Properties

Label 1849.4.a.g.1.20
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $30$
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] = [1849,4,Mod(1,1849)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1849, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1849.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.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: \(109.094531601\)
Analytic rank: \(1\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) \(=\) 1849.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.56619 q^{2} +5.42418 q^{3} -5.54704 q^{4} -8.12470 q^{5} +8.49531 q^{6} +30.7370 q^{7} -21.2173 q^{8} +2.42171 q^{9} +O(q^{10})\) \(q+1.56619 q^{2} +5.42418 q^{3} -5.54704 q^{4} -8.12470 q^{5} +8.49531 q^{6} +30.7370 q^{7} -21.2173 q^{8} +2.42171 q^{9} -12.7249 q^{10} +9.06614 q^{11} -30.0881 q^{12} -16.1006 q^{13} +48.1400 q^{14} -44.0698 q^{15} +11.1459 q^{16} -13.4280 q^{17} +3.79286 q^{18} +76.7126 q^{19} +45.0680 q^{20} +166.723 q^{21} +14.1993 q^{22} -38.4314 q^{23} -115.086 q^{24} -58.9893 q^{25} -25.2166 q^{26} -133.317 q^{27} -170.499 q^{28} -251.400 q^{29} -69.0219 q^{30} -120.743 q^{31} +187.195 q^{32} +49.1764 q^{33} -21.0308 q^{34} -249.729 q^{35} -13.4333 q^{36} +391.816 q^{37} +120.147 q^{38} -87.3323 q^{39} +172.384 q^{40} -338.720 q^{41} +261.120 q^{42} -50.2902 q^{44} -19.6756 q^{45} -60.1911 q^{46} +120.370 q^{47} +60.4574 q^{48} +601.761 q^{49} -92.3886 q^{50} -72.8358 q^{51} +89.3103 q^{52} -480.889 q^{53} -208.800 q^{54} -73.6597 q^{55} -652.155 q^{56} +416.103 q^{57} -393.741 q^{58} +357.983 q^{59} +244.457 q^{60} +592.360 q^{61} -189.106 q^{62} +74.4359 q^{63} +204.016 q^{64} +130.812 q^{65} +77.0197 q^{66} -672.610 q^{67} +74.4856 q^{68} -208.459 q^{69} -391.123 q^{70} -417.963 q^{71} -51.3820 q^{72} +922.640 q^{73} +613.660 q^{74} -319.968 q^{75} -425.528 q^{76} +278.666 q^{77} -136.779 q^{78} -107.902 q^{79} -90.5571 q^{80} -788.521 q^{81} -530.501 q^{82} -1003.60 q^{83} -924.817 q^{84} +109.098 q^{85} -1363.64 q^{87} -192.359 q^{88} +188.544 q^{89} -30.8158 q^{90} -494.882 q^{91} +213.181 q^{92} -654.930 q^{93} +188.522 q^{94} -623.267 q^{95} +1015.38 q^{96} +481.540 q^{97} +942.474 q^{98} +21.9555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9} - 27 q^{10} + 80 q^{11} + 36 q^{12} - 13 q^{13} + 36 q^{14} + 16 q^{15} + 318 q^{16} + 66 q^{17} - 80 q^{18} - 254 q^{19} - 312 q^{20} - 548 q^{21} - 305 q^{22} - 105 q^{23} + 123 q^{24} + 523 q^{25} - 549 q^{26} + 10 q^{27} - 578 q^{28} - 793 q^{29} - 1560 q^{30} - 359 q^{31} - 676 q^{32} - 208 q^{33} - 1007 q^{34} - 514 q^{35} + 776 q^{36} - 510 q^{37} - 2066 q^{38} - 898 q^{39} - 1248 q^{40} - 270 q^{41} + 915 q^{42} + 3256 q^{44} - 807 q^{45} - 1960 q^{46} + 1421 q^{47} + 632 q^{48} + 386 q^{49} + 141 q^{50} - 209 q^{51} + 2825 q^{52} - 21 q^{53} + 2368 q^{54} - 2258 q^{55} + 2521 q^{56} - 1723 q^{57} - 347 q^{58} + 1752 q^{59} + 2711 q^{60} - 1759 q^{61} - 395 q^{62} - 2204 q^{63} + 222 q^{64} - 1151 q^{65} + 160 q^{66} - 3001 q^{67} + 1921 q^{68} - 1660 q^{69} - 1597 q^{70} - 727 q^{71} - 9100 q^{72} - 4623 q^{73} - 2649 q^{74} - 1027 q^{75} - 874 q^{76} - 3556 q^{77} - 4979 q^{78} + 546 q^{79} - 5809 q^{80} - 410 q^{81} + 4397 q^{82} - 492 q^{83} - 10611 q^{84} + 1723 q^{85} + 5937 q^{87} - 3974 q^{88} - 5218 q^{89} + 10492 q^{90} - 1104 q^{91} + 1060 q^{92} - 1997 q^{93} + 2134 q^{94} + 6346 q^{95} - 11984 q^{96} + 2590 q^{97} - 6270 q^{98} - 2693 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.56619 0.553733 0.276867 0.960908i \(-0.410704\pi\)
0.276867 + 0.960908i \(0.410704\pi\)
\(3\) 5.42418 1.04388 0.521942 0.852981i \(-0.325208\pi\)
0.521942 + 0.852981i \(0.325208\pi\)
\(4\) −5.54704 −0.693380
\(5\) −8.12470 −0.726695 −0.363348 0.931654i \(-0.618366\pi\)
−0.363348 + 0.931654i \(0.618366\pi\)
\(6\) 8.49531 0.578033
\(7\) 30.7370 1.65964 0.829820 0.558031i \(-0.188443\pi\)
0.829820 + 0.558031i \(0.188443\pi\)
\(8\) −21.2173 −0.937680
\(9\) 2.42171 0.0896928
\(10\) −12.7249 −0.402395
\(11\) 9.06614 0.248504 0.124252 0.992251i \(-0.460347\pi\)
0.124252 + 0.992251i \(0.460347\pi\)
\(12\) −30.0881 −0.723808
\(13\) −16.1006 −0.343499 −0.171750 0.985141i \(-0.554942\pi\)
−0.171750 + 0.985141i \(0.554942\pi\)
\(14\) 48.1400 0.918998
\(15\) −44.0698 −0.758585
\(16\) 11.1459 0.174155
\(17\) −13.4280 −0.191574 −0.0957872 0.995402i \(-0.530537\pi\)
−0.0957872 + 0.995402i \(0.530537\pi\)
\(18\) 3.79286 0.0496659
\(19\) 76.7126 0.926267 0.463134 0.886288i \(-0.346725\pi\)
0.463134 + 0.886288i \(0.346725\pi\)
\(20\) 45.0680 0.503876
\(21\) 166.723 1.73247
\(22\) 14.1993 0.137605
\(23\) −38.4314 −0.348413 −0.174207 0.984709i \(-0.555736\pi\)
−0.174207 + 0.984709i \(0.555736\pi\)
\(24\) −115.086 −0.978829
\(25\) −58.9893 −0.471914
\(26\) −25.2166 −0.190207
\(27\) −133.317 −0.950255
\(28\) −170.499 −1.15076
\(29\) −251.400 −1.60979 −0.804893 0.593420i \(-0.797777\pi\)
−0.804893 + 0.593420i \(0.797777\pi\)
\(30\) −69.0219 −0.420054
\(31\) −120.743 −0.699549 −0.349775 0.936834i \(-0.613742\pi\)
−0.349775 + 0.936834i \(0.613742\pi\)
\(32\) 187.195 1.03412
\(33\) 49.1764 0.259409
\(34\) −21.0308 −0.106081
\(35\) −249.729 −1.20605
\(36\) −13.4333 −0.0621911
\(37\) 391.816 1.74092 0.870462 0.492235i \(-0.163820\pi\)
0.870462 + 0.492235i \(0.163820\pi\)
\(38\) 120.147 0.512905
\(39\) −87.3323 −0.358573
\(40\) 172.384 0.681408
\(41\) −338.720 −1.29022 −0.645112 0.764088i \(-0.723189\pi\)
−0.645112 + 0.764088i \(0.723189\pi\)
\(42\) 261.120 0.959327
\(43\) 0 0
\(44\) −50.2902 −0.172308
\(45\) −19.6756 −0.0651793
\(46\) −60.1911 −0.192928
\(47\) 120.370 0.373568 0.186784 0.982401i \(-0.440194\pi\)
0.186784 + 0.982401i \(0.440194\pi\)
\(48\) 60.4574 0.181797
\(49\) 601.761 1.75440
\(50\) −92.3886 −0.261315
\(51\) −72.8358 −0.199981
\(52\) 89.3103 0.238175
\(53\) −480.889 −1.24633 −0.623163 0.782092i \(-0.714153\pi\)
−0.623163 + 0.782092i \(0.714153\pi\)
\(54\) −208.800 −0.526188
\(55\) −73.6597 −0.180587
\(56\) −652.155 −1.55621
\(57\) 416.103 0.966915
\(58\) −393.741 −0.891392
\(59\) 357.983 0.789923 0.394961 0.918698i \(-0.370758\pi\)
0.394961 + 0.918698i \(0.370758\pi\)
\(60\) 244.457 0.525987
\(61\) 592.360 1.24334 0.621672 0.783278i \(-0.286454\pi\)
0.621672 + 0.783278i \(0.286454\pi\)
\(62\) −189.106 −0.387364
\(63\) 74.4359 0.148858
\(64\) 204.016 0.398469
\(65\) 130.812 0.249619
\(66\) 77.0197 0.143644
\(67\) −672.610 −1.22645 −0.613227 0.789907i \(-0.710129\pi\)
−0.613227 + 0.789907i \(0.710129\pi\)
\(68\) 74.4856 0.132834
\(69\) −208.459 −0.363703
\(70\) −391.123 −0.667831
\(71\) −417.963 −0.698635 −0.349318 0.937004i \(-0.613587\pi\)
−0.349318 + 0.937004i \(0.613587\pi\)
\(72\) −51.3820 −0.0841032
\(73\) 922.640 1.47927 0.739636 0.673007i \(-0.234998\pi\)
0.739636 + 0.673007i \(0.234998\pi\)
\(74\) 613.660 0.964008
\(75\) −319.968 −0.492623
\(76\) −425.528 −0.642255
\(77\) 278.666 0.412427
\(78\) −136.779 −0.198554
\(79\) −107.902 −0.153670 −0.0768350 0.997044i \(-0.524481\pi\)
−0.0768350 + 0.997044i \(0.524481\pi\)
\(80\) −90.5571 −0.126557
\(81\) −788.521 −1.08165
\(82\) −530.501 −0.714440
\(83\) −1003.60 −1.32722 −0.663609 0.748080i \(-0.730976\pi\)
−0.663609 + 0.748080i \(0.730976\pi\)
\(84\) −924.817 −1.20126
\(85\) 109.098 0.139216
\(86\) 0 0
\(87\) −1363.64 −1.68043
\(88\) −192.359 −0.233017
\(89\) 188.544 0.224558 0.112279 0.993677i \(-0.464185\pi\)
0.112279 + 0.993677i \(0.464185\pi\)
\(90\) −30.8158 −0.0360919
\(91\) −494.882 −0.570085
\(92\) 213.181 0.241583
\(93\) −654.930 −0.730248
\(94\) 188.522 0.206857
\(95\) −623.267 −0.673114
\(96\) 1015.38 1.07950
\(97\) 481.540 0.504051 0.252026 0.967721i \(-0.418903\pi\)
0.252026 + 0.967721i \(0.418903\pi\)
\(98\) 942.474 0.971472
\(99\) 21.9555 0.0222890
\(100\) 327.216 0.327216
\(101\) −1370.30 −1.35000 −0.674999 0.737818i \(-0.735856\pi\)
−0.674999 + 0.737818i \(0.735856\pi\)
\(102\) −114.075 −0.110736
\(103\) −823.195 −0.787494 −0.393747 0.919219i \(-0.628821\pi\)
−0.393747 + 0.919219i \(0.628821\pi\)
\(104\) 341.610 0.322092
\(105\) −1354.57 −1.25898
\(106\) −753.166 −0.690132
\(107\) −1936.29 −1.74943 −0.874713 0.484640i \(-0.838950\pi\)
−0.874713 + 0.484640i \(0.838950\pi\)
\(108\) 739.515 0.658887
\(109\) −1765.66 −1.55155 −0.775777 0.631007i \(-0.782642\pi\)
−0.775777 + 0.631007i \(0.782642\pi\)
\(110\) −115.365 −0.0999969
\(111\) 2125.28 1.81732
\(112\) 342.591 0.289034
\(113\) −1935.29 −1.61112 −0.805559 0.592515i \(-0.798135\pi\)
−0.805559 + 0.592515i \(0.798135\pi\)
\(114\) 651.698 0.535413
\(115\) 312.244 0.253190
\(116\) 1394.52 1.11619
\(117\) −38.9908 −0.0308094
\(118\) 560.671 0.437406
\(119\) −412.736 −0.317945
\(120\) 935.042 0.711310
\(121\) −1248.81 −0.938246
\(122\) 927.751 0.688481
\(123\) −1837.28 −1.34684
\(124\) 669.764 0.485053
\(125\) 1494.86 1.06963
\(126\) 116.581 0.0824275
\(127\) 216.846 0.151511 0.0757556 0.997126i \(-0.475863\pi\)
0.0757556 + 0.997126i \(0.475863\pi\)
\(128\) −1178.03 −0.813470
\(129\) 0 0
\(130\) 204.877 0.138222
\(131\) 629.545 0.419875 0.209937 0.977715i \(-0.432674\pi\)
0.209937 + 0.977715i \(0.432674\pi\)
\(132\) −272.783 −0.179869
\(133\) 2357.91 1.53727
\(134\) −1053.44 −0.679128
\(135\) 1083.16 0.690545
\(136\) 284.906 0.179636
\(137\) 1799.38 1.12212 0.561062 0.827774i \(-0.310393\pi\)
0.561062 + 0.827774i \(0.310393\pi\)
\(138\) −326.487 −0.201394
\(139\) 1364.33 0.832524 0.416262 0.909245i \(-0.363340\pi\)
0.416262 + 0.909245i \(0.363340\pi\)
\(140\) 1385.25 0.836252
\(141\) 652.906 0.389962
\(142\) −654.611 −0.386857
\(143\) −145.970 −0.0853610
\(144\) 26.9921 0.0156204
\(145\) 2042.55 1.16982
\(146\) 1445.03 0.819122
\(147\) 3264.06 1.83139
\(148\) −2173.42 −1.20712
\(149\) 61.1081 0.0335984 0.0167992 0.999859i \(-0.494652\pi\)
0.0167992 + 0.999859i \(0.494652\pi\)
\(150\) −501.132 −0.272782
\(151\) −1992.02 −1.07357 −0.536784 0.843720i \(-0.680361\pi\)
−0.536784 + 0.843720i \(0.680361\pi\)
\(152\) −1627.63 −0.868543
\(153\) −32.5186 −0.0171828
\(154\) 436.445 0.228375
\(155\) 980.998 0.508359
\(156\) 484.435 0.248627
\(157\) −517.016 −0.262818 −0.131409 0.991328i \(-0.541950\pi\)
−0.131409 + 0.991328i \(0.541950\pi\)
\(158\) −168.996 −0.0850922
\(159\) −2608.43 −1.30102
\(160\) −1520.90 −0.751487
\(161\) −1181.27 −0.578241
\(162\) −1234.98 −0.598944
\(163\) −1550.46 −0.745041 −0.372521 0.928024i \(-0.621506\pi\)
−0.372521 + 0.928024i \(0.621506\pi\)
\(164\) 1878.89 0.894615
\(165\) −399.543 −0.188512
\(166\) −1571.83 −0.734924
\(167\) −2645.09 −1.22565 −0.612825 0.790219i \(-0.709967\pi\)
−0.612825 + 0.790219i \(0.709967\pi\)
\(168\) −3537.40 −1.62450
\(169\) −1937.77 −0.882008
\(170\) 170.869 0.0770887
\(171\) 185.775 0.0830795
\(172\) 0 0
\(173\) −2035.37 −0.894488 −0.447244 0.894412i \(-0.647594\pi\)
−0.447244 + 0.894412i \(0.647594\pi\)
\(174\) −2135.72 −0.930509
\(175\) −1813.15 −0.783208
\(176\) 101.050 0.0432782
\(177\) 1941.76 0.824587
\(178\) 295.297 0.124345
\(179\) 3581.88 1.49566 0.747828 0.663893i \(-0.231097\pi\)
0.747828 + 0.663893i \(0.231097\pi\)
\(180\) 109.141 0.0451940
\(181\) 142.547 0.0585384 0.0292692 0.999572i \(-0.490682\pi\)
0.0292692 + 0.999572i \(0.490682\pi\)
\(182\) −775.081 −0.315675
\(183\) 3213.07 1.29791
\(184\) 815.411 0.326700
\(185\) −3183.39 −1.26512
\(186\) −1025.75 −0.404362
\(187\) −121.740 −0.0476070
\(188\) −667.694 −0.259025
\(189\) −4097.76 −1.57708
\(190\) −976.157 −0.372726
\(191\) −613.948 −0.232585 −0.116292 0.993215i \(-0.537101\pi\)
−0.116292 + 0.993215i \(0.537101\pi\)
\(192\) 1106.62 0.415956
\(193\) −251.603 −0.0938381 −0.0469190 0.998899i \(-0.514940\pi\)
−0.0469190 + 0.998899i \(0.514940\pi\)
\(194\) 754.185 0.279110
\(195\) 709.548 0.260573
\(196\) −3337.99 −1.21647
\(197\) −2046.10 −0.739992 −0.369996 0.929033i \(-0.620641\pi\)
−0.369996 + 0.929033i \(0.620641\pi\)
\(198\) 34.3866 0.0123422
\(199\) −602.944 −0.214782 −0.107391 0.994217i \(-0.534250\pi\)
−0.107391 + 0.994217i \(0.534250\pi\)
\(200\) 1251.59 0.442505
\(201\) −3648.36 −1.28027
\(202\) −2146.15 −0.747539
\(203\) −7727.27 −2.67167
\(204\) 404.023 0.138663
\(205\) 2752.00 0.937599
\(206\) −1289.28 −0.436061
\(207\) −93.0696 −0.0312502
\(208\) −179.455 −0.0598220
\(209\) 695.488 0.230181
\(210\) −2121.52 −0.697138
\(211\) −4349.61 −1.41914 −0.709572 0.704633i \(-0.751112\pi\)
−0.709572 + 0.704633i \(0.751112\pi\)
\(212\) 2667.51 0.864176
\(213\) −2267.11 −0.729294
\(214\) −3032.61 −0.968716
\(215\) 0 0
\(216\) 2828.63 0.891035
\(217\) −3711.26 −1.16100
\(218\) −2765.36 −0.859147
\(219\) 5004.56 1.54419
\(220\) 408.593 0.125215
\(221\) 216.198 0.0658057
\(222\) 3328.60 1.00631
\(223\) 2364.75 0.710113 0.355057 0.934845i \(-0.384462\pi\)
0.355057 + 0.934845i \(0.384462\pi\)
\(224\) 5753.80 1.71626
\(225\) −142.855 −0.0423273
\(226\) −3031.03 −0.892130
\(227\) −6612.47 −1.93341 −0.966707 0.255886i \(-0.917633\pi\)
−0.966707 + 0.255886i \(0.917633\pi\)
\(228\) −2308.14 −0.670439
\(229\) 2165.70 0.624950 0.312475 0.949926i \(-0.398842\pi\)
0.312475 + 0.949926i \(0.398842\pi\)
\(230\) 489.034 0.140200
\(231\) 1511.53 0.430526
\(232\) 5334.02 1.50946
\(233\) 2568.88 0.722286 0.361143 0.932510i \(-0.382387\pi\)
0.361143 + 0.932510i \(0.382387\pi\)
\(234\) −61.0671 −0.0170602
\(235\) −977.966 −0.271470
\(236\) −1985.75 −0.547716
\(237\) −585.280 −0.160414
\(238\) −646.424 −0.176057
\(239\) 952.472 0.257784 0.128892 0.991659i \(-0.458858\pi\)
0.128892 + 0.991659i \(0.458858\pi\)
\(240\) −491.198 −0.132111
\(241\) −2756.95 −0.736892 −0.368446 0.929649i \(-0.620110\pi\)
−0.368446 + 0.929649i \(0.620110\pi\)
\(242\) −1955.87 −0.519538
\(243\) −677.520 −0.178860
\(244\) −3285.85 −0.862109
\(245\) −4889.13 −1.27492
\(246\) −2877.53 −0.745792
\(247\) −1235.12 −0.318172
\(248\) 2561.83 0.655954
\(249\) −5443.69 −1.38546
\(250\) 2341.24 0.592291
\(251\) −725.383 −0.182413 −0.0912067 0.995832i \(-0.529072\pi\)
−0.0912067 + 0.995832i \(0.529072\pi\)
\(252\) −412.898 −0.103215
\(253\) −348.425 −0.0865822
\(254\) 339.622 0.0838968
\(255\) 591.769 0.145326
\(256\) −3477.15 −0.848915
\(257\) 3866.44 0.938452 0.469226 0.883078i \(-0.344533\pi\)
0.469226 + 0.883078i \(0.344533\pi\)
\(258\) 0 0
\(259\) 12043.2 2.88931
\(260\) −725.620 −0.173081
\(261\) −608.816 −0.144386
\(262\) 985.989 0.232499
\(263\) −1275.72 −0.299103 −0.149552 0.988754i \(-0.547783\pi\)
−0.149552 + 0.988754i \(0.547783\pi\)
\(264\) −1043.39 −0.243243
\(265\) 3907.08 0.905699
\(266\) 3692.95 0.851238
\(267\) 1022.70 0.234412
\(268\) 3730.99 0.850398
\(269\) −702.374 −0.159199 −0.0795994 0.996827i \(-0.525364\pi\)
−0.0795994 + 0.996827i \(0.525364\pi\)
\(270\) 1696.44 0.382378
\(271\) 6411.84 1.43724 0.718619 0.695404i \(-0.244775\pi\)
0.718619 + 0.695404i \(0.244775\pi\)
\(272\) −149.667 −0.0333636
\(273\) −2684.33 −0.595102
\(274\) 2818.17 0.621358
\(275\) −534.805 −0.117273
\(276\) 1156.33 0.252184
\(277\) −2089.16 −0.453161 −0.226580 0.973992i \(-0.572755\pi\)
−0.226580 + 0.973992i \(0.572755\pi\)
\(278\) 2136.80 0.460996
\(279\) −292.403 −0.0627445
\(280\) 5298.56 1.13089
\(281\) −3390.64 −0.719817 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(282\) 1022.58 0.215935
\(283\) 6836.92 1.43609 0.718044 0.695998i \(-0.245038\pi\)
0.718044 + 0.695998i \(0.245038\pi\)
\(284\) 2318.46 0.484419
\(285\) −3380.71 −0.702653
\(286\) −228.617 −0.0472672
\(287\) −10411.2 −2.14131
\(288\) 453.331 0.0927527
\(289\) −4732.69 −0.963299
\(290\) 3199.03 0.647770
\(291\) 2611.96 0.526171
\(292\) −5117.92 −1.02570
\(293\) 6814.59 1.35875 0.679373 0.733793i \(-0.262252\pi\)
0.679373 + 0.733793i \(0.262252\pi\)
\(294\) 5112.15 1.01410
\(295\) −2908.51 −0.574033
\(296\) −8313.28 −1.63243
\(297\) −1208.67 −0.236142
\(298\) 95.7071 0.0186046
\(299\) 618.767 0.119680
\(300\) 1774.88 0.341575
\(301\) 0 0
\(302\) −3119.90 −0.594470
\(303\) −7432.75 −1.40924
\(304\) 855.032 0.161314
\(305\) −4812.75 −0.903532
\(306\) −50.9305 −0.00951471
\(307\) 1604.28 0.298244 0.149122 0.988819i \(-0.452355\pi\)
0.149122 + 0.988819i \(0.452355\pi\)
\(308\) −1545.77 −0.285969
\(309\) −4465.16 −0.822052
\(310\) 1536.43 0.281495
\(311\) −2318.26 −0.422690 −0.211345 0.977412i \(-0.567784\pi\)
−0.211345 + 0.977412i \(0.567784\pi\)
\(312\) 1852.95 0.336227
\(313\) 4380.98 0.791142 0.395571 0.918435i \(-0.370547\pi\)
0.395571 + 0.918435i \(0.370547\pi\)
\(314\) −809.748 −0.145531
\(315\) −604.769 −0.108174
\(316\) 598.537 0.106552
\(317\) 9306.43 1.64890 0.824450 0.565934i \(-0.191484\pi\)
0.824450 + 0.565934i \(0.191484\pi\)
\(318\) −4085.31 −0.720417
\(319\) −2279.23 −0.400038
\(320\) −1657.57 −0.289566
\(321\) −10502.8 −1.82620
\(322\) −1850.09 −0.320191
\(323\) −1030.10 −0.177449
\(324\) 4373.96 0.749993
\(325\) 949.760 0.162102
\(326\) −2428.33 −0.412554
\(327\) −9577.24 −1.61964
\(328\) 7186.72 1.20982
\(329\) 3699.79 0.619989
\(330\) −625.762 −0.104385
\(331\) 5183.44 0.860749 0.430374 0.902651i \(-0.358382\pi\)
0.430374 + 0.902651i \(0.358382\pi\)
\(332\) 5566.99 0.920266
\(333\) 948.864 0.156148
\(334\) −4142.73 −0.678683
\(335\) 5464.75 0.891258
\(336\) 1858.28 0.301718
\(337\) 3000.12 0.484947 0.242474 0.970158i \(-0.422041\pi\)
0.242474 + 0.970158i \(0.422041\pi\)
\(338\) −3034.93 −0.488397
\(339\) −10497.3 −1.68182
\(340\) −605.173 −0.0965297
\(341\) −1094.67 −0.173841
\(342\) 290.960 0.0460039
\(343\) 7953.52 1.25204
\(344\) 0 0
\(345\) 1693.67 0.264301
\(346\) −3187.78 −0.495307
\(347\) −9148.76 −1.41536 −0.707682 0.706531i \(-0.750259\pi\)
−0.707682 + 0.706531i \(0.750259\pi\)
\(348\) 7564.15 1.16518
\(349\) −2096.34 −0.321531 −0.160766 0.986993i \(-0.551396\pi\)
−0.160766 + 0.986993i \(0.551396\pi\)
\(350\) −2839.75 −0.433688
\(351\) 2146.48 0.326412
\(352\) 1697.14 0.256982
\(353\) 319.106 0.0481142 0.0240571 0.999711i \(-0.492342\pi\)
0.0240571 + 0.999711i \(0.492342\pi\)
\(354\) 3041.18 0.456601
\(355\) 3395.82 0.507695
\(356\) −1045.86 −0.155704
\(357\) −2238.75 −0.331897
\(358\) 5609.92 0.828194
\(359\) 9112.72 1.33970 0.669848 0.742498i \(-0.266359\pi\)
0.669848 + 0.742498i \(0.266359\pi\)
\(360\) 417.463 0.0611174
\(361\) −974.175 −0.142029
\(362\) 223.257 0.0324147
\(363\) −6773.74 −0.979419
\(364\) 2745.13 0.395285
\(365\) −7496.17 −1.07498
\(366\) 5032.29 0.718694
\(367\) −2726.93 −0.387860 −0.193930 0.981015i \(-0.562123\pi\)
−0.193930 + 0.981015i \(0.562123\pi\)
\(368\) −428.353 −0.0606779
\(369\) −820.280 −0.115724
\(370\) −4985.81 −0.700540
\(371\) −14781.1 −2.06845
\(372\) 3632.92 0.506339
\(373\) 13249.6 1.83924 0.919622 0.392804i \(-0.128495\pi\)
0.919622 + 0.392804i \(0.128495\pi\)
\(374\) −190.669 −0.0263616
\(375\) 8108.37 1.11657
\(376\) −2553.92 −0.350288
\(377\) 4047.68 0.552960
\(378\) −6417.89 −0.873282
\(379\) 8778.93 1.18982 0.594912 0.803791i \(-0.297187\pi\)
0.594912 + 0.803791i \(0.297187\pi\)
\(380\) 3457.28 0.466724
\(381\) 1176.21 0.158160
\(382\) −961.562 −0.128790
\(383\) 11141.2 1.48639 0.743194 0.669076i \(-0.233310\pi\)
0.743194 + 0.669076i \(0.233310\pi\)
\(384\) −6389.85 −0.849168
\(385\) −2264.07 −0.299709
\(386\) −394.058 −0.0519613
\(387\) 0 0
\(388\) −2671.12 −0.349499
\(389\) −11673.9 −1.52157 −0.760786 0.649003i \(-0.775186\pi\)
−0.760786 + 0.649003i \(0.775186\pi\)
\(390\) 1111.29 0.144288
\(391\) 516.057 0.0667471
\(392\) −12767.7 −1.64507
\(393\) 3414.76 0.438300
\(394\) −3204.58 −0.409758
\(395\) 876.672 0.111671
\(396\) −121.788 −0.0154548
\(397\) 9942.51 1.25693 0.628464 0.777839i \(-0.283684\pi\)
0.628464 + 0.777839i \(0.283684\pi\)
\(398\) −944.327 −0.118932
\(399\) 12789.7 1.60473
\(400\) −657.489 −0.0821861
\(401\) 4884.85 0.608323 0.304162 0.952620i \(-0.401624\pi\)
0.304162 + 0.952620i \(0.401624\pi\)
\(402\) −5714.03 −0.708930
\(403\) 1944.02 0.240295
\(404\) 7601.10 0.936061
\(405\) 6406.50 0.786028
\(406\) −12102.4 −1.47939
\(407\) 3552.26 0.432627
\(408\) 1545.38 0.187519
\(409\) −10707.9 −1.29455 −0.647276 0.762256i \(-0.724092\pi\)
−0.647276 + 0.762256i \(0.724092\pi\)
\(410\) 4310.16 0.519180
\(411\) 9760.14 1.17137
\(412\) 4566.29 0.546032
\(413\) 11003.3 1.31099
\(414\) −145.765 −0.0173043
\(415\) 8153.92 0.964483
\(416\) −3013.94 −0.355218
\(417\) 7400.36 0.869058
\(418\) 1089.27 0.127459
\(419\) 5076.87 0.591936 0.295968 0.955198i \(-0.404358\pi\)
0.295968 + 0.955198i \(0.404358\pi\)
\(420\) 7513.86 0.872950
\(421\) 3748.60 0.433956 0.216978 0.976176i \(-0.430380\pi\)
0.216978 + 0.976176i \(0.430380\pi\)
\(422\) −6812.33 −0.785827
\(423\) 291.500 0.0335064
\(424\) 10203.2 1.16865
\(425\) 792.107 0.0904067
\(426\) −3550.73 −0.403834
\(427\) 18207.4 2.06350
\(428\) 10740.7 1.21302
\(429\) −791.767 −0.0891069
\(430\) 0 0
\(431\) −472.408 −0.0527960 −0.0263980 0.999652i \(-0.508404\pi\)
−0.0263980 + 0.999652i \(0.508404\pi\)
\(432\) −1485.94 −0.165491
\(433\) −500.116 −0.0555058 −0.0277529 0.999615i \(-0.508835\pi\)
−0.0277529 + 0.999615i \(0.508835\pi\)
\(434\) −5812.56 −0.642884
\(435\) 11079.1 1.22116
\(436\) 9794.17 1.07582
\(437\) −2948.18 −0.322724
\(438\) 7838.12 0.855068
\(439\) −15113.3 −1.64309 −0.821544 0.570144i \(-0.806887\pi\)
−0.821544 + 0.570144i \(0.806887\pi\)
\(440\) 1562.86 0.169333
\(441\) 1457.29 0.157357
\(442\) 338.608 0.0364388
\(443\) −8694.15 −0.932441 −0.466221 0.884668i \(-0.654385\pi\)
−0.466221 + 0.884668i \(0.654385\pi\)
\(444\) −11789.0 −1.26009
\(445\) −1531.86 −0.163185
\(446\) 3703.65 0.393213
\(447\) 331.461 0.0350729
\(448\) 6270.84 0.661316
\(449\) 15475.4 1.62657 0.813287 0.581863i \(-0.197676\pi\)
0.813287 + 0.581863i \(0.197676\pi\)
\(450\) −223.738 −0.0234380
\(451\) −3070.88 −0.320626
\(452\) 10735.1 1.11712
\(453\) −10805.1 −1.12068
\(454\) −10356.4 −1.07060
\(455\) 4020.77 0.414278
\(456\) −8828.57 −0.906658
\(457\) −7336.32 −0.750938 −0.375469 0.926835i \(-0.622518\pi\)
−0.375469 + 0.926835i \(0.622518\pi\)
\(458\) 3391.91 0.346056
\(459\) 1790.18 0.182045
\(460\) −1732.03 −0.175557
\(461\) −1976.62 −0.199698 −0.0998488 0.995003i \(-0.531836\pi\)
−0.0998488 + 0.995003i \(0.531836\pi\)
\(462\) 2367.35 0.238397
\(463\) −7644.41 −0.767313 −0.383656 0.923476i \(-0.625335\pi\)
−0.383656 + 0.923476i \(0.625335\pi\)
\(464\) −2802.08 −0.280352
\(465\) 5321.11 0.530668
\(466\) 4023.36 0.399954
\(467\) 4008.28 0.397176 0.198588 0.980083i \(-0.436365\pi\)
0.198588 + 0.980083i \(0.436365\pi\)
\(468\) 216.283 0.0213626
\(469\) −20674.0 −2.03547
\(470\) −1531.69 −0.150322
\(471\) −2804.39 −0.274351
\(472\) −7595.43 −0.740695
\(473\) 0 0
\(474\) −916.662 −0.0888263
\(475\) −4525.22 −0.437119
\(476\) 2289.46 0.220456
\(477\) −1164.57 −0.111786
\(478\) 1491.76 0.142743
\(479\) −10236.8 −0.976473 −0.488236 0.872711i \(-0.662360\pi\)
−0.488236 + 0.872711i \(0.662360\pi\)
\(480\) −8249.65 −0.784465
\(481\) −6308.46 −0.598006
\(482\) −4317.92 −0.408042
\(483\) −6407.39 −0.603616
\(484\) 6927.17 0.650560
\(485\) −3912.37 −0.366292
\(486\) −1061.13 −0.0990406
\(487\) 4057.87 0.377576 0.188788 0.982018i \(-0.439544\pi\)
0.188788 + 0.982018i \(0.439544\pi\)
\(488\) −12568.3 −1.16586
\(489\) −8410.00 −0.777737
\(490\) −7657.32 −0.705964
\(491\) 3884.88 0.357072 0.178536 0.983933i \(-0.442864\pi\)
0.178536 + 0.983933i \(0.442864\pi\)
\(492\) 10191.4 0.933874
\(493\) 3375.80 0.308394
\(494\) −1934.43 −0.176182
\(495\) −178.382 −0.0161973
\(496\) −1345.79 −0.121830
\(497\) −12846.9 −1.15948
\(498\) −8525.87 −0.767176
\(499\) 8005.88 0.718222 0.359111 0.933295i \(-0.383080\pi\)
0.359111 + 0.933295i \(0.383080\pi\)
\(500\) −8292.03 −0.741662
\(501\) −14347.5 −1.27944
\(502\) −1136.09 −0.101008
\(503\) −3167.37 −0.280767 −0.140384 0.990097i \(-0.544834\pi\)
−0.140384 + 0.990097i \(0.544834\pi\)
\(504\) −1579.33 −0.139581
\(505\) 11133.3 0.981037
\(506\) −545.701 −0.0479434
\(507\) −10510.8 −0.920714
\(508\) −1202.85 −0.105055
\(509\) −2437.53 −0.212262 −0.106131 0.994352i \(-0.533846\pi\)
−0.106131 + 0.994352i \(0.533846\pi\)
\(510\) 926.825 0.0804716
\(511\) 28359.2 2.45506
\(512\) 3978.34 0.343398
\(513\) −10227.1 −0.880190
\(514\) 6055.60 0.519652
\(515\) 6688.21 0.572268
\(516\) 0 0
\(517\) 1091.29 0.0928332
\(518\) 18862.1 1.59991
\(519\) −11040.2 −0.933741
\(520\) −2775.48 −0.234063
\(521\) −12207.6 −1.02654 −0.513268 0.858228i \(-0.671565\pi\)
−0.513268 + 0.858228i \(0.671565\pi\)
\(522\) −953.525 −0.0799514
\(523\) 18236.1 1.52468 0.762340 0.647177i \(-0.224050\pi\)
0.762340 + 0.647177i \(0.224050\pi\)
\(524\) −3492.11 −0.291133
\(525\) −9834.85 −0.817577
\(526\) −1998.02 −0.165623
\(527\) 1621.33 0.134016
\(528\) 548.115 0.0451774
\(529\) −10690.0 −0.878608
\(530\) 6119.25 0.501515
\(531\) 866.930 0.0708504
\(532\) −13079.4 −1.06591
\(533\) 5453.58 0.443191
\(534\) 1601.74 0.129802
\(535\) 15731.8 1.27130
\(536\) 14271.0 1.15002
\(537\) 19428.8 1.56129
\(538\) −1100.05 −0.0881537
\(539\) 5455.65 0.435977
\(540\) −6008.33 −0.478810
\(541\) −4277.33 −0.339920 −0.169960 0.985451i \(-0.554364\pi\)
−0.169960 + 0.985451i \(0.554364\pi\)
\(542\) 10042.2 0.795846
\(543\) 773.202 0.0611073
\(544\) −2513.65 −0.198110
\(545\) 14345.4 1.12751
\(546\) −4204.18 −0.329528
\(547\) 10280.6 0.803596 0.401798 0.915728i \(-0.368385\pi\)
0.401798 + 0.915728i \(0.368385\pi\)
\(548\) −9981.21 −0.778058
\(549\) 1434.52 0.111519
\(550\) −837.609 −0.0649377
\(551\) −19285.5 −1.49109
\(552\) 4422.93 0.341037
\(553\) −3316.58 −0.255037
\(554\) −3272.03 −0.250930
\(555\) −17267.3 −1.32064
\(556\) −7567.98 −0.577255
\(557\) −19282.4 −1.46683 −0.733413 0.679784i \(-0.762074\pi\)
−0.733413 + 0.679784i \(0.762074\pi\)
\(558\) −457.960 −0.0347437
\(559\) 0 0
\(560\) −2783.45 −0.210040
\(561\) −660.340 −0.0496962
\(562\) −5310.40 −0.398586
\(563\) 14877.6 1.11371 0.556854 0.830610i \(-0.312008\pi\)
0.556854 + 0.830610i \(0.312008\pi\)
\(564\) −3621.69 −0.270391
\(565\) 15723.6 1.17079
\(566\) 10707.9 0.795209
\(567\) −24236.8 −1.79515
\(568\) 8868.04 0.655096
\(569\) −10794.5 −0.795307 −0.397654 0.917536i \(-0.630175\pi\)
−0.397654 + 0.917536i \(0.630175\pi\)
\(570\) −5294.85 −0.389082
\(571\) −7995.91 −0.586022 −0.293011 0.956109i \(-0.594657\pi\)
−0.293011 + 0.956109i \(0.594657\pi\)
\(572\) 809.700 0.0591875
\(573\) −3330.16 −0.242792
\(574\) −16306.0 −1.18571
\(575\) 2267.04 0.164421
\(576\) 494.067 0.0357398
\(577\) −11565.4 −0.834444 −0.417222 0.908805i \(-0.636996\pi\)
−0.417222 + 0.908805i \(0.636996\pi\)
\(578\) −7412.31 −0.533411
\(579\) −1364.74 −0.0979560
\(580\) −11330.1 −0.811132
\(581\) −30847.5 −2.20270
\(582\) 4090.83 0.291358
\(583\) −4359.81 −0.309717
\(584\) −19575.9 −1.38708
\(585\) 316.788 0.0223890
\(586\) 10673.0 0.752382
\(587\) −6354.81 −0.446833 −0.223417 0.974723i \(-0.571721\pi\)
−0.223417 + 0.974723i \(0.571721\pi\)
\(588\) −18105.8 −1.26985
\(589\) −9262.49 −0.647970
\(590\) −4555.28 −0.317861
\(591\) −11098.4 −0.772465
\(592\) 4367.15 0.303190
\(593\) 21747.7 1.50602 0.753010 0.658009i \(-0.228601\pi\)
0.753010 + 0.658009i \(0.228601\pi\)
\(594\) −1893.01 −0.130760
\(595\) 3353.35 0.231049
\(596\) −338.969 −0.0232965
\(597\) −3270.47 −0.224207
\(598\) 969.110 0.0662706
\(599\) 10874.2 0.741750 0.370875 0.928683i \(-0.379058\pi\)
0.370875 + 0.928683i \(0.379058\pi\)
\(600\) 6788.86 0.461923
\(601\) −16092.5 −1.09223 −0.546113 0.837712i \(-0.683893\pi\)
−0.546113 + 0.837712i \(0.683893\pi\)
\(602\) 0 0
\(603\) −1628.86 −0.110004
\(604\) 11049.8 0.744390
\(605\) 10146.2 0.681819
\(606\) −11641.1 −0.780344
\(607\) 23235.8 1.55372 0.776862 0.629671i \(-0.216810\pi\)
0.776862 + 0.629671i \(0.216810\pi\)
\(608\) 14360.2 0.957868
\(609\) −41914.1 −2.78891
\(610\) −7537.70 −0.500316
\(611\) −1938.02 −0.128320
\(612\) 180.382 0.0119142
\(613\) −14997.4 −0.988153 −0.494076 0.869418i \(-0.664494\pi\)
−0.494076 + 0.869418i \(0.664494\pi\)
\(614\) 2512.61 0.165148
\(615\) 14927.3 0.978745
\(616\) −5912.53 −0.386725
\(617\) 2016.56 0.131578 0.0657890 0.997834i \(-0.479044\pi\)
0.0657890 + 0.997834i \(0.479044\pi\)
\(618\) −6993.30 −0.455197
\(619\) 17984.1 1.16776 0.583879 0.811841i \(-0.301534\pi\)
0.583879 + 0.811841i \(0.301534\pi\)
\(620\) −5441.63 −0.352486
\(621\) 5123.57 0.331081
\(622\) −3630.85 −0.234058
\(623\) 5795.27 0.372685
\(624\) −973.397 −0.0624472
\(625\) −4771.61 −0.305383
\(626\) 6861.46 0.438082
\(627\) 3772.45 0.240282
\(628\) 2867.91 0.182233
\(629\) −5261.31 −0.333517
\(630\) −947.185 −0.0598996
\(631\) 7502.61 0.473335 0.236667 0.971591i \(-0.423945\pi\)
0.236667 + 0.971591i \(0.423945\pi\)
\(632\) 2289.39 0.144093
\(633\) −23593.1 −1.48142
\(634\) 14575.7 0.913051
\(635\) −1761.80 −0.110103
\(636\) 14469.1 0.902100
\(637\) −9688.68 −0.602637
\(638\) −3569.71 −0.221515
\(639\) −1012.18 −0.0626625
\(640\) 9571.14 0.591145
\(641\) 17658.5 1.08809 0.544047 0.839055i \(-0.316891\pi\)
0.544047 + 0.839055i \(0.316891\pi\)
\(642\) −16449.4 −1.01123
\(643\) −12908.4 −0.791694 −0.395847 0.918316i \(-0.629549\pi\)
−0.395847 + 0.918316i \(0.629549\pi\)
\(644\) 6552.52 0.400940
\(645\) 0 0
\(646\) −1613.33 −0.0982595
\(647\) 20030.3 1.21712 0.608558 0.793510i \(-0.291748\pi\)
0.608558 + 0.793510i \(0.291748\pi\)
\(648\) 16730.3 1.01424
\(649\) 3245.53 0.196299
\(650\) 1487.51 0.0897613
\(651\) −20130.6 −1.21195
\(652\) 8600.48 0.516597
\(653\) 3145.07 0.188478 0.0942391 0.995550i \(-0.469958\pi\)
0.0942391 + 0.995550i \(0.469958\pi\)
\(654\) −14999.8 −0.896849
\(655\) −5114.86 −0.305121
\(656\) −3775.34 −0.224699
\(657\) 2234.36 0.132680
\(658\) 5794.60 0.343308
\(659\) −16132.7 −0.953627 −0.476813 0.879005i \(-0.658208\pi\)
−0.476813 + 0.879005i \(0.658208\pi\)
\(660\) 2216.28 0.130710
\(661\) −9134.09 −0.537481 −0.268740 0.963213i \(-0.586607\pi\)
−0.268740 + 0.963213i \(0.586607\pi\)
\(662\) 8118.28 0.476625
\(663\) 1172.70 0.0686935
\(664\) 21293.6 1.24451
\(665\) −19157.3 −1.11713
\(666\) 1486.10 0.0864645
\(667\) 9661.66 0.560871
\(668\) 14672.4 0.849841
\(669\) 12826.8 0.741275
\(670\) 8558.86 0.493519
\(671\) 5370.42 0.308976
\(672\) 31209.7 1.79158
\(673\) 5116.23 0.293041 0.146520 0.989208i \(-0.453193\pi\)
0.146520 + 0.989208i \(0.453193\pi\)
\(674\) 4698.78 0.268531
\(675\) 7864.27 0.448439
\(676\) 10748.9 0.611567
\(677\) 4972.81 0.282305 0.141153 0.989988i \(-0.454919\pi\)
0.141153 + 0.989988i \(0.454919\pi\)
\(678\) −16440.9 −0.931280
\(679\) 14801.1 0.836543
\(680\) −2314.77 −0.130540
\(681\) −35867.2 −2.01826
\(682\) −1714.47 −0.0962614
\(683\) −7581.73 −0.424754 −0.212377 0.977188i \(-0.568120\pi\)
−0.212377 + 0.977188i \(0.568120\pi\)
\(684\) −1030.50 −0.0576056
\(685\) −14619.4 −0.815443
\(686\) 12456.8 0.693296
\(687\) 11747.2 0.652375
\(688\) 0 0
\(689\) 7742.58 0.428112
\(690\) 2652.61 0.146352
\(691\) 7653.70 0.421361 0.210681 0.977555i \(-0.432432\pi\)
0.210681 + 0.977555i \(0.432432\pi\)
\(692\) 11290.3 0.620219
\(693\) 674.846 0.0369918
\(694\) −14328.7 −0.783734
\(695\) −11084.8 −0.604991
\(696\) 28932.7 1.57571
\(697\) 4548.33 0.247174
\(698\) −3283.27 −0.178043
\(699\) 13934.0 0.753982
\(700\) 10057.6 0.543060
\(701\) 5942.69 0.320189 0.160094 0.987102i \(-0.448820\pi\)
0.160094 + 0.987102i \(0.448820\pi\)
\(702\) 3361.80 0.180745
\(703\) 30057.3 1.61256
\(704\) 1849.64 0.0990213
\(705\) −5304.66 −0.283383
\(706\) 499.782 0.0266424
\(707\) −42118.8 −2.24051
\(708\) −10771.0 −0.571752
\(709\) −5113.78 −0.270877 −0.135439 0.990786i \(-0.543244\pi\)
−0.135439 + 0.990786i \(0.543244\pi\)
\(710\) 5318.52 0.281127
\(711\) −261.307 −0.0137831
\(712\) −4000.39 −0.210563
\(713\) 4640.31 0.243732
\(714\) −3506.32 −0.183783
\(715\) 1185.96 0.0620314
\(716\) −19868.8 −1.03706
\(717\) 5166.38 0.269096
\(718\) 14272.3 0.741835
\(719\) 36844.4 1.91108 0.955538 0.294869i \(-0.0952761\pi\)
0.955538 + 0.294869i \(0.0952761\pi\)
\(720\) −219.303 −0.0113513
\(721\) −25302.5 −1.30696
\(722\) −1525.75 −0.0786460
\(723\) −14954.2 −0.769230
\(724\) −790.715 −0.0405893
\(725\) 14829.9 0.759681
\(726\) −10609.0 −0.542337
\(727\) 21420.6 1.09277 0.546387 0.837533i \(-0.316003\pi\)
0.546387 + 0.837533i \(0.316003\pi\)
\(728\) 10500.1 0.534558
\(729\) 17615.1 0.894939
\(730\) −11740.5 −0.595252
\(731\) 0 0
\(732\) −17823.0 −0.899942
\(733\) 35541.1 1.79091 0.895457 0.445148i \(-0.146849\pi\)
0.895457 + 0.445148i \(0.146849\pi\)
\(734\) −4270.90 −0.214771
\(735\) −26519.5 −1.33087
\(736\) −7194.17 −0.360300
\(737\) −6097.98 −0.304779
\(738\) −1284.72 −0.0640801
\(739\) −5111.15 −0.254420 −0.127210 0.991876i \(-0.540602\pi\)
−0.127210 + 0.991876i \(0.540602\pi\)
\(740\) 17658.4 0.877209
\(741\) −6699.49 −0.332135
\(742\) −23150.0 −1.14537
\(743\) −37864.8 −1.86962 −0.934808 0.355153i \(-0.884429\pi\)
−0.934808 + 0.355153i \(0.884429\pi\)
\(744\) 13895.8 0.684739
\(745\) −496.485 −0.0244158
\(746\) 20751.4 1.01845
\(747\) −2430.42 −0.119042
\(748\) 675.297 0.0330098
\(749\) −59515.8 −2.90342
\(750\) 12699.3 0.618283
\(751\) −4814.57 −0.233937 −0.116968 0.993136i \(-0.537318\pi\)
−0.116968 + 0.993136i \(0.537318\pi\)
\(752\) 1341.63 0.0650587
\(753\) −3934.60 −0.190418
\(754\) 6339.45 0.306192
\(755\) 16184.6 0.780156
\(756\) 22730.4 1.09352
\(757\) 13342.1 0.640589 0.320295 0.947318i \(-0.396218\pi\)
0.320295 + 0.947318i \(0.396218\pi\)
\(758\) 13749.5 0.658845
\(759\) −1889.92 −0.0903817
\(760\) 13224.0 0.631166
\(761\) 17924.5 0.853826 0.426913 0.904293i \(-0.359601\pi\)
0.426913 + 0.904293i \(0.359601\pi\)
\(762\) 1842.17 0.0875785
\(763\) −54271.0 −2.57502
\(764\) 3405.59 0.161270
\(765\) 264.204 0.0124867
\(766\) 17449.2 0.823062
\(767\) −5763.73 −0.271338
\(768\) −18860.7 −0.886168
\(769\) 4472.87 0.209748 0.104874 0.994486i \(-0.466556\pi\)
0.104874 + 0.994486i \(0.466556\pi\)
\(770\) −3545.98 −0.165959
\(771\) 20972.3 0.979635
\(772\) 1395.65 0.0650654
\(773\) −33168.7 −1.54333 −0.771665 0.636029i \(-0.780576\pi\)
−0.771665 + 0.636029i \(0.780576\pi\)
\(774\) 0 0
\(775\) 7122.52 0.330127
\(776\) −10217.0 −0.472639
\(777\) 65324.7 3.01610
\(778\) −18283.6 −0.842544
\(779\) −25984.1 −1.19509
\(780\) −3935.89 −0.180676
\(781\) −3789.31 −0.173614
\(782\) 808.245 0.0369601
\(783\) 33515.9 1.52971
\(784\) 6707.17 0.305538
\(785\) 4200.60 0.190988
\(786\) 5348.18 0.242701
\(787\) 6452.90 0.292276 0.146138 0.989264i \(-0.453316\pi\)
0.146138 + 0.989264i \(0.453316\pi\)
\(788\) 11349.8 0.513095
\(789\) −6919.72 −0.312229
\(790\) 1373.04 0.0618361
\(791\) −59484.8 −2.67388
\(792\) −465.837 −0.0209000
\(793\) −9537.33 −0.427088
\(794\) 15571.9 0.696002
\(795\) 21192.7 0.945444
\(796\) 3344.55 0.148925
\(797\) 8676.60 0.385622 0.192811 0.981236i \(-0.438240\pi\)
0.192811 + 0.981236i \(0.438240\pi\)
\(798\) 20031.2 0.888593
\(799\) −1616.32 −0.0715661
\(800\) −11042.5 −0.488014
\(801\) 456.598 0.0201412
\(802\) 7650.62 0.336849
\(803\) 8364.79 0.367605
\(804\) 20237.6 0.887716
\(805\) 9597.43 0.420205
\(806\) 3044.72 0.133059
\(807\) −3809.80 −0.166185
\(808\) 29074.0 1.26587
\(809\) 10277.3 0.446637 0.223319 0.974745i \(-0.428311\pi\)
0.223319 + 0.974745i \(0.428311\pi\)
\(810\) 10033.8 0.435250
\(811\) −31979.8 −1.38466 −0.692332 0.721579i \(-0.743416\pi\)
−0.692332 + 0.721579i \(0.743416\pi\)
\(812\) 42863.4 1.85248
\(813\) 34779.0 1.50031
\(814\) 5563.53 0.239560
\(815\) 12597.1 0.541418
\(816\) −811.821 −0.0348277
\(817\) 0 0
\(818\) −16770.7 −0.716837
\(819\) −1198.46 −0.0511325
\(820\) −15265.4 −0.650112
\(821\) 4889.45 0.207848 0.103924 0.994585i \(-0.466860\pi\)
0.103924 + 0.994585i \(0.466860\pi\)
\(822\) 15286.3 0.648625
\(823\) −27193.5 −1.15177 −0.575884 0.817531i \(-0.695342\pi\)
−0.575884 + 0.817531i \(0.695342\pi\)
\(824\) 17466.0 0.738417
\(825\) −2900.88 −0.122419
\(826\) 17233.3 0.725937
\(827\) −5190.02 −0.218228 −0.109114 0.994029i \(-0.534801\pi\)
−0.109114 + 0.994029i \(0.534801\pi\)
\(828\) 516.260 0.0216682
\(829\) −22048.5 −0.923733 −0.461867 0.886949i \(-0.652820\pi\)
−0.461867 + 0.886949i \(0.652820\pi\)
\(830\) 12770.6 0.534066
\(831\) −11332.0 −0.473047
\(832\) −3284.78 −0.136874
\(833\) −8080.44 −0.336099
\(834\) 11590.4 0.481226
\(835\) 21490.6 0.890674
\(836\) −3857.89 −0.159603
\(837\) 16097.1 0.664750
\(838\) 7951.36 0.327775
\(839\) 24727.9 1.01752 0.508761 0.860908i \(-0.330104\pi\)
0.508761 + 0.860908i \(0.330104\pi\)
\(840\) 28740.3 1.18052
\(841\) 38812.9 1.59141
\(842\) 5871.03 0.240296
\(843\) −18391.4 −0.751405
\(844\) 24127.4 0.984006
\(845\) 15743.8 0.640951
\(846\) 456.545 0.0185536
\(847\) −38384.5 −1.55715
\(848\) −5359.95 −0.217054
\(849\) 37084.7 1.49911
\(850\) 1240.59 0.0500612
\(851\) −15058.1 −0.606562
\(852\) 12575.7 0.505677
\(853\) −23989.1 −0.962922 −0.481461 0.876468i \(-0.659894\pi\)
−0.481461 + 0.876468i \(0.659894\pi\)
\(854\) 28516.3 1.14263
\(855\) −1509.37 −0.0603735
\(856\) 41082.9 1.64040
\(857\) −10224.3 −0.407534 −0.203767 0.979019i \(-0.565319\pi\)
−0.203767 + 0.979019i \(0.565319\pi\)
\(858\) −1240.06 −0.0493414
\(859\) 20256.5 0.804590 0.402295 0.915510i \(-0.368213\pi\)
0.402295 + 0.915510i \(0.368213\pi\)
\(860\) 0 0
\(861\) −56472.3 −2.23527
\(862\) −739.882 −0.0292349
\(863\) 21978.0 0.866905 0.433453 0.901176i \(-0.357295\pi\)
0.433453 + 0.901176i \(0.357295\pi\)
\(864\) −24956.3 −0.982673
\(865\) 16536.8 0.650020
\(866\) −783.278 −0.0307354
\(867\) −25670.9 −1.00557
\(868\) 20586.5 0.805014
\(869\) −978.256 −0.0381876
\(870\) 17352.1 0.676197
\(871\) 10829.4 0.421286
\(872\) 37462.5 1.45486
\(873\) 1166.15 0.0452097
\(874\) −4617.42 −0.178703
\(875\) 45947.4 1.77521
\(876\) −27760.5 −1.07071
\(877\) −11390.5 −0.438575 −0.219287 0.975660i \(-0.570373\pi\)
−0.219287 + 0.975660i \(0.570373\pi\)
\(878\) −23670.3 −0.909833
\(879\) 36963.5 1.41837
\(880\) −821.004 −0.0314500
\(881\) 36825.6 1.40827 0.704135 0.710066i \(-0.251335\pi\)
0.704135 + 0.710066i \(0.251335\pi\)
\(882\) 2282.39 0.0871340
\(883\) −27495.1 −1.04789 −0.523943 0.851754i \(-0.675539\pi\)
−0.523943 + 0.851754i \(0.675539\pi\)
\(884\) −1199.26 −0.0456283
\(885\) −15776.2 −0.599224
\(886\) −13616.7 −0.516324
\(887\) 39402.6 1.49155 0.745777 0.666195i \(-0.232078\pi\)
0.745777 + 0.666195i \(0.232078\pi\)
\(888\) −45092.7 −1.70407
\(889\) 6665.17 0.251454
\(890\) −2399.20 −0.0903610
\(891\) −7148.85 −0.268794
\(892\) −13117.3 −0.492378
\(893\) 9233.86 0.346024
\(894\) 519.132 0.0194210
\(895\) −29101.7 −1.08689
\(896\) −36209.1 −1.35007
\(897\) 3356.30 0.124932
\(898\) 24237.5 0.900687
\(899\) 30354.7 1.12612
\(900\) 792.420 0.0293489
\(901\) 6457.38 0.238764
\(902\) −4809.60 −0.177541
\(903\) 0 0
\(904\) 41061.5 1.51071
\(905\) −1158.15 −0.0425396
\(906\) −16922.9 −0.620557
\(907\) −842.302 −0.0308359 −0.0154180 0.999881i \(-0.504908\pi\)
−0.0154180 + 0.999881i \(0.504908\pi\)
\(908\) 36679.6 1.34059
\(909\) −3318.46 −0.121085
\(910\) 6297.30 0.229399
\(911\) 7588.34 0.275975 0.137987 0.990434i \(-0.455937\pi\)
0.137987 + 0.990434i \(0.455937\pi\)
\(912\) 4637.84 0.168393
\(913\) −9098.75 −0.329819
\(914\) −11490.1 −0.415819
\(915\) −26105.2 −0.943182
\(916\) −12013.2 −0.433328
\(917\) 19350.3 0.696841
\(918\) 2803.77 0.100804
\(919\) 1126.30 0.0404278 0.0202139 0.999796i \(-0.493565\pi\)
0.0202139 + 0.999796i \(0.493565\pi\)
\(920\) −6624.97 −0.237412
\(921\) 8701.89 0.311332
\(922\) −3095.78 −0.110579
\(923\) 6729.44 0.239981
\(924\) −8384.53 −0.298518
\(925\) −23113.0 −0.821567
\(926\) −11972.6 −0.424887
\(927\) −1993.54 −0.0706325
\(928\) −47060.8 −1.66470
\(929\) −12483.2 −0.440861 −0.220431 0.975403i \(-0.570746\pi\)
−0.220431 + 0.975403i \(0.570746\pi\)
\(930\) 8333.89 0.293848
\(931\) 46162.6 1.62505
\(932\) −14249.6 −0.500818
\(933\) −12574.7 −0.441239
\(934\) 6277.74 0.219929
\(935\) 989.102 0.0345958
\(936\) 827.279 0.0288894
\(937\) −50746.3 −1.76927 −0.884636 0.466282i \(-0.845593\pi\)
−0.884636 + 0.466282i \(0.845593\pi\)
\(938\) −32379.5 −1.12711
\(939\) 23763.2 0.825860
\(940\) 5424.82 0.188232
\(941\) 21706.6 0.751983 0.375991 0.926623i \(-0.377302\pi\)
0.375991 + 0.926623i \(0.377302\pi\)
\(942\) −4392.22 −0.151917
\(943\) 13017.5 0.449531
\(944\) 3990.05 0.137569
\(945\) 33293.1 1.14606
\(946\) 0 0
\(947\) 16594.0 0.569412 0.284706 0.958615i \(-0.408104\pi\)
0.284706 + 0.958615i \(0.408104\pi\)
\(948\) 3246.57 0.111228
\(949\) −14855.0 −0.508129
\(950\) −7087.37 −0.242047
\(951\) 50479.8 1.72126
\(952\) 8757.13 0.298131
\(953\) −5575.02 −0.189499 −0.0947495 0.995501i \(-0.530205\pi\)
−0.0947495 + 0.995501i \(0.530205\pi\)
\(954\) −1823.95 −0.0618998
\(955\) 4988.14 0.169018
\(956\) −5283.40 −0.178742
\(957\) −12362.9 −0.417593
\(958\) −16032.8 −0.540705
\(959\) 55307.4 1.86232
\(960\) −8990.96 −0.302273
\(961\) −15212.2 −0.510631
\(962\) −9880.27 −0.331136
\(963\) −4689.13 −0.156911
\(964\) 15292.9 0.510946
\(965\) 2044.20 0.0681917
\(966\) −10035.2 −0.334242
\(967\) 25230.5 0.839046 0.419523 0.907745i \(-0.362197\pi\)
0.419523 + 0.907745i \(0.362197\pi\)
\(968\) 26496.3 0.879775
\(969\) −5587.43 −0.185236
\(970\) −6127.52 −0.202828
\(971\) 37852.9 1.25104 0.625519 0.780209i \(-0.284887\pi\)
0.625519 + 0.780209i \(0.284887\pi\)
\(972\) 3758.23 0.124018
\(973\) 41935.3 1.38169
\(974\) 6355.41 0.209076
\(975\) 5151.67 0.169216
\(976\) 6602.39 0.216534
\(977\) −24861.4 −0.814111 −0.407055 0.913404i \(-0.633444\pi\)
−0.407055 + 0.913404i \(0.633444\pi\)
\(978\) −13171.7 −0.430659
\(979\) 1709.37 0.0558035
\(980\) 27120.2 0.884002
\(981\) −4275.90 −0.139163
\(982\) 6084.47 0.197722
\(983\) 38582.6 1.25187 0.625937 0.779873i \(-0.284717\pi\)
0.625937 + 0.779873i \(0.284717\pi\)
\(984\) 38982.0 1.26291
\(985\) 16623.9 0.537748
\(986\) 5287.15 0.170768
\(987\) 20068.3 0.647196
\(988\) 6851.23 0.220614
\(989\) 0 0
\(990\) −279.381 −0.00896900
\(991\) 45385.9 1.45483 0.727413 0.686200i \(-0.240723\pi\)
0.727413 + 0.686200i \(0.240723\pi\)
\(992\) −22602.4 −0.723415
\(993\) 28115.9 0.898521
\(994\) −20120.8 −0.642044
\(995\) 4898.74 0.156081
\(996\) 30196.3 0.960650
\(997\) −55850.4 −1.77412 −0.887061 0.461653i \(-0.847257\pi\)
−0.887061 + 0.461653i \(0.847257\pi\)
\(998\) 12538.8 0.397703
\(999\) −52235.8 −1.65432
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 1849.4.a.g.1.20 30
43.2 odd 14 43.4.e.a.4.4 60
43.22 odd 14 43.4.e.a.11.4 yes 60
43.42 odd 2 1849.4.a.h.1.11 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.e.a.4.4 60 43.2 odd 14
43.4.e.a.11.4 yes 60 43.22 odd 14
1849.4.a.g.1.20 30 1.1 even 1 trivial
1849.4.a.h.1.11 30 43.42 odd 2