Properties

Label 6026.2.a.f.1.15
Level $6026$
Weight $2$
Character 6026.1
Self dual yes
Analytic conductor $48.118$
Analytic rank $1$
Dimension $20$
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] = [6026,2,Mod(1,6026)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6026, 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("6026.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6026 = 2 \cdot 23 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6026.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: \(48.1178522580\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 5 x^{19} - 17 x^{18} + 115 x^{17} + 78 x^{16} - 1083 x^{15} + 248 x^{14} + 5359 x^{13} - 3723 x^{12} - 14776 x^{11} + 14837 x^{10} + 21886 x^{9} - 28084 x^{8} - 14682 x^{7} + \cdots - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Root \(-1.29878\) of defining polynomial
Character \(\chi\) \(=\) 6026.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.29878 q^{3} +1.00000 q^{4} +2.87965 q^{5} +1.29878 q^{6} -3.57798 q^{7} +1.00000 q^{8} -1.31316 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.29878 q^{3} +1.00000 q^{4} +2.87965 q^{5} +1.29878 q^{6} -3.57798 q^{7} +1.00000 q^{8} -1.31316 q^{9} +2.87965 q^{10} -1.21143 q^{11} +1.29878 q^{12} -1.04790 q^{13} -3.57798 q^{14} +3.74004 q^{15} +1.00000 q^{16} -6.73345 q^{17} -1.31316 q^{18} -2.53015 q^{19} +2.87965 q^{20} -4.64701 q^{21} -1.21143 q^{22} +1.00000 q^{23} +1.29878 q^{24} +3.29240 q^{25} -1.04790 q^{26} -5.60186 q^{27} -3.57798 q^{28} -2.36709 q^{29} +3.74004 q^{30} -4.49280 q^{31} +1.00000 q^{32} -1.57338 q^{33} -6.73345 q^{34} -10.3033 q^{35} -1.31316 q^{36} +3.59289 q^{37} -2.53015 q^{38} -1.36099 q^{39} +2.87965 q^{40} -4.03401 q^{41} -4.64701 q^{42} -1.35679 q^{43} -1.21143 q^{44} -3.78146 q^{45} +1.00000 q^{46} +8.78952 q^{47} +1.29878 q^{48} +5.80191 q^{49} +3.29240 q^{50} -8.74529 q^{51} -1.04790 q^{52} +3.01840 q^{53} -5.60186 q^{54} -3.48849 q^{55} -3.57798 q^{56} -3.28612 q^{57} -2.36709 q^{58} -2.93882 q^{59} +3.74004 q^{60} -0.722314 q^{61} -4.49280 q^{62} +4.69847 q^{63} +1.00000 q^{64} -3.01759 q^{65} -1.57338 q^{66} -12.0647 q^{67} -6.73345 q^{68} +1.29878 q^{69} -10.3033 q^{70} -1.34627 q^{71} -1.31316 q^{72} -14.3307 q^{73} +3.59289 q^{74} +4.27611 q^{75} -2.53015 q^{76} +4.33446 q^{77} -1.36099 q^{78} +8.17446 q^{79} +2.87965 q^{80} -3.33611 q^{81} -4.03401 q^{82} -2.57120 q^{83} -4.64701 q^{84} -19.3900 q^{85} -1.35679 q^{86} -3.07434 q^{87} -1.21143 q^{88} -8.39887 q^{89} -3.78146 q^{90} +3.74936 q^{91} +1.00000 q^{92} -5.83517 q^{93} +8.78952 q^{94} -7.28596 q^{95} +1.29878 q^{96} -1.82015 q^{97} +5.80191 q^{98} +1.59080 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{2} - 5 q^{3} + 20 q^{4} - 6 q^{5} - 5 q^{6} - 12 q^{7} + 20 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{2} - 5 q^{3} + 20 q^{4} - 6 q^{5} - 5 q^{6} - 12 q^{7} + 20 q^{8} - q^{9} - 6 q^{10} - 3 q^{11} - 5 q^{12} - 13 q^{13} - 12 q^{14} - 10 q^{15} + 20 q^{16} - 14 q^{17} - q^{18} - 21 q^{19} - 6 q^{20} - 8 q^{21} - 3 q^{22} + 20 q^{23} - 5 q^{24} - 14 q^{25} - 13 q^{26} - 5 q^{27} - 12 q^{28} - 27 q^{29} - 10 q^{30} - 27 q^{31} + 20 q^{32} - 12 q^{33} - 14 q^{34} - 23 q^{35} - q^{36} - 19 q^{37} - 21 q^{38} - 35 q^{39} - 6 q^{40} - 17 q^{41} - 8 q^{42} - 27 q^{43} - 3 q^{44} + 4 q^{45} + 20 q^{46} - 28 q^{47} - 5 q^{48} - 10 q^{49} - 14 q^{50} + 6 q^{51} - 13 q^{52} - 47 q^{53} - 5 q^{54} - 4 q^{55} - 12 q^{56} - 16 q^{57} - 27 q^{58} - 16 q^{59} - 10 q^{60} - 9 q^{61} - 27 q^{62} - 9 q^{63} + 20 q^{64} + 9 q^{65} - 12 q^{66} - 8 q^{67} - 14 q^{68} - 5 q^{69} - 23 q^{70} - 30 q^{71} - q^{72} - 26 q^{73} - 19 q^{74} - 18 q^{75} - 21 q^{76} - 50 q^{77} - 35 q^{78} - 35 q^{79} - 6 q^{80} - 60 q^{81} - 17 q^{82} + 2 q^{83} - 8 q^{84} - 62 q^{85} - 27 q^{86} + q^{87} - 3 q^{88} - 25 q^{89} + 4 q^{90} + 22 q^{91} + 20 q^{92} - 21 q^{93} - 28 q^{94} - 14 q^{95} - 5 q^{96} + 2 q^{97} - 10 q^{98} - 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.29878 0.749852 0.374926 0.927055i \(-0.377668\pi\)
0.374926 + 0.927055i \(0.377668\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.87965 1.28782 0.643910 0.765101i \(-0.277311\pi\)
0.643910 + 0.765101i \(0.277311\pi\)
\(6\) 1.29878 0.530226
\(7\) −3.57798 −1.35235 −0.676174 0.736742i \(-0.736363\pi\)
−0.676174 + 0.736742i \(0.736363\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.31316 −0.437721
\(10\) 2.87965 0.910626
\(11\) −1.21143 −0.365259 −0.182630 0.983182i \(-0.558461\pi\)
−0.182630 + 0.983182i \(0.558461\pi\)
\(12\) 1.29878 0.374926
\(13\) −1.04790 −0.290635 −0.145318 0.989385i \(-0.546420\pi\)
−0.145318 + 0.989385i \(0.546420\pi\)
\(14\) −3.57798 −0.956254
\(15\) 3.74004 0.965675
\(16\) 1.00000 0.250000
\(17\) −6.73345 −1.63310 −0.816551 0.577273i \(-0.804117\pi\)
−0.816551 + 0.577273i \(0.804117\pi\)
\(18\) −1.31316 −0.309516
\(19\) −2.53015 −0.580457 −0.290229 0.956957i \(-0.593731\pi\)
−0.290229 + 0.956957i \(0.593731\pi\)
\(20\) 2.87965 0.643910
\(21\) −4.64701 −1.01406
\(22\) −1.21143 −0.258277
\(23\) 1.00000 0.208514
\(24\) 1.29878 0.265113
\(25\) 3.29240 0.658479
\(26\) −1.04790 −0.205510
\(27\) −5.60186 −1.07808
\(28\) −3.57798 −0.676174
\(29\) −2.36709 −0.439558 −0.219779 0.975550i \(-0.570534\pi\)
−0.219779 + 0.975550i \(0.570534\pi\)
\(30\) 3.74004 0.682835
\(31\) −4.49280 −0.806931 −0.403466 0.914995i \(-0.632194\pi\)
−0.403466 + 0.914995i \(0.632194\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.57338 −0.273890
\(34\) −6.73345 −1.15478
\(35\) −10.3033 −1.74158
\(36\) −1.31316 −0.218861
\(37\) 3.59289 0.590668 0.295334 0.955394i \(-0.404569\pi\)
0.295334 + 0.955394i \(0.404569\pi\)
\(38\) −2.53015 −0.410445
\(39\) −1.36099 −0.217933
\(40\) 2.87965 0.455313
\(41\) −4.03401 −0.630006 −0.315003 0.949091i \(-0.602006\pi\)
−0.315003 + 0.949091i \(0.602006\pi\)
\(42\) −4.64701 −0.717049
\(43\) −1.35679 −0.206908 −0.103454 0.994634i \(-0.532989\pi\)
−0.103454 + 0.994634i \(0.532989\pi\)
\(44\) −1.21143 −0.182630
\(45\) −3.78146 −0.563706
\(46\) 1.00000 0.147442
\(47\) 8.78952 1.28208 0.641042 0.767506i \(-0.278503\pi\)
0.641042 + 0.767506i \(0.278503\pi\)
\(48\) 1.29878 0.187463
\(49\) 5.80191 0.828844
\(50\) 3.29240 0.465615
\(51\) −8.74529 −1.22459
\(52\) −1.04790 −0.145318
\(53\) 3.01840 0.414609 0.207304 0.978277i \(-0.433531\pi\)
0.207304 + 0.978277i \(0.433531\pi\)
\(54\) −5.60186 −0.762317
\(55\) −3.48849 −0.470388
\(56\) −3.57798 −0.478127
\(57\) −3.28612 −0.435257
\(58\) −2.36709 −0.310815
\(59\) −2.93882 −0.382602 −0.191301 0.981531i \(-0.561271\pi\)
−0.191301 + 0.981531i \(0.561271\pi\)
\(60\) 3.74004 0.482837
\(61\) −0.722314 −0.0924828 −0.0462414 0.998930i \(-0.514724\pi\)
−0.0462414 + 0.998930i \(0.514724\pi\)
\(62\) −4.49280 −0.570586
\(63\) 4.69847 0.591951
\(64\) 1.00000 0.125000
\(65\) −3.01759 −0.374286
\(66\) −1.57338 −0.193670
\(67\) −12.0647 −1.47394 −0.736970 0.675925i \(-0.763744\pi\)
−0.736970 + 0.675925i \(0.763744\pi\)
\(68\) −6.73345 −0.816551
\(69\) 1.29878 0.156355
\(70\) −10.3033 −1.23148
\(71\) −1.34627 −0.159773 −0.0798864 0.996804i \(-0.525456\pi\)
−0.0798864 + 0.996804i \(0.525456\pi\)
\(72\) −1.31316 −0.154758
\(73\) −14.3307 −1.67728 −0.838641 0.544685i \(-0.816649\pi\)
−0.838641 + 0.544685i \(0.816649\pi\)
\(74\) 3.59289 0.417665
\(75\) 4.27611 0.493762
\(76\) −2.53015 −0.290229
\(77\) 4.33446 0.493957
\(78\) −1.36099 −0.154102
\(79\) 8.17446 0.919699 0.459849 0.887997i \(-0.347903\pi\)
0.459849 + 0.887997i \(0.347903\pi\)
\(80\) 2.87965 0.321955
\(81\) −3.33611 −0.370679
\(82\) −4.03401 −0.445482
\(83\) −2.57120 −0.282226 −0.141113 0.989994i \(-0.545068\pi\)
−0.141113 + 0.989994i \(0.545068\pi\)
\(84\) −4.64701 −0.507031
\(85\) −19.3900 −2.10314
\(86\) −1.35679 −0.146306
\(87\) −3.07434 −0.329604
\(88\) −1.21143 −0.129139
\(89\) −8.39887 −0.890279 −0.445139 0.895461i \(-0.646846\pi\)
−0.445139 + 0.895461i \(0.646846\pi\)
\(90\) −3.78146 −0.398600
\(91\) 3.74936 0.393040
\(92\) 1.00000 0.104257
\(93\) −5.83517 −0.605079
\(94\) 8.78952 0.906570
\(95\) −7.28596 −0.747524
\(96\) 1.29878 0.132556
\(97\) −1.82015 −0.184808 −0.0924041 0.995722i \(-0.529455\pi\)
−0.0924041 + 0.995722i \(0.529455\pi\)
\(98\) 5.80191 0.586081
\(99\) 1.59080 0.159882
\(100\) 3.29240 0.329240
\(101\) 5.20144 0.517562 0.258781 0.965936i \(-0.416679\pi\)
0.258781 + 0.965936i \(0.416679\pi\)
\(102\) −8.74529 −0.865913
\(103\) 17.7702 1.75095 0.875473 0.483267i \(-0.160550\pi\)
0.875473 + 0.483267i \(0.160550\pi\)
\(104\) −1.04790 −0.102755
\(105\) −13.3818 −1.30593
\(106\) 3.01840 0.293173
\(107\) 6.79008 0.656422 0.328211 0.944604i \(-0.393554\pi\)
0.328211 + 0.944604i \(0.393554\pi\)
\(108\) −5.60186 −0.539039
\(109\) 8.16821 0.782373 0.391186 0.920311i \(-0.372065\pi\)
0.391186 + 0.920311i \(0.372065\pi\)
\(110\) −3.48849 −0.332614
\(111\) 4.66639 0.442914
\(112\) −3.57798 −0.338087
\(113\) −3.26557 −0.307199 −0.153600 0.988133i \(-0.549087\pi\)
−0.153600 + 0.988133i \(0.549087\pi\)
\(114\) −3.28612 −0.307773
\(115\) 2.87965 0.268529
\(116\) −2.36709 −0.219779
\(117\) 1.37606 0.127217
\(118\) −2.93882 −0.270540
\(119\) 24.0921 2.20852
\(120\) 3.74004 0.341418
\(121\) −9.53244 −0.866586
\(122\) −0.722314 −0.0653952
\(123\) −5.23930 −0.472412
\(124\) −4.49280 −0.403466
\(125\) −4.91731 −0.439817
\(126\) 4.69847 0.418573
\(127\) 22.2684 1.97600 0.987998 0.154464i \(-0.0493650\pi\)
0.987998 + 0.154464i \(0.0493650\pi\)
\(128\) 1.00000 0.0883883
\(129\) −1.76217 −0.155150
\(130\) −3.01759 −0.264660
\(131\) −1.00000 −0.0873704
\(132\) −1.57338 −0.136945
\(133\) 9.05283 0.784980
\(134\) −12.0647 −1.04223
\(135\) −16.1314 −1.38837
\(136\) −6.73345 −0.577389
\(137\) −13.2723 −1.13393 −0.566964 0.823743i \(-0.691882\pi\)
−0.566964 + 0.823743i \(0.691882\pi\)
\(138\) 1.29878 0.110560
\(139\) −4.60002 −0.390169 −0.195084 0.980786i \(-0.562498\pi\)
−0.195084 + 0.980786i \(0.562498\pi\)
\(140\) −10.3033 −0.870790
\(141\) 11.4157 0.961373
\(142\) −1.34627 −0.112976
\(143\) 1.26945 0.106157
\(144\) −1.31316 −0.109430
\(145\) −6.81641 −0.566072
\(146\) −14.3307 −1.18602
\(147\) 7.53541 0.621510
\(148\) 3.59289 0.295334
\(149\) 11.8855 0.973695 0.486847 0.873487i \(-0.338147\pi\)
0.486847 + 0.873487i \(0.338147\pi\)
\(150\) 4.27611 0.349143
\(151\) −16.0358 −1.30497 −0.652487 0.757800i \(-0.726274\pi\)
−0.652487 + 0.757800i \(0.726274\pi\)
\(152\) −2.53015 −0.205223
\(153\) 8.84213 0.714844
\(154\) 4.33446 0.349281
\(155\) −12.9377 −1.03918
\(156\) −1.36099 −0.108967
\(157\) 9.20977 0.735020 0.367510 0.930020i \(-0.380210\pi\)
0.367510 + 0.930020i \(0.380210\pi\)
\(158\) 8.17446 0.650325
\(159\) 3.92024 0.310895
\(160\) 2.87965 0.227656
\(161\) −3.57798 −0.281984
\(162\) −3.33611 −0.262109
\(163\) −7.37833 −0.577916 −0.288958 0.957342i \(-0.593309\pi\)
−0.288958 + 0.957342i \(0.593309\pi\)
\(164\) −4.03401 −0.315003
\(165\) −4.53079 −0.352722
\(166\) −2.57120 −0.199564
\(167\) −9.50905 −0.735832 −0.367916 0.929859i \(-0.619929\pi\)
−0.367916 + 0.929859i \(0.619929\pi\)
\(168\) −4.64701 −0.358525
\(169\) −11.9019 −0.915531
\(170\) −19.3900 −1.48714
\(171\) 3.32251 0.254078
\(172\) −1.35679 −0.103454
\(173\) 7.14326 0.543092 0.271546 0.962425i \(-0.412465\pi\)
0.271546 + 0.962425i \(0.412465\pi\)
\(174\) −3.07434 −0.233065
\(175\) −11.7801 −0.890493
\(176\) −1.21143 −0.0913148
\(177\) −3.81689 −0.286895
\(178\) −8.39887 −0.629522
\(179\) 1.15119 0.0860443 0.0430222 0.999074i \(-0.486301\pi\)
0.0430222 + 0.999074i \(0.486301\pi\)
\(180\) −3.78146 −0.281853
\(181\) 10.6770 0.793614 0.396807 0.917902i \(-0.370118\pi\)
0.396807 + 0.917902i \(0.370118\pi\)
\(182\) 3.74936 0.277921
\(183\) −0.938128 −0.0693485
\(184\) 1.00000 0.0737210
\(185\) 10.3463 0.760674
\(186\) −5.83517 −0.427856
\(187\) 8.15709 0.596505
\(188\) 8.78952 0.641042
\(189\) 20.0433 1.45794
\(190\) −7.28596 −0.528579
\(191\) 6.95720 0.503406 0.251703 0.967805i \(-0.419009\pi\)
0.251703 + 0.967805i \(0.419009\pi\)
\(192\) 1.29878 0.0937316
\(193\) −18.3198 −1.31869 −0.659345 0.751841i \(-0.729166\pi\)
−0.659345 + 0.751841i \(0.729166\pi\)
\(194\) −1.82015 −0.130679
\(195\) −3.91919 −0.280659
\(196\) 5.80191 0.414422
\(197\) −6.99960 −0.498701 −0.249350 0.968413i \(-0.580217\pi\)
−0.249350 + 0.968413i \(0.580217\pi\)
\(198\) 1.59080 0.113053
\(199\) −9.68250 −0.686374 −0.343187 0.939267i \(-0.611506\pi\)
−0.343187 + 0.939267i \(0.611506\pi\)
\(200\) 3.29240 0.232808
\(201\) −15.6694 −1.10524
\(202\) 5.20144 0.365972
\(203\) 8.46940 0.594436
\(204\) −8.74529 −0.612293
\(205\) −11.6165 −0.811334
\(206\) 17.7702 1.23811
\(207\) −1.31316 −0.0912712
\(208\) −1.04790 −0.0726588
\(209\) 3.06510 0.212017
\(210\) −13.3818 −0.923430
\(211\) −0.926514 −0.0637839 −0.0318919 0.999491i \(-0.510153\pi\)
−0.0318919 + 0.999491i \(0.510153\pi\)
\(212\) 3.01840 0.207304
\(213\) −1.74851 −0.119806
\(214\) 6.79008 0.464160
\(215\) −3.90707 −0.266460
\(216\) −5.60186 −0.381158
\(217\) 16.0751 1.09125
\(218\) 8.16821 0.553221
\(219\) −18.6125 −1.25771
\(220\) −3.48849 −0.235194
\(221\) 7.05598 0.474637
\(222\) 4.66639 0.313187
\(223\) −25.2594 −1.69150 −0.845749 0.533582i \(-0.820846\pi\)
−0.845749 + 0.533582i \(0.820846\pi\)
\(224\) −3.57798 −0.239064
\(225\) −4.32346 −0.288230
\(226\) −3.26557 −0.217223
\(227\) 12.0184 0.797692 0.398846 0.917018i \(-0.369411\pi\)
0.398846 + 0.917018i \(0.369411\pi\)
\(228\) −3.28612 −0.217629
\(229\) 21.5872 1.42653 0.713263 0.700897i \(-0.247217\pi\)
0.713263 + 0.700897i \(0.247217\pi\)
\(230\) 2.87965 0.189879
\(231\) 5.62952 0.370395
\(232\) −2.36709 −0.155407
\(233\) 0.772477 0.0506066 0.0253033 0.999680i \(-0.491945\pi\)
0.0253033 + 0.999680i \(0.491945\pi\)
\(234\) 1.37606 0.0899561
\(235\) 25.3108 1.65109
\(236\) −2.93882 −0.191301
\(237\) 10.6168 0.689638
\(238\) 24.0921 1.56166
\(239\) 11.3578 0.734677 0.367339 0.930087i \(-0.380269\pi\)
0.367339 + 0.930087i \(0.380269\pi\)
\(240\) 3.74004 0.241419
\(241\) −8.33944 −0.537191 −0.268595 0.963253i \(-0.586559\pi\)
−0.268595 + 0.963253i \(0.586559\pi\)
\(242\) −9.53244 −0.612769
\(243\) 12.4727 0.800125
\(244\) −0.722314 −0.0462414
\(245\) 16.7075 1.06740
\(246\) −5.23930 −0.334045
\(247\) 2.65135 0.168701
\(248\) −4.49280 −0.285293
\(249\) −3.33943 −0.211628
\(250\) −4.91731 −0.310998
\(251\) 1.91642 0.120963 0.0604815 0.998169i \(-0.480736\pi\)
0.0604815 + 0.998169i \(0.480736\pi\)
\(252\) 4.69847 0.295976
\(253\) −1.21143 −0.0761618
\(254\) 22.2684 1.39724
\(255\) −25.1834 −1.57704
\(256\) 1.00000 0.0625000
\(257\) 8.91386 0.556031 0.278016 0.960577i \(-0.410323\pi\)
0.278016 + 0.960577i \(0.410323\pi\)
\(258\) −1.76217 −0.109708
\(259\) −12.8553 −0.798789
\(260\) −3.01759 −0.187143
\(261\) 3.10838 0.192404
\(262\) −1.00000 −0.0617802
\(263\) −20.1270 −1.24108 −0.620542 0.784174i \(-0.713087\pi\)
−0.620542 + 0.784174i \(0.713087\pi\)
\(264\) −1.57338 −0.0968349
\(265\) 8.69193 0.533941
\(266\) 9.05283 0.555064
\(267\) −10.9083 −0.667578
\(268\) −12.0647 −0.736970
\(269\) −4.97063 −0.303065 −0.151532 0.988452i \(-0.548421\pi\)
−0.151532 + 0.988452i \(0.548421\pi\)
\(270\) −16.1314 −0.981727
\(271\) 0.105264 0.00639432 0.00319716 0.999995i \(-0.498982\pi\)
0.00319716 + 0.999995i \(0.498982\pi\)
\(272\) −6.73345 −0.408275
\(273\) 4.86960 0.294722
\(274\) −13.2723 −0.801808
\(275\) −3.98850 −0.240516
\(276\) 1.29878 0.0781775
\(277\) 2.27158 0.136486 0.0682431 0.997669i \(-0.478261\pi\)
0.0682431 + 0.997669i \(0.478261\pi\)
\(278\) −4.60002 −0.275891
\(279\) 5.89979 0.353211
\(280\) −10.3033 −0.615741
\(281\) 26.1385 1.55929 0.779646 0.626221i \(-0.215399\pi\)
0.779646 + 0.626221i \(0.215399\pi\)
\(282\) 11.4157 0.679794
\(283\) −5.64096 −0.335320 −0.167660 0.985845i \(-0.553621\pi\)
−0.167660 + 0.985845i \(0.553621\pi\)
\(284\) −1.34627 −0.0798864
\(285\) −9.46288 −0.560533
\(286\) 1.26945 0.0750644
\(287\) 14.4336 0.851987
\(288\) −1.31316 −0.0773789
\(289\) 28.3394 1.66702
\(290\) −6.81641 −0.400273
\(291\) −2.36398 −0.138579
\(292\) −14.3307 −0.838641
\(293\) −25.3398 −1.48037 −0.740183 0.672406i \(-0.765261\pi\)
−0.740183 + 0.672406i \(0.765261\pi\)
\(294\) 7.53541 0.439474
\(295\) −8.46277 −0.492722
\(296\) 3.59289 0.208833
\(297\) 6.78625 0.393778
\(298\) 11.8855 0.688506
\(299\) −1.04790 −0.0606016
\(300\) 4.27611 0.246881
\(301\) 4.85455 0.279811
\(302\) −16.0358 −0.922756
\(303\) 6.75554 0.388095
\(304\) −2.53015 −0.145114
\(305\) −2.08001 −0.119101
\(306\) 8.84213 0.505471
\(307\) 30.2571 1.72687 0.863433 0.504464i \(-0.168310\pi\)
0.863433 + 0.504464i \(0.168310\pi\)
\(308\) 4.33446 0.246979
\(309\) 23.0796 1.31295
\(310\) −12.9377 −0.734812
\(311\) 10.5052 0.595694 0.297847 0.954614i \(-0.403731\pi\)
0.297847 + 0.954614i \(0.403731\pi\)
\(312\) −1.36099 −0.0770511
\(313\) 19.7062 1.11386 0.556930 0.830559i \(-0.311979\pi\)
0.556930 + 0.830559i \(0.311979\pi\)
\(314\) 9.20977 0.519737
\(315\) 13.5300 0.762327
\(316\) 8.17446 0.459849
\(317\) −23.6058 −1.32583 −0.662916 0.748694i \(-0.730681\pi\)
−0.662916 + 0.748694i \(0.730681\pi\)
\(318\) 3.92024 0.219836
\(319\) 2.86756 0.160553
\(320\) 2.87965 0.160977
\(321\) 8.81884 0.492219
\(322\) −3.57798 −0.199393
\(323\) 17.0367 0.947945
\(324\) −3.33611 −0.185339
\(325\) −3.45010 −0.191377
\(326\) −7.37833 −0.408648
\(327\) 10.6087 0.586664
\(328\) −4.03401 −0.222741
\(329\) −31.4487 −1.73382
\(330\) −4.53079 −0.249412
\(331\) 24.5531 1.34956 0.674779 0.738020i \(-0.264239\pi\)
0.674779 + 0.738020i \(0.264239\pi\)
\(332\) −2.57120 −0.141113
\(333\) −4.71806 −0.258548
\(334\) −9.50905 −0.520312
\(335\) −34.7422 −1.89817
\(336\) −4.64701 −0.253515
\(337\) 0.827698 0.0450876 0.0225438 0.999746i \(-0.492823\pi\)
0.0225438 + 0.999746i \(0.492823\pi\)
\(338\) −11.9019 −0.647378
\(339\) −4.24127 −0.230354
\(340\) −19.3900 −1.05157
\(341\) 5.44271 0.294739
\(342\) 3.32251 0.179661
\(343\) 4.28675 0.231463
\(344\) −1.35679 −0.0731530
\(345\) 3.74004 0.201357
\(346\) 7.14326 0.384024
\(347\) −1.61910 −0.0869177 −0.0434589 0.999055i \(-0.513838\pi\)
−0.0434589 + 0.999055i \(0.513838\pi\)
\(348\) −3.07434 −0.164802
\(349\) −10.1191 −0.541661 −0.270831 0.962627i \(-0.587298\pi\)
−0.270831 + 0.962627i \(0.587298\pi\)
\(350\) −11.7801 −0.629673
\(351\) 5.87019 0.313328
\(352\) −1.21143 −0.0645693
\(353\) −20.1340 −1.07162 −0.535812 0.844337i \(-0.679994\pi\)
−0.535812 + 0.844337i \(0.679994\pi\)
\(354\) −3.81689 −0.202865
\(355\) −3.87679 −0.205759
\(356\) −8.39887 −0.445139
\(357\) 31.2904 1.65606
\(358\) 1.15119 0.0608425
\(359\) −18.7206 −0.988035 −0.494017 0.869452i \(-0.664472\pi\)
−0.494017 + 0.869452i \(0.664472\pi\)
\(360\) −3.78146 −0.199300
\(361\) −12.5983 −0.663070
\(362\) 10.6770 0.561170
\(363\) −12.3806 −0.649811
\(364\) 3.74936 0.196520
\(365\) −41.2674 −2.16004
\(366\) −0.938128 −0.0490368
\(367\) 27.6525 1.44345 0.721723 0.692182i \(-0.243350\pi\)
0.721723 + 0.692182i \(0.243350\pi\)
\(368\) 1.00000 0.0521286
\(369\) 5.29731 0.275767
\(370\) 10.3463 0.537878
\(371\) −10.7997 −0.560695
\(372\) −5.83517 −0.302540
\(373\) −16.0419 −0.830621 −0.415310 0.909680i \(-0.636327\pi\)
−0.415310 + 0.909680i \(0.636327\pi\)
\(374\) 8.15709 0.421793
\(375\) −6.38651 −0.329798
\(376\) 8.78952 0.453285
\(377\) 2.48048 0.127751
\(378\) 20.0433 1.03092
\(379\) 6.99216 0.359163 0.179582 0.983743i \(-0.442526\pi\)
0.179582 + 0.983743i \(0.442526\pi\)
\(380\) −7.28596 −0.373762
\(381\) 28.9217 1.48171
\(382\) 6.95720 0.355962
\(383\) 18.2335 0.931688 0.465844 0.884867i \(-0.345751\pi\)
0.465844 + 0.884867i \(0.345751\pi\)
\(384\) 1.29878 0.0662782
\(385\) 12.4817 0.636128
\(386\) −18.3198 −0.932455
\(387\) 1.78168 0.0905680
\(388\) −1.82015 −0.0924041
\(389\) −18.8141 −0.953913 −0.476956 0.878927i \(-0.658260\pi\)
−0.476956 + 0.878927i \(0.658260\pi\)
\(390\) −3.91919 −0.198456
\(391\) −6.73345 −0.340525
\(392\) 5.80191 0.293040
\(393\) −1.29878 −0.0655149
\(394\) −6.99960 −0.352635
\(395\) 23.5396 1.18441
\(396\) 1.59080 0.0799409
\(397\) 16.8205 0.844196 0.422098 0.906550i \(-0.361294\pi\)
0.422098 + 0.906550i \(0.361294\pi\)
\(398\) −9.68250 −0.485340
\(399\) 11.7577 0.588619
\(400\) 3.29240 0.164620
\(401\) 24.0199 1.19950 0.599749 0.800188i \(-0.295267\pi\)
0.599749 + 0.800188i \(0.295267\pi\)
\(402\) −15.6694 −0.781521
\(403\) 4.70801 0.234523
\(404\) 5.20144 0.258781
\(405\) −9.60683 −0.477367
\(406\) 8.46940 0.420329
\(407\) −4.35253 −0.215747
\(408\) −8.74529 −0.432956
\(409\) −1.45554 −0.0719718 −0.0359859 0.999352i \(-0.511457\pi\)
−0.0359859 + 0.999352i \(0.511457\pi\)
\(410\) −11.6165 −0.573700
\(411\) −17.2378 −0.850279
\(412\) 17.7702 0.875473
\(413\) 10.5150 0.517410
\(414\) −1.31316 −0.0645385
\(415\) −7.40416 −0.363456
\(416\) −1.04790 −0.0513775
\(417\) −5.97442 −0.292569
\(418\) 3.06510 0.149919
\(419\) 33.6359 1.64322 0.821612 0.570047i \(-0.193075\pi\)
0.821612 + 0.570047i \(0.193075\pi\)
\(420\) −13.3818 −0.652964
\(421\) −29.1473 −1.42055 −0.710276 0.703923i \(-0.751430\pi\)
−0.710276 + 0.703923i \(0.751430\pi\)
\(422\) −0.926514 −0.0451020
\(423\) −11.5421 −0.561195
\(424\) 3.01840 0.146586
\(425\) −22.1692 −1.07536
\(426\) −1.74851 −0.0847156
\(427\) 2.58442 0.125069
\(428\) 6.79008 0.328211
\(429\) 1.64875 0.0796022
\(430\) −3.90707 −0.188416
\(431\) −28.2169 −1.35916 −0.679581 0.733601i \(-0.737838\pi\)
−0.679581 + 0.733601i \(0.737838\pi\)
\(432\) −5.60186 −0.269520
\(433\) −1.85854 −0.0893158 −0.0446579 0.999002i \(-0.514220\pi\)
−0.0446579 + 0.999002i \(0.514220\pi\)
\(434\) 16.0751 0.771631
\(435\) −8.85303 −0.424470
\(436\) 8.16821 0.391186
\(437\) −2.53015 −0.121034
\(438\) −18.6125 −0.889338
\(439\) 28.4457 1.35764 0.678819 0.734306i \(-0.262492\pi\)
0.678819 + 0.734306i \(0.262492\pi\)
\(440\) −3.48849 −0.166307
\(441\) −7.61885 −0.362803
\(442\) 7.05598 0.335619
\(443\) −12.7941 −0.607867 −0.303934 0.952693i \(-0.598300\pi\)
−0.303934 + 0.952693i \(0.598300\pi\)
\(444\) 4.66639 0.221457
\(445\) −24.1858 −1.14652
\(446\) −25.2594 −1.19607
\(447\) 15.4366 0.730127
\(448\) −3.57798 −0.169043
\(449\) −3.35017 −0.158104 −0.0790521 0.996870i \(-0.525189\pi\)
−0.0790521 + 0.996870i \(0.525189\pi\)
\(450\) −4.32346 −0.203810
\(451\) 4.88691 0.230115
\(452\) −3.26557 −0.153600
\(453\) −20.8270 −0.978538
\(454\) 12.0184 0.564053
\(455\) 10.7969 0.506164
\(456\) −3.28612 −0.153887
\(457\) −9.79457 −0.458170 −0.229085 0.973406i \(-0.573573\pi\)
−0.229085 + 0.973406i \(0.573573\pi\)
\(458\) 21.5872 1.00871
\(459\) 37.7199 1.76061
\(460\) 2.87965 0.134264
\(461\) −3.92003 −0.182574 −0.0912869 0.995825i \(-0.529098\pi\)
−0.0912869 + 0.995825i \(0.529098\pi\)
\(462\) 5.62952 0.261909
\(463\) 30.5027 1.41758 0.708791 0.705419i \(-0.249241\pi\)
0.708791 + 0.705419i \(0.249241\pi\)
\(464\) −2.36709 −0.109890
\(465\) −16.8033 −0.779233
\(466\) 0.772477 0.0357843
\(467\) 38.6037 1.78637 0.893183 0.449692i \(-0.148466\pi\)
0.893183 + 0.449692i \(0.148466\pi\)
\(468\) 1.37606 0.0636086
\(469\) 43.1673 1.99328
\(470\) 25.3108 1.16750
\(471\) 11.9615 0.551156
\(472\) −2.93882 −0.135270
\(473\) 1.64365 0.0755750
\(474\) 10.6168 0.487648
\(475\) −8.33027 −0.382219
\(476\) 24.0921 1.10426
\(477\) −3.96365 −0.181483
\(478\) 11.3578 0.519495
\(479\) 14.5801 0.666180 0.333090 0.942895i \(-0.391909\pi\)
0.333090 + 0.942895i \(0.391909\pi\)
\(480\) 3.74004 0.170709
\(481\) −3.76499 −0.171669
\(482\) −8.33944 −0.379851
\(483\) −4.64701 −0.211446
\(484\) −9.53244 −0.433293
\(485\) −5.24139 −0.237999
\(486\) 12.4727 0.565773
\(487\) −27.0085 −1.22387 −0.611935 0.790908i \(-0.709609\pi\)
−0.611935 + 0.790908i \(0.709609\pi\)
\(488\) −0.722314 −0.0326976
\(489\) −9.58285 −0.433351
\(490\) 16.7075 0.754767
\(491\) 3.42553 0.154592 0.0772960 0.997008i \(-0.475371\pi\)
0.0772960 + 0.997008i \(0.475371\pi\)
\(492\) −5.23930 −0.236206
\(493\) 15.9387 0.717843
\(494\) 2.65135 0.119290
\(495\) 4.58096 0.205899
\(496\) −4.49280 −0.201733
\(497\) 4.81692 0.216068
\(498\) −3.33943 −0.149643
\(499\) 24.9775 1.11815 0.559073 0.829119i \(-0.311157\pi\)
0.559073 + 0.829119i \(0.311157\pi\)
\(500\) −4.91731 −0.219909
\(501\) −12.3502 −0.551766
\(502\) 1.91642 0.0855338
\(503\) −1.55078 −0.0691460 −0.0345730 0.999402i \(-0.511007\pi\)
−0.0345730 + 0.999402i \(0.511007\pi\)
\(504\) 4.69847 0.209286
\(505\) 14.9783 0.666527
\(506\) −1.21143 −0.0538545
\(507\) −15.4580 −0.686513
\(508\) 22.2684 0.987998
\(509\) −17.3850 −0.770575 −0.385287 0.922797i \(-0.625898\pi\)
−0.385287 + 0.922797i \(0.625898\pi\)
\(510\) −25.1834 −1.11514
\(511\) 51.2749 2.26827
\(512\) 1.00000 0.0441942
\(513\) 14.1736 0.625778
\(514\) 8.91386 0.393174
\(515\) 51.1719 2.25490
\(516\) −1.76217 −0.0775752
\(517\) −10.6479 −0.468293
\(518\) −12.8553 −0.564829
\(519\) 9.27754 0.407239
\(520\) −3.01759 −0.132330
\(521\) 10.6051 0.464619 0.232309 0.972642i \(-0.425372\pi\)
0.232309 + 0.972642i \(0.425372\pi\)
\(522\) 3.10838 0.136050
\(523\) −23.4166 −1.02394 −0.511969 0.859004i \(-0.671084\pi\)
−0.511969 + 0.859004i \(0.671084\pi\)
\(524\) −1.00000 −0.0436852
\(525\) −15.2998 −0.667738
\(526\) −20.1270 −0.877578
\(527\) 30.2521 1.31780
\(528\) −1.57338 −0.0684726
\(529\) 1.00000 0.0434783
\(530\) 8.69193 0.377553
\(531\) 3.85915 0.167473
\(532\) 9.05283 0.392490
\(533\) 4.22724 0.183102
\(534\) −10.9083 −0.472049
\(535\) 19.5531 0.845353
\(536\) −12.0647 −0.521117
\(537\) 1.49515 0.0645205
\(538\) −4.97063 −0.214299
\(539\) −7.02859 −0.302743
\(540\) −16.1314 −0.694185
\(541\) −20.1763 −0.867445 −0.433723 0.901046i \(-0.642800\pi\)
−0.433723 + 0.901046i \(0.642800\pi\)
\(542\) 0.105264 0.00452147
\(543\) 13.8671 0.595093
\(544\) −6.73345 −0.288694
\(545\) 23.5216 1.00756
\(546\) 4.86960 0.208400
\(547\) −41.1622 −1.75997 −0.879984 0.475004i \(-0.842447\pi\)
−0.879984 + 0.475004i \(0.842447\pi\)
\(548\) −13.2723 −0.566964
\(549\) 0.948517 0.0404817
\(550\) −3.98850 −0.170070
\(551\) 5.98911 0.255145
\(552\) 1.29878 0.0552799
\(553\) −29.2480 −1.24375
\(554\) 2.27158 0.0965104
\(555\) 13.4376 0.570393
\(556\) −4.60002 −0.195084
\(557\) 1.30671 0.0553670 0.0276835 0.999617i \(-0.491187\pi\)
0.0276835 + 0.999617i \(0.491187\pi\)
\(558\) 5.89979 0.249758
\(559\) 1.42178 0.0601347
\(560\) −10.3033 −0.435395
\(561\) 10.5943 0.447291
\(562\) 26.1385 1.10259
\(563\) −37.6195 −1.58547 −0.792737 0.609563i \(-0.791345\pi\)
−0.792737 + 0.609563i \(0.791345\pi\)
\(564\) 11.4157 0.480687
\(565\) −9.40371 −0.395617
\(566\) −5.64096 −0.237107
\(567\) 11.9365 0.501286
\(568\) −1.34627 −0.0564882
\(569\) −4.25748 −0.178483 −0.0892414 0.996010i \(-0.528444\pi\)
−0.0892414 + 0.996010i \(0.528444\pi\)
\(570\) −9.46288 −0.396356
\(571\) −14.2901 −0.598021 −0.299011 0.954250i \(-0.596657\pi\)
−0.299011 + 0.954250i \(0.596657\pi\)
\(572\) 1.26945 0.0530786
\(573\) 9.03590 0.377480
\(574\) 14.4336 0.602446
\(575\) 3.29240 0.137302
\(576\) −1.31316 −0.0547152
\(577\) −16.7062 −0.695490 −0.347745 0.937589i \(-0.613053\pi\)
−0.347745 + 0.937589i \(0.613053\pi\)
\(578\) 28.3394 1.17876
\(579\) −23.7935 −0.988823
\(580\) −6.81641 −0.283036
\(581\) 9.19968 0.381667
\(582\) −2.36398 −0.0979900
\(583\) −3.65657 −0.151440
\(584\) −14.3307 −0.593008
\(585\) 3.96259 0.163833
\(586\) −25.3398 −1.04678
\(587\) −23.1764 −0.956591 −0.478295 0.878199i \(-0.658745\pi\)
−0.478295 + 0.878199i \(0.658745\pi\)
\(588\) 7.53541 0.310755
\(589\) 11.3675 0.468389
\(590\) −8.46277 −0.348407
\(591\) −9.09096 −0.373952
\(592\) 3.59289 0.147667
\(593\) −36.2015 −1.48662 −0.743308 0.668949i \(-0.766744\pi\)
−0.743308 + 0.668949i \(0.766744\pi\)
\(594\) 6.78625 0.278443
\(595\) 69.3769 2.84418
\(596\) 11.8855 0.486847
\(597\) −12.5755 −0.514679
\(598\) −1.04790 −0.0428518
\(599\) −46.1700 −1.88645 −0.943227 0.332148i \(-0.892227\pi\)
−0.943227 + 0.332148i \(0.892227\pi\)
\(600\) 4.27611 0.174571
\(601\) 14.5421 0.593186 0.296593 0.955004i \(-0.404150\pi\)
0.296593 + 0.955004i \(0.404150\pi\)
\(602\) 4.85455 0.197857
\(603\) 15.8430 0.645175
\(604\) −16.0358 −0.652487
\(605\) −27.4501 −1.11601
\(606\) 6.75554 0.274425
\(607\) 26.1879 1.06293 0.531466 0.847080i \(-0.321641\pi\)
0.531466 + 0.847080i \(0.321641\pi\)
\(608\) −2.53015 −0.102611
\(609\) 10.9999 0.445739
\(610\) −2.08001 −0.0842172
\(611\) −9.21054 −0.372619
\(612\) 8.84213 0.357422
\(613\) 15.2030 0.614045 0.307022 0.951702i \(-0.400667\pi\)
0.307022 + 0.951702i \(0.400667\pi\)
\(614\) 30.2571 1.22108
\(615\) −15.0874 −0.608381
\(616\) 4.33446 0.174640
\(617\) −35.3715 −1.42400 −0.712001 0.702178i \(-0.752211\pi\)
−0.712001 + 0.702178i \(0.752211\pi\)
\(618\) 23.0796 0.928397
\(619\) −40.6458 −1.63369 −0.816846 0.576855i \(-0.804280\pi\)
−0.816846 + 0.576855i \(0.804280\pi\)
\(620\) −12.9377 −0.519591
\(621\) −5.60186 −0.224795
\(622\) 10.5052 0.421219
\(623\) 30.0510 1.20397
\(624\) −1.36099 −0.0544834
\(625\) −30.6221 −1.22488
\(626\) 19.7062 0.787618
\(627\) 3.98090 0.158982
\(628\) 9.20977 0.367510
\(629\) −24.1926 −0.964621
\(630\) 13.5300 0.539046
\(631\) −7.47816 −0.297701 −0.148850 0.988860i \(-0.547557\pi\)
−0.148850 + 0.988860i \(0.547557\pi\)
\(632\) 8.17446 0.325163
\(633\) −1.20334 −0.0478285
\(634\) −23.6058 −0.937505
\(635\) 64.1251 2.54473
\(636\) 3.92024 0.155448
\(637\) −6.07982 −0.240891
\(638\) 2.86756 0.113528
\(639\) 1.76787 0.0699360
\(640\) 2.87965 0.113828
\(641\) −26.4093 −1.04310 −0.521552 0.853220i \(-0.674647\pi\)
−0.521552 + 0.853220i \(0.674647\pi\)
\(642\) 8.81884 0.348052
\(643\) −25.1848 −0.993193 −0.496597 0.867982i \(-0.665417\pi\)
−0.496597 + 0.867982i \(0.665417\pi\)
\(644\) −3.57798 −0.140992
\(645\) −5.07444 −0.199806
\(646\) 17.0367 0.670299
\(647\) 17.5582 0.690286 0.345143 0.938550i \(-0.387830\pi\)
0.345143 + 0.938550i \(0.387830\pi\)
\(648\) −3.33611 −0.131055
\(649\) 3.56017 0.139749
\(650\) −3.45010 −0.135324
\(651\) 20.8781 0.818277
\(652\) −7.37833 −0.288958
\(653\) −2.55328 −0.0999174 −0.0499587 0.998751i \(-0.515909\pi\)
−0.0499587 + 0.998751i \(0.515909\pi\)
\(654\) 10.6087 0.414834
\(655\) −2.87965 −0.112517
\(656\) −4.03401 −0.157502
\(657\) 18.8186 0.734182
\(658\) −31.4487 −1.22600
\(659\) −35.2149 −1.37178 −0.685889 0.727706i \(-0.740586\pi\)
−0.685889 + 0.727706i \(0.740586\pi\)
\(660\) −4.53079 −0.176361
\(661\) 13.5417 0.526712 0.263356 0.964699i \(-0.415171\pi\)
0.263356 + 0.964699i \(0.415171\pi\)
\(662\) 24.5531 0.954282
\(663\) 9.16419 0.355908
\(664\) −2.57120 −0.0997818
\(665\) 26.0690 1.01091
\(666\) −4.71806 −0.182821
\(667\) −2.36709 −0.0916542
\(668\) −9.50905 −0.367916
\(669\) −32.8065 −1.26837
\(670\) −34.7422 −1.34221
\(671\) 0.875031 0.0337802
\(672\) −4.64701 −0.179262
\(673\) 10.5669 0.407322 0.203661 0.979041i \(-0.434716\pi\)
0.203661 + 0.979041i \(0.434716\pi\)
\(674\) 0.827698 0.0318817
\(675\) −18.4435 −0.709892
\(676\) −11.9019 −0.457766
\(677\) −24.6734 −0.948274 −0.474137 0.880451i \(-0.657240\pi\)
−0.474137 + 0.880451i \(0.657240\pi\)
\(678\) −4.24127 −0.162885
\(679\) 6.51245 0.249925
\(680\) −19.3900 −0.743572
\(681\) 15.6093 0.598151
\(682\) 5.44271 0.208412
\(683\) −44.9833 −1.72124 −0.860618 0.509251i \(-0.829923\pi\)
−0.860618 + 0.509251i \(0.829923\pi\)
\(684\) 3.32251 0.127039
\(685\) −38.2196 −1.46029
\(686\) 4.28675 0.163669
\(687\) 28.0371 1.06968
\(688\) −1.35679 −0.0517270
\(689\) −3.16298 −0.120500
\(690\) 3.74004 0.142381
\(691\) −17.4954 −0.665558 −0.332779 0.943005i \(-0.607986\pi\)
−0.332779 + 0.943005i \(0.607986\pi\)
\(692\) 7.14326 0.271546
\(693\) −5.69185 −0.216216
\(694\) −1.61910 −0.0614601
\(695\) −13.2465 −0.502467
\(696\) −3.07434 −0.116533
\(697\) 27.1628 1.02886
\(698\) −10.1191 −0.383012
\(699\) 1.00328 0.0379475
\(700\) −11.7801 −0.445246
\(701\) −12.4027 −0.468445 −0.234222 0.972183i \(-0.575254\pi\)
−0.234222 + 0.972183i \(0.575254\pi\)
\(702\) 5.87019 0.221556
\(703\) −9.09057 −0.342858
\(704\) −1.21143 −0.0456574
\(705\) 32.8732 1.23808
\(706\) −20.1340 −0.757753
\(707\) −18.6106 −0.699924
\(708\) −3.81689 −0.143447
\(709\) 14.7913 0.555500 0.277750 0.960653i \(-0.410411\pi\)
0.277750 + 0.960653i \(0.410411\pi\)
\(710\) −3.87679 −0.145493
\(711\) −10.7344 −0.402572
\(712\) −8.39887 −0.314761
\(713\) −4.49280 −0.168257
\(714\) 31.2904 1.17101
\(715\) 3.65559 0.136711
\(716\) 1.15119 0.0430222
\(717\) 14.7514 0.550900
\(718\) −18.7206 −0.698646
\(719\) 36.8121 1.37286 0.686431 0.727195i \(-0.259177\pi\)
0.686431 + 0.727195i \(0.259177\pi\)
\(720\) −3.78146 −0.140927
\(721\) −63.5812 −2.36789
\(722\) −12.5983 −0.468861
\(723\) −10.8311 −0.402814
\(724\) 10.6770 0.396807
\(725\) −7.79341 −0.289440
\(726\) −12.3806 −0.459486
\(727\) 5.97159 0.221474 0.110737 0.993850i \(-0.464679\pi\)
0.110737 + 0.993850i \(0.464679\pi\)
\(728\) 3.74936 0.138961
\(729\) 26.2077 0.970654
\(730\) −41.2674 −1.52738
\(731\) 9.13585 0.337902
\(732\) −0.938128 −0.0346742
\(733\) 5.02248 0.185509 0.0927547 0.995689i \(-0.470433\pi\)
0.0927547 + 0.995689i \(0.470433\pi\)
\(734\) 27.6525 1.02067
\(735\) 21.6994 0.800393
\(736\) 1.00000 0.0368605
\(737\) 14.6155 0.538370
\(738\) 5.29731 0.194997
\(739\) −15.0999 −0.555459 −0.277729 0.960659i \(-0.589582\pi\)
−0.277729 + 0.960659i \(0.589582\pi\)
\(740\) 10.3463 0.380337
\(741\) 3.44352 0.126501
\(742\) −10.7997 −0.396471
\(743\) −22.4975 −0.825353 −0.412676 0.910878i \(-0.635406\pi\)
−0.412676 + 0.910878i \(0.635406\pi\)
\(744\) −5.83517 −0.213928
\(745\) 34.2260 1.25394
\(746\) −16.0419 −0.587337
\(747\) 3.37640 0.123536
\(748\) 8.15709 0.298253
\(749\) −24.2947 −0.887710
\(750\) −6.38651 −0.233202
\(751\) −6.19706 −0.226134 −0.113067 0.993587i \(-0.536067\pi\)
−0.113067 + 0.993587i \(0.536067\pi\)
\(752\) 8.78952 0.320521
\(753\) 2.48901 0.0907045
\(754\) 2.48048 0.0903337
\(755\) −46.1775 −1.68057
\(756\) 20.0433 0.728969
\(757\) 40.3843 1.46779 0.733897 0.679261i \(-0.237699\pi\)
0.733897 + 0.679261i \(0.237699\pi\)
\(758\) 6.99216 0.253967
\(759\) −1.57338 −0.0571101
\(760\) −7.28596 −0.264290
\(761\) −23.0516 −0.835620 −0.417810 0.908534i \(-0.637202\pi\)
−0.417810 + 0.908534i \(0.637202\pi\)
\(762\) 28.9217 1.04772
\(763\) −29.2257 −1.05804
\(764\) 6.95720 0.251703
\(765\) 25.4622 0.920589
\(766\) 18.2335 0.658803
\(767\) 3.07959 0.111197
\(768\) 1.29878 0.0468658
\(769\) 37.1987 1.34142 0.670709 0.741721i \(-0.265990\pi\)
0.670709 + 0.741721i \(0.265990\pi\)
\(770\) 12.4817 0.449810
\(771\) 11.5772 0.416942
\(772\) −18.3198 −0.659345
\(773\) −38.0295 −1.36783 −0.683913 0.729564i \(-0.739723\pi\)
−0.683913 + 0.729564i \(0.739723\pi\)
\(774\) 1.78168 0.0640413
\(775\) −14.7921 −0.531347
\(776\) −1.82015 −0.0653395
\(777\) −16.6962 −0.598974
\(778\) −18.8141 −0.674518
\(779\) 10.2067 0.365691
\(780\) −3.91919 −0.140329
\(781\) 1.63091 0.0583585
\(782\) −6.73345 −0.240788
\(783\) 13.2601 0.473878
\(784\) 5.80191 0.207211
\(785\) 26.5209 0.946573
\(786\) −1.29878 −0.0463260
\(787\) 24.2669 0.865022 0.432511 0.901629i \(-0.357628\pi\)
0.432511 + 0.901629i \(0.357628\pi\)
\(788\) −6.99960 −0.249350
\(789\) −26.1406 −0.930629
\(790\) 23.5396 0.837501
\(791\) 11.6841 0.415440
\(792\) 1.59080 0.0565267
\(793\) 0.756913 0.0268788
\(794\) 16.8205 0.596936
\(795\) 11.2889 0.400377
\(796\) −9.68250 −0.343187
\(797\) 25.9410 0.918879 0.459439 0.888209i \(-0.348050\pi\)
0.459439 + 0.888209i \(0.348050\pi\)
\(798\) 11.7577 0.416216
\(799\) −59.1838 −2.09377
\(800\) 3.29240 0.116404
\(801\) 11.0291 0.389694
\(802\) 24.0199 0.848173
\(803\) 17.3606 0.612642
\(804\) −15.6694 −0.552619
\(805\) −10.3033 −0.363144
\(806\) 4.70801 0.165832
\(807\) −6.45577 −0.227254
\(808\) 5.20144 0.182986
\(809\) −46.6891 −1.64150 −0.820750 0.571287i \(-0.806444\pi\)
−0.820750 + 0.571287i \(0.806444\pi\)
\(810\) −9.60683 −0.337550
\(811\) −23.9964 −0.842629 −0.421314 0.906915i \(-0.638431\pi\)
−0.421314 + 0.906915i \(0.638431\pi\)
\(812\) 8.46940 0.297218
\(813\) 0.136715 0.00479480
\(814\) −4.35253 −0.152556
\(815\) −21.2470 −0.744251
\(816\) −8.74529 −0.306146
\(817\) 3.43288 0.120101
\(818\) −1.45554 −0.0508917
\(819\) −4.92352 −0.172042
\(820\) −11.6165 −0.405667
\(821\) −21.3105 −0.743743 −0.371872 0.928284i \(-0.621284\pi\)
−0.371872 + 0.928284i \(0.621284\pi\)
\(822\) −17.2378 −0.601238
\(823\) 33.3082 1.16105 0.580525 0.814243i \(-0.302847\pi\)
0.580525 + 0.814243i \(0.302847\pi\)
\(824\) 17.7702 0.619053
\(825\) −5.18019 −0.180351
\(826\) 10.5150 0.365864
\(827\) 48.1541 1.67448 0.837242 0.546833i \(-0.184167\pi\)
0.837242 + 0.546833i \(0.184167\pi\)
\(828\) −1.31316 −0.0456356
\(829\) 53.8596 1.87062 0.935311 0.353827i \(-0.115120\pi\)
0.935311 + 0.353827i \(0.115120\pi\)
\(830\) −7.40416 −0.257002
\(831\) 2.95029 0.102345
\(832\) −1.04790 −0.0363294
\(833\) −39.0668 −1.35359
\(834\) −5.97442 −0.206877
\(835\) −27.3828 −0.947619
\(836\) 3.06510 0.106009
\(837\) 25.1681 0.869935
\(838\) 33.6359 1.16193
\(839\) 7.76511 0.268081 0.134041 0.990976i \(-0.457205\pi\)
0.134041 + 0.990976i \(0.457205\pi\)
\(840\) −13.3818 −0.461715
\(841\) −23.3969 −0.806789
\(842\) −29.1473 −1.00448
\(843\) 33.9482 1.16924
\(844\) −0.926514 −0.0318919
\(845\) −34.2733 −1.17904
\(846\) −11.5421 −0.396825
\(847\) 34.1068 1.17193
\(848\) 3.01840 0.103652
\(849\) −7.32638 −0.251441
\(850\) −22.1692 −0.760397
\(851\) 3.59289 0.123163
\(852\) −1.74851 −0.0599030
\(853\) −27.8503 −0.953575 −0.476788 0.879019i \(-0.658199\pi\)
−0.476788 + 0.879019i \(0.658199\pi\)
\(854\) 2.58442 0.0884371
\(855\) 9.56766 0.327207
\(856\) 6.79008 0.232080
\(857\) −39.2785 −1.34173 −0.670865 0.741580i \(-0.734077\pi\)
−0.670865 + 0.741580i \(0.734077\pi\)
\(858\) 1.64875 0.0562873
\(859\) −31.8716 −1.08745 −0.543723 0.839265i \(-0.682986\pi\)
−0.543723 + 0.839265i \(0.682986\pi\)
\(860\) −3.90707 −0.133230
\(861\) 18.7461 0.638865
\(862\) −28.2169 −0.961072
\(863\) −35.5983 −1.21178 −0.605890 0.795549i \(-0.707183\pi\)
−0.605890 + 0.795549i \(0.707183\pi\)
\(864\) −5.60186 −0.190579
\(865\) 20.5701 0.699405
\(866\) −1.85854 −0.0631558
\(867\) 36.8067 1.25002
\(868\) 16.0751 0.545626
\(869\) −9.90277 −0.335928
\(870\) −8.85303 −0.300146
\(871\) 12.6426 0.428379
\(872\) 8.16821 0.276611
\(873\) 2.39015 0.0808945
\(874\) −2.53015 −0.0855837
\(875\) 17.5940 0.594786
\(876\) −18.6125 −0.628857
\(877\) 11.3252 0.382426 0.191213 0.981549i \(-0.438758\pi\)
0.191213 + 0.981549i \(0.438758\pi\)
\(878\) 28.4457 0.959995
\(879\) −32.9109 −1.11006
\(880\) −3.48849 −0.117597
\(881\) −35.6260 −1.20027 −0.600134 0.799899i \(-0.704886\pi\)
−0.600134 + 0.799899i \(0.704886\pi\)
\(882\) −7.61885 −0.256540
\(883\) 20.4132 0.686960 0.343480 0.939160i \(-0.388394\pi\)
0.343480 + 0.939160i \(0.388394\pi\)
\(884\) 7.05598 0.237318
\(885\) −10.9913 −0.369469
\(886\) −12.7941 −0.429827
\(887\) 28.9673 0.972627 0.486314 0.873784i \(-0.338341\pi\)
0.486314 + 0.873784i \(0.338341\pi\)
\(888\) 4.66639 0.156594
\(889\) −79.6756 −2.67223
\(890\) −24.1858 −0.810711
\(891\) 4.04145 0.135394
\(892\) −25.2594 −0.845749
\(893\) −22.2388 −0.744194
\(894\) 15.4366 0.516278
\(895\) 3.31504 0.110810
\(896\) −3.57798 −0.119532
\(897\) −1.36099 −0.0454423
\(898\) −3.35017 −0.111797
\(899\) 10.6349 0.354693
\(900\) −4.32346 −0.144115
\(901\) −20.3242 −0.677098
\(902\) 4.88691 0.162716
\(903\) 6.30500 0.209817
\(904\) −3.26557 −0.108611
\(905\) 30.7460 1.02203
\(906\) −20.8270 −0.691931
\(907\) 58.2775 1.93507 0.967537 0.252730i \(-0.0813285\pi\)
0.967537 + 0.252730i \(0.0813285\pi\)
\(908\) 12.0184 0.398846
\(909\) −6.83034 −0.226548
\(910\) 10.7969 0.357912
\(911\) 23.8413 0.789899 0.394949 0.918703i \(-0.370762\pi\)
0.394949 + 0.918703i \(0.370762\pi\)
\(912\) −3.28612 −0.108814
\(913\) 3.11482 0.103086
\(914\) −9.79457 −0.323975
\(915\) −2.70148 −0.0893083
\(916\) 21.5872 0.713263
\(917\) 3.57798 0.118155
\(918\) 37.7199 1.24494
\(919\) −8.95728 −0.295474 −0.147737 0.989027i \(-0.547199\pi\)
−0.147737 + 0.989027i \(0.547199\pi\)
\(920\) 2.87965 0.0949393
\(921\) 39.2974 1.29489
\(922\) −3.92003 −0.129099
\(923\) 1.41076 0.0464356
\(924\) 5.62952 0.185198
\(925\) 11.8292 0.388943
\(926\) 30.5027 1.00238
\(927\) −23.3351 −0.766427
\(928\) −2.36709 −0.0777037
\(929\) 6.77862 0.222399 0.111200 0.993798i \(-0.464531\pi\)
0.111200 + 0.993798i \(0.464531\pi\)
\(930\) −16.8033 −0.551001
\(931\) −14.6797 −0.481108
\(932\) 0.772477 0.0253033
\(933\) 13.6439 0.446683
\(934\) 38.6037 1.26315
\(935\) 23.4896 0.768191
\(936\) 1.37606 0.0449781
\(937\) 9.72972 0.317856 0.158928 0.987290i \(-0.449196\pi\)
0.158928 + 0.987290i \(0.449196\pi\)
\(938\) 43.1673 1.40946
\(939\) 25.5941 0.835231
\(940\) 25.3108 0.825546
\(941\) −57.7289 −1.88191 −0.940955 0.338532i \(-0.890070\pi\)
−0.940955 + 0.338532i \(0.890070\pi\)
\(942\) 11.9615 0.389726
\(943\) −4.03401 −0.131365
\(944\) −2.93882 −0.0956504
\(945\) 57.7178 1.87756
\(946\) 1.64365 0.0534396
\(947\) −19.1054 −0.620843 −0.310421 0.950599i \(-0.600470\pi\)
−0.310421 + 0.950599i \(0.600470\pi\)
\(948\) 10.6168 0.344819
\(949\) 15.0171 0.487477
\(950\) −8.33027 −0.270270
\(951\) −30.6588 −0.994178
\(952\) 24.0921 0.780830
\(953\) 2.67308 0.0865896 0.0432948 0.999062i \(-0.486215\pi\)
0.0432948 + 0.999062i \(0.486215\pi\)
\(954\) −3.96365 −0.128328
\(955\) 20.0343 0.648296
\(956\) 11.3578 0.367339
\(957\) 3.72434 0.120391
\(958\) 14.5801 0.471061
\(959\) 47.4879 1.53346
\(960\) 3.74004 0.120709
\(961\) −10.8147 −0.348862
\(962\) −3.76499 −0.121388
\(963\) −8.91649 −0.287330
\(964\) −8.33944 −0.268595
\(965\) −52.7547 −1.69823
\(966\) −4.64701 −0.149515
\(967\) 23.3728 0.751617 0.375809 0.926697i \(-0.377365\pi\)
0.375809 + 0.926697i \(0.377365\pi\)
\(968\) −9.53244 −0.306384
\(969\) 22.1269 0.710819
\(970\) −5.24139 −0.168291
\(971\) 7.56455 0.242758 0.121379 0.992606i \(-0.461268\pi\)
0.121379 + 0.992606i \(0.461268\pi\)
\(972\) 12.4727 0.400062
\(973\) 16.4588 0.527643
\(974\) −27.0085 −0.865407
\(975\) −4.48093 −0.143505
\(976\) −0.722314 −0.0231207
\(977\) −7.26805 −0.232526 −0.116263 0.993218i \(-0.537091\pi\)
−0.116263 + 0.993218i \(0.537091\pi\)
\(978\) −9.58285 −0.306426
\(979\) 10.1746 0.325182
\(980\) 16.7075 0.533701
\(981\) −10.7262 −0.342461
\(982\) 3.42553 0.109313
\(983\) 8.27376 0.263892 0.131946 0.991257i \(-0.457877\pi\)
0.131946 + 0.991257i \(0.457877\pi\)
\(984\) −5.23930 −0.167023
\(985\) −20.1564 −0.642237
\(986\) 15.9387 0.507592
\(987\) −40.8450 −1.30011
\(988\) 2.65135 0.0843506
\(989\) −1.35679 −0.0431433
\(990\) 4.58096 0.145592
\(991\) 51.1780 1.62572 0.812861 0.582457i \(-0.197909\pi\)
0.812861 + 0.582457i \(0.197909\pi\)
\(992\) −4.49280 −0.142647
\(993\) 31.8891 1.01197
\(994\) 4.81692 0.152783
\(995\) −27.8822 −0.883926
\(996\) −3.33943 −0.105814
\(997\) −21.9378 −0.694776 −0.347388 0.937721i \(-0.612931\pi\)
−0.347388 + 0.937721i \(0.612931\pi\)
\(998\) 24.9775 0.790648
\(999\) −20.1269 −0.636787
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 6026.2.a.f.1.15 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6026.2.a.f.1.15 20 1.1 even 1 trivial