Properties

Label 475.2.a.d.1.1
Level $475$
Weight $2$
Character 475.1
Self dual yes
Analytic conductor $3.793$
Analytic rank $1$
Dimension $3$
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] = [475,2,Mod(1,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, 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("475.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.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: \(3.79289409601\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.80194 q^{2} -0.554958 q^{3} +5.85086 q^{4} +1.55496 q^{6} +3.04892 q^{7} -10.7899 q^{8} -2.69202 q^{9} +O(q^{10})\) \(q-2.80194 q^{2} -0.554958 q^{3} +5.85086 q^{4} +1.55496 q^{6} +3.04892 q^{7} -10.7899 q^{8} -2.69202 q^{9} -2.93900 q^{11} -3.24698 q^{12} -3.24698 q^{13} -8.54288 q^{14} +18.5308 q^{16} +2.15883 q^{17} +7.54288 q^{18} -1.00000 q^{19} -1.69202 q^{21} +8.23490 q^{22} -1.19806 q^{23} +5.98792 q^{24} +9.09783 q^{26} +3.15883 q^{27} +17.8388 q^{28} -1.77479 q^{29} -9.34481 q^{31} -30.3424 q^{32} +1.63102 q^{33} -6.04892 q^{34} -15.7506 q^{36} +1.15883 q^{37} +2.80194 q^{38} +1.80194 q^{39} +8.57002 q^{41} +4.74094 q^{42} -5.27413 q^{43} -17.1957 q^{44} +3.35690 q^{46} -2.35690 q^{47} -10.2838 q^{48} +2.29590 q^{49} -1.19806 q^{51} -18.9976 q^{52} -8.82371 q^{53} -8.85086 q^{54} -32.8974 q^{56} +0.554958 q^{57} +4.97285 q^{58} -5.70171 q^{59} -9.96077 q^{61} +26.1836 q^{62} -8.20775 q^{63} +47.9560 q^{64} -4.57002 q^{66} -4.98254 q^{67} +12.6310 q^{68} +0.664874 q^{69} +2.70171 q^{71} +29.0465 q^{72} -13.7778 q^{73} -3.24698 q^{74} -5.85086 q^{76} -8.96077 q^{77} -5.04892 q^{78} +5.66487 q^{79} +6.32304 q^{81} -24.0127 q^{82} +3.00969 q^{83} -9.89977 q^{84} +14.7778 q^{86} +0.984935 q^{87} +31.7114 q^{88} -10.2838 q^{89} -9.89977 q^{91} -7.00969 q^{92} +5.18598 q^{93} +6.60388 q^{94} +16.8388 q^{96} -3.24698 q^{97} -6.43296 q^{98} +7.91185 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{6} - 9 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{6} - 9 q^{8} - 3 q^{9} + q^{11} - 5 q^{12} - 5 q^{13} - 7 q^{14} + 18 q^{16} - 2 q^{17} + 4 q^{18} - 3 q^{19} + q^{22} - 8 q^{23} - q^{24} + 9 q^{26} + q^{27} + 21 q^{28} - 7 q^{29} - 5 q^{31} - 27 q^{32} - 10 q^{33} - 9 q^{34} - 11 q^{36} - 5 q^{37} + 4 q^{38} + q^{39} + q^{41} - 5 q^{43} - 15 q^{44} + 6 q^{46} - 3 q^{47} + 2 q^{48} - 7 q^{49} - 8 q^{51} - 16 q^{52} - 19 q^{53} - 13 q^{54} - 35 q^{56} + 2 q^{57} + 21 q^{58} + 10 q^{59} - 17 q^{61} + 23 q^{62} - 7 q^{63} + 49 q^{64} + 11 q^{66} + q^{67} + 23 q^{68} + 3 q^{69} - 19 q^{71} + 37 q^{72} + q^{73} - 5 q^{74} - 4 q^{76} - 14 q^{77} - 6 q^{78} + 18 q^{79} - q^{81} - 6 q^{82} - 13 q^{83} - 7 q^{84} + 2 q^{86} + 28 q^{87} + 46 q^{88} + 2 q^{89} - 7 q^{91} + q^{92} + q^{93} + 11 q^{94} + 18 q^{96} - 5 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.80194 −1.98127 −0.990635 0.136540i \(-0.956402\pi\)
−0.990635 + 0.136540i \(0.956402\pi\)
\(3\) −0.554958 −0.320405 −0.160203 0.987084i \(-0.551215\pi\)
−0.160203 + 0.987084i \(0.551215\pi\)
\(4\) 5.85086 2.92543
\(5\) 0 0
\(6\) 1.55496 0.634809
\(7\) 3.04892 1.15238 0.576191 0.817315i \(-0.304538\pi\)
0.576191 + 0.817315i \(0.304538\pi\)
\(8\) −10.7899 −3.81479
\(9\) −2.69202 −0.897340
\(10\) 0 0
\(11\) −2.93900 −0.886142 −0.443071 0.896486i \(-0.646111\pi\)
−0.443071 + 0.896486i \(0.646111\pi\)
\(12\) −3.24698 −0.937322
\(13\) −3.24698 −0.900550 −0.450275 0.892890i \(-0.648674\pi\)
−0.450275 + 0.892890i \(0.648674\pi\)
\(14\) −8.54288 −2.28318
\(15\) 0 0
\(16\) 18.5308 4.63270
\(17\) 2.15883 0.523594 0.261797 0.965123i \(-0.415685\pi\)
0.261797 + 0.965123i \(0.415685\pi\)
\(18\) 7.54288 1.77787
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.69202 −0.369229
\(22\) 8.23490 1.75569
\(23\) −1.19806 −0.249813 −0.124907 0.992169i \(-0.539863\pi\)
−0.124907 + 0.992169i \(0.539863\pi\)
\(24\) 5.98792 1.22228
\(25\) 0 0
\(26\) 9.09783 1.78423
\(27\) 3.15883 0.607918
\(28\) 17.8388 3.37121
\(29\) −1.77479 −0.329570 −0.164785 0.986329i \(-0.552693\pi\)
−0.164785 + 0.986329i \(0.552693\pi\)
\(30\) 0 0
\(31\) −9.34481 −1.67838 −0.839189 0.543840i \(-0.816970\pi\)
−0.839189 + 0.543840i \(0.816970\pi\)
\(32\) −30.3424 −5.36383
\(33\) 1.63102 0.283925
\(34\) −6.04892 −1.03738
\(35\) 0 0
\(36\) −15.7506 −2.62510
\(37\) 1.15883 0.190511 0.0952555 0.995453i \(-0.469633\pi\)
0.0952555 + 0.995453i \(0.469633\pi\)
\(38\) 2.80194 0.454534
\(39\) 1.80194 0.288541
\(40\) 0 0
\(41\) 8.57002 1.33841 0.669206 0.743077i \(-0.266634\pi\)
0.669206 + 0.743077i \(0.266634\pi\)
\(42\) 4.74094 0.731543
\(43\) −5.27413 −0.804297 −0.402148 0.915575i \(-0.631736\pi\)
−0.402148 + 0.915575i \(0.631736\pi\)
\(44\) −17.1957 −2.59234
\(45\) 0 0
\(46\) 3.35690 0.494947
\(47\) −2.35690 −0.343789 −0.171894 0.985115i \(-0.554989\pi\)
−0.171894 + 0.985115i \(0.554989\pi\)
\(48\) −10.2838 −1.48434
\(49\) 2.29590 0.327985
\(50\) 0 0
\(51\) −1.19806 −0.167762
\(52\) −18.9976 −2.63449
\(53\) −8.82371 −1.21203 −0.606015 0.795453i \(-0.707233\pi\)
−0.606015 + 0.795453i \(0.707233\pi\)
\(54\) −8.85086 −1.20445
\(55\) 0 0
\(56\) −32.8974 −4.39610
\(57\) 0.554958 0.0735060
\(58\) 4.97285 0.652968
\(59\) −5.70171 −0.742299 −0.371150 0.928573i \(-0.621036\pi\)
−0.371150 + 0.928573i \(0.621036\pi\)
\(60\) 0 0
\(61\) −9.96077 −1.27535 −0.637673 0.770307i \(-0.720103\pi\)
−0.637673 + 0.770307i \(0.720103\pi\)
\(62\) 26.1836 3.32532
\(63\) −8.20775 −1.03408
\(64\) 47.9560 5.99450
\(65\) 0 0
\(66\) −4.57002 −0.562531
\(67\) −4.98254 −0.608714 −0.304357 0.952558i \(-0.598442\pi\)
−0.304357 + 0.952558i \(0.598442\pi\)
\(68\) 12.6310 1.53174
\(69\) 0.664874 0.0800415
\(70\) 0 0
\(71\) 2.70171 0.320634 0.160317 0.987066i \(-0.448748\pi\)
0.160317 + 0.987066i \(0.448748\pi\)
\(72\) 29.0465 3.42317
\(73\) −13.7778 −1.61257 −0.806283 0.591530i \(-0.798524\pi\)
−0.806283 + 0.591530i \(0.798524\pi\)
\(74\) −3.24698 −0.377454
\(75\) 0 0
\(76\) −5.85086 −0.671139
\(77\) −8.96077 −1.02117
\(78\) −5.04892 −0.571677
\(79\) 5.66487 0.637348 0.318674 0.947864i \(-0.396762\pi\)
0.318674 + 0.947864i \(0.396762\pi\)
\(80\) 0 0
\(81\) 6.32304 0.702560
\(82\) −24.0127 −2.65176
\(83\) 3.00969 0.330356 0.165178 0.986264i \(-0.447180\pi\)
0.165178 + 0.986264i \(0.447180\pi\)
\(84\) −9.89977 −1.08015
\(85\) 0 0
\(86\) 14.7778 1.59353
\(87\) 0.984935 0.105596
\(88\) 31.7114 3.38045
\(89\) −10.2838 −1.09008 −0.545041 0.838409i \(-0.683486\pi\)
−0.545041 + 0.838409i \(0.683486\pi\)
\(90\) 0 0
\(91\) −9.89977 −1.03778
\(92\) −7.00969 −0.730811
\(93\) 5.18598 0.537761
\(94\) 6.60388 0.681138
\(95\) 0 0
\(96\) 16.8388 1.71860
\(97\) −3.24698 −0.329681 −0.164840 0.986320i \(-0.552711\pi\)
−0.164840 + 0.986320i \(0.552711\pi\)
\(98\) −6.43296 −0.649827
\(99\) 7.91185 0.795171
\(100\) 0 0
\(101\) 5.09246 0.506718 0.253359 0.967372i \(-0.418465\pi\)
0.253359 + 0.967372i \(0.418465\pi\)
\(102\) 3.35690 0.332382
\(103\) 14.3110 1.41010 0.705051 0.709157i \(-0.250924\pi\)
0.705051 + 0.709157i \(0.250924\pi\)
\(104\) 35.0344 3.43541
\(105\) 0 0
\(106\) 24.7235 2.40136
\(107\) 8.18598 0.791369 0.395684 0.918387i \(-0.370507\pi\)
0.395684 + 0.918387i \(0.370507\pi\)
\(108\) 18.4819 1.77842
\(109\) −11.3274 −1.08496 −0.542482 0.840067i \(-0.682515\pi\)
−0.542482 + 0.840067i \(0.682515\pi\)
\(110\) 0 0
\(111\) −0.643104 −0.0610407
\(112\) 56.4989 5.33864
\(113\) −3.43535 −0.323171 −0.161585 0.986859i \(-0.551661\pi\)
−0.161585 + 0.986859i \(0.551661\pi\)
\(114\) −1.55496 −0.145635
\(115\) 0 0
\(116\) −10.3840 −0.964134
\(117\) 8.74094 0.808100
\(118\) 15.9758 1.47069
\(119\) 6.58211 0.603381
\(120\) 0 0
\(121\) −2.36227 −0.214752
\(122\) 27.9095 2.52680
\(123\) −4.75600 −0.428834
\(124\) −54.6752 −4.90997
\(125\) 0 0
\(126\) 22.9976 2.04879
\(127\) −1.46144 −0.129681 −0.0648407 0.997896i \(-0.520654\pi\)
−0.0648407 + 0.997896i \(0.520654\pi\)
\(128\) −73.6848 −6.51288
\(129\) 2.92692 0.257701
\(130\) 0 0
\(131\) −13.2295 −1.15587 −0.577934 0.816083i \(-0.696141\pi\)
−0.577934 + 0.816083i \(0.696141\pi\)
\(132\) 9.54288 0.830601
\(133\) −3.04892 −0.264375
\(134\) 13.9608 1.20603
\(135\) 0 0
\(136\) −23.2935 −1.99740
\(137\) −16.1739 −1.38183 −0.690915 0.722936i \(-0.742792\pi\)
−0.690915 + 0.722936i \(0.742792\pi\)
\(138\) −1.86294 −0.158584
\(139\) 13.0978 1.11094 0.555472 0.831535i \(-0.312538\pi\)
0.555472 + 0.831535i \(0.312538\pi\)
\(140\) 0 0
\(141\) 1.30798 0.110152
\(142\) −7.57002 −0.635262
\(143\) 9.54288 0.798015
\(144\) −49.8853 −4.15711
\(145\) 0 0
\(146\) 38.6045 3.19493
\(147\) −1.27413 −0.105088
\(148\) 6.78017 0.557326
\(149\) 11.0640 0.906397 0.453198 0.891410i \(-0.350283\pi\)
0.453198 + 0.891410i \(0.350283\pi\)
\(150\) 0 0
\(151\) 10.4426 0.849811 0.424905 0.905238i \(-0.360307\pi\)
0.424905 + 0.905238i \(0.360307\pi\)
\(152\) 10.7899 0.875173
\(153\) −5.81163 −0.469842
\(154\) 25.1075 2.02322
\(155\) 0 0
\(156\) 10.5429 0.844106
\(157\) −2.48427 −0.198266 −0.0991332 0.995074i \(-0.531607\pi\)
−0.0991332 + 0.995074i \(0.531607\pi\)
\(158\) −15.8726 −1.26276
\(159\) 4.89679 0.388341
\(160\) 0 0
\(161\) −3.65279 −0.287880
\(162\) −17.7168 −1.39196
\(163\) −3.16852 −0.248178 −0.124089 0.992271i \(-0.539601\pi\)
−0.124089 + 0.992271i \(0.539601\pi\)
\(164\) 50.1420 3.91543
\(165\) 0 0
\(166\) −8.43296 −0.654525
\(167\) −4.74632 −0.367281 −0.183640 0.982993i \(-0.558788\pi\)
−0.183640 + 0.982993i \(0.558788\pi\)
\(168\) 18.2567 1.40853
\(169\) −2.45712 −0.189009
\(170\) 0 0
\(171\) 2.69202 0.205864
\(172\) −30.8582 −2.35291
\(173\) −3.96316 −0.301314 −0.150657 0.988586i \(-0.548139\pi\)
−0.150657 + 0.988586i \(0.548139\pi\)
\(174\) −2.75973 −0.209214
\(175\) 0 0
\(176\) −54.4620 −4.10523
\(177\) 3.16421 0.237837
\(178\) 28.8146 2.15975
\(179\) −14.0858 −1.05282 −0.526409 0.850231i \(-0.676462\pi\)
−0.526409 + 0.850231i \(0.676462\pi\)
\(180\) 0 0
\(181\) −12.2513 −0.910631 −0.455316 0.890330i \(-0.650474\pi\)
−0.455316 + 0.890330i \(0.650474\pi\)
\(182\) 27.7385 2.05612
\(183\) 5.52781 0.408628
\(184\) 12.9269 0.952985
\(185\) 0 0
\(186\) −14.5308 −1.06545
\(187\) −6.34481 −0.463979
\(188\) −13.7899 −1.00573
\(189\) 9.63102 0.700554
\(190\) 0 0
\(191\) 20.1468 1.45777 0.728884 0.684637i \(-0.240039\pi\)
0.728884 + 0.684637i \(0.240039\pi\)
\(192\) −26.6136 −1.92067
\(193\) 9.52781 0.685827 0.342913 0.939367i \(-0.388586\pi\)
0.342913 + 0.939367i \(0.388586\pi\)
\(194\) 9.09783 0.653186
\(195\) 0 0
\(196\) 13.4330 0.959497
\(197\) −4.96316 −0.353611 −0.176805 0.984246i \(-0.556576\pi\)
−0.176805 + 0.984246i \(0.556576\pi\)
\(198\) −22.1685 −1.57545
\(199\) 5.42221 0.384370 0.192185 0.981359i \(-0.438443\pi\)
0.192185 + 0.981359i \(0.438443\pi\)
\(200\) 0 0
\(201\) 2.76510 0.195035
\(202\) −14.2687 −1.00395
\(203\) −5.41119 −0.379791
\(204\) −7.00969 −0.490776
\(205\) 0 0
\(206\) −40.0984 −2.79379
\(207\) 3.22521 0.224168
\(208\) −60.1691 −4.17198
\(209\) 2.93900 0.203295
\(210\) 0 0
\(211\) 15.9638 1.09899 0.549495 0.835497i \(-0.314820\pi\)
0.549495 + 0.835497i \(0.314820\pi\)
\(212\) −51.6262 −3.54570
\(213\) −1.49934 −0.102733
\(214\) −22.9366 −1.56791
\(215\) 0 0
\(216\) −34.0834 −2.31908
\(217\) −28.4916 −1.93413
\(218\) 31.7385 2.14961
\(219\) 7.64609 0.516675
\(220\) 0 0
\(221\) −7.00969 −0.471523
\(222\) 1.80194 0.120938
\(223\) 23.2150 1.55459 0.777297 0.629134i \(-0.216590\pi\)
0.777297 + 0.629134i \(0.216590\pi\)
\(224\) −92.5115 −6.18119
\(225\) 0 0
\(226\) 9.62565 0.640288
\(227\) 17.4795 1.16015 0.580077 0.814562i \(-0.303022\pi\)
0.580077 + 0.814562i \(0.303022\pi\)
\(228\) 3.24698 0.215036
\(229\) −2.32544 −0.153669 −0.0768346 0.997044i \(-0.524481\pi\)
−0.0768346 + 0.997044i \(0.524481\pi\)
\(230\) 0 0
\(231\) 4.97285 0.327190
\(232\) 19.1497 1.25724
\(233\) 18.9342 1.24042 0.620211 0.784435i \(-0.287047\pi\)
0.620211 + 0.784435i \(0.287047\pi\)
\(234\) −24.4916 −1.60106
\(235\) 0 0
\(236\) −33.3599 −2.17154
\(237\) −3.14377 −0.204210
\(238\) −18.4426 −1.19546
\(239\) 11.1685 0.722432 0.361216 0.932482i \(-0.382362\pi\)
0.361216 + 0.932482i \(0.382362\pi\)
\(240\) 0 0
\(241\) 7.42998 0.478607 0.239303 0.970945i \(-0.423081\pi\)
0.239303 + 0.970945i \(0.423081\pi\)
\(242\) 6.61894 0.425482
\(243\) −12.9855 −0.833022
\(244\) −58.2790 −3.73093
\(245\) 0 0
\(246\) 13.3260 0.849637
\(247\) 3.24698 0.206600
\(248\) 100.829 6.40266
\(249\) −1.67025 −0.105848
\(250\) 0 0
\(251\) −14.7409 −0.930440 −0.465220 0.885195i \(-0.654025\pi\)
−0.465220 + 0.885195i \(0.654025\pi\)
\(252\) −48.0224 −3.02512
\(253\) 3.52111 0.221370
\(254\) 4.09485 0.256934
\(255\) 0 0
\(256\) 110.548 6.90927
\(257\) −3.34721 −0.208793 −0.104397 0.994536i \(-0.533291\pi\)
−0.104397 + 0.994536i \(0.533291\pi\)
\(258\) −8.20105 −0.510575
\(259\) 3.53319 0.219542
\(260\) 0 0
\(261\) 4.77777 0.295737
\(262\) 37.0683 2.29009
\(263\) 16.6853 1.02886 0.514430 0.857532i \(-0.328003\pi\)
0.514430 + 0.857532i \(0.328003\pi\)
\(264\) −17.5985 −1.08311
\(265\) 0 0
\(266\) 8.54288 0.523797
\(267\) 5.70709 0.349268
\(268\) −29.1521 −1.78075
\(269\) 15.5961 0.950911 0.475456 0.879740i \(-0.342283\pi\)
0.475456 + 0.879740i \(0.342283\pi\)
\(270\) 0 0
\(271\) −13.2131 −0.802640 −0.401320 0.915938i \(-0.631449\pi\)
−0.401320 + 0.915938i \(0.631449\pi\)
\(272\) 40.0049 2.42565
\(273\) 5.49396 0.332510
\(274\) 45.3183 2.73778
\(275\) 0 0
\(276\) 3.89008 0.234156
\(277\) −12.9758 −0.779642 −0.389821 0.920891i \(-0.627463\pi\)
−0.389821 + 0.920891i \(0.627463\pi\)
\(278\) −36.6993 −2.20108
\(279\) 25.1564 1.50608
\(280\) 0 0
\(281\) 24.8901 1.48482 0.742409 0.669947i \(-0.233683\pi\)
0.742409 + 0.669947i \(0.233683\pi\)
\(282\) −3.66487 −0.218240
\(283\) 13.5821 0.807372 0.403686 0.914898i \(-0.367729\pi\)
0.403686 + 0.914898i \(0.367729\pi\)
\(284\) 15.8073 0.937992
\(285\) 0 0
\(286\) −26.7385 −1.58108
\(287\) 26.1293 1.54236
\(288\) 81.6824 4.81318
\(289\) −12.3394 −0.725849
\(290\) 0 0
\(291\) 1.80194 0.105631
\(292\) −80.6118 −4.71745
\(293\) −1.27652 −0.0745751 −0.0372875 0.999305i \(-0.511872\pi\)
−0.0372875 + 0.999305i \(0.511872\pi\)
\(294\) 3.57002 0.208208
\(295\) 0 0
\(296\) −12.5036 −0.726760
\(297\) −9.28382 −0.538702
\(298\) −31.0006 −1.79582
\(299\) 3.89008 0.224969
\(300\) 0 0
\(301\) −16.0804 −0.926857
\(302\) −29.2597 −1.68370
\(303\) −2.82610 −0.162355
\(304\) −18.5308 −1.06281
\(305\) 0 0
\(306\) 16.2838 0.930884
\(307\) 32.2295 1.83944 0.919718 0.392580i \(-0.128417\pi\)
0.919718 + 0.392580i \(0.128417\pi\)
\(308\) −52.4282 −2.98737
\(309\) −7.94198 −0.451804
\(310\) 0 0
\(311\) 12.4983 0.708712 0.354356 0.935111i \(-0.384700\pi\)
0.354356 + 0.935111i \(0.384700\pi\)
\(312\) −19.4426 −1.10072
\(313\) 20.9390 1.18354 0.591771 0.806106i \(-0.298429\pi\)
0.591771 + 0.806106i \(0.298429\pi\)
\(314\) 6.96077 0.392819
\(315\) 0 0
\(316\) 33.1444 1.86452
\(317\) −15.8562 −0.890575 −0.445287 0.895388i \(-0.646898\pi\)
−0.445287 + 0.895388i \(0.646898\pi\)
\(318\) −13.7205 −0.769407
\(319\) 5.21611 0.292046
\(320\) 0 0
\(321\) −4.54288 −0.253559
\(322\) 10.2349 0.570369
\(323\) −2.15883 −0.120121
\(324\) 36.9952 2.05529
\(325\) 0 0
\(326\) 8.87800 0.491707
\(327\) 6.28621 0.347628
\(328\) −92.4693 −5.10576
\(329\) −7.18598 −0.396176
\(330\) 0 0
\(331\) −13.2349 −0.727456 −0.363728 0.931505i \(-0.618496\pi\)
−0.363728 + 0.931505i \(0.618496\pi\)
\(332\) 17.6093 0.966433
\(333\) −3.11960 −0.170953
\(334\) 13.2989 0.727682
\(335\) 0 0
\(336\) −31.3545 −1.71053
\(337\) 26.0780 1.42056 0.710279 0.703920i \(-0.248569\pi\)
0.710279 + 0.703920i \(0.248569\pi\)
\(338\) 6.88471 0.374479
\(339\) 1.90648 0.103546
\(340\) 0 0
\(341\) 27.4644 1.48728
\(342\) −7.54288 −0.407872
\(343\) −14.3424 −0.774418
\(344\) 56.9071 3.06822
\(345\) 0 0
\(346\) 11.1045 0.596984
\(347\) −8.98254 −0.482208 −0.241104 0.970499i \(-0.577510\pi\)
−0.241104 + 0.970499i \(0.577510\pi\)
\(348\) 5.76271 0.308914
\(349\) 0.599564 0.0320939 0.0160470 0.999871i \(-0.494892\pi\)
0.0160470 + 0.999871i \(0.494892\pi\)
\(350\) 0 0
\(351\) −10.2567 −0.547460
\(352\) 89.1764 4.75312
\(353\) −31.8896 −1.69731 −0.848656 0.528945i \(-0.822588\pi\)
−0.848656 + 0.528945i \(0.822588\pi\)
\(354\) −8.86592 −0.471218
\(355\) 0 0
\(356\) −60.1691 −3.18896
\(357\) −3.65279 −0.193326
\(358\) 39.4674 2.08592
\(359\) −18.3763 −0.969863 −0.484931 0.874552i \(-0.661155\pi\)
−0.484931 + 0.874552i \(0.661155\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 34.3274 1.80421
\(363\) 1.31096 0.0688077
\(364\) −57.9221 −3.03594
\(365\) 0 0
\(366\) −15.4886 −0.809601
\(367\) 14.6377 0.764083 0.382042 0.924145i \(-0.375221\pi\)
0.382042 + 0.924145i \(0.375221\pi\)
\(368\) −22.2010 −1.15731
\(369\) −23.0707 −1.20101
\(370\) 0 0
\(371\) −26.9028 −1.39672
\(372\) 30.3424 1.57318
\(373\) 1.02715 0.0531837 0.0265918 0.999646i \(-0.491535\pi\)
0.0265918 + 0.999646i \(0.491535\pi\)
\(374\) 17.7778 0.919267
\(375\) 0 0
\(376\) 25.4306 1.31148
\(377\) 5.76271 0.296795
\(378\) −26.9855 −1.38799
\(379\) −22.5284 −1.15721 −0.578603 0.815609i \(-0.696402\pi\)
−0.578603 + 0.815609i \(0.696402\pi\)
\(380\) 0 0
\(381\) 0.811035 0.0415506
\(382\) −56.4499 −2.88823
\(383\) −32.6698 −1.66935 −0.834674 0.550745i \(-0.814344\pi\)
−0.834674 + 0.550745i \(0.814344\pi\)
\(384\) 40.8920 2.08676
\(385\) 0 0
\(386\) −26.6963 −1.35881
\(387\) 14.1981 0.721728
\(388\) −18.9976 −0.964457
\(389\) −8.16421 −0.413942 −0.206971 0.978347i \(-0.566361\pi\)
−0.206971 + 0.978347i \(0.566361\pi\)
\(390\) 0 0
\(391\) −2.58642 −0.130801
\(392\) −24.7724 −1.25120
\(393\) 7.34183 0.370346
\(394\) 13.9065 0.700598
\(395\) 0 0
\(396\) 46.2911 2.32622
\(397\) −37.5478 −1.88447 −0.942235 0.334954i \(-0.891279\pi\)
−0.942235 + 0.334954i \(0.891279\pi\)
\(398\) −15.1927 −0.761541
\(399\) 1.69202 0.0847070
\(400\) 0 0
\(401\) 32.2519 1.61058 0.805291 0.592880i \(-0.202009\pi\)
0.805291 + 0.592880i \(0.202009\pi\)
\(402\) −7.74764 −0.386417
\(403\) 30.3424 1.51146
\(404\) 29.7952 1.48237
\(405\) 0 0
\(406\) 15.1618 0.752468
\(407\) −3.40581 −0.168820
\(408\) 12.9269 0.639978
\(409\) −31.5459 −1.55984 −0.779921 0.625878i \(-0.784741\pi\)
−0.779921 + 0.625878i \(0.784741\pi\)
\(410\) 0 0
\(411\) 8.97584 0.442745
\(412\) 83.7314 4.12515
\(413\) −17.3840 −0.855413
\(414\) −9.03684 −0.444136
\(415\) 0 0
\(416\) 98.5212 4.83040
\(417\) −7.26875 −0.355952
\(418\) −8.23490 −0.402782
\(419\) −14.6866 −0.717490 −0.358745 0.933436i \(-0.616795\pi\)
−0.358745 + 0.933436i \(0.616795\pi\)
\(420\) 0 0
\(421\) 15.1142 0.736622 0.368311 0.929703i \(-0.379936\pi\)
0.368311 + 0.929703i \(0.379936\pi\)
\(422\) −44.7294 −2.17740
\(423\) 6.34481 0.308495
\(424\) 95.2065 4.62364
\(425\) 0 0
\(426\) 4.20105 0.203541
\(427\) −30.3696 −1.46969
\(428\) 47.8950 2.31509
\(429\) −5.29590 −0.255688
\(430\) 0 0
\(431\) 1.67696 0.0807761 0.0403881 0.999184i \(-0.487141\pi\)
0.0403881 + 0.999184i \(0.487141\pi\)
\(432\) 58.5357 2.81630
\(433\) 13.1545 0.632166 0.316083 0.948732i \(-0.397632\pi\)
0.316083 + 0.948732i \(0.397632\pi\)
\(434\) 79.8316 3.83204
\(435\) 0 0
\(436\) −66.2747 −3.17398
\(437\) 1.19806 0.0573111
\(438\) −21.4239 −1.02367
\(439\) 6.45580 0.308118 0.154059 0.988062i \(-0.450765\pi\)
0.154059 + 0.988062i \(0.450765\pi\)
\(440\) 0 0
\(441\) −6.18060 −0.294314
\(442\) 19.6407 0.934213
\(443\) 25.3709 1.20541 0.602704 0.797965i \(-0.294090\pi\)
0.602704 + 0.797965i \(0.294090\pi\)
\(444\) −3.76271 −0.178570
\(445\) 0 0
\(446\) −65.0471 −3.08007
\(447\) −6.14005 −0.290414
\(448\) 146.214 6.90795
\(449\) −3.59956 −0.169874 −0.0849370 0.996386i \(-0.527069\pi\)
−0.0849370 + 0.996386i \(0.527069\pi\)
\(450\) 0 0
\(451\) −25.1873 −1.18602
\(452\) −20.0998 −0.945413
\(453\) −5.79523 −0.272284
\(454\) −48.9764 −2.29858
\(455\) 0 0
\(456\) −5.98792 −0.280410
\(457\) 12.2784 0.574361 0.287181 0.957876i \(-0.407282\pi\)
0.287181 + 0.957876i \(0.407282\pi\)
\(458\) 6.51573 0.304460
\(459\) 6.81940 0.318302
\(460\) 0 0
\(461\) 39.4935 1.83939 0.919697 0.392628i \(-0.128434\pi\)
0.919697 + 0.392628i \(0.128434\pi\)
\(462\) −13.9336 −0.648251
\(463\) −18.3991 −0.855079 −0.427540 0.903997i \(-0.640620\pi\)
−0.427540 + 0.903997i \(0.640620\pi\)
\(464\) −32.8883 −1.52680
\(465\) 0 0
\(466\) −53.0525 −2.45761
\(467\) 36.2282 1.67644 0.838220 0.545332i \(-0.183596\pi\)
0.838220 + 0.545332i \(0.183596\pi\)
\(468\) 51.1420 2.36404
\(469\) −15.1914 −0.701472
\(470\) 0 0
\(471\) 1.37867 0.0635256
\(472\) 61.5206 2.83172
\(473\) 15.5007 0.712721
\(474\) 8.80864 0.404594
\(475\) 0 0
\(476\) 38.5109 1.76515
\(477\) 23.7536 1.08760
\(478\) −31.2935 −1.43133
\(479\) −34.3129 −1.56780 −0.783898 0.620890i \(-0.786771\pi\)
−0.783898 + 0.620890i \(0.786771\pi\)
\(480\) 0 0
\(481\) −3.76271 −0.171565
\(482\) −20.8183 −0.948249
\(483\) 2.02715 0.0922384
\(484\) −13.8213 −0.628242
\(485\) 0 0
\(486\) 36.3846 1.65044
\(487\) 24.4722 1.10894 0.554470 0.832203i \(-0.312921\pi\)
0.554470 + 0.832203i \(0.312921\pi\)
\(488\) 107.475 4.86518
\(489\) 1.75840 0.0795175
\(490\) 0 0
\(491\) 18.3067 0.826168 0.413084 0.910693i \(-0.364452\pi\)
0.413084 + 0.910693i \(0.364452\pi\)
\(492\) −27.8267 −1.25452
\(493\) −3.83148 −0.172561
\(494\) −9.09783 −0.409331
\(495\) 0 0
\(496\) −173.167 −7.77542
\(497\) 8.23729 0.369493
\(498\) 4.67994 0.209713
\(499\) −25.3424 −1.13448 −0.567241 0.823552i \(-0.691989\pi\)
−0.567241 + 0.823552i \(0.691989\pi\)
\(500\) 0 0
\(501\) 2.63401 0.117679
\(502\) 41.3032 1.84345
\(503\) −14.1105 −0.629156 −0.314578 0.949232i \(-0.601863\pi\)
−0.314578 + 0.949232i \(0.601863\pi\)
\(504\) 88.5605 3.94480
\(505\) 0 0
\(506\) −9.86592 −0.438594
\(507\) 1.36360 0.0605596
\(508\) −8.55065 −0.379374
\(509\) −17.8019 −0.789057 −0.394529 0.918884i \(-0.629092\pi\)
−0.394529 + 0.918884i \(0.629092\pi\)
\(510\) 0 0
\(511\) −42.0073 −1.85829
\(512\) −162.380 −7.17625
\(513\) −3.15883 −0.139466
\(514\) 9.37867 0.413675
\(515\) 0 0
\(516\) 17.1250 0.753885
\(517\) 6.92692 0.304646
\(518\) −9.89977 −0.434971
\(519\) 2.19939 0.0965425
\(520\) 0 0
\(521\) 13.3948 0.586837 0.293418 0.955984i \(-0.405207\pi\)
0.293418 + 0.955984i \(0.405207\pi\)
\(522\) −13.3870 −0.585934
\(523\) 1.46921 0.0642439 0.0321219 0.999484i \(-0.489774\pi\)
0.0321219 + 0.999484i \(0.489774\pi\)
\(524\) −77.4040 −3.38141
\(525\) 0 0
\(526\) −46.7512 −2.03845
\(527\) −20.1739 −0.878789
\(528\) 30.2241 1.31534
\(529\) −21.5646 −0.937593
\(530\) 0 0
\(531\) 15.3491 0.666095
\(532\) −17.8388 −0.773409
\(533\) −27.8267 −1.20531
\(534\) −15.9909 −0.691994
\(535\) 0 0
\(536\) 53.7609 2.32212
\(537\) 7.81700 0.337329
\(538\) −43.6993 −1.88401
\(539\) −6.74764 −0.290642
\(540\) 0 0
\(541\) −42.7144 −1.83643 −0.918217 0.396077i \(-0.870371\pi\)
−0.918217 + 0.396077i \(0.870371\pi\)
\(542\) 37.0224 1.59025
\(543\) 6.79895 0.291771
\(544\) −65.5042 −2.80847
\(545\) 0 0
\(546\) −15.3937 −0.658791
\(547\) −10.6866 −0.456928 −0.228464 0.973552i \(-0.573370\pi\)
−0.228464 + 0.973552i \(0.573370\pi\)
\(548\) −94.6311 −4.04244
\(549\) 26.8146 1.14442
\(550\) 0 0
\(551\) 1.77479 0.0756086
\(552\) −7.17390 −0.305341
\(553\) 17.2717 0.734469
\(554\) 36.3575 1.54468
\(555\) 0 0
\(556\) 76.6335 3.24999
\(557\) 34.2083 1.44945 0.724727 0.689036i \(-0.241966\pi\)
0.724727 + 0.689036i \(0.241966\pi\)
\(558\) −70.4868 −2.98394
\(559\) 17.1250 0.724310
\(560\) 0 0
\(561\) 3.52111 0.148661
\(562\) −69.7405 −2.94182
\(563\) 5.46788 0.230444 0.115222 0.993340i \(-0.463242\pi\)
0.115222 + 0.993340i \(0.463242\pi\)
\(564\) 7.65279 0.322241
\(565\) 0 0
\(566\) −38.0562 −1.59962
\(567\) 19.2784 0.809618
\(568\) −29.1511 −1.22315
\(569\) −17.4819 −0.732878 −0.366439 0.930442i \(-0.619423\pi\)
−0.366439 + 0.930442i \(0.619423\pi\)
\(570\) 0 0
\(571\) −7.21877 −0.302096 −0.151048 0.988526i \(-0.548265\pi\)
−0.151048 + 0.988526i \(0.548265\pi\)
\(572\) 55.8340 2.33454
\(573\) −11.1806 −0.467076
\(574\) −73.2127 −3.05584
\(575\) 0 0
\(576\) −129.099 −5.37911
\(577\) −1.80625 −0.0751952 −0.0375976 0.999293i \(-0.511971\pi\)
−0.0375976 + 0.999293i \(0.511971\pi\)
\(578\) 34.5743 1.43810
\(579\) −5.28754 −0.219743
\(580\) 0 0
\(581\) 9.17629 0.380697
\(582\) −5.04892 −0.209284
\(583\) 25.9329 1.07403
\(584\) 148.660 6.15160
\(585\) 0 0
\(586\) 3.57673 0.147753
\(587\) −2.39075 −0.0986767 −0.0493384 0.998782i \(-0.515711\pi\)
−0.0493384 + 0.998782i \(0.515711\pi\)
\(588\) −7.45473 −0.307428
\(589\) 9.34481 0.385046
\(590\) 0 0
\(591\) 2.75435 0.113299
\(592\) 21.4741 0.882580
\(593\) 42.2650 1.73562 0.867808 0.496899i \(-0.165528\pi\)
0.867808 + 0.496899i \(0.165528\pi\)
\(594\) 26.0127 1.06731
\(595\) 0 0
\(596\) 64.7338 2.65160
\(597\) −3.00910 −0.123154
\(598\) −10.8998 −0.445725
\(599\) 8.95838 0.366029 0.183015 0.983110i \(-0.441414\pi\)
0.183015 + 0.983110i \(0.441414\pi\)
\(600\) 0 0
\(601\) −19.8998 −0.811729 −0.405864 0.913933i \(-0.633029\pi\)
−0.405864 + 0.913933i \(0.633029\pi\)
\(602\) 45.0562 1.83635
\(603\) 13.4131 0.546224
\(604\) 61.0984 2.48606
\(605\) 0 0
\(606\) 7.91856 0.321669
\(607\) −26.0084 −1.05565 −0.527823 0.849354i \(-0.676992\pi\)
−0.527823 + 0.849354i \(0.676992\pi\)
\(608\) 30.3424 1.23055
\(609\) 3.00298 0.121687
\(610\) 0 0
\(611\) 7.65279 0.309599
\(612\) −34.0030 −1.37449
\(613\) −41.3763 −1.67117 −0.835586 0.549360i \(-0.814872\pi\)
−0.835586 + 0.549360i \(0.814872\pi\)
\(614\) −90.3051 −3.64442
\(615\) 0 0
\(616\) 96.6854 3.89557
\(617\) −43.9845 −1.77075 −0.885374 0.464880i \(-0.846098\pi\)
−0.885374 + 0.464880i \(0.846098\pi\)
\(618\) 22.2529 0.895145
\(619\) 23.4553 0.942749 0.471374 0.881933i \(-0.343758\pi\)
0.471374 + 0.881933i \(0.343758\pi\)
\(620\) 0 0
\(621\) −3.78448 −0.151866
\(622\) −35.0194 −1.40415
\(623\) −31.3545 −1.25619
\(624\) 33.3913 1.33672
\(625\) 0 0
\(626\) −58.6698 −2.34492
\(627\) −1.63102 −0.0651368
\(628\) −14.5351 −0.580014
\(629\) 2.50173 0.0997505
\(630\) 0 0
\(631\) −34.8877 −1.38886 −0.694429 0.719562i \(-0.744343\pi\)
−0.694429 + 0.719562i \(0.744343\pi\)
\(632\) −61.1232 −2.43135
\(633\) −8.85922 −0.352122
\(634\) 44.4282 1.76447
\(635\) 0 0
\(636\) 28.6504 1.13606
\(637\) −7.45473 −0.295367
\(638\) −14.6152 −0.578622
\(639\) −7.27306 −0.287718
\(640\) 0 0
\(641\) −38.5314 −1.52190 −0.760949 0.648812i \(-0.775266\pi\)
−0.760949 + 0.648812i \(0.775266\pi\)
\(642\) 12.7289 0.502368
\(643\) 18.6353 0.734906 0.367453 0.930042i \(-0.380230\pi\)
0.367453 + 0.930042i \(0.380230\pi\)
\(644\) −21.3720 −0.842173
\(645\) 0 0
\(646\) 6.04892 0.237991
\(647\) −19.2282 −0.755938 −0.377969 0.925818i \(-0.623377\pi\)
−0.377969 + 0.925818i \(0.623377\pi\)
\(648\) −68.2247 −2.68012
\(649\) 16.7573 0.657783
\(650\) 0 0
\(651\) 15.8116 0.619706
\(652\) −18.5386 −0.726026
\(653\) 3.98553 0.155966 0.0779828 0.996955i \(-0.475152\pi\)
0.0779828 + 0.996955i \(0.475152\pi\)
\(654\) −17.6136 −0.688745
\(655\) 0 0
\(656\) 158.809 6.20046
\(657\) 37.0901 1.44702
\(658\) 20.1347 0.784931
\(659\) 33.5719 1.30778 0.653889 0.756591i \(-0.273136\pi\)
0.653889 + 0.756591i \(0.273136\pi\)
\(660\) 0 0
\(661\) 36.9909 1.43878 0.719390 0.694607i \(-0.244422\pi\)
0.719390 + 0.694607i \(0.244422\pi\)
\(662\) 37.0834 1.44129
\(663\) 3.89008 0.151078
\(664\) −32.4741 −1.26024
\(665\) 0 0
\(666\) 8.74094 0.338704
\(667\) 2.12631 0.0823310
\(668\) −27.7700 −1.07445
\(669\) −12.8834 −0.498100
\(670\) 0 0
\(671\) 29.2747 1.13014
\(672\) 51.3400 1.98048
\(673\) −12.0747 −0.465447 −0.232723 0.972543i \(-0.574764\pi\)
−0.232723 + 0.972543i \(0.574764\pi\)
\(674\) −73.0689 −2.81451
\(675\) 0 0
\(676\) −14.3763 −0.552934
\(677\) −41.9135 −1.61087 −0.805434 0.592686i \(-0.798067\pi\)
−0.805434 + 0.592686i \(0.798067\pi\)
\(678\) −5.34183 −0.205152
\(679\) −9.89977 −0.379918
\(680\) 0 0
\(681\) −9.70038 −0.371719
\(682\) −76.9536 −2.94671
\(683\) −9.17092 −0.350915 −0.175458 0.984487i \(-0.556141\pi\)
−0.175458 + 0.984487i \(0.556141\pi\)
\(684\) 15.7506 0.602240
\(685\) 0 0
\(686\) 40.1866 1.53433
\(687\) 1.29052 0.0492364
\(688\) −97.7338 −3.72606
\(689\) 28.6504 1.09149
\(690\) 0 0
\(691\) −0.111244 −0.00423193 −0.00211596 0.999998i \(-0.500674\pi\)
−0.00211596 + 0.999998i \(0.500674\pi\)
\(692\) −23.1879 −0.881472
\(693\) 24.1226 0.916341
\(694\) 25.1685 0.955384
\(695\) 0 0
\(696\) −10.6273 −0.402827
\(697\) 18.5013 0.700785
\(698\) −1.67994 −0.0635867
\(699\) −10.5077 −0.397438
\(700\) 0 0
\(701\) −27.7832 −1.04936 −0.524678 0.851301i \(-0.675814\pi\)
−0.524678 + 0.851301i \(0.675814\pi\)
\(702\) 28.7385 1.08467
\(703\) −1.15883 −0.0437062
\(704\) −140.943 −5.31198
\(705\) 0 0
\(706\) 89.3527 3.36283
\(707\) 15.5265 0.583933
\(708\) 18.5133 0.695774
\(709\) 37.4668 1.40710 0.703548 0.710648i \(-0.251598\pi\)
0.703548 + 0.710648i \(0.251598\pi\)
\(710\) 0 0
\(711\) −15.2500 −0.571918
\(712\) 110.961 4.15844
\(713\) 11.1957 0.419281
\(714\) 10.2349 0.383031
\(715\) 0 0
\(716\) −82.4137 −3.07994
\(717\) −6.19806 −0.231471
\(718\) 51.4892 1.92156
\(719\) −21.0513 −0.785081 −0.392541 0.919735i \(-0.628404\pi\)
−0.392541 + 0.919735i \(0.628404\pi\)
\(720\) 0 0
\(721\) 43.6329 1.62498
\(722\) −2.80194 −0.104277
\(723\) −4.12333 −0.153348
\(724\) −71.6805 −2.66399
\(725\) 0 0
\(726\) −3.67324 −0.136327
\(727\) 44.7023 1.65792 0.828958 0.559310i \(-0.188934\pi\)
0.828958 + 0.559310i \(0.188934\pi\)
\(728\) 106.817 3.95891
\(729\) −11.7627 −0.435656
\(730\) 0 0
\(731\) −11.3860 −0.421125
\(732\) 32.3424 1.19541
\(733\) −25.0301 −0.924509 −0.462254 0.886747i \(-0.652959\pi\)
−0.462254 + 0.886747i \(0.652959\pi\)
\(734\) −41.0140 −1.51385
\(735\) 0 0
\(736\) 36.3521 1.33996
\(737\) 14.6437 0.539407
\(738\) 64.6426 2.37953
\(739\) −23.9982 −0.882788 −0.441394 0.897313i \(-0.645516\pi\)
−0.441394 + 0.897313i \(0.645516\pi\)
\(740\) 0 0
\(741\) −1.80194 −0.0661958
\(742\) 75.3798 2.76728
\(743\) −10.4601 −0.383744 −0.191872 0.981420i \(-0.561456\pi\)
−0.191872 + 0.981420i \(0.561456\pi\)
\(744\) −55.9560 −2.05145
\(745\) 0 0
\(746\) −2.87800 −0.105371
\(747\) −8.10215 −0.296442
\(748\) −37.1226 −1.35734
\(749\) 24.9584 0.911959
\(750\) 0 0
\(751\) −29.9420 −1.09260 −0.546299 0.837590i \(-0.683964\pi\)
−0.546299 + 0.837590i \(0.683964\pi\)
\(752\) −43.6752 −1.59267
\(753\) 8.18060 0.298118
\(754\) −16.1468 −0.588030
\(755\) 0 0
\(756\) 56.3497 2.04942
\(757\) −27.2174 −0.989235 −0.494617 0.869111i \(-0.664692\pi\)
−0.494617 + 0.869111i \(0.664692\pi\)
\(758\) 63.1232 2.29274
\(759\) −1.95407 −0.0709281
\(760\) 0 0
\(761\) 18.6813 0.677195 0.338598 0.940931i \(-0.390047\pi\)
0.338598 + 0.940931i \(0.390047\pi\)
\(762\) −2.27247 −0.0823229
\(763\) −34.5362 −1.25029
\(764\) 117.876 4.26459
\(765\) 0 0
\(766\) 91.5387 3.30743
\(767\) 18.5133 0.668478
\(768\) −61.3497 −2.21377
\(769\) 6.34780 0.228907 0.114454 0.993429i \(-0.463488\pi\)
0.114454 + 0.993429i \(0.463488\pi\)
\(770\) 0 0
\(771\) 1.85756 0.0668984
\(772\) 55.7458 2.00634
\(773\) −49.7851 −1.79064 −0.895322 0.445419i \(-0.853055\pi\)
−0.895322 + 0.445419i \(0.853055\pi\)
\(774\) −39.7821 −1.42994
\(775\) 0 0
\(776\) 35.0344 1.25766
\(777\) −1.96077 −0.0703423
\(778\) 22.8756 0.820130
\(779\) −8.57002 −0.307053
\(780\) 0 0
\(781\) −7.94033 −0.284127
\(782\) 7.24698 0.259151
\(783\) −5.60627 −0.200352
\(784\) 42.5448 1.51946
\(785\) 0 0
\(786\) −20.5714 −0.733756
\(787\) 9.19865 0.327897 0.163948 0.986469i \(-0.447577\pi\)
0.163948 + 0.986469i \(0.447577\pi\)
\(788\) −29.0388 −1.03446
\(789\) −9.25965 −0.329652
\(790\) 0 0
\(791\) −10.4741 −0.372416
\(792\) −85.3678 −3.03341
\(793\) 32.3424 1.14851
\(794\) 105.207 3.73364
\(795\) 0 0
\(796\) 31.7245 1.12445
\(797\) −13.5084 −0.478493 −0.239247 0.970959i \(-0.576900\pi\)
−0.239247 + 0.970959i \(0.576900\pi\)
\(798\) −4.74094 −0.167827
\(799\) −5.08815 −0.180006
\(800\) 0 0
\(801\) 27.6843 0.978175
\(802\) −90.3678 −3.19100
\(803\) 40.4929 1.42896
\(804\) 16.1782 0.570562
\(805\) 0 0
\(806\) −85.0176 −2.99462
\(807\) −8.65519 −0.304677
\(808\) −54.9469 −1.93302
\(809\) 31.9560 1.12351 0.561756 0.827303i \(-0.310126\pi\)
0.561756 + 0.827303i \(0.310126\pi\)
\(810\) 0 0
\(811\) 40.7012 1.42921 0.714607 0.699526i \(-0.246606\pi\)
0.714607 + 0.699526i \(0.246606\pi\)
\(812\) −31.6601 −1.11105
\(813\) 7.33273 0.257170
\(814\) 9.54288 0.334478
\(815\) 0 0
\(816\) −22.2010 −0.777192
\(817\) 5.27413 0.184518
\(818\) 88.3895 3.09047
\(819\) 26.6504 0.931240
\(820\) 0 0
\(821\) −13.4300 −0.468709 −0.234355 0.972151i \(-0.575298\pi\)
−0.234355 + 0.972151i \(0.575298\pi\)
\(822\) −25.1497 −0.877198
\(823\) 26.7275 0.931663 0.465832 0.884873i \(-0.345755\pi\)
0.465832 + 0.884873i \(0.345755\pi\)
\(824\) −154.413 −5.37924
\(825\) 0 0
\(826\) 48.7090 1.69480
\(827\) −33.3435 −1.15947 −0.579733 0.814806i \(-0.696843\pi\)
−0.579733 + 0.814806i \(0.696843\pi\)
\(828\) 18.8702 0.655786
\(829\) 12.2849 0.426672 0.213336 0.976979i \(-0.431567\pi\)
0.213336 + 0.976979i \(0.431567\pi\)
\(830\) 0 0
\(831\) 7.20105 0.249802
\(832\) −155.712 −5.39835
\(833\) 4.95646 0.171731
\(834\) 20.3666 0.705237
\(835\) 0 0
\(836\) 17.1957 0.594725
\(837\) −29.5187 −1.02032
\(838\) 41.1511 1.42154
\(839\) 36.9922 1.27711 0.638557 0.769575i \(-0.279532\pi\)
0.638557 + 0.769575i \(0.279532\pi\)
\(840\) 0 0
\(841\) −25.8501 −0.891383
\(842\) −42.3491 −1.45945
\(843\) −13.8130 −0.475743
\(844\) 93.4016 3.21502
\(845\) 0 0
\(846\) −17.7778 −0.611212
\(847\) −7.20237 −0.247477
\(848\) −163.510 −5.61497
\(849\) −7.53750 −0.258686
\(850\) 0 0
\(851\) −1.38835 −0.0475922
\(852\) −8.77240 −0.300537
\(853\) −7.16959 −0.245482 −0.122741 0.992439i \(-0.539168\pi\)
−0.122741 + 0.992439i \(0.539168\pi\)
\(854\) 85.0936 2.91184
\(855\) 0 0
\(856\) −88.3256 −3.01891
\(857\) −2.55124 −0.0871486 −0.0435743 0.999050i \(-0.513875\pi\)
−0.0435743 + 0.999050i \(0.513875\pi\)
\(858\) 14.8388 0.506587
\(859\) 53.9493 1.84073 0.920363 0.391065i \(-0.127893\pi\)
0.920363 + 0.391065i \(0.127893\pi\)
\(860\) 0 0
\(861\) −14.5007 −0.494181
\(862\) −4.69873 −0.160039
\(863\) −21.4698 −0.730840 −0.365420 0.930843i \(-0.619075\pi\)
−0.365420 + 0.930843i \(0.619075\pi\)
\(864\) −95.8467 −3.26077
\(865\) 0 0
\(866\) −36.8582 −1.25249
\(867\) 6.84787 0.232566
\(868\) −166.700 −5.65817
\(869\) −16.6491 −0.564781
\(870\) 0 0
\(871\) 16.1782 0.548178
\(872\) 122.221 4.13891
\(873\) 8.74094 0.295836
\(874\) −3.35690 −0.113549
\(875\) 0 0
\(876\) 44.7362 1.51149
\(877\) 38.9638 1.31571 0.657856 0.753144i \(-0.271463\pi\)
0.657856 + 0.753144i \(0.271463\pi\)
\(878\) −18.0887 −0.610465
\(879\) 0.708415 0.0238942
\(880\) 0 0
\(881\) −13.4373 −0.452713 −0.226357 0.974045i \(-0.572681\pi\)
−0.226357 + 0.974045i \(0.572681\pi\)
\(882\) 17.3177 0.583116
\(883\) 27.8799 0.938234 0.469117 0.883136i \(-0.344572\pi\)
0.469117 + 0.883136i \(0.344572\pi\)
\(884\) −41.0127 −1.37941
\(885\) 0 0
\(886\) −71.0877 −2.38824
\(887\) 28.9135 0.970821 0.485410 0.874286i \(-0.338670\pi\)
0.485410 + 0.874286i \(0.338670\pi\)
\(888\) 6.93900 0.232858
\(889\) −4.45580 −0.149443
\(890\) 0 0
\(891\) −18.5834 −0.622568
\(892\) 135.828 4.54785
\(893\) 2.35690 0.0788705
\(894\) 17.2040 0.575389
\(895\) 0 0
\(896\) −224.659 −7.50533
\(897\) −2.15883 −0.0720814
\(898\) 10.0858 0.336566
\(899\) 16.5851 0.553144
\(900\) 0 0
\(901\) −19.0489 −0.634611
\(902\) 70.5733 2.34983
\(903\) 8.92394 0.296970
\(904\) 37.0670 1.23283
\(905\) 0 0
\(906\) 16.2379 0.539467
\(907\) −30.7127 −1.01980 −0.509900 0.860234i \(-0.670317\pi\)
−0.509900 + 0.860234i \(0.670317\pi\)
\(908\) 102.270 3.39395
\(909\) −13.7090 −0.454699
\(910\) 0 0
\(911\) 39.3690 1.30435 0.652176 0.758067i \(-0.273856\pi\)
0.652176 + 0.758067i \(0.273856\pi\)
\(912\) 10.2838 0.340531
\(913\) −8.84548 −0.292743
\(914\) −34.4034 −1.13796
\(915\) 0 0
\(916\) −13.6058 −0.449548
\(917\) −40.3357 −1.33200
\(918\) −19.1075 −0.630642
\(919\) −7.87023 −0.259615 −0.129808 0.991539i \(-0.541436\pi\)
−0.129808 + 0.991539i \(0.541436\pi\)
\(920\) 0 0
\(921\) −17.8860 −0.589365
\(922\) −110.658 −3.64434
\(923\) −8.77240 −0.288747
\(924\) 29.0954 0.957170
\(925\) 0 0
\(926\) 51.5532 1.69414
\(927\) −38.5254 −1.26534
\(928\) 53.8514 1.76776
\(929\) 39.2924 1.28914 0.644572 0.764544i \(-0.277036\pi\)
0.644572 + 0.764544i \(0.277036\pi\)
\(930\) 0 0
\(931\) −2.29590 −0.0752450
\(932\) 110.781 3.62876
\(933\) −6.93602 −0.227075
\(934\) −101.509 −3.32148
\(935\) 0 0
\(936\) −94.3135 −3.08273
\(937\) 26.7554 0.874061 0.437031 0.899447i \(-0.356030\pi\)
0.437031 + 0.899447i \(0.356030\pi\)
\(938\) 42.5652 1.38980
\(939\) −11.6203 −0.379213
\(940\) 0 0
\(941\) −41.9191 −1.36653 −0.683263 0.730173i \(-0.739440\pi\)
−0.683263 + 0.730173i \(0.739440\pi\)
\(942\) −3.86294 −0.125861
\(943\) −10.2674 −0.334353
\(944\) −105.657 −3.43885
\(945\) 0 0
\(946\) −43.4319 −1.41209
\(947\) −55.4868 −1.80308 −0.901539 0.432698i \(-0.857562\pi\)
−0.901539 + 0.432698i \(0.857562\pi\)
\(948\) −18.3937 −0.597401
\(949\) 44.7362 1.45220
\(950\) 0 0
\(951\) 8.79954 0.285345
\(952\) −71.0200 −2.30177
\(953\) −8.48081 −0.274720 −0.137360 0.990521i \(-0.543862\pi\)
−0.137360 + 0.990521i \(0.543862\pi\)
\(954\) −66.5561 −2.15483
\(955\) 0 0
\(956\) 65.3454 2.11342
\(957\) −2.89472 −0.0935731
\(958\) 96.1426 3.10623
\(959\) −49.3129 −1.59240
\(960\) 0 0
\(961\) 56.3256 1.81695
\(962\) 10.5429 0.339916
\(963\) −22.0368 −0.710127
\(964\) 43.4717 1.40013
\(965\) 0 0
\(966\) −5.67994 −0.182749
\(967\) 7.63102 0.245397 0.122699 0.992444i \(-0.460845\pi\)
0.122699 + 0.992444i \(0.460845\pi\)
\(968\) 25.4886 0.819234
\(969\) 1.19806 0.0384873
\(970\) 0 0
\(971\) −19.3599 −0.621288 −0.310644 0.950526i \(-0.600545\pi\)
−0.310644 + 0.950526i \(0.600545\pi\)
\(972\) −75.9764 −2.43695
\(973\) 39.9342 1.28023
\(974\) −68.5695 −2.19711
\(975\) 0 0
\(976\) −184.581 −5.90829
\(977\) −55.8883 −1.78802 −0.894012 0.448042i \(-0.852121\pi\)
−0.894012 + 0.448042i \(0.852121\pi\)
\(978\) −4.92692 −0.157546
\(979\) 30.2241 0.965968
\(980\) 0 0
\(981\) 30.4935 0.973582
\(982\) −51.2941 −1.63686
\(983\) 4.58450 0.146223 0.0731114 0.997324i \(-0.476707\pi\)
0.0731114 + 0.997324i \(0.476707\pi\)
\(984\) 51.3166 1.63591
\(985\) 0 0
\(986\) 10.7356 0.341890
\(987\) 3.98792 0.126937
\(988\) 18.9976 0.604394
\(989\) 6.31873 0.200924
\(990\) 0 0
\(991\) 27.5666 0.875681 0.437840 0.899053i \(-0.355743\pi\)
0.437840 + 0.899053i \(0.355743\pi\)
\(992\) 283.544 9.00254
\(993\) 7.34481 0.233081
\(994\) −23.0804 −0.732065
\(995\) 0 0
\(996\) −9.77240 −0.309650
\(997\) −15.9119 −0.503933 −0.251967 0.967736i \(-0.581077\pi\)
−0.251967 + 0.967736i \(0.581077\pi\)
\(998\) 71.0079 2.24772
\(999\) 3.66056 0.115815
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 475.2.a.d.1.1 3
3.2 odd 2 4275.2.a.bn.1.3 3
4.3 odd 2 7600.2.a.bw.1.2 3
5.2 odd 4 475.2.b.c.324.1 6
5.3 odd 4 475.2.b.c.324.6 6
5.4 even 2 475.2.a.h.1.3 yes 3
15.14 odd 2 4275.2.a.z.1.1 3
19.18 odd 2 9025.2.a.be.1.3 3
20.19 odd 2 7600.2.a.bn.1.2 3
95.94 odd 2 9025.2.a.w.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.d.1.1 3 1.1 even 1 trivial
475.2.a.h.1.3 yes 3 5.4 even 2
475.2.b.c.324.1 6 5.2 odd 4
475.2.b.c.324.6 6 5.3 odd 4
4275.2.a.z.1.1 3 15.14 odd 2
4275.2.a.bn.1.3 3 3.2 odd 2
7600.2.a.bn.1.2 3 20.19 odd 2
7600.2.a.bw.1.2 3 4.3 odd 2
9025.2.a.w.1.1 3 95.94 odd 2
9025.2.a.be.1.3 3 19.18 odd 2