Properties

Label 1003.2.a.f.1.1
Level $1003$
Weight $2$
Character 1003.1
Self dual yes
Analytic conductor $8.009$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1003,2,Mod(1,1003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1003, 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("1003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1003 = 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1003.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.00899532273\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.75660\) of defining polynomial
Character \(\chi\) \(=\) 1003.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.75660 q^{2} +1.84224 q^{3} +5.59883 q^{4} -0.0856374 q^{5} -5.07830 q^{6} -0.236068 q^{7} -9.92054 q^{8} +0.393832 q^{9} +O(q^{10})\) \(q-2.75660 q^{2} +1.84224 q^{3} +5.59883 q^{4} -0.0856374 q^{5} -5.07830 q^{6} -0.236068 q^{7} -9.92054 q^{8} +0.393832 q^{9} +0.236068 q^{10} -2.52053 q^{11} +10.3144 q^{12} -0.243402 q^{13} +0.650745 q^{14} -0.157764 q^{15} +16.1493 q^{16} -1.00000 q^{17} -1.08564 q^{18} +0.722871 q^{19} -0.479470 q^{20} -0.434893 q^{21} +6.94809 q^{22} -7.44107 q^{23} -18.2760 q^{24} -4.99267 q^{25} +0.670961 q^{26} -4.80118 q^{27} -1.32171 q^{28} +1.99267 q^{29} +0.434893 q^{30} +0.670961 q^{31} -24.6760 q^{32} -4.64341 q^{33} +2.75660 q^{34} +0.0202163 q^{35} +2.20500 q^{36} -8.56511 q^{37} -1.99267 q^{38} -0.448403 q^{39} +0.849569 q^{40} -2.34192 q^{41} +1.19882 q^{42} +5.36277 q^{43} -14.1120 q^{44} -0.0337268 q^{45} +20.5120 q^{46} -5.67829 q^{47} +29.7508 q^{48} -6.94427 q^{49} +13.7628 q^{50} -1.84224 q^{51} -1.36277 q^{52} +6.85575 q^{53} +13.2349 q^{54} +0.215852 q^{55} +2.34192 q^{56} +1.33170 q^{57} -5.49298 q^{58} -1.00000 q^{59} -0.883296 q^{60} +2.52671 q^{61} -1.84957 q^{62} -0.0929712 q^{63} +35.7232 q^{64} +0.0208443 q^{65} +12.8000 q^{66} +1.90703 q^{67} -5.59883 q^{68} -13.7082 q^{69} -0.0557281 q^{70} -2.04458 q^{71} -3.90703 q^{72} +9.29064 q^{73} +23.6106 q^{74} -9.19767 q^{75} +4.04724 q^{76} +0.595016 q^{77} +1.23607 q^{78} +14.9241 q^{79} -1.38298 q^{80} -10.0264 q^{81} +6.45574 q^{82} -6.73638 q^{83} -2.43489 q^{84} +0.0856374 q^{85} -14.7830 q^{86} +3.67096 q^{87} +25.0050 q^{88} -12.3892 q^{89} +0.0929712 q^{90} +0.0574594 q^{91} -41.6613 q^{92} +1.23607 q^{93} +15.6528 q^{94} -0.0619049 q^{95} -45.4590 q^{96} -13.6636 q^{97} +19.1426 q^{98} -0.992666 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} - 2 q^{3} + 5 q^{4} + q^{5} - 2 q^{6} + 8 q^{7} - 12 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} - 2 q^{3} + 5 q^{4} + q^{5} - 2 q^{6} + 8 q^{7} - 12 q^{8} + 2 q^{9} - 8 q^{10} - 11 q^{11} + 14 q^{12} - 9 q^{13} - q^{14} - 10 q^{15} + 11 q^{16} - 4 q^{17} - 3 q^{18} + 10 q^{19} - q^{20} - 4 q^{21} + 14 q^{22} - 3 q^{23} - 20 q^{24} - 3 q^{25} - 4 q^{26} - 8 q^{27} + 5 q^{28} - 9 q^{29} + 4 q^{30} - 4 q^{31} - 17 q^{32} + 2 q^{33} + 3 q^{34} - 3 q^{35} - 9 q^{36} - 32 q^{37} + 9 q^{38} + 8 q^{39} + 11 q^{40} + 4 q^{41} + 16 q^{42} + 13 q^{43} - 23 q^{44} + 15 q^{45} + 20 q^{46} - 33 q^{47} + 24 q^{48} + 8 q^{49} + 18 q^{50} + 2 q^{51} + 3 q^{52} + 6 q^{53} - 2 q^{54} - 5 q^{55} - 4 q^{56} - 12 q^{57} - 9 q^{58} - 4 q^{59} + 4 q^{60} - 18 q^{61} - 15 q^{62} - 16 q^{63} + 34 q^{64} + 5 q^{65} - 6 q^{66} - 8 q^{67} - 5 q^{68} - 28 q^{69} - 36 q^{70} - 5 q^{71} + 18 q^{73} + 11 q^{74} - 2 q^{75} - 11 q^{76} - 37 q^{77} - 4 q^{78} + 27 q^{79} + 6 q^{80} - 8 q^{81} + 33 q^{82} - 22 q^{83} - 12 q^{84} - q^{85} - 19 q^{86} + 8 q^{87} + 25 q^{88} - 9 q^{89} + 16 q^{90} - 23 q^{91} - 51 q^{92} - 4 q^{93} + 27 q^{94} - 7 q^{95} - 60 q^{96} - 31 q^{97} + 14 q^{98} + 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.75660 −1.94921 −0.974605 0.223932i \(-0.928110\pi\)
−0.974605 + 0.223932i \(0.928110\pi\)
\(3\) 1.84224 1.06362 0.531808 0.846865i \(-0.321513\pi\)
0.531808 + 0.846865i \(0.321513\pi\)
\(4\) 5.59883 2.79942
\(5\) −0.0856374 −0.0382982 −0.0191491 0.999817i \(-0.506096\pi\)
−0.0191491 + 0.999817i \(0.506096\pi\)
\(6\) −5.07830 −2.07321
\(7\) −0.236068 −0.0892253 −0.0446127 0.999004i \(-0.514205\pi\)
−0.0446127 + 0.999004i \(0.514205\pi\)
\(8\) −9.92054 −3.50744
\(9\) 0.393832 0.131277
\(10\) 0.236068 0.0746512
\(11\) −2.52053 −0.759968 −0.379984 0.924993i \(-0.624071\pi\)
−0.379984 + 0.924993i \(0.624071\pi\)
\(12\) 10.3144 2.97750
\(13\) −0.243402 −0.0675075 −0.0337537 0.999430i \(-0.510746\pi\)
−0.0337537 + 0.999430i \(0.510746\pi\)
\(14\) 0.650745 0.173919
\(15\) −0.157764 −0.0407346
\(16\) 16.1493 4.03732
\(17\) −1.00000 −0.242536
\(18\) −1.08564 −0.255887
\(19\) 0.722871 0.165838 0.0829190 0.996556i \(-0.473576\pi\)
0.0829190 + 0.996556i \(0.473576\pi\)
\(20\) −0.479470 −0.107213
\(21\) −0.434893 −0.0949014
\(22\) 6.94809 1.48134
\(23\) −7.44107 −1.55157 −0.775785 0.630997i \(-0.782646\pi\)
−0.775785 + 0.630997i \(0.782646\pi\)
\(24\) −18.2760 −3.73057
\(25\) −4.99267 −0.998533
\(26\) 0.670961 0.131586
\(27\) −4.80118 −0.923987
\(28\) −1.32171 −0.249779
\(29\) 1.99267 0.370029 0.185014 0.982736i \(-0.440767\pi\)
0.185014 + 0.982736i \(0.440767\pi\)
\(30\) 0.434893 0.0794002
\(31\) 0.670961 0.120508 0.0602541 0.998183i \(-0.480809\pi\)
0.0602541 + 0.998183i \(0.480809\pi\)
\(32\) −24.6760 −4.36214
\(33\) −4.64341 −0.808314
\(34\) 2.75660 0.472753
\(35\) 0.0202163 0.00341717
\(36\) 2.20500 0.367500
\(37\) −8.56511 −1.40809 −0.704047 0.710153i \(-0.748626\pi\)
−0.704047 + 0.710153i \(0.748626\pi\)
\(38\) −1.99267 −0.323253
\(39\) −0.448403 −0.0718020
\(40\) 0.849569 0.134329
\(41\) −2.34192 −0.365747 −0.182873 0.983136i \(-0.558540\pi\)
−0.182873 + 0.983136i \(0.558540\pi\)
\(42\) 1.19882 0.184983
\(43\) 5.36277 0.817814 0.408907 0.912576i \(-0.365910\pi\)
0.408907 + 0.912576i \(0.365910\pi\)
\(44\) −14.1120 −2.12747
\(45\) −0.0337268 −0.00502769
\(46\) 20.5120 3.02434
\(47\) −5.67829 −0.828264 −0.414132 0.910217i \(-0.635915\pi\)
−0.414132 + 0.910217i \(0.635915\pi\)
\(48\) 29.7508 4.29415
\(49\) −6.94427 −0.992039
\(50\) 13.7628 1.94635
\(51\) −1.84224 −0.257965
\(52\) −1.36277 −0.188982
\(53\) 6.85575 0.941709 0.470855 0.882211i \(-0.343946\pi\)
0.470855 + 0.882211i \(0.343946\pi\)
\(54\) 13.2349 1.80104
\(55\) 0.215852 0.0291054
\(56\) 2.34192 0.312952
\(57\) 1.33170 0.176388
\(58\) −5.49298 −0.721264
\(59\) −1.00000 −0.130189
\(60\) −0.883296 −0.114033
\(61\) 2.52671 0.323512 0.161756 0.986831i \(-0.448284\pi\)
0.161756 + 0.986831i \(0.448284\pi\)
\(62\) −1.84957 −0.234896
\(63\) −0.0929712 −0.0117133
\(64\) 35.7232 4.46540
\(65\) 0.0208443 0.00258542
\(66\) 12.8000 1.57557
\(67\) 1.90703 0.232981 0.116490 0.993192i \(-0.462836\pi\)
0.116490 + 0.993192i \(0.462836\pi\)
\(68\) −5.59883 −0.678958
\(69\) −13.7082 −1.65027
\(70\) −0.0557281 −0.00666078
\(71\) −2.04458 −0.242647 −0.121323 0.992613i \(-0.538714\pi\)
−0.121323 + 0.992613i \(0.538714\pi\)
\(72\) −3.90703 −0.460448
\(73\) 9.29064 1.08739 0.543694 0.839284i \(-0.317025\pi\)
0.543694 + 0.839284i \(0.317025\pi\)
\(74\) 23.6106 2.74467
\(75\) −9.19767 −1.06206
\(76\) 4.04724 0.464250
\(77\) 0.595016 0.0678084
\(78\) 1.23607 0.139957
\(79\) 14.9241 1.67909 0.839544 0.543292i \(-0.182822\pi\)
0.839544 + 0.543292i \(0.182822\pi\)
\(80\) −1.38298 −0.154622
\(81\) −10.0264 −1.11404
\(82\) 6.45574 0.712917
\(83\) −6.73638 −0.739414 −0.369707 0.929148i \(-0.620542\pi\)
−0.369707 + 0.929148i \(0.620542\pi\)
\(84\) −2.43489 −0.265669
\(85\) 0.0856374 0.00928868
\(86\) −14.7830 −1.59409
\(87\) 3.67096 0.393568
\(88\) 25.0050 2.66554
\(89\) −12.3892 −1.31325 −0.656624 0.754218i \(-0.728016\pi\)
−0.656624 + 0.754218i \(0.728016\pi\)
\(90\) 0.0929712 0.00980002
\(91\) 0.0574594 0.00602338
\(92\) −41.6613 −4.34349
\(93\) 1.23607 0.128174
\(94\) 15.6528 1.61446
\(95\) −0.0619049 −0.00635130
\(96\) −45.4590 −4.63964
\(97\) −13.6636 −1.38733 −0.693666 0.720297i \(-0.744005\pi\)
−0.693666 + 0.720297i \(0.744005\pi\)
\(98\) 19.1426 1.93369
\(99\) −0.992666 −0.0997667
\(100\) −27.9531 −2.79531
\(101\) 11.2490 1.11931 0.559656 0.828725i \(-0.310933\pi\)
0.559656 + 0.828725i \(0.310933\pi\)
\(102\) 5.07830 0.502827
\(103\) −7.22140 −0.711546 −0.355773 0.934572i \(-0.615782\pi\)
−0.355773 + 0.934572i \(0.615782\pi\)
\(104\) 2.41468 0.236779
\(105\) 0.0372431 0.00363455
\(106\) −18.8985 −1.83559
\(107\) 5.00618 0.483965 0.241983 0.970281i \(-0.422202\pi\)
0.241983 + 0.970281i \(0.422202\pi\)
\(108\) −26.8810 −2.58662
\(109\) −11.1977 −1.07254 −0.536271 0.844046i \(-0.680168\pi\)
−0.536271 + 0.844046i \(0.680168\pi\)
\(110\) −0.595016 −0.0567326
\(111\) −15.7789 −1.49767
\(112\) −3.81233 −0.360231
\(113\) 7.45863 0.701648 0.350824 0.936441i \(-0.385901\pi\)
0.350824 + 0.936441i \(0.385901\pi\)
\(114\) −3.67096 −0.343817
\(115\) 0.637234 0.0594224
\(116\) 11.1566 1.03586
\(117\) −0.0958595 −0.00886221
\(118\) 2.75660 0.253765
\(119\) 0.236068 0.0216403
\(120\) 1.56511 0.142874
\(121\) −4.64693 −0.422448
\(122\) −6.96512 −0.630592
\(123\) −4.31437 −0.389014
\(124\) 3.75660 0.337352
\(125\) 0.855746 0.0765403
\(126\) 0.256284 0.0228316
\(127\) −9.83842 −0.873018 −0.436509 0.899700i \(-0.643785\pi\)
−0.436509 + 0.899700i \(0.643785\pi\)
\(128\) −49.1226 −4.34187
\(129\) 9.87948 0.869840
\(130\) −0.0574594 −0.00503952
\(131\) −8.72553 −0.762353 −0.381177 0.924502i \(-0.624481\pi\)
−0.381177 + 0.924502i \(0.624481\pi\)
\(132\) −25.9977 −2.26281
\(133\) −0.170647 −0.0147970
\(134\) −5.25691 −0.454128
\(135\) 0.411160 0.0353870
\(136\) 9.92054 0.850679
\(137\) 12.2451 1.04617 0.523086 0.852280i \(-0.324781\pi\)
0.523086 + 0.852280i \(0.324781\pi\)
\(138\) 37.7880 3.21673
\(139\) 16.3317 1.38524 0.692618 0.721304i \(-0.256457\pi\)
0.692618 + 0.721304i \(0.256457\pi\)
\(140\) 0.113187 0.00956609
\(141\) −10.4608 −0.880955
\(142\) 5.63608 0.472969
\(143\) 0.613501 0.0513036
\(144\) 6.36011 0.530009
\(145\) −0.170647 −0.0141714
\(146\) −25.6106 −2.11954
\(147\) −12.7930 −1.05515
\(148\) −47.9546 −3.94184
\(149\) 5.82820 0.477464 0.238732 0.971085i \(-0.423268\pi\)
0.238732 + 0.971085i \(0.423268\pi\)
\(150\) 25.3543 2.07017
\(151\) −15.9408 −1.29724 −0.648620 0.761112i \(-0.724654\pi\)
−0.648620 + 0.761112i \(0.724654\pi\)
\(152\) −7.17127 −0.581667
\(153\) −0.393832 −0.0318395
\(154\) −1.64022 −0.132173
\(155\) −0.0574594 −0.00461525
\(156\) −2.51054 −0.201004
\(157\) −2.38766 −0.190556 −0.0952778 0.995451i \(-0.530374\pi\)
−0.0952778 + 0.995451i \(0.530374\pi\)
\(158\) −41.1396 −3.27289
\(159\) 12.6299 1.00162
\(160\) 2.11319 0.167062
\(161\) 1.75660 0.138439
\(162\) 27.6387 2.17150
\(163\) 4.21585 0.330211 0.165105 0.986276i \(-0.447204\pi\)
0.165105 + 0.986276i \(0.447204\pi\)
\(164\) −13.1120 −1.02388
\(165\) 0.397650 0.0309570
\(166\) 18.5695 1.44127
\(167\) −12.4519 −0.963558 −0.481779 0.876293i \(-0.660009\pi\)
−0.481779 + 0.876293i \(0.660009\pi\)
\(168\) 4.31437 0.332861
\(169\) −12.9408 −0.995443
\(170\) −0.236068 −0.0181056
\(171\) 0.284690 0.0217708
\(172\) 30.0252 2.28940
\(173\) −2.93902 −0.223450 −0.111725 0.993739i \(-0.535638\pi\)
−0.111725 + 0.993739i \(0.535638\pi\)
\(174\) −10.1194 −0.767147
\(175\) 1.17861 0.0890944
\(176\) −40.7047 −3.06823
\(177\) −1.84224 −0.138471
\(178\) 34.1519 2.55980
\(179\) 20.6862 1.54616 0.773080 0.634309i \(-0.218715\pi\)
0.773080 + 0.634309i \(0.218715\pi\)
\(180\) −0.188831 −0.0140746
\(181\) −5.55044 −0.412561 −0.206280 0.978493i \(-0.566136\pi\)
−0.206280 + 0.978493i \(0.566136\pi\)
\(182\) −0.158392 −0.0117408
\(183\) 4.65479 0.344092
\(184\) 73.8194 5.44204
\(185\) 0.733494 0.0539275
\(186\) −3.40734 −0.249838
\(187\) 2.52053 0.184319
\(188\) −31.7918 −2.31866
\(189\) 1.13340 0.0824430
\(190\) 0.170647 0.0123800
\(191\) −11.3252 −0.819464 −0.409732 0.912206i \(-0.634378\pi\)
−0.409732 + 0.912206i \(0.634378\pi\)
\(192\) 65.8106 4.74947
\(193\) 4.21937 0.303717 0.151858 0.988402i \(-0.451474\pi\)
0.151858 + 0.988402i \(0.451474\pi\)
\(194\) 37.6651 2.70420
\(195\) 0.0384001 0.00274989
\(196\) −38.8798 −2.77713
\(197\) −15.8478 −1.12911 −0.564554 0.825396i \(-0.690952\pi\)
−0.564554 + 0.825396i \(0.690952\pi\)
\(198\) 2.73638 0.194466
\(199\) 11.1988 0.793864 0.396932 0.917848i \(-0.370075\pi\)
0.396932 + 0.917848i \(0.370075\pi\)
\(200\) 49.5299 3.50230
\(201\) 3.51320 0.247802
\(202\) −31.0088 −2.18177
\(203\) −0.470405 −0.0330159
\(204\) −10.3144 −0.722150
\(205\) 0.200556 0.0140074
\(206\) 19.9065 1.38695
\(207\) −2.93053 −0.203686
\(208\) −3.93076 −0.272549
\(209\) −1.82202 −0.126032
\(210\) −0.102664 −0.00708451
\(211\) 13.1701 0.906668 0.453334 0.891341i \(-0.350234\pi\)
0.453334 + 0.891341i \(0.350234\pi\)
\(212\) 38.3842 2.63624
\(213\) −3.76659 −0.258083
\(214\) −13.8000 −0.943350
\(215\) −0.459253 −0.0313208
\(216\) 47.6302 3.24083
\(217\) −0.158392 −0.0107524
\(218\) 30.8675 2.09061
\(219\) 17.1155 1.15656
\(220\) 1.20852 0.0814783
\(221\) 0.243402 0.0163730
\(222\) 43.4962 2.91927
\(223\) −15.9948 −1.07109 −0.535545 0.844506i \(-0.679894\pi\)
−0.535545 + 0.844506i \(0.679894\pi\)
\(224\) 5.82521 0.389213
\(225\) −1.96627 −0.131085
\(226\) −20.5604 −1.36766
\(227\) −20.3454 −1.35037 −0.675187 0.737647i \(-0.735937\pi\)
−0.675187 + 0.737647i \(0.735937\pi\)
\(228\) 7.45596 0.493783
\(229\) 15.7818 1.04289 0.521446 0.853284i \(-0.325393\pi\)
0.521446 + 0.853284i \(0.325393\pi\)
\(230\) −1.75660 −0.115827
\(231\) 1.09616 0.0721221
\(232\) −19.7683 −1.29785
\(233\) 11.2247 0.735354 0.367677 0.929954i \(-0.380153\pi\)
0.367677 + 0.929954i \(0.380153\pi\)
\(234\) 0.264246 0.0172743
\(235\) 0.486275 0.0317211
\(236\) −5.59883 −0.364453
\(237\) 27.4936 1.78590
\(238\) −0.650745 −0.0421815
\(239\) 4.13639 0.267561 0.133780 0.991011i \(-0.457288\pi\)
0.133780 + 0.991011i \(0.457288\pi\)
\(240\) −2.54778 −0.164458
\(241\) 25.2416 1.62596 0.812978 0.582295i \(-0.197845\pi\)
0.812978 + 0.582295i \(0.197845\pi\)
\(242\) 12.8097 0.823439
\(243\) −4.06745 −0.260927
\(244\) 14.1466 0.905644
\(245\) 0.594690 0.0379933
\(246\) 11.8930 0.758269
\(247\) −0.175948 −0.0111953
\(248\) −6.65629 −0.422675
\(249\) −12.4100 −0.786452
\(250\) −2.35895 −0.149193
\(251\) 3.30704 0.208738 0.104369 0.994539i \(-0.466718\pi\)
0.104369 + 0.994539i \(0.466718\pi\)
\(252\) −0.520530 −0.0327903
\(253\) 18.7554 1.17914
\(254\) 27.1206 1.70170
\(255\) 0.157764 0.00987958
\(256\) 63.9648 3.99780
\(257\) 4.12318 0.257197 0.128599 0.991697i \(-0.458952\pi\)
0.128599 + 0.991697i \(0.458952\pi\)
\(258\) −27.2338 −1.69550
\(259\) 2.02195 0.125638
\(260\) 0.116704 0.00723766
\(261\) 0.784776 0.0485764
\(262\) 24.0528 1.48599
\(263\) −30.7273 −1.89472 −0.947362 0.320164i \(-0.896262\pi\)
−0.947362 + 0.320164i \(0.896262\pi\)
\(264\) 46.0651 2.83511
\(265\) −0.587108 −0.0360658
\(266\) 0.470405 0.0288424
\(267\) −22.8238 −1.39679
\(268\) 10.6771 0.652210
\(269\) 6.53670 0.398550 0.199275 0.979944i \(-0.436141\pi\)
0.199275 + 0.979944i \(0.436141\pi\)
\(270\) −1.13340 −0.0689768
\(271\) 11.7276 0.712399 0.356199 0.934410i \(-0.384072\pi\)
0.356199 + 0.934410i \(0.384072\pi\)
\(272\) −16.1493 −0.979194
\(273\) 0.105854 0.00640656
\(274\) −33.7549 −2.03921
\(275\) 12.5842 0.758854
\(276\) −76.7500 −4.61980
\(277\) −23.5035 −1.41219 −0.706096 0.708116i \(-0.749545\pi\)
−0.706096 + 0.708116i \(0.749545\pi\)
\(278\) −45.0199 −2.70012
\(279\) 0.264246 0.0158200
\(280\) −0.200556 −0.0119855
\(281\) 23.0865 1.37723 0.688613 0.725129i \(-0.258220\pi\)
0.688613 + 0.725129i \(0.258220\pi\)
\(282\) 28.8361 1.71717
\(283\) −9.04724 −0.537802 −0.268901 0.963168i \(-0.586661\pi\)
−0.268901 + 0.963168i \(0.586661\pi\)
\(284\) −11.4472 −0.679269
\(285\) −0.114043 −0.00675534
\(286\) −1.69118 −0.100001
\(287\) 0.552853 0.0326339
\(288\) −9.71820 −0.572650
\(289\) 1.00000 0.0588235
\(290\) 0.470405 0.0276231
\(291\) −25.1716 −1.47559
\(292\) 52.0167 3.04405
\(293\) −29.5020 −1.72353 −0.861764 0.507309i \(-0.830640\pi\)
−0.861764 + 0.507309i \(0.830640\pi\)
\(294\) 35.2651 2.05670
\(295\) 0.0856374 0.00498600
\(296\) 84.9705 4.93881
\(297\) 12.1015 0.702201
\(298\) −16.0660 −0.930678
\(299\) 1.81117 0.104743
\(300\) −51.4962 −2.97314
\(301\) −1.26598 −0.0729697
\(302\) 43.9423 2.52859
\(303\) 20.7232 1.19052
\(304\) 11.6738 0.669541
\(305\) −0.216381 −0.0123899
\(306\) 1.08564 0.0620618
\(307\) 10.0443 0.573261 0.286631 0.958041i \(-0.407465\pi\)
0.286631 + 0.958041i \(0.407465\pi\)
\(308\) 3.33140 0.189824
\(309\) −13.3035 −0.756811
\(310\) 0.158392 0.00899608
\(311\) 22.6044 1.28178 0.640888 0.767634i \(-0.278566\pi\)
0.640888 + 0.767634i \(0.278566\pi\)
\(312\) 4.44840 0.251841
\(313\) −15.3936 −0.870098 −0.435049 0.900407i \(-0.643269\pi\)
−0.435049 + 0.900407i \(0.643269\pi\)
\(314\) 6.58181 0.371433
\(315\) 0.00796181 0.000448597 0
\(316\) 83.5573 4.70047
\(317\) −0.376276 −0.0211338 −0.0105669 0.999944i \(-0.503364\pi\)
−0.0105669 + 0.999944i \(0.503364\pi\)
\(318\) −34.8156 −1.95236
\(319\) −5.02258 −0.281210
\(320\) −3.05924 −0.171017
\(321\) 9.22256 0.514753
\(322\) −4.84224 −0.269847
\(323\) −0.722871 −0.0402216
\(324\) −56.1361 −3.11867
\(325\) 1.21522 0.0674085
\(326\) −11.6214 −0.643650
\(327\) −20.6287 −1.14077
\(328\) 23.2331 1.28283
\(329\) 1.34046 0.0739021
\(330\) −1.09616 −0.0603417
\(331\) −11.1262 −0.611550 −0.305775 0.952104i \(-0.598915\pi\)
−0.305775 + 0.952104i \(0.598915\pi\)
\(332\) −37.7159 −2.06993
\(333\) −3.37322 −0.184851
\(334\) 34.3249 1.87818
\(335\) −0.163313 −0.00892274
\(336\) −7.02320 −0.383147
\(337\) −22.3740 −1.21879 −0.609394 0.792868i \(-0.708587\pi\)
−0.609394 + 0.792868i \(0.708587\pi\)
\(338\) 35.6725 1.94033
\(339\) 13.7405 0.746284
\(340\) 0.479470 0.0260029
\(341\) −1.69118 −0.0915824
\(342\) −0.784776 −0.0424358
\(343\) 3.29180 0.177740
\(344\) −53.2015 −2.86843
\(345\) 1.17394 0.0632026
\(346\) 8.10171 0.435551
\(347\) 36.6939 1.96983 0.984915 0.173036i \(-0.0553578\pi\)
0.984915 + 0.173036i \(0.0553578\pi\)
\(348\) 20.5531 1.10176
\(349\) −13.9924 −0.748994 −0.374497 0.927228i \(-0.622185\pi\)
−0.374497 + 0.927228i \(0.622185\pi\)
\(350\) −3.24895 −0.173664
\(351\) 1.16861 0.0623760
\(352\) 62.1966 3.31509
\(353\) −12.1757 −0.648045 −0.324023 0.946049i \(-0.605035\pi\)
−0.324023 + 0.946049i \(0.605035\pi\)
\(354\) 5.07830 0.269909
\(355\) 0.175092 0.00929293
\(356\) −69.3648 −3.67633
\(357\) 0.434893 0.0230170
\(358\) −57.0236 −3.01379
\(359\) −14.6095 −0.771058 −0.385529 0.922696i \(-0.625981\pi\)
−0.385529 + 0.922696i \(0.625981\pi\)
\(360\) 0.334588 0.0176343
\(361\) −18.4775 −0.972498
\(362\) 15.3003 0.804168
\(363\) −8.56073 −0.449322
\(364\) 0.321705 0.0168619
\(365\) −0.795626 −0.0416450
\(366\) −12.8314 −0.670707
\(367\) 9.93694 0.518704 0.259352 0.965783i \(-0.416491\pi\)
0.259352 + 0.965783i \(0.416491\pi\)
\(368\) −120.168 −6.26418
\(369\) −0.922324 −0.0480143
\(370\) −2.02195 −0.105116
\(371\) −1.61842 −0.0840243
\(372\) 6.92054 0.358813
\(373\) 3.90703 0.202298 0.101149 0.994871i \(-0.467748\pi\)
0.101149 + 0.994871i \(0.467748\pi\)
\(374\) −6.94809 −0.359277
\(375\) 1.57649 0.0814094
\(376\) 56.3317 2.90509
\(377\) −0.485018 −0.0249797
\(378\) −3.12434 −0.160699
\(379\) −28.6285 −1.47055 −0.735274 0.677770i \(-0.762947\pi\)
−0.735274 + 0.677770i \(0.762947\pi\)
\(380\) −0.346595 −0.0177799
\(381\) −18.1247 −0.928556
\(382\) 31.2191 1.59731
\(383\) 18.4504 0.942773 0.471387 0.881927i \(-0.343754\pi\)
0.471387 + 0.881927i \(0.343754\pi\)
\(384\) −90.4954 −4.61807
\(385\) −0.0509557 −0.00259694
\(386\) −11.6311 −0.592008
\(387\) 2.11203 0.107361
\(388\) −76.5004 −3.88372
\(389\) 8.95894 0.454236 0.227118 0.973867i \(-0.427070\pi\)
0.227118 + 0.973867i \(0.427070\pi\)
\(390\) −0.105854 −0.00536011
\(391\) 7.44107 0.376311
\(392\) 68.8909 3.47952
\(393\) −16.0745 −0.810851
\(394\) 43.6860 2.20087
\(395\) −1.27806 −0.0643061
\(396\) −5.55777 −0.279289
\(397\) 30.8281 1.54722 0.773610 0.633662i \(-0.218449\pi\)
0.773610 + 0.633662i \(0.218449\pi\)
\(398\) −30.8707 −1.54741
\(399\) −0.314372 −0.0157383
\(400\) −80.6279 −4.03140
\(401\) 3.17243 0.158424 0.0792118 0.996858i \(-0.474760\pi\)
0.0792118 + 0.996858i \(0.474760\pi\)
\(402\) −9.68447 −0.483018
\(403\) −0.163313 −0.00813520
\(404\) 62.9810 3.13342
\(405\) 0.858634 0.0426659
\(406\) 1.29672 0.0643550
\(407\) 21.5886 1.07011
\(408\) 18.2760 0.904795
\(409\) 28.5540 1.41190 0.705952 0.708260i \(-0.250520\pi\)
0.705952 + 0.708260i \(0.250520\pi\)
\(410\) −0.552853 −0.0273034
\(411\) 22.5584 1.11272
\(412\) −40.4314 −1.99191
\(413\) 0.236068 0.0116161
\(414\) 8.07830 0.397027
\(415\) 0.576886 0.0283182
\(416\) 6.00618 0.294477
\(417\) 30.0868 1.47336
\(418\) 5.02258 0.245662
\(419\) 24.2074 1.18261 0.591303 0.806449i \(-0.298614\pi\)
0.591303 + 0.806449i \(0.298614\pi\)
\(420\) 0.208518 0.0101746
\(421\) −15.0173 −0.731900 −0.365950 0.930635i \(-0.619256\pi\)
−0.365950 + 0.930635i \(0.619256\pi\)
\(422\) −36.3047 −1.76729
\(423\) −2.23630 −0.108732
\(424\) −68.0127 −3.30299
\(425\) 4.99267 0.242180
\(426\) 10.3830 0.503057
\(427\) −0.596475 −0.0288654
\(428\) 28.0288 1.35482
\(429\) 1.13021 0.0545673
\(430\) 1.26598 0.0610508
\(431\) 16.3024 0.785257 0.392629 0.919697i \(-0.371566\pi\)
0.392629 + 0.919697i \(0.371566\pi\)
\(432\) −77.5355 −3.73043
\(433\) −25.2725 −1.21452 −0.607258 0.794504i \(-0.707731\pi\)
−0.607258 + 0.794504i \(0.707731\pi\)
\(434\) 0.436624 0.0209586
\(435\) −0.314372 −0.0150730
\(436\) −62.6939 −3.00249
\(437\) −5.37894 −0.257309
\(438\) −47.1807 −2.25438
\(439\) 1.02778 0.0490532 0.0245266 0.999699i \(-0.492192\pi\)
0.0245266 + 0.999699i \(0.492192\pi\)
\(440\) −2.14137 −0.102086
\(441\) −2.73488 −0.130232
\(442\) −0.670961 −0.0319144
\(443\) −30.5595 −1.45193 −0.725963 0.687734i \(-0.758606\pi\)
−0.725963 + 0.687734i \(0.758606\pi\)
\(444\) −88.3437 −4.19261
\(445\) 1.06098 0.0502951
\(446\) 44.0912 2.08778
\(447\) 10.7369 0.507838
\(448\) −8.43311 −0.398427
\(449\) 35.1296 1.65787 0.828934 0.559347i \(-0.188948\pi\)
0.828934 + 0.559347i \(0.188948\pi\)
\(450\) 5.42023 0.255512
\(451\) 5.90288 0.277956
\(452\) 41.7596 1.96421
\(453\) −29.3666 −1.37976
\(454\) 56.0842 2.63216
\(455\) −0.00492067 −0.000230685 0
\(456\) −13.2112 −0.618670
\(457\) −35.7939 −1.67437 −0.837183 0.546923i \(-0.815799\pi\)
−0.837183 + 0.546923i \(0.815799\pi\)
\(458\) −43.5042 −2.03282
\(459\) 4.80118 0.224100
\(460\) 3.56777 0.166348
\(461\) 13.3921 0.623734 0.311867 0.950126i \(-0.399046\pi\)
0.311867 + 0.950126i \(0.399046\pi\)
\(462\) −3.02167 −0.140581
\(463\) 35.0042 1.62678 0.813391 0.581718i \(-0.197619\pi\)
0.813391 + 0.581718i \(0.197619\pi\)
\(464\) 32.1801 1.49392
\(465\) −0.105854 −0.00490885
\(466\) −30.9420 −1.43336
\(467\) 31.7710 1.47019 0.735093 0.677966i \(-0.237139\pi\)
0.735093 + 0.677966i \(0.237139\pi\)
\(468\) −0.536701 −0.0248090
\(469\) −0.450188 −0.0207878
\(470\) −1.34046 −0.0618310
\(471\) −4.39862 −0.202678
\(472\) 9.92054 0.456630
\(473\) −13.5170 −0.621513
\(474\) −75.7889 −3.48110
\(475\) −3.60906 −0.165595
\(476\) 1.32171 0.0605803
\(477\) 2.70001 0.123625
\(478\) −11.4024 −0.521532
\(479\) −5.85956 −0.267730 −0.133865 0.991000i \(-0.542739\pi\)
−0.133865 + 0.991000i \(0.542739\pi\)
\(480\) 3.89299 0.177690
\(481\) 2.08476 0.0950570
\(482\) −69.5810 −3.16933
\(483\) 3.23607 0.147246
\(484\) −26.0174 −1.18261
\(485\) 1.17012 0.0531323
\(486\) 11.2123 0.508602
\(487\) 5.13519 0.232698 0.116349 0.993208i \(-0.462881\pi\)
0.116349 + 0.993208i \(0.462881\pi\)
\(488\) −25.0663 −1.13470
\(489\) 7.76659 0.351217
\(490\) −1.63932 −0.0740569
\(491\) −29.6469 −1.33795 −0.668974 0.743286i \(-0.733266\pi\)
−0.668974 + 0.743286i \(0.733266\pi\)
\(492\) −24.1555 −1.08901
\(493\) −1.99267 −0.0897452
\(494\) 0.485018 0.0218220
\(495\) 0.0850094 0.00382089
\(496\) 10.8355 0.486530
\(497\) 0.482659 0.0216502
\(498\) 34.2094 1.53296
\(499\) −9.62164 −0.430724 −0.215362 0.976534i \(-0.569093\pi\)
−0.215362 + 0.976534i \(0.569093\pi\)
\(500\) 4.79118 0.214268
\(501\) −22.9394 −1.02486
\(502\) −9.11617 −0.406875
\(503\) 1.16776 0.0520678 0.0260339 0.999661i \(-0.491712\pi\)
0.0260339 + 0.999661i \(0.491712\pi\)
\(504\) 0.922324 0.0410836
\(505\) −0.963331 −0.0428677
\(506\) −51.7012 −2.29840
\(507\) −23.8399 −1.05877
\(508\) −55.0837 −2.44394
\(509\) 32.0068 1.41867 0.709337 0.704869i \(-0.248994\pi\)
0.709337 + 0.704869i \(0.248994\pi\)
\(510\) −0.434893 −0.0192574
\(511\) −2.19322 −0.0970224
\(512\) −78.0802 −3.45069
\(513\) −3.47063 −0.153232
\(514\) −11.3660 −0.501331
\(515\) 0.618422 0.0272509
\(516\) 55.3136 2.43504
\(517\) 14.3123 0.629455
\(518\) −5.57370 −0.244894
\(519\) −5.41438 −0.237665
\(520\) −0.206787 −0.00906820
\(521\) 25.8464 1.13235 0.566176 0.824285i \(-0.308422\pi\)
0.566176 + 0.824285i \(0.308422\pi\)
\(522\) −2.16331 −0.0946856
\(523\) 39.6079 1.73193 0.865966 0.500102i \(-0.166704\pi\)
0.865966 + 0.500102i \(0.166704\pi\)
\(524\) −48.8528 −2.13414
\(525\) 2.17127 0.0947622
\(526\) 84.7027 3.69321
\(527\) −0.670961 −0.0292275
\(528\) −74.9877 −3.26342
\(529\) 32.3695 1.40737
\(530\) 1.61842 0.0702998
\(531\) −0.393832 −0.0170909
\(532\) −0.955423 −0.0414228
\(533\) 0.570028 0.0246906
\(534\) 62.9159 2.72264
\(535\) −0.428716 −0.0185350
\(536\) −18.9188 −0.817166
\(537\) 38.1089 1.64452
\(538\) −18.0191 −0.776857
\(539\) 17.5032 0.753918
\(540\) 2.30202 0.0990631
\(541\) 5.02147 0.215890 0.107945 0.994157i \(-0.465573\pi\)
0.107945 + 0.994157i \(0.465573\pi\)
\(542\) −32.3282 −1.38861
\(543\) −10.2252 −0.438806
\(544\) 24.6760 1.05797
\(545\) 0.958939 0.0410765
\(546\) −0.291796 −0.0124877
\(547\) 33.7551 1.44327 0.721633 0.692276i \(-0.243392\pi\)
0.721633 + 0.692276i \(0.243392\pi\)
\(548\) 68.5585 2.92867
\(549\) 0.995099 0.0424698
\(550\) −34.6895 −1.47916
\(551\) 1.44044 0.0613649
\(552\) 135.993 5.78824
\(553\) −3.52309 −0.149817
\(554\) 64.7898 2.75266
\(555\) 1.35127 0.0573581
\(556\) 91.4385 3.87786
\(557\) −4.49735 −0.190559 −0.0952795 0.995451i \(-0.530374\pi\)
−0.0952795 + 0.995451i \(0.530374\pi\)
\(558\) −0.728420 −0.0308365
\(559\) −1.30531 −0.0552086
\(560\) 0.326478 0.0137962
\(561\) 4.64341 0.196045
\(562\) −63.6402 −2.68450
\(563\) 17.4255 0.734399 0.367200 0.930142i \(-0.380317\pi\)
0.367200 + 0.930142i \(0.380317\pi\)
\(564\) −58.5680 −2.46616
\(565\) −0.638737 −0.0268719
\(566\) 24.9396 1.04829
\(567\) 2.36691 0.0994009
\(568\) 20.2833 0.851069
\(569\) −8.28770 −0.347438 −0.173719 0.984795i \(-0.555579\pi\)
−0.173719 + 0.984795i \(0.555579\pi\)
\(570\) 0.314372 0.0131676
\(571\) 28.2431 1.18194 0.590969 0.806694i \(-0.298746\pi\)
0.590969 + 0.806694i \(0.298746\pi\)
\(572\) 3.43489 0.143620
\(573\) −20.8637 −0.871595
\(574\) −1.52399 −0.0636102
\(575\) 37.1508 1.54929
\(576\) 14.0690 0.586207
\(577\) −20.2009 −0.840974 −0.420487 0.907299i \(-0.638141\pi\)
−0.420487 + 0.907299i \(0.638141\pi\)
\(578\) −2.75660 −0.114659
\(579\) 7.77307 0.323038
\(580\) −0.955423 −0.0396718
\(581\) 1.59024 0.0659744
\(582\) 69.3880 2.87623
\(583\) −17.2801 −0.715669
\(584\) −92.1682 −3.81395
\(585\) 0.00820916 0.000339407 0
\(586\) 81.3253 3.35952
\(587\) 41.6693 1.71988 0.859938 0.510399i \(-0.170502\pi\)
0.859938 + 0.510399i \(0.170502\pi\)
\(588\) −71.6258 −2.95380
\(589\) 0.485018 0.0199848
\(590\) −0.236068 −0.00971876
\(591\) −29.1954 −1.20094
\(592\) −138.320 −5.68493
\(593\) 44.1769 1.81413 0.907063 0.420994i \(-0.138319\pi\)
0.907063 + 0.420994i \(0.138319\pi\)
\(594\) −33.3590 −1.36874
\(595\) −0.0202163 −0.000828786 0
\(596\) 32.6311 1.33662
\(597\) 20.6309 0.844365
\(598\) −4.99267 −0.204165
\(599\) 32.5816 1.33125 0.665625 0.746287i \(-0.268165\pi\)
0.665625 + 0.746287i \(0.268165\pi\)
\(600\) 91.2458 3.72510
\(601\) 8.11906 0.331183 0.165592 0.986194i \(-0.447047\pi\)
0.165592 + 0.986194i \(0.447047\pi\)
\(602\) 3.48979 0.142233
\(603\) 0.751050 0.0305851
\(604\) −89.2496 −3.63152
\(605\) 0.397951 0.0161790
\(606\) −57.1256 −2.32057
\(607\) −23.3707 −0.948586 −0.474293 0.880367i \(-0.657296\pi\)
−0.474293 + 0.880367i \(0.657296\pi\)
\(608\) −17.8376 −0.723409
\(609\) −0.866596 −0.0351163
\(610\) 0.596475 0.0241506
\(611\) 1.38211 0.0559141
\(612\) −2.20500 −0.0891319
\(613\) 0.363394 0.0146773 0.00733867 0.999973i \(-0.497664\pi\)
0.00733867 + 0.999973i \(0.497664\pi\)
\(614\) −27.6882 −1.11741
\(615\) 0.369472 0.0148985
\(616\) −5.90288 −0.237834
\(617\) 13.7417 0.553220 0.276610 0.960982i \(-0.410789\pi\)
0.276610 + 0.960982i \(0.410789\pi\)
\(618\) 36.6725 1.47518
\(619\) −36.5267 −1.46813 −0.734065 0.679079i \(-0.762379\pi\)
−0.734065 + 0.679079i \(0.762379\pi\)
\(620\) −0.321705 −0.0129200
\(621\) 35.7259 1.43363
\(622\) −62.3112 −2.49845
\(623\) 2.92468 0.117175
\(624\) −7.24139 −0.289888
\(625\) 24.8900 0.995602
\(626\) 42.4340 1.69600
\(627\) −3.35659 −0.134049
\(628\) −13.3681 −0.533445
\(629\) 8.56511 0.341513
\(630\) −0.0219475 −0.000874410 0
\(631\) −40.9431 −1.62992 −0.814960 0.579517i \(-0.803241\pi\)
−0.814960 + 0.579517i \(0.803241\pi\)
\(632\) −148.055 −5.88930
\(633\) 24.2625 0.964346
\(634\) 1.03724 0.0411942
\(635\) 0.842537 0.0334351
\(636\) 70.7127 2.80394
\(637\) 1.69025 0.0669701
\(638\) 13.8452 0.548138
\(639\) −0.805220 −0.0318540
\(640\) 4.20673 0.166286
\(641\) −13.9640 −0.551543 −0.275772 0.961223i \(-0.588933\pi\)
−0.275772 + 0.961223i \(0.588933\pi\)
\(642\) −25.4229 −1.00336
\(643\) −48.3429 −1.90646 −0.953229 0.302249i \(-0.902262\pi\)
−0.953229 + 0.302249i \(0.902262\pi\)
\(644\) 9.83490 0.387549
\(645\) −0.846053 −0.0333133
\(646\) 1.99267 0.0784004
\(647\) 27.9262 1.09789 0.548946 0.835858i \(-0.315029\pi\)
0.548946 + 0.835858i \(0.315029\pi\)
\(648\) 99.4672 3.90744
\(649\) 2.52053 0.0989395
\(650\) −3.34988 −0.131393
\(651\) −0.291796 −0.0114364
\(652\) 23.6039 0.924398
\(653\) −48.6822 −1.90508 −0.952540 0.304413i \(-0.901540\pi\)
−0.952540 + 0.304413i \(0.901540\pi\)
\(654\) 56.8652 2.22360
\(655\) 0.747232 0.0291968
\(656\) −37.8203 −1.47664
\(657\) 3.65895 0.142749
\(658\) −3.69512 −0.144051
\(659\) −8.01235 −0.312117 −0.156058 0.987748i \(-0.549879\pi\)
−0.156058 + 0.987748i \(0.549879\pi\)
\(660\) 2.22637 0.0866615
\(661\) −43.0909 −1.67604 −0.838021 0.545638i \(-0.816287\pi\)
−0.838021 + 0.545638i \(0.816287\pi\)
\(662\) 30.6704 1.19204
\(663\) 0.448403 0.0174145
\(664\) 66.8285 2.59345
\(665\) 0.0146138 0.000566697 0
\(666\) 9.29860 0.360313
\(667\) −14.8276 −0.574126
\(668\) −69.7162 −2.69740
\(669\) −29.4662 −1.13923
\(670\) 0.450188 0.0173923
\(671\) −6.36864 −0.245859
\(672\) 10.7314 0.413973
\(673\) −3.34903 −0.129096 −0.0645478 0.997915i \(-0.520560\pi\)
−0.0645478 + 0.997915i \(0.520560\pi\)
\(674\) 61.6760 2.37567
\(675\) 23.9707 0.922631
\(676\) −72.4531 −2.78666
\(677\) −49.1335 −1.88836 −0.944178 0.329436i \(-0.893141\pi\)
−0.944178 + 0.329436i \(0.893141\pi\)
\(678\) −37.8772 −1.45466
\(679\) 3.22554 0.123785
\(680\) −0.849569 −0.0325795
\(681\) −37.4811 −1.43628
\(682\) 4.66190 0.178513
\(683\) −22.2710 −0.852176 −0.426088 0.904682i \(-0.640109\pi\)
−0.426088 + 0.904682i \(0.640109\pi\)
\(684\) 1.59393 0.0609455
\(685\) −1.04864 −0.0400665
\(686\) −9.07416 −0.346453
\(687\) 29.0739 1.10924
\(688\) 86.6048 3.30178
\(689\) −1.66870 −0.0635724
\(690\) −3.23607 −0.123195
\(691\) 41.6463 1.58430 0.792150 0.610326i \(-0.208962\pi\)
0.792150 + 0.610326i \(0.208962\pi\)
\(692\) −16.4551 −0.625529
\(693\) 0.234337 0.00890172
\(694\) −101.150 −3.83961
\(695\) −1.39860 −0.0530521
\(696\) −36.4179 −1.38042
\(697\) 2.34192 0.0887066
\(698\) 38.5713 1.45995
\(699\) 20.6785 0.782134
\(700\) 6.59883 0.249412
\(701\) −35.6854 −1.34782 −0.673910 0.738813i \(-0.735387\pi\)
−0.673910 + 0.738813i \(0.735387\pi\)
\(702\) −3.22140 −0.121584
\(703\) −6.19147 −0.233516
\(704\) −90.0415 −3.39356
\(705\) 0.895832 0.0337390
\(706\) 33.5634 1.26318
\(707\) −2.65552 −0.0998710
\(708\) −10.3144 −0.387638
\(709\) 0.673948 0.0253107 0.0126553 0.999920i \(-0.495972\pi\)
0.0126553 + 0.999920i \(0.495972\pi\)
\(710\) −0.482659 −0.0181139
\(711\) 5.87758 0.220426
\(712\) 122.907 4.60614
\(713\) −4.99267 −0.186977
\(714\) −1.19882 −0.0448649
\(715\) −0.0525387 −0.00196484
\(716\) 115.819 4.32834
\(717\) 7.62021 0.284582
\(718\) 40.2724 1.50295
\(719\) −37.5874 −1.40177 −0.700887 0.713273i \(-0.747212\pi\)
−0.700887 + 0.713273i \(0.747212\pi\)
\(720\) −0.544663 −0.0202984
\(721\) 1.70474 0.0634879
\(722\) 50.9349 1.89560
\(723\) 46.5010 1.72939
\(724\) −31.0760 −1.15493
\(725\) −9.94872 −0.369486
\(726\) 23.5985 0.875823
\(727\) 11.4772 0.425664 0.212832 0.977089i \(-0.431731\pi\)
0.212832 + 0.977089i \(0.431731\pi\)
\(728\) −0.570028 −0.0211266
\(729\) 22.5860 0.836517
\(730\) 2.19322 0.0811748
\(731\) −5.36277 −0.198349
\(732\) 26.0614 0.963257
\(733\) −15.0467 −0.555763 −0.277881 0.960615i \(-0.589632\pi\)
−0.277881 + 0.960615i \(0.589632\pi\)
\(734\) −27.3921 −1.01106
\(735\) 1.09556 0.0404103
\(736\) 183.616 6.76816
\(737\) −4.80672 −0.177058
\(738\) 2.54248 0.0935899
\(739\) −22.3369 −0.821678 −0.410839 0.911708i \(-0.634764\pi\)
−0.410839 + 0.911708i \(0.634764\pi\)
\(740\) 4.10671 0.150966
\(741\) −0.324138 −0.0119075
\(742\) 4.46134 0.163781
\(743\) −12.9134 −0.473748 −0.236874 0.971540i \(-0.576123\pi\)
−0.236874 + 0.971540i \(0.576123\pi\)
\(744\) −12.2625 −0.449564
\(745\) −0.499112 −0.0182860
\(746\) −10.7701 −0.394322
\(747\) −2.65300 −0.0970684
\(748\) 14.1120 0.515987
\(749\) −1.18180 −0.0431820
\(750\) −4.34574 −0.158684
\(751\) −25.5796 −0.933413 −0.466706 0.884412i \(-0.654559\pi\)
−0.466706 + 0.884412i \(0.654559\pi\)
\(752\) −91.7003 −3.34397
\(753\) 6.09234 0.222017
\(754\) 1.33700 0.0486907
\(755\) 1.36513 0.0496820
\(756\) 6.34574 0.230792
\(757\) 28.0754 1.02042 0.510208 0.860051i \(-0.329568\pi\)
0.510208 + 0.860051i \(0.329568\pi\)
\(758\) 78.9173 2.86641
\(759\) 34.5519 1.25416
\(760\) 0.614130 0.0222768
\(761\) −49.7060 −1.80184 −0.900920 0.433984i \(-0.857107\pi\)
−0.900920 + 0.433984i \(0.857107\pi\)
\(762\) 49.9625 1.80995
\(763\) 2.64341 0.0956979
\(764\) −63.4080 −2.29402
\(765\) 0.0337268 0.00121939
\(766\) −50.8604 −1.83766
\(767\) 0.243402 0.00878873
\(768\) 117.838 4.25212
\(769\) −14.5900 −0.526129 −0.263065 0.964778i \(-0.584733\pi\)
−0.263065 + 0.964778i \(0.584733\pi\)
\(770\) 0.140464 0.00506198
\(771\) 7.59587 0.273559
\(772\) 23.6235 0.850230
\(773\) 21.7414 0.781984 0.390992 0.920394i \(-0.372132\pi\)
0.390992 + 0.920394i \(0.372132\pi\)
\(774\) −5.82202 −0.209268
\(775\) −3.34988 −0.120331
\(776\) 135.551 4.86598
\(777\) 3.72490 0.133630
\(778\) −24.6962 −0.885402
\(779\) −1.69291 −0.0606547
\(780\) 0.214996 0.00769809
\(781\) 5.15342 0.184404
\(782\) −20.5120 −0.733509
\(783\) −9.56714 −0.341902
\(784\) −112.145 −4.00518
\(785\) 0.204473 0.00729794
\(786\) 44.3109 1.58052
\(787\) 38.3701 1.36775 0.683874 0.729600i \(-0.260294\pi\)
0.683874 + 0.729600i \(0.260294\pi\)
\(788\) −88.7291 −3.16084
\(789\) −56.6069 −2.01526
\(790\) 3.52309 0.125346
\(791\) −1.76074 −0.0626048
\(792\) 9.84778 0.349926
\(793\) −0.615005 −0.0218395
\(794\) −84.9808 −3.01586
\(795\) −1.08159 −0.0383601
\(796\) 62.7004 2.22236
\(797\) 35.1878 1.24642 0.623208 0.782056i \(-0.285829\pi\)
0.623208 + 0.782056i \(0.285829\pi\)
\(798\) 0.866596 0.0306772
\(799\) 5.67829 0.200884
\(800\) 123.199 4.35574
\(801\) −4.87925 −0.172400
\(802\) −8.74512 −0.308801
\(803\) −23.4173 −0.826380
\(804\) 19.6698 0.693700
\(805\) −0.150431 −0.00530198
\(806\) 0.450188 0.0158572
\(807\) 12.0421 0.423904
\(808\) −111.596 −3.92592
\(809\) −13.8123 −0.485616 −0.242808 0.970074i \(-0.578069\pi\)
−0.242808 + 0.970074i \(0.578069\pi\)
\(810\) −2.36691 −0.0831648
\(811\) 12.2833 0.431325 0.215663 0.976468i \(-0.430809\pi\)
0.215663 + 0.976468i \(0.430809\pi\)
\(812\) −2.63372 −0.0924254
\(813\) 21.6049 0.757718
\(814\) −59.5111 −2.08586
\(815\) −0.361035 −0.0126465
\(816\) −29.7508 −1.04149
\(817\) 3.87659 0.135625
\(818\) −78.7118 −2.75209
\(819\) 0.0226294 0.000790734 0
\(820\) 1.12288 0.0392127
\(821\) −16.6387 −0.580693 −0.290347 0.956922i \(-0.593771\pi\)
−0.290347 + 0.956922i \(0.593771\pi\)
\(822\) −62.1845 −2.16893
\(823\) 7.19288 0.250728 0.125364 0.992111i \(-0.459990\pi\)
0.125364 + 0.992111i \(0.459990\pi\)
\(824\) 71.6402 2.49570
\(825\) 23.1830 0.807128
\(826\) −0.650745 −0.0226423
\(827\) 12.5974 0.438056 0.219028 0.975719i \(-0.429711\pi\)
0.219028 + 0.975719i \(0.429711\pi\)
\(828\) −16.4076 −0.570202
\(829\) −29.8496 −1.03672 −0.518360 0.855163i \(-0.673457\pi\)
−0.518360 + 0.855163i \(0.673457\pi\)
\(830\) −1.59024 −0.0551982
\(831\) −43.2991 −1.50203
\(832\) −8.69509 −0.301448
\(833\) 6.94427 0.240605
\(834\) −82.9373 −2.87188
\(835\) 1.06635 0.0369026
\(836\) −10.2012 −0.352815
\(837\) −3.22140 −0.111348
\(838\) −66.7300 −2.30515
\(839\) −39.1941 −1.35313 −0.676565 0.736383i \(-0.736532\pi\)
−0.676565 + 0.736383i \(0.736532\pi\)
\(840\) −0.369472 −0.0127480
\(841\) −25.0293 −0.863079
\(842\) 41.3967 1.42663
\(843\) 42.5308 1.46484
\(844\) 73.7373 2.53814
\(845\) 1.10821 0.0381237
\(846\) 6.16457 0.211942
\(847\) 1.09699 0.0376930
\(848\) 110.715 3.80198
\(849\) −16.6671 −0.572015
\(850\) −13.7628 −0.472059
\(851\) 63.7336 2.18476
\(852\) −21.0885 −0.722481
\(853\) −27.5883 −0.944606 −0.472303 0.881436i \(-0.656577\pi\)
−0.472303 + 0.881436i \(0.656577\pi\)
\(854\) 1.64424 0.0562648
\(855\) −0.0243801 −0.000833783 0
\(856\) −49.6640 −1.69748
\(857\) −31.1275 −1.06330 −0.531648 0.846966i \(-0.678427\pi\)
−0.531648 + 0.846966i \(0.678427\pi\)
\(858\) −3.11555 −0.106363
\(859\) 18.8182 0.642069 0.321035 0.947067i \(-0.395969\pi\)
0.321035 + 0.947067i \(0.395969\pi\)
\(860\) −2.57128 −0.0876801
\(861\) 1.01848 0.0347099
\(862\) −44.9391 −1.53063
\(863\) 32.6988 1.11308 0.556541 0.830820i \(-0.312128\pi\)
0.556541 + 0.830820i \(0.312128\pi\)
\(864\) 118.474 4.03056
\(865\) 0.251690 0.00855773
\(866\) 69.6660 2.36735
\(867\) 1.84224 0.0625656
\(868\) −0.886813 −0.0301004
\(869\) −37.6165 −1.27605
\(870\) 0.866596 0.0293804
\(871\) −0.464174 −0.0157279
\(872\) 111.087 3.76188
\(873\) −5.38118 −0.182125
\(874\) 14.8276 0.501550
\(875\) −0.202014 −0.00682933
\(876\) 95.8271 3.23770
\(877\) 13.6701 0.461607 0.230803 0.973000i \(-0.425865\pi\)
0.230803 + 0.973000i \(0.425865\pi\)
\(878\) −2.83317 −0.0956149
\(879\) −54.3497 −1.83317
\(880\) 3.48585 0.117508
\(881\) −52.2246 −1.75949 −0.879746 0.475443i \(-0.842288\pi\)
−0.879746 + 0.475443i \(0.842288\pi\)
\(882\) 7.53896 0.253850
\(883\) 40.5188 1.36357 0.681783 0.731555i \(-0.261205\pi\)
0.681783 + 0.731555i \(0.261205\pi\)
\(884\) 1.36277 0.0458348
\(885\) 0.157764 0.00530319
\(886\) 84.2403 2.83011
\(887\) 38.3375 1.28725 0.643623 0.765342i \(-0.277430\pi\)
0.643623 + 0.765342i \(0.277430\pi\)
\(888\) 156.536 5.25299
\(889\) 2.32254 0.0778953
\(890\) −2.92468 −0.0980356
\(891\) 25.2718 0.846638
\(892\) −89.5522 −2.99843
\(893\) −4.10468 −0.137358
\(894\) −29.5974 −0.989883
\(895\) −1.77151 −0.0592152
\(896\) 11.5963 0.387404
\(897\) 3.33660 0.111406
\(898\) −96.8382 −3.23153
\(899\) 1.33700 0.0445915
\(900\) −11.0088 −0.366961
\(901\) −6.85575 −0.228398
\(902\) −16.2719 −0.541794
\(903\) −2.33223 −0.0776117
\(904\) −73.9936 −2.46099
\(905\) 0.475325 0.0158003
\(906\) 80.9520 2.68945
\(907\) 11.0883 0.368181 0.184090 0.982909i \(-0.441066\pi\)
0.184090 + 0.982909i \(0.441066\pi\)
\(908\) −113.911 −3.78026
\(909\) 4.43020 0.146940
\(910\) 0.0135643 0.000449653 0
\(911\) −18.1245 −0.600491 −0.300245 0.953862i \(-0.597069\pi\)
−0.300245 + 0.953862i \(0.597069\pi\)
\(912\) 21.5060 0.712134
\(913\) 16.9793 0.561931
\(914\) 98.6693 3.26369
\(915\) −0.398624 −0.0131781
\(916\) 88.3599 2.91949
\(917\) 2.05982 0.0680212
\(918\) −13.2349 −0.436817
\(919\) 20.3942 0.672741 0.336371 0.941730i \(-0.390801\pi\)
0.336371 + 0.941730i \(0.390801\pi\)
\(920\) −6.32171 −0.208420
\(921\) 18.5041 0.609729
\(922\) −36.9168 −1.21579
\(923\) 0.497654 0.0163805
\(924\) 6.13722 0.201900
\(925\) 42.7627 1.40603
\(926\) −96.4924 −3.17094
\(927\) −2.84402 −0.0934099
\(928\) −49.1710 −1.61412
\(929\) 11.3149 0.371230 0.185615 0.982623i \(-0.440572\pi\)
0.185615 + 0.982623i \(0.440572\pi\)
\(930\) 0.291796 0.00956837
\(931\) −5.01982 −0.164518
\(932\) 62.8452 2.05856
\(933\) 41.6426 1.36332
\(934\) −87.5798 −2.86570
\(935\) −0.215852 −0.00705911
\(936\) 0.950978 0.0310837
\(937\) −44.0194 −1.43805 −0.719025 0.694984i \(-0.755411\pi\)
−0.719025 + 0.694984i \(0.755411\pi\)
\(938\) 1.24099 0.0405197
\(939\) −28.3586 −0.925449
\(940\) 2.72257 0.0888005
\(941\) −10.5419 −0.343658 −0.171829 0.985127i \(-0.554968\pi\)
−0.171829 + 0.985127i \(0.554968\pi\)
\(942\) 12.1252 0.395062
\(943\) 17.4264 0.567482
\(944\) −16.1493 −0.525614
\(945\) −0.0970618 −0.00315742
\(946\) 37.2610 1.21146
\(947\) −21.7442 −0.706591 −0.353296 0.935512i \(-0.614939\pi\)
−0.353296 + 0.935512i \(0.614939\pi\)
\(948\) 153.932 4.99949
\(949\) −2.26136 −0.0734068
\(950\) 9.94872 0.322779
\(951\) −0.693190 −0.0224782
\(952\) −2.34192 −0.0759021
\(953\) −15.7927 −0.511575 −0.255788 0.966733i \(-0.582335\pi\)
−0.255788 + 0.966733i \(0.582335\pi\)
\(954\) −7.44285 −0.240971
\(955\) 0.969863 0.0313840
\(956\) 23.1590 0.749015
\(957\) −9.25277 −0.299100
\(958\) 16.1525 0.521862
\(959\) −2.89068 −0.0933450
\(960\) −5.63585 −0.181896
\(961\) −30.5498 −0.985478
\(962\) −5.74685 −0.185286
\(963\) 1.97159 0.0635337
\(964\) 141.324 4.55173
\(965\) −0.361336 −0.0116318
\(966\) −8.92054 −0.287014
\(967\) −33.9655 −1.09226 −0.546128 0.837702i \(-0.683899\pi\)
−0.546128 + 0.837702i \(0.683899\pi\)
\(968\) 46.1000 1.48171
\(969\) −1.33170 −0.0427804
\(970\) −3.22554 −0.103566
\(971\) 17.6534 0.566525 0.283262 0.959042i \(-0.408583\pi\)
0.283262 + 0.959042i \(0.408583\pi\)
\(972\) −22.7730 −0.730444
\(973\) −3.85539 −0.123598
\(974\) −14.1557 −0.453576
\(975\) 2.23873 0.0716967
\(976\) 40.8045 1.30612
\(977\) −19.1541 −0.612793 −0.306396 0.951904i \(-0.599123\pi\)
−0.306396 + 0.951904i \(0.599123\pi\)
\(978\) −21.4094 −0.684596
\(979\) 31.2273 0.998027
\(980\) 3.32957 0.106359
\(981\) −4.41000 −0.140801
\(982\) 81.7247 2.60794
\(983\) 39.0710 1.24617 0.623085 0.782154i \(-0.285879\pi\)
0.623085 + 0.782154i \(0.285879\pi\)
\(984\) 42.8009 1.36444
\(985\) 1.35716 0.0432428
\(986\) 5.49298 0.174932
\(987\) 2.46945 0.0786035
\(988\) −0.985105 −0.0313404
\(989\) −39.9047 −1.26890
\(990\) −0.234337 −0.00744771
\(991\) −38.2233 −1.21420 −0.607101 0.794625i \(-0.707668\pi\)
−0.607101 + 0.794625i \(0.707668\pi\)
\(992\) −16.5566 −0.525673
\(993\) −20.4970 −0.650453
\(994\) −1.33050 −0.0422008
\(995\) −0.959039 −0.0304036
\(996\) −69.4815 −2.20161
\(997\) −46.2303 −1.46413 −0.732065 0.681235i \(-0.761443\pi\)
−0.732065 + 0.681235i \(0.761443\pi\)
\(998\) 26.5230 0.839571
\(999\) 41.1226 1.30106
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 1003.2.a.f.1.1 4
3.2 odd 2 9027.2.a.i.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1003.2.a.f.1.1 4 1.1 even 1 trivial
9027.2.a.i.1.4 4 3.2 odd 2