Properties

Label 475.2.a.i.1.4
Level $475$
Weight $2$
Character 475.1
Self dual yes
Analytic conductor $3.793$
Analytic rank $0$
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] = [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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.28734\) 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)
 
\(f(q)\) \(=\) \(q+2.63010 q^{2} -3.04306 q^{3} +4.91744 q^{4} -8.00355 q^{6} +0.574672 q^{7} +7.67316 q^{8} +6.26020 q^{9} +O(q^{10})\) \(q+2.63010 q^{2} -3.04306 q^{3} +4.91744 q^{4} -8.00355 q^{6} +0.574672 q^{7} +7.67316 q^{8} +6.26020 q^{9} +2.57467 q^{11} -14.9641 q^{12} +0.468387 q^{13} +1.51145 q^{14} +10.3463 q^{16} +4.08612 q^{17} +16.4650 q^{18} +1.00000 q^{19} -1.74876 q^{21} +6.77165 q^{22} -1.51145 q^{23} -23.3499 q^{24} +1.23191 q^{26} -9.92099 q^{27} +2.82591 q^{28} -4.08612 q^{29} -9.92099 q^{31} +11.8656 q^{32} -7.83488 q^{33} +10.7469 q^{34} +30.7842 q^{36} +8.30326 q^{37} +2.63010 q^{38} -1.42533 q^{39} -1.83488 q^{41} -4.59942 q^{42} +0.574672 q^{43} +12.6608 q^{44} -3.97526 q^{46} -7.09508 q^{47} -31.4845 q^{48} -6.66975 q^{49} -12.4343 q^{51} +2.30326 q^{52} -4.30326 q^{53} -26.0932 q^{54} +4.40955 q^{56} -3.04306 q^{57} -10.7469 q^{58} -2.68553 q^{59} +12.4095 q^{61} -26.0932 q^{62} +3.59756 q^{63} +10.5150 q^{64} -20.6065 q^{66} +2.70570 q^{67} +20.0932 q^{68} +4.59942 q^{69} -7.40058 q^{71} +48.0356 q^{72} -12.0861 q^{73} +21.8384 q^{74} +4.91744 q^{76} +1.47959 q^{77} -3.74876 q^{78} -6.68553 q^{79} +11.4095 q^{81} -4.82591 q^{82} +6.66079 q^{83} -8.59942 q^{84} +1.51145 q^{86} +12.4343 q^{87} +19.7559 q^{88} +14.6065 q^{89} +0.269169 q^{91} -7.43244 q^{92} +30.1902 q^{93} -18.6608 q^{94} -36.1076 q^{96} -17.4526 q^{97} -17.5421 q^{98} +16.1180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{7} + 12 q^{8} + 8 q^{9} + 4 q^{11} - 6 q^{12} - 2 q^{13} - 8 q^{14} + 4 q^{16} - 4 q^{17} + 34 q^{18} + 4 q^{19} - 4 q^{21} - 4 q^{22} + 8 q^{23} - 24 q^{24} + 4 q^{26} + 4 q^{27} + 8 q^{28} + 4 q^{29} + 4 q^{31} + 6 q^{32} - 8 q^{33} - 4 q^{34} + 40 q^{36} + 6 q^{37} + 2 q^{38} - 12 q^{39} + 16 q^{41} - 28 q^{42} - 4 q^{43} + 24 q^{44} + 12 q^{47} - 38 q^{48} + 20 q^{49} - 36 q^{51} - 18 q^{52} + 10 q^{53} - 20 q^{54} - 12 q^{56} - 2 q^{57} + 4 q^{58} + 20 q^{61} - 20 q^{62} - 20 q^{63} - 4 q^{64} - 28 q^{66} + 18 q^{67} - 4 q^{68} + 28 q^{69} - 20 q^{71} + 52 q^{72} - 28 q^{73} + 32 q^{74} + 8 q^{76} + 40 q^{77} - 12 q^{78} - 16 q^{79} + 16 q^{81} - 16 q^{82} - 44 q^{84} - 8 q^{86} + 36 q^{87} + 12 q^{88} + 4 q^{89} - 36 q^{91} + 28 q^{92} + 40 q^{93} - 48 q^{94} - 52 q^{96} - 30 q^{97} - 38 q^{98} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.63010 1.85976 0.929882 0.367859i \(-0.119909\pi\)
0.929882 + 0.367859i \(0.119909\pi\)
\(3\) −3.04306 −1.75691 −0.878455 0.477825i \(-0.841425\pi\)
−0.878455 + 0.477825i \(0.841425\pi\)
\(4\) 4.91744 2.45872
\(5\) 0 0
\(6\) −8.00355 −3.26744
\(7\) 0.574672 0.217205 0.108603 0.994085i \(-0.465362\pi\)
0.108603 + 0.994085i \(0.465362\pi\)
\(8\) 7.67316 2.71287
\(9\) 6.26020 2.08673
\(10\) 0 0
\(11\) 2.57467 0.776293 0.388146 0.921598i \(-0.373116\pi\)
0.388146 + 0.921598i \(0.373116\pi\)
\(12\) −14.9641 −4.31975
\(13\) 0.468387 0.129907 0.0649536 0.997888i \(-0.479310\pi\)
0.0649536 + 0.997888i \(0.479310\pi\)
\(14\) 1.51145 0.403951
\(15\) 0 0
\(16\) 10.3463 2.58658
\(17\) 4.08612 0.991029 0.495514 0.868600i \(-0.334980\pi\)
0.495514 + 0.868600i \(0.334980\pi\)
\(18\) 16.4650 3.88083
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.74876 −0.381611
\(22\) 6.77165 1.44372
\(23\) −1.51145 −0.315158 −0.157579 0.987506i \(-0.550369\pi\)
−0.157579 + 0.987506i \(0.550369\pi\)
\(24\) −23.3499 −4.76627
\(25\) 0 0
\(26\) 1.23191 0.241596
\(27\) −9.92099 −1.90930
\(28\) 2.82591 0.534047
\(29\) −4.08612 −0.758773 −0.379386 0.925238i \(-0.623865\pi\)
−0.379386 + 0.925238i \(0.623865\pi\)
\(30\) 0 0
\(31\) −9.92099 −1.78186 −0.890931 0.454138i \(-0.849947\pi\)
−0.890931 + 0.454138i \(0.849947\pi\)
\(32\) 11.8656 2.09755
\(33\) −7.83488 −1.36388
\(34\) 10.7469 1.84308
\(35\) 0 0
\(36\) 30.7842 5.13069
\(37\) 8.30326 1.36505 0.682524 0.730863i \(-0.260882\pi\)
0.682524 + 0.730863i \(0.260882\pi\)
\(38\) 2.63010 0.426659
\(39\) −1.42533 −0.228235
\(40\) 0 0
\(41\) −1.83488 −0.286560 −0.143280 0.989682i \(-0.545765\pi\)
−0.143280 + 0.989682i \(0.545765\pi\)
\(42\) −4.59942 −0.709705
\(43\) 0.574672 0.0876366 0.0438183 0.999040i \(-0.486048\pi\)
0.0438183 + 0.999040i \(0.486048\pi\)
\(44\) 12.6608 1.90869
\(45\) 0 0
\(46\) −3.97526 −0.586119
\(47\) −7.09508 −1.03492 −0.517462 0.855706i \(-0.673123\pi\)
−0.517462 + 0.855706i \(0.673123\pi\)
\(48\) −31.4845 −4.54439
\(49\) −6.66975 −0.952822
\(50\) 0 0
\(51\) −12.4343 −1.74115
\(52\) 2.30326 0.319405
\(53\) −4.30326 −0.591099 −0.295549 0.955327i \(-0.595503\pi\)
−0.295549 + 0.955327i \(0.595503\pi\)
\(54\) −26.0932 −3.55084
\(55\) 0 0
\(56\) 4.40955 0.589251
\(57\) −3.04306 −0.403063
\(58\) −10.7469 −1.41114
\(59\) −2.68553 −0.349627 −0.174813 0.984602i \(-0.555932\pi\)
−0.174813 + 0.984602i \(0.555932\pi\)
\(60\) 0 0
\(61\) 12.4095 1.58888 0.794440 0.607343i \(-0.207765\pi\)
0.794440 + 0.607343i \(0.207765\pi\)
\(62\) −26.0932 −3.31384
\(63\) 3.59756 0.453250
\(64\) 10.5150 1.31438
\(65\) 0 0
\(66\) −20.6065 −2.53649
\(67\) 2.70570 0.330554 0.165277 0.986247i \(-0.447148\pi\)
0.165277 + 0.986247i \(0.447148\pi\)
\(68\) 20.0932 2.43666
\(69\) 4.59942 0.553705
\(70\) 0 0
\(71\) −7.40058 −0.878288 −0.439144 0.898417i \(-0.644718\pi\)
−0.439144 + 0.898417i \(0.644718\pi\)
\(72\) 48.0356 5.66104
\(73\) −12.0861 −1.41457 −0.707286 0.706927i \(-0.750081\pi\)
−0.707286 + 0.706927i \(0.750081\pi\)
\(74\) 21.8384 2.53867
\(75\) 0 0
\(76\) 4.91744 0.564069
\(77\) 1.47959 0.168615
\(78\) −3.74876 −0.424463
\(79\) −6.68553 −0.752181 −0.376091 0.926583i \(-0.622732\pi\)
−0.376091 + 0.926583i \(0.622732\pi\)
\(80\) 0 0
\(81\) 11.4095 1.26773
\(82\) −4.82591 −0.532933
\(83\) 6.66079 0.731117 0.365558 0.930788i \(-0.380878\pi\)
0.365558 + 0.930788i \(0.380878\pi\)
\(84\) −8.59942 −0.938273
\(85\) 0 0
\(86\) 1.51145 0.162983
\(87\) 12.4343 1.33310
\(88\) 19.7559 2.10598
\(89\) 14.6065 1.54829 0.774144 0.633009i \(-0.218180\pi\)
0.774144 + 0.633009i \(0.218180\pi\)
\(90\) 0 0
\(91\) 0.269169 0.0282165
\(92\) −7.43244 −0.774885
\(93\) 30.1902 3.13057
\(94\) −18.6608 −1.92471
\(95\) 0 0
\(96\) −36.1076 −3.68522
\(97\) −17.4526 −1.77204 −0.886022 0.463643i \(-0.846542\pi\)
−0.886022 + 0.463643i \(0.846542\pi\)
\(98\) −17.5421 −1.77202
\(99\) 16.1180 1.61992
\(100\) 0 0
\(101\) 14.2831 1.42122 0.710611 0.703586i \(-0.248419\pi\)
0.710611 + 0.703586i \(0.248419\pi\)
\(102\) −32.7035 −3.23813
\(103\) −4.79182 −0.472152 −0.236076 0.971735i \(-0.575861\pi\)
−0.236076 + 0.971735i \(0.575861\pi\)
\(104\) 3.59401 0.352421
\(105\) 0 0
\(106\) −11.3180 −1.09930
\(107\) −9.22611 −0.891922 −0.445961 0.895052i \(-0.647138\pi\)
−0.445961 + 0.895052i \(0.647138\pi\)
\(108\) −48.7859 −4.69442
\(109\) 4.89810 0.469153 0.234577 0.972098i \(-0.424630\pi\)
0.234577 + 0.972098i \(0.424630\pi\)
\(110\) 0 0
\(111\) −25.2673 −2.39827
\(112\) 5.94574 0.561819
\(113\) −1.61773 −0.152183 −0.0760916 0.997101i \(-0.524244\pi\)
−0.0760916 + 0.997101i \(0.524244\pi\)
\(114\) −8.00355 −0.749602
\(115\) 0 0
\(116\) −20.0932 −1.86561
\(117\) 2.93220 0.271082
\(118\) −7.06323 −0.650223
\(119\) 2.34818 0.215257
\(120\) 0 0
\(121\) −4.37107 −0.397370
\(122\) 32.6384 2.95494
\(123\) 5.58364 0.503459
\(124\) −48.7859 −4.38110
\(125\) 0 0
\(126\) 9.46196 0.842938
\(127\) 13.1292 1.16503 0.582513 0.812821i \(-0.302070\pi\)
0.582513 + 0.812821i \(0.302070\pi\)
\(128\) 3.92440 0.346871
\(129\) −1.74876 −0.153970
\(130\) 0 0
\(131\) −8.17223 −0.714011 −0.357006 0.934102i \(-0.616202\pi\)
−0.357006 + 0.934102i \(0.616202\pi\)
\(132\) −38.5275 −3.35339
\(133\) 0.574672 0.0498304
\(134\) 7.11627 0.614752
\(135\) 0 0
\(136\) 31.3534 2.68853
\(137\) 14.6065 1.24792 0.623960 0.781456i \(-0.285523\pi\)
0.623960 + 0.781456i \(0.285523\pi\)
\(138\) 12.0969 1.02976
\(139\) −2.07219 −0.175761 −0.0878804 0.996131i \(-0.528009\pi\)
−0.0878804 + 0.996131i \(0.528009\pi\)
\(140\) 0 0
\(141\) 21.5907 1.81827
\(142\) −19.4643 −1.63341
\(143\) 1.20594 0.100846
\(144\) 64.7701 5.39751
\(145\) 0 0
\(146\) −31.7877 −2.63077
\(147\) 20.2964 1.67402
\(148\) 40.8308 3.35627
\(149\) 8.91203 0.730102 0.365051 0.930988i \(-0.381052\pi\)
0.365051 + 0.930988i \(0.381052\pi\)
\(150\) 0 0
\(151\) 11.4572 0.932372 0.466186 0.884687i \(-0.345628\pi\)
0.466186 + 0.884687i \(0.345628\pi\)
\(152\) 7.67316 0.622376
\(153\) 25.5799 2.06801
\(154\) 3.89148 0.313584
\(155\) 0 0
\(156\) −7.00896 −0.561166
\(157\) 6.60653 0.527258 0.263629 0.964624i \(-0.415081\pi\)
0.263629 + 0.964624i \(0.415081\pi\)
\(158\) −17.5836 −1.39888
\(159\) 13.0951 1.03851
\(160\) 0 0
\(161\) −0.868585 −0.0684541
\(162\) 30.0083 2.35767
\(163\) −20.2444 −1.58567 −0.792833 0.609439i \(-0.791395\pi\)
−0.792833 + 0.609439i \(0.791395\pi\)
\(164\) −9.02289 −0.704569
\(165\) 0 0
\(166\) 17.5186 1.35970
\(167\) 5.89372 0.456069 0.228035 0.973653i \(-0.426770\pi\)
0.228035 + 0.973653i \(0.426770\pi\)
\(168\) −13.4185 −1.03526
\(169\) −12.7806 −0.983124
\(170\) 0 0
\(171\) 6.26020 0.478730
\(172\) 2.82591 0.215474
\(173\) 3.53161 0.268504 0.134252 0.990947i \(-0.457137\pi\)
0.134252 + 0.990947i \(0.457137\pi\)
\(174\) 32.7035 2.47924
\(175\) 0 0
\(176\) 26.6384 2.00794
\(177\) 8.17223 0.614263
\(178\) 38.4167 2.87945
\(179\) 7.18801 0.537257 0.268629 0.963244i \(-0.413430\pi\)
0.268629 + 0.963244i \(0.413430\pi\)
\(180\) 0 0
\(181\) 15.5433 1.15532 0.577662 0.816276i \(-0.303965\pi\)
0.577662 + 0.816276i \(0.303965\pi\)
\(182\) 0.707941 0.0524761
\(183\) −37.7630 −2.79152
\(184\) −11.5976 −0.854984
\(185\) 0 0
\(186\) 79.4032 5.82213
\(187\) 10.5204 0.769329
\(188\) −34.8896 −2.54459
\(189\) −5.70131 −0.414710
\(190\) 0 0
\(191\) 13.3216 0.963915 0.481958 0.876194i \(-0.339926\pi\)
0.481958 + 0.876194i \(0.339926\pi\)
\(192\) −31.9978 −2.30924
\(193\) −18.9959 −1.36736 −0.683678 0.729784i \(-0.739621\pi\)
−0.683678 + 0.729784i \(0.739621\pi\)
\(194\) −45.9021 −3.29558
\(195\) 0 0
\(196\) −32.7981 −2.34272
\(197\) −2.17223 −0.154765 −0.0773826 0.997001i \(-0.524656\pi\)
−0.0773826 + 0.997001i \(0.524656\pi\)
\(198\) 42.3919 3.01266
\(199\) −1.87355 −0.132812 −0.0664061 0.997793i \(-0.521153\pi\)
−0.0664061 + 0.997793i \(0.521153\pi\)
\(200\) 0 0
\(201\) −8.23361 −0.580754
\(202\) 37.5660 2.64313
\(203\) −2.34818 −0.164810
\(204\) −61.1449 −4.28100
\(205\) 0 0
\(206\) −12.6030 −0.878091
\(207\) −9.46196 −0.657651
\(208\) 4.84608 0.336015
\(209\) 2.57467 0.178094
\(210\) 0 0
\(211\) −17.1090 −1.17783 −0.588916 0.808194i \(-0.700445\pi\)
−0.588916 + 0.808194i \(0.700445\pi\)
\(212\) −21.1610 −1.45335
\(213\) 22.5204 1.54307
\(214\) −24.2656 −1.65876
\(215\) 0 0
\(216\) −76.1254 −5.17968
\(217\) −5.70131 −0.387030
\(218\) 12.8825 0.872514
\(219\) 36.7788 2.48528
\(220\) 0 0
\(221\) 1.91388 0.128742
\(222\) −66.4556 −4.46021
\(223\) −5.12918 −0.343475 −0.171737 0.985143i \(-0.554938\pi\)
−0.171737 + 0.985143i \(0.554938\pi\)
\(224\) 6.81880 0.455600
\(225\) 0 0
\(226\) −4.25480 −0.283025
\(227\) 7.31223 0.485330 0.242665 0.970110i \(-0.421978\pi\)
0.242665 + 0.970110i \(0.421978\pi\)
\(228\) −14.9641 −0.991019
\(229\) −8.40955 −0.555719 −0.277859 0.960622i \(-0.589625\pi\)
−0.277859 + 0.960622i \(0.589625\pi\)
\(230\) 0 0
\(231\) −4.50248 −0.296242
\(232\) −31.3534 −2.05845
\(233\) 14.1722 0.928454 0.464227 0.885716i \(-0.346332\pi\)
0.464227 + 0.885716i \(0.346332\pi\)
\(234\) 7.71198 0.504148
\(235\) 0 0
\(236\) −13.2059 −0.859634
\(237\) 20.3445 1.32152
\(238\) 6.17594 0.400327
\(239\) −14.1902 −0.917885 −0.458943 0.888466i \(-0.651772\pi\)
−0.458943 + 0.888466i \(0.651772\pi\)
\(240\) 0 0
\(241\) 27.8807 1.79595 0.897976 0.440045i \(-0.145038\pi\)
0.897976 + 0.440045i \(0.145038\pi\)
\(242\) −11.4964 −0.739013
\(243\) −4.95694 −0.317988
\(244\) 61.0232 3.90661
\(245\) 0 0
\(246\) 14.6855 0.936315
\(247\) 0.468387 0.0298027
\(248\) −76.1254 −4.83397
\(249\) −20.2692 −1.28451
\(250\) 0 0
\(251\) −26.1902 −1.65311 −0.826554 0.562857i \(-0.809702\pi\)
−0.826554 + 0.562857i \(0.809702\pi\)
\(252\) 17.6908 1.11441
\(253\) −3.89148 −0.244655
\(254\) 34.5311 2.16667
\(255\) 0 0
\(256\) −10.7084 −0.669276
\(257\) −9.01831 −0.562547 −0.281273 0.959628i \(-0.590757\pi\)
−0.281273 + 0.959628i \(0.590757\pi\)
\(258\) −4.59942 −0.286347
\(259\) 4.77165 0.296496
\(260\) 0 0
\(261\) −25.5799 −1.58336
\(262\) −21.4938 −1.32789
\(263\) 9.00896 0.555517 0.277758 0.960651i \(-0.410409\pi\)
0.277758 + 0.960651i \(0.410409\pi\)
\(264\) −60.1183 −3.70002
\(265\) 0 0
\(266\) 1.51145 0.0926727
\(267\) −44.4485 −2.72020
\(268\) 13.3051 0.812739
\(269\) −30.6136 −1.86655 −0.933273 0.359167i \(-0.883061\pi\)
−0.933273 + 0.359167i \(0.883061\pi\)
\(270\) 0 0
\(271\) 24.1180 1.46506 0.732531 0.680733i \(-0.238339\pi\)
0.732531 + 0.680733i \(0.238339\pi\)
\(272\) 42.2763 2.56338
\(273\) −0.819096 −0.0495739
\(274\) 38.4167 2.32084
\(275\) 0 0
\(276\) 22.6173 1.36140
\(277\) −4.56075 −0.274029 −0.137014 0.990569i \(-0.543751\pi\)
−0.137014 + 0.990569i \(0.543751\pi\)
\(278\) −5.45007 −0.326874
\(279\) −62.1074 −3.71828
\(280\) 0 0
\(281\) −6.11563 −0.364828 −0.182414 0.983222i \(-0.558391\pi\)
−0.182414 + 0.983222i \(0.558391\pi\)
\(282\) 56.7859 3.38155
\(283\) 19.0547 1.13269 0.566344 0.824169i \(-0.308358\pi\)
0.566344 + 0.824169i \(0.308358\pi\)
\(284\) −36.3919 −2.15946
\(285\) 0 0
\(286\) 3.17175 0.187550
\(287\) −1.05445 −0.0622423
\(288\) 74.2808 4.37704
\(289\) −0.303649 −0.0178617
\(290\) 0 0
\(291\) 53.1093 3.11332
\(292\) −59.4327 −3.47804
\(293\) 6.43887 0.376163 0.188081 0.982153i \(-0.439773\pi\)
0.188081 + 0.982153i \(0.439773\pi\)
\(294\) 53.3817 3.11329
\(295\) 0 0
\(296\) 63.7123 3.70320
\(297\) −25.5433 −1.48217
\(298\) 23.4395 1.35782
\(299\) −0.707941 −0.0409413
\(300\) 0 0
\(301\) 0.330247 0.0190351
\(302\) 30.1336 1.73399
\(303\) −43.4643 −2.49696
\(304\) 10.3463 0.593402
\(305\) 0 0
\(306\) 67.2778 3.84602
\(307\) 6.77389 0.386606 0.193303 0.981139i \(-0.438080\pi\)
0.193303 + 0.981139i \(0.438080\pi\)
\(308\) 7.27580 0.414577
\(309\) 14.5818 0.829529
\(310\) 0 0
\(311\) 20.6205 1.16928 0.584639 0.811293i \(-0.301236\pi\)
0.584639 + 0.811293i \(0.301236\pi\)
\(312\) −10.9368 −0.619173
\(313\) −19.3711 −1.09492 −0.547459 0.836833i \(-0.684405\pi\)
−0.547459 + 0.836833i \(0.684405\pi\)
\(314\) 17.3758 0.980575
\(315\) 0 0
\(316\) −32.8757 −1.84940
\(317\) −3.37360 −0.189480 −0.0947401 0.995502i \(-0.530202\pi\)
−0.0947401 + 0.995502i \(0.530202\pi\)
\(318\) 34.4414 1.93138
\(319\) −10.5204 −0.589030
\(320\) 0 0
\(321\) 28.0756 1.56703
\(322\) −2.28447 −0.127308
\(323\) 4.08612 0.227358
\(324\) 56.1057 3.11699
\(325\) 0 0
\(326\) −53.2449 −2.94896
\(327\) −14.9052 −0.824260
\(328\) −14.0793 −0.777399
\(329\) −4.07734 −0.224791
\(330\) 0 0
\(331\) −32.7788 −1.80168 −0.900842 0.434148i \(-0.857050\pi\)
−0.900842 + 0.434148i \(0.857050\pi\)
\(332\) 32.7540 1.79761
\(333\) 51.9801 2.84849
\(334\) 15.5011 0.848181
\(335\) 0 0
\(336\) −18.0932 −0.987066
\(337\) −8.74915 −0.476596 −0.238298 0.971192i \(-0.576590\pi\)
−0.238298 + 0.971192i \(0.576590\pi\)
\(338\) −33.6143 −1.82838
\(339\) 4.92285 0.267372
\(340\) 0 0
\(341\) −25.5433 −1.38325
\(342\) 16.4650 0.890324
\(343\) −7.85562 −0.424164
\(344\) 4.40955 0.237747
\(345\) 0 0
\(346\) 9.28850 0.499353
\(347\) 18.5028 0.993281 0.496640 0.867956i \(-0.334567\pi\)
0.496640 + 0.867956i \(0.334567\pi\)
\(348\) 61.1449 3.27771
\(349\) 3.54330 0.189668 0.0948342 0.995493i \(-0.469768\pi\)
0.0948342 + 0.995493i \(0.469768\pi\)
\(350\) 0 0
\(351\) −4.64686 −0.248031
\(352\) 30.5499 1.62832
\(353\) 3.41140 0.181571 0.0907853 0.995870i \(-0.471062\pi\)
0.0907853 + 0.995870i \(0.471062\pi\)
\(354\) 21.4938 1.14238
\(355\) 0 0
\(356\) 71.8267 3.80681
\(357\) −7.14564 −0.378187
\(358\) 18.9052 0.999172
\(359\) −1.70609 −0.0900438 −0.0450219 0.998986i \(-0.514336\pi\)
−0.0450219 + 0.998986i \(0.514336\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 40.8805 2.14863
\(363\) 13.3014 0.698143
\(364\) 1.32362 0.0693765
\(365\) 0 0
\(366\) −99.3205 −5.19157
\(367\) −37.6155 −1.96351 −0.981756 0.190144i \(-0.939105\pi\)
−0.981756 + 0.190144i \(0.939105\pi\)
\(368\) −15.6379 −0.815182
\(369\) −11.4867 −0.597974
\(370\) 0 0
\(371\) −2.47296 −0.128390
\(372\) 148.458 7.69720
\(373\) −13.4031 −0.693987 −0.346994 0.937868i \(-0.612797\pi\)
−0.346994 + 0.937868i \(0.612797\pi\)
\(374\) 27.6698 1.43077
\(375\) 0 0
\(376\) −54.4417 −2.80762
\(377\) −1.91388 −0.0985700
\(378\) −14.9950 −0.771262
\(379\) −9.37107 −0.481359 −0.240680 0.970605i \(-0.577370\pi\)
−0.240680 + 0.970605i \(0.577370\pi\)
\(380\) 0 0
\(381\) −39.9528 −2.04685
\(382\) 35.0371 1.79265
\(383\) 2.09917 0.107263 0.0536314 0.998561i \(-0.482920\pi\)
0.0536314 + 0.998561i \(0.482920\pi\)
\(384\) −11.9422 −0.609422
\(385\) 0 0
\(386\) −49.9612 −2.54296
\(387\) 3.59756 0.182874
\(388\) −85.8221 −4.35696
\(389\) 1.07238 0.0543718 0.0271859 0.999630i \(-0.491345\pi\)
0.0271859 + 0.999630i \(0.491345\pi\)
\(390\) 0 0
\(391\) −6.17594 −0.312331
\(392\) −51.1781 −2.58488
\(393\) 24.8686 1.25445
\(394\) −5.71320 −0.287827
\(395\) 0 0
\(396\) 79.2591 3.98292
\(397\) 23.5341 1.18114 0.590572 0.806985i \(-0.298902\pi\)
0.590572 + 0.806985i \(0.298902\pi\)
\(398\) −4.92762 −0.246999
\(399\) −1.74876 −0.0875475
\(400\) 0 0
\(401\) 18.6469 0.931180 0.465590 0.885001i \(-0.345842\pi\)
0.465590 + 0.885001i \(0.345842\pi\)
\(402\) −21.6552 −1.08006
\(403\) −4.64686 −0.231477
\(404\) 70.2362 3.49438
\(405\) 0 0
\(406\) −6.17594 −0.306507
\(407\) 21.3782 1.05968
\(408\) −95.4103 −4.72351
\(409\) −6.88017 −0.340203 −0.170101 0.985427i \(-0.554410\pi\)
−0.170101 + 0.985427i \(0.554410\pi\)
\(410\) 0 0
\(411\) −44.4485 −2.19248
\(412\) −23.5635 −1.16089
\(413\) −1.54330 −0.0759408
\(414\) −24.8859 −1.22308
\(415\) 0 0
\(416\) 5.55767 0.272487
\(417\) 6.30580 0.308796
\(418\) 6.77165 0.331212
\(419\) 13.4796 0.658521 0.329261 0.944239i \(-0.393201\pi\)
0.329261 + 0.944239i \(0.393201\pi\)
\(420\) 0 0
\(421\) −5.83488 −0.284374 −0.142187 0.989840i \(-0.545414\pi\)
−0.142187 + 0.989840i \(0.545414\pi\)
\(422\) −44.9984 −2.19049
\(423\) −44.4167 −2.15961
\(424\) −33.0196 −1.60357
\(425\) 0 0
\(426\) 59.2310 2.86975
\(427\) 7.13142 0.345113
\(428\) −45.3688 −2.19298
\(429\) −3.66975 −0.177177
\(430\) 0 0
\(431\) 29.2039 1.40670 0.703351 0.710843i \(-0.251686\pi\)
0.703351 + 0.710843i \(0.251686\pi\)
\(432\) −102.646 −4.93855
\(433\) 12.5229 0.601814 0.300907 0.953653i \(-0.402711\pi\)
0.300907 + 0.953653i \(0.402711\pi\)
\(434\) −14.9950 −0.719785
\(435\) 0 0
\(436\) 24.0861 1.15352
\(437\) −1.51145 −0.0723022
\(438\) 96.7319 4.62203
\(439\) 15.6769 0.748216 0.374108 0.927385i \(-0.377949\pi\)
0.374108 + 0.927385i \(0.377949\pi\)
\(440\) 0 0
\(441\) −41.7540 −1.98829
\(442\) 5.03371 0.239429
\(443\) 29.8281 1.41717 0.708587 0.705624i \(-0.249333\pi\)
0.708587 + 0.705624i \(0.249333\pi\)
\(444\) −124.250 −5.89667
\(445\) 0 0
\(446\) −13.4903 −0.638782
\(447\) −27.1198 −1.28272
\(448\) 6.04267 0.285489
\(449\) 29.9668 1.41422 0.707110 0.707104i \(-0.249999\pi\)
0.707110 + 0.707104i \(0.249999\pi\)
\(450\) 0 0
\(451\) −4.72420 −0.222454
\(452\) −7.95509 −0.374176
\(453\) −34.8649 −1.63809
\(454\) 19.2319 0.902598
\(455\) 0 0
\(456\) −23.3499 −1.09346
\(457\) −17.6698 −0.826556 −0.413278 0.910605i \(-0.635616\pi\)
−0.413278 + 0.910605i \(0.635616\pi\)
\(458\) −22.1180 −1.03350
\(459\) −40.5383 −1.89217
\(460\) 0 0
\(461\) 22.3445 1.04069 0.520343 0.853957i \(-0.325804\pi\)
0.520343 + 0.853957i \(0.325804\pi\)
\(462\) −11.8420 −0.550939
\(463\) 6.83302 0.317557 0.158779 0.987314i \(-0.449244\pi\)
0.158779 + 0.987314i \(0.449244\pi\)
\(464\) −42.2763 −1.96263
\(465\) 0 0
\(466\) 37.2744 1.72670
\(467\) −9.00896 −0.416885 −0.208443 0.978035i \(-0.566839\pi\)
−0.208443 + 0.978035i \(0.566839\pi\)
\(468\) 14.4189 0.666514
\(469\) 1.55489 0.0717981
\(470\) 0 0
\(471\) −20.1040 −0.926345
\(472\) −20.6065 −0.948492
\(473\) 1.47959 0.0680317
\(474\) 53.5080 2.45771
\(475\) 0 0
\(476\) 11.5470 0.529256
\(477\) −26.9393 −1.23347
\(478\) −37.3216 −1.70705
\(479\) 9.26731 0.423434 0.211717 0.977331i \(-0.432094\pi\)
0.211717 + 0.977331i \(0.432094\pi\)
\(480\) 0 0
\(481\) 3.88914 0.177329
\(482\) 73.3290 3.34004
\(483\) 2.64315 0.120268
\(484\) −21.4944 −0.977020
\(485\) 0 0
\(486\) −13.0373 −0.591382
\(487\) 38.7694 1.75681 0.878405 0.477917i \(-0.158608\pi\)
0.878405 + 0.477917i \(0.158608\pi\)
\(488\) 95.2205 4.31043
\(489\) 61.6050 2.78587
\(490\) 0 0
\(491\) 21.6877 0.978751 0.489376 0.872073i \(-0.337225\pi\)
0.489376 + 0.872073i \(0.337225\pi\)
\(492\) 27.4572 1.23787
\(493\) −16.6964 −0.751966
\(494\) 1.23191 0.0554260
\(495\) 0 0
\(496\) −102.646 −4.60893
\(497\) −4.25291 −0.190769
\(498\) −53.3100 −2.38888
\(499\) −27.8372 −1.24616 −0.623082 0.782156i \(-0.714120\pi\)
−0.623082 + 0.782156i \(0.714120\pi\)
\(500\) 0 0
\(501\) −17.9349 −0.801273
\(502\) −68.8828 −3.07439
\(503\) 41.2449 1.83902 0.919510 0.393067i \(-0.128586\pi\)
0.919510 + 0.393067i \(0.128586\pi\)
\(504\) 27.6047 1.22961
\(505\) 0 0
\(506\) −10.2350 −0.455000
\(507\) 38.8922 1.72726
\(508\) 64.5619 2.86447
\(509\) 0.416364 0.0184550 0.00922751 0.999957i \(-0.497063\pi\)
0.00922751 + 0.999957i \(0.497063\pi\)
\(510\) 0 0
\(511\) −6.94555 −0.307253
\(512\) −36.0131 −1.59157
\(513\) −9.92099 −0.438023
\(514\) −23.7191 −1.04620
\(515\) 0 0
\(516\) −8.59942 −0.378568
\(517\) −18.2675 −0.803404
\(518\) 12.5499 0.551412
\(519\) −10.7469 −0.471737
\(520\) 0 0
\(521\) 24.0458 1.05346 0.526732 0.850031i \(-0.323417\pi\)
0.526732 + 0.850031i \(0.323417\pi\)
\(522\) −67.2778 −2.94467
\(523\) 5.19736 0.227265 0.113632 0.993523i \(-0.463751\pi\)
0.113632 + 0.993523i \(0.463751\pi\)
\(524\) −40.1865 −1.75555
\(525\) 0 0
\(526\) 23.6945 1.03313
\(527\) −40.5383 −1.76588
\(528\) −81.0621 −3.52778
\(529\) −20.7155 −0.900675
\(530\) 0 0
\(531\) −16.8120 −0.729578
\(532\) 2.82591 0.122519
\(533\) −0.859432 −0.0372261
\(534\) −116.904 −5.05894
\(535\) 0 0
\(536\) 20.7613 0.896751
\(537\) −21.8735 −0.943913
\(538\) −80.5170 −3.47133
\(539\) −17.1724 −0.739669
\(540\) 0 0
\(541\) −1.93863 −0.0833481 −0.0416741 0.999131i \(-0.513269\pi\)
−0.0416741 + 0.999131i \(0.513269\pi\)
\(542\) 63.4327 2.72467
\(543\) −47.2992 −2.02980
\(544\) 48.4841 2.07874
\(545\) 0 0
\(546\) −2.15431 −0.0921958
\(547\) 12.9075 0.551883 0.275941 0.961174i \(-0.411010\pi\)
0.275941 + 0.961174i \(0.411010\pi\)
\(548\) 71.8267 3.06828
\(549\) 77.6863 3.31557
\(550\) 0 0
\(551\) −4.08612 −0.174074
\(552\) 35.2921 1.50213
\(553\) −3.84199 −0.163378
\(554\) −11.9952 −0.509628
\(555\) 0 0
\(556\) −10.1899 −0.432147
\(557\) 34.1040 1.44503 0.722517 0.691353i \(-0.242985\pi\)
0.722517 + 0.691353i \(0.242985\pi\)
\(558\) −163.349 −6.91511
\(559\) 0.269169 0.0113846
\(560\) 0 0
\(561\) −32.0142 −1.35164
\(562\) −16.0847 −0.678494
\(563\) 14.4911 0.610727 0.305363 0.952236i \(-0.401222\pi\)
0.305363 + 0.952236i \(0.401222\pi\)
\(564\) 106.171 4.47061
\(565\) 0 0
\(566\) 50.1159 2.10653
\(567\) 6.55674 0.275357
\(568\) −56.7859 −2.38268
\(569\) −39.2110 −1.64381 −0.821906 0.569624i \(-0.807089\pi\)
−0.821906 + 0.569624i \(0.807089\pi\)
\(570\) 0 0
\(571\) 21.9915 0.920316 0.460158 0.887837i \(-0.347793\pi\)
0.460158 + 0.887837i \(0.347793\pi\)
\(572\) 5.93015 0.247952
\(573\) −40.5383 −1.69351
\(574\) −2.77331 −0.115756
\(575\) 0 0
\(576\) 65.8261 2.74275
\(577\) −23.8735 −0.993869 −0.496934 0.867788i \(-0.665541\pi\)
−0.496934 + 0.867788i \(0.665541\pi\)
\(578\) −0.798628 −0.0332185
\(579\) 57.8057 2.40232
\(580\) 0 0
\(581\) 3.82777 0.158803
\(582\) 139.683 5.79004
\(583\) −11.0795 −0.458866
\(584\) −92.7387 −3.83756
\(585\) 0 0
\(586\) 16.9349 0.699574
\(587\) −17.8281 −0.735843 −0.367921 0.929857i \(-0.619930\pi\)
−0.367921 + 0.929857i \(0.619930\pi\)
\(588\) 99.8065 4.11595
\(589\) −9.92099 −0.408787
\(590\) 0 0
\(591\) 6.61023 0.271909
\(592\) 85.9082 3.53081
\(593\) 21.2446 0.872412 0.436206 0.899847i \(-0.356322\pi\)
0.436206 + 0.899847i \(0.356322\pi\)
\(594\) −67.1815 −2.75649
\(595\) 0 0
\(596\) 43.8244 1.79512
\(597\) 5.70131 0.233339
\(598\) −1.86196 −0.0761411
\(599\) 24.3374 0.994397 0.497199 0.867637i \(-0.334362\pi\)
0.497199 + 0.867637i \(0.334362\pi\)
\(600\) 0 0
\(601\) 15.7891 0.644051 0.322025 0.946731i \(-0.395636\pi\)
0.322025 + 0.946731i \(0.395636\pi\)
\(602\) 0.868585 0.0354009
\(603\) 16.9382 0.689779
\(604\) 56.3400 2.29244
\(605\) 0 0
\(606\) −114.316 −4.64375
\(607\) −7.81471 −0.317189 −0.158595 0.987344i \(-0.550696\pi\)
−0.158595 + 0.987344i \(0.550696\pi\)
\(608\) 11.8656 0.481212
\(609\) 7.14564 0.289556
\(610\) 0 0
\(611\) −3.32324 −0.134444
\(612\) 125.788 5.08467
\(613\) −12.9547 −0.523235 −0.261618 0.965172i \(-0.584256\pi\)
−0.261618 + 0.965172i \(0.584256\pi\)
\(614\) 17.8160 0.718996
\(615\) 0 0
\(616\) 11.3531 0.457431
\(617\) −18.8873 −0.760373 −0.380187 0.924910i \(-0.624140\pi\)
−0.380187 + 0.924910i \(0.624140\pi\)
\(618\) 38.3516 1.54273
\(619\) 29.2673 1.17635 0.588176 0.808733i \(-0.299846\pi\)
0.588176 + 0.808733i \(0.299846\pi\)
\(620\) 0 0
\(621\) 14.9950 0.601730
\(622\) 54.2339 2.17458
\(623\) 8.39396 0.336297
\(624\) −14.7469 −0.590349
\(625\) 0 0
\(626\) −50.9479 −2.03629
\(627\) −7.83488 −0.312895
\(628\) 32.4872 1.29638
\(629\) 33.9281 1.35280
\(630\) 0 0
\(631\) −5.25309 −0.209122 −0.104561 0.994518i \(-0.533344\pi\)
−0.104561 + 0.994518i \(0.533344\pi\)
\(632\) −51.2992 −2.04057
\(633\) 52.0637 2.06935
\(634\) −8.87291 −0.352388
\(635\) 0 0
\(636\) 64.3942 2.55340
\(637\) −3.12402 −0.123778
\(638\) −27.6698 −1.09546
\(639\) −46.3292 −1.83275
\(640\) 0 0
\(641\) −13.0021 −0.513554 −0.256777 0.966471i \(-0.582661\pi\)
−0.256777 + 0.966471i \(0.582661\pi\)
\(642\) 73.8417 2.91430
\(643\) 17.3534 0.684353 0.342176 0.939636i \(-0.388836\pi\)
0.342176 + 0.939636i \(0.388836\pi\)
\(644\) −4.27121 −0.168309
\(645\) 0 0
\(646\) 10.7469 0.422831
\(647\) −12.4848 −0.490830 −0.245415 0.969418i \(-0.578924\pi\)
−0.245415 + 0.969418i \(0.578924\pi\)
\(648\) 87.5473 3.43918
\(649\) −6.91437 −0.271413
\(650\) 0 0
\(651\) 17.3494 0.679978
\(652\) −99.5507 −3.89871
\(653\) 28.3532 1.10955 0.554774 0.832001i \(-0.312805\pi\)
0.554774 + 0.832001i \(0.312805\pi\)
\(654\) −39.2022 −1.53293
\(655\) 0 0
\(656\) −18.9842 −0.741209
\(657\) −75.6616 −2.95184
\(658\) −10.7238 −0.418058
\(659\) 45.2202 1.76153 0.880764 0.473556i \(-0.157030\pi\)
0.880764 + 0.473556i \(0.157030\pi\)
\(660\) 0 0
\(661\) −14.6086 −0.568207 −0.284104 0.958794i \(-0.591696\pi\)
−0.284104 + 0.958794i \(0.591696\pi\)
\(662\) −86.2115 −3.35070
\(663\) −5.82406 −0.226188
\(664\) 51.1093 1.98343
\(665\) 0 0
\(666\) 136.713 5.29752
\(667\) 6.17594 0.239133
\(668\) 28.9820 1.12135
\(669\) 15.6084 0.603455
\(670\) 0 0
\(671\) 31.9505 1.23344
\(672\) −20.7500 −0.800449
\(673\) 0.440534 0.0169813 0.00849067 0.999964i \(-0.497297\pi\)
0.00849067 + 0.999964i \(0.497297\pi\)
\(674\) −23.0111 −0.886356
\(675\) 0 0
\(676\) −62.8479 −2.41723
\(677\) −32.8057 −1.26083 −0.630414 0.776259i \(-0.717115\pi\)
−0.630414 + 0.776259i \(0.717115\pi\)
\(678\) 12.9476 0.497249
\(679\) −10.0295 −0.384898
\(680\) 0 0
\(681\) −22.2515 −0.852681
\(682\) −67.1815 −2.57251
\(683\) 39.6092 1.51561 0.757803 0.652484i \(-0.226273\pi\)
0.757803 + 0.652484i \(0.226273\pi\)
\(684\) 30.7842 1.17706
\(685\) 0 0
\(686\) −20.6611 −0.788844
\(687\) 25.5907 0.976348
\(688\) 5.94574 0.226679
\(689\) −2.01559 −0.0767879
\(690\) 0 0
\(691\) −19.8962 −0.756889 −0.378444 0.925624i \(-0.623541\pi\)
−0.378444 + 0.925624i \(0.623541\pi\)
\(692\) 17.3665 0.660175
\(693\) 9.26254 0.351855
\(694\) 48.6642 1.84727
\(695\) 0 0
\(696\) 95.4103 3.61652
\(697\) −7.49752 −0.283989
\(698\) 9.31924 0.352738
\(699\) −43.1269 −1.63121
\(700\) 0 0
\(701\) −14.1251 −0.533497 −0.266748 0.963766i \(-0.585949\pi\)
−0.266748 + 0.963766i \(0.585949\pi\)
\(702\) −12.2217 −0.461279
\(703\) 8.30326 0.313163
\(704\) 27.0727 1.02034
\(705\) 0 0
\(706\) 8.97234 0.337678
\(707\) 8.20809 0.308697
\(708\) 40.1865 1.51030
\(709\) −41.1815 −1.54660 −0.773302 0.634038i \(-0.781396\pi\)
−0.773302 + 0.634038i \(0.781396\pi\)
\(710\) 0 0
\(711\) −41.8528 −1.56960
\(712\) 112.078 4.20031
\(713\) 14.9950 0.561569
\(714\) −18.7938 −0.703338
\(715\) 0 0
\(716\) 35.3466 1.32097
\(717\) 43.1815 1.61264
\(718\) −4.48718 −0.167460
\(719\) −18.0227 −0.672133 −0.336067 0.941838i \(-0.609097\pi\)
−0.336067 + 0.941838i \(0.609097\pi\)
\(720\) 0 0
\(721\) −2.75372 −0.102554
\(722\) 2.63010 0.0978823
\(723\) −84.8425 −3.15533
\(724\) 76.4332 2.84062
\(725\) 0 0
\(726\) 34.9841 1.29838
\(727\) 21.1266 0.783544 0.391772 0.920062i \(-0.371862\pi\)
0.391772 + 0.920062i \(0.371862\pi\)
\(728\) 2.06537 0.0765479
\(729\) −19.1444 −0.709051
\(730\) 0 0
\(731\) 2.34818 0.0868504
\(732\) −185.697 −6.86356
\(733\) 27.6660 1.02187 0.510934 0.859620i \(-0.329300\pi\)
0.510934 + 0.859620i \(0.329300\pi\)
\(734\) −98.9326 −3.65167
\(735\) 0 0
\(736\) −17.9341 −0.661061
\(737\) 6.96629 0.256607
\(738\) −30.2112 −1.11209
\(739\) −14.2987 −0.525986 −0.262993 0.964798i \(-0.584710\pi\)
−0.262993 + 0.964798i \(0.584710\pi\)
\(740\) 0 0
\(741\) −1.42533 −0.0523607
\(742\) −6.50415 −0.238775
\(743\) −49.9438 −1.83226 −0.916130 0.400881i \(-0.868704\pi\)
−0.916130 + 0.400881i \(0.868704\pi\)
\(744\) 231.654 8.49285
\(745\) 0 0
\(746\) −35.2516 −1.29065
\(747\) 41.6979 1.52565
\(748\) 51.7335 1.89156
\(749\) −5.30198 −0.193730
\(750\) 0 0
\(751\) 9.69216 0.353672 0.176836 0.984240i \(-0.443414\pi\)
0.176836 + 0.984240i \(0.443414\pi\)
\(752\) −73.4080 −2.67691
\(753\) 79.6982 2.90436
\(754\) −5.03371 −0.183317
\(755\) 0 0
\(756\) −28.0359 −1.01965
\(757\) −46.6889 −1.69694 −0.848469 0.529245i \(-0.822475\pi\)
−0.848469 + 0.529245i \(0.822475\pi\)
\(758\) −24.6469 −0.895214
\(759\) 11.8420 0.429837
\(760\) 0 0
\(761\) −38.7361 −1.40418 −0.702091 0.712087i \(-0.747750\pi\)
−0.702091 + 0.712087i \(0.747750\pi\)
\(762\) −105.080 −3.80665
\(763\) 2.81480 0.101903
\(764\) 65.5080 2.37000
\(765\) 0 0
\(766\) 5.52104 0.199483
\(767\) −1.25787 −0.0454190
\(768\) 32.5864 1.17586
\(769\) −5.38666 −0.194248 −0.0971239 0.995272i \(-0.530964\pi\)
−0.0971239 + 0.995272i \(0.530964\pi\)
\(770\) 0 0
\(771\) 27.4433 0.988345
\(772\) −93.4112 −3.36194
\(773\) −7.20137 −0.259015 −0.129508 0.991578i \(-0.541340\pi\)
−0.129508 + 0.991578i \(0.541340\pi\)
\(774\) 9.46196 0.340103
\(775\) 0 0
\(776\) −133.917 −4.80733
\(777\) −14.5204 −0.520917
\(778\) 2.82047 0.101119
\(779\) −1.83488 −0.0657413
\(780\) 0 0
\(781\) −19.0541 −0.681808
\(782\) −16.2434 −0.580861
\(783\) 40.5383 1.44872
\(784\) −69.0074 −2.46455
\(785\) 0 0
\(786\) 65.4069 2.33299
\(787\) 33.5231 1.19497 0.597485 0.801880i \(-0.296167\pi\)
0.597485 + 0.801880i \(0.296167\pi\)
\(788\) −10.6818 −0.380524
\(789\) −27.4148 −0.975993
\(790\) 0 0
\(791\) −0.929664 −0.0330550
\(792\) 123.676 4.39463
\(793\) 5.81247 0.206407
\(794\) 61.8972 2.19665
\(795\) 0 0
\(796\) −9.21305 −0.326548
\(797\) 10.4979 0.371855 0.185927 0.982563i \(-0.440471\pi\)
0.185927 + 0.982563i \(0.440471\pi\)
\(798\) −4.59942 −0.162818
\(799\) −28.9913 −1.02564
\(800\) 0 0
\(801\) 91.4398 3.23087
\(802\) 49.0432 1.73177
\(803\) −31.1178 −1.09812
\(804\) −40.4882 −1.42791
\(805\) 0 0
\(806\) −12.2217 −0.430492
\(807\) 93.1591 3.27936
\(808\) 109.596 3.85559
\(809\) −13.0724 −0.459600 −0.229800 0.973238i \(-0.573807\pi\)
−0.229800 + 0.973238i \(0.573807\pi\)
\(810\) 0 0
\(811\) −10.0790 −0.353922 −0.176961 0.984218i \(-0.556627\pi\)
−0.176961 + 0.984218i \(0.556627\pi\)
\(812\) −11.5470 −0.405221
\(813\) −73.3924 −2.57398
\(814\) 56.2268 1.97075
\(815\) 0 0
\(816\) −128.649 −4.50362
\(817\) 0.574672 0.0201052
\(818\) −18.0956 −0.632697
\(819\) 1.68505 0.0588804
\(820\) 0 0
\(821\) 40.2987 1.40643 0.703217 0.710975i \(-0.251746\pi\)
0.703217 + 0.710975i \(0.251746\pi\)
\(822\) −116.904 −4.07750
\(823\) 5.07715 0.176978 0.0884892 0.996077i \(-0.471796\pi\)
0.0884892 + 0.996077i \(0.471796\pi\)
\(824\) −36.7684 −1.28089
\(825\) 0 0
\(826\) −4.05904 −0.141232
\(827\) 42.8077 1.48857 0.744285 0.667862i \(-0.232791\pi\)
0.744285 + 0.667862i \(0.232791\pi\)
\(828\) −46.5286 −1.61698
\(829\) 24.2018 0.840562 0.420281 0.907394i \(-0.361932\pi\)
0.420281 + 0.907394i \(0.361932\pi\)
\(830\) 0 0
\(831\) 13.8786 0.481444
\(832\) 4.92509 0.170747
\(833\) −27.2534 −0.944274
\(834\) 16.5849 0.574288
\(835\) 0 0
\(836\) 12.6608 0.437883
\(837\) 98.4261 3.40210
\(838\) 35.4527 1.22469
\(839\) −8.63975 −0.298277 −0.149139 0.988816i \(-0.547650\pi\)
−0.149139 + 0.988816i \(0.547650\pi\)
\(840\) 0 0
\(841\) −12.3036 −0.424264
\(842\) −15.3463 −0.528869
\(843\) 18.6102 0.640971
\(844\) −84.1325 −2.89596
\(845\) 0 0
\(846\) −116.820 −4.01637
\(847\) −2.51193 −0.0863109
\(848\) −44.5229 −1.52892
\(849\) −57.9847 −1.99003
\(850\) 0 0
\(851\) −12.5499 −0.430206
\(852\) 110.743 3.79398
\(853\) 13.0229 0.445895 0.222948 0.974830i \(-0.428432\pi\)
0.222948 + 0.974830i \(0.428432\pi\)
\(854\) 18.7564 0.641829
\(855\) 0 0
\(856\) −70.7934 −2.41967
\(857\) 2.82367 0.0964548 0.0482274 0.998836i \(-0.484643\pi\)
0.0482274 + 0.998836i \(0.484643\pi\)
\(858\) −9.65182 −0.329508
\(859\) 24.1227 0.823057 0.411529 0.911397i \(-0.364995\pi\)
0.411529 + 0.911397i \(0.364995\pi\)
\(860\) 0 0
\(861\) 3.20876 0.109354
\(862\) 76.8092 2.61613
\(863\) −32.7307 −1.11417 −0.557084 0.830456i \(-0.688080\pi\)
−0.557084 + 0.830456i \(0.688080\pi\)
\(864\) −117.718 −4.00485
\(865\) 0 0
\(866\) 32.9366 1.11923
\(867\) 0.924022 0.0313814
\(868\) −28.0359 −0.951599
\(869\) −17.2131 −0.583913
\(870\) 0 0
\(871\) 1.26731 0.0429413
\(872\) 37.5839 1.27275
\(873\) −109.257 −3.69779
\(874\) −3.97526 −0.134465
\(875\) 0 0
\(876\) 180.857 6.11060
\(877\) 49.5072 1.67174 0.835869 0.548929i \(-0.184964\pi\)
0.835869 + 0.548929i \(0.184964\pi\)
\(878\) 41.2318 1.39150
\(879\) −19.5939 −0.660884
\(880\) 0 0
\(881\) 40.3152 1.35826 0.679128 0.734020i \(-0.262358\pi\)
0.679128 + 0.734020i \(0.262358\pi\)
\(882\) −109.817 −3.69774
\(883\) 29.3347 0.987192 0.493596 0.869691i \(-0.335682\pi\)
0.493596 + 0.869691i \(0.335682\pi\)
\(884\) 9.41140 0.316540
\(885\) 0 0
\(886\) 78.4508 2.63561
\(887\) −43.3214 −1.45459 −0.727295 0.686325i \(-0.759223\pi\)
−0.727295 + 0.686325i \(0.759223\pi\)
\(888\) −193.880 −6.50619
\(889\) 7.54496 0.253050
\(890\) 0 0
\(891\) 29.3758 0.984128
\(892\) −25.2224 −0.844508
\(893\) −7.09508 −0.237428
\(894\) −71.3279 −2.38556
\(895\) 0 0
\(896\) 2.25524 0.0753424
\(897\) 2.15431 0.0719302
\(898\) 78.8157 2.63011
\(899\) 40.5383 1.35203
\(900\) 0 0
\(901\) −17.5836 −0.585796
\(902\) −12.4251 −0.413712
\(903\) −1.00496 −0.0334431
\(904\) −12.4131 −0.412854
\(905\) 0 0
\(906\) −91.6982 −3.04647
\(907\) −14.0731 −0.467288 −0.233644 0.972322i \(-0.575065\pi\)
−0.233644 + 0.972322i \(0.575065\pi\)
\(908\) 35.9574 1.19329
\(909\) 89.4151 2.96571
\(910\) 0 0
\(911\) 30.0725 0.996346 0.498173 0.867078i \(-0.334004\pi\)
0.498173 + 0.867078i \(0.334004\pi\)
\(912\) −31.4845 −1.04255
\(913\) 17.1493 0.567560
\(914\) −46.4733 −1.53720
\(915\) 0 0
\(916\) −41.3534 −1.36636
\(917\) −4.69635 −0.155087
\(918\) −106.620 −3.51898
\(919\) 29.6518 0.978123 0.489062 0.872249i \(-0.337339\pi\)
0.489062 + 0.872249i \(0.337339\pi\)
\(920\) 0 0
\(921\) −20.6133 −0.679233
\(922\) 58.7682 1.93543
\(923\) −3.46634 −0.114096
\(924\) −22.1407 −0.728375
\(925\) 0 0
\(926\) 17.9715 0.590582
\(927\) −29.9978 −0.985256
\(928\) −48.4841 −1.59157
\(929\) 58.3803 1.91540 0.957698 0.287775i \(-0.0929154\pi\)
0.957698 + 0.287775i \(0.0929154\pi\)
\(930\) 0 0
\(931\) −6.66975 −0.218592
\(932\) 69.6911 2.28281
\(933\) −62.7492 −2.05432
\(934\) −23.6945 −0.775308
\(935\) 0 0
\(936\) 22.4992 0.735410
\(937\) 3.15850 0.103184 0.0515918 0.998668i \(-0.483571\pi\)
0.0515918 + 0.998668i \(0.483571\pi\)
\(938\) 4.08952 0.133528
\(939\) 58.9473 1.92367
\(940\) 0 0
\(941\) −9.66975 −0.315225 −0.157612 0.987501i \(-0.550380\pi\)
−0.157612 + 0.987501i \(0.550380\pi\)
\(942\) −52.8757 −1.72278
\(943\) 2.77331 0.0903116
\(944\) −27.7854 −0.904337
\(945\) 0 0
\(946\) 3.89148 0.126523
\(947\) −31.3714 −1.01943 −0.509716 0.860343i \(-0.670250\pi\)
−0.509716 + 0.860343i \(0.670250\pi\)
\(948\) 100.043 3.24923
\(949\) −5.66098 −0.183763
\(950\) 0 0
\(951\) 10.2661 0.332900
\(952\) 18.0179 0.583964
\(953\) 13.1224 0.425075 0.212537 0.977153i \(-0.431827\pi\)
0.212537 + 0.977153i \(0.431827\pi\)
\(954\) −70.8531 −2.29395
\(955\) 0 0
\(956\) −69.7792 −2.25682
\(957\) 32.0142 1.03487
\(958\) 24.3740 0.787488
\(959\) 8.39396 0.271055
\(960\) 0 0
\(961\) 67.4261 2.17504
\(962\) 10.2288 0.329791
\(963\) −57.7573 −1.86120
\(964\) 137.101 4.41574
\(965\) 0 0
\(966\) 6.95177 0.223669
\(967\) 44.4400 1.42910 0.714548 0.699587i \(-0.246633\pi\)
0.714548 + 0.699587i \(0.246633\pi\)
\(968\) −33.5399 −1.07801
\(969\) −12.4343 −0.399447
\(970\) 0 0
\(971\) −5.69927 −0.182898 −0.0914491 0.995810i \(-0.529150\pi\)
−0.0914491 + 0.995810i \(0.529150\pi\)
\(972\) −24.3755 −0.781843
\(973\) −1.19083 −0.0381762
\(974\) 101.968 3.26725
\(975\) 0 0
\(976\) 128.393 4.10977
\(977\) 23.9164 0.765154 0.382577 0.923924i \(-0.375037\pi\)
0.382577 + 0.923924i \(0.375037\pi\)
\(978\) 162.027 5.18106
\(979\) 37.6070 1.20193
\(980\) 0 0
\(981\) 30.6631 0.978998
\(982\) 57.0408 1.82025
\(983\) 45.5302 1.45219 0.726095 0.687595i \(-0.241333\pi\)
0.726095 + 0.687595i \(0.241333\pi\)
\(984\) 42.8441 1.36582
\(985\) 0 0
\(986\) −43.9131 −1.39848
\(987\) 12.4076 0.394938
\(988\) 2.30326 0.0732766
\(989\) −0.868585 −0.0276194
\(990\) 0 0
\(991\) 27.5521 0.875220 0.437610 0.899165i \(-0.355825\pi\)
0.437610 + 0.899165i \(0.355825\pi\)
\(992\) −117.718 −3.73756
\(993\) 99.7477 3.16540
\(994\) −11.1856 −0.354785
\(995\) 0 0
\(996\) −99.6724 −3.15824
\(997\) −16.8557 −0.533826 −0.266913 0.963721i \(-0.586004\pi\)
−0.266913 + 0.963721i \(0.586004\pi\)
\(998\) −73.2147 −2.31757
\(999\) −82.3766 −2.60628
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.i.1.4 4
3.2 odd 2 4275.2.a.bo.1.1 4
4.3 odd 2 7600.2.a.cf.1.4 4
5.2 odd 4 475.2.b.e.324.8 8
5.3 odd 4 475.2.b.e.324.1 8
5.4 even 2 95.2.a.b.1.1 4
15.14 odd 2 855.2.a.m.1.4 4
19.18 odd 2 9025.2.a.bf.1.1 4
20.19 odd 2 1520.2.a.t.1.1 4
35.34 odd 2 4655.2.a.y.1.1 4
40.19 odd 2 6080.2.a.ch.1.4 4
40.29 even 2 6080.2.a.cc.1.1 4
95.94 odd 2 1805.2.a.p.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.b.1.1 4 5.4 even 2
475.2.a.i.1.4 4 1.1 even 1 trivial
475.2.b.e.324.1 8 5.3 odd 4
475.2.b.e.324.8 8 5.2 odd 4
855.2.a.m.1.4 4 15.14 odd 2
1520.2.a.t.1.1 4 20.19 odd 2
1805.2.a.p.1.4 4 95.94 odd 2
4275.2.a.bo.1.1 4 3.2 odd 2
4655.2.a.y.1.1 4 35.34 odd 2
6080.2.a.cc.1.1 4 40.29 even 2
6080.2.a.ch.1.4 4 40.19 odd 2
7600.2.a.cf.1.4 4 4.3 odd 2
9025.2.a.bf.1.1 4 19.18 odd 2