Properties

Label 1470.3.f.a.391.1
Level $1470$
Weight $3$
Character 1470.391
Analytic conductor $40.055$
Analytic rank $0$
Dimension $8$
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] = [1470,3,Mod(391,1470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1470, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1470.391");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1470.f (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(40.0545988610\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 391.1
Root \(1.01575 - 1.40294i\) of defining polynomial
Character \(\chi\) \(=\) 1470.391
Dual form 1470.3.f.a.391.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949i q^{6} -2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949i q^{6} -2.82843 q^{8} -3.00000 q^{9} +3.16228i q^{10} +12.3241 q^{11} -3.46410i q^{12} +7.26007i q^{13} -3.87298 q^{15} +4.00000 q^{16} -9.29122i q^{17} +4.24264 q^{18} +6.07644i q^{19} -4.47214i q^{20} -17.4289 q^{22} -2.25029 q^{23} +4.89898i q^{24} -5.00000 q^{25} -10.2673i q^{26} +5.19615i q^{27} +42.2122 q^{29} +5.47723 q^{30} +1.21852i q^{31} -5.65685 q^{32} -21.3460i q^{33} +13.1398i q^{34} -6.00000 q^{36} +35.0593 q^{37} -8.59339i q^{38} +12.5748 q^{39} +6.32456i q^{40} +57.8811i q^{41} -34.0190 q^{43} +24.6482 q^{44} +6.70820i q^{45} +3.18238 q^{46} +57.0897i q^{47} -6.92820i q^{48} +7.07107 q^{50} -16.0929 q^{51} +14.5201i q^{52} +14.5402 q^{53} -7.34847i q^{54} -27.5575i q^{55} +10.5247 q^{57} -59.6971 q^{58} -57.8582i q^{59} -7.74597 q^{60} -5.86193i q^{61} -1.72325i q^{62} +8.00000 q^{64} +16.2340 q^{65} +30.1878i q^{66} +49.4709 q^{67} -18.5824i q^{68} +3.89761i q^{69} +101.986 q^{71} +8.48528 q^{72} -82.3051i q^{73} -49.5813 q^{74} +8.66025i q^{75} +12.1529i q^{76} -17.7835 q^{78} +111.706 q^{79} -8.94427i q^{80} +9.00000 q^{81} -81.8562i q^{82} +91.6237i q^{83} -20.7758 q^{85} +48.1102 q^{86} -73.1137i q^{87} -34.8578 q^{88} +127.795i q^{89} -9.48683i q^{90} -4.50057 q^{92} +2.11055 q^{93} -80.7370i q^{94} +13.5873 q^{95} +9.79796i q^{96} +61.4455i q^{97} -36.9723 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{4} - 24 q^{9} + 8 q^{11} + 32 q^{16} - 48 q^{22} - 24 q^{23} - 40 q^{25} + 72 q^{29} - 48 q^{36} + 192 q^{37} - 48 q^{39} - 112 q^{43} + 16 q^{44} - 16 q^{46} - 168 q^{51} - 64 q^{53} + 216 q^{57} - 208 q^{58} + 64 q^{64} - 40 q^{65} + 240 q^{67} + 8 q^{71} + 32 q^{74} - 192 q^{78} - 24 q^{79} + 72 q^{81} + 120 q^{85} + 80 q^{86} - 96 q^{88} - 48 q^{92} + 264 q^{93} + 80 q^{95} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −0.707107
\(3\) − 1.73205i − 0.577350i
\(4\) 2.00000 0.500000
\(5\) − 2.23607i − 0.447214i
\(6\) 2.44949i 0.408248i
\(7\) 0 0
\(8\) −2.82843 −0.353553
\(9\) −3.00000 −0.333333
\(10\) 3.16228i 0.316228i
\(11\) 12.3241 1.12037 0.560187 0.828366i \(-0.310729\pi\)
0.560187 + 0.828366i \(0.310729\pi\)
\(12\) − 3.46410i − 0.288675i
\(13\) 7.26007i 0.558467i 0.960223 + 0.279233i \(0.0900803\pi\)
−0.960223 + 0.279233i \(0.909920\pi\)
\(14\) 0 0
\(15\) −3.87298 −0.258199
\(16\) 4.00000 0.250000
\(17\) − 9.29122i − 0.546542i −0.961937 0.273271i \(-0.911894\pi\)
0.961937 0.273271i \(-0.0881056\pi\)
\(18\) 4.24264 0.235702
\(19\) 6.07644i 0.319813i 0.987132 + 0.159906i \(0.0511192\pi\)
−0.987132 + 0.159906i \(0.948881\pi\)
\(20\) − 4.47214i − 0.223607i
\(21\) 0 0
\(22\) −17.4289 −0.792224
\(23\) −2.25029 −0.0978385 −0.0489192 0.998803i \(-0.515578\pi\)
−0.0489192 + 0.998803i \(0.515578\pi\)
\(24\) 4.89898i 0.204124i
\(25\) −5.00000 −0.200000
\(26\) − 10.2673i − 0.394896i
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) 42.2122 1.45559 0.727797 0.685793i \(-0.240544\pi\)
0.727797 + 0.685793i \(0.240544\pi\)
\(30\) 5.47723 0.182574
\(31\) 1.21852i 0.0393072i 0.999807 + 0.0196536i \(0.00625634\pi\)
−0.999807 + 0.0196536i \(0.993744\pi\)
\(32\) −5.65685 −0.176777
\(33\) − 21.3460i − 0.646848i
\(34\) 13.1398i 0.386464i
\(35\) 0 0
\(36\) −6.00000 −0.166667
\(37\) 35.0593 0.947548 0.473774 0.880647i \(-0.342891\pi\)
0.473774 + 0.880647i \(0.342891\pi\)
\(38\) − 8.59339i − 0.226142i
\(39\) 12.5748 0.322431
\(40\) 6.32456i 0.158114i
\(41\) 57.8811i 1.41173i 0.708345 + 0.705867i \(0.249442\pi\)
−0.708345 + 0.705867i \(0.750558\pi\)
\(42\) 0 0
\(43\) −34.0190 −0.791140 −0.395570 0.918436i \(-0.629453\pi\)
−0.395570 + 0.918436i \(0.629453\pi\)
\(44\) 24.6482 0.560187
\(45\) 6.70820i 0.149071i
\(46\) 3.18238 0.0691823
\(47\) 57.0897i 1.21467i 0.794444 + 0.607337i \(0.207762\pi\)
−0.794444 + 0.607337i \(0.792238\pi\)
\(48\) − 6.92820i − 0.144338i
\(49\) 0 0
\(50\) 7.07107 0.141421
\(51\) −16.0929 −0.315546
\(52\) 14.5201i 0.279233i
\(53\) 14.5402 0.274343 0.137172 0.990547i \(-0.456199\pi\)
0.137172 + 0.990547i \(0.456199\pi\)
\(54\) − 7.34847i − 0.136083i
\(55\) − 27.5575i − 0.501046i
\(56\) 0 0
\(57\) 10.5247 0.184644
\(58\) −59.6971 −1.02926
\(59\) − 57.8582i − 0.980647i −0.871540 0.490324i \(-0.836879\pi\)
0.871540 0.490324i \(-0.163121\pi\)
\(60\) −7.74597 −0.129099
\(61\) − 5.86193i − 0.0960972i −0.998845 0.0480486i \(-0.984700\pi\)
0.998845 0.0480486i \(-0.0153002\pi\)
\(62\) − 1.72325i − 0.0277944i
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) 16.2340 0.249754
\(66\) 30.1878i 0.457391i
\(67\) 49.4709 0.738372 0.369186 0.929356i \(-0.379637\pi\)
0.369186 + 0.929356i \(0.379637\pi\)
\(68\) − 18.5824i − 0.273271i
\(69\) 3.89761i 0.0564871i
\(70\) 0 0
\(71\) 101.986 1.43643 0.718214 0.695822i \(-0.244960\pi\)
0.718214 + 0.695822i \(0.244960\pi\)
\(72\) 8.48528 0.117851
\(73\) − 82.3051i − 1.12747i −0.825957 0.563733i \(-0.809365\pi\)
0.825957 0.563733i \(-0.190635\pi\)
\(74\) −49.5813 −0.670017
\(75\) 8.66025i 0.115470i
\(76\) 12.1529i 0.159906i
\(77\) 0 0
\(78\) −17.7835 −0.227993
\(79\) 111.706 1.41400 0.707001 0.707213i \(-0.250048\pi\)
0.707001 + 0.707213i \(0.250048\pi\)
\(80\) − 8.94427i − 0.111803i
\(81\) 9.00000 0.111111
\(82\) − 81.8562i − 0.998246i
\(83\) 91.6237i 1.10390i 0.833877 + 0.551950i \(0.186116\pi\)
−0.833877 + 0.551950i \(0.813884\pi\)
\(84\) 0 0
\(85\) −20.7758 −0.244421
\(86\) 48.1102 0.559420
\(87\) − 73.1137i − 0.840387i
\(88\) −34.8578 −0.396112
\(89\) 127.795i 1.43589i 0.696097 + 0.717947i \(0.254918\pi\)
−0.696097 + 0.717947i \(0.745082\pi\)
\(90\) − 9.48683i − 0.105409i
\(91\) 0 0
\(92\) −4.50057 −0.0489192
\(93\) 2.11055 0.0226940
\(94\) − 80.7370i − 0.858904i
\(95\) 13.5873 0.143025
\(96\) 9.79796i 0.102062i
\(97\) 61.4455i 0.633459i 0.948516 + 0.316729i \(0.102585\pi\)
−0.948516 + 0.316729i \(0.897415\pi\)
\(98\) 0 0
\(99\) −36.9723 −0.373458
\(100\) −10.0000 −0.100000
\(101\) − 62.1392i − 0.615239i −0.951509 0.307620i \(-0.900468\pi\)
0.951509 0.307620i \(-0.0995325\pi\)
\(102\) 22.7587 0.223125
\(103\) − 177.942i − 1.72759i −0.503843 0.863795i \(-0.668081\pi\)
0.503843 0.863795i \(-0.331919\pi\)
\(104\) − 20.5346i − 0.197448i
\(105\) 0 0
\(106\) −20.5629 −0.193990
\(107\) 84.9711 0.794122 0.397061 0.917792i \(-0.370030\pi\)
0.397061 + 0.917792i \(0.370030\pi\)
\(108\) 10.3923i 0.0962250i
\(109\) −172.858 −1.58586 −0.792928 0.609316i \(-0.791444\pi\)
−0.792928 + 0.609316i \(0.791444\pi\)
\(110\) 38.9723i 0.354293i
\(111\) − 60.7244i − 0.547067i
\(112\) 0 0
\(113\) −82.0616 −0.726209 −0.363105 0.931748i \(-0.618283\pi\)
−0.363105 + 0.931748i \(0.618283\pi\)
\(114\) −14.8842 −0.130563
\(115\) 5.03179i 0.0437547i
\(116\) 84.4244 0.727797
\(117\) − 21.7802i − 0.186156i
\(118\) 81.8238i 0.693422i
\(119\) 0 0
\(120\) 10.9545 0.0912871
\(121\) 30.8837 0.255237
\(122\) 8.29002i 0.0679510i
\(123\) 100.253 0.815065
\(124\) 2.43705i 0.0196536i
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −193.480 −1.52346 −0.761732 0.647892i \(-0.775651\pi\)
−0.761732 + 0.647892i \(0.775651\pi\)
\(128\) −11.3137 −0.0883883
\(129\) 58.9227i 0.456765i
\(130\) −22.9583 −0.176603
\(131\) − 181.114i − 1.38255i −0.722590 0.691276i \(-0.757049\pi\)
0.722590 0.691276i \(-0.242951\pi\)
\(132\) − 42.6920i − 0.323424i
\(133\) 0 0
\(134\) −69.9625 −0.522108
\(135\) 11.6190 0.0860663
\(136\) 26.2795i 0.193232i
\(137\) 1.26333 0.00922140 0.00461070 0.999989i \(-0.498532\pi\)
0.00461070 + 0.999989i \(0.498532\pi\)
\(138\) − 5.51205i − 0.0399424i
\(139\) 12.1327i 0.0872857i 0.999047 + 0.0436428i \(0.0138964\pi\)
−0.999047 + 0.0436428i \(0.986104\pi\)
\(140\) 0 0
\(141\) 98.8822 0.701292
\(142\) −144.231 −1.01571
\(143\) 89.4739i 0.625691i
\(144\) −12.0000 −0.0833333
\(145\) − 94.3894i − 0.650961i
\(146\) 116.397i 0.797239i
\(147\) 0 0
\(148\) 70.1185 0.473774
\(149\) 289.725 1.94447 0.972233 0.234015i \(-0.0751866\pi\)
0.972233 + 0.234015i \(0.0751866\pi\)
\(150\) − 12.2474i − 0.0816497i
\(151\) 117.303 0.776842 0.388421 0.921482i \(-0.373021\pi\)
0.388421 + 0.921482i \(0.373021\pi\)
\(152\) − 17.1868i − 0.113071i
\(153\) 27.8737i 0.182181i
\(154\) 0 0
\(155\) 2.72470 0.0175787
\(156\) 25.1496 0.161215
\(157\) − 183.737i − 1.17030i −0.810924 0.585151i \(-0.801035\pi\)
0.810924 0.585151i \(-0.198965\pi\)
\(158\) −157.976 −0.999850
\(159\) − 25.1843i − 0.158392i
\(160\) 12.6491i 0.0790569i
\(161\) 0 0
\(162\) −12.7279 −0.0785674
\(163\) 121.571 0.745833 0.372917 0.927865i \(-0.378358\pi\)
0.372917 + 0.927865i \(0.378358\pi\)
\(164\) 115.762i 0.705867i
\(165\) −47.7311 −0.289279
\(166\) − 129.576i − 0.780575i
\(167\) 94.6539i 0.566790i 0.959003 + 0.283395i \(0.0914608\pi\)
−0.959003 + 0.283395i \(0.908539\pi\)
\(168\) 0 0
\(169\) 116.291 0.688115
\(170\) 29.3814 0.172832
\(171\) − 18.2293i − 0.106604i
\(172\) −68.0380 −0.395570
\(173\) − 260.692i − 1.50689i −0.657511 0.753445i \(-0.728391\pi\)
0.657511 0.753445i \(-0.271609\pi\)
\(174\) 103.398i 0.594244i
\(175\) 0 0
\(176\) 49.2964 0.280093
\(177\) −100.213 −0.566177
\(178\) − 180.729i − 1.01533i
\(179\) 306.529 1.71245 0.856225 0.516603i \(-0.172803\pi\)
0.856225 + 0.516603i \(0.172803\pi\)
\(180\) 13.4164i 0.0745356i
\(181\) − 314.885i − 1.73970i −0.493318 0.869849i \(-0.664216\pi\)
0.493318 0.869849i \(-0.335784\pi\)
\(182\) 0 0
\(183\) −10.1532 −0.0554817
\(184\) 6.36477 0.0345911
\(185\) − 78.3949i − 0.423756i
\(186\) −2.98476 −0.0160471
\(187\) − 114.506i − 0.612331i
\(188\) 114.179i 0.607337i
\(189\) 0 0
\(190\) −19.2154 −0.101134
\(191\) −90.7891 −0.475336 −0.237668 0.971346i \(-0.576383\pi\)
−0.237668 + 0.971346i \(0.576383\pi\)
\(192\) − 13.8564i − 0.0721688i
\(193\) −229.655 −1.18992 −0.594961 0.803754i \(-0.702833\pi\)
−0.594961 + 0.803754i \(0.702833\pi\)
\(194\) − 86.8971i − 0.447923i
\(195\) − 28.1181i − 0.144195i
\(196\) 0 0
\(197\) 227.989 1.15730 0.578652 0.815574i \(-0.303579\pi\)
0.578652 + 0.815574i \(0.303579\pi\)
\(198\) 52.2868 0.264075
\(199\) − 11.0461i − 0.0555082i −0.999615 0.0277541i \(-0.991164\pi\)
0.999615 0.0277541i \(-0.00883554\pi\)
\(200\) 14.1421 0.0707107
\(201\) − 85.6862i − 0.426299i
\(202\) 87.8781i 0.435040i
\(203\) 0 0
\(204\) −32.1857 −0.157773
\(205\) 129.426 0.631346
\(206\) 251.648i 1.22159i
\(207\) 6.75086 0.0326128
\(208\) 29.0403i 0.139617i
\(209\) 74.8867i 0.358310i
\(210\) 0 0
\(211\) 151.187 0.716526 0.358263 0.933621i \(-0.383369\pi\)
0.358263 + 0.933621i \(0.383369\pi\)
\(212\) 29.0804 0.137172
\(213\) − 176.646i − 0.829322i
\(214\) −120.167 −0.561529
\(215\) 76.0688i 0.353809i
\(216\) − 14.6969i − 0.0680414i
\(217\) 0 0
\(218\) 244.458 1.12137
\(219\) −142.557 −0.650943
\(220\) − 55.1151i − 0.250523i
\(221\) 67.4549 0.305226
\(222\) 85.8773i 0.386835i
\(223\) − 308.586i − 1.38379i −0.721997 0.691896i \(-0.756776\pi\)
0.721997 0.691896i \(-0.243224\pi\)
\(224\) 0 0
\(225\) 15.0000 0.0666667
\(226\) 116.053 0.513507
\(227\) − 100.422i − 0.442389i −0.975230 0.221195i \(-0.929004\pi\)
0.975230 0.221195i \(-0.0709956\pi\)
\(228\) 21.0494 0.0923220
\(229\) 319.997i 1.39736i 0.715432 + 0.698682i \(0.246230\pi\)
−0.715432 + 0.698682i \(0.753770\pi\)
\(230\) − 7.11603i − 0.0309392i
\(231\) 0 0
\(232\) −119.394 −0.514630
\(233\) 310.789 1.33386 0.666929 0.745121i \(-0.267608\pi\)
0.666929 + 0.745121i \(0.267608\pi\)
\(234\) 30.8019i 0.131632i
\(235\) 127.656 0.543219
\(236\) − 115.716i − 0.490324i
\(237\) − 193.481i − 0.816374i
\(238\) 0 0
\(239\) 136.263 0.570139 0.285069 0.958507i \(-0.407983\pi\)
0.285069 + 0.958507i \(0.407983\pi\)
\(240\) −15.4919 −0.0645497
\(241\) − 386.179i − 1.60240i −0.598394 0.801202i \(-0.704194\pi\)
0.598394 0.801202i \(-0.295806\pi\)
\(242\) −43.6761 −0.180480
\(243\) − 15.5885i − 0.0641500i
\(244\) − 11.7239i − 0.0480486i
\(245\) 0 0
\(246\) −141.779 −0.576338
\(247\) −44.1154 −0.178605
\(248\) − 3.44651i − 0.0138972i
\(249\) 158.697 0.637337
\(250\) − 15.8114i − 0.0632456i
\(251\) − 99.3717i − 0.395903i −0.980212 0.197952i \(-0.936571\pi\)
0.980212 0.197952i \(-0.0634289\pi\)
\(252\) 0 0
\(253\) −27.7328 −0.109616
\(254\) 273.622 1.07725
\(255\) 35.9847i 0.141117i
\(256\) 16.0000 0.0625000
\(257\) 21.1178i 0.0821705i 0.999156 + 0.0410852i \(0.0130815\pi\)
−0.999156 + 0.0410852i \(0.986918\pi\)
\(258\) − 83.3292i − 0.322982i
\(259\) 0 0
\(260\) 32.4680 0.124877
\(261\) −126.637 −0.485198
\(262\) 256.134i 0.977612i
\(263\) −380.517 −1.44683 −0.723417 0.690411i \(-0.757430\pi\)
−0.723417 + 0.690411i \(0.757430\pi\)
\(264\) 60.3756i 0.228695i
\(265\) − 32.5128i − 0.122690i
\(266\) 0 0
\(267\) 221.347 0.829014
\(268\) 98.9419 0.369186
\(269\) − 16.7910i − 0.0624202i −0.999513 0.0312101i \(-0.990064\pi\)
0.999513 0.0312101i \(-0.00993609\pi\)
\(270\) −16.4317 −0.0608581
\(271\) 369.672i 1.36410i 0.731303 + 0.682052i \(0.238912\pi\)
−0.731303 + 0.682052i \(0.761088\pi\)
\(272\) − 37.1649i − 0.136636i
\(273\) 0 0
\(274\) −1.78662 −0.00652051
\(275\) −61.6205 −0.224075
\(276\) 7.79522i 0.0282435i
\(277\) 485.825 1.75388 0.876940 0.480600i \(-0.159581\pi\)
0.876940 + 0.480600i \(0.159581\pi\)
\(278\) − 17.1582i − 0.0617203i
\(279\) − 3.65557i − 0.0131024i
\(280\) 0 0
\(281\) 360.234 1.28197 0.640986 0.767553i \(-0.278526\pi\)
0.640986 + 0.767553i \(0.278526\pi\)
\(282\) −139.841 −0.495889
\(283\) − 270.502i − 0.955836i −0.878404 0.477918i \(-0.841392\pi\)
0.878404 0.477918i \(-0.158608\pi\)
\(284\) 203.973 0.718214
\(285\) − 23.5340i − 0.0825753i
\(286\) − 126.535i − 0.442431i
\(287\) 0 0
\(288\) 16.9706 0.0589256
\(289\) 202.673 0.701292
\(290\) 133.487i 0.460299i
\(291\) 106.427 0.365728
\(292\) − 164.610i − 0.563733i
\(293\) − 9.63230i − 0.0328747i −0.999865 0.0164374i \(-0.994768\pi\)
0.999865 0.0164374i \(-0.00523241\pi\)
\(294\) 0 0
\(295\) −129.375 −0.438559
\(296\) −99.1626 −0.335009
\(297\) 64.0379i 0.215616i
\(298\) −409.734 −1.37494
\(299\) − 16.3372i − 0.0546395i
\(300\) 17.3205i 0.0577350i
\(301\) 0 0
\(302\) −165.892 −0.549310
\(303\) −107.628 −0.355209
\(304\) 24.3058i 0.0799532i
\(305\) −13.1077 −0.0429760
\(306\) − 39.4193i − 0.128821i
\(307\) − 46.0412i − 0.149971i −0.997185 0.0749857i \(-0.976109\pi\)
0.997185 0.0749857i \(-0.0238911\pi\)
\(308\) 0 0
\(309\) −308.204 −0.997425
\(310\) −3.85331 −0.0124300
\(311\) 436.662i 1.40406i 0.712148 + 0.702029i \(0.247722\pi\)
−0.712148 + 0.702029i \(0.752278\pi\)
\(312\) −35.5669 −0.113997
\(313\) 286.522i 0.915405i 0.889105 + 0.457703i \(0.151328\pi\)
−0.889105 + 0.457703i \(0.848672\pi\)
\(314\) 259.844i 0.827529i
\(315\) 0 0
\(316\) 223.412 0.707001
\(317\) −1.36884 −0.00431809 −0.00215905 0.999998i \(-0.500687\pi\)
−0.00215905 + 0.999998i \(0.500687\pi\)
\(318\) 35.6160i 0.112000i
\(319\) 520.228 1.63081
\(320\) − 17.8885i − 0.0559017i
\(321\) − 147.174i − 0.458487i
\(322\) 0 0
\(323\) 56.4575 0.174791
\(324\) 18.0000 0.0555556
\(325\) − 36.3003i − 0.111693i
\(326\) −171.927 −0.527384
\(327\) 299.399i 0.915594i
\(328\) − 163.712i − 0.499123i
\(329\) 0 0
\(330\) 67.5019 0.204551
\(331\) −384.034 −1.16022 −0.580111 0.814537i \(-0.696991\pi\)
−0.580111 + 0.814537i \(0.696991\pi\)
\(332\) 183.247i 0.551950i
\(333\) −105.178 −0.315849
\(334\) − 133.861i − 0.400781i
\(335\) − 110.620i − 0.330210i
\(336\) 0 0
\(337\) −141.948 −0.421211 −0.210606 0.977571i \(-0.567544\pi\)
−0.210606 + 0.977571i \(0.567544\pi\)
\(338\) −164.461 −0.486571
\(339\) 142.135i 0.419277i
\(340\) −41.5516 −0.122211
\(341\) 15.0172i 0.0440388i
\(342\) 25.7802i 0.0753806i
\(343\) 0 0
\(344\) 96.2203 0.279710
\(345\) 8.71532 0.0252618
\(346\) 368.674i 1.06553i
\(347\) −250.352 −0.721475 −0.360737 0.932667i \(-0.617475\pi\)
−0.360737 + 0.932667i \(0.617475\pi\)
\(348\) − 146.227i − 0.420194i
\(349\) 195.188i 0.559277i 0.960105 + 0.279639i \(0.0902147\pi\)
−0.960105 + 0.279639i \(0.909785\pi\)
\(350\) 0 0
\(351\) −37.7244 −0.107477
\(352\) −69.7157 −0.198056
\(353\) − 142.199i − 0.402831i −0.979506 0.201416i \(-0.935446\pi\)
0.979506 0.201416i \(-0.0645541\pi\)
\(354\) 141.723 0.400348
\(355\) − 228.049i − 0.642390i
\(356\) 255.589i 0.717947i
\(357\) 0 0
\(358\) −433.497 −1.21089
\(359\) −656.886 −1.82977 −0.914884 0.403718i \(-0.867718\pi\)
−0.914884 + 0.403718i \(0.867718\pi\)
\(360\) − 18.9737i − 0.0527046i
\(361\) 324.077 0.897720
\(362\) 445.315i 1.23015i
\(363\) − 53.4921i − 0.147361i
\(364\) 0 0
\(365\) −184.040 −0.504218
\(366\) 14.3587 0.0392315
\(367\) 513.152i 1.39823i 0.715007 + 0.699117i \(0.246424\pi\)
−0.715007 + 0.699117i \(0.753576\pi\)
\(368\) −9.00114 −0.0244596
\(369\) − 173.643i − 0.470578i
\(370\) 110.867i 0.299641i
\(371\) 0 0
\(372\) 4.22109 0.0113470
\(373\) −632.524 −1.69577 −0.847887 0.530177i \(-0.822126\pi\)
−0.847887 + 0.530177i \(0.822126\pi\)
\(374\) 161.936i 0.432984i
\(375\) 19.3649 0.0516398
\(376\) − 161.474i − 0.429452i
\(377\) 306.463i 0.812900i
\(378\) 0 0
\(379\) −617.180 −1.62844 −0.814222 0.580553i \(-0.802836\pi\)
−0.814222 + 0.580553i \(0.802836\pi\)
\(380\) 27.1747 0.0715123
\(381\) 335.117i 0.879573i
\(382\) 128.395 0.336113
\(383\) 173.947i 0.454170i 0.973875 + 0.227085i \(0.0729195\pi\)
−0.973875 + 0.227085i \(0.927081\pi\)
\(384\) 19.5959i 0.0510310i
\(385\) 0 0
\(386\) 324.781 0.841402
\(387\) 102.057 0.263713
\(388\) 122.891i 0.316729i
\(389\) 196.192 0.504350 0.252175 0.967682i \(-0.418854\pi\)
0.252175 + 0.967682i \(0.418854\pi\)
\(390\) 39.7650i 0.101962i
\(391\) 20.9079i 0.0534729i
\(392\) 0 0
\(393\) −313.699 −0.798217
\(394\) −322.425 −0.818338
\(395\) − 249.782i − 0.632361i
\(396\) −73.9447 −0.186729
\(397\) − 398.701i − 1.00428i −0.864785 0.502142i \(-0.832545\pi\)
0.864785 0.502142i \(-0.167455\pi\)
\(398\) 15.6216i 0.0392502i
\(399\) 0 0
\(400\) −20.0000 −0.0500000
\(401\) 347.199 0.865832 0.432916 0.901434i \(-0.357485\pi\)
0.432916 + 0.901434i \(0.357485\pi\)
\(402\) 121.179i 0.301439i
\(403\) −8.84657 −0.0219518
\(404\) − 124.278i − 0.307620i
\(405\) − 20.1246i − 0.0496904i
\(406\) 0 0
\(407\) 432.074 1.06161
\(408\) 45.5175 0.111562
\(409\) − 338.786i − 0.828327i −0.910203 0.414163i \(-0.864074\pi\)
0.910203 0.414163i \(-0.135926\pi\)
\(410\) −183.036 −0.446429
\(411\) − 2.18815i − 0.00532398i
\(412\) − 355.884i − 0.863795i
\(413\) 0 0
\(414\) −9.54715 −0.0230608
\(415\) 204.877 0.493679
\(416\) − 41.0691i − 0.0987239i
\(417\) 21.0145 0.0503944
\(418\) − 105.906i − 0.253363i
\(419\) 369.514i 0.881894i 0.897533 + 0.440947i \(0.145357\pi\)
−0.897533 + 0.440947i \(0.854643\pi\)
\(420\) 0 0
\(421\) −217.571 −0.516797 −0.258398 0.966038i \(-0.583195\pi\)
−0.258398 + 0.966038i \(0.583195\pi\)
\(422\) −213.811 −0.506661
\(423\) − 171.269i − 0.404891i
\(424\) −41.1259 −0.0969949
\(425\) 46.4561i 0.109308i
\(426\) 249.815i 0.586420i
\(427\) 0 0
\(428\) 169.942 0.397061
\(429\) 154.973 0.361243
\(430\) − 107.578i − 0.250180i
\(431\) −285.831 −0.663181 −0.331591 0.943423i \(-0.607585\pi\)
−0.331591 + 0.943423i \(0.607585\pi\)
\(432\) 20.7846i 0.0481125i
\(433\) − 643.490i − 1.48612i −0.669224 0.743060i \(-0.733374\pi\)
0.669224 0.743060i \(-0.266626\pi\)
\(434\) 0 0
\(435\) −163.487 −0.375833
\(436\) −345.716 −0.792928
\(437\) − 13.6737i − 0.0312900i
\(438\) 201.605 0.460286
\(439\) 570.437i 1.29940i 0.760190 + 0.649700i \(0.225106\pi\)
−0.760190 + 0.649700i \(0.774894\pi\)
\(440\) 77.9445i 0.177147i
\(441\) 0 0
\(442\) −95.3956 −0.215827
\(443\) −66.0149 −0.149018 −0.0745089 0.997220i \(-0.523739\pi\)
−0.0745089 + 0.997220i \(0.523739\pi\)
\(444\) − 121.449i − 0.273533i
\(445\) 285.758 0.642152
\(446\) 436.406i 0.978489i
\(447\) − 501.819i − 1.12264i
\(448\) 0 0
\(449\) 515.072 1.14715 0.573577 0.819152i \(-0.305555\pi\)
0.573577 + 0.819152i \(0.305555\pi\)
\(450\) −21.2132 −0.0471405
\(451\) 713.333i 1.58167i
\(452\) −164.123 −0.363105
\(453\) − 203.175i − 0.448510i
\(454\) 142.019i 0.312816i
\(455\) 0 0
\(456\) −29.7684 −0.0652815
\(457\) 79.0271 0.172926 0.0864629 0.996255i \(-0.472444\pi\)
0.0864629 + 0.996255i \(0.472444\pi\)
\(458\) − 452.543i − 0.988086i
\(459\) 48.2786 0.105182
\(460\) 10.0636i 0.0218774i
\(461\) − 9.58316i − 0.0207878i −0.999946 0.0103939i \(-0.996691\pi\)
0.999946 0.0103939i \(-0.00330854\pi\)
\(462\) 0 0
\(463\) −232.103 −0.501303 −0.250652 0.968077i \(-0.580645\pi\)
−0.250652 + 0.968077i \(0.580645\pi\)
\(464\) 168.849 0.363898
\(465\) − 4.71932i − 0.0101491i
\(466\) −439.522 −0.943181
\(467\) 685.536i 1.46796i 0.679172 + 0.733979i \(0.262339\pi\)
−0.679172 + 0.733979i \(0.737661\pi\)
\(468\) − 43.5604i − 0.0930778i
\(469\) 0 0
\(470\) −180.533 −0.384114
\(471\) −318.243 −0.675674
\(472\) 163.648i 0.346711i
\(473\) −419.254 −0.886372
\(474\) 273.623i 0.577264i
\(475\) − 30.3822i − 0.0639626i
\(476\) 0 0
\(477\) −43.6206 −0.0914477
\(478\) −192.705 −0.403149
\(479\) 629.101i 1.31336i 0.754168 + 0.656682i \(0.228041\pi\)
−0.754168 + 0.656682i \(0.771959\pi\)
\(480\) 21.9089 0.0456435
\(481\) 254.533i 0.529174i
\(482\) 546.140i 1.13307i
\(483\) 0 0
\(484\) 61.7673 0.127618
\(485\) 137.396 0.283291
\(486\) 22.0454i 0.0453609i
\(487\) −498.768 −1.02417 −0.512083 0.858936i \(-0.671126\pi\)
−0.512083 + 0.858936i \(0.671126\pi\)
\(488\) 16.5800i 0.0339755i
\(489\) − 210.567i − 0.430607i
\(490\) 0 0
\(491\) 166.583 0.339274 0.169637 0.985507i \(-0.445741\pi\)
0.169637 + 0.985507i \(0.445741\pi\)
\(492\) 200.506 0.407532
\(493\) − 392.203i − 0.795543i
\(494\) 62.3886 0.126293
\(495\) 82.6726i 0.167015i
\(496\) 4.87410i 0.00982681i
\(497\) 0 0
\(498\) −224.431 −0.450665
\(499\) 36.3062 0.0727579 0.0363790 0.999338i \(-0.488418\pi\)
0.0363790 + 0.999338i \(0.488418\pi\)
\(500\) 22.3607i 0.0447214i
\(501\) 163.945 0.327236
\(502\) 140.533i 0.279946i
\(503\) − 634.940i − 1.26231i −0.775658 0.631153i \(-0.782582\pi\)
0.775658 0.631153i \(-0.217418\pi\)
\(504\) 0 0
\(505\) −138.947 −0.275143
\(506\) 39.2200 0.0775100
\(507\) − 201.423i − 0.397283i
\(508\) −386.960 −0.761732
\(509\) 578.347i 1.13624i 0.822945 + 0.568121i \(0.192330\pi\)
−0.822945 + 0.568121i \(0.807670\pi\)
\(510\) − 50.8901i − 0.0997845i
\(511\) 0 0
\(512\) −22.6274 −0.0441942
\(513\) −31.5741 −0.0615480
\(514\) − 29.8651i − 0.0581033i
\(515\) −397.890 −0.772602
\(516\) 117.845i 0.228382i
\(517\) 703.579i 1.36089i
\(518\) 0 0
\(519\) −451.532 −0.870003
\(520\) −45.9167 −0.0883013
\(521\) 636.210i 1.22113i 0.791965 + 0.610566i \(0.209058\pi\)
−0.791965 + 0.610566i \(0.790942\pi\)
\(522\) 179.091 0.343087
\(523\) 453.645i 0.867390i 0.901060 + 0.433695i \(0.142791\pi\)
−0.901060 + 0.433695i \(0.857209\pi\)
\(524\) − 362.229i − 0.691276i
\(525\) 0 0
\(526\) 538.133 1.02307
\(527\) 11.3216 0.0214831
\(528\) − 85.3839i − 0.161712i
\(529\) −523.936 −0.990428
\(530\) 45.9801i 0.0867549i
\(531\) 173.575i 0.326882i
\(532\) 0 0
\(533\) −420.220 −0.788406
\(534\) −313.032 −0.586202
\(535\) − 190.001i − 0.355142i
\(536\) −139.925 −0.261054
\(537\) − 530.923i − 0.988684i
\(538\) 23.7461i 0.0441377i
\(539\) 0 0
\(540\) 23.2379 0.0430331
\(541\) −576.318 −1.06528 −0.532641 0.846341i \(-0.678800\pi\)
−0.532641 + 0.846341i \(0.678800\pi\)
\(542\) − 522.796i − 0.964568i
\(543\) −545.397 −1.00442
\(544\) 52.5591i 0.0966159i
\(545\) 386.523i 0.709216i
\(546\) 0 0
\(547\) 481.306 0.879901 0.439950 0.898022i \(-0.354996\pi\)
0.439950 + 0.898022i \(0.354996\pi\)
\(548\) 2.52666 0.00461070
\(549\) 17.5858i 0.0320324i
\(550\) 87.1446 0.158445
\(551\) 256.500i 0.465517i
\(552\) − 11.0241i − 0.0199712i
\(553\) 0 0
\(554\) −687.060 −1.24018
\(555\) −135.784 −0.244656
\(556\) 24.2654i 0.0436428i
\(557\) 108.811 0.195352 0.0976762 0.995218i \(-0.468859\pi\)
0.0976762 + 0.995218i \(0.468859\pi\)
\(558\) 5.16976i 0.00926480i
\(559\) − 246.980i − 0.441825i
\(560\) 0 0
\(561\) −198.330 −0.353530
\(562\) −509.448 −0.906491
\(563\) − 602.195i − 1.06962i −0.844973 0.534809i \(-0.820384\pi\)
0.844973 0.534809i \(-0.179616\pi\)
\(564\) 197.764 0.350646
\(565\) 183.495i 0.324771i
\(566\) 382.547i 0.675878i
\(567\) 0 0
\(568\) −288.461 −0.507854
\(569\) −9.53369 −0.0167552 −0.00837759 0.999965i \(-0.502667\pi\)
−0.00837759 + 0.999965i \(0.502667\pi\)
\(570\) 33.2820i 0.0583896i
\(571\) 982.205 1.72015 0.860075 0.510168i \(-0.170417\pi\)
0.860075 + 0.510168i \(0.170417\pi\)
\(572\) 178.948i 0.312846i
\(573\) 157.251i 0.274435i
\(574\) 0 0
\(575\) 11.2514 0.0195677
\(576\) −24.0000 −0.0416667
\(577\) 524.235i 0.908553i 0.890861 + 0.454276i \(0.150102\pi\)
−0.890861 + 0.454276i \(0.849898\pi\)
\(578\) −286.623 −0.495888
\(579\) 397.774i 0.687002i
\(580\) − 188.779i − 0.325481i
\(581\) 0 0
\(582\) −150.510 −0.258608
\(583\) 179.195 0.307367
\(584\) 232.794i 0.398620i
\(585\) −48.7020 −0.0832513
\(586\) 13.6221i 0.0232460i
\(587\) 651.322i 1.10958i 0.831991 + 0.554789i \(0.187201\pi\)
−0.831991 + 0.554789i \(0.812799\pi\)
\(588\) 0 0
\(589\) −7.40429 −0.0125710
\(590\) 182.964 0.310108
\(591\) − 394.889i − 0.668170i
\(592\) 140.237 0.236887
\(593\) − 705.411i − 1.18956i −0.803887 0.594782i \(-0.797239\pi\)
0.803887 0.594782i \(-0.202761\pi\)
\(594\) − 90.5633i − 0.152464i
\(595\) 0 0
\(596\) 579.451 0.972233
\(597\) −19.1325 −0.0320477
\(598\) 23.1043i 0.0386360i
\(599\) −593.819 −0.991351 −0.495676 0.868508i \(-0.665080\pi\)
−0.495676 + 0.868508i \(0.665080\pi\)
\(600\) − 24.4949i − 0.0408248i
\(601\) 12.1644i 0.0202403i 0.999949 + 0.0101202i \(0.00322140\pi\)
−0.999949 + 0.0101202i \(0.996779\pi\)
\(602\) 0 0
\(603\) −148.413 −0.246124
\(604\) 234.606 0.388421
\(605\) − 69.0579i − 0.114145i
\(606\) 152.209 0.251170
\(607\) 806.583i 1.32880i 0.747376 + 0.664401i \(0.231313\pi\)
−0.747376 + 0.664401i \(0.768687\pi\)
\(608\) − 34.3736i − 0.0565354i
\(609\) 0 0
\(610\) 18.5370 0.0303886
\(611\) −414.475 −0.678355
\(612\) 55.7473i 0.0910904i
\(613\) −794.474 −1.29604 −0.648021 0.761622i \(-0.724403\pi\)
−0.648021 + 0.761622i \(0.724403\pi\)
\(614\) 65.1121i 0.106046i
\(615\) − 224.172i − 0.364508i
\(616\) 0 0
\(617\) 108.982 0.176633 0.0883164 0.996092i \(-0.471851\pi\)
0.0883164 + 0.996092i \(0.471851\pi\)
\(618\) 435.867 0.705286
\(619\) 358.795i 0.579636i 0.957082 + 0.289818i \(0.0935948\pi\)
−0.957082 + 0.289818i \(0.906405\pi\)
\(620\) 5.44941 0.00878937
\(621\) − 11.6928i − 0.0188290i
\(622\) − 617.533i − 0.992819i
\(623\) 0 0
\(624\) 50.2992 0.0806077
\(625\) 25.0000 0.0400000
\(626\) − 405.203i − 0.647289i
\(627\) 129.708 0.206870
\(628\) − 367.475i − 0.585151i
\(629\) − 325.743i − 0.517875i
\(630\) 0 0
\(631\) −612.351 −0.970446 −0.485223 0.874391i \(-0.661262\pi\)
−0.485223 + 0.874391i \(0.661262\pi\)
\(632\) −315.953 −0.499925
\(633\) − 261.864i − 0.413687i
\(634\) 1.93583 0.00305335
\(635\) 432.634i 0.681314i
\(636\) − 50.3687i − 0.0791960i
\(637\) 0 0
\(638\) −735.713 −1.15316
\(639\) −305.959 −0.478810
\(640\) 25.2982i 0.0395285i
\(641\) 919.411 1.43434 0.717169 0.696899i \(-0.245437\pi\)
0.717169 + 0.696899i \(0.245437\pi\)
\(642\) 208.136i 0.324199i
\(643\) 835.879i 1.29997i 0.759948 + 0.649984i \(0.225224\pi\)
−0.759948 + 0.649984i \(0.774776\pi\)
\(644\) 0 0
\(645\) 131.755 0.204271
\(646\) −79.8430 −0.123596
\(647\) 569.191i 0.879738i 0.898062 + 0.439869i \(0.144975\pi\)
−0.898062 + 0.439869i \(0.855025\pi\)
\(648\) −25.4558 −0.0392837
\(649\) − 713.051i − 1.09869i
\(650\) 51.3364i 0.0789791i
\(651\) 0 0
\(652\) 243.142 0.372917
\(653\) 392.324 0.600803 0.300402 0.953813i \(-0.402879\pi\)
0.300402 + 0.953813i \(0.402879\pi\)
\(654\) − 423.414i − 0.647423i
\(655\) −404.984 −0.618296
\(656\) 231.524i 0.352933i
\(657\) 246.915i 0.375822i
\(658\) 0 0
\(659\) 505.063 0.766408 0.383204 0.923664i \(-0.374821\pi\)
0.383204 + 0.923664i \(0.374821\pi\)
\(660\) −95.4621 −0.144640
\(661\) 295.390i 0.446884i 0.974717 + 0.223442i \(0.0717293\pi\)
−0.974717 + 0.223442i \(0.928271\pi\)
\(662\) 543.105 0.820401
\(663\) − 116.835i − 0.176222i
\(664\) − 259.151i − 0.390288i
\(665\) 0 0
\(666\) 148.744 0.223339
\(667\) −94.9895 −0.142413
\(668\) 189.308i 0.283395i
\(669\) −534.486 −0.798933
\(670\) 156.441i 0.233494i
\(671\) − 72.2430i − 0.107665i
\(672\) 0 0
\(673\) −624.569 −0.928038 −0.464019 0.885825i \(-0.653593\pi\)
−0.464019 + 0.885825i \(0.653593\pi\)
\(674\) 200.745 0.297841
\(675\) − 25.9808i − 0.0384900i
\(676\) 232.583 0.344057
\(677\) − 781.556i − 1.15444i −0.816589 0.577220i \(-0.804138\pi\)
0.816589 0.577220i \(-0.195862\pi\)
\(678\) − 201.009i − 0.296474i
\(679\) 0 0
\(680\) 58.7628 0.0864159
\(681\) −173.937 −0.255414
\(682\) − 21.2376i − 0.0311401i
\(683\) 101.705 0.148909 0.0744546 0.997224i \(-0.476278\pi\)
0.0744546 + 0.997224i \(0.476278\pi\)
\(684\) − 36.4587i − 0.0533021i
\(685\) − 2.82490i − 0.00412393i
\(686\) 0 0
\(687\) 554.250 0.806769
\(688\) −136.076 −0.197785
\(689\) 105.563i 0.153211i
\(690\) −12.3253 −0.0178628
\(691\) 732.870i 1.06059i 0.847812 + 0.530297i \(0.177920\pi\)
−0.847812 + 0.530297i \(0.822080\pi\)
\(692\) − 521.384i − 0.753445i
\(693\) 0 0
\(694\) 354.051 0.510160
\(695\) 27.1296 0.0390353
\(696\) 206.797i 0.297122i
\(697\) 537.786 0.771572
\(698\) − 276.037i − 0.395469i
\(699\) − 538.303i − 0.770104i
\(700\) 0 0
\(701\) −795.928 −1.13542 −0.567709 0.823229i \(-0.692170\pi\)
−0.567709 + 0.823229i \(0.692170\pi\)
\(702\) 53.3504 0.0759977
\(703\) 213.036i 0.303038i
\(704\) 98.5929 0.140047
\(705\) − 221.107i − 0.313627i
\(706\) 201.100i 0.284845i
\(707\) 0 0
\(708\) −200.427 −0.283089
\(709\) 1029.06 1.45143 0.725715 0.687995i \(-0.241509\pi\)
0.725715 + 0.687995i \(0.241509\pi\)
\(710\) 322.509i 0.454239i
\(711\) −335.118 −0.471334
\(712\) − 361.458i − 0.507666i
\(713\) − 2.74203i − 0.00384576i
\(714\) 0 0
\(715\) 200.070 0.279818
\(716\) 613.057 0.856225
\(717\) − 236.015i − 0.329170i
\(718\) 928.978 1.29384
\(719\) − 157.673i − 0.219295i −0.993971 0.109648i \(-0.965028\pi\)
0.993971 0.109648i \(-0.0349723\pi\)
\(720\) 26.8328i 0.0372678i
\(721\) 0 0
\(722\) −458.314 −0.634784
\(723\) −668.883 −0.925149
\(724\) − 629.771i − 0.869849i
\(725\) −211.061 −0.291119
\(726\) 75.6492i 0.104200i
\(727\) − 13.5224i − 0.0186003i −0.999957 0.00930013i \(-0.997040\pi\)
0.999957 0.00930013i \(-0.00296037\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 260.271 0.356536
\(731\) 316.078i 0.432391i
\(732\) −20.3063 −0.0277409
\(733\) 394.471i 0.538159i 0.963118 + 0.269080i \(0.0867195\pi\)
−0.963118 + 0.269080i \(0.913280\pi\)
\(734\) − 725.707i − 0.988701i
\(735\) 0 0
\(736\) 12.7295 0.0172956
\(737\) 609.685 0.827253
\(738\) 245.569i 0.332749i
\(739\) 950.161 1.28574 0.642869 0.765976i \(-0.277744\pi\)
0.642869 + 0.765976i \(0.277744\pi\)
\(740\) − 156.790i − 0.211878i
\(741\) 76.4101i 0.103118i
\(742\) 0 0
\(743\) 425.883 0.573194 0.286597 0.958051i \(-0.407476\pi\)
0.286597 + 0.958051i \(0.407476\pi\)
\(744\) −5.96953 −0.00802356
\(745\) − 647.846i − 0.869592i
\(746\) 894.524 1.19909
\(747\) − 274.871i − 0.367967i
\(748\) − 229.012i − 0.306166i
\(749\) 0 0
\(750\) −27.3861 −0.0365148
\(751\) −631.747 −0.841208 −0.420604 0.907244i \(-0.638182\pi\)
−0.420604 + 0.907244i \(0.638182\pi\)
\(752\) 228.359i 0.303669i
\(753\) −172.117 −0.228575
\(754\) − 433.405i − 0.574807i
\(755\) − 262.298i − 0.347414i
\(756\) 0 0
\(757\) 882.903 1.16632 0.583159 0.812358i \(-0.301816\pi\)
0.583159 + 0.812358i \(0.301816\pi\)
\(758\) 872.825 1.15148
\(759\) 48.0346i 0.0632866i
\(760\) −38.4308 −0.0505668
\(761\) − 1132.82i − 1.48859i −0.667851 0.744295i \(-0.732786\pi\)
0.667851 0.744295i \(-0.267214\pi\)
\(762\) − 473.927i − 0.621952i
\(763\) 0 0
\(764\) −181.578 −0.237668
\(765\) 62.3274 0.0814737
\(766\) − 245.998i − 0.321146i
\(767\) 420.054 0.547659
\(768\) − 27.7128i − 0.0360844i
\(769\) − 41.6421i − 0.0541510i −0.999633 0.0270755i \(-0.991381\pi\)
0.999633 0.0270755i \(-0.00861944\pi\)
\(770\) 0 0
\(771\) 36.5771 0.0474411
\(772\) −459.310 −0.594961
\(773\) − 1286.33i − 1.66408i −0.554715 0.832041i \(-0.687173\pi\)
0.554715 0.832041i \(-0.312827\pi\)
\(774\) −144.330 −0.186473
\(775\) − 6.09262i − 0.00786145i
\(776\) − 173.794i − 0.223962i
\(777\) 0 0
\(778\) −277.458 −0.356629
\(779\) −351.711 −0.451490
\(780\) − 56.2362i − 0.0720977i
\(781\) 1256.89 1.60934
\(782\) − 29.5682i − 0.0378110i
\(783\) 219.341i 0.280129i
\(784\) 0 0
\(785\) −410.850 −0.523375
\(786\) 443.638 0.564425
\(787\) 997.472i 1.26744i 0.773564 + 0.633718i \(0.218472\pi\)
−0.773564 + 0.633718i \(0.781528\pi\)
\(788\) 455.978 0.578652
\(789\) 659.075i 0.835330i
\(790\) 353.246i 0.447146i
\(791\) 0 0
\(792\) 104.574 0.132037
\(793\) 42.5580 0.0536671
\(794\) 563.848i 0.710136i
\(795\) −56.3139 −0.0708351
\(796\) − 22.0923i − 0.0277541i
\(797\) 971.547i 1.21901i 0.792784 + 0.609503i \(0.208631\pi\)
−0.792784 + 0.609503i \(0.791369\pi\)
\(798\) 0 0
\(799\) 530.433 0.663871
\(800\) 28.2843 0.0353553
\(801\) − 383.384i − 0.478632i
\(802\) −491.013 −0.612236
\(803\) − 1014.34i − 1.26318i
\(804\) − 171.372i − 0.213150i
\(805\) 0 0
\(806\) 12.5109 0.0155223
\(807\) −29.0829 −0.0360383
\(808\) 175.756i 0.217520i
\(809\) −1304.22 −1.61214 −0.806072 0.591817i \(-0.798411\pi\)
−0.806072 + 0.591817i \(0.798411\pi\)
\(810\) 28.4605i 0.0351364i
\(811\) − 210.763i − 0.259880i −0.991522 0.129940i \(-0.958521\pi\)
0.991522 0.129940i \(-0.0414785\pi\)
\(812\) 0 0
\(813\) 640.291 0.787566
\(814\) −611.045 −0.750670
\(815\) − 271.841i − 0.333547i
\(816\) −64.3714 −0.0788866
\(817\) − 206.715i − 0.253017i
\(818\) 479.115i 0.585716i
\(819\) 0 0
\(820\) 258.852 0.315673
\(821\) −724.506 −0.882467 −0.441234 0.897392i \(-0.645459\pi\)
−0.441234 + 0.897392i \(0.645459\pi\)
\(822\) 3.09452i 0.00376462i
\(823\) −446.037 −0.541964 −0.270982 0.962584i \(-0.587348\pi\)
−0.270982 + 0.962584i \(0.587348\pi\)
\(824\) 503.296i 0.610796i
\(825\) 106.730i 0.129370i
\(826\) 0 0
\(827\) −702.737 −0.849743 −0.424871 0.905254i \(-0.639681\pi\)
−0.424871 + 0.905254i \(0.639681\pi\)
\(828\) 13.5017 0.0163064
\(829\) 416.873i 0.502863i 0.967875 + 0.251431i \(0.0809013\pi\)
−0.967875 + 0.251431i \(0.919099\pi\)
\(830\) −289.740 −0.349084
\(831\) − 841.473i − 1.01260i
\(832\) 58.0805i 0.0698083i
\(833\) 0 0
\(834\) −29.7189 −0.0356342
\(835\) 211.653 0.253476
\(836\) 149.773i 0.179155i
\(837\) −6.33164 −0.00756468
\(838\) − 522.571i − 0.623593i
\(839\) − 359.231i − 0.428166i −0.976815 0.214083i \(-0.931324\pi\)
0.976815 0.214083i \(-0.0686763\pi\)
\(840\) 0 0
\(841\) 940.871 1.11875
\(842\) 307.692 0.365430
\(843\) − 623.943i − 0.740146i
\(844\) 302.374 0.358263
\(845\) − 260.036i − 0.307734i
\(846\) 242.211i 0.286301i
\(847\) 0 0
\(848\) 58.1607 0.0685858
\(849\) −468.523 −0.551852
\(850\) − 65.6988i − 0.0772927i
\(851\) −78.8933 −0.0927066
\(852\) − 353.291i − 0.414661i
\(853\) 988.948i 1.15938i 0.814838 + 0.579688i \(0.196826\pi\)
−0.814838 + 0.579688i \(0.803174\pi\)
\(854\) 0 0
\(855\) −40.7620 −0.0476749
\(856\) −240.334 −0.280765
\(857\) − 361.298i − 0.421584i −0.977531 0.210792i \(-0.932396\pi\)
0.977531 0.210792i \(-0.0676043\pi\)
\(858\) −219.165 −0.255437
\(859\) − 702.593i − 0.817920i −0.912552 0.408960i \(-0.865892\pi\)
0.912552 0.408960i \(-0.134108\pi\)
\(860\) 152.138i 0.176904i
\(861\) 0 0
\(862\) 404.226 0.468940
\(863\) 1102.42 1.27742 0.638711 0.769447i \(-0.279468\pi\)
0.638711 + 0.769447i \(0.279468\pi\)
\(864\) − 29.3939i − 0.0340207i
\(865\) −582.925 −0.673902
\(866\) 910.033i 1.05085i
\(867\) − 351.040i − 0.404891i
\(868\) 0 0
\(869\) 1376.68 1.58421
\(870\) 231.206 0.265754
\(871\) 359.162i 0.412356i
\(872\) 488.917 0.560684
\(873\) − 184.337i − 0.211153i
\(874\) 19.3376i 0.0221254i
\(875\) 0 0
\(876\) −285.113 −0.325472
\(877\) 668.121 0.761826 0.380913 0.924611i \(-0.375610\pi\)
0.380913 + 0.924611i \(0.375610\pi\)
\(878\) − 806.720i − 0.918815i
\(879\) −16.6836 −0.0189802
\(880\) − 110.230i − 0.125262i
\(881\) 15.3854i 0.0174636i 0.999962 + 0.00873178i \(0.00277945\pi\)
−0.999962 + 0.00873178i \(0.997221\pi\)
\(882\) 0 0
\(883\) 51.0884 0.0578577 0.0289289 0.999581i \(-0.490790\pi\)
0.0289289 + 0.999581i \(0.490790\pi\)
\(884\) 134.910 0.152613
\(885\) 224.084i 0.253202i
\(886\) 93.3591 0.105371
\(887\) 30.4440i 0.0343225i 0.999853 + 0.0171612i \(0.00546286\pi\)
−0.999853 + 0.0171612i \(0.994537\pi\)
\(888\) 171.755i 0.193417i
\(889\) 0 0
\(890\) −404.122 −0.454070
\(891\) 110.917 0.124486
\(892\) − 617.171i − 0.691896i
\(893\) −346.902 −0.388468
\(894\) 709.679i 0.793825i
\(895\) − 685.419i − 0.765831i
\(896\) 0 0
\(897\) −28.2969 −0.0315462
\(898\) −728.422 −0.811160
\(899\) 51.4366i 0.0572154i
\(900\) 30.0000 0.0333333
\(901\) − 135.096i − 0.149940i
\(902\) − 1008.80i − 1.11841i
\(903\) 0 0
\(904\) 232.105 0.256754
\(905\) −704.105 −0.778017
\(906\) 287.333i 0.317144i
\(907\) −690.016 −0.760767 −0.380384 0.924829i \(-0.624208\pi\)
−0.380384 + 0.924829i \(0.624208\pi\)
\(908\) − 200.845i − 0.221195i
\(909\) 186.418i 0.205080i
\(910\) 0 0
\(911\) −264.542 −0.290386 −0.145193 0.989403i \(-0.546380\pi\)
−0.145193 + 0.989403i \(0.546380\pi\)
\(912\) 42.0988 0.0461610
\(913\) 1129.18i 1.23678i
\(914\) −111.761 −0.122277
\(915\) 22.7031i 0.0248122i
\(916\) 639.993i 0.698682i
\(917\) 0 0
\(918\) −68.2762 −0.0743750
\(919\) 538.135 0.585566 0.292783 0.956179i \(-0.405419\pi\)
0.292783 + 0.956179i \(0.405419\pi\)
\(920\) − 14.2321i − 0.0154696i
\(921\) −79.7457 −0.0865860
\(922\) 13.5526i 0.0146992i
\(923\) 740.428i 0.802198i
\(924\) 0 0
\(925\) −175.296 −0.189510
\(926\) 328.244 0.354475
\(927\) 533.825i 0.575864i
\(928\) −238.788 −0.257315
\(929\) − 889.199i − 0.957157i −0.878045 0.478579i \(-0.841152\pi\)
0.878045 0.478579i \(-0.158848\pi\)
\(930\) 6.67413i 0.00717649i
\(931\) 0 0
\(932\) 621.578 0.666929
\(933\) 756.321 0.810633
\(934\) − 969.495i − 1.03800i
\(935\) −256.043 −0.273843
\(936\) 61.6037i 0.0658159i
\(937\) − 997.355i − 1.06441i −0.846615 0.532206i \(-0.821363\pi\)
0.846615 0.532206i \(-0.178637\pi\)
\(938\) 0 0
\(939\) 496.270 0.528509
\(940\) 255.313 0.271609
\(941\) 1107.47i 1.17691i 0.808531 + 0.588454i \(0.200263\pi\)
−0.808531 + 0.588454i \(0.799737\pi\)
\(942\) 450.063 0.477774
\(943\) − 130.249i − 0.138122i
\(944\) − 231.433i − 0.245162i
\(945\) 0 0
\(946\) 592.915 0.626760
\(947\) 587.602 0.620488 0.310244 0.950657i \(-0.399589\pi\)
0.310244 + 0.950657i \(0.399589\pi\)
\(948\) − 386.961i − 0.408187i
\(949\) 597.540 0.629653
\(950\) 42.9669i 0.0452284i
\(951\) 2.37089i 0.00249305i
\(952\) 0 0
\(953\) 441.771 0.463558 0.231779 0.972768i \(-0.425545\pi\)
0.231779 + 0.972768i \(0.425545\pi\)
\(954\) 61.6888 0.0646633
\(955\) 203.011i 0.212577i
\(956\) 272.526 0.285069
\(957\) − 901.061i − 0.941548i
\(958\) − 889.684i − 0.928688i
\(959\) 0 0
\(960\) −30.9839 −0.0322749
\(961\) 959.515 0.998455
\(962\) − 359.963i − 0.374182i
\(963\) −254.913 −0.264707
\(964\) − 772.359i − 0.801202i
\(965\) 513.524i 0.532150i
\(966\) 0 0
\(967\) 1581.63 1.63561 0.817804 0.575497i \(-0.195191\pi\)
0.817804 + 0.575497i \(0.195191\pi\)
\(968\) −87.3522 −0.0902398
\(969\) − 97.7873i − 0.100916i
\(970\) −194.308 −0.200317
\(971\) 597.019i 0.614849i 0.951572 + 0.307425i \(0.0994672\pi\)
−0.951572 + 0.307425i \(0.900533\pi\)
\(972\) − 31.1769i − 0.0320750i
\(973\) 0 0
\(974\) 705.365 0.724194
\(975\) −62.8740 −0.0644862
\(976\) − 23.4477i − 0.0240243i
\(977\) −211.442 −0.216420 −0.108210 0.994128i \(-0.534512\pi\)
−0.108210 + 0.994128i \(0.534512\pi\)
\(978\) 297.787i 0.304485i
\(979\) 1574.96i 1.60874i
\(980\) 0 0
\(981\) 518.575 0.528618
\(982\) −235.584 −0.239903
\(983\) 592.844i 0.603096i 0.953451 + 0.301548i \(0.0975034\pi\)
−0.953451 + 0.301548i \(0.902497\pi\)
\(984\) −283.558 −0.288169
\(985\) − 509.799i − 0.517562i
\(986\) 554.659i 0.562534i
\(987\) 0 0
\(988\) −88.2308 −0.0893024
\(989\) 76.5525 0.0774039
\(990\) − 116.917i − 0.118098i
\(991\) −379.941 −0.383391 −0.191696 0.981454i \(-0.561399\pi\)
−0.191696 + 0.981454i \(0.561399\pi\)
\(992\) − 6.89301i − 0.00694860i
\(993\) 665.166i 0.669855i
\(994\) 0 0
\(995\) −24.6999 −0.0248240
\(996\) 317.394 0.318669
\(997\) − 206.634i − 0.207256i −0.994616 0.103628i \(-0.966955\pi\)
0.994616 0.103628i \(-0.0330451\pi\)
\(998\) −51.3447 −0.0514476
\(999\) 182.173i 0.182356i
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 1470.3.f.a.391.1 8
7.2 even 3 210.3.o.a.31.3 8
7.3 odd 6 210.3.o.a.61.3 yes 8
7.6 odd 2 inner 1470.3.f.a.391.4 8
21.2 odd 6 630.3.v.b.451.2 8
21.17 even 6 630.3.v.b.271.2 8
35.2 odd 12 1050.3.q.c.199.4 16
35.3 even 12 1050.3.q.c.649.4 16
35.9 even 6 1050.3.p.b.451.2 8
35.17 even 12 1050.3.q.c.649.5 16
35.23 odd 12 1050.3.q.c.199.5 16
35.24 odd 6 1050.3.p.b.901.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.3.o.a.31.3 8 7.2 even 3
210.3.o.a.61.3 yes 8 7.3 odd 6
630.3.v.b.271.2 8 21.17 even 6
630.3.v.b.451.2 8 21.2 odd 6
1050.3.p.b.451.2 8 35.9 even 6
1050.3.p.b.901.2 8 35.24 odd 6
1050.3.q.c.199.4 16 35.2 odd 12
1050.3.q.c.199.5 16 35.23 odd 12
1050.3.q.c.649.4 16 35.3 even 12
1050.3.q.c.649.5 16 35.17 even 12
1470.3.f.a.391.1 8 1.1 even 1 trivial
1470.3.f.a.391.4 8 7.6 odd 2 inner