Properties

Label 71.2.a.a.1.3
Level $71$
Weight $2$
Character 71.1
Self dual yes
Analytic conductor $0.567$
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] = [71,2,Mod(1,71)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(71, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("71.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 71.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: \(0.566937854351\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.91223\) of defining polynomial
Character \(\chi\) \(=\) 71.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.91223 q^{2} -1.91223 q^{3} +1.65662 q^{4} +3.25561 q^{5} -3.65662 q^{6} -3.82446 q^{7} -0.656620 q^{8} +0.656620 q^{9} +O(q^{10})\) \(q+1.91223 q^{2} -1.91223 q^{3} +1.65662 q^{4} +3.25561 q^{5} -3.65662 q^{6} -3.82446 q^{7} -0.656620 q^{8} +0.656620 q^{9} +6.22547 q^{10} +1.31324 q^{11} -3.16784 q^{12} -3.31324 q^{13} -7.31324 q^{14} -6.22547 q^{15} -4.56885 q^{16} +5.13770 q^{17} +1.25561 q^{18} +5.48108 q^{19} +5.39331 q^{20} +7.31324 q^{21} +2.51122 q^{22} -4.00000 q^{23} +1.25561 q^{24} +5.59899 q^{25} -6.33568 q^{26} +4.48108 q^{27} -6.33568 q^{28} +4.59899 q^{29} -11.9045 q^{30} +4.00000 q^{31} -7.42345 q^{32} -2.51122 q^{33} +9.82446 q^{34} -12.4509 q^{35} +1.08777 q^{36} -5.65662 q^{37} +10.4811 q^{38} +6.33568 q^{39} -2.13770 q^{40} -9.64892 q^{41} +13.9846 q^{42} +1.43115 q^{43} +2.17554 q^{44} +2.13770 q^{45} -7.64892 q^{46} +7.13770 q^{47} +8.73669 q^{48} +7.62648 q^{49} +10.7065 q^{50} -9.82446 q^{51} -5.48878 q^{52} -8.62648 q^{53} +8.56885 q^{54} +4.27540 q^{55} +2.51122 q^{56} -10.4811 q^{57} +8.79432 q^{58} -4.51122 q^{59} -10.3132 q^{60} -3.64892 q^{61} +7.64892 q^{62} -2.51122 q^{63} -5.05763 q^{64} -10.7866 q^{65} -4.80202 q^{66} -5.31324 q^{67} +8.51122 q^{68} +7.64892 q^{69} -23.8091 q^{70} +1.00000 q^{71} -0.431150 q^{72} +16.3933 q^{73} -10.8168 q^{74} -10.7065 q^{75} +9.08007 q^{76} -5.02243 q^{77} +12.1153 q^{78} +5.08007 q^{79} -14.8744 q^{80} -10.5387 q^{81} -18.4509 q^{82} -10.1678 q^{83} +12.1153 q^{84} +16.7263 q^{85} +2.73669 q^{86} -8.79432 q^{87} -0.862301 q^{88} -3.22547 q^{89} +4.08777 q^{90} +12.6714 q^{91} -6.62648 q^{92} -7.64892 q^{93} +13.6489 q^{94} +17.8442 q^{95} +14.1953 q^{96} +11.8245 q^{97} +14.5836 q^{98} +0.862301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + q^{3} + 3 q^{4} + 5 q^{5} - 9 q^{6} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + q^{3} + 3 q^{4} + 5 q^{5} - 9 q^{6} + 2 q^{7} + 8 q^{10} + 2 q^{12} - 6 q^{13} - 18 q^{14} - 8 q^{15} - 5 q^{16} - 2 q^{17} - q^{18} + q^{19} - 6 q^{20} + 18 q^{21} - 2 q^{22} - 12 q^{23} - q^{24} + 14 q^{25} + 4 q^{26} - 2 q^{27} + 4 q^{28} + 11 q^{29} - 4 q^{30} + 12 q^{31} - 6 q^{32} + 2 q^{33} + 16 q^{34} - 16 q^{35} + 10 q^{36} - 15 q^{37} + 16 q^{38} - 4 q^{39} + 11 q^{40} - 2 q^{41} - 8 q^{42} + 13 q^{43} + 20 q^{44} - 11 q^{45} + 4 q^{46} + 4 q^{47} + 6 q^{48} + 15 q^{49} + 6 q^{50} - 16 q^{51} - 26 q^{52} - 18 q^{53} + 17 q^{54} - 22 q^{55} - 2 q^{56} - 16 q^{57} + 7 q^{58} - 4 q^{59} - 27 q^{60} + 16 q^{61} - 4 q^{62} + 2 q^{63} - 16 q^{64} + 12 q^{65} - 20 q^{66} - 12 q^{67} + 16 q^{68} - 4 q^{69} - 8 q^{70} + 3 q^{71} - 10 q^{72} + 27 q^{73} + 6 q^{74} - 6 q^{75} + 9 q^{76} + 4 q^{77} + 38 q^{78} - 3 q^{79} - 7 q^{80} - 17 q^{81} - 34 q^{82} - 19 q^{83} + 38 q^{84} - 6 q^{85} - 12 q^{86} - 7 q^{87} - 20 q^{88} + q^{89} + 19 q^{90} - 8 q^{91} - 12 q^{92} + 4 q^{93} + 14 q^{94} + 10 q^{95} + 26 q^{96} + 22 q^{97} - 9 q^{98} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.91223 1.35215 0.676075 0.736833i \(-0.263679\pi\)
0.676075 + 0.736833i \(0.263679\pi\)
\(3\) −1.91223 −1.10403 −0.552013 0.833835i \(-0.686140\pi\)
−0.552013 + 0.833835i \(0.686140\pi\)
\(4\) 1.65662 0.828310
\(5\) 3.25561 1.45595 0.727976 0.685602i \(-0.240461\pi\)
0.727976 + 0.685602i \(0.240461\pi\)
\(6\) −3.65662 −1.49281
\(7\) −3.82446 −1.44551 −0.722755 0.691105i \(-0.757124\pi\)
−0.722755 + 0.691105i \(0.757124\pi\)
\(8\) −0.656620 −0.232150
\(9\) 0.656620 0.218873
\(10\) 6.22547 1.96867
\(11\) 1.31324 0.395957 0.197979 0.980206i \(-0.436562\pi\)
0.197979 + 0.980206i \(0.436562\pi\)
\(12\) −3.16784 −0.914476
\(13\) −3.31324 −0.918928 −0.459464 0.888196i \(-0.651958\pi\)
−0.459464 + 0.888196i \(0.651958\pi\)
\(14\) −7.31324 −1.95455
\(15\) −6.22547 −1.60741
\(16\) −4.56885 −1.14221
\(17\) 5.13770 1.24608 0.623038 0.782192i \(-0.285898\pi\)
0.623038 + 0.782192i \(0.285898\pi\)
\(18\) 1.25561 0.295950
\(19\) 5.48108 1.25745 0.628723 0.777629i \(-0.283578\pi\)
0.628723 + 0.777629i \(0.283578\pi\)
\(20\) 5.39331 1.20598
\(21\) 7.31324 1.59588
\(22\) 2.51122 0.535393
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.25561 0.256300
\(25\) 5.59899 1.11980
\(26\) −6.33568 −1.24253
\(27\) 4.48108 0.862384
\(28\) −6.33568 −1.19733
\(29\) 4.59899 0.854011 0.427005 0.904249i \(-0.359569\pi\)
0.427005 + 0.904249i \(0.359569\pi\)
\(30\) −11.9045 −2.17346
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −7.42345 −1.31229
\(33\) −2.51122 −0.437147
\(34\) 9.82446 1.68488
\(35\) −12.4509 −2.10459
\(36\) 1.08777 0.181295
\(37\) −5.65662 −0.929943 −0.464971 0.885326i \(-0.653935\pi\)
−0.464971 + 0.885326i \(0.653935\pi\)
\(38\) 10.4811 1.70026
\(39\) 6.33568 1.01452
\(40\) −2.13770 −0.338000
\(41\) −9.64892 −1.50691 −0.753454 0.657501i \(-0.771614\pi\)
−0.753454 + 0.657501i \(0.771614\pi\)
\(42\) 13.9846 2.15787
\(43\) 1.43115 0.218248 0.109124 0.994028i \(-0.465195\pi\)
0.109124 + 0.994028i \(0.465195\pi\)
\(44\) 2.17554 0.327975
\(45\) 2.13770 0.318669
\(46\) −7.64892 −1.12777
\(47\) 7.13770 1.04114 0.520570 0.853819i \(-0.325719\pi\)
0.520570 + 0.853819i \(0.325719\pi\)
\(48\) 8.73669 1.26103
\(49\) 7.62648 1.08950
\(50\) 10.7065 1.51413
\(51\) −9.82446 −1.37570
\(52\) −5.48878 −0.761157
\(53\) −8.62648 −1.18494 −0.592469 0.805593i \(-0.701847\pi\)
−0.592469 + 0.805593i \(0.701847\pi\)
\(54\) 8.56885 1.16607
\(55\) 4.27540 0.576495
\(56\) 2.51122 0.335576
\(57\) −10.4811 −1.38825
\(58\) 8.79432 1.15475
\(59\) −4.51122 −0.587310 −0.293655 0.955911i \(-0.594872\pi\)
−0.293655 + 0.955911i \(0.594872\pi\)
\(60\) −10.3132 −1.33143
\(61\) −3.64892 −0.467196 −0.233598 0.972333i \(-0.575050\pi\)
−0.233598 + 0.972333i \(0.575050\pi\)
\(62\) 7.64892 0.971413
\(63\) −2.51122 −0.316384
\(64\) −5.05763 −0.632204
\(65\) −10.7866 −1.33792
\(66\) −4.80202 −0.591088
\(67\) −5.31324 −0.649116 −0.324558 0.945866i \(-0.605215\pi\)
−0.324558 + 0.945866i \(0.605215\pi\)
\(68\) 8.51122 1.03214
\(69\) 7.64892 0.920821
\(70\) −23.8091 −2.84573
\(71\) 1.00000 0.118678
\(72\) −0.431150 −0.0508116
\(73\) 16.3933 1.91869 0.959346 0.282233i \(-0.0910752\pi\)
0.959346 + 0.282233i \(0.0910752\pi\)
\(74\) −10.8168 −1.25742
\(75\) −10.7065 −1.23629
\(76\) 9.08007 1.04156
\(77\) −5.02243 −0.572360
\(78\) 12.1153 1.37178
\(79\) 5.08007 0.571552 0.285776 0.958296i \(-0.407749\pi\)
0.285776 + 0.958296i \(0.407749\pi\)
\(80\) −14.8744 −1.66301
\(81\) −10.5387 −1.17097
\(82\) −18.4509 −2.03757
\(83\) −10.1678 −1.11607 −0.558033 0.829819i \(-0.688444\pi\)
−0.558033 + 0.829819i \(0.688444\pi\)
\(84\) 12.1153 1.32188
\(85\) 16.7263 1.81423
\(86\) 2.73669 0.295105
\(87\) −8.79432 −0.942850
\(88\) −0.862301 −0.0919216
\(89\) −3.22547 −0.341899 −0.170950 0.985280i \(-0.554684\pi\)
−0.170950 + 0.985280i \(0.554684\pi\)
\(90\) 4.08777 0.430889
\(91\) 12.6714 1.32832
\(92\) −6.62648 −0.690858
\(93\) −7.64892 −0.793156
\(94\) 13.6489 1.40778
\(95\) 17.8442 1.83078
\(96\) 14.1953 1.44880
\(97\) 11.8245 1.20059 0.600296 0.799778i \(-0.295049\pi\)
0.600296 + 0.799778i \(0.295049\pi\)
\(98\) 14.5836 1.47316
\(99\) 0.862301 0.0866645
\(100\) 9.27540 0.927540
\(101\) 2.34338 0.233175 0.116587 0.993180i \(-0.462804\pi\)
0.116587 + 0.993180i \(0.462804\pi\)
\(102\) −18.7866 −1.86015
\(103\) −16.1876 −1.59501 −0.797507 0.603309i \(-0.793848\pi\)
−0.797507 + 0.603309i \(0.793848\pi\)
\(104\) 2.17554 0.213329
\(105\) 23.8091 2.32353
\(106\) −16.4958 −1.60221
\(107\) 0.351083 0.0339405 0.0169703 0.999856i \(-0.494598\pi\)
0.0169703 + 0.999856i \(0.494598\pi\)
\(108\) 7.42345 0.714321
\(109\) 0.598988 0.0573727 0.0286863 0.999588i \(-0.490868\pi\)
0.0286863 + 0.999588i \(0.490868\pi\)
\(110\) 8.17554 0.779507
\(111\) 10.8168 1.02668
\(112\) 17.4734 1.65108
\(113\) 3.13770 0.295170 0.147585 0.989049i \(-0.452850\pi\)
0.147585 + 0.989049i \(0.452850\pi\)
\(114\) −20.0422 −1.87713
\(115\) −13.0224 −1.21435
\(116\) 7.61878 0.707386
\(117\) −2.17554 −0.201129
\(118\) −8.62648 −0.794132
\(119\) −19.6489 −1.80121
\(120\) 4.08777 0.373161
\(121\) −9.27540 −0.843218
\(122\) −6.97757 −0.631719
\(123\) 18.4509 1.66367
\(124\) 6.62648 0.595076
\(125\) 1.95007 0.174420
\(126\) −4.80202 −0.427798
\(127\) −4.33568 −0.384729 −0.192365 0.981324i \(-0.561616\pi\)
−0.192365 + 0.981324i \(0.561616\pi\)
\(128\) 5.17554 0.457458
\(129\) −2.73669 −0.240952
\(130\) −20.6265 −1.80906
\(131\) 14.3330 1.25228 0.626141 0.779710i \(-0.284633\pi\)
0.626141 + 0.779710i \(0.284633\pi\)
\(132\) −4.16013 −0.362093
\(133\) −20.9622 −1.81765
\(134\) −10.1601 −0.877702
\(135\) 14.5886 1.25559
\(136\) −3.37352 −0.289277
\(137\) 14.2754 1.21963 0.609815 0.792544i \(-0.291244\pi\)
0.609815 + 0.792544i \(0.291244\pi\)
\(138\) 14.6265 1.24509
\(139\) −4.45094 −0.377524 −0.188762 0.982023i \(-0.560447\pi\)
−0.188762 + 0.982023i \(0.560447\pi\)
\(140\) −20.6265 −1.74326
\(141\) −13.6489 −1.14945
\(142\) 1.91223 0.160471
\(143\) −4.35108 −0.363856
\(144\) −3.00000 −0.250000
\(145\) 14.9725 1.24340
\(146\) 31.3478 2.59436
\(147\) −14.5836 −1.20283
\(148\) −9.37087 −0.770281
\(149\) −8.45094 −0.692328 −0.346164 0.938174i \(-0.612516\pi\)
−0.346164 + 0.938174i \(0.612516\pi\)
\(150\) −20.4734 −1.67164
\(151\) 17.9122 1.45768 0.728838 0.684686i \(-0.240061\pi\)
0.728838 + 0.684686i \(0.240061\pi\)
\(152\) −3.59899 −0.291916
\(153\) 3.37352 0.272733
\(154\) −9.60405 −0.773916
\(155\) 13.0224 1.04599
\(156\) 10.4958 0.840337
\(157\) −5.97251 −0.476658 −0.238329 0.971184i \(-0.576600\pi\)
−0.238329 + 0.971184i \(0.576600\pi\)
\(158\) 9.71425 0.772824
\(159\) 16.4958 1.30820
\(160\) −24.1678 −1.91064
\(161\) 15.2978 1.20564
\(162\) −20.1524 −1.58332
\(163\) −11.3735 −0.890843 −0.445421 0.895321i \(-0.646946\pi\)
−0.445421 + 0.895321i \(0.646946\pi\)
\(164\) −15.9846 −1.24819
\(165\) −8.17554 −0.636465
\(166\) −19.4432 −1.50909
\(167\) 9.08007 0.702637 0.351318 0.936256i \(-0.385733\pi\)
0.351318 + 0.936256i \(0.385733\pi\)
\(168\) −4.80202 −0.370484
\(169\) −2.02243 −0.155572
\(170\) 31.9846 2.45311
\(171\) 3.59899 0.275222
\(172\) 2.37087 0.180777
\(173\) 1.64892 0.125365 0.0626824 0.998034i \(-0.480034\pi\)
0.0626824 + 0.998034i \(0.480034\pi\)
\(174\) −16.8168 −1.27487
\(175\) −21.4131 −1.61868
\(176\) −6.00000 −0.452267
\(177\) 8.62648 0.648406
\(178\) −6.16784 −0.462299
\(179\) 5.19533 0.388317 0.194159 0.980970i \(-0.437802\pi\)
0.194159 + 0.980970i \(0.437802\pi\)
\(180\) 3.54136 0.263957
\(181\) 14.0999 1.04803 0.524017 0.851708i \(-0.324433\pi\)
0.524017 + 0.851708i \(0.324433\pi\)
\(182\) 24.2305 1.79609
\(183\) 6.97757 0.515796
\(184\) 2.62648 0.193627
\(185\) −18.4157 −1.35395
\(186\) −14.6265 −1.07247
\(187\) 6.74704 0.493392
\(188\) 11.8245 0.862387
\(189\) −17.1377 −1.24658
\(190\) 34.1223 2.47549
\(191\) −6.98965 −0.505753 −0.252877 0.967499i \(-0.581377\pi\)
−0.252877 + 0.967499i \(0.581377\pi\)
\(192\) 9.67135 0.697970
\(193\) 7.93972 0.571514 0.285757 0.958302i \(-0.407755\pi\)
0.285757 + 0.958302i \(0.407755\pi\)
\(194\) 22.6111 1.62338
\(195\) 20.6265 1.47709
\(196\) 12.6342 0.902442
\(197\) −3.48878 −0.248565 −0.124283 0.992247i \(-0.539663\pi\)
−0.124283 + 0.992247i \(0.539663\pi\)
\(198\) 1.64892 0.117183
\(199\) 17.4811 1.23920 0.619600 0.784917i \(-0.287295\pi\)
0.619600 + 0.784917i \(0.287295\pi\)
\(200\) −3.67641 −0.259961
\(201\) 10.1601 0.716641
\(202\) 4.48108 0.315288
\(203\) −17.5886 −1.23448
\(204\) −16.2754 −1.13951
\(205\) −31.4131 −2.19399
\(206\) −30.9545 −2.15670
\(207\) −2.62648 −0.182553
\(208\) 15.1377 1.04961
\(209\) 7.19798 0.497894
\(210\) 45.5284 3.14176
\(211\) −4.23582 −0.291606 −0.145803 0.989314i \(-0.546577\pi\)
−0.145803 + 0.989314i \(0.546577\pi\)
\(212\) −14.2908 −0.981497
\(213\) −1.91223 −0.131024
\(214\) 0.671352 0.0458927
\(215\) 4.65927 0.317759
\(216\) −2.94237 −0.200203
\(217\) −15.2978 −1.03848
\(218\) 1.14540 0.0775765
\(219\) −31.3478 −2.11829
\(220\) 7.08271 0.477516
\(221\) −17.0224 −1.14505
\(222\) 20.6841 1.38823
\(223\) −19.5259 −1.30755 −0.653777 0.756687i \(-0.726817\pi\)
−0.653777 + 0.756687i \(0.726817\pi\)
\(224\) 28.3907 1.89693
\(225\) 3.67641 0.245094
\(226\) 6.00000 0.399114
\(227\) −20.0396 −1.33007 −0.665037 0.746811i \(-0.731584\pi\)
−0.665037 + 0.746811i \(0.731584\pi\)
\(228\) −17.3632 −1.14990
\(229\) −1.65156 −0.109138 −0.0545691 0.998510i \(-0.517379\pi\)
−0.0545691 + 0.998510i \(0.517379\pi\)
\(230\) −24.9019 −1.64198
\(231\) 9.60405 0.631900
\(232\) −3.01979 −0.198259
\(233\) 23.5809 1.54484 0.772419 0.635113i \(-0.219046\pi\)
0.772419 + 0.635113i \(0.219046\pi\)
\(234\) −4.16013 −0.271956
\(235\) 23.2376 1.51585
\(236\) −7.47338 −0.486475
\(237\) −9.71425 −0.631009
\(238\) −37.5732 −2.43551
\(239\) 0.571495 0.0369669 0.0184835 0.999829i \(-0.494116\pi\)
0.0184835 + 0.999829i \(0.494116\pi\)
\(240\) 28.4432 1.83600
\(241\) 25.4734 1.64088 0.820442 0.571730i \(-0.193727\pi\)
0.820442 + 0.571730i \(0.193727\pi\)
\(242\) −17.7367 −1.14016
\(243\) 6.70919 0.430395
\(244\) −6.04487 −0.386983
\(245\) 24.8288 1.58626
\(246\) 35.2824 2.24953
\(247\) −18.1601 −1.15550
\(248\) −2.62648 −0.166782
\(249\) 19.4432 1.23217
\(250\) 3.72898 0.235842
\(251\) 20.1076 1.26918 0.634589 0.772850i \(-0.281169\pi\)
0.634589 + 0.772850i \(0.281169\pi\)
\(252\) −4.16013 −0.262064
\(253\) −5.25296 −0.330251
\(254\) −8.29081 −0.520211
\(255\) −31.9846 −2.00295
\(256\) 20.0121 1.25076
\(257\) −24.2754 −1.51426 −0.757129 0.653266i \(-0.773399\pi\)
−0.757129 + 0.653266i \(0.773399\pi\)
\(258\) −5.23317 −0.325803
\(259\) 21.6335 1.34424
\(260\) −17.8693 −1.10821
\(261\) 3.01979 0.186920
\(262\) 27.4080 1.69327
\(263\) 9.54641 0.588657 0.294329 0.955704i \(-0.404904\pi\)
0.294329 + 0.955704i \(0.404904\pi\)
\(264\) 1.64892 0.101484
\(265\) −28.0844 −1.72521
\(266\) −40.0844 −2.45774
\(267\) 6.16784 0.377466
\(268\) −8.80202 −0.537669
\(269\) −11.8847 −0.724625 −0.362313 0.932057i \(-0.618013\pi\)
−0.362313 + 0.932057i \(0.618013\pi\)
\(270\) 27.8968 1.69775
\(271\) −3.51628 −0.213599 −0.106799 0.994281i \(-0.534060\pi\)
−0.106799 + 0.994281i \(0.534060\pi\)
\(272\) −23.4734 −1.42328
\(273\) −24.2305 −1.46650
\(274\) 27.2978 1.64912
\(275\) 7.35282 0.443392
\(276\) 12.6714 0.762726
\(277\) 5.99230 0.360042 0.180021 0.983663i \(-0.442383\pi\)
0.180021 + 0.983663i \(0.442383\pi\)
\(278\) −8.51122 −0.510469
\(279\) 2.62648 0.157243
\(280\) 8.17554 0.488582
\(281\) −6.51122 −0.388427 −0.194213 0.980959i \(-0.562215\pi\)
−0.194213 + 0.980959i \(0.562215\pi\)
\(282\) −26.0999 −1.55422
\(283\) 11.4734 0.682021 0.341011 0.940059i \(-0.389231\pi\)
0.341011 + 0.940059i \(0.389231\pi\)
\(284\) 1.65662 0.0983023
\(285\) −34.1223 −2.02123
\(286\) −8.32027 −0.491988
\(287\) 36.9019 2.17825
\(288\) −4.87439 −0.287226
\(289\) 9.39595 0.552703
\(290\) 28.6309 1.68126
\(291\) −22.6111 −1.32548
\(292\) 27.1575 1.58927
\(293\) −20.6265 −1.20501 −0.602506 0.798114i \(-0.705831\pi\)
−0.602506 + 0.798114i \(0.705831\pi\)
\(294\) −27.8871 −1.62641
\(295\) −14.6868 −0.855096
\(296\) 3.71425 0.215887
\(297\) 5.88474 0.341467
\(298\) −16.1601 −0.936131
\(299\) 13.2530 0.766439
\(300\) −17.7367 −1.02403
\(301\) −5.47338 −0.315480
\(302\) 34.2523 1.97100
\(303\) −4.48108 −0.257431
\(304\) −25.0422 −1.43627
\(305\) −11.8794 −0.680215
\(306\) 6.45094 0.368776
\(307\) 11.8245 0.674857 0.337429 0.941351i \(-0.390443\pi\)
0.337429 + 0.941351i \(0.390443\pi\)
\(308\) −8.32027 −0.474091
\(309\) 30.9545 1.76094
\(310\) 24.9019 1.41433
\(311\) −18.3830 −1.04240 −0.521201 0.853434i \(-0.674516\pi\)
−0.521201 + 0.853434i \(0.674516\pi\)
\(312\) −4.16013 −0.235521
\(313\) −5.30554 −0.299887 −0.149943 0.988695i \(-0.547909\pi\)
−0.149943 + 0.988695i \(0.547909\pi\)
\(314\) −11.4208 −0.644513
\(315\) −8.17554 −0.460640
\(316\) 8.41574 0.473423
\(317\) −5.88474 −0.330520 −0.165260 0.986250i \(-0.552846\pi\)
−0.165260 + 0.986250i \(0.552846\pi\)
\(318\) 31.5438 1.76889
\(319\) 6.03958 0.338152
\(320\) −16.4657 −0.920459
\(321\) −0.671352 −0.0374712
\(322\) 29.2530 1.63020
\(323\) 28.1601 1.56687
\(324\) −17.4586 −0.969925
\(325\) −18.5508 −1.02901
\(326\) −21.7488 −1.20455
\(327\) −1.14540 −0.0633409
\(328\) 6.33568 0.349829
\(329\) −27.2978 −1.50498
\(330\) −15.6335 −0.860596
\(331\) 12.7316 0.699794 0.349897 0.936788i \(-0.386217\pi\)
0.349897 + 0.936788i \(0.386217\pi\)
\(332\) −16.8442 −0.924448
\(333\) −3.71425 −0.203540
\(334\) 17.3632 0.950070
\(335\) −17.2978 −0.945082
\(336\) −33.4131 −1.82283
\(337\) 28.7712 1.56727 0.783634 0.621223i \(-0.213364\pi\)
0.783634 + 0.621223i \(0.213364\pi\)
\(338\) −3.86736 −0.210357
\(339\) −6.00000 −0.325875
\(340\) 27.7092 1.50274
\(341\) 5.25296 0.284464
\(342\) 6.88209 0.372141
\(343\) −2.39595 −0.129369
\(344\) −0.939723 −0.0506664
\(345\) 24.9019 1.34067
\(346\) 3.15311 0.169512
\(347\) 1.02243 0.0548872 0.0274436 0.999623i \(-0.491263\pi\)
0.0274436 + 0.999623i \(0.491263\pi\)
\(348\) −14.5688 −0.780972
\(349\) 2.46635 0.132021 0.0660103 0.997819i \(-0.478973\pi\)
0.0660103 + 0.997819i \(0.478973\pi\)
\(350\) −40.9468 −2.18870
\(351\) −14.8469 −0.792469
\(352\) −9.74877 −0.519611
\(353\) 8.74175 0.465276 0.232638 0.972563i \(-0.425264\pi\)
0.232638 + 0.972563i \(0.425264\pi\)
\(354\) 16.4958 0.876742
\(355\) 3.25561 0.172790
\(356\) −5.34338 −0.283199
\(357\) 37.5732 1.98859
\(358\) 9.93466 0.525063
\(359\) −7.59128 −0.400653 −0.200326 0.979729i \(-0.564200\pi\)
−0.200326 + 0.979729i \(0.564200\pi\)
\(360\) −1.40366 −0.0739792
\(361\) 11.0422 0.581170
\(362\) 26.9622 1.41710
\(363\) 17.7367 0.930935
\(364\) 20.9916 1.10026
\(365\) 53.3702 2.79352
\(366\) 13.3427 0.697434
\(367\) 30.1025 1.57134 0.785669 0.618647i \(-0.212319\pi\)
0.785669 + 0.618647i \(0.212319\pi\)
\(368\) 18.2754 0.952671
\(369\) −6.33568 −0.329822
\(370\) −35.2151 −1.83075
\(371\) 32.9916 1.71284
\(372\) −12.6714 −0.656979
\(373\) 8.39331 0.434589 0.217295 0.976106i \(-0.430277\pi\)
0.217295 + 0.976106i \(0.430277\pi\)
\(374\) 12.9019 0.667140
\(375\) −3.72898 −0.192564
\(376\) −4.68676 −0.241701
\(377\) −15.2376 −0.784774
\(378\) −32.7712 −1.68557
\(379\) −10.4389 −0.536208 −0.268104 0.963390i \(-0.586397\pi\)
−0.268104 + 0.963390i \(0.586397\pi\)
\(380\) 29.5611 1.51645
\(381\) 8.29081 0.424751
\(382\) −13.3658 −0.683855
\(383\) −4.97757 −0.254342 −0.127171 0.991881i \(-0.540590\pi\)
−0.127171 + 0.991881i \(0.540590\pi\)
\(384\) −9.89682 −0.505045
\(385\) −16.3511 −0.833328
\(386\) 15.1826 0.772772
\(387\) 0.939723 0.0477688
\(388\) 19.5886 0.994462
\(389\) −15.3528 −0.778419 −0.389209 0.921149i \(-0.627252\pi\)
−0.389209 + 0.921149i \(0.627252\pi\)
\(390\) 39.4426 1.99725
\(391\) −20.5508 −1.03930
\(392\) −5.00770 −0.252927
\(393\) −27.4080 −1.38255
\(394\) −6.67135 −0.336098
\(395\) 16.5387 0.832153
\(396\) 1.42851 0.0717851
\(397\) −3.93972 −0.197729 −0.0988645 0.995101i \(-0.531521\pi\)
−0.0988645 + 0.995101i \(0.531521\pi\)
\(398\) 33.4278 1.67559
\(399\) 40.0844 2.00673
\(400\) −25.5809 −1.27905
\(401\) −11.4734 −0.572953 −0.286477 0.958087i \(-0.592484\pi\)
−0.286477 + 0.958087i \(0.592484\pi\)
\(402\) 19.4285 0.969006
\(403\) −13.2530 −0.660177
\(404\) 3.88209 0.193141
\(405\) −34.3099 −1.70487
\(406\) −33.6335 −1.66920
\(407\) −7.42851 −0.368217
\(408\) 6.45094 0.319369
\(409\) −24.6232 −1.21754 −0.608768 0.793348i \(-0.708336\pi\)
−0.608768 + 0.793348i \(0.708336\pi\)
\(410\) −60.0690 −2.96660
\(411\) −27.2978 −1.34650
\(412\) −26.8168 −1.32117
\(413\) 17.2530 0.848963
\(414\) −5.02243 −0.246839
\(415\) −33.1025 −1.62494
\(416\) 24.5957 1.20590
\(417\) 8.51122 0.416796
\(418\) 13.7642 0.673228
\(419\) −5.06534 −0.247458 −0.123729 0.992316i \(-0.539485\pi\)
−0.123729 + 0.992316i \(0.539485\pi\)
\(420\) 39.4426 1.92460
\(421\) −22.0396 −1.07414 −0.537072 0.843537i \(-0.680470\pi\)
−0.537072 + 0.843537i \(0.680470\pi\)
\(422\) −8.09986 −0.394295
\(423\) 4.68676 0.227878
\(424\) 5.66432 0.275084
\(425\) 28.7659 1.39535
\(426\) −3.65662 −0.177164
\(427\) 13.9551 0.675336
\(428\) 0.581612 0.0281133
\(429\) 8.32027 0.401706
\(430\) 8.90958 0.429658
\(431\) −16.4278 −0.791301 −0.395650 0.918401i \(-0.629481\pi\)
−0.395650 + 0.918401i \(0.629481\pi\)
\(432\) −20.4734 −0.985026
\(433\) 33.6885 1.61897 0.809483 0.587143i \(-0.199748\pi\)
0.809483 + 0.587143i \(0.199748\pi\)
\(434\) −29.2530 −1.40419
\(435\) −28.6309 −1.37274
\(436\) 0.992296 0.0475224
\(437\) −21.9243 −1.04878
\(438\) −59.9441 −2.86424
\(439\) 9.70390 0.463142 0.231571 0.972818i \(-0.425613\pi\)
0.231571 + 0.972818i \(0.425613\pi\)
\(440\) −2.80731 −0.133833
\(441\) 5.00770 0.238462
\(442\) −32.5508 −1.54828
\(443\) −22.1601 −1.05286 −0.526430 0.850219i \(-0.676470\pi\)
−0.526430 + 0.850219i \(0.676470\pi\)
\(444\) 17.9193 0.850410
\(445\) −10.5009 −0.497789
\(446\) −37.3381 −1.76801
\(447\) 16.1601 0.764348
\(448\) 19.3427 0.913857
\(449\) −29.0224 −1.36965 −0.684827 0.728706i \(-0.740122\pi\)
−0.684827 + 0.728706i \(0.740122\pi\)
\(450\) 7.03014 0.331404
\(451\) −12.6714 −0.596671
\(452\) 5.19798 0.244492
\(453\) −34.2523 −1.60931
\(454\) −38.3203 −1.79846
\(455\) 41.2530 1.93397
\(456\) 6.88209 0.322283
\(457\) 19.5491 0.914466 0.457233 0.889347i \(-0.348840\pi\)
0.457233 + 0.889347i \(0.348840\pi\)
\(458\) −3.15817 −0.147571
\(459\) 23.0224 1.07460
\(460\) −21.5732 −1.00586
\(461\) 6.04487 0.281538 0.140769 0.990042i \(-0.455043\pi\)
0.140769 + 0.990042i \(0.455043\pi\)
\(462\) 18.3651 0.854424
\(463\) −21.1300 −0.981994 −0.490997 0.871161i \(-0.663367\pi\)
−0.490997 + 0.871161i \(0.663367\pi\)
\(464\) −21.0121 −0.975462
\(465\) −24.9019 −1.15480
\(466\) 45.0922 2.08885
\(467\) 40.8262 1.88921 0.944606 0.328208i \(-0.106445\pi\)
0.944606 + 0.328208i \(0.106445\pi\)
\(468\) −3.60405 −0.166597
\(469\) 20.3203 0.938303
\(470\) 44.4355 2.04966
\(471\) 11.4208 0.526243
\(472\) 2.96216 0.136344
\(473\) 1.87945 0.0864170
\(474\) −18.5759 −0.853218
\(475\) 30.6885 1.40808
\(476\) −32.5508 −1.49196
\(477\) −5.66432 −0.259352
\(478\) 1.09283 0.0499848
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 46.2144 2.10939
\(481\) 18.7417 0.854550
\(482\) 48.7109 2.21872
\(483\) −29.2530 −1.33106
\(484\) −15.3658 −0.698446
\(485\) 38.4958 1.74800
\(486\) 12.8295 0.581959
\(487\) 10.9622 0.496743 0.248371 0.968665i \(-0.420105\pi\)
0.248371 + 0.968665i \(0.420105\pi\)
\(488\) 2.39595 0.108460
\(489\) 21.7488 0.983514
\(490\) 47.4784 2.14486
\(491\) 23.4734 1.05934 0.529669 0.848204i \(-0.322316\pi\)
0.529669 + 0.848204i \(0.322316\pi\)
\(492\) 30.5662 1.37803
\(493\) 23.6282 1.06416
\(494\) −34.7263 −1.56241
\(495\) 2.80731 0.126179
\(496\) −18.2754 −0.820590
\(497\) −3.82446 −0.171550
\(498\) 37.1799 1.66607
\(499\) 7.11021 0.318297 0.159148 0.987255i \(-0.449125\pi\)
0.159148 + 0.987255i \(0.449125\pi\)
\(500\) 3.23053 0.144474
\(501\) −17.3632 −0.775729
\(502\) 38.4503 1.71612
\(503\) −31.8365 −1.41952 −0.709761 0.704443i \(-0.751197\pi\)
−0.709761 + 0.704443i \(0.751197\pi\)
\(504\) 1.64892 0.0734486
\(505\) 7.62913 0.339492
\(506\) −10.0449 −0.446549
\(507\) 3.86736 0.171755
\(508\) −7.18257 −0.318675
\(509\) −11.0224 −0.488561 −0.244280 0.969705i \(-0.578552\pi\)
−0.244280 + 0.969705i \(0.578552\pi\)
\(510\) −61.1619 −2.70829
\(511\) −62.6955 −2.77349
\(512\) 27.9166 1.23375
\(513\) 24.5611 1.08440
\(514\) −46.4201 −2.04750
\(515\) −52.7006 −2.32227
\(516\) −4.53365 −0.199583
\(517\) 9.37352 0.412247
\(518\) 41.3682 1.81762
\(519\) −3.15311 −0.138406
\(520\) 7.08271 0.310597
\(521\) −29.4105 −1.28850 −0.644248 0.764817i \(-0.722829\pi\)
−0.644248 + 0.764817i \(0.722829\pi\)
\(522\) 5.77453 0.252744
\(523\) −20.8623 −0.912245 −0.456122 0.889917i \(-0.650762\pi\)
−0.456122 + 0.889917i \(0.650762\pi\)
\(524\) 23.7444 1.03728
\(525\) 40.9468 1.78706
\(526\) 18.2549 0.795953
\(527\) 20.5508 0.895207
\(528\) 11.4734 0.499315
\(529\) −7.00000 −0.304348
\(530\) −53.7039 −2.33275
\(531\) −2.96216 −0.128547
\(532\) −34.7263 −1.50558
\(533\) 31.9692 1.38474
\(534\) 11.7943 0.510390
\(535\) 1.14299 0.0494158
\(536\) 3.48878 0.150692
\(537\) −9.93466 −0.428712
\(538\) −22.7263 −0.979802
\(539\) 10.0154 0.431394
\(540\) 24.1678 1.04002
\(541\) 14.7417 0.633797 0.316899 0.948459i \(-0.397359\pi\)
0.316899 + 0.948459i \(0.397359\pi\)
\(542\) −6.72393 −0.288817
\(543\) −26.9622 −1.15706
\(544\) −38.1394 −1.63521
\(545\) 1.95007 0.0835319
\(546\) −46.3343 −1.98293
\(547\) −26.5189 −1.13387 −0.566934 0.823763i \(-0.691870\pi\)
−0.566934 + 0.823763i \(0.691870\pi\)
\(548\) 23.6489 1.01023
\(549\) −2.39595 −0.102257
\(550\) 14.0603 0.599532
\(551\) 25.2074 1.07387
\(552\) −5.02243 −0.213769
\(553\) −19.4285 −0.826184
\(554\) 11.4586 0.486831
\(555\) 35.2151 1.49480
\(556\) −7.37352 −0.312707
\(557\) 44.1274 1.86974 0.934868 0.354996i \(-0.115518\pi\)
0.934868 + 0.354996i \(0.115518\pi\)
\(558\) 5.02243 0.212617
\(559\) −4.74175 −0.200554
\(560\) 56.8865 2.40389
\(561\) −12.9019 −0.544718
\(562\) −12.4509 −0.525211
\(563\) 4.84689 0.204272 0.102136 0.994770i \(-0.467432\pi\)
0.102136 + 0.994770i \(0.467432\pi\)
\(564\) −22.6111 −0.952098
\(565\) 10.2151 0.429753
\(566\) 21.9397 0.922195
\(567\) 40.3049 1.69265
\(568\) −0.656620 −0.0275512
\(569\) −37.9665 −1.59164 −0.795820 0.605533i \(-0.792960\pi\)
−0.795820 + 0.605533i \(0.792960\pi\)
\(570\) −65.2496 −2.73301
\(571\) −20.7789 −0.869570 −0.434785 0.900534i \(-0.643176\pi\)
−0.434785 + 0.900534i \(0.643176\pi\)
\(572\) −7.20809 −0.301386
\(573\) 13.3658 0.558365
\(574\) 70.5649 2.94532
\(575\) −22.3960 −0.933976
\(576\) −3.32094 −0.138373
\(577\) 37.9665 1.58057 0.790284 0.612741i \(-0.209933\pi\)
0.790284 + 0.612741i \(0.209933\pi\)
\(578\) 17.9672 0.747338
\(579\) −15.1826 −0.630966
\(580\) 24.8038 1.02992
\(581\) 38.8865 1.61328
\(582\) −43.2376 −1.79225
\(583\) −11.3286 −0.469185
\(584\) −10.7642 −0.445425
\(585\) −7.08271 −0.292834
\(586\) −39.4426 −1.62936
\(587\) −9.03452 −0.372895 −0.186447 0.982465i \(-0.559697\pi\)
−0.186447 + 0.982465i \(0.559697\pi\)
\(588\) −24.1595 −0.996319
\(589\) 21.9243 0.903376
\(590\) −28.0844 −1.15622
\(591\) 6.67135 0.274423
\(592\) 25.8442 1.06219
\(593\) 9.44588 0.387896 0.193948 0.981012i \(-0.437871\pi\)
0.193948 + 0.981012i \(0.437871\pi\)
\(594\) 11.2530 0.461715
\(595\) −63.9692 −2.62248
\(596\) −14.0000 −0.573462
\(597\) −33.4278 −1.36811
\(598\) 25.3427 1.03634
\(599\) 39.9089 1.63063 0.815317 0.579015i \(-0.196563\pi\)
0.815317 + 0.579015i \(0.196563\pi\)
\(600\) 7.03014 0.287004
\(601\) 45.5130 1.85651 0.928256 0.371942i \(-0.121308\pi\)
0.928256 + 0.371942i \(0.121308\pi\)
\(602\) −10.4663 −0.426576
\(603\) −3.48878 −0.142074
\(604\) 29.6738 1.20741
\(605\) −30.1971 −1.22769
\(606\) −8.56885 −0.348086
\(607\) −40.4509 −1.64185 −0.820927 0.571034i \(-0.806542\pi\)
−0.820927 + 0.571034i \(0.806542\pi\)
\(608\) −40.6885 −1.65014
\(609\) 33.6335 1.36290
\(610\) −22.7162 −0.919753
\(611\) −23.6489 −0.956733
\(612\) 5.58864 0.225907
\(613\) −22.4683 −0.907487 −0.453743 0.891132i \(-0.649912\pi\)
−0.453743 + 0.891132i \(0.649912\pi\)
\(614\) 22.6111 0.912509
\(615\) 60.0690 2.42222
\(616\) 3.29783 0.132873
\(617\) −3.95778 −0.159334 −0.0796670 0.996822i \(-0.525386\pi\)
−0.0796670 + 0.996822i \(0.525386\pi\)
\(618\) 59.1920 2.38105
\(619\) −4.80202 −0.193010 −0.0965048 0.995333i \(-0.530766\pi\)
−0.0965048 + 0.995333i \(0.530766\pi\)
\(620\) 21.5732 0.866402
\(621\) −17.9243 −0.719278
\(622\) −35.1524 −1.40948
\(623\) 12.3357 0.494218
\(624\) −28.9468 −1.15880
\(625\) −21.6463 −0.865851
\(626\) −10.1454 −0.405492
\(627\) −13.7642 −0.549688
\(628\) −9.89418 −0.394821
\(629\) −29.0620 −1.15878
\(630\) −15.6335 −0.622854
\(631\) −32.0449 −1.27569 −0.637843 0.770166i \(-0.720173\pi\)
−0.637843 + 0.770166i \(0.720173\pi\)
\(632\) −3.33568 −0.132686
\(633\) 8.09986 0.321940
\(634\) −11.2530 −0.446912
\(635\) −14.1153 −0.560147
\(636\) 27.3273 1.08360
\(637\) −25.2684 −1.00117
\(638\) 11.5491 0.457232
\(639\) 0.656620 0.0259755
\(640\) 16.8495 0.666036
\(641\) −40.2677 −1.59048 −0.795239 0.606296i \(-0.792655\pi\)
−0.795239 + 0.606296i \(0.792655\pi\)
\(642\) −1.28378 −0.0506667
\(643\) 12.6714 0.499709 0.249855 0.968283i \(-0.419617\pi\)
0.249855 + 0.968283i \(0.419617\pi\)
\(644\) 25.3427 0.998642
\(645\) −8.90958 −0.350815
\(646\) 53.8486 2.11865
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 6.91993 0.271841
\(649\) −5.92432 −0.232550
\(650\) −35.4734 −1.39138
\(651\) 29.2530 1.14651
\(652\) −18.8416 −0.737894
\(653\) 31.6643 1.23912 0.619560 0.784949i \(-0.287311\pi\)
0.619560 + 0.784949i \(0.287311\pi\)
\(654\) −2.19027 −0.0856464
\(655\) 46.6627 1.82326
\(656\) 44.0844 1.72121
\(657\) 10.7642 0.419951
\(658\) −52.1997 −2.03496
\(659\) −38.0570 −1.48249 −0.741244 0.671235i \(-0.765764\pi\)
−0.741244 + 0.671235i \(0.765764\pi\)
\(660\) −13.5438 −0.527191
\(661\) 41.1672 1.60122 0.800609 0.599188i \(-0.204510\pi\)
0.800609 + 0.599188i \(0.204510\pi\)
\(662\) 24.3458 0.946226
\(663\) 32.5508 1.26417
\(664\) 6.67641 0.259095
\(665\) −68.2446 −2.64641
\(666\) −7.10250 −0.275216
\(667\) −18.3960 −0.712294
\(668\) 15.0422 0.582001
\(669\) 37.3381 1.44357
\(670\) −33.0774 −1.27789
\(671\) −4.79191 −0.184989
\(672\) −54.2895 −2.09426
\(673\) 14.9622 0.576749 0.288374 0.957518i \(-0.406885\pi\)
0.288374 + 0.957518i \(0.406885\pi\)
\(674\) 55.0171 2.11918
\(675\) 25.0895 0.965696
\(676\) −3.35041 −0.128862
\(677\) −39.5457 −1.51987 −0.759933 0.650001i \(-0.774768\pi\)
−0.759933 + 0.650001i \(0.774768\pi\)
\(678\) −11.4734 −0.440632
\(679\) −45.2221 −1.73547
\(680\) −10.9829 −0.421173
\(681\) 38.3203 1.46844
\(682\) 10.0449 0.384638
\(683\) −25.2925 −0.967792 −0.483896 0.875125i \(-0.660779\pi\)
−0.483896 + 0.875125i \(0.660779\pi\)
\(684\) 5.96216 0.227969
\(685\) 46.4751 1.77572
\(686\) −4.58161 −0.174927
\(687\) 3.15817 0.120492
\(688\) −6.53871 −0.249286
\(689\) 28.5816 1.08887
\(690\) 47.6181 1.81279
\(691\) −12.2600 −0.466392 −0.233196 0.972430i \(-0.574918\pi\)
−0.233196 + 0.972430i \(0.574918\pi\)
\(692\) 2.73163 0.103841
\(693\) −3.29783 −0.125274
\(694\) 1.95513 0.0742157
\(695\) −14.4905 −0.549657
\(696\) 5.77453 0.218883
\(697\) −49.5732 −1.87772
\(698\) 4.71622 0.178512
\(699\) −45.0922 −1.70554
\(700\) −35.4734 −1.34077
\(701\) −7.83987 −0.296108 −0.148054 0.988979i \(-0.547301\pi\)
−0.148054 + 0.988979i \(0.547301\pi\)
\(702\) −28.3907 −1.07154
\(703\) −31.0044 −1.16935
\(704\) −6.64189 −0.250326
\(705\) −44.4355 −1.67354
\(706\) 16.7162 0.629123
\(707\) −8.96216 −0.337057
\(708\) 14.2908 0.537081
\(709\) −52.6352 −1.97676 −0.988379 0.152009i \(-0.951426\pi\)
−0.988379 + 0.152009i \(0.951426\pi\)
\(710\) 6.22547 0.233638
\(711\) 3.33568 0.125098
\(712\) 2.11791 0.0793720
\(713\) −16.0000 −0.599205
\(714\) 71.8486 2.68887
\(715\) −14.1654 −0.529757
\(716\) 8.60669 0.321647
\(717\) −1.09283 −0.0408125
\(718\) −14.5163 −0.541743
\(719\) 1.71161 0.0638322 0.0319161 0.999491i \(-0.489839\pi\)
0.0319161 + 0.999491i \(0.489839\pi\)
\(720\) −9.76683 −0.363988
\(721\) 61.9089 2.30561
\(722\) 21.1153 0.785829
\(723\) −48.7109 −1.81158
\(724\) 23.3581 0.868097
\(725\) 25.7497 0.956319
\(726\) 33.9166 1.25876
\(727\) 15.0378 0.557723 0.278861 0.960331i \(-0.410043\pi\)
0.278861 + 0.960331i \(0.410043\pi\)
\(728\) −8.32027 −0.308370
\(729\) 18.7866 0.695801
\(730\) 102.056 3.77726
\(731\) 7.35282 0.271954
\(732\) 11.5592 0.427239
\(733\) 45.7334 1.68920 0.844600 0.535398i \(-0.179838\pi\)
0.844600 + 0.535398i \(0.179838\pi\)
\(734\) 57.5629 2.12468
\(735\) −47.4784 −1.75127
\(736\) 29.6938 1.09453
\(737\) −6.97757 −0.257022
\(738\) −12.1153 −0.445969
\(739\) −31.5284 −1.15979 −0.579895 0.814691i \(-0.696906\pi\)
−0.579895 + 0.814691i \(0.696906\pi\)
\(740\) −30.5079 −1.12149
\(741\) 34.7263 1.27570
\(742\) 63.0875 2.31602
\(743\) 0.471638 0.0173027 0.00865136 0.999963i \(-0.497246\pi\)
0.00865136 + 0.999963i \(0.497246\pi\)
\(744\) 5.02243 0.184131
\(745\) −27.5130 −1.00800
\(746\) 16.0499 0.587630
\(747\) −6.67641 −0.244277
\(748\) 11.1773 0.408682
\(749\) −1.34270 −0.0490613
\(750\) −7.13067 −0.260375
\(751\) −16.5059 −0.602310 −0.301155 0.953575i \(-0.597372\pi\)
−0.301155 + 0.953575i \(0.597372\pi\)
\(752\) −32.6111 −1.18920
\(753\) −38.4503 −1.40121
\(754\) −29.1377 −1.06113
\(755\) 58.3152 2.12231
\(756\) −28.3907 −1.03256
\(757\) −32.8918 −1.19547 −0.597736 0.801693i \(-0.703933\pi\)
−0.597736 + 0.801693i \(0.703933\pi\)
\(758\) −19.9615 −0.725034
\(759\) 10.0449 0.364606
\(760\) −11.7169 −0.425017
\(761\) 2.39595 0.0868532 0.0434266 0.999057i \(-0.486173\pi\)
0.0434266 + 0.999057i \(0.486173\pi\)
\(762\) 15.8539 0.574327
\(763\) −2.29081 −0.0829327
\(764\) −11.5792 −0.418921
\(765\) 10.9829 0.397086
\(766\) −9.51825 −0.343908
\(767\) 14.9468 0.539696
\(768\) −38.2677 −1.38087
\(769\) 53.6335 1.93407 0.967037 0.254636i \(-0.0819557\pi\)
0.967037 + 0.254636i \(0.0819557\pi\)
\(770\) −31.2670 −1.12679
\(771\) 46.4201 1.67178
\(772\) 13.1531 0.473391
\(773\) 30.5059 1.09722 0.548611 0.836078i \(-0.315157\pi\)
0.548611 + 0.836078i \(0.315157\pi\)
\(774\) 1.79696 0.0645906
\(775\) 22.3960 0.804486
\(776\) −7.76418 −0.278718
\(777\) −41.3682 −1.48408
\(778\) −29.3581 −1.05254
\(779\) −52.8865 −1.89485
\(780\) 34.1703 1.22349
\(781\) 1.31324 0.0469915
\(782\) −39.2978 −1.40529
\(783\) 20.6084 0.736485
\(784\) −34.8442 −1.24444
\(785\) −19.4441 −0.693991
\(786\) −52.4105 −1.86942
\(787\) 38.6936 1.37928 0.689638 0.724154i \(-0.257770\pi\)
0.689638 + 0.724154i \(0.257770\pi\)
\(788\) −5.77959 −0.205889
\(789\) −18.2549 −0.649893
\(790\) 31.6258 1.12520
\(791\) −12.0000 −0.426671
\(792\) −0.566204 −0.0201192
\(793\) 12.0897 0.429319
\(794\) −7.53365 −0.267359
\(795\) 53.7039 1.90468
\(796\) 28.9595 1.02644
\(797\) −35.4450 −1.25553 −0.627763 0.778405i \(-0.716029\pi\)
−0.627763 + 0.778405i \(0.716029\pi\)
\(798\) 76.6507 2.71340
\(799\) 36.6714 1.29734
\(800\) −41.5638 −1.46950
\(801\) −2.11791 −0.0748327
\(802\) −21.9397 −0.774719
\(803\) 21.5284 0.759719
\(804\) 16.8315 0.593601
\(805\) 49.8038 1.75535
\(806\) −25.3427 −0.892659
\(807\) 22.7263 0.800005
\(808\) −1.53871 −0.0541317
\(809\) 38.9622 1.36984 0.684918 0.728620i \(-0.259838\pi\)
0.684918 + 0.728620i \(0.259838\pi\)
\(810\) −65.6084 −2.30525
\(811\) 36.1421 1.26912 0.634560 0.772874i \(-0.281181\pi\)
0.634560 + 0.772874i \(0.281181\pi\)
\(812\) −29.1377 −1.02253
\(813\) 6.72393 0.235818
\(814\) −14.2050 −0.497885
\(815\) −37.0277 −1.29702
\(816\) 44.8865 1.57134
\(817\) 7.84425 0.274436
\(818\) −47.0851 −1.64629
\(819\) 8.32027 0.290734
\(820\) −52.0396 −1.81730
\(821\) 14.8788 0.519273 0.259636 0.965706i \(-0.416397\pi\)
0.259636 + 0.965706i \(0.416397\pi\)
\(822\) −52.1997 −1.82067
\(823\) 24.2151 0.844086 0.422043 0.906576i \(-0.361313\pi\)
0.422043 + 0.906576i \(0.361313\pi\)
\(824\) 10.6291 0.370283
\(825\) −14.0603 −0.489516
\(826\) 32.9916 1.14793
\(827\) 24.4114 0.848866 0.424433 0.905459i \(-0.360473\pi\)
0.424433 + 0.905459i \(0.360473\pi\)
\(828\) −4.35108 −0.151211
\(829\) −14.2228 −0.493979 −0.246990 0.969018i \(-0.579441\pi\)
−0.246990 + 0.969018i \(0.579441\pi\)
\(830\) −63.2996 −2.19716
\(831\) −11.4586 −0.397496
\(832\) 16.7572 0.580950
\(833\) 39.1826 1.35760
\(834\) 16.2754 0.563571
\(835\) 29.5611 1.02301
\(836\) 11.9243 0.412411
\(837\) 17.9243 0.619555
\(838\) −9.68608 −0.334600
\(839\) 10.6241 0.366784 0.183392 0.983040i \(-0.441292\pi\)
0.183392 + 0.983040i \(0.441292\pi\)
\(840\) −15.6335 −0.539407
\(841\) −7.84931 −0.270666
\(842\) −42.1447 −1.45240
\(843\) 12.4509 0.428833
\(844\) −7.01714 −0.241540
\(845\) −6.58426 −0.226505
\(846\) 8.96216 0.308125
\(847\) 35.4734 1.21888
\(848\) 39.4131 1.35345
\(849\) −21.9397 −0.752969
\(850\) 55.0070 1.88673
\(851\) 22.6265 0.775626
\(852\) −3.16784 −0.108528
\(853\) −56.0415 −1.91883 −0.959413 0.282005i \(-0.909001\pi\)
−0.959413 + 0.282005i \(0.909001\pi\)
\(854\) 26.6854 0.913156
\(855\) 11.7169 0.400709
\(856\) −0.230528 −0.00787930
\(857\) 11.8968 0.406388 0.203194 0.979139i \(-0.434868\pi\)
0.203194 + 0.979139i \(0.434868\pi\)
\(858\) 15.9103 0.543167
\(859\) −53.2221 −1.81592 −0.907958 0.419061i \(-0.862359\pi\)
−0.907958 + 0.419061i \(0.862359\pi\)
\(860\) 7.71863 0.263203
\(861\) −70.5649 −2.40484
\(862\) −31.4138 −1.06996
\(863\) −45.0620 −1.53393 −0.766964 0.641690i \(-0.778234\pi\)
−0.766964 + 0.641690i \(0.778234\pi\)
\(864\) −33.2650 −1.13170
\(865\) 5.36823 0.182525
\(866\) 64.4201 2.18908
\(867\) −17.9672 −0.610199
\(868\) −25.3427 −0.860187
\(869\) 6.67135 0.226310
\(870\) −54.7488 −1.85616
\(871\) 17.6040 0.596490
\(872\) −0.393308 −0.0133191
\(873\) 7.76418 0.262778
\(874\) −41.9243 −1.41811
\(875\) −7.45797 −0.252125
\(876\) −51.9313 −1.75460
\(877\) −57.0647 −1.92694 −0.963468 0.267822i \(-0.913696\pi\)
−0.963468 + 0.267822i \(0.913696\pi\)
\(878\) 18.5561 0.626238
\(879\) 39.4426 1.33036
\(880\) −19.5337 −0.658479
\(881\) 22.4329 0.755783 0.377892 0.925850i \(-0.376649\pi\)
0.377892 + 0.925850i \(0.376649\pi\)
\(882\) 9.57588 0.322437
\(883\) −10.3805 −0.349333 −0.174667 0.984628i \(-0.555885\pi\)
−0.174667 + 0.984628i \(0.555885\pi\)
\(884\) −28.1997 −0.948459
\(885\) 28.0844 0.944048
\(886\) −42.3753 −1.42362
\(887\) −29.4734 −0.989619 −0.494810 0.869001i \(-0.664762\pi\)
−0.494810 + 0.869001i \(0.664762\pi\)
\(888\) −7.10250 −0.238344
\(889\) 16.5816 0.556129
\(890\) −20.0801 −0.673085
\(891\) −13.8399 −0.463653
\(892\) −32.3471 −1.08306
\(893\) 39.1223 1.30918
\(894\) 30.9019 1.03351
\(895\) 16.9140 0.565372
\(896\) −19.7936 −0.661259
\(897\) −25.3427 −0.846168
\(898\) −55.4975 −1.85198
\(899\) 18.3960 0.613539
\(900\) 6.09042 0.203014
\(901\) −44.3203 −1.47652
\(902\) −24.2305 −0.806788
\(903\) 10.4663 0.348298
\(904\) −2.06028 −0.0685238
\(905\) 45.9036 1.52589
\(906\) −65.4982 −2.17603
\(907\) 13.5939 0.451379 0.225690 0.974199i \(-0.427537\pi\)
0.225690 + 0.974199i \(0.427537\pi\)
\(908\) −33.1980 −1.10171
\(909\) 1.53871 0.0510358
\(910\) 78.8851 2.61502
\(911\) −20.3203 −0.673241 −0.336620 0.941640i \(-0.609284\pi\)
−0.336620 + 0.941640i \(0.609284\pi\)
\(912\) 47.8865 1.58568
\(913\) −13.3528 −0.441914
\(914\) 37.3823 1.23650
\(915\) 22.7162 0.750975
\(916\) −2.73601 −0.0904004
\(917\) −54.8161 −1.81019
\(918\) 44.0242 1.45301
\(919\) 17.4734 0.576393 0.288197 0.957571i \(-0.406944\pi\)
0.288197 + 0.957571i \(0.406944\pi\)
\(920\) 8.55080 0.281911
\(921\) −22.6111 −0.745060
\(922\) 11.5592 0.380681
\(923\) −3.31324 −0.109057
\(924\) 15.9103 0.523409
\(925\) −31.6714 −1.04135
\(926\) −40.4054 −1.32780
\(927\) −10.6291 −0.349106
\(928\) −34.1403 −1.12071
\(929\) 7.36581 0.241665 0.120832 0.992673i \(-0.461444\pi\)
0.120832 + 0.992673i \(0.461444\pi\)
\(930\) −47.6181 −1.56146
\(931\) 41.8013 1.36998
\(932\) 39.0647 1.27961
\(933\) 35.1524 1.15084
\(934\) 78.0690 2.55450
\(935\) 21.9657 0.718356
\(936\) 1.42851 0.0466921
\(937\) 8.96745 0.292954 0.146477 0.989214i \(-0.453207\pi\)
0.146477 + 0.989214i \(0.453207\pi\)
\(938\) 38.8570 1.26873
\(939\) 10.1454 0.331083
\(940\) 38.4958 1.25559
\(941\) 11.0455 0.360075 0.180037 0.983660i \(-0.442378\pi\)
0.180037 + 0.983660i \(0.442378\pi\)
\(942\) 21.8392 0.711559
\(943\) 38.5957 1.25685
\(944\) 20.6111 0.670833
\(945\) −55.7936 −1.81497
\(946\) 3.59393 0.116849
\(947\) −36.1729 −1.17546 −0.587731 0.809057i \(-0.699978\pi\)
−0.587731 + 0.809057i \(0.699978\pi\)
\(948\) −16.0928 −0.522671
\(949\) −54.3150 −1.76314
\(950\) 58.6834 1.90394
\(951\) 11.2530 0.364902
\(952\) 12.9019 0.418152
\(953\) −25.7307 −0.833500 −0.416750 0.909021i \(-0.636831\pi\)
−0.416750 + 0.909021i \(0.636831\pi\)
\(954\) −10.8315 −0.350682
\(955\) −22.7556 −0.736353
\(956\) 0.946750 0.0306201
\(957\) −11.5491 −0.373328
\(958\) −45.8935 −1.48275
\(959\) −54.5957 −1.76299
\(960\) 31.4861 1.01621
\(961\) −15.0000 −0.483871
\(962\) 35.8385 1.15548
\(963\) 0.230528 0.00742868
\(964\) 42.1997 1.35916
\(965\) 25.8486 0.832097
\(966\) −55.9384 −1.79979
\(967\) 0.862301 0.0277297 0.0138649 0.999904i \(-0.495587\pi\)
0.0138649 + 0.999904i \(0.495587\pi\)
\(968\) 6.09042 0.195753
\(969\) −53.8486 −1.72987
\(970\) 73.6128 2.36356
\(971\) 52.7505 1.69284 0.846422 0.532512i \(-0.178752\pi\)
0.846422 + 0.532512i \(0.178752\pi\)
\(972\) 11.1146 0.356501
\(973\) 17.0224 0.545714
\(974\) 20.9622 0.671671
\(975\) 35.4734 1.13606
\(976\) 16.6714 0.533637
\(977\) 25.8761 0.827851 0.413925 0.910311i \(-0.364157\pi\)
0.413925 + 0.910311i \(0.364157\pi\)
\(978\) 41.5886 1.32986
\(979\) −4.23582 −0.135377
\(980\) 41.1320 1.31391
\(981\) 0.393308 0.0125574
\(982\) 44.8865 1.43238
\(983\) 50.4881 1.61032 0.805160 0.593057i \(-0.202079\pi\)
0.805160 + 0.593057i \(0.202079\pi\)
\(984\) −12.1153 −0.386221
\(985\) −11.3581 −0.361900
\(986\) 45.1826 1.43891
\(987\) 52.1997 1.66154
\(988\) −30.0844 −0.957114
\(989\) −5.72460 −0.182032
\(990\) 5.36823 0.170613
\(991\) −6.74175 −0.214159 −0.107079 0.994250i \(-0.534150\pi\)
−0.107079 + 0.994250i \(0.534150\pi\)
\(992\) −29.6938 −0.942779
\(993\) −24.3458 −0.772590
\(994\) −7.31324 −0.231962
\(995\) 56.9116 1.80422
\(996\) 32.2101 1.02061
\(997\) 55.5706 1.75994 0.879969 0.475031i \(-0.157563\pi\)
0.879969 + 0.475031i \(0.157563\pi\)
\(998\) 13.5963 0.430385
\(999\) −25.3478 −0.801968
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 71.2.a.a.1.3 3
3.2 odd 2 639.2.a.h.1.1 3
4.3 odd 2 1136.2.a.h.1.3 3
5.4 even 2 1775.2.a.f.1.1 3
7.6 odd 2 3479.2.a.k.1.3 3
8.3 odd 2 4544.2.a.u.1.1 3
8.5 even 2 4544.2.a.r.1.3 3
11.10 odd 2 8591.2.a.g.1.1 3
71.70 odd 2 5041.2.a.a.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
71.2.a.a.1.3 3 1.1 even 1 trivial
639.2.a.h.1.1 3 3.2 odd 2
1136.2.a.h.1.3 3 4.3 odd 2
1775.2.a.f.1.1 3 5.4 even 2
3479.2.a.k.1.3 3 7.6 odd 2
4544.2.a.r.1.3 3 8.5 even 2
4544.2.a.u.1.1 3 8.3 odd 2
5041.2.a.a.1.3 3 71.70 odd 2
8591.2.a.g.1.1 3 11.10 odd 2