Properties

Label 1849.4.a.c.1.5
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1849,4,Mod(1,1849)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1849, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1849.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(109.094531601\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 32x^{4} - 16x^{3} + 251x^{2} + 276x + 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(4.15653\) of defining polynomial
Character \(\chi\) \(=\) 1849.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.15653 q^{2} -7.20925 q^{3} +1.96369 q^{4} -1.36370 q^{5} -22.7562 q^{6} -13.0131 q^{7} -19.0538 q^{8} +24.9733 q^{9} +O(q^{10})\) \(q+3.15653 q^{2} -7.20925 q^{3} +1.96369 q^{4} -1.36370 q^{5} -22.7562 q^{6} -13.0131 q^{7} -19.0538 q^{8} +24.9733 q^{9} -4.30455 q^{10} +64.7677 q^{11} -14.1567 q^{12} -19.2944 q^{13} -41.0763 q^{14} +9.83123 q^{15} -75.8534 q^{16} -54.1213 q^{17} +78.8289 q^{18} +69.0659 q^{19} -2.67787 q^{20} +93.8149 q^{21} +204.441 q^{22} +29.6031 q^{23} +137.364 q^{24} -123.140 q^{25} -60.9032 q^{26} +14.6112 q^{27} -25.5537 q^{28} -13.1279 q^{29} +31.0326 q^{30} +185.439 q^{31} -87.0032 q^{32} -466.927 q^{33} -170.836 q^{34} +17.7460 q^{35} +49.0397 q^{36} +369.949 q^{37} +218.009 q^{38} +139.098 q^{39} +25.9836 q^{40} -294.860 q^{41} +296.129 q^{42} +127.183 q^{44} -34.0560 q^{45} +93.4431 q^{46} +367.319 q^{47} +546.846 q^{48} -173.659 q^{49} -388.696 q^{50} +390.174 q^{51} -37.8881 q^{52} +708.046 q^{53} +46.1207 q^{54} -88.3236 q^{55} +247.950 q^{56} -497.914 q^{57} -41.4385 q^{58} +116.159 q^{59} +19.3054 q^{60} -218.910 q^{61} +585.344 q^{62} -324.980 q^{63} +332.199 q^{64} +26.3116 q^{65} -1473.87 q^{66} -133.114 q^{67} -106.277 q^{68} -213.416 q^{69} +56.0156 q^{70} +926.738 q^{71} -475.836 q^{72} -455.867 q^{73} +1167.75 q^{74} +887.749 q^{75} +135.624 q^{76} -842.831 q^{77} +439.067 q^{78} -620.178 q^{79} +103.441 q^{80} -779.614 q^{81} -930.734 q^{82} -1317.85 q^{83} +184.223 q^{84} +73.8050 q^{85} +94.6419 q^{87} -1234.07 q^{88} -509.295 q^{89} -107.499 q^{90} +251.080 q^{91} +58.1312 q^{92} -1336.88 q^{93} +1159.45 q^{94} -94.1850 q^{95} +627.228 q^{96} +965.870 q^{97} -548.159 q^{98} +1617.46 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} - 7 q^{3} + 22 q^{4} - 43 q^{5} - 3 q^{6} - 8 q^{7} - 54 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} - 7 q^{3} + 22 q^{4} - 43 q^{5} - 3 q^{6} - 8 q^{7} - 54 q^{8} + 81 q^{9} + 57 q^{10} - 28 q^{11} + 157 q^{12} + 56 q^{13} - 184 q^{14} - 124 q^{15} - 54 q^{16} + 19 q^{17} + 81 q^{18} + 75 q^{19} - 135 q^{20} - 18 q^{21} + 504 q^{22} + 131 q^{23} - 567 q^{24} + 105 q^{25} - 44 q^{26} - 238 q^{27} + 404 q^{28} - 515 q^{29} + 396 q^{30} + 237 q^{31} - 558 q^{32} - 540 q^{33} + 107 q^{34} + 198 q^{35} + 73 q^{36} - 269 q^{37} + 527 q^{38} - 290 q^{39} + 613 q^{40} + 471 q^{41} - 362 q^{42} - 428 q^{44} - 334 q^{45} + 67 q^{46} + 415 q^{47} + 989 q^{48} + 350 q^{49} - 1335 q^{50} + 1241 q^{51} - 8 q^{52} + 450 q^{53} + 402 q^{54} + 1732 q^{55} - 780 q^{56} - 1000 q^{57} - 1055 q^{58} + 356 q^{59} - 2732 q^{60} + 1328 q^{61} - 1603 q^{62} + 2290 q^{63} + 466 q^{64} + 62 q^{65} + 156 q^{66} - 632 q^{67} + 571 q^{68} + 1130 q^{69} + 1902 q^{70} + 144 q^{71} - 567 q^{72} - 864 q^{73} + 1207 q^{74} + 2494 q^{75} - 1005 q^{76} - 2660 q^{77} + 2222 q^{78} - 1613 q^{79} - 2399 q^{80} - 102 q^{81} - 1673 q^{82} - 682 q^{83} + 3758 q^{84} - 84 q^{85} + 449 q^{87} + 608 q^{88} - 3378 q^{89} + 930 q^{90} + 3900 q^{91} + 3491 q^{92} - 1879 q^{93} - 3197 q^{94} - 79 q^{95} - 591 q^{96} - 55 q^{97} - 2398 q^{98} - 1612 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.15653 1.11600 0.558001 0.829840i \(-0.311569\pi\)
0.558001 + 0.829840i \(0.311569\pi\)
\(3\) −7.20925 −1.38742 −0.693710 0.720254i \(-0.744025\pi\)
−0.693710 + 0.720254i \(0.744025\pi\)
\(4\) 1.96369 0.245461
\(5\) −1.36370 −0.121973 −0.0609864 0.998139i \(-0.519425\pi\)
−0.0609864 + 0.998139i \(0.519425\pi\)
\(6\) −22.7562 −1.54836
\(7\) −13.0131 −0.702643 −0.351321 0.936255i \(-0.614268\pi\)
−0.351321 + 0.936255i \(0.614268\pi\)
\(8\) −19.0538 −0.842067
\(9\) 24.9733 0.924936
\(10\) −4.30455 −0.136122
\(11\) 64.7677 1.77529 0.887646 0.460527i \(-0.152339\pi\)
0.887646 + 0.460527i \(0.152339\pi\)
\(12\) −14.1567 −0.340557
\(13\) −19.2944 −0.411638 −0.205819 0.978590i \(-0.565986\pi\)
−0.205819 + 0.978590i \(0.565986\pi\)
\(14\) −41.0763 −0.784151
\(15\) 9.83123 0.169227
\(16\) −75.8534 −1.18521
\(17\) −54.1213 −0.772138 −0.386069 0.922470i \(-0.626167\pi\)
−0.386069 + 0.922470i \(0.626167\pi\)
\(18\) 78.8289 1.03223
\(19\) 69.0659 0.833938 0.416969 0.908921i \(-0.363092\pi\)
0.416969 + 0.908921i \(0.363092\pi\)
\(20\) −2.67787 −0.0299395
\(21\) 93.8149 0.974861
\(22\) 204.441 1.98123
\(23\) 29.6031 0.268377 0.134189 0.990956i \(-0.457157\pi\)
0.134189 + 0.990956i \(0.457157\pi\)
\(24\) 137.364 1.16830
\(25\) −123.140 −0.985123
\(26\) −60.9032 −0.459389
\(27\) 14.6112 0.104145
\(28\) −25.5537 −0.172471
\(29\) −13.1279 −0.0840614 −0.0420307 0.999116i \(-0.513383\pi\)
−0.0420307 + 0.999116i \(0.513383\pi\)
\(30\) 31.0326 0.188858
\(31\) 185.439 1.07438 0.537191 0.843461i \(-0.319486\pi\)
0.537191 + 0.843461i \(0.319486\pi\)
\(32\) −87.0032 −0.480629
\(33\) −466.927 −2.46308
\(34\) −170.836 −0.861707
\(35\) 17.7460 0.0857032
\(36\) 49.0397 0.227035
\(37\) 369.949 1.64376 0.821881 0.569659i \(-0.192925\pi\)
0.821881 + 0.569659i \(0.192925\pi\)
\(38\) 218.009 0.930676
\(39\) 139.098 0.571115
\(40\) 25.9836 0.102709
\(41\) −294.860 −1.12315 −0.561577 0.827424i \(-0.689805\pi\)
−0.561577 + 0.827424i \(0.689805\pi\)
\(42\) 296.129 1.08795
\(43\) 0 0
\(44\) 127.183 0.435764
\(45\) −34.0560 −0.112817
\(46\) 93.4431 0.299509
\(47\) 367.319 1.13998 0.569989 0.821652i \(-0.306947\pi\)
0.569989 + 0.821652i \(0.306947\pi\)
\(48\) 546.846 1.64438
\(49\) −173.659 −0.506293
\(50\) −388.696 −1.09940
\(51\) 390.174 1.07128
\(52\) −37.8881 −0.101041
\(53\) 708.046 1.83505 0.917524 0.397680i \(-0.130185\pi\)
0.917524 + 0.397680i \(0.130185\pi\)
\(54\) 46.1207 0.116226
\(55\) −88.3236 −0.216537
\(56\) 247.950 0.591672
\(57\) −497.914 −1.15702
\(58\) −41.4385 −0.0938127
\(59\) 116.159 0.256316 0.128158 0.991754i \(-0.459094\pi\)
0.128158 + 0.991754i \(0.459094\pi\)
\(60\) 19.3054 0.0415387
\(61\) −218.910 −0.459484 −0.229742 0.973252i \(-0.573788\pi\)
−0.229742 + 0.973252i \(0.573788\pi\)
\(62\) 585.344 1.19901
\(63\) −324.980 −0.649899
\(64\) 332.199 0.648827
\(65\) 26.3116 0.0502086
\(66\) −1473.87 −2.74880
\(67\) −133.114 −0.242723 −0.121361 0.992608i \(-0.538726\pi\)
−0.121361 + 0.992608i \(0.538726\pi\)
\(68\) −106.277 −0.189529
\(69\) −213.416 −0.372352
\(70\) 56.0156 0.0956450
\(71\) 926.738 1.54906 0.774532 0.632535i \(-0.217985\pi\)
0.774532 + 0.632535i \(0.217985\pi\)
\(72\) −475.836 −0.778858
\(73\) −455.867 −0.730893 −0.365446 0.930832i \(-0.619084\pi\)
−0.365446 + 0.930832i \(0.619084\pi\)
\(74\) 1167.75 1.83444
\(75\) 887.749 1.36678
\(76\) 135.624 0.204699
\(77\) −842.831 −1.24740
\(78\) 439.067 0.637365
\(79\) −620.178 −0.883234 −0.441617 0.897204i \(-0.645595\pi\)
−0.441617 + 0.897204i \(0.645595\pi\)
\(80\) 103.441 0.144563
\(81\) −779.614 −1.06943
\(82\) −930.734 −1.25344
\(83\) −1317.85 −1.74281 −0.871405 0.490564i \(-0.836791\pi\)
−0.871405 + 0.490564i \(0.836791\pi\)
\(84\) 184.223 0.239290
\(85\) 73.8050 0.0941797
\(86\) 0 0
\(87\) 94.6419 0.116629
\(88\) −1234.07 −1.49492
\(89\) −509.295 −0.606575 −0.303287 0.952899i \(-0.598084\pi\)
−0.303287 + 0.952899i \(0.598084\pi\)
\(90\) −107.499 −0.125904
\(91\) 251.080 0.289234
\(92\) 58.1312 0.0658760
\(93\) −1336.88 −1.49062
\(94\) 1159.45 1.27222
\(95\) −94.1850 −0.101718
\(96\) 627.228 0.666835
\(97\) 965.870 1.01102 0.505511 0.862820i \(-0.331304\pi\)
0.505511 + 0.862820i \(0.331304\pi\)
\(98\) −548.159 −0.565024
\(99\) 1617.46 1.64203
\(100\) −241.809 −0.241809
\(101\) −1501.80 −1.47955 −0.739776 0.672853i \(-0.765068\pi\)
−0.739776 + 0.672853i \(0.765068\pi\)
\(102\) 1231.60 1.19555
\(103\) 1312.60 1.25567 0.627837 0.778345i \(-0.283940\pi\)
0.627837 + 0.778345i \(0.283940\pi\)
\(104\) 367.631 0.346627
\(105\) −127.935 −0.118906
\(106\) 2234.97 2.04792
\(107\) −611.656 −0.552627 −0.276313 0.961068i \(-0.589113\pi\)
−0.276313 + 0.961068i \(0.589113\pi\)
\(108\) 28.6918 0.0255636
\(109\) −946.664 −0.831871 −0.415935 0.909394i \(-0.636546\pi\)
−0.415935 + 0.909394i \(0.636546\pi\)
\(110\) −278.796 −0.241656
\(111\) −2667.05 −2.28059
\(112\) 987.090 0.832779
\(113\) −2109.50 −1.75615 −0.878076 0.478521i \(-0.841173\pi\)
−0.878076 + 0.478521i \(0.841173\pi\)
\(114\) −1571.68 −1.29124
\(115\) −40.3696 −0.0327347
\(116\) −25.7790 −0.0206338
\(117\) −481.843 −0.380739
\(118\) 366.660 0.286049
\(119\) 704.287 0.542537
\(120\) −187.322 −0.142501
\(121\) 2863.86 2.15166
\(122\) −690.996 −0.512785
\(123\) 2125.72 1.55829
\(124\) 364.144 0.263718
\(125\) 338.388 0.242131
\(126\) −1025.81 −0.725289
\(127\) 788.583 0.550988 0.275494 0.961303i \(-0.411159\pi\)
0.275494 + 0.961303i \(0.411159\pi\)
\(128\) 1744.62 1.20472
\(129\) 0 0
\(130\) 83.0535 0.0560329
\(131\) −1873.57 −1.24957 −0.624787 0.780795i \(-0.714814\pi\)
−0.624787 + 0.780795i \(0.714814\pi\)
\(132\) −916.897 −0.604588
\(133\) −898.764 −0.585960
\(134\) −420.177 −0.270879
\(135\) −19.9252 −0.0127029
\(136\) 1031.22 0.650192
\(137\) 1860.27 1.16010 0.580050 0.814581i \(-0.303033\pi\)
0.580050 + 0.814581i \(0.303033\pi\)
\(138\) −673.654 −0.415545
\(139\) −2822.52 −1.72232 −0.861161 0.508333i \(-0.830262\pi\)
−0.861161 + 0.508333i \(0.830262\pi\)
\(140\) 34.8475 0.0210368
\(141\) −2648.09 −1.58163
\(142\) 2925.28 1.72876
\(143\) −1249.65 −0.730777
\(144\) −1894.31 −1.09624
\(145\) 17.9024 0.0102532
\(146\) −1438.96 −0.815678
\(147\) 1251.95 0.702442
\(148\) 726.463 0.403479
\(149\) −1615.30 −0.888126 −0.444063 0.895996i \(-0.646463\pi\)
−0.444063 + 0.895996i \(0.646463\pi\)
\(150\) 2802.21 1.52533
\(151\) −1399.96 −0.754483 −0.377241 0.926115i \(-0.623127\pi\)
−0.377241 + 0.926115i \(0.623127\pi\)
\(152\) −1315.97 −0.702232
\(153\) −1351.59 −0.714178
\(154\) −2660.42 −1.39210
\(155\) −252.882 −0.131045
\(156\) 273.144 0.140186
\(157\) −2933.25 −1.49108 −0.745539 0.666462i \(-0.767808\pi\)
−0.745539 + 0.666462i \(0.767808\pi\)
\(158\) −1957.61 −0.985691
\(159\) −5104.48 −2.54598
\(160\) 118.646 0.0586237
\(161\) −385.229 −0.188573
\(162\) −2460.88 −1.19349
\(163\) 166.212 0.0798695 0.0399347 0.999202i \(-0.487285\pi\)
0.0399347 + 0.999202i \(0.487285\pi\)
\(164\) −579.012 −0.275690
\(165\) 636.746 0.300428
\(166\) −4159.85 −1.94498
\(167\) 2143.42 0.993191 0.496595 0.867982i \(-0.334583\pi\)
0.496595 + 0.867982i \(0.334583\pi\)
\(168\) −1787.53 −0.820899
\(169\) −1824.73 −0.830554
\(170\) 232.968 0.105105
\(171\) 1724.80 0.771339
\(172\) 0 0
\(173\) −119.018 −0.0523050 −0.0261525 0.999658i \(-0.508326\pi\)
−0.0261525 + 0.999658i \(0.508326\pi\)
\(174\) 298.740 0.130158
\(175\) 1602.44 0.692189
\(176\) −4912.85 −2.10409
\(177\) −837.420 −0.355618
\(178\) −1607.61 −0.676939
\(179\) −2572.05 −1.07399 −0.536995 0.843585i \(-0.680441\pi\)
−0.536995 + 0.843585i \(0.680441\pi\)
\(180\) −66.8752 −0.0276921
\(181\) 1429.30 0.586956 0.293478 0.955966i \(-0.405187\pi\)
0.293478 + 0.955966i \(0.405187\pi\)
\(182\) 792.541 0.322786
\(183\) 1578.18 0.637498
\(184\) −564.052 −0.225992
\(185\) −504.498 −0.200494
\(186\) −4219.89 −1.66353
\(187\) −3505.31 −1.37077
\(188\) 721.299 0.279820
\(189\) −190.137 −0.0731769
\(190\) −297.298 −0.113517
\(191\) −411.149 −0.155758 −0.0778788 0.996963i \(-0.524815\pi\)
−0.0778788 + 0.996963i \(0.524815\pi\)
\(192\) −2394.91 −0.900195
\(193\) −3108.57 −1.15938 −0.579689 0.814838i \(-0.696826\pi\)
−0.579689 + 0.814838i \(0.696826\pi\)
\(194\) 3048.80 1.12830
\(195\) −189.687 −0.0696604
\(196\) −341.011 −0.124275
\(197\) 1960.75 0.709125 0.354563 0.935032i \(-0.384630\pi\)
0.354563 + 0.935032i \(0.384630\pi\)
\(198\) 5105.57 1.83251
\(199\) −456.187 −0.162504 −0.0812519 0.996694i \(-0.525892\pi\)
−0.0812519 + 0.996694i \(0.525892\pi\)
\(200\) 2346.29 0.829540
\(201\) 959.650 0.336759
\(202\) −4740.48 −1.65118
\(203\) 170.834 0.0590651
\(204\) 766.179 0.262957
\(205\) 402.099 0.136994
\(206\) 4143.27 1.40133
\(207\) 739.286 0.248232
\(208\) 1463.54 0.487877
\(209\) 4473.24 1.48048
\(210\) −403.831 −0.132700
\(211\) 1742.50 0.568524 0.284262 0.958747i \(-0.408251\pi\)
0.284262 + 0.958747i \(0.408251\pi\)
\(212\) 1390.38 0.450432
\(213\) −6681.08 −2.14920
\(214\) −1930.71 −0.616732
\(215\) 0 0
\(216\) −278.399 −0.0876974
\(217\) −2413.14 −0.754906
\(218\) −2988.17 −0.928370
\(219\) 3286.46 1.01406
\(220\) −173.440 −0.0531514
\(221\) 1044.24 0.317841
\(222\) −8418.63 −2.54514
\(223\) −3149.26 −0.945697 −0.472848 0.881144i \(-0.656774\pi\)
−0.472848 + 0.881144i \(0.656774\pi\)
\(224\) 1132.18 0.337711
\(225\) −3075.22 −0.911175
\(226\) −6658.71 −1.95987
\(227\) −2108.03 −0.616365 −0.308182 0.951327i \(-0.599721\pi\)
−0.308182 + 0.951327i \(0.599721\pi\)
\(228\) −977.746 −0.284003
\(229\) 1529.66 0.441409 0.220704 0.975341i \(-0.429164\pi\)
0.220704 + 0.975341i \(0.429164\pi\)
\(230\) −127.428 −0.0365320
\(231\) 6076.18 1.73066
\(232\) 250.136 0.0707854
\(233\) −2999.86 −0.843466 −0.421733 0.906720i \(-0.638578\pi\)
−0.421733 + 0.906720i \(0.638578\pi\)
\(234\) −1520.95 −0.424905
\(235\) −500.912 −0.139046
\(236\) 228.100 0.0629154
\(237\) 4471.02 1.22542
\(238\) 2223.10 0.605472
\(239\) 3421.34 0.925976 0.462988 0.886364i \(-0.346777\pi\)
0.462988 + 0.886364i \(0.346777\pi\)
\(240\) −745.732 −0.200570
\(241\) 4496.86 1.20194 0.600971 0.799271i \(-0.294781\pi\)
0.600971 + 0.799271i \(0.294781\pi\)
\(242\) 9039.86 2.40126
\(243\) 5225.93 1.37960
\(244\) −429.870 −0.112785
\(245\) 236.818 0.0617540
\(246\) 6709.89 1.73905
\(247\) −1332.58 −0.343280
\(248\) −3533.32 −0.904701
\(249\) 9500.74 2.41801
\(250\) 1068.13 0.270219
\(251\) 1478.05 0.371688 0.185844 0.982579i \(-0.440498\pi\)
0.185844 + 0.982579i \(0.440498\pi\)
\(252\) −638.159 −0.159525
\(253\) 1917.33 0.476448
\(254\) 2489.19 0.614904
\(255\) −532.079 −0.130667
\(256\) 2849.36 0.695645
\(257\) −677.467 −0.164433 −0.0822164 0.996614i \(-0.526200\pi\)
−0.0822164 + 0.996614i \(0.526200\pi\)
\(258\) 0 0
\(259\) −4814.19 −1.15498
\(260\) 51.6678 0.0123242
\(261\) −327.845 −0.0777514
\(262\) −5913.97 −1.39453
\(263\) 7753.56 1.81789 0.908944 0.416917i \(-0.136890\pi\)
0.908944 + 0.416917i \(0.136890\pi\)
\(264\) 8896.73 2.07408
\(265\) −965.560 −0.223826
\(266\) −2836.97 −0.653933
\(267\) 3671.63 0.841574
\(268\) −261.393 −0.0595789
\(269\) −300.965 −0.0682161 −0.0341081 0.999418i \(-0.510859\pi\)
−0.0341081 + 0.999418i \(0.510859\pi\)
\(270\) −62.8946 −0.0141765
\(271\) −2176.33 −0.487833 −0.243917 0.969796i \(-0.578432\pi\)
−0.243917 + 0.969796i \(0.578432\pi\)
\(272\) 4105.28 0.915145
\(273\) −1810.10 −0.401290
\(274\) 5872.00 1.29467
\(275\) −7975.52 −1.74888
\(276\) −419.082 −0.0913978
\(277\) −2947.64 −0.639373 −0.319687 0.947523i \(-0.603578\pi\)
−0.319687 + 0.947523i \(0.603578\pi\)
\(278\) −8909.36 −1.92211
\(279\) 4631.02 0.993734
\(280\) −338.128 −0.0721679
\(281\) 459.209 0.0974879 0.0487440 0.998811i \(-0.484478\pi\)
0.0487440 + 0.998811i \(0.484478\pi\)
\(282\) −8358.79 −1.76510
\(283\) 6190.86 1.30038 0.650192 0.759770i \(-0.274688\pi\)
0.650192 + 0.759770i \(0.274688\pi\)
\(284\) 1819.82 0.380234
\(285\) 679.003 0.141125
\(286\) −3944.56 −0.815549
\(287\) 3837.05 0.789176
\(288\) −2172.75 −0.444551
\(289\) −1983.89 −0.403803
\(290\) 56.5095 0.0114426
\(291\) −6963.20 −1.40271
\(292\) −895.179 −0.179405
\(293\) 5693.95 1.13530 0.567652 0.823269i \(-0.307852\pi\)
0.567652 + 0.823269i \(0.307852\pi\)
\(294\) 3951.81 0.783927
\(295\) −158.406 −0.0312635
\(296\) −7048.93 −1.38416
\(297\) 946.333 0.184888
\(298\) −5098.75 −0.991150
\(299\) −571.173 −0.110474
\(300\) 1743.26 0.335491
\(301\) 0 0
\(302\) −4419.01 −0.842005
\(303\) 10826.9 2.05276
\(304\) −5238.89 −0.988391
\(305\) 298.527 0.0560446
\(306\) −4266.32 −0.797024
\(307\) −5108.18 −0.949639 −0.474819 0.880083i \(-0.657487\pi\)
−0.474819 + 0.880083i \(0.657487\pi\)
\(308\) −1655.05 −0.306187
\(309\) −9462.87 −1.74215
\(310\) −798.231 −0.146247
\(311\) −1048.11 −0.191102 −0.0955511 0.995425i \(-0.530461\pi\)
−0.0955511 + 0.995425i \(0.530461\pi\)
\(312\) −2650.34 −0.480917
\(313\) −4279.92 −0.772893 −0.386446 0.922312i \(-0.626298\pi\)
−0.386446 + 0.922312i \(0.626298\pi\)
\(314\) −9258.91 −1.66405
\(315\) 443.175 0.0792700
\(316\) −1217.83 −0.216799
\(317\) 5929.95 1.05066 0.525330 0.850899i \(-0.323942\pi\)
0.525330 + 0.850899i \(0.323942\pi\)
\(318\) −16112.4 −2.84132
\(319\) −850.261 −0.149234
\(320\) −453.019 −0.0791392
\(321\) 4409.58 0.766725
\(322\) −1215.99 −0.210448
\(323\) −3737.94 −0.643915
\(324\) −1530.92 −0.262503
\(325\) 2375.91 0.405514
\(326\) 524.653 0.0891345
\(327\) 6824.73 1.15415
\(328\) 5618.20 0.945772
\(329\) −4779.97 −0.800997
\(330\) 2009.91 0.335278
\(331\) −509.174 −0.0845521 −0.0422761 0.999106i \(-0.513461\pi\)
−0.0422761 + 0.999106i \(0.513461\pi\)
\(332\) −2587.85 −0.427792
\(333\) 9238.83 1.52037
\(334\) 6765.77 1.10840
\(335\) 181.527 0.0296056
\(336\) −7116.18 −1.15541
\(337\) −10856.3 −1.75484 −0.877419 0.479724i \(-0.840737\pi\)
−0.877419 + 0.479724i \(0.840737\pi\)
\(338\) −5759.81 −0.926900
\(339\) 15207.9 2.43652
\(340\) 144.930 0.0231174
\(341\) 12010.5 1.90734
\(342\) 5444.39 0.860816
\(343\) 6723.34 1.05839
\(344\) 0 0
\(345\) 291.035 0.0454168
\(346\) −375.684 −0.0583725
\(347\) −8973.74 −1.38829 −0.694143 0.719837i \(-0.744217\pi\)
−0.694143 + 0.719837i \(0.744217\pi\)
\(348\) 185.847 0.0286277
\(349\) 5.70408 0.000874877 0 0.000437439 1.00000i \(-0.499861\pi\)
0.000437439 1.00000i \(0.499861\pi\)
\(350\) 5058.15 0.772485
\(351\) −281.913 −0.0428702
\(352\) −5635.00 −0.853257
\(353\) −1504.40 −0.226830 −0.113415 0.993548i \(-0.536179\pi\)
−0.113415 + 0.993548i \(0.536179\pi\)
\(354\) −2643.34 −0.396870
\(355\) −1263.79 −0.188944
\(356\) −1000.10 −0.148890
\(357\) −5077.38 −0.752727
\(358\) −8118.76 −1.19857
\(359\) −1296.86 −0.190657 −0.0953285 0.995446i \(-0.530390\pi\)
−0.0953285 + 0.995446i \(0.530390\pi\)
\(360\) 648.896 0.0949995
\(361\) −2088.90 −0.304548
\(362\) 4511.63 0.655044
\(363\) −20646.3 −2.98526
\(364\) 493.042 0.0709957
\(365\) 621.664 0.0891490
\(366\) 4981.56 0.711449
\(367\) 12908.3 1.83599 0.917995 0.396592i \(-0.129807\pi\)
0.917995 + 0.396592i \(0.129807\pi\)
\(368\) −2245.50 −0.318083
\(369\) −7363.61 −1.03885
\(370\) −1592.46 −0.223752
\(371\) −9213.89 −1.28938
\(372\) −2625.20 −0.365888
\(373\) 1371.48 0.190382 0.0951908 0.995459i \(-0.469654\pi\)
0.0951908 + 0.995459i \(0.469654\pi\)
\(374\) −11064.6 −1.52978
\(375\) −2439.52 −0.335937
\(376\) −6998.83 −0.959939
\(377\) 253.293 0.0346029
\(378\) −600.174 −0.0816656
\(379\) −458.768 −0.0621776 −0.0310888 0.999517i \(-0.509897\pi\)
−0.0310888 + 0.999517i \(0.509897\pi\)
\(380\) −184.950 −0.0249677
\(381\) −5685.09 −0.764452
\(382\) −1297.80 −0.173826
\(383\) −9513.45 −1.26923 −0.634614 0.772829i \(-0.718841\pi\)
−0.634614 + 0.772829i \(0.718841\pi\)
\(384\) −12577.4 −1.67145
\(385\) 1149.37 0.152148
\(386\) −9812.31 −1.29387
\(387\) 0 0
\(388\) 1896.66 0.248166
\(389\) −5643.25 −0.735538 −0.367769 0.929917i \(-0.619878\pi\)
−0.367769 + 0.929917i \(0.619878\pi\)
\(390\) −598.754 −0.0777412
\(391\) −1602.16 −0.207224
\(392\) 3308.86 0.426333
\(393\) 13507.0 1.73369
\(394\) 6189.17 0.791385
\(395\) 845.735 0.107730
\(396\) 3176.19 0.403054
\(397\) −13816.5 −1.74668 −0.873340 0.487110i \(-0.838051\pi\)
−0.873340 + 0.487110i \(0.838051\pi\)
\(398\) −1439.97 −0.181355
\(399\) 6479.41 0.812973
\(400\) 9340.62 1.16758
\(401\) 3544.45 0.441400 0.220700 0.975342i \(-0.429166\pi\)
0.220700 + 0.975342i \(0.429166\pi\)
\(402\) 3029.16 0.375823
\(403\) −3577.93 −0.442256
\(404\) −2949.06 −0.363172
\(405\) 1063.16 0.130441
\(406\) 539.244 0.0659168
\(407\) 23960.7 2.91816
\(408\) −7434.30 −0.902090
\(409\) 2736.88 0.330880 0.165440 0.986220i \(-0.447096\pi\)
0.165440 + 0.986220i \(0.447096\pi\)
\(410\) 1269.24 0.152886
\(411\) −13411.2 −1.60955
\(412\) 2577.54 0.308219
\(413\) −1511.59 −0.180098
\(414\) 2333.58 0.277027
\(415\) 1797.15 0.212575
\(416\) 1678.67 0.197845
\(417\) 20348.2 2.38958
\(418\) 14119.9 1.65222
\(419\) −9280.65 −1.08208 −0.541038 0.840998i \(-0.681968\pi\)
−0.541038 + 0.840998i \(0.681968\pi\)
\(420\) −251.224 −0.0291869
\(421\) 7308.00 0.846010 0.423005 0.906127i \(-0.360975\pi\)
0.423005 + 0.906127i \(0.360975\pi\)
\(422\) 5500.26 0.634475
\(423\) 9173.16 1.05441
\(424\) −13491.0 −1.54523
\(425\) 6664.51 0.760650
\(426\) −21089.0 −2.39852
\(427\) 2848.70 0.322853
\(428\) −1201.10 −0.135648
\(429\) 9009.05 1.01390
\(430\) 0 0
\(431\) −953.510 −0.106564 −0.0532818 0.998580i \(-0.516968\pi\)
−0.0532818 + 0.998580i \(0.516968\pi\)
\(432\) −1108.31 −0.123434
\(433\) −3447.00 −0.382569 −0.191284 0.981535i \(-0.561265\pi\)
−0.191284 + 0.981535i \(0.561265\pi\)
\(434\) −7617.15 −0.842477
\(435\) −129.063 −0.0142255
\(436\) −1858.95 −0.204192
\(437\) 2044.57 0.223810
\(438\) 10373.8 1.13169
\(439\) −9245.65 −1.00517 −0.502586 0.864527i \(-0.667618\pi\)
−0.502586 + 0.864527i \(0.667618\pi\)
\(440\) 1682.90 0.182339
\(441\) −4336.82 −0.468289
\(442\) 3296.16 0.354711
\(443\) −3885.76 −0.416745 −0.208372 0.978050i \(-0.566817\pi\)
−0.208372 + 0.978050i \(0.566817\pi\)
\(444\) −5237.25 −0.559795
\(445\) 694.524 0.0739856
\(446\) −9940.75 −1.05540
\(447\) 11645.1 1.23220
\(448\) −4322.95 −0.455893
\(449\) 14413.1 1.51491 0.757455 0.652887i \(-0.226443\pi\)
0.757455 + 0.652887i \(0.226443\pi\)
\(450\) −9707.02 −1.01687
\(451\) −19097.4 −1.99393
\(452\) −4142.40 −0.431066
\(453\) 10092.6 1.04679
\(454\) −6654.06 −0.687864
\(455\) −342.397 −0.0352787
\(456\) 9487.15 0.974291
\(457\) 1765.21 0.180685 0.0903424 0.995911i \(-0.471204\pi\)
0.0903424 + 0.995911i \(0.471204\pi\)
\(458\) 4828.41 0.492613
\(459\) −790.776 −0.0804145
\(460\) −79.2733 −0.00803508
\(461\) 7626.64 0.770517 0.385258 0.922809i \(-0.374112\pi\)
0.385258 + 0.922809i \(0.374112\pi\)
\(462\) 19179.6 1.93142
\(463\) 5954.18 0.597655 0.298827 0.954307i \(-0.403405\pi\)
0.298827 + 0.954307i \(0.403405\pi\)
\(464\) 995.792 0.0996304
\(465\) 1823.09 0.181815
\(466\) −9469.16 −0.941310
\(467\) −153.388 −0.0151991 −0.00759954 0.999971i \(-0.502419\pi\)
−0.00759954 + 0.999971i \(0.502419\pi\)
\(468\) −946.189 −0.0934564
\(469\) 1732.22 0.170547
\(470\) −1581.14 −0.155176
\(471\) 21146.6 2.06875
\(472\) −2213.27 −0.215835
\(473\) 0 0
\(474\) 14112.9 1.36757
\(475\) −8504.80 −0.821531
\(476\) 1383.00 0.133171
\(477\) 17682.2 1.69730
\(478\) 10799.6 1.03339
\(479\) −11334.3 −1.08117 −0.540584 0.841290i \(-0.681797\pi\)
−0.540584 + 0.841290i \(0.681797\pi\)
\(480\) −855.348 −0.0813357
\(481\) −7137.92 −0.676635
\(482\) 14194.5 1.34137
\(483\) 2777.21 0.261630
\(484\) 5623.72 0.528148
\(485\) −1317.15 −0.123317
\(486\) 16495.8 1.53964
\(487\) −2845.89 −0.264804 −0.132402 0.991196i \(-0.542269\pi\)
−0.132402 + 0.991196i \(0.542269\pi\)
\(488\) 4171.07 0.386917
\(489\) −1198.26 −0.110813
\(490\) 747.522 0.0689176
\(491\) 8345.94 0.767102 0.383551 0.923520i \(-0.374701\pi\)
0.383551 + 0.923520i \(0.374701\pi\)
\(492\) 4174.24 0.382499
\(493\) 710.496 0.0649070
\(494\) −4206.34 −0.383101
\(495\) −2205.73 −0.200283
\(496\) −14066.2 −1.27337
\(497\) −12059.8 −1.08844
\(498\) 29989.4 2.69851
\(499\) −17591.1 −1.57813 −0.789064 0.614311i \(-0.789434\pi\)
−0.789064 + 0.614311i \(0.789434\pi\)
\(500\) 664.488 0.0594336
\(501\) −15452.5 −1.37797
\(502\) 4665.51 0.414805
\(503\) 7975.83 0.707008 0.353504 0.935433i \(-0.384990\pi\)
0.353504 + 0.935433i \(0.384990\pi\)
\(504\) 6192.11 0.547259
\(505\) 2048.00 0.180465
\(506\) 6052.10 0.531717
\(507\) 13154.9 1.15233
\(508\) 1548.53 0.135246
\(509\) −9016.94 −0.785204 −0.392602 0.919708i \(-0.628425\pi\)
−0.392602 + 0.919708i \(0.628425\pi\)
\(510\) −1679.52 −0.145825
\(511\) 5932.25 0.513556
\(512\) −4962.89 −0.428380
\(513\) 1009.14 0.0868507
\(514\) −2138.45 −0.183507
\(515\) −1789.99 −0.153158
\(516\) 0 0
\(517\) 23790.4 2.02379
\(518\) −15196.1 −1.28896
\(519\) 858.030 0.0725691
\(520\) −501.337 −0.0422790
\(521\) 3086.13 0.259512 0.129756 0.991546i \(-0.458581\pi\)
0.129756 + 0.991546i \(0.458581\pi\)
\(522\) −1034.85 −0.0867707
\(523\) −8338.32 −0.697149 −0.348575 0.937281i \(-0.613334\pi\)
−0.348575 + 0.937281i \(0.613334\pi\)
\(524\) −3679.10 −0.306722
\(525\) −11552.4 −0.960358
\(526\) 24474.3 2.02877
\(527\) −10036.2 −0.829570
\(528\) 35418.0 2.91926
\(529\) −11290.7 −0.927974
\(530\) −3047.82 −0.249790
\(531\) 2900.87 0.237076
\(532\) −1764.89 −0.143830
\(533\) 5689.13 0.462333
\(534\) 11589.6 0.939199
\(535\) 834.114 0.0674054
\(536\) 2536.32 0.204389
\(537\) 18542.6 1.49008
\(538\) −950.004 −0.0761293
\(539\) −11247.5 −0.898818
\(540\) −39.1269 −0.00311806
\(541\) −14924.6 −1.18606 −0.593029 0.805181i \(-0.702068\pi\)
−0.593029 + 0.805181i \(0.702068\pi\)
\(542\) −6869.66 −0.544423
\(543\) −10304.2 −0.814355
\(544\) 4708.72 0.371112
\(545\) 1290.96 0.101466
\(546\) −5713.63 −0.447840
\(547\) 10658.3 0.833120 0.416560 0.909108i \(-0.363236\pi\)
0.416560 + 0.909108i \(0.363236\pi\)
\(548\) 3652.99 0.284759
\(549\) −5466.90 −0.424994
\(550\) −25175.0 −1.95175
\(551\) −906.687 −0.0701020
\(552\) 4066.39 0.313545
\(553\) 8070.45 0.620598
\(554\) −9304.31 −0.713542
\(555\) 3637.05 0.278170
\(556\) −5542.53 −0.422762
\(557\) 4303.28 0.327354 0.163677 0.986514i \(-0.447665\pi\)
0.163677 + 0.986514i \(0.447665\pi\)
\(558\) 14617.9 1.10901
\(559\) 0 0
\(560\) −1346.09 −0.101576
\(561\) 25270.7 1.90183
\(562\) 1449.51 0.108797
\(563\) −13280.8 −0.994173 −0.497086 0.867701i \(-0.665597\pi\)
−0.497086 + 0.867701i \(0.665597\pi\)
\(564\) −5200.02 −0.388228
\(565\) 2876.72 0.214203
\(566\) 19541.7 1.45123
\(567\) 10145.2 0.751427
\(568\) −17657.9 −1.30442
\(569\) 11641.6 0.857720 0.428860 0.903371i \(-0.358915\pi\)
0.428860 + 0.903371i \(0.358915\pi\)
\(570\) 2143.29 0.157496
\(571\) −21799.0 −1.59765 −0.798825 0.601563i \(-0.794545\pi\)
−0.798825 + 0.601563i \(0.794545\pi\)
\(572\) −2453.92 −0.179377
\(573\) 2964.07 0.216101
\(574\) 12111.8 0.880723
\(575\) −3645.34 −0.264384
\(576\) 8296.10 0.600123
\(577\) −6854.44 −0.494548 −0.247274 0.968946i \(-0.579535\pi\)
−0.247274 + 0.968946i \(0.579535\pi\)
\(578\) −6262.20 −0.450645
\(579\) 22410.5 1.60855
\(580\) 35.1547 0.00251676
\(581\) 17149.4 1.22457
\(582\) −21979.5 −1.56543
\(583\) 45858.5 3.25775
\(584\) 8686.00 0.615461
\(585\) 657.088 0.0464397
\(586\) 17973.1 1.26700
\(587\) 16741.7 1.17718 0.588589 0.808433i \(-0.299684\pi\)
0.588589 + 0.808433i \(0.299684\pi\)
\(588\) 2458.43 0.172422
\(589\) 12807.5 0.895967
\(590\) −500.012 −0.0348902
\(591\) −14135.5 −0.983855
\(592\) −28061.9 −1.94820
\(593\) −21885.0 −1.51553 −0.757764 0.652529i \(-0.773708\pi\)
−0.757764 + 0.652529i \(0.773708\pi\)
\(594\) 2987.13 0.206336
\(595\) −960.434 −0.0661747
\(596\) −3171.95 −0.218000
\(597\) 3288.77 0.225461
\(598\) −1802.92 −0.123289
\(599\) −14440.9 −0.985039 −0.492519 0.870301i \(-0.663924\pi\)
−0.492519 + 0.870301i \(0.663924\pi\)
\(600\) −16915.0 −1.15092
\(601\) −2570.21 −0.174444 −0.0872221 0.996189i \(-0.527799\pi\)
−0.0872221 + 0.996189i \(0.527799\pi\)
\(602\) 0 0
\(603\) −3324.28 −0.224503
\(604\) −2749.08 −0.185196
\(605\) −3905.44 −0.262444
\(606\) 34175.3 2.29089
\(607\) 11683.5 0.781249 0.390624 0.920550i \(-0.372259\pi\)
0.390624 + 0.920550i \(0.372259\pi\)
\(608\) −6008.96 −0.400815
\(609\) −1231.59 −0.0819482
\(610\) 942.309 0.0625458
\(611\) −7087.18 −0.469258
\(612\) −2654.09 −0.175303
\(613\) 12739.0 0.839356 0.419678 0.907673i \(-0.362143\pi\)
0.419678 + 0.907673i \(0.362143\pi\)
\(614\) −16124.1 −1.05980
\(615\) −2898.83 −0.190069
\(616\) 16059.1 1.05039
\(617\) 4380.70 0.285835 0.142918 0.989735i \(-0.454352\pi\)
0.142918 + 0.989735i \(0.454352\pi\)
\(618\) −29869.8 −1.94424
\(619\) −5157.68 −0.334902 −0.167451 0.985880i \(-0.553554\pi\)
−0.167451 + 0.985880i \(0.553554\pi\)
\(620\) −496.582 −0.0321665
\(621\) 432.536 0.0279502
\(622\) −3308.39 −0.213270
\(623\) 6627.52 0.426205
\(624\) −10551.0 −0.676891
\(625\) 14931.1 0.955589
\(626\) −13509.7 −0.862550
\(627\) −32248.7 −2.05405
\(628\) −5759.99 −0.366001
\(629\) −20022.1 −1.26921
\(630\) 1398.89 0.0884655
\(631\) 27679.9 1.74631 0.873154 0.487444i \(-0.162071\pi\)
0.873154 + 0.487444i \(0.162071\pi\)
\(632\) 11816.8 0.743743
\(633\) −12562.1 −0.788783
\(634\) 18718.1 1.17254
\(635\) −1075.39 −0.0672055
\(636\) −10023.6 −0.624939
\(637\) 3350.63 0.208409
\(638\) −2683.88 −0.166545
\(639\) 23143.7 1.43278
\(640\) −2379.14 −0.146943
\(641\) 16453.5 1.01384 0.506921 0.861992i \(-0.330783\pi\)
0.506921 + 0.861992i \(0.330783\pi\)
\(642\) 13919.0 0.855667
\(643\) −29203.6 −1.79110 −0.895551 0.444958i \(-0.853219\pi\)
−0.895551 + 0.444958i \(0.853219\pi\)
\(644\) −756.468 −0.0462873
\(645\) 0 0
\(646\) −11798.9 −0.718610
\(647\) 16835.6 1.02299 0.511495 0.859286i \(-0.329092\pi\)
0.511495 + 0.859286i \(0.329092\pi\)
\(648\) 14854.6 0.900532
\(649\) 7523.36 0.455035
\(650\) 7499.64 0.452554
\(651\) 17396.9 1.04737
\(652\) 326.388 0.0196048
\(653\) 13735.1 0.823117 0.411558 0.911383i \(-0.364985\pi\)
0.411558 + 0.911383i \(0.364985\pi\)
\(654\) 21542.5 1.28804
\(655\) 2554.98 0.152414
\(656\) 22366.1 1.33117
\(657\) −11384.5 −0.676029
\(658\) −15088.1 −0.893915
\(659\) −5708.76 −0.337453 −0.168727 0.985663i \(-0.553966\pi\)
−0.168727 + 0.985663i \(0.553966\pi\)
\(660\) 1250.37 0.0737433
\(661\) 18339.5 1.07916 0.539578 0.841936i \(-0.318584\pi\)
0.539578 + 0.841936i \(0.318584\pi\)
\(662\) −1607.22 −0.0943604
\(663\) −7528.15 −0.440979
\(664\) 25110.1 1.46756
\(665\) 1225.64 0.0714712
\(666\) 29162.7 1.69674
\(667\) −388.625 −0.0225602
\(668\) 4209.00 0.243789
\(669\) 22703.8 1.31208
\(670\) 572.995 0.0330399
\(671\) −14178.3 −0.815719
\(672\) −8162.19 −0.468547
\(673\) 482.234 0.0276207 0.0138104 0.999905i \(-0.495604\pi\)
0.0138104 + 0.999905i \(0.495604\pi\)
\(674\) −34268.3 −1.95840
\(675\) −1799.23 −0.102596
\(676\) −3583.19 −0.203868
\(677\) −31086.0 −1.76474 −0.882372 0.470553i \(-0.844055\pi\)
−0.882372 + 0.470553i \(0.844055\pi\)
\(678\) 48004.3 2.71916
\(679\) −12569.0 −0.710388
\(680\) −1406.27 −0.0793057
\(681\) 15197.3 0.855157
\(682\) 37911.4 2.12860
\(683\) 31381.0 1.75807 0.879034 0.476758i \(-0.158188\pi\)
0.879034 + 0.476758i \(0.158188\pi\)
\(684\) 3386.97 0.189333
\(685\) −2536.85 −0.141501
\(686\) 21222.4 1.18116
\(687\) −11027.7 −0.612420
\(688\) 0 0
\(689\) −13661.3 −0.755375
\(690\) 918.660 0.0506852
\(691\) −19856.9 −1.09319 −0.546593 0.837398i \(-0.684076\pi\)
−0.546593 + 0.837398i \(0.684076\pi\)
\(692\) −233.714 −0.0128388
\(693\) −21048.2 −1.15376
\(694\) −28325.9 −1.54933
\(695\) 3849.06 0.210076
\(696\) −1803.29 −0.0982091
\(697\) 15958.2 0.867230
\(698\) 18.0051 0.000976365 0
\(699\) 21626.8 1.17024
\(700\) 3146.69 0.169905
\(701\) 2722.80 0.146703 0.0733514 0.997306i \(-0.476631\pi\)
0.0733514 + 0.997306i \(0.476631\pi\)
\(702\) −889.868 −0.0478432
\(703\) 25550.9 1.37079
\(704\) 21515.8 1.15186
\(705\) 3611.20 0.192916
\(706\) −4748.67 −0.253143
\(707\) 19543.1 1.03960
\(708\) −1644.43 −0.0872901
\(709\) −1225.30 −0.0649043 −0.0324521 0.999473i \(-0.510332\pi\)
−0.0324521 + 0.999473i \(0.510332\pi\)
\(710\) −3989.19 −0.210861
\(711\) −15487.9 −0.816935
\(712\) 9704.01 0.510777
\(713\) 5489.57 0.288339
\(714\) −16026.9 −0.840045
\(715\) 1704.15 0.0891349
\(716\) −5050.70 −0.263622
\(717\) −24665.3 −1.28472
\(718\) −4093.59 −0.212774
\(719\) −28875.1 −1.49772 −0.748858 0.662730i \(-0.769398\pi\)
−0.748858 + 0.662730i \(0.769398\pi\)
\(720\) 2583.26 0.133712
\(721\) −17081.0 −0.882290
\(722\) −6593.67 −0.339876
\(723\) −32419.0 −1.66760
\(724\) 2806.70 0.144075
\(725\) 1616.57 0.0828108
\(726\) −65170.6 −3.33155
\(727\) 25072.0 1.27905 0.639525 0.768770i \(-0.279131\pi\)
0.639525 + 0.768770i \(0.279131\pi\)
\(728\) −4784.03 −0.243555
\(729\) −16625.5 −0.844660
\(730\) 1962.30 0.0994905
\(731\) 0 0
\(732\) 3099.04 0.156481
\(733\) −1017.07 −0.0512503 −0.0256251 0.999672i \(-0.508158\pi\)
−0.0256251 + 0.999672i \(0.508158\pi\)
\(734\) 40745.5 2.04897
\(735\) −1707.28 −0.0856787
\(736\) −2575.56 −0.128990
\(737\) −8621.47 −0.430904
\(738\) −23243.5 −1.15935
\(739\) 1001.77 0.0498658 0.0249329 0.999689i \(-0.492063\pi\)
0.0249329 + 0.999689i \(0.492063\pi\)
\(740\) −990.675 −0.0492134
\(741\) 9606.92 0.476274
\(742\) −29083.9 −1.43895
\(743\) 2115.35 0.104448 0.0522238 0.998635i \(-0.483369\pi\)
0.0522238 + 0.998635i \(0.483369\pi\)
\(744\) 25472.6 1.25520
\(745\) 2202.78 0.108327
\(746\) 4329.11 0.212466
\(747\) −32911.1 −1.61199
\(748\) −6883.33 −0.336470
\(749\) 7959.56 0.388299
\(750\) −7700.43 −0.374907
\(751\) −39172.1 −1.90334 −0.951671 0.307119i \(-0.900635\pi\)
−0.951671 + 0.307119i \(0.900635\pi\)
\(752\) −27862.4 −1.35111
\(753\) −10655.6 −0.515688
\(754\) 799.528 0.0386169
\(755\) 1909.12 0.0920263
\(756\) −373.370 −0.0179621
\(757\) −5944.10 −0.285392 −0.142696 0.989767i \(-0.545577\pi\)
−0.142696 + 0.989767i \(0.545577\pi\)
\(758\) −1448.12 −0.0693904
\(759\) −13822.5 −0.661033
\(760\) 1794.58 0.0856531
\(761\) 1954.89 0.0931205 0.0465603 0.998915i \(-0.485174\pi\)
0.0465603 + 0.998915i \(0.485174\pi\)
\(762\) −17945.2 −0.853130
\(763\) 12319.1 0.584508
\(764\) −807.367 −0.0382324
\(765\) 1843.15 0.0871102
\(766\) −30029.5 −1.41646
\(767\) −2241.21 −0.105509
\(768\) −20541.7 −0.965152
\(769\) −39312.9 −1.84351 −0.921756 0.387770i \(-0.873245\pi\)
−0.921756 + 0.387770i \(0.873245\pi\)
\(770\) 3628.01 0.169798
\(771\) 4884.03 0.228138
\(772\) −6104.26 −0.284582
\(773\) 5102.38 0.237413 0.118706 0.992929i \(-0.462125\pi\)
0.118706 + 0.992929i \(0.462125\pi\)
\(774\) 0 0
\(775\) −22835.0 −1.05840
\(776\) −18403.5 −0.851349
\(777\) 34706.7 1.60244
\(778\) −17813.1 −0.820862
\(779\) −20364.8 −0.936641
\(780\) −372.486 −0.0170989
\(781\) 60022.7 2.75004
\(782\) −5057.26 −0.231262
\(783\) −191.813 −0.00875460
\(784\) 13172.6 0.600064
\(785\) 4000.07 0.181871
\(786\) 42635.3 1.93480
\(787\) 21898.3 0.991855 0.495928 0.868364i \(-0.334828\pi\)
0.495928 + 0.868364i \(0.334828\pi\)
\(788\) 3850.30 0.174062
\(789\) −55897.3 −2.52218
\(790\) 2669.59 0.120227
\(791\) 27451.2 1.23395
\(792\) −30818.8 −1.38270
\(793\) 4223.73 0.189141
\(794\) −43612.3 −1.94930
\(795\) 6960.96 0.310541
\(796\) −895.809 −0.0398883
\(797\) 18262.4 0.811652 0.405826 0.913950i \(-0.366984\pi\)
0.405826 + 0.913950i \(0.366984\pi\)
\(798\) 20452.5 0.907280
\(799\) −19879.8 −0.880220
\(800\) 10713.6 0.473479
\(801\) −12718.8 −0.561043
\(802\) 11188.2 0.492604
\(803\) −29525.5 −1.29755
\(804\) 1884.45 0.0826610
\(805\) 525.335 0.0230008
\(806\) −11293.8 −0.493559
\(807\) 2169.73 0.0946444
\(808\) 28615.0 1.24588
\(809\) 26097.8 1.13418 0.567089 0.823656i \(-0.308069\pi\)
0.567089 + 0.823656i \(0.308069\pi\)
\(810\) 3355.89 0.145573
\(811\) −1067.58 −0.0462243 −0.0231122 0.999733i \(-0.507357\pi\)
−0.0231122 + 0.999733i \(0.507357\pi\)
\(812\) 335.465 0.0144982
\(813\) 15689.7 0.676830
\(814\) 75632.8 3.25667
\(815\) −226.663 −0.00974190
\(816\) −29596.0 −1.26969
\(817\) 0 0
\(818\) 8639.05 0.369263
\(819\) 6270.29 0.267523
\(820\) 789.596 0.0336267
\(821\) −20127.0 −0.855585 −0.427793 0.903877i \(-0.640709\pi\)
−0.427793 + 0.903877i \(0.640709\pi\)
\(822\) −42332.7 −1.79626
\(823\) −3011.70 −0.127559 −0.0637797 0.997964i \(-0.520315\pi\)
−0.0637797 + 0.997964i \(0.520315\pi\)
\(824\) −25010.1 −1.05736
\(825\) 57497.5 2.42643
\(826\) −4771.39 −0.200990
\(827\) −13138.4 −0.552437 −0.276219 0.961095i \(-0.589081\pi\)
−0.276219 + 0.961095i \(0.589081\pi\)
\(828\) 1451.73 0.0609311
\(829\) −19550.4 −0.819075 −0.409538 0.912293i \(-0.634310\pi\)
−0.409538 + 0.912293i \(0.634310\pi\)
\(830\) 5672.77 0.237235
\(831\) 21250.2 0.887079
\(832\) −6409.57 −0.267082
\(833\) 9398.63 0.390928
\(834\) 64229.8 2.66678
\(835\) −2922.98 −0.121142
\(836\) 8784.05 0.363400
\(837\) 2709.48 0.111892
\(838\) −29294.7 −1.20760
\(839\) −3227.69 −0.132816 −0.0664078 0.997793i \(-0.521154\pi\)
−0.0664078 + 0.997793i \(0.521154\pi\)
\(840\) 2437.65 0.100127
\(841\) −24216.7 −0.992934
\(842\) 23067.9 0.944149
\(843\) −3310.55 −0.135257
\(844\) 3421.72 0.139550
\(845\) 2488.38 0.101305
\(846\) 28955.4 1.17672
\(847\) −37267.8 −1.51185
\(848\) −53707.7 −2.17492
\(849\) −44631.5 −1.80418
\(850\) 21036.7 0.848887
\(851\) 10951.6 0.441148
\(852\) −13119.5 −0.527545
\(853\) 12419.8 0.498530 0.249265 0.968435i \(-0.419811\pi\)
0.249265 + 0.968435i \(0.419811\pi\)
\(854\) 8992.01 0.360305
\(855\) −2352.11 −0.0940823
\(856\) 11654.4 0.465349
\(857\) 18592.3 0.741075 0.370537 0.928818i \(-0.379174\pi\)
0.370537 + 0.928818i \(0.379174\pi\)
\(858\) 28437.3 1.13151
\(859\) −6443.09 −0.255920 −0.127960 0.991779i \(-0.540843\pi\)
−0.127960 + 0.991779i \(0.540843\pi\)
\(860\) 0 0
\(861\) −27662.2 −1.09492
\(862\) −3009.78 −0.118925
\(863\) 22085.1 0.871132 0.435566 0.900157i \(-0.356548\pi\)
0.435566 + 0.900157i \(0.356548\pi\)
\(864\) −1271.22 −0.0500553
\(865\) 162.304 0.00637979
\(866\) −10880.6 −0.426948
\(867\) 14302.3 0.560245
\(868\) −4738.65 −0.185300
\(869\) −40167.5 −1.56800
\(870\) −407.391 −0.0158757
\(871\) 2568.34 0.0999139
\(872\) 18037.5 0.700491
\(873\) 24120.9 0.935131
\(874\) 6453.73 0.249772
\(875\) −4403.49 −0.170131
\(876\) 6453.57 0.248911
\(877\) 39786.3 1.53191 0.765956 0.642893i \(-0.222266\pi\)
0.765956 + 0.642893i \(0.222266\pi\)
\(878\) −29184.2 −1.12177
\(879\) −41049.1 −1.57514
\(880\) 6699.64 0.256642
\(881\) 38834.0 1.48508 0.742538 0.669804i \(-0.233622\pi\)
0.742538 + 0.669804i \(0.233622\pi\)
\(882\) −13689.3 −0.522611
\(883\) −5483.65 −0.208992 −0.104496 0.994525i \(-0.533323\pi\)
−0.104496 + 0.994525i \(0.533323\pi\)
\(884\) 2050.55 0.0780175
\(885\) 1141.99 0.0433756
\(886\) −12265.5 −0.465088
\(887\) −21289.7 −0.805905 −0.402953 0.915221i \(-0.632016\pi\)
−0.402953 + 0.915221i \(0.632016\pi\)
\(888\) 50817.5 1.92041
\(889\) −10261.9 −0.387148
\(890\) 2192.29 0.0825681
\(891\) −50493.8 −1.89855
\(892\) −6184.17 −0.232131
\(893\) 25369.2 0.950671
\(894\) 36758.2 1.37514
\(895\) 3507.50 0.130997
\(896\) −22703.0 −0.846488
\(897\) 4117.73 0.153274
\(898\) 45495.3 1.69064
\(899\) −2434.41 −0.0903140
\(900\) −6038.76 −0.223658
\(901\) −38320.4 −1.41691
\(902\) −60281.5 −2.22523
\(903\) 0 0
\(904\) 40194.0 1.47880
\(905\) −1949.13 −0.0715927
\(906\) 31857.7 1.16821
\(907\) 16875.4 0.617794 0.308897 0.951095i \(-0.400040\pi\)
0.308897 + 0.951095i \(0.400040\pi\)
\(908\) −4139.51 −0.151293
\(909\) −37504.9 −1.36849
\(910\) −1080.79 −0.0393711
\(911\) −17586.7 −0.639597 −0.319799 0.947486i \(-0.603615\pi\)
−0.319799 + 0.947486i \(0.603615\pi\)
\(912\) 37768.4 1.37131
\(913\) −85354.4 −3.09400
\(914\) 5571.94 0.201645
\(915\) −2152.15 −0.0777574
\(916\) 3003.77 0.108349
\(917\) 24381.0 0.878005
\(918\) −2496.11 −0.0897428
\(919\) −43891.7 −1.57547 −0.787733 0.616017i \(-0.788745\pi\)
−0.787733 + 0.616017i \(0.788745\pi\)
\(920\) 769.196 0.0275648
\(921\) 36826.1 1.31755
\(922\) 24073.7 0.859898
\(923\) −17880.8 −0.637653
\(924\) 11931.7 0.424810
\(925\) −45555.6 −1.61931
\(926\) 18794.5 0.666984
\(927\) 32779.9 1.16142
\(928\) 1142.16 0.0404024
\(929\) 41161.8 1.45369 0.726844 0.686803i \(-0.240987\pi\)
0.726844 + 0.686803i \(0.240987\pi\)
\(930\) 5754.65 0.202906
\(931\) −11993.9 −0.422217
\(932\) −5890.79 −0.207038
\(933\) 7556.08 0.265139
\(934\) −484.175 −0.0169622
\(935\) 4780.18 0.167197
\(936\) 9180.95 0.320608
\(937\) −41754.1 −1.45576 −0.727880 0.685705i \(-0.759494\pi\)
−0.727880 + 0.685705i \(0.759494\pi\)
\(938\) 5467.82 0.190331
\(939\) 30855.0 1.07233
\(940\) −983.633 −0.0341304
\(941\) −37587.5 −1.30214 −0.651071 0.759016i \(-0.725680\pi\)
−0.651071 + 0.759016i \(0.725680\pi\)
\(942\) 66749.8 2.30873
\(943\) −8728.76 −0.301429
\(944\) −8811.06 −0.303788
\(945\) 259.289 0.00892559
\(946\) 0 0
\(947\) −25689.0 −0.881500 −0.440750 0.897630i \(-0.645287\pi\)
−0.440750 + 0.897630i \(0.645287\pi\)
\(948\) 8779.67 0.300792
\(949\) 8795.66 0.300863
\(950\) −26845.7 −0.916830
\(951\) −42750.5 −1.45771
\(952\) −13419.4 −0.456853
\(953\) 10527.3 0.357831 0.178916 0.983864i \(-0.442741\pi\)
0.178916 + 0.983864i \(0.442741\pi\)
\(954\) 55814.5 1.89419
\(955\) 560.682 0.0189982
\(956\) 6718.44 0.227291
\(957\) 6129.75 0.207050
\(958\) −35777.2 −1.20659
\(959\) −24207.9 −0.815136
\(960\) 3265.93 0.109799
\(961\) 4596.60 0.154295
\(962\) −22531.1 −0.755126
\(963\) −15275.1 −0.511144
\(964\) 8830.41 0.295029
\(965\) 4239.15 0.141413
\(966\) 8766.35 0.291980
\(967\) 4292.71 0.142755 0.0713776 0.997449i \(-0.477260\pi\)
0.0713776 + 0.997449i \(0.477260\pi\)
\(968\) −54567.5 −1.81184
\(969\) 26947.7 0.893380
\(970\) −4157.63 −0.137622
\(971\) −41459.8 −1.37025 −0.685123 0.728427i \(-0.740252\pi\)
−0.685123 + 0.728427i \(0.740252\pi\)
\(972\) 10262.1 0.338638
\(973\) 36729.8 1.21018
\(974\) −8983.13 −0.295522
\(975\) −17128.6 −0.562618
\(976\) 16605.1 0.544585
\(977\) 3577.72 0.117156 0.0585779 0.998283i \(-0.481343\pi\)
0.0585779 + 0.998283i \(0.481343\pi\)
\(978\) −3782.36 −0.123667
\(979\) −32985.9 −1.07685
\(980\) 465.035 0.0151582
\(981\) −23641.3 −0.769427
\(982\) 26344.2 0.856088
\(983\) −50532.3 −1.63960 −0.819802 0.572647i \(-0.805917\pi\)
−0.819802 + 0.572647i \(0.805917\pi\)
\(984\) −40503.0 −1.31218
\(985\) −2673.87 −0.0864939
\(986\) 2242.70 0.0724363
\(987\) 34460.0 1.11132
\(988\) −2616.77 −0.0842618
\(989\) 0 0
\(990\) −6962.45 −0.223516
\(991\) −46280.6 −1.48350 −0.741752 0.670674i \(-0.766005\pi\)
−0.741752 + 0.670674i \(0.766005\pi\)
\(992\) −16133.8 −0.516379
\(993\) 3670.76 0.117309
\(994\) −38067.0 −1.21470
\(995\) 622.101 0.0198210
\(996\) 18656.5 0.593527
\(997\) −53334.8 −1.69421 −0.847107 0.531423i \(-0.821658\pi\)
−0.847107 + 0.531423i \(0.821658\pi\)
\(998\) −55526.9 −1.76119
\(999\) 5405.39 0.171190
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1849.4.a.c.1.5 6
43.42 odd 2 43.4.a.b.1.2 6
129.128 even 2 387.4.a.h.1.5 6
172.171 even 2 688.4.a.i.1.2 6
215.214 odd 2 1075.4.a.b.1.5 6
301.300 even 2 2107.4.a.c.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.a.b.1.2 6 43.42 odd 2
387.4.a.h.1.5 6 129.128 even 2
688.4.a.i.1.2 6 172.171 even 2
1075.4.a.b.1.5 6 215.214 odd 2
1849.4.a.c.1.5 6 1.1 even 1 trivial
2107.4.a.c.1.2 6 301.300 even 2