Properties

Label 667.2.a.d.1.4
Level $667$
Weight $2$
Character 667.1
Self dual yes
Analytic conductor $5.326$
Analytic rank $0$
Dimension $16$
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] = [667,2,Mod(1,667)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(667, 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("667.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 667 = 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 667.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: \(5.32602181482\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} - 22 x^{14} + 68 x^{13} + 187 x^{12} - 597 x^{11} - 795 x^{10} + 2592 x^{9} + 1860 x^{8} - 5877 x^{7} - 2496 x^{6} + 6612 x^{5} + 1842 x^{4} - 3011 x^{3} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.63671\) of defining polynomial
Character \(\chi\) \(=\) 667.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.63671 q^{2} -0.492566 q^{3} +0.678809 q^{4} -1.90242 q^{5} +0.806186 q^{6} +0.625725 q^{7} +2.16240 q^{8} -2.75738 q^{9} +O(q^{10})\) \(q-1.63671 q^{2} -0.492566 q^{3} +0.678809 q^{4} -1.90242 q^{5} +0.806186 q^{6} +0.625725 q^{7} +2.16240 q^{8} -2.75738 q^{9} +3.11371 q^{10} +1.86394 q^{11} -0.334358 q^{12} -3.61185 q^{13} -1.02413 q^{14} +0.937068 q^{15} -4.89684 q^{16} -0.946611 q^{17} +4.51302 q^{18} -0.390297 q^{19} -1.29138 q^{20} -0.308211 q^{21} -3.05072 q^{22} +1.00000 q^{23} -1.06513 q^{24} -1.38079 q^{25} +5.91154 q^{26} +2.83589 q^{27} +0.424747 q^{28} -1.00000 q^{29} -1.53371 q^{30} +5.69517 q^{31} +3.68988 q^{32} -0.918114 q^{33} +1.54932 q^{34} -1.19039 q^{35} -1.87173 q^{36} +5.54547 q^{37} +0.638801 q^{38} +1.77907 q^{39} -4.11380 q^{40} +11.6520 q^{41} +0.504450 q^{42} -0.753478 q^{43} +1.26526 q^{44} +5.24570 q^{45} -1.63671 q^{46} +8.82078 q^{47} +2.41201 q^{48} -6.60847 q^{49} +2.25995 q^{50} +0.466268 q^{51} -2.45176 q^{52} +3.41241 q^{53} -4.64152 q^{54} -3.54600 q^{55} +1.35307 q^{56} +0.192247 q^{57} +1.63671 q^{58} +10.8373 q^{59} +0.636090 q^{60} -6.58204 q^{61} -9.32133 q^{62} -1.72536 q^{63} +3.75442 q^{64} +6.87126 q^{65} +1.50268 q^{66} +7.92908 q^{67} -0.642568 q^{68} -0.492566 q^{69} +1.94832 q^{70} -4.28721 q^{71} -5.96256 q^{72} -7.27027 q^{73} -9.07632 q^{74} +0.680129 q^{75} -0.264937 q^{76} +1.16631 q^{77} -2.91182 q^{78} +15.1128 q^{79} +9.31585 q^{80} +6.87528 q^{81} -19.0710 q^{82} +16.1616 q^{83} -0.209216 q^{84} +1.80085 q^{85} +1.23322 q^{86} +0.492566 q^{87} +4.03059 q^{88} -16.1681 q^{89} -8.58567 q^{90} -2.26002 q^{91} +0.678809 q^{92} -2.80525 q^{93} -14.4370 q^{94} +0.742509 q^{95} -1.81751 q^{96} +12.8667 q^{97} +10.8161 q^{98} -5.13959 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 3 q^{2} + 5 q^{3} + 21 q^{4} + 16 q^{5} + 6 q^{6} + q^{7} + 9 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 3 q^{2} + 5 q^{3} + 21 q^{4} + 16 q^{5} + 6 q^{6} + q^{7} + 9 q^{8} + 23 q^{9} - 14 q^{10} + 4 q^{11} + 3 q^{12} + 15 q^{13} + 8 q^{14} + 8 q^{15} + 23 q^{16} + 20 q^{17} + 2 q^{18} - 4 q^{19} + 25 q^{20} + 5 q^{21} + 13 q^{22} + 16 q^{23} - 24 q^{24} + 30 q^{25} + 25 q^{26} + 8 q^{27} - 13 q^{28} - 16 q^{29} - 45 q^{30} - 19 q^{32} + q^{33} - 23 q^{34} + 5 q^{35} + 37 q^{36} + 5 q^{37} + 38 q^{38} - 10 q^{39} - 20 q^{40} + 7 q^{41} + 14 q^{42} - 17 q^{43} - 21 q^{44} + 48 q^{45} + 3 q^{46} + 29 q^{47} + 35 q^{48} + 31 q^{49} - 44 q^{50} - 14 q^{51} + 20 q^{52} + 63 q^{53} - 13 q^{54} + q^{55} - 19 q^{56} - 22 q^{57} - 3 q^{58} + 11 q^{59} - 3 q^{60} + 33 q^{62} - 33 q^{63} + 29 q^{64} + 53 q^{65} - 43 q^{66} - 13 q^{67} + 63 q^{68} + 5 q^{69} - 46 q^{70} - 23 q^{71} + 46 q^{72} - 38 q^{73} - 47 q^{74} + 37 q^{75} - 56 q^{76} + 97 q^{77} - 24 q^{78} - 27 q^{79} + 8 q^{80} + 40 q^{81} + 9 q^{82} + 36 q^{83} + 22 q^{84} + 6 q^{85} - 11 q^{86} - 5 q^{87} - 24 q^{88} - 16 q^{89} - 95 q^{90} - 47 q^{91} + 21 q^{92} + 62 q^{93} + 37 q^{94} - 12 q^{95} - 74 q^{96} - 30 q^{97} - 27 q^{98} - 42 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.63671 −1.15733 −0.578663 0.815567i \(-0.696425\pi\)
−0.578663 + 0.815567i \(0.696425\pi\)
\(3\) −0.492566 −0.284383 −0.142192 0.989839i \(-0.545415\pi\)
−0.142192 + 0.989839i \(0.545415\pi\)
\(4\) 0.678809 0.339404
\(5\) −1.90242 −0.850789 −0.425395 0.905008i \(-0.639865\pi\)
−0.425395 + 0.905008i \(0.639865\pi\)
\(6\) 0.806186 0.329124
\(7\) 0.625725 0.236502 0.118251 0.992984i \(-0.462271\pi\)
0.118251 + 0.992984i \(0.462271\pi\)
\(8\) 2.16240 0.764525
\(9\) −2.75738 −0.919126
\(10\) 3.11371 0.984641
\(11\) 1.86394 0.561999 0.281000 0.959708i \(-0.409334\pi\)
0.281000 + 0.959708i \(0.409334\pi\)
\(12\) −0.334358 −0.0965208
\(13\) −3.61185 −1.00175 −0.500873 0.865521i \(-0.666988\pi\)
−0.500873 + 0.865521i \(0.666988\pi\)
\(14\) −1.02413 −0.273710
\(15\) 0.937068 0.241950
\(16\) −4.89684 −1.22421
\(17\) −0.946611 −0.229587 −0.114793 0.993389i \(-0.536621\pi\)
−0.114793 + 0.993389i \(0.536621\pi\)
\(18\) 4.51302 1.06373
\(19\) −0.390297 −0.0895402 −0.0447701 0.998997i \(-0.514256\pi\)
−0.0447701 + 0.998997i \(0.514256\pi\)
\(20\) −1.29138 −0.288762
\(21\) −0.308211 −0.0672571
\(22\) −3.05072 −0.650417
\(23\) 1.00000 0.208514
\(24\) −1.06513 −0.217418
\(25\) −1.38079 −0.276158
\(26\) 5.91154 1.15935
\(27\) 2.83589 0.545767
\(28\) 0.424747 0.0802697
\(29\) −1.00000 −0.185695
\(30\) −1.53371 −0.280015
\(31\) 5.69517 1.02288 0.511442 0.859318i \(-0.329112\pi\)
0.511442 + 0.859318i \(0.329112\pi\)
\(32\) 3.68988 0.652285
\(33\) −0.918114 −0.159823
\(34\) 1.54932 0.265707
\(35\) −1.19039 −0.201213
\(36\) −1.87173 −0.311956
\(37\) 5.54547 0.911670 0.455835 0.890064i \(-0.349341\pi\)
0.455835 + 0.890064i \(0.349341\pi\)
\(38\) 0.638801 0.103627
\(39\) 1.77907 0.284880
\(40\) −4.11380 −0.650449
\(41\) 11.6520 1.81974 0.909871 0.414892i \(-0.136181\pi\)
0.909871 + 0.414892i \(0.136181\pi\)
\(42\) 0.504450 0.0778384
\(43\) −0.753478 −0.114904 −0.0574522 0.998348i \(-0.518298\pi\)
−0.0574522 + 0.998348i \(0.518298\pi\)
\(44\) 1.26526 0.190745
\(45\) 5.24570 0.781983
\(46\) −1.63671 −0.241319
\(47\) 8.82078 1.28664 0.643322 0.765596i \(-0.277556\pi\)
0.643322 + 0.765596i \(0.277556\pi\)
\(48\) 2.41201 0.348144
\(49\) −6.60847 −0.944067
\(50\) 2.25995 0.319605
\(51\) 0.466268 0.0652906
\(52\) −2.45176 −0.339997
\(53\) 3.41241 0.468730 0.234365 0.972149i \(-0.424699\pi\)
0.234365 + 0.972149i \(0.424699\pi\)
\(54\) −4.64152 −0.631630
\(55\) −3.54600 −0.478143
\(56\) 1.35307 0.180811
\(57\) 0.192247 0.0254637
\(58\) 1.63671 0.214910
\(59\) 10.8373 1.41089 0.705447 0.708762i \(-0.250746\pi\)
0.705447 + 0.708762i \(0.250746\pi\)
\(60\) 0.636090 0.0821189
\(61\) −6.58204 −0.842744 −0.421372 0.906888i \(-0.638451\pi\)
−0.421372 + 0.906888i \(0.638451\pi\)
\(62\) −9.32133 −1.18381
\(63\) −1.72536 −0.217375
\(64\) 3.75442 0.469303
\(65\) 6.87126 0.852275
\(66\) 1.50268 0.184967
\(67\) 7.92908 0.968691 0.484345 0.874877i \(-0.339058\pi\)
0.484345 + 0.874877i \(0.339058\pi\)
\(68\) −0.642568 −0.0779228
\(69\) −0.492566 −0.0592980
\(70\) 1.94832 0.232869
\(71\) −4.28721 −0.508798 −0.254399 0.967099i \(-0.581878\pi\)
−0.254399 + 0.967099i \(0.581878\pi\)
\(72\) −5.96256 −0.702695
\(73\) −7.27027 −0.850921 −0.425461 0.904977i \(-0.639888\pi\)
−0.425461 + 0.904977i \(0.639888\pi\)
\(74\) −9.07632 −1.05510
\(75\) 0.680129 0.0785346
\(76\) −0.264937 −0.0303903
\(77\) 1.16631 0.132914
\(78\) −2.91182 −0.329699
\(79\) 15.1128 1.70032 0.850161 0.526523i \(-0.176505\pi\)
0.850161 + 0.526523i \(0.176505\pi\)
\(80\) 9.31585 1.04154
\(81\) 6.87528 0.763919
\(82\) −19.0710 −2.10603
\(83\) 16.1616 1.77396 0.886981 0.461806i \(-0.152798\pi\)
0.886981 + 0.461806i \(0.152798\pi\)
\(84\) −0.209216 −0.0228273
\(85\) 1.80085 0.195330
\(86\) 1.23322 0.132982
\(87\) 0.492566 0.0528086
\(88\) 4.03059 0.429662
\(89\) −16.1681 −1.71382 −0.856909 0.515468i \(-0.827618\pi\)
−0.856909 + 0.515468i \(0.827618\pi\)
\(90\) −8.58567 −0.905009
\(91\) −2.26002 −0.236915
\(92\) 0.678809 0.0707707
\(93\) −2.80525 −0.290891
\(94\) −14.4370 −1.48907
\(95\) 0.742509 0.0761799
\(96\) −1.81751 −0.185499
\(97\) 12.8667 1.30641 0.653205 0.757181i \(-0.273424\pi\)
0.653205 + 0.757181i \(0.273424\pi\)
\(98\) 10.8161 1.09259
\(99\) −5.13959 −0.516548
\(100\) −0.937292 −0.0937292
\(101\) 14.9233 1.48493 0.742463 0.669887i \(-0.233658\pi\)
0.742463 + 0.669887i \(0.233658\pi\)
\(102\) −0.763144 −0.0755626
\(103\) −18.6970 −1.84227 −0.921133 0.389249i \(-0.872735\pi\)
−0.921133 + 0.389249i \(0.872735\pi\)
\(104\) −7.81027 −0.765860
\(105\) 0.586347 0.0572216
\(106\) −5.58511 −0.542474
\(107\) 19.2587 1.86181 0.930904 0.365263i \(-0.119021\pi\)
0.930904 + 0.365263i \(0.119021\pi\)
\(108\) 1.92503 0.185236
\(109\) 7.69095 0.736659 0.368330 0.929695i \(-0.379930\pi\)
0.368330 + 0.929695i \(0.379930\pi\)
\(110\) 5.80377 0.553367
\(111\) −2.73151 −0.259264
\(112\) −3.06407 −0.289527
\(113\) 2.99643 0.281881 0.140940 0.990018i \(-0.454987\pi\)
0.140940 + 0.990018i \(0.454987\pi\)
\(114\) −0.314652 −0.0294698
\(115\) −1.90242 −0.177402
\(116\) −0.678809 −0.0630258
\(117\) 9.95924 0.920732
\(118\) −17.7375 −1.63287
\(119\) −0.592318 −0.0542977
\(120\) 2.02632 0.184977
\(121\) −7.52572 −0.684157
\(122\) 10.7729 0.975329
\(123\) −5.73939 −0.517504
\(124\) 3.86593 0.347171
\(125\) 12.1390 1.08574
\(126\) 2.82391 0.251574
\(127\) −12.9406 −1.14829 −0.574147 0.818752i \(-0.694666\pi\)
−0.574147 + 0.818752i \(0.694666\pi\)
\(128\) −13.5246 −1.19542
\(129\) 0.371138 0.0326768
\(130\) −11.2462 −0.986361
\(131\) −12.3946 −1.08292 −0.541461 0.840726i \(-0.682129\pi\)
−0.541461 + 0.840726i \(0.682129\pi\)
\(132\) −0.623224 −0.0542447
\(133\) −0.244218 −0.0211764
\(134\) −12.9776 −1.12109
\(135\) −5.39506 −0.464333
\(136\) −2.04695 −0.175525
\(137\) 1.05173 0.0898552 0.0449276 0.998990i \(-0.485694\pi\)
0.0449276 + 0.998990i \(0.485694\pi\)
\(138\) 0.806186 0.0686271
\(139\) −21.9528 −1.86201 −0.931005 0.365007i \(-0.881067\pi\)
−0.931005 + 0.365007i \(0.881067\pi\)
\(140\) −0.808049 −0.0682926
\(141\) −4.34482 −0.365900
\(142\) 7.01690 0.588845
\(143\) −6.73227 −0.562981
\(144\) 13.5024 1.12520
\(145\) 1.90242 0.157988
\(146\) 11.8993 0.984794
\(147\) 3.25511 0.268477
\(148\) 3.76432 0.309425
\(149\) −12.7909 −1.04787 −0.523934 0.851759i \(-0.675536\pi\)
−0.523934 + 0.851759i \(0.675536\pi\)
\(150\) −1.11317 −0.0908901
\(151\) −1.98814 −0.161793 −0.0808963 0.996723i \(-0.525778\pi\)
−0.0808963 + 0.996723i \(0.525778\pi\)
\(152\) −0.843979 −0.0684557
\(153\) 2.61017 0.211019
\(154\) −1.90891 −0.153825
\(155\) −10.8346 −0.870258
\(156\) 1.20765 0.0966895
\(157\) 2.97702 0.237592 0.118796 0.992919i \(-0.462097\pi\)
0.118796 + 0.992919i \(0.462097\pi\)
\(158\) −24.7352 −1.96783
\(159\) −1.68084 −0.133299
\(160\) −7.01971 −0.554957
\(161\) 0.625725 0.0493140
\(162\) −11.2528 −0.884104
\(163\) 2.96244 0.232036 0.116018 0.993247i \(-0.462987\pi\)
0.116018 + 0.993247i \(0.462987\pi\)
\(164\) 7.90950 0.617628
\(165\) 1.74664 0.135976
\(166\) −26.4517 −2.05305
\(167\) 5.29393 0.409657 0.204828 0.978798i \(-0.434336\pi\)
0.204828 + 0.978798i \(0.434336\pi\)
\(168\) −0.666475 −0.0514197
\(169\) 0.0454596 0.00349689
\(170\) −2.94747 −0.226061
\(171\) 1.07620 0.0822988
\(172\) −0.511468 −0.0389990
\(173\) 12.7744 0.971220 0.485610 0.874176i \(-0.338598\pi\)
0.485610 + 0.874176i \(0.338598\pi\)
\(174\) −0.806186 −0.0611168
\(175\) −0.863994 −0.0653118
\(176\) −9.12741 −0.688005
\(177\) −5.33808 −0.401235
\(178\) 26.4625 1.98345
\(179\) −0.984747 −0.0736034 −0.0368017 0.999323i \(-0.511717\pi\)
−0.0368017 + 0.999323i \(0.511717\pi\)
\(180\) 3.56083 0.265408
\(181\) 1.37675 0.102333 0.0511664 0.998690i \(-0.483706\pi\)
0.0511664 + 0.998690i \(0.483706\pi\)
\(182\) 3.69900 0.274188
\(183\) 3.24209 0.239662
\(184\) 2.16240 0.159414
\(185\) −10.5498 −0.775639
\(186\) 4.59137 0.336655
\(187\) −1.76443 −0.129028
\(188\) 5.98763 0.436693
\(189\) 1.77448 0.129075
\(190\) −1.21527 −0.0881650
\(191\) −18.3610 −1.32856 −0.664279 0.747485i \(-0.731261\pi\)
−0.664279 + 0.747485i \(0.731261\pi\)
\(192\) −1.84930 −0.133462
\(193\) 19.1865 1.38108 0.690539 0.723296i \(-0.257374\pi\)
0.690539 + 0.723296i \(0.257374\pi\)
\(194\) −21.0589 −1.51194
\(195\) −3.38455 −0.242373
\(196\) −4.48589 −0.320420
\(197\) −14.3075 −1.01937 −0.509683 0.860362i \(-0.670237\pi\)
−0.509683 + 0.860362i \(0.670237\pi\)
\(198\) 8.41200 0.597815
\(199\) 12.0238 0.852348 0.426174 0.904641i \(-0.359861\pi\)
0.426174 + 0.904641i \(0.359861\pi\)
\(200\) −2.98582 −0.211129
\(201\) −3.90559 −0.275479
\(202\) −24.4251 −1.71854
\(203\) −0.625725 −0.0439173
\(204\) 0.316507 0.0221599
\(205\) −22.1671 −1.54822
\(206\) 30.6014 2.13210
\(207\) −2.75738 −0.191651
\(208\) 17.6866 1.22635
\(209\) −0.727490 −0.0503215
\(210\) −0.959678 −0.0662240
\(211\) −10.6373 −0.732302 −0.366151 0.930555i \(-0.619325\pi\)
−0.366151 + 0.930555i \(0.619325\pi\)
\(212\) 2.31637 0.159089
\(213\) 2.11173 0.144693
\(214\) −31.5208 −2.15472
\(215\) 1.43343 0.0977594
\(216\) 6.13233 0.417252
\(217\) 3.56361 0.241914
\(218\) −12.5878 −0.852555
\(219\) 3.58109 0.241988
\(220\) −2.40706 −0.162284
\(221\) 3.41902 0.229988
\(222\) 4.47068 0.300053
\(223\) 7.75259 0.519152 0.259576 0.965723i \(-0.416417\pi\)
0.259576 + 0.965723i \(0.416417\pi\)
\(224\) 2.30885 0.154266
\(225\) 3.80736 0.253824
\(226\) −4.90428 −0.326228
\(227\) 7.50827 0.498341 0.249171 0.968460i \(-0.419842\pi\)
0.249171 + 0.968460i \(0.419842\pi\)
\(228\) 0.130499 0.00864250
\(229\) 10.5239 0.695439 0.347719 0.937599i \(-0.386956\pi\)
0.347719 + 0.937599i \(0.386956\pi\)
\(230\) 3.11371 0.205312
\(231\) −0.574486 −0.0377984
\(232\) −2.16240 −0.141969
\(233\) −3.79993 −0.248942 −0.124471 0.992223i \(-0.539723\pi\)
−0.124471 + 0.992223i \(0.539723\pi\)
\(234\) −16.3004 −1.06559
\(235\) −16.7809 −1.09466
\(236\) 7.35645 0.478864
\(237\) −7.44404 −0.483543
\(238\) 0.969451 0.0628402
\(239\) −5.77321 −0.373438 −0.186719 0.982413i \(-0.559785\pi\)
−0.186719 + 0.982413i \(0.559785\pi\)
\(240\) −4.58867 −0.296197
\(241\) 11.7005 0.753694 0.376847 0.926276i \(-0.377008\pi\)
0.376847 + 0.926276i \(0.377008\pi\)
\(242\) 12.3174 0.791793
\(243\) −11.8942 −0.763013
\(244\) −4.46795 −0.286031
\(245\) 12.5721 0.803202
\(246\) 9.39370 0.598921
\(247\) 1.40969 0.0896966
\(248\) 12.3153 0.782020
\(249\) −7.96063 −0.504485
\(250\) −19.8679 −1.25656
\(251\) −18.4434 −1.16414 −0.582068 0.813140i \(-0.697756\pi\)
−0.582068 + 0.813140i \(0.697756\pi\)
\(252\) −1.17119 −0.0737780
\(253\) 1.86394 0.117185
\(254\) 21.1800 1.32895
\(255\) −0.887039 −0.0555486
\(256\) 14.6270 0.914190
\(257\) −1.32683 −0.0827652 −0.0413826 0.999143i \(-0.513176\pi\)
−0.0413826 + 0.999143i \(0.513176\pi\)
\(258\) −0.607443 −0.0378178
\(259\) 3.46994 0.215612
\(260\) 4.66427 0.289266
\(261\) 2.75738 0.170677
\(262\) 20.2863 1.25329
\(263\) 10.6066 0.654030 0.327015 0.945019i \(-0.393957\pi\)
0.327015 + 0.945019i \(0.393957\pi\)
\(264\) −1.98533 −0.122189
\(265\) −6.49184 −0.398791
\(266\) 0.399714 0.0245080
\(267\) 7.96387 0.487381
\(268\) 5.38233 0.328778
\(269\) 17.9977 1.09734 0.548670 0.836039i \(-0.315134\pi\)
0.548670 + 0.836039i \(0.315134\pi\)
\(270\) 8.83013 0.537384
\(271\) 25.3228 1.53825 0.769127 0.639097i \(-0.220692\pi\)
0.769127 + 0.639097i \(0.220692\pi\)
\(272\) 4.63540 0.281062
\(273\) 1.11321 0.0673745
\(274\) −1.72137 −0.103992
\(275\) −2.57371 −0.155200
\(276\) −0.334358 −0.0201260
\(277\) 29.9578 1.79999 0.899995 0.435900i \(-0.143570\pi\)
0.899995 + 0.435900i \(0.143570\pi\)
\(278\) 35.9302 2.15495
\(279\) −15.7038 −0.940159
\(280\) −2.57411 −0.153832
\(281\) 13.7978 0.823109 0.411555 0.911385i \(-0.364986\pi\)
0.411555 + 0.911385i \(0.364986\pi\)
\(282\) 7.11119 0.423465
\(283\) −5.14314 −0.305728 −0.152864 0.988247i \(-0.548850\pi\)
−0.152864 + 0.988247i \(0.548850\pi\)
\(284\) −2.91019 −0.172688
\(285\) −0.365735 −0.0216643
\(286\) 11.0188 0.651553
\(287\) 7.29096 0.430372
\(288\) −10.1744 −0.599532
\(289\) −16.1039 −0.947290
\(290\) −3.11371 −0.182843
\(291\) −6.33767 −0.371521
\(292\) −4.93513 −0.288806
\(293\) −2.75536 −0.160970 −0.0804848 0.996756i \(-0.525647\pi\)
−0.0804848 + 0.996756i \(0.525647\pi\)
\(294\) −5.32765 −0.310715
\(295\) −20.6171 −1.20037
\(296\) 11.9915 0.696995
\(297\) 5.28593 0.306721
\(298\) 20.9349 1.21272
\(299\) −3.61185 −0.208879
\(300\) 0.461678 0.0266550
\(301\) −0.471470 −0.0271751
\(302\) 3.25400 0.187247
\(303\) −7.35071 −0.422288
\(304\) 1.91122 0.109616
\(305\) 12.5218 0.716997
\(306\) −4.27208 −0.244218
\(307\) 17.5805 1.00337 0.501687 0.865049i \(-0.332713\pi\)
0.501687 + 0.865049i \(0.332713\pi\)
\(308\) 0.791704 0.0451115
\(309\) 9.20948 0.523909
\(310\) 17.7331 1.00717
\(311\) 12.4105 0.703733 0.351867 0.936050i \(-0.385547\pi\)
0.351867 + 0.936050i \(0.385547\pi\)
\(312\) 3.84707 0.217798
\(313\) 3.10520 0.175517 0.0877583 0.996142i \(-0.472030\pi\)
0.0877583 + 0.996142i \(0.472030\pi\)
\(314\) −4.87251 −0.274972
\(315\) 3.28236 0.184940
\(316\) 10.2587 0.577097
\(317\) 29.2431 1.64246 0.821229 0.570598i \(-0.193289\pi\)
0.821229 + 0.570598i \(0.193289\pi\)
\(318\) 2.75104 0.154270
\(319\) −1.86394 −0.104361
\(320\) −7.14250 −0.399278
\(321\) −9.48618 −0.529467
\(322\) −1.02413 −0.0570724
\(323\) 0.369459 0.0205573
\(324\) 4.66700 0.259278
\(325\) 4.98720 0.276640
\(326\) −4.84865 −0.268542
\(327\) −3.78830 −0.209493
\(328\) 25.1964 1.39124
\(329\) 5.51938 0.304293
\(330\) −2.85874 −0.157368
\(331\) −10.2179 −0.561628 −0.280814 0.959762i \(-0.590604\pi\)
−0.280814 + 0.959762i \(0.590604\pi\)
\(332\) 10.9706 0.602090
\(333\) −15.2910 −0.837940
\(334\) −8.66462 −0.474107
\(335\) −15.0845 −0.824152
\(336\) 1.50926 0.0823367
\(337\) 4.19076 0.228285 0.114143 0.993464i \(-0.463588\pi\)
0.114143 + 0.993464i \(0.463588\pi\)
\(338\) −0.0744041 −0.00404705
\(339\) −1.47594 −0.0801621
\(340\) 1.22244 0.0662959
\(341\) 10.6155 0.574860
\(342\) −1.76142 −0.0952465
\(343\) −8.51515 −0.459775
\(344\) −1.62932 −0.0878472
\(345\) 0.937068 0.0504501
\(346\) −20.9080 −1.12402
\(347\) −6.16002 −0.330687 −0.165344 0.986236i \(-0.552873\pi\)
−0.165344 + 0.986236i \(0.552873\pi\)
\(348\) 0.334358 0.0179235
\(349\) 7.85979 0.420725 0.210362 0.977623i \(-0.432536\pi\)
0.210362 + 0.977623i \(0.432536\pi\)
\(350\) 1.41410 0.0755870
\(351\) −10.2428 −0.546720
\(352\) 6.87772 0.366584
\(353\) 11.6365 0.619351 0.309675 0.950842i \(-0.399780\pi\)
0.309675 + 0.950842i \(0.399780\pi\)
\(354\) 8.73687 0.464359
\(355\) 8.15608 0.432880
\(356\) −10.9751 −0.581677
\(357\) 0.291756 0.0154413
\(358\) 1.61174 0.0851832
\(359\) −2.33879 −0.123437 −0.0617184 0.998094i \(-0.519658\pi\)
−0.0617184 + 0.998094i \(0.519658\pi\)
\(360\) 11.3433 0.597845
\(361\) −18.8477 −0.991983
\(362\) −2.25333 −0.118433
\(363\) 3.70691 0.194563
\(364\) −1.53412 −0.0804099
\(365\) 13.8311 0.723955
\(366\) −5.30635 −0.277367
\(367\) 26.2052 1.36790 0.683951 0.729528i \(-0.260260\pi\)
0.683951 + 0.729528i \(0.260260\pi\)
\(368\) −4.89684 −0.255265
\(369\) −32.1291 −1.67257
\(370\) 17.2670 0.897668
\(371\) 2.13523 0.110856
\(372\) −1.90423 −0.0987296
\(373\) 17.5926 0.910909 0.455454 0.890259i \(-0.349477\pi\)
0.455454 + 0.890259i \(0.349477\pi\)
\(374\) 2.88785 0.149327
\(375\) −5.97923 −0.308766
\(376\) 19.0741 0.983671
\(377\) 3.61185 0.186020
\(378\) −2.90431 −0.149382
\(379\) −24.9604 −1.28213 −0.641064 0.767488i \(-0.721507\pi\)
−0.641064 + 0.767488i \(0.721507\pi\)
\(380\) 0.504022 0.0258558
\(381\) 6.37411 0.326555
\(382\) 30.0516 1.53757
\(383\) −7.00053 −0.357710 −0.178855 0.983875i \(-0.557239\pi\)
−0.178855 + 0.983875i \(0.557239\pi\)
\(384\) 6.66178 0.339957
\(385\) −2.21882 −0.113082
\(386\) −31.4027 −1.59836
\(387\) 2.07762 0.105612
\(388\) 8.73400 0.443402
\(389\) −29.5510 −1.49829 −0.749147 0.662404i \(-0.769536\pi\)
−0.749147 + 0.662404i \(0.769536\pi\)
\(390\) 5.53952 0.280504
\(391\) −0.946611 −0.0478722
\(392\) −14.2902 −0.721763
\(393\) 6.10516 0.307965
\(394\) 23.4172 1.17974
\(395\) −28.7509 −1.44662
\(396\) −3.48880 −0.175319
\(397\) 10.9239 0.548253 0.274126 0.961694i \(-0.411611\pi\)
0.274126 + 0.961694i \(0.411611\pi\)
\(398\) −19.6795 −0.986444
\(399\) 0.120294 0.00602221
\(400\) 6.76150 0.338075
\(401\) 5.68048 0.283669 0.141835 0.989890i \(-0.454700\pi\)
0.141835 + 0.989890i \(0.454700\pi\)
\(402\) 6.39231 0.318819
\(403\) −20.5701 −1.02467
\(404\) 10.1301 0.503990
\(405\) −13.0797 −0.649934
\(406\) 1.02413 0.0508266
\(407\) 10.3364 0.512358
\(408\) 1.00826 0.0499163
\(409\) −33.6508 −1.66393 −0.831963 0.554832i \(-0.812783\pi\)
−0.831963 + 0.554832i \(0.812783\pi\)
\(410\) 36.2810 1.79179
\(411\) −0.518045 −0.0255533
\(412\) −12.6917 −0.625273
\(413\) 6.78116 0.333679
\(414\) 4.51302 0.221803
\(415\) −30.7461 −1.50927
\(416\) −13.3273 −0.653424
\(417\) 10.8132 0.529524
\(418\) 1.19069 0.0582385
\(419\) −3.58358 −0.175070 −0.0875348 0.996161i \(-0.527899\pi\)
−0.0875348 + 0.996161i \(0.527899\pi\)
\(420\) 0.398017 0.0194213
\(421\) −31.5947 −1.53983 −0.769915 0.638147i \(-0.779701\pi\)
−0.769915 + 0.638147i \(0.779701\pi\)
\(422\) 17.4102 0.847513
\(423\) −24.3222 −1.18259
\(424\) 7.37900 0.358356
\(425\) 1.30707 0.0634022
\(426\) −3.45629 −0.167458
\(427\) −4.11854 −0.199310
\(428\) 13.0730 0.631906
\(429\) 3.31609 0.160102
\(430\) −2.34611 −0.113139
\(431\) −9.32434 −0.449138 −0.224569 0.974458i \(-0.572097\pi\)
−0.224569 + 0.974458i \(0.572097\pi\)
\(432\) −13.8869 −0.668133
\(433\) −31.0162 −1.49054 −0.745272 0.666761i \(-0.767680\pi\)
−0.745272 + 0.666761i \(0.767680\pi\)
\(434\) −5.83258 −0.279973
\(435\) −0.937068 −0.0449290
\(436\) 5.22069 0.250025
\(437\) −0.390297 −0.0186704
\(438\) −5.86119 −0.280059
\(439\) 23.7687 1.13442 0.567210 0.823573i \(-0.308023\pi\)
0.567210 + 0.823573i \(0.308023\pi\)
\(440\) −7.66789 −0.365552
\(441\) 18.2221 0.867717
\(442\) −5.59593 −0.266171
\(443\) 25.0563 1.19046 0.595230 0.803555i \(-0.297061\pi\)
0.595230 + 0.803555i \(0.297061\pi\)
\(444\) −1.85417 −0.0879952
\(445\) 30.7586 1.45810
\(446\) −12.6887 −0.600828
\(447\) 6.30034 0.297996
\(448\) 2.34923 0.110991
\(449\) 40.3089 1.90230 0.951148 0.308734i \(-0.0999054\pi\)
0.951148 + 0.308734i \(0.0999054\pi\)
\(450\) −6.23153 −0.293757
\(451\) 21.7187 1.02269
\(452\) 2.03401 0.0956716
\(453\) 0.979290 0.0460111
\(454\) −12.2888 −0.576744
\(455\) 4.29952 0.201565
\(456\) 0.415715 0.0194676
\(457\) −1.07682 −0.0503713 −0.0251856 0.999683i \(-0.508018\pi\)
−0.0251856 + 0.999683i \(0.508018\pi\)
\(458\) −17.2245 −0.804849
\(459\) −2.68448 −0.125301
\(460\) −1.29138 −0.0602110
\(461\) 36.1413 1.68327 0.841635 0.540047i \(-0.181594\pi\)
0.841635 + 0.540047i \(0.181594\pi\)
\(462\) 0.940265 0.0437451
\(463\) −17.3235 −0.805090 −0.402545 0.915400i \(-0.631874\pi\)
−0.402545 + 0.915400i \(0.631874\pi\)
\(464\) 4.89684 0.227330
\(465\) 5.33677 0.247487
\(466\) 6.21937 0.288107
\(467\) −11.1144 −0.514311 −0.257156 0.966370i \(-0.582785\pi\)
−0.257156 + 0.966370i \(0.582785\pi\)
\(468\) 6.76042 0.312500
\(469\) 4.96142 0.229097
\(470\) 27.4653 1.26688
\(471\) −1.46638 −0.0675671
\(472\) 23.4346 1.07866
\(473\) −1.40444 −0.0645762
\(474\) 12.1837 0.559617
\(475\) 0.538917 0.0247272
\(476\) −0.402071 −0.0184289
\(477\) −9.40930 −0.430822
\(478\) 9.44906 0.432190
\(479\) −26.5340 −1.21237 −0.606184 0.795324i \(-0.707301\pi\)
−0.606184 + 0.795324i \(0.707301\pi\)
\(480\) 3.45767 0.157820
\(481\) −20.0294 −0.913263
\(482\) −19.1502 −0.872270
\(483\) −0.308211 −0.0140241
\(484\) −5.10853 −0.232206
\(485\) −24.4778 −1.11148
\(486\) 19.4673 0.883055
\(487\) −0.676829 −0.0306701 −0.0153350 0.999882i \(-0.504881\pi\)
−0.0153350 + 0.999882i \(0.504881\pi\)
\(488\) −14.2330 −0.644298
\(489\) −1.45920 −0.0659872
\(490\) −20.5768 −0.929567
\(491\) 5.74000 0.259043 0.129521 0.991577i \(-0.458656\pi\)
0.129521 + 0.991577i \(0.458656\pi\)
\(492\) −3.89595 −0.175643
\(493\) 0.946611 0.0426332
\(494\) −2.30725 −0.103808
\(495\) 9.77767 0.439474
\(496\) −27.8883 −1.25222
\(497\) −2.68261 −0.120332
\(498\) 13.0292 0.583853
\(499\) −10.2276 −0.457851 −0.228926 0.973444i \(-0.573521\pi\)
−0.228926 + 0.973444i \(0.573521\pi\)
\(500\) 8.24003 0.368505
\(501\) −2.60761 −0.116499
\(502\) 30.1864 1.34728
\(503\) −6.43144 −0.286764 −0.143382 0.989667i \(-0.545798\pi\)
−0.143382 + 0.989667i \(0.545798\pi\)
\(504\) −3.73092 −0.166188
\(505\) −28.3904 −1.26336
\(506\) −3.05072 −0.135621
\(507\) −0.0223919 −0.000994457 0
\(508\) −8.78421 −0.389736
\(509\) −6.86016 −0.304071 −0.152036 0.988375i \(-0.548583\pi\)
−0.152036 + 0.988375i \(0.548583\pi\)
\(510\) 1.45182 0.0642878
\(511\) −4.54919 −0.201244
\(512\) 3.10912 0.137405
\(513\) −1.10684 −0.0488681
\(514\) 2.17163 0.0957864
\(515\) 35.5695 1.56738
\(516\) 0.251931 0.0110907
\(517\) 16.4414 0.723093
\(518\) −5.67927 −0.249533
\(519\) −6.29224 −0.276198
\(520\) 14.8584 0.651586
\(521\) 37.1452 1.62736 0.813680 0.581313i \(-0.197461\pi\)
0.813680 + 0.581313i \(0.197461\pi\)
\(522\) −4.51302 −0.197530
\(523\) 42.2124 1.84582 0.922910 0.385017i \(-0.125804\pi\)
0.922910 + 0.385017i \(0.125804\pi\)
\(524\) −8.41357 −0.367549
\(525\) 0.425574 0.0185736
\(526\) −17.3599 −0.756926
\(527\) −5.39111 −0.234841
\(528\) 4.49585 0.195657
\(529\) 1.00000 0.0434783
\(530\) 10.6252 0.461531
\(531\) −29.8825 −1.29679
\(532\) −0.165778 −0.00718737
\(533\) −42.0854 −1.82292
\(534\) −13.0345 −0.564059
\(535\) −36.6382 −1.58401
\(536\) 17.1459 0.740588
\(537\) 0.485053 0.0209316
\(538\) −29.4570 −1.26998
\(539\) −12.3178 −0.530565
\(540\) −3.66221 −0.157597
\(541\) −18.0359 −0.775423 −0.387711 0.921781i \(-0.626734\pi\)
−0.387711 + 0.921781i \(0.626734\pi\)
\(542\) −41.4461 −1.78026
\(543\) −0.678139 −0.0291017
\(544\) −3.49288 −0.149756
\(545\) −14.6314 −0.626742
\(546\) −1.82200 −0.0779743
\(547\) 1.10461 0.0472299 0.0236149 0.999721i \(-0.492482\pi\)
0.0236149 + 0.999721i \(0.492482\pi\)
\(548\) 0.713922 0.0304972
\(549\) 18.1492 0.774588
\(550\) 4.21241 0.179618
\(551\) 0.390297 0.0166272
\(552\) −1.06513 −0.0453348
\(553\) 9.45644 0.402129
\(554\) −49.0322 −2.08318
\(555\) 5.19649 0.220579
\(556\) −14.9017 −0.631974
\(557\) −10.4532 −0.442917 −0.221459 0.975170i \(-0.571082\pi\)
−0.221459 + 0.975170i \(0.571082\pi\)
\(558\) 25.7024 1.08807
\(559\) 2.72145 0.115105
\(560\) 5.82916 0.246327
\(561\) 0.869097 0.0366933
\(562\) −22.5830 −0.952606
\(563\) −19.6296 −0.827287 −0.413644 0.910439i \(-0.635744\pi\)
−0.413644 + 0.910439i \(0.635744\pi\)
\(564\) −2.94930 −0.124188
\(565\) −5.70048 −0.239821
\(566\) 8.41781 0.353827
\(567\) 4.30203 0.180668
\(568\) −9.27067 −0.388989
\(569\) −5.92670 −0.248460 −0.124230 0.992253i \(-0.539646\pi\)
−0.124230 + 0.992253i \(0.539646\pi\)
\(570\) 0.598600 0.0250726
\(571\) 39.4368 1.65038 0.825190 0.564855i \(-0.191068\pi\)
0.825190 + 0.564855i \(0.191068\pi\)
\(572\) −4.56993 −0.191078
\(573\) 9.04401 0.377819
\(574\) −11.9332 −0.498081
\(575\) −1.38079 −0.0575829
\(576\) −10.3524 −0.431348
\(577\) −26.3371 −1.09643 −0.548213 0.836339i \(-0.684692\pi\)
−0.548213 + 0.836339i \(0.684692\pi\)
\(578\) 26.3574 1.09632
\(579\) −9.45063 −0.392755
\(580\) 1.29138 0.0536217
\(581\) 10.1127 0.419545
\(582\) 10.3729 0.429971
\(583\) 6.36053 0.263426
\(584\) −15.7213 −0.650550
\(585\) −18.9467 −0.783349
\(586\) 4.50971 0.186294
\(587\) 25.2199 1.04094 0.520468 0.853881i \(-0.325757\pi\)
0.520468 + 0.853881i \(0.325757\pi\)
\(588\) 2.20959 0.0911221
\(589\) −2.22281 −0.0915892
\(590\) 33.7442 1.38922
\(591\) 7.04738 0.289890
\(592\) −27.1553 −1.11608
\(593\) −16.8992 −0.693965 −0.346983 0.937872i \(-0.612794\pi\)
−0.346983 + 0.937872i \(0.612794\pi\)
\(594\) −8.65151 −0.354976
\(595\) 1.12684 0.0461959
\(596\) −8.68255 −0.355651
\(597\) −5.92253 −0.242393
\(598\) 5.91154 0.241741
\(599\) −17.5628 −0.717596 −0.358798 0.933415i \(-0.616813\pi\)
−0.358798 + 0.933415i \(0.616813\pi\)
\(600\) 1.47071 0.0600416
\(601\) 23.3210 0.951282 0.475641 0.879640i \(-0.342216\pi\)
0.475641 + 0.879640i \(0.342216\pi\)
\(602\) 0.771658 0.0314504
\(603\) −21.8635 −0.890349
\(604\) −1.34957 −0.0549131
\(605\) 14.3171 0.582073
\(606\) 12.0310 0.488725
\(607\) 2.37912 0.0965657 0.0482828 0.998834i \(-0.484625\pi\)
0.0482828 + 0.998834i \(0.484625\pi\)
\(608\) −1.44015 −0.0584057
\(609\) 0.308211 0.0124893
\(610\) −20.4945 −0.829800
\(611\) −31.8594 −1.28889
\(612\) 1.77180 0.0716209
\(613\) −2.43882 −0.0985031 −0.0492516 0.998786i \(-0.515684\pi\)
−0.0492516 + 0.998786i \(0.515684\pi\)
\(614\) −28.7742 −1.16123
\(615\) 10.9187 0.440286
\(616\) 2.52204 0.101616
\(617\) 1.46026 0.0587877 0.0293938 0.999568i \(-0.490642\pi\)
0.0293938 + 0.999568i \(0.490642\pi\)
\(618\) −15.0732 −0.606334
\(619\) 13.7842 0.554034 0.277017 0.960865i \(-0.410654\pi\)
0.277017 + 0.960865i \(0.410654\pi\)
\(620\) −7.35464 −0.295369
\(621\) 2.83589 0.113800
\(622\) −20.3123 −0.814449
\(623\) −10.1168 −0.405321
\(624\) −8.71183 −0.348752
\(625\) −16.1895 −0.647579
\(626\) −5.08231 −0.203130
\(627\) 0.358337 0.0143106
\(628\) 2.02083 0.0806398
\(629\) −5.24941 −0.209308
\(630\) −5.37227 −0.214036
\(631\) 41.7386 1.66159 0.830794 0.556580i \(-0.187887\pi\)
0.830794 + 0.556580i \(0.187887\pi\)
\(632\) 32.6799 1.29994
\(633\) 5.23957 0.208254
\(634\) −47.8625 −1.90086
\(635\) 24.6185 0.976956
\(636\) −1.14097 −0.0452423
\(637\) 23.8688 0.945716
\(638\) 3.05072 0.120779
\(639\) 11.8215 0.467650
\(640\) 25.7296 1.01705
\(641\) −33.0749 −1.30638 −0.653190 0.757194i \(-0.726570\pi\)
−0.653190 + 0.757194i \(0.726570\pi\)
\(642\) 15.5261 0.612766
\(643\) −4.39408 −0.173285 −0.0866427 0.996239i \(-0.527614\pi\)
−0.0866427 + 0.996239i \(0.527614\pi\)
\(644\) 0.424747 0.0167374
\(645\) −0.706060 −0.0278011
\(646\) −0.604696 −0.0237915
\(647\) −23.0920 −0.907839 −0.453920 0.891043i \(-0.649975\pi\)
−0.453920 + 0.891043i \(0.649975\pi\)
\(648\) 14.8671 0.584035
\(649\) 20.2001 0.792922
\(650\) −8.16259 −0.320163
\(651\) −1.75531 −0.0687961
\(652\) 2.01093 0.0787541
\(653\) 21.3521 0.835573 0.417786 0.908545i \(-0.362806\pi\)
0.417786 + 0.908545i \(0.362806\pi\)
\(654\) 6.20033 0.242452
\(655\) 23.5798 0.921338
\(656\) −57.0581 −2.22774
\(657\) 20.0469 0.782104
\(658\) −9.03361 −0.352167
\(659\) −26.4261 −1.02941 −0.514707 0.857366i \(-0.672099\pi\)
−0.514707 + 0.857366i \(0.672099\pi\)
\(660\) 1.18563 0.0461508
\(661\) −20.4354 −0.794846 −0.397423 0.917636i \(-0.630095\pi\)
−0.397423 + 0.917636i \(0.630095\pi\)
\(662\) 16.7237 0.649987
\(663\) −1.68409 −0.0654047
\(664\) 34.9478 1.35624
\(665\) 0.464606 0.0180167
\(666\) 25.0268 0.969770
\(667\) −1.00000 −0.0387202
\(668\) 3.59357 0.139039
\(669\) −3.81866 −0.147638
\(670\) 24.6888 0.953813
\(671\) −12.2685 −0.473621
\(672\) −1.13726 −0.0438708
\(673\) −6.31825 −0.243551 −0.121775 0.992558i \(-0.538859\pi\)
−0.121775 + 0.992558i \(0.538859\pi\)
\(674\) −6.85905 −0.264201
\(675\) −3.91576 −0.150718
\(676\) 0.0308584 0.00118686
\(677\) 13.2844 0.510561 0.255280 0.966867i \(-0.417832\pi\)
0.255280 + 0.966867i \(0.417832\pi\)
\(678\) 2.41568 0.0927737
\(679\) 8.05098 0.308968
\(680\) 3.89417 0.149335
\(681\) −3.69832 −0.141720
\(682\) −17.3744 −0.665300
\(683\) 28.2726 1.08182 0.540911 0.841080i \(-0.318080\pi\)
0.540911 + 0.841080i \(0.318080\pi\)
\(684\) 0.730531 0.0279326
\(685\) −2.00083 −0.0764478
\(686\) 13.9368 0.532110
\(687\) −5.18371 −0.197771
\(688\) 3.68966 0.140667
\(689\) −12.3251 −0.469549
\(690\) −1.53371 −0.0583872
\(691\) −9.71176 −0.369453 −0.184726 0.982790i \(-0.559140\pi\)
−0.184726 + 0.982790i \(0.559140\pi\)
\(692\) 8.67138 0.329636
\(693\) −3.21597 −0.122165
\(694\) 10.0821 0.382713
\(695\) 41.7634 1.58418
\(696\) 1.06513 0.0403735
\(697\) −11.0299 −0.417789
\(698\) −12.8642 −0.486916
\(699\) 1.87171 0.0707947
\(700\) −0.586486 −0.0221671
\(701\) −47.1272 −1.77997 −0.889984 0.455991i \(-0.849285\pi\)
−0.889984 + 0.455991i \(0.849285\pi\)
\(702\) 16.7645 0.632734
\(703\) −2.16438 −0.0816312
\(704\) 6.99802 0.263748
\(705\) 8.26568 0.311303
\(706\) −19.0456 −0.716791
\(707\) 9.33789 0.351187
\(708\) −3.62353 −0.136181
\(709\) 24.4222 0.917194 0.458597 0.888644i \(-0.348352\pi\)
0.458597 + 0.888644i \(0.348352\pi\)
\(710\) −13.3491 −0.500983
\(711\) −41.6717 −1.56281
\(712\) −34.9620 −1.31026
\(713\) 5.69517 0.213286
\(714\) −0.477518 −0.0178707
\(715\) 12.8076 0.478978
\(716\) −0.668455 −0.0249813
\(717\) 2.84369 0.106199
\(718\) 3.82792 0.142857
\(719\) 15.6219 0.582599 0.291300 0.956632i \(-0.405912\pi\)
0.291300 + 0.956632i \(0.405912\pi\)
\(720\) −25.6873 −0.957310
\(721\) −11.6991 −0.435699
\(722\) 30.8481 1.14805
\(723\) −5.76325 −0.214338
\(724\) 0.934549 0.0347322
\(725\) 1.38079 0.0512812
\(726\) −6.06713 −0.225172
\(727\) −41.8405 −1.55178 −0.775890 0.630868i \(-0.782699\pi\)
−0.775890 + 0.630868i \(0.782699\pi\)
\(728\) −4.88708 −0.181127
\(729\) −14.7672 −0.546932
\(730\) −22.6375 −0.837852
\(731\) 0.713251 0.0263805
\(732\) 2.20076 0.0813423
\(733\) 26.7436 0.987797 0.493899 0.869519i \(-0.335571\pi\)
0.493899 + 0.869519i \(0.335571\pi\)
\(734\) −42.8903 −1.58311
\(735\) −6.19259 −0.228417
\(736\) 3.68988 0.136011
\(737\) 14.7793 0.544404
\(738\) 52.5859 1.93571
\(739\) 8.37011 0.307899 0.153950 0.988079i \(-0.450801\pi\)
0.153950 + 0.988079i \(0.450801\pi\)
\(740\) −7.16132 −0.263255
\(741\) −0.694367 −0.0255082
\(742\) −3.49474 −0.128296
\(743\) −35.0079 −1.28432 −0.642158 0.766572i \(-0.721961\pi\)
−0.642158 + 0.766572i \(0.721961\pi\)
\(744\) −6.06607 −0.222393
\(745\) 24.3336 0.891514
\(746\) −28.7939 −1.05422
\(747\) −44.5635 −1.63049
\(748\) −1.19771 −0.0437926
\(749\) 12.0506 0.440321
\(750\) 9.78625 0.357343
\(751\) 13.3621 0.487591 0.243796 0.969827i \(-0.421607\pi\)
0.243796 + 0.969827i \(0.421607\pi\)
\(752\) −43.1939 −1.57512
\(753\) 9.08457 0.331060
\(754\) −5.91154 −0.215286
\(755\) 3.78228 0.137651
\(756\) 1.20454 0.0438086
\(757\) 6.34165 0.230491 0.115246 0.993337i \(-0.463234\pi\)
0.115246 + 0.993337i \(0.463234\pi\)
\(758\) 40.8528 1.48384
\(759\) −0.918114 −0.0333254
\(760\) 1.60560 0.0582414
\(761\) −19.1744 −0.695070 −0.347535 0.937667i \(-0.612981\pi\)
−0.347535 + 0.937667i \(0.612981\pi\)
\(762\) −10.4325 −0.377931
\(763\) 4.81242 0.174221
\(764\) −12.4636 −0.450918
\(765\) −4.96564 −0.179533
\(766\) 11.4578 0.413988
\(767\) −39.1427 −1.41336
\(768\) −7.20478 −0.259980
\(769\) 12.8445 0.463183 0.231591 0.972813i \(-0.425607\pi\)
0.231591 + 0.972813i \(0.425607\pi\)
\(770\) 3.63156 0.130872
\(771\) 0.653550 0.0235370
\(772\) 13.0240 0.468744
\(773\) 17.6744 0.635704 0.317852 0.948140i \(-0.397038\pi\)
0.317852 + 0.948140i \(0.397038\pi\)
\(774\) −3.40046 −0.122227
\(775\) −7.86383 −0.282477
\(776\) 27.8229 0.998783
\(777\) −1.70917 −0.0613163
\(778\) 48.3663 1.73401
\(779\) −4.54775 −0.162940
\(780\) −2.29746 −0.0822623
\(781\) −7.99110 −0.285944
\(782\) 1.54932 0.0554037
\(783\) −2.83589 −0.101346
\(784\) 32.3606 1.15574
\(785\) −5.66355 −0.202141
\(786\) −9.99236 −0.356416
\(787\) −13.0310 −0.464506 −0.232253 0.972655i \(-0.574610\pi\)
−0.232253 + 0.972655i \(0.574610\pi\)
\(788\) −9.71205 −0.345977
\(789\) −5.22444 −0.185995
\(790\) 47.0568 1.67421
\(791\) 1.87494 0.0666653
\(792\) −11.1139 −0.394914
\(793\) 23.7733 0.844216
\(794\) −17.8792 −0.634507
\(795\) 3.19766 0.113409
\(796\) 8.16189 0.289291
\(797\) 51.3069 1.81738 0.908691 0.417469i \(-0.137083\pi\)
0.908691 + 0.417469i \(0.137083\pi\)
\(798\) −0.196885 −0.00696966
\(799\) −8.34985 −0.295397
\(800\) −5.09494 −0.180134
\(801\) 44.5817 1.57522
\(802\) −9.29727 −0.328298
\(803\) −13.5514 −0.478217
\(804\) −2.65115 −0.0934989
\(805\) −1.19039 −0.0419558
\(806\) 33.6672 1.18588
\(807\) −8.86507 −0.312065
\(808\) 32.2702 1.13526
\(809\) −39.9620 −1.40499 −0.702495 0.711689i \(-0.747931\pi\)
−0.702495 + 0.711689i \(0.747931\pi\)
\(810\) 21.4076 0.752186
\(811\) 41.7664 1.46662 0.733308 0.679897i \(-0.237976\pi\)
0.733308 + 0.679897i \(0.237976\pi\)
\(812\) −0.424747 −0.0149057
\(813\) −12.4732 −0.437453
\(814\) −16.9177 −0.592966
\(815\) −5.63581 −0.197414
\(816\) −2.28324 −0.0799294
\(817\) 0.294080 0.0102886
\(818\) 55.0765 1.92570
\(819\) 6.23174 0.217755
\(820\) −15.0472 −0.525471
\(821\) 31.4508 1.09764 0.548820 0.835941i \(-0.315077\pi\)
0.548820 + 0.835941i \(0.315077\pi\)
\(822\) 0.847888 0.0295735
\(823\) 3.09308 0.107818 0.0539090 0.998546i \(-0.482832\pi\)
0.0539090 + 0.998546i \(0.482832\pi\)
\(824\) −40.4303 −1.40846
\(825\) 1.26772 0.0441364
\(826\) −11.0988 −0.386176
\(827\) −13.6474 −0.474566 −0.237283 0.971441i \(-0.576257\pi\)
−0.237283 + 0.971441i \(0.576257\pi\)
\(828\) −1.87173 −0.0650472
\(829\) 38.3562 1.33217 0.666083 0.745877i \(-0.267970\pi\)
0.666083 + 0.745877i \(0.267970\pi\)
\(830\) 50.3224 1.74672
\(831\) −14.7562 −0.511887
\(832\) −13.5604 −0.470123
\(833\) 6.25565 0.216745
\(834\) −17.6980 −0.612832
\(835\) −10.0713 −0.348532
\(836\) −0.493827 −0.0170794
\(837\) 16.1509 0.558256
\(838\) 5.86528 0.202613
\(839\) 25.1781 0.869243 0.434622 0.900613i \(-0.356882\pi\)
0.434622 + 0.900613i \(0.356882\pi\)
\(840\) 1.26792 0.0437473
\(841\) 1.00000 0.0344828
\(842\) 51.7112 1.78209
\(843\) −6.79634 −0.234078
\(844\) −7.22070 −0.248547
\(845\) −0.0864834 −0.00297512
\(846\) 39.8084 1.36864
\(847\) −4.70903 −0.161804
\(848\) −16.7100 −0.573824
\(849\) 2.53334 0.0869438
\(850\) −2.13929 −0.0733771
\(851\) 5.54547 0.190096
\(852\) 1.43346 0.0491096
\(853\) 18.3174 0.627176 0.313588 0.949559i \(-0.398469\pi\)
0.313588 + 0.949559i \(0.398469\pi\)
\(854\) 6.74085 0.230667
\(855\) −2.04738 −0.0700189
\(856\) 41.6451 1.42340
\(857\) 6.45885 0.220630 0.110315 0.993897i \(-0.464814\pi\)
0.110315 + 0.993897i \(0.464814\pi\)
\(858\) −5.42746 −0.185291
\(859\) −26.8098 −0.914739 −0.457370 0.889277i \(-0.651208\pi\)
−0.457370 + 0.889277i \(0.651208\pi\)
\(860\) 0.973027 0.0331800
\(861\) −3.59128 −0.122390
\(862\) 15.2612 0.519799
\(863\) 36.8892 1.25572 0.627861 0.778325i \(-0.283931\pi\)
0.627861 + 0.778325i \(0.283931\pi\)
\(864\) 10.4641 0.355995
\(865\) −24.3023 −0.826304
\(866\) 50.7644 1.72504
\(867\) 7.93224 0.269393
\(868\) 2.41901 0.0821065
\(869\) 28.1693 0.955580
\(870\) 1.53371 0.0519975
\(871\) −28.6386 −0.970383
\(872\) 16.6309 0.563194
\(873\) −35.4782 −1.20076
\(874\) 0.638801 0.0216078
\(875\) 7.59564 0.256780
\(876\) 2.43087 0.0821316
\(877\) 5.06724 0.171108 0.0855542 0.996334i \(-0.472734\pi\)
0.0855542 + 0.996334i \(0.472734\pi\)
\(878\) −38.9025 −1.31289
\(879\) 1.35719 0.0457770
\(880\) 17.3642 0.585347
\(881\) 49.1495 1.65589 0.827945 0.560809i \(-0.189510\pi\)
0.827945 + 0.560809i \(0.189510\pi\)
\(882\) −29.8242 −1.00423
\(883\) 37.3623 1.25734 0.628671 0.777671i \(-0.283599\pi\)
0.628671 + 0.777671i \(0.283599\pi\)
\(884\) 2.32086 0.0780589
\(885\) 10.1553 0.341366
\(886\) −41.0098 −1.37775
\(887\) −28.0049 −0.940312 −0.470156 0.882583i \(-0.655802\pi\)
−0.470156 + 0.882583i \(0.655802\pi\)
\(888\) −5.90663 −0.198213
\(889\) −8.09726 −0.271574
\(890\) −50.3428 −1.68750
\(891\) 12.8151 0.429322
\(892\) 5.26253 0.176203
\(893\) −3.44272 −0.115206
\(894\) −10.3118 −0.344878
\(895\) 1.87340 0.0626210
\(896\) −8.46270 −0.282719
\(897\) 1.77907 0.0594015
\(898\) −65.9739 −2.20158
\(899\) −5.69517 −0.189945
\(900\) 2.58447 0.0861489
\(901\) −3.23022 −0.107614
\(902\) −35.5471 −1.18359
\(903\) 0.232230 0.00772813
\(904\) 6.47949 0.215505
\(905\) −2.61916 −0.0870637
\(906\) −1.60281 −0.0532498
\(907\) 15.4437 0.512798 0.256399 0.966571i \(-0.417464\pi\)
0.256399 + 0.966571i \(0.417464\pi\)
\(908\) 5.09668 0.169139
\(909\) −41.1492 −1.36483
\(910\) −7.03705 −0.233276
\(911\) 41.4852 1.37447 0.687233 0.726437i \(-0.258825\pi\)
0.687233 + 0.726437i \(0.258825\pi\)
\(912\) −0.941401 −0.0311729
\(913\) 30.1242 0.996965
\(914\) 1.76243 0.0582960
\(915\) −6.16782 −0.203902
\(916\) 7.14371 0.236035
\(917\) −7.75561 −0.256113
\(918\) 4.39371 0.145014
\(919\) −56.7005 −1.87038 −0.935188 0.354152i \(-0.884770\pi\)
−0.935188 + 0.354152i \(0.884770\pi\)
\(920\) −4.11380 −0.135628
\(921\) −8.65957 −0.285343
\(922\) −59.1528 −1.94809
\(923\) 15.4848 0.509687
\(924\) −0.389966 −0.0128290
\(925\) −7.65713 −0.251765
\(926\) 28.3535 0.931752
\(927\) 51.5546 1.69327
\(928\) −3.68988 −0.121126
\(929\) −25.5957 −0.839768 −0.419884 0.907578i \(-0.637929\pi\)
−0.419884 + 0.907578i \(0.637929\pi\)
\(930\) −8.73472 −0.286423
\(931\) 2.57926 0.0845320
\(932\) −2.57942 −0.0844919
\(933\) −6.11297 −0.200130
\(934\) 18.1910 0.595226
\(935\) 3.35669 0.109775
\(936\) 21.5359 0.703922
\(937\) −35.6726 −1.16537 −0.582687 0.812697i \(-0.697999\pi\)
−0.582687 + 0.812697i \(0.697999\pi\)
\(938\) −8.12039 −0.265140
\(939\) −1.52952 −0.0499139
\(940\) −11.3910 −0.371533
\(941\) 43.2631 1.41034 0.705169 0.709039i \(-0.250871\pi\)
0.705169 + 0.709039i \(0.250871\pi\)
\(942\) 2.40003 0.0781972
\(943\) 11.6520 0.379442
\(944\) −53.0684 −1.72723
\(945\) −3.37582 −0.109815
\(946\) 2.29865 0.0747357
\(947\) 45.7018 1.48511 0.742555 0.669785i \(-0.233614\pi\)
0.742555 + 0.669785i \(0.233614\pi\)
\(948\) −5.05308 −0.164117
\(949\) 26.2591 0.852408
\(950\) −0.882050 −0.0286175
\(951\) −14.4042 −0.467087
\(952\) −1.28083 −0.0415119
\(953\) −53.7722 −1.74185 −0.870926 0.491414i \(-0.836480\pi\)
−0.870926 + 0.491414i \(0.836480\pi\)
\(954\) 15.4003 0.498602
\(955\) 34.9304 1.13032
\(956\) −3.91891 −0.126747
\(957\) 0.918114 0.0296784
\(958\) 43.4283 1.40311
\(959\) 0.658092 0.0212509
\(960\) 3.51815 0.113548
\(961\) 1.43500 0.0462903
\(962\) 32.7823 1.05694
\(963\) −53.1035 −1.71124
\(964\) 7.94238 0.255807
\(965\) −36.5009 −1.17501
\(966\) 0.504450 0.0162304
\(967\) 40.5056 1.30257 0.651287 0.758832i \(-0.274230\pi\)
0.651287 + 0.758832i \(0.274230\pi\)
\(968\) −16.2736 −0.523055
\(969\) −0.181983 −0.00584614
\(970\) 40.0630 1.28635
\(971\) 19.0454 0.611195 0.305597 0.952161i \(-0.401144\pi\)
0.305597 + 0.952161i \(0.401144\pi\)
\(972\) −8.07388 −0.258970
\(973\) −13.7364 −0.440368
\(974\) 1.10777 0.0354953
\(975\) −2.45653 −0.0786718
\(976\) 32.2312 1.03169
\(977\) −50.0749 −1.60204 −0.801019 0.598639i \(-0.795708\pi\)
−0.801019 + 0.598639i \(0.795708\pi\)
\(978\) 2.38828 0.0763687
\(979\) −30.1364 −0.963165
\(980\) 8.53405 0.272610
\(981\) −21.2069 −0.677083
\(982\) −9.39469 −0.299797
\(983\) −22.5598 −0.719547 −0.359773 0.933040i \(-0.617146\pi\)
−0.359773 + 0.933040i \(0.617146\pi\)
\(984\) −12.4109 −0.395644
\(985\) 27.2189 0.867265
\(986\) −1.54932 −0.0493406
\(987\) −2.71866 −0.0865359
\(988\) 0.956912 0.0304434
\(989\) −0.753478 −0.0239592
\(990\) −16.0032 −0.508615
\(991\) 38.5400 1.22426 0.612132 0.790755i \(-0.290312\pi\)
0.612132 + 0.790755i \(0.290312\pi\)
\(992\) 21.0145 0.667211
\(993\) 5.03300 0.159717
\(994\) 4.39065 0.139263
\(995\) −22.8744 −0.725168
\(996\) −5.40375 −0.171224
\(997\) −14.3246 −0.453665 −0.226833 0.973934i \(-0.572837\pi\)
−0.226833 + 0.973934i \(0.572837\pi\)
\(998\) 16.7396 0.529883
\(999\) 15.7263 0.497560
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 667.2.a.d.1.4 16
3.2 odd 2 6003.2.a.q.1.13 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
667.2.a.d.1.4 16 1.1 even 1 trivial
6003.2.a.q.1.13 16 3.2 odd 2