Properties

Label 165.4.a.g.1.1
Level $165$
Weight $4$
Character 165.1
Self dual yes
Analytic conductor $9.735$
Analytic rank $0$
Dimension $3$
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] = [165,4,Mod(1,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, 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("165.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.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: \(9.73531515095\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1957.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.91150\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.38835 q^{2} -3.00000 q^{3} +11.2577 q^{4} +5.00000 q^{5} +13.1651 q^{6} -11.7304 q^{7} -14.2958 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.38835 q^{2} -3.00000 q^{3} +11.2577 q^{4} +5.00000 q^{5} +13.1651 q^{6} -11.7304 q^{7} -14.2958 q^{8} +9.00000 q^{9} -21.9418 q^{10} +11.0000 q^{11} -33.7730 q^{12} -72.8298 q^{13} +51.4772 q^{14} -15.0000 q^{15} -27.3264 q^{16} -9.89921 q^{17} -39.4952 q^{18} +0.0238576 q^{19} +56.2883 q^{20} +35.1912 q^{21} -48.2719 q^{22} +73.0456 q^{23} +42.8873 q^{24} +25.0000 q^{25} +319.603 q^{26} -27.0000 q^{27} -132.057 q^{28} +202.097 q^{29} +65.8253 q^{30} +181.642 q^{31} +234.284 q^{32} -33.0000 q^{33} +43.4413 q^{34} -58.6521 q^{35} +101.319 q^{36} +299.887 q^{37} -0.104695 q^{38} +218.489 q^{39} -71.4788 q^{40} +88.5438 q^{41} -154.432 q^{42} -146.114 q^{43} +123.834 q^{44} +45.0000 q^{45} -320.550 q^{46} +185.746 q^{47} +81.9792 q^{48} -205.397 q^{49} -109.709 q^{50} +29.6976 q^{51} -819.893 q^{52} -347.001 q^{53} +118.486 q^{54} +55.0000 q^{55} +167.695 q^{56} -0.0715727 q^{57} -886.874 q^{58} +691.824 q^{59} -168.865 q^{60} +491.854 q^{61} -797.108 q^{62} -105.574 q^{63} -809.510 q^{64} -364.149 q^{65} +144.816 q^{66} +715.379 q^{67} -111.442 q^{68} -219.137 q^{69} +257.386 q^{70} +541.957 q^{71} -128.662 q^{72} -159.702 q^{73} -1316.01 q^{74} -75.0000 q^{75} +0.268580 q^{76} -129.035 q^{77} -958.809 q^{78} -212.257 q^{79} -136.632 q^{80} +81.0000 q^{81} -388.562 q^{82} +413.252 q^{83} +396.171 q^{84} -49.4961 q^{85} +641.202 q^{86} -606.292 q^{87} -157.253 q^{88} -1099.40 q^{89} -197.476 q^{90} +854.323 q^{91} +822.322 q^{92} -544.925 q^{93} -815.121 q^{94} +0.119288 q^{95} -702.852 q^{96} +567.999 q^{97} +901.357 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 27 q^{9} + 5 q^{10} + 33 q^{11} - 51 q^{12} - 20 q^{13} + 144 q^{14} - 45 q^{15} + 25 q^{16} + 32 q^{17} + 9 q^{18} + 116 q^{19} + 85 q^{20} - 18 q^{21} + 11 q^{22} + 240 q^{23} + 9 q^{24} + 75 q^{25} + 302 q^{26} - 81 q^{27} + 160 q^{28} + 238 q^{29} - 15 q^{30} + 92 q^{31} + 197 q^{32} - 99 q^{33} + 354 q^{34} + 30 q^{35} + 153 q^{36} - 90 q^{37} + 324 q^{38} + 60 q^{39} - 15 q^{40} - 46 q^{41} - 432 q^{42} - 134 q^{43} + 187 q^{44} + 135 q^{45} - 240 q^{46} - 220 q^{47} - 75 q^{48} - 457 q^{49} + 25 q^{50} - 96 q^{51} - 1530 q^{52} - 798 q^{53} - 27 q^{54} + 165 q^{55} + 688 q^{56} - 348 q^{57} - 978 q^{58} + 1236 q^{59} - 255 q^{60} + 342 q^{61} - 1792 q^{62} + 54 q^{63} - 1919 q^{64} - 100 q^{65} - 33 q^{66} + 764 q^{67} + 1074 q^{68} - 720 q^{69} + 720 q^{70} + 1816 q^{71} - 27 q^{72} + 100 q^{73} - 1874 q^{74} - 225 q^{75} + 396 q^{76} + 66 q^{77} - 906 q^{78} - 96 q^{79} + 125 q^{80} + 243 q^{81} - 910 q^{82} + 858 q^{83} - 480 q^{84} + 160 q^{85} + 188 q^{86} - 714 q^{87} - 33 q^{88} + 838 q^{89} + 45 q^{90} + 332 q^{91} - 688 q^{92} - 276 q^{93} - 3112 q^{94} + 580 q^{95} - 591 q^{96} - 1322 q^{97} + 1017 q^{98} + 297 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.38835 −1.55152 −0.775759 0.631029i \(-0.782633\pi\)
−0.775759 + 0.631029i \(0.782633\pi\)
\(3\) −3.00000 −0.577350
\(4\) 11.2577 1.40721
\(5\) 5.00000 0.447214
\(6\) 13.1651 0.895769
\(7\) −11.7304 −0.633383 −0.316691 0.948529i \(-0.602572\pi\)
−0.316691 + 0.948529i \(0.602572\pi\)
\(8\) −14.2958 −0.631789
\(9\) 9.00000 0.333333
\(10\) −21.9418 −0.693860
\(11\) 11.0000 0.301511
\(12\) −33.7730 −0.812452
\(13\) −72.8298 −1.55380 −0.776898 0.629627i \(-0.783208\pi\)
−0.776898 + 0.629627i \(0.783208\pi\)
\(14\) 51.4772 0.982705
\(15\) −15.0000 −0.258199
\(16\) −27.3264 −0.426975
\(17\) −9.89921 −0.141230 −0.0706150 0.997504i \(-0.522496\pi\)
−0.0706150 + 0.997504i \(0.522496\pi\)
\(18\) −39.4952 −0.517173
\(19\) 0.0238576 0.000288069 0 0.000144034 1.00000i \(-0.499954\pi\)
0.000144034 1.00000i \(0.499954\pi\)
\(20\) 56.2883 0.629322
\(21\) 35.1912 0.365684
\(22\) −48.2719 −0.467800
\(23\) 73.0456 0.662220 0.331110 0.943592i \(-0.392577\pi\)
0.331110 + 0.943592i \(0.392577\pi\)
\(24\) 42.8873 0.364764
\(25\) 25.0000 0.200000
\(26\) 319.603 2.41074
\(27\) −27.0000 −0.192450
\(28\) −132.057 −0.891301
\(29\) 202.097 1.29409 0.647043 0.762453i \(-0.276005\pi\)
0.647043 + 0.762453i \(0.276005\pi\)
\(30\) 65.8253 0.400600
\(31\) 181.642 1.05238 0.526191 0.850367i \(-0.323620\pi\)
0.526191 + 0.850367i \(0.323620\pi\)
\(32\) 234.284 1.29425
\(33\) −33.0000 −0.174078
\(34\) 43.4413 0.219121
\(35\) −58.6521 −0.283257
\(36\) 101.319 0.469069
\(37\) 299.887 1.33246 0.666231 0.745746i \(-0.267907\pi\)
0.666231 + 0.745746i \(0.267907\pi\)
\(38\) −0.104695 −0.000446943 0
\(39\) 218.489 0.897085
\(40\) −71.4788 −0.282545
\(41\) 88.5438 0.337274 0.168637 0.985678i \(-0.446063\pi\)
0.168637 + 0.985678i \(0.446063\pi\)
\(42\) −154.432 −0.567365
\(43\) −146.114 −0.518191 −0.259096 0.965852i \(-0.583424\pi\)
−0.259096 + 0.965852i \(0.583424\pi\)
\(44\) 123.834 0.424289
\(45\) 45.0000 0.149071
\(46\) −320.550 −1.02745
\(47\) 185.746 0.576466 0.288233 0.957560i \(-0.406932\pi\)
0.288233 + 0.957560i \(0.406932\pi\)
\(48\) 81.9792 0.246514
\(49\) −205.397 −0.598826
\(50\) −109.709 −0.310304
\(51\) 29.6976 0.0815392
\(52\) −819.893 −2.18651
\(53\) −347.001 −0.899325 −0.449662 0.893199i \(-0.648456\pi\)
−0.449662 + 0.893199i \(0.648456\pi\)
\(54\) 118.486 0.298590
\(55\) 55.0000 0.134840
\(56\) 167.695 0.400164
\(57\) −0.0715727 −0.000166316 0
\(58\) −886.874 −2.00780
\(59\) 691.824 1.52657 0.763287 0.646060i \(-0.223584\pi\)
0.763287 + 0.646060i \(0.223584\pi\)
\(60\) −168.865 −0.363339
\(61\) 491.854 1.03238 0.516192 0.856473i \(-0.327349\pi\)
0.516192 + 0.856473i \(0.327349\pi\)
\(62\) −797.108 −1.63279
\(63\) −105.574 −0.211128
\(64\) −809.510 −1.58107
\(65\) −364.149 −0.694879
\(66\) 144.816 0.270085
\(67\) 715.379 1.30444 0.652220 0.758030i \(-0.273838\pi\)
0.652220 + 0.758030i \(0.273838\pi\)
\(68\) −111.442 −0.198740
\(69\) −219.137 −0.382333
\(70\) 257.386 0.439479
\(71\) 541.957 0.905893 0.452947 0.891538i \(-0.350373\pi\)
0.452947 + 0.891538i \(0.350373\pi\)
\(72\) −128.662 −0.210596
\(73\) −159.702 −0.256051 −0.128026 0.991771i \(-0.540864\pi\)
−0.128026 + 0.991771i \(0.540864\pi\)
\(74\) −1316.01 −2.06734
\(75\) −75.0000 −0.115470
\(76\) 0.268580 0.000405372 0
\(77\) −129.035 −0.190972
\(78\) −958.809 −1.39184
\(79\) −212.257 −0.302288 −0.151144 0.988512i \(-0.548296\pi\)
−0.151144 + 0.988512i \(0.548296\pi\)
\(80\) −136.632 −0.190949
\(81\) 81.0000 0.111111
\(82\) −388.562 −0.523286
\(83\) 413.252 0.546510 0.273255 0.961942i \(-0.411900\pi\)
0.273255 + 0.961942i \(0.411900\pi\)
\(84\) 396.171 0.514593
\(85\) −49.4961 −0.0631600
\(86\) 641.202 0.803983
\(87\) −606.292 −0.747141
\(88\) −157.253 −0.190492
\(89\) −1099.40 −1.30939 −0.654697 0.755891i \(-0.727204\pi\)
−0.654697 + 0.755891i \(0.727204\pi\)
\(90\) −197.476 −0.231287
\(91\) 854.323 0.984148
\(92\) 822.322 0.931880
\(93\) −544.925 −0.607593
\(94\) −815.121 −0.894397
\(95\) 0.119288 0.000128828 0
\(96\) −702.852 −0.747235
\(97\) 567.999 0.594553 0.297276 0.954791i \(-0.403922\pi\)
0.297276 + 0.954791i \(0.403922\pi\)
\(98\) 901.357 0.929090
\(99\) 99.0000 0.100504
\(100\) 281.441 0.281441
\(101\) 944.617 0.930623 0.465311 0.885147i \(-0.345942\pi\)
0.465311 + 0.885147i \(0.345942\pi\)
\(102\) −130.324 −0.126510
\(103\) 1832.42 1.75295 0.876475 0.481448i \(-0.159889\pi\)
0.876475 + 0.481448i \(0.159889\pi\)
\(104\) 1041.16 0.981672
\(105\) 175.956 0.163539
\(106\) 1522.76 1.39532
\(107\) −706.148 −0.637999 −0.319000 0.947755i \(-0.603347\pi\)
−0.319000 + 0.947755i \(0.603347\pi\)
\(108\) −303.957 −0.270817
\(109\) −1830.83 −1.60882 −0.804411 0.594074i \(-0.797519\pi\)
−0.804411 + 0.594074i \(0.797519\pi\)
\(110\) −241.360 −0.209207
\(111\) −899.660 −0.769297
\(112\) 320.550 0.270439
\(113\) 654.436 0.544815 0.272408 0.962182i \(-0.412180\pi\)
0.272408 + 0.962182i \(0.412180\pi\)
\(114\) 0.314086 0.000258043 0
\(115\) 365.228 0.296154
\(116\) 2275.14 1.82105
\(117\) −655.468 −0.517932
\(118\) −3035.97 −2.36851
\(119\) 116.122 0.0894527
\(120\) 214.436 0.163127
\(121\) 121.000 0.0909091
\(122\) −2158.43 −1.60176
\(123\) −265.631 −0.194725
\(124\) 2044.86 1.48092
\(125\) 125.000 0.0894427
\(126\) 463.295 0.327568
\(127\) 814.399 0.569025 0.284513 0.958672i \(-0.408168\pi\)
0.284513 + 0.958672i \(0.408168\pi\)
\(128\) 1678.15 1.15882
\(129\) 438.343 0.299178
\(130\) 1598.01 1.07812
\(131\) 2320.98 1.54798 0.773989 0.633199i \(-0.218259\pi\)
0.773989 + 0.633199i \(0.218259\pi\)
\(132\) −371.503 −0.244963
\(133\) −0.279859 −0.000182458 0
\(134\) −3139.34 −2.02386
\(135\) −135.000 −0.0860663
\(136\) 141.517 0.0892277
\(137\) 2014.15 1.25606 0.628030 0.778189i \(-0.283861\pi\)
0.628030 + 0.778189i \(0.283861\pi\)
\(138\) 961.650 0.593196
\(139\) 1832.51 1.11821 0.559106 0.829096i \(-0.311144\pi\)
0.559106 + 0.829096i \(0.311144\pi\)
\(140\) −660.285 −0.398602
\(141\) −557.239 −0.332823
\(142\) −2378.30 −1.40551
\(143\) −801.128 −0.468487
\(144\) −245.938 −0.142325
\(145\) 1010.49 0.578733
\(146\) 700.830 0.397268
\(147\) 616.192 0.345733
\(148\) 3376.02 1.87505
\(149\) −3583.75 −1.97042 −0.985208 0.171365i \(-0.945182\pi\)
−0.985208 + 0.171365i \(0.945182\pi\)
\(150\) 329.127 0.179154
\(151\) −2822.48 −1.52112 −0.760562 0.649265i \(-0.775077\pi\)
−0.760562 + 0.649265i \(0.775077\pi\)
\(152\) −0.341062 −0.000181999 0
\(153\) −89.0929 −0.0470767
\(154\) 566.249 0.296297
\(155\) 908.209 0.470639
\(156\) 2459.68 1.26238
\(157\) 2052.04 1.04313 0.521564 0.853212i \(-0.325349\pi\)
0.521564 + 0.853212i \(0.325349\pi\)
\(158\) 931.459 0.469005
\(159\) 1041.00 0.519225
\(160\) 1171.42 0.578806
\(161\) −856.855 −0.419439
\(162\) −355.457 −0.172391
\(163\) −742.552 −0.356817 −0.178409 0.983956i \(-0.557095\pi\)
−0.178409 + 0.983956i \(0.557095\pi\)
\(164\) 996.796 0.474614
\(165\) −165.000 −0.0778499
\(166\) −1813.50 −0.847920
\(167\) −699.537 −0.324143 −0.162071 0.986779i \(-0.551817\pi\)
−0.162071 + 0.986779i \(0.551817\pi\)
\(168\) −503.086 −0.231035
\(169\) 3107.18 1.41428
\(170\) 217.206 0.0979939
\(171\) 0.214718 9.60228e−5 0
\(172\) −1644.91 −0.729203
\(173\) −734.967 −0.322997 −0.161499 0.986873i \(-0.551633\pi\)
−0.161499 + 0.986873i \(0.551633\pi\)
\(174\) 2660.62 1.15920
\(175\) −293.260 −0.126677
\(176\) −300.590 −0.128738
\(177\) −2075.47 −0.881367
\(178\) 4824.56 2.03155
\(179\) −2687.28 −1.12211 −0.561053 0.827780i \(-0.689604\pi\)
−0.561053 + 0.827780i \(0.689604\pi\)
\(180\) 506.595 0.209774
\(181\) −3766.56 −1.54677 −0.773387 0.633934i \(-0.781439\pi\)
−0.773387 + 0.633934i \(0.781439\pi\)
\(182\) −3749.07 −1.52692
\(183\) −1475.56 −0.596047
\(184\) −1044.24 −0.418383
\(185\) 1499.43 0.595895
\(186\) 2391.33 0.942691
\(187\) −108.891 −0.0425825
\(188\) 2091.07 0.811207
\(189\) 316.721 0.121895
\(190\) −0.523477 −0.000199879 0
\(191\) −1784.15 −0.675899 −0.337950 0.941164i \(-0.609733\pi\)
−0.337950 + 0.941164i \(0.609733\pi\)
\(192\) 2428.53 0.912834
\(193\) −4463.74 −1.66480 −0.832402 0.554173i \(-0.813035\pi\)
−0.832402 + 0.554173i \(0.813035\pi\)
\(194\) −2492.58 −0.922459
\(195\) 1092.45 0.401188
\(196\) −2312.29 −0.842673
\(197\) 1166.19 0.421765 0.210883 0.977511i \(-0.432366\pi\)
0.210883 + 0.977511i \(0.432366\pi\)
\(198\) −434.447 −0.155933
\(199\) −1747.63 −0.622542 −0.311271 0.950321i \(-0.600755\pi\)
−0.311271 + 0.950321i \(0.600755\pi\)
\(200\) −357.394 −0.126358
\(201\) −2146.14 −0.753119
\(202\) −4145.31 −1.44388
\(203\) −2370.68 −0.819652
\(204\) 334.326 0.114743
\(205\) 442.719 0.150833
\(206\) −8041.31 −2.71973
\(207\) 657.410 0.220740
\(208\) 1990.18 0.663432
\(209\) 0.262433 8.68559e−5 0
\(210\) −772.158 −0.253733
\(211\) −415.081 −0.135428 −0.0677141 0.997705i \(-0.521571\pi\)
−0.0677141 + 0.997705i \(0.521571\pi\)
\(212\) −3906.42 −1.26554
\(213\) −1625.87 −0.523018
\(214\) 3098.83 0.989867
\(215\) −730.572 −0.231742
\(216\) 385.986 0.121588
\(217\) −2130.73 −0.666560
\(218\) 8034.32 2.49611
\(219\) 479.107 0.147831
\(220\) 619.171 0.189748
\(221\) 720.957 0.219443
\(222\) 3948.03 1.19358
\(223\) −72.9064 −0.0218932 −0.0109466 0.999940i \(-0.503484\pi\)
−0.0109466 + 0.999940i \(0.503484\pi\)
\(224\) −2748.25 −0.819755
\(225\) 225.000 0.0666667
\(226\) −2871.90 −0.845291
\(227\) 2362.35 0.690727 0.345363 0.938469i \(-0.387756\pi\)
0.345363 + 0.938469i \(0.387756\pi\)
\(228\) −0.805741 −0.000234042 0
\(229\) 4534.32 1.30846 0.654228 0.756297i \(-0.272993\pi\)
0.654228 + 0.756297i \(0.272993\pi\)
\(230\) −1602.75 −0.459488
\(231\) 387.104 0.110258
\(232\) −2889.13 −0.817590
\(233\) 231.076 0.0649713 0.0324856 0.999472i \(-0.489658\pi\)
0.0324856 + 0.999472i \(0.489658\pi\)
\(234\) 2876.43 0.803581
\(235\) 928.732 0.257803
\(236\) 7788.32 2.14821
\(237\) 636.771 0.174526
\(238\) −509.584 −0.138787
\(239\) 4463.18 1.20795 0.603973 0.797004i \(-0.293583\pi\)
0.603973 + 0.797004i \(0.293583\pi\)
\(240\) 409.896 0.110244
\(241\) −2097.51 −0.560633 −0.280316 0.959908i \(-0.590439\pi\)
−0.280316 + 0.959908i \(0.590439\pi\)
\(242\) −530.991 −0.141047
\(243\) −243.000 −0.0641500
\(244\) 5537.12 1.45278
\(245\) −1026.99 −0.267803
\(246\) 1165.68 0.302119
\(247\) −1.73754 −0.000447600 0
\(248\) −2596.71 −0.664883
\(249\) −1239.76 −0.315528
\(250\) −548.544 −0.138772
\(251\) 3400.42 0.855110 0.427555 0.903989i \(-0.359375\pi\)
0.427555 + 0.903989i \(0.359375\pi\)
\(252\) −1188.51 −0.297100
\(253\) 803.501 0.199667
\(254\) −3573.87 −0.882853
\(255\) 148.488 0.0364655
\(256\) −888.218 −0.216850
\(257\) 4994.11 1.21215 0.606077 0.795406i \(-0.292742\pi\)
0.606077 + 0.795406i \(0.292742\pi\)
\(258\) −1923.60 −0.464180
\(259\) −3517.80 −0.843958
\(260\) −4099.46 −0.977838
\(261\) 1818.88 0.431362
\(262\) −10185.3 −2.40172
\(263\) 6869.81 1.61069 0.805344 0.592808i \(-0.201981\pi\)
0.805344 + 0.592808i \(0.201981\pi\)
\(264\) 471.760 0.109980
\(265\) −1735.00 −0.402190
\(266\) 1.22812 0.000283086 0
\(267\) 3298.20 0.755979
\(268\) 8053.49 1.83562
\(269\) −2019.60 −0.457758 −0.228879 0.973455i \(-0.573506\pi\)
−0.228879 + 0.973455i \(0.573506\pi\)
\(270\) 592.428 0.133533
\(271\) 5191.61 1.16372 0.581859 0.813289i \(-0.302325\pi\)
0.581859 + 0.813289i \(0.302325\pi\)
\(272\) 270.510 0.0603017
\(273\) −2562.97 −0.568198
\(274\) −8838.80 −1.94880
\(275\) 275.000 0.0603023
\(276\) −2466.97 −0.538021
\(277\) −456.134 −0.0989402 −0.0494701 0.998776i \(-0.515753\pi\)
−0.0494701 + 0.998776i \(0.515753\pi\)
\(278\) −8041.70 −1.73493
\(279\) 1634.78 0.350794
\(280\) 838.476 0.178959
\(281\) 3004.92 0.637930 0.318965 0.947767i \(-0.396665\pi\)
0.318965 + 0.947767i \(0.396665\pi\)
\(282\) 2445.36 0.516380
\(283\) −409.156 −0.0859428 −0.0429714 0.999076i \(-0.513682\pi\)
−0.0429714 + 0.999076i \(0.513682\pi\)
\(284\) 6101.16 1.27478
\(285\) −0.357864 −7.43790e−5 0
\(286\) 3515.63 0.726866
\(287\) −1038.66 −0.213623
\(288\) 2108.56 0.431416
\(289\) −4815.01 −0.980054
\(290\) −4434.37 −0.897915
\(291\) −1704.00 −0.343265
\(292\) −1797.87 −0.360317
\(293\) −7410.14 −1.47749 −0.738745 0.673985i \(-0.764581\pi\)
−0.738745 + 0.673985i \(0.764581\pi\)
\(294\) −2704.07 −0.536410
\(295\) 3459.12 0.682704
\(296\) −4287.11 −0.841835
\(297\) −297.000 −0.0580259
\(298\) 15726.7 3.05713
\(299\) −5319.89 −1.02895
\(300\) −844.324 −0.162490
\(301\) 1713.98 0.328213
\(302\) 12386.0 2.36005
\(303\) −2833.85 −0.537295
\(304\) −0.651941 −0.000122998 0
\(305\) 2459.27 0.461696
\(306\) 390.971 0.0730403
\(307\) 4985.77 0.926883 0.463442 0.886127i \(-0.346614\pi\)
0.463442 + 0.886127i \(0.346614\pi\)
\(308\) −1452.63 −0.268737
\(309\) −5497.26 −1.01207
\(310\) −3985.54 −0.730205
\(311\) −6238.99 −1.13756 −0.568779 0.822491i \(-0.692584\pi\)
−0.568779 + 0.822491i \(0.692584\pi\)
\(312\) −3123.47 −0.566768
\(313\) −9821.30 −1.77359 −0.886794 0.462166i \(-0.847072\pi\)
−0.886794 + 0.462166i \(0.847072\pi\)
\(314\) −9005.10 −1.61843
\(315\) −527.869 −0.0944191
\(316\) −2389.52 −0.425382
\(317\) −1747.26 −0.309577 −0.154789 0.987948i \(-0.549470\pi\)
−0.154789 + 0.987948i \(0.549470\pi\)
\(318\) −4568.29 −0.805588
\(319\) 2223.07 0.390182
\(320\) −4047.55 −0.707078
\(321\) 2118.44 0.368349
\(322\) 3760.18 0.650766
\(323\) −0.236171 −4.06839e−5 0
\(324\) 911.870 0.156356
\(325\) −1820.74 −0.310759
\(326\) 3258.58 0.553608
\(327\) 5492.48 0.928853
\(328\) −1265.80 −0.213086
\(329\) −2178.88 −0.365124
\(330\) 724.079 0.120785
\(331\) −7209.01 −1.19711 −0.598554 0.801082i \(-0.704258\pi\)
−0.598554 + 0.801082i \(0.704258\pi\)
\(332\) 4652.25 0.769053
\(333\) 2698.98 0.444154
\(334\) 3069.82 0.502913
\(335\) 3576.90 0.583363
\(336\) −961.650 −0.156138
\(337\) 53.0763 0.00857938 0.00428969 0.999991i \(-0.498635\pi\)
0.00428969 + 0.999991i \(0.498635\pi\)
\(338\) −13635.4 −2.19428
\(339\) −1963.31 −0.314549
\(340\) −557.210 −0.0888792
\(341\) 1998.06 0.317305
\(342\) −0.942259 −0.000148981 0
\(343\) 6432.93 1.01267
\(344\) 2088.82 0.327388
\(345\) −1095.68 −0.170984
\(346\) 3225.30 0.501136
\(347\) 3326.14 0.514572 0.257286 0.966335i \(-0.417172\pi\)
0.257286 + 0.966335i \(0.417172\pi\)
\(348\) −6825.42 −1.05138
\(349\) 12042.6 1.84706 0.923530 0.383526i \(-0.125290\pi\)
0.923530 + 0.383526i \(0.125290\pi\)
\(350\) 1286.93 0.196541
\(351\) 1966.40 0.299028
\(352\) 2577.12 0.390231
\(353\) 10994.7 1.65775 0.828876 0.559432i \(-0.188981\pi\)
0.828876 + 0.559432i \(0.188981\pi\)
\(354\) 9107.91 1.36746
\(355\) 2709.78 0.405128
\(356\) −12376.7 −1.84259
\(357\) −348.366 −0.0516455
\(358\) 11792.8 1.74097
\(359\) 2385.44 0.350694 0.175347 0.984507i \(-0.443895\pi\)
0.175347 + 0.984507i \(0.443895\pi\)
\(360\) −643.309 −0.0941816
\(361\) −6859.00 −1.00000
\(362\) 16529.0 2.39985
\(363\) −363.000 −0.0524864
\(364\) 9617.68 1.38490
\(365\) −798.511 −0.114510
\(366\) 6475.29 0.924778
\(367\) 3227.78 0.459098 0.229549 0.973297i \(-0.426275\pi\)
0.229549 + 0.973297i \(0.426275\pi\)
\(368\) −1996.07 −0.282751
\(369\) 796.894 0.112425
\(370\) −6580.05 −0.924542
\(371\) 4070.46 0.569617
\(372\) −6134.58 −0.855009
\(373\) −5948.20 −0.825700 −0.412850 0.910799i \(-0.635467\pi\)
−0.412850 + 0.910799i \(0.635467\pi\)
\(374\) 477.854 0.0660675
\(375\) −375.000 −0.0516398
\(376\) −2655.39 −0.364205
\(377\) −14718.7 −2.01075
\(378\) −1389.88 −0.189122
\(379\) 10332.2 1.40034 0.700170 0.713976i \(-0.253107\pi\)
0.700170 + 0.713976i \(0.253107\pi\)
\(380\) 1.34290 0.000181288 0
\(381\) −2443.20 −0.328527
\(382\) 7829.50 1.04867
\(383\) −6686.90 −0.892127 −0.446063 0.895001i \(-0.647174\pi\)
−0.446063 + 0.895001i \(0.647174\pi\)
\(384\) −5034.44 −0.669043
\(385\) −645.173 −0.0854053
\(386\) 19588.5 2.58297
\(387\) −1315.03 −0.172730
\(388\) 6394.34 0.836659
\(389\) 6099.57 0.795014 0.397507 0.917599i \(-0.369875\pi\)
0.397507 + 0.917599i \(0.369875\pi\)
\(390\) −4794.04 −0.622451
\(391\) −723.094 −0.0935254
\(392\) 2936.31 0.378332
\(393\) −6962.94 −0.893725
\(394\) −5117.67 −0.654376
\(395\) −1061.28 −0.135187
\(396\) 1114.51 0.141430
\(397\) 2640.37 0.333795 0.166897 0.985974i \(-0.446625\pi\)
0.166897 + 0.985974i \(0.446625\pi\)
\(398\) 7669.20 0.965885
\(399\) 0.839578 0.000105342 0
\(400\) −683.160 −0.0853950
\(401\) 8787.59 1.09434 0.547171 0.837021i \(-0.315705\pi\)
0.547171 + 0.837021i \(0.315705\pi\)
\(402\) 9418.01 1.16848
\(403\) −13228.9 −1.63519
\(404\) 10634.2 1.30958
\(405\) 405.000 0.0496904
\(406\) 10403.4 1.27170
\(407\) 3298.75 0.401752
\(408\) −424.550 −0.0515156
\(409\) −5870.27 −0.709698 −0.354849 0.934924i \(-0.615468\pi\)
−0.354849 + 0.934924i \(0.615468\pi\)
\(410\) −1942.81 −0.234021
\(411\) −6042.44 −0.725187
\(412\) 20628.8 2.46676
\(413\) −8115.38 −0.966905
\(414\) −2884.95 −0.342482
\(415\) 2066.26 0.244407
\(416\) −17062.9 −2.01100
\(417\) −5497.53 −0.645600
\(418\) −1.15165 −0.000134759 0
\(419\) 14208.8 1.65667 0.828337 0.560231i \(-0.189288\pi\)
0.828337 + 0.560231i \(0.189288\pi\)
\(420\) 1980.85 0.230133
\(421\) −12301.9 −1.42413 −0.712063 0.702115i \(-0.752239\pi\)
−0.712063 + 0.702115i \(0.752239\pi\)
\(422\) 1821.52 0.210119
\(423\) 1671.72 0.192155
\(424\) 4960.64 0.568184
\(425\) −247.480 −0.0282460
\(426\) 7134.89 0.811471
\(427\) −5769.65 −0.653895
\(428\) −7949.57 −0.897797
\(429\) 2403.38 0.270481
\(430\) 3206.01 0.359552
\(431\) −5640.08 −0.630331 −0.315166 0.949037i \(-0.602060\pi\)
−0.315166 + 0.949037i \(0.602060\pi\)
\(432\) 737.813 0.0821714
\(433\) 10122.9 1.12350 0.561749 0.827308i \(-0.310129\pi\)
0.561749 + 0.827308i \(0.310129\pi\)
\(434\) 9350.41 1.03418
\(435\) −3031.46 −0.334132
\(436\) −20610.8 −2.26394
\(437\) 1.74269 0.000190765 0
\(438\) −2102.49 −0.229363
\(439\) −10753.2 −1.16907 −0.584534 0.811369i \(-0.698723\pi\)
−0.584534 + 0.811369i \(0.698723\pi\)
\(440\) −786.267 −0.0851905
\(441\) −1848.58 −0.199609
\(442\) −3163.82 −0.340469
\(443\) −10346.6 −1.10967 −0.554834 0.831961i \(-0.687218\pi\)
−0.554834 + 0.831961i \(0.687218\pi\)
\(444\) −10128.1 −1.08256
\(445\) −5497.00 −0.585579
\(446\) 319.939 0.0339676
\(447\) 10751.2 1.13762
\(448\) 9495.89 1.00143
\(449\) 9569.21 1.00579 0.502894 0.864348i \(-0.332269\pi\)
0.502894 + 0.864348i \(0.332269\pi\)
\(450\) −987.380 −0.103435
\(451\) 973.982 0.101692
\(452\) 7367.42 0.766668
\(453\) 8467.43 0.878222
\(454\) −10366.8 −1.07167
\(455\) 4271.62 0.440124
\(456\) 1.02319 0.000105077 0
\(457\) 7446.31 0.762197 0.381098 0.924535i \(-0.375546\pi\)
0.381098 + 0.924535i \(0.375546\pi\)
\(458\) −19898.2 −2.03009
\(459\) 267.279 0.0271797
\(460\) 4111.61 0.416750
\(461\) 10934.9 1.10475 0.552373 0.833597i \(-0.313722\pi\)
0.552373 + 0.833597i \(0.313722\pi\)
\(462\) −1698.75 −0.171067
\(463\) −9889.15 −0.992630 −0.496315 0.868142i \(-0.665314\pi\)
−0.496315 + 0.868142i \(0.665314\pi\)
\(464\) −5522.59 −0.552543
\(465\) −2724.63 −0.271724
\(466\) −1014.04 −0.100804
\(467\) −100.396 −0.00994808 −0.00497404 0.999988i \(-0.501583\pi\)
−0.00497404 + 0.999988i \(0.501583\pi\)
\(468\) −7379.04 −0.728838
\(469\) −8391.69 −0.826210
\(470\) −4075.61 −0.399987
\(471\) −6156.13 −0.602250
\(472\) −9890.15 −0.964473
\(473\) −1607.26 −0.156241
\(474\) −2794.38 −0.270780
\(475\) 0.596439 5.76137e−5 0
\(476\) 1307.26 0.125878
\(477\) −3123.01 −0.299775
\(478\) −19586.0 −1.87415
\(479\) −4413.85 −0.421032 −0.210516 0.977590i \(-0.567514\pi\)
−0.210516 + 0.977590i \(0.567514\pi\)
\(480\) −3514.26 −0.334174
\(481\) −21840.7 −2.07037
\(482\) 9204.62 0.869832
\(483\) 2570.56 0.242163
\(484\) 1362.18 0.127928
\(485\) 2840.00 0.265892
\(486\) 1066.37 0.0995299
\(487\) −16795.6 −1.56279 −0.781396 0.624035i \(-0.785492\pi\)
−0.781396 + 0.624035i \(0.785492\pi\)
\(488\) −7031.43 −0.652250
\(489\) 2227.66 0.206008
\(490\) 4506.78 0.415502
\(491\) 19003.5 1.74667 0.873336 0.487118i \(-0.161952\pi\)
0.873336 + 0.487118i \(0.161952\pi\)
\(492\) −2990.39 −0.274018
\(493\) −2000.60 −0.182764
\(494\) 7.62495 0.000694459 0
\(495\) 495.000 0.0449467
\(496\) −4963.61 −0.449340
\(497\) −6357.37 −0.573777
\(498\) 5440.49 0.489547
\(499\) −19969.5 −1.79150 −0.895748 0.444562i \(-0.853359\pi\)
−0.895748 + 0.444562i \(0.853359\pi\)
\(500\) 1407.21 0.125864
\(501\) 2098.61 0.187144
\(502\) −14922.2 −1.32672
\(503\) 15076.5 1.33643 0.668217 0.743967i \(-0.267058\pi\)
0.668217 + 0.743967i \(0.267058\pi\)
\(504\) 1509.26 0.133388
\(505\) 4723.09 0.416187
\(506\) −3526.05 −0.309787
\(507\) −9321.53 −0.816536
\(508\) 9168.23 0.800737
\(509\) 12404.7 1.08022 0.540108 0.841596i \(-0.318384\pi\)
0.540108 + 0.841596i \(0.318384\pi\)
\(510\) −651.619 −0.0565768
\(511\) 1873.37 0.162178
\(512\) −9527.35 −0.822370
\(513\) −0.644154 −5.54388e−5 0
\(514\) −21915.9 −1.88068
\(515\) 9162.11 0.783943
\(516\) 4934.72 0.421005
\(517\) 2043.21 0.173811
\(518\) 15437.3 1.30942
\(519\) 2204.90 0.186483
\(520\) 5205.79 0.439017
\(521\) −18075.1 −1.51993 −0.759964 0.649965i \(-0.774784\pi\)
−0.759964 + 0.649965i \(0.774784\pi\)
\(522\) −7981.87 −0.669266
\(523\) 9878.08 0.825886 0.412943 0.910757i \(-0.364501\pi\)
0.412943 + 0.910757i \(0.364501\pi\)
\(524\) 26128.8 2.17833
\(525\) 879.781 0.0731367
\(526\) −30147.2 −2.49901
\(527\) −1798.11 −0.148628
\(528\) 901.771 0.0743268
\(529\) −6831.34 −0.561465
\(530\) 7613.81 0.624005
\(531\) 6226.42 0.508858
\(532\) −3.15056 −0.000256756 0
\(533\) −6448.63 −0.524054
\(534\) −14473.7 −1.17292
\(535\) −3530.74 −0.285322
\(536\) −10226.9 −0.824131
\(537\) 8061.85 0.647848
\(538\) 8862.71 0.710220
\(539\) −2259.37 −0.180553
\(540\) −1519.78 −0.121113
\(541\) −4740.76 −0.376749 −0.188374 0.982097i \(-0.560322\pi\)
−0.188374 + 0.982097i \(0.560322\pi\)
\(542\) −22782.6 −1.80553
\(543\) 11299.7 0.893030
\(544\) −2319.23 −0.182787
\(545\) −9154.14 −0.719487
\(546\) 11247.2 0.881569
\(547\) 20441.4 1.59782 0.798912 0.601448i \(-0.205409\pi\)
0.798912 + 0.601448i \(0.205409\pi\)
\(548\) 22674.6 1.76754
\(549\) 4426.69 0.344128
\(550\) −1206.80 −0.0935600
\(551\) 4.82155 0.000372786 0
\(552\) 3132.73 0.241554
\(553\) 2489.86 0.191464
\(554\) 2001.68 0.153507
\(555\) −4498.30 −0.344040
\(556\) 20629.8 1.57356
\(557\) 3587.32 0.272890 0.136445 0.990648i \(-0.456432\pi\)
0.136445 + 0.990648i \(0.456432\pi\)
\(558\) −7173.98 −0.544263
\(559\) 10641.5 0.805164
\(560\) 1602.75 0.120944
\(561\) 326.674 0.0245850
\(562\) −13186.6 −0.989759
\(563\) 8.82275 0.000660452 0 0.000330226 1.00000i \(-0.499895\pi\)
0.000330226 1.00000i \(0.499895\pi\)
\(564\) −6273.21 −0.468351
\(565\) 3272.18 0.243649
\(566\) 1795.52 0.133342
\(567\) −950.163 −0.0703759
\(568\) −7747.68 −0.572334
\(569\) 17601.2 1.29681 0.648403 0.761297i \(-0.275437\pi\)
0.648403 + 0.761297i \(0.275437\pi\)
\(570\) 1.57043 0.000115400 0
\(571\) 10826.5 0.793477 0.396738 0.917932i \(-0.370142\pi\)
0.396738 + 0.917932i \(0.370142\pi\)
\(572\) −9018.82 −0.659258
\(573\) 5352.46 0.390231
\(574\) 4557.99 0.331440
\(575\) 1826.14 0.132444
\(576\) −7285.59 −0.527025
\(577\) −20294.7 −1.46426 −0.732131 0.681163i \(-0.761474\pi\)
−0.732131 + 0.681163i \(0.761474\pi\)
\(578\) 21130.0 1.52057
\(579\) 13391.2 0.961175
\(580\) 11375.7 0.814398
\(581\) −4847.62 −0.346150
\(582\) 7477.75 0.532582
\(583\) −3817.01 −0.271157
\(584\) 2283.07 0.161770
\(585\) −3277.34 −0.231626
\(586\) 32518.3 2.29235
\(587\) 1338.81 0.0941375 0.0470687 0.998892i \(-0.485012\pi\)
0.0470687 + 0.998892i \(0.485012\pi\)
\(588\) 6936.88 0.486517
\(589\) 4.33353 0.000303158 0
\(590\) −15179.8 −1.05923
\(591\) −3498.58 −0.243506
\(592\) −8194.82 −0.568928
\(593\) −11326.3 −0.784345 −0.392173 0.919892i \(-0.628276\pi\)
−0.392173 + 0.919892i \(0.628276\pi\)
\(594\) 1303.34 0.0900282
\(595\) 580.609 0.0400045
\(596\) −40344.6 −2.77278
\(597\) 5242.88 0.359425
\(598\) 23345.6 1.59644
\(599\) −13279.5 −0.905822 −0.452911 0.891556i \(-0.649614\pi\)
−0.452911 + 0.891556i \(0.649614\pi\)
\(600\) 1072.18 0.0729528
\(601\) 6423.07 0.435944 0.217972 0.975955i \(-0.430056\pi\)
0.217972 + 0.975955i \(0.430056\pi\)
\(602\) −7521.56 −0.509229
\(603\) 6438.41 0.434813
\(604\) −31774.5 −2.14054
\(605\) 605.000 0.0406558
\(606\) 12435.9 0.833623
\(607\) 4767.58 0.318798 0.159399 0.987214i \(-0.449044\pi\)
0.159399 + 0.987214i \(0.449044\pi\)
\(608\) 5.58945 0.000372832 0
\(609\) 7112.05 0.473226
\(610\) −10792.1 −0.716330
\(611\) −13527.9 −0.895710
\(612\) −1002.98 −0.0662467
\(613\) 26476.3 1.74448 0.872240 0.489078i \(-0.162667\pi\)
0.872240 + 0.489078i \(0.162667\pi\)
\(614\) −21879.3 −1.43808
\(615\) −1328.16 −0.0870837
\(616\) 1844.65 0.120654
\(617\) −23097.5 −1.50708 −0.753542 0.657400i \(-0.771656\pi\)
−0.753542 + 0.657400i \(0.771656\pi\)
\(618\) 24123.9 1.57024
\(619\) 8579.27 0.557076 0.278538 0.960425i \(-0.410150\pi\)
0.278538 + 0.960425i \(0.410150\pi\)
\(620\) 10224.3 0.662287
\(621\) −1972.23 −0.127444
\(622\) 27378.9 1.76494
\(623\) 12896.4 0.829348
\(624\) −5970.53 −0.383033
\(625\) 625.000 0.0400000
\(626\) 43099.4 2.75175
\(627\) −0.787300 −5.01463e−5 0
\(628\) 23101.2 1.46790
\(629\) −2968.64 −0.188184
\(630\) 2316.47 0.146493
\(631\) 17464.3 1.10181 0.550905 0.834568i \(-0.314282\pi\)
0.550905 + 0.834568i \(0.314282\pi\)
\(632\) 3034.37 0.190982
\(633\) 1245.24 0.0781895
\(634\) 7667.60 0.480314
\(635\) 4072.00 0.254476
\(636\) 11719.2 0.730658
\(637\) 14959.0 0.930454
\(638\) −9755.62 −0.605374
\(639\) 4877.61 0.301964
\(640\) 8390.73 0.518239
\(641\) 13105.0 0.807513 0.403756 0.914867i \(-0.367704\pi\)
0.403756 + 0.914867i \(0.367704\pi\)
\(642\) −9296.49 −0.571500
\(643\) 2748.22 0.168553 0.0842763 0.996442i \(-0.473142\pi\)
0.0842763 + 0.996442i \(0.473142\pi\)
\(644\) −9646.18 −0.590237
\(645\) 2191.72 0.133796
\(646\) 1.03640 6.31219e−5 0
\(647\) −2332.01 −0.141701 −0.0708506 0.997487i \(-0.522571\pi\)
−0.0708506 + 0.997487i \(0.522571\pi\)
\(648\) −1157.96 −0.0701988
\(649\) 7610.06 0.460279
\(650\) 7990.07 0.482148
\(651\) 6392.20 0.384839
\(652\) −8359.40 −0.502116
\(653\) −4241.59 −0.254190 −0.127095 0.991891i \(-0.540565\pi\)
−0.127095 + 0.991891i \(0.540565\pi\)
\(654\) −24103.0 −1.44113
\(655\) 11604.9 0.692277
\(656\) −2419.58 −0.144007
\(657\) −1437.32 −0.0853504
\(658\) 9561.71 0.566496
\(659\) 32050.8 1.89457 0.947285 0.320391i \(-0.103814\pi\)
0.947285 + 0.320391i \(0.103814\pi\)
\(660\) −1857.51 −0.109551
\(661\) −27446.2 −1.61503 −0.807513 0.589850i \(-0.799187\pi\)
−0.807513 + 0.589850i \(0.799187\pi\)
\(662\) 31635.7 1.85734
\(663\) −2162.87 −0.126695
\(664\) −5907.76 −0.345279
\(665\) −1.39930 −8.15975e−5 0
\(666\) −11844.1 −0.689113
\(667\) 14762.3 0.856970
\(668\) −7875.15 −0.456136
\(669\) 218.719 0.0126400
\(670\) −15696.7 −0.905098
\(671\) 5410.39 0.311276
\(672\) 8244.74 0.473286
\(673\) 24175.8 1.38471 0.692354 0.721558i \(-0.256574\pi\)
0.692354 + 0.721558i \(0.256574\pi\)
\(674\) −232.918 −0.0133111
\(675\) −675.000 −0.0384900
\(676\) 34979.5 1.99019
\(677\) −30852.6 −1.75150 −0.875748 0.482769i \(-0.839631\pi\)
−0.875748 + 0.482769i \(0.839631\pi\)
\(678\) 8615.69 0.488029
\(679\) −6662.87 −0.376579
\(680\) 707.584 0.0399038
\(681\) −7087.06 −0.398791
\(682\) −8768.19 −0.492304
\(683\) −15562.9 −0.871883 −0.435941 0.899975i \(-0.643585\pi\)
−0.435941 + 0.899975i \(0.643585\pi\)
\(684\) 2.41722 0.000135124 0
\(685\) 10070.7 0.561727
\(686\) −28230.0 −1.57117
\(687\) −13603.0 −0.755438
\(688\) 3992.78 0.221255
\(689\) 25272.0 1.39737
\(690\) 4808.25 0.265285
\(691\) 19228.0 1.05856 0.529281 0.848446i \(-0.322462\pi\)
0.529281 + 0.848446i \(0.322462\pi\)
\(692\) −8274.01 −0.454524
\(693\) −1161.31 −0.0636574
\(694\) −14596.3 −0.798367
\(695\) 9162.55 0.500080
\(696\) 8667.40 0.472036
\(697\) −876.514 −0.0476332
\(698\) −52847.1 −2.86575
\(699\) −693.228 −0.0375112
\(700\) −3301.42 −0.178260
\(701\) 15681.1 0.844891 0.422446 0.906388i \(-0.361172\pi\)
0.422446 + 0.906388i \(0.361172\pi\)
\(702\) −8629.28 −0.463948
\(703\) 7.15457 0.000383840 0
\(704\) −8904.61 −0.476712
\(705\) −2786.20 −0.148843
\(706\) −48248.5 −2.57203
\(707\) −11080.7 −0.589440
\(708\) −23365.0 −1.24027
\(709\) 21226.9 1.12439 0.562196 0.827004i \(-0.309957\pi\)
0.562196 + 0.827004i \(0.309957\pi\)
\(710\) −11891.5 −0.628563
\(711\) −1910.31 −0.100763
\(712\) 15716.8 0.827262
\(713\) 13268.1 0.696908
\(714\) 1528.75 0.0801290
\(715\) −4005.64 −0.209514
\(716\) −30252.5 −1.57904
\(717\) −13389.5 −0.697408
\(718\) −10468.2 −0.544107
\(719\) −17928.4 −0.929928 −0.464964 0.885330i \(-0.653933\pi\)
−0.464964 + 0.885330i \(0.653933\pi\)
\(720\) −1229.69 −0.0636497
\(721\) −21495.1 −1.11029
\(722\) 30099.7 1.55152
\(723\) 6292.53 0.323682
\(724\) −42402.6 −2.17663
\(725\) 5052.43 0.258817
\(726\) 1592.97 0.0814336
\(727\) −11876.7 −0.605890 −0.302945 0.953008i \(-0.597970\pi\)
−0.302945 + 0.953008i \(0.597970\pi\)
\(728\) −12213.2 −0.621774
\(729\) 729.000 0.0370370
\(730\) 3504.15 0.177664
\(731\) 1446.42 0.0731842
\(732\) −16611.4 −0.838762
\(733\) −25344.9 −1.27713 −0.638563 0.769569i \(-0.720471\pi\)
−0.638563 + 0.769569i \(0.720471\pi\)
\(734\) −14164.7 −0.712299
\(735\) 3080.96 0.154616
\(736\) 17113.4 0.857077
\(737\) 7869.17 0.393303
\(738\) −3497.05 −0.174429
\(739\) −12511.6 −0.622795 −0.311398 0.950280i \(-0.600797\pi\)
−0.311398 + 0.950280i \(0.600797\pi\)
\(740\) 16880.1 0.838548
\(741\) 5.21263 0.000258422 0
\(742\) −17862.6 −0.883771
\(743\) −32707.2 −1.61495 −0.807477 0.589899i \(-0.799168\pi\)
−0.807477 + 0.589899i \(0.799168\pi\)
\(744\) 7790.12 0.383871
\(745\) −17918.7 −0.881196
\(746\) 26102.8 1.28109
\(747\) 3719.27 0.182170
\(748\) −1225.86 −0.0599224
\(749\) 8283.41 0.404098
\(750\) 1645.63 0.0801200
\(751\) 20409.8 0.991694 0.495847 0.868410i \(-0.334858\pi\)
0.495847 + 0.868410i \(0.334858\pi\)
\(752\) −5075.78 −0.246137
\(753\) −10201.3 −0.493698
\(754\) 64590.9 3.11971
\(755\) −14112.4 −0.680268
\(756\) 3565.54 0.171531
\(757\) 5746.29 0.275895 0.137948 0.990440i \(-0.455949\pi\)
0.137948 + 0.990440i \(0.455949\pi\)
\(758\) −45341.3 −2.17265
\(759\) −2410.50 −0.115278
\(760\) −1.70531 −8.13923e−5 0
\(761\) −2595.93 −0.123656 −0.0618282 0.998087i \(-0.519693\pi\)
−0.0618282 + 0.998087i \(0.519693\pi\)
\(762\) 10721.6 0.509715
\(763\) 21476.4 1.01900
\(764\) −20085.4 −0.951130
\(765\) −445.465 −0.0210533
\(766\) 29344.5 1.38415
\(767\) −50385.4 −2.37198
\(768\) 2664.66 0.125199
\(769\) 12449.7 0.583807 0.291903 0.956448i \(-0.405711\pi\)
0.291903 + 0.956448i \(0.405711\pi\)
\(770\) 2831.25 0.132508
\(771\) −14982.3 −0.699838
\(772\) −50251.3 −2.34272
\(773\) 21032.1 0.978621 0.489310 0.872110i \(-0.337249\pi\)
0.489310 + 0.872110i \(0.337249\pi\)
\(774\) 5770.81 0.267994
\(775\) 4541.04 0.210476
\(776\) −8119.98 −0.375632
\(777\) 10553.4 0.487260
\(778\) −26767.1 −1.23348
\(779\) 2.11244 9.71579e−5 0
\(780\) 12298.4 0.564555
\(781\) 5961.52 0.273137
\(782\) 3173.19 0.145106
\(783\) −5456.63 −0.249047
\(784\) 5612.77 0.255684
\(785\) 10260.2 0.466501
\(786\) 30555.9 1.38663
\(787\) 31784.6 1.43964 0.719821 0.694160i \(-0.244224\pi\)
0.719821 + 0.694160i \(0.244224\pi\)
\(788\) 13128.6 0.593511
\(789\) −20609.4 −0.929931
\(790\) 4657.29 0.209746
\(791\) −7676.80 −0.345077
\(792\) −1415.28 −0.0634972
\(793\) −35821.6 −1.60411
\(794\) −11586.9 −0.517888
\(795\) 5205.01 0.232205
\(796\) −19674.2 −0.876046
\(797\) 30740.5 1.36623 0.683115 0.730311i \(-0.260625\pi\)
0.683115 + 0.730311i \(0.260625\pi\)
\(798\) −3.68436 −0.000163440 0
\(799\) −1838.74 −0.0814143
\(800\) 5857.10 0.258850
\(801\) −9894.60 −0.436465
\(802\) −38563.0 −1.69789
\(803\) −1756.72 −0.0772023
\(804\) −24160.5 −1.05979
\(805\) −4284.27 −0.187579
\(806\) 58053.2 2.53702
\(807\) 6058.79 0.264287
\(808\) −13504.0 −0.587958
\(809\) 821.592 0.0357054 0.0178527 0.999841i \(-0.494317\pi\)
0.0178527 + 0.999841i \(0.494317\pi\)
\(810\) −1777.28 −0.0770955
\(811\) 35151.8 1.52201 0.761003 0.648749i \(-0.224707\pi\)
0.761003 + 0.648749i \(0.224707\pi\)
\(812\) −26688.3 −1.15342
\(813\) −15574.8 −0.671873
\(814\) −14476.1 −0.623326
\(815\) −3712.76 −0.159573
\(816\) −811.529 −0.0348152
\(817\) −3.48593 −0.000149275 0
\(818\) 25760.8 1.10111
\(819\) 7688.91 0.328049
\(820\) 4983.98 0.212254
\(821\) 5435.02 0.231040 0.115520 0.993305i \(-0.463147\pi\)
0.115520 + 0.993305i \(0.463147\pi\)
\(822\) 26516.4 1.12514
\(823\) 36291.6 1.53711 0.768557 0.639781i \(-0.220975\pi\)
0.768557 + 0.639781i \(0.220975\pi\)
\(824\) −26195.9 −1.10749
\(825\) −825.000 −0.0348155
\(826\) 35613.2 1.50017
\(827\) −18370.7 −0.772443 −0.386222 0.922406i \(-0.626220\pi\)
−0.386222 + 0.922406i \(0.626220\pi\)
\(828\) 7400.90 0.310627
\(829\) 33861.4 1.41864 0.709322 0.704885i \(-0.249001\pi\)
0.709322 + 0.704885i \(0.249001\pi\)
\(830\) −9067.49 −0.379201
\(831\) 1368.40 0.0571231
\(832\) 58956.5 2.45667
\(833\) 2033.27 0.0845723
\(834\) 24125.1 1.00166
\(835\) −3497.69 −0.144961
\(836\) 2.95438 0.000122224 0
\(837\) −4904.33 −0.202531
\(838\) −62353.3 −2.57036
\(839\) −22131.3 −0.910676 −0.455338 0.890319i \(-0.650482\pi\)
−0.455338 + 0.890319i \(0.650482\pi\)
\(840\) −2515.43 −0.103322
\(841\) 16454.3 0.674660
\(842\) 53985.0 2.20956
\(843\) −9014.75 −0.368309
\(844\) −4672.84 −0.190576
\(845\) 15535.9 0.632486
\(846\) −7336.09 −0.298132
\(847\) −1419.38 −0.0575803
\(848\) 9482.28 0.383989
\(849\) 1227.47 0.0496191
\(850\) 1086.03 0.0438242
\(851\) 21905.4 0.882383
\(852\) −18303.5 −0.735994
\(853\) −34402.7 −1.38092 −0.690460 0.723370i \(-0.742592\pi\)
−0.690460 + 0.723370i \(0.742592\pi\)
\(854\) 25319.3 1.01453
\(855\) 1.07359 4.29427e−5 0
\(856\) 10094.9 0.403081
\(857\) −26140.0 −1.04192 −0.520960 0.853581i \(-0.674426\pi\)
−0.520960 + 0.853581i \(0.674426\pi\)
\(858\) −10546.9 −0.419656
\(859\) 38993.9 1.54884 0.774421 0.632671i \(-0.218042\pi\)
0.774421 + 0.632671i \(0.218042\pi\)
\(860\) −8224.53 −0.326109
\(861\) 3115.97 0.123335
\(862\) 24750.7 0.977970
\(863\) −43742.0 −1.72537 −0.862685 0.505741i \(-0.831219\pi\)
−0.862685 + 0.505741i \(0.831219\pi\)
\(864\) −6325.67 −0.249078
\(865\) −3674.84 −0.144449
\(866\) −44422.8 −1.74313
\(867\) 14445.0 0.565834
\(868\) −23987.1 −0.937988
\(869\) −2334.83 −0.0911433
\(870\) 13303.1 0.518411
\(871\) −52100.9 −2.02683
\(872\) 26173.1 1.01644
\(873\) 5111.99 0.198184
\(874\) −7.64754 −0.000295975 0
\(875\) −1466.30 −0.0566515
\(876\) 5393.62 0.208029
\(877\) −3327.46 −0.128119 −0.0640596 0.997946i \(-0.520405\pi\)
−0.0640596 + 0.997946i \(0.520405\pi\)
\(878\) 47188.7 1.81383
\(879\) 22230.4 0.853030
\(880\) −1502.95 −0.0575733
\(881\) −2068.21 −0.0790915 −0.0395458 0.999218i \(-0.512591\pi\)
−0.0395458 + 0.999218i \(0.512591\pi\)
\(882\) 8112.21 0.309697
\(883\) −24764.3 −0.943812 −0.471906 0.881649i \(-0.656434\pi\)
−0.471906 + 0.881649i \(0.656434\pi\)
\(884\) 8116.29 0.308801
\(885\) −10377.4 −0.394160
\(886\) 45404.7 1.72167
\(887\) −34771.6 −1.31625 −0.658125 0.752908i \(-0.728650\pi\)
−0.658125 + 0.752908i \(0.728650\pi\)
\(888\) 12861.3 0.486034
\(889\) −9553.24 −0.360411
\(890\) 24122.8 0.908536
\(891\) 891.000 0.0335013
\(892\) −820.755 −0.0308082
\(893\) 4.43146 0.000166062 0
\(894\) −47180.2 −1.76504
\(895\) −13436.4 −0.501821
\(896\) −19685.3 −0.733975
\(897\) 15959.7 0.594067
\(898\) −41993.1 −1.56050
\(899\) 36709.3 1.36187
\(900\) 2532.97 0.0938138
\(901\) 3435.03 0.127012
\(902\) −4274.18 −0.157777
\(903\) −5141.94 −0.189494
\(904\) −9355.66 −0.344209
\(905\) −18832.8 −0.691738
\(906\) −37158.1 −1.36258
\(907\) −12892.0 −0.471965 −0.235983 0.971757i \(-0.575831\pi\)
−0.235983 + 0.971757i \(0.575831\pi\)
\(908\) 26594.6 0.971995
\(909\) 8501.55 0.310208
\(910\) −18745.4 −0.682860
\(911\) −20530.9 −0.746673 −0.373337 0.927696i \(-0.621786\pi\)
−0.373337 + 0.927696i \(0.621786\pi\)
\(912\) 1.95582 7.10130e−5 0
\(913\) 4545.78 0.164779
\(914\) −32677.1 −1.18256
\(915\) −7377.81 −0.266561
\(916\) 51045.9 1.84127
\(917\) −27226.1 −0.980462
\(918\) −1172.91 −0.0421699
\(919\) −19937.9 −0.715658 −0.357829 0.933787i \(-0.616483\pi\)
−0.357829 + 0.933787i \(0.616483\pi\)
\(920\) −5221.21 −0.187107
\(921\) −14957.3 −0.535136
\(922\) −47986.1 −1.71403
\(923\) −39470.6 −1.40757
\(924\) 4357.88 0.155156
\(925\) 7497.17 0.266492
\(926\) 43397.1 1.54008
\(927\) 16491.8 0.584317
\(928\) 47348.1 1.67487
\(929\) −55075.7 −1.94508 −0.972538 0.232746i \(-0.925229\pi\)
−0.972538 + 0.232746i \(0.925229\pi\)
\(930\) 11956.6 0.421584
\(931\) −4.90028 −0.000172503 0
\(932\) 2601.38 0.0914280
\(933\) 18717.0 0.656769
\(934\) 440.571 0.0154346
\(935\) −544.457 −0.0190435
\(936\) 9370.41 0.327224
\(937\) −29820.6 −1.03970 −0.519848 0.854259i \(-0.674011\pi\)
−0.519848 + 0.854259i \(0.674011\pi\)
\(938\) 36825.7 1.28188
\(939\) 29463.9 1.02398
\(940\) 10455.3 0.362783
\(941\) 32262.4 1.11767 0.558833 0.829280i \(-0.311249\pi\)
0.558833 + 0.829280i \(0.311249\pi\)
\(942\) 27015.3 0.934401
\(943\) 6467.73 0.223349
\(944\) −18905.1 −0.651808
\(945\) 1583.61 0.0545129
\(946\) 7053.22 0.242410
\(947\) −20302.5 −0.696666 −0.348333 0.937371i \(-0.613252\pi\)
−0.348333 + 0.937371i \(0.613252\pi\)
\(948\) 7168.55 0.245594
\(949\) 11631.1 0.397851
\(950\) −2.61739 −8.93887e−5 0
\(951\) 5241.78 0.178734
\(952\) −1660.05 −0.0565153
\(953\) 7243.13 0.246199 0.123100 0.992394i \(-0.460717\pi\)
0.123100 + 0.992394i \(0.460717\pi\)
\(954\) 13704.9 0.465106
\(955\) −8920.76 −0.302271
\(956\) 50245.0 1.69983
\(957\) −6669.21 −0.225272
\(958\) 19369.6 0.653238
\(959\) −23626.8 −0.795567
\(960\) 12142.7 0.408232
\(961\) 3202.72 0.107506
\(962\) 95844.7 3.21222
\(963\) −6355.33 −0.212666
\(964\) −23613.1 −0.788927
\(965\) −22318.7 −0.744523
\(966\) −11280.5 −0.375720
\(967\) −35402.4 −1.17732 −0.588658 0.808382i \(-0.700344\pi\)
−0.588658 + 0.808382i \(0.700344\pi\)
\(968\) −1729.79 −0.0574354
\(969\) 0.708513 2.34889e−5 0
\(970\) −12462.9 −0.412536
\(971\) −4236.86 −0.140028 −0.0700140 0.997546i \(-0.522304\pi\)
−0.0700140 + 0.997546i \(0.522304\pi\)
\(972\) −2735.61 −0.0902724
\(973\) −21496.1 −0.708256
\(974\) 73704.9 2.42470
\(975\) 5462.23 0.179417
\(976\) −13440.6 −0.440802
\(977\) 31833.5 1.04242 0.521210 0.853429i \(-0.325481\pi\)
0.521210 + 0.853429i \(0.325481\pi\)
\(978\) −9775.75 −0.319626
\(979\) −12093.4 −0.394797
\(980\) −11561.5 −0.376855
\(981\) −16477.5 −0.536274
\(982\) −83394.1 −2.70999
\(983\) −36137.8 −1.17255 −0.586274 0.810113i \(-0.699406\pi\)
−0.586274 + 0.810113i \(0.699406\pi\)
\(984\) 3797.40 0.123025
\(985\) 5830.96 0.188619
\(986\) 8779.36 0.283562
\(987\) 6536.65 0.210804
\(988\) −19.5607 −0.000629866 0
\(989\) −10673.0 −0.343157
\(990\) −2172.24 −0.0697355
\(991\) 13795.5 0.442207 0.221103 0.975250i \(-0.429034\pi\)
0.221103 + 0.975250i \(0.429034\pi\)
\(992\) 42555.8 1.36204
\(993\) 21627.0 0.691151
\(994\) 27898.4 0.890225
\(995\) −8738.13 −0.278409
\(996\) −13956.8 −0.444013
\(997\) 52392.7 1.66429 0.832144 0.554560i \(-0.187113\pi\)
0.832144 + 0.554560i \(0.187113\pi\)
\(998\) 87633.2 2.77954
\(999\) −8096.94 −0.256432
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 165.4.a.g.1.1 3
3.2 odd 2 495.4.a.i.1.3 3
5.2 odd 4 825.4.c.m.199.2 6
5.3 odd 4 825.4.c.m.199.5 6
5.4 even 2 825.4.a.p.1.3 3
11.10 odd 2 1815.4.a.q.1.3 3
15.14 odd 2 2475.4.a.z.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.g.1.1 3 1.1 even 1 trivial
495.4.a.i.1.3 3 3.2 odd 2
825.4.a.p.1.3 3 5.4 even 2
825.4.c.m.199.2 6 5.2 odd 4
825.4.c.m.199.5 6 5.3 odd 4
1815.4.a.q.1.3 3 11.10 odd 2
2475.4.a.z.1.1 3 15.14 odd 2