(319\) −0.836038 −0.0468091
\(320\) −3.30104 −0.184534
\(321\) 43.5462 2.43051
\(322\) −4.13708 −0.230550
\(323\) 4.06627 0.226253
\(324\) 29.6700 1.64833
\(325\) −6.86374 −0.380732
\(326\) 19.8494 1.09935
\(327\) 53.1502 2.93921
\(328\) 7.80915 0.431188
\(329\) 1.01961 0.0562128
\(330\) 9.11020 0.501500
\(331\) −1.10313 −0.0606333 −0.0303167 0.999540i \(-0.509652\pi\)
−0.0303167 + 0.999540i \(0.509652\pi\)
\(332\) 9.19166 0.504458
\(333\) −22.0554 −1.20863
\(334\) −8.00000 −0.437741
\(335\) 2.86917 0.156760
\(336\) −13.6567 −0.745032
\(337\) −7.25793 −0.395365 −0.197682 0.980266i \(-0.563341\pi\)
−0.197682 + 0.980266i \(0.563341\pi\)
\(338\) 11.6452 0.633415
\(339\) 7.68458 0.417369
\(340\) 3.73291 0.202446
\(341\) −5.98934 −0.324341
\(342\) 28.3958 1.53547
\(343\) −12.8887 −0.695927
\(344\) 6.35563 0.342672
\(345\) 10.8969 0.586668
\(346\) −18.5483 −0.997164
\(347\) −13.0825 −0.702305 −0.351153 0.936318i \(-0.614210\pi\)
−0.351153 + 0.936318i \(0.614210\pi\)
\(348\) −3.30104 −0.176954
\(349\) −16.8152 −0.900097 −0.450048 0.893004i \(-0.648593\pi\)
−0.450048 + 0.893004i \(0.648593\pi\)
\(350\) 24.3958 1.30401
\(351\) −18.8152 −1.00428
\(352\) −0.836038 −0.0445610
\(353\) −2.61206 −0.139026 −0.0695129 0.997581i \(-0.522144\pi\)
−0.0695129 + 0.997581i \(0.522144\pi\)
\(354\) 5.51959 0.293363
\(355\) −45.5337 −2.41668
\(356\) −12.3225 −0.653091
\(357\) 15.4433 0.817349
\(358\) 3.39501 0.179432
\(359\) −33.5115 −1.76867 −0.884334 0.466856i \(-0.845387\pi\)
−0.884334 + 0.466856i \(0.845387\pi\)
\(360\) 26.0679 1.37390
\(361\) −6.06999 −0.319473
\(362\) −22.9523 −1.20634
\(363\) −34.0042 −1.78476
\(364\) 4.81540 0.252396
\(365\) 38.9825 2.04044
\(366\) 33.4629 1.74913
\(367\) 29.1833 1.52336 0.761679 0.647955i \(-0.224376\pi\)
0.761679 + 0.647955i \(0.224376\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −61.6679 −3.21030
\(370\) −9.21957 −0.479303
\(371\) 29.5741 1.53541
\(372\) −23.6485 −1.22612
\(373\) −0.698959 −0.0361907 −0.0180954 0.999836i \(-0.505760\pi\)
−0.0180954 + 0.999836i \(0.505760\pi\)
\(374\) 0.945415 0.0488862
\(375\) −9.77311 −0.504681
\(376\) 0.246456 0.0127100
\(377\) 1.16396 0.0599471
\(378\) 66.8750 3.43968
\(379\) 24.5175 1.25938 0.629690 0.776847i \(-0.283182\pi\)
0.629690 + 0.776847i \(0.283182\pi\)
\(380\) 11.8700 0.608918
\(381\) 58.4533 2.99465
\(382\) −3.67208 −0.187880
\(383\) −15.4504 −0.789479 −0.394740 0.918793i \(-0.629165\pi\)
−0.394740 + 0.918793i \(0.629165\pi\)
\(384\) −3.30104 −0.168456
\(385\) 11.4175 0.581889
\(386\) −18.7113 −0.952378
\(387\) −50.1896 −2.55128
\(388\) 13.2042 0.670340
\(389\) −25.1608 −1.27571 −0.637853 0.770158i \(-0.720177\pi\)
−0.637853 + 0.770158i \(0.720177\pi\)
\(390\) −12.6836 −0.642257
\(391\) 1.13083 0.0571884
\(392\) −10.1154 −0.510906
\(393\) −46.8679 −2.36417
\(394\) −15.2042 −0.765975
\(395\) 54.4431 2.73933
\(396\) 6.60208 0.331767
\(397\) −29.8423 −1.49774 −0.748871 0.662716i \(-0.769404\pi\)
−0.748871 + 0.662716i \(0.769404\pi\)
\(398\) −6.32792 −0.317190
\(399\) 49.1071 2.45843
\(400\) 5.89687 0.294844
\(401\) 4.55191 0.227311 0.113656 0.993520i \(-0.463744\pi\)
0.113656 + 0.993520i \(0.463744\pi\)
\(402\) 2.86917 0.143101
\(403\) 8.33858 0.415374
\(404\) 8.40126 0.417978
\(405\) −97.9418 −4.86677
\(406\) −4.13708 −0.205320
\(407\) −2.33499 −0.115741
\(408\) 3.73291 0.184807
\(409\) −38.5329 −1.90533 −0.952665 0.304023i \(-0.901670\pi\)
−0.952665 + 0.304023i \(0.901670\pi\)
\(410\) −25.7783 −1.27310
\(411\) 65.5408 3.23289
\(412\) 14.8100 0.729635
\(413\) 6.91751 0.340388
\(414\) 7.89687 0.388110
\(415\) −30.3421 −1.48943
\(416\) 1.16396 0.0570680
\(417\) −30.4342 −1.49037
\(418\) 3.00625 0.147041
\(419\) −10.2917 −0.502781 −0.251391 0.967886i \(-0.580888\pi\)
−0.251391 + 0.967886i \(0.580888\pi\)
\(420\) 45.0812 2.19974
\(421\) 12.9375 0.630533 0.315267 0.949003i \(-0.397906\pi\)
0.315267 + 0.949003i \(0.397906\pi\)
\(422\) 13.5627 0.660222
\(423\) −1.94623 −0.0946291
\(424\) 7.14855 0.347165
\(425\) −6.66835 −0.323463
\(426\) −45.5337 −2.20612
\(427\) 41.9379 2.02952
\(428\) 13.1917 0.637643
\(429\) −3.21230 −0.155091
\(430\) −20.9802 −1.01175
\(431\) −7.61831 −0.366961 −0.183480 0.983023i \(-0.558736\pi\)
−0.183480 + 0.983023i \(0.558736\pi\)
\(432\) 16.1648 0.777728
\(433\) 5.75586 0.276609 0.138305 0.990390i \(-0.455835\pi\)
0.138305 + 0.990390i \(0.455835\pi\)
\(434\) −29.6379 −1.42266
\(435\) 10.8969 0.522465
\(436\) 16.1010 0.771100
\(437\) 3.59583 0.172012
\(438\) 38.9825 1.86266
\(439\) −2.06084 −0.0983583 −0.0491792 0.998790i \(-0.515661\pi\)
−0.0491792 + 0.998790i \(0.515661\pi\)
\(440\) 2.75979 0.131568
\(441\) 79.8802 3.80382
\(442\) −1.31624 −0.0626072
\(443\) −15.9254 −0.756638 −0.378319 0.925675i \(-0.623498\pi\)
−0.378319 + 0.925675i \(0.623498\pi\)
\(444\) −9.21957 −0.437542
\(445\) 40.6771 1.92828
\(446\) 5.20416 0.246424
\(447\) 21.4327 1.01373
\(448\) −4.13708 −0.195459
\(449\) 18.7821 0.886380 0.443190 0.896428i \(-0.353847\pi\)
0.443190 + 0.896428i \(0.353847\pi\)
\(450\) −46.5669 −2.19518
\(451\) −6.52875 −0.307427
\(452\) 2.32792 0.109496
\(453\) −47.4800 −2.23080
\(454\) −2.62746 −0.123313
\(455\) −15.8958 −0.745209
\(456\) 11.8700 0.555863
\(457\) −37.7342 −1.76513 −0.882565 0.470191i \(-0.844185\pi\)
−0.882565 + 0.470191i \(0.844185\pi\)
\(458\) 8.80543 0.411451
\(459\) −18.2796 −0.853218
\(460\) 3.30104 0.153912
\(461\) 0.462477 0.0215397 0.0107699 0.999942i \(-0.496572\pi\)
0.0107699 + 0.999942i \(0.496572\pi\)
\(462\) 11.4175 0.531190
\(463\) −13.3316 −0.619574 −0.309787 0.950806i \(-0.600258\pi\)
−0.309787 + 0.950806i \(0.600258\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 78.0648 3.62017
\(466\) 21.3590 0.989435
\(467\) 7.22770 0.334458 0.167229 0.985918i \(-0.446518\pi\)
0.167229 + 0.985918i \(0.446518\pi\)
\(468\) −9.19166 −0.424885
\(469\) 3.59583 0.166040
\(470\) −0.813562 −0.0375268
\(471\) −32.3392 −1.49011
\(472\) 1.67208 0.0769635
\(473\) −5.31354 −0.244317
\(474\) 54.4431 2.50065
\(475\) −21.2042 −0.972914
\(476\) 4.67833 0.214431
\(477\) −56.4512 −2.58472
\(478\) −15.6666 −0.716576
\(479\) 36.4406 1.66501 0.832506 0.554016i \(-0.186905\pi\)
0.832506 + 0.554016i \(0.186905\pi\)
\(480\) 10.8969 0.497372
\(481\) 3.25086 0.148227
\(482\) 20.1369 0.917209
\(483\) 13.6567 0.621400
\(484\) −10.3010 −0.468229
\(485\) −43.5875 −1.97921
\(486\) −49.4475 −2.24298
\(487\) 23.6717 1.07267 0.536333 0.844006i \(-0.319809\pi\)
0.536333 + 0.844006i \(0.319809\pi\)
\(488\) 10.1371 0.458884
\(489\) −65.5235 −2.96308
\(490\) 33.3914 1.50847
\(491\) 10.3960 0.469167 0.234583 0.972096i \(-0.424627\pi\)
0.234583 + 0.972096i \(0.424627\pi\)
\(492\) −25.7783 −1.16218
\(493\) 1.13083 0.0509299
\(494\) −4.18541 −0.188311
\(495\) −21.7937 −0.979556
\(496\) −7.16396 −0.321671
\(497\) −57.0658 −2.55975
\(498\) −30.3421 −1.35966
\(499\) 38.5358 1.72510 0.862550 0.505972i \(-0.168866\pi\)
0.862550 + 0.505972i \(0.168866\pi\)
\(500\) −2.96062 −0.132403
\(501\) 26.4083 1.17984
\(502\) 10.5436 0.470582
\(503\) 14.4827 0.645753 0.322877 0.946441i \(-0.395350\pi\)
0.322877 + 0.946441i \(0.395350\pi\)
\(504\) 32.6700 1.45524
\(505\) −27.7329 −1.23410
\(506\) 0.836038 0.0371664
\(507\) −38.4413 −1.70724
\(508\) 17.7075 0.785645
\(509\) 1.80895 0.0801802 0.0400901 0.999196i \(-0.487236\pi\)
0.0400901 + 0.999196i \(0.487236\pi\)
\(510\) −12.3225 −0.545649
\(511\) 48.8554 2.16124
\(512\) −1.00000 −0.0441942
\(513\) −58.1258 −2.56632
\(514\) −3.54275 −0.156264
\(515\) −48.8883 −2.15428
\(516\) −20.9802 −0.923601
\(517\) −0.206047 −0.00906192
\(518\) −11.5546 −0.507678
\(519\) 61.2288 2.68764
\(520\) −3.84229 −0.168495
\(521\) 29.4627 1.29078 0.645392 0.763851i \(-0.276694\pi\)
0.645392 + 0.763851i \(0.276694\pi\)
\(522\) 7.89687 0.345637
\(523\) 8.69335 0.380134 0.190067 0.981771i \(-0.439130\pi\)
0.190067 + 0.981771i \(0.439130\pi\)
\(524\) −14.1979 −0.620239
\(525\) −80.5316 −3.51469
\(526\) −0.777728 −0.0339106
\(527\) 8.10121 0.352894
\(528\) 2.75979 0.120105
\(529\) 1.00000 0.0434783
\(530\) −23.5977 −1.02502
\(531\) −13.2042 −0.573012
\(532\) 14.8762 0.644967
\(533\) 9.08956 0.393713
\(534\) 40.6771 1.76027
\(535\) −43.5462 −1.88267
\(536\) 0.869172 0.0375425
\(537\) −11.2071 −0.483621
\(538\) 25.6358 1.10524
\(539\) 8.45687 0.364263
\(540\) −53.3606 −2.29627
\(541\) 13.6112 0.585193 0.292596 0.956236i \(-0.405481\pi\)
0.292596 + 0.956236i \(0.405481\pi\)
\(542\) −10.2590 −0.440660
\(543\) 75.7664 3.25145
\(544\) 1.13083 0.0484839
\(545\) −53.1502 −2.27670
\(546\) −15.8958 −0.680280
\(547\) 37.2442 1.59245 0.796223 0.605004i \(-0.206828\pi\)
0.796223 + 0.605004i \(0.206828\pi\)
\(548\) 19.8546 0.848146
\(549\) −80.0512 −3.41650
\(550\) −4.93001 −0.210216
\(551\) 3.59583 0.153188
\(552\) 3.30104 0.140502
\(553\) 68.2316 2.90150
\(554\) 7.88249 0.334895
\(555\) 30.4342 1.29186
\(556\) −9.21957 −0.390997
\(557\) −30.3750 −1.28703 −0.643515 0.765434i \(-0.722525\pi\)
−0.643515 + 0.765434i \(0.722525\pi\)
\(558\) 56.5729 2.39492
\(559\) 7.39771 0.312890
\(560\) 13.6567 0.577100
\(561\) −3.12085 −0.131763
\(562\) −11.1102 −0.468655
\(563\) −40.4131 −1.70321 −0.851604 0.524186i \(-0.824370\pi\)
−0.851604 + 0.524186i \(0.824370\pi\)
\(564\) −0.813562 −0.0342571
\(565\) −7.68458 −0.323293
\(566\) −11.7762 −0.494992
\(567\) −122.747 −5.15489
\(568\) −13.7937 −0.578773
\(569\) −15.0088 −0.629201 −0.314600 0.949224i \(-0.601870\pi\)
−0.314600 + 0.949224i \(0.601870\pi\)
\(570\) −39.1833 −1.64121
\(571\) −1.94460 −0.0813789 −0.0406894 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(572\) −0.973116 −0.0406880
\(573\) 12.1217 0.506390
\(574\) −32.3071 −1.34847
\(575\) −5.89687 −0.245917
\(576\) 7.89687 0.329036
\(577\) 3.68748 0.153512 0.0767560 0.997050i \(-0.475544\pi\)
0.0767560 + 0.997050i \(0.475544\pi\)
\(578\) 15.7212 0.653917
\(579\) 61.7666 2.56693
\(580\) 3.30104 0.137068
\(581\) −38.0266 −1.57761
\(582\) −43.5875 −1.80676
\(583\) −5.97646 −0.247520
\(584\) 11.8092 0.488666
\(585\) 30.3421 1.25449
\(586\) 2.20335 0.0910194
\(587\) 24.7421 1.02121 0.510607 0.859814i \(-0.329421\pi\)
0.510607 + 0.859814i \(0.329421\pi\)
\(588\) 33.3914 1.37704
\(589\) 25.7604 1.06144
\(590\) −5.51959 −0.227238
\(591\) 50.1896 2.06452
\(592\) −2.79293 −0.114789
\(593\) 25.6614 1.05379 0.526894 0.849931i \(-0.323356\pi\)
0.526894 + 0.849931i \(0.323356\pi\)
\(594\) −13.5144 −0.554501
\(595\) −15.4433 −0.633115
\(596\) 6.49271 0.265952
\(597\) 20.8887 0.854919
\(598\) −1.16396 −0.0475980
\(599\) −41.4094 −1.69194 −0.845970 0.533230i \(-0.820978\pi\)
−0.845970 + 0.533230i \(0.820978\pi\)
\(600\) −19.4658 −0.794689
\(601\) 36.5405 1.49052 0.745258 0.666776i \(-0.232326\pi\)
0.745258 + 0.666776i \(0.232326\pi\)
\(602\) −26.2937 −1.07165
\(603\) −6.86374 −0.279513
\(604\) −14.3833 −0.585249
\(605\) 34.0042 1.38247
\(606\) −27.7329 −1.12657
\(607\) −33.2094 −1.34793 −0.673964 0.738764i \(-0.735410\pi\)
−0.673964 + 0.738764i \(0.735410\pi\)
\(608\) 3.59583 0.145830
\(609\) 13.6567 0.553396
\(610\) −33.4629 −1.35487
\(611\) 0.286866 0.0116053
\(612\) −8.93001 −0.360974
\(613\) 13.2860 0.536617 0.268309 0.963333i \(-0.413535\pi\)
0.268309 + 0.963333i \(0.413535\pi\)
\(614\) 21.3335 0.860950
\(615\) 85.0954 3.43138
\(616\) 3.45875 0.139357
\(617\) −7.84751 −0.315929 −0.157965 0.987445i \(-0.550493\pi\)
−0.157965 + 0.987445i \(0.550493\pi\)
\(618\) −48.8883 −1.96658
\(619\) −24.0856 −0.968083 −0.484042 0.875045i \(-0.660832\pi\)
−0.484042 + 0.875045i \(0.660832\pi\)
\(620\) 23.6485 0.949748
\(621\) −16.1648 −0.648670
\(622\) 10.6575 0.427326
\(623\) 50.9791 2.04244
\(624\) −3.84229 −0.153815
\(625\) −19.7113 −0.788450
\(626\) 13.7227 0.548471
\(627\) −9.92376 −0.396317
\(628\) −9.79665 −0.390929
\(629\) 3.15832 0.125931
\(630\) −107.845 −4.29665
\(631\) −34.9316 −1.39061 −0.695303 0.718716i \(-0.744730\pi\)
−0.695303 + 0.718716i \(0.744730\pi\)
\(632\) 16.4927 0.656045
\(633\) −44.7710 −1.77949
\(634\) −3.99457 −0.158645
\(635\) −58.4533 −2.31965
\(636\) −23.5977 −0.935709
\(637\) −11.7740 −0.466502
\(638\) 0.836038 0.0330990
\(639\) 108.927 4.30910
\(640\) 3.30104 0.130485
\(641\) 38.2867 1.51223 0.756116 0.654438i \(-0.227095\pi\)
0.756116 + 0.654438i \(0.227095\pi\)
\(642\) −43.5462 −1.71863
\(643\) −41.1879 −1.62429 −0.812145 0.583455i \(-0.801700\pi\)
−0.812145 + 0.583455i \(0.801700\pi\)
\(644\) 4.13708 0.163024
\(645\) 69.2565 2.72697
\(646\) −4.06627 −0.159985
\(647\) 0.170006 0.00668362 0.00334181 0.999994i \(-0.498936\pi\)
0.00334181 + 0.999994i \(0.498936\pi\)
\(648\) −29.6700 −1.16555
\(649\) −1.39792 −0.0548731
\(650\) 6.86374 0.269218
\(651\) 97.8359 3.83449
\(652\) −19.8494 −0.777361
\(653\) −1.49664 −0.0585679 −0.0292840 0.999571i \(-0.509323\pi\)
−0.0292840 + 0.999571i \(0.509323\pi\)
\(654\) −53.1502 −2.07834
\(655\) 46.8679 1.83128
\(656\) −7.80915 −0.304896
\(657\) −93.2554 −3.63824
\(658\) −1.01961 −0.0397485
\(659\) 10.7290 0.417942 0.208971 0.977922i \(-0.432989\pi\)
0.208971 + 0.977922i \(0.432989\pi\)
\(660\) −9.11020 −0.354614
\(661\) −37.1937 −1.44667 −0.723334 0.690498i \(-0.757391\pi\)
−0.723334 + 0.690498i \(0.757391\pi\)
\(662\) 1.10313 0.0428742
\(663\) 4.34497 0.168745
\(664\) −9.19166 −0.356706
\(665\) −49.1071 −1.90429
\(666\) 22.0554 0.854630
\(667\) 1.00000 0.0387202
\(668\) 8.00000 0.309529
\(669\) −17.1792 −0.664185
\(670\) −2.86917 −0.110846
\(671\) −8.47498 −0.327173
\(672\) 13.6567 0.526817
\(673\) 43.2139 1.66577 0.832887 0.553443i \(-0.186686\pi\)
0.832887 + 0.553443i \(0.186686\pi\)
\(674\) 7.25793 0.279565
\(675\) 95.3217 3.66893
\(676\) −11.6452 −0.447892
\(677\) −17.1658 −0.659735 −0.329868 0.944027i \(-0.607004\pi\)
−0.329868 + 0.944027i \(0.607004\pi\)
\(678\) −7.68458 −0.295124
\(679\) −54.6267 −2.09638
\(680\) −3.73291 −0.143151
\(681\) 8.67337 0.332364
\(682\) 5.98934 0.229344
\(683\) 27.7092 1.06026 0.530131 0.847916i \(-0.322143\pi\)
0.530131 + 0.847916i \(0.322143\pi\)
\(684\) −28.3958 −1.08574
\(685\) −65.5408 −2.50418
\(686\) 12.8887 0.492094
\(687\) −29.0671 −1.10898
\(688\) −6.35563 −0.242306
\(689\) 8.32065 0.316992
\(690\) −10.8969 −0.414837
\(691\) 13.7429 0.522804 0.261402 0.965230i \(-0.415815\pi\)
0.261402 + 0.965230i \(0.415815\pi\)
\(692\) 18.5483 0.705101
\(693\) −27.3133 −1.03755
\(694\) 13.0825 0.496605
\(695\) 30.4342 1.15443
\(696\) 3.30104 0.125126
\(697\) 8.83081 0.334491
\(698\) 16.8152 0.636465
\(699\) −70.5068 −2.66681
\(700\) −24.3958 −0.922076
\(701\) −41.5743 −1.57024 −0.785120 0.619344i \(-0.787399\pi\)
−0.785120 + 0.619344i \(0.787399\pi\)
\(702\) 18.8152 0.710134
\(703\) 10.0429 0.378775
\(704\) 0.836038 0.0315094
\(705\) 2.68560 0.101146
\(706\) 2.61206 0.0983061
\(707\) −34.7567 −1.30716
\(708\) −5.51959 −0.207439
\(709\) 14.3910 0.540465 0.270232 0.962795i \(-0.412899\pi\)
0.270232 + 0.962795i \(0.412899\pi\)
\(710\) 45.5337 1.70885
\(711\) −130.241 −4.88441
\(712\) 12.3225 0.461805
\(713\) 7.16396 0.268293
\(714\) −15.4433 −0.577953
\(715\) 3.21230 0.120133
\(716\) −3.39501 −0.126878
\(717\) 51.7162 1.93138
\(718\) 33.5115 1.25064
\(719\) 4.21294 0.157116 0.0785581 0.996910i \(-0.474968\pi\)
0.0785581 + 0.996910i \(0.474968\pi\)
\(720\) −26.0679 −0.971493
\(721\) −61.2700 −2.28181
\(722\) 6.06999 0.225902
\(723\) −66.4726 −2.47214
\(724\) 22.9523 0.853015
\(725\) −5.89687 −0.219004
\(726\) 34.0042 1.26201
\(727\) 8.81250 0.326837 0.163419 0.986557i \(-0.447748\pi\)
0.163419 + 0.986557i \(0.447748\pi\)
\(728\) −4.81540 −0.178471
\(729\) 74.2183 2.74883
\(730\) −38.9825 −1.44281
\(731\) 7.18712 0.265825
\(732\) −33.4629 −1.23682
\(733\) −28.5454 −1.05435 −0.527174 0.849757i \(-0.676749\pi\)
−0.527174 + 0.849757i \(0.676749\pi\)
\(734\) −29.1833 −1.07718
\(735\) −110.226 −4.06576
\(736\) 1.00000 0.0368605
\(737\) −0.726660 −0.0267669
\(738\) 61.6679 2.27003
\(739\) 36.4306 1.34012 0.670061 0.742306i \(-0.266268\pi\)
0.670061 + 0.742306i \(0.266268\pi\)
\(740\) 9.21957 0.338918
\(741\) 13.8162 0.507552
\(742\) −29.5741 −1.08570
\(743\) −34.6366 −1.27070 −0.635348 0.772226i \(-0.719143\pi\)
−0.635348 + 0.772226i \(0.719143\pi\)
\(744\) 23.6485 0.866997
\(745\) −21.4327 −0.785233
\(746\) 0.698959 0.0255907
\(747\) 72.5854 2.65576
\(748\) −0.945415 −0.0345678
\(749\) −54.5750 −1.99413
\(750\) 9.77311 0.356864
\(751\) −3.25957 −0.118943 −0.0594717 0.998230i \(-0.518942\pi\)
−0.0594717 + 0.998230i \(0.518942\pi\)
\(752\) −0.246456 −0.00898733
\(753\) −34.8047 −1.26836
\(754\) −1.16396 −0.0423890
\(755\) 47.4800 1.72797
\(756\) −66.8750 −2.43222
\(757\) 31.9458 1.16109 0.580545 0.814228i \(-0.302839\pi\)
0.580545 + 0.814228i \(0.302839\pi\)
\(758\) −24.5175 −0.890516
\(759\) −2.75979 −0.100174
\(760\) −11.8700 −0.430570
\(761\) −25.2695 −0.916020 −0.458010 0.888947i \(-0.651438\pi\)
−0.458010 + 0.888947i \(0.651438\pi\)
\(762\) −58.4533 −2.11754
\(763\) −66.6113 −2.41149
\(764\) 3.67208 0.132851
\(765\) 29.4783 1.06579
\(766\) 15.4504 0.558246
\(767\) 1.94623 0.0702744
\(768\) 3.30104 0.119116
\(769\) −2.08458 −0.0751720 −0.0375860 0.999293i \(-0.511967\pi\)
−0.0375860 + 0.999293i \(0.511967\pi\)
\(770\) −11.4175 −0.411458
\(771\) 11.6948 0.421176
\(772\) 18.7113 0.673433
\(773\) −53.1704 −1.91241 −0.956204 0.292701i \(-0.905446\pi\)
−0.956204 + 0.292701i \(0.905446\pi\)
\(774\) 50.1896 1.80403
\(775\) −42.2450 −1.51748
\(776\) −13.2042 −0.474002
\(777\) 38.1421 1.36834
\(778\) 25.1608 0.902060
\(779\) 28.0804 1.00608
\(780\) 12.6836 0.454144
\(781\) 11.5321 0.412651
\(782\) −1.13083 −0.0404383
\(783\) −16.1648 −0.577682
\(784\) 10.1154 0.361265
\(785\) 32.3392 1.15423
\(786\) 46.8679 1.67172
\(787\) 48.5679 1.73126 0.865629 0.500686i \(-0.166919\pi\)
0.865629 + 0.500686i \(0.166919\pi\)
\(788\) 15.2042 0.541626
\(789\) 2.56731 0.0913988
\(790\) −54.4431 −1.93700
\(791\) −9.63081 −0.342432
\(792\) −6.60208 −0.234595
\(793\) 11.7992 0.419001
\(794\) 29.8423 1.05906
\(795\) 77.8969 2.76272
\(796\) 6.32792 0.224287
\(797\) −37.9692 −1.34494 −0.672469 0.740126i \(-0.734766\pi\)
−0.672469 + 0.740126i \(0.734766\pi\)
\(798\) −49.1071 −1.73837
\(799\) 0.278700 0.00985968
\(800\) −5.89687 −0.208486
\(801\) −97.3092 −3.43825
\(802\) −4.55191 −0.160733
\(803\) −9.87290 −0.348407
\(804\) −2.86917 −0.101188
\(805\) −13.6567 −0.481334
\(806\) −8.33858 −0.293714
\(807\) −84.6249 −2.97894
\(808\) −8.40126 −0.295555
\(809\) 26.9795 0.948549 0.474274 0.880377i \(-0.342710\pi\)
0.474274 + 0.880377i \(0.342710\pi\)
\(810\) 97.9418 3.44133
\(811\) 44.9333 1.57782 0.788910 0.614508i \(-0.210646\pi\)
0.788910 + 0.614508i \(0.210646\pi\)
\(812\) 4.13708 0.145183
\(813\) 33.8652 1.18771
\(814\) 2.33499 0.0818415
\(815\) 65.5235 2.29519
\(816\) −3.73291 −0.130678
\(817\) 22.8538 0.799552
\(818\) 38.5329 1.34727
\(819\) 38.0266 1.32876
\(820\) 25.7783 0.900219
\(821\) 26.3960 0.921227 0.460613 0.887601i \(-0.347629\pi\)
0.460613 + 0.887601i \(0.347629\pi\)
\(822\) −65.5408 −2.28600
\(823\) 32.9525 1.14865 0.574326 0.818627i \(-0.305264\pi\)
0.574326 + 0.818627i \(0.305264\pi\)
\(824\) −14.8100 −0.515930
\(825\) 16.2742 0.566593
\(826\) −6.91751 −0.240691
\(827\) 26.9927 0.938628 0.469314 0.883031i \(-0.344501\pi\)
0.469314 + 0.883031i \(0.344501\pi\)
\(828\) −7.89687 −0.274435
\(829\) 12.5358 0.435387 0.217693 0.976017i \(-0.430147\pi\)
0.217693 + 0.976017i \(0.430147\pi\)
\(830\) 30.3421 1.05319
\(831\) −26.0204 −0.902639
\(832\) −1.16396 −0.0403531
\(833\) −11.4388 −0.396331
\(834\) 30.4342 1.05385
\(835\) −26.4083 −0.913898
\(836\) −3.00625 −0.103973
\(837\) −115.804 −4.00277
\(838\) 10.2917 0.355520
\(839\) −41.5089 −1.43305 −0.716524 0.697563i \(-0.754268\pi\)
−0.716524 + 0.697563i \(0.754268\pi\)
\(840\) −45.0812 −1.55545
\(841\) 1.00000 0.0344828
\(842\) −12.9375 −0.445854
\(843\) 36.6752 1.26316
\(844\) −13.5627 −0.466847
\(845\) 38.4413 1.32242
\(846\) 1.94623 0.0669129
\(847\) 42.6162 1.46431
\(848\) −7.14855 −0.245482
\(849\) 38.8738 1.33415
\(850\) 6.66835 0.228723
\(851\) 2.79293 0.0957404
\(852\) 45.5337 1.55996
\(853\) −47.8387 −1.63797 −0.818983 0.573818i \(-0.805462\pi\)
−0.818983 + 0.573818i \(0.805462\pi\)
\(854\) −41.9379 −1.43509
\(855\) 93.7358 3.20570
\(856\) −13.1917 −0.450882
\(857\) −19.8593 −0.678382 −0.339191 0.940718i \(-0.610153\pi\)
−0.339191 + 0.940718i \(0.610153\pi\)
\(858\) 3.21230 0.109666
\(859\) −11.4561 −0.390876 −0.195438 0.980716i \(-0.562613\pi\)
−0.195438 + 0.980716i \(0.562613\pi\)
\(860\) 20.9802 0.715418
\(861\) 106.647 3.63452
\(862\) 7.61831 0.259481
\(863\) 21.5325 0.732974 0.366487 0.930423i \(-0.380560\pi\)
0.366487 + 0.930423i \(0.380560\pi\)
\(864\) −16.1648 −0.549937
\(865\) −61.2288 −2.08184
\(866\) −5.75586 −0.195592
\(867\) −51.8964 −1.76249
\(868\) 29.6379 1.00598
\(869\) −13.7885 −0.467744
\(870\) −10.8969 −0.369439
\(871\) 1.01168 0.0342796
\(872\) −16.1010 −0.545250
\(873\) 104.272 3.52906
\(874\) −3.59583 −0.121631
\(875\) 12.2483 0.414068
\(876\) −38.9825 −1.31710
\(877\) 42.9106 1.44899 0.724494 0.689281i \(-0.242073\pi\)
0.724494 + 0.689281i \(0.242073\pi\)
\(878\) 2.06084 0.0695498
\(879\) −7.27334 −0.245324
\(880\) −2.75979 −0.0930327
\(881\) −55.2237 −1.86053 −0.930267 0.366882i \(-0.880425\pi\)
−0.930267 + 0.366882i \(0.880425\pi\)
\(882\) −79.8802 −2.68971
\(883\) −12.2234 −0.411350 −0.205675 0.978620i \(-0.565939\pi\)
−0.205675 + 0.978620i \(0.565939\pi\)
\(884\) 1.31624 0.0442700
\(885\) 18.2204 0.612472
\(886\) 15.9254 0.535024
\(887\) −43.1248 −1.44799 −0.723995 0.689806i \(-0.757696\pi\)
−0.723995 + 0.689806i \(0.757696\pi\)
\(888\) 9.21957 0.309389
\(889\) −73.2574 −2.45698
\(890\) −40.6771 −1.36350
\(891\) 24.8052 0.831006
\(892\) −5.20416 −0.174248
\(893\) 0.886215 0.0296560
\(894\) −21.4327 −0.716816
\(895\) 11.2071 0.374611
\(896\) 4.13708 0.138210
\(897\) 3.84229 0.128290
\(898\) −18.7821 −0.626766
\(899\) 7.16396 0.238932
\(900\) 46.5669 1.55223
\(901\) 8.08379 0.269310
\(902\) 6.52875 0.217383
\(903\) 86.7967 2.88841
\(904\) −2.32792 −0.0774256
\(905\) −75.7664 −2.51856
\(906\) 47.4800 1.57742
\(907\) 11.7150 0.388989 0.194495 0.980904i \(-0.437693\pi\)
0.194495 + 0.980904i \(0.437693\pi\)
\(908\) 2.62746 0.0871955
\(909\) 66.3437 2.20048
\(910\) 15.8958 0.526942
\(911\) 29.2160 0.967969 0.483984 0.875077i \(-0.339189\pi\)
0.483984 + 0.875077i \(0.339189\pi\)
\(912\) −11.8700 −0.393055
\(913\) 7.68458 0.254322
\(914\) 37.7342 1.24814
\(915\) 110.462 3.65178
\(916\) −8.80543 −0.290940
\(917\) 58.7379 1.93970
\(918\) 18.2796 0.603316
\(919\) 18.6117 0.613943 0.306971 0.951719i \(-0.400684\pi\)
0.306971 + 0.951719i \(0.400684\pi\)
\(920\) −3.30104 −0.108832
\(921\) −70.4227 −2.32051
\(922\) −0.462477 −0.0152309
\(923\) −16.0554 −0.528470
\(924\) −11.4175 −0.375608
\(925\) −16.4695 −0.541515
\(926\) 13.3316 0.438105
\(927\) 116.952 3.84122
\(928\) 1.00000 0.0328266
\(929\) 28.0255 0.919486 0.459743 0.888052i \(-0.347942\pi\)
0.459743 + 0.888052i \(0.347942\pi\)
\(930\) −78.0648 −2.55985
\(931\) −36.3734 −1.19209
\(932\) −21.3590 −0.699636
\(933\) −35.1808 −1.15177
\(934\) −7.22770 −0.236498
\(935\) 3.12085 0.102063
\(936\) 9.19166 0.300439
\(937\) 18.4158 0.601617 0.300809 0.953685i \(-0.402743\pi\)
0.300809 + 0.953685i \(0.402743\pi\)
\(938\) −3.59583 −0.117408
\(939\) −45.2993 −1.47829
\(940\) 0.813562 0.0265355
\(941\) −29.9006 −0.974732 −0.487366 0.873198i \(-0.662042\pi\)
−0.487366 + 0.873198i \(0.662042\pi\)
\(942\) 32.3392 1.05367
\(943\) 7.80915 0.254301
\(944\) −1.67208 −0.0544214
\(945\) 220.757 7.18123
\(946\) 5.31354 0.172758
\(947\) 45.1527 1.46727 0.733633 0.679546i \(-0.237823\pi\)
0.733633 + 0.679546i \(0.237823\pi\)
\(948\) −54.4431 −1.76823
\(949\) 13.7454 0.446195
\(950\) 21.2042 0.687954
\(951\) 13.1862 0.427593
\(952\) −4.67833 −0.151625
\(953\) 17.0235 0.551445 0.275723 0.961237i \(-0.411083\pi\)
0.275723 + 0.961237i \(0.411083\pi\)
\(954\) 56.4512 1.82768
\(955\) −12.1217 −0.392248
\(956\) 15.6666 0.506695
\(957\) −2.75979 −0.0892115
\(958\) −36.4406 −1.17734
\(959\) −82.1400 −2.65244
\(960\) −10.8969 −0.351695
\(961\) 20.3224 0.655560
\(962\) −3.25086 −0.104812
\(963\) 104.173 3.35692
\(964\) −20.1369 −0.648565
\(965\) −61.7666 −1.98834
\(966\) −13.6567 −0.439396
\(967\) −9.32642 −0.299918 −0.149959 0.988692i \(-0.547914\pi\)
−0.149959 + 0.988692i \(0.547914\pi\)
\(968\) 10.3010 0.331088
\(969\) 13.4229 0.431206
\(970\) 43.5875 1.39951
\(971\) −28.2433 −0.906372 −0.453186 0.891416i \(-0.649713\pi\)
−0.453186 + 0.891416i \(0.649713\pi\)
\(972\) 49.4475 1.58603
\(973\) 38.1421 1.22278
\(974\) −23.6717 −0.758490
\(975\) −22.6575 −0.725620
\(976\) −10.1371 −0.324480
\(977\) 2.86063 0.0915195 0.0457598 0.998952i \(-0.485429\pi\)
0.0457598 + 0.998952i \(0.485429\pi\)
\(978\) 65.5235 2.09521
\(979\) −10.3021 −0.329255
\(980\) −33.3914 −1.06665
\(981\) 127.148 4.05952
\(982\) −10.3960 −0.331751
\(983\) −4.45936 −0.142232 −0.0711158 0.997468i \(-0.522656\pi\)
−0.0711158 + 0.997468i \(0.522656\pi\)
\(984\) 25.7783 0.821783
\(985\) −50.1896 −1.59917
\(986\) −1.13083 −0.0360129
\(987\) 3.36577 0.107134
\(988\) 4.18541 0.133156
\(989\) 6.35563 0.202097
\(990\) 21.7937 0.692651
\(991\) −0.388761 −0.0123494 −0.00617469 0.999981i \(-0.501965\pi\)
−0.00617469 + 0.999981i \(0.501965\pi\)
\(992\) 7.16396 0.227456
\(993\) −3.64147 −0.115558
\(994\) 57.0658 1.81002
\(995\) −20.8887 −0.662218
\(996\) 30.3421 0.961425
\(997\) −2.94289 −0.0932022 −0.0466011 0.998914i \(-0.514839\pi\)
−0.0466011 + 0.998914i \(0.514839\pi\)
\(998\) −38.5358 −1.21983
\(999\) −45.1471 −1.42839
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 1334.2.a.d.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1334.2.a.d.1.4 4 1.1 even 1 trivial