Properties

Label 1045.4.a.d.1.1
Level $1045$
Weight $4$
Character 1045.1
Self dual yes
Analytic conductor $61.657$
Analytic rank $1$
Dimension $22$
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] = [1045,4,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1045.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: \(61.6569959560\)
Analytic rank: \(1\)
Dimension: \(22\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.35702 q^{2} -10.2482 q^{3} +20.6977 q^{4} +5.00000 q^{5} +54.9000 q^{6} -17.5818 q^{7} -68.0218 q^{8} +78.0263 q^{9} +O(q^{10})\) \(q-5.35702 q^{2} -10.2482 q^{3} +20.6977 q^{4} +5.00000 q^{5} +54.9000 q^{6} -17.5818 q^{7} -68.0218 q^{8} +78.0263 q^{9} -26.7851 q^{10} -11.0000 q^{11} -212.115 q^{12} +22.3565 q^{13} +94.1861 q^{14} -51.2412 q^{15} +198.813 q^{16} +122.147 q^{17} -417.989 q^{18} -19.0000 q^{19} +103.488 q^{20} +180.182 q^{21} +58.9273 q^{22} -136.576 q^{23} +697.104 q^{24} +25.0000 q^{25} -119.764 q^{26} -522.929 q^{27} -363.903 q^{28} -107.494 q^{29} +274.500 q^{30} +66.8030 q^{31} -520.871 q^{32} +112.731 q^{33} -654.345 q^{34} -87.9090 q^{35} +1614.96 q^{36} -152.006 q^{37} +101.783 q^{38} -229.115 q^{39} -340.109 q^{40} -228.172 q^{41} -965.241 q^{42} -282.305 q^{43} -227.675 q^{44} +390.131 q^{45} +731.640 q^{46} +133.238 q^{47} -2037.48 q^{48} -33.8802 q^{49} -133.926 q^{50} -1251.79 q^{51} +462.728 q^{52} +402.016 q^{53} +2801.34 q^{54} -55.0000 q^{55} +1195.95 q^{56} +194.716 q^{57} +575.847 q^{58} +78.0966 q^{59} -1060.57 q^{60} +665.770 q^{61} -357.865 q^{62} -1371.84 q^{63} +1199.81 q^{64} +111.783 q^{65} -603.900 q^{66} -431.056 q^{67} +2528.16 q^{68} +1399.66 q^{69} +470.931 q^{70} -388.228 q^{71} -5307.49 q^{72} +1225.28 q^{73} +814.300 q^{74} -256.206 q^{75} -393.256 q^{76} +193.400 q^{77} +1227.37 q^{78} +935.437 q^{79} +994.065 q^{80} +3252.39 q^{81} +1222.33 q^{82} -475.989 q^{83} +3729.36 q^{84} +610.736 q^{85} +1512.32 q^{86} +1101.62 q^{87} +748.240 q^{88} -1267.32 q^{89} -2089.94 q^{90} -393.068 q^{91} -2826.80 q^{92} -684.613 q^{93} -713.757 q^{94} -95.0000 q^{95} +5338.01 q^{96} +476.201 q^{97} +181.497 q^{98} -858.289 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9} - 20 q^{10} - 242 q^{11} - 196 q^{12} - q^{13} - 63 q^{14} - 105 q^{15} + 6 q^{16} + 187 q^{17} - 361 q^{18} - 418 q^{19} + 370 q^{20} - 107 q^{21} + 44 q^{22} - 361 q^{23} + 208 q^{24} + 550 q^{25} - 365 q^{26} - 1467 q^{27} - 773 q^{28} - 319 q^{29} - 45 q^{30} - 402 q^{31} - 873 q^{32} + 231 q^{33} - 717 q^{34} - 205 q^{35} + 725 q^{36} - 838 q^{37} + 76 q^{38} - 607 q^{39} - 390 q^{40} - 392 q^{41} - 1350 q^{42} - 610 q^{43} - 814 q^{44} + 1045 q^{45} - 605 q^{46} - 1866 q^{47} - 1637 q^{48} + 379 q^{49} - 100 q^{50} - 2659 q^{51} - 638 q^{52} - 1303 q^{53} + 2338 q^{54} - 1210 q^{55} + 727 q^{56} + 399 q^{57} + 44 q^{58} - 2417 q^{59} - 980 q^{60} + 918 q^{61} - 1634 q^{62} - 374 q^{63} - 1716 q^{64} - 5 q^{65} + 99 q^{66} - 2339 q^{67} + 4940 q^{68} + 127 q^{69} - 315 q^{70} - 2370 q^{71} - 3306 q^{72} + 2207 q^{73} + 2051 q^{74} - 525 q^{75} - 1406 q^{76} + 451 q^{77} + 1380 q^{78} + 586 q^{79} + 30 q^{80} + 1950 q^{81} - 1566 q^{82} - 2870 q^{83} + 3076 q^{84} + 935 q^{85} - 1246 q^{86} - 1811 q^{87} + 858 q^{88} - 1768 q^{89} - 1805 q^{90} - 2195 q^{91} - 6728 q^{92} - 2916 q^{93} + 672 q^{94} - 2090 q^{95} + 6022 q^{96} - 4022 q^{97} + 1162 q^{98} - 2299 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.35702 −1.89399 −0.946997 0.321243i \(-0.895899\pi\)
−0.946997 + 0.321243i \(0.895899\pi\)
\(3\) −10.2482 −1.97227 −0.986137 0.165935i \(-0.946936\pi\)
−0.986137 + 0.165935i \(0.946936\pi\)
\(4\) 20.6977 2.58721
\(5\) 5.00000 0.447214
\(6\) 54.9000 3.73547
\(7\) −17.5818 −0.949328 −0.474664 0.880167i \(-0.657430\pi\)
−0.474664 + 0.880167i \(0.657430\pi\)
\(8\) −68.0218 −3.00617
\(9\) 78.0263 2.88986
\(10\) −26.7851 −0.847020
\(11\) −11.0000 −0.301511
\(12\) −212.115 −5.10269
\(13\) 22.3565 0.476968 0.238484 0.971146i \(-0.423350\pi\)
0.238484 + 0.971146i \(0.423350\pi\)
\(14\) 94.1861 1.79802
\(15\) −51.2412 −0.882027
\(16\) 198.813 3.10645
\(17\) 122.147 1.74265 0.871324 0.490707i \(-0.163262\pi\)
0.871324 + 0.490707i \(0.163262\pi\)
\(18\) −417.989 −5.47338
\(19\) −19.0000 −0.229416
\(20\) 103.488 1.15704
\(21\) 180.182 1.87233
\(22\) 58.9273 0.571061
\(23\) −136.576 −1.23818 −0.619088 0.785322i \(-0.712497\pi\)
−0.619088 + 0.785322i \(0.712497\pi\)
\(24\) 697.104 5.92899
\(25\) 25.0000 0.200000
\(26\) −119.764 −0.903374
\(27\) −522.929 −3.72732
\(28\) −363.903 −2.45611
\(29\) −107.494 −0.688314 −0.344157 0.938912i \(-0.611835\pi\)
−0.344157 + 0.938912i \(0.611835\pi\)
\(30\) 274.500 1.67055
\(31\) 66.8030 0.387038 0.193519 0.981097i \(-0.438010\pi\)
0.193519 + 0.981097i \(0.438010\pi\)
\(32\) −520.871 −2.87743
\(33\) 112.731 0.594663
\(34\) −654.345 −3.30057
\(35\) −87.9090 −0.424552
\(36\) 1614.96 7.47669
\(37\) −152.006 −0.675396 −0.337698 0.941255i \(-0.609648\pi\)
−0.337698 + 0.941255i \(0.609648\pi\)
\(38\) 101.783 0.434512
\(39\) −229.115 −0.940711
\(40\) −340.109 −1.34440
\(41\) −228.172 −0.869136 −0.434568 0.900639i \(-0.643099\pi\)
−0.434568 + 0.900639i \(0.643099\pi\)
\(42\) −965.241 −3.54619
\(43\) −282.305 −1.00119 −0.500595 0.865682i \(-0.666885\pi\)
−0.500595 + 0.865682i \(0.666885\pi\)
\(44\) −227.675 −0.780074
\(45\) 390.131 1.29239
\(46\) 731.640 2.34510
\(47\) 133.238 0.413504 0.206752 0.978393i \(-0.433711\pi\)
0.206752 + 0.978393i \(0.433711\pi\)
\(48\) −2037.48 −6.12678
\(49\) −33.8802 −0.0987760
\(50\) −133.926 −0.378799
\(51\) −1251.79 −3.43698
\(52\) 462.728 1.23402
\(53\) 402.016 1.04191 0.520954 0.853585i \(-0.325576\pi\)
0.520954 + 0.853585i \(0.325576\pi\)
\(54\) 2801.34 7.05953
\(55\) −55.0000 −0.134840
\(56\) 1195.95 2.85384
\(57\) 194.716 0.452471
\(58\) 575.847 1.30366
\(59\) 78.0966 0.172327 0.0861637 0.996281i \(-0.472539\pi\)
0.0861637 + 0.996281i \(0.472539\pi\)
\(60\) −1060.57 −2.28199
\(61\) 665.770 1.39743 0.698714 0.715401i \(-0.253756\pi\)
0.698714 + 0.715401i \(0.253756\pi\)
\(62\) −357.865 −0.733047
\(63\) −1371.84 −2.74343
\(64\) 1199.81 2.34339
\(65\) 111.783 0.213307
\(66\) −603.900 −1.12629
\(67\) −431.056 −0.785998 −0.392999 0.919539i \(-0.628562\pi\)
−0.392999 + 0.919539i \(0.628562\pi\)
\(68\) 2528.16 4.50860
\(69\) 1399.66 2.44202
\(70\) 470.931 0.804100
\(71\) −388.228 −0.648932 −0.324466 0.945897i \(-0.605185\pi\)
−0.324466 + 0.945897i \(0.605185\pi\)
\(72\) −5307.49 −8.68741
\(73\) 1225.28 1.96450 0.982251 0.187570i \(-0.0600611\pi\)
0.982251 + 0.187570i \(0.0600611\pi\)
\(74\) 814.300 1.27920
\(75\) −256.206 −0.394455
\(76\) −393.256 −0.593547
\(77\) 193.400 0.286233
\(78\) 1227.37 1.78170
\(79\) 935.437 1.33221 0.666107 0.745856i \(-0.267960\pi\)
0.666107 + 0.745856i \(0.267960\pi\)
\(80\) 994.065 1.38925
\(81\) 3252.39 4.46144
\(82\) 1222.33 1.64614
\(83\) −475.989 −0.629477 −0.314738 0.949178i \(-0.601917\pi\)
−0.314738 + 0.949178i \(0.601917\pi\)
\(84\) 3729.36 4.84413
\(85\) 610.736 0.779336
\(86\) 1512.32 1.89625
\(87\) 1101.62 1.35754
\(88\) 748.240 0.906394
\(89\) −1267.32 −1.50939 −0.754693 0.656078i \(-0.772214\pi\)
−0.754693 + 0.656078i \(0.772214\pi\)
\(90\) −2089.94 −2.44777
\(91\) −393.068 −0.452799
\(92\) −2826.80 −3.20342
\(93\) −684.613 −0.763344
\(94\) −713.757 −0.783175
\(95\) −95.0000 −0.102598
\(96\) 5338.01 5.67509
\(97\) 476.201 0.498463 0.249232 0.968444i \(-0.419822\pi\)
0.249232 + 0.968444i \(0.419822\pi\)
\(98\) 181.497 0.187081
\(99\) −858.289 −0.871326
\(100\) 517.442 0.517442
\(101\) −64.5129 −0.0635571 −0.0317786 0.999495i \(-0.510117\pi\)
−0.0317786 + 0.999495i \(0.510117\pi\)
\(102\) 6705.88 6.50962
\(103\) 1946.30 1.86189 0.930945 0.365160i \(-0.118986\pi\)
0.930945 + 0.365160i \(0.118986\pi\)
\(104\) −1520.73 −1.43385
\(105\) 900.912 0.837334
\(106\) −2153.61 −1.97337
\(107\) 581.019 0.524946 0.262473 0.964939i \(-0.415462\pi\)
0.262473 + 0.964939i \(0.415462\pi\)
\(108\) −10823.4 −9.64338
\(109\) −974.937 −0.856715 −0.428358 0.903609i \(-0.640908\pi\)
−0.428358 + 0.903609i \(0.640908\pi\)
\(110\) 294.636 0.255386
\(111\) 1557.79 1.33206
\(112\) −3495.49 −2.94904
\(113\) −1475.54 −1.22838 −0.614190 0.789158i \(-0.710517\pi\)
−0.614190 + 0.789158i \(0.710517\pi\)
\(114\) −1043.10 −0.856976
\(115\) −682.879 −0.553729
\(116\) −2224.87 −1.78081
\(117\) 1744.40 1.37837
\(118\) −418.365 −0.326387
\(119\) −2147.57 −1.65435
\(120\) 3485.52 2.65152
\(121\) 121.000 0.0909091
\(122\) −3566.54 −2.64672
\(123\) 2338.36 1.71417
\(124\) 1382.67 1.00135
\(125\) 125.000 0.0894427
\(126\) 7348.99 5.19603
\(127\) 300.551 0.209997 0.104998 0.994472i \(-0.466516\pi\)
0.104998 + 0.994472i \(0.466516\pi\)
\(128\) −2260.47 −1.56093
\(129\) 2893.13 1.97462
\(130\) −598.822 −0.404001
\(131\) 2328.92 1.55327 0.776636 0.629949i \(-0.216924\pi\)
0.776636 + 0.629949i \(0.216924\pi\)
\(132\) 2333.26 1.53852
\(133\) 334.054 0.217791
\(134\) 2309.18 1.48868
\(135\) −2614.65 −1.66691
\(136\) −8308.67 −5.23870
\(137\) 1875.98 1.16990 0.584948 0.811071i \(-0.301115\pi\)
0.584948 + 0.811071i \(0.301115\pi\)
\(138\) −7498.02 −4.62517
\(139\) 3061.96 1.86843 0.934216 0.356709i \(-0.116101\pi\)
0.934216 + 0.356709i \(0.116101\pi\)
\(140\) −1819.51 −1.09841
\(141\) −1365.45 −0.815544
\(142\) 2079.75 1.22907
\(143\) −245.922 −0.143811
\(144\) 15512.6 8.97722
\(145\) −537.469 −0.307823
\(146\) −6563.88 −3.72076
\(147\) 347.212 0.194813
\(148\) −3146.18 −1.74739
\(149\) −2524.81 −1.38819 −0.694094 0.719884i \(-0.744195\pi\)
−0.694094 + 0.719884i \(0.744195\pi\)
\(150\) 1372.50 0.747095
\(151\) 599.046 0.322845 0.161423 0.986885i \(-0.448392\pi\)
0.161423 + 0.986885i \(0.448392\pi\)
\(152\) 1292.42 0.689663
\(153\) 9530.69 5.03601
\(154\) −1036.05 −0.542124
\(155\) 334.015 0.173089
\(156\) −4742.15 −2.43382
\(157\) −2324.10 −1.18142 −0.590711 0.806883i \(-0.701153\pi\)
−0.590711 + 0.806883i \(0.701153\pi\)
\(158\) −5011.16 −2.52320
\(159\) −4119.95 −2.05493
\(160\) −2604.36 −1.28683
\(161\) 2401.25 1.17543
\(162\) −17423.1 −8.44994
\(163\) −2448.46 −1.17655 −0.588276 0.808660i \(-0.700193\pi\)
−0.588276 + 0.808660i \(0.700193\pi\)
\(164\) −4722.64 −2.24864
\(165\) 563.653 0.265941
\(166\) 2549.88 1.19222
\(167\) 416.578 0.193029 0.0965144 0.995332i \(-0.469231\pi\)
0.0965144 + 0.995332i \(0.469231\pi\)
\(168\) −12256.3 −5.62856
\(169\) −1697.19 −0.772502
\(170\) −3271.72 −1.47606
\(171\) −1482.50 −0.662980
\(172\) −5843.07 −2.59029
\(173\) 1241.65 0.545668 0.272834 0.962061i \(-0.412039\pi\)
0.272834 + 0.962061i \(0.412039\pi\)
\(174\) −5901.41 −2.57118
\(175\) −439.545 −0.189866
\(176\) −2186.94 −0.936631
\(177\) −800.352 −0.339877
\(178\) 6789.05 2.85877
\(179\) −3277.94 −1.36874 −0.684370 0.729135i \(-0.739923\pi\)
−0.684370 + 0.729135i \(0.739923\pi\)
\(180\) 8074.82 3.34368
\(181\) −368.648 −0.151389 −0.0756945 0.997131i \(-0.524117\pi\)
−0.0756945 + 0.997131i \(0.524117\pi\)
\(182\) 2105.67 0.857599
\(183\) −6822.97 −2.75611
\(184\) 9290.14 3.72216
\(185\) −760.030 −0.302046
\(186\) 3667.49 1.44577
\(187\) −1343.62 −0.525428
\(188\) 2757.71 1.06982
\(189\) 9194.04 3.53845
\(190\) 508.917 0.194320
\(191\) 3566.75 1.35121 0.675605 0.737264i \(-0.263883\pi\)
0.675605 + 0.737264i \(0.263883\pi\)
\(192\) −12296.0 −4.62180
\(193\) 2635.17 0.982818 0.491409 0.870929i \(-0.336482\pi\)
0.491409 + 0.870929i \(0.336482\pi\)
\(194\) −2551.02 −0.944086
\(195\) −1145.57 −0.420699
\(196\) −701.241 −0.255554
\(197\) −3188.41 −1.15312 −0.576561 0.817054i \(-0.695606\pi\)
−0.576561 + 0.817054i \(0.695606\pi\)
\(198\) 4597.87 1.65029
\(199\) 1975.30 0.703644 0.351822 0.936067i \(-0.385562\pi\)
0.351822 + 0.936067i \(0.385562\pi\)
\(200\) −1700.55 −0.601234
\(201\) 4417.56 1.55020
\(202\) 345.597 0.120377
\(203\) 1889.94 0.653436
\(204\) −25909.2 −8.89220
\(205\) −1140.86 −0.388689
\(206\) −10426.4 −3.52641
\(207\) −10656.5 −3.57816
\(208\) 4444.77 1.48168
\(209\) 209.000 0.0691714
\(210\) −4826.21 −1.58590
\(211\) 1868.49 0.609631 0.304816 0.952411i \(-0.401405\pi\)
0.304816 + 0.952411i \(0.401405\pi\)
\(212\) 8320.80 2.69564
\(213\) 3978.65 1.27987
\(214\) −3112.53 −0.994244
\(215\) −1411.53 −0.447745
\(216\) 35570.6 11.2050
\(217\) −1174.52 −0.367426
\(218\) 5222.76 1.62261
\(219\) −12557.0 −3.87454
\(220\) −1138.37 −0.348860
\(221\) 2730.78 0.831188
\(222\) −8345.14 −2.52292
\(223\) 720.605 0.216391 0.108196 0.994130i \(-0.465493\pi\)
0.108196 + 0.994130i \(0.465493\pi\)
\(224\) 9157.86 2.73163
\(225\) 1950.66 0.577972
\(226\) 7904.49 2.32654
\(227\) 933.659 0.272992 0.136496 0.990641i \(-0.456416\pi\)
0.136496 + 0.990641i \(0.456416\pi\)
\(228\) 4030.18 1.17064
\(229\) 3861.15 1.11420 0.557100 0.830445i \(-0.311914\pi\)
0.557100 + 0.830445i \(0.311914\pi\)
\(230\) 3658.20 1.04876
\(231\) −1982.01 −0.564530
\(232\) 7311.93 2.06919
\(233\) −2597.68 −0.730386 −0.365193 0.930932i \(-0.618997\pi\)
−0.365193 + 0.930932i \(0.618997\pi\)
\(234\) −9344.77 −2.61063
\(235\) 666.188 0.184925
\(236\) 1616.42 0.445847
\(237\) −9586.57 −2.62749
\(238\) 11504.6 3.13332
\(239\) −4138.45 −1.12006 −0.560030 0.828473i \(-0.689210\pi\)
−0.560030 + 0.828473i \(0.689210\pi\)
\(240\) −10187.4 −2.73998
\(241\) −4421.41 −1.18178 −0.590888 0.806753i \(-0.701223\pi\)
−0.590888 + 0.806753i \(0.701223\pi\)
\(242\) −648.200 −0.172181
\(243\) −19212.2 −5.07186
\(244\) 13779.9 3.61544
\(245\) −169.401 −0.0441740
\(246\) −12526.7 −3.24663
\(247\) −424.774 −0.109424
\(248\) −4544.06 −1.16350
\(249\) 4878.04 1.24150
\(250\) −669.628 −0.169404
\(251\) −1157.13 −0.290986 −0.145493 0.989359i \(-0.546477\pi\)
−0.145493 + 0.989359i \(0.546477\pi\)
\(252\) −28394.0 −7.09783
\(253\) 1502.33 0.373324
\(254\) −1610.06 −0.397732
\(255\) −6258.96 −1.53706
\(256\) 2510.85 0.613000
\(257\) 2158.82 0.523982 0.261991 0.965070i \(-0.415621\pi\)
0.261991 + 0.965070i \(0.415621\pi\)
\(258\) −15498.6 −3.73992
\(259\) 2672.54 0.641172
\(260\) 2313.64 0.551869
\(261\) −8387.34 −1.98913
\(262\) −12476.1 −2.94189
\(263\) 3522.37 0.825851 0.412926 0.910765i \(-0.364507\pi\)
0.412926 + 0.910765i \(0.364507\pi\)
\(264\) −7668.14 −1.78766
\(265\) 2010.08 0.465955
\(266\) −1789.54 −0.412494
\(267\) 12987.8 2.97692
\(268\) −8921.87 −2.03354
\(269\) −3275.36 −0.742388 −0.371194 0.928555i \(-0.621051\pi\)
−0.371194 + 0.928555i \(0.621051\pi\)
\(270\) 14006.7 3.15712
\(271\) −8668.59 −1.94310 −0.971549 0.236838i \(-0.923889\pi\)
−0.971549 + 0.236838i \(0.923889\pi\)
\(272\) 24284.4 5.41346
\(273\) 4028.25 0.893043
\(274\) −10049.7 −2.21578
\(275\) −275.000 −0.0603023
\(276\) 28969.8 6.31802
\(277\) 5182.43 1.12412 0.562061 0.827096i \(-0.310009\pi\)
0.562061 + 0.827096i \(0.310009\pi\)
\(278\) −16403.0 −3.53880
\(279\) 5212.39 1.11849
\(280\) 5979.73 1.27628
\(281\) 236.777 0.0502666 0.0251333 0.999684i \(-0.491999\pi\)
0.0251333 + 0.999684i \(0.491999\pi\)
\(282\) 7314.75 1.54463
\(283\) 3240.59 0.680683 0.340341 0.940302i \(-0.389457\pi\)
0.340341 + 0.940302i \(0.389457\pi\)
\(284\) −8035.42 −1.67892
\(285\) 973.582 0.202351
\(286\) 1317.41 0.272378
\(287\) 4011.68 0.825095
\(288\) −40641.6 −8.31539
\(289\) 10006.9 2.03683
\(290\) 2879.23 0.583015
\(291\) −4880.22 −0.983105
\(292\) 25360.6 5.08258
\(293\) 745.107 0.148565 0.0742826 0.997237i \(-0.476333\pi\)
0.0742826 + 0.997237i \(0.476333\pi\)
\(294\) −1860.02 −0.368975
\(295\) 390.483 0.0770671
\(296\) 10339.7 2.03035
\(297\) 5752.22 1.12383
\(298\) 13525.4 2.62922
\(299\) −3053.36 −0.590570
\(300\) −5302.87 −1.02054
\(301\) 4963.43 0.950457
\(302\) −3209.10 −0.611467
\(303\) 661.143 0.125352
\(304\) −3777.45 −0.712669
\(305\) 3328.85 0.624949
\(306\) −51056.1 −9.53818
\(307\) 6652.85 1.23680 0.618401 0.785863i \(-0.287781\pi\)
0.618401 + 0.785863i \(0.287781\pi\)
\(308\) 4002.93 0.740546
\(309\) −19946.1 −3.67216
\(310\) −1789.33 −0.327829
\(311\) −9396.95 −1.71335 −0.856675 0.515856i \(-0.827474\pi\)
−0.856675 + 0.515856i \(0.827474\pi\)
\(312\) 15584.8 2.82794
\(313\) −3328.41 −0.601063 −0.300532 0.953772i \(-0.597164\pi\)
−0.300532 + 0.953772i \(0.597164\pi\)
\(314\) 12450.3 2.23761
\(315\) −6859.21 −1.22690
\(316\) 19361.4 3.44672
\(317\) 1827.59 0.323810 0.161905 0.986806i \(-0.448236\pi\)
0.161905 + 0.986806i \(0.448236\pi\)
\(318\) 22070.7 3.89202
\(319\) 1182.43 0.207534
\(320\) 5999.07 1.04800
\(321\) −5954.42 −1.03534
\(322\) −12863.5 −2.22627
\(323\) −2320.80 −0.399791
\(324\) 67317.0 11.5427
\(325\) 558.913 0.0953936
\(326\) 13116.4 2.22838
\(327\) 9991.38 1.68968
\(328\) 15520.7 2.61277
\(329\) −2342.56 −0.392551
\(330\) −3019.50 −0.503691
\(331\) 4218.31 0.700482 0.350241 0.936660i \(-0.386100\pi\)
0.350241 + 0.936660i \(0.386100\pi\)
\(332\) −9851.87 −1.62859
\(333\) −11860.5 −1.95180
\(334\) −2231.62 −0.365595
\(335\) −2155.28 −0.351509
\(336\) 35822.6 5.81632
\(337\) 1969.47 0.318350 0.159175 0.987250i \(-0.449117\pi\)
0.159175 + 0.987250i \(0.449117\pi\)
\(338\) 9091.87 1.46311
\(339\) 15121.7 2.42270
\(340\) 12640.8 2.01631
\(341\) −734.833 −0.116696
\(342\) 7941.78 1.25568
\(343\) 6626.23 1.04310
\(344\) 19202.9 3.00974
\(345\) 6998.30 1.09210
\(346\) −6651.53 −1.03349
\(347\) 3326.07 0.514561 0.257280 0.966337i \(-0.417174\pi\)
0.257280 + 0.966337i \(0.417174\pi\)
\(348\) 22801.0 3.51225
\(349\) −5500.77 −0.843694 −0.421847 0.906667i \(-0.638618\pi\)
−0.421847 + 0.906667i \(0.638618\pi\)
\(350\) 2354.65 0.359604
\(351\) −11690.9 −1.77781
\(352\) 5729.58 0.867579
\(353\) −4985.29 −0.751672 −0.375836 0.926686i \(-0.622644\pi\)
−0.375836 + 0.926686i \(0.622644\pi\)
\(354\) 4287.51 0.643724
\(355\) −1941.14 −0.290211
\(356\) −26230.5 −3.90510
\(357\) 22008.8 3.26282
\(358\) 17560.0 2.59238
\(359\) −4940.66 −0.726346 −0.363173 0.931722i \(-0.618307\pi\)
−0.363173 + 0.931722i \(0.618307\pi\)
\(360\) −26537.5 −3.88513
\(361\) 361.000 0.0526316
\(362\) 1974.86 0.286730
\(363\) −1240.04 −0.179298
\(364\) −8135.60 −1.17149
\(365\) 6126.42 0.878552
\(366\) 36550.8 5.22005
\(367\) 1863.31 0.265025 0.132512 0.991181i \(-0.457696\pi\)
0.132512 + 0.991181i \(0.457696\pi\)
\(368\) −27153.1 −3.84633
\(369\) −17803.5 −2.51168
\(370\) 4071.50 0.572073
\(371\) −7068.16 −0.989112
\(372\) −14169.9 −1.97493
\(373\) −11611.1 −1.61180 −0.805900 0.592051i \(-0.798318\pi\)
−0.805900 + 0.592051i \(0.798318\pi\)
\(374\) 7197.79 0.995158
\(375\) −1281.03 −0.176405
\(376\) −9063.07 −1.24306
\(377\) −2403.19 −0.328304
\(378\) −49252.7 −6.70181
\(379\) 10339.8 1.40137 0.700686 0.713470i \(-0.252877\pi\)
0.700686 + 0.713470i \(0.252877\pi\)
\(380\) −1966.28 −0.265442
\(381\) −3080.11 −0.414171
\(382\) −19107.2 −2.55918
\(383\) −5035.18 −0.671765 −0.335882 0.941904i \(-0.609034\pi\)
−0.335882 + 0.941904i \(0.609034\pi\)
\(384\) 23165.8 3.07858
\(385\) 966.999 0.128007
\(386\) −14116.7 −1.86145
\(387\) −22027.2 −2.89330
\(388\) 9856.27 1.28963
\(389\) −10846.6 −1.41374 −0.706870 0.707343i \(-0.749893\pi\)
−0.706870 + 0.707343i \(0.749893\pi\)
\(390\) 6136.87 0.796801
\(391\) −16682.3 −2.15770
\(392\) 2304.59 0.296937
\(393\) −23867.3 −3.06348
\(394\) 17080.4 2.18401
\(395\) 4677.18 0.595784
\(396\) −17764.6 −2.25431
\(397\) 3844.91 0.486072 0.243036 0.970017i \(-0.421857\pi\)
0.243036 + 0.970017i \(0.421857\pi\)
\(398\) −10581.7 −1.33270
\(399\) −3423.47 −0.429543
\(400\) 4970.33 0.621291
\(401\) −10086.3 −1.25608 −0.628039 0.778182i \(-0.716142\pi\)
−0.628039 + 0.778182i \(0.716142\pi\)
\(402\) −23665.0 −2.93608
\(403\) 1493.48 0.184605
\(404\) −1335.27 −0.164436
\(405\) 16262.0 1.99522
\(406\) −10124.4 −1.23760
\(407\) 1672.07 0.203639
\(408\) 85149.2 10.3321
\(409\) 9080.10 1.09776 0.548878 0.835903i \(-0.315055\pi\)
0.548878 + 0.835903i \(0.315055\pi\)
\(410\) 6111.63 0.736175
\(411\) −19225.5 −2.30736
\(412\) 40283.9 4.81710
\(413\) −1373.08 −0.163595
\(414\) 57087.1 6.77700
\(415\) −2379.94 −0.281510
\(416\) −11644.9 −1.37244
\(417\) −31379.7 −3.68506
\(418\) −1119.62 −0.131010
\(419\) −689.620 −0.0804060 −0.0402030 0.999192i \(-0.512800\pi\)
−0.0402030 + 0.999192i \(0.512800\pi\)
\(420\) 18646.8 2.16636
\(421\) 4657.73 0.539202 0.269601 0.962972i \(-0.413108\pi\)
0.269601 + 0.962972i \(0.413108\pi\)
\(422\) −10009.6 −1.15464
\(423\) 10396.0 1.19497
\(424\) −27345.8 −3.13215
\(425\) 3053.68 0.348530
\(426\) −21313.7 −2.42407
\(427\) −11705.4 −1.32662
\(428\) 12025.8 1.35815
\(429\) 2520.26 0.283635
\(430\) 7561.58 0.848027
\(431\) −4942.88 −0.552413 −0.276207 0.961098i \(-0.589077\pi\)
−0.276207 + 0.961098i \(0.589077\pi\)
\(432\) −103965. −11.5788
\(433\) −13852.0 −1.53738 −0.768691 0.639620i \(-0.779092\pi\)
−0.768691 + 0.639620i \(0.779092\pi\)
\(434\) 6291.92 0.695902
\(435\) 5508.11 0.607112
\(436\) −20178.9 −2.21650
\(437\) 2594.94 0.284057
\(438\) 67268.1 7.33835
\(439\) 6315.04 0.686561 0.343281 0.939233i \(-0.388462\pi\)
0.343281 + 0.939233i \(0.388462\pi\)
\(440\) 3741.20 0.405352
\(441\) −2643.54 −0.285449
\(442\) −14628.9 −1.57426
\(443\) −15593.0 −1.67234 −0.836170 0.548471i \(-0.815210\pi\)
−0.836170 + 0.548471i \(0.815210\pi\)
\(444\) 32242.7 3.44633
\(445\) −6336.59 −0.675018
\(446\) −3860.30 −0.409844
\(447\) 25874.8 2.73789
\(448\) −21094.9 −2.22464
\(449\) 14240.7 1.49680 0.748399 0.663249i \(-0.230823\pi\)
0.748399 + 0.663249i \(0.230823\pi\)
\(450\) −10449.7 −1.09468
\(451\) 2509.90 0.262054
\(452\) −30540.2 −3.17808
\(453\) −6139.16 −0.636739
\(454\) −5001.63 −0.517044
\(455\) −1965.34 −0.202498
\(456\) −13245.0 −1.36020
\(457\) −3433.25 −0.351424 −0.175712 0.984442i \(-0.556223\pi\)
−0.175712 + 0.984442i \(0.556223\pi\)
\(458\) −20684.3 −2.11029
\(459\) −63874.3 −6.49542
\(460\) −14134.0 −1.43261
\(461\) −2067.32 −0.208860 −0.104430 0.994532i \(-0.533302\pi\)
−0.104430 + 0.994532i \(0.533302\pi\)
\(462\) 10617.7 1.06922
\(463\) 754.808 0.0757644 0.0378822 0.999282i \(-0.487939\pi\)
0.0378822 + 0.999282i \(0.487939\pi\)
\(464\) −21371.2 −2.13822
\(465\) −3423.06 −0.341378
\(466\) 13915.9 1.38335
\(467\) −2552.61 −0.252935 −0.126468 0.991971i \(-0.540364\pi\)
−0.126468 + 0.991971i \(0.540364\pi\)
\(468\) 36105.0 3.56614
\(469\) 7578.74 0.746170
\(470\) −3568.79 −0.350246
\(471\) 23817.9 2.33009
\(472\) −5312.28 −0.518045
\(473\) 3105.36 0.301870
\(474\) 51355.5 4.97645
\(475\) −475.000 −0.0458831
\(476\) −44449.7 −4.28014
\(477\) 31367.8 3.01097
\(478\) 22169.8 2.12138
\(479\) −4618.53 −0.440555 −0.220278 0.975437i \(-0.570696\pi\)
−0.220278 + 0.975437i \(0.570696\pi\)
\(480\) 26690.0 2.53798
\(481\) −3398.33 −0.322142
\(482\) 23685.6 2.23828
\(483\) −24608.6 −2.31828
\(484\) 2504.42 0.235201
\(485\) 2381.01 0.222919
\(486\) 102920. 9.60606
\(487\) 10808.0 1.00567 0.502833 0.864384i \(-0.332291\pi\)
0.502833 + 0.864384i \(0.332291\pi\)
\(488\) −45286.9 −4.20091
\(489\) 25092.4 2.32048
\(490\) 907.484 0.0836652
\(491\) 382.870 0.0351908 0.0175954 0.999845i \(-0.494399\pi\)
0.0175954 + 0.999845i \(0.494399\pi\)
\(492\) 48398.8 4.43493
\(493\) −13130.1 −1.19949
\(494\) 2275.52 0.207248
\(495\) −4291.45 −0.389669
\(496\) 13281.3 1.20232
\(497\) 6825.74 0.616049
\(498\) −26131.8 −2.35139
\(499\) −8075.28 −0.724447 −0.362224 0.932091i \(-0.617982\pi\)
−0.362224 + 0.932091i \(0.617982\pi\)
\(500\) 2587.21 0.231407
\(501\) −4269.19 −0.380705
\(502\) 6198.77 0.551125
\(503\) 11916.7 1.05634 0.528171 0.849138i \(-0.322878\pi\)
0.528171 + 0.849138i \(0.322878\pi\)
\(504\) 93315.3 8.24721
\(505\) −322.564 −0.0284236
\(506\) −8048.04 −0.707073
\(507\) 17393.2 1.52358
\(508\) 6220.71 0.543306
\(509\) −16121.3 −1.40386 −0.701931 0.712245i \(-0.747679\pi\)
−0.701931 + 0.712245i \(0.747679\pi\)
\(510\) 33529.4 2.91119
\(511\) −21542.7 −1.86496
\(512\) 4633.05 0.399910
\(513\) 9935.65 0.855107
\(514\) −11564.8 −0.992419
\(515\) 9731.50 0.832662
\(516\) 59881.1 5.10876
\(517\) −1465.61 −0.124676
\(518\) −14316.9 −1.21438
\(519\) −12724.7 −1.07621
\(520\) −7603.66 −0.641236
\(521\) −4625.91 −0.388992 −0.194496 0.980903i \(-0.562307\pi\)
−0.194496 + 0.980903i \(0.562307\pi\)
\(522\) 44931.2 3.76740
\(523\) −12487.1 −1.04402 −0.522010 0.852940i \(-0.674817\pi\)
−0.522010 + 0.852940i \(0.674817\pi\)
\(524\) 48203.3 4.01865
\(525\) 4504.56 0.374467
\(526\) −18869.4 −1.56416
\(527\) 8159.80 0.674471
\(528\) 22412.3 1.84729
\(529\) 6485.96 0.533078
\(530\) −10768.0 −0.882516
\(531\) 6093.59 0.498002
\(532\) 6914.15 0.563471
\(533\) −5101.14 −0.414550
\(534\) −69575.7 −5.63827
\(535\) 2905.09 0.234763
\(536\) 29321.2 2.36284
\(537\) 33593.0 2.69953
\(538\) 17546.2 1.40608
\(539\) 372.682 0.0297821
\(540\) −54117.1 −4.31265
\(541\) −137.057 −0.0108920 −0.00544598 0.999985i \(-0.501734\pi\)
−0.00544598 + 0.999985i \(0.501734\pi\)
\(542\) 46437.9 3.68022
\(543\) 3777.99 0.298580
\(544\) −63622.9 −5.01436
\(545\) −4874.68 −0.383135
\(546\) −21579.4 −1.69142
\(547\) −10320.1 −0.806681 −0.403341 0.915050i \(-0.632151\pi\)
−0.403341 + 0.915050i \(0.632151\pi\)
\(548\) 38828.5 3.02677
\(549\) 51947.6 4.03837
\(550\) 1473.18 0.114212
\(551\) 2042.38 0.157910
\(552\) −95207.5 −7.34113
\(553\) −16446.7 −1.26471
\(554\) −27762.4 −2.12908
\(555\) 7788.97 0.595718
\(556\) 63375.5 4.83403
\(557\) −13798.5 −1.04966 −0.524830 0.851207i \(-0.675871\pi\)
−0.524830 + 0.851207i \(0.675871\pi\)
\(558\) −27922.9 −2.11841
\(559\) −6311.36 −0.477535
\(560\) −17477.5 −1.31885
\(561\) 13769.7 1.03629
\(562\) −1268.42 −0.0952046
\(563\) −8764.57 −0.656097 −0.328048 0.944661i \(-0.606391\pi\)
−0.328048 + 0.944661i \(0.606391\pi\)
\(564\) −28261.7 −2.10998
\(565\) −7377.69 −0.549348
\(566\) −17359.9 −1.28921
\(567\) −57182.9 −4.23537
\(568\) 26408.0 1.95080
\(569\) 22241.5 1.63869 0.819344 0.573302i \(-0.194338\pi\)
0.819344 + 0.573302i \(0.194338\pi\)
\(570\) −5215.50 −0.383251
\(571\) 1963.65 0.143916 0.0719581 0.997408i \(-0.477075\pi\)
0.0719581 + 0.997408i \(0.477075\pi\)
\(572\) −5090.01 −0.372070
\(573\) −36552.9 −2.66495
\(574\) −21490.7 −1.56272
\(575\) −3414.40 −0.247635
\(576\) 93617.1 6.77207
\(577\) −18689.9 −1.34848 −0.674238 0.738514i \(-0.735528\pi\)
−0.674238 + 0.738514i \(0.735528\pi\)
\(578\) −53607.3 −3.85773
\(579\) −27005.9 −1.93839
\(580\) −11124.4 −0.796404
\(581\) 8368.74 0.597580
\(582\) 26143.5 1.86200
\(583\) −4422.17 −0.314147
\(584\) −83346.1 −5.90563
\(585\) 8721.98 0.616426
\(586\) −3991.55 −0.281381
\(587\) 12745.7 0.896206 0.448103 0.893982i \(-0.352100\pi\)
0.448103 + 0.893982i \(0.352100\pi\)
\(588\) 7186.48 0.504023
\(589\) −1269.26 −0.0887926
\(590\) −2091.83 −0.145965
\(591\) 32675.6 2.27427
\(592\) −30220.8 −2.09809
\(593\) 8483.09 0.587451 0.293726 0.955890i \(-0.405105\pi\)
0.293726 + 0.955890i \(0.405105\pi\)
\(594\) −30814.8 −2.12853
\(595\) −10737.8 −0.739846
\(596\) −52257.6 −3.59154
\(597\) −20243.3 −1.38778
\(598\) 16356.9 1.11854
\(599\) 11916.3 0.812831 0.406416 0.913688i \(-0.366778\pi\)
0.406416 + 0.913688i \(0.366778\pi\)
\(600\) 17427.6 1.18580
\(601\) 15408.8 1.04582 0.522910 0.852388i \(-0.324847\pi\)
0.522910 + 0.852388i \(0.324847\pi\)
\(602\) −26589.2 −1.80016
\(603\) −33633.7 −2.27143
\(604\) 12398.9 0.835269
\(605\) 605.000 0.0406558
\(606\) −3541.76 −0.237416
\(607\) 9832.07 0.657449 0.328724 0.944426i \(-0.393381\pi\)
0.328724 + 0.944426i \(0.393381\pi\)
\(608\) 9896.55 0.660129
\(609\) −19368.5 −1.28875
\(610\) −17832.7 −1.18365
\(611\) 2978.73 0.197228
\(612\) 197263. 13.0292
\(613\) −9526.36 −0.627677 −0.313838 0.949476i \(-0.601615\pi\)
−0.313838 + 0.949476i \(0.601615\pi\)
\(614\) −35639.5 −2.34249
\(615\) 11691.8 0.766601
\(616\) −13155.4 −0.860466
\(617\) 16852.9 1.09963 0.549815 0.835286i \(-0.314698\pi\)
0.549815 + 0.835286i \(0.314698\pi\)
\(618\) 106852. 6.95504
\(619\) 26515.7 1.72174 0.860869 0.508827i \(-0.169921\pi\)
0.860869 + 0.508827i \(0.169921\pi\)
\(620\) 6913.34 0.447817
\(621\) 71419.5 4.61508
\(622\) 50339.7 3.24507
\(623\) 22281.7 1.43290
\(624\) −45551.0 −2.92228
\(625\) 625.000 0.0400000
\(626\) 17830.4 1.13841
\(627\) −2141.88 −0.136425
\(628\) −48103.5 −3.05659
\(629\) −18567.1 −1.17698
\(630\) 36745.0 2.32374
\(631\) 13284.1 0.838083 0.419041 0.907967i \(-0.362366\pi\)
0.419041 + 0.907967i \(0.362366\pi\)
\(632\) −63630.1 −4.00486
\(633\) −19148.7 −1.20236
\(634\) −9790.45 −0.613294
\(635\) 1502.75 0.0939133
\(636\) −85273.5 −5.31653
\(637\) −757.442 −0.0471130
\(638\) −6334.32 −0.393069
\(639\) −30292.0 −1.87532
\(640\) −11302.3 −0.698068
\(641\) −2618.02 −0.161319 −0.0806595 0.996742i \(-0.525703\pi\)
−0.0806595 + 0.996742i \(0.525703\pi\)
\(642\) 31897.9 1.96092
\(643\) 18996.3 1.16507 0.582535 0.812805i \(-0.302061\pi\)
0.582535 + 0.812805i \(0.302061\pi\)
\(644\) 49700.3 3.04110
\(645\) 14465.6 0.883076
\(646\) 12432.6 0.757202
\(647\) −24160.5 −1.46808 −0.734039 0.679108i \(-0.762367\pi\)
−0.734039 + 0.679108i \(0.762367\pi\)
\(648\) −221234. −13.4118
\(649\) −859.063 −0.0519587
\(650\) −2994.11 −0.180675
\(651\) 12036.7 0.724664
\(652\) −50677.4 −3.04399
\(653\) −9656.17 −0.578675 −0.289338 0.957227i \(-0.593435\pi\)
−0.289338 + 0.957227i \(0.593435\pi\)
\(654\) −53524.0 −3.20024
\(655\) 11644.6 0.694645
\(656\) −45363.7 −2.69993
\(657\) 95604.4 5.67714
\(658\) 12549.1 0.743490
\(659\) 29035.7 1.71634 0.858171 0.513364i \(-0.171601\pi\)
0.858171 + 0.513364i \(0.171601\pi\)
\(660\) 11666.3 0.688046
\(661\) 12259.2 0.721375 0.360688 0.932687i \(-0.382542\pi\)
0.360688 + 0.932687i \(0.382542\pi\)
\(662\) −22597.6 −1.32671
\(663\) −27985.7 −1.63933
\(664\) 32377.6 1.89231
\(665\) 1670.27 0.0973990
\(666\) 63536.8 3.69670
\(667\) 14681.1 0.852253
\(668\) 8622.21 0.499406
\(669\) −7384.92 −0.426783
\(670\) 11545.9 0.665756
\(671\) −7323.47 −0.421340
\(672\) −93851.8 −5.38752
\(673\) −13899.6 −0.796123 −0.398062 0.917359i \(-0.630317\pi\)
−0.398062 + 0.917359i \(0.630317\pi\)
\(674\) −10550.5 −0.602952
\(675\) −13073.2 −0.745465
\(676\) −35127.8 −1.99863
\(677\) 14551.4 0.826078 0.413039 0.910713i \(-0.364467\pi\)
0.413039 + 0.910713i \(0.364467\pi\)
\(678\) −81007.1 −4.58858
\(679\) −8372.48 −0.473205
\(680\) −41543.4 −2.34282
\(681\) −9568.35 −0.538414
\(682\) 3936.52 0.221022
\(683\) −18376.3 −1.02950 −0.514750 0.857341i \(-0.672115\pi\)
−0.514750 + 0.857341i \(0.672115\pi\)
\(684\) −30684.3 −1.71527
\(685\) 9379.90 0.523194
\(686\) −35496.9 −1.97562
\(687\) −39570.0 −2.19751
\(688\) −56126.0 −3.11015
\(689\) 8987.67 0.496956
\(690\) −37490.1 −2.06844
\(691\) 3707.73 0.204123 0.102061 0.994778i \(-0.467456\pi\)
0.102061 + 0.994778i \(0.467456\pi\)
\(692\) 25699.2 1.41176
\(693\) 15090.3 0.827175
\(694\) −17817.8 −0.974575
\(695\) 15309.8 0.835588
\(696\) −74934.3 −4.08100
\(697\) −27870.6 −1.51460
\(698\) 29467.7 1.59795
\(699\) 26621.7 1.44052
\(700\) −9097.57 −0.491223
\(701\) −11773.4 −0.634342 −0.317171 0.948368i \(-0.602733\pi\)
−0.317171 + 0.948368i \(0.602733\pi\)
\(702\) 62628.3 3.36717
\(703\) 2888.12 0.154946
\(704\) −13198.0 −0.706558
\(705\) −6827.25 −0.364722
\(706\) 26706.3 1.42366
\(707\) 1134.25 0.0603366
\(708\) −16565.5 −0.879333
\(709\) −8345.33 −0.442053 −0.221026 0.975268i \(-0.570941\pi\)
−0.221026 + 0.975268i \(0.570941\pi\)
\(710\) 10398.7 0.549658
\(711\) 72988.7 3.84991
\(712\) 86205.3 4.53747
\(713\) −9123.68 −0.479221
\(714\) −117901. −6.17976
\(715\) −1229.61 −0.0643143
\(716\) −67845.7 −3.54122
\(717\) 42411.8 2.20906
\(718\) 26467.2 1.37569
\(719\) 15608.5 0.809595 0.404798 0.914406i \(-0.367342\pi\)
0.404798 + 0.914406i \(0.367342\pi\)
\(720\) 77563.2 4.01474
\(721\) −34219.5 −1.76754
\(722\) −1933.89 −0.0996839
\(723\) 45311.7 2.33079
\(724\) −7630.16 −0.391675
\(725\) −2687.35 −0.137663
\(726\) 6642.90 0.339588
\(727\) −32543.1 −1.66019 −0.830094 0.557624i \(-0.811713\pi\)
−0.830094 + 0.557624i \(0.811713\pi\)
\(728\) 26737.2 1.36119
\(729\) 109076. 5.54165
\(730\) −32819.4 −1.66397
\(731\) −34482.8 −1.74472
\(732\) −141220. −7.13064
\(733\) −15281.4 −0.770027 −0.385014 0.922911i \(-0.625803\pi\)
−0.385014 + 0.922911i \(0.625803\pi\)
\(734\) −9981.81 −0.501955
\(735\) 1736.06 0.0871231
\(736\) 71138.4 3.56277
\(737\) 4741.62 0.236987
\(738\) 95373.5 4.75711
\(739\) −33491.3 −1.66711 −0.833556 0.552435i \(-0.813699\pi\)
−0.833556 + 0.552435i \(0.813699\pi\)
\(740\) −15730.9 −0.781457
\(741\) 4353.18 0.215814
\(742\) 37864.3 1.87337
\(743\) −11232.3 −0.554608 −0.277304 0.960782i \(-0.589441\pi\)
−0.277304 + 0.960782i \(0.589441\pi\)
\(744\) 46568.6 2.29474
\(745\) −12624.0 −0.620817
\(746\) 62201.1 3.05274
\(747\) −37139.6 −1.81910
\(748\) −27809.8 −1.35939
\(749\) −10215.4 −0.498346
\(750\) 6862.50 0.334111
\(751\) −12976.0 −0.630493 −0.315246 0.949010i \(-0.602087\pi\)
−0.315246 + 0.949010i \(0.602087\pi\)
\(752\) 26489.4 1.28453
\(753\) 11858.5 0.573903
\(754\) 12873.9 0.621805
\(755\) 2995.23 0.144381
\(756\) 190295. 9.15473
\(757\) −8251.46 −0.396175 −0.198087 0.980184i \(-0.563473\pi\)
−0.198087 + 0.980184i \(0.563473\pi\)
\(758\) −55390.5 −2.65419
\(759\) −15396.3 −0.736297
\(760\) 6462.08 0.308426
\(761\) −20356.7 −0.969684 −0.484842 0.874602i \(-0.661123\pi\)
−0.484842 + 0.874602i \(0.661123\pi\)
\(762\) 16500.2 0.784437
\(763\) 17141.1 0.813304
\(764\) 73823.5 3.49587
\(765\) 47653.4 2.25217
\(766\) 26973.6 1.27232
\(767\) 1745.97 0.0821946
\(768\) −25731.8 −1.20900
\(769\) −11065.1 −0.518878 −0.259439 0.965760i \(-0.583538\pi\)
−0.259439 + 0.965760i \(0.583538\pi\)
\(770\) −5180.24 −0.242445
\(771\) −22124.1 −1.03344
\(772\) 54542.0 2.54276
\(773\) −36895.8 −1.71675 −0.858376 0.513021i \(-0.828526\pi\)
−0.858376 + 0.513021i \(0.828526\pi\)
\(774\) 118000. 5.47989
\(775\) 1670.08 0.0774076
\(776\) −32392.1 −1.49846
\(777\) −27388.8 −1.26457
\(778\) 58105.5 2.67761
\(779\) 4335.28 0.199393
\(780\) −23710.7 −1.08844
\(781\) 4270.51 0.195660
\(782\) 89367.7 4.08668
\(783\) 56211.7 2.56557
\(784\) −6735.82 −0.306843
\(785\) −11620.5 −0.528348
\(786\) 127858. 5.80221
\(787\) −36859.2 −1.66949 −0.834745 0.550637i \(-0.814385\pi\)
−0.834745 + 0.550637i \(0.814385\pi\)
\(788\) −65992.8 −2.98337
\(789\) −36098.1 −1.62880
\(790\) −25055.8 −1.12841
\(791\) 25942.6 1.16614
\(792\) 58382.4 2.61935
\(793\) 14884.3 0.666528
\(794\) −20597.3 −0.920617
\(795\) −20599.8 −0.918991
\(796\) 40884.1 1.82048
\(797\) −35271.7 −1.56761 −0.783807 0.621005i \(-0.786725\pi\)
−0.783807 + 0.621005i \(0.786725\pi\)
\(798\) 18339.6 0.813552
\(799\) 16274.6 0.720593
\(800\) −13021.8 −0.575487
\(801\) −98884.0 −4.36192
\(802\) 54032.7 2.37900
\(803\) −13478.1 −0.592320
\(804\) 91433.4 4.01071
\(805\) 12006.2 0.525670
\(806\) −8000.62 −0.349640
\(807\) 33566.7 1.46419
\(808\) 4388.28 0.191063
\(809\) 24015.7 1.04369 0.521846 0.853040i \(-0.325244\pi\)
0.521846 + 0.853040i \(0.325244\pi\)
\(810\) −87115.7 −3.77893
\(811\) −16204.2 −0.701611 −0.350806 0.936448i \(-0.614092\pi\)
−0.350806 + 0.936448i \(0.614092\pi\)
\(812\) 39117.3 1.69058
\(813\) 88837.8 3.83232
\(814\) −8957.30 −0.385692
\(815\) −12242.3 −0.526170
\(816\) −248873. −10.6768
\(817\) 5363.80 0.229689
\(818\) −48642.3 −2.07914
\(819\) −30669.6 −1.30853
\(820\) −23613.2 −1.00562
\(821\) 13747.4 0.584395 0.292197 0.956358i \(-0.405614\pi\)
0.292197 + 0.956358i \(0.405614\pi\)
\(822\) 102991. 4.37012
\(823\) −23783.7 −1.00735 −0.503674 0.863894i \(-0.668019\pi\)
−0.503674 + 0.863894i \(0.668019\pi\)
\(824\) −132391. −5.59716
\(825\) 2818.26 0.118933
\(826\) 7355.62 0.309848
\(827\) 33359.2 1.40268 0.701339 0.712828i \(-0.252586\pi\)
0.701339 + 0.712828i \(0.252586\pi\)
\(828\) −220565. −9.25745
\(829\) 30807.7 1.29071 0.645353 0.763884i \(-0.276710\pi\)
0.645353 + 0.763884i \(0.276710\pi\)
\(830\) 12749.4 0.533179
\(831\) −53110.7 −2.21708
\(832\) 26823.7 1.11772
\(833\) −4138.36 −0.172132
\(834\) 168102. 6.97948
\(835\) 2082.89 0.0863251
\(836\) 4325.82 0.178961
\(837\) −34933.2 −1.44262
\(838\) 3694.31 0.152288
\(839\) 40537.6 1.66807 0.834036 0.551711i \(-0.186025\pi\)
0.834036 + 0.551711i \(0.186025\pi\)
\(840\) −61281.7 −2.51717
\(841\) −12834.1 −0.526224
\(842\) −24951.6 −1.02124
\(843\) −2426.54 −0.0991395
\(844\) 38673.5 1.57725
\(845\) −8485.93 −0.345473
\(846\) −55691.8 −2.26327
\(847\) −2127.40 −0.0863026
\(848\) 79926.0 3.23664
\(849\) −33210.3 −1.34249
\(850\) −16358.6 −0.660113
\(851\) 20760.4 0.836258
\(852\) 82348.9 3.31130
\(853\) −10993.1 −0.441262 −0.220631 0.975357i \(-0.570812\pi\)
−0.220631 + 0.975357i \(0.570812\pi\)
\(854\) 62706.3 2.51261
\(855\) −7412.50 −0.296494
\(856\) −39522.0 −1.57808
\(857\) −5644.19 −0.224973 −0.112486 0.993653i \(-0.535881\pi\)
−0.112486 + 0.993653i \(0.535881\pi\)
\(858\) −13501.1 −0.537203
\(859\) −46178.8 −1.83422 −0.917112 0.398629i \(-0.869486\pi\)
−0.917112 + 0.398629i \(0.869486\pi\)
\(860\) −29215.3 −1.15841
\(861\) −41112.7 −1.62731
\(862\) 26479.1 1.04627
\(863\) −15945.7 −0.628966 −0.314483 0.949263i \(-0.601831\pi\)
−0.314483 + 0.949263i \(0.601831\pi\)
\(864\) 272379. 10.7251
\(865\) 6208.23 0.244030
\(866\) 74205.7 2.91179
\(867\) −102553. −4.01718
\(868\) −24309.8 −0.950609
\(869\) −10289.8 −0.401677
\(870\) −29507.1 −1.14987
\(871\) −9636.91 −0.374896
\(872\) 66317.0 2.57543
\(873\) 37156.2 1.44049
\(874\) −13901.2 −0.538002
\(875\) −2197.73 −0.0849105
\(876\) −259901. −10.0242
\(877\) −3561.97 −0.137149 −0.0685743 0.997646i \(-0.521845\pi\)
−0.0685743 + 0.997646i \(0.521845\pi\)
\(878\) −33829.8 −1.30034
\(879\) −7636.03 −0.293011
\(880\) −10934.7 −0.418874
\(881\) −14389.6 −0.550282 −0.275141 0.961404i \(-0.588725\pi\)
−0.275141 + 0.961404i \(0.588725\pi\)
\(882\) 14161.5 0.540639
\(883\) 8926.08 0.340189 0.170094 0.985428i \(-0.445593\pi\)
0.170094 + 0.985428i \(0.445593\pi\)
\(884\) 56520.9 2.15046
\(885\) −4001.76 −0.151997
\(886\) 83532.1 3.16740
\(887\) 16462.1 0.623160 0.311580 0.950220i \(-0.399142\pi\)
0.311580 + 0.950220i \(0.399142\pi\)
\(888\) −105964. −4.00441
\(889\) −5284.23 −0.199356
\(890\) 33945.2 1.27848
\(891\) −35776.3 −1.34518
\(892\) 14914.9 0.559850
\(893\) −2531.51 −0.0948644
\(894\) −138612. −5.18554
\(895\) −16389.7 −0.612119
\(896\) 39743.1 1.48183
\(897\) 31291.5 1.16477
\(898\) −76288.0 −2.83493
\(899\) −7180.91 −0.266404
\(900\) 40374.1 1.49534
\(901\) 49105.1 1.81568
\(902\) −13445.6 −0.496329
\(903\) −50866.4 −1.87456
\(904\) 100369. 3.69272
\(905\) −1843.24 −0.0677032
\(906\) 32887.6 1.20598
\(907\) −41412.3 −1.51607 −0.758033 0.652216i \(-0.773840\pi\)
−0.758033 + 0.652216i \(0.773840\pi\)
\(908\) 19324.6 0.706287
\(909\) −5033.70 −0.183671
\(910\) 10528.4 0.383530
\(911\) 6059.30 0.220366 0.110183 0.993911i \(-0.464856\pi\)
0.110183 + 0.993911i \(0.464856\pi\)
\(912\) 38712.2 1.40558
\(913\) 5235.88 0.189794
\(914\) 18392.0 0.665594
\(915\) −34114.8 −1.23257
\(916\) 79916.9 2.88267
\(917\) −40946.6 −1.47457
\(918\) 342176. 12.3023
\(919\) −14328.2 −0.514301 −0.257151 0.966371i \(-0.582784\pi\)
−0.257151 + 0.966371i \(0.582784\pi\)
\(920\) 46450.7 1.66460
\(921\) −68179.9 −2.43931
\(922\) 11074.7 0.395580
\(923\) −8679.42 −0.309520
\(924\) −41023.0 −1.46056
\(925\) −3800.15 −0.135079
\(926\) −4043.53 −0.143497
\(927\) 151863. 5.38060
\(928\) 55990.4 1.98058
\(929\) −26144.7 −0.923336 −0.461668 0.887053i \(-0.652749\pi\)
−0.461668 + 0.887053i \(0.652749\pi\)
\(930\) 18337.4 0.646568
\(931\) 643.723 0.0226608
\(932\) −53766.1 −1.88966
\(933\) 96302.1 3.37919
\(934\) 13674.4 0.479057
\(935\) −6718.09 −0.234979
\(936\) −118657. −4.14362
\(937\) 25227.3 0.879551 0.439776 0.898108i \(-0.355058\pi\)
0.439776 + 0.898108i \(0.355058\pi\)
\(938\) −40599.5 −1.41324
\(939\) 34110.3 1.18546
\(940\) 13788.6 0.478440
\(941\) −12630.8 −0.437570 −0.218785 0.975773i \(-0.570209\pi\)
−0.218785 + 0.975773i \(0.570209\pi\)
\(942\) −127593. −4.41317
\(943\) 31162.8 1.07614
\(944\) 15526.6 0.535327
\(945\) 45970.2 1.58244
\(946\) −16635.5 −0.571740
\(947\) 4795.83 0.164565 0.0822827 0.996609i \(-0.473779\pi\)
0.0822827 + 0.996609i \(0.473779\pi\)
\(948\) −198420. −6.79787
\(949\) 27393.1 0.937005
\(950\) 2544.59 0.0869024
\(951\) −18729.6 −0.638642
\(952\) 146081. 4.97324
\(953\) −23179.9 −0.787903 −0.393951 0.919131i \(-0.628892\pi\)
−0.393951 + 0.919131i \(0.628892\pi\)
\(954\) −168038. −5.70276
\(955\) 17833.8 0.604279
\(956\) −85656.4 −2.89783
\(957\) −12117.8 −0.409315
\(958\) 24741.6 0.834409
\(959\) −32983.1 −1.11062
\(960\) −61479.9 −2.06693
\(961\) −25328.4 −0.850202
\(962\) 18204.9 0.610135
\(963\) 45334.7 1.51702
\(964\) −91513.0 −3.05751
\(965\) 13175.9 0.439530
\(966\) 131829. 4.39080
\(967\) −3000.93 −0.0997967 −0.0498983 0.998754i \(-0.515890\pi\)
−0.0498983 + 0.998754i \(0.515890\pi\)
\(968\) −8230.64 −0.273288
\(969\) 23784.1 0.788497
\(970\) −12755.1 −0.422208
\(971\) 75.1115 0.00248243 0.00124122 0.999999i \(-0.499605\pi\)
0.00124122 + 0.999999i \(0.499605\pi\)
\(972\) −397648. −13.1220
\(973\) −53834.8 −1.77375
\(974\) −57898.9 −1.90472
\(975\) −5727.87 −0.188142
\(976\) 132364. 4.34105
\(977\) 21697.9 0.710520 0.355260 0.934767i \(-0.384392\pi\)
0.355260 + 0.934767i \(0.384392\pi\)
\(978\) −134420. −4.39498
\(979\) 13940.5 0.455097
\(980\) −3506.21 −0.114287
\(981\) −76070.7 −2.47579
\(982\) −2051.04 −0.0666511
\(983\) −16460.1 −0.534075 −0.267037 0.963686i \(-0.586045\pi\)
−0.267037 + 0.963686i \(0.586045\pi\)
\(984\) −159060. −5.15309
\(985\) −15942.1 −0.515692
\(986\) 70338.0 2.27183
\(987\) 24007.1 0.774219
\(988\) −8791.84 −0.283103
\(989\) 38556.1 1.23965
\(990\) 22989.4 0.738030
\(991\) 1402.86 0.0449681 0.0224840 0.999747i \(-0.492843\pi\)
0.0224840 + 0.999747i \(0.492843\pi\)
\(992\) −34795.8 −1.11368
\(993\) −43230.2 −1.38154
\(994\) −36565.7 −1.16679
\(995\) 9876.49 0.314679
\(996\) 100964. 3.21202
\(997\) 58337.2 1.85312 0.926559 0.376149i \(-0.122752\pi\)
0.926559 + 0.376149i \(0.122752\pi\)
\(998\) 43259.5 1.37210
\(999\) 79488.4 2.51742
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 1045.4.a.d.1.1 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.4.a.d.1.1 22 1.1 even 1 trivial