Properties

Label 4009.2.a.c.1.62
Level $4009$
Weight $2$
Character 4009.1
Self dual yes
Analytic conductor $32.012$
Analytic rank $1$
Dimension $71$
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] = [4009,2,Mod(1,4009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4009, 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("4009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4009 = 19 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4009.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: \(32.0120261703\)
Analytic rank: \(1\)
Dimension: \(71\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.62
Character \(\chi\) \(=\) 4009.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.09813 q^{2} +1.29704 q^{3} +2.40214 q^{4} +1.04726 q^{5} +2.72135 q^{6} -4.44368 q^{7} +0.843733 q^{8} -1.31769 q^{9} +O(q^{10})\) \(q+2.09813 q^{2} +1.29704 q^{3} +2.40214 q^{4} +1.04726 q^{5} +2.72135 q^{6} -4.44368 q^{7} +0.843733 q^{8} -1.31769 q^{9} +2.19729 q^{10} +2.24351 q^{11} +3.11567 q^{12} -6.49092 q^{13} -9.32340 q^{14} +1.35834 q^{15} -3.03401 q^{16} +0.862835 q^{17} -2.76468 q^{18} +1.00000 q^{19} +2.51567 q^{20} -5.76363 q^{21} +4.70718 q^{22} +0.555918 q^{23} +1.09436 q^{24} -3.90324 q^{25} -13.6188 q^{26} -5.60021 q^{27} -10.6743 q^{28} -3.63048 q^{29} +2.84998 q^{30} +2.77163 q^{31} -8.05321 q^{32} +2.90993 q^{33} +1.81034 q^{34} -4.65371 q^{35} -3.16527 q^{36} -2.37924 q^{37} +2.09813 q^{38} -8.41898 q^{39} +0.883612 q^{40} -6.25508 q^{41} -12.0928 q^{42} +4.44636 q^{43} +5.38923 q^{44} -1.37997 q^{45} +1.16639 q^{46} +6.66058 q^{47} -3.93524 q^{48} +12.7463 q^{49} -8.18949 q^{50} +1.11913 q^{51} -15.5921 q^{52} -8.77722 q^{53} -11.7500 q^{54} +2.34955 q^{55} -3.74928 q^{56} +1.29704 q^{57} -7.61720 q^{58} +1.01193 q^{59} +3.26293 q^{60} -10.1469 q^{61} +5.81523 q^{62} +5.85538 q^{63} -10.8286 q^{64} -6.79771 q^{65} +6.10540 q^{66} +13.2545 q^{67} +2.07265 q^{68} +0.721048 q^{69} -9.76407 q^{70} +5.89404 q^{71} -1.11178 q^{72} +4.79546 q^{73} -4.99194 q^{74} -5.06265 q^{75} +2.40214 q^{76} -9.96945 q^{77} -17.6641 q^{78} +0.749154 q^{79} -3.17742 q^{80} -3.31063 q^{81} -13.1239 q^{82} +8.32026 q^{83} -13.8450 q^{84} +0.903617 q^{85} +9.32902 q^{86} -4.70887 q^{87} +1.89293 q^{88} -3.92777 q^{89} -2.89535 q^{90} +28.8436 q^{91} +1.33539 q^{92} +3.59491 q^{93} +13.9747 q^{94} +1.04726 q^{95} -10.4453 q^{96} -9.57864 q^{97} +26.7433 q^{98} -2.95625 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.09813 1.48360 0.741800 0.670621i \(-0.233972\pi\)
0.741800 + 0.670621i \(0.233972\pi\)
\(3\) 1.29704 0.748846 0.374423 0.927258i \(-0.377841\pi\)
0.374423 + 0.927258i \(0.377841\pi\)
\(4\) 2.40214 1.20107
\(5\) 1.04726 0.468351 0.234176 0.972194i \(-0.424761\pi\)
0.234176 + 0.972194i \(0.424761\pi\)
\(6\) 2.72135 1.11099
\(7\) −4.44368 −1.67955 −0.839776 0.542933i \(-0.817314\pi\)
−0.839776 + 0.542933i \(0.817314\pi\)
\(8\) 0.843733 0.298305
\(9\) −1.31769 −0.439229
\(10\) 2.19729 0.694846
\(11\) 2.24351 0.676445 0.338223 0.941066i \(-0.390174\pi\)
0.338223 + 0.941066i \(0.390174\pi\)
\(12\) 3.11567 0.899415
\(13\) −6.49092 −1.80026 −0.900129 0.435623i \(-0.856528\pi\)
−0.900129 + 0.435623i \(0.856528\pi\)
\(14\) −9.32340 −2.49178
\(15\) 1.35834 0.350723
\(16\) −3.03401 −0.758503
\(17\) 0.862835 0.209268 0.104634 0.994511i \(-0.466633\pi\)
0.104634 + 0.994511i \(0.466633\pi\)
\(18\) −2.76468 −0.651641
\(19\) 1.00000 0.229416
\(20\) 2.51567 0.562522
\(21\) −5.76363 −1.25773
\(22\) 4.70718 1.00357
\(23\) 0.555918 0.115917 0.0579585 0.998319i \(-0.481541\pi\)
0.0579585 + 0.998319i \(0.481541\pi\)
\(24\) 1.09436 0.223384
\(25\) −3.90324 −0.780647
\(26\) −13.6188 −2.67086
\(27\) −5.60021 −1.07776
\(28\) −10.6743 −2.01726
\(29\) −3.63048 −0.674162 −0.337081 0.941476i \(-0.609440\pi\)
−0.337081 + 0.941476i \(0.609440\pi\)
\(30\) 2.84998 0.520332
\(31\) 2.77163 0.497799 0.248899 0.968529i \(-0.419931\pi\)
0.248899 + 0.968529i \(0.419931\pi\)
\(32\) −8.05321 −1.42362
\(33\) 2.90993 0.506553
\(34\) 1.81034 0.310470
\(35\) −4.65371 −0.786620
\(36\) −3.16527 −0.527545
\(37\) −2.37924 −0.391144 −0.195572 0.980689i \(-0.562656\pi\)
−0.195572 + 0.980689i \(0.562656\pi\)
\(38\) 2.09813 0.340361
\(39\) −8.41898 −1.34812
\(40\) 0.883612 0.139711
\(41\) −6.25508 −0.976879 −0.488439 0.872598i \(-0.662434\pi\)
−0.488439 + 0.872598i \(0.662434\pi\)
\(42\) −12.0928 −1.86596
\(43\) 4.44636 0.678063 0.339032 0.940775i \(-0.389901\pi\)
0.339032 + 0.940775i \(0.389901\pi\)
\(44\) 5.38923 0.812457
\(45\) −1.37997 −0.205714
\(46\) 1.16639 0.171974
\(47\) 6.66058 0.971546 0.485773 0.874085i \(-0.338538\pi\)
0.485773 + 0.874085i \(0.338538\pi\)
\(48\) −3.93524 −0.568002
\(49\) 12.7463 1.82090
\(50\) −8.18949 −1.15817
\(51\) 1.11913 0.156710
\(52\) −15.5921 −2.16223
\(53\) −8.77722 −1.20564 −0.602822 0.797876i \(-0.705957\pi\)
−0.602822 + 0.797876i \(0.705957\pi\)
\(54\) −11.7500 −1.59897
\(55\) 2.34955 0.316814
\(56\) −3.74928 −0.501018
\(57\) 1.29704 0.171797
\(58\) −7.61720 −1.00019
\(59\) 1.01193 0.131742 0.0658711 0.997828i \(-0.479017\pi\)
0.0658711 + 0.997828i \(0.479017\pi\)
\(60\) 3.26293 0.421242
\(61\) −10.1469 −1.29918 −0.649588 0.760286i \(-0.725059\pi\)
−0.649588 + 0.760286i \(0.725059\pi\)
\(62\) 5.81523 0.738534
\(63\) 5.85538 0.737709
\(64\) −10.8286 −1.35358
\(65\) −6.79771 −0.843153
\(66\) 6.10540 0.751522
\(67\) 13.2545 1.61930 0.809650 0.586912i \(-0.199657\pi\)
0.809650 + 0.586912i \(0.199657\pi\)
\(68\) 2.07265 0.251345
\(69\) 0.721048 0.0868040
\(70\) −9.76407 −1.16703
\(71\) 5.89404 0.699494 0.349747 0.936844i \(-0.386268\pi\)
0.349747 + 0.936844i \(0.386268\pi\)
\(72\) −1.11178 −0.131024
\(73\) 4.79546 0.561266 0.280633 0.959815i \(-0.409456\pi\)
0.280633 + 0.959815i \(0.409456\pi\)
\(74\) −4.99194 −0.580301
\(75\) −5.06265 −0.584585
\(76\) 2.40214 0.275544
\(77\) −9.96945 −1.13612
\(78\) −17.6641 −2.00006
\(79\) 0.749154 0.0842865 0.0421432 0.999112i \(-0.486581\pi\)
0.0421432 + 0.999112i \(0.486581\pi\)
\(80\) −3.17742 −0.355246
\(81\) −3.31063 −0.367848
\(82\) −13.1239 −1.44930
\(83\) 8.32026 0.913267 0.456634 0.889655i \(-0.349055\pi\)
0.456634 + 0.889655i \(0.349055\pi\)
\(84\) −13.8450 −1.51062
\(85\) 0.903617 0.0980110
\(86\) 9.32902 1.00597
\(87\) −4.70887 −0.504844
\(88\) 1.89293 0.201787
\(89\) −3.92777 −0.416343 −0.208172 0.978092i \(-0.566751\pi\)
−0.208172 + 0.978092i \(0.566751\pi\)
\(90\) −2.89535 −0.305197
\(91\) 28.8436 3.02363
\(92\) 1.33539 0.139224
\(93\) 3.59491 0.372775
\(94\) 13.9747 1.44138
\(95\) 1.04726 0.107447
\(96\) −10.4453 −1.06607
\(97\) −9.57864 −0.972564 −0.486282 0.873802i \(-0.661647\pi\)
−0.486282 + 0.873802i \(0.661647\pi\)
\(98\) 26.7433 2.70148
\(99\) −2.95625 −0.297115
\(100\) −9.37611 −0.937611
\(101\) −5.98188 −0.595219 −0.297610 0.954688i \(-0.596189\pi\)
−0.297610 + 0.954688i \(0.596189\pi\)
\(102\) 2.34808 0.232495
\(103\) 1.75148 0.172579 0.0862893 0.996270i \(-0.472499\pi\)
0.0862893 + 0.996270i \(0.472499\pi\)
\(104\) −5.47661 −0.537025
\(105\) −6.03604 −0.589057
\(106\) −18.4157 −1.78869
\(107\) −12.6277 −1.22076 −0.610381 0.792108i \(-0.708984\pi\)
−0.610381 + 0.792108i \(0.708984\pi\)
\(108\) −13.4525 −1.29446
\(109\) −7.70014 −0.737540 −0.368770 0.929521i \(-0.620221\pi\)
−0.368770 + 0.929521i \(0.620221\pi\)
\(110\) 4.92966 0.470025
\(111\) −3.08596 −0.292907
\(112\) 13.4822 1.27395
\(113\) 15.5042 1.45851 0.729256 0.684241i \(-0.239866\pi\)
0.729256 + 0.684241i \(0.239866\pi\)
\(114\) 2.72135 0.254878
\(115\) 0.582194 0.0542898
\(116\) −8.72090 −0.809715
\(117\) 8.55301 0.790726
\(118\) 2.12316 0.195453
\(119\) −3.83416 −0.351477
\(120\) 1.14608 0.104622
\(121\) −5.96664 −0.542422
\(122\) −21.2895 −1.92746
\(123\) −8.11308 −0.731532
\(124\) 6.65783 0.597890
\(125\) −9.32405 −0.833968
\(126\) 12.2853 1.09446
\(127\) −22.1737 −1.96759 −0.983797 0.179287i \(-0.942621\pi\)
−0.983797 + 0.179287i \(0.942621\pi\)
\(128\) −6.61342 −0.584549
\(129\) 5.76710 0.507765
\(130\) −14.2625 −1.25090
\(131\) 5.35571 0.467931 0.233965 0.972245i \(-0.424830\pi\)
0.233965 + 0.972245i \(0.424830\pi\)
\(132\) 6.99004 0.608405
\(133\) −4.44368 −0.385316
\(134\) 27.8097 2.40239
\(135\) −5.86491 −0.504771
\(136\) 0.728003 0.0624257
\(137\) 16.8479 1.43942 0.719709 0.694276i \(-0.244275\pi\)
0.719709 + 0.694276i \(0.244275\pi\)
\(138\) 1.51285 0.128782
\(139\) −1.71682 −0.145618 −0.0728092 0.997346i \(-0.523196\pi\)
−0.0728092 + 0.997346i \(0.523196\pi\)
\(140\) −11.1788 −0.944784
\(141\) 8.63904 0.727538
\(142\) 12.3665 1.03777
\(143\) −14.5625 −1.21778
\(144\) 3.99788 0.333157
\(145\) −3.80207 −0.315745
\(146\) 10.0615 0.832695
\(147\) 16.5324 1.36357
\(148\) −5.71525 −0.469790
\(149\) −9.81425 −0.804015 −0.402007 0.915636i \(-0.631687\pi\)
−0.402007 + 0.915636i \(0.631687\pi\)
\(150\) −10.6221 −0.867290
\(151\) −1.90938 −0.155383 −0.0776915 0.996977i \(-0.524755\pi\)
−0.0776915 + 0.996977i \(0.524755\pi\)
\(152\) 0.843733 0.0684358
\(153\) −1.13695 −0.0919168
\(154\) −20.9172 −1.68555
\(155\) 2.90263 0.233145
\(156\) −20.2235 −1.61918
\(157\) 15.4067 1.22959 0.614793 0.788689i \(-0.289240\pi\)
0.614793 + 0.788689i \(0.289240\pi\)
\(158\) 1.57182 0.125047
\(159\) −11.3844 −0.902842
\(160\) −8.43385 −0.666754
\(161\) −2.47032 −0.194689
\(162\) −6.94613 −0.545739
\(163\) −0.0626218 −0.00490492 −0.00245246 0.999997i \(-0.500781\pi\)
−0.00245246 + 0.999997i \(0.500781\pi\)
\(164\) −15.0255 −1.17330
\(165\) 3.04746 0.237245
\(166\) 17.4570 1.35492
\(167\) 3.12383 0.241729 0.120864 0.992669i \(-0.461433\pi\)
0.120864 + 0.992669i \(0.461433\pi\)
\(168\) −4.86296 −0.375186
\(169\) 29.1321 2.24093
\(170\) 1.89590 0.145409
\(171\) −1.31769 −0.100766
\(172\) 10.6808 0.814400
\(173\) 5.99542 0.455824 0.227912 0.973682i \(-0.426810\pi\)
0.227912 + 0.973682i \(0.426810\pi\)
\(174\) −9.87981 −0.748986
\(175\) 17.3447 1.31114
\(176\) −6.80685 −0.513086
\(177\) 1.31252 0.0986547
\(178\) −8.24097 −0.617687
\(179\) 14.1110 1.05471 0.527354 0.849646i \(-0.323184\pi\)
0.527354 + 0.849646i \(0.323184\pi\)
\(180\) −3.31487 −0.247076
\(181\) −24.7124 −1.83685 −0.918427 0.395590i \(-0.870540\pi\)
−0.918427 + 0.395590i \(0.870540\pi\)
\(182\) 60.5175 4.48585
\(183\) −13.1609 −0.972883
\(184\) 0.469047 0.0345786
\(185\) −2.49169 −0.183193
\(186\) 7.54258 0.553049
\(187\) 1.93578 0.141558
\(188\) 15.9996 1.16689
\(189\) 24.8855 1.81016
\(190\) 2.19729 0.159408
\(191\) −0.0355414 −0.00257168 −0.00128584 0.999999i \(-0.500409\pi\)
−0.00128584 + 0.999999i \(0.500409\pi\)
\(192\) −14.0452 −1.01362
\(193\) −22.6374 −1.62948 −0.814739 0.579828i \(-0.803120\pi\)
−0.814739 + 0.579828i \(0.803120\pi\)
\(194\) −20.0972 −1.44289
\(195\) −8.81691 −0.631392
\(196\) 30.6183 2.18702
\(197\) 15.6341 1.11388 0.556942 0.830551i \(-0.311975\pi\)
0.556942 + 0.830551i \(0.311975\pi\)
\(198\) −6.20259 −0.440799
\(199\) −20.7116 −1.46821 −0.734104 0.679037i \(-0.762398\pi\)
−0.734104 + 0.679037i \(0.762398\pi\)
\(200\) −3.29329 −0.232871
\(201\) 17.1917 1.21261
\(202\) −12.5507 −0.883067
\(203\) 16.1327 1.13229
\(204\) 2.68831 0.188219
\(205\) −6.55072 −0.457522
\(206\) 3.67483 0.256038
\(207\) −0.732527 −0.0509141
\(208\) 19.6935 1.36550
\(209\) 2.24351 0.155187
\(210\) −12.6644 −0.873926
\(211\) 1.00000 0.0688428
\(212\) −21.0841 −1.44806
\(213\) 7.64481 0.523814
\(214\) −26.4944 −1.81112
\(215\) 4.65652 0.317572
\(216\) −4.72509 −0.321501
\(217\) −12.3162 −0.836079
\(218\) −16.1559 −1.09421
\(219\) 6.21990 0.420302
\(220\) 5.64395 0.380515
\(221\) −5.60060 −0.376737
\(222\) −6.47474 −0.434556
\(223\) −14.6037 −0.977939 −0.488969 0.872301i \(-0.662627\pi\)
−0.488969 + 0.872301i \(0.662627\pi\)
\(224\) 35.7859 2.39104
\(225\) 5.14325 0.342883
\(226\) 32.5298 2.16385
\(227\) 9.00004 0.597354 0.298677 0.954354i \(-0.403455\pi\)
0.298677 + 0.954354i \(0.403455\pi\)
\(228\) 3.11567 0.206340
\(229\) 18.6656 1.23345 0.616727 0.787177i \(-0.288458\pi\)
0.616727 + 0.787177i \(0.288458\pi\)
\(230\) 1.22152 0.0805444
\(231\) −12.9308 −0.850783
\(232\) −3.06315 −0.201106
\(233\) 0.777953 0.0509654 0.0254827 0.999675i \(-0.491888\pi\)
0.0254827 + 0.999675i \(0.491888\pi\)
\(234\) 17.9453 1.17312
\(235\) 6.97539 0.455024
\(236\) 2.43080 0.158231
\(237\) 0.971683 0.0631176
\(238\) −8.04456 −0.521451
\(239\) 22.8760 1.47972 0.739862 0.672758i \(-0.234891\pi\)
0.739862 + 0.672758i \(0.234891\pi\)
\(240\) −4.12123 −0.266025
\(241\) −9.49697 −0.611754 −0.305877 0.952071i \(-0.598950\pi\)
−0.305877 + 0.952071i \(0.598950\pi\)
\(242\) −12.5188 −0.804737
\(243\) 12.5066 0.802300
\(244\) −24.3742 −1.56040
\(245\) 13.3487 0.852819
\(246\) −17.0223 −1.08530
\(247\) −6.49092 −0.413008
\(248\) 2.33851 0.148496
\(249\) 10.7917 0.683897
\(250\) −19.5630 −1.23727
\(251\) −0.187156 −0.0118132 −0.00590658 0.999983i \(-0.501880\pi\)
−0.00590658 + 0.999983i \(0.501880\pi\)
\(252\) 14.0654 0.886039
\(253\) 1.24721 0.0784115
\(254\) −46.5231 −2.91912
\(255\) 1.17203 0.0733952
\(256\) 7.78147 0.486342
\(257\) 20.2010 1.26010 0.630051 0.776554i \(-0.283034\pi\)
0.630051 + 0.776554i \(0.283034\pi\)
\(258\) 12.1001 0.753320
\(259\) 10.5726 0.656947
\(260\) −16.3290 −1.01268
\(261\) 4.78383 0.296112
\(262\) 11.2370 0.694222
\(263\) −5.53122 −0.341070 −0.170535 0.985352i \(-0.554550\pi\)
−0.170535 + 0.985352i \(0.554550\pi\)
\(264\) 2.45520 0.151107
\(265\) −9.19207 −0.564665
\(266\) −9.32340 −0.571654
\(267\) −5.09448 −0.311777
\(268\) 31.8392 1.94489
\(269\) 7.80601 0.475941 0.237970 0.971272i \(-0.423518\pi\)
0.237970 + 0.971272i \(0.423518\pi\)
\(270\) −12.3053 −0.748878
\(271\) −1.28339 −0.0779601 −0.0389801 0.999240i \(-0.512411\pi\)
−0.0389801 + 0.999240i \(0.512411\pi\)
\(272\) −2.61785 −0.158731
\(273\) 37.4112 2.26423
\(274\) 35.3491 2.13552
\(275\) −8.75697 −0.528065
\(276\) 1.73206 0.104257
\(277\) −23.4206 −1.40721 −0.703603 0.710593i \(-0.748427\pi\)
−0.703603 + 0.710593i \(0.748427\pi\)
\(278\) −3.60210 −0.216040
\(279\) −3.65214 −0.218648
\(280\) −3.92649 −0.234653
\(281\) 2.16613 0.129220 0.0646102 0.997911i \(-0.479420\pi\)
0.0646102 + 0.997911i \(0.479420\pi\)
\(282\) 18.1258 1.07938
\(283\) −13.4163 −0.797515 −0.398757 0.917057i \(-0.630558\pi\)
−0.398757 + 0.917057i \(0.630558\pi\)
\(284\) 14.1583 0.840140
\(285\) 1.35834 0.0804614
\(286\) −30.5539 −1.80669
\(287\) 27.7955 1.64072
\(288\) 10.6116 0.625296
\(289\) −16.2555 −0.956207
\(290\) −7.97722 −0.468439
\(291\) −12.4239 −0.728300
\(292\) 11.5194 0.674119
\(293\) −20.9770 −1.22549 −0.612745 0.790281i \(-0.709935\pi\)
−0.612745 + 0.790281i \(0.709935\pi\)
\(294\) 34.6871 2.02299
\(295\) 1.05976 0.0617016
\(296\) −2.00744 −0.116680
\(297\) −12.5642 −0.729046
\(298\) −20.5915 −1.19284
\(299\) −3.60842 −0.208680
\(300\) −12.1612 −0.702126
\(301\) −19.7582 −1.13884
\(302\) −4.00612 −0.230526
\(303\) −7.75873 −0.445728
\(304\) −3.03401 −0.174013
\(305\) −10.6265 −0.608470
\(306\) −2.38546 −0.136368
\(307\) −25.0664 −1.43062 −0.715309 0.698809i \(-0.753714\pi\)
−0.715309 + 0.698809i \(0.753714\pi\)
\(308\) −23.9480 −1.36456
\(309\) 2.27174 0.129235
\(310\) 6.09008 0.345893
\(311\) −20.4264 −1.15827 −0.579137 0.815230i \(-0.696611\pi\)
−0.579137 + 0.815230i \(0.696611\pi\)
\(312\) −7.10338 −0.402149
\(313\) −13.0662 −0.738547 −0.369273 0.929321i \(-0.620393\pi\)
−0.369273 + 0.929321i \(0.620393\pi\)
\(314\) 32.3251 1.82421
\(315\) 6.13214 0.345507
\(316\) 1.79957 0.101234
\(317\) 5.77241 0.324211 0.162105 0.986773i \(-0.448172\pi\)
0.162105 + 0.986773i \(0.448172\pi\)
\(318\) −23.8859 −1.33946
\(319\) −8.14502 −0.456034
\(320\) −11.3404 −0.633950
\(321\) −16.3786 −0.914163
\(322\) −5.18305 −0.288840
\(323\) 0.862835 0.0480094
\(324\) −7.95259 −0.441811
\(325\) 25.3356 1.40537
\(326\) −0.131389 −0.00727694
\(327\) −9.98739 −0.552304
\(328\) −5.27761 −0.291408
\(329\) −29.5975 −1.63176
\(330\) 6.39397 0.351976
\(331\) −34.6799 −1.90618 −0.953090 0.302688i \(-0.902116\pi\)
−0.953090 + 0.302688i \(0.902116\pi\)
\(332\) 19.9864 1.09690
\(333\) 3.13509 0.171802
\(334\) 6.55418 0.358629
\(335\) 13.8810 0.758401
\(336\) 17.4869 0.953990
\(337\) −17.0961 −0.931283 −0.465641 0.884974i \(-0.654176\pi\)
−0.465641 + 0.884974i \(0.654176\pi\)
\(338\) 61.1228 3.32464
\(339\) 20.1095 1.09220
\(340\) 2.17061 0.117718
\(341\) 6.21819 0.336734
\(342\) −2.76468 −0.149497
\(343\) −25.5346 −1.37874
\(344\) 3.75154 0.202270
\(345\) 0.755128 0.0406547
\(346\) 12.5792 0.676260
\(347\) 19.2223 1.03191 0.515955 0.856616i \(-0.327437\pi\)
0.515955 + 0.856616i \(0.327437\pi\)
\(348\) −11.3113 −0.606352
\(349\) 30.7423 1.64560 0.822800 0.568331i \(-0.192411\pi\)
0.822800 + 0.568331i \(0.192411\pi\)
\(350\) 36.3914 1.94520
\(351\) 36.3505 1.94025
\(352\) −18.0675 −0.963001
\(353\) 30.4596 1.62120 0.810600 0.585600i \(-0.199141\pi\)
0.810600 + 0.585600i \(0.199141\pi\)
\(354\) 2.75382 0.146364
\(355\) 6.17262 0.327609
\(356\) −9.43505 −0.500057
\(357\) −4.97306 −0.263202
\(358\) 29.6067 1.56477
\(359\) 12.6836 0.669415 0.334708 0.942322i \(-0.391362\pi\)
0.334708 + 0.942322i \(0.391362\pi\)
\(360\) −1.16433 −0.0613653
\(361\) 1.00000 0.0526316
\(362\) −51.8496 −2.72516
\(363\) −7.73897 −0.406191
\(364\) 69.2862 3.63158
\(365\) 5.02212 0.262870
\(366\) −27.6133 −1.44337
\(367\) 13.2861 0.693531 0.346766 0.937952i \(-0.387280\pi\)
0.346766 + 0.937952i \(0.387280\pi\)
\(368\) −1.68666 −0.0879234
\(369\) 8.24224 0.429074
\(370\) −5.22788 −0.271785
\(371\) 39.0031 2.02494
\(372\) 8.63546 0.447728
\(373\) 10.7372 0.555952 0.277976 0.960588i \(-0.410337\pi\)
0.277976 + 0.960588i \(0.410337\pi\)
\(374\) 4.06152 0.210016
\(375\) −12.0937 −0.624514
\(376\) 5.61975 0.289817
\(377\) 23.5651 1.21367
\(378\) 52.2130 2.68555
\(379\) 2.77982 0.142790 0.0713950 0.997448i \(-0.477255\pi\)
0.0713950 + 0.997448i \(0.477255\pi\)
\(380\) 2.51567 0.129051
\(381\) −28.7601 −1.47342
\(382\) −0.0745703 −0.00381535
\(383\) −0.345409 −0.0176496 −0.00882478 0.999961i \(-0.502809\pi\)
−0.00882478 + 0.999961i \(0.502809\pi\)
\(384\) −8.57787 −0.437738
\(385\) −10.4407 −0.532105
\(386\) −47.4962 −2.41749
\(387\) −5.85891 −0.297825
\(388\) −23.0092 −1.16812
\(389\) −3.14417 −0.159416 −0.0797079 0.996818i \(-0.525399\pi\)
−0.0797079 + 0.996818i \(0.525399\pi\)
\(390\) −18.4990 −0.936733
\(391\) 0.479666 0.0242577
\(392\) 10.7545 0.543182
\(393\) 6.94657 0.350408
\(394\) 32.8023 1.65256
\(395\) 0.784563 0.0394756
\(396\) −7.10132 −0.356855
\(397\) 17.0151 0.853963 0.426982 0.904260i \(-0.359577\pi\)
0.426982 + 0.904260i \(0.359577\pi\)
\(398\) −43.4556 −2.17823
\(399\) −5.76363 −0.288542
\(400\) 11.8425 0.592124
\(401\) 1.84999 0.0923841 0.0461921 0.998933i \(-0.485291\pi\)
0.0461921 + 0.998933i \(0.485291\pi\)
\(402\) 36.0703 1.79902
\(403\) −17.9904 −0.896167
\(404\) −14.3693 −0.714899
\(405\) −3.46711 −0.172282
\(406\) 33.8484 1.67987
\(407\) −5.33785 −0.264587
\(408\) 0.944248 0.0467473
\(409\) −0.703759 −0.0347986 −0.0173993 0.999849i \(-0.505539\pi\)
−0.0173993 + 0.999849i \(0.505539\pi\)
\(410\) −13.7442 −0.678780
\(411\) 21.8525 1.07790
\(412\) 4.20730 0.207279
\(413\) −4.49670 −0.221268
\(414\) −1.53693 −0.0755362
\(415\) 8.71352 0.427730
\(416\) 52.2728 2.56288
\(417\) −2.22678 −0.109046
\(418\) 4.70718 0.230236
\(419\) 30.9871 1.51382 0.756909 0.653520i \(-0.226708\pi\)
0.756909 + 0.653520i \(0.226708\pi\)
\(420\) −14.4994 −0.707498
\(421\) −6.04667 −0.294696 −0.147348 0.989085i \(-0.547074\pi\)
−0.147348 + 0.989085i \(0.547074\pi\)
\(422\) 2.09813 0.102135
\(423\) −8.77657 −0.426731
\(424\) −7.40563 −0.359649
\(425\) −3.36785 −0.163365
\(426\) 16.0398 0.777130
\(427\) 45.0895 2.18203
\(428\) −30.3334 −1.46622
\(429\) −18.8881 −0.911927
\(430\) 9.76996 0.471149
\(431\) 7.43875 0.358312 0.179156 0.983821i \(-0.442663\pi\)
0.179156 + 0.983821i \(0.442663\pi\)
\(432\) 16.9911 0.817486
\(433\) −7.50896 −0.360858 −0.180429 0.983588i \(-0.557749\pi\)
−0.180429 + 0.983588i \(0.557749\pi\)
\(434\) −25.8410 −1.24041
\(435\) −4.93143 −0.236444
\(436\) −18.4968 −0.885835
\(437\) 0.555918 0.0265932
\(438\) 13.0501 0.623560
\(439\) −6.96888 −0.332607 −0.166303 0.986075i \(-0.553183\pi\)
−0.166303 + 0.986075i \(0.553183\pi\)
\(440\) 1.98240 0.0945070
\(441\) −16.7956 −0.799791
\(442\) −11.7508 −0.558927
\(443\) −2.64701 −0.125763 −0.0628816 0.998021i \(-0.520029\pi\)
−0.0628816 + 0.998021i \(0.520029\pi\)
\(444\) −7.41290 −0.351801
\(445\) −4.11342 −0.194995
\(446\) −30.6405 −1.45087
\(447\) −12.7295 −0.602083
\(448\) 48.1190 2.27341
\(449\) −32.3865 −1.52841 −0.764206 0.644972i \(-0.776869\pi\)
−0.764206 + 0.644972i \(0.776869\pi\)
\(450\) 10.7912 0.508702
\(451\) −14.0334 −0.660805
\(452\) 37.2432 1.75177
\(453\) −2.47654 −0.116358
\(454\) 18.8832 0.886234
\(455\) 30.2069 1.41612
\(456\) 1.09436 0.0512479
\(457\) −32.3720 −1.51430 −0.757148 0.653243i \(-0.773408\pi\)
−0.757148 + 0.653243i \(0.773408\pi\)
\(458\) 39.1627 1.82995
\(459\) −4.83206 −0.225541
\(460\) 1.39851 0.0652058
\(461\) 4.80178 0.223641 0.111821 0.993728i \(-0.464332\pi\)
0.111821 + 0.993728i \(0.464332\pi\)
\(462\) −27.1304 −1.26222
\(463\) −23.2991 −1.08280 −0.541399 0.840766i \(-0.682105\pi\)
−0.541399 + 0.840766i \(0.682105\pi\)
\(464\) 11.0149 0.511354
\(465\) 3.76482 0.174589
\(466\) 1.63224 0.0756123
\(467\) −23.6997 −1.09669 −0.548345 0.836252i \(-0.684742\pi\)
−0.548345 + 0.836252i \(0.684742\pi\)
\(468\) 20.5455 0.949716
\(469\) −58.8989 −2.71970
\(470\) 14.6353 0.675074
\(471\) 19.9830 0.920770
\(472\) 0.853800 0.0392993
\(473\) 9.97547 0.458673
\(474\) 2.03871 0.0936412
\(475\) −3.90324 −0.179093
\(476\) −9.21018 −0.422148
\(477\) 11.5656 0.529554
\(478\) 47.9967 2.19532
\(479\) −21.3331 −0.974735 −0.487367 0.873197i \(-0.662043\pi\)
−0.487367 + 0.873197i \(0.662043\pi\)
\(480\) −10.9390 −0.499296
\(481\) 15.4434 0.704160
\(482\) −19.9259 −0.907598
\(483\) −3.20410 −0.145792
\(484\) −14.3327 −0.651486
\(485\) −10.0314 −0.455501
\(486\) 26.2405 1.19029
\(487\) 4.37734 0.198356 0.0991781 0.995070i \(-0.468379\pi\)
0.0991781 + 0.995070i \(0.468379\pi\)
\(488\) −8.56127 −0.387550
\(489\) −0.0812230 −0.00367303
\(490\) 28.0073 1.26524
\(491\) −2.63593 −0.118958 −0.0594788 0.998230i \(-0.518944\pi\)
−0.0594788 + 0.998230i \(0.518944\pi\)
\(492\) −19.4887 −0.878620
\(493\) −3.13250 −0.141081
\(494\) −13.6188 −0.612738
\(495\) −3.09598 −0.139154
\(496\) −8.40915 −0.377582
\(497\) −26.1912 −1.17484
\(498\) 22.6424 1.01463
\(499\) 18.8424 0.843500 0.421750 0.906712i \(-0.361416\pi\)
0.421750 + 0.906712i \(0.361416\pi\)
\(500\) −22.3976 −1.00165
\(501\) 4.05173 0.181018
\(502\) −0.392676 −0.0175260
\(503\) −21.7574 −0.970113 −0.485056 0.874483i \(-0.661201\pi\)
−0.485056 + 0.874483i \(0.661201\pi\)
\(504\) 4.94038 0.220062
\(505\) −6.26461 −0.278772
\(506\) 2.61681 0.116331
\(507\) 37.7855 1.67811
\(508\) −53.2641 −2.36321
\(509\) −19.8113 −0.878121 −0.439061 0.898457i \(-0.644689\pi\)
−0.439061 + 0.898457i \(0.644689\pi\)
\(510\) 2.45906 0.108889
\(511\) −21.3095 −0.942676
\(512\) 29.5533 1.30609
\(513\) −5.60021 −0.247255
\(514\) 42.3842 1.86949
\(515\) 1.83427 0.0808274
\(516\) 13.8534 0.609861
\(517\) 14.9431 0.657197
\(518\) 22.1826 0.974646
\(519\) 7.77630 0.341342
\(520\) −5.73546 −0.251516
\(521\) 11.3441 0.496994 0.248497 0.968633i \(-0.420063\pi\)
0.248497 + 0.968633i \(0.420063\pi\)
\(522\) 10.0371 0.439312
\(523\) −10.3078 −0.450728 −0.225364 0.974275i \(-0.572357\pi\)
−0.225364 + 0.974275i \(0.572357\pi\)
\(524\) 12.8652 0.562017
\(525\) 22.4968 0.981841
\(526\) −11.6052 −0.506011
\(527\) 2.39146 0.104174
\(528\) −8.82876 −0.384222
\(529\) −22.6910 −0.986563
\(530\) −19.2861 −0.837736
\(531\) −1.33341 −0.0578651
\(532\) −10.6743 −0.462790
\(533\) 40.6012 1.75863
\(534\) −10.6889 −0.462552
\(535\) −13.2245 −0.571745
\(536\) 11.1833 0.483045
\(537\) 18.3026 0.789814
\(538\) 16.3780 0.706106
\(539\) 28.5964 1.23174
\(540\) −14.0883 −0.606264
\(541\) 24.8746 1.06944 0.534721 0.845029i \(-0.320417\pi\)
0.534721 + 0.845029i \(0.320417\pi\)
\(542\) −2.69271 −0.115662
\(543\) −32.0529 −1.37552
\(544\) −6.94859 −0.297918
\(545\) −8.06409 −0.345427
\(546\) 78.4935 3.35921
\(547\) −19.2194 −0.821761 −0.410881 0.911689i \(-0.634779\pi\)
−0.410881 + 0.911689i \(0.634779\pi\)
\(548\) 40.4711 1.72884
\(549\) 13.3704 0.570636
\(550\) −18.3732 −0.783437
\(551\) −3.63048 −0.154663
\(552\) 0.608372 0.0258940
\(553\) −3.32900 −0.141564
\(554\) −49.1393 −2.08773
\(555\) −3.23182 −0.137183
\(556\) −4.12403 −0.174898
\(557\) −8.97320 −0.380207 −0.190103 0.981764i \(-0.560882\pi\)
−0.190103 + 0.981764i \(0.560882\pi\)
\(558\) −7.66265 −0.324386
\(559\) −28.8610 −1.22069
\(560\) 14.1194 0.596654
\(561\) 2.51079 0.106006
\(562\) 4.54481 0.191711
\(563\) −27.4628 −1.15742 −0.578710 0.815534i \(-0.696443\pi\)
−0.578710 + 0.815534i \(0.696443\pi\)
\(564\) 20.7521 0.873823
\(565\) 16.2370 0.683095
\(566\) −28.1490 −1.18319
\(567\) 14.7114 0.617820
\(568\) 4.97300 0.208662
\(569\) −29.6879 −1.24458 −0.622291 0.782786i \(-0.713798\pi\)
−0.622291 + 0.782786i \(0.713798\pi\)
\(570\) 2.84998 0.119372
\(571\) −35.5672 −1.48844 −0.744220 0.667935i \(-0.767179\pi\)
−0.744220 + 0.667935i \(0.767179\pi\)
\(572\) −34.9811 −1.46263
\(573\) −0.0460986 −0.00192580
\(574\) 58.3186 2.43417
\(575\) −2.16988 −0.0904903
\(576\) 14.2688 0.594532
\(577\) −32.0389 −1.33380 −0.666898 0.745149i \(-0.732378\pi\)
−0.666898 + 0.745149i \(0.732378\pi\)
\(578\) −34.1061 −1.41863
\(579\) −29.3616 −1.22023
\(580\) −9.13309 −0.379231
\(581\) −36.9726 −1.53388
\(582\) −26.0669 −1.08051
\(583\) −19.6918 −0.815552
\(584\) 4.04609 0.167428
\(585\) 8.95727 0.370338
\(586\) −44.0124 −1.81814
\(587\) −2.08587 −0.0860929 −0.0430465 0.999073i \(-0.513706\pi\)
−0.0430465 + 0.999073i \(0.513706\pi\)
\(588\) 39.7131 1.63774
\(589\) 2.77163 0.114203
\(590\) 2.22351 0.0915405
\(591\) 20.2781 0.834128
\(592\) 7.21863 0.296684
\(593\) 39.3869 1.61743 0.808713 0.588203i \(-0.200165\pi\)
0.808713 + 0.588203i \(0.200165\pi\)
\(594\) −26.3612 −1.08161
\(595\) −4.01538 −0.164615
\(596\) −23.5752 −0.965677
\(597\) −26.8638 −1.09946
\(598\) −7.57093 −0.309598
\(599\) 10.2820 0.420109 0.210055 0.977690i \(-0.432636\pi\)
0.210055 + 0.977690i \(0.432636\pi\)
\(600\) −4.27153 −0.174384
\(601\) 1.57055 0.0640639 0.0320320 0.999487i \(-0.489802\pi\)
0.0320320 + 0.999487i \(0.489802\pi\)
\(602\) −41.4552 −1.68959
\(603\) −17.4654 −0.711245
\(604\) −4.58659 −0.186626
\(605\) −6.24866 −0.254044
\(606\) −16.2788 −0.661281
\(607\) 9.66958 0.392476 0.196238 0.980556i \(-0.437127\pi\)
0.196238 + 0.980556i \(0.437127\pi\)
\(608\) −8.05321 −0.326601
\(609\) 20.9247 0.847912
\(610\) −22.2957 −0.902727
\(611\) −43.2333 −1.74903
\(612\) −2.73110 −0.110398
\(613\) 22.0410 0.890229 0.445114 0.895474i \(-0.353163\pi\)
0.445114 + 0.895474i \(0.353163\pi\)
\(614\) −52.5926 −2.12246
\(615\) −8.49654 −0.342614
\(616\) −8.41156 −0.338911
\(617\) −20.8105 −0.837797 −0.418899 0.908033i \(-0.637584\pi\)
−0.418899 + 0.908033i \(0.637584\pi\)
\(618\) 4.76640 0.191733
\(619\) 2.59263 0.104206 0.0521032 0.998642i \(-0.483408\pi\)
0.0521032 + 0.998642i \(0.483408\pi\)
\(620\) 6.97251 0.280023
\(621\) −3.11326 −0.124931
\(622\) −42.8572 −1.71842
\(623\) 17.4538 0.699270
\(624\) 25.5433 1.02255
\(625\) 9.75144 0.390057
\(626\) −27.4146 −1.09571
\(627\) 2.90993 0.116211
\(628\) 37.0089 1.47682
\(629\) −2.05289 −0.0818540
\(630\) 12.8660 0.512594
\(631\) −32.4017 −1.28989 −0.644946 0.764228i \(-0.723120\pi\)
−0.644946 + 0.764228i \(0.723120\pi\)
\(632\) 0.632086 0.0251430
\(633\) 1.29704 0.0515527
\(634\) 12.1112 0.480999
\(635\) −23.2217 −0.921525
\(636\) −27.3469 −1.08437
\(637\) −82.7351 −3.27808
\(638\) −17.0893 −0.676572
\(639\) −7.76651 −0.307238
\(640\) −6.92600 −0.273774
\(641\) 37.7273 1.49014 0.745069 0.666987i \(-0.232416\pi\)
0.745069 + 0.666987i \(0.232416\pi\)
\(642\) −34.3643 −1.35625
\(643\) 27.3168 1.07727 0.538635 0.842539i \(-0.318940\pi\)
0.538635 + 0.842539i \(0.318940\pi\)
\(644\) −5.93405 −0.233834
\(645\) 6.03968 0.237812
\(646\) 1.81034 0.0712268
\(647\) −26.5713 −1.04462 −0.522312 0.852754i \(-0.674930\pi\)
−0.522312 + 0.852754i \(0.674930\pi\)
\(648\) −2.79329 −0.109731
\(649\) 2.27028 0.0891164
\(650\) 53.1573 2.08500
\(651\) −15.9746 −0.626095
\(652\) −0.150426 −0.00589114
\(653\) 39.0140 1.52674 0.763368 0.645964i \(-0.223545\pi\)
0.763368 + 0.645964i \(0.223545\pi\)
\(654\) −20.9548 −0.819398
\(655\) 5.60885 0.219156
\(656\) 18.9780 0.740966
\(657\) −6.31892 −0.246525
\(658\) −62.0993 −2.42088
\(659\) −2.95550 −0.115130 −0.0575649 0.998342i \(-0.518334\pi\)
−0.0575649 + 0.998342i \(0.518334\pi\)
\(660\) 7.32042 0.284947
\(661\) −27.9159 −1.08580 −0.542902 0.839796i \(-0.682674\pi\)
−0.542902 + 0.839796i \(0.682674\pi\)
\(662\) −72.7628 −2.82801
\(663\) −7.26419 −0.282118
\(664\) 7.02008 0.272432
\(665\) −4.65371 −0.180463
\(666\) 6.57782 0.254885
\(667\) −2.01825 −0.0781469
\(668\) 7.50386 0.290333
\(669\) −18.9416 −0.732326
\(670\) 29.1241 1.12516
\(671\) −22.7647 −0.878821
\(672\) 46.4157 1.79052
\(673\) 38.2268 1.47353 0.736767 0.676146i \(-0.236351\pi\)
0.736767 + 0.676146i \(0.236351\pi\)
\(674\) −35.8697 −1.38165
\(675\) 21.8590 0.841352
\(676\) 69.9792 2.69151
\(677\) −46.1987 −1.77556 −0.887780 0.460268i \(-0.847753\pi\)
−0.887780 + 0.460268i \(0.847753\pi\)
\(678\) 42.1924 1.62039
\(679\) 42.5644 1.63347
\(680\) 0.762412 0.0292371
\(681\) 11.6734 0.447326
\(682\) 13.0465 0.499578
\(683\) −33.2989 −1.27415 −0.637074 0.770803i \(-0.719855\pi\)
−0.637074 + 0.770803i \(0.719855\pi\)
\(684\) −3.16527 −0.121027
\(685\) 17.6443 0.674153
\(686\) −53.5748 −2.04549
\(687\) 24.2100 0.923668
\(688\) −13.4903 −0.514313
\(689\) 56.9723 2.17047
\(690\) 1.58435 0.0603154
\(691\) 10.6975 0.406952 0.203476 0.979080i \(-0.434776\pi\)
0.203476 + 0.979080i \(0.434776\pi\)
\(692\) 14.4018 0.547475
\(693\) 13.1366 0.499019
\(694\) 40.3309 1.53094
\(695\) −1.79796 −0.0682006
\(696\) −3.97303 −0.150597
\(697\) −5.39710 −0.204430
\(698\) 64.5013 2.44141
\(699\) 1.00904 0.0381653
\(700\) 41.6644 1.57477
\(701\) 13.4978 0.509804 0.254902 0.966967i \(-0.417957\pi\)
0.254902 + 0.966967i \(0.417957\pi\)
\(702\) 76.2681 2.87855
\(703\) −2.37924 −0.0897346
\(704\) −24.2942 −0.915622
\(705\) 9.04736 0.340743
\(706\) 63.9081 2.40521
\(707\) 26.5815 0.999702
\(708\) 3.15284 0.118491
\(709\) 29.4137 1.10466 0.552328 0.833627i \(-0.313740\pi\)
0.552328 + 0.833627i \(0.313740\pi\)
\(710\) 12.9509 0.486040
\(711\) −0.987152 −0.0370211
\(712\) −3.31399 −0.124197
\(713\) 1.54080 0.0577033
\(714\) −10.4341 −0.390487
\(715\) −15.2508 −0.570346
\(716\) 33.8966 1.26678
\(717\) 29.6711 1.10809
\(718\) 26.6118 0.993144
\(719\) −19.7348 −0.735985 −0.367993 0.929829i \(-0.619955\pi\)
−0.367993 + 0.929829i \(0.619955\pi\)
\(720\) 4.18684 0.156034
\(721\) −7.78302 −0.289855
\(722\) 2.09813 0.0780842
\(723\) −12.3179 −0.458109
\(724\) −59.3624 −2.20619
\(725\) 14.1706 0.526283
\(726\) −16.2373 −0.602624
\(727\) 33.3056 1.23523 0.617617 0.786479i \(-0.288098\pi\)
0.617617 + 0.786479i \(0.288098\pi\)
\(728\) 24.3363 0.901962
\(729\) 26.1535 0.968647
\(730\) 10.5370 0.389993
\(731\) 3.83647 0.141897
\(732\) −31.6143 −1.16850
\(733\) −1.18908 −0.0439196 −0.0219598 0.999759i \(-0.506991\pi\)
−0.0219598 + 0.999759i \(0.506991\pi\)
\(734\) 27.8760 1.02892
\(735\) 17.3138 0.638630
\(736\) −4.47693 −0.165022
\(737\) 29.7368 1.09537
\(738\) 17.2933 0.636574
\(739\) 19.1641 0.704962 0.352481 0.935819i \(-0.385338\pi\)
0.352481 + 0.935819i \(0.385338\pi\)
\(740\) −5.98538 −0.220027
\(741\) −8.41898 −0.309279
\(742\) 81.8335 3.00420
\(743\) −24.4294 −0.896229 −0.448114 0.893976i \(-0.647904\pi\)
−0.448114 + 0.893976i \(0.647904\pi\)
\(744\) 3.03315 0.111200
\(745\) −10.2781 −0.376561
\(746\) 22.5280 0.824810
\(747\) −10.9635 −0.401134
\(748\) 4.65001 0.170021
\(749\) 56.1133 2.05033
\(750\) −25.3740 −0.926528
\(751\) −31.7586 −1.15889 −0.579444 0.815012i \(-0.696730\pi\)
−0.579444 + 0.815012i \(0.696730\pi\)
\(752\) −20.2083 −0.736921
\(753\) −0.242748 −0.00884624
\(754\) 49.4426 1.80059
\(755\) −1.99963 −0.0727738
\(756\) 59.7785 2.17412
\(757\) −2.39216 −0.0869445 −0.0434722 0.999055i \(-0.513842\pi\)
−0.0434722 + 0.999055i \(0.513842\pi\)
\(758\) 5.83243 0.211843
\(759\) 1.61768 0.0587181
\(760\) 0.883612 0.0320520
\(761\) 14.9245 0.541013 0.270506 0.962718i \(-0.412809\pi\)
0.270506 + 0.962718i \(0.412809\pi\)
\(762\) −60.3424 −2.18597
\(763\) 34.2169 1.23874
\(764\) −0.0853752 −0.00308877
\(765\) −1.19069 −0.0430493
\(766\) −0.724711 −0.0261849
\(767\) −6.56837 −0.237170
\(768\) 10.0929 0.364195
\(769\) −26.9105 −0.970419 −0.485210 0.874398i \(-0.661257\pi\)
−0.485210 + 0.874398i \(0.661257\pi\)
\(770\) −21.9058 −0.789431
\(771\) 26.2015 0.943623
\(772\) −54.3782 −1.95711
\(773\) 48.3216 1.73801 0.869004 0.494805i \(-0.164761\pi\)
0.869004 + 0.494805i \(0.164761\pi\)
\(774\) −12.2927 −0.441854
\(775\) −10.8183 −0.388605
\(776\) −8.08182 −0.290120
\(777\) 13.7130 0.491952
\(778\) −6.59687 −0.236509
\(779\) −6.25508 −0.224111
\(780\) −21.1794 −0.758345
\(781\) 13.2234 0.473169
\(782\) 1.00640 0.0359888
\(783\) 20.3314 0.726586
\(784\) −38.6724 −1.38116
\(785\) 16.1349 0.575878
\(786\) 14.5748 0.519865
\(787\) −25.6110 −0.912932 −0.456466 0.889741i \(-0.650885\pi\)
−0.456466 + 0.889741i \(0.650885\pi\)
\(788\) 37.5553 1.33785
\(789\) −7.17421 −0.255409
\(790\) 1.64611 0.0585661
\(791\) −68.8956 −2.44965
\(792\) −2.49429 −0.0886307
\(793\) 65.8627 2.33885
\(794\) 35.6998 1.26694
\(795\) −11.9225 −0.422847
\(796\) −49.7522 −1.76342
\(797\) −37.6890 −1.33501 −0.667506 0.744605i \(-0.732638\pi\)
−0.667506 + 0.744605i \(0.732638\pi\)
\(798\) −12.0928 −0.428081
\(799\) 5.74698 0.203314
\(800\) 31.4336 1.11135
\(801\) 5.17558 0.182870
\(802\) 3.88151 0.137061
\(803\) 10.7587 0.379666
\(804\) 41.2967 1.45642
\(805\) −2.58708 −0.0911826
\(806\) −37.7462 −1.32955
\(807\) 10.1247 0.356407
\(808\) −5.04711 −0.177557
\(809\) 3.80843 0.133897 0.0669486 0.997756i \(-0.478674\pi\)
0.0669486 + 0.997756i \(0.478674\pi\)
\(810\) −7.27444 −0.255598
\(811\) 48.4893 1.70269 0.851344 0.524607i \(-0.175788\pi\)
0.851344 + 0.524607i \(0.175788\pi\)
\(812\) 38.7529 1.35996
\(813\) −1.66460 −0.0583801
\(814\) −11.1995 −0.392542
\(815\) −0.0655816 −0.00229722
\(816\) −3.39546 −0.118865
\(817\) 4.44636 0.155558
\(818\) −1.47658 −0.0516273
\(819\) −38.0068 −1.32807
\(820\) −15.7357 −0.549515
\(821\) −23.5608 −0.822277 −0.411138 0.911573i \(-0.634869\pi\)
−0.411138 + 0.911573i \(0.634869\pi\)
\(822\) 45.8492 1.59918
\(823\) −14.8910 −0.519069 −0.259535 0.965734i \(-0.583569\pi\)
−0.259535 + 0.965734i \(0.583569\pi\)
\(824\) 1.47778 0.0514810
\(825\) −11.3581 −0.395439
\(826\) −9.43464 −0.328273
\(827\) 33.9453 1.18039 0.590197 0.807259i \(-0.299050\pi\)
0.590197 + 0.807259i \(0.299050\pi\)
\(828\) −1.75963 −0.0611514
\(829\) 0.768540 0.0266925 0.0133462 0.999911i \(-0.495752\pi\)
0.0133462 + 0.999911i \(0.495752\pi\)
\(830\) 18.2821 0.634580
\(831\) −30.3774 −1.05378
\(832\) 70.2878 2.43679
\(833\) 10.9979 0.381056
\(834\) −4.67207 −0.161780
\(835\) 3.27147 0.113214
\(836\) 5.38923 0.186390
\(837\) −15.5217 −0.536509
\(838\) 65.0149 2.24590
\(839\) 14.6577 0.506041 0.253020 0.967461i \(-0.418576\pi\)
0.253020 + 0.967461i \(0.418576\pi\)
\(840\) −5.09281 −0.175719
\(841\) −15.8196 −0.545505
\(842\) −12.6867 −0.437212
\(843\) 2.80955 0.0967662
\(844\) 2.40214 0.0826849
\(845\) 30.5090 1.04954
\(846\) −18.4144 −0.633099
\(847\) 26.5138 0.911026
\(848\) 26.6302 0.914485
\(849\) −17.4014 −0.597216
\(850\) −7.06618 −0.242368
\(851\) −1.32266 −0.0453402
\(852\) 18.3639 0.629136
\(853\) −0.189713 −0.00649564 −0.00324782 0.999995i \(-0.501034\pi\)
−0.00324782 + 0.999995i \(0.501034\pi\)
\(854\) 94.6035 3.23726
\(855\) −1.37997 −0.0471939
\(856\) −10.6544 −0.364159
\(857\) 36.6495 1.25192 0.625961 0.779854i \(-0.284707\pi\)
0.625961 + 0.779854i \(0.284707\pi\)
\(858\) −39.6297 −1.35293
\(859\) −33.4942 −1.14281 −0.571403 0.820670i \(-0.693601\pi\)
−0.571403 + 0.820670i \(0.693601\pi\)
\(860\) 11.1856 0.381425
\(861\) 36.0519 1.22865
\(862\) 15.6074 0.531592
\(863\) 43.4514 1.47910 0.739551 0.673101i \(-0.235038\pi\)
0.739551 + 0.673101i \(0.235038\pi\)
\(864\) 45.0997 1.53432
\(865\) 6.27880 0.213486
\(866\) −15.7547 −0.535368
\(867\) −21.0840 −0.716052
\(868\) −29.5852 −1.00419
\(869\) 1.68074 0.0570151
\(870\) −10.3468 −0.350789
\(871\) −86.0343 −2.91516
\(872\) −6.49686 −0.220012
\(873\) 12.6217 0.427179
\(874\) 1.16639 0.0394536
\(875\) 41.4331 1.40069
\(876\) 14.9411 0.504812
\(877\) 21.5313 0.727060 0.363530 0.931582i \(-0.381571\pi\)
0.363530 + 0.931582i \(0.381571\pi\)
\(878\) −14.6216 −0.493455
\(879\) −27.2080 −0.917703
\(880\) −7.12858 −0.240304
\(881\) 33.6439 1.13349 0.566746 0.823892i \(-0.308202\pi\)
0.566746 + 0.823892i \(0.308202\pi\)
\(882\) −35.2393 −1.18657
\(883\) −40.5906 −1.36598 −0.682991 0.730427i \(-0.739321\pi\)
−0.682991 + 0.730427i \(0.739321\pi\)
\(884\) −13.4534 −0.452487
\(885\) 1.37455 0.0462050
\(886\) −5.55376 −0.186582
\(887\) 34.2297 1.14932 0.574660 0.818392i \(-0.305134\pi\)
0.574660 + 0.818392i \(0.305134\pi\)
\(888\) −2.60373 −0.0873754
\(889\) 98.5326 3.30468
\(890\) −8.63048 −0.289294
\(891\) −7.42745 −0.248829
\(892\) −35.0802 −1.17457
\(893\) 6.66058 0.222888
\(894\) −26.7081 −0.893251
\(895\) 14.7780 0.493974
\(896\) 29.3879 0.981781
\(897\) −4.68027 −0.156270
\(898\) −67.9509 −2.26755
\(899\) −10.0623 −0.335597
\(900\) 12.3548 0.411826
\(901\) −7.57329 −0.252303
\(902\) −29.4437 −0.980370
\(903\) −25.6271 −0.852818
\(904\) 13.0814 0.435081
\(905\) −25.8804 −0.860293
\(906\) −5.19610 −0.172629
\(907\) −43.2753 −1.43693 −0.718466 0.695562i \(-0.755155\pi\)
−0.718466 + 0.695562i \(0.755155\pi\)
\(908\) 21.6193 0.717463
\(909\) 7.88225 0.261438
\(910\) 63.3778 2.10095
\(911\) −55.7370 −1.84665 −0.923324 0.384021i \(-0.874539\pi\)
−0.923324 + 0.384021i \(0.874539\pi\)
\(912\) −3.93524 −0.130309
\(913\) 18.6666 0.617775
\(914\) −67.9205 −2.24661
\(915\) −13.7830 −0.455651
\(916\) 44.8372 1.48146
\(917\) −23.7991 −0.785914
\(918\) −10.1383 −0.334613
\(919\) 47.1638 1.55579 0.777895 0.628394i \(-0.216287\pi\)
0.777895 + 0.628394i \(0.216287\pi\)
\(920\) 0.491216 0.0161949
\(921\) −32.5122 −1.07131
\(922\) 10.0747 0.331794
\(923\) −38.2578 −1.25927
\(924\) −31.0615 −1.02185
\(925\) 9.28672 0.305345
\(926\) −48.8844 −1.60644
\(927\) −2.30791 −0.0758016
\(928\) 29.2370 0.959751
\(929\) 5.13009 0.168313 0.0841565 0.996453i \(-0.473180\pi\)
0.0841565 + 0.996453i \(0.473180\pi\)
\(930\) 7.89908 0.259021
\(931\) 12.7463 0.417742
\(932\) 1.86875 0.0612129
\(933\) −26.4939 −0.867370
\(934\) −49.7249 −1.62705
\(935\) 2.02728 0.0662991
\(936\) 7.21646 0.235877
\(937\) −14.1273 −0.461518 −0.230759 0.973011i \(-0.574121\pi\)
−0.230759 + 0.973011i \(0.574121\pi\)
\(938\) −123.577 −4.03495
\(939\) −16.9474 −0.553058
\(940\) 16.7558 0.546515
\(941\) −56.0024 −1.82563 −0.912814 0.408376i \(-0.866095\pi\)
−0.912814 + 0.408376i \(0.866095\pi\)
\(942\) 41.9270 1.36605
\(943\) −3.47731 −0.113237
\(944\) −3.07021 −0.0999270
\(945\) 26.0618 0.847789
\(946\) 20.9298 0.680487
\(947\) 19.0292 0.618365 0.309182 0.951003i \(-0.399945\pi\)
0.309182 + 0.951003i \(0.399945\pi\)
\(948\) 2.33411 0.0758085
\(949\) −31.1270 −1.01042
\(950\) −8.18949 −0.265702
\(951\) 7.48704 0.242784
\(952\) −3.23501 −0.104847
\(953\) 34.7501 1.12567 0.562833 0.826570i \(-0.309711\pi\)
0.562833 + 0.826570i \(0.309711\pi\)
\(954\) 24.2662 0.785647
\(955\) −0.0372212 −0.00120445
\(956\) 54.9512 1.77725
\(957\) −10.5644 −0.341499
\(958\) −44.7596 −1.44612
\(959\) −74.8668 −2.41758
\(960\) −14.7090 −0.474731
\(961\) −23.3181 −0.752196
\(962\) 32.4023 1.04469
\(963\) 16.6393 0.536195
\(964\) −22.8130 −0.734758
\(965\) −23.7074 −0.763168
\(966\) −6.72262 −0.216297
\(967\) 8.59986 0.276553 0.138276 0.990394i \(-0.455844\pi\)
0.138276 + 0.990394i \(0.455844\pi\)
\(968\) −5.03426 −0.161807
\(969\) 1.11913 0.0359517
\(970\) −21.0471 −0.675781
\(971\) 47.3606 1.51987 0.759937 0.649996i \(-0.225230\pi\)
0.759937 + 0.649996i \(0.225230\pi\)
\(972\) 30.0426 0.963617
\(973\) 7.62898 0.244574
\(974\) 9.18422 0.294281
\(975\) 32.8613 1.05240
\(976\) 30.7858 0.985429
\(977\) −15.9673 −0.510840 −0.255420 0.966830i \(-0.582214\pi\)
−0.255420 + 0.966830i \(0.582214\pi\)
\(978\) −0.170416 −0.00544931
\(979\) −8.81202 −0.281633
\(980\) 32.0655 1.02429
\(981\) 10.1464 0.323949
\(982\) −5.53051 −0.176486
\(983\) 56.6749 1.80765 0.903825 0.427903i \(-0.140748\pi\)
0.903825 + 0.427903i \(0.140748\pi\)
\(984\) −6.84528 −0.218219
\(985\) 16.3731 0.521689
\(986\) −6.57239 −0.209307
\(987\) −38.3891 −1.22194
\(988\) −15.5921 −0.496050
\(989\) 2.47181 0.0785990
\(990\) −6.49576 −0.206449
\(991\) −26.8142 −0.851782 −0.425891 0.904775i \(-0.640039\pi\)
−0.425891 + 0.904775i \(0.640039\pi\)
\(992\) −22.3205 −0.708677
\(993\) −44.9812 −1.42744
\(994\) −54.9525 −1.74299
\(995\) −21.6906 −0.687637
\(996\) 25.9232 0.821407
\(997\) −12.1263 −0.384043 −0.192021 0.981391i \(-0.561504\pi\)
−0.192021 + 0.981391i \(0.561504\pi\)
\(998\) 39.5336 1.25142
\(999\) 13.3242 0.421560
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 4009.2.a.c.1.62 71
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4009.2.a.c.1.62 71 1.1 even 1 trivial