Properties

Label 1075.2.a.l.1.2
Level $1075$
Weight $2$
Character 1075.1
Self dual yes
Analytic conductor $8.584$
Analytic rank $0$
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] = [1075,2,Mod(1,1075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1075, 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("1075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1075 = 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1075.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: \(8.58391821729\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 215)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.239123\) of defining polynomial
Character \(\chi\) \(=\) 1075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.760877 q^{2} -0.239123 q^{3} -1.42107 q^{4} -0.181943 q^{6} -1.94282 q^{7} -2.60301 q^{8} -2.94282 q^{9} +O(q^{10})\) \(q+0.760877 q^{2} -0.239123 q^{3} -1.42107 q^{4} -0.181943 q^{6} -1.94282 q^{7} -2.60301 q^{8} -2.94282 q^{9} +5.66019 q^{11} +0.339810 q^{12} +0.478247 q^{13} -1.47825 q^{14} +0.861564 q^{16} -3.52175 q^{17} -2.23912 q^{18} +7.88564 q^{19} +0.464574 q^{21} +4.30671 q^{22} +7.88564 q^{23} +0.622440 q^{24} +0.363887 q^{26} +1.42107 q^{27} +2.76088 q^{28} +0.478247 q^{29} +1.57893 q^{31} +5.86156 q^{32} -1.35348 q^{33} -2.67962 q^{34} +4.18194 q^{36} -0.339810 q^{37} +6.00000 q^{38} -0.114360 q^{39} -0.603010 q^{41} +0.353483 q^{42} +1.00000 q^{43} -8.04351 q^{44} +6.00000 q^{46} +12.8421 q^{47} -0.206020 q^{48} -3.22545 q^{49} +0.842133 q^{51} -0.679620 q^{52} -2.47825 q^{53} +1.08126 q^{54} +5.05718 q^{56} -1.88564 q^{57} +0.363887 q^{58} -6.96690 q^{59} +8.24953 q^{61} +1.20137 q^{62} +5.71737 q^{63} +2.73680 q^{64} -1.02983 q^{66} +7.88564 q^{67} +5.00465 q^{68} -1.88564 q^{69} -13.2060 q^{71} +7.66019 q^{72} -6.60301 q^{73} -0.258554 q^{74} -11.2060 q^{76} -10.9967 q^{77} -0.0870138 q^{78} +3.06758 q^{79} +8.48865 q^{81} -0.458816 q^{82} +4.56526 q^{83} -0.660190 q^{84} +0.760877 q^{86} -0.114360 q^{87} -14.7335 q^{88} +14.2495 q^{89} -0.929147 q^{91} -11.2060 q^{92} -0.377560 q^{93} +9.77128 q^{94} -1.40164 q^{96} +9.77128 q^{97} -2.45417 q^{98} -16.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - q^{3} + 4 q^{4} + 8 q^{6} + 3 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} - q^{3} + 4 q^{4} + 8 q^{6} + 3 q^{7} + 9 q^{8} + 9 q^{11} + 9 q^{12} + 2 q^{13} - 5 q^{14} + 10 q^{16} - 10 q^{17} - 7 q^{18} + 6 q^{19} - 8 q^{21} - 13 q^{22} + 6 q^{23} + 9 q^{24} - 16 q^{26} - 4 q^{27} + 8 q^{28} + 2 q^{29} + 13 q^{31} + 25 q^{32} - 22 q^{33} - 24 q^{34} + 4 q^{36} - 9 q^{37} + 18 q^{38} - 18 q^{39} + 15 q^{41} + 19 q^{42} + 3 q^{43} - 23 q^{44} + 18 q^{46} + 22 q^{47} + 33 q^{48} - 14 q^{51} - 18 q^{52} - 8 q^{53} - 13 q^{54} + 24 q^{56} + 12 q^{57} - 16 q^{58} + 13 q^{59} - 10 q^{61} + 19 q^{62} + 18 q^{63} + 33 q^{64} + 8 q^{66} + 6 q^{67} - 34 q^{68} + 12 q^{69} - 6 q^{71} + 15 q^{72} - 3 q^{73} - 25 q^{74} + 12 q^{77} + 2 q^{78} - 17 q^{79} - 9 q^{81} + 22 q^{82} + 12 q^{83} + 6 q^{84} + 2 q^{86} - 18 q^{87} - 24 q^{88} + 8 q^{89} + 16 q^{91} + 6 q^{93} - 6 q^{94} + 28 q^{96} - 6 q^{97} - 33 q^{98} + 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.760877 0.538021 0.269011 0.963137i \(-0.413303\pi\)
0.269011 + 0.963137i \(0.413303\pi\)
\(3\) −0.239123 −0.138058 −0.0690289 0.997615i \(-0.521990\pi\)
−0.0690289 + 0.997615i \(0.521990\pi\)
\(4\) −1.42107 −0.710533
\(5\) 0 0
\(6\) −0.181943 −0.0742781
\(7\) −1.94282 −0.734317 −0.367158 0.930158i \(-0.619669\pi\)
−0.367158 + 0.930158i \(0.619669\pi\)
\(8\) −2.60301 −0.920303
\(9\) −2.94282 −0.980940
\(10\) 0 0
\(11\) 5.66019 1.70661 0.853306 0.521411i \(-0.174594\pi\)
0.853306 + 0.521411i \(0.174594\pi\)
\(12\) 0.339810 0.0980947
\(13\) 0.478247 0.132642 0.0663209 0.997798i \(-0.478874\pi\)
0.0663209 + 0.997798i \(0.478874\pi\)
\(14\) −1.47825 −0.395078
\(15\) 0 0
\(16\) 0.861564 0.215391
\(17\) −3.52175 −0.854151 −0.427075 0.904216i \(-0.640456\pi\)
−0.427075 + 0.904216i \(0.640456\pi\)
\(18\) −2.23912 −0.527766
\(19\) 7.88564 1.80909 0.904545 0.426378i \(-0.140211\pi\)
0.904545 + 0.426378i \(0.140211\pi\)
\(20\) 0 0
\(21\) 0.464574 0.101378
\(22\) 4.30671 0.918193
\(23\) 7.88564 1.64427 0.822135 0.569293i \(-0.192783\pi\)
0.822135 + 0.569293i \(0.192783\pi\)
\(24\) 0.622440 0.127055
\(25\) 0 0
\(26\) 0.363887 0.0713640
\(27\) 1.42107 0.273484
\(28\) 2.76088 0.521757
\(29\) 0.478247 0.0888082 0.0444041 0.999014i \(-0.485861\pi\)
0.0444041 + 0.999014i \(0.485861\pi\)
\(30\) 0 0
\(31\) 1.57893 0.283585 0.141792 0.989896i \(-0.454713\pi\)
0.141792 + 0.989896i \(0.454713\pi\)
\(32\) 5.86156 1.03619
\(33\) −1.35348 −0.235611
\(34\) −2.67962 −0.459551
\(35\) 0 0
\(36\) 4.18194 0.696991
\(37\) −0.339810 −0.0558644 −0.0279322 0.999610i \(-0.508892\pi\)
−0.0279322 + 0.999610i \(0.508892\pi\)
\(38\) 6.00000 0.973329
\(39\) −0.114360 −0.0183122
\(40\) 0 0
\(41\) −0.603010 −0.0941743 −0.0470872 0.998891i \(-0.514994\pi\)
−0.0470872 + 0.998891i \(0.514994\pi\)
\(42\) 0.353483 0.0545436
\(43\) 1.00000 0.152499
\(44\) −8.04351 −1.21260
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 12.8421 1.87322 0.936609 0.350377i \(-0.113946\pi\)
0.936609 + 0.350377i \(0.113946\pi\)
\(48\) −0.206020 −0.0297364
\(49\) −3.22545 −0.460779
\(50\) 0 0
\(51\) 0.842133 0.117922
\(52\) −0.679620 −0.0942464
\(53\) −2.47825 −0.340413 −0.170207 0.985408i \(-0.554444\pi\)
−0.170207 + 0.985408i \(0.554444\pi\)
\(54\) 1.08126 0.147140
\(55\) 0 0
\(56\) 5.05718 0.675794
\(57\) −1.88564 −0.249759
\(58\) 0.363887 0.0477807
\(59\) −6.96690 −0.907013 −0.453506 0.891253i \(-0.649827\pi\)
−0.453506 + 0.891253i \(0.649827\pi\)
\(60\) 0 0
\(61\) 8.24953 1.05624 0.528122 0.849169i \(-0.322896\pi\)
0.528122 + 0.849169i \(0.322896\pi\)
\(62\) 1.20137 0.152575
\(63\) 5.71737 0.720321
\(64\) 2.73680 0.342100
\(65\) 0 0
\(66\) −1.02983 −0.126764
\(67\) 7.88564 0.963384 0.481692 0.876341i \(-0.340022\pi\)
0.481692 + 0.876341i \(0.340022\pi\)
\(68\) 5.00465 0.606903
\(69\) −1.88564 −0.227004
\(70\) 0 0
\(71\) −13.2060 −1.56727 −0.783633 0.621224i \(-0.786636\pi\)
−0.783633 + 0.621224i \(0.786636\pi\)
\(72\) 7.66019 0.902762
\(73\) −6.60301 −0.772824 −0.386412 0.922326i \(-0.626286\pi\)
−0.386412 + 0.922326i \(0.626286\pi\)
\(74\) −0.258554 −0.0300562
\(75\) 0 0
\(76\) −11.2060 −1.28542
\(77\) −10.9967 −1.25319
\(78\) −0.0870138 −0.00985237
\(79\) 3.06758 0.345130 0.172565 0.984998i \(-0.444794\pi\)
0.172565 + 0.984998i \(0.444794\pi\)
\(80\) 0 0
\(81\) 8.48865 0.943183
\(82\) −0.458816 −0.0506678
\(83\) 4.56526 0.501102 0.250551 0.968103i \(-0.419388\pi\)
0.250551 + 0.968103i \(0.419388\pi\)
\(84\) −0.660190 −0.0720326
\(85\) 0 0
\(86\) 0.760877 0.0820474
\(87\) −0.114360 −0.0122607
\(88\) −14.7335 −1.57060
\(89\) 14.2495 1.51045 0.755223 0.655467i \(-0.227528\pi\)
0.755223 + 0.655467i \(0.227528\pi\)
\(90\) 0 0
\(91\) −0.929147 −0.0974011
\(92\) −11.2060 −1.16831
\(93\) −0.377560 −0.0391511
\(94\) 9.77128 1.00783
\(95\) 0 0
\(96\) −1.40164 −0.143054
\(97\) 9.77128 0.992123 0.496062 0.868287i \(-0.334779\pi\)
0.496062 + 0.868287i \(0.334779\pi\)
\(98\) −2.45417 −0.247909
\(99\) −16.6569 −1.67408
\(100\) 0 0
\(101\) −6.67059 −0.663749 −0.331874 0.943324i \(-0.607681\pi\)
−0.331874 + 0.943324i \(0.607681\pi\)
\(102\) 0.640760 0.0634447
\(103\) −13.7713 −1.35692 −0.678462 0.734635i \(-0.737353\pi\)
−0.678462 + 0.734635i \(0.737353\pi\)
\(104\) −1.24488 −0.122071
\(105\) 0 0
\(106\) −1.88564 −0.183150
\(107\) −19.5699 −1.89189 −0.945947 0.324321i \(-0.894864\pi\)
−0.945947 + 0.324321i \(0.894864\pi\)
\(108\) −2.01943 −0.194320
\(109\) 6.05718 0.580173 0.290086 0.957000i \(-0.406316\pi\)
0.290086 + 0.957000i \(0.406316\pi\)
\(110\) 0 0
\(111\) 0.0812565 0.00771253
\(112\) −1.67386 −0.158165
\(113\) 11.3776 1.07031 0.535155 0.844754i \(-0.320253\pi\)
0.535155 + 0.844754i \(0.320253\pi\)
\(114\) −1.43474 −0.134376
\(115\) 0 0
\(116\) −0.679620 −0.0631012
\(117\) −1.40739 −0.130114
\(118\) −5.30095 −0.487992
\(119\) 6.84213 0.627217
\(120\) 0 0
\(121\) 21.0377 1.91252
\(122\) 6.27687 0.568281
\(123\) 0.144194 0.0130015
\(124\) −2.24377 −0.201496
\(125\) 0 0
\(126\) 4.35021 0.387548
\(127\) 16.2495 1.44191 0.720956 0.692981i \(-0.243703\pi\)
0.720956 + 0.692981i \(0.243703\pi\)
\(128\) −9.64076 −0.852131
\(129\) −0.239123 −0.0210536
\(130\) 0 0
\(131\) −7.52175 −0.657179 −0.328589 0.944473i \(-0.606573\pi\)
−0.328589 + 0.944473i \(0.606573\pi\)
\(132\) 1.92339 0.167410
\(133\) −15.3204 −1.32845
\(134\) 6.00000 0.518321
\(135\) 0 0
\(136\) 9.16716 0.786077
\(137\) 11.8090 1.00891 0.504457 0.863437i \(-0.331693\pi\)
0.504457 + 0.863437i \(0.331693\pi\)
\(138\) −1.43474 −0.122133
\(139\) 9.53216 0.808507 0.404254 0.914647i \(-0.367531\pi\)
0.404254 + 0.914647i \(0.367531\pi\)
\(140\) 0 0
\(141\) −3.07085 −0.258612
\(142\) −10.0482 −0.843222
\(143\) 2.70697 0.226368
\(144\) −2.53543 −0.211286
\(145\) 0 0
\(146\) −5.02408 −0.415796
\(147\) 0.771280 0.0636141
\(148\) 0.482893 0.0396935
\(149\) −6.92915 −0.567658 −0.283829 0.958875i \(-0.591605\pi\)
−0.283829 + 0.958875i \(0.591605\pi\)
\(150\) 0 0
\(151\) 12.6134 1.02647 0.513233 0.858250i \(-0.328448\pi\)
0.513233 + 0.858250i \(0.328448\pi\)
\(152\) −20.5264 −1.66491
\(153\) 10.3639 0.837871
\(154\) −8.36716 −0.674245
\(155\) 0 0
\(156\) 0.162513 0.0130115
\(157\) 2.12476 0.169575 0.0847873 0.996399i \(-0.472979\pi\)
0.0847873 + 0.996399i \(0.472979\pi\)
\(158\) 2.33405 0.185687
\(159\) 0.592606 0.0469967
\(160\) 0 0
\(161\) −15.3204 −1.20742
\(162\) 6.45882 0.507453
\(163\) 17.9097 1.40280 0.701399 0.712769i \(-0.252559\pi\)
0.701399 + 0.712769i \(0.252559\pi\)
\(164\) 0.856917 0.0669140
\(165\) 0 0
\(166\) 3.47360 0.269604
\(167\) 5.20602 0.402854 0.201427 0.979504i \(-0.435442\pi\)
0.201427 + 0.979504i \(0.435442\pi\)
\(168\) −1.20929 −0.0932987
\(169\) −12.7713 −0.982406
\(170\) 0 0
\(171\) −23.2060 −1.77461
\(172\) −1.42107 −0.108355
\(173\) 16.4782 1.25282 0.626409 0.779495i \(-0.284524\pi\)
0.626409 + 0.779495i \(0.284524\pi\)
\(174\) −0.0870138 −0.00659650
\(175\) 0 0
\(176\) 4.87661 0.367589
\(177\) 1.66595 0.125220
\(178\) 10.8421 0.812652
\(179\) −8.24953 −0.616599 −0.308299 0.951289i \(-0.599760\pi\)
−0.308299 + 0.951289i \(0.599760\pi\)
\(180\) 0 0
\(181\) −15.1683 −1.12745 −0.563724 0.825963i \(-0.690632\pi\)
−0.563724 + 0.825963i \(0.690632\pi\)
\(182\) −0.706966 −0.0524038
\(183\) −1.97265 −0.145823
\(184\) −20.5264 −1.51323
\(185\) 0 0
\(186\) −0.287276 −0.0210641
\(187\) −19.9338 −1.45770
\(188\) −18.2495 −1.33098
\(189\) −2.76088 −0.200824
\(190\) 0 0
\(191\) −17.6843 −1.27959 −0.639794 0.768546i \(-0.720980\pi\)
−0.639794 + 0.768546i \(0.720980\pi\)
\(192\) −0.654433 −0.0472296
\(193\) −18.1352 −1.30540 −0.652699 0.757617i \(-0.726363\pi\)
−0.652699 + 0.757617i \(0.726363\pi\)
\(194\) 7.43474 0.533783
\(195\) 0 0
\(196\) 4.58358 0.327399
\(197\) −4.72777 −0.336840 −0.168420 0.985715i \(-0.553866\pi\)
−0.168420 + 0.985715i \(0.553866\pi\)
\(198\) −12.6739 −0.900692
\(199\) 2.56526 0.181846 0.0909232 0.995858i \(-0.471018\pi\)
0.0909232 + 0.995858i \(0.471018\pi\)
\(200\) 0 0
\(201\) −1.88564 −0.133003
\(202\) −5.07550 −0.357111
\(203\) −0.929147 −0.0652133
\(204\) −1.19673 −0.0837877
\(205\) 0 0
\(206\) −10.4782 −0.730054
\(207\) −23.2060 −1.61293
\(208\) 0.412040 0.0285698
\(209\) 44.6342 3.08741
\(210\) 0 0
\(211\) −25.4074 −1.74912 −0.874559 0.484920i \(-0.838849\pi\)
−0.874559 + 0.484920i \(0.838849\pi\)
\(212\) 3.52175 0.241875
\(213\) 3.15787 0.216373
\(214\) −14.8903 −1.01788
\(215\) 0 0
\(216\) −3.69905 −0.251689
\(217\) −3.06758 −0.208241
\(218\) 4.60877 0.312145
\(219\) 1.57893 0.106694
\(220\) 0 0
\(221\) −1.68427 −0.113296
\(222\) 0.0618262 0.00414950
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) −11.3880 −0.760890
\(225\) 0 0
\(226\) 8.65692 0.575850
\(227\) −4.87197 −0.323364 −0.161682 0.986843i \(-0.551692\pi\)
−0.161682 + 0.986843i \(0.551692\pi\)
\(228\) 2.67962 0.177462
\(229\) 13.5458 0.895134 0.447567 0.894250i \(-0.352291\pi\)
0.447567 + 0.894250i \(0.352291\pi\)
\(230\) 0 0
\(231\) 2.62957 0.173013
\(232\) −1.24488 −0.0817304
\(233\) 11.3308 0.742304 0.371152 0.928572i \(-0.378963\pi\)
0.371152 + 0.928572i \(0.378963\pi\)
\(234\) −1.07085 −0.0700039
\(235\) 0 0
\(236\) 9.90042 0.644463
\(237\) −0.733531 −0.0476479
\(238\) 5.20602 0.337456
\(239\) 9.35021 0.604815 0.302408 0.953179i \(-0.402210\pi\)
0.302408 + 0.953179i \(0.402210\pi\)
\(240\) 0 0
\(241\) 2.22872 0.143564 0.0717822 0.997420i \(-0.477131\pi\)
0.0717822 + 0.997420i \(0.477131\pi\)
\(242\) 16.0071 1.02898
\(243\) −6.29303 −0.403698
\(244\) −11.7231 −0.750496
\(245\) 0 0
\(246\) 0.109714 0.00699509
\(247\) 3.77128 0.239961
\(248\) −4.10998 −0.260984
\(249\) −1.09166 −0.0691811
\(250\) 0 0
\(251\) −1.68427 −0.106310 −0.0531550 0.998586i \(-0.516928\pi\)
−0.0531550 + 0.998586i \(0.516928\pi\)
\(252\) −8.12476 −0.511812
\(253\) 44.6342 2.80613
\(254\) 12.3639 0.775779
\(255\) 0 0
\(256\) −12.8090 −0.800564
\(257\) −2.70370 −0.168652 −0.0843260 0.996438i \(-0.526874\pi\)
−0.0843260 + 0.996438i \(0.526874\pi\)
\(258\) −0.181943 −0.0113273
\(259\) 0.660190 0.0410222
\(260\) 0 0
\(261\) −1.40739 −0.0871155
\(262\) −5.72313 −0.353576
\(263\) 3.54583 0.218645 0.109323 0.994006i \(-0.465132\pi\)
0.109323 + 0.994006i \(0.465132\pi\)
\(264\) 3.52313 0.216834
\(265\) 0 0
\(266\) −11.6569 −0.714732
\(267\) −3.40739 −0.208529
\(268\) −11.2060 −0.684517
\(269\) 28.1866 1.71857 0.859283 0.511500i \(-0.170910\pi\)
0.859283 + 0.511500i \(0.170910\pi\)
\(270\) 0 0
\(271\) −22.4887 −1.36609 −0.683044 0.730377i \(-0.739344\pi\)
−0.683044 + 0.730377i \(0.739344\pi\)
\(272\) −3.03421 −0.183976
\(273\) 0.222181 0.0134470
\(274\) 8.98522 0.542817
\(275\) 0 0
\(276\) 2.67962 0.161294
\(277\) −5.57893 −0.335206 −0.167603 0.985855i \(-0.553603\pi\)
−0.167603 + 0.985855i \(0.553603\pi\)
\(278\) 7.25280 0.434994
\(279\) −4.64652 −0.278180
\(280\) 0 0
\(281\) 28.6375 1.70837 0.854185 0.519970i \(-0.174057\pi\)
0.854185 + 0.519970i \(0.174057\pi\)
\(282\) −2.33654 −0.139139
\(283\) 13.1190 0.779844 0.389922 0.920848i \(-0.372502\pi\)
0.389922 + 0.920848i \(0.372502\pi\)
\(284\) 18.7666 1.11359
\(285\) 0 0
\(286\) 2.05967 0.121791
\(287\) 1.17154 0.0691538
\(288\) −17.2495 −1.01644
\(289\) −4.59725 −0.270427
\(290\) 0 0
\(291\) −2.33654 −0.136970
\(292\) 9.38332 0.549117
\(293\) 11.6361 0.679789 0.339894 0.940464i \(-0.389609\pi\)
0.339894 + 0.940464i \(0.389609\pi\)
\(294\) 0.586849 0.0342257
\(295\) 0 0
\(296\) 0.884529 0.0514122
\(297\) 8.04351 0.466732
\(298\) −5.27223 −0.305412
\(299\) 3.77128 0.218099
\(300\) 0 0
\(301\) −1.94282 −0.111982
\(302\) 9.59725 0.552260
\(303\) 1.59509 0.0916358
\(304\) 6.79398 0.389661
\(305\) 0 0
\(306\) 7.88564 0.450792
\(307\) −7.09166 −0.404742 −0.202371 0.979309i \(-0.564865\pi\)
−0.202371 + 0.979309i \(0.564865\pi\)
\(308\) 15.6271 0.890436
\(309\) 3.29303 0.187334
\(310\) 0 0
\(311\) 5.53216 0.313700 0.156850 0.987622i \(-0.449866\pi\)
0.156850 + 0.987622i \(0.449866\pi\)
\(312\) 0.297680 0.0168528
\(313\) −19.1488 −1.08236 −0.541178 0.840908i \(-0.682022\pi\)
−0.541178 + 0.840908i \(0.682022\pi\)
\(314\) 1.61668 0.0912347
\(315\) 0 0
\(316\) −4.35924 −0.245226
\(317\) 12.7004 0.713327 0.356664 0.934233i \(-0.383914\pi\)
0.356664 + 0.934233i \(0.383914\pi\)
\(318\) 0.450900 0.0252852
\(319\) 2.70697 0.151561
\(320\) 0 0
\(321\) 4.67962 0.261191
\(322\) −11.6569 −0.649615
\(323\) −27.7713 −1.54524
\(324\) −12.0629 −0.670163
\(325\) 0 0
\(326\) 13.6271 0.754735
\(327\) −1.44841 −0.0800974
\(328\) 1.56964 0.0866689
\(329\) −24.9500 −1.37554
\(330\) 0 0
\(331\) 5.49441 0.302000 0.151000 0.988534i \(-0.451751\pi\)
0.151000 + 0.988534i \(0.451751\pi\)
\(332\) −6.48754 −0.356050
\(333\) 1.00000 0.0547997
\(334\) 3.96114 0.216744
\(335\) 0 0
\(336\) 0.400260 0.0218360
\(337\) −22.4782 −1.22447 −0.612234 0.790677i \(-0.709729\pi\)
−0.612234 + 0.790677i \(0.709729\pi\)
\(338\) −9.71737 −0.528555
\(339\) −2.72064 −0.147765
\(340\) 0 0
\(341\) 8.93706 0.483969
\(342\) −17.6569 −0.954777
\(343\) 19.8662 1.07267
\(344\) −2.60301 −0.140345
\(345\) 0 0
\(346\) 12.5379 0.674042
\(347\) −15.8227 −0.849407 −0.424704 0.905332i \(-0.639622\pi\)
−0.424704 + 0.905332i \(0.639622\pi\)
\(348\) 0.162513 0.00871161
\(349\) −21.7713 −1.16539 −0.582695 0.812691i \(-0.698002\pi\)
−0.582695 + 0.812691i \(0.698002\pi\)
\(350\) 0 0
\(351\) 0.679620 0.0362754
\(352\) 33.1776 1.76837
\(353\) −24.1352 −1.28459 −0.642293 0.766459i \(-0.722017\pi\)
−0.642293 + 0.766459i \(0.722017\pi\)
\(354\) 1.26758 0.0673711
\(355\) 0 0
\(356\) −20.2495 −1.07322
\(357\) −1.63611 −0.0865923
\(358\) −6.27687 −0.331743
\(359\) 27.7713 1.46571 0.732856 0.680384i \(-0.238187\pi\)
0.732856 + 0.680384i \(0.238187\pi\)
\(360\) 0 0
\(361\) 43.1833 2.27281
\(362\) −11.5412 −0.606591
\(363\) −5.03062 −0.264039
\(364\) 1.32038 0.0692067
\(365\) 0 0
\(366\) −1.50095 −0.0784557
\(367\) 8.36389 0.436591 0.218296 0.975883i \(-0.429950\pi\)
0.218296 + 0.975883i \(0.429950\pi\)
\(368\) 6.79398 0.354161
\(369\) 1.77455 0.0923794
\(370\) 0 0
\(371\) 4.81479 0.249971
\(372\) 0.536538 0.0278182
\(373\) −18.3067 −0.947885 −0.473943 0.880556i \(-0.657170\pi\)
−0.473943 + 0.880556i \(0.657170\pi\)
\(374\) −15.1672 −0.784275
\(375\) 0 0
\(376\) −33.4282 −1.72393
\(377\) 0.228720 0.0117797
\(378\) −2.10069 −0.108048
\(379\) 1.88891 0.0970268 0.0485134 0.998823i \(-0.484552\pi\)
0.0485134 + 0.998823i \(0.484552\pi\)
\(380\) 0 0
\(381\) −3.88564 −0.199067
\(382\) −13.4555 −0.688446
\(383\) 33.8687 1.73061 0.865305 0.501246i \(-0.167125\pi\)
0.865305 + 0.501246i \(0.167125\pi\)
\(384\) 2.30533 0.117643
\(385\) 0 0
\(386\) −13.7986 −0.702332
\(387\) −2.94282 −0.149592
\(388\) −13.8856 −0.704937
\(389\) −7.13052 −0.361532 −0.180766 0.983526i \(-0.557858\pi\)
−0.180766 + 0.983526i \(0.557858\pi\)
\(390\) 0 0
\(391\) −27.7713 −1.40445
\(392\) 8.39588 0.424056
\(393\) 1.79863 0.0907287
\(394\) −3.59725 −0.181227
\(395\) 0 0
\(396\) 23.6706 1.18949
\(397\) −1.56991 −0.0787914 −0.0393957 0.999224i \(-0.512543\pi\)
−0.0393957 + 0.999224i \(0.512543\pi\)
\(398\) 1.95185 0.0978372
\(399\) 3.66346 0.183402
\(400\) 0 0
\(401\) 6.89931 0.344535 0.172268 0.985050i \(-0.444891\pi\)
0.172268 + 0.985050i \(0.444891\pi\)
\(402\) −1.43474 −0.0715583
\(403\) 0.755119 0.0376152
\(404\) 9.47936 0.471616
\(405\) 0 0
\(406\) −0.706966 −0.0350861
\(407\) −1.92339 −0.0953389
\(408\) −2.19208 −0.108524
\(409\) −24.8421 −1.22836 −0.614182 0.789164i \(-0.710514\pi\)
−0.614182 + 0.789164i \(0.710514\pi\)
\(410\) 0 0
\(411\) −2.82381 −0.139288
\(412\) 19.5699 0.964140
\(413\) 13.5354 0.666035
\(414\) −17.6569 −0.867790
\(415\) 0 0
\(416\) 2.80327 0.137442
\(417\) −2.27936 −0.111621
\(418\) 33.9611 1.66109
\(419\) −34.6134 −1.69098 −0.845488 0.533994i \(-0.820690\pi\)
−0.845488 + 0.533994i \(0.820690\pi\)
\(420\) 0 0
\(421\) 14.4509 0.704294 0.352147 0.935945i \(-0.385452\pi\)
0.352147 + 0.935945i \(0.385452\pi\)
\(422\) −19.3319 −0.941062
\(423\) −37.7921 −1.83751
\(424\) 6.45090 0.313283
\(425\) 0 0
\(426\) 2.40275 0.116413
\(427\) −16.0273 −0.775618
\(428\) 27.8101 1.34425
\(429\) −0.647299 −0.0312519
\(430\) 0 0
\(431\) −14.2599 −0.686877 −0.343438 0.939175i \(-0.611592\pi\)
−0.343438 + 0.939175i \(0.611592\pi\)
\(432\) 1.22434 0.0589060
\(433\) −22.4750 −1.08008 −0.540039 0.841640i \(-0.681591\pi\)
−0.540039 + 0.841640i \(0.681591\pi\)
\(434\) −2.33405 −0.112038
\(435\) 0 0
\(436\) −8.60766 −0.412232
\(437\) 62.1833 2.97463
\(438\) 1.20137 0.0574039
\(439\) 30.8389 1.47186 0.735929 0.677058i \(-0.236746\pi\)
0.735929 + 0.677058i \(0.236746\pi\)
\(440\) 0 0
\(441\) 9.49192 0.451996
\(442\) −1.28152 −0.0609556
\(443\) −27.9338 −1.32717 −0.663587 0.748099i \(-0.730967\pi\)
−0.663587 + 0.748099i \(0.730967\pi\)
\(444\) −0.115471 −0.00548001
\(445\) 0 0
\(446\) −3.04351 −0.144114
\(447\) 1.65692 0.0783696
\(448\) −5.31711 −0.251210
\(449\) 8.20137 0.387047 0.193523 0.981096i \(-0.438008\pi\)
0.193523 + 0.981096i \(0.438008\pi\)
\(450\) 0 0
\(451\) −3.41315 −0.160719
\(452\) −16.1683 −0.760491
\(453\) −3.01616 −0.141712
\(454\) −3.70697 −0.173977
\(455\) 0 0
\(456\) 4.90834 0.229854
\(457\) 10.1248 0.473616 0.236808 0.971556i \(-0.423899\pi\)
0.236808 + 0.971556i \(0.423899\pi\)
\(458\) 10.3067 0.481601
\(459\) −5.00465 −0.233597
\(460\) 0 0
\(461\) −16.3606 −0.761990 −0.380995 0.924577i \(-0.624418\pi\)
−0.380995 + 0.924577i \(0.624418\pi\)
\(462\) 2.00078 0.0930848
\(463\) −0.738177 −0.0343060 −0.0171530 0.999853i \(-0.505460\pi\)
−0.0171530 + 0.999853i \(0.505460\pi\)
\(464\) 0.412040 0.0191285
\(465\) 0 0
\(466\) 8.62133 0.399375
\(467\) 11.5426 0.534126 0.267063 0.963679i \(-0.413947\pi\)
0.267063 + 0.963679i \(0.413947\pi\)
\(468\) 2.00000 0.0924500
\(469\) −15.3204 −0.707429
\(470\) 0 0
\(471\) −0.508080 −0.0234111
\(472\) 18.1349 0.834726
\(473\) 5.66019 0.260256
\(474\) −0.558126 −0.0256356
\(475\) 0 0
\(476\) −9.72313 −0.445659
\(477\) 7.29303 0.333925
\(478\) 7.11436 0.325403
\(479\) 11.3776 0.519854 0.259927 0.965628i \(-0.416302\pi\)
0.259927 + 0.965628i \(0.416302\pi\)
\(480\) 0 0
\(481\) −0.162513 −0.00740996
\(482\) 1.69578 0.0772407
\(483\) 3.66346 0.166693
\(484\) −29.8960 −1.35891
\(485\) 0 0
\(486\) −4.78822 −0.217198
\(487\) 25.0917 1.13701 0.568506 0.822679i \(-0.307522\pi\)
0.568506 + 0.822679i \(0.307522\pi\)
\(488\) −21.4736 −0.972064
\(489\) −4.28263 −0.193667
\(490\) 0 0
\(491\) 31.3685 1.41564 0.707821 0.706392i \(-0.249678\pi\)
0.707821 + 0.706392i \(0.249678\pi\)
\(492\) −0.204909 −0.00923801
\(493\) −1.68427 −0.0758555
\(494\) 2.86948 0.129104
\(495\) 0 0
\(496\) 1.36035 0.0610816
\(497\) 25.6569 1.15087
\(498\) −0.830619 −0.0372209
\(499\) 18.4991 0.828131 0.414066 0.910247i \(-0.364108\pi\)
0.414066 + 0.910247i \(0.364108\pi\)
\(500\) 0 0
\(501\) −1.24488 −0.0556172
\(502\) −1.28152 −0.0571970
\(503\) 36.8539 1.64323 0.821617 0.570039i \(-0.193072\pi\)
0.821617 + 0.570039i \(0.193072\pi\)
\(504\) −14.8824 −0.662913
\(505\) 0 0
\(506\) 33.9611 1.50976
\(507\) 3.05391 0.135629
\(508\) −23.0917 −1.02453
\(509\) −4.98633 −0.221015 −0.110508 0.993875i \(-0.535248\pi\)
−0.110508 + 0.993875i \(0.535248\pi\)
\(510\) 0 0
\(511\) 12.8285 0.567498
\(512\) 9.53543 0.421410
\(513\) 11.2060 0.494758
\(514\) −2.05718 −0.0907383
\(515\) 0 0
\(516\) 0.339810 0.0149593
\(517\) 72.6889 3.19685
\(518\) 0.502323 0.0220708
\(519\) −3.94033 −0.172961
\(520\) 0 0
\(521\) 6.11436 0.267875 0.133937 0.990990i \(-0.457238\pi\)
0.133937 + 0.990990i \(0.457238\pi\)
\(522\) −1.07085 −0.0468700
\(523\) −16.9532 −0.741313 −0.370656 0.928770i \(-0.620867\pi\)
−0.370656 + 0.928770i \(0.620867\pi\)
\(524\) 10.6889 0.466947
\(525\) 0 0
\(526\) 2.69794 0.117636
\(527\) −5.56061 −0.242224
\(528\) −1.16611 −0.0507485
\(529\) 39.1833 1.70362
\(530\) 0 0
\(531\) 20.5023 0.889725
\(532\) 21.7713 0.943905
\(533\) −0.288387 −0.0124914
\(534\) −2.59261 −0.112193
\(535\) 0 0
\(536\) −20.5264 −0.886605
\(537\) 1.97265 0.0851263
\(538\) 21.4465 0.924625
\(539\) −18.2567 −0.786370
\(540\) 0 0
\(541\) −29.0884 −1.25061 −0.625304 0.780381i \(-0.715025\pi\)
−0.625304 + 0.780381i \(0.715025\pi\)
\(542\) −17.1111 −0.734984
\(543\) 3.62709 0.155653
\(544\) −20.6430 −0.885061
\(545\) 0 0
\(546\) 0.169052 0.00723476
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −16.7814 −0.716867
\(549\) −24.2769 −1.03611
\(550\) 0 0
\(551\) 3.77128 0.160662
\(552\) 4.90834 0.208913
\(553\) −5.95976 −0.253435
\(554\) −4.24488 −0.180348
\(555\) 0 0
\(556\) −13.5458 −0.574471
\(557\) −36.0963 −1.52945 −0.764725 0.644357i \(-0.777125\pi\)
−0.764725 + 0.644357i \(0.777125\pi\)
\(558\) −3.53543 −0.149667
\(559\) 0.478247 0.0202277
\(560\) 0 0
\(561\) 4.76663 0.201247
\(562\) 21.7896 0.919139
\(563\) 7.43474 0.313337 0.156668 0.987651i \(-0.449925\pi\)
0.156668 + 0.987651i \(0.449925\pi\)
\(564\) 4.36389 0.183753
\(565\) 0 0
\(566\) 9.98195 0.419573
\(567\) −16.4919 −0.692596
\(568\) 34.3754 1.44236
\(569\) −5.16035 −0.216333 −0.108167 0.994133i \(-0.534498\pi\)
−0.108167 + 0.994133i \(0.534498\pi\)
\(570\) 0 0
\(571\) 17.5426 0.734133 0.367067 0.930195i \(-0.380362\pi\)
0.367067 + 0.930195i \(0.380362\pi\)
\(572\) −3.84678 −0.160842
\(573\) 4.22872 0.176657
\(574\) 0.891397 0.0372062
\(575\) 0 0
\(576\) −8.05391 −0.335580
\(577\) −19.9130 −0.828988 −0.414494 0.910052i \(-0.636041\pi\)
−0.414494 + 0.910052i \(0.636041\pi\)
\(578\) −3.49794 −0.145495
\(579\) 4.33654 0.180220
\(580\) 0 0
\(581\) −8.86948 −0.367968
\(582\) −1.77782 −0.0736930
\(583\) −14.0273 −0.580953
\(584\) 17.1877 0.711232
\(585\) 0 0
\(586\) 8.85365 0.365741
\(587\) 6.89931 0.284765 0.142383 0.989812i \(-0.454524\pi\)
0.142383 + 0.989812i \(0.454524\pi\)
\(588\) −1.09604 −0.0452000
\(589\) 12.4509 0.513030
\(590\) 0 0
\(591\) 1.13052 0.0465034
\(592\) −0.292768 −0.0120327
\(593\) −1.74720 −0.0717491 −0.0358745 0.999356i \(-0.511422\pi\)
−0.0358745 + 0.999356i \(0.511422\pi\)
\(594\) 6.12012 0.251111
\(595\) 0 0
\(596\) 9.84678 0.403340
\(597\) −0.613413 −0.0251053
\(598\) 2.86948 0.117342
\(599\) 23.2085 0.948274 0.474137 0.880451i \(-0.342760\pi\)
0.474137 + 0.880451i \(0.342760\pi\)
\(600\) 0 0
\(601\) −8.41204 −0.343134 −0.171567 0.985172i \(-0.554883\pi\)
−0.171567 + 0.985172i \(0.554883\pi\)
\(602\) −1.47825 −0.0602488
\(603\) −23.2060 −0.945022
\(604\) −17.9245 −0.729338
\(605\) 0 0
\(606\) 1.21367 0.0493020
\(607\) −24.9877 −1.01422 −0.507110 0.861881i \(-0.669286\pi\)
−0.507110 + 0.861881i \(0.669286\pi\)
\(608\) 46.2222 1.87456
\(609\) 0.222181 0.00900322
\(610\) 0 0
\(611\) 6.14171 0.248467
\(612\) −14.7278 −0.595335
\(613\) −16.0755 −0.649283 −0.324642 0.945837i \(-0.605244\pi\)
−0.324642 + 0.945837i \(0.605244\pi\)
\(614\) −5.39588 −0.217760
\(615\) 0 0
\(616\) 28.6246 1.15332
\(617\) 15.2722 0.614837 0.307418 0.951574i \(-0.400535\pi\)
0.307418 + 0.951574i \(0.400535\pi\)
\(618\) 2.50559 0.100790
\(619\) 12.8512 0.516532 0.258266 0.966074i \(-0.416849\pi\)
0.258266 + 0.966074i \(0.416849\pi\)
\(620\) 0 0
\(621\) 11.2060 0.449682
\(622\) 4.20929 0.168777
\(623\) −27.6843 −1.10915
\(624\) −0.0985283 −0.00394429
\(625\) 0 0
\(626\) −14.5699 −0.582331
\(627\) −10.6731 −0.426242
\(628\) −3.01943 −0.120488
\(629\) 1.19673 0.0477166
\(630\) 0 0
\(631\) −0.842133 −0.0335248 −0.0167624 0.999860i \(-0.505336\pi\)
−0.0167624 + 0.999860i \(0.505336\pi\)
\(632\) −7.98495 −0.317624
\(633\) 6.07550 0.241479
\(634\) 9.66346 0.383785
\(635\) 0 0
\(636\) −0.842133 −0.0333928
\(637\) −1.54256 −0.0611185
\(638\) 2.05967 0.0815430
\(639\) 38.8629 1.53739
\(640\) 0 0
\(641\) 18.1833 0.718198 0.359099 0.933299i \(-0.383084\pi\)
0.359099 + 0.933299i \(0.383084\pi\)
\(642\) 3.56061 0.140526
\(643\) 49.3893 1.94773 0.973863 0.227137i \(-0.0729367\pi\)
0.973863 + 0.227137i \(0.0729367\pi\)
\(644\) 21.7713 0.857909
\(645\) 0 0
\(646\) −21.1305 −0.831369
\(647\) 7.07334 0.278082 0.139041 0.990287i \(-0.455598\pi\)
0.139041 + 0.990287i \(0.455598\pi\)
\(648\) −22.0960 −0.868014
\(649\) −39.4340 −1.54792
\(650\) 0 0
\(651\) 0.733531 0.0287493
\(652\) −25.4509 −0.996734
\(653\) −6.01832 −0.235515 −0.117758 0.993042i \(-0.537571\pi\)
−0.117758 + 0.993042i \(0.537571\pi\)
\(654\) −1.10206 −0.0430941
\(655\) 0 0
\(656\) −0.519531 −0.0202843
\(657\) 19.4315 0.758094
\(658\) −18.9838 −0.740067
\(659\) 0.401636 0.0156455 0.00782276 0.999969i \(-0.497510\pi\)
0.00782276 + 0.999969i \(0.497510\pi\)
\(660\) 0 0
\(661\) 36.8662 1.43393 0.716965 0.697109i \(-0.245531\pi\)
0.716965 + 0.697109i \(0.245531\pi\)
\(662\) 4.18057 0.162482
\(663\) 0.402747 0.0156414
\(664\) −11.8834 −0.461166
\(665\) 0 0
\(666\) 0.760877 0.0294834
\(667\) 3.77128 0.146025
\(668\) −7.39810 −0.286241
\(669\) 0.956493 0.0369802
\(670\) 0 0
\(671\) 46.6939 1.80260
\(672\) 2.72313 0.105047
\(673\) −48.3333 −1.86311 −0.931555 0.363600i \(-0.881548\pi\)
−0.931555 + 0.363600i \(0.881548\pi\)
\(674\) −17.1032 −0.658790
\(675\) 0 0
\(676\) 18.1488 0.698032
\(677\) 31.4796 1.20986 0.604930 0.796279i \(-0.293201\pi\)
0.604930 + 0.796279i \(0.293201\pi\)
\(678\) −2.07007 −0.0795006
\(679\) −18.9838 −0.728533
\(680\) 0 0
\(681\) 1.16500 0.0446429
\(682\) 6.80000 0.260386
\(683\) 7.88564 0.301736 0.150868 0.988554i \(-0.451793\pi\)
0.150868 + 0.988554i \(0.451793\pi\)
\(684\) 32.9773 1.26092
\(685\) 0 0
\(686\) 15.1157 0.577122
\(687\) −3.23912 −0.123580
\(688\) 0.861564 0.0328468
\(689\) −1.18521 −0.0451530
\(690\) 0 0
\(691\) −30.7759 −1.17077 −0.585386 0.810755i \(-0.699057\pi\)
−0.585386 + 0.810755i \(0.699057\pi\)
\(692\) −23.4167 −0.890169
\(693\) 32.3614 1.22931
\(694\) −12.0391 −0.456999
\(695\) 0 0
\(696\) 0.297680 0.0112835
\(697\) 2.12365 0.0804391
\(698\) −16.5653 −0.627004
\(699\) −2.70945 −0.102481
\(700\) 0 0
\(701\) −14.7310 −0.556384 −0.278192 0.960526i \(-0.589735\pi\)
−0.278192 + 0.960526i \(0.589735\pi\)
\(702\) 0.517107 0.0195170
\(703\) −2.67962 −0.101064
\(704\) 15.4908 0.583832
\(705\) 0 0
\(706\) −18.3639 −0.691134
\(707\) 12.9598 0.487402
\(708\) −2.36742 −0.0889732
\(709\) −24.6238 −0.924767 −0.462383 0.886680i \(-0.653006\pi\)
−0.462383 + 0.886680i \(0.653006\pi\)
\(710\) 0 0
\(711\) −9.02735 −0.338552
\(712\) −37.0917 −1.39007
\(713\) 12.4509 0.466290
\(714\) −1.24488 −0.0465885
\(715\) 0 0
\(716\) 11.7231 0.438114
\(717\) −2.23585 −0.0834995
\(718\) 21.1305 0.788584
\(719\) −41.4315 −1.54513 −0.772567 0.634934i \(-0.781027\pi\)
−0.772567 + 0.634934i \(0.781027\pi\)
\(720\) 0 0
\(721\) 26.7551 0.996413
\(722\) 32.8572 1.22282
\(723\) −0.532939 −0.0198202
\(724\) 21.5551 0.801090
\(725\) 0 0
\(726\) −3.82768 −0.142058
\(727\) −41.4588 −1.53762 −0.768811 0.639476i \(-0.779151\pi\)
−0.768811 + 0.639476i \(0.779151\pi\)
\(728\) 2.41858 0.0896385
\(729\) −23.9611 −0.887450
\(730\) 0 0
\(731\) −3.52175 −0.130257
\(732\) 2.80327 0.103612
\(733\) −24.4120 −0.901679 −0.450840 0.892605i \(-0.648875\pi\)
−0.450840 + 0.892605i \(0.648875\pi\)
\(734\) 6.36389 0.234895
\(735\) 0 0
\(736\) 46.2222 1.70377
\(737\) 44.6342 1.64412
\(738\) 1.35021 0.0497021
\(739\) −0.613413 −0.0225648 −0.0112824 0.999936i \(-0.503591\pi\)
−0.0112824 + 0.999936i \(0.503591\pi\)
\(740\) 0 0
\(741\) −0.901801 −0.0331285
\(742\) 3.66346 0.134490
\(743\) 38.1182 1.39842 0.699211 0.714915i \(-0.253535\pi\)
0.699211 + 0.714915i \(0.253535\pi\)
\(744\) 0.982792 0.0360309
\(745\) 0 0
\(746\) −13.9291 −0.509982
\(747\) −13.4347 −0.491551
\(748\) 28.3272 1.03575
\(749\) 38.0208 1.38925
\(750\) 0 0
\(751\) −8.11436 −0.296097 −0.148049 0.988980i \(-0.547299\pi\)
−0.148049 + 0.988980i \(0.547299\pi\)
\(752\) 11.0643 0.403474
\(753\) 0.402747 0.0146769
\(754\) 0.174028 0.00633771
\(755\) 0 0
\(756\) 3.92339 0.142692
\(757\) −32.6044 −1.18503 −0.592513 0.805561i \(-0.701864\pi\)
−0.592513 + 0.805561i \(0.701864\pi\)
\(758\) 1.43723 0.0522024
\(759\) −10.6731 −0.387408
\(760\) 0 0
\(761\) 36.0000 1.30500 0.652499 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) −2.95649 −0.107102
\(763\) −11.7680 −0.426031
\(764\) 25.1305 0.909190
\(765\) 0 0
\(766\) 25.7699 0.931104
\(767\) −3.33189 −0.120308
\(768\) 3.06294 0.110524
\(769\) 13.7141 0.494543 0.247272 0.968946i \(-0.420466\pi\)
0.247272 + 0.968946i \(0.420466\pi\)
\(770\) 0 0
\(771\) 0.646517 0.0232837
\(772\) 25.7713 0.927529
\(773\) 32.2794 1.16101 0.580504 0.814257i \(-0.302856\pi\)
0.580504 + 0.814257i \(0.302856\pi\)
\(774\) −2.23912 −0.0804836
\(775\) 0 0
\(776\) −25.4347 −0.913054
\(777\) −0.157867 −0.00566344
\(778\) −5.42545 −0.194512
\(779\) −4.75512 −0.170370
\(780\) 0 0
\(781\) −74.7486 −2.67471
\(782\) −21.1305 −0.755626
\(783\) 0.679620 0.0242876
\(784\) −2.77893 −0.0992475
\(785\) 0 0
\(786\) 1.36853 0.0488140
\(787\) 43.6777 1.55694 0.778471 0.627680i \(-0.215995\pi\)
0.778471 + 0.627680i \(0.215995\pi\)
\(788\) 6.71848 0.239336
\(789\) −0.847890 −0.0301857
\(790\) 0 0
\(791\) −22.1046 −0.785947
\(792\) 43.3581 1.54066
\(793\) 3.94531 0.140102
\(794\) −1.19451 −0.0423914
\(795\) 0 0
\(796\) −3.64541 −0.129208
\(797\) 4.18057 0.148083 0.0740416 0.997255i \(-0.476410\pi\)
0.0740416 + 0.997255i \(0.476410\pi\)
\(798\) 2.78744 0.0986743
\(799\) −45.2268 −1.60001
\(800\) 0 0
\(801\) −41.9338 −1.48166
\(802\) 5.24953 0.185367
\(803\) −37.3743 −1.31891
\(804\) 2.67962 0.0945029
\(805\) 0 0
\(806\) 0.574553 0.0202378
\(807\) −6.74007 −0.237262
\(808\) 17.3636 0.610850
\(809\) −21.0058 −0.738523 −0.369262 0.929326i \(-0.620389\pi\)
−0.369262 + 0.929326i \(0.620389\pi\)
\(810\) 0 0
\(811\) −3.69578 −0.129776 −0.0648882 0.997893i \(-0.520669\pi\)
−0.0648882 + 0.997893i \(0.520669\pi\)
\(812\) 1.32038 0.0463362
\(813\) 5.37756 0.188599
\(814\) −1.46346 −0.0512943
\(815\) 0 0
\(816\) 0.725551 0.0253994
\(817\) 7.88564 0.275884
\(818\) −18.9018 −0.660886
\(819\) 2.73431 0.0955446
\(820\) 0 0
\(821\) 31.6673 1.10520 0.552599 0.833448i \(-0.313636\pi\)
0.552599 + 0.833448i \(0.313636\pi\)
\(822\) −2.14857 −0.0749401
\(823\) −40.8903 −1.42535 −0.712673 0.701497i \(-0.752516\pi\)
−0.712673 + 0.701497i \(0.752516\pi\)
\(824\) 35.8468 1.24878
\(825\) 0 0
\(826\) 10.2988 0.358341
\(827\) 4.07550 0.141719 0.0708595 0.997486i \(-0.477426\pi\)
0.0708595 + 0.997486i \(0.477426\pi\)
\(828\) 32.9773 1.14604
\(829\) −23.4074 −0.812972 −0.406486 0.913657i \(-0.633246\pi\)
−0.406486 + 0.913657i \(0.633246\pi\)
\(830\) 0 0
\(831\) 1.33405 0.0462778
\(832\) 1.30887 0.0453767
\(833\) 11.3592 0.393574
\(834\) −1.73431 −0.0600543
\(835\) 0 0
\(836\) −63.4282 −2.19371
\(837\) 2.24377 0.0775560
\(838\) −26.3365 −0.909781
\(839\) −6.45090 −0.222710 −0.111355 0.993781i \(-0.535519\pi\)
−0.111355 + 0.993781i \(0.535519\pi\)
\(840\) 0 0
\(841\) −28.7713 −0.992113
\(842\) 10.9954 0.378925
\(843\) −6.84789 −0.235854
\(844\) 36.1056 1.24281
\(845\) 0 0
\(846\) −28.7551 −0.988621
\(847\) −40.8726 −1.40440
\(848\) −2.13517 −0.0733219
\(849\) −3.13706 −0.107664
\(850\) 0 0
\(851\) −2.67962 −0.0918562
\(852\) −4.48754 −0.153741
\(853\) 9.82597 0.336435 0.168217 0.985750i \(-0.446199\pi\)
0.168217 + 0.985750i \(0.446199\pi\)
\(854\) −12.1948 −0.417299
\(855\) 0 0
\(856\) 50.9407 1.74112
\(857\) −42.0093 −1.43501 −0.717505 0.696553i \(-0.754716\pi\)
−0.717505 + 0.696553i \(0.754716\pi\)
\(858\) −0.492514 −0.0168142
\(859\) −23.0643 −0.786944 −0.393472 0.919337i \(-0.628726\pi\)
−0.393472 + 0.919337i \(0.628726\pi\)
\(860\) 0 0
\(861\) −0.280142 −0.00954723
\(862\) −10.8500 −0.369554
\(863\) −42.3822 −1.44271 −0.721354 0.692567i \(-0.756480\pi\)
−0.721354 + 0.692567i \(0.756480\pi\)
\(864\) 8.32967 0.283381
\(865\) 0 0
\(866\) −17.1007 −0.581105
\(867\) 1.09931 0.0373345
\(868\) 4.35924 0.147962
\(869\) 17.3631 0.589003
\(870\) 0 0
\(871\) 3.77128 0.127785
\(872\) −15.7669 −0.533935
\(873\) −28.7551 −0.973213
\(874\) 47.3138 1.60041
\(875\) 0 0
\(876\) −2.24377 −0.0758099
\(877\) −16.0755 −0.542831 −0.271416 0.962462i \(-0.587492\pi\)
−0.271416 + 0.962462i \(0.587492\pi\)
\(878\) 23.4646 0.791891
\(879\) −2.78247 −0.0938502
\(880\) 0 0
\(881\) 2.23121 0.0751713 0.0375856 0.999293i \(-0.488033\pi\)
0.0375856 + 0.999293i \(0.488033\pi\)
\(882\) 7.22218 0.243183
\(883\) −52.2977 −1.75996 −0.879979 0.475013i \(-0.842443\pi\)
−0.879979 + 0.475013i \(0.842443\pi\)
\(884\) 2.39345 0.0805006
\(885\) 0 0
\(886\) −21.2542 −0.714048
\(887\) −32.4991 −1.09121 −0.545606 0.838042i \(-0.683700\pi\)
−0.545606 + 0.838042i \(0.683700\pi\)
\(888\) −0.211511 −0.00709786
\(889\) −31.5699 −1.05882
\(890\) 0 0
\(891\) 48.0474 1.60965
\(892\) 5.68427 0.190323
\(893\) 101.268 3.38882
\(894\) 1.26071 0.0421645
\(895\) 0 0
\(896\) 18.7303 0.625734
\(897\) −0.901801 −0.0301103
\(898\) 6.24023 0.208239
\(899\) 0.755119 0.0251846
\(900\) 0 0
\(901\) 8.72777 0.290764
\(902\) −2.59699 −0.0864702
\(903\) 0.464574 0.0154600
\(904\) −29.6159 −0.985010
\(905\) 0 0
\(906\) −2.29493 −0.0762438
\(907\) −7.69578 −0.255534 −0.127767 0.991804i \(-0.540781\pi\)
−0.127767 + 0.991804i \(0.540781\pi\)
\(908\) 6.92339 0.229761
\(909\) 19.6304 0.651098
\(910\) 0 0
\(911\) −31.3412 −1.03838 −0.519190 0.854659i \(-0.673766\pi\)
−0.519190 + 0.854659i \(0.673766\pi\)
\(912\) −1.62460 −0.0537958
\(913\) 25.8402 0.855187
\(914\) 7.70370 0.254816
\(915\) 0 0
\(916\) −19.2495 −0.636022
\(917\) 14.6134 0.482577
\(918\) −3.80792 −0.125680
\(919\) −16.9784 −0.560066 −0.280033 0.959990i \(-0.590345\pi\)
−0.280033 + 0.959990i \(0.590345\pi\)
\(920\) 0 0
\(921\) 1.69578 0.0558779
\(922\) −12.4484 −0.409967
\(923\) −6.31573 −0.207885
\(924\) −3.73680 −0.122932
\(925\) 0 0
\(926\) −0.561662 −0.0184573
\(927\) 40.5264 1.33106
\(928\) 2.80327 0.0920219
\(929\) −50.3847 −1.65307 −0.826534 0.562887i \(-0.809691\pi\)
−0.826534 + 0.562887i \(0.809691\pi\)
\(930\) 0 0
\(931\) −25.4347 −0.833590
\(932\) −16.1018 −0.527432
\(933\) −1.32287 −0.0433087
\(934\) 8.78247 0.287371
\(935\) 0 0
\(936\) 3.66346 0.119744
\(937\) −27.1488 −0.886914 −0.443457 0.896296i \(-0.646248\pi\)
−0.443457 + 0.896296i \(0.646248\pi\)
\(938\) −11.6569 −0.380612
\(939\) 4.57893 0.149428
\(940\) 0 0
\(941\) −0.640760 −0.0208882 −0.0104441 0.999945i \(-0.503325\pi\)
−0.0104441 + 0.999945i \(0.503325\pi\)
\(942\) −0.386587 −0.0125957
\(943\) −4.75512 −0.154848
\(944\) −6.00242 −0.195362
\(945\) 0 0
\(946\) 4.30671 0.140023
\(947\) −30.5587 −0.993025 −0.496513 0.868030i \(-0.665386\pi\)
−0.496513 + 0.868030i \(0.665386\pi\)
\(948\) 1.04240 0.0338554
\(949\) −3.15787 −0.102509
\(950\) 0 0
\(951\) −3.03697 −0.0984804
\(952\) −17.8101 −0.577230
\(953\) −20.4484 −0.662389 −0.331195 0.943562i \(-0.607452\pi\)
−0.331195 + 0.943562i \(0.607452\pi\)
\(954\) 5.54910 0.179659
\(955\) 0 0
\(956\) −13.2873 −0.429741
\(957\) −0.647299 −0.0209242
\(958\) 8.65692 0.279692
\(959\) −22.9428 −0.740862
\(960\) 0 0
\(961\) −28.5070 −0.919580
\(962\) −0.123652 −0.00398671
\(963\) 57.5907 1.85583
\(964\) −3.16716 −0.102007
\(965\) 0 0
\(966\) 2.78744 0.0896844
\(967\) −13.6181 −0.437927 −0.218964 0.975733i \(-0.570268\pi\)
−0.218964 + 0.975733i \(0.570268\pi\)
\(968\) −54.7615 −1.76010
\(969\) 6.64076 0.213332
\(970\) 0 0
\(971\) 17.5789 0.564135 0.282067 0.959395i \(-0.408980\pi\)
0.282067 + 0.959395i \(0.408980\pi\)
\(972\) 8.94282 0.286841
\(973\) −18.5193 −0.593700
\(974\) 19.0917 0.611736
\(975\) 0 0
\(976\) 7.10749 0.227505
\(977\) −32.7004 −1.04618 −0.523090 0.852278i \(-0.675221\pi\)
−0.523090 + 0.852278i \(0.675221\pi\)
\(978\) −3.25855 −0.104197
\(979\) 80.6550 2.57775
\(980\) 0 0
\(981\) −17.8252 −0.569115
\(982\) 23.8676 0.761645
\(983\) 29.1819 0.930759 0.465380 0.885111i \(-0.345918\pi\)
0.465380 + 0.885111i \(0.345918\pi\)
\(984\) −0.375338 −0.0119653
\(985\) 0 0
\(986\) −1.28152 −0.0408119
\(987\) 5.96611 0.189904
\(988\) −5.35924 −0.170500
\(989\) 7.88564 0.250749
\(990\) 0 0
\(991\) −3.58796 −0.113975 −0.0569877 0.998375i \(-0.518150\pi\)
−0.0569877 + 0.998375i \(0.518150\pi\)
\(992\) 9.25502 0.293847
\(993\) −1.31384 −0.0416935
\(994\) 19.5218 0.619192
\(995\) 0 0
\(996\) 1.55132 0.0491555
\(997\) 31.8889 1.00993 0.504966 0.863139i \(-0.331505\pi\)
0.504966 + 0.863139i \(0.331505\pi\)
\(998\) 14.0755 0.445552
\(999\) −0.482893 −0.0152781
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 1075.2.a.l.1.2 3
3.2 odd 2 9675.2.a.bp.1.2 3
5.2 odd 4 1075.2.b.g.474.4 6
5.3 odd 4 1075.2.b.g.474.3 6
5.4 even 2 215.2.a.b.1.2 3
15.14 odd 2 1935.2.a.r.1.2 3
20.19 odd 2 3440.2.a.l.1.2 3
215.214 odd 2 9245.2.a.h.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
215.2.a.b.1.2 3 5.4 even 2
1075.2.a.l.1.2 3 1.1 even 1 trivial
1075.2.b.g.474.3 6 5.3 odd 4
1075.2.b.g.474.4 6 5.2 odd 4
1935.2.a.r.1.2 3 15.14 odd 2
3440.2.a.l.1.2 3 20.19 odd 2
9245.2.a.h.1.2 3 215.214 odd 2
9675.2.a.bp.1.2 3 3.2 odd 2