Properties

Label 1849.4.a.g.1.24
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 1.24
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)
 
\(f(q)\) \(=\) \(q+3.68353 q^{2} +8.25521 q^{3} +5.56841 q^{4} +12.2925 q^{5} +30.4083 q^{6} -28.5576 q^{7} -8.95684 q^{8} +41.1485 q^{9} +O(q^{10})\) \(q+3.68353 q^{2} +8.25521 q^{3} +5.56841 q^{4} +12.2925 q^{5} +30.4083 q^{6} -28.5576 q^{7} -8.95684 q^{8} +41.1485 q^{9} +45.2798 q^{10} -55.8522 q^{11} +45.9684 q^{12} -70.9474 q^{13} -105.193 q^{14} +101.477 q^{15} -77.5401 q^{16} +20.9256 q^{17} +151.572 q^{18} -25.4332 q^{19} +68.4497 q^{20} -235.749 q^{21} -205.733 q^{22} +103.780 q^{23} -73.9407 q^{24} +26.1057 q^{25} -261.337 q^{26} +116.799 q^{27} -159.020 q^{28} -94.0840 q^{29} +373.795 q^{30} +16.0997 q^{31} -213.967 q^{32} -461.072 q^{33} +77.0801 q^{34} -351.044 q^{35} +229.132 q^{36} +23.8504 q^{37} -93.6840 q^{38} -585.686 q^{39} -110.102 q^{40} -75.5892 q^{41} -868.388 q^{42} -311.008 q^{44} +505.819 q^{45} +382.278 q^{46} -5.56130 q^{47} -640.110 q^{48} +472.534 q^{49} +96.1613 q^{50} +172.745 q^{51} -395.064 q^{52} +622.211 q^{53} +430.234 q^{54} -686.564 q^{55} +255.786 q^{56} -209.956 q^{57} -346.562 q^{58} +189.542 q^{59} +565.067 q^{60} +371.320 q^{61} +59.3037 q^{62} -1175.10 q^{63} -167.832 q^{64} -872.121 q^{65} -1698.37 q^{66} -620.843 q^{67} +116.522 q^{68} +856.729 q^{69} -1293.08 q^{70} -644.449 q^{71} -368.561 q^{72} -488.895 q^{73} +87.8536 q^{74} +215.508 q^{75} -141.622 q^{76} +1595.00 q^{77} -2157.39 q^{78} -1110.49 q^{79} -953.162 q^{80} -146.808 q^{81} -278.435 q^{82} -351.896 q^{83} -1312.74 q^{84} +257.228 q^{85} -776.684 q^{87} +500.260 q^{88} +429.648 q^{89} +1863.20 q^{90} +2026.08 q^{91} +577.891 q^{92} +132.906 q^{93} -20.4852 q^{94} -312.638 q^{95} -1766.34 q^{96} +1183.38 q^{97} +1740.59 q^{98} -2298.24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.68353 1.30233 0.651163 0.758938i \(-0.274282\pi\)
0.651163 + 0.758938i \(0.274282\pi\)
\(3\) 8.25521 1.58872 0.794358 0.607450i \(-0.207807\pi\)
0.794358 + 0.607450i \(0.207807\pi\)
\(4\) 5.56841 0.696051
\(5\) 12.2925 1.09948 0.549738 0.835337i \(-0.314728\pi\)
0.549738 + 0.835337i \(0.314728\pi\)
\(6\) 30.4083 2.06903
\(7\) −28.5576 −1.54196 −0.770981 0.636858i \(-0.780234\pi\)
−0.770981 + 0.636858i \(0.780234\pi\)
\(8\) −8.95684 −0.395840
\(9\) 41.1485 1.52402
\(10\) 45.2798 1.43187
\(11\) −55.8522 −1.53092 −0.765458 0.643486i \(-0.777488\pi\)
−0.765458 + 0.643486i \(0.777488\pi\)
\(12\) 45.9684 1.10583
\(13\) −70.9474 −1.51364 −0.756818 0.653626i \(-0.773247\pi\)
−0.756818 + 0.653626i \(0.773247\pi\)
\(14\) −105.193 −2.00814
\(15\) 101.477 1.74675
\(16\) −77.5401 −1.21156
\(17\) 20.9256 0.298541 0.149271 0.988796i \(-0.452307\pi\)
0.149271 + 0.988796i \(0.452307\pi\)
\(18\) 151.572 1.98477
\(19\) −25.4332 −0.307093 −0.153547 0.988141i \(-0.549070\pi\)
−0.153547 + 0.988141i \(0.549070\pi\)
\(20\) 68.4497 0.765291
\(21\) −235.749 −2.44974
\(22\) −205.733 −1.99375
\(23\) 103.780 0.940856 0.470428 0.882438i \(-0.344099\pi\)
0.470428 + 0.882438i \(0.344099\pi\)
\(24\) −73.9407 −0.628878
\(25\) 26.1057 0.208846
\(26\) −261.337 −1.97125
\(27\) 116.799 0.832520
\(28\) −159.020 −1.07328
\(29\) −94.0840 −0.602447 −0.301224 0.953554i \(-0.597395\pi\)
−0.301224 + 0.953554i \(0.597395\pi\)
\(30\) 373.795 2.27484
\(31\) 16.0997 0.0932771 0.0466386 0.998912i \(-0.485149\pi\)
0.0466386 + 0.998912i \(0.485149\pi\)
\(32\) −213.967 −1.18201
\(33\) −461.072 −2.43219
\(34\) 77.0801 0.388798
\(35\) −351.044 −1.69535
\(36\) 229.132 1.06080
\(37\) 23.8504 0.105972 0.0529862 0.998595i \(-0.483126\pi\)
0.0529862 + 0.998595i \(0.483126\pi\)
\(38\) −93.6840 −0.399935
\(39\) −585.686 −2.40474
\(40\) −110.102 −0.435217
\(41\) −75.5892 −0.287928 −0.143964 0.989583i \(-0.545985\pi\)
−0.143964 + 0.989583i \(0.545985\pi\)
\(42\) −868.388 −3.19036
\(43\) 0 0
\(44\) −311.008 −1.06560
\(45\) 505.819 1.67562
\(46\) 382.278 1.22530
\(47\) −5.56130 −0.0172596 −0.00862978 0.999963i \(-0.502747\pi\)
−0.00862978 + 0.999963i \(0.502747\pi\)
\(48\) −640.110 −1.92483
\(49\) 472.534 1.37765
\(50\) 96.1613 0.271985
\(51\) 172.745 0.474297
\(52\) −395.064 −1.05357
\(53\) 622.211 1.61259 0.806294 0.591515i \(-0.201470\pi\)
0.806294 + 0.591515i \(0.201470\pi\)
\(54\) 430.234 1.08421
\(55\) −686.564 −1.68320
\(56\) 255.786 0.610371
\(57\) −209.956 −0.487884
\(58\) −346.562 −0.784582
\(59\) 189.542 0.418242 0.209121 0.977890i \(-0.432940\pi\)
0.209121 + 0.977890i \(0.432940\pi\)
\(60\) 565.067 1.21583
\(61\) 371.320 0.779388 0.389694 0.920944i \(-0.372581\pi\)
0.389694 + 0.920944i \(0.372581\pi\)
\(62\) 59.3037 0.121477
\(63\) −1175.10 −2.34998
\(64\) −167.832 −0.327798
\(65\) −872.121 −1.66420
\(66\) −1698.37 −3.16751
\(67\) −620.843 −1.13206 −0.566030 0.824384i \(-0.691521\pi\)
−0.566030 + 0.824384i \(0.691521\pi\)
\(68\) 116.522 0.207800
\(69\) 856.729 1.49475
\(70\) −1293.08 −2.20790
\(71\) −644.449 −1.07721 −0.538606 0.842558i \(-0.681049\pi\)
−0.538606 + 0.842558i \(0.681049\pi\)
\(72\) −368.561 −0.603269
\(73\) −488.895 −0.783847 −0.391924 0.919998i \(-0.628190\pi\)
−0.391924 + 0.919998i \(0.628190\pi\)
\(74\) 87.8536 0.138010
\(75\) 215.508 0.331797
\(76\) −141.622 −0.213753
\(77\) 1595.00 2.36062
\(78\) −2157.39 −3.13175
\(79\) −1110.49 −1.58151 −0.790755 0.612132i \(-0.790312\pi\)
−0.790755 + 0.612132i \(0.790312\pi\)
\(80\) −953.162 −1.33208
\(81\) −146.808 −0.201382
\(82\) −278.435 −0.374976
\(83\) −351.896 −0.465368 −0.232684 0.972552i \(-0.574751\pi\)
−0.232684 + 0.972552i \(0.574751\pi\)
\(84\) −1312.74 −1.70515
\(85\) 257.228 0.328239
\(86\) 0 0
\(87\) −776.684 −0.957118
\(88\) 500.260 0.605999
\(89\) 429.648 0.511715 0.255857 0.966715i \(-0.417642\pi\)
0.255857 + 0.966715i \(0.417642\pi\)
\(90\) 1863.20 2.18221
\(91\) 2026.08 2.33397
\(92\) 577.891 0.654884
\(93\) 132.906 0.148191
\(94\) −20.4852 −0.0224776
\(95\) −312.638 −0.337642
\(96\) −1766.34 −1.87788
\(97\) 1183.38 1.23870 0.619352 0.785114i \(-0.287396\pi\)
0.619352 + 0.785114i \(0.287396\pi\)
\(98\) 1740.59 1.79415
\(99\) −2298.24 −2.33315
\(100\) 145.367 0.145367
\(101\) −1990.58 −1.96109 −0.980547 0.196283i \(-0.937113\pi\)
−0.980547 + 0.196283i \(0.937113\pi\)
\(102\) 636.313 0.617690
\(103\) −412.891 −0.394984 −0.197492 0.980305i \(-0.563280\pi\)
−0.197492 + 0.980305i \(0.563280\pi\)
\(104\) 635.465 0.599158
\(105\) −2897.94 −2.69343
\(106\) 2291.93 2.10011
\(107\) 47.8596 0.0432408 0.0216204 0.999766i \(-0.493117\pi\)
0.0216204 + 0.999766i \(0.493117\pi\)
\(108\) 650.386 0.579476
\(109\) 684.904 0.601852 0.300926 0.953647i \(-0.402704\pi\)
0.300926 + 0.953647i \(0.402704\pi\)
\(110\) −2528.98 −2.19208
\(111\) 196.890 0.168360
\(112\) 2214.36 1.86819
\(113\) 503.311 0.419005 0.209502 0.977808i \(-0.432816\pi\)
0.209502 + 0.977808i \(0.432816\pi\)
\(114\) −773.381 −0.635384
\(115\) 1275.72 1.03445
\(116\) −523.898 −0.419334
\(117\) −2919.38 −2.30681
\(118\) 698.185 0.544688
\(119\) −597.584 −0.460340
\(120\) −908.916 −0.691436
\(121\) 1788.47 1.34371
\(122\) 1367.77 1.01502
\(123\) −624.005 −0.457436
\(124\) 89.6497 0.0649256
\(125\) −1215.66 −0.869854
\(126\) −4328.53 −3.06044
\(127\) −1896.38 −1.32501 −0.662507 0.749056i \(-0.730507\pi\)
−0.662507 + 0.749056i \(0.730507\pi\)
\(128\) 1093.52 0.755111
\(129\) 0 0
\(130\) −3212.49 −2.16734
\(131\) 57.7668 0.0385275 0.0192638 0.999814i \(-0.493868\pi\)
0.0192638 + 0.999814i \(0.493868\pi\)
\(132\) −2567.44 −1.69293
\(133\) 726.310 0.473527
\(134\) −2286.90 −1.47431
\(135\) 1435.76 0.915335
\(136\) −187.427 −0.118175
\(137\) 2080.41 1.29738 0.648692 0.761051i \(-0.275316\pi\)
0.648692 + 0.761051i \(0.275316\pi\)
\(138\) 3155.79 1.94666
\(139\) −1936.42 −1.18162 −0.590809 0.806811i \(-0.701191\pi\)
−0.590809 + 0.806811i \(0.701191\pi\)
\(140\) −1954.76 −1.18005
\(141\) −45.9097 −0.0274206
\(142\) −2373.85 −1.40288
\(143\) 3962.57 2.31725
\(144\) −3190.66 −1.84645
\(145\) −1156.53 −0.662376
\(146\) −1800.86 −1.02082
\(147\) 3900.87 2.18869
\(148\) 132.809 0.0737622
\(149\) −488.602 −0.268643 −0.134322 0.990938i \(-0.542885\pi\)
−0.134322 + 0.990938i \(0.542885\pi\)
\(150\) 793.832 0.432107
\(151\) 3294.70 1.77562 0.887810 0.460211i \(-0.152226\pi\)
0.887810 + 0.460211i \(0.152226\pi\)
\(152\) 227.801 0.121560
\(153\) 861.058 0.454983
\(154\) 5875.24 3.07429
\(155\) 197.906 0.102556
\(156\) −3261.34 −1.67382
\(157\) 561.132 0.285243 0.142622 0.989777i \(-0.454447\pi\)
0.142622 + 0.989777i \(0.454447\pi\)
\(158\) −4090.51 −2.05964
\(159\) 5136.48 2.56195
\(160\) −2630.19 −1.29959
\(161\) −2963.71 −1.45077
\(162\) −540.771 −0.262265
\(163\) −647.402 −0.311095 −0.155547 0.987828i \(-0.549714\pi\)
−0.155547 + 0.987828i \(0.549714\pi\)
\(164\) −420.912 −0.200413
\(165\) −5667.73 −2.67414
\(166\) −1296.22 −0.606061
\(167\) 2604.15 1.20668 0.603339 0.797485i \(-0.293837\pi\)
0.603339 + 0.797485i \(0.293837\pi\)
\(168\) 2111.56 0.969707
\(169\) 2836.53 1.29109
\(170\) 947.508 0.427474
\(171\) −1046.54 −0.468017
\(172\) 0 0
\(173\) 1969.05 0.865344 0.432672 0.901552i \(-0.357571\pi\)
0.432672 + 0.901552i \(0.357571\pi\)
\(174\) −2860.94 −1.24648
\(175\) −745.516 −0.322033
\(176\) 4330.79 1.85480
\(177\) 1564.71 0.664469
\(178\) 1582.62 0.666419
\(179\) 1327.24 0.554203 0.277102 0.960841i \(-0.410626\pi\)
0.277102 + 0.960841i \(0.410626\pi\)
\(180\) 2816.61 1.16632
\(181\) −2504.61 −1.02854 −0.514272 0.857627i \(-0.671938\pi\)
−0.514272 + 0.857627i \(0.671938\pi\)
\(182\) 7463.14 3.03959
\(183\) 3065.33 1.23823
\(184\) −929.544 −0.372429
\(185\) 293.181 0.116514
\(186\) 489.565 0.192993
\(187\) −1168.74 −0.457042
\(188\) −30.9676 −0.0120135
\(189\) −3335.50 −1.28371
\(190\) −1151.61 −0.439719
\(191\) −1610.67 −0.610179 −0.305090 0.952324i \(-0.598686\pi\)
−0.305090 + 0.952324i \(0.598686\pi\)
\(192\) −1385.49 −0.520777
\(193\) −3546.59 −1.32274 −0.661371 0.750059i \(-0.730025\pi\)
−0.661371 + 0.750059i \(0.730025\pi\)
\(194\) 4359.02 1.61319
\(195\) −7199.54 −2.64395
\(196\) 2631.26 0.958914
\(197\) 2862.77 1.03535 0.517674 0.855578i \(-0.326798\pi\)
0.517674 + 0.855578i \(0.326798\pi\)
\(198\) −8465.63 −3.03852
\(199\) −1776.37 −0.632781 −0.316390 0.948629i \(-0.602471\pi\)
−0.316390 + 0.948629i \(0.602471\pi\)
\(200\) −233.825 −0.0826696
\(201\) −5125.19 −1.79852
\(202\) −7332.38 −2.55398
\(203\) 2686.81 0.928951
\(204\) 961.916 0.330135
\(205\) −929.181 −0.316570
\(206\) −1520.90 −0.514398
\(207\) 4270.41 1.43388
\(208\) 5501.27 1.83387
\(209\) 1420.50 0.470134
\(210\) −10674.7 −3.50772
\(211\) 5127.05 1.67280 0.836400 0.548120i \(-0.184656\pi\)
0.836400 + 0.548120i \(0.184656\pi\)
\(212\) 3464.72 1.12244
\(213\) −5320.06 −1.71138
\(214\) 176.292 0.0563136
\(215\) 0 0
\(216\) −1046.15 −0.329545
\(217\) −459.768 −0.143830
\(218\) 2522.87 0.783807
\(219\) −4035.93 −1.24531
\(220\) −3823.07 −1.17160
\(221\) −1484.62 −0.451883
\(222\) 725.250 0.219260
\(223\) 557.733 0.167482 0.0837412 0.996488i \(-0.473313\pi\)
0.0837412 + 0.996488i \(0.473313\pi\)
\(224\) 6110.36 1.82262
\(225\) 1074.21 0.318285
\(226\) 1853.96 0.545680
\(227\) −2998.43 −0.876709 −0.438355 0.898802i \(-0.644439\pi\)
−0.438355 + 0.898802i \(0.644439\pi\)
\(228\) −1169.12 −0.339592
\(229\) 946.354 0.273087 0.136543 0.990634i \(-0.456401\pi\)
0.136543 + 0.990634i \(0.456401\pi\)
\(230\) 4699.16 1.34719
\(231\) 13167.1 3.75035
\(232\) 842.696 0.238473
\(233\) 993.000 0.279200 0.139600 0.990208i \(-0.455418\pi\)
0.139600 + 0.990208i \(0.455418\pi\)
\(234\) −10753.6 −3.00422
\(235\) −68.3624 −0.0189765
\(236\) 1055.45 0.291118
\(237\) −9167.29 −2.51257
\(238\) −2201.22 −0.599512
\(239\) −2419.96 −0.654954 −0.327477 0.944859i \(-0.606198\pi\)
−0.327477 + 0.944859i \(0.606198\pi\)
\(240\) −7868.56 −2.11630
\(241\) −58.0078 −0.0155046 −0.00775231 0.999970i \(-0.502468\pi\)
−0.00775231 + 0.999970i \(0.502468\pi\)
\(242\) 6587.89 1.74994
\(243\) −4365.51 −1.15246
\(244\) 2067.66 0.542494
\(245\) 5808.62 1.51469
\(246\) −2298.54 −0.595731
\(247\) 1804.42 0.464827
\(248\) −144.202 −0.0369229
\(249\) −2904.97 −0.739338
\(250\) −4477.92 −1.13283
\(251\) −1283.94 −0.322876 −0.161438 0.986883i \(-0.551613\pi\)
−0.161438 + 0.986883i \(0.551613\pi\)
\(252\) −6543.45 −1.63571
\(253\) −5796.36 −1.44037
\(254\) −6985.38 −1.72560
\(255\) 2123.47 0.521478
\(256\) 5370.67 1.31120
\(257\) 2588.04 0.628162 0.314081 0.949396i \(-0.398304\pi\)
0.314081 + 0.949396i \(0.398304\pi\)
\(258\) 0 0
\(259\) −681.108 −0.163405
\(260\) −4856.33 −1.15837
\(261\) −3871.42 −0.918142
\(262\) 212.786 0.0501754
\(263\) 3698.37 0.867116 0.433558 0.901126i \(-0.357258\pi\)
0.433558 + 0.901126i \(0.357258\pi\)
\(264\) 4129.75 0.962760
\(265\) 7648.53 1.77300
\(266\) 2675.38 0.616686
\(267\) 3546.84 0.812970
\(268\) −3457.11 −0.787972
\(269\) −2299.84 −0.521278 −0.260639 0.965436i \(-0.583933\pi\)
−0.260639 + 0.965436i \(0.583933\pi\)
\(270\) 5288.65 1.19206
\(271\) 7070.12 1.58479 0.792397 0.610006i \(-0.208833\pi\)
0.792397 + 0.610006i \(0.208833\pi\)
\(272\) −1622.57 −0.361702
\(273\) 16725.7 3.70802
\(274\) 7663.26 1.68962
\(275\) −1458.06 −0.319726
\(276\) 4770.62 1.04043
\(277\) 708.917 0.153771 0.0768857 0.997040i \(-0.475502\pi\)
0.0768857 + 0.997040i \(0.475502\pi\)
\(278\) −7132.87 −1.53885
\(279\) 662.479 0.142156
\(280\) 3144.25 0.671088
\(281\) 8502.44 1.80503 0.902514 0.430660i \(-0.141719\pi\)
0.902514 + 0.430660i \(0.141719\pi\)
\(282\) −169.110 −0.0357105
\(283\) −813.075 −0.170786 −0.0853928 0.996347i \(-0.527215\pi\)
−0.0853928 + 0.996347i \(0.527215\pi\)
\(284\) −3588.56 −0.749794
\(285\) −2580.89 −0.536417
\(286\) 14596.2 3.01781
\(287\) 2158.64 0.443974
\(288\) −8804.42 −1.80141
\(289\) −4475.12 −0.910873
\(290\) −4260.11 −0.862628
\(291\) 9769.07 1.96795
\(292\) −2722.37 −0.545598
\(293\) −4761.50 −0.949385 −0.474693 0.880152i \(-0.657441\pi\)
−0.474693 + 0.880152i \(0.657441\pi\)
\(294\) 14369.0 2.85039
\(295\) 2329.95 0.459847
\(296\) −213.624 −0.0419481
\(297\) −6523.50 −1.27452
\(298\) −1799.78 −0.349861
\(299\) −7362.94 −1.42411
\(300\) 1200.04 0.230948
\(301\) 0 0
\(302\) 12136.1 2.31243
\(303\) −16432.7 −3.11562
\(304\) 1972.09 0.372063
\(305\) 4564.46 0.856918
\(306\) 3171.73 0.592536
\(307\) −8092.16 −1.50438 −0.752189 0.658948i \(-0.771002\pi\)
−0.752189 + 0.658948i \(0.771002\pi\)
\(308\) 8881.63 1.64311
\(309\) −3408.50 −0.627517
\(310\) 728.992 0.133561
\(311\) 2093.98 0.381797 0.190899 0.981610i \(-0.438860\pi\)
0.190899 + 0.981610i \(0.438860\pi\)
\(312\) 5245.90 0.951892
\(313\) −5734.10 −1.03550 −0.517748 0.855533i \(-0.673230\pi\)
−0.517748 + 0.855533i \(0.673230\pi\)
\(314\) 2066.95 0.371480
\(315\) −14444.9 −2.58375
\(316\) −6183.64 −1.10081
\(317\) −9549.73 −1.69201 −0.846003 0.533178i \(-0.820998\pi\)
−0.846003 + 0.533178i \(0.820998\pi\)
\(318\) 18920.4 3.33649
\(319\) 5254.80 0.922296
\(320\) −2063.08 −0.360405
\(321\) 395.091 0.0686973
\(322\) −10916.9 −1.88937
\(323\) −532.205 −0.0916801
\(324\) −817.485 −0.140172
\(325\) −1852.13 −0.316116
\(326\) −2384.73 −0.405147
\(327\) 5654.03 0.956173
\(328\) 677.041 0.113974
\(329\) 158.817 0.0266136
\(330\) −20877.3 −3.48259
\(331\) 1228.97 0.204079 0.102040 0.994780i \(-0.467463\pi\)
0.102040 + 0.994780i \(0.467463\pi\)
\(332\) −1959.50 −0.323920
\(333\) 981.408 0.161504
\(334\) 9592.47 1.57149
\(335\) −7631.72 −1.24467
\(336\) 18280.0 2.96802
\(337\) −2889.26 −0.467027 −0.233514 0.972354i \(-0.575022\pi\)
−0.233514 + 0.972354i \(0.575022\pi\)
\(338\) 10448.4 1.68142
\(339\) 4154.94 0.665679
\(340\) 1432.35 0.228471
\(341\) −899.204 −0.142799
\(342\) −3854.96 −0.609510
\(343\) −3699.17 −0.582321
\(344\) 0 0
\(345\) 10531.3 1.64345
\(346\) 7253.07 1.12696
\(347\) 1936.22 0.299544 0.149772 0.988721i \(-0.452146\pi\)
0.149772 + 0.988721i \(0.452146\pi\)
\(348\) −4324.89 −0.666203
\(349\) −1821.71 −0.279410 −0.139705 0.990193i \(-0.544615\pi\)
−0.139705 + 0.990193i \(0.544615\pi\)
\(350\) −2746.13 −0.419391
\(351\) −8286.60 −1.26013
\(352\) 11950.5 1.80956
\(353\) −845.304 −0.127453 −0.0637267 0.997967i \(-0.520299\pi\)
−0.0637267 + 0.997967i \(0.520299\pi\)
\(354\) 5763.67 0.865354
\(355\) −7921.89 −1.18437
\(356\) 2392.46 0.356180
\(357\) −4933.18 −0.731349
\(358\) 4888.92 0.721753
\(359\) −8399.94 −1.23491 −0.617454 0.786607i \(-0.711836\pi\)
−0.617454 + 0.786607i \(0.711836\pi\)
\(360\) −4530.54 −0.663279
\(361\) −6212.15 −0.905694
\(362\) −9225.83 −1.33950
\(363\) 14764.2 2.13477
\(364\) 11282.1 1.62456
\(365\) −6009.75 −0.861821
\(366\) 11291.2 1.61257
\(367\) −6905.35 −0.982170 −0.491085 0.871112i \(-0.663399\pi\)
−0.491085 + 0.871112i \(0.663399\pi\)
\(368\) −8047.14 −1.13991
\(369\) −3110.39 −0.438808
\(370\) 1079.94 0.151739
\(371\) −17768.8 −2.48655
\(372\) 740.077 0.103148
\(373\) 8411.92 1.16770 0.583850 0.811861i \(-0.301545\pi\)
0.583850 + 0.811861i \(0.301545\pi\)
\(374\) −4305.10 −0.595217
\(375\) −10035.5 −1.38195
\(376\) 49.8117 0.00683203
\(377\) 6675.01 0.911885
\(378\) −12286.4 −1.67181
\(379\) 6112.46 0.828433 0.414216 0.910178i \(-0.364056\pi\)
0.414216 + 0.910178i \(0.364056\pi\)
\(380\) −1740.89 −0.235016
\(381\) −15655.0 −2.10507
\(382\) −5932.97 −0.794652
\(383\) −10015.6 −1.33622 −0.668110 0.744062i \(-0.732897\pi\)
−0.668110 + 0.744062i \(0.732897\pi\)
\(384\) 9027.22 1.19966
\(385\) 19606.6 2.59544
\(386\) −13064.0 −1.72264
\(387\) 0 0
\(388\) 6589.55 0.862201
\(389\) −1981.11 −0.258216 −0.129108 0.991631i \(-0.541211\pi\)
−0.129108 + 0.991631i \(0.541211\pi\)
\(390\) −26519.8 −3.44328
\(391\) 2171.67 0.280884
\(392\) −4232.41 −0.545329
\(393\) 476.877 0.0612094
\(394\) 10545.1 1.34836
\(395\) −13650.6 −1.73883
\(396\) −12797.5 −1.62399
\(397\) 3781.76 0.478088 0.239044 0.971009i \(-0.423166\pi\)
0.239044 + 0.971009i \(0.423166\pi\)
\(398\) −6543.31 −0.824086
\(399\) 5995.84 0.752300
\(400\) −2024.24 −0.253030
\(401\) 13602.5 1.69395 0.846976 0.531632i \(-0.178421\pi\)
0.846976 + 0.531632i \(0.178421\pi\)
\(402\) −18878.8 −2.34226
\(403\) −1142.23 −0.141188
\(404\) −11084.4 −1.36502
\(405\) −1804.64 −0.221415
\(406\) 9896.95 1.20980
\(407\) −1332.10 −0.162235
\(408\) −1547.25 −0.187746
\(409\) −9450.00 −1.14248 −0.571238 0.820785i \(-0.693537\pi\)
−0.571238 + 0.820785i \(0.693537\pi\)
\(410\) −3422.67 −0.412277
\(411\) 17174.2 2.06118
\(412\) −2299.14 −0.274929
\(413\) −5412.86 −0.644914
\(414\) 15730.2 1.86738
\(415\) −4325.68 −0.511661
\(416\) 15180.4 1.78913
\(417\) −15985.6 −1.87726
\(418\) 5232.46 0.612268
\(419\) −5265.77 −0.613961 −0.306980 0.951716i \(-0.599319\pi\)
−0.306980 + 0.951716i \(0.599319\pi\)
\(420\) −16136.9 −1.87477
\(421\) −708.957 −0.0820723 −0.0410361 0.999158i \(-0.513066\pi\)
−0.0410361 + 0.999158i \(0.513066\pi\)
\(422\) 18885.7 2.17853
\(423\) −228.840 −0.0263039
\(424\) −5573.04 −0.638328
\(425\) 546.278 0.0623491
\(426\) −19596.6 −2.22878
\(427\) −10604.0 −1.20179
\(428\) 266.502 0.0300978
\(429\) 32711.8 3.68145
\(430\) 0 0
\(431\) 12530.7 1.40042 0.700210 0.713937i \(-0.253090\pi\)
0.700210 + 0.713937i \(0.253090\pi\)
\(432\) −9056.63 −1.00865
\(433\) −13633.0 −1.51307 −0.756534 0.653954i \(-0.773109\pi\)
−0.756534 + 0.653954i \(0.773109\pi\)
\(434\) −1693.57 −0.187313
\(435\) −9547.39 −1.05233
\(436\) 3813.82 0.418920
\(437\) −2639.47 −0.288931
\(438\) −14866.5 −1.62180
\(439\) −16838.6 −1.83067 −0.915336 0.402691i \(-0.868075\pi\)
−0.915336 + 0.402691i \(0.868075\pi\)
\(440\) 6149.45 0.666280
\(441\) 19444.1 2.09957
\(442\) −5468.63 −0.588498
\(443\) −7973.21 −0.855121 −0.427561 0.903987i \(-0.640627\pi\)
−0.427561 + 0.903987i \(0.640627\pi\)
\(444\) 1096.36 0.117187
\(445\) 5281.45 0.562618
\(446\) 2054.43 0.218117
\(447\) −4033.51 −0.426798
\(448\) 4792.88 0.505452
\(449\) 248.826 0.0261533 0.0130766 0.999914i \(-0.495837\pi\)
0.0130766 + 0.999914i \(0.495837\pi\)
\(450\) 3956.90 0.414511
\(451\) 4221.83 0.440794
\(452\) 2802.64 0.291649
\(453\) 27198.4 2.82096
\(454\) −11044.8 −1.14176
\(455\) 24905.6 2.56614
\(456\) 1880.55 0.193124
\(457\) −18522.3 −1.89593 −0.947964 0.318379i \(-0.896862\pi\)
−0.947964 + 0.318379i \(0.896862\pi\)
\(458\) 3485.93 0.355648
\(459\) 2444.09 0.248542
\(460\) 7103.73 0.720029
\(461\) 3120.99 0.315313 0.157656 0.987494i \(-0.449606\pi\)
0.157656 + 0.987494i \(0.449606\pi\)
\(462\) 48501.4 4.88418
\(463\) 2643.00 0.265293 0.132647 0.991163i \(-0.457652\pi\)
0.132647 + 0.991163i \(0.457652\pi\)
\(464\) 7295.28 0.729903
\(465\) 1633.75 0.162932
\(466\) 3657.75 0.363609
\(467\) 14644.0 1.45106 0.725528 0.688192i \(-0.241595\pi\)
0.725528 + 0.688192i \(0.241595\pi\)
\(468\) −16256.3 −1.60566
\(469\) 17729.8 1.74560
\(470\) −251.815 −0.0247135
\(471\) 4632.26 0.453171
\(472\) −1697.70 −0.165557
\(473\) 0 0
\(474\) −33768.0 −3.27219
\(475\) −663.952 −0.0641352
\(476\) −3327.59 −0.320420
\(477\) 25603.1 2.45762
\(478\) −8913.99 −0.852964
\(479\) 16033.1 1.52938 0.764689 0.644399i \(-0.222892\pi\)
0.764689 + 0.644399i \(0.222892\pi\)
\(480\) −21712.8 −2.06468
\(481\) −1692.12 −0.160404
\(482\) −213.674 −0.0201921
\(483\) −24466.1 −2.30486
\(484\) 9958.94 0.935287
\(485\) 14546.7 1.36192
\(486\) −16080.5 −1.50088
\(487\) −15972.8 −1.48624 −0.743119 0.669159i \(-0.766654\pi\)
−0.743119 + 0.669159i \(0.766654\pi\)
\(488\) −3325.86 −0.308513
\(489\) −5344.44 −0.494242
\(490\) 21396.3 1.97262
\(491\) −13730.1 −1.26197 −0.630987 0.775793i \(-0.717350\pi\)
−0.630987 + 0.775793i \(0.717350\pi\)
\(492\) −3474.71 −0.318399
\(493\) −1968.76 −0.179855
\(494\) 6646.63 0.605356
\(495\) −28251.1 −2.56524
\(496\) −1248.37 −0.113011
\(497\) 18403.9 1.66102
\(498\) −10700.6 −0.962859
\(499\) 15534.6 1.39364 0.696820 0.717246i \(-0.254598\pi\)
0.696820 + 0.717246i \(0.254598\pi\)
\(500\) −6769.28 −0.605463
\(501\) 21497.8 1.91707
\(502\) −4729.45 −0.420489
\(503\) −8754.61 −0.776041 −0.388021 0.921651i \(-0.626841\pi\)
−0.388021 + 0.921651i \(0.626841\pi\)
\(504\) 10525.2 0.930218
\(505\) −24469.3 −2.15617
\(506\) −21351.1 −1.87583
\(507\) 23416.1 2.05118
\(508\) −10559.8 −0.922277
\(509\) 14111.2 1.22882 0.614408 0.788989i \(-0.289395\pi\)
0.614408 + 0.788989i \(0.289395\pi\)
\(510\) 7821.88 0.679134
\(511\) 13961.6 1.20866
\(512\) 11034.9 0.952495
\(513\) −2970.58 −0.255661
\(514\) 9533.13 0.818071
\(515\) −5075.46 −0.434275
\(516\) 0 0
\(517\) 310.611 0.0264230
\(518\) −2508.88 −0.212807
\(519\) 16255.0 1.37479
\(520\) 7811.45 0.658759
\(521\) 5869.85 0.493595 0.246797 0.969067i \(-0.420622\pi\)
0.246797 + 0.969067i \(0.420622\pi\)
\(522\) −14260.5 −1.19572
\(523\) −16310.7 −1.36370 −0.681851 0.731491i \(-0.738825\pi\)
−0.681851 + 0.731491i \(0.738825\pi\)
\(524\) 321.669 0.0268171
\(525\) −6154.39 −0.511618
\(526\) 13623.1 1.12927
\(527\) 336.896 0.0278471
\(528\) 35751.6 2.94676
\(529\) −1396.64 −0.114789
\(530\) 28173.6 2.30902
\(531\) 7799.39 0.637410
\(532\) 4044.39 0.329599
\(533\) 5362.85 0.435818
\(534\) 13064.9 1.05875
\(535\) 588.315 0.0475422
\(536\) 5560.80 0.448115
\(537\) 10956.6 0.880472
\(538\) −8471.54 −0.678874
\(539\) −26392.1 −2.10907
\(540\) 7994.88 0.637120
\(541\) −6431.70 −0.511129 −0.255564 0.966792i \(-0.582261\pi\)
−0.255564 + 0.966792i \(0.582261\pi\)
\(542\) 26043.0 2.06392
\(543\) −20676.1 −1.63407
\(544\) −4477.38 −0.352879
\(545\) 8419.18 0.661722
\(546\) 61609.8 4.82904
\(547\) 7509.24 0.586969 0.293484 0.955964i \(-0.405185\pi\)
0.293484 + 0.955964i \(0.405185\pi\)
\(548\) 11584.6 0.903045
\(549\) 15279.3 1.18780
\(550\) −5370.82 −0.416387
\(551\) 2392.86 0.185008
\(552\) −7673.59 −0.591684
\(553\) 31712.7 2.43863
\(554\) 2611.32 0.200260
\(555\) 2420.27 0.185108
\(556\) −10782.8 −0.822467
\(557\) −16812.6 −1.27895 −0.639474 0.768813i \(-0.720848\pi\)
−0.639474 + 0.768813i \(0.720848\pi\)
\(558\) 2440.26 0.185134
\(559\) 0 0
\(560\) 27220.0 2.05402
\(561\) −9648.21 −0.726110
\(562\) 31319.0 2.35073
\(563\) −10965.3 −0.820840 −0.410420 0.911897i \(-0.634618\pi\)
−0.410420 + 0.911897i \(0.634618\pi\)
\(564\) −255.644 −0.0190861
\(565\) 6186.95 0.460685
\(566\) −2994.99 −0.222418
\(567\) 4192.47 0.310524
\(568\) 5772.23 0.426404
\(569\) −12831.2 −0.945364 −0.472682 0.881233i \(-0.656714\pi\)
−0.472682 + 0.881233i \(0.656714\pi\)
\(570\) −9506.79 −0.698589
\(571\) −9947.46 −0.729051 −0.364526 0.931193i \(-0.618769\pi\)
−0.364526 + 0.931193i \(0.618769\pi\)
\(572\) 22065.2 1.61292
\(573\) −13296.5 −0.969402
\(574\) 7951.43 0.578199
\(575\) 2709.26 0.196494
\(576\) −6906.06 −0.499570
\(577\) 1301.43 0.0938978 0.0469489 0.998897i \(-0.485050\pi\)
0.0469489 + 0.998897i \(0.485050\pi\)
\(578\) −16484.2 −1.18625
\(579\) −29277.8 −2.10146
\(580\) −6440.02 −0.461047
\(581\) 10049.3 0.717581
\(582\) 35984.7 2.56291
\(583\) −34751.8 −2.46874
\(584\) 4378.96 0.310278
\(585\) −35886.5 −2.53628
\(586\) −17539.1 −1.23641
\(587\) −9125.56 −0.641656 −0.320828 0.947137i \(-0.603961\pi\)
−0.320828 + 0.947137i \(0.603961\pi\)
\(588\) 21721.6 1.52344
\(589\) −409.467 −0.0286448
\(590\) 8582.44 0.598871
\(591\) 23632.7 1.64487
\(592\) −1849.36 −0.128392
\(593\) 20848.4 1.44375 0.721873 0.692025i \(-0.243281\pi\)
0.721873 + 0.692025i \(0.243281\pi\)
\(594\) −24029.5 −1.65984
\(595\) −7345.80 −0.506132
\(596\) −2720.73 −0.186989
\(597\) −14664.3 −1.00531
\(598\) −27121.6 −1.85466
\(599\) −794.230 −0.0541759 −0.0270879 0.999633i \(-0.508623\pi\)
−0.0270879 + 0.999633i \(0.508623\pi\)
\(600\) −1930.28 −0.131339
\(601\) −1489.20 −0.101075 −0.0505373 0.998722i \(-0.516093\pi\)
−0.0505373 + 0.998722i \(0.516093\pi\)
\(602\) 0 0
\(603\) −25546.8 −1.72528
\(604\) 18346.2 1.23592
\(605\) 21984.8 1.47737
\(606\) −60530.4 −4.05755
\(607\) −20951.2 −1.40096 −0.700480 0.713672i \(-0.747031\pi\)
−0.700480 + 0.713672i \(0.747031\pi\)
\(608\) 5441.86 0.362987
\(609\) 22180.2 1.47584
\(610\) 16813.3 1.11599
\(611\) 394.560 0.0261247
\(612\) 4794.72 0.316691
\(613\) −17473.3 −1.15129 −0.575646 0.817699i \(-0.695249\pi\)
−0.575646 + 0.817699i \(0.695249\pi\)
\(614\) −29807.7 −1.95919
\(615\) −7670.59 −0.502940
\(616\) −14286.2 −0.934427
\(617\) −5000.48 −0.326275 −0.163138 0.986603i \(-0.552161\pi\)
−0.163138 + 0.986603i \(0.552161\pi\)
\(618\) −12555.3 −0.817232
\(619\) 8385.16 0.544472 0.272236 0.962231i \(-0.412237\pi\)
0.272236 + 0.962231i \(0.412237\pi\)
\(620\) 1102.02 0.0713841
\(621\) 12121.5 0.783281
\(622\) 7713.26 0.497224
\(623\) −12269.7 −0.789045
\(624\) 45414.1 2.91349
\(625\) −18206.7 −1.16523
\(626\) −21121.7 −1.34855
\(627\) 11726.5 0.746910
\(628\) 3124.61 0.198544
\(629\) 499.083 0.0316371
\(630\) −53208.4 −3.36488
\(631\) −18872.3 −1.19064 −0.595320 0.803489i \(-0.702975\pi\)
−0.595320 + 0.803489i \(0.702975\pi\)
\(632\) 9946.45 0.626026
\(633\) 42324.9 2.65761
\(634\) −35176.7 −2.20354
\(635\) −23311.3 −1.45682
\(636\) 28602.0 1.78325
\(637\) −33525.0 −2.08526
\(638\) 19356.2 1.20113
\(639\) −26518.1 −1.64169
\(640\) 13442.1 0.830226
\(641\) 1488.27 0.0917054 0.0458527 0.998948i \(-0.485400\pi\)
0.0458527 + 0.998948i \(0.485400\pi\)
\(642\) 1455.33 0.0894663
\(643\) 4204.04 0.257840 0.128920 0.991655i \(-0.458849\pi\)
0.128920 + 0.991655i \(0.458849\pi\)
\(644\) −16503.2 −1.00981
\(645\) 0 0
\(646\) −1960.39 −0.119397
\(647\) 19185.2 1.16576 0.582882 0.812557i \(-0.301925\pi\)
0.582882 + 0.812557i \(0.301925\pi\)
\(648\) 1314.93 0.0797153
\(649\) −10586.4 −0.640294
\(650\) −6822.39 −0.411686
\(651\) −3795.48 −0.228505
\(652\) −3605.00 −0.216538
\(653\) −18907.5 −1.13309 −0.566544 0.824031i \(-0.691720\pi\)
−0.566544 + 0.824031i \(0.691720\pi\)
\(654\) 20826.8 1.24525
\(655\) 710.099 0.0423601
\(656\) 5861.19 0.348843
\(657\) −20117.3 −1.19460
\(658\) 585.008 0.0346596
\(659\) 24740.8 1.46247 0.731234 0.682127i \(-0.238945\pi\)
0.731234 + 0.682127i \(0.238945\pi\)
\(660\) −31560.2 −1.86133
\(661\) 11817.8 0.695400 0.347700 0.937606i \(-0.386963\pi\)
0.347700 + 0.937606i \(0.386963\pi\)
\(662\) 4526.95 0.265778
\(663\) −12255.8 −0.717913
\(664\) 3151.88 0.184212
\(665\) 8928.17 0.520631
\(666\) 3615.05 0.210331
\(667\) −9764.07 −0.566816
\(668\) 14501.0 0.839910
\(669\) 4604.21 0.266082
\(670\) −28111.7 −1.62097
\(671\) −20739.1 −1.19318
\(672\) 50442.4 2.89562
\(673\) 9905.49 0.567353 0.283676 0.958920i \(-0.408446\pi\)
0.283676 + 0.958920i \(0.408446\pi\)
\(674\) −10642.7 −0.608221
\(675\) 3049.13 0.173868
\(676\) 15794.9 0.898666
\(677\) 1165.33 0.0661555 0.0330778 0.999453i \(-0.489469\pi\)
0.0330778 + 0.999453i \(0.489469\pi\)
\(678\) 15304.9 0.866931
\(679\) −33794.5 −1.91003
\(680\) −2303.95 −0.129930
\(681\) −24752.7 −1.39284
\(682\) −3312.25 −0.185971
\(683\) 6889.70 0.385984 0.192992 0.981200i \(-0.438181\pi\)
0.192992 + 0.981200i \(0.438181\pi\)
\(684\) −5827.56 −0.325763
\(685\) 25573.5 1.42644
\(686\) −13626.0 −0.758371
\(687\) 7812.35 0.433857
\(688\) 0 0
\(689\) −44144.2 −2.44087
\(690\) 38792.5 2.14030
\(691\) 6806.51 0.374720 0.187360 0.982291i \(-0.440007\pi\)
0.187360 + 0.982291i \(0.440007\pi\)
\(692\) 10964.5 0.602323
\(693\) 65632.1 3.59763
\(694\) 7132.13 0.390104
\(695\) −23803.5 −1.29916
\(696\) 6956.64 0.378866
\(697\) −1581.75 −0.0859584
\(698\) −6710.33 −0.363882
\(699\) 8197.43 0.443570
\(700\) −4151.34 −0.224151
\(701\) 32393.6 1.74535 0.872674 0.488304i \(-0.162384\pi\)
0.872674 + 0.488304i \(0.162384\pi\)
\(702\) −30524.0 −1.64110
\(703\) −606.591 −0.0325434
\(704\) 9373.81 0.501831
\(705\) −564.346 −0.0301482
\(706\) −3113.71 −0.165986
\(707\) 56846.2 3.02393
\(708\) 8712.95 0.462504
\(709\) 6305.19 0.333987 0.166993 0.985958i \(-0.446594\pi\)
0.166993 + 0.985958i \(0.446594\pi\)
\(710\) −29180.5 −1.54243
\(711\) −45694.9 −2.41025
\(712\) −3848.29 −0.202557
\(713\) 1670.83 0.0877604
\(714\) −18171.5 −0.952454
\(715\) 48709.9 2.54776
\(716\) 7390.60 0.385754
\(717\) −19977.3 −1.04054
\(718\) −30941.5 −1.60825
\(719\) −8766.36 −0.454701 −0.227351 0.973813i \(-0.573006\pi\)
−0.227351 + 0.973813i \(0.573006\pi\)
\(720\) −39221.2 −2.03012
\(721\) 11791.2 0.609050
\(722\) −22882.7 −1.17951
\(723\) −478.867 −0.0246324
\(724\) −13946.7 −0.715919
\(725\) −2456.13 −0.125819
\(726\) 54384.5 2.78016
\(727\) −2282.96 −0.116465 −0.0582327 0.998303i \(-0.518547\pi\)
−0.0582327 + 0.998303i \(0.518547\pi\)
\(728\) −18147.3 −0.923879
\(729\) −32074.4 −1.62955
\(730\) −22137.1 −1.12237
\(731\) 0 0
\(732\) 17069.0 0.861869
\(733\) −11976.6 −0.603500 −0.301750 0.953387i \(-0.597571\pi\)
−0.301750 + 0.953387i \(0.597571\pi\)
\(734\) −25436.1 −1.27910
\(735\) 47951.4 2.40642
\(736\) −22205.5 −1.11210
\(737\) 34675.5 1.73309
\(738\) −11457.2 −0.571471
\(739\) −18648.1 −0.928255 −0.464127 0.885769i \(-0.653632\pi\)
−0.464127 + 0.885769i \(0.653632\pi\)
\(740\) 1632.55 0.0810997
\(741\) 14895.9 0.738479
\(742\) −65452.0 −3.23830
\(743\) −841.949 −0.0415722 −0.0207861 0.999784i \(-0.506617\pi\)
−0.0207861 + 0.999784i \(0.506617\pi\)
\(744\) −1190.42 −0.0586599
\(745\) −6006.14 −0.295366
\(746\) 30985.6 1.52073
\(747\) −14480.0 −0.709231
\(748\) −6508.03 −0.318124
\(749\) −1366.75 −0.0666757
\(750\) −36966.2 −1.79975
\(751\) −24798.5 −1.20494 −0.602469 0.798142i \(-0.705816\pi\)
−0.602469 + 0.798142i \(0.705816\pi\)
\(752\) 431.224 0.0209111
\(753\) −10599.2 −0.512958
\(754\) 24587.6 1.18757
\(755\) 40500.1 1.95225
\(756\) −18573.4 −0.893531
\(757\) 16188.4 0.777248 0.388624 0.921396i \(-0.372950\pi\)
0.388624 + 0.921396i \(0.372950\pi\)
\(758\) 22515.4 1.07889
\(759\) −47850.2 −2.28834
\(760\) 2800.25 0.133652
\(761\) −3997.42 −0.190416 −0.0952078 0.995457i \(-0.530352\pi\)
−0.0952078 + 0.995457i \(0.530352\pi\)
\(762\) −57665.8 −2.74149
\(763\) −19559.2 −0.928034
\(764\) −8968.89 −0.424716
\(765\) 10584.6 0.500243
\(766\) −36892.7 −1.74019
\(767\) −13447.5 −0.633066
\(768\) 44336.0 2.08312
\(769\) 11416.7 0.535367 0.267683 0.963507i \(-0.413742\pi\)
0.267683 + 0.963507i \(0.413742\pi\)
\(770\) 72221.5 3.38011
\(771\) 21364.8 0.997971
\(772\) −19748.9 −0.920695
\(773\) 25632.5 1.19267 0.596337 0.802734i \(-0.296622\pi\)
0.596337 + 0.802734i \(0.296622\pi\)
\(774\) 0 0
\(775\) 420.294 0.0194805
\(776\) −10599.4 −0.490329
\(777\) −5622.69 −0.259605
\(778\) −7297.47 −0.336282
\(779\) 1922.47 0.0884208
\(780\) −40090.0 −1.84032
\(781\) 35993.9 1.64912
\(782\) 7999.40 0.365803
\(783\) −10988.9 −0.501549
\(784\) −36640.3 −1.66911
\(785\) 6897.72 0.313618
\(786\) 1756.59 0.0797145
\(787\) −1740.19 −0.0788198 −0.0394099 0.999223i \(-0.512548\pi\)
−0.0394099 + 0.999223i \(0.512548\pi\)
\(788\) 15941.0 0.720655
\(789\) 30530.9 1.37760
\(790\) −50282.6 −2.26453
\(791\) −14373.3 −0.646089
\(792\) 20585.0 0.923554
\(793\) −26344.2 −1.17971
\(794\) 13930.2 0.622626
\(795\) 63140.2 2.81680
\(796\) −9891.54 −0.440448
\(797\) −22762.5 −1.01165 −0.505827 0.862635i \(-0.668813\pi\)
−0.505827 + 0.862635i \(0.668813\pi\)
\(798\) 22085.9 0.979739
\(799\) −116.374 −0.00515269
\(800\) −5585.76 −0.246858
\(801\) 17679.4 0.779864
\(802\) 50105.1 2.20608
\(803\) 27305.9 1.20000
\(804\) −28539.2 −1.25186
\(805\) −36431.5 −1.59508
\(806\) −4207.44 −0.183872
\(807\) −18985.7 −0.828163
\(808\) 17829.4 0.776280
\(809\) 9835.50 0.427439 0.213719 0.976895i \(-0.431442\pi\)
0.213719 + 0.976895i \(0.431442\pi\)
\(810\) −6647.43 −0.288354
\(811\) −28928.9 −1.25257 −0.626283 0.779595i \(-0.715425\pi\)
−0.626283 + 0.779595i \(0.715425\pi\)
\(812\) 14961.3 0.646597
\(813\) 58365.3 2.51779
\(814\) −4906.82 −0.211283
\(815\) −7958.20 −0.342041
\(816\) −13394.7 −0.574642
\(817\) 0 0
\(818\) −34809.4 −1.48787
\(819\) 83370.4 3.55702
\(820\) −5174.06 −0.220349
\(821\) −7886.08 −0.335233 −0.167616 0.985852i \(-0.553607\pi\)
−0.167616 + 0.985852i \(0.553607\pi\)
\(822\) 63261.9 2.68432
\(823\) 8877.58 0.376006 0.188003 0.982168i \(-0.439799\pi\)
0.188003 + 0.982168i \(0.439799\pi\)
\(824\) 3698.20 0.156351
\(825\) −12036.6 −0.507953
\(826\) −19938.5 −0.839888
\(827\) 14776.4 0.621312 0.310656 0.950522i \(-0.399451\pi\)
0.310656 + 0.950522i \(0.399451\pi\)
\(828\) 23779.4 0.998057
\(829\) −40753.4 −1.70739 −0.853695 0.520774i \(-0.825643\pi\)
−0.853695 + 0.520774i \(0.825643\pi\)
\(830\) −15933.8 −0.666349
\(831\) 5852.26 0.244299
\(832\) 11907.3 0.496166
\(833\) 9888.05 0.411285
\(834\) −58883.3 −2.44480
\(835\) 32011.5 1.32671
\(836\) 7909.93 0.327238
\(837\) 1880.43 0.0776551
\(838\) −19396.6 −0.799577
\(839\) 32492.3 1.33702 0.668509 0.743704i \(-0.266933\pi\)
0.668509 + 0.743704i \(0.266933\pi\)
\(840\) 25956.4 1.06617
\(841\) −15537.2 −0.637058
\(842\) −2611.46 −0.106885
\(843\) 70189.4 2.86768
\(844\) 28549.5 1.16435
\(845\) 34868.0 1.41952
\(846\) −842.938 −0.0342563
\(847\) −51074.4 −2.07194
\(848\) −48246.3 −1.95375
\(849\) −6712.11 −0.271330
\(850\) 2012.23 0.0811988
\(851\) 2475.20 0.0997048
\(852\) −29624.3 −1.19121
\(853\) 9047.76 0.363176 0.181588 0.983375i \(-0.441876\pi\)
0.181588 + 0.983375i \(0.441876\pi\)
\(854\) −39060.2 −1.56512
\(855\) −12864.6 −0.514573
\(856\) −428.671 −0.0171164
\(857\) 5782.99 0.230505 0.115253 0.993336i \(-0.463232\pi\)
0.115253 + 0.993336i \(0.463232\pi\)
\(858\) 120495. 4.79445
\(859\) −46957.7 −1.86516 −0.932582 0.360959i \(-0.882450\pi\)
−0.932582 + 0.360959i \(0.882450\pi\)
\(860\) 0 0
\(861\) 17820.1 0.705349
\(862\) 46157.1 1.82380
\(863\) −35843.8 −1.41383 −0.706915 0.707298i \(-0.749914\pi\)
−0.706915 + 0.707298i \(0.749914\pi\)
\(864\) −24991.2 −0.984047
\(865\) 24204.6 0.951424
\(866\) −50217.5 −1.97051
\(867\) −36943.1 −1.44712
\(868\) −2560.18 −0.100113
\(869\) 62023.1 2.42116
\(870\) −35168.1 −1.37047
\(871\) 44047.2 1.71353
\(872\) −6134.58 −0.238237
\(873\) 48694.4 1.88781
\(874\) −9722.55 −0.376282
\(875\) 34716.2 1.34128
\(876\) −22473.7 −0.866800
\(877\) −10540.2 −0.405837 −0.202918 0.979196i \(-0.565043\pi\)
−0.202918 + 0.979196i \(0.565043\pi\)
\(878\) −62025.7 −2.38413
\(879\) −39307.2 −1.50830
\(880\) 53236.2 2.03931
\(881\) 9958.34 0.380823 0.190411 0.981704i \(-0.439018\pi\)
0.190411 + 0.981704i \(0.439018\pi\)
\(882\) 71622.9 2.73432
\(883\) −4226.86 −0.161093 −0.0805465 0.996751i \(-0.525667\pi\)
−0.0805465 + 0.996751i \(0.525667\pi\)
\(884\) −8266.95 −0.314533
\(885\) 19234.2 0.730567
\(886\) −29369.6 −1.11365
\(887\) −28768.5 −1.08901 −0.544504 0.838758i \(-0.683282\pi\)
−0.544504 + 0.838758i \(0.683282\pi\)
\(888\) −1763.51 −0.0666437
\(889\) 54156.0 2.04312
\(890\) 19454.4 0.732711
\(891\) 8199.54 0.308300
\(892\) 3105.69 0.116576
\(893\) 141.442 0.00530030
\(894\) −14857.6 −0.555829
\(895\) 16315.1 0.609333
\(896\) −31228.2 −1.16435
\(897\) −60782.6 −2.26251
\(898\) 916.559 0.0340601
\(899\) −1514.72 −0.0561945
\(900\) 5981.66 0.221543
\(901\) 13020.1 0.481424
\(902\) 15551.2 0.574057
\(903\) 0 0
\(904\) −4508.08 −0.165859
\(905\) −30788.0 −1.13086
\(906\) 100186. 3.67380
\(907\) −17393.0 −0.636742 −0.318371 0.947966i \(-0.603136\pi\)
−0.318371 + 0.947966i \(0.603136\pi\)
\(908\) −16696.5 −0.610234
\(909\) −81909.7 −2.98875
\(910\) 91740.7 3.34195
\(911\) 24542.3 0.892560 0.446280 0.894893i \(-0.352749\pi\)
0.446280 + 0.894893i \(0.352749\pi\)
\(912\) 16280.0 0.591103
\(913\) 19654.2 0.712440
\(914\) −68227.7 −2.46911
\(915\) 37680.6 1.36140
\(916\) 5269.69 0.190082
\(917\) −1649.68 −0.0594080
\(918\) 9002.90 0.323682
\(919\) 47170.9 1.69317 0.846586 0.532252i \(-0.178654\pi\)
0.846586 + 0.532252i \(0.178654\pi\)
\(920\) −11426.4 −0.409476
\(921\) −66802.5 −2.39003
\(922\) 11496.3 0.410640
\(923\) 45722.0 1.63051
\(924\) 73319.7 2.61044
\(925\) 622.632 0.0221319
\(926\) 9735.58 0.345498
\(927\) −16989.9 −0.601964
\(928\) 20130.8 0.712098
\(929\) 42375.1 1.49654 0.748268 0.663397i \(-0.230886\pi\)
0.748268 + 0.663397i \(0.230886\pi\)
\(930\) 6017.98 0.212191
\(931\) −12018.0 −0.423067
\(932\) 5529.43 0.194338
\(933\) 17286.3 0.606568
\(934\) 53941.6 1.88975
\(935\) −14366.8 −0.502506
\(936\) 26148.4 0.913129
\(937\) 2978.79 0.103856 0.0519278 0.998651i \(-0.483463\pi\)
0.0519278 + 0.998651i \(0.483463\pi\)
\(938\) 65308.1 2.27333
\(939\) −47336.2 −1.64511
\(940\) −380.670 −0.0132086
\(941\) 18777.6 0.650511 0.325255 0.945626i \(-0.394550\pi\)
0.325255 + 0.945626i \(0.394550\pi\)
\(942\) 17063.1 0.590176
\(943\) −7844.67 −0.270899
\(944\) −14697.1 −0.506727
\(945\) −41001.7 −1.41141
\(946\) 0 0
\(947\) 53490.4 1.83548 0.917742 0.397177i \(-0.130010\pi\)
0.917742 + 0.397177i \(0.130010\pi\)
\(948\) −51047.2 −1.74888
\(949\) 34685.8 1.18646
\(950\) −2445.69 −0.0835249
\(951\) −78835.0 −2.68812
\(952\) 5352.46 0.182221
\(953\) 2203.86 0.0749108 0.0374554 0.999298i \(-0.488075\pi\)
0.0374554 + 0.999298i \(0.488075\pi\)
\(954\) 94309.7 3.20062
\(955\) −19799.2 −0.670877
\(956\) −13475.3 −0.455882
\(957\) 43379.5 1.46527
\(958\) 59058.5 1.99175
\(959\) −59411.5 −2.00052
\(960\) −17031.2 −0.572582
\(961\) −29531.8 −0.991299
\(962\) −6232.98 −0.208898
\(963\) 1969.35 0.0658998
\(964\) −323.011 −0.0107920
\(965\) −43596.5 −1.45432
\(966\) −90121.6 −3.00167
\(967\) −13488.8 −0.448572 −0.224286 0.974523i \(-0.572005\pi\)
−0.224286 + 0.974523i \(0.572005\pi\)
\(968\) −16019.1 −0.531893
\(969\) −4393.46 −0.145654
\(970\) 53583.3 1.77367
\(971\) 11027.9 0.364473 0.182237 0.983255i \(-0.441666\pi\)
0.182237 + 0.983255i \(0.441666\pi\)
\(972\) −24308.9 −0.802171
\(973\) 55299.4 1.82201
\(974\) −58836.4 −1.93556
\(975\) −15289.8 −0.502219
\(976\) −28792.2 −0.944279
\(977\) 4624.99 0.151450 0.0757250 0.997129i \(-0.475873\pi\)
0.0757250 + 0.997129i \(0.475873\pi\)
\(978\) −19686.4 −0.643663
\(979\) −23996.8 −0.783392
\(980\) 32344.8 1.05430
\(981\) 28182.8 0.917235
\(982\) −50575.2 −1.64350
\(983\) 34064.0 1.10526 0.552631 0.833426i \(-0.313624\pi\)
0.552631 + 0.833426i \(0.313624\pi\)
\(984\) 5589.12 0.181072
\(985\) 35190.6 1.13834
\(986\) −7252.01 −0.234230
\(987\) 1311.07 0.0422815
\(988\) 10047.7 0.323544
\(989\) 0 0
\(990\) −104064. −3.34077
\(991\) 16168.7 0.518280 0.259140 0.965840i \(-0.416561\pi\)
0.259140 + 0.965840i \(0.416561\pi\)
\(992\) −3444.80 −0.110254
\(993\) 10145.4 0.324224
\(994\) 67791.3 2.16319
\(995\) −21836.0 −0.695727
\(996\) −16176.1 −0.514617
\(997\) −25391.2 −0.806568 −0.403284 0.915075i \(-0.632131\pi\)
−0.403284 + 0.915075i \(0.632131\pi\)
\(998\) 57222.4 1.81497
\(999\) 2785.71 0.0882241
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.1.24 30
43.2 odd 14 43.4.e.a.4.3 60
43.22 odd 14 43.4.e.a.11.3 yes 60
43.42 odd 2 1849.4.a.h.1.7 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.e.a.4.3 60 43.2 odd 14
43.4.e.a.11.3 yes 60 43.22 odd 14
1849.4.a.g.1.24 30 1.1 even 1 trivial
1849.4.a.h.1.7 30 43.42 odd 2