Properties

Label 65.4.a.c.1.2
Level $65$
Weight $4$
Character 65.1
Self dual yes
Analytic conductor $3.835$
Analytic rank $0$
Dimension $2$
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] = [65,4,Mod(1,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("65.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 65.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: \(3.83512415037\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 65.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.56155 q^{2} +8.24621 q^{3} -1.43845 q^{4} -5.00000 q^{5} +21.1231 q^{6} +24.8769 q^{7} -24.1771 q^{8} +41.0000 q^{9} +O(q^{10})\) \(q+2.56155 q^{2} +8.24621 q^{3} -1.43845 q^{4} -5.00000 q^{5} +21.1231 q^{6} +24.8769 q^{7} -24.1771 q^{8} +41.0000 q^{9} -12.8078 q^{10} -38.8769 q^{11} -11.8617 q^{12} -13.0000 q^{13} +63.7235 q^{14} -41.2311 q^{15} -50.4233 q^{16} -79.9697 q^{17} +105.024 q^{18} +128.354 q^{19} +7.19224 q^{20} +205.140 q^{21} -99.5852 q^{22} -101.939 q^{23} -199.369 q^{24} +25.0000 q^{25} -33.3002 q^{26} +115.447 q^{27} -35.7841 q^{28} +117.723 q^{29} -105.616 q^{30} -298.756 q^{31} +64.2547 q^{32} -320.587 q^{33} -204.847 q^{34} -124.384 q^{35} -58.9763 q^{36} +254.216 q^{37} +328.786 q^{38} -107.201 q^{39} +120.885 q^{40} +219.633 q^{41} +525.477 q^{42} -104.462 q^{43} +55.9224 q^{44} -205.000 q^{45} -261.123 q^{46} +308.816 q^{47} -415.801 q^{48} +275.860 q^{49} +64.0388 q^{50} -659.447 q^{51} +18.6998 q^{52} -0.0909300 q^{53} +295.723 q^{54} +194.384 q^{55} -601.451 q^{56} +1058.44 q^{57} +301.555 q^{58} -349.710 q^{59} +59.3087 q^{60} +442.000 q^{61} -765.278 q^{62} +1019.95 q^{63} +567.978 q^{64} +65.0000 q^{65} -821.201 q^{66} +228.074 q^{67} +115.032 q^{68} -840.614 q^{69} -318.617 q^{70} +964.328 q^{71} -991.260 q^{72} +662.682 q^{73} +651.187 q^{74} +206.155 q^{75} -184.631 q^{76} -967.136 q^{77} -274.600 q^{78} -65.2614 q^{79} +252.116 q^{80} -155.000 q^{81} +562.600 q^{82} -92.4185 q^{83} -295.083 q^{84} +399.848 q^{85} -267.585 q^{86} +970.773 q^{87} +939.930 q^{88} -178.614 q^{89} -525.118 q^{90} -323.400 q^{91} +146.634 q^{92} -2463.60 q^{93} +791.049 q^{94} -641.771 q^{95} +529.858 q^{96} -1707.46 q^{97} +706.630 q^{98} -1593.95 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 7 q^{4} - 10 q^{5} + 34 q^{6} + 58 q^{7} - 3 q^{8} + 82 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 7 q^{4} - 10 q^{5} + 34 q^{6} + 58 q^{7} - 3 q^{8} + 82 q^{9} - 5 q^{10} - 86 q^{11} + 34 q^{12} - 26 q^{13} + 12 q^{14} - 39 q^{16} - 28 q^{17} + 41 q^{18} + 166 q^{19} + 35 q^{20} - 68 q^{21} - 26 q^{22} + 60 q^{23} - 374 q^{24} + 50 q^{25} - 13 q^{26} - 220 q^{28} + 120 q^{29} - 170 q^{30} - 78 q^{31} - 123 q^{32} + 68 q^{33} - 286 q^{34} - 290 q^{35} - 287 q^{36} + 360 q^{37} + 270 q^{38} + 15 q^{40} - 72 q^{41} + 952 q^{42} - 44 q^{43} + 318 q^{44} - 410 q^{45} - 514 q^{46} + 362 q^{47} - 510 q^{48} + 1030 q^{49} + 25 q^{50} - 1088 q^{51} + 91 q^{52} - 396 q^{53} + 476 q^{54} + 430 q^{55} + 100 q^{56} + 748 q^{57} + 298 q^{58} + 18 q^{59} - 170 q^{60} + 884 q^{61} - 1110 q^{62} + 2378 q^{63} + 769 q^{64} + 130 q^{65} - 1428 q^{66} + 1322 q^{67} - 174 q^{68} - 2176 q^{69} - 60 q^{70} + 634 q^{71} - 123 q^{72} - 60 q^{73} + 486 q^{74} - 394 q^{76} - 2528 q^{77} - 442 q^{78} - 180 q^{79} + 195 q^{80} - 310 q^{81} + 1018 q^{82} + 714 q^{83} + 1224 q^{84} + 140 q^{85} - 362 q^{86} + 952 q^{87} - 58 q^{88} - 852 q^{89} - 205 q^{90} - 754 q^{91} - 754 q^{92} - 4284 q^{93} + 708 q^{94} - 830 q^{95} + 2074 q^{96} + 32 q^{97} - 471 q^{98} - 3526 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56155 0.905646 0.452823 0.891601i \(-0.350417\pi\)
0.452823 + 0.891601i \(0.350417\pi\)
\(3\) 8.24621 1.58698 0.793492 0.608581i \(-0.208261\pi\)
0.793492 + 0.608581i \(0.208261\pi\)
\(4\) −1.43845 −0.179806
\(5\) −5.00000 −0.447214
\(6\) 21.1231 1.43725
\(7\) 24.8769 1.34323 0.671613 0.740902i \(-0.265602\pi\)
0.671613 + 0.740902i \(0.265602\pi\)
\(8\) −24.1771 −1.06849
\(9\) 41.0000 1.51852
\(10\) −12.8078 −0.405017
\(11\) −38.8769 −1.06562 −0.532810 0.846235i \(-0.678864\pi\)
−0.532810 + 0.846235i \(0.678864\pi\)
\(12\) −11.8617 −0.285349
\(13\) −13.0000 −0.277350
\(14\) 63.7235 1.21649
\(15\) −41.2311 −0.709721
\(16\) −50.4233 −0.787864
\(17\) −79.9697 −1.14091 −0.570456 0.821328i \(-0.693233\pi\)
−0.570456 + 0.821328i \(0.693233\pi\)
\(18\) 105.024 1.37524
\(19\) 128.354 1.54981 0.774907 0.632075i \(-0.217797\pi\)
0.774907 + 0.632075i \(0.217797\pi\)
\(20\) 7.19224 0.0804116
\(21\) 205.140 2.13168
\(22\) −99.5852 −0.965075
\(23\) −101.939 −0.924167 −0.462083 0.886837i \(-0.652898\pi\)
−0.462083 + 0.886837i \(0.652898\pi\)
\(24\) −199.369 −1.69567
\(25\) 25.0000 0.200000
\(26\) −33.3002 −0.251181
\(27\) 115.447 0.822881
\(28\) −35.7841 −0.241520
\(29\) 117.723 0.753817 0.376909 0.926250i \(-0.376987\pi\)
0.376909 + 0.926250i \(0.376987\pi\)
\(30\) −105.616 −0.642756
\(31\) −298.756 −1.73091 −0.865453 0.500990i \(-0.832969\pi\)
−0.865453 + 0.500990i \(0.832969\pi\)
\(32\) 64.2547 0.354961
\(33\) −320.587 −1.69112
\(34\) −204.847 −1.03326
\(35\) −124.384 −0.600709
\(36\) −58.9763 −0.273039
\(37\) 254.216 1.12954 0.564768 0.825250i \(-0.308966\pi\)
0.564768 + 0.825250i \(0.308966\pi\)
\(38\) 328.786 1.40358
\(39\) −107.201 −0.440150
\(40\) 120.885 0.477842
\(41\) 219.633 0.836606 0.418303 0.908308i \(-0.362625\pi\)
0.418303 + 0.908308i \(0.362625\pi\)
\(42\) 525.477 1.93055
\(43\) −104.462 −0.370473 −0.185236 0.982694i \(-0.559305\pi\)
−0.185236 + 0.982694i \(0.559305\pi\)
\(44\) 55.9224 0.191605
\(45\) −205.000 −0.679102
\(46\) −261.123 −0.836967
\(47\) 308.816 0.958415 0.479207 0.877702i \(-0.340924\pi\)
0.479207 + 0.877702i \(0.340924\pi\)
\(48\) −415.801 −1.25033
\(49\) 275.860 0.804256
\(50\) 64.0388 0.181129
\(51\) −659.447 −1.81061
\(52\) 18.6998 0.0498692
\(53\) −0.0909300 −0.000235664 0 −0.000117832 1.00000i \(-0.500038\pi\)
−0.000117832 1.00000i \(0.500038\pi\)
\(54\) 295.723 0.745238
\(55\) 194.384 0.476560
\(56\) −601.451 −1.43522
\(57\) 1058.44 2.45953
\(58\) 301.555 0.682691
\(59\) −349.710 −0.771668 −0.385834 0.922568i \(-0.626086\pi\)
−0.385834 + 0.922568i \(0.626086\pi\)
\(60\) 59.3087 0.127612
\(61\) 442.000 0.927743 0.463871 0.885903i \(-0.346460\pi\)
0.463871 + 0.885903i \(0.346460\pi\)
\(62\) −765.278 −1.56759
\(63\) 1019.95 2.03971
\(64\) 567.978 1.10933
\(65\) 65.0000 0.124035
\(66\) −821.201 −1.53156
\(67\) 228.074 0.415876 0.207938 0.978142i \(-0.433325\pi\)
0.207938 + 0.978142i \(0.433325\pi\)
\(68\) 115.032 0.205143
\(69\) −840.614 −1.46664
\(70\) −318.617 −0.544029
\(71\) 964.328 1.61190 0.805948 0.591986i \(-0.201656\pi\)
0.805948 + 0.591986i \(0.201656\pi\)
\(72\) −991.260 −1.62252
\(73\) 662.682 1.06248 0.531240 0.847221i \(-0.321726\pi\)
0.531240 + 0.847221i \(0.321726\pi\)
\(74\) 651.187 1.02296
\(75\) 206.155 0.317397
\(76\) −184.631 −0.278666
\(77\) −967.136 −1.43137
\(78\) −274.600 −0.398620
\(79\) −65.2614 −0.0929428 −0.0464714 0.998920i \(-0.514798\pi\)
−0.0464714 + 0.998920i \(0.514798\pi\)
\(80\) 252.116 0.352343
\(81\) −155.000 −0.212620
\(82\) 562.600 0.757669
\(83\) −92.4185 −0.122220 −0.0611099 0.998131i \(-0.519464\pi\)
−0.0611099 + 0.998131i \(0.519464\pi\)
\(84\) −295.083 −0.383288
\(85\) 399.848 0.510231
\(86\) −267.585 −0.335517
\(87\) 970.773 1.19630
\(88\) 939.930 1.13860
\(89\) −178.614 −0.212730 −0.106365 0.994327i \(-0.533921\pi\)
−0.106365 + 0.994327i \(0.533921\pi\)
\(90\) −525.118 −0.615026
\(91\) −323.400 −0.372544
\(92\) 146.634 0.166171
\(93\) −2463.60 −2.74692
\(94\) 791.049 0.867984
\(95\) −641.771 −0.693098
\(96\) 529.858 0.563317
\(97\) −1707.46 −1.78728 −0.893640 0.448785i \(-0.851857\pi\)
−0.893640 + 0.448785i \(0.851857\pi\)
\(98\) 706.630 0.728371
\(99\) −1593.95 −1.61816
\(100\) −35.9612 −0.0359612
\(101\) −1364.04 −1.34383 −0.671915 0.740628i \(-0.734528\pi\)
−0.671915 + 0.740628i \(0.734528\pi\)
\(102\) −1689.21 −1.63977
\(103\) 355.167 0.339763 0.169882 0.985464i \(-0.445661\pi\)
0.169882 + 0.985464i \(0.445661\pi\)
\(104\) 314.302 0.296345
\(105\) −1025.70 −0.953316
\(106\) −0.232922 −0.000213428 0
\(107\) 325.905 0.294453 0.147226 0.989103i \(-0.452965\pi\)
0.147226 + 0.989103i \(0.452965\pi\)
\(108\) −166.064 −0.147959
\(109\) 1738.93 1.52807 0.764034 0.645176i \(-0.223216\pi\)
0.764034 + 0.645176i \(0.223216\pi\)
\(110\) 497.926 0.431594
\(111\) 2096.32 1.79256
\(112\) −1254.37 −1.05828
\(113\) −1628.89 −1.35605 −0.678024 0.735040i \(-0.737164\pi\)
−0.678024 + 0.735040i \(0.737164\pi\)
\(114\) 2711.24 2.22746
\(115\) 509.697 0.413300
\(116\) −169.339 −0.135541
\(117\) −533.000 −0.421161
\(118\) −895.801 −0.698857
\(119\) −1989.40 −1.53250
\(120\) 996.847 0.758327
\(121\) 180.413 0.135547
\(122\) 1132.21 0.840206
\(123\) 1811.14 1.32768
\(124\) 429.744 0.311227
\(125\) −125.000 −0.0894427
\(126\) 2612.66 1.84726
\(127\) 165.333 0.115519 0.0577596 0.998331i \(-0.481604\pi\)
0.0577596 + 0.998331i \(0.481604\pi\)
\(128\) 940.868 0.649702
\(129\) −861.417 −0.587934
\(130\) 166.501 0.112332
\(131\) −1867.49 −1.24552 −0.622760 0.782413i \(-0.713989\pi\)
−0.622760 + 0.782413i \(0.713989\pi\)
\(132\) 461.148 0.304074
\(133\) 3193.05 2.08175
\(134\) 584.223 0.376636
\(135\) −577.235 −0.368003
\(136\) 1933.43 1.21905
\(137\) 1740.80 1.08560 0.542798 0.839864i \(-0.317365\pi\)
0.542798 + 0.839864i \(0.317365\pi\)
\(138\) −2153.28 −1.32825
\(139\) −985.511 −0.601367 −0.300683 0.953724i \(-0.597215\pi\)
−0.300683 + 0.953724i \(0.597215\pi\)
\(140\) 178.920 0.108011
\(141\) 2546.56 1.52099
\(142\) 2470.18 1.45981
\(143\) 505.400 0.295550
\(144\) −2067.35 −1.19639
\(145\) −588.617 −0.337117
\(146\) 1697.49 0.962230
\(147\) 2274.80 1.27634
\(148\) −365.676 −0.203097
\(149\) −2496.90 −1.37285 −0.686423 0.727202i \(-0.740820\pi\)
−0.686423 + 0.727202i \(0.740820\pi\)
\(150\) 528.078 0.287449
\(151\) 357.710 0.192782 0.0963909 0.995344i \(-0.469270\pi\)
0.0963909 + 0.995344i \(0.469270\pi\)
\(152\) −3103.23 −1.65595
\(153\) −3278.76 −1.73250
\(154\) −2477.37 −1.29631
\(155\) 1493.78 0.774085
\(156\) 154.203 0.0791416
\(157\) −3218.32 −1.63599 −0.817993 0.575228i \(-0.804913\pi\)
−0.817993 + 0.575228i \(0.804913\pi\)
\(158\) −167.170 −0.0841732
\(159\) −0.749828 −0.000373995 0
\(160\) −321.274 −0.158743
\(161\) −2535.94 −1.24136
\(162\) −397.041 −0.192558
\(163\) −2901.51 −1.39425 −0.697127 0.716948i \(-0.745539\pi\)
−0.697127 + 0.716948i \(0.745539\pi\)
\(164\) −315.930 −0.150427
\(165\) 1602.94 0.756293
\(166\) −236.735 −0.110688
\(167\) 531.036 0.246065 0.123032 0.992403i \(-0.460738\pi\)
0.123032 + 0.992403i \(0.460738\pi\)
\(168\) −4959.69 −2.27767
\(169\) 169.000 0.0769231
\(170\) 1024.23 0.462089
\(171\) 5262.52 2.35342
\(172\) 150.263 0.0666132
\(173\) 809.826 0.355895 0.177948 0.984040i \(-0.443054\pi\)
0.177948 + 0.984040i \(0.443054\pi\)
\(174\) 2486.69 1.08342
\(175\) 621.922 0.268645
\(176\) 1960.30 0.839564
\(177\) −2883.78 −1.22462
\(178\) −457.528 −0.192658
\(179\) −1831.87 −0.764919 −0.382459 0.923972i \(-0.624923\pi\)
−0.382459 + 0.923972i \(0.624923\pi\)
\(180\) 294.882 0.122107
\(181\) 1046.76 0.429863 0.214931 0.976629i \(-0.431047\pi\)
0.214931 + 0.976629i \(0.431047\pi\)
\(182\) −828.405 −0.337393
\(183\) 3644.83 1.47231
\(184\) 2464.60 0.987459
\(185\) −1271.08 −0.505144
\(186\) −6310.65 −2.48774
\(187\) 3108.97 1.21578
\(188\) −444.216 −0.172329
\(189\) 2871.96 1.10531
\(190\) −1643.93 −0.627701
\(191\) 759.973 0.287904 0.143952 0.989585i \(-0.454019\pi\)
0.143952 + 0.989585i \(0.454019\pi\)
\(192\) 4683.67 1.76049
\(193\) −411.618 −0.153518 −0.0767588 0.997050i \(-0.524457\pi\)
−0.0767588 + 0.997050i \(0.524457\pi\)
\(194\) −4373.74 −1.61864
\(195\) 536.004 0.196841
\(196\) −396.810 −0.144610
\(197\) −805.204 −0.291210 −0.145605 0.989343i \(-0.546513\pi\)
−0.145605 + 0.989343i \(0.546513\pi\)
\(198\) −4082.99 −1.46548
\(199\) 2932.02 1.04445 0.522225 0.852808i \(-0.325102\pi\)
0.522225 + 0.852808i \(0.325102\pi\)
\(200\) −604.427 −0.213697
\(201\) 1880.75 0.659988
\(202\) −3494.05 −1.21703
\(203\) 2928.59 1.01255
\(204\) 948.580 0.325558
\(205\) −1098.16 −0.374142
\(206\) 909.778 0.307705
\(207\) −4179.51 −1.40336
\(208\) 655.503 0.218514
\(209\) −4990.01 −1.65151
\(210\) −2627.39 −0.863366
\(211\) −3922.57 −1.27982 −0.639908 0.768452i \(-0.721027\pi\)
−0.639908 + 0.768452i \(0.721027\pi\)
\(212\) 0.130798 4.23738e−5 0
\(213\) 7952.05 2.55805
\(214\) 834.824 0.266670
\(215\) 522.311 0.165680
\(216\) −2791.17 −0.879237
\(217\) −7432.11 −2.32500
\(218\) 4454.37 1.38389
\(219\) 5464.61 1.68614
\(220\) −279.612 −0.0856883
\(221\) 1039.61 0.316432
\(222\) 5369.83 1.62342
\(223\) 6115.27 1.83636 0.918181 0.396162i \(-0.129658\pi\)
0.918181 + 0.396162i \(0.129658\pi\)
\(224\) 1598.46 0.476792
\(225\) 1025.00 0.303704
\(226\) −4172.50 −1.22810
\(227\) 4216.13 1.23275 0.616376 0.787452i \(-0.288600\pi\)
0.616376 + 0.787452i \(0.288600\pi\)
\(228\) −1522.50 −0.442238
\(229\) −299.451 −0.0864116 −0.0432058 0.999066i \(-0.513757\pi\)
−0.0432058 + 0.999066i \(0.513757\pi\)
\(230\) 1305.62 0.374303
\(231\) −7975.21 −2.27156
\(232\) −2846.21 −0.805443
\(233\) 1223.17 0.343916 0.171958 0.985104i \(-0.444991\pi\)
0.171958 + 0.985104i \(0.444991\pi\)
\(234\) −1365.31 −0.381423
\(235\) −1544.08 −0.428616
\(236\) 503.040 0.138750
\(237\) −538.159 −0.147499
\(238\) −5095.95 −1.38790
\(239\) −48.7253 −0.0131874 −0.00659368 0.999978i \(-0.502099\pi\)
−0.00659368 + 0.999978i \(0.502099\pi\)
\(240\) 2079.01 0.559163
\(241\) −4499.83 −1.20274 −0.601368 0.798972i \(-0.705378\pi\)
−0.601368 + 0.798972i \(0.705378\pi\)
\(242\) 462.137 0.122757
\(243\) −4395.23 −1.16031
\(244\) −635.794 −0.166814
\(245\) −1379.30 −0.359674
\(246\) 4639.32 1.20241
\(247\) −1668.60 −0.429841
\(248\) 7223.04 1.84945
\(249\) −762.103 −0.193961
\(250\) −320.194 −0.0810034
\(251\) −110.288 −0.0277343 −0.0138671 0.999904i \(-0.504414\pi\)
−0.0138671 + 0.999904i \(0.504414\pi\)
\(252\) −1467.15 −0.366753
\(253\) 3963.09 0.984811
\(254\) 423.510 0.104620
\(255\) 3297.23 0.809729
\(256\) −2133.74 −0.520933
\(257\) 7453.76 1.80916 0.904578 0.426308i \(-0.140186\pi\)
0.904578 + 0.426308i \(0.140186\pi\)
\(258\) −2206.56 −0.532460
\(259\) 6324.10 1.51722
\(260\) −93.4991 −0.0223022
\(261\) 4826.66 1.14469
\(262\) −4783.67 −1.12800
\(263\) −225.156 −0.0527897 −0.0263948 0.999652i \(-0.508403\pi\)
−0.0263948 + 0.999652i \(0.508403\pi\)
\(264\) 7750.86 1.80694
\(265\) 0.454650 0.000105392 0
\(266\) 8179.17 1.88533
\(267\) −1472.89 −0.337600
\(268\) −328.072 −0.0747769
\(269\) −2742.09 −0.621517 −0.310758 0.950489i \(-0.600583\pi\)
−0.310758 + 0.950489i \(0.600583\pi\)
\(270\) −1478.62 −0.333281
\(271\) 6749.69 1.51297 0.756484 0.654012i \(-0.226916\pi\)
0.756484 + 0.654012i \(0.226916\pi\)
\(272\) 4032.34 0.898883
\(273\) −2666.82 −0.591221
\(274\) 4459.15 0.983164
\(275\) −971.922 −0.213124
\(276\) 1209.18 0.263710
\(277\) 1511.65 0.327893 0.163946 0.986469i \(-0.447578\pi\)
0.163946 + 0.986469i \(0.447578\pi\)
\(278\) −2524.44 −0.544625
\(279\) −12249.0 −2.62841
\(280\) 3007.25 0.641849
\(281\) −5593.69 −1.18751 −0.593757 0.804645i \(-0.702356\pi\)
−0.593757 + 0.804645i \(0.702356\pi\)
\(282\) 6523.16 1.37748
\(283\) −3221.02 −0.676572 −0.338286 0.941043i \(-0.609847\pi\)
−0.338286 + 0.941043i \(0.609847\pi\)
\(284\) −1387.13 −0.289828
\(285\) −5292.18 −1.09994
\(286\) 1294.61 0.267664
\(287\) 5463.78 1.12375
\(288\) 2634.44 0.539014
\(289\) 1482.15 0.301679
\(290\) −1507.77 −0.305309
\(291\) −14080.1 −2.83638
\(292\) −953.233 −0.191040
\(293\) 7809.84 1.55719 0.778594 0.627528i \(-0.215933\pi\)
0.778594 + 0.627528i \(0.215933\pi\)
\(294\) 5827.02 1.15591
\(295\) 1748.55 0.345100
\(296\) −6146.20 −1.20689
\(297\) −4488.22 −0.876878
\(298\) −6395.94 −1.24331
\(299\) 1325.21 0.256318
\(300\) −296.543 −0.0570698
\(301\) −2598.69 −0.497628
\(302\) 916.294 0.174592
\(303\) −11248.1 −2.13264
\(304\) −6472.04 −1.22104
\(305\) −2210.00 −0.414899
\(306\) −8398.71 −1.56903
\(307\) 8581.07 1.59527 0.797634 0.603141i \(-0.206084\pi\)
0.797634 + 0.603141i \(0.206084\pi\)
\(308\) 1391.17 0.257369
\(309\) 2928.78 0.539199
\(310\) 3826.39 0.701047
\(311\) 9251.84 1.68689 0.843447 0.537213i \(-0.180523\pi\)
0.843447 + 0.537213i \(0.180523\pi\)
\(312\) 2591.80 0.470294
\(313\) 4386.17 0.792081 0.396040 0.918233i \(-0.370384\pi\)
0.396040 + 0.918233i \(0.370384\pi\)
\(314\) −8243.89 −1.48162
\(315\) −5099.76 −0.912188
\(316\) 93.8750 0.0167117
\(317\) −587.686 −0.104125 −0.0520626 0.998644i \(-0.516580\pi\)
−0.0520626 + 0.998644i \(0.516580\pi\)
\(318\) −1.92072 −0.000338707 0
\(319\) −4576.72 −0.803283
\(320\) −2839.89 −0.496109
\(321\) 2687.48 0.467292
\(322\) −6495.93 −1.12424
\(323\) −10264.4 −1.76820
\(324\) 222.959 0.0382303
\(325\) −325.000 −0.0554700
\(326\) −7432.36 −1.26270
\(327\) 14339.6 2.42502
\(328\) −5310.07 −0.893902
\(329\) 7682.39 1.28737
\(330\) 4106.00 0.684934
\(331\) −6559.71 −1.08929 −0.544644 0.838668i \(-0.683335\pi\)
−0.544644 + 0.838668i \(0.683335\pi\)
\(332\) 132.939 0.0219759
\(333\) 10422.9 1.71522
\(334\) 1360.28 0.222847
\(335\) −1140.37 −0.185985
\(336\) −10343.8 −1.67947
\(337\) 3316.98 0.536165 0.268083 0.963396i \(-0.413610\pi\)
0.268083 + 0.963396i \(0.413610\pi\)
\(338\) 432.902 0.0696651
\(339\) −13432.2 −2.15203
\(340\) −575.161 −0.0917426
\(341\) 11614.7 1.84449
\(342\) 13480.2 2.13137
\(343\) −1670.24 −0.262928
\(344\) 2525.59 0.395845
\(345\) 4203.07 0.655900
\(346\) 2074.41 0.322315
\(347\) −3729.06 −0.576906 −0.288453 0.957494i \(-0.593141\pi\)
−0.288453 + 0.957494i \(0.593141\pi\)
\(348\) −1396.41 −0.215101
\(349\) 3748.22 0.574892 0.287446 0.957797i \(-0.407194\pi\)
0.287446 + 0.957797i \(0.407194\pi\)
\(350\) 1593.09 0.243297
\(351\) −1500.81 −0.228226
\(352\) −2498.02 −0.378253
\(353\) −3353.90 −0.505694 −0.252847 0.967506i \(-0.581367\pi\)
−0.252847 + 0.967506i \(0.581367\pi\)
\(354\) −7386.97 −1.10908
\(355\) −4821.64 −0.720862
\(356\) 256.926 0.0382502
\(357\) −16405.0 −2.43206
\(358\) −4692.44 −0.692746
\(359\) −1041.87 −0.153169 −0.0765843 0.997063i \(-0.524401\pi\)
−0.0765843 + 0.997063i \(0.524401\pi\)
\(360\) 4956.30 0.725611
\(361\) 9615.79 1.40192
\(362\) 2681.33 0.389303
\(363\) 1487.72 0.215111
\(364\) 465.193 0.0669856
\(365\) −3313.41 −0.475155
\(366\) 9336.41 1.33339
\(367\) 1099.95 0.156449 0.0782247 0.996936i \(-0.475075\pi\)
0.0782247 + 0.996936i \(0.475075\pi\)
\(368\) 5140.12 0.728117
\(369\) 9004.93 1.27040
\(370\) −3255.94 −0.457481
\(371\) −2.26206 −0.000316550 0
\(372\) 3543.76 0.493913
\(373\) 2967.14 0.411884 0.205942 0.978564i \(-0.433974\pi\)
0.205942 + 0.978564i \(0.433974\pi\)
\(374\) 7963.80 1.10106
\(375\) −1030.78 −0.141944
\(376\) −7466.28 −1.02405
\(377\) −1530.41 −0.209071
\(378\) 7356.68 1.00102
\(379\) −9221.67 −1.24983 −0.624915 0.780693i \(-0.714866\pi\)
−0.624915 + 0.780693i \(0.714866\pi\)
\(380\) 923.153 0.124623
\(381\) 1363.37 0.183327
\(382\) 1946.71 0.260739
\(383\) −11048.0 −1.47395 −0.736976 0.675918i \(-0.763747\pi\)
−0.736976 + 0.675918i \(0.763747\pi\)
\(384\) 7758.60 1.03107
\(385\) 4835.68 0.640128
\(386\) −1054.38 −0.139032
\(387\) −4282.95 −0.562570
\(388\) 2456.09 0.321363
\(389\) −201.382 −0.0262480 −0.0131240 0.999914i \(-0.504178\pi\)
−0.0131240 + 0.999914i \(0.504178\pi\)
\(390\) 1373.00 0.178268
\(391\) 8152.06 1.05439
\(392\) −6669.49 −0.859337
\(393\) −15399.7 −1.97662
\(394\) −2062.57 −0.263733
\(395\) 326.307 0.0415653
\(396\) 2292.82 0.290955
\(397\) −4354.09 −0.550443 −0.275221 0.961381i \(-0.588751\pi\)
−0.275221 + 0.961381i \(0.588751\pi\)
\(398\) 7510.53 0.945902
\(399\) 26330.6 3.30370
\(400\) −1260.58 −0.157573
\(401\) 4241.31 0.528182 0.264091 0.964498i \(-0.414928\pi\)
0.264091 + 0.964498i \(0.414928\pi\)
\(402\) 4817.63 0.597715
\(403\) 3883.82 0.480067
\(404\) 1962.10 0.241629
\(405\) 775.000 0.0950866
\(406\) 7501.75 0.917009
\(407\) −9883.12 −1.20366
\(408\) 15943.5 1.93461
\(409\) −1622.25 −0.196124 −0.0980622 0.995180i \(-0.531264\pi\)
−0.0980622 + 0.995180i \(0.531264\pi\)
\(410\) −2813.00 −0.338840
\(411\) 14355.0 1.72282
\(412\) −510.889 −0.0610914
\(413\) −8699.70 −1.03652
\(414\) −10706.0 −1.27095
\(415\) 462.093 0.0546584
\(416\) −835.311 −0.0984483
\(417\) −8126.73 −0.954359
\(418\) −12782.2 −1.49569
\(419\) −6344.39 −0.739723 −0.369861 0.929087i \(-0.620595\pi\)
−0.369861 + 0.929087i \(0.620595\pi\)
\(420\) 1475.42 0.171412
\(421\) −7050.55 −0.816206 −0.408103 0.912936i \(-0.633810\pi\)
−0.408103 + 0.912936i \(0.633810\pi\)
\(422\) −10047.9 −1.15906
\(423\) 12661.5 1.45537
\(424\) 2.19842 0.000251804 0
\(425\) −1999.24 −0.228182
\(426\) 20369.6 2.31669
\(427\) 10995.6 1.24617
\(428\) −468.798 −0.0529444
\(429\) 4167.63 0.469033
\(430\) 1337.93 0.150048
\(431\) −2749.74 −0.307309 −0.153655 0.988125i \(-0.549104\pi\)
−0.153655 + 0.988125i \(0.549104\pi\)
\(432\) −5821.22 −0.648318
\(433\) −12718.7 −1.41160 −0.705798 0.708413i \(-0.749411\pi\)
−0.705798 + 0.708413i \(0.749411\pi\)
\(434\) −19037.7 −2.10562
\(435\) −4853.86 −0.535000
\(436\) −2501.36 −0.274756
\(437\) −13084.3 −1.43229
\(438\) 13997.9 1.52704
\(439\) 17656.7 1.91961 0.959807 0.280660i \(-0.0905535\pi\)
0.959807 + 0.280660i \(0.0905535\pi\)
\(440\) −4699.65 −0.509198
\(441\) 11310.3 1.22128
\(442\) 2663.01 0.286575
\(443\) 18355.7 1.96864 0.984319 0.176397i \(-0.0564444\pi\)
0.984319 + 0.176397i \(0.0564444\pi\)
\(444\) −3015.44 −0.322312
\(445\) 893.068 0.0951360
\(446\) 15664.6 1.66309
\(447\) −20590.0 −2.17869
\(448\) 14129.5 1.49008
\(449\) −2675.78 −0.281242 −0.140621 0.990063i \(-0.544910\pi\)
−0.140621 + 0.990063i \(0.544910\pi\)
\(450\) 2625.59 0.275048
\(451\) −8538.63 −0.891504
\(452\) 2343.08 0.243825
\(453\) 2949.75 0.305942
\(454\) 10799.9 1.11644
\(455\) 1617.00 0.166607
\(456\) −25589.9 −2.62797
\(457\) 6114.51 0.625875 0.312937 0.949774i \(-0.398687\pi\)
0.312937 + 0.949774i \(0.398687\pi\)
\(458\) −767.059 −0.0782583
\(459\) −9232.26 −0.938834
\(460\) −733.172 −0.0743137
\(461\) −13983.6 −1.41275 −0.706377 0.707836i \(-0.749671\pi\)
−0.706377 + 0.707836i \(0.749671\pi\)
\(462\) −20428.9 −2.05723
\(463\) −5068.98 −0.508802 −0.254401 0.967099i \(-0.581878\pi\)
−0.254401 + 0.967099i \(0.581878\pi\)
\(464\) −5936.01 −0.593905
\(465\) 12318.0 1.22846
\(466\) 3133.21 0.311466
\(467\) 2449.21 0.242689 0.121345 0.992610i \(-0.461279\pi\)
0.121345 + 0.992610i \(0.461279\pi\)
\(468\) 766.692 0.0757273
\(469\) 5673.77 0.558615
\(470\) −3955.25 −0.388174
\(471\) −26538.9 −2.59628
\(472\) 8454.97 0.824516
\(473\) 4061.16 0.394783
\(474\) −1378.52 −0.133582
\(475\) 3208.85 0.309963
\(476\) 2861.64 0.275553
\(477\) −3.72813 −0.000357860 0
\(478\) −124.813 −0.0119431
\(479\) −14106.1 −1.34556 −0.672780 0.739842i \(-0.734900\pi\)
−0.672780 + 0.739842i \(0.734900\pi\)
\(480\) −2649.29 −0.251923
\(481\) −3304.81 −0.313277
\(482\) −11526.6 −1.08925
\(483\) −20911.9 −1.97003
\(484\) −259.514 −0.0243721
\(485\) 8537.29 0.799296
\(486\) −11258.6 −1.05083
\(487\) 11324.4 1.05371 0.526855 0.849955i \(-0.323371\pi\)
0.526855 + 0.849955i \(0.323371\pi\)
\(488\) −10686.3 −0.991280
\(489\) −23926.4 −2.21266
\(490\) −3533.15 −0.325737
\(491\) −11824.8 −1.08686 −0.543430 0.839455i \(-0.682875\pi\)
−0.543430 + 0.839455i \(0.682875\pi\)
\(492\) −2605.22 −0.238725
\(493\) −9414.31 −0.860039
\(494\) −4274.22 −0.389284
\(495\) 7969.76 0.723665
\(496\) 15064.2 1.36372
\(497\) 23989.5 2.16514
\(498\) −1952.17 −0.175660
\(499\) 11462.3 1.02830 0.514150 0.857700i \(-0.328108\pi\)
0.514150 + 0.857700i \(0.328108\pi\)
\(500\) 179.806 0.0160823
\(501\) 4379.03 0.390501
\(502\) −282.508 −0.0251174
\(503\) −13165.2 −1.16701 −0.583506 0.812109i \(-0.698320\pi\)
−0.583506 + 0.812109i \(0.698320\pi\)
\(504\) −24659.5 −2.17941
\(505\) 6820.19 0.600979
\(506\) 10151.7 0.891890
\(507\) 1393.61 0.122076
\(508\) −237.823 −0.0207710
\(509\) 8849.94 0.770661 0.385331 0.922779i \(-0.374087\pi\)
0.385331 + 0.922779i \(0.374087\pi\)
\(510\) 8446.04 0.733327
\(511\) 16485.5 1.42715
\(512\) −12992.6 −1.12148
\(513\) 14818.1 1.27531
\(514\) 19093.2 1.63845
\(515\) −1775.83 −0.151947
\(516\) 1239.10 0.105714
\(517\) −12005.8 −1.02131
\(518\) 16199.5 1.37407
\(519\) 6677.99 0.564800
\(520\) −1571.51 −0.132529
\(521\) 20509.2 1.72461 0.862306 0.506387i \(-0.169019\pi\)
0.862306 + 0.506387i \(0.169019\pi\)
\(522\) 12363.8 1.03668
\(523\) 434.056 0.0362906 0.0181453 0.999835i \(-0.494224\pi\)
0.0181453 + 0.999835i \(0.494224\pi\)
\(524\) 2686.28 0.223952
\(525\) 5128.50 0.426336
\(526\) −576.748 −0.0478087
\(527\) 23891.4 1.97481
\(528\) 16165.1 1.33237
\(529\) −1775.36 −0.145916
\(530\) 1.16461 9.54479e−5 0
\(531\) −14338.1 −1.17179
\(532\) −4593.04 −0.374311
\(533\) −2855.22 −0.232033
\(534\) −3772.88 −0.305746
\(535\) −1629.53 −0.131683
\(536\) −5514.16 −0.444357
\(537\) −15106.0 −1.21391
\(538\) −7024.00 −0.562874
\(539\) −10724.6 −0.857032
\(540\) 830.322 0.0661692
\(541\) −5500.16 −0.437099 −0.218549 0.975826i \(-0.570132\pi\)
−0.218549 + 0.975826i \(0.570132\pi\)
\(542\) 17289.7 1.37021
\(543\) 8631.82 0.682185
\(544\) −5138.43 −0.404979
\(545\) −8694.66 −0.683373
\(546\) −6831.20 −0.535437
\(547\) 8536.10 0.667235 0.333617 0.942709i \(-0.391731\pi\)
0.333617 + 0.942709i \(0.391731\pi\)
\(548\) −2504.05 −0.195196
\(549\) 18122.0 1.40879
\(550\) −2489.63 −0.193015
\(551\) 15110.3 1.16828
\(552\) 20323.6 1.56708
\(553\) −1623.50 −0.124843
\(554\) 3872.18 0.296955
\(555\) −10481.6 −0.801655
\(556\) 1417.61 0.108129
\(557\) −4701.29 −0.357630 −0.178815 0.983883i \(-0.557226\pi\)
−0.178815 + 0.983883i \(0.557226\pi\)
\(558\) −31376.4 −2.38041
\(559\) 1358.01 0.102751
\(560\) 6271.87 0.473277
\(561\) 25637.2 1.92942
\(562\) −14328.5 −1.07547
\(563\) 14613.9 1.09396 0.546981 0.837145i \(-0.315777\pi\)
0.546981 + 0.837145i \(0.315777\pi\)
\(564\) −3663.10 −0.273483
\(565\) 8144.47 0.606443
\(566\) −8250.82 −0.612735
\(567\) −3855.92 −0.285597
\(568\) −23314.6 −1.72229
\(569\) −10789.3 −0.794921 −0.397461 0.917619i \(-0.630108\pi\)
−0.397461 + 0.917619i \(0.630108\pi\)
\(570\) −13556.2 −0.996152
\(571\) −1553.39 −0.113848 −0.0569240 0.998379i \(-0.518129\pi\)
−0.0569240 + 0.998379i \(0.518129\pi\)
\(572\) −726.991 −0.0531416
\(573\) 6266.90 0.456900
\(574\) 13995.8 1.01772
\(575\) −2548.48 −0.184833
\(576\) 23287.1 1.68454
\(577\) 11828.6 0.853431 0.426715 0.904386i \(-0.359671\pi\)
0.426715 + 0.904386i \(0.359671\pi\)
\(578\) 3796.61 0.273215
\(579\) −3394.29 −0.243630
\(580\) 846.695 0.0606157
\(581\) −2299.09 −0.164169
\(582\) −36066.8 −2.56876
\(583\) 3.53507 0.000251128 0
\(584\) −16021.7 −1.13525
\(585\) 2665.00 0.188349
\(586\) 20005.3 1.41026
\(587\) 2823.97 0.198565 0.0992825 0.995059i \(-0.468345\pi\)
0.0992825 + 0.995059i \(0.468345\pi\)
\(588\) −3272.18 −0.229494
\(589\) −38346.5 −2.68258
\(590\) 4479.01 0.312539
\(591\) −6639.89 −0.462146
\(592\) −12818.4 −0.889921
\(593\) 2016.22 0.139623 0.0698115 0.997560i \(-0.477760\pi\)
0.0698115 + 0.997560i \(0.477760\pi\)
\(594\) −11496.8 −0.794141
\(595\) 9946.99 0.685356
\(596\) 3591.66 0.246846
\(597\) 24178.1 1.65753
\(598\) 3394.60 0.232133
\(599\) 9247.60 0.630796 0.315398 0.948960i \(-0.397862\pi\)
0.315398 + 0.948960i \(0.397862\pi\)
\(600\) −4984.23 −0.339134
\(601\) 8105.84 0.550156 0.275078 0.961422i \(-0.411296\pi\)
0.275078 + 0.961422i \(0.411296\pi\)
\(602\) −6656.69 −0.450675
\(603\) 9351.03 0.631515
\(604\) −514.547 −0.0346633
\(605\) −902.065 −0.0606184
\(606\) −28812.7 −1.93141
\(607\) 11059.6 0.739534 0.369767 0.929124i \(-0.379437\pi\)
0.369767 + 0.929124i \(0.379437\pi\)
\(608\) 8247.36 0.550123
\(609\) 24149.8 1.60690
\(610\) −5661.03 −0.375752
\(611\) −4014.61 −0.265816
\(612\) 4716.32 0.311513
\(613\) −12579.3 −0.828831 −0.414416 0.910088i \(-0.636014\pi\)
−0.414416 + 0.910088i \(0.636014\pi\)
\(614\) 21980.9 1.44475
\(615\) −9055.68 −0.593757
\(616\) 23382.5 1.52940
\(617\) 16915.6 1.10372 0.551862 0.833935i \(-0.313917\pi\)
0.551862 + 0.833935i \(0.313917\pi\)
\(618\) 7502.22 0.488323
\(619\) 10313.7 0.669697 0.334849 0.942272i \(-0.391315\pi\)
0.334849 + 0.942272i \(0.391315\pi\)
\(620\) −2148.72 −0.139185
\(621\) −11768.6 −0.760479
\(622\) 23699.1 1.52773
\(623\) −4443.35 −0.285745
\(624\) 5405.41 0.346778
\(625\) 625.000 0.0400000
\(626\) 11235.4 0.717344
\(627\) −41148.7 −2.62093
\(628\) 4629.38 0.294160
\(629\) −20329.6 −1.28870
\(630\) −13063.3 −0.826119
\(631\) −20558.9 −1.29705 −0.648524 0.761194i \(-0.724613\pi\)
−0.648524 + 0.761194i \(0.724613\pi\)
\(632\) 1577.83 0.0993080
\(633\) −32346.4 −2.03105
\(634\) −1505.39 −0.0943006
\(635\) −826.666 −0.0516618
\(636\) 1.07859 6.72465e−5 0
\(637\) −3586.18 −0.223061
\(638\) −11723.5 −0.727490
\(639\) 39537.4 2.44769
\(640\) −4704.34 −0.290555
\(641\) −5651.22 −0.348221 −0.174111 0.984726i \(-0.555705\pi\)
−0.174111 + 0.984726i \(0.555705\pi\)
\(642\) 6884.13 0.423201
\(643\) 2197.26 0.134761 0.0673805 0.997727i \(-0.478536\pi\)
0.0673805 + 0.997727i \(0.478536\pi\)
\(644\) 3647.81 0.223205
\(645\) 4307.08 0.262932
\(646\) −26292.9 −1.60136
\(647\) 23529.6 1.42974 0.714872 0.699255i \(-0.246485\pi\)
0.714872 + 0.699255i \(0.246485\pi\)
\(648\) 3747.45 0.227182
\(649\) 13595.6 0.822305
\(650\) −832.505 −0.0502362
\(651\) −61286.8 −3.68974
\(652\) 4173.66 0.250695
\(653\) 26577.6 1.59275 0.796373 0.604806i \(-0.206749\pi\)
0.796373 + 0.604806i \(0.206749\pi\)
\(654\) 36731.6 2.19621
\(655\) 9337.44 0.557014
\(656\) −11074.6 −0.659132
\(657\) 27170.0 1.61340
\(658\) 19678.8 1.16590
\(659\) −454.595 −0.0268718 −0.0134359 0.999910i \(-0.504277\pi\)
−0.0134359 + 0.999910i \(0.504277\pi\)
\(660\) −2305.74 −0.135986
\(661\) −21367.3 −1.25732 −0.628662 0.777679i \(-0.716397\pi\)
−0.628662 + 0.777679i \(0.716397\pi\)
\(662\) −16803.0 −0.986508
\(663\) 8572.81 0.502173
\(664\) 2234.41 0.130590
\(665\) −15965.3 −0.930987
\(666\) 26698.7 1.55338
\(667\) −12000.7 −0.696653
\(668\) −763.867 −0.0442439
\(669\) 50427.8 2.91428
\(670\) −2921.12 −0.168437
\(671\) −17183.6 −0.988622
\(672\) 13181.2 0.756662
\(673\) −18710.7 −1.07169 −0.535843 0.844318i \(-0.680006\pi\)
−0.535843 + 0.844318i \(0.680006\pi\)
\(674\) 8496.63 0.485576
\(675\) 2886.17 0.164576
\(676\) −243.098 −0.0138312
\(677\) −29181.5 −1.65662 −0.828312 0.560267i \(-0.810699\pi\)
−0.828312 + 0.560267i \(0.810699\pi\)
\(678\) −34407.3 −1.94897
\(679\) −42476.3 −2.40072
\(680\) −9667.17 −0.545175
\(681\) 34767.1 1.95636
\(682\) 29751.6 1.67045
\(683\) 21953.8 1.22992 0.614961 0.788557i \(-0.289172\pi\)
0.614961 + 0.788557i \(0.289172\pi\)
\(684\) −7569.86 −0.423159
\(685\) −8704.00 −0.485493
\(686\) −4278.40 −0.238120
\(687\) −2469.33 −0.137134
\(688\) 5267.32 0.291882
\(689\) 1.18209 6.53614e−5 0
\(690\) 10766.4 0.594013
\(691\) 2242.59 0.123462 0.0617309 0.998093i \(-0.480338\pi\)
0.0617309 + 0.998093i \(0.480338\pi\)
\(692\) −1164.89 −0.0639921
\(693\) −39652.6 −2.17356
\(694\) −9552.18 −0.522472
\(695\) 4927.56 0.268939
\(696\) −23470.4 −1.27823
\(697\) −17563.9 −0.954493
\(698\) 9601.25 0.520649
\(699\) 10086.5 0.545788
\(700\) −894.602 −0.0483040
\(701\) 16013.4 0.862794 0.431397 0.902162i \(-0.358021\pi\)
0.431397 + 0.902162i \(0.358021\pi\)
\(702\) −3844.41 −0.206692
\(703\) 32629.7 1.75057
\(704\) −22081.2 −1.18213
\(705\) −12732.8 −0.680207
\(706\) −8591.19 −0.457980
\(707\) −33933.0 −1.80507
\(708\) 4148.17 0.220195
\(709\) −34333.3 −1.81864 −0.909318 0.416102i \(-0.863396\pi\)
−0.909318 + 0.416102i \(0.863396\pi\)
\(710\) −12350.9 −0.652845
\(711\) −2675.72 −0.141135
\(712\) 4318.36 0.227300
\(713\) 30455.0 1.59965
\(714\) −42022.3 −2.20258
\(715\) −2527.00 −0.132174
\(716\) 2635.05 0.137537
\(717\) −401.799 −0.0209281
\(718\) −2668.79 −0.138717
\(719\) −4359.43 −0.226119 −0.113059 0.993588i \(-0.536065\pi\)
−0.113059 + 0.993588i \(0.536065\pi\)
\(720\) 10336.8 0.535040
\(721\) 8835.44 0.456379
\(722\) 24631.4 1.26965
\(723\) −37106.5 −1.90872
\(724\) −1505.71 −0.0772919
\(725\) 2943.09 0.150763
\(726\) 3810.88 0.194814
\(727\) −9626.47 −0.491095 −0.245547 0.969385i \(-0.578968\pi\)
−0.245547 + 0.969385i \(0.578968\pi\)
\(728\) 7818.86 0.398058
\(729\) −32059.0 −1.62877
\(730\) −8487.47 −0.430322
\(731\) 8353.80 0.422677
\(732\) −5242.89 −0.264731
\(733\) −11224.3 −0.565590 −0.282795 0.959180i \(-0.591262\pi\)
−0.282795 + 0.959180i \(0.591262\pi\)
\(734\) 2817.58 0.141688
\(735\) −11374.0 −0.570797
\(736\) −6550.09 −0.328043
\(737\) −8866.81 −0.443165
\(738\) 23066.6 1.15053
\(739\) 20190.6 1.00504 0.502519 0.864566i \(-0.332407\pi\)
0.502519 + 0.864566i \(0.332407\pi\)
\(740\) 1828.38 0.0908279
\(741\) −13759.7 −0.682151
\(742\) −5.79437 −0.000286682 0
\(743\) −19756.1 −0.975479 −0.487739 0.872989i \(-0.662178\pi\)
−0.487739 + 0.872989i \(0.662178\pi\)
\(744\) 59562.7 2.93505
\(745\) 12484.5 0.613956
\(746\) 7600.49 0.373021
\(747\) −3789.16 −0.185593
\(748\) −4472.09 −0.218604
\(749\) 8107.51 0.395517
\(750\) −2640.39 −0.128551
\(751\) 17597.3 0.855040 0.427520 0.904006i \(-0.359387\pi\)
0.427520 + 0.904006i \(0.359387\pi\)
\(752\) −15571.5 −0.755100
\(753\) −909.456 −0.0440138
\(754\) −3920.21 −0.189345
\(755\) −1788.55 −0.0862146
\(756\) −4131.17 −0.198742
\(757\) 29920.3 1.43655 0.718277 0.695757i \(-0.244931\pi\)
0.718277 + 0.695757i \(0.244931\pi\)
\(758\) −23621.8 −1.13190
\(759\) 32680.4 1.56288
\(760\) 15516.1 0.740565
\(761\) 13040.7 0.621190 0.310595 0.950542i \(-0.399472\pi\)
0.310595 + 0.950542i \(0.399472\pi\)
\(762\) 3492.35 0.166030
\(763\) 43259.2 2.05254
\(764\) −1093.18 −0.0517669
\(765\) 16393.8 0.774796
\(766\) −28299.9 −1.33488
\(767\) 4546.23 0.214022
\(768\) −17595.3 −0.826712
\(769\) 23856.2 1.11869 0.559347 0.828934i \(-0.311052\pi\)
0.559347 + 0.828934i \(0.311052\pi\)
\(770\) 12386.9 0.579729
\(771\) 61465.3 2.87110
\(772\) 592.090 0.0276034
\(773\) −20701.3 −0.963224 −0.481612 0.876384i \(-0.659949\pi\)
−0.481612 + 0.876384i \(0.659949\pi\)
\(774\) −10971.0 −0.509489
\(775\) −7468.89 −0.346181
\(776\) 41281.4 1.90968
\(777\) 52149.9 2.40781
\(778\) −515.851 −0.0237714
\(779\) 28190.8 1.29658
\(780\) −771.013 −0.0353932
\(781\) −37490.1 −1.71767
\(782\) 20881.9 0.954906
\(783\) 13590.8 0.620302
\(784\) −13909.8 −0.633644
\(785\) 16091.6 0.731635
\(786\) −39447.2 −1.79012
\(787\) −4082.21 −0.184899 −0.0924493 0.995717i \(-0.529470\pi\)
−0.0924493 + 0.995717i \(0.529470\pi\)
\(788\) 1158.24 0.0523613
\(789\) −1856.68 −0.0837764
\(790\) 835.852 0.0376434
\(791\) −40521.8 −1.82148
\(792\) 38537.1 1.72899
\(793\) −5746.00 −0.257310
\(794\) −11153.2 −0.498506
\(795\) 3.74914 0.000167256 0
\(796\) −4217.56 −0.187798
\(797\) 17269.3 0.767514 0.383757 0.923434i \(-0.374630\pi\)
0.383757 + 0.923434i \(0.374630\pi\)
\(798\) 67447.2 2.99199
\(799\) −24695.9 −1.09347
\(800\) 1606.37 0.0709921
\(801\) −7323.16 −0.323035
\(802\) 10864.3 0.478346
\(803\) −25763.0 −1.13220
\(804\) −2705.35 −0.118670
\(805\) 12679.7 0.555155
\(806\) 9948.62 0.434771
\(807\) −22611.8 −0.986337
\(808\) 32978.5 1.43586
\(809\) −35166.8 −1.52831 −0.764153 0.645034i \(-0.776843\pi\)
−0.764153 + 0.645034i \(0.776843\pi\)
\(810\) 1985.20 0.0861147
\(811\) −40718.1 −1.76302 −0.881509 0.472167i \(-0.843472\pi\)
−0.881509 + 0.472167i \(0.843472\pi\)
\(812\) −4212.63 −0.182062
\(813\) 55659.3 2.40106
\(814\) −25316.1 −1.09009
\(815\) 14507.5 0.623529
\(816\) 33251.5 1.42651
\(817\) −13408.1 −0.574164
\(818\) −4155.47 −0.177619
\(819\) −13259.4 −0.565715
\(820\) 1579.65 0.0672729
\(821\) −30057.8 −1.27774 −0.638870 0.769314i \(-0.720598\pi\)
−0.638870 + 0.769314i \(0.720598\pi\)
\(822\) 36771.1 1.56027
\(823\) 4548.10 0.192633 0.0963163 0.995351i \(-0.469294\pi\)
0.0963163 + 0.995351i \(0.469294\pi\)
\(824\) −8586.89 −0.363032
\(825\) −8014.68 −0.338225
\(826\) −22284.7 −0.938724
\(827\) 9335.94 0.392554 0.196277 0.980548i \(-0.437115\pi\)
0.196277 + 0.980548i \(0.437115\pi\)
\(828\) 6012.01 0.252333
\(829\) −22765.0 −0.953751 −0.476876 0.878971i \(-0.658231\pi\)
−0.476876 + 0.878971i \(0.658231\pi\)
\(830\) 1183.67 0.0495011
\(831\) 12465.4 0.520361
\(832\) −7383.72 −0.307673
\(833\) −22060.4 −0.917585
\(834\) −20817.1 −0.864312
\(835\) −2655.18 −0.110043
\(836\) 7177.87 0.296952
\(837\) −34490.4 −1.42433
\(838\) −16251.5 −0.669927
\(839\) 28521.3 1.17362 0.586808 0.809726i \(-0.300384\pi\)
0.586808 + 0.809726i \(0.300384\pi\)
\(840\) 24798.4 1.01860
\(841\) −10530.2 −0.431760
\(842\) −18060.3 −0.739193
\(843\) −46126.7 −1.88456
\(844\) 5642.41 0.230118
\(845\) −845.000 −0.0344010
\(846\) 32433.0 1.31805
\(847\) 4488.11 0.182070
\(848\) 4.58499 0.000185671 0
\(849\) −26561.2 −1.07371
\(850\) −5121.16 −0.206652
\(851\) −25914.6 −1.04388
\(852\) −11438.6 −0.459953
\(853\) 5037.10 0.202189 0.101094 0.994877i \(-0.467766\pi\)
0.101094 + 0.994877i \(0.467766\pi\)
\(854\) 28165.8 1.12859
\(855\) −26312.6 −1.05248
\(856\) −7879.44 −0.314619
\(857\) −27178.6 −1.08332 −0.541658 0.840599i \(-0.682203\pi\)
−0.541658 + 0.840599i \(0.682203\pi\)
\(858\) 10675.6 0.424778
\(859\) 21916.5 0.870525 0.435262 0.900304i \(-0.356656\pi\)
0.435262 + 0.900304i \(0.356656\pi\)
\(860\) −751.316 −0.0297903
\(861\) 45055.4 1.78337
\(862\) −7043.60 −0.278313
\(863\) 38533.9 1.51994 0.759971 0.649957i \(-0.225213\pi\)
0.759971 + 0.649957i \(0.225213\pi\)
\(864\) 7418.01 0.292090
\(865\) −4049.13 −0.159161
\(866\) −32579.6 −1.27841
\(867\) 12222.1 0.478761
\(868\) 10690.7 0.418048
\(869\) 2537.16 0.0990417
\(870\) −12433.4 −0.484520
\(871\) −2964.96 −0.115343
\(872\) −42042.3 −1.63272
\(873\) −70005.8 −2.71402
\(874\) −33516.2 −1.29714
\(875\) −3109.61 −0.120142
\(876\) −7860.56 −0.303178
\(877\) 4812.41 0.185295 0.0926475 0.995699i \(-0.470467\pi\)
0.0926475 + 0.995699i \(0.470467\pi\)
\(878\) 45228.7 1.73849
\(879\) 64401.6 2.47123
\(880\) −9801.51 −0.375464
\(881\) −18910.3 −0.723160 −0.361580 0.932341i \(-0.617763\pi\)
−0.361580 + 0.932341i \(0.617763\pi\)
\(882\) 28971.8 1.10605
\(883\) −35174.6 −1.34057 −0.670283 0.742106i \(-0.733827\pi\)
−0.670283 + 0.742106i \(0.733827\pi\)
\(884\) −1495.42 −0.0568963
\(885\) 14418.9 0.547669
\(886\) 47019.2 1.78289
\(887\) 38502.7 1.45749 0.728745 0.684785i \(-0.240104\pi\)
0.728745 + 0.684785i \(0.240104\pi\)
\(888\) −50682.9 −1.91532
\(889\) 4112.98 0.155168
\(890\) 2287.64 0.0861595
\(891\) 6025.92 0.226572
\(892\) −8796.49 −0.330189
\(893\) 39637.9 1.48536
\(894\) −52742.3 −1.97312
\(895\) 9159.36 0.342082
\(896\) 23405.9 0.872696
\(897\) 10928.0 0.406772
\(898\) −6854.14 −0.254706
\(899\) −35170.6 −1.30479
\(900\) −1474.41 −0.0546077
\(901\) 7.27164 0.000268872 0
\(902\) −21872.2 −0.807387
\(903\) −21429.4 −0.789728
\(904\) 39381.9 1.44892
\(905\) −5233.81 −0.192240
\(906\) 7555.95 0.277075
\(907\) −52947.3 −1.93835 −0.969176 0.246368i \(-0.920763\pi\)
−0.969176 + 0.246368i \(0.920763\pi\)
\(908\) −6064.69 −0.221656
\(909\) −55925.5 −2.04063
\(910\) 4142.03 0.150887
\(911\) 11565.8 0.420627 0.210314 0.977634i \(-0.432551\pi\)
0.210314 + 0.977634i \(0.432551\pi\)
\(912\) −53369.8 −1.93778
\(913\) 3592.94 0.130240
\(914\) 15662.6 0.566821
\(915\) −18224.1 −0.658438
\(916\) 430.744 0.0155373
\(917\) −46457.3 −1.67302
\(918\) −23648.9 −0.850251
\(919\) −31800.9 −1.14148 −0.570738 0.821132i \(-0.693343\pi\)
−0.570738 + 0.821132i \(0.693343\pi\)
\(920\) −12323.0 −0.441605
\(921\) 70761.3 2.53167
\(922\) −35819.6 −1.27945
\(923\) −12536.3 −0.447060
\(924\) 11471.9 0.408440
\(925\) 6355.40 0.225907
\(926\) −12984.5 −0.460794
\(927\) 14561.8 0.515937
\(928\) 7564.29 0.267575
\(929\) 39306.1 1.38815 0.694075 0.719903i \(-0.255814\pi\)
0.694075 + 0.719903i \(0.255814\pi\)
\(930\) 31553.2 1.11255
\(931\) 35407.8 1.24645
\(932\) −1759.46 −0.0618380
\(933\) 76292.6 2.67707
\(934\) 6273.78 0.219791
\(935\) −15544.9 −0.543713
\(936\) 12886.4 0.450005
\(937\) 2554.39 0.0890589 0.0445294 0.999008i \(-0.485821\pi\)
0.0445294 + 0.999008i \(0.485821\pi\)
\(938\) 14533.7 0.505907
\(939\) 36169.3 1.25702
\(940\) 2221.08 0.0770677
\(941\) 17871.8 0.619133 0.309567 0.950878i \(-0.399816\pi\)
0.309567 + 0.950878i \(0.399816\pi\)
\(942\) −67980.9 −2.35131
\(943\) −22389.2 −0.773163
\(944\) 17633.5 0.607969
\(945\) −14359.8 −0.494312
\(946\) 10402.9 0.357534
\(947\) 13013.0 0.446532 0.223266 0.974758i \(-0.428328\pi\)
0.223266 + 0.974758i \(0.428328\pi\)
\(948\) 774.113 0.0265211
\(949\) −8614.86 −0.294679
\(950\) 8219.65 0.280716
\(951\) −4846.18 −0.165245
\(952\) 48097.8 1.63746
\(953\) −11984.0 −0.407345 −0.203672 0.979039i \(-0.565288\pi\)
−0.203672 + 0.979039i \(0.565288\pi\)
\(954\) −9.54980 −0.000324094 0
\(955\) −3799.87 −0.128755
\(956\) 70.0888 0.00237117
\(957\) −37740.6 −1.27480
\(958\) −36133.5 −1.21860
\(959\) 43305.7 1.45820
\(960\) −23418.3 −0.787316
\(961\) 59463.9 1.99604
\(962\) −8465.44 −0.283718
\(963\) 13362.1 0.447132
\(964\) 6472.77 0.216259
\(965\) 2058.09 0.0686551
\(966\) −53566.8 −1.78415
\(967\) 34643.6 1.15208 0.576041 0.817421i \(-0.304597\pi\)
0.576041 + 0.817421i \(0.304597\pi\)
\(968\) −4361.86 −0.144830
\(969\) −84642.8 −2.80611
\(970\) 21868.7 0.723879
\(971\) 11097.6 0.366776 0.183388 0.983041i \(-0.441294\pi\)
0.183388 + 0.983041i \(0.441294\pi\)
\(972\) 6322.31 0.208630
\(973\) −24516.5 −0.807771
\(974\) 29008.0 0.954288
\(975\) −2680.02 −0.0880300
\(976\) −22287.1 −0.730935
\(977\) 6609.39 0.216431 0.108215 0.994127i \(-0.465486\pi\)
0.108215 + 0.994127i \(0.465486\pi\)
\(978\) −61288.8 −2.00389
\(979\) 6943.94 0.226690
\(980\) 1984.05 0.0646716
\(981\) 71296.2 2.32040
\(982\) −30290.0 −0.984309
\(983\) 30211.3 0.980254 0.490127 0.871651i \(-0.336950\pi\)
0.490127 + 0.871651i \(0.336950\pi\)
\(984\) −43788.0 −1.41861
\(985\) 4026.02 0.130233
\(986\) −24115.3 −0.778891
\(987\) 63350.6 2.04303
\(988\) 2400.20 0.0772880
\(989\) 10648.8 0.342378
\(990\) 20415.0 0.655384
\(991\) 36910.1 1.18313 0.591567 0.806256i \(-0.298509\pi\)
0.591567 + 0.806256i \(0.298509\pi\)
\(992\) −19196.5 −0.614403
\(993\) −54092.7 −1.72868
\(994\) 61450.3 1.96085
\(995\) −14660.1 −0.467092
\(996\) 1096.24 0.0348753
\(997\) −42912.3 −1.36313 −0.681567 0.731755i \(-0.738701\pi\)
−0.681567 + 0.731755i \(0.738701\pi\)
\(998\) 29361.2 0.931275
\(999\) 29348.5 0.929473
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 65.4.a.c.1.2 2
3.2 odd 2 585.4.a.h.1.1 2
4.3 odd 2 1040.4.a.k.1.1 2
5.2 odd 4 325.4.b.f.274.4 4
5.3 odd 4 325.4.b.f.274.1 4
5.4 even 2 325.4.a.g.1.1 2
13.12 even 2 845.4.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.4.a.c.1.2 2 1.1 even 1 trivial
325.4.a.g.1.1 2 5.4 even 2
325.4.b.f.274.1 4 5.3 odd 4
325.4.b.f.274.4 4 5.2 odd 4
585.4.a.h.1.1 2 3.2 odd 2
845.4.a.d.1.1 2 13.12 even 2
1040.4.a.k.1.1 2 4.3 odd 2